diff --git a/data_all_eng_slimpj/shuffled/split2/finalzerg b/data_all_eng_slimpj/shuffled/split2/finalzerg new file mode 100644 index 0000000000000000000000000000000000000000..08852a90bad55ecd7df29f17098ad98aeafdf11a --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzerg @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\vspace{-0.15cm}\n\\label{sec_intro}\n\nDeep neural networks (DNNs) achieve state-of-the-art performance in various tasks in machine learning and artificial intelligence, such as image classification, speech recognition, machine translation and game-playing. Despite their effectiveness, recent studies have illustrated the vulnerability of DNNs to adversarial examples~\\citep{szegedy2013intriguing,goodfellow2014explaining}. For instance, a carefully designed perturbation to an image can lead a well-trained DNN to misclassify. Even worse, effective adversarial examples can also be made virtually indistinguishable to human perception. Adversarial examples crafted to evade a specific model can even be used to mislead other models trained for the same task, exhibiting a property known as transferability.\n\nTo address this problem, the adversarial robustness of neural networks was studied through the lens of robust optimization in~\\citep{madry2017towards}, leading to an adversarially robust model for image classification we term as the ``Madry Defense Model''. An attack challenge\\footnote{\\url{https:\/\/github.com\/MadryLab\/mnist_challenge}} was proposed for the MNIST dataset to test the defense, however, attacks were constrained to perturb each pixel by at most $\\epsilon$ = 0.3, a scaled maximal $L_\\infty$ distortion. This rule greatly reduces the power of attacks which are not optimized for the $L_\\infty$ distortion metric, and imposes an unrealistic constraint on the attacker.\n\nTo justify the limitation of the $L_\\infty$ constraint, we conduct extensive experiments on the Madry Defense Model to investigate the transferability properties of the state-of-the-art adversarial attacks without the $L_\\infty$ constraint. We find that $L_1$-based adversarial examples generated by EAD, which is short for \\textbf{E}lastic-net \\textbf{A}ttack to \\textbf{D}NNs~\\citep{ead}, readily transfer in both the targeted and non-targeted cases, and despite the high $L_\\infty$ distortion, the visual distortion on the adversarial examples is minimal. These results call into question the use of $L_\\infty$ to quantify visual distortion, and further demonstrate the state-of-the-art transferability properties of EAD.\n\n\\section{Experiment Setup}\n\\label{sec_EAD}\n\\vspace{-0.25cm}\n\n\n\nThe Madry Defense Model is a sufficiently high capacity network trained against the strongest possible adversary, which they deem to be Projected Gradient Descent (PGD) starting from random perturbations around the natural examples~\\citep{madry2017towards}. We evaluate using the MNIST model, as generating visually imperceptible adversarial examples using this dataset is difficult due to its simple, grayscale nature. If this attack objective can be achieved, our methodology can be applied to more realistic, color datasets, like CIFAR-10 and ImageNet~\\citep{deng2009imagenet}. For the MNIST model, 40 iterations of PGD were run, with a step size of 0.01. Gradient steps were taken in the $L_\\infty$ norm. The network was trained and evaluated against perturbations of size no greater than $\\epsilon$ = 0.3. We experimentally show in the supplementary material that this is the maximum $\\epsilon$ for which the Madry Defense Model can be successfully adversarially trained. \n\nWe test the transferability properties of attacks in both the targeted case and non-targeted case against the adversarially trained model. We test the ability for attacks to generate transferable adversarial examples using a single undefended model as well as an ensemble of undefended models, where the ensemble size is set to 3. We expect that if an adversarial example remains adversarial to multiple models, it is more likely to transfer to other unseen models~\\citep{liu2016delving,papernot2016transferability}. We discuss results transferring from an ensemble in the following sections, and provide results transferring from a single undefended model in the supplementary material. The undefended models we use are naturally trained networks of the same architecture as the defended model; the architecture was provided in the competition. For generating adversarial examples, we compare the optimization-based approach and the fast gradient-based approach using EAD and PGD, respectively. \n\nEAD generalizes the state-of-the-art C\\&W $L_2$ attack \\citep{carlini2017towards} by performing elastic-net regularization, linearly combining the $L_1$ and $L_2$ penalty functions~\\citep{ead}. The hyperparameter $\\beta$ controls the trade-off between $L_1$ and $L_2$ minimization. We test EAD in both the general case and the special case where $\\beta$ is set to 0, which is equivalent to the C\\&W $L_2$ attack. We tune the $\\kappa$ ``confidence'' hyperparameter, which controls the necessary margin between the predicted probability of the target class and that of the rest, in our experiments. Full detail on the implementation is provided in the supplementary material.\n\n \nFor $L_\\infty$ attacks, which we will consider, fast gradient methods (FGM) use the sign of the gradient $\\nabla J$ of the training loss $J$ with respect to the input for crafting adversarial examples~\\citep{goodfellow2014explaining}. Iterative fast gradient methods (I-FGM) iteratively use FGM with a finer distortion, followed by an $\\epsilon$-ball clipping~\\citep{kurakin2016adversarial_ICLR}. We note that this is essentially projected gradient descent on the negative loss function. We test PGD both with and without random starts, and as the classic I-FGM formulation does not include random starts, PGD without random starts shall be termed I-FGM for the remainder of the paper. We perform grid search to tune $\\epsilon$, which controls the allowable $L_\\infty$ distortion. Full detail on the implementation is provided in the supplementary material.\n\n\\section{Experiment Results}\n\\vspace{-0.25cm}\n\\begin{table*}[t]\n\\centering\n\\caption{Comparison of tuned PGD, I-FGM, C\\&W, and EAD adversarial examples at various confidence levels. ASR means attack success rate (\\%).\tThe distortion metrics are averaged over successful examples. }\n\\label{my-label}\n\\begin{tabular}{ll|llll|llll} \\hline\n & & \\multicolumn{4}{c|}{Targeted} & \\multicolumn{4}{c}{Non-Targeted} \\\\ \\hline\nAttack Method & Confidence & ASR & $L_1$ & $L_2$ & $L_\\infty$ & ASR & $L_1$ & $L_2$ & $L_\\infty$ \\\\ \\hline\nPGD & None & 68.5 & 188.3 & 8.947 & 0.6 & 99.9 & 270.5 & 13.27 & 0.8 \\\\\nI-FGM & None & 75.1 & 144.5 & 7.406 & 0.915 & 99.8 & 199.4 & 10.66 & 0.9 \\\\ \\hline\n\\multirow{4}{*}{C\\&W} & 10 & 1.1 & 34.15 & 2.482 & 0.548 & 4.9 & 23.23 & 1.702 & 0.424 \\\\\n & 30 & 69.4 & 68.14 & 4.864 & 0.871 & 71.3 & 51.04 & 3.698 & 0.756 \\\\\n & 50 & 92.9 & 117.45 & 8.041 & 0.987 & 99.1 & 78.65 & 5.598 & 0.937 \\\\\n & 70 & 34.8 & 169.7 & 10.88 & 0.994 & 99 & 119.4 & 8.097 & 0.99 \\\\ \\hline\n\\multirow{4}{*}{EAD} & 10 & 27.4 & 25.79 & 3.209 & 0.876 & 39.9 & 19.19 & 2.636 & 0.8 \\\\\n & 30 & 85.8 & 49.64 & 5.179 & 0.995 & 94.5 & 34.28 & 4.192 & 0.971 \\\\\n & 50 & 98.5 & 93.46 & 7.711 & 1 & 99.6 & 57.68 & 5.839 & 0.999 \\\\\n & 70 & 67.2 & 148.9 & 10.36 & 1 & 99.8 & 90.84 & 7.719 & 1 \\\\ \\hline \n\\end{tabular}\n \\vspace{-2mm}\n\\end{table*}\n\nIn our experiment, 1000 random samples from the MNIST test set were used. For the targeted case, a target class that is different from the original one was randomly selected for each input image. The results of the $\\beta$ tuning for EAD is provided in the supplementary material. It is observed that the highest attack success rate (ASR) in both the targeted and non-targeted cases was yielded at $\\beta$ = 0.01. This was in fact the largest $\\beta$ tested, indicating the importance of minimizing the $L_1$ distortion for generating transferable adversarial examples. Furthermore, as can be seen in the supplementary material, the improvement in ASR with increasing $\\beta$ is more significant at lower $\\kappa$, indicating the importance of minimizing the $L_1$ distortion for generating transferable adversarial examples with minimal visual distortion. For PGD and I-FGM, $\\epsilon$ was tuned from 0.1 to 1.0 at 0.1 increments, full results of which are provided in the supplementary material\n\nIn Table \\ref{my-label}, the results for tuning $\\kappa$ for C\\&W and EAD at $\\beta$ = 0.01 are provided, and are presented with the results for PGD and I-FGM at the lowest $\\epsilon$ values at which the highest ASR was yielded, for comparison. It is observed that in both the targeted case and the non-targeted case, EAD outperforms C\\&W at all $\\kappa$. Furthermore, in the targeted case, at the optimal $\\kappa$ = 50, EAD's adversarial examples surprisingly have lower average $L_2$ distortion.\n\nIn the targeted case, EAD outperforms PGD and I-FGM at $\\kappa$ = 30 with much lower $L_1$ and $L_2$ distortion. In the non-targeted case, PGD and I-FGM yield similar ASR at lower $L_\\infty$ distortion. However, we argue that in the latter case the drastic increase in induced $L_1$ and $L_2$ distortion of PGD and I-FGM to generate said adversarial examples indicates greater visual distortion, and thus that the generated examples are less adversarial in nature. We examine this claim in the following section. \n\n\n\n\\section{Visual Comparison}\n\\vspace{-0.25cm}\nAdversarial examples generated by EAD and PGD were analyzed in the non-targeted case, to understand if the successful examples are visually adversarial. A similar analysis in the targeted case is provided in the supplementary material. \nWe find that adversarial examples generated by PGD even at low $\\epsilon$ such as 0.3, at which the attack performance is weak, have visually apparent noise.\n\n\\begin{figure*}[h]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{EAD}\n\t\\caption{Visual illustration of adversarial examples crafted in the non-targeted case by EAD and PGD with similar average $L_\\infty$ distortion.}.\n\t\\label{Fig_ostrich_all_attack_2}\n \\vspace{-2mm}\n\\end{figure*}\n\n\n\nIn Figure \\ref{Fig_ostrich_all_attack_2}, adversarial examples generated by EAD are directly compared to those generated by PGD with similar average $L_\\infty$ distortion. EAD tuned to the optimal $\\beta$ (0.01) was used to generate adversarial examples with $\\kappa$ = 10. As can be seen in Table \\ref{my-label}, the average $L_\\infty$ distortion under this configuration is 0.8. Therefore, adversarial examples were generated by PGD with $\\epsilon$ = 0.8. Clearly, performing elastic-net minimization aids in minimizing visual distortion, even when the $L_\\infty$ distortion is large. Furthermore, these results also emphasize the issues in solely using $L_\\infty$ to measure visual distortion, and how setting such a distortion constraint for adversarial training is not sufficient for characterizing the subspaces of visually similar adversarial examples.\n\n\\section{Conclusion}\n\\vspace{-0.15cm}\nThe Madry Lab developed a defense model by focusing on training a sufficiently high-capacity network and using the strongest possible adversary. By their estimations, this adversary was PGD, which they motivate as the strongest attack utilizing the local first-order information about the network. Through extensive experiments,\nwe demonstrate that EAD is able to outperform PGD in transferring in the targeted case. We also show that EAD is able to generate much less visually distorted transferable adversarial examples than PGD with comparable $L_\\infty$ distortion, due to the drastic reduction in $L_1$ and $L_2$ distortion. \nThese results demonstrate the power of EAD, particularly in its transferability capabilities. Furthermore, these results indicate the drawbacks of using $L_\\infty$ as the sole distortion metric, and suggest incorporating $L_1$ or $L_2$ to complement the subspace analysis of adversarial examples for possible defenses such as adversarial training and adversary detection.\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Science questions}\n\\label{sec:science-questions}\n\nEurope has long been a pioneer of astrometry, from the time of ancient\nGreece to Tycho Brahe, Johannes Kepler, the Copernican revolution and\nFriedrich Bessel. ESA's Hipparcos \\citep{1997A&A...323L..49P,\n 2000A&A...355L..27H} and Gaia \\citep{2016A&A...595A...1G} satellites\ncontinued this tradition, revolutionizing our view of the Solar\nNeighborhood and Milky Way, and providing a crucial foundation for\nmany disciplines of astronomy.\n\nAn unprecedented microarcsecond relative precision mission will\nadvance European astrometry still further, setting the stage for\nbreakthroughs on the most critical questions of cosmology, astronomy,\nand particle physics.\n\n\\subsection{Dark matter}\n\\label{sec:dark-matter}\n\nThe current hypothesis of cold dark matter (CDM) urgently\n needs verification. Dark matter (DM) is essential to the $\\Lambda$\n+ CDM cosmological model ($\\Lambda$CDM), which successfully describes\nthe large-scale distribution of galaxies and the angular fluctuations\nof the Cosmic Microwave Background, as confirmed by ESA's Planck\nmission \\citep{2011A&A...536A...1P}. Dark matter is the dominant form\nof matter ($\\sim 85\\%$) in the Universe, and ensures the formation and\nstability of enmeshed galaxies and clusters of galaxies. The current\nparadigm is that dark matter is made of heavy, hence cold, particles,\notherwise galaxies will not form. However, the nature of dark\n matter is still unknown.\n\nThere are a number of open issues regarding $\\Lambda$CDM on\nsmall scales. Simulations based on DM-only predict 1) a large number\nof small objects orbiting the Milky Way, 2) a steep DM distribution in\ntheir centre and 3) a prolate Milky Way halo. However, hydrodynamical\nsimulations, which include dissipative gas and powerful astrophysical\nphenomena (such as supernovae explosions and jets from galactic\nnuclei) can change this picture. Quantitative predictions are based on\nvery poorly understood sub-grid physics and there is no consensus yet\non the results. Answers are buried at small-scales, which are\nextremely difficult to probe. A new high precision astrometric mission\nappears to be the best way to settle the nature of DM and will allow\nus to validate or refute key predictions of $\\Lambda$CDM, such as\n\\begin{itemize}\n\\item the DM distribution in dwarf spheroidal galaxies \n\\item the outer shape of the Milky Way DM halo \n\\item the lowest masses of the Milky Way satellites and subhalos\n\\item the power spectrum of density perturbations\n\\end{itemize}\nThese observations will significantly advance research in\\-to DM. They\nmay indicate that DM is warmer than $\\Lambda$CDM predicts. Or we may\nfind that DM is prone to self-interac\\-tions that reduces its density\nin the central part of the satellites of the Milky Way. We may\ndiscover that DM has small interactions that reduce the number of\nsatellite companions. Alternatively, measurement of the Milky Way DM\nhalo could reveal that DM is a sophisticated manifestation of a\nmodification of Einstein's gravity.\n\n\\subsubsection{ The DM distribution in dwarf spheroidal galaxies}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.8\\hsize]{Nvsmoverl_4Oct.pdf}\n \\caption{Number of dwarf spheroidal galaxy stars within the\n field of view of Theia, a high precision astrometry concept, with expected\n plane-of-sky errors lower than half the galaxy's velocity\n dispersion as a function of the galaxy's estimated mass-to-light\n ratio within the effective (half-projected-light) radius of the\n galaxy. Luminosities and total masses within the half-light radii\n are mainly from \\citet{Walker+09}.}\n \\label{fig:NvsMoverL}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.45\\hsize,clip]{plum_core_iso_dra_pcs.pdf}\n\\includegraphics[width=0.45\\hsize,clip]{plum_cusp_iso_dra_pcs.pdf}\n\\includegraphics[width=0.45\\hsize,clip]{plum_core_om_dra_pcs.pdf}\n\\includegraphics[width=0.45\\hsize,clip]{plum_cusp_om_dra_pcs.pdf}\n \\caption{Reconstruction of the DM halo profile of the Draco dSph\n without (\\textit{blue}) and with (\\textit{red}) proper motions using\n the mass-orbit modeling algorithm of \\citet{2013MNRAS.436.2598W}. Four\n mocks of Draco \\citep{2011ApJ...742...20W} were used, with cored\n (\\textit{left}) and cuspy (\\textit{right}) DM halos, and with\n isotropic velocities everywhere (\\textit{top}) or only in the inner\n regions with increasingly radial motions in the outer regions\n (\\textit{bottom}). The effective (half-projected light) radii of\n each mock is shown with the \\textit{arrows}. The stellar proper\n motions in the mocks were perturbed with apparent magnitude\n dependent errors as expected with 1000 hours of observations spread\n over 4 years.}\n\\label{fig:rhoofrPM}\n\\end{figure*}\n\nBecause they are DM-dominated (see Fig.~\\ref{fig:NvsMoverL} where the\nnumber of stars versus the mass-to-light ratio is present\\-ed), dwarf\nSpheroidal galaxies (dSphs) are excellent laboratories to test the\ndistribution of DM within the central part of small galaxies and\ndisentangle the influence of complex baryonic processes from that of\nDM at these scales.\n\nSimulations from \\citet{Onorbe+15} or \\citet{Read+16} for example show\nthat the DM distribution (referred to as DM profile) in dSphs strongly\ndepends on their star formation history. More specifically, these\nsimulations find that CDM can be heated by bursty star formation\ninside the stellar half-light radius $r_{1\/2}$, if star formation\nproceeds for long enough. As a result, some dSphs like Fornax have\nformed stars for almost a Hubble time and so should have large central\nDM cores, while others, like Draco and Ursa Major 2, had their star\nformation truncated after just $\\sim 1-2$\\,Gyrs and should retain\ntheir steep central DM cusp.\n\nLarge DM cores could also be attributed however to strong\nself-interactions. Hence finding evidence for such cores in the\nfaintest dSphs (which are even more DM dominated \\citep{Wolf+10} than\nthe classical ones), will bring tremendous insights about the history\nof baryonic process\\-es in these objects and could even dramatically\nchange our understanding of the nature of DM. Indeed,\nself-inter\\-acting DM \\citep{Spergel&Steinhardt00} is expected to\nscatter in the dense inner regions of dSphs, and thus leads to\nhomogeneous cores. Finding such a core DM distribution in dSphs could\nthen reveal a new type of particle forces in the DM sector and provide\nus with new directions to build extensions of the Standard Model of\nparticle physics. On the other hand, finding cuspy DM profiles in all\ndSphs (including the faintest ones) will confirm $\\Lambda$CDM and\nplace strong constraints on galaxy formation. As shown in\nFigs.~\\ref{fig:vposerrvsDist} and \\ref{fig:astroComparisons}, a\ntelescope with microarcsecond astrometric precision allows us to\ndetermine whether the DM distribution in dSphs is cuspy or has a core,\nand hence can lead to a very significant breakthrough regarding the\nnature of DM.\n \nTo determine the inner DM distribution in dSphs, one needs to remove\nthe degeneracy between the radial DM profile and orbital anisotropy\nthat quantifies whether stellar orbits are more radial or more\ntangential in the Jeans equation \\citep{Binney&Mamon82}. This can be\ndone by adding the proper motions of stars in dSphs.\nFig.~\\ref{fig:rhoofrPM} shows that for the Draco dSph (which was\nobtained using single-component spherical mock datasets from the Gaia\nChallenge Spherical and Triaxial Systems working group,\\footnote{See\n \\url{http:\/\/astrowiki.ph.surrey.ac.uk\/dokuwiki\/doku.php?id=tests:sphtri}}\nand the number of stars expected to be observed by a high precision\nastrometry mission), the inclusion of proper motions lifts the cusp \/\ncore degeneracy that line-of-sight-only kinematics cannot disentangle.\n\nWe remark in addition that a high precision astrometric mission is\nable to perform follow-ups of Gaia's observations of dSphs streams of\nstars if needed. Not only will such a mission provide the missing\ntangential velocities for stars with existing radial velocities, but\nit will also provide crucial membership information - and tangential\nvelocities - for stars in the outer regions of the satellite galaxies\nthat are tidally disrupted by the Milky Way.\n\n\\subsubsection{ The triaxiality of the Milky Way dark matter halo \\label{triaxial} }\n\nFor almost two decades cosmological simulations have shown that Milky\nWay-like DM halos have triaxial sha\\-pes, with the degree of triaxiality\nvarying with radius \\citep[][for example]{dubins_94, kkzanm04}: halos\nare more round or oblate at the center, become triaxial at\nintermediate radii, and prolate at large radii \\citep{zemp_etal_11}.\n\nPrecise measurement of the velocity of distant Hyper Velocity Stars\n(hereafter HVS) can test these departures from spherical symmetry,\nindependently of any other technique attempted so far (such as the\ntidal streams). HVSs were first discovered serendipitously\n\\citep{Brown+05,Hirsch+05,Edelmann+05}, and later discovered in a\ntargeted survey of blue main-sequence stars \\citep[][and references\nthere\\-in]{Brown15_ARAA}. Gaia measurements demonstrate that candidate\nhyper velocity stars include unbound disc runaways \\citep{2019A&A...628L...5I},\nunbound white dwarfs ejected from double-degenerate type Ia supernovae\n\\citep{Shen2018}, and runaways from the LMC \\citep{Erkal2018}, however\nthe highest-velocity main sequence stars in the Milky Way halo have\ntrajectories that point from the Galactic Center \\citep{Brown2018,\n 2020MNRAS.491.2465K}. \\vspace{5mm}\n\n\\begin{figure}[thb]\n\\centering\n\\fbox{\\includegraphics[width=0.9\\hsize,clip]{HVS_cartoon}}\n \\caption{Illustration of the trajectories of hyper velocity stars\n ejected from Galactic Center for 3 different outer DM halo shapes:\n oblate (\\textit{left}), spherical (\\textit{middle}), and prolate\n (\\textit{right}).}\n\\label{Fig:triaxial_halos}\n\\end{figure}\n\nBecause these velocities exceed the plausible limit for a runaway star\nejected from a binary, in which one component has undergone a\nsupernova explosion, the primary mechanism for a star to obtain such\nan extreme velocity is assumed to be a three-body interaction and\nejection from the deep potential well of the supermassive black hole\nat the Galactic Center \\citep{Hills88,Yu&Tremaine03}.\n\n\\vspace{8mm}\n\\begin{figure}[bht]\n \\centering\n \\includegraphics[width=0.8\\hsize]{pm_hvs5.png}\n \\caption{Expected proper motions of HVS5 under different\n assumptions about the shape and orientation of the DM halo. The\n families of models are shown with the halo major axis along the\n Galactic X- (\\textit{red squares}), Y- (\\textit{blue triangles}),\n and Z- (\\textit{green circles}) coordinates. The \\textit{solid\n line} shows how the centroid of the proper motions will shift\n with a different distance to HVS5.}\n\\label{fig:pm_hvs5} \n\\end{figure}\n\nBy measuring the three-dimensional velocity of these stars, we will\nreconstruct the triaxiality of the Galactic potential. In a spherical\npotential, unbound HVS ejected from the Galactic Center should travel\nin nearly a straight line, as depicted in\nFig.\\ref{Fig:triaxial_halos}. However, for triaxial halos, the\npresent velocity vector should not point exactly from the Galactic\nCenter because of the small curvature of the orbit caused by\nnon-spherically symmetric part of the potential\n\\citep{Gnedin+05,Yu&madau07}. While both the halo and stellar disc\ninduce transverse motions, the effect is dominated by halo triaxiality\nat the typical distance of HVSs. The deflection contributed by the\ndisc peaks around 10 kpc but quickly declines at larger distances,\nwhile the deflection due to the triaxial halo continues to accumulate\nalong the whole trajectory. Fig.~\\ref{fig:pm_hvs5} shows the\nspread of proper motion for one star, HVS5, for different halo shapes\n(different halo axis ratios and different orientations of the major\naxis).\n\nProper motions of several HVSs were measured with the Hubble Space\nTelescope (HST) by \\cite{Brown+15}, using an astrometric frame based\non background galaxies. However, these measurements were not\nsufficiently accurate to constrain the halo shape or the origin of\neach HVS. A high precision astrometric mission with a sufficiently\nlarge field of view could include about 10 known quasars from the SDSS\ncatalog around most HVSs. This will provide a much more stable and\naccurate astrometric frame, and will allow us to constrain the halo\naxis ratios to about 5$\\%$.\n\n\\begin{figure}[bht]\n\\centering\n\\includegraphics[width=0.9\\hsize, trim=0.cm 0.3cm 0.8cm 1cm, clip]{axis_hvs5_d20_theia_new.png}\n \\caption{Example of a reconstruction of the Galactic halo shape\n from a high precision astrometry mission (Theia) measurement of\n proper motion of HVS5. The assumed proper motions correspond to a\n prolate model with $q_X = q_Y = 0.8\\, q_Z$, marked by a \\textit{red\n square}. \\textit{Shaded contours} represent confidence limits\n corresponding to the expected 1, 2, and $3\\, \\sigma_\\mu$ proper\n motion errors. The \\textit{outer blue contours} show the accuracy\n that will be achieved by Gaia at the end of its mission, even if its\n expected error was reduced by a factor of\n 2.} \\label{Fig:axishvs5d20theia}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0. \\hsize]{Fig6-meanz.jpg}\n \\caption{Face-on view of the evolution of the perturbation of a\n Galactic Disc due to a DM subhalo of mass 3$\\%$ of the mass of the\n disc crossing the disc from above. The \\textit{upper} and\n \\textit{lower} panels are before and after the crossing,\n respectively, for different times 125, 75 and 25 Myr before the\n crossing and 25,75,125 Myr after (from \\textit{left} to\n \\textit{right}). The mean displacement amplitude is indicated in\n the color bar, while the \\textit{contours} indicate the amplitude of\n the bending mode in velocity space, using plain lines for positive\n values and dashed lines for negative values. The \\textit{green line}\n shows the projected orbit of the subhalo (dashed line after the\n impact with the disc). The \\textit{green triangle} shows the current\n location of the subhalo on its orbit. The \\textit{red lines} are\n our potential lines of sight for Theia, a high precision astrometry\n mission concept, spaced by 10$^\\circ$ in longitude with one pointing\n above the plane and one below the plane, that will allow us to map\n the disc perturbation behind the Galactic Center. }\n\\label{fig:dm:1}\n\\end{figure*}\n\nFig.\\ref{Fig:axishvs5d20theia} shows that with a precision of\n$4 \\, \\mu$as\/yr one can constrain the orientation of the halo major\naxis and measure the axis ratios to an accuracy of\n$\\delta (q_Z\/q_X) < 0.05$ for the typical HVS distance of 50 kpc. For\ncomparison, Gaia at the end of its mission will achieve only\n$40 - 150 \\, \\mu$as\/yr, which is insufficient to provide useful\nconstraints on the axis ratios.\n\nStatistical studies of high-precision proper motions of HVSs can also\nconstrain departures of the halo shape from spherical \\citep[][ in\npreparation]{Gallo2021}. Indeed, numerical simulations of the\ntrajectories of synthetic HVSs ejected through the Hills mechanism\nshow that the distributions of the HVS tangential velocities in the\nGalactocentric reference frame are significantly different from\nspherical and non-spherical halos: the significance is\n$P \\le 1.3 \\times 10^{-6}$ for oblate halos with $q_Z\/q_X \\le 0.9$ and\n$P \\le 2.2 \\times 10^{-5}$ for prolate halos with $q_Z\/q_X \\ge\n1.1$. The median tangential velocity of a sample of $\\sim 100$ HVSs\nlocated at heliocentric distances $\\sim 50\\,\\mbox{kpc}$ can differ by\n$\\sim 5-10\\,\\mbox{km\/s}$, implying differences in proper motions of\n$\\sim 20-40\\,\\mbox{$\\mu$as\/yr}$ between spherical and non-spherical\nhalos.\n\nFinally, an accurate measurement of HVS velocities may lead to\nimproved understanding of the black hole(s) at the Galactic\nCenter. Indeed, theoretical models show that HVSs will have a\ndifferent spectrum of ejection velocities from a binary black hole\nversus a single massive black hole. Gaia has led to the discovery of\nseveral candidate hypervelocity stars \\citep[ejection velocities of\nover $550\\,\\mbox{km\/s}$:][]{2018A&A...620A..48I, 2018ApJ...869...33H,\n 2019ApJ...881..106M}, that were definitely not ejected from the\nGalactic Center but were ejected from spiral arms in the Milky Way\ndisc. These possibly point to intermediate mass black holes of mass\n$~100\\,\\mbox{M}_\\odot$ - these could be local remnants of binary black\nhole mergers of the kind discovered by LIGO\/Virgo\n\\citep{2009RPPh...72g6901A, 2015CQGra..32b4001A} and could have\nimportant implications for our understanding of stellar evolution.\n\n\\subsubsection{Orbital distribution of Dark Matter from the orbits of halo stars}\n\nThe orbits of DM particles in halos\\footnote{For an analysis of\n orbital content of DM halos see \\cite{valluri_etal_10,\n valluri_etal_12, valluri_etal_13, bryan_etal_12}.} cannot be\ndetected directly since DM particles interact only weakly with normal\nmatter. However, in a triaxial potential such as described above, it\nis expected that a large fraction of the DM orbits do not have any net\nangular momentum. Hence these particles should get arbitrarily close\nto the center of the cusp, regardless of how far from the center they\nwere originally. This allows DM particles, which annihilate within the\ncusp to be replenished on a timescale $10^4$ times shorter than in a\nspherical halo \\citep[analogous to loss cone filling in the case of\nbinary black holes, see][]{merritt_poon_04}.\n\nRecent work on the orbital properties and kinematic distributions of\nhalo stars and DM particles show that halo stars, especially the ones\nwith lowest metallicities, are relatively good tracers of DM particles\n\\citep{2013ApJ...767...93V, 2018PhRvL.120d1102H, 2018JCAP...04..052H}\nand observations with Gaia Data Release 2 (DR2) may have already led\nto the kinematic discovery of dark substructure \\citep{Necib2019}.\nThe orbits reflect both the accretion\/formation history and the\ncurrent shape of the potential because DM halos are dynamically young\n(i.e. they are still growing and have not attained a long term\nequilibrium configuration where all orbits are fully phase mixed).\nThis opens up the very exciting possibility that one can infer the\nkinematical distribution of DM particles by assuming that they are\nrepresented by the kinematics of halo stars.\n\n\\subsubsection{Perturbations by Dark Matter subhalos}\n\nA central prediction of $\\Lambda$CDM in contrast to many alternatives\nof DM, such as warm DM \\citep[e.g.][]{Schaeffer:1984bt} or interacting\nDM \\citep[e.g.][]{Boehm:2014vja}, is the existence of numerous $10^6$\nto $10^8$ M$_\\odot$ DM \\emph{subhalos} in the Milky Way halo. Their\ndetection is extremely challenging, as they are very faint and lighter\nthan dSphs. However, N-body simulations of the Galactic Disc show that\nsuch a DM halo passing through the Milky Way disc will warp the disc\nand produce a motion (bending mode), as shown in\nFig.~\\ref{fig:dm:1}. This opens new avenues for detection as such\nperturbations of the disc will result in anomalous motions of the\nstars in the disc \\citep[e.g.][for recent\nanalysis]{2015MNRAS.446.1000F}, that could give rise to an astrometric\nsignal.\n\n\\begin{figure*}[ht]\n \\centering\n \n \n \\includegraphics[width=0.9\\hsize]{l-+25b+2_astrometry}\n \\caption{Astrometric signatures in the proper motion along\n Galactic latitude of the perturbation of disc stars by a\n subhalo. The \\textit{left} and \\textit{right} panels show lines of\n sight as a function of distance along the line of sight and time,\n for $\\ell=-25^\\circ$ and $\\ell=+25^\\circ$ respectively for\n $b=+2^\\circ$. The color codes the time in Myr, \\textit{red} for\n times prior to the crossing of the plane by the satellite,\n \\textit{blue} for later times. The \\textit{green line} is Gaia's\n expected end of mission performance for a population of red clump\n stars along these lines of sight. The \\textit{vertical dashed\n line} is Gaia's detection limit ($G$=20) for the same\n population. The \\textit{red lines} are Theia's expected 1$\\sigma$\n accuracy for the same stars and for a 400~h exposure of the field\n over the course of the mission. }\n\\label{fig:dm:2}\n\\end{figure*}\n\n\nThese anomalous bulk motions develop both in the solar vicinity\n\\citep{2012ApJ...750L..41W} and on larger scales\n\\citep{2015MNRAS.446.1000F}, see Fig.\\ref{fig:dm:2}. Therefore,\nmeasuring very small proper motions of individual faint stars in\ndifferent directions towards the Galactic Disc could prove the\nexistence of these subhalos and confirm the CDM\nscenario. Alternatively, in case they are not found, high precision\nastrometric observations will provide tantalizing evidence for\nalternative DM scenarios, the most popular today being a warmer form\nof DM particle, though these results could also indicate DM\ninteractions \\citep{Boehm:2014vja}.\n\nA field of view of $1^\\circ\\times 1^\\circ$ in the direction of the\nGalactic Disc has $\\sim 10^6$ stars with an apparent magnitude of\n$R \\leq 20$ (given by the confusion limit). Given the astrometric\nprecisions per field of view of Fig.~\\ref{fig:astroComparisons}, a\nhigh precision astrometric instrument could detect up to 7 impacts on\nthe disc from sub-halos as small as a few $10^6\\,\\mbox{M}_\\odot$.\n\nGaia DR2 astrometry has led to the discovery of gaps in tidal streams\n\\citep{2018ApJ...863L..20P} like the GD1 stream. The gaps and\noff-stream stars (spur) are consistent with gravitational interactions\nwith compact DM subhalos. Furthermore, Gaia DR2 data have revealed\nthat globular cluster streams (GD1 and Jhelum) show evidence for\ncocoon-like structures that most likely arise from evolution inside a\n(dark) subhalo prior to their tidal disruption by the Milky Way itself\n\\citep{2020ApJ...889..107C, 2019ApJ...881..106M, 2019ApJ...881L..37B}.\nThe high astrometric precision of a {Theia}-like mission will enable\nus to measure the small velocity perturbations around the gaps in\nstreams and allow for a much more accurate determination of both the\nmasses and density structures of the perturbing dark subhalos.\n\n\\subsubsection{ Ultra-compact minihalos of dark matter in the Milky Way} \n\\label{sec:science_minihalos}\n\nIn the $\\Lambda$CDM model, galaxies and other large-scale structures\nformed from tiny fluctuations in the distribution of matter in the\nearly Universe. Inflation predicts a spectrum of primordial\nfluctuations in the curvature of spacetime, which directly leads to\nthe power spectrum of initial density fluctuations. This spectrum is\nobserved on large scales in the cosmic microwave background and the\nlarge scale structure of galaxies, but is very poorly constrained on\nscales smaller than 2\\,Mpc. This severely restricts our ability to\nprobe the physics of the early Universe. A high precision astrometric\nmission could provide a new window on these small scales by searching\nfor astrometric microlensing events caused by \\emph{ultra-compact\n minihalos} (UCMHs) of DM.\n\nUCMHs form shortly after matter domination (at $z\\sim1000$), in\nregions that are initially overdense \\citep[e.g.\n$\\delta\\rho\/\\rho > 0.001$ in][]{2009ApJ...707..979R}. UCMHs only form\nfrom fluctuations about a factor of 100 larger than their regular\ncosmological counterparts, so their discovery will indicate that the\nprimordial power spectrum is not scale invariant. This will rule out\nthe single-field models of inflation that have dominated the\ntheoretical landscape for the past thirty years. Conversely, the\nabsence of UCMHs can be used to establish upper bounds on the\namplitude of the primordial power spectrum on small scales\n\\citep{Bringmann11}, which will rule out inflationary models that\npredict enhanced small-scale structure \\citep{2016PhRvL.117n1102A}.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.8\\hsize]{Theia_UCMH_limits}\n \\caption{Projected sensitivity of a high precision astrometry\n mission (Theia) to the fraction of dark matter in the form of\n ultra-compact minihalos (UCMHs) of mass $M_i$ at the time of\n matter-radiation equality. Smaller masses probe smaller scales,\n which correspond to earlier formation times (and therefore to\n \\textit{later} stages of inflation). A UCMH mass of 0.1\\,M$_\\odot$\n corresponds to a scale of just 700\\,pc. Expected constraints from\n Gaia are given for comparison, showing that a {Theia}-like mission\n will provide much stronger sensitivity, as well as probe smaller\n scales and earlier formation times than ever reached before.}\n \\label{fig:ucmh}\n\\end{figure}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.8\\hsize,trim = 0cm 6.3cm 0cm 7.5cm,clip]{PKBOUNDS_mod}\n \\caption{Limits on the power of primordial cosmological\n perturbations at all scales, from a range of different sources. A\n {Theia}-like mission will provide far stronger sensitivity to\n primordial fluctuations on small scales than Gaia, spectral\n distortions or primordial black holes (PBHs). Unlike gamma-ray\n UCMH limits, a high precision astrometry mission's sensitivity to\n cosmological perturbations will also be independent of the\n specific particle nature of dark matter.}\n \\label{fig:ucmh2}\n\\end{figure*}\n\nLike standard DM halos, UCMHs are too diffuse to be detected by\nregular photometric microlensing sear\\-ches for MAssive Compact Halo\nObjects (MACHOs). Because they are far more compact than standard DM\nhalos, they however produce much stronger \\textit{astrometric}\nmicrolensing signatures \\citep{2012PhRvD..86d3519L}. By searching for\nmicrolensing events due to UCMHs in the Milky Way, a high precision\nastrometric mission will provide a new probe of inflation. A search\nfor astrometric signatures of UCMHs in the Gaia dataset could\nconstrain the amplitude of the primordial power spectrum to be less\nthan about $10^{-5}$ on scales around 2\\,kpc\n\\citep{2012PhRvD..86d3519L}. Fig.\\ \\ref{fig:ucmh} shows that higher\nastrometric precision (corresponding to that of\nFig.~\\ref{fig:astroComparisons}) will provide more than \\textit{an\n order of magnitude higher sensitivity} to UCMHs, and around\n\\textit{four orders of magnitude greater mass coverage} than Gaia.\nThese projections are based on 8000\\,hr of observations of 10 fields\nin the Milky Way disc, observed three times a year, assuming that the\nfirst year of data is reserved for calibrating stellar proper motions\nagainst which to look for lensing perturbations. Fig.\\\n\\ref{fig:ucmh2} shows that a high precision astrometric mission will\ntest the primordial spectrum of perturbations down to scales as small\nas 700\\,pc, and improve on the expected limits from Gaia by over an\norder of magnitude at larger scales.\n\nThe results will be independent of the DM nature, as astrometric\nmicrolensing depends on gravity only, unlike other constraints at\nsimilar scales based on DM annihilation, from the Fermi Gamma-Ray\nSpace Telescope \\citep{Bringmann11}. An astrometric mission with\nhigher precision (shown in Fig.~\\ref{fig:astroComparisons}) will have\nsensitivity four orders of magnitude better than constraints from the\nabsence of primordial black holes\n\\citep[PBHs;][]{2009PhRvD..79j3520J}, and more than an order of\nmagnitude better than CMB spectral distortions\n\\citep{2012ApJ...758...76C}, which give the current best\nmodel-independent limit on the primordial power spectrum at similar\nscales.\n\n\\nocite{2011MNRAS.413.1717B, 2012ApJ...749...90H, 2009PhRvD..79j3520J}\n\n\\subsubsection{Directly Testing Gravity} \n\nUsing the nearest star, Proxima Centauri, astrometry could measure the\nbehaviour of gravity at low accelerations. A high precision\nastrometry mission with an extended baseline of 10 years and a\nprecision of 0.5 $\\mu$as could measure the wide binary orbit of\nProxima Centauri around Alpha Centauri A and B to distinguish between\nNewtonian gravity and Milgromian dynamics (MOND). The separation\nbetween Proxima Centauri and the Alpha Centauri system suggests orbital\nacceleration that is significantly less than the MOND acceleration\nconstant $a_0 \\sim 1.2 \\times 10^{-10}$ m\/s$^2$ \\citep{Banik2019}. It\nwould be the first direct measurement of the departure from Newtonian\ngravity in the very weak field limit, as expected in MOND, and the\nresults could have profound implications for fundamental physics.\n\n\\subsection{Exoplanets}\n\\label{sec:exoplanets}\n\n\\subsubsection{The Frontier of Exoplanet Astrophysics}\n\nThe ultimate exoplanetary science goal is to answer the enigmatic and\nancient question, ``\\emph{Are we alone?'}' via unambiguous detection\nof biogenic gases and molecules in the atmosphere of an Earth twin\naround a Sun-like star \\citep{Schwieterman+16}. Directly addressing\nthis age-old question related to the uniqueness of the Earth as a\nhabitat for complex biology constitutes today the vanguard of the\nfield, and it is clearly recognized as an unprecedented,\ncross-technique, interdisciplinary endeavor.\n\nSince the discovery of the first Jupiter-mass companion to a\nsolar-type star \\citep{Mayor&Queloz95}, tremendous progress has been\nmade in the field of exo\\-planets. Our knowledge is expanding quickly\ndue to the discovery of thousands of planets, and the skillful\ncombination of high-sensitivity space-borne and ground-based programs\nthat have unveiled the variety of planetary systems architectures that\nexist in the Galaxy\n\\citep[e.g.][]{2013Sci...340..572H,2011arXiv1109.2497M}. Preliminary\nestimates \\citep[e.g.][]{Winn&Fabrycky15} are now also available for\nthe occurrence rate $\\eta_\\uplus$ of terrestrial-type planets in the\nHabitable Zone (HZ) of stars more like the Sun ($\\eta_\\uplus\\sim10\\%$)\nand low-mass M dwarfs ($\\eta_\\uplus\\sim50\\%$).\n\nHowever, transiting or Doppler-detected HZ terrestrial planet\ncandidates \\citep[including the discovery of the $m_{\\rm p}\\sin i=1.3$\n$M_\\oplus$ HZ-planet orbiting Proxima\nCentauri;][]{2016Natur.536..437A} lack determinations of their bulk\ndensities $\\varrho_{\\rm p}$. Thus, the HZ terrestrial planets known\nto date are not amenable to making clear statements on their\nhabitability. The K2 \\citep{2014PASP..126..398H}, TESS\n\\citep{2015JATIS...1a4003R}, and PLATO \\citep{2018haex.bookE..86R}\nmissions are bound to provide tens of Earths and Super-Earths in the\nHZ around bright M dwarfs and solar-type stars for which the estimates\nof their bulk densities $\\varrho_{\\rm p}$ might be obtained in\nprinciple, but atmospheric characterization for the latter sample\nmight be beyond the capabilities of JWST and the ground-based\ntelescopes with very large aperture diameters. The nearest stars to\nthe Sun are thus the most natural reservoir for the identification of\npotentially habitable rocky planets that might be characterized via a\ncombination of high-dispersion spectroscopy and high-contrast imaging\nwith the ground-based telescopes \\citep{Snellen+15} or via\ncoronagraphic or interferometric observations in space\n\\citep{Leger15}.\n\nUnlike the Doppler and transit methods, astrometry alone can determine\nreliably and precisely the true mass and three-dimensional orbital\ngeometry of an exoplanet, which are fundamental inputs to models of\nplanetary evolution, biosignature identification, and habitability.\nBy determining the times, angular separation and position angle at\nperiastron and apoastron passage, exquisitely precise astrometric\nposition measurements will allow the prediction of where and when a\nplanet will be at its brightest (and even the likelihood of a transit\nevent), thus (a) crucially helping in the optimization of direct\nimaging observations and (b) relaxing important model degeneracies in\npredictions of the planetary phase function in terms of orbit\ngeometry, companion mass, system age, orbital phase, cloud cover,\nscattering mechanisms, and degree of polarization\n\\citep[e.g.][]{Madhusudhan&Burrows12}. \\emph{Only a high precision\n astrometric mission's observations will have the potential to 1)\n discover most of the potentially habitable planets around the\n nearest stars to the Sun, 2) directly measure their masses and\n system architectures, and 3) provide the most complete target list\n and vastly improve the efficiency of detection of potential habitats\n for complex exo-life with the next generation of space telescopes\n and ground-based very large aperture telescopes.}\n\n\\subsubsection{Fundamental Program} \n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.65 \\hsize, trim = 0cm 0cm 0cm 1cm, clip]\n{detect_limit_theia80cm_2000h_125nas_v2.pdf}\n \\caption{Minimum masses of planets that can be detected at the\n center of the HZ of their star for the 63 best nearby A, F, G, K, M\n target systems. The target systems (either single or binary stars),\n are ranked from left to right with increasing minimum detectable\n mass in HZ around the primary system component, assuming equal\n observing time per system. Thus for binary stars, A and B components\n are aligned vertically, as they belong to the same system therefore\n they share the same rank. When the A and B mass thresholds are close\n the name is usually not explicitly written down to avoid\n overcrowding. B components that have mass thresholds above 2.2\n $M_\\oplus$ are named in gray and binaries that are estimated too\n close for follow-up spectroscopy are named in gray and in\n parenthesis. These binaries are expected to be only part of the\n secondary science program (planet formation around binaries). The\n star sample that is best for astrometry is similar to that of the\n best stars for spectroscopy in the visible, or in thermal IR (see\n text for explanations). Earths and Super-Earths with\n $M_{\\rm p} \\geq 1.5$ $M_\\oplus$ can be detected and characterized\n (actual mass and full orbit) around 22 stars. All Super-Earths with\n $M_{\\rm p} < 2.2$ $M_\\oplus$ can be detected and characterized\n around 59 stars. }\n\\label{fig:exoplan1}\n\\end{figure*}\n\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.95\\hsize]{Figdegeneracy_bis.png}\n \\caption{An example where astrometry breaks the degeneracy. Two\n simulated planetary systems are around a solar-type star at 10 pc,\n with two Jupiter-like planets at 0.5 and 2.5 AU (\\textit{left}). One\n is co-planar (\\textit{dotted black line}), the other has a mutual\n inclination of 30$^\\circ$ (\\textit{full red line}). The two\n corresponding RV curves are shown (\\textit{middle}), as well as the\n two astrometric ones (\\textit{right}). Curves are identical in the\n former case, but clearly separated in the latter revealing the\n inclined orbits.}\n\\label{fig:exoplan2}\n\\end{figure*}\n\nSurgical single-point positional precision measurements in pointed,\ndifferential astrometric mode ($<1\\mu$as), could exploit a high\nprecision astrometric mission's unique capability to search for the\nnearest Earth-like planets to the Sun. The amplitude $\\alpha$ of the\nastrometric motion of a star due to an orbiting planet is (in\nmicroarcseconds):\n\\begin{equation}\n \\alpha = 3\\left(\\frac{M_{\\rm p}}{M_\\oplus}\\right)\n \\left(\\frac{a_{\\rm p}}{1\\,\\mathrm{AU}}\\right)\n \\left(\\frac{M_\\star}{M_\\odot}\\right)^{-1}\n \\left(\\frac{D}{1\\,\\mathrm{pc}}\\right)^{-1}\\,\\,\\mu\\mathrm{as}\n\\end{equation}\nwhere $M_\\star$ is the stellar mass, $M_{\\rm p}$ is the mass of the\nplanet, $a_{\\rm p}$ is the semi-major axis of the orbit of the planet,\nand $D$ is the distance to the star. For a terrestrial planet in the\nHZ of a nearby sun-like star, a typical value is 0.3 $\\mu$as (an Earth\nat 1.0 AU of a Sun, at 10 pc). This very small motion (the size of a\ncoin thickness on the Moon as measured from the Earth) will be\naccessible to a high precision astrometric instrument by measuring the\ndifferential motion of the star with respect to far-away reference\nsources.\n\nA core exoplanet program could be comprised of 63 of the nearest A, F,\nG, K, and M stars (Fig. \\ref{fig:exoplan1}). Many of them are found in\nbinary and multiple systems. Binary stars are compelling for a high\nprecision astrometry space mission for a number of reasons. They are easier\ntargets than single stars.\n\nFurthermore, as the photon noise from the reference stars is the\ndominant factor of the error budget, the accuracy for binaries\nincreases faster with telescope staring time than around single\nstars. For binaries, the reference stars only need to provide the plate\nscale and the reference direction of the local frame, the origin point\ncoordinates are constrained by the secondary\/primary component of the\nbinary. Finally, when observing a binary, the astrometry on both\ncomponents is obtained simultaneously: the staring time is only spent\nonce as both components are within the same field of view (FoV). These\ntwo effects combined cause the observation of stars in binary systems\nto be much more efficient (as measured in $\\mbox{$\\mu$as} \\times\n\\mbox{h}^{-1\/2}$) than that of single stars.\n\nWe further stress that the complete census of small and nearby planets\naround solar-type stars is unique to high-precision astrometry. On\nthe one hand, Sun-like stars have typical activity levels producing\nDoppler noise of $\\sim1$ m\/s (or larger), which is still 10 times the\nsignal expected from an Earth-analog \\citep{2011arXiv1107.5325L}.\nHigh precision space astrometry will be almost insensitive to the\ndisturbances (spots, plages) due to stellar activity, having typical\nactivity-induced astrometric signals with amplitude below 0.1 $\\mu$as\n\\citep{2011A&A...528L...9L}.\n\nFor the full sample of the nearest stars considered in Fig.~\n\\ref{fig:exoplan1} we achieve sensitivity (at the $6\\sigma$ level) to\nplanets with $M_p \\leq 3$ $M_\\oplus$. If we consider\n$\\eta_\\uplus\\sim10\\%$, for the sample of 63 stars closest to our Solar\nSystem we thus expect to detect $\\sim6$ HZ terrestrial planets. Of\nthese, 5 will be amenable for further spectroscopic characterization\nof their atmospheres\\footnote{One target is a binary which is too\n close for follow-up spectroscopy}. A high precision astrometry\nmission could perform the measurements of the relevant stars and make\na thorough census (95\\% completeness) of these planets by using less\nthan 10\\% of a four-year mission. As indicated above, this program\nwill also be valuable for understanding planetary diversity, the\narchitecture of planetary systems (2-d information plus Kepler's laws,\nresults in 3-d knowledge) including the mutual inclination of the\norbits, a piece of information that is often missing in our\nexploration of planetary systems.\n\n\\subsubsection{Additional Exoplanet Investigations}\n\nA secondary program can help elucidate other important questions in\nexoplanetary science.\n\\begin{enumerate}\n %\n\\item {\\bf Planetary systems in S-Type binary systems}. A high\n precision astrometry mission's performance for exoplanet detection\n around nearby binaries will be of crucial importance in revealing\n planet formation in stellar systems, the environment in which\n roughly half of main-sequence stars are born. The discovery of giant\n planets in binaries has sparked a string of theoretical studies,\n aimed at understanding how planets can form and evolve in highly\n perturbed environments \\citep{2015pes..book..309T}. Giant planets\n around one component of a binary (S-type orbits) have often been\n found in orbits very close to theoretical stability limits\n \\citep[e.g.][]{2004AIPC..713..269H, 2011CeMDA.111...29T,\n 2016AN....337..300S}, and as for most of the binary targets the HZ\n of each component is stable, finding other and smaller bodies in\n their HZs is a real possibility. The contribution of a high\n precision astrometric mission could be decisive for these ongoing\n studies, by allowing the exploration of a crucial range of\n exoplanetary architectures in binaries.\n\\item {\\bf Follow-up of known Doppler systems}. Another unique use of\n a high precision astrometry mission will be the study of\n non-transiting, low-mass mul\\-ti\\-ple-planet systems that have\n already been detected with Radial Velocities (RV). High precision\n astrometry will confirm or refute controversial detections, remove\n the $\\sin i$ ambiguity and measure actual planetary\n masses. Furthermore, it will directly determine mutual inclination\n angles, which are critical to study (i) the habitability of\n exoplanets in multiple systems, since they modify the orientation of\n the spin axes and hence the way the climates change across time\n \\citep[e.g.][]{1993Natur.361..608L,2013MNRAS.428.1673B,2014MNRAS.444.1873A}\n and (ii) the dynamical evolution history of multiple systems, as\n e.g. coplanar orbits are indicative of smooth evolution, while large\n mutual inclinations and eccentricities point toward episodes of\n strong interactions, such as planet-planet\n scattering. Fig.~\\ref{fig:exoplan2} illustrates a case where\n degeneracy in RV can be removed by astrometry. Using the proper\n motion difference technique or diagnostics representing 'excess'\n residuals to a single-star fit, there are a few Gaia-based results\n worth mentioning, such as mass constraints on the cool Super-Earth\n orbiting Proxima Centauri \\citep{2020A&A...635L..14K}, the inferred\n true mass for HD\\,114762b \\citep{2019A&A...632L...9K}, and the first\n high-quality measurement of highly mutually inclined orbits in the\n Pi Mensae system \\citep{2020A&A...642A..31D, 2020MNRAS.497.2096X}.\n\\item {\\bf Planetary systems on and off the main sequen\\-ce}. Gaia\n will be able to detect thousands of giant planetary companions\n around stars of all ages (including pre- and post-main-sequence),\n spectral type, chemical abundance, and multiplicity with results\n expected in the DR4 and DR5 data releases\n \\citep{2018haex.bookE..81S, 2018A&A...614A..30R,\n 2020AJ....160...16S}.\n \n \n \n \n \n \n A high precision astrometry mission could cherry-pick from Gaia\n discoveries and identify systems amenable to follow-up to search for\n additional low-mass components in such systems, particularly in the\n regime of stellar parameters difficult for radial velocity work like\n early spectral types, young ages, very low metallicity, white\n dwarfs. Some of the systems selected might also contain transiting\n companions identified by TESS and PLATO (and possibly even Gaia\n itself), or planets directly imaged by SPHERE on the VLT or European\n Extremely Large Telescope.\n\\item {\\bf Terrestrial planets around Brown Dwarfs}. So far, among the\n few planetary mass objects that have been associated with brown dwarf\n (BD) hosts using direct imaging and microlensing techniques, only\n one is likely to be a low-mass planet \\citep[][and references\n therein)]{2015ApJ...812...47U}. However, there are both\n observational\n \\citep{2008ApJ...681L..29S,2012ApJ...761L..20R,2014ApJ...791...20R}\n as well as theoretical\n \\citep{2007MNRAS.381.1597P,2013ApJ...774L...4M} reasons to believe\n that such systems could also be frequent around BDs. The recent\n identification of a trio of short-period Earth-size planets\n transiting a nearby star with a mass only $\\sim10\\%$ more massive\n than the hydrogen-burn\\-ing limit \\citep{2016Natur.533..221G} is a\n tantalizing element in this direction. In its all-sky survey, Gaia\n will observe thousands of ultra-cool dwarfs in the backyard of the\n Sun with sufficient astrometric precision to reveal any orbiting\n companions with masses as low as that of Jupiter\n \\citep{2014MmSAI..85..643S}. A high precision astrometry mission\n could push detection limits of companions down to terrestrial\n mass. If the occurrence rate of $P\\leq1.3$ d, Earth-sized planets\n around BDs is $\\eta=27\\%$ as suggested by\n \\citet{2017MNRAS.464.2687H} based on extrapolations from transit\n detections around late M dwarfs, the high precision measurements,\n probing for the first time a much larger range of separations with\n respect to transit surveys with sensitivity to low-mass planets,\n will unveil a potentially large number of such companions, and place\n the very first upper limits on their occurrence rates in case of\n null detection.\n\\item \\textbf{Astrometric effect of disks.} As pointed out by\n \\citet{2016A&A...592A..39K}, the photocenter of a star+disc system\n will have an elliptic motion for asymmetric discs. The latter is\n likely to mimic a planet or to perburb the characteriscs of an\n existing planet. Additional measurements (e.g.\\ infrared flux) will\n be necessary to disentangle the disk asymmetry from a real planet.\n\n\\end{enumerate}\n\\subsection{Compact objects}\n\\label{sec:compact-objects}\n\n\n\\subsubsection{X-ray Binaries}\nThe brightest Galactic X-ray sources are accreting compact objects in\nbinary systems. Precise optical astrometry of these X-ray binaries\nprovides a unique opportunity to obtain quantities which are very\ndifficult to obtain otherwise. In particular, it is possible to\ndetermine the distances to the systems via parallax measurements and\nthe masses of the compact objects by detecting orbital motion to\nmeasure the binary inclination and the mass function. With a high\nprecision astrometric mission, distance measurements are feasible for\n$>$50 X-ray binaries\\footnote{with 2\\,000 hours of observation}, and\norbital measurements will be obtained for dozens of systems. This\nwill revolutionize the studies of X-ray binaries in several ways: here\nwe discuss goals for neutron stars (NSs), including constraining their\nequation of state (EoS), and for black holes (BHs).\n\nMatter in the NS interior is compressed to densities exceeding\nthose in the center of atomic nuclei, opening the possibility to probe\nthe nature of the strong interaction under conditions dramatically\ndifferent from those in terrestrial experiments and to determine the\nNS composition. NSs might be composed of nucleons only;\nstrange baryons (hyperons) or mesons might appear in the core or even\ndeconfined quark matter, forming then a hybrid star with a quark\nmatter core and hadronic matter outer layers; or of pure strange quark\nmatter (a quark star). \nA sketch of the different possibilities is given in\nFig.~\\ref{fig:nsstructure}. Via the equation of state (EoS), matter\nproperties determine the star's radius for a given mass. In\nparticular, since general relativity limits the mass for a given EoS,\nthe observation of a massive NS can exclude EoS models. Presently, the\nmain constraint stems from the measurements of very massive NSs in\nradio pulsar\/white dwarf systems which have been reported with high\nprecision \\citep{Demorest2010, Antoniadis2013, 2018ApJS..235...37A, 2020NatAs...4...72C}.\n\nThe key to constraining the NS EoS is to measure the masses and radii\nof NSs. While masses have been measured for a number of X-ray binary\nand radio pulsar binary systems \\citep[e.g., ][]{Lattimer2016,\n Ozel2016}, the errors on the mass measurements for most X-ray\nbinaries are large (see Fig.~\\ref{fig:nsmasses}, left). The ultimate\nconstraint on the EoS will be a determination of radius and mass of\nthe same object, and a small number of such objects might be\nsufficient to pin down the entire EoS (e.g. \\citet{Ozel2009}), see\nFig.~\\ref{fig:nsmasses} (right), where several $M$-$R$ relations for\ndifferent EoSs are shown. Current techniques to determine radii rely\non spectroscopic measurements of accreting neutron stars, either in\nquiescence \\citep{Heinke2014} or during thermonuclear (type I) X-ray\nbursts \\citep{Ozel2016}, and also timing observations of surface\ninhomogeneities of rotating NSs \\citep{Miller2016, Haensel2016}.\n\nA high precision astrometric mission will contribute by obtaining\nprecise mass constraints with orbital measurements \\citep{Tomsick2010}\nand by improving distance measurements. Distances must be known\naccurately to determine the NS radii. For that purpose, new high\nprecision data can be combined with existing and future X-ray data,\ne.g., from Athena, which is scheduled as the second large-class (L2)\nmission in ESA's Cosmic Vision. The Athena Science Working Group on\nthe endpoints of stellar evolution has observations of quiescent\nneutron star X-ray binaries to determine the NS EoS as its first\nscience goal; however, their target list is restricted to systems that\nare in globular clusters. A high precision astrometric mission will\nenable distance measurements for many more NS X-ray binaries, allowing\nAthena to expand their target list.\n\nOther techniques for constraining the NS EoS might also be possible in\nthe future:\ndetecting redshifted absorption lines; determining the NS moment of\ninertia of systems like the double pulsar J0737$-$3039; and more\ndetections of gravitational wave (GW) emission by LIGO, Virgo or Kagra\nfrom the inspiral of a binary neutron star merger like for GW170817\n\\citep{Abbott2017}.\nGWs from the post-merger phase could strongly constrain the EoS, too.\nHowever, the mass and distance measurements that\na high precision astrometric mission will obtain use techniques that\nare already well-established, providing the most certain opportunity\nfor greatly increasing the numbers of NSs with mass or radius\ndeterminations.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\hsize]{nscrosfw.png}\n \\caption{Sketch of the different existing possibilities for the\n internal structure of a neutron star. Figure courtesy of\n \\citet{Weber_2001}.}\n\\label{fig:nsstructure}\n\\end{figure}\n\n\n\\begin{figure*}[htb]\n \\centering\n \n \n \\hfill\n \\includegraphics[width=0.35\\hsize]{masses_xray-eps-converted-to.pdf}\n \\hfill\n \\includegraphics[width=0.45\\hsize]{MRtheia.pdf}\n \\hfill~\\\\\n \\caption{Left: Neutron star mass measurements in X-ray\n binaries, update from \\citet{Lattimer2005},\n \\texttt{http:\/\/stellarcollapse.org}. Right: $M$-$R$ relation\n for different EoS models (adapted from \\citet{Fortin2016}): NS\n with a purely nucleonic core (in blue), with a core containing\n hyperons at high density (in red), and pure strange quark stars\n (in green). The horizontal grey bars indicate the masses of PSR\n J1614$-$2230 and PSR J0348+0432. The models indicated by dotted\n or dashed lines are either not compatible with NS masses or\n nuclear physics constraints. Note that a transition to matter\n containing hyperons is not excluded by present constraints.}\n\\label{fig:nsmasses}\n\\end{figure*}\n\nIn addition to the goal of constraining the NS EoS, NS masses are also\nrelevant to NS formation and binary evolution. Current evolutionary\nscenarios predict that the amount of matter accreted, even during\nlong-lived X-ray binary phases, is small compared to the NS mass.\nThis means that the NS mass distribution is mainly determined by birth\nmasses. Determining the masses of NSs in X-ray binaries, therefore,\nalso provides a test of current accretion models and evolutionary\nscenarios, including the creation of the NSs in supernovae.\n\nBHs are, according to the theory of general relativity, remarkably\nsimple objects. They are fully described by just two parameters,\ntheir mass and their spin. Precise masses are available for very few\nBHs in X-ray binaries. The recent detection of gravitational waves\n\\citep{Abbott2016c} found in the binary BH mergers\n\\citep{Abbott2016b,2019PhRvX...9c1040A} show that they have, on\naverage, higher masses and probably lower spins than the BHs in X-ray\nbinaries. These measurements are difficult to explain based on our\nunderstanding of stellar evolution and the fate of massive\nstars. Although BHs leave few clues about their origin, one more\nparameter that can be determined is the proper motion of BHs in X-ray\nbinaries. Measurements of proper motions provides information about\ntheir birthplaces and formation. It includes whether they were\nproduced in a supernova (or hypernova) or whether it is possible for\nmassive stars to collapse directly to BHs. A few BH X-ray binaries\nhave proper motion measurements (e.g., \\citet{Mirabel2001}), but this\nnumber will rise dramatically with the astrometry measurements that a\nhigh precision astrometry mission will provide.\n\nCurrently, the cutting edge of research in BH X-ray binaries involves\nconstraining BH spins, including the rate of spin and the orientation\nof the spin axis. Techniques for determining the rate of spin include\nmeasuring the relativistic broadening of the fluorescent iron\n$K_\\alpha$ line in the X-ray emission and the study of the thermal\ncontinuum X-ray spectra \\citep{Remillard2006, Miller2007}. Concerning\nthe direction of their spin axes, there is evidence that the standard\nassumption of alignment between the BH spin and orbital angular\nmomentum axes is incorrect in some, if not many, cases\n\\citep{Maccarone2002, Tomsick2014, Walton2016}, likely requiring a\nwarped accretion disc. Theoretical studies show that such\nmisalignments should be common \\citep{King2016}. However, binary\ninclination measurements rely on modeling the ellipsoidal modulations\nseen in the optical light curves \\citep{Orosz2011}, which is subject\nto systematic uncertainties, and a high precision astrometry mission\nwill be able to provide direct measurements of orbital inclination for\nmany of the BH X-ray binaries that show evidence for misalignments and\nwarped discs.\n\n\\subsubsection{Astrometric microlensing}\n\\label{sec:Compact objects in the GC}\n\nIn 1986 Bohdan Paczy{\\'n}ski \\citep{Paczynski1986} proposed a new\nmethod for finding compact dark objects, via photometric gravitational\nmicrolensing. This technique relies on continuous monitoring of\nmillions of stars in order to spot the temporal brightening due to\nspace-time curvature caused by the presence and motion of a dark\nmassive object. Microlensing reveals itself also in astrometry, since\nthe centre of light of both unresolved images (separated by $\\sim$1\nmas) changes its position while the relative brightness of the images\nchanges in the course of the event. Astrometric time-series at sub-mas\nprecision over the course of a couple of years will provide\nmeasurement of the size of the Einstein Ring, which combined with the\nphotometric light curve, will directly yield the lens distance and\nmass. Most microlensing events are detected by large-scale surveys,\ne.g., OGLE and, in future possibly also the Rubin Observatory\n(previously known as the LSST). At typical\nbrightness of V=19-20mag only a high-precision astrometry mission will\nbe capable of providing good-enough astrometric follow-up of\nphotometrically detected microlensing events\n(Fig.~\\ref{fig:microlensing}). Among 2000 events found every year, at\nleast a couple should have a black hole as the lens, for which the\nmass measurement via astrometric microlensing will be possible.\n\\let\\figmicrolensig=\\ref{fig:microlensing}\n\\begin{figure*}[htb]\n \\centering\n \\includegraphics[width=.98 \\hsize]{OGLE3-ULENS-PAR-02}\n \\caption{Microlensing event, OGLE3-ULENS-PAR-02, the best\n candidate for a $\\sim$10M$_\\odot$ single black hole. Left:\n photometric data from OGLE-III survey from 2001-2008. Parallax\n model alone can only provide mass measurement accuracy of\n 50-100$\\%$. Right: simulated astrometric microlensing path for a\n similar event if observed with Theia, a high-precision astrometry\n mission. Combining this mission's superb astrometric accuracy with\n long-term photometric data will yield mass measurements of black\n holes and other dark compact object to 1$\\%$ even at faint\n magnitudes.}\n \\label{fig:microlensing}\n\\end{figure*}\n\n\\begin{figure*}[t] \n\\centering\n\\includegraphics[width=0.75\\hsize]{skyplot_4Oct.pdf}\n\\caption{Sky map of the targets considered for observations with a\n high precision astrometric mission.}\n\\label{fig:skyplot}\n\\end{figure*}\n\nDetection of isolated black holes and a complete census of masses of\nstellar remnants will for the first time allow for a robust\nverification of theoretical predictions of stellar\nevolution. Additionally, it will yield a mass distribution of lensing\nstars as well as hosts of planets detected via microlensing.\n\n\\subsection{Cosmic distance ladder}\n\\label{sec:cosm-dist-ladd}\n\nThe measurement of cosmological distances has revolutionized modern\ncosmology and will continue to be a major pathway to explore the\nphysics of the early Universe. The age of the Universe ($H_0^{-1}$) is\na key measurement in non-standard DM scenarios. Its exact value is\ncurrently strong\\-ly debated, with a number of scientific papers\npointing at discrepancies between measurement methods at the\n2-3$\\sigma$ level. But the most serious tension with a discrepancy at\nthe 3-4 $\\sigma$ level appears between CMB estimates\n($H_0 = 67.8 \\pm 0.9$\\,km\/s\/Mpc) or for that matter BAO results from\nthe SDSS-III DR12 data combined with SNIa which indicate\n$H_0 = 67.3 \\pm 1.0$\\,km\/s\/ Mpc \\citep[see][]{Alam2016b} and\nmeasurements based on Cephe\\-ids and SNIa \\citep{2016ApJ...826...56R}\ngiving $H_0 = 73.24 \\pm 1.74$\\,km\/s\/Mpc.\n\nThe tension between the methods can be due to unknown sources of\nsystematics, to degeneracies between cosmological parameters, or to\nnew physics \\citep[e.g.][]{Karwal&Kamionkowski16}. It is therefore of\ncrucial importance to consider methods capable of measuring $H_0$ with\nno or little sensitivity to other cosmological\nparameters. Uncertainties can be drastically reduced by measuring time\ndelays (TD) in gravitationally lensed quasars \\citep{Refsdal64}, as\nthis technique only relies on well-known physics that is General\nRelativity. With enough statistics, and good modeling of the mass\ndistribution in the lensing galaxy, TD measurements can lead to\npercent-level accuracy on $H_0$, independently of any other\ncosmological probe \\citep[e.g. ][]{Bonvin2016a, Suyu2013, Suyu2014}.\nIn practice, TDs can be measured by following the photometric\nvariations in the images of lensed quasars. As the optical paths to\nthe quasar images have different lengths and they intersect the lens\nplane at different impact parameters, the wavefronts along each of\nthese paths reach the observer at different times. Hence the notion of\nTD.\n\nSignificant improvements in lens modeling, combined with long-term\nlens monitoring, should allow measuring $H_0$ at the percent\nlevel. The H0LiCOW program ($H_0$ Lenses in COSMOGRAIL's Wellspring),\nwhich focuses on improving the detailed modeling of the lens galaxy\nand of the mass along the line of the sight to the background quasar,\nled to $H_0 = 73.3 ^{ +1.7}_{-1.8}$\\,km\/s\/Mpc (that is 2.4\\% precision)\nin a flat $\\Lambda$CMD Universe by using deep HST imaging, Keck\nspectroscopy and AO imaging and wide field Subaru imaging \\citep{\n Bonvin2016a, Rusu2016, Sluse2016, Suyu2016, Wong2016,\n 2020MNRAS.498.1420W}. This value is in excellent agreement with the\nmost recent measurements using the distance ladder (though in tension\nwith the CMB measurements from Planck) but still lacks precision.\n\nBy performing photometric measurements with the required sensitivity\nand no interruption, the combination of a high precision astrometric\nmission and excellent modeling of the lens galaxy, will enable\nmeasurement of $H_0$ at the percent level and remove any possible\ndegeneracies between $H_0$ and other cosmological parameters. This\nwill open up new avenues to test the nature of DM. An alternative\ntechnique consists of using trigonometric parallaxes. This is the only\n(non-statistical and model-independent) direct measurement method and\nthe foundation of the distance scale. A high precision astrometric\nmission has the potential to extend the ``standard candles'' - the more\ndistant pulsating variables: Cepheids, RR Lyrae, Miras and also\nStellar Twin stars - well beyond the reach of Gaia.\n\nThese distance measurements can be transferred to nearby galaxies\nallowing us to convert observable quantities, such as angular size and\nflux, into physical qualities such as energy and\nluminosity. Importantly, these distances scale linearly with $H_0$,\nwhich gives the temporal and spatial scale of the Universe. With this\nimproved knowledge, we will then be able to better understand the\nstructure and evolution of both our own and more distant galaxies, and\nthe {\\it age} of our Universe.\n\n\\subsection{Position of the science targets in the sky}\n\\label{sec:posit-science-tar}\n\nThe different targets considered for observations with a high\nprecision astrometry mission have been located in\nFig.~\\ref{fig:skyplot} on a sky map.\n\n\\section{Possible space mission}\n\\label{sec:poss-space-miss}\n\nSeveral mission profiles have been considered in the last few years\nfocused on differential astrometry, for instance NEAT, micro-NEAT and\nTheia. Additional new differential astrometry mission configurations\nadapted with technological innovations will certainly be envisioned to\npursue accurate measurements of the extremely small motions required\nby the science cases in this White Paper.\n\n\\subsection{Scientific requirements}\n\\label{sec:scient-requ}\n\nTo address the science described in this white paper, a high precision\nastrometry mission should stare towards:\n\\begin{itemize}\n\\setlength\\itemsep{0em} \n\\item dwarf spheroidal galaxies to probe their dark matter inner\n structure;\n\\item hyper-velocity stars to probe the triaxiality of the halo, the\n existence of compact minihalo objects and the time delay of quasars;\n\\item the Galactic Disc, to probe DM subhalos and compact minihalo\n objects; \n\\item star systems in the vicinity of the Sun, to find the nearest\n potentially habitable terrestrial planets; \n\\item known X-ray binaries hosting neutron stars or black holes.\n\\end{itemize}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\hsize,trim = 0.5cm 0cm 0cm 0cm, clip]{vposerrdist3_4Oct.pdf}\n \\caption{Expected plane-of-sky velocity errors from a high\n precision astrometry mission's proper motions as a function of\n distance from Earth. These errors respectively correspond to 40 and\n 1000 cumulative hours of exposures for exo\\-planets (\\textit{green})\n and more distant objects (\\textit{cyan} and \\textit{blue}), during a\n 4 year interval for observations, including the systematic limit\n from calibration on Gaia reference stars. The expected precision for\n specific objects are highlighted. The accuracy for the 5-year Gaia\n mission is shown in \\textit{magenta}.}\n\\label{fig:vposerrvsDist}\n\\end{figure}\n\n\\begin{figure*}[t] \n \\centering\n \n \n \n \n \n \\hfill\n \\includegraphics[width=0.85\\hsize,clip]{Fig-17-astrometric-EoM-precision-v2.jpg}\n \\hfill~\\\\ \\caption{Estimated RMS precision on a high precision astrometry\n mission relative parallax (\\textit{left}, for ecliptic latitude\n $0^\\circ$) and proper motion (\\textit{right}) in the\n $R$-band. Also shown for comparison are the estimated accuracies\n for 10~years LSST \\citep{2009arXiv0912.0201L} as well as the\n 5-year nominal Gaia mission \\citep{deBruijne+15} (vertical spread\n caused by position on the sky, star color, and bright-star\n observing conditions). Small-scale spatial correlations\n ($<$1$^\\circ$) between Gaia reference sources will limit the\n maximum reachable absolute parallax and proper motion calibration\n for a high precision astrometry mission\n \\citep{2012A&A...543A..14H, 2012A&A...543A..15H}, indicated by the\n light blue band for a range of assumed spatial correlations\n \\citep[expected to be much below $r=0.5$\\%;][]{2009IAU...261.1703H}\n as a function of reference star magnitude. Bright stars ($V<13$) and\n low star-density regions will have the highest correlations.}\n\\label{fig:astroComparisons}\n\\end{figure*}\n\n\n\n\\begin{table*}[t]\n \\small\n \\caption{Summary of science cases with most stringent\n performance requirements set in each case. Figures are based on a\n 4 year mission, thermal stabilisation ($+$slew time) is assumed to\n take 30$\\%$ of the mission time. \n \\label{tab:tech.summary.science}}\n \\centering\n \\begin{tabular}{lrrrrr} \n \n \\hline \nProgram \n& Used\n& Mission \n& Number of\n& Benchmark target \n& EoM precision \\\\\n& time (h)\n& fraction \n& objects per field \n& $R$ mag (and range) \n& (at ref.\\ mag.) \\\\\n \\hline \n \\hline\n Dark Matter & \n 17\\,000 &\n 0.69 &\n 10$^2$--10$^5$ &\n 20 (14--22) &\n 10 $\\mu$as \\\\ \n \n $\\&$ compact objects \n \\\\\n \\hline\n Nearby Earth-like planets \n & 3\\,500 &\n 0.14 &\n $<$20 &\n 5 (1--18) &\n 0.15 $\\mu$as \\\\ \n $\\&$ follow-up \\\\\n \n \\hline\n Open observatory & \n 4\\,000 &\n 0.17 &\n 10-10$^5$ &\n\n 6 (1-22) &\n 1.0 $\\mu$as \\\\\n \n \n \\hline \n\n Overall requirements& \n 24 \\,500 &\n 1.00 &\n 10$^1$-10$^5$ &\n 6 (1-22) &\n 0.15-10 $\\mu$as \\\\ \n \n \n \\hline \n \\end{tabular}\n\\end{table*}\nFor a targeted mission, the objects of interest must be sampled\nthroughout the lifetime of the mission. After re-pointing the\ntelescope and while waiting for stabilization, photometric surveys,\ne.g. for measurements of $H_0$ using lensed quasar time delays could\nbe performed, thus optimizing the mission scientific\nthroughput. Fig.~\\ref{fig:skyplot} shows a sample sky map with\npotential targets.\n\nAs illustrated in Fig.~\\ref{fig:vposerrvsDist}, high precision\nastrometric missions could measure the plane-of-sky velocities of the\nfaintest objects in the local Universe, with errors as small as a few\nmm\/s in the case of the hosts of Earth-mass exoplanets in the\nhabitable zone of nearby stars, a few m\/s for stars in the Milky Way\ndisc, i.e. for kinematical searches for DM sub-halos, micro-lensing\nsearches for ultra-compact minihaloes, and for the companions of\nneutron stars and black holes in X-ray binaries, 200m\/s for\nhyper-velocity stars whose line of sight velocities are typically\n$>500$\\,km\/s, and finally 1\\,km\/s for $R=20$\\, mag stars for dwarf\nspheroidal galaxies.\n\nA mission concept with an expected Theia-like astrometric precision,\nas shown in Fig.~\\ref{fig:astroComparisons}, surpasses what will be\nachieved by other approved space astrometric surveys and ground\nsurveys, thus unlocking science cases that are still unreachable.\n\nTable \\ref{tab:tech.summary.science} summarizes the science cases with\nthe most stringent performance requirements. To cover the science\nquestions from this White Paper, any mission concept must be flexible,\nallowing for observing modes covering a wide flux dynamical\nrange. This requires the concepts to cope with \\emph{Deep Field\n Modes}, aimed towards objects such as dwarf galaxies, and\n\\emph{Bright Star Modes}, focused on the study of planetary systems\naround nearby stars.\n\n\\subsection{Example of a medium-size mission}\n\\label{sec:prop-scient-instr}\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.9\\hsize]{theiam5-plm-concept} \n \\caption{Overall layout of the Theia Payload Module concept. Volume\n is estimated at $1.6\\times1.9\\times2.2$m$^3$.}\n \\label{fig:theiam5-plmconcept}\n\\end{figure*}\n\nThe Payload Module (PLM) of a high precision astrometric mission must\nbe simple. It is essentially composed of four subsystems: telescope,\ncamera, focal plane array metrology and telescope metrology. In the\ncase of the Theia\/M5 concept, they were designed applying heritage\nfrom space missions and concepts like Gaia, HST\/FGS, SIM, NEAT\n(proposed for the ESA M3 opportunity),\nTheia (proposed for the ESA M4 opportunity), and Euclid.\n\nHowever, achieving microarcsecond differential astrometric precision\nrequires the control of all effects that can impact the determination\nof the relative positions of the point spread function. The typical\napparent size of an unresolved star corresponds to 0.2~arcseconds for\na 0.8~m telescope operating in visible wavelengths. The challenge is\ntherefore to control systematic effects to the level of 1 part per\n200\\,000. The precision of relative position determination in the\nFocal Plane Array (FPA) depends on i) the photon noise, which can be\neither dominated by the target or by the reference stars; ii) the\ngeometrical stability of the instrument, iii) the stability of the\noptical aberrations, and iv) the variation of the detector quantum\nefficiency between pixels. The control of these effects impairs other\nmissions that otherwise could perform microarcsecond differential\nastrometry measurements, like HST, Kepler, the Roman Space Observatory\n(previously known as WFIRST), or Euclid, posing\nfundamental limits to their astrometric accuracy. All these effects\nmust be taken into account in any high precision differential\nastrometry mission concept.\n\nTo address the challenges and fulfil the requirements from section\n\\ref{sec:scient-requ}, two different possible concepts can be\ninvestigated. A NEAT-like mission consisting of a formation flight\nconfiguration \\citep{2012ExA....34..385M} or an Euclid-like\nmission,\\footnote{Euclid red book:\n \\url{http:\/\/sci.esa.int\/euclid\/48983-euclid}\n \\url{-definition-study-report-esa-sre-2011-12}.} but with a single\nfocal plane and additional metrology subsystems. Both concepts are\nbased on adopting a long focal length, diffraction-limited telescope,\nand additional metrological control of the focal plane array. The\nproposed Theia\/M5 mission concept was the result of a trade-off\nanalysis between both concepts.\n\n\\subsubsection{Telescope concept}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width = 0.9\\hsize]{theiam5-onaxis-korsch}\\\\*[1em]\n\\caption{On-axis Korsch TMA option. Ray-tracing and spot diagrams for\n the entire FoV. This design was adopted as the baseline for the\n Theia\/M5 proposal. EFL is Effective Focal Length.}\n\\vspace{-0.4cm}\n\\label{fig:theiam5-onaxis-korsch}\n\\end{figure}\n\nThe Theia PLM concept consists of a single Three Mirror Anastigmatic\n(TMA) telescope with a single focal plane (see\nFig.~\\ref{fig:theiam5-plmconcept}) covering a $0.5^\\circ$\nfield-of-view with a mosaic of detectors. To monitor the mosaic\ngeometry and its quantum efficiency, the PLM includes a focal plane\nmetrology subsystem, while to monitor the telescope geometry, a\ndedicated telescope metrology subsystem is used.\n\nTo reach sub-microarcsecond differential astrometry a\ndiffraction-limited telescope, with all aberrations controlled, is\nnecessary. A trade-off analysis was performed between different\noptical designs, which resulted in two optical concepts that could\nfulfil all requirements. Both are based on a Korsch TMA telescope;\none is an on-axis solution while the second is an off-axis\ntelescope. In both cases only three of the mirrors are powered\nmirrors. While the on-axis solution adopts a single folding mirror,\nthe off-axis solution adopts two folding mirrors. The on-axis design\nwas the Theia\/M5 baseline (Fig.~\\ref{fig:theiam5-onaxis-korsch}). More\nrecently, however, studies from NASA\/JPL show that a customized and\ncorrected Rit\\-chey-Chr\u00e9tien can reach 5 $\\mu$as over a $0.5^\\circ$ FoV,\nwhich even if not capable of addressing habitable exoplanet science\ncases, would provide a valuable instrument for Dark Matter studies.\n\nTo achieve the precision by centroiding as many stars as possible, a\nmosaic of detectors (in principle CCD or CMOS) must be assembled on\nthe focal plane (Fig.~\\ref{fig:theiam5-fpaconcept}). The detectors\nmust feature small pixels ($\\sim 10\\,\\mu$m) and well controlled\nsystematic errors along the lifetime of the mission. Detailed in-orbit\ncalibration of the focal plane and detector geometry and response must\nbe monitored, and in the Theia concept this is addressed via a\ndedicated laser metrology (see Sect.~\\ref{sec:metrology}).\n\n\\begin{figure}[t]\n\\centering %\n\\includegraphics[width = \\hsize]{theiam5-camera-concept-lateral}\\\\*[1em]\n\\vspace{-0.4cm}\n\\caption{Concept for the Theia\/M5 Camera. Left: concept for the FPA detector\n plate. Right: overall view of the camera concept.}\n\\label{fig:theiam5-fpaconcept}\n\\end{figure}\n\n\\subsubsection{Mission configuration and profile}\n\\label{sec:miss-conf-prof}\n\n\\begin{figure*}[p]\n \\centering\n \\includegraphics[width=0.98\\hsize]{theiam5-satellite-key-features}\n \\caption{Proposed Theia satellite concept (Thales Alenia Space).\n FGS: Fine Guidance Sensor; FOG: Fiber Optics Gyroscope; AOCS:\n Attitude and Orbit Control System,TT\\&C: Telemetry, Tracking \\&\n Control; TWTA: Travelling Wave Tube amplifier Assembly; LGA: Low Gain Antenna;\n HGA: High Gain Antenna.}\n \\label{fig:mission.sc.concept}\n\\end{figure*}\n\n\\begin{table}[t]\n \\centering\n \\caption{Theia's mission main characteristics.}\n \\smallskip\n \\includegraphics[width=\\hsize]{theiam5-mission-main-characteristics}\n \\vspace{-0.4cm}\n \\label{tab:mission-main-characteristics}\n\\end{table}\n\nThe time baseline (Table~\\ref{tab:mission-main-characteristics}) to\nproperly investigate the science topics of this White Paper would be\nat least 4 years, including time devoted to orbit maintenance. A total\nof approximately 6 months has been estimated for the orbit transfer\nincluding the spacecraft and instrument commissioning. This estimate\nis made from the total of $\\sim 35\\,000$ h corresponding to the total\ntime for the scientific program (Table \\ref{tab:tech.summary.science})\nand considering that about 15 min per slew will be dedicated to\nreconfiguration and station-keeping, while thermal stabilization time\nis in addition to the slew time.\n\nSome instrument key features of the Theia concept are presented in\nFig.~\\ref{fig:mission.sc.concept}. The concept is inspired by the\nEuclid service module with a downscaled size to minimize mass and\nimprove mechanical properties. Similar to the Euclid and Herschel\nsatellites, Theia's Korsch telescope is accommodated on top of the\nservice module in a vertical position leading to a spacecraft height\nof about 5m. This concept optimizes the payload size.\n\n\\section{Worldwide context of ground-based and space science}\n\nObservations carried out with a mission dedicated to high precision\nastrometry will add significant value and will benefit from a number\nof other ground-based and space missions operating in the 2030s and\nbeyond, including ESA's Athena, PLATO, Euclid and Gaia missions, ESO's MICADO\nand Gravity instruments, CTA, SKA, the NASA\/ESA\/CSA JWST and the Rubin\nObservatory (previously known as LSST). For example:\n\\begin{itemize}\n\\item \\textbf{JWST:} Estimates suggest that JWST will be able to\n detect Lyman Break galaxies with absolute magnitudes as faint as\n $M_{\\rm{UV}} \\sim -15$ at $z\\sim7$ \\citep{2013MNRAS.434.1486D},\n corresponding to halo masses of about $10^{9.5} \\ M_{\\odot}$. The\n combination of a high precision astrometry mission and JWST's\n observations will enable unambiguous tests of DM.\n\\item \\textbf{PLATO:} this mission will look at planetary transits and\n star oscillations in two fields (each covering 2250 deg$^2$), for\n 2-3 years each, in host stars brighter than 16 mag. PLATO high\n cadence continuous monitoring of its target stars will provide\n information on the internal structure of the stars, allowing\n determination of their stellar ages and masses. A high precision\n astrometry mission will benefit from PLATO characterization of many\n of the astrometry mission's core star samples. For close PLATO\n stars where transits were observed this astrometry mission can\n measure additional inclined planets.\n\\item \\textbf{SKA:} SKA aims to use radio signals to look for building\n blocks of life (e.g. amino acids) in Earth-sized planets\n \\citep{2015aska.confE.120Z, 2015aska.confE.115H}. A high precision\n astrometry will identify target planets from their astrometric\n ``wobble'' that can be followed-up spectroscopically with the\n SKA. Furthermore, SKA aims to use its immensely fast sky coverage to\n detect transients \\citep{2015aska.confE..51F}, such as supernovae and\n gamma-ray bursts. With its precise astrometry, Theia will help study\n the specific locations of such events in stellar clusters.\n\\item \\textbf{CTA: } The Cherenkov Telescope Array (CTA) in the\n Northern and Southern Hemispheres will carry out measurements of the\n gamma-ray flux with almost complete sky coverage and unprecedented\n energy and angular resolution, in the $\\sim$ 0.02-100\\,TeV energy\n range \\citep{2011ExA....32..193A}. The sub-micro\\-arc-sec\\-ond\n performance of a high precision astrometry mission allow us\n investigating the so-called J-factor which corresponds to the\n brightness of the gamma-ray flux in dwarf galaxies and thus\n determines the prime candidates for CTA's observations. CTA also\n aims at observing star forming systems over six orders of magnitude\n in formation rate, to measure the fraction of interacting cosmic\n rays as a function of the star-formation rate. By combining high\n precision astrometry and CTA measurements, we will better understand\n the relative importance of cosmic rays and DM in places where\n star-formation is important. Furthermore, a small number of black\n hole and neutron star binary systems in our Galaxy are known to emit\n gamma rays. The mechanism by which the particle acceleration is\n achieved is not well understood. The sub-micro\\-arcsecond\n performance of a high precision astrometry mission will al\\-low us\n to probe the velocity structure of the nearby gam\\-ma-ray bright\n radio galaxies of NGC\\,1275, IC\\,310, M\\,87, and Cen\\,A, which\n combined with CTA's observations will enable important astrophysics\n breakthroughs.\n\\end{itemize}\n\n\\section{Technology challenges for high precision astrometry}\n\\label{sec:cont-other-miss}\n\n\n\\subsection{Spacecraft technology and cost}\n\nThere have been several propositions for a space mission dedicated to\nhigh precision astrometry: a 6 meter baseline visible interferometer\non a single satellite like SIM or SIM-Lite\n\\citep{2008SPIE.7013E..4TG}; a single mirror off-axis parabola\n1\\,m-diameter telescope based on two spacecraft, one carrying the\ntelescope mirror and the other the focal plane like the NEAT telescope\n\\citep{2012ExA....34..385M}; or a single-mirror telescope like Theia\n\\citep{2016SPIE.9904E..2FM, Boehm2017}. The variety of the concepts\nshows that there are areas of progress on spacecraft technologies,\nespecially concerning formation flying, actively-controlled\n large structure interferometers.\n\nOne interesting potential solution to be considered is the nanosat\ntechnology and the cost reduction that is linked to it. There is a\nhuge cost difference between cubesats ($< 10$\\,M\\euro{}) and an ESA\nM-class mission ($400-500$\\,M\\euro{}) or NASA MIDEX\/Discovery mission\n($300-500$\\,M\\$). The cubesat technology has matured and many hundreds\nare launched every year. That technology has now crept into micro-sats\nthat are up to 200\\,kg and spacecraft bus of this category are now\n$<5$\\,M\\euro{}, while only a few years ago they were\n$\\sim40$\\,M\\euro{}. Because of their low cost and the high number of\nflying satellites, this technology has now demonstrated 5 year typical\nlifetime, comparable to a more expensive traditional spacecraft. In\nany case, all the price scales will change between now and the epoch\nwhen Voyage 2050 will be implemented and that includes flying heavier\npayloads because of the decrease of launch costs \\citep{Jones2018}.\n \n\\subsection{Detection}\n\nPresently, two detector technologies are used: CCD or CMOS. CMOS\ndetectors present a high quantum efficiency over a large visible\nspectral band that can also reach infrared wavelengths depending on\nthe sensitive layer. CMOS detectors also have programmable readout\nmodes, faster read\\-out, lower pow\\-er, better radiation hardness, and\nthe ability to put specialized processing within each pixel. On the\nother hand there are many known detector systematics, even for\nad\\-van\\-ced detectors like the Teledyne H4RG10. The main challenging\neffects are the following: fluence-dependent PSF, correlated read\nnoise, inhomogeneity in electric field lines and persistence effects\n\\citep[e.g.][]{Simms2009}. All mission proposals so far were based on\nCCD technology, but detector evolution will certainly take place on\nthe context of any mission concept to answer the challenges being\nposed by the Voyage 2050 White Papers.\n\nIf a Theia-like mission is selected for the 2040's, detector\ntechnology might be different from anything we have in place\nnowadays. The main requirements are small pixels, low read-out noise (RON)\non large format focal plane and mastering intrapixels effects in order\nto reach the highest precision astrometry. It should be noticed that\nthe development of European detector technology for low-RON and\nlarge-format IR and visible detector matrices, like the Alfa detector\nthat ESA is undertaking with Lynred, is of high interest for our\nscience cases.\n \n\\subsection{Metrology}\n\\label{sec:metrology}\n\nTraditionally systematic errors have been the major challenge for\n$\\mu$as-level astrometry from space. Astrometric accuracy has a lot in\ncommon with photometric accuracy, and the technology development that\nproceeded the Kepler mission demonstrated $\\sim 10^{-5}$ relative\nphotometry. Similar advances have been made in detector calibration\nfor astrometry \\citep{2016SPIE.9904E..5GC}. Photons from stars carry\nthe astrometric information at exquisite precision, systematic errors\nare imparted when those photons strike the telescope optics and also\nwhen they are detected by the focal plane array. The calibration of\noptical field distortion using reference stars is a technique that is\nperhaps a century old and used on ground and space-based telescopes.\n\nMetrology laser-feed optical fibers placed at the back of the nearest\nmirror to the detectors can be used to monitor distortions of the\nfocal plane array, and to allow the associated systematic errors to be\ncorrected \\citep{2016SPIE.9904E..5GC}. Such detector calibration at\n$10^{-6}$ pixel levels should be continued. In addition to measuring\nthe FPA physical shape, the rest of the telescope needs monitoring to\ncontrol time-variable aberrations at sub $\\mu$as level. Even at very\nstable space environments such as L2, the telescope geometry is\nexpected to vary for different reasons: structural lattice\nreorganization (such as the micro-clanks observed in ESA's Gaia mission), outgassing\nand most importantly, thermo-elastic effects due to the necessary\nvariation of the Solar Aspect Angle during the mission for pointings\nto the different science targets.\n\nIn the case of Theia, the telescope metrology subsystem to monitor\nperturbations to the telescope geometry is based on a concept of a\nseries of simple and independent linear displacement interferometers\ninstalled between the telescope mirrors and organized in a virtual\nhexapod configuration. Existing space-based interferometers from TNO,\nas the Gaia Basic Angle Monitor (BAM) are already capable of reaching\nmore precise measurements than those required by Theia\/M5 -- BAM can\nperform $\\sim1.5$ pm optical path difference measurements\n\\citep{doi:10.1117\/12.2026928}. A Thales telemeter developed for CNES\ncan reach $\\sim100$ pm, and the Thales interferometer produced for the\nMTG (Meteosat Third Generation) satellite can reach 1 nm per\nmeasurement \\citep{Scheidel2011} -- higher precisions can be reached\nby averaging over many measurements.\n\nFor telescopes that do not have high stability levels, there are\nsome alternatives. One is the diffractive pupil concept that puts a\nprecision array of dots on the primary, which produces a regular\npattern of dots in the focal plane. One way to use the diffractive\npupil is to look at a very bright star (0 mag) and record the\ndiffraction pattern interspersed with observations of a much dimmer\ntarget star ($\\sim7$\\,mag). The diffractive pupil can also be used\nduring science observations, but when the target star is $\\sim7$\\,mag\nphoton noise of the diffracted light can be significantly higher than\nthe photon noise of the reference stars ($\\sim11-14$\\,mag).\n\n\\section*{Conclusion}\n\nTo solve fundamental questions like\n\\begin{itemize}\n\\item ``\\emph{What is the nature of dark matter?}''\n\\item ``\\emph{Are there habitable exo-Earths nearby?}''\n\\item ``\\emph{What is the equation of state of matter in extreme environments}?''\n\\item ``\\emph{Can we put direct constraints on cosmological models and dark energy parameters?}''\n\\end{itemize}\nmany branches of astronomy need to monitor the motion of faint objects\nwith significantly higher precision than what is accessible\ntoday. Through ultra-precise micro\\-arcsecond relative\n astrometry, a high precision astrometry space mission will address\n the large number of important open questions that have been detailed in\n this White Paper.\n\nThe scientific requirements points toward a space mission that is\nrelatively simple: a single telescope, with metrology subsystems and a\ncamera. Such a mission can fit as an M-class mission, or even at a\nsmaller mission class depending on the final accuracy which is desired.\n\nSome technological challenges must be tackled and advanced: the\nspacecraft, the focal plane detector and the metrology. We believe\nthat these challenges can be mastered well before 2050 and that they\nwill open the compelling scientific window of the faint objects in\nmotion.\n\n\\begin{acknowledgements}\n The authors would like to thank the researchers and engineers who\n are not co-authors of this paper but who have taken part and have\n brought their contribution to the proposed missions to ESA\n successive calls: NEAT (M3), micro-NEAT (S1), and Theia (M4 and\n M5). An extensive list of supporters for the science objectives is\n given in \\cite{Boehm2017}. We thank also Arianna Gallo for her\n contribution in our investigation of the shape of the Milky Way dark\n matter halo and Krzysztof A.\\ Rybicki who generated the\n plots from Fig.~\\ref{fig:microlensing}.\n\n We are grateful to the anonymous referee who helped to improve the\n quality of the paper with his\/her remarks.\n\n Concerning the funding of our work, we would like to acknowledge the\n support of many agencies or programs.\n R.B.\\ acknowledges support from NASA's Virtual Planetary Laboratory lead\n team under cooperative agreements NNA\\-13\\-AA\\-93A.\n A.C.M.C.\\ acknowledges support from CFisUC strategic project\n (UID\/FIS\/04564\/2019). \n F.C.\\ acknowledges support by the Swiss National Science Foundation (SNSF)\n and by the European Research Council (ERC) under the European Union's\n Horizon 2020 research and innovation program (COSMICLENS: grant agreement\n No.~787886).\n M.F.\\ received support from Polish National Science Centre (NCN) under\n Grant No.~2017\/26\/D\/ST9\/00591.\n M.F.\\ gratefully acknowledge the support of the Swedish National Space\n Agency (DNR 65\/19, 174\/18).\n D.H.\\ thanks the Swedish National Space Agency (SNSA\/Rymdstyrelsen) for\n their support.\n A.M.\\ thanks the Portugese Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)\n through the Strategic Programme UID\/FIS\/00099\/2019 for\n CENTRA.\n P.S.\\ acknowledges support from the Australian Research Council under\n grant FT190100814.\n L.W.\\ acknowledges support from the Polish NCN grants: Harmonia\n No.\\ 2018\/06\\\/M\/ST9\/00311 and Daina No.\\ 2017\/27\/L\/ST9\/03221.\n The OATo team acknowledges partial funding by the Italian Space Agency\n (ASI) under contracts 2014-025-R.1.2015 and 2018-24-HH.0, and by a grant\n from the Italian Ministry of Foreign Affairs and International\n Cooperation (ASTRA).\n A.C.\\ and F.M.\\ acknowledge support by the LabEx FOCUS ANR-11-LABX-0013.\n The work of C.J., X.L.\\ and J.P.\\ was supported by the Spanish Ministry of\n Science, Innovation and University (MICIU\/FEDER, UE) through grants\n RTI2018-095076-B-C21, ESP2016-80079-C2-1-R, and the Institute of Cosmos\n Sciences University of Barcelona (ICCUB, Unidad de Excelencia 'Mar\\'{\\i}a\n de Maeztu') through grants MDM-2014-0369 and\n CEX2019-000918-M.\n A.K.-M., A.A., V.C., P.G., P.G., A.M.A., A.M., M.S.\\ were supported by\n Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e a Tecnologia, with grants reference\n UIDB\/00099\/ 2020 and SFRH\/BSAB\/142940\/2018 (P.G.\\ only).\n A.D. and L.O.\\ also acknowledge partial support from the Italian\n Ministry of Education, University and Research (MIUR) under the\n Departments of Excellence grant L.232\/2016, and from the INFN grant\n InDark.\n G.J.W. gratefully acknowledges support of an Emeritus Fellowship\n from The Leverhulme Trust.\n E.V. is supported by Spanish grant PGC2018-101950-B-100.\n\n %\n This research has made use of NASA's Astrophysics Data System\n Bibliographic Services. \n\n\\end{acknowledgements}\n\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\part{SCIENCE CASE}\n\n\n\n\\clearpage\n\\phantomsection\n\\addcontentsline{toc}{section}{Bibliography}\n\\bibliographystyle{et_dsd}\n\n\n\n\\subsection{Existing infrastructure}\n\nCreated by the University of Li\u00e8ge, the Centre Spatial de Li\u00e8ge (CSL) is a research center focused on space instrumentation with an environmental test facility serving the European Space Agency (ESA), the space industry and regional companies.\n\nCSL activities inherit more than 50 years of experience focused on scientific space instrumentation.\nThe common thread of these activities is the optical heart of space missions. From design to qualification testing, CSL is involved in each phase of engineering.\n\nAssembly, integration and verification of space instrumentation are operated on clean rooms fully compatible with the requirements of ESA programs. A majority of scientific optical instruments was tested in our facilities.\n\nCSL provides a customized service to expose instruments, systems and satellites to extreme conditions encountered in space with a unique expertise in optics and thermal regulation, including cryogenics.\n\nThe huge size of vacuum chambers (up to 6.5 m diameter) enables the qualification of full satellites. In the recent past, Herschel, Plank, Ga\u00efa, Aeolus and EUCLID were successfully tested.\nThe expertise is maintained thanks to the research and development activities shared with our university and with numerous partners (academics and industrials) at regional and international levels.\n\nRegarding the E-Test project, the available CSL's facilities are particularly convenient for the cryogenic test implementation. Indeed, CSL is equipped with cleanrooms, vacuum chambers and He liquefiers. \n\n\\subsection{Cool down strategy}\n\nThe E-test experiment will take place at CSL in our biggest vacuum chamber: Focal 6.5. One goal of the experiment is the cooling down to cryogenic temperature of a 100 kg silicon mirror. As explained before in the document, the mirror is suspended in order to avoid any vibration. This vibration isolation implies that no contact can be occurs between the suspended parts and external (to the pendulum) elements. \n\nThis very important constraint forced us to develop an innovative concept allowing the contactless cooling of the mirror. This concept is based on radiative exchanges between two intertwined radiators-like structures called inner and outer cryostats. \n\nFigure~\\ref{fig:concept} presents the scheme of the cryogenic concept developed for the E-test project in close collaboration with the partners of the consortium. It is based on the radiative exchanges only between the inner cryostat (in green on the Figure~\\ref{fig:concept}) directly attached on the cold platform supporting the suspended mirror (this cryostat is consequently also suspended), and the outer cryostat (in blue on the Figure~\\ref{fig:concept}) in direct connection with the He shrouds and pre-cooled by the LN2 shield. It is important to mention that the LN2 shield and the He shrouds as well as the outer cryostat have no contact with any part of the suspended elements in order to avoid vibration transmission to the mirror. \n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.75\\linewidth]{Figures\/FiguresWP1\/Cryostat_concept.JPG}\n \\caption{Design of the cooling concept}\n \\label{fig:concept}\n\\end{figure}\n\nThe required size of the radiative heat exchange area depends on two main factors: the target final temperature of the payload and the total heat load to be extracted at that temperature. A third important aspect is the transient behavior of the system: the total mass of the cold suspended. The cooling time should be minimized and so the mass of the cold payload (inner cryostat, mirror, cold platform). Assuming the total heat load to be extracted is about 0.3W (including IR loading through openings, harness heat leaks), the required area then depends on the emissivity at that temperature. As the emissivity decreases drastically with temperature, even for black paint, one can assume a 0.5 emissivity. Under these assumptions and with a sink temperature at 20K (average temperature of the external cryostat), the required flat area of the heat exchanger is about 70m\u00b2 for a target payload temperature of 25K. Now if the external cryostat is fed with liquid helium at 4K, this reduces to 41m\u00b2 still for a target of 25K. Because it depends of the fourth power of the temperature, the required area significantly increases if the required target temperature is lower: for 20K, it goes up to 100m\u00b2 instead of 41m\u00b2 and to more than 300m\u00b2 for a 15K target. \n\nTo increase the exchange area while remaining in a compact envelope, the chosen strategy is to work with a stack of horizontal fins for both cryostats: the suspended inner one and the outer one linked to the gaseous Helium fed shrouds. These fins are spaced in order to leave 2cm gap above and bottom each fin when the inner and outer cryostats are in place (interlaced), avoiding any contact between fins and consequently cryostat structure. In order to maximize the emissivity and the cooling efficiency, the fins will be coated with thick layer of black paint on both faces. Finally, the fins are designed in order to minimize their weight while ensuring sufficient rigidity in order to keep the natural bending in the range of 1.5mm. This intertwined radiator geometry improves the radiative exchange area by about a factor 16 compared to the projected flat surface. This means that to reach 25K on the inner payload with an outer cryostat at an average temperature of 20K, the projected area can be reduced from 70m\u00b2 to 4m\u00b2. \n\nFigure~\\ref{fig:proto_cooling} shows the results of the latest thermal simulations which give a stabilized (dT\/dt << 1 K\/day) temperature of the mirror of 27K after 14 days. The whole cryostat is modelled with ESATAN-TMS. The geometrical mathematical model (GMM) is based on the latest CAO available to compute the view-factors between the surfaces. The thermo-optical properties of the materials are obtained by heritage from previous verified projects. For the black paint of the radiator fins however, a worst-case approach was preferred to be conservative regarding its dimensions.\n\nThen the thermal mathematical model (TMM) is solved by taking into account both thermal conductivities and specific heat in function of the temperature. Some of those properties, including the thermal conductivity of the fibres in sapphire were provided by Nikhef.\n\nWith a worst-case approach, the external cryotat assumes:\n\n\\begin{itemize}\n \\item The LN2 (liquid nitrogen) panels (orange in Figure~\\ref{fig:CAO_proto}) is supporting the GHe (gaseous Helium) panels supporting themselves the external fins.\n \\item The three apertures in front of the mirror are assumed to be with 30mm diameter through the LN2 shield and GHe panels and are aligned such that only one internal fin is removed (keeping two consecutive external fins). In addition, these three holes are equipped with LN2 shield and GHe tubes in order to minimize the incoming heat load in the system, see Figure~\\ref{fig:CAO_proto}.\n \\item The three suspension apertures are assumed with 50mm diameter on the top of the cryostat for the Titanium wired and cold harness, also equipped with LN2 shield + GHe tubes protruding inside the cryostat to radiative enhance thermalisation of the wires \\& harnesses.\n\\end{itemize}\n\nFor the inner\/suspended cryostat, the assumptions are the following:\n\n\\begin{itemize}\n \\item Three Titanium wires coming from the marionette are assumed in Ti6Al4V with 4mm in diameter.\n \\item 20 AWG24 phosphore bronze wires for sensors.\n \\item 10 AWG32 phosphore bronze wires for temperature sensors (cooling down monitoring).\n \\item 4 sapphire rods with 10mm diameter but with thinner ends with 3mm in diameter, supporting the mirror from the cold platform.\n \\item The interfaces between these sapphire rods and the CuCrZr blades of the cold platform are improved with indium sheets.\n \\item Flexible thermal straps linked the fin frames (T profile) directly to the cold platform.\n \\item No continuous power dissipation.\n\\end{itemize}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.75\\linewidth]{Figures\/FiguresWP1\/CAO_cryostat.PNG}\n \\caption{CAD and GMM design views of the cryostat}\n \\label{fig:CAO_proto}\n\\end{figure}\n\n\nTo these numbers should be added some uncertainties due to the presence of large uncertainties in various parameters such as the emissivity of the paint, contact conductances through the screws and straps, thermal properties of materials at cryogenic temperatures, etc. Therefore, Figure~\\ref{fig:proto_cooling_uncrt} depicts a range of possible temperature profiles for the cooling of the mirror, using a worst case approach.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.75\\linewidth]{Figures\/FiguresWP1\/CSL_transient_cooling.png}\n \\caption{Temperature profiles of silicon mirror (grey), cold platform (purple), inner cryostat (green), outer cryostat (blue), with the temperature of the helium thermal tent cooled to 20K in 12 hours and emissivity of black paint assumed at 0.5.}\n \\label{fig:proto_cooling}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.75\\linewidth]{Figures\/FiguresWP1\/CSL_transient_cooling_uncertainties.png}\n \\caption{Temperature profiles of the mirror compared with worst case and best case assumptions.}\n \\label{fig:proto_cooling_uncrt}\n\\end{figure} \n\\subsection{Future work} \nIn the next few months, specifically in close collaboration with the PML Lab (ULi\u00e8ge) and Nikhef, we will produce the final design of the prototype which will take place in the vacuum chamber, see Figure~\\ref{fig:proto_design}. This design includes the pendulum, the cryostat, but also all of the external devices necessary for the experiments (thermal tents, interfaces, sensors,\u2026). \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Proto_design.jpg}\n \\caption{Preliminary design of the prototype on the optical bench of Focal 6.5 vacuum chamber (sectional view)}\n \\label{fig:proto_design}\n\\end{figure} \nIn parallel to the mechanical design, CSL will finalize the thermal simulations of the thermal tent\/cryostat\/mirror system. These simulations will provide the best and worst case based on realistic assumptions and taking into account some uncertainties base on our experience in such problem (mainly thermal contact, emissivity dependence with temperature). The only way to reduce the uncertainties would be to carry out specific cryogenic tests on representative elements in smaller chamber. Unfortunately, due to schedule and cost constraints, such characterizations cannot be foreseen in this project. Consequently, it is expected that large uncertainties will remain on the projected final temperature after a constrained cooling time of about 20 days. The E-test project will be an experimental concept validation for the cooling down method. \n\n\\subsection{Design and use case of the test bench}\n\nWithin the E-Test Project a variety of cryogenic components are developed. The main E-TEST prototype will take multiple days to reach its vacuum and temperature goals. Quickly adjusting or repairing components inside the chamber is thus not possible. Therefore, testing them prior to using them inside of the E-TEST prototype is necessary. That is one of the use cases of this cryogenic test bench. The design parameters such as temperature and pressure level are chosen within the specification of the ET-LF configuration displayed in Tab. \\ref{tab:Parameters_Cryostat}.\nThe cooling power is provided by a two-stage Gifford-McMahon (GM) cryocooler. This closed-cycle cryostat is driven by compressed helium provided by a water cooled compressor. \n\n\\begin{table}[htb]\n \\centering\n \\begin{tabular}{|c|c|c|}\n \\hline\n Parameter & ET-LF design goal & Cryogenic Test Bench \\\\\n \\hline\n Minimum temperature & $ 10 - 20 \\,\\text{K} $ & $ 10 \\,\\text{K} $ \\\\\n Achievable vacuum level & $ > 10^{-10} \\,\\text{mbar} $ & $ > 10^{-10} \\,\\text{mbar} $\\\\\n Maximum power load & $ \\text{several } \\,\\text{W} \\text{ at } 10 \\,\\text{K} $ & $ \\sim 1 \\,\\text{W} \\text{ at } 10 \\,\\text{K} $ \\\\\n Cooldown time & several months & $ < 2 \\,\\text{h} $ \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison of the chosen parameters for the Cryogenic Test Bench to ET-LF from \\cite{ET20202} }\n \\label{tab:Parameters_Cryostat}\n\\end{table}\n\nThe cryocooler is placed inside an UHV vessel with 4 window ports. A heat shield is mounted on the first stage of the cryocooler. This reduces the power introduced in the cold environment by thermal radiation. The usable volume is cylindrical and has a diameter of $150 \\,\\text{mm}$ and a height of $150 \\,\\text{mm}$. The sample plate, which provides the mounting points for the tested components, is fastened to the second stage, seen in Figure \\ref{fig:Cryostat}.\n\nThe whole cryogenic test bench is affixed inside a holding structure as can be seen in Figure \\ref{fig:Cryostat_setup}. Among the improved seismic isolation, this allows an easier access to the insides of the cryostat. This structure is rigidly mounted onto the concrete floor of the room. By filling the hollow aluminium extrusions with sand, high frequency resonances are dampened. Next to the cryostat, the control electronics and the fore vacuum system are installed. The helium compressor is placed on a vibration dampening matt next to the water cooling system.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width = 0.75\\linewidth]{Figures\/FiguresWP1\/Cryogenic Test Bench\/Cryostat.png}\n \\caption{Left: Cryostat with turbomolecular pump attached; Middle: Open cryostat without heat shield, on the second stage, a heater cartridge and two temperature sensors (LS-DT-670B-CU) are located on the sample plate; Right: Open cryostat with heat shield}\n \\label{fig:Cryostat}\n\\end{figure}\n\nWhile the cryocooler is running, it induces vibrations to the sample plate. These vibrations are in an order of a few $ \\, \\mu\\text{m} $ in amplitude with a frequency of $ \\sim 1 \\,\\text{Hz} $. To reduce these vibrations, the cryocooler can be turned off and measurements can be performed while the structure is slowly warming up. Depending on the power output of the sample, these measurements can last up to $ 10 \\,\\text{min} $ below $20 \\,\\text{K} $. To further reduce the vibration level, a valve can be closed and the turbomolecular pump shut down in addition.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width = 0.75\\linewidth]{Figures\/FiguresWP1\/Cryogenic Test Bench\/Cryostat_setup.png}\n \\caption{The cryostat is placed inside a holding structure mounted to the concrete floor. The aluminium extrusions are filled with sand to dampening higher frequency resonances. The control electronics are installed on a lab table next to the compressor and its coolant cycle.}\n \\label{fig:Cryostat_setup}\n\\end{figure}\n\n\\subsection{Commissioning of the test bench}\n\n\nAfter the cryogenic test bench was successfully installed, different commissioning measurements were performed. Alongside with a vacuum leakage test, long-term vacuum tests over a few days and multiple temperature cycle measurements were performed. The minimum temperature achieved, $ T_{min}=6.7 \\,\\text{K} $, was reached at a minimal pressure level of $ P=6 \\cdot 10^{-9} \\,\\text{mbar} $. With a different pre-pump the setup was able to reach a minimal pressure level of $ P_{min}=4 \\cdot 10^{-10} \\,\\text{mbar} $ while cooling.\n\nA typical cool down of the cryostat takes approximately $ 95 \\,\\text{min} $ as can be seen in Figure \\ref{fig:Typical_Temp_Curve_new}. The pressure level decreases with lower temperature due to the freeze-out of residual gases. Once cooled down, the temperature is stable within $ \\pm 0.2 \\,\\text{K} $. The fluctuations are induced by the cyclic nature of the cooling process. Heating the setup to ambient temperatures takes approximately $ 60 \\,\\text{min}$ using the heater at full power.\n\nAs an additional measurement, the reachable temperature for a specific heat load was measured. As seen in Figure \\ref{fig:Power_over_Temp}, the setup is capable of reaching the target temperature of $ 10\\,\\text{K}$ at an additional heat load of $ 3.66 \\,\\text{W} $ which is far above the required $ 1 \\,\\text{W} $ design value. The used heater cartridge has a resistance of $ 40 \\Omega$.\n\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width = 0.98\\linewidth]{Figures\/FiguresWP1\/Cryogenic Test Bench\/Typical_Temp_Curve_new.png}\n \\caption{Complete Temperature cycle of cooling down to $ 7 \\, \\text{K} $ and heating back up. Sensor 2 is mounted on the sample plate, while sensor 1 is located next to the plate. The pressure spices in the beginning of the heating due to frozen out gases on the heater cartridge itself.}\n \\label{fig:Typical_Temp_Curve_new}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width = 0.98\\linewidth]{Figures\/FiguresWP1\/Cryogenic Test Bench\/Power_over_Temp.png}\n \\caption{Comparison between the measured temperature of the sample plate with different heating power levels. The heater cartridge is operated by a Lake-shore control unit. At an additional $ 3.66 \\,\\text{W} $ our design goal of $ 10 \\,\\text{K} $ is reached. The lowest achieved temperature is $ 6.7 \\,\\text{K} $ with the cryostat.}\n \n \\label{fig:Power_over_Temp}\n\\end{figure}\n\\chapter{Ultra--cold vibration control}\n\\label{sec:cryo}\n\n\n\n\\section{Review of ET Superattenuator}\n\\label{sei_concept}\n\n\nMany investigation campaigns have been conducted since 2010 to define the conceptual design of the ET \\cite{sathyaprakash2012scientific,Thoughts2019}. The baseline suspension system is based on the configuration of the Superattenuator isolation system that has been developed by Virgo group but with an upgraded model in order to achieve the desired sensitivity requirement of the ET \\cite{accadia2011seismic}. The selection of Virgo Superattenuator as a reference for the ET is because it is more complaint with the ET requirements particularly above 3 Hz and more importantly because its behavior is well confirmed experimentally over several years \\cite{abernathy2011einstein}. The desired cross-over frequency (between the antenna sensitivity and horizontal seismic noise) of the ET in the horizontal direction was set to be around 1.8 Hz whereas the one obtained in the Virgo is about 3 Hz \\cite{abernathy2011einstein,accadia2011seismic}. Therefore, a maximization algorithm has been applied to attain the performance which ends up with having 17-m long Superattenuator, as shown in Figure \\ref{fig:HybridSuperattenuator} (right), that involves six standard multi-cascaded pendulums as well as a payload (marionette and test mirror) \\cite{abernathy2011einstein,press1992numerical}. It was noticed that the only way to reduce cross-over frequency so as to extend ET bandwidth below 3 Hz, upon proposed approach, is by increasing the length of the Superattenuator and adds extra pendulum. \nThe Virgo Superattenuator however contains one Inverted Pendulum Platform (IPP) stage (7m long with resonance frequency of 40 mHz) \\cite{accadia2011seismic}, five cascaded pendulum-masses of standard Geometric Anti-Spring (GAS) filters (each reduces seismic noise by 40 dB in the horizontal and in the vertical Degree Of Freedom (DOF)) and three cascaded pendulum-masses for the payload with an overall height of about 10.5m \\cite{acernese2014advanced}. Principally, the Superattenuator was built on the idea of adding pendulums where a typical amount of isolation above a certain cut-off frequency can be achieved just by cascading multiple pendulums. Consequently, the horizontal displacement of the suspension points in an N-stage pendulums is transmitted to the last stage with an attenuation that is proportional to $f^{-2N}$ above its resonance frequencies. Therefore, the ratio between the linear spectral density of the last suspended mass displacement and the linear spectral density of the suspension point displacement decreases as $A\/f^{-2N}$ where A = $f_0^{2}$.$f_1^{2}$.$f_2^{2}$.$f_3^{2}$....etc. Hence, cascading multi-pendulums is an efficient way to passively isolate seismic noise above the resonance frequency where the better attenuation is obtained by employing longer pendulums as the resulted resonances are lower \\cite{abernathy2011einstein,finn2013gravitational}. \n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/ET-Schematic.png}\n \\caption{Isolation Systems: ET Superattenuator (Right) and Hybrid Isolation System (Left).}\n \\label{fig:HybridSuperattenuator}\n\\end{figure} \nAn IP stage is implemented to limit the amount of seismic noise by offering very low frequency horizontal filtering stage. The IP stage is also used to provide the isolation system with a suspension point positioning system as well as to allow for low-noise control of the mirror by reducing its swinging \\cite{losurdo1999inverted,abernathy2011einstein,takamori2002low}. These objectives are achieved by implementing IP which is somehow can be considered (ideal IP) as a massless vertical bar with a certain length that is supporting a mass on its top end and connected to the ground at its bottom end by means of a flexure. The resulting resonance then can be tuned to reach up to 40 mHz \\cite{losurdo1999inverted}. The IP however should be carefully designed in order to avoid bistability state if the IP payload is overloaded, and from other side, to avoid flexure's creep or buckling if materials and total weight are not properly selected \\cite{takamori2002low}. \n\n\\section{E-TEST Isolation Approach}\n\\label{sei_concept}\n\\subsection{System architecture}\n\nThere is a high interest among ET community to reduce the overall height of the proposed Superattenuator of the ET while maintaining the aforementioned performance requirements. This is mainly to reduce the cost and overall complexity of the infrastructure \\cite{ET20202} and also the possibility to obtain better isolation performance if other approaches are engaged in the suspension system as highlighted in \\cite{Thoughts2019}. In this reference, it has been proposed to involve an active platform in the suspension system or to design a suspension system with two IP platforms instead of only using one IP platform \\cite{Thoughts2019}. \nThese early researches inspired the suspension concept proposed in E-TEST. It consists of inserting active inertial platform below the Superattenuator suspension system. In doing so, the new isolation system of the E-TEST prototype, shown in Figure \\ref{fig:HybridSuperattenuator} (left), combines the LIGO and Virgo approaches, by having active control platform as well as having IP alongside the standard multi-cascaded pendulums, in one unique system. \nAs shown in Figure \\ref{fig:HybridSuperattenuator}, the new proposed hybrid suspension system contains an Active Platform (AP) that moves freely in 6 DOFs. This active platform is designed on the success of HAM-ISI active platform that was developed for LIGO \\cite{kissel2010calibrating}. Then three IPs are mounted on the AP using flexures that move in rotational directions only. Similarly, the upper top of the IPs are attached with the Inverted Pendulum Platform (IPP) by flexures that also move in rotational directions only.The horizontal motion of the IPP is resulted from the mechanism behavior of the IP and therefore IPP is not really restricted from moving in horizontal direction. The Geometric Anti-Spring (GAS) filter afterwards is mounted on the IPP, forming the first standard GAS filter (GF-0) which then followed by other four standard stages of GAS filter. The Cold Platform (CP) is suspended from last standard GAS filter (GF-4). On the other side, the CP is attached to the next stage that is called Intermediate Mass (IM) by three connected wires. This platform is mainly proposed to provide a means for compensating the drift as well as steering the mirror in the desired position. Lastly, the Mirror (Mir) is hanging from IM via 4 wires as shown in Figure \\ref{fig:HybridSuperattenuator}. An analytical model using lumped masses connected by springs can be derived to study the dynamics of the system.\n\nThe individual stiffnesses that are used in the model for the vertical direction are further defined as follows (the variables are defined in Figure \\ref{fig:HybridSuperattenuator} and Table \\ref{tab:E-TEST_abbreviation}): \n\n\\begin{equation}\\label{APzStiffness}\nK_{AP_z} = m_{AP}(2\\times\\pi\\times1.7)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{IPLzStiffness}\nK_{IPL_z} = m_{IPL}(2\\times\\pi\\times100)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{IPPzStiffness}\nK_{IPP_z} = m_{IPP}(2\\times\\pi\\times100)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{GF0zStiffness}\nK_{GF0_z} = (m_{ST1}+m_{ST2}+m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(2\\times\\pi\\times0.25)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{GF1zStiffness}\nK_{GF1_z} = (m_{ST2}+m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(2\\times\\pi\\times0.25)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{GF2zStiffness}\nK_{GF2_z} = (m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(2\\times\\pi\\times0.25)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{GF3zStiffness}\nK_{GF3_z} = (m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(2\\times\\pi\\times0.25)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{GF4zStiffness}\nK_{GF4_z} = (m_{CP}+m_{IM}+m_{Mir})(2\\times\\pi\\times0.25)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{IMzStiffness}\nK_{IM_z} = \\frac{n\\times E\\times\\pi\\times d^{2}}{4\\times l}\n\\end{equation}\n\n\\begin{equation}\\label{MirzStiffness}\nK_{Mir_z} = \\frac{1}{\\frac{1}{K_{Mir_{Wire}}}+\\frac{1}{K_{Mir_{Cantilever}}}}\n\\end{equation}\n\n\\begin{equation}\\label{MirWirezStiffness}\nK_{Mir_{Wire}} = \\frac{n\\times E\\times\\pi\\times d^{2}}{4\\times l}\n\\end{equation}\n\n\\begin{equation}\\label{MirCantileverStiffness}\n K_{Mir_{Cantilever}}=m_{Mir}(2\\times\\pi\\times13)^{2}\n\\end{equation}\n\nIt is noted from equation \\ref{APzStiffness} that the resonance frequency of the active platform is selected to be 1.7 Hz. This is achieved by applying proper designed flexure similar to those implemented by LIGO in HAM-ISI active platform \\cite{HAMISIBrian}. However, the resonances of the IP (equation \\ref{IPLzStiffness} and equation \\ref{IPPzStiffness}) are set to 100 Hz. This is because the IP is stiff in the vertical direction. On the other side, equations \\ref{GF0zStiffness}-\\ref{GF4zStiffness} show the values of the stiffnesses for the GAS filters which depend on the suspended masses from the filters and also depend on their individual resonance frequency which is here set to $0.25$ Hz. Such resonance frequency of the GAS filter is experimentally achieved in \\cite{BERTOLINI1999475}. Furthermore, the stiffness value of the intermediate stage depends mainly on the wire property as seen in equation \\ref{IMzStiffness}. So far, the wire selected for that is based on fused silica with a Young modulus E of $67 \\times 10^9$ N\/m$^{2}$ and wire diameter d of $1$ X $10^{-3}$ m. Lastly, equation \\ref{MirzStiffness} shows that the stiffness value relies also on the wire property as well as triangular cantilever spring. This cantilever spring is attached in parallel to the suspended wire which adds extra isolation in the vertical direction. The experimental result shows that this cantilever spring can be designed to achieve a resonance frequency up to $13$ Hz. This cantilever spring is a property of the payload and not of the seismic platform (pre-isolator). The mirror is suspended by means of machined silicon rods that are very rigid vertically. Therefore, in order to make the mirror suspension assembly possible, metal cantilever springs have been introduced at the top end of each rod to provide some vertical compliance, resulting in a 13 Hz bounce mode of the last suspension stage. \n\nSimilarly, the individual stiffnesses for the horizontal direction can be further presented as follows: \n\\begin{equation}\\label{APxStiffness}\nK_{AP_x} = m_{AP}(2\\times\\pi\\times1.7)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{IPLxStiffness}\nK_{IPL_x} = m_{IPL}(2\\times\\pi\\times30)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{IPPxStiffness}\nK_{IPP_x} = (m_{IPL}+m_{IPP}+m_{ST1}+m_{ST2}+m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(2\\times\\pi\\times0.07)^{2}\n\\end{equation}\n\n\\begin{equation}\\label{GF0xStiffness}\nK_{GF0_x} = (m_{ST1}+m_{ST2}+m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(\\frac{g}{l_{(IPP-SF1)}})\n\\end{equation}\n\n\\begin{equation}\\label{GF1xStiffness}\nK_{GF1_x} = (m_{ST2}+m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(\\frac{g}{l_{(SF1-SF2)}})\n\\end{equation}\n\n\\begin{equation}\\label{GF2xStiffness}\nK_{GF2_x} = (m_{ST3}+m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(\\frac{g}{l_{(SF2-SF3)}})\n\\end{equation}\n\n\\begin{equation}\\label{GF3xStiffness}\nK_{GF3_x} = (m_{ST4}+m_{CP}+m_{IM}+m_{Mir})(\\frac{g}{l_{(SF3-SF4)}})\n\\end{equation}\n\n\\begin{equation}\\label{GF4xStiffness}\nK_{GF4_x} = (m_{CP}+m_{IM}+m_{Mir})(\\frac{g}{l_{(SF4-CP)}})\n\\end{equation}\n\n\\begin{equation}\\label{IMxStiffness}\nK_{IM_x} = (m_{IM}+m_{Mir})(\\frac{g}{l_{(CP-IM)}})\n\\end{equation}\n\n\\begin{equation}\\label{MirxStiffness}\nK_{Mir_x} = (m_{Mir})(\\frac{g}{l_{(IM-Mir)}})\n\\end{equation}\n\n\nSimilarly, the stiffness value presented in equation \\ref{APxStiffness} shows that the resonance frequency of the AP in the horizontal direction is set to 1.7 Hz. This can be achieved by implementing cantilever spring similar to those employed by LIGO in HAM-ISI suspension system \\cite{HAMISIBrian}. On the other side, the stiffness value between the AP and IP leg is selected to be 30 Hz as depicted in equation \\ref{IPLxStiffness}. Again, this resonance frequency is selected to be large enough since the motion is restricted in the horizontal direction. However, the resonance frequency of the IPP, shown in equation \\ref{IPPxStiffness}, is chosen to be 0.04 Hz as this value is experimentally obtained in Virgo Superattenuator as presented in \\cite{accadia2011seismic}. Lastly, equations \\ref{GF0xStiffness}-\\ref{MirxStiffness} show that the stiffness values are depending on the cascaded pendulum masses and wire length under gravity force effects. \n\\subsection{Closed-Loop Performance}\nIn order to examine the closed loop performance of the E-TEST suspension system, two controllers are designed; one for horizontal, shown in equation \\ref{ControllerH}, and one for vertical, shown in equation \\ref{ControllerV}. Although quite aggressive, it is believed that these controllers remain realistic with a careful mechanical design and low noise instruments. \n\\begin{equation}\\label{ControllerH}\nC_{h} = 8.9691\\times10^{7}\\times\\frac{(s+69.59)(s+0.446)}{(s+3809)(s+0.009174)}\n\\end{equation}\n\\begin{equation}\\label{ControllerV}\nC_{v} = 8.9675\\times10^{7}\\times\\frac{(s+54.99)(s+0.1318)}{(s+1563)(s+0.005429)}\n\\end{equation}\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = 0.50\\linewidth]{Figures\/Transmissibility_Horizontal.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/Transmissibility_Vertical.png}\n \\caption{Transfer Function (Ground\/Mirror) of Superattenuator and Hybrid Isolation System.}\n \\label{fig:Transmissibility}\n\\end{figure}\n\nFigure \\ref{fig:Transmissibility} presents the output performance of the transmissibility (from ground to the last stage (mirror)) for the open loop (blue curve) as well as for the closed loop (red curve) of the E-TEST suspension system. In addition, the transmissibility behavior of the ET Superattenuator isolation system (yellow curve) is presented in the same figure. It is clearly seen from Figure \\ref{fig:Transmissibility} that the ET Superattenuator obtains the desired cross over frequency (1.8 Hz @ 10$^{-9}$) with on overall suspension chain of about 17m. However, the E-TEST suspension system can achieve the same cross over frequency with only 11m suspension chain.\nFurthermore, the E-TEST approach introduces extra low frequency isolation of about two orders of magnitude around 1 Hz for the horizontal and also for the vertical directions. This isolation is only attainable by implementing active inertial control such as in the E-TEST approach. This is because ET Superattenuator approach depends only on the passive isolation strategy which can add a certain amount of isolation but only above the resonance of the structure. \nThe Amplitude Spectral Densities (ASD) of the active inertial platform motion is shown in Figure \\ref{fig:ASDAP}. For comparison, the figure also shows the sensor resolution as well as the ground motion. The sensor resolution which is considered in this study is based on the Horizontal Inertial Sensor (HINS) and Vertical Inertial Sensor (VINS) that were developed at Precision Mechatronics Laboratory (PML) \u2013 Belgium \\cite{ding2021development}. Moreover, the ground motion used in this study was measured in Terziet, 250m below the surface of the ground. In addition to seismic isolation, Figure \\ref{fig:ASDAP} clearly shows that the active control successfully damp the resonance of the platform. \n \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.50\\linewidth]{Figures\/ASD_AP_Horizontal.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/ASD_AP_Vertical.png}\n \\caption{Active Platform Motion of E-TEST Isolation System.}\n \\label{fig:ASDAP}\n\\end{figure}\nFurthermore, the ASD for the last stage (test mirror) when considering the entire suspension chain of the ET Superattenuator as well as the entire E-TEST suspension system are further shown in Figure \\ref{fig:ASDMirror}. It is shown that the isolation performance of the E-TEST system in the closed loop (green curve) is superior compared to the performance of the open loop (black curve) at low frequency. Consequently, using the E-TEST approach,\nthe mirror is also more stable in that frequency range.\n \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.50\\linewidth]{Figures\/ASD_Mirror_Horizontal.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/ASD_Mirror_Vertical.png}\n \\caption{Test Mirror Motion of Superattenuator and Hybrid Isolation System.}\n \\label{fig:ASDMirror}\n\\end{figure}\n\n\n\n\\section{E-TEST Prototype Modeling}\n\\label{sei_concept}\n\\subsection{Isolation System}\n\nIn order to validate experimentally the isolation concept presented in the previous section, it is proposed to build a prototype of hybrid isolation system. The prototype will be full scale with a 100kg cryogenic silicon mirror. However, it will have less stages due to space and budget limitations. A sketch of the prototype is shown in Figure \\ref{fig:ETestSuspensionSystem6DOF}. \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/E_Test_Suspension_System_6DOF.png}\n \\caption{Analytical Schematic Model of E-Test Isolation Prototype.}\n \\label{fig:ETestSuspensionSystem6DOF}\n\\end{figure}\n\nThe notations of the schematic model are provided in Table \\ref{tab:E-TEST_abbreviation}. Overall, the E-TEST suspension system contains an Active Platform (AP) that moves freely in 6 DOFs. It contains three springs aligned horizontally (x-axis or y-axis) and three springs aligned vertically (z-axis). Then three IPs are mounted on the AP through flexural hinges. Similarly, the upper top of the IPs are attached to the IPP by flexures which also move in rotational directions only. The GAS filter is then mounted on the IPP. Thereafter, a Marionette (Mar) platform is suspended from the GAS filter by one wire. This platform is mainly used for the alignment and for drift compensation of the cryogenic part. The cryogenic part involves a Cold Platform (CP) and a large Mirror (Mir) which will be operated at cryogenic temperature (20 Kelvin). The CP is designed to mount the cryogenic sensors. The cryogenic part however is suspended from Mar platform via 3 wires. Lastly, the test mirror is hanging from CP via 4 wires as shown in the Figure \\ref{fig:ETestSuspensionSystem6DOF}. Overall, the entire suspension system will be less than 5m high. \n\n\\begin{table}[H]\n \\centering\n \\begin{tabular}{|c|c|c|}\n \\hline\n AP & Active control platform \\\\\n \\hline\n SF & Standard filter \\\\\n \\hline\n IPL & Inverted pendulum leg \\\\\n \\hline\n IPP & Inverted pendulum platform \\\\\n \\hline\n CP & Cold platform \\\\\n \\hline\n IM & Intermediate Mass\\\\\n \\hline\n Mir & Mirror \\\\\n \\hline\n GAS & Geometric-Anti Spring \\\\\n \\hline\n $m_{AP}$ & Mass of first active platform \\\\\n \\hline\n $m_{IPL}$ & Mass of inverted pendulum leg \\\\\n \\hline\n $m_{IPP}$ & Mass of inverted pendulum Platform \\\\\n \\hline\n $m_{ST}$ & Mass of standard filter \\\\\n \\hline\n $m_{CP}$ & Mass of cold mass \\\\\n \\hline\n $m_{IM}$ & Mass of intermediate mass \\\\\n \\hline\n $m_{Mir}$ & Mass of mirror \\\\\n \\hline\n $z_{i}$ & vertical displacement of stage $i$ \\\\\n \\hline\n $x_{i}$ & Horizontal displacement of stage $i$ \\\\\n \\hline\n $K_{AP}$ & Stiffness of AP \\\\\n \\hline\n $K_{IPL}$ & Stiffness of IPL \\\\\n \\hline\n $K_{IPP}$ & Stiffness of IPP \\\\\n \\hline\n $K_{ST}$ & Stiffness of ST \\\\\n \\hline\n $K_{IM}$ & Stiffness of IM \\\\\n \\hline\n $K_{Mir}$ & Stiffness of Mir \\\\\n \\hline\n $f_{a}$ & External force \\\\\n \\hline\n E & Young modulus of the wire \\\\\n \\hline\n l & Pendulum wire length \\\\\n \\hline\n $w_{0}$ & Ground motion \\\\\n \\hline\n g & Gravity acceleration \\\\\n \\hline\n \\end{tabular}\n \\caption{Notations Used in the Sketch of the E-TEST Isolation System.}\n \\label{tab:E-TEST_abbreviation}\n\\end{table}\n\n\n\n\n\n\n\n\n\\label{sei_concept}\n\\subsection{Dynamics and Modal Analysis}\n\nIn order to study the dynamics of the prototype in 3D, a multi-body model has been developed using Simscape. It is a Matlab toolbox allowing to study lumped mass systems under gravity. Since it works under the Simulink environment, it is also a convenient tool for implementing feedback control strategies. \nA 3D view of the Simscape model is shown in Figure \\ref{fig:Simscape}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Simscape3d.png}\n \\caption{3D View of the E-TEST Simscape Model (Left: Front View) and (Right: Isometric View).}\n \\label{fig:Simscape}\n\\end{figure}\nA deeper understanding of the model dynamics is provided by computing the mode shapes. This is achieved by building Finite Element (FE) model on the Structural Dynamic Toolbox (SDT) \\cite{SDT}. From the Simscape model, the mode shapes have been obtained in three steps: 1) extraction of a state space model from Simscape, 2) calculation of eigenvalues and eignvectors in Matlab and 3) projection of these modes on a finite element representation of the system. \n\nAs an illustration, horizontal mode shapes (x-axis) are shown in Figure \\ref{fig:ModeShapeHorizontal} and vertical (z-axis) mode shapes are shown in Figure \\ref{fig:ModeShapeVertical}. \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Mode_Shape_Horizontal.png}\n \\caption{Horizontal Mode Shapes (x-axis).}\n \\label{fig:ModeShapeHorizontal}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Mode_Shape_Vertical.png}\n \\caption{Vertical Mode Shapes (z-axis).}\n \\label{fig:ModeShapeVertical}\n\\end{figure}\n\n\\subsection{Closed-Loop Performance}\nA schematic diagram for the active platform of the E-TEST is further shown in Figure \\ref{fig:ActivePlatform} (left). It contains six actuators (3 for horizontal directions and 3 for vertical directions) and six inertial sensors (3 for horizontal directions (HINS) and 3 for vertical directions (VINS)). The interferometer used in these inertial sensors are re-calibrated very regularly. In addition, High pass filters (with a very low corner frequency, around 10mHz) are used in order to avoid drifts in the control signals. A block diagram of the input\/output is depicted in Figure \\ref{fig:ActivePlatform} (right). \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.8\\linewidth]{Figures\/FiguresWP1\/Active_Platform.png}\n \\caption{Active Platform (left) and Input\/Output Block Representation (right) of the Active Platform.}\n \\label{fig:ActivePlatform}\n\\end{figure}\nThe transfer function of each pair of sensor\/actuator are shown in Figure \\ref{fig:TransferFunctionSensorActuator}. These curves have been extracted from the full model. \nDue to the symmetry, the three vertical ones are identical and the three horizontal ones are identical as well. \n \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/SensorActuator_Horizontal.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/SensorActuator_Vertical.png}\n \\caption{Transfer Function of Each Sensor\/Actuator Pair of the Active Platform.}\n \\label{fig:TransferFunctionSensorActuator}\n\\end{figure}\n The off-diagonal elements of the transfer matrix (not shown in this report) are of the same order of magnitude as the diagonal ones. It means that, in order to control the platform, a decoupling strategy has to be applied. \n As an example, a decoupling strategy based on a Jacobian approach has been used to minimize the coupling and move from the local frame to the Cartesian frame of the AP. This is schematically represented in Figure \\ref{fig:ControllerSchematic}, where $J_a$ is the actuator Jacobian and $J_s$ is sensor Jacobian. These Jacobian matrices are mainly depending on the geometry of the active platform as well as on the position of the sensors and actuators on the active platform. \n\nOnce the plant is decoupled, each degree of freedom can be controlled individually as shown in the same figure. The poles and zeros of the controllers are designed based on a manual tuning (loop shaping) in the SISO tool - Matlab. This method is efficient in terms of understanding the behavior of the system and also the positions of the poles\/zeros of the controller. Such a method is considered time consuming and the controller needs to be re-designed in case of any modification in the parameters of the system. Therefore, a more advanced method based on H-infinity loop shaping is going to be implemented later which can define the optimal controller automatically based on the given specifications and restrictions. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.8\\linewidth]{Figures\/FiguresWP1\/Controller_Schematic.png}\n \\caption{Schematic Representation of the Centralized Control of the AP.}\n \\label{fig:ControllerSchematic}\n\\end{figure}\nFigure \\ref{fig:TransmissibilityAPETEST} compares the open loop and the closed loop transmissibilities from the ground to the active platform in vertical (z-axis) and horizontal (x-axis) directions. The figure shows that the transmitted motion is reduced by about two orders of magnitude when the inertial control is switched on. Not surprisingly, the same reduction is also visible on the transmissibilities from the ground to the mirror as shown in Figure \\ref{fig:TransmissibilityMirrorETEST}.\n\n \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/Transmissibilityx_AP_ETEST.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/Transmissibilityz_AP_ETEST.png}\n \\caption{Transmissibilities from Ground to AP.}\n \\label{fig:TransmissibilityAPETEST}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/Transmissibilityx_Mirror_ETEST.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/Transmissibilityz_Mirror_ETEST.png}\n \\caption{Transmissibilities from Ground to Mirror.}\n \\label{fig:TransmissibilityMirrorETEST}\n\\end{figure}\n\nFurthermore, the ASD of the active platform as well as of the mirror are illustrated respectively in Figures \\ref{fig:ASDAPETEST} and \\ref{fig:ASDMirrorETEST}. For reference, the noise floor of the in-loop sensors is also shown in the same figure. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/ASDx_AP_ETEST.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/ASDz_AP_ETEST.png}\n \\caption{ASD of the Active Platform Motion, the Sensor Noise and the Ground Motion at CSL.}\n \\label{fig:ASDAPETEST}\n\\end{figure}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/ASDx_Mirror_ETEST.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/ASDz_Mirror_ETEST.png}\n \\caption{ASD of the Mirror and the Ground Motions at CSL.}\n \\label{fig:ASDMirrorETEST}\n\\end{figure}\n\n\n\n\\subsection{Effect of the center of mass altitude }\n\nIn the previous section, it was assumed that the Center of Mass (COM) of the AP was close from the Center of Stiffness (COK), which facilitates the natural decoupling between the different directions. However, in practice, the center of mass of the AP will be much higher. This is mainly due to the safety frames and tubes which are part of the AP body. \n\nIn this section, we illustrate the effect of a modification of the altitude of the COM on the controller design. To this purpose, additional simulations have been conducted where the center of mass of the AP has been shifted 65cm higher than the top of the AP. \n\nIn order to better understand the effect, Figure \\ref{fig:APWithoutIP} shows the control plant of the AP obtained after decoupling with respect to the COM, when the COM coincide with the COK. One sees an excellent decoupling, except for two terms, which is inherent to the nature of inertial sensors. Horizontal sensors are very sensitive to rotation at low frequency. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/AP_Decoupled_Without_IP.png}\n \\caption{Transfer Matrix of Decoupled Active Platform System only with Implementing Sensor Dynamics (Inertial Sensor).}\n \\label{fig:APWithoutIP}\n\\end{figure}\n\n\nFigure \\ref{fig:APWithIP} compares the decoupling obtained after shifting the COM, while still using the same decoupling technique (with respect to the center of stiffness). When the COM is higher, one sees that larger couplings exist between horizontal forces and cross rotations. However, similar performances can still be obtained using PID-like controller on each element of the diagonal control matrix. As an example, the control filters shown in Figure \\ref{fig:ControllerDesignETEST} have been used to obtain the closed loop transfer matrix shown in Figure \\ref{fig:TransferMatrixETEST} and the transmissibilities shown in Figure \\ref{fig:TransmissibilityALL}. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/AP_Decoupled_With_IP.png}\n \\caption{Transfer Matrix of the Decoupled Systems (AP alone and Entire E-TEST) with Implementing Sensor Dynamics (Inertial Sensor).}\n \\label{fig:APWithIP}\n\\end{figure}\n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Controller_Design_ETEST.png}\n \\caption{Controller Design for the Closed Loop System of the E-TEST.}\n \\label{fig:ControllerDesignETEST}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Tranfer_Matrix_ETEST.png}\n \\caption{Transfer Matrix Of the Open Loop and Closed Loop of the E-TEST.}\n \\label{fig:TransferMatrixETEST}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Transmissibility_ALL.png}\n \\caption{Transmissibility (AP\/Ground) where COM is 0.65m above AP.}\n \\label{fig:TransmissibilityALL}\n\\end{figure}\n\n\n\nThrough this example, one concludes that, although the altitude of the COM is inducing some additional couplings, it is not inducing any additional difficulty in the design of the AP controllers. \n\n\n\n\n\\section{Prototype conceptual design}\nA 3D view of the E-TEST suspension system is shown in Figure \\ref{fig:E-TESTprtOvvw}. It consists of one active platform that provides inertial control in the 6 degrees of freedom (detailed in section \\ref{sec:ActivePlatformCAD}). The three legs of the IPs are mounted on the active platform and support the top stage (IPP). The IPP provides a large amount of isolation in the horizontal direction as the resonance frequency can be tuned to extremely low values (about 70 mHz in KAGRA \\cite{okutomi2019development} and 30 mHz in Virgo \\cite{accadia2011seismic}). Additionally, the IPP stage provides means for positioning the parts suspended from the top stage, which can compensate for the tidal drift\\;\\cite{takamori2002low}. \n\n\\begin{figure}[H]\n \\centering\n\\includegraphics[width=17cm]{Figures\/FiguresWP1\/ETESTProtoOverview.png}\n\\caption{Overview of the E-TEST prototype design. A large vacuum tank (left) hosts the cryogenic mirror suspension (right). From top to bottom we can see 1) the top GAS filter, 2) the top stage, 3) the marionette and 4) the inverted pendulum legs within pipes that support a reference ring below the top stage. The cryogenic part features 5) the inner cryostat which has the interlacing fin type heat exchanger. The whole cryostat features (6) three access points for outside experiments to interact with the cryogenic mirror. The inner cryostat is attached to 7) the cold platform. The inner cryostat fins interlace into the fins of the 8) outer cryostat which provides a cold environment and houses the (9) 100\\;kg silicon mirror. All of this is supported by 10) an active platform, which provides a stable and quiet environment. In turn, the active platform hangs from three large blades with have a (11) support pillar on the ground.}\\label{fig:E-TESTprtOvvw}\n\\end{figure}\n\nThe top stage (IPP) houses a large GAS filter that adds vertical isolation (shown in Figure \\ref{fig:GASFilterCAD}). The GAS filter is composed of twelve springs. From the GAS filter, a marionette, shown in Figure \\ref{fig:MarionetteCAD}, is suspended, which is mainly used to position a cryogenic payload (presented in section \\ref{cryo_rad}). This cryogenic part contains the cold platform and the large mirror which will be operated at cryogenic temperature (20\\;K). \nThe cold platform is a cryogenic test-bed for cryogenic electronics and sensors presented separately in sections \\ref{cryo_cmos} to \\ref{cryo_sens}. Before installing sensors or other equipment in the suspended cryostat, a smaller cryostat has been developed with a cylindrical cold volume -- down to 6\\;K -- of 15\\;cm diameter and 15\\;cm height for testing these instruments (presented in section \\ref{cryo_test}).\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/GAS_CAD.png}\n \\caption{CAD View of Geometric-Anti Spring (GAS) Filter}\n \\label{fig:GASFilterCAD}\n\\end{figure}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/Marionette1_CAD.png}\n \\includegraphics[width = 0.49\\linewidth]{Figures\/FiguresWP1\/Marionette2_CAD.png}\n \\caption{CAD View of Marionette: Top view (left) and Bottom View (right) }\n \\label{fig:MarionetteCAD}\n\\end{figure}\n\n \n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/FEM_CAD.png}\n \\caption{Stress Map Under Gravity in One blade and one Rod Used To Suspend the Mirror.}\n \\label{fig:FEMStress}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/Graph_FEM_CAD.png}\n \\caption{graph FEM}\n \\label{fig:FEMGraphe}\n\\end{figure}\n\n\n\n\\section{Active Platform}\\label{sec:ActivePlatformCAD}\n\\subsection{Mechanical Design}\nThe active platform developed for the E-TEST prototype is shown in Figure \\ref{fig:ActivePlatformCAD}.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/AP_CAD.png}\n \\caption{CAD View of Active Platform.}\n \\label{fig:ActivePlatformCAD}\n\\end{figure}\n\nIt is a completely redesigned and up-scaled version of the LIGO HAM-ISI platform. In order to host the large cryostat, the platform diameter has been increased to 2.5m.\nThe sandwich structure of the payload using vertical panels has been reinforced for maintaining the first flexible mode above 300 Hz. The shape of this mode is shown in Figure \\ref{fig:FirstModeAP}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/FEMAB2112.png}\n \\caption{Different Views of Finite Element Analysis of the Active Platform: First Flexible Mode Appears Above 300 Hz }\n \\label{fig:FirstModeAP}\n\\end{figure}\nA particular attention has also been paid to the mass and stiffness repartition in order to facilitate the decoupling of the plant in view of its control. Thanks to this careful design, the rotation modes around the horizontal axis correspond to pure rotations around the center of mass (see Figure \\ref{fig:FirstModeAP} (right)). The blade springs used for vertical isolation have been stiffened for supporting a total payload of 1700kg, while maintaining the suspension mode below 2Hz. The material chosen for the blades is steel PH13-8 due to its high yield strength. The tip of each blade is connected to a flexure in steel C250 ensuring an adequate compliance in the horizontal direction. The blades have been designed in such a way that they are inclined at equilibrium, with the aim of changing the load in pure traction to overcome the high stress that occurs in the flexures due to bending. This is shown in Figure \\ref{fig:FluxureAP}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP1\/FEMAB21122.png}\n \\caption{Finite Element Analysis of the Flexure.}\n \\label{fig:FluxureAP}\n\\end{figure}\n\nThe positions of the horizontal sensors as well as the horizontal actuators are carefully chosen in order to facilitate the plant decoupling.\nThe next section presents some recent development in inertial sensors that will be adapted for controlling the active platform. \n\n\n\n\\subsection{Inertial Sensors}\\label{sec:ActivePlatformSensor}\n\\input{ETEST_subsystems\/WP1_Ultracold_vibration_control\/HINSandVINS}\n\\section{Radiative cooling strategy}\n\\label{cryo_rad}\n\\input{ETEST_subsystems\/WP1_Ultracold_vibration_control\/CryoStrat}\n\n\\section{Cryogenic test bench}\n\\label{cryo_test}\n\\input{ETEST_subsystems\/WP1_Ultracold_vibration_control\/CryogenicTestBench}\n\n\n\\section{Cryo-CMOS electronics}\n\\label{cryo_cmos}\n\\input{ETEST_subsystems\/WP1_Ultracold_vibration_control\/CryoCMOS}\n\n\\section{Cryogenic MEMS}\n\\label{cryo_iner}\n\\input{ETEST_subsystems\/WP1_Ultracold_vibration_control\/CryogenicMEMS}\n\n\\section{Development of cryogenic sensors}\n\\input{ETEST_subsystems\/WP1_Ultracold_vibration_control\/CryoACC}\n\n\\label{cryo_sens}\n\n\n\n\\chapter{Optical engineering}\n\\label{sec:optics}\n\n\\section{Overview}\n\\label{opt_over}\nBesides the technology mentioned in the earlier chapters, also the necessary optical technology for ET-LF shall be developed and demonstrated within the E-TEST project. Since the hanging mirror from the suspensions in the cryogenic environment is a silicon mirror, which is opaque at the conventionally used wavelength of 1064 nm, new challenges arise: First of all, a new laser wavelength in the range of app. 1500-2000 nm has to be used, which will not be absorbed within the silicon mirror. Resulting from this, also new photodiodes and coatings have to be developed. Since the mirror shape might change when cooled down to the required 10 K, also a new interferometer to measure this behaviour is in development. \nWithin E-TEST, a laser at a wavelength of 2090 nm, consisting of a crystal based Ho:YAG seeder and an ultra-stable holmium-doped fiber amplifier shall be developed. Therefore, e.g. the sensors and the coatings are also optimized to fit to the laser wavelength to enable a high level of synergy. In the following chapters, information on all the key technologies will be given. Afterwards the three big key experiments for the technology validation will be presented, which also covers the collaboration and shared frameworks of the WPT2 experiments.\n\n\\section{Laser at 2~$\\mathrm{\\mu}$m}\n\\label{sec:laser2um}\nTo achieve the very well-defined spectral parameters and in addition a high power, a MOPA (Master Oscillator, Power Amplifier) configuration is a well-established and typically used approach. In this approach, a seed laser at low power will be used to define the spectral parameters, e.g. the wavelength and the linewidth. Since such a seed laser typically has low power in the mW range, the radiation has to be amplified in e.g. a fiber-based power amplifier. By separating the power parameters from the spectral parameters, overall better parameters can be achieved, which is the reason why the MOPA concept will be used for the development of the E-TEST laser source. The development of such a system will be performed by Fraunhofer Institute for Laser Technology.\nAt one of the most common fiber laser wavelengths of 1064 nm, which is also used in most GWD, the lasers are a well-established technology. In ytterbium-doped fiber lasers, output power of up to 10 kW are achievable, solid state laser seed lasers with very well defined spectral properties in the kHz range are commercially available, the stabilization techniques of ultra-high stability lasers are demonstrated, and fiber laser components are well developed and at low prices affordable.\nHowever, at a wavelength of 2090 nm, the development of lasers is not that advanced. Typically, holmium is used as an active dopant in crystals and fibers to enable lasing at a wavelength of 2090 nm. While at 1064 nm, e.g. NPROs (Non-Planar Ring Oscillators) are a commercially available low-bandwidth seed laser, there are no such lasers commercially available at ca. 2000 nm. For amplification in a to-be-used Holmium doped fiber laser even low output powers are a high challenge. The fiber laser components needed for the build-up of a fiber amplifier at 2090 nm are based on well-established technologies, but the unusual wavelength results in expensive, non off-the-shelf components, which have typically worse parameters (e.g. transmission loss or power capability) in comparison to 1064 nm fiber components.\nFor LF-ET a laser system with the following parameters is needed:\n\n\n- Wavelength: 2090 nm\n\n- Power: 5-10 W\n\n- Spectral linewidth: 1 kHz\n\n- Power Noise (RIN): < 1e-6 Hz\/sqrt(Hz) @ 100 Hz\n\n- Frequency Noise: 100 Hz\/sqrt(Hz) @ 100 Hz (assumes 1\/f spectrum)\n\n- Linear polarization: > 98%\n\n\nFor such a system, a typical approach is the development of a MOPA system as described above. Therefore, it is planned to develop a solid-state laser based on Ho:YAG crystals as a seed laser, which defines the spectral properties. For the power amplification, a two-stage fiber amplifier based on holmium-doped fiber shall be developed. Even though the power noise is only defined at a single frequency, it is planned to measure it in different frequency bands to gather more information on the laser performance.\n\\ref{fig:LaserConceptx} shows the scheme of the full laser concept. In addition to the mentioned seed laser (Seeder) and the two amplifier stages (Ho1 $\\&$ Ho2), pump lasers for Ho1 $\\&$ Ho2 have to be used and developed (Tm1 $\\&$ Tm2). These pump lasers are additional thulium-doped fiber lasers at a wavelength of 1950 nm. For the pumping of the Ho:YAG seeder, an ILT developed Tm:YLF laser will be used. Here, it will be tested whether the stability of the pump laser is high enough to achieve stable lasing of the seeder, or if another highly stable fiber based pump source has to be developed. Within the development, different concepts are tested and compared to find the most practical and satisfying concept.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 0.7\\linewidth]{Figures\/FiguresWP2\/laser_conceptx.png}\n \\caption{ Model of the laser concept. Within the Ho:YAG seeder, the spectral properties are defined. This will result in a low bandwidth and a wavelength of 2090 nm. Within the Ho-doped amplifier 1 $\\&$ 2 the power will be amplified}\n \\label{fig:LaserConceptx}\n\\end{figure}\n\nWhile the goal is to develop an NPRO seed laser, the development is divided in two phases. Since the NPRO concept at 1064 nm shows satisfying results with respect to stability and linewidth, it is very hard to manufacture such a laser crystal and a lot of information on the crystal material itself are needed. Therefore, in the first phase, a ring oscillator will be developed. Such a concept has a significantly higher success probability and a lot of information on Ho:YAG and the power amplification within a fiber resonator can be gathered. For illustration, \\ref{fig:YAGCrystalx} shows the to-be-used Ho:YAG crystals for the ring oscillator. After the demonstration of the ring oscillator, an NPRO concept shall be tested. To enable active controlling of the laser parameters, it is planned to implement actuators to control these.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 0.3\\linewidth]{Figures\/FiguresWP2\/YAGCrystal.png}\n \\caption{Ho:YAG crystals to be used for the ring oscillator}\n \\label{fig:YAGCrystalx}\n\\end{figure}\n\nSince the seed laser is not available in the early stages of the project, a diode laser will be used as a seed source in the early stages of the fiber laser development, to prevent delay. With this diode laser, the power amplification can be demonstrated. Since the required relative intensity noise is very ambitious, the output power of the fiber amplifier will be actively controlled. This is done by adjusting the output power of the pump lasers. Based on the chosen concept, 4-7 individual fiber lasers are developed, which are all actively controlled by control algorithms. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 0.5\\linewidth]{Figures\/FiguresWP2\/YAGCrystalxx.png}\n \\caption{Ho:YAG crystals to be used for the ring oscillator}\n \\label{fig:YAGCrystalxx}\n\\end{figure}\n\nWhile a very useful characteristic of a fiber laser is the all-in-fiber concept, which results in e.g. no de-adjustment over time in comparison to other laser concepts, a process that needs to be perfected is the splicing of fibers. This way, two different (or same) types of fibers can be connected to enable the guidance of light within the fiber. A good splice is very important e.g. for a high efficiency and low noise laser. Especially for Thulium-doped fibers, the current state-of-the-art fibers are not as easy to splice as e.g. ytterbium-doped fibers, which results in higher losses and lower damage thresholds if the splice parameters are not optimized. Therefore, it is planned to perform studies to optimize the splices.\n \\ref{fig:YAGCrystalxx} shows an exemplary splice for illustration.\n\n\\section{Low noise coatings}\n\nMany crystalline materials show lower optical absorption and mechanical loss than amorphous materials. However, the low and high refractive-index materials have to be lattice matched, which limits the number of possible materials significantly. Crystalline coatings made of AlGaAs and GaAs are under investigation and show promising performance. The requirement of a lattice matched substrate -- GaAs wafers in case of AlGaAs coatings -- for the growth procedure requires a transfer of the coating to a suitable substrate (e.g. SiO$_2$ or cSi) after production and also can limit the maximum coating size (to $\\approx$ 20\\,cm in case of GaAs). Defects in the bond for attaching the coating to the new substrate can lead to the coating detaching during the test-mass cooling procedure or to thermal noise from a change in bulk and shear loss. Alternative materials may solve some of these problems (e.g. AlGaP and GaP coatings can directly be grown on cSi). \n\nPart of the approved E-Test budget proposal is a contribution to the capital funding for the construction of a novel MBE coating facility at KU Leuven to investigate new crystalline coatings for application in future GW detectors. In the following sections a detailed motivation and description of this activity is given. \n\nCurrently, the main methods to create state of the art oxide mirror coatings are based on Sputtering and Ion Beam (Assisted) Deposition (IBAD) techniques. There is already quite a large group of laboratories worldwide that focuses their efforts on using these techniques for oxide mirror coatings. \n\n\n\\subsection{Molecular Beam Epitaxy} \n\n\nThe Oxide Molecular Beam Epitaxy (MBE) technique is an interesting addition to the portfolio of techniques for mirror coatings and presents several unique and complimentary advantages. This technique has matured well over the past 30 years and the participants in the E-Test project have build 3 generations of small or large area oxide MBE systems. \n\n\\underline{Overall principle}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=8cm]{Figures\/MBE_principle.png}\n \\caption{Schematic drawing of the main elements in a standard molecular beam epitaxy system.}\n \\label{fig:MBE principle}\n\\end{figure}\n\nThe overall principle of MBE is illustrated in Figure~\\ref{fig:MBE principle}. In short, a rotating and heated substrate is bombarded with a beam of atomic of molecular (precursors) species emanating from for instance an effusion cell. An effusion cell is essentially a crucible in which the material to be evaporated is inserted combined with a heating element wrapped around it and a shutter that opens and closes the path to the substrate. The deposited species on the substrate then react with oxygen species (atomic oxygen, molecular oxygen or ozone) emanation for instance from a gas inlet, to form on oxide film. The structure of the growing film is then monitored in real-time using an electron diffraction pattern created on a screen with a technique called Reflection High Energy Electron Diffraction (RHEED). \n\nCompared to other techniques, the main advantages of oxide MBE include: \n\n\\begin{itemize}\n \\item A low background (10$^{-11}$ Torr) and operational pressure (10$^{-8}$-10$^{-5}$ Torr). This enables an excellent control on (gaseous) contamination as well as oxygen content in the layers. For instance it allows to grow suboxides, i.e. oxides that do not have the highest oxidation degree; but many of which have a higher refractive index. In addition, it permits a good control on the oxidation modulation in a coating. A good example would be the case of combining aSi with an oxide as mentioned above. Nevertheless if a higher oxidation state is required then atomic oxygen or ozone gases are added.\n \\item A low energy deposition process in the sense that energetic species - as is typical in sputtering in IBAD - are not present during growth. Indeed MBE uses elemental thermal evaporation from sources operating at a maximum of 3000C or about 0.75 meV. Typically higher energetic growth processes are more prone to the appearance of 3D features and particles on the growing surface. In addition, diffusion reactions are also enhanced which can lead to a poorer control on interface sharpness and roughness. \n \\item An excellent uniformity control at or below 1 $\\%$. The main sources - effusion cells and electron beam guns - are essentially point sources. This means that a simple geometric relation exist between the source and the evaporation profile. Hence, to improve uniformity over large areas requires only to change the geometrical position of the source vis-a-vis the substrate. Interestingly, it does not need the uses of larger targets as might be required in large area IBAD or sputtering processes. \n \\item A directed \"beam\" deposition. The low pressure operational conditions ensure that the evaporating beam has molecular beam properties. This means that atoms evaporating from the hot source surface are not scattered on their path to the substrate. This has several advantages. First, it means that the beam can be directed only towards the substrate and that no atoms hit - and contaminate - the walls of the chamber. Accumulation of such \"debris\" deposits on chamber walls can be a main source of subsequent particle contamination in the coatings which then necessitates frequent cleaning of the chamber. Second, it enable to easily shape the coatings in a well defined manner, through the use of shutters and the substrate rotation. For instance composition-graded or thickness-graded profiles can be grown while conserving rotational symmetry. As an example, a rotational symmetric \"curved\" coating can be made whereby more substrate material is deposited at the edges. \n \\item In-situ thin film characterisation tools. Because of the low operational pressure several unique tools can be used to monitor the deposition process such as the beam fluxes and the growing film surface. The beam flux techniques include Quadruple Mass Spectrometry (QMS), Electron Impact Emission Spectroscopy (EIES) and and ionisation gauges. With these the composition of the evaporants to the substrate can be controlled with high precision ($\\approx 1\\%$). The film structure (roughness, crystallinity) can be measured in real-time using Reflection High Energy Electron Diffraction (RHEED) while the film composition can be measured using the x-rays emitted from the RHEED gun with a technique called Low Angle x-ray Spectroscopy (LAXS). The RHEED technique becomes very important if the goal is to grow crystalline oxides. \\\\ \n\\end{itemize} \n\nThe unique advantages of MBE apply to all types of coatings that can be made whether metals, semiconductors, oxides, nitrides, etc. Furthermore, they also apply to both amorphous as well as crystalline oxides. Nevertheless given the availability of in-situ characterisation tools, the biggest impact that can be made in this field with a new coating tool is related to exploring crystalline oxides. \n\n\\underline{Crystalline coatings}\n\nThe motivation to explore crystalline oxides is similar to that used by the Cole et al.~\\cite{Cole2016}, namely that the thermal noise of single crystals is currently still several orders of magnitude better than that of amorphous coatings. While single crystals can have very low mechanical loss angles of the order of $2 \\times 10^{-9}$ (Al$_2$O$_3$) or $5\\times 10^{-9}$ (SiO$_2$), the values for their amorphous coating counterparts is at best around $5 \\times 10^{-5}$ for state of the art materials. Furthermore, there are a number of interesting features of oxide single crystals that may offer unique advantages in specific situations such as: \\\\\n\n\\begin{itemize}\n \\item Explore tensor properties. The unique features of crystals are often tensor properties in contrast to those of amorphous materials. While the latter can typically be described by values averaged over the 3 crystal axes, this is often not the case for the crystalline materials. This can be a blessing, for instance when the anisotropy of the thin film properties can be used to compensate those of the substrate. The relevant properties here are the elastic, thermo-optic and thermal expansion coefficients. \n \\item Tailoring thermal expansion. While most materials have positive thermal expansion coefficients along the three crystallographic axes, there are quite a number of materials that have anisotropic coefficients whereby the thermal expansion coefficient is negative (NTE) along one direction. To date the highest NTE reported is $-1.3 \\times 10^{-4}$\/K which is about 200 times larger - in absolute value - than the PTE of fused quartz. That means that in principle such a 500 $\\mu$m thick NTE coating may completely compensate the PTE of a 10 cm thick fused quartz mirror. \n \\item Controlling the uniformity of elastic properties. While the origin of the mechanical losses in coatings are not fully understood or identified, it is reasonable to estimate that non-uniformity in structure and composition are some of the main culprits. Both have an immediate effect on the local elastic properties. Although on a macroscopic scale the elastic properties of thin films are usually considered uniform they are most definitely not uniform on a microscopic \/ atomic scale. Such local non-uniformity are for instance structural defects like dislocations, anti-phase grain boundaries, vacancies, interstitials, variations in local composition, density variations, etc. Hereby not only elasticity in terms of its static components - with properties such as Young's modulus, Poisson's ratio, etc. - must be considered but also its dynamic components like the velocities at which elastic waves propagate in solid media.\n \\item Large family of interesting compounds. The amount of interesting oxide binary, ternary, quarternary, etc. materials is quite large compared to the crystalline AlGaAs and AlGaP compounds. Two families stand out namely the corundum type (R$_2$O$_3$) as well as the perovskite (ABO$_3$) type. The hexagonal corundum compounds include for instance Al$_2$O$_3$, Cr$_2$O$_3$, Fe$_2$O$_3$, Ti$_2$O$_3$, V$_2$O$_3$, Ga$_2$O$_3$ and In$_2$O$_3$. Within these materials it is also easily possible to create intermediary alloys, such as (Cr$_x$Fe$_{1-x}$)$_2$O$_3$ or (Ga$_x$In$_{1-x}$)$_2$O$_3$. Here, the in-plane lattice parameters spans a broad range between 0.476 nm - 0.549 nm, while the refractive index (at 1064 nm) varies between 1.75 to 2.80. Epitaxial films of these materials can be grown on sapphire or on Si(111) substrates. The perovskite family of compounds - like SrTiO$_3$ - is even much larger and can be well grown on Si(100) substrates. Note that there is also quite some structural variability within the perovskites that can be either cubic, tetragonal or orthorhombic. \n \\item Use of strain engineering. Thin crystal films can be quite easily deformed in a uniform and homogeneous manner through the use of epitaxial strain. This means that the lattice parameters of a substrate - or of a thick buffer layer - can be imposed on the subsequent films. In such a case, the film lattice parameters - and all its other properties - can be tailored within a well defined range until optimal features are obtained. \n\\end{itemize}\n\n\\subsection{Infrastructure components} \n\nBased on the above challenges, there are quite a few elements that need to be considered when defining the appropriate tooling to grow the thin film coatings. Both the individual elements as well as their combination will contain a number of technological innovations necessary to produce the high quality films needed for this application. \n\n\\underline{Sources}\n\nTypically each element to be deposited requires its own source. Such a source can be an effusion cell - as illustrated above - an electron beam evaporator or a gas inlet system for gaseous (molecular\/chemical) precursors. Each of these sources can come in different flavours. For instance under the umbrella of effusion cells, we have high temperature cells, low temperature cells, dual filament cells, cold-lip cells, valved crackers, etc. \nAmong the different constructions of electron beam evaporators we have single crucible, multiple crucible as well as vertical evaporators. Depending on the amount of heat these sources produce, they are contained with a water cooled enclosure. A schematic drawing of an electron beam evaporator is shown in Figure~\\ref{fig:Ebeam and gas manifold} (left). \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=6cm]{Figures\/Ebeam_gas_injector.png}\n \\caption{Schematic drawing of an electron beam evaporator (left) and a gas injection manifold (right).}\n \\label{fig:Ebeam and gas manifold}\n\\end{figure}\n\nAlso for the gaseous species injection there are a range of possible designs and options. One such gas manifold is illustrated in Figure~\\ref{fig:Ebeam and gas manifold} (right). Depending on the type of gas to be used, instead of a gas cylinder a gas bubbler may be used and the gas lines might required to be heated. The bubblers are typically used when the gaseous precursors have a low vapor pressure. In that case the mode of operation is to use a (inert) carrier gas - such as nitrogen or argon - that is flown through the heated bubbler that contains the precursor. The overall principle of a bubbler system - with the Mass Flow Fontroller (MFC) that controls the gas flow rate - is illustrated in Figure~\\ref{fig:Bubbler and atomic} (left). Note that in a typical system, several gas lines and manifolds will be put together in a rather complex system. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=6cm]{Figures\/Bubbler_atomic.png}\n \\caption{Schematic drawing of a precursor bubbler system (left) and an atomic beam source (right).}\n \\label{fig:Bubbler and atomic}\n\\end{figure}\n\nAnother important element is a source of reactive and\/or highly energetic species. This can be performed by a range of different sources, mostly involving the creation of plasma's and\/or high energy ion beams. One such radio frequency (RF) plasma source is depicted in Figure~\\ref{fig:Bubbler and atomic} (right) where an oxygen plasma is generated inside a crucible leading to the generation of a relatively high amount of atomic oxygen (up to 30$\\%$) which then flows out through an aperture on top. Other implementations make use of a electron cyclotron resonance (ECR) to create a plasma. Furthermore when grids are added on top of these sources, then the application of a voltage allows to extract electrons or ions from the plasma leading to an ion source. \n\n\\underline{Vacuum and pressure control}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=5cm]{Figures\/Pumps.png}\n \\caption{Pictures of a cryopump (left), a turbomolecular pump (middle) and an ion pump with titanium sublimation filaments (right).}\n \\label{fig:Pumps}\n\\end{figure}\n\nThe base pressure inside an MBE system is typically of the order of 10$^{-11}$ Torr. This requires a range of pumping systems including cryopumps, turbomolecular pumps and their forepumps, ion pumps equipped with titanium sublimation filaments as well as liquid nitrogen cooled cryopanels. In standby operation, mostly the ion pumps with titanium sublimation and the cryopumps are active. However during film growth where the temperature inside the chamber increases and a gaseous oxidizing flow is necessary then the turbomolecular pumps are active while the others are valved off. Schematic pictures of the different systems are shown in Figure~\\ref{fig:Pumps}.\n\nThe liquid nitrogen that feeds the cryopanels of the system require a specific set of double walled vacuum tubings that transport the liquid nitrogen and a phase separator. The latter recovers part of the liquid and separates it from the hotter gas that comes back after circulating through the panels. This concept is illustrated in Figure~\\ref{fig:Liquid nitrogen pressure gauges} (left). \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=5cm]{Figures\/Liquid_nitrogen_pressure_gauges.png}\n \\caption{Pictures of a liquid nitrogen feed lines coming from a phase separator (left), an ion gauge (middle) and a Pirani gauge filaments (right).}\n \\label{fig:Liquid nitrogen pressure gauges}\n\\end{figure}\n\nThe pressure in the different vacuum chambers is measured using a variety of instruments including ion gauges, Pirani gauges, baratrons, and residual gas analysers. The latter is used to monitor the composition of the base pressure as well as to detect small vacuum leaks and contaminations. The ion gauge and Pirani gauges are shown in Figure~\\ref{fig:Liquid nitrogen pressure gauges} (middle and right). \n\n\\underline{Flux control}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=6cm]{Figures\/Quartz_EIES.png}\n \\caption{Examples of quartz crystal monitors (left) and an EIES system (right).}\n \\label{fig:Quartz and EIES}\n\\end{figure}\n\nIn an MBE system, thin films of compound materials are formed by the combination of evaporants from different sources on the substrate. Hence it is quite essential to control the evaporation rate of each of the evaporants precisely. This is done by a combination of several techniques each with their advantages and disadvantages. \n\nQuartz crystal monitors use the measurement of a resonant frequency of a quartz crystal that shifts when material is added to its surface. Such monitors are used in many deposition processes and provide a good estimation of the flux rate. However since the conditions of deposition on the quartz surface are not the same as on the substrate - difference in temperature, sticking coefficient, oxidation coefficient etc. - it is not always the best technique. Furthermore its sensitivity is a strong function of temperature which can induce additional errors. Some examples of quartz crystal monitors are shown in Figure~\\ref{fig:Quartz and EIES} (left).\n\nHence in most MBE system addition flux monitors are added for instance based on optical spectroscopy. In the case of Electron Impact Emission Spectroscopy (EIES), the flux of atoms or molecules is ionised under electron bombardment which results into the emission of light. In this case, different elements will give rise to a different emission spectrum. Using a combination of fiber optic beam splitters and optical detectors, the spectrum can be collected and used to control the deposition process as illustrated in Figure~\\ref{fig:Quartz and EIES} (right). \n\nOther optical techniques are based on laser fluorescence or on Atomic Absorption Spectroscopy (AAS). Both of these methods have the advantage that no equipment - besides optical windows - needs to be added inside the deposition chamber since only interactions with light are important. An example of an implementation using AAs is shown in Figure~\\ref{fig:AAS and XBS} (left). \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=6cm]{Figures\/AAS_XBS.png}\n \\caption{An implementation of AAS (left) and a picture of a cross-beam ion source as part of a QMS system (right).}\n \\label{fig:AAS and XBS}\n\\end{figure}\n\nAlthough optical methods provide quite a flexible implementation, they are not the most sensitive instruments and for instance in the case of growing films with small doping levels, that sensitivity is not sufficient. An additional method is then to use Quadrupole Mass Spectrometry (QMS) systems equipped with specialized ion sources. The ion sources have a cross-beam design to ensure that no evaporants are deposited on the inside of the source. Using special pulse counting techniques, such systems are the most sensitive tools for this application. \n\n\\underline{In-situ thin film \/ crystal formation control} \n\nWhile the flux controls mentioned above enable to control the impinging amount of material on the substrate, this does not provide a characterisation of the growing films. Many factors play a role in determining the crystal phases, orientations, roughness, etc. These are a function of the temperature, oxidation coefficients, sticking coefficients, re-evaporation rates, etc. Hence it becomes important to use methods to monitor the growing material on top of the substrate.\n\nThere are multiple methods that are being used for this that are based on electron, photon and\/or x-ray beams, impinging upon or emitted from the growing films. One technique using electron diffraction (RHEED) is already mentioned and illustrated above in Figure~\\ref{fig:MBE principle}. This technique enables to control the crystal phases that are formed but does not give direct information about the composition of the films. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=4.8cm]{Figures\/LAXS_Ellipsometry.png}\n \\caption{An implementation of LAXS (left) and a schematic picture of an ellipsometry measurement configuration. (right).}\n \\label{fig:LAXS and SE}\n\\end{figure}\n\nAnother technique that uses the same electron source to bombard the surface and to collect the emitted element specific x-rays is LAXS. This technique is specifically designed to measure the surface chemical composition and to detect in-real time deviations from the overall stochiometry and is illustrated in Figure~\\ref{fig:LAXS and SE} (left). \n\nThe above techniques monitor what happens at or near the surface of the growing film. However during deposition process - certainly at high temperatures - a number of processes can take place below the growing surface such as reactions with the substrate or among the different layers or further oxidation processes, etc. To monitor those, one uses optical methods that have a much higher penetration depth than the electron beams from the preceding paragraph. \n\nOne such method is Spectroscopic Ellipsometry (SE), a technique that measures the light polarization rotation as a function of the interaction with a thin film. The principle is illustrated in Figure~\\ref{fig:LAXS and SE} (right) and this method allows to extract and monitor the refractive index and extinction coefficient as a function of the thin film thickness. Both of these quantities are essential parameters in the design of mirror coatings and can thus be accessed in real-time during the growth. \n\n\\underline{Thin film temperature and strain control}\n\nIn most thin film deposition system, the substrate and thin film temperature is induced by light radiation for instance from a a hot filament or from a laser source. To measure the substrate temperature itself is not as straightforward since in most cases the substrate rotates which makes the use of thermocouples or resistive temperature sensors impossible. The most common method is to make use of light emission or pyrometry. This requires knowledge of the spectral emissivity of the substrate which is known or can be calibrated. One of the issues is that the filament radiation - as a function of wavelength - is partially absorbed and thus can reach the pyrometer and can interfere with the temperature measurement. \n\nAs thin films grow, part of the radiation is now also absorbed in the film and the spectral emissivity becomes dominated by the thin film itself instead of the substrate. One method to then monitor the substrate temperature is to measure the band gap of the film, which is also a function of the temperature. In principle this can be done by SE but there are also dedicated tools for this on the market such as the EpiTT from Laytec as depicted in Figure~\\ref{fig:Temperature and bow} (left). \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=4.3cm]{Figures\/Temperature_Bow.png}\n \\caption{An implementation of a substrate\/thin film temperature pyrometer (left) and a substrate bow measurement device (right).}\n \\label{fig:Temperature and bow}\n\\end{figure}\n\nAnother critical parameter - certainly for thick films - is that of strain control. A large lattice and thermal expansion coefficient mismatch between film and substrate often leads to films which show cracks and a high density of other defects. To minimize the amount of strain being build up during growth and cooling it is important to be able to measure this in-situ. This is typically done by measuring how the entire wafer (substrate + film) curves up or downwards or in other words to determine the amount of bow. Such an instrument is shown in Figure~\\ref{fig:Temperature and bow}. The amount of strain can then be tuned partially for instance through changes in composition. \n\n\\underline{Beam of energetic particles}\n\nAn alternative method to control the strain and the overall density of thin films is to make use of ion bombardment. For instance in plasma assisted deposition, the properties of the plasma can be used to control the amount and even the sign of strain in the thin films. A part of this is related to the energy transfer from the energetic beam to the film and another part can be due to the implantation of extra species (oxygen, nitrogen, argon, etc.).\n\nA further advantage of using energetic species is that they can improve the density of thin films (amorphous as well as crystalline). This principle is illustrated in Figure~\\ref{fig:Ion Beam} and is one of the main features of the Ion Beam Assisted Deposition (IBAD) technique, which is the standard technique used for GW mirror coatings.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=6cm]{Figures\/Ion_beam.png}\n \\caption{An implementation of an ion beam inside an evaporator (left) and the densification process that it induces (right).}\n \\label{fig:Ion Beam}\n\\end{figure}\n\n\\underline{In-situ annealing}\n\nCrystal growth of thin films in general requires higher temperatures than their amorphous counterparts. This high temperature step can then either be applied during the growth or after the growth is finished. To minimize diffusion and reaction process either among the different layers or with the substrate, the goal is to expose the layers to a high temperature set for an as short a time as possible. Such approaches aiming to minimize the thermal budget to which the films are exposed are commonly called Rapid Thermal Annealing (RTA). \n\nThere are a number of different annealing implementations available on the market. One approach uses for instance high intensity pulsed lasers and another uses flash lamps. This variety is necessary to control precisely the annealing duration as well as the photon wavelength that is used. Depending on the band gap and the optical properties of the materials to be annealed light of different wavelengths will be absorbed differently. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[height=6cm]{Figures\/RTA_cleaning.png}\n \\caption{A rapid thermal annealing chamber with heating lamps (left) and a wet processing unit to clean wafers (right).}\n \\label{fig:RTA and cleaning}\n\\end{figure}\n\nThe RTA process could in principle be done in the deposition chamber. However, the specifications of the heater and chamber environments for RTA are quite different from those of a typical substrate heater specially since a much higher amount of thermal power needs to be supplied and evacuated. Also the gaseous environment in which the annealing is performed plays a crucial role since it may need to be performed in an oxygen, nitrogen, or an inert ambient pressure for which a deposition system is not designed. As a consequence in most systems, RTA takes place in a separate and optimized chamber. Such a chamber equipped with heating lamps is shown in Figure~\\ref{fig:RTA and cleaning} (left). \n\n\\underline{In-situ substrate cleaning}\n\nBefore the thin film growth process starts it is important to clean the silicon substrates in order to remove an amorphous silicon oxide surface layer, but also to eliminate organic residues, particles, etc. \nThis is typically done using a series of wet chemistry processes such as dipping in cleaning solvents as well as in a buffered HF solution for a short time. After cleaning the wafer needs to be transferred to the vacuum system as soon as possible to avoid re-oxidation and contamination for instance due to the adsorption of water vapour. The Figure~\\ref{fig:RTA and cleaning} (right) shows a wet chemistry processor that can be used for these purposes. Note that these are not optimal solutions since it usually takes quite some time before the wet solution is removed from the wafer and it is dried. Recently some in-situ dry cleaning processes have been developed and this will be the preferred solution if available. \n\n\\underline{Load lock, Wafer Transfer and Storage} \n\nAn ultrahigh vacuum deposition system can never be an isolated chamber otherwise any time a sample is transferred inside the chamber, it would need to be vented then pumped and backed out before the required base pressure is obtained again. \n\nTherefore, the main deposition chamber is connected with a set of ultrahigh vacuum chambers that enable to transfer a wafer from the ambient atmosphere through a series of glove boxes, load locks, pumping stages, transfer systems and wafer storage units. Depending on the final system configuration, the transfer systems also connect to the RTA chamber, the in-situ cleaning unit etc. \n\nNote that these wafer transfer requirements can turn out to be quite complicated and expensive. For instance in most RTA systems the wafer is facing upwards, while in an MBE system it faces downwards. To transfer the wafer from the RTA to the MBE then requires an 180 degrees flip operation under ultra high vacuum conditions. This can be performed either in a vacuum flipping station or as part of a standard atmospheric robot handler called a front opening universal pod (FOUP). \n\nThe use of all this equipment is however critical for the final thin film \nquality. Atmospheric reactions (oxidation, water layer, carbonisation, etc. ...)and the presence of particles, dust can be avoided in this way, and therefore the contamination of the inside of the coaters can be avoided. Note that contamination has recently been identified within the Virgo coating community as one of the main issues to be addressed for the improvement also of the current generation of the amorphous mirror coatings.\n\nThere is no final layout available of the experimental setup yet as this will be part of the design process as well as depending on total cost. \n\n\n\\section{Photodiodes}\nAn important active part of the interferometer is the light detector. The photodetector detects photons exiting the interferometer at the dark fringe and thus it is susceptible to the photon number variations caused by the interaction of the GW with the interferometer. The photon detection has been developed to a perfection in previous GW collaborations ( LIGO, VIRGO, KAGRA), however these work at a different laser wavelength as planned for the Einstein telescope. In particular, the infrared wavelength $\\sim$ 2 microns is one of the main innovation of the E-TEST development as it couples to requirements imposed by the low temperature operated Si mirrors and accompanying optics. This allows for a significant reduction of the vibrational noises at temperatures 5 \u2013 20 K. To match the wavelength of the laser ( 2 micron) a detector by optimal spectral dependence and as well the quantum efficiency needs to be chosen. The other GW collaborations use the laser wavelength at typically 1064 nm and these detectors don't have therefore sufficient detectivity at 2 microns. Also, for novel future quantum-based methodologies using light squeezing, the quantum efficiency needs to be as high as possible, ideally close to 99\n\nSecondly, the low temperature operation allows for a significant reduction of the interferometer noise and thus improving the signal to noise (S\/N) ratio as well as allowing to widen operation frequency for the GW detection. There are several components that enter into the play for reduction of the detection noise. In particular, in addition to vibrations and dealt with in the WP1, there are two types of detection noises that need to be optimized, this in particular the shot noise ( quantum noise) and the radiation pressure back action pressure noise, induced on the mirrors. These two noise inputs represent the fundamental physical limits of the interferometer sensitivity. The work of UHasselt contributes precisely to this main goal. e.g. the reduction of the detection noise. However, the physical limits of the detection can only be achieved if the all possible sources of the technical noises are reduced. This concerns for example the circuitry noise, amplifier noise, photodetector device noise (a pin diode) which has been addressed by UHasselt work.\n\n\\begin{enumerate}\n\\item Optimization of the detection wavelength:\n\nBased on 3.2 developments of the highly stabilized laser at ILF, UHasselt was matching the laser wavelength by searching for a suitable detector devices. We have carried out set of measurements on different commercial devices, trying to find the best match with operation close to 2 micron range. PbS and InGaAs diode have been tested and their absolute quantum efficiency has been measured ( see table 1). Figure \\ref{fig:HamamatsuDiodes} summarized the available detectors. In collaboration with Hamamatsu several commercial detectors were identified and are beeing evaluted in particular PbS detector and InGaAs G12181, G12182, G12183. From those IR-enhanced InGaAs ( series varies by different size and the dark current) is most interesting due to higher QE The peak sensitivity of the QE can also be slightly tuned ($\\sim$ 10 nm) by changing the temperature. The parameters are summarized in table below. Photocurrent measurement setup has been build and the detectors purchased for these measurements. We have built two different detection setups, a classical dc lock-in detection and homodyne detection. These systems are operational and being used for extensive detector testing. The testing involves the characterization of the dark current of the device that contributes to the noise level.\n\n\\begin{table}[htb]\n \\centering\n \\footnotesize{\n \\begin{tabular}{|c|c|c|c|c|c|}\n \\hline\n Material & Si & InGaAs & InAs & PbS & HgCdTe \\\\\n \\hline\n Detectivity (cm $H^{1\/2}$ $W^{-1}$) & & 5e12 & 5e11 & 6e10 & 7e10\\\\\n QE ($\\lambda$) & 70 & 99 &&&\\\\\n Spectral range (nm) & UV-1000 &900-2600$^{*}$, *extended & 1200-3000 & 1000-3000 & 1800-6000 \\\\\n Maximum response & 1000 & 2000 & 3000 & 2200 & Tuanable by composition \\\\\n \\hline\n \\end{tabular}\n }\n \\caption{The detectors tested in E-TEST project and their main specifications }\n \\label{tab:detectorstested }\n\\end{table}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 0.9\\linewidth]{Figures\/FiguresWP2\/Hamamatsu_diodes.png}\n \\caption{The spectral ranges of commercial Hamamatsu diodes relevant for the GW detection in IR 2 micron range}\n \\label{fig:HamamatsuDiodes}\n\\end{figure}\n\n\n\\item The detector noise:\n\nExtensive testing has been executed with the InGaAs detector, more specifically UHasselt optimized the detector circuitry. The tabulated pin device noise correspond 5 10$^{-13}$ W\/sqrtHz. The most important part of the detection circuit is the current preamplifier. First tests started with the commercial preamplifier SRS 570. Further on, to improve the S\/N ratio and the frequency range we and we have developed in collaboration with University of Stuttgart and tested novel trans-impedance preamplifier using multi-element pseudo resistors. In particular based on the University Stuttgart design we modified the preamplifier for our needs and constructed it on a PCB board. The developed pre-amplifier has a bandwidth of 10 kHz and the noise level of a 500 aW\/sqrt(Hz). The preamplifier outperforms SRS 570 when using test currents and these especially for higher bandwidth. This will give theoretical > 10$^{11}$ noise suppression in expected currents measured by the photodiode. The design of the PCB board allows a direct integration with PIN diode directly to minimise parasitic capacitance and limits the noise. This work will be carried out in the next period.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.9\\linewidth]{Figures\/FiguresWP2\/Input_referred_noise_feedback_resistor.png}\n \\caption{Left) Input referred noise for various dc feedback resistor values Rdc. Blue: Rdc = 150 G$\\Omega$. Red: Rdc = 1.2 G$\\Omega$. Yellow: Rdc = 10 M$\\Omega$. For comparison, the dashed lines show the theoretical thermal noise of ideal resistors with noise- equivalent values of 1.2 G$\\Omega$ (blue), 600 M$\\Omega$ (red), and 10 M$\\Omega$ (yellow). Right) Comparison of commercial SRS 570 and tailor made ST1 trans-impedance preamplifier performance expressed on detection voltage as a function of the frequency. The ST1 is still more sensitive to higher frequency and has the current noise $\\sim 10^{-14}$ A\/$\\sqrt{}$Hz.}\n \\label{fig:Inputreferrednoisefeedback}\n\\end{figure}\n\n\\item Further perspectives:\n\nUHasselt was further involved in theoretical studies aimed at mathematical description of quantum detection, in particular squeezing and entanglement based method. This modeling allowed to calculate the theoretical limits of the detection taking into the account the shot noise and squeezing ratio and the back action noise. The article ( Entanglement Limits in Hybrid Spin-Mechanical Systems) has been submitted to the journal and it is available at \\url{https:\/\/arxiv.org\/abs\/2108.13216}.\n\nBased on these calculations UHasselt is designing the system for the squeezing. The squeezing setup will involve the tunable laser, that was procured, to find optimal squeezing configurations ( for at 2 micron wavelengths). We plan to apply the squeezed light configuration with heterodyne detection and devise the noise improvements. For this the performed calculations will be extended and used.\n\n\n\\end{enumerate}\n\n\n\\section{Technology validation and collaboration}\nWhile all of the developed optical technologies are an important step towards the technological realization of ET-LF, it is also important that they are adapted to each other. One example for this is e.g. the sensitivity of the photo diodes, which has to be designed to fit to the laser wavelength to deliver the best results. While the earlier sections of this chapter described the technologies developed by the specific research facilities, this section will give an overview on their collaboration, and especially on the collaboration with the prototype developed in WPT1. To demonstrate this, three key experiments are planned, namely\n\n- Measurement of the thermo-optic properties of the silicon test mass\n\n- Mechanical quality factor measurement on the silicon substrate\n\n- White light interferometry to characterize the mirror\n\n\\noindent\nwhich will be described in detail in the following sections. A schematic representation of these three experiments is shown in Figure \\ref{fig:E-TESToptexps}. All of the experiments require a high collaboration between the different project partners, e.g. photo diodes are needed from UHasselt, the Laser is needed from Fraunhofer ILT and the white light interferometer from CSL. In addition, all experiments shall be performed on the prototype developed in WPT1. \n\n\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=.93\\textwidth]{Figures\/FiguresWP2\/WP2exps.png}\n \\caption{The 3 E-TEST prototype optical experiments: a) the white light interferometer, b) the temperature measurement and c) the quality factor measurement. Contrary to the suggestions of this figure, all experiments probe the same side of the mirror, having access through the 3 tubes into the cryostat visible in Figure~\\ref{fig:E-TESTprtOvvw}. Experiment a) will also probe the full mirror surface, but on an optical bench outside the cryostat prior to cold suspension.}\n \\label{fig:E-TESToptexps}\n\\end{figure}\n\n\\subsection{Measurement of the thermo-optic properties of the silicon test mass}\n\nDue to its high refractive index of around 3.5 at a wavelength of 1550\\,nm, the boundary between silicon and air reflects around 30\\% of light. The two surfaces of the silicon test mass therefore form a Fabry-P\u00e9rot etalon, where light bounces back and forth. The individual light fields interfere constructively or destructively, depending on the optical path for one round-trip. As the silicon mirror gets cooled down, this optical path changes due to two effects: an expansion of the material due to temperature, described by the thermo-elastic coefficient $\\alpha=dL\/dT$, and a change in refractive index, described by the thermo-refractive coefficient $\\beta=dn\/dT$. Together, they form the thermo-optic coefficient of the silicon material. This thermo-optic coefficient can be measured by monitoring the interference fringes of the etalon, together with the substrate temperature \\cite{Komma2012}. Conversely, knowing the thermo-optic coefficient, the observed interference fringes can be used to obtain the temperature evolution of the silicon substrate, which allows for a contactless measurement and monitoring of the silicon temperature at the location of the laser beam. This is especially relevant for E-TEST, where one is interested in the temperature gradient during cool-down, but at the same time cannot place conventional temperature sensors all over the test mass without impacting the performance of the seismic isolation, and more importantly, destroying the optical and mechanical quality of the mirror itself.\n\n\\begin{figure}[h]\n \\begin{center}\n \\includegraphics{Figures\/FiguresWP2\/to_experiment_layout.pdf}\n \\end{center}\n \\caption{Schematic layout of the experiment to measure thermo-optic properties of the silicon test mass.}\n \\label{fig:to_experiment_layout}\n\\end{figure}\n\nThe experiment itself is straight-forward to setup, a schematic is given in Fig.~\\ref{fig:to_experiment_layout}. A pick-off from the laser source at 2\u00b5m\\footnote{alternatively, a different source at e.g. 1550nm can be used, as long as the wavelength sits within the transparency window of silicon}, see Section \\ref{sec:laser2um}, will be sent orthogonally onto the silicon test mass. Thus, the beam will interfere with itself when reflected off the surfaces of the test mass. The reflected light is detected with a suitable photo detector and recorded with a data-acquisition system during the cool-down phase of the mirror.\n\nThe experiment requires the following components:\n\\begin{itemize}\n \\item fibre-coupled laser source,\n \\item fibre outcoupler\/collimator,\n \\item polarizing beam splitter and half-wave plate for power adjustment,\n \\item pair of steering mirrors and mounts,\n \\item beam splitting optic,\n \\item photodetector,\n \\item two-channel DAQ system,\n \\item optical breadboard mounted at vacuum tank.\n\\end{itemize}\nAs shown in Figure~\\ref{fig:surface_misalignment}, the two faces of the silicon test mass need to be parallel to within 30\\,\u00b5rad, to achieve sufficient interference contrast.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=9cm]{Figures\/FiguresWP2\/surface_misalignment.pdf}\n \\caption{Loss of interference contrast for misalignment of the two surfaces of the silicon test mass. To keep loss to less than 10\\%, any wedge of the test mass must remain less than around 30\\,\u00b5rad (ca.\\ 0.10\\,arcmin).}\n \\label{fig:surface_misalignment}\n\\end{figure}\n\n\\subsection{A mechanical quality factor measurement of the silicon substrate}\nAnother way of probing the quality of the mirror substrate is by exciting its mechanical internal modes and see how much energy they dissipate per cycle of the kHz mode. Using a tiny \\textit{hammer}, \\textit{i.e.} a small mass with a spring element that can be made with \\textit{e.g.} a coil magnet actuator to tap the test mass ever so slightly, can excite all modes that do not have a node at the position of hammer impact. Once the modes are excited and ringing down it is up to an optical measurement to determine the envelop of the exponentially decaying ring-down.\n\nThe optical setup to do just that is depicted in Figure~\\ref{fig:E-TESToptexps}b). An interferometer that uses polarizing optics to divert the light of both arms to two photodiodes~\\cite{Badaracco2021}. The error signal is generated by subtracting the two out-of-phase photodiode output signals. This also ensures all common mode noises - typically laser noises such as amplitude fluctuation - are decreased to shot noise levels. The error signal is used to make the mirror attached to a piezo actuator follow to probed mirror point, such that the signal sent to the piezo actuator will show a ring down indicative of the mechanical losses of the mirror. Such interferometric readout systems can reach fm\/$\\sqrt{\\rm{Hz}}$ sensitivity~\\cite{JorisPhD2018}. \n\nThe different modes can be isolated using a bandpass filter. By finite element analysis the frequency of each mode can be compared and the mode shape identified. This will also show which part of the mirror substrate (face) will move with a large enough amplitude for our sensitivity and an estimation of the expected signal strength can be made. The laser probe position on the mirror front surface can be varied by using the three different access tubes through the cryostat. \n\n\\subsection{White Light Interferometry to characterize the mirror}\nIn the following paragraphs, we describe one of the optical experiments for the E-TEST project and the interest in implementing it on the Einstein Telescope \\cite{jesus}.\n\n\\subsubsection{Purpose of the optical experiment}\nWe develop a metrology instrument to characterize the induced vibrations and wavefront or topology change of the silicon mirror at cryogenic temperatures.\n\n\\subsubsection{The added value of the system}\nThe instrument measures relative values of vibration and deformation of different points of the mirror surface instead of a single absolute value. We target a high precision on the measurements in the order of sub-nanometers. \n\n\\subsubsection{Implementation in E-TEST and the future Einstein Telescope }\nFor the E-TEST project, we aim to build a proof of concept of the metrology instrument to test its innovative features. The final implementation of our optical experiment on the vacuum chamber for E-TEST depends on the state of the research and the results of the laboratory validation. In the framework of the Einstein Telescope, the mirror characterization done by the metrology instrument will be essential to ensure the accuracy of the results on the detection of gravitational waves.\n\n\\subsubsection{Working principle and optical setup of the instrument}\nThe induced vibrations and wavefront or topology change are two issues that may appear consequence of cooling down the suspended mirror to 10 K - 20 K. Reference \\cite{Kaneda2004} already detected the permanent deformations of a mirror after the cooling down. As a result, we need to study these two side effects in detail on the Einstein Telescope framework. \n\nTo fulfil the accuracy requirements needed for the characterization, the metrology instrument is based on an interferometer optical layout. It is, therefore, a non-contact measurement solution. In addition, it does not implement movable components to avoid additional unwanted vibrations. In specific, we design a Michelson Interferometer that implements two sources of light. The primary light source for the mirror surface characterization is a white light source of short coherence. The second light source is a laser of high coherence that works as a complementary source during this first research phase. Figure \\ref{fig:WLI_interferometer} a) represents a schematic representation of the metrology instrument setup.\n\n\\begin{figure}[h]\n \\centering\n \\begin{minipage}[b]{0.57\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/FiguresWP2\/CSL_WP2_schema.png}\n \\end{minipage}\n \\hfill\n \\begin{minipage}[b]{0.40\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/FiguresWP2\/CSL_WP2_phase_mask.png}\n \\end{minipage}\n \\caption{a) Concept view of the phase mask designed to have single frame white light interferometry. b) Concept optical layout of the metrology instrument.}\n \\label{fig:WLI_interferometer}\n\\end{figure}\n\nThe metrology instrument we are developing is a dynamic interferometer that captures a whole interference pattern in a single frame. To achieve this milestone, we design an innovative phase mask. The phase mask has a step design, where each step introduces a different Optical Path Difference (OPD) on the interferometer. The OPD is the inequality in the path followed by the light rays after the beamsplitter in each of the two arms of the interferometer. Therefore the phase mask simulates the changes in OPD introduced by the displacement of one of the interferometer mirrors by a PZT device in conventional interferometers. As a result, we also avoid additional unwanted vibrations from movable components. A schematic representation of the phase mask appears in Fig. \\ref{fig:WLI_interferometer} b). The phase mask allows capturing Fields of View, which corresponds to different surface points, at different OPD's in a single camera frame. Each of the interferograms measured contains the information of the relative displacement and deformation of this specific point. \n\nThe polychromatic source that we use has a central wavelength of around 550 nm and at least a 100 nm bandwidth. The characteristic feature of low coherence interferometry interferograms is a well-differentiated central fringe. This feature allows an unambiguous determination of the position of the mirror since this zero-order fringe is visible only when the two arms of the interferometer have an equal path length \\cite{Taylor}. Moreover, low coherence interferometry avoids spurious unwanted interferometry signals from scattered rays, dust, imperfections on the optical surfaces that may interfere with the measurement beam. Compared with monochromatic laser-based systems, white light interferometry creates a lower coherent noise and lower phase noise \\cite{Deng2017}. The white light interference pattern is represented in Fig. \\ref{fig:interference_patterns} a), which plots the intensity value captured for a specific measurement point at different OPD. Figure \\ref{fig:interference_patterns} b) is the sinusoidal pattern created by a source with a high coherence like a laser for comparison.\n\n\\begin{figure}[h!]\n \\centering\n \\begin{minipage}[b]{0.47\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/FiguresWP2\/CSL_WP2_white_light_pattern.png}\n \\end{minipage}\n \\begin{minipage}[b]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{Figures\/FiguresWP2\/CSL_WP2_laser_pattern.png}\n \\end{minipage}\n \\caption{a)Low coherence interferometry pattern vs b) high coherence interferometry pattern}\n \\label{fig:interference_patterns}\n\\end{figure}\n\n\nThe next step is to proceed with the interferogram analysis to determine the peak of the coherence envelope of the pattern. For this, we study the phase of the white light interference pattern. Besides, we implement new approaches for the interferograms analysis to avoid limitations from conventional Phase Shifting Interferometry algorithms. The number and height of the mask steps depend on the smallest vibration and deformation that we would like to measure. The mask material and the tolerance in manufacturing also influence the mask design. Finally, we perform the pattern analysis at the simultaneous interferograms captured at successive camera frames. As a result, we can determine the relative displacement of the white light interference pattern associated with the relative value of the mirror induced vibration and wavefront deformation. \n\n\\subsubsection{Performance of the metrology device} \nThe lateral resolution of the instrument will be between 10 $\\mu$m to 20 $\\mu$m, according to the detector pixel size. The phase resolution and, therefore, the height difference between consecutive steps of the phase mask is intended to be maximum $\\lambda$\/8, where $\\lambda$ is 550 nm. Since the optical experiments are located outside the vacuum chamber, the relative motion between the metrology instrument and the mirror will be in the order of micrometres. The vibration of the mirror is expected to be in the order of a few nanometers. There will certainly be aberrations from the fact that one of the arms of the interferometers is inside the vacuum chamber and the other one outside. We are looking for solutions to minimize this effect.\n\n\n\\subsubsection{Development of the instrument} \n\\begin{itemize}\n \\item \\textbf{Planned case:} we are building a prototype of the metrology instrument at the laboratory. The aim is to test the performance. In particular, to test the innovative concept of the phase mask for dynamic or single frame interferometry with a polychromatic source.\n\n \\item \\textbf{Ideal case:} to perform measurements on the suspended mirror used for E-TEST. We consider two options. The first option is to locate the metrology instrument outside the vacuum chamber, according to Fig. \\ref{fig:option1_CSL_WP2}. It uses a 50 cm x 3 m space on the optical table. Nevertheless, we can compact the instrument by modifying the optical path of the light on the 3 m arm of the Michelson Interferometer. The second option is to place the instrument inside the vacuum chamber (and outside the cooling architecture of the mirror) over the optical table. That reduces an unstable environment (outside the vacuum chamber) and reduces induced aberrations from the optical window, Ref \\cite{Vandenrijt2016}.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[scale = 0.39]{Figures\/FiguresWP2\/CSL_WP2_setup.png}\n \\caption{Concept view of the phase mask designed to have single frame white light interferometry.}\n \\label{fig:option1_CSL_WP2}\n \\end{figure}\n\\end{itemize}\n\n\\subsubsection{Component List}\n\\begin{itemize}\n \\item White Light Interferometry\n \\begin{itemize}\n \\item 1xHalogen low coherence light source (in the visible and with 200 nm bandpass).\n \\item Optical lenses for collimating the source.\n \\item Optical filters to limit the source bandpass.\n \\item 1xBeamsplitter (50\/50).\n \\item 1xPhase mask (on design).\n \\item 1xPixelated photodetector.\n \\item 1xAnalysis unit (CPU).\n \\item 1xPZT if the instrument is located inside the vacuum chamber.\n \\end{itemize}\n \\item Large coherence interferometry\n \\begin{itemize}\n \\item We use similar components to the white light experiment except for the source. We use the Laser source developed by ILT for the E-TEST project. \n \\end{itemize}\n\\end{itemize}\n\n\n\n\n\\chapter{Organisational Overview of ETpathfinder}\n\\label{sec:orga}\n\n\\begin{figure}[htbp!]\n\\centering\n\\includegraphics[width=17.5cm]{Figures\/ETpathfinder_ornigram_v2.pdf}\n\\label{fig:wp4_gantt}\n\\end{figure}\n\n\\chapter{Aims of the E--TEST Prototype Project}\n\\label{sec:aims}\n\n\n\n\n\n\\section{Context}\nWith the recent detections of Gravitational Wave (GW), a new window has been opened on the universe, one century after these waves were predicted by Einstein's general theory of relativity. Together with these detections, gravitational wave astronomy was born, offering an exciting future for the exploration of the universe through large Michelson interferometers used as gravitational wave detectors, and calling for sensitivity improvement. Although existing instruments do have margins for upgrades, their performance will be ultimately limited by their location and design. A significant improvement of sensitivity can only be obtained using a disruptive technology like it is proposed in 3rd generation gravitational wave detectors e.g. Einstein Telescope (ET) and Cosmic Explorer (CE). \n\nA simplified sketch of ET is shown in Figure \\ref{fig:EinsteinTelescope}. For maximizing the detection capability, it is constituted of 3 detectors (red, blue and green). Each detector consists of two Michelson interferometers whose arms are 10 km long: one designed to maximize the sensitivity at high frequency (HF) and one to maximize the sensitivity at low frequency (LF). ET-HF will be around room temperature (290K), and use very high power laser. ET-LF will be cryogenic (10K) especially for reducing thermal noise and increase drastically the sensitivity. \n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP2\/Einstein_Telescope.png}\n \\caption{General scheme of the Einstein telescope representing the geometry of the telescope .}\n \\label{fig:EinsteinTelescope}\n\\end{figure}\n\n\nThe concept proposed for ET is very different from currently operating GW detectors Advanced LIGO, Advanced Virgo and KAGRA. For example, Advanced Virgo and Advanced LIGO operate with limited size fused silica mirrors at room temperature and use a laser wavelength of 1064\\,nm. KAGRA (sometimes referred to as detector of generation 2.5) operates with cryogenic, sapphire mirrors, which allows to stay at 1064\\,nm laser wavelength.\nIn contrast, both ET and CE plan to operate large cryogenic mirrors made from silicon and read out by lasers with wavelength further into the infrared (about 1550\\,nm to about 2100\\,nm). Another characteristic of 3rd generation detectors is to be much more sensitive at low frequency, which implies to improve the isolation of the mirror from seismic motion and gravity fluctuations. \n\nIn order to prepare and validate the technology necessary for these future instruments, several prototypes and facilities are being constructed all over the world. The prototype presented in this report is one of them. Next section gives an overview of the project which is funding the prototype, namely the \\textit{E-TEST} project. \n\n\\section{The E-TEST project}\n\nE-TEST (Einstein Telescope Euregio-Meuse-Rhin Site and Technology) is a project recently funded by the European program Ineterreg Euregio Meuse-Rhine. This program is dedicated to innovative cross boarder activities between Belgium, The Netherlands and Germany. With a total budget of 15M\u20ac and a consortium of 11 partners from the three countries, the objective of the project is twofold. Firstly, develop an eco-friendly and non-invasive imaging of the geological conditions as well as the development of an observatory of the underground in the EMR region.\nSecondly, develop technologies necessary for 3rd generation gravitational wave detectors. In particular, it is proposed to develop a prototype of large suspended cryogenic silicon mirror, isolated from seismic vibrations at low frequency. The total budget of the project is equally spread over the two activities. The first activity is not discussed at all in this report.\n\n\nThe E-TEST prototype will have some key unique features: \n\\begin{itemize}\n \\item A silicon mirror of 100\\,kg\n \\item A radiative cooling strategy (non contact)\n \\item A low frequency hybrid isolation stage\n \\item Cryogenic sensors and electronics\n \\item A laser and optics at 2 microns\n \\item A low thermal noise coating\n\\end{itemize}\n\n \n\n A sketch of the initial proposal is shown in Fig.\\ref{fig:ETestPrototype}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP2\/E-Test.png}\n \\caption{Sketch of E-TEST prototype.}\n \\label{fig:ETestPrototype}\n\\end{figure}\nThe activities were broken down in two workpackages, WP1 and WP2. WP1 will focus on the design and the realization of a full scale seismic attenuator, combining passive filtering with at least one active pre-isolation stage, with the goal of boosting the performance in the 0.1-10 Hz frequency band. \nThe development and characterization of cryogenic sensors and electronics is also part of WP1. The activities in WP2 are focused on optical engineering, and encompass the development of a large silicon mirror, laser and optics at 2 microns, photodetectors and low noise coating technologies. \nThe work proposed in WP1 and WP2 are described in details in Chapter 2 and 3. \n\nIn order to take advantages of the existing expertise and infrastructure available in the EMR region, it has been proposed to mount the E-TEST prototype in existing facilities at the Space Centre of Li\u00e8ge (CSL), which has a strong know-how in cryogenic qualification of Space Instruments.\n\n\\section{Ecosystem in the Euregio Meuse Rhin}\n\nEinstein telescope is very strategic for the EMR region as it is one site candidate for hosting ET. More than the scientific and technologic return, the operational aspects of such facility will cover a huge spectrum of economical domains.\n\nTwo other large scale Interreg projects have been recently funded in the EMR: ETPathfinder and ET2SME. \n\n\nETpathfinder will provide a new facility to \\emph{holistically prototype} the following aspects:\n\\begin{itemize}\n \\item \\textbf{New Temperature, i.e. Cyogenics:} Cryogenic temperature of the main test masses (120\\,K and 15\\,K); low-noise cryo-coolers, decoupling of cryo-coolers from test masses while providing sufficient cooling power, cryo-shields etc;\n \\item \\textbf{New Mirror Material, i.e. Silicon:} Changing from fused silica (amorphous material, electrically insulating) to silicon (crystalline semi-conductor) will be a major change with many implications for optical aspects, polishing, coatings etc as well as mechanical aspects.\n \\item \\textbf{New Wavelength, i.e. 1550 - 2100\\,nm:} Moving to silicon as test mass material requires to go to longer laser wavelength to reduce optical absorption. Lasers, optical and electro-optical components (such as high-efficiency photo detectors) need to be developed and qualified for GW detector relevant aspects. Laser stabilisation loops (power, frequency, jitter, etc) need to be developed. \n\\end{itemize}\nAll of these strands are interlinked and ultimately will be tested altogether in ETpathfinder, at displacement sensitivity not too dissimilar to a the Einstein Telescope. A sketch of ETpathfinder is shown in Figure \\ref{fig:ETPathfinder}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 1\\linewidth]{Figures\/FiguresWP2\/ET_Pathfinder.png}\n \\caption{Sketch of ETpathfinder. More information on ETpathfinder can be found here: \\url{https:\/\/www.einsteintelescope.nl\/etpathfinder\/.}}\n \\label{fig:ETPathfinder}\n\\end{figure}\n\n\\subsection{Synergies between ETpathfinder and E-TEST}\n\nTowards the development of the full Einstein Telescope, ET-Pathfinder and E-TEST are two independent but very complementary building blocks. The focus of both projects is clearly different as they are dedicated to different parts of the full telescope but with the final objective to provide the innovative technologies required for ET. \n\nBoth initiatives can be seen as the development of expertise centers to make the links between gravitational wave science, its technological aspects and the huge number of economical actors involved into such development, present in the EMR region. \n\nThe main difference between the two projects is that ETpathfinder results in a full facility laser-interferometer that can be used by knowledge institutes and (SME) companies for developing and testing new technologies. On the other side E-TEST will result in a prototype in a vacuum chamber and optical instruments working at 20 Kelvin, which corresponds to 20 degrees above the absolute zero. E-TEST also results in a seismic model that can be used to prove that ET can be built in the EMR region. Knowledge institutes and companies can use E-TEST results to develop and test new products or gain new insights\/knowledge. \nET-pathfinder will focus on the high temperature technologies of the Einstein Telescope while the E-TEST will develop innovative and challenging components and practices for ultra-low temperature. \nIt is important to note that the technologies developed in E-TEST could also be transposed to other gravitational wave detectors, placing the E-TEST actors in the forehand of this community. \nDue to their complementarity, both projects will work in synergy. Moreover, at the end, the outputs of both will be merged and implemented into the ET. The region will gain a massive competitive edge in this field via ETpathfinder and E-TEST. EMR based companies can use this advantage to create and deliver products and services to ET, even if ET could not be built in the EMR region. \n\n\n\n\n \n \n\n\\textbf{Added value of ETpathfinder for E-TEST:}\n\nAs ETpathfinder started more than half a year before E-TEST, the last one will benefit from the expertise and lessons learnt from ETpathfinder consortium. From the technical point of view, some of the groups are involved in the two projects (Nikhef, UMaastricht, KULeuven). Their experience will be particularly useful for ensuring the feasibility of E-TEST. E-TEST will also benefit from ETpathfinder in the management of an Interreg project, whose aim is to validate experimentally technology for ET. \n\n\\textbf{Added value of E-TEST for ETpathfinder:}\n\nIn E-TEST, we will have only one suspended mirror, which means that we will be able to measure its stability with a precision limited to $10^{-14}m$. After the completion of the E-TEST project, the performance of the suspended mirror and the whole vibration control chain would be tested and even transferred to the ETpathfinder facility in order to assess precisely the residual vibrations of the cryogenic suspension in a full interferometer. And finally the ETEST will bring to the whole community the geophysical model mandatory the build the Einstein Telescope. \n\n\n\\subsection{ET2SME}\n\nET2SMEs is an Interreg project which supports the development of new innovative products and services by promoting transnational R$\\&$D projects established by SMEs collaborating across borders in the EMR. It can be seen as the economic counterpart of the E-TEST project. With the conviction that the technology of 3rd generation gravitational wave detectors need a close cooperation between research institutes and industries, ET2SME provides innovation vouchers for relevant cross boarder projects launched by SME. \nInterested companies will also receive direct support in expanding their business network into neighbouring countries, especially in finding their suitable SME partner. They will also have the opportunity to present their know-how and special competences in a virtual 3D model of the Einstein Telescope (Mapping). Of particular interest here are the Einstein Telescope operating technologies in cryogenics, vacuum, precision mechanics\/mechatronics, sensors, optics and optical metrology, mirrors coating, lasers and advanced control algorithms.\nFinally, there is the possibility for companies of participating in an ET Industrial Advisory Board, organized jointly by E-TEST, ETpathfinder and ET2SMEs, between Business and Research, for the large-scale facility as a dialogue partner and advisor from now and in the coming years.\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{Introduction}\n\\chapter{Budget Summary}\n\\label{chap:BudgetSummary}\n\nThe overall equipment budget of the ETpathfinder phase 1a\/b project is composed of the individual budgets of work packages WP3 to WP7 as follows. We additionally list a budget contingency of 10\\%, which is foreseen to cover unexpected expenses, especially for items that have to go through a tendering process and thus precise costings cannot be given in advance.\n\n\\begin{table}[htbp]\n\\centering\n \\begin{tabular}{lSS}\n \\toprule\n Work Package & {Budget in \u20ac, incl VAT} & {Budget in \u20ac, excl VAT}\\\\\n \\midrule\n WP3: Cleanroom & 3250000 & 2685950 \\\\\n WP4: Vacuum System & 4481840 & 3704000 \\\\\n WP5: Seismic Isolation & 1300000 & 1074380 \\\\\n WP6: Optics & 3408320 & 2816793 \\\\\n WP7: Controls & 2321108 & 1918271 \\\\\n \\midrule\n Subtotal & 14761268 & 12199395 \\\\\n Contingency 10\\% & 1476127 & 1219940 \\\\\n Total & 16237394 & 13419335 \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\begin{figure}\n \\centering\n \n \\includegraphics[width=0.95\\linewidth]{Figures\/budget.pdf}\n\\end{figure}\n\n\n\nThe subtotal without contingency (14.76\\,MEuro) matches pretty well the earlier cost estimates of the proposal (14.50\\,MEuro) and we assume the deviation of less than 2\\,\\% is well within the noise of the accuracy of our knowledge of the costs and hence no cause for major concern. \nHowever, the Interreg grant did not consider any contingency. If an contingency is found to be necessary, additional funds have to be found. \n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nPerturbative Quantum Chromodynamics (pQCD) at high energies can be formulated in\ncoordinate space in the dipole picture \\cite{Mueller}. If we in particular focus on Deep Inelastic Scattering the scattering process\ncan be described in this picture as interaction of virtual photon which has just enough energy to dissociate into a 'color\ndipole' with the hadronic target carrying most of the total energy. The interaction process is described here by the \ndipole-nucleus scattering amplitude. \nThis amplitude can be modeled and we will focus here on \nthe Golec-Biernat W\\\"usthoff saturation model \\cite{GBW} which \nincludes saturation effects. It was motivated by requirements that at the high energy limit of QCD the total cross section for hadronic processes\nshould obey unitarity requirements. At present there are much more sophisticated approaches to introduce these requirements\nin a description of scatterings at high energies \\cite{JalilianMarian:1997jx,JalilianMarian:1997gr,JalilianMarian:1997dw,Bal,Kov,Avsar:2006jy,BartKut}. \nHowever, one can still ask the question if there is any dynamics \nbehind the GBW model or to put it differently is there any equation to which formula proposed by Golec-Biernat and W\\\"usthoff is a solution? \nAnd what is the role of the initial conditions? In this article we report on answer to these questions provided in \\cite{Kutak:2009zk}.\\\\\n\\section{GBW model and a transport equation}\nThe GBW amplitude following form GBW cross section and related to it by $\\sigma(x,r)=2\\int d^2b N(x,r,b)$ reads (here we are interested in the original \nformulation):\n\\begin{equation}\nN(x,r,b)=\\theta(b_0-b)\\Bigg[1-\\exp\\left(-\\frac{r^2}{4R_0^2}\\right)\\Bigg]\n\\label{eq:GBW98}\n\\end{equation} \nwhere $b$ is the impact parameter of the collision defined as distance between center of the proton with radius $b_0$ and center of a dipole \nscattering on it, \n$r$ is a transversal size of the dipole, $x$ is the Bjorken variable, \n$R_0(x)\\!=\\!\\frac{1}{Q_0}\\left(\\frac{x}{x_0}\\right)^{\\lambda\/2}$ \nis the so called saturation radius and its inverse defines saturation scale, $Q_s(x)\\!=\\!1\/R_0(x)$ and $x_0$, $\\lambda$ are free parameters. \nThis amplitude saturates for large dipoles $r\\!\\gg\\!2R_0$ and exhibits geometrical scaling which has been confirmed by data \\cite{Stasto:2000er}.\n\\subsection{Transport equation for unintegrated gluon density}\nThe dipole amplitude (\\ref{eq:GBW98}) can be related to the unintegrated gluon density which convoluted with the $k_T$ dependent \noff-shell matrix elements allows to calculate observables in the high energy limit of QCD. This relation is the following (after assumption that the dipole is \nmuch smaller than the target):\n\\begin{equation}\nf(x,k^2,b)=\\frac{N_c}{4\\alpha_s\\pi^2}k^4\\nabla^2_k\\int \\frac{d^2{\\bold r}}{2\\pi}\\exp(-i{\\bold k}\\cdot{\\bold r})\\frac{N(x,r,b)}{r^2}\n\\label{eq:ugd}\n\\end{equation}\nwhere $r$ and $k$ are two-dimensional vectors in transversal plane of the collision and $r\\equiv|\\bold r|$, $k\\equiv|\\bold k|$\\\\ \nPerforming this transformation we obtain the known result \\cite{GBF}:\n\\begin{equation}\nf(x,k^2,b)=\\frac{N_c}{2\\pi^2\\alpha_s}\\theta(b_0-b)R_0^2(x)k^4\\exp\\left[-R_0^2(x)k^2\\right]\n\\label{eq:ugd2}\n\\end{equation}\n Now motivated by the fact that function $f(x,k^2,b)$ exhibits a maximum both as a function of $x$ for fixed $k^2$ and as a function\nof $k^2$ for fixed $x$, we differentiate $f(x,k^2,b)$ with respect to $x$ and $f(x,k^2,b)\/k^2$ with respect to $k^2$.\nWe obtain:\n\\begin{equation}\n\\partial_x f(x,k^2,b)=\\frac{\\lambda f(x,k^2,b)(1-R_0^2(x)k^2)}{x Q_0^2}\n\\label{eq:poch1}\n\\end{equation}\n\\begin{equation}\n\\partial_{k^2} \\frac{f(x,k^2,b)}{k^2}=\\frac{f(x,k^2,b)(1-R_0^2(x)k^2)}{k^4Q_0^2}\n\\label{eq:poch2}\n\\end{equation}\nDividing eqn. (\\ref{eq:poch1}) by (\\ref{eq:poch2}) and rearranging the terms and defining ${\\cal F}(x,k^2,b)=f(x,k^2,b)\/k^2$, $Y=\\ln x_0\/x$, $L=\\ln k^2\/Q_0^2$\nwe obtain:\n\\begin{equation}\n\\partial_Y {\\cal F}(Y,L,b)+\\lambda\\partial_L {\\cal F}(Y,L,b)=0\n\\label{eq:trans}\n\\end{equation}\nwhich is the first order linear wave equation also known as the transport equation. As it is linear it cannot generate saturation dynamically but it can propagate well the initial condition leading to a successful phenomenology \\cite{GBW}. \nIt describes the change (wave) in the particle distribution flowing into and out of the phase space volume with velocity $\\lambda$. \nThis wave propagates in one direction. The quantity ${\\cal F}(x,k^2,b)$ gains here the interpretation of a number density of gluons with momentum \nfraction $x$ with the transversal momentum $k^2$ at distance $b$ from the center of the proton. \nThe general solution of (\\ref{eq:trans}) can be found by the method of characteristics and is given by:\n\\begin{equation}\n{\\cal F}(Y,L,b)={\\cal F}_0(L-\\lambda Y,b)\n\\label{eq:sol}\n\\end{equation}\nOne can go back from (\\ref{eq:trans}) to (\\ref{eq:ugd2}) using following initial condition at $x=x_0$: \n\\begin{equation}\n{\\cal F}(x\\!=\\!x_0,k^2,b)=\n\\frac{N_c}{2\\pi^2\\alpha_s}\\theta(b_0-b)k^2\\exp(-k^2)\\\\\n\\label{eq:gbwini}\n\\end{equation}\nThis initial condition has saturation built in, since the gluon density vanishes for small $k^2$.\nKnowing the properties of the linear first order partial differential equation we see that the property of saturation\nof GBW was a consequence of the wave solution which relates $x$ and $k^2$ supplemented by initial conditions with saturation built in. \nWe also see that the {\\it critical line} of the GBW saturation model visualizing, the dependence of the saturation scale on $x$, \n$Q_s(x)=Q_0\\left(\\frac{x_0}{x}\\right)^{\\lambda\/2}$ is in fact from the mathematical point of view the characteristics of the \ntransport equation.\n\\subsection{Transport equation for the dipole amplitude in momentum space}\nSimilar investigations can be repeated for the momentum space representation of the dipole amplitude $N(x,r,b)$ \nwhich we denote by $\\phi(x,k^2,b)$. \n\\begin{equation}\n\\phi(x,k^2,b)=\\int\\frac{d^2{\\bold r}}{2\\pi}\\exp(-i{\\bold k}\\cdot{\\bold r})\\frac{N(x,r,b)}{r^2}\n\\label{eq:ugd}\n\\end{equation}\nA nonlinear pQCD evolution equation like the Balitksy-Kovchegov (BK) equation written for $\\phi$ (in large target approximation) takes quite simple form and \ncan be related directly to the statistical formulation of the high energy limit of QCD (see \\cite{SM2} and references therein). \nApplying this transformation to (\\ref{eq:GBW98}) we proceed with differentiation similarly as before and we obtain:\n\\begin{equation}\n\\partial_Y\\phi(Y,L,b)+\\lambda \\partial_L \\phi(Y,L,b)=0\n\\end{equation}\nwhich is, as before, the transport equation.\n\\subsection{Relation to pQCD}\nIt is tempting to investigate the relation between found transport equation and the high energy pQCD evolution equations like \\cite{Bal,Kov,BartKut}. \nLet us focus here in particular on the form of the BK equation in \nlarge cylindrical target approximation for the dipole amplitude in momentum space for which the nonlinear term is just a simple local quadratic expression.\nThe BK equation for the dipole amplitude in the momentum space reads:\n\\begin{equation}\n\\partial_Y\\phi(Y,k^2,b)=\\overline\\alpha\\chi\\left(-\\frac{\\partial}{\\partial_{\\log k^2}}\\right)\\phi(Y,k^2,b)-\\overline\\alpha\\phi^2(Y,k^2,b)\n\\label{eq:BKdip}\n\\end{equation}\nwhere $\\overline\\alpha=\\frac{N_c\\alpha_s}{2\\pi}$ and $\\chi(\\gamma)=2\\psi(1)-\\psi(\\gamma)-\\psi(1-\\gamma)$ is the characteristic function of the BFKL \nkernel which allows for emission of dipoles and therefore drives the rise of the amplitude. The role of the nonlinear term is \nroughly to allow for multiple scatterings of dipoles which contributes with negative sign and slows down the rise of the amplitude. This equation \nprovides unitarization of the dipole amplitude \\cite{BKphen} for fixed impact parameter and admits traveling wave solution in the diffusion approximation\\cite{SM1}.\nThe analytic solution of (\\ref{eq:BKdip}) within the diffusion approximation relying on expanding the kernel of (\\ref{eq:BKdip}) up to second \norder and mapping it to the Fisher-Kolmogorov equation has been obtained by Munier and Peschanski \\cite{MP3}. It reads:\n\\begin{equation}\n\\phi(Y,k^2,b)=\\theta(b_0-b)\\sqrt{\\frac{2}{\\overline\\alpha\\chi''(\\gamma_c)}}\n\\ln\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\left(\\frac{k^2}{Q_s^2(Y)}\\right)^{\\gamma_c-1}\n\\exp\\left[-\\frac{1}{2\\overline\\alpha\\chi''(\\gamma_c)Y}\\ln^2\\left(\\frac{k^2}{Q_s^2(Y)}\\right) \n\\right]\n\\label{eq:solBK1}\n\\end{equation}\nwhere $\\gamma_c=0.373$ and $Q_s^2(Y)$ is emergent\nsaturation scale given by:\n\\begin{equation}\nQ_s^2(Y)=Q_0^2e^{-\\bar\\alpha\\chi'(\\gamma_c) Y-\\frac{3}{2\\gamma_c}\\log Y\n-\\frac{3}{(1-\\gamma_c)^2}\n\\sqrt{\\frac{2\\pi}{\\bar\\alpha\\chi^{\\prime\\prime}(\\gamma_c)}}\\frac{1}{\\sqrt{Y}}\n+{\\cal O}(1\/Y)}\\ \n\\label{eq:satscal}\n\\end{equation}\nBy inspection we see that (\\ref{eq:solBK1}) does not obey the transport equation. The problem is caused by the diffusion term. \nHowever, we can consider the asymptotic regime called \"front interior\" \\cite{MP3}, region where transverse momenta $k$ is close to the saturation scale \n$Q_s(Y)$ and rapidity $Y$ is large and where the condition $\\ln^2\\left(\\frac{k^2}{Q_s^2(Y)}\\right)\/2\\overline\\alpha\\chi''(\\gamma_c)Y\\!<\\!\\!\\!<\\!1$ is satisfied.\nIn this regime (\\ref{eq:solBK1}) simplifies and after taking derivatives as in the previous sections we obtain the following wave equation:\n\\begin{equation}\n\\partial_Y {\\phi}(Y,L,b)+\\lambda_{BK}\\partial_L {\\phi}(Y,L,b)=0\n\\label{eq:transBK}\n\\end{equation} \n where $\\lambda_{BK}=\\partial \\log Q_s^2(Y)\/\\partial Y$. In the limit where $\\lambda_{BK}$ does not depends on energy \\cite{IJM} we obtain:\n\\begin{equation}\n\\lambda_{BK}=-\\overline\\alpha\\chi'(\\gamma_c)\n\\label{eq:asymlimit}\n\\end{equation}\n\n\\section{Conclusions}\nIn this note we have shown that the GBW saturation model is the exact solution of a one-dimensional linear transport equation of the form (\\ref{eq:trans}).\nWe conclude that since (\\ref{eq:trans}) is a linear equation the saturation property has to be provided in the initial condition.\nWe found that for the GBW model this equation is universal for the unintegrated gluon density $f(x,k^2,b)$ and the dipole amplitude in momentum space \n$\\phi(x,k^2,b)$ but the details of the shape of the wave depends on the initial condition which is different for each of them. We also studied the \nrelation of the transport equation to the BK equation in the diffusion approximation. We have shown that in the \nregion of phase space where diffusion and splitting processes are of the same order as the nonlinear term, the GBW model is consistent with the \nBK equation. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nHomogeneous cosmological models have successfully explained many important features of our universe. However, we know that the distribution of matter is not homogeneous and so these models are only an approximation. In the last decade modelling of inhomogeneity in cosmology has become a topic of substantial interest. There are several different approaches to the problem. Due to the nonlinear nature of Einstein equations one should not rely on perturbation theory completely therefore exact models with nonuniform distribution of matter should be considered as well. Among the most studied exact inhomogeneous solutions of Einstein equations belong Lemaitre-Tolman-Bondi (LTB) metric \\cite{1}, Szekeres solution \\cite{2}, Szafron family of solutions \\cite{3}, Stephani solution \\cite{4} or Lemaitre metric \\cite{1}, which is a generalization of LTB for a fluid with nonzero pressure. An overview of inhomogeneous cosmological solutions can be found in \\cite{6}. \n\nLTB metric was extensively studied by Krasinski and Hellaby to model structure formation \\cite{7} - \\cite{9}. Together with Bolejko they used the solution to describe formation of voids in the universe \\cite{10}. \nLTB metric is a special case of the Szekeres solution that was discovered by Szekeres \\cite{2} and was developed by Hellaby and Krasinski in a series of papers \\cite{11} - \\cite{14}. They gave a geometrical interpretation of the metric functions in all three different types of geometry that are quasi-spherical, quasi-pseudospherical and quasi-planar. The quasi-spherical case is currently the best understood of all three types. It has found a cosmological application in the study of Bolejko who focused on structure formation \\cite{15}, \\cite{16} and constructed models of a void with an adjourning supercluster. Bolejko also investigated Buchert averaging of the quasispherical Szekeres metric \\cite{17}, constructed a Szekeres-Swiss-cheese model that he used to estimate the impact of inhomogeneity on the propagation of light \\cite{18} and CMB observations \\cite{19}.\n\nWalters and Hellaby showed in a recent paper \\cite{20} how to model inhomogeneity in quasispherical Szekeres metric using initial and final data. They constructed three models where they specified initial and final radial density profile and one extreme value of the density on the final time slice in terms of a deviation function along with the position of the extreme. In this way they set up all 5 degrees of freedom in the Szekeres metric and worked out an algorithm to calculate all other metric functions in the model.\n\nFor practical purposes it would be desirable to consider the evolution of inhomogeneities in Szekeres spacetime determined only by initial data. In our approach we specify the curvature function $f$ and the radial density profile with one extreme value at the initial time. Thus we set up only 3 degrees of freedom which is however sufficient to model the density contrast. To make an appropriate choice for the initial density extreme we investigate the shell crossing conditions in terms of the density extremes and show that the resulting constraints are preserved throughout the evolution (for the assumed value of $f$). \n\nThe paper is organized as follows. In the next section we give an overview of the Szekeres metric and describe its basic properties. In section 3 we derive conditions that the density extremes have to satisfy in order to avoid a shell crossing singularity. In section 4 we show that if these conditions are met at the initial time, then they are fulfilled at later times if the curvature function vanishes. In section 5 we derive an analytical formula for the density contrast defined as the difference of the extreme values and for the case $f=0$ we show that if the initial radial inhomogeneity is small, then the density contrast is proportional to ${t^{-3}}$ and the radial dependence of the function does not evolve substantially in time. In section 6 we set up the model by specifying two more functions and show the time evolution of the density contrast in a specific example. We conclude in section 7.\n\n\\section{Szekeres spacetime}\n\nThe Szekeres spacetime is an exact dust solution of Einstein equations without any symmetries. It was found by Szekeres \\cite{2} and generalized by Szafron \\cite{3} for an energy momentum tensor describing a perfect fluid.\nThe LT type of Szekeres metric can be written as \\cite{21}\n\\begin{equation}\n\\label{9}\n{\\rm d}s^{2}=-{\\rm d}t^{2}+\\frac{\\left(R'-R\\frac{E'}{E}\\right)^2}{\\epsilon+f\\left(r\\right)}{\\rm d}r^2+\\frac{R^2}{E^2}\\left({\\rm d}p^2+{\\rm d}q^2\\right),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{10}\nE\\left(r,p,q\\right)\\equiv\\frac{S}{2}\\left[\\left(\\frac{p-P}{S}\\right)^2+\\left(\\frac{q-Q}{S}\\right)^2+\\epsilon\\right],\n\\end{equation}\n\\begin{equation}\\label{25}\nE'=\\frac{S'}{2}\\left[1-\\frac{\\left(p-P\\right)^2}{S^2}-\\frac{\\left(q-Q\\right)^2}{S^2}\\right]-\\frac{P'}{S}\\left(p-P\\right)-\\frac{Q'}{S}\\left(q-Q\\right)\n\\end{equation}\nand $f$, $P$, $Q$, $S$ are arbitrary functions of $r$. The parameter $\\epsilon$ can have only three values $-1, 0, 1$ and it determines the geometry of the two-spaces of constant $t$ and $r$.\nFrom Einstein equations it follows a dynamical equation for the function $R$ \n\\begin{equation}\n\\label{11}\n{\\dot{R}}^2=\\frac{2M\\left(r\\right)}{R}+f\\left(r\\right)+\\frac{\\Lambda R^2}{3}\n\\end{equation}\nand an equation for the density evolution\n\\begin{equation}\n\\label{12}\n\\rho=\\frac{2}{\\kappa c^{2}}\\frac{M'-3M\\frac{E'}{E}}{R^2\\left(R'-R\\frac{E'}{E}\\right)},\n\\end{equation}\nwhere prime denotes a derivative with respect to $r$, $M$ is another arbitrary function and $\\Lambda$ is cosmological constant. So we have two Einstein equations (\\ref{11}), (\\ref{12}) and by solving the first one we obtain one more arbitrary function $t_B\\left(r\\right)$, which enters the solution in the form $t-t_B$. From now on we will use a redefined form of the density $\\bar{\\rho}\\equiv\\kappa c^{2}\\rho$ and for the sake of simplicity we will drop the bar.\n\nAs mentioned above, the metric does not have any symmetries, in other words it has no Killing vectors. Nevertheless the three spaces of constant $t$ are conformally flat \\cite{22}.\n\nThe parametrization of the metric involves 6 arbitrary functions however the number of physical degrees of freedom is 5, because we can still rescale the radial coordinate (the metric is invariant with respect to the transformation $r'=g(r)$). \n\n\\subsection{Interpretation of $f$ and $t_B$}\n\nThe sign of the function $f$ effects the solution of the Einstein equation (\\ref{11}). It is a dynamical equation for $R$ and it looks very similar to the Friedman equation, except that here the functions $f$ and $M$ depend on $r$. We will assume only the case when $\\Lambda=0$. There are three different types of evolution \\cite{23}. If $f<0$ the evolution is elliptic which means that the universe is first expanding and at some point the expansion stops and the universe is collapsing to a final singularity. The solution is given in a parametric form\n\\begin{equation}\n\\label{13}\nR=\\frac{M}{\\left(-f\\right)}\\left(1-\\cos\\eta\\right), \\quad \\eta-\\sin\\eta=\\frac{\\left(-f\\right)^{\\frac{3}{2}}\\left(t-t_B\\right)}{M}.\n\\end{equation}\nFor $f>0$ the evolution is hyperbolic in the sense that the sign of the expansion does not change, the universe either expands or collapses depending on initial conditions and the parametric solution to the equation (\\ref{11}) is\n\\begin{equation}\n\\label{14}\nR=\\frac{M}{\\left(-f\\right)}\\left(\\cosh\\eta-1\\right), \\quad \\sinh\\eta-\\eta=\\frac{f^{\\frac{3}{2}}\\left(t-t_B\\right)}{M}.\n\\end{equation}\nFor $f=0$ the evolution is parabolic and is given by equation\n\\begin{equation}\n\\label{15}\nR=\\left(\\frac{9}{2}M\\right)^{\\frac{1}{3}}\\left(t-t_B\\right)^{\\frac{2}{3}}.\n\\end{equation}\nSince $f$ is a function of $r$, the universe may have different evolution in different regions. The parabolic evolution can be on the boundary between two regions, one having elliptic evolution and the other one hyperbolic. So the sign of $f$ determines the sign of scalar curvature of the three-spaces of constant $t$ and when $f=0=f'$, the three-spaces are flat which corresponds to a presence of pure decaying modes as was shown in \\cite{sussman}.\n\nThe function $t_B$ is called the bang time function, because $t=t_B$ is the moment when big bang happened. So unlike in homogeneous models, the initial moment of evolution is position dependent. \n\nThe interpretation of the other metric functions ($R$, $M$, $P$, $Q$, $S$ ) depends on $\\epsilon$ and from now on we will only consider the case $\\epsilon=+1$ which is often called the quasi-spherical case. \n\n\n\\subsection{Coordinate transformation and interpretation of $R$}\n\nWe can make a coordinate transformation \\cite{11}\n\\begin{equation}\n\\label{16}\n\\frac{p-P}{S}=\\cot\\frac{\\theta}{2}\\cos\\phi, \\quad \\frac{q-Q}{S}=\\cot\\frac{\\theta}{2}\\sin\\phi\n\\end{equation}\nand rewrite the induced two-metric of the surfaces of constant $t$ and $r$\ninto coordinates $\\theta$ and $\\phi$. After applying the transformation we get\n\\begin{equation}\n\\label{18}\n{\\rm d}s^2=\\frac{4}{S^2}\\sin^4\\frac{\\theta}{2}\\left({\\rm d}p^2+{\\rm d}q^2\\right)={\\rm d}\\theta^2+\\sin^2\\theta{\\rm d}\\phi^2,\n\\end{equation}\nwhich is a metric on a unit sphere. The transformation (\\ref{16}) is nothing but a stereographic projection and the function $E$ describes how the $(p,q)$ plane is mapped onto a unit sphere. Every sphere is multiplied by $R^2$ and the function is sometimes called the areal radius, because it is actually the radius of a sphere on a comoving coordinate $r$. \n\n\\subsection{Geometrical meaning of $E$}\n\nIn order to understand more about the geometrical properties of the metric (\\ref{9}) it is important to investigate the function $E$. We can see that this function appears in the metric and in the equation for the density (\\ref{12}) in the form $\\frac{E'}{E}$. Particularly, we can investigate when the function is equal to zero and what are its extreme values. \nUsing the transformation (\\ref{16}) in (\\ref{10}) and (\\ref{25}) we can write $\\frac{E'}{E}$ as\n\\begin{equation}\n\\label{31}\n\\frac{E'}{E}=-\\frac{S'\\cos\\theta+\\sin\\theta\\left(P'\\cos\\phi+Q'\\sin\\phi\\right)}{S}.\n\\end{equation} \nThe equation $\\frac{E'}{E}=0$ now becomes\n\\begin{equation}\n\\label{28}\nS'\\cos\\theta+P'\\sin\\theta\\cos\\phi+Q'\\sin\\theta\\sin\\phi=0\n\\end{equation}\nand after realizing that \n$\\cos\\theta=z$, $\\sin\\theta\\cos\\phi=y$ and $\\sin\\theta\\sin\\phi=x$,\nthe equation (\\ref{28}) becomes\n\\begin{equation}\n\\label{30}\nS'z+P'x+Q'y=0,\n\\end{equation}\nwhich is an equation of a plane that goes through the origin of the spherical coordinate system and intersects the $r=t=const$ sphere in a great circle. \nNow calculating the derivatives of (\\ref{31}) with respect to $\\theta$ and $\\phi$ and putting it equal to zero we can find that there are two extreme values and they are \nlocated at opposite sites on the sphere, in other words if one extreme is at the coordinates $\\left(\\theta_{1},\\phi_{1}\\right)$, the other one is at $\\left(\\pi-\\theta_{1}, \\phi_{1}+\\pi\\right)$ and they are located symmetrically with respect to the plane $\\frac{E'}{E}=0$, because the coordinates of the extremes are exactly the components of the unit normal to the plane $\\frac{E'}{E}=0$. The values of the extremes are\n\\begin{eqnarray}\n\\label{33}\n\\left(\\frac{E'}{E}\\right)_{max}&=&\\frac{\\sqrt{\\left(S'\\right)^2+\\left(P'\\right)^2+\\left(Q'\\right)^2}}{S}, \\nonumber\\\\ \n\\left(\\frac{E'}{E}\\right)_{min}&=&-\\frac{\\sqrt{\\left(S'\\right)^2+\\left(P'\\right)^2+\\left(Q'\\right)^2}}{S},\n\\end{eqnarray}\nwhere max and min refers to maximum and minimum respectively. So the extremes have opposite values and the function $\\frac{E'}{E}$ behaves on the sphere like a dipole \\cite{11}. \n\n\\section{Shell crossing conditions in terms of the density extremes}\n\nWe are now going to study the formula for the density (\\ref{12}). We can see that under certain circumstances the density can change sign or possibly diverge if the denominator becomes zero. Those points where this happens are called the shell crossing singularity. The conditions that the metric functions $M$, $f$, $t_{B}$ and $R$ have to satisfy in order to avoid shell crossing singularity can be found in \\cite{11}. In this section we will derive the conditions that the extreme values of the density $\\rho_{max}$ and $\\rho_{min}$ have to satisfy in order to avoid the shell crossing singularity. \n\nWe can rewrite (\\ref{12}) as\n\\begin{equation}\n\\label{34}\n\\rho\\left(t,r,\\theta, \\varphi\\right)=2\\frac{M'-3M\\frac{E'}{E}}{R^{2}\\left(R'-R\\frac{E'}{E}\\right)}=\\frac{R'\\rho_{LT}-R\\frac{E'}{E}\\rho_{AV}}{R'-R\\frac{E'}{E}},\n\\end{equation}\nwhere we define\n\\begin{equation}\n\\label{35}\n\\rho_{LT}\\left(t,r\\right)\\equiv\\frac{2M'}{R^{2}R'}\n\\end{equation}\nand\n\\begin{equation}\n\\label{36}\n\\rho_{AV}\\left(t,r\\right)\\equiv\\frac{6M}{R^{3}}.\n\\end{equation}\nHere $\\rho_{LT}$ is the density that we get from (\\ref{12}) if we set $\\frac{E'}{E}=0$. It is also a radial density in the sense that on a given $r$ it is the value of the density around the great circle that lies in the plane that defines the dipole on the sphere. The index $LT$ refers to Lemaitre-Tolman-Bondi metric, because its density is given with exactly the same formula as (\\ref{35}). \nIf we make the choice \n\\begin{equation}\n\\label{ic}\nR\\left(t_{i}\\right)\\equiv r\n\\end{equation} \nand use the fact that the function $M$ depends only on $r$, we can express it on the initial time slice when $\\frac{E'}{E}=0$ as\n\\begin{equation}\n\\label{37}\nM=\\frac{1}{2}\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r',\n\\end{equation}\nwhere the index $0$ refers to the value of the function at the initial time $t_{i}$.\nIn that case the definitions $(\\ref{35})$ and $(\\ref{36})$ can be further rewritten as\n\\begin{equation}\n\\label{den}\n\\rho_{LT}\\left(t,r\\right) = \\frac{1}{R^{2}R'}\\rho_{LT0}r^{2}, \\quad \\quad \\rho_{AV}\\left(t,r\\right) = \\frac{3}{R^{3}}\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'.\n\\end{equation}\nThe initial condition (\\ref{ic}) and formulas (\\ref{37}) - (\\ref{den}) will however not be used in the following calculations (with the exception of equation (\\ref{68}) giving critical point position) and the main results of this section are independent of it, thus giving us generally applicable criteria. \n\nIt is good to take a look at the derivative of the density with respect to $\\frac{E'}{E}$\n\\begin{equation}\n\\label{38}\n\\frac{\\partial\\rho}{\\partial\\frac{E'}{E}}=RR'\\frac{\\rho_{LT}-\\rho_{AV}}{\\left(R'-R\\frac{E'}{E}\\right)^{2}}.\n\\end{equation}\nFrom the derivative we can see, that if $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)>0$ the derivative is positive and the density is growing as $\\frac{E'}{E}$ increases. On the other hand if $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)<0$ the density will decrease as $\\frac{E'}{E}$ increases. Either way we can see that the density behaves on each sphere also like a dipole in the sense that it has two extreme values, maximum and minimum and they are located at the same position as extreme values of the function $\\frac{E'}{E}$. In the case when $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)<0$ we can write for the density maximum\n\\begin{equation}\n\\label{39}\n\\rho_{max}=\\frac{R'\\rho_{LT}-R\\left(\\frac{E'}{E}\\right)_{min}\\rho_{AV}}{R'-R\\left(\\frac{E'}{E}\\right)_{min}}\n\\end{equation}\nand for the minimum\n\\begin{equation}\n\\label{40}\n\\rho_{min}=\\frac{R'\\rho_{LT}-R\\left(\\frac{E'}{E}\\right)_{max}\\rho_{AV}}{R'-R\\left(\\frac{E'}{E}\\right)_{max}}.\n\\end{equation}\nIn the case when $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)>0$ the role of $\\left(\\frac{E'}{E}\\right)_{max}$ interchanges with $\\left(\\frac{E'}{E}\\right)_{min}$ in the last two equations.\n\nWe split the investigation of the density into three parts, in the first part we investigate when both the numerator and the denominator are positive, in the second case we check when they are both negative and in the last case we take a look at the special case when they are both zero. \nIt will be convenient to split the first part into three more subcases depending on the value of $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)$. For the investigation we will also need a formula for the function $\\frac{E'}{E}$ given in terms of $\\rho_{LT}$ and $\\rho_{AV}$ which we can derive from (\\ref{34})\n\\begin{equation}\n\\label{41}\n\\frac{E'}{E}=\\frac{R'}{R}\\frac{\\rho-\\rho_{LT}}{\\rho-\\rho_{AV}}.\n\\end{equation}\n\n\\vspace{10px}\n\\noindent\\textsc{First case: numerator and denominator of the density are both positive.} \\\\\n\\noindent\\textsc{Subcase A: $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)<0$}\n\nFrom the positivity of the denominator in (\\ref{34}) we get \n\\begin{equation}\n\\label{42}\n\\frac{R'}{R}>\\left(\\frac{E'}{E}\\right)_{max}=\\frac{R'}{R}\\frac{\\rho_{min}-\\rho_{LT}}{\\rho_{min}-\\rho_{AV}},\n\\end{equation}\nbecause as follows from (\\ref{38}) if $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)<0$ we can see that $\\frac{E'}{E}$ has maximum where the density has minimum. \nFrom the dipole property\n\\begin{equation}\n\\label{44}\n\\left(\\frac{E'}{E}\\right)_{max}=-\\left(\\frac{E'}{E}\\right)_{min}\n\\end{equation}\nwe know that $\\left(\\frac{E'}{E}\\right)_{max}$ is non-negative and (\\ref{42}) implies that $R'>0$ and therefore we have to consider $\\rho_{LT}<\\rho_{AV}$. We can now set conditions for $\\rho_{min}$ so that the right hand side in (\\ref{42}) is nonnegative. \nInequality (\\ref{42}) simplifies to\n\\begin{equation}\n\\label{43}\n1>\\frac{\\rho_{min}-\\rho_{LT}}{\\rho_{min}-\\rho_{AV}}\n\\end{equation}\nIt is reasonable to require $\\rho_{min}\\leq\\rho_{LT}$, because the minimum should be the smallest value on each sphere ($\\rho_{min}=\\rho_{LT}$ corresponds to the case when the density is homogeneously distributed on the whole sphere). Since we consider $\\rho_{LT}<\\rho_{AV}$ it follows that $\\rho_{min}<\\rho_{AV}$ and the right hand side of (\\ref{42}) is nonnegative. It means that besides the obvious condition $\\rho_{min}\\leq\\rho_{LT}$ we don't have any other restriction for the minimum. \n\n\nDue to the dipole property we can try to derive a condition for the maximum\n\\begin{equation}\n\\label{45}\n\\frac{R'}{R}>\\left(\\frac{E'}{E}\\right)_{max}=-\\frac{R'}{R}\\frac{\\rho_{max}-\\rho_{LT}}{\\rho_{max}-\\rho_{AV}}.\n\\end{equation} \nSince $\\rho_{max}$ should be greater than or equal to $\\rho_{LT}$ it implies that $\\rho_{max}<\\rho_{AV}$ in order for the right hand side in (\\ref{45}) to be non-negative. This condition is reasonable since $\\rho_{AV}$ is not true value of the density as follows from its definition (\\ref{36}). Taking this into consideration we can solve the inequality (\\ref{45}) and the solution is\n\\begin{equation}\n\\label{46}\n\\rho_{max}<\\frac{1}{2}\\left(\\rho_{LT}+\\rho_{AV}\\right)\n\\end{equation}\nand this condition is clearly more restrictive than $\\rho_{max}<\\rho_{AV}$.\n\nFor the numerator of (\\ref{34}) we need\n\\begin{equation}\n\\label{47}\n\\frac{R'}{R}\\rho_{LT}>\\frac{E'}{E}\\rho_{AV}\n\\end{equation}\nwhich can be rewritten as\n\\begin{equation}\n\\label{48}\n\\frac{R'}{R}\\frac{\\rho_{LT}}{\\rho_{AV}}>\\left(\\frac{E'}{E}\\right)_{max}=\\frac{R'}{R}\\frac{\\rho_{min}-\\rho_{LT}}{\\rho_{min}-\\rho_{AV}}.\n\\end{equation}\nIt again follows that $R'>0$ and $\\rho_{LT}<\\rho_{AV}$. The right-hand side of (\\ref{48}) has to be non-negative and we know that $\\rho_{min}\\leq\\rho_{LT}$ and since $\\rho_{LT}<\\rho_{AV}$ we also have $\\rho_{min}<\\rho_{AV}$ so the inequality (\\ref{48}) is always true and we don't get any new condition for $\\rho_{min}$. We now use the property (\\ref{44}) and look for a condition for $\\rho_{max}$\n\\begin{equation}\n\\label{49}\n\\frac{\\rho_{LT}}{\\rho_{AV}}>-\\frac{\\rho_{max}-\\rho_{LT}}{\\rho_{max}-\\rho_{AV}}.\n\\end{equation}\nWe know that $\\rho_{max}\\geq\\rho_{LT}$ so to make the right hand side non-negative we need $\\rho_{max}<\\rho_{AV}$. The solution to the inequality (\\ref{49}) is\n\\begin{equation}\n\\label{50}\n\\rho_{max}<2\\frac{\\rho_{AV}\\rho_{LT}}{\\rho_{LT}+\\rho_{AV}}.\n\\end{equation}\nThe conditions for the denominator and the numerator have to be satisfied simultaneously, so for the maximum we have (\\ref{46}) and (\\ref{50}) and (as can be seen from figure 1) the condition (\\ref{50}) is stronger in the case of $\\rho_{LT}<\\rho_{AV}$ and has to be fulfilled in order for the density to be positive. As far as the minimum is concerned, apart from the obvious condition $\\rho_{min}\\leq\\rho_{LT}$ it is not constrained at all.\n \n\\vspace{10px}\n\\noindent\\textsc{First case: numerator and denominator of the density are both positive.} \\\\\n\\noindent\\textsc{Subcase B: $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)>0$}\n\nThe subcase B will be investigated similarly. The only difference is that from (\\ref{38}) we know that if $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)<0$ the function $\\frac{E'}{E}$ has maximum where the density has maximum and this modifies (\\ref{39}) and (\\ref{40}) accordingly. From the denominator of (\\ref{34}) we have\n\\begin{equation}\n\\label{51}\n\\frac{R'}{R}>\\left(\\frac{E'}{E}\\right)_{max}=\\frac{R'}{R}\\frac{\\rho_{max}-\\rho_{LT}}{\\rho_{max}-\\rho_{AV}},\n\\end{equation} \nwhich again implies $R'>0$ and $\\rho_{LT}>\\rho_{AV}$ because $\\left(\\frac{E'}{E}\\right)_{max}\\geq 0$ and it simplifies to\n\\begin{equation}\n\\label{52}\n1>\\frac{\\rho_{max}-\\rho_{LT}}{\\rho_{max}-\\rho_{AV}}.\n\\end{equation}\nThe analysis is basically the same, we again need the right-hand side to be non-negative. We know that $\\rho_{max}\\geq\\rho_{LT}$ and therefore $\\rho_{max}>\\rho_{AV}$ as well, so the inequality (\\ref{52}) will be always true as long as $\\rho_{LT}>\\rho_{AV}$ and we don't get any further condition for the maximum. For the minimum we have\n\\begin{equation}\n\\label{53}\n\\frac{R'}{R}>\\left(\\frac{E'}{E}\\right)_{max}=-\\frac{R'}{R}\\frac{\\rho_{min}-\\rho_{LT}}{\\rho_{min}-\\rho_{AV}}.\n\\end{equation} \nIn order for the right-hand side to be non-negative we need $\\rho_{min}>\\rho_{AV}$ which is a valid condition for the density minimum, because as mentioned above $\\rho_{AV}$ is not true value of the density. The inequality (\\ref{53}) has the solution\n\\begin{equation}\n\\label{54}\n\\rho_{min}>\\frac{1}{2}\\left(\\rho_{AV}+\\rho_{LT}\\right).\n\\end{equation}\n\nFrom the numerator of (\\ref{34}) we have the condition\n\\begin{equation}\n\\label{55}\n\\frac{R'}{R}\\frac{\\rho_{LT}}{\\rho_{AV}}>\\left(\\frac{E'}{E}\\right)_{max}=\\frac{R'}{R}\\frac{\\rho_{max}-\\rho_{LT}}{\\rho_{max}-\\rho_{AV}}.\n\\end{equation}\nIt again implies $R'>0$ and $\\rho_{LT}>\\rho_{AV}$ and it does not give us any constraint for the maximum, because we know that $\\rho_{max}\\geq\\rho_{LT}$ and therefore $\\rho_{max}>\\rho_{AV}$. So (\\ref{55}) holds as long as $R'>0$ and $\\rho_{LT}>\\rho_{AV}$.\nFor the minimum we can write\n\\begin{equation}\n\\label{56}\n\\frac{R'}{R}\\frac{\\rho_{LT}}{\\rho_{AV}}>\\left(\\frac{E'}{E}\\right)_{max}=-\\frac{R'}{R}\\frac{\\rho_{min}-\\rho_{LT}}{\\rho_{min}-\\rho_{AV}}.\n\\end{equation}\nThe right-hand side will be positive if $\\rho_{min}>\\rho_{AV}$ and the solution to (\\ref{56}) is\n\\begin{equation}\n\\label{57}\n\\rho_{min}>2\\frac{\\rho_{LT}\\rho_{AV}}{\\rho_{LT}+\\rho_{AV}}.\n\\end{equation}\nAll together in the case of $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)>0$ we have two conditions (\\ref{54}) and (\\ref{57}) for the minimum and as can be seen from figure 1 the first one is stronger. For the maximum, except for the obvious $\\rho_{max}\\geq\\rho_{LT}$, we don't have any constraint so the density will be positive as long as (\\ref{54}) holds and $\\rho_{max}\\geq\\rho_{LT}$.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.5]{conditions.eps}\n\\caption{In this figure we can see 4 different surfaces that constrain the density extremes. In the right part of the figure they are from the bottom to the top: $\\rho_{AV}$, $2\\frac{\\rho_{AV}\\rho_{LT}}{\\rho_{AV}+\\rho_{LT}}$, $\\frac{1}{2}\\left(\\rho_{AV}+\\rho_{LT}\\right)$, $\\rho_{LT}$.}\n\\end{figure}\n\n\n\\vspace{10px}\n\\noindent\\textsc{First case: numerator and denominator of the density are both positive.}\\\\\n\\noindent\\textsc{Subcase C: $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)=0$}\n\nBefore we consider this special case we can first rewrite (\\ref{34}) as\n\\begin{equation}\n\\rho=\\frac{R'\\left(\\rho_{LT}-\\rho_{AV}\\right)+\\rho_{AV}\\left(R'-R\\frac{E'}{E}\\right)}{R'-R\\frac{E'}{E}}\n\\end{equation}\nNow using $R'\\left(\\rho_{LT}-\\rho_{AV}\\right)=0$ we can see that the formula for the density simplifies\n\\begin{equation}\n\\rho=\\rho_{AV}\\frac{R'-R\\frac{E'}{E}}{R'-R\\frac{E'}{E}}=\\rho_{AV},\n\\end{equation}\nin other words the value of the density is independent of $\\theta$ and $\\phi$ and the density is homogeneously distributed on the whole sphere as follows also from (\\ref{38}) because the derivative of the density is now zero. Using the same argument as in subcase A and B we have $R'>0$ and therefore $\\rho_{LT}=\\rho_{AV}$. For the extreme values of the density we obviously need in this case\n\\begin{equation}\n\\rho_{max}=\\rho_{min}=\\rho_{AV}=\\rho_{LT}.\n\\end{equation}\n\n\nThe point where $\\rho_{LT}=\\rho_{AV}$ is especially interesting because the shell-crossing conditions for $\\rho_{max}$ and $\\rho_{min}$ change here and the location of $\\rho_{max}$ and $\\rho_{min}$ interchanges by passing through the plane $\\frac{E'}{E}=0$. The position of this critical point $r_{c}$ generally evolves and we can get its value as the solution to the equation\n\\begin{equation}\n\\label{68}\n3\\frac{R'}{R}\\int_{0}^{r_{c}}\\rho_{LT0}r^{2}{\\rm d}r=\\rho_{LT0}r_{c}^{2},\n\\end{equation}\nwhere we used the initial condition (\\ref{ic}) and formulas (\\ref{den}) in order to express $\\rho_{LT}$ and $\\rho_{AV}$. So the time evolution of the critical point will depend on the time evolution of the function $\\frac{R'}{R}$.\n\n\n\\vspace{10px}\n\\noindent\\textsc{Second case: numerator and denominator of the density are both negative.}\n\nIn order for the denominator of the density (\\ref{34}) to be negative, we need\n\\begin{equation}\n\\label{58}\n\\frac{R'}{R}<\\left(\\frac{E'}{E}\\right)_{min}.\n\\end{equation}\nThis condition will not be fulfilled unless $R'<0$ because $\\left(\\frac{E'}{E}\\right)_{min}\\leq 0$. The investigation is than similar as in the first case and leads to the same conditions for $\\rho_{min}$ and $\\rho_{max}$ as can be easily verified. \n\nOn the other hand if $R'>0$ and we allow the denominator to be negative on a given $r$, it will be also positive on that $r$ for some specific $\\theta$ and $\\phi$. This behaviour may or may not be correct, depending on the behaviour of the numerator and specifically depending on whether or not the numerator changes sign at the same $\\theta$, $\\phi$. For the numerator we have inequality\n\\begin{equation}\n\\label{59}\nR'\\rho_{LT}-R\\frac{E'}{E}\\rho_{AV}<0.\n\\end{equation}\nThis can be rewritten as\n\\begin{equation}\n\\label{60}\n\\frac{R'}{R}\\frac{\\rho_{LT}}{\\rho_{AV}}<\\left(\\frac{E'}{E}\\right)_{min}.\n\\end{equation}\nThe same argument as in the case of denominator tells us that this will be satisfied only if $R'<0$. So if the numerator is negative on a given $r$ but $R'>0$, there is a region on this $r$ where it is positive too.\nFor the positiveness of the denominator in a region $(\\theta, \\phi)$ we have\n\\begin{equation}\n\\label{61}\n\\frac{R'}{R}>\\frac{E'}{E}\n\\end{equation}\nand for the numerator the condition is\n\\begin{equation}\n\\label{62}\n\\frac{R'}{R}\\frac{\\rho_{AV}}{\\rho_{LT}}>\\frac{E'}{E}.\n\\end{equation}\nClearly the conditions are not the same unless $\\rho_{LT}=\\rho_{AV}$, so except for this special case, there will always be a region on a given $r$ where the numerator and the denominator will have different signs. The places where the sign changes are in both cases circles on the sphere, that are parallel to the great circle that defines the plane of the dipole. The order in which the circles go are seen in figure 2, where the parallel circles are mapped as parallel lines. \n\nSo if in the second case $R'>0$ there are no conditions for the density extremes that would prevent shell crossing.\n\n\\begin{figure}[!h]\n\\centering\n\\subfloat[]{\\includegraphics[trim = 10mm 170mm 110mm 20mm, clip, width=6cm]{obr.pdf}}\n\\subfloat[]{\\includegraphics[trim = 10mm 170mm 110mm 20mm, clip, width=6cm]{obr2.pdf}}\n\\caption{The behaviour of the density for $\\rho_{AV}>\\rho_{LT}$ in panel (a) and $\\rho_{AV}<\\rho_{LT}$ in panel (b) when the conditions for no shell-crossing are not met. The circle represents a sphere on a given $r$, the solid line represents the plane $\\frac{E'}{E}=0$. The dashed line represents a plane $\\frac{E'}{E}\\frac{\\rho_{LT}}{\\rho_{AV}}=\\frac{R'}{R}$, that is parallel to the plane $\\frac{E'}{E}=0$, the density becomes zero here. The dotted line is the plane $\\frac{E'}{E}=\\frac{R'}{R}$ and it is the plane where the density diverges. The density is negative between the dashed and dotted line.}\n\\label{fig_2}\n\\end{figure}\n\n\\vspace{10px}\n\\noindent\\textsc{Third case: numerator and denominator of the density are both zero.}\n\nThe denominator is zero when\n\\begin{equation}\n\\label{63}\n\\frac{R'}{R}=\\frac{E'}{E},\n\\end{equation}\nbut this equation will be satisfied on a given $r$ for all $\\theta$ and $\\phi$ only if both $\\frac{R'}{R}$ and $\\frac{E'}{E}$ are zero. This consequently means that $E'=P'=Q'=S'=R'=0$. The same arguments are valid for the numerator of the density. As was discussed in \\cite{11} and in the context of LTB metric in \\cite{23} this point will not be a singularity if also $M'=0$. Since $M'$ and $E'$ are independent of time it implies that $R'$ cannot evolve and we need stationary configuration in order to avoid shell crossing. \n\n\\begin{table}[!ht]\n\\label{tab1}\n\\centering\n\\begin{tabular}{|c| c|}\n\\hline\nConditions for $\\rho_{min}$ & Conditions for $\\rho_{max}$ \\\\\n\\hline\n\\multicolumn{2}{|c|}{$\\rho_{AV}>\\rho_{LT}$} \\\\\n\\hline\n$\\rho_{min}\\leq\\rho_{LT}<\\rho_{AV}$ & $\\rho_{max} \\in\\langle\\left.\\rho_{LT}; 2\\frac{\\rho_{AV}\\rho_{LT}}{\\rho_{LT}+\\rho_{AV}}\\right) $\\\\\n\n\\hline\n\\multicolumn{2}{|c|}{$\\rho_{AV}<\\rho_{LT}$} \\\\\n\\hline\n$\\rho_{min}\\in\\left(\\frac{\\rho_{AV}+\\rho_{LT}}{2}; \\rho_{LT}\\right.\\rangle$ & $\\rho_{max}\\geq\\rho_{LT}>\\rho_{AV}$\\\\\n\\hline\n\\multicolumn{2}{|c|}{$\\rho_{AV}=\\rho_{LT}$} \\\\\n\\hline\n$\\rho_{min}=\\rho_{LT}$ & $\\rho_{max}=\\rho_{LT}$\\\\\n\\hline\n\\end{tabular}\n\\caption{The list of shell crossing conditions for density extremes in terms of $\\rho_{LT}$ and $\\rho_{AV}$.}\n\\end{table}\n\nThe table 1 summarizes the conditions for the extreme values of the density that we obtained so that shell crossing would be avoided. From the dipole property (\\ref{44}) it follows that there is a constraint between the density extremes\n\\begin{equation}\n\\label{69}\n\\frac{\\rho_{min}-\\rho_{LT}}{\\rho_{min}-\\rho_{AV}}=-\\frac{\\rho_{max}-\\rho_{LT}}{\\rho_{max}-\\rho_{AV}},\n\\end{equation}\nwhich means that knowing one extreme value allows us calculate the other one with a formula\n\\begin{equation}\n\\label{70}\n\\rho_{max}=\\frac{\\rho_{min}\\left(\\rho_{AV}+\\rho_{LT}\\right)-2\\rho_{LT}\\rho_{AV}}{2\\rho_{min}-\\rho_{LT}-\\rho_{AV}}\n\\end{equation}\nthat follows from (\\ref{69}). And for the other extreme we just interchange min for max in the last equation. Consequently it is sufficient to ensure that just one of the extremes satisfies the conditions in table 1 and the constraint (\\ref{69}) ensures that the other density extreme has a value that does not break the conditions in table 1.\n\n\n\n\n\n\\section{Time dependence of the shell crossing conditions, case $f=0$}\n\nIn the previous section we derived conditions that $\\rho_{max}$ and $\\rho_{min}$ have to satisfy in order to avoid shell crossing singularity. To find out if a shell crossing occurs at a given time, we need to calculate $\\rho_{max}$ or $\\rho_{min}$ at that time and then check if the conditions are met. It would be useful to have conditions in terms of the initial $\\rho_{LT0}$ and $\\rho_{min0}$ or $\\rho_{max0}$ that would ensure no shell crossing at any time during the evolution. In this section we show that in the special case when $f=0$, if we avoid shell crossing at the initial time, it is guaranteed that no shell crossing occurs during the time evolution. \n\nFirst we show that if on the initial time slice $\\rho_{AV0}>\\rho_{LT0}$, then this condition holds at any later time. We will use the initial condition (\\ref{ic}) and start with the inequality that we want to prove $\\rho_{AV}>\\rho_{LT}$\n\\begin{equation}\n\\label{71}\n\\frac{3}{R^{3}}\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'>\\frac{1}{R^{2}R'}\\rho_{LT0}r^2,\n\\end{equation}\nthis can be rewritten as\n\\begin{equation}\n\\label{72}\n\\frac{R'}{R}r\\,\\frac{3\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}{\\rho_{LT0}r^{3}}>1\n\\end{equation}\nand the inequality sign depends on the sign of $R'$ and here we assumed $R'>0$. We can see that the time dependence in the last inequality is hidden in the function $\\frac{R'}{R}r$. \nThe choice $f=0$ has parabolic evolution and the solution to the equation (\\ref{11}) is given by (\\ref{15}). If we substitute in (\\ref{15}) for $M$ from (\\ref{37}) we get\n\\begin{equation}\n\\label{73}\nR\\left(t,r\\right)=\\left(\\frac{9}{4}\\right)^{\\frac{1}{3}}\\left(\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'\\right)^{\\frac{1}{3}}\\left(t-t_{B}\\right)^{\\frac{2}{3}}.\n\\end{equation}\nWe can fix the bang time function $t_{B}$ using (\\ref{ic}) to be\n\\begin{equation}\n\\label{74}\nt_{B}=\\left[t_{i}-\\frac{2}{3}\\left(\\frac{r^{3}}{\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}\\right)^{\\frac{1}{2}}\\right],\n\\end{equation}\nwhere $t_{i}$ is the initial moment of the evolution. By calculating the radial derivative of (\\ref{73}) we can express $\\frac{R'}{R}r$ as\n\\begin{equation}\n\\label{75}\n\\frac{R'}{R}r=\\frac{\\rho_{LT0}r^{3}}{3\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}-\\frac{2}{3}\\frac{t_{B}'}{t-t_{B}}r.\n\\end{equation}\nAfter calculating radial derivative of the bang time function (\\ref{74}) and substituting it in the last formula we obtain\n\\begin{equation}\\label{76}\n\\frac{R'}{R}r=\\frac{\\rho_{LT0}r^{3}}{3\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}+\\frac{2}{3}\\frac{\\frac{3\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}{\\rho_{LT0}r^{3}}-1}{\\left(\\frac{3\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}{\\rho_{LT0}r^{3}}\\right)^{\\frac{3}{2}}}\\,\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}.\n\\end{equation}\nNow we define\n\\begin{equation}\n\\label{77}\nA\\equiv\\frac{\\rho_{AV0}}{\\rho_{LT0}}=\\frac{3\\int_{0}^{r}\\rho_{LT0}r'^{2}{\\rm d}r'}{\\rho_{LT0}r^{3}}\n\\end{equation}\nand using this definition we can rewrite (\\ref{76}) as\n\\begin{equation}\n\\label{78}\n\\frac{R'}{R}r=\\frac{1}{A}+\\frac{2}{3}\\frac{A-1}{A^{\\frac{3}{2}}}\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}.\n\\end{equation}\nFrom this formula it can be shown that $\\frac{R'}{R}r$ is a monotonic function of $t$. For $t=t_{i}$, its value is 1, which follows from our initial condition $R\\left(t_{i}\\right)=r$ and for $t\\to\\infty$, $\\frac{R'}{R}r\\to\\frac{1}{A}$ so it is increasing if $\\rho_{AV0}>\\rho_{LT0}$ and it is decreasing if $\\rho_{AV0}<\\rho_{LT0}$. In any case the function is positive which implies $R'>0$ so our assumption of the inequality sign in (\\ref{72}) is justified.\nUsing the definition (\\ref{77}) and substituting for $\\frac{R'}{R}r$ into (\\ref{72}) we get\n\\begin{equation}\n\\label{79}\n\\frac{2}{3}\\frac{A-1}{\\sqrt{A}}\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}>0.\n\\end{equation} \nSince $t>t_{B}$ the last inequality will be satisfied as long as $A>1$, which is equivalent to $\\rho_{AV0}>\\rho_{LT0}$ as can be seen from the definition of $A$. So we can see exactly what we wanted to prove, if $\\rho_{AV0}>\\rho_{LT0}$ than $\\rho_{AV}>\\rho_{LT}$ at any time. Similarly we could prove that if $\\rho_{AV0}<\\rho_{LT0}$ than $\\rho_{AV}<\\rho_{LT}$ at any time. Next we will assume that $\\rho_{AV0}>\\rho_{LT0}$ and we will express the shell crossing condition $\\rho_{min}<\\rho_{LT}$ in terms of initial data.\nFor the function $\\frac{E'}{E}$ we have formula (\\ref{41}) and since the function does not depend on time, we can express it on the initial time slice\n\\begin{equation}\n\\label{80}\n\\frac{E'}{E}_{max}=\\frac{1}{r}\\frac{\\rho_{min0}-\\rho_{LT0}}{\\rho_{min0}-\\rho_{AV0}}.\n\\end{equation}\nWe plug the last formula into (\\ref{40}) and after some calculations we get\n\\begin{equation}\n\\label{81}\n\\rho_{min}=\\frac{r^{3}}{R^{3}}\\rho_{LT0}\\frac{\\frac{\\rho_{min0}}{\\rho_{LT0}}\\left(1-\\frac{\\rho_{AV0}}{\\rho_{LT0}}\\right)}{\\frac{R'}{R}r\\left(\\frac{\\rho_{min0}}{\\rho_{LT0}}-\\frac{\\rho_{AV0}}{\\rho_{LT0}}\\right)-\\frac{\\rho_{min0}}{\\rho_{LT0}}+1}\n\\end{equation}\nand we used $\\frac{\\rho_{AV}}{\\rho_{LT}}=\\frac{R'}{R}r\\frac{\\rho_{AV0}}{\\rho_{LT0}}$ which follows from its definitions (\\ref{35}) and (\\ref{36}).\nWe will now define\n\\begin{equation}\n\\label{82}\nC\\equiv\\frac{\\rho_{min0}}{\\rho_{LT0}}\n\\end{equation}\nand using this definition with (\\ref{77}) we can rewrite (\\ref{81}) as \n\\begin{equation}\n\\label{83}\n\\rho_{min}=\\frac{r^{3}}{R^{3}}\\rho_{LT0}\\frac{C\\left(1-A\\right)}{\\frac{R'}{R}r\\left(C-A\\right)+\\left(1-C\\right)}.\n\\end{equation}\nThe inequality $\\rho_{min}<\\rho_{LT}$ can now be rewritten as\n\\begin{equation}\n\\label{86}\n\\frac{R'}{R}r\\frac{\\left(1-A\\right)C}{\\frac{R'}{R}r\\left(C-A\\right)+\\left(1-C\\right)}<1.\n\\end{equation}\nWe need to multiply the last inequality by the denominator, but in order to do that we need to find out its sign. \nFrom (\\ref{77}) and (\\ref{82}) it follows\n\\begin{equation}\nC-A=-\\frac{1}{\\rho_{LT0}}\\left(\\rho_{AV0}-\\rho_{min0}\\right)<0,\n\\end{equation}\nnext using the minimal value of $\\frac{R'}{R}r$ to be $\\frac{1}{A}$ we can write for the denominator of (\\ref{86})\n\\begin{equation}\\label{87}\n\\left(1-C\\right)-\\frac{R'}{R}r\\left(A-C\\right)\\leq 1-C-\\frac{1}{A}\\left(A-C\\right)=C\\left(\\frac{1}{A}-1\\right)<0,\n\\end{equation}\nbecause we assume $\\rho_{AV0}>\\rho_{LT0}$, i.e $A>1$. So the denominator is negative and when we multiply by it in (\\ref{86}), we get after some calculations\n\\begin{equation}\n\\label{88}\n\\frac{R'}{R}rA>1,\n\\end{equation}\nwhich is inequality (\\ref{72}) written in terms of $A$ and we already proved that it holds as long as $A>1$. So we can see that if $\\rho_{AV0}>\\rho_{LT0}$ than $\\rho_{min}<\\rho_{LT}$ is fulfilled and the shell crossing will be avoided at any time. Similarly it could be shown that in the case $\\rho_{AV0}<\\rho_{LT0}$, i.e. $A<1$, if $\\rho_{max0}>\\rho_{LT0}$ than $\\rho_{max}>\\rho_{LT}$ at any time.\n\nWe can also see that if $\\rho_{AV0}=\\rho_{LT0}$ at some point $r_{c}$, it means that we have $A=1$ and from (\\ref{78}) it follows $\\frac{R'}{R}r=1$. So the position of the critical point $r_{c}$ as defined in (\\ref{68}) is time independent.\n\n\n\\section{Density contrast, case $f=0$}\n\nWe will choose $\\rho_{LT0}$ in such a way that $\\rho_{LT0}<\\rho_{AV0}$. We already expressed $\\rho_{min}$ in terms of $\\rho_{LT}$ and $\\rho_{min0}$ in (\\ref{83}). Similarly we can express $\\rho_{max}$ in terms of $\\rho_{LT0}$ and $\\rho_{max0}$ as\n\\begin{equation}\n\\label{90}\n\\rho_{max}=\\frac{r^{3}}{R^{3}}\\rho_{LT0}\\frac{D\\left(1-A\\right)}{\\frac{R'}{R}r\\left(D-A\\right)-\\left(D-1\\right)},\n\\end{equation}\nwhere we define\n\\begin{equation}\n\\label{91}\nD\\equiv\\frac{\\rho_{max0}}{\\rho_{LT0}}.\n\\end{equation}\n\nWe are now interested in the time evolution of the difference of the extreme values of the density, so we define\n\\begin{eqnarray}\n\\label{92}\n\\Delta\\rho\\equiv\\rho_{max}-\\rho_{min}=\\frac{r^{3}}{R^{3}}\\rho_{LT0}\\left(1-A\\right)\\times\\nonumber\\\\\n\\left[\\frac{D}{\\frac{R'}{R}r\\left(D-A\\right)-\\left(D-1\\right)}-\\frac{C}{\\frac{R'}{R}r\\left(C-A\\right)-\\left(C-1\\right)}\\right]\\ \n\\end{eqnarray}\nWe can now substitute for $R$ from (\\ref{73}) and after some more calculations it can be rewritten in the form\n\\begin{equation}\n\\label{93}\n\\Delta\\rho\\left(t,r\\right)=\\frac{4}{3}\\frac{1}{\\left(t-t_{B}\\right)^{2}}\\frac{\\Delta\\rho_{0}}{\\rho_{LT0}}h\\left(t,r\\right)\\, ,\n\\end{equation}\nwhere we have defined $\\Delta\\rho_{0}\\equiv\\rho_{max0}-\\rho_{min0}$,\n\\begin{equation}\\label{94}\nh\\left(t,r\\right)\\equiv\\frac{1-A}{A}\\ \\frac{1-\\frac{R'}{R}rA}{\\left(\\frac{R'}{R}r\\right)^{2}\\left(D-A\\right)\\left(C-A\\right)+\\left(D-1\\right)\\left(C-1\\right)},\n\\end{equation}\nand we used\n\\begin{equation}\n\\label{95}\n2\\left(CD+A\\right)-\\left(C+D\\right)\\left(A+1\\right)=0,\n\\end{equation}\nwhich can be derived from the constraint (\\ref{70}). We will now investigate the behaviour of the function $h$. \n\nWe set $A=1+\\epsilon$ and take a look at how the function $h$ behaves in the case when $\\epsilon<<1$, which corresponds to the situation when the radial derivative of $\\rho_{LT0}$ is small. First we approximate $\\frac{R'}{R}r$ that is given by (\\ref{78}),\n\\begin{eqnarray}\n\\label{96}\n\\frac{R'}{R}r&=&\\frac{1}{1+\\epsilon}\\left(1+\\frac{2}{3}\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}\\frac{\\epsilon}{\\sqrt{1+\\epsilon}}\\right)\\approx\\\\\n&\\approx& 1+\\left(\\frac{2}{3}\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}-1\\right)\\epsilon+\\left(2-\\frac{1}{3}\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}\\right)\\epsilon^{2}+o\\left(\\epsilon^{3}\\right).\\nonumber\n\\end{eqnarray}\nSo in the numerator on the right hand side of (\\ref{94}) we get\n\\begin{equation}\\label{97}\n\\left(1-A\\right)(1-\\frac{R'}{R}rA)=-\\left(1-\\frac{R'}{R}r\\right)\\epsilon+\\frac{R'}{R}r\\epsilon^{2}\\approx\\frac{2}{3}\\frac{1}{t-t_{B}}\\sqrt{\\frac{3}{\\rho_{LT0}}}\\epsilon^{2}+o\\left(\\epsilon^{3}\\right).\n\\end{equation}\nIn order to simplify the denominator in (\\ref{94}) we set $D=1+\\delta$ and $C=1-\\xi$ and we assume that $\\Delta\\rho_{0}<<\\rho_{LT0}$ in which case $\\delta<<1$ and $\\xi<<1$. In this case the approximation of the denominator reads\n\\begin{eqnarray}\n\\label{98}\n&&A\\left[\\left(\\frac{R'}{R}r\\right)^2\\left(C-A\\right)\\left(D-A\\right)+\\left(C-1\\right)\\left(D-1\\right)\\right]\\approx\\nonumber\\\\\n&&\\quad\\approx\\left(C-A\\right)\\left(D-A\\right)+\\left(C-1\\right)\\left(D-1\\right)\n\\end{eqnarray}\nand for the function $h$ we get\n\\begin{equation}\n\\label{99}\nh\\approx \\frac{2}{\\sqrt{3}}\\frac{1}{t-t_{B}}\\frac{1}{\\sqrt{\\rho_{LT0}}}\\frac{\\left(A-1\\right)^{2}}{\\left(C-A\\right)\\left(D-A\\right)+\\left(C-1\\right)\\left(D-1\\right)}.\n\\end{equation}\nFrom (\\ref{95}) it follows\n\\begin{equation}\n\\label{101}\n2CD=AC+AD-2A+C+D.\n\\end{equation}\nUsing (\\ref{91}) we can rewrite the denominator in (\\ref{99})\n\\begin{equation}\\label{102}\n\\left(C-A\\right)\\left(D-A\\right)+\\left(C-1\\right)\\left(D-1\\right)=A^{2}-2A+1=\\left(A-1\\right)^{2}.\n\\end{equation}\nSo for $h$ we can write\n\\begin{equation}\n\\label{103}\nh\\approx\\frac{2}{\\sqrt{3}}\\frac{1}{t-t_{B}}\\frac{1}{\\sqrt{\\rho_{LT0}}}\n\\end{equation}\nand for $\\Delta\\rho$ we have\n\\begin{equation}\n\\label{104}\n\\Delta\\rho\\approx\\frac{8\\sqrt{3}}{9}\\frac{1}{\\left(t-t_{B}\\right)^{3}}\\frac{\\Delta\\rho_{0}}{\\rho_{LT0}^{\\frac{3}{2}}}.\n\\end{equation}\nFrom the last equation we can see that if the radial derivative of $\\rho_{LT0}^{-\\frac{3}{2}}$ is small then it does not effect much the shape of the initial difference of the density extremes. Also if at late time $t$ the bang time function is small compared to $t$ we can see that $\\Delta\\rho$ is proportional to $\\frac{1}{t^{3}}$. We can try to evaluate (\\ref{104}) at the initial time $t=t_{i}$\n\\begin{equation}\n\\label{105}\n\\Delta\\rho\\left(t=t_{i}\\right)\\approx\\frac{8\\sqrt{3}}{9}\\frac{27}{8}\\frac{\\left(\\int\\rho_{LT0}r^{2}{\\rm d}r\\right)^{\\frac{3}{2}}}{r^{\\frac{9}{2}}}\\frac{\\Delta\\rho_{0}}{\\rho_{LT0}^{\\frac{3}{2}}}=A^{\\frac{3}{2}}\\Delta\\rho_{0},\n\\end{equation}\nso we can see that if A is close to 1 the approximation gives us what we expect.\n\n\\section{Model specification}\n\nThe Szekeres spacetime has 5 degrees of freedom, so to fully specify the model we need to set up 5 functions. But since we are only interested in the density contrast, it is sufficient to specify just three functions, which is the curvature function $f$ that we assume to be zero, the initial radial profile $\\rho_{LT0}$ and one extreme value of the density. We will set up two models $a$ and $b$. For the initial radial profile we choose \n\\begin{eqnarray}\n\\label{106}\n&a:&\\quad\\rho_{LT0}=\\rho_{b0}\\left(1+\\frac{1}{10}e^{-\\frac{r^{2}}{500}}\\right),\\nonumber\\\\\n&b:&\\quad\\rho_{LT0}=\\rho_{b0}\\left(1+\\frac{1}{5}e^{-\\frac{r^{2}}{500}}\\right).\n\\end{eqnarray}\nBoth of them are peaked at the origin and $\\rho_{b0}$ is the background density at the initial time. The $\\rho_{LT0}$ is chosen in such a way that besides the origin it is less then $\\rho_{AV0}$, therefore for the extreme value we specify the density minimum since the only requirement for it is that it has to be equal to $\\rho_{LT0}$ at the origin and less everywhere else. We choose $\\rho_{min0}$ as\n\\begin{eqnarray}\n\\label{107}\n&a:&\\quad\\rho_{min0}=\\rho_{b0}\\left(1+\\frac{1}{10}e^{-\\frac{r^{2}}{400}}\\right), \\nonumber\\\\\n&b:&\\quad \\rho_{min0}=\\rho_{b0}\\left(1+\\frac{1}{5}e^{-\\frac{r^{2}}{400}}\\right).\n\\end{eqnarray}\nThe models differ by the size of the radial inhomogeneity that is in the model $b$ twice as big as in the model $a$ and we want to demonstrate, that the approximation formula will work better for the model $a$, because the function $A$ is closer to 1 than in the model $b$. The functions $\\rho_{LT0}$, $\\rho_{min0}$, $\\rho_{max0}$ and $\\rho_{AV0}$ for the model $a$ are shown in figure 3a. The function $A$ for both models is shown in figure 3b. The initial time is chosen as $t_{i}=5\\cdot 10^{5}y$ that approximately corresponds to the time of last scattering. The final time $t_{f}=13.7\\cdot 10^{9}y$ which is approximately present time. The initial density contrast $\\frac{\\Delta\\rho_{0}}{\\rho_{b0}}$ and the final density contrast $\\frac{\\Delta\\rho}{\\rho_{b}}$ for both models are shown in figures 4, $\\rho_{b}$ denotes the background density at the final time. The impact of the time evolution on the initial shape is minimal as we expected since $A$ is close to $1$, on the other hand\n the magnitude of the density contrast drops significantly because of the factor $t^{-3}$ in (\\ref{104}). We can also see in the figure 4 that the formula (\\ref{104}) approximates the density contrast better for the model $a$, that corresponds to the lower peak, because we chose the radial inhomogeneity lower than in model $b$. \n\n\n\\begin{figure}[]\n\\subfloat[]{\\includegraphics[scale=0.3]{plot1.eps}}\n\\subfloat[]{\\includegraphics[scale=0.3]{A.eps}}\n\\caption{Panel (a): The functions $\\rho_{min0}$, $\\rho_{max0}$, $\\rho_{LT0}$ and $\\rho_{AV0}$ for the model $a$. The dotted line represents the chosen initial density minimum $\\rho_{min0}$ according to (\\ref{107}). The dashed line is the initial density maximum $\\rho_{max0}$ computed according to (\\ref{70}). The solid line between the dotted and dashed lines is the chosen initial radial density profile $\\rho_{LT0}$ as specified in (\\ref{106}). The top solid line is the computed $\\rho_{AV0}$. All values on the vertical axis are divided by $\\rho_{b}$. Panel (b): The function $A$ as defined in (\\ref{77}) for both models. The lower curve corresponds to the model $a$, the upper curve corresponds to the model $b$.\n}\n\\end{figure}\n\n\\begin{figure}[]\n\\subfloat[]{\\includegraphics[scale=0.3]{kontrast_pred.eps}}\n\\subfloat[]{\\includegraphics[scale=0.3]{kontrast_po.eps}}\n\\caption{Panel (a): The behaviour of the initial density contrast $\\frac{\\Delta\\rho_{0}}{\\rho_{b0}}$ for both models. Panel (b): the behaviour of the final density contrast $\\frac{\\Delta\\rho}{\\rho_{b}}$ for both models. The solid curve represents the exact formula (\\ref{93}). The dotted curve is the density contrast as calculated according to the approximation formula (\\ref{104}). The lower and upper curves correspond to the model $a$ and $b$ respectively. The lower curve is apparently approximated better, because the radial inhomogeneity was chosen smaller.}\n\\end{figure} \n\n\\section{Conclusion}\n\nWe studied model of inhomogeneity in quasispherical Szekeres model. We set up only 3 of 5 degrees of freedom, which is sufficient for studying the evolution of the density contrast. The lack of specification of the last two degrees of freedom means that we do not have the detailed information about the density distribution, particularly we do not know the position of the extreme values on the spheres. We specify the initial radial density profile $\\rho_{LT0}$ which is the value of the density around the great circle that lies in the plane that defines the dipole. Next, we choose one extreme value of the density at the initial time, either $\\rho_{min0}$ or $\\rho_{max0}$, and the last function that we choose is the curvature function $f$. In order to choose an appropriate value for the density extremes, we investigated the shell crossing conditions in terms of density $\\rho_{max}$, $\\rho_{min}$, $\\rho_{LT}$ and $\\rho_{AV}$. We derived conditions that $\\rho_{max}$ and $\\rho_{min\n }$ have to satisfy in order to avoid shell crossing and we showed that in the special case $f=0$, if the conditions are fulfilled on the initial time slice, then they will hold at any time. \n\nNext, we derived an analytical formula for the density contrast $\\Delta\\rho$ as a function of $t$ and $r$. In the special case $f=0$, we derived an approximation formula that is valid if the initial inhomogeneity is small and we showed that in this approximation the density contrast is proportional to the initial density contrast and depends on time as $t^{-3}$, so there is a decrease in magnitude during time evolution, however the shape of the function is preserved. It shows that the dynamics is very simple and close to homogeneous one for small inhomogeneity confirming the expected behaviour. In this sense one may argue that a small inhomogeneity can be successfully treated in perturbation theory and the influence of nonlinearity is negligible.\n\nThe next research will be focused on the situation when the curvature function is chosen more generally and is not equal to zero.\n\n\n\\begin{acknowledgements}\nD. V. was supported by grant GAUK 398911 and project SVV-267301. O. S. was supported by grant GA\\v{C}R 14-37086G.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzgni b/data_all_eng_slimpj/shuffled/split2/finalzgni new file mode 100644 index 0000000000000000000000000000000000000000..3dc650a2beac0a6c0797db8434c5dff75ae9b789 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzgni @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\nCentrality measures play an important role in network analysis \nby quantifying the importance of each node \\cite{BV14,Estradabook}. \nHow to make the notion of importance mathematically precise is highly context dependent, and hence a range of centrality measures exist. \nWalk-based centralities constitute \na large and widely-used class, based on\ncounting the \nnumber of walks beginning from each node, with some form of\ndownweighting based on the walk length \\cite{BEK13,EHSiamRev}. Different types of weights lead to different families of centralities, such as Katz versus exponential; one or more parameters allow for even further freedom within each family. Moreover, in recent years it has been proposed that one might wish to ignore walks that immediately backtrack, to obtain centrality measures that are better suited for certain applications. Such centrality measures \nare said to be \\emph{non-backtracking} and are known to offer tangible benefits \\cite{AGHN17a,AGHN17b,Beyond,GHN18}. In this paper, we present a\nframework for analyzing and computing walk-based\ncentrality measures, including \nnon-backtracking versions, in the case of \n time-evolving networks.\n \n Key contributions in this work are as follows.\n \\begin{itemize}\n \\item A proof that Katz (or resolvent-based) centrality is unique among\n matrix function-based versions, in that a combinatorially \n correct expression exists in the form of products of functions of node-level adjacency matrices (Theorem~\\ref{thm:katzunique});\n thereby showing that dynamic Katz has a major computational \n advantage.\n \\item Edge-level expressions for other matrix function-based versions ((\\ref{eq:f1}) and (\\ref{eq:f2})) using a block matrix construction, with an accompanying convergence theory\n (Proposition~\\ref{prop:generalf}).\n \\item Corresponding computable expressions for non-backtracking versions of matrix function-based temporal centrality measures\n \n \\eqref{eq:nbtf1} and \\eqref{eq:nbtf2}) along with an accompanying convergence theory \n (Propositions~\\ref{prop:radius} and \\ref{prop:fnbt}).\n \\end{itemize}\n\n\nThe material is organized as follows.\nSection~\\ref{sec:background}\nsets up some notation and explains what we mean by a temporal\nnetwork. \nIn section~\\ref{sec:expono} we \nstate and prove a unique property of dynamic Katz.\nSections~\\ref{sec:dyncom} and \\ref{sec:measures_temporal}\ndevelop a framework in which \ngeneral matrix function-based centralities can be handled.\nIn section~\\ref{sec:nbt} we \nintroduce and study new non-backtracking versions of these measures,\nfocusing on three main variations where backtracking is disallowed \nwithin timesteps, across timesteps, or both.\nSection~\\ref{sec:conc} concludes with a brief discussion. \n\n\n\n\n\n\\section{Background}\\label{sec:background}\nIn this section we review some definitions and notation associated with graphs and \nsome of their matrix representations. \n\n\nA {\\it network}, or {\\it graph}, is an ordered pair of sets $G = (V,E)$, \nwhere $V$ is the set of {\\it nodes} \nand $E\\subset V\\times V$ is the set of \\emph{edges} among the nodes \\cite{Estradabook,Newmanbook}. \nWe consider graphs with \nunweighted edges and no self-loops.\nGiven an edge $i\\to j$ we will call $i$ its {\\it source} and $j$ its {\\it target}. If the edge from \n$j\\to i$ also exists, we will refer to this as the {\\it reciprocal} of the edge $i\\to j$ and we will say \nthat edge $i\\to j$ is {\\it reciprocated}.\n\nThroughout this work $I$ denotes the identity matrix, ${\\bf 1}$ denotes the vector of all ones; the size of both, unless otherwise stated, is chosen such that the resulting expressions in which they appear are coherent. The symbol $\\star$ will be used as a placeholder to denote an arbitrary vertex, where appropriate.\n\nTwo definitions that are central to our work those of \na {\\it walk} and a {\\it (non-)backtracking walk} around a graph. \nWe review them in the following. \n\\begin{definition}\nA {\\rm walk of length $r$} is a sequence of $r$ \nedges $(e_1,e_2,\\ldots,e_r)$ such that the target of $e_\\ell$ coincides with \nthe source of $e_{\\ell+1}$ for all $\\ell=1,2,\\ldots, r-1$.\n\\end{definition}\n\\begin{remark}\nAn equivalent definition can be given in terms of the nodes: namely, a walk of length $r$ can also be seen as a sequence of $r+1$ nodes \n$(i_1,i_2,\\ldots,i_{r+1})$ such that $(i_\\ell,i_{\\ell+1})\\in E$ for all $\\ell = 1,2,\\ldots, r$. \n\\end{remark}\n\n\\begin{definition}\\label{def:nbt}\nLet $G$ be a graph. A walk in $G$ is said to be {\\rm backtracking} if any \ntwo consecutive edges in it are reciprocated. The walk is said to be {\\rm non-backtracking (NBT)} otherwise.\n\\end{definition}\n\n\n\\subsection{Graphs and matrices}\\label{ssec:matrices}\nA widely used linear algebraic representation of a graph is its adjacency matrix.\n\\begin{definition}\\label{def:A}\nLet $G = (V,E)$ be an unweighted graph with $n$ nodes. Its {\\rm adjacency matrix} $A \\in \\mathbb{R}^{n \\times n}$ is entrywise \ndefined as:\n$$ A_{ij}=\\begin{cases}\n1 \\ \\ \\text{if}\\,\\, (i,j) \\in E\\\\\n0 \\ \\ \\text{otherwise}\n\\end{cases}\n$$\nfor all $ i,j = 1,2,\\ldots, n$.\n\\end{definition}\n\nOther matrices of interest for the purpose of our work are the {\\it source} and {\\it target} (or {\\it terminal}) \nmatrices associated with $G$ and the adjacency matrix of the line graph of $G$; see, e.g., \\cite{Estradabook,VDF09,ZV08}. \n\n\n\\begin{definition}\\label{def:LR}\nLet $G$ be an unweighted graph with $n$ nodes and $m$ edges. Its {\\rm source} and {\\rm target} (or {\\rm terminal}) {\\rm matrices} $L,R \\in \\mathbb{R}^{m \\times n}$ are entrywise defined as:\n$$L_{ei}=\\begin{cases}\n1 \\ \\ \\text{if edge~} e \\text{~has the form~} i\\to\\star\\\\\n0 \\ \\ \\text{otherwise}\n\\end{cases}\n$$\nand\n$$\nR_{ej}=\\begin{cases}\n1 \\ \\ \\text{if edge~} e \\text{~has the form~} \\star\\to j\\\\\n0 \\ \\ \\text{otherwise}\n\\end{cases}$$\nrespectively, for all $e = 1,2,\\ldots,m$ and for all $ i,j = 1,2,\\ldots, n$.\n\\end{definition}\n\n\nThe {\\it line graph} (or {\\it interchange graph} or {\\it dual graph}) of $G$ is constructed from $G$ as follows: edges in the original graph are regarded as nodes \nand two nodes\n$i \\to j$ and $k \\to \\ell$ \n in this new line graph \nare connected if $ j = k$, that is, if, together, they represent \na walk of length two in the original graph $G$; see, e.g.,~\\cite{ore62}.\nThe adjacency matrix $W\\in\\mathbb{R}^{m\\times m}$ of the line graph can then be entrywise \ndefined in terms of elements of $G$ as \n\\\nW_{i\\to j,k\\to \\ell} = \\delta_{jk}, \n\\\nwhere $\\delta_{jk}$ is the Kronecker delta. \n\n\n\n \n\n\\subsubsection{Walk-based centrality measures}\n\nKatz centrality \\cite{Katz53} is a widely used centrality measure that assigns to each node $i$ the $i$th element of the vector\n\\begin{equation}\\label{eq:Katz}\n {\\bf y}(\\alpha) = \\sum_{r=0}^\\infty \\alpha^rA^r{\\bf 1}.\n\\end{equation}\nCombinatorially, Katz centrality can be understood as assigning to node $i$ the sum over all possible walk lengths of the number of walks originating from $i$ of length $r$ scaled by $\\alpha^r$, where $\\alpha>0$ is a downweighting parameter. \nIndeed, it is easy to show that $(A^r)_{ij}$ is the number of walks of length $r$ originating from node $i$ and ending at node $j$ and hence the above interpretation of the entries of $A^r{\\bf 1}$ readily follows. \n\nFurthermore, whenever $0 < \\alpha < 1\/\\rho(A)$, where $\\rho(A)$ denotes the spectral radius of $A$, the series appearing in~\\eqref{eq:Katz} converges and the vector of Katz centralities can be equivalently defined as \n\\\n {\\bf y}(\\alpha) = (I-\\alpha A)^{-1}{\\bf 1}.\n\\\n\n\nThe idea underlying Katz centrality can be generalized to obtain other centrality measures for nodes defined in terms of matrix functions and their entries, or sums thereof; see, e.g.,~\\cite{EHSiamRev} and references therein. \nSuppose that the analytic function \n\\begin{equation}\nf(z) = \\sum_{r=0}^\\infty c_r z^r\n\\label{eq:fz}\n\\end{equation}\nhas nonnegative Maclaurin coefficients $c_r \\geq 0$ and is convergent for $|z|0$,\nwas extended to the time-dependent setting \nusing a quantum physics motivation \nto give \\cite{estrada2013temporal}\n\\[\n\\prod_{\\tau = 1}^Ne^{\\beta A^{[\\tau]}} = e^{\\beta A^{[1]}}e^{\\beta A^{[2]}}\\cdots e^{\\beta A^{[N]}}.\n\\]\nHowever, \\emph{from a combinatorial viewpoint}\nthis generalization has the drawback of not weighting walks \nconsistently with their length. Consider for example a walk of length five from $i$ to $j$ that is realized by traveling two edges at time $t=t_1$, one edge at time $t=t_2$, and two edges at time $t=t_3$. \nBecause of its length, we would expect this walk to be weighted as $\\frac{\\beta^5}{5!}$ in keeping with the theory that we have seen for static networks. \nHowever, it is straightforward to check that the $(i,j)$ entry of $e^{\\beta A^{[1]}} e^{\\beta A^{[2]}}e^{\\beta A^{[3]}}$ is weighted as $\\frac{\\beta^2}{2!}\\frac{\\beta^1}{1!}\\frac{\\beta^2}{2!}=\\frac{\\beta^5}{4}$. \nFurthermore, this walk of length five is weighted differently from another walk of length \nfive that travels, say, \none edge at times $t=t_1, t_2$ and three edges at time $t=t_3$. Indeed, the weight of \nthis second walk is $\\frac{\\beta}{1!}\\frac{\\beta}{1!}\\frac{\\beta^3}{3!}=\\frac{\\beta^5}{6}$. \nThus, a simple generalization to the temporal setting as the one presented in \\cite{estrada2013temporal} \nnot only fails to respect the combinatorics of walks, with walks of length $r$ not weighted as \n$\\frac{\\beta^r}{r!}$, but it may also be inconsistent \nin weighting different walks of the same length. \n\n\\smallskip\n\nIn the following subsection we show that resolvent-based \ncentrality measures are unique in this regard: they are the only \nfunctions that respect the combinatorics of walks when translated to the \ntemporal setting with the simple expression\n\\[\nf(\\alpha A^{[1]})f(\\alpha A^{[2]})\\cdots f(\\alpha A^{[N]}).\n\\]\n\n\\subsection{A remarkable property of Katz centrality}\\label{ssec:Katz_unique}\n\nIs it possible for choices of $f$ other than the resolvent (including the popular $f(z)=e^z$) to compute $f$-centralities on a time-evolving network directly using products of functions of adjacency matrices? \n\nWhile the example above clarified that the exponential subgraph and total communicability on a time-evolving network cannot be computed by simply multiplying the matrix exponential of each adjacency matrix, it is still {\\it a priori} possible that they could be computed by multiplying some \\emph{other} function of each adjacency matrix, say, via $\\prod_{\\tau=1}^N g(\\alpha A^{[\\tau]})$ for some $g(z)$. More generally, posing this question for $f$-centrality measures is equivalent to the following combinatorial problem: Given the function $f(z) = \\sum\\limits_{r=0}^\\infty c_r z^r$, $c_r \\geq 0$, find a function $g(z)$ such that\n \\begin{equation}\\label{eq:Estradaiswrong}\n\\sum_{r=0}^\\infty c_r \\alpha^r h_r(x_1,\\dots,x_N)= \\prod_{i=1}^N g(\\alpha x_i),\n\\end{equation} \nwhere \n\\[ h_r(x_1,\\dots,x_N)= \\sum_{i_1+\\dots+i_N=r} x_1^{i_1} \\dots x_N^{i_N}\\]\nis the $r$th complete homogeneous symmetric polynomial in $N\\geq 2$ variables. Indeed, finding a solution to this scalar problem would imply that the weights assigned to a walk in the matrix $\\prod_{\\tau=1}^N g(\\alpha A^{[\\tau]})$ \n\\begin{itemize}\n \\item depend only on the total walk length and not, for example, on how it is split between time frames; and\n \\item hold for any possible time evolving graph.\n\\end{itemize}\n\n\n \nIt turns out that this is essentially (up to a multiplicative constant and a linear change of variable) only possible if $f(z)$ is the resolvent. Since multiplication by a constant does not change the ranking while a linear change of variable is tantamount to changing the Katz parameter $\\alpha$, it follows that no $f$-centrality other than Katz (i.e., resolvent subgraph) centrality is applicable to time-evolving networks via an expression only based on products of functions of the adjacency matrices. For example, there is no way to correctly compute the combinatorics of exponential centrality on time-evolving graphs by directly multiplying some functions of the adjacency matrix.\\footnote{Of course, this does not mean that computing exponential centralities on time evolving graphs is not possible: indeed, using the multilayer approach proposed in this paper is for example a combinatorially exact, albeit potentially demanding in terms of computational complexity, way to compute it.} The theorem below gives a formal proof of this claim.\n\n\\begin{theorem}\\label{thm:katzunique}\nGiven the sequence of nonnegative real numbers $(c_r)_r$, let $f(z)=\\sum_{r=0}^\\infty c_r z^r$ and let $N \\geq 2$. The functional equation \\eqref{eq:Estradaiswrong} has a solution $g(z)$ if and only if there exist nonnegative constants $\\gamma,\\delta \\geq 0$ such that $c_r=\\gamma \\delta^r$ for all $r$, i.e, if and only if $f(z)=\\gamma(1-\\delta z)^{-1}$. Moreover, in that case the solution is \\[ g(z)= \\sqrt[N]{\\gamma}\\ (1-\\delta z)^{-1}. \\]\n\\end{theorem}\n\n\\begin{proof}\nSuppose first that $c_r=\\gamma\\delta^r$ for some $\\gamma,\\delta\\geq 0$, then (see also \\cite{GHPE11} and \\cite[Ch. 7]{Stanley})\n\\[\n\\sum_{r=0}^\\infty c_r \\alpha^r h_r(x_1,\\dots,x_N) = \\gamma \\sum_{r=0}^\\infty \\alpha^r h_r(\\delta x_1,\\dots,\\delta x_N) = \n\\gamma \\prod_{i=1}^N \\sum_{r=0}^\\infty (\\delta x_i \\alpha)^r = \\gamma \\prod_{i=1}^N \\frac{1}{1-\\alpha \\delta x_i},\n\\]\nso taking\n$ g(\\alpha x_i) = \\sqrt[N]{\\gamma} (1-x_i \\delta \\alpha)^{-1}$\nyields a solution to \\eqref{eq:Estradaiswrong}. \n\n\nConversely suppose that \\eqref{eq:Estradaiswrong} has a solution $g(z)$, and assume that $f(0)=1$. In doing so we do not lose generality, for it is clear that $g(z)$ solves \\eqref{eq:Estradaiswrong} for $f(z)$ if and only if, for all $\\gamma \\geq 0$, $\\sqrt[N]{\\gamma}\\ g(z)$ solves it for $\\gamma f(z)$. Recall now the following (classic) property of complete homogeneous symmetric polynomials \\cite[Sec. 7.5]{Stanley}:\n\\[ h_r(\\underbrace{1,\\dots,1}_{\\ell \\text{ variables}},\\underbrace{0,\\dots,0}_{(N-\\ell) \\text{ variables}}) = \\#\\{ \\text{monomials of degree } r \\text{ in } \\ell \\text{ variables } \\} = {\\ell+r-1 \\choose r}\\]\nThen, evaluating \\eqref{eq:Estradaiswrong} at $(x_1,\t\\dots,x_N)=(0,\\dots,0)$ yields $g(0)=1$. \nNext, we evaluate \\eqref{eq:Estradaiswrong} at $(x_1,x_2,\\dots,x_N)=(1,0,\\dots,0)$ and conclude that $f(z)=g(z)$. Finally, by evaluating \\eqref{eq:Estradaiswrong} at $(x_1,x_2,x_3,\\dots,x_N)=(1,1,0,\\dots,0)$, we obtain\n\\begin{equation}\\label{eq:Eir} \n\\sum_{r=0}^\\infty (r+1) c_r \\alpha^r = \\sum_{r=0}^\\infty \\alpha^r \\sum_{k=0}^r c_k c_{r-k}. \n\\end{equation}\nNow define $\\delta:=c_1$ and proceed by induction.\nWe begin with $c_1=\\delta^1$. Let us then assume that $c_k=\\delta^k$ for all $1 \\leq k \\leq r-1$; equating the coefficients of $\\alpha^r$ in~\\eqref{eq:Eir} we get\n\\[ (r+1) c_r = \\sum_{k=0}^r c_k c_{r-k} = 2 c_r + \\sum_{k=1}^{r-1} \\delta^r \\quad\\Longrightarrow\\quad c_r = \\delta^r.\\]\n\\end{proof}\n\n\n\nMotivated by this result, in sections~\\ref{sec:dyncom} and \\ref{sec:measures_temporal} we introduce a \nblock upper-triangular matrix $M\\in\\mathbb{R}^{m\\times m}$ which will allow us to \ngeneralize $f$-subgraph centrality and $f$-total communicability to \ntreat temporal networks while respecting the combinatorics of walks. \nThese results will then be generalized to the non-backtracking setting in section~\\ref{sec:nbt}.\n\n\n\n\n\n\\section{The global temporal transition matrix}\n\\label{sec:dyncom}\n\nWe start by re-interpreting the results in \\cite{GHPE11} via the line graph construction. \n\nTo build intuition, suppose for simplicity that $N=2$ and let $A^{[1]}$ and $A^{[2]}$ be the adjacency matrices of the graphs appearing at time \nstamps $t_1 < t_2$ in the temporal network $\\mathcal{G}$. \nFor any adjacency matrix $A$ and adjacency matrix $W$ of the associated \nline graph, it holds that $A = L^T R$ and $W = RL^T$. \nThese immediately yield\n\n\\begin{equation}\\label{eq:Ar}\n(A^r)_{ij} = (L^TW^{r-1}R)_{ij}\n\\end{equation} \nfor all $i,j\\in V$ and for all $r\\geq 1$; see~\\cite{Beyond,VDF09,ZV08}. \n\\begin{remark}\nEquation~\\eqref{eq:Ar} above summarizes the fact that taking $r\\geq 1$\nsteps in the node space, i.e., on the original \ngraph $G$, corresponds to taking $r-1$ steps in the edge space, i.e., on the line graph associated with $G$. \n\\end{remark}\nEquation~\\eqref{eq:Ar} together with~\\eqref{eq:dcm} and the assumption that \n$0<\\alpha<(\\max_{\\tau = 1,2}\\{\\rho(A^{[\\tau]})\\})^{-1}$, yields \n\\begin{align*}\n \\mathcal{Q} &= (I-\\alpha A^{[1]})^{-1} (I-\\alpha A^{[2]})^{-1}\\\\\n &= \\left(I + \\sum_{r = 1}^\\infty \\alpha^r {A^{[1]}}^r\\right)\\left(I + \\sum_{r = 1}^\\infty \\alpha^r {A^{[2]}}^r\\right)\\\\\n &=(I + \\sum_{r = 1}^\\infty \\alpha^r {L^{[1]}}^T {W^{[1]}}^{r-1} R^{[1]})(I + \\sum_{r = 1}^\\infty \\alpha^r {L^{[2]}}^T {W^{[2]}}^{r-1} R^{[2]})\\\\\n\n &= (I_n - \\alpha {L^{[1]}}^T (I_m-\\alpha W^{[1]})^{-1}R^{[1]})(I_n - \\alpha {L^{[2]}}^T (I_m-\\alpha W^{[2]})^{-1}R^{[2]}),\n\\end{align*}\nand hence\n\\begin{equation}\\label{eq:goodexample}\n \\mathcal{Q}= I_n + \\alpha \\left( \\sum_{\\tau=1}^2 {L^{[\\tau]}}^T (I_m - \\alpha W^{[\\tau]})^{-1} R^{[\\tau]}\\right) + \n\\alpha^2 {L^{[1]}}^T (I_m-\\alpha W^{[1]})^{-1} W^{[1,2]} (I_m- \\alpha W^{[2]})^{-1} R^{[2]},\n\\end{equation}\nwhere $W^{[1,2]}:=R^{[1]} (L^{[2]})^T$.\nThe combinatorial role of each term appearing in~\\eqref{eq:goodexample} is as follows:\n\\begin{itemize}\n\\item $I_n$ counts walks that do not involve any edge within either graph, i.e., walks of length zero.\n\\item $\\alpha \\left(\\sum_{\\tau=1}^2 {L^{[\\tau]}}^T (I - \\alpha W^{[\\tau]})^{-1} R^{[\\tau]}\\right)$ counts walks that use at least one \nedge within one of the two graphs, and no edges within the other: note that necessarily these walks have length at least one.\n\\item $\\alpha^2 {L^{[1]}}^T (I_m - \\alpha W^{[1]})^{-1} W^{[1,2]} (I_m- \\alpha W^{[2]})^{-1} R^{[2]}$ counts walks that use at least one edge within each graph; note that necessarily these walks have length at least two, and, moreover, that at some point necessarily the walk has passed over an edge that exists at time $t_1$ to a subsequent edge that exists at time $t_2$, with the target of the time $t_1$ edge \nmatching the source of the time $t_2$ edge (this is encoded in $W^{[1,2]}$).\n\\end{itemize}\n\n\nConsider now the following block-triangular matrix:\n\\begin{equation}\\label{eq:Mzero}\n M^{[1,2]} = \\begin{bmatrix} W^{[1]} & W^{[1,2]} \\\\ 0 & W^{[2]}\\end{bmatrix}.\n\\end{equation}\nThe following result holds. \n\\begin{prop}\\label{prop:properblocksW}\nSuppose $M^{[1,2]}$ is defined as in~\\eqref{eq:Mzero}. Then, when $0<\\alpha<(\\max_{\\tau = 1,2}\\{\\rho(A^{[\\tau]})\\})^{-1}$ ,\n \\[ (I - \\alpha M^{[1,2]})^{-1}\n = \\begin{bmatrix} (I-\\alpha W^{[1]})^{-1} & \\alpha(I-\\alpha W^{[1]})^{-1}W^{[1,2]}(I-\\alpha W^{[2]})^{-1} \\\\ 0 & (I-\\alpha W^{[2]})^{-1}\\end{bmatrix}.\\]\n\\end{prop}\n\\begin{proof}\nNote that the $r$th power of $M^{[1,2]}$ is given by \n\\[\n {M^{[1,2]}}^r = \\begin{bmatrix} W^{[1]} & W^{[1,2]}\\\\\n0 & W^{[2]}\n\\end{bmatrix}^r = \\begin{pmatrix}\n{W^{[1]}}^r & \\sum_{s = 0}^{r-1} {W^{[1]}}^s W^{[1,2]}{W^{[2]}}^{r-1-s} \\\\ \n0 & {W^{[2]}}^r\\end{pmatrix}. \\]\n Thus\n\\[\n\\sum_{r = 0}^\\infty \\alpha^r {M^{[1,2]}}^r = I_{2m}+ \\sum_{r = 1}^\\infty \\begin{pmatrix}\\alpha^r{W^{[1]}}^r & \\alpha^r\\sum_{s = 0}^{r-1} {W^{[1]}}^s W^{[1,2]}{W^{[2]}}^{r-1-s} \\\\ 0 & \\alpha^r{W^{[2]}}^r \\end{pmatrix}.\n\\]\nSince the spectral radius of $W^{[\\tau]}$ coincides with that of $A^{[\\tau]}$ by Flanders' theorem, it follows that, when $0<\\alpha<(\\max_{\\tau = 1,2}\\{\\rho(A^{[\\tau]})\\})^{-1}$, the diagonal blocks of the matrix on the right-hand side equal $(I-\\alpha W^{[\\tau]})^{-1}$, for $\\tau = 1,2$, respectively. \n\nFinally, the non-zero off-diagonal block is such that\n\\begin{align*}\n\\sum_{r = 1}^\\infty \\alpha^r\\sum_{s = 0}^{r-1} {W^{[1]}}^s W^{[1,2]}{W^{[2]}}^{r-1-s} &= \n\\alpha\\sum_{r = 0}^\\infty \\alpha^r\\sum_{s = 0}^{r} {W^{[1]}}^s W^{[1,2]}{W^{[2]}}^{r-s} \\\\\n&= \\alpha \\left(\\sum_{s = 0}^\\infty \\alpha^s {W^{[1]}}^s \\right) W^{[1,2]}\\left(\\sum_{s = 0}^\\infty \\alpha^s {W^{[2]}}^s \\right)\\\\ \n&= (I_m-\\alpha W^{[1]})^{-1}\\alpha W^{[1,2]}(I_m-\\alpha W^{[2]})^{-1}.\n\\end{align*}\nThe above, together with the fact that \n\\[(I_{2m}-\\alpha M^{[1,2]})^{-1} = I_{2m} + \\sum_{r = 1}^\\infty \\alpha^r {M^{[1,2]}}^r,\\] \nconcludes the proof.\n\\end{proof}\n\nIf we now let \n\\[\\mathcal{L} = \\begin{bmatrix}L^{[1]} \\\\ L^{[2]} \\end{bmatrix}\\quad \\text{ and }\\quad \\mathcal{R} = \\begin{bmatrix}R^{[1]} \\\\ R^{[2]} \\end{bmatrix}\n\\]\nit follows from \n(\\ref{eq:Ar}) and Proposition~\\ref{prop:properblocksW}\n that \n\\[\n{\\mathcal{Q}} = I + \\alpha {\\mathcal{L}}^T (I-\\alpha M^{[1,2]})^{-1}{\\mathcal{R}}.\n\\]\n\nThis alternative description of ${\\mathcal{Q}}$ is not necessary in practice, since ${\\mathcal{Q}}$ can be \neasily computed by working in the node space, i.e., \ndirectly from the adjacency matrices $A^{[\\tau]}$; however, \nthis construction, appropriately generalized, will allow us to\ndefine and compute \n $f$-centrality measures for temporal networks for all functions $f$ \nand NBT versions for temporal networks. \n\nThese two points will be addressed in sections~\\ref{sec:measures_temporal} and \\ref{sec:nbt} below. \nIn the reminder of this section, we will extend the definition of the matrices $M=M^{[1,2]}$, ${\\mathcal{L}}$, \nand ${\\mathcal{R}}$ to treat the case of $N \\ge 2$ time stamps. \nTo gain further intuition, we will also describe how the generalization of the matrix $M$ can be interpreted as the \nadjacency matrix of a static multilayer network. \n\n\n\n\\smallskip\n\nWe begin by extending the definitions and results above to the case of $N \\ge 2$. \n\n\\begin{definition}\\label{def:Wt1t2}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps $t_1 < t_2 < \\cdots < t_N$, with $G^{[\\tau]} = ({\\mathcal{V}}, E^{[\\tau]})$. \nThe {\\rm backtrack-allowing transition matrix} $W^{[\\tau_1,\\tau_2]}$\nwhere $1\\leq\\tau_1 < \\tau_2\\leq N$, is defined entrywise as follows:\n\\[\n(W^{[\\tau_1,\\tau_2]})_{i \\rightarrow j, k \\rightarrow \\ell} = \\delta_{jk}\n\\]\nfor all $i\\to j\\in E^{[\\tau_1]}$ and $k\\to\\ell\\in E^{[\\tau_2]}$. \n\\end{definition}\nThese matrices are generally non-square and encode in their entries the presence \nof temporal walks of length two \nthat take place across two distinct, but not necessarily consecutive, time stamps.\n\n\\begin{prop}\\label{prop:Wt1t2}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps $t_1< t_2<\\cdots < t_N$. \nLet moreover $L^{[\\tau]}$ and $R^{[\\tau]}$ be the source and target matrices associated with graph $G^{[\\tau]}$. \nThen for all $\\tau_1 < \\tau_2$ we have \n\\[\nW^{[\\tau_1,\\tau_2]} = R^{[\\tau_1]}{L^{[\\tau_2]}}^T.\n\\]\n\\end{prop}\n\\begin{proof}\nThe result may be established straightforwardly by working entrywise.\n\\end{proof}\n\n\nWe now extend the definition of $M^{[1,2]}$ to the case of $N\\geq 2$ as follows. \n\\begin{definition}\\label{def:Mnolla}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps \n$t_1< t_2<\\cdots< t_N$.\nThe {\\rm global temporal transition matrix} associated with ${\\mathcal{G}}$ is the $m\\times m$ block matrix:\n\\[\nM:=M^{[1,2,\\dots,N]} = \\begin{bmatrix} \nW^{[1]} & W^{[1,2]} & W^{[1,3]} & \\dots & W^{[1,N]} \\\\ \n0 & W^{[2]} & W^{[2,3]} & \\dots & W^{[2,N]} \\\\\n\\vdots & & \\ddots & & \\vdots \\\\\n\\vdots & & & \\ddots & \\vdots\\\\\n0 & \\dots & \\dots & 0 & W^{[N]}\n\\end{bmatrix}.\n\\]\n\\end{definition}\nWe mention in passing that this is the adjacency matrix of a multilayer graph associated with the line graph of ${\\mathcal{G}}$; see subsection~\\ref{sec:multilayer} for more details. \nSimilar constructions, although at node level, have been used over the years as a means to incorporate the temporal aspect in block-matrix \nrepresentations of ${\\mathcal{G}}$; \nexamples include~\\cite{al2021block,fenu2017block,MRMPO10,TMCPM17}. \nWe note however that the adjacency matrix presented in Definition~\\ref{def:Mnolla} does not correspond to the adjacency matrix of the line graph of any of the networks underlying the matrices presented in these references. Indeed, the number of nonzeros in the matrices presented in these references is strictly larger than the number of edges in the temporal graph, due to the presence of ``artificial edges\" described in the off-diagonal blocks. Therefore the adjacency matrix of the associated line graph would be of larger dimension than the matrix in Definition~\\ref{def:Mnolla}\n\n \\begin{definition}\\label{def:globalLR}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps $t_1< t_2<\\cdots< t_N$ \nand let $L^{[\\tau]}$ and $R^{[\\tau]}$ be the source and target matrices of $G^{[\\tau]}$ for $\\tau = 1,2,\\ldots, N$. \nThe {\\rm global source} and {\\rm global target} matrices to $N$ time stamps associated to ${\\mathcal{G}}$ are\n\\[\\mathcal{L}=\\mathcal{L}^{[1,\\dots,N]} = \\begin{pmatrix} L^{[1]} \\\\ L^{[2]} \\\\ \\vdots \\\\ L^{[N]}\\end{pmatrix} \\quad \\text{ and } \\quad \\mathcal{R}=\\mathcal{R}^{[1,\\dots,N]} = \\begin{pmatrix}R^{[1]} \\\\ R^{[2]} \\\\ \\vdots \\\\ R^{[N]}\\end{pmatrix},\\]\nrespectively.\n\\end{definition}\n\nIt can be verified that, for all $\\alpha\\in(0,(\\max_{\\tau}\\{\\rho(A^{[\\tau]})\\})^{-1})$,\n Katz centrality \n \\eqref{eq:Qx} \n on a time-evolving network with global temporal transition matrix $M=M^{[1,\\dots, N]}$ can be rewritten as \n\\begin{equation}\\label{eq:ysym_N}\n\\boldsymbol{y}(\\alpha) = \n{\\mathcal{Q}}{\\bf 1} = [I + \\alpha \\mathcal{L}^T (I-\\alpha M)^{-1} \\mathcal{R}] {\\bf 1} = {\\bf 1} + \\alpha \\mathcal{L}^T (I-\\alpha M)^{-1}{\\bf 1}, \n\\end{equation}\nwhere we have used the fact that ${\\mathcal{R}}{\\bf 1} = {\\bf 1}$. \n\nAs we mentioned earlier in the section, this construction is not actually needed \nto compute $\\boldsymbol{y}(\\alpha) ={\\mathcal{Q}}{\\bf 1}$. However, it will be instrumental to \nthe generalizations to $f$-centrality measures and their NBT equivalent. \nOne observation that emerges from the above is that walks around the temporal graph \n${\\mathcal{G}}$ can be enumerated by counting walks around the graph underlying $M$ and then \nprojecting to the node space appropriately. \nWe explore this connection in the next subsection. \n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\centering\n \\begin{tikzpicture}[]\n \\node[circle,draw=black,thick] (11) at (0,1.5) {1};\n \\node[circle,draw=black,thick] (21) at (1.5,1.5) {2};\n \\node[circle,draw=black,thick] (31) at (1.5,0) {3};\n \\node[circle,draw=black,thick] (41) at (0,0) {4};\n\\path[->, thick] (41)edge[] node[]{} (11) (11)edge[] node[]{} (21) (21)edge[] node[]{}(31);\n\\end{tikzpicture}\n \\caption{$t=t_1$}\n \\label{fig:small_t1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\centering\n\\begin{tikzpicture}[]\n \\node[circle,draw=black,thick] (12) at (0,1.5) {1};\n \\node[circle,draw=black,thick] (22) at (1.5,1.5) {2};\n \\node[circle,draw=black,thick] (32) at (1.5,0) {3};\n \\node[circle,draw=black,thick] (42) at (0,0) {4};\n\\path[->, thick] (32)edge[] node[]{} (22) (32)edge[] node[]{} (42);\n\\end{tikzpicture}\n \\caption{$t = t_2$}\n \\label{fig:small_t2}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\centering\n\\begin{tikzpicture}[]\n \\node[circle,draw=black,thick] (13) at (0,1.5) {1};\n \\node[circle,draw=black,thick] (23) at (1.5,1.5) {2};\n \\node[circle,draw=black,thick] (33) at (1.5,0) {3};\n \\node[circle,draw=black,thick] (43) at (0,0) {4};\n\\path[->, thick] (43)edge[bend right] node[]{} (13) (13)edge[bend right] node[]{} (43); \n\\end{tikzpicture}\n \\caption{$t=t_3$}\n \\label{fig:small_t3}\n \\end{subfigure}\n \\caption{Small temporal network ${\\mathcal{G}} = (G^{[1]}, G^{[2]},G^{[3]})$.}\n \\label{fig:small_graph}\n\\end{figure}\n\n\\subsection{Multilayer interpretation of $M$}\\label{sec:multilayer} \nThe motivation for the definition of the block matrices above is that we can view walks across multiple time-frames in the following way. \nRecall that given any graph $G$, one can construct the line graph associated with $G$ whose adjacency matrix is the matrix $W$. \nThe nodes of the line graph are the edges of $G$ and any two nodes are connected if the concatenation of the corresponding edges in $G$ forms a walk of length two across the original graph. Katz centrality of a node can then be computed by counting walks on the line graph, summing them using appropriate weights, and then using the source and target matrices to ``translate\" the obtained result from the edge space (represented by the line graph) to the node space (represented by $G$). \nThe key result that allows this approach is~\\eqref{eq:Ar} or, \nequivalently, the fact that a walk of length $r+1$ in the line graph corresponds to a walk of length $r$ in $G$.\n\nTo mirror this construction in the case of temporal networks, one begins by constructing the line graphs of each $G^{[\\tau]}$ appearing in ${\\mathcal{G}}$. \nThe new graph representation of ${\\mathcal{G}}$ in the edge space will now contain $m=\\sum_\\tau m_\\tau$ nodes, labelled as $i\\xrightarrow[]{t_\\tau}j$ and hence identified by the source and target of each edge and by the time stamp at which the connection occurs. \nTwo nodes in this new graph are connected by a directed edge if their concatenation forms a walk of length two in the graph corresponding to the time stamp $t=t_\\tau$, i.e., if they form a walk of length two in $G^{[\\tau]}$. \nConsider for example the graph in Figure~\\ref{fig:small_graph}. The construction built so far is presented in the left-hand panel of \nFigure~\\ref{fig:multi_small_graph}. Clearly, this construction is incomplete, as the walks that we are able to represent, and hence count, at this stage are only those that take place entirely in one of the $G^{[\\tau]}$, for some $\\tau = 1,2,\\ldots, N$. \nWalks of length two in temporal networks may also take place {\\it across} time stamps, and these are not accounted for by the construction of the line graphs $W^{[\\tau]}$. \nFor example, the temporal walk $4 \\xrightarrow[]{t_1} 1 \\xrightarrow[]{t_3} 4$ in Figure~\\ref{fig:small_graph} is not accounted for in the edge space representation of Figure~\\ref{fig:a}.\n\n The graph built so far, whose adjacency matrix is block-diagonal with $W^{[\\tau]}$, $\\tau = 1,2,\\ldots, N$ as diagonal blocks, thus needs to be adapted to allow for walks that happen across time stamps: directed edges are added to connect nodes of the form $i\\xrightarrow[]{t_{\\tau_1}} j$ to edges of the form $j\\xrightarrow[]{t_{\\tau_2} }k$ if $\\tau_1<\\tau_2$.\n The resulting graph is now a multilayer network which encodes all possible temporal walks of length two taking place in ${\\mathcal{G}}$. \n For example, in the (partial) temporal line-graph construction presented in Figure~\\ref{fig:a} associated with the graph in Figure~\\ref{fig:small_graph} one needs to add the dashed and dotted edges pictured in Figure~\\ref{fig:b}. \n Here, dashed lines represent connections that happen across two consecutive time stamps, while dotted lines represent walks of length two that start at time $t_1$ and finish at time $t_3$. \n\n \nIf we now turn our attention to the adjacency matrix of this multilayer, it is easy to see that this is indeed the matrix $M$ that we presented in Definition~\\ref{def:Mnolla}. \n\n \n\n\n\n\n\\begin{figure}[]\n\\centering\n\\subfloat[]{\\label{fig:a}\\input{small_graph_partial.tikz}}\n\\hspace{1cm}\n\\subfloat[]{\\label{fig:b}\\input{small_graph2.tikz}}\n\\caption{(a) Partial line graph construction associated to ${\\mathcal{G}}$ in Figure \\ref{fig:small_graph}, as described in subsection~\\ref{sec:multilayer}. (b) Multilayer graph underlying the matrix $M$ in Definition~\\ref{def:Mnolla} associated with the graph in Figure~\\ref{fig:small_graph}}\n\\label{fig:multi_small_graph}\n\\end{figure}\n\n \n\\begin{remark}\\label{rem:Mr}\nAs in the case of static networks, a temporal walk of length $r$ in the graph associated to $M$ corresponds to a temporal walk of length $r+1$ in ${\\mathcal{G}}$.\n\\end{remark}\nBecause of this interpretation of $M$ as the adjacency matrix of a network associated with ${\\mathcal{G}}$, one can expect to be able to compute dynamic walk-based centrality measures using $M$ by counting walks in its associated multilayer graph, summing these walks with appropriate weights, and then projecting the result to the node space using the global source and target matrices. This was achieved in \\eqref{eq:ysym_N} for Katz centrality, and will be generalized to other matrix \nfunctions in the next section.\n\n\n\n\n\n\\section{$f$-total communicability and $f$-subgraph centrality on time-evolving networks}\\label{sec:measures_temporal}\nIn this section, we consider generalization of the proposed block-matrix approach to walk-based centralities with weights induced by some suitable analytic functions $f$ via their Maclaurin series. \n\n\n\nSuppose that $\\mathcal{G}$ is a temporal network and let $M=M^{[1,\\dots, N]}$ be its global temporal transition matrix. Suppose moreover that, for all $|z|0$ can be computed as the diagonal entries of\n\\begin{equation}\\label{eq:genfunexp3}\nI + \\beta \\mathcal{L}^T \\psi_1 \\left( \\beta \\begin{pmatrix} W^{[1]} & W^{[1,2]} & W^{[1,3]}\\\\\n0&W^{[2]}&W^{[2,3]}\\\\\n0&0&W^{[3]} \\end{pmatrix} \\right) \\mathcal{R} \n\\end{equation} \nwhile the total communicability vector is obtained by multiplying \\eqref{eq:genfunexp3} by $\\mathbf{1}$.\nWe emphasize that the generating function \\eqref{eq:genfunexp3} is generally \\emph{not} equal to $e^{\\beta A^{[1]}} e^{\\beta A^{[2]}}e^{\\beta A^{[3]}}$. \n\nIndeed, consider for example the temporal graph ${\\mathcal{G}}$ represented by the following adjacency matrices\n\\[\nA^{[1]}= \\begin{bmatrix}\n0 & 1 & 0 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \n\\end{bmatrix}, \\, \nA^{[2]}= \\begin{bmatrix}\n0 & 0 & 0 & 0 \\\\ \n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0 \n\\end{bmatrix}, \\text{ and } \nA^{[3]}= \\begin{bmatrix}\n0 & 0 & 0 & 0 \\\\ \n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 1 \\\\\n0 & 0 & 0 & 0 \n\\end{bmatrix}.\n\\]\nClearly, all the admissible walks in this temporal graph are \n\\begin{itemize}\n \\item walks of length one, weighted by $\\beta$:\n \\[1 \\xrightarrow[]{t_1} 2, \\; 2 \\xrightarrow[]{t_2} 3,\\; \\text{ and } \\; 3 \\xrightarrow[]{t_3} 4;\\]\n \\item walks of length two, weighted by $\\frac{\\beta^2}{2}$:\n \\[1 \\xrightarrow[]{t_1} 2 \\xrightarrow[]{t_2} 3\\; \\text{and }\\; 2 \\xrightarrow[]{t_2} 3 \\xrightarrow[]{t_3} 4;\\]\n and finally, \n \\item one walk of length three, weighted by a factor $\\frac{\\beta^3}{3!}$\n\\[1 \\xrightarrow[]{t_1} 2 \\xrightarrow[]{t_2} 3 \\xrightarrow[]{t_3} 4.\\]\n\\end{itemize}\nSince these are the only admissible walks, together with walks of length zero, weighted by $1$, \nwe expect the entries of the $4\\times 4$ matrix~\\eqref{eq:genfunexp3} to encode these weights in the appropriate entries; \nfor example, in the entry $(1,3)$ we expect to read the weight $\\frac{\\beta^2}{2}$ since the only walk in the \nnetwork originating at node $1$ and terminating at node $3$ is of length two. \n\nIt is easy to verify that, up to permutation similarity, \n\\[\nM = \\begin{bmatrix}\n0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0\n\\end{bmatrix}\n\\]\nwhich is nilpotent, and hence \n\\[\\psi_1(\\beta M) = I_3 + \\frac{\\beta}{2} M + \\frac{\\beta^2}{3!}M^2 = \n\\begin{bmatrix}\n1 & \\frac{\\beta}{2} & \\frac{\\beta^2}{3!} \\\\\n0 & 1 & \\frac{\\beta}{2} \\\\\n0 & 0 & 1\n\\end{bmatrix}.\\]\nWith the labelling of edges induced by $M$ above it holds that\n\\[\n{\\mathcal{L}} = \n\\begin{bmatrix}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\ \n0 & 0 & 1 & 0\n\\end{bmatrix}\\quad \\text{ and }\\quad\n{\\mathcal{R}} =\n\\begin{bmatrix}\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 1 & 0 \\\\ \n0 & 0 & 0 & 1\n\\end{bmatrix}.\n\\]\nIt immediately follows that the matrix in \\eqref{eq:genfunexp3} is \n\\[\nI_4 + \\beta {\\mathcal{L}}^T\\psi_1(\\beta M){\\mathcal{R}} = \n\\begin{bmatrix}\n1 & \\beta & \\frac{\\beta^2}{2} & \\frac{\\beta^3}{3!} \\\\\n0 & 1 & \\beta & \\frac{\\beta^2}{2} \\\\\n0 & 0 & 1 & \\beta \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix},\n\\]\nwhich encodes in its entries exactly the required weights. \nOn the other hand, multiplying the exponentials of the adjacency matrices gives \n\\[\ne^{\\beta A^{[1]}}e^{\\beta A^{[2]}}e^{\\beta A^{[3}} = \n\\begin{bmatrix}\n1 & \\beta & \\beta^2 &\\beta^3 \\\\\n0 & 1 & \\beta & \\beta^2 \\\\\n0 & 0 & 1 & \\beta \\\\\n0 & 0 & 0 & 1\n\\end{bmatrix},\n\\]\nwhere we have used $(A^{[\\tau]})^r = 0$ for all $r\\geq 2$ and $A^{[1]}A^{[3]} = 0$. \nThis matrix fails \nto correctly weight walks of length two and three. \n\\end{example}\n\\smallskip\n\nIn the next section we move on to the problem \nof generalizing the results obtained so far to the \nnon-backtracking setting, where backtracking is \ndisallowed either within time stamps or across \ntime stamps. \nThe construction presented mirrors the one described \nabove and will build on modified versions of the global \ntemporal transition matrix $M$.\n\n\n\n\\section{Non-backtracking walks}\\label{sec:nbt} \n\nNon-backtracking walks, as defined in\nDefinition~\\ref{def:nbt}, \nmay be studied using the \n{\\it Hashimoto matrix}; see~\\cite{Hash90}.\nWe also note that the \nHashimoto matrix plays a key role in related studies of \n\\emph{random} NBT walks \\cite{FTT21}.\n\\begin{definition}\nThe {\\rm Hashimoto matrix} associated with a graph $G=(V,E)$ with nodes $V = \\{1,2,\\ldots, n\\}$ and $m$ edges is the matrix $B\\in\\mathbb{R}^{m\\times m}$ defined as\n\\[\nB_{i\\to j,k\\to \\ell} = \\delta_{jk}(1-\\delta_{i\\ell})\n\\] \nfor all $i\\to j,k\\to \\ell\\in E$. \n\\end{definition}\nIt is easily verified that \n\\[\nB = W - W\\circ W^T\n\\]\nwhere $\\circ$ denotes the Schur (or entrywise) product and $W$ is the adjacency matrix of the \nline graph of $G$.\nIn words, rows and columns of $B$ correspond to edges in $G$, and \nthe nonzero entries record pairs of edges that form a \nNBT walk of length two in $G$.\n\nTaking powers of this matrix and then projecting back to the node space using the source and target \nmatrices allows us to count NBT walks taking place in $G$. Specifically, \n\\begin{equation}\\label{eq:Br}\n(L^TB^{r-1}R)_{ij} = (p_r(A))_{ij}\n\\end{equation}\nis the number of NBT walks of length $r$ originating at node $i$ and ending at node $j$; see~\\cite{Beyond}.\nHere and in the following we denote by $p_r(A)\\in\\mathbb{R}^{n\\times n}$ the matrix whose entries record the number \nof NBT walks of length $r$ between any two nodes in the network with adjacency matrix $A$. \nThese matrices satisfy a four-term recurrence that allows the computation of the number \nof NBT walks of any length in the network. \nWe refer the interested reader to~\\cite{AGHN17a,GHN18} and references therein.\n \n\\subsection{Non-backtracking walk-based centrality measure}\nA NBT version of Katz centrality was proposed and studied in~\\cite{AGHN17a,GHN18}. \nGeneralization to other $f$-centrality measures was presented in~\\cite{AGHN17b}. \nLater, in~\\cite{Beyond} it was shown how NBT Katz centrality, as well as other NBT versions of $f$-subgraph centrality and $f$-total communicability, could also be derived via the Hashimoto matrix. \n\n\nWe may \ndefine the NBT Katz centrality by changing \n(\\ref{eq:Katz}) to \n\\\n\\widehat{{\\bf y}}(\\alpha) =\n\\sum_{r=0}^\\infty \\alpha^r p_r(A){\\bf 1}.\n\\\n \nAs shown in \\cite{Beyond}, we may also obtain $\\widehat{{\\bf y}}(\\alpha)$ \nby computing Katz centrality for the graph of the Hashimoto matrix and then projecting via the source and target matrices; that is, \n\\[\n \\widehat{{\\bf y}}(\\alpha) = \\left(I + \\alpha L^T(I-\\alpha B)^{-1}R\\right) {\\bf 1}, \n\\]\nfor $\\alpha<1\/\\rho(B)$. \nSimilarly, NBT $f$-total communicability is obtained by computing $\\partial f$-total communicability for the Hashimoto matrix and then projecting, where $\\partial$ is the operator defined in \\eqref{eq:partial}, \nthat is, \n\\begin{equation*}\n \\widehat{{\\bf y}}(\\alpha) = \\left(f(0)I + \\alpha L^T \\partial f (\\alpha B) R\\right) \\boldsymbol{1}.\n\\end{equation*}\nNBT versions of $f$-subgraph centrality can also be defined analogously using the diagonal elements of $f(0)I + \\alpha L^T \\partial f (\\alpha B) R$.\n\n\n\\subsection{Temporal NBT centrality measures}\n\nIn the remainder of this section we \ncarry through concepts and results\nfrom sections~\\ref{sec:dyncom} and \\ref{sec:measures_temporal}\nto the \nNBT setting.\nThis produces, for the first time,\ndefinitions and computable \nexpressions for temporal NBT centrality measures. \nWe first notice that there is not just one type of backtracking for temporal networks; indeed, there are three:\n\\begin{itemize}\n\\item Backtracking happens within a certain time stamp; we will refer to this as {\\it backtracking in space}.\n\\item Backtracking happens across time stamps; we will refer to this as {\\it backtracking in time}.\n\\item Backtracking happens both within a time stamp and across time stamps (not necessarily in that order); we will refer to this as {\\it backtracking in time and space}.\n\\end{itemize}\nTo explain these different types, consider the temporal graph ${\\mathcal{G}}$ in Figure~\\ref{fig:small_graph}. \nThe walks $1 \\xrightarrow[]{t_3} 4 \\xrightarrow[]{t_3} 1$ and $2 \\xrightarrow[]{t_1} 3 \\xrightarrow[]{t_2} 2$ showcase two different types of backtracking: \nthe first is backtracking in space and the second is backtracking in time; additionally, the walk $4 \\xrightarrow[]{t_1} 1 \\xrightarrow[]{t_3} 4 $ provides an example of a walk that backtracks in both space \\textit{and} time. \n\n\n\n \\subsubsection{Non-backtracking Katz centrality for time-evolving networks: the case of $2$ time-frames}\n\nAs in section~\\ref{sec:measures_temporal}, we begin by considering the case of $\\mathcal{G} = (G^{[1]},G^{[2]})$ at times $t_1 < t_2$. \nIn the following, we make use of the notation $B^{[1,2]} := W^{[1,2]} - W^{[1,2]}\\circ {W^{[2,1]}}^T$. \n\\begin{remark}\nRecall that the matrix $W^{[1,2]} = R^{[1]}(L^{[2]})^T\\in\\mathbb{R}^{m_1\\times m_2}$ is a possibly rectangular matrix that encodes in its entries whether two edges, one appearing at time $t=t_1$ and a second appearing at time $t=t_2$, can be concatenated; see Definition~\\ref{def:Wt1t2} and Proposition~\\ref{prop:Wt1t2}. \nAnalogously, the matrix \n$B^{[1,2]}\\in\\mathbb{R}^{m_1\\times m_2}$ records whether a walk of length two that occurs across time stamps \n$t_1$ and $t_2$ is non-backtracking, since we have \n\\[\n(B^{[1,2]})_{i\\to j,k\\to \\ell} = \\delta_{jk}(1-\\delta_{i\\ell})\n\\]\nfor all $i\\to j\\in E^{[1]}$ and for all $k\\to\\ell\\in E^{[2]}$. \nWe will formalize this concept in Definition~\\ref{def:Bt1t2} and Proposition~\\ref{prop:Bt1t2} below.\n\\end{remark}\n\nWith consideration to the previous interpretation of the terms appearing in~\\eqref{eq:goodexample}, if one wants to forbid backtracking, in full or in part, it is appropriate to generalize the expression~\\eqref{eq:goodexample} in three possible ways, depending on the type of backtracking behavior that we want to forbid.\n\\begin{itemize}\n\\item Backtracking in space forbidden: we replace the $W^{[\\tau]}$ with the Hashimoto matrix $B^{[\\tau]}$ for $\\tau=1,2$, giving \n\\begin{eqnarray}\nI_n &+& \\alpha \\left( \\sum_{\\tau=1}^2 (L^{[\\tau]})^T (I_m - \\alpha B^{[\\tau]})^{-1} R^{[\\tau]} \\right) \\nonumber\\\\\n&&\n+ \\alpha^2 (L^{[1]})^T (I_m-\\alpha B^{[1]})^{-1} W^{[1,2]} (I_m- \\alpha B^{[2]})^{-1} R^{[2]}. \\label{eq:one}\n\\end{eqnarray} \n\\item Backtracking in time forbidden: we replace $W^{[1,2]}$ with $B^{[1,2]}$, giving \n\\begin{eqnarray}\nI_n &+& \\alpha \\left( \\sum_{\\tau=1}^2 (L^{[\\tau]})^T (I_m - \\alpha W^{[\\tau]})^{-1} R^{[\\tau]} \\right) \\nonumber \\\\\n&&\n+ \\alpha^2 (L^{[1]})^T (I_m-\\alpha W^{[1]})^{-1} B^{[1,2]} (I_m- \\alpha W^{[2]})^{-1} R^{[2]}. \n\\label{eq:two}\n\\end{eqnarray} \n\\item Backtracking in time and space both forbidden: we perform both replacements, giving \n\\begin{eqnarray}\nI_n &+& \\alpha \\left( \\sum_{\\tau=1}^2 (L^{[\\tau]})^T (I_m - \\alpha B^{[\\tau]})^{-1} R^{[\\tau]} \\right) \\nonumber \\\\\n&& + \\alpha^2 (L^{[1]})^T (I_m-\\alpha B^{[1]})^{-1} B^{[1,2]} (I_m- \\alpha B^{[2]})^{-1} R^{[2]}. \\label{eq:three}\n\\end{eqnarray} \n\\end{itemize}\n\n\nIn the case of \\eqref{eq:one}, we observe that for $\\tau = 1,2$ \n\\[ \\alpha(L^{[\\tau]})^T (I_m - \\alpha B^{[\\tau]})^{-1} R^{[\\tau]} = \\alpha\\sum_{r=0}^\\infty \\alpha^r p_{r+1}(A^{[\\tau]}), \\] \nwhere we have used \\eqref{eq:Br}.\n\nMoreover, for small enough values of $\\alpha$,\n\\[ \\sum_{r=0}^\\infty \\alpha^r p_r(A^{[\\tau]}) = (1-\\alpha^2) [I - \\alpha A^{[\\tau]} + \\alpha^2 (D^{[\\tau]} - I) + \\alpha^3 (A^{[\\tau]}-S^{[\\tau]})]^{-1}, \\] where\n\\begin{equation}\\label{eq:dands}\nD^{[\\tau]} = \\diag \\diag (A^{[\\tau]})^2, \\qquad S^{[\\tau]} = A^{[\\tau]} \\circ (A^{[\\tau]})^T\n\\end{equation}\n (both formulae are proved in the existing literature \\cite{AGHN17a,Beyond}). \n Hence, all the terms appearing in \\eqref{eq:one} can be reformulated \n in terms of matrices of order $n$,\n thus obtaining a \\emph{non-backtracking (in space only) dynamic communicability matrix}:\n\\begin{equation*}\\label{eq:nbtisodcm}\n \\mathcal{Q} = (1 - \\alpha^2)^2 \\prod_{\\tau=1}^2 [I - \\alpha A^{[\\tau]} + \\alpha^2 (D^{[\\tau]} - I) + \\alpha^3 (A^{[\\tau]}-S^{[\\tau]})]^{-1}. \n\\end{equation*} \n\nWhen possible, it is clearly preferable to work at node-level, since for large values of $n$ it may easily happen that $n \\ll m$. We observe that this may be true even if, individually, each graph has $O(n)$ edges, since in total there may be up to $O(Nn)$ edges. However, a direct node-level formulation appears to be infeasible when backtracking is also forbidden in time, due to the fact that a memory of the last traveled edge in a given time frame is necessary. Thus, below we will focus on developing further the edge level formulations \\eqref{eq:one}--\\eqref{eq:three}.\n\n\nOur goal is to generalize these formulae to the case of $N>2$ time stamps. \nWhen doing so, one issue that arises is that the above formulae for the dynamic communicability matrices become unwieldy. \nThe solution we propose builds on the fact that \\eqref{eq:one}--\\eqref{eq:three} can be rewritten using block matrices, just like we did with~\\eqref{eq:goodexample}. \nIndeed, depending on what type of backtracking is forbidden, one can generalize \\eqref{eq:Mzero} as follows:\n\\begin{equation}\\label{eq:M2general}\nM^{[1,2]} = \\begin{bmatrix} C^{[1]} & C^{[1,2]}\\\\\n0 & C^{[2]}\n\\end{bmatrix},\n\\end{equation}\nwhere\n\\begin{itemize}\n \\item $C^{[\\tau]} = B^{[\\tau]}$ for $\\tau = 1,2$ and $C^{[1,2]} = W^{[1,2]}$, if one wants to recover \\eqref{eq:one}, i.e., avoid backtracking in space but not in time; \n \\item $C^{[\\tau]} = W^{[\\tau]}$ for $\\tau = 1,2$ and $C^{[1,2]} = B^{[1,2]}$, if one wants to recover \\eqref{eq:two}, i.e., avoid backtracking in time but not in space; and\n \\item $C^{[\\tau]} = B^{[\\tau]}$ for $\\tau = 1,2$ and $C^{[1,2]} = B^{[1,2]}$, if one wants to recover \\eqref{eq:three}, i.e., avoid backtracking in space and time.\n\\end{itemize}\nThe following result then holds.\n\\begin{prop}\\label{prop:properblocks}\nSuppose $M^{[1,2]}$ is defined as in~\\eqref{eq:M2general}. Then, within the radius of convergence,\n \\[ (I_{m} - \\alpha {M^{[1,2]}})^{-1} = \\begin{pmatrix}(I_m-\\alpha C^{[1]})^{-1} & (I_m-\\alpha C^{[1]})^{-1}\\alpha C^{[1,2]}(I_m-\\alpha C^{[2]})^{-1} \\\\ 0 & (I_m-\\alpha C^{[2]})^{-1}\\end{pmatrix}.\\]\n\\end{prop}\nOur proof of this result mirrors the proof of Proposition ~\\ref{prop:properblocksW} and is therefore omitted.\n\nBy using the above proposition and the fact that \n\\[(I_m-\\alpha M^{[1,2]})^{-1} = I_{m} + \\sum_{r = 1}^\\infty \\alpha^r {M^{[1,2]}}^r,\\] \nwe can easily recover~\\eqref{eq:one}--\\eqref{eq:three} by projecting back to the node-level, multiplying by $\\alpha$ to account for the fact that a walk on the edge level is one step longer than its equivalent on the node level, and adding back the identity matrix $I_n$; such a projection is achieved by the global source and target matrices associated with ${\\mathcal{G}}$: \n\\[\n I_n + \\alpha {\\mathcal{L}^{[1,2]}}^T (I_m-\\alpha M^{[1,2]})^{-1} \\mathcal{R}^{[1,2]}.\\]\n\n\n\n\n\n\\smallskip\n\nWe now move to the goal of extending this line-graph interpretation of Katz centrality for evolving networks to possibly more than two graphs. In doing so, we will allow for backtracking to be forbidden (in time or space or both).\nWe adopt the same strategy used in sections~\\ref{sec:dyncom} and \\ref{sec:measures_temporal}. \n\n\n\n\n\n\\subsubsection{Non-backtracking Katz centrality for time-evolving networks: more than $2$ time-frames}\nWe begin by extending Definition~\\ref{def:Wt1t2} and Definition~\\ref{def:Mnolla} to the NBT framework. The latter \nmay be reformulated in different ways, depending on the type of backtracking that one wants to avoid. \n\\begin{definition}\\label{def:Bt1t2}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps. Let moreover $G^{[\\tau]} = ({\\mathcal{V}}, E^{[\\tau]})$ for all $\\tau = 1,2,\\ldots, N$. \nThe {\\rm non-backtracking transition matrix} $B^{[\\tau_1,\\tau_2]}$ representing walks of length two that traverse the first edge at time $t_{\\tau_1}$ and the second at time $t_{\\tau_2}$, $\\tau_1 < \\tau_2$, is defined entry-wise as follows:\n\\[\n(B^{[\\tau_1,\\tau_2]})_{i \\rightarrow j, k \\rightarrow \\ell} = \\delta_{jk}(1-\\delta_{i\\ell})\n\\]\nfor all $i\\to j\\in E^{[\\tau_1]}$ and $k\\to\\ell\\in E^{[\\tau_2]}$. \n\\end{definition}\n\n\nThese matrices, which are generally non-square, record information about NBT walks of length two where the first edge is traversed at time $\\tau_1$ and the second is traversed at time $\\tau_2$ (after staying idle for some time, in the case where $\\tau_2 > \\tau_1+1$). \nThe next result extends Proposition~\\ref{prop:Wt1t2}.\n\\begin{prop}\\label{prop:Bt1t2}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network. \nLet moreover $L^{[\\tau]}$ and $R^{[\\tau]}$ be the source and target matrices associated with graph $G^{[\\tau]}$. \nThen for all $1\\leq\\tau_1<\\tau_2\\leq N$ we have \n\\[\n B^{[\\tau_1,\\tau_2]} = W^{[\\tau_1,\\tau_2]} - W^{[\\tau_1,\\tau_2]}\\circ {W^{[\\tau_2,\\tau_1]}}^T= R^{[\\tau_1]}(L^{[\\tau_2]})^T - (R^{[\\tau_1]}L^{[\\tau_2]^T}) \\circ (L^{[\\tau_1]}R^{[\\tau_2]^T}).\n\\]\n\\end{prop}\n\\begin{proof}\nWe proceed entrywise. Let $i\\to j\\in E^{[\\tau_1]}$ and $k\\to\\ell\\in E^{[\\tau_2]}$ \nbe two edges in ${\\mathcal{G}}$. Then\n\\[\n(R^{[\\tau_1]}L^{[\\tau_2]^T})_{i\\to j,k\\to \\ell} = \n\\sum_{h = 1}^n (R^{[\\tau_1]})_{i\\to j,h} (L^{[\\tau_2]})_{k\\to \\ell, h}= \\delta_{jk}.\n\\]\nSimilarly,\n\\[\n(L^{[\\tau_1]}R^{[\\tau_2]^T})_{i\\to j,k\\to\\ell} = \n\\sum_{h = 1}^n (L^{[\\tau_1]})_{i\\to j,h} (R^{[\\tau_2]})_{k\\to \\ell, h}= \\delta_{i\\ell}.\n\\]\nThe conclusion then follows from Definition~\\ref{def:Bt1t2} and Proposition~\\ref{prop:Wt1t2}.\n\n\\end{proof}\n\nAn advantage of the block-matrix approach is that it is readily expanded to deal with $N$ time-frames. This is done by defining $M^{[1,\\dots,N]}$ as one of the following block upper triangular block matrices.\n\\begin{definition}\\label{def:Mkolme}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps. \nThe {\\rm NBT global temporal transition matrix} associated with ${\\mathcal{G}}$ is the $m\\times m$ block matrix \n\\begin{equation}\\label{eq:Mkolme}\nM = M^{[1,\\dots,N]} = \n\\begin{bmatrix} \nC^{[1]} & C^{[1,2]} & C^{[1,3]} & \\dots & C^{[1,N]} \\\\ \n0 & C^{[2]} & C^{[2,3]} & \\dots & C^{[2,N]} \\\\\n\\vdots & & \\ddots & & \\vdots \\\\\n\\vdots & & & \\ddots & \\vdots\\\\\n0 & \\dots & \\dots & 0 & C^{[N]}\n\\end{bmatrix}, \n\\end{equation}\nwhere \n\\begin{itemize}\n\\item[(i)] $C^{[\\tau_1]} = B^{[\\tau_1]}$ and $C^{[\\tau_1,\\tau_2]} = W^{[\\tau_1,\\tau_2]}$ for all $\\tau_1,\\tau_2=1,2,\\ldots,N$ ($\\tau_1< \\tau_2$) if backtracking in space is forbidden;\n\\item[(ii)] $C^{[\\tau_1]} = W^{[\\tau_1]}$ and $C^{[\\tau_1,\\tau_2]} = B^{[\\tau_1,\\tau_2]}$ for all $\\tau_1,\\tau_2=1,2,\\ldots,N$ ($\\tau_1< \\tau_2$) if backtracking in time is forbidden; and \n\\item[(iii)] $C^{[\\tau_1]} = B^{[\\tau_1]}$ and $C^{[\\tau_1,\\tau_2]} = B^{[\\tau_1,\\tau_2]}$ for all $\\tau_1,\\tau_2=1,2,\\ldots,N$ ($\\tau_1< \\tau_2$) if backtracking in both time and space is forbidden.\n\\end{itemize} \n\\end{definition}\n\nSuch a formulation allows for non-backtracking across multiple graphs to be calculated. \n\n\n\\begin{remark}\nIn the third case, one can calculate $M$ from the global source and target matrices as $\\mathcal{R}\\mathcal{L}^T - \\mathcal{R}\\mathcal{L}^T \\circ \\mathcal{L}\\mathcal{R}^T$ by setting all elements below the block diagonal to zero.\n\\end{remark}\n\nWe can now define Katz centrality on a time-evolving network with NBT global temporal transition matrix $M=M^{[1,\\dots, N]}$ equal to any of the three options in Definition~\\ref{def:Mkolme} (depending on the type of backtracking allowed)\nas\n\\[ \\widehat{\\boldsymbol{y}}(\\alpha) = \\sum_{r=0}^\\infty (\\alpha M)^r {\\bf 1}.\\]\n\nThe following result gives an upper bound for the possible choices of $\\alpha>0$.\n\\begin{prop}\\label{prop:radius}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]}, \\ldots, G^{[N]})$ be a temporal network with \nadjacency matrices $A^{[1]}, A^{[2]}, \\ldots, A^{[N]}$ and let $M = M^{[1,\\dots, N]}$ be \ndefined as in Definition~\\ref{def:Mkolme}. \nMoreover, let $\\rho=\\max_\\tau \\rho(A^{[\\tau]})$ and $\\lambda=\\min_\\tau \\lambda_\\tau$, where $\\lambda_\\tau$ is the smallest eigenvalue of the matrix polynomial $P^{[\\tau]}(z)=I-A^{[\\tau]} z + (D^{[\\tau]}-I) z^2 + (A^{[\\tau]}-S^{[\\tau]}) z^3$, for all $\\tau = 1,2,\\ldots, N$, where $D^{[\\tau]}$ and $S^{[\\tau]}$ are as in \\eqref{eq:dands}. \n\nThen, the series $\\sum_{r=0}^\\infty (\\alpha M)^r$ converges and moreover\n\\[\n\\sum_{r=0}^\\infty (\\alpha M^{[1,\\dots,N]})^r = (I_m-\\alpha M^{[1,\\dots, N]})^{-1}\n\\]\nfor all $\\alpha\\in (0,\\ell)$, where\n\\begin{itemize}\n\\item $\\ell=\\lambda$, if $M$ is as in {\\it (i)} or {\\it (iii)} from Definition~\\ref{def:Mkolme};\n\\item $\\ell=\\rho^{-1}$, if $M$ is as in {\\it (ii)} from Definition~\\ref{def:Mkolme}. \n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nThe rightmost endpoint of the interval of allowed values for $\\alpha$ is the radius of convergence of $\\sum_{r=0}^\\infty M^r$, which is equal to the inverse of the spectral radius of $M$. Since the latter is a block triangular matrix, its spectral radius is equal to the maximum of the spectral radii of the diagonal blocks. It follows by the analysis in \\cite{AGHN17a, Beyond, GHN18} that the spectral radius of $B^{[i]}$ is $\\lambda^{-1}_i$, while by Flanders' theorem the spectral radius of $W^{[i]}=R^{[i]}(L^{[i]})^T$ is equal to the spectral radius of $A^{[i]}=(L^{[i]})^T R^{[i]}$, which is $\\rho_i$.\n\\end{proof}\n\nThis convergence result allows us to thus define NBT Katz centrality for temporal networks as follows.\n\\begin{definition}\\label{def:katznbt}\nLet ${\\mathcal{G}} = (G^{[1]},G^{[2]},\\ldots,G^{[N]})$ be a temporal network with $N$ time stamps.\nLet $M$ be defined as in Definition~\\ref{def:Mkolme} and let $\\alpha\\in (0,\\ell)$, where $\\ell$ is defined according to Proposition~\\ref{prop:radius}.\nWe define {\\rm NBT temporal Katz centrality} as the vector \n\\[\n \\widehat{\\boldsymbol{y}}(\\alpha) = \\sum_{r=0}^\\infty (\\alpha M)^r {\\bf 1} = (I_n + \\alpha (\\mathcal{L}^T (I_m-\\alpha M)^{-1} \\mathcal{R}) {\\bf 1}_n.\n\\]\n\\end{definition}\n\n\\begin{remark}\nNote that, since $\\mathcal{R}{\\bf 1}_n={\\bf 1}_m$, Katz centrality is also equal to\n\\begin{equation*}\\label{eq:k1}\n \\widehat{\\boldsymbol{y}}(\\alpha) = {\\bf 1}_n+ \\alpha \\mathcal{L}^T (I-\\alpha M)^{-1} {\\bf 1}_m.\n\\end{equation*}\n\\end{remark}\n\\begin{remark}\nOne may also define subgraph centrality indices by considering the \ndiagonal entries of the matrix under study:\n\\[\n\\widehat{\\boldsymbol{x}}(\\alpha)_i = 1 + \\alpha (\\mathcal{L}^T (I_m-\\alpha M)^{-1} \\mathcal{R})_{ii}.\n\\]\n\\end{remark}\n\n\n\n\nThe results presented so far for the matrix resolvent can also be extended to treat other analytic matrix functions. Indeed, the following result is an analogue of Proposition~\\ref{prop:generalf} in the non-backtracking framework. \n\\begin{prop}\\label{prop:fnbt}\nIf the power series $f(z) =\\sum_{r=0}^\\infty c_rz^r$ converges with radius of convergence $R$, then the series\n$\\sum_{r=0}^\\infty c_r \\alpha^r M^r$ also converges for all $\\alpha \\in [0,R\/\\ell)$, with $M$ is as in Definition~\\ref{def:Mkolme} and \n$\\ell$ is as in Proposition~\\ref{prop:radius}. \nMoreover, \n\\[\\sum_{r=0}^\\infty c_r \\alpha^r M^r =\\partial f(\\alpha M),\\] where $\\partial$ is the functional operator defined in \\eqref{eq:partial}.\n\\end{prop}\nUsing this result, Definition~\\ref{def:katznbt} can straightforwardly be generalized to describe centrality indices that avoid different types of backtracking, according to how the matrix $M$ in \\eqref{eq:Mkolme} is selected. \n\\begin{definition}\nLet $\\mathcal{G}$ be a temporal network and let $M=M^{[1,\\dots, N]}$ be defined as in Definition~\\ref{def:Mkolme} Let $f$ be a function which is analytic in a neighborhood of zero with Maclaurin series $f(z) = \\sum_{r=0}^\\infty c_r z^r$, such that $c_r \\geq 0$ for all $r$, and let $\\alpha\\in(0,\\ell)$, where $\\ell$ is as in Proposition~\\ref{prop:radius}. \nMoreover, let $\\partial$ be the functional operator defined in \\eqref{eq:partial}. \nThe {\\rm NBT temporal $f$-subgraph centrality} of node $i$ is defined as\n\\begin{equation}\n \\widehat{\\boldsymbol{x}}(\\alpha)_i\n= \\left(c_0 I + \\alpha\\mathcal{L}^T \\partial f(\\alpha M) \\mathcal{R}\\right)_{ii}.\n\\label{eq:nbtf1}\n\\end{equation}\nAdditionally, the {\\rm NBT temporal $f$-total communicability} of node $i$ is defined as the $i$th entry of the following vector:\n\\begin{equation}\n\\widehat{\\boldsymbol{y}}(\\alpha\n= c_0 {\\bf 1}_n + \\alpha\\mathcal{L}^T \\partial f(\\alpha M) {\\bf 1}_m.\n\\label{eq:nbtf2}\n\\end{equation}\nwhere $\\mathcal{L}$ and $\\mathcal{R}$ be defined as in Definition~\\ref{def:globalLR}. \n\\end{definition}\n\n\\section{Conclusions}\\label{sec:conc}\n\nOur aim in this work was to study combinatorically-motivated centrality measures for temporal networks.\nThis required us to extend existing work involving\nmatrix functions applied to (a) the adjacency matrix of a network,\n(b) the adjacency matrix of the line graph, and (c) the Hashimoto matrix.\nFor the Katz, or matrix resolvent, case, it is known \nfrom \\cite{GHPE11} that dynamic \ncentrality may be computed at the node level; \nthat is, by using $n$ by $n$ matrices, where $n$ is the number of nodes. \nHere, we proved that any other matrix function-based measure\ndoes not have such a straightforward \nrepresentation, but can be expressed in terms of a higher\ndimensional edge-level formula.\nIn the case of non-backtracking temporal centrality,\nwe identified three types of constraint \nand showed how these may be computed from block matrix \nextensions of the \nHashimoto construction.\n\n\n\n\n ------------------------------------------------------------------\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\n\nIn the last twenty years single-molecule magnets (SMMs) have been\nwidely studied both for their fundamental physical\nproperties~\\cite{Gatteschi2006}, and for possible appli\\-cations in\nmagnetic storage and quantum\ninformation.\\cite{Leuenberger2001,Lehmann2007} Unlike traditional\nbulk magnetic mate\\-rials, molecular magnetic materials can be\nmagnetized in a magnetic field without any interaction between the\nindividual molecules. This magnetization is a property of the\nmolecules themselves. The magnetization occurs because of the large\nground-state spin and the large easy-axis magnetic anisotropy\nbarrier separating spin-up and the spin-down states. In principle it\nis possible to store and manipulate information in one SMM.\nFurthermore the two quantum states representing the two possible\nspin orientations can be used to build a quantum qubit. Whether used\nas classical magnetic storage units or as quantum coherent elements,\nthe crucial requirement in both cases is the ability to control and\nmanipulate the magnetic states of the SMM in an efficient way.\nManipulation by magnetic fields is straightforward but, in practice,\ncannot be realized with molecular-size spatial resolution and at\nfast temporal scales. Unlike magnetic fields, electric fields are\neasy to produce, quickly switched and can be applied locally at the\nnano and molecular scale. Therefore manipulation of the properties\nof SMMs by external electric fields is an attractive and promising\nalternative.\\cite{Trif2008}\n\nAlthough electric fields do not directly couple to spins, electric\nmanipulation of the spin states is possible indirectly via\nspin-orbit coupling. This requires the presence of a strong\nspin-orbit coupling such that the electric field can effectively\nflip the spin states by acting on the the orbital part of the\nspin-orbitals. When SMMs are involved, this is not the most\nefficient mechanism, since the relative strength of spin-orbit\ninteraction scales like the volume of the molecule.\n\nRecently, a different mechanism of spin-electric coupling in\nantiferromagnetic SMMs, characterized by lack of inversion symmetry\nand spin frustration, has been proposed.\\cite{Trif2008} The best\nexample of such a system is a triangular spin $s = 1\/2$ ring with\nantiferromagnetic coupling, realized for example in the \\{Cu$_3$\\}\nSMM. The low energy physics of this system can be described by a\nthree-site spin $s= 1\/2$ Heisenberg Hamiltonian whose ground-state\nmanifold is composed of two degenerate (total) spin $S=1\/2 $\ndoublets, with wave functions represented by\n\\begin{equation}\n|\\chi_{\\pm}, S_z= +\\frac{1}{2}\\rangle =\\frac{1}{\\sqrt{3}} \\big (|\\downarrow\n\\uparrow \\uparrow\\rangle + \\epsilon_{\\pm}\n|\\uparrow\\downarrow\\uparrow\\rangle + \\epsilon_{\\mp}\n|\\uparrow\\uparrow\\downarrow\\rangle\\big ) \\label{eq:chi_pm_sz+}\\;,\n\\end{equation}\n\\begin{equation}\n|\\chi_ {\\pm}, S_z= -\\frac{1}{2}\\rangle =\\frac{1}{\\sqrt{3}} \\big\n(|\\uparrow\\downarrow \\downarrow \\rangle + \\epsilon_{\\pm}\n|\\downarrow\\uparrow\\downarrow\\rangle + \\epsilon_{\\mp}\n|\\downarrow\\downarrow\\uparrow\\rangle\\big ) \\label{eq:chi_pm_sz-}\\;,\n\\end{equation}\nwhere the many-body states $|\\sigma_1\\sigma_2\\sigma_3\\rangle$\nare products of spin-orbital states $\\sigma_i =(\\uparrow,\n\\downarrow), i = 1,2,3$ localized on the three magnetic ions of the\nmolecules, and $\\epsilon_{\\pm}= \\exp\\left( \\pm 2\\pi i \/ 3\\right)$.\nThe four states $|\\chi_{\\pm}, S_z = \\pm 1\/2 \\rangle$ in\nEqs.~(\\ref{eq:chi_pm_sz+}), (\\ref{eq:chi_pm_sz-}) are labeled by the\neigenvalues $S_z=\\pm 1\/2$ of the $z$-component of the total spin,\nand by the chirality quantum number $\\chi_{\\pm} = \\pm 1$, that is,\nthe eigenvalues of the chiral operator\n\\begin{equation}\nC_z= \\frac{4}{\\sqrt{3}} {\\mathbf s}_1\\cdot {\\mathbf s}_2 \\times\n{\\mathbf s}_3 \\label{eq:chiraloperator}\\;.\n\\end{equation}\n\nAn electric field couples to the SMM through $e {\\bf E}\\cdot {\\bf\nR}$, where $e$ is the electron charge and ${\\bf R} = \\sum_{i= 1}^3\n{\\bf r}_i$. The two spin-orbital states $|\\chi_{\\pm}, S_z\\rangle$,\ncha\\-racterized by opposite chirality and equal spin projection, form\nthe basis of a two-dimensional $E'$ irreducible representation of\n$D_{3h}$. General group theory arguments then guarantee that the\nmatrix elements, $e\\langle \\chi_{+1}, S_z| X_-|\\chi_{-1}, S_z\\rangle =\ne\\langle \\chi_{-1}, S_z| X_+|\\chi_{+1}, S_z\\rangle = 2id\\neq 0$, where\n$X_{\\pm}\\equiv \\pm X + iY$ are the in-plane components of $\\bf R$,\nwhich also transform as the two-dimensional irreducible representation $E'$. Here $d$ is a\nreal number that is refereed to as spin electric-dipole coupling.\nIt follows that, due to these non-zero matrix ele\\-ments, an electric\nfield can cause transitions bet\\-ween two ground state wavefunctions of opposite\nchirality but with same $S_z$.\n\nThe observation of such electric-field induced transitions from one\nchiral state to another requires that the degeneracies between these\nstates be lifted. The anisotropic Dzyaloshinskii-Moriya (DM)\ninteraction plays a crucial role in that, it provides one possible\nmechanism that lifts the degeneracies between states of different\nchirality without mixing them, as shown in Fig.~\\ref{spin-elec}.\nMore in general, the presence of DM interaction provides a\nmechanism to control the size of quantum entanglement in magnetic\ntrimers as a function of the temperature and external magnetic\nfield\\cite{florez_JMMM_2012}. Experimentally the DM-induced\nsplitting in \\{Cu$_3$\\} is estimated to be small (approximately 0.5\nK\\cite{choi06}).\n\\begin{figure}[h]\n{\\resizebox{3in}{1.7in}{\\includegraphics{sp-el-transition.eps}}}\n\\caption{ (Color online) Schematic diagram of electric-field-induced transitions between states of\ndifferent chirality belonging to the spin $S= 1\/2$ ground-state manifold of a triangular\nantiferromagnet.\n$\\Delta$ represents zero-field splitting of the\nchiral states due to Dzyaloshinskii-Moriya interaction.}\n\\label{spin-elec}\n\\end{figure}\n\nRecently\\cite{islam2010} we have investigated the details of the\nelectronic properties of the \\{Cu$_3$\\} SMM, which is one of the\nmost promising triangular spin 1\/2 molecules where the spin-electric\neffect can be rea\\-lized. In particular, we introduced a scheme to\nevaluate the strength of spin electric-dipole coupling $d$ using\n{\\it ab-initio} methods. However, the value of the anisotropic\nDM-exchange constant interaction, which is responsible of the GS\nzero-field splitting, has not yet been calculated. The purpose of\nthis work is to calculate this splitting by {\\it ab-initio} me\\-thods.\nIn order to achieve this goal, we analyze the microscopic origin of\nthe DM interaction via a Hubbard model approach in the presence of\nspin-orbit integration, which is the correct minimal model to\ndescribe both spin and charge fluctuations of these strongly\ncorrelated electron systems. At half-filling and in the large\nHubbard $U$ limit, spin-dependent virtual ho\\-pping processes, induced\nby the spin-orbit interaction, give rise to an anisotropic exchange\ninteraction.\\cite{Trif2010} There is a close analogy with the\nisotropic Heisenberg exchange interaction obtained in second-order\nperturbation theo\\-ry in the spin-independent ho\\-pping perturbation.\nBeside elucidating the physical mechanism leading to the anisotropic\nDM exchange interaction, this approach provides a very convenient\nprescription on how to extract the DM exchange cons\\-tant from\nfirst-principle calculations, which we have ca\\-rried out for\n\\{Cu$_3$\\}.\n\nThis paper is organized as follows. In\nSec.~\\ref{sec:GeneralpropertiesoftheDMinteraction} we discuss the\ngeneral properties of the DM interaction. The Hubbard model approach\nfor calculating DM vector, adopted in this work, is discussed in\nSec.~\\ref{methods:3}. In\nSec.~\\ref{sec:AbinitiocalculationoftheDMvector} we discuss details\nof extracting Hubbard model parameters from our {\\it ab-initio}\ncalculations. In Sec.~\\ref{sec:othermethods} we discuss other\nmethods that are usually employed for calculating the DM vector.\nFinally in Sec.~\\ref{summary} we present a summary of our work.\n\n\\section{The Dzyaloshinskii-Moriya interaction in frustrated antiferromagnetic spin rings}\n\\label{DM_theory}\n\n\\subsection{General properties of the DM interaction}\n\\label{sec:GeneralpropertiesoftheDMinteraction}\n\n\nThe Dzyaloshinskii-Moriya (DM) interaction is an anisotropic\nexchange interaction resulting from the interplay of the Coulomb\ninteraction and the spin-orbit coupling in systems of low crystal\nsymmetry. The DM interaction is an important effect for many\nmagnetic systems and plays a crucial role in determining the\nzero-field splitting of energy levels. An anisotropic exchange\ninteraction of the form\n\\begin{equation}\n{\\bf D}_{12}\\cdot {\\bf S}_1 \\times {\\bf S}_2\\;,\n\\label{dmi}\n\\end{equation}\nwhich is linear in the spin-orbit interaction, was first put forward\nby Dzyaloshinskii on the basis of symme\\-try\nconsiderations.\\cite{Dzyaloshinskii_1958} Later\nMoriya\\cite{Moriya_prl_1960, Moriya1960} provided a mechanism for\nthis interaction by exten\\-ding Anderson's theory of\nsuperexchange\\cite{PW_Anderson_1959} to include the effect of\nspin-orbit coupling. Let us consider for simplicity two ``magnetic\nions\", $\\bf R$ and $\\bf R'$, each occupied by a single electron in\nthe ground state. Second-order perturbation theory in the ho\\-pping\nHamiltonian $H_t$ coupling the two sites gives rise to an isotropic\nantiferromagnetic interaction with exchange constant $J = 2 t_{\\bf R\nR'}^2\/U$, where $t_{\\bf R R'}$ is a spin-independent ho\\-pping\nintegral and $U$ is the energy required to transfer an electron from\n$\\bf R$ to $\\bf R'$. When spin-orbit interaction $H_{\\rm SOI}$ is\nincluded, similar second-order processes can gene\\-rate an anisotropic\nexchange interaction in the form of Eq.~(\\ref{dmi}), with $D \\sim\nt_{\\bf R R'} b_{\\bf R R'}\/U$ where $b_{\\bf R R'}$ is a SOI-induced\nor SOI-dependent ho\\-pping integral. To lowest-order, $b_{\\bf R R'}$\nis just the matrix elements of the $H_{\\rm SOI}$ between two\norbitals localized at $\\bf R$ and $\\bf R'$. This is the dominant\ncontribution to $D$. In case that at each site more than one orbital\n$|{\\bf R}, \\mu\\rangle\\;, \\mu = 1, 2, \\cdots $ plays a role,\nhigher-order terms such as $b_{\\bf R R'} = t_{\\bf R R'} \\langle {\\bf\nR}, \\mu |H_{\\rm SOI}|{\\bf R}', \\mu' \\rangle\/\\Delta E_{\\mu, \\mu'}$ are\npossible, making the corresponding $D$ effectively a third-order\ncoupling in the perturbations $H_t$ and $H_{\\rm SOI}$. It turns out\nthat $D \\simeq (\\Delta g\/g)$, where $g$ is the free-electron\ngyromagnetic ratio and $\\Delta g$ the deviation from $g$ induced by\nSOI.\\cite{Moriya1960}\n\nAs shown by Moriya, other terms linear in the SOI contribute to the\nanisotropic exchange of the form of Eq.~(\\ref{dmi}). The second most\nimportant contribution is also a second-order term resulting from\nSOI and {\\it direct} inter-atomic exchange interaction $J^{\\rm\nex}({\\bf R, R'})$. In antiferromagnetic crystals this term is\n$J^{\\rm ex}({\\bf R, R'})\/J$ times smaller than the second-order\ncontribution proportional to $t_{\\bf R R'} b_{\\bf R R'}$. Finally,\nthird-order contributions to $D$ include the ho\\-pping terms twice\nand the intra-atomic exchange constant $J_0$. They are $J_0\/U$\nsmaller than second-order terms.\n\nThe DM exchange vector $ \\bf D$ vanishes when the symmetry of the\ncrystal is high. This is the case, for example, when the point\nlocated halfway between the two magnetic ions in a unit cell is a\ncenter of inversion. In low-dimensional crystals where $D \\neq 0$,\nthe anisotropic exchange is typically the most important anisotropic\ncontribution between spins. The DM interaction favors non-collinear\nspin configurations, with typical canted spins. As such, it\ndetermines the spin arrangements and it is responsible for the weak\nferromagnetism observed in some predo\\-minantly antiferromagnetic\ncrystals such as $\\alpha$-Fe$_2$O$_2$. The tendency toward canted\nspin configurations is most-easily seen by minimizing the energy in\nEq.~(\\ref{dmi}) for two classical spins, when the DM vector $\\bf D$\nis, for example, perpendicular to the line joining the two ions. It\ncan be shown that the minimum ener\\-gy corresponds to a spin\nconfiguration where both spin are perpendicular to each other and to\nthe direction of $\\bf D$. Similar conclusions can be obtained by\nanalyzing the same system quantum mechanically. The DM interactions\nis also responsible for proposed non-collinear spin configurations\nin magnetic clusters engineered by STM techniques on insulating\nsurfaces.\\cite{cyrus07,lichtenstein09}\n\n\n\\subsection{The DM interaction for antiferromagnetic spin rings within a Hubbard model approach}\n\\label{methods:3}\n\n\nIn this section we specialize the previous discussion to the\ncase of an antiferromagnetic spin triangle,\nand show how the DM interaction can be derived microscopically from\na Hubbard model at half filling, in the presence of spin-orbit\ninteraction.\n\nAs mentioned in the introduction, the low-energy magnetic properties\nof \\{Cu$_3$\\} are well-described by an isotropic antiferromagnetic\nHeisenberg model\n\\begin{equation}\nH_{\\rm H} = \\sum_{\\left\\langle i,j \\right\\rangle } J_{ij}{\\bf\ns}_i \\cdot {\\bf s}_j\\;, \\ \\ \\ J_{ij}>0\\;, \\label{eq:afmh}\n\\end{equation}\nwhere ${\\bf s}_i$ are spin vector operators of magnitude $s_i =1\/2$,\npredominately localized at the three Cu sites. If the small\ndistortion from a perfect equilateral arrangement of the three Cu\natoms is neglected, the three exchange constants are the same,\n$J_{ij} = J$. DFT calculations\\cite{islam2010} find $J\\approx $ 3.7\nmeV. The GS manifold comprises two spin $S=1\/2$ doublets, which can\nbe represented by the two chiral states given in\nEqs.~(\\ref{eq:chi_pm_sz+}),~(\\ref{eq:chi_pm_sz-}), or any two\northogonal linear combination of these. The spin $S=3\/2$\nexcited-state multiplet is separated by the GS by an energy of order\n$J$.\n\nIt is well-known that the AFM Heisenberg model represents an\neffective low-energy spin model that can be derived from an\nunderlying Hubbard model at half-filling in the large $t\/U$ limit.\nThe choice of the best minimal model capturing the essential\nmicroscopic features of the electronic system is often a complex\ntask, particularly when the exchange interaction between the\nmagnetic ions is mediated via several paths involving non-magnetic\nions, as for the case of \\{Cu$_3$\\}. We will neglect these\ncomplications and assume that an effective one-band Hubbard model\nsuffices for this purpose. We will see that our first-principles\ncalculations corroborate this choice, showing that one localized orbital at\neach magnetic ion indeed is enough to describe the low energy physics of the system.\nWe will comment later on the possibility of considering a more\ncomplex Hubbard model to describe the non-magnetic bridges between\nCu atoms, as well as the need of including more than one orbital at\nthe Cu sites.\n\nThe second quantized one-band Hubbard Hamiltonian reads\n\\begin{equation}\nH_U = -t\\sum_{i,j }\\sum_{\\scriptstyle \\alpha}\n\\Big\\{c_{i\\alpha}^{\\dagger}c_{j\\alpha}^{\\phantom{\\dagger}}+\n{\\rm h. c.}\\Big\\} +\\frac{1}{2}U\\sum_i n_{i\\uparrow}\\, n_{i\\downarrow}\\;,\n\\label{eq:Hu}\n\\end{equation}\nwhere\n$c_{i\\alpha}^{\\dagger}$\n($c_{i\\alpha}^{\\dagger}$) creates (destroys) an electron with spin\n$\\alpha$ at site $i$ and $n_{i\\alpha}=c_{i\\alpha}^{\\dagger}c_{i\\alpha}$ is the particle\nnumber operator. More precisely the index $i$ labels a Wannier function localized at site $i$.\nThe first term represents the kinetic energy, characterized by a spin-independent ho\\-pping parameter $t$,\nwhich is the same for all pairs of site due to the $C_3$ symmetry of the Cu$_{3}$\nmolecule magnet.\nThe second-term is an on-site repulsion energy of strength $U$,\nwhich has an effect only when two electrons of opposite spins reside\non the same site. It is the on-site repulsion energy.\n\nThe spin-orbit interaction in the Hubbard model is described by\nadding the following spin-dependent ho\\-pping term~\\cite{friedel_1964, kaplan_1982, bonesteel1992, Trif2010}\n\\begin{equation}\nH_{\\rm SOI} = \\sum_{i,j}\\sum_{\\scriptstyle \\alpha, \\beta}\n\\Big\\{ c_{i\\alpha}^{\\dagger}\\Big(i\\frac {{\\bf P}_{ij}}{2}\n\\cdot {\\pmb{ \\sigma}}_{\\alpha\\beta}\\;\\Big) c_{j\\beta}^{\\phantom{\\dagger}}+\n{\\rm h. c.}\\Big\\}\\;,\n\\label{eq:hubbard_soi_gen}\n\\end{equation}\nwhere $\\pmb{\\sigma}$ is the vector of the three Pauli matrices.\nHere the vector ${\\bf P}_{ij}$ is proportional to the matrix element\nof $ {\\pmb{ \\nabla}} V\\times {\\bf p}$ between the orbital parts of\nthe Wannier functions at sites $i$ and $j$; $V$ is the one-electron\npotential and $\\bf p$ is the momentum operator. Clearly the\nspin-orbit term has the form of a spin-dependent ho\\-pping, which is\nadded to the usual spin-independent ho\\-pping proportional to $t$.\nThis form of the spin-orbit interaction is a special case of\nMoriya's ho\\-pping terms\\cite{Moriya1960} in the limit that all but\none orbital energy is taken to infinity.\\cite{kaplan_1982}\n\nIn contrast to the spin-independent ho\\-pping term, the spin-depending\nho\\-pping parameters are related by both the full symmetry of the\nmolecule and the {\\it local symmetry} of localized\norbitals.\\cite{Trif2010} Now, because of the $\\sigma_v$ symmetry,\n${\\bf P}_{ij} = P {\\bf e}_z$. The final expression of the Hubbard\nmodel, including the spin-orbit interaction is\n\\begin{eqnarray}\nH_{U+{\\rm SOI}} &=& \\sum_{\\scriptstyle i,\\alpha}\n\\Big\\{ c_{i\\alpha}^{\\dagger}\\big (-t + i\\lambda_{\\rm SOI}\\alpha\\big )\nc_{i+1\\alpha}^{\\phantom{\\dagger}}+\n{\\rm h. c.}\\Big\\} \\nonumber \\\\\n&& + \\frac{1}{2}U \\sum_{\\scriptstyle i}n_{i\\uparrow}\\, n_{i\\downarrow}\\;,\n\\label{eq:hubbard_soi}\n\\end{eqnarray}\nwhere $\\lambda_{\\rm SOI} \\equiv P\/2 = {\\bf P}_{ij}\/ 2 \\cdot {\\bf e}_z$ is the\nspin-orbit parameter.\n\nWe want to treat the two ho\\-pping terms perturbatively on the same\nfooting, by doing an expansion around the atomic limit $t\/U\\;, \\\n\\lambda_{\\rm SOI}\/U \\to 0$. In many molecular magnets $t\n\\gg\\lambda_{\\rm SOI}$. This turns out to be the case also for\n\\{Cu$_3$\\}. In other molecules the two ho\\-pping parameters are of the\nsame order of magnitude.\n\nWe are interested in the half-filling regime. We\nknow that second-order perturbation theory in $t$ results in an\nantiferromagnetic isotropic exchange term that splits the spin\ndegeneracy of the low-energy sector of the Hubbard model, defined by\nthe singly-occupied states. This action can be represented with an\neffective spin Hamiltonian, the isotropic Heisenberg model, with\nexchange constant $J = 4t^2\/|U|$.\\cite{Fradkin1991} Similarly Loss\n{\\it et al}. showed that another second-order term proportional to\n$t\\lambda_{\\rm SOI}\/U$ generates an anisotropic exchange term that\ncan be identified with the DM interaction.\\cite{Trif2010} They write\napproximate adapted many-body states to first-order in the\nperturbation $|t|,\\lambda_{\\rm SOI} \\ll U$, corresponding to\nsingly-occupied states. In particular there are two independent\ndoublets,\n\\begin{equation}\n|\\psi_{E'_{\\pm}}^{1\\, \\alpha}\\rangle =\n\\frac{1}{\\sqrt{3}}\\big (|\\downarrow \\uparrow \\uparrow\\rangle +\n\\epsilon_{\\pm} |\\uparrow\\downarrow\\uparrow\\rangle + \\epsilon_{\\mp} |\\uparrow\\uparrow\\downarrow\\rangle\\big )\\;,\n\\label{eq:chiral_h_p}\n\\end{equation}\nand\n\\begin{equation}\n|\\psi_{E'_{\\pm}}^{1\\, \\alpha}\\rangle=\n\\frac{1}{\\sqrt{3}}\\big (|\\uparrow\\downarrow \\downarrow \\rangle +\n\\epsilon_{\\pm} |\\downarrow\\uparrow\\downarrow\\rangle + \\epsilon_{\\mp} |\\downarrow\\downarrow\\uparrow\\rangle\\big )\n\\;,\n\\label{eq:chiral_h_m}\n\\end{equation}\nwith $\\epsilon_{\\pm}=\\exp\\left( \\pm 2\\pi i\/3 \\right)$.\nThese are states with $S=1\/2$ and $S_z = \\pm 1\/2$. These\nstates are formally identical to the chiral states given in Eqs.\n(\\ref{eq:chi_pm_sz+}) and (\\ref{eq:chi_pm_sz-}). Now, each of the\nterms appearing in these equations is a single Slater determinant\nobtained by three creation operators acting on the vacuum, e.g.,\n\\begin{equation}\n|\\uparrow \\uparrow\\downarrow\\rangle \\equiv\nc_{1\\,\\uparrow}^{\\dagger} c_{2\\,\\uparrow}^{\\dagger} c_{3\\,\\downarrow}^{\\dagger}\n|0\\rangle\\;.\n\\end{equation}\nThe states $|\\psi_{E'_+}^{1\\, \\alpha}\\rangle$ and $|\\psi_{E'_-}^{1\\,\n\\alpha}\\rangle$ are eigenstates of Hubbard Hamiltonian when $t=\n\\lambda_{\\rm SOI} =0$. The tunneling and SOI mix the singly-occupied\nand double-occupied states. The first-order correction is obtained\nby mixing in doubly-occupied states\n\\begin{eqnarray}\n|\\Phi_{E'_{\\pm}}^{1\\, \\alpha}\\rangle\n&\\equiv &|\\psi_{E'_{\\pm}}^{1\\, \\alpha}\\rangle\n+ \\frac {({\\epsilon_-} -1)(t\\pm \\alpha \\lambda_{\\rm SOI})}{\\sqrt{2}U}\n |\\psi_{E^{'1}_{\\pm}}^{2\\, \\alpha}\\rangle \\nonumber \\\\\n &&\n+ \\frac {3{\\epsilon_+} (t\\pm \\alpha \\lambda_{\\rm SOI})}{\\sqrt{2}U}\n |\\psi_{E^{'2}_{\\pm}}^{2\\, \\alpha}\\rangle\\;,\n\\label{eq:chiral_u1st}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n|\\psi_{E^{'1}_{\\pm}}^{2\\, \\alpha}\\rangle= \\frac{1}{\\sqrt{6}}\\sum_{i=1}^3 \\epsilon_{1,2}^{i-1}\\left(\n|\\psi_{i1}^{\\alpha}\\rangle +|\\psi_{i2}^{\\alpha}\\rangle\n \\right)\\;,\n\\end{equation}\nand\n\\begin{equation}\n|\\psi_{E^{'2}_{\\pm}}^{2\\, \\alpha}\\rangle = \\frac{1}{\\sqrt{6}}\\sum_{i=1}^3 \\epsilon_{1,2}^{i-1}\\left(\n|\\psi_{i1}^{\\alpha}\\rangle - |\\psi_{i2}^{\\alpha}\\rangle\n \\right)\\;,\n\\end{equation}\nwith $|\\psi_{ij}^{\\alpha}\\rangle = c_{i\\uparrow}^{\\dagger}\nc_{i\\downarrow}^{\\dagger} c_{j\\alpha}^{\\dagger}|0\\rangle$ ($i=1,2,3$\nand $j\\neq i$) representing the double-occupied sites.\n\nThe next step is to take the expectation value of the spin-orbit\npart of Eq.(\\ref{eq:hubbard_soi}) in these approximated states. The\nresult is\\cite{Trif2010}\n\\begin{equation}\n\\langle\\Phi_{E'_{\\pm}}^{1\\,\\alpha}|H_{\\rm SOI}|\\Phi_{E'_{\\pm}}^{1\\,\\alpha}\\rangle\n= \\pm \\frac {5 \\sqrt{3} \\lambda_{\\rm SOI} t} {2U} {\\rm sgn} (\\alpha)\\;.\n\\end{equation}\n\nNote that off-diagonal matrix elements of $H_{\\rm SOI}$ vanish; in\nother words, SOI splits but does not mix the chiral states.\n\nIn the small $t\/U \\;, \\ \\lambda_{\\rm SOI}\/U $ limit, we can resort\nto a spin-only description of the low-energy physics of the system.\nThe ground-state manifold (corresponding to the states\nEq.~(\\ref{eq:chiral_u1st})) is given by the two chiral spin states\nEqs.~(\\ref{eq:chi_pm_sz+}),~(\\ref{eq:chi_pm_sz-}).\n\nThe anisotropic DM spin exchange Hamiltonian in $D_{3\\text{h}}$\nsymmetry is given by~\\cite{Trif2010}\n\\begin{equation}\nH_{\\rm DM}= \\frac{iD_z}{2}\\sum_1^3(s^i_+\\; s^{i+1}_- - s^i_-\\;\ns^{i+1}_+)\n\\label{eq:dm_spin}\n\\end{equation}\n\nNow, in the low energy regime corresponding to a $D_{3 \\text h}$\nsymmetric molecule magnet, the spin-orbit interaction can be reduced\nto the effective form\n\\begin{eqnarray}\nH_{SOI}&=& \\Delta_{SOI} C_z S_z~\\;,\n\\label{eq:Hsoitriangle}\n\\end{eqnarray}\nwhere $\\Delta_{SOI}$ is the effective SOI coupling constant.\n\nThe DM interaction expressed in this form clearly shows that it\nsplits but does not mix the two chiral states.\\footnote{Note also\nthat the Hamiltonian in Eq.~\\ref{eq:dm_spin} does not mix the GS\nspin $S=1\/2$ manifold with the excited-state spin $S= 3\/2$ multiplet\nand leaves the latter unchanged.} The splitting is exactly\nproportional to $D_z$ and allows us, in the low-energy regime, to\nmake the identification\n\\begin{equation}\nD_z = \\frac {5 \\lambda_{\\rm SOI} t} {U}~.\n\\label{eq:Dz}\n\\end{equation}\n\nThis Hubbard model analysis suggests an avenue to extract the DM\nparameters from an ab-initio calculation. Only three parameters are\nneeded, namely the spin-orbit interaction $\\lambda_{SOI}$, the\nho\\-pping parameter $t$ and the on-site repulsion energy $U$.\n\n\n\\subsection{Semiclassical analysis of the DM interaction in frustrated spin systems}\n\\label{sec:SemiclassicalanalysisoftheDMinfrustratedspinsystems}\nThe quantum mechanical frustration present in an antiferromagnetic\nspin triangle and the DM interaction both tend to favor\nnon-collinear spin configurations. It is instructive to study their\ninterplay in a semiclassical approach, where non-collinearity is a\nmore intuitive concept.\n\nThe {\\it classical} Heisenberg model with energy functional given by\nEq.~(\\ref{eq:afmh}) has two degenerate ``ground states'', given by\nthe two non-collinear spin configurations shown in\nFig.~\\ref{fig:triangleSpin}.\nClassically these two states are the best way to by-pass the\nfrustration present for any collinear spin configuration in a\ntriangular antiferromagnet. Quantum mechanically the two\nnon-collinear spin configurations can be re\\-presented by the states\n\\begin{eqnarray}\n|\\psi_{{\\rm nc}\\, \\pm} \\rangle &=& \\Big[ (\\alpha_1\n|\\uparrow\\rangle_1 + \\beta_1|\\downarrow\\rangle_1 )\\otimes\n (\\alpha_2 |\\uparrow\\rangle_2 + \\beta_2|\\downarrow\\rangle_2 ) \\nonumber \\\\\n && \\otimes\n (\\alpha_3 |\\uparrow\\rangle_3 + \\beta_3|\\downarrow\\rangle_3 ) \\Big]\\;,\n\\label{eq:ncs}\n\\end{eqnarray}\nwhere $\\alpha= \\cos(\\theta\/2)$ and $\\beta=\\exp\\{ i\\phi\n\\}\\sin(\\theta\/2)$. Here $\\theta$ is the elevation angle and $\\phi$\nis the azimuth angle. The three spinors $(\\alpha_i\n|\\uparrow\\rangle_i + \\beta_i|\\downarrow\\rangle_i )\\;, \\ i = 1, 2,3$\nare three spin-$1\/2$ coherent states defined by the three\nnon-collinear directions obtained by rotating consecutively by the\nangle $\\pm 240^0$ (see Fig. \\ref{fig:triangleSpin}). Anticlockwise rotations (by $- 240^0$)\ndefine a left-handed helical state (Fig. \\ref{fig:Lefthanded}); clockwise\nrotations (by $+ 240^0$) define a right-hand helical state (Fig.\n\\ref{fig:Righthanded}).\n\\begin{figure}[h]\n\\subfigure[Left-handed]{\\label{fig:Lefthanded}{\\resizebox{1.5in}{1.5in}{\\includegraphics{triangleSpin.eps}}}}\n\\subfigure[Right-handed]{\\label{fig:Righthanded}{\\resizebox{1.5in}{1.5in}{\\includegraphics{triangleSpin2.eps}}}}\n\\caption{Non-collinear helical system of three Cu-atom spins. Anticlockwise rotations (by $- 240^0$)\ndefine a left-handed helical state (a); clockwise\nrotations (by $+ 240^0$) define a right-hand helical state (b).} \\label{fig:triangleSpin}\n\\end{figure}\n\nIn contrast to the true GS given in Eqs.~(\\ref{eq:chi_pm_sz+}),~(\\ref{eq:chi_pm_sz-})\nthe non-collinear states defined in Eq.~(\\ref{eq:ncs})\nare neither eigenstates of the quantum Hamiltonian\nEq.~(\\ref{eq:afmh}) nor of ${\\bf S}^2$ and $S_z$. The expectation\nvalue of the Hamiltonian $H_H$ at these states is defined by\n\\begin{equation}\n\\langle\\psi_{{\\rm nc}\\, \\pm} | H_{\\rm H}| \\psi _{{\\rm nc}\\, \\pm}\n\\rangle= 3J\/4\\;. \\label{eq:e_ncs}\n\\end{equation}\n\nThe fact that the energy of the collinear states is higher than the\nenergy of the chiral states by $3J\/8$ is not surprising, since the\nnoncollinear states defined in Eq.~(\\ref{eq:ncs}) are a mixture of\n$S=1\/2$ and $S=3\/2$ components.\n\nWhen rewritten in term of the electronic states for the\ncorresponding Hubbard model at half-filling in the small $t\/U$\nlimit, the non-collinear spin-coherent states defined in\nEq.~(\\ref{eq:ncs}) can be considered to be the ``best'' energy states\ngiven by a single Slater determinant (Note that the chiral states\ncannot be written as a single Slater determinant).\n\nIt is now interesting to examine the effect of the DM interaction on\nthese states. A straightforward calculations shows that for the DM\ninteraction of Eq.~(\\ref{eq:dm_spin}), where only the $z$-component\nof ${\\bf D}$ is nonzero\n\\begin{equation}\n\\langle\\psi_{{\\rm nc}\\, \\pm} | H_{\\rm DM}| \\psi _{{\\rm nc}\\, \\pm}\n\\rangle = \\pm \\frac{3}{4} \\frac{\\sqrt{3}}{2} D_z\\;.\n\\label{eq:ev_dm_ncs}\n\\end{equation}\n\nTherefore, as for the GS manifold of the exact eigenstates, the DM\ninteraction {\\it splits} but does {\\it not couple} the two\nnoncollinear states.\nThe DM para\\-meter $D_z$ is, by Eq.~(\\ref{eq:ev_dm_ncs}), related to\nthe DM interaction-induced energy-gap between the two noncollinear states\n\\begin{eqnarray}\n\\Delta E_{\\rm nc} &=& \\langle\\psi_{{\\rm nc}\\, +} | H_{\\rm DM}| \\psi\n_{{\\rm nc}\\, +} \\rangle -\n\\langle\\psi_{{\\rm nc}\\, -} | H_{\\rm DM}| \\psi _{{\\rm nc}\\, -} \\rangle \\nonumber \\\\\n&=& \\frac{3\\sqrt{3}}{4} D_z\\;.\n\\label{eq:ncs_gap}\n\\end{eqnarray}\n\nThis result suggests a way of extracting the DM vector parameter\n${\\bf D}$ similar in spirit to the method used to calculate the\nisotropic exchange parameter $J$ by comparing the energy difference\nof states with ferromagnetic and antiferromagnetic spin\nconfigurations respectively. In the next section we will see that\nthis procedure can also be carried out by first-principle methods.\n\n\\section{Ab-initio calculation of the DM vector}\n\\label{sec:AbinitiocalculationoftheDMvector}\n\nAll the calculations in this work are carried out by using {\\it ab\ninitio} package NRLMOL \\cite{Pederson1990, Jackson1990}, which uses a\nGaussian basis set to solve the Kohn-Sham equations within PBE-GGA\napproximation.\\cite{Perdew} For more computational details and the\nelectronic properties of \\{Cu$_3$\\} we refer the reader to our\nprevious work.\\cite{islam2010}\n\n\n\\subsection{Calculation of the ho\\-pping term $t$}\n\\label{sec:tCalculation}\n\nAs discussed in the section \\ref{methods:3}, the Hubbard model approach is\nbased on allowing the localized electrons to hop to its nearest\nneighbor sites and in the present case of the \\{Cu$_3$\\} molecule, these\nlocalized electrons are $d$ electrons. Therefore, for calculating\nho\\-pping parameter $t$, the relevant states are those $d$ electron\nstates that lie close to the Fermi level. Let $|K,\\alpha\\rangle$ be\nthe three relevant Kohn-Sham eigenstates calculated from NRLMOL. We\ncan write them as a linear combination of the localized atomic\norbitals, centered at the three Cu sites, $\\{ \\left| \\phi_a\n\\right\\rangle, \\left| \\phi_b \\right\\rangle , \\left| \\phi_c\n\\right\\rangle\\}\\otimes \\left| \\chi_{\\alpha} \\right\\rangle $, with\n$\\alpha=\\uparrow,\\downarrow$ for spin up and down, respectively:\n\n\\begin{equation}\n\\left|K,\\alpha \\right\\rangle = \\sum_{i} C^i_{K\\alpha} \\left| \\phi_i \\right\\rangle \\left| \\chi_{\\alpha} \\right\\rangle\\;.\n\\label{eq:lc}\n\\end{equation}\nwhere $C^i_{K\\alpha}$ is the weight of the localized $\\left| \\phi_i \\right\\rangle \\left| \\chi_{\\alpha} \\right\\rangle$ wavefunction.\n\nFor the $\\left| \\uparrow \\uparrow \\uparrow \\right\\rangle$ spin\nconfiguration, in the absence of spin-orbit interaction, the\nrelevant three levels around the Fermi level are doubly and singly\ndegenerate. These levels are sketched in Fig. \\ref{fig:levels}\n\\begin{figure}[h]\n{\\resizebox{2.0in}{1.0in}{\\includegraphics{levels.eps}}}\n\\caption{Schematic diagram of the Kohn-Sham energy levels around the Fermi level}\n \\label{fig:levels}\n\\end{figure}\n\nWe obtain the level structure by diagonalizing the three-site\nHamiltonian in the absence of the SOI:\n\\begin{equation}\nH_0 = \\varepsilon_0 \\sum_{i} \\left| \\phi_i \\right\\rangle \\left\\langle \\phi_i \\right| -t\\sum_{i \\neq j } \\left| \\phi_i \\right\\rangle \\left\\langle \\phi_j \\right|\\;,\n\\label{eq:H0}\n\\end{equation}\nwhere $\\varepsilon_0$ is the onsite energy, $t$ is the ho\\-pping term\nand $i,j=a,b,c$ represent the copper sites. We get the eigenvalues\n$\\varepsilon_0+t$ and $\\varepsilon_0-2t$ for the two-fold and one-fold\ndegenerate states, respectively. The Kohn-Sham eigenvectors can be\ndefined as a linear combination of the localized wavefunctions,\n\\begin{eqnarray}\n\\left|E_1,\\uparrow \\right\\rangle &=& \\frac{1}{\\sqrt{2}}\\left( \\left| \\phi_a \\right\\rangle -\\left| \\phi_b \\right\\rangle \\right)\\left|\\uparrow \\right\\rangle \\;,\\nonumber \\\\\n\\left|E_2,\\uparrow \\right\\rangle &=& \\frac{1}{\\sqrt{6}}\\left( \\left|\n\\phi_a \\right\\rangle +\\left| \\phi_b \\right\\rangle -2 \\left| \\phi_c\n\\right\\rangle \\right)\\left|\\uparrow\\right\\rangle \\;,\n\\label{eq:evectors} \\\\\n\\left|A,\\uparrow \\right\\rangle &=& \\frac{1}{\\sqrt{3}}\\left( \\left|\n\\phi_a \\right\\rangle +\\left| \\phi_b \\right\\rangle +\\left| \\phi_c\n\\right\\rangle \\right)\\left|\\uparrow \\right\\rangle\\;. \\nonumber\n\\end{eqnarray}\n\nNow the localized states can be written in term of the Kohn-Sham functions\n\\begin{eqnarray}\n\\left|\\phi_a \\right\\rangle \\left|\\uparrow \\right\\rangle&=&\n \\frac{\\left| A, \\uparrow \\right\\rangle}{\\sqrt{3}}\n+\\frac{\\left| E_1,\\uparrow \\right\\rangle}{\\sqrt{2}}\n+\\frac{\\left| E_2,\\uparrow \\right\\rangle}{\\sqrt{6}}\\;, \\nonumber \\\\\n\\left|\\phi_b \\right\\rangle \\left|\\uparrow \\right\\rangle&=&\n \\frac{\\left| A, \\uparrow \\right\\rangle}{\\sqrt{3}}\n-\\frac{\\left| E_1,\\uparrow \\right\\rangle}{\\sqrt{2}}\n+\\frac{\\left| E_2,\\uparrow \\right\\rangle}{\\sqrt{6}}\\;,\n\\label{eq:linearKohnSham} \\\\\n\\left|\\phi_c \\right\\rangle \\left|\\uparrow \\right\\rangle&=&\n \\frac{\\left| A, \\uparrow \\right\\rangle}{\\sqrt{3}}\n-2\\frac{\\left| E_2,\\uparrow \\right\\rangle}{\\sqrt{6}}\\;. \\nonumber\n\\end{eqnarray}\n\nOur calculations showed that these states are primarily localized on\nthe Cu atoms and have $d$ character. We have obtained the Kohn-Sham\neigenenergies for the one-fold and two-fold degenerate states\n\\begin{eqnarray}\n\\left\\langle E_1,\\uparrow \\right|H_0 \\left|E_1,\\uparrow \\right\\rangle &=&\n\\frac{1}{2}\\left( \\left\\langle \\phi_a \\right| -\\left\\langle \\phi_b \\right| \\right) H_0\n\\left( \\left| \\phi_a \\right\\rangle -\\left| \\phi_b \\right\\rangle \\right) \\nonumber \\\\\n&=&\\varepsilon_0+t\\;, \\nonumber \\\\\n\\left\\langle A,\\uparrow \\right| H_0 \\left|A,\\uparrow \\right\\rangle &=&\n\\frac{1}{3}\\left( \\left\\langle \\phi_a \\right| +\\left\\langle \\phi_b \\right| +\\left\\langle \\phi_c \\right| \\right)H_0\n\\nonumber \\\\\n&&\\left( \\left| \\phi_a \\right\\rangle +\\left| \\phi_b \\right\\rangle +\\left| \\phi_c \\right\\rangle \\right)\n\\nonumber \\\\\n&=&\\varepsilon_0-2t \\;. \\label{eq:system}\n\\end{eqnarray}\n\nFrom Eqs.~(\\ref{eq:system}) we can finally evaluate the value of the parameter $t$ as:\n\\begin{eqnarray}\nt&=&\\frac{1}{3}\\left( \\left\\langle E_1,\\uparrow \\right|H_0 \\left|E_1,\\uparrow \\right\\rangle-\\left\\langle A,\\uparrow \\right| H_0 \\left|A,\\uparrow \\right\\rangle \\right) \\nonumber \\\\\n&=&50.84 \\text{meV} \\;. \\label{eq:t}\n\\end{eqnarray}\n\n\n\\subsection{Calculation of the spin-orbit interaction parameter $\\lambda_{SOI}$}\n\\label{sec:lambdaSOCalculation}\n\nStandard spin-orbit interaction representation for spherical systems\nis given by\n\\begin{equation}\nU_{so}(r,{\\textbf L},{\\textbf S}) = \\frac{1}{2c^2} {\\textbf S} \\cdot {\\textbf L} \\frac{1}{r} \\frac{d{\\bf \\Phi} (r)}{dr}\\;,\n\\label{eq:cso}\n\\end{equation}\nwhere $r$ is the position, ${\\textbf L}$ is the angular momentum,\n${\\textbf S}$ is the spin moment, $c$ is the speed of light, and\n$\\Phi$ is a spherically symmetric potential. The above\nequation is exact for spherical systems. For a multicenter system a superposition of such terms needs to be considered.\nHowever, this\napproximation could miss non-spherical correlations important for\nanisotropic energies. Instead of using Eq.~(\\ref{eq:cso}), a\ngeneralization of the spin-orbit interaction for non-spherical or\nmulticenter systems is given by\n\\begin{equation}\nU_{so}({\\textbf r},{\\textbf p},{\\textbf S}) = - \\frac{1}{2c^2} {\\textbf S} \\cdot {\\textbf p} \\times {\\bf \\nabla\\Phi({\\textbf r})}\\;,\n\\label{eq:qso}\n\\end{equation}\nwhere $\\textbf p$ is the momentum operator and a external electric\nfield is given by ${\\textbf E}=-{\\bf \\nabla\\Phi}$.\n\nPederson {\\it et. al} (see Ref. \\onlinecite{Pederson1999}) have\nshown an exact simplified method for incorporating spin-orbit\ncoupling into density-functional calculations. In order to get the\nbasis-set for the spin-orbit coupling the single-electron wave\nfunction can be expressed as\n\\begin{equation}\n\\psi_{is}({\\textbf r}) = \\sum_{j\\alpha} C_{j\\alpha}^{is} f_{j}({\\textbf r}) \\chi_{\\alpha}\\;,\n\\label{eq:basisset}\n\\end{equation}\nwhere $ f_{j}({\\textbf r})$ is a spatial basis function,\n$\\chi_{\\alpha}$ is either a majority or minority spin spinor, and $\nC_{j\\alpha}^{is}$ are determined by effectively diagonalizing the\nHamiltonian matrix. In order to calculate the effect of the SOI (Eq.\n(\\ref{eq:qso})) it is necessary to calculate matrix elements of the\nform\n\\begin{eqnarray}\nU_{j\\alpha,k\\alpha'} &=& \\left\\langle f_j \\chi_{\\alpha} \\right| U({\\textbf r},{\\textbf p},{\\textbf S}) \\left| f_k \\chi_{\\alpha'} \\right\\rangle\n\\nonumber \\\\\n&=& \\sum_x \\frac{1}{i}\n\\left\\langle f_j \\right| V_x \\left| f_k \\right\\rangle\n\\left\\langle \\chi_{\\alpha} \\right| S_x \\left| \\chi_{\\alpha'} \\right\\rangle \\;,\n\\label{eq:matrixelements}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\left\\langle f_j \\right| V_x \\left| f_k \\right\\rangle = \\frac{1}{2c^2} \\left(\n\\left\\langle \\frac{df_j}{dz} \\right| { \\mathbf \\Phi} \\left| \\frac{df_k}{dy} \\right\\rangle-\n\\left\\langle \\frac{df_j}{dy} \\right| { \\mathbf \\Phi} \\left| \\frac{df_k}{dz} \\right\\rangle\n\\right) \\;.\n\\label{eq:Vx}\n\\end{equation}\n\nThe matrix elements for $V_y$ and $V_z$ are obtained by cyclical\npermutations of $x,y $ and $z$ in Eq.~(\\ref{eq:Vx}). This\nmethodology for the SOI matrix gives several advantages, namely, it\ndoes not require the determination of the electric field; it is\nspecially ideal for basis functions constructed from\nGaussian-type orbitals, Slater-type functions, and plane waves.\n\nWe are interested in the matrix elements in the localized basis-set,\nEq.~(\\ref{eq:linearKohnSham}):\n\\begin{eqnarray}\n\\left\\langle \\phi_i \\right| \\left\\langle \\chi_{\\uparrow} \\right| U_{so} \\left| \\phi_k \\right\\rangle \\left| \\chi_{\\uparrow} \\right\\rangle &=& - \\frac{1}{2c^2}\n\\left\\langle \\phi_i \\right| {\\textbf p} \\times {\\bf \\nabla\\Phi({\\textbf r})} \\left| \\phi_k \\right\\rangle\n\\nonumber \\\\\n&&\\cdot \\left\\langle \\chi_{\\uparrow} \\right| {\\textbf S} \\left| \\chi_{\\uparrow} \\right\\rangle \\nonumber \\\\\n&=& \\frac{1}{2i}\n\\left\\langle \\phi_i \\right| V_z \\left| \\phi_k \\right\\rangle \\nonumber \\\\\n&=& - \\frac{i}{2} p_{ik}^z \\equiv -i \\lambda_{SOI}\\;.\n\\label{eq:solbs}\n\\end{eqnarray}\n\nWe can write these matrix elements in the Kohn-Sham basis set\n\\begin{eqnarray}\n\\left\\langle \\phi_i \\right| \\left\\langle \\chi_{\\uparrow} \\right| U_{so} \\left| \\phi_k \\right\\rangle \\left| \\chi_{\\uparrow} \\right\\rangle &=&\n\\sum_{KK'}\\left( \\tilde{C}_{K\\uparrow}^i \\right)^* \\tilde{C}_{K'\\uparrow}^i \\nonumber \\\\\n&&\\times \\left\\langle K,\\uparrow \\right| U_{SOI} \\left| K',\\uparrow\n\\right\\rangle ~. \\label{eq:sokhbs}\n\\end{eqnarray}\n\nWe have obtained the matrix elements for the spin-orbit\ninteraction in the Kohn-Sham basis, $\\{ \\left| E_1 \\right\\rangle ,\n\\left|E_2 \\right\\rangle , \\left| A \\right\\rangle \\}\\otimes\n\\left| \\chi_{\\alpha} \\right\\rangle$ (Eq.~(\\ref{eq:sokhbs})), and\nused Eqs.~(\\ref{eq:linearKohnSham}) to obtain the matrix elements:\n\\begin{equation}\np^z=\n\\begin{pmatrix}\n0&0.85&0.85\\\\\n0.85&0&0.85\\\\\n0.85&0.85&0\\\\\n\\end{pmatrix}\\;.\n\\end{equation}\n\nFrom Eq.~(\\ref{eq:solbs}) we have $\\lambda_{SOI}=p_{ik}^z\/2=0.43$ meV.\n\n\\subsection{Calculation of the Hubbard $U$ and evaluation of $D_z$ and $J$}\n\\label{U-Dz-Calc}\n\nThe most common approach for calculating $U$ involves calculation of\nenergy, $E$, of the molecule with $N$, $N+1$ and $N-1$ electron and extracting\nU from the equation below,\n\\begin{eqnarray}\nU&=&E(N+1)+E(N-1)-2E(N) \\nonumber \\\\\n &=&[E(N+1)-E(N)]-[E(N)-E(N-1)] \\nonumber \\\\\n &=& A-I \\;.\n\\label{eq:Uconventional}\n\\end{eqnarray}\n\nIn the above equation A is (minus) the electron\naffinity\\footnote{Note that usually, the electron\naffinity is defined as $[E(N)-E(N+1)]$, where $E(N)]$ is the energy of the neutral system.}\nand I is the\nionization energy. For systems that are not closed shell, such as\nthose considered here, the $U$ value is essentially the second\nderivative of energy with respect to charge and it is possible to\ndetermine $U$ by calculating the energy as a function of charge.\n\nFor the single-band Hubbard-model corresponding to the \\{Cu$_3$\\}\nmolecule, we are interested in obtaining energies for the\ncharge-transfer excitations involving the transfer of a localized\n{\\it d}-electron on one copper site to a localized {\\it d}-electron\non another site. Specifically we wish to know the energy of $\\left|\nX \\right\\rangle\n=\\left|\\uparrow_a\\downarrow_a\\uparrow_c\\right\\rangle$ relative to\n$\\left| \\uparrow_a\\downarrow_b\\uparrow_c \\right\\rangle $. There are\na total of twelve charge-transfer excitations that can be made with\none-site doubly occupied and one electron on one of the other sites.\nFor the half-filled case of interest here, the energy difference\ndepends upon the electron affinity of the state on site $a$, the\nionization energy of the state on site $b$ and the residual\nlong-range coulomb interaction between the negatively charged\nelectron added to site $a$ and the positively charged hole that is\nleft behind on site $b$. Since site $b$ and site $a$ are equivalent,\nit follows that we simply need to calculate $U$ for any one of the\ncopper sites in the half filled case. A very rough estimate of the\ncharge transfer energy may be determined by calculating the PBE-GGA\nenergy of the Cu atom with an electron configuration of\n$1s^22s^23s^24s^2p^63p^63d^n$ with $n$=8,9,10. Using $n$=9 as the\nreference state, one finds a bare $U$ value of 13.76 eV which, after\naccounting for the particle-hole interaction\n($27.2116\/R_{Cu-Cu}=2.95 eV$, where $R_{Cu-Cu}=4.87$ Bohr is the\ndistance between magnetic centers) is shifted to 10.8 eV.\n\nIn the \\{Cu$_3$\\} molecule, we have chosen to calculate $U$\nquasi-analytically by gradually adding (or subtracting) a small\nfraction of electronic charge $\\delta q$ to one of the half-filled\nCu $d$-states. The energy of the system as a function of $\\delta q$\nis shown in Fig. \\ref{fig:Evsq}, where we can see that it can well be reproduced by a\nquadratic fitting curve.\nThe figure shows that, upon adding a fractional charge to a localized orbital, the\ntotal energy initially decreases, since the orbital energy is negative. Eventually, however, the\ncompeting Coulomb repulsion takes over and the net change in total energy for adding one electron\nto a localized orbital is positive. In contrast, with one extra electron delocalized throughout the molecule\nthe total energy is usually smaller than the energy of the neutral molecule.\n\\begin{figure}[h]\n{\\resizebox{3.2in}{2.2in}{\\includegraphics{U.eps}}}\n\\caption{(Color online) Dependence of the total energy on added fractional\ncharge $\\delta q$. The (blue) circle represent the results of NRLMOL calculations and the\ndashed (red) line represents a quadratic fit.}\n\\label{fig:Evsq}\n\\end{figure}\n\nThe difference in the energy of the system before and after adding a\nfraction of electronic charge $\\delta q$ is given by $\\Delta E =\nU_{eff} = U\\delta q^2 - e^2 \\delta q^2 \/ R_{\\text{Cu-Cu}}$, where\n$U=\\partial^2 E(q)\/\\partial q^2$. We have calculated the effective\nparameter $U_{eff}$ by setting $\\delta q = 1$:\n\\begin{eqnarray}\nU_{eff}&=& \\delta q^2 \\left( \\frac{\\partial^2 E(q)}{\\partial q^2}-\\frac{e^2}{R_{{\\text{Cu-Cu}}}} \\right) \\nonumber \\\\\n&=& 9.06 {\\text{ eV}} \\;, \\label{eq:ourU}\n\\end{eqnarray}\nwhere $E(q)=E_0+(U\/2)(q-q_0)^2$ with $E_0$ being a constant.\n\n\\subsubsection{Evaluation of $D_z$ and $J$}\nHaving calculated the parameters $t$, $\\lambda_{SO}$, and\n$U_{eff}$, we are now able to use Eq.~(\\ref{eq:Dz}) and evaluate the\nDzyaloshinskii-Moriya parameter $D_z$. We obtain\n\n\\begin{equation}\nD_z= 5 \\frac{\\lambda_{\\rm SOI} t} {U_{\\rm eff}}=0.01 {\\rm meV}\\;.\n\\end{equation}\n\nThis value of $D_z$ yields a small splitting of the chiral state, $\\Delta \\approx 0.02$ meV\n$\\approx 0.3$ K. Experimental estimates of the DM parameter\nfind a splitting 3-4 times larger than this value. Considering the\nsmallness of this energy and the uncertainty in the experimental\nmeasurements, the two estimates are consistent with each other. On\nthe other hand, it is also possible that part of the discrepancy\nbetween theory and experiment is due to the fact that other\nmechanisms, different from the DM interaction, contribute to the\nsplitting. In particular Ref.\\cite{furukawa_PRB_2007} pointed out\nthat small deformations of the triangular molecule can lift the\nchiral degeneracy and this contribution to the splitting could be\neven more important than the DM interaction. If this is indeed the\ncase, our results would imply that our method of computing the DM\nparameter is actually rather accurate.\n\nFrom a computational point of view, it is interesting at this point\nto evaluate the isotropic exchange constant $J$ from the Hubbard\nmodel perturbative approach, which gives\n\n\\begin{equation}\nJ = 4t^2\/U \\approx 1 {\\rm meV}.\n\\end{equation}\n\nThis estimate of $J$ is considerably closer to the experimental\nvalue of 0.5 meV than the value of 3.7 meV obtained by computing the\nenergy difference between states with ferromagnetic and\nantiferromagnetic spin configurations.\\cite{islam2010}\n\n\n\n\n\n\\subsection{Comparison with other methods}\n\\label{sec:othermethods}\n\n\nIn a recent work Takeda {\\it et. al.}\\cite{takeda2007} have used a\nnon-collinear approach to estimate the DM interaction. Instead of\nthe use of simple product functions, this work capitalizes on the\nuse of generalized orbitals which are composed of a linear\ncombination of both spinors with different and variable spatial\nfunctions. By using such a representation it is possible to develop\nsingle-determinants which are composed of a linear combination of\nthe chiral spin 1\/2 states and the non-chiral spin 3\/2 states. For\nexample, the states associated with the system depicted in Fig\n\\ref{fig:Lefthanded} would be represented according to:\\\\\n\\begin{eqnarray}\n\\left| \\psi_{nc\\pm} \\right\\rangle& = & \\left| X_{\\pm}^aX_{\\pm}^bX_{\\pm}^c \\right> \\nonumber \\\\\n & = & \\frac{1}{2\\sqrt{2}} \\left(\n \\left|\\uparrow\\uparrow\\uparrow\\right>\n\\pm i\\left|\\downarrow\\uparrow\\uparrow\\right>\n\\mp(-1)^{1\/6}\\left|\\uparrow\\downarrow\\uparrow\\right> \\right. \\nonumber \\\\\n&&\\left. \\mp(-1)^{5\/6}\\left|\\uparrow\\uparrow\\downarrow\\right>\n\\mp i\\left|\\downarrow\\downarrow\\downarrow\\right>\n-\\left|\\uparrow\\downarrow\\downarrow\\right> \\right. \\nonumber \\\\\n&&\\left.\n+(-1)^{1\/3}\\left|\\downarrow\\uparrow\\downarrow\\right>\n-(-1)^{2\/3}\\left|\\downarrow\\uparrow\\downarrow\\right> \\right) \\label{eq:noncollstates}\n\\end{eqnarray}\nwhere $X_{+}(\\theta,\\phi)=\\cos(\\theta\/2)\\left|\\uparrow \\right> +\\exp\\{i\\phi\\}\\sin(\\theta\/2)\\left|\\downarrow \\right>$ and\n$X_{-}(\\theta,\\phi)=\\sin(\\theta\/2)\\left|\\uparrow \\right> -\\exp\\{i\\phi\\}\\cos(\\theta\/2)\\left|\\downarrow \\right>$, with $\\theta=\\pi\/2$ and $\\phi=\\pi\/2,7\\pi\/2,-\\pi\/2$.\nThey further claim that $\\Delta E_{\\rm nc} = 3\\sqrt{3}\/4 D_z$\n(see Eq.~\\ref{eq:ncs_gap}) can be\nestimated by a perturbational treatment of the SOI, as follows\n\\begin{equation}\n\\Delta E_{\\rm nc} = \\langle\\psi_{{\\rm nc}\\, +} | H_{\\rm SOI}| \\psi\n_{{\\rm nc}\\, +} \\rangle - \\langle\\psi_{{\\rm nc}\\, -} | H_{\\rm SOI}|\n\\psi _{{\\rm nc}\\, -} \\rangle\\;,\n\\end{equation}\nwhere $H_{\\rm SOI}$ is the one-electron spin-orbit interaction.\nThese expectation values can be calculated by DFT.\n\nIt is clear from the expression Eq. (\\ref{eq:noncollstates}) that\nthe expectation value of the spin-orbit interaction for this and\nother states would be linear so, without other considerations, one\ncan not extract an interaction that depends upon the excitations of\ninterest to the Hubbard Hamiltonian. However, in analogy to the\nexpansion of the many-electron wavefunction for molecular hydrogen\nin regions intermediate between the bonding and separated-atom\nlimit, a self-consistent optimization of such a starting determinant\nallows the spin-orbitals to be intermediate between the doubly\noccupied and single occupied representations. While the resulting\nnoncollinear wavefunction is still a single Slater determinant in\ncharacter, expansion of the noncollinear state in terms of the\nHubbard states would show a wavefunction comprised primarily of the\n8$\\times$8 half-filled determinants but would also contain small\ncontributions of the ionic contributions which are shifted upward\nby $U_{eff}$. It is the small admixture of these states that allow\nTakeda {\\it et. al.} to extract both the exchange parameters and the DM\ninteraction through the use of noncollinear representations. This\napproach could have advantages from an operational viewpoint since\nit effectively addresses the potential role of other excited states\nthat are routinely excluded from the Hubbard Hamiltonian. However,\nthe precise interactions which ultimately mediate the appearance of\nthe DM interaction require additional analysis which is every bit as\narduous as that presented here.\n\n\n\nAn alternative method to calculate the DM vector, based on\nAndersen's ``local force theorem\" \\cite{andersen80}, was developed\nby Solovyev {\\it et al}.\\cite{solovyev96} More recently this method\nwas utilized in conjunction with DFT to study the DM interaction\nbetween magnetic atoms inserted in different crystalline systems and\nsurfaces. \\cite{anisimov05,lichtenstein09}\nEssentially this method expresses the DM vector in terms of the\nGreen's functions of the system, modified by the spin-orbit\ninteraction. Although computationally sophisticated, the Green's\nfunction method is physically less transparent than the one adopted\nhere, particularly for a finite system such a triangular SMM, where\nthe crucial ingredients leading to the anisotropic DM exchange can\nbe reduced to a few parameters that have a direct physical\ninterpretation within the Hubbard model.\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{summary}\n\n\nWe carried out a first-principles investigation of the zero-field\nsplitting of the chiral ground states of a \\{Cu$_3$\\}\nsingle-molecule magnet (SMM), caused by the Dzyaloshinskii-Moriya\ninteraction. Our approach relies on the perturbative analysis of a\nHubbard model, which includes spin-orbit interaction. In the large\n$U$ limit, appropriate for \\{Cu$_3$\\}, it is possible to express the\nDzyaloshinskii-Moriya constant in terms of the parameters that\ndefine the Hubbard model, such as the effective ho\\-pping integral\nbetween magnetic sites $t$, the on-site repulsion energy $U$, and\nthe strength of the spin-orbit $\\lambda_{\\rm SOI}$. We then carried\nout an approximate method to extract the values of these parameters\nfrom our spin density functional theory calculations of the SMM. The\nvalue of the Dzyaloshinskii-Moriya constant $D$ that we found is of\nthe order of $0.01$ meV, which is a factor of 5 smaller than the\nvalue measured experimentally. Given the uncertainty of the\nexperimental result and the fact that other effects might contribute\nto the zero-field spin splitting of the chiral states, our estimate\nshould be considered consistent with experiment.\n\nThe method of computing the DM parameter by effectively extending\nAnderson's theory of superexchange to include spin-orbit interaction\nis very close to Moriya's original formulation of anisotropic\nexchange. It is interesting to note that if we use this approach to\ncalculate the isotropic superexchange constant $J$ of the Heisenberg\nmodel describing \\{Cu$_3$\\}, we obtain a value that is closer to\nexperimental result than the estimates based on total energy\ncalculations of ferromagnetic vs antiferromagnetic spin\nconfigurations. This seems to suggest that this approach is not only\nphysically very intuitive, but it might also bear promise of good\nnumerical accuracy.\n\nWhile the methods discussed here provide physical insight into the\nnature of the DM interaction, we note that for future calculations it\nwould be desirable to consider excitations that are not normally\nincluded in the single-band Hubbard model. For such an approach it\nwould be necessary to include methodologies that allow for the\ncalculation of all excitations in such systems.\n\n\n\n\n\\section*{Acknowledgment}\n\\label{sec:Acknowledgment} This work was supported by the School of\nComputer Science, Physics and Mathematics at Linnaeus University,\nthe Swedish Research Council under Grants No: 621-2007-5019 and 621-2010-3761,\nand the NordForsk research network 080134 ``Nanospintronics: theory and simulations\".\nWe would like to thank D. Loss and D. Stepanenko for very a helpful explanation of their Hubbard model\napproach to the DM interaction in molecular antiferromagnets.\nThe early parts of this collaboration were\nsupported in part by NRL.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nComplex behavior is a fundamental trait of autonomous agents that often arises as a response to conflicting requirements. When learning these behaviors using a reinforcement learning~(RL) approach, the agent receives a reward signaling the level of attainment of each of the task requirements~\\cite{sutton2018reinforcement}. Typically, the ensuing rewards are weighted and combined resulting in a regularized multi-objective reinforcement learning problem~\\cite{mannor2004geometric}. While popular and sometimes effective, this approach suffers from several drawbacks, not the least of which is the fact that combination coefficients are problem-specific and must be manually selected. A time-consuming calibration process, which causes severe computational overhead~\\cite{mania2018simple,peng2018deepmimic}. \n\nAlternatively, requirements can be made explicit in a constrained reinforcement learning (CRL) problem \\cite{paternain2019constrained}. This is not altogether unrelated to regularized RL approaches because algorithms to solve CRL rely on Lagrangian dual formulations and Lagrangians \\emph{are} weighted combinations of rewards (Section \\ref{S:constrainedRL}). The difference is that dual methods incorporate rules to adjust the value of the multipliers, thereby eliminating the design overhead of their manual selection \\cite{bhatnagar2012online, chow2015risk, tessler2018reward, paternain2019safe,ding2020natural}. \n\n\\begin{figure}[t]\n \\centering\n\n\t\\includegraphics[scale=0.75]{img\/pdf\/example_real.pdf} \t\t\t\t\t\n\n\t\\includegraphics[scale=0.75]{img\/pdf\/example_mdp.pdf} \t\t\t\t\t\n\t\\caption{\\emph{Monitoring problem}. An agent must spend $c_1$ of its time at $R_1$ and $c_2$ of its time at $R_2$. Continuous state representation on the left and discrete state abstraction on the right. Solving this problem requires a state augmentation method we develop here. }\n \\label{fig:example}\n\\end{figure}\n\nHowever related, CRL and regularized RL formulations are not equivalent. In fact, the motivation for this paper is the following observation:\n\\vspace{-0.5\\baselineskip}\n\\begin{description}\n\\item[(O1)] CRL can solve problems regularized RL cannot solve.\n\\end{description}\n\\vspace{-0.5\\baselineskip}\nThe precise meaning of this statement is that there is a class of problems for which no regularization parameters enable learning a policy that satisfies the problem requirements. There is, moreover, nothing contrived about this class. A case in point is the \\emph{monitoring problem} illustrated in Figure~\\ref{fig:example}. The agent is tasked with monitoring different regions of the environment; it collects unit rewards whenever it visits a target region and it must spend a certain fraction of its time in each of them. No fixed combination of these rewards can ensure that the agent learns a policy that satisfies the monitoring requirements~(Example~\\ref{ex_monitoring}).\n\nThis observation foreshadows a central role for CRL in autonomy. Even simple tasks cannot be specified by formulations without constraints. It also casts some doubt on dual formulations given that Lagrangians are weighted combinations of rewards. This worry turns out to be well founded since the following is also true:\n\\vspace{-0.5\\baselineskip}\n\\begin{description}\n\\item[(O2)] Dual methods do not, in general, converge to optimal CRL policies.\n\\end{description}\n\\vspace{-0.5\\baselineskip}\nThe monitoring problem of Figure~\\ref{fig:example} is an example of a problem for which dual methods fail (Sections \\ref{sec_policy_switching} and \\ref{sec_primal_dual}). Indeed, as dual methods rely on the solution of regularized (weighted) problems, the inability to obtain solutions to such problems poses a fundamental obstacle to solving CRL problems using dual descent methods (Section \\ref{sec_primal_recovery}).\n\nObservation (O1) highlights the fundamental importance of CRL and Observation (O2) the lack of an algorithm that can provably solve CRL. The main contribution of this paper is the following:\n\\vspace{-0.5\\baselineskip}\n\\begin{description}\n\\item[(C1)] We develop a state augmentation procedure that is guaranteed to solve CRL (Section \\ref{sec_algorithm}).\n\\end{description}\n\\vspace{-0.5\\baselineskip}\nThe proposed state augmentation is based on reinterpreting Lagrangian weights as part of the state space of the Markov decision process (MDP). As such, this procedure is systematic and independent of the problem instance. Contribution (C1) hinges upon the following technical contribution:\n\\vspace{-0.5\\baselineskip}\n\\begin{description}\n\\item[(C2)] We prove that dual gradient descent dynamics generate \\emph{trajectories} that are almost surely feasible and have expected costs that are near-optimal (Theorem \\ref{theo_main_intro}). \n\\end{description}\n\\vspace{-0.5\\baselineskip}\nContribution (C2) is the technical result that makes contribution (C1) possible. It claims convergence of \\emph{trajectories} that are generated by dual gradient descent dynamics, but it does not claim convergence to an optimal policy. The latter can be shown impossible through counterexamples~[cf.~(O2)]. The fact that trajectories are feasible and near-optimal without converging to optimal policies is the reason why state augmentation is necessary to solve CRL~[cf.~(C1)]. This peculiar situation is a unique aspect of CRL that is due to the unusual combination of a problem in which (i) there is no duality gap despite presence of constraints and objectives that are non-convex (Section \\ref{sec_relationship}) and (ii) recovery of optimal primal variables from optimal dual variables is not always possible despite absence of duality gap (Section \\ref{sec_primal_recovery}).\n\n\\subsection{Related Work}\n\nMDPs and CRL have diverse applications ranging from highway management \\cite{golabi1982statewide} to smart grids \\cite{koutsopoulos2011control}, finance \\cite{krokhmal2002portfolio,di2012policy} and robotics \\cite{chow2015risk,gu2017deep,achiam2017constrained}. To tackle these problems, there are several distinct approaches. The most common are regularized approaches \\cite{di2012policy,peng2018deepmimic,tamar2013variance}, which attempt to approximate the constrained reinforcement learning problem by the maximization of an unconstrained Lagrangian with specific multiplier values \\cite{borkar2005actor}. Primal-dual approaches have also shown success in the past, providing a methodology to dynamically select the Lagrange multipliers \\cite{bhatnagar2012online,chow2015risk,tessler2018reward,paternain2019safe,ding2020natural}. \n\n\\section{Constrained Reinforcement Learning}\n\\label{S:constrainedRL}\n\nLet~$t \\in \\mathbb{N} \\cup \\{0\\}$ denote a time index and~$\\ensuremath{\\mathcal{S}} \\subset \\ensuremath{\\mathbb{R}}^n$ and~$\\ensuremath{\\mathcal{A}} \\subset \\ensuremath{\\mathbb{R}}^d$ be compact sets denoting the possible states and actions of an agent described by an MDP with transition probability $p\\left(s_{t+1} \\mid s_{t},a_t\\right)$. The agent chooses actions sequentially based on a policy~$\\pi(a|s)$. In a constrained MDP the hypothesis is that the action taken by the agent at each state results in the collection of several rewards defined by the functions $r_i: \\ensuremath{\\mathcal{S}} \\times \\ensuremath{\\mathcal{A}} \\to \\mathbb{R}$ for~$i = 0,\\ldots, m$. Thus, when the agent is in state $s_t$ and takes action $a_t$, it collects rewards $r_i(s_t,a_t)$. As in any MDP, the interest is on the rewards the agent accumulates over time which are represented by the value functions\n\\begin{align}\\label{eqn_value_function}\n V_i(\\pi) \\triangleq \\lim_{T\\to\\infty} \\frac{1}{T} \\E_{s,a\\sim\\pi} \\left[ \\sum_{t = 0}^T r_i(s_t,a_t) \\right] .\n\\end{align}\nTypically, the value functions $V_i(\\pi)$ are in conflict with each other. Policies $\\pi$ that are optimal for some $V_i$ are not good for some other $V_j$. We therefore interpret the first of these rewards, $i=0$ as an objective to maximize and the remaining rewards as \\emph{requirements} to satisfy. We then introduce the constrained optimization problem\n\\begin{subequations}\n\\begin{align*}\n\\tag{P-CRL}\\label{P:constrainedRL}\n \\max_{\\pi} \\enskip & \\lim_{T\\to\\infty} \\frac{1}{T}\\E_{s,a\\sim\\pi} \\left[ \\sum_{t = 0}^T r_0(s_t,a_t) \\right] \\\\\n \\sto \n\t\t\\enskip & \\lim_{T\\to\\infty} \\frac{1}{T} \\E_{s,a\\sim\\pi} \\left[ \\sum_{t = 0}^T r_i(s_t,a_t) \\right] \\geq c_i \\text{,} \\, i = 1, \\ldots, m\\text{.}\n\\end{align*}\n\\end{subequations}\nwhere the constant $c_i \\in \\ensuremath{\\mathbb{R}}$ represents the~$i$-th reward \\emph{specification}. We denote by $ \\Pi^\\star $ the optimal solutions of \\eqref{P:constrainedRL}. It is important for forthcoming discussions that $\\Pi^\\star$ is a set. There may be several policies $\\pi^\\star \\in \\Pi^\\star$ that satisfy the constraints and attain the optimal cost $P^\\star \\triangleq V_0(\\pi^\\star)$. Our goal is to find at least one element of this set.\n\nA customary approach to solving \\eqref{P:constrainedRL} is to consider a penalized version. Formally, introduce penalty coefficients $\\ensuremath{\\lambda}_i$ associated with each of the constraints in \\eqref{P:constrainedRL}\nand define the Lagrangian $\\ensuremath{\\mathcal{L}}(\\pi,\\ensuremath{\\lambda})$ as\n\\begin{align}\\label{eqn_lagrangian}\n \\ensuremath{\\mathcal{L}}(\\pi,\\lambda) & \\triangleq V_0(\\pi) + \\sum_{i=1}^m \\lambda_i \\Big(V_i(\\pi) - c_i\\Big) .\n\\end{align}\nIn lieu of the constrained optimization \\eqref{P:constrainedRL}, we settle for policies $\\Pi(\\ensuremath{\\lambda})$ that maximize the Lagrangian in \\eqref{eqn_lagrangian},\n\\begin{align}\\label{eqn_lagrangian_maximizers}\n \\Pi(\\lambda) \\triangleq \\argmax_{\\pi} V_0(\\pi) + \\sum_{i=1}^m \\lambda_i \\Big(V_i(\\pi) - c_i\\Big).\n\\end{align}\nThe advantage of replacing \\eqref{P:constrainedRL} with its Lagrangian form in \\eqref{eqn_lagrangian_maximizers} is that the latter is a plain, unconstrained MDP which we can solve with standard reinforcement learning algorithms. Indeed, define the weighted reward\n\\begin{align}\\label{eqn_lagrangian_reward}\n r_\\lambda(s_t,a_t) = r_0(s_t,a_t) + \\sum_{i=1}^m \\ensuremath{\\lambda}_i \\Big( r_i(s_t,a_t) - c_i \\Big).\n\\end{align}\nGiven this definition we can reorder terms in \\eqref{eqn_lagrangian_maximizers} to write $\\Pi(\\ensuremath{\\lambda})$ as the maximizer of the value function $V(\\pi,\\lambda)$ that is associated with this reward,\n\\begin{align}\\label{eqn_lagrangian_maximizers_rl} \n \\Pi(\\lambda) &= \n \\argmax_{\\pi} \\lim_{T\\to\\infty} \\frac{1}{T} \\E_{s,a\\sim\\pi} \\left[ \\sum_{t = 0}^T r_\\lambda(s_t,a_t) \\right]\\nonumber\\\\\n & \\triangleq \\argmax_{\\pi} V(\\pi,\\lambda). \\tag{R-CRL}\n\\end{align}\nIt is important for forthcoming discussions that $\\Pi(\\ensuremath{\\lambda})$ is a set. There may be several policies that maximize value function $V(\\pi,\\lambda)$. This is the same observation we made of \\eqref{P:constrainedRL}. We use $\\pi(\\ensuremath{\\lambda})$ to denote individual Lagrangian maximizing policies and $\\pi(s, \\ensuremath{\\lambda})$ to denote their evaluation at state $s$.\n\nThe drawback of using \\eqref{eqn_lagrangian_maximizers_rl} is that appropriate choice of the penalty coefficients $\\ensuremath{\\lambda}_i$ is difficult. Among other challenges, coefficients $\\ensuremath{\\lambda}_i$ that make \\eqref{eqn_lagrangian_maximizers_rl} and \\eqref{P:constrainedRL} close depend on the transition probability of the MDP, which is assumed unknown. We can circumvent this problem by adapting the $\\ensuremath{\\lambda}_i$ penalties. To do that, we introduce an iteration index $k$, a step size $\\eta_\\lambda$ and an epoch duration $T_0$. Coefficients $\\ensuremath{\\lambda}_i$ are updated as determined by the iteration, \n\\begin{align}\\label{eqn_stochastic_dual_descent_specified}\n \\lambda_{i, k+1} \n = \\left [ \\lambda_{i,k} - \n \\frac{\\eta_\\lambda}{T_0}\n \\sum_{t= kT_0}^{(k+1)T_0-1} \n \\Big( r_i(s_t,a_t)-c_i \\Big) \n \\right]_+\\hspace{-0.5ex},\\hspace{-0.5ex} \n\\end{align}\nwhere actions $a_t \\sim \\pi(s_t, \\ensuremath{\\lambda}_k)$ for some policy $\\pi(\\ensuremath{\\lambda}_k)\\in\\Pi(\\ensuremath{\\lambda}_k)$. According to this expression, time is separated in epochs of duration $T_0$. During the $k$-th epoch, the penalty coefficients are $\\ensuremath{\\lambda}_{k}$ and we consequently execute the policy $ \\pi(\\ensuremath{\\lambda}_k)$ between times $kT_0$ and $(k+1)T_0 -1$. During this epoch, the agent collects rewards $r_i(s_t,a_t)$ that are attributed to this policy. The penalty coefficient $\\ensuremath{\\lambda}_{i,k}$ is updated according to whether the accumulated reward exceeded the target $c_i$ or not. If the accumulated reward exceeds the target $c_i$, the penalty coefficient $\\ensuremath{\\lambda}_{i,k}$ is reduced. If the accumulated reward falls below target, the coefficient is increased. Beyond a sensible update rule, \\eqref{eqn_stochastic_dual_descent_specified} can be seen as a stochastic gradient descent update on the dual function. Leveraging this fact, this paper establishes the following theorem, the proof of which can be found in Appendix \\ref{sec_unbiased}. \n\n\\begin{theorem}\\label{theo_main_intro}\nAn agent chooses actions according to $a_t \\sim \\pi(\\lambda_k)$ with $\\lambda_k$ updates following \\eqref{eqn_stochastic_dual_descent_specified}. Assume that\n\\vspace{-0.5\\baselineskip}\n\\begin{description}\n\\item[(A1)] There exists a strictly feasible policy $\\pi^\\dagger$ such that $V_i(\\pi^\\dagger)-c_i \\geq C > 0$ for all constraints $i$.\n\\item[(A2)] There exists $B$ such that $|r_i(s,a) - c_i| \\leq B$ for all states $s$, actions $a$ and constraints $i$.\n\\item[(A3)] The accumulated reward in \\eqref{eqn_stochastic_dual_descent_specified} is an unbiased estimate of the value function $V_i(\\pi(\\ensuremath{\\lambda}_k))$,\n\\begin{align}\\label{eqn_unbiased}\n \\E\\left[\n \\frac{1}{T_0} \n \\sum_{t= kT_0}^{(k+1)T_0-1} \n r_i(s_t,a_t)\n ~\\Big |~ \\ensuremath{\\lambda}_k\n \\right] \n = V_i \\bigl(\\pi(\\ensuremath{\\lambda}_k)\\bigr).\n\\end{align}\n\\end{description}\n\\vspace{-0.5\\baselineskip}\nThen, the state-action sequences $(s_t,a_t)$ are feasible with probability one,\n\\begin{align}\\label{eqn_theo_feasibility_informal} \n \\lim_{T\\to\\infty} \\frac{1}{T}\\sum_{t=0}^{T-1} r_i(s_t,a_t) \\geq c_i, \n \\quad \\text{~a.s.}\n\\end{align}\nAnd they are near-optimal in the following sense\n\\begin{align}\\label{eqn_theo_optimality_informal}\n \\lim_{T\\to\\infty} \\mathbb{E}\\left[\\frac{1}{T} \\sum_{t=0}^{T-1} r_0(s_t,a_t) \\right]\\geq P^\\star-\\eta_\\lambda \\frac{B^2}{2}.\n\\end{align}\n\\end{theorem}\n\nTheorem \\ref{theo_main_intro} claims that if the penalty coefficients are \\emph{continuously updated} as in \\eqref{eqn_stochastic_dual_descent_specified} with actions being chosen as $a_t \\sim \\pi(\\ensuremath{\\lambda}_k)$, then the state-action \\emph{trajectory} satisfies the constraints in \\eqref{P:constrainedRL} and attains a reward that is within $\\eta_\\lambda B^2 \/ 2$ of optimal. This claim produces an \\emph{algorithm} whose execution results in actions that are feasible and nearly optimal in the long term. This is notable because \\eqref{P:constrainedRL} is not a convex optimization problem. Although not completely unexpected since recent results have shown that \\eqref{P:constrainedRL} has null duality gap \\cite{paternain2019constrained}. And indeed, a result analogous to Theorem \\ref{theo_main_intro} for discounted constrained MDPs is part of \\cite{ding2020natural}. \n\nThe assumptions made in Theorem \\ref{theo_main_intro} are customary. Assumption~(A1) merely asks for the existence of a feasible policy, while (A2) requires the rewards to be bounded, a necessary condition for producing feasible iterates. Assumption~(A3) is guaranteed if one can sample from the stationary distribution of the MDP. A common hypothesis in RL, specially for continuing tasks \\cite{sutton2018reinforcement}. However, it is possible that (A3) might not hold in practice. Appendix \\ref{app:sec_realizable} addresses this, replacing (A3) in a more technical variation of Theorem \\ref{theo_main_intro}, for which the same guarantees are shown to hold with a judicious choice of the time horizon $T_0$.\n\nHypotheses aside, there is however a caveat to Theorem \\ref{theo_main_intro}. While notable for what it claims, it is also notable for what it does \\emph{not} claim. Theorem \\ref{theo_main_intro} does not guarantee that we are able to find an optimal policy $\\pi^\\star$. It is not possible to stop the iteration in \\eqref{eqn_stochastic_dual_descent_specified} at some $k$ and claim that the policy $\\pi(\\ensuremath{\\lambda}_k)$ is close to optimal. This is a subtle yet fundamental issue because it precludes a claim that \\eqref{eqn_stochastic_dual_descent_specified} finds the optimal policy. Instead, \\eqref{eqn_stochastic_dual_descent_specified} generates \\emph{trajectories} that are feasible and near-optimal by the \\emph{continuous execution} of the primal iteration in \\eqref{eqn_lagrangian_maximizers_rl} and the dual iteration in \\eqref{eqn_stochastic_dual_descent_specified}. Section \\ref{sec_discussions} covers a simple example to further clarify this point.\n\nThe algorithmic implication is that to guarantee a solution of \\eqref{P:constrainedRL} we must choose the policies $\\pi(\\ensuremath{\\lambda}_k)$ in \\eqref{eqn_lagrangian_maximizers_rl} while we run the dual iteration in \\eqref{eqn_stochastic_dual_descent_specified}; see Algorithm~\\ref{algorithm_exec}. Then, the purpose of a CRL algorithm can be modified as learning the policies $\\pi(\\ensuremath{\\lambda}_k)$ that solve \\eqref{eqn_lagrangian_maximizers_rl}; see Algorithm~\\ref{algorithm}. Both of these algorithmic modifications are easiest to understand as a form of state augmentation as we explain in the following section. \n\n\n\\subsection{Augmented Constrained Reinforcement Learning} \\label{sec_algorithm}\n\nThe \\emph{State Augmented Constrained Reinforcement Learning (A-CRL)} algorithm reinterprets $\\ensuremath{\\lambda}_k$ as part of the state space. This is possible because the dual variable update in \\eqref{eqn_stochastic_dual_descent_specified} defines a Markov process. If we let $\\mathbf{s}_k$ and $\\mathbf{a}_k$ denote the states and actions that are observed during the $k$-th epoch, the update in \\eqref{eqn_stochastic_dual_descent_specified} defines a memoryless probability distribution,\n\\begin{align}\\label{eqn_dual_pd}\n \\lambda_{k+1} \\sim\n p(\\ensuremath{\\lambda}_{k+1} | \\ensuremath{\\lambda}_k, \\mathbf{s}_k, \\mathbf{a}_k). \n\\end{align}\nThis state augmentation is summarized in Figure \\ref{fig_blcok_diagram}. The top part of the diagram represents the transition probability of the given MDP. The bottom part of the diagram represents state augmentation with Lagrange multipliers that evolve according to the probability distribution in \\eqref{eqn_dual_pd} defined by the update in \\eqref{eqn_stochastic_dual_descent_specified}. Put together, we have a state \\emph{augmented} MDP whose state is made of of the \\emph{pair} $(s_t, \\ensuremath{\\lambda}_k)$. Observe that the time scales at which $s_t$ and $\\ensuremath{\\lambda}_k$ are updated are different. This is important in both theory (see Appendix) and practice (see Section \\ref{sec_example}).\n\n\\begin{figure}[t]\n \\centering\n \n\t\\includegraphics[scale=1]{img\/pdf\/block_diagram.pdf} \t\t\t\t\t \t\t\t\n\t\\caption{To solve CRL we must augment the state to incorporate multipliers $\\ensuremath{\\lambda}_k$ (red). During training we learn a policy for this augmented MDP (blue). During online execution we must run dual dynamics\n\t(red) to guarantee convergence (Theorem \\ref{theo_main_intro}).}\n \\label{fig_blcok_diagram}\n\\end{figure}\n\nIn the middle of the diagram in Figure \\ref{fig_blcok_diagram} we have the policy $\\pi (s_t,\\ensuremath{\\lambda}_k)$ which generates actions $a_t$. This policy is a function of the MDP's state and the multiplier $\\ensuremath{\\lambda}_k$. Our goal is to learn a policy for which $\\pi (\\,\\cdot\\,, \\ensuremath{\\lambda}) = \\pi(\\ensuremath{\\lambda})$. That is, a policy $\\pi$ that when evaluated at a specific state $\\lambda$ yields a Lagrangian maximizing policy $\\pi(\\ensuremath{\\lambda})$ [cf. \\eqref{eqn_lagrangian_maximizers_rl}]. Remarkably, this is just a conventional RL problem with instantaneous rewards given by \\eqref{eqn_lagrangian_reward}. Thus, the training step of A-CRL entails the use of any RL algorithm of choice to solve this augmented MDP. \n\nThis training methodology is summarized in Algorithm \\ref{algorithm}. For practical reasons we introduce a parameter $\\theta \\in \\mathbb{R}^d$ and consider parameterized policies $\\pi_\\theta$ so that actions are drawn as $a\\sim\\pi_\\theta(s,\\ensuremath{\\lambda})$. To train this policy we sample the augmented state space ${\\cal S}\\times\\Lambda$ (Step 1) and evaluate the rewards $ r_\\lambda(s_t,a_t)$ (Step 2). These rewards define an (augmented) MDP that we solve with the chosen RL method (Step 3). Since the rewards used in Step 3 are the ones in \\eqref{eqn_lagrangian_reward}, the policy $\\pi_{\\theta^\\star}$ is such that when evaluating $\\pi_{\\theta^\\star}(\\,\\cdot\\, , \\ensuremath{\\lambda})$ for a given $\\lambda$ the policy is a Lagrangian maximizing policy $\\pi(\\lambda)$ if the learning parametrization is sufficiently expressive. We point out that numerical results in Section \\ref{sec_example} use policy gradient in Step 3, but A-CRL is agnostic to this choice.\n\n\\begin{algorithm}[t]\n{\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\caption{A-CRL Training Phase} \\label{algorithm} \n\\begin{algorithmic}[1] \n\\Ensure Trained policy $\\pi_{\\theta^\\star}$\n\\State Sample $(s,\\ensuremath{\\lambda})$ from augmented space $\\ensuremath{\\mathcal{S}}\\times\\Lambda$ \n\\State Construct the augmented rewards [cf. \\eqref{eqn_lagrangian_reward}]\n$$\\displaystyle{\n r_\\lambda(s,a) = r_0(s,a) + \\sum_{i=1}^m \\ensuremath{\\lambda}_i \\Big( r_i(s,a) - c_i \\Big)}\n$$\n\\State Use the RL algorithm to obtain policy\n$$\\displaystyle{\n\\theta^\\star = \n\\argmax\\limits_{\\theta \\in \\ensuremath{\\mathbb{R}}^d} \\lim\\limits_{T\\to\\infty} \\frac{1}{T}\\E_{s,a\\sim\\pi_\\theta} \\left[ \\sum_{t = 0}^T r_{\\lambda}(s_t,a_t)\\right]}\n$$\n\\end{algorithmic}}\n\\end{algorithm}\n\n\\begin{algorithm}[t]\n{\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\caption{A-CRL Execution Phase} \\label{algorithm_exec} \n\\begin{algorithmic}[1] \n\\Require Policy $\\pi_{\\theta^\\star}$, step $\\eta_\\lambda$, requirements $c_i$, epoch $T_0$\n\\Ensure Trajectories $(s_t,a_t)$ satisfying Theorem \\ref{theo_main_intro}\n\\setcounter{ALG@line}{0}\n\\State \\textit{Initialize}: Initial state $s_0$, dual variable $\\lambda_0 = 0$\n\\For {$k=0,1\\ldots$}\n\\State Rollout $T_0$ steps with actions $a_t\\sim\\pi_{\\theta^\\star}(s_t,\\lambda_k)$\n\\State Update dual dynamics [cf. \\eqref{eqn_stochastic_dual_descent_specified}]\n$$\n\\displaystyle{\\lambda_{i, k+1} = \n \\left [ \\lambda_{i,k} - \n \\frac{\\eta_\\lambda}{T_0} \n \\sum_{t= kT_0}^{(k+1)T_0-1} \n \\Big( r_i(s_t,a_t)-c_i \\Big)\n \\right]_+} \n$$\n\\EndFor\n\\end{algorithmic}}\n\\end{algorithm}\n\nThe more important aspect to highlight of A-CRL is that during execution, we use the trained policy $\\pi_{\\theta^\\star}(s, \\lambda)$, and continuously sample $\\lambda_k$ by using the dual dynamics in \\eqref{eqn_stochastic_dual_descent_specified}. This is summarized in Algorithm \\ref{algorithm_exec}. Time is divided in epochs of duration $T_0$ (Step 2). During each epoch we rollout $T_0$ steps where actions are drawn as $a_t\\sim\\pi_{\\theta^\\star}(s_t,\\lambda_k)$ (Step 3). We then proceed to update the state of the dual variables (Step 4). This algorithm is unusual because we do not use \\eqref{eqn_stochastic_dual_descent_specified} as part of a learning process. We use \\eqref{eqn_stochastic_dual_descent_specified} as an online computation that controls a switching across the policies $\\pi_{\\theta^\\star}(\\,\\cdot\\, ,\\lambda_k)$ that are learned during the training process. Online execution of Algorithm \\ref{algorithm_exec} guarantees that the claims of Theorem \\ref{theo_main_intro} hold for the MDP trajectory $(s_t,a_t)$.\n\nA-CRL redefines the purpose of CRL as learning the Lagrangian maximizing policies $\\pi(\\ensuremath{\\lambda})$. This is a good premise because we know how to find policies $\\pi(\\ensuremath{\\lambda})$. It is just an MDP with an augmented state space that we solve with Algorithm 1. We further know that executing policies $\\pi(\\ensuremath{\\lambda}_k)$ while updating $\\ensuremath{\\lambda}_k$ with the dynamics in \\eqref{eqn_stochastic_dual_descent_specified} guarantees the creation of trajectories that satisfy the feasibility and near-optimality guarantees of Theorem \\ref{theo_main_intro}. This is what Algorithm 2 does. One may think that this is a cumbersome redefinition of the stated goal of finding an optimal policy $\\pi^\\star$. However, as the next section explains, this is not an artifact but a fundamental aspect of CRL. \n\n\\section{Discussions and Implications}\\label{sec_discussions}\n\nTo explain the meaning of Theorem \\ref{theo_main_intro} and the algorithm in Section \\ref{sec_algorithm} it is instructive to go back to the monitoring task introduced in Figure \\ref{fig:example}. We have a moving agent agent that must spend some of its time in each of two regions $R_1$ and $R_2$. A minimal abstraction of this problem is the MDP with 3 states that is also illustrated in Figure \\ref{fig:example} and which we formally define next. \n\n\\begin{example}[\\textit{Monitoring Problem}]\\label{ex_monitoring}\nConsider an MDP with states $R_0$, $R_1$ and $R_2$. In each state the choice of action determines the next state, $s_{t+1} = a_t$. When in state $R_0$ possible actions are $\\{R_1, R_2\\}$. When in sate $R_i$ with $i\\neq0$, possible actions are $\\{R_0, R_i\\}$. We want to spend at least $c_i$ fraction of time in each of the states $R_1$ and $R_2$. Otherwise, we want to spend as much time as possible at $R_0$. We set $c_1=c_2=c=1\/3$.\n\\end{example}\n\nTo learn policies that solve the monitoring task in Example~\\ref{ex_monitoring} we define the reward functions\n\\begin{align}\\label{eqn_monitoring_rewards}\n r_i(s_t, a_t) \n = \\indicator \n ( s_t = R_i ) \n \\text{\\quad for~} i=0, 1, 2,\n\\end{align}\nto formulate a constrained MDP [cf. \\eqref{eqn_value_function}] in which we have: (i) Two constraints $V_i(\\pi)\\geq c_i$ for $i=1,2$ where the value functions $V_i(\\pi)$ are associated with the corresponding rewards $r_i(s_t, a_t)$ of \\eqref{eqn_monitoring_rewards}. (ii) The objective $V_0(\\pi)$ is the value function associated with the reward $r_0(s_t, a_t)$ of \\eqref{eqn_monitoring_rewards}. \n\nThe choice $c=1\/3$ makes an optimal policy $\\pi^\\star$ that solves this constrained MDP ready to find. Just choose with probability $1\/2$ between the two actions that are allowed at each state. This yields a symmetric Markov chain in which the ergodic limits are $1\/3$ for all states. This satisfies the requirement of spending at least $c=1\/3$ fraction of time at $R_1$ and $R_2$ and maximizes the time spent at $R_0$. More to the point, this policy is optimal because $V_1(\\pi^\\star)\\geq c=1\/3$, $V_2(\\pi^\\star)\\geq c=1\/3$ and $V_0(\\pi^\\star) = 1\/3 = P^\\star$.\n\nTo design an algorithm that finds a solution we write the Lagrangian [cf. \\eqref{eqn_lagrangian}] to transform the constrained MDP into a plain MDP. As we saw in \\eqref{eqn_lagrangian_reward}, doing so is equivalent to defining a weighted reward as in \\eqref{eqn_lagrangian_maximizers_rl}. In this particular example, the weighted reward takes the form\n\\begin{align}\\label{eqn_monitoring_weighted_reward}\n r_{\\lambda} (s_t, a_t)\n = \\indicator ( s_t = R_0 ) \n &+ \\lambda_1 \\Big(\\indicator ( s_t = R_1 ) - c \\Big) \\nonumber\\\\\n &+ \\lambda_2 \\Big( \\indicator ( s_t = R_2 ) - c \\Big).\n\\end{align}\nAn unconstrained MDP with this reward has elementary solutions. If $1>\\ensuremath{\\lambda}_1$ and $1>\\ensuremath{\\lambda}_2$, we maximize the accumulated reward $(1\/T)\\sum_{t = 0}^T r_{\\lambda}(s_t, a_t)$ by staying at the region $R_0$ as much as possible; which is half of the time. If $\\ensuremath{\\lambda}_1>1$ and $\\ensuremath{\\lambda}_1>\\ensuremath{\\lambda}_2$, we maximize the accumulated reward by staying at $R_1$ all of the time. Similarly, if $\\ensuremath{\\lambda}_2>1$ and $\\ensuremath{\\lambda}_2>\\ensuremath{\\lambda}_1$, we maximize the accumulated reward by staying at $R_2$ all of the time. Clearly, none of these three policies are optimal for the constrained MDP, as they do not satisfy the constraints. The remaining option is to make $\\ensuremath{\\lambda}_1=\\ensuremath{\\lambda}_2=1$. This choice of weights is such that \\emph{any} policy maximizes the accumulated reward of the unconstrained MDP, since the agent collects the same reward in all states. The set $\\Pi(\\ensuremath{\\lambda}_1=1, \\ensuremath{\\lambda}_2=1)$ includes all policies. We therefore see that a constrained MDP that uses the rewards in \\eqref{eqn_monitoring_rewards} as constraints and its penalized version using the reward in \\eqref{eqn_monitoring_weighted_reward} are not equivalent. We summarize this observation in the following remark.\n\n\\begin{remark}\\label{rmk_cmdp_pmdp_not_equivalent}\nThe constrained MDP in \\eqref{P:constrainedRL} and the regularized MDP in \\eqref{eqn_lagrangian_maximizers_rl} are not equivalent representations of the same problem. There exist constrained MDPs for which there is no unconstrained MDP with regularizer $\\lambda$, such that $\\Pi^\\star \\equiv \\Pi(\\lambda)$.\n\\end{remark}\n\nRemark \\ref{rmk_cmdp_pmdp_not_equivalent} appears to indicate that the algorithm defined by~\\eqref{eqn_stochastic_dual_descent_specified} should not be able to find solutions of the constrained MDP defined in \\eqref{P:constrainedRL}. Any choice of $\\lambda$ results in an optimal set $\\Pi(\\lambda)$ that is not equivalent to the optimal set $\\Pi^\\star$. Whenever we solve \\eqref{eqn_lagrangian_maximizers_rl} we obtain a policy that is not optimal. Further note that, if anything, Remark \\ref{rmk_cmdp_pmdp_not_equivalent} understates the magnitude of the problem. The policies $\\pi(\\ensuremath{\\lambda})$ for the monitoring problem bear no resemblance to the optimal policy $\\pi^\\star$ unless $\\ensuremath{\\lambda}_1=\\ensuremath{\\lambda}_2=1$. And when $\\ensuremath{\\lambda}_1=\\ensuremath{\\lambda}_2=1$ all policies belong to the set $\\Pi(\\ensuremath{\\lambda}_1=1, \\ensuremath{\\lambda}_2=1)$. One of these is the optimal policy $\\pi^\\star$. This is a tautology that provides no information about how to find the optimal policy. \n\nHow come, then, that Theorem \\ref{theo_main_intro} holds? As we explained in Section \\ref{S:constrainedRL}, Theorem \\ref{theo_main_intro} does not claim that we find the optimal policy. Rather, that we can provide an \\emph{algorithm} whose execution results in the solution of \\eqref{P:constrainedRL}. This is weaker, but still difficult to reconcile with the statement of Remark \\ref{rmk_cmdp_pmdp_not_equivalent}. To reconcile these statements, we dwell in the duality properties of the constrained MDP in \\eqref{P:constrainedRL}.\n\n\\subsection{Constrained Reinforcement Learning and Duality} \\label{sec_relationship}\n\nA candidate culprit for the observation in Remark \\ref{rmk_cmdp_pmdp_not_equivalent} is lack of convexity. The functions involved in the constrained MDP in \\eqref{P:constrainedRL} are not concave in the policy $\\pi$. We will see here that this is not a problem because despite their lack of convexity, constrained MDPs have null duality gap~\\cite{paternain2019constrained}. Formally, recall the definition of the dual function $d(\\lambda)$ as the maximum of the Lagrangian \n\\begin{align}\\label{eqn_crl_dual_function}\n d(\\lambda) & \\triangleq \\max_\\pi \\ensuremath{\\mathcal{L}}(\\pi,\\lambda) \n = \\ensuremath{\\mathcal{L}}(\\pi(\\lambda), \\lambda),\n\\end{align}\nwhere the Lagrangian maximizers are denoted by $\\pi(\\lambda)$ [cf. \\eqref{eqn_lagrangian_maximizers} and \\eqref{eqn_lagrangian_maximizers_rl}]. By its definition, for any $\\lambda \\in \\ensuremath{\\mathbb{R}}^m_+$, the dual function provides an upper bound on the optimal value~$P^\\star$ of \\eqref{P:constrainedRL}. The dual problem aims to find the tightest of these bounds by minimizing the dual function,\n\\begin{align}\\label{eqn_crl_dual_problem}\\tag{D-CRL}\n \\lambda^\\star \n \\in \\argmin_{\\lambda \\in \\ensuremath{\\mathbb{R}}^m_+} d(\\lambda)\n = \\argmin_{\\lambda \\in \\ensuremath{\\mathbb{R}}^m_+} \\ensuremath{\\mathcal{L}}(\\pi(\\lambda), \\lambda), \n\\end{align}\nand wherein the optimal dual value is denoted by $D^\\star \\triangleq d(\\lambda^\\star)$. In convex optimization problems we know that the primal and dual problems are equivalent in the sense that they attain the same optimal values. The constrained MDP in \\eqref{P:constrainedRL} is not a convex optimization problem. Still, it has a particular structure that renders it equivalent to \\eqref{eqn_crl_dual_problem} as we formally assert next. \n\n\\begin{theorem}[Strong Duality \\cite{paternain2019constrained}]\\label{theo_no_gap}\nThe constrained reinforcement learning problem in \\eqref{P:constrainedRL} and its dual form in \\eqref{eqn_crl_dual_problem} are such that \n\\begin{align}\n V_0(\\pi^\\star) = P^\\star = D^\\star = d(\\lambda^\\star).\n\\end{align} \\end{theorem}\n\nThe equivalence of Theorem \\ref{theo_no_gap} implies that the constrained MDP in \\eqref{P:constrainedRL} is equivalent to the unconstrained Lagrangian form in \\eqref{eqn_lagrangian_maximizers_rl}. Thus, at a fundamental level, it seems that constrained RL is not more difficult to solve than an unconstrained RL. One need only find the optimal Lagrange multiplier $\\lambda^\\star$ that renders \\eqref{P:constrainedRL} and \\eqref{eqn_lagrangian_maximizers_rl} equivalent and proceed to solve the unconstrained MDP in \\eqref{eqn_lagrangian_maximizers_rl} with $\\lambda=\\lambda^\\star$. This is a relatively simple task because the dual problem in \\eqref{eqn_crl_dual_problem} is a convex program \\cite{boyd2004convex} and therefore easy to minimize. Thus, solving a constrained reinforcement learning problem is not more difficult than conventional reinforcement learning save for the relatively simple task of finding optimal dual variables. However, as we saw in Example \\ref{ex_monitoring} and concluded in Remark \\ref{rmk_cmdp_pmdp_not_equivalent}, \\emph{this is not the case}. The optimal dual variable may be uninformative about the optimal policy. We explain why in the next section.\n\n\\subsection{Recovering Optimal Policies from Dual Variables}\\label{sec_primal_recovery}\n\nThe key nuance that limits the use of the equivalence between \\eqref{P:constrainedRL} and \\eqref{eqn_lagrangian_maximizers_rl} that is implied by Theorem \\ref{theo_no_gap} is this: Our goal is not to determine the optimal value $P^\\star$ but to find an optimal policy $\\pi^\\star\\in\\Pi^\\star$. The latter is not always easy to recover from \\eqref{eqn_lagrangian_maximizers_rl} even if the optimal dual variable $\\lambda^\\star$ is known. To explain the complications that may arise, we introduce a proposition that relates optimal policies of \\eqref{P:constrainedRL} to optimal policies of \\eqref{eqn_lagrangian_maximizers_rl}.\n\n\\begin{proposition}\\label{prop_primal_recovery} The set $\\Pi^\\star$ of optimal policies of \\eqref{P:constrainedRL} is included in the set $\\Pi(\\ensuremath{\\lambda}^\\star)$ of Lagrangian maximizing policies of \\eqref{eqn_lagrangian_maximizers_rl} with $\\ensuremath{\\lambda}=\\ensuremath{\\lambda}^\\star$,\n\\begin{align}\\label{eqn_set_maximizers}\n \\Pi^\\star \\subseteq \\Pi(\\ensuremath{\\lambda}^\\star).\n\\end{align}\n\\end{proposition}\n\\vspace{-0.5\\baselineskip} \nAs per Proposition \\ref{prop_primal_recovery}, the issue is that $\\Pi(\\ensuremath{\\lambda}^\\star)$ is a set. Optimal policies $\\pi^\\star\\in\\Pi^\\star$ are included in this set. But the two sets are not equivalent. There may be policies in $\\Pi(\\ensuremath{\\lambda}^\\star)$ that are not optimal. This is the case of Example \\ref{ex_monitoring} in which the set $\\Pi(\\ensuremath{\\lambda}^\\star)$ contains all policies. This illustrates that, in general, finding a policy in the set $\\Pi(\\ensuremath{\\lambda}^\\star)$ that is optimal for \\eqref{P:constrainedRL} is as difficult as solving \\eqref{P:constrainedRL} itself. The two problems are equivalent in this particular example.\n\nAn optimal policy $\\pi^\\star$ is readily recoverable from $\\Pi(\\ensuremath{\\lambda}^\\star)$ when the latter set is a singleton. In such case \\eqref{eqn_set_maximizers} implies the equivalence $\\Pi^\\star \\equiv \\Pi(\\ensuremath{\\lambda}^\\star)$ and policies $\\pi(\\ensuremath{\\lambda}_k)$ approach $\\pi^\\star$ when $\\ensuremath{\\lambda}_k$ is updated as per \\eqref{eqn_stochastic_dual_descent_specified}. The A-CRL algorithm is not necessary in this case, although it would still work. \n\nHowever, given that there is nothing contrived about Example \\ref{ex_monitoring} it is reasonable to expect that problems where $\\Pi(\\ensuremath{\\lambda}^\\star)$ is not a singleton are widespread. This expectation can be asserted by recalling that: (i) unconstrained MDPs \\emph{always} admit a deterministic policy $\\pi$ as optimal \\cite{puterman1994markov}; (ii) constrained MDPs \\emph{often} admit nondeterministic policies as solutions \\cite{altman1999constrained}. The only way to reconcile the existence of a deterministic policy in the set $\\Pi(\\ensuremath{\\lambda}^\\star)$ and the presence of a nondeterministic optimal policy $\\Pi^\\star$ with Proposition \\ref{prop_primal_recovery} is for the set $\\Pi(\\ensuremath{\\lambda}^\\star)$ to contain multiple policies. In these situations, A-CRL is necessary to find solutions of \\eqref{P:constrainedRL}, in the sense of generating trajectories that satisfy \\eqref{eqn_theo_feasibility_informal} and \\eqref{eqn_theo_optimality_informal}. We illustrate this point next.\n\n\\begin{figure*}[t!]\n \\centering\n \\subfigure[{Constraint satisfaction $V_1(\\pi)$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/example_sim_dual_variablev.pdf} \t\t\t\t\t\n \\label{fig:example_sim_dual_v}\n\t } \\hspace{-0.325cm}\n \\subfigure[{Dual variable $\\lambda_{1,k}$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/example_sim_dual_dual.pdf} \t\t\t\t\t\n \\label{fig:example_sim_dual_dual}\n\t } \\hspace{-0.325cm}\n \\subfigure[{Policy $\\pi(s_t=R_1, \\ensuremath{\\lambda}_k)$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/example_sim_dual_policy.pdf} \t\t\t\t\t\n \\label{fig:example_sim_dual_policy}\n\t }\n\t\\caption{Execution of A-CRL for Example \\ref{ex_monitoring}. Values for average accumulated reward $r_1$ and dual variables $\\ensuremath{\\lambda}_1$ are shown in Figures \\ref{fig:example_sim_dual_v} and \\ref{fig:example_sim_dual_dual}.\n\tFigure \\ref{fig:example_sim_dual_policy} shows the policy $\\pi(s_t=R_1, \\ensuremath{\\lambda}_k)$. Probability of staying at $R_1$ (blue circles) and the complementary probability of jumping to $R_0$ (gray squares) are shown. We run $1{,}000$ rollouts with length $T_0=10$ and step size $\\eta_\\lambda=0.5$. Figures show last 20 rollouts. A-CRL does not converge to $\\pi^\\star$ but $\\pi(\\ensuremath{\\lambda}_k)$ generates a trajectory that is feasible [cf. \\eqref{eqn_theo_feasibility_informal}] and nearly optimal [cf. \\eqref{eqn_theo_optimality_informal}].\n\t}\t \n\t\\label{fig:example_sim_dual} \n\\end{figure*}\n\n\\vspace{-0.5\\baselineskip} \n\\subsection{Policy Switching and Memory}\\label{sec_policy_switching}\n\nIn the training stage of A-CRL, Algorithm \\ref{algorithm} solves the augmented MDP of Figure \\ref{fig_blcok_diagram}. Assuming that training succeeds, this is equivalent to finding Lagrangian maximizing policies $\\pi(\\ensuremath{\\lambda})$. As explained in Section \\ref{sec_primal_recovery} there is no guarantee that the optimal policy $\\pi^\\star$ is recoverable from $\\pi(\\ensuremath{\\lambda}^\\star)$. Thus, no guarantee that A-CRL converges to $\\pi^\\star$ either. Nevertheless, A-CRL guarantees that \\eqref{eqn_theo_feasibility_informal} and \\eqref{eqn_theo_optimality_informal} hold. This is attained through \\emph{policy switching.} Step 4 of Algorithm \\ref{algorithm_exec} updates multipliers as per \\eqref{eqn_stochastic_dual_descent_specified} in order to effect the execution of different policies $\\pi(\\ensuremath{\\lambda}_k)$ in Step 3.\n\n Figure \\ref{fig:example_sim_dual} illustrates policy switching for Example \\ref{ex_monitoring}, showing variables of A-CRL related to state $R_1$. Figure \\ref{fig:example_sim_dual_v} shows the average accumulated reward $r_1$ and Figure \\ref{fig:example_sim_dual_dual} shows the value of the Lagrange multiplier $\\ensuremath{\\lambda}_1$. Figure \\ref{fig:example_sim_dual_policy} illustrates the policy $\\pi(s_t=R_1, \\ensuremath{\\lambda}_k)$. It shows the probability of staying at $R_1$ (blue circles) and the complementary probability of jumping to $R_0$ (gray squares). In this latter figure, policy $\\pi(s_t=R_1, \\ensuremath{\\lambda}_k)$ jumps from staying at $R_1$ with probability $0$ to staying at $R_1$ with probability $1$. None of these two policies solve Example \\ref{ex_monitoring}. This fact is consistent with the impossibility of recovering the optimal policy from any given set of multipliers (Section \\ref{sec_primal_recovery}). This is also something we have already highlighted in Remark \\ref{rmk_cmdp_pmdp_not_equivalent}. Maximizing \\eqref{eqn_monitoring_weighted_reward} requires maximizing the time spent at a particular state $R_i$ depending on the values of the multipliers $\\ensuremath{\\lambda}_i$. \n \nThat the policies in Figure \\ref{fig:example_sim_dual_policy} do not solve Example \\ref{ex_monitoring} is attested by the accumulated reward plot (solid blue) in Figure \\ref{fig:example_sim_dual_v}. The requirement is for this reward to accumulate to at least $c=1\/3$ and the optimal policy requires that this reward accumulate to exactly $c=1\/3$. This does not happen at any iteration. However, the time average of the accumulated reward across all rollouts does accumulate to exactly $1\/3$ (dashed red). This is as it should be. It is the claim of Theorem \\ref{theo_main_intro} that this time average should satisfy the constraints with probability $1$ if we execute A-CRL.\n\nFigure \\ref{fig:example_sim_dual_dual} illustrates how the Lagrange multiplier controls the policy switch. As it follows from \\eqref{eqn_monitoring_weighted_reward}, when $\\ensuremath{\\lambda}_1<1$ the Lagrangian maximizing policy requires jumping back to $R_0$ with probability $1$. When the multiplier switches to $\\ensuremath{\\lambda}_1>1$ the Lagrangian maximizing policy switches to one that requires staying at $R_1$ with probability $1$\\textemdash we must also have $\\ensuremath{\\lambda}_2>\\ensuremath{\\lambda}_1$, this holds but is not shown. In Figure \\ref{fig:example_sim_dual_dual} the switching from $\\ensuremath{\\lambda}_1<1$ to $\\ensuremath{\\lambda}_1>1$ happens for $k=989$. This is precisely the same moment when the policies switch in Figure \\ref{fig:example_sim_dual_policy}. This highlights the instrumental role of state augmentation in solving CRL problems. It is the policy switching controlled by $\\ensuremath{\\lambda}_i$ that generates trajectories that guarantee feasibility and near-optimality (Theorem \\ref{theo_main_intro}).\n\n\\textbf{Memory.~} The Lagrange multipliers $\\lambda_{i, k}$ in A-CRL accept an interpretation in terms of memory. To make this clearest assume that $\\lambda_{i, 0} = 0$ and that the nonnegative projection operators in \\eqref{eqn_stochastic_dual_descent_specified} is never applied\\textemdash which is true in Figure \\ref{fig:example_sim_dual_dual}. With these assumptions we can rewrite \\eqref{eqn_stochastic_dual_descent_specified} as\n\\begin{align}\\label{eqn_dd_ex1}\n \\lambda_{i, k} \n = \\frac{\\eta_\\lambda}{T_0}\n \\sum_{t= 0}^{kT_0-1} \n \\Big(c - \\indicator ( s_t = R_i )\\Big) .\n\\end{align}\nThus, the multiplier $\\ensuremath{\\lambda}_i$ records the deficit in the amount of time the agent spends at $R_i$ relative to the requirement $c$. When this deficit grows large, the multiplier triggers a policy switch to reduce it. This observation holds true in general. The multiplier update in \\eqref{eqn_stochastic_dual_descent_specified} is such that $\\ensuremath{\\lambda}_{i,k}$ memorizes the constraint deficit. As these deficits rise and fall, they control the choice of policies $\\pi(\\ensuremath{\\lambda}_k)$ to generate trajectories that satisfy \\eqref{eqn_theo_feasibility_informal} and \\eqref{eqn_theo_optimality_informal}. The nonnegative projection operator limits the future credit that an agent is given for over-satisfying the constraint at a particular rollout. \n\n\n\\textbf{Primal Averaging. } The optimal policy $\\pi^\\star$ can be recovered from primal averaging \\cite{anstreicher2009two,nedic2009approximate}. Indeed, the average policy\n\\begin{align}\\label{eqn_primal_averaging}\n \\bar\\pi_K = \\frac{1}{K} \\sum_{k=0}^{K-1} \\pi(\\ensuremath{\\lambda}_k),\n\\end{align}\nconsists of the combination up to a time horizon $K$ of the Lagrangian maximizing policies sampled while running dual dynamics \\eqref{eqn_stochastic_dual_descent_specified}. If the horizon is sufficiently large, drawing actions from this average policy generates a sequence with the same ergodic properties of the sequence generated by A-CRL. Since the guarantees of Theorem \\ref{theo_main_intro} hold for the latter, they also hold for a trajectory that is sampled from $\\bar\\pi_K$. Primal averaging provides an alternative to A-CRL that is applicable to some problems but not all. In general, policies $\\pi(\\ensuremath{\\lambda}_k)$ are parametrized as in Algorithms \\ref{algorithm} and \\ref{algorithm_exec}. However, \\eqref{eqn_primal_averaging} is averaged over \\emph{policies}, not \\emph{parameterizations}. Finding a parametric description of the average policy $\\bar\\pi_K$ other than the impractical storage of the whole sequence of parameters is a difficult problem in itself.\n\n\\begin{figure*}[t!]\n \\centering\n \\subfigure[{Constraint satisfaction $V_1(\\pi)$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/example_sim_pd_variablev.pdf} \t\t\t\t\t\n \\label{fig:example_pd_v}\n\t }\n \\subfigure[{Dual variable $\\lambda_{1,k}$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/example_sim_pd_dual.pdf} \t\t\t\t\t\n \\label{fig:example_pd_dual}\n\t }\n \\subfigure[{Policy $\\pi(s_t=R_1, \\ensuremath{\\lambda}_k)$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/example_sim_pd_policy.pdf} \t\t\t\t\t\n \\label{fig:example_pd_policy}\n\t }\n\t \n\t\\caption{Primal-dual method for Example \\ref{ex_monitoring}. Figures \\ref{fig:example_pd_v}\\textendash\\ref{fig:example_pd_policy} are analogous to Figures \\ref{fig:example_sim_dual_v}\\textendash\\ref{fig:example_sim_dual_policy}. We run $400{,}000$ rollouts with length $T_0=10$, dual step $\\eta_\\theta=0.0025$ and primal step $\\eta_\\theta=0.025$. Primal-dual does not learn the optimal policy although it generates \\emph{trajectories} that are optimal on average if executed online, as is the case of A-CRL. Executing primal-dual methods online is challenging.}\t \n \\label{fig:example_sim}\n\\end{figure*}\n\n\\subsection{Primal-Dual Methods}\\label{sec_primal_dual}\n\nA-CRL uses gradient descent in the dual domain and policy optimization in the primal domain. Primal-dual methods use gradient descent in the dual domain and gradient ascent in the primal domain. That is, the maximization step in \\eqref{eqn_lagrangian_maximizers_rl} is replaced by the gradient ascent step,\n\\begin{align}\\label{eq:pd_equations}\n \\pi_{\\theta,k+1}&= \\pi_{\\theta,k} + \\eta_\\theta \\widehat{\\nabla}_{\\pi_\\theta} \\ensuremath{\\mathcal{L}}(\\pi_{\\theta,k},\\lambda_k) .\n\\end{align}\nThe trajectories generated by primal-dual methods solve CRL in the same time average sense of Theorem \\ref{theo_main_intro}\\cite{tessler2018reward, ding2020natural}. This is illustrated in Figure \\ref{fig:example_sim} where we show the same variables of Figure \\ref{fig:example_sim_dual} for a primal-dual method. The same switching behavior is apparent. Multipliers rise and fall [Figure \\ref{fig:example_pd_dual}] triggering policy switches [Figure \\ref{fig:example_pd_policy}]. These policies are instantaneously not optimal, but they satisfy the constraints on average [Figure \\ref{fig:example_pd_v}]. The important consequence is that, in general, we cannot stop a primal-dual method at iteration $k$ and claim that the policy $\\pi_{\\theta,k}$ is close to optimal. As is the case of A-CRL, the primal-dual method must be run online to ensure that \\emph{trajectories} are optimal on average.\n\nWhile possible, running \\eqref{eq:pd_equations} online is challenging. This is because online estimation of the gradient in \\eqref{eq:pd_equations} is difficult. We could attempt to mitigate this problem with an actor-critic method, but one has remember that the usefulness of actor-critic hinges upon convergence to a stationary policy\\textemdash something that does not hold for some CRL problems. A-CRL circumvents these challenges by augmenting the state and learning instead the Lagrangian maximizing policies in \\eqref{eqn_lagrangian_maximizers_rl}.\n\nWe also call attention to the significant delay between the time at which $\\ensuremath{\\lambda}_1>1$ [$k\\approx 3.82 \\times 10^5$ in Figure \\ref{fig:example_pd_dual}] and the time at which the policy switches [$k\\approx 3.90 \\times 10^5$ in Figure \\ref{fig:example_pd_policy}]. This is because as $\\ensuremath{\\lambda}_1$ switches, the primal-dual method needs to relearn the Lagrangian maximizing policy. This delay results on $\\ensuremath{\\lambda}_1$ growing larger than in Figure \\ref{fig:example_sim_dual}. As it follows from \\eqref{eqn_dd_ex1} this implies that the transient violation of constraints is larger for primal-dual methods than it is for A-CRL. Even if both satisfy the constraints on average. \n\n\n\n\n\\section{Numerical Example}\\label{sec_example}\n\n\\begin{figure*}[t!]\n \\centering\n \\subfigure[{Policy for $\\lambda=[5,0,0,0]$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/fig_quiver_high_low.pdf} \t\n \\label{fig:quiver_high_low}\n\t \n \\subfigure[{Policy for $\\lambda=[0,5,5,0]$.}]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/fig_quiver_mid_mid.pdf} \t\t\n \\label{fig:quiver_mid_mid}\n\t \n \\subfigure[Occupation measure.]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/heatmap.pdf} \t\t\t\n \\label{fig:heatmap}\n\t }\t \n\t\\caption{Policy $\\pi_{\\theta}(\\lambda)$ after $1{,}000{,}000$ training iterations, shown for two possible values of $\\lambda$. The state occupation measure resulting from an agent executing $\\pi_{\\theta}(\\lambda)$ coupled with the dual dynamics \\eqref{eqn_stochastic_dual_descent_specified} is shown on the right.}\n \\label{fig:quiver}\n \\vspace{-0.5\\baselineskip} \n\\end{figure*}\n\nLet us consider again the \\emph{monitoring problem}, first introduced in Fig.~\\ref{fig:example}, in which an agent must spend a specific portion of its time in certain regions of the environment. As we discussed in the simpler Example~\\ref{ex_monitoring}, this is a problem for which classical reinforcement learning is unable to learn a solution. For the numerical results in this section, we study a slightly more complex version of this problem. More formally, let us consider a MDP formed by a state space $\\mathcal{S}=[0,10] \\times [0,10]$ representing the $x$- and $y$-axis position of the agent. The agent takes actions, following a Gaussian policy, the mean of which is constructed by a function approximator given by a weighted linear combination of Gaussian kernels. The weights of this function approximator are the parameters to be learned by the agent. The task is specified as a CRL problem. Specifically, setting $r_0(s_t,a_t)=0$ and $r_i(s_t,a_t)=1$ if the agent is inside the $i$-th region to be monitored. As illustrated in Figure \\ref{fig:quiver}, we consider four regions to be monitored, indexed by $i\\in \\{1,2,3,4\\}=\\{ \\texttt{red}, \\texttt{blue}, \\texttt{green}, \\texttt{orange}\\}$, where the colors are matched in the plots. The task requirements $c_i$ of the agent are to monitor each region at least $20\\%$, $15\\%$, $10\\%$ and $5\\%$ of its time, respectively. The agent is trained via the A-CRL algorithm described in Section \\ref{sec_algorithm}. During the training phase, the RL algorithm used is a policy gradient \\cite{sutton2000policy} with rollouts of fixed horizon $T=20$ and step size of $\\eta_{\\theta}=0.001$, for $1{,}000{,}000$ iterations.\n\n\\begin{figure}[t!]\n \\centering\n \\subfigure[Time average value of the dual variables.]{\n \\hspace{2.15ex}\n\n\t\\includegraphics[scale=1]{img\/pdf\/fig_dual_variables.pdf} \t\t\t\n \\label{fig:dual_variables}\n }\n \\subfigure[Constraint satisfaction.]{\n\n\t\\includegraphics[scale=1]{img\/pdf\/fig_constraints.pdf} \t\t\t\t\n \\label{fig:constraints}\n\t }\t \n\t\\caption{Dual variables and constraint satisfaction when executing the trained policy $\\pi_{\\theta}(\\lambda)$ coupled with the dual dynamics \\eqref{eqn_stochastic_dual_descent_specified}. The dual variables are shown for a single realization. The constraint satisfaction is averaged (solid line) over $100$ independent runs, where the shadowed band covers to the worst realization. The requirements are shown as dashed lines and the colors are set accordingly to the region they represent, following Fig. \\ref{fig:quiver}.}\n \\label{fig:dual_constraints}\n \\vspace{-0.5\\baselineskip}\n\\end{figure}\n\nThe policy $\\pi_\\theta(\\lambda)$ after training is shown in Figure \\ref{fig:quiver}. Specifically, Fig. \\ref{fig:quiver_high_low} and Fig. \\ref{fig:quiver_mid_mid} show the resulting policy for different values of the Lagrange multipliers $\\lambda$. In the A-CRL method, the policy includes the Lagrangian variables as part of the augmented state space. When the Lagrange multiplier associated to one of the regions takes a higher value compared to the other regions, the policy moves the agent towards that region. This is the case shown in Fig. \\ref{fig:quiver_high_low}, where all dual variables but $\\lambda_1=5$ are set to zero. For such case, the policy is to move towards the red ($i=1$) monitoring area. Similarly, we show in Fig. \\ref{fig:quiver_mid_mid} the policy when two of the Lagrangian multipliers take large values, and the other regions are kept to zero. In this case, the policy is to move into any of the two regions, blue ($i=2$) or green ($i=3$). Similar behavior can be observed when setting the Lagrange multipliers of the other regions accordingly. \n\nIt is important to remember that no dual dynamics are used during training. Recall that the policy acquired during training maximizes the Lagrangian of the problem [cf. \\eqref{eqn_lagrangian}] for all possible Lagrange multipliers. Now, consider an agent executing this policy. We use the requirements $c_i$ and generate Lagrange multipliers $\\lambda_{i,k}$ following the dual dynamics, as in \\eqref{eqn_stochastic_dual_descent_specified}, using a step size $\\eta_{\\lambda}=0.01$ and an epoch length of $T_0=1$. We compute the state occupation measure shown in Fig. \\ref{fig:heatmap}. When executing the A-CRL algorithm, the agent spends more in in areas with higher demands, while barely spending any time in the areas with no demands (outside of the monitoring regions).\n\nA more detailed understanding regarding the underpinnings of the A-CRL approach can be attained by examining the values of the dual variables and the constraint satisfaction (the percentage of time spent in each of the monitoring areas). These are shown in Figure \\ref{fig:dual_constraints}. For the dual variables, we show in Fig. \\ref{fig:dual_variables} their time averages for a single run. Requirements more difficult to satisfy lead to higher values of their respective dual variables. More precise information is shown by the constraint satisfaction, plotted in Fig. \\ref{fig:constraints}. In this case, we plot the results of $100$ independent realizations, each one over $200{,}000$ time steps. The monitoring requirements are shown by a dashed line, the average constraint satisfaction by a solid line and the shaded area represents trajectories covering up to the worst realization. In general, the average constraint satisfaction for each monitoring region is well over their respective specification. More importantly, all trajectories satisfy the problem requirements, as Theorem \\ref{theo_main_intro} guarantees.\n\n\\vspace{-0.5\\baselineskip} \n\\section{Conclusions}\n\nIn this work, we have studied CRL problems. The motivation being that CRL and regularized RL problems are not equivalent, in the sense that the latter are unable to specify tasks that the former can. A concern that also extends to dual approaches, precluding their guaranteed converge to optimal CRL policies. To overcome these issues we have introduced the A-CRL algorithm. A state augmentation procedure that learns Lagrangian maximizing policies. During execution, these policies are coupled with dual dynamics, producing trajectories which are feasible and near-optimal.\n\n\n\\subsubsection{#1}\\vspace{-3\\baselineskip}\\color{black}\\bigskip{\\noindent \\bf \\thesubsubsection. #1.}}\n\n\\newcommand{\\myparagraph}[1]{\\needspace{1\\baselineskip}\\medskip\\noindent {\\it #1.}}\n\n\\newcommand{\\myparagraphtc}[1]{\\needspace{1\\baselineskip}\\medskip\\noindent {\\it #1.}\\addcontentsline{toc}{subsubsection}{\\qquad\\qquad\\quad#1}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Problem Specification.}In this paper, we consider the solution of the $N \\times\nN$ linear\nsystem\n\\begin{equation} \\label{e1.1}\nA x = b\n\\end{equation}\nwhere $A$ is large, sparse, symmetric, and positive definite. We consider\nthe direct solution of (\\ref{e1.1}) by means of general sparse Gaussian\nelimination. In such a procedure, we find a permutation matrix $P$, and\ncompute the decomposition\n\\[\nP A P^{t} = L D L^{t}\n\\]\nwhere $L$ is unit lower triangular and $D$ is diagonal.\n\n\n\\section{Design Considerations.}Several good ordering algorithms (nested dissection and\nminimum degree)\nare available for computing $P$ \\cite{GEORGELIU}, \\cite{ROSE72}.\nSince our interest here does not\nfocus directly on the ordering, we assume for convenience that $P=I$,\nor that $A$ has been preordered to reflect an appropriate choice of $P$.\n\nOur purpose here is to examine the nonnumerical complexity of the\nsparse elimination algorithm given in \\cite{BANKSMITH}.\nAs was shown there, a general sparse elimination scheme based on the\nbordering algorithm requires less storage for pointers and\nrow\/column indices than more traditional implementations of general\nsparse elimination. This is accomplished by exploiting the m-tree,\na particular spanning tree for the graph of the filled-in matrix.\n\n\\begin{theorem} The method was extended to three\ndimensions. For the standard multigrid\ncoarsening\n(in which, for a given grid, the next coarser grid has $1\/8$\nas many points), anisotropic problems require plane\nrelaxation to\nobtain a good smoothing factor.\\end{theorem}\n\nOur purpose here is to examine the nonnumerical complexity of the\nsparse elimination algorithm given in \\cite{BANKSMITH}.\nAs was shown there, a general sparse elimination scheme based on the\nbordering algorithm requires less storage for pointers and\nrow\/column indices than more traditional implementations of general\nsparse elimination. This is accomplished by exploiting the m-tree,\na particular spanning tree for the graph of the filled-in matrix.\nSeveral good ordering algorithms (nested dissection and minimum degree)\nare available for computing $P$ \\cite{GEORGELIU}, \\cite{ROSE72}.\nSince our interest here does not\nfocus directly on the ordering, we assume for convenience that $P=I$,\nor that $A$ has been preordered to reflect an appropriate choice of $P$.\n\n\\begin{proof} In this paper we consider two methods. The first method\nis\nbasically the method considered with two differences:\nfirst, we perform plane relaxation by a two-dimensional\nmultigrid method, and second, we use a slightly different\nchoice of\ninterpolation operator, which improves performance\nfor nearly singular problems. In the second method coarsening\nis done by successively coarsening in each of the three\nindependent variables and then ignoring the intermediate\ngrids; this artifice simplifies coding considerably.\n\\end{proof}\n\nOur purpose here is to examine the nonnumerical complexity of the\nsparse elimination algorithm given in \\cite{BANKSMITH}.\nAs was shown there, a general sparse elimination scheme based on the\nbordering algorithm requires less storage for pointers and\nrow\/column indices than more traditional implementations of general\nsparse elimination. This is accomplished by exploiting the m-tree,\na particular spanning tree for the graph of the filled-in matrix.\n\n\\begin{Definition}{\\rm We describe the two methods in \\S 1.2. In \\S\\ 1.3. we\ndiscuss\nsome remaining details.}\n\\end{Definition}\n\nOur purpose here is to examine the nonnumerical complexity of the\nsparse elimination algorithm given in \\cite{BANKSMITH}.\nAs was shown there, a general sparse elimination scheme based on the\nbordering algorithm requires less storage for pointers and\nrow\/column indices than more traditional implementations of general\nsparse elimination. This is accomplished by exploiting the m-tree,\na particular spanning tree for the graph of the filled-in matrix.\nSeveral good ordering algorithms (nested dissection and minimum degree)\nare available for computing $P$ \\cite{GEORGELIU}, \\cite{ROSE72}.\nSince our interest here does not\nfocus directly on the ordering, we assume for convenience that $P=I$,\nor that $A$ has been preordered to reflect an appropriate choice of $P$.\n\nOur purpose here is to examine the nonnumerical complexity of the\nsparse elimination algorithm given in \\cite{BANKSMITH}.\nAs was shown there, a general sparse elimination scheme based on the\nbordering algorithm requires less storage for pointers and\nrow\/column indices than more traditional implementations of general\nsparse elimination.\n\n\\begin{lemma} We discuss first the choice for $I_{k-1}^k$\nwhich is a generalization. We assume that $G^{k-1}$ is\nobtained\nfrom $G^k$\nby standard coarsening; that is, if $G^k$ is a tensor product\ngrid $G_{x}^k \\times G_{y}^k \\times G_{z}^k$,\n$G^{k-1}=G_{x}^{k-1} \\times G_{y}^{k-1} \\times G_{z}^{k-1}$,\nwhere $G_{x}^{k-1}$ is obtained by deleting every other grid\npoint of $G_x^k$ and similarly for $G_{y}^k$ and $G_{z}^k$.\n\\end{lemma}\n\nTo our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6]. In \\S 1.3., we analyze the complexity of the old and new\napproaches to the intersection problem for the special case of\nan $n \\times n$ grid ordered by nested dissection. The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. To our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6].\n\nIn \\S 1.2, we review the bordering algorithm, and introduce\nthe sorting and intersection problems that arise in the\nsparse formulation of the algorithm.\nIn \\S 1.3., we analyze the complexity of the old and new\napproaches to the intersection problem for the special case of\nan $n \\times n$ grid ordered by nested dissection. The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. To our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6].\n\n\nFor the old approach, we show that the\ncomplexity of the intersection problem is $O(n^{3})$, the same\nas the complexity of the numerical computations. For the\nnew approach, the complexity of the second part is reduced to\n$O(n^{2} (\\log n)^{2})$.\n\nTo our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6]. In \\S 1.3., we analyze the complexity of the old and new\napproaches to the intersection problem for the special case of\nan $n \\times n$ grid ordered by nested dissection. The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. To our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6].\nThis is accomplished by exploiting the m-tree,\na particular spanning tree for the graph of the filled-in matrix.\nTo our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n\\cite{EISENSTAT} - \\cite{LIU2}, \\cite{ROSE76}, \\cite{SCHREIBER}.\n\n\\subsection{Robustness.}We do not\nattempt to present an overview\nhere, but rather attempt to focus on those results that\nare relevant to our particular algorithm.\nThis section assumes prior knowledge of the role of graph theory\nin sparse Gaussian elimination; surveys of this role are\navailable in \\cite{ROSE72} and \\cite{GEORGELIU}. More general\ndiscussions of elimination trees are given in\n\\cite{LAW} - \\cite{LIU2}, \\cite{SCHREIBER}.\nThus, at the $k$th stage, the bordering algorithm consists of\nsolving the lower triangular system\n\\begin{equation} \\label{1.2}\n L_{k-1}v = c\n\\end{equation}\nand setting\n\\begin{eqnarray}\n\\ell &=& D^{-1}_{k-1}v , \\\\\n\\delta &=& \\alpha - \\ell^{t} v .\n\\end{eqnarray}\n\n\\begin{figure}\n\\vspace{14pc}\n\\caption{This is a figure 1.1.}\n\\end{figure}\n\n\\section{Robustness.} We do not\nattempt to present an overview\nhere, but rather attempt to focus on those results that\nare relevant to our particular algorithm.\n\n\\subsection{Versatility.}The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. For the old approach, we show that the\ncomplexity of the intersection problem is $O(n^{3})$, the same\nas the complexity of the numerical computations\n\\cite{GEORGELIU}, \\cite{ROSEWHITTEN}. For the\nnew approach, the complexity of the second part is reduced to\n$O(n^{2} (\\log n)^{2})$.\n\nTo our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6]. In \\S 1.3., we analyze the complexity of the old and new\napproaches to the intersection problem for the special case of\nan $n \\times n$ grid ordered by nested dissection. The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. To our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6].\n\nIn \\S 1.2, we review the bordering algorithm, and introduce\nthe sorting and intersection problems that arise in the\nsparse formulation of the algorithm.\nIn \\S 1.3., we analyze the complexity of the old and new\napproaches to the intersection problem for the special case of\nan $n \\times n$ grid ordered by nested dissection. The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. To our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6].\n\n\nFor the old approach, we show that the\ncomplexity of the intersection problem is $O(n^{3})$, the same\nas the complexity of the numerical computations. For the\nnew approach, the complexity of the second part is reduced to\n$O(n^{2} (\\log n)^{2})$.\n\nTo our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6]. In \\S 1.3., we analyze the complexity of the old and new\napproaches to the intersection problem for the special case of\nan $n \\times n$ grid ordered by nested dissection. The special\nstructure of this problem allows us to make exact estimates of\nthe complexity. To our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n[4] - [10], [5], [6].\nThis is accomplished by exploiting the m-tree,\na particular spanning tree for the graph of the filled-in matrix.\nTo our knowledge, the m-tree previously has not been applied in this\nfashion to the numerical factorization, but it has been used,\ndirectly or indirectly, in several optimal order algorithms for\ncomputing the fill-in during the symbolic factorization phase\n\\cite{EISENSTAT} - \\cite{LIU2}, \\cite{ROSE76}, \\cite{SCHREIBER}.\n\n\n\\section{Introduction}\n\\label{sect:introduction}\n\\input{sections\/introduction.tex}\n\nBelow we discuss connections to existing work in Section \\ref{sect:other work} before presenting our method in Section \\ref{sect:method}. Numerical experiments follow in Section \\ref{sect:experiments} before concluding in Section \\ref{sect:conclusion}.\n\n\\section{Connections to Existing Work}\n\\label{sect:other work}\n\\input{sections\/other_work.tex}\n \n\\section{Method}\n\\label{sect:method}\n\n \\subsection{Testing the clusterability of a dataset}\n \\label{sect:clusterability}\n \\input{sections\/clusterability.tex}\n \n \\subsection{The $n$-TARP method}\n \\label{sect:method nTARP}\n \\input{sections\/method_nTARP.tex}\n \n \\subsection{Comments and insights on $n$-TARP}\n \\label{sect:method comments}\n \\input{sections\/method_comments.tex}\n\n\\section{Experiments}\n\\label{sect:experiments}\n \n \\subsection{Data sets considered}\n \\label{subsect:data_sets}\n Our main focus for $n$-TARP is on small data problems, wherein we have small number of data samples in a high-dimensional space. It is not immediately clear what types of structures we should expect to be able to uncover from such a dataset. With that in mind, the most reasonable test for the $n$-TARP method is how it performs on ``real-world'' data sets, i.e. clustering problems for which data is readily available. We will be using the following data sets. Unless otherwise specified, these were provided by the UCI Machine Learning Repository \\cite{Dua:2019}.\n \n \\begin{description}\n \\item[m-feat:] the ``Multiple Features Data Set''. This consists of several different lists of features drawn from a set of 2000 handwritten numerals. For a detailed description of each, consult \\cite{Dua:2019}. It is relevant that each sub-collection has a different number of dimensions for each data point, as summarized in this table:\n \n \\begin{tabular}{r|llllll}\n mfeat- & fou & fac & kar & pix & zer & mor \\\\\\hline\n dimensionality & 76 & 216 & 64 & 240 & 47 & 6\n \\end{tabular}\n \n Notice in particular that the mfeat-mor dataset has only six dimensions. In the experiments which follow it shows how the $n$-TARP method behaves differently when one of its key assumptions, i.e. that the data points lie in high dimensions, is violated.\n \n \\item[libras:] the ``Libras Movement Data Set''. This consists of 360 points in 90 dimensions, representing measurements of hand position in recordings of Brazilian sign language.\n \n \\item[mushrooms:] the ``Mushroom Data Set''. This consists of 8124 points, but is different from the other data sets in that it contains only categorical data. Each point is a list of 24 attributes measured from a mushroom. In order to embed the data into $\\mathbb{R}^d$ a one-hot encoding is used, giving the resulting points a dimensionality of 119.\n \n \\item[gaussian:] This is not a ``real'' dataset, rather it consists of 2000 points drawn from a standard Gaussian in 100 dimensions. This is provided as a null model, to show what happens in each experiment when given a collection with no structure.\n \\end{description}\n \n \n \n \n \n\n \n\n \\subsection{Experiments to determine how the parameter $n$ affects the successfulness of $n$-TARP.}\\label{sec:experiment_n}\n \\input{sections\/experiment_n_dependence}\n \n \\subsection{Experiments to determine how the number of data points affects the successfulness of $n$-TARP.}\\label{sec:experiment_size}\n \\input{sections\/experiment_size_dependence}\n \n \\subsection{Feature Space Extension}\n \\label{sect:ExtFeat}\n \\input{sections\/extensions}\n\n \\subsection{Comparison with other methods}\n \\input{sections\/comparison}\n \n\n\\section{Conclusions}\n\\label{sect:conclusion}\n\\input{sections\/conclusion.tex}\n\n\n\n\n\n\n\\subsubsection{Are projected clusters likely?}\n\\label{sect:likely_clusters}\n\nTo see if clusters are likely to appear in projections of our data we follow the approach proposed in \\cite{yellamraju2018clusterability}, summarized here. Given a collection of points $a_1,a_2,\\dots,a_m\\in \\mathbb{R}$, and taking $a=\\langle a_1,\\dots,a_m\\rangle^T$, we define ``normalized withinss\" ($W$) as\n\\[\nW=W(a)=\\left(\\frac{1}{\\hat{\\sigma}^{2} m}\\right) \\underset{C_{1},C_{2}}{\\min} \\left(\n\\sum_{a_i\\in C_{1}}\\left(a_{i}-\\mu_{1}\\right)^{2}+\\sum_{a_i\\in C_{2}}\\left(a_{i}-\\mu_{2}\\right)^{2}\\right),\n\\]\nwhere $C_1$ and $C_2$ partition the set $\\{a_1,\\dots,a_m\\}$ into groups with means $\\mu_1$ and $\\mu_2$ respectively, and $\\hat{\\sigma}^2$ is the empirical variance of $\\{a_1,\\dots,a_m\\}$. Dividing by $\\hat{\\sigma}^2$ serves the purpose of obtaining scale invariance. The question of how to efficiently compute $W$ is considered in \\cref{sect:compute_W}.\n\nGiven a data set packed into a matrix $X$, and a random projection vector $\\mathbf{v}$, \\cite{yellamraju2018clusterability} proposes studying the random variable $\\mathbf{W}=W(X\\mathbf{v})$. The idea is that low values of $W$ indicate a group of points which is well-split into two groups, so the data $X$ will be likely to be clustered by random projection if $\\mathbf{W}$ takes on small values frequently.\n\n\n\\subsubsection{Are projected clusters meaningful?}\n\\label{sect:meaningful_clusters}\n\nThe fact that a (projection of a) sample set $X$ exhibits clusters does not necessarily imply that there are clusters in the distribution from which $X$ was drawn. As an example, consider \\cref{fig:strange_normal}, which shows a kernel density estimate for a sample of 50 points drawn from a normal distribution. This particular sample has a normalized withinss of $W=0.147$. This was generated, for demonstration purposes, by simply drawing 1,000,000 such sample distributions and choosing the distribution with the smallest $W$. \nAlthough the empirical data is bimodal, the underlying normal distribution is not. Because of our small sample size, however, a bimodal structure was obtained ``by luck.\" This shows that it is not enough to simply find a projection with a small $W$, because the clustering generated by this separation does not necessarily generalize to other data drawn from the underlying distribution.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{example_strange_normal_projection}\n\\caption{A kernel density estimate for an unusual sample from a normal distribution. The pictured distribution has 50 points sampled from a standard Gaussian, and was chosen as the most-clustered among 1,000,000 such distributions.}\n\\label{fig:strange_normal}\n\\end{figure}\n\nAs was shown in \\cite{yellamraju2018clusterability} the distribution of $\\mathbf{W}$ depends heavily on the type of data considered. The question of finding the statistical significance of a given projection was discussed in \\cite{SunJ.1991SLiE}, where they focus on classical projection pursuit. We address this issue by reserving some data for validation in a ``hold out\" fashion; if a given projection cluster represents true structure in the underlying distribution of $X$, then a vector $v$ for which $W(Xv)$ is small should also cause $W(X'v)$ to be small, for another set of data $X'$ from the same source.\n\n\\subsubsection{Computing $W$}\n\\label{sect:compute_W}\n\n\\newcommand{\\indi}{\\ensuremath{\\mathbbm{1}_{C_1}}}\n\nIt will be helpful for computations later on to have a more thorough understanding of $W$. We can understand $W$ better by considering a random variable $Y$ which takes on the values $a_1,a_2,\\dots,a_m$ with equal probability $\\frac{1}{m}$. Then let $\\indi$ denote a random variable indicating $Y\\in C_1$. That is, $\\indi=1$ if $Y\\in C_1$ while $\\indi=0$ otherwise. Now it is a direct consequence of the definition of conditional variance that \n\\[\nW = \\left(\\frac{1}{\\text{Var}(Y)}\\right)\\underset{C_{1}}{\\min} E[\\text{Var}(Y|\\indi)].\n\\]\nNotice that the contribution of $C_2$ to $W$ is handled by the case where $\\indi=0$. This allows us to understand $W$ in the context of the law of total variance which in this case states\n\\[\n\\text{Var}(Y)=E[\\text{Var}(Y|\\indi)]+\\text{Var}(E[Y|\\indi]).\n\\]\nFrom this perspective, $W$ is precisely the fraction of the total variance in $Y$ that cannot be explained by a classifier of the form $\\indi$. More familiar to the reader may be the complement of $W$, \n\\[\n\\frac{\\text{Var}(E[Y|\\indi])}{\\text{Var}(Y)},\n\\]\nwhich is precisely the coefficient of determination $R^2$ used in regression analysis, in this case treating $Y$ as a function of $\\indi$. That is,\n\\[\nW=1-\\underset{C_1,C_2}{\\max} R^2.\n\\]\nBesides connecting these two quantities, this also gives us a significantly easier way of computing $W$. Notice that $E[Y|\\indi=1]=\\mu_1$ while $E[Y|\\indi=0]=\\mu_2$. Hence,\n\\begin{align*}\n R^2\n &= \\text{Var}(\\mu_1 \\indi + \\mu_2 (1-\\indi)),\\\\\n &= (\\mu_1-\\mu_2)^2\\text{Var}(\\indi),\\\\\n &= (\\mu_1-\\mu_2)^2 \\tfrac{|C_1||C_2|}{m^2}.\n\\end{align*}\nIn dimensions 2 or greater, the problem of minimizing $W$ is NP-Hard; a good algorithm is presented in \\cite{fast_k_means} in one dimension. Since we are interested only in the specific case of a binary clustering in one dimension and our number of data points $m$ is small, it is fast and practical to simply compute $R^2$ for every possible clustering $C_1,C_2$ and choose the best value. The clusters $C_1$ and $C_2$ must be convex, so in one dimension they must each lie within an interval; this means that the possible clusterings can be found by sorting the points $a_i$ and dividing by taking a threshold. In this way we can use an online method to compute all of the means, so computing all the $R^2$ values takes only $O(m)$ time. The most costly aspect of the procedure is sorting, which takes $O(m\\log m)$ time.\n\n\\subsubsection{Typical values of $W$} \\label{sect:typical_w_values}\nIn order to assess whether a given value of $W$ is significant we need to know what values would be typical given a sample from a typical distribution of points $X$. Here we provide a good approximation for the distribution of $\\mathbf{W}$ assuming the points $X$ were drawn from a Gaussian distribution.\n\nNotice that if $Z$ is a standard Gaussian random variable, then $E[\\text{Var}(Z|Z20$ the quantity $\\left(\\mathbf{W} - \\sigma^2+\\frac{1}{n}\\right)\/\\sqrt{\\frac{\\kappa^2}{n} - \\frac{0.4}{n^{1.9}}}$ is very nearly distributed as a standard Gaussian.}\\label{fig:w_limiting_distribution}\n\\end{figure}\n\nThis allows us to approximate the distribution of $W$ accurately. To see this, compare the distributions shown in \\cref{fig:w_limiting_distribution}. In this demonstration for each value of $n$ and 100,000 times, $n$ samples were drawn from a standard Gaussian distribution and the labeled quantity computed. We can see in \\cref{fig:w_limiting_distribution_half_normal} and \\cref{fig:w_limiting_distribution_unadjusted} that $\\text{Var}(|Y|)\/\\text{Var}(Y)$ is in fact very close to $\\mathbf{W}$ in terms of its limiting distribution. We can also see that while $\\frac{\\sqrt{n}}{\\kappa}\\left[\\mathbf{W} - \\sigma^2\\right]$ does approach a standard Gaussian, it is biased for low values of $n$. We can correct for this bias by adding a factor which disappears as $n$ increases; this is shown in \\cref{fig:w_limiting_distribution_centered}. We can go a bit further in correcting the variance for low $n$, accelerating this convergence with another small-$n$ correction demonstrated in \\cref{fig:w_limiting_distribution_scaled}. This gives us a good rule for approximating $\\mathbf{W}$ as a Gaussian random variable with mean $\\sigma^2-\\frac{1}{n}$ and variance $\\frac{\\kappa^2}{n}-\\frac{0.4}{n^{1.9}}$.\n\nSome justification is due for these empirically determined corrections. They were found in the following manner. First, several distributions of $\\mathbf{W}$ were simulated, with 100,000 samples for each value of $n$ from 5 to 99. The mean and the variance were computed for each value of $n$. The deviation from the theoretical limiting mean of $\\sigma^2$ and the theoretical limiting variance of $\\kappa^2\/n$ were recorded, and a linear regression was performed on a log-log scale. The corrections used are based on the best-fit lines found, with constants reported to one significant figure. The regressions for the mean and variance had $r^2$ values of $0.999$ and $0.998$ respectively.\n\nThis gives us a way to evaluate a $W$ score on other, possibly non-Gaussian data. For any value $w$ we can use this approximation to give a $p$ value for the probability $\\mathbf{W}0}, where we classify the variants of the form $\\PBd^{\\alpha}_{n}$, where the sandwich element $\\alpha$ has non-zero rank. The remaining case, when the rank of the sandwich element is zero, is considered in Section \\ref{s:rank_zero}.\n\n\n\\section{Preliminaries}\n\\label{s:Prelimi}\n\nLet $S$ be a semigroup. Recall that Green's preorders on $S$ are defined, for $x,y\\in S$ by\n\\begin{equation*}\n \\begin{gathered}\n x\\leq_{\\Rr} y \\Leftrightarrow xS^{1}\\subseteq yS^{1}, \\hspace{1cm}\n x\\leq_{\\Ll} y \\Leftrightarrow S^{1}x\\subseteq S^{1}y, \\hspace{1cm}\n x\\leq_{\\Jj} y \\Leftrightarrow S^{1}xS^{1}\\subseteq S^{1}yS^{1},\\\\\n \\end{gathered}\n\\end{equation*}\nwhere $S^{1}$ is the monoid obtained from $S$ by adjoining an identity element 1, if necessary. Then, Green's relations of $S$ are defined as follows: for $\\Kk\\in\\{\\Rr,\\Ll,\\Jj\\}$, we define $\\Kk=\\leq_{\\Kk}\\cap\\geq_{\\Kk}$, and we combine these to obtain $\\Hh=\\Rr\\cap\\Ll$ and $\\Dd=\\Rr\\circ\\Ll=\\Ll\\circ\\Rr$. These relations are clearly equivalences. For $x\\in S$, and for $\\Kk\\in\\{\\Rr,\\Ll,\\Hh,\\Dd,\\Jj\\}$, let $\\Kkk_{x}$ denote the $\\Kk$-class of $S$ containing $x$. \n\nFor $T\\subseteq S$, let $\\Ee(T)=\\{x\\in T:x=x^{2}\\}$ denote the set of all idempotents of $S$ that belong to $T$. An element $x\\in S$ is \\emph{regular} if $x=xyx$ and $y=yxy$ for some $y\\in S$, and $\\Reg(S)$ denotes the set of all regular elements in $S$. It is well-known that for $x\\in\\Reg(S)$ we have $\\Ddd_{x}\\subseteq\\Reg(S)$.\n\nIt is easily seen that any semigroup homomorphism preserves Green's classes and maps idempotents to idempotents, as well as regular elements to regular elements.\nIf $S$ is a monoid with identity $1$, let\n$G(S)=\\{x\\in S: (\\exists y\\in S)\\ xy=yx=1\\}$ denote the group of units of $S$. \n\nNext, let $S$ be a semigroup, and consider the variant $S^{a}=(S,\\star_{a})$, where\n$x\\star_{a}y=xay$ for $x,y\\in S$. To avoid confusion, we denote the Green's relations of $S^{a}$ by $\\Rr^{a}$, $\\Ll^{a}$, $\\Hh^{a}$, $\\Dd^{a}$ and $\\Jj^{a}$. Similarly, for each $\\Kk\\in\\{\\Rr,\\Ll,\\Hh,\\Dd,\\Jj\\}$ and each $x\\in S$, $\\Kkk^{a}_{x}$ denotes the $\\Kk^{a}$-class of $S^{a}$ containing $x$. As in \\cite{SandwichI}, we define the \\emph{P-sets} of $S^{a}$:\n\\begin{equation*}\n \\begin{aligned}\n P^{a}_{1}=&\\{x\\in S:xa \\Rr x\\} \\hspace{2cm} P^{a}_{3}=\\{x\\in S:axa\\Jj x\\}\\\\\n P^{a}_{2}=&\\{x\\in S:ax \\Ll x\\} \\hspace{2cm} P^{a}=P^{a}_{1} \\cap P^{a}_{2}\n \\end{aligned}\n\\end{equation*}\nThese sets shape the Green's classes of $S^{a}$, as proved in \\cite[Theorem 2.13]{DolinkaEastMat} (the partial semigroup in the proof is simply the semigroup $S$). We state the result for convenience.\n\\begin{thm}\n\\label{t:GreenSandSgp}\n Let $S$ be a semigroup with $a\\in S$. In the variant $S^{a}$, we have\n \\setlength{\\columnsep}{-0.7cm}\n \\begin{multicols}{2}\n \\begin{enumerate}[(i)]\n \\item $\\Rrr^{a}_{x}=\\left\\{\n \\begin{array}{ll}\n \\Rrr_{x}\\cap \\Pp^{a}_{1}, & \\hbox{\\ if $x\\in \\Pp^{a}_{1}$} \\\\\n \\{x\\}, & \\hbox{\\ if $x\\in S\\setminus \\Pp^{a}_{1}$,}\n \\end{array}\n \\right.$\n \\item $\\Lll^{a}_{x}=\\left\\{\n \\begin{array}{ll}\n \\Lll_{x}\\cap \\Pp^{a}_{2}, & \\hbox{\\ if $x\\in \\Pp^{a}_{2}$} \\\\\n \\{x\\}, & \\hbox{\\ if $x\\in S\\setminus \\Pp^{a}_{2}$,}\n \\end{array}\n \\right.$\n \\item $\\Hhh^{a}_{x}=\\left\\{\n \\begin{array}{ll}\n \\Hhh_{x}, & \\hbox{\\ \\ \\ \\ \\ if $x\\in \\Pp^{a}$} \\\\\n \\{x\\}, & \\hbox{\\ \\ \\ \\ \\ if $x\\in S\\setminus \\Pp^{a}$,}\n \\end{array}\n \\right.$\n \\item $\\Ddd_{x}^{a}=\\left\\{\n \\begin{array}{ll}\n \\Ddd_{x}\\cap \\Pp^{a}, & \\hbox{if $x\\in \\Pp^{a}$} \\\\\n \\Lll^{a}_{x}, & \\hbox{if $x\\in \\Pp^{a}_{2}\\setminus \\Pp^{a}_{1}$} \\\\\n \\Rrr^{a}_{x}, & \\hbox{if $x\\in \\Pp^{a}_{1}\\setminus \\Pp^{a}_{2}$} \\\\\n \\{x\\}, & \\hbox{if $x\\in S\\setminus (\\Pp^{a}_{1}\\cup \\Pp^{a}_{2})$,}\n \\end{array}\n \\right.$\n \\item $\\Jjj^{a}_{x}=\\left\\{\n \\begin{array}{ll}\n \\Jjj_{x}\\cap \\Pp^{a}_{3}, & \\hbox{\\ \\ if $x\\in \\Pp^{a}_{3}$} \\\\\n \\Ddd^{a}_{x}, & \\hbox{\\ \\ if $x\\in S\\setminus \\Pp^{a}_{3}$.}\n \\end{array}\n \\right.$ \n \\end{enumerate}\n \\end{multicols}\n If $x\\in S\\setminus \\Pp^{a}$, then $\\Hhh^{a}_{x}=\\{x\\}$ is a non-group $\\Hh^{a}$-class in $S^{a}$.\n\\end{thm}\n\n\n\\section{Partial Brauer monoids}\n\\label{s:PartialBrauerMonoids}\n\n\nLet $\\mathbb{N}=\\{0,1,2,\\ldots\\}$ denote the set of all natural numbers. For an integer $n\\geq 1$ write $[n]=\\{1,\\ldots,n\\}$, and write $[0]=\\emptyset$. For $A\\subseteq \\mathbb{N}$, let $A'=\\{a':a\\in A\\}$. Now, for $n\\in\\mathbb{N}$, let \n$\\PBd_{n}$ denote the set of all partitions of the set $[n]\\cup [n]'$ into blocks of size at most 2. A partition $\\alpha\\in\\PBd_{n}$ may be visually presented in the form of a diagram, consisting of two rows of $n$ vertices corresponding the elements of $[n]$ and $[n]'$ (increasing from left to right), where the vertex $i$ is directly above the vertex $i'$, and the elements of the same block are connected by a line drawn inside the rectangle formed by these vertices. In Figure \\ref{f:exampleDiagMult}, we present such diagrams for the partitions:\n\\begin{equation*}\n \\begin{aligned}\n \\alpha&=\\{ \\{1,5\\},\\{2\\},\\{3,2'\\},\\{4\\},\\{6,5'\\},\\{7,7'\\},\\{1',6'\\},\\{3',4'\\} \\}\\in\\PBd_{7},\\\\\n \\beta&=\\{ \\{1,2\\},\\{3,2'\\},\\{4\\},\\{5,7\\},\\{6,6'\\},\\{1',3'\\},\\{4',7'\\},\\{5'\\} \\}\\in\\PBd_{7}.\n \\end{aligned}\n\\end{equation*}\nIt is easily seen that the diagram representing a partial Brauer partition is unique in terms of vertices and edges it contains (which is not necessarily true for partitions of other types). Note that we identify the elements of the symmetric group $\\mathcal{S}_{n}$ with the corresponding elements of the group of units $G(\\PBd_{n})$.\n\n\nBlocks containing elements of both sets ($[n]$ and $[n]'$) are called \\emph{transversals}. All other blocks are \\emph{non-transversals}. Non-singleton, non-transversal blocks are called \\emph{hooks} (upper or lower, if their elements belong to $[n]$ or $[n]'$, respectively). The number of transversals in $\\alpha$ is the \\emph{rank} of $\\alpha$, denoted $\\rank(\\alpha)$. Further, for $\\alpha\\in\\PBd_{n}$, we define the \\emph{domain}, \\emph{codomain}, \\emph{kernel}, \\emph{cokernel} and the sets of all nontransversal upper and lower blocks:\n\\begin{equation*}\n \\begin{aligned}\n \\dom(\\alpha) &= \\{x\\in [n]:x \\text{ belongs to a transversal of }\\alpha\\},\\\\\n \\codom(\\alpha) &= \\{x\\in [n]:x' \\text{ belongs to a transversal of }\\alpha\\},\\\\\n \\ker(\\alpha) &= \\{(x,y)\\in [n]\\times[n]: x \\text{ and } y \\text{ belong to the same block of } \\alpha\\},\\\\\n \\coker(\\alpha) &= \\{(x,y)\\in [n]\\times[n]: x' \\text{ and } y' \\text{ belong to the same block of } \\alpha\\},\\\\\n \\Nu(\\alpha) &= \\{X\\in\\alpha: X \\text{ is an upper non-transversal block of }\\alpha\\},\\\\\n \\Nl(\\alpha) &= \\{X\\in\\alpha: X' \\text{ is a lower non-transversal block of }\\alpha\\}.\n \\end{aligned}\n\\end{equation*}\nFor example, in the partition $\\alpha$ from Figure \\ref{f:exampleDiagMult}, we have $\\rank(\\alpha)=3$, $\\dom(\\alpha)=\\{3,6,7\\}$, $\\codom(\\alpha)=\\{2,5,7\\}$, and the non-trivial kernel class (the only upper hook) is $\\{1,5\\}$, while the non-trivial cokernel classes (i.e.\\ the lower hooks) are $\\{1,6\\}$ and $\\{3,4\\}$.\n\n\nFor $n\\in\\mathbb{N}$ and partitions $\\alpha,\\beta\\in\\PBd_{n}$, we define the product diagram $\\Pi(\\alpha,\\beta)$ in the following way: \n\\begin{itemize}\n \\item we modify the diagram representing $\\alpha$ by renaming each lower vertex $x'\\in[n]'$ to $x''$, hence obtaining the graph $\\alpha_{\\downarrow}$ on $[n]\\cup [n]''$;\n \\item we modify the diagram representing $\\beta$ by renaming each upper vertex $x\\in[n]$ to $x''$, hence obtaining the graph $\\beta^{\\uparrow}$ on $[n]''\\cup [n]$;\n \\item we identify the vertices of the set $[n]''$ in $\\alpha_{\\downarrow}$ with the corresponding vertices of $[n]''$ in $\\beta^{\\uparrow}$, and obtain the graph $\\Pi(\\alpha,\\beta)$.\n\\end{itemize}\nFinally, the product partition $\\alpha\\beta$ of $\\alpha$ and $\\beta$ is the partial Brauer partition on $[n]\\cup[n]'$ defined in the following way: for distinct $i,j\\in[n]\\cup[n]'$ we have $\\{i,j\\}\\in\\alpha\\beta$ if, and only if, vertices $i$ and $j$ in $\\Pi(\\alpha,\\beta)$ are connected by a path. In Figure \\ref{f:exampleDiagMult}, we provide an example illustrating this calculation.\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.7\\textwidth]{Multiplication_of_Partitiions_NEW.pdf}\n \\caption{Multiplication of partitions $\\alpha$ and $\\beta$ via the product diagram $\\Pi(\\alpha,\\beta)$.}\n \\label{f:exampleDiagMult}\n \\end{center}\n\\end{figure}\n\nIn addition to these standard notions, we will also need some novel ones, in order to present our results effectively. As in \\cite{SandwichIII}, we say an equivalence $\\varepsilon$ is a \\emph{1-2-equivalence} if each $\\varepsilon$-class has size at most $2$.\nWe introduce a new term, tailored to partial Brauer partitions. Let $\\varepsilon$ be an equivalence on a set $T$, and let $X\\subseteq T$. The pair $(\\varepsilon,X)$ is a \\emph{PB-pair} on $T$, if $\\varepsilon$ is a 1-2-equivalence, and each element of $X$ belongs to a singleton $\\varepsilon$-class. Note that any PB-pair on $[n]$ is a kernel-domain pair $(\\ker(\\alpha),\\dom(\\alpha))$, for some $\\alpha\\in\\PBd_{n}$ (and similarly, there exists $\\beta\\in\\PBd_{n}$ such that $(\\coker(\\beta),\\codom(\\beta))$ is the targeted pair). Thus, the elements of $X$ are called the \\emph{domain} elements of the PB-pair, and $|X|$ is the \\emph{rank} of the PB-pair. \n\nLet $(\\varepsilon_{1},X_{1})$ and $(\\varepsilon_{2},X_{2})$ be PB-pairs on a set $T$ and consider $(\\varepsilon_{1}\\vee\\varepsilon_{2}, Z)$, where \n$$Z=\\{(x,y)\\in X_{1}\\times X_{2}:x \\text{ and } y \\text{ belong to the same class of } \\varepsilon_{1}\\vee\\varepsilon_{2}\\}.$$\nWe say that $(\\varepsilon_{1}\\vee\\varepsilon_{2}, Z)$ is the \\emph{join} of the PB-pairs $(\\varepsilon_{1},X_{1})$ and $(\\varepsilon_{2},X_{2})$, and we denote it $(\\varepsilon_{1},X_{1})\\vee(\\varepsilon_{2},X_{2})$.\nNote that, in general, the join of two PB-pairs is not a PB-pair. The set $Z$ is called the \\emph{domain} of the join, and $|Z|$ is the \\emph{rank} of the join, denoted $\\rank((\\varepsilon_{1},X_{1})\\vee(\\varepsilon_{2},X_{2}))$. Note that the rank of the join does not depend on the order of PB-pairs (in other words, the join $(\\varepsilon_{2},X_{2})\\vee(\\varepsilon_{1},X_{1})$ has the same rank). Furthermore, each element of $X_{1}$ ($X_{2}$) occurs in at most one pair of $Z$, as $\\varepsilon_{1}$ and $\\varepsilon_{2}$ are 1-2-equivalences, and elements of the domains belong to singleton classes of the corresponding equivalence. Just as a PB-pair represents a half of some partition, the join of PB-pairs represents identifying vertices in some product diagram (where $\\varepsilon_{1}\\vee\\varepsilon_{2}$ is the resulting equivalence on the middle row and $Z$ contains the terminal vertices of the paths that will determine the transversals of the product). \n\nTo visually present a PB-pair $(\\varepsilon, X)$ on a finite set $T$, we will use the same technique as for diagrams. We arrange $|T|$ vertices in a row, and identify them with the elements of $T$ (if $T=[n]$ for some $n\\in\\mathbb{N}$, we arrange them in the ascending order). Then, elements belonging to the same $\\varepsilon$-class are connected by a line drawn above the vertices. Finally, each of the elements of the domain is the starting point of an upward straight line. Effectively, it means drawing a half-diagram, as in Figure \\ref{f:sigmacoker}.\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.7\\textwidth]{PB_pair_visual_presentation.pdf}\n \\caption{A visual presentation of a PB-pair.} \n \\label{f:sigmacoker}\n \\end{center}\n\\end{figure}\nWe may also visually present the join of two PB-pairs. This will correspond to the middle part of a product diagram. Namely, in the join of the PB-pairs $(\\varepsilon_{1},X_{1})$ and $(\\varepsilon_{2},X_{2})$ on a set $T$, the first PB-pair is drawn in the way described above and the second one is drawn on the same set of vertices, but all the lines corresponding to the $\\varepsilon_{2}$-connections and elements of $X_{2}$ will be drawn below the vertices, as in Figure \\ref{f:joinOfPB-pairs}.\n\\begin{figure}\n \\begin{center}\n \\includegraphics[width=0.7\\textwidth]{Join_of_PB_pairs.pdf}\n \\caption{A visual presentation of the join of two PB-pairs.} \n \\label{f:joinOfPB-pairs}\n \\end{center}\n\\end{figure}\nClearly, the elements of the domain of the join correspond to the \\emph{domain paths} - paths connecting elements of $X_{1}$ and $X_{2}$ (including the trivial paths, as well). Such a path connects an upward straight line and a downward straight line, while the rest of the path is made up by hooks. In fact, such a component of the graph is of form\n\\begin{description}\n \\item[(P)\\label{(P)}] $u\\xlongleftrightarrow{\\varepsilon_{2}} w_{1}\\xlongleftrightarrow{\\varepsilon_{1}} \\cdots \\xlongleftrightarrow{\\varepsilon_{2}} w_{2k-1} \\xlongleftrightarrow{\\varepsilon_{1}} v$, for some $u\\in X_{1}$, $v\\in X_{2}$, $k\\geq 1$ and $w_{1},\\ldots,w_{2k-1}\\in[n],$\\\\[-10pt]\n\\end{description}\n(If $u=v$, we have a trivial domain path.) Note that each of the vertices $w_{1},\\ldots,w_{2k-1}$ belongs to a non-singleton $\\varepsilon_{1}$-class and to a non-singleton $\\varepsilon_{2}$-class.\n\nThis graphical construction is quite similar to graphs $\\Lambda(\\alpha)$ and $\\Gamma_{\\alpha}$ in \\cite{TwAlg2} and \\cite{TwAlg7}, respectively. In both cases the graph was to determine whether $\\alpha$ is an idempotent, which boils down to whether $\\rank(\\alpha^{2})=\\rank(\\alpha)$. Here, our graphs will be used to determine if $\\rank(\\alpha\\xi)=\\rank(\\alpha)$, which is a more general problem.\n\nIn our discussion, we will rely on the notions and conclusions made here, even without explicitly mentioning the visual presentations. \n\n\n\\section{Variants of $\\PBd_{n}$}\n\\label{s:Variants}\n\nLet $n\\in\\mathbb{N}$ and let $\\alpha\\in\\PBd_{n}$. Put $r=\\rank \\alpha$ and consider the variant $\\PBd^{\\alpha}_{n}$. In \\cite[Proposition 5.7]{SandwichIII} it was proved that\n\\begin{equation}\n\\label{e:PsetsPB}\n \\begin{aligned}\n \\Pp^{\\alpha}_{1}&=\\{\\xi:\\PBd_{n}:\\rank(\\xi\\alpha)=\\rank(\\xi)\\},\\\\\n \\Pp^{\\alpha}_{2}&=\\{\\xi:\\PBd_{n}:\\rank(\\alpha\\xi)=\\rank(\\xi)\\},\\\\\n \\Reg(\\PBd^{\\alpha}_{n})=\\Pp^{\\alpha}=\\Pp^{\\alpha}_{3}&=\n \\{\\xi\\in\\PBd_{n}:\\rank(\\xi\\alpha)=\\rank(\\alpha\\xi)=\\rank(\\xi)\\}\\\\\n &=\\{\\xi\\in\\PBd_{n}:\\rank(\\alpha\\xi\\alpha)=\\rank(\\xi)\\}.\n \\end{aligned}\n\\end{equation}\nFurthermore, we have\n\\begin{prop}\n\\label{p:regClassesPB}\n If $\\xi\\in\\Pp^{\\alpha}=\\Reg(\\PBd^{\\alpha}_{n})$, then\n \\begin{enumerate}[(i)]\n \\item $\\Rrr^{\\alpha}_{\\xi}=\\Rrr_{\\xi}\\cap\\Pp^{\\alpha}=\\{\\sigma\\in\\Pp^{\\alpha}:\\ker(\\sigma)=\\ker(\\xi), \\text{ and } \\dom(\\sigma)=\\dom(\\xi)\\}$,\n \\item $\\Lll^{\\alpha}_{\\xi}=\\Lll_{\\xi}\\cap\\Pp^{\\alpha}=\\{\\sigma\\in\\Pp^{\\alpha}:\\coker(\\sigma)=\\coker(\\xi), \\text{ and } \\codom(\\sigma)=\\codom(\\xi)\\}$,\n \\item $\\Ddd^{\\alpha}_{\\xi}=\\Ddd_{\\xi}\\cap\\Pp^{\\alpha}=\\{\\sigma\\in\\Pp^{\\alpha}:\\rank(\\sigma)=\\rank(\\xi)\\}$.\n \\end{enumerate}\n Thus, the regular $\\Jj^{\\alpha}=\\Dd^{\\alpha}$-classes of $\\PBd^{\\alpha}_{n}$ are precisely the sets\n \\begin{equation*}\n \\Ddd^{\\alpha}_{k}=\\Ddd_{k}\\cap\\Pp^{\\alpha}=\\{\\xi\\in\\Pp^{\\alpha}:\n \\rank(\\xi)=k\\}\\hspace{1cm} \\text{ for each } 0\\leq k\\leq r.\n \\end{equation*}\n These form a chain under the usual ordering $\\Jj^{\\alpha}$-classes: $\\Ddd^{\\alpha}_{k}\\leq\\Ddd^{\\alpha}_{l}\\Leftrightarrow k\\leq l$.\n\\end{prop}\n\\begin{proof}\n Part $(iii)$ follows from \\cite[Proposition 5.8]{SandwichIII}. We prove only $(i)$, as the proof for $(ii)$ is dual. Let $\\xi\\in\\Pp^{\\alpha}$. Since $\\Rrr^{\\alpha}_{\\xi}\\subseteq\\Ddd^{\\alpha}_{\\xi}\\subseteq\\Reg(\\PBd^{\\alpha}_{n})=\\Pp^{\\alpha}$, Theorem \\ref{t:GreenSandSgp}(i) gives\n $$\\Rrr^{\\alpha}_{\\xi}=\\Rrr^{\\alpha}_{\\xi}\\cap\\Pp^{\\alpha}=\\Rrr_{\\xi}\\cap\\Pp^{\\alpha}_{1}\\cap\\Pp^{\\alpha}=\\Rrr_{\\xi}\\cap\\Pp^{\\alpha}.$$\n The last equality in $(i)$ follows from \\cite[Theorem 4.9(iv)]{SandwichIII}. The statement about the regular $\\Dd^{\\alpha}$-classes was proved in \\cite[Proposition 5.8(i)]{SandwichIII}.\n\\end{proof}\n\nNow, we consider a regular $\\Dd^{\\alpha}$-class, with the aim to calculate the number of $\\Ll^{\\alpha}$-classes in it. Let $0\\leq q\\leq r=\\rank(\\alpha)$, and consider an $\\Ll^{\\alpha}$-class in $\\Ddd^{\\alpha}_{q}$. By Lemma \\ref{p:regClassesPB}(ii), such a class is uniquely determined by the properties of the lower row of its elements (the cokernel-codomain combination). If we want to enumerate these classes, we need to know which cokernel-codomain pairs occur in elements of $\\Ddd^{\\alpha}_{q}$. Such a pair is a PB-pair on $[n]=\\{1,\\ldots,n\\}$. \n\nIn order to calculate the number of these PB-pairs, we will introduce additional notation. For $m\\in\\mathbb{N}$ and $0\\leq k\\leq m$ with $k\\equiv m\\ (\\mod1 2)$, let us fix the equivalence $\\varepsilon_{m,k}$ with classes $\\{1\\},\\ldots,\\{k\\},\\{k+1,k+2\\},\\ldots,\\{m-1,m\\}$. Furthermore, for for $m,k,t,q\\in\\mathbb{N}$, let $\\mu(m,k,t,q)$ be the number of PB-pairs $(\\eta,X)$ such that $|X|=q$ and $\\rank((\\varepsilon_{m,k},[t])\\vee(\\eta,X))=q$. If the equivalence $\\varepsilon_{m,k}$ is undefined, or $(\\varepsilon_{m,k},[t])$ is not a PB-pair, we fix $\\mu(m,k,t,q)=0$. Note that, by symmetry, $\\mu(m,k,t,q)$ is also the number of PB-pairs $(\\eta,X)$ such that $|X|=q$ and the rank of $(\\eta,X)\\vee(\\varepsilon_{m,k},[t])$ is $q$.\nNow, we may prove the following two lemmas.\n\n\\begin{lem}\\label{l:ddd\/ll}\n Let $n\\in\\mathbb{N}$ and $\\alpha\\in\\PBd_{n}$. Put $r=\\rank \\alpha$ and let $k$ denote the number of singleton classes in $\\ker(\\alpha)$. For $0\\leq q\\leq r$, in the variant $\\PBd^{\\alpha}_{n}$, we have $|\\Ddd^{\\alpha}_{q}\/\\Ll^{\\alpha}|=\\mu(n,k,r,q)$.\n\\end{lem}\n\\begin{proof}\n Let $0\\leq q\\leq r=\\rank(\\alpha)$. As we noted above, $|\\Ddd^{\\alpha}_{q}\/\\Ll^{\\alpha}|$ is the number of all PB-pairs on $[n]$ that occur as cokernel-codomain pairs in the elements of $\\Ddd^{\\alpha}_{q}$. Let $(\\eta,X)$ be such a PB-pair. Firstly, note that $|X|=q$, since $\\Ddd^{\\alpha}_{q}$ contains only elements of rank $q$. Secondly, note that the join $(\\eta,X)\\vee(\\ker(\\alpha),\\dom(\\alpha))$ has rank $q$. Let us elaborate. Since $\\eta=\\coker(\\xi)$ and $X=\\codom(\\xi)$ for some $\\xi\\in\\Pp^{\\alpha}\\subseteq\\Pp^{\\alpha}_{1}$, from (\\ref{e:PsetsPB}) follows $\\rank(\\xi\\alpha)=\\rank(\\xi)$, which means that the join $(\\eta,X)\\vee(\\ker(\\alpha),\\dom(\\alpha))$ has rank $|X|$.\n \n Thirdly, we show that each PB-pair $(\\eta, X)$ on $[n]$ such that\n \\begin{enumerate}[(a)]\n \\item\\label{(a1)} $|X|=q$, and\n \\item\\label{(b1)} the join $(\\eta,X)\\vee(\\ker(\\alpha),\\dom(\\alpha))$ has rank $q$,\n \\end{enumerate}\n occurs as a cokernel-codomain pair of an element from $\\Ddd^{\\alpha}_{q}$. Let $(\\eta,X)$ be a PB-pair on $[n]$ that satisfies these requirements. Let $\\beta\\in\\Ddd^{\\alpha}_{q}$, and consider a partition $\\gamma\\in\\PBd_{n}$ with %\n $$\\ker(\\gamma)=\\ker(\\beta),\\ \\dom(\\gamma)=\\dom(\\beta),\\ \\ \\text{and}\\ \\ \\coker(\\gamma)=\\eta,\\ \\codom(\\gamma)=X.$$\n %\n (such a partial Brauer partition clearly exists). Then, the first two properties and Proposition (\\ref{e:PsetsPB}) imply $\\gamma\\in\\Pp^{\\alpha}_{2}$ (because $\\rank(\\alpha\\beta)=\\rank(\\beta)$, so $\\rank(\\alpha\\gamma)=\\rank(\\gamma)$). Similarly, the last two properties and (\\ref{e:PsetsPB}) imply $\\gamma\\in\\Pp^{\\alpha}_{1}$ (because the join $(\\eta,X)\\vee(\\ker(\\alpha),\\dom(\\alpha))$ has rank $q$). Thus, we have $\\gamma\\in\\Pp^{\\alpha}$. Since $\\rank(\\gamma)=|X|=q$, from Proposition \\ref{p:regClassesPB} we deduce $\\gamma\\in\\Ddd^{\\alpha}_{q}$.\n \n We have proved that $|\\Ddd^{\\alpha}_{q}\/\\Ll^{\\alpha}|$ is the number of all PB-pairs on $[n]$ satisfying (\\ref{(a1)}) and (\\ref{(b1)}). We claim that the number of such pairs is $\\mu(m,k,r,q)$. Recall that $k$ denotes the number of singletons in the partition corresponding $\\ker(\\alpha)$. Note that the PB-pairs $(\\ker(\\alpha),\\dom(\\alpha))$ and $(\\varepsilon_{n,k},[r])$ have equal ranks and their equivalences have the same number of singletons. Thus, there exists a bijection $\\phi\\in\\Ss_{n}$ mapping $\\dom(\\alpha)$ to $[r]$, and the classes of $\\ker(\\alpha)$ to the classes of $\\varepsilon_{n,k}$. Hence, for a PB-pair $(\\eta,X)$ on $[n]$ satisfying (\\ref{(a1)}) and (\\ref{(b1)}), we may define a PB-pair $(\\eta_{\\phi},X_{\\phi})$ on $[n]$ with $\\eta_{\\phi}=\\{(x\\phi,y\\phi):(x,y)\\in\\eta\\}$ and $X_{\\phi}=\\{x\\phi:x\\in X\\}$. Since $\\phi$ maps $\\dom(\\alpha)$ to $[r]$, and since it maps the classes of $\\ker(\\alpha)$ to the classes of $\\varepsilon_{n,k}$, the rank of the join $(\\eta_{\\phi},X_{\\phi})\\vee(\\varepsilon_{n,k},[r])$ is $q$. It easily seen that $(\\eta,X)\\mapsto (\\eta_{\\phi},X_{\\phi})$ is a bijection mapping PB-pairs on $[n]$ satisfying (\\ref{(a1)}) and (\\ref{(b1)}) to PB-pairs on $[n]$ of rank $q$ such that the rank of their join with $(\\varepsilon_{m,k},[r])$ is $q$. Therefore, $|\\Ddd^{\\alpha}_{q}\/\\Ll^{\\alpha}|=\\mu(m,k,r,q)$.\n\\end{proof}\n\n\nOf course, the dual statement immediately follows: \n\\begin{lem}\n\\label{l:ddd\/rr}\n Let $n\\in\\mathbb{N}$ and $\\alpha\\in\\PBd_{n}$. Put $r=\\rank \\alpha$ and let $k$ denote the number of singleton classes in $\\coker(\\alpha)$. For $0\\leq q\\leq r$, in $\\PBd^{\\alpha}_{n}$ holds:\n $$|\\Ddd^{\\alpha}_{q}\/\\Rr^{\\alpha}|=\\mu(n,k,r,q).$$\n\\end{lem}\n\n\nIn the following lemma, we will prove a recurrence describing the numbers $\\mu(m,k,r,q)$.\n\\begin{lem}\n\\label{l:muRecurrence}\n For $n,k,r,q\\in\\mathbb{N}$, the numbers $\\mu(n,k,r,q)$ satisfy the following recurrence: \n \\begin{enumerate}[(i)]\n \\item $\\begin{aligned}[t]\\mu(n,k,r,q)=&(n-k)\\mu(n-2,k,r,q)+\\mu(n-1,k-1,r-1,q-1)+\\mu(n-1,k-1,r-1,q)+\\\\\n &(k-r)\\mu(n-2,k-2,r-1,q)+(r-1)\\mu(n-2,k-2,r-2,q)\n \\end{aligned}$\\\\\n if $n\\geq k\\geq r\\geq q>0$ and $n\\equiv k(\\mod1 2)$.\n \\item $\\mu(n,k,r,0)=\\sum^{\\lfloor \\frac{n}{2}\\rfloor}_{i=0}\\binom{n}{2i}(2i-1)!!$ if $n\\geq k\\geq r$ and $n\\equiv k(\\mod1 2)$.\n \\item $\\mu(n,k,r,q)=0$, otherwise.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\n Recall that, for $n,k,r,q\\in\\mathbb{N}$, $\\mu(n,k,r,q)$ is the number of PB-pairs $(\\eta,X)$ such that $|X|=q$ and the rank of the join $(\\varepsilon_{n,k},[r])\\vee(\\eta,X)$ is $q$. Consider the PB-pair $(\\varepsilon_{n,k},[r])$. We may present it visually as in the example in Figure \\ref{f:sigmacoker}.\n \\begin{itemize}\n \\item If $r\\geq q>0$, consider the element $r$. The question is: what role can be played by this element in $\\eta$? We have five cases:\n \\begin{description}\n \\item[Case 1:] The element $r$ is an element of $X$ (the domain of our pair). Then, we have chosen one of the $q$ elements of $X$, so the remaining $n-1$ elements can be connected to construct a suitable PB-pair in $\\lambda(n-1,k-1,r-1,q-1)$ ways.\n \\item[Case 2:] The element $r$ is a member of a singleton $\\eta$-class, outside of $X$. Then, the remaining $n-1$ elements can be connected to construct a suitable PB-pair in $\\lambda(n-1,k-1,r-1,q)$ ways.\n \\item[Case 3:] The element $r$ is connected to an element $b$ belonging to a two-element $\\varepsilon_{n,k}$-class (i.e.\\ $b\\in\\{k+1,\\ldots,n\\}$). There are $n-k$ such elements. In that case, these three elements ($r$, $b$, and $b$'s pair) may be considered as a single domain element. This element and the remaining $n-3$ elements can be connected to form a suitable PB-pair in $\\lambda(n-2,k,r,q)$ ways.\n \\item[Case 4:] The element $r$ is connected to an element $b$, which forms a singleton $\\varepsilon_{n,k}$-class and is outside of $[r]$ (i.e.\\ $b\\in\\{r+1,\\ldots,k\\}$). There are $k-r$ such elements. Then, the remaining $n-2$ elements can be connected to construct a suitable PB-pair in $\\lambda(n-2,k-2,r-1,q)$ ways.\n \\item[Case 5:] The element $r$ is connected to an element $b$, which belongs to $[r-1]$. Obviously, there are $r-1$ such elements). Then, the remaining $n-2$ elements can be connected to construct a suitable PB-pair in $\\lambda(n-2,k-2,r-2,q)$ ways.\n \\end{description}\n These cases are depicted in Figure \\ref{f:fiveCases}. We add up the values in these five cases, and obtain the same recurrence as in the case $(i)$.\n \\item If $q=0$, we fix $X=\\emptyset$, and count all the possible 1-2-partitions of the set $[n]$. Note that the number of two-element classes can be any number between $0$ and $\\lfloor \\frac{n}{2}\\rfloor$. If there are $i$ such classes, then the vertices belonging to these classes can be chosen in $\\binom{n}{2i}$ ways (the remaining vertices form singleton $\\varepsilon$-classes). These $2i$ elements can be paired in $(2i-1)!!$ ways. We obtain the same formula as in the case $(ii)$.\n \\item If $r\\mu(n,k+2,r,q).$$\n\\end{lem}\n\\begin{proof}\n Suppose $n,k,r,q\\in\\mathbb{N}$ satisfy all the assumptions of the lemma. Again, recall that $\\mu(n,k,r,q)$ is the number of PB-pairs $(\\eta,X)$ such that:\n \\begin{itemize}\n \\item[(I)] $|X|=q$ and $\\rank((\\varepsilon_{n,k},[r])\\vee(\\eta,X))=q$.\n \\end{itemize}\n Similarly, $\\mu(n,k+2,r,q)$ is the number of PB-pairs $(\\eta,X)$ such that:\n \\begin{itemize}\n \\item[(II)] $|X|=q$ and $\\rank((\\varepsilon_{n,k+2},[r])\\vee(\\eta,X))=q$.\n \\end{itemize}\n The PB pairs $(\\varepsilon_{n,k},[r])$ and $(\\varepsilon_{n,k+2},[r])$ are illustrated in Figure \\ref{f:Epsilon_nk}, using an example.\n \n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=\\textwidth]{Epsilon_nk_Blue.pdf}\n \\caption{PB-pairs $(\\varepsilon_{13,7},[4])$ (left) and $(\\varepsilon_{13,9},[4])$ (right). Note that the only difference is the upper hook connecting $8$ and $9$ (coloured blue) in the first pair.} \n \\label{f:Epsilon_nk}\n\\end{center}\n\\end{figure}\n \n First, we will prove that all PB-pairs on $[n]$ satisfying (II) also satisfy (I). Recall from the discussion in Section \\ref{s:PartialBrauerMonoids} that the domain elements of the join of two PB-pairs correspond to the paths connecting elements of their domains. Thus, in both cases we consider the domain paths connecting the elements of $[r]$ and $X$. Hence, these paths are of the form \\ref{(P)}, so none of them contains elements that belong to non-domain singletons in either of the factors. Let $(\\eta,X)$ be a PB-pair on $[n]$ satisfying (II). By the previous discussion, elements $k+1$ and $k+2$ do not belong to any of these domain paths in the join $(\\varepsilon_{n,k+2},[r])\\vee(\\eta, X)$. So, all the $q$ domain paths exist also in the join $(\\varepsilon_{n,k},[r])\\vee(\\eta, X)$. Furthermore, the connection of $k+1$ and $k+2$ in $\\varepsilon_{n,k}$ does not generate a new domain path in the join $(\\varepsilon_{n,k},[r])\\vee(\\eta, X)$, as we have $|X|=q$ and no element of $X$ can appear in two domain paths, so the join has at most $q$ domain paths. Thus, we have\n $$\\mu(n,k,r,q)\\geq\\mu(n,k+2,r,q).$$\n \n We need to prove the strict inequality. Consider the PB-pair $(\\eta,X)$, where $X=\\{1,\\ldots,q-1,k+2\\}$ and $\\eta$ is the equivalence on $[n]$ with a unique non-trivial class $\\{q,k+1\\}$. It is easily seen that $(\\eta,X)$ is a PB-pair of rank $q$. In Figure \\ref{f:lemma44}, we present the joins of $(\\eta,X)$ with $\\varepsilon_{n,k}$ and $\\varepsilon_{n,k+2}$, respectively. \n \\begin{figure}\n \\includegraphics[width=\\textwidth]{Lema_4_4_illustration.pdf}\n \\caption{The joins of PB-pairs $(\\varepsilon_{13,7},[4])$ (left) and $(\\varepsilon_{13,9},[4])$ (right) with the pair $(\\eta,X)$ in the case $q=3$. The connections belonging to the PB-pairs $(\\varepsilon_{13,7},[4])$ and $(\\varepsilon_{13,9},[4])$ are coloured black, and those that belong to $(\\eta,X)$ are red.} \n \\label{f:lemma44}\n \\end{figure}\n Note that the ranks of these joins are $q$ and $q-1$, respectively. Thus, the PB-pair $(\\xi,Y)$ satisfies (I), but not (II).\n Therefore, we may conclude that\n $$\\mu(n,k,r,q)>\\mu(n,k+2,r,q).\\qedhere$$\n\\end{proof}\n\n\n\nNote that Lemma \\ref{l:muInEq} applies only in the case when $r\\geq 1$. Namely, it is false when $r=0$, since Lemma \\ref{l:muRecurrence}(ii) implies that $\\mu(n,k,r,0)$ depends only on $n$. Thus, the case $r=0$ will be discussed separately. In order to prove the classification result for $r\\geq1$, we prove the following:\n\\begin{lem}\n\\label{l:Eq->isom}\n Let $n\\in\\mathbb{N}$, and let $\\alpha,\\beta\\in\\PBd_{n}$ with $\\rank(\\alpha)=\\rank(\\beta)$. In addition, write $k$ and $l$ for the number of singleton classes in $\\ker(\\alpha)$ and $\\ker(\\beta)$, respectively. Similarly, write $p$ and $w$ for the number of singleton classes in $\\coker(\\alpha)$ and $\\coker(\\beta)$, respectively. If $k=l$ and $p=w$, then $\\PBd^{\\alpha}_{n}\\cong\\PBd^{\\beta}_{n}$.\n\\end{lem}\n\\begin{proof}\n Suppose that $k=l$ and $p=w$. \n \n From these equalities and $\\rank(\\alpha)=\\rank(\\beta)$, we have $\\beta=\\pi_{1}\\alpha\\pi_{2}$ for some permutations $\\pi_{1},\\pi_{2}\\in\\mathcal{S}_{n}=G(\\PBd_{n})$. Then, $\\alpha\\mapsto \\pi^{-1}_{2}\\alpha\\pi^{-1}_{1}$ determines an isomorphism $\\PBd^{\\alpha}_{n}\\rightarrow\\PBd^{\\beta}_{n}$.\n\\end{proof}\n \nNow, we want to prove the classification result. In order to to that, we will need the size of the underlying set of the variant $\\PBd^{\\alpha}_{n}$. As in \\cite{SandwichIII}, we define the numbers $a(k)$ by $a(0)=a(1)=1$ and $a(k)=a(k-1)+(k-1)a(k-2)$ for $k\\geq 2$. It is easy to see that $a(k)\\geq 1$ for all $k\\in\\mathbb{N}$. Thus, $a(k)>a(k-1)$ for all $k\\geq 2$.\nIn \\cite[Proposition 4.4]{SandwichIII} it was noted that \n\\begin{equation}\n\\label{e:cardPBm}\n |\\PBd_{n}|=a(2n).\n\\end{equation}\n\n\\begin{thm}\n\\label{t:IsomClass>0}\n Let $m,n\\in\\mathbb{N}$, and let $\\alpha\\in\\PBd_{m}$ and $\\beta\\in\\PBd_{n}$ with $r=\\rank(\\alpha)\\geq 1$ and $s=\\rank(\\beta)\\geq 1$. In addition, write $k$ and $l$ for the number of singleton classes in $\\ker(\\alpha)$ and $\\ker(\\beta)$, respectively. Similarly, write $p$ and $w$ for the number of singleton classes in $\\coker(\\alpha)$ and $\\coker(\\beta)$, respectively. Then $\\PBd^{\\alpha}_{m}\\cong\\PBd^{\\beta}_{n}$ if and only if $m=n$, $k=l$, $p=w$, and $r=s$.\n\\end{thm}\n\\begin{proof}\n The reverse implication follows from Lemma \\ref{l:Eq->isom}.\n \n Conversely, suppose that $\\PBd^{\\alpha}_{m}\\cong\\PBd^{\\beta}_{n}$. \n As $|\\PBd_m|=a(2m)$ and $|\\PBd_n|=a(2n)$, and $\\mathbb{N}\\rightarrow\\mathbb{N}:x\\mapsto a(2x)$ is an increasing function, we have $m=n$.\n Further, by Lemma \\ref{p:regClassesPB}, the variants $\\PBd^{\\alpha}_{m}$ and $\\PBd^{\\beta}_{n}$ have $r+1$ regular $\\Dd^{\\alpha}$-classes and $s+1$ regular $\\Dd^{\\beta}$-classes, respectively. Since isomorphisms preserve regularity and Green's classes, we have $r=s$.\n By Lemma \\ref{l:ddd\/ll}, in variants $\\PBd^{\\alpha}_{m}$ and $\\PBd^{\\beta}_{n}$, for $q=r$ we have\n $$\\mu(n,k,r,r)=|\\Ddd^{\\alpha}_{r}\/\\Ll^{\\alpha}|\\ \\ \\ \\text{and}\\ \\ \\ |\\Ddd^{\\beta}_{q}\/\\Ll^{\\beta}|=\\mu(n,l,r,r),$$\n respectively. Since $r\\geq 1$ and isomorphism preserves the number of these $\\Ll$-classes, Lemma \\ref{l:muInEq} gives $k=l$.\n Similarly, Lemma \\ref{l:ddd\/rr} and Lemma \\ref{l:muInEq} give $p=w$.\n\\end{proof}\n\n\\begin{rem}\n Note that, in the proof of Theorem \\ref{t:IsomClass>0}, we have shown that $\\PBd^{\\alpha}_{m}\\cong\\PBd^{\\beta}_{n}$ implies $m=n$ and $r=s$ (this part of the proof does not require the assumption $r,s\\geq 1$).\n\\end{rem}\n\n\\section{Sandwich element of rank zero}\n\\label{s:rank_zero}\n\nOur proof of the classification result does not apply in the case when the sandwich element has rank $0$. We consider this case separately. It is natural to expect that the same criterion holds:\n\\begin{hyp}\n\\label{h:rank0}\n Let $n\\in\\mathbb{N}$, and let $\\alpha,\\beta\\in\\PBd_{n}$ with $\\rank(\\alpha)=\\rank(\\beta)=0$. In addition, write $k$ and $l$ for the number of singleton classes in $\\ker(\\alpha)$ and $\\ker(\\beta)$, respectively. Similarly, write $p$ and $w$ for the number of singleton classes in $\\coker(\\alpha)$ and $\\coker(\\beta)$, respectively. Then $\\PBd^{\\alpha}_{n}\\cong\\PBd^{\\beta}_{n}$ if and only if $k=l$ and $p=q$.\n\\end{hyp}\n\nThe reverse implication was proved in Lemma \\ref{l:Eq->isom}. However, for the direct implication, our previous approach does not work, since we have $\\mu(n,k,0,0)=\\mu(n,l,0,0)$ even if $k\\neq l$. Furthermore, since $\\rank(\\alpha)=\\rank(\\beta)=0$, by (\\ref{e:PsetsPB}) we have\n\\begin{equation*}\n \\Pp^{\\alpha}_{1}=\\Pp^{\\alpha}_{2}=\\Pp^{\\alpha}=\\{\\sigma\\in\\PBd_{n}:\\rank(\\sigma)=0\\}=\\Pp^{\\beta}=\\Pp^{\\beta}_{1}=\\Pp^{\\beta}_{2}, \n\\end{equation*}\nwhich is the unique regular $\\Dd^{\\alpha}$- and $\\Dd^{\\beta}$-class in $\\PBd^{\\alpha}_{n}$ and $\\PBd^{\\beta}_{n}$, respectively. It is easily seen that this set is a rectangular band, where the sandwich operation coincides with the original operation (for any sandwich element).\nTherefore, we cannot hope for any progress by considering the regular subsemigroup. In addition, all the elements of $\\PBd_{n}\\setminus\\Pp^{\\alpha}$ form singleton $\\Jj^{\\alpha}$-classes ($\\Jj^{\\beta}$-classes), which are above the regular $\\Dd^{\\alpha}=\\Jj^{\\alpha}$-class ($\\Dd^{\\beta}=\\Jj^{\\beta}$-class), and no other pair of $\\Jj^{\\alpha}$-classes ($\\Jj^{\\beta}$-classes) are related. For this reason, even if $k\\neq l$, the $\\leq_{\\Jj}$-structure of the semigroups is the same.\n\nIn spite of these similarities, these variants are not necessarily isomorphic. To show that, we need to consider the Green's preorders $\\leq_{\\Ll^{\\alpha}}$ and $\\leq_{\\Rr^{\\alpha}}$ on $\\PBd^{\\alpha}_{n}$.\nSince $\\id_{n}$ is the left- and right- identity for $\\PBd_{n}$, \\cite[Remark 3.8]{SandwichIII} gives \n\\begin{equation}\n\\label{e:pom1}\n \\sigma\\leq_{\\Rr^{\\alpha}}\\tau\\Leftrightarrow \\sigma\\leq_{\\Rr}\\tau\\alpha \\ \\ \\ \\ \\ \\text{and}\\ \\ \\ \\ \\ \\sigma\\leq_{\\Rr^{\\beta}}\\tau\\Leftrightarrow \\sigma\\leq_{\\Rr}\\tau\\beta,\n\\end{equation}\nfor $\\sigma\\in\\Pp^{\\alpha}_{1}=\\Pp^{\\alpha}=\\Pp^{\\beta}=\\Pp^{\\beta}_{1}$ and $\\tau\\in\\PBd_{n}$. From \\cite[Theorem 8]{PartBr6} follows the characterisation of relation $\\leq_{\\Rr}$ in $\\PBd_{n}$:\n\\begin{equation}\n\\label{e:pom2}\n \\sigma\\leq_{\\Rr}\\tau\\Leftrightarrow \\Nu(\\sigma)\\supseteq\\Nu(\\tau) \n\\end{equation}\nfor $\\sigma,\\tau\\in\\PBd_{n}$.\nNow, we provide a simple example in which the difference in the case $k\\neq l$ is easily seen:\n\\begin{exm}\\label{ex:kraj}\n Let $n=2$ and consider $\\alpha=\\disppartnx{2}{2}{\n \\darcx12{.2}\n }$ and\n $\\beta=\\disppartnx{2}{2}{\n \\uarcx12{.2}\n \\darcx12{.2}\n }$. \n Then, $k=2$, $l=0$ and $p=w=0$. Using (\\ref{e:pom1}) and (\\ref{e:pom2}), one can easily verify that Figure \\ref{f:nonisomRank0} depicts the $\\leq_{\\Rr^{\\alpha}}$-relations and $\\leq_{\\Rr^{\\beta}}$-relations in $\\PBd^{\\alpha}_{2}$ and $\\PBd^{\\beta}_{2}$, respectively.\n Since isomorphisms preserve Green's preorders and Green's classes, we clearly have $\\PBd^{\\alpha}_{2}\\not\\cong\\PBd^{\\beta}_{2}$.\n %\n \\begin{figure}[h]\n \\begin{center}\n \\includegraphics[width=0.85\\textwidth]{NonisomorphicRank0.pdf}\n \\caption{The relations among the $\\Rr^{\\alpha}$-classes of $\\PBd^{\\alpha}_{2}$ (left) and the relations among the $\\Rr^{\\beta}$-classes of $\\PBd^{\\beta}_{2}$ (right).}\n \\label{f:nonisomRank0}\n \\end{center}\n \\end{figure}\n %\n\\end{exm}\n\nThus, in order to prove Hypothesis \\ref{h:rank0}, one should consider the distribution of non-regular $\\Rr^{\\alpha}$-classes above the regular ones and symmetrically, the distribution of non-regular $\\Ll^{\\alpha}$-classes above the regular ones. \n\nWe give an additional result which might prove useful in this direction of investigation. Recall from \\cite{Tamura} that a semigroup $S$ is an \\emph{inflation} of a semigroup $T$ if $T$ is a subsemigroup of $S$ and there exists a mapping $\\phi:S\\rightarrow T$ such that $\\phi(\\sigma)=\\sigma$ for $\\sigma\\in T$ and\n\\begin{equation*}\n \\sigma\\tau=\\phi(\\sigma)\\phi(\\tau) \\hspace{2cm} \\text{for } \\sigma,\\tau\\in S.\n\\end{equation*}\nNow, we may prove:\n\\begin{prop}\\label{p:Walter}\n Let $n\\in\\mathbb{N}$ and let $\\alpha\\in\\PBd_{n}$ with $\\rank(\\alpha)=0$. Then, the variant $\\PBd^{\\alpha}_{n}$ is the inflation of the rectangular band\n \\begin{equation}\n \\label{e:regPart}\n \\Reg(\\PBd^{\\alpha}_{n})=\\Pp^{\\alpha}=\\{\\sigma\\in\\PBd_{n}:\\rank(\\sigma)=0\\}\n \\end{equation}\n along the map $\\phi:\\PBd_{n}\\rightarrow\\Pp^{\\alpha}:\\xi\\mapsto \\xi\\alpha\\xi$.\n\\end{prop}\n\\begin{proof}\n From the discussion following Hypothesis \\ref{h:rank0} we have (\\ref{e:regPart}) and it follows that $\\Pp^{\\alpha}$ is a subsemigroup of $\\PBd^{\\alpha}_{n}$. Furthermore, since $\\rank(\\alpha)=0$, it is easily seen that $\\sigma\\alpha\\sigma=\\sigma$ for any $\\sigma\\in\\Pp^{\\alpha}$, and $\\alpha\\tau\\alpha=\\alpha$ for any $\\tau\\in\\PBd_{n}$.\n Thus, for $\\sigma,\\tau\\in\\PBd_{n}$ we have\n \\begin{equation*}\n \\phi(\\sigma)\\star_{\\sigma}\\phi(\\tau)=\n (\\sigma\\alpha\\sigma)\\alpha(\\tau\\alpha\\tau)\n =\\sigma(\\alpha\\sigma\\alpha\\tau\\alpha)\\tau\n =\\sigma\\alpha\\tau=\\sigma\\star_{\\alpha}\\tau.\\qedhere\n \\end{equation*}\n\\end{proof}\n\nIt is easily seen that any isomorphism of variants preserves the sizes of the pre-images under $\\phi$ of the elements of $\\Pp^{\\alpha}$. Thus, if variants $\\PBd^{\\alpha}_{n}$ and $\\PBd^{\\beta}_{n}$ have different sets, counted with multiplicities, of sizes of pre-images under $\\phi_{\\alpha}:\\xi\\mapsto\\xi\\star_{\\alpha}\\xi$ and $\\phi_{\\beta}:\\xi\\mapsto \\xi\\star_{\\beta}\\xi$, respectively, of the set $\\Pp^{\\alpha}=\\Pp^{\\beta}$, they are non-isomorphic. However, the reverse implication does not hold. Namely, in Example \\ref{ex:kraj}, both $\\PBd^{\\alpha}_{n}$ and $\\PBd^{\\beta}_{n}$ have the same set of sizes of the pre-images under $\\phi_{\\alpha}$ and $\\phi_{\\beta}$, respectively, $\\{1,1,3,5\\}$, but are proved to be non-isomorphic.\n\n\\section*{Acknowledgements}\n\nThis work was supported by the Serbian Ministry of Education, Science and\nTechnological Development through the Mathematical Institute of the Serbian\nAcademy of Sciences and Arts.\nThe author would like to thank the referee for their suggestions, which greatly improved the readability of the paper.\nAdditionally, the author thanks Prof.\\ James East for his immensely helpful comments, and Prof.\\ Volodymyr Mazorchuk, for suggesting Proposition \\ref{p:Walter}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzmzb b/data_all_eng_slimpj/shuffled/split2/finalzmzb new file mode 100644 index 0000000000000000000000000000000000000000..d72bf739a363aa3d78e0c4f792fd232cc655563b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzmzb @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\nSmall Unmanned Aerial Vehicle (sUAV) and photogrammetry technologies have witnessed dramatic development over the past few years, enabling rapid reconstruction of large terrain with several square kilometers. Compared with the airborne LiDAR mapping \\cite{18,19,20}, aerial photogrammetry offers an affordable solution for 3D mapping, hence attracting widespread attention from both researchers and industry practitioners for various applications \\cite{hou2021approach,chen2020semantic,32,shi2021spatial}. Recently, a handful of works \\cite{23,46,57,58,54,94,hu2021learning,hu2021sqn,56,chen2020photogrammetric,chen2020fully} have started to explore the semantic understanding of large-scale 3D point clouds, with promising results and insightful conclusions achieved. \n\n\nAlthough a number of 3D datasets \\cite{1,3,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25,26,hu2022sensaturban} have been proposed in the last decades, it remains highly challenging for individuals to complete the whole pipeline of the customized dataset production independently for three reasons. 1) The annotation of large-scale 3D data is labor-intensive and time-consuming. In contrast to 2D data annotation, annotating 3D data such as point clouds requires extensive training to navigate and operate in the 3D environment \\cite{3}. 2) Due to the limitations of hardware configurations (\\textit{e}.\\textit{g}., availability of gimbal) and survey constraints (\\textit{e}.\\textit{g}., flight altitudes and overlaps between images), the reconstructed point clouds are usually relatively small in size or have low-quality data samples (non-uniform density, holes, outliers, \\textit{etc.}), which may have a negative impact on the execution of subsequent tasks. 3) Considering the long-tail distribution of objects in the real world, the created datasets are likely to suffer from extremely imbalanced class distribution, which poses extra challenges for downstream tasks such as semantic understanding \\cite{23}.\n\nMotivated by this, we develop a fully automatic pipeline for controllable, high-quality, and photorealistic synthetic aerial photogrammetry 3D data generation. In particular, the rich annotations, including semantic and instance labels, can be generated effortlessly as byproducts of our pipeline. Specifically, the proposed data generation pipeline has the following appealing advantages: 1) Unlike other virtual gaming engine-based generation approaches \\cite{1,63}, where only limited gaming environments created by artists are used, our pipeline fully exploits existing open geospatial data sources to set up the 3D environment, with a large variety of authentic terrain shapes and building densities. 2) Considering the homogeneous architectural styles and construction materials in real-world environments, we leverage procedural modeling tools to create building models with variations and adapted different material databases to enrich the diversity of building appearances. 3) We explicitly balance the class distribution in the real world by heuristically placing 3D models of underrepresented objects in virtual environments. 4) Lastly, instead of random points sampling or ray casting \\cite{63,64,72} on the 3D surfaces, we simulate similar UAV paths over the virtual terrain as the real-world survey, followed by the photogrammetry steps to reconstruct the 3D point clouds. This ensures that the generated 3D point clouds from our pipeline have similar quality and even comparable noise as real-world aerial photogrammetry data since the exact same data collection and reconstruction processes are executed.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Figures\/Teaser2.png}\n\\vspace{-0.2cm}\n\\caption{Example point clouds in STPLS3D{} dataset. Top row: synthetic point clouds with point-wise semantic and instance annotations. Bottom row: real point clouds captured from USC.}\n\\vspace{-0.5cm}\n\\label{fig: teaser}\n\\end{figure}\n\nWith the proposed synthetic data generation pipeline, we have further built a large-scale photogrammetry 3D point cloud dataset, termed Semantic Terrain Points Labeling - Synthetic 3D (\\href{www.stpls3d.com}{STPLS3D}), which is composed of high-quality, rich-annotated point clouds from real and synthetic environments, as shown in Figure \\ref{fig: teaser}. Specifically, we first collect real-world aerial images using photogrammetry best practices with quadcopter drone flight at a low altitude with significant overlaps between adjacent photos. We then reconstructed point clouds with a 1.27 $km^2$ landscape following the standard photogrammetry pipeline. Next, we follow the same UAV path and flying pattern to generate 62 synthetic point clouds with different architectural styles, vegetation types, and terrain shapes. The synthetic dataset covers about 16 $km^2$ of the city landscape, with up to 18 fine-grained semantic classes and 14 instance classes. Extensive experiments were conducted on our STPLS3D{} dataset to validate the quality and function of the synthetic dataset. In particular, by incorporating our synthetic dataset into the training pipeline, existing deep neural architectures can achieve visible improvement on the real data, even without adopting any domain adaptation techniques. To summarize, the main contributions of our paper are listed as follows:\n\n\\vspace{-0.1cm}\n\\begin{itemize}\n\\setlength{\\itemsep}{0pt}\n\\setlength{\\parsep}{0pt}\n\\setlength{\\parskip}{0pt}\n \\item We built a unique, richly-annotated large-scale photogrammetry point clouds dataset with synthetic and real subsets, covering more than 17 $km^2$ of the city landscape.\n \\item We introduce a fully automatic pipeline for controllable, high-quality, and photorealistic synthetic aerial photogrammetry 3D data generation.\n \\item Extensive experiments demonstrate the quality and function of the generated synthetic data.\n\\end{itemize}\n\\vspace{-0.3cm}\n\n\\section{Related works}\n\\label{sec:related_work}\nHere, we provide a brief overview of existing 3D datasets; for comprehensive surveys, please refer to \\cite{2,19,20,23,24,guo2020deep}. \\noindent\\textbf{3D Real-World Datasets.} Thanks to the development of remote sensing technologies, considerable efforts have been devoted to building 3D datasets and benchmarks for semantic understanding. To capture 3D rich geometry of the real environments, previous works usually adopted RGB-D sensors \\cite{6,7,8,9,75,76} for indoor 3D scenes and utilized terrestrial scanners \\cite{3,11,77}, mobile scanners \\cite{10,12,13,14,15,16,22,79,80,81,82,83}, and aerial laser scanners\\cite{17,18,19,20,KOLLE2021100001} for outdoor environments. Additionally, researchers from remote sensing communities also collected large-scale 3D scene-based datasets (\\textit{e}.\\textit{g}., construction sites) \\cite{30,31} through photogrammetry techniques with quadcopter drones and fixed-wing UAVs as the main platform. In particular, a handful of recent works have started to mount multiple sensors together on UAVs for efficient data collection in a large district \\cite{32,33,34}. Overall, the scale of recent datasets has become increasingly large, and the content covers sufficient information for multiple purposes. However, due to the survey configuration and the specific photogrammetry software \\cite{23,24} used, noticeable drawbacks could be found in the existing released datasets, such as missing points on the vertical surfaces, large holes, and non-uniform point density, \\textit{etc.} In addition, insufficient and incorrect annotations are another common issue that could deteriorate the quality of the dataset, further leading to the inability to fairly and comprehensively evaluate the performance of deep neural models for subsequent tasks.\n\n\\begin{table}\n\\begin{center}\n\\caption{Comparison with the representative aerial datasets used for segmentation of 3D point clouds. \\protect\\textsuperscript{1}The number of categories with instance labels, \\protect\\textsuperscript{2}Labeled area.}\n\\vspace{-0.2cm}\n\\setlength{\\tabcolsep}{5pt}\n\\resizebox{1.0\\textwidth}{!}{%\n\\begin{tabular}{rcccccc}\n \\Xhline{2.0\\arrayrulewidth}\nName and Reference & \\# Semantic & \\# Instance\\textsuperscript{1} & \\# Views \/ scenes & 2D Annotations & Area\\textsuperscript{2} ($km^2$) & Sensor \\\\ \\hline\nDublinCity \\cite{18}& 13 & No & 8,504 \/ 2 & No & 2& \\multirow{3}{*}{Aerial LiDAR} \\\\\nDALES \\cite{19} & 8 & No & 1 large scene & - & 10 & \\\\\nLASDU \\cite{20} & 5 & No & 1 scene & - & 1.02 & \\\\\nSwiss3DCities \\cite{25} & 5 & No & 3 scenes & No & 2.7 & quadcopter + photogrammetry \\\\\nCampus3D \\cite{24} & 14 & 4 classes & 6 scenes & No & 1.58 & quadcopter + photogrammetry \\\\\nSensatUrban \\cite{23} & 13 & No & 3 scenes & No & 4.4 & fixed wing + photogrammetry \\\\ \\cdashline{1-7}\n\\textbf{STPLS3D - Real} & 6 & No & 16,376 \/ 4 & Yes& 1.27 & quadcopter + photogrammetry \\\\\n\\textbf{STPLS3D - SyntheticV1} & 5 & No & \\textbf{17,164} \/ 14 & Yes& 4.22 & Synthetic Aerial photogrammetry \\\\\n\\textbf{STPLS3D - SyntheticV2} & 17 & 14 classes & 13,229 \/ 24 & Yes& 5.76 & Synthetic Aerial photogrammetry \\\\\n\\textbf{STPLS3D - SyntheticV3} & \\textbf{18} & \\textbf{14 classes} & 15,888 \/ \\textbf{25} & Yes& \\textbf{6} & Synthetic Aerial photogrammetry \\\\\n\\Xhline{2.0\\arrayrulewidth}\n\\end{tabular}%\n}\n\\vspace{-0.7cm}\n\\label{tab:Overview-dataset-1}\n\\end{center}\n\\end{table}\n\n\\noindent\\textbf{3D Synthetic Datasets.} Due to the expensive data collection and annotation costs, several works have explored the possibility of creating replaceable 3D synthetic data. Specifically, earlier works typically focused on creating synthetic point clouds for individual objects \\cite{4,5,65,73,74}, while recent works have started to investigate the synthetic generation of the outdoor 3D point clouds in virtual gaming environments \\cite{1,26,63,64,70,71,72,chen2020generating,SynLiDAR,le21wacv}. However, the geometrical structure, noise pattern, and sampling scheme of these datasets are still different from the real environment, leading to visible domain gaps. Additionally, since the gaming environments were manually created by artists and designers, the spatial scale of existing synthetic datasets is also limited. By contrast, we explore the outdoor large-scale 3D scene synthesis from aerial views and photogrammetry techniques with procedurally generated virtual environments. Table \\ref{tab:Overview-dataset-1} compares the statistics of the proposed STPLS3D{} with a number of existing aerial 3D datasets.\n\n\n\n\n\n\n\n\\section{Synthetic Data Generation Pipeline}\n\\label{section:3}\n\nThe synthetic data generation pipeline is illustrated in Figure \\ref{fig: pipeline}. Overall, the main idea is to replicate the steps one would take when creating aerial photogrammetry point clouds in the real world. In particular, we focused on bringing 3D virtual assets in the simulation that is close to reality and reconstructing point clouds with similar quality as the real ones to minimize the domain gap between synthetic and real data as much as possible. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.0\\linewidth]{Figures\/Fig2_3_MC.pdf}\n \\vspace{-0.3cm}\n\\caption{The proposed synthetic data generation pipeline.}\n \\vspace{-0.2cm}\n\\label{fig: pipeline}\n\\end{figure}\n\n\n\\subsection{Procedural 3D Environment Generation}\n\\label{sec3.1}\n\nTo ensure the placements of objects in the virtual environments roughly follow the form of a real city block, we built 3D virtual environments based on the Geographic Information System (GIS) data sources \\cite{68} (\\textit{i}.\\textit{e}., building footprints, road networks, and digital surface models) that are publicly available. Specifically, 3D road segments are placed and extended along the road vectors obtained from the Open Street Map (OSM) \\cite{69}. 3D trees, vehicles, and other city furniture are placed in the scene using predefined strategies to increase its realism and diversity. Distance constraints are also heuristically introduced during object placements to avoid unrealistic situations such as the intersection between objects. Additionally, such constraints are also used to ensure the locations of each object are contextually reasonable, \\textit{i}.\\textit{e}., the vehicles, street signs, and light poles will be on or near the roads. Besides, placing the trees pure randomly throughout the environment may produce unnatural results. Therefore, trees are placed in clusters within polygonal areas procedurally generated as boundaries to simulate forests. In addition, individual trees are also placed around the buildings within a buffer to simulate the residential blocks. Please refer to Appendix \\ref{Object placement principles for creating the synthetic environment layouts} for more details on our designed object placement principles.\n\nThis study used Computer Generated Architectural (CGA) shape grammar of CityEngine-based tools to create 3D building models based on the OSM building footprints. The procedural tool automatically extruded the footprints and added architectural elements. The overall fa\u00e7ades generation and architectural element placements allow various types of 3D buildings to be generated from the same building footprint with different predefined CGA rules. Both the building types and heights were randomly assigned during the building generation process to ensure the synthetic environments cover a large spectrum of building variations.\n\n\\subsection{2D Image Rendering and 3D Reconstructions}\n\\label{2D Image Rendering and 3D Reconstructions}\n\nThe naive solution to generating a point cloud with the created 3D environments would be either directly sampling points on the 3D model surface or using a ray casting approach with predefined camera parameters. However, it produces point clouds that perfectly match the 3D virtual environment, which does not have the same quality and noise level as the data that was collected from the real world. To reduce the domain gap that exists between the sampled or ray-casted points and the real-world aerial photogrammetry point clouds, we propose to first render the 2D images in Unreal Engine 4 (UE4) using the AirSim simulator \\cite{59}. In particular, we utilized weather effects to simulate fog, wind and changing sunlight directions, so as to generate more realistic 2D images from the virtual environment. With the rendered 2D images, we then reconstructed the 3D point clouds using the off-the-shelf commercial photogrammetry software (\\textit{i.e.}, Bentley ContextCapture). In particular, we keep the software consistent with that of reconstructing point clouds from real-world photos. Please refer to Appendix \\ref{sec:ComparisonofDataQuality} for the intuitive quality comparison between the ray-casting 3D points, the synthetic photogrammetric points, and the real-world photogrammetric point clouds.\n\n\\subsection{Semantic and Instance Annotation}\nFinally, the generated synthetic point clouds are enriched with semantic and instance annotations that are automatically generated while rendering the 2D images. Note that, due to the noises introduced from the photogrammetry reconstruction process, directly casting the 2D labels to the photogrammetry point cloud will create misaligned annotations. To this end, we transfer the rendered 2D annotations to the photogrammetry point clouds with the following two steps. First, we create a proxy 3D point cloud using the ray casting method with the known intrinsic and extrinsic camera parameters and depth maps. Next, we transfer the labels from the proxy 3D point cloud to the photogrammetry points through a nearest-neighbor search algorithm, with the constraint that ground points are connected to form a large connected component and reduce the inconsistent projections due to the simulated wind effects. Though a small amount of mislabeled points may still occur at the boundaries between different objects, they did not have a significant impact while training the segmentation models in our experiments.\n\nAs shown in Figure \\ref{fig: pipeline}, the proposed pipeline can generate synthetic point clouds with semantic and instance annotations. It is worth mentioning that the instance annotations are very useful for tasks such as vegetation identification (\\textit{e}.\\textit{g}., tree segmentation \\cite{erikson2003segmentation}) and forest management (\\textit{e}.\\textit{g}., automatic tree counting \\cite{li2017deep}) since it is highly challenging or even infeasible to obtain precise instance labels in the real data (\\textit{e}.\\textit{g}., hundreds of thousands of overlapped trees need to be manually segmented from forest areas).\n\n\n\\section{Datasets}\n\nWe first conducted surveys on four real-world sites, including the University of Southern California Park Campus (USC), Wrigley Marine Science Center (WMSC) located on Catalina Island, Orange County Convention Center (OCCC), and a residential area (RA). The aerial images were collected using a crosshatch-type flight pattern with predefined overlaps ranging from 75\\%$\\sim$85\\% and flight altitudes ranging from 25m$\\sim$70m. The 3D data were reconstructed using a standard photogrammetric process and manually annotated with one of the six semantic class labels. Following that, we used our designed synthetic data generation pipeline with the same UAV flight pattern to generate an extra 62 synthetic point clouds in a wide variety of synthetic environments. In particular, three versions of the synthetic datasets were generated with different focuses. Examples of different versions of the synthetic data and real data are shown in Figure \\ref{fig: ExampleData}. Please refer to Appendix \\ref{sec:Data generation details} and \\ref{sec:Definition of Semantic Categories} for detailed discussions of our released datasets and the available semantic labels.\n\n\\begin{figure*}\n\\centering\n\t\\includegraphics[width=1.0\\linewidth]{Figures\/Fig3.pdf}\n \\vspace{-0.5cm}\n\t\\caption{Examples of our STPLS3D dataset, including the proposed Synthetic V1, Synthetic V2, Synthetic V3, and the real-world subsets. Different semantic classes are shown in different colors, as illustrated in the color legend. Note that different instances are displayed in different random colors. Best viewed in color. }\n \\vspace{-0.1cm}\n\t\\label{fig: ExampleData}\n\\end{figure*}\n\n\n\\subsection{Comparison}\n\\label{sec:Comparison}\nWe provide an empirical comparison of the cost to collect the 3D data in real-world and virtual environments. Specifically, the real-world data (1.27 $km^2$) was collected with \\textbf{over four months of team efforts} for data collection (including getting flight permits, planning and repeatedly executing the data collection process), processing, and annotation. By contrast, the synthetic data ($>$16 $km^2$) was generated by \\textbf{a single person using one desktop PC within a month} (with an Intel Core\u2122 i9-10900X CPU and an NVIDIA RTX 3090 with 24G memory). In particular, the time cost for synthetic data generation is not constrained by available workforce talent and can be parallel accelerated with additional computing resources.\n\n\\section{Experiments}\n\\label{sec:Experiments}\n\n\\subsection{Evaluation of 3D Semantic segmentation}\n\\label{Evaluation of 3D Semantic segmentation}\n\nWe selected five representative approaches, including PointTransformer \\cite{zhao2021point}, RandLA-Net \\cite{57}, SCF-Net \\cite{58}, MinkowskiNet \\cite{84}, and KPConv \\cite{46}, as the baselines to build a semantic segmentation benchmark in our STPLS3D{}. Specifically, we used the original architectures of these approaches and only adapted the data-related hyperparameters to our dataset (see Appendix \\ref{Data-Related Hyperparameters of Benchmark Methods}). The mean Intersection-over-Union (mIoU) and Overall Accuracy (oAcc) are used as the evaluation metrics. Note that the semantic categories of the synthetic datasets are inconsistent with the real-world dataset (18 vs. 6); see Appendix \\ref{Class mapping between synthetic and real data} for details of the class mapping.\n\nThree groups of experiments were conducted to investigate whether and how synthetic data impact the semantic segmentation performance of real-world data. Note that all three groups of experiments are tested on the test set of the real-world dataset (\\textit{i}.\\textit{e}., WMSC split), but trained with different settings: 1) Train in the real-world training set. 2) Train in synthetic datasets (V1-V3) only. 3) Train in both real and synthetic datasets.\n\n\\begin{table*}[!t]\n\\centering\n\\caption{Quantitative evaluation of the baselines on the \\textit{WMSC} dataset.}\n\\label{tab:WMSC}\n\\resizebox{1.0\\textwidth}{!}{%\n\\begin{tabular}{crcccccccc}\n\\Xhline{2.0\\arrayrulewidth}\n\\multirow{2}{*}{Training sets} & \\multirow{2}{*}{Methods} & \\multirow{2}{*}{\\textbf{mIoU (\\%)}} & \\multirow{2}{*}{oAcc (\\%)} & \\multicolumn{6}{c}{Per Class IoU (\\%)} \\\\ \\cline{5-10} \n & &&& Ground & Building & Tree & Car& Light pole & Fence \\\\ \\Xhline{2.0\\arrayrulewidth}\n \n\\multirow{3}{*}{Real subsets} \n & PointTransformer \\cite{zhao2021point}& 36.27 & 54.31 & 39.95 & 20.88 & 62.57 & 36.13 & 49.32 & 8.76 \\\\\n & RandLA-Net \\cite{57}& 42.33 & 60.19 & 46.13 & 24.23 & \\textbf{72.46} & 53.37 & 44.82 & 12.95 \\\\\n & SCF-Net \\cite{58}& 45.93 & \\textbf{75.75} & \\textbf{68.77} & \\textbf{37.27} & 65.49 & 51.50 & 31.22 & \\textbf{21.34} \\\\\n & MinkowskiNet \\cite{84}& \\textbf{46.52} & 70.44 & 64.22 & 29.95 & 61.33 & 45.96 & \\textbf{65.25} & 12.43 \\\\\n & KPConv \\cite{46}& 45.22 & 70.67 & 60.87 & 32.13 & 69.05 & \\textbf{53.80} & 52.08 & 3.40 \\\\ \n \\hdashline\n \n\\multirow{3}{*}{Synthetic subsets} \n & PointTransformer \\cite{zhao2021point}& 45.73 & 86.76 & 84.12 & \\textbf{73.37} & 60.60 & 16.96 & 27.23 & 12.10 \\\\\n & RandLA-Net \\cite{57}& 45.03 & 81.30 & 76.78 & 57.74 & 56.08 & 28.44 & 40.36 & 10.78 \\\\\n & SCF-Net \\cite{58}& 47.82 & 82.69 & 77.51 & 68.68 & 56.81 & 29.87 & \\textbf{42.53} & 11.52 \\\\\n & MinkowskiNet \\cite{84}& \\textbf{50.78} & 87.64 & 85.23 & 72.66 & \\textbf{64.80} & \\textbf{31.31} & 36.85 & \\textbf{13.83} \\\\\n & KPConv \\cite{46}& 49.16 & \\textbf{88.08} & \\textbf{85.50} & 70.65 & 63.84 & 28.75 & 32.97 & 13.22 \\\\ \n \\hdashline\n \n\\multirow{3}{*}{Real+Synthetic}\n & PointTransformer \\cite{zhao2021point}& 47.64 & 84.37 & 80.19 & 76.35 & 57.13 & 36.35 & 23.72 & 12.10 \\\\\n & RandLA-Net \\cite{57}& 50.53 & 86.25 & 82.90 & 66.59 & 63.77 & 33.91 & 41.84 & 14.19 \\\\\n & SCF-Net \\cite{58}& 50.65 & 83.32 & 77.80 & 58.98 & 64.86 & \\textbf{46.37} & 40.50 & 15.41 \\\\\n & MinkowskiNet \\cite{84}& 51.35 & 84.90 & 80.86 & 74.03 & 59.21 & 31.72 & \\textbf{45.51} & \\textbf{16.79} \\\\\n & KPConv \\cite{46}& \\textbf{53.73} & \\textbf{89.87} & \\textbf{87.40} & \\textbf{78.51} & \\textbf{66.18} & 39.63 & 41.30 & 9.34 \\\\ \n \\Xhline{2.0\\arrayrulewidth}\n\\end{tabular}%\n}\n\\end{table*}\n\n\nThe quantitative performance of baselines is reported in Table \\ref{tab:WMSC}. It can be seen that: 1) MinkowskiNet achieves the best overall performance with a mIoU score of 46.52\\% when only trained on the real-world dataset. 2) All baselines achieved better mIoU when trained on the synthetic dataset compared with training on the real-world dataset, despite the fact that there is inevitably a domain gap that exists. This is likely because the synthetic dataset is much larger than real data in spatial scale and contains more variations of terrain shapes and building styles. 3) All baselines achieved the best mIoU when trained on real and synthetic datasets. In particular, KPConv achieved an improvement of nearly 8\\% in mIoU score by training on the synthetic + real-world data. These results clearly validate that the synthetic datasets could have a positive impact on the performance of real-world 3D understanding. On the other hand, we also noticed that the addition of synthetic subsets into training sets leads to significant performance improvement for categories such as \\textit{ground} and \\textit{building} but with limited improvement or even worse results for small objects. This is likely due to the domain discrepancy between the real data and synthetic data. In particular, two issues need to be further addressed in future work: 1) Although we randomly assigned various materials to different objects, limited geometrical variations of 3D game objects were adopted when creating the synthetic subsets. 2) There is a lack of enforcing comprehensive contextual relationships between specific objects. For instance, cars placed off the road have random orientations in synthetic datasets, but vehicles in a parking lot are usually heading in the same direction in rows in real-world environments.\n\n\\begin{table*}[!b]\n\\centering\n\\caption{Quantitative generalization performance of baselines on the FDc dataset.}\n\\label{tab:FDc}\n\\resizebox{1.0\\textwidth}{!}{%\n\\begin{tabular}{crcccccccc}\n\\Xhline{2.0\\arrayrulewidth}\n\\multirow{2}{*}{Training sets} & \\multirow{2}{*}{Methods} & \\multicolumn{1}{c}{\\multirow{2}{*}{\\textbf{mIoU (\\%)}}} & \\multicolumn{1}{c}{\\multirow{2}{*}{oAcc (\\%)}} & \\multicolumn{6}{c}{Per Class IoU (\\%)} \\\\ \\cline{5-10} \n & & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{} & \\multicolumn{1}{c}{Ground} & \\multicolumn{1}{c}{Building} & \\multicolumn{1}{c}{Tree} & \\multicolumn{1}{c}{Car} & \\multicolumn{1}{c}{Light pole} & \\multicolumn{1}{c}{Fence} \\\\ \\Xhline{2.0\\arrayrulewidth}\n\n\\multirow{3}{*}{Real subsets}\n\n& PointTransformer \\cite{zhao2021point} & 49.40 & 85.85 & 85.23 & 47.77 & \\textbf{76.72} & 39.51& 28.61 & 18.56 \\\\\n& RandLA-Net \\cite{57} & 51.84 & 84.79 & 88.14& 46.88 & 61.40 & 48.72 & 46.04& 19.83 \\\\\n& SCF-Net \\cite{58} & 53.79 & 86.66 & \\textbf{89.19} & 53.12 & 65.28 & \\textbf{48.91}& 46.59 & 19.63 \\\\\n& MinkowskiNet \\cite{84} & 52.85 & 83.28 & 82.76& 40.30 & 71.68 & 47.00 & 49.33 & 26.04 \\\\\n& KPConv \\cite{46} & \\textbf{57.80} & \\textbf{87.20} & 86.69& \\textbf{63.41} & 66.32 & 46.36 & \\textbf{56.08} & \\textbf{27.95} \\\\ \n \n \\hdashline\n\\multirow{3}{*}{Synthetic subsets} \n& PointTransformer \\cite{zhao2021point} & 58.65 & 92.01 & 90.42& \\textbf{74.54} & 85.18 & 31.76& 42.36& 27.67 \\\\\n& RandLA-Net \\cite{57} & 59.38 & 91.33 & 90.15& 69.20 & 82.21 & 50.13 & 40.36& 24.20 \\\\\n& SCF-Net \\cite{58} & 58.82 & 90.49 & 89.53& 62.39 & 81.55 & \\textbf{52.99}& 44.10& 22.36 \\\\\n& MinkowskiNet \\cite{84} & 56.17 & 90.55 & 90.74& 66.11 & 78.63 & 36.86 & 36.41& \\textbf{28.26} \\\\\n& KPConv \\cite{46} & \\textbf{61.92} & \\textbf{92.35} & \\textbf{91.41} & 68.31 & \\textbf{86.00} & 48.97 & \\textbf{51.99} & 24.82 \\\\ \n \n \\hdashline\n\n\\multirow{3}{*}{Real+Synthetic}\n& PointTransformer \\cite{zhao2021point} & 62.14 & 91.96 & 89.74& \\textbf{74.79} & 84.73 & 45.10& 46.75& \\textbf{31.72} \\\\\n& RandLA-Net \\cite{57} & 61.38 & 92.31 & 91.25& 68.71 & 84.35 & 55.04 & 43.30& 23.83 \\\\\n& SCF-Net \\cite{58} & 61.89 & 92.10 & 90.99& 68.69 & 84.99 & \\textbf{55.58}& 45.36& 25.71 \\\\\n& MinkowskiNet \\cite{84} & 62.59 & \\textbf{93.16} & 91.66& 74.70 & \\textbf{87.97} & 48.80 & 43.95& 28.49 \\\\\n& KPConv \\cite{46} & \\textbf{65.01} & 93.03 & \\textbf{91.86} & 71.44 & 87.12 & 54.77 & \\textbf{55.39} & 29.48 \n\n \\\\ \\Xhline{2.0\\arrayrulewidth}\n\\end{tabular}%\n}\n\\end{table*}\n\n\\smallskip\\noindent\\textbf{Cross Datasets Generalization.} We further verified the generalization ability of the trained model on the photogrammetry Fort Drum cantonment (FDc) dataset (\\textit{i}.\\textit{e}., dataset \\#7 in \\cite{62}). Note that the main differences between the FDc and our STPLS3D real data are that the cold weather tree dominates the vegetation types, the aerial images were collected with smaller overlaps (50\\% to 60\\%), resulting in lower quality 3D data, and FDc contains various vehicle types including military vehicles that do not exist in the STPLS3D real data. Please refer to the Appendix \\ref{Visualization of the FDc dataset} for data visualization. \n\n\nAs shown in the quantitative results reported in Table \\ref{tab:FDc}, we can see that 1) KPConv consistently achieved the best generalization performance on the FDc dataset, regardless of the variation of training sets. 2) Similarly, all baselines achieved better generalization performance when trained on the synthetic dataset and achieved the best performance when trained on real and synthetic datasets. In particular, the generalization performance (\\textit{i}.\\textit{e}., mIoU score) of PointTransformer achieved an improvement of nearly 13\\% when augmented with synthetic datasets during training. This clearly shows that the proposed synthetic dataset is helpful for improving the generalization capacity of the trained deep learning model.\n\n\n\n\\subsection{Evaluation of 3D Instance Segmentation}\n\\label{sec:Evaluation_of_3D_Instance_Segmentation}\n\n\\qy{For instance segmentation, we selected two representative voxel-based approaches, including PointGroup \\cite{PointGroup} and HAIS \\cite{HAIS}, as the baselines to build an instance segmentation benchmark in our STPLS3D{}. Considering the large spatial size of our dataset, we first tuned the data-related hyperparameters (\\textit{i}.\\textit{e}., voxel size and cluster radius) to adapt to our dataset and then utilized the weighted loss to mitigate the class imbalance issue. We followed the common practice of using the mAP, mAP50, and mAP25 as the main evaluation metrics. The SyntheticV3 dataset was selected again for evaluation, and the quantitative results achieved by different baselines are shown in Table \\ref{tab:instance_segmentation}.}\n\n\\begin{table*}[b]\n\\setlength{\\tabcolsep}{3pt}\n\\centering\n\\caption{Quantitative evaluation of two instance segmentation baselines on the synthetic v3 subset.}\n\\label{tab:instance_segmentation}\n\\resizebox{1.0\\textwidth}{!}{%\n\\begin{tabular}{r|cc|cccccccccccccc}\n\\hline\n & \\multicolumn{1}{l}{Metric} & \\rotatebox{90}{mean (\\%)} & \\rotatebox{90}{Build.} & \\rotatebox{90}{LowVeg.} & \\rotatebox{90}{MediumVeg.} & \\rotatebox{90}{HighVeg.} & \\rotatebox{90}{Vehicle} & \\rotatebox{90}{Truck} & \\rotatebox{90}{Aircraft} & \\rotatebox{90}{MilitaryVeh.} & \\rotatebox{90}{Bike} & \\rotatebox{90}{Motorcycle} & \\rotatebox{90}{LightPole} & \\rotatebox{90}{StreetSign} & \\rotatebox{90}{Clutter} & \\rotatebox{90}{Fence} \\\\ \\hline\n\\multirow{3}{*}{HAIS\\cite{HAIS}} & AP & \\textbf{35.1} & 66.8 & 20.9 & 17.6 & 23.2 & 75.7 & 51.9 & 42.6 & 31.1 & 7.4 & 50.8 & 47.0 & 8.3 & 22.6 & 25.7 \\\\\n & AP50 & \\textbf{46.7} & 73.9 & 35.7 & 25.0 & 29.2 & 86.9 & 61.3 & 65.2 & 39.2 & 17.0 & 69.0 & 62.9 & 13.7 & 27.9 & 46.5 \\\\\n & AP25 & \\textbf{52.8} & 75.9 & 46.8 & 31.9 & 32.1 & 89.0 & 66.0 & 72.0 & 44.5 & 22.1 & 75.4 & 68.1 & 15.0 & 31.7 & 68.4 \\\\ \\hdashline\n\\multirow{3}{*}{PointGroup\\cite{PointGroup}} & AP & 23.3 & 60.0 & 11.6 & 10.7 & 19.2 & 58.7 & 39.8 & 27.6 & 21.2 & 2.2 & 12.0 & 23.7 & 8.1 & 13.9 & 18.1 \\\\\n & AP50 & 38.5 & 70.4 & 28.3 & 19.0 & 25.4 & 83.9 & 57.9 & 47.9 & 35.3 & 7.9 & 44.0 & 46.8 & 14.7 & 19.6 & 38.4 \\\\\n & AP25 & 48.6 & 73.7 & 43.8 & 23.7 & 29.5 & 87.9 & 61.4 & 59.8 & 42.3 & 19.4 & 68.1 & 66.8 & 16.6 & 22.6 & 64.9 \\\\ \\hline\n\\end{tabular}%\n}\n\n\\end{table*}\n\nIt can be seen that HAIS outperformed PointGroup and achieved the best mAP, mAP50, and mAP25 with 40.4, 51.9, and 57.3, respectively. We also noticed that the performance of both baselines on our dataset is still far inferior to that of existing indoor datasets (\\textit{i}.\\textit{e}., Scannet \\cite{9} and S3DIS \\cite{8}). We attribute this performance gap to the natural differences between the large-scale outdoor and the indoor scenes, where the size of the objects in the outdoor environments are dramatically different (\\textit{i}.\\textit{e}., buildings vs. bikes) compared to the indoor scenes. In addition, the limitation of the aerial photogrammetry technique may pose extra challenges to 3D instance segmentation, where objects that are physically close to each other may not have a clear boundary in terms of geometry and texture (\\textit{i}.\\textit{e}., 3D reconstruction of forests may result in solid blobs). With the identified challenges posed by large-scale outdoor scenes, we hope our STPLS3D{} will pave the way for future works on designing and developing more general and effective instance segmentation techniques that can also achieve satisfactory performance on outdoor scenes.\n\n\n\\section{Discussions and Limitations}\n\n \nTo facilitate the research in the community, we will release not only all of the 3D point clouds but also all byproducts and relevant data, including 2D source images, annotation masks, intrinsic and extrinsic camera parameters, depth maps, and meshes. Thus STPLS3D{} also holds great potential for supporting other computer vision-related tasks beyond 3D semantic and instance segmentation. Tasks such as neural rendering for large outdoor scenes \\cite{CityNeRF,NeRF_in_the_Wild}, style transfer for both 2D and 3D aerial data \\cite{MUNIT,UNIT,DRIT++,CUT,GCGAN,CycleGAN}, 3D scene reconstruction, and object detection can all be supported.\n\n\\smallskip\\noindent\\textbf{Limitations.} The proposed STPLS3D{} has been demonstrated to have good data quality and functions; it also has limitations. First, the generated synthetic 3D environments do not have sufficient high-level contextual priors between objects, such as generating realistic site plans for houses or placing vehicles, bikes, and motorcycles in parking lots, \\textit{etc.} Second, there is a visible domain gap between the synthetic and real-world data since the 2D appearance of the rendered images does not have the same style as the real-world images. We leave these domain adaptation issues for future exploration.\n\n\\section{Conclusion}\n \nIn this paper, we present STPLS3D{}, a large-scale aerial photogrammetry dataset with real and synthetic 3D point clouds. In particular, a fully automatic synthetic data generation pipeline is introduced to produce high-quality, richly-annotated 3D synthetic point clouds. Extensive experiments demonstrated the quality and functions of the generated synthetic datasets. Additionally, we also show that incorporating the synthetic data into the training set could be a good way of data augmentation, and the learning capacity and generalization ability of existing deep neural models could be further strengthened. Overall, synthetic data is easy to acquire and free of annotation, and potentially helpful for avoiding overfitting and generalized representation learning. We believe this is a promising research avenue for future research and hope our STPLS3D{} will inspire more research works on other tasks such as domain adaptation and pretraining.\n\\\\\n\\\\\n\\\\\n\\noindent\\textbf{Acknowledgements} The authors would like to thank the two primary sponsors of this research: US Army Simulation and Training Technology Center (STTC), and the Office of Naval Research (ONR). They would also like to acknowledge the assistance provided by Army Futures Command (AFC) and Synthetic Training Environment (STE). This work is supported by University Affiliated Research Center (UARC) award W911NF-14-D-0005. Statements and opinions expressed and content included do not necessarily reflect the position or the policy of the Government, and no official endorsement should be inferred.\n\n\\clearpage\n\n\n\n\n\n\n \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nGiven $(M^n, g)$ a compact Riemannian manifold and $E \\to M$ a vector\nbundle, a one parameter family of connections $\\nabla_t$ on $E$ is a solution to\nYang-Mills flow if\n\\begin{align*}\n\\frac{ \\partial \\nabla_t}{\\partial t} &=\\ - D_{\\nabla_t}^* F_{\\nabla_t}.\n\\end{align*}\nThis is the negative gradient flow for the Yang-Mills energy, and is a natural\ntool for investigating its variational structure. Global existence and\nconvergence of the flow in dimensions $n=2,3$ was established in \\cite{Rade}. \nFinite time singularities in dimension $n=4$ can only occur via energy\nconcentration, as established in \\cite{Struwe}. More recently this result has\nbeen refined in \\cite{Feehan, Waldron} to show concentration of the self-dual\nand antiself-dual energies. Preliminary investigations into Yang-Mills flow in\nhigher dimensions have been made in \\cite{Gastel, Naito,Weinkove}.\n\nIn this paper we establish structure theorems on the singular set for Yang-Mills\nflow in dimensions $n \\geq 4$. Our results are inspired generally by results on\nharmonic map flow, specifically \\cite{LW1, LW2, LW3}. The first main\nresult is a weak compactness theorem for solutions to Yang-Mills flow which\nincludes a rough description of the singular set of a sequence of solutions. A\nsimilar result for harmonic map flow was established in \\cite{LW1}. \nMoreover, a related result on the singularity formation at infinity for a global\nsolution of Yang-Mills flow was established in \\cite{HongTian}. We include a\nrough statement here, see Theorem \\ref{fullweakcompactthm} for the precise\nstatement.\n\n\\begin{thm} \\label{thm:weakcompactness} Fix $n\\geq 4$ and let $E \\to (M^n,g)$ be\na vector\nbundle over a closed Riemannian manifold. Weak $H^{1,2}$ limits of\nsequences of smooth solutions to Yang-Mills flow are weak solutions to\nYang-Mills flow which are smooth outside of a closed set $\\Sigma$ of locally\nfinite\n$(n-2)$-dimensional parabolic Hausdorff measure.\n\\end{thm}\n\nThe first key ingredients of the proof are localized entropy monotonicities for\nthe\nYang-Mills flow, defined in \\cite{HongTian}, together with a low-entropy\nregularity theorem \\cite{HongTian}. Fairly general methods allow for the\nexistence of\nthe weak limit claimed in Theorem \\ref{thm:weakcompactness}, and the entropy\nmonotonicities are the key to showing that the singular set is small enough to\nensure that the weak limit is a weak solution to Yang-Mills flow. The arguments\nare closely related to those appearing in\n\\cite{HongTian,LW1,Tian}.\n\nThe second main result is a stratification of the singular set. This involves\ninvestigating tangent measures associated to solutions of Yang-Mills flow. In\nparticular we are able to establish the existence of a density for these\nmeasures together with certain parabolic scaling invariance properties. One\nimmediate\nconsequence is that we can apply the general results of \\cite{White} to obtain a\nstratification of the singular set. \nSee \\S \\ref{sec:tmstrat} for the relevant definitions.\n\\begin{thm} \\label{dimredthm}\nFor $0 \\leq k \\leq n-2$ let\n\\begin{equation*}\n\\Sigma_k := \\left\\{ z_0 \\in \\Sigma \\mid \\dim \\prs{ \\Theta^0 \\prs{\\mu^*, \\cdot}}\n\\leq\nk, \\forall \\mu^* \\in T_{z_0}(\\mu) \\right\\}.\n\\end{equation*}\nThen $\\dim_{\\mathcal{P}}\\prs{\\Sigma_k} \\leq k$ and $\\Sigma_0$ is countable.\n\\end{thm}\n\nThe third main theorem characterizes the failure of strong convergence in the\nstatement of Theorem \\ref{thm:weakcompactness} in terms of the bubbling off of\nYang-Mills connections. \nAgain, an analogous result for harmonic maps was established in \\cite{LW1}. The\nproof requires significant further analysis on tangent measures, leading to the\nexistence of a refined blowup sequence which yields the Yang-Mills connection. \nWe give a rough statement below, see\nTheorem \\ref{weaktostrongH12} for the precise statement. \n\n\\begin{thm}\\label{thm:weaktostrong} Fix $n\\geq4$ and let $E \\to\n(M^n,g)$ be a vector bundle over a closed Riemannian manifold. A\nsequence of\nsolutions to Yang-Mills flow converging weakly in $H^{1,2}$ either converges\nstrongly in $H^{1,2}$, and the $(n-2)$-dimensional parabolic Hausdorff measure\nof\n$\\Sigma$ vanishes, or it admits a blowup limit which is a Yang-Mills connection\non $S^4$.\n\\end{thm}\n\nA corollary of these theorems is the existence of a either Yang-Mills connection\nor Yang-Mills soliton as a blowup limit of arbitrary finite time singularities. \nFor type I singularities the existence of soliton blowup limits was established\nin \\cite{Weinkove}, following from the entropy monotonicity for Yang-Mills flow\ndemonstrated in \\cite{Hamilton}. The existence of soliton blowup limits for\narbitrary singularities of mean curvature flow was established in\n\\cite{Ilmanen}, relying on the structure theory associated with Brakke's weak\nsolutions. A preliminary investigation into the entropy-stability of Yang-Mills\nsolitons\nwas undertaken in \\cite{CZ} and \\cite{KS}. Those results now apply to studying\narbitrary\nfinite-time singularities of Yang-Mills flow, as all admit singularity models\nwhich are either Yang-Mills connections or Yang-Mills solitons.\n\n\\begin{cor}\\label{cor:weaktostrong} Fix $n\\geq4$ and let $E \\to (M^n,g)$ be a\nvector\nbundle over\na closed Riemannian manifold. Let\n$\\nabla_t$ a smooth solution to Yang-Mills flow on $[0,T)$ such that $\\limsup_{t \\to\nT}\n\\brs{F_{\\nabla_t}}_{C^0} = \\infty$. There exist a sequence $\\{(x_i,t_i,\\lambda_i)\\}\n\\subset M \\times [0,T)\\times [0,\\infty)$ such that the corresponding blowup\nsequence converges modulo gauge transformations to\neither\n\\begin{enumerate}\n\\item A Yang-Mills connection on $S^4$.\n\\item A Yang-Mills soliton.\n\\end{enumerate}\n\\end{cor}\n\n\\subsection*{Acknowledgements}\n\nThe first author gratefully thanks Osaka University mathematics department, in\nparticular Toshiki Mabuchi and Ryushi Goto where much of the preliminary work\nwas performed, for their warm hospitality. The first author also sincerely thanks Gang Tian and all of Princeton University, where much of the intermediate and final work was conducted, for providing such a friendly and productive atmosphere. The first author was supported by an\nNSF Graduate Research Fellowship DGE-1321846. The second author was supported\nby the NSF via DMS-1341836, DMS-1454854 and by the Alfred P. Sloan Foundation\nthrough a Sloan Research Fellowship.\n\n\\section{Background}\n\nWe will begin with a discussion of notation and conventions that are used\nthroughout the paper. We will then provide general analytic\nbackground as well as a review of Yang-Mills flow and its key properties.\n\n\\subsection{Notation and conventions}\\label{ss:notconv}\n\nLet $(E,h) \\to (M,g)$ be a vector bundle over a closed Riemannian manifold. Let\n$S(E)$ denote the smooth\nsections of $E$. For each point $x \\in M$ choose a local orthonormal basis of\n$TM$ given by $\\{ \\partial_i \\}$ with dual basis $\\{ e^i \\}$ and a local basis for\n$E$ given by $\\{ \\mu_{\\alpha} \\}$ with dual basis $\\{ (\\mu^*)^{\\alpha} \\}$ for the\ndual $E^*$. Let\n$\\Lambda^p(M)$ denote the set of smooth $p$-forms over $M$ and set $\\Lambda^p(E)\n:= \\Lambda^p(M) \\otimes S(E)$. Next set $\\End E := E \\otimes E^*$, where $E^*$\ndenotes the dual space of $E$ and take\n\\begin{equation*}\n\\Lambda^p( \\Ad E) := \\{ \\omega \\in \\Lambda^p(\\End E) \\mid h_{\\alpha \\gg}\n\\omega_{\\beta}^{\\gg} = -\nh_{\\beta \\gg} \\omega_{\\alpha}^{\\gg} \\}.\n\\end{equation*}\nThe set of all bundle metric compatible connections on $E$ will be denoted by\n$\\mathcal{A}_{E}(M)$. Thus, given a chart\ncontaining $p \\in M$ the action of a\nconnection $\\nabla$ on $E$ is captured by the \\textit{coefficient\nmatrices} $\\Gamma = ( \\Gamma_{i \\alpha}^{\\beta} e^i \\otimes \\mu_{\\beta} \\otimes\n\\mu_{\\alpha}^* )$, where\n\\begin{align*}\n\\nabla \\mu_{\\beta} &= \\Gamma_{i \\beta}^{\\delta} e^i \\otimes \\mu_{\\delta}.\n\\end{align*}\nWhen\nsequences of one-parameter families of connections $\\{\\nabla_t^i \\}$ are in play we\nwill at times drop the explicit dependence on $t$ and $i$ for notational\nsimplicity.\n\n\\subsection{Weak solutions of Yang-Mills flow}\n\nWe first recall here the definitions of Sobolev spaces relevant to discussing\nconvergence of connections. Refer to (\\cite{Struwe} \\S 1.3) for further\ninformation. Using this we give the definition of a weak solution to Yang-Mills\nflow.\n\\begin{defn}\\label{defn:SobAdE} Fix $\\nabla_{\\refc}$ a background connection on $E$.\n The space $H^{l,p}(\\Lambda^i(\\Ad E))$ is the\ncompletion of the space of smooth sections of $\\Lambda^i(\\Ad E)$ with respect to\nthe norm\n\\begin{equation*}\n\\brs{\\brs{ \\Upsilon }}_{H^{l,p}(\\Lambda^i(\\Ad E))} := \\prs{\\sum_{k=0}^l\n\\brs{\\brs{ \\nabla^{(k)}_{\\refc} \\Upsilon }}_{L^p(\\Lambda^i(\\Ad E))}^p }^{1\/p} <\n\\infty.\n\\end{equation*}\nWe will say that a connection $\\nabla$ is of\n\\emph{Sobolev class $H^{l,p}$}, and write $\\nabla \\in H^{l,p}$, if $\\nabla = \\nabla_{\\refc}\n+ \\Upsilon$ where $\\Upsilon\n\\in H^{l,p}\\prs{\\Lambda^1(\\Ad E)}$.\n\\end{defn}\nNow, for a vector bundle $E \\to (M,g)$ over a Riemannian manifold, recall that\nthe\n\\emph{Yang-Mills energy} of a smooth connection $\\nabla$ on $E$ with curvature\n$F_{\\nabla}$ is\n\\begin{equation*}\\label{eq:YMe}\n\\mathcal{YM}(\\nabla) := \\frac{1}{2} \\int_{M} \\brs{F_{\\nabla}}^2 \\, dV.\n\\end{equation*}\nFrom this we can consider the corresponding negative gradient flow, which is\neasily shown to be the \n\\emph{Yang-Mills flow}:\n\\begin{gather*}\n\\begin{split}\n\\left. \\frac{\\partial \\nabla_t}{\\partial t} \\right. =&\\ - D_{\\nabla_t}^* F_{\\nabla_t}.\n\\end{split}\n\\end{gather*}\nWith these definitions in place we can now define the notion of a weak solution\nto the flow.\n\n\\begin{defn}\\label{def:YMfwksol} A one-parameter\nfamily $\\nabla_t =\n\\nabla_{0} + \\Upsilon_t$ is a \\emph{weak solution of Yang-Mills flow} on $[0,T]$ if\n\\begin{align*}\n\\Upsilon_t \\in L^1([0,T]; L^2(\\Lambda^1(\\Ad E))), \\qquad F_{\\nabla_t} \\in\nL^{\\infty}([0,T];\nL^2(\\Lambda^2(\\Ad E))),\n\\end{align*}\nand if for all $\\alpha_t \\in C^{\\infty}([0,T]; H_1^2(\\Lambda^2(\\Ad E)))$ which\nvanish at\n$t=0,\nt=T$, one has\n\\begin{align} \\label{weaksoln}\n\\int_0^T \\int_M \\ip{\\Upsilon_t, \\frac{\\partial \\alpha_t}{\\partial t}} - \\ip{F_{\\nabla_t}, \\nabla_t\n\\alpha_t}\n\\, dV\\, dt = 0.\n\\end{align}\n\\end{defn}\n\n\\subsection{Blowup constructions}\\label{ss:blowupconst}\n\nHere we will give a discussion of the construction of blowup limits in the\nsetting of Yang-Mills flow. First we define the fundamental scaling law.\n\\begin{defn}\\label{defn:rescaledconn} Fix $U \\subset \\mathbb R^n$ and consider\nthe restricted bundle $E \\to U$. Suppose $\\nabla_t$ is a smooth solution to\nYang-Mills flow over $U$ on $[0,T)$. Fixing a basis for $E$, $\\nabla_t$ is\ndescribed by local coefficient matrices\n $\\Gamma_t$. Given $z_0 = (x_0,t_0) \\in U \\times [0,T)$ and $\\lambda \\in \\mathbb{R}$ we\ndefine a connection $\\nabla^{\\lambda,z_0}_t$ via\ncoefficient matrices\n %\n\\begin{align\n \\Gamma^{\\lambda,z_0}_{t}\\prs{x} = \\lambda \\Gamma_{\\lambda^2 t + t_0} \\prs{ \\lambda x + x_0}.\n \\end{align}\n %\n Typically the basepoint $z_0$ will be suppressed notationally when understood.\n\\end{defn}\n\nNow consider a sequence $\\sqg{(x_i,t_i,\\lambda_i)} \\subset M \\times \\mathbb{R} \\times\n[0,\\infty)$ with $\\lambda_i \\to 0$. Assuming $M$ is\ncompact there exists a subsequence such that $\\{x_i\\} \\to x_{\\infty} \\in M$. \nMoreover,\nwe can pick a chart around $x_{\\infty}$ so that the tail of the\nsequence $\\{ x_i \\}$ is contained within this chart, identified with $B_1\n\\subset \\mathbb R^n$.\nFor sufficiently large $i$, define a connection $\\nabla_t^i$ via coefficient\nmatrices\n\\begin{equation*}\n\\Gamma^i_t(x):= \\Gamma^{\\lambda_i,z_i}_t(x).\n\\end{equation*}\nWe call $\\sqg{\\nabla_t^i}$ an \\emph{$(x_i,t_i,\\lambda_i)$-blowup sequence}. Note the\ncorresponding curvatures are scaled in the following manner,\n\\begin{align}\\label{eq:curvrescale}\nF_{\\nabla_t^i}(x) = \\lambda_i^2 F_{\\nabla_{\\lambda_i^2 t + t_i}} \\prs{\\lambda_ix + x_i}.\n\\end{align}\nObserve that the domain of $\\nabla^i_t$ contains $B_{\\lambda_i^{-1}}(x_i) \\times [\n\\frac{-t_i}{\\lambda_i^2},\n\\frac{T - t_i}{\\lambda_i^2}]$, so that the limiting domain is $\\mathbb R^n \\times\n(-\\infty,0]$. If the points are chosen as a maximal blowup sequence so that the\ncurvatures are bounded, then these blowup solutions converge to a smooth ancient\nsolution to Yang-Mills flow. However, in our analysis though we will be\nchoosing very general sequences and taking weak limits.\n\n\\subsection{Parabolic Hausdorff measures}\n\nFor any $0 \\leq k \\leq n$ and any $\\Omega \\subset \\mathbb{R}^n$, the\n$k$-dimensional Hausdorff measure of $\\Omega$ is defined by\n\\begin{align*}\n\\mathcal{H}^{k}(\\Omega)\n&= \\lim_{\\delta \\to 0} \\mathcal{H}_{\\delta}^{k}(\\Omega)= \\liminf_{\\delta \\to 0}\n\\left\\{ \\sum_i r_i^{k} \\mid \\Omega \\subset \\bigcup_i B_{r_i}(z_i), z_i \\in\n\\Omega,\nr_i \\leq \\delta \\right\\}.\n\\end{align*}\nThis leads to the definition of \\emph{Hausdorff dimension}, i.e.\n\\begin{align*}\n\\dim_{\\mathcal H}(\\Omega) =&\\ \\inf\\ \\{d \\geq 0\\ |\\ \\mathcal H^d(\\Omega) = 0 \\}.\n\\end{align*}\nNext, we define the \\emph{parabolic metric} $\\varrho$ on $\\mathbb{R}^n \\times \\mathbb{R}$ given\nby, for $\\prs{x,t},\\prs{y,s} \\in \\mathbb{R}^n \\times \\mathbb{R}$,\n\\begin{equation*}\n\\varrho \\prs{\\prs{x,t},\\prs{y,s}} := \\max \\left\\{ \\brs{x-y}, \\sqrt{\\brs{t-s}}\n\\right\\}.\n\\end{equation*}\nUsing this metric we can obtain the notion of parabolic Hausdorff dimension by\nusing covers by balls with respect to this metric. In particular, for any $0\n\\leq \\ell \\leq n+2$ and any $\\Omega \\subset \\mathbb{R}^n\n\\times \\mathbb{R}$, the $\\ell$-dimensional parabolic Hausdorff measure of\n$\\Omega$ is given by\n\\begin{align*}\n\\mathcal{P}^{\\ell}(\\Omega)\n&= \\lim_{\\delta \\to 0} \\mathcal{P}_{\\delta}^{\\ell}(\\Omega)= \\liminf_{\\delta \\to\n0} \\left\\{ \\sum_i r_i^{\\ell} \\mid \\Omega \\subset \\bigcup_i P_{r_i}(z_i), z_i \\in\n\\Omega, r_i \\leq \\delta \\right\\},\n\\end{align*}\nwhere, for $z_0 = (x_0,t_0) \\in \\mathbb{R}^n \\times \\mathbb{R}$,\n\\begin{equation*}\nP_r(z_0) := \\left\\{ z = (x,t) \\in \\mathbb{R}^n \\times \\mathbb{R} \\mid \\brs{x-x_0} < r,\n\\brs{t-t_0} < r^2\n\\right\\}.\n\\end{equation*}\nUsing this we can then define the \\emph{parabolic Hausdorff dimension}\n\\begin{align*}\n\\dim_{\\mathcal P}(\\Omega) := \\inf\\ \\{ d \\geq 0\\ |\\ \\mathcal P^d(\\Omega) = 0 \\}.\n\\end{align*}\n\n\n\n\n\n\n\\section{Monotonicity Formulas} \\label{s:monotonform}\n\nIn this section we observe some energy and entropy monotonicity formulas for\nsolutions to Yang-Mills flow which are central to the analysis below.\n\n\\subsection{Energy monotonicity}\n\n\\begin{lemma}\\label{lem:8.2.1LWYM} Let $\\nabla_t$ be a solution to Yang-Mills flow\non $M \\times [t_1,t_2]$. For any $\\phi \\in\nC^1_0(M,[0,\\infty))$,\n\\begin{align*}\n\\frac{1}{4}\\int_M \\prs{ \\brs{F_{\\nabla_{t_1}}}^2 - \\brs{F_{\\nabla_{t_2}}}^2 } \\phi^2 \\,\ndV \n&= \\int_{t_1}^{t_2} \\int_M \\prs{ \\brs{\\frac{\\partial \\nabla_t}{\\partial t}}^2 + \\ip{\\frac{2\n\\nabla_t\n\\phi}{\\phi} \\hook F_{\\nabla_t}, \\frac{\\partial \\nabla_t}{\\partial t}} }\\phi^2 \\, dV \\, dt.\n\\end{align*}\n\\begin{proof}\nWe differentiate and find that\n\\begin{align*}\n\\tfrac{d}{dt} \\brk{\\tfrac{1}{2}\\int_M \\brs{F}^2 \\phi^2 \\, dV }\n&= \\int_M \\ip{ F,\\tfrac{\\partial F}{\\partial t} } \\phi^2 \\, dV \\\\\n&= \\int_M \\ip{ F, D \\brk{ \\tfrac{\\partial \\nabla}{\\partial t} } } \\phi^2 \\, dV \\\\\n&= 2 \\int_M \\ip{ F,\\nabla \\brk{ \\tfrac{\\partial \\nabla}{\\partial t} } } \\phi^2 \\, dV \\\\\n&= 2 \\int_M \\ip{ D^* F - 2 \\tfrac{\\nabla \\phi}{\\phi} \\hook F, \n\\tfrac{\\partial \\nabla}{\\partial t} } \\phi^2 \\, dV \\\\\n&= 2 \\int_M \\ip{ - \\tfrac{\\partial \\nabla}{\\partial t} - 2 \\tfrac{\\nabla \\phi}{\\phi} \\hook F, \n\\tfrac{\\partial \\nabla}{\\partial t} } \\phi^2 \\, dV \\\\\n&= -2 \\int_M \\prs{ \\ip{ \\tfrac{\\partial \\nabla}{\\partial t}, 2 \\tfrac{\\nabla \\phi}{\\phi} \\hook\nF} +\n\\brs{\\tfrac{\\partial \\nabla}{\\partial t}}^2} \\phi^2 \\, dV.\n\\end{align*}\nIntegrating both sides over $\\brk{t_1,t_2}$ yields the result.\n\\end{proof}\n\\end{lemma}\n\n\n\\subsection{Entropy setup and scaling laws} \\label{entropysetup}\n\n\nLet $(M,g)$ be a Riemannian manifold. Let $\\iota_M > 0$ be a lower bound for the\ninjectivity radius of $M$. Note that if $\\nabla_t$ is a smooth solution to\nYang-Mills flow on $M \\times [0,T)$, we can restrict it to any coordinate\nneighborhood $B_{\\iota_M} \\subset \\mathbb{R}^n$ is the Euclidean ball in $\\mathbb{R}^n$ centered\nat the\norigin. \nNow fix $z_0 := \\prs{x_0,t_0} \\in\n\\mathbb{R}^n \\times [0,\\infty)$, and define\n\\begin{equation*}\nG_{z_0}\\prs{x,t} = \\frac{e^{-\\frac{\\brs{x-x_0}^2}{4\\brs{t-t_0}}}}{\\prs{4 \\pi\n\\brs{t-t_0}}^{n\/2}}.\n\\end{equation*} \nWe need to move this function onto the manifold $M$, and so we must localize. \nFor $x_0 \\in M$ we let $\\mathcal{B}_{x_0}$ denote the set of \\emph{cutoff\nfunctions}, that is, all $\\phi \\in\nC_0^{\\infty}\\prs{B_{\\iota_M}(x_0),[0,\\infty)}$ such that\n\\begin{align*}\n\\phi \\in \\brk{0,1}, \\qquad \\phi \\equiv& 1 \\text{ on }\nB_{\\tfrac{\\iota_M}{2}}(x_0), \\qquad \\supp \\phi \\subset B_{\\iota_M}(x_0).\n\\end{align*}\nIn this sense, given $z_0 = (x_0,t_0) \\in M \\times \\mathbb{R}$, for $\\phi \\in\n\\mathcal{B}_{x_0}$ one may consider the globally defined function $\\phi G_{z_0}:\nM \\times \\mathbb R \\to [0,\\infty)$. Lastly, given $z_0 =\n(x_0,t_0) \\in M \\times \\mathbb{R}$ and $R \\in (0,\\infty)$, we define\n\\begin{align*}\nS_R(t_0) &:= M \\times \\{ t_0 - R^2 \\},\\\\\nP_R(z_0) &:= B_{R}(x_0) \\times \\prs{\\brk{t_0 - R^2, t_0} \\cap\n(0,\\infty)},\\\\\nT_R(t_0) &:= M \\times \\prs{ \\brk{t_0 - 4 R^2, t_0 - R^2} \\cap (0,\\infty)}.\n\\end{align*}\n\\begin{defn} \\label{def:PhiPsifunc} Assume $\\nabla_t$ is a solution to\nYang-Mills flow on $M \\times [0,T)$. For $z_0 = (x_0,t_0) \\in\nM \\times [0,T)$, $\\phi \\in \\mathcal{B}_{x_0}$, and $R \\in [0,\n\\min\\{\\iota_M, \\sqrt{t_0}\/2\\} ]$, let\n\\begin{align*}\n\\Phi_{z_0}( R; \\nabla_t) &:= \\frac{R^4}{2} \\int_{S_{R}(t_0)} \\brs{F_{\\nabla_t}}^2\\phi^2 \nG_{z_0} \\, dV,\\\\\n\\Psi_{z_0}(R; \\nabla_t) &:= \\frac{R^2}{2}\\int_{T_{R}(t_0)} \\brs{F_{\\nabla_t}}^2\n\\phi^2 G_{z_0} \\, dV \\, dt.\n\\end{align*}\n\\end{defn}\n\nNext we record a fundamental scaling law for the entropy functionals which is\nutilized in deriving the monotonicity formulas under Yang-Mills flow. These\nmonotonicity formulas are shown in (\\cite{HongTian}), but we include some brief\ndisussion of some properties for convenience, and also because we utilize some\nof the calculations in the sequel. We restrict the lemma to flat space for\nconvenience.\n\n\\begin{lemma} \\label{lem:entropyscaling} Fix $\\nabla_t$ a solution to\nYang-Mills flow on $\\prs{\\mathbb{R}^n, g_{\\Euc}} \\times [0,T)$.\nFor all $z_0 = (x_0,t_0) \\in \\mathbb{R}^n \\times [0,T)$, and $\\prs{0 < R \\leq\n\\sqrt{t_0}\/2}$, setting $\\phi\n\\equiv 1$ in Definition \\ref{def:PhiPsifunc} yields\n\\begin{align*}\n \\Phi_{z_0}(R;\\nabla_t) &=\\ \\Phi_{z_0}\\prs{1; \\nabla^R_t},\\\\\n \\Psi_{z_0}(R;\\nabla_t) &=\\ \\Psi_{z_0}\\prs{1; \\nabla^R_t},\n\\end{align*}\nwhere here $\\nabla^R_t$ is the rescaled connection as defined in Definition\n\\ref{defn:rescaledconn}.\n \\begin{proof} Without loss of generality we may take $z_0 = 0$. For notational\nconvenience we suppress the subscripts on $\\Phi$, $\\Psi$, and $G$. We fix $R >\n0$\nand consider a change of coordinates\n\\begin{align*}\nx = R y, \\qquad t = R^2 s.\n\\end{align*}\nThen, rescaling coordinates and recalling the rescaling of the curvature tensor\n(\\ref{eq:curvrescale}),\n\\begin{align*}\ndx = R^n \\, dy, \\qquad dt = R^2 \\, ds, \\qquad G(x,t) = R^{-n} G(y,s), \\qquad\nF_{\\nabla^R_{s}} (y) = R^{2} F_{\\nabla_{R^2 s}} (R y).\n\\end{align*}\n It follows that\n %\n\\begin{align*}\n\\Phi(R;\\nabla_t) &=\\ \\frac{R^4}{2} \\int_{S_1} \\brs{F_{\\nabla_{R^2 s}}(Ry)}^2 \\phi(y)G(y,\ns) \\,\ndy\\\\\n&=\\ \\frac{1}{2} \\int_{S_1} \\brs{F_{\\nabla^R_s}(y)}^2 \\phi(y) G(y,s) \\, dy\\\\\n&=\\ \\Phi(1; \\nabla^R_t).\n\\end{align*}\nSimilarly,\n\\begin{align*}\n\\Psi(R;\\nabla_t) &=\\ \\frac{R^4}{2} \\int_{T_1} \\brs{F_{\\nabla_{R^2 s}}(R y)}^2\n\\phi(y)G(y, s) \\, dy\n\\, ds\\\\\n&=\\ \\frac{1}{2} \\int_{T_1} \\brs{F_{\\nabla^R_s}(y)}^2 \\phi(y)G(y,s) \\, dy\\, ds\\\\\n&=\\ \\Psi\\prs{1 ; \\nabla^R_t}.\n\\end{align*}\nThe result follows.\n\\end{proof}\n\\end{lemma}\n\n\n\\subsection{Entropy monotonicities}\\label{ss:entropmon}\n\nIn this section we recall the monotonicity formulae for $\\Phi$ and $\\Psi$,\nestablished in \\cite{HongTian}. Again we record the proof on $\\mathbb R^n$ for\nconvenience and as we will use parts of argument in the sequel.\n\n\\begin{prop}\\label{prop:energymonform} Let $\\nabla_t$ to be a smooth solution to\nYang-Mills flow for $\\prs{\\mathbb{R}^n,\ng_{\\Euc}} \\times [0,T)$.\nFor all $z_0 = (x_0,t_0) \\in \\mathbb{R}^n\n\\times [0,T)$, and $0\n< \\rho \\leq r < \\sqrt{t_0}\/2$, setting $\\phi\n\\equiv 1$ in Definition \\ref{def:PhiPsifunc} yields\n\\begin{align*}\n\\Phi_{z_0}(\\rho;\\nabla_t) &\\leq \\Phi_{z_0}(r;\\nabla_t)\\\\\n\\Psi_{z_0}(\\rho;\\nabla_t) &\\leq \\Psi_{z_0}(r;\\nabla_t).\n\\end{align*}\n\n\\begin{proof} We begin with the monotonicity statement for $\\Phi$. \nWe will include a generic cutoff function for purposes of a later Lemma. We fix\n$R > 0$ and consider a\nchange of coordinates as in Lemma\n\\ref{lem:entropyscaling}.\nAs described there, it follows that\n\\begin{align*}\n\\Phi(R;\\nabla_t) &=\\ \\frac{R^4}{2} \\int_{S_1} \\brs{F_{\\nabla_{R^2 s}}(Ry)}^2\n\\phi^2(Ry)G(y, s) \\,\ndy.\n\\end{align*}\nA crucial point here is that we are \\emph{not} rescaling the connection as well.\nOne now differentiates and rescales back to obtain \n\\begin{align*}\n \\frac{\\partial}{\\partial R} \\brk{\\Phi(R;\\nabla_t)} &=\\ \\frac{4}{R} \\Phi(R;\\nabla_t) + \\brk{\nR^3 \\int_{S_R} \\ip{F_{\\nabla_t}, x \\hook \\partial\nF_{\\nabla_t}} \\phi^2 G \\, dx}_{I_1} \\\\\n\\hspace{0.5cm} & + \\brk{ 2 R^3 \\int_{S_R} \\ip{F_{\\nabla_t}, t \\prs{\\tfrac{\\partial F_{\\nabla_t}}{\\partial\nt}}} \\phi^2 G \\, dx}_{I_2} + \\brk{R^3 \\int_{S_R} \\brs{F_{\\nabla}}^2 \\phi x \\hook \\nabla\n\\phi\\, dx}.\n\\end{align*}\nTo address $I_1$, we recall some coordinate formulas.\n\\begin{align*}\n\\nabla_i F_{jk \\alpha}^{\\beta}\n&= \\partial_i F_{jk \\alpha}^{\\beta} + \\Gamma_{i \\mu}^{\\beta}F_{jk \\alpha}^{\\beta} - F_{jk\n\\mu}^{\\beta}\\Gamma_{i \\alpha}^{\\mu},\\\\\n\\nabla_i F_{jk \\alpha}^{\\beta} &=- \\prs{ \\nabla_k F_{ij \\alpha}^{\\beta} + \\nabla_j F_{ki \\alpha}^{\\beta}}.\n\\end{align*}\nCombining these we conclude that\n\\begin{align*}\n\\partial_i F_{jk \\alpha}^{\\beta} = - \\prs{ \\nabla_k F_{ij \\alpha}^{\\beta} + \\nabla_j F_{ki \\alpha}^{\\beta}}\n- \\Gamma_{i \\mu}^{\\beta}F_{jk \\alpha}^{\\mu} + F_{jk \\mu}^{\\beta}\\Gamma_{i \\alpha}^{\\mu}.\n\\end{align*}\nWith this in mind we manipulate $I_1$,\n\\begin{align*}\nI_1\n&= R^3 \\int_{S_R } x^i\\prs{ \\nabla_k F_{ij \\alpha}^{\\beta} + \\nabla_j F_{ki\n\\alpha}^{\\beta}}F_{jk \\beta}^{\\alpha} \\phi^2 G \\, dx \\\\\n& \\hspace{0.5cm} + R^3 \\int_{S_R } x^i \\Gamma_{i \\mu}^{\\beta}F_{jk \\alpha}^{\\mu} F_{jk\n\\beta}^{\\alpha} \\phi^2 G \\, dx - R \\int_{S_R } x^i F_{jk \\mu}^{\\beta}\\Gamma_{i\n\\alpha}^{\\mu}F_{jk\n\\beta}^{\\alpha} \\phi^2 G \\, dx \\\\\n&= 2 R^3 \\int_{S_R } x^i\\prs{ \\nabla_k F_{ij \\alpha}^{\\beta}}F_{jk \\beta}^{\\alpha} \\phi^2 G\n\\, \ndx\\\\\n&= - 2 R^3 \\int_{S_R } F_{ij \\alpha}^{\\beta} \\nabla_k \\brk{x^iF_{jk \\beta}^{\\alpha} \\phi^2 G\n} \\,\ndx \\\\\n&= 2 R^3 \\int_{S_R } \\left[ F_{ij \\alpha}^{\\beta} F_{ij \\beta}^{\\alpha} + F_{ij\n\\alpha}^{\\beta}x^i \\prs{\\nabla_k F_{kj \\beta}^{\\alpha}} - \\tfrac{1}{2 t} x^i F_{ij\n\\alpha}^{\\beta} x^k F_{kj \\beta}^{\\alpha} \\right] \\phi^2 G \\, dx\\\\\n& \\hspace{0.5cm} - 4 R^3 \\int_{S_R } F_{ij \\alpha}^{\\beta} x^iF_{jk \\beta}^{\\alpha} \\prs{\\nabla_k\\phi}\n\\phi G \\, dx \\\\\n&= \\ -\\tfrac{4}{R} \\Phi(R;\\nabla) + R^3 \\int_{S_R } \\left[ \\tfrac{1}{t} \\brs{x \\hook\nF}^2 - 2 \\ip{x \\hook F, D^* F} \\right] \\phi^2 G \\, dx \\\\\n& \\hspace{0.5cm} - 4 R^3 \\int_{S_R } F_{ij \\alpha}^{\\beta} x^iF_{jk \\beta}^{\\alpha} \\prs{\\nabla_k\\phi}\n \\phi G \\, dx.\n\\end{align*}\nAlso we have\n\\begin{align*}\nI_2\n&= 2 R^3\\int_{S_R} t \\ip{F, \\tfrac{\\partial F}{\\partial t}} \\phi^2 G \\, dx\\\\\n& = - 2 R^3 \\int_{S_R} t \\ip{F, DD^* F} \\phi^2 G \\, dx\\\\\n& = 4 R^3 \\int_{S_R} t F_{ij \\alpha}^{\\beta} \\nabla_i (D^* F)_{j \\beta}^{\\alpha} \\phi^2 G \\,\ndx\\\\\n& = R^3 \\int_{S_R} \\left[ 4 t \\brs{D^* F}^2 - 2 \\ip{x \\hook F, D^* F}\n\\right] \\phi G \\, dx - 8 R^3 \\int_{S_R} t F_{ij \\alpha}^{\\beta} (D^* F)_{j\n\\beta}^{\\alpha} \\prs{\\nabla_i\\phi} \\phi G \\, dx.\n\\end{align*}\nCombining these calculations gives\n\\begin{align}\n\\begin{split}\\label{eq:delRmonotonicity}\n \\frac{\\partial}{\\partial R} \\brk{\\Phi(R;\\nabla_t)} &=\\ \\brs{t} R^3 \\int_{S_R}\n\\brs{\\frac{x}{t}\n\\hook F_{\\nabla_t} - 2 D^*_{\\nabla_t} F_{\\nabla_t}}^2 \\phi^2 G \\,dx \\\\\n& \\hspace{0.5cm} + 4 R^3 \\int_{S_R} \\prs{x^k F_{kj\\beta}^{\\alpha} - 2 t \\prs{D^* F}_{j\n\\beta}^{\\alpha} }F_{ij \\alpha}^{\\beta} \\prs{\\nabla_i \\phi} \\phi G \\, dx\\\\\n& \\hspace{0.5cm} + R^3 \\int_{S_R} \\brs{F_{\\nabla}}^2 \\phi x \\hook \\nabla \\phi\\, dx.\n\\end{split}\n\\end{align}\nIn particular, when $\\phi \\equiv 1$, we have monotonicity, which yields the\nfirst claim.\n\nNext we prove the monotonicity of $\\Psi$, only considering the case where $\\phi\n\\equiv 1$. We fix $R > 0$ and use the coordinate\nchange as in Lemma \\ref{lem:entropyscaling} once more, and it follows that\n\\begin{align*}\n\\Psi(R;\\nabla_t) &=\\ R^4 \\int_{T_1} \\brs{F_{\\nabla_{R^2 s}}(Ry)}^2 \\phi^2(y) G(y, s) \\,\ndy\n\\, ds.\n\\end{align*}\nOnce again, crucially, we are \\emph{not} rescaling the connection. One now\nobtains\n\\begin{align*}\n\\frac{\\partial }{\\partial R}\\brk{ \\Psi(R ;\\nabla_t) } &= \\frac{4}{R} \\Psi(R;\\nabla_t) + \\brk{ 2\nR \\int_{T_{R}} \\ip{F_{\\nabla_t}, x \\hook \\partial F_{\\nabla_t}} \\phi^2 G \\, dx \\,\ndt}_{I_1}\\\\\n& \\hspace{0.5cm} + \\brk{4 R \\int_{T_R} \\ip{F_{\\nabla_t}, t \\prs{\\tfrac{ \\partial F_{\\nabla_t} }{\\partial\nt}}} \\phi^2 G \\, dx \\, dt}_{I_2}.\n\\end{align*}\nNearly identical estimates for $I_1$ and $I_2$ as in the case of $\\Phi$ above\nyield\n\\begin{align*}\n\\frac{\\partial}{\\partial R} \\brk{\\Psi(R;\\nabla_t)} &=\\ 2 R \\int_{T_R} \\brs{t}\n\\brs{\\frac{x}{t} \\hook F_{\\nabla_t} -\n2 D^*_{\\nabla_t} F_{\\nabla_t}}^2 \\phi^2 G \\, dx \\, dt.\n\\end{align*}\nThe result follows.\n\\end{proof}\n\\end{prop}\n\nNext we state the general monotonicity formula for $\\Phi$ and $\\Psi$ on\narbitrary Riemannian manifolds. The proof is similar to that of Proposition\n\\ref{prop:energymonform}, incorporating further estimates due to the presence of\nthe cutoff function. We state here the result of (\\cite{HongTian} Theorem 2),\nwhich applies to Yang-Mills-Higgs flow, and we just restrict the result to\nYang-Mills flow. We point out that a similar result was claimed\nin \\cite{CS}, but uses definitions of $\\Phi$ and $\\Psi$ with incorrect scaling. \nNote that the notation for $\\Phi$ and $\\Psi$ agrees with various other\nliterature, but is reversed from that chosen in \\cite{HongTian}. Moreover, we\nstate an improved statement which is clearly implicit in \\cite{HongTian}, simply\nincluding an extra term in the inequality which is dropped in the statement in\n\\cite{HongTian}.\n\n\\begin{thm}[\\cite{HongTian} Theorem 2, pp.448] \\label{thm:8.2.2LWYM} Let $\\nabla_t$\nbe a smooth\nsolution to Yang-Mills flow on $M \\times [0,T)$. Then for $z_0 = (x_0,t_0) \\in M\n\\times [0,T]$ and\n$0 < R_1 \\leq R_2 \\leq \\min \\{\\iota_M, \\sqrt{t_0}\/2\n\\}$, we have\n\\begin{align}\n\\begin{split}\\label{Psimono}\n\\Psi_{z_0}(R_1;\\nabla_t) & + \\int_{R_1}^{R_2} r \\int_{T_r(t_0)} \\brs{t - t_0}\n\\brs{\\frac{x - x_0}{2 \\brs{t - t_0}} \\hook F_{\\nabla_t} - D^*_{\\nabla_t} F_{\\nabla_t}}^2\n\\phi^2 G_{z_0} \\, dV \\, dt \\, dr\n\\end{split}\n\\end{align}\n\\begin{align} \\label{Phimono}\n\\begin{split}\n& \\leq e^{C(R_2-R_1)} \\Psi_{z_0}(R_2;\\nabla_t) + C (R_2 - R_1)\\mathcal{YM}(\\nabla_0),\\\\\n\\Phi_{z_0}(R_1;\\nabla_t) & + \\int_{R_1}^{R_2} r^3 \\int_{S_r(t_0)} \\brs{t - t_0}\n\\brs{\\frac{x - x_0}{2 \\brs{t - t_0}} \\hook F_{\\nabla_t} - D^*_{\\nabla_t} F_{\\nabla_t}}^2\n\\phi^2 G_{z_0} \\, dV \\, dr\\\\\n& \\leq e^{C(R_2-R_1)} \\Phi_{z_0}(R_2;\\nabla_t) + C \\prs{R_2 - R_1}\n\\mathcal{YM}(\\nabla_0).\n\\end{split}\n\\end{align}\n\\end{thm}\n\nAs the statement above makes clear, the functionals $\\Phi$ and $\\Psi$ are fixed\nif\nthe connection satisfies a certain modified Yang-Mills type equation:\n\n\\begin{defn} Let $\\nabla_t$\nbe a nontrivial smooth\none-parameter family of connections on\n$\\mathbb R^n \\times (- \\infty,0]$. Then $\\nabla_t$ is a \\emph{soliton} if\n\\begin{equation*}\nD_{\\nabla_t}^*F_{\\nabla_t} = \\frac{x }{2t} \\hook F_{\\nabla_t}.\n\\end{equation*}\n\\end{defn}\n\nWe end with a useful technical observation showing that the different entropies\n$\\Phi$ and $\\Psi$ are uniformly equivalent, which exploits the monotonicity\n\n\\begin{lemma} \\label{lem:cute} Let $\\nabla_t$ be a solution to Yang-Mills\nflow on $M \\times [0,T)$. There\nexists a uniform constant $C$ such that for $z_0 = (x_0,t_0) \\in M \\times [0,T)$\nand for $R$ with $0 < R \\leq \\min \\{\\iota_M, \\sqrt{t_0}\/2 \\}$, we have\n\\begin{align*}\nC^{-1} \\Psi_{z_0}(R; \\nabla_t) \\leq \\Phi_{z_0}(2 R;\\nabla_t) \\leq C \\Psi_{z_0}(2\nR;\\nabla_t).\n\\end{align*}\n\\begin{proof} We give the proof on $\\mathbb R^n$, in which case the monotonicity\ndoes not involve the error term involving the Yang-Mills energy, with the\ngeneralization to manifolds a straightforward extension. Without loss of\ngenerality we can consider the time interval to be $[-1,0]$ and choose $z_0 =\n(0,0)$. Then we have, using the monotonicity of $\\Phi$ and a change of\nvariables,\n\\begin{align*}\n\\Phi(2R) \n&\\geq\\ \\tfrac{1}{R} \\int_{R}^{2R} \\Phi(s) \\, ds\\\\\n&=\\ \\tfrac{R^3}{2} \\int_{s=R}^{s=2R} \\int_{M \\times \\{-s^2\\}} \\brs{F_s}^2 \\phi^2\nG \\, dV \\, ds\\\\\n&=\\ \\tfrac{R^3}{2} \\int_{t=-R^2}^{t=-4R^2} \\tfrac{1}{2\\sqrt{-t}}\\int_{M \\times\n\\{t\\}} \\brs{F_t}^2 \\phi^2 G \\, dV \\, dt\\\\\n&\\geq\\ c R^2 \\int_{T_{R}(0)} \\brs{F_t}^2 \\phi^2G \\, dV \\, dt\\\\\n&=\\ c \\Psi (R).\n\\end{align*}\nAnalogously we have\n\\begin{align*}\n\\Phi(R)\n&\\leq\\ \\tfrac{1}{R} \\int_{R}^{2R} \\Phi(s) \\, ds\\\\\n&=\\ \\tfrac{R^3}{4} \\int_{s=R}^{s=2R} \\int_{M \\times \\{-s^2\\}} \\brs{F_s}^2 \\phi^2\nG \\, dV \\, ds\\\\\n&=\\ \\tfrac{R^3}{4} \\int_{t=-R^2}^{t=-4R^2} \\tfrac{1}{2\\sqrt{-t}}\\int_{M \\times\n\\{t\\}} \\brs{F_t}^2 \\phi^2 G \\, dV \\, dt\\\\\n&\\leq\\ C R^2 \\int_{T_{R}(0)} \\brs{F_t}^2 \\phi^2 G \\, dV \\, dt\\\\\n&=\\ C \\Psi (R).\n\\end{align*}\nThe result follows.\n\\end{proof}\n\\end{lemma}\n\n\\subsection{Epsilon-regularity}\n\nA central phenomenon in understanding the singularity formation of geometric\nflows is that of $\\ge$-regularity. A result of this kind for Yang-Mills flow is\nshown in \\cite{HongTian}, relying centrally on the monotonicity formula for\n$\\Psi$ and the evolution equation for the curvature. Once again we only state\nthe result for solutions to Yang-Mills flow though the result is shown for\nYang-Mills-Higgs flow in \\cite{HongTian}. We also point out that a similar\nresult is claimed in \\cite{CS}, although it relies on the incorrectly defined\n$\\Psi$ functional.\n\n\\begin{thm}[\\cite{HongTian} Theorem 4, pp.454]\\label{thm:eregularity}Suppose\n$\\nabla_t$ is a solution to\nYang-Mills flow on $M \\times [0,T)$. There exist\nconstants $C,\\delta,\\ge_0 > 0$ depending on $(M, g)$ and $\\mathcal{YM}(\\nabla_0)$ so that given\n$z_0 = (x_0,t_0) \\in M \\times [0,T)$ and $0 < R < \\min \\{ \\iota_M, \\sqrt{t_0}\/2\n\\}$ such that\n\\begin{equation*}\n\\Psi_{z_0}(R;\\nabla_t) < \\ge_0,\n\\end{equation*}\none has\n\\begin{align*}\n\\sup_{P_{\\delta R}(z_0)} \\brs{F_{\\nabla_t}}^2 \\leq \\frac{C}{ (\\delta R)^{4}}.\n\\end{align*}\n\\end{thm}\n\n\n\\section{Weak Compactness and limit measures}\\label{s:weakcompt}\n\nIn this section we establish a weak compactness result for solutions to\nYang-Mills flow satisfying certain weak convergence hypotheses. In the first\nsubsection below we establish this theorem, and in the following subsection we\nrefine the analysis to show a number of properties of the limiting energy\ndensities and defect measures.\n\n\\subsection{Weak compactness theorem}\n\n\\begin{thm}\\label{fullweakcompactthm} Suppose $\\{\\nabla^i_t\\}$ is a sequence of\nsmooth solutions to Yang-Mills flow over $M \\times [-1,0]$ with $\\mathcal{YM}(\\nabla^i_t)\n\\leq\n\\mathcal{YM} (\\nabla^i_{-1}) <\nC$. \nMoreover, suppose $\\{\\nabla_{-1}^i\\} \\to \\nabla$ weakly in\n$H^{1,2}_{loc}(\\mathcal{A}_E(M))$,\nand\n\\begin{itemize}\n\\item $\\nabla^i_t \\to \\nabla_t$ in $L^2_{\\mbox{loc}}(M \\times [-1,0])$,\n \\item $\\frac{\\partial \\nabla^i_t}{\\partial t} \\to \\frac{\\partial \\nabla_t}{\\partial t}$ weakly in\n$L^2_{\\mbox{loc}}(M \\times [-1,0])$,\n \\item $F_{\\nabla^i_t} \\to F_{\\nabla_t}$ weakly in $L^2_{\\mbox{loc}}(M \\times [-1,0])$.\n\\end{itemize}\nThen $\\nabla_t$ is gauge equivalent to a weak solution to Yang-Mills flow, and there\nexists a closed set $\\Sigma$ of locally finite $\\prs{n-2}$-dimensional parabolic\nHausdorff measure such that $\\nabla_t$ is a smooth solution on $(M \\times (-1,0))\n\\backslash \\Sigma$.\n\n\\begin{proof}\nSet\n\\begin{equation*}\n\\Phi^i_{z_0}(r) := \n\\begin{cases}\n\\Phi_{z_0}\\prs{r;\\nabla_t^i} & r \\in \\prs{0, \\sqrt{1+t_0}}\\\\\n\\Phi_{z_0}\\prs{\\sqrt{1+t_0};\\nabla_t^i}& \\text{ otherwise.}\n\\end{cases}\n\\end{equation*}\nNow define the concentration set\n\\begin{equation*}\n\\Sigma := \\bigcap_{r>0}\\sqg{z \\in M \\times \\brk{-1,0} \\mid \\liminf_{k\\to\\infty}\n\\Phi_z^k\\prs{r} \\geq \\ge_0},\n\\end{equation*}\nwhere $\\ge_0$ is the constant of Theorem \\ref{thm:eregularity}. To address the\ntheorem, we divide the proof up into three pieces: Lemma \\ref{lem:Sigmacl},\nLemma \\ref{weakcompactlemma3}, and Lemma \\ref{weakcompactlemma10}.\n\\begin{lemma}\\label{lem:Sigmacl} $\\Sigma$ is closed.\n\\begin{proof}\nLet $\\bar{z}$ lie in the closure of $\\Sigma$ and $\\{z_k\\}_{k \\in \\mathbb{N}} \\in\n\\Sigma$ with $z_k \\to \\bar{z}$. By the definition of $\\Sigma$,\n\\begin{equation*}\n\\liminf_{k \\to \\infty} \\liminf_{i \\to \\infty} \\Phi^i_{z_k}(r) =\\liminf_{k \\to\n\\infty} \\liminf_{i \\to \\infty}\\brk{\\frac{r^4}{2} \\int_{\\mathbb{R}^n \\times \\{ t_k -r^2\n\\}} \\brs{ F^i_t}^2 \\phi^2G_{z_k} \\, dV}\\geq \\ge_0. \\end{equation*}\nNote that $G_{z_k} \\to G_{\\bar{z}}$ on any closed sets not containing $\\bar{z}$.\n Moreover, for fixed $i$ the function $\\brs{F^i_t}^2$ is in $L^1$. Therefore we\ncan fix $r > 0$, apply the dominated convergence theorem and interchange\n$\\liminf$ ordering by an elementary argument to conclude\n\\begin{align*}\n\\liminf_{i \\to \\infty} \\frac{r^4}{2}\\int_{M \\times \\sqg{ \\bar{t} - r^2 }} \\brs{\nF^i_t}^2 \\phi^2G_{\\bar{z}} \\, dV =&\\ \\liminf_{i \\to \\infty} \\lim_{k \\to \\infty}\n\\frac{r^4}{2} \\int_{M \\times \\{\\bar{t} - r^2\\}} \\brs{F_t^i}^2 \\phi^2 G_{z_k} \\,\ndV\\\\\n=&\\ \\liminf_{i \\to \\infty} \\liminf_{k \\to \\infty} \\frac{r^4}{2} \\int_{M \\times\n\\{\\bar{t} - r^2\\}} \\brs{F_t^i}^2 \\phi^2 G_{z_k} \\, dV\\\\\n\\geq&\\ \\ge_0.\n\\end{align*}\nTherefore $\\bar{z} \\in \\Sigma$, so we conclude $\\Sigma$ is closed.\n\\end{proof}\n\\end{lemma}\n\\begin{lemma} \\label{weakcompactlemma3} $\\nabla_t$ is gauge equivalent to a smooth\nsolution to Yang-Mills flow on $\\prs{M \\times (-1,0]} \\backslash \\Sigma$.\n\\begin{proof} Given $z \\in \\prs{\\mathbb R^n \\times (-1,0]} \\backslash \\Sigma$,\nby construction there exists $r_0 > 0$ such that\n\\begin{equation*}\n\\liminf_{k \\to \\infty} \\Phi^k_z (r_0) \\leq \\ge_0.\n\\end{equation*}\nPassing to a subsequence and applying Lemma \\ref{lem:cute}, we obtain an $\\ge_0$\nupper bound for $\\Psi$, and by Theorem\n\\ref{thm:eregularity}, we conclude that\n\\begin{align*}\n\\sup_{P_{\\delta r_0}(z)} \\brs{F^k_t}^2 \\leq \\frac{C}{(\\delta r_0)^{4}},\n\\end{align*}\nfor some universal constant $\\delta > 0$.\nApplying (\\cite{Weinkove}, Theorem 2.2) we conclude uniform estimates on all\nderivatives of curvature on a parabolic ball of radius $\\frac{\\delta r_0}{2}$.\n\nUsing the Uhlenbeck gauge-fixing Theorem (\\cite{Uhlenbeck} Theorem 1.3) and the\ngauge-patching argument of (\\cite{DonB} Corollary 4.4.8) we can obtain a Coloumb\ngauge on $B_{\\frac{\\delta r_0}{4}}$. Moreover, by applying elliptic regularity\nestimates (\\cite{DonB} Lemma 2.3.11) and the Sobolev inequality we obtain\nuniform pointwise estimates for the connection in the Coloumb gauge on\n$B_{\\frac{\\delta r_0}{8}}$. By applying the Yang-Mills flow PDE directly to this\ngauge-fixed connection and using the previous estimates on the derivatives of\ncurvature we obtain uniform pointwise estimates for the gauge fixed connections\non $P_{\\frac{\\delta r_0}{8}}$. Thus for each point $z_0$ we have constructed a\nradius $\\frac{\\delta r_0}{8}$ and a sequence of gauge transformations for which the\nparabolic ball of that radius has uniform control along some subsequence of\ngauge-fixed connections.\n\nFix a compact set $K$ such that $K \\cap \\Sigma = \\varnothing$. For each $z \\in\nK$ there exist arbitrarily large values of $k$ and parabolic balls centered at\n$z$ of the type described above. This collection of parabolic balls covers $K$,\nand since $K$ is compact we can choose a finite subcover, and also pass to a\nsubsequence of connections all of which have the bounds described above. A\nfurther application of the gauge-patching result (\\cite{DonB} Corollary 4.4.8)\nallows us to conclude the existence of a single gauge transformation, which,\nwhen applied to our sequence, yields a sequence of connections with uniform\n$C^{l,\\alpha}$ bounds. By the Arzela-Ascoli Theorem we obtain a further subsequence\nconverging on $K$.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma} \\label{weakcompactlemma5} $\\Sigma$\nhas locally finite $(n-2)$-dimensional parabolic Hausdorff measure.\n\\begin{proof} Fix a compact set $K$, and some $r_0 > 0$. By Vitali's covering\nlemma there exists some $l \\in \\mathbb{N}$, $\\{ z_k \\}_{k =1}^l \\subset K \\cap\n\\Sigma$ and $\\{ r_k \\}_{k = 1}^l \\subset \\prs{0,r_0}$ so that the sets $\\{\nP_{r_k}\\prs{z_k} \\}_{k=1}^l$, are mutually disjoint and $K \\cap \\Sigma$ is\ncovered by $\\{ P_{5r_k}\\prs{z_k} \\}_{k=1}^l$. Let $\\bar{z}_k := z_k +\n\\prs{0,r_k^2}$ and fix some $\\delta > 0$ to be determined later. \n\nThe proof requires two different estimates on $G$ on different domains. First,\non $(M \\times \\brk{t_k - 4 \\delta^2 r_k^2 , t_k - \\delta^2 r_k}) \\backslash\nP_{r_k}\\prs{z_k}$ one has\n\\begin{equation*}\nG_{z_k} \\leq \\delta^{-n} e^{- 1 \/ \\prs{4 \\delta}^2} G_{\\bar{z}_k}.\n\\end{equation*}\nAlso, for points in $B_{r_k}(x^k) \\times [t_k - 4 \\delta^2 r_k^2, t_k - \\delta^2\nr_k^2]$ one has\n\\begin{align*}\nG_{z_k} \\leq C_{\\delta} r^{-n}.\n\\end{align*}\nWe will also employ the estimate of Lemma \\ref{lem:cute}, in particular\n\\begin{align*}\n\\Phi_{z_0}(R;\\nabla_t) \\leq C \\Psi_{z_0}(R;\\nabla_t).\n\\end{align*}\nCombining the observations above we obtain, for all $k$, $i$\n\\begin{align*}\n\\begin{split}\n\\ge_0\n&\\leq \\Phi_{z_k}^i \\prs{\\delta r_k}\\\\\n&\\leq C \\Psi_{z_k}\\prs{\\delta r_k;\\nabla^i_t} \\\\\n&= C \\delta^2 r_k^2 \\int_{t_k - 4 \\delta^2 r_k^2}^{t_k - \\delta^2 r_k^2} \\int_{M\n\\backslash B_{r_k}(x_k)}\\brs{F^{i}_t}^2 G_{z_k} dV dt + C \\delta^2 r_k^2\n\\int_{t_k - 4 \\delta^2 r_k^2}^{t_k - \\delta^2 r_k^2}\n\\int_{B_{r_k}(x_k)}\\brs{F^{i}_t}^2 G_{z_k} dV dt \\\\\n&\\leq C \\tfrac{e^{-1\/(4 \\delta)^2}}{4 \\delta^{n}} \\delta^2 r_k^2 \\int_{t_k - 4\n\\delta^2 r_k^2}^{t_k - \\delta^2 r_k^2} \\int_{M \\backslash B_{\nr_k}(x_k)}\\brs{F^{i}_t}^2 G_{\\bar{z}_k} dV dt + C_{\\delta} r_k^{2-n} \\int_{t_k\n- 4 \\delta^2 r_k^2}^{t_k - \\delta^2 r_k^2} \\int_{B_{r_k}(x_k)}\\brs{F^{i}_t}^2\ndV dt \\\\\n&\\leq \\brk{ C \\tfrac{e^{-1\/(4 \\delta)^2}}{4\\delta^{n}} \\delta^2 r_k^2 \\int_{t_k - 4\n\\delta^2 r_k^2}^{t_k - \\delta^2 r_k^2} \\int_{M}\\brs{F^{i}_t}^2 G_{\\bar{z}_k}\ndV dt}_{I_1} + \\brk{C_\\delta r_k^{2-n} \\int_{P_{r_k}(z_k)} \\brs{F^{i}_t}^2 dV\ndt}_{I_2}.\n\\end{split}\n\\end{align*}\nObserve that we can estimate $I_1$ using Theorem \\ref{thm:8.2.2LWYM} via\n\\begin{align*}\n\\delta^2 r_k^2 \\int_{t_k - 4 \\delta^2 r_k^2}^{t_k - \\delta^2 r_k^2} \\int_M\n\\brs{F^{i}_t}^2 G_{\\bar{z}_k} \\,dV dt\n& =\\ \\delta^2 r_k^2 \\int_{t_k + r_k^2 -\n4 r_k^2 (1 + \\delta^2)}^{t_k + r_k^2 - r_k^2(1 + \\delta^2)} \\int_M \\brs{F^{i}_t}^2\nG_{\\bar{z}_k} \\, dV dt\\\\\n&=\\ \\Psi_{\\bar{z}_k} \\prs{r_k \\sqrt{1 + \\delta^2}; \\nabla^i_t}\\\\\n&\\leq\\ C \\Psi_{\\bar{z}_k} \\prs{r_0;\\nabla_t^i} + C(\\prs{\\mathcal{YM}\\prs{\\nabla_{-1}}})\\\\\n&\\leq\\ C\\prs{\\mathcal{YM} \\prs{\\nabla_{-1}}}.\n\\end{align*}\nHence, since $\\lim_{\\delta \\to 0} \\frac{e^{-1\/(4\\delta)^2}}{4\\delta^n} = 0$, we can\nchoose $\\delta > 0$ sufficiently small so that $I_1 \\leq\n\\frac{\\ge_0}{2}$, which then implies that $I_2 \\geq \\frac{\\ge_0}{2}$, which by\nelementary manipulations gives\n\\begin{equation*}\nr_k^{n-2}\\leq \\frac{C}{\\ge_0} \\int_{P_{r_k}(z_k))} \\brs{F^{i}_t}^2 \\, dV \\, dt.\n\\end{equation*}\nTherefore we have\n\\begin{align*}\n\\mathcal{P}_{5 r_0}^{n-2} \\prs{P_R \\cap \\Sigma}\n& \\leq \\sum_{k=1}^l \\prs{5 r_k}^{n-2} \\\\\n& \\leq C \\sum_{k=1}^l \\int_{P_{r_k}(z_k)} \\brs{F^{i}_t}^2 dV dt\n\\\\\n& \\leq C \\mathcal{YM}\\prs{\\nabla_{-1}}.\n\\end{align*}\nSending $r_0 \\to 0$ allows us to conclude that $\\mathcal{P}^{n-2}(\\Sigma \\cap\nK) < \\infty$ for any compact set $K$. The result follows.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma} \\label{weakcompactlemma10} $\\nabla_t$ is a\nweak solution to Yang-Mills flow.\n\\begin{proof} We verify\n(\\ref{weaksoln}) by approximating via cutoff functions which excise the singular\nset $\\Sigma$. To construct these functions, first consider the coverings\nconstructed in Lemma \\ref{weakcompactlemma5}. In particular, given any\n$r_0>0$ there is some finite cover $\\{ P_{r_i}(z_i)\\}_{i=1}^{l}$ of\n$\\Sigma$, for some $l \\in \\mathbb{N}$ with $r_i < r_0$ satisfying\n\\begin{equation} \\label{weakcompactlemma105}\n\\sum_{i=1}^{l} r_i^{-4} \\brs{P_{r_i}(z_i)} \\approx \\mathcal\nP_{5r_0}^{n-2}(K\n\\cap \\Sigma) \\leq C \\mathcal{YM} \\prs{\\nabla_{-1}},\n\\end{equation}\nwhere here $\\brs{\\cdot}$ denotes the Lebesgue measure on $\\mathbb{R}^n \\times\n\\mathbb{R}$.\n\nLet $\\phi \\in C_0^{\\infty}(P_2, [0,\\infty))$ be a standard bump function\nsatisfying\n$0 \\leq \\phi \\leq 1$ and $\\phi \\equiv 1$ on $P_1$. For all $i \\in\n\\mathbb{N}$, define\n\\begin{equation*}\n\\phi_i (x,t) := \\phi \\prs{\\tfrac{x-x_i}{r_i},\\tfrac{t-t_i}{r_i^2}}.\n\\end{equation*}\nLet $\\alpha \\in C^{\\infty}([0,T]; L^2(\\Lambda^2(\\Ad E)))$ and arbitrary and set\n\\begin{align*}\n\\eta\n:= \\alpha \\inf_i \\prs{1-\\phi_i} \\in C_0^{\\infty}\\prs{ \\prs{\\mathbb{R}^n \\times\n(-1,0)} \\backslash \\Sigma}.\n\\end{align*}\nNote that by definition, $\\eta \\to \\alpha$\nalmost\neverywhere as $r_0 \\to 0$. Furthermore, observing that $\\supp \\eta \\subset\n\\prs{\\mathbb{R}^n \\times \\prs{-1,0}} \\backslash \\Sigma$, it follows from Lemma\n\\ref{weakcompactlemma3} that, setting $\\Upsilon_t = \\nabla_{\\refc} - \\nabla_t$, we have\n\\begin{align*}\n\\int_{-1}^0 \\int_{M} \\ip{\\Upsilon, \\tfrac{\\partial \\eta}{\\partial t}} - \\ip{F, D \\eta} \\, dV\n\\, dt =\n0.\n\\end{align*}\nUsing this we can estimate\n\\begin{align*}\n\\int_{-1}^0 & \\int_M \\ip{ \\Upsilon, \\tfrac{\\partial \\alpha}{\\partial t}} - \\ip{F, D \\alpha}\ndV\n\\, dt\\\\\n&=\\ \\brs{ \\int_{-1}^0 \\int_M \\ip{ \\Upsilon, \\tfrac{\\partial (\\alpha - \\eta)}{\\partial t}} -\n\\ip{F,\nD(\\alpha - \\eta)} \\, dV \\, dt }\\\\\n&=\\ \\brs{\\int_{-1}^0 \\int_M \\ip{ \\tfrac{\\partial \\Upsilon}{\\partial t}, \\alpha - \\eta} - \\ip{F,\n[1\n- \\inf_i (1 - \\phi_i)] D \\alpha} - \\ip{F, \\alpha \\wedge d( \\inf_i (1 - \\phi_i))}\n\\, dV \\, dt }\\\\\n&=\\ \\brs{ I_1 + I_2 + I_3 } \\\\\n& \\leq \\sum_{j=1}^3 \\brs{I_j}.\n\\end{align*}\nFirst, since we have almost everywhere convergence of $\\alpha$ to $\\eta$ and\n$\\frac{\\partial \\Upsilon}{\\partial t}$ is in $L^2$ we have $\\lim_{r_0 \\to 0} I_1 = 0.$ \nSimilarly since $[1 - \\inf_i (1 - \\phi_i)]$ goes to zero uniformly one has that\n$\\lim_{r_0 \\to 0} I_2 = 0$. For the final term, we observe using H\\\"older's\ninequality and (\\ref{weakcompactlemma105}) that\n\\begin{align*}\n\\lim_{r_0 \\to 0} \\brs{I_3} &\\leq\\ C \\lim_{r_0 \\to 0} \\nm{F}{L^2( \\cup_i\nP_{r_i}(z_i))} \\left[ \\int_{-1}^0 \\int_M \\brs{\\nabla \\inf_{1 \\leq i \\leq l}\n(1 - \\phi_i)}^2 \\, dV \\, dt \n\\right]^{\\frac{1}{2}}\\\\\n&\\leq\\ C \\lim_{r_0 \\to 0} \\nm{F}{L^2( \\cup_i P_{r_i}(z_i))} \\left[\n\\sum_{i=1}^{l}\nr_i^{-2} \\brs{P_{r_i}(z_i)} \\right]^{\\frac{1}{2}}\\\\\n&\\leq\\ C \\lim_{r_0 \\to 0} r_0 \\left[ \\sum_{i=1}^l r_i^{-4} \\brs{P_{r_i}(z_i)}\n\\right]^{\\frac{1}{2}}\\\\\n&=\\ 0.\n\\end{align*}\nThe lemma follows.\n\\end{proof}\n\\end{lemma}\n\nCombining the result of Lemma \\ref{lem:Sigmacl}, Lemma \\ref{weakcompactlemma3},\nand Lemma \\ref{weakcompactlemma10}, the results of Theorem\n\\ref{fullweakcompactthm} follow.\n\\end{proof}\n\\end{thm}\n\n\\subsection{Structure of limit measures} \\label{limmeasstruct}\n\nAssume the setup of Theorem \\ref{fullweakcompactthm}. Observe that the measures\n\\begin{equation*}\n\\sqg{ \\brs{F_{\\nabla_t^i}}^2 dV \\, dt } \\text{ and }\\sqg{\\brs{\\frac{\\partial\n\\nabla^i_t}{\\partial t}}^2\ndV \\, dt }\n\\end{equation*}\nadmit subsequences converging in the sense of Radon measures to some\nlimit measures. We can compare these to the measures induced by the weak\n$H_1^2$ limit $\\nabla$ to define measures $\\mu,$ $\\nu$ and $\\eta$ via\n\\begin{align*}\n\\brs{F_{\\nabla^i_t}}^2 \\, dV \\, dt \\to&\\ \\brs{F_{\\nabla^{\\infty}_t}}^2 \\, dV \\, dt + \\nu\n\\equiv \\mu,\\\\\n\\brs{\\frac{ \\partial \\nabla^i_t}{\\partial t}}^2 \\, dV \\, dt \\to&\\ \\brs{ \\frac{\\partial\n\\nabla^{\\infty}_t}{\\partial t}}^2 \\, dV \\, dt + \\eta.\n\\end{align*}\n\nThe remainder of the section consists of a series of lemmas further refining the\nnature of these measures.\n\n\\begin{lemma}\\label{lem:8.2.5LWYM} Fix $z = (x,t) \\in M \\times [-1 ,0]$\nand $\\phi \\in \\mathcal{B}_x$. Then\n\\begin{equation*}\n\\Theta(\\mu,z) := \\lim_{R \\to 0} R^2\\int_{T_R(z)} \\phi^2(x) G_z(x,t) \\, d\n\\mu(x,t)\n\\end{equation*}\nexists and is upper semicontinuous for all $z \\in M \\times [0, \\infty)$.\n Moreover,\n %\n\\begin{align*}\n \\Sigma = \\left\\{ z \\in M \\times (0,\\infty) \\mid \\ge_0 \\leq \\Theta(\\mu,z) <\n\\infty \\right\\}.\n\\end{align*}\n\n\\begin{proof} We consider the limit as $i \\to \\infty$ in the monotonicity\ninequality (\\ref{Psimono}). In particular, for $0 < R \\leq R_0$, let\n\\begin{align*}\nf(R,d\\mu) =&\\ e^{CR} \\left[ \\frac{R^2}{2} \\int_{T_{R}} \\phi^2 G_z \\, d \\mu + C\ne^{CR} R \\mathcal{YM}(\\nabla_{-1}) \\right].\n\\end{align*}\nWe observe that (\\ref{Psimono}) implies that\n\\begin{align*}\nf(R,\\brs{F_{\\nabla^i_t}}^2 dV) =&\\ e^{C R} \\left[ \\Psi_{z_0}(R,\\nabla^i_t) + C R\n\\mathcal{YM}(\\nabla_{-1}) \\right]\\\\\n\\leq&\\ e^{CR} \\left[ e^{C(R_0 - R)} \\Psi_{z_0}(R,\\nabla_t^i) + C(R_0 - R)\n\\mathcal{YM}(\\nabla_{-1}) + C R \\mathcal{YM}(\\nabla_{-1}) \\right]\\\\\n=&\\ f(R_0,\\brs{F_{\\nabla^i_t}}^2 dV).\n\\end{align*}\nUsing that $\\brs{F_{\\nabla_t^i}}^2 dV$ converges to $d\\mu$, it follows that $f(R,d\n\\mu)$ is monotone nondecreasing as well. It follows that $\\lim_{R \\to 0} f(R,d\n\\mu)$ exists, and by elementary arguments the limit defining $\\Theta$ also\nexists, and is upper semicontinuous.\n\\end{proof}\n\\end{lemma}\n\n\n\\begin{lemma} \\label{weaktostronglem10} For $\\mathcal P^{n-2}$-almost everywhere $z \\in\n\\Sigma$, one\nhas\n\\begin{align*}\n \\lim_{R \\to 0} R^{2-n} \\int_{P_R(z)} \\brs{F_{\\nabla_t}}^2 \\, dV \\, dt = 0,\n\\qquad\n\\Theta(\\mu,z) = \\Theta(\\nu,z) \\geq \\ge_0.\n\\end{align*}\n\\begin{proof} To show the first claim, let\n\\begin{align*}\n K_j = \\sqg{ z \\in \\Sigma \\ |\\ \\limsup_{R \\to 0} R^{2-n} \\int_{P_R(z)}\n\\brs{F_t}^2 \\,\ndV \\, dt > j^{-1}\n}.\n\\end{align*}\nWe will show that the $(n-2)$-parabolic Hausdorff measure of $K_j$ is zero for\neach $j$, which suffices. Fixing some $\\delta > 0$ we can apply Vitali's covering\nlemma to obtain a covering of $K_j$ by disjoint parabolic balls\n$P_{r_{k}}(z_{k})$\nwith $z_{k} \\in K_j, 5 r_{k} \\leq \\delta$, such that $K_j \\subset \\bigcup P_{5\nr_{k}}(z_{k})$. It follows that there\nexists $C > 0$ such that\n\\begin{align*}\n \\mathcal P^{n-2}(K_j) &\\leq\\ \\lim_{\\delta \\to 0} \\sum_{k} (5 r_{k})^{n-2}\\\\\n &\\leq\\ C j \\lim_{\\delta \\to 0} \\int_{N_{\\delta}(\\Sigma)} \\brs{F_t}^2 \\, dV \\, dt\\\\\n &=\\ 0,\n\\end{align*}\nwhere $N_{\\delta}(\\Sigma)$ indicates the parabolic $\\delta$-tubular neighborhood of\n$\\Sigma$, and the last line follows by the dominated convergence theorem. The\nsecond claim now follows from the first and the definitions of $\\mu,\\nu$.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma} \\label{wktostronglemma5} For\n$\\mathcal P^{n-2}$-almost everywhere $z \\in \\Sigma$.\n\\begin{align*}\n\\lim_{r \\to 0} \\lim_{i \\to \\infty} r^{4-n} \\int_{P_r(z)} \\brs{ \\frac{\\partial\n\\nabla^i_t}{\\partial t}}^2 \\, dV \\, dt = 0.\n\\end{align*}\n\\begin{proof} We will show that for any $\\ge > 0$, the set\n\\begin{align*}\n\\mathcal C_{\\ge} := \\sqg{z \\in \\Sigma \\mid \\liminf_{r \\to 0} \\liminf_{i \\to \\infty}\nr^{4-n}\n\\int_{P_r(z)} \\brs{\\frac{ \\partial \\nabla^i}{\\partial t}}^2 \\, dV \\, dt \\geq \\ge }\n\\end{align*}\nsatisfies $\\mathcal P^{n-4}(\\mathcal C_{\\ge}) < \\infty$. Given this, we can express\n\\begin{align*}\n\\Sigma' := \\left\\{ z \\in \\Sigma \\mid \\liminf_{r \\to 0} \\liminf_{i \\to \\infty}\nr^{4-n}\n\\int_{P_r(z)} \\brs{\\frac{\\partial \\nabla^i}{\\partial t}}^2 \\, dV \\, dt = 0 \\right\\} =\n\\Sigma\\\n\\backslash\\ \\prs{ \\bigcup_{n \\in \\mathbb N} \\mathcal C_{2^{-n}}}.\n\\end{align*}\nIn particular, $\\Sigma'$ can be obtained from $\\Sigma$ by removing a countable\nunion of sets of finite $\\mathcal P^{n-4}$ measure, which has zero $\\mathcal P^{n-2}$ measure\nby a\nstandard argument.\n\nTo show $\\mathcal P^{n-4}(\\mathcal C_{\\ge}) < \\infty$, fix a $\\delta > 0$, and apply Vitali's\ncovering lemma to obtain a collection $\\{ z_k \\}_{i \\in \\mathbb{N}} \\subset\n\\Sigma$ and $r_k \\in (0,\\delta)$ satisfying that $\\{ P_{r_k}(z_k) \\}$ are\nmutually disjoint, $\\{ P_{5 r_k}(z_k) \\}$ cover $\\Sigma$, and furthermore there\nis some subsequence $\\{ \\nabla^{i}_t \\}$ so that for all $k,i$,\n\\begin{equation*}\nr_k^{4-n} \\int_{P_{r_k}(z_k)} \\brs{\\tfrac{\\partial \\nabla^{i}}{\\partial t}}^2 \\, dV \\, dt\n\\geq \\ge.\n\\end{equation*}\nUsing this we obtain\n\\begin{align*}\n\\mathcal{P}^{n-4}_{5 \\delta} \\prs{\\mathcal C_{\\ge}}\n&\\leq \\sum_{k=1}^{\\infty} \\prs{5 r_k}^{n-4} \\\\\n&= 5^{n-4} \\sum_{k=1}^{\\infty} r_k^{n-4} \\\\\n&\\leq \\tfrac{5^{n-4}}{\\ge} \\sum_{k=1}^{\\infty} \\int_{P_{r_k}(z_k)} \\brs{\n\\tfrac{\\partial \\nabla^{i}}{\\partial t} }^2 \\, dV \\, dt \\\\\n&\\leq \\tfrac{5^{n-4}}{\\ge} \\int_{\\bigcup_{k=1}^{\\infty} P_{r_k}(z_k)}\n\\brs{ \\tfrac{\\partial \\nabla^{i}}{\\partial t}}^2 \\, dV \\, dt \\\\\n&\\leq C(n,\\ge) \\int_0^2 \\int_{B_2} \\brs{ \\tfrac{\\partial \\nabla^{i}}{\\partial t}}^2 \\, dV \\,\ndt\\\\\n&\\leq C(n,\\ge, \\mathcal{YM}(\\nabla^i_{-1})),\n\\end{align*}\nwhere the last line follows via the Yang-Mills energy monotonicity. Sending\n$\\delta$ to zero\nproves that $\\mathcal P^{n-4}(\\mathcal C_{\\ge}) < \\infty$, finishing the proof.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma} \\label{wktostronglem15} The density function $\\Theta(\\mu,x)$ is\n$\\mathcal P^{n-2}$-approximately continuous at $\\mathcal P^{n-2}$-almost every $x \\in \\Sigma$.\n That is, for all $\\mathcal P^{n-2}$-a.e. $z \\in \\Sigma$ one has that for all $\\ge >\n0$,\n\\begin{align*}\n \\lim_{r \\to 0} r^{2-n} \\mathcal P^{n-2} \\left( \\{ w \\in P_r(x) \\cap \\Sigma\\ |\\\n\\brs{\\Theta(\\mu,w) - \\Theta(\\mu,z)} > \\ge \\} \\right) = 0.\n\\end{align*}\n\\begin{proof} Note that for a given $x \\in \\Sigma$, the density $\\Theta(\\mu,x)$\nis upper semicontinuous, so the set\n\\begin{equation*}\nA_c := \\sqg{ z \\mid \\Theta (\\mu, z) < c}\n\\end{equation*}\n is open. Therefore for any $c_1, c_2 \\in [0,\\infty)$ with $c_1 < c_2$, the set\n$A_{c_2} \\backslash A_{c_1}$ is a Borel set and thus measurable. Hence\n %\n \\begin{equation*}\n E_i := \\sqg{ z \\in \\Sigma \\mid \\frac{(i-1) \\ge}{2} \\leq \\Theta(\\mu,z) < \\frac{i\n\\ge}{2} } = A_{ \\frac{i \\ge}{2}} \\backslash A_{\\frac{(i-1)\\ge}{2}},\n \\end{equation*}\n is a Borel set. Note that, by the definition of $E_i$,\n \\begin{equation*}\n \\mathcal{P}^{n-2} \\prs{\\Sigma \\backslash \\bigcup_i E_i } = 0.\n \\end{equation*}\n %\n For all $x \\in E_i$, by applying Theorem 3.5 of \\cite{Simon} to the measure\n$\\mathcal P^{n-2}$ we have that\n %\n \\begin{align*}\n \\lim_{R \\to 0} R^{2-n} \\mathcal{P}^{n-2} &\\prs{ \\sqg{ y \\in P_r(x) \\cap\n\\Sigma \\mid \\brs{ \\Theta(\\mu, w) - \\Theta(\\mu,z)} > \\ge}}\\\\\n &= \\limsup_{R \\to 0} R^{2-n} \\mathcal{P}\\prs{ P_r(z) \\cap \\prs{\\Sigma\n\\backslash E_i} } \\\\\n &= 0.\n \\end{align*}\nThe result follows.\n\\end{proof}\n\\end{lemma}\n\n\n\\begin{lemma} \\label{wktostronglemma17} One has that $\\sqg{\\nabla_t^i}$ does not\nconverge\nto $\\nabla_t$ strongly in $H^{1,2}_{loc}$ if and only if $\\mathcal{P}^{n-2}(\\Sigma)\n> 0$\nand $\\nu\\prs{M \\times [-1,0]} > 0$.\n\n\\begin{proof}\nIt follows from Lemma \\ref{weaktostronglem10} that if $\\mathcal{P}^{n-2}(\\Sigma)\n>\n0$ then for $\\mathcal{P}^{n-2}$ almost everywhere $z \\in \\Sigma$ one has\n\\begin{equation*}\n\\Theta\\prs{\\nu,z} = \\Theta \\prs{\\mu,z} \\geq \\ge_0,\n\\end{equation*}\nhence $\\nu\\prs{M \\times [-1,0]} = \\nu\\prs{\\Sigma} >0$, and\n$\\tfrac{1}{2}\\brs{F_{\\nabla_t^i}}^2 \\, dV \\, dt$ does not converge to\n$\\tfrac{1}{2}\\brs{F_{\\nabla_t}}^2 \\, dV \\, dt$. Therefore $\\sqg{\\nabla_t^i}$ doesn't\nconverge to\n$\\nabla_t$ strongly in $H^{1,2}_{loc}$. \nConversely, directly from the definition of $\\nu$, if $\\nu\\prs{M \\times\n[-1,0] } > 0$ then $\\{\\nabla^i_t\\}$ cannot converge strongly to $\\nabla$ in\n$H^{1,2}_{loc}$.\n\\end{proof}\n\\end{lemma}\n\n\n\n\\section{Tangent measures and stratification} \\label{sec:tmstrat}\n\nIn this section we establish results on the structure of tangent measures along\nYang-Mills flow which will be central in the sequel. First we discuss the space\n$T_z\\mu$ of all tangent measures of $\\mu$ for $z \\in \\Sigma$. We first show that\nevery tangent measure is invariant under parabolic dilations. Building upon\nthis, we will associate to each tangent measure a nonnegative integer which is\nthe\ndimension of the largest parabolic dilation invariant subspace which is a subset\nof the points of maximal density. Using this dimension we can then stratify the\nset $\\Sigma$ accordingly. In\nparticular, we demonstrate enough structure on the tangent measures to apply a\nstratification result of White \\cite{White}, which generalizes Federer's\ndimension reduction argument \\cite{FZ}.\n\n\\subsection{Setup}\n\nFor the following we set\n\\begin{equation*}\n\\mathbb{R}_+^{n+1} : = \\mathbb{R}^n \\times [0,\\infty), \\quad \\mathbb{R}_-^{n+1} : = \\mathbb{R}^n \\times\n(-\\infty,0].\n\\end{equation*}\n\\begin{defn}\nFor $z_0 = \\prs{x_0, t_0} \\in \\mathbb{R}^n \\times \\mathbb{R}$ and $\\lambda >0$, define\n\\emph{parabolic dilation} and \\emph{Euclidean dilation} respectively by,\n\\begin{align*}\n\\mathsf{P}_{z_0,\\lambda}(x,t) &:= \\prs{\\lambda(x-x_0), \\lambda^2 (t-t_0)},\\\\\n\\mathsf{D}_{x_0,\\lambda}(x) &:= \\lambda(x-x_0).\n\\end{align*}\nMoreover, we may apply parabolic rescaling to a measure as follows. For all $A\n\\subset \\mathbb{R}^n \\times \\mathbb{R}$, we have\n\\begin{align*}\n\\mathsf{P}_{z_0,\\lambda}(\\mu)(A) :=&\\ \\lambda^{2-n} \\mu \\prs{\\mathsf{P}_{z_0,\\lambda}(A)},\\\\\n\\mathsf{D}_{x_0,\\lambda}\\prs{\\mu} \\prs{A} :=&\\ \\lambda^{4-n} \\mu\n\\prs{\\mathsf{D}_{x_0,\\lambda}\nA}.\n\\end{align*}\nWe note that this scaling law reflects the scaling properties for Yang-Mills\nflow densities, and not a pure parabolic rescaling of say Euclidean measure.\n\\end{defn}\n\n\\begin{defn} For any $z_0 \\in \\Sigma$, the \\emph{tangent measure cone of $\\mu$\nat $z_0$}, $T_{z_0}(\\mu)$, consists of all nonnegative Radon measures on\n$\\mathbb{R}^{n+1}$ that are given by\n\\begin{equation*}\nT_{z_0}(\\mu) := \\left\\{ \\mu^* \\mid \\exists r_i \\to 0, \\text{ such that\n}\\mathsf{P}_{z_0,r_i}(\\mu) \\to \\mu^* \\right\\}.\n\\end{equation*}\n\\end{defn}\nFixing $z_0 \\in \\Sigma$ and $\\mu^* = \\mu^*_s ds \\in T_{z_0}(\\mu)$, we set,\nfor any $z = (x,t) \\in \\mathbb{R}^{n+1}$,\n\\begin{equation*}\n\\Theta \\prs{\\mu^*, z , r} := r^4 \\int_{M \\times \\{{t-r^2}\\}} G_z(y,s) \\, d\n\\mu_s^*(y).\n\\end{equation*}\nThis is monotonically nondecreasing with respect to $r$ so that the\n\\emph{$\\mu^*$\ndensity at $z$}, given by\n\\begin{equation*}\n\\Theta(\\mu^*,z) := \\lim_{r \\to 0} \\Theta\\prs{\\mu^*,z,r},\n\\end{equation*}\nexists and is upper semicontinuous for $z = (x,t) \\in \\mathbb{R}^{n+1}$.\nMoreover, for any $z_0 \\in \\Sigma$ and $\\mu^* \\in T_{z_0}(\\mu)$, we set\n\\begin{align*}\nU\\prs{\\Theta\\prs{\\mu^*}} &:= \\left\\{ z \\in\n\\mathbb{R}^{n+1} \\mid \\Theta\\prs{\\mu^*,z} = \\Theta\\prs{\\mu^*,0} \\right\\},\\\\\nV \\prs {\\Theta\\prs{\\mu^*}} &:= U\\prs{\\Theta\\prs{\\mu^*}} \\cap\n\\prs{\\mathbb{R}^{n} \\times \\{0\\} },\\\\\nW \\prs{\\Theta\\prs{\\mu^*}} &:= \\left\\{(x,0) \\in \\mathbb{R}^n \\times \\mathbb{R} \\mid \\forall\n(y,s) \\in \\mathbb{R}^{n+1}_- ,\\Theta \\prs{\\mu^*, \\prs{y,s}} = \\Theta\n\\prs{\\mu^*,\\prs{x+y,s}} \\right\\}.\n\\end{align*}\n\\begin{defn} For $z_0 \\in \\Sigma$ and $\\mu^* \\in T_{z_0}(\\Sigma)$, let\n\\begin{equation*}\n\\dim \\prs{\\Theta \\prs{\\mu^*}} = \n\\begin{cases}\n\\dim \\prs{V\\prs{\\Theta\\prs{\\mu^*}}} + 2, &\\text{ if }\nU\\prs{\\Theta(\\mu^*)} = V\\prs{\\Theta \\prs{\\mu^*}} \\times\n\\mathbb{R},\\\\\n\\dim \\prs{V\\prs{\\Theta\\prs{\\mu^*}}} &\\text{ otherwise.}\n\\end{cases}\n\\end{equation*}\n\\end{defn}\n\n\n\\subsection{Preliminary results}\n\nIn this subsection we show various preliminary results on the structure of\ntangent measures. First we establish the existence of at least one tangent\nmeasure in Lemma \\ref{tangmeasexist}. We then establish parabolic scaling\ninvariance of tangent measures in Lemma \\ref{lem:8.3.2LWYM}. \n\n\\begin{lemma} \\label{tangmeasexist} Given a weak\nlimit measure $\\mu$, $z_0 \\in \\Sigma$, and $\\lambda_i \\to 0$ there exists a\nsubsequence $\\{ \\lambda_{i_j} \\}$ and some nonnegative Radon measure $\\mu^*$ on\n$\\mathbb{R}^{n+1}$ such that $\\mathsf{P}_{z_0,\\lambda_{i_j}}(\\mu) \\to \\mu^*$ as weak\nconvergence of Radon measures on $\\mathbb{R}^{n+1}$. \n\\begin{proof} We fix some small radius $r_0$ and claim that\n\\begin{align} \\label{tangmeasexist10}\n\\sup_{(z,r) \\in M \\times [-1,0] \\times (0,r_0)} r^{2-n} \\, \\mu(P_r(z)) < \\infty.\n\\end{align}\nIn particular, we use a change of variables and Theorem \\ref{thm:8.2.2LWYM} to\nyield\n\\begin{align*}\nr^{2-n} \\mu(P_r(z)) =&\\ r^{2-n} \\lim_{i \\to \\infty} \\int_{P_r(z)}\n\\brs{F_{\\nabla^i}}^2 \\, dV \\, dt\\\\\n=&\\ \\lim_{i \\to \\infty} r^{2-n} \\int_{t=0}^{r^2} \\int_{S_{\\sqrt{t}}}\n\\brs{F_{\\nabla^i}}^2 \\phi \\, dV \\, dt\\\\\n=&\\ \\lim_{i \\to \\infty} r^{2-n} \\int_{s=0}^r s \\int_{S_{s}} \\brs{F_{\\nabla^i}}^2\n\\phi \\, dV \\, ds\\\\\n\\leq&\\ \\lim_{i \\to \\infty} C r^{-2} \\int_{s=0}^r s \\Phi(s) \\, ds\\\\\n\\leq&\\ \\lim_{i \\to \\infty} C (\\Phi(r_0)) r^{-2} \\int_{s=0}^r s \\, ds\\\\\n\\leq&\\ C.\n\\end{align*}\nHence, using (\\ref{tangmeasexist10}), for any $\\lambda_i$ the sequence of dilated\nmeasures $\\mathsf{P}_{z_0,\\lambda_i}(\\mu)$ is\nuniformly bounded on all Borel sets in $\\mathbb R^{n+1}$, hence by the weak\ncompactness of families of uniformly bounded Radon measures we obtain the\nexistence of the subsequential limiting measure $\\mu$.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma}\\label{tangmeasureconelemma} For any\n$z_0 \\in \\Sigma$, $0 < r_1 < r_2 < \\infty$ a sequence $\\lambda_i \\to 0$ and a\nblowup\nsequence $\\bar{\\nabla}^i_t$ one has\n\\begin{equation*}\n\\lim_{i \\to \\infty} \\int_{-r_1^2}^{-r_2^2} \\int_{\\mathbb{R}^n} \\brs{x \\hook\nF_{\\bar{\\nabla}_t^i} + 2t \\partial_t \\bar{\\nabla}_t^i}^2 G_{z_0} \\, dx \\, dt = 0.\n\\end{equation*} \n\n\\begin{proof} First recall that as\nconvergence of Radon measures on $\\mathbb{R}^n$ we have\n\\begin{align*}\n\\tfrac{1}{2}\\brs{{\\overline{F_t}^i}}^2\\, dV \\to \\mu_t^* \\mbox{ for all } t \\in\n\\mathbb (-\\infty,0].\n\\end{align*}\nHence, for any $R > 0$, applying a change of variables we obtain\n\\begin{gather} \\label{tangmeascone10}\n\\begin{split}\nR^4 \\int_{\\mathbb{R}^n \\times \\{- R^2\\}} G_{(0,0)} \\, d\\mu_t^* \\, dt\n&= \\lim_{i \\to \\infty} \\int_{\\mathbb{R}^n \\times \\{- R^2\\}}\n\\tfrac{R^4}{2}\\brs{{\\overline{F}_t^i}}^2 G_{(0,0)} \\, dV_x \\, dt \\\\\n&= \\lim_{i \\to \\infty} \\int_{\\mathbb{R}^n \\times \\{- R^2\\}} \\tfrac{R^4 \\lambda_i^4}{2\n}\\brs{{F_{t_0 + \\lambda_i^2 t}^i}(x_0 + \\lambda_i x)}^2 G_{(0,0)} \\, dV_x \\, dt \\\\\n&=\\lim_{i \\to \\infty}\\int_{\\mathbb{R}^n \\times \\{t_0- R^2\\lambda_i^2\\}} \\tfrac{(\\lambda_i\nR)^4}{2 \\lambda_i^2}\\brs{{F_{t}^i}}^2 G_{z_0} \\, dV_y \\, ds\\\\\n&= \\lim_{\\lambda_i \\to 0} \\left[ \\left. \\int_{\\mathbb{R}^n} (\\lambda_i R)^4 \\, \\mu_t\n\\right|_{t = t_0- R^2\\lambda_i^2} \\right]\\\\\n&= \\Theta\\prs{\\mu,z_0},\n\\end{split}\n\\end{gather}\nwhere the last line follows from Lemma \\ref{lem:cute}.\nIn particular, the $\\Phi$ functional is approximately constant in $R$ for the\nconnections $\\bar{\\nabla}^i_t$, and hence using \\eqref{Phimono} we obtain the\nresult.\n\\end{proof}\n\\end{lemma}\n\nFor $\\Omega \\subset \\mathbb{R}^n \\times \\mathbb{R}$ we will use $\\left. \\mu^* \\right\\lfloor\n{\\Omega}$ to\ndenote the restriction of the tangent measure to $\\Omega$.\n\n\\begin{lemma} \\label{lem:8.3.2LWYM} For any $z_0 \\in \\Sigma$ and $\\mu^* \\in\nT_{z_0}(\\mu)$, the quantity $\\left. \\mu^* \\right\\lfloor {\\mathbb{R}^{n+1}_-}$ is\ninvariant under all parabolic dilation, i.e.\n\\begin{equation*}\n\\mathsf{P}_{\\kappa}\\prs{\\left. \\mu^* \\right\\lfloor {\\mathbb{R}^{n+1}_-}} = \\left.\n\\mu^* \\right\\lfloor {\\mathbb{R}^{n+1}_-}.\n\\end{equation*}\n\\begin{proof} First we observe that\n\\begin{align*}\n\\mathsf{P}_{\\kappa}\\prs{\\left. \\mu^* \\right\\lfloor {\\mathbb{R}^{n+1}_-}}\n&= \\mathsf{P}_{\\kappa}\\prs{ \\left\\{ \\prs{\\mu^*_t,t} \\mid t \\in (-\\infty,0]\n\\right\\} }\\\\\n&= \\left\\{\\prs{ \\mathsf{D}_{\\kappa}(\\mu_t^*), \\kappa^2 t } \\mid t \\in (-\\infty,0]\n\\right\\} \\\\\n&=\\left\\{\\prs{ \\mathsf{D}_{\\kappa}\\prs{\\mu_{\\tfrac{t}{\\kappa^2}}^*}, \\kappa^2 t }\\mid t\n\\in\n(-\\infty,0] \\right\\}.\n\\end{align*}\nThus, to prove the lemma it suffices to show that for all $\\kappa < 0$, for all $t\n\\in (-\\infty,0]$,\n\\begin{equation*}\n\\mathsf{D}_{\\kappa} \\prs{\\mu^*_{\\tfrac{t}{\\kappa^2}}} = \\mu^*_t.\n\\end{equation*}\nSince $\\kappa$ is arbitrary this is equivalent to demonstrating this at $t=-1$. To\nprove this it suffices to show the result for $\\mu_t^*$ multiplied by an\narbitrary smooth positive function. We will take advantage of this by inserting\na factor of the Greens function $G = G_{(0,0)}$, then multiplying by an\narbitrary compactly supported positive function. This will allow us to take\nadvantage of monotonicity formulae to obtain the result. In particular, we will\nshow that\n\\begin{equation}\\label{eq:8.27}\n\\kappa^{n-4} \\int_{\\mathbb{R}^n} \\phi(\\kappa x) G\\prs{\\kappa x, -1} d\\mu^*_{-\\kappa^{-2}} =\n\\int_{\\mathbb{R}^n} \\phi(x) G(x,-1) \\, d\\mu_{-1}^*,\n\\end{equation} \nfor any $\\phi \\in C_0^1(\\mathbb{R}^n)$.\nWe attain the claim \\eqref{eq:8.27} if we can show that\n\\begin{equation}\\label{eq:lambdaderivvanish}\n\\lim_{i \\to \\infty} \\frac{d}{d \\kappa} \\brk{\\frac{\\kappa^{n-4}}{2}\\int_{\\mathbb{R}^n}\n\\phi(\\kappa x) G\\prs{\\kappa x, -1} \\brs{\\overline{F}_{- \\kappa^{-2}}^i}^2 \\, dx } =\n0.\n\\end{equation}\nFor notational simplicity we will remove both the sequence index $i$ and the bar\nfrom the connection. Manipulating\nthe integrand by applying the change of coordinates $\\kappa x = y$ yields,\n\\begin{align*}\n\\frac{\\kappa^{n-4}}{2} \\int_{\\mathbb{R}^n \\times \\sqg{-1}} \\phi(\\kappa x) & \\brs{F_{-\\kappa^{-2}}\n\\prs{x}}^2 G(\\kappa x,-1) \\, dx \\\\\n&= \\frac{\\kappa^{n-4}}{2} \\int_{\\mathbb{R}^n \\times \\sqg{-1}} \\phi(y)\n\\brs{F_{-\\kappa^{-2}}\\prs{\\tfrac{y}{\\kappa}}}^2 G(y,-1) \\, d \\prs{\\frac{y}{\\kappa}}\\\\\n&= \\frac{\\kappa^{-4}}{2} \\int_{\\mathbb{R}^n \\times \\sqg{-1}} \\phi(y)\n\\brs{F_{-\\kappa^{-2}}\\prs{\\tfrac{y}{\\kappa}}}^2 G(y,-1) \\, dy\\\\\n&= \\left[ \\Phi \\prs{\\tfrac{1}{\\kappa};\\nabla_t} \\right|_{t=-1}.\n\\end{align*}\nSet $R(\\kappa) : = \\frac{1}{\\kappa}$. Then by a calculation similar to\n\\eqref{eq:delRmonotonicity}, where the final term vanishes since the cutoff\nfunction $\\phi$ no longer depends on the parameter $R$, we see\nthat\n\\begin{align*}\n\\frac{\\partial}{\\partial \\kappa} \\brk{\\Phi \\prs{\\tfrac{1}{\\kappa};\\nabla_t}}\n&= \\frac{-1}{\\kappa^2} \\frac{\\partial}{\\partial R} \\brk{\\Phi \\prs{R(\\kappa);\\nabla_t}}\\\\\n&= \\frac{-1}{\\kappa^5} \\int_{S_{\\kappa^{-1}}}\n\\brs{t}\\brs{\\frac{x}{t}\\hook F - 2 D^* F}^2 \\phi G \\, dx + \\frac{4}{\\kappa^5}\n\\int_{S_{\\kappa^{-1}}} \\ip{ \\prs{x \\hook F - 2 t \\prs{D^* F} }, \\nabla \\phi \\hook F } \nG \\, dx\n\\end{align*}\nTaking the limit as $i \\to \\infty$, we have that the first quantity vanishes by\nLemma \\ref{tangmeasureconelemma}. For the second we apply weighted H\\\"{o}lder's\ninequality for an arbitrary $\\ge > 0$,\n\\begin{align*}\n\\frac{1}{\\kappa^5}& \\int_{S_{\\kappa^{-1}}} \\ip{ \\prs{x \\hook F - 2 t \\prs{D^* F} }, \\nabla\n\\phi \\hook F } G \\, dx\\\\\n&\\leq \\frac{C}{\\ge \\kappa^5} \\int_{S_{\\kappa^{-1}}} \\brs{ \\prs{x \\hook F - 2 t\n\\prs{D^* F} } }^2 G \\, dx + \\frac{\\ge}{\\kappa^5} \\int_{S_{\\kappa^{-1}}} \\brs{ \\nabla\n\\phi}^2 \\brs{F }^2 G \\, dx.\n\\end{align*}\nThe first factor vanishes with another application of\nLemma \\ref{tangmeasureconelemma}. The integrand of the second term is bounded\nby the monotonicity of $\\Phi$, using an argument similar to\n(\\ref{tangmeascone10}). Sending $\\ge \\to 0$ therefore yields\n\\eqref{eq:lambdaderivvanish}. The result follows.\n\\end{proof}\n\\end{lemma}\n\n\\subsection{Stratification of tangent measures}\n\n\\begin{lemma}\\label{lem:8.3.3LWYM} For $z_0 \\in\n\\Sigma$ and $\\mu^* \\in T_{z_0}(\\mu)$, the following hold.\n\\begin{enumerate}\n\\item For all $z \\in \\mathbb{R}^{n+1}$, $\\Theta(\\mu^*,z) \\leq\n\\Theta(\\mu^*,0)$.\n\\item If $z \\in \\mathbb{R}^{n+1}$ satisfies $\\Theta\\prs{\\mu^*,z} =\n\\Theta\\prs{\\mu^*,0}$, then for all $\\lambda >0$ and $v \\in\n\\mathbb{R}^{n+1}_{-}$,\n\\begin{equation*}\n\\Theta\\prs{\\mu^*, z+v} = \\Theta\\prs{\\mu^*, z + \\mathsf{P}_{\\lambda}v}.\n\\end{equation*}\n\\end{enumerate}\n\\begin{proof} For $\\mu^* \\in T_{z_0}(\\mu)$, there exists some sequence $r_i\n\\to 0$ such that $\\mathsf{P}_{z_0,r_i}(\\mu) \\to \\mu^*$. We first observe how\nthe rescaling law for $\\Phi$ is reflected in the definition of $\\Theta$. In\nparticular, since we are integrating over a space slice we apply the scaling law\nfor $\\mathsf{D}_{\\lambda}$ and change variables to yield\n\\begin{align*}\n \\Theta(\\mathsf{P}_{\\lambda}(\\mu),z,r) =&\\ \\frac{r^4}{2} \\int_{S_r} G_z\n\\mathsf{P}_{\\lambda}(\\mu)\\\\\n =&\\ \\frac{r^4}{2} \\int_{S_r} \\left( \\lambda^n \\mathsf{P}_{\\lambda}^*\nG_{\\mathsf{P}_{\\lambda}(z)} \\right) \\left( \\lambda^{4-n} \\mathsf{P}_{\\lambda}^* \\mu\n\\right)\\\\\n =&\\ \\frac{(\\lambda r)^4}{2} \\int_{\\mathsf{P}_{\\lambda}(S_r)} G_{\\mathsf{P}_{\\lambda}(z)}\n\\mu\\\\\n =&\\ \\Theta(\\mu,\\mathsf{P}_{\\lambda}(z), \\lambda r).\n\\end{align*}\nUsing this, for any $r>0$, and\n$z = \\prs{x,t} \\in \\mathbb{R}^{n+1}$,\n\\begin{align}\n\\begin{split}\\label{eq:Thetanmu0ineq}\n\\Theta\\prs{\\mu^*,z}\n&\\leq \\Theta\\prs{\\mu^*,z,r}\\\\\n&= \\lim_{r_i \\to 0} \\Theta \\prs{ \\mathsf{P}_{z_0,r_i}(\\mu),z,r} \\\\\n&= \\lim_{r_i \\to 0} \\Theta\\prs{\\mu,z_0 + \\prs{r_i x,r_i^2 t}, r_i r} \\\\\n&\\leq \\Theta\\prs{\\mu,z_0}\\\\\n&=\\Theta\\prs{\\mu^*,0},\n\\end{split}\n\\end{align}\nwhere we have applied the upper semicontinuity of $\\Theta(\\mu,\\cdot,\\cdot)$\nwith respect to the last two variables. Thus claim (1) follows.\n\nTo prove claim (2), observe that the hypothesis $\\Theta(\\mu^*,z) =\n\\Theta(\\mu^*,0)$ implies that the inequalities of \\eqref{eq:Thetanmu0ineq} are\nequalities. This implies that $\\Theta\\prs{\\mu^*,z,r}=\\Theta(\\mu,z_0)$,\nnamely, it is constant with respect to $r$. By an argument similar to that of\nLemma \\ref{lem:8.3.2LWYM}, we have that $\\Theta\\prs{\\mu^*, z+v} =\n\\Theta\\prs{z+\\mathsf{P}_{\\lambda}(v)}$ for any $v \\in \\mathbb{R}^{n+1}_-$ and $\\lambda\n>0$. The result follows.\n\\end{proof}\n\\end{lemma}\n\n\n\\begin{prop} \\label{prop:8.3.4LWYM} For $z_0 \\in\n\\Sigma$ and $\\mu^* \\in T_{z_0}(\\mu)$,\n\\begin{equation*}\nV\\prs{\\Theta \\prs{\\mu^*, \\cdot}} = W \\prs{\\Theta \\prs{\\mu^*, \\cdot}}.\n\\end{equation*}\nIn particular, both $V\\prs{\\Theta \\prs{\\mu^*, \\cdot}}$ and $W\\prs{\\Theta\n\\prs{\\mu^*, \\cdot}}$ are linear subspaces of $\\mathbb{R}^n$. Moreover,\n$U\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ is either $V\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$,\nor $V\\prs{\\Theta\\prs{\\mu^*,\\cdot}} \\times (-\\infty,a]$ for some $0 \\leq a \\leq\n\\infty$ and $\\Theta\\prs{\\mu^*,\\cdot}$ is time-independent on $(-\\infty,a]$.\n\n\\begin{proof}\nFirst we show that $W(\\Theta(\\mu^*, \\cdot)) \\subset V(\\Theta(\\mu^*,\\cdot))$.\n Fix $(x,0) \\in W\\prs{\\Theta \\prs{\\mu^*, \\cdot}}$. Since the second component\nis identically zero it suffices to verify that $(x,0) \\in\nU\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$. Note that by definition of $W\\prs{\\Theta\n\\prs{\\mu^*, \\cdot}}$, choosing $y = -x$ as in its definition,\n\\begin{align*}\n\\Theta\\prs{\\mu^*,(x,0)}\n&= \\Theta\\prs{\\mu^*,(x - x,0)}= \\Theta\\prs{\\mu^*,0}.\n\\end{align*}\nIt follows that $W\\prs{\\Theta \\prs{\\mu^*, \\cdot}} \\subset V\\prs{\\Theta\n\\prs{\\mu^*, \\cdot}}$.\n\nNow we show the containment $V\\prs{\\Theta \\prs{\\mu^*, \\cdot}} \\subset\nW\\prs{\\Theta \\prs{\\mu^*, \\cdot}}$. First note that $V\\prs{\\Theta\n\\prs{\\mu^*, \\cdot}}$ is closed under scalar multiplications from Lemma\n\\ref{lem:8.3.2LWYM}. Next, for any nonzero $x \\in\nV\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ we have that for all $\\lambda > 0$ and all $v \\in\n\\mathbb{R}^{n+1}_{-}$, by applying Lemma \\ref{lem:8.3.3LWYM} (2), and using the\nparabolic scaling invariance of $\\Theta$ from Lemma \\ref{lem:8.3.2LWYM},\n\\begin{align}\n\\begin{split}\\label{eq:8.33LW}\n\\Theta \\prs{\\mu^*, \\prs{x,0} + v}\n&= \\Theta \\prs{\\mu^*, \\prs{x,0}+\\mathsf{P}_{\\lambda} v}\\\\\n&=\\Theta\\prs{\\mu^*,\\mathsf{P}_{\\lambda^{-1}}\\prs{\\prs{x,0} + \\mathsf{P}_{\\lambda} v}}\\\\\n&=\\Theta\\prs{\\mu^*,\\mathsf{P}_{\\lambda^{-1}}\\prs{x,0} + v}.\n\\end{split}\n\\end{align}\nBy the upper semicontinuity of $\\Theta$, sending $\\lambda \\to \\infty$ yields\n\\begin{align*}\n\\Theta(\\mu^*, (x,0) + v) \\leq \\Theta(\\mu^*, v).\n\\end{align*}\nOn the other hand, since $v - \\mathsf{P}_{\\lambda^{-1}}\\prs{x,0} \\in \\mathbb{R}^{n+1}_-$,\nwe can replace $v\n\\mapsto v - \\mathsf{P}_{\\lambda^{-1}}\\prs{x,0}$ throughout the equalities in\n\\eqref{eq:8.33LW} and obtain that\n\\begin{equation*}\n\\Theta\\prs{\\mu^*,\\prs{x,0} + v - \\mathsf{P}_{\\lambda^{-1}}\\prs{x,0}} =\n\\Theta\\prs{\\mu^*, v}.\n\\end{equation*}\nAgain sending $\\lambda \\to \\infty$ and utilizing the upper semicontinuity of\n$\\Theta\\prs{\\mu^*,\\cdot}$ yields\n\\begin{align*}\n\\Theta(\\mu^*, (x,0) + v) \\geq \\Theta(\\mu^*, v).\n\\end{align*}\nHence we have $\\Theta\\prs{\\mu^*, v} = \\Theta \\prs{\\mu^*, \\prs{x,0} + v}$, and so\nwe conclude $V\\prs{\\Theta\\prs{\\mu^*,\\cdot}} \\subset\nW\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ so that $V\\prs{\\Theta\\prs{\\mu^*,\\cdot}} =\nW\\prs{\\Theta\\prs{\\mu^*,\\cdot}} $.\n\nNote that by definition of $W\\prs{\\Theta\\prs{\\mu^*,\\cdot}} $ we have that it\nis closed under linear combinations since for all $\\prs{x,0}$, $\\prs{v,0}$ in\n$W\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ we have that for all $\\prs{y,s} \\in\n\\mathbb{R}_-^{n+1}$, just iterating its definition twice\n\\begin{align*}\n\\Theta \\prs{\\mu^*, \\prs{\\prs{x+v} + y, s}}\n&= \\Theta \\prs{\\mu^*, \\prs{x + y, s}} \\\\\n&= \\Theta \\prs{\\mu^*, \\prs{y, s}}.\n\\end{align*}\nTherefore by equality of $V\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ to\n$W\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$, with the combined scaling invariance and\nlinear combinations invariance both are linear subspaces of $\\mathbb{R}^n$.\n\nNow we prove the remaining statement of the proposition concerning the structure\nof $U\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$. \n Suppose that $z := \\prs{x,t} \\in U\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ with $t <\n0$. Then for all $w := \\prs{y,s} \\in \\mathbb{R}^{n+1}$ with $s \\leq t$ and for all $\\lambda\n> 0$, using Lemma \\ref{lem:8.3.3LWYM} (b)\n\\begin{align}\n\\begin{split}\\label{eq:8.3.4LWYMc}\n\\Theta \\prs{\\mu^*, \\mathsf{P}_{\\lambda^{-1}}(w)}\n&= \\Theta \\prs{\\mu^*, w}\\\\\n&= \\Theta \\prs{\\mu^*, z + w - z}\\\\\n&= \\Theta \\prs{\\mu^*, z + \\mathsf{P}_{\\lambda^{-1}}(w-z)}.\n\\end{split}\n\\end{align}\nIn particular, take $\\lambda \\in \\prs{0,1}$, and note that consequently\n$\\tfrac{s}{\\lambda^{2}} \\leq s \\leq t$. So taking \\eqref{eq:8.3.4LWYMc} and\nreplacing $w \\mapsto \\mathsf{P}_{\\lambda}\\prs{w}$ in yields\n\\begin{equation}\\label{eq:8.3.4LWYMb}\n\\Theta \\prs{\\mu^*,w} :=\\Theta \\prs{\\mu^*, z+w-\\mathsf{P}_{\\lambda^{-1}}(z)}.\n\\end{equation}\nTaking $\\lambda \\to 0$, we see that $\\Theta\\prs{\\mu^*,w} \\leq\n\\Theta\\prs{\\mu^*,z+w}$. Taking \\eqref{eq:8.3.4LWYMb} again and instead\nreplacing $w \\mapsto w + \\mathsf{P}_{\\lambda^{-1}}(z)$, we conclude that\n\\begin{equation*}\n\\Theta\\prs{\\mu^*,w+\\mathsf{P}_{\\lambda^{-1}}\\prs{z}} = \\Theta\\prs{\\mu^*,z+w}.\n\\end{equation*}\nAgain sending $\\lambda \\to 0$ we obtain that\n\\begin{equation*}\n\\Theta\\prs{\\mu^*,z} \\leq \\Theta\\prs{\\mu^*,z+w}.\n\\end{equation*}\nTherefore, we conclude that for any $z := \\prs{x,t} \\in\nU\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ with $t < 0$, for all $w :=\n\\prs{y,s}$ with $s \\leq t$,\n\\begin{equation}\\label{eq:8.3.4LWYMa}\n\\Theta\\prs{\\mu^*,w} = \\Theta\\prs{\\mu^*,z+w}.\n\\end{equation}\nThen choosing $w \\equiv z$, iterating \\eqref{eq:8.3.4LWYMa}, applying the\nparabolic\nscaling invariance of $\\Theta$ from Lemma \\ref{lem:8.3.2LWYM}, and the upper\nsemicontinuity of $\\Theta \\prs{\\mu^*, \\cdot }$, one has\n\\begin{align*}\n\\Theta\\prs{\\mu^*,0} &=\\ \\Theta(\\mu^*, z) = \\Theta(\\mu^*, z + z) = \\dots =\n\\Theta(\\mu^*, mz)\\\\\n&=\\Theta \\prs{\\mu^*, \\prs{mx,mt}}\\\\\n&= \\Theta \\prs{\\mu^*, \\mathsf{P}_{\\frac{1}{m}}\\prs{mx,mt}}\\\\\n&= \\Theta \\prs{\\mu^*, \\prs{x,\\tfrac{t}{m}}}\\\\\n&\\leq \\Theta \\prs{\\mu^*, \\prs{x,0}}.\n\\end{align*}\nCombining this with Lemma \\ref{lem:8.3.3LWYM} (1) we conclude that $\\prs{x,0}\n\\in V\\prs{\\Theta\\prs{\\mu^*,\\cdot}} = W\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$.\nTherefore\n\\begin{align*}\n\\Theta \\prs{ \\mu^*, \\prs{0,t}} = \\Theta(\\mu^*, (x,0) + (0,t)) =\n\\Theta(\\mu^*,0)\n\\end{align*}\nIt follows that $\\prs{0,t}\\in U\\prs{\\Theta\\prs{\\mu^*,\\cdot}}$. It follows\nthat $\\Theta\\prs{\\mu^*,\\cdot}$ is actually time independent for $t \\leq 0$.\nTherefore for all $t \\leq 0$,\n\\begin{equation*}\nV \\prs{\\Theta \\prs{\\mu^*,\\cdot}} := U \\prs{\\Theta \\prs{\\mu^* , \\cdot }} \\cap\n\\prs{\\mathbb{R}^n \\times \\sqg{t}}.\n\\end{equation*}\nLastly, if $z = (x,t) \\in U \\prs{\\Theta\\prs{\\mu^*,\\cdot}}$ with $t > 0$, then we\ncan repeat the argument above to show that\n$\\Theta\\prs{\\mu^*,\\cdot}$ is time-independent up to $t$. We set $a$ to be the\nvalue of the maximal time $t \\geq 0$ for which this time independence exists on.\nThen we have $U \\prs{\\Theta\\prs{\\mu^*,\\cdot}} = V\\prs{\\Theta\n\\prs{\\mu^*,\\cdot}} \\times (-\\infty, a]$, which concludes the proof.\n\\end{proof}\n\\end{prop}\n\n\n\nWe can now establish Theorem \\ref{dimredthm}, which we restate for convenience.\n\n\\begin{thm*} \nFor $0 \\leq k \\leq n-2$ let\n\\begin{equation*}\n\\Sigma_k = \\left\\{ z_0 \\in \\Sigma \\mid \\dim \\prs{ \\Theta \\prs{\\mu^*, \\cdot}}\n\\leq\nk, \\forall \\mu^* \\in T_{z_0}(\\mu) \\right\\}.\n\\end{equation*}\nThen $\\dim_{\\mathcal{P}}\\prs{\\Sigma_k} \\leq k$ and $\\Sigma_0$ is countable.\n\\begin{proof}[Proof of Theorem \\ref{dimredthm}] This is a direct consequence of\n(\\cite{White} Theorem 8.2). To connect directly to the notation of that paper,\nthe function $f$ is given by the density function. Hypothesis (1), the\nsubsequential compactness of blowup limits, is established in Lemma\n\\ref{tangmeasexist}. Hypothesis (2) is clear from the construction of blowup\nlimits. Hypothesis (3), the parabolic scaling invariance of the limit\nfunctions, is established in Lemma \\ref{lem:8.3.2LWYM}. The theorem thus\napplies to give the claimed statement.\n\\end{proof}\n\\end{thm*}\n\n\n\\section{Characterization of strong convergence}\\label{s:charstrconv}\n\nIn this section we prove Theorem \\ref{thm:weaktostrong} (stated more precisely\nas Theorem \\ref{weaktostrongH12} below), which characterizes when the weak\nconvergence in $H^{1,2}$ for\nsequences as in Theorem \\ref{fullweakcompactthm} can be improved to strong\nconvergence. In particular, we know this means that the defect measure is\nnontrivial, and we use this to obtain refined estimates on tangent measures,\neventually leading to a further blowup sequence which yields the required\nYang-Mills connection.\n\n\\begin{thm} \\label{weaktostrongH12} Suppose $\\{\\nabla^i_t\\}$ is a sequence of smooth\nsolutions to Yang-Mills flow on $[-1,0]$ with \n\\begin{align*}\n\\sup_i \\int_{M \\times [-1,0]} \\left( \\brs{\\frac{\\partial \\nabla^i_t}{\\partial t} }^2 +\n\\brs{F_{\\nabla_t^i}}^2 \\right) dV dt < \\infty.\n\\end{align*}\nFurthermore, suppose $\\{\\nabla^i_t\\} \\to \\nabla^{\\infty}_t$ weakly in $H^{1,2}_{loc}$. \nThen\nexactly\none of the following holds:\n\\begin{itemize}\n\\item There exists a blowup sequence converging to a\nYang-Mills connection on $S^4$.\n\\item One has\n\\begin{align*}\n\\brs{F_{\\nabla^i_t}}^2 \\, dV \\, dt \\to \\brs{F_{\\nabla^{\\infty}_t}}^2 \\, dV \\, dt\n\\end{align*}\nas convergence of Radon measures, and hence $\\{\\nabla^i_t\\} \\to \\nabla^{\\infty}_t$\nstrongly\nin $H^{1,2}_{loc}$. Thus $\\nabla^{\\infty}_t$ is a weak solution of Yang-Mills flow\nsatisfying\n$\\mathcal P^{n-2}(\\Sigma) = 0$.\n\\end{itemize}\n\\begin{proof} \n\nWe adopt the setup of the previous sections in this proof. In particular, we\nassume we have a particular blowup sequence together with a limiting tangent\nmeasure $\\mu^*$. Moreover, various results from \\S \\ref{limmeasstruct} were\nestablished which apply to almost every point in the singular set. We will\nassume without loss of generality that our tangent measure arises from a blowup\nsequence around one of these points, so that the Lemmas of \\S\n\\ref{limmeasstruct} apply. In particular, in the discussion below we will refer\nto a sequence $\\{\\nabla_t^i\\}$ but this will refer to a \\emph{blowup} sequence, not\nthe original given sequence of the statement.\n\n\\begin{lemma} \\label{mustarmeasure} For $t \\in\n(-4,0]$, we have $\\mathcal{H}^{n-4} \\brk{\\Sigma_t^*} > 0$.\n\\begin{proof}\nSuppose to the contrary there is some $t_0\n\\in (-4,0]$ such that $\\mathcal H^{n-4}(\\Sigma_{t_0}^*) = 0$. Then for all $\\ge\n> 0$, there exists some $\\delta_{\\ge} >0$ and a covering of $\\Sigma_{t_0}^*$ of\nthe form $\\{ B_{r_j}(x_j) \\}_{i \\in \\mathbb{N}}$, with $x \\in \\Sigma_{t_0}^*$\nand $0 < r_j \\leq \\delta_{\\ge}$ satisfying\n\\begin{equation*}\n\\sum_{j=1}^{\\infty}r_j^{n-4} < \\ge.\n\\end{equation*}\nNow, because $\\mu^*_{t_0}\\brk{ B_1 \\backslash \\prs{\\bigcup_{j \\in \\mathbb{N}}\nB_{r_j}(x_j) }} = 0,$ then by a diagonalization argument we may choose a\nsubsequence $\\{ \\nabla^{i}_t \\}$ such that\n\\begin{equation} \\label{mustarmeasure10}\n\\lim_{{i} \\to \\infty} \\tfrac{1}{2}\\int_{B_1 \\backslash \\bigcup_{j \\in\n\\mathbb{N}} B_{r_j}(x_j)} \\brs{ F_{\\nabla^{i}_{t_0}}}^2 \\, dV = 0.\n\\end{equation}\n\nFurthermore we will use (\\ref{Phimono}) to estimate the curvature on balls in\nthe cover. We choose a cutoff function $\\phi$ for a ball of radius $1$, and\nfurther fix some radius $R$. Note that for the compact set $\\supp \\phi$ there\nis a uniform estimate for the $L^2$ norm of the Yang-Mills energy. This follows\nfrom the argument of Lemma \\ref{tangmeasexist}, which shows that the sequence of\nblowup measures is uniformly locally finite. In particular, there exists $K <\n\\infty$ such that\n\\begin{align*}\n\\int_{\\supp \\phi} \\brs{F_{\\nabla^i_{t_0}}}^2 dV \\leq K.\n\\end{align*}\nHence using (\\ref{Phimono}) we estimate for all $i,j \\in \\mathbb N$,\n\\begin{align*}\n\\tfrac{r_j^{4-n}}{2} \\int_{B_{r_j}(x_j)} \\brs{F_{\\nabla_{t_0}^{i}}}^2 \\, dV\n&\\leq \\tfrac{1}{2 e} r_{j}^4 \\int_{B_{r_j}(x_j) \\times\\{ (t_0 + r_j^2) - r_j^2\n\\}}\n\\brs{F_{\\nabla^{i}_t}}^2 \\phi^2 G_{(x_i,t_0 + r_j^2)} dV\\\\\n&\\leq \\tfrac{1}{2 e} \\Phi\\prs{\\nabla_t^{i}, \\prs{x_j, t_0 + r_j^2}, r_j} \\\\\n&\\leq \\tfrac{1}{2 e} \\Phi\\prs{\\nabla^{i}_t, \\prs{x_j, t_0 + r_j^2}, R} + C K \\prs{R\n-\nr_j} \\\\\n&\\leq \\tfrac{R^{4-n}}{2e} \\int_{\\mathbb{R}^n \\times \\{t_0+r_j^2-R^2\\}}\n\\brs{F_{\\nabla_t^{i}}}^2 \\phi^2 \\, dV + CK \\prs{R-r_j}\\\\\n&\\leq C\\prs{R,K}.\n\\end{align*}\nTherefore we have that\n\\begin{align*}\n\\tfrac{1}{2}\\int_{\\bigcup_{j \\in \\mathbb{N}}B_{r_j}(x_j)}\n\\brs{F_{\\nabla_{t_0}^{i}}}^2 dV\n&\\leq \\tfrac{1}{2}\\sum_{j\\in \\mathbb{N}} \\int_{B_{r_j}(x_j)}\n\\brs{F_{\\nabla_{t_0}^{i}}}^2 dV\\\\\n&\\leq C \\sum_{j \\in \\mathbb{N}} r_j^{n-4}\\\\\n&\\leq C \\ge.\n\\end{align*}\nChoosing $\\ge < \\tfrac{\\ge_0}{16C}$ and combining with (\\ref{mustarmeasure10})\nyields, for $i$ sufficiently large,\n\\begin{equation} \\label{mustarmeasure30}\n\\int_{B_1} \\brs{F_{\\nabla^i_{t_0}}}^2 \\, dV \\leq \\frac{\\ge_0}{2}.\n\\end{equation}\nAlso, using Lemma \\ref{wktostronglemma5} we have\n\\begin{equation} \\label{mustarmeasure40}\n\\lim_{i \\to \\infty} \\int_{P_2}\\brs{\\frac{\\partial \\nabla_t^i}{\\partial t}}^2 dV = 0.\n\\end{equation}\nUsing Lemma \\ref{lem:8.2.1LWYM} we find that for any $\\phi \\in\nC_0^{\\infty}(B_2)$ and $-4 < t_1 < t_2 \\leq 0$ one has\n\\begin{equation} \\label{mustarmeasure50}\n\\tfrac{1}{2} \\int_{B_2}\\prs{\\brs{{F_{t_2}^i}}^2-\\brs{{F_{t_1}^i}}^2} \\phi \\, dV\n= - \\int_{t_1}^{t_2}\\int_{B_2}\\prs{\\brs{\\partial_t \\nabla_t^i}^2 \\phi + \\ip{\\nabla \\phi\n\\hook {F_t^i},\\partial_t \\nabla_t^i}} \\, dV\\, dt.\n\\end{equation}\nCombining (\\ref{mustarmeasure40})-(\\ref{mustarmeasure50}) shows that the\nlimiting measure $\\mu_t^*(\\phi)$ is independent of time for $t \\in (-4,0]$. \nApplying (\\ref{mustarmeasure30}) again yields\n\\begin{align*}\n\\tfrac{1}{2}\\int_{P_1}\\brs{{F_t^i}}^2 dV \\, dt\n&\\leq \\tfrac{1}{2}\\sup_{t \\in \\brk{-1,0}} \\int_{B_1} \\brs{{F_t^i}}^2 \\, dV\\\\\n&\\leq \\tfrac{1}{2} \\int_{B_1} \\brs{{F_{t_0}^i}}^2 \\, dV + o(i)\\\\\n&\\leq \\frac{\\ge_0}{4}. \n\\end{align*}\nThis is a contradiction to the assumption that $(0,0) \\in \\Sigma^*$. \n\\end{proof}\n\\end{lemma}\n\n\n\n\\begin{prop} \\label{singularsetisplane} There is a linear subspace $\\mathscr{P}$\nof dimension $\\prs{n-4}$ such that for all $t < 0$ one has that $\\supp(\\mu^*_t)\n= \\mathscr{P}$\n \\begin{proof} \nFirst, by Lemma \\ref{wktostronglem15}, we have that $\\Theta\\prs{\\nu,z} =\n\\Theta\\prs{\\mu,z}$ is $\\mathcal{P}^{n-2}$-approximately continuous at $z_0$ for\n$z\n\\in \\Sigma$ and $\\Theta\\prs{\\mu,z}$ is upper semicontinuous with respect to\n$z$, we conclude that for $\\mathcal{P}^{n-2}$-almost all $z \\in\n\\Sigma^*$,\nwe have\n\\begin{equation*}\n\\Theta\\prs{\\mu^*,z} \\geq \\Theta\\prs{\\mu,z_0}.\n\\end{equation*}\nAlso, it follows from Lemma \\ref{lem:8.3.3LWYM} that\n\\begin{equation*}\n\\Theta\\prs{\\mu^*,z} \\leq \\Theta\\prs{\\mu^*,0} = \\Theta\\prs{\\mu,z_0}.\n\\end{equation*}\nHence for $\\mathcal{P}^{n-2}$ a.e. $z \\in \\Sigma_*$,\n\\begin{equation} \\label{singsetplaneloc10}\n\\Theta\\prs{\\mu^*,z} = \\Theta\\prs{\\mu^*,0}.\n\\end{equation}\nWe now show that in fact all points in $\\Sigma_*$ have maximal density.\nIn particular, by Proposition \\ref{prop:8.3.4LWYM} there is some set\n$\\mathscr{S} \\subset \\mathbb{R}^n$ with $\\mathcal{H}^{n-4}\\prs{\\mathscr{S}} = 0$ and an\n$(n-4)$-dimensional plane $\\mathscr{P} \\subset \\mathbb{R}^n$ such that\n$\\mathscr{S} \\cap \\mathscr{P} = \\varnothing$, and\n\\begin{align*}\n\\Sigma_t^* = \\mathscr{S} \\cap \\mathscr{P}\n\\end{align*}\nfor all $t$. We claim that in fact $\\mathscr{S} = \\varnothing$. \nSuppose to the contrary we had some $z \\in\n\\mathscr{S}$. Note that by construction, it must hold that $0 < \\Theta(\\mu^*,z)\n< \\Theta(\\mu^*,0)$. By Lemma \\ref{lem:8.3.3LWYM} (2) we have that for all $w\n\\in\n\\mathscr{P}$,\n\\begin{equation*}\n\\Theta(\\mu^*,z) = \\Theta(\\mu^*, w + (z-w))= \\Theta\\prs{\\mu^*,w +\n\\mathsf{P}_{\\lambda}(z - w)}.\n\\end{equation*}\nApplying this for $w \\in B_{\\ge}^{n-4}(0) \\subset \\mathscr{P}$ and $\\lambda \\in\n[1-\\ge,1]$ yields a set of positive $\\mathcal P^{n-2}$-measure in $\\Sigma^*$ on which\n$\\Theta(\\mu^*, \\cdot) = \\Theta(\\mu^*, z) < \\Theta(\\mu^*,0)$, contradicting\n(\\ref{singsetplaneloc10}).\n \\end{proof}\n\\end{prop}\n\n\n\n\nUsing this characterization of the singular set of the blowup limit, we can\nrefine our estimates on the blowup sequence to obtain further structure on the\nblowup limit. Without loss of generality we can assume that $\\mathscr{P} =\n\\mathbb R^{n-4} \\subset \\mathbb R^n$ is the standard embedding in the first\n$n-4$ coordinates, and we express a general point as $X = (x,y)$ where $x \\in\n\\mathbb R^4, y \\in \\mathbb R^{n-4}$. We first show two lemmas which give\nimproved vanishing results for the time derivative of the connection as well as\nfor the curvature in directions along the singular locus.\n\n\\begin{lemma} \\label{wktostronglemma10} Given the setup above and $0 < t_1 <\nt_2\n\\leq 1$, one has\n\\begin{gather*}\n\\lim_{i \\to \\infty} \\int_{-t_2}^{-t_1} \\int_{B_1^{n}} \\prs{\\brs{ \\frac{\\partial\n\\nabla^i_t}{\\partial t}}^2 + \\sum_{j=1}^{n-4} \\brs{\\frac{\\partial}{\\partial y_j} \\hook\nF_{\\nabla^i_t}}^2} \\, dV \\, dt = 0.\n\\end{gather*}\n\\begin{proof} We first observe that by rescaling the result of Lemma\n\\ref{wktostronglemma5} we observe that\n\\begin{align} \\label{wktostronglemma1020}\n\\lim_{i \\to \\infty} \\int_{-t_2}^{-t_1} \\int_{B_1^{n}} \\brs{ \\frac{\\partial\n\\nabla^i_t}{\\partial t}}^2 \\, dV \\, dt = 0.\n\\end{align}\nNow let $\\xi_j = 2 \\frac{\\partial}{\\partial y_j}$. Since we know that the limiting\ndensity $\\Theta$ is a\nmultiple of the Hausdorff measure of the given $\\mathbb R^{n-4}$ on each time\nslice, applying the monotonicity formula (\\ref{Psimono}) with centers $(X_0,t_0)\n= (\\prs{0,\\xi_j},0)$ implies that for any $\\rho > 0$ we have\n\\begin{align*}\n0&= \\lim_{i \\to \\infty} \\int_{\\rho}^1 r \\int_M\\int_{-4r^2}^{-r^2} \n\\frac{\\brs{\\prs{X-\\xi_j} \\hook F_t^i + 2t \\partial_t {\\nabla}^i_t}^2}{\\brs{t}} \\phi^2\nG_{0,\\xi_j} \\, dV \\, dt \\, dr\\\\\n& \\geq \\lim_{i \\to \\infty} C_{\\rho} \\int_{\\rho}^1 r\n\\int_{B_1}\\int_{-4r^2}^{-r^2} \\frac{\\brs{\\prs{X-\\xi_j} \\hook F^i_t + 2t \\partial_t\n{\\nabla}^i_t}^2}{\\brs{t}} \\, dV \\, dt \\, dr.\n\\end{align*}\nNote that the second equality follows since, for a given $\\rho$, on $B_1$ we\nhave that there is a constant $C_{\\rho} \\in (0,\\infty)$ dependent solely on\n$\\rho$ such that $G_{0,\\xi_j} \\phi^2 \\geq C_{\\rho} \\in (0,\\infty)$.\n\nNext we apply Fubini's theorem to switch the integration bounds $dr \\, dt$ to\n$dt \\, dr$. In that case we have that if we set \n\\begin{equation*}\nI_i(r,t) := C_{\\rho} r \\int_{B_1} \\frac{\\brs{(X-\\xi_j) \\hook\nF_{{\\nabla}^i_t} + 2t \\partial_t {\\nabla}^i_t }^2}{\\brs{t}} \\, dV\n\\end{equation*}\nthen applying Fubini's theorem to the regions corresponding to the variables $r$\nand $t$ give that\n\\begin{align*}\n0= \\lim_{i \\to \\infty} \\left[ \\int_{- 4 \\rho^2}^{- \\rho^2}\n\\int_{\\rho}^{\\sqrt{-t}} I_i(r,t) \\, dr \\, dt + \\int_{-1}^{- 4 \\rho^2}\n\\int_{\\tfrac{1}{4} \\sqrt{-t}}^{\\sqrt{-t}} I_i(r,t) \\, dr \\, dt + \\int_{-4}^{-1}\n\\int_{\\tfrac{1}{4} \\sqrt{-t}}^1 I_i(r,t) \\, dr \\, dt. \\right]\n\\end{align*}\n\\color{black}\nThen for $0 < t_1 < t_2 < 1$, if we choose $\\rho \\leq t_1$, so that $[-t_2^2, -\nt_1^2] \\subset \\brk{ -1, - \\rho^2}$ (the unions of the temporal domains of the\nfirst two integrals) then we can conclude that since the sums are $0$ and the\narguments of each integral are positive,\n\\begin{align*}\n0 &=\\ \\lim_{i \\to \\infty} \\int_{-t_2^2}^{-t_1^2} \\int_{B_1}\\frac{\\brs{(X-\\xi_j)\n\\hook F_{{\\nabla}_t^i} + 2t \\partial_t {\\nabla}_t^i }^2}{\\brs{t}} \\, dV \\, dt \\\\\n&\\geq\\ \\lim_{i \\to \\infty} \\brs{t_1}^{-1} \\int_{-t_2^2}^{-t_1^2} \\int_{B_1}\n\\frac{1}{2} \\brs{(X-\\xi_j) \\hook F_{\\nabla_t^i}}^2 - C \\brs{t \\partial_t\n\\nabla_t^i }^2 \\, dV \\, dt \\\\\n&=\\ \\lim_{i \\to \\infty} \\brs{t_1}^{-1} \\int_{-t_2^2}^{-t_1^2} \\int_{B_1}\n\\frac{1}{2} \\brs{(X-\\xi_j) \\hook F_{{\\nabla}_t^i}}^2\\, dV \\, dt.\n\\end{align*}\nThe second inequality follows using the Cauchy-Schwarz inequality, and the final\nline follows from (\\ref{wktostronglemma1020}). Now observing that $\\xi_j = 2\n\\frac{\\partial}{\\partial y_j}$ we see that for all $X = (x,y) \\in B_1$ we have that\n$\\brs{\\ip{(X -\n\\xi_j),\\frac{\\partial}{\\partial y_j}}} \\geq 1$. The result follows.\n\\end{proof}\n\\end{lemma}\n\n\\begin{lemma} \\label{wktostronglemma20} Given the\nsetup above, there exists $(y,t) \\in B_{1\/2}^{n-4} \\times\n[-\\frac{1}{2},-\\frac{1}{4}]$ such that\n\\begin{align*}\n\\lim_{i \\to \\infty} & \\sup_{0 < r \\leq 1} r^{4-n} \\int_{B_r^{n-4}(y)}\n\\int_{B_1^{4} \\times \\{y\\} \\times [-1,0]} \\brs{\\frac{\\partial \\nabla^i_t}{\\partial t}}^2 \\,\ndx \\, dt \\, dy = 0,\\\\\n\\lim_{i \\to \\infty} & \\sup_{0 < r \\leq 1} r^{2-n} \\int_{B_r^{n-4}(y) \\times [t -\nr^2,t]} \\int_{B_1^{4} \\times \\{y\\}} \\sum_{j=1}^{n-4} \\brs{\\frac{\\partial}{\\partial\ny_j} \\hook F_{\\nabla^i_t}}^2 \\, dx \\, dt \\, dy = 0.\n\\end{align*}\n\\begin{proof} To begin we show a preliminary statement using maximal functions. \nIn particular, let\n\\begin{align*}\nf_i\n&: B_1^{n-4} \\subset \\mathbb{R}^{n-4} \\to [0,\\infty), \\qquad f_i({y}) =\n\\int_{\\prs{B_1^{4} \\times \\sqg{ {y}} } \\times \\brk{-1,0}} \\brs{\\partial_t\n{\\nabla}_i} ^2 \\, dx\\, dt \\\\\ng_i &: B_1^{n-4} \\times \\brk{-1, -\\tfrac{1}{8}} \\to [0,\\infty), \\qquad\ng_i({y},{t}) = \\int_{B_1^{4} \\times \\sqg{y}} \\sum_{j=1}^{n-4}\n\\brs{\\xi_{j} \\hook {F}_{{t}}^i }^2 \\, dx.\n\\end{align*}\nUsing these two quantities, we define two local Hardy-Littlewood maximal\nfunctions of $f_i$ on $B_1^{n-4}$ and $g_i$ on $B_1^{n-4} \\times\n\\brk{-1,-\\tfrac{1}{8}}$ by, for $y \\in B_1^{n-4}$,\n\\begin{align*}\nM(f_i)(y)\n&= \\sup_{0 \\leq r \\leq 1} r^{4-n} \\int_{B_r^{n-4}(y)} f_i \\, d{y}.\n\\end{align*}\nFurthermore for $\\prs{y,t} \\in B_1^{n-4} \\times \\brk{-1,-\\tfrac{1}{8}}$,\n\\begin{align*}\nM (g_i) (y, t)&= \\sup_{0 0$, there is a $\\delta_0 > 0$, depending on $\\delta_1$\nand $\\brs{\\brs{\\psi}}_{L^1(B_1)}$ such that, whenever $\\delta \\leq \\delta_0$,\n\\begin{equation*}\n\\brs{\\brs{\\psi - \\bar{\\psi}}}_{L^1(B_1)} \\leq \\delta_1.\n\\end{equation*}\nwhere $\\bar{\\psi}$ denotes the average value of $\\psi$ on $B_1$.\n\\end{lemma}\n\n\\begin{lemma} \\label{wktostronglemma40} Given a $(y,t) \\in B_{1\/2}^{n-4} \\times\n[-\\frac{1}{2},-\\frac{1}{4}]$ as in Lemma \\ref{wktostronglemma20}, there exists a\nuniversal constant $\\Lambda$\nand sequences $x_i \\to 0, \\delta_i \\to 0$ such that\n\\begin{gather*}\n\\begin{split}\n\\frac{\\ge_0}{\\Lambda} &=\\ \\delta_i^{-2} \\int_{B_{\\delta_i}^{4}(x_i) \\times [t -\n\\delta_i^2,t]} \\brs{F_{{\\nabla}^i}}^2(x,y,t) \\, dx \\, dt\\\\\n&=\\ \\max \\left\\{ \\delta_i^{-2} \\int_{B_{\\delta_i}^{4} (\\til{x}) \\times [t -\n\\delta_i^2,t]} \\brs{F_{{\\nabla}^i}}^2(\\til{x},y,t) \\, dx\\, dt \\mid \\til{x} \\in\nB_{\\frac{1}{2}}^{4} \\right\\}.\n\\end{split}\n\\end{gather*}\n\\begin{proof} Given $(y,t)$, we fix some $\\Lambda > 0$, then as each blowup\nconnection\n${\\nabla}^i_t$ is smooth we may first choose a constant $\\delta_i$ which is the\nsmallest positive number such that\n\\begin{align} \\label{wktostronglemma4011}\n\\max \\left\\{ \\delta_i^{-2} \\int_{B_{\\delta_i}^{4}(\\til{x}) \\times [t -\n\\delta_i^2,t]} \\brs{{{F}^i}}^2(\\til{x},y,t) \\ |\\ \\til{x} \\in\nB_{\\frac{1}{2}}^{4} \\right\\} = \\frac{\\ge_0}{\\Lambda}.\n\\end{align}\nChoosing $x_i$ as some point in realizing the maximum defined above, all that\nremains to check is that $\\delta_i \\to 0$, $x_i \\to 0$. First, suppose $\\delta_i \\geq\n\\delta_0 > 0$. Fix $\\phi \\in C_0^{\\infty}(B_{\\delta_0}^{4}, [0,1])$, and let\n\\begin{align*}\n\\psi(y) := \\delta_0^{-2} \\int_{t - \\delta_0^2}^{t} \\int_{B_{\\delta_0}^4} \\phi^2\n\\brs{{{F}^i}}^2 (x,y,s) \\, dx \\, ds.\n\\end{align*}\nNow observe that the result of Lemma \\ref{wktostronglemma30} can be interpreted\nas $\\nabla \\psi = f + \\divg Z$, with $f$ and $Z$ defined by the equality. It\nfollows from Lemma \\ref{wktostronglemma20} that\n\\begin{align*}\n\\lim_{i \\to \\infty} \\brs{\\brs{f}}_{L^1(B_{\\delta_0}^{n-4})} +\n\\brs{\\brs{Z}}_{L^1(B_{\\delta_0}^{n-4})} = 0.\n\\end{align*}\nThen we observe using Lemma \\ref{allardlemma} and (\\ref{wktostronglemma4011})\nthat\n\\begin{gather}\\label{wktostronglemma4015}\n\\begin{split}\n\\lim_{i \\to \\infty} \\delta_0^{2-n} \\int_{P_{\\delta_0}((0,y),t)} \\brs{{{F}^i_t}}^2 dV\ndt\n&=\\ \\lim_{i \\to \\infty} {\\bar{\\psi}}\\\\\n&=\\ \\lim_{i \\to \\infty} \\delta_0^{-4} \\int_{B_{\\delta_0}^{4}} {\\bar{\\psi}} \\, dx\\\\\n&=\\ \\lim_{i \\to \\infty} \\delta_0^{-4} \\int_{B_{\\delta_0}^{4}} \\prs{\\bar{\\psi} - \\psi +\n\\psi} \\, dx\\\\\n&\\leq\\ \\lim_{i \\to \\infty} \\left[ \\delta_0^{-4} \\brs{\\brs{\\psi -\n\\bar{\\psi}}}_{L^1(\\mathbb{R}^4)} + \\sup_{B_{\\delta_0}^{4}} \\psi \\right]\\\\\n&\\leq\\ \\frac{\\ge_0}{\\Lambda}.\n\\end{split}\n\\end{gather}\nThis contradicts that $((0,y),t) \\in \\Sigma^*$, hence $\\delta_i \\to 0$. Now we\nnote that the sequence $((x_i,y),t)$ develops concentration of $\\brs{F^i_t}$,\nand\nhence must limit to a singular point, which forces $x_i \\to 0$.\n\\end{proof}\n\\end{lemma}\n\nWith this sequence we can perform a further rescaling to finally obtain a\nYang-Mills connection as blowup limit. In particular, define the blowup\nsequence\n\\begin{align*}\n{\\til{\\Gamma}} ^i (x,y,t) &=\\ \\delta_i \\Gamma ((x_i,y_i) + (\\delta_i x, \\delta_i y), t_i +\n\\delta_i^2 t).\n\\end{align*}\nLet us observe some basic properties of this blowup sequence. In particular, by\nrescaling the estimates of Lemmas \\ref{wktostronglemma20} and\n\\ref{wktostronglemma40} we obtain\n\\begin{gather} \\label{wktostronglemma4020}\n\\begin{split}\n\\frac{\\ge_0}{\\Lambda} &=\\ \\int_{B_{1}^{4} \\times [-1,0]}\n\\brs{\\til{F}^i}^2(0,0,t) \\, dx \\, dt = \\max \\left\\{ \\int_{B_1^{4}\n(\\til{x}) \\times [-1,0]} \\brs{\\til{F}^i}^2(x,0,t)\\, dx \\, dt \\ |\\ x \\in\n\\delta_i^{-1}\nB_{\\frac{1}{2}}^{4} \\right\\},\\\\\n0 &=\\ \\lim_{i \\to \\infty} \\sup_{r \\in (0,\\frac{1}{4 \\delta_i})} r^{4-n}\n\\int_{B_r^{n-4}(0)} \\int_{B_{\\frac{1}{2 \\delta_i}}^{4} \\times \\{0\\} \\times\n[-\\delta_i^{-2},0]} \\brs{\\frac{\\partial \\til{\\nabla}^i}{\\partial t}}^2 \\, dx \\, dt \\, dy,\\\\\n0 &=\\ \\lim_{i \\to \\infty} \\sup_{r \\in (0,\\frac{1}{4 \\delta_i})} r^{2-n}\n\\int_{B_r^{n-4}(0) \\times\n[- r^2,0]} \\int_{B_{\\frac{1}{2 \\delta_i}}^{4} \\times \\{y\\}} \\sum_{j=1}^{n-4}\n\\brs{\\frac{\\partial}{\\partial y_j} \\hook \\til{F}^i_t}^2 \\, dx \\, dt \\, dy.\n\\end{split}\n\\end{gather}\n\n\\begin{lemma} \\label{wktostronglemma50} The sequence $\\{\\til{\\nabla}^i_t\\}$\nconverges\nstrongly to a nonflat Yang-Mills connection on $S^4$.\n\\begin{proof} We use the estimates of (\\ref{wktostronglemma4020}) and argue as\nin the estimate (\\ref{wktostronglemma4015}) to show an\nenergy estimate of the form\n\\begin{align} \\label{wktostronglemma5010}\n\\int_{P_{\\frac{3}{2}} ((\\til{x},0),0)} \\brs{{\\til{F}^i_t}}^2 \\, dV \\, dt \\leq\n\\frac{\\ge_0}{2} \\quad \\mbox{for all } \\til{x} \\in \\delta_i^{-1}\n\\prs{B_{\\frac{1}{2}}^{4}}.\n\\end{align}\nGiven this, we can complete the proof as follows. There is a local $H^{1,2}$\nestimate for $\\til{\\nabla}^i_t$ and hence we can choose a subsequence so that\n$\\til{\\nabla}^i_t \\to \\til{\\nabla}^{\\infty}_t$ weakly in $H_{\\mbox{loc}}^{1,2}(\\mathbb\nR^n\n\\times \\mathbb (- \\infty, 0])$. However, using (\\ref{wktostronglemma4020}) we\nhave that\n\\begin{align*}\n\\int_{\\mathbb R^n \\times (-\\infty,0]} \\prs{ \\brs{ \\frac{\\partial \n\\til{\\nabla}^{\\infty}_t}{\\partial t}}^2 +\n\\sum_{j=1}^{n-4} \\brs{ \\frac{\\partial}{\\partial y_j} \\hook \\til{F}^{\\infty}_t}^2\n} \\, dV \\, dt= 0.\n\\end{align*}\nUsing (\\ref{wktostronglemma5010}) and Theorem \\ref{thm:eregularity} we obtain\nconvergence of $\\til{\\nabla}^i_t$ to $\\til{\\nabla}^{\\infty}_t$ in $C^{k,\\alpha}(K)$ for any\ncompact set $K \\subset \\mathbb R^n \\times (-\\infty, 0]$. In particular, using\n\\eqref{wktostronglemma4020} we obtain\n\\begin{align*}\n\\frac{\\ge_0}{\\Lambda} \\leq \\int_{\\mathbb R^4} \\brs{\\til{F}^{\\infty}_t}^2 \\, dx <\n\\infty,\n\\end{align*}\nhence $\\til{\\nabla}^{\\infty}_t$ is not flat. The result follows.\n\\end{proof}\n\\end{lemma}\n\\noindent Lemma \\ref{wktostronglemma50} finishes the proof of the theorem.\n\\end{proof}\n\\end{thm}\n\n\\begin{proof} [Proof of Corollary \\ref{cor:weaktostrong}] Without loss of\ngenerality by an overall rescaling we assume $T \\geq 2$. Choose any sequence\n$\\{t_i\\} \\to T$, and observe that the sequence of solutions given by restricting\nthe given solution to $[t_i - 1, t_i]$ satisfies the hypotheses of Theorem\n\\ref{fullweakcompactthm}. By hypothesis that $T$ is maximal we know that\n$\\Sigma \\neq \\varnothing$. As shown in Theorem \\ref{fullweakcompactthm} the\npoint\n$z \\in \\Sigma$ is a point of entropy concentration. Thus we can choose a\nsequence of radii $r_i \\to 0$ and rescale the parabolic balls $P_{r_i}(z_0)$ to\nunit\nsize, to obtain a sequence of solutions with finite, nonzero entropy. It\nfollows easily that the hypotheses of Theorem \\ref{fullweakcompactthm} hold for\nthis sequence. If the sequence does not converge strongly in $H^{1,2}$, Theorem\n\\ref{thm:weaktostrong} yields the further blowup sequence which converges to a\nYang-Mills connection on $S^4$. If this sequence does converge strongly in\n$H^{1,2}$, as the $\\Psi$ functional is becoming constant along the blowup\nsequence, the \nsecond term of the entropy monotonicity formula of (\\ref{Psimono}) converges to\nzero, which implies that the\nblowup limit is a soliton.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFirst discovered over thirty years ago~\\citep[e.g.,][]{Gregory1978}, \ncosmic voids are emerging as a novel and innovative probe of both \ncosmology and astrophysics through their\nintrinsic properties~\\citep[e.g.,][]{Bos2012}, \nexploitation of their statistical isotropy via the \nAlcock-Paczy\\'{n}ski~test~\\citep{Sutter2014b}, \nor cross-correlation with the cosmic microwave \nbackground \\citep{Ilic2013, Planck2013b, Cai2014}.\nAdditionally, fifth forces and modified gravity are unscreened \nin void environments, making them singular probes \nof exotic \nphysics~\\citep[e.g.,][]{Li2012,Clampitt2013,Carlesi2013a}.\n\nVoids are also fascinating objects to study by themselves. \nFor example, there appears to be \na self-similar relationship between voids in dark \nmatter distributions and voids in galaxies~\\citep{Sutter2013a} \nvia a universal density \nprofile~\\citep[]{Hamaus2014} (hereafter HSW).\nObservationally, the anti-lensing shear \nsignal~\\citep{Melchior2014,Clampitt2014} and\nLy-alpha absorption measurements~\\citep{Tejos2012} have illuminated the \ndark matter properties in voids.\n\nHowever, there are only a few public catalogs of voids identified \nin galaxy redshift surveys, primarily the \nSDSS~\\citep[e.g.,][]{Pan2011, Sutter2012a}.\nAnd while there are many published methods for finding voids based \non a variety of techniques, most codes remain private. \nIn order to accelerate the application of voids, it is essential \nto provide easy-to-use, flexible, and strongly supported \nvoid-finding codes.\n\nIn this work we discuss \\textsc{vide},\nfor Void IDentification and Examination, \na toolkit based on the publicly-available \nwatershed code \\textsc{zobov}~\\citep{Neyrinck2008} for finding voids \nbut considerably enhanced and extended to handle a variety of simulations \nand observations~\\citep{Sutter2014d}. \n\\textsc{vide}~also provides an extensive interface for \nanalyzing void catalogs. \n\n\\section{Void Finding}\n\\label{sec:finding}\n\n\\textsc{vide}~is able to identify voids in $N$-body simulation snapshots produced by\n\\textsc{Gadget}~\\citep{Gadget},\n\\textsc{FLASH}~\\citep{Dubey2008}, \\textsc{RAMSES}~\\citep{ramses} \n\\textsc{2HOT}~\\citep{warren2013}, and generic ASCII \nfiles.\n\\textsc{vide}~can either find voids directly in the full simulation, \nor randomly subsampled populations, as well as halo catalogs and \nmock galaxy samples using the Halo Occupation Distribution\nformalism~\\citep{Berlind2002}.\nFor observational datasets, to find voids in galaxy surveys\nthe user must provide a list of galaxy positions and a\npixelization of the survey mask\nusing \\textsc{healpix} \\citep{Gorski2005}\\footnote{http:\/\/healpix.jpl.nasa.gov}.\n\nThe core of our void finding algorithm is\n\\textsc{zobov}~\\citep{Neyrinck2008}, \nwhich creates a Voronoi tessellation of the tracer particle\npopulation and uses the watershed transform to group Voronoi\ncells into zones and subsequently voids~\\citep{Platen2007}.\nThe Voronoi tessellation provides a density field estimator \nbased on the underlying particle positions.\n\n\\textsc{zobov} first groups nearby Voronoi\ncells into {\\emph zones}, then merges adjacent zones into voids \nby finding minimum-density barriers between them.\nWe impose a density-based threshold within \\textsc{zobov} where\nadjacent zones are only added to a void\nif the density of the wall between them is\nless than $0.2$ times the mean particle density $\\bar{n}$.\nThis density criterion \nprevents voids from expanding deeply into overdense structures and\nlimits the depth of the void hierarchy~\\citep{Neyrinck2008}.\n\nIn this picture, a void is simply a basin in the Voronoi density field \nbounded by a common set of higher-density ridgelines,\nas demonstrated by Figure~\\ref{fig:void}, which shows a typical \n20 $h^{-1}$Mpc~void identified in the SDSS DR7 galaxy survey~\\citep{Sutter2012a}.\nThis also means that voids may have any \\emph{mean} density, since \nthe watershed includes in the void definition all wall particles \nall the way up to the very highest-density separating ridgeline.\n\n\\begin{figure}\n \\centering\n {\\includegraphics[type=png,ext=.png,read=.png,width=0.49\\columnwidth]{f2a}}\n {\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=0.49\\columnwidth]{dmgv}}\n \\caption{Watershed void finding in \\textsc{vide}. \n The left panel shows the Voronoi cells belonging to a \n void embedded in an observed galaxy population.\n \\emph{Reproduced from~\\citet{Sutter2012a}}.\n The right panel shows projections of dark matter, \n mock galaxies, and void positions in a simulation.\n \\emph{Reproduced from~\\citet{Hamaus2013}}.\n }\n \\label{fig:void}\n\\end{figure}\n\nWe have made several modifications and improvements to the original\n\\textsc{zobov} algorithm. First, we have strengthened the algorithm \nwith respect to numerical precision by\nrewriting large portions in a \ntemplated C++ framework.\nWe enforce bijectivity in the Voronoi graph, so that the tessellation \nis self-consistent, and\nuse an improved volume-splitting technique to minimize the number of \ndifference operators at the edge of the box when\njoining subregions.\nFinally, we have optimized several portions of the central \nwatershed algorithm, enabling the identification of voids in simulations \nwith up to $1024^3$ particles in $\\sim10$ hours using \n16 cores.\n\nTo prevent the growth of voids outside survey boundaries in observational \ndatasets,\nwe place a large number of mock particles\nalong any identified edge and along \nthe redshift caps. \nWe assign essentially infinite density to these mock particles, \npreventing the watershed from including zones external \nto the survey.\nSince the local volumes of the edge galaxies are arbitrary, \nwe prevent these mock particles from participating in any \nvoid by disconnecting their adjacency links in the Voronoi graph.\n\nWe use the mean particle spacing to set a lower size limit for voids\nbecause of Poisson shot noise. \\textsc{vide}~does not include any void with \neffective radius smaller than $\\bar{n}^{-1\/3}$, where $\\bar{n}$ is the \nmean number density of the sample.\nWe define the effective radius as the radius of a sphere with \nthe same total volume of all the Voronoi cells that make up the void.\n\nFigure~\\ref{fig:void} shows an example void population with\n\\textsc{vide}, taken from \nthe analysis of~\\citet{Hamaus2013}. In this figure we show a slice \nfrom an $N$-body simulation, a set of mock galaxies painted onto the \nsimulation using the HOD formalism discussed above, and the distribution \nof voids identified in the mock galaxies.\n\n\\textsc{vide}~automatically provides some basic derived void information, such as \nthe \\emph{macrocenter}, or volume-weighted\ncenter of all the Voronoi cells in the void, and the ellipticity.\n\n\\section{Post-Processing \\& Analysis}\n\\label{sec:analysis}\n\n\\textsc{vide}~provides a Python-based application programming interface (API) \nfor loading and manipulating the void catalog and performing \nanalysis and plotting.\n\nVia the API the user has immediate access to all basic and derived void properties\n(ID, macrocenter, radius, density contrast, RA, Dec, hierarchy \nlevel, ellipticity, etc.)\nas well as the positions ($x$, $y$, $z$, RA, Dec, redshift),\nvelocities (if known), and local Voronoi volumes of all void member particles. \nThe user can also access all particle and galaxy sample \ninformation, such as redshift extents, the mask, simulation extents \nand cosmological parameters.\nUpon request the user can also load all particles in the simulation \nor observation. \n\n\\textsc{vide}~includes several built-in plotting routines.\nFirst, users may plot cumulative number functions of multiple \ncatalogs on a logarithmic scale. \nVolume normalizations are handled automatically, \nand 1$\\sigma$ Poisson uncertainties are shown as shaded regions, \nas demonstrated in Figure~\\ref{fig:numberfunc}. \nSecondly, users may plot a slice from a single void and its \nsurrounding environment.\nIn these plots we bin the background particles onto a \ntwo-dimensional grid and plot the density on a logarithmic scale. \nWe draw the void member galaxies as small semi-transparent disks with\nradii equal to the effective radii of their corresponding Voronoi\ncells. \n\nThe user can directly compare two void catalogs by using a built-in \nfunction for computing the overlap. This function attempts to \nfind a one-to-one match between voids in one catalog and another \nand reports the relative properties (size, ellipticity, etc.) \nbetween them. It also records which voids cannot find a reliable match \nand how many potential matches a void may have. \nA more detailed discussion of the matching process can be \nfound in~\\citet{Sutter2013b}.\n\n\\textsc{vide}~allows the user to compute simple two-point clustering statistics, i.e. power spectra and correlation functions. \n\nThree-dimensional stacks of voids are useful in a variety of \nscenarios, and the user may construct these with built-in functions.\nThe stacked void \nmay optionally contain only void member particles or all particles \nwithin a certain radius. \nWith this stacked profile \\textsc{vide}~can build a \nspherically averaged radial density profile and fit\nthe universal HSW void profile to it. All proper \nnormalizations are handled internally.\nFigure~\\ref{fig:numberfunc} shows an example of \\textsc{vide}-produced \ndensity profiles and best-fit HSW profile curves. \n\n\\section{Applications}\n\\label{sec:applications}\n\nThroughout its development \\textsc{vide}~has been used in a variety of applications.\nInitially, the enhancements to \\textsc{zobov} were used to create a set \nof public catalogs from both observations~\\citep{Sutter2012a, Sutter2013c}\nand simulations~\\citep{Sutter2013a}. The toolkit presented in this work \nis fully compatible with those public releases.\n\nThe publication of simple void properties, such as sky position and \neffective radius, enabled direct cross correlations with other datasets\nleading to several interesting results. Examples of such \ncorrelations include: \ngalaxy shear maps leading to a detection of a weak lensing effect \ninside voids~\\citep{Melchior2014}, \nthe cosmic microwave background giving rise to a detection of the \nintegrated Sachs-Wolfe effect~\\citep{Planck2013b, Ilic2013}, and \npositions of high-energy blazars putting limits on the intergalactic\nmagnetic field~\\citep{Furniss2014}. The latter correlation is shown \nin the right panel of Figure~\\ref{fig:apps}.\nAdditionally, cross-correlations of void centers with galaxy positions\n have recently been demonstrated\n to allow the construction of \na static ruler~\\citep{Hamaus2013}, and the three-dimensional \narrangement of galaxies around\nstacked voids has been exploited for an Alcock-Paczy\\'{n}ski~\ntest~\\citep{Sutter2014b}, as shown in the left-hand panel of \nFigure~\\ref{fig:apps}.\n\n\\begin{figure} \n \\centering \n {\\includegraphics[type=png,ext=.png,read=.png,width=0.49\\columnwidth]{hubble_combined_sdss_drAll_central_debiased}}\n {\\includegraphics[type=png,ext=.png,read=.png,width=0.49\\columnwidth]{Relations-png}}\n \\caption{ \n Left panel: a detection of the Alcock-Paczy\\'{n}ski~effect using stacked cosmic voids \n in the SDSS DR7 and DR10. The slight deviation from unity in the \n expected void stretch can be used to extract cosmological \n parameters. \n \\emph{Reproduced from~\\citet{Sutter2014b}}.\n Right panel: GeV luminosity versus void fraction along the line of sight for Fermi-detected blazars.\n \\emph{Reproduced from~\\citet{Furniss2014}}.\n }\n\\label{fig:apps}\n\\end{figure}\n\nDerived void statistical information,\nsuch as radial density profiles (as shown in Figure~\\ref{fig:numberfunc}), have \nled to the discovery of a universal density \nprofile~\\citep{Hamaus2014,Sutter2013a} and a method for constructing \nreal-space profiles in a parameter-independent manner~\\citep{Pisani2013}.\nThe abundance of voids is potentially a very powerful probe of \ncosmology, as shown in~\\citet{Sutter2014c} and highlighted\nin Figure~\\ref{fig:numberfunc}.\n\n\\begin{figure} \n \\centering \n {\\includegraphics[type=png,ext=.png,read=.png,width=0.49\\columnwidth]{fig_1dprofile_20-25_1024}}\n {\\includegraphics[type=png,ext=.png,read=.png,width=0.49\\columnwidth]{fig_numberfunc_compare_ss}}\n \\caption{ \n Left panel: example one-dimensional radial density profiles of stacked voids\n (points with error bars) and best-fit curves (thin lines)\n using the HSW profile.\n \\emph{Reproduced from~\\citet{Sutter2013a}}.\n Right panel: example cumulative void number functions from simulations.\n \\emph{Reproduced from~\\citet{Sutter2014c}}.\n }\n\\label{fig:numberfunc}\n\\end{figure}\n\nLastly, the ability to match and compare void properties from one catalog \nto another (a built-in feature of \\textsc{vide}) has been used to examine the \nrelationship of voids identified in sparse, biased galaxy samples to \ndark matter underdensities~\\citep{Sutter2013a} and to reconstruct \nthe life history of voids using merger trees~\\citep{Sutter2014a}.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have presented and discussed the capabilities of \\textsc{vide}, a new \nPython\/C++ toolkit for identifying cosmic voids in cosmological \nsimulations and galaxy redshift surveys, as well as a brief overview \nof its current applications. \\textsc{vide}~performs void \nidentification using a substantially modified and enhanced version \nof the watershed code \\textsc{zobov}. \nFurthermore, \\textsc{vide}~is able to support a variety of mock and real datasets,\nand provides extensive and flexible tools for \nloading and analyzing void catalogs. We have highlighted these \nanalysis tools (e.g., filtering, plotting, clustering statistics, stacking, \nprofile fitting) using examples from previous and current \nresearch using \\textsc{vide}.\n\nThe analysis toolkit bundled in \\textsc{vide}~enables a wide variety of \nboth theoretical \nand observational void research. \nAs its most basic, the void properties made available to the \nuser, such as sky positions, shapes, and sizes, permit simple explorations \nof void properties and cross-correlation with other datasets.\nThe user may also use void member particles and their associated \nvolumes for examining galaxy properties in low-density \nenvironments. Cross-matching by overlapping catalogs can be\n useful for understanding the \nimpacts of peculiar velocities or galaxy bias, as well as providing \na platform for studying the effects of modified gravity or \nfifth forces on a void-by-void basis. \nVoid power spectra, \nshape distributions, number functions, and density profiles, \neasily accessible via \\textsc{vide}, \nare sensitive probes of cosmology. Users may also access \nHSW density profiles, enabling theoretical predictions\nof the ISW or gravitational lensing signals. \n\nThe past few years have seen an enormous growth in void \ninterest and research. This research includes new void-finding \nalgorithms, studies of void properties, investigations and forecasts \nof cosmological \nprobes, and explorations into the nature of void themselves from theoretical \nand numerical viewpoints. \nPut simply, \\textsc{vide}~is designed to meet the growing demand of next-generation \nvoid science.\n\nThe \\textsc{vide}~code and documentation is currently hosted at\\linebreak {\\tt http:\/\/bitbucket.org\/cosmicvoids\/vide\\_public}, with links to numbered versions at\\linebreak {\\tt http:\/\/www.cosmicvoids.net}.\n\n\\section*{Acknowledgments}\nThe authors acknowledge\nsupport from NSF Grant NSF AST 09-08693 ARRA. BDW\nacknowledges funding from an ANR Chaire d'Excellence (ANR-10-CEXC-004-01),\nthe UPMC Chaire Internationale in Theoretical Cosmology, and NSF grants AST-0908\n902 and AST-0708849.\nThis work made in the ILP LABEX (under reference ANR-10-LABX-63) was supported by French state funds managed by the ANR within the Investissements d'Avenir programme under reference ANR-11-IDEX-0004-02.\n\n\\bibliographystyle{mn2e\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nA classical numerical invariant associated to a point of an algebraic variety is the (Hilbert-Samuel) multiplicity. This invariant has \nnumerous applications on algebraic geometry and commutative algebra (for instance, it plays a fundamental role in the problem of \nresolution of singularities). On the other hand, it is known that this invariant can be computed using the theory of Gr\\\"obner bases \n(see \\cite[Chapter 5]{GP}).\n\nIn the context of toric varieties the computation of this invariant is simpler. Assuming that an affine toric variety contains the origin, \nits multiplicity can be computed in a combinatorial way by looking at the semigroup defining the toric variety (see \n\\cite[Chapter 5, Theorem 3.14]{GKZ}). The proof of this fact, although not too difficult, requires some tools from topology, \ncommutative algebra, and combinatorics. The first goal of this note is to give a completely elementary proof of that fact in the case\nof toric curves.\n\nLet $S\\subset\\N$ be a numerical semigroup generated by ${a_1,\\ldots,a_r}$, where $a_1<\\cdots0$. Then $j-\\beta_2 >0.$ \nSince $a_1 \\mid j-\\beta_2,$ we have $j-\\beta_2 \\geq a_1$. Thus $\\beta_2\\leq j-a_1\\leq -1$, a contradiction. Finally, suppose \n$\\beta_1-n+j <0$. Then $0 \\leq \\beta_1 F(\\Snu)$.\nBy lemma \\ref{division}, $F(\\Snu)=F(\\Sn)+a_1$ and so, for some $l\\geq1$, $s=F(\\Sn)+a_1+l=(F(\\Sn)+l)+a_1\\in\\Sn+a_1$.\n\nNow suppose that $s(n+1)a_1$, let $s=pa_1+t$, where \n$0\\leq t < a_1$ and $n+1\\leq p$. Notice that $t=0$ implies $s=pa_1\\in S^{(p)} \\subset S^{(n+1)}$, a contradiction. Thus, \n$0[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$ form an adjoint pair (notice how the counit is obvious: given an $\\Oh$-algebra structure on $V$, there is a local dg $\\op{Ass}_\\circ(\\Oh)$-algebra structure on $C^*(\\Delta_1;V)$ via the tensor product operations, and by defninition this is the same as an $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$-algebra structure on $V$).\n\nThe trick to compute $\\sC om\\bullet-$ and $pre\\sL ie\\bullet-$ (and dually $\\sL ie\\circ-$ and $\\sP erm\\circ-$) is similar: we replace the dg associative algebra $(C^*(\\Delta_1;\\mathbb{K}),\\cup)$ in the above discussion with the dg Lie algebra $(C^*(\\Delta_1;\\mathbb{K}),[-,-]=\\cup-\\cup^{op})$ in the first case, and with the dg (right) permutative algebra $(C^*(\\Delta_1,v_l;\\mathbb{K}),\\cup)$ (where $v_l\\subset\\Delta_1$ is the left vertex) in the second case.\n\n\\begin{ack} The author is grateful to Domenico Fiorenza for some useful discussions.\n\\end{ack}\n\n\\bigskip\n\n\\subsection*{Preliminary remarks.} The author is not an actual expert on operads (and is afraid this might be painfully evident throughout the reading to anyone who is), accordingly, we will think of operads n\\\"aively, as defined by the corresponding type of algebras. Moreover, although the constructions seem to make sense in more general settings (for instance, dg operads), we limit ourselves to work in the category $\\mathbf{Op}$ of finitely generated binary quadratic symmetric operads on vector spaces over a field $\\mathbb{K}$.\n\nGiven a vector space $V$, we denote by $V^{\\ten n}$, $V^{\\odot n}$, $V^{\\wedge n}$, $n\\geq0$, the tensor powers, symmetric powers and exterior powers of $V$ respectively. We shall always denote by $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ an operad in $\\mathbf{Op}$ generated by non-symmetric products $\\cdot_i$, $i=1,\\ldots,p$, commutative products~$\\bullet_j$, $j=1,\\ldots,q$, and anti-commutative brackets $[-,-]_k$, $k=1,\\ldots,r$: the relations shall be omitted from the notation. For the definitions of the Koszul duality functor $-^!:\\mathbf{Op}\\to\\mathbf{Op}$ and the Manin white and black products $-\\circ-,-\\bullet-:\\mathbf{Op}\\times\\mathbf{Op}\\to\\mathbf{Op}$ we refer to \\cite{V,LV}. For the definitions of the various operads we will consider, where not already specified, we refer to \\cite{LV,zinb}.\n\nWe denote by $\\Delta_1$ the standard $1$-simplex, which shall be represented as an arrow $\\to$, by $v_i\\subset\\Delta_1$, $i=l,r$, the left and right vertex respectively, and by $\\de\\Delta_1=v_l\\sqcup v_r\\subset\\Delta_1$ the boundary. Given a vector space $V$, we shall denote by $C^\\ast(\\Delta_1;V)$ (resp.: $C^*(\\Delta_1\\de\\Delta_1;V)$, $C^*(\\Delta_1,v_i;V)$, $i=l,r$) the usual complex of non-degenerate (resp.: relative) cochains with coefficients in $V$. We shall depict a $0$-cochain in $C^*(\\Delta_1;V)$ as $_x\\to_y$, $x,y\\in V$ (moreover, we write $_x\\to$ for $_x\\to_0$ and $\\to_y$ for $_0\\to_y$), and a $1$-cochain as $\\xrightarrow{x}$, similarly in the relative cases. The differential $d:C^0(\\Delta_1;V)\\to C^1(\\Delta_1;V)$ is the usual one $d(_x\\to_y)=\\xrightarrow{y-x}$.\n\\section{A trick to compute Manin black products}\n\nIn this section we shall introduce, and compute in several cases, endofunctors $\\mathbf{Op}\\to\\mathbf{Op}$ which we denote by $\\op{Ass}_\\bullet(-)$, $\\op{Com}_\\bullet(-)$, $\\op{preLie}_\\bullet(-)$, later we shall prove that these coincide with the functors $\\sA ss\\bullet-$, $\\sC om\\bullet-$, $pre\\sL ie\\bullet-$ respectively.\n\n\n\n\\begin{definition}\\label{def:ManinBlack} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ in $\\mathbf{Op}$ and a vector space $V$, a structure of $\\op{Ass}_\\bullet(\\Oh)$-algebra on $V$ is the datum of operations \n\\begin{multline*} \\cdot'_i:C^\\ast(\\Delta_1;V)^{\\ten 2}\\to C^*(\\Delta_1;V),\\quad\\bullet'_j:C^\\ast(\\Delta_1;V)^{\\odot2}\\to C^*(\\Delta_1;V)\\\\\\mbox{and}\\quad[-,-]'_k:C^\\ast(\\Delta_1;V)^{\\wedge2}\\to C^*(\\Delta_1;V)\\quad\\mbox{such that}\\end{multline*}\n\\begin{enumerate}\n\\item\\label{item:dg} $(C^*(\\Delta_1;V),d,\\cdot'_i,\\bullet'_j,[-,-]'_k)$ is a dg $\\Oh$-algebra structure on $C^*(\\Delta_1;V)$, and moreover the following locality assumption holds:\\\\\n\n\\item\\label{item:locality} for all closed subsets $X\\subset\\Delta_1$ (i.e., $X=v_l,v_r,\\de\\Delta_1$, cf. the preliminary remarks) the complex of relative cochains $C^*(\\Delta_1,X;V)\\subset C^*(\\Delta_1;V)$ is a dg $\\Oh$-ideal.\n\\end{enumerate}\n\\end{definition}\nThe functor $\\op{preLie}_\\bullet(-)$ is defined similarly. \n\\begin{definition} A $\\op{preLie}_\\bullet(\\Oh)$-algebra structure on $V$ is the datum of a dg $\\Oh$-algebra structure $(C^*(\\Delta_1,v_l;V),d,\\cdot'_i,\\bullet'_j,[-,-]'_k)$ on $C^*(\\Delta_1,v_l;V)$ (we notice that in this case the locality assumption, that is, the fact that $C^*(\\Delta_1,\\de\\Delta_1;V)\\subset C^*(\\Delta_1,v_l;V)$ is a dg $\\Oh$-ideal, is satisfied for trivial degree reasons).\n\\end{definition} \n\\begin{remark}\\label{rem:prelie}\n\tThis actually defines $\\op{preLie}_\\bullet(-)=\\op{preLie}_{r,\\bullet}(-)$, by replacing the vertex $v_l$ with the one $v_r$ in the previous definition we get a second endofunctor $\\op{preLie}_{l,\\bullet}(-)$ : more on this in Remark~\\ref{rem:leftcase}.\n\\end{remark} \n\\newcommand{\\w}{\\widetilde}\n\nIt is not immediately obvious that this defines endofunctors $\\mathbf{Op}\\to\\mathbf{Op}$: we consider first the case of $\\op{Ass}_\\bullet(-)$. Given the datum $(C^\\ast(\\Delta_1;V),d,\\cdot_i',\\bullet_j',[-,-]_k')$ of a local dg $\\Oh$-algebra structure on $(C^\\ast(\\Delta_1;V),d)$, we notice that the locality assumption implies $_x\\to\\cdot_i'\\to_y\\:=\\:\\to_x\\cdot_i'\\,\\,_y\\!\\to=0$, thus, applying the differential $d$ and Leibniz rule we see that $_x\\to\\cdot_i'\\xrightarrow{y}\\:=\\:\\xrightarrow{x}\\cdot_i'\\to_y$, and similarly $\\to_x\\cdot_i'\\xrightarrow{y}\\:=\\:\\xrightarrow{x}\\cdot_i'\\,\\,_y\\to$. We define non-symmetric products $\\prec_i,\\succ_i:V^{\\ten 2}\\to V$ on $V$ by \n\\[ _x\\to\\cdot_i'\\xrightarrow{y}\\,\\,=:\\,\\,\\xrightarrow{x\\prec_iy}\\,\\,:=\\,\\,\\xrightarrow{x}\\cdot_i'\\to_y,\\qquad\\to_x\\cdot_i'\\xrightarrow{y}\\,\\,=:\\,\\,-\\xrightarrow{y\\succ_i x}\\,\\,:=\\,\\,\\xrightarrow{x}\\cdot_i'\\,\\,_y\\to.\\]\nAlways by locality and Leibniz rule we have a product $\\cdot_i:V^{\\ten2}\\to V$ defined equivalently by $\\to_x\\cdot_i'\\to_y=\\to_{x\\cdot_iy}$ or $_x\\to\\cdot_i'\\,_y\\to=_{x\\cdot_i y}\\to$, and moreover $x\\cdot_i y= x\\prec_i y - y\\succ_i x$. In fact, \n\\[\\xrightarrow{x\\cdot_iy}=d\\left(\\to_x\\cdot_i'\\to_y\\right)=\\xrightarrow{x}\\cdot_i'\\to_y+\\to_x\\cdot_i'\\xrightarrow{y}=\\xrightarrow{x\\prec_iy-y\\succ_ix}=-d\\left(_x\\to\\cdot_i'\\,_y\\to\\right).\\]\nIn the same way, there are non-symmetric products $\\circ_j,\\ast_k:V^{\\ten2}\\to V$ on $V$, defined by the formulas \n\\begin{multline*} _x\\to\\bullet_j'\\xrightarrow{y}\\,\\,=\\,\\,\\xrightarrow{x}\\bullet_j'\\to_y\\,\\,=\\,\\,\\xrightarrow{y}\\bullet_j'\\,\\,_x\\to\\,\\,=\\,\\,\\to_y\\bullet_j'\\xrightarrow{x}\\,\\,=:\\,\\,\\xrightarrow{x\\circ_j y},\\\\ \\left[ _x\\to,\\xrightarrow{y}\\right]_k'\\,\\,=\\,\\, \\left[ \\xrightarrow{x},\\to_y\\right]_k' \\,\\,=\\,\\, -\\left[\\xrightarrow{y},\\,_x\\to\\right]_k' \\,\\,=\\,\\, -\\left[ \\to_y,\\xrightarrow{x}\\right]_k'\\,\\,=:\\,\\, \\xrightarrow{x\\ast_k y},\\end{multline*}\nproducts $\\bullet_j:V^{\\odot 2}\\to V$, $[-,-]_k:V^{\\wedge2}\\to V$ defined by \n\\[\\to_x\\bullet_j'\\to_y=\\to_{x\\bullet_j y},\\qquad_x\\to\\bullet_j'\\,_y\\to=_{x\\bullet_jy}\\to,\\qquad \\left[ \\to_x,\\to_y\\right]_k'=\\to_{[x,y]_k},\\qquad\\left[ _x\\to,\\,_y\\to \\right]_k'=_{[x,y]_k}\\to, \\]\nand moreover by Leibniz rule\n\\[ x\\bullet_j y = x\\circ_j y + y \\circ_j x, \\qquad [x,y]_k=x\\ast_k y - y\\ast_k x.\\]\n\n\\emph{Warning:} the previous formulas will be used extensively troughout the computations of this section without further mention.\n \nIn other words, the datum of an $\\op{Ass}_\\bullet(\\Oh)$-algebra structure on $V$ is the same as the one of operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k:V^{\\ten 2}\\to V$ on $V$, inducing operations $\\cdot_i',\\bullet_j',[-,-]_k'$ on $C^\\ast(\\Delta_1;V)$ via the previous formulas: the requirement that these make the latter into a (by construction, local dg) $\\Oh$-algebra translates into a finite set of terniary relations on the operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$. More precisely, for every relation $R(x,y,z)=0$ satisfied in an $\\Oh$-algebra, we get six (in general not independent) relations in the operad $\\op{Ass}_\\bullet(\\Oh)$ as in the next lemma.\n\n\\begin{lemma}\\label{lem:rel} Given operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $V$ inducing operations $\\cdot_i',\\bullet_j',[-,-]_k'$ on $C^\\ast(\\Delta_1;V)$ as above, then the latter is a local dg $\\Oh$-algebra if and only if for every relation $R(x,y,z)=0$ in the operad $\\Oh$ the six relations \n\\begin{multline*}0=R(\\xrightarrow{x},\\to_y,\\,_z\\to)=R(\\xrightarrow{x},\\,_y\\to,\\to_z)=R(\\to_x,\\xrightarrow{y},\\,_z\\to)=\\\\=R(_x\\to,\\xrightarrow{y},\\to_z)=R(\\to_x,\\,_y\\to,\\xrightarrow{z})=R(_x\\to,\\to_y,\\xrightarrow{z})=0,\\end{multline*}\nare satisfied.\n\\end{lemma}\n\\begin{proof} The only if part is clear. For the if part we have to show that the given relations imply all the others. First of all, we notice that the only relations not trivially satisfied are those in total degree 0 or 1. Moreover, in total degree 0 all necessary relations follow by the locality assumption but the ones $R(_x\\to,\\,_y\\to,\\,_z\\to)=0=R(\\to_x,\\to_y,\\to_z)$. In total degree 1, all the necessary relations are satisfied by hypothesis but the ones\n\\begin{multline*}0=R(_x\\to,\\,_y\\to,\\xrightarrow{z})=R(\\to_x,\\to_y,\\xrightarrow{z})=R(_x\\to,\\xrightarrow{y},\\,_z\\to)=\\\\=R(\\to_x,\\xrightarrow{y},\\to_z)=R(\\xrightarrow{x},\\,_y\\to,\\,_z\\to)=R(\\xrightarrow{x},\\to_y,\\to_z)=0.\\end{multline*}\nFor instance, to prove the first one we apply $d$ to the relation $0=R(_x\\to,\\,_y\\to,\\to_z)$ (which follows by locality), and by Leibniz rule and the hypothesis of the lemma\n\\[0 = -R(\\xrightarrow{x},\\,_y\\to,\\to_z)-R(_x\\to,\\xrightarrow{y},\\to_z)+R(_x\\to,\\,_y\\to,\\xrightarrow{z})=R(_x\\to,\\,_y\\to,\\xrightarrow{z}). \\]\nThe others are proved similarly. Finally, if we denote by $R(x,y,z)$ the relation $R$ computed in the operations $\\cdot_i$, $\\bullet_j$, $[-,-]_k$ on $V$, we have $R(\\to_x,\\to_y,\\to_z)=\\to_{R(x,y,z)}$, and by Leibniz rule\n\\[\\xrightarrow{R(x,y,z)}=d\\left(\\to_{R(x,y,z)}\\right)=R(\\xrightarrow{x},\\to_y,\\to_z)+R(\\to_x,\\xrightarrow{y},\\to_z)+R(\\to_x,\\to_y,\\xrightarrow{z})=0,\\]\nhence $R(\\to_x,\\to_y,\\to_z)=0$, and $R(_x\\to,\\,_y\\to,\\,_z\\to)=0$ is proved similarly.\n\\end{proof}\n\\begin{example}\\label{ex:ass} We consider the operad $\\sL ie$ of Lie algebras. By the previous argument, an $\\op{Ass}_\\bullet(\\sL ie)$-algebra structure on $V$ is the datum of $\\ast:V^{\\ten 2}\\to V$ such that\n\\begin{multline*} \\left[\\left[_x\\to,\\to_y\\right],\\xrightarrow{z}\\right]=0=\\left[ _x\\to,\\left[\\to_y,\\xrightarrow{z}\\right]\\right]+\\left[\\left[_x\\to,\\xrightarrow{z}\\right],\\to_y\\right]=\\\\=\\left[_x\\to,\\xrightarrow{-z\\ast y}\\right]+\\left[\\xrightarrow{x\\ast z},\\to_y\\right]=\\xrightarrow{-x\\ast(z\\ast y)+(x\\ast z)\\ast y},\\qquad\\forall x,y,z\\in V.\n\\end{multline*}\nIn other words, $\\ast$ is an associative product on $V$. The other five relations to be checked in the previous lemma follow from this one by symmetry of the Jacobi identity, thus, we see that $\\op{Ass}_\\bullet(\\sL ie)=\\sA ss$, the operad of associative algebras. \\end{example}\n\nWe may repeat the previous considerations in the case of $\\op{preLie}_\\bullet(-)$, showing that a dg $\\Oh$-algebra structure $(C^\\ast(\\Delta_1,v_l;V),d,\\cdot_i'.\\bullet_j',[-,-]_k')$ on the complex $C^\\ast(\\Delta_1,v_l;V)$ is determined, via the previous formulas, by operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k:V^{\\ten 2}\\to V$ on $V$. The analog of Lemma \\ref{lem:rel} is the following\n\\begin{lemma}\\label{lem:relpl}\u25cb Given operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $V$ inducing operations $\\cdot_i',\\bullet_j',[-,-]_k'$ on $C^\\ast(\\Delta_1,v_l;V)$ as above, then the latter is a local dg $\\Oh$-algebra if and only if for every relation $R(x,y,z)=0$ in the operad $\\Oh$ the three relations \n\\begin{equation*}R(\\xrightarrow{x},\\to_y,\\to_z)=R(\\to_x,\\xrightarrow{y},\\to_z)=R(\\to_x,\\to_y,\\xrightarrow{z})=0,\\end{equation*}\nare satisfied. \n\\end{lemma}\n\\begin{proof} It follows from the proof of Lemma \\ref{lem:rel}.\\end{proof}\n\\begin{example}\\label{ex:prelie} A $\\op{preLie}_\\bullet(\\sL ie)$-algebra structure on $V$ is the datum of $\\ast:V^{\\ten 2}\\to V$ such that, where as usual $[x,y]=x\\ast y - y \\ast x$,\n\\begin{multline*} \\left[\\xrightarrow{x},\\left[\\to_y,\\to_z\\right]\\right]=\\left[\\xrightarrow{x},\\to_{[y,z]} \\right]=\\xrightarrow{x\\ast[y,z]}=\\\\=\\left[ \\left[\\xrightarrow{x},\\to_y\\right],\\to_z\\right]+\\left[\\to_y,\\left[\\xrightarrow{x},\\to_z\\right]\\right]=\\left[\\xrightarrow{x\\ast y},\\to_z\\right]+\\left[\\to_y,\\xrightarrow{x\\ast z}\\right]=\\xrightarrow{(x\\ast y)\\ast z- (x\\ast z)\\ast y},\\qquad\\forall x,y,z\\in V.\n\\end{multline*}\nIn other words, $\\ast$ is a right pre-Lie product on $V$. Since the remaining two relations to be checked in the previous lemma follow from this one by symmetry of the Jacobi identity, we see that $\\op{preLie}_\\bullet(\\sL ie)=pre\\sL ie$, the operad of (right) pre-Lie algebras. \\end{example}\n\\begin{remark}\\label{rem:leftcase} As was said in Remark \\ref{rem:prelie} the above construction is actually the one of $\\op{preLie}_\\bullet(\\sL ie)=\\op{preLie}_{r,\\bullet}(\\sL ie)$: applying the functor $\\op{preLie}_{l,\\bullet}(-)$ introduced there we find by similar computations as in the above example that $\\op{preLie}_{l,\\bullet}(\\sL ie)$ is the operad of \\emph{left} pre-Lie algebras. More in general, it is easy to see that $\\op{preLie}_{l,\\bullet}(\\Oh)$ is always the opposite of the operad $\\op{preLie}_{r,\\bullet}(\\Oh)=\\op{preLie}_{\\bullet}(\\Oh)$ for any $\\Oh\\in\\mathbf{Op}$. We won't insist further on this point, and restrict our computations to the right case.\n\\end{remark}\n\nBy the proof of Lemma \\ref{lem:rel} we have morphisms of operads $\\Oh\\to \\op{Ass}_\\bullet(\\Oh)$, $\\Oh\\to \\op{preLie}_\\bullet(\\Oh)$, sending an $\\op{Ass}_\\bullet(\\Oh)$-algebra structure $(V,\\prec_i,\\succ_i,\\circ_j,\\ast_k)$ on $V$ to the associated $\\Oh$-algebra structure \\[(V\\:\\:,\\:\\:x\\cdot_iy=x\\prec_i y-y\\succ_ix\\:\\:,\\:\\:x\\bullet_j y=x\\circ_j y+y\\circ_j x\\:\\:,\\:\\:[x,y]_k=x\\ast_k y-y\\ast_k x),\\] \nand similarly in the other case.\n\n\\begin{definition} A $\\op{Com}_\\bullet(\\Oh)$-algebra is an $\\op{Ass}_\\bullet(\\Oh)$-algebra with trivial associated $\\Oh$-algebra. In other words, the operad $\\op{Com}_\\bullet(\\Oh)$ is generated by non-symmetric products $\\star_i$, anti-commutative brackets $\\{-,-\\}_j$ and commutative products $\\circledast_k$, together with the relations obtained from the ones of $\\op{Ass}_\\bullet(\\Oh)$-algebra by further imposing the identities \\[x\\prec_i y= y\\succ_i x=: x\\star_i y,\\qquad x\\circ_j y=-y\\circ_j x =:\\{x,y\\}_j,\\qquad x\\ast_k y= y\\ast_k x=: x\\circledast_k y.\\]\n\\end{definition}\n\n\\begin{example} We see immediately by Example \\ref{ex:ass} that $\\op{Com}_\\bullet(\\sL ie)=\\sC om$, the operad of commutative and associative algebras. \n\\end{example}\n\n\\begin{theorem}\\label{th:black} For any operad $\\Oh$ in $\\mathbf{Op}$, we have natural isomorphisms\n$\\op{Ass}_\\bullet(\\Oh)=\\sA ss\\bullet\\Oh$, $\\op{Com}_\\bullet(\\Oh)=\\sC om\\bullet\\Oh$, $\\op{preLie}_\\bullet(\\Oh)=pre\\sL ie\\bullet\\Oh$.\n\\end{theorem}\n\nThe proof is postponed to the next section (it follows from theorems \\ref{th:asswhite}, \\ref{th:permwhite}, \\ref{th:liewhite} and \\ref{th:adj}). In the remaining of this section we shall illustrate the result by several explicit computations. \n\n\\begin{remark}\\label{rem:curlyeqprec}\n\tIn some of the following computations it is conventient to choose a different basis for the operations of $\\op{Ass}_\\bullet(\\Oh)$ and $\\op{preLie}_\\bullet(\\Oh)$, by replacing the products $\\succ_i$ with the opposite products $\\curlyeqsucc_i:=-\\succ^{op}_i$: with the previous notations $\\to_x\\cdot'_i\\xrightarrow{y}=:\\xrightarrow{x\\curlyeqsucc_i y}:=\\xrightarrow{x}\\cdot'_i\\,_y\\to$.\n\\end{remark}\n\n\\begin{example} Given non-negative integers $p,q,r$, we denote by $\\sM ag_{p,q,r}$ the operad generated by $p$ magmatic non-symmetric operations, $q$ magmatic commutative operations and $r$ magmatic anti-commutative operations with no relations among them. The previous arguments show that $\\op{Ass}_\\bullet(\\sM ag_{p,q,r} )=\\op{preLie}_\\bullet (\\sM ag_{p,q,r})=\\sM ag_{2p+q+r,0,0}$, whereas $\\op{Com}_\\bullet(\\sM ag_{p,q,r})=\\sM ag_{p,r,q}$.\n\\end{example}\n\nWe introduce a further notation.\n\n\\begin{definition}\\label{def:+x} Given operads $\\mathcal{O},\\mathcal{P}\\in\\mathbf{Op}$, we denote by $\\mathcal{O}+\\mathcal{P}$ the operad defined by saying that an $(\\mathcal{O}+\\mathcal{P})$-algebra structure on $V$ is the datum of both an $\\Oh$-algebra structure and a $\\mathcal{P}$-algebra structure on $V$ with no relations among them, while we denote by $\\Oh\\times\\mathcal{P}$ the operad defined in the same way and by further imposing that every triple product involving an operation from $\\Oh$ and one from $\\mathcal{P}$ vanishes. It is easy to see that the functors $-+-,-\\times-:\\mathbf{Op}\\to\\mathbf{Op}$ are Koszul dual in the sense that $(\\Oh+\\sP)^!=\\Oh^!\\times\\sP^!$ for any pair of operads $\\Oh,\\sP\\in\\mathbf{Op}$. \\end{definition}\n\n\\begin{example} We consider the operad $(\\sA ss,\\cdot)$ of associative algebras. Given an $\\op{Ass}_\\bullet(\\sA ss)$-algebra structure $(V,\\prec,\\succ)$ on a vector space, we shall denote the associator of the associated product $\\cdot'$ on $C^\\ast(\\Delta_1;V)$ by $A_{\\cdot'}(-,-,-)$. Straightforward computations show that\n\\[ 0=A_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})\\:=\\: \\left(_x\\to\\cdot'\\to_y\\right)\\cdot'\\xrightarrow{z}-\\,_x\\to\\cdot'(\\to_y\\cdot'\\xrightarrow{z}) \\:=\\: 0+\\,_x\\to\\cdot'\\xrightarrow{z\\succ y}\\:=\\:\\xrightarrow{x\\prec(z\\succ y)},\\]\n\\begin{eqnarray} \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,{}_y\\xrightarrow{},\\xrightarrow{z} \\right) &=& \\xrightarrow{(y\\prec z)\\succ x}, \\\\\n\\nonumber 0=A_{\\cdot'}\\left({}_x\\xrightarrow{},\\xrightarrow{y},\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-x\\prec(y\\prec z)}, \\\\\n\\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},{}_z\\xrightarrow{} \\right) &=& \\xrightarrow{z\\succ(y\\succ x)-(z\\succ y)\\succ x}, \\\\\n\\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{x},{}_y\\xrightarrow{},\\xrightarrow{}_z \\right) &=& \\xrightarrow{-(y\\succ x)\\prec z},\\\\ \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{x},\\xrightarrow{}_y,{}_z\\xrightarrow{} \\right) &=& \\xrightarrow{-z\\succ(x\\prec y)}. \\end{eqnarray}\nConversely, according to Lemma \\ref{lem:rel}, the above six relations on $\\prec,\\succ$ imply the vanishing of $A_{\\cdot'}(-,-,-)$ on $C^*(\\Delta_1;V)$. Hence, we see that an $\\op{Ass}_\\bullet( \\sA ss)$-algebra structure on $V$ is the datum of two non-symmetric products $\\prec,\\succ:V^{\\ten 2}\\to V$ which are associative and such that moreover\n\\[x\\prec(y\\succ z)=(x\\prec y)\\succ z =x\\succ(y\\prec z)=(x\\succ y)\\prec z =0,\\qquad\\forall x,y,z\\in V.\\]\nWith the notations from the previous definition we found that that $\\op{Ass}_\\bullet( \\sA ss)=\\sA ss\\times\\sA ss$. By further imposing $x\\prec y-y\\succ x =: x\\cdot y= 0$, that is, $x \\prec y = y\\succ x=:x\\star y$, we see that the operad $\\op{Com}_\\bullet(\\sA ss)$ is generated by a non symmetric product $\\star$ such that $(x\\star y)\\star z=x\\star (y\\star z)=0$: in other words $\\op{Com}_\\bullet(\\sA ss)= nil\\sA ss$, the operad of two step nilpotent associative algebras, in accord with Theorem \\ref{th:black} and the computations in \\cite{GK}. To compute the relations of $\\op{preLie}_\\bullet(\\sA ss)$ it is convenient to replace the generating set of operations $\\prec,\\succ$ with the one $\\prec,\\curlyeqsucc=-\\succ^{op}$ as explained in Remark \\ref{rem:curlyeqprec}, then we get the following relations according to Lemma \\ref{lem:relpl} (notice that $x\\cdot y:=x\\prec y-y\\succ x=x\\prec y + x\\curlyeqsucc y$)\n\\begin{eqnarray} \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{x},\\xrightarrow{}_y,\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-x\\prec(y\\cdot z)}, \\\\ \n\\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},\\xrightarrow{}_z \\right) & = & \\xrightarrow{(x\\curlyeqsucc y)\\prec z-x\\curlyeqsucc(y\\prec z)}, \\\\ \\nonumber 0=A_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{}_y,\\xrightarrow{z} \\right) &=& \\xrightarrow{(x\\cdot y)\\curlyeqsucc z - x\\curlyeqsucc(y\\curlyeqsucc z)}, \\end{eqnarray}\nwhich are exactly the dendirform relations for the products $\\prec,\\curlyeqsucc$, thus $\\op{preLie}_\\bullet(\\sA ss)=\\sD end$, the operad of dendriform algebras, which as well known is also $\\sD end=pre\\sL ie\\bullet\\sA ss$.\n\\end{example}\n\\begin{example} Next we consider the operad $(\\sC om,\\bullet)$ of commutative and associative algebras. Given an $\\op{Ass}_\\bullet(\\sC om)$ algebra structure $(V,\\circ)$ on a vector space, together with the asociated commutative product $\\bullet'$ on $C^*(\\Delta_1;V)$, similar computations as in the previous example show that the vanishing of the associator $A_{\\bullet'}(-,-,-)$ is equivalent to the identities $(x\\circ y)\\circ z = x\\circ(y\\circ z)=0$, thus recovering once again $\\op{Com}_\\bullet(\\sA ss)= \\sC om\\bullet \\sA ss = nil\\sA ss$. If we further impose that the commutative product $x\\bullet y = x\\circ y + y\\circ x$ on $V$ vanishes, we see that $\\op{Com}_\\bullet(\\sC om)$ is the operad generated by an anti-commutative bracket $\\{x,y\\}:=x\\circ y=-y\\circ x$ such that $\\{ \\{ x,y \\}, z \\}=0$, or in other words $\\op{Com}_\\bullet(\\sC om)=nil\\sL ie$, the operad of two step nilpotent Lie algebras. Finally, a $\\op{preLie}_\\bullet(\\sC om)$-algebra structure on $V$ is the datum of a non-symmetric product $\\circ:V^{\\ten 2}\\to V$ such that, where as usual we denote by $x\\bullet y=x\\circ y + y\\circ x$, \\[ 0=A_{\\bullet'}(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{x\\circ(y\\bullet z)-(x\\circ y) \\circ z},\\qquad\\forall x,y,z\\in V, \\] \nand a second relation coming from Lemma \\ref{lem:relpl}, namely, $(x\\circ y)\\circ z= (x\\circ z)\\circ y$, already follows from this one. We find that $\\op{preLie}_\\bullet(\\sC om)=\\sZ inb$, the operad of (right) Zinbiel algebra, according to the well known fact $pre\\sL ie\\bullet \\sC om=\\sZ inb$ \\cite{V}.\n\\end{example}\n\n\\begin{example} We consider the operad $pre\\sL ie$ of (right) pre-Lie algebras. Given an $\\op{Ass}_\\bullet(pre\\sL ie)$-algebra structure $(V,\\prec,\\succ)$ on $V$, together with the associated dg pre-Lie algebra $(C^\\ast(\\Delta_1;V),d,\\cdot')$, the associator $A_{\\cdot'}(-,-,-)$ on $C^\\ast(\\Delta_1;V)$ is computed as in Example \\ref{ex:ass}. Writing the generating relation in $pre\\sL ie$ as $0=R(\\alpha,\\beta,\\gamma)=A_{\\cdot'}(\\alpha,\\beta,\\gamma)-A_{\\cdot'}(\\alpha,\\gamma,\\beta)$, $\\forall \\alpha,\\beta,\\gamma\\in C^*(\\Delta_1;V)$, we find that according to Lemma \\ref{lem:rel} this is equivalent to the following relations on $\\prec,\\succ$ (the remaining three follow from these ones by symmetry of $R$) \n\t\\[ 0 = R(_x\\to,\\to_y,\\xrightarrow{z})=\\xrightarrow{x\\prec(z\\succ y)-(x\\prec z)\\prec y+x\\prec(z\\prec y)}\\]\n\t\\[ 0 = R(\\to_x,\\,_y\\to,\\xrightarrow{z})=\\xrightarrow{(y \\prec z)\\succ x -y\\succ(z\\succ x)+(y\\succ z)\\succ x}\\]\n\t\\[ 0 = R(\\xrightarrow{x},\\,_y\\to,\\to_z)=\\xrightarrow{-(y\\succ x)\\prec z+ y\\succ(x\\prec z)}\\]\n\t\t\nThese are precisely the relations of dendriform algebra, thus once again $\\op{Ass}_\\bullet(pre\\sL ie)=\\sD end$. If we impose that the right pre-Lie product $x \\cdot y = x\\prec y - y\\succ x$ on $V$ vanishes, we get as well known \\cite{LV} the Zinbiel relation for $x\\star y:=x\\prec y=y\\succ x$, that is, once again $\\op{Com}_\\bullet(pre\\sL ie)=\\sZ inb$. Finally, in a $\\op{preLie}_\\bullet(pre\\sL ie)$-algebra we get the relations (the remaining one follows from the first one and simmetry of $R$)\n\\[0 = R(\\to_x,\\to_y,\\xrightarrow{z})=\\xrightarrow{-z\\succ (x\\cdot y)-(z\\succ y)\\succ x+(z\\succ x)\\prec y-(z\\prec y)\\succ x}\\]\n\\[0 = R(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{(x\\prec y)\\prec z -(x\\prec z)\\prec y- x\\prec(y\\cdot z -z\\cdot y)}\\]\nThe reader will recognize the relations of $L$-dendriform algebra \\cite{bai2,V}. We recall the proof of the following fact, giving the two forgetful functors from the category of $(pre\\sL ie\\bullet pre\\sL ie)$-algebras to the one of $pre\\sL ie$-algebras. \n\\begin{proposition} Let $V$ be a vector space, together with non symmetric products $\\prec,\\succ:V^{\\ten2}\\to V$ such that the following relations, where we put $x\\cdot y:= x\\prec y-y\\succ x$, $x\\triangleleft y := x\\prec y + x\\succ y$, $[x,y]:=x\\cdot y-y\\cdot x=x\\triangleleft y-y\\triangleleft x$, are verified\n\t\\[ x\\prec [y,z]=(x\\prec y)\\prec z - (x\\prec z)\\prec y,\\qquad (x\\succ y)\\prec z=x\\succ(y\\cdot z)+(x\\triangleleft z)\\succ y,\\qquad\\forall x,y,z\\in V. \\]\nThen the products $\\triangleleft,\\cdot$ are right pre-Lie products on $V$ and $[-,-]$ is a Lie bracket.\n\\end{proposition}\n\t\n\\begin{proof} The relations in the claim of the lemma imply the pre-Lie relation for $\\cdot$ according to the previous computations and the proof of Lemma \\ref{lem:rel}. The fact that $[-,-]$ is a Lie bracket follows. Finally, we may write the second relation as $A_{\\succ}(x,z,y)=(x\\succ y)\\prec z - (x\\prec z)\\succ y -x\\succ(y\\prec z)$: substituting in \n\t\t\\[ A_{\\triangleleft}(x,y,z)=A_{\\prec}(x,y,z)+A_{\\succ}(x,y,z)+(x\\prec y)\\succ z-x\\prec(y\\succ z)+(x\\succ y)\\prec z-x\\succ(y\\prec z)\\]\n\t\twe find that \n\t\t\\begin{multline*}\n\t\tA_{\\triangleleft}(x,y,z)= A_\\prec(x,y,z)-x\\prec(y\\succ z)+\\\\+(x\\succ z)\\prec y-(x\\prec y)\\succ z-x\\succ(z\\prec y)+(x\\prec y)\\succ z+(x\\succ y)\\prec z-x\\succ(y\\prec z).\n\t\t\\end{multline*}\n\t\tThe bottom row is clearly symmetric in $y$ and $z$, while the same statement for the right hand side of the top row is just another way of writing the first relation in the claim of the lemma.\\end{proof}\n\\end{example}\n\n\\begin{example}\\label{ex:leib} An interesting example is the operad $\\sL eib$ of (right) Leibniz algebras: this is the operad generated by a single non symmetric product $\\cdot$ and the relation $0=R_\\cdot(x,y,z):=(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y$. From Lemma \\ref{lem:rel} we get the following five indipendent relations on $\\prec,\\succ$\n\t\\begin{multline*} 0=R_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})\\:=\\: \\left(_x\\to\\cdot'\\to_y\\right)\\cdot'\\xrightarrow{z}-\\,_x\\to\\cdot'(\\to_y\\cdot'\\xrightarrow{z})-(_x\\to\\cdot'\\xrightarrow{z})\\cdot'\\to_y \\:= \\\\ 0+\\,_x\\to\\cdot'\\xrightarrow{z\\succ y}-\\xrightarrow{x\\prec z}\\cdot'\\to_y\\:=\\:\\xrightarrow{x\\prec(z\\succ y)-(x\\prec z)\\prec y},\\end{multline*}\n\t\n\t\\begin{eqnarray} \\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,{}_y\\xrightarrow{},\\xrightarrow{z} \\right) &=& \\xrightarrow{(y\\prec z)\\succ x-y\\succ(z\\succ x)}, \\\\\n\t\\nonumber 0=R_{\\cdot'}\\left({}_x\\xrightarrow{},\\xrightarrow{y},\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-x\\prec(y\\prec z)}, \\\\\n\t\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},{}_z\\xrightarrow{} \\right) &=& \\xrightarrow{z\\succ(y\\succ x)-(z\\succ y)\\succ x}, \\\\\n\t\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{x},{}_y\\xrightarrow{},\\xrightarrow{}_z \\right) &=& \\xrightarrow{-(y\\succ x)\\prec z+y\\succ(x\\prec z)},\n\t\\end{eqnarray}\n\tand the remaining one is equivalent to the last one.\nThese are the precisely the relations of diassociative algebra on $(V,\\prec,\\succ)$, in other words we found that $\\op{Ass}_\\bullet(\\sL eib)=di\\sA ss$, the operad of diassociative algebras. If we further impose $0=x\\cdot y=x\\prec y -y\\succ x$ and we put as usual $x\\star y :=x\\prec y=y\\succ x$, the previous relations reduce to the two independent ones $(x\\star y)\\star z=x\\star(y\\star z)$, $(x\\star y)\\star z=x\\star(z\\star y)$: in other words, we found that $\\op{Com}_\\bullet(\\sL eib)=\\sP erm$, the operad of (right) permutative algebras. Finally, we consider the operad $\\op{preLie}_\\bullet(\\sL eib)$: this is generated by non-symmetric products $\\prec,\\succ$ and the relations, where as usual $x\\cdot y= x\\prec y- y\\succ x$, \n\\begin{eqnarray} \\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{x},\\xrightarrow{}_y,\\xrightarrow{}_z \\right) &=& \\xrightarrow{(x\\prec y)\\prec z-(x\\prec z)\\prec y-x\\prec(y\\cdot z)}, \\\\\n\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{y},\\xrightarrow{}_z \\right) &=& \\xrightarrow{-(y\\succ x)\\prec z+(y\\prec z)\\succ x+y\\succ(x\\cdot z)}, \\\\\n\\nonumber 0=R_{\\cdot'}\\left(\\xrightarrow{}_x,\\xrightarrow{}_y,\\xrightarrow{z} \\right) &=& \\xrightarrow{-z\\succ(x\\cdot y)-(z\\succ y)\\succ x+(z\\succ x)\\prec y}.\n\\end{eqnarray}\nThese are equivalent to the following relations for $\\prec,\\succ$\n\\begin{multline*} x\\prec(y\\cdot z)=(x\\prec y)\\prec z -(x\\prec z)\\prec y,\\qquad (x\\succ y)\\succ z=(x\\prec y)\\succ z, \\\\ x\\succ(y\\cdot z)=(x\\succ y)\\prec z-(x\\prec z)\\succ y. \\end{multline*}\n\n\\end{example}\n\n\\begin{example}\\label{ex:pois} We consider the operad $(\\sP ois,\\bullet, [-,-])$ of commutative Poisson algebras. By the previous computations, an $\\operatorname{Ass}_\\bullet(\\sP ois)$-algebra structure on $V$ is the datum $(V,\\circ,\\ast)$ of an $\\op{Ass}_\\bullet(\\sC om)=nil\\sA ss$-algebra structure $(V,\\circ)$ and an $\\op{Ass}_\\bullet(\\sL ie)=\\sA ss$-algebra structure $(V,\\ast)$ satisfying the additional relations induced, as in Lemma \\ref{lem:rel}, by the Poisson identity $0=[x,y\\bullet z]-[x,y]\\bullet z - y\\bullet[x,z]$ in the operad $\\sP ois$. These additional relations are easily computed, for instance\n\t\\begin{multline*} 0=\\left[\\xrightarrow{x},\\,_y\\to\\bullet'\\to_z\\right]' - \\left[\\xrightarrow{x},\\,_y\\to\\right]'\\bullet'\\to_z-\\,_y\\to\\bullet'\\left[\\xrightarrow{x},\\to_z\\right]' = \\\\=\\,0\\,-\\xrightarrow{-y\\ast x}\\cdot'\\to_z -\\,_y\\to\\cdot'\\xrightarrow{x\\ast z}\\:=\\:\\xrightarrow{(y\\ast x)\\circ z - y\\circ (x\\ast z)}.\n\t\\end{multline*} \n\tProceeding in this way we find the following relations for $\\circ,\\ast$\n\t\\[ (x\\ast y)\\circ z= x\\ast(y\\circ z)=x\\circ(y\\ast z)= (x\\circ y)\\ast z.\\] \n\tSimilarly, a $\\operatorname{Com}_\\bullet(\\sP ois)$-algebra structure on $V$ is the datum $(V,\\{-,-\\},\\circledast)$ of a $\\operatorname{Com}_\\bullet(\\sC om)=nil\\sL ie$-algebra structure $(V,\\{-,-\\})$ and a $\\operatorname{Com}_\\bullet(\\sL ie)=\\sC om$-algebra structure $(V,\\circledast)$ such that moreover the above identities hold. Consider the one $\\{x,y\\}\\circledast z=\\{x\\circledast y,z \\} $: as the left and right hand side are respectively symmetric and anti-symmetric in $x$ and $y$ both vanish, thus we have the relations\n\t\\[ \\{x\\circledast y,z \\}=x\\circledast\\{y,z\\}=0,\\qquad\\forall x,y,z\\in V.\\]\n\tIn other words, we found $\\operatorname{Com}_\\bullet(\\sP ois)=nil\\sL ie\\times\\sC om$. Finally, a $\\op{preLie}_\\bullet(\\sP ois)$-algebra structure on $V$ is the datum $(V,\\circ,\\ast)$ of a $\\op{preLie}_\\bullet(\\sC om)=\\sZ inb$-algebra structure $(V,\\circ)$ and a $\\op{preLie}_\\bullet(\\sL ie)=pre\\sL ie$-algebra structure $(V,\\ast)$, such that moreover, where as usual we denote by $x\\bullet y =x\\circ y+ y\\circ x$ and $[x,y]=x\\ast y-y\\ast x$ the associated $\\sC om$- and $\\sL ie$-algebra structures on $V$ respectively,\n\t\\begin{multline*} 0=\\left[\\xrightarrow{x},\\to_y\\bullet'\\to_z\\right]' - \\left[\\xrightarrow{x},\\,\\to_y\\right]'\\bullet'\\to_z-\\to_y\\bullet'\\left[\\xrightarrow{x},\\to_z\\right]' = \\\\=\\left[\\xrightarrow{x},\\to_{y\\bullet z}\\right]'-\\xrightarrow{x\\ast y}\\bullet'\\to_z -\\to_y\\bullet'\\xrightarrow{x\\ast z}=\\xrightarrow{x\\ast(y\\bullet z) -(x\\ast y)\\circ z - (x\\ast z)\\circ y},\n\t\\end{multline*} \n\t\\begin{multline*} 0=\\left[\\to_x,\\xrightarrow{y}\\bullet'\\to_z\\right]' - \\left[\\to_x,\\xrightarrow{y}\\right]'\\bullet'\\to_z-\\xrightarrow{y}\\bullet'\\large[\\to_x,\\to_z\\large]' = \\\\=\\left[\\to_x,\\xrightarrow{y\\circ z}\\right]'-\\xrightarrow{-y\\ast x}\\bullet'\\to_z -\\xrightarrow{y}\\bullet'\\to_{[x,z]}=\\xrightarrow{-(y\\circ z)\\ast x+(y\\ast x)\\circ z - y\\circ [x,z]}.\n\t\\end{multline*} \n\tWe conclude that $\\op{preLie}_\\bullet(\\sP ois)=pre\\sP ois$, Aguiar's operad of right pre-Poisson algebras \\cite{A}, in accord with the fact \\cite{U} that $pre\\sL ie\\bullet\\sP ois = pre\\sP ois$.\n\\end{example}\n\n\n\\begin{example}\\label{ex:perm} We consider the operad $\\sP erm$: recall that a $\\sP erm$-algebra structure, or (right) permutative algebra structure, on $V$ is the datum of an associative product $\\cdot$ which is commutative on the right hand side whenever there are three or more variables, or, in other words, satisfies the relations $(x\\cdot y)\\cdot z=x\\cdot(y\\cdot z)=x\\cdot(z\\cdot y)$. Thus, an $\\op{Ass}_\\bullet(\\sP erm)$-algebra structure on $V$ is the datum of an $\\op{Ass}_\\bullet(\\sA ss)=\\sA ss\\times\\sA ss$ algebra structure $(V,\\prec,\\prec)$, as in Example \\ref{ex:ass}, such that moreover\n\\[ 0 \\:=\\: _x\\to\\cdot'\\left( \\to_y\\cdot'\\xrightarrow{z}\\right)\\,-\\,_x\\to\\cdot'\\left( \\xrightarrow{z}\\cdot'\\to_y\\right) =\\xrightarrow{-x\\prec(z\\prec y)-x\\prec(z\\prec y)}=\\xrightarrow{-x\\prec(y\\prec z)}\\]\n\\[ 0 \\:=\\: \\to_x\\cdot'\\left( _y\\to\\cdot'\\xrightarrow{z}\\right)-\\to_x\\cdot'\\left( \\xrightarrow{z}\\cdot'\\,_y\\to \\right) =\\xrightarrow{-(y\\prec z)\\prec x-(y\\prec z)\\prec x}=\\xrightarrow{-(y\\prec z)\\prec x}.\\]\nIn other words, $\\op{Ass}_\\bullet(\\sP erm)=nil\\sA ss\\times nil\\sA ss$. Further imposing $0=x\\prec y-y\\prec x=x\\cdot y$, we find that $\\op{Com}_\\bullet(\\sP erm)=nil\\sA ss$. Finally, as in Remark \\ref{rem:curlyeqprec} and Example \\ref{ex:ass}, it is convenient in the compuation of $\\op{preLie}_\\bullet(\\sP erm)$ to replace the product $\\prec$ by the opposite one $\\curlyeqsucc:=-\\prec^{op}$, satisfying the formulas $\\to_x\\cdot'\\xrightarrow{y}=\\xrightarrow{x}\\cdot'\\,_y\\to=:\\xrightarrow{x\\curlyeqsucc y}$. In fact, as we saw in \\ref{ex:ass} the relations of $\\op{preLie}_\\bullet(\\sP erm)$-algebra translate in the relations of $\\op{preLie}_\\bullet(\\sA ss)=\\sD end$-algebra for the products $\\prec,\\curlyeqsucc$, and we get the additional relations (where as usual $x\\cdot y=x\\prec y-y\\prec x=x\\prec y+ x\\curlyeqsucc y$)\n\\[ 0= \\xrightarrow{x}\\cdot'\\left( \\to_y\\cdot'\\to_z\\right)-\\xrightarrow{x}\\cdot'\\left(\\to_z\\cdot'\\to_y\\right)=\\xrightarrow{x\\prec(y\\cdot z) - x\\prec(z\\cdot y)}\\]\n\\[ 0= \\to_x\\cdot'(\\xrightarrow{y}\\cdot'\\to_z)-\\to_x\\cdot'\\left( \\to_z\\cdot'\\xrightarrow{y}\\right)= \\xrightarrow{x\\curlyeqsucc(y\\prec z)-x\\curlyeqsucc(z\\curlyeqsucc y)}.\\]\nThis is in accord with \\cite[Theorem 21]{V}.\n\\end{example}\n\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$, an $\\Oh_{adm}$-algebra structure on $V$ is the datum of non-symmetric operations $\\prec_i,\\succ_i,\\circ_j.\\ast_k:V^{\\ten 2}\\to V$ such that the induced operations $x\\cdot_i y:=x\\prec_i y+x\\succ_iy$, $x\\bullet_j y= x\\circ_j y+y\\circ_j x$, $[x,y]_k=x\\ast_k y-y\\ast_k x$ define an $\\Oh$-algebra structure on $(V,\\cdot_i,\\bullet_j,[-,-]_k)$. This defines the operad $(\\Oh_{adm},\\prec_i,\\succ_i,\\circ_j,\\ast_k)$ of $\\Oh$ admissible algebras.\n\\end{definition}\n\n\\begin{example}\\label{ex:lieadm} We consider the operad $\\sL ie_{adm}$ of Lie admissible algebras, that is, the operad generated by a non-symmetric operation $\\cdot$ such that the commutator is a Lie bracket. We shall write the only relation in $\\sL ie_{adm}$ as $0=R_\\cdot(x_1,x_2,x_3)=\\sum_{\\sigma\\in S_3}\\varepsilon(\\sigma)A_\\cdot(x_{\\sigma(1)},x_{\\sigma(2)},x_{\\sigma(3)})$, where $\\varepsilon(\\sigma)$ is the sign of the permutation $\\sigma$ and as usual $A_\\cdot(-,-,-)$ is the asociator of $\\cdot$. Given an $\\operatorname{Ass}_\\bullet(\\sL ie_{adm})$-algebra structure $(V,\\prec,\\succ)$ on $V$, together with the associated $\\sL ie_{adm}$-algebra structure on $C^*(\\Delta_1;V)$, the associator $A_{\\cdot'}(-,-,-)$ is computed as in Example \\ref{ex:ass}. By Lemma \\ref{lem:rel} the vanishing of $R_\\cdot'(-,-,-)$ is equivalent to the following relation on $\\prec,\\succ$ (the others follow from this one by symmetry of $R$) \n\\begin{multline*} 0= R_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})=A_{\\cdot'}(_x\\to,\\to_y,\\xrightarrow{z})+ A_{\\cdot'}(\\to_y,\\xrightarrow{z},\\,_x\\to)+A_{\\cdot'}(\\xrightarrow{z},\\,_x\\to,\\to_y)-\\\\-A_{\\cdot'}(\\to_y,\\,_x\\to,\\xrightarrow{z})-A_{\\cdot'}(_x\\to,\\xrightarrow{z},\\to_y)-A_{\\cdot'}(\\xrightarrow{z},\\to_y,\\,_x\\to)=\n\\end{multline*} \n\\[=\\xrightarrow{x\\prec(z\\succ y) + x\\succ(z\\succ y)-(x\\succ z)\\succ y - (x\\succ z)\\prec y-(x\\prec z)\\succ y-(x\\prec z)\\prec y -x\\prec(z\\prec y)+x\\succ(z\\prec y)}\\]\nThis is exactly the relation saying that the product $x\\cup y:=x\\prec y + x\\succ y$ on $V$ (so denoted to distinguish it from the Lie admissible product $x\\cdot y=x\\prec y-y\\succ x$) is associative. Hence, we found $\\op{Ass}_\\bullet(\\sL ie_{adm})=\\sA ss_{adm}$. Further imposing $x\\prec y=y\\succ x=:x\\star y$, we see that $x\\cup y= x\\star y + y\\star x$ is an associative and commutative product, and thus that $\\op{Com}_\\bullet(\\sL ie_{adm})=\\sC om_{adm}$. For the operad $(\\operatorname{preLie}_\\bullet(\\sL ie_{adm}),\\prec,\\succ)$ we find the relation (and the others follow by symmetry of $R$), where as usual $x\\cdot y=x\\prec y-y\\succ x$,\n\\begin{multline*} 0= R_{\\cdot'}(\\to_x,\\to_y,\\xrightarrow{z})=A_{\\cdot'}(\\to_x,\\to_y,\\xrightarrow{z})+ A_{\\cdot'}(\\to_y,\\xrightarrow{z},\\to_x)+A_{\\cdot'}(\\xrightarrow{z},\\to_x,\\to_y)-\\\\-A_{\\cdot'}(\\to_y,\\to_x,\\xrightarrow{z})-A_{\\cdot'}(\\to_x,\\xrightarrow{z},\\to_y)-A_{\\cdot'}(\\xrightarrow{z},\\to_y,\\to_x)= \n\\end{multline*} \n\\[=\\xrightarrow{-z\\succ(x\\cdot y) - (z\\succ y)\\succ x - (z\\succ y)\\prec x + (z\\prec x)\\succ y + (z\\prec x)\\prec y-z\\prec(x\\cdot y)+z\\succ (y\\cdot x) + (z\\succ x)\\succ y + (z\\succ x)\\prec y -(z\\prec y)\\succ x -(z\\prec y)\\prec x+ z\\prec(y\\cdot x)}\\]\nPutting $x\\cup y = x\\prec y + x\\succ y$, the reader will check that the above relation can be rewritten as $0=A_\\cup(z,x,y)-A_\\cup(z,y,x)$, and thus we found $\\op{preLie}_\\bullet(\\sL ie_{adm})=pre\\sL ie_{adm}$.\\end{example} \n\n\\begin{remark}\n\tThe latter example suggests $\\Oh\\bullet\\sL ie_{adm}=\\Oh_{adm}$ for every operad $\\Oh$ in $\\mathbf{Op}$.\t\n\\end{remark}\n\n\\begin{example}\\label{ex:postlie} We consider the operad $(post\\sL ie,\\cdot,[-,-])$ of post-Lie algebras: this is the operad generated by a non-symmetric product $\\cdot$ and a Lie bracket $[-,-]$ satisfying the relations $0=R_1(x,y,z)=[x,y]\\cdot z-[x,y\\cdot z]-[x\\cdot z,y]$ and $0=R_2(x,y,z)=x\\cdot(y\\cdot z +z\\cdot y + [z,y])-(x\\cdot y)\\cdot z+(x\\cdot z)\\cdot y$. In the operad $(\\op{Ass}_\\bullet(\\Oh),\\prec,\\succ,\\ast)$ we get the associativity relation for the product $\\ast$ and the six independent relations \n\\[ 0=R_1(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{(x\\ast y)\\prec z-x\\ast(y\\prec z)}, \\]\n\\[ 0=R_1(\\to_x,\\xrightarrow{y},_z\\to)=\\xrightarrow{z\\succ(y\\ast x)-(z\\succ y)\\ast x},\\]\n\\[ 0=R_1(_x\\to,\\to_y,\\xrightarrow{z})=\\xrightarrow{x\\ast(z\\succ y)-(x\\prec z)\\ast y},\\]\n\\[ 0=R_2(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{x\\prec(y\\prec z+z\\prec y+y\\ast z)-(x\\prec y)\\prec z},\\]\n\\[ 0=R_2(\\to_x,\\xrightarrow{y},_z\\to)= \\xrightarrow{(z\\succ y+ y\\prec z +z\\star y)\\succ x-z\\succ(y\\succ x)},\\]\n\\[ 0=R_2( \\xrightarrow{x},_y\\to,\\to_z)=\\xrightarrow{(y\\succ x)\\prec z-y\\succ(x\\prec z)}.\\]\nThese are the relations of dendriform trialgebra \\cite{LR}, hence we found $\\op{Ass}_\\bullet(post\\sL ie)=\\mathcal{T}ridend$. If we further impose $x\\prec y= y\\succ x=: y\\star x$, $x\\star y=y\\star x=:x\\circledast y$, we find that the operad $(\\op{Com}_\\bullet(post\\sL ie),\\star,\\circledast)$ is defined by the following relations\n\\[ (x\\circledast y)\\circledast z=x\\circledast(y\\circledast z),\\quad(x\\circledast y)\\star z=x\\circledast(y\\star z),\\quad(x\\star y)\\star z =x\\star(y\\star z+z\\star y +y\\circledast z). \\]\nWe shall denote this operad by $post\\sC om:=\\op{Com}_\\bullet(post\\sL ie)$. We leave to the interested reader the computation of $\\op{preLie}_\\bullet(post\\sL ie)$. We have morphism of operads \n\\[ (\\sL ie,\\{-,-\\})\\to(post\\sL ie,\\cdot,[-,-]):\\{-,-\\}\\:\\:\\longrightarrow\\:\\:\\cdot\\:-\\:\\cdot^{op}\\:+\\:[-,-],\\]\n\\[ (\\sA ss,\\cup)\\to(\\mathcal{T}ridend,\\prec,\\succ,\\ast):\\cup\\:\\:\\longrightarrow\\:\\:\\prec+\\succ+\\ast,\\]\n\\[(\\sC om,\\bullet)\\to(post\\sC om,\\star,\\circledast):\\bullet\\:\\:\\longrightarrow\\:\\: \\star+\\star^{op}+\\circledast, \\] \nin the first (as well as the second) case this is well known, the other two follow by functoriality. \n\\end{example}\n\n\\begin{example} We consider the operad generated by a Lie bracket $[-,-]$ and a right Leibniz product $\\cdot$ satisfying the additional relations \n\t\\[ [x,y]\\cdot z=[x,y\\cdot z]+[x\\cdot z,y],\\qquad x\\cdot[y,z]=x\\cdot(y\\cdot z). \\]\n\tIt can be checked that this is the Koszul dual $(post\\sC om^!, \\cdot,[-,-])$ of the operad $post\\sC om:=\\op{Com}_\\bullet(post\\sL ie)$ from the previous example. To aid in the computations, we notice that this is the same, up to changing the sign of the operation $\\cdot$, as the operad we obtain from $(post\\sL ie,\\cdot,[-,-])$ by further imposing the right Leibniz identity for $\\cdot$. By the previous example and Example \\ref{ex:leib}, we see that the operad $\\op{Ass}_\\bullet(post\\sC om^!)$ is the same as the one we obtain from the operad $\\mathcal{T}ridend$ of dendriform trialgebras by further imposing the diassociativity relations for $\\prec,\\succ$, and then changing the signs of $\\prec,\\succ$: the reader will readily verify that this is the operad $\\op{Ass}_\\bullet(post\\sC om^!)=\\mathcal{T}riass$ of triassociative algebras by Loday and Ronco \\cite{LR}. Likewise, $\\op{Com}_\\bullet(post\\sC om^!)$ is the operad we obtain from $(post\\sC om,\\star,\\circledast)$, defined as in the previous example, by further imposing that $\\star$ is a right permutative product, and then changing the sign of $\\star$: the reader will readily verify that this is the same as the operad $\\sC omtrias$ of commutative trialgebras \\cite{Vpos}. \n\\end{example}\n\\newcommand{\\sR}{\\mathcal{R}}\n\n\\section{A Koszul dual trick to compute Manin white products}\n\nWe denote by $-\\circ-:\\mathbf{Op}\\times\\mathbf{Op}\\to\\mathbf{Op}$ the Manin white product of operads \\cite{V}, it gives $\\mathbf{Op}$ a structure of symmetric monoidal category, and by $-^!:\\mathbf{Op}\\to\\mathbf{Op}$ the Koszul duality functor. Recall that the Manin white and black products are Koszul dual, in the sense that $(\\Oh\\bullet\\sP)^!=\\Oh^!\\circ\\sP^!$ for every pair of operads $\\Oh$ and $\\sP$ in $\\mathbf{Op}$, and moreover for every operad $\\Oh$ the functors $\\xymatrix{\\Oh\\bullet-:\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\Oh^!\\circ-\\ar@<2pt>[l]}$ form an adjoint pair. recall that $\\sA ss^!=\\sA ss$, $\\sC om^!=\\sL ie$ and $pre\\sL ie^!=\\sP erm$. In the previous section we introduced easily computable functors $\\op{Ass}_\\bullet(-)$, $\\op{Com}_\\bullet(-),\\op{preLie}_\\bullet(-):\\mathbf{Op}\\to\\mathbf{Op}$ and claimed that they coincide with the respective Manin black products $\\sA ss\\bullet-,\\sC om\\bullet-, pre\\sL ie\\bullet-$. The aim of this section is to introduce the respective right adjoint functors $\\op{Ass}_\\circ(-),\\op{Lie}_\\circ(-),\\op{Perm}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$ (the adjointness relation will be shown at the very end of the section) and prove that they in fact coincide with the Manin white products $\\sA ss\\circ-,\\sL ie\\circ-,\\sP erm\\circ-$:\nthis will also complete the proof of Theorem \\ref{th:black}.\n\nWe consider first the case of $\\operatorname{Ass}_\\circ(-)$. Given an (as usual, binary, quadratic and finitely generated) operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$, the operad $\\op{Ass}_\\circ(\\Oh)$ is generated by non symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$. We want a morphism of operads $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ corresponding to the counit of the adjunction $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$: in other words, by definition of $\\op{Ass}_\\bullet(-)$, given an $\\Oh$-algebra structure $(V,\\cdot_i,\\bullet_j,[-,-]_k)$ on a vector space $V$ we want an induced local dg $\\op{Ass}_\\circ(\\Oh)$-algebra structure on the complex $C^*(\\Delta_1;V)=C^*(\\Delta_1;\\mathbb{K})\\otimes V$. Denoting by $\\cup$ the usual cup product of cochains on $C^*(\\Delta_1;\\mathbb{K})$, the $\\op{Ass}_\\circ(\\Oh)$-algebra structure on $C^*(\\Delta_1;V)$ will be given by the tensor product operations $\\prec_i=\\cup\\otimes\\cdot_i$, $\\succ_i=\\cup\\otimes\\cdot_i^{op}$, $\\circ_j=\\cup\\ten\\bullet_j$, $\\ast_k=\\cup\\ten[-,-]_k$: explicitly, with the same notations as in the previous section, \n\\[ \\to_x\\prec_i\\to_y\\:=\\:\\to_{x\\cdot_i y}\\:=\\:\\to_y\\succ_i\\to_x,\\qquad \\to_x\\circ_j\\to_y\\:=\\:\\to_{x\\bullet_j y},\\qquad\\to_x\\ast_k\\to_y\\:=\\: \\to_{[x,y]_k}, \\] \\[ _x\\to\\prec_i\\,_y\\to\\:=\\:_{x\\cdot_i y}\\to\\:=\\:_y\\to\\succ_i\\,_x\\to,\\qquad _x\\to\\circ_j\\,_y\\to\\:=\\:_{x\\bullet_j y}\\to,\\qquad_x\\to\\ast_k\\,_y\\to\\:=\\: _{[x,y]_k}\\to, \\] \\[ _x\\to\\prec_i\\xrightarrow{y}=_y\\to\\succ_i\\xrightarrow{x}\\:=\\:\\xrightarrow{x\\cdot_i y}\\:=\\:\\xrightarrow{x}\\prec_i\\to_y= \\xrightarrow{y}\\succ_i\\to_{x},\\] \\[ _x\\to\\circ_j\\xrightarrow{y}\\:=\\:\\xrightarrow{x\\bullet_j y}\\:=\\:\\xrightarrow{x}\\circ_j\\to_y,\\qquad _x\\to\\ast_k\\xrightarrow{y}\\:=\\:\\xrightarrow{[x,y]_k}\\:=\\:\\xrightarrow{x}\\ast_k\\to_y, \\] \nand the remaining products vanish. It is immediately seen that these operations satisfy the Leibniz identity with respect to $d$ and the locality assumption from Definition \\ref{def:ManinBlack}. We will generically call the operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$ on $C^*(\\Delta_1;V)$ the cup products. \n\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\op{Ass}_\\circ(\\Oh)$ is generated by non-symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, together with the larger set of relations making $C^*(\\Delta_1;V)$ with the cup products a local dg $\\op{Ass}_\\circ(\\Oh)$-algebra for every $\\Oh$-algebra $V$.\\end{definition}\n\nNext we want to describe a generating set of relations of $\\op{Ass}_\\circ(\\Oh)$. Given a terniary operation $R(-,-,-):C^*(\\Delta_1;V)^{\\ten3}\\to C^*(\\Delta_1;V)$ in the cup products, we write $R=\\sum_{\\sigma\\in S_3}R_\\sigma$, where $S_3$ is the symmetric group and $R_\\sigma$ is spanned by triple products permuting the variables according to $\\sigma$. As in the proof of Lemma \\ref{lem:rel}, we see that the relation $R(-,-,-)=0$ holds in $C^*(\\Delta_1;V)$ if and only if\n\\begin{multline*}0=R(\\xrightarrow{x},\\to_y,\\,_z\\to)=R(\\xrightarrow{x},\\,_y\\to,\\to_z)=R(\\to_x,\\xrightarrow{y},\\,_z\\to)=\\\\=R(_x\\to,\\xrightarrow{y},\\to_z)=R(\\to_x,\\,_y\\to,\\xrightarrow{z})=R(_x\\to,\\to_y,\\xrightarrow{z})=0\\end{multline*}\nfor all $x,y,z\\in V$. We claim that this is true if and only if $R_\\sigma(-,-,-)=0$ holds in $C^*(\\Delta_1;V)$ for every $\\sigma\\in S_3$, where the \\vr if'' implication is obvious. To fix the ideas, we consider the identical permutation $\\id\\in S_3$ and prove $R(-,-,-)=0\\solose R_{\\id}(-,-,-)=0$: the remaining implications are proved similarly. We obtain the desired conclusion again by Lemma \\ref{lem:rel}, the point is that the only non-vanishing triple products that we can form out of the cochains $_1\\to,\\to_1,\\xrightarrow{1}\\in C^*(\\Delta_1;\\mathbb{K})$ are $(_1\\to\\cup\\xrightarrow{1})\\cup\\to_1\\:=\\:\\xrightarrow{1}\\:=\\:_1\\to\\cup(\\xrightarrow{1}\\cup\\to_1)$: this shows at once that $R_{\\id}(\\xrightarrow{x},\\to_y,\\,_z\\to)=R_{\\id}(\\xrightarrow{x},\\,_y\\to,\\to_z)=R_{\\id}(\\to_x,\\xrightarrow{y},\\,_z\\to)=R_{\\id}(\\to_x,\\,_y\\to,\\xrightarrow{z})=R_{\\id}(_x\\to,\\to_y,\\xrightarrow{z})=0$, whereas $R_{\\id}(_x\\to,\\xrightarrow{y},\\to_z)=R(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{R'(x,y,z)}=0$ by hypothesis, where $R'(-,-,-):V^{\\ten3}\\to V$ is a terniary operation in $\\Oh$; finally, if this holds for a generic $\\Oh$-algebra $V$, we conclude that $R'(-,-,-)=0$ is a relation in the operad $\\Oh$. We sum up the previous discussion in the following lemma.\n\\begin{lemma}\\label{lem:relwhite} The operad $\\op{Ass}_\\circ(\\Oh)$ has a generating set of relations $R(-,-,-)=0$ which are non-symmetric (where a relation is non-symmetric if $R=R_{\\id}$), consisting of all the possible splittings, as in the following \n\t\\[ R(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{R'(x,y,z)}=0,\\]\nof a relation $R'(-,-,-)=0$ in $\\Oh$ via the cup products on $C^*(\\Delta_1;V)$ (where $V$ is a generic $\\Oh$-algebra).\\end{lemma}\nIt is best to illustrate the previous lemma by some examples.\n\\begin{example} We consider the operad $(\\sC om,\\bullet)$: in this case $R'$ as in the previous lemma has to be non-symmetric itself, and since the space of non-symmetric relators of $\\sC om$ is one-dimensional, spanned by the associativity relation, we only get the splitting\n\t\\[ (_x\\to \\circ \\xrightarrow{y})\\circ\\to_y - _x\\to\\circ(\\xrightarrow{y}\\circ\\to_z) =\\xrightarrow{(x\\bullet y)\\bullet z-x\\bullet (y\\bullet z)}=0,\\]\n\ttelling us that the cup product $\\circ$ is associative: thus $\\op{Ass}_\\circ(\\sC om)=\\sA ss$ as expected. In the case of the operad $(\\sL ie,[-,-])$, again $R'$ as in the previous Lemma has to be non-symmetric, and since the space of relators of the operad $\\sL ie$ is one-dimensional generated by the Jacobi identity, and this can't be written in a non-symmetric form, we find $\\op{Ass}_\\circ(\\sL ie)=\\sM ag_{1,0,0}$, in accord with our computation of the Koszul dual $\\op{Ass}_\\bullet(\\sC om)=nil\\sA ss$. \n\t\\end{example}\n\n\\begin{remark} Whereas the computation of the functor $\\op{Ass}_\\bullet(-)$ is completely mechanical, as we apply Lemma \\ref{lem:rel} to a generating set of relations of $\\Oh$ to get a generating set of relations of $\\op{Ass}_\\bullet(\\Oh)$, the previous lemma is not as effective in the compution of $\\op{Ass}_\\circ(\\Oh)$: we illustrate this fact by considering the operad $(\\sP ois,\\bullet,[-,-])$. By the previous example, the operad $\\op{Ass}_\\circ(\\sP ois)$ is generated by an associative product $\\circ$ and a magmatic product $\\ast$. Since there is no way to split the Poisson identity $[x\\bullet y, z] -x\\bullet[y,z] - [x,z]\\bullet y=0$ as in the previous lemma, as this can't be written in a non-symmetric form, we may be tempted to conclude that $\\op{Ass}_\\circ(\\sP ois)=\\sA ss+\\sM ag_{1,0,0}$, but this is in contrast with our computation of the Koszul dual $\\op{Ass}_\\bullet(\\sP ois)$ in Example \\ref{ex:pois}. In fact, looking closely we find the splitting\n\\begin{multline*} (_x\\to\\circ \\xrightarrow{y})\\ast\\to_z-_x\\to\\circ(\\xrightarrow{y}\\ast\\to_z) + (_x\\to\\ast\\xrightarrow{y})\\circ\\to_z-_x\\to\\ast(\\xrightarrow{y}\\circ\\to_z) = \\\\ = \\xrightarrow{[x\\bullet y,z ]-x\\bullet[y,z]+[x,y]\\bullet z-[x,y\\bullet z]}=\\xrightarrow{[x,z]\\bullet y - y\\bullet[x,z]}=0.\n\\end{multline*}\nSo we find the relation $(\\alpha\\circ\\beta)\\ast\\gamma-\\alpha\\circ(\\beta\\ast\\gamma)+(\\alpha\\ast\\beta)\\circ\\gamma-\\alpha\\ast(\\beta\\circ\\gamma)=0$, $\\forall \\alpha,\\beta,\\gamma\\in C^*(\\Delta_1;V)$, in the operad $\\op{Ass}_\\circ(\\sP ois)$. Together with the associativity relation for $\\circ$, these generate the relations of $\\op{Ass}_\\circ(\\sP ois)$: the easiest way to see this is to check that the operad $\\op{Ass}_\\circ(\\sP ois)$ defined in this way is in fact the Koszul dual of $\\op{Ass}_\\bullet(\\sP ois)$ from \\ref{ex:pois}. Returning to the initial remark, the difficulty in applying Lemma \\ref{lem:relwhite} is that we have to look for the splittings of any relation $R'$ in $\\Oh$, and we can't limit ourselves to let $R'$ vary in a generating set of these relations, so there is no telling us in general if we are missing some of the possible splittings. As in the previous computation, the easiest way around this, and what we will do in practice in the following examples, is to take advantage of the computation of the Koszul dual $\\op{Ass}_\\bullet(\\Oh^!)$ from the previous section. Of course, this depends on the yet to be given proofs of theorems \\ref{th:black}, \\ref{th:asswhite},\\ref{th:permwhite} and \\ref{th:liewhite}.\n\\end{remark}\n\\begin{example} We consider the operad $(\\sA ss,\\cdot)$: in this case we know from Example \\ref{ex:ass} that $\\op{Ass}_\\circ(\\sA ss)=(\\sA ss\\bullet\\sA ss)^!=(\\sA ss\\times\\sA ss)^!=\\sA ss+\\sA ss$ (with the functors $-\\times-$ and $-+-$ as in Definition \\ref{def:+x}), and in fact we find the relations\n\t\\[ (_x\\to\\prec\\xrightarrow{y})\\prec\\to_z-_x\\to\\prec(\\xrightarrow{y}\\prec \\to_z)=\\xrightarrow{(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)}=0, \\]\n\t\\[ (_x\\to\\succ\\xrightarrow{y})\\succ\\to_z-_x\\to\\succ(\\xrightarrow{y}\\succ \\to_z)=\\xrightarrow{z\\cdot(y\\cdot x)-(z\\cdot y)\\cdot x}=0, \\] \n\tin the operad $(\\op{Ass}_\\circ(\\sA ss),\\prec,\\succ)$.\n\t\n\tWe consider the operad $(pre\\sL ie, \\cdot)$. The right pre-Lie relation $(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y+x\\cdot(z\\cdot y)=0$ can't be splitted as in Lemma \\ref{lem:relwhite}, for instance because there is no argument remaining inside every parenthesis. As the previous remark illustrates, this alone wouldn't be enough to conclude $\\op{Ass}_\\circ(pre\\sL ie)=\\sM ag_{2,0,0}$: on the other hand, this is true since it agrees with the computation of the Koszul dual $\\op{Ass}_\\bullet(\\sP erm)=nil\\sA ss\\times nil\\sA ss$ in Example \\ref{ex:perm}.\n\t\n\tWe consider the operad $(\\sL eib,\\cdot)$. Again, the right Leibniz relation $(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y=0$ can't be splitted as in Lemma \\ref{lem:relwhite}, on the other hand, this imply the relation $x\\cdot(y\\cdot z+z\\cdot y) = (x\\cdot y)\\cdot z-(x\\cdot z)\\cdot y +(x\\cdot z)\\cdot y-(x\\cdot y)\\cdot z=0$ in $\\sL eib$: for the latter, we find the splittings\n\t\\[ _x\\to\\prec (\\xrightarrow{y}\\prec\\to_z + \\xrightarrow{y}\\succ\\to_z ) =\\xrightarrow{x\\cdot(y\\cdot z+z\\cdot y)}=0,\\]\n\t\\[ (_x\\to\\prec\\xrightarrow{y} + \\,_x\\to\\succ\\xrightarrow{y})\\succ\\to_z = \\xrightarrow{z\\cdot (x\\cdot y + y\\cdot x)}=0, \\]\n\thence the relations $\\alpha\\prec(\\beta\\prec\\gamma+\\beta\\succ\\gamma)=0=(\\alpha\\prec\\beta +\\alpha\\succ\\beta)\\succ\\gamma$ in $(\\op{Ass}_\\circ(\\sL eib),\\prec,\\succ)$. We leave to the reader to check that this is a generating set of relations, for instance by computing the Koszul dual $\\op{Ass}_\\bullet(\\sZ inb)$ with the method of the previous section.\n\n As a final example, we consider $(\\sZ inb,\\cdot)$. In this case, we know form Example \\ref{ex:leib} that $\\op{Ass}_\\circ(\\sZ inb)=(\\sA ss\\bullet\\sL eib)^!=di\\sA ss^!=\\sD end$: in fact, we get the dendriform relations on the operad $(\\op{Ass}_\\circ(\\sZ inb),\\prec,\\succ)$, corresponding to the splittings as in Lemma \\ref{lem:relwhite} (notice that the relation $(x\\cdot y)\\cdot z-(x\\cdot z)\\cdot y=0$ holds in $\\sZ inb$)\n \\[ _x\\to\\prec( \\xrightarrow{y}\\prec\\to_z+\\xrightarrow{y}\\succ\\to_z)-(_x\\to\\prec\\xrightarrow{y})\\prec\\to_z=\\xrightarrow{x\\cdot(y\\cdot z+ z\\cdot y)-(x\\cdot y)\\cdot z}=0\\]\n \\[ (_x\\to\\prec\\xrightarrow{y}+\\,_x\\to\\succ\\xrightarrow{y})\\succ\\to_z -\\,_x\\to\\succ(\\xrightarrow{y}\\succ\\to_z)=\\xrightarrow{z\\cdot(x\\cdot y+y\\cdot x)-(z\\cdot y)\\cdot x}=0\\]\n \\[ (_x\\to\\succ\\xrightarrow{y})\\prec\\to_z -\\, _x\\to\\succ(\\xrightarrow{y}\\prec\\to_z)=\\xrightarrow{(y\\cdot x)\\cdot z-(y\\cdot z)\\cdot x}=0.\\]\n\\end{example}\n\n\\begin{theorem}\\label{th:asswhite} There is a natural isomorphism $\\op{Ass}_\\circ(-)\\xrightarrow{\\cong}\\sA ss\\circ-$ of functors $\\mathbf{Op}\\to\\mathbf{Op}$.\n\\end{theorem}\n\\begin{proof} We denote by $(\\sA ss,\\cup)$ the associative operad with its generating product. Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\sA ss\\circ\\Oh$ is generated by the tensor product operations (which are non-symmetric) $\\cup\\otimes\\cdot_i, \\cup\\otimes\\cdot_i^{op}$, $\\cup\\otimes\\bullet_j$, $\\cup\\otimes[-,-]_k$, together with the larger set of relations holding in the tensor product $A\\otimes V$ of a generic $\\sA ss$-algebra $(A,\\cup)$ and a generic $\\Oh$-algebra $(V,\\cdot_i,\\bullet_j,[-,-]_k)$. The isomorphism of operads $\\op{Ass}_\\circ(\\Oh)\\leftrightarrow \\sA ss\\circ\\Oh$ is given by $\\prec_i\\leftrightarrow \\cup\\otimes\\cdot_i$, $\\succ_i\\leftrightarrow\\cup\\otimes\\cdot_i^{op}$, $\\circ_j\\leftrightarrow\\cup\\otimes\\bullet_j$ and finally $\\ast_k\\leftrightarrow\\cup\\otimes[-,-]_k$. This defines a morphism of operads $\\sA ss\\circ\\Oh\\to\\op{Ass}_\\circ(\\Oh)$: in fact, by construction of the cup products $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, these satisfy the relations of $\\sA ss\\circ\\Oh$-algebra in the tensor product $C^*(\\Delta_1;V)$ of the $\\sA ss$-algebra $(C^*(\\Delta_1;\\mathbb{K}),\\cup)$ (where we may forget the gradation) and the generic $\\Oh$-algebra $V$. Conversely, we must show that if we evaluate a generating relation $R(-,-,-)=0$ of $\\op{Ass}_\\circ(\\Oh)$, given as in Lemma \\ref{lem:relwhite}, in the operations $\\cup\\otimes\\cdot_i, \\cup\\otimes\\cdot_i^{op}$, $\\cup\\otimes\\bullet_j$, $\\cup\\otimes[-,-]_k$, we get a relation of the operad $\\sA ss\\circ\\Oh$: but this is also clear, since given a generic $\\sA ss$-algebra $(A,\\cup)$ we find $R(a\\otimes x,b\\otimes y,c\\otimes z)= (a\\cup b\\cup c)\\otimes R'(x,y,z)=0$, $\\forall a\\otimes x,b\\otimes y,c\\otimes z\\in A\\otimes V$, where $a\\cup b\\cup c:=(a \\cup b)\\cup c=a\\cup(b\\cup c)$.\n\\end{proof}\n\nThe construction of the functor $\\op{Perm}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$ is similar. Given a generic $\\Oh$-algebra $V$, the cup products on the complex $C^*(\\Delta_1,v_l;V)$ are defined by the same formulas as in the associative case.\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\op{Perm}_\\circ(\\Oh)$ is generated by non-symmetric operations $\\prec_i,\\succ_i,\\circ_j,\\ast_k$, together with the larger set of relations making $C^*(\\Delta_1,v_l;V)$ with the cup products a local dg $\\op{Perm}_\\circ(\\Oh)$-algebra for every $\\Oh$-algebra $V$.\\end{definition}\n\nWe want to describe a generating set of relations of $\\op{Perm}_\\circ(\\Oh)$ as in the associative case. We denote the elements of the symmetric group $S_3=\\{\\id,(123),(132),(12),(13),(23)\\}$ according to their cycle decomposition. Given a terniary operation $R(-,-,-):C^*(\\Delta_1,v_l;V)^{\\otimes 3}\\to C^*(\\Delta_1,v_l;V)$ in the cup products, we write $R=R_1+R_2+R_3$, where $R_1:=R_{\\id}+R_{(23)}$, $R_2:=R_{(123)}+ R_{(12)}$ and $R_3:=R_{(132)}+R_{(13)}$. By Lemma \\ref{lem:relpl}, the relation $R(-,-,-)=0$ holds in $C^*(\\Delta_1,v_l;V)$ if and only if \n\\[ R(\\xrightarrow{x},\\to_y,\\to_z)=R(\\to_x,\\xrightarrow{y},\\to_z)=R(\\to_x,\\to_y,\\xrightarrow{z})=0\\]\nfor all $x,y,z\\in V$. We claim that this is true if and only if so is $R_i(-,-,-)=0$ for $i=1,2,3$: again, we limit ourselves to show $R(-,-,-)=0\\solose R_1(-,-,-)=0$. This time, the point is that the only way to form a non-vanishing triple product out of $\\xrightarrow{1},\\to_1,\\to_1\\in (C^*(\\Delta_1,v_l;\\mathbb{K}),\\cup)$ is by putting the degree one cochain $\\xrightarrow{1}$ on the left: this implies $R_1(\\to_x,\\xrightarrow{y},\\to_z)=R_1(\\to_x,\\to_y,\\xrightarrow{z})=0$, whereas $R_1(\\xrightarrow{x},\\to_y,\\to_z)=R(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{R'(x,y,z)}=0$ by hypothesis, where $R'(-,-,-)=0$ is a relation of $\\Oh$ since $V$ is generic, so we may conclude again by Lemma \\ref{lem:relpl}. To sum up\n\\begin{lemma}\\label{lem:relpm} The operad $\\op{Perm}_\\circ(\\Oh)$ has a generating set of relations $R(-,-,-)=0$ of the form $R=R_{\\id}+R_{(23)}$, consisting of all the possible splittings, as in the following \n\t\\[ R(\\xrightarrow{x},\\to_y,\\to_z)=\\xrightarrow{R'(x,y,z)}=0,\\]\n\tof a relation $R'(-,-,-)=0$ in $\\Oh$ via the cup products on $C^*(\\Delta_1,v_l;V)$.\\end{lemma}\n\n\\begin{theorem}\\label{th:permwhite} There is a natural isomorphism $\\op{Perm}_\\circ(-)\\xrightarrow{\\cong}\\sP erm\\circ-$ of functors $\\mathbf{Op}\\to\\mathbf{Op}$.\n\\end{theorem}\n\\begin{proof}\n\tThis is done as in Theorem \\ref{th:asswhite} by noticing that the relations of $\\sP erm\\circ\\Oh$-algebra are satisfied by the cup products on the tensor product $C^*(\\Delta_1,v_l;V)$ of the right permutative algebra $(C^*(\\Delta_1,v_l;\\mathbb{K}),\\cup)$ and a generic $\\Oh$-algebra $V$. Conversely, given a generating relation $R(-,-,-)=0$ of $\\op{Perm}_\\circ(\\Oh)$ as in the previous lemma, it is straightforward to check that this holds more in general in the tensor product $A\\otimes V$ of a generic $\\sP erm$-algebra $A$ and a generic $\\Oh$-algebra $V$. \n\\end{proof}\n\\begin{example} We consider the operad $(\\sL ie,[-,-])$: by the previous theorem $\\op{Perm}_\\circ(\\sL ie)=\\sP erm\\circ\\sL ie=\\sL eib$, where the right Leibniz relation corresponds to the splitting as in Lemma \\ref{lem:relpm}\n\t\\[ (\\xrightarrow{x}\\ast \\to_y)\\ast\\to_z-\\xrightarrow{x}\\ast(\\to_y\\ast\\to_z)-(\\xrightarrow{x}\\ast\\to_z)\\ast\\to_y=\\xrightarrow{[[x,y],z]-[x,[y,z]]-[[x,z],y]}=0.\\]\n\nWe consider the operad $(\\sP ois,\\bullet,[-,-])$. This is generated by a right permutative product $\\circ$ and a right Leibniz product $\\ast$: we have the following splittings of the Poisson identity\n\\begin{multline*}\n0=\\xrightarrow{[x\\bullet y,z]-x\\bullet [y,z]-[x,z]\\bullet y} = \\left( \\xrightarrow{x}\\circ\\to_y\\right)\\ast\\to_z -\\xrightarrow{x}\\circ(\\to_y\\ast\\to_z)-\\left( \\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to{y}=\\\\= \\left( \\xrightarrow{x}\\circ\\to_y\\right)\\ast\\to_z +\\xrightarrow{x}\\circ(\\to_z\\ast\\to_y)-\\left( \\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to{y}, \\end{multline*}\n\\begin{multline*} 0=\\xrightarrow{[x,y\\bullet z]-[x,y]\\bullet z-[x,z]\\bullet y}= \\xrightarrow{x}\\ast\\left( \\to_y\\circ\\to_z\\right) -\\left(\\xrightarrow{x}\\ast\\to_y\\right)\\circ\\to_z-\\left(\\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to_y =\\\\ = \\xrightarrow{x}\\ast\\left( \\to_z\\circ\\to_y\\right) -\\left(\\xrightarrow{x}\\ast\\to_y\\right)\\circ\\to_z-\\left(\\xrightarrow{x}\\ast\\to_z\\right)\\circ\\to_y,\\end{multline*}\ncorresponding to the three independent relations\n\\[ (\\alpha\\circ \\beta)\\ast\\gamma=\\alpha\\circ(\\beta\\ast\\gamma)+(\\alpha\\ast\\gamma)\\circ\\beta,\\quad\\alpha\\ast(\\beta\\circ\\gamma)=(\\alpha\\ast\\beta)\\circ\\gamma+(\\alpha\\ast\\gamma)\\circ\\beta,\\quad \\alpha\\circ(\\beta\\ast\\gamma)=-\\alpha\\circ(\\gamma\\ast\\beta), \\]\nin the operad $\\op{Perm}_\\circ(\\sP ois)$. In fact, these are the relations defining Aguiar's operad $\\sP erm\\circ\\sP ois=(pre\\sL ie\\bullet\\sP ois)^!$ of dual pre-Poisson algebras \\cite{A,U}.\n\nWe consider the operad $(\\sA ss,\\cdot)$: then we know $\\sP erm\\circ\\sA ss=di\\sA ss$. In this case, the relations we find as in Lemma \\ref{lem:relpm} turn out to be the diassociative relations for the generating products $\\prec$, $\\curlyeqsucc:=\\succ^{op}$ of $\\op{Ass}_\\circ(\\Oh)$: in fact, we find\n\\begin{equation*} 0=\\xrightarrow{(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)}=(\\xrightarrow{x}\\prec\\to_y)\\prec\\to_z -\\xrightarrow{x}\\prec(\\to_y\\prec\\to_z)=(\\xrightarrow{x}\\prec\\to_y)\\prec\\to_z -\\xrightarrow{x}\\prec(\\to_z\\succ\\to_y), \\end{equation*}\n\\begin{equation*} 0=\\xrightarrow{z\\cdot(y\\cdot x)-(z\\cdot y)\\cdot x}=(\\xrightarrow{x}\\succ\\to_y)\\succ\\to_z -\\xrightarrow{x}\\succ(\\to_y\\succ\\to_z)=\\\\=(\\xrightarrow{x}\\succ\\to_y)\\succ\\to_z -\\xrightarrow{x}\\succ(\\to_z\\prec\\to_y), \\end{equation*}\n\\[ 0=\\xrightarrow{(y\\cdot x)\\cdot z-y\\cdot(x\\cdot z)} =(\\xrightarrow{x}\\succ\\to_y)\\prec\\to_x-(\\xrightarrow{x}\\prec\\to_z)\\succ\\to_y.\\]\nThe reader may compare this with the computation of $\\op{preLie}_\\bullet(\\sA ss)$ in Example \\ref{ex:ass}, as well as what we said in Remark \\ref{rem:curlyeqprec}.\n\nWe consider the operad $(pre\\sL ie,\\cdot)$. We split the right pre-Lie identity as \n\\[0=\\xrightarrow{(x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-(x\\cdot z)\\cdot y +x\\cdot(z\\cdot y)}= (\\xrightarrow{x}\\prec\\to_y)\\prec\\to_z-\\xrightarrow{x}\\prec\\to_{y\\cdot z}-(\\xrightarrow{x}\\prec\\to_z)\\prec\\to_y+\\xrightarrow{x}\\prec\\to_{z\\cdot y},\\]\nSince we may split $\\to_{y\\cdot z}$ both as $\\to_y\\prec\\to_z=\\to_{y\\cdot z}=\\to_z\\succ\\to_y$, and similarly for $\\to_{z\\cdot y}$, we get the two independent relations $(\\alpha\\prec\\beta)\\prec\\gamma-\\alpha\\prec(\\beta\\prec\\gamma)=(\\alpha\\prec\\gamma)\\prec\\beta-\\alpha\\prec(\\gamma\\prec\\beta)$ and $\\alpha\\prec(\\beta\\prec\\gamma)=\\alpha\\prec(\\gamma\\succ\\beta)$ in the operad $\\op{Perm}_\\circ(pre\\sL ie)$. We also have the splitting\n\\[0=\\xrightarrow{z\\cdot (y\\cdot x)-(z\\cdot y)\\cdot x-z\\cdot(x\\cdot y) +(z\\cdot x)\\cdot y}= (\\xrightarrow{x}\\succ\\to_y)\\succ\\to_z-\\xrightarrow{x}\\succ\\to_{z\\cdot y}-(\\xrightarrow{x}\\prec\\to_y)\\succ\\to_z+(\\xrightarrow{x}\\succ\\to_{z})\\prec\\to_y,\\]\ngiving the relations $(\\alpha\\succ\\beta)\\succ\\gamma-\\alpha\\succ(\\beta\\succ\\gamma)=(\\alpha\\prec\\beta)\\succ\\gamma-(\\alpha\\succ\\gamma)\\prec\\beta$ and $\\alpha\\succ(\\beta\\succ\\gamma)=\\alpha\\succ(\\gamma\\prec\\beta)$. We leave to the reader to check that this is a generating set of relations, and in fact the operad $(\\op{Perm}_\\circ(pre\\sL ie),\\prec,\\curlyeqsucc:=\\succ^{op})$ with the relations\n\\begin{eqnarray} \\nonumber\n(\\alpha\\prec\\beta)\\prec\\gamma-\\alpha\\prec(\\beta\\prec\\gamma)\\:=\\:(\\alpha\\prec\\gamma)\\prec\\beta-\\alpha\\prec(\\gamma\\prec\\beta), & \\alpha\\prec(\\beta\\prec\\gamma)\\:=\\:\\alpha\\prec(\\beta\\curlyeqsucc\\gamma), \\\\ \\nonumber (\\alpha\\curlyeqsucc\\beta)\\curlyeqsucc\\gamma-\\alpha\\curlyeqsucc(\\beta\\curlyeqsucc\\gamma)\\:=\\:(\\alpha\\curlyeqsucc\\gamma)\\prec\\beta-\\alpha\\curlyeqsucc(\\gamma\\prec\\beta), & (\\alpha\\curlyeqsucc\\beta)\\curlyeqsucc\\gamma \\:=\\:(\\alpha\\prec\\beta)\\curlyeqsucc\\gamma,\n\\end{eqnarray}\nis the Koszul dual of the operad $\\op{preLie}_\\bullet(\\sP erm)$ from Example \\ref{ex:perm}.\n\\end{example}\n\nWe finally come to the definition of $\\op{Lie}_\\circ(-):\\mathbf{Op}\\to\\mathbf{Op}$. In this case, we consider the complex $C^*(\\Delta_1;\\mathbb{K})$ as a $\\sL ie$-algebra via the bracket $[-,-]=\\cup-\\cup^{op}$, then given a generic $\\Oh$-algebra $(V,\\cdot_i,\\bullet_j,\\ast_k)$ we shall call the tensor product operations $\\star_i := [-,-]\\otimes\\cdot_i=(\\cup-\\cup^{op})\\otimes\\cdot_i=\\prec_i-\\succ_i^{op}$, $\\{-,-\\}_j:=[-,-]\\otimes\\bullet_j=\\circ_j-\\circ_j^{op}$, $\\circledast_k:=[-,-]\\otimes[-,-]_k=\\ast_k+\\ast_k^{op}$ the cup brackets on $C^*(\\Delta_1;V)$.\n\\begin{definition} Given an operad $(\\Oh,\\cdot_i,\\bullet_j,[-,-]_k)$ as usual, the operad $\\op{Lie}_\\circ(\\Oh)$ is generated by non-symmetric operations $\\star_i$, anti-symmetric operations $\\{-,-\\}_j$ and symmetric operations $\\circledast_k$ together with the larger set of relations making $C^*(\\Delta_1;V)$ with the cup brackets a local dg $\\op{Lie}_\\circ(\\Oh)$-algebra for every $\\Oh$-algebra $V$.\\end{definition}\n\\begin{theorem}\\label{th:liewhite} There is a natural isomorphism $\\op{Lie}_\\circ(-)\\xrightarrow{\\cong}\\sL ie\\circ-$ of functors $\\mathbf{Op}\\to\\mathbf{Op}$.\n\\end{theorem}\n\\begin{proof} This follows from Theorem \\ref{th:asswhite}: in fact, by construction $\\op{Lie}_\\circ(\\Oh)$ is the suboperad of $\\op{Ass}_\\circ(\\Oh)$ generated by the operations $\\star_i=\\prec_i-\\succ_i^{op}$, $\\{-,-\\}_j=\\circ_j-\\circ_j^{op}$, $\\circledast_k=\\ast_k+\\ast_k^{op}$, and similarly $\\sL ie\\circ\\Oh$ is the suboperad of $\\sA ss\\circ\\Oh$ generated by the operations $(\\cup-\\cup^{op})\\otimes\\cdot_i$, $(\\cup-\\cup^{op})\\otimes\\bullet_j$, $(\\cup-\\cup^{op})\\otimes[-,-]_k$.\n\\end{proof}\n\n \n\\begin{example}\\label{ex:liewhite} The computations from the previous section, together with the above theorem and the fact that $\\sL ie\\circ\\Oh=(\\sC om\\bullet\\Oh^!)^!$, imply for instance $\\op{Lie}_\\circ(\\sA ss)=\\op{Lie}_\\circ(pre\\sL ie)=\\sM ag_{1,0,0}$, $\\op{Lie}_\\circ(\\sL ie)=\\sM ag_{0,1,0}$, $\\op{Lie}_\\circ(\\sP ois)=\\sL ie+\\sM ag_{0,1,0}$, $\\op{Lie}_\\circ(\\sP erm)=\\sL eib$, $\\op{Lie}_\\circ(\\sZ inb)=pre\\sL ie$. To illustrate the latter, we show that the (right) pre-Lie relation holds in the tensor product $L\\otimes V$ of a generic Lie algebra $(L,[-,-])$ and a generic $\\sZ inb$-algebra $(V,\\cdot)$, equipped with the tensor product operation $\\star:=[-,-]\\otimes\\cdot$. The associator of $\\star$ is given by $A_\\star(l\\ten x,m\\ten y,n\\ten z)=[[l,m],n]\\ten (x\\cdot y)\\cdot z\\,-\\,[l,[m,n]]\\ten x\\cdot(y\\cdot z)$: to show that this is graded symmetric in the last two arguments, we compute\n\t\\[ [[l,m],n]\\ten (x\\cdot y)\\cdot z\\,-\\,[l,[m,n]]\\ten x\\cdot(y\\cdot z)\\,-\\,[[l,n],m]\\ten (x\\cdot z)\\cdot y\\,+\\,[l,[n,m]]\\ten x\\cdot(z\\cdot y) = \\]\n\t\\[ =[l,[m,n]]\\ten\\left( (x\\cdot y)\\cdot z-x\\cdot(y\\cdot z)-x\\cdot(z\\cdot y) \\right)\\,+\\,[[l,n],m]\\ten\\left( (x\\cdot y)\\cdot z -(x\\cdot z)\\cdot y \\right)\\:=\\:0. \\] \n\t\n\n\t\n\t\n\\end{example}\n\nWe are finally ready to complete the proof of Theorem \\ref{th:black}: this will follow from the following theorem and theorems \\ref{th:asswhite}, \\ref{th:permwhite}, \\ref{th:liewhite}.\n\\begin{theorem}\\label{th:adj} The following $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$, $\\xymatrix{\\op{Com}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Lie}_\\circ(-)\\ar@<2pt>[l]}$, $\\xymatrix{\\op{preLie}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Perm}_\\circ(-)\\ar@<2pt>[l]}$ are pairs of adjoint functors.\n\t\\end{theorem}\n\n\\begin{proof} We consider the case of $\\xymatrix{\\op{Ass}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Ass}_\\circ(-)\\ar@<2pt>[l]}$ in detail.\n\t\nSo far we used overlapping notations for the generating sets of operations of $\\op{Ass}_\\bullet(\\Oh)$ and $\\op{Ass}_\\circ(\\Oh)$: this isn't practical to prove adjointness, thus, only for this proof, we will have to change notations; moreover, we will use slightly different generating sets than the ones we used before. We consider first the case of a non-symmetric operation $\\cdot_i$ of $\\Oh$. Corresponding to $\\cdot_i$ there are two generating operations of $\\op{Ass}_\\bullet(\\Oh)$ which we denote by $\\underline{\\cdot_i}$ and $\\overline{\\cdot_i}$ respectively: we use the generating set of operations from Remark \\ref{rem:curlyeqprec}, explicitly, $\\underline{\\cdot_i}$ and $\\overline{\\cdot_i}$ are defined by the formulas $_x\\to\\cdot_i\\xrightarrow{y}\\:=\\:\\xrightarrow{x\\underline{\\cdot_i} y}\\:=\\:\\xrightarrow{x}\\cdot_i\\to_y$ and $\\to_x\\cdot_i\\xrightarrow{y}\\:=\\:\\xrightarrow{x\\overline{\\cdot_i}y}\\:=\\:\\xrightarrow{x}\\cdot_i\\,_y\\to$ (with the previous notations from remark \\ref{rem:curlyeqprec} we have $\\underline{\\cdot_i}=\\prec_i$ and $\\overline{\\cdot_i}=\\curlyeqsucc_i=-\\succ_i^{op}$). Likewise, corresponding to $\\cdot_i$ there are two generating operations of $\\op{Ass}_\\circ(\\Oh)$ which we denote by $\\cup\\otimes\\cdot_i$ and $\\cup^{op}\\otimes\\cdot_i$ respectively: explicitly, these are defined by the formulas $_x\\to(\\cup\\otimes\\cdot_i)\\xrightarrow{y}=\\xrightarrow{x\\cdot_i y}=\\xrightarrow{x}(\\cup\\otimes\\cdot_i)\\to_y$ and $\\to_x(\\cup^{op}\\otimes\\cdot_i)\\xrightarrow{y}=\\xrightarrow{x\\cdot_i y}=\\xrightarrow{x}(\\cup^{op}\\otimes\\cdot_i)\\,_y\\to$ (with the previous notations from this section we have $\\cup\\otimes\\cdot_i=\\prec_i$ and $\\cup^{op}\\otimes\\cdot_i=\\succ_i^{op}$). Finally, corresponding to $\\cdot_i$ there are four generating operations of the operad $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$, which we denote by $\\cup\\otimes\\underline{\\cdot_i}$, $\\cup^{op}\\otimes\\underline{\\cdot_i}$, $\\cup\\otimes\\overline{\\cdot_i}$ and $\\cup^{op}\\otimes\\overline{\\cdot_i}$ respectively, and four generating operations of $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$, which we denote by $\\underline{\\cup\\otimes\\cdot_i}$, $\\overline{\\cup\\otimes\\cdot_i}$, $\\underline{\\cup^{op}\\otimes\\cdot_i}$ and $\\overline{\\cup^{op}\\otimes\\cdot_i}$ respectively. Similarly, to a generating symmetric operation $\\bullet_j$ of $\\Oh$ correspond a generating (non-symmetric) operation $\\underline{\\bullet_j}$ of $\\op{Ass}_\\bullet(\\Oh)$ and a generating (non-symmetric) operation $\\cup\\otimes\\bullet_j$ of $\\op{Ass}_\\circ(\\Oh)$, as well as two generating (non-symmetric) operations $\\cup\\otimes\\underline{\\bullet_j}$, $\\cup^{op}\\otimes\\underline{\\bullet_j}$ of $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$ and two generating (non-symmetric) operations $\\underline{\\cup\\otimes\\bullet_j}$, $\\overline{\\cup\\otimes\\bullet_j}$ of $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$. Finally, to a generating antisymmetric operation $[-,-]_k$ of $\\Oh$ correspond generating (non-symmetric) operations $\\underline{[-,-]_k}$ and $\\cup\\otimes[-,-]_k$ of $\\op{Ass}_\\bullet(\\Oh)$ and $\\op{Ass}_\\circ(\\Oh)$ respectively, as well as generating operations $\\cup\\otimes\\underline{[-,-]_k}$, $\\cup^{op}\\otimes\\underline{[-,-]_k}$ of $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$ and $\\underline{\\cup\\otimes[-,-]_k}$, $\\overline{\\cup\\otimes[-,-]_k}$ of $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$ respectively.\n\t\nHaving established the previous notations, we will prove the theorem by explicitly exihibiting the unit $\\varepsilon_\\Oh:\\Oh\\to\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$ and the counit $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ of the adjunction. The construction of the counit is implicit in the very definition of the functors $\\op{Ass}_\\circ(-)$ and $\\op{Ass}_\\bullet(-)$: given an $\\Oh$-algebra $V$, the tensor product operations induce a local dg $\\op{Ass}_\\circ(\\Oh)$-algebra structure on the complex of $V$-valued cochains $C^*(\\Delta_1;V)=C^*(\\Delta_1;\\mathbb{K})\\otimes V$, but by definition this is the same as an $\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))$-algebra structure on the space $V$. Unraveling the definitions, we find that the counit $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ is explicitly given by \n\\[ \\underline{\\cup\\otimes\\cdot_i},\\:\\overline{\\cup^{op}\\otimes\\cdot_i}\\to\\cdot_i,\\qquad\\overline{\\cup\\otimes\\cdot_i},\\: \\underline{\\cup^{op}\\otimes\\cdot_i}\\to0, \\] \n\\[ \\underline{\\cup\\otimes\\bullet_j}\\to\\bullet_j,\\qquad\\overline{\\cup\\otimes\\bullet_j}\\to0,\\qquad\\underline{\\cup\\otimes[-,-]_k}\\to[-,-]_k,\\qquad\\overline{\\cup\\otimes[-,-]_k}\\to0. \\]\nAs already remarked, this defines a morphism of operads $\\mu_{\\Oh}:\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh))\\to\\Oh$ by construction of the functors $\\op{Ass}_\\circ(-)$ and $\\op{Ass}_\\bullet(-)$. It remains to define the unit $\\varepsilon_\\Oh:\\Oh\\to\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$, this is explicitly given by\n\\[\\cdot_i\\to \\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i}, \\qquad\\bullet_j\\to\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op},\\qquad[-,-]_k\\to\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op}. \\]\nWe have to show that this is a morphism of operads. Given a generic $\\op{Ass}_\\bullet(\\Oh)$-algebra $(V,\\underline{\\cdot_i},\\overline{\\cdot_i},\\underline{\\bullet_j},\\underline{[-,-]_k})$, the complex $C^*(\\Delta_1;V)$ carries both an $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$-algebra structure via the tensor product operations and an $\\Oh$-algebra structure $(C^*(\\Delta_1;V),\\cdot_i,\\bullet_j,[-,-]_k)$ by definition of $\\op{Ass}_\\bullet(-)$. An easy verification shows $\\alpha(\\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i})\\beta=\\alpha\\cdot_i\\beta$ for all $\\alpha,\\beta\\in C^*(\\Delta_1;V)$. For instance,\n\\[ \\to_x (\\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i}) \\xrightarrow{y}=\\to_x(\\cup^{op}\\otimes\\overline{\\cdot_i})\\xrightarrow{y}=\\xrightarrow{x\\overline{\\cdot_i} y}=\\to_x\\cdot_i\\xrightarrow{y}.\\]\nSimilarly, one verifies $\\alpha(\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op})\\beta=\\alpha\\bullet_j\\beta$ and $\\alpha(\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op})\\beta=[\\alpha,\\beta]_k$ for all $\\alpha,\\beta\\in C^*(\\Delta_1;V)$. Finally, given a relation $R(-,-,-)=0$ of $\\Oh$, this induces a relation $R'(-,-,-)=0$ of $\\op{Ass}_\\bullet(\\Oh)$ as in Lemma \\ref{lem:rel}\n\\[ R(_x\\to,\\xrightarrow{y},\\to_z)=\\xrightarrow{R'(x,y,z)}=0.\\]\nOn the other hand, by the previous considerations $R(-,-,-)$ in the left hand side can be computed equivalently either in the products $\\cdot_i,\\bullet_j,[-,-]_k$ of the $\\Oh$-algebra structure or in the products $\\cup\\otimes\\underline{\\cdot_i}+\\cup^{op}\\otimes\\overline{\\cdot_i}, \\:\\:\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op},\\:\\:\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op}$ of the $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$-algebra structure, which shows that $\\varepsilon_\\Oh$ sends an $\\Oh$-algebra relation to an $\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh))$-algebra relation, and is thus a well defined morphism of operads.\n\nTo complete the proof, it remains to show that $\\varepsilon_{-}$ and $\\mu_{-}$ satisfy the conditions to be the unit and the counit of an adjunction. More precisely, we have to show that the compositions\n\\[ \\op{Ass}_\\bullet(\\Oh)\\xrightarrow{\\op{Ass}_\\bullet(\\varepsilon_\\Oh)}\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\Oh)))\\xrightarrow{\\mu_{\\op{Ass}_\\bullet(\\Oh)}}\\op{Ass}_\\bullet(\\Oh), \\]\n\\[ \\op{Ass}_\\circ(\\Oh)\\xrightarrow{\\varepsilon_{\\op{Ass}_\\circ(\\Oh)}}\\op{Ass}_\\circ(\\op{Ass}_\\bullet(\\op{Ass}_\\circ(\\Oh)))\\xrightarrow{\\op{Ass}_\\circ(\\mu_\\Oh)}\\op{Ass}_\\circ(\\Oh),\\]\nare the respective identities. We consider the first one: notice that given a morphism $f:\\Oh\\to\\sP$ of operads, the morhism $\\op{Ass}_\\bullet(f)$ is defined by $\\op{Ass}_\\bullet(f)(\\underline{\\cdot_i})=\\underline{f(\\cdot_i)}$, $\\op{Ass}_\\bullet(f)(\\overline{\\cdot_i})=\\overline{f(\\cdot_i)}$, $\\op{Ass}_\\bullet(f)(\\underline{\\bullet_j})=\\underline{f(\\bullet_j)}$ and $\\op{Ass}_\\bullet(f)(\\underline{[-,-]_k})=\\underline{f([-,-]_k)}$. Now it is easy to compute the first composition using the previous formulas (for a non-symmetric operation $\\#$ we notice that $\\underline{(\\#^{op})}=(\\overline{\\#})^{op}$, in the following computation we apply this for $\\#=\\cup\\otimes\\underline{\\bullet_j}$ and $\\#=\\cup\\otimes\\underline{[-,-]_k}$)\n\\[ \\underline{\\cdot_i}\\to\\underline{\\cup\\otimes\\underline{\\cdot_i}}+\\underline{\\cup^{op}\\otimes\\overline{\\cdot_i}}\\to\\underline{\\cdot_i}+0,\\qquad\\overline{\\cdot_i}\\to\\overline{\\cup\\otimes\\underline{\\cdot_i}}+\\overline{\\cup^{op}\\otimes\\overline{\\cdot_i}}\\to0+\\overline{\\cdot_i},\\]\\[ \\underline{\\bullet_j}\\to\\underline{\\cup\\otimes\\underline{\\bullet_j}+(\\cup\\otimes\\underline{\\bullet_j})^{op}}=\\underline{\\cup\\otimes\\underline{\\bullet_j}}+(\\overline{\\cup\\otimes\\underline{\\bullet_j}})^{op}\\to\\underline{\\bullet_j}+0, \\]\\[ \\underline{[-,-]_k}\\to\\underline{\\cup\\otimes\\underline{[-,-]_k}-(\\cup\\otimes\\underline{[-,-]_k})^{op}}=\\underline{\\cup\\otimes\\underline{[-,-]_k}}-(\\overline{\\cup\\otimes\\underline{[-,-]_k}})^{op}\\to\\underline{[-,-]_k}-0. \\]\nNext we consider the second composition. We have $\\op{Ass}_\\circ(f)(\\cup\\otimes\\cdot_i)=\\cup\\otimes f(\\cdot_i)$, $\\op{Ass}_\\circ(f)(\\cup^{op}\\otimes \\cdot_i)=\\cup^{op}\\otimes f(\\cdot_i)$, and similarly for the other cases. As desired, the second composition is\n\\[\\cup\\otimes\\cdot_i\\to\\cup\\otimes\\underline{\\cup\\otimes\\cdot_i} +\\cup^{op}\\otimes\\overline{\\cup\\otimes\\cdot_i}\\to\\cup\\otimes\\cdot_i+0,\\]\\[ \\cup^{op}\\otimes\\cdot_i\\to\\cup\\otimes\\underline{\\cup^{op}\\otimes\\cdot_i} +\\cup^{op}\\otimes\\overline{\\cup^{op}\\otimes\\cdot_i}\\to0+\\cup^{op}\\otimes\\cdot_i, \\]\n\\[\\cup\\otimes\\bullet_j\\to\\cup\\otimes\\underline{\\cup\\otimes\\bullet_j} +\\cup^{op}\\otimes\\overline{\\cup\\otimes\\bullet_j}\\to\\cup\\otimes\\bullet_j+0,\\]\n\\[ \\cup\\otimes[-,-]_k\\to\\cup\\otimes\\underline{\\cup\\otimes[-,-]_k} +\\cup^{op}\\otimes\\overline{\\cup\\otimes[-,-]_k}\\to\\cup\\otimes[-,-]_k+0.\\] \n\nThe remaining cases of $\\xymatrix{\\op{Com}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Lie}_\\circ(-)\\ar@<2pt>[l]}$ and $\\xymatrix{\\op{preLie}_\\bullet(-):\\mathbf{Op}\\ar@<2pt>[r]& \\mathbf{Op}:\\op{Perm}_\\circ(-)\\ar@<2pt>[l]}$ are proved by the same argument: in the former, we notice that we may identify $\\op{Com}_\\bullet(\\op{Lie}_\\circ(\\Oh))$, $\\Oh$ and $\\op{Lie}_\\circ(\\op{Com}_\\bullet(\\Oh))$ with quotients of the same free operad $(\\sM ag_{p,q,r},\\cdot_i,\\bullet_j,[-.-]_k)$ in such a way that the unit and the counit are the identities on the generating operations.\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe work over the complex number field $\\C$. A K3 surface is a compact simply connected, in the classical topology, smooth complex surface with nowhere vanishing global holomorphic $2$-form. An Enriques surface is a smooth complex surface which is isomorphic to a non-trivial \\'etale quotient of a K3 surface. The quotient map is necessarily of degree two and every Enriques surface is projective.\n\nOur main theorem is the following:\n\n\\begin{theorem}\\label{thm1}\n\nThere is a smooth projective surface $Y$ birational to some Enriques surface such that ${\\rm Aut}\\, (Y)$ is not finitely generated.\n\n\\end{theorem}\n\n\\begin{remark}\\label{rem1} Let $Y$ be a smooth projective surface birational to an Enriques surface $S$ and let $\\tilde{S}$ be the universal covering K3 surface of $S$.\n\\begin{enumerate}\n\\item ${\\rm Aut}^0(S) = \\{\\id_S\\}$, i.e., ${\\rm Aut}\\,(S)$ is discrete. This is because $H^0(S, T_{S}) = 0$ by $H^0(\\tilde{S}, T_{\\tilde{S}}) = 0$. On the other hand, ${\\rm Aut}\\, (S)$ itself is finitely generated. This is because, \"up to finite kernel and cokernel\", ${\\rm Aut}\\, (S)$ is isomorphic to the quotient group ${\\rm O}({\\rm NS}\\, (S)\/{\\rm torsion})\/W(S)$ of the arithmetic subgroup ${\\rm O}({\\rm NS}\\, (S)\/{\\rm torsion})$ by the Weyl group $W(S)$ generated by the reflections corresponding to the smooth rational curves on $S$ (see \\cite[Theorem]{Do84} for a more precise statement) and ${\\rm O}({\\rm NS}\\, (S)\/{\\rm torsion})$ is finitely generated by a general result on arithmetic subgroups of linear algebraic groups \\cite[Theorem 6.12]{BH62} (See also Theorem \\ref{thm11}). So, $S$ itself is not a candidate surface in Theorem \\ref{thm1}.\n\\item $S$ is the unique minimal model of $Y$ up to isomorphisms. So, we have a birational morphism $\\nu : Y \\to S$, which is a finite composition of blowings up at points. Therefore, we have $H^0(Y, T_Y) = 0$ and also an injective group homomorphism\n$${\\rm Aut}\\, (Y) \\subset {\\rm Bir}\\, (S) = {\\rm Aut}\\, (S)\\,\\, ; \\,\\, f \\mapsto \\nu \\circ f \\circ \\nu^{-1}\\,\\, ,$$\nvia $\\nu$. Note that a subgroup of a finitely generated group is not necessarily finitely generated (cf. Theorem \\ref{thm11}).\n\\end{enumerate}\n\\end{remark}\n\nWe show Theorem \\ref{thm1} by constructing $Y$ explicitly. Our construction is inspired by \\cite{Le18} for 6-dimensional examples, \\cite{DO19} and \\cite{Og19} for exmaples birational to K3 surfaces and also \\cite{Mu10} for his new construction of an Enriques surface with a numerically trivial involution, which is missed in an earlier paper \\cite{MN84}. As usual in study of Enriques surfaces, our construction is more involved than \\cite{DO19} and \\cite{Og19} for K3 surfaces, whereas the basic strategy of the construction is essentially the same.\n\nAs in \\cite{DO19} and \\cite{Og19}, the following purely group theoretical theorem (see eg. \\cite{Su82}) will be frequently used in this paper.\n\n\\begin{thm}\\label{thm11}\nLet $G$ be a group and $H \\subset G$ a subgroup of $G$. Assume that $H$ is of finite index, i.e., $[G :H] < \\infty$. Then, the group $H$\n is finitely generated if and only if $G$ is finitely generated.\n\\end{thm}\n\nIn this paper, for a variety $V$ we denote\nthe group of biregular automorphisms of $V$ and the group of birational automorphisms of $V$ by\n$${\\rm Aut}\\, (V),\\,\\, \\,\\, {\\rm Bir}\\, (V)$$\nrespectively, and for closed subsets $W_1$, $W_2$, $\\ldots$, $W_n$ of $V$ the decomposition group and the inertia group by\n$${\\rm Dec}\\, (W_1, \\ldots, W_n) := {\\rm Dec}\\, (V, W_1, \\ldots, W_n) := \\{ f \\in {\\rm Aut}\\,(V)\\,\\, |\\,\\, f(W_i) = W_i (\\forall i)\\}\\,\\, ,$$\n$${\\rm Ine}\\, (W_1, \\ldots, W_n) := {\\rm Ine}\\, (V, W_1, \\ldots, W_n) := \\{ f \\in {\\rm Dec}\\, (V, W_1, \\ldots, W_n)\\, |\\, f_{W_i} = \\id_{W_i} (\\forall i)\\}.$$\nFor basic properties of surfaces, we refer to \\cite{BHPV04} and \\cite{CD89}.\n\nWe believe that large part of our construction should work also in positive characteristic $\\ge 3$ if the based field is carefully chosen (see e.g. for some sensitive aspect of the base field in positive characteristic \\cite{Og19}). We leave it to the readers who are interested in this generalization.\n\n\\medskip\\noindent\n{\\bf Acknowledgements.} We would like to thank Professor Jun-Muk Hwang for organizing one day workshop at KIAS,\nwhich made our collaboration possible, and Professor Yuya Matsumoto for very kind help concerning Figure \\ref{fig1}.\n\n\\section{Preliminaries}\\label{sect2}\n\nIn this section, first we fix some basic notation concerning a Kummer surface ${\\rm Km}\\, (E \\times F)$ of the product of two non-isogenous elliptic curves. Our notation follows \\cite{DO19} and \\cite{Og19}. Then we recall Mukai's construction of Enriques surfaces with a numerically trivial involution of odd type \\cite{Mu10} arising from ${\\rm Km}\\, (E \\times F)$. His construction is very crucial in our construction in Section \\ref{sect3}.\n\n\\subsection{Kummer surfaces of product type}\\label{sub1}\n\nLet $E$ be the elliptic curve defined by the Weierstrass equation\n$$y^2 = x(x-1)(x-t)\\,\\, ,$$\nand $F$ be the elliptic curve defined by the Weierstrass equation\n$$v^2 = u(u-1)(u-s)\\,\\, .$$\nNote that $E\/\\langle -1_E \\rangle = \\P^1$, the associated quotient map $E \\to \\P^1$ is given by $(x,y)\\mapsto x$ and the points $0$, $1$, $t$ and $\\infty$ of $\\P^1$ are exactly the branch points of this quotient map. The same holds for $F$ if we replace $t$ by $s$.\n\nThroughout this paper, we make the following assumption:\n\n\\begin{assumption}\\label{ass21} $t$ and $s$ are transcendental over $\\Q$ and the two elliptic curves\n$E$ and $F$ are not isogenous.\n\\end{assumption}\nAssumption \\ref{ass21} is satisfied if $s \\in \\C$ is generic with respect to a transcendental number $t \\in \\C$.\n\nLet\n$$X := {\\rm Km} (E \\times F)$$\nbe the Kummer K3 surface accociated to the product abelian surface $E \\times F$, that is, the minimal resolution of the quotient surface $E \\times F\/\\langle -1_{E\\times F} \\rangle$. We write $H^0(X, \\Omega_X^2) = \\C \\omega_X$. Since $E$ and $F$ are not isogenous, the Picard number $\\rho(X)$ of $X$ is $18$ (See eg.\\cite[Prop. 1 and Appendix]{Sh75}).\n\nLet $\\{a_i\\}_{i=1}^{4}$ and $\\{b_i\\}_{i=1}^{4}$ be the $2$-torsion subgroups of $F$ and $E$ respectively. Then $X$ contains 24 smooth rational curves which form the so called double Kummer pencil on $X$, as in Figure \\ref{fig1}. Here smooth rational curves $E_i$, $F_i$ ($1 \\le i \\le 4$) are arising from the elliptic curves $E \\times \\{a_i\\}$, $\\{b_i\\} \\times F$ on $E \\times F$. Smooth rational curves $C_{ij}$ ($1\\le i,j \\le 4$) are the exceptional curves over the $A_1$-singular points of the quotient surface $E \\times F\/\\langle -1_{E\\times F} \\rangle$. Throughout this paper, we will freely use the names of curves in Figure \\ref{fig1}.\n\n\\begin{figure}\n\n\\unitlength 0.1in\n\\begin{picture}(25.000000,24.000000)(-1.000000,-23.500000)\n\\put(4.500000, -22.000000){\\makebox(0,0)[rb]{$F_1$}\n\\put(9.500000, -22.000000){\\makebox(0,0)[rb]{$F_2$}\n\\put(14.500000, -22.000000){\\makebox(0,0)[rb]{$F_3$}\n\\put(19.500000, -22.000000){\\makebox(0,0)[rb]{$F_4$}\n\\put(0.250000, -18.500000){\\makebox(0,0)[lb]{$E_1$}\n\\put(0.250000, -13.500000){\\makebox(0,0)[lb]{$E_2$}\n\\put(0.250000, -8.500000){\\makebox(0,0)[lb]{$E_3$}\n\\put(0.250000, -3.500000){\\makebox(0,0)[lb]{$E_4$}\n\\put(6.000000, -16.000000){\\makebox(0,0)[lt]{$C_{11}$}\n\\put(6.000000, -11.000000){\\makebox(0,0)[lt]{$C_{12}$}\n\\put(6.000000, -6.000000){\\makebox(0,0)[lt]{$C_{13}$}\n\\put(6.000000, -1.000000){\\makebox(0,0)[lt]{$C_{14}$}\n\\put(11.000000, -16.000000){\\makebox(0,0)[lt]{$C_{21}$}\n\\put(11.000000, -11.000000){\\makebox(0,0)[lt]{$C_{22}$}\n\\put(11.000000, -6.000000){\\makebox(0,0)[lt]{$C_{23}$}\n\\put(11.000000, -1.000000){\\makebox(0,0)[lt]{$C_{24}$}\n\\put(16.000000, -16.000000){\\makebox(0,0)[lt]{$C_{31}$}\n\\put(16.000000, -11.000000){\\makebox(0,0)[lt]{$C_{32}$}\n\\put(16.000000, -6.000000){\\makebox(0,0)[lt]{$C_{33}$}\n\\put(16.000000, -1.000000){\\makebox(0,0)[lt]{$C_{34}$}\n\\put(21.000000, -16.000000){\\makebox(0,0)[lt]{$C_{41}$}\n\\put(21.000000, -11.000000){\\makebox(0,0)[lt]{$C_{42}$}\n\\put(21.000000, -6.000000){\\makebox(0,0)[lt]{$C_{43}$}\n\\put(21.000000, -1.000000){\\makebox(0,0)[lt]{$C_{44}$}\n\\special{pa 500 2200\n\\special{pa 500 0\n\\special{fp\n\\special{pa 1000 2200\n\\special{pa 1000 0\n\\special{fp\n\\special{pa 1500 2200\n\\special{pa 1500 0\n\\special{fp\n\\special{pa 2000 2200\n\\special{pa 2000 0\n\\special{fp\n\\special{pa 0 1900\n\\special{pa 450 1900\n\\special{fp\n\\special{pa 550 1900\n\\special{pa 950 1900\n\\special{fp\n\\special{pa 1050 1900\n\\special{pa 1450 1900\n\\special{fp\n\\special{pa 1550 1900\n\\special{pa 1950 1900\n\\special{fp\n\\special{pa 0 1400\n\\special{pa 450 1400\n\\special{fp\n\\special{pa 550 1400\n\\special{pa 950 1400\n\\special{fp\n\\special{pa 1050 1400\n\\special{pa 1450 1400\n\\special{fp\n\\special{pa 1550 1400\n\\special{pa 1950 1400\n\\special{fp\n\\special{pa 0 900\n\\special{pa 450 900\n\\special{fp\n\\special{pa 550 900\n\\special{pa 950 900\n\\special{fp\n\\special{pa 1050 900\n\\special{pa 1450 900\n\\special{fp\n\\special{pa 1550 900\n\\special{pa 1950 900\n\\special{fp\n\\special{pa 0 400\n\\special{pa 450 400\n\\special{fp\n\\special{pa 550 400\n\\special{pa 950 400\n\\special{fp\n\\special{pa 1050 400\n\\special{pa 1450 400\n\\special{fp\n\\special{pa 1550 400\n\\special{pa 1950 400\n\\special{fp\n\\special{pa 200 2000\n\\special{pa 600 1600\n\\special{fp\n\\special{pa 200 1500\n\\special{pa 600 1100\n\\special{fp\n\\special{pa 200 1000\n\\special{pa 600 600\n\\special{fp\n\\special{pa 200 500\n\\special{pa 600 100\n\\special{fp\n\\special{pa 700 2000\n\\special{pa 1100 1600\n\\special{fp\n\\special{pa 700 1500\n\\special{pa 1100 1100\n\\special{fp\n\\special{pa 700 1000\n\\special{pa 1100 600\n\\special{fp\n\\special{pa 700 500\n\\special{pa 1100 100\n\\special{fp\n\\special{pa 1200 2000\n\\special{pa 1600 1600\n\\special{fp\n\\special{pa 1200 1500\n\\special{pa 1600 1100\n\\special{fp\n\\special{pa 1200 1000\n\\special{pa 1600 600\n\\special{fp\n\\special{pa 1200 500\n\\special{pa 1600 100\n\\special{fp\n\\special{pa 1700 2000\n\\special{pa 2100 1600\n\\special{fp\n\\special{pa 1700 1500\n\\special{pa 2100 1100\n\\special{fp\n\\special{pa 1700 1000\n\\special{pa 2100 600\n\\special{fp\n\\special{pa 1700 500\n\\special{pa 2100 100\n\\special{fp\n\\end{picture\n \\caption{Curves $E_i$, $F_j$ and $C_{ij}$}\n \\label{fig1}\n\\end{figure}\n\nWe denote the unique point $E_j \\cap C_{ij}$ by $P_{ij}$ and the unique point $F_i \\cap C_{ij}$ by $P_{ij}'$. We may and do adapt $x$ (resp. $u$) the affine coordinate of $E_j$ and $F_i$ so that\n$$P_{1j} = 1\\,\\, ,\\,\\, P_{2j} = t\\,\\, ,\\,\\, P_{3j} = \\infty\\,\\, ,\\,\\, P_{4j} = 0$$\non $E_j$ with respect to the coordinate $x$ and\n$$P_{i1}' = 1\\,\\, ,\\,\\, P_{i2}' = s\\,\\, ,\\,\\, P_{i3}' = \\infty\\,\\, ,\\,\\,\nP_{i4}' = 0$$\non $F_i$ with respect to the coordinate $u$.\n\nSet\n$$\\theta := [(1_E, -1_F)] = [(-1_E, 1_F)] \\in {\\rm Aut}\\, (X)\\,\\, .$$\nThen $\\theta$ is an involution of $X$, i.e., an automorphism of $X$ of order $2$. The following lemma was proved in \\cite[Lemmas (1.3), (1.4)]{Og89} (See also \\cite{Og19}).\n\n\\begin{lemma}\\label{lem21}\n\\begin{enumerate}\n\\item $\\theta^* = \\id$ on ${\\rm Pic}\\, (X)$ and $\\theta^* \\omega_X = -\\omega_X$.\\item $f \\circ \\theta = \\theta \\circ f$ for all $f \\in {\\rm Aut}\\,(X)$.\n\\item Let $X^{\\theta}$ be the fixed locus of $\\theta$. Then $X^{\\theta} = \\cup_{i=1}^{4} (E_i \\cup F_i)$.\n\\item ${\\rm Aut}\\,(X) = {\\rm Dec}\\, (X, \\cup_{i=1}^{4} (E_i \\cup F_i))$.\n\\end{enumerate}\n\\end{lemma}\n\n\\subsection{Enriques surfaces with a numerically trivial involution of odd type}\\label{sub2}\n\nWe employ the same notation as in Subsection \\ref{sub1}. By Assumption \\ref{ass21}, the two ordered sets\n$$\\{P_{i1}', P_{i2}', P_{i3}', P_{i4}'\\}\\subset F_i\\cong \\P^1\\,\\, ,\\,\\, \\{P_{1j}, P_{2j}, P_{3j}, P_{4j}\\} \\subset E_j\\cong \\P^1$$\nare not projectively equivalent, i.e., not in the same orbit of the action of ${\\rm Aut}\\, (\\P^1) = {\\rm PGL}\\, (2, \\C)$ on $\\P^1$.\n\nWe recall the construction of\n Mukai \\cite{Mu10} for our $X = {\\rm Km}\\, (E \\times F)$.\n\nLet $$T := X\/\\langle \\theta \\rangle$$ be the quotient surface and $$q : X \\to T$$ be the quotient morphism. Then $T$ is a smooth projective surface such that $q(C_{ij})$ ($1 \\le i, j \\le 4$) is a $(-1)$-curve, i.e., a smooth rational curve with self interesection number $-1$. Then $T$ is obtained by the blowings up of $\\P^1 \\times \\P^1$ at the 16 points $p_{ij}$ ($1 \\le i, j \\le 4$) of $\\P^1 \\times \\P^1$. We may assume that $p_{ij}$ is the image of $C_{ij}$ under the composite morphism\n$$X \\to T \\to \\P^1 \\times \\P^1\\,\\, .$$\nLet us consider the Segre embedding\n$$\\P^1 \\times \\P^1 \\subset \\P^3\\,\\, ,$$\nand identify $\\P^1 \\times \\P^1$ with a smooth quadric surface\n$Q$ in $\\P^3$. Since the four points $p_{11}, p_{22}, p_{33}, p_{44} \\in Q$ are not coplaner in $\\P^3$, we may adjust coordinates $[x_1 : x_2 : x_3 : x_4]$ of $\\P^3$ so that the $4$ points are\n$$p_{11}=[1:0:0:0],\\,\\, \\, p_{22}= [0:1:0:0],\\,\\, \\, p_{33}=[0:0:1:0],\\,\\, \\, p_{44}=[0:0:0:1]\\,\\, .$$\nThen the equation of $Q$ is of the form\n$$\\alpha_1x_2x_3 + \\alpha_2x_1x_3 + \\alpha_3x_1x_2 + (x_1+x_2+x_3)x_4 = 0$$\nfor some complex numbers $\\alpha_i$ satisfying non-degeneracy condition.\nThen the Cremona involution of $\\P^3$\n$$\\tilde{\\tau}' : [x_1 :x_2 : x_3 : x_4] \\mapsto [\\frac{\\alpha_1}{x_1} : \\frac{\\alpha_2}{x_2} : \\frac{\\alpha_3}{x_3} : \\frac{\\alpha_1\\alpha_2\\alpha_3}{x_4}]$$ satisfies\n $\\tilde{\\tau}'(Q) = Q$, hence induces a birational automorphism of $Q$\n$$\\tau' := \\tilde{\\tau}'|_{Q} \\in {\\rm Bir}\\, (Q)\\,\\, .$$\nLet $I(\\tau')$ be the indeterminacy locus of $\\tau'$.\nBy the definition of $\\tau'$, we readily check the following (\\cite[Section 2]{Mu10}):\n\\begin{lemma}\\label{lem22}\n\\begin{enumerate}\n\\item $I(\\tau') = \\{p_{ii}\\}_{i=1}^{4}$ and $\\tau'$ contracts the (smooth) conic curve $C'_{i} := Q \\cap (x_i= 0)$ to $p_{ii}$.\n\\item $\\tau'$ interchanges the two lines through $p_{ii}$ for each $i=1$, $2$, $3$, $4$.\n\\item $\\mu^{-1}\\circ \\tau' \\circ \\mu \\in {\\rm Aut}\\, (B)$, where $\\mu : B \\to \\P^1 \\times \\P^1$ is the blowing up at the four points $p_{ii}$ ($1 \\le i \\le 4$).\n\\end{enumerate}\n\\end{lemma}\n\nBy the property (2), $\\tau'(p_{ij}) = p_{ji}$ if $1 \\le i \\not= j \\le 4$. Therefore $\\tau'$ lifts to $$\\tau \\in {\\rm Aut}\\, (T).$$ Since $q : X \\to T$ is the finite double cover branched along the unique anti-bicanonical divisor $$\\sum_{i=1}^{4} (q(E_i)+q(F_i))\\in |-2K_T|,$$ it follows that $\\tau$ lifts to an involution $$\\epsilon \\in {\\rm Aut}\\, (X).$$ Apriori, there are exactly the two choices of the lifting $\\epsilon$; if we denote one lifting by $\\epsilon_0$ then the other is $\\epsilon_0 \\circ \\theta$. Recall that $\\theta^*\\omega_X = -\\omega_X$. Thus, we may and do choose the unique lift $\\epsilon$ with $\\epsilon^* \\omega_X = -\\omega_X$.\nSet\n$$Z := X\/\\langle \\epsilon \\rangle\\,\\, .$$\nand denote the quotient morphism by\n$$\\pi : X \\to Z\\,\\, .$$\nThe following discovery due to\nMukai \\cite[Proposition 2]{Mu10} is also crucial for us:\n\\begin{proposition}\\label{prop21}\nThe involution $\\epsilon$ acts on $X$ freely. In particular,\n$Z$ is an Enriques surface with a numerically trivial involution $\\theta_Z \\in {\\rm Aut}\\, (Z)$ induced from $\\theta \\in {\\rm Aut}\\, (X)$.\n\\end{proposition}\nSet $$C_i := \\epsilon (C_{ii})\\,\\, (i = 1,\\,2,\\,3,\\, 4).$$\nThen, $C_i$ is the proper transform of the curve $C_i'$ in Lemma \\ref{lem22} under the morphism\n$$X \\to T \\to B \\to \\P^1 \\times \\P^1 = Q\\,\\, .$$\n\\begin{corollary}\\label{cor21}\n\\begin{enumerate}\n\\item $\\epsilon (E_i) = F_i$, $\\epsilon(F_i) = E_{i}$ for all $i =1$, $2$, $3$, $4$.\n\\item $\\epsilon(C_{ij}) = C_{ji}$ for all $i$, $j$ such that $i \\not= j$.\n\\item $(C_i, E_i) = (C_i, F_i) = 1$, $(C_i, C_{ii}) = 0$, $(C_i, C_{kj}) = 0$\nfor all $i$, $j$, $k$ such that $k \\not= j$.\n\\item $(C_i, E_j) = (C_i, F_j) = 0$ for all $i$, $j$ such that $j \\not= i$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{proof} The assertions (1) and (2) follow from the description of $\\tau$. Then the assertions (3) and (4) follow from $\\epsilon(C_{ii}) = C_{i}$ and the assertions (1) and (2), except possibly $(C_i, C_{ii}) = 0$. The latter follows from the fact that the conic curve $C'_{i} \\subset Q$ that is contracted to $p_{ii}$ by $\\tau'$ does not pass through $p_{ii}$ (See Lemma \\ref{lem22} (1)).\n\n\\end{proof}\n\n\\section{Construction and proof of Theorem \\ref{thm1}}\\label{sect3}\n\nWe employ the same notation and the assumption (Assumption \\ref{ass21}) as in Section \\ref{sect2}. For instance,\n$$X = {\\rm Km}\\, (E \\times F)\\,\\, ,\\,\\, Z = X\/\\langle \\epsilon \\rangle\\,\\, ,\\,\\, \\pi : X \\to Z\\,\\, .$$\nWe also use the following notation for curves and points on the Enriques surface $Z$:\n$$H_j := \\pi(E_j)\\,\\, ,\n\\,\\, D_{ij} := \\pi(C_{ij})\\,\\, , \\,\\, Q_{ij} := \\pi(P_{ij})\\,\\, ,$$\nand via the isomorphism $\\pi|_{E_j} : E_j \\to H_j$, we also regard $x$ as the affine coordinate of $H_j$. Then $Q_{ij} \\in H_j$ and\n$$x(Q_{1j}) = 1\\,\\, ,\\,\\, x(Q_{2j}) = t\\,\\, ,\\,\\, x(Q_{3j}) = \\infty\\,\\, ,\\,\\, x(Q_{4j}) = 0\\,\\, .$$\nBy Corollary \\ref{cor21}, we have\n$$\\pi^{-1}(H_j) = E_j \\cup F_j$$\nfor each $j =1$, $2$, $3$, $4$ and\n$$\\pi^{-1}(D_{ij}) = C_{ij} \\cup C_{ji}\\,\\, ,\\,\\, \\pi^{-1}(Q_{ij}) = \\{P_{ij},\nP_{ji}'\\}$$\nif $i \\not= j$, while\n$$\\pi^{-1}(D_{ii}) = C_{ii} \\cup C_i\\,\\, ,\\,\\, \\pi^{-1}(Q_{ii}) = \\{P_{ii} \\cup P_{i}\\}\\,\\, ,$$\nagain for each $i$.\nHere $P_{i}$ is the unique intersection point of $C_i \\cap F_i$.\n\nLet $\\mu_1 : Z_1 \\to Z$ be the blowing up at the point $Q_{32} \\in H_2$, i.e., the blowing up at $\\infty$ under the coordinate $x$ of $H_2$. Let\n$$E_{\\infty} := \\P(T_{Z,Q_{32}}) \\simeq \\P^1$$\nbe the exceptional divisor of $\\mu_1$. We then choose three mutually different points on $\\P(T_{Z,Q_{32}})$, say $Q_{32k}$ ($k=1$, $2$, $3$). Let $\\mu_2 : Z_2 \\to Z_1$ be the blowings up of $Z_1$ at the three points $Q_{32k}$.\n\nOur main theorem is Theorem \\ref{thm31} below. Clearly, Theorem \\ref{thm1} follows from Theorem \\ref{thm31} by taking $Y = Z_2$:\n\\begin{theorem}\\label{thm31}\n${\\rm Aut}\\, (Z_2)$ is not finitely generated.\n\\end{theorem}\n\nIn the rest of this section, we prove Theorem \\ref{thm31}.\n\nWe denote\n$$\\mu := \\mu_1 \\circ \\mu_2 : Z_2 \\to Z_1 \\to Z\\,\\, .$$\nBy $E_{32k}$, we denote the exceptional curve over $Q_{32k}$ under $\\mu_2$ and\nby $E_{\\infty}'$ the proper transform of $E_{\\infty}$ under $\\mu_2$.\n\nFirst we reduce the proof to $Z$. For this, we recall that\n$${\\rm Aut}\\, (Z_2) \\subset {\\rm Aut}\\, (Z)$$\nvia $\\mu$ (See Remark \\ref{rem1}).\nWe define\n$${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}}) := \\{ f \\in {\\rm Dec}\\,(Z, Q_{32})\\,|\\, df|_{T_{Z, Q_{32}}} = \\id_{T_{Z, Q_{32}}}\\}\\,\\, .$$\n\n\\begin{proposition}\\label{prop31}\n\\begin{enumerate}\n\\item There is a subgroup $K$ of\n${\\rm Aut}\\, (Z_2)$ such that $[{\\rm Aut}\\, (Z_2) : K] < \\infty$, ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}}) \\subset K$ via $\\mu$ and $[K : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})] < \\infty$.\n\n\\item If ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated, then ${\\rm Aut}\\, (Z_2)$ is not finitely generated.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof} First we show (1). By the canonical bundle formula, we have\n$$|2K_{Z_2}| = \\{2E_{\\infty}' + 4(E_{321} + E_{322} + E_{323})\\}\\,\\, .$$\nSince ${\\rm Aut}\\, (Z_2)$ preserves $|2K_{Z_2}|$, it follows that\n$${\\rm Aut}\\, (Z_2) = {\\rm Dec}\\,(Z_2, E_{\\infty}', E_{321} \\cup E_{322} \\cup E_{323})\\,\\, .$$\nTherefore, via $\\tau_2$, we have\n$${\\rm Aut}\\, (Z_2) = {\\rm Dec}\\,(Z_1, E_{\\infty}, \\{Q_{321}, Q_{322}, Q_{323}\\}) \\subset {\\rm Aut}\\, (Z_1)\\,\\, .$$\nThus, the group\n$$K :=\n{\\rm Dec}\\,(Z_1, E_{\\infty}, \\{Q_{321}\\}, \\{Q_{322}\\}, \\{Q_{323}\\})$$\nis a subgroup of ${\\rm Aut}\\, (Z_2)$ with $[{\\rm Aut}\\, (Z_2) : K] \\le 6 = |{\\rm Aut}_{{\\rm set}}\\,(\\{Q_{321}, Q_{322}, Q_{323}\\})|$.\n\nWe will show that $K$ satisfies the requirement.\n\nSince only $\\id_{\\P^1}$ is the automorphism of $\\P^1$ pointwisely fixes three points, it follows that\n$$K = {\\rm Ine}\\,(Z_1, E_{\\infty})\\,\\, .$$\nSince $E_{\\infty} = \\P(T_{Z, Q_{32}})$, we deduce that\n$$K = \\{f \\in {\\rm Dec}\\, (Z, Q_{32})\\,|\\, df|_{T_{Z, Q_{32}}} = \\alpha(f)\n\\id_{T_{Z, Q_{32}}}\\,\\, (\\exists\\alpha(f) \\in \\C^{\\times})\\} \\subset {\\rm Dec}\\, (Z, Q_{32})\\,\\, .$$\nObserve that if $df|_{T_{Z, Q_{32}}} = \\alpha(f) \\id_{T_{Z, Q_{32}}}$ for $f \\in K$, then\n$$(df \\wedge df)^{\\otimes 2}|_{(\\wedge^2 T_{Z, Q_{32}})^{\\otimes 2}} = \\alpha(f)^4 \\id_{(\\wedge^2 T_{Z, Q_{32}})^{\\otimes 2}}\\,\\, .$$\nSince the line bundle $(\\Omega_Z^2)^{\\otimes 2}$ admits a nowhere vanishing global section, it follows that $\\alpha(f)^4$ is in the image ${\\rm Im}\\, r_2$ of the bicanonical representation\n$$r_2 : {\\rm Aut}\\, (Z) \\to {\\rm GL}(H^0(Z, (\\Omega_Z^2)^{\\otimes 2})) \\simeq \\C^{\\times}$$\nof ${\\rm Aut}\\, (Z)$ (\\cite[Section 14]{Ue75}). Since ${\\rm Im}\\, r_2$\nis finite by \\cite[Theorem 14.10]{Ue75}, it follows that\n$\\{\\alpha(f)\\, |\\, f \\in K\\}$ is also finite.\nHence $[K : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})] < \\infty$ as well.\n\nLet us show (2). Recall Theorem \\ref{thm11}. Then, if ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated, then $K$\nis not finitely generated by $[K : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})] < \\infty$. Hence ${\\rm Aut}\\, (Z_2)$ is not finitely generated, again by $[{\\rm Aut}\\, (Z_2) : K] < \\infty$.\n\\end{proof}\n\nIn what follows, we will show that ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated. This is a problem on the Enriques surface $Z$.\n\n\\begin{lemma}\\label{lem31}\n\\begin{enumerate}\n\\item Let $f \\in {\\rm Dec}\\, (Z, Q_{32})$. Then $f(H_2) = H_2$, i.e., $f \\in {\\rm Dec}\\, (Z, H_2)$.\n\n\\item The differential maps $df|_{T_{Z, Q_{32}}}$ for all $f \\in {\\rm Dec}\\, (Z, Q_{32})$ are simultaneously diagonalizable.\n\n\\item Let $f \\in {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$. Then $f \\in {\\rm Dec}\\, (Z, H_2)$ by (1)\nand\n$$d(f|_{H_2})|_{T_{H_2, Q_{32}}} = \\id_{T_{H_2, Q_{32}}}$$\nfor the induced action.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet $f \\in {\\rm Dec}\\, (Z, Q_{32})$. Then, the one of the two lifts of $f$, say $\\tilde{f}$, satisfies $\\tilde{f}(P_{32}) = P_{32}$. Therefore the result follows from the corresponding result on $X$ (see eg. \\cite{DO19}).\n\nFor the convenience of the readers, we recall the proof here from \\cite{DO19}. Since $\\tilde{f} \\in {\\rm Dec}(X, \\cup_{j=1}^{4} (E_j \\cup F_j))$ by Lemma \\ref{lem21} (4) and $E_{2}$ is the unique component of $\\cup_{j=1}^{4} (E_j \\cup F_j)$, containinig $P_{32}$, it follows that $\\tilde{f} \\in {\\rm Dec}\\,(X, E_2)$. This shows (1).\n\nBy Lemma \\ref{lem21} (1), (3), one has $\\theta(R) = R$ for any smooth rational curve $R$ on $X$\nand\n$$d(\\theta|_{E_2})_{P_{32}} = 1\\,\\, ,\\,\\, d(\\theta|_{C_{32}})_{P_{32}} = -1\\,\\, .$$\nIn particular,\n$$T_{X, P_{32}} = T_{E_2, P_{32}} \\oplus T_{C_{32}, P_{32}}\\,\\, .$$\nNote that $\\tilde{f}(E_2) = E_2$ as observed above. Let $C_{32}' := \\tilde{f}(C_{32})$. Then $P_{32} \\in C_{32}' \\simeq \\P^1$ and the induced action $\\theta|_{C_{32}'}$ satisfies\n$$d(\\theta|_{C_{32}'})_{P_{32}} = -1$$\nby Lemma \\ref{lem21} (1). Thus, $d\\tilde{f}|_{T_{X, P_{32}}}$ for all $\\tilde{f}$ preserve both $T_{E_2, P_{32}}$ and $T_{C_{32}, P_{32}}$. This implies (2).\n\nThe assertion (3) is now obvious.\n\\end{proof}\nRecall that for $Q \\in \\P^1$,\n$${\\rm Ine}\\, (\\P^1, Q, T_{\\P^1, Q}) := \\{f \\in {\\rm Ine}(\\P^1, Q)\\, |\\, df|_{T_{\\P^1, Q}} = \\id_{T_{\\P^1, Q}}\\} \\simeq (\\C, +)\\,\\, .$$\nHere $(\\C, +)$ is the additive group, in particular, an abelian group. The last isomorphism is given by\n$$\\C \\ni c \\mapsto (z \\mapsto z + c) \\in {\\rm Ine}\\, (\\P^1, Q, T_{\\P^1, Q})\\,\\, ,$$\nif we choose an affine coordinate $z$ of $\\P^1$ such that $z(Q) = \\infty$.\nBy Lemma \\ref{lem31} (3), we have then a representation\n$$\\rho : {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}}) \\to {\\rm Ine}\\, (H_2, Q_{32}, T_{H_{2}, Q_{32}}) \\simeq (\\C, +)\\,\\, .$$\nHere, for the last isomorphism, we can use the affine coordinate $x$ of $H_2$ fixed at the beginning of this section.\n\n\\begin{proposition}\\label{prop32}\n\\begin{enumerate}\n\\item There is $a \\in \\C \\setminus \\{0\\}$ such that $t^{-2n}a \\in {\\rm Im}\\,\\rho$ for all positive integers $n$.\n\\item ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$\nis not finitely generated.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe assertion (2) follows from the assertion (1).\nIndeed, the additive subgroup $M$ generated by $\\{t^{-2n}a\\, |\\, n \\in \\Z_{\\ge 0}\\}$ is not finitely generated as $a \\not= 0$ and $t$ is transcendental over $\\Q$ by our assumption (Assumption \\ref{ass21}). The assertion (1) says that $M \\subset {\\rm Im}\\,\\rho$. Since ${\\rm Im}\\, \\rho \\subset (\\C, +)$, the group ${\\rm Im}\\, \\rho$ is also an abelian group. It follows that the abelian group ${\\rm Im}\\,\\rho$ is not finitely generated, either,\nregardless of $[{\\rm Im}\\, \\rho : M]$.\nHence ${\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$ is not finitely generated as claimed.\n\nIn the rest, we will show the assertion (1) by constructing two genus one fibrations on $Z$ and by considering their Jacobian fibrations.\n\nConsider the following two divisors $M_1$ and $M_2$ of Kodaira's type $I_8$ and $IV^*$ on $Z$:\n$$M_1 := H_2 + D_{32} + H_3 + D_{31} + H_1 + D_{41} + H_4 + D_{42}\\,\\, ,$$\n$$M_2 := H_2 + 2D_{32} + H_1 + 2D_{31} + H_4 + 2D_{34} + 3H_3\\,\\, .$$\nThen $|M_1|$ and $|M_2|$ define genus one fibrations\n$$\\varphi_{M_1} : Z \\to \\P^1\\,\\, ,\\,\\, \\varphi_{M_2} : Z \\to \\P^1\\,\\, .$$\n$\\varphi_{M_1}$ is the genus one fibration induced from an elliptic fibration\n$\\Phi_1 : X \\to \\P^1$ on $X$ given by the divisor of Kodaira's type $I_8$\n$$N_1 := E_2 + C_{32} + F_3 + C_{31} + E_1 + C_{41} + F_4 + C_{42}\\,\\, ,$$\nand $\\varphi_{M_2}$ is the genus one fibration induced from an elliptic fibration $\\Phi_2 : X \\to \\P^1$ on $X$ given by the divisor of Kodaira's type $IV^*$\n$$N_2 := E_2 + 2C_{32} + E_1 + 2C_{31} + E_4 + 2C_{34} + 3F_3\\,\\, .$$\nBy the classification of \\cite[Theorem 2.1]{Og89}, $\\Phi_1$ then belongs to Type ${\\mathcal J}_1$ and $\\Phi_2$ belongs to Type ${\\mathcal J}_3$ in \\cite[Theorem 2.1]{Og89}. By the definition of the action of our Enriques involution $\\epsilon$ on $X$ and the classification of \\cite[Theorem 2.1]{Og89}, it follows that the reducible fibers of $\\Phi_1$ are exactly $N_1$ and $\\epsilon(N_1)$, and the reducible fibers of $\\Phi_2$ are exactly $N_2$ and $\\epsilon(N_2)$. Thus, $\\varphi_{M_1}$ has no reducible fibers other than $M_1$ and $\\varphi_{M_2}$ has also no reducible fibers other than $M_2$.\n\nLet us consider the (proper non-singular, relatively minimal) Jacobian fibration $\\varphi_i : R_i \\to \\P^1$ of $\\varphi_{M_i}$ for $i=1$ and $2$. Then the fiber $R_{i, p}$ of $\\varphi_i$ over general $p \\in \\P^1$ is ${\\rm Pic}^0\\,(Z_{i, p})$, i.e., the identity component of the Picard group of the corresponding fiber $Z_{i, t}$ of $\\varphi_{M_i}$. Therefore, the Mordell-Weil group ${\\rm MW}\\,(\\varphi_i)$ of $\\varphi_i$ acts on $\\varphi_{M_i}$, which is the unique biregular extension of the translation action of ${\\rm Pic}^0\\,(Z_{i, p})$ on $Z_{i, p}$ where $p \\in \\P^1$ runs through general points. Note also that the types of singular fibers are the same for $\\varphi_{M_i}$ and $\\varphi_i$ up to multiplicities \\cite[Theorem 5.3.1]{CD89}. Therefore $c_2(R_i) = c_2(Z) = 12$. In particular, $R_i \\to \\P^1$ are rational elliptic surfaces.\n\nHere and hereafter, we will use basic notions and properties of Mordell-Weil lattices due to Shioda \\cite{Sh90}.\n\nLet us consider first the action of ${\\rm MW}\\,(\\varphi_2)$ on $\\varphi_{M_2} : Z \\to \\P^1$. From the fact that $\\varphi_{M_2}$ has also no reducible fibers other than $M_2$, we see that $\\varphi_2 : R_2 \\to \\P^1$ belongs to\nNo. 27 in the classfication of \\cite[Main Theorem]{OS91}. Then, the narrow Mordell-Weil lattice ${\\rm MW}^0\\,(\\varphi_2)$ of $\\varphi_2$ is isomorphic to the positive definite root lattice $A_2$. In particular, there is $P \\in {\\rm MW}^0\\,(\\varphi_2)$ such that $\\langle P, P\\rangle =2$ for the height pairing of ${\\rm MW}^0(\\varphi_2)$ \\cite[Section 8]{Sh90}. For this $P$, we have $(P) \\cap (O) = \\emptyset$ by \\cite[Formula 8.19]{Sh90}. Here $(P)$ is the divisor on $R_2$ corresponding to $P$. The action $t_P$ of $P$ on $\\varphi_{M_2} : Z \\to \\P^1$ then preserves each irreducible component of $M_2$ as $P \\in {\\rm MW}^0\\,(\\varphi_2)$, particularly the curve $H_2$ and the point $Q_{32} \\in H_2$, and the action $t_P|_{H_2}$ is of the form\n$$x \\mapsto x + a$$\nfor some $a \\not= 0$ under the affine coordinate $x$ of $H_2$. Recall that the action of $d(t_P)$ on $T_{Z, Q_{32}}$ is diagonalizable (Lemma \\ref{lem31}). Then, by the finiteness of bicanonical representation \\cite[Theorem 14.10]{Ue75}, by replacing $t_P$ by some power $t_P^k$ ($k \\not= 0$) and $a$ by $ka$ if necessary, we obtain an element\n$$f_2 \\in {\\rm Ine}\\,(Z, Q_{32}, T_{Q_{32}})$$\nsuch that $\\rho(f_2) = a \\not= 0$.\n\nNext we consider the Jacobian fibration $\\varphi_1 : R_1 \\to \\P^1$.\nWe need an explicit geometric construction of $\\varphi_1$ from $\\varphi_{M_1}$ explained by \\cite[Lemma 2.6]{Ko86} and \\cite[Section 3]{HS11}. Note that $D_{21}$ is a $2$-section of $\\varphi_{M_1}$ and $\\pi^{-1}(D_{21}) = C_{12} \\cup C_{21}$. The curves $C_{12}$ and $C_{21}$ are sections of $\\Phi_1$.\nWe may and do choose $C_{21}$ as the zero section of $\\Phi_1$ and set\n$$0 := [C_{21}] \\in {\\rm MW}(\\Phi_1)\\,\\, .$$\nHere and hereafter, we use the following notation:\n\n\\medskip\\noindent\n{\\bf Notation.} \n\\begin{enumerate}\\item For a section $D$ of $\\Phi_1$, we denote by $[D]$ the corresponding element of ${\\rm MW}(\\Phi_1)$ with respect to the zero section $C_{21}$. \n\\item We denote by $T(R) \\in {\\rm Aut}\\, (X)$ the automorphism corresponding to $R \\in {\\rm MW}\\, (\\Phi_1)$.\n\\end{enumerate}\n\n\\medskip\\noindent\nThen the element $[C_{12}] \\in {\\rm MW}(\\Phi_1)$ is a $2$-torsion, because\n$$\\langle [C_{12}], [C_{12}] \\rangle = 2\\cdot2 + 2\\cdot2 - \\frac{4(8-4)}{8} - \\frac{4(8-4)}{8} =0$$\nfor the height pairing \\cite[Theorem 8.6, Formula 8.10]{Sh90} and ${\\rm MW}(\\Phi_1) \\simeq \\Z^{\\oplus 2} \\oplus \\Z\/2$ by \\cite[Theorem 2.1, Case\n${\\mathcal J}_1$]{Og89}. \nSet\n$$\\iota := T([C_{12}]) \\circ \\epsilon \\in {\\rm Aut}\\, (X)\\,\\, .$$\nThen $\\iota$ is an involution on $X$ (\\cite[Lemma 2.6]{Ko86}) such that $X^{\\iota}$ consists of two elliptic curves corresponding to the multiple fibers of $\\varphi_{M_{2}}$ by Assumption \\ref{ass21}. Then, by \\cite[Lemma 2.6]{Ko86} (see also \\cite[Section 3]{HS11}), the Jacobian fibration $\\varphi_1$ of $\\varphi_{M_1}$ is given by\n$$\\varphi_1 : R_1 = X\/\\langle \\iota \\rangle \\to \\P^1\/\\langle \\epsilon \\rangle\n\\,\\, . $$\nHere $\\P^1\/\\langle \\epsilon \\rangle$ is the quotient of the base space $\\P^1$ of $\\Phi_1$ on which $\\epsilon$ acts equivariantly as an involution. Let us denote by $\\pi_{R_1} : X \\to R_1$ the quotient morphism and the fibers $\\pi_{R_1}(N_1)$ by $N_{1, R_1}$ and $\\pi_{R_1}(X_p)$ by $R_{1, \\overline{p}}$.\n\nWe may and do identify both $X_{p}$ and $X_{\\epsilon(p)}$ with $R_{1, \\overline{p}}$ for general $p \\in \\P^1$ via $\\pi_{R_1}$.\n\nSince $\\iota$, $T([C_{12}])$ and $\\epsilon$ are involutions, we have\n$$\\iota := T([C_{12}]) \\circ \\epsilon = \\epsilon \\circ T([C_{12}])\\,\\, .$$\nAlso by the construction, we find that\n$$\\iota(C_{21}) = \\epsilon \\circ T([C_{12}])(C_{21}) = \\epsilon (C_{12}) = C_{21}\\,\\, ,$$\ni.e., preservation of the zero section $C_{21}$ under $\\iota$. Therefore, $Q \\in {\\rm MW}\\,(\\Phi_1)$ is induced from some element $Q'\\in {\\rm MW}(\\varphi_1)$ exactly when\n$$\\iota \\circ T([Q]) = T([Q]) \\circ \\iota\\,\\, {\\rm ,i.e.,}\\,\\, \\iota \\circ T([Q]) \\circ \\iota = T([Q])\\,\\, .$$\n\\begin{lemma}\\label{lem32}\n\\begin{enumerate}\n\\item $\\iota(C_{11}) = C_2$ and $\\iota(C_2) = C_{11}$.\n\\item $[C_{11}]+[C_2]\\in {\\rm MW}\\,(\\Phi_1)$ is induced from some element $Q'\\in {\\rm MW}(\\varphi_1)$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} By preservation of the zero section $C_{21}$ under $\\iota$, we obtain that\n$$\\iota \\circ T([C_{11}]+[C_2]) \\circ \\iota (x) = \\iota (\\iota(x) + [C_{11}] + [C_2]) = x + [\\iota(C_{11})] + [\\iota(C_2)] \\,\\, ,$$\nfor any $x \\in X_p$ on each smooth fiber $X_p$. Hence\n$$\\iota \\circ T(([C_{11}]+[C_2])) \\circ \\iota = T([\\iota(C_{11})] + [\\iota(C_2)])\\,\\, .$$\nSo, the assertion (2) follows from the assertion (1). We show the assertion (1).\nNote that the torsion group of ${\\rm MW}\\, (\\Phi_1)$ is isomorphic to $\\Z\/2$ by \\cite[Theorem 2.1, Case ${\\mathcal J}_1$]{Og89}. In particular, the non-zero torsion element is only $[C_{12}]$.\n\nIf we choose $C_{11}$ (instead of $C_{21}$) as the zero section of $\\Phi_1$, then, the height pairing of the section $C_{22}$ with respect to the zero section $C_{11}$ is computed as\n$$\\langle C_{22}, C_{22}\\rangle = 2\\cdot2 + 2\\cdot2 - \\frac{4(8-4)}{8} - \\frac{4(8-4)}{8} =0\\,\\, .$$\nThus $[C_{22}] - [C_{11}]$ is a non-zero torsion element in ${\\rm MW}\\, (\\Phi_1)$ and therefore coincides with $[C_{12}]$, i.e.,\n$$[C_{22}] = [C_{11}] + [C_{12}]$$\nin ${\\rm MW}\\, (\\Phi_1)$.\nSince $\\iota = \\epsilon \\circ T([C_{12}])$, it follows that\n$$\\iota (C_{11}) = \\epsilon \\circ T([C_{12}])(C_{11}) = \\epsilon (C_{22}) = C_{2}\\,\\, $$\nas claimed. Then\n$$\\iota(C_2) = \\iota (\\iota(C_{11})) = C_{11}\\,\\, ,$$\nas $\\iota$ is an involution. This completes the proof of Lemma \\ref{lem32}.\n\\end{proof}\nLet $Q' \\in {\\rm MW}\\,({\\rm MW}(\\varphi_1)$ be as in Lemma \\ref{lem32}. Then $Q'$ induces an automorphism $f_2 \\in {\\rm Aut}\\, (Z)$ preserving each fiber of $\\varphi_{M_1}$. The action of $f_2$ on $M_1 \\setminus {\\rm Sing}\\, M_1 = \\C^{\\times} \\times \\Z\/8$ \\cite[Page 604]{Ko63} is then the same action of $Q'$ on $N_{1, R} \\setminus {\\rm Sing}\\, (N_{1, R})$ and therefore also the same action of $[C_{11}] + [C_2]$ on $N_1 \\setminus {\\rm Sing}\\, N_1$ under the identifications of these\nthree fibers by $\\pi$ and $\\pi_{R_1}$.\nThus, representing points on $M_1 \\setminus {\\rm Sing}\\, M_1 = \\C^{\\times} \\times \\Z\/8$\nby $(x, m\\,{\\rm mod}\\, 8)$, we have by \\cite[Theorem 9.1, Page 604]{Ko63}\n$$f_2 : (x,m\\,{\\rm mod}\\, 8) \\mapsto (tx, m\\,{\\rm mod}\\, 8) \\mapsto (tx, m + 4 \\,{\\rm mod}\\, 8)\\,\\, .$$\nHere we recall that $C_{22} \\cap E_2 = t$ (resp. $C_2 \\cap F_2 = t$) with respect to the affine coordinate $x$ on $E_2$ (resp. $u$ on $F_2$). Hence\n$f_2^2(H_2) = H_2$, $f_2^2(Q_{32}) = Q_{32}$ and\n$$f_2^2(x) = t^2x$$\non $H_2$. Then\n$$(f_2^2)^{-n} \\circ f_1 \\circ (f_2^2)^{n} \\in {\\rm Ine}\\,(Z, Q_{32}, T_{Z, Q_{32}})$$\nand\n$$(f_2^2)^{-n} \\circ f_1 \\circ (f_2^2)^{n}|_{H_2} : x \\mapsto t^{2n}x \\mapsto t^{2n}x +a \\mapsto x + t^{-2n}a\\,\\,$$\non $H_2$. Thus\n$$t^{-2n}a = \\rho((f_2^2)^{-n} \\circ f_1 \\circ (f_2^2)^{n}) \\in {\\rm Im}\\, \\rho\\,\\, ,$$\nas claimed.\n This completes the proof.\n\\end{proof}\n\nTheorem \\ref{thm31}, hence Theorem \\ref{thm1}, now follows from Propositions \\ref{prop31} (2) and \\ref{prop32} (2).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Electron nuclei coupled system in a coupled double QD \\label{sec_dqd}}\n We consider an electrically gated double quantum dot(QD)\noccupied by two electrons. Under a high magnetic field, s.t. the\nelectron Zeeman splitting is much greater than the hyperfine fields\nand the exchange energy, dynamics takes place in the spin singlet\nground state $|S\\rangle$ and triplet state of zero magnetic quantum\nnumber $|T\\rangle$,\n\\begin{eqnarray} H_e=JS_z+ r\\delta h_z S_x,\n\\label{eq_hf} \\end{eqnarray} where ${\\bf S}$ is the pseudospin\noperator with $|T\\rangle$ and $|S\\rangle$ forming the $S_z$ basis.\n $\\delta h_z=h_{1z}-h_{2z}$, where\n$h_{1z}$ and $h_{2z}$ are the components of nuclear HF field along\nthe external magnetic field in the first and second dot,\nrespectively\\cite{Coish05,Merkulov02}.\n$r=t\/\\sqrt{t^2+(\\delta\/2+\\sqrt{\\delta^2\/4+t^2})^2}$ is the amplitude\nof the hyperfine coupling, which is determined by the gate voltages.\n$\\delta$ is the detuning which is a linear function of gate voltage\ndifferences, and $t$ is the tunneling coefficient. When $\\delta\\gg\nt$, the ground state singlet state corresponds to the case where\nboth electrons are localized in the same dot and HF coupling is\nswitched off, $r\\rightarrow 0$. The opposite limit $\\delta\\ll -t$\ncorresponds to the singlet state where electrons are located in\ndifferent dots, and HF coupling is maximized $r\\rightarrow 1$.\n\n\\subsection{Bunching in electron spin measurements}\nNow we show that by electron spin measurements the coherent behavior\nof nuclear spins can be demonstrated.\n Electron spins are initialized in the singlet state and the nuclear spin states are initially in a mixture of $\\delta h_z$ eigenstates,\n$\\rho(t=0)=\\sum_n p_n\\rho_n|S\\rangle\\langle S|$, where $\\rho_n$ is a\nnuclear state with an eigenvalue of $\\delta h_z=h_n$ and satisfies\n$Tr(\\rho_n)=1$. $p_n$ is the probability of the hyperfine field\n$\\delta h_z$ having the value $h_n$.\n\nIn the unbiased regime $\\delta\\ll -t$, the nuclear spins and the\nelectron spins interact for a time span of $\\tau$. Then the gate\nvoltage is swept adiabatically to a high value(s.t. $\\delta\\gg t$),\nin a time scale much shorter than HF interaction time, leading to\nthe state, $ \\rho=\\sum_n p_n\\rho_n|\\Psi_n\\rangle\\langle \\Psi_n|$,\n where $|\\Psi_n\\rangle=\\alpha_n|S\\rangle+\\beta_n|T\\rangle$, with\n $\n \\alpha_n=\\cos(\\Omega_n \\tau\/2)+iJ\/\\Omega_n \\sin(\\Omega_n \\tau\/ 2)$,\n $\\beta_n=\n -ih_n\/\\Omega_n\\sin(\\Omega_n \\tau\/2)\\label{ampl}\n $\n and $\\Omega_n=\\sqrt{J^2+h_n^2}$ is the Rabi frequency.\n\nNext a charge state measurement is performed which detects a singlet\nor triplet state\\cite{Petta05}. Probability to detect the singlet\nstate is $\\sum_n p_n|\\alpha_n|^2$, and the triplet state is $\\sum_n\np_n|\\beta_n|^2$. Subsequently one can again initialize the system in\nthe singlet state of electron spins, and turn on the hyperfine\ninteraction for a time span of $\\tau$, and perform a second\nmeasurement. In general over $N$ measurements, the probability of\n$k$ times singlet outcomes is\n \\begin{eqnarray}\n P_{N,k}=\\bigl(^N_{\\,k}\\bigr)\\langle\n|\\alpha|^{2k}|\\beta|^{2(N-k)}\\rangle.\\label{Pqm}\n \\end{eqnarray}\nwhere $\\langle\\ldots\\rangle$ is the ensemble averaging over the\nhyperfine field $h_n$\\cite{Merkulov02}.\n Here the key assumption is\nthat nuclear states preserve their coherence over $N$ measurements,\nthus the measurements are not independent due to nuclear memory. One\ncan easily contrast this result with the semiclassical(SC) result\nfor which nuclear spins are assumed to be purely classical, whereas\nelectron spins are taken to obey quantum mechanics\\cite{Merkulov02}.\nIn SC case results of successive measurements\n are independent and the probability for obtaining $k$ times singlet results over $N$ measurements is given by,\n\\begin{eqnarray}\n P'_{N,k}=\\bigl(^N_{\\,k}\\bigr)\\langle|\\alpha|^2\\rangle^{k}\\langle|\\beta|^2\\rangle^{(N-k)}.\\label{Psc}\n\\end{eqnarray}\n\nIn the SC case the probability distribution (\\ref{Psc}) obeys\nsimply a Gaussian distribution with mean\n$k=N\\langle|\\alpha|^2\\rangle$, and variance\n$N\\langle|\\alpha|^2\\rangle\\langle|\\beta|^2\\rangle$, as\n$N\\rightarrow\\infty$. However, in quantum mechanical(QM) treatment\nof nuclear spins, the probability distribution (\\ref{Pqm}) may\nexhibit different statistics depending on the initial nuclear state.\nIf the SC distribution of $h_n$ is characterized by the same\ndistribution as in QM case, the two probability distributions\n(\\ref{Pqm}) and (\\ref{Psc}) yield the same mean value, $\n\\overline{k}=N\\langle|\\alpha|^2\\rangle$, however with distinct\nhigher order moments. If the distribution of initial nuclear state\n$p_n$ has a width $\\Delta$, then for HF interaction time $\\tau\\geq\n1\/\\Delta$, the SC and QM distributions start to deviate from each\nother. They yield the same distribution only when the initial\nnuclear state is in a well defined eigenstate of $\\delta h_z$, i.e.\nwhen $\\Delta=0$.\n\nIn particular we are going to consider the case when the nuclear\nspins are initially randomly oriented; probability distribution for\nhyperfine fields obeying a Gaussian distribution $p_n\\rightarrow\np[h]=1\/\\sqrt{2\\pi\\sigma^2}\\exp[-h^2\/2\\sigma^2]$. In Fig.\n\\ref{Fig_20meas}, for $N=20$ measurements, $P_{N,k}$ is shown for HF\ninteraction times $\\sigma\\tau=0.5, 1.5, \\infty$. For $\\tau=0$, the\nprobability for both SC and QM cases is peaked at $k=20$. However,\nimmediately after the HF interaction is introduced, the probability\ndistributions show distinct behavior. The SC distribution converges\nto a Gaussian distribution. In QM case the probabilities bunch at\n$k=0,20$ for $J=0$, and when $J\/\\sigma=0.5$ those bunch at $k=20$\nonly. As $J$ is increased above some critical value, no bunching\ntakes place at $k=0$ singlet measurement.\n\\begin{figure}[h!]\n\\begin{minipage}{18pc}\n\\includegraphics[width=18pc]{fig_2}\n\\caption{Double QD: Probability distribution at $N=20$ measurements\nfor $k={0,1,\\ldots,20}$ times singlet detections, for QM(solid\nlines), SC(dashed lines). Two cases of the exchange energy are\nconsidered a) $J=0$ b) $J\/\\sigma=0.5$ for HF interaction times\n$\\sigma\\tau=$ i)$0.5$, ii)$1.5$, iii)$\\infty$. \\label{Fig_20meas}}\n\\end{minipage}\\hspace{2pc}%\n\\begin{minipage}{18pc}\n\\psfrag{a}{$k$} \\psfrag{b}{$P$}\n\\includegraphics[width=18pc]{figSQD}\n\\caption{Single QD: Probability distribution at $N=40$ measurements\nfor $k={0,1,\\ldots,20}$ times $|+\\rangle$ detections at HF\ninteraction times $\\sigma\\tau=$ i)$0.3$, ii)$0.6$, iii)$0.9$\niv)$\\infty$. \\label{figSQD}}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Electron spin revivals\\label{sec_esr}}\n\nThe modified nuclear spectrum leads to correlations between the\nsuccessive electron spin measurements. Depending on the results of\nprevious measurement, one may decrease the singlet-triplet mixing.\n As a particular example consider the\ncase: Starting from a random spin configuration, $N$ successive\nelectron spin measurements are performed, each following\ninitialization of electron spins in the spin singlet state and a HF\ninteraction of duration $\\{\\tau_i,i=1\\ldots N\\}$ and all outcomes\nturn out to be singlet. Then again HF interaction is switched on for\na time $t$, and the $(N+1)$th measurement is carried out. The\nconditional probability to detect the singlet state is given by\n \\begin{eqnarray}\n P=\\frac{\\sum(^{~2}_{s_1})(^{~2}_{s_2})\\ldots(^{~2}_{s_{N+1}})e^{-\\frac{1}{2}\\bigl[(s_1-1)\\tilde{\\tau}_1+(s_2-1)\\tilde{\\tau}_2+\\ldots\n+(s_N-1)\\tilde{\\tau}_N+(s_{N+1}-1)\\tilde{t}\\bigr]^2}}{\n4\\sum(^{~2}_{s_1})(^{~2}_{s_2})\\ldots(^{~2}_{s_N})e^{-\\frac{1}{2}\\bigl[(s_1-1)\\tilde{\\tau}_1+(s_2-1)\\tilde{\\tau}_2+\\ldots\n+(s_N-1)\\tilde{\\tau}_N\\bigr]^2} },\n \\label{condprob}\n \\end{eqnarray}\n where the sums run over $s_i=0\\ldots 2$ and $\\tilde{\\tau}_i=\\sigma\\tau_i$.\nFor the particular case $\\tau_1=\\tau_2=\\ldots=\\tau_N=\\tau\\gg\n1\/\\sigma$,\n the initial state is revived at $t=n\\tau, \\,(n=1,2,\\ldots,N)$ with\n a decreasing amplitude, $P\\simeq1\/2+\\sum_{s=0}^{N}(\n^{2N}_{~s})e^{\\frac{-\\sigma^2}{2}(t-(N-s)\\tau)^2}\/4(^{2N}_{~N})$. In\n Fig. \\ref{Fig-cond} the conditional probabilities(\\ref{condprob})\n are shown for $\\sigma\\tau=1.0, 3.0, 6.0$ subject to $N=0,1,2,5,10$\n prior singlet measurements in each. Revivals are observable only\n for $\\sigma\\tau>1$, because the modulation period of the nuclear\n state spectrum characterized by $1\/\\tau$ should be smaller than the\n variance $\\sigma$.\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.9\\textwidth]{fig_3.eps}\n\\caption{Conditional probability for singlet state detection as a\nfunction of HF interaction time $\\sigma t$, subject to\n$N=0,1,2,5,10$ times prior singlet state measurements and for HF\ninteraction times a)$\\sigma\\tau=1.0$, b)$\\sigma\\tau=3.0$,\nc)$\\sigma\\tau=6.0$. \\label{Fig-cond}}\n\\end{figure}\n\nFrom (\\ref{condprob}), number of revivals can be increased with\nvarious choices for the ratios of HF interaction times $\\tau_i$.\n The underlying mechanism of revivals is purification of nuclear\nspins by the electron spin measurements. The purity of a system\ncharacterized by the density matrix $\\hat{\\rho}$ is given by ${\\cal\nP}=Tr\\rho^2$. As an example we are again going to consider the\nnuclear state prepared by $N$ successive electron spin measurements\nwith singlet outcomes, each followed by HF interaction times\n$\\tau_{1}\\ldots \\tau_N$,\n\\begin{eqnarray}\n{\\cal P}=\\frac{1}{\\cal D}\\frac{\\sum_{s_i=0}^\n4(^{~4}_{s_1})(^{~4}_{s_2})\\ldots(^{~4}_{s_N})e^{-\\frac{1}{2}\\bigl[(s_1-2)\\tilde{\\tau}_1+(s_2-2)\\tilde{\\tau}_2+\\ldots\n+(s_N-2)\\tilde{\\tau}_N\\bigr]^2}}{\\bigl[ \\sum_{s_i=0}^\n2(^{~2}_{s_1})(^{~2}_{s_2})\\ldots(^{~2}_{s_N})e^{-\\frac{1}{2}\\bigl[(s_1-1)\\tilde{\\tau}_1+(s_2-1)\\tilde{\\tau}_2+\\ldots\n+(s_N-1)\\tilde{\\tau}_N\\bigr]^2}\\bigr]^2 }.\\label{purity}\n\\end{eqnarray}\n${\\cal D}$ is the dimension of the Hilbert space for the nuclear spins. For a fixed ratio of $\\tau_1:\\tau_2:\\ldots:\\tau_N$, purity\n(\\ref{purity}) is a monotonically increasing function of time. For\n$\\sigma\\tau_i\\gg 1$, one can attain various asymptotic limits for\nthe purity. For instance for $N=2$, there are three asymptotic\nlimits;when a)$\\tau_1=2\\tau_2$ then ${\\cal P}=11\/4{\\cal D}$,\nb)$\\tau_1=\\tau_2$ then ${\\cal P}=35\/18{\\cal D}$, c)otherwise ${\\cal P}=9\/4{\\cal D}$.\nFor $N=2$ with $\\tau_1=2\\tau_2=2\\tau\\gg 1\/\\sigma$, the conditional\nprobability (\\ref{condprob}) is given as follows,\n\\begin{eqnarray}\nP\\simeq\\frac{1}{2}+\\frac{1}{8}\\bigl\\{ e^{-\\frac{(\\tilde{t}-3\\tilde{\\tau})^2}{2} }+\n2e^{-\\frac{(\\tilde{t}-2\\tilde{\\tau})^2}{2} }+ 3e^{-\\frac{(\\tilde{t}-\\tilde{\\tau})^2}{2} }+\n4e^{-\\frac{\\tilde{t}^2}{2} } \\bigr\\},\\label{cond2}\n\\end{eqnarray}\nwhereas for $\\tau_2=\\tau_1=\\tau\\gg 1\/\\sigma$,\n\\begin{eqnarray}\nP\\simeq\\frac{1}{2}+\\frac{1}{12}\\bigl\\{ e^{-\\frac{(\\tilde{t}-2\\tilde{\\tau})^2}{2} }+\n4e^{-\\frac{(\\tilde{t}-\\tilde{\\tau})^2}{2} }+6e^{-\\frac{\\tilde{t}^2}{2} }\n\\bigr\\}.\\label{cond3}\n\\end{eqnarray}\nAs the purity of nuclear spins increase, more revivals are present\nwith an increased amplitude.\n\n\\section{Electron spin bunching and revivals in a single QD\\label{sec_sqd}}\nNow we are going\nto consider a single electron on a single QD. Under external field\n$B$, the system is governed by the Hamiltonian,\n\\begin{eqnarray}\nH=g_e\\mu_BB S_z +g_n\\mu_n B\\sum I^{(j)}_z + {\\bf h}\\cdot{\\bf\nS}.\\label{hfsqd}\n\\end{eqnarray}\nIn (\\ref{hfsqd}), the first two terms are electron and nuclear\nZeeman energies respectively, and the last term is the HF\ninteraction where ${\\bf h}$ is the HF field. When electron Zeeman\nenergy is much greater than rms value of HF fields, viz.\n$g_e\\mu_BB\\gg\\sqrt{\\langle h^2\\rangle} $, flip-flop terms are\nsuppressed and the Hamiltonian (\\ref{hfsqd}) becomes, $H\\simeq\ng_e\\mu_BB S_z + h_z S_z$. Up to a unitary rotation, with $B=0$, this\nis equivalent to the Hamiltonian (\\ref{eq_hf}) with $J=0$.\n$|\\pm\\rangle=(|\\uparrow\\rangle)\\pm|\\downarrow\\rangle\/\\sqrt{2}$ states\nare coupled by HF interaction with $|\\uparrow(\\downarrow)\\rangle$\nbeing the eigenstates of $S_z$. Each time the electron is prepared\nin $|+\\rangle$. Next it is loaded onto the QD, then removed from the\nQD after some dwelling time $\\tau$. Next spin measurement is\nperformed in $|\\pm\\rangle$ basis. Essentially the same predictions\nas that of double QD can be made for this system, namely electron\nspin bunching and revival.\n\nIn Fig. \\ref{figSQD}, for $N=40$ measurements, the QM probability\ndistribution of $P_{N,k}$ is shown at electron Zeeman energy\n$\\epsilon=g_e\\mu_B\\hbar\/2=3\\sigma$, for $\\sigma\\tau=0.3,~,0.6,~0.9,~\\infty$. It is\nseen that contrary to the double QD, the population bunches at\n$|-\\rangle$ states at times $\\tau\\sim \\pi\/\\epsilon$, but then relaxes to the\nequilibrium distribution cf. Fig. \\ref{Fig_20meas}.\n\nNext we are going to consider electron spin revivals. For instance\nafter $N$ times HF interaction of duration $\\tau\\gg 1\/\\sigma$, each\nfollowed by $|+\\rangle$ measurement, the conditional probability for\nobtaining $|+\\rangle$ in the $(N+1)$th step followed by a HF\ninteraction of duration $t$ is given as, $ P\\simeq\n1\/2+\\sum_{s=0}^{N}(\n^{2N}_{~s})e^{-\\sigma^2(t-(N-s)\\tau)^2\/2}\\cos\\epsilon[t-(N-s)\\tau]\/4(^{2N}_{~N})$.\n This is\nessentially the same result for that of a double QD discussed in\nsection-\\ref{sec_esr}.\n\n\n\\section{Discussion and conclusion}\n\nThe randomization of nuclear spins will lead to loss of memory\neffects described above. The nuclear state conditioned on the\nelectron spin measurements will decohere during time interval\nbetween the successive measurements, i.e., when the HF interaction\nis switched off. Thus, the main decoherence mechanism of nuclear\nspins is due to intrinsic nuclear dipole-dipole interactions. In\ndouble quantum dots the duration of the cycle involving electron\nspin initialization and measurement is about $10\n\\mu$s\\cite{Petta05}. Since the nuclear spin coherence time\ndetermined mostly by the nuclear spin diffusion is longer than about\nseveral tens of ms\\cite{Paget82,Giedke05,Paget77}, the bunching for\n$N$ successive measurements up to $N>1000$ can be observed. The same\nholds for the number of revivals that can be observed.\n\nWe have studied the quantum dynamics of the electron-nuclei coupled\nsystem in QD's. The bunching of results of the electron spin\nmeasurements and the revival in the conditional probabilities are\nemerging features of coherence of nuclear spins. The underlying\nmechanism is the correlations between successive measurements\ninduced via nuclear spins and the increase in the purity of the\nnuclear spin state through the electron spin measurements. This\nmechanism is expected to lead to the extension of the electron spin\ncoherence time.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA priori, the boundary of a graph is not a meaningful concept, as graphs do not have an interior or complement. However, given that the usual boundary for subsets in $\\mathbb{R}^n$ is induced by a metric, one interesting question is if it is possible in a finite metric space to find an axiomatic approach to defining a subset that behaves in a boundary-like manner. Chartrand, Erwins, Johns, and Zhang proposed one definition for graphs in \\cite{chartrand}, and Steinerberger proposed another in \\cite{steinerberger}.\n\n\\begin{figure}[h!]\n \\centering\n \\scalebox{0.3}{\n \\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n \\node[shape=circle,draw=black, fill=blue] (02) at (0,4) {};\n \\node[shape=circle,draw=black, fill=blue] (12) at (2,4) {};\n \\node[shape=circle,draw=black, fill=blue] (22) at (4,4) {};\n \\node[shape=circle,draw=black, fill=blue] (30) at (6,0) {};\n \\node[shape=circle,draw=black, fill=blue] (31) at (6,2) {};\n \\node[shape=circle,draw=black, fill=blue] (32) at (6,4) {};\n 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\\node[shape=circle,draw=black, fill=red] (44) at (8,8) {};\n \\node at (0,-1) {};\n \\path (00) edge node {} (10);\n \\path (10) edge node {} (20);\n \\path (20) edge node {} (30);\n \\path (30) edge node {} (40);\n \n \\path (01) edge node {} (11);\n \\path (11) edge node {} (21);\n \\path (21) edge node {} (31);\n \\path (31) edge node {} (41);\n \n \\path (02) edge node {} (12);\n \\path (32) edge node {} (42);\n \n \\path (03) edge node {} (13);\n \\path (13) edge node {} (23);\n \\path (23) edge node {} (33);\n \\path (33) edge node {} (43);\n \n \\path (04) edge node {} (14);\n \\path (14) edge node {} (24);\n \\path (24) edge node {} (34);\n \\path (34) edge node {} (44);\n \n \\path (01) edge node {} (02);\n \\path (02) edge node {} (03);\n \\path (03) edge node {} (04);\n \\path (00) edge node {} (01);\n \n \\path (11) edge node {} (12);\n \\path (12) edge node {} (13);\n \\path (13) edge node {} (14);\n \\path (10) edge node {} (11);\n \n \\path (21) edge node {} (20);\n \\path (23) edge node {} (24);\n \n \\path (31) edge node {} (32);\n \\path (32) edge node {} (33);\n \\path (33) edge node {} (34);\n \\path (30) edge node {} (31);\n \n \\path (41) edge node {} (42);\n \\path (42) edge node {} (43);\n \\path (43) edge node {} (44);\n \\path (40) edge node {} (41);\n\\end{tikzpicture} \n}\n \\caption{The CEJZ (top) and the Steinerberger (bottom) boundaries of three graphs, with boundary vertices shown in red.}\n \\label{boundaryexamples}\n\\end{figure}\n\n\nIn the Chartrand, Erwin, Johns, and Zhang definition, a vertex $v$ is in the boundary of $G$ if there is a vertex $u$ such that no neighbor of $v$ is farther away from $u$ than $v$. \nHowever, when considering subsets in $\\mathbb{R}^n$ with the Euclidean metric, many boundary points do not satisfy a condition analogous to this. Motivated by this, Steinerberger proposed an alternative notion, where $v$ is a boundary vertex if there is a vertex $u$ such that on average, the neighbors of $v$ are closer to $u$ than $v$ is. \nExamples of both boundaries are shown in \\cref{boundaryexamples}.\n\n\nFormally, the \\emph{Chartrand-Erwin-Johns-Zhang (CEJZ) boundary} of a connected graph $G=(V,E)$ is\n\\begin{equation}\n (\\partial G)'=\\left\\{v \\in V\\; \\bigg\\vert \\;\\exists\\;u\\in V \\text{ such that } d(w,u) \\leq d(v,u) \\text{ for all } (v,w)\\in E \\right\\}.\n\\end{equation}\nThe \\emph{Steinerberger boundary} of a connected graph $G = (V, E)$ is defined as\n\\begin{equation}\\label{partialG}\n \\partial G = \\bigg\\{v \\in V\\; \\bigg\\vert \\;\\exists\\;u\\in V \\text{ such that }\\frac{1}{\\text{deg}(v)}\\sum_{(v, w) \\in E} d(w,u) < d(v,u)\\bigg\\},\n\\end{equation}\nwhere we use the convention $\\partial G=V$ when $|V|=1$. We call any $u\\in V$ establishing that $v\\in \\partial G$ or $v\\in (\\partial G)'$ a \\emph{witness} for $v$. \nIt is an immediate consequence of the definitions that the condition for $\\partial G$ is a relaxation of the condition for $(\\partial G)'$.\n\n\\begin{proposition}[Steinerberger \\cite{steinerberger}, Proposition 1]\\label{containment}\nFor any connected graph $G$, $(\\partial G)'\\subseteq \\partial G$. \n\\end{proposition}\n\nThe CEJZ boundary has been studied in a variety of contexts, and we refer the reader to \\cite{allgeier, caceres, caceres3, caceres2, chartrand2, hernando, kang, rodriguez, zejnilovic} for details. When discussing notions of boundary, one natural question is what sets minimize or maximize the size of the boundary. Our work will parallel prior work on characterizing graphs with small CEJZ boundary. The cases of $|(\\partial G)'|=2$ or $|(\\partial G)'|=3$ were classified by Hasegawa and Saito \\cite{hasegawa}, and graphs with $|(\\partial G)'|=4$ were classified by M\\\"{u}ller, P\\'{o}r, and Sereni \\cite{mullerfour}. The main result of our paper is the following theorem classifying graphs with Steinerberger boundary size at most 4. \n\n\\begin{theorem}\\label{maintheorem}\nLet $G=(V,E)$ be a connected graph. \n\\begin{enumerate}[label=(\\alph*)]\n \\item $|\\partial G|=2$ if and only if $G$ is a path graph with at least two vertices.\n \\item $|\\partial G|=3$ if and only if $G$ is a tree with three leaves or a tripod.\n \\item $|\\partial G|=4$ if and only if $G$ is a tree with four leaves or one of the graphs in \\cref{partialG4} with paths of arbitrary lengths attached to boundary vertices $v$ with boundary stability number $\\max_{u\\in V} \\sum_{w\\in N(v)}[d(v,u)-d(w,u)]=1.$\n\\end{enumerate}\n\\end{theorem}\n\n\n\\begin{figure}[h!]\n \\centering\n \\scalebox{0.3}{\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n \\node at (0,2.5) {\\Huge{$3$}};\n \\node at (2.5,0) {\\Huge{$3$}};\n \\node at (0,-2.5) {\\Huge{$3$}};\n \\node at (-2.5,0) {\\Huge{$3$}};\n \\node at (0.5,0.5) {\\Huge{$0$}};\n \\path (A) edge node {} (M);\n \\path (M) edge node {} (D);\n \\path (A) edge node {} (D);\n \\path (A) edge node {} (B);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \\path (B) edge node {} (M);\n \\path (C) edge node {} (M);\n \n\\end{tikzpicture}\\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$2$}};\n \\node at (2.5,0) {\\Huge{$2$}};\n \\node at (0,-2.5) {\\Huge{$2$}};\n \\node at (-2.5,0) {\\Huge{$2$}};\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \n\\end{tikzpicture}\\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$1$}};\n \\node at (2.5,0) {\\Huge{$1$}};\n \\node at (0,-2.5) {\\Huge{$1$}};\n \\node at (-2.5,0) {\\Huge{$1$}};\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (C);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (D);\n \\path (C) edge node {} (D);\n \n\\end{tikzpicture} \\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$2$}};\n \\node at (2.5,0) {\\Huge{$2$}};\n \\node at (0,-2.5) {\\Huge{$2$}};\n \\node at (-2.5,0) {\\Huge{$2$}};\n \\node at (0.5,0.5) {\\Huge{$0$}};\n \\path (A) edge node {} (M);\n \\path (M) edge node {} (D);\n \\path (A) edge node {} (D);\n \\path (A) edge node {} (B);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (M);\n \\path (C) edge node {} (M);\n \n\\end{tikzpicture}\\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\\node at (0,2.5) {\\Huge{$1$}};\n \\node at (2.5,0) {\\Huge{$2$}};\n \\node at (0,-2.5) {\\Huge{$1$}};\n \\node at (-2.5,0) {\\Huge{$2$}};\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (C);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \n\\end{tikzpicture}}\\\\\n\\phantom{-}\\\\\n\\scalebox{0.3}{\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (-4,2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (-4,-2) {};\n \\node[shape=circle,draw=black, fill=blue] (M1) at (-2,0) {};\n \\node (P) at (0,0) {$\\cdots$};\n \\node[shape=circle,draw=black, fill=blue] (M2) at (2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (4,-2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (4,2) {};\n \\node at (-4.5,2) {\\Huge{$1$}};\n \\node at (4.5,2) {\\Huge{$1$}};\n \\node at (-4.5,-2) {\\Huge{$1$}};\n \\node at (4.5,-2) {\\Huge{$1$}};\n \\node at (1.75,0.5) {\\Huge{$0$}};\n \\node at (-1.75,0.5) {\\Huge{$0$}};\n \\path (A) edge node {} (M1);\n \\path (M1) edge node {} (D);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (M2);\n \\path (C) edge node {} (M2);\n \\path (M1) edge node {} (P);\n \\path (P) edge node {} (M2);\n \n\\end{tikzpicture}\\qquad\n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (-4,2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (-4,-2) {};\n \\node[shape=circle,draw=black, fill=blue] (M1) at (-2,0) {};\n \\node (P) at (0,0) {$\\cdots$};\n \\node[shape=circle,draw=black, fill=blue] (M2) at (2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (4,-2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (4,2) {};\n \\node at (-4.5,2) {\\Huge{$1$}};\n \\node at (4.5,2) {\\Huge{$1$}};\n \\node at (-4.5,-2) {\\Huge{$1$}};\n \\node at (4.5,-2) {\\Huge{$1$}};\n \\node at (1.75,0.5) {\\Huge{$0$}};\n \\node at (-1.75,0.5) {\\Huge{$0$}};\n \\path (A) edge node {} (M1);\n \\path (M1) edge node {} (D);\n \n \\path (B) edge node {} (C);\n \\path (B) edge node {} (M2);\n \\path (C) edge node {} (M2);\n \\path (M1) edge node {} (P);\n \\path (P) edge node {} (M2);\n \n\\end{tikzpicture}}\n\\caption{Non-tree graphs with $|\\partial G|=4$, where $\\partial G$ is shown in red. The boundary stability number is given beside each vertex.}\n\\label{partialG4}\n\\end{figure}\n\nThe boundary stability number is a new parameter that we introduce to establish this theorem. For any $v\\in \\partial G$, this parameter is an integer that measures how stable the condition in \\cref{partialG} is under certain operations. It allows us to more accurately describe what happens when we build larger graphs from smaller ones, which is one of the techniques used in the classification of graphs with small CEJZ boundary.\n\nObserve that in each of the graphs in Theorem \\ref{maintheorem}, the maximum degree is small, and few vertices are not on long paths. In Euclidean space, larger regions have larger boundary. One formal statement of this is the isoperimetric \ninequality, which gives a lower bound on the surface area or perimeter of a region in terms of its volume. Steinerberger established a corresponding isoperimetric inequality for $\\partial G$ in \\cite[Theorem 1]{steinerberger} given by\n\\[|\\partial G|\\geq \\frac{1}{2\\Delta}\\cdot \\frac{|V|}{\\diam(G)},\\]\nwhere $\\Delta$ is the maximum degree of $G$. Hence, graphs with a large number of vertices have more boundary vertices unless these graphs contain large paths. We see that the results of Theorem \\ref{maintheorem} are consistent with the above isoperimetric inequality for $\\partial G$. \n\nNote that the CEJZ boundary does not exhibit this general behavior. From generalizations of the first graph in \\cref{boundaryexamples} and other grid-like graphs, $(\\partial G)'$ can remain very small for very large graphs that are not formed from paths. A lower bound on $|(\\partial G)'|$ in terms of the maximum degree was previously studied in \\cite{mullerbound}, where the authors established a bound that is logarithmic with respect to $\\Delta$ and showed that this bound is sharp up to constants. However, this bound is entirely independent of $|V|$.\n\n\nWe start in \\cref{preliminaries} by summarizing previous results for the CEJZ and Steinerberger boundaries. In particular, we will summarize the characterizations of graphs with small CEJZ boundary. In \\cref{stability}, we will introduce the boundary stability number of a vertex and establish several lemmas. We then apply these results in \\cref{boundarysize4} to prove \\cref{maintheorem}. In \\cref{largeboundary}, we apply some of our results to describe some graphs with large Steinerberger boundary. We then conclude with open questions in \\cref{futurework}.\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\n\nIn this section, we outline necessary definitions and notation, and we then summarize previous results for both the CEJZ and Steinerberger boundaries. Throughout this paper, all graphs are assumed to be simple and undirected. We assume basic familiarity with these graphs and refer the reader to \\cite{bona} or \\cite{bondy} for this information. We summarize the definitions and notation relevant for this paper. \n\nGraphs will be denoted $G=(V,E)$. An edge in $E$ will be denoted as a pair, such as $(v,w)$. Note that graphs in this paper are undirected, so this is equivalent to $(w,v)$. We use $N_G(v)$ to denote the set of neighbors of $v$ in the graph $G$ and $\\deg_G(v)$ to denote the degree of $v$ in $G$. \n\nFor any graph $G=(V,E)$, a \\emph{walk} is a sequence of vertices $W=v_0v_1\\ldots v_\\ell$ such that $(v_i,v_{i+1})\\in E$ for $i=0,1,\\ldots,\\ell-1$. We call $v_0$ the \\emph{initial vertex} and $v_\\ell$ the \\emph{terminal vertex} of $W$. We also call $\\ell$ the \\emph{length} of the walk $W$. \nA walk is a \\emph{path} if all vertices are distinct, and a \\emph{shortest} path from $v$ to $w$ is a path of minimum length with $v$ as its initial vertex and $w$ as its terminal vertex. A walk is \\emph{closed} if $v_0=v_{\\ell}$, and a closed walk is called a \\emph{cycle} if no vertices are repeated except for $v_0=v_{\\ell}$. \n\nA graph $G=(V,E)$ is \\emph{connected} if for every $v,w\\in V$, there exists a path with $v$ as its initial vertex and $w$ as its terminal vertex. The \\emph{connected components} of $G$ are the maximal connected subgraphs of $G$. For a connected graph $G=(V,E)$, a vertex $v\\in V$ is a \\emph{cut vertex} if $G-v$, the graph obtained by deleting $v$ and all edges incident to it, is not connected.\n\nFor a connected graph $G=(V,E)$ and any $v,w\\in V$, the \\emph{distance} from $v$ to $w$, denoted $d_G(v,w)$, is the length of a shortest path with $v$ as its initial vertex and $w$ as its terminal vertex. It is straightforward to verify that $d_G$ satisfies the properties of a metric. For any $v\\in V$, the \\emph{eccentricity} of $v$ is\n$\\ecc_G(v)=\\max_{w\\in V} \\{d_G(v,w)\\}$. The \\emph{diameter} of a graph is $\\diam(G)=\\max_{v\\in V} \\{\\ecc_G(v)\\}$.\nIf $v\\in V$ satisfies $\\ecc_G(v)=\\diam(G)$, then we call $v$ a \\emph{peripheral} vertex. When the context is clear, we omit the subscript $G$ in $N_G,\\deg_G,d_G$, and $\\ecc_G$. An example of these definitions is given in \\cref{graphexample}.\n\n\n\\begin{figure}[h!]\n\\centering\n \\scalebox{0.4}{\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=blue] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=blue] (D) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (2) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (3) at (6,0) {};\n \\node[shape=circle,draw=black, fill=blue] (4) at (8,0) {};\n \\node[shape=circle,draw=black, fill=blue] (5) at (10,0) {};\n \\node[shape=circle,draw=black, fill=blue] (6) at (8,2) {};\n \\node[shape=circle,draw=black, fill=blue] (7) at (8,-2) {};\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (C);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (D);\n \\path (C) edge node {} (D);\n \\path (D) edge node {} (2);\n \\path (2) edge node {} (3);\n \\path (3) edge node {} (4);\n \\path (4) edge node {} (5);\n \\path (4) edge node {} (6);\n \\path (4) edge node {} (7);\n \\node at (-2.75,0) {\\Huge{$v_1$}};\n \\node at (0,2.75) {\\Huge{$v_2$}};\n \\node at (0,-2.75) {\\Huge{$v_3$}};\n \\node at (2,-0.75) {\\Huge{$v_4$}};\n \\node at (4,-0.75) {\\Huge{$v_5$}};\n \\node at (6,-0.75) {\\Huge{$v_6$}};\n \\node at (8,2.75) {\\Huge{$v_7$}};\n \\node at (8.75,-0.75) {\\Huge{$v_8$}};\n \\node at (8,-2.75) {\\Huge{$v_9$}};\n \\node at (11,0) {\\Huge{$v_{10}$}};\n\\end{tikzpicture}}\n\\caption{A graph $G$ with $\\diam(G)=5$, where $v_1,v_2,v_3,v_7,v_9$, and $v_{10}$ are all peripheral vertices. The vertices $v_4,v_5,v_6$, and $v_8$ are cut vertices, as removing any of them disconnects $G$.}\n\\label{graphexample}\n\\end{figure}\n\nWe now summarize previous results. We start with the following observation, which is used throughout the study of both the CEJZ and Steinerberger boundaries.\n\n\\begin{observation}\\label{peripheral}\nFor any connected graph $G=(V,E)$, peripheral vertices are always in $(\\partial G)'$. Combined with \\cref{containment}, this implies that peripheral vertices are always in $\\partial G$. \n\\end{observation}\n\nFor $\\partial G$, the following results were established by Steinerberger in \\cite{steinerberger}. The first result immediately implies part (a) of \\cref{maintheorem}. Hence, it remains to show parts (b) and (c).\n\n\\begin{proposition}[Steinerberger \\cite{steinerberger}, Proposition 3]\\label{boundary2}\nFor any connected graph with at least\ntwo vertices, we have $|\\partial G|\\geq 2$. If $|\\partial G|=2$, then $G$ is a path.\n\\end{proposition}\n\n\\begin{proposition}[Steinerberger \\cite{steinerberger}, Proposition 2]\\label{treeandleaves}\nIf $G$ is a tree, then $\\partial G$ are the vertices of\ndegree $1$. For any connected graph, vertices of degree $1$ are in $\\partial G$. \n\\end{proposition}\n\nTo establish the remaining parts of our main theorem, we will use the characterization of graphs with small CEJZ boundary, which we summarize here. We start with the results of Hasegawa and Saito characterizing graphs with CEJZ boundary size $2$ or $3$. Note that a \\emph{tripod} is a graph formed by starting with the complete graph on three vertices $K_3$ and attaching a path of arbitrary length (possibly 0) to each vertex.\n\n\n\n\\begin{theorem}[Hasegawa and Saito \\cite{hasegawa}, Theorem 7]\\label{CEJZ2}\nLet $G$ be a connected graph. If $|(\\partial G)'|=2$, then $G$ is a path.\n\\end{theorem}\n\n\\begin{theorem}[Hasegawa and Saito \\cite{hasegawa}, Theorem 9]\\label{CEJZ3}\nA connected graph $G$ has $|(\\partial G)'|=3$ if and only if $G$ is either a tree with three leaves or a\ntripod.\n\\end{theorem}\n\nThe characterization of graphs with CEJZ boundary size $4$ is significantly more complex. We start by defining several families of graphs and axis slice convex sets in $\\mathbb{R}^2$.\n\n\\begin{definition}[M\\\"{u}ller, P\\'{o}r, and Sereni, \\cite{mullerfour}, Definition 2]\nLet $a$ and $c$ be two positive integers. Define the vertex sets \n\\[V_{a\\times c}^0=\\{(x,y)\\in \\mathbb{N}^2\\, | \\, 0\\leq x\\leq a \\text{ and } 0\\leq y\\leq c\\}\\]\n\\[ V_{a\\times c}^1=\\left\\{\\left(x+\\frac{1}{2},y+\\frac{1}{2}\\right)\\, \\bigg| \\, (x,y)\\in \\mathbb{N}^2,0\\leq x< a, \\text{ and } 0\\leq y< c\\right\\}.\\]\n\\begin{enumerate}[label=(\\alph*)]\n \\item The grid graph $G_{a\\times c}$ has vertex set $V_{a\\times c}^0$ and edges between any vertices of Euclidean distance 1. Note that $|V_{a\\times c}^0|=(a+1)(c+1)$.\n \\item The graph $N_{a\\times c}$ has vertex set $V_{a\\times c}=V_{a\\times c}^0\\cup V_{a\\times c}^1$ and edges between vertices of Euclidean distance at most $1$.\n \\item For $a>2$, the graph $X_{a\\times c}$ is the subgraph of $N_{(a-1)\\times c}$ induced by \\[V_{(a-1)\\times c}\\setminus \\{(x,y)\\in \\mathbb{N}^2\\, | \\, 01$, define $X_{a\\times c}=X_{c\\times a}$. Finally, define $X_{1\\times 1}=K_4$.\n \\item The graph $T_{a\\times c}$ is the subgraph of $N_{a\\times (c+1)}$ induced by \n \\[V_{a\\times (c+1)}\\setminus (\\{(0,y)\\, |\\, y\\in \\mathbb{N}\\}\\cup \\{(x,y)\\, |\\, x1$ and $|V_2|>1$. Then \\cref{graphedgeattachment} implies that for any vertices $v\\in V_i\\setminus \\{v_i\\}$, we have $v\\in \\partial G_i$ if and only if $v\\in \\partial G$. For $v_i$, observe that the same is true except when $\\beta_{G_i}(v_i)=1$, and in this case $v_i\\in \\partial G_i$ and $v_i\\notin \\partial G$. The theorem then follows from the various cases.\n\nNow suppose $|V_1|>1$ and $|V_2|=1$. In this case, $\\partial G_2=\\{v_2\\}$ and $\\beta_{G_2}(v_2)=0$, so it suffices to show (a) and (b) with $\\partial G_2=\\{v_2\\}$. Since $v_2$ is a leaf in $G$, \\cref{treeandleaves} implies $v_2\\in \\partial G$. Applying the above argument with \\cref{graphedgeattachment} on $V_1$, we conclude (a) and (b). The case $|V_1|=1$ and $|V_2|>1$ is similar. Finally, when $|V_1|=|V_2|=1$, we see that $\\partial G_i=\\{v_i\\}$ and $\\beta_{G_i}(v_i)=0$ for all $i$. In this case, $G$ is a path graph on two vertices, and (a) is clear.\n\\end{proof}\n}\n\nWe conclude this section with results on subgraphs. Using the boundary stability number, we can sometimes determine when boundary vertices in a subgraph of $G$ are also boundary vertices in $G$ itself.\n\n\\begin{lemma}\\label{boundarysubgraph}\nLet $G$ be a connected graph with a connected subgraph $H$ on at least two vertices. Suppose $v\\in \\partial H$, and let $u$ be a witness for $v$. If $N_{H}(v)=N_G(v)$ and $d_{H}(v,u)=d_G(v,u)$, then $v \\in \\partial G$.\n\\end{lemma}\n\n\\begin{proof}\nNotice that for any $w\\in N_G(v)=N_{H}(v)$, we have that $d_G(w,u)\\leq d_{H}(w,u)$. Then a direct calculation shows \n \\begin{align*}\n \\beta_{G}(v)&\\geq \\beta_G(v,u) \\\\\n & =\\sum_{w\\in N_G(v)}[d_G(v,u)-d_G(w,u)] \\\\\n \n & \\geq \\sum_{w\\in N_{H}(v)}[d_{H}(v,u)-d_{H}(w,u)]\\\\\n & = \\beta_{H}(v,u) \\\\\n & \\geq 1. \\qedhere\n \\end{align*}\n\\end{proof}\n\n\n\\begin{corollary}\\label{boundarysubgraphcor}\nLet $G$ be a connected graph with a connected subgraph $H$ on at least two vertices. Suppose $v\\in \\partial H$, and let $u$ be a witness for $v$. If $N_{H}(v)=N_G(v)$ and $d_{H}(v,u)=2$, then $v \\in \\partial G$.\n\\end{corollary}\n\\begin{proof}\nSince $H$ is a subgraph of $G$, we have that $d_G(v,u)\\leq d_H(v,u)=2$. Observe that $d_H(v,u)=2$ implies $u\\notin N_H(v)$, and the assumption $N_{H}(v)=N_G(v)$ implies $u\\notin N_G(v)$. Then $d_G(v,u)>1$, and combined, we conclude that $d_G(v,u)=2$. The result then follows from \\cref{boundarysubgraph}.\n\\end{proof}\n\n\n\n\n\\section{Graphs with at most four boundary vertices}\\label{boundarysize4}\n\nIn this section, we establish the proof of \\cref{maintheorem}. We start by applying the results of \\cref{stability} to establish lemmas for graphs with $|\\partial G|=3$ or $|\\partial G|=4$.\n\n\\begin{lemma}\\label{boundary3lemma}\nLet $G=(V,E)$ be a connected graph. If $|(\\partial G)'|=3$, then $|\\partial G|=3$. \n\\end{lemma}\n\n\\begin{proof}\nWe use the characterization of $|(\\partial G)'|=3$ given in \\cref{CEJZ3}. If $G$ is a tree on three leaves, then \\cref{treeandleaves} implies that $|\\partial G|=3$. For tripods, we start by considering the complete graph $K_3$. A direct calculation shows that $|\\partial K_3|=3$, and each vertex $v\\in V$ has $\\beta(v)=1$. Tripods are formed by attaching arbitrary length paths to the vertices of $K_3$. By applying \\cref{graphedgeattachment} for each nontrivial path attached, we conclude that tripods have three boundary vertices. \n\\end{proof}\n\n\n\n\\begin{lemma}\\label{boundary4lemma}\nLet $G=(V,E)$ be a connected graph with $|(\\partial G)'|=4$. Then $|\\partial G|=4$ if and only if $G$ is one of the following graphs:\n\\begin{enumerate}[label=(\\alph*)]\n \\item a subdivision of the star graph $K_{1,4}$,\n \\item a subdivision of the tree with exactly four leaves and two vertices of degree~3,\n \\item a graph obtained from one of the trees in (b) by removing a vertex of degree 3 and adding edges between all of its neighbors, or\n \\item one of the graphs in \\cref{partialG4base} with a path of arbitrary length attached to each $v\\in \\partial G$ with $\\beta(v)=1$. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{figure}[h!]\n \\centering\n \\scalebox{0.4}{\n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n \\path (A) edge node {} (M);\n \\path (M) edge node {} (D);\n \\path (A) edge node {} (D);\n \\path (A) edge node {} (B);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \\path (B) edge node {} (M);\n \\path (C) edge node {} (M);\n \\node at (0,-3) {\\Huge{$N_{1,1}$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \\node at (0,-3) {\\Huge{$C_4$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (C);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (D);\n \\path (C) edge node {} (D);\n \\node at (0,-3) {\\Huge{$K_4=X_{1,1}$}};\n\\end{tikzpicture}} \n\\\\\n\\phantom{-}\\\\\n\\scalebox{0.4}{\n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (-4,2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (-4,-2) {};\n \\node[shape=circle,draw=black, fill=blue] (M1) at (-2,0) {};\n \\node (P) at (0,0) {$\\cdots$};\n \\node[shape=circle,draw=black, fill=blue] (M2) at (2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (4,-2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (4,2) {};\n\n \\path (A) edge node {} (M1);\n \\path (M1) edge node {} (D);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (M2);\n \\path (C) edge node {} (M2);\n \\path (M1) edge node {} (P);\n \\path (P) edge node {} (M2);\n \\node at (0,-3) {\\Huge{$X_{1\\times c}$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (M) at (0,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n \\path (A) edge node {} (M);\n \\path (M) edge node {} (D);\n \\path (A) edge node {} (D);\n \\path (A) edge node {} (B);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (M);\n \\path (C) edge node {} (M);\n \\node at (0,-3) {\\Huge{$T_{1,1}$}};\n\\end{tikzpicture}\\qquad \\qquad \n\\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,2) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-2,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-2) {};\n \\node[shape=circle,draw=black, fill=red] (D) at (2,0) {};\n\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (C);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (C) edge node {} (D);\n \\node at (0,-3) {\\Huge{$D_{1,1}=L_{1,1}$}};\n\\end{tikzpicture}}\n\n\\caption{Graphs from \\cref{mullerthm} (d)-(h) with $|\\partial G|=4$.}\n\\label{partialG4base}\n\\end{figure}\n\n\n\\begin{proof}\nWe show this by considering each case of \\cref{mullerthm}. Note that graphs (a), (b), and (c) match \\cref{mullerthm}, and the results for (a) and (b) follow directly from \\cref{treeandleaves}. For (c), notice that these graphs $G$ can alternatively be constructed by starting with a tripod $H$ and attaching two nontrivial paths to some $v\\in \\partial H$. Observe that $H$ is constructed by attaching paths of arbitrary lengths to $K_3$. Using \\cref{graphedgeattachment} on $K_3$ with paths attached, the boundary stability number of any vertex in $\\partial H$ is 1. Using \\cref{graphedgeattachment} once for each path attached to $v\\in \\partial H$, we conclude that $|\\partial G|=4$.\n\nNow consider the remaining cases in \\cref{mullerthm}. Notice that if $G$ is constructed as a subgraph of $N_{a\\times c}$, $X_{a\\times c}$, $T_{a\\times c}$, $D_{a\\times c}$, or $L_{a\\times c}$ and $|\\partial G|\\geq |(\\partial G)'|\\geq 5$, then by \\cref{edgeattachment}, a graph $H$ formed by attaching paths to $G$ has $|\\partial H|\\geq |\\partial G|\\geq 5$. Hence, we must consider constructions where $|(\\partial G)'|=4$ to obtain a graph $H$ with $|\\partial H|=4$. We do this using the graphs in \\cref{basecases}, which will allow us to show the existence of additional Steinerberger boundary vertices in addition to the four CEJZ boundary vertices described in \\cref{mullerremark}. Notice that for each $G_i$ in the figure, $v\\in \\partial G_i$ with witness $u$, and $d_{G_i}(v,u)=2$.\n\n\\begin{figure}[h!]\n \\centering\n \\scalebox{0.45}{\n \\begin{tikzpicture}\n \\node at (-1,0) {};\n \\node at (5,0) {};\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n \\path (00) edge node {} (10);\n \\path (20) edge node {} (10);\n \\path (01) edge node {} (11);\n \\path (21) edge node {} (11);\n \\path (00) edge node {} (01);\n \\path (10) edge node {} (11);\n \\path (20) edge node {} (21);\n \\node[shape=circle,draw=black, fill=blue] (05) at (1,1) {};\n \\node[shape=circle,draw=black, fill=blue] (15) at (3,1) {};\n \\path (00) edge node {} (05);\n \\path (10) edge node {} (05);\n \\path (01) edge node {} (05);\n \\path (11) edge node {} (05);\n \\path (10) edge node {} (15);\n \\path (11) edge node {} (15);\n \\path (20) edge node {} (15);\n \\path (21) edge node {} (15);\n \\path (05) edge node {} (15);\n \\node at (2,-0.5) {\\huge{$v$}};\n \\node at (4,2.5) {\\huge{$u$}};\n \\node at (2,-1.75) {\\Huge{$G_1$}};\n \\end{tikzpicture}\n \\quad \n \\begin{tikzpicture}\n \\node at (-1,0) {};\n \\node at (5,0) {};\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n \\path (00) edge node {} (10);\n \\path (20) edge node {} (10);\n \\path (01) edge node {} (11);\n \\path (21) edge node {} (11);\n \\path (00) edge node {} (01);\n \\path (10) edge node {} (11);\n \\path (20) edge node {} (21);\n \\node[shape=circle,draw=black, fill=blue] (05) at (1,1) {};\n \\path (00) edge node {} (05);\n \\path (10) edge node {} (05);\n \\path (01) edge node {} (05);\n \\path (11) edge node {} (05);\n \\node at (2,-0.5) {\\huge{$v$}};\n \\node at (4,2.5) {\\huge{$u$}};\n \\node at (2,-1.75) {\\Huge{$G_2$}};\n \\node at (-0.5,2) {};\n \\end{tikzpicture}\n \\quad \n \\begin{tikzpicture}\n \n \\node at (-1,0) {};\n \\node at (5,0) {};\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n \\path (00) edge node {} (10);\n \\path (20) edge node {} (10);\n \\path (01) edge node {} (11);\n \\path (21) edge node {} (11);\n \\path (00) edge node {} (01);\n \\path (10) edge node {} (11);\n \\path (20) edge node {} (21);\n \\node at (2,-0.5) {\\huge{$v$}};\n \\node at (4,2.5) {\\huge{$u$}};\n \\node at (2,-1.75) {\\Huge{$G_3$}};\n \\end{tikzpicture} \n }\n \\\\\n \\phantom{-}\\\\\n \\scalebox{0.45}{\n \\begin{tikzpicture}\n \\node at (-1,0) {};\n \\node at (5,0) {};\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (10) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (01) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (11) at (4,2) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (0,4) {};\n \\node[shape=circle,draw=black, fill=blue] (21) at (4,4) {};\n \\path (00) edge node {} (10);\n \\path (20) edge node {} (10);\n \\path (01) edge node {} (11);\n \\path (21) edge node {} (11);\n \n \\path (10) edge node {} (11);\n \n \\node[shape=circle,draw=black, fill=blue] (05) at (2,1) {};\n \\node[shape=circle,draw=black, fill=blue] (15) at (2,3) {};\n \\path (00) edge node {} (05);\n \\path (10) edge node {} (05);\n \\path (01) edge node {} (05);\n \\path (11) edge node {} (05);\n \\path (10) edge node {} (15);\n \\path (11) edge node {} (15);\n \\path (20) edge node {} (15);\n \\path (21) edge node {} (15);\n \\path (05) edge node {} (15);\n \\node at (-0.5,2) {\\huge{$v$}};\n \\node at (4.5,0) {\\huge{$u$}};\n \\node at (2,-0.75) {\\Huge{$G_4$}};\n \\end{tikzpicture}\n \\quad \n \\begin{tikzpicture}\n \\node at (-1,0) {};\n \\node at (5,0) {};\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (02) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (22) at (2,2) {};\n \\node[shape=circle,draw=black, fill=blue] (4n) at (4,-1) {};\n \\node[shape=circle,draw=black, fill=blue] (41) at (4,1) {};\n \\node[shape=circle,draw=black, fill=blue] (43) at 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(12);\n \\path (00) edge node {} (11);\n \\path (10) edge node {} (21);\n \\path (11') edge node {} (21);\n \\node at (-0.5,2) {\\huge{$v$}};\n \\node at (1.625,0.325) {\\huge{$u$}};\n \\node at (2,-0.75) {\\Huge{$G_8$}};\n \\end{tikzpicture}\n \\quad \n \\begin{tikzpicture}\n \\node at (-1,0) {};\n \\node at (5,0) {};\n \\node[shape=circle,draw=black, fill=blue] (00) at (0,0) {};\n \\node[shape=circle,draw=black, fill=blue] (10) at (2,0) {};\n \\node[shape=circle,draw=black, fill=blue] (01) at (0,2) {};\n \\node[shape=circle,draw=black, fill=blue] (11) at (2,2) {};\n \\node[shape=circle,draw=black, fill=blue] (20) at (4,0) {};\n \\node[shape=circle,draw=black, fill=blue] (21) at (4,2) {};\n \\path (00) edge node {} (10);\n \\path (20) edge node {} (10);\n \\path (01) edge node {} (11);\n \\path (21) edge node {} (11);\n \\path (00) edge node {} (01);\n \\path (10) edge node {} (11);\n \\path (20) edge node {} (21);\n \\path (00) edge node {} (11);\n \\path (10) edge node {} (21);\n \\node at (2,-0.5) {\\huge{$v$}};\n \\node at (0,2.5) {\\huge{$u$}};\n \\node at (2,-1.75) {\\Huge{$G_9$}};\n \\end{tikzpicture}\n }\n \\caption{In each $G_i$, observe that $v\\in \\partial G_i$ with witness $u$, and $d_{G_i}(v,u)=2$.}\n \\label{basecases}\n\\end{figure}\n\n\nFirst, consider $N_{a\\times c}$, and suppose $G$ is the subgraph induced by $V_{a\\times c}^0\\cup (W\\cap V_{a\\times c}^1)$ for axis slice convex $W$. \nIf $a>1$ and $W$ contains both, one, or none of the vertices in $\\{(1\/2,1\/2),(3\/2,1\/2)\\}$, then $G$ will respectively contain $G_1$, $G_2$, or $G_3$ from \\cref{basecases} as a subgraph with $N_G(v)=N_{G_i}(v)$, where in each case, $v$ corresponds to the vertex $(1,0)$ in $N_{a\\times c}$. A similar argument applies if $c>1$, where $v$ corresponds to the vertex $(0,1)$.\nIn these cases, \\cref{boundarysubgraphcor} implies that $v\\in \\partial G$ and $|\\partial G|\\geq 5$. From this, we see that the only graphs $G$ that have four boundary vertices must be constructed using $N_{1\\times 1}$, which are $N_{1\\times 1}$ itself or the cycle graph $C_4$. A direct verification shows $|\\partial G|=4$ for both of these graphs, and they are shown in \\cref{partialG4base}.\n\nNow consider $G=X_{a\\times c}$. Recall that $X_{a\\times c}$ is isomorphic to $X_{c\\times a}$, so assume without loss of generality that $a\\leq c$. \nIf $c\\geq 2$ and $a=2$, then $X_{a\\times c}$ contains $G_4$ as a subgraph with $N_G(v)=N_{G_4}(v)$, where $v$ corresponds to the vertex $(0,1)$ in $X_{a\\times c}$. If $c\\geq a\\geq 2$, then $X_{a\\times c}$ contains $G_5$ as a subgraph with $N_G(v)=N_{G_5}(v)$, where $v$ corresponds to the vertex at $(1\/2,1\/2)$. In these cases, \\cref{boundarysubgraphcor} implies $|\\partial G|\\geq 5$. The graphs $X_{1\\times 1}=K_4$ and $X_{1\\times c}$ are depicted in \\cref{partialG4base}, and a direct calculation shows that when $G$ is one of these graphs, $|\\partial G|=4$.\n\n\nNext, consider $T_{a\\times c}$, and suppose that $G$ is the subgraph of $T_{a\\times c}$ induced by $V_{a\\times (c+1)}^1\\cup (W\\cap V_{a\\times (c+1)}^0)$ for some axis slice convex $W$ that contains $(a,0)$ and $(a,c+1)$. If $a>1$ and $(a-1,c)\\in W$, then $G$ will contain $G_5$ from \\cref{basecases} as a subgraph with $N_G(v)=N_{G_5}(v)$, where $v$ corresponds to $(a-1\/2,c+1\/2)$. If $a>1$ and $(a-1,c)\\not\\in W$, then a similar statement is true for $G_6$, where $v$ again corresponds to $(a-1\/2,c+1\/2)$. If $a=1$ and $c>1$, then $G$ contains $G_7$ from \\cref{basecases} as a subgraph with $N_G(v)=N_{G_7}(v)$, where $v$ corresponds to $(1,1)$. In all of these cases, \\cref{boundarysubgraphcor} implies that $|\\partial G|\\geq 5$. Finally, when $a=c=1$,\nnote that any axis slice convex set containing $(1,0)$ and $(1,2)$ must also contain $(1,1)$. Hence, the only graph in this case is $T_{1\\times 1}$. Direct verification shows $|\\partial T_{1\\times 1}|=4$, and this graph is shown in \\cref{partialG4base}.\n\n\nNow consider $G=D_{a\\times c}$. If $a\\geq 2$ and $c\\geq 2$, then the graph $G_8$ from \\cref{basecases} is a subgraph with $N_G(v)=N_{G_8}(v)$, where $v$ corresponds to the $(0,1)$ in $D_{a\\times c}$.\nIf either $a=1$ and $c\\geq 2$, or $a\\geq 2$ and $c=1$, then $G_9$ from \\cref{basecases} is a subgraph of $G$ with $N_G(v)=N_{G_9}(v)$. Thus in each of these cases, \\cref{boundarysubgraphcor} implies $v\\in \\partial G$ and $|\\partial G|\\geq 5$. Finally, if $a=c=1$ then $|\\partial G|=4$, and $D_{1\\times 1}$ is depicted in \\cref{partialG4base}. \n\n\nFinally, consider $G=L_{a\\times c}$. If $a>1$, then $G$ contains $G_{9}$ as a subgraph with $N_G(v)=N_{G_{9}}(v)$, where $v$ corresponds to $(1,0)$. When $a=1$, observe that by definition, $D_{1\\times c}=L_{1\\times c}$, and we have already considered these cases above.\n\nCombined, we see that the only graphs described in \\cref{mullerthm} parts (d)-(h) with $|\\partial G|=4$ are those formed by attaching paths to the graphs in \\cref{partialG4base} at the vertices in $\\partial G=(\\partial G)'$. By applying \\cref{graphedgeattachment} repeatedly, we conclude that attaching nontrivial paths at $v\\in \\partial G$ preserves the number of boundary vertices if and only if $\\beta(v)=~1$.\n\\end{proof}\n\n\nWe are now able to prove our main theorem characterizing graphs with small Steinerberger boundary.\n\n\\begin{proof}[Proof of \\cref{maintheorem}] \nBy \\cref{CEJZ2} and \\cref{boundary2}, $|\\partial G|=2$ and $|(\\partial G')|=2$ are equivalent, corresponding precisely to paths. This establishes \\cref{maintheorem}(a). Combined with $(\\partial G)'\\subseteq \\partial G$, we also see that a necessary condition for $|\\partial G|=3$ is that $|(\\partial G)'|=3$. Applying \\cref{boundary3lemma}, we conclude that $|\\partial G|=3$ if and only if $|(\\partial G)'|=3$, implying \\cref{maintheorem}(b). \n\nUsing similar reasoning, we see that a necessary condition for $|\\partial G|=4$ is that $|(\\partial G)'|=4$. We consider each case in \\cref{boundary4lemma}. By \\cref{treeandleaves}, any tree on four leaves has $|\\partial G|=4$, and this accounts for \\cref{boundary4lemma} (a) and (b). For \\cref{boundary4lemma} (c), observe that this is the last graph in \\cref{partialG4}. The remaining graphs in \\cref{boundary4lemma} (d) are also given in \\cref{partialG4}. Hence, we conclude \\cref{maintheorem}(c). \n\\end{proof}\n\n\n\n\\section{Some graphs with large boundary}\\label{largeboundary}\n\nIn this section, we consider some graphs with large Steinerberger boundary. Observe that the cycle and complete graphs consist entirely of boundary vertices, as each vertex is peripheral. In the case of $\\diam(G)=2$, Chartrand, Erwin, Johns, and Zhang showed the following result on the CEJZ boundary.\n\n\\begin{lemma}[Chartrand, Erwin, Johns, and Zhang \\cite{chartrand}, Lemma 2.1]\\label{diameter2}\nLet $G=(V,E)$ be connected graph of diameter 2. Then every vertex $v$ is in $(\\partial G)'$ unless $v$ is the unique vertex of $G$ having eccentricity $1$.\n\\end{lemma}\n\nThe same result holds for the Steinerberger boundary. We now show this, with some additional results.\n\n\\begin{theorem}\\label{diameter}\nLet $G=(V,E)$ be a connected graph on at least two vertices with $\\diam(G)\\leq 2$. If $G$ has a single vertex $v$ with eccentricity 1, then $\\partial G=V\\setminus \\{v\\}$. Otherwise, $\\partial G=V$. Furthermore, the bound $\\diam(G)\\leq 2$ is sharp.\n\\end{theorem}\n\n\n\\begin{proof}\nIf $G$ has diameter 1, then $G$ is the complete graph $K_n$, which satisfies $\\partial G=V$. Otherwise, by \\cref{containment} and \\cref{diameter2}, every vertex in G with eccentricity greater than 1 is a Steinerberger boundary vertex. If $G$ has no vertices of eccentricity 1, then $\\partial G=V$. If $G$ has two or more vertices of eccentricity 1, then by \\cref{diameter2}, these vertices are in $\\partial G$. Thus, $\\partial G=V$. \n\nNow suppose that $G$ has a single vertex $v$ of eccentricity $1$. Then $N_G(v)=V\\setminus \\{v\\}$, implying $d(v,w)=1$ for all $w\\in V\\setminus \\{v\\}$.\nLet $u\\in N_G(v)$ satisfy $\\beta(v)=\\beta(v,u)$. Because no vertex other than $v$ has eccentricity $1$, we know that $\\ecc(u)\\geq 2$. In particular, there exists at least one vertex $x\\in N(v)$ such that $d(u,x)= 2$. Therefore,\n\\begin{align*}\n \\beta_{G}(v) & =\\beta_{G}(v,u)\\\\\n &=\\sum_{w\\in N_G(v)}[d(v,u)-d(w,u) ]\\\\\n \n & = [d(v,u)-d(u,u)]+[d(v,u)-d(x,u)] +\\sum_{w\\in N_G(v)\\setminus \\{u,x\\}}[d(v,u)-d(w,u)] \\\\\n &= 1 + (-1) +\\sum_{w\\in N_G(v)\\setminus \\{u,x\\}}[1-d(w,u)]\\\\\n \n &\\leq 0. \\qedhere\n\\end{align*}\n\nTo show that our bound is sharp, let $G=(V,E)$ be the $n$-barbell graph for $n\\geq 2$, which is formed by adding an edge between two disjoint copies of $K_n$, as shown in \\cref{barbell}. Observe that $\\diam(G)=3$. Denote the two $K_n$ graphs $G_1=(V_1,E_1),G_2=(V_2,E_2)$, and let $v_1\\in G_1$, $v_2\\in G_2$ be the vertices where an edge is added. Since $\\beta_{G_1}(v_1)=\\beta_{G_2}(v_2)=1$, \\cref{edgeattachment}, implies that $\\partial G = (\\partial G_1\\cup\\partial G_2)\\setminus \\{v_1,v_2\\}$. Hence, $|\\partial G|=|V|-2$.\n\n\\begin{figure}[h!]\n \\centering\n \\scalebox{0.5}{\n \\begin{tikzpicture}\n \\node[shape=circle,draw=black, fill=red] (A) at (0,1) {};\n \\node[shape=circle,draw=black, fill=red] (B) at (-1,0) {};\n \\node[shape=circle,draw=black, fill=red] (C) at (0,-1) {};\n \\node[shape=circle,draw=black, fill=blue] (D) at (1,0) {};\n \\node[shape=circle,draw=black, fill=red] (A2) at (4,1) {};\n \\node[shape=circle,draw=black, fill=blue] (B2) at (3,0) {};\n \\node[shape=circle,draw=black, fill=red] (C2) at (4,-1) {};\n \\node[shape=circle,draw=black, fill=red] (D2) at (5,0) {};\n \\path (A) edge node {} (B);\n \\path (A) edge node {} (C);\n \\path (A) edge node {} (D);\n \\path (B) edge node {} (C);\n \\path (B) edge node {} (D);\n \\path (C) edge node {} (D);\n \\path (D) edge node {} (B2);\n \\path (A2) edge node {} (B2);\n \\path (A2) edge node {} (C2);\n \\path (A2) edge node {} (D2);\n \\path (B2) edge node {} (C2);\n \\path (B2) edge node {} (D2);\n \\path (C2) edge node {} (D2);\n\\end{tikzpicture}}\n \\caption{The 4-barbell graph with $\\partial G$ shown in red.}\n \\label{barbell}\n\\end{figure}\n\\end{proof}\n\n\n\\begin{corollary}\nLet $G=(V,E)$ be a graph on at least two vertices with minimum degree $\\delta(G)\\geq \\frac{1}{2}(|V|-1)$. If $G$ has a single vertex $u$ of eccentricity $1$, then $\\partial G=V\\setminus \\{u\\}$. Otherwise, $\\partial G=V$. Furthermore, the bound $\\frac{1}{2}(|V|-1)$ is sharp.\n\\end{corollary}\n\n\\begin{proof}\nSuppose $v\\in G$ has $\\ecc(v)>1$. We claim that $\\ecc(v)=2$ by showing that $d(v,u)\\leq 2$ for any $u\\in V$. Let $u\\in V$ be any vertex. If $u\\in N_G(v)$, then $d(v,u)=1$. Otherwise, $u\\notin N_G(v)$ and $v\\notin N_G(u)$. The assumption $\\delta(G)\\geq \\frac{1}{2}(|V|-1)$ implies $|N_G(v)|\\geq \\frac{1}{2}(|V|-1)$ and $|N_G(u)|\\geq \\frac{1}{2}(|V|-1)$. If $N_G(v)\\cap N_G(u)=\\emptyset$, then\n\\[|V|\\geq |N_G(u)\\cup N_G(v)\\cup \\{u,v\\}|=|N_G(u)|+|N_G(v)|+2 \\geq |V|+1,\\]\nwhich is a contradiction. Hence, $d(v,u)\\leq 2$ for all $u$ and $\\ecc(v)=2$. We conclude that when $\\diam(G)\\neq 1$, it must be that $\\diam(G)=2$. The result now follows from \\cref{diameter}, where the $n$-barbell graph also establishes that the bound $\\frac{1}{2}(|V|-1|)$ is sharp, as the minimum degree in the $n$-barbell graph is $\\frac{1}{2}|V|-1<\\frac{1}{2}(|V|-1)$.\n\\end{proof}\n\n\n\n\\section{Open questions}\\label{futurework}\n\nWe have characterized the graphs with $|\\partial G|$ at most 4. Hence, we propose the natural next step.\n\n\\begin{problem}\nClassify connected graphs $G=(V,E)$ for which $|\\partial G|=5$.\n\\end{problem}\n\nOur characterization of $|\\partial G|=4$ relied on a characterization of $|(\\partial G)'|=4$. However, a characterization of graphs with $|(\\partial G)'|=5$ is not currently known. Hence, this problem requires new methods for studying Steinerberger boundary vertices. Since the Steinberberger boundary has more vertices, we expect the characterization problem to be easier than for the CEJZ boundary. \n\nIn \\cref{largeboundary}, we described some graphs where all or almost all vertices are in $\\partial G$. We propose identifying some additional cases when this occurs. Note that a complete characterization is likely difficult. Data on random graphs suggests that they often consist entirely of boundary vertices. \n\\begin{problem}\nDescribe some additional cases when $|\\partial G|=|V|-1$ or $|\\partial G|=|V|$.\n\\end{problem}\n\nIn \\cref{stability}, we characterized when vertices of degree 2 are in $\\partial G$, and one case of our results in \\cref{largeboundary} was on graphs with large minimum degree. We propose a problem related to vertices with low degree.\n\n\\begin{problem}\nCharacterize when vertices of degree $3$ or $4$ are in $\\partial G$. Apply this characterization to find properties of $\\partial G$ for graphs with maximum degree $\\Delta =3$ or $\\Delta= 4$.\n\\end{problem}\n\n\n\nFinally, we propose a question of the boundary stability number, a central tool in establishing our results. To prove our characterization of graphs with small boundary, we explicitly characterized the effect on boundary stability number when adding an edge between two graphs, and we described a case when boundary vertices of a subgraph are also boundary vertices in the graph itself. One natural question is the effect on boundary stability number for other operations.\n\n\\begin{problem}\nDescribe the effect on the boundary stability number of a vertex under other graph operations, such as edge contraction, Cartesian products, etc.\n\\end{problem}\n\n\n\\section*{Acknowledgements}\nWe would like to thank Stefan Steinerberger for suggesting this problem, helpful discussions, and valuable feedback. We would also like to thank Catherine Babecki for valuable feedback and Sam Millard for helpful discussions. Finally, we would like to thank the Washington eXperimental Mathematics Lab for organizing and supporting this project.\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Prolog language was invented 50 years ago and today in 2022 it is still widely used.\nMany possible extensions were and are being defined for Prolog to allow\nit to evolve as a living language.\nThe Oz language was developed starting in 1991\nand it led to a major system release in 1999 called Mozart \\cite{hopl}.\nThe main goal of Oz was to cleanly support many programming paradigms.\nThe Oz language and Mozart system saw wide use for about one decade,\nbut then its use declined.\nAfter 2009 and up to the present day, it is mainly used only for programming education.\nThis is unlike Prolog, which still enjoys wide use up to the present day.\n\nDespite its decline in usage, Oz pioneered many important concepts that\nhave moved into the mainstream since its release.\nWe mention deterministic dataflow computation, actors that return futures,\ndeep support for constraint programming through computation spaces,\nand deep embedding of distributed computing.\nThis is relevant for Prolog because of the\nclose semantic relationship between Oz and Prolog.\nThe semantic foundation of Oz is concurrent constraint programming,\nwhich is a generalization of Prolog's execution model \\cite{saraswat90}.\nProlog programs with logical semantics can be directly\ntranslated into Oz, as explained in Section \\ref{syntax}.\nGiven these facts, we believe that\nthe design of Oz can be an inspiration for the future evolution of Prolog,\nso that Prolog can enjoy another 50 years of success.\nThe purpose of this article is to present three important extensions\nto Prolog-style logic programming made by Oz and explain why\nthey can be an inspiration to future Prolog development:\n\\begin{itemize}\n\\item {\\em Deterministic logic programming} (Section \\ref{dlp}).\nWe show how Oz supports writing logic programs that support\nmanaging deterministic execution while maintaining their logical semantics.\nFor a given logical semantics, we show how Oz can express different operational\nsemantics.\n\n\\item {\\em Lazy concurrent functional programming} (Section \\ref{lcfp}).\nWe show how the logic programming core of Oz can be the foundation of a series\nof functional programming paradigms, culminating in lazy deterministic dataflow.\nAll these paradigms keep the strong confluence properties of pure functional\nprogramming as well as their origin in deterministic logic programming.\n\n\\item {\\em Purely functional distributed computing} (Section \\ref{deep}).\nThe Oz community built a distributed implementation of Oz based on a deep embedding\napproach, where each language entity has its own distributed algorithm.\nAs part of this work, we designed and proved correctness of distributed\nrational tree unification.\nBased on the use of logic variables for dataflow synchronization,\nwe showed that asynchronous message passing can be purely functional.\nFinally, we showed that most practical distributed systems used today\ngreatly overuse nondeterminism, and that it would be better to design\nthem as mostly functional, adding nondeterminism only in the few places where\nit is needed.\n\\end{itemize}\nEach of these extensions is presented separately in its own section,\nIn each case, we explain the main ideas and we give references for readers\nwishing to investigate further.\nAs a preliminary step, Section \\ref{syntax} explains how to translate Prolog\nprograms into Oz, to justify our claim that Oz is a syntactic variant of Prolog.\n\n\\section{Oz is a syntactic variant of Prolog}\n\\label{syntax}\n\nWe explain how to translate any pure Prolog program\nextended with green or blue cuts and\nthe \\verb+bagof\/3+ or \\verb+setof\/3+ predicates\ninto an Oz program with\nthe same logical and operational semantics as the Prolog program.\nWe assume the reader has enough knowledge of Prolog to understand our examples.\nFor brevity, we do not explain the semantics of the Oz statements in detail.\nTheir meaning should be clear from the examples.\nFor a more complete and formal explanation of the relationship\nbetween Oz and Prolog, we refer the reader to Chapter 9 of \\cite{ctm}.\n\nBoth Oz and Prolog support symbolic data structures that are bound\nusing unification.\nSimilar to many modern Prolog systems, Oz uses rational tree unification.\nThere are slight differences in syntax and semantics\nbetween Oz and Prolog data structures, mainly because Oz supports additional\ndata structures such as records.\nWe do not explain these differences here;\nwe refer the reader to \\cite{ctm} for a precise definition of Oz data structures.\n\n\\subsection{Deterministic predicates}\nA predicate is deterministic if all its clauses are logically disjoint.\nThis is the case when the clauses have disjoint guards.\nDeterministic predicates are translated using the Oz statements\n\\OzInline{\\OzKeyword{if}} and \\OzInline{\\OzKeyword{case}}, which both have a logical semantics.\nConsider the following deterministic Prolog predicate:\n\\begin{verbatim}\n place_queens(I, _, _, _) :- I==0, !.\n place_queens(I, Cs, Us, [_|Ds]) :-\n I>0, J is I-1,\n place_queens(J, Cs, [_|Us], Ds),\n place_queen(I, Cs, Us, Ds).\n\\end{verbatim}\nThis predicate has a\n\\index{blue cut (in Prolog)|emph}blue cut according\nto \\index{O'Keefe, Richard}O'Keefe~\\cite{craft},\ni.e., the cut is needed \nto inform naive implementations that the\npredicate is deterministic, so they can improve efficiency,\nbut it does not change the program's results.\nThe predicate \\verb+place_queens\/4+ is translated into the following Oz code: \n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueens\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{if}\\OzSpace{1}N==0\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}\\OzKeyword{skip}\\OzEol\n\\OzSpace{4}\\OzKeyword{elseif}\\OzSpace{1}N>0\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}Ds2\\OzSpace{1}Us2=\\OzChar\\_|Us\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}Ds=\\OzChar\\_|Ds2\\OzEol\n\\OzSpace{7}\\OzChar\\{PlaceQueens\\OzSpace{1}N-1\\OzSpace{1}Cs\\OzSpace{1}Us2\\OzSpace{1}Ds2\\OzChar\\}\\OzEol\n\\OzSpace{7}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzSpace{1}\\OzKeyword{fail}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\n\n\\subsection{Nondeterministic predicates}\nA predicate is nondeterministic when its clauses are not logically disjoint.\nSuch predicates can be backtracked into, possibly giving multiple solutions.\nNondeterministic predicates are translated using the Oz statement \\OzInline{\\OzKeyword{choice}}.\nConsider the following nondeterministic Prolog predicate:\n\\begin{verbatim}\n place_queen(N, [N|_], [N|_], [N|_]).\n place_queen(N, [_|Cs2], [_|Us2], [_|Ds2]) :-\n place_queen(N, Cs2, Us2, Ds2).\n\\end{verbatim}\nThis predicate is translated into a nondeterministic Oz procedure:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{choice}\\OzSpace{1}N|\\OzChar\\_\\OzSpace{2}=Cs\\OzSpace{1}N|\\OzChar\\_\\OzSpace{2}=Us\\OzSpace{1}N|\\OzChar\\_\\OzSpace{2}=Ds\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}\\OzChar\\_|Cs2=Cs\\OzSpace{1}\\OzChar\\_|Us2=Us\\OzSpace{1}\\OzChar\\_|Ds2=Ds\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs2\\OzSpace{1}Us2\\OzSpace{1}Ds2\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\n\n\\subsection{First-class Prolog top level}\nThe previous two predicates are part of a program that computes solutions\nto the $n$-queens problem: how to place $n$ queens on an $n \\times n$ chessboard so that\nno queen attacks another.\nFor completeness, we give the full program here:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Queens\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{MakeList\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{7}\\OzKeyword{if}\\OzSpace{1}N==0\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{else}\\OzSpace{1}\\OzChar\\_|\\OzChar\\{MakeList\\OzSpace{1}N-1\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}Qs=\\OzChar\\{MakeList\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueens\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzSpace{1}\\OzComment{\\OzSpace{1}as\\OzSpace{1}before\\OzSpace{1}}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{PlaceQueen\\OzSpace{1}N\\OzSpace{1}Cs\\OzSpace{1}Us\\OzSpace{1}Ds\\OzChar\\}\\OzSpace{1}\\OzComment{\\OzSpace{1}as\\OzSpace{1}before\\OzSpace{1}}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{4}\\OzChar\\{PlaceQueens\\OzSpace{1}N\\OzSpace{1}Qs\\OzSpace{1}\\OzChar\\_\\OzSpace{1}\\OzChar\\_\\OzChar\\}\\OzEol\n\\OzSpace{4}Qs\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThis program is run by calling the system procedure\n\\OzInline{SolveOne} which corresponds to\na Prolog top-level query:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzChar\\{Browse\\OzSpace{1}\\OzChar\\{SolveOne\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}\\OzChar\\{Queens\\OzSpace{1}8\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzChar\\}\\end{oz2texdisplay}\nThe syntax \\OzInline{\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}...\\OzSpace{1}\\OzKeyword{end}} defines a zero-argument\nfunction that calls the one-argument function \\OzInline{Queens}.\nThe system procedure \\OzInline{Browse} displays its argument.\nThe answer displayed by \\OzInline{Browse} is a list giving the first solution:\n\\begin{oz2texdisplay}\\OzSpace{1}[[1\\OzSpace{1}7\\OzSpace{1}5\\OzSpace{1}8\\OzSpace{1}2\\OzSpace{1}4\\OzSpace{1}6\\OzSpace{1}3]]\\end{oz2texdisplay}\nOz also provides \\OzInline{SolveAll} to compute\nall solutions and \\OzInline{Solve} to compute a lazy list of solutions.\nThe \\OzInline{Solve} corresponds closely to a Prolog top level where solutions\nare computed on demand.\n\n\\subsection{Predicates with green or blue cuts}\n\nIf your Prolog program uses cut ``!'', then the translation to Oz is simple if the\ncut is a green or blue cut as defined by O'Keefe \\cite{craft}. A green cut removes\nirrelevant solutions. A blue cut indicates to the\ncompiler that the program is deterministic.\nTo show the translation scheme, we translate the following predicate:\n\\begin{verbatim}\n foo(X, Z) :- guard1(X, Y), !, body1(Y, Z).\n foo(X, Z) :- guard2(X, Y), !, body2(Y, Z).\n\\end{verbatim}\nThe two guards must not bind any head variables, i.e., they are quiet guards.\nIt is good Prolog style to postpone\nbinding head variables until after the cut.\nThe translation has two cases, depending on\nwhether the guards are deterministic or not.\nIf a guard is deterministic (it has no \\OzInline{\\OzKeyword{choice}}),\nthen it can be written as a deterministic boolean function.\nThis gives the following simple translation:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Foo\\OzSpace{1}X\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{if}\\OzSpace{5}Y\\OzSpace{1}\\OzKeyword{in}\\OzSpace{1}\\OzChar\\{Guard1\\OzSpace{1}X\\OzSpace{1}Y\\OzChar\\}\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}\\OzChar\\{Body1\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{elseif}\\OzSpace{1}Y\\OzSpace{1}\\OzKeyword{in}\\OzSpace{1}\\OzChar\\{Guard2\\OzSpace{1}X\\OzSpace{1}Y\\OzChar\\}\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}\\OzChar\\{Body2\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzSpace{1}\\OzKeyword{fail}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nIf a guard is nondeterministic (it uses \\OzInline{\\OzKeyword{choice}}),\nthen it can be written with one input\nand one output argument, like this: \\OzInline{\\OzChar\\{Guard1\\OzSpace{1}In\\OzSpace{1}Out\\OzChar\\}}.\nIt should not bind the input argument.\nThis gives the following translation:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Foo\\OzSpace{1}X\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{case}\\OzSpace{1}\\OzChar\\{SolveOne\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}\\OzChar\\{Guard1\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzSpace{1}\\OzKeyword{of}\\OzSpace{1}[Y]\\OzSpace{1}\\OzKeyword{then}\\OzEol\n\\OzSpace{7}\\OzChar\\{Body1\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{elsecase}\\OzSpace{1}\\OzChar\\{SolveOne\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzChar\\}\\OzSpace{1}\\OzChar\\{Guard2\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzSpace{1}\\OzKeyword{of}\\OzSpace{1}[Y]\\OzSpace{1}\\OzKeyword{then}\\OzEol\n\\OzSpace{7}\\OzChar\\{Body2\\OzSpace{1}Y\\OzSpace{1}Z\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzSpace{1}\\OzKeyword{fail}\\OzSpace{1}\\OzKeyword{then}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nIf neither of these two cases apply to your Prolog program, e.g.,\neither your guards bind head variables\nor you use cuts in other ways (i.e., as red cuts),\nthen your program likely does not have a logical semantics.\nA program with red cuts is defined only by its operational semantics\nand this is outside the scope of our translation scheme.\n\n\\subsection{The {\\tt bagof\/3} and {\\tt setof\/3} predicates}\n\nProlog's \\verb+bagof\/3+ predicate\ncorresponds to using \\OzInline{SolveAll} inside an Oz program.\nIts extension \\verb+setof\/3+ sorts the result and removes duplicates.\nThis can be done with the Oz built-in \\OzInline{Sort} operation.\nWe show how to translate \\verb+bagof\/3+ both without and with existential quantification.\nConsider the following small biblical database\n(inspired by~\\cite{artofprolog}):\n\\begin{verbatim}\n father(terach, abraham).\n father(abraham, isaac).\n father(haran, milcah).\n father(haran, yiscah).\n\\end{verbatim}\nThis can be written as follows in Oz:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}F\\OzSpace{1}C\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{choice}\\OzSpace{1}F=terach\\OzSpace{2}C=abraham\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}F=abraham\\OzSpace{1}C=isaac\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}F=haran\\OzSpace{3}C=milcah\\OzEol\n\\OzSpace{4}[]\\OzSpace{5}F=haran\\OzSpace{3}C=yiscah\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay} \nCalling \\verb+bagof\/3+ without existential quantification, e.g.:\n\\begin{verbatim}\n children1(X, Kids) :- bagof(K, father(X,K), Kids).\n\\end{verbatim}\nis defined as follows with \\OzInline{SolveAll}:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Children1\\OzSpace{1}X\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{SolveAll\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe \\OzInline{Children1} definition is deterministic;\nit assumes \\OzInline{X} is known and it returns \\OzInline{Kids}.\nTo search over different values of \\OzInline{X}\nthe following definition should be used instead:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Children1\\OzSpace{1}X\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}\\OzChar\\_\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{SolveAll\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe call \\OzInline{\\OzChar\\{Father\\OzSpace{1}X\\OzSpace{1}\\OzChar\\_\\OzChar\\}} creates a choice point on \\OzInline{X}.\nThe ``\\OzInline{\\OzChar\\_}'' is syntactic sugar for \\OzInline{\\OzKeyword{local}\\OzSpace{1}X\\OzSpace{1}\\OzKeyword{in}\\OzSpace{1}X\\OzSpace{1}\\OzKeyword{end}},\nwhich is just a new variable with a very small scope. \n\n\\noindent\nCalling \\verb+bagof\/3+ with existential quantification, e.g.:\n\\begin{verbatim}\n children2(Kids) :- bagof(K, X^father(X,K), Kids).\n\\end{verbatim}\nis defined as follows with \\OzInline{SolveAll}:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Children2\\OzSpace{1}?Kids\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{SolveAll\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzChar\\{Father\\OzSpace{1}\\OzChar\\_\\OzSpace{1}K\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzSpace{1}Kids\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe Oz solution uses \\OzInline{\\OzChar\\_} to add a new existentially scoped variable.\nThe Prolog solution, on the other hand, introduces\na new syntactic concept, namely the\n``existential quantifier'' \\verb+X^+,\nwhich only has meaning in terms of \\verb+setof\/3+ and \\verb+bagof\/3+.\nThe fact that this notation denotes an existential quantifier\nis defined explicitly in the Prolog semantics.\nThe Oz solution, on the other hand, requires no new semantics.\n\nIn addition to doing all-solutions \\verb+bagof\/3+,\nOz programs can do a lazy \\verb+bagof\/3+,\ni.e., where each new solution is calculated on demand.\nLazy \\verb+bagof\/3+ is done by \\OzInline{Solve}, which\nreturns a lazy list of solutions.\n\n\\section{Deterministic logic programming}\n\\label{dlp}\n\nWriting a logic program in Prolog or another logic language consists in defining\nthe logical semantics and then choosing an operational semantics that gives a\nsatisfactory efficiency. \nThis follows Kowalski's equation ``Algorithm = Logic + Control''.\nLogic and control need to be balanced.\nThe art of logic programming consists in balancing two \nconflicting tensions: the logical semantics should be simple and the operational\nsemantics should be efficient. \nWhen done well, this gives an elegant style in Prolog \\cite{craft}.\nOz supports this kind of program design in logic programming by supporting both\ndeterministic and nondeterministic control flow.\nFor example, in Prolog we can define a list append predicate as follows:\n\\begin{verbatim}\n append([], L2, L2).\n append([X|M1], L2, [X|M3]) :- append(M1, L2, M3).\n\\end{verbatim}\nThis definition follows Prolog's operational semantics.\nCompilers can optimize this to make it deterministic in certain cases,\nfor example if the first argument is bound to a list.\nIn Oz we can define the operational semantics more precisely.\nLet us show three ways that the append can be defined in Oz.\n\n\\subsection{Nondeterministic append}\n\nIn our first definition, we define\nthe append nondeterministically:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}L1\\OzSpace{1}L2\\OzSpace{1}L3\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{choice}\\OzEol\n\\OzSpace{7}L1=nil\\OzSpace{2}L3=L2\\OzEol\n\\OzSpace{4}[]\\OzSpace{1}X\\OzSpace{1}M1\\OzSpace{1}M3\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}L1=X|M1\\OzSpace{1}L3=X|M3\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}M1\\OzSpace{1}L2\\OzSpace{1}M3\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThis has the same operational semantics as Prolog.\n\n\\subsection{Deterministic append (first version)}\n\nWe give another definition of append that has the same logical semantics\nas before but a deterministic operational semantics:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}A\\OzSpace{1}B\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{case}\\OzSpace{1}A\\OzEol\n\\OzSpace{4}\\OzKeyword{of}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}B\\OzEol\n\\OzSpace{4}[]\\OzSpace{1}X|As\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}X|\\OzChar\\{Append\\OzSpace{1}As\\OzSpace{1}B\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nIn this case, argument \\OzInline{A} is bound to a list so execution is directional.\nOz allows to use a functional syntax for such definitions.\n\n\\subsection{Deterministic append (second version)}\n\nWe give yet another definition that again has the same logical semantics\nbut a second deterministic operational semantics:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}B\\OzSpace{1}C\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{if}\\OzSpace{1}B==C\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}nil\\OzEol\n\\OzSpace{4}\\OzKeyword{else}\\OzEol\n\\OzSpace{7}\\OzKeyword{case}\\OzSpace{1}C\\OzSpace{1}\\OzKeyword{of}\\OzSpace{1}X|Cs\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}X|\\OzChar\\{Append\\OzSpace{1}B\\OzSpace{1}Cs\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThis version of \\OzInline{Append} takes the last two arguments as inputs\nand returns the first argument as its output.\nFor example, \\OzInline{\\OzChar\\{Append\\OzSpace{1}[3]\\OzSpace{1}[1\\OzSpace{1}2\\OzSpace{1}3]\\OzChar\\}} returns \\OzInline{[1\\OzSpace{1}2]}.\nCorrect execution requires that the second argument is a suffix of the third argument.\n\n\\section{Lazy concurrent functional programming}\n\\label{lcfp}\n\nAn important insight of the Oz project was that the logic programming core\nof Oz can also support functional programming paradigms.\nWe show this in four steps:\n\\begin{itemize}\n\\item {\\em Functional programming with values}.\nWidely used functional programming languages, such as Scheme or Haskell, compute\nwith values.\nThis can be done in Oz simply by not using unbound logic variables.\nTo support higher-order programming,\nwe extend the Oz computation model with function values.\nVariables can be bound to function values, giving traditional eager functional programming.\n\\item {\\em Functional programming with logic variables}.\nWe add logic variables to functional programming with values. \nThe bind operation is unification.\nThis gives exactly the deterministic logic programming paradigm of Section \\ref{dlp}.\nThis paradigm has a surprising benefit: many more function definitions become tail-recursive.\n\\item {\\em Deterministic dataflow}.\nWe add concurrency to functional programming with logic variables.\nWe allow any statement to execute in its own thread where\nthreads synchronize on variable binding.\nThis is exactly the synchronization model of concurrent logic programming.\nCompared to concurrent logic programming, however, we are still purely functional.\n\\item {\\em Lazy deterministic dataflow}.\nThe final extension adds on-demand computation. We add a new synchronization operation\nthat waits until another thread waits on a variable being bound.\nWith this new primitive operation (called \\OzInline{WaitNeeded}), we have added all the power\nof lazy evaluation to deterministic dataflow.\n\\end{itemize}\nWe examine these four steps in more detail.\n\n\\subsection{Functional programming with values}\n\nWe extend the computation model with function values.\nWe allow variables to be bound to lexically scoped closures.\nThis gives pure functional programming with eager evaluation.\nPure functional programming is the foundation of higher-order\nprogramming, which underlies most of the development of data abstraction, such\nas object-oriented programming, abstract data types,\ncomponents, templates, metaclasses, and so forth.\n\nIn addition to being a foundation for data abstraction,\nthis model has strong formal properties, \nwhich are different from but analogous to the strong properties of logic programming. \nThe main property is known as the Church-Rosser property, often referred to\nas confluence, which states that the final result of an expression reduction\nis independent of the reduction choices made during the reduction.\nIn what follows, we show how Oz exploits the synergy that is obtained by combining\nconfluence with logic variables.\n\n\\subsection{Functional programming with logic variables}\n\nWe now extend functional programming with logic variables.\nThis corresponds exactly to the \ndeterministic logic programming of Section \\ref{dlp},\nso this form of functional programming is in fact doing logic programming.\nWhat have we gained from this extension?\nFirst of all, it is now possible to write functional programs\nwith partially instantiated data structures, just like in Prolog.\n\nBut we have gained much more than just the use of partial data structures.\nAllowing unbound logic variables makes possible reduction orders that\nare impossible in a functional language based on values only.\nFor example, the two deterministic append predicates defined in\nSection \\ref{dlp} are both tail-recursive,\nwhich is not possible in a functional language\nbased on values without doing complex program transformation.\n\nLet us examine precisely how adding logic variables makes functions tail-recursive.\nWe start with the first deterministic append from the previous section.\nSection \\ref{dlp} defines it syntactically as a function,\nbut in Oz functions are just syntactic sugar for procedures.\nThe deterministic append is actually defined as a procedure:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Append\\OzSpace{1}A\\OzSpace{1}B\\OzSpace{1}C\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{case}\\OzSpace{1}A\\OzEol\n\\OzSpace{4}\\OzKeyword{of}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}C=B\\OzEol\n\\OzSpace{4}[]\\OzSpace{1}X|As\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}Cs\\OzSpace{1}\\OzKeyword{in}\\OzEol\n\\OzSpace{7}C=X|Cs\\OzEol\n\\OzSpace{7}\\OzChar\\{Append\\OzSpace{1}As\\OzSpace{1}B\\OzSpace{1}Cs\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nLook at the recursive call, \\OzInline{\\OzChar\\{Append\\OzSpace{1}As\\OzSpace{1}B\\OzSpace{1}Cs\\OzChar\\}}.\nThis call comes after the binding \\OzInline{C=X|Cs} which\nincrementally builds the output of the append.\n\\OzInline{C} is bound to a new cons cell consisting of \\OzInline{X} paired with\na new unbound variable \\OzInline{Cs}.\nThe recursive call can be a tail call because we pass it the unbound variable \\OzInline{Cs}.\nThis is not possible in functional programming with values only.\n\n\\subsection{Deterministic dataflow}\n\nWe extend the previous paradigm by adding threads and using logic variables\nfor synchronization.\nTo understand this paradigm, we first explain the dataflow \nbehavior of logic variables. Assume we declare a logic variable \\OzInline{X}.\nIn a first thread, we do \\OzInline{X=10}. In a second thread, we do \\OzInline{Y=X+1}.\nBecause of the scheduler, we do not know which thread will run first.\nThe second thread, if it is scheduled first, will suspend when it attempts\nto do the addition. It will wait until \\OzInline{X} is bound.\nWhen the first thread binds \\OzInline{X}, then the second thread becomes runnable again\nand the addition will complete, binding \\OzInline{Y} to 11.\nTo summarize, this is a declarative form of dataflow\nwith two basic operations, namely binding a logic variable and waiting\nuntil the variable is bound.\n\nDataflow synchronization is such an important use of logic variables\nthat we call them {\\em dataflow variables}\nwhen we explain the history of Oz \\cite{hopl}.\nThe Oz community did not invent this; it was pioneered in concurrent logic programming,\nwhich was an early development of Prolog for concurrent systems \\cite{lncs259*30,tr003}.\nThe Oz contributions to this concept are its use for purely functional\nconcurrent and distributed computing and its extension to lazy evaluation.\n\nIn deterministic dataflow,\nany instruction can execute in its own thread.\nThis lets us define networks of concurrent agents that communicate through streams,\nwhere we define a {\\em stream} as a list\nthat is built incrementally and that may have an unbound tail.\nFor example, consider the following sequential functional program:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{Gen\\OzSpace{1}L\\OzSpace{1}H\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzChar\\{Delay\\OzSpace{1}1000\\OzChar\\}\\OzSpace{1}\\OzEolComment{\\OzSpace{1}Suspend\\OzSpace{1}thread\\OzSpace{1}execution\\OzSpace{1}for\\OzSpace{1}1000\\OzSpace{1}ms}\\OzSpace{4}\\OzKeyword{if}\\OzSpace{1}L>H\\OzSpace{1}\\OzKeyword{then}\\OzSpace{1}nil\\OzSpace{1}\\OzKeyword{else}\\OzSpace{1}L|\\OzChar\\{Gen\\OzSpace{1}L+1\\OzSpace{1}H\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzEol\n\\OzSpace{1}Xs=\\OzChar\\{Gen\\OzSpace{1}1\\OzSpace{1}10\\OzChar\\}\\OzEol\n\\OzSpace{1}Ys=\\OzChar\\{Map\\OzSpace{1}Xs\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}X*X\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzChar\\{Browse\\OzSpace{1}Ys\\OzChar\\}\\end{oz2texdisplay}\nThis computes a list of successive integers, squares each element of this list,\nand displays the result.\nTo follow the execution during an interactive session,\nwe have slowed down the generation so it takes\none second per element.\nWe can make this concurrent by doing\nthe generation and mapping in their own threads:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{thread}\\OzSpace{1}Xs=\\OzChar\\{Gen\\OzSpace{1}1\\OzSpace{1}10\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{thread}\\OzSpace{1}Ys=\\OzChar\\{Map\\OzSpace{1}Xs\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}\\OzChar\\{\\OzChar\\$\\OzSpace{1}X\\OzChar\\}\\OzSpace{1}X*X\\OzSpace{1}\\OzKeyword{end}\\OzChar\\}\\OzSpace{1}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzChar\\{Browse\\OzSpace{1}Ys\\OzChar\\}\\end{oz2texdisplay}\nThis uses the \\OzInline{\\OzKeyword{thread}} statement to create new threads.\nWhat is the difference between the concurrent and\nthe sequential versions?\nThe result of the calculation is the same in both\ncases, namely \\OzInline{[1\\OzSpace{1}4\\OzSpace{1}9\\OzSpace{1}16\\OzSpace{1}...\\OzSpace{1}81\\OzSpace{1}100]}.\nSo what is different?\nIn the sequential version, \\OzInline{Gen} calculates\nthe whole list before \\OzInline{Map} starts.\nThe final result is displayed all at once\nwhen the calculation is complete, which happens after ten seconds.\nIn the concurrent version, \\OzInline{Gen} and \\OzInline{Map} both\nexecute concurrently.\nWhenever \\OzInline{Gen} adds an element to its list,\n\\OzInline{Map} will immediately calculate its square\nbefore the next element is added.\nThe result is displayed incrementally\nas the elements are generated,\none element each second.\nConcurrency has converted a batch computation into an incremental computation\nwithout changing the functional semantics.\n\nThe concurrent executions of \\OzInline{Gen} and \\OzInline{Map} can be considered as\nfunctional agents, where we define an {\\em agent} as a concurrent computation that\nreads zero or more input streams and writes zero or more output streams.\nBecause of logic variables, both \\OzInline{Gen} and \\OzInline{Map} are tail-recursive.\nThis means that the agents execute with constant stack size.\nThis justifies calling these computations ``agents''.\nThe concurrent execution is memory efficient as well as being purely functional.\nThis shows clearly the synergy between logic variables and concurrency.\n\n\\subsection{Lazy deterministic dataflow}\n\nThe final extension adds the ability to do lazy evaluation to deterministic dataflow.\nIn the previous section,\nwe explained how logic variables are used for dataflow synchronization.\nWe now extend this dataflow paradigm to do lazy evaluation.\nWe do this by adding one new operation, \\OzInline{WaitNeeded}, which does by-need synchronization \non logic variables.\nThe operation \\OzInline{\\OzChar\\{WaitNeeded\\OzSpace{1}X\\OzChar\\}} suspends the\ncurrent thread if \\OzInline{X} is unbound and no other thread is waiting for \\OzInline{X} to be bound.\nOtherwise, if another thread is suspended on \\OzInline{X} or if \\OzInline{X} is bound,\nthe operation succeeds.\nWe say that \\OzInline{WaitNeeded} ``waits until \\OzInline{X} is needed''.\nAdding this single operation to the language lets us fully define lazy evaluation.\nOz introduces a syntax sugar to make this easy for the programmer.\nFor example, we define the function \\OzInline{\\OzChar\\{Ints\\OzSpace{1}N\\OzChar\\}}\nthat returns a lazy list of successive integers starting with \\OzInline{N}:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{fun}\\OzSpace{1}lazy\\OzSpace{1}\\OzChar\\{Ints\\OzSpace{1}N\\OzChar\\}\\OzEol\n\\OzSpace{4}N|\\OzChar\\{Ints\\OzSpace{1}N+1\\OzChar\\}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nThe ``\\OzInline{lazy}'' annotation means that the function does lazy evaluation.\nIt is syntactic sugar for the following procedure:\n\\begin{oz2texdisplay}\\OzSpace{1}\\OzKeyword{proc}\\OzSpace{1}\\OzChar\\{Ints\\OzSpace{1}N\\OzSpace{1}R\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{thread}\\OzEol\n\\OzSpace{7}\\OzChar\\{WaitNeeded\\OzSpace{1}R\\OzChar\\}\\OzEol\n\\OzSpace{7}R=N|\\OzChar\\{Ints\\OzSpace{1}N+1\\OzChar\\}\\OzEol\n\\OzSpace{4}\\OzKeyword{end}\\OzEol\n\\OzSpace{1}\\OzKeyword{end}\\end{oz2texdisplay}\nCalling \\OzInline{S=\\OzChar\\{Ints\\OzSpace{1}1\\OzChar\\}} will suspend in the \\OzInline{WaitNeeded} call\nuntil another thread needs the first element of \\OzInline{S} to run. When\nthis happens, the \\OzInline{WaitNeeded} call succeeds and \\OzInline{R} is bound to a list\nwith one new element.\nThe recursive call of \\OzInline{Ints} continues this behavior for the next elements.\nWe note that the compiler can optimize the above definition by reusing the thread,\nto avoid the creation of a new thread for each recursive call.\nThe combination of lazy evaluation and concurrency has been known at least since \n1977 \\cite{kahn77}, but as far as we know,\nthe definition of lazy evaluation in a dataflow paradigm and\nits connection to logic programming are both original with Oz.\n\n\\paragraph{Summary} We summarize the paradigms introduced in this section.\nWe start with a first-order paradigm that computes with values.\nWe extend this paradigm in four steps.\nFirst, we add function values, i.e., lexically scoped closures,\nwhich gives pure functional programming with eager evaluation.\nSecond, we add logic variables,\nwhich gives deterministic logic programming as seen in Section \\ref{dlp}.\nA side benefit is that functions that compute recursive data structures such as lists\nbecome tail-recursive.\nThird, we add threads and use\nthe dataflow behavior of logic variables to do synchronization,\nwhich gives deterministic dataflow.\nThis supports concurrent networks of functional agents that communicate\nthrough shared lists used as communication channels, which we call streams.\nNote that because list functions are tail-recursive, these agents use constant stack space.\nFourth, in the final step, we add by-need synchronization using \\OzInline{WaitNeeded}.\nThis gives a paradigm that supports both lazy evaluation and concurrency,\nwhich we call lazy deterministic dataflow.\nAll four paradigms keep the strong confluence properties\nof pure functional programming, as well as being conservative extensions to a\nlanguage supporting deterministic logic programming.\nSince Prolog is identical to a subset of Oz, we propose that some form of these extensions\ncould become part of a future evolution of Prolog.\n\n\\section{Purely functional distributed computing}\n\\label{deep}\n\nThe Oz research community realized early on that \nthe Oz language design would be a good starting point\nfor building a distributed programming system.\nBecause the design cleanly separates immutable, dataflow, and mutable language entities,\nit would be possible to use a {\\em deep embedding approach},\nwhere each language entity would be implemented with its own distributed algorithm.\nThe distribution support was part of the Mozart system at its first release in 1999.\nThis design had many important innovations including the following:\n\\begin{itemize}\n\\item {\\em Deep embedding}.\nEach language concept was implemented with a distributed algorithm.\nThis means that applications written for one distribution structure\ncan easily be ported to another distribution structure without changing the source code.\nThe only differences are failure behavior and timing. Failure behavior can be handled\nin a modular way, inspired by techniques from Erlang \\cite{collet,erlang}.\n\\item {\\em Distributed rational tree unification}.\nThe concurrent examples of Section \\ref{lcfp} can all be run with unchanged source\ncode if distributed across different compute nodes.\nConcurrent pipelines become asynchronous distributed pipelines, while maintaining\nthe semantics of pure functional programming.\nA key part of this system is the distributed rational tree unification algorithm.\nThe first formal specification and proof of distributed unification was done during\nthis work and published in \\cite{toplas99}.\n\\item {\\em Global references}, {\\em distributed lexical scoping},\n{\\em distributed garbage collection}, and {\\em migratory state}.\nThese are some of the other properties of this design.\n\\end{itemize}\nIn the rest of this section we will focus on what we consider to be\nthe most significant discovery that was made in the work on distributed\ncomputing for Oz. This discovery is intimately tied to the logic programming\norigins of Oz and it may open a window of opportunity for Prolog.\n\n\\paragraph{Asynchronous message passing can be pure}\nThe deterministic data\\-flow paradigm and its lazy extension\ncan be implemented in a distributed setting.\nA dataflow variable can be read on one compute node and bound on another node.\nThis does a distributed synchronization, which in its general form is implemented\nby distributed unification.\nThe salient property of this distributed synchronization is that it enjoys all\nthe strong properties of pure functional programming.\nSpecifically, it means that {\\em asynchronous message passing between compute nodes\ncan be completely pure}.\nThis property remains mostly unknown in the distributed computing community.\nIn our experience as members of this community,\nwe observe that many often insist that asynchronous message passing\nis intrinsically impure, which is false.\n\n\\paragraph{Overuse of nondeterminism}\nMost of today's large distributed systems do not take advantage of this property.\nIn fact, the contrary is true: {\\em today's systems greatly overuse nondeterminism}.\nThis is one of the main reasons\nwhy it is still difficult to build and debug such systems.\nLarge distributed systems, such as used daily by Google, Facebook, Twitter, and so on,\nmust be continually babysitted by experts.\nThis is not an inevitable property of such systems.\nIn our view, it is mainly due to the massive overuse of nondeterminism in the design\nof these systems.\nEvery single library that is a part of such a system and that has its own API, which\nis true of most libraries, is a point of nondeterminism.\n\n\\paragraph{Purely functional distributed computing}\nThe problem of overuse of nondeterminism and a possible solution\nare explained in a keynote talk given by one of the authors at CodeBEAM 2019,\n{\\em Why Time is Evil in Distributed Systems and What To Do About It} \\cite{evil}.\nThe solution we propose is to use deterministic dataflow programming\nas the default paradigm for distributed systems and to add nondeterminism\nonly where it is needed and nowhere else.\nWith this solution, most distributed systems become mostly functional\nwith a limited use of nondeterminism.\nThis greatly simplifies their development, maintenance, and evolution.\nWe observe that this solution is intimately tied to logic programming,\nsince the basic concept that makes it work is the dataflow variable,\nwhich is simply a logic variable used in a concurrent setting.\n\n\\paragraph{Role of logic programming}\nWe believe that an appropriate extension of Prolog or another logic language\ncan potentially solve this problem\nand play a significant role in distributed computing.\nIf such an extension is not done by the logic programing community,\nthen the distributed computing community will\nreinvent concepts from logic programming as it solves these problems.\nThis reinvention has already started with the development of CRDTs (Conflict-Free\nReplicated Data Types), which have monotonic properties and can replace consensus\nalgorithms, which are nondeterministic,\nby purely functional operations in many cases \\cite{crdt}.\n\n\\section{Conclusions}\n\\label{conclusions}\n\nProlog has enjoyed a relatively large popularity since its conception in 1972\nup to the present day,\nunlike Oz, which despite having many innovations at its release in 1999,\nhas seen a decline since 2009 and is only used today for education.\nOz declined for sociological reasons that have nothing to do with its\ninnovations; in fact we observe that the innovations are all becoming\nadopted in modern programming systems despite the decline of Oz.\nAs members of both the Prolog and Oz communities,\nwe see this as an opportunity to help Prolog evolve\nin order to maintain its popularity.\n\nThis paper presents three innovations of Oz,\nnamely deterministic logic programming, lazy concurrent functional programming,\nand purely functional distributed computing.\nWe propose that it would be straightforward to add these innovations to Prolog\nbecause the core logic languages of Oz and Prolog are quite similar.\nMost of pure Prolog has a direct syntactic translation to Oz.\nThis means that the hard work of formulating and understanding\nthese innovations in a logic programming context has already been done.\nWe hope that the ideas presented in this paper will help the future of Prolog\nas well as help Oz regain some recognition as a source of innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzbxlv b/data_all_eng_slimpj/shuffled/split2/finalzzbxlv new file mode 100644 index 0000000000000000000000000000000000000000..ddb91780c046b317248cb6b895914cc508f8b230 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzbxlv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe spinfoam formalism \\cite{Reisenberger:1994aw,spinfoam,spinfoam11,BC1,BC2,spinfoams} and\ncanonical loop quantum gravity (LQG) \\cite{alrev,libro,lqg} can ideally be viewed as the covariant and the canonical versions, respectively, of a background-independent quantum theory of gravity \\cite{Rovelli:2010wq}. This scenario is nicely realized in three dimensions \\cite{alex}, and there are recent attempts to implement it in quantum cosmology \\cite{Ashtekar:2009dn,sfc}. An important step ahead towards the realization of this scenario in the complete four dimensional theory has been taken with the recent introduction of two spin-foam models whose kinematics appears to match the one of LQG rather well, which we refer to as the new model \\cite{epr1,EPR,EPRL,P,LS2} and the Freidel-Krasnov-Livine-Speziale (FKLS) model \\cite{LS1,FK}. The \\emph{kinematics} of canonical loop quantum gravity, indeed, is rather well understood; in particular, the properties of the geometrical operators, including the area and the volume operators\\cite{RS,area,volume} are well established. (On the volume operator, see also \\cite{semi}.) The area operator of the new spinfoam model has been derived in \\cite{EPRL,EP} and shown to match the LQG one. Does the volume do so as well?\n\nThe volume observable in the covariant spinfoam language has not been constructed yet. The essential property of the volume operator is that it has contribution only from the nodes of a spin network state. Thus the only possible action of the volume operator is on the intertwiners. That's the reason why there is no generic well-defined volume operator in the old Barrett-Crane (BC) model\\cite{BC2}, based on the vertex amplitude introduced by Barrett and Crane \\cite{BC1}, where intertwiners are fixed. In fact, the absence of the volume operator in the Barrett-Crane theory can be traced precisely to the key problem of the BC model: the fact that \\emph{intertwiner} quantum numbers are fully constrained. This follows from the $SO(4) \\rightarrow SU(2)$ gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. The new model \\cite{epr1,EPR,EPRL,P} imposes second class simplicity constraints weakly, rather than strongly as in the BC theory. This choice frees intertwiner degrees of freedom, and the volume operator can be nontrivial. In this model, the state space where the constraints vanish weakly turns also out to match that of LQG, providing a solution to the problem of connecting the covariant $SO(4)$ spinfoam formalism with the canonical $SU(2)$ spin-network one. To complete this identification, we construct here the volume operator in the covariant spinfoam picture, and show explicitly that it matches the corresponding LQG canonical operator.\n\nAs a first step, in the next section we review the derivation of the boundary state space of the spinfoam theory. We do so for completeness, but also in order to clean up and simplify previous derivations in the literature. In particular, we show explicitly and directly that all the constraints vanish in a weak sense in the physical boundary space. The form of the constraints that we write turns out to strongly simplify the study of the volume operator.\n\nWe work only in the Euclidean theory, on a fixed triangulation, and assume here that the Barbero-Immirzi parameter $\\gamma$ is positive. The paper is organized as follows. In section 2, we review the definition of the physical boundary Hilbert space. The volume operator is constructed and shown to match the LQG operator in Section 3.\n\n\n\\section{The boundary space}\n\\subsection{Classical theory}\nFollowing \\cite{EPR,EPRL}, we start with a Regge geometry \\cite{regge}\non a fixed triangulation. Consider a 4d triangulation, which is formed by oriented\n4-simplices, tetrahedra, triangles, segments and points. We call\n$v,t$ and $f$ respectively the 4-simplices, the tetrahedra and the\ntriangles of the triangulation. For each simplex $v$, we introduce a variable $e_{\\mu}^I(v)$: a right-handed tetrad one-form, constant over a coordinate patch covering the simplex $v$, with the determinant $\\mathrm{det}(e)>0$ positive. Here $\\mu=(0,a)$\nand $a=1,2,3$ are spacetime indices, while $I=(0,i)$\nand $i=1,2,3$ are internal indices (the value 0 instead of 4 is for later convenience and does not indicate a Lorentzian metric).\nWithout loss of generality, we can choose a linear coordinate system with basis vectors $\\overrightarrow{X}_{\\mu}$ parallel with four edges of $v$ emanating from the same point, and where the (coordinate) length of the four segments is 1. Consider in particular the tetrahedron $t$ spanned by the three\nvectors $\\overrightarrow{X}_a$. To each triangle $f_a$\n(coordinate-)normal to the coordinate basis vector\n$\\overrightarrow{X}_a$, we associate a bivector\n${}^*B_a(t)$ defined by: %\n\\begin{equation} %\n{}^*B^{IJ}_a=\\frac{1}{2}\\epsilon_a^{\\ bc}e^I_be^J_c.%\n\\label{B}\n\\end{equation}\n$B_f(t)$ can be seen as elements in the algebra $\\mathfrak{g}=\\mathfrak{so}(4)$, in the Euclidean case, and $*$ stands for the Hodge dual in the internal indices.\nIf we choose $B_f(t)$ as independent variables instead of the\ntetrads, and $n_I$ denotes the normal to the tetrahedron $t$, the\nsimplicity constraints on $B_f(t)$, which assure that a tetrad field exist, can be stated as follows \\cite{EPR,EPRL}:\n\\begin{equation}\nC_{f}^J:=n_I\\ ({}^* B_f(t))^{IJ} = 0. \\label{c2}\n\\end{equation}\nThe usual quadratic diagonal\n\\begin{equation}\nC_{ff}:={}^* B_f(t)\\cdot B_f(t) = 0\n\\label{c1}\n\\end{equation}\nand off-diagonal\n\\begin{equation}\nC_{ff'}:={}^* B_f(t)\\cdot B_{f'}(t) = 0\n\\label{c0}\n\\end{equation}\nsimplicity constraints can be easily shown to follow from (\\ref{c2}).\nHere the dot stands for the scalar product in the $\\mathfrak{so}(4)$ algebra.\nIn addition, we should impose the closure constraint\n\\begin{equation}\\sum_{f \\in \\partial t} B_f(t) = 0.\n\\label{c3}\n\\end{equation}\nThe new linear simplicity constraint (\\ref{c2}) selects the solution of the quadratic constraints where $B_f=\\int_f{}^*(e\\wedge e)$. This reformulation is central for the new model \\cite{epr1,EPR,EPRL,P}.\nIn particular, if we choose a ``time\" gauge where $n^I=(0,0,0,1)$, the\n simplicity constraint (\\ref{c2}) turns out to be\n\\begin{eqnarray}\n{}^*B^{0i}_f(t) = 0\\label{c2g}.\n\\end{eqnarray}\n\nThe classical discrete action is \\cite{EP,EPRL}\n\\begin{align}\n\\nonumber\nS=&-\\sum_{f \\in int\\Delta} \\mathrm{Tr}\\left[B_f(t)U_f(t)\n+ \\frac{1}{\\gamma}\\,^* B_f(t) U_f(t) \\right] \\\\\n\\label{discaction}\n&-\\sum_{f \\in \\partial\\Delta} \\mathrm{Tr}\\left[B_f(t)U_f(t,t')\n+ \\frac{1}{\\gamma}\\,^* B_f(t) U_f(t,t') \\right],\n\\end{align}\nwhere $U_f(t, t')$ is the group element of $SO(4)$,\ngiving the parallel transport across each triangle $f$ bounding $t$\nand $t'$ and $U_f(t):= U_f(t,t)$ is the holonomy around the full link,\nstarting at $t$. We use here unites where $2\\kappa = 16\\pi G=1$ and $\\gamma$ the Barbero-Immirzi parameter. This action, plus the simplicity and closure constraints defines a discretization of general relativity \\cite{EPR,EPRL}. From the action, we can read off the boundary variables as\n$B_f(t) \\in \\mathfrak{so}(4)$, $U_f(t,t') \\in SO(4)$.\nOne can also see that the variable conjugate to $U_f(t,t')$ is\n\\begin{align}\nJ_f(t):=&B_f(t)+\\frac{1}{\\gamma}{}^*B_f(t),\n\\end{align}\ninverting which gives\n\\begin{align}\n\\,^*B_f(t)=\\frac{\\gamma^2}{1-\\gamma^2}\\Big(\\frac{1}{\\gamma}J_f(t)-^*J_f(t)\\Big)\\label{inver}.\n\\end{align}\n\nThus to each boundary triangle $f$ in the boundary of the triangulation, we have an $SO(4)$ group element $U_f$ and,\nas conjugate variable an $\\mathfrak{so}(4)$ algebra element $J_f$. It is convenient to think these variables as associated with the\nlinks of the graph formed by the one-skeleton of the cellular complex dual to the boundary triangulation. Notice\nthat these define precisely the same boundary phase space as the one of an $SO(4)$ lattice Yang-Mills theory. As in\nYang-Mills theory, the symplectic structure can be taken to be \\cite{EPR}\n\\begin{align}\n\\{ U_f ,U_{f'} \\} &=0 , \\nonumber \\\\\n\\{ (J_f)^{IJ} ,U_{f'} \\} &=\\delta_{ff'}\\ U_f\\ \\tau^{IJ} ,\\\\\n\\{ (J_f)^{IJ} , (J_f')^{KL} \\} &= \\delta_{ff'}\\\n\\lambda^{IJ\\,KL}_{MN} (J_f)^{MN}\\nonumber,\n\\end{align}\nwhere $\\tau^{IJ}$ and $ \\lambda^{IJ\\,KL}_{MN} $ are, respectively,\nthe generators and the structure constants of $SO(4)$.\n\nIn terms of the momentum variable $J_f$, the constraints (\\ref{c2}) and (\\ref{c3}) read respectively:\n\\begin{align}\n &C_f^J=n_I\\left((^*\nJ_f)^{IJ}-\\frac{1}{\\gamma}J_f^{IJ}\\right) = 0 \\label{CJ},\\\\\n&\\sum_{f \\in \\partial t} J_f(t) = 0.\\label{sumJ}\n\\end{align}\n\nFor the gauge-fixed version, introduce $L_f^j:=\\frac{1}{2}\\epsilon^{j}{}_{kl}J_f^{kl}$ and $K_f^j:=J_f^{0j}$ , which are respectively the generators of the $SO(3)$ subgroup that leaves $n_I$ invariant, and the generators of the corresponding boosts. Then the simplicity constraint (\\ref{c2}) becomes simply\n\\begin{equation}\n K^i_f=\\gamma\\ L^i_f.\n \\label{key}\n\\end{equation}\nThis is the key constraint.\nIn terms of $(K_f^i,\\,L_f^i)$, the closure constraint (\\ref{sumJ}) turns out to be\n\\begin{subequations}\\label{closure}\n\\begin{align}\n\\sum_{f\\in\\partial t}L_f^i&=0,\\label{closureL}\\\\\n\\mathrm{and}\\quad\\sum_{f\\in\\partial t}K_f^i&=0.\\label{closureK}\n\\end{align}\n\\end{subequations}\nIf we further make the self-dual\/anti-self-dual decomposition of $J_f^{IJ}$:\n\\begin{align}\n J_f^{(\\pm)i}:=&\\frac{1}{2}(L_f^i\\pm\nK_f^i),\\label{JLK}\n\\end{align}\nthe simplicity constraint (\\ref{CJ}) implies\n\\begin{align}\n&C^i_{f}=(1-\\gamma)J_f^{(+)i}-(1+\\gamma)J_f^{(-)i}\\label{co2}.\n\\end{align}\nIn terms of $J^{(\\pm)i}$, the usual quadratic diagonal simplicity constraint (\\ref{c1}), which follows from the new simplicity constraint (\\ref{c2}) or (\\ref{co2}), can be reexpressed as\n\\begin{align}\n&C_{ff}=(1-\\gamma)^2J_f^{(+)2}-(1+\\gamma)^2J_f^{(-)2}\\label{co1}.\n\\end{align}\n\n\\subsection{Quantization}\nFrom the discrete boundary variables and their symplectic\nstructure, we can construct the Hilbert space associated\nwith a boundary or 3-slice.\nTo do this, it is simpler to\nswitch to the dual, 2-complex picture, $\\Delta^*$.\nFor each 3-surface $\\Sigma$ intersecting no vertices of $\\Delta^*$,\nlet $\\gamma_{\\Sigma} := \\Sigma \\cap \\Delta^*$. The Hilbert space\nassociated with $\\Sigma$ is then\n\\begin{equation}\n\\mathcal{H}_{\\Sigma} = L^2\\left({Spin}(4)^{|L(\\gamma_\\Sigma)|},\\mathrm{d}\\mu_{\\mathrm{Haar}}\\right),\n\\end{equation}\nwhere we replace $SO(4)$ with its covering group $Spin(4)=SU(2)\\times SU(2)$ and $\\mu_{\\mathrm{Haar}}$ is the Haar measure on the group $Spin(4)$; $|L(\\gamma_\\Sigma)|$ denotes the number of links in\n$\\gamma_\\Sigma$. Let $\\hat{J}_f(t)^{IJ}$ denote the\nright-invariant vector fields, determined by the basis $J^{IJ}$ of\n$\\mathfrak{su}(2)\\oplus\\mathfrak{su}(2)$, on the copy of $Spin(4)$ associated with the link\n$l = f \\cap \\Sigma$ determined by $f$, with orientation such that\nthe node $n = t \\cap \\Sigma$ is the source of $l$.\n\nBy Peter-Weyl theorem, $\\mathcal{H}_{\\Sigma}$ can be decomposed as follows\n\\begin{equation}\n\\mathcal{H}_{\\Sigma} = \\bigoplus_{j_l}\\bigotimes_l \\left(\\mathcal{H}_{j_l}^* \\otimes \\mathcal{H}_{j_l}\\right),\n\\end{equation}\nwhere $j_l$ is an assignment of a $Spin(4)$ representation to each link $l$ and $\\mathcal{H}_{j}$ is the carrier space of the representation $j$. The two Hilbert spaces associated to the link $l$ are naturally associated to the two nodes that bound the link $l$, because they transform under the action of a gauge transformation at one end of the link. Regrouping the four Hilbert spaces associated to each node $n$, the last equation can be rewritten in the form\n\\begin{equation}\n\\mathcal{H}_{\\Sigma} = \\bigoplus_{j_l}\\bigotimes_n \\mathcal{H}_{n}.\n\\end{equation}\nHere the Hilbert space associated to a node $n$ is\n\\begin{equation}\n\\mathcal{H}_{n} = \\bigotimes_{a=1}^4 \\mathcal{H}_{j_{a}},\n\\end{equation}\nwhere $a=1,2,3,4$ runs here over the four edges that join at the node $n$ (that is, the four faces of the boundary tetrahedron),\nand we have identified the Hilbert space carrying a representation and its dual.\n We restrict our attention to a single boundary tetrahedron $t$, and its associated Hilbert space $\\mathcal{H}_{n}$, which we call simply $\\mathcal{H}$ in the following.\n\nThe irreducible unitary representations of $Spin(4)$ are labelled by a couple of spins $(j^+,j^-)$ and are given by the tensor product of two $SU(2)$ irreducibles. That is $\\mathcal{H}:=\\mathcal{H}_n$ has the structure\n\\begin{equation}\n\\mathcal{H} = \\bigotimes_{a=1}^{4}\\mathcal{H}_{(j_a^+,j_a^-)}=\\bigotimes_{a=1}^{4}\\big(\\mathcal{H}_{j_a^+}\\otimes\\mathcal{H}_{j_a^-}\\big).\n\\end{equation}\nThe physical intertwiner state space $\\mathcal{K}_{\\mathrm{ph}}$ is a subspace of this space, where the constraints hold in a suitable sense.\n\nAs a first step to impose the constraints, let us restrict the representations to the ones that satisfy\n\\begin{equation}%\nj^+=\\Big|\\frac{1+\\gamma}{1-\\gamma}\\Big|j^-,\n\\label{gammasimple}\n\\end{equation}%\nwhich satisfies the usual quadratic diagonal simplicity constraint (\\ref{co1}) in the classical limit, and is what we need to recover the correct classical theory in the limit. We call $\\gamma$-simple the $Spin(4)$ representations that satisfy this relation.\n\nNext, the Clebsch-Gordan decomposition for the single component of $\\mathcal{H}$ associated with a single boundary face $f$ gives\n\\begin{equation}\n\\mathcal{H}_{j^+ \\otimes j^-} = \\mathcal{H}_{j^+} \\bigotimes\n\\mathcal{H}_{j^-} = \\bigoplus_{p=|j^+-j^-|}^{j^++j^-}\\mathcal{H}_p.\\label{CG}\n\\end{equation}\nConsider the highest spin term in each factor for $\\gamma<1$ and the lowest for $\\gamma>1$ respectively; this selects the ``extremum'' subspace\n\\begin{align}\n\\mathcal{H}^{\\mathrm{max}}&=\\bigotimes_{a=1}^4 \\mathcal{H}_{j^++j^-},\\quad \\mathrm{for}\\ \\gamma<1;\\\\\n\\mathcal{H}^{\\mathrm{min}}&=\\bigotimes_{a=1}^4 \\mathcal{H}_{j^+-j^-},\\quad \\mathrm{for}\\ \\gamma>1.\n\\end{align}\nWe are now going to show that in this space (with (\\ref{gammasimple}) holding), the simplicity constraint (\\ref{co2}) is satisfied weakly. That is, the action of the constraints on the states in\n${\\mathcal{H}}^{\\mathrm{ext}}$ results in states orthogonal to\n${\\mathcal{H}}^{\\mathrm{ext}}$. Namely, $\\bra{\\Psi}\\hat{C}^{i}_{f}\\ket{\\Phi}=0,\\ \\forall \\,\\Psi,\\,\\Phi\\,\\in{\\mathcal{H}}^{\\mathrm{ext}}$. This follows from the following considerations. \n$\\forall \\Psi,\\,\\Phi\\,\\in {\\mathcal{H}}^{\\mathrm{ext}}$, consider the matrix element of the form (\\ref{co2}) of the simplicity constraints\n\\begin{equation}\n\\bra{\\Psi}\\hat{C}^{i}_{f}\\ket{\\Phi}=(1-\\gamma)\\bra{\\Psi}\\vec{{J}_f^+}\\ket{\\Phi}-(1+\\gamma)\\bra{\\Psi}\\vec{{J}_f^-}\\ket{\\Phi}\\label{mepm}\n\\end{equation}\nand write the r.h.s. of this equation in a representation where elements of $\\mathcal{H}_j$ are symmetric spinors with $2j$ indices. The generators of $SU(2)$ are then Pauli matrices $\\sigma^A_i{}_B$ acting on each index. {For} $\\gamma<1$,\n\\begin{align}\\label{proof}\n\\bra{\\Psi}\\hat{C}^{i}_{f}\\ket{\\Phi}\n=&(1-\\gamma)\\Psi_{(A_1...A_{2j^+}B_1...B_{2j^-})}\\sum_{p=1}^{2j^+}\\sigma^{A_p}_i{}_{\\widetilde{A}_p}\\Phi^{(A_1...\\widetilde{A}_p...A_{2j^+}B_1...B_{2j^-})}\\nonumber\\\\\n&-(1+\\gamma)\\Psi_{(A_1...A_{2j^+}B_1...B_{2j^-})}\\sum_{p=1}^{2j^-}\\sigma^{B_p}_i{}_{\\widetilde{B}_p}\\Phi^{(A_1...A_{2j^+}B_1...\\widetilde{B}_p...B_{2j^-})}\\nonumber\\\\\n=&2\\big((1-\\gamma)j^+-(1+\\gamma)j^-\\big)\\Psi_{(A_1...A_{2j^+}B_1...B_{2j^-})}\\sigma^{A_1}_i{}_{\\widetilde{A}_1}\\Phi^{(\\widetilde{A}_1...A_{2j^+}B_1...B_{2j^-})}\\nonumber\\\\\n=&0.\n\\end{align}\nThe first step is obtained by the symmetry of the highest spin states, and the last follows from (\\ref{gammasimple}). Therefore the simplicity constraint is implemented weakly in ${\\mathcal{H}}^{\\mathrm{max}}$ for $\\gamma<1$. For $\\gamma>1$, the state $\\ket{\\Psi}$ in $\\mathcal{H}^{\\mathrm{min}}$ can be expressed as $\\Phi^{(A_1...A_{2j^+})(B_1...B_{2j_-})}=\\epsilon^{A_1B_1}...\\epsilon^{A_{2j^-}B_{2j^-}}\\Phi^{(A_{2j^-+1}...A_{2j^+})}$, on which the action of $J^{(-)}$ can be obtained as\n\\begin{align}\n\\vec{J}^{-}\\Phi^{(A_1...A_{2j^+})(B_1...B_{2j_-})}=&\\sum_{p=1}^{2j^-}\\sigma_i^{B_p}{}_{\\widetilde{B}_p}\\epsilon^{A_1B_1}...\\epsilon^{A_p\\widetilde{B}_p}...\\epsilon^{A_{2j^-}B_{2j^-}}\\Phi^{(A_{2j^-+1}...A_{2j^+})}\\nonumber\\\\\n=&-\\sum_{p=1}^{2j^-}\\sigma_i^{A_p}{}_{\\widetilde{A}_p}\\epsilon^{A_1B_1}...\\epsilon^{\\widetilde{A}_pB_p}...\\epsilon^{A_{2j^-}B_{2j^-}}\\Phi^{(A_{2j^-+1}...A_{2j^+})}\\nonumber\\\\\n=&-\\sum_{p=1}^{2j^-}\\sigma_i^{A_p}{}_{\\widetilde{A}_p}\\Phi^{(A_1...\\widetilde{A_{p}}...A_{2j^+})(B_1...B_{2j_-})}.\n\\end{align}\nHence the matrix elements of the simplicity constraint can be obtained as\n\\begin{align}\n\\bra{\\Psi}\\hat{C}^{i}_{f}\\ket{\\Phi}&=2\\big((1-\\gamma)j^++(1+\\gamma)j^-\\big)\\Psi_{(A_1...A_{2j^+})(B_1...B_{2j^-})}\\sigma^{A_1}_i{}_{\\widetilde{A}_1}\\Phi^{(\\widetilde{A}_1...A_{2j^+})(B_1...B_{2j^-})}=0\\nonumber\n\\end{align}\nThe last step follows again from (\\ref{gammasimple}). Therefore the space $\\mathcal{H}^{\\mathrm{ert}}$ solves the simplicity constraint.\nThe physical intertwiner space associated with a single node $n$ is then obtained by solving the closure\nconstraint (\\ref{closure}) weakly in the space ${\\mathcal{H}}^{\\mathrm{ext}}$, which turns out\nto be\n\\begin{equation}%\n{\\cal\nK}_{\\mathrm{ph}}= \\mathrm{Inv}_{\\mathrm{SU}(2)}[{\\mathcal{H}}^{\\mathrm{ext}}].\n\\label{inv}\n\\end{equation}%\nTo show that the closure constraints (\\ref{closure}) hold weakly on this space, observe the matrix elements (\\ref{mepm}) of the simplicity constraint implies\n\\begin{align}\n\\bra{\\Psi}K^{i}_{f}\\ket{\\Phi}=\\gamma\\bra{\\Psi}L^{i}_{f}\\ket{\\Phi}.\\label{meKL}\n\\end{align}\nThe l.h.s of (\\ref{closureL}) is the generator of $SU(2)$ transformations at the node and vanishes strongly on (\\ref{inv}) by definition; the l.h.s of (\\ref{closureK}) is proportional weakly to the one of (\\ref{closureL}) by (\\ref{meKL}) and therefore vanishes weakly. Hence the $SU(2)$-invariant space turns out to be $SO(4)$-invariant space in the weak sense.\nThus $\\mathcal{K}_{\\mathrm{ph}}$ is the intertwiner space as a solution of \\emph{all} the constraints: all the constraints hold weakly.\n\nThe total\nphysical boundary space ${\\rd H}_{\\mathrm{ph}}$ of the theory is then\nobtained as the span of spin-networks in $L^2[{Spin}(4)^L\/{Spin}(4)^N,\\mathrm{d}{\\mu}_{\\mathrm{Haar}}]$ with\n$\\gamma$-simple representations on edges and with intertwiners in the spaces\n${\\rd K}_{\\mathrm{ph}}$ at each node.\n\nWe have then the remarkable result that $\\mathcal{K}_{\\mathrm{ph}}$ is naturally isomorphic to the $SU(2)$ intertwiner space, and therefore the\nconstrained boundary space ${\\rd H}_{\\mathrm{ph}}$ can be identified with the $SU(2)$\nLQG state space ${\\rd H}_{{SU}(2)}$ associated to the graph which is\ndual to the boundary of the triangulation, namely the space of the\n$SU(2)$ spin networks on this graph.\n\nSince we have not proven that the physical Hilbert space considered is the \\emph{maximal} space where the constraints hold weakly, one might worry that the physically correct quantization of the degrees of freedom of general relativity could need a larger space. Also, it has been pointed out that imposing second class constraints weakly might lead to inconsistencies in some cases\\cite{Alexandrov:2010pg}. In the present case, however, these worries are not relevant, since the space obtained is directly related to the one of the canonical theory, which we can trust to capture the degrees of freedom of gravity correctly.\n\nLet us now consider the geometrical operators in these two versions. Classically, the area $A(f)$ of a triangle $f$ is given by $A(f)^2={\\frac{1}{2}({}^*B_f)^{IJ}\\cdot({}^*B_f)_{IJ}}$. If we fix the time gauge, we have\n$A_3(f)^2={\\frac{1}{2}({}^*B_f)^{ij}\\cdot({}^*B_f)_{ij}}$. These two quantities are equal up to a constrained term. As shown in \\cite{EP,EPRL}, using the constraints, the operator related to $A_{3}(f)^2$ can be obtained as $A_3(f)^2=\\kappa^2\\gamma^2L_f^2$, which matches three-dimensional area as determined by LQG, including the correct Barbero-Immirzi parameter proportionality factor. Let us now turn to study the volume operator on this space ${\\rd H}_{\\mathrm{ph}}$ and its relation with the $SU(2)$ volume in LQG.\n\n\\section{The Volume}\n\nIt is easy to see from the definition of $e_a^i(v)$ given at the beginning of the previous section, that the volume of the tetrahedron $t$ is given by\n\\begin{equation}\nV(t)=\\frac{1}{6}{\\det (e(v))}.\\label{volume}\n\\e\nIn terms of the variables ${}^*B$ defined in (\\ref{B}), the volume of a boundary tetrahedron $t$ reads\n$V$\nrelated to the tetrahedra $t$ as %\n\\begin{equation}%\nV(t)=\\sqrt{\\frac{1}{27}\\epsilon^{abc}\\mathrm{Tr}[{}^*B_a{}^*B_b{}^*B_c]}\\label{v4}\n\\end{equation}%\nTo see this, let the gauge-fixed simplicity constraint\n(\\ref{c2g}) hold, then the ${}^*B^{0i}_f(t)$ vanish and the above\nquantity is equal to\n\\begin{align}\nV_3(t)=\\sqrt{\\frac{1}{27}\\epsilon^{abc}{}^*B^{ij}_a{}^*B^{jk}_b{}^*B^{ki}_c}\n=\\frac{1}{6}{\\det (e)},\\label{v3t}\n\\end{align}\nwhich is exactly the expression (\\ref{volume}) of the discrete volume.\nNote that the ${SO}(4)$ volume $V_{{SO}(4)}(t)$ is gauge invariant, hence we can obtain eq (\\ref{v4}) by the gauge-fixed version (\\ref{v3t}) without loss of generality.\nGoing to the variables $J$, and using (\\ref{inver}), the volume reads\n\\begin{align}\n&V(t)=\\sqrt{\\frac{1}{27}\\Big(\\frac{\\gamma^2}{1-\\gamma^2}\\Big)^3\\epsilon^{abc}\\mathrm{Tr}\\Big[\\Big(\\frac{1}{\\gamma}J_a-^*J_a\\Big)\\Big(\\frac{1}{\\gamma}J_b-^*J_b\\Big)\\Big(\\frac{1}{\\gamma}J_c-^*J_c\\Big)\\Big]}\n\\label{v4j}\n\\end{align}\n\nThe volume operator $\\hat{V}(t)$ of the tetrahedron $t$ is\n then formally given by (\\ref{v4j}) with $J^{IJ}$ replaced by the corresponding operators:\n\\begin{align}\n\\hat{V}(t)=\\sqrt{\\frac{1}{27}\\Big(\\frac{\\gamma^2}{1-\\gamma^2}\\Big)^3\\epsilon^{abc}\\mathrm{Tr}\\Big[\\Big(\\frac{1}{\\gamma}\\hat{J}_a-^*\\hat{J}_a\\Big)\\Big(\\frac{1}{\\gamma}\\hat{J}_b-^*\\hat{J}_b\\Big)\\Big(\\frac{1}{\\gamma}\\hat{J}_c-^*\\hat{J}_c\\Big)\\Big]}.\n\\label{hatv}\n\\end{align}\nHowever, the physical volume should be defined on the physical boundary space $\\cal{H}_{\\mathrm{ph}}$, satisfying the constraints.\nSince the volume operator does not change the graph of the spin network sates, nor the coloring of the links, its action can be studied on the Hilbert space associated to a single node. Consider the matrix element of the square of the volume operator between two states in the physical Hilbert space (we drop the hats):\n\\begin{equation}\n\\bra{i} V(t)^2 \\ket{j}\n=\\frac{1}{27}\\big(\\frac{\\gamma^2}{1-\\gamma^2}\\big)^3\\epsilon^{abc}\\bra{i} \\big(\\frac{1}{\\gamma}J_a^{ij}-{}^*J^{ij}_a\\big)\\big(\\frac{1}{\\gamma}J_b^{jk}-{}^*J^{jk}_b\\big)\\big(\\frac{1}{\\gamma}J_c^{ki}-{}^*J_c^{ki}\\big) \\ket{j}.\n\\end{equation}\nWriting this in terms of $L$ and $K$ components gives\n\\begin{equation}\n\\bra{i} V(t)^2 \\ket{j}\n=\\frac{1}{27\\cdot 8}\\big(\\frac{\\gamma^2}{1-\\gamma^2}\\big)^3\\epsilon^{abc}\\epsilon^{ij}_{\\ \\ m}\\epsilon^{jk}_{\\ \\ n}\\epsilon^{ki}_{\\ \\ p}\\bra{i} \\big(\\frac{1}{\\gamma}L_a^{m}-K^{m}_a\\big)\\big(\\frac{1}{\\gamma}L_b^{n}-K^{n}_b\\big)\\big(\\frac{1}{\\gamma}L_c^{p}-K_c^{p}\\big) \\ket{j}.\\label{simplify}\n\\end{equation}\nNotice that the intertwiner space is the subspace of the product of the space $\\mathcal{H}_{a}$ associated to the link $a$, and the action of $(K_a,\\,L_a)$ is in fact on $\\mathcal{H}_a$. Hence we can use the form (\\ref{meKL}) of the simplicity constraint to simplify Eq. (\\ref{simplify}), although the r.h.s seems a polynomial.\nUsing the form (\\ref{meKL}) of the constraint, we can rewrite it as\n\\begin{align}\n\\bra{i} V(t)^2 \\ket{j}\n=&\\frac{1}{27\\cdot 8}\\big(\\frac{\\gamma^2}{1-\\gamma^2}\\big)^3\\big(\\frac{1}{\\gamma}-\\gamma\\big)^3\\epsilon^{abc}\\epsilon_{ijk}\\bra{i} L_a^{i}L_b^{j}L_c^{k} \\ket{j}\n\\end{align}\nand a little algebra gives\n\\begin{equation}\n\\bra{i} V(t)^2 \\ket{j}\n=\\gamma^3 \\ \\bra{i} \\epsilon^{abc}\\epsilon_{ijk}L_a^{i}L_b^{j}L_c^{k} \\ket{j}.\n\\end{equation}\nThat is\n\\begin{equation}\nV(t)\n=\\gamma^{\\frac32} \\sqrt{\\epsilon^{abc}\\epsilon_{ijk}L_a^{i}L_b^{j}L_c^{k}}\n\\end{equation}\nNow, the operator on the r.h.s. is precisely the LQG volume operator $V_{\\mathrm{LQC}}$, as it acts on ${\\cal\nK}_{\\mathrm{ph}}$ including the correct dependence on the Barbero-Immirzi parameter $\\gamma$.\n\\section*{Acknowledgments}\nThanks to Eugenio Bianchi for enlightening discussions. We wish also to thank Jonathan Engle, Simone Speziale, Franck Hellmann, Roberto Pereira and Muxin Han for helpful comments and suggestions. Y. D. is supported by CSC scholarship No. 2008604080.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCan cats or dogs do barter? Trade is one of the essential feature of human intelligence. \\cite{10.1371\/journal.pone.0001518} reported that chimpanzees who are an intelligent species like human have the ability to trade, but are reluctant to trade. This subsequently led to a major divergence in the fates of the two species. The market is nothing but an expression of human intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\\cite{malone2015handbook}). The securities market, including crypto-assets, is the ultimate expression of human intelligence. As considered by \\cite{fama1965behavior}, if the security market is efficient in the strong or semi-strong sense, information on securities instantaneously changes the traders' subjective equilibrium and the differences in the speed with which they respond to the information decides who are the winners and losers. If the security market is efficient in the weak sense, the market equilibrium trends according to changes in the traders' subjective equilibrium because of noise. The efficiency of security market can be summarized as follows. \n\\begin{itemize}\n\\item Information is incorporated correctly into the price.\n\\item Information is incorporated rapidly into the price.\n\\item Arbitrage deals profit if there are errors or delays.\n\\item Countless traders are always looking for arbitrage opportunities.\n\\item The arbitrage opportunity is gradually lost and the market becomes more efficient.\n\\end{itemize}\nAlthough the standard theory draws a story above, transaction is not established if there is no noise and price will stay distorted if bias is strong.\n\nIn an efficient market where noise and bias have no effect and information is perfectly symmetrical, security prices should accurately reflect only information. However, if traders had perfectly and simultaneously symmetrical knowledge and information, including the asset valuation model, the transaction will not function because the traders' valuation of the asset would be the same. They must be doing some rational calculations via intelligence in natural and digital computing. In this sense, a rational representative agent in macroeconomics corresponds to the perfect symmetry of information~(\\cite{Lucas1976EconometricPE, Kirman1992WhomOW, 10.2307\/2138488}). \n\nWhile \\cite{black1986noise} treated this ``symmetry breaking'' as noise, the effect of noise on a security's price is expected to be symmetrical based on its nature. But if there exist so many irrational noise traders synchronizing erroneous stochastic beliefs, both affect prices and earn higher expected returns, the unpredictability of noise traders' beliefs creates a risk in the price of the asset that deters rational arbitrageurs from aggressively betting against them. As a result, prices can diverge significantly from fundamental values even in the absence of fundamental risk. Moreover, bearing a disproportionate amount of risk that they themselves create enables noise traders to earn a higher expected return than rational investors do~(\\cite{doi:10.1086\/261703}).\n\nOn the other hand, \\cite{doi:10.1126\/science.185.4157.1124} found the effect of bias is asymmetric. They described three heuristics that are employed in making judgments under uncertainty: 1) representativeness; 2) availability of instances or scenarios; and 3) adjustment from an anchor. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. But the effect of these heuristics has not been detected as a global bias in the security market. Consequently, identifying the specific effects of noise and bias on security price is challenging.\n\nIf attention is paid to any statistical property in any complex system, the log-normal distribution is the most natural and appropriate among the standard or ''normal'' statistics to overview the whole system~(\\cite{doi:10.1143\/JPSJ.80.072001}). The log-normality emerges as familiar and typical examples of statistical aspects in various complex systems. Since every member of any complex system has its own history, each member is in the process of growth (or retrogression). The log-normal distribution is realized as a result of Gibrat' law, or Mathew effect. It is applied to cities size and growth rate, where proportionate growth process may give rise to a distribution of city sizes that is log-normal. When considering the entire size distribution, not just the largest cities, then the city size distribution is log-normal~(\\cite{10.2307\/2296055}). However, it has been argued that it is problematic to define cities through their fairly arbitrary legal boundaries. According to \\cite{10.1162\/003355399556133},\nZipf's law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1\/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf's law.\n\nGibrat's law of proportionate effect also states that the proportional change in the size of a firm is independent of its absolute size. An implication of this is that large and small firms have the same average proportionate rates of growth. Against this law, \\cite{10.2307\/2296055} shows large firms are growing faster significantly. \\cite{10.1257\/mac.20150051} construct a tractable neoclassical growth model that generates Pareto's law of income distribution and Zipf's law of the firm size distribution from idiosyncratic, firm-level productivity shocks. Executives and entrepreneurs invest in risk-free assets, as well as their own firms' risky stocks, through which their wealth and income depend on firm-level shocks. By using the model, they evaluate how changes in tax rates can account for the evolution of top incomes in the United States. The model matches the decline in the Pareto exponent of the income distribution and the trend of the top 1 percent income share in recent decades. In the same research direction, \\cite{NIREI201625} construct a neoclassical growth model with heterogeneous households that accounts for the Pareto distributions of income and wealth in the upper tail. In an standard Bewley model~(\\cite{BEWLEY1977252}), they feature households' business productivity risks and borrowing constraints, which they find generate the Pareto distributions. Households with low productivity rely on wages and returns from safe assets, while high productivity households choose not to diversify their business risks. Their model can quantitatively account for the observed income distribution in the U.S. under reasonable calibrations. Furthermore, they conduct several comparative statics to examine how changes in parameters affect the Pareto distributions. In particular, they find that the change in the top tax rates in the 1980s potentially accounts for much of the observed increase in top income dispersion in the last decades. Their analytical result provides a coherent interpretation for the numerical comparative statics.\n\nIn this article, I present the true fundamentals hypothesis based on rational expectations~(\\cite{10.2307\/1909635}) and, using a log-normal distribution model, detect global bias components from the price-earnings (P\/E), price-to-book (P\/B), and price-to-cash flow (P\/CF) ratios. The traditional theory of the firm is based on the assumption that the firm acts in the stockholders' interests and that stockholders are interested in profit, so that the object of the firm is to maximize profit. However, in fact, there is a certain range in the profit concept~(\\cite{10.2307\/2977477}). The analysis results strongly support the true fundamentals hypothesis as the detected biases show similar characteristics. Additionally, the results show that the cash flow indicators contain relatively few bias components and are closer to the true fundamentals. I further demonstrate and examine why the positive P\/IC ratio among the indicators analyzed is a proxy for the true fundamentals that does not include bias components.\n\n\\section{Hypothesis}\n\nWhen the true fundamentals of listed companies at time $t$ is denoted as $X_t$ and their growth rate is denoted as $R_t$, the growth of those companies can be expressed by the following Gibrat's process\n\\begin{equation}\n X_t=R_tX_{t-1} \\label{1}.\n\\end{equation}\nIt is important to note, however, that I assume that the growth rates\n$R_t$s are mutually independent random variables that follow the same\ndistribution with finite variance. The initial value of the fundamentals\nset as $X_0$ yields\n\\begin{equation}\n X_T=X_0\\prod_{t=1}^T R_t\n\\label{2}\n\\end{equation}\nat time $T$. Taking the log of both sides of the equation results in\n\\begin{equation}\n \\log X_T =\\log X_0+\\log R_1+\\cdots +\\log R_T.\n\\label{3}\n\\end{equation}\nTherefore,\n\\begin{equation}\n \\log X_T\\sim LN(\\mu, \\sigma^2)\n\\label{4}\n\\end{equation}\nwould hold true for a sufficiently large $T$ based on the central limit theorem. Essentially, the true fundamentals $X_T$ of listed companies follows the log-normal distribution.\n\nFurthermore, by assuming rational expectations through the future point in time $T$ as of the present point in time 0 on the premise of a going concern, the following equation becomes true:\n\\begin{equation}\n X_0=\\mathbb{E}[X_T]\\prod_{t=1}^T \\mathbb{E}[R_t^{-1}].\n\\label{5}\n\\end{equation}\n Therefore, the rational expectations $X_0$ for true fundamentals also follow the log-normal distribution. In other words, the true fundamentals $X_t$ at a given point in time $t$ follow the log-normal distribution.\n\n\\section{Analysis}\n\n\\subsection{Data}\n\nAmong the actual fundamental indicators, the price-earnings (P\/E), forward price-earnings (P\/FE), price-to-book (P\/B), price-to-cash flows from operating activities (P\/OC), price-to-cash flows from investing activities (P\/IC), price-to-cash flows from financing activities (P\/FC), and the price-to-cash equivalents at the year-end (P\/CE) ratios are analyzed. The cash flow ratios were divided into positive and negative data. The daily data with absolute values less than 1,000 were extracted for all companies listed in Japan for the period spanning 1,817 business days from January 2007 to May 2014. \n\n\\subsection{Test}\n\nIf the true fundamentals hypothesis is true, the actual fundamental indicators should follow the log-normal distribution. Therefore, the $p$-values were calculated using the Kolmogorov-Smirnov test, Pearson's $\\chi^2$ test, and Anderson-Darling test, to test the goodness of fit to the log-normal distribution. It is recognized that the results of these tests are strongly affected by the sample size.\n\nSince the sample size differs by indicator, samples at the 300 quartiles were extracted to calculate the $p$-value for testing the goodness of fit of the overall average of each indicator's samples to the log-normal distribution with variance. In all cases, the null hypothesis is that ``the indicator follows the log-normal distribution,'' or the otherwise worded ``the indicator is a proxy of true fundamentals.'' The indicator reflects the true fundamentals more when the $p$-value is closer to 1. When the $p$-value is closer to 0, the bias is stronger. The lowest $p$-value for each test is shown in Table~\\ref{tab1} for the data spanning 1,817 business days.\n\nThe test results indicate that bias has a strong effect on the P\/E, P\/FE, and P\/B ratios. It is consistent with our instincts that forward price-earnings ratio (P\/FE) is most biased. Interestingly, among all price-to-cash flow ratios, the bias is significant only for P\/OC ratio. It can be easier to manipulate than others. I illustrate the time series of $p$-values since the $p$-value is an indicator of the strength of bias. From table~\\ref{tab1}, Figure~\\ref{bias1} and Figure~\\ref{bias2}, it is clear that the positive P\/IC ratio is the best proxy for the true fundamentals that does not include bias components. Probably, it should be difficult to manipulate according to the opinions of management and accounting.\n\\begin{table}[ht] \n\\caption{The lowest $p$-values for the data spanning 1,817 business days. Samples at the 300 quantile. Significant (bias) levels: 1\\%(***), 3\\%(**), 5\\%(*)\\label{tab1}}\n\\newcolumntype{C}{>{\\centering\\arraybackslash}X}\n\\begin{tabularx}{\\textwidth}{llll}\n\\toprule\n\\textbf{Index}\t&\\qquad \\textbf{Kolmogorov-Smirnov}\t&\\qquad\\quad \\textbf{Peason's $\\chi^2$} &\\qquad \\textbf{Anderson-Darling}\\\\\n\\midrule\nP\/OC$+$ &\\qquad\\qquad\\qquad 0.0986 &\\qquad\\qquad 0.2843 &\\qquad\\qquad\\qquad 0.0398$^{*}$ \\\\\nP\/OC$-$ &\\qquad\\qquad\\qquad 0.0634 &\\qquad\\qquad 0.0045$^{***}$ &\\qquad\\qquad\\qquad 0.0850 \\\\\nP\/IC$+$ &\\qquad\\qquad\\qquad 0.6205 &\\qquad\\qquad 0.4441 &\\qquad\\qquad\\qquad 0.7201 \\\\\nP\/IC$-$ &\\qquad\\qquad\\qquad 0.2787 &\\qquad\\qquad 0.7012 &\\qquad\\qquad\\qquad 0.1849 \\\\\nP\/FC$+$ &\\qquad\\qquad\\qquad 0.2042 &\\qquad\\qquad 0.2287 &\\qquad\\qquad\\qquad 0.1093 \\\\\nP\/FC$-$ &\\qquad\\qquad\\qquad 0.2522 &\\qquad\\qquad 0.5763 & \\qquad\\qquad\\qquad 0.1269 \\\\\nP\/CE &\\qquad\\qquad\\qquad 0.4611 &\\qquad\\qquad 0.8740 &\\qquad\\qquad\\qquad 0.2376 \\\\\nP\/E &\\qquad\\qquad\\qquad 0.0170$^{**}$ &\\qquad\\qquad 0.0000$^{***}$ &\\qquad\\qquad\\qquad 0.0018$^{***}$ \\\\\nP\/FE &\\qquad\\qquad\\qquad 0.0107$^{**}$ &\\qquad\\qquad 0.0040$^{***}$ &\\qquad\\qquad\\qquad 0.0019$^{***}$ \\\\\nP\/B &\\qquad\\qquad \\qquad0.0032$^{***}$ &\\qquad\\qquad 0.0023$^{***}$ &\\qquad\\qquad\\qquad 0.0005$^{***}$ \\\\\n\\bottomrule\n\\end{tabularx}\n\\end{table}\n\\unskip\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=11 cm]{bias1.pdf}\n\\caption{Time series of the goodness of fit test to the log-normal distribution. Blue: Kolmogorov-Smirnov, Yellow: Peason's $\\chi^2$, Blue: Anderson-Darling. The higher the value, the less bias there is, vice versa. \\label{bias1}}\n\\end{center}\n\\end{figure} \n\\unskip\n\n\\subsection{Shape of Bias}\n\nThe detected biases exhibit similar characteristics. To facilitate an intuitive understanding of these results, each indicator is shown in figure~\\ref{bias2} and is compared to the log-normal distribution. While the figures only show the data for the first day of the period, January 4, 2007, the characteristics of the biases themselves remain essentially unchanged throughout the period although the levels fluctuated. It is worth noting that the shape of the curve, and in particular the strength of the bias, is independent of whether the economy is doing well or not. The stock market, which is clearly artificial, appears to have a natural intelligence.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=10 cm]{bias2.pdf}\n\\caption{The more the data deviates from a straight line showing a log-normal distribution, the stronger the bias. These graphs are drawn with data from the first day of the time series. The shape of the bias varies proportionally to Figure~\\ref{bias1}, but is stable throughout the time. \\url{https:\/\/figshare.com\/projects\/Intelligence_and_Global_Bias_in_the_Stock_Market\/131711}\\label{bias2}}\n\\end{center}\n\\end{figure} \n\\unskip\n\n\n\\section{Discussion and Conclusion}\n\nThe results indicate that the major fundamental indicators including the P\/E, P\/FE, and P\/B ratios are strongly affected by bias. Bias also has a significant effect on the positive and negative P\/OC ratios. Additionally, there may be a weak effect of bias on the negative P\/IC, positive and negative P\/FC, and P\/CE ratios.\n\nWhen the test results are compared, the positive P\/IC ratio is the stable proxy of true fundamentals among all these indicators, and it is important to consider why this occurs.\n\nPositive cash flows from investing activities represent the realized gain or loss from past investments such as marketable securities, tangible fixed assets, the sales of investment securities, and income from the collection on loans declared at the end of accounting period. In other words, positive cash flow is the indicator that most directly reflects the past business decisions that determine the growth of a company. The year-end cash equivalent ratio and other cash flow ratios can be interpreted to strongly reflect the true fundamentals because they have less bias components due to the strong characteristic of having definite value.\n\nOn the other hand, although the P\/E ratio and P\/B ratio are definite values, investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. The P\/OC ratio might also have a lower credibility compared to other cash flow indicators.\n\nWhere do these biases come from? The answer is extremely simple: ``Cash is a fact, profit is an opinion.'' Namely, opinions of management and accountant are added as noise to true fundamentals. As a result, Kesten process~(\\cite{10.1007\/BF02392040})\n\\begin{equation}\n X_t = R_t X_{t-1}+\\epsilon_t,\\quad E[\\epsilon_t]>0\n\\end{equation}\nis realized and the Pareto distribution is to be obtained. This means their opinions are accompanied by a positive bias.\n\nIn fact, these biases fit the Pareto distribution quite well. The generalized Pareto distribution (GPD) is represented by the following functions. \n\n\\paragraph{Generalized Pareto Distribution (GPD)}\n\\[\nF(x)=1-\\left(1+\\left(\\frac{x-\\mu}{\\kappa}\\right)^{1\/\\gamma}\\right)^{-\\alpha} \n\\]\n\\[\nf(x)=\\frac{\\alpha}{\\gamma}\\kappa^{-1\/\\gamma}(x-\\mu)^{-1+\\frac{1}{\\gamma}}\\left(1+\\left(\\frac{x-\\mu}{\\kappa}\\right)^{1\/\\gamma}\\right)^{-\\alpha-1}\n\\]\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=11cm]{bias3.pdf}\n\\caption{Comparison of three distributions: Log-Normal(blue), GDP-1(Yellow), GDP-2(Green)\\label{bias3}}\n\\end{center}\n\\end{figure} \n\\unskip\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=13cm]{bias4.pdf}\n\\caption{Left: GDP-1 to fit P\/FE ($\\kappa=18.82,\\alpha=1, \\gamma=0.385, \\mu=0.993$), Right: GDP-2 to fit P\/FE ($\\kappa=13.70,\\alpha=0.515, \\gamma=0.238, \\mu=0.993$)\\label{bias4}. Note that P\/FE has the strongest bias.}\n\\end{center}\n\\end{figure} \n\n\nWhat are the implications of the results? Does the existence of bias negate the efficient market hypothesis? From the view point of intelligence in natural and digital computing, how should we deal with this problem?\n\nThe stock market, which is clearly artificial, and where countless traders are using digital computers to trade, appears to have a natural intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\\cite{malone2015handbook}). If the true fundamentals hypothesis is really true, the major fundamental indicators including the P\/E, P\/FE, and P\/B ratios are strongly affected by bias. This means that investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. Globally, this implies that the efficient market hypothesis holds~(\\cite{fama1965behavior}). The shape of the curve, and in particular the strength of the bias, is stable throughout the time and independent of whether the economy is doing well or not. The curve with bias is the path of the ants~(\\cite{kirman1993ants, app12147019}) to avoid obstacles, so to speak, and traders move back and forth along it in response to noise~(\\cite{black1986noise, doi:10.1086\/261703}). This noise, among other things, plays a role in activating the market system, which knows true fundamentals of firms and represents as a stable global bias in the stock market.\n\n\n\n\\newpage\n\\bibliographystyle{unsrtnat}\n\n\\section{Introduction}\n\nCan cats or dogs do barter? Trade is one of the essential feature of human intelligence. \\cite{10.1371\/journal.pone.0001518} reported that chimpanzees who are an intelligent species like human have the ability to trade, but are reluctant to trade. This subsequently led to a major divergence in the fates of the two species. The market is nothing but an expression of human intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\\cite{malone2015handbook}). The securities market, including crypto-assets, is the ultimate expression of human intelligence. As considered by \\cite{fama1965behavior}, if the security market is efficient in the strong or semi-strong sense, information on securities instantaneously changes the traders' subjective equilibrium and the differences in the speed with which they respond to the information decides who are the winners and losers. If the security market is efficient in the weak sense, the market equilibrium trends according to changes in the traders' subjective equilibrium because of noise. The efficiency of security market can be summarized as follows. \n\\begin{itemize}\n\\item Information is incorporated correctly into the price.\n\\item Information is incorporated rapidly into the price.\n\\item Arbitrage deals profit if there are errors or delays.\n\\item Countless traders are always looking for arbitrage opportunities.\n\\item The arbitrage opportunity is gradually lost and the market becomes more efficient.\n\\end{itemize}\nAlthough the standard theory draws a story above, transaction is not established if there is no noise and price will stay distorted if bias is strong.\n\nIn an efficient market where noise and bias have no effect and information is perfectly symmetrical, security prices should accurately reflect only information. However, if traders had perfectly and simultaneously symmetrical knowledge and information, including the asset valuation model, the transaction will not function because the traders' valuation of the asset would be the same. They must be doing some rational calculations via intelligence in natural and digital computing. In this sense, a rational representative agent in macroeconomics corresponds to the perfect symmetry of information~(\\cite{Lucas1976EconometricPE, Kirman1992WhomOW, 10.2307\/2138488}). \n\nWhile \\cite{black1986noise} treated this ``symmetry breaking'' as noise, the effect of noise on a security's price is expected to be symmetrical based on its nature. But if there exist so many irrational noise traders synchronizing erroneous stochastic beliefs, both affect prices and earn higher expected returns, the unpredictability of noise traders' beliefs creates a risk in the price of the asset that deters rational arbitrageurs from aggressively betting against them. As a result, prices can diverge significantly from fundamental values even in the absence of fundamental risk. Moreover, bearing a disproportionate amount of risk that they themselves create enables noise traders to earn a higher expected return than rational investors do~(\\cite{doi:10.1086\/261703}).\n\nOn the other hand, \\cite{doi:10.1126\/science.185.4157.1124} found the effect of bias is asymmetric. They described three heuristics that are employed in making judgments under uncertainty: 1) representativeness; 2) availability of instances or scenarios; and 3) adjustment from an anchor. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. But the effect of these heuristics has not been detected as a global bias in the security market. Consequently, identifying the specific effects of noise and bias on security price is challenging.\n\nIf attention is paid to any statistical property in any complex system, the log-normal distribution is the most natural and appropriate among the standard or ''normal'' statistics to overview the whole system~(\\cite{doi:10.1143\/JPSJ.80.072001}). The log-normality emerges as familiar and typical examples of statistical aspects in various complex systems. Since every member of any complex system has its own history, each member is in the process of growth (or retrogression). The log-normal distribution is realized as a result of Gibrat' law, or Mathew effect. It is applied to cities size and growth rate, where proportionate growth process may give rise to a distribution of city sizes that is log-normal. When considering the entire size distribution, not just the largest cities, then the city size distribution is log-normal~(\\cite{10.2307\/2296055}). However, it has been argued that it is problematic to define cities through their fairly arbitrary legal boundaries. According to \\cite{10.1162\/003355399556133},\nZipf's law is a very tight constraint on the class of admissible models of local growth. It says that for most countries the size distribution of cities strikingly fits a power law: the number of cities with populations greater than S is proportional to 1\/S. Suppose that, at least in the upper tail, all cities follow some proportional growth process (this appears to be verified empirically). This automatically leads their distribution to converge to Zipf's law.\n\nGibrat's law of proportionate effect also states that the proportional change in the size of a firm is independent of its absolute size. An implication of this is that large and small firms have the same average proportionate rates of growth. Against this law, \\cite{10.2307\/2296055} shows large firms are growing faster significantly. \\cite{10.1257\/mac.20150051} construct a tractable neoclassical growth model that generates Pareto's law of income distribution and Zipf's law of the firm size distribution from idiosyncratic, firm-level productivity shocks. Executives and entrepreneurs invest in risk-free assets, as well as their own firms' risky stocks, through which their wealth and income depend on firm-level shocks. By using the model, they evaluate how changes in tax rates can account for the evolution of top incomes in the United States. The model matches the decline in the Pareto exponent of the income distribution and the trend of the top 1 percent income share in recent decades. In the same research direction, \\cite{NIREI201625} construct a neoclassical growth model with heterogeneous households that accounts for the Pareto distributions of income and wealth in the upper tail. In an standard Bewley model~(\\cite{BEWLEY1977252}), they feature households' business productivity risks and borrowing constraints, which they find generate the Pareto distributions. Households with low productivity rely on wages and returns from safe assets, while high productivity households choose not to diversify their business risks. Their model can quantitatively account for the observed income distribution in the U.S. under reasonable calibrations. Furthermore, they conduct several comparative statics to examine how changes in parameters affect the Pareto distributions. In particular, they find that the change in the top tax rates in the 1980s potentially accounts for much of the observed increase in top income dispersion in the last decades. Their analytical result provides a coherent interpretation for the numerical comparative statics.\n\nIn this article, I present the true fundamentals hypothesis based on rational expectations~(\\cite{10.2307\/1909635}) and, using a log-normal distribution model, detect global bias components from the price-earnings (P\/E), price-to-book (P\/B), and price-to-cash flow (P\/CF) ratios. The traditional theory of the firm is based on the assumption that the firm acts in the stockholders' interests and that stockholders are interested in profit, so that the object of the firm is to maximize profit. However, in fact, there is a certain range in the profit concept~(\\cite{10.2307\/2977477}). The analysis results strongly support the true fundamentals hypothesis as the detected biases show similar characteristics. Additionally, the results show that the cash flow indicators contain relatively few bias components and are closer to the true fundamentals. I further demonstrate and examine why the positive P\/IC ratio among the indicators analyzed is a proxy for the true fundamentals that does not include bias components.\n\n\\section{Hypothesis}\n\nWhen the true fundamentals of listed companies at time $t$ is denoted as $X_t$ and their growth rate is denoted as $R_t$, the growth of those companies can be expressed by the following Gibrat's process\n\\begin{equation}\n X_t=R_tX_{t-1} \\label{1}.\n\\end{equation}\nIt is important to note, however, that I assume that the growth rates\n$R_t$s are mutually independent random variables that follow the same\ndistribution with finite variance. The initial value of the fundamentals\nset as $X_0$ yields\n\\begin{equation}\n X_T=X_0\\prod_{t=1}^T R_t\n\\label{2}\n\\end{equation}\nat time $T$. Taking the log of both sides of the equation results in\n\\begin{equation}\n \\log X_T =\\log X_0+\\log R_1+\\cdots +\\log R_T.\n\\label{3}\n\\end{equation}\nTherefore,\n\\begin{equation}\n \\log X_T\\sim LN(\\mu, \\sigma^2)\n\\label{4}\n\\end{equation}\nwould hold true for a sufficiently large $T$ based on the central limit theorem. Essentially, the true fundamentals $X_T$ of listed companies follows the log-normal distribution.\n\nFurthermore, by assuming rational expectations through the future point in time $T$ as of the present point in time 0 on the premise of a going concern, the following equation becomes true:\n\\begin{equation}\n X_0=\\mathbb{E}[X_T]\\prod_{t=1}^T \\mathbb{E}[R_t^{-1}].\n\\label{5}\n\\end{equation}\n Therefore, the rational expectations $X_0$ for true fundamentals also follow the log-normal distribution. In other words, the true fundamentals $X_t$ at a given point in time $t$ follow the log-normal distribution.\n\n\\section{Analysis}\n\n\\subsection{Data}\n\nAmong the actual fundamental indicators, the price-earnings (P\/E), forward price-earnings (P\/FE), price-to-book (P\/B), price-to-cash flows from operating activities (P\/OC), price-to-cash flows from investing activities (P\/IC), price-to-cash flows from financing activities (P\/FC), and the price-to-cash equivalents at the year-end (P\/CE) ratios are analyzed. The cash flow ratios were divided into positive and negative data. The daily data with absolute values less than 1,000 were extracted for all companies listed in Japan for the period spanning 1,817 business days from January 2007 to May 2014. \n\n\\subsection{Test}\n\nIf the true fundamentals hypothesis is true, the actual fundamental indicators should follow the log-normal distribution. Therefore, the $p$-values were calculated using the Kolmogorov-Smirnov test, Pearson's $\\chi^2$ test, and Anderson-Darling test, to test the goodness of fit to the log-normal distribution. It is recognized that the results of these tests are strongly affected by the sample size.\n\nSince the sample size differs by indicator, samples at the 300 quartiles were extracted to calculate the $p$-value for testing the goodness of fit of the overall average of each indicator's samples to the log-normal distribution with variance. In all cases, the null hypothesis is that ``the indicator follows the log-normal distribution,'' or the otherwise worded ``the indicator is a proxy of true fundamentals.'' The indicator reflects the true fundamentals more when the $p$-value is closer to 1. When the $p$-value is closer to 0, the bias is stronger. The lowest $p$-value for each test is shown in Table~\\ref{tab1} for the data spanning 1,817 business days.\n\nThe test results indicate that bias has a strong effect on the P\/E, P\/FE, and P\/B ratios. It is consistent with our instincts that forward price-earnings ratio (P\/FE) is most biased. Interestingly, among all price-to-cash flow ratios, the bias is significant only for P\/OC ratio. It can be easier to manipulate than others. I illustrate the time series of $p$-values since the $p$-value is an indicator of the strength of bias. From table~\\ref{tab1}, Figure~\\ref{bias1} and Figure~\\ref{bias2}, it is clear that the positive P\/IC ratio is the best proxy for the true fundamentals that does not include bias components. Probably, it should be difficult to manipulate according to the opinions of management and accounting.\n\\begin{table}[ht] \n\\caption{The lowest $p$-values for the data spanning 1,817 business days. Samples at the 300 quantile. Significant (bias) levels: 1\\%(***), 3\\%(**), 5\\%(*)\\label{tab1}}\n\\newcolumntype{C}{>{\\centering\\arraybackslash}X}\n\\begin{tabularx}{\\textwidth}{llll}\n\\toprule\n\\textbf{Index}\t&\\qquad \\textbf{Kolmogorov-Smirnov}\t&\\qquad\\quad \\textbf{Peason's $\\chi^2$} &\\qquad \\textbf{Anderson-Darling}\\\\\n\\midrule\nP\/OC$+$ &\\qquad\\qquad\\qquad 0.0986 &\\qquad\\qquad 0.2843 &\\qquad\\qquad\\qquad 0.0398$^{*}$ \\\\\nP\/OC$-$ &\\qquad\\qquad\\qquad 0.0634 &\\qquad\\qquad 0.0045$^{***}$ &\\qquad\\qquad\\qquad 0.0850 \\\\\nP\/IC$+$ &\\qquad\\qquad\\qquad 0.6205 &\\qquad\\qquad 0.4441 &\\qquad\\qquad\\qquad 0.7201 \\\\\nP\/IC$-$ &\\qquad\\qquad\\qquad 0.2787 &\\qquad\\qquad 0.7012 &\\qquad\\qquad\\qquad 0.1849 \\\\\nP\/FC$+$ &\\qquad\\qquad\\qquad 0.2042 &\\qquad\\qquad 0.2287 &\\qquad\\qquad\\qquad 0.1093 \\\\\nP\/FC$-$ &\\qquad\\qquad\\qquad 0.2522 &\\qquad\\qquad 0.5763 & \\qquad\\qquad\\qquad 0.1269 \\\\\nP\/CE &\\qquad\\qquad\\qquad 0.4611 &\\qquad\\qquad 0.8740 &\\qquad\\qquad\\qquad 0.2376 \\\\\nP\/E &\\qquad\\qquad\\qquad 0.0170$^{**}$ &\\qquad\\qquad 0.0000$^{***}$ &\\qquad\\qquad\\qquad 0.0018$^{***}$ \\\\\nP\/FE &\\qquad\\qquad\\qquad 0.0107$^{**}$ &\\qquad\\qquad 0.0040$^{***}$ &\\qquad\\qquad\\qquad 0.0019$^{***}$ \\\\\nP\/B &\\qquad\\qquad \\qquad0.0032$^{***}$ &\\qquad\\qquad 0.0023$^{***}$ &\\qquad\\qquad\\qquad 0.0005$^{***}$ \\\\\n\\bottomrule\n\\end{tabularx}\n\\end{table}\n\\unskip\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=11 cm]{bias1.pdf}\n\\caption{Time series of the goodness of fit test to the log-normal distribution. Blue: Kolmogorov-Smirnov, Yellow: Peason's $\\chi^2$, Blue: Anderson-Darling. The higher the value, the less bias there is, vice versa. \\label{bias1}}\n\\end{center}\n\\end{figure} \n\\unskip\n\n\\subsection{Shape of Bias}\n\nThe detected biases exhibit similar characteristics. To facilitate an intuitive understanding of these results, each indicator is shown in figure~\\ref{bias2} and is compared to the log-normal distribution. While the figures only show the data for the first day of the period, January 4, 2007, the characteristics of the biases themselves remain essentially unchanged throughout the period although the levels fluctuated. It is worth noting that the shape of the curve, and in particular the strength of the bias, is independent of whether the economy is doing well or not. The stock market, which is clearly artificial, appears to have a natural intelligence.\n\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=10 cm]{bias2.pdf}\n\\caption{The more the data deviates from a straight line showing a log-normal distribution, the stronger the bias. These graphs are drawn with data from the first day of the time series. The shape of the bias varies proportionally to Figure~\\ref{bias1}, but is stable throughout the time. \\url{https:\/\/figshare.com\/projects\/Intelligence_and_Global_Bias_in_the_Stock_Market\/131711}\\label{bias2}}\n\\end{center}\n\\end{figure} \n\\unskip\n\n\n\\section{Discussion and Conclusion}\n\nThe results indicate that the major fundamental indicators including the P\/E, P\/FE, and P\/B ratios are strongly affected by bias. Bias also has a significant effect on the positive and negative P\/OC ratios. Additionally, there may be a weak effect of bias on the negative P\/IC, positive and negative P\/FC, and P\/CE ratios.\n\nWhen the test results are compared, the positive P\/IC ratio is the stable proxy of true fundamentals among all these indicators, and it is important to consider why this occurs.\n\nPositive cash flows from investing activities represent the realized gain or loss from past investments such as marketable securities, tangible fixed assets, the sales of investment securities, and income from the collection on loans declared at the end of accounting period. In other words, positive cash flow is the indicator that most directly reflects the past business decisions that determine the growth of a company. The year-end cash equivalent ratio and other cash flow ratios can be interpreted to strongly reflect the true fundamentals because they have less bias components due to the strong characteristic of having definite value.\n\nOn the other hand, although the P\/E ratio and P\/B ratio are definite values, investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. The P\/OC ratio might also have a lower credibility compared to other cash flow indicators.\n\nWhere do these biases come from? The answer is extremely simple: ``Cash is a fact, profit is an opinion.'' Namely, opinions of management and accountant are added as noise to true fundamentals. As a result, Kesten process~(\\cite{10.1007\/BF02392040})\n\\begin{equation}\n X_t = R_t X_{t-1}+\\epsilon_t,\\quad E[\\epsilon_t]>0\n\\end{equation}\nis realized and the Pareto distribution is to be obtained. This means their opinions are accompanied by a positive bias.\n\nIn fact, these biases fit the Pareto distribution quite well. The generalized Pareto distribution (GPD) is represented by the following functions. \n\n\\paragraph{Generalized Pareto Distribution (GPD)}\n\\[\nF(x)=1-\\left(1+\\left(\\frac{x-\\mu}{\\kappa}\\right)^{1\/\\gamma}\\right)^{-\\alpha} \n\\]\n\\[\nf(x)=\\frac{\\alpha}{\\gamma}\\kappa^{-1\/\\gamma}(x-\\mu)^{-1+\\frac{1}{\\gamma}}\\left(1+\\left(\\frac{x-\\mu}{\\kappa}\\right)^{1\/\\gamma}\\right)^{-\\alpha-1}\n\\]\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=11cm]{bias3.pdf}\n\\caption{Comparison of three distributions: Log-Normal(blue), GDP-1(Yellow), GDP-2(Green)\\label{bias3}}\n\\end{center}\n\\end{figure} \n\\unskip\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=13cm]{bias4.pdf}\n\\caption{Left: GDP-1 to fit P\/FE ($\\kappa=18.82,\\alpha=1, \\gamma=0.385, \\mu=0.993$), Right: GDP-2 to fit P\/FE ($\\kappa=13.70,\\alpha=0.515, \\gamma=0.238, \\mu=0.993$)\\label{bias4}. Note that P\/FE has the strongest bias.}\n\\end{center}\n\\end{figure} \n\n\nWhat are the implications of the results? Does the existence of bias negate the efficient market hypothesis? From the view point of intelligence in natural and digital computing, how should we deal with this problem?\n\nThe stock market, which is clearly artificial, and where countless traders are using digital computers to trade, appears to have a natural intelligence. Intelligence does not arise only in individual brains; it also arises in groups of individuals~(\\cite{malone2015handbook}). If the true fundamentals hypothesis is really true, the major fundamental indicators including the P\/E, P\/FE, and P\/B ratios are strongly affected by bias. This means that investors might not view them as indicators that reflect the true fundamentals since there is a high degree of freedom in accounting. Globally, this implies that the efficient market hypothesis holds~(\\cite{fama1965behavior}). The shape of the curve, and in particular the strength of the bias, is stable throughout the time and independent of whether the economy is doing well or not. The curve with bias is the path of the ants~(\\cite{kirman1993ants, app12147019}) to avoid obstacles, so to speak, and traders move back and forth along it in response to noise~(\\cite{black1986noise, doi:10.1086\/261703}). This noise, among other things, plays a role in activating the market system, which knows true fundamentals of firms and represents as a stable global bias in the stock market.\n\n\n\n\\newpage\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nThe Hamilton--Jacobi theory is a topic of interest in mathematical physics\nsince it is a way to integrate systems of first-order ordinary differential equations\n(Hamilton equations in the standard case).\nThe classical method in Hamiltonian mechanics consists in \nobtaining a suitable canonical transformation which leads \nthe system to equilibrium \\cite{Ar,JS,LL-mec,Sa71},\nand is given by its generating function. This function is \nthe solution to the so-called {\\sl Hamilton--Jacobi equation},\nwhich is a partial differential equation.\nThe {\\sl ``classical'' Hamilton--Jacobi problem} consists in finding this canonical transformation.\nBecause of its interest, the method was generalized in other\nkinds of physical systems; such as, for instance,\nsingular Lagrangian systems \\cite{Gomis1} or higher-order dynamics \\cite{art:Constantelos84},\nand different types of solutions have been proposed and studied\n\\cite{CEL-84,CL-83}.\n\nNevertheless, in recent times, a lot of research has been done to understand \nthe Hamilton--Jacobi equation from a more general geometric approach,\nand some geometric descriptions to the theory were done in\n\\cite{Ab,BT-80,EMS-2004,LM-87,MMMcq}.\nFrom a geometric way, the above mentioned canonical transformation\nis associated with a foliation in the the phase space of the system\nwhich is represented by the cotangent bundle $\\Tan^*Q$\nof a manifold (the configuration manifold $Q$).\nThis foliation has some characteristic geometric properties:\nit is invariant by the dynamics, transversal to the fibers of the cotangent bundle,\nand Lagrangian with respect to the\ncanonical symplectic structure of $\\Tan^*Q$\n(although this last property could be ignored in some particular situations).\nThe restriction of the dynamical vector field in $\\Tan^*Q$ to\neach leaf $S_\\lambda$ of this foliation \nprojects onto another vector field $X_\\lambda$ on $Q$,\nand the integral curves of these vector fields are one-to-one related.\nHence, the complete set of dynamical trajectories are recovered from the integral curves\nof the complete family $\\{ X_\\lambda\\}$ of all these vector fields in the base.\nThese geometric considerations can be done in an analogous way \nin the Lagrangian formalism and hence this geometrical picture \nof the Hamilton--Jacobi theory can be also stated for this formalism.\nThe {\\sl geometric Hamilton--Jacobi problem} consists in finding\nthis foliation and these vector fields $\\{ X_\\lambda\\}$.\n\nFollowing these ideas, the Lagrangian and Hamiltonian versions\nof the Hamilton--Jacobi theory, for autonomous and non-autonomous mechanical systems, \nwas formulated in another geometrical way\n in \\cite{art:Carinena_Gracia_Marmo_Martinez_Munoz_Roman06}. \nThe foundations of this geometric generalization are similar \nto those given in \\cite{Ab,KV-1993}.\nLater, this framework has been used to develop the Hamilton--Jacobi theory\nfor many other kinds of systems in physics. \nFor instance, other applications of the theory are to the case of\nsingular Lagrangian and Hamiltonian systems \\cite{HJteam-K,LMV-12,LMV-13,LOS-12},\nhigher-order dynamical systems \\cite{CLRP-13,CLRP-14},\nholonomic and non-holonomic mechanics\n\\cite{BFS-14,HJTeam2,leones2,leones1,LOS-12,OFB-11,blo},\nand control theory \\cite{BLMMM-12,Wang2}.\nThe theory is also been extended \nfor dynamical systems described using other geometric structures, \nsuch as Poisson manifolds \\cite{LMV-13,GP-2016}, Lie algebroids \\cite{BMMP-10,LS-12},\ncontact manifolds (which model dissipative systems) \\cite{Ll-2020,GP-2020},\nand other geometric applications and generalizations:\n\\cite{BLM-12,HJTeam3,Wang3}.\nFurthermore, in \\cite{BDM-12,OBL-11}\nthe geometric discretization of the Hamilton--Jacobi equation was analyzed.\nFinally, the Hamilton--Jacobi theory \nis developed for the usual covariant formulations of first-order classical field theories,\nthe $k$-symplectic and $k$-cosymplectic \\cite{LMMSV-12,DeLeon_Vilarino}\nand the multisymplectic ones \\cite{LMM-09,LPRV-2017},\nfor higher-order field theories \\cite{Vi-10,art:Vitagliano12},\nfor the formulation in the {\\sl Cauchy data space\\\/} \\cite{CLMV-14},\nand for partial differential equations in general \\cite{Vi-11,Vi-14}.\n\nThis review paper is devoted, first of all, to present,\nin Section \\ref{firstsec}, the foundations\nof this modern geometric formulation of the Hamilton--Jacobi theory,\nstarting from the most general problem and explaining how to derive the standard\nHamilton--Jacobi equation for the Hamiltonian and the Lagrangian formalisms of autonomous mechanics.\nAfter this, the notion of complete solution allows us to establish the\nrelation with the ``classical'' Hamilton--Jacobi theory based on canonical transformations,\nwhich is summarized in Section \\ref{clasic},\nwhere this relation is also analyzed \n(this topic had been already discussed in \\cite{Vi-14}).\nWe also present briefly a more general geometric framework\nfor the Hamilton--Jacobi theory which was stated in \\cite{HJTeam3},\nfrom which we can derive the majority of the applications\nof the theory to other kinds of physical systems,\nincluding the case of autonomous dynamical systems.\nThis is done in Section \\ref{co}.\nFinally, among all the extensions of the theory, \nwe have selected two of them for reviewing:\nthe case of higher-order autonomous dynamical systems\n(Lagrangian and Hamiltonian formalisms), \nwhich is a direct application of the above general framework,\nand the generalization to the Lagrangian and Hamiltonian multisymplectic\nformalisms of first-order classical field theories, which can also be interpreted as a\nspecial case of the general framework.\nBoth of them are treated in Section \\ref{others}.\n\nThroughout the work it is considered that\nthe manifolds are real, smooth and second countable. \nIn the same way, all the maps are assumed to be smooth. \nThe summation convention for repeated cross indices is also assumed.\n\n\n \\section{The geometric Hamilton--Jacobi theory}\n\\label{firstsec}\n\n\nWe summarize the main features of the geometric Hamilton--Jacobi theory\n for the Hamiltonian and Lagrangian formalisms\nof autonomous dynamical systems as it is stated in\n\\cite{art:Carinena_Gracia_Marmo_Martinez_Munoz_Roman06}\n(see also \\cite{Ab,KV-1993}).\n\n\n \\subsection{Hamiltonian Hamilton--Jacobi problem}\n\n\nTypically, a {\\sl (regular autonomous) Hamiltonian system} is a triad\n$(\\Tan^*Q,\\omega,H )$, where the bundle\n$\\pi_Q\\colon\\Tan^*Q\\to Q$ represents the {\\sl phase space} of a dynamical system\n($Q$ is the {\\sl configuration space\\\/}),\n$\\omega=-\\d\\theta \\in\\df^2(\\Tan^*Q)$ is the natural symplectic form in $\\Tan^*Q$,\nand $H \\in \\Cinfty (\\Tan^*Q)$ is the {\\sl Hamiltonian function}.\nThe dynamical trajectories are the integral curves \n$\\sigma \\colon I\\subseteq\\Real\\to \\Tan^*Q$\nof the {\\sl Hamiltonian vector field} $Z_H\\in\\vf(\\Tan^* Q)$ associated with $H$,\nwhich is the solution to the {\\sl Hamiltonian equation}\n\\beq\n\\inn(Z_H)\\omega=\\d H \\ .\n\\label{hameq}\n\\eeq\n(Here, $\\df^k(\\Tan^*Q)$ and $\\vf(\\Tan^* Q)$ are the sets of differentiable $k$-forms and vector fields in $\\Tan^*Q$, and $\\inn(Z_H)\\omega$ denotes the inner contraction of $Z_H$ and $\\omega$). \nIn a chart of natural coordinates $(q^i,p_i)$ in $\\Tan^*Q$\nwe have that $\\omega=\\d q^i\\wedge\\d p_i$,\nand the curves $\\sigma(t)=(q^i(t),p_i(t))$\nare the solution to the {\\sl Hamilton equations}\n$$\n\\frac{\\d q^i}{\\d t} = \\derpar{H}{p_i}(q(t),p(t)) \\quad , \\quad \n\\frac{\\d p_i}{\\d t} = -\\derpar{H}{q^i}(q(t),p(t)) \\ .\n$$\n\n\\begin{definition}\n\\label{genHJh}\nThe \\textbf{generalized Hamiltonian Hamilton--Jacobi problem}\nfor a Hamiltonian system $(\\Tan^*Q,\\omega,H )$\nis to find a vector field $X\\in\\vf(Q)$ and\na $1$-form $\\alpha\\in\\df^1(Q)$\nsuch that, if $\\gamma\\colon\\Real\\to Q$ is an integral curve\nof $X$, then $\\alpha\\circ\\gamma\\colon\\Real\\to\\Tan^*Q$\nis an integral curve of $Z_H$; that is,\nif $X\\circ\\gamma=\\dot\\gamma$, then\n$\\dot{\\overline{\\alpha\\circ\\gamma}}=Z_H\\circ(\\alpha\\circ\\gamma)$.\nThen, the couple $(X,\\alpha)$ is a \\textbf{solution to the generalized Hamiltonian Hamilton--Jacobi problem}.\n\\end{definition}\n\n\\begin{teor}\n\\label{HJhcns}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item\nThe couple $(X,\\alpha)$ is a solution to the generalized Hamiltonian Hamilton--Jacobi problem.\n\\item \nThe vector fields $X$ and $Z_H$ are $\\alpha$-related; that is,\n$Z_H\\circ\\alpha=\\Tan\\alpha\\circ X$.\nAs a consequence\n$X=\\Tan\\pi_Q\\circ Z_H\\circ\\alpha$ and it is called the\n{\\rm vector field associated with the form $\\alpha$}.\n\\item\nThe submanifold ${\\rm Im}\\,\\alpha$ of $\\Tan^*Q$\nis invariant by the Hamiltonian vector field $Z_H$\n(or, what means the same thing, $Z_H$ is tangent to ${\\rm Im}\\,\\alpha$).\n\\item\nThe integral curves of $Z_H$ which have their initial conditions\nin $\\mathrm{Im}\\,\\alpha$ project onto the integral curves of \n$X$.\n\\item\nThe equation\n\\ $\\inn(X)\\d\\alpha=-\\d(\\alpha^*H)$ holds for the $1$-form $\\alpha$.\n\\end{enumerate}\n$$\n\\xymatrix{\n\\ & \\Tan Q \\ar@\/_1.5pc\/[rr]_{\\Tan\\alpha} & \\ & \\ar[ll]_{\\Tan\\pi_Q} \\Tan(\\Tan^*Q) \\\\\n\\ & \\ & \\ & \\ \\\\\n\\Real \\ar[r]_{\\gamma} \\ & Q \\ar[uu]^{X} \\ar@\/_1.5pc\/[rr]_{\\alpha} & \\ & \\ar[ll]_{\\pi_Q} \\Tan^*Q \\ar[uu]_{Z_H}\n}\n$$\n\\end{teor}\n\\proof\n(Guidelines for the proof):\n\nThe equivalence between 1 and 2 is a consequence of the Definition \\ref{genHJh} and the definition of integral curves.\nThen, the expression $X=\\Tan\\pi_Q\\circ Z_H\\circ\\alpha$ is obtained by composing both members of the equality \n$Z_H\\circ\\alpha=\\Tan\\alpha\\circ X$ with $\\Tan\\pi_Q$\nand taking into account that $\\pi_Q\\circ\\alpha={\\rm Id}_Q$.\n\nItems 3 and 4 follow from 2.\n\nItem 5 is obtained from Definition \\ref{genHJh} and using the dynamical equation \\eqref{hameq}.\n\\qed\n\nIn order to solve the generalized Hamilton--Jacobi problem,\nit is usual to state a less general version of it,\nwhich constitutes the standard Hamilton--Jacobi problem.\n\n\\begin{definition}\n\\label{HJh}\nThe \\textbf{Hamiltonian Hamilton--Jacobi problem}\nfor a Hamiltonian system $(\\Tan^*Q,\\omega,H )$ \nis to find a $1$-form $\\alpha\\in\\df^1(\\Tan^*Q)$\nsuch that it is a solution to the generalized Hamiltonian Hamilton--Jacobi problem and is closed.\nThen, the form $\\alpha$ is a \\textbf{solution to the Hamiltonian Hamilton--Jacobi problem}.\n\\end{definition}\n\nAs $\\alpha$ is closed, for every point in $Q$,\nthere is a function $S$ in a neighbourhood $U\\subset Q$\nsuch that $\\alpha=\\d S$.\nIt is called a {\\sl local generating function} of the solution $\\alpha$.\n\n\\begin{teor}\n\\label{cnsh}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item\nThe form $\\alpha\\in\\df^1(Q)$ is a solution to the Hamiltonian Hamilton--Jacobi problem.\n\\item\n$\\mathrm{Im}\\,\\alpha$ is a Lagrangian submanifold of\n$\\Tan^*Q$ which is invariant by $Z_H$,\nand $S$ is a local generating function of this Lagrangian submanifold.\n\\item\nThe equation $\\d(\\alpha^*H)=0$ holds for $\\alpha$ \nor, what is equivalent, the function\n$H\\circ\\d S\\colon Q\\to\\Real$ is locally constant.\n\\end{enumerate}\n\\end{teor}\n\\proof\n(Guidelines for the proof):\nThey are consequences of Theorem \\ref{HJhcns} and Definition \\ref{HJh}.\n\\qed\n\nThe last condition, written in natural coordinates, gives the classical form of the\nHamiltonian Hamilton--Jacobi equation, which is\n\\beq\nH\\left(q^i,\\derpar{S}{q^i}\\right)= E\\ (ctn.) \\ .\n\\label{tres}\n\\eeq\n\nThese forms $\\alpha$ are particular solutions to the (generalized) Hamilton--Jacobi problem, \nbut we are also interested in finding {\\sl complete solutions} to the problem.\nThen:\n\n\\begin{definition}\n\\label{completeHJ}\nLet $\\Lambda\\subseteq\\Real^n$.\nA family of solutions $\\{ \\alpha_\\lambda; \\lambda\\in\\Lambda\\}$,\n depending on $n$ parameters\n$\\lambda\\equiv (\\lambda_1,\\ldots,\\lambda_n)\\in\\Lambda$, \nis a \\textbf{complete solution} to the Hamiltonian Hamilton--Jacobi problem\nif the map \n$$\n\\begin{array}{ccccc}\n\\Phi & \\colon &Q\\times\\Lambda & \\longrightarrow & \\Tan^*Q \\\\\n & & (q,\\lambda) & \\mapsto & \\alpha_\\lambda(q)\n\\end{array}\n$$\nis a local diffeomorphism.\n\\end{definition}\n\n\\begin{remark}{\\rm\nGiven a complete solution $\\{ \\alpha_\\lambda; \\lambda\\in\\Lambda\\}$,\nas $\\d\\alpha_\\lambda=0$, $\\forall\\lambda\\in\\Lambda$, \nthere is a family of functions $\\{ S_\\lambda\\}$ defined in \nneighbourhoods $U_\\lambda\\subset Q$ of every point\nsuch that $\\alpha_\\lambda=\\d S_\\lambda$.\nTherefore we have a locally defined function\n$$\n\\begin{array}{ccccc}\n{\\cal S} & \\colon &\\bigcap U_\\lambda\\times\\Lambda\\subset Q\\times\\Lambda & \\longrightarrow & \\Real \\\\\n & & (q,\\lambda) & \\mapsto & S_\\lambda(q)\n\\end{array}\n$$\nwhich is called a {\\sl local generating function} of the \ncomplete solution $\\{ \\alpha_\\lambda; \\lambda\\in\\Lambda\\}$.\n\n\\noindent A complete solution defines a Lagrangian foliation in $\\Tan^*Q$\nwhich is transverse to the fibers,\nand such that $Z_H$ is tangent to the leaves.\nThe functions that define locally this foliation are the components of a map\n$F\\colon \\Tan^*Q\\stackrel{\\Phi^{-1}}{\\longrightarrow} Q\\times\\Lambda \n\\stackrel{pr_2}{\\longrightarrow} \\Lambda\\subset\\Real^n$,\nand give a family of constants of motion of $Z_H$.\nConversely, if we have $n$ first integrals $f_1,\\ldots,f_n$ \nof $Z_H$ in involution, such that $\\d f_1\\wedge\\ldots\\wedge \\d f_n\\not=0$;\nthen $f_i=\\lambda_i$, with $\\lambda_i\\in\\Real$,\ndefine this transversal Lagrangian foliation and hence a\nlocal complete solution $\\{\\alpha_\\lambda,\\lambda\\in\\Lambda\\}$.\nThus we can locally isolate\n$p_i=p_i(q,\\lambda)$, replace them in $Z_H$\nand project to the basis, then obtaining the family of vector fields \n$\\{X_\\lambda\\}$ associated with the local complete solution.\nIf $\\{ \\alpha_\\lambda;\\lambda\\in\\Lambda\\}$ is a complete solution,\nthen all the integral curves of $Z_H$ are obtained from\nthe integral curves of the associated vector fields $\\{ X_\\lambda\\}$.}\n\\end{remark}\n\n\n\\subsection{Lagrangian Hamilton--Jacobi problem}\n\nThe above framework for the Hamilton--Jacobi theory can\nbe easily translated to the Lagrangian formalism of mechanics.\nNow, the phase space is\nthe tangent bundle $\\tau_Q\\colon\\Tan Q\\to Q$ of the configuration bundle $Q$ and\nthe dynamics is given by the Lagrangian function of the system,\n$\\L\\in\\Cinfty(\\Tan Q)$.\nUsing the canonical structures in $\\Tan Q$; that is,\nthe {\\sl vertical endomorphism} $S\\in{\\mathfrak T}^1_1(\\Tan Q)$,\nand the {\\sl Liouville vector field} $\\Delta\\in\\vf(\\Tan Q)$,\nthe Lagrangian forms\n$\\theta_\\L:=\\d\\L\\circ S\\in\\df^1(\\Tan Q)$, $\\omega_\\L=-d\\theta_\\L\\in\\df^2(\\Tan Q)$,\nand the Lagrangian energy\n$E_\\L:=\\Delta(\\L)-\\L\\in\\Cinfty(\\Tan Q)$ are constructed.\nThen the {\\sl Lagrangian equation} is\n\\begin{equation}\n\\inn(\\Gamma_\\L)\\omega_\\L=\\d E_\\L \\ ,\n\\label{ELeq}\n\\end{equation}\nand $(\\Tan Q,\\omega_\\L,E_\\L)$ is a {\\sl Lagrangian dynamical system}.\nFurthermore, the {\\sl Legendre transformation} associated with $\\L$,\ndenoted by $\\FL\\colon \\Tan Q\\to \\Tan^*Q$, is defined\nas the fiber derivative of the Lagrangian function.\nWe assume that $\\L$ is regular; that is,\n$\\FL$ is a local diffeomorphism or, equivalently,\n$\\omega_\\L$ is a symplectic form\n(the Lagrangian is {\\sl hyper-regular} if $\\FL$ is a global diffeomorphism).\nIn that case, the Lagrangian equation (\\ref{ELeq})\nhas a unique solution $\\Gamma_\\L\\in\\vf(\\Tan Q)$,\nwhich is called the {\\sl Lagrangian vector field},\nwhose integral curves are holonomic ,\nand are the solutions to the Euler-Lagrange equations.\n(See \\cite{Cr-83} for details).\n\n\\begin{definition}\n\\label{genHJl}\nThe \\textbf{generalized Lagrangian Hamilton--Jacobi problem} \nfor a Lagrangian system $(\\Tan Q,\\omega_\\L,E_\\L)$ \nis to find a vector field $X\\in\\vf(Q)$\nsuch that, if $\\gamma\\colon\\Real\\to Q$ is an integral curve\nof $X$, then $X\\circ\\gamma=\\dot\\gamma\\colon\\Real\\to \\Tan Q$\nis an integral curve of $\\Gamma_\\L$; that is,\nif $X\\circ\\gamma=\\dot\\gamma$, then\n$\\Gamma_\\L\\circ\\dot\\gamma=\\dot{\\overline{X\\circ\\gamma}}$.\nThen, the vector field $X$ is a \\textbf{solution to the generalized \nLagrangian Hamilton--Jacobi problem}.\n\\end{definition}\n\n\\begin{teor}\n\\label{HJlcns}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item \nThe vector field $X$ is a solution to the generalized Lagrangian\nHamilton--Jacobi problem.\n\\item\nThe vector fields $X$ and $\\Gamma_\\L$ are $X$-related; that is,\n$\\Gamma_\\L\\circ X=TX\\circ X$.\n\\item\nThe submanifold ${\\rm Im}\\,X$ of $\\Tan Q$\nis invariant by the Lagrangian vector field $\\Gamma_\\L$\n(or, what means the same thing, $\\Gamma_\\L$ is tangent to ${\\rm Im}\\,X$).\n\\item\nThe integral curves of $\\Gamma_\\L$ which have their initial conditions\nin ${\\rm Im}\\, X$ project onto the integral curves of $X$.\n\\item\nThe equation\n\\ $\\inn(X)(X^*\\omega_\\L)=\\d(X^*E_\\L)$ holds for the vector field $X$.\n\\end{enumerate}\n$$\n\\xymatrix{\n\\ & \\Tan Q \\ar@\/_1.5pc\/[rr]_{\\Tan X} & \\ & \\ar[ll]_{\\Tan\\tau_Q} \\Tan(\\Tan Q) \\\\\n\\ & \\ & \\ & \\ \\\\\n\\Real \\ar[r]_{\\gamma} \\ & Q \\ar[uu]^{X} \\ar@\/_1.5pc\/[rr]_{X} & \\ & \\ar[ll]_{\\tau_Q} \\Tan Q \\ar[uu]_{\\Gamma_\\Lag}\n}\n$$\n\\end{teor}\n\\proof\n(Guidelines for the proof):\nThe proof follows the same patterns as Theorem \\ref{HJhcns}.\n\\qed\n\nAs in the Hamiltonian formalism, we consider the following simpler case:\n\n\\begin{definition}\n\\label{HJl}\nThe \\textbf{Lagrangian Hamilton--Jacobi problem} for a Lagrangian system \n$(\\Tan Q,\\omega_\\L,E_\\L)$ is to find a vector field $X$ such that\nit is a solution to the generalized Lagrangian Hamilton--Jacobi problem\nand satisfies that $X^*\\omega_\\L=0$.\nThen, this vector field $X$ is a \\textbf{solution to the Lagrangian Hamilton--Jacobi problem}.\n\\end{definition}\n\nSince $0=X^*\\omega_\\L=-X^*\\d\\theta_\\L=-d(X^*\\theta_\\L)$\nthen, for every point of $Q$, there is a neighbourhood $U\\subset Q$\nand a function $S$ such that\n$X^*\\theta_\\L=\\d S$ (in $U$).\n\n\\begin{teor}\n\\label{cnsl}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item \nThe vector field $X$ is a solution to the Lagrangian Hamilton--Jacobi problem.\n\\item\n${\\rm Im}\\,X$ is a Lagrangian submanifold of $\\Tan Q$ which\nis invariant by the Lagrangian vector field $\\Gamma_\\L$\n(and $S$ is a local generating function of this Lagrangian submanifold).\n\\item\nThe equation\n$\\d(X^*E_\\L)=0$ holds for $X$ or, what is equivalent, the function\n$E_\\Lag\\circ\\d S\\colon Q\\to\\Real$ is locally constant.\n\\end{enumerate}\n\\end{teor}\n\\proof\n(Guidelines for the proof):\nThey are consequences of Theorem \\ref{HJlcns} and Definition \\ref{HJl}.\n\\qed\n\nThe last condition leads to the following expression\nwhich is the form of the Lagrangian Hamilton--Jacobi equation in natural coordinates,\n\\beq\n\\label{eqn:LagHJCharFunctLocalClassic}\n\\derpar{S}{q^i} = \\derpar{\\Lag}{v^i}(q^i,X^i) \\ .\n\\eeq\n\nAs in the Hamiltonian Hamilton--Jacobi theory,\nwe are interested in the complete solutions to the problem,\nwhich are defined as:\n\n\\begin{definition}\n\\label{lagcomplete}\nLet $\\Lambda\\subseteq\\Real^n$.\nA family of solutions$\\{ X_\\lambda; \\lambda\\in\\Lambda\\}$\ndepending on $n$ parameters \n$\\lambda\\equiv (\\lambda_1,\\ldots,\\lambda_n)\\in\\Lambda$, \nis a \\textbf{complete solution} to the Lagrangian Hamilton--Jacobi problem\nif the map \n$$\n\\begin{array}{ccccc}\n\\Psi & \\colon &Q\\times\\Lambda & \\longrightarrow & \\Tan Q \\\\\n & & (q,\\lambda) & \\mapsto & X_\\lambda(q)\n\\end{array}\n$$\nis a local diffeomorphism.\n\\end{definition}\n\nIf we have a complete solution to the Lagrangian Hamilton--Jacobi problem,\nall the integral curves of the Lagrangian vector field $\\Gamma_\\L$\nare obtained from the integral curves of all the vector fields$X_\\lambda$.\n\nThe equivalence between the Lagrangian and the Hamiltonian \nHamilton--Jacobi problems is stated as follows:\n\n\\begin{teor}\n\\label{relation}\nLet \\ $(\\Tan Q,\\omega_\\L,E_\\L)$ \\ be a (hyper)regular Lagrangian system, and\n$(\\Tan^*Q,\\omega,H)$ its associated Hamiltonian system.\nIf $\\alpha\\in\\df^1(Q)$ is a solution to the \n(generalized) Hamiltonian Hamilton--Jacobi\nproblem, then $X=\\FL^{-1}\\circ\\alpha$ is a solution to the\n(generalized) Lagrangian Hamilton--Jacobi problem\nand conversely,\nIf $X\\in\\vf(Q)$ is a solution to the\n(generalized) Lagrangian Hamilton--Jacobi\nproblem, then $\\alpha=\\FL\\circ X$ is a solution to the\n(generalized) Hamiltonian Hamilton--Jacobi problem.\n\\end{teor} \n\\proof\n(Guidelines for the proof):\nIt can be proven that $\\alpha={\\cal FL}\\circ X$; then,\nbearing in mind that $\\Tan{\\cal FL}\\circ\\Gamma_\\L=Z_H\\circ{\\cal FL}$,\nthe proof follows using items 2 and 5 of Theorems \\ref{HJhcns} and \\ref{HJlcns} (or item 3 of Theorems \\ref{cnsh} and \\ref{cnsl}).\n\\qed\n\n\n \\section{The ``classical'' Hamilton--Jacobi theory}\n\\label{clasic}\n\n\nIn this section we review the geometric description\nof the classical Hamiltonian Hamilton--Jacobi theory (for autonomous systems),\nbased on using canonical transformations \\cite{Ab,Ar,JS,LM-87,MMMcq}.\nIt is stated in the Hamiltonian formalism .\n\n\n \\subsection{Canonical transformations and the classical Hamiltonian Hamilton--Jacobi problem}\n\n\n\nFirst we remind the following well-known results \\cite{Ab}:\n\n\\begin{prop}\nLet $(M_1,\\omega_1)$, $(M_2,\\omega_2)$ be symplectic manifolds\nand $\\pi_j\\colon M_1\\times M_2\\to M_j$, $j=1,2$.\nThen $(M_1\\times M_2,\\pi_1^*\\omega_1-\\pi_2^*\\omega_2)$\nis a symplectic manifold.\n\\end{prop}\n\n\\begin{prop}\nLet $\\Phi\\colon M_1\\to M_2$ be a diffeomorphism and\n${\\jmath}\\colon graph\\,\\Phi\\hookrightarrow M_1\\times M_2$.\\\\\n$\\Phi$ is a symplectomorphism (i.e., $\\Phi^*\\omega_2=\\omega_1$) if, and only if,\n$graph\\,\\Phi$ is a Lagrangian submanifold of $(M_1\\times M_2,\\pi_1^*\\omega_1-\\pi_2^*\\omega_2)$.\n\\end{prop}\n\nIf $\\omega_j=-\\d\\theta_j$, $j=1,2$;\nbeing $graph\\,\\Phi$ a Lagrangian submanifold we have\n\\beq\n0=\\jmath^*(\\pi_1^*\\omega_1-\\pi_2^*\\omega_2)=\\d\\jmath^*(\\pi_2^*\\theta_2-\\pi_1^*\\theta_1)\n\\ \\Longleftrightarrow\\ \\jmath^*(\\pi_2^*\\theta_2-\\pi_1^*\\theta_1)\\vert_{\\cal W}=-\\d{\\cal S} \\ .\n\\label{Wlocal}\n\\eeq\n${\\cal S}$ is a function defined in \nan open neighbourhood ${\\cal W}\\subset graph\\,\\Phi$ of every point, which \ndepends on the choice of $\\theta_1$ and $\\theta_2$.\n\n\\begin{definition}\n${\\cal S}$ is called a\n\\textbf{generating function} of the Lagrangian submanifold $graph\\,\\Phi$\nand hence of the symplectomorphism $\\Phi$.\n\\end{definition}\n\nIf $({\\cal U}_1; q^i,p_i)$, $({\\cal U}_2; \\tilde q^i,\\tilde p_i)$ are Darboux charts such that\n${\\cal W}\\subset {\\cal U}_1\\times {\\cal U}_2$, local coordinates in ${\\cal W}$ can be chosen in several ways.\nThis leads to different possible choices for ${\\cal S}$.\nThus, for instance, if $({\\cal W}; q^i\\tilde q^i)$ is a chart,\nthen \\eqref{Wlocal} gives the symplectomorphism explicitly as\n$$\n\\tilde p_i\\d\\tilde q^i-p_i\\d q^i=-\\d {\\cal S}(q,\\tilde q)\n\\quad \\Longleftrightarrow\\ \\quad\n\\tilde p_i=-\\derpar{{\\cal S}}{\\tilde q^i}(q,\\tilde q)\\ , \\ p_i=\\derpar{{\\cal S}}{q^i}(q,\\tilde q)\\ .\n$$\n\nNow, let $(\\Tan^*Q,\\omega,H )$ be a Hamiltonian system.\n\n\\begin{definition}\nA \\textbf{canonical transformation} for a Hamiltonian system\n$(\\Tan^*Q,\\omega,H )$ is a symplectomorphism\n$\\Phi\\colon \\Tan^*Q\\to \\Tan^*Q$.\nAs a consequence, $\\Phi$ transforms Hamiltonian vector fields into Hamiltonian vector fields.\n\\end{definition}\n\n\n\\begin{definition}\nThe \\textbf{Hamilton--Jacobi problem}\nfor a Hamiltonian system $(\\Tan^*Q,\\omega,H )$ consists in finding\na canonical transformation $\\Phi\\colon \\Tan^*Q\\to \\Tan^*Q$ leading the\nsystem to equilibrium; that is, such that\n$H\\circ\\Phi=E\\ (ctn.)$.\n\\end{definition}\nThe canonical transformation $\\Psi$ is given by a generating function ${\\cal S}$:\n\\beq\n\\derpar{{\\cal S}}{q^i}(q,\\tilde q)=p_i \\quad , \\quad\n-\\derpar{{\\cal S}}{\\tilde q^i}(q,\\tilde q)=\\tilde p_i \\ ,\n\\label{uno}\n\\eeq\nwhere ${\\cal S}$ the general solution to the {\\sl Hamilton--Jacobi equation\\\/}\n\\beq\nH\\left(q^i,\\derpar{{\\cal S}}{q^i}\\right)= E\\ (ctn.) \\ .\n\\label{unobis}\n\\eeq\nThen, the Hamilton equations for the transformed Hamiltonian function $H\\circ\\Phi\\equiv\\tilde H$ are\n\\beq\n\\frac{\\d\\tilde q^i}{\\d t} = \\derpar{\\tilde H}{\\tilde p_i}(\\tilde q(t),\\tilde p(t))=0 \\quad , \\quad \n\\frac{\\d\\tilde p_i}{\\d t} = -\\derpar{\\tilde H}{\\tilde q^i}(\\tilde q(t),\\tilde p(t))=0 \\ ;\n\\label{dos}\n\\eeq\nand solving \\eqref{unobis}, from (\\ref{dos}) and (\\ref{uno}), the dynamical curves $(q^i(t),p_i(t))$\nof the original Hamiltonian system $(\\Tan^*Q,\\omega,H )$ are obtained.\n\n\n \\subsection{Relation between the ``classical'' and the geometric Hamilton--Jacobi theories}\n\nThe relation between the ``classical'' and the geometric Hamilton--Jacobi theories\nis established through the equivalence of complete solutions \nand canonical transformations (see also \\cite{Vi-14}).\n\n\\begin{teor}\nLet $(\\Tan^*Q,\\omega,H )$ be a Hamiltonian system.\nA complete solution $\\{ \\alpha_\\lambda; \\lambda\\in\\Lambda\\}$\nto the Hamilton--Jacobi problem provides a canonical transformation \n$\\Phi\\colon \\Tan^*Q\\to \\Tan^*Q$ leading the system to equilibrium,\nand conversely.\n\\end{teor}\n\\proof\nIn a neighbourhood of every point, consider a complete solution\n$\\{ \\alpha_\\lambda; \\lambda\\in\\Lambda\\}$, and\nlet ${\\cal S}$ be a generating function of it.\nAs ${\\cal S}={\\cal S}(q^i,\\lambda^i)$, we can identify $\\lambda^i$ \nwith a subset of coordinates $\\lambda^i\\equiv\\tilde q^i$ in $\\Tan^*Q\\times\\Tan^*Q$, and then\n${\\cal S}={\\cal S}(q^i,\\tilde q^i)$ can be thought as a generating function\nof a local canonical transformation $\\Phi$ and hence\nof an open set ${\\cal W}$ of the Lagrangian submanifold \n$graph\\,\\Phi\\hookrightarrow \\Tan^*Q\\times\\Tan^*Q$.\nMaking this construction in every chart, we have the transformation $\\Phi$\nand the submanifold $graph\\,\\Psi$.\nNow, as (\\ref{tres}) holds for every particular solution $S_\\lambda$, we have that\n$$\nE=H\\left(q^i(\\tilde q,\\tilde p),\\derpar{{\\cal S}}{q^i}(q(\\tilde q,\\tilde p),\\tilde q)\\right)= \n\\tilde H(\\tilde q^i,\\tilde p_i) \\ .\n$$\nConversely, if we have the canonical transformation $\\Psi$, from a\ngenerating function ${\\cal S}={\\cal S}(q^i,\\tilde q^i)$, taking\n$\\tilde q\\equiv (\\tilde q^i)=(\\lambda^i)\\equiv\\lambda$, we obtain\na family of functions $\\{ S_\\lambda\\}$ and, hence a\nlocal complete solution $\\{\\alpha_\\lambda=\\d S_\\lambda; \\lambda\\in\\Lambda\\}$\nto the Hamiltonian Hamilton--Jacobi problem.\nMaking this construction in every chart, we have the complete solution.\n\nGeometrically, this means that, on each local chart of $\\Tan^*Q$,\nfixing the coordinates $\\tilde q^i=\\lambda^i$ of a point,\nwe obtain a local submanifold whose image by $\\Phi^{-1}$ gives\nthe image of a local section $\\alpha_\\lambda\\colon Q\\to \\Tan^*Q$\nwhich is a particular solution to the Hamiltonian Hamilton--Jacobi problem.\n\\qed\n\n\n\n\n\\section{General geometric framework for the Hamilton--Jacobi theory}\n\\protect\\label{co}\n\n\nThe geometric Hamilton--Jacobi theory can be stated in a more general framework\nwhich allows us to extend the theory to a wide variety of systems and situations.\nNext we present a summary of this general framework as it is stated in \\cite{HJTeam3} \n(see also \\cite{BMMP-10} for another similar approach).\n\n\n\\subsection{Slicing problems}\n\n\nIn general, a {\\sl dynamical system} is just a couple $(P,Z)$,\nwhere $P$ is a manifold and $Z\\in\\vf(P)$ is a vector field\nwhich defines the dynamical equation on~$P$.\nThen, in order to state the analogous to the Hamilton--Jacobi problem for this system\nin a more general context, consider a manifold~${\\cal M}$,\na vector field $X\\in\\vf({\\cal M})$, and a map $\\alpha \\colon {\\cal M} \\to P$,\nas it is showed in the following diagram:\n$$\n\\xymatrix{\n*++{\\Tan {\\cal M}} \\ar[r]^{\\Tan \\alpha} \\ar[d]_{} &\n*++{\\Tan P} \\ar[d]_{}\n\\\\\n*++{{\\cal M}} \\ar[r]^{\\alpha} \\ar@\/^3mm\/[u]^{X} &\n*++{P} \\ar@\/_3mm\/[u]_{Z}\n}\n$$\n\n\\begin{prop}\n\\label{firstprop}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\itemsep 0pt plus 1pt\n\\item\nIf $\\gamma$ is an integral curve of $X$, then\n$\\zeta= \\alpha\\circ\\gamma$ is an integral curve of~$Z$.\n\\item\nThe vector fields $X$ and $Z$ are $\\alpha$-related:\n\\beq\n\\label{slicing}\n\\Tan \\alpha\\circ X = Z\\circ\\alpha \\ ,\n\\eeq\n\\end{enumerate}\nFurthermore, if $\\alpha$ is an injective immersion,\n(inducing a diffeomorphism\n$\\alpha_o \\colon {\\cal M} \\to \\alpha({\\cal M})$),\nthen these properties are equivalent to:\n\\begin{enumerate}\n\\itemsep 0pt plus 1pt\n\\setcounter{enumi}{2}\n\\item\nThe vector field $Z$ is tangent to $\\alpha({\\cal M})$, \nand, if $Z_o= Z|_{\\alpha({\\cal M})}$,\nthen $X = \\alpha_o^*(Z_o)$.\n\nThen, the map $\\gamma\\mapsto \\alpha\\circ\\xi$\nis a bijection between the integral curves of~$X$\nand the integral curves of~$Z$ in $\\alpha({\\cal M})$.\n\\end{enumerate}\n\\end{prop}\n\\proof\nThey are immediate, bearing in mind the commutativity of the above diagram.\n\\qed\n\n\\begin{definition}\nA \\textbf{slicing} of a dynamical system $(P,Z)$ is\na triple $({\\cal M},\\alpha,X)$ which is a solution to the\n\\textbf{slicing equation} \\eqref{slicing}.\n\\end{definition}\n\nIf $(x^i)$ and $(z^j)$ are coordinates in~${\\cal M}$ and $P$, respectively;\nand $\\alpha(x) = (a^j(x))$, $\\ds X = X^i \\derpar{}{x^i}$,\nand $\\ds Z = Z^j\\derpar{}{z^j}$, then\n$\\ds (\\Tan \\alpha \\circ X - Z \\circ \\alpha)(x^i)=\\Big(a^j(x) ,\\derpar{a^j}{x^i} \\, X^i -Z^j(\\alpha(x))\\Big)$,\nand $({\\cal M},\\alpha,X)$ is a solution to the slicing equation if, and only if,\n$$\n\\ds \\derpar{a^j}{x^i} \\, X^i(x) = Z^j (\\alpha(x)) \\ .\n$$\n\nWe say that the vector field $X$ gives a ``partial dynamics'' or a ``slice'' \nof the ``whole dynamics'' which is given by $Z$, \nand the whole dynamics can be recovered from these slices.\nIn fact, the integral curves of $Z$ contained in $\\alpha({\\cal M}) \\subset P$\ncan be described by a solution $(\\alpha,X)$ to the slicing equation; \nbut we need a {\\sl complete solution} to describe all the integral curves of $Z$ \nand it can be defined as a family of solutions depending on the parameters\nof a space $\\Lambda\\subseteq{\\mathbb R}^n$.\n\n\\begin{definition}\n\\label{compslice}\nA \\textbf{complete slicing} of a dynamical system $(P,Z)$ is a map \n$\\overline\\alpha\\colon {\\cal M}\\times\\Lambda\\to P$\nand a vector field $\\overline X \\colon {\\cal M} \\times\\Lambda \\to \\Tan {\\cal M}$\nalong the projection ${\\cal M} \\times\\Lambda \\to {\\cal M}$\nsuch that:\n\\ben\n\\item\nThe map $\\overline\\alpha$ is surjective,\n\\item\nfor every $\\lambda\\in\\Lambda$, the map $\\alpha_\\lambda\\colon {\\cal M} \\to P$\nand $X_\\lambda\\colon {\\cal M} \\to \\Tan {\\cal M}$ are a slicing of $Z$.\n\\een\n$$\n\\xymatrix{\n*++{\\Tan {\\cal M} \\times\\Lambda} \\ar[r]^{\\Tan_1 \\overline\\alpha} \\ar[d]_{} &\n*++{\\Tan P} \\ar[d]_{}\n\\\\\n*++{{\\cal M} \\times\\Lambda} \\ar[r]^{\\overline\\alpha} \\ar@\/^3mm\/[u]^{\\overline X} &\n*++{P} \\ar@\/_3mm\/[u]_{Z}\n}\n$$\n\\end{definition}\n\nThus, a complete slicing is a family of maps\n$\\alpha_\\lambda \\equiv \\overline\\alpha(\\cdot,\\lambda) \\colon {\\cal M} \\to P$\nand vector fields\n$X_\\lambda \\equiv \\overline X(\\cdot,\\lambda) \\colon {\\cal M} \\to \\Tan {\\cal M}$\nsatisfying the above conditions.\n\nAs for every point $p\\in P$ there exits $(x,\\lambda) \\in {\\cal M} \\times\\Lambda$ \nsuch that $\\overline\\alpha(x,\\lambda)=p$;\nthe integral curve of $Z$ through~$p$\nis described by the integral curve of $X_\\lambda$ through $x$\nby means of the map $\\alpha_\\lambda$.\nIn addition, if each $\\alpha_\\lambda$ is an immersion\n(for instance,when it is a diffeomorphism)\nthen $X_\\lambda$ are determined by the $\\alpha_\\lambda$.\n\nThe hypothesis of $\\alpha$ being an embedding holds in many situations;\nfor instance, for the sections of a fiber bundle $\\pi \\colon P \\to {\\cal M}$.\nThen we can consider the slicing problem\nfor sections $\\alpha \\colon {\\cal M} \\to P$ of~$\\pi$, as before.\nIn this case, as $\\alpha$ is an embedding,\nthe equation \\eqref{slicing} determines $X$, and\n$X$ is given from~$\\alpha$ by the equation\n$$\nX = \\Tan \\pi \\circ Z \\circ \\alpha \\ .\n$$\nIn this case, Proposition \\ref{firstprop} states that\na section $\\alpha$ of $\\pi \\colon P \\to {\\cal M}$\nis a solution to the slicing equation for $(P,Z)$ if, and only if,\n$$\n\\Tan \\alpha \\circ \\Tan \\pi \\circ Z \\circ \\alpha = Z \\circ \\alpha \\ .\n$$\n\n\n\\subsection{Recovering the Hamilton--Jacobi equation for Hamiltonian and Lagrangian dynamical systems}\n\n\nConsider the case of a Hamiltonian system $(P,\\omega,H)$,\nwhere $(P,\\omega)$ is a symplectic manifold, $H\\in\\Cinfty(P)$ is a\nHamiltonian function, and $Z = Z_H$ is its Hamiltonian vector field; that is,\nthe solution to \\eqref{hameq}. Then:\n\n\\begin{teor}\n\\label{prop-ham}\nIf $({\\cal M},\\alpha,X)$ is a solution to the slicing equation \\eqref{slicing} for $(P,Z_H)$, then\n$$\n\\inn(X)\\,\\alpha^*\\omega - \\d \\,\\alpha^*H = 0 \\ .\n$$\nIn addition, if $\\alpha \\colon {\\cal M} \\to P$ is an embedding\nsatisfying the condition $\\alpha^*\\omega= 0$, then\n$$\n\\d\\,(\\alpha^*H)= 0 \\ ;\n$$\nand conversely, if $\\dim P = 2 \\dim {\\cal M}$\nand $\\alpha$ satisfies this equation and $\\alpha^*\\omega= 0$,\nthen $\\alpha$ is a solution to the slicing equation \\eqref{slicing}.\n\\end{teor}\n\nIn the particular case where $\\pi \\colon P \\to {\\cal M}$ is a fiber bundle\n(for instance, ${\\cal M}=Q$ and $P=\\Tan^*Q$),\nwe can consider the slicing problem as before,\nbut only for sections of~$\\pi$,\n$$\n\\xymatrix{\n*++{\\Tan {\\cal M}} \\ar[r]^{\\Tan \\alpha} \\ar[d]_{} &\n*++{\\Tan P} \\ar[d]_{}\n\\\\\n*++{{\\cal M}} \\ar[r]^{\\alpha} \\ar@\/^3mm\/[u]^{X} &\n*++{P} \\ar@\/^3mm\/[l]^{\\pi} \\ar@\/_3mm\/[u]_{Z}\n}\n$$\nBeing $\\alpha$ an embedding, the equation (\\ref{slicing}) determines~$X$,\nand composing this equation with $\\Tan\\pi$, we obtain that\n$X = \\Tan\\pi \\circ Z_H \\circ \\alpha$.\nTherefore the slicing equation \\eqref{slicing} reads\n$$\n\\Tan\\alpha \\circ\\Tan\\pi \\circ Z \\circ\\alpha = Z \\circ\\alpha \\ .\n$$\n\nIn this way, the equation \\eqref{slicing} can be considered as a\ngeneralization of the Hamilton--Jacobi equation in the\nHamiltonian formalism, which\nis just the slicing equation for a closed $1$-form~$\\alpha$ in $Q$.\nTherefore, $\\alpha = \\d S$ locally,\nand the slicing equation looks in the ordinary form\n$H \\circ\\d S = \\mathrm{const}$.\n\nThe same applies to the Lagrangian formalism.\nIn this case $P = \\Tan Q$ and, if $\\Lag\\in\\Cinfty(\\Tan Q)$ \nis a regular Lagrangian function, $Z=\\Gamma_\\Lag$\nis the Lagrangian vector field solution to the Lagrangian equation \\eqref{ELeq}.\nThen all proceeds as in the Hamiltonian case.\n\nThe definitions \\ref{completeHJ} and \\ref{lagcomplete} of complete solutions to the \nHamiltonian and Lagrangian Hamilton--Jacobi problems respectively\nare particular cases of the definition \\ref{compslice} of complete slicings.\n\n\n\n\\section{The Hamilton--Jacobi problem for other physical systems}\n\\label{others}\n\n\n\nUsing the general framework presented in the above section,\nthe Hamilton--Jacobi problem can be stated for a wide kind of physical systems.\nNext we review two of them.\n(Other applications of the theory are listed in detail in the Introduction). \n\n\n\n\\subsection{Higher-order (autonomous) dynamical systems}\n\n\n\nLet $Q$ be a $n$-dimensional manifold) and let\n$\\Tan^kQ$ the $k$th-order tangent bundle of $Q$,\nwhich is endowed with natural coordinates\n$\\left(q_{0}^A,q_{1}^A,\\ldots,q_{k}^A\\right) = \\left(q_i^A\\right)$,\n$0 \\leqslant i \\leqslant k$, $1\\leqslant A\\leqslant n$.\nIf $\\Lag \\in\\Cinfty(\\Tan^{k}Q)$ is the Lagrangian function of\nan autonomous $k$th-order Lagrangian system, \nusing the canonical structures of the higher-order tangent bundles, \nwe can construct the Poincar\\'{e}-Cartan\nforms and the Lagrangian energy whose coordinate expressions are\n\\begin{align*}\n&\\omega_\\Lag =-\\d \\theta_\\L=\n\\sum_{r=1}^k \\sum_{i=0}^{k-r}(-1)^{i+1} d_T^i\\,\\d\\left(\\derpar{\\Lag}{q_{r+i}^A}\\right) \\wedge \\d q_{r-1}^A\\in\\df^{2}(\\Tan^{2k-1}Q) \\, , \\\\\n&E_\\Lag = \\sum_{r=1}^{k}\nq_{r}^A \\sum_{i=0}^{k-r} (-1)^i d_T^i\\left( \\derpar{\\Lag}{q_{r+i}^A}\n\\right)- \\Lag\\in\\Cinfty(\\Tan^{2k-1}Q) \\ ,\n\\end{align*}\nwhere\n$\\ds d_Tf\\left(q_0^A,\\ldots,q_{k+1}^A\\right) =\n \\sum_{i=0}^{k}q_{i+1}^A \\derpar{f}{q_i^A}(q_0^A,\\ldots,q_{k}^A)$.\nThus we have the higher-order Lagrangian system $(\\Tan^{2k-1}Q,\\omega_\\Lag,E_\\Lag)$.\nAssuming that the Lagrangian function is regular; that is,\n$\\omega_\\Lag$ is a symplectic form;\nthe Lagrangian equation\n$\\inn(X_\\Lag) \\, \\omega_\\Lag = \\d E_\\Lag$\nhas a unique solution $X_\\Lag \\in \\vf(\\Tan^{2k-1}Q)$ (the Lagrangian vector field)\nwhose integral curves are holonomic (that is, they are canonical liftings \n$j^{2k-1}\\phi\\colon\\Real\\to\\Tan^{2k-1}Q$\nof curves $\\phi\\colon\\Real\\to Q$)\nand are the solutions to the {\\sl Otrogradskii} or \n{\\sl higher-order Euler-Lagrange equations}\n(see\n\\cite{book:DeLeon_Rodrigues85,art:Gracia_Pons_Roman91,art:Prieto_Roman11}\nfor details).\n\nThe Hamilton--Jacobi problem for higher-order \nLagrangian dynamical systems is just the {\\sl slicing problem}\nfor the particular situation represented in the diagram\n$$\n\\xymatrix{\n\\Tan(\\Tan^{k-1}Q) \\ar@\/_1.5pc\/[rr]_{\\Tan s} & \\ & \\ar[ll]_{\\Tan\\rho^{2k-1}_{k-1}} \\Tan(\\Tan^{2k-1}Q) \\\\\n\\ & \\ & \\ \\\\\n\\Tan^{k-1}Q \\ar[uu]^{X} \\ar@\/_1.5pc\/[rr]_{s} & \\ & \\ar[ll]_{\\rho^{2k-1}_{k-1}} \\Tan^{2k-1}Q \\ar[uu]_{X_\\Lag}\n}\n$$\nthat is, for sections of the natural projection\n$\\rho^{2k-1}_{k-1}\\colon\\Tan^{2k-1}Q\\to\\Tan^{k-1}Q$,\n$s \\in \\Gamma(\\rho^{2k-1}_{k-1})$;\nand thus we have the following settings\n(see \\cite{CLRP-13,CLRP-14} for the details and proofs):\n\n\\begin{definition}\n\\label{def:GenLagHJDef}\nThe \\textbf{generalized $k$th-order Lagrangian Hamilton--Jacobi problem}\nfor the higher-order Lagrangian system $(\\Tan^{2k-1}Q,\\omega_\\Lag,E_\\Lag)$\nis to find a section $s \\in \\Gamma(\\rho^{2k-1}_{k-1})$ \nand a vector field $X \\in \\vf(\\Tan^{k-1}Q)$ such that,\nif $\\gamma ~\\colon~\\R~\\to~\\Tan^{k-1}Q$ is an integral curve of $X$,\nthen $s\\circ\\gamma \\colon \\R \\to \\Tan^{2k-1}Q$ is an integral curve of $X_\\Lag$;\nthat is, if $X \\circ \\gamma = \\dot \\gamma$, then $X_\\Lag \\circ (s \\circ \\gamma) = \\dot{\\overline{s \\circ \\gamma}}$.\nThen, the couple $(s,X)$ is a \\textbf{solution to the generalized $k$th-order\nLagrangian Hamilton--Jacobi problem}.\n\\end{definition}\n\n\\begin{teor}\n\\label{higher-gen-cns}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item \nThe couple $(s,X)$ is a solution to the generalized $k$th-order Lagrangian\nHamilton--Jacobi problem.\n\\item\nThe vector fields $X$ and $X_\\Lag$ are $s$-related; that is,\n$X_\\Lag \\circ s = \\Tan s \\circ X$.\nAs a consequence, $X = \\Tan\\rho^{2k-1}_{k-1} \\circ X_\\Lag \\circ s$,\nand $X$ is said to be the {\\rm vector field associated with the section $s$}.\n\\item \nThe submanifold ${\\rm Im}(s) $ of $\\Tan^{2k-1}Q$\nis invariant by the Lagrangian vector field $X_\\Lag$ \n(or, what means the same thing, $X_\\Lag$ is tangent to $s(\\Tan^{k-1}Q)$).\n\\item \nThe integral curves of $X_\\Lag$ which have initial conditions in $\\textnormal{Im}(s)$ \nproject onto the integral curves of $X$.\n\\item \nThe equation\n$\\inn(X)(s^*\\omega_\\Lag) = \\d(s^*E_\\Lag)$ holds for $\\alpha$.\n\\end{enumerate}\n\\end{teor}\n\\proof\n(Guidelines for the proof):\nThe proof follows a pattern similar to that of Theorem \\ref{HJhcns},\nbut now using Definition \\ref{def:GenLagHJDef}.\n\\qed\n\n\\begin{definition}\n\\label{higherHJl}\nThe \\textbf{$k$th-order Lagrangian Hamilton--Jacobi problem} \nfor the higher-order Lagrangian system\n$(\\Tan^{2k-1}Q,\\omega_\\Lag,E_\\Lag)$ is to find\na section $s \\in \\Gamma(\\rho^{2k-1}_{k-1})$ such that it is a solution to the generalized\n$k$th-order Lagrangian Hamilton--Jacobi problem and satisfies that\n$s^*\\omega_\\Lag = 0$. \nThen, this section $s$ is a \\textbf{solution to the $k$th-order Lagrangian Hamilton--Jacobi problem}.\n\\end{definition}\n\nObserve that that $0=s^*\\omega_\\Lag = -s^*(\\d\\theta_\\Lag) =- \\d(s^*\\theta_\\Lag) = 0$;\nthat is, $s^*\\theta_\\Lag$ is a closed $1$-form and then\nthere exists $S \\in \\Cinfty(U)$, $U \\subset \\Tan^{k-1}Q$, such that $s^*\\theta_\\Lag\\vert_U = \\d S$.\n\n\\begin{teor}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item \nThe section $s$ is a solution to the generalized $k$th-order Lagrangian\nHamilton--Jacobi problem.\n\\item \n${\\rm Im}(s)$ is a Lagrangian submanifold of $\\Tan^{2k-1}Q$,\nwhich is invariant by the Lagrangian vector field $X_\\Lag$\n(and $S$ is a local generating function of this Lagrangian submanifold).\n\\item \nThe equation $\\d(s^*E_\\Lag) = 0$ holds for $s$\nor, what is equivalent, the function\n$E_\\Lag\\circ\\d S\\colon Q\\to\\Real$ is locally constant.\n\\end{enumerate}\n\\end{teor}\n\\proof\n(Guidelines for the proof):\nThey are consequences of Theorem \\ref{higher-gen-cns} and Definition \\ref{higherHJl}.\n\\qed\n\nIn natural coordinates, from this last condition we obtain that\n$$\n\\derpar{S}{q_i^A} =\n\\sum_{l=0}^{k-i-1}(-1)^ld_T^l\\left( \\derpar{\\Lag}{q_{i+1+l}^A} \\right)\\Big\\vert_{{\\rm Im}(s)} \\ .\n$$\nThis system of $kn$ partial differential equations for $S$\ngeneralizes the equation \\eqref{eqn:LagHJCharFunctLocalClassic}\nto higher-order systems.\n\n\\begin{definition}\nLet $\\Lambda\\subseteq\\Real^n$.\nA family of solutions$\\{ s_\\lambda; \\lambda\\in\\Lambda\\}$,\ndepending on $n$ parameters \n$\\lambda\\equiv (\\lambda_1,\\ldots,\\lambda_n)\\in\\Lambda$, is a \n\\textbf{complete solution to the $k$th-order Lagrangian Hamilton--Jacobi problem} \nif the map \n$$\n\\begin{array}{ccccc}\n\\Phi & \\colon &\\Tan^{k-1}Q\\times\\Lambda & \\longrightarrow & \\Tan^{2k-1}Q \\\\\n & & (q,\\lambda) & \\mapsto & s_\\lambda(q)\n\\end{array}\n$$\nis a local diffeomorphism.\n\\end{definition}\n\nFor the Hamiltonian formalism, let \n$h \\in \\Cinfty(\\Tan^*(\\Tan^{k-1}Q))$ be the Hamiltonian function \nof a (regular) higher-order dynamical system. \nUsing the canonical Liouville forms of the cotangent bundle,\n$\\theta_{k-1} = p_{A}^{i}\\d q_{i}^{A} \\in\n\\df^{1}(\\Tan^*(\\Tan^{k-1}Q))$ and $\\omega_{k-1} = \\d q_{i}^{A}\\wedge\n\\d p_{A}^{i} \\in \\df^{2}(\\Tan^*(\\Tan^{k-1}Q))$, where\n$(q_{i}^{A},p_{i}^{A})$ ($1\\leq A\\leq n,$ $0\\leq i\\leq k-1$) are\ncanonical coordinates in $\\Tan^{*}(\\Tan^{k-1}Q)$; the\ndynamical equation for the Hamiltonian system \n$(\\Tan^{*}(\\Tan^{k-1}Q),\\omega_k,h))$ is\n$\\inn(X_h)\\,\\omega_{k-1} = \\d h$,\nand it has a unique solution $X_h \\in \\vf(\\Tan^*(\\Tan^{k-1}Q))$.\nAs we are working in the cotangent bundle $\\Tan^*(\\Tan^{k-1}Q)$,\n the Hamiltonian Hamilton--Jacobi problems for higher-order\nsystems is stated in the same way as in the first-order case\nand, hence, it is the {\\sl slicing problem} for the particular situation\nrepresented in the diagram\n$$\n\\xymatrix{\n\\Tan(\\Tan^{k-1}Q) \\ar@\/_1.5pc\/[rrr]_{\\Tan \\alpha} & \\ & \\ &\n \\ar[lll]_-{\\Tan\\pi_{\\Tan^{k-1}Q}} \\Tan(\\Tan^*(\\Tan^{k-1}Q)) \\\\\n\\ & \\ & \\ & \\ \\\\\n\\Tan^{k-1}Q \\ar[uu]^{X} \\ar@\/_1.5pc\/[rrr]_{\\alpha} & \\ & \\ &\n \\ar[lll]_{\\pi_{\\Tan^{k-1}Q}} \\Tan^*(\\Tan^{k-1}Q) \\ar[uu]_{X_h}\n}\n$$\nTherefore, all the definitions and results are like in the first-order case,\nand the relation between both the Lagrangian and the Hamiltonian Hamilton--Jacobi problems\nis stated as in Theorem \\ref{relation}.\n\n\n\n\\subsection{Multisymplectic field theories}\n\n\nThe Hamilton--Jacobi theory for multisymplectic field theories\nhas been studied in \\cite{LMM-09,LPRV-2017,art:Vitagliano12}.\nNext we state the Lagrangian and the Hamiltonian problems for these systems.\n(For details on multisymplectic field theories see, for instance, \\cite{proc:Carinena_Crampin_Ibort91,EMR-96,art:Roman09} and the references therein).\n\n\n\\subsubsection{Multisymplectic Lagrangian Hamilton--Jacobi problem}\n\n\nLet $\\pi\\colon E\\to M$ a bundle, \nwhere $M$ is an oriented manifold with $\\dim\\,M=m$ and $\\dim\\,E=n+m$.\nThe Lagrangian description of multisymplectic classical field theories\nis stated in the first-order jet bundle $\\pi^1\\colon J^1\\pi\\to E$,\nwhich is also a bundle $\\bar\\pi^1\\colon J^1\\pi\\longrightarrow M$.\nNatural coordinates in $J^1\\pi$ adapted to the bundle structure are \n$(x^i,y^\\alpha,y^\\alpha_i)$ ($i= 1,\\ldots,m$; $\\alpha= 1,\\ldots,n$).\nGiving a Lagrangian density associated to a Lagrangian function $\\Lag$\nand using the canonical structures of $J^1\\pi$\nwe can define the {\\sl Poincar\\'e--Cartan forms}\nassociated with~$\\Lag$,\n$\\Theta_{\\Lag}\\in\\df^{m}(J^1\\pi)$ and\n$\\Omega_{\\Lag}:= -\\d\\Theta_{\\Lag}\\in\\df^{m+1}(J^1\\pi)$,\nwhose local expression is\n$$\n\\Omega_{\\Lag}=-\\d\\Theta_\\L=\n-\\d\\left(\\derpar{L}{y_i^\\alpha}\\d y^\\alpha \\wedge \\d^{m-1}x_i -\n\\Big( \\derpar{L}{y_i^\\alpha}y_i^\\alpha - L \\Big)\\d^{m}x \\right) \\ ,\n$$\nwhere $\\d^mx = \\d x^1 \\wedge \\ldots \\wedge \\d x^m$ and \n$\\ds\\d^{m-1}x_i = \\inn\\Big(\\derpar{}{x^i}\\Big)\\d^m x$.\nThe Lagrangian function is regular if $\\Omega_{\\Lag}$ is a multisymplectic\n$(m+1)$-form (i.e., $1$-non\\-degenerate).\nThen the couple $(J^1\\pi,\\Omega_\\Lag)$ is a {\\sl multisymplectic Lagrangian system}.\nThe {\\sl Lagrangian problem} consists in finding $m$-dimensional, \n$\\bar{\\pi}^1$-transverse, and holonomic distributions \n${\\cal D}_\\L$ in $J^1\\pi$ such that their integral sections\n$\\psi_\\L \\in \\Gamma(\\bar{\\pi}^1)$ are canonical liftings $j^1\\phi$ of sections \n$\\phi\\in\\Gamma(\\pi)$ that are solutions to the\nLagrangian field equation\n\\beq\n(j^1\\phi)^*\\inn (X)\\Omega_\\Lag= 0\\, , \\ \\mbox{\\textit{for every} } X \\in \\vf(J^1\\pi) \\ .\n\\label{fieldlag}\n\\eeq\nIn coordinates, the components of\n$j^1\\phi {=}\\Big(x^i ,y^\\alpha,\\ds\\derpar{y^\\alpha}{x^i}\\Big)$\nsatisfy the {\\sl Euler--Lagrange equations}\n$$\n \\derpar{\\Lag}{y^A}\\circ j^1\\phi-\n\\derpar{}{x^\\mu}\\Big(\\derpar{\\Lag}{y_\\mu^A}\\circ j^1\\phi\\Big)= 0 .\n$$\n\n\\begin{definition}\n\\label{multigenHJl}\nThe \\textbf{generalized Lagrangian Hamilton--Jacobi problem} \nfor the multisymplectic Lagrangian system $(J^1\\pi,\\Omega_\\L)$ \nis to find a section\n$\\Psi \\in \\Gamma(\\pi^1)$ (which is called a {\\sl jet field\\\/}) \nand an $m$-dimensional integrable distribution ${\\cal D}$ in $E$ such that, if\n$\\gamma \\in \\Gamma(\\pi)$ is an integral section of ${\\cal D}$, then\n$\\psi_\\L=\\Psi \\circ \\gamma \\in \\Gamma(\\bar{\\pi}^1)$ \nis an integral section of ${\\cal D}_\\Lag$;\nthat is, if\n$\\Tan_{u}{\\rm Im}(\\gamma) = {\\cal D}_{u}$, for every $u \\in {\\rm Im}(\\gamma)$,\nthen\n$\\Tan_{\\bar{u}}{\\rm Im}(\\Psi \\circ \\gamma) = ({\\cal D}_\\Lag)_{\\bar{u}}$,\nfor every $\\bar{u} \\in {\\rm Im}(\\Psi \\circ \\gamma)$.\nThen, the couple $(\\Psi,{\\cal D})$ is a \\textbf{solution to the generalized \nHamiltonian Hamilton--Jacobi problem}.\n\\end{definition}\n\n\\begin{remark}{\\rm\nThe Hamilton--Jacobi problem can also be stated\nassociating the distributions ${\\cal D}$ and ${\\cal D}_\\L$ with {\\sl multivector fields}.\nAn \\textsl{$m$-multivector field}, on a manifold ${\\cal M}$ is a section of the bundle \n$\\Lambda^m(\\Tan {\\cal M}) \\to {\\cal M}$,\nwhere $\\Lambda^m(\\Tan {\\cal M})=\\Tan {\\cal M}\\wedge^{\\,(m)}_{\\ldots\\ldots}\\wedge\\Tan {\\cal M}$\n (i.e, a skew-symmetric contravariant tensor field). \nIf ${\\bf X}$ is an $m$-multivector field in ${\\cal M}$ then,\nfor every $p\\in{\\cal M}$, there is a neighbourhood \n$U_p \\subset {\\cal M}$ and local vector fields\n$X_1,\\ldots,X_m \\in \\vf(U_p)$ such that \n${\\bf X}\\vert_{U_p}=X_1\\wedge^{\\,(m)}_{\\ldots\\ldots}\\wedge X_m$.\nThen, if ${\\cal D}$ is an $m$-dimensional distribution in ${\\cal M}$, \nsections of $\\Lambda^m{\\cal D} \\to {\\cal M}$ \nare $m$-multivector fields in ${\\cal M}$, and a multivector field is\n{\\sl integrable} if its associated distribution is also.\n\n\\noindent \nNow, if ${\\cal M}=J^1\\pi$, let ${\\bf X}$ and ${\\bf X}_\\L$ be the $m$-multivector fields\nassociated with the distributions ${\\cal D}$ and ${\\cal D}_\\L$, respectively;\nthen the Lagrangian Hamilton--Jacobi problem can be represented by the diagram\n$$\n\\xymatrix{\n\\Lambda^m\\Tan E \\ar@\/_1.5pc\/[rrr]_{\\Lambda^m\\Tan\\Psi} & \\ & \\ &\n \\ar[lll]_-{\\Lambda^m\\Tan\\pi^1} \\Lambda^m\\Tan J^1\\pi \\\\\n\\ & \\ & \\ & \\ \\\\\nE \\ar[uu]^{{\\bf X}} \\ar@\/_1.5pc\/[rrr]_{\\Psi} & \\ & \\ &\n \\ar[lll]_{\\pi^1}J^1\\pi \\ar[uu]_{{\\bf X}_\\L}\n}\n$$\n(where $\\Lambda^m\\Tan\\Psi$ and $\\Lambda^m\\Tan\\pi^1$ denote the natural extensions of the maps $\\psi$ and $\\pi^1$ to the multitangent bundles),\nand thus this problem can be considered as a special case of a {\\sl slicing problem}.\n}\n\\end{remark}\n\n\\begin{teor}\n\\label{prop:LagGenHJEquivalences}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item \nThe couple $(\\Psi,{\\cal D})$ is a solution to the generalized Lagrangian Hamilton--Jacobi problem.\n\\item\nThe distributions ${\\cal D}$ and ${\\cal D}_\\Lag$ are $\\Psi$-related.\nAs a consequence,\n${\\cal D} = \\Tan\\pi^1({\\cal D}_\\L\\vert_{{\\rm Im}(\\Psi)})$, and is called the\n{\\rm distribution associated with $\\Psi$}.\n\\item \nThe distribution ${\\cal D}_\\Lag$ is tangent to the submanifold ${\\rm Im}(\\Psi)$\nof $J^1\\pi$.\n\\item\nThe integral sections of ${\\cal D}_\\Lag$ which have boundary conditions in\n${\\rm Im}(\\Psi)$ project onto the integral sections of ${\\cal D} $.\n\\item \nIf $\\gamma$ is an integral section of the distribution\n${\\cal D}$ associated with the jet field $\\Psi$ then, for every $Y \\in \\vf(E)$, the equation $\\gamma^*\\inn(Y)(\\Psi^*\\Omega_\\Lag) = 0$\nholds for $\\Psi$.\n\\end{enumerate}\n\\end{teor}\n\\proof\n(Guidelines for the proof):\n\nThe equivalence between 1 and 2 is a consequence of the Definition \\ref{multigenHJl}, the equivalence between distributions and multivector fields, and the definition of integral sections.\n\nItems 3 and 4 follow from 2.\n\nItem 5 is obtained from Definition \\ref{multigenHJl} and using the field equation \\eqref{fieldlag}.\n\\qed\n\n\\begin{definition}\\label{def:LagHJDef}\nThe \\textbf{Lagrangian Hamilton--Jacobi problem} \nfor the multisymplectic Lagrangian system $(J^1\\pi,\\Omega_\\L)$ \nis to find a jet field\n$\\Psi \\in \\Gamma(\\pi^1)$ such that it is solution to the generalized Lagrangian Hamilton--Jacobi problem and\nsatisfies that $\\Psi^*\\Omega_\\Lag = 0$. \nThen, the jet field $\\Psi$ is a \\textbf{solution to the Lagrangian Hamilton--Jacobi problem}.\n\\end{definition}\n\nThe condition $\\Psi^*\\Omega_\\Lag= -\\d(\\Psi^*\\Theta_\\Lag)=0$ \nis equivalent to ask that the form \n$\\Psi^*\\Theta_\\Lag $ is closed and \nthen there exists a $(m-1)$-form $\\omega \\in \\df^{m-1}(U)$, with $U \\subset E$, \nsuch that $\\Psi^*\\Theta_\\Lag=\\d\\omega$. Furthermore, \n$\\omega$ is $\\pi$-semibasic, since $\\Theta_\\Lag$, \nand hence $\\Psi^*\\Theta_\\Lag$, are also.\n\n\\begin{teor}\nThe following statements are equivalent:\n\\begin{enumerate}\n\\item \nThe jet field $\\Psi$ is a solution to the Lagrangian Hamilton--Jacobi problem.\n\\item \n${\\rm Im}(\\Psi)$ is an $m$-Lagrangian submanifold of $J^1\\pi$\nand the distribution ${\\cal D}_\\Lag$ is tangent to it.\n\\item \nThe form $\\Psi^*\\Theta_\\Lag$ is closed.\n\\end{enumerate}\n\\end{teor}\n\nIn coordinates, $\\omega = W^i \\d^{m-1}x_i$, and the\nHamilton--Jacobi equation in the Lagrangian formalism has the form\n$$\n\\sum_{i=1}^{m}\\derpar{W^i}{x^i} + \\psi_i^\\alpha\\derpar{W^i}{u^\\alpha} - L(x^i,u^\\alpha,\\psi^\\alpha_i) = 0 \\, ,\n$$\n\n\\begin{definition}\nLet $\\Lambda\\subseteq\\Real^{mn}$.\nA family of solutions$\\{\\Psi_\\lambda; \\lambda\\in\\Lambda\\}$,\ndepending on $n$ parameters\n$\\lambda\\equiv (\\lambda_1,\\ldots,\\lambda_n)\\in\\Lambda$, is a \n\\textbf{complete solution to the Lagrangian Hamilton--Jacobi problem} \nif the map \n$$\n\\begin{array}{ccccc}\n\\Phi & \\colon &E\\times\\Lambda & \\longrightarrow & J^1\\pi \\\\\n & & (p,\\lambda) & \\mapsto & \\Psi_\\lambda(p)\n\\end{array}\n$$\nis a local diffeomorphism.\n\\end{definition}\n\nA complete solution defines an $(m-n)$-dimensional foliation in $J^1\\pi$ which is transverse to the fibers\nand such that the distribution ${\\cal D}_\\L$ is tangent to it.\nThen, all the sections which are solutions to the Euler--Lagrange equations \n(that is, all the integral sections of the distribution ${\\cal D}_\\L$)\nare recovered from a complete solution. \n\n\n\\subsubsection{Multisymplectic Hamiltonian Hamilton--Jacobi problem}\n\n\nThe Hamiltonian formalism for a regular first-order multisymplectic field theory\nis developed in the so-called {\\sl reduced dual jet bundle of $J^1\\pi$},\n$J^1\\pi^*= \\Lambda^{m}_{2}(\\Tan^*E)\/\\Lambda^{m}_{1}(\\Tan^*E)$\n(where $\\Lambda^m_2(\\Tan^*E)$ is the bundle of \n$m$-forms over $E$ vanishing when they act on $\\pi$-vertical bivectors).\nIt is endowed with the canonical projections $\\pi_E \\colon J^{1}\\pi^* \\to E$ and\n$\\bar{\\pi}_E \\colon J^{1}\\pi^* \\to M$,\nand natural coordinates in $J^1\\pi^*$ are denoted $(x^i,y^\\alpha,p_\\alpha^i)$.\nThe physical information is given by a {\\sl Hamiltonian section} $h$ of the \nnatural projection \n$\\mu \\colon \\Lambda^{m}_{2}(\\Tan^*E) \\to J^{1}\\pi^*$,\nwhich is associated with a {\\sl local Hamiltonian function} $H \\in C^\\infty(J^1\\pi^*)$ \nsuch that $h(x^i,y^\\alpha,p_\\alpha^i) = (x^i,y^\\alpha,-H,p_\\alpha^i)$. \nThen, from the canonical\nform $\\Omega \\in\\df^{m+1}(\\Lambda^m_2(\\Tan^*E))$, \nwe construct the Hamilton-Cartan multisymplectic form\n$\\Omega_h = h^*\\Omega\\in \\df^{m+1}(J^1\\pi^*)$ whose coordinate expression is\n$$\n\\Omega_h = -\\d p_\\alpha^i \\wedge \\d y^\\alpha \\wedge \\d^{m-1}x_i + \\d H \\wedge \\d^{m}x \\ ,\n$$\nand the couple $(J^1\\pi^*,\\Omega_h)$ is a {\\sl multisymplectic Hamiltonian system}. \nThen, the {\\sl Hamiltonian problem} consists in finding integrable $m$-dimensional\n$\\bar{\\pi}_E$-transverse distributions ${\\cal D}_h$ in $J^1\\pi^*$ such that their integral sections\n$\\psi_h \\in \\Gamma(\\bar{\\pi}_E)$ are solutions to the\nHamiltonian field equation\n$$\n\\psi_h^*\\inn(X)\\Omega_h = 0 \\, , \\ \\mbox{\\textit{for every} } X \\in \\vf(J^1\\pi^*) \\ .\n$$\nThe existence of such distributions ${\\cal D}_h$ is assured.\nIn coordinates, this equation gives the\n{\\sl Hamilton--De Donder--Weyl equations}\n$$\n \\derpar{(y^A\\circ\\psi_h)}{x^\\nu}=\n \\derpar{{\\rm h}}{p^\\nu_A}\\circ\\psi_h\n\\quad ,\\quad\n \\derpar{(p_A^\\nu\\circ\\psi_h)}{x^\\nu}=\n - \\derpar{{\\rm h}}{y^A}\\circ\\psi_h \\ .\n$$\n\n\\begin{definition}\n\\label{multigenHJh}\nThe \\textbf{generalized Hamiltonian Hamilton--Jacobi problem} \nfor the multisymplectic Hamiltonian system $(J^1\\pi^*,\\Omega_h)$ \nis to find a section $s \\in \\Gamma(\\pi_E)$ \nand an integrable $m$-dimensional distribution ${\\cal D}$ in $E$ such that,\nif $\\gamma \\in \\Gamma(\\pi)$ is an integral section of ${\\cal D}$, then\n$\\psi_h=s \\circ \\gamma \\in \\Gamma(\\bar{\\pi}_E)$ is an integral section of ${\\cal D}_h$; that is, if\n$\\Tan_{u}{\\rm Im}(\\gamma) = {\\cal D}_{u}$, for every $u\\in{\\rm Im}(\\gamma)$,\nthen $\\Tan_{\\bar u}{\\rm Im}(s \\circ \\gamma) = ({\\cal D}_h)_{\\bar u}$,\nfor every $\\bar u \\in {\\rm Im}(s \\circ \\gamma)$.\nThen, the couple $(s,{\\cal D})$ is a \\textbf{solution to the generalized \nHamiltonian Hamilton--Jacobi problem}.\n\\end{definition}\n\n\\begin{remark}{\\rm\nAs in the Lagrangian case, the Hamiltonian Hamilton--Jacobi problem\ncan be considered as a special case of the following {\\sl slicing problem}\n$$\n\\xymatrix{\n\\Lambda^m\\Tan E \\ar@\/_1.5pc\/[rrr]_{\\Lambda^m\\Tan s} & \\ & \\ &\n \\ar[lll]_-{\\Lambda^m\\Tan\\pi_E} \\Lambda^m\\Tan J^1\\pi^* \\\\\n\\ & \\ & \\ & \\ \\\\\nE \\ar[uu]^{{\\bf X}} \\ar@\/_1.5pc\/[rrr]_{s} & \\ & \\ &\n \\ar[lll]_{\\pi_E}J^1\\pi^* \\ar[uu]_{{\\bf X}_h}\n}\n$$\nwhere ${\\bf X}$ and ${\\bf X}_h$ are $m$-multivector fields\nassociated with the distributions ${\\cal D}$ and ${\\cal D}_h$, respectively.\n}\n\\end{remark}\n\nThe following Theorems and Definitions are analogous to those of the Lagrangian case.\n\n\\begin{teor}\n\\label{prop:HamGenHJEquivalences}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item\nThe couple $(s,{\\cal D})$ is a solution to the generalized \nHamiltonian Hamilton--Jacobi problem.\n\\item\nThe distributions ${\\cal D}$ and ${\\cal D}_h$ are $s$-related.\nAs a consequence, the distribution ${\\cal D}$ is given by\n${\\cal D} = \\Tan\\pi_E({\\cal D}_h\\vert_{{\\rm Im}(s)})$, and it is called the\n{\\rm distribution associated with $s$}.\n\\item \nThe distribution ${\\cal D}_h$ is tangent to the submanifold ${\\rm Im}(s)$\nof $J^1\\pi^*$.\n\\item\nThe integral sections of ${\\cal D}_h$ which have boundary conditions in ${\\rm Im}(s)$ \nproject onto the integral sections of ${\\cal D}$.\n\\item \nIf $\\gamma$ is an integral section of the distribution\n${\\cal D}$ associated with $s$ then, for every $Y \\in \\vf(E)$, the equation \n$\\gamma^*\\inn(Y)\\d(h \\circ s) = 0$\nholds for $s$.\n\\end{enumerate}\n\\end{teor}\n\n\\begin{definition}\\label{def:HamHJDef}\nThe \\textbf{Hamiltonian Hamilton--Jacobi problem} \nfor the multisymplectic Hamiltonian system $(J^1\\pi^*,\\Omega_h)$ \nis to find a section\n$s \\in \\Gamma(\\pi_E)$ such that it is a solution to the generalized Hamilton--Jacobi problem\nand satisfies that $s^*\\Omega_h = 0$.\nThen, the section $s$ is a \\textbf{solution to the Hamiltonianian Hamilton--Jacobi problem}.\n\\end{definition}\n\n\\begin{teor}\n\\label{prop:HamHJEquivalences}\nThe following conditions are equivalent.\n\\begin{enumerate}\n\\item\nThe couple $(s,{\\cal D})$ is a solution to the generalized \nHamiltonian Hamilton--Jacobi problem.\n\\item \n${\\rm Im}(s)$ is an $m$-Lagrangian submanifold of $J^1\\pi^*$ and the distribution ${\\cal D}_h$ is tangent to it.\n\\item \nThe form $h \\circ s \\in \\df^{m}(E)$ is closed.\n\\end{enumerate}\n\\end{teor}\n\nAs the $\\pi_E$-semibasic $m$-form $h \\circ s$ is closed,\nthere exists a local $\\pi$-semibasic $(m-1)$-form $\\omega \\in \\df^{m-1}(E)$, such that\n$h \\circ s=\\d\\omega$. In coordinates, if $\\omega = W^i \\d^{m-1}x_i$, where $W^i \\in C^\\infty(E)$\nare local functions, we obtain that\n$$\n-H(x^i,y^\\alpha,s_\\alpha^i) = \\sum_{i=1}^{m} \\derpar{W^i}{x^i} \\quad ; \\quad\n\\derpar{W^i}{y^\\alpha} = s_\\alpha^i \\, ,\n$$\nfrom where we obtain the classical Hamiltonian Hamilton--Jacobi equation\n$$\n\\sum_{i=1}^{m} \\derpar{W^i}{x^i} + H\\left(x^i,y^\\alpha,\\derpar{W^i}{y^\\alpha}\\right) = 0 \\, .\n$$\n\n\nThe definition and the characteristics of complete solution are similar to those\nof the Lagrangian case.\n\n\n\\subsubsection{Relation between the multisymplectic Hamilton--Jacobi problems}\n\n\nFinally, let ${\\cal FL}\\colon J^1\\pi\\to J^1\\pi^*$ be the Legendre transformation\ndefined by the Lagrangian $\\L$, which is locally given by\n\\begin{equation*}\n{\\cal FL}^*x^i = x^i \\quad , \\quad \n{\\cal FL}^*y^\\alpha = y^\\alpha \\quad , \\quad\n{\\cal FL}^*p_\\alpha^i = \\derpar{\\L}{y_i^\\alpha} \\, .\n\\end{equation*}\nIf $\\L$ is a regular or a hyperregular Lagrangian (i.e., ${\\cal FL}$ is\na local or global diffeomorphism),\nthen ${\\cal FL}^*\\Theta_h = \\Theta_\\Lag$ and ${\\cal FL}^*\\Omega_h = \\Omega_\\Lag$.\nIn addition, the integral sections of the distributions\n${\\cal D}_\\Lag$ and ${\\cal D}_h$, which are the solution to the Lagrangian and \nthe Hamiltonian problems respectively,\nare in one-to-one correspondence through ${\\cal FL}$. \n(See \\cite{LPRV-2017} for definitions and details).\nThen we have:\n\n\\begin{teor}\\label{thm:EquivalenceLagHam}\nLet $\\Lag \\in \\df^{m}(J^1\\pi)$ be a regular or a hyperregular Lagrangian.\nThen, if $\\Psi \\in \\Gamma(\\pi^1)$ is a jet field solution to the (generalized) Lagrangian\nHamilton--Jacobi problem, then the section $s={\\cal FL}\\circ \\Psi \\in \\Gamma(\\pi_E)$ is a solution to\nthe (generalized) Hamiltonian Hamilton--Jacobi problem.\nConversely, if $s \\in \\Gamma(\\pi_E)$ is a solution to the (generalized) Hamiltonian\nHamilton--Jacobi problem, then the jet field \n$\\Psi ={\\cal FL}^{-1} \\circ s \\in \\Gamma(\\pi^1)$ is\na solution to the (generalized) Lagrangian Hamilton--Jacobi problem.\n\\end{teor}\n\\proof\n(Guidelines for the proof):\nThe proof follows the same patterns as Theorem \\ref{relation},\nbut using multivector fields.\n\\qed\n\n\\begin{remark}{\\rm\nAs a final remark, notice that the Hamilton--Jacobi theory for {\\sl non-autonomous} \n(i.e., {\\sl time-dependent\\\/}) {\\sl dynamical systems}\ncan be recovered from the multisymplectic Hamilton--Jacobi theory\nas a particular case taking $M=\\Real$ and identifying\nthe distributions ${\\cal D}$, ${\\cal D}_\\L$, ${\\cal D}_h$ and their\nassociated multivector fields ${\\bf X}$, ${\\bf X}_\\L$, ${\\bf X}_h$\nwith time-dependent vector fields (see \\cite{LPRV-2017}).\n}\\end{remark}\n\n\n\\section{Conclusions and outlook}\n\\protect\\label{conc}\n\n\nIn this work, the Lagrangian and the Hamiltonian versions of the\nHamilton--Jacobi theory has been reviewed from a modern geometric perspective.\n\nFirst, this formulation is done for autonomous dynamical systems\nand, in particular, the Hamiltonian case is compared with the \n ``classical'' Hamiltonian Hamilton--Jacobi theory which \nis based in using canonical transformations.\n\nThere is also a general framework for the theory, \nwhich is also reviewed in the work.\nIt contains the above standard theory\nfor autonomous dynamical systems as a particular case, and\nallows us to extend the Hamilton--Jacobi theory to a \nwide range of physical systems.\nIn particular two of these extensions have been analyzed here:\nthe higher-order (autonomous) dynamical systems \nand the (first-order) classical Lagrangian and Hamiltonian field theories,\nusing their multisymplectic formulation.\n\nThis geometric model has been extended and applied\nto many kinds of physical systems (as it is mentioned in the Introduction\nand cited in the bibliography).\nAs a future line of research that has not been explored yet,\nthe application of this geometric framework to state\nthe Hamilton--Jacoby equation for dissipative systems in classical field theories\nshould be explored, using an extension of the\ncontact formalism which has been recently introduced to\ndescribe geometrically these kinds of dissipative field theories\n\\cite{GGMRR-2019,GGMRR-2020}.\n\n\n\n\\section*{Acknowledgments}\n\n\nI acknowledge the financial support from\nproject PGC2018-098265-B-C33 of the Spanish\nMinisterio de Ciencia, Innovaci\\'on y Universidades \nand the project 2017--SGR--932 of the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction.}\\label{sec:intro}\n\\subsection{Ultrametrics.}\nUltrametrics are used to model any system that can be represented\nby a bifurcating hierarchical tree.\nThe is a relationship between trees annd ultrametrics is as follows.\nAn $N$-leaf edge(node)-weighted tree corresponds to an $N\\times N$ \nsquare matrix $M$ in which $M_{ij}=$ the sum of the weights of the edges(nodes)\nin the shortest path between $i$ and $j$.\nWhen the weights are non-negative, $M$ ia a measure in the usual sense when\n\\begin{eqnarray}\n&&\\forall x,y,z~~~M_{xy}=0~~~ {\\rm if}~~~ x=y\\\\\n&&M_{xy}>0~~~ {\\rm for}~~~ x\\ne y\\\\\n&&M_{xy}=M_{yx}\\\\\n&&M_{xy}\\le M_{xz}+M{zy},\\label{triaineq}\\\\\n\\nonumber\n\\end{eqnarray}\nif the traingle inequality \\ref{triaineq} is replaced by \n\\begin{equation}\nM_{xy}\\le {\\rm max}\\{M_{xz},M_{zy}\\}.\n\\label{ultramax}\n\\ee\nthen $M$ is an ultrametric.\nTo briefly go through some areas where ultrametrics \nhave been applied.\nPerhaps the most important application is to taxonomy,\nJardine and Sibson (1971) Ch.7 \\cite{bi:JS}, \nand Sneath and Sokal (1973) \\cite{bi:SS}.\nHere the end of a branch of the tree represents a species and the ultrametric\ndistance between them shows how closely the species are related.\nThe actual technique is called the hierarchical cluster method, \nthe method classifies species and also shows how closely species are related.\nThis technique has also been used in semantics,\nShepard and Arabie (1979) \\cite{bi:SA}.\nThe technique can become quite complex because they \ninvolve statistical analysis with continuous variates.\nUltrametrics have been applied frequently in the theory of spin glass,\nWeissman \\cite{bi:weissman}.\nUltrametrics have been used for description of \nslowly driven dissipative systems,\nwhich exhibit avalanche-like behaviour, these include earthquakes,\nextinction events in biological evolution, and landscape formation,\nBoettcher and Paiginski (1997) \\cite{bi:BP},\nalso ultrametrics can describe systems with fast relaxation,\nVlad (1994) \\cite{bi:vlad}.\nUltrametrics are frequently used in the theory of neural nets,\nParga and Virasoro \\cite{bi:PV}.\nThe dynamics of random walks on ultrametric spaces have been studied,\nOgielchi and Stein (1985) \\cite{bi:OS}.\nUltrametrics have been applied to the thermodynamics of macromolecules \nsuch as RNA, Higgs (1996) \\cite{bi:higgs}.\nBounds on the size of ultrametric structure have been discussed by\nBaldi and Baun (1986) \\cite{bi:BB}.\nFrom a more theoretical angle, a category theory approach \nhas been elucidated by Rutten (1996) \\cite{bi:rutten}, \nand a model theoretic approach to them given Delon (1984) \\cite{bi:delon}.\nThe relationship between ultrametric distance and hierarchy is further \ndiscussed in Gu\\'{e}noche (1997) \\cite{bi:guenoche}.\nConstruction of optimal ultrametric trees is discussed by\nYoung and DeSarbo (1995) \\cite{bi:YDS}.\nUltrametrics are related to p-adelic quantities, \nKarwowski and Mendes (1994) \\cite{bi:KM}.\nP-adelic quantities are used in string theory, \nthe way that ultrametrics enters here is explained in \\S10\\&\\S13.4 of\nBekke and Freund (1993) \\cite{bi:BF}.\nThere does not seem to be any straightforward connection of any of the above\nto the optimization techniques of\nPrince and Smolensky (1997) \\cite{bi:smolensky}.\nAs well as ultrametric trees, \nthere are also {\\sl decision trees} Hammer (1998) \\cite{bi:hammer},\nand the connection between them is still not known.\nSome of the above ultrametric applications have been reviewed by\nRammal {\\it et al} (1986) \\cite{bi:rammal}\n\\subsection{Syntactic Phase Trees.}\nFor the analysis of syntactic phrase \ntrees the necessary technique is quite simple and \nis illustrated by the examples in section\n\\ref{sec:sec2}. The examples here mainly follow the examples in \nLockward (1972) \\cite{bi:lockward}, Kayne (1981) \\cite{bi:kayne},\nMcCloskey (1988) \\cite{bi:mccloskey}, \nand especially Haegeman (1994) \\cite{bi:haegeman}.\nThere are at least {\\bf five} reasons for introducing an ultrametric \ndescription of syntax.\\\\\nThe {\\sf first} is to completely specify tree \n(also called {\\sl dendrogram}) structure.\nConsider the following example illustrated by {\\bf Figure 1}:\n\\begin{figure}\n{\\epsfig{figure=udfig1.eps,height=3in}}\n\\caption{Different syntactic descriptions of ``the man ate a dog''}\n\\label{DiagramOne}\n\\end{figure}\nFor current syntactic models the two trees usually are equivalent\n(perhaps not always McCloskey (1988) \\cite{bi:mccloskey} footnote 6): \nhowever consider the ultrametric distance between 'the' and 'man',\n\\begin{equation}\nA(the,man)=1,~~~~~~\nB(the,man)=2,\n\\label{eq:udtheman}\n\\ee\nthis difference does not occur in current syntactic models,\nand a purpose of an ultrametric model is to disambiguate this.\\\\\nThe {\\sf second} is it gives a measure of the complexity of a sentence:\nthe greater the ultrametric distance required the more complex a sentence is.\nThe above can also be viewed in terms of 'closeness'.\nThe example illustrates that current syntactic models give no notion of how \n'close' determiners and nouns are. However ultrametrics do give an \nindication of closeness and this can be compared:\n{\\sl firstly} to the closeness indicated by features,\n{\\sl secondly} to the idea that if a sentence is not sufficiently close\nthen there is a {\\sc barrier} Chomsky (1986b) \\cite{bi:chomsky2}.\nOnly the first is looked at here. \nIn traditional syntax phrases can be iteratively \nembedded to give sentences of unbounded length and complexity.\nA degree of sentence complexity perhaps corresponds to the height of the tree\nrepresenting the sentence. As people can only process a finite amount\nof information this height must be finite,\nin the traditional theoretical framework\nthere is no finite bound on sentence length.\nAn upper bound could perhaps be found by experiment,\ninspection of phrase trees suggests\na first guess of $h=12$.\\\\\nThe {\\sf third} is that it means that syntax is described in the same \nformalism as a lot of other science, \nfor example those topics described in the first paragraph,\nso that there is the possibility of techniques being used in one area\nbeing deployed in another.\\\\\nThe {\\sf fourth} is that an ultrametric formulation might allow a \ngeneralization so that ideas in syntax can be applied to other \ncognitive processes.\\\\\nThe {\\sf fifth}, and perhaps the most important,\nis that it might be possible to use some sort of minimum distance principle\nin syntax, \nindeed it could be this minimum description which would have application in\nother cognitive processes.\\\\\n\\subsection{Ockams Razor.}\nMinimum description in science go back several hundred years\nto ``{\\em Ockams razor}'' or perhaps further, \nsee for example Sorton (1947) \\cite{sorton} page 552.\nThe principle of least action\n(see for example Bjorken and Drell (1965) \\cite{bi:BD} \\S 11.2), \nin physics is that minimal variation of a given action gives\nfield equations which describe the dynamics of a system. For example,\nMaxwell's equations can be derived from a simple action by varying it.\nIn the present context one would hope that syntax allows for\na minimum encoding of semantic information,\nthe minimum encoding being given by some ultrametric measure.\nA different approach along these lines is that of \nRissanen (1982) \\cite{bi:rissanen} and Zadrozny (2000) \\cite{bi:zadrozny}.\nBriefly they assign a length of $1$ to each symbol in a sentence,\nthen the {\\sc minimum description length} states that the best theory \nto explain a set of data is the one which minimizes both the sum of:\ni) the length, in bits, of the description of the theory, and\nii) the length, in bits, of data when encoded with the help of the theory.\nChristiansen (2001) \\cite{christiansen} discusses how constraint handling\nrules (CHR) can be applied to grammars, this can be thought of as a\nminimizing procedure.\n\\subsection{Recticulate \\& N-ary Trees.}\nA {\\sc reticulate} tree is a tree in which the branches reconverge, \nillustrated by {\\bf Figure 2},\n\\begin{figure}\n {\\epsfig{figure=udfig2.eps,height=3in}}\n \\caption{A {\\sc reticulate} tree}\n\\end{figure}\na {\\sc non-reticulate} tree is a tree in which the branches do not reconverge. \n{\\sc N-ary} branching is illustrated by {\\bf Figure 3}. \n\\begin{figure}\n {\\epsfig{figure=udfig3.eps,height=3in}}\n \\caption{{\\sc N-ary} branching}\n\\end{figure}\n{\\sc Binary} branching is {\\sc N-ary} branching with $N=2$.\n{\\sc N-ary} branching can be replaced by binary branching \nif additional layers are used. \nA {\\sc switched} tree is a tree in which all the branches are binary.\nSyntactic phrase trees are {\\sc non-reticulate} and {\\sc switched}.\nIn most linguistic theories all syntactic phrase trees have \\={X} structure,\nJackendoff (1977) \\cite{bi:jackendoff},\nhere attention is restricted to theory which has \\={X} structure.\n{\\subsection{Sectional Contents.}\nIn section \\ref{sec:sec2} it is shown how to represent trees by matrices \nand triangles. All \\={X} triangles are isosceles but not equilateral.\nIn section \\ref{sec:sec3} the matrix {\\bf U} for the minimum ultrametric \ndistance for lexical categories is given.\nFor simplicity discussion is limited \nto active voice sentences with only determiners, nouns, transitive verbs,\nadjectives, and prepositions. \nInclusion of case theory, COMP, INFL,.. might be of interest \nbut would complicate matters.\nIn section \\ref{sec:sec4} the singular matrix {\\bf F} for features is given.\n{\\bf F} is not an ultrametric matrix and there appears to be no relation \nto {\\bf U}. In section \\ref{sec:sec5}\nit is shown that the notion of {\\sc c-command} is equivalent to \nan ultrametric minimum distance. This allows a new definition of \ngovernment to be given. \nIn appendix \\ref{sec:app} other linguistic hierarchies are discussed; \nin particular there appears to be at least two separate \noccurrences of culturally determined partial ordered hierarchies \n- the {\\sl accessibility hierarchy} for relative clauses \nand the {\\sl universal colour ordering}. \nFor completion in appendix \\ref{sec:app} there is a very briefly\naccount what these hierarchies are, a comparison and contrasting of them, \nand the speculation that they are specific examples of a \n{\\sl grand cultural hierarchy}.\nThe question arises of why such hierarchies should exist,\nand it might be because they reduce the amount of memory \nneeded to process information by clumping information together \nin the style of Miller (1956) \\cite{bi:miller}, \nfor a more recent reference see Cowan (2001) \\cite{cowan}.\nA {\\sl hierarchy} is an example of a representation as\ndiscussed by Roberts (1998) \\cite{bi:roberts}.\n\\section{ \\={X} Structure Implies No Equilateral Triangles.}\n\\label{sec:sec2}\n\\subsection{Binary and N-ary Branching for simple sentences.}\n\\={X} structure implies {\\sc binary branching} Haegeman (1994) p.139 \n\\cite{bi:haegeman}.\nTo see what this implies for ultrametric \ndistances consider all five species of {\\sc binary branched} tree,\nthe {\\sl first} has diagram {\\bf Figure 4} and corresponding matrix:\n\\begin{figure}\n{\\epsfig{figure=udfig4.eps,height=2.5in}}\n \\caption{The simplest {\\sc binary} tree for ``Alf must jump high''}\n\\end{figure}\n\\begin{eqnarray}\nFirst=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 1 & 2 & 2\\\\\nM & . & 0 & 2 & 2\\\\\nJ & . & . & 0 & 1\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq2}\n\\end{eqnarray}\nrespectively. The matrices corresponding to the other {\\sl four}\n{\\sc binary branched} trees are:\n\\begin{eqnarray}\nSecond=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 3 & 3 & 3\\\\\nM & . & 0 & 2 & 2\\\\\nJ & . & . & 0 & 1\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq3}\n\\end{eqnarray}\n\\begin{eqnarray}\nThird=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 3 & 3 & 3\\\\\nM & . & 0 & 1 & 2\\\\\nJ & . & . & 0 & 1\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq4}\n\\end{eqnarray}\n\\begin{eqnarray}\nFourth=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 2 & 2 & 3\\\\\nM & . & 0 & 1 & 3\\\\\nJ & . & . & 0 & 3\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq5}\n\\end{eqnarray}\n\\begin{eqnarray}\nFifth=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 1 & 2 & 3\\\\\nM & . & 0 & 2 & 3\\\\\nJ & . & . & 0 & 3\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq6}\n\\end{eqnarray}\nThere are {\\sl two} {\\sc 3-ary} trees with matrices:\n\\begin{eqnarray}\nSixth=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 1 & 1 & 3\\\\\nM & . & 0 & 1 & 2\\\\\nJ & . & . & 0 & 2\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq7}\n\\end{eqnarray}\n\\begin{eqnarray}\nSeventh=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 2 & 2 & 2\\\\\nM & . & 0 & 1 & 1\\\\\nJ & . & . & 0 & 1\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq8}\n\\end{eqnarray}\nand finally there is {\\sl one} {\\sc 4-ary} tree with diagram {\\bf Figure 5} \nand matrix:\n\\begin{figure}\n {\\epsfig{figure=udfig5.eps,height=2.5in}}\n \\caption{The {\\sc 4-ary} tree for ``Alf must jump high''}\n\\end{figure}\n\\begin{eqnarray}\nEighth=\n\\begin{array}{ccccc}\n\\bullet & A & M & J & H\\\\\nA & 0 & 1 & 1 & 1\\\\\nM & . & 0 & 1 & 1\\\\\nJ & . & . & 0 & 1\\\\\nH & . & . & . & 0\\\\\n\\end{array}\n\\label{eq9}\n\\end{eqnarray}\n\\subsection{Triangle representation of the proceeding.}\nAll ultrametric triangles are isosceles, but only some are equilateral.\nThe above suggests that binary branching implies that there are no equilateral \ntriangles. For example from matrix \\ref{eq2}, \n$d(A,M)=1, d(A,J)=2, d(M,J)=2$\ngiving in the triangle representation {\\bf Figure 6},\n\\begin{figure}\n {\\epsfig{figure=udfig6,height=3in}}\n \\caption{The Isosceles Triangle Representation.}\n\\end{figure}\nand from matrix \\ref{eq9}, $d(A,M)=1, d(A,J)=1, d(M,J)=1$ \ngiving in the triangle representation {\\bf Figure 7}.\n\\begin{figure}\n {\\epsfig{figure=udfig7,height=2in}}\n \\caption{The Equilateral Triangle Representation.}\n\\end{figure} \nFormally it is proved that \\=X structure implies that there are no \nequilateral triangles.\n\\subsection{The \\=X Template.}\nConsider the \\=X template {\\bf Figure 8}.\n\\begin{figure}\n {\\epsfig{figure=udfig8,height=3in}} \n \\caption{The \\=X Template}\n\\end{figure} \n\\newpage\nThe matrix representation of this is:\n\\begin{eqnarray}\n{\\bf \\bar{X}}=\n\\begin{array}{cccc}\n\\bullet & Spec & X & YP\\\\\nSpec & 0 & i+2 & i+2\\\\\nX & . & 0& i+1\\\\\nYP & . & . & 0\\\\\n\\end{array}\n\\end{eqnarray}\nFrom this the triangle representation is {\\bf Figure 9}.\n\\begin{figure}\n {\\epsfig{figure=udfig9,height=3in}} \n \\caption{The Triangle Representation of \\=X structure}\n\\end{figure} \nThis must be isosceles but not equilateral.\n\\section{The minimum ultrametric distance between lexical categories.}\n\\label{sec:sec3}\n\\subsection{The minimum distance principle.}\nWe assume that it is the minimum distance between lexical categories that\nis important, and refer to this as the {\\sc minimum distance principle}.\nIn part this is motivated by the principle of least action in physics,\nsee for example Bjorken and Drell (1965) \\cite{bi:BD} \\S 11.2,\nsee also the introduction \\S1.3 above and Roberts (1998) \\cite{mdr}\\S3.\nA current psycholinguistic model of sentence production is the garden path\nmodel, see for example Frazier (1987) \\cite{frazier} \nand Roberts (1998) \\cite{mdr}\\S5.4. Part of this model requires \n{\\it the minimal attachment principle}, \nwhich is ``do not postulate unnecessary modes.'':\nthis can be thought of as a minimum principle.\nThe {\\sc minimum distance principle} implies that the correct tree\nfor the example in the introduction is {\\bf Figure 10},\n\\begin{figure}\n {\\epsfig{figure=udfig10,height=4in}}\n \\caption{The Correct Tree for the Example in the Introduction.}\n\\end{figure} \nso that all entries occur at the lowest possible height.\nThus in particular tree {\\bf A} is preferred to tree {\\bf B}.\nThis assumption does not effect the matrix {\\bf U} given below,\nbut will have an effect when the analysis is extended to ${\\bf\\theta}$-theory.\nFrom the above $d(N,D)=1, d(N,V)=d(D,V)=2$. Similarly from {\\bf Figure 11},\n\\begin{figure}\n {\\epsfig{figure=udfig11,height=3.6in}}\n \\caption{The Distance Between Verbs and Adjectives.}\n\\end{figure} \n$d(V,A)=4$.\nConstructing other examples gives the ultrametric distance matrix\n\\begin{eqnarray}\n{\\bf U}=\n\\begin{array}{cccccc}\n\\bullet & D & N & V & A & P\\\\\nD & 0 & 1 & 2 & 2 & 2\\\\\nN & . & 0 & 2 & 2 & 2\\\\\nV & . & . & 0 & 4 & 3\\\\\nA & . & . & . & 0 & 3\\\\\nP & . & . & . & . & 0\\\\\n\\end{array}\n\\label{matrixU}\n\\end{eqnarray}\nIgnoring D and ordering the matrix NPVA (noun, pronoun, verb, adjective)\nsuggests the pattern\n\\begin{eqnarray}\n{\\bf I}=\n\\begin{array}{ccccc}\n0 & i & i & i & \\dots\\\\\n. & 0 & i+1 & i+1 & \\dots\\\\\n. & . & 0 & i+2 & \\dots\\\\\n. & . & . & 0 & \\dots\\\\\n\\end{array}\n\\label{matrixI}\n\\end{eqnarray}\nwhich is compatible with the \\={X} matrix of the last section;\nhowever it does not follow by necessity as the \\={X} \ncase holds for a single sentence and {\\bf U} is constructed from several.\n\\section{Features}\\label{sec:sec4}\n\\subsection{No square matrix representation of Features.}\nThis section investigates whether there is a general framework\nwhich can describe the preceding and also features.\nHaegeman (1994) \\cite{bi:haegeman} p.146 \ngives the following diagram for features:\n\\begin{eqnarray}\n{\\rm Features~ diagram} =\n\\begin{array}{ccc}\nNoun & +N & -V\\\\\nVerb & -N & +V\\\\\nAdj. & +N & +V\\\\\nPre. & -N & -V\\\\\n\\end{array}\n\\end{eqnarray}\nThis can be represented by the matrix\n\\begin{eqnarray}\n{\\rm Features~ matrix} =\n\\begin{array}{ccc}\n\\cdot & Noun & Verb\\\\\nNoun & +1 & -1 \\\\\nVerb & -1 & +1 \\\\\nAdj. & -1 & +1 \\\\\nPre. & -1 & -1 \\\\\n\\end{array}\n\\end{eqnarray}\nA square matrix can be constructed by assuming that the matrix is symmetric.\nThis leaves only one unknown $F(A,P)$. Taking $F(A,P)= -1$\ngives equal number of positive and negative entries in the matrix\n\\begin{eqnarray}\n{\\bf F} =\n\\begin{array}{ccccc}\n\\bullet & N & V & A & P\\\\\nN & +1 & -1 & +1 & -1\\\\\nV & -1 & +1 & +1 & -1\\\\\nA & +1 & +1 & +1 & -1\\\\\nP & -1 & -1 & -1 & +1\\\\\n\\end{array}\n\\label{matrixF}\n\\end{eqnarray}\nwhich is singular as its determinant vanishes. There appears to be no \nrelation between matrix {\\bf F} \\ref{matrixF} and matrix {\\bf U} \\ref{matrixU}.\nUsing the Pauli matrices \n(see for example Bjorken and Drell (1965) \\cite{bi:BD} p.378)\n\\begin{eqnarray}\nI=\n\\begin{array}{cc}\n1 & 0\\\\\n0 & 1\\\\\n\\end{array}~~~\n\\sigma^1=\n\\begin{array}{cc}\n0 & 1\\\\\n1 & 0\\\\\n\\end{array}~~~\n\\sigma^2=\n\\begin{array}{cc}\n0 & -i\\\\\ni & 0\\\\\n\\end{array}~~~\n\\sigma^3=\n\\begin{array}{cc}\n1 & 0\\\\\n0 & -1\\\\\n\\end{array}\n\\end{eqnarray}\n{\\bf F} can be expressed as\n\\begin{eqnarray}\n{\\bf F}=\n\\begin{array}{cc}\nI-\\sigma^1 & -i\\sigma^2+\\sigma^3\\\\\n+i\\sigma^2+\\sigma^3 & I-\\sigma^1\\\\\n\\end{array}\n\\end{eqnarray}\nhowever this does not correspond in any straightforward way to any of the \nDirac matrices (see for example Bjorken and Drell (1965) \\cite{bi:BD} page 378)\nin standard representations.\n\\section{Ultrametric Approach to Government.}\\label{sec:sec5}\nRecall the following definitions in Haegeman \\cite{bi:haegeman}:\n\\subsection{Definition of {\\sc dominates}.}\n{\\bf Definition} \\cite{bi:haegeman} p.85 \\newline\nNode {\\bf A} {\\sc dominates} node {\\bf B} iff:\\newline\n i)$~~h(A)$ is higher up or at the same height on the tree as $h(B)$\\\\\n i.e.$h(A)\\geq h(B)$\\newline\n ii) it is possible to trace a line from {\\bf A} to {\\bf B} \n going only downward,\\newline \n $~~~~$ or at most going to one higher node.\\newline\n{\\bf Remarks}\\newline\nThe {\\sl first} requirement is that {\\bf A} is at a greater height than\n{\\bf B}. The {\\sl second} requirement restricts the possible downward route \nfrom {\\bf A} to {\\bf B} so that it contains at most one upward segment.\n\\newline {\\bf Example} (compare \\cite{bi:haegeman} p.83)\\newline\n the phrase tree {\\bf Figure 12}\n\\begin{figure}\n {\\epsfig{figure=udfig12,height=3in}} \n \\caption{Illustration of {\\sc dominates}}\n\\end{figure} \ngives the 'dominates' matrix\n\\begin{eqnarray}\n{\\bf D}=\n\\begin{array}{cccccccccc}\n\\bullet & S & NP(S) & N(S) & AUX & VP & V & NP(E) & Det & N(e)\\\\\nS & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\nNP(S) & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nN(S) & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\nAUX & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\nVP & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 \\\\\nV & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\\nNP(E) & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\\\nDet & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\nN(E) & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ \n\\end{array}\n\\end{eqnarray}\nwhere 1 indicates {\\bf A} dominates {\\bf B} and 0 indicates that it does not.\n\\subsection{Definition of {\\sc C-command}.}\n{\\bf Definition} \\cite{bi:haegeman} p.134 \\newline\nNode {\\bf A} {\\sc c-commands} (constituent-commands) node {\\bf B} iff:\\newline\n i) {\\bf A} does not dominate {\\bf B} \n and {\\bf B} does not dominate {\\bf A},\\newline\n ii) The first branching node dominating {\\bf A} \n also dominates {\\bf B}.\\newline\n{\\bf Remarks} \\newline\nThe {\\sl first} requirement is that there is no direct route up or \ndown from {\\bf A} to {\\bf B} passing more than one higher node.\nThe {\\sl second} requirement restricts {\\bf A} and {\\bf B} to be 'close'. \nHaegeman's first criteria for dominance needs to be adjusted,\nif it is correct then $h(A)>h(B)$ and $h(B)>h(A)$ so that the set of \nall {\\sc c-commands} is empty, therefore greater than or equal $\\geq$ \nis used here instead of greater than $>$. \nHaegeman's second criteria for dominance also needs to be adjusted,\nif no higher node is allowed the set of {\\sc c-commands} is again empty. \nChomsky (1986a) \\cite{bi:chomsky} p.161 approaches the \nsubject in a different manner using maximal projections.\\\\\n{\\bf Example}:\n{\\bf Figure 13}\n\\begin{figure}\n {\\epsfig{figure=udfig13,height=3in}}\n \\caption{Example of {\\sc c-commands}.}\n\\end{figure} \nin the figure $0$\" are more accessible \nthan those on the right. SU is short for subject, \nDO for direct object, IO for indirect object, \nOBL for major oblique case NP, GEN for genitive NP, \nOCOMP for object of comparison. \nThe properties of the accessible hierarchy \nare contained in two sets of constraints.\\\\ \nThe accessible hierarchy constraints ({\\sl AHC's}) are:\\\\\n{\\bf AHC1)} A language must be able to relativize subjects.\\\\\n{\\bf ACH2)} Any RC forming strategy must apply to a \ncontinuous segment of the AH.\\\\\n{\\bf ACH3)} Strategies that apply at one point of the \nAH may in principle cease to apply \nat any lower point.\\\\\nThe primary relativization constraints ({\\sl PRC's}) are\\\\\n{\\bf PRC1}) A language must have a primary RC-forming strategy.\\\\\n{\\bf PRC2)} If a primary strategy in a given language \ncan apply to a low position on the AH, then it \ncan apply to all higher positions.\\\\\n{\\bf PRC3)} A primary strategy may cut off at any point on the AH.\\newline\nFor a given language a deployment that can be used to relativize a clause \nat a specified place on the AH can also be used to relativize all more\naccessible clauses. The type of relativization varies from language to \nlanguage. There appears to be nothing known on how the skill \nto deploy a relativization develops in an individual. \nOne would expect that when a given method is applied the\nless accessible would take longer to process, there seems to be no \npsycholinguistic tests done to see if this is indeed the case.\n\\subsection{The Berlin-Kay Universal Colour Partial Ordering.}\nThe perception of colour often involves the deployment of a colour name\nstrategy. The effect of this is to alter the way the colour is perceived.\nThe five principles of colour perception are:\\\\\n{\\bf CP1)}The communicability of a referent in an array and for a particular\ncommunity is \nvery closely related to the memorability of that referent in\nthe same array and for members of the same community.\\\\\n{\\bf CP2)} In the total domain of colour there are \neleven small focal areas in which are found \nthe best instances of the colour categories named in \nany particular language. \nThe focal areas are human universals, \nbut languages differ in the number of \nbasic colour terms they have: \nthey vary from two to eleven.\\\\\n{\\bf CP3)} Colour terms appear to evolve in a language \naccording to the Berlin-Kay (1969) \n\\cite{bi:BK}\nuniversal partial ordering illustrated by {\\bf Figure 16},\n\\begin{figure}\n {\\epsfig{figure=udfig16,height=1.2in}}\n \\caption{The Berlin-Kay Universal Colour Partial Ordering}\n\\end{figure} \n{\\bf CP4)} Focal colours are more memorable and easier \nto recognize than any other colours, \nwhether or not the subject speak a \nlanguage having a name for the colour.\\\\\n{\\bf CP5)} The structure of the colour space determined by\nmulti-dimensional scaling of \nperceptual data is probably the \nsame for all human communities and it is \nunrelated to the space yielded by naming data.\\\\\nAgain there is a {\\sl culturally determined linguistic partial ordering} (or\nhierarchy). On this occasion it determines the semantic content of \nindividual words rather than syntax rules. Again there appears to be\nnothing known on how the skill develops in an individual, or any timing\ntests on the possession of a colour name strategy. The existence of two \nseparate hierarchical {\\sl partial orderings} suggests that there is a general \nmechanism for there construction. Most members of a community seem to develop\nthese culturally determined skills suggesting that the capacity to develop them\nis usually innate but their manifestation depends on environment.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{The True Fundamental Value Times Series}\n\nA common feature of agent-based financial simulation is the presence of an exogenous (\\emph{defined:} ``derived from external factors'') price time series, representing some global consensus value for an asset at any time, arising from the accumulation of all available news and information which could influence the valuation of the asset. A simple intuition for the need of such an extrinsic value can be obtained by its omission. Imagine a simulated stock entering its opening auction period, such that there is not yet a limit order book to examine. In the absence of an extrinsic value of some kind, what would cause market agents to place bids or offers with any particular limit price? We could imagine using the prior day's closing price as an anchor, but then what about the first day? There must be \\emph{some} concept of ``what this company is really worth'' for at least \\emph{some} of the market participants, or there is no reason for the stock to trade in the \\$5 range versus the \\$5,000 range. In real-world terms, we can imagine consultation of this extrinsic series to be the distinguishing factor between value investors versus purely technical traders.\n\nThis ``fundamental value series'' requires an omniscient perspective that is not available to any market participant. Thus at \\emph{best}, market participants may obtain a noisy observation of this value at some particular point in time. This paper focuses on the set of mean-reverting fundamental value series with characteristics as described in Section 3 of Chakraborty and Kearns \\cite{chakraborty2011market}.\n\n\\subsection*{Properties of the Fundamental}\n\nIntuitively, think of the fundamental as a consensus valuation of all people everywhere that somehow reflects the ``real'' value of a company. This value changes over time in response to events, and of course no one can ever know precisely what the global consensus value is.\n\n\\begin{itemize}\n \\item There is exactly one fundamental value time series for each equity in the simulation.\n \\item It is a property of the equity and does not vary per agent.\n \\item It is \\emph{predetermined}; agent actions do \\emph{not} affect it.\n\\end{itemize}\n\nMost any type of numeric time series data can serve as a fundamental value series for an equity. For example, one could simply use historical market data. We consider three possibilities: a discrete mean reverting series, a continuous mean reverting series based on the Ornstein-Uhlenbeck process, and a modification of that OU process with ``megashock events''.\n\n\\subsection*{Discrete Mean Reverting Fundamental}\n\nWe first consider a discrete mean reverting fundamental time series as presented in Section 4 of Wah et al. \\cite{wah2017welfare}\n\nIn this case, the fundamental value series for an equity is defined at all times $t\\in[0,T]$ by:\n$$r_t=\\mathbf{max}\\{0,\\kappa\\bar{r}+(1-\\kappa)r_{t-1}+u_t\\};r_0=\\bar{r}$$\n\nThe mean fundamental value for an equity ($\\bar{r}$) is chosen as part of experimental configuration. The fundamental value series $r$ begins at this mean ($r_0=\\bar{r}$) and is never allowed to become negative.\n\nAt each time step $t \\in [0,T]$, the series makes a single-step reversion to the mean using parameter $\\kappa\\in[0,1]$, the mean reversion rate. This functions similarly to the learning rate in many machine learning algorithms: $\\alpha\\times\\mathtt{new\\_value}+(1-\\alpha)\\times\\mathtt{old\\_value}$. Here, \\texttt{new\\_value} is just $\\bar{r}$ (to produce mean reversion). Thus the series is always drifting back toward the mean at a predictable rate, faster for high $\\kappa$ and slower for low $\\kappa$. If $\\kappa = 0$, of course, the time series does not revert to the mean.\n\nSo far this looks \\emph{very} unlike a stock price. When the price deviates from the mean, it slowly and predictably returns to the mean, then stays there forever. In fact, nothing thus far would make it leave the mean in the first place.\n\nEnter $u_t\\sim\\mathcal{N}(0,\\sigma^2_s)$, the ``shock variance''. This is a random number drawn at each time step $t$ to perturb the otherwise completely predictable fundamental value series. It erratically pushes the series around (including away from the mean) to introduce noise into the price series and create a random walk.\n\n\\textbf{Important:} This $u_t$ ``noise'' is part of producing the single, true fundamental value series for an equity. This is \\emph{not} the observation noise experienced by agents.\n\nAgents are sometimes required to infer the fixed parameters of the mean reverting fundamental, but for simplicity it is often assumed that agents have become ``tuned'' to their environment. In either case, agents \\emph{do} know the above equations, but do not know the true fundamental value after time $t=0$, because of the random draws for $u_t$. In the ``tuned'' case, agents additionally know that $u_t\\sim\\mathcal{N}(0,\\sigma^2_s)$, and know the correct values for $\\bar{r}$, $\\kappa$, and $\\sigma^2_s$.\n\n\\subsection*{Ornstein-Uhlenbeck Process Fundamental}\n\nWe next consider a continuous mean reverting time series as the fundamental, based on the Ornstein-Uhlenbeck (OU) Process as described in Section 3.1 of Chakraborty and Kearns. \\cite{chakraborty2011market}\n\nA potential weakness in the discrete mean reverting series is that it must be computed at every discrete time step, in order, for $t \\in [0,T]$. Time steps at which no agent activity occurs must still be evaluated to obtain future values. If there are many time steps, and agents tend to arrive at the market infrequently, this can become a significant computational burden for the simulation.\n\nThe OU Process, with values represented by $Q$, has the advantage that $Q_t$ can be computed for any future time $t$, given value $Q_0$. Because OU has the Markov property (\\emph{i.e.} $Q_{t+1}$ is conditionally independent of $Q_{t-1}$ given $Q_t$) and $Q_t$ is normally distributed, we can determine the expectation and variance of $Q_t$ and obtain the value by sampling, without computing any of the values ``in between''. In the case of sparse agent arrivals, this can save considerable computation effort for the simulation.\n\nThe OU process requires similar parameters and values to the discrete mean reverting process. The equivalent discrete series parameter is given in parenthesis after each OU compoment. Specifically, the OU process requires a mean fundamental $\\mu$ (equiv: $\\bar{r}$), a prior value $Q_0$ (equiv: $r_{t-1}$), a mean reversion rate $\\gamma$ (equiv: $\\kappa$), and a volatility value $\\sigma$ (equiv: $\\sigma^2_s$).\n\nThe value $Q_t$ of the OU process at time $t$ can then be sampled from a normal distribution with mean: $\\mu + (Q_0 - \\mu)e^{-\\gamma t}$, and variance: $\\frac{\\sigma^2}{2\\gamma}(1-e^{-2\\gamma t})$.\n\nLike the discrete mean reverting process, the OU process can only move away from the mean through its volatility value. If this value is kept small to represent high-frequency ``noise'' in the process, both processes will appear as essentially single-scale accumulating white noise around the fundamental mean, and will not look very much like a plot of, for example, a day of real transactions (or quotes) for some equity traded on a US exchange.\n\n\\subsection*{Megashock OU Fundamental}\n\nTo obtain the simulation computation time improvements of the OU Process Fundamental, while producing a series that looks more like a real market equity price time series, we introduce the concept of ``megashock events''. Megashock events are intended to represent extrinsic news of a substantial nature that occur relatively infrequently, but have the potential to significantly alter the consensus valuation of a stock.\n\nIn technical terms, megashock events are layered on top of the preliminary OU fundamental, arriving via a Poisson process, and are drawn from a bimodal distribution with mean zero. Each sub-distribution is a Gaussian with mean significantly different from zero, and a high variance relative to the OU process base variance. The sub-distributions are centered in positive and negative territory with mean equally far from zero, producing the desired overall mean zero for the bimodal distribution.\n\n\n\n\\section*{Estimating the Final Fundamental Value}\n\nIn the simulation described in Wang and Wellman \\cite{wang2017spoofing}, agents attempt to maximize the value of their portfolio at the end of the simulation relative to the final fundamental value. This is what the closing price ``should'' be. It is \\emph{not} the same as marking to market. Regardless, predicting this final fundamental value is an important part of making trading decisions.\n\nThe agents arrive according to a Poisson distribution and a discrete mean reverting process is used for the fundamental. Assume some agent wakes at time $t$ and receives new (noisy) fundamental observation $o_t$. This observation is unique to each agent with $o_t=r_t+n_t$, where $r_t$ is the true fundamental value and $n_t$ is random observation noise drawn from $\\mathcal{N}(0,\\sigma^2_n)$. The agent \\emph{does} know $\\sigma^2_n$.\n\nThe agent maintains an estimate $\\tilde{r}_t$ of the current fundamental value $r_t$, and an estimated variance $\\tilde{\\sigma}^2_t$ \\emph{of this estimate}. \\textbf{Important:} Again, $\\tilde{\\sigma}^2_t$ is the agent estimating the variance \\emph{of its $\\tilde{r}_t$ estimate}. Thus, $\\tilde{\\sigma}^2_t$ is like an internal error metric (or loosely an ``inverse confidence''), not an estimate of any market or simulation parameter.\n\nLet $t'$ be the last time the agent woke. At that time, it had some estimates ($\\tilde{r}_{t'}$ and $\\tilde{\\sigma}_{t'}$) of $r_{t'}$ and its own estimation error. First it must mentally advance time from $t'$ to $t$. With no observations while asleep, it can only apply mean reversion at each time step:\n$$\\tilde{r}_{t'} \\leftarrow (1-(1-\\kappa)^\\delta)\\bar{r} + (1-\\kappa)^\\delta \\tilde{r}_{t'}$$\nIn the above update, $\\delta=t-t'$, the number of time steps since the agent last woke.\nNote that with $\\delta=1$, it is exactly the mean reversion equation used to compute the true fundamental value series $r_t$, except: the agent does not know $r_{t-1}$ and so uses its previous estimate $\\tilde{r}_{t'}$ instead, and the agent does not know the random perturbations $u_t$ and so uses the known mean value of zero instead.\n\nWith $\\delta>1$, this is equivalent to applying mean reversion $\\delta$ times. The first time the agent wakes, its previous wake time is effectively $t'=0$. Thus it should assume $\\tilde{r}_{t'}=\\bar{r}$, as it knows the value of $\\bar{r}$ and that $r_0=\\bar{r}$.\n\nTo complete the advancement of time from $t'$ to $t$, the agent must also update its internal error estimate $\\tilde{\\sigma}_{t'}$, which is initially zero. It does so as follows:\n$$\\tilde{\\sigma}^2_{t'} \\leftarrow (1-\\kappa)^{2\\delta} \\tilde{\\sigma}^2_{t'} + \\frac{1-(1-\\kappa)^{2\\delta}}{1-(1-\\kappa)^2} \\sigma^2_s$$\nThis update is a weighted mixture of the previous internal error estimate $\\tilde{\\sigma}^2_{t'}$ and the shock variance $\\sigma^2_s$. The weight term for shock variance $\\sigma^2_s$ starts at $1$ when $\\delta=1$ and increases with $\\delta$. The weight term for previous internal error estimate $\\tilde{\\sigma}^2_{t'}$ starts at $(1-\\kappa)^2$ when $\\delta=1$ and decreases as $\\delta$ increases. Thus when computing the new error estimate, the more time that has passed since the agent's last observation, the more weight is placed on shock variance $\\sigma^2_s$ and the less weight is placed on the prior error estimate $\\tilde{\\sigma}^2_{t'}$.\n\nHaving now advanced time from $t'$ to $t$, the agent must apply its new observation $o_t$. This process is not unlike a Dynamic Bayes Network, in which the agent alternates stepping forward in time (transition model) and applying new evidence (sensor model). The current estimate of the fundamental value is a weighted combination of the prior estimate with the new observation:\n$$\\tilde{r}_t=\\frac{\\sigma^2_n}{\\sigma^2_n+\\tilde{\\sigma}^2_{t'}}\\tilde{r}_{t'} + \\frac{\\tilde{\\sigma}^2_{t'}}{\\sigma^2_n+\\tilde{\\sigma}^2_{t'}}o_t$$\nThe weights are simply the observation noise $\\sigma^2_n$ and the agent's internal estimate of its error $\\tilde{\\sigma}^2_{t'}$, normalized to sum to one. The observation noise weights the agent's prior estimate of fundamental value $\\tilde{r}_{t'}$ and the internal error estimate weights the new observation $o_t$.\n\n\\textbf{Important:} The above weights are reversed from what might be initially expected. This is appropriate because a higher variance term is resulting in a \\emph{higher} weight, which is the opposite of what we would want, so we switch the weights. The higher our estimation of our internal error, the more weight we put on the new observation. The higher the observation noise, the more weight we put on our previous estimates.\n\nThe agent must also finish updating its internal error estimate for use the next time it wakes:\n$$\\tilde{\\sigma}^2_t=\\frac{\\sigma^2_n\\tilde{\\sigma}^2_{t'}}{\\sigma^2_n+\\tilde{\\sigma}^2_{t'}}$$\nSimilarly to applying the new observation to $\\tilde{r}_t$, here we apply one ``step'' of the observation noise $\\sigma^2_n$ to our internal error estimate $\\tilde{\\sigma}^2_t$. Making an observation of course tends to reduce our internal error estimate (or increase our ``certainty'') but with higher observation noise, this effect is diminished.\n\nThe agent now has updated estimates $\\tilde{r}_t$ and $\\tilde{\\sigma}^2_t$ for the current fundamental value and its internal error estimation. At last, it can compute $\\hat{r}_t$, the final fundamental value $r_T$ as estimated at current time $t$. Having no future observations (past time $t$) it can only advance time as before:\n$$\\hat{r}_{t} \\leftarrow (1-(1-\\kappa)^{T-t})\\bar{r} + (1-\\kappa)^{T-t} \\tilde{r}_{t}$$\nThis matches the previous ``time advancement'' process with $\\delta=T-t$, the number of time steps remaining in the simulation. Again the agent must assume the random perturbations $u_t$ take on their mean value of zero.\n\nNow the agent can use its estimate of the final fundamental value (or what the closing price \\emph{should} be) to make its trading decisions.\n\nNote that the above estimation method is not substantially different when the fundamental follows the OU process, because the two processes map very closely.\n\n\\section*{Agent Preferences}\n\nIt is important that each agent have individual preferences for holding certain quantities of stock, as we would want each agent to behave differently even without forcing arbitrary randomness into the decision process. In the real world, independent of any ``consensus value'' for INTC, some people will just \\emph{really} want to own Intel. Here we describe the method as put forth in Wah and Wellman. \\cite{wah2017welfare}\n\nThese preferences are codified as a vector of \\emph{incremental private values} placed on the acquisition or release of one additional unit of stock, given the agent's current holdings. If $q_{max}$ is the holding limit, then the preferences for trading agent $i$ are the elements $\\theta^q_i$ in:\n$$\\Theta_i=(\\theta^{-q_{max}+1}_i,\\dots,\\theta^0_i,\\theta^1_i,\\dots,\\theta^{q_{max}}_i)$$\nwhere $q$ is the quantity of stock currently held. $\\Theta_i$ is drawn randomly from $\\mathcal{N}(0,\\sigma^2_{PV})$, where $\\sigma^2_{PV}$ is a selected experimental parameter. The values are sorted in descending order, ensuring that each additional unit of stock acquired is valued less than the one before it.\n\nFor example, consider a private value vector for some particular agent:\n$$\\Theta^q=(0.5,0.3,0.2,0.1,-0.2,-0.4)$$\nfor $q\\in[-2,3]$, and assume this agent's $\\hat{r}_t$ estimate of the final fundamental value is $100$. Since each agent's total valuation for its $q\\textsuperscript{th}$ unit of stock is $\\hat{r}_t + \\theta^q$, we can compute the following total valuations:\n$$V^q=(100.5,100.3,100.2,100.1,99.8,99.6)$$\n\nIf the agent currently holds zero units of stock, it would pay (\\emph{see caveat in next paragraph}) \\$100.10 to go long one unit, or demand \\$100.20 to go short one unit. If the agent is currently short two units, it would pay \\$100.30 to buy back a unit, or demand \\$100.50 to sell another unit. If the agent is long two units, it would pay \\$99.60 to buy a third unit, or require \\$99.80 to sell a unit.\n\nIt is this total valuation of a unit (incremental private preference plus estimated final fundamental value) that governs each trader's limit prices on its orders. However an agent should not bid\/offer \\emph{exactly} the limit prices listed above, as that would result in zero surplus (gain) versus its valuation. Instead it will \\emph{shade} its bids lower or offers higher to ensure a positive surplus if the order is filled.\n\n\\section*{Zero Intelligence Agent}\n\nThe Zero Intelligence (ZI) agent has been a part of the financial literature from at least 1993, when Gode and Sunder studied the allocative efficiency of constrained or unconstrained ZI agents with that of a population of human traders, finding that efficiency arose as a consequence of market structure rather than human intelligence or motivation. \\cite{gode1993allocative} Today, the term Zero Intelligence (or Zero Intelligence Plus) is extended to more complex agents, so long as they place orders priced substantially at random, at times selected substantially at random, and do not possess significant memory, intelligence, or visibility into the order stream.\n\nHere we again describe the particular flavor of ZI agent used in Wah and Wellman. \\cite{wah2017welfare} We note that these agents are not \\emph{strictly} zero intelligence, because they (1) do observe the current best bid or ask as part of a final rejection process when placing an order and (2) the same agent arrives at the market multiple times, versus only once in ``classic'' ZI.\n\nUpon waking, a ZI trader cancels any outstanding order and places a new, single-unit limit order with equal probability to buy or sell.\n\nThe total valuation used by a ZI trader for stock unit $q$ is always as explained in the previous section: $\\hat{r}_t+\\theta^q$. If the ZI agent currently holds $q$ units, then it could contemplate selling unit $q$ with valuation $\\hat{r}_t+\\theta^q$ or buying unit $q+1$ with valuation $\\hat{r}_t+\\theta^{q+1}_i$.\n\nTo ensure a surplus (gain) from the trade, the agent must \\emph{shade} its limit prices away from this valuation, as pricing at the valuation would produce zero net gain even if the order is executed. Bids are shaded downward (bid less than your valuation) and asks are shaded upward (ask more than your valuation).\n\nCall the trader's total valuation for the share in question $v$. The range $[R_{\\mathrm{min}},R_{\\mathrm{max}}]$ represents the minimum and maximum surplus (gain) the agent will demand on a filled order. The agent draws a random number $R$, called the \\emph{requested surplus}, from that range. $R$ is the difference between the agent's total valuation of the stock unit and the ``shaded'' limit price it actually offers. The agent will gain $R$ surplus if the order trades, or nothing if it does not. Thus the limit price is $v-R$ if buying, and $v+R$ if selling.\n\nThis particular (Wah and Wellman \\cite{wah2017welfare}) agent also has a specified experimental parameter $\\eta\\in[0,1]$ called the ``strategic threshold''. If an agent can immediately secure an executed surplus (gain) of $\\eta R$ by taking the current best bid or offer, it will do that instead of placing the order previously described.\n\n\\textbf{Shortcoming:} Note that the ZI agent selected a ``requested surplus'' $R$ at random with some fixed parameter range. This may well result in its order not being executed at all (too far from the spread). \\emph{This} is the problem solved by the Heuristic Belief Learner (HBL) agent, which improves overall performance by estimating the likelihood that any particular limit price will successfully trade, therefore allowing it to bid less or ask more \\emph{without} too badly damaging the odds that the order will be filled.\n\n\\section*{Example of ZI Limit Price Determination}\n\nHere we extend the private and total valuations example from the \\emph{Agent Preferences} section, using private value vector:\n$$\\Theta^q=(0.5,0.3,0.2,0.1,-0.2,-0.4)$$\nand total value vector:\n$$V^q=(100.5,100.3,100.2,100.1,99.8,99.6)$$\nfor $q\\in[-2,3]$, and assuming $\\hat{r}_t=100$ as before.\n\nAn agent holds (long) $q=1$ unit of stock. It has been randomly selected to buy on this market arrival. The agent estimates the final fundamental at 100. Its private valuation for stock unit $q+1=2$ is $-0.20$, thus its total valuation for stock unit 2, buying one unit while already holding one unit, is 99.8.\n\nAssume $R_{\\mathrm{min}}=0.1$ and $R_{\\mathrm{max}}=0.5$. This means the agent will place orders with limit prices chosen to achieve at least 10 cents gain (for one unit) but no more than 50 cents gain relative to its total valuation of the stock unit. Assume the agent randomly selects $R$ to be 0.25 this time. Since the agent is buying, its BID limit price will be $99.8-0.25=99.55$, which will net a gain of 0.25 if the order is filled.\n\nThe strategic threshold $\\eta$ \\emph{could} cause the agent not to place the above order. Assume $\\eta=0.5$, meaning the agent will accept one-half of its desired gain if the order can be executed immediately. If the agent can get at least $\\eta R=0.5(0.25)=0.125$ by accepting the current best offer, it will do that. In this example, if the best offer is at most 99.67, the agent will place a bid to trade with that offer. Otherwise, it will place the bid with limit price 99.55 as planned.\n\n\\section*{Heuristic Belief Learning Agent}\n\nGD \\cite{gjerstad1998price}, named for its authors Gjerstad and Dickhaut, is at the root of a family tree of strategies that maintain a belief function representing the probability that a particular auction bid will be accepted based on its price. Heuristic Belief Learning (HBL) is the name used for a generalized form of GD presented later by Gjerstad. \\cite{gjerstad2007competitive} Here we present HBL as described in Wang and Wellman. \\cite{wang2017spoofing}\n\nThese HBL agents work exactly like the previously described ZI agents (having private preferences, arriving according to a Poisson distribution, estimating the final fundamental value, etc) except as noted herein.\n\nUsing its belief function (likelihood of bid success conditioned on price), an HBL agent places the bid that maximizes expected surplus, which accounts for both the surplus \\emph{if} the bid is accepted and the likelihood the bid will \\emph{be} accepted. This belief function is constructed from an accurately observed history of all bids (accepted or rejected) leading to the last $L$ (memory length) transactions. In the context of a more complex market with an order book that allows delayed execution and order cancellation, we can instead estimate the probability that a limit order will be successfully transacted within some maximum time period based on its limit price.\n\n\\textbf{Important:} This belief function has \\emph{nothing} to do with price prediction, \\emph{nothing} to do with the ``true value'' of an item, and \\emph{nothing} to do with the profitability of a transaction. It simply asks: ``If I were to place a bid at this price, what are the odds I would receive the item I bid on?''\n\nGD and HBL are considered heuristic agents because, as we will see, their estimation of bid success probability uses an arbitrary (but intuitive) formulation.\n\n\\section*{Intuitive Formulation of HBL Success Likelihood Estimation}\n\nAt any given time, the HBL agent can use its memory of observed limit orders and their results to estimate the probability of a successful transaction given any observed price.\n\nIf the HBL wished to buy, we could construct a simple ratio of successful to unsuccessful bids at exactly the proposed limit price $p$, but this ratio is unbounded and we want a probability. Normalizing the denominator restricts the range to $[0,1]$ and permits our desired interpretation:\n$$Pr(p)=\\frac{S_p}{S_p+U_p}$$\nwhere $S_p$ is the number of successful bid orders at price $p$, and $U_p$ is the number of unsuccessful bid orders at price $p$. The agent thus asks: ``Of all bids at price $p$, what proportion were successful?''\n\nThis approach has an obvious limitation: data sparsity. If there have been ten unsuccessful bids at $p=5$, ten successful bids at $p=7$, and ten successful bids at $p=9$, the value $Pr(8)=\\frac{0}{0+0}$ is undefined. If we extend $S$ and $U$ to include bids equal or less than $p$:\n$$Pr(p)=\\frac{S_{\\leq p}}{S_{\\leq p}+U_{\\leq p}}$$\nthe agent now asks: ``Of all bids at or below price $p$, what proportion were successful?''. Now $Pr(8)=\\frac{10}{10+10}=0.5$, which is at least \\emph{something}.\n\nIncluding unsuccessful bids at prices \\emph{lower} than our proposed price is unhelpful, though. If we are offering 8, why should failed bids at 5 affect our estimation of success? $Pr(8)=0.5$ seems unrealistic; \\emph{all} bids at $p=7$ were \\emph{accepted}!\n\nInstead, we should evaluate successful bids at or below $p$, but unsuccessful bids at or \\emph{above} $p$. The agent now considers two different factors: ``How many bids succeeded offering no more than this?'' and ``How many bids failed \\emph{despite} offering at least this much?''\n$$Pr(p)=\\frac{S_{\\leq p}}{S_{\\leq p}+U_{\\geq p}}$$\nWe now appropriately disregard the failed bids at 5, because we are offering more than that, and estimate $Pr(8)=\\frac{10}{10+0}=1$, because we have never observed the failure of a bid with a price of at least 8.\n\nThere is still one problem. Orders in our type of auction can be cancelled. Consider the ten unsuccessful bids at $p=5$. Were they unsuccessful because there has never been an ask at $p<=5$? Or were they merely unlucky, and there have been offers that low, but not at the same time as those bids? In the second case, our estimation of $Pr(5)$ should be greater than zero, because although no bid succeeded, one clearly \\emph{could} have.\n\nOur agent also has limited memory. What if it should happen that within this memory, there were no bids at or below $p$ (successful or not), but there were ask orders below $p$? Our heuristic might end up zero or undefined, when again, the chance of fulfillment is near certain due to the volume of unmatched ask orders.\n\nGenerally speaking then, the volume of asks in our considered price range ($<=p$) should be a factor. In addition to considering bids and their success rate, the more shares that have been \\emph{available} at or below $p$, the more likely we are to succeed with a bid at $p$. The fewer shares that have been available, the less likely we are to succeed.\n\nThus we arrive at the final formulation for the HBL agent's heuristic estimation of success, with this rough interpretation: ``Seeing ask order volume, or successful bid orders, at prices no more than $p$ increases the likelihood of success. Seeing unsuccessful bid orders at prices of at least $p$ decreases the likelihood of success.'' The ratio is formalized and normalized as:\n$$Pr(p)=\\frac{A_{\\le p}+S_{\\leq p}}{A_{\\le p}+S_{\\leq p}+U_{\\geq p}}$$\nwhere $A_{\\leq p}$ is the total number of ask orders at price $p$ or lower, $S_{\\leq p}$ is the number of successful bid orders at price $p$ or lower, and $U_{\\geq p}$ is the number of unsuccessful bid orders at price $p$ or higher. The heuristic calculation is exactly mirrored when the agent considers selling a unit.\n\nThe agent can now estimate the probability of success for any candidate limit order price $p$. If it is desired to estimate probability for all possible prices without bound, cubic spline interpolation can be employed as suggested by Wang and Wellman. \\cite{wang2017spoofing}\n\n\\newpage\n\n\\section*{HBL Considerations and Strategy}\n\nIn a market with an order book, orders are almost never rejected. They simply enter the order book to await a later match. The HBL, though, depends on the idea of rejected or unsuccessful orders. A simple approach to this problem would be to place a time limit on each order. If the order is not matched within this time limit, it is considered unsuccessful.\n\nThe approach taken by Wang and Wellman is a little more complex. \\cite{wang2017spoofing} They define a grace period relative to the frequency of trader arrivals (the Poisson distribution mentioned earlier). An order is considered rejected if it existed in the order book for a length of time greater than the grace period. All orders not accepted immediately are considered ``partially rejected'' and assigned a fractional weight depending on the length of time they spent in the order book.\n\nThe HBL agent selects the limit price that maximizes \\emph{expected surplus}, where surplus is simply the difference between the limit price $p$ and the agent's total valuation of a share (see ``Agent Preferences''), and expected surplus is the same, weighted by the probability of successful transaction computed in the previous section.\n\nFor example, an HBL agent contemplating buying unit $q+1$ would consider total valuation $\\hat{r}_t+\\theta^{q+1}_i$, limit price $p$, and estimation of successful transaction $Pr(p)$, and select its optimal bid $p^*$ as:\n$$p^*=\\mathrm{argmax}_p(\\hat{r}_t+\\theta^{q+1}_i-p)Pr(p)$$\n\nAnytime the HBL agent does not have enough information to enact the above strategy, it temporarily behaves as a ZI agent instead.\n\n\\textbf{Limitation:} HBL assumes the agent can observe a complete order stream from all agents, including order execution (even if delayed), cancellation, and the length of time each order was ``alive'' in the order book. This may not be practical in many environments.\n\n\\section*{HBL Limit Price Example}\n\nHere we reuse the private and total valuations example from the \\emph{Example of ZI Limit Price Determination} section, using private value vector:\n$$\\Theta^q=(0.5,0.3,0.2,0.1,-0.2,-0.4)$$\nand total value vector:\n$$V^q=(100.5,100.3,100.2,100.1,99.8,99.6)$$\nfor $q\\in[-2,3]$, and assuming $\\hat{r}_t=100$ as before.\n\nAn agent is short one unit of stock ($q=-1$). It has been randomly selected to buy on this market arrival. The agent estimates the final fundamental at 100. Its private valuation for stock unit $q+1=0$ is $0.20$, thus its total valuation for stock unit 0, buying one unit while being short one unit, is 100.2.\n\nAssume the HBL agent has access to the following order memory of length $L=4$, with superscripts denoting pairs of matched orders, and that no orders were cancelled:\n\\begin{center}\n\\begin{tabular}{c|D{.}{.}{-1}|c|D{.}{.}{-1}|c|D{.}{.}{-1}|c|D{.}{.}{-1}}\n\\multicolumn{2}{c|}{Transaction 1} & \\multicolumn{2}{c|}{Transaction 2} & \\multicolumn{2}{c}{Transaction 3} & \\multicolumn{2}{c}{Transaction 4} \\\\\n\\hline\nASK & 100.0^1 & ASK & 100.2^2 & BID & 100.1 & BID & 100.2 \\\\\nBID & 99.8 & BID & 100.0 & ASK & 99.9^3 & ASK & 100.4 \\\\\nASK & 100.3^4 & ASK & 100.3 & & & BID & 100.4^4 \\\\\nBID & 99.6 & BID & 100.1^3 & & \\\\\nBID & 100.0^1 & BID & 100.2^2 & & \\\\\n\\end{tabular}\n\\end{center}\n\nThe agent can compute the estimated probability of transaction for a proposed limit price $p$ as described in the previous section:\n\\begin{equation*}\n \\begin{aligned}\n Pr(100.5)&=\\frac{6+4}{6+4+0}=1 \\\\\n Pr(100.4)&=\\frac{6+4}{6+4+0}=1 \\\\\n Pr(100.3)&=\\frac{5+3}{5+3+0}=1 \\\\\n Pr(100.2)&=\\frac{3+3}{3+3+1}=0.86 \\\\\n Pr(100.1)&=\\frac{2+2}{2+2+2}=0.67\n \\end{aligned}\n \\hspace{1cm}\n \\begin{aligned}\n Pr(100.0)&=\\frac{2+1}{2+1+3}=0.50 \\\\\n Pr(99.9)&=\\frac{1+0}{1+0+3}=0.25 \\\\\n Pr(99.8)&=\\frac{0+0}{0+0+4}=0 \\\\\n Pr(99.7)&=\\frac{0+0}{0+0+4}=0 \\\\\n Pr(99.6)&=\\frac{0+0}{0+0+5}=0\n \\end{aligned}\n\\end{equation*}\n\nNow the agent can use its total valuation, the proposed limit price, and the estimated probability of transaction to compute expected surplus $\\mathbb{E}(s)$:\n\\begin{equation*}\n \\begin{aligned}\n \\mathbb{E}(s|p=100.5)&=(100.2-100.5)(1)= -0.3 \\\\\n \\mathbb{E}(s|p=100.4)&=(100.2-100.4)(1)= -0.2 \\\\\n \\mathbb{E}(s|p=100.3)&=(100.2-100.3)(1)= -0.1 \\\\\n \\mathbb{E}(s|p=100.2)&=(100.2-100.2)(0.86)= 0 \\\\\n \\mathbb{E}(s|p=100.1)&=(100.2-100.1)(0.67)= 0.067 \\\\\n \\mathbb{E}(s|p=100.0)&=(100.2-100.0)(0.5)= 0.1 \\\\\n \\mathbb{E}(s|p=99.9)&=(100.2-99.9)(0.25)= 0.075 \\\\\n \\mathbb{E}(s|p=99.8)&=(100.2-99.8)(0)= 0 \\\\\n \\mathbb{E}(s|p=99.7)&=(100.2-99.7)(0)= 0 \\\\\n \\mathbb{E}(s|p=99.6)&=(100.2-99.6)(0)= 0\n \\end{aligned}\n\\end{equation*}\n\nIntuitively, the agent can see that any price above \\$100.2 would almost certainly be accepted, but the agent would lose money relative to its valuation of the stock unit. The agent would love to achieve the surpluses associated with prices less than \\$99.9, but it can guess that those bids would never transact. Only the prices $p\\in [99.9,100.1]$ have a nonzero chance of transaction \\emph{and} a positive surplus if transacted.\n\nIn this example, the agent will place a limit buy order at limit price $p=100.0$, the price that produces the maximum expected surplus of \\$0.10.\n\n\n\\section*{The ABIDES Simulator}\n\nOur simulation platform, ABIDES (Agent-Based Interactive Discrete Event Simulation) provides the fundamental value series and agents discussed herein ``out of the box'', including the novel ``Megashock OU Fundamental''. \\cite{byrd2019abides} It also provides the ability to use \\emph{real} market data as the fundamental value series. \\cite{balch2019evaluate} The ABIDES platform is available under a BSD-style license at \\url{https:\/\/github.com\/abides-sim\/abides}. The distribution includes relevant example configurations which are further explained in the project's wiki at \\url{https:\/\/github.com\/abides-sim\/abides\/wiki}.\n\nThe ABIDES simulator uses the OU process (plus megashock events) to provide a ``sparse discrete'' fundamental process that provides extremely fine time resolution (nanoseconds) while still permitting quick simulation of reasonable time scales (days) due to its ability to completely skip computation of time periods during which nothing happens. The Megashock OU fundamental additionally produces intraday time series that more closely resemble typical real-world intraday stock charts.\n\nThe ABIDES implementation of Zero Intelligence agents reproduces the strategy described herein, including the strategic threshold parameter. The implementation of Heuristic Belief Learning agents treats an order as successful if any part of it is transacted within the observed order stream, and unsuccessful otherwise.\n\n\\section*{Acknowledgements}\nThis material is based on research supported in part by the National Science Foundation under Grant no. 1741026, and by a JPMorgan AI Research Fellowship.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzdttx b/data_all_eng_slimpj/shuffled/split2/finalzzdttx new file mode 100644 index 0000000000000000000000000000000000000000..e0b75a7445ade6b57ec96e11f6e0159095df90df --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzdttx @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe purpose of this article is to initiate the study of the slice Dirac operator over octonions. The Dirac operator for quaternions\n \\begin{equation}\\label{Dirac1}\n D=\\frac{\\partial}{\\partial x_0}+i \\frac{\\partial}{\\partial x_1}+j \\frac{\\partial}{\\partial x_2}+k \\frac{\\partial}{\\partial x_3},\n \\end{equation}\nhas its root in mathematical physics, quantum mechanics, special relativity, and engineering (see \\cite{A_1, Adler_1, Girard_1}) and it plays a key role in the Atiyah-Singer index theorem (ref. \\cite{Bismut_1}). It may be called Dirac operator since it factorizes the $4$-dimensional Laplacian. However, we note that in the literature \\eqref{Dirac1} is often called generalized Cauchy-Riemann operator or Cauchy-Fueter operator, see e.g. \\cite{Brackx_1, Gurlebeck_1, Sudbery_1}, even though it has been originally introduced in a paper by Moisil, see \\cite{Moisil}.\n\nBased on the Dirac operator for quaternions in \\eqref{Dirac1}, we shall introduce what we call the slice Dirac operator over octonions, using the slice technique.\nThis technique was used by Gentili and Struppa for quaternions in \\cite{Gentili_1, Gentili_2} and for octonions in \\cite{Gentili_3} based on Cullen's approach \\cite{Cullen}.\nThis technique makes it possible to extend some properties of holomorphic functions in one complex variable to the high dimensional and non-commutative case of quaternions.\nIt has found significant applications especially in operator theory \\cite{Alpay_1, CSS_book, Colombo_5}, differential geometry \\cite{Gentili_4}, geometric function theory \\cite{Ren_1, Ren_2} and it can be generalized to other higher dimensional settings like Clifford algebras \\cite{Colombo_1, Colombo_2} and real alternative algebras \\cite{Ghiloni_1, Ghiloni_2, Ghiloni_5, Ren_3}.\n\nThe heart of the slice technique comes from the slice structure of quaternions $\\mathbb H$, namely the fact that $\\mathbb H$ can be expressed as union of complex half planes as\n$$ \\mathbb H=\\bigcup_{I \\in \\mathbb S}\\mathbb C_I^+,$$\nwhere $ \\mathbb S $ denotes the set of imaginary unit in $ \\mathbb H $, and $ \\mathbb C_I^+ $ is the upper half plane\n$$ \\{ x+yI:x\\in \\mathbb R, y\\ge 0 \\} .$$\nFrom this decomposition, it is then natural to say that quaternions have a \\textit{book structure} since\n $ \\mathbb C_I^+ $ plays the role of a page in a book for any $ I \\in \\mathbb S $. The real axis $ \\mathbb R $ plays the role of the edge of the book in which all the pages of the book intersect, i.e.,\n$$\\mathbb C_I^+ \\cap \\mathbb C_J^+=\\mathbb R$$\nfor any $I\\neq J$.\n\n\nThe book structure for quaternions plays the same role as the sheaf or fiber bundle structure in differential geometry.\n\nIt is remarkable that the topology in the book structure is no longer the Euclidean one.\nIndeed, the distance compatible with the topology is given by the Euclidean one in a plane, otherwise the distance between any two points from distinct half planes is measured through the path of light via the real axis.\n\n\nFollowing the Fueter's construction \\cite{Fueter_1}, see also \\cite{Sce}, when one considers an open set $O$ in the upper half complex plane $\\mathbb C^+$ minus the real line and a holomorphic function $f(x+\\iota y)=F_1(x,y)+\\iota F_2(x,y)$, one may define a function defined over the quaternions using the book structure. In fact, if we consider $q=x+Iy$, $y>0$, for some suitable $I$, we may set $f(q)=f(x+ I y)=F_1(x,y)+I F_2(x,y)$. Note that $\\bar q=x-Iy$, $y>0$ and so, by definition,\n$f(\\bar q)=f(x -I y)=F_1(x,y)-I F_2(x,y)$. Note also that the pair $(F_1,F_2)$ satisfies the Cauchy-Riemann system and thus $f(x+Iy)$ is in the kernel of the Cauchy-Riemann operator $\\partial_x+I\\partial_y$. If one is willing to extend the definition to the points of the real line, there is a problem since if $q\\in\\mathbb R$ then $q=x+I0$ and the imaginary unit $I$ is no longer unique.\n\n To solve this problem, one may consider a weaker notion of book structure and observe that\n$$ \\mathbb H=\\bigcup_{I \\in \\mathbb S}\\mathbb C_I,$$\nin other words, we may consider $\\mathbb H$ as the union of complex planes.\n\n\nFollowing a slight modification of the Fueter's construction, see \\cite{Qian},\n let $O$ an open set symmetric with respect to the real axis (possibly intersecting the real axis) and a holomorphic function $f(x+\\iota y)=F_1(x,y)+\\iota F_2(x,y)$. If $F_1, F_2$ are an even-odd pair in the second variable, namely if they satisfy\n\\begin{equation}\\label{eq:odd-even}\\begin{cases}\nF_1(x+iy)=F_1(x-iy)\n\\\\\nF_2(x+iy)=-F_2(x-iy)\n\\end{cases} \\forall\\ x,y\\in \\mathbb R,\n\\end{equation}\nwe may define a function on an open set in $\\mathbb H$ (suitably constructed using $ O$)\nNote that these conditions immediately imply that\n$$\nf(x+Iy)=f(x+(-I)(-y))\n$$\nso that $f$ is well defined. Moreover, the fact that $F_2$ is odd in the second variable imply that $F_2(x,0)=0$, thus $f$ is well defined also at real points.\nThis second approach is the one that we will generalize to the octonionic case.\n\nTo this end, we set\n$$F\\equiv\\left(\n\\begin{array}{c}\nF_1\\\\\nF_2\\\\\n\\end{array}\\right), \\qquad z=x+iy\\equiv\\left(\n\\begin{array}{c}\nx\\\\\ny\\\\\n\\end{array}\\right),\n$$\nand we consider\n\\begin{equation*}\ng=\\left(\n\\begin{array}{cc}\n1&0\\\\\n0&-1\\\\\n\\end{array}\\right)\\in O(1)\n\\end{equation*}\nwhere $O(1)$ is identified with the group of matrices $\\left\\{\\left(\n\\begin{array}{cc}\n1&0\\\\\n0&1\\\\\n\\end{array}\\right), \\left(\n\\begin{array}{cc}\n1&0\\\\\n0&-1\\\\\n\\end{array}\\right)\\right\\}$.\nThen\nwe have\n\\begin{equation*}\ngz=\\left(\n\\begin{array}{c}\nx\\\\\n-y\\\\\n\\end{array}\\right), \\qquad\n g^{-1}=\\left(\n\\begin{array}{cc}\n1&0\\\\\n0&-1\\\\\n\\end{array}\\right),\n\\end{equation*}\n\\\\\nso (\\ref{eq:odd-even}) can be rewritten as\n\\begin{equation}\n \\left(\\begin{array}{c}\nF_1(z)\\\\\nF_2(z)\\\\\n\\end{array}\\right)\n=\ng^{-1} \\left(\\begin{array}{c}\nF_1(gz)\\\\\nF_2(gz)\\\\\n\\end{array}\\right).\n\\end{equation}\nThus, following \\cite{Rin}, we impose that\n$$ F(z)=g^{-1}F(gz),\\quad \\forall\\ g \\in O(1), $$\nand any $F$ satisfying this condition is called {\\em stem function}.\n\n\n\nWe can regard at this construction as the commutative stem function theory since $ F $ is invariant under the commutative group $ O(1) $.\n\nAs we shall see, the significant property of the slice regular function in $\\mathbb H$ (non-commutative counterpart of holomorphic functions, i.e. holomorphic maps depending on the parameter $I\\in\\mathbb S$) is given by\n the representation formula, which demonstrates that any slice regular function is completely determined by its evaluation at any two distinct half planes, or {\\em pages} in this description.\n\n\n\nIn order to extend the slice theory for the Cauchy-Riemann operator over quaternions into the slice theory for the slice Dirac operator over octonions, we need to introduce a modified theory of stem functions. It turns out that the corresponding notion of stem function is invariant under the non-commutative group $ O(3) $.\nIt will result in a new form of the representation formula, expressed in term of a quaternionic matrix.\n\nWe point out that the non-commutative and non-associative setting of octonions has found significant applications in the universal model of $M$-theory, in which the universe is given by the product of the $4$-dimensional Minkowski space with a $G_2$-manifold of very small scale.\nHere the exceptional Lie group $G_2$ is an automorphism group of octonions (ref. \\cite{Baez_1, Grigolian_1}).\n\n\n\n\n We conclude this introduction with a remark about our definition of intrinsic and stem functions. Rinehart \\cite{Rin} studied the intrinsic functions as self-mappings of an associative algebra. In contrast, our intrinsic functions have distinct dimensions for their definition and target domains, and are constructed in the non-associative setting;\n see also \\cite{DSce, Sce}. Fueter \\cite{Fueter_1} initiated the study of stem functions for complex-valued functions in his construction of radially holomorphic functions on the space of quaternions; see \\cite{Ghiloni_1} for its recent development. However, their considerations are all restricted to the commutative $O(1)$ setting. In this paper we initiate the study in the non-commutative $ O(3)$ setting. It is interesting to note that the procedure we followed in this paper may lead to further generalizations to higher dimensional algebras.\n\nThe structure of this paper is as follow:\nin section 2, we recall some important properties of octonionic algebra $ \\mathbb O $.\nIn section 3, we introduce the book structure in the octonionic algebra in terms of quaternionic subspaces and the stem function for the non-commutative group $ O(3)$; we also provide the representation formula which can be written via a quaternionic matrix.\nIn section 4 we introduce the slice Dirac operator and a splitting property for slice Dirac functions. Section 5 contains the Cauchy-Pompeiu integral formula for slice functions and the Cauchy integral formula for slice Dirac-regular functions.\nFinally, in section 6 we give the expansion of slice Dirac-regular as Taylor series as well as Laurent series.\n\n\n\n\\section{The algebra of octonions}\nThe algebra of octonions $ \\mathbb O $ is a real, alternative, non-commutative and non-associative division algebra (see for example \\cite{Schafer_1}).\nIt is isomorphic to $ \\mathbb R^8 $ as a real inner product vector space and it can be equipped with the standard orthogonal basis : $ e_0=1,e_1,\\dots,e_7 $.\n\nThe multiplication between elements in the basis $ e_0,e_1,\\dots,e_7 $ is defined by\n$$ e_ie_j=-\\delta_{ij}+\\varepsilon_{ijk}e_k, \\ \\ \\ \\forall\\ i, j, k \\in \\{ 1,2,\\dots,7 \\}. $$\nHere $ \\delta_{ij} $ is the Kronecker symbol and\n\\[ \\varepsilon_{ijk}=\n\\begin{cases} (-1)^{\\sigma(\\pi)} &\\textrm{if}\\ \\ (i, j, k) \\in \\pi(\\Sigma),\n\\\\\n0 & \\textrm{otherwise}.\n\\end{cases} \\]\nwhere $ \\pi $ is a permutation, $\\sigma(\\pi)$ its sign, and\n $$ \\Sigma=\\{ (1, 2, 3), (1, 4, 5), (2, 4, 6), (3, 4, 7), (5, 3, 6), (6, 1,7), (7, 2, 5) \\}. $$\n\n\n\n\nThe octonionic algebra $ \\mathbb O $ also can be generated from the quaternions algebra $ \\mathbb H $ by the famous Cayley-Dickson process.\nLet $ \\{e_0=1, e_1, e_2, e_3:=e_1e_2 \\} $ be a basis of $ \\mathbb H $. Then\nevery $ x \\in \\mathbb O $ can be expressed as $ x=a+e_4b $, where $ a, b \\in \\mathbb H $, and $ e_4 $ is a fixed imaginary unit in $ \\mathbb O $ not belonging to $\\mathbb H=\\{e_0,e_1,e_2,e_3\\}$.\nThe addition and multiplication are defined as follow: for any $ x=a+e_4b, \\ y=c+e_4d \\in \\mathbb O$,\n\\[ x+y:=(a+c)+e_4(b+d), \\]\n\\[ xy:=ac-d\\overline b+e_4(\\overline a d + b c ). \\]\nThis means the following relations hold:\n\\begin{lemma}\\label{ee}\nLet $ a,b,e_4 $ as above, then\n\\begin{enumerate}\n\\item $a(e_4b)=e_4(\\overline a b)$,\n\\item $(e_4a)b=e_4(ba)$,\n\\item $(e_4a)(e_4b)=-b\\overline a$.\n\\end{enumerate}\n\\end{lemma}\n\nThe two definitions above of the octonionic algebra $ \\mathbb O $ are equivalent by setting $$ e_5:=e_4e_1,\\quad e_6:=e_4e_2, \\quad e_7:=e_4e_3. $$\n\n\nEvery $ x \\in \\mathbb O $ can be written as\n$$ x=x_0+\\sum_{k=1}^7e_k x_k, \\qquad\\forall\\\n x_k\\in\\mathbb R. $$\nWe can introduce its conjugate $$ \\overline x:=x_0-\\sum_{k=1}^7e_k x_k,$$\nand then set $$|x|^2:=x \\overline x=\\sum_{k=0}^7x_k^2. $$\nThe modulus is multiplicative, i.e.\n$$ |xy|=|x||y|, \\qquad \\forall\\ x,\\ y \\in \\mathbb O.$$\n\nIn the sequel, given $ x \\in \\mathbb O $ we introduce a left multiplication operator $ L_x: \\mathbb O \\to \\mathbb O $, defined as\n$$ L_x z= x z, \\quad \\forall\\ z \\in \\mathbb O. $$ In general, for any $ x,\\ y \\in \\mathbb O, \\ L_xL_y \\ne L_{xy} $, but equality may hold when suitable assumptions hold:\n\n\\begin{theorem}(Artin theorem, \\cite{Schafer_1})\\label{Artin}.\nThe subalgebra generated by any two elements of an alternative algebra is associative.\nIn particular, for all $ r \\in \\mathbb R $, and for all $ x \\in \\mathbb O $.\n\\begin{enumerate}\n\\item $ L_xL_{\\overline x}=L_{x\\overline x} $,\n\\item $ L_rL_x =L_{rx} $.\n\\end{enumerate}\n\\end{theorem}\nIt is also useful to recall the following well known result:\n\\begin{theorem}(Moufang Laws, \\cite{Schafer_1}).\nLet $ x, y, z \\in \\mathbb O $, then\n\\begin{enumerate}\n\\item $L_{xyx}z=L_xL_yL_x z$,\n\\item $L_zL_{xy}x=L_{zx}L_yx$,\n\\item $L_xL_{yz}x=L_{xy}L_z x$.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Stem function in the octonionic setting}\nLet $ \\mathbb O $ be the algebra of octonions. The set of its imaginary units is a sphere of dimension six\n$$\\mathbb S^6 := \\{ x \\in \\mathbb O:x^2=-1 \\}.$$\nLet\n$$ \\mathbb I:=(1,I,J,K)\\in \\mathbb O^4,$$\nwith\nthe triple $I,J, K$ satisfying\n$$I, J \\in \\mathbb S^6, \\qquad I \\perp J, \\qquad K=IJ. $$\nThe set of all such row vectors $\\mathbb I$ is denoted by $ \\mathcal N$.\nFor any $ \\mathbb I:=(1,I,J,K)\\in \\mathcal N,$\nwe consider the algebra of quaternions generated by it, i.e.,\n$$\\mathbb H_{\\mathbb I}=\\textrm{span}_{\\mathbb R}\\{1,I,J,K\\}.$$\nWe can endow the octonionic algebra with a structure that we still call \\textit{book} structure\n$$ \\mathbb O=\\bigcup_{\\mathbb I \\in \\mathcal N} \\mathbb H_{\\mathbb I}, $$\nas we prove in the following result:\n\n\\begin{proposition}\\label{bookOct} The octonionic algebra has the structure:\n$$ \\mathbb O=\\bigcup_{\\mathbb I \\in \\mathcal N} \\mathbb H_{\\mathbb I}. $$\n\\end{proposition}\n\\begin{proof} Any $ x \\in \\mathbb O $ can be written as the sum of its real part $x_0$ and its imaginary part ${\\rm Im}(x)=\\sum_{k=1}^7e_kx_k$. Therefore, it can be further expressed as\n$$ x=x_0+Iy $$\nwith $ x_0,\\ y \\in \\mathbb R $ and $I={\\rm Im}(x)\/|{\\rm Im}(x)|$. We have that\n$$\nI^2=\\frac{1}{|{\\rm Im}(x)|^2}(\\sum_{k=1}^7e_kx_k)(\\sum_{k=1}^7e_kx_k)=-\\frac{1}{|{\\rm Im}(x)|^2}\\sum_{k=1}^7 x_k^2=-1,\n$$\nthus $I \\in \\mathbb S^6 $. Now we can choose $ J,\\ K \\in \\mathbb S^6 $\nsuch that $ \\mathbb I:=(1,\\ I,\\ J,\\ K)\\in\\mathcal N $. Hence $ x_0+ I y\\in \\mathbb H_{\\mathbb I} $.\n\\end{proof}\nWe note that, in general, any $x\\in\\mathbb O$ belongs to more than one quaternionic space, as the following\nexample shows.\n\\begin{example} Let $\\{1, e_1, \\cdots, e_7\\}$ be a standard basis of $\\mathbb O$ and consider $$x=1+e_1+e_2+e_3+e_4 .$$ According to Proposition \\ref{bookOct}, we have $x=1+I\\sqrt 4$ where $I=\\frac{1}{\\sqrt 4}(e_1+e_2+e_3+e_4)$, so $x\\in\\mathbb H_{\\mathbb I}$, $\\mathbb I=(1,\\ I,\\ J,\\ K)$ where $J$, $K$ are any two elements orthogonal to $I$ such that $\\mathbb I\\in\\mathcal N$.\nTake now $$ I'=e_1,\\ J'=\\frac{e_2+e_3+e_4}{\\sqrt{3}} ,\\ K'=I'J'=e_1\\frac{e_2+e_3+e_4}{\\sqrt{3}}.$$\nIt is easy to see that $$I',\\ J', \\ K' \\in \\mathbb S^6, \\qquad {I'}^2={J'}^2={K'}^2=-1, \\qquad J'K'=-K'J',\\quad K'I'=-I'K'. $$\nSince $x=1+I'+\\sqrt 3 J'$, we have $ x \\in \\mathbb H_{\\mathbb I'}$, where\n $ \\mathbb I:=(1,\\ I',\\ J',\\ K') $.\n\\end{example}\n\n\n\n\n\n\n Let $ O(4) $ be the group of orthogonal transformations of $ \\mathbb R^4 $, and let $ O(3) $ be its subgroup keeping\nthe real axis invariant.\n{\nTherefore, any $ g\\in O(3) $ can be regarded as matrix in the form\n\\begin{equation*}\n\\begin{aligned}\ng={\\left( \\begin{array}{cc}\n1&0\\\\\n0&P\\\\\n\\end{array} \\right)}.\n\\end{aligned}\n\\end{equation*}\nwhere $ P$ is an orthogonal transformations of $ \\mathbb R^3 $. The transformation $g: \\mathbb R^4\\longrightarrow \\mathbb R^4$ can be naturally extended to be a map\n$g: \\mathbb O^4\\longrightarrow \\mathbb O^4$ via\n\\begin{equation*}\n\\begin{aligned}\nga=\ng{\\left( \\begin{array}{c}\na_0\\\\\na_1\\\\\na_2\\\\\na_3\\end{array} \\right)}.\n\\end{aligned}\n\\end{equation*}\n for any $ a=(a_0,a_1,a_2,a_3) \\in \\mathbb O^4 $.\n}\n\n\n\\begin{definition}\nLet $ [\\Omega] $ be an open subset of $ \\mathbb R^4 $. If $ F :[\\Omega] \\to \\mathbb O^4 $ is a\n{\\em $ O(3)-$intrinsic function}, i.e. for any $ x \\in [\\Omega] $ and for any $ g \\in O(3) $ such that $ g x \\in [\\Omega] $, it satisfies\n\\begin{equation} \\label{eq:variant-1}\nF(x)= g^{-1}F(g x),\n\\end{equation}\nthen F is called an {\\em $\\mathbb O-$stem function} on $ [\\Omega] $.\n\\end{definition}\n\\begin{remark}\n{\\rm We point out that it is not reductive to assume that $[\\Omega]\\subseteq\\mathbb R^4$ is $ O(3)$-intrinsic, otherwise in the previous definition we should consider the subset $[\\Omega]'$ of $[\\Omega]$ such that $x\\in[\\Omega]'$ if and only if $ g x \\in [\\Omega]'$ for any $ g \\in O(3)$. But this is equivalent to assume that $[\\Omega]'$ is $ O(3)$-intrinsic.\\\\\n We also recall that in the quaternionic case, the stem function is complex intrinsic i.e. it is invariant under the commutative group $O(1)$. In other words, $f(\\bar z)=\\overline{f(z)}$ where $\\bar z$ denotes the complex conjugation.\nIn our setting, the stem function is intrinsic under the non-commutative group $ O(3)$ and this set is evidently non-empty since it contains,\ne.g., $F_i(x)= x_i c, \\ i=1,2,3,4 $ where $ c $ is a constant in $\\mathbb O $.}\n\\end{remark}\n\n\nWith the book structure, we can define a slice function by lifting a stem function.\nIn fact, if a $ \\mathbb O^4-$valued function $F$ defined on $ \\mathbb R^4 $ is an $ O(3)$-intrinsic function,\nthen there exists a slice function $ f:\\mathbb O \\to \\mathbb O $ such that the following diagram commutes for all $ \\mathbb I=(1, I, J, K) \\in \\mathcal N $:\n\\[\n\\xymatrix{\n\\mathbb O \\ar[r]^f &\\mathbb O \\\\\n \\mathbb R^4 \\ar [u]^{\\phi_{\\mathbb I}} \\ar [r]^F &\\mathbb O^4 \\ar [u]^{\\widetilde{\\phi}_{\\mathbb I}} }\n\\]\nwhere\n$$ \\phi_{\\mathbb I}(x)=x_0+Ix_1+Jx_2+Kx_3=:\\mathbb I x^T, \\quad \\forall\\ x=(x_0,x_1,x_2,x_3) \\in \\mathbb R^4, $$\nand\n$$ \\widetilde{\\phi}_{\\mathbb I}(y)=\\mathbb I y^T, \\quad \\forall\\ y \\in \\mathbb O^4. $$\nHere we denote by $x^T$\n the transpose of the row vector $x=(x_0,x_1,x_2,x_3)$ and similarly for $y^T$.\n\n\n\nGiven an open subset $ [\\Omega] $ of $ \\mathbb R^4 $, we consider the axially symmetric open set in $ \\mathbb O $ generated by $[\\Omega]$, defined as\n$$ \\Omega:=\\Big\\{q=\\mathbb I x^T \\in \\mathbb O : \\mathbb I \\in \\mathcal N, x \\in [\\Omega] \\Big\\}. $$\nIf $ [\\Omega] $ is a domain, then $ \\Omega $ is an axially symmetric domain.\n\nFor any $ x=(x_0,x_1,x_2,x_3) \\in \\mathbb R^4 $, we consider the three involutions\n$$ \\alpha(x)=(x_0,x_1,-x_2,-x_3), $$\n$$ \\beta(x)=(x_0,-x_1,x_2,-x_3), $$\n$$ \\gamma(x)=(x_0,-x_1,-x_2,x_3). $$\n{\nLet $\\mathbb I\\in\\mathcal N$ be fixed arbitrarily. By virtue of the identification of $\\mathbb H_{\\mathbb I}$ with $\\mathbb R^4$, the map $\\alpha$ can be identified with the map\n$$\\alpha_{\\mathbb I}: \\mathbb H_{\\mathbb I}\\longrightarrow \\mathbb O$$\ndefined by\n$$\\alpha_{\\mathbb I}(\\mathbb I x^T):=\\mathbb I \\alpha(x)^T.$$\nTo ease the notation we still write $\\alpha$ instead of\n $ \\alpha_{\\mathbb I} $.\nThe same convention is adopted in the sequel for\n $ \\beta,\\gamma, \\mathcal F, V, V_{\\alpha}, \\mathbb V, \\mathcal V, P_{\\alpha},\\mathbb P_{\\alpha}, \\mathcal P_{\\alpha} , A_{\\alpha}$, and $B_{\\alpha}$ which also depend on $ \\mathbb I \\in \\mathcal N $.\n}\n\n\\begin{definition}\nFor any open subset $ [\\Omega] $ of $ \\mathbb R^4 $,\nwe define the symmetrized set $ \\widetilde \\Omega$ as\n$$ \\widetilde \\Omega:=\\{ \\mathbb I x^T \\in \\mathbb O: \\mathbb I \\in \\mathcal N, x \\in [\\Omega] \\ \\textrm{such that} \\ \\alpha(x),\\beta(x),\\gamma(x) \\in [\\Omega] \\}. $$\n\\end{definition}\nIt is easy to check that $$ \\widetilde \\Omega \\subset \\Omega.$$\n\n\\begin{definition}\\label{slice}\nLet $\\Omega$ be an open set in $\\mathbb O$.\nAny $ \\mathbb O $-stem function $ F: [\\Omega] \\to \\mathbb O^4 $ induces a function $$ f=\\mathcal L(F): \\Omega \\to \\mathbb O $$\n defined by $$ f(q)=\\mathbb I F(x)^T$$ for any\n $ q\\in \\Omega $ with\n $ q=\\mathbb I x^T$ for some $\\mathbb I\\in\\mathcal N$. We say that $f$ is a (left) slice function (induced by $F$).\n\\end{definition}\nSince, in general, any element in $\\mathbb O$ may belong to more than one $\\mathbb H_{\\mathbb I}$ we need to prove the following:\n\\begin{proposition}\nDefinition \\ref{slice} is well-posed.\n\\end{proposition}\n\\begin{proof}\nAssume that $ q\\in\\mathbb O $ can be written in two ways as $$q=\\mathbb I x^T=\\mathbb I' {x'}^T,$$ we have to show that $$f(\\mathbb I x^T)=f(\\mathbb I^\\prime {x^\\prime}^T).$$\nWe divide the proof into various cases.\\\\\nCase 1: Assume that $\\mathbb I\\not=\\mathbb I'$ but $\\mathbb H_{\\mathbb I}=\\mathbb H_{\\mathbb I'}$. Then there exist $g\\in O(3)$ such that $\\mathbb I'=\\mathbb I g$. This means that $g{x'}^T=x^T$. Since $F$ is an $\\mathbb O$-stem function, we have\n$$\nF({x'}^T)=F(g^{-1}x^T)=g^{-1}F(x^T)\n$$\nso that $$\\mathbb I'F({x'}^T)=\\mathbb I'g^{-1}F(x^T)$$\n which yields $\\mathbb I'F({x'}^T)=\\mathbb IF(x^T)$ and the assertion follows.\n\\\\\nCase 2: If $ \\mathbb H_{\\mathbb I}\\ne \\mathbb H_{\\mathbb I'} $, we claim that $\\mathbb H_{\\mathbb I} $ and $ \\mathbb H_{\\mathbb I'} $ intersect at $ \\mathbb C_{I} $ for some\n$ I \\in \\mathbb S^6 $. Indeed, since $q=\\mathbb I x^T=\\mathbb I' {x'}^T$, there exists $ y_0,y_1 \\in \\mathbb R $ and $ I \\in \\mathbb S^6 $\n such that $ q=y_0+Iy_1\\in \\mathbb C_I $ and the claim follows. Therefore, we can choose $J_1, J_1^{\\prime} \\in \\mathbb S^6 $ respectively such that\n$$ \\mathbb H_{\\mathbb I}= \\mathbb H_{\\mathbb I_1},\\qquad\n \\mathbb H_{\\mathbb I'}= \\mathbb H_{\\mathbb I_1^{\\prime}} $$\n where $ \\mathbb I_1=(1, I, J_1 , IJ_1) $ and $ \\mathbb I_1^{\\prime}=(1, I, J_1^{\\prime} , IJ_1^{\\prime}) $.\n Since $q= \\mathbb I x^T= \\mathbb I' {x'}^T \\in \\mathbb C_I $, it can be written as\n $$ q=\\mathbb Ix^T=\\mathbb I_1 y^T=\\mathbb I_1^{\\prime}y^T=\\mathbb I'x'^T$$ for some $ y=(y_0,y_1,0,0) \\in \\mathbb R^4 $.\n The computations in Case 1 then shows\n$$ \\mathbb IF(x)^T=\\mathbb I_1 F(y)^T=\\mathbb I_1^{\\prime}F(y)^T=\\mathbb I'F(x')^T.$$\nIn conclusion, Definition \\ref{slice} is well--posed.\n\\end{proof}\n\n\\begin{definition}\nLet $\\Omega$ be an open set in $\\mathbb O$. We set\n$$ S(\\Omega):=\\Big\\{ f:\\Omega \\to \\mathbb O\\ |\\ f=\\mathcal L(F),\\ F:[\\Omega] \\to \\mathbb O^4 \\ {\\rm is \\ an} \\ \\mathbb O-{\\rm stem\\ function} \\Big\\}, $$\nin other words, $ S(\\Omega)$ denotes the collection of slice functions on $ \\Omega$.\n\\end{definition}\n Now we provide the representation formula of slice functions in terms of a quaternion matrix:\n\n\\begin{theorem}\\label{rep.s formula}\nLet $ f $ be a slice function on an axially symmetric set $ \\Omega $ in $\\mathbb O$. Let $q\\in\\mathbb O$ and let $q=\\mathbb I x^T$, for\n $\\mathbb I \\in\\mathcal N$ and $x\\in\\mathbb R^4$. Then for any $p:=\\mathbb I^\\prime x^T$ with $\\mathbb I^\\prime \\in\\mathcal N$ the following formula holds:\n\\begin{equation}\n\\begin{aligned}\nf(p)=\n&{\\left( \\begin{array}{cccc}\n1&I'&J'&K'\n\\end{array} \\right )}\n\\left(\\frac{1}{4}{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n-I&-I&I&I\\\\\n-J&J&-J&J\\\\\n-K&K&K&-K\n\\end{array} \\right )}\n {\\left( \\begin{array}{c}\nf(q)\\\\\nf(\\alpha(q))\\\\\nf(\\beta(q))\\\\\nf(\\gamma(q))\\end{array} \\right )}\\right).\n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nSince $ \\Omega $ is an axially symmetric set, we have $ \\alpha(q),\\beta(q),\\gamma(q) \\in \\Omega $ for any $ q \\in \\Omega $. By definition,\n\\begin{equation}\n{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n1&1&-1&-1\\\\\n1&-1&1&-1\\\\\n1&-1&-1&1\\end{array} \\right )}\n{\\left( \\begin{array}{c}\nF_0(x)\\\\\nIF_1(x)\\\\\nJF_2(x)\\\\\nKF_3(x)\\end{array} \\right )}\n=\n{\\left( \\begin{array}{c}\nf(q)\\\\\nf(\\alpha(q))\\\\\nf(\\beta(q))\\\\\nf(\\gamma(q))\n\\end{array} \\right )}\n\\end{equation}\nso that\n\\begin{equation}\n\\begin{aligned}\n{\\left( \\begin{array}{c}\nF_0(x)\\\\\nIF_1(x)\\\\\nJF_2(x)\\\\\nKF_3(x)\n\\end{array} \\right )}\n=&\n{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n1&1&-1&-1\\\\\n1&-1&1&-1\\\\\n1&-1&-1&1\n\\end{array} \\right )}^{-1}\n{\\left( \\begin{array}{c}\nf(q)\\\\\nf(\\alpha(q))\\\\\nf(\\beta(q))\\\\\nf(\\gamma(q))\\end{array} \\right )}\n\\\\\n=&\\frac{1}{4}{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n1&1&-1&-1\\\\\n1&-1&1&-1\\\\\n1&-1&-1&1\n\\end{array} \\right )}\n{\\left( \\begin{array}{c}\nf(q)\\\\\nf(\\alpha(q))\\\\\nf(\\beta(q))\\\\\nf(\\gamma(q))\\end{array} \\right )}\n\\end{aligned}\n\\end{equation}\nThanks to Artin Theorem, see Theorem \\ref{Artin}, we get\n\\begin{equation}\\label{eq35}\n{\\left( \\begin{array}{c}\nF_0(x)\\\\\nF_1(x)\\\\\nF_2(x)\\\\\nF_3(x)\\end{array} \\right )}\n=\n\\frac{1}{4}{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n-I&-I&I&I\\\\\n-J&J&-J&J\\\\\n-K&K&K&-K\n\\end{array} \\right )}\n{\\left( \\begin{array}{c}\nf(q)\\\\\nf(\\alpha(q))\\\\\nf(\\beta(q))\\\\\nf(\\gamma(q))\\end{array} \\right )}.\n\\end{equation}\nBy the definition of slice functions, for any $ \\mathbb I'=(1,I',J',K') \\in \\mathcal N $ we then have\n\\begin{equation*}\n\\begin{aligned}\n&f(x_0+I'x_1+J'x_2+K'x_3)=F_0(x)+I'F_1(x)+J'F_2(x)+K'F_3(x)\n\\\\\n&={\\left( \\begin{array}{cccc}\n1&I'&J'&K'\n\\end{array} \\right )}\n\\left(\\frac{1}{4}{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n-I&-I&I&I\\\\\n-J&J&-J&J\\\\\n-K&K&K&-K\n\\end{array} \\right )}\n{\\left( \\begin{array}{c}\nf(q)\\\\\nf(\\alpha(q))\\\\\nf(\\beta(q))\\\\\nf(\\gamma(q))\\end{array} \\right )}\\right).\n\\end{aligned}\n\\end{equation*}\n\\end{proof}\n\n\\begin{remark}\nThe representation formula can be briefly expressed as\n$$ f(\\mathbb I'x^T)=\\mathbb I'(M_{\\mathbb I}\\mathcal F(q)),$$\n where $ \\mathbb I'=(1,I',J',K') \\in \\mathcal N $, $q=\\mathbb I x^T$,\n$$ \\mathcal F(q)=(f(q),f(\\alpha(q)),f(\\beta(q)),f(\\gamma(q)))^T,$$ and\n\\[M_{\\mathbb I}=\n\\frac{1}{4}{\\left( \\begin{array}{cccc}\n1&1&1&1\\\\\n-I&-I&I&I\\\\\n-J&J&-J&J\\\\\n-K&K&K&-K\n\\end{array} \\right )}.\n\\]\nThis representation is very useful to prove further properties of slice functions.\n\\\\\nMoreover, notice that $2 M_{\\mathbb I} $ is an orthogonal matrix with elements in $ \\mathbb H_{\\mathbb I } $, i.e. $$ 2 M_{\\mathbb I } \\in O(\\mathbb H_{\\mathbb I}).$$\n\\end{remark}\n\n\\noindent\nThe following result shows that\nthe slice function $f(\\mathbb Ix^T)$ is a linear function of $\\mathbb I$.\n\n\n\n\\begin{theorem}\\label{rep-formula}\nLet $ f $ be a slice function on an axially symmetric set $ \\Omega $. Then\n the octonionic-valued vector function $ M_{\\mathbb I}\\mathcal F(q) $ depends only on $ x $ but not on $ \\mathbb I, \\mathbb I'$ and $f(\\mathbb I'x)=\\mathbb I' (M_{\\mathbb I}\\mathcal F(q)) $ is a linear function in $\\mathbb I'$.\n\\end{theorem}\n\n\\begin{proof} By construction, $ M_{\\mathbb I}\\mathcal F(q) $ is independent of $\\mathbb I'$. Theorem \\ref{rep.s formula} shows that\n$$ f(\\mathbb I'x^T)=\\mathbb I'(M_{\\mathbb I}\\mathcal F(q)) $$\nholds for any $\\mathbb I$, which implies that $ M_{\\mathbb I}\\mathcal F(q) $ is independent of $\\mathbb I$.\n(In alternative, one can prove the assertion noting that \\eqref{eq35} shows that $$M_{\\mathbb I}\\mathcal F(q)=\\left( \\begin{array}{c}\nF_0(x)\\\\\n\\vdots \\\\\nF_3(x)\\end{array} \\right )$$ and so $M_{\\mathbb I}\\mathcal F(q)$ is independent of $\\mathbb I$). Moreover, the linearity in $\\mathbb I'$ is immediate.\n\\end{proof}\n\n\n\n\\begin{remark}\nAlso the representation formula for quaternionic slice regular functions can be written in matrix form. In fact, for any $ I,J \\in \\mathbb S $ where $ \\mathbb S $ is the set of imaginary unit of quaternions $ \\mathbb H $, and for any $ x, y\\in \\mathbb R $, the representation formula can be written as\n\\begin{equation}\nf(x+Jy)=\n\\frac{1}{2}{\\left( \\begin{array}{cc}\n1&J\n\\end{array} \\right )}\n{\\left( \\begin{array}{cc}\n1&1\\\\\n-I&I\n\\end{array} \\right )}\n{\\left( \\begin{array}{c}\nf(x+Iy)\\\\\nf(x-Iy)\\end{array} \\right )}\n\\end{equation}\n\\end{remark}\n\n\n\n\n\\section{Slice Dirac operator}\nIn this section, we introduce the slice Dirac operator in $ \\mathbb O $ and establish the corresponding splitting lemma. We begin by recalling the Dirac operator \\eqref{Dirac1} introduced in Section 1:\n$$ D=\\frac{\\partial}{\\partial x_0}+i\\frac{\\partial}{\\partial x_1}+j\\frac{\\partial}{\\partial x_2}+k\\frac{\\partial}{\\partial x_3}= {\\left( \\begin{array}{cccc}\n1&i & j &k\n\\end{array} \\right )}\n{\\left( \\begin{array}{cc}\n\\dfrac{\\partial}{\\partial x_0}\\\\\n\\vdots \\\\\n\\dfrac{\\partial}{\\partial x_3}\n\\end{array} \\right )}= {\\left( \\begin{array}{cccc}\n1&i & j &k\n\\end{array} \\right )} \\mathsf D\n$$\nwhere $\\mathsf D=\\left( \\dfrac{\\partial}{\\partial x_0}\\ \\,\\dfrac{\\partial}{\\partial x_1}\\ \\,\\dfrac{\\partial}{\\partial x_2}\\ \\,\\dfrac{\\partial}{\\partial x_3}\\right)^T$,\nand its conjugate operator\n$$ \\overline D =\\frac{\\partial}{\\partial x_0}-i\\frac{\\partial}{\\partial x_1}-j\\frac{\\partial}{\\partial x_2}-k\\frac{\\partial}{\\partial x_3}= {\\left( \\begin{array}{cccc}\n1&-i & -j &-k\n\\end{array} \\right )} \\mathsf D.$$\nFor any fixed $\\mathbb I=(1, I, J, K)\\in\\mathcal N$,\nwe define the {\\em slice Dirac operator} in $ \\mathbb O $ as\n \\begin{equation}\\label{duebolli}\n D_{\\mathbb I}=\\frac{\\partial}{\\partial x_0}+I\\frac{\\partial}{\\partial x_1}+J\\frac{\\partial}{\\partial x_2}+K\\frac{\\partial}{\\partial x_3} ={\\left( \\begin{array}{cccc}\n1& I & J & K\n\\end{array} \\right )} \\mathsf D.\n \\end{equation}\nIn the sequel, the restriction $f\\left.\\right|_{\\mathbb H_{\\mathbb I}}$ of a function $f$ to $\\mathbb H_{\\mathbb I}$ shall be denoted by $f_{\\mathbb I}$:\n$$f_{\\mathbb I}=f\\left.\\right|_{\\mathbb H_{\\mathbb I}}.$$\n\nWe now introduce a main definition:\n\n\n\n\\begin{definition}\\label{slice Dirac}\nLet $ \\Omega $ be an axially symmetric domain in $ \\mathbb O $ and let $f \\in S(\\Omega) \\cap C^1(\\Omega)$ so that $f=\\mathcal L(F)$, $f(q)=\\mathbb I F(x)^T$, where $q=\\mathbb I x^T$, $F=[F_0,F_1,F_2,F_3]$. If $F$ satisfies\n\\begin{equation}\\label{CFcondition}\n\\left( \\begin{array}{cccc}\n\\partial_{x_0}&-\\partial_{x_1}&-\\partial_{x_2}&-\\partial_{x_3}\\\\\n\\partial_{x_1}&\\partial_{x_0}&-\\partial_{x_3}&\\partial_{x_2}\\\\\n\\partial_{x_2}&\\partial_{x_3}&\\partial_{x_0}&-\\partial_{x_1}\\\\\n\\partial_{x_3}&-\\partial_{x_2}&\\partial_{x_1}&\\partial_{x_0}\n\\end{array} \\right )\n\\left( \\begin{array}{c} F_0 \\\\ F_1\\\\ F_2\\\\ F_3\n\\end{array} \\right )=\\left( \\begin{array}{c} 0 \\\\ 0\\\\ 0\\\\ 0\n\\end{array} \\right )\n\\end{equation}\nthen $f$ is called a (left) slice Dirac-regular function in $ \\Omega $.\n\\end{definition}\n\nWe denote the set of slice Dirac-regular functions on the axially symmetric set $ \\Omega $ by\n$ SR(\\Omega)$.\n\n\\begin{proposition}\nLet $ \\Omega $ be an axially symmetric domain in $ \\mathbb O $ and let $f \\in S(\\Omega) \\cap C^1(\\Omega)$. Then $f$ is (left) slice Dirac-regular if and only if\n \\begin{equation}\\label{DiracI} D_{\\mathbb I}f(q)=0, \\qquad \\forall\\ q \\in \\Omega\\cap\\mathbb H_{\\mathbb I}=:\\Omega_{\\mathbb I}\n \\end{equation}\n and for all $\\mathbb I\\in\\mathcal N$.\n\\end{proposition}\n\\begin{proof}\nLet $f$ be slice Dirac-regular and let $q \\in \\Omega\\cap\\mathbb H_{\\mathbb I}$, $q=\\mathbb Ix^T$. Then using \\eqref{CFcondition} we have:\n\\begin{equation}\\label{comp}\n\\begin{split}\nD_{\\mathbb I}f(q)&=D_{\\mathbb I} \\mathbb I F(x^T)=D_{\\mathbb I}(F_0+IF_1+JF_2+KF_3)\\\\\n&=(\\partial_{x_0}F_0-\\partial_{x_1}F_1-\\partial_{x_2}F_2-\\partial_{x_3}F_3)+I(\n\\partial_{x_1}F_0+\\partial_{x_0}F_1-\\partial_{x_3}F_2+\\partial_{x_2}F_3)\\\\\n&+J(\\partial_{x_2}F_0+\\partial_{x_3}F_1+\\partial_{x_0}F_2-\\partial_{x_1}F_3)+K(\n\\partial_{x_3}F_0-\\partial_{x_2}F_1+\\partial_{x_1}F_2+\\partial_{x_0}F_3)\\\\\n&=0.\n\\end{split}\n\\end{equation}\nConversely, let us assume that the slice function $f$ is such that \\eqref{DiracI} holds for all $\\mathbb I\\in\\mathcal N$. Let us fix an arbitrary $\\mathbb I\\in\\mathcal N$ and $q=\\mathbb I x^T$ and let us impose that $f(q)=\\mathbb I F(x^T)$ satisfies \\eqref{DiracI}. Computations as in \\eqref{comp} show that \\eqref{CFcondition} holds, by arbitrarity of $\\mathbb I$.\n\\end{proof}\n\n\\begin{remark}\\label{DDD}\nWe note that \\eqref{DiracI} is well-defined.\nIndeed, for any $ q \\in \\mathbb O $ there exist $ \\mathbb I \\in \\mathcal N $ such that $ q \\in \\mathbb H_{\\mathbb I} $ and $ q=(1,I,J,K)(x_0,x_1,x_2,x_3)^T$.\n It can be also written as\n$ q=((1,I,J,K)g^{-1})(y_0,y_1,y_2,y_3)^T$ for any $ g \\in O(3) $ and $ y^T=g x^T $.\nBy the chain rule, it is direct to show that\n$$ (1,I,J,K)(\\partial_{x_0}, \\partial_{x_1}, \\partial_{x_2}, \\partial_{x_3})^Tf(q)= ((1,I,J,K)g^{-1})(\\partial_{y_0},\\partial_{y_1},\\partial_{y_2},\\partial_{y_3})^Tf(q) $$\nwhich implies the claim.\n\\end{remark}\n\n\n\n\n{\n\\begin{example}\\label{eg1}\nWe consider the function $ F=(F_0,F_1,F_2,F_3):\\mathbb R^4 \\to \\mathbb O^4 $ defined by\n\\[\n\\begin{cases} F_0(x)=3x_0,\n\\\\\n F_1(x)=x_1,\n \\\\\nF_2(x)=x_2,\n\\\\\nF_3(x)=x_3,\n \\end{cases} \\]\nwhere $ x=(x_0,x_1,x_2,x_3)\\in \\mathbb R^4 $.\nIt is evident that $ F $ is an $ \\mathbb O-$stem function since (\\ref{eq:variant-1}) holds true.\nThen it induces a slice function $ f: \\mathbb O \\to \\mathbb O $ given by\n\\[f(x_0+Ix_0+Jx_2+Kx_3)=3x_0+Ix_1+Jx_2+Kx_3,\\]\nfor any $ \\mathbb I=(1,I,J,K)\\in \\mathcal N $.\nIt is easy to verify $ f $ is a solution of equations (\\ref{DiracI}) by direct calculation, which means that $ f $ is a slice Dirac-regular function on $ \\mathbb O $.\n\\end{example}\n}\n\n{\n\\begin{example}\nWe further generalize the above example, by constructing a function $ F=(F_0,F_1,F_2,F_3):\\mathbb R^4 \\to \\mathbb O^4 $ such that\n\\[\n\\begin{cases} F_0(x)=S(x_0,r),\n\\\\\n F_1(x)=x_1h(x_0,r),\n \\\\\nF_2(x)=x_2h(x_0,r),\n\\\\\nF_3(x)=x_3h(x_0,r),\n \\end{cases} \\]\nwhere $ x=(x_0,x_1,x_2,x_3)\\in \\mathbb R^4 $ and $ r=\\sqrt{x_1^2+x_2^2+x_3^2}$.\nHere $ S, h:\\mathbb R^2 \\to \\mathbb O $ satisfy the differential equations\n\\begin{equation}\\label{diff:equ}\n\\begin{cases} y\\partial_yh(x,y)+3h(x,y)=\\partial_xS(x,y),\n\\\\\ny\\partial_xh(x,y)=-\\partial_yS(x,y).\n \\end{cases}\n \\end{equation}\nIt is direct to verify that (\\ref{eq:variant-1}) holds so that $ F $ is an $ \\mathbb O-$stem function.\nThis stem function $F$ induces a slice function $ f: \\mathbb O \\to \\mathbb O $ defined by\n\\[f(x_0+Ix_0+Jx_2+Kx_3)=S(x_0,r)+Ix_1h(x_0,r)+Jx_2h(x_0,r)+Kx_3h(x_0,r)\\]\nfor any $ \\mathbb I=(1,I,J,K)\\in \\mathcal N $.\nSince $ S,h $ satisfy equations (\\ref{diff:equ}), it is easy to verify that $ F $ is a solution of equations (\\ref{CFcondition}).\nThis means that $ f $ is a slice Dirac-regular function on $ \\mathbb O $.\n\\\\\n\\indent An explicit example for $F$ is given by\n\\[\n\\begin{cases} F_0(x)=\\frac{3}{5}x_0^5-2x_0^3r^2+\\frac{3}{5}x_0r^4,\n\\\\\n F_1(x)=x_1(x_0^4-\\frac{6}{5}x_0^2r^2+\\frac{3}{35}r^4),\n \\\\\nF_2(x)=x_2(x_0^4-\\frac{6}{5}x_0^2r^2+\\frac{3}{35}r^4),\n\\\\\nF_3(x)=x_3(x_0^4-\\frac{6}{5}x_0^2r^2+\\frac{3}{35}r^4),\n \\end{cases} \\]\n but many others can be easily written.\n\\end{example}\n}\n\n\n\n\n\n\nThe restriction of a slice Dirac-regular function to a quaternionic space $\\mathbb H_{\\mathbb I} $ satisfy the following splitting property:\n\n\\begin{lemma}\\label{splitting lemma}\nLet $ f $ be a slice Dirac-regular function defined on an axially symmetric domain $\\Omega$.\nThen for any $ \\mathbb I \\in \\mathcal N $ and any $ e_4 \\in \\mathbb S^6 $ with $ e_4 \\perp \\mathbb H_{\\mathbb I} $,\nthere exist two functions $ G_1,\\ G_2: \\Omega_{\\mathbb I} \\to \\mathbb H_{\\mathbb I} $ with $ D_{\\mathbb I} G_1=0, \\ \\overline D_{\\mathbb I} G_2=0 $\nsuch that\n\\begin{equation}\nf(q)=G_1(q) +e_4G_2(q),\\ \\ \\ \\forall\\ q \\in \\Omega_{\\mathbb I}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince $ f $ is an octonion-valued function, there exists an element $ e_4 \\in \\mathbb S^6 $ with $ e_4 \\perp \\mathbb H_{\\mathbb I} $ such that\n\\begin{equation}\\label{aaa}\nf=G_1+e_4G_2,\n\\end{equation}\nwhere $ G_1,\\ G_2 \\in \\mathbb H_{\\mathbb I} $.\nHence, also using Lemma \\ref{ee}, it follows that\n\\begin{eqnarray*}\n0=D_{\\mathbb I} f&=&D_{\\mathbb I} G_1+D_{\\mathbb I} (e_4G_2)\n\\\\\n&=&D_{\\mathbb I} G_1+e_4(\\overline D_{\\mathbb I} G_2),\n\\end{eqnarray*}\nwhich implies $$ D_{\\mathbb I} G_1=\\overline D_{\\mathbb I} G_2=0 ,$$\nand the assertion follows.\n\\end{proof}\n\\begin{remark}\nWe note that, in principle, one could have written $$f=G_1+G_2e_4$$ and the condition\nof being slice Dirac regular would translate into $$D_{\\mathbb I} G_1=0, \\qquad\nG_2 D_{\\mathbb I} =0,$$ obtaining that $G_2$ is right regular.\n\\end{remark}\n\n\n\n\\section{Cauchy integral formula}\nIn this section, we present the Cauchy integral theory for slice Dirac operator.\n\n\nThroughout, we let $[\\Omega]$ be an open subset in $\\mathbb R^4$ and recall the notations\n$$ \\Omega:=\\Big\\{q=\\mathbb I x^T \\in \\mathbb O : \\mathbb I \\in \\mathcal N, x \\in [\\Omega] \\Big\\}, \\qquad \\Omega_{\\mathbb I}=\\Omega\\cap\\mathbb H_{\\mathbb I}. $$\nWe shall consider the function $f:\\Omega\\longrightarrow \\mathbb O$ and its restrictions $f_{\\mathbb I }:=\\left.f\\right|_{\\mathbb H_{\\mathbb I}}$.\n\n\n\nWe let\n $$ {\\bf n}(\\xi)=n_0+In_1+Jn_2+Kn_3 $$ denote the unit exterior normal to the boundary $ \\partial \\Omega_{\\mathbb I} $ at $ \\xi $.\nWe consider the Cauchy kernel in $\\mathbb H_{\\mathbb I}$ defined by\n\\begin{equation}\\label{dues} V(\\xi-q)=\\frac{1}{2\\pi^2}\\frac{\\overline{\\xi-q}}{|\\xi-q|^4},\\qquad \\forall\\ \\xi, q\\in \\mathbb H_{\\mathbb I},\n\\end{equation}\nand we finally let $$ dm=dx_0 \\land dx_1 \\land dx_2 \\land dx_3 $$ be the Lebesgue volume element in $ \\mathbb R^4 $, and $ dS $ the induced surface element.\n\n\n\n\n\\begin{theorem}\\label{Pompeiu formula}\nLet $f:\\Omega\\longrightarrow \\mathbb O$ be a slice function on a bounded axially symmetric set $ \\Omega\\subset\\mathbb O$. Suppose that $ f_{\\mathbb I } \\in C^1( \\overline \\Omega_{\\mathbb I}) $ and $ \\partial \\Omega_{\\mathbb I} $ is piecewise smooth for some given $ \\mathbb I \\in \\mathcal N $.\nThen for all $ q \\in \\Omega_{\\mathbb I} $, we have\n\\begin{equation}\nf_{\\mathbb I}(q)=\\int_{\\partial \\Omega_{\\mathbb I}}V(\\xi-q)({\\bf n}(\\xi) f_{\\mathbb I}(\\xi)) dS(\\xi)-\\int_{\\Omega_{\\mathbb I}} V(\\xi-q)(D_{\\mathbb I} f_{\\mathbb I}(\\xi))dm(\\xi).\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof} The classical divergence theorem shows that\n$$ \\int_{\\Omega_{\\mathbb I}} \\frac{ \\partial }{ \\partial x_j} \\mu dm= \\int_{ \\partial \\Omega_{\\mathbb I} }n_j \\mu dS, \\qquad j=0,1,2,3. $$\nfor any real-valued function $ \\mu \\in C^1(\\Omega_{\\mathbb I}) \\cap C(\\overline \\Omega_{\\mathbb I}) $.\nThus for the octonion-valued function $ f_{\\mathbb I} \\in C^1(\\Omega) \\cap C(\\overline \\Omega) $, we have\n \\begin{equation}\\label{asterisco}\n \\int_{\\Omega_{\\mathbb I}} \\frac{ \\partial }{ \\partial x_j} f_{\\mathbb I} dm= \\int_{ \\partial \\Omega_{\\mathbb I} } n_j f_{\\mathbb I} dS, \\qquad j=0,1,2,3. \\end{equation}\nBy Lemma \\ref{splitting lemma}, there exist $ e_4 \\in \\mathbb S^6 $ with $ e_4 \\perp \\mathbb H_{\\mathbb I} $ such that $ f_{\\mathbb I}=G_1+e_4G_2 $ with $ G_1,\\ G_2$ which are $\\mathbb H_{\\mathbb I} $-valued. Hence for any map $ V:\\Omega_{\\mathbb I} \\to \\mathbb H_{\\mathbb I}$ such that $V \\in C^1(\\Omega_{\\mathbb I}) \\cap C(\\overline \\Omega_{\\mathbb I}) $, we have\n\\begin{equation}\\label{G1}\n\\begin{aligned}\n\\int_{\\Omega_{\\mathbb I}}& (V D_{\\mathbb I}) G_1 +V( D_{\\mathbb I} G_1) dm\n\\\\\n=&\\int_{\\Omega_{\\mathbb I}} \\frac{\\partial}{\\partial x_1}( V G_1)+\\frac{\\partial}{\\partial x_2}(V I G_1)+\\frac{\\partial}{\\partial x_3}(V J G_1)+\\frac{\\partial}{\\partial x_3}(V K G_1)dm\n\\\\\n=&\\int_{\\partial \\Omega_{\\mathbb I}} V {\\bf n} G_1 dS,\n\\end{aligned}\n\\end{equation}\nwhere we have used associativity in $\\mathbb H_{\\mathbb I}$.\nSimilarly, we have\n\\begin{equation}\\label{G2}\n\\int_{\\Omega_{\\mathbb I}} (\\overline{V} \\ \\overline {D_{\\mathbb I}}) G_2 +\\overline{V} ( \\overline D_{\\mathbb I} G_2) dm = \\int_{\\partial \\Omega_{\\mathbb I}} \\overline{V} \\overline {\\bf n} G_2 dS .\n\\end{equation}\nThe equalities \\eqref{G1} and \\eqref{G2} hold, in particular, when $V$ is the Cauchy kernel in \\eqref{dues}.\nWe now fix $ q \\in \\Omega_{\\mathbb I} $ and note that $$ V(\\xi-q)=-\\frac{1}{4\\pi^2}\\overline D_\\xi \\frac{1}{|\\xi-q|^2},$$\nwhere $D_{\\xi}$ denotes the Dirac operator with respect to the variable $\\xi$.\nIndeed,\n\\begin{equation}\\label{V}\n\\begin{aligned}\nV(\\xi-q)=&\\frac{-1}{4\\pi^2}(\\frac{\\partial}{\\partial x_0}-I\\frac{\\partial}{\\partial x_1}-J\\frac{\\partial}{\\partial x_2}-K\\frac{\\partial}{\\partial x_3})\\frac{1}{|\\xi-q|^2}\n\\\\\n =&\\frac{1}{2\\pi^2}\\frac{\\overline{\\xi-q}}{|\\xi-q|^4}.\n\\end{aligned}\n \\end{equation}\nStraightforward calculations show that\n$$ (V D_\\xi)(q)=(D_\\xi V)(q)=0$$ for any $\\xi\\neq q$.\n\n\nTake a sufficient small $ \\varepsilon $ such that the ball $ B_\\varepsilon(q)$ centered at $q$ and with radius $\\varepsilon$ is contained in $\\Omega_{\\mathbb I} $. From (\\ref{G1}), (\\ref{G2}), and Lemma \\ref{splitting lemma}\nwe have\n\\begin{equation}\\label{f_{I,J}}\n\\int_{\\Omega_{\\mathbb I} \\setminus B_\\varepsilon (q)} V( D_\\xi f_{\\mathbb I}) dm = \\int_{\\partial (\\Omega_{\\mathbb I}\\setminus B_\\varepsilon (q))} V ({\\bf n } f_{\\mathbb I}) dS .\n\\end{equation}\nHence we can calculate this integral as follows:\n\\begin{equation}\\label{bbb}\n\\begin{aligned}\n\\int_{\\Omega_{\\mathbb I} \\setminus B_\\varepsilon(q)}V(\\xi-q)(D_\\xi f_{\\mathbb I})dm=&\\int_{\\partial \\Omega_{\\mathbb I}}V(\\xi-q)({\\bf n } f_{\\mathbb I})dS-\\int_{\\{|\\xi-q|= \\varepsilon\\}} V(\\xi-q)({\\bf n} f_{\\mathbb I})dS\n\\\\\n:=&I_{\\partial \\Omega_{\\mathbb I}}-I_\\varepsilon\n\\end{aligned}\n\\end{equation}\nBy Equation (\\ref{V}) and Artin Theorem \\ref{Artin}, we can evaluate the limit of $ I_\\varepsilon $:\n\\begin{equation}\\label{1}\n\\begin{aligned}\n\\lim_{\\varepsilon \\to 0 }I_\\varepsilon=&\n \\frac{1}{2\\pi^2}\\lim_{\\varepsilon \\to 0}\\int_{\\{|\\xi-q|=\\varepsilon\\}}\\frac{\\overline{\\xi-q}}{|\\xi-q|^4} \\frac{\\xi-q}{|\\xi-q|}f_{\\mathbb I}(\\xi)dS\n\\\\\n=&\\frac{1}{2\\pi^2}\\lim_{\\varepsilon \\to 0}\\int_{\\{|\\xi-q|=\\varepsilon\\}}\\frac{1}{|\\xi-q|^3}f_{\\mathbb I}(\\xi)dS\n\\\\\n=&f_{\\mathbb I}(q).\n\\end{aligned}\n\\end{equation}\nLet $ \\varepsilon \\to 0 $ in (\\ref{bbb}), we get\n$$ f_{\\mathbb I}(q)=\\int_{\\partial \\Omega_{\\mathbb I}}V(\\xi-q)({\\bf n}(\\xi) f_{\\mathbb I}(\\xi)) dS-\\int_{\\Omega_{\\mathbb I}} V(\\xi-q)(D_\\xi f_{\\mathbb I}(\\xi))dm,$$\nas desired.\n\\end{proof}\n\n\n\\begin{corollary}\\label{Cauchy formula} Let $ f $ be a slice Dirac-regular function on a bounded axially symmetric set $\\Omega $. Suppose that $ f_{\\mathbb I } \\in C^1( \\overline \\Omega_{\\mathbb I}) $ and $ \\partial \\Omega_{\\mathbb I} $ is piecewise smooth for some given $ \\mathbb I \\in \\mathcal N $.\n then\n\\begin{equation}\\label{eqC1}\nf_{\\mathbb I}(q)=\\int_{ \\partial \\Omega_{\\mathbb I} } V(\\xi-q)({\\bf n }(\\xi)f_{\\mathbb I}(\\xi))dS,\\ \\ \\ \\forall\\ q \\in \\Omega_{\\mathbb I}\n\\end{equation}\nand\n\\begin{equation}\\label{eqC2}\n\\int_{ \\partial \\Omega_{\\mathbb I} } V(\\xi-q)({\\bf n} (\\xi)f_{\\mathbb I}(\\xi))dS=0, \\ \\ \\ \\forall\\ q \\notin \\overline{\\Omega}_{\\mathbb I}.\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof}\nIf $ q \\in \\Omega_{\\mathbb I} $, then (\\ref{eqC1}) follows from Theorem \\ref{Pompeiu formula} since $ D_{\\xi} f_{\\mathbb I}(\\xi)=0 $.\nFor $ q \\notin \\overline{\\Omega}_{\\mathbb I} $, the integral at the left hand side of (\\ref{eqC2}) is a proper integral so that after limit process, (\\ref{f_{I,J}}) becomes\n$$ \\int_{\\Omega_{\\mathbb I}}V(\\xi-q)(D_\\xi f_{\\mathbb I})dm=\\int_{\\partial \\Omega_{\\mathbb I}}V(\\xi-q)({\\bf n } f_{\\mathbb I})dS $$\nSince $f $ is slice Dirac-regular in $ \\Omega $, the left hand side vanishes and we obtain (\\ref{eqC2}).\n\\end{proof}\n\n\n\nUsing the representation formula, we can introduce another kernel which allows to write a Cauchy formula of more general validity.\n\nDenote $ M_{n\\times m}(\\mathbb O) $ the set of octonion matrix of $m$ rows and $n$ columns where $n, m$ are positive integer.\nGiven an octonion matrix $ A \\in M_{n\\times m}(\\mathbb O) $,\nthe left multiplication operator $ L_A: M_{m\\times k}(\\mathbb O) \\to M_{n\\times k}(\\mathbb O) $ defined as:\n$$ L_AB:=AB,\\ \\ \\ \\forall B\\in M_{m\\times k}(\\mathbb O)$$\nIn general $ L_AL_B\\neq L_{AB} $.\n\n\\begin{definition}\n Let $ \\mathbb I' \\in \\mathcal N $ and $ \\xi,q \\in \\mathbb H_{\\mathbb I }$ be given. We define the slice Cauchy kernel $\\mathcal V(\\xi, q,\\mathbb I'): \\mathbb O\\to \\mathbb O $\nvia\n\\begin{equation}\\label{enne} \\mathcal V(\\xi, q,\\mathbb I')=L_{\\mathbb I'}L_{M_{\\mathbb I}}L_{\\mathbb V(\\xi-q)},\\end{equation}\nwhere\n$$ \\mathbb V(\\xi-q)=(V(\\xi-q),V(\\xi-\\alpha(q)),V(\\xi-\\beta(q)),V(\\xi-\\gamma(q)))^T.$$\n\\end{definition}\n\n\n\n\n\\begin{theorem}\\label{slice Cauchy} Let $ f $ be a slice function on a bounded axially symmetric set $\\Omega$,\nsuppose that $ f_{\\mathbb I } \\in C^1( \\overline \\Omega_{\\mathbb I}) $ and $ \\partial \\Omega_{\\mathbb I} $ is piecewise smooth for some given $ \\mathbb I \\in \\mathcal N $.\nThen for any $p\\in\\Omega$, there exists $ \\mathbb I' \\in \\mathcal N $ such that $p=\\mathbb I^\\prime x^T$ with $x\\in\\mathbb R^4$ and\n\\begin{equation}\nf(p)=\\int_{ \\partial \\Omega_{\\mathbb I} }\\mathcal V(\\xi, q,\\mathbb I')({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))dS-\\int_{ \\Omega_{\n\\mathbb I} }\\mathcal V(\\xi, q,\\mathbb I')(D_{\\mathbb I}f_{\\mathbb I}(\\xi))dm,\n\\end{equation}\nwhere $ q=\\mathbb I x^T \\in \\Omega_{\\mathbb I}$ and $\\mathcal V$ is as in \\eqref{enne}.\n\\end{theorem}\n\\begin{proof}\nBy the representation formula in Theorem \\ref{rep.s formula}, for any $ q=\\mathbb I x^T \\in \\Omega_{\\mathbb I} $ and any $ \\mathbb I' \\in \\mathcal N $ we have\n\\begin{equation}\\label{M}\nf(p)=f(\\mathbb I'x^T)=\\mathbb I'(M_{\\mathbb I}\\mathcal F(q)).\n\\end{equation}\nSince $ \\Omega $ is an axially symmetric set, it follows that $ \\alpha(q),\\beta(q),\\gamma(q)\\in \\Omega_{\\mathbb I} $ for any $ q \\in \\Omega_{\\mathbb I}$. Theorem \\ref{Pompeiu formula} gives\n\\begin{equation}\\label{MM}\n\\mathcal F(q)=\\int_{\\partial \\Omega_{\\mathbb I}} \\mathbb V(\\xi-q)({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))dS-\\int_{\\Omega_{\\mathbb I}} \\mathbb V(\\xi-q)(D_{\\mathbb I}f_{\\mathbb I}(\\xi))dm.\n\\end{equation}\nSubstituting (\\ref{MM}) into (\\ref{M}) and moving the integral out, we finally get\n\\begin{equation*}\nf(\\mathbb I'x^T)=\\int_{\\partial \\Omega_{\\mathbb I}}\\mathbb I'(M_{\\mathbb I}(\\mathbb V(\\xi-q)({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))))dS-\\int_{\\Omega_{\\mathbb I}} \\mathbb I'(M_{\\mathbb I}(\\mathbb V(\\xi-q)(D_{\\mathbb I}f_{\\mathbb I}(\\xi))))dm,\n\\end{equation*}\nand \\eqref{enne} allows to conclude.\n\\end{proof}\n\n\n\n\n\\section{Slice Dirac-regular power series}\nIn this section, we provide the Taylor series for the slice Dirac-regular function and the Laurent series for the slice Dirac-regular function near an isolated singularity.\n\nFor any $ \\alpha=(\\alpha_1, \\alpha_2, \\alpha_3) \\in \\mathbb N^3 $, we set\n$ n=|\\alpha|=\\alpha_1+\\alpha_2+\\alpha_3 $,\n$ \\alpha!=\\alpha_1!\\alpha_2!\\alpha_3! $,\n $$ \\partial_\\alpha=\\frac{\\partial^n}{\\partial x_1^{\\alpha_1}\\partial x_2^{\\alpha_2}\\partial x_3^{\\alpha_3}}, $$\nand\n{for any $ q=x_0+ix_1+jx_2+kx_3\\in \\mathbb H $, denote}\n$$ V_\\alpha(q)=\\partial_\\alpha V(q), $$\nwhere $$ V(q)=\\frac{1}{2\\pi^2}\\frac{\\overline q}{|q|^4}.$$\n It is well known that $ V_\\alpha(q) $ is left and right Dirac-regular except at zero since $ \\partial_\\alpha $ commutes with the Dirac operator $ D $.\n\nNote that the monomials $ f(q)=q^n$ are not Dirac-regular. Their Dirac-regular counterparts are\nthe homogeneous left and right Dirac-regular polynomials $ P_\\alpha $, defined by\n$$ P_\\alpha(q)=\\frac{\\alpha !}{n!}\\sum(x_{\\beta_1}-i_{\\beta_1}x_0)\\cdots(x_{\\beta_n}-i_{\\beta_n}x_0), $$\nwhere $ q=x_0+ix_1+jx_2+kx_3 $.\nHere the sum runs over all $ \\frac{n!}{a!} $ different orderings of $ \\alpha_1 \\ 1's $, $ \\alpha_2 \\ 2's $ and $ \\alpha_3 \\ 3's $\nand $ i_{\\beta_l} \\in \\{ i , j, k \\}$ for any $l=1,2,\\cdots,n$.\n\n\n\n\nThe polynomials $ P_\\alpha $ are homogeneous of degree $ n $, while $ V_\\alpha $ is homogeneous of degree $ -n-3 $ (see \\cite{Brackx_1}).\n\n\n\\noindent Let $ U_n $ be the right quaternionic vector space of homogeneous Dirac-regular functions of degree $ n\\in\\mathbb N $.\nThen, the polynomials $ P_\\alpha \\ (\\alpha \\in \\mathbb N^3) $ are Dirac-regular and form a basis for $ U_n $.\n\n\\begin{theorem}\\label{0}\n Let $ f $ be a slice Dirac-regular function in the unit ball $ B \\subset \\mathbb O $ centered at the origin and let $ f \\in C^1(\\overline{B}) $. For any $ q \\in B $, there exist $ \\mathbb I \\in \\mathcal N $ such that\n $ q \\in \\mathbb H_{\\mathbb I} $, and\n \\begin{equation}\\label{tt}\n f(q)=\\sum_{n=0}^{+\\infty} \\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}} P_{\\alpha}(q)\\frac{\\partial_\\alpha f_{\\mathbb I}(0)}{\\alpha!},\n \\end{equation}\nwhere the power series is uniformly convergent over $ B_{\\mathbb I}$.\n \\end{theorem}\n\n \\begin{proof\n Let $q\\in B$, then there exists $ \\mathbb I \\in \\mathcal N $ such that\n $ q \\in \\mathbb H_{\\mathbb I} $. Moreover, there exists a closed ball $ B_{\\rho} $ with $ \\rho < 1 $ such that $ q \\in B_{\\rho} $.\nBy Lemma \\ref{splitting lemma}, we can pick $e_4\\in \\mathbb S^6 $ with $ e_4 \\perp \\mathbb H_{\\mathbb I} $, and write\n$$ f_{\\mathbb I}(q)=G_1(q)+e_4G_2(q),$$ where\n$ G_1 $ and $ G_2 $ are $\\mathbb H_{\\mathbb I}-$valued Dirac-regular and conjugate Dirac-regular, respectively. The integral formula (\\ref{eqC1}) gives\n \\begin{equation}\\label{3}\n \\begin{aligned}\n f_{\\mathbb I}(q)=&\\int_{\\partial B_{\\mathbb I}}V(\\xi-q)({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))dS\n \\\\\n =&\\int_{\\partial B_{\\mathbb I}}V(\\xi-q){\\bf n}(\\xi)G_1(\\xi)dS+e_4\\Big(\\int_{\\partial B_{\\mathbb I}}\\overline{V(\\xi-q)} \\ \\overline{{\\bf n}(\\xi)}G_2(\\xi)dS \\Big)\n \\\\\n :=&I_1+e_4I_2\n \\end{aligned}\n \\end{equation}\n By Theorem 28 in \\cite{Sudbery_1}, we can expand $ V(\\xi-q) $ in power series for any $ |q| < |\\xi| $\n \\begin{equation}\\label{VV}\n \\begin{aligned}\nV(\\xi-q)=&\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}(-1)^nP_\\alpha(q)V_\\alpha(\\xi)\n \\\\\n =&\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}(-1)^nV_\\alpha(\\xi)P_\\alpha(q)\n\\end{aligned}\n\\end{equation}\nand the right hand side converges uniformly\n in any region $ \\{ (\\xi, q): |q| \\le r|\\xi| \\} $ with $ r < 1 $.\nSince $ q \\in B_{\\rho} $ and $ \\xi \\in \\partial B $, we have $ |q | \\le r|\\xi| $ with $ r<1$. Using the rightmost expression in (\\ref{VV}), we get\n\\begin{equation}\\label{1}\n\\begin{aligned}\n I_2=&\\int_{\\partial B_{\\mathbb I}}\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}(-1)^n\\overline{V_\\alpha(\\xi)P_\\alpha(q)}\\ \\overline {{\\bf n}(\\xi)} G_2(\\xi)dS\n \\\\\n =&\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}\\overline {P_\\alpha(q)}(-1)^n\\int_{\\partial B_{\\mathbb I}} \\overline{V_\\alpha(\\xi)}\\ \\overline{{\\bf n}(\\xi)} G_2(\\xi)dS.\n \\end{aligned}\n \\end{equation}\nUsing the first expression in (\\ref{VV}) and repeating the procedure, we have\n\\begin{equation}\\label{2}\nI_1=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}P_\\alpha(q)(-1)^n\\int_{\\partial B_{\\mathbb I}} V_\\alpha(\\xi){\\bf n}(\\xi)G_2(\\xi)dS.\n\\end{equation}\nSubstituting (\\ref{1}) and (\\ref{2}) into (\\ref{3}) we obtain\n$$ f_{\\mathbb I}=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}P_\\alpha(q)((-1)^n\\int_{\\partial B_{\\mathbb I}} V_\\alpha(\\xi)({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))dS). $$\n\n\\noindent Differentiating both sides of the integral formula (\\ref{3}), we have\n $$ \\partial_\\alpha f_{\\mathbb I}(q)=(-1)^n\\int_{\\partial B_{\\mathbb I}}V_\\alpha(\\xi-q)({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))dS. $$\nIn particular, letting $ q \\to 0 $ we conclude that\n $$ \\partial_\\alpha f_{\\mathbb I}(0)=(-1)^n\\int_{\\partial B_{\\mathbb I}} V_\\alpha(\\xi)({\\bf n}(\\xi)f_{\\mathbb I}(\\xi))dS. $$\n \\end{proof}\n\n\n{\n\\begin{remark} We point out that although the functions $ P_\\alpha : \\mathbb H_{\\mathbb I} \\to \\mathbb O $ are homogeneous left and right Dirac-regular polynomials, they do not extend, in general, to a slice Dirac-regular function on the whole $ \\mathbb O $.\nFor example, we consider the special case $ n=2 $ and set\n\\begin{equation*}\n\\begin{aligned}\nf(q):=\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=2}}P_\\alpha(q).\n\\end{aligned}\n\\end{equation*}\nFor any $q=x_0+Ix_1+Jx_2+Kx_3\\in\\mathbb H_{\\mathbb I}$, by direct calculation we have\n\\begin{equation*}\n\\begin{aligned}\nf(q)=F_0(x)+IF_1(x)+JF_2(x)+KF_3(x),\n\\end{aligned}\n\\end{equation*}\nwhere\n\\begin{eqnarray*}\n F_0(x)&=&-6x_0^2+(x_1+x_2+x_3)^2+x_1^2+x_2^2+x_3^2\n \\\\\n F_1(x)&=&-2x_0(x_1+x_2+x_3)-2x_0x_1\n \\\\\n F_2(x)&=&-2x_0(x_1+x_2+x_3)-2x_0x_2\n \\\\\n F_3(x)&=&-2x_0(x_1+x_2+x_3)-2x_0x_3\n \\end{eqnarray*}\nNote that $ F_0 $ does not satisfy the compatibility conditions (\\ref{eq:variant-1}).\nTherefore, not all $ P_\\alpha $ can be extended to a slice Dirac-regular function on the whole $ \\mathbb O $.\n\\end{remark}\n}\n\n\n\n\n\\begin{remark} We still do not know if the series in (\\ref{tt}) is convergent uniformly on the whole unit ball $B$, besides on the subsets $B_{\\mathbb I}$.\nOur proof on $B_{\\mathbb I}$ depends on the explicit formula of the kernel $V$ and the associativity of quaternions. This technique obviously fails in the setting of octonions and to consider the uniform convergence over $B$, one should follow a different approach. In fact,\nfor any $ f \\in C^1(\\overline{B}) $, one needs the estimate\n $$ |f(q)-f(q')|\\le|f(\\mathbb I x^T)-f(\\mathbb I' x^T)|+| f(\\mathbb I'x^T)-f(\\mathbb I'x'^T)|,$$\nand it is problematic to show that $ |f(\\mathbb I x^T)-f(\\mathbb I' x^T)| $ is small enough.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\nWe now come to study the power series at any point $ q_0 \\in \\mathbb O $ for slice Dirac-regular functions.\nWith the same approach used in Theorem \\ref{0},\none can show that\n \\begin{equation}\\label{q_0}\n f(q)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}P_{\\alpha}(q-q_0)\\frac{\\partial_\\alpha f_{\\mathbb I}(q_0)}{\\alpha!}\n \\end{equation}\nfor any $ q_0 \\in \\mathbb O \\cap \\mathbb H_{\\mathbb I} $ and $q\\in B_{\\mathbb I}(q_0,R,\\mathbb O):=B(q_0,R,\\mathbb O)\\cap \\mathbb H_{\\mathbb I}$.\nHere $ B(q_0,R, \\mathbb O)\\subset \\mathbb O $ denotes the ball of radius $ R $ centered at $ q_0 $.\n\n\nLet $ B(x_0,R)\\subset \\mathbb R^4 $ be the ball of radius $ R $ centered at $ x_0 $ and denote by $\\widetilde B(x_0, R)\\subset \\mathbb O $ the symmetrized set of $ B(x_0, R) $.\n\nWe set\n $$ \\mathbb P_{\\alpha}(q-q_0)=(P_{\\alpha}(q-q_0),\\, P_{\\alpha}(\\alpha(q)-q_0),\\, P_{\\alpha}(\\beta(q)-q_0),\\,\n P_{\\alpha}(\\gamma(q)-q_0))^T$$\n and consider the operator\n $$ \\mathcal P_\\alpha(q ,q_0,\\mathbb I')=L_{\\mathbb I'}L_{M_{\\mathbb I}}L_{\\mathbb P_{\\alpha}(q-q_0)}. $$\n\n\n\n\n\n \\begin{theorem} Assume that $ f \\in C^1(\\overline{B(q_0,R, \\mathbb O)})$ with $ R>0 $ and let $ q_0=\\mathbb I x_0^T \\in \\mathbb H_{\\mathbb I} $.\nIf $ f $ is slice Dirac-regular on $ B(q_0,R, \\mathbb O)$,\nthen\n $$ f( \\mathbb I'x^T)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}\\mathcal P_{\\alpha}(q, q_0, \\mathbb I')\\frac{\\partial_{\\alpha} f(q_0)}{\\alpha!} $$\n for any $ q=\\mathbb I x^T \\in B_{\\mathbb I}(q_0,R,\\mathbb O)\\cup \\widetilde B(x_0, R)$.\n \\end{theorem}\n\n \\begin{proof}\n\nBy virtue of (\\ref{q_0}), $f$ admits the power series expansion\n $$ f(q)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}P_{\\alpha}(q-q_0)\\frac{\\partial_{\\alpha} f(q_0)}{\\alpha!}. $$\n for any $q\\in B_{\\mathbb I}(q_0,R,\\mathbb O)$ and $ q_0=\\mathbb I x_0^T \\in \\mathbb H_{\\mathbb I} $.\nIf $ q \\in \\widetilde B(x_0, R) $, then $ \\alpha(q),\\beta(q),\\gamma(q) \\in \\widetilde B(x_0, R)$ so that\n $$ \\mathcal F(q)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}\\mathbb P_{\\alpha}(q-q_0)\\frac{\\partial_{\\alpha} f(q_0)}{\\alpha!}. $$\nBy the representation formula, we have\n$$ f(\\mathbb I' x^T)=\\mathbb I'(M_{\\mathbb I}\\mathcal F(q)). $$\nCombining these two formulas and taking the sum we conclude that\n$$ f(\\mathbb I'x^T)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}\\mathcal P_{\\alpha}(q, q_0, K)\\frac{\\partial_{\\alpha} f(q_0)}{\\alpha!}. $$\n\\end{proof}\n\n\nFinally we study the Laurent power series. We need to introduce some notation.\nLet $ q_0 \\in \\mathbb O $ and $0 \\le R_1 |q-q_0| $.\nTherefore, the same approach as in the proof of Theorem \\ref{0} shows\n$$ \\int_{S_2}V(\\xi-q)({\\bf n}(\\xi)f(\\xi))dS=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}P_\\alpha(q-q_0)A_\\alpha. $$\n\nFor any $ \\xi \\in S_1 $, we have $ |\\xi-q_0|<|q-q_0| $. Now we use the second series in (\\ref{VV}) and repeat the procedure in the proof Theorem \\ref{0} to deduce that\n\\begin{equation}\n\\begin{aligned}\n-\\int_{S_1}&V(\\xi-q)({\\bf n}(\\xi)f(\\xi))dS\n\\\\\n=&\\int_{S_1}V(q-\\xi)({\\bf n}(\\xi)f(\\xi))dS\n\\\\\n=&\\int_{S_1}\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}(-1)^nV_\\alpha(q-q_0)P_\\alpha(\\xi-q_0)({\\bf n}(\\xi)f(\\xi))dS\n\\\\\n=&\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}V_\\alpha(q-q_0)B_\\alpha.\n\\end{aligned}\n\\end{equation}\n\n\n\\noindent This means\n$$ f(q)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}[P_\\alpha(q-q_0)A_\\alpha+V_\\alpha(q-q_0)B_\\alpha]. $$\nFor any $ q \\in B_{\\mathbb I}(q_0,R_1,R_2,\\mathbb O) \\cap \\widetilde B(x_0,R_1,R_2) $, we have $$ \\alpha(q),\\beta(q),\\gamma(q) \\in B_{\\mathbb I}(q_0,R_1,R_2,\\mathbb O) \\cap \\widetilde B(x_0,R_1,R_2)$$ so that\n$$ \\mathcal F(q)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}[\\mathbb P_\\alpha(q-q_0)A_\\alpha+\\mathbb V_\\alpha(q-q_0)B_\\alpha]. $$\nBy the representation formula\n$$ f(\\mathbb I'x^T)=\\mathbb I'(M_{\\mathbb I}\\mathcal F_{\\mathbb I}(q)). $$\nwe get the stated result\n$$ f(\\mathbb I'x^T)=\\sum_{n=0}^{+\\infty}\\sum\\limits_{\\stackrel{\\alpha \\in \\mathbb N^3}{|\\alpha|=n}}[\\mathcal P_\\alpha(q, q_0, K)A_\\alpha+\\mathcal V_\\alpha(q, q_0, K)B_\\alpha]. $$\n\\end{proof}\n\n\n\n\n\\bibliographystyle{amsplain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nCharmonium is usually studied experimentally through $e^+e^-$ annihilation\nor hadronic production, notably in $p\\bar p$ annihilation.\nThe $p\\bar p$ annihilation process was employed by the fixed target experiments\nE760 and E835 at Fermilab, which despite small production cross sections succeeded in\ngiving very accurate results for the masses and total widths of the narrow charmonium states\n$J\/\\psi$, $\\psi'$, $\\chi_1$ and $\\chi_2$. A future experimental program of\ncharmonium and charmonium hybrid production using $p\\bar p$ annihilation that is planned by the\nPANDA collaboration \\cite{PandaTechnicalProgress} at GSI is one of the principal motivations \nfor this study.\n\nObviously the strengths and detailed forms of the couplings of charmonium states\nto $p\\bar p$ are crucial questions for any experimental program that uses $p\\bar p$\nannihilation to study charmonium; see for example the predictions for the associated\nproduction processes $p\\bar p \\to \\pi^0\\Psi$ in\nRefs.\\cite{Gaillard:1982zm,Lundborg:2005am,Barnes:2006ck}.\n(We use $\\Psi$ to denote a generic charmonium or charmonium hybrid state, and $\\psi$\nif the state has $J^{PC} = 1^{--}$.)\n\nUnfortunately these low to moderate energy production reactions involve obscure and\npresumably rather complicated QCD processes, so for the present they are best\ninferred from experiment. In Ref.\\cite{Barnes:2006ck} we carried out this exercise\nby using the measured $p\\bar p$ partial widths to estimate\nthe coupling constants of the $J\/\\psi, \\psi', \\eta_c, \\eta_c', \\chi_0$ and $\\chi_1$\nto $p\\bar p$, assuming that the simplest Dirac couplings were dominant.\nThese $\\Psi p\\bar p$ couplings were then used in a PCAC-like model to give numerical\npredictions for several associated charmonium production cross sections of the\ntype $p\\bar p \\to \\pi^0 \\Psi$.\n\nIn this paper we generalize these results for the $J\/\\psi$ and $\\psi'$ by relaxing\nthe assumption of $\\gamma_{\\mu}$ dominance of the $\\psi p\\bar p$ vertex.\nWe assume a $\\psi p\\bar p$ vertex with both Dirac ($\\gamma_{\\mu}$) and\nPauli ($\\sigma_{\\mu\\nu}$) couplings, and derive\nthe differential and total cross sections for\n$e^+ e^- \\to \\psi \\to p\\bar p$ given this more general vertex.\nBoth unpolarized and polarized processes are treated.\n\nA comparison of our theoretical unpolarized angular\ndistributions to recent experimental $J\/\\psi$ results\nallows estimates of both the Dirac and Pauli \n$J\/\\psi p\\bar p$ couplings. There is a phase ambiguity \nthat precludes a precise determination of the \n(complex) ratio of the Pauli and Dirac\n$J\/\\psi p\\bar p$ couplings from the unpolarized data; \nwe shall see that an importance interference effect \nbetween the Pauli and Dirac terms leads to a strong dependence of the \nunpolarized $p\\bar p \\to \\pi^0 J\/\\psi$ cross section near threshold \non the currently unknown phase between these terms.\n\nDetermining these couplings is evidently quite\nimportant for PANDA, and can be accomplished through studies of the \npolarized process $e^+ e^- \\to J\/\\psi \\to p\\bar p$. The \nangular distribution of the unpolarized, self-analyzing process \n$e^+ e^- \\to J\/\\psi \\to \\Lambda\\bar\\Lambda$ may also provide complementary \ninformation regarding the closely related $J\/\\psi \\Lambda\\bar\\Lambda$ vertex. \nBoth of these processes should be accessible at the upgraded BES-III facility.\n\n\\section{Unpolarized cross section}\nThe Feynman diagram used to model this process is shown in Fig.\\ref{fig:diag}.\nWe assume a vertex for the coupling of a generic $1^{--}$ vector charmonium state\n$\\psi$ to $p\\bar p$ of the form\n\\begin{equation}\n\\Gamma_{\\mu}^{(\\psi p \\bar p)} =\ng \\Big( \\gamma_{\\mu} + \\frac{i\\kappa}{2m}\\, \\sigma_{\\mu\\nu} q_{\\nu}\\Big).\n\\label{eq:vertex}\n\\end{equation}\nIn this paper $m$ and $M$ are the proton and charmonium mass,\n$\\Gamma$ is the charmonium total width, and\nwe assume massless initial leptons. Following DIS conventions, $q_{\\nu}$\nis the four momentum transfer {\\it from} the nucleon to\nthe electron; thus in our reaction $e^+e^-\\to p \\bar p$ in the c.m. frame,\nwe have $q = (-\\sqrt{s},\\vec 0)$.\nThe couplings $g$ and $\\kappa$ are actually momentum dependent form factors,\nbut since we only access them very close to the kinematic point $q^2 = M^2$ in the reactions\n$e^+e^-\\to (J\/\\psi, \\psi') \\to p \\bar p$, we will treat them as constants.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.7\\linewidth]{blr_fig1.eps}\n\\caption{The Feynman diagram assumed in this model of\nthe generic reaction $e^+e^- \\to \\psi \\to p\\bar p$.}\n\\label{fig:diag}\n\\end{figure}\n\nThe unpolarized differential and total cross sections for\n$e^+e^-\\to \\psi \\to p \\bar p$\nmay be expressed succinctly in terms of the strong $\\psi p \\bar p$ Sachs form factors\n${\\cal G}_E = g(1+\\kappa s\/4m^2)$ and ${\\cal G}_M = g(1+\\kappa)$.\nBoth ${\\cal G}_E$ and ${\\cal G}_M$ are complex above $p\\bar p$ threshold, in part because\nphases are induced by $p\\bar p$ rescattering. If we assume that the lowest-order Feynman diagram \nof Fig.\\ref{fig:diag} is dominant, the phase of $g$ itself\nis irrelevant, so here we take $g$ to be real and positive. \n$\\kappa$ however has a nontrivial phase. \nWe express this by introducing a Sachs form factor ratio, with magnitude $\\rho \\geq 0$ \nand phase $\\chi$;\n\\begin{equation}\n{\\cal G}_E \/ {\\cal G}_M \\equiv \\rho e^{i\\chi}.\n\\label{eq:rho_chi_defn}\n\\end{equation}\nThe corresponding relation between the Pauli coupling constant $\\kappa$ and this Sachs \nform factor ratio is\n\\begin{equation}\n\\kappa \\equiv |\\kappa|\\, e^{i\\phi_{\\kappa}} = \n\\frac {\\rho e^{i\\chi} - 1}{(M^2\/4m^2 - \\rho e^{i\\chi})}\n\\label{eq:kappa}\n\\end{equation}\nwhere we have assumed that we are on a narrow resonance, so we\ncan replace $s$ by $M^2$.\n\nWe will first consider the unpolarized process \n$e^+e^-\\to \\psi \\to p \\bar p$, and establish what the differential and total \ncross sections imply regarding the $\\psi p \\bar p$ vertex.\nThe unpolarized total cross section predicted by Fig.\\ref{fig:diag} is \n\\begin{equation}\n\\langle \\sigma \\rangle =\n\\frac{4\\pi\\alpha^2}{3f_{\\psi}^2}\n\\frac{M^4}{s^2}\n\\frac{(1-4m^2\/s)^{1\/2}} {[ (s-M^2)^2 + \\Gamma^2 M^2 ]}\\,\n(2m^2 {|\\cal G}_E|^2 + s {|\\cal G}_M|^2).\n\\label{eq:sigma}\n\\end{equation}\n(We use angle brackets to denote a polarization averaged quantity.)\nExactly on resonance (at $s=M^2$) this can be\nexpressed in terms of the $\\psi$ partial widths\n\\begin{equation}\n{\\hskip -4.7cm}\n\\Gamma_{\\psi \\to e^+e^-} = \\frac{4\\pi \\alpha^2 M}{3 f_{\\psi}^2}\n\\label{eq:psi_ee_width}\n\\end{equation}\nand\n\\begin{equation}\n\\Gamma_{\\psi \\to p\\bar p } = \\frac{(1-4m^2\/M^2)^{1\/2}}{12\\pi M}\\, (2m^2 |{\\cal G}_E|^2 + M^2 |{\\cal G}_M|^2),\n\\label{eq:psi_pp_width}\n\\end{equation}\nwhich gives the familiar result\n\\begin{equation}\n\\langle \\sigma \\rangle \\big|_{s=M^2} = \\frac{12\\pi}{M^2} \\, B_{e^+e^-} B_{p\\bar p }.\n\\label{eq:sigma_onres}\n\\end{equation}\nHere $B_{e^+e^-}$ and $B_{p\\bar p }$ are the\n$\\psi \\to e^+e^-$ and $\\psi \\to p\\bar p$ branching fractions.\n\nSince the (unpolarized) $p\\bar p$ width and total cross section on\nresonance involve only the\nsingle linear combination $(2m^2 |{\\cal G}_E|^2 + M^2 |{\\cal G}_M|^2)$,\nseparating these two strong form factors requires additional information,\nsuch as the angular distribution.\nThe unpolarized $e^+e^- \\to \\psi \\to p\\bar p$ differential cross\nsection in the c.m. frame is given by\n\\begin{displaymath}\n\\langle \\frac{d\\sigma}{d\\Omega}\\rangle =\n\\frac{\\alpha^2}{4f_{\\psi}^2}\n\\frac{M^4}{s^2}\n\\frac{(1-4m^2\/s)^{1\/2}}\n{[ (s-M^2)^2 + \\Gamma^2 M^2 ] }\n\\end{displaymath}\n\\begin{equation}\n\\cdot \\ \\bigg[ 4 m^2 |{\\cal G}_E|^2(1-\\mu^2) + s |{\\cal G}_M|^2 (1+\\mu^2)\\bigg],\n\\label{eq:dsigma_dOmega}\n\\end{equation}\nwhere $\\mu = \\cos(\\theta_{c.m.}).$\nThis angular distribution is often expressed as\n$1+\\alpha \\mu^2$, where\n\\begin{equation}\n\\alpha =\n\\frac{1-(4m^2\/s)\\big|{\\cal G}_E\/{\\cal G}_M\\big|^2}\n {1+(4m^2\/s)\\big|{\\cal G}_E\/{\\cal G}_M\\big|^2}.\n\\label{eq:alpha_general_Sachs}\n\\end{equation}\nInspection of Eqs.(\\ref{eq:dsigma_dOmega},\\ref{eq:alpha_general_Sachs}) shows that \none can determine the magnitude $\\rho = | {\\cal G}_E \/ {\\cal G}_M |$\nof the Sachs form factor ratio from the unpolarized\ndifferential cross section, but that the phase $\\chi$ of ${\\cal G}_E \/ {\\cal G}_M$ \nis unconstrained. \n\nThe undetermined phase $\\chi$ implies an unavoidable ambiguity in determining the magnitude and phase \nof the Dirac and Pauli $\\psi p \\bar p$ couplings $g$ and $\\kappa$\nfrom the unpolarized $e^+e^- \\to (J\/\\psi, \\psi') \\to p\\bar p $ angular distribution. We will discuss \nthis ambiguity in the next section.\n\n\\section{Comparison with Experiment}\n\n\\subsection{Summary of the data}\n\nExperimental values of $\\alpha$ have been reported by several collaborations. \nThe results for the $J\/\\psi$ are\n\\begin{equation}\n\\alpha =\n\\cases\n{\n1.45\\pm 0.56, &MarkI~\\cite{Peruzzi:1977pb} \\cr\n1.7\\pm 1.7, &DASP~\\cite{Brandelik:1979hy} \\cr\n0.61\\pm 0.23, &MarkII~\\cite{Eaton:1983kb} \\cr\n0.56\\pm 0.14, &MarkIII~\\cite{Brown:1984jq} \\cr\n0.62\\pm 0.11, &DM2~\\cite{Pallin:1987py} \\cr\n0.676\\pm 0.036 \\pm 0.042, &BES~\\cite{Bai:2004jg}.\n}\n\\label{eq:alpha_Jpsi_expt}\n\\end{equation}\nand for the $\\psi'$\n\\begin{equation}\n\\hskip -1.1cm\n\\alpha =\n\\cases\n{\n0.67\\pm 0.15 \\pm 0.04, {\\hskip 0.5cm} &E835~\\cite{Ambrogiani:2004uj} \\cr\n0.85\\pm 0.24 \\pm 0.04, &BES~\\cite{Ablikim:2006aw}.\n}\n\\label{eq:alpha_psip_expt}\n\\end{equation}\nFor our comparison with experiment we use the statistically most\naccurate measurement for each charmonium state, and combine the errors in quadrature.\nThis gives experimental estimates for $\\alpha$ of $0.676\\pm 0.055$ and\n$0.67\\pm 0.155$ for the $J\/\\psi$ and $\\psi'$ respectively.\n\n\\subsection{Testing the pure Dirac hypothesis}\n\n\\begin{figure}[b]\n\\vskip 0.5cm\n\\includegraphics[width=0.8\\linewidth]{blr_fig2.eps}\n\\caption{The coefficient $\\alpha$ observed in the unpolarized\n$e^+e^-\\to (J\/\\psi, \\psi')\\to p\\bar p$ angular distributions,\ntogether with the theoretical result $\\alpha = (1-r)\/(1+r)$ predicted by a pure\nDirac $(\\gamma_{\\mu})$ $\\psi p\\bar p$ coupling.}\n\\label{fig:alpha_nokappa}\n\\end{figure}\n\nWe first examine these experimental numbers using the\n``null hypothesis\" of no Pauli term, $\\kappa = 0$, in which case\n$\\alpha = (1-r)\/(1+r)$, where $r=4m^2\/M^2$.\nThis $\\kappa = 0$ formula was previously given by Claudson, Glashow and\nWise~\\cite{Claudson:1981fj} and by Carimalo~\\cite{Carimalo:1985mw};\nthe value of $\\alpha$ under various theoretical assumptions has been\ndiscussed by these references and by Brodsky and LePage~\\cite{Brodsky:1981kj},\nwho predicted $\\alpha=1$.\nFig.\\ref{fig:alpha_nokappa} shows these two experimental values\ntogether with the pure Dirac $(\\gamma_{\\mu})$ formula for $\\alpha$.\nThe $\\psi'$ case is evidently consistent with a Dirac\n$(\\gamma_{\\mu})$ $\\psi'p\\bar p$ coupling at present accuracy,\nbut the better determined $J\/\\psi$ angular distribution is\ninconsistent with a pure Dirac $J\/\\psi\\, p\\bar p$ coupling at the $4\\sigma$ level.\n\nThe discrepancy evident in Fig.\\ref{fig:alpha_nokappa}\nmay imply the presence of a Pauli term $(\\kappa \\neq 0)$ in the\n$J\/\\psi\\, p \\bar p$ vertex. Inspection of our result\nfor $\\alpha$ in the general case (Eq.\\ref{eq:alpha_general_Sachs})\nshows that one can certainly accommodate this discrepancy by introducing a Pauli term.\n\n\\subsection{Determining $\\rho = |{\\cal G}_E\/{\\cal G}_M|$ from $\\alpha$}\n\nThe dependence of the predicted $\\alpha$ on $\\rho$ at the $J\/\\psi$ mass\n(from Eq.\\ref{eq:alpha_general_Sachs})\nis shown in Fig.\\ref{fig:alpha_kappa}. \nThe experimental value $\\alpha = 0.676\\pm 0.055$ (shown) is consistent with the Sachs form factor\nmagnitude ratio of\n\\begin{equation}\n\\rho = |{\\cal G}_E\/{\\cal G}_M| = 0.726\\pm 0.074.\n\\label{eq:rho_solns}\n\\end{equation}\n\n\\begin{figure}[h]\n\\vskip 0.5cm\n\\includegraphics[width=0.8\\linewidth]{blr_fig3.eps}\n\\caption{The experimental value of the unpolarized $e^+e^-\\to J\/\\psi \\to p\\bar p$ \nangular coefficient, $\\alpha = 0.676\\pm 0.055$ (shaded), and the resulting Sachs \n$J\/\\psi p\\bar p$ strong form factor magnitude ratio $\\rho = |{\\cal G}_E\/{\\cal G}_M|$ \n(Eq.\\ref{eq:rho_solns}).}\n\\label{fig:alpha_kappa}\n\\end{figure}\n\nIn terms of $\\rho$ and $\\chi$ this completes our discussion: Given the \nunpolarized angular distribution, one obtains a result for \n$\\rho = | {\\cal G}_E\/{\\cal G}_M |$ from Eq.\\ref{eq:alpha_general_Sachs}, \nbut the phase $\\chi$ of ${\\cal G}_E\/{\\cal G}_M$ is undetermined. \nHowever one may ask the more fundamental question of what values of \nthe Dirac and Pauli coupling constants $g$ and $\\kappa$ in Eq.\\ref{eq:vertex} \nare consistent with a given experimental unpolarized angular distribution. \n\n\\subsection{Determining $\\kappa$}\n\nFirst we consider the experimentally allowed values of $\\kappa$.\nThe unpolarized angular distribution provides us with a range of values \nof $\\rho$ (Eq.\\ref{eq:rho_solns}), but $\\chi$ is unconstrained; \nwe may combine this information through Eq.\\ref{eq:rho_chi_defn} \nto determine the locus of allowed (complex) values of $\\kappa$. \nThis is shown in Fig.\\ref{fig:blr_fig4}. \n\n\\begin{figure}[h]\n\\vskip 0.5cm\n\\includegraphics[width=0.8\\linewidth]{blr_fig4.eps}\n\\caption{The locus of complex $\\kappa$ (the $J\/\\psi p \\bar p$ Pauli coupling) \nallowed by the experimental constraint $\\rho = 0.726 \\pm 0.074$, taken from the \nunpolarized differential cross section for $e^+e^- \\to J\/\\psi \\to p \\bar p$.} \n\\label{fig:blr_fig4}\n\\end{figure}\n\nFor $\\chi = 0$, Eq.\\ref{eq:rho_chi_defn} implies that $\\kappa$ is real \nand negative, and takes on the smallest allowed magnitude. \nAs we increase $\\chi$ from $0$, the allowed $\\kappa$ values proceed \nclockwise, since $\\kappa$ initially acquires a \nnegative imaginary part. The extreme values of $\\kappa$ on the real axis in \nFig.\\ref{fig:blr_fig4} are for $\\chi = 0,\\pi$, and are \n\\begin{equation}\n\\kappa =\n\\cases\n{\n-0.137 \\pm 0.032, & $\\chi = 0$ \\cr\n-0.500 \\pm 0.011, & $\\chi = \\pi$.\n}\n\\label{eq:kappa_solns}\n\\end{equation}\n\n\\subsection{Determining $g$}\n\nNext we consider the determination of the overall \n$J\/\\psi \\to p \\bar p$ vertex strength $g$.\nSince the differential and total cross sections for \n$e^+e^- \\to J\/\\psi \\to p \\bar p$ only involve $g$ through the ratio \n$g\/f_{\\psi}$,\nwe must introduce additional experimental data to constrain $g$. \nThe partial width for $J\/\\psi \\to p \\bar p$ is especially \nconvenient in this regard,\nsince it only involves the strong $J\/\\psi p \\bar p$ vertex, and thus depends \nonly on $g$ and $\\kappa$ (and kinematic factors). This partial width was\ngiven in terms of the strong Sachs form factors in Eq.\\ref{eq:psi_pp_width};\nas a function of $g$ and $\\kappa$ it is\n\\begin{equation}\n\\Gamma_{\\psi \\to p\\bar p} = \\frac{1}{3}\\frac{g^2}{4\\pi} M \\sqrt{1-r}\\,\n\\bigg[\n1 + \\frac{r}{2} + 3\\Re(\\kappa) + \\Big(1+\\frac{1}{2r}\\Big)|\\kappa|^2\n\\bigg].\n\\label{eq:psi_ppbar}\n\\end{equation}\n\nThis generalizes the $\\kappa=0$ result given in Eq.27 of \nRef.\\cite{Barnes:2006ck} to a nonzero Pauli coupling.\nUsing the PDG values \\cite{Yao:2006px}\nof $\\Gamma_{J\/\\psi} = 93.4 \\pm 2.1$~keV\nand\n$B_{J\/\\psi\\to p\\bar p}=(2.17 \\pm 0.07)\\cdot 10^{-3}$,\nEq.\\ref{eq:psi_ppbar} imples a range of values of the overall vertex \nstrength $g$ for each value of the (unknown) phase $\\chi$. \nThis is shown in Fig.\\ref{fig:blr_fig5}. There is a range of \nuncertainty in $g$ at each $\\chi$ (not shown in the figure), \ndue to the experimental errors in $\\Gamma_{J\/\\psi}$, \n$B_{J\/\\psi\\to p\\bar p}$ and $\\rho$, which is at most $\\sim \\pm 5 \\%$. \n\n\\begin{figure}[h]\n\\vskip 0.5cm\n\\includegraphics[width=0.8\\linewidth]{blr_fig5.eps}\n\\caption{The value of the overall $J\/\\psi p \\bar p$ vertex strength \n$g$ implied by the experimental $\\Gamma_{J\/\\psi\\to p\\bar p}$ and \n$\\rho$ as a function of the unknown $J\/\\psi p \\bar p$ Sachs phase $\\chi$.}\n\\label{fig:blr_fig5}\n\\end{figure}\n\nNote that $g$ is bounded by the limits at $\\chi = 0$ and $\\pi$, \nfor which $g \\approx 2.0 \\cdot 10^{-3}$ and $\\approx 3.4 \\cdot 10^{-3}$ \nrespectively. The allowed values of $g$ are somewhat larger than \nour previous estimate of \n$g = (1.62 \\pm 0.03)\\cdot 10^{-3}$~\\cite{Barnes:2006ck} \nassuming only a Dirac $J\/\\psi p \\bar p$ coupling, as a result of\ndestructive interference between the Pauli and Dirac terms.\n\n\\section{Effect on $\\sigma(p\\bar p \\to \\pi^0 J\/\\psi)$}\n\nThe effect of a $J\/\\psi p\\bar p$ Pauli term on the $p\\bar p \\to \\pi^0 J\/\\psi$ \ncross section may be of considerable interest for the PANDA project, since \none might use this as a ``calibration\" reaction for associated \ncharmonium production, and the Pauli term may be numerically important.\nAlthough we have carried out this calculation with the vertex\nof Eq.\\ref{eq:vertex} for general masses, the full result is rather complicated;\nhere for illustration we discuss the much simpler massless pion limit.\n\n\\begin{figure}[b]\n\\vskip 0.5cm\n\\includegraphics[width=0.8\\linewidth]{blr_fig6.eps}\n\\caption{The dependence of the unpolarized, near-threshold cross section \n$\\langle \\sigma (p\\bar p \\to \\pi^0 J\/\\psi )\\rangle$ \non the (complex) Pauli coupling \n$\\kappa = |\\kappa| e^{i\\phi_{\\kappa}}$ \n(from Eq.\\ref{eq:kappa_effect}).}\n\\label{fig:sigma_chi}\n\\end{figure}\nFor a massless pion the ratio of the unpolarized cross section\n$\\langle \\sigma(p\\bar p \\to \\pi^0 J\/\\psi) \\rangle$\nwith a Pauli term to the pure Dirac result \n($\\gamma_{\\mu}$ only, denoted by $D$) is\n\n\\begin{displaymath}\n\\frac{\\langle \\sigma(p\\bar p \\to \\pi^0 J\/\\psi )\\rangle_{\\phantom{D}}}\n{\\langle \\sigma(p\\bar p \\to \\pi^0 J\/\\psi )\\rangle_D}\\bigg|_{m_{\\pi}=0}\n\\!\\!\\! = \\bigg[1 + 2\\Re(\\kappa) + \n\\Big(\\frac{1}{2}+\\frac{M^2}{8m^2}\\Big) |\\kappa|^2\n\\end{displaymath}\n\\begin{equation}\n+\\,\n\\frac{(s-M^2)}{4m^2}\n\\,\n\\frac{\\beta}{\\ln\\big[(1+\\beta)\/(1-\\beta)\\big]}|\\kappa|^2\\,\\bigg],\n\\label{eq:kappa_effect}\n\\end{equation}\nwhere $\\beta = \\sqrt{1-4m^2\/s}$ is the velocity of the annihilating $p$ and\n$\\bar p$ in the c.m. frame. The limit of this cross section ratio \nat threshold is shown in Fig.\\ref{fig:sigma_chi} for a range of \ncomplex~$\\kappa$. \n\nEvidently there is destructive interference for a $\\kappa$ with a dominant\nnegative real part, as is suggested by the unpolarized data. For the value\n$\\kappa = -0.50$ (the larger solution in Eq.\\ref{eq:kappa_solns}) there is\nroughly a factor of two suppression in the cross section over the prediction\nfor a pure Dirac coupling. The suppression however depends strongly on the \nphase of $\\kappa$, and for imaginary $\\kappa$ has become a moderate \nenhancement. Thus, the near-threshold cross section for \n$p\\bar p \\to \\pi^0 J\/\\psi$ is quite sensitive to the strength and phase of the \nPauli coupling; it will therefore be important for PANDA to have an accurate estimate\nof this quantity. In the next section we will show how both the magnitude \nand phase of $\\kappa$ can be determined in polarized \n$e^+e^- \\to J\/\\psi \\to p\\bar p$ scattering, and may be accessible at BES. \n \n\\section{Polarization Observables}\n\nThe relative phase $\\chi$ of the \n$J\/\\psi p\\bar p$ Sachs strong form factors \n${\\cal G}_E$ and ${\\cal G}_M$ \nmay be determined\nexperimentally through a study\nof the polarized process $e^+e^- \\to J\/\\psi \\to p\\bar p$. \nAs each of the external particles in this reaction has two \npossible helicity states, there are 16 helicity amplitudes in total.\nAll the helicity amplitudes to the final $p\\bar p$ helicity states\n$|p(\\pm)\\bar p (\\pm)\\rangle $ are proportional to ${\\cal G}_E$, and\nall to $|p(\\pm)\\bar p (\\mp)\\rangle $ are proportional to ${\\cal G}_M$.\nIn the unpolarized case these are squared and summed,\nwhich leads to a cross section proportional to a weighted sum of\n$|{\\cal G}_E|^2$ and $|{\\cal G}_M|^2$. As we stressed earlier, this implies that \nthe phase $\\chi$ of ${\\cal G}_E \/ {\\cal G}_M$ is not determined by the unpolarized data.\n\n\\begin{figure}[h]\n\\includegraphics[width=0.8\\linewidth]{blr_fig7.eps}\n\\caption{Axes used to define the polarization observables.}\n\\label{fig:axes}\n\\end{figure}\n\nTo show how $\\chi$ can be measured in polarized scattering, \nit is useful to introduce the polarization observables discussed by \nPaschke and Quinn~\\cite{Paschke:2000mx}. These are angular asymmetries \nthat arise when the polarizations of particles are aligned or \nanti-aligned along particular directions. For example, for our reaction\n$e^+ e^- \\to p\\bar p$, $Q(0,0,z,0)$ is the difference of \ntwo angular distributions, $(d\\sigma \/ d\\Omega)_{p\\uparrow} - \n(d\\sigma \/ d\\Omega)_{p\\downarrow}$.\nHere we will use $x$ and $y$ for the two transverse axes and \n$z$ for the longitudinal axis (see Fig.\\ref{fig:axes}). \n$\\hat x$ and $\\hat z$ vary with the particle, and\n$\\hat y$ is chosen to be common to all. An entry of $0$ signifies an \nunpolarized particle. Since there are four possible arguments for \neach particle, $0,x,y$ and $z$, there are $4^4 = 256$ polarization \nobservables for this process. Of course there is considerable redundancy, \nsince they are all determined by the 16 helicity amplitudes. \nThe constraints of parity and charge conjugation reduce this set to \n6 independent helicity amplitudes, and for massless leptons (as we \nassume here) this is further reduced to 3 independent nonzero helicity \namplitudes.\n\nWe introduce the normalized polarization observables \n${\\cal Q}_{\\epsilon_{e^+}\\epsilon_{e^-}\\epsilon_{\\bar p}\\epsilon_p}$\n$\\equiv $\n$Q({\\epsilon_{e^+},\\epsilon_{e^-},\\epsilon_{\\bar p}, \\epsilon_p})$ \/\n$Q(0,0,0,0)$,\nwhere $Q(0,0,0,0)$ is the unpolarized differential cross section.\nThe (nonzero) polarization observables for this process satisfy the relations\n\\begin{itemize}\n\\renewcommand{\\labelitemi}{$(a)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{0 0 0 0} =\n\\phantom{-} {\\cal Q}_{x x y y} =\n\\phantom{-} {\\cal Q}_{y y y y} =\n-{\\cal Q}_{z z 0 0} =\n1,\n$\n\\renewcommand{\\labelitemi}{$(b)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{0 0 y 0}=\n\\phantom{-} {\\cal Q}_{x x 0 y}=\n\\phantom{-} {\\cal Q}_{y y 0 y}=\n\\phantom{-} {\\cal Q}_{z z 0 y}=\n$\n\\renewcommand{\\labelitemi}{}\n\\item\n$\n-{\\cal Q}_{0 0 0 y}=\n-{\\cal Q}_{x x y 0}=\n-{\\cal Q}_{y y y 0}=\n-{\\cal Q}_{z z y 0},\n$\n\\renewcommand{\\labelitemi}{$(c)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{x x 0 0} =\n\\phantom{-} {\\cal Q}_{y y 0 0} =\n\\phantom{-} {\\cal Q}_{0 0 y y} =\n-{\\cal Q}_{z z y y},\n$\n\\renewcommand{\\labelitemi}{$(d)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{z 0 x 0} =\n\\phantom{-} {\\cal Q}_{z 0 0 x} =\n\\phantom{-} {\\cal Q}_{y x y z} =\n\\phantom{-} {\\cal Q}_{y x z y} =\n$\n\\renewcommand{\\labelitemi}{}\n\\item\n$\n-{\\cal Q}_{0 z x 0} =\n-{\\cal Q}_{0 z 0 x} =\n-{\\cal Q}_{x y y z} =\n-{\\cal Q}_{x y z y},\n$\n\\renewcommand{\\labelitemi}{$(e)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{z 0 z 0} =\n\\phantom{-} {\\cal Q}_{0 z 0 z} =\n\\phantom{-} {\\cal Q}_{x y x y} =\n\\phantom{-} {\\cal Q}_{y x y x} =\n$\n\\renewcommand{\\labelitemi}{}\n\\item\n$\n-{\\cal Q}_{0 z z 0} =\n-{\\cal Q}_{z 0 0 z} =\n-{\\cal Q}_{y x x y} =\n-{\\cal Q}_{x y y x},\n$\n\\renewcommand{\\labelitemi}{$(f)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{x x x x} =\n\\phantom{-} {\\cal Q}_{y y x x} =\n\\phantom{-} {\\cal Q}_{z z z z} =\n-{\\cal Q}_{0 0 z z},\n$\n\\renewcommand{\\labelitemi}{$(g)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{0 0 x x} =\n-{\\cal Q}_{z z x x} =\n-{\\cal Q}_{x x z z} =\n-{\\cal Q}_{y y z z},\n$\n\\renewcommand{\\labelitemi}{$(h)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{0 0 x z} =\n\\phantom{-} {\\cal Q}_{x x z x} =\n\\phantom{-} {\\cal Q}_{y y z x} =\n\\phantom{-} {\\cal Q}_{z z z x} =\n$\n\\renewcommand{\\labelitemi}{}\n\\item\n$\n-{\\cal Q}_{x x x z}=\n-{\\cal Q}_{y y x z}=\n-{\\cal Q}_{z z x z}=\n-{\\cal Q}_{0 0 z x},\n$\n\\renewcommand{\\labelitemi}{$(i)$}\n\\item\n$\n\\phantom{-} {\\cal Q}_{x y x 0}=\n\\phantom{-} {\\cal Q}_{x y 0 x}=\n\\phantom{-} {\\cal Q}_{z 0 y z}=\n\\phantom{-} {\\cal Q}_{z 0 z y}=\n$\n\\renewcommand{\\labelitemi}{}\n\\item\n$\n-{\\cal Q}_{y x x 0}=\n-{\\cal Q}_{y x 0 x}=\n-{\\cal Q}_{0 z y z}=\n-{\\cal Q}_{0 z z y}.\n$\n\\end{itemize}\n\n\\vskip -1.5cm\n\\begin{equation}\n\\label{eq:relations}\n\\end{equation}\nExplicit expressions for these observables are given in \nTable~\\ref{table:observables}.\n\n\\begin{table}[h]\n\\caption{Nonzero inequivalent polarization observables in $e^+e^-\\to J\/\\psi\\to p{\\bar p}$.\nThe function $F$ is $4 - 2(1-\\rho^2) \\sin^2{\\theta}$.}\n\\begin{center}\n\\begin{tabular}{c|c}\n\\hline\nPol. Observable & Result \\\\\n\\hline\n${\\cal Q}_{0 0 0 0}$ & $1$ \\\\\n${\\cal Q}_{0 0 y 0}$ & $4\\rho\\,\\sin{\\chi} \\sin{\\theta} \\cos{\\theta}\/F$\\\\\n${\\cal Q}_{x x 0 0}$ & $2(1-\\rho^2)\\sin^2{\\theta}\/F$ \\\\\n${\\cal Q}_{z 0 x 0}$ & $4\\rho\\,\\cos{\\chi} \\sin{\\theta}\/F$ \\\\\n${\\cal Q}_{z 0 z 0}$ & $4\\cos{\\theta}\/F$ \\\\\n${\\cal Q}_{0 0 x z}$ & $-4\\rho\\,\\cos{\\chi} \\sin{\\theta} \\cos{\\theta}\/F$ \\\\\n${\\cal Q}_{x y x 0}$ & $-4\\rho\\,\\sin{\\chi} \\sin{\\theta}\/F$ \\\\\n${\\cal Q}_{x x x x}$ & $[4 - 2(1+\\rho^2) \\sin^2{\\theta}]\/F$ \\\\\n${\\cal Q}_{z z x x}$ & $-2(1 + \\rho^2)\\sin^2{\\theta}\/F$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{table:observables}\n\\end{table}\n\nThe results in Table~\\ref{table:observables} suggest how we may\ndetermine $\\chi$ experimentally. \nInspection of the table shows that only four of the independent \npolarization observables depend on $\\chi$; two are proportional to \n$\\sin{\\chi}$ and two to $\\cos{\\chi}$. \nAssuming that one knows $\\rho$ with sufficient accuracy from the unpolarized data, \none may then determine $\\chi$ unambiguously by extracting $\\sin{\\chi}$ and $\\cos{\\chi}$\nfrom the measurement of two of these polarization observables. \n\nThe determination of \n$\\sin{\\chi}$ is the most straightforward, since it only requires the detection\nof a single final polarized particle (for example the proton, through ${\\cal Q}_{0 0 y 0}$).\nIf $\\kappa$ is close to real, which corresponds to $\\chi \\approx 0$ or $\\approx \\pi$,\nthis observable may be relatively small. The other polarization observables that are \nproportional to $\\sin{\\chi}$ involve asymmetries with either one or three particles\npolarized; these are given in relations ($b$) and ($i$) of Eq.\\ref{eq:relations}.\n\nDetermining $\\cos{\\chi}$ involves measuring double or quadruple polarization \nobservables, which are given in relations ($d$) and ($h$) in Eq.\\ref{eq:relations}.\nIn the double polarization case, either one initial and one final polarization \nare measured (such as $e^-$ and $p$) or the polarizations of both final particles \n($p$ and ${\\bar p}$) are measured.\nIn the first case the relevant observables (such as ${\\cal Q}_{z 0 x 0}$)\nrequire the initial lepton to have longitudinal ($\\pm \\hat z$) polarization,\nwhich is difficult to achieve experimentally. In the second case the initial $e^+e^-$ \nbeams are unpolarized, and the longitudinal polarization of one final particle and \nthe transverse polarization of the other must be measured. Determining the ${\\bar p}$ \npolarization may prove to be an experimental challenge.\n\nOf these two general possibilities, the most attractive ``next experiment\" beyond \nunpolarized $e^+ e^- \\to J\/\\psi \\to p\\, \\bar p$ scattering may be a \nmeasurement of the differential cross section with unpolarized leptons and\nonly the final $p$ polarization detected. This will determine $\\sin{\\chi}$, \nwhich specifies $\\chi$ up to the usual trigonometric ambiguities. \n\nAnother interesting experimental possibility is to resolve the phase ambiguity \nin unpolarized $e^+ e^- \\to J\/\\psi \\to p\\, \\bar p$ scattering through a study of the \nclosely related reaction $e^+ e^- \\to J\/\\psi \\to \\Lambda \\bar \\Lambda $,\nwhich has recently been observed by BABAR~\\cite{Aubert:2007uf} using the ISR technique.\nSince the $\\psi p \\bar p$ and $\\psi \\Lambda \\bar \\Lambda $ couplings are identical in the SU(3)\nflavor symmetry limit, a determination of $J\/\\psi \\Lambda \\bar \\Lambda $ couplings\nwould suggest plausible $J\/\\psi p \\bar p$ couplings.\nThis approach has some experimental advantages; as the $\\Lambda$ and ${\\bar \\Lambda}$\ndecays are self-analyzing, no rescattering of the final baryons is required\nto determine their polarization. In addition no beam polarization is required,\nsince it suffices to measure the (odd-$\\rho$) polarization observables ${\\cal Q}_{0 0 y 0}$ and \n${\\cal Q}_{0 0 x z}$. One may also measure the even-$\\rho$ observables\n${\\cal Q}_{0 0 x x}$ and ${\\cal Q}_{0 0 z z}$ as a cross-check of the\nresult for $\\rho$. \n\nFinally, we note in passing that it may also be possible to measure \nthe appropriate polarization observables in the time-reversed reaction \n$p\\bar p\\to J\/\\psi \\to e^+e^-$.\n\n\n\\section{Summary and Conclusions}\n\nThe unpolarized angular distribution for the process \n$e^+ e^- \\to J\/\\psi \\to p\\, \\bar p$, measured recently by the \nBES Collaboration, is inconsistent with theoretical expectations\nfor a pure Dirac $J\/\\psi p \\bar p$ coupling. In this paper we have \nderived the effect of an additional Pauli-type $J\/\\psi p\\bar p$ coupling, \nand find that this can accommodate the observed angular distribution. \nThe $J\/\\psi p \\bar p$ Pauli coupling may significantly affect the cross section\nfor the charmonium production reaction $ p\\bar p \\to \\pi^0 J\/\\psi$, which will be\nstudied at PANDA.\nThere is an ambiguity in determining the relative Dirac and Pauli\n$J\/\\psi p \\bar p$ couplings from the unpolarized\n$e^+ e^- \\to J\/\\psi \\to p\\, \\bar p$ data; \nwe noted that this ambiguity can be fully resolved through measurements \nof the polarized reaction. The most attractive polarized process to study \ninitially appears to be the case of unpolarized initial $e^+e^-$ beams,\nwith only the final $p$ (transversely) polarized. \nAlternatively, measurement of the required polarization observables may also \nbe possible using the time-reversed reaction\n$p\\bar p \\to J\/\\psi \\to e^+e^-$. \nIt may also be possible to use self-analyzing processes such as\n$e^+ e^- \\to J\/\\psi \\to \\Lambda \\bar \\Lambda $ to estimate the Dirac and Pauli\ncouplings in the closely related $J\/\\psi \\Lambda \\bar \\Lambda $ vertex. \n\n\\section{Acknowledgements}\n\nWe are happy to acknowledge useful communications with\nW.M.Bugg, V.Ciancolo, F.E.Close, S.Olsen, J.-M.Richard, K.Seth, S.Spanier, \nE.S.Swanson, U.Wiedner,C.Y.Wong and Q.Zhao\nregarding this research. \nWe also gratefully acknowledge the support of the Institute of High Energy Physics \n(Beijing) of the Chinese Academy of Sciences, the Department of Physics and \nAstronomy at the University of Tennessee, and the Department of Physics, \nthe College of Arts and Sciences, \nand the Office of Research at Florida State University.\nThis research was supported in part by the U.S. Department of Energy \nunder contract DE-AC05-00OR22725 at Oak Ridge National Laboratory.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMany questions in physics are easy to pose but difficult to answer.\nOne such question is: how many microscopic states of a given system\nare compatible with its macroscopic properties? In statistical\nmechanics, knowledge of this number allows us to compute the entropy,\nand thereby predict the macroscopic properties of a system from\nknowledge of the interaction between atoms or molecules.\n\nIn granular matter we can similarly ask how many microstates are\ncompatible with a given set of macroscopic properties. However, the\ncomputation of the corresponding absolute entropy has thus far proven\nto be extremely challenging. Without such knowledge, it is not\npossible to explore the analogies and differences between granular and\nBoltzmann entropy. Being able to compute the configurational entropy\nis therefore clearly important. The more so as granular materials are\nubiquitous in everyday life (sand, soil, powders). Many industrial\nprocesses involve granular materials. In the natural world, the\nEarth's surface contains vast granular assemblies such as dunes, which\ninteract with wind, water, and vegetation \\cite{Kok12}. Packings of\nparticles that are soft or biological in nature, such as cells,\nhydrogels and foams are also known to undergo jamming\n\\cite{van2010jamming} and their behaviour to be ``granular''\n\\emph{viz} not subject to thermal motion. Moreover, as glasses and\ngranular materials share many properties it has been proposed that\ntheir physics may be controlled by the same underlying principles\n\\cite{Liu98}.\n\nThe study of granular materials is complicated by the fact that these\nmaterials are intrinsically out-of-equilibrium. In fact, thermal\nmotion plays no role in granular matter. It maintains its\nconfiguration unless driven by external forces. As a consequence, the\nproperties of granular materials depend upon their preparation\nprotocol.\n\nGranular materials are athermal and cannot therefore be described by\nstatistical mechanics. However, these materials can exists in a very\nlarge number of distinct states and this fact inspired Edwards and\nOakeshott \\cite{Edwards89} well over two decades ago to propose a\nstatistical-mechanics-like formalism to describe the properties of\ngranular matter. In its original version, the Edwards theory assumed\nthat all mechanically stable configurations (`jammed' states) are\nequally probable and that the logarithm of the number of these states\nplays a role similar to that of entropy. In this theoretical\nframework, the volume of the system and its \\emph{compactivity}\n(i.e. the derivative of volume with respect to the configurational\nentropy) are the analogues of the energy and temperature in thermal\nsystems.\n\nIn the absence of explicit calculations (or measurements) of the\nabsolute configurational entropy, a direct test of the Edwards\nhypothesis has proven difficult, and different authors have arrived at\ndifferent conclusions based on indirect tests in either\nsimulations~\\cite{Makse02, Song05, Barrat01, Gao06} or\nexperiments~\\cite{Lechenault06, Daniels08}. In addition, alternative\ndefinitions of entropy have been proposed to characterise the\ncomplexity of granular systems while circumventing explicit\nenumeration of states \\cite{Brujic07, Baranau13}.\n\nNumerous tests of the Edwards volume ensemble have focused on the\ndetermination of the compactivity~\\cite{Nowak98, Schroter05,\n Mcnamara09, Dean03, Zhao12, Aste08, Blumenfeld03, Blumenfeld12,\n Zhao14}. However, the role of compactivity as a temperature-like\nquantity is problematic as Puckett and Daniels \\cite{Puckett13,Bi15}\nhave shown that it does not satisfy the equivalent of the zero-th law\nof Thermodynamics - the law that is the basis of all thermometry.\n\nEdwards' theory has been generalised to include the distribution of\nstresses within the system through the force-moment tensor\n\\cite{Briscoe08, Blumenfeld09, Henkes07, Henkes09} and another\nanalogue of temperature emerged, known as \\emph{angoricity}, which is\na measure of the change in entropy with stress. The experiments by\nPuckett and Daniels \\cite{Puckett13} showed that angoricity, unlike\ncompactivity, is a temperature-like quantity as it satisfies the\nzero-th law.\n\nTo date only a few examples of numerical tests of the generalised\nEdwards ensemble are available \\cite{Henkes07, Wang10, Wang12,\n Puckett13}. Numerical tests of the stress ensemble focus on systems\nof soft spheres near jamming where the compactivity $X \\to \\infty$ and\nfluctuations in volume are negligible compared to stress fluctuations\n\\cite{Henkes07, Henkes09}. Wang \\emph{et al.} \\cite{Wang10, Wang12}\nproposed a unified test that compared ensemble averaged results over\nvolume and stress with predictions for the jamming transition, finding\nagreement; we note, however, that in the latter approach the results\nrely significantly on the equiprobability assumption.\n\nWhen the system is composed of very stiff grains, or is close to\njamming, any small deformation will lead to a large change in the\ncontact forces. In these limits the geometric and the force degrees of\nfreedom can be decoupled, giving rise to the force network ensemble\n\\cite{Snoeijer04} (FNE). In this framework, force networks are\nconstructed on a given geometry and each force state is assumed to be\nequiprobable. The FNE has been utilised as a testing ground for\nstatistical frameworks \\cite{Van09, Tighe10, Tighe11}.\n\nMore than two decades after its introduction many fundamental\nquestions concerning the Edwards hypothesis remain unanswered. This\nunsatisfactory state of affairs is at least partly due to the fact\nthat no efficient methods existed to measure or compute the absolute\nconfigurational entropy directly. Until recently, the only way to\ndetermine the configurational entropy was by direct enumeration of the\ndistinct jammed states of a system. This method is inefficient and\ncannot be used for systems that contain more than 10-20\nparticles. Over the past few years, the situation on the numerical\nfront has changed: recent numerical work by Asenjo \\emph{et al.}\n\\cite{Asenjo14,Paillusson15}, based on an approach introduced by Xu\n\\emph{et al.} \\cite{Xu11}, has demonstrated that it is possible to\ncompute the number of distinct jammed states of a system, even when\nthis number is far too large (e.g. 10$^{250}$) to allow direct\nenumeration. The approach of Refs.~\\cite{Xu11, Asenjo14, Paillusson15}\nreplaces an intractable enumeration problem by a tractable scheme to\nsample the (absolute) volume of the basins of attraction of stable\nstates in the potential energy landscape.\n\nThe approach described herein is completely general and it extends to\nany energy landscape problem that aims to find the extrema of a scalar\ncost function that depends on many degrees of freedom. Enumerating the\nnumber of solutions or stationary points, and their distribution, for\ncertain classes of random functions is a classical problem in\nmathematics and statistics \\cite{kac1948average, farahmand1986average,\n bogomolny1992distribution, edelman1995many, rojas1996average,\n kostlan2000expected, malajovich2004high, azais2005roots,\n armentano2009random, fyodorov2012freezing, nicolaescu2012complexity,\n fyodorov2013high, cheng2015explicit, fyodorov2015number}. In\nstatistical physics, \\emph{ad hoc} numerical and theoretical methods\nhave been developed in the realms of random Gaussian and polynomial\nfields \\cite{fyodorov2004complexity, bray2007statistics,\n fyodorov2012critical, auffinger2013random, fyodorov2014topology,\n hughes2014inversion, mehta2014communication, cleveland2014certified,\n mehta2015statistics}. In this sense, particular attention has been\ndevoted to the mean-field p-spin spherical model\nof a spin glass with quenched disorder\n\\cite{cugliandolo1993analytical, kurchan1993barriers, mehta2013energy,\n auffinger2013complexity, subag2015complexity}. A related area is\nthe computation of the configurational contribution to the entropy of\nstructural glasses \\cite{sastry2001relationship,\n banerjee2014role}. The physical significance of this method goes\neven further to encompass string theory \\cite{distler2005random,\n douglas2007flux, greene2013tumbling}, cosmology\n\\cite{aazami2006cosmology, tye2009multi, frazer2011exploring,\n frazer2012multi, battefeld2012unlikeliness} and machine learning\n\\cite{wainrib2013topological, choromanska2014loss,\n sagun2014explorations, chaudhari2015trivializing}.\n\nWe note that the geometrical structure of the basins of attraction of\njammed states had been studied by O'Hern and co-workers \\cite{OHern03,\n Gao06, Ashwin12}. O'Hern also reported direct enumeration estimates\nof the number of jammed states of small systems. A rather different\ntechnique (`basin sampling') to count the number of energy minima in\nthe potential energy landscape of small clusters had been reported by\nWales and co-workers ~\\cite{Wales03, Ballard15}.\n\nWe note that, for the system (and protocol) considered by Asenjo\n\\emph{et al.}, not all packings are equally probable. However, as\nshown in Ref.~\\cite{Asenjo14}, the equal-probability hypothesis is not\nneeded to arrive at a meaningful definition of an extensive granular\nentropy. When, in the remainder of the present paper, we mention the\nconfigurational (granular) entropy, we refer to the definition of\nRef.~\\cite{Asenjo14}. \n\nWe stress that, even though the approach of Refs.~\\cite{Xu11,\n Asenjo14, Paillusson15} allows to solve enumeration problems which\nwere far from possible using direct enumeration, it is still\ncomputationally expensive. Thus far, it had only been applied to\ntwo-dimensional packings. Substantial `technical' improvements were\nneeded to make the method fast enough to deal with three-dimensional\nsystems. \n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{compare_entropy.pdf}\n \\caption{\\label{fig::compare_entropy} Entropy as a function of\n system size $N$ for two (Ref.~\\cite{Asenjo14}) and\n three-dimensional (this work) jammed sphere packings. Dashed\n curves are lines of best fit of the form $S=aN$.}\n\\end{figure}\n\nIn the present paper, we present the first enumeration of the number\nof jammed packings for three-dimensional systems consisting of up to\n$128$ soft spheres. A direct comparison of the entropy measured as a\nfunction of system size for two and three-dimensional jammed sphere\npackings is shown in Fig.~\\ref{fig::compare_entropy}. The potential of\nthe method presented herein can be verified unequivocally from\nFig.~\\ref{fig::compare_entropy}: we are able of tackling problems at\nleast $500$ million times more complex, and of greater computational\ncost, than the already spectacularly difficult questions confronted by\nAsenjo et al. \\cite{Asenjo14}. Furthermore we show how our improved\nprocedure allows first-principles computation of configurational\nentropy as a function of system size and pressure. The method and the\ntechnical improvements needed to overcome this numerical challenge are\npresented alongside the main results.\n\nThe remainder of this work is organised as follows.\nSection~\\ref{sec::basic_principle} describes the basic principle of\nthe mean basin volume method for counting, and explains how that\nstrategy can be applied to enumerate granular packings. The\nenumeration and entropy results for three-dimensional jammed sphere\npackings as a function of system size and pressure are reported in\nSec.~\\ref{sec::r3d}. The rest of the manuscript is dedicated to the\ndescription of the improved numerical\nmethod. Section~\\ref{sec::packing_sampling_protocol} outlines our\nprotocol for sampling different granular packings, and it describes\nthe corresponding potential energy landscape and minimisation\ntechniques. Application of thermodynamic integration to compute the\nvolume of a basin of attraction in such a landscape is described in\nSec.~\\ref{sec::basin_volume_thermodynamic_integration}. Aspects of\nthe data analysis tools used on the histograms of sampled basin\nvolumes, and related configurational entropy definitions, are described in\nSec.~\\ref{sec::volume_distributions_data_analysis}. Conclusions are\ndrawn in Sec.~\\ref{sec::conclusion}. Further technical background is\ngiven in the appendices.\n \n\\makeatletter{}\\begin{figure*}\n\\centering\n\\begin{minipage}[b]{.45\\textwidth}\n\\includegraphics[width=\\columnwidth]{packing3marble_crop.png}\n\\end{minipage}\\qquad\n\\begin{minipage}[b]{.45\\textwidth}\n\\includegraphics[width=\\columnwidth]{jammed_packing3marble_crop.png}\n\\end{minipage}\n\\caption{\\label{fig:3d_packings} Hard sphere fluid at $\\phi_\\text{HS}\n = 0.5$, left, and HS-WCA jammed packing at $\\phi_\\text{SS} = 0.7$,\n right, for a system of $44$ polydisperse hard spheres with mean\n radius $\\langle r_h \\rangle=1$ and standard deviation\n $\\sigma_\\text{HS}=0.05$. We prepare the polydisperse HS fluid\n configurations at fixed packing fraction $\\phi_{\\mathrm{HS}}=0.5$ by\n a Monte Carlo simulation. Particles are then inflated by the same\n factor, proportional to their radius (spheres are coloured according\n to their radius), to obtain an over-compressed soft spheres jammed\n packing at $\\phi_{\\mathrm{SS}}=0.7$ by an infinitely fast quench\n (energy minimisation).}\n\\end{figure*}\n\n\\section{\\label{sec::basic_principle}Basic principle: counting by\n sampling}\nIn this section, we briefly review the numerical approach that we use\nto compute the number of distinct jammed states. We stress that the\napproach that we use has much wider applicability than the counting of\ngranular packings~\\cite{fyodorov2004complexity, bray2007statistics,\n fyodorov2012critical, auffinger2013random, fyodorov2014topology,\n hughes2014inversion, mehta2014communication, cleveland2014certified,\n mehta2015statistics, cugliandolo1993analytical, kurchan1993barriers,\n mehta2013energy, auffinger2013complexity, subag2015complexity,\n sastry2001relationship, banerjee2014role, distler2005random,\n douglas2007flux, aazami2006cosmology, tye2009multi,\n frazer2011exploring, frazer2012multi, battefeld2012unlikeliness,\n wainrib2013topological, choromanska2014loss, sagun2014explorations,\n chaudhari2015trivializing, Frenkel13}. In the context of granular\npackings, our aim is to compute the number of ways $\\Omega$ in which\n$N$ spheres can be arranged in a given volume $V_\\text{box}$ of\nEuclidean dimension $d$. Knowledge of $\\Omega$ allows us to compute\nconfigurational entropies and related quantities from first principles\n\\cite{Edwards89, Asenjo14}. Our approach is based on a rigorous\nmapping of the enumeration problem onto counting the number of minima\nof a potential energy landscape \\cite{Xu11}. The approach makes no use\nof a harmonic~\\cite{Speedy99} or quasi-harmonic \\cite{Wales13}\napproximation. For a system of hard particles the potential energy\nfunction is discontinuous, that is, the energy of the system is either\nzero, if no two particles overlap, or infinity otherwise. Then, at\njamming, in the absence of rattlers, basins of attraction are single\npoints in configuration space and they have no associated volume.\nThis does not mean that we cannot sample the energy minima of a system\nof hard particles. The reason is that all jammed structures of hard\nparticles correspond to the zero potential energy minima of a system\nwith a continuous repulsive potential with the same range as the\nhard-core diameter of the hard particles. In what follows, we focus on\nthis class of systems, but we generalise the problem by also\nconsidering minima with a non-zero potential energy. In particular, we\nconsider spherical particles with a hard core and a short-ranged\ncontinuous repulsive interaction. Under conditions where this system\nis jammed, a system with {\\em only} the hard-core interactions would\nstill be fluid and would sample the accessible configuration space\nuniformly. This remaining accessible volume is partitioned in basins\nof attraction defined by the soft shells. The HS-WCA potential used to\nsimulate hard-core plus soft-shell interactions and the packing\npreparation protocol are described in Sec~\\ref{sec:wca_pot}. For an\nillustration of the packing preparation protocol refer to\nFig.~\\ref{fig:3d_packings}. As we argue below, using an HS-WCA model\ngreatly improves the efficiency of determination of basin volumes.\n\nLet us denote the total available volume in $dN$-dimensional space as\n$\\mathcal{V}$. Note that $\\mathcal{V}$ is not the total volume of\nconfiguration space ($V^N$), but just that part of the volume that is\nfree of hard-core overlaps. It is the configurational part of the\npartition function of the hard-core system at the number density under\nconsideration. Since the accessible configuration space is tiled by\nthe basins of attraction of the distinct energy minima\n\\cite{Stillinger84, Stillinger85, Stillinger95, Speedy99} we can\nwrite:\n\\begin{equation}\n \\mathcal{V} = \\sum_{i=1}^{\\Omega} v_i,\n\\end{equation}\nwhere $v_i$ is the volume of the $i$-th basin of attraction and\n$\\Omega$ is the total number of distinct minima. We thus make the\nsimple observation:\n\\begin{equation}\n \\sum_{i=1}^{\\Omega} v_i = \\frac{\\Omega}{\\Omega}\\sum_{i=1}^{\\Omega}\n v_i = \\Omega \\langle v \\rangle,\n\\end{equation}\nwhere $\\langle v \\rangle$ is the \\emph{mean basin volume}, from which\nit follows immediately that\n\\begin{equation}\n \\label{eq:bv_omega}\n \\Omega = \\frac{\\mathcal{V}}{\\langle v \\rangle}.\n\\end{equation}\nWe note that, for sphere packings, $\\mathcal{V}$ is known from the\nequation of state of the underlying hard sphere fluid (see\nAppendix~\\ref{sec::poly_fluid}) and we can measure $\\langle v \\rangle$\nby thermodynamic integration, as discussed in detail in the\nSec~\\ref{sec::packing_sampling_protocol}. The approach of~\\cite{Xu11,\n Asenjo14} has thus turned the intractable enumeration problem of\nfinding $\\Omega$ into a sampling one, namely measuring $\\langle v\n\\rangle$.\n \n\\makeatletter{}\\section{\\label{sec::r3d}Results: counting disordered 3D sphere packings}\n\nThe mean basin volume method for enumerating the number of\nmechanically stable packings was introduced by Xu \\emph{et al.}\n\\cite{Xu11}, and tested on a small system of soft disks. Asenjo\n\\emph{et al.} \\cite{Asenjo14} then made a number of modifications to\nthe algorithm that allowed them to apply it to larger systems of up to\n128 disks. As was the case with Ref.~\\cite{Xu11}, the calculations of\nRef.~\\cite{Asenjo14} focused on two-dimensional systems because of the\nhigh computational costs involved in studying 3D systems. Here we\npresent results for systems of three-dimensional soft spheres. We are\nthus in a position to compute the configurational entropy of a real\n(but idealised) three-dimensional system.\n\nWe first describe an analysis similar to the one reported by Asenjo\n\\emph{et al.} \\cite{Asenjo14} to verify the extensivity of the entropy\n$S(V)$ at constant packing fraction. Next, we extend our approach to\nthe generalised Edwards ensemble, i.e.~one based on a description\nof the system in terms of its volume and\npressure, to compute the generalised entropy $S(V, \\mathcal{P})$.\n\nWe investigate three-dimensional packings with system sizes ranging\nfrom $24$ to $128$ HS-WCA particles, see Eq.~(\\ref{eq:hswca}), at\n$\\phi_\\text{HS}=0.5$ hard-sphere fluid packing fraction and\n$\\phi_\\text{SS}=0.7$ soft sphere packing fraction, corresponding to a\nratio of the soft and hard-sphere radii ratio $r_\\text{SS}\/r_\\text{HS}\n= 1.12$, prepared following the protocol outlined in\nSec.~\\ref{sec::packing_sampling_protocol}. For each system size we\ncompute the volume of the basin of attraction of approximately $1000$\npackings. Each PT run (see Sec.~\\ref{sec::basin_volume_thermodynamic_integration}) was performed on $15$ parallel threads of a\nsingle eight-core dual Xeon E$5-2670$ ($2.6$GHz, Westmere) node. Our\nchoice of convergence criterion was such that when the uncorrelated\nstatistical error for each of the replicas' mean square displacement\nfell below $5\\%$ the calculations were terminated. This set-up\ntranslated in run times ranging from $10$ to $300$ hours per basin\ndepending on system size, which amounts to $\\mathcal{O}(10^6)$ hours\nof total run time and $\\mathcal{O}(10^7)$ total cpu hours. We then\nanalyse the corresponding distribution of dimensionless free energies\nfollowing the protocol described in\nSec.~\\ref{sec::basin_volume_thermodynamic_integration} and\n\\ref{sec::volume_distributions_data_analysis} and summarised in\nAppendix~\\ref{sec::method_summary}.\n\n\\subsection{Extensivity of the entropy}\nWe first computed two alternative definitions of entropy: the Gibbs\nentropy $S_G = -\\sum_{i = 1}^{\\Omega}p_i\\ln(p_i)-\\ln(N!)$ and\nEdwards (Boltzmann) entropy $S_B= \\ln(\\Omega)-\\ln(N!)$, where $p_i$\nis the probability to sample packing $i$ and $\\Omega$ is the total\nnumber of mechanically stable states (or minima in the energy\nlandscape). A detailed discussion of these definitions is outlined in\nSec.~\\ref{sec::volume_distributions_data_analysis}. The results of\nthese calculations are summarised in Fig.~\\ref{fig::compare_all}. Our\nresults strongly suggest that, also in three dimensions, the entropy\nthus defined is extensive. Note that extensivity requires not only\nthat the entropy scales linearly with system size, but also that it\ncrosses zero at the origin. The slightly higher value of the Edwards\nentropy compared to the Gibbs entropy is consistent with the\nobservation that Edwards' equiprobability corresponds to the maximum\npossible entropy of a system with $\\Omega$ states. We also show that\nour estimates for the Edwards' entropy are relatively insensitive to\nthe precise strategy used to compute it. In\nFig.~\\ref{fig::compare_all}, we compare three methods: a parametric\nfit to a generalised Gaussian cumulative distribution function\n(c.d.f.) using a non-linear least squares method, a fit to the\ncorresponding probability density function (p.d.f.) using maximum\nlikelihood, and a non-parametric fit by kernel density estimation,\nwhich makes no a priori assumption about the shape of the\ndistribution, other than the choice of the kernel function. We note,\nonce again, that no post-processing is needed to compute the Gibbs\nversion of the configurational entropy. Our results are in line with\nthose reported by Asenjo \\emph{et al.} \\cite{Asenjo14} for\ntwo-dimensional systems.\n\nThe number of mechanically stable states $\\Omega$ required by the\nEdwards' definition of entropy is obtained subsequently to fitting the\nnumerically obtained distribution of log-basin volumes (dimensionless\nfree energies) to a generalised Gaussian distribution and unbiasing it\nappropriately, as described in\nSec~\\ref{sec::volume_distributions_data_analysis}. We observe that the\nbest-fit mean and scale parameters of the generalised Gaussian for the\ndistribution of dimensionless free energies, $\\mu$ and $\\sigma$ in\nEq.~(\\ref{eq:gen_gaussian}) respectively, are also extensive, which\nalthough in line with what was found in two dimensions, is not {\\em a\n priori} obvious. Finally we find that the shape parameter, $\\zeta$\nin Eq.~(\\ref{eq:gen_gaussian}), appears to depend only weakly on\nsystem size. The statistics are poor, but the data are compatible with\nthe assumption that $\\zeta\\rightarrow 2$ (Gaussian distribution) as\n$N\\rightarrow\\infty$. In 2D, the same limiting distribution of\n$\\zeta$, but with a much stronger size dependence, was observed.\n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{entropy_all_paper.pdf}\n \\caption{\\label{fig::compare_all} Top left, entropy as a function of\n the system size $N$ computed, in order, according to the Gibbs\n configurational entropy and the Edwards configurational entropy\n using a non-parametric fit by kernel density estimation (KDE), a\n parametric fit to a generalised Gaussian c.d.f. using a non-linear\n least squares method and a fit to the corresponding p.d.f. using\n maximum likelihood (ML). Comparison of generalised Gaussian\n best-fit parameters for 2D (see Ref.~\\cite{Asenjo14}) and 3D\n sphere packings: scale parameter $\\sigma$ (bottom left) and mean\n log-volume $\\mu$ (top right) scale linearly with system size $N$;\n distributions are more peaked for 2D packings. In 2D we observe\n much stronger dependence of the shape parameter $\\zeta$ (bottom\n right) as a function of system size than in 3D. }\n\\end{figure}\n\n\n\\subsection{\\label{sec::r3dgen}Entropy in the generalised Edwards ensemble}\nWe next consider the situation where the configurational entropy is a\nfunction of both the volume $V$ and the stress tensor $\\hat{\\Sigma}$\nof the system. The number of packings with fixed $V$ and\n$\\hat{\\Sigma}$ is denoted by $\\Omega(\\hat{\\Sigma}, V) $.\n\nIn the generalised Edwards ensemble~\\cite{Henkes07,Henkes09,Bi15}, we\nfix the variables conjugate to $V$ and $\\hat{\\Sigma}$, {\\em viz.}\\ the\ncompactivity $X$ and the inverse angoricity tensor $\\hat{\\alpha}$. The\ngeneralised `partition function' can then be written as~\\cite{Bi15}:\n\\begin{equation}\n Z_{\\text{dyn}} = \\sum_{\\nu} \\omega(\\hat{\\Sigma}_{\\nu}, V_{\\nu})\n e^{-V_{\\nu}\/X} e^{-\\text{Tr}(\\hat{\\alpha} \\hat{\\Sigma}_{\\nu})},\n\\end{equation}\nwhere $V_{\\nu}$ and $\\hat{\\Sigma}_{\\nu}$ are the volume and the\nforce-moment (stress) tensor for state $\\nu$. The weights $\\omega$\naccount for the protocol dependence of the probability to generate a\nstate, and the sum runs over all mechanically stable states $\\nu$.\n\nWe can rewrite this partition function in terms of the density of states: \n\\begin{equation}\nZ_{\\text{dyn}} = \\prod_{l,k>l} \\iint \\dif \\hat{\\Sigma}^{lk} \\dif V\n\\Omega_{\\text{dyn}}(\\hat{\\Sigma}, V) e^{-V\/X}\ne^{-\\text{Tr}(\\hat{\\alpha} \\hat{\\Sigma})}.\n\\end{equation}\nFor a system under hydrostatic pressure, and in the absence of shear,\nwe can write the force-moment tensor as $\\hat{\\Sigma}=\\hat{I} \\Gamma$,\nwhere $\\Gamma=\\mathcal{P}V=\\mathrm{Tr}(\\hat{\\Sigma})\/3$ is the\ninternal Virial of the system. The inverse angoricity tensor\n$\\hat{\\alpha}$ becomes a scalar $\\alpha = \\partial S \/ \\partial\n\\Gamma$ \\cite{Henkes09}. This result allows to simplify the notation\nsignificantly and at fixed volume, through the mean basin volume\nmethod, we obtain the number of states integrated over all pressure\nstates, $\\Omega(V) = \\int \\dif{\\mathcal{P}} \\Omega(V,\\mathcal{P})$. We\nnow discuss how to generalise this procedure so that one can compute\n$\\Omega(V,\\mathcal{P})$, and therefore the configurational entropy, in\nthe context of the generalised Edwards ensemble.\n\n\\subsubsection{Pressure to basin volume power-law relation}\n\\begin{figure}[t]\n \\includegraphics[width=\\columnwidth]{plot_f_logp.pdf}\n \\caption{\\label{fig::f_logp} Top: dimensionless free energy versus\n pressure of mechanically stable states at fixed volume for\n several system sizes. Best fit lines are in black. In the bottom\n left and right plots we show slope and intercept for each of the\n best fit lines as a function of system size. Both slope and\n intercept scale linearly with system size.}\n\\end{figure}\nTo compute $Z(X,\\alpha)$ directly, we would have to evaluate\n$\\Omega(\\mathcal{P}, V) $ as a function of both $\\mathcal{P}$ and\n$V$. Whilst, with the tools that we have, this calculation is in\nprinciple possible, the computational costs would be several orders of\nmagnitude larger than the, already quite substantial, costs of\ncomputing $\\Omega(V)$. This would suggest that the computation of\n$Z(X,\\alpha)$ is not possible at present.\n\nHowever, it turns out that we can still estimate the generalised\nconfigurational entropy because, as we discuss below, we observe a\nsurprisingly strong correlation between pressure and basin volume.\n\nFrom Fig.~\\ref{fig::f_logp} we see that the basin volume for a given\npressure state at fixed volume is strongly correlated with the\npressure $\\mathcal{P}$. As the figure suggests, the relation between\n$-\\ln (v)$ and $\\ln (\\mathcal{P})$ is approximately linear, and\nhence\n\\begin{equation}\n\\label{eq:pressures_f_law}\nF(\\mathcal{P} | N, \\phi_\\text{SS}) \\equiv -\\ln(v) =\n\\frac{N}{\\kappa}\\ln(\\mathcal{P}) + \\mathcal{C}(N),\n\\end{equation}\nwhere $\\kappa$ denotes the slope of the linear fit, and\n$\\mathcal{C}(N)$ its value at ${\\mathcal P}=1$ (see\nFig.~\\ref{fig::f_logp}). The value of $\\kappa$ is not known a\npriori. It seems likely that $\\kappa$ depends on the functional form\nof the potential. $\\mathcal{C}(N)$ is a even less universal linear\nfunction of $N$, as it depends on the choice of units. \n\nWe anticipate that this power law relationship survives for packings\nin two dimensions for a wide spectrum of packing fractions $\\phi >\n\\phi_J$ \\emph{viz.} as long as the system is jammed and sufficiently\nover-compressed \\footnote{Results to appear elsewhere.}.\n\n\\subsubsection{Gibbs configurational entropy}\n\\begin{figure}[t]\n \\includegraphics[width=\\columnwidth]{plot_p_cdf_fit.pdf}\n \\caption{\\label{fig::gen_fits} Empirical cumulative distribution\n functions of the pressures for several system sizes. Dashed lines\n in the corresponding colour are curves of best fit to a\n generalised log-normal distribution. The curves are mostly\n indistinguishable. Inset: best fit parameters for the generalised\n log-normal distribution as a function of system size. The mean\n $\\mu$ and scale parameter $\\sigma$ scale linearly with $1\/N$,\n while the shape parameter $\\zeta$ is approximately insensitive\n with respect to system size.}\n\\end{figure}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{plot_p_S.pdf}\n \\caption{\\label{fig::p_S} Top: generalised Edwards entropy at\n fixed volume fraction for various system sizes. The curves show\n a well defined maximum for all sizes, while their shape depends\n on the specific parameters of the generalised log-normal that\n best fits the underlying distribution of pressures. Bottom left:\n comparison between the Edwards entropy and the maximum value\n attained by each curve: $\\max[S_B(\\mathcal{P},V)]$ scales\n linearly with size and its value is progressively closer to the\n marginal (total) Edwards entropy $S_B(V)$, consistent with the\n fact that $S_B(\\mathcal{P},V)$ is a negative exponential\n function, and the area under the curve is dominated by the mode\n for increasing system size. $S_B(V)$ should constitute an upper\n limit to $\\max[S_B(\\mathcal{P},V)] \\leq S_B(V)$ and the two\n should be equivalent only in the thermodynamic limit. Bottom\n right: ensemble average of the pressure computed as a function\n of inverse angoricity $\\alpha$ and system size. The curves, in\n the same colour as the top figure, do not diverge and the arrows\n indicates their value at $\\alpha=0$.}\n\\end{figure}\nUsing our approximate relation between pressure and basin volume, we\ncan now rewrite Eq.~(\\ref{eq:pressures_f_law}) in terms of the\nprobabilities for each jammed state\n\\begin{equation}\n\\label{eq:pressures_f_law_prob}\n\\ln(p_i) = -\\frac{N}{\\kappa}\\ln(\\mathcal{P}_i) - \\mathcal{C}(N) -\n\\ln(\\mathcal{V}),\n\\end{equation}\nwhich when substituted in the definition for the Gibbs entropy\nEq.~(\\ref{eq::apf_shannon}), gives the configurational entropy at a\ngiven volume in terms of the biased mean log-pressure\n\\begin{equation}\n S_G = \\frac{N}{\\kappa} \\left\\langle \\ln(\\mathcal{P})\n \\right\\rangle_{\\mathcal{B}} + \\mathcal{C}(N) + \\ln(\\mathcal{V}) -\n \\ln(N!).\n\\end{equation}\nThe significance of this equation should be apparent: for a\nsufficiently over-compressed packings of soft spheres at a given\npacking fraction, the Gibbs configurational entropy can be\napproximately computed from sole knowledge of the average pressure,\nprovided that $\\kappa$ is known.\n\n\\subsubsection{Generalised Edwards configurational entropy}\n\nTo recover the number of states as a function of volume and stress we\nnote that\n\\begin{equation}\n \\label{eq:omega_volume_stress}\n \\Omega(V, \\mathcal{P}) = \\Omega(V)\n \\int_{\\mathcal{P}}^{\\mathcal{P}+\\delta\\mathcal{P}}\\mathcal{U}(x|V)\\dif\n x,\n\\end{equation}\nwhere $\\mathcal{U}(\\mathcal{P}|V)$ is the unbiased probability\ndistribution function of stresses at some specified volume. The\ndirectly measured distribution of pressures depends on the protocol\nwith which packings are generated.\n\nWe distinguish between the biased,\n$\\mathcal{B}(\\mathcal{P}|N,\\phi_\\text{SS})$ (as sampled by the packing\nprotocol), and the unbiased,\n$\\mathcal{U}(\\mathcal{P}|N,\\phi_\\text{SS})$, pressure distributions.\nSince the configurations were sampled proportional to the volume of\ntheir basin of attraction, using Eq.~(\\ref{eq:pressures_f_law}) we can\ncompute the unbiased distribution analogously to\nEq.~(\\ref{eq:unbiasing}) as\n\\begin{equation}\n \\label{eq:unbiased_p_dist}\n \\mathcal{U}(\\mathcal{P}|N,\\phi_\\text{SS}) = \\mathcal{Q}(N,\\phi_\\text{SS})\n \\mathcal{B}(\\mathcal{P}|N,\\phi_\\text{SS}) e^{\\mathcal{C}(N)}\n \\mathcal{P}^{N \/\\kappa},\n\\end{equation}\nwhere $\\mathcal{Q}(N,\\phi_\\text{SS}) = \\langle v\n\\rangle(N,\\phi_\\text{SS})$ is the normalisation constant.\n\nUpon substitution of $\\ln\\Omega(V) = \\ln(\\mathcal{V}) - \\ln(\\langle\nv \\rangle)$ and of Eq.~(\\ref{eq:unbiased_p_dist}) for\n$\\mathcal{U}(x|V)$, we write an expression for the Edwards entropy as\na function of volume and pressure\n\\begin{equation}\n\\begin{aligned}\n \\label{eq:gen_edw_entropy}\n S_B(V, \\mathcal{P}) = & \\ln \\left(\n \\int_{\\mathcal{P}}^{\\mathcal{P}+\\delta\\mathcal{P}}\\mathcal{B}(x|V)x^{N\/\\kappa}\\dif\n x\\right) \\\\& + \\ln(\\mathcal{V}) + \\mathcal{C}(N) - \\ln(N!).\n\\end{aligned}\n\\end{equation}\nWe fit the empirical cumulative distribution function (c.d.f.)~of\n$\\mathcal{B}(\\mathcal{P})$ with the generalised log-normal\nc.d.f. corresponding to Eq.~(\\ref{eq:gen_lognormal}) (see\nFig.~\\ref{fig::gen_fits}). We then numerically evaluate the\ngeneralised Edwards entropy $S_B(\\mathcal{P},V)$ at fixed volume, as\nshown in Fig.~\\ref{fig::p_S}.\n\nIn the thermodynamic limit we find\n\\begin{equation}\n\\label{eq:entropy_per_particle2}\ns_B(\\phi_\\text{SS}) = 1 + c + \\frac{\\langle \\ln (\\mathcal{P})\n \\rangle_{\\mathcal{B}}}{\\kappa} - \\ln(\\phi_\\text{SS}) -\nf_\\text{ex}(\\phi_\\text{HS}),\n\\end{equation}\nwhere $c=\\mathcal{C}(N)\/N$, see Appendix \\ref{sec::gen_edwards} for\ndetails of the derivation and further discussion.\n\nIn Fig.~\\ref{fig::p_S} we also show the predicted expectation value\nfor the pressure obtained via the ensemble average at arbitrary\ninverse angoricity $\\alpha$,\n\\begin{equation}\n\\label{eq:ens_avg}\n\\langle \\mathcal{P} \\rangle_{\\alpha}^{(ens)} =\n\\frac{\\displaystyle\\int_0^{\\infty}\\mathcal{P}\n \\mathcal{B}(\\mathcal{P}|V)\\mathcal{P}^{N\/\\kappa}e^{-\\alpha\\mathcal{P}V}\\dif\n \\mathcal{P}}{\\displaystyle\\int_0^{\\infty}\\mathcal{B}(\\mathcal{P}|V)\\mathcal{P}^{N\/\\kappa}e^{-\\alpha\\mathcal{P}V}\n \\dif \\mathcal{P}}.\n\\end{equation}\n \n\\makeatletter{}\\section{\\label{sec::packing_sampling_protocol}Packing preparation\n protocol}\n\\subsection{\\label{sec::sampling_packings}Sampling packings}\n\nThe physical properties of granular packings may depend strongly on\nthe preparation protocol. This is illustrated by the\nLubachevsky-Stillinger algorithm (LSA) procedure to prepare jammed\npackings of hard particles~\\cite{Lubachevsky91} by compression (or,\nequivalently, by `inflation' of the particles). If a monodisperse HS\nfluid is compressed rapidly the LSA will generate a low\nvolume-fraction disordered packing. However, for (very) slow\ncompression rates, LSA will produce dense crystals\n\\cite{Lubachevsky91, Torquato00}.\n\nIn the present work, we study a fluid of polydisperse spheres. We use\na protocol related to a Stillinger-Weber quench that maps each fluid\nstate to a local minimum, or ``inherent structure'', connected by a\npath of steepest descent \\cite{Stillinger82, Stillinger84}.\n\nTo prepare the polydisperse fluid, we draw $N$ particle radii\n$\\{r\\}_N$ from a Gaussian distribution\n$\\mathrm{Normal}(1,\\sigma_\\text{HS}) > 0$, truncated at $r=0$ (note\nthat in our application the standard deviation $\\sigma_\\text{HS}$ is\nsufficiently small that it is extremely improbable to ever sample a\nnegative radius). We set the box size to meet the target packing\nfraction of the hard sphere fluid $\\phi_\\text{HS}$ and then place the\nparticles in a random valid initial hard spheres configuration. The\ninitial configuration is then evolved by a MC simulation \\cite{mcpele}\nconsisting of single particle random displacements and\nparticle-particle swaps, and after equilibration, new configurations\nare recorded at regular intervals. We choose the length of these\nintervals such that, on average, each particle diffuses over a\ndistance equal to the diameter of the largest particle. As long as\n$\\phi_\\text{HS}$ is well below the volume fraction where the fluid\nundergoes structural arrest, the allowed configurations of the fluid\ncan be sampled uniformly. Importantly, this volume fraction is well\nbelow the random close packing ($\\phi_\\text{HS}^{(\\text{RCP, 3D})}\n\\approx 0.64$ and $\\phi_\\text{HS}^{(\\text{RCP, 2D})} \\approx 0.82$\n\\cite{OHern03}).\n\nGiven these HS fluid configurations, we now switch on the soft,\nrepulsive interaction to generate over-compressed jammed packings of\nthe particles (see Fig.~\\ref{fig:3d_packings}). The particles are\ninflated with a WCA-like potential \\cite{Weeks71} to reach the target\nsoft packing fraction $\\phi_\\text{SS} > \\phi_\\text{HS}^{(\\text{RCP})}\n> \\phi_\\text{HS}$. The hard spheres are inflated proportional to\ntheir radius, so that the soft sphere radius is\n\\begin{equation}\n r_s\n =\\left(\\frac{\\phi_\\text{SS}}{\\phi_\\text{HS}}\\right)^{1\/d}\n r_h,\n\\end{equation}\nwhere $d$ is the dimensionality of the box. Clearly, this procedure\ndoes not change the polydispersity of the sample.\n\\subsection{\\label{sec:wca_pot}Soft shells and minimisation}\nWe define the WCA-like potential around a hard core as follows:\nconsider two spherical particles with hard core distance $r_h$ and\nsoft core contact distance $r_s=r_h(1+\\theta)$, with $\\theta =\n(\\phi_\\text{SS}\/\\phi_\\text{HS})^{1\/d}-1$. We can then write a\nhorizontally shifted hard-sphere plus WCA (HS-WCA) potential as\n\\begin{equation}\n \\label{eq:hswca}\n v_\\text{HS-WCA}(r) = \n \\left\\{ \n \\begin{array}{l l}\n \\infty &\\quad r \\leq r_h,\\\\\n \\begin{array}{l}\n 4\\epsilon \\left[\n \\left(\\frac{\\displaystyle\\sigma(r_h)}{\\displaystyle\n r^2-r_h^2} \\right)^{12}\n \\right. \\\\ \\left. -\\left(\\frac{\\displaystyle\\sigma(r_h)}{\\displaystyle\n r^2-r_h^2} \\right)^6 \\right] + \\epsilon\n \\end{array}\n &\\quad r_h < r < r_{s}, \\\\\n 0 &\\quad r \\geq r_s\n \\end{array}\n \\right.\n\\end{equation}\nwhere $\\sigma(r_h) = (2\\theta +\\theta^2) r_h^2\/2^{1\/6}$ guarantees\nthat the potential goes to zero at $r_s$. For computational\nconvenience (avoidance of square-root evaluations), the potential in\nEq.~\\ref{eq:hswca} differs from the WCA form in that the\ninter-particle distance in the denominator of the WCA potential has\nbeen replaced with a difference of squares. Note that this implies\nthat our potential resembles a 6-3 potential more than a 12-6\npotential. For our purpose, this difference is immaterial: we just\nneed a short-ranged repulsive potential that diverges at the hard-core\ndiameter and vanishes continuously at the soft-core diameter. The\nfunctional form of this potential is very similar to the HS-WCA\npotential used by Asenjo \\emph{et al.} \\cite{Asenjo14}, but cheaper\nto compute. We note that this potential is a $C^{1}$ type function,\nthat is, its first derivative is continuous but not differentiable and\nits second derivative is discontinuous at $r_s$. We take advantage of\nthis property for the identification of rattlers (non-jammed\nparticles) in our packings.\n\nNumerically evaluating this potential, we match the gradient and\nlinearly continue the function $v_\\text{HS-WCA}(r)$ for $r \\leq r_h +\n\\varepsilon$, with $\\varepsilon > 0$ an arbitrary small constant, such\nthat minimisation is still meaningful if overlaps do occur.\n\nThe HS-WCA pair-potential was implemented using cell-lists\n\\cite{allen1989computer, pele} with periodic boundary conditions,\nguaranteeing $\\mathcal{O}(N)$ time complexity to the energy and\ngradient evaluations. Energy minimisations were performed with the\nCG$\\_$DESCENT algorithm \\cite{Hager05, Hager06, PyCG_DESCENT} which,\ncompared to FIRE \\cite{Bitzek06, pele}, reduces the average number of\nfunction evaluations for our system by a factor of $5$, while\npreserving many of its desirable properties.\n\n \n\\makeatletter{}\\section{\\label{sec::basin_volume_thermodynamic_integration}Basin\n volume by thermodynamic integration}\nThe basin of attraction of a given minimum-energy configuration is the\ncollection of all points connected to that minimum via a path of\nsteepest descent \\cite{Mezey82, Wales03}. To measure the volume of a\nbasin of attraction in the PES, we use thermodynamic integration\n\\cite{Frenkel84, Polson00} and parallel tempering (PT)\n\\cite{Lyubartsev92, Marinari92, Geyer95, mcpele}.\n\nThe basic idea behind the method is that, but for the sign, the\nlogarithm of the basin volume can be viewed as a dimensionless free\nenergy. We cannot determine this free energy directly. We now switch\non an increasingly harmonic potential that has its minimum at the\nminimum of the basin. In the limit of very large coupling constants\n(how large depends on the shape of the basin) the boundaries of the\nbasin no longer affect the free energy of the system, which has\neffectively been reduced to a $dN$ dimensional harmonic oscillator\nwith known free energy (for more details, see\nAppendix~\\ref{sec::Einstein_crystal_CM}). For zero coupling constant,\ninstead, the system is completely unconstrained and therefore in the\nstate of interest. Thermodynamic integration allows us to compute the\nfree energy difference between a reference state of known free energy\nand the (unknown) free energy associated with the original basin of\nattraction.\n\nA closely related approach is often used to compute the free energy of\ncrystals of particles with a discontinuous potential, such as hard\nspheres~\\cite{Frenkel84, Polson00, Frenkel02}. Details of that method\nare summarised in the Appendix~\\ref{sec::flm}, and the extension of\nthe technique to basin volume measurement is described below. Details\nof the Hamiltonian PT are discussed in\nAppendix~\\ref{sec::sampling_th_integrand}.\n\\subsection{\\label{sec::flm_for_basins}Free energy calculation for\n basin volumes}\n\nTo measure the volume of a basin by thermodynamic integration, we\nperform a walk inside the basin, that is, we start the MCMC random\nwalk from the minimum energy configuration $\\mathbf{r}_i$ and we\nreject every move that takes us outside the basin \\cite{Xu11,\n Frenkel13, Asenjo14}. This procedure can be cast in normal Monte\nCarlo language by defining an effective potential energy function\n(oracle) $U_{B}(\\mathbf{r} | \\mathbf{r}_i)$ which is zero inside the\nbasin and infinite outside. We can then write the volume of the\nbasin:\n\\begin{equation} \n v_i = \\int \\dif \\mathbf{r} e^{-U_{B}(\\mathbf{r}| \\mathbf{r}_i)}.\n\\end{equation}\nIn order for the oracle to test whether a proposed configuration is\ninside or outside the basin, a full energy minimisation must be\nperformed. The numerous potential energy calls required for a full\nenergy minimisation represent the major obstacle to the scalability of\nthe method.\n\nWe view the negative log-basin-volume as a dimensionless free energy\n$F_i \\equiv -\\ln(v_i)$~\\cite{Xu11} and compute it by thermodynamic\nintegration, as described in Appendix \\ref{sec::flm}. Therefore, we\nwrite, analogously to Eq.~(\\ref{eq:int_frenkelladd}):\n\\begin{equation}\n \\label{eq:bv_1}\n - \\ln v_i = F_{\\mathrm{har}}(k_\\text{max}) - \\frac{1}{2}\n \\int_0^{k_\\text{max}} \\dif k \\left \\langle \\vert \\mathbf{r} -\n \\mathbf{r}_i \\vert \\right \\rangle_k ,\n\\end{equation}\nwhere $\\mathbf{r}_i$ denotes the coordinates of the $i$-th energy\nminimum. Unless $k_\\text{max}$, the maximum spring constant of the\nharmonic reference system, is very large, a finite fraction of the\npoints belonging to the purely harmonic reference system will be\nlocated in the region where $U_B=\\infty$.\n\nWe can correct for this effect in our calculation of\n$F_{\\mathrm{har}}(k_\\text{max})$ by computing the ratio of the\npartition functions of a system with a harmonic spring constant\n$k_\\text{max}$, both with and without the basin potential energy\nfunction $U_B$. This ratio is given by\n\\begin{equation}\n \\label{eqn::rho_kmax}\n\\mathcal{R} \\equiv \\frac {\\displaystyle\\int \\dif \\mathbf{r} \\exp[-V(\\mathbf{r}\n | \\mathbf{r}_i, k_\\text{max}) - U_B(\\mathbf{r} |\n \\mathbf{r}_i)]} {\\displaystyle\\int \\dif \\mathbf{r}\n \\exp[-V(\\mathbf{r} | \\mathbf{r}_i, k_\\text{max})]},\n\\end{equation}\nwhere $V$ is the sum of the hard-core potential and the harmonic\npotential with spring constant $k_\\text{max}$, see\nEq.~(\\ref{eq::hs_plus_harmonic}). We note that $\\mathcal{R}$ can be\ncomputed using a `static' (i.e. non-Markov chain) Monte Carlo\nsimulation, sampling directly from the Boltzmann distribution of the\nharmonic oscillator with spring constant $k_\\text{max}$. Since the\nintegral in the denominator is known [see Eq.~(\\ref{eq:z_har})], we\nwrite the dimensionless free energy of the harmonic reference state\nfor basin $i$ as\n\\begin{equation}\n \\label{eq:f_har_cor}\n F_{\\text{har}}(k_\\text{max}) = -\\frac{dN}{2}\\ln\\left(\n \\frac{2\\pi}{k_\\text{max}}\\right) - \\ln \\mathcal{R}.\n\\end{equation}\n\nWe note that, in order to avoid a singularity in the integrand, it is\nuseful to perform the simulations fixing the centre of mass. It\nfollows that the same corrections to the free energy as derived in\nRefs.\\cite{Frenkel84, Polson00, Frenkel02} must be applied: similarly\nto Eq.~(\\ref{eq:full_f2}), but with the additional correction in\nEq.~(\\ref{eq:f_har_cor}), we write the basin volume as:\n\\begin{equation}\n \\label{eq:bv_cm}\n \\begin{aligned}\n -\\ln v_i &= \\Delta F^{(\\text{CM})} -\\ln\n \\left(V_\\text{box}\\right) \\\\ &- \\frac{(N-1)d}{2}\\ln \\left(\n \\frac{2\\pi}{k_\\text{max}}\\right) -\\ln\\mathcal{R},\n \\end{aligned}\n\\end{equation}\nwhere $\\Delta F^{(\\text{CM})}$ is the integral in Eq.~(\\ref{eq:bv_1}),\nand the ensemble averages have been computed with a constrained centre\nof mass and it is evaluated as in Eq.~(\\ref{eq:int_cov3}).\n\nFigure~\\ref{fig::u2_vs_k} shows an example of the mean squared\ndisplacement $\\left \\langle |\\mathbf{r}-\\mathbf{r}_0|^2 \\right\n\\rangle_{k}$, as a function of the spring constant $k$, along with the\napproximate expression in Eq.~(\\ref{eq:rms_approx_k}) used to\nconstruct the change of variables in Eq.~(\\ref{eq:int_cov3}). The\nresulting integrand, after the variable transform, is shown in the\ninset of Fig.~\\ref{fig::u2_vs_k}.\n\n\\begin{figure}[t]\n \\includegraphics[width=\\columnwidth]{u2_vs_k.pdf}\n \\caption{\\label{fig::u2_vs_k} Average squared displacement $\\left\n \\langle |\\mathbf{r}-\\mathbf{r}_0|^2 \\right \\rangle_{k}$ as a\n function of the spring constant $k$ (symbols). The dashed line\n shows the expression in Eq.~(\\ref{eq:rms_approx_k}). The data\n is measured for a packing of $N=32$ spheres, with\n $\\phi_\\text{HS}=0.5$ and $\\phi_\\text{SS}=0.7$ via Hamiltonian\n PT. Inset: corresponding integrand for the thermodynamic\n integration, resulting from the change of variables in\n Eq.~(\\ref{eq:int_cov3}).}\n\\end{figure}\n \n\\makeatletter{}\\section{\\label{sec::volume_distributions_data_analysis}Basin volume\n distributions and data analysis}\nOnce the volumes of multiple basins have been sampled, these data can\nbe used to compute the number of distinct packings \\cite{Asenjo14},\nand from that, the Edwards entropy~\\cite{Edwards89}. Furthermore we\nanalyse the distribution of pressures of the different energy minima\nat given volume. In this work, we express pressure and volume in\nreduced units $\\mathcal{P}\/\\mathcal{P}^{\\ast}$ and $v\/v^{\\ast}$\neverywhere with $v^{\\ast} \\equiv (4\\pi\/3)\\langle r_h ^3\\rangle$ and\n$\\mathcal{P}^{\\ast}\\equiv \\epsilon\/v^{\\ast}$ being the units of volume\nand pressure, respectively.\n\\subsection{Gibbs configurational entropy}\nLet us first consider the `Gibbs' configurational entropy, $S_G$, defined by\nAsenjo \\emph{et al.} \\cite{Asenjo14}:\n\\begin{equation}\n \\label{eq::basic_shannon}\n S_G = -\\sum_{i = 1}^{\\Omega} p_i \\ln(p_i) - \\ln(N!),\n\\end{equation}\nwhere $p_i$ is the probability to sample packing $i$. For our\npreparation protocol, packings are sampled according to the volume of\ntheir basin of attraction, such that $p_i = v_i \/ \\mathcal{V}$. Then\nEq.~(\\ref{eq::basic_shannon}) gives\n\\begin{equation}\n \\begin{aligned}\n \\label{eq::apf_shannon}\n S_G &= -\\sum_{i = 1}^{\\Omega} p_i\\ln(v_i) + \\ln(\\mathcal{V}) -\n \\ln(N!) \\\\\n &= \\langle F \\rangle_{\\mathcal{B}} + \\ln(\\mathcal{V}) -\n \\ln(N!).\n \\end{aligned}\n\\end{equation}\nThe sum in Eq.~(\\ref{eq::apf_shannon}) is the mean of the negative\nlog-basin volumes (dimensionless free energies), as computed above,\nand weighted by the probabilities of preparing the corresponding\nbasins. Therefore, the entropy can be obtained directly, and without\napproximation, from the sampled mean basin dimensionless free energy.\n\nFrom Eq.~(\\ref{eq::apf_shannon}) we can also write the entropy per\nparticle in the thermodynamic limit as\n\\begin{equation}\n\\label{eq:entropy_per_particle1}\ns_G(\\phi_\\text{SS}) = 1 + \\langle f \\rangle_{\\mathcal{B}} +\n\\ln(\\phi_\\text{SS}) - f_\\text{ex}(\\phi_\\text{HS}) ,\n\\end{equation}\nwhere $f_\\text{ex}(\\phi_\\text{HS})$ is the excess free energy of the\nhard spheres fluid. In deriving this results we used Stirling's\napproximation for large $N$ and the fact that\n$V_{\\text{box}}\/v^{\\ast}=N \/ \\phi_\\text{SS}$.\n\\subsection{Edwards configurational entropy}\nEdwards~\\cite{Edwards89} suggested a Boltzmann-like entropy, where $S$\nequals the logarithm of\\enskip $\\Omega$, the total number of packings.\nAsenjo \\emph{et al.}~\\cite{Asenjo14} showed that, even for\npolydisperse particles, indistinguishability of macrostates requires\nthat\n\\begin{equation}\n\\label{eq:boltzmann_entropy}\nS_B = \\ln(\\Omega) - \\ln(N!),\n\\end{equation}\nThe subtraction of $\\ln(N!)$ is necessary to guarantee extensivity of\nthe entropy. Unlike the Gibbs definition of entropy,\nEq.~(\\ref{eq:boltzmann_entropy}) makes the explicit assumption of\nequiprobability of states.\n\nFor a direct computation of the number of packings $\\Omega$, using\nEq.~(\\ref{eq:bv_omega}), we need the average basin volume $\\langle v\n\\rangle$. Since our preparation protocol samples each minimum with a\nprobability proportional to the volume of its basin of attraction, our\nsamples of $v$ are biased accordingly. Therefore, to obtain the\nunbiased average basin volume $\\langle v \\rangle$, the sampled basin\nvolume distribution needs to be unbiased \\cite{Frenkel13, Asenjo14,\n Paillusson15}. The unbiasing method used in the following work\nrequires an analytical (or at least numerically integrable)\ndescription of the biased basin free energy distribution function.\nDifferent approaches to modelling this distribution give rise to\nsomewhat different analysis methods, which all yield consistent\nresults. Again, we stress that no such additional steps are needed to\ncompute the `Gibbs' version of the configurational entropy.\n\nWe distinguish between the biased, $\\mathcal{B}(F|N,\\phi_\\text{SS})$\n(as sampled by the packing protocol), and the unbiased,\n$\\mathcal{U}(F|N,\\phi_\\text{SS})$, free energy distributions. Since\nthe configurations were sampled proportional to the volume of their\nbasin of attraction, we can compute the unbiased distribution as\n\\begin{equation}\n \\label{eq:unbiasing}\n \\mathcal{U}(F|N,\\phi_\\text{SS}) = \\mathcal{Q}(N,\\phi_\\text{SS})\n \\mathcal{B}(F|N,\\phi_\\text{SS}) e^F\n\\end{equation}\nwhere $\\mathcal{Q}(N,\\phi_\\text{SS})$ is the normalisation constant\n\\begin{equation}\n \\label{eq::un_bias_integral_c_omega}\n \\mathcal{Q}(N, \\phi_\\text{SS}) = \\left [\n \\int_{F_\\text{min}}^{\\infty} \\dif F\n \\mathcal{B}(F|N,\\phi_\\text{SS})e^F \\right]^{-1} = \\langle v\n \\rangle(N,\\phi_\\text{SS}).\n\\end{equation}\nFrom Eq.~(\\ref{eq:unbiasing}), unbiasing of the raw free energy\ndistribution seems straightforward, however Asenjo \\emph{at al.}\n\\cite{Asenjo14} noted that the most probable basins are about\n$O(10^3)$ more probable than the small ones. Upon unbiasing, this\nfactor is multiplied by a factor of about $e^{-20}$, hence they\nobserve that small basins are much more numerous than large ones and\ngrossly under-sampled.\n\nTo overcome this problem, one can fit the biased measured free energy\ndistribution $\\mathcal{B}(F|N,\\phi_\\text{SS})$ and perform the\nunbiasing via Eq.~(\\ref{eq::un_bias_integral_c_omega}) on the best\nfitting distribution. $\\mathcal{B}(F|N,\\phi_\\text{SS})$ must be\nbounded, hence it should decay with a functional form $\\exp(-F^{\\nu})$\nwhere $\\nu > 1$.\n\nBefore performing the fit we remove outliers from the free energy\ndistribution following the distance-based outlier removal method\nintroduced by Knorr and Ng \\cite{Knorr98}. This is a form of\nclustering for which we choose to keep only those points for which at\nleast half of the remaining data set is within $3\\sigma$ from the\npoint, where $\\sigma$ is the standard deviation computed for the raw\ndata set. This procedure typically results in the exclusion of one or\ntwo points and it is essential for a successful fit to a generalised\nGaussian model.\n\\subsubsection{Generalised Gaussian}\nAssuming that $\\mathcal{U}(F|N,\\phi_\\text{SS})$ is unimodal, which has\nbeen verified for very small systems \\cite{Xu11}, one can fit the raw\ndistribution $\\mathcal{B}(F|N,\\phi_\\text{SS})$ with a three-parameter\ngeneralised normal distribution\n\\begin{equation}\n \\label{eq:gen_gaussian}\n p(F|\\overline{F},\\sigma,\\zeta) \\equiv\n \\frac{\\zeta}{2\\sigma\\Gamma(1\/\\zeta)}\\exp\\left[-\\left(\\frac{|F-\\overline{F}|}{\\sigma}\\right)^{\\zeta}\\right],\n\\end{equation}\nwhere $\\Gamma(x)$ is the gamma function, $\\sigma$ is the scale\nparameter, $\\zeta$ is the shape parameter and $\\overline{F}$ is the\nmean (free energy) with variance $\\sigma^2\n\\Gamma(3\/\\zeta)\/\\Gamma(1\/\\zeta)$. In the limit $\\zeta \\rightarrow 2$\nwe recover the Gaussian distribution with standard deviation $\\sigma$.\nIn practice it appears to be most stable to fit the empirical biased\ncumulative distribution function, rather than the histogram shape\n\\cite{Asenjo14}. Alternatively, we also tested fitting to the\nobserved p.d.f. with the maximum-likelihood method, obtaining\nconsistent, but more scattered, results (see also\nSec.~\\ref{sec::r3d}).\n\\subsubsection{Kernel density estimate}\nTo relax the assumption that the empirical distributions can be fitted\nby a generalised Gaussian, one can also describe the distributions by\nkernel density estimation \\cite{Bishop09, scikitKDE}. Bandwidth\nselection is then done using Silverman's rule of thumb as the initial\nguess for integrated squared error cross-validation \\cite{Bowman84}.\nThe numerical integration step is performed, as for the generalised\nGaussian description, via Eq.~(\\ref{eq::un_bias_integral_c_omega}).\n\\subsection{Distribution of pressures}\nIn Sec.~\\ref{sec::r3dgen} we have established a link between the\npressure of a packing and the volume of its basin of attraction. In\norder to compute the entropy as a function of volume and pressure it\nis necessary to unbias the distribution of pressures with respect to\nthe sampling bias $\\exp(-F)$, analogous to the previous section. We\nchoose to describe the distribution of pressures $\\mathcal{P}$ using\nthe generalised log-normal distribution \\cite{Singh12}\n\\begin{equation}\n\\label{eq:gen_lognormal}\n\\begin{aligned}\np(\\mathcal{P}|\\overline{\\ln(\\mathcal{P})}, \\sigma, \\zeta) =&\n\\frac{\\zeta\/\\mathcal{P}}{2^{(\\zeta+1)\/\\zeta}\\sigma\\Gamma(1\/\\zeta)} \\\\ & \\exp\\left(-\\frac{1}{2}\\left|\\frac{\\ln(\\mathcal{P})-\\overline{\\ln(\\mathcal{P})}}{\\sigma}\\right|^\\zeta\n\\right),\n\\end{aligned}\n\\end{equation}\nwith the first term on the r.h.s. being the normalisation constant and\nthe remaining notation analogous to that of\nEq.~(\\ref{eq:gen_gaussian}). For $\\zeta=2$ this distribution reduces\nto the log-normal distribution.\n \n\\makeatletter{}\\section{\\label{sec::conclusion}Conclusions}\n\nThe study of a statistical mechanics of granular materials has been\ncomplicated by the impossibility of directly computing fundamental\nthermodynamic quantities. In the present paper we have shown that\nconfigurational entropies of three-dimensional packings can, in fact,\nbe computed.\n\nWe have presented a method for the direct enumeration of the\nmechanically stable states of systems consisting of up to $128$\nfrictionless soft three-dimensional spheres and we have shown that a\ndefinition of extensive entropy is possible, in line with the results\nfor two dimensional systems reported by Asenjo \\emph{et\n al.}~\\cite{Asenjo14}, with very minor differences in our\nobservations. The study of 3D packings is computationally demanding:\nthe computational time required for each packing ranged between $10$\nand $10^4$ cpu hours, depending on system size. The present study\ntherefore required substantial algorithmic optimisation.\n\nWe find that there is an approximately linear relationship between the\nlogarithm of the pressure of a mechanically stable configuration and\nthe logarithm of the volume of its basin of attraction.\n\nThe unexpected power law relationship between pressure and basin\nvolume provides a way to extend our approach to the generalised\nEdwards ensemble. We can analytically unbias the observed distribution\nof pressures and compute the entropy as a function of pressure at a\ngiven volume. Hence we have obtained consistent expressions for the\nentropy in the thermodynamic limit. Knowledge of this distribution\nenables the first direct computation of angoricity.\n\nTackling the study of granular materials from the energy landscapes\npoint of view is rather advantageous, although this does not come\nwithout burdens. This sort of approach is limited to soft frictionless\nparticles, and we expect it to be reliable only at $\\phi>\\phi_J$ when\nthe system is at least slightly over-compressed. Other theoretical\napproaches are useful in more limiting situations, see for instance\nthe discussion of the stress ensemble in the limit $\\phi \\rightarrow\n\\phi_J$ by Henkes and Chakraborty \\cite{Henkes07, Henkes09} and the\nwork on the force network ensemble for systems of almost hard grains\n\\cite{Van09, Tighe10, Tighe11}.\n \n\\begin{acknowledgments}\nWe acknowledge useful discussions with Daniel Asenjo, Carl Goodrich,\nSilke Henkes, and Fabien Paillusson. S.M. acknowledges financial\nsupport by the Gates Cambridge Scholarship. K.J.S. acknowledges\nsupport by the Swiss National Science Foundation under Grant\nNo. P2EZP2-152188 and No. P300P2-161078. J.D.S. acknowledges support\nby Marie Curie Grant 275544. D.F. and D.J.W. acknowledge support by\nEPSRC Programme Grant EP\/I001352\/1, by EPSRC grant EP\/I000844\/1 (D.F.)\nand ERC Advanced Grant RG59508 (D.J.W.)\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn \\cite{Kh1} Khovanov introduced a categorification of the Jones polynomial by assigning a chain complex $CKh(D)$ of graded abelian groups to a diagram $D$ of an oriented link $L\\subset \\mathbb{R}^3$. Reidemeister moves between link diagrams induce chain homotopy equivalences between the chain complexes, and the graded Euler characteristic of $CKh(D)$ is the Jones polynomial of $L$. The chain complex is built by forming the so-called \\emph{cube of resolutions} and applying the two-dimensional TQFT corresponding to the Frobenius algebra \n\\[\nH^*(S^2;\\mathbb{Z}) = \\mathbb{Z}[X]\/(X^2).\n\\]\nA crossingless diagram $D$ is assigned a chain complex supported in homological degree zero by applying the TQFT directly to $D$. In particular, the empty link is assigned $H^*(\\{*\\};\\mathbb{Z})=\\mathbb{Z}$, while the unknot is assigned $\\mathbb{Z}[X]\/(X^2)$. \n\nVarying the TQFT has been explored extensively, \\cite{BN2, Kh3, Le, KR}, and has proven to be fruitful for topological applications, \\cite{Ra}. Of particular interest is the \\emph{equivariant} or \\emph{universal} theory, built using the Frobenius algebra \n\\[\nA = \\mathbb{Z}[E_1, E_2, X]\/(X^2 - E_1 X + E_2)\n\\]\nwith ground ring $R = \\mathbb{Z}[E_1, E_2]$. This is the Frobenius algebra associated with $U(2)$-equivariant cohomology of $\\mathbb{CP}^1$ \\cite{Kh3}. It specializes to the original theory by setting $E_1 = E_2 = 0$ and to the Lee deformation \\cite{Le} by setting $E_1= 0, E_2=-1$. An equivariant version of $\\sl_3$-homology was constructed in \\cite{MV}, and a generalization to $\\sl_n$-homology can be found in \\cite{Kr}. \n \n\nIn another direction, Asaeda-Przytycki-Sikora \\cite{APS} defined homology for links in $I$-bundles over surfaces. The present paper concerns links in the solid torus, identified with $\\mathbb{A} \\times [0,1]$ where $\\mathbb{A} = S^1 \\times [0,1]$ is the annulus. The APS construction in this case is known as \\emph{annular Khovanov homology} or \\emph{annular APS homology}. It is a triply graded theory; in addition to homological and quantum gradings, there is a third grading arising from the presence of non-contractible circles in $\\mathbb{A}$. The APS annular chain complex may be obtained by applying to the cube of resolutions the \\emph{annular TQFT} \n\\[\n\\mathcal{G} \\colon \\mathscr{BN}(\\mathbb{A}) \\to \\mathbb{Z}{\\rm -}\\operatorname{ggmod},\n\\]\nwhere $\\mathscr{BN}(\\mathbb{A})$ is the Bar-Natan category of the annulus, and $\\k {\\rm -}\\operatorname{ggmod}$ denotes the category of bigraded modules over a ring $\\k$.\n\n\nThis paper extends annular Khovanov homology to the equivariant setting. We work with the Frobenius algebra \n\\[\nA_\\alpha = \\mathbb{Z}[\\alpha_0, \\alpha_1, X]\/((X-\\alpha_0)(X-\\alpha_1))\n\\]\nwith ground ring $R_\\alpha = \\mathbb{Z}[\\alpha_0, \\alpha_1]$, which are the $U(1)\\times U(1)$-equivariant cohomology of $\\mathbb{CP}^1$ and of a point, respectively \\cite{KR}. The Frobenius pair $(R_\\alpha, A_\\alpha)$ is an extension of $(R,A)$ by identifying $E_1, E_2$ with elementary symmetric polynomials in $\\alpha_0, \\alpha_1$,\n\\[\nE_1\\mapsto \\alpha_0 + \\alpha_1, E_2\\mapsto \\alpha_0\\alpha_1,\n\\]\nso that the polynomial $X^2 - E_1X+E_2 \\in R[X]$ splits as $(X-\\alpha_0)(X-\\alpha_1)$ in $R_\\alpha[X]$. We observe in Section \\ref{sec:preliminary observation} that working over $(R,A)$ cannot produce an equivariant version of annular APS homology. There is a natural $U(1)\\times U(1)$-equivariant analogue $\\mathscr{BN}_\\alpha(\\mathbb{A})$ of $\\mathscr{BN}(\\mathbb{A})$ where the local relations are dictated by the structure of $A_\\alpha$. \n\nOur main construction is a TQFT $\\mathcal{G}_{\\alpha}$, which, when applied to the cube of resolutions of an annular link diagram, gives a $U(1) \\times U(1)$-equivariant version of annular homology.\n\\begin{theorem}\nThere exists a functor $\\mathcal{G}_{\\alpha} \\colon \\mathscr{BN}_\\alpha(\\mathbb{A}) \\to R_\\alpha{\\rm -}\\operatorname{ggmod}$ such that the following diagram commutes \n\\begin{equation*}\n\\begin{tikzcd}\n\\mathscr{BN}_\\alpha(\\mathbb{A}) \\ar[r, \"\\mathcal{G}_{\\alpha}\"] \\ar[d] & R_\\alpha{\\rm -}\\operatorname{ggmod} \\ar[d]\\\\\n\\mathscr{BN}(\\mathbb{A}) \\ar[r, \"\\mathcal{G}\"] & \\mathbb{Z}{\\rm -}\\operatorname{ggmod}\n\\end{tikzcd}\n\\end{equation*}\nwhere the vertical arrows are obtained by setting $\\alpha_0 = \\alpha_1 = 0$. \n\\end{theorem}\\noindent\n\nWe define $\\mathcal{G}_{\\alpha}$ by choosing a suitable basis and using a filtration induced by an additional annular grading, as in \\cite{Ro}. Given a collection of disjoint simple closed curves $\\mathscr{C} \\subset \\mathbb{A}$, each circle in $\\mathscr{C}$ is assigned the module $A_\\alpha$, with the module assigned to a trivial circle concentrated in annular degree zero. The essential circles in $\\mathscr{C}$ are naturally ordered. For each essential circle $C$ in $\\mathscr{C}$ we equip its module $A_\\alpha$ with a distinguished homogeneous basis, either $\\{1, X-\\alpha_0\\}$ or $\\{1, X-\\alpha_1\\}$, depending in an alternating manner on the position of $C$. We show that maps assigned to cobordisms are non-decreasing with respect to the annular grading.\n\nA feature of the equivariant theory is that the dotted product cobordism on a non-contractible circle in $\\mathbb{A}$, \n\\begin{equation*}\n\\begin{gathered}\n\\includegraphics{AD3.pdf}\n\\end{gathered}\\, ,\n\\end{equation*}\nis not sent to the zero map. Algebraically, this says that multiplication by $X$ on an essential circle is nonzero in the equivariant theory. On the other hand, this cobordism evaluates to zero in APS homology and also in the quantum annular homology \\cite{BPW}. \n\n\nThe paper is organized as follows. In Section \\ref{sec:Khovanov homology} we review Khovanov homology using the framework of the Bar-Natan cobordism category \\cite{BN2}. The remaining parts of Section \\ref{sec:some link homology theories} give an overview of Frobenius algebras $A$, $A_\\alpha$, and $A_{\\alpha\\mathcal{D}}$, following \\cite{KR}. Section \\ref{sec:annular Khovanov homology} reviews annular Khovanov homology. Our equivariant theory is defined in Section \\ref{sec:defining the functor}. In Section \\ref{sec:inverting D equiv annular} we study a further extension appearing in \\cite{KR}, which is obtained by inverting an element $\\mathcal{D} \\in A_\\alpha$. We prove an analogue of \\cite[Theorem 4.2]{Le}, that for a $k$-component annular link, the homology obtained by inverting $\\mathcal{D}$ is free of rank $2^k$. In Section \\ref{sec:dotted TL} we recall the Temperley-Lieb category and its relation to annular Khovanov homology, following observations in \\cite{BPW}. This perspective leads to a natural equivariant analogue of the Temperley-Lieb category and algebra, where strands may carry dots. \n\\vskip1em\n\\textbf{Acknowledgements.} I would like to thank Mikhail Khovanov for suggesting this project, for many helpful discussions, and for comments on earlier versions of the paper. I would also like to thank my advisor Slava Krushkal. The author was supported by NSF RTG Grant DMS-1839968. \n\n\n\\section{Some link homology theories}\\label{sec:some link homology theories}\nWe review Bar-Natan's approach to Khovanov homology and describe four Frobenius algebras, all of which have appeared in the literature and yield homology for links in $\\mathbb{R}^3$. \n\n\\subsection{Khovanov homology}\\label{sec:Khovanov homology}\n\nWe start with a brief overview of the Bar-Natan category $\\mathscr{BN}$ and the construction of the chain complex $[[D]]$ assigned to a link diagram $D$; for a complete treatment see \\cite{BN2}. \n\nFirst, recall the (dotted) Bar-Natan category $\\mathscr{BN}$. Let $I :=[0,1]$ denote the unit interval. Objects of $\\mathscr{BN}$ are formal direct sums of formally graded collections of simple closed curves in the plane $\\mathbb{R}^2$. Morphisms are matrices whose entries are formal $\\mathbb{Z}$-linear combinations of dotted cobordisms properly embedded in $\\mathbb{R}^2 \\times I$, modulo isotopy relative to the boundary, and subject to the local relations shown in Figure \\ref{fig:BN relations}. For the remainder of the paper, all cobordisms are assumed to possibly carry dots, unless specified otherwise.\n\n\n\nLet $A_0= \\mathbb{Z}[X]\/(X^2)$. The trace\n\\[\n\\varepsilon_0 \\colon A_0 \\to \\mathbb{Z},\\, 1\\mapsto 0,\\, X\\mapsto 1\n\\]\nmakes $A_0$ a Frobenius algebra, with comultiplication\n\\begin{align*}\nA_0 & \\to A_0 \\o A_0\\\\\n1 & \\mapsto X \\o 1 + 1\\o X\\\\\nX & \\mapsto X\\o X \n\\end{align*}\nThis is the Frobenius algebra underlying $\\sl_2$ link homology \\cite{Kh1}.\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{BN_rel1.pdf}\n\\end{center}\n\\caption{Sphere}\\label{fig:sphere}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.4\\textwidth}\n\\begin{center}\n\\includegraphics{BN_rel2.pdf}\n\\end{center}\n\\caption{Neck-cutting}\\label{fig:neck-cutting}\n\\end{subfigure}\\\\\n\\vskip1em\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{BN_rel3.pdf}\n\\end{center}\n\\caption{Dotted sphere}\\label{fig:dotted sphere}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.35\\textwidth}\n\\begin{center}\n\\includegraphics{BN_rel4.pdf}\n\\end{center}\n\\caption{Two dots}\\label{fig:two dot}\n\\end{subfigure}\n\\caption{Relations in $\\mathscr{BN}$}\\label{fig:BN relations}\n\\end{figure}\n\nThe Bar-Natan relations (Figure \\ref{fig:BN relations}) can be seen as arising from the structure of $A_0$ in the following way. Interpret the cup cobordism as the unit map\n\\[\n\\eta_0\\colon \\mathbb{Z} \\to A_0,\\, 1\\mapsto 1,\n\\]\nthe cap as the trace $\\varepsilon_0$, and a dot as multiplication by $X\\in A_0$. Then the sphere relation corresponds to \n\\[\n\\varepsilon_0(\\eta_0(1))=0\n\\]\nwhile the dotted sphere comes from $\\varepsilon_0(X) = 1$. The two dots relation corresponds to the relation $X^2 =0$ in $A_0$. Neck-cutting is a topological incarnation of the algebraic relation\n\\[\ny = X \\varepsilon_0(y) + \\varepsilon_0(Xy),\n\\]\nwhich holds for every $y\\in A_0$. \n\nFor a cobordism $S\\subset \\mathbb{R}^2 \\times I$, let $d(S)$ denote the number of dots on $S$, and set the degree of $S$ to be \n\\begin{equation}\\label{eq:grading on cobs}\n\\deg(S) = -\\chi(S) + 2d(S).\n\\end{equation}\nNote that the relations in Figure \\ref{fig:BN relations} are homogeneous. Define the quantum grading $\\operatorname{qdeg}$ on $A_0$ by setting \n\\begin{equation}\\label{eq:1 and X gradings}\n\\operatorname{qdeg}(1) = -1 \\hskip2em \\operatorname{qdeg}(X) = 1.\n\\end{equation}\n\n\\begin{remark}\\label{rmk:gradings A_0}\nThe grading elsewhere in the literature \\cite{Kh1, BN1, BN2} is opposite that of the one appearing here. Moreover, viewing $A_0$ as an algebra, it is more natural to set $1$ and $X$ in degrees $0$ and $2$, respectively, to make the multiplication grading-preserving. However, when viewing $A_0$ as a module, degrees are balanced around $0$ as above. \n\\end{remark}\n\n\nFor a ring $\\k$, let $\\k{\\rm -}\\operatorname{gmod}$ denote the category of $\\mathbb{Z}$-graded $\\k$-modules and graded maps (of any degree) between them. \nThe Frobenius algebra $A_0$ defines a $(1+1)$-dimensional TQFT, and it descends to a graded, additive functor \n\\begin{equation}\\label{eq:tqft F}\n\\mathcal{F} \\colon \\mathscr{BN} \\to \\mathbb{Z}{\\rm -}\\operatorname{gmod}.\n\\end{equation}\nwhich is $\\mathbb{Z}$-linear on each morphism space. In fact, due to delooping \\cite{BN3}, any such functor is determined by its value on the empty diagram.\n\nLet $D$ be a diagram for an oriented link $L\\subset \\mathbb{R}^3$. We recall the construction $[[D]]$ from \\cite{BN2}, which is a chain complex over the additive category $\\mathscr{BN}$. One first forms the \\emph{cube of resolutions} as follows. Label the crossings of $D$ by $1,\\ldots, n$. Every crossing may be resolved in two ways, called the \\emph{0-smoothing} and \\emph{1-smoothing}, shown in Figure \\ref{fig:0 and 1 smoothings}. For each $u = (u_1,\\ldots, u_n)\\in \\{0,1\\}^n$, perform the $u_i$-smoothing at the $i$-th crossing. The resulting diagram is a collection of disjoint simple closed curves in the plane $\\mathbb{R}^2$, and we denote it by $D_u$. Thinking of elements of $\\{0,1\\}^n$ as vertices of an $n$-dimensional cube, decorate the vertex $u$ by the smoothing $D_u$. \n\n\\begin{figure}\n\\centering\n\\includegraphics{QAH65.pdf}\n\\caption{The two smoothings at a crossing}\\label{fig:0 and 1 smoothings}\n\\end{figure}\n\nNext, let $u=(u_1,\\ldots, u_n)$ and $v=(v_1,\\ldots, v_n)$ be vertices which differ only in the $i$-th entry, where $u_i = 0$ and $v_i=1$. Then the diagrams $D_u$ and $D_v$ are the same outside of a small disk around the $i$-th crossing. There is a cobordism from $D_u$ to $D_v$, which is the obvious saddle ($1$-handle attachment) near the $i$-th crossing and the identity (product cobordism) elsewhere. Denote this cobordism by $d_{u,v}$, and decorate each edge of the $n$-dimensional cube by these saddle cobordisms. This forms a commutative cube in the category $\\mathscr{BN}$. There is a way to assign $s_{u,v} \\in \\{0,1\\}$ to each edge in the cube so that multiplying the edge map $d_{u,v}$ by $(-1)^{s_{u,v}}$ results in an anti-commutative cube (see \\cite[Section 2.7]{BN2}). \n\n\nFor $u=(u_1,\\ldots, u_n) \\in \\{0,1\\}^n$, set $\\lr{u} = \\sum_i u_i$. Now, form the chain complex $[[D]]$ over $\\mathscr{BN}$ by setting \n\\[\n[[D]]^i = \\bigoplus_{\\vert u \\vert = i+ n_-} D_{u}\\{n_- - n_+ -i \\} \n\\] \nin homological degree $i$, where $n_-$, $n_+$ denote the number of negative and positive crossings in $D$, and $\\{-\\}$ is the upwards grading shift in $\\mathscr{BN}$. The differential is given on each summand by the edge map $(-1)^{s_{u,v}}d_{u,v}$. Anti-commutativity of the cube ensures that $[[D]]$ is a chain complex. \n\nThe relations in Figure \\ref{fig:BN relations} imply the $S$, $T$, and $4Tu$ relations from \\cite{BN2}. \n\n\\begin{theorem}\\emph{(\\cite[Theorem 1]{BN2})}\\label{thm:[[D]] is invariant}\nIf diagrams $D$ and $D'$ are related by a Reidemeister move, then $[[D]]$ and $[[D']]$ are chain homotopy equivalent.\n\\end{theorem}\n\nThus to obtain link homology, it suffices to apply a functor from $\\mathscr{BN}$ into an abelian category, and Theorem \\ref{thm:[[D]] is invariant} guarantees that the homotopy class of the resulting chain complex is a link invariant. In particular, the TQFT \\eqref{eq:tqft F} yields a chain complex \n\\[\nCKh(D) := \\mathcal{F}([[D]])\n\\]\nof graded abelian groups. After reversing the quantum grading, this is the chain complex appearing in \\cite[Section 7]{Kh1}. \n\n\\subsection{$U(2)$-equivariant Khovanov homology}\\label{sec:equivariant Khovanov homology}\n\n\nThis section reviews the so-called $U(2)$-equivariant Frobenius pair, denoted $(R,A)$. Although this extension is of general importance, it is not necessary for our construction in Section \\ref{sec:equivariant annular Khovanov homology}; in fact, in Section \\ref{sec:preliminary observation} we note that an analogue of annular APS homology using $(R,A)$ is not possible. \n\nConsider the graded ring\n\\[\nR = \\mathbb{Z}[E_1, E_2] \n\\]\nwith $\\deg(E_1) = 2$, $\\deg(E_2) = 4$. The $R$-algebra\n\\[\nA = R[X]\/(X^2 - E_1 X + E_2)\n\\]\nequipped with the trace \n\\[\n\\varepsilon \\colon A\\to R,\\, 1 \\mapsto 0, X\\mapsto 1\n\\]\nis a Frobenius algebra over $R$. The rings $R$ and $A$ are the $U(2)$-equivariant cohomology with $\\mathbb{Z}$ coefficients of a point and $\\mathbb{CP}^1$, respectively \\cite{Kh3}. The Frobenius algebra $A$ determines a link homology theory as in Section \\ref{sec:Khovanov homology}, obtained by applying the corresponding TQFT to the formal complex $[[D]]$. \n\n\n\n\n\\subsection{$U(1) \\times U(1)$-equivariant Khovanov homology}\\label{sec:equivariant Khovanov homology extension}\n\nIn this section we review an extension of the Frobenius pair $(R,A)$. This extension was studied in \\cite{KR} and is central to our construction in Section \\ref{sec:equivariant annular Khovanov homology}. \n\nLet \n\\[\nR_\\alpha = \\mathbb{Z}[\\alpha_0, \\alpha_1],\n\\]\nand consider the $R_\\alpha$-algebra\n\\[\nA_\\alpha = R_\\alpha[X]\/((X-\\alpha_0)(X-\\alpha_1)).\n\\]\nThe trace \n\\[\n\\varepsilon_\\alpha \\colon A_\\alpha \\to R_\\alpha,\\, 1\\mapsto 0,\\, X\\mapsto 1.\n\\]\nmakes $A_\\alpha$ into a Frobenius algebra, with comultiplication\n\\begin{align*}\n\\Delta \\colon A_\\alpha & \\to A_\\alpha\\otimes A_\\alpha\\\\\n 1 & \\mapsto (X-\\alpha_0)\\otimes 1 + 1\\otimes (X-\\alpha_1)\\\\\n X & \\mapsto X\\otimes X - \\alpha_0 \\alpha_1 1\\otimes 1 .\n\\end{align*}\nThere is an inclusion $(R, A) \\hookrightarrow (R_\\alpha, A_\\alpha)$ given by identifying $E_1, E_2\\in R$ with the elementary symmetric polynomials in $R_\\alpha$,\n\\[\nE_1 \\mapsto \\alpha_0 + \\alpha_1 \\hskip2em E_2 \\mapsto \\alpha_0\\alpha_1.\n\\]\nAs noted in \\cite{KR}, $R_\\alpha$ and $A_\\alpha$ are the $U(1) \\times U(1)$-equivariant cohomology with $\\mathbb{Z}$ coefficients of a point and $2$-sphere $S^2$, respectively. \n\n\n\nLet $\\mathscr{BN}_\\alpha$ denote the Bar-Natan category subject to relations coming from $A_\\alpha$. Objects of $\\mathscr{BN}_\\alpha$ are formal direct sums of formally graded collections of simple closed curves in the plane $\\mathbb{R}^2$. Morphisms are matrices whose entries are formal $R_\\alpha$-linear combinations of dotted cobordisms properly embedded in $\\mathbb{R}^2 \\times I$, modulo isotopy relative to the boundary, and subject to the local relations shown in Figure \\ref{fig:BNa relations}. As outlined in Section \\ref{sec:Khovanov homology}, these topological relations correspond to algebraic relations in the Frobenius algebra $A_\\alpha$. \n\n\\begin{remark}\\label{rmk:cob cat BNa is induced from U(2)}\nWe note that $\\mathscr{BN}_\\alpha$ is induced from the corresponding $U(2)$-equivariant cobordism category with relations dictated by $(R,A)$, since the relations involve only symmetric polynomials in $\\alpha_0, \\alpha_1$. \n\\end{remark}\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{BN_rel1.pdf}\n\\end{center}\n\\caption{Sphere}\\label{fig:alpha sphere}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.7\\textwidth}\n\\begin{center}\n\\includegraphics{BNalpha_neck_cutting.pdf}\n\\end{center}\n\\caption{Neck-cutting}\\label{fig:alpha neck-cutting}\n\\end{subfigure}\\\\\n\\vskip1em\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{BN_rel3.pdf}\n\\end{center}\n\\caption{Dotted sphere}\\label{fig:alpha dotted sphere}\n\\end{subfigure}%\n\\begin{subfigure}[b]{.65\\textwidth}\n\\begin{center}\n\\includegraphics{BNalpha_two_dots.pdf}\n\\end{center}\n\\caption{Two dots}\\label{fig:alpha two dot}\n\\end{subfigure}\n\\caption{Relations in $\\mathscr{BN}_\\alpha$}\\label{fig:BNa relations}\n\\end{figure}\n\n\n\nFor a cobordism $S\\subset \\mathbb{R}^2\\times I$, define the degree of $S$ as in \\eqref{eq:grading on cobs}, and put $\\alpha_0, \\alpha_1\\in R_\\alpha$ in degree $2$. Note that the relations in Figure \\ref{fig:BNa relations} are homogeneous. The algebra $A_\\alpha$ is a free $R_\\alpha$-module with basis $\\{1, X\\}$. Using the same notation as in \\eqref{eq:1 and X gradings}, define a grading $\\operatorname{qdeg}$ on $A_\\alpha$ by setting \n\\begin{equation}\\label{eq:gradings A alpha}\n\\operatorname{qdeg}(1) = -1 \\hskip2em \\operatorname{qdeg}(X) = 1.\n\\end{equation}\n\n\\begin{remark}\\label{rmk:gradings}\nViewing $A_\\alpha$ as an $R_\\alpha$-algebra, it is more natural to set $1$ and $X$ in degrees $0$ and $2$, respectively, so that multiplication in $A_\\alpha$ is grading-preserving. When viewing $A_\\alpha$ as an $R_\\alpha$-module with homogeneous basis $\\{1, X\\}$ according to the grading \\eqref{eq:gradings A alpha}, the elements $X-\\alpha_0$ and $X-\\alpha_1$ should be interpreted as $X - \\alpha_0 \\cdot 1$ and $X-\\alpha_1 \\cdot 1$, which are homogeneous of degree $1$. In either case, multiplication by $X$ is a degree $2$ endomorphism of $A_\\alpha$. \n\\end{remark}\n\nThe Frobenius algebra $A_\\alpha$ defines a two-dimensional TQFT, and it descends to a graded, additive functor \n\\begin{equation}\\label{eq:tqft Falpha}\n\\mathcal{F}_{\\alpha} \\colon \\mathscr{BN}_\\alpha \\to R_\\alpha{\\rm -}\\operatorname{gmod}\n\\end{equation}\nwhich is $R_\\alpha$-linear on each morphism space. Moreover, the following diagram commutes\n\\begin{equation}\\label{eq:diagram1}\n\\begin{tikzcd}\n\\mathscr{BN}_\\alpha \\ar[r,\"\\mathcal{F}_{\\alpha}\"] \\ar[d] & R_\\alpha{\\rm -}\\operatorname{gmod} \\ar[d]\\\\\n\\mathscr{BN} \\ar[r, \"\\mathcal{F}\"] \\ar[r] & \\mathbb{Z}{\\rm -}\\operatorname{gmod}\n\\end{tikzcd}\n\\end{equation}\nwhere the vertical maps are obtained by setting $\\alpha_0 = \\alpha_1=0$.\n\nGiven a diagram $D$ for an oriented link $L \\subset \\mathbb{R}^3$, form the chain complex $[[D]]$ as described in Section \\ref{sec:Khovanov homology}. We may view $[[D]]$ as a chain complex over $\\mathscr{BN}_\\alpha$. The relations in Figure \\ref{fig:BNa relations} imply the $S$, $T$, and $4Tu$ relations from \\cite{BN2}, so by \\cite[Theorem 1]{BN2}, the homotopy class of $[[D]]$ is an invariant of $L$. It follows that the chain complex obtained by applying $\\mathcal{F}_{\\alpha}$ to $[[D]]$ is an invariant of $L$ up to chain homotopy equivalence. \n\n\n\\subsection{Inverting the discriminant and Lee homology}\\label{sec:inverting D} \n\nWe recall from \\cite{KR} a further extension of the Frobenius pair $(R_\\alpha, A_\\alpha)$. Let \n\\begin{equation}\\label{eq:D}\n\\mathcal{D} = (\\alpha_0 -\\alpha_1)^2\n\\end{equation}\ndenote the discriminant of the quadratic polynomial $(X-\\alpha_0)(X-\\alpha_1) \\in R_\\alpha[X]$. Let \n\\[\nR_{\\alpha\\mathcal{D}} = R_\\alpha [ \\mathcal{D}^{-1}]\n\\]\ndenote the ring obtained by inverting $\\mathcal{D}$ (equivalently, one may invert $\\alpha_0 - \\alpha_1$) and let\n\\[\nA_{\\alpha\\mathcal{D}} = A_\\alpha \\o_{R_\\alpha} R_{\\alpha\\mathcal{D}}\n\\]\nbe the extension of $A_\\alpha$ to an $R_{\\alpha\\mathcal{D}}$-algebra. Let $\\mathcal{F}_{\\alpha \\mathcal{D}}$ denote the composition \n\\[\n\\mathscr{BN}_\\alpha \\rar{\\mathcal{F}_{\\alpha}} R_\\alpha {\\rm -}\\operatorname{gmod} \\to R_{\\alpha\\mathcal{D}} {\\rm -}\\operatorname{gmod}\n\\]\nwhere the second functor is extension of scalars, $(-)\\o_{R_\\alpha} R_{\\alpha\\mathcal{D}}$. For a link $L\\subset \\mathbb{R}^3$ with diagram $D$, let \n\\[\nCKh_{{\\alpha\\mathcal{D}}}(D) := \\mathcal{F}_{\\alpha \\mathcal{D}}([[D]])\n\\]\ndenote the resulting chain complex. It is an invariant of $L$ up to chain homotopy equivalence, and we will denote its homology by $Kh_{{\\alpha\\mathcal{D}}}(L)$. \n\nThe elements \n\\begin{equation}\\label{eq:e_0 and e_1}\ne_0 = \\frac{X-\\alpha_0}{\\alpha_1-\\alpha_0},\\hskip1em e_1 = \\frac{X - \\alpha_1}{\\alpha_0 - \\alpha_1} \\in A_{\\alpha\\mathcal{D}}. \n\\end{equation}\nform a basis for $A_{\\alpha\\mathcal{D}}$ and satisfy \n\\[\ne_0 + e_1 = 1,\\ e_0^2 = e_0,\\ e_1^2 = e_1,\\ e_0e_1=0,\n\\]\nso that the algebra $A_{\\alpha\\mathcal{D}}$ decomposes as a product, $A_{\\alpha\\mathcal{D}} = R_{\\alpha\\mathcal{D}} e_0 \\times R_{\\alpha\\mathcal{D}} e_1$. With respect to the basis $\\{e_0, e_1\\}$, comultiplication in $A_{\\alpha\\mathcal{D}}$ is simply given by\n\\begin{equation}\\label{eq:comultiplication D}\n\\begin{split}\n\\Delta(e_0) &= (\\alpha_1 - \\alpha_0) e_0 \\o e_0 \\\\\n\\Delta(e_1) &= (\\alpha_0 - \\alpha_1) e_1\\o e_1.\n\\end{split}\n\\end{equation}\n\n\nAs noted in \\cite[Section 1.2]{KR}, the TQFT $\\mathcal{F}_{\\alpha \\mathcal{D}}$ is essentially the Lee deformation \\cite{Le}. By \\cite[Theorem 4.2]{Le}, the Lee homology of a $k$-component link is free (over $\\mathbb{Q}$) of rank $2^k$. A quick alternate proof can be found in the final remark in \\cite{We}. The following is stated in \\cite{KR} without proof, but the arguments in \\cite{We} apply without modification. \n\n\\begin{proposition}\\label{prop:Lee rank}\nFor a link $L\\subset \\mathbb{R}^3$ with $k$ components, the homology $Kh_{{\\alpha\\mathcal{D}}}(L)$ is a free $R_{\\alpha\\mathcal{D}}$-module of rank $2^k$.\n\\end{proposition}\n\n\n\n\n\\section{Annular Khovanov homology}\\label{sec:annular Khovanov homology}\nWe give an overview of annular Khovanov homology, also known as annular Asaeda-Przytycki-Sikora (APS) homology. It was originally defined in \\cite{APS} as part of a broader categorification of the Kauffman bracket skein module of $I$-bundles over surfaces. A convenient reference for the annular setting is \\cite{GLW}. \n\nLet $\\mathbb{A} = S^1\\times I$ denote the annulus. An \\textit{annular link} is a link in the thickened annulus $\\mathbb{A}\\times I$, and its diagram is a projection onto the first factor of $\\mathbb{A} \\times I$. Embed $\\mathbb{A}$ standardly in $\\mathbb{R}^2$ as \n\\[\n\\mathbb{A} = \\{ x\\in \\mathbb{R}^2 \\mid 1\\leq \\lr{x} \\leq 2\\},\n\\]\nso that an annular link diagram and all of its smoothings are drawn in the punctured plane $\\mathbb{R}^2\\setminus (0,0)$. We represent the annulus in the plane by simply indicating the puncture using the symbol $\\times$. Figure \\ref{fig:annular link} illustrates an example of an annular link diagram. By a \\emph{circle} in $\\mathbb{A}$ we mean a smoothly and properly embedded $S^1$ in $\\mathbb{A}$. There are two kinds of circles in $\\mathbb{A}$: \\emph{trivial} circles, which are contractible in $\\mathbb{A}$, and \\emph{essential} ones, which are not contractible. \n \n \n\\begin{figure}\n\\centering\n\\includegraphics{QAH49.pdf}\n\\caption{An annular link diagram}\\label{fig:annular link}\n\\end{figure}\n\n\nLet $\\mathscr{BN}(\\mathbb{A})$ denote the Bar-Natan category of the annulus. Its objects are formal direct sums of formally bigraded collections of simple closed curves in $\\mathbb{A}$. Morphisms are matrices whose entries are formal $\\mathbb{Z}$-linear combinations of dotted cobordisms properly embedded in $\\mathbb{A} \\times I$, modulo isotopy relative to the boundary, and subject to the local relations shown in Figure \\ref{fig:BN relations}. The bidegree of a cobordism $S\\subset \\mathbb{A} \\times I$ is defined to be\n\\begin{equation}\\label{eq:annular bidegree}\n(-\\chi(S) + d(S), 0),\n\\end{equation}\nwhere $d(S)$ is the number of dots on $S$. \n\nFor a ring $\\k$, denote by $\\k{\\rm -}\\operatorname{ggmod}$ the category of $\\mathbb{Z} \\times \\mathbb{Z}$-graded $\\k$-modules and graded maps (of any bidegree) between them. We now describe the annular TQFT\n\\[\n\\mathcal{G} \\colon \\mathscr{BN}(\\mathbb{A}) \\to \\mathbb{Z}{\\rm -}\\operatorname{ggmod},\n\\]\nwhich will be additive, graded, and $\\mathbb{Z}$-linear on each morphism space. \n\nLet $\\mathscr{C} \\subset \\mathbb{A}$ be a collection of $n$ trivial and $m$ essential circles. Embed $\\mathbb{A} \\times I$ standardly into $\\mathbb{R}^2 \\times I$, and apply the TQFT $\\mathcal{F}$ from Section \\ref{sec:Khovanov homology},\n\\[\n\\mathcal{F}(\\mathscr{C}) = A_0^{\\o n} \\o A_0^{\\o m}. \n\\]\nDefine a second grading, called the \\emph{annular} grading and denoted $\\operatorname{adeg}$, on $\\mathcal{F}(\\mathscr{C})$ in the following way. A tensor factor $A_0$ corresponding to a trivial circle is concentrated in annular degree $0$. For a factor $A_0$ corresponding to an essential circle, let\n\\[\nv_0 = 1,\\hskip2em v_1 = X.\n\\]\ndenote a basis for this copy of $A_0$, and set \n\\[\n\\operatorname{adeg}(v_0) = -1 \\hskip2em \\operatorname{adeg}(v_1) = 1.\n\\]\nBigradings are summarized in Figure \\ref{fig:bigradings}. \n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\draw[->] (-1.5,0) -- (1.5,0) node[right] {$\\operatorname{qdeg}$};\n\\foreach \\x in {-1, 1}\n\\draw[shift={(\\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {\\tiny $\\x$};\n\\draw[->] (0,-1.5) -- (0,1.5) node[above] {$\\operatorname{adeg}$};\n\\foreach \\y in {-1,1}\n\\draw[shift={(0,\\y)}] (2pt,0pt) -- (-2pt,0pt) node[left] {\\tiny $\\y$};\n\\filldraw[black] (1,1) circle (2pt) node[anchor=west] {$v_1$};\n\\filldraw[black] (-1,-1) circle (2pt) node[anchor=east] {$v_0$};\n\\filldraw[black] (1,0) circle (2pt) node[anchor=south] {$X$};\n\\filldraw[black] (-1,0) circle (2pt) node[anchor=south] {$1$};\n\\end{tikzpicture}\n\\caption{Bigradings, where $\\{1, X\\}$ corresponds to trivial circles, and $\\{v_0, v_1\\}$ correspond to essential ones. See also Remark \\ref{rmk:gradings A_0}.}\\label{fig:bigradings}\n\\end{figure}\n\n\nThe underlying abelian group of $\\mathcal{G}(\\mathscr{C})$ is $\\mathcal{F}(\\mathscr{C})$, and the bigrading is given by $(\\operatorname{qdeg}, \\operatorname{adeg})$. For a cobordism $S\\subset \\mathbb{A}\\times I$, first view $S$ as a surface in $\\mathbb{R}^2 \\times I$ and consider the map $\\mathcal{F}(S)$. It is shown in \\cite[Section 2]{Ro} that $\\mathcal{F}(S)$ splits as a sum\n\\begin{equation}\\label{eq:non alpha cob split}\n\\mathcal{F}(S) = \\mathcal{F}(S)_0 + \\mathcal{F}(S)_+\n\\end{equation}\nwhere $\\mathcal{F}(S)_0$ preserves $\\operatorname{adeg}$ and $\\mathcal{F}(S)_+$ increases $\\operatorname{adeg}$. Set \n\\[\n\\mathcal{G}(S) := \\mathcal{F}(S)_0\n\\]\nto be the $\\operatorname{adeg}$-preserving part. It follows from \\eqref{eq:non alpha cob split} that $\\mathcal{G}$ is functorial with respect to composition of cobordisms. By construction, $\\mathcal{G}(S)$ is a map of bidegree \n\\[\n(-\\chi(S)+2d(S), 0)\n\\]\nso the functor $\\mathcal{G}$ is degree preserving on morphism spaces. We will refer to $\\mathcal{G}$ as the \\emph{annular TQFT}. \n\nTo distinguish the bigraded modules assigned to trivial and essential circles, write \n\\[\nV = \\mathcal{G}(C)\n\\]\nif $C$ is an essential circle, with basis written as $\\{v_0, v_1\\}$, and keep the notation $A_0 = \\mathcal{G}(C)$ when $C$ is trivial. Then if $\\mathscr{C} \\subset \\mathbb{A}$ consists of $n$ trivial and $m$ essential circles, the module assigned to $\\mathscr{C}$ is\n\\[\n\\mathcal{G}(\\mathscr{C}) = A_0^{\\o n} \\o V^{\\o m}.\n\\]\n\nGiven a diagram $D$ for an oriented annular link $L$, form the chain complex $[[D]]$ as described in Section \\ref{sec:Khovanov homology}. The construction is completely local and crossings are away from the puncture $\\times$. Thus we may view $[[D]]$ as a chain complex over $\\mathscr{BN}(\\mathbb{A})$, with $\\mathbb{Z}$-grading shifts $\\{-\\}$ in $\\mathscr{BN}$ rewritten as a $\\mathbb{Z}\\times \\mathbb{Z}$-grading shifts $\\{-,0\\}$ in $\\mathscr{BN}(\\mathbb{A})$. Isotopies of annular links are described by Reidemeister moves away from the puncture, and it follows that the homotopy class of $[[D]]$, viewed as a chain complex over $\\mathscr{BN}(\\mathbb{A})$, is an invariant of $L$. Therefore the chain complex \n\\begin{equation}\\label{eq:annular chain complex}\nCKh^\\mathbb{A}(D) := \\mathcal{G}([[D]])\n\\end{equation}\nis an invariant of $L$ up to chain homotopy equivalence. \n\n\nAn \\emph{elementary cobordism} is one that has a single non-degenerate critical point with respect to the height function $\\mathbb{A} \\times I \\to I$. It consists of a union of a product cobordism and a single cup, cap, or saddle. An elementary cobordism $S$ with $\\d S$ consisting of trivial circles in $\\mathbb{A}$ is assigned the same map by $\\mathcal{F}$ and $\\mathcal{G}$. We record the maps assigned to the four elementary saddles involving at least one essential circle, Figure \\ref{fig:el saddles}. The vertical red arc is the central axis of $\\mathbb{A} \\times I \\subset \\mathbb{R}^2\\times I$. \n\n\n\\noindent\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\\label{eq:formula1}\n \\begin{split}\n V\\o A_0 & \\rar{\\hyperref[fig:type1]{\\operatorname{(I)}}} V \\\\ v_0 \\o 1 & \\mapsto v_0 \\\\ v_1\\o 1 &\\mapsto v_1 \\\\\n v_0 \\o X & \\mapsto 0 \\\\\n v_1 \\o X & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.35\\linewidth}\n\\begin{equation}\\label{eq:formula2}\n \\begin{split}\n V \\o V & \\rar{\\hyperref[fig:type2]{\\operatorname{(II)}}} A_0 \\\\ v_0 \\o v_0 &\\mapsto 0 \\\\\nv_1 \\o v_0 & \\mapsto X \\\\\nv_0 \\o v_1 & \\mapsto X \\\\\nv_1 \\o v_1 & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\n\\vskip1em\n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation}\\label{eq:formula3}\n \\begin{split}\n V & \\rar{\\hyperref[fig:type3]{\\operatorname{(III)}}} V \\o A_0 \\\\\nv_0 & \\mapsto v_0 \\o X \\\\\nv_1 & \\mapsto v_1 \\o X\n \\end{split}\n\\end{equation}\n\\end{minipage} \n\\begin{minipage}{.4\\linewidth}\n\\begin{equation}\\label{eq:formula4}\n \\begin{split}\n A_0 & \\rar{\\hyperref[fig:type4]{\\operatorname{(IV)}}} V\\o V \\\\\n1 &\\mapsto v_0 \\o v_1 + v_1 \\o v_0\\\\\nX & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\\vskip1em\n\n\n\\begin{figure}\n\\begin{subfigure}[b]{.2\\textwidth}\n\\begin{center}\n\\includegraphics{AD_type1.pdf}\n\\end{center}\n\\caption{Type (I)}\\label{fig:type1}\n\\end{subfigure}\n\\begin{subfigure}[b]{.2\\textwidth}\n\\begin{center}\n\\includegraphics{AD_type2b.pdf}\n\\end{center}\n\\caption{Type (II)}\\label{fig:type2}\n\\end{subfigure}\n\\begin{subfigure}[b]{.2\\textwidth}\n\\begin{center}\n\\includegraphics{AD_type3.pdf}\n\\end{center}\n\\caption{Type (III)}\\label{fig:type3}\n\\end{subfigure}\n\\begin{subfigure}[b]{.2\\textwidth}\n\\begin{center}\n\\includegraphics{AD_type4b.pdf}\n\\end{center}\n\\caption{Type (IV)}\\label{fig:type4}\n\\end{subfigure}\n\\caption{Saddles involving essential circles}\\label{fig:el saddles}\n\\end{figure}\n\n\n\nFrom \\eqref{eq:formula1}, we see that $X$ acts trivially on any essential circle. It follows that a cobordism with a component that carries a dot and a closed curve which is nonzero in $\\pi_1(\\mathbb{A} \\times I)$ is assigned the zero map. Thus $\\mathcal{G}$ factors through the relation shown in Figure \\ref{fig:Boerner}, called Boerner's relation \\cite{Bo}. Indeed, for an essential circle $C\\subset \\mathbb{A}$, there are no nonzero endomorphisms of $\\mathcal{G}(C)$ with bidegree $(2,0)$. \n\n\\begin{figure}\n\\centering\n\\includegraphics{Boerner_relation.pdf}\n\\caption{Boerner's relation}\\label{fig:Boerner}\n\\end{figure}\n\nThe category $\\mathscr{BN}(\\mathbb{A})$ is monoidal, with monoidal product given by taking two copies $\\mathbb{A}_1, \\mathbb{A}_2$ of $\\mathbb{A}$ and gluing the boundary component $S^1\\times \\{1\\}$ of $\\mathbb{A}_1$ to the boundary component $S^1 \\times \\{0\\}$ of $\\mathbb{A}_2$. The annular TQFT $\\mathcal{G}$ is evidently monoidal.\n\n\n\\section{Equivariant annular Khovanov homology}\\label{sec:equivariant annular Khovanov homology}\n\n \nWe are interested in an annular version of the theory outlined in Section \\ref{sec:equivariant Khovanov homology extension}. Precisely, the goal is to fill in the dashed arrow in the diagram\n\\[\n\\begin{tikzcd}\\label{eq:dashed diagram}\n\\mathscr{BN}_\\alpha(\\mathbb{A}) \\ar[r, dashed, \"\\mathcal{G}_{\\alpha}\"] \\ar[d] & R_\\alpha{\\rm -}\\operatorname{ggmod} \\ar[d]\\\\\n\\mathscr{BN}(\\mathbb{A}) \\ar[r, \"\\mathcal{G}\"] & \\mathbb{Z}{\\rm -}\\operatorname{ggmod}\n\\end{tikzcd}\n\\]\nwhere the vertical arrows are obtained by setting $\\alpha_0 = \\alpha_1 = 0$. Section \\ref{sec:preliminary observation} justifies working with the extension $(R_\\alpha, A_\\alpha)$ rather than $(R,A)$. The desired functor $\\mathcal{G}_{\\alpha}$ is defined in Section \\ref{sec:defining the functor}. Maps assigned to saddle cobordisms can be found in \\eqref{eq:equiv formula1}--\\eqref{eq:equiv formula4}. In Section \\ref{sec:inverting D equiv annular} we invert $\\mathcal{D}$ in the annular theory and show that the rank of the resulting homology depends only on the number of components.\n\n\\subsection{A preliminary observation}\\label{sec:preliminary observation}\n\nBefore defining our equivariant annular TQFT, we note that the $U(2)$-equivariant Frobenius pair $(R,A)$ from Section \\ref{sec:equivariant Khovanov homology} does not admit such a lift, under the minor assumption that modules assigned to circles are free. \n\nThe ring $R=\\mathbb{Z}[E_1, E_2]$ can be made bigraded, with bidegrees of $E_1$ and $E_2$ given by $(2,0)$ and $(4,0$), respectively. Let $M$ be a free $\\mathbb{Z} \\times \\mathbb{Z}$-graded $R$-module with basis $m_-, m_+$ in bidegrees $(-1,-1)$ and $(1,1)$, respectively. Suppose $g \\colon M\\to M$ is an $R$-linear map of bidegree $(2,0)$. Then necessarily \n\\begin{equation}\\label{eq:map must be}\ng(m_-) = n E_1 m_-\n\\end{equation}\nfor some $n\\in \\mathbb{Z}$. In particular, if $M$ is the module assigned to a single essential circle and $g$ is the map assigned to the cobordism in Figure \\ref{fig:dotted id}, then the relation $X^2 - E_1X + E_2 = 0$\nin $A$ implies \n\\begin{equation}\\label{eq:two dots must be}\ng^2(m_-) - E_1 g(m_-) + E_2 m_- = 0.\n\\end{equation}\nHowever, \\eqref{eq:map must be} and \\eqref{eq:two dots must be} are incompatible. \n\n\\begin{figure}\n\\centering\n\\includegraphics{AD3.pdf}\n\\caption{Dotted product cobordism on an essential circle}\\label{fig:dotted id}\n\\end{figure}\n\n\n\n\\subsection{The equivariant annular TQFT $\\mathcal{G}_\\alpha$}\\label{sec:defining the functor}\n\nLet $\\mathscr{BN}_\\alpha(\\mathbb{A})$ denote the Bar-Natan category of the annulus subject to the relations determined by $A_\\alpha$. Its objects are formal direct sums of formally bigraded collections of simple closed curves in $\\mathbb{A}$. Morphisms are matrices whose entries are formal $R_\\alpha$-linear combinations of dotted cobordisms properly embedded in $\\mathbb{A} \\times I$, modulo isotopy relative to the boundary, and subject to the local relations shown in Figure \\ref{fig:BNa relations}. The bidegree of a cobordism $S\\subset \\mathbb{A} \\times I$ is given by \\eqref{eq:annular bidegree}. For an oriented annular link $L$ with diagram $D$, the formal complex $[[D]]$ over $\\mathscr{BN}_\\alpha(\\mathbb{A})$ is an invariant of $L$ up to chain homotopy equivalence. \n\n\n\nLet $\\mathscr{C} \\subset \\mathbb{A}$ be a collection of circles, and view $\\mathscr{C}$ as embedded in $\\mathbb{R}^2$. Consider $\\mathcal{F}_{\\alpha}(\\mathscr{C})$ with the following additional annular grading, denoted $\\operatorname{adeg}$ as in Section \\ref{sec:equivariant Khovanov homology extension}. Define elements of $A_\\alpha$, \n\\begin{align*}\n\\v_0 = 1, \\hskip2em \\v_1 = X - \\alpha_0, \\\\\n\\v_0'= 1, \\hskip2em \\v_1'= X - \\alpha_1,\n\\end{align*}\nwith the annular gradings \n\\begin{equation}\n\\operatorname{adeg}(\\v_0) = \\operatorname{adeg}(\\v_0') = -1, \\hskip2em \\operatorname{adeg}(\\v_1) = \\operatorname{adeg}(\\v_1') = 1. \n\\end{equation}\n\n\\begin{remark}\nThe notation $v_0, v_1$ was also used in Section \\ref{sec:annular Khovanov homology}. Setting $\\alpha_0 = \\alpha_1 = 0$ in the above expressions recovers $v_0, v_1$ in the non-equivariant setting. \n\\end{remark}\n\n\nBoth $\\{\\v_0, \\v_1\\} = \\{1, X - \\alpha_0\\}$ and $\\{\\v_0', \\v_1'\\} = \\{1, X - \\alpha_1\\}$ is an $R_\\alpha$-basis for $A_\\alpha$. Together with the quantum grading, these equip $A_\\alpha$ with two (isomorphic) structures of a bigraded $R_\\alpha$-module, with the bigrading given by $(\\operatorname{qdeg}, \\operatorname{adeg})$. The ground ring $R_\\alpha$ lies in annular degree $0$. \n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\draw[->] (-2.5,0) -- (2.5,0) node[right] {$\\operatorname{qdeg}$};\n\\foreach \\x in {-2, -1, 1, 2}\n\\draw[shift={(\\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {\\tiny $\\x$};\n\\draw[->] (0,-1.5) -- (0,1.5) node[above] {$\\operatorname{adeg}$};\n\\foreach \\y in {-1,1}\n\\draw[shift={(0,\\y)}] (2pt,0pt) -- (-2pt,0pt) node[left] {\\tiny $\\y$};\n\\filldraw[black] (1,1) circle (2pt) node[anchor=west] {$\\v_1$, $\\v_1'$};\n\\filldraw[black] (-1,-1) circle (2pt) node[anchor=east] {$\\v_0$, $\\v_0'$};\n\\filldraw[black] (1,0) circle (2pt) node[anchor=south] {$X$};\n\\filldraw[black] (-1,0) circle (2pt) node[anchor=south] {$1$};\n\\filldraw[black] (2,0) circle (2pt) node[anchor=south] {$\\alpha_0, \\alpha_1$};\n\\end{tikzpicture}\n\\caption{Bigradings, where $\\{1, X\\}$ corresponds to trivial circles, and $\\{v_0, v_1\\}$, $\\{v_0', v_1'\\}$ correspond to essential ones. See also Remark \\ref{rmk:gradings}.}\\label{fig:bigradings alpha}\n\\end{figure}\n\n\n\nLet $\\mathscr{C} \\subset \\mathbb{A}$ consist of $n$ trivial and $m$ essential circles, with the essential circles ordered from innermost (closest to the puncture $\\times$) to outermost. Define the annular grading on \n\\[\n\\mathcal{F}_{\\alpha}(\\mathscr{C}) = A_\\alpha^{\\o n} \\o A_\\alpha^{\\o m}\n\\]\nby declaring that every copy of $A_\\alpha$ corresponding to a trivial circle is concentrated in annular degree $0$ and that the copy of $A_\\alpha$ corresponding to the $i$-th essential circle $( 1\\leq i \\leq m)$ is given the homogeneous basis \n\\[\n\\{\\v_0, \\v_1\\} = \\{1, X - \\alpha_0\\}\n\\]\nif $i$ is odd and \n\\[\n\\{\\v_0', \\v_1'\\} = \\{1, X - \\alpha_1\\}\n\\]\nif $i$ is even. In other words, the essential circles are assigned the homogeneous bases $\\{ 1, X- \\alpha_0\\}$ or $\\{1, X - \\alpha_1\\}$ in an alternating manner, with the innermost circle assigned $\\{1, X- \\alpha_0\\}$. Bigradings are summarized in Figure \\ref{fig:bigradings alpha}\n\nAs in Section \\ref{sec:annular Khovanov homology}, it is convenient to distinguish the modules assigned to essential and trivial circles. Let $V_\\alpha$ and $V'_\\alpha$ denote the module $A_\\alpha$ with homogeneous bases $\\{\\v_0, \\v_1\\}$ and $\\{\\v_0', \\v_1'\\}$, respectively. Then for a collection of circles $\\mathscr{C} \\subset \\mathbb{A}$, the $i$-th essential circle in $\\mathscr{C}$ is assigned $V_\\alpha$ if $i$ is odd and $V_\\alpha'$ if $i$ is even. We reserve the notation $A_\\alpha$ for the module assigned to a trivial circle. Note that interchanging $\\alpha_0 \\leftrightarrow \\alpha_1$ also interchanges $\\v_0 \\leftrightarrow \\v_0'$ and $\\v_1 \\leftrightarrow \\v_1'$.\n\n\n\\begin{lemma}\\label{lem:el cobs split}\n\nLet $S\\subset \\mathbb{A}\\times I$ be an elementary cobordism. Viewing $S$ as a cobordism in $\\mathbb{R}^2 \\times I$, the map $\\mathcal{F}_{\\alpha}(S)$ splits as a sum\n\\[\n\\mathcal{F}_{\\alpha}(S) = \\mathcal{F}_{\\alpha}(S)_0 + \\mathcal{F}_{\\alpha}(S)_2\n\\]\nwhere $\\mathcal{F}_{\\alpha}(S)_0$ preserves $\\operatorname{adeg}$ and $\\mathcal{F}_{\\alpha}(S)_2$ increases $\\operatorname{adeg}$ by $2$.\n\\end{lemma}\n\n\\begin{proof}\n \nIf the saddle component of $S$ involves only trivial circles then the claim is immediate, since $\\mathcal{F}_{\\alpha}(S) = \\mathcal{F}_{\\alpha}(S)_0$ in this case. We verify the claim for the four elementary cobordisms in Figure \\ref{fig:el saddles} by rewriting $\\mathcal{F}_{\\alpha}(S)$ in terms of the bases for the circles involved. Terms where $\\operatorname{adeg}$ is increased by $2$ are boxed.\n\n\n\\noindent\\begin{minipage}{.5\\textwidth}\n\\begin{equation*}\n \\begin{split}\n V_\\alpha \\o A_\\alpha & \\rar{\\hyperref[fig:type1]{\\operatorname{(I)}}} V_\\alpha \\\\ \\v_0 \\o 1 & \\mapsto \\v_0 \\\\ \\v_1\\o 1 &\\mapsto \\v_1 \\\\\n \\v_0 \\o X & \\mapsto \\alpha_0 \\v_0 +\\fbox{$\\v_1$} \\\\\n \\v_1 \\o X & \\mapsto \\alpha_1 \\v_1\n \\end{split}\n\\end{equation*}\n\\end{minipage}%\n\\begin{minipage}{.38\\textwidth}\n\\begin{equation*}\n \\begin{split}\n V_\\alpha \\o V_\\alpha' & \\rar{\\hyperref[fig:type2]{\\operatorname{(II)}}} A_\\alpha \\\\ \\v_0 \\o \\v_0' &\\mapsto \\fbox{$1$} \\\\\n\\v_1 \\o \\v_0' & \\mapsto X - \\alpha_0 \\\\\n\\v_0 \\o \\v_1' & \\mapsto X - \\alpha_1 \\\\\n\\v_1 \\o \\v_1' & \\mapsto 0\n \\end{split}\n\\end{equation*}\n\\end{minipage}\n\n\\vskip1em\n \n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation*}\n \\begin{split}\n V_\\alpha & \\rar{\\hyperref[fig:type3]{\\operatorname{(III)}}} V_\\alpha \\o A_\\alpha \\\\\n\\v_0 & \\mapsto \\v_0 \\o X - \\alpha_1 \\v_0 \\o 1 + \\fbox{$\\v_1 \\o 1$}\\\\\n\\v_1 & \\mapsto \\v_1 \\o X - \\alpha_0 \\v_1 \\o 1\n \\end{split}\n\\end{equation*}\n\\end{minipage} \n\\begin{minipage}{.5\\linewidth}\n\\begin{equation*}\n \\begin{split}\n A_\\alpha & \\rar{\\hyperref[fig:type4]{\\operatorname{(IV)}}} V_\\alpha\\o V_\\alpha' \\\\\n1 &\\mapsto \\v_0 \\o \\v_1' + \\v_1 \\o \\v_0'\\\\\nX & \\mapsto \\alpha_0 \\v_0 \\o \\v_1' + \\alpha_1 \\v_1 \\o \\v_0' + \\fbox{$\\v_1 \\o v_1'$}\n \\end{split}\n\\end{equation*}\n\\end{minipage}\n\\vskip1em\n\\noindent\n\nOur assignment for essential circles depends on nesting, so strictly speaking the above calculations do not handle all cases. However, note that for types \\hyperref[fig:type3]{(I)} and \\hyperref[fig:type3]{(II)}, the position of the essential circle does not change, and for types \\hyperref[fig:type3]{(III)} and \\hyperref[fig:type3]{(IV)}, the two essential circles involved in the saddle must be consecutive in the ordering. Thus a full verification amounts to interchanging $\\v_0 \\leftrightarrow \\v_0'$, $\\v_1 \\leftrightarrow \\v_1'$ in the input of above maps. One may check that this amounts to interchanging $\\v_0 \\leftrightarrow \\v_0'$, $\\v_1 \\leftrightarrow \\v_1'$, and $\\alpha_0 \\leftrightarrow \\alpha_1$ in the output.\n\\end{proof}\n\n\n\\begin{corollary}\\label{cor:cobs split} \n\\begin{enumerate}[label= \\emph{(\\arabic*)}]\n\\item Let $S\\subset \\mathbb{A} \\times I$ be a cobordism. Viewing $S$ as a cobordism in $\\mathbb{R}^2 \\times I$, the map $\\mathcal{F}_{\\alpha}(S)$ splits as a sum\n\\[\n\\mathcal{F}_{\\alpha}(S) = \\mathcal{F}_{\\alpha}(S)_0 + \\mathcal{F}_{\\alpha}(S)_+\n\\]\nwhere $\\mathcal{F}_{\\alpha}(S)_0$ preserves $\\operatorname{adeg}$ and $\\mathcal{F}_{\\alpha}(S)_+$ increases $\\operatorname{adeg}$. \n\\item Let $S_1, S_2 \\subset \\mathbb{A} \\times I$ be composable cobordisms. Then\n\\[\n\\mathcal{F}_{\\alpha}(S_2 S_1)_0 = \\mathcal{F}_{\\alpha}(S_2)_0 \\mathcal{F}_{\\alpha}(S_1)_0.\n\\] \n\\end{enumerate}\n\\end{corollary}\n\n\n\\begin{proof}\nFor $(1)$, write $S$ as a composition $S = S_n \\cdots S_1$ where each $S_i$ is an elementary cobordism. Functoriality of $\\mathcal{F}_{\\alpha}$ and Lemma \\ref{lem:el cobs split} yield\n\n\\begin{align*}\n\\mathcal{F}_{\\alpha}(S) &= \\mathcal{F}_{\\alpha}(S_n) \\cdots \\mathcal{F}_{\\alpha}(S_1)\\\\\n& = \\left( \\mathcal{F}_{\\alpha}(S_n)_0 + \\mathcal{F}_{\\alpha}(S_n)_2\\right) \\cdots \\left( \\mathcal{F}_{\\alpha}(S_1)_0 + \\mathcal{F}_{\\alpha}(S_1)_2\\right)\\\\\n& = \\mathcal{F}_{\\alpha}(S_n)_0 \\cdots \\mathcal{F}_{\\alpha}(S_1)_0 + \\text{terms that increase } \\operatorname{adeg} .\n\\end{align*}\nTherefore \n\\[\n\\mathcal{F}_{\\alpha}(S)_0 = \\mathcal{F}_{\\alpha}(S_n)_0 \\cdots \\mathcal{F}_{\\alpha}(S_1)_0\n\\]\nis the desired $\\operatorname{adeg}$-preserving part, and the remaining terms constitute $\\mathcal{F}_{\\alpha}(S)_+$. Statement (2) follows from (1) in a similar fashion.\n\\end{proof}\n\nWe are now ready for the main theorem. \n\n\\begin{theorem}\\label{thm:equivariant annular}\nThere exists a functor $\\mathcal{G}_{\\alpha} \\colon \\mathscr{BN}_\\alpha(\\mathbb{A}) \\to R_\\alpha{\\rm -}\\operatorname{ggmod}$ such that the following diagram commutes \n\\begin{equation*}\n\\begin{tikzcd}\n\\mathscr{BN}_\\alpha(\\mathbb{A}) \\ar[r, \"\\mathcal{G}_{\\alpha}\"] \\ar[d] & R_\\alpha{\\rm -}\\operatorname{ggmod} \\ar[d]\\\\\n\\mathscr{BN}(\\mathbb{A}) \\ar[r, \"\\mathcal{G}\"] & \\mathbb{Z}{\\rm -}\\operatorname{ggmod}\n\\end{tikzcd}\n\\end{equation*}\nwhere the vertical arrows are obtained by setting $\\alpha_0 = \\alpha_1 = 0$. \n\\end{theorem}\n\n\\begin{proof}\nFor a collection of circles $\\mathscr{C}\\subset \\mathbb{A}$, set \n\\[\n\\mathcal{G}_{\\alpha}(S) := \\mathcal{F}_{\\alpha}(\\mathscr{C}),\n\\]\nwith the bigrading $(\\operatorname{qdeg}, \\operatorname{adeg})$ as defined earlier in this section. For a cobordism $S\\subset \\mathbb{A} \\times I$, set \n\\[\n\\mathcal{G}_{\\alpha}(S) := \\mathcal{F}_{\\alpha}(S)_0\n\\]\nas in Corollary \\ref{cor:cobs split} (1). That $\\mathcal{G}_{\\alpha}$ is well-defined on cobordisms and factors through the relations in $\\mathscr{BN}_\\alpha(\\mathbb{A})$ follows from the analogous statements for $\\mathcal{F}_{\\alpha}$. Corollary \\ref{cor:cobs split} (2) implies functoriality of $\\mathcal{G}_{\\alpha}$. Finally, commutativity of the diagram follows from deleting the boxed terms and setting $\\alpha_0 = \\alpha_1 =0$ in the maps appearing in the proof of Lemma \\ref{lem:el cobs split}, and comparing the result with the maps \\eqref{eq:formula1}--\\eqref{eq:formula4}. \n\\end{proof}\n\n\nMaps assigned to the four elementary saddles in Figure \\ref{fig:el saddles} are recorded below. The full set of maps -- that is, if other essential circles are present -- can be obtained by interchanging $\\alpha_0 \\leftrightarrow \\alpha_1$. \n\n\\noindent\\begin{minipage}{.5\\textwidth}\n\\begin{equation}\\label{eq:equiv formula1}\n \\begin{split}\n V_\\alpha \\o A_\\alpha & \\rar{\\hyperref[fig:type1]{\\operatorname{(I)}}} V_\\alpha \\\\ \\v_0 \\o 1 & \\mapsto \\v_0 \\\\ \\v_1\\o 1 &\\mapsto \\v_1 \\\\\n \\v_0 \\o X & \\mapsto \\alpha_0 \\v_0 \\\\\n \\v_1 \\o X & \\mapsto \\alpha_1 \\v_1\n \\end{split}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.38\\textwidth}\n\\begin{equation}\\label{eq:equiv formula2}\n \\begin{split}\n V_\\alpha \\o V_\\alpha' & \\rar{\\hyperref[fig:type2]{\\operatorname{(II)}}} A_\\alpha \\\\ \\v_0 \\o \\v_0' &\\mapsto 0 \\\\\n\\v_1 \\o \\v_0' & \\mapsto X - \\alpha_0 \\\\\n\\v_0 \\o \\v_1' & \\mapsto X - \\alpha_1 \\\\\n\\v_1 \\o \\v_1' & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\n\\vskip1em\n \n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation}\\label{eq:equiv formula3}\n \\begin{split}\n V_\\alpha & \\rar{\\hyperref[fig:type3]{\\operatorname{(III)}}} V_\\alpha \\o A_\\alpha \\\\\n\\v_0 & \\mapsto \\v_0 \\o X - \\alpha_1 \\v_0 \\o 1 \\\\\n\\v_1 & \\mapsto \\v_1 \\o X - \\alpha_0 \\v_1 \\o 1\n \\end{split}\n\\end{equation}\n\\end{minipage} \n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\\label{eq:equiv formula4}\n \\begin{split}\n A_\\alpha & \\rar{\\hyperref[fig:type4]{\\operatorname{(IV)}}} V_\\alpha\\o V_\\alpha' \\\\\n1 &\\mapsto \\v_0 \\o \\v_1' + \\v_1 \\o \\v_0'\\\\\nX & \\mapsto \\alpha_0 \\v_0 \\o \\v_1' + \\alpha_1 \\v_1 \\o \\v_0'\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\\vskip1em\n\\noindent\n \nLet $\\mathscr{C} \\subset \\mathbb{A}$ consist of $m>0$ essential circles, and let $C$ be the $i$-th essential circle in $\\mathscr{C}$. Consider the cobordism $S$ whose underlying surface is the identity cobordism $\\mathscr{C} \\times I$, with a single dot on the component $C\\times I$, as shown in Figure \\ref{fig:single dot on identity}. Then $\\mathcal{G}_{\\alpha}(S)$ is the identity on all tensor factors except the one corresponding to $C$, and on $C$ it is given by the left-hand side of \\eqref{eq:X on odd and even} if $i$ is odd, and the right-hand side if $i$ is even. \n \n \\begin{figure}\n\\centering\n\\includegraphics{AD7.pdf}\n\\caption{Product cobordism on $m>0$ essential circles, with the $i$-th component dotted}\\label{fig:single dot on identity}\n\\end{figure}\n\n\\begin{align}\\label{eq:X on odd and even} \n\\begin{aligned}\n V_\\alpha & \\to V_\\alpha \\\\\n\\v_0 & \\mapsto \\alpha_0 \\v_0 \\\\\n\\v_1 & \\mapsto \\alpha_1 \\v_1\n\\end{aligned}\n&&\n\\begin{aligned}\n V_\\alpha' & \\to V_\\alpha' \\\\\n\\v_0' & \\mapsto \\alpha_1 \\v_0' \\\\\n\\v_1' & \\mapsto \\alpha_0 \\v_1'\n\\end{aligned}\n\\end{align}\nObserve that the functor $\\mathcal{G}_{\\alpha}$ is not monoidal, since the action of $X$ on an essential circle depends on its nestedness. \n\nLet $L\\subset \\mathbb{A} \\times I$ be an oriented link with diagram $D$. Let \n\\[\nCKh^\\mathbb{A}_\\alpha(D) := \\mathcal{G}_{\\alpha}([[D]])\n\\]\ndenote the chain complex obtained by applying $\\mathcal{G}_{\\alpha}$ to the formal complex $[[D]]$. The differential preserves bidegree, and the complex is an invariant of $L$ up to bidegree-preserving chain homotopy equivalence. \n\nThe remainder of this section discusses variants of $\\mathcal{G}_{\\alpha}$. Instead of setting both $\\alpha_0 = \\alpha_1=0$, it is possible to set only $\\alpha_0 = 0$ and rename the remaining parameter $\\alpha_1$ to $\\alpha_1 = h$. Denote the resulting Frobenius pair by $(R_h, A_h)$. Explicitly,\n\\[\nR_h = \\mathbb{Z}[h],\\ A_h = R_h [X]\/(X^2-hX).\n\\]\nIt may also be obtained from $(R,A)$ by setting $E_1 =h$, $E_2=0$; note that the obstruction in Section \\ref{sec:preliminary observation} disappears when $E_2=0$. Collapsing $(R_h, A_h)$ further to characteristic $2$ (that is, applying $(-) \\o_{R_h} \\mathbb{Z}_2 [h])$ recovers Bar-Natan's theory \\cite[Section 9.3]{BN2}. We expect that the resulting annular homology is related to \\cite{TW}. \n\n\nLet $L\\subset \\mathbb{A} \\times I$ be an oriented link with diagram $D$. Viewing $D$ as a diagram in $\\mathbb{R}^2$ and applying $\\mathcal{F}_{\\alpha}$ to $[[D]]$ yields a chain complex $CKh_\\alpha(D)$ of bigraded $R_\\alpha$-modules. Letting $\\d$ denote the differential, Lemma \\ref{lem:el cobs split} implies that $\\d$ splits as \n\\[\n\\d = \\d_0 + \\d_2\n\\]\nwhere $\\d_0$ is of bidegree $(0,0)$ and $\\d_2$ is of bidegree $(0,2)$. As in \\cite{HKLM}, we can introduce an extra parameter $\\beta$ to account for $\\d_2$. Let $R_{\\alpha \\beta} = R_\\alpha[\\beta]$ with $\\beta$ in bidegree $(0,-2)$, and let $CKh_{\\alpha \\beta}(D)$ be the chain complex over $R_{\\alpha \\beta}$ with\n\\[\nCKh_{\\alpha \\beta}^i (D) := CKh_{\\alpha}^i(D) \\o_{R_\\alpha} R_{\\alpha \\beta}\n\\]\nin homological degree $i$ and differential $\\d_\\beta$ given by \n\\[\n\\d_\\beta := \\d_0 + \\beta \\d_2.\n\\]\nNote that $\\d_\\beta$ preserves bidegree. Maps assigned to the four elementary saddles in Figure \\ref{fig:el saddles} are given below.\n\n\n\\noindent\\begin{minipage}{.5\\textwidth}\n\\begin{equation*}\n \\begin{split}\n V_\\alpha \\o A_\\alpha & \\rar{\\hyperref[fig:type1]{\\operatorname{(I)}}} V_\\alpha \\\\ \\v_0 \\o 1 & \\mapsto \\v_0 \\\\ \\v_1\\o 1 &\\mapsto \\v_1 \\\\\n \\v_0 \\o X & \\mapsto \\alpha_0 \\v_0 +\\ \\beta v_1 \\\\\n \\v_1 \\o X & \\mapsto \\alpha_1 \\v_1\n \\end{split}\n\\end{equation*}\n\\end{minipage}%\n\\begin{minipage}{.38\\textwidth}\n\\begin{equation*}\n \\begin{split}\n V_\\alpha \\o V_\\alpha' & \\rar{\\hyperref[fig:type2]{\\operatorname{(II)}}} A_\\alpha \\\\ \\v_0 \\o \\v_0' &\\mapsto \\beta\\\\\n\\v_1 \\o \\v_0' & \\mapsto X - \\alpha_0 \\\\\n\\v_0 \\o \\v_1' & \\mapsto X - \\alpha_1 \\\\\n\\v_1 \\o \\v_1' & \\mapsto 0\n \\end{split}\n\\end{equation*}\n\\end{minipage}\n\n\\vskip1em\n \n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation*}\n \\begin{split}\n V_\\alpha & \\rar{\\hyperref[fig:type3]{\\operatorname{(III)}}} V_\\alpha \\o A_\\alpha \\\\\n\\v_0 & \\mapsto \\v_0 \\o X - \\alpha_1 \\v_0 \\o 1 + \\beta \\v_1 \\o 1\\\\\n\\v_1 & \\mapsto \\v_1 \\o X - \\alpha_0 \\v_1 \\o 1\n \\end{split}\n\\end{equation*}\n\\end{minipage} \n\\begin{minipage}{.5\\linewidth}\n\\begin{equation*}\n \\begin{split}\n A_\\alpha & \\rar{\\hyperref[fig:type4]{\\operatorname{(IV)}}} V_\\alpha\\o V_\\alpha' \\\\\n1 &\\mapsto \\v_0 \\o \\v_1' + \\v_1 \\o \\v_0'\\\\\nX & \\mapsto \\alpha_0 \\v_0 \\o \\v_1' + \\alpha_1 \\v_1 \\o \\v_0' + \\beta \\v_1 \\o \\v_1'\n \\end{split}\n\\end{equation*}\n\\end{minipage}\n\\vskip1em\n\\noindent\n\n\n\n\\subsection{Inverting $\\mathcal{D}$ in equivariant annular homology}\\label{sec:inverting D equiv annular}\n\nRecall the Frobenius pair $(R_{\\alpha\\mathcal{D}}, A_{\\alpha\\mathcal{D}})$ from \\cite{KR}, which was reviewed in Section \\ref{sec:inverting D}. Let $\\mathcal{G}_{\\alpha \\mathcal{D}}$ denote the composition \n\\[\n\\mathscr{BN}_\\alpha(\\mathbb{A}) \\rar{\\mathcal{G}_{\\alpha}} R_\\alpha {\\rm -}\\operatorname{ggmod} \\to R_{\\alpha\\mathcal{D}} {\\rm -}\\operatorname{ggmod}\n\\]\nwhere the second functor is extension of scalars. Consider the following elements of $A_{\\alpha\\mathcal{D}}$, \n\\begin{align*}\n\\b{v}_0 := \\v_0 = 1, \\hskip2em \\b{v}_1 := \\frac{v_1}{\\alpha_1 - \\alpha_0} = \\frac{X - \\alpha_0}{\\alpha_1 - \\alpha_0}, \\\\\n\\b{v}_0':= \\v_0' = 1, \\hskip2em \\b{v}_1' := \\frac{v_1'}{\\alpha_0 - \\alpha_1} = \\frac{X - \\alpha_1}{\\alpha_0 - \\alpha_1}.\n\\end{align*}\n\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\draw[->] (-1.5,0) -- (1.5,0) node[right] {$\\operatorname{qdeg}$};\n\\foreach \\x in { -1, 1}\n\\draw[shift={(\\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {\\tiny $\\x$};\n\\draw[->] (0,-1.5) -- (0,1.5) node[above] {$\\operatorname{adeg}$};\n\\foreach \\y in {-1,1}\n\\draw[shift={(0,\\y)}] (2pt,0pt) -- (-2pt,0pt) node[left] {\\tiny $\\y$};\n\\filldraw[black] (-1,1) circle (2pt) node[anchor=east] {$\\b{v}_1$, $\\b{v}_1'$};\n\\filldraw[black] (-1,-1) circle (2pt) node[anchor=east] {$\\b{v}_0$, $\\b{v}_0'$};\n\\filldraw[black] (-1,0) circle (2pt) node[anchor=south] {$e_0, e_1$};\n\\end{tikzpicture}\n\\caption{Bigradings}\\label{fig:bigradings alpha D}\n\\end{figure}\n\n\nAs in Section \\ref{sec:defining the functor}, let $V_{\\alpha\\mathcal{D}}$ and $V_{\\alpha\\mathcal{D}}'$ denote the module $A_{\\alpha\\mathcal{D}}$ with distinguished homogeneous bases $\\{\\b{v}_0, \\b{v}_1\\}$ and $\\{\\b{v}_0', \\b{v}_1'\\}$, respectively. For a collection of circles $\\mathscr{C} \\subset \\mathbb{A}$, the $i$-th essential circle in $\\mathscr{C}$ is assigned $V_{\\alpha\\mathcal{D}}$ if $i$ is odd and $V_{\\alpha\\mathcal{D}}'$ if $i$ is even. The notation $A_{\\alpha\\mathcal{D}}$ is reserved for trivial circles, with distinguished basis $\\{e_0, e_1\\}$, see \\eqref{eq:e_0 and e_1}. Bigradings are summarized in Figure \\ref{fig:bigradings alpha D}. \n\nWith respect to these bases, the maps assigned to the four elementary saddles in Figure \\ref{fig:el saddles} are recorded below.\n\n\\noindent\n\\begin{minipage}{.5\\textwidth}\n\\begin{equation}\\label{eq:equiv D formula1}\n \\begin{split}\n V_{\\alpha\\mathcal{D}} \\o A_{\\alpha\\mathcal{D}} & \\rar{\\hyperref[fig:type1]{\\operatorname{(I)}}} V_{\\alpha\\mathcal{D}} \\\\ \n \\b{v}_0 \\o e_0 & \\mapsto 0 \\\\ \n \\b{v}_1\\o e_0 &\\mapsto \\b{v}_1 \\\\\n \\b{v}_0 \\o e_1 & \\mapsto \\b{v}_0 \\\\\n \\b{v}_1 \\o e_1 & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.38\\textwidth}\n\\begin{equation}\\label{eq:equiv D formula2}\n \\begin{split}\n V_{\\alpha\\mathcal{D}} \\o V_{\\alpha\\mathcal{D}}' & \\rar{\\hyperref[fig:type2]{\\operatorname{(II)}}} A_{\\alpha\\mathcal{D}} \\\\ \n \\b{v}_0 \\o \\b{v}_0' &\\mapsto 0 \\\\\n \\b{v}_1 \\o \\b{v}_0' & \\mapsto e_0 \\\\\n \\b{v}_0 \\o \\b{v}_1' & \\mapsto e_1 \\\\\n \\b{v}_1 \\o \\b{v}_1' & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\n\\vskip1em\n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation}\\label{eq:equiv D formula3}\n \\begin{split}\n V_{\\alpha\\mathcal{D}} & \\rar{\\hyperref[fig:type3]{\\operatorname{(III)}}} V_{\\alpha\\mathcal{D}} \\o A_{\\alpha\\mathcal{D}} \\\\\n\\b{v}_0 & \\mapsto (\\alpha_0 - \\alpha_1) \\b{v}_0 \\o e_1 \\\\\n\\b{v}_1 & \\mapsto (\\alpha_1 - \\alpha_0) \\b{v}_1 \\o e_0 \n \\end{split}\n\\end{equation}\n\\end{minipage} \n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\\label{eq:equiv D formula4}\n \\begin{split}\n A_{\\alpha\\mathcal{D}} & \\rar{\\hyperref[fig:type4]{\\operatorname{(IV)}}} V_{\\alpha\\mathcal{D}}\\o V_{\\alpha\\mathcal{D}}' \\\\\ne_0 &\\mapsto (\\alpha_1 - \\alpha_0) \\b{v}_1 \\o \\b{v}_0'\\\\\ne_1 & \\mapsto (\\alpha_0 - \\alpha_1) \\b{v}_0 \\o \\b{v}_1'\n \\end{split}\n\\end{equation}\n\\vskip1em\n\\end{minipage}\nTo obtain the full set of maps -- that is, if other essential circles are present -- one interchanges $\\alpha_0 \\leftrightarrow \\alpha_1$, which has the effect of interchanging $\\b{v}_0 \\leftrightarrow \\b{v}_0'$, $\\b{v}_1 \\leftrightarrow \\b{v}_1'$, and $e_0 \\leftrightarrow e_1$. They are recorded below for convenience. \n\n\\vskip1em\n\\noindent\n\\begin{minipage}{.5\\textwidth}\n\\begin{equation}\\label{eq:equiv D formula1'}\n \\begin{split}\n V_{\\alpha\\mathcal{D}}' \\o A_{\\alpha\\mathcal{D}} & \\to V_{\\alpha\\mathcal{D}}' \\\\ \n \\b{v}_0' \\o e_0 & \\mapsto \\b{v}_0' \\\\ \n \\b{v}_1'\\o e_0 &\\mapsto 0 \\\\\n \\b{v}_0' \\o e_1 & \\mapsto 0 \\\\\n \\b{v}_1' \\o e_1 & \\mapsto \\b{v}_1'\n \\end{split}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.38\\textwidth}\n\\begin{equation}\\label{eq:equiv D formula2'}\n \\begin{split}\n V_{\\alpha\\mathcal{D}}' \\o V_{\\alpha\\mathcal{D}} & \\to A_{\\alpha\\mathcal{D}} \\\\ \n \\b{v}_0' \\o \\b{v}_0 &\\mapsto 0 \\\\\n \\b{v}_1' \\o \\b{v}_0 & \\mapsto e_1 \\\\\n \\b{v}_0' \\o \\b{v}_1 & \\mapsto e_0 \\\\\n \\b{v}_1' \\o \\b{v}_1 & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\n\\vskip1em\n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation}\\label{eq:equiv formula3'}\n \\begin{split}\n V_{\\alpha\\mathcal{D}}' & \\to V_{\\alpha\\mathcal{D}}' \\o A_{\\alpha\\mathcal{D}} \\\\\n\\b{v}_0' & \\mapsto (\\alpha_1 - \\alpha_0) \\b{v}_0' \\o e_0 \\\\\n\\b{v}_1' & \\mapsto (\\alpha_0 - \\alpha_1) \\b{v}_1' \\o e_1 \n \\end{split}\n\\end{equation}\n\\end{minipage} \n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\\label{eq:equiv formula4'}\n \\begin{split}\n A_{\\alpha\\mathcal{D}} & \\to V_{\\alpha\\mathcal{D}}'\\o V_{\\alpha\\mathcal{D}} \\\\\ne_0 & \\mapsto (\\alpha_1 - \\alpha_0) \\b{v}_0' \\o \\b{v}_1\\\\\ne_1 & \\mapsto (\\alpha_0 - \\alpha_1) \\b{v}_1' \\o \\b{v}_0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\\vskip1em \n\nThese maps may be written uniformly in the following way. Let $\\mathscr{C} \\subset \\mathbb{A}$ be a collection of circles, and label each circle in $\\mathscr{C}$ by one of the letters $\\mathbf{a}$ or $\\mathbf{b}$. From such a labeling we obtain a distinguished basis element of $\\mathcal{G}_{\\alpha \\mathcal{D}}(\\mathscr{C})$ by using the correspondence\n\\begin{equation}\\label{eq:a b on trivial}\n\\mathbf{a} \\leftrightarrow e_0,\\ \\mathbf{b} \\leftrightarrow e_1\n\\end{equation}\nfor a trivial circle, and \n\\begin{equation}\\label{eq:a b on essential}\n\\mathbf{a} \\leftrightarrow \n\\begin{cases}\n\\b{v}_1 & i \\text{ is odd} \\\\\n\\b{v}_0' & i \\text{ is even}\n\\end{cases},\\ \n\\mathbf{b} \\leftrightarrow \n\\begin{cases}\n\\b{v}_0 & i \\text{ is odd} \\\\\n\\b{v}_1' & i \\text{ is even}\n\\end{cases}\n\\end{equation}\non the $i$-th essential circle. Then the saddle maps are \n\n\\noindent\n\\begin{minipage}{.5\\textwidth}\n\\begin{equation}\\label{eq:equiv D formula1 ab}\n \\begin{split}\n V_{\\alpha\\mathcal{D}} \\o A_{\\alpha\\mathcal{D}} & \\rar{\\hyperref[fig:type1]{\\operatorname{(I)}}} V_{\\alpha\\mathcal{D}} \\\\ \n \\mathbf{b} \\o \\mathbf{a} & \\mapsto 0 \\\\ \n \\mathbf{a} \\o \\mathbf{a} &\\mapsto \\mathbf{a} \\\\\n \\mathbf{b} \\o \\mathbf{b} & \\mapsto \\mathbf{b} \\\\\n \\mathbf{a} \\o \\mathbf{b} & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}%\n\\begin{minipage}{.38\\textwidth}\n\\begin{equation}\\label{eq:equiv D formula2 ab}\n \\begin{split}\n V_{\\alpha\\mathcal{D}} \\o V_{\\alpha\\mathcal{D}}' & \\rar{\\hyperref[fig:type2]{\\operatorname{(II)}}} A_{\\alpha\\mathcal{D}} \\\\ \n \\mathbf{b} \\o \\mathbf{a} &\\mapsto 0 \\\\\n \\mathbf{a} \\o \\mathbf{a} & \\mapsto \\mathbf{a} \\\\\n \\mathbf{b} \\o \\mathbf{b} & \\mapsto \\mathbf{b} \\\\\n \\mathbf{a} \\o \\mathbf{b} & \\mapsto 0\n \\end{split}\n\\end{equation}\n\\end{minipage}\n\n\\vskip1em\n \n\\noindent\\begin{minipage}{.49\\linewidth}\n\\begin{equation}\\label{eq:equiv D formula3 ab}\n \\begin{split}\n V_{\\alpha\\mathcal{D}} & \\rar{\\hyperref[fig:type3]{\\operatorname{(III)}}} V_{\\alpha\\mathcal{D}} \\o A_{\\alpha\\mathcal{D}} \\\\\n\\mathbf{b} & \\mapsto (\\alpha_0 - \\alpha_1) \\mathbf{b} \\o \\mathbf{b} \\\\\n\\mathbf{a} & \\mapsto (\\alpha_1 - \\alpha_0) \\mathbf{a} \\o \\mathbf{a}\n \\end{split}\n\\end{equation}\n\\end{minipage} \n\\begin{minipage}{.5\\linewidth}\n\\begin{equation}\\label{eq:equiv D formula4 ab}\n \\begin{split}\n A_{\\alpha\\mathcal{D}} & \\rar{\\hyperref[fig:type4]{\\operatorname{(IV)}}} V_{\\alpha\\mathcal{D}}\\o V_{\\alpha\\mathcal{D}}' \\\\\n\\mathbf{a} &\\mapsto (\\alpha_1 - \\alpha_0) \\mathbf{a} \\o \\mathbf{a}\\\\\n\\mathbf{b} & \\mapsto (\\alpha_0 - \\alpha_1) \\mathbf{b} \\o \\mathbf{b}\n \\end{split}\n\\end{equation}\n\\vskip1em\n\\end{minipage}\nMoreover, the same formulas hold with $V_{\\alpha\\mathcal{D}}$ and $V_{\\alpha\\mathcal{D}}'$ interchanged. \n\nFor an annular link $L$ with diagram $D$, let \n\\[\nCKh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(D) := \\mathcal{G}_{\\alpha \\mathcal{D}}([[D]])\n\\]\ndenote the chain complex obtained by applying $\\mathcal{G}_{\\alpha \\mathcal{D}}$ to $[[D]]$. It is an invariant of $L$ up to chain homotopy equivalence, so we may write $Kh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(L)$ to denote the homology of $CKh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(D)$, for any diagram $D$ of $L$. \n\n\n\\begin{theorem}\nLet $L \\subset \\mathbb{A} \\times I$ be a link with diagram $D$. Viewing $L$ as a link in $\\mathbb{R}^3$, there is a $\\operatorname{qdeg}$-preserving isomorphism \n\\[\n\\varphi\\: CKh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(D) \\rar{\\sim} CKh_{{\\alpha\\mathcal{D}}}(D). \n\\]\n\\end{theorem}\n\\begin{proof}\nFor a smoothing $D_u$, the inclusion $\\mathbb{A} \\hookrightarrow \\mathbb{R}^2$ induces an isomorphism\n\\[\n\\varphi_u\\: \\mathcal{G}_{\\alpha \\mathcal{D}}(D_u) \\to \\mathcal{F}_{\\alpha \\mathcal{D}}(D_u),\n\\]\ndefined in terms of the basis elements labeled by $\\mathbf{a}$ and $\\mathbf{b}$ by \n\\[\n\\mathbf{a} \\mapsto e_0,\\ \\mathbf{b}\\mapsto e_1.\n\\]\nComparing the formulas \\eqref{eq:equiv D formula1 ab}--\\eqref{eq:equiv D formula4 ab} with multiplication and comultiplication in $A_{\\alpha\\mathcal{D}}$, we see that each of the maps $\\varphi_u$ commute with cobordism maps and thus assemble into an isomorphism $\\varphi\\: CKh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(D) \\to CKh_{{\\alpha\\mathcal{D}}}(D)$. It is evident from Figure \\ref{fig:bigradings alpha D} that each $\\varphi_u$ preserves $\\operatorname{qdeg}$. Quantum grading shifts in both chain complexes are the same, so the isomorphism $\\varphi$ preserves $\\operatorname{qdeg}$ as well. \n\\end{proof}\n\nThe following is immediate from Proposition \\ref{prop:Lee rank}. \n\\begin{corollary}\nFor a link $L\\subset \\mathbb{A}\\times I$ with $k$ components, the homology $Kh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(L)$ is a free $R_{\\alpha\\mathcal{D}}$-module of rank $2^k$. \n\\end{corollary}\n\n\n\nWe recall the \\emph{canonical generators} for Lee homology, following \\cite{Le} and \\cite{Ra}. Let $L\\subset \\mathbb{A} \\times I$ be a link with diagram $D$. Given an orientation $o$ on $L$, let $D_o\\subset \\mathbb{A}$ denote the result of performing the oriented resolution at each crossing,\n\\begin{equation*}\\label{fig:oriented resolution}\n\\begin{gathered}\n\\includegraphics{AD_oriented_resolution.pdf}\n\\end{gathered}\\, .\n\\end{equation*}\nEach of the resulting circles is naturally oriented. Assign a mod $2$ invariant to each circle $C$ as follows. First, consider the number of circles in $D_o$ separating $C$ from infinity, mod $2$. Add $1$ if $C$ is standardly (counterclockwise) oriented, and add $0$ otherwise. Now that each circle in $D_o$ is labeled by $0$ or $1$, use the correspondence $0 \\leftrightarrow \\mathbf{a}$, $1\\leftrightarrow \\mathbf{b}$ to label each circle by $\\mathbf{a}$ or $\\mathbf{b}$, and finally use \\eqref{eq:a b on trivial} and \\eqref{eq:a b on essential} to obtain a generator $\\mathfrak{s}_o$ in $CKh^\\mathbb{A}_{{\\alpha\\mathcal{D}}}(D)$. \n\nFor a collection of oriented circles $\\mathscr{C}\\subset \\mathbb{A}$, let $w(\\mathscr{C})$ denote the winding number of $\\mathscr{C}$. That is, $w(\\mathscr{C})$ equals the number of counterclockwise essential circles minus the number of clockwise ones. If $C_1,\\ldots , C_m$ are the essential circles in $\\mathscr{C}$, then \n\\[\nw(\\mathscr{C}) = \\sum_{i=1}^m w(C_i). \n\\]\n\n\n\\begin{proposition}\\label{prop:adeg of canonical generators}\nLet $L\\subset \\mathbb{A} \\times I$ be a link with diagram $D$, and let $o$ be an orientation of $L$. Let $m$ be the number of essential circles in the oriented resolution $D_o$. Then \n\\[\n\\operatorname{adeg}(\\mathfrak{s}_o) = (-1)^m w(L,o)\n\\]\nwhere $w(L,o)$ is the winding number of $L$ with respect to the orientation $o$. \n\\end{proposition}\n\n\\begin{proof}\nFirst note that $w(L,o) = w(D_o)$. It is straightforward to verify that each essential circle $C$ in $D_o$ contributes $(-1)^m w(C)$ to the annular degree of $\\mathfrak{s}_o$. The claim follows, since trivial circles do not contribute to the annular degree or the winding number.\n\\end{proof}\n\n\n\n\n\n\n\\section{Dotted Temperley-Lieb category}\\label{sec:dotted TL}\n\nThis section reviews the Temperley-Lieb category $\\mathscr{TL}$ and its relation to annular Khovanov homology, following observations in \\cite{BPW}. We then introduce a natural equivariant analogue. \n\nFor each $n\\geq 0$, fix a collection $\\u{n} \\subset (0,1)$ of $n$ points in the interior of the unit interval. A \\emph{planar} $(n,m)$\\emph{-tangle} is a smooth embedding of a compact $1$-manifold $M$ into $I\\times I$, such that the boundary of $M$ maps to $I \\times \\{0\\} \\cup I \\times \\{1\\}$, with $n$ points in $\\d M$ mapping to $\\u{n} \\subset I\\times \\{0\\}$, and the remaining $m$ points in $\\d M$ mapping to $\\u{m} \\subset \\times \\{1\\}$. Planar tangles are always considered up to isotopy of $I\\times I$ fixing the boundary. \n\nLet $\\mathscr{TL}$ denote the $\\Z[q,q^{-1}]$-linear category whose objects are nonnegative integers. The morphism space $\\mathscr{TL}(n,m)$ is freely generated over $\\Z[q,q^{-1}]$ by planar $(n,m)$-tangles, modulo the local relation that an innermost circle can be removed at the cost of multiplying the remaining tangle by $q+q^{-1}$, \n\\begin{equation}\\label{eq:TL rel}\n\\begin{gathered}\n\\includegraphics{AD_rel_TL.pdf}\n\\end{gathered}\\, .\n\\end{equation}\nComposition is defined by vertically stacking planar tangles. Denote the space of morphisms from $n$ to $m$ by\n\\[\n\\mathscr{TL}(n,m).\n\\]\nThe Temperley-Lieb algebra $TL_n$ can then be identified with the endomorphism space $\\mathscr{TL}(n,n)$. Let $\\mathscr{TL}_{q=1}$ denote the category obtained by setting $q=1$. \n\n\n\nIt was observed in \\cite[Section 6.1]{BPW} that $\\mathscr{TL}$ is closely tied to annular Khovanov homology. By spinning planar tangles in the $S^1$ direction, one obtains a functor\n\\begin{equation}\\label{eq:spinning 1}\nS^1 \\times (-) \\: \\mathscr{TL}_{q=1} \\to \\mathscr{BN}(\\mathbb{A}).\n\\end{equation}\nExplicitly, $n$ is sent to the essential circles $S^1\\times \\u{n} \\subset \\mathbb{A}$, and a planar tangle $T$ is sent to the cobordism $S^1\\times T$. That $S^1\\times (-)$ factors through the relation \\eqref{eq:TL rel} when $q=1$ follows from the fact that a torus in $\\mathbb{A}\\times I$ evaluates to $2$ in $\\mathscr{BN}(\\mathbb{A})$. Let \n\\[\n\\mathscr{BBN}(\\mathbb{A})\n\\]\ndenote the category obtained from $\\mathscr{BN}(\\mathbb{A})$ by imposing Boerner's relation, Figure \\ref{fig:Boerner}. Recall that the non-equivariant annular TQFT $\\mathcal{G}$ factors through this relation, so no information is lost from the point of view of link homology. \n\nIt is shown in \\cite[Section 4.2]{GLW} that $\\mathcal{G}$ can be made to take values in the representation category of $\\sl_2$. On the other hand, it is well-known that Temperley-Lieb diagrams (planar tangles modulo relation \\eqref{eq:TL rel}) can be interpreted as $U_q(\\sl_2)$-linear maps between tensor powers of the fundamental representation of $U_q(\\sl_2)$; a convenient reference is \\cite[Appendix A.1]{BPW}. Thus for $q=1$, both $\\mathscr{TL}_{q=1}$ and $\\mathscr{BBN}(\\mathbb{A})$ admit functors $\\mathcal{F}_{\\mathscr{TL}}$ and $\\mathcal{G}$ to $U(\\sl_2){\\rm -}\\operatorname{mod}$. It was observed in \\cite[(6.1)]{BPW} that the following diagram commutes.\n\n\\begin{equation}\\label{eq:commuting triangle}\n\\begin{tikzcd}[column sep=tiny]\n\\mathscr{TL}_{q=1} \\arrow[rr, \"S^1\\times (-)\"] \\arrow[rd, \"\\mathcal{F}_{\\mathscr{TL}}\"'] & & \\mathscr{BBN}(\\mathbb{A}) \\arrow[ld, \"\\mathcal{G}\"] \\\\\n & U(\\sl_2){\\rm -}\\operatorname{mod} & \n\\end{tikzcd}\n\\end{equation}\nMoreover, if $\\mathscr{TL}_{q=1}$ is made additive and graded by introducing formal direct sums \\cite[Definition 3.2]{BN2} and formal grading shifts \\cite[Section 6]{BN2}, then the horizontal functor in \\eqref{eq:commuting triangle} becomes an equivalence of categories, \\cite[Proposition 6.1]{BPW}.\n\nThus $\\mathscr{TL}_{q=1}$ characterizes the skein category $\\mathscr{BBN}(\\mathbb{A})$ and the functor $\\mathcal{F}_{\\mathscr{TL}}$ characterizes the annular TQFT $\\mathcal{G}$. On the other hand, $\\mathscr{BN}(\\mathbb{A})$ was similarly described using planar diagrams in \\cite{Ru}, which we restate now. A \\emph{dotted} planar $(n,m)$-tangle is a planar $(n,m)$ tangle whose components may be decorated by some number of dots, which are allowed to float freely along a component. Let \n\\[\n\\mathscr{TL}_\\bullet\n\\]\ndenote the category whose objects are nonnegative integers and whose morphisms are $\\mathbb{Z}$-linear combinations of dotted planar tangles, modulo the additional local relations in Figure \\ref{fig:TLd rels}. \n\n\\begin{figure}\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{AD_rel_TL1.pdf}\n\\end{center}\n\\caption{Undotted circle}\\label{fig:undotted circle TLd}\n\\end{subfigure}\n\\begin{subfigure}[b]{.2\\textwidth}\n\\begin{center}\n\\includegraphics{AD_rel_TLd.pdf}\n\\end{center}\n\\caption{Dotted circle}\\label{fig:dotted circle TLd}\n\\end{subfigure}\n\\begin{subfigure}[b]{.4\\textwidth}\n\\begin{center}\n\\includegraphics{AD_2dotrel_TLd.pdf}\n\\end{center}\n\\caption{Two dots relation}\\label{fig:two dots TLd}\n\\end{subfigure}\n\\caption{Relations in $\\mathscr{TL}_\\bullet$.}\\label{fig:TLd rels}\n\\end{figure}\n\nFor a dotted planar tangle $T$, let $T^u$ denote the tangle obtained by removing all dots ($u$ stands for undotted). Consider the functor \n\\begin{equation}\\label{eq:spinning d}\nS^1 \\times (-) \\: \\mathscr{TL}_\\bullet \\to \\mathscr{BN}(\\mathbb{A})\n\\end{equation} which sends a dotted planar tangle $T$ to the cobordism whose underlying surface is $S^1 \\times T^u$, and which carries $k$ dots on a component if $T$ carried $k$ dots on the corresponding component. The relations in Figure \\ref{fig:TLd rels} are planar analogues of relations in $\\mathscr{BN}(\\mathbb{A})$, Figure \\ref{fig:BN relations}. Figures \\ref{fig:undotted circle TLd} and \\ref{fig:dotted circle TLd} correspond to an undotted torus and a once-dotted torus evaluating to $2$ and $0$ in $\\mathscr{BN}(\\mathbb{A})$, respectively. Figure \\ref{fig:two dots TLd} corresponds to the two dots relation in Figure \\ref{fig:two dot}. \n\nUpon introducing formal direct sums and formal grading shifts, the argument in \\cite[Proposition 6.1]{BPW} shows that the functor \\eqref{eq:spinning d} is essentially surjective and full. It is not faithful, but by \\cite[Theorem 3.1]{Ru}, its kernel is generated by the local relations shown in Figure \\ref{fig:Russel relations nonequiv}. Note that the second follows from the first by adding a dot near one of the endpoints of the strands and simplifying using the two dots relation. \n\\begin{figure}\n\\includegraphics{AD_Russel_rels.pdf}\n\\caption{Russel relations.}\\label{fig:Russel relations nonequiv}\n\\end{figure}\nTo see that the relations hold, consider two annuli embedded in $\\mathbb{A} \\times I$ with a tube joining them, Figure \\ref{fig:tubed annuli}, and perform neck-cutting along the two disks shown in Figure \\ref{fig:tubed annuli2} and Figure \\ref{fig:tubed annuli1}. Denote by \n\\[\n\\til{\\mathscr{TL}}_\\bullet\n\\]\nthe category obtained by imposing the Russel relations. It follows from \\cite[Theorem 3.1]{Ru} that the induced functor \n\\[\nS^1 \\times (-) \\: \\widetilde{\\mathscr{TL}}_\\bullet \\to \\mathscr{BN}(\\mathbb{A})\n\\]\nbecomes an equivalence of categories after introducing formal direct sums and formal grading shifts to $\\widetilde{\\mathscr{TL}}_\\bullet$. \n\n\n\n\\begin{figure}\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{AD_tubed_annuli.pdf}\n\\end{center}\n\\caption{}\\label{fig:tubed annuli}\n\\end{subfigure}\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{AD_tubed_annuli2.pdf}\n\\end{center}\n\\caption{}\\label{fig:tubed annuli2}\n\\end{subfigure}\n\\begin{subfigure}[b]{.3\\textwidth}\n\\begin{center}\n\\includegraphics{AD_tubed_annuli1.pdf}\n\\end{center}\n\\caption{}\\label{fig:tubed annuli1}\n\\end{subfigure}\n\\caption{Two compression disks.}\\label{fig:Russel rels explained}\n\\end{figure}\n\nAn equivariant version of $\\mathscr{TL}_\\bullet$ follows from considering the skein category $\\mathscr{BN}_\\alpha(\\mathbb{A})$. Arguing as in \\cite[Proposition 6.1]{BPW}, any object of $\\mathscr{BN}_\\alpha(\\mathbb{A})$ is isomorphic to a direct sum of grading-shifted essential circles. Any cobordism in $\\mathscr{BN}_\\alpha(\\mathbb{A})$ can be expressed, in a non-unique way, as an $R_\\alpha$-linear combination of cobordisms of the form $S^1\\times T$, where $T$ is a dotted planar tangle. It follows that any additive functor out of $\\mathscr{BN}_\\alpha(\\mathbb{A})$ is determined by its value on each collection of $n\\geq 0$ essential circles and on cobordisms of the form $S^1\\times T$. This naturally leads to the following definition.\n\n\\begin{definition}\nLet $\\mathscr{TL}_\\alpha$ denote the category whose objects are nonnegative integers, and whose morphisms are formal $R_\\alpha$-linear combinations of dotted planar tangles, modulo the local relations shown in Figure \\ref{fig:TLa rels}. \n\\end{definition}\n\nFor a dotted planar tangle $T$ with $d(T)$ dots, define its degree to be \n\\begin{equation*}\\label{eq:deg tangle}\n\\deg(T) = 2 d(T).\n\\end{equation*}\nNote that the relations in Figure \\ref{fig:TLa rels} are homogeneous.\n\n\\begin{figure}\n\\begin{subfigure}[b]{.25\\textwidth}\n\\begin{center}\n\\includegraphics{AD_rel_TL1.pdf}\n\\end{center}\n\\caption{Undotted circle}\\label{fig:undotted circle}\n\\end{subfigure}\n\\begin{subfigure}[b]{.25\\textwidth}\n\\begin{center}\n\\includegraphics{AD_rel_TLa.pdf}\n\\end{center}\n\\caption{Dotted circle}\\label{fig:dotted circle}\n\\end{subfigure}\n\\begin{subfigure}[b]{.4\\textwidth}\n\\begin{center}\n\\includegraphics{AD_2dotrel_TL.pdf}\n\\end{center}\n\\caption{Two dots relation}\\label{fig:two dots TLa}\n\\end{subfigure}\n\\caption{Relations in $\\mathscr{TL}_\\alpha$.}\\label{fig:TLa rels}\n\\end{figure}\n\nTo motivate the relations, consider the functor \n\\begin{equation}\\label{eq:spinning alpha}\nS^1 \\times (-) \\: \\mathscr{TL}_\\alpha \\to \\mathscr{BN}_\\alpha(\\mathbb{A})\n\\end{equation}\ndefined as in \\eqref{eq:spinning d}. An undotted torus and a once-dotted torus in $\\mathbb{A} \\times I$ evaluate to $2$ and $\\alpha_0 + \\alpha_1$ in $\\mathscr{BN}_\\alpha(\\mathbb{A})$, respectively, which explains the relations in Figure \\ref{fig:undotted circle} and Figure \\ref{fig:dotted circle}. The relation in Figure \\ref{fig:two dots TLa} is a planar analogue of the two dots relation in $\\mathscr{BN}_\\alpha(\\mathbb{A})$, see Figure \\ref{fig:alpha two dot}. A straightforward induction argument also shows that an innermost circle with $k\\geq 0$ dots evaluates to $\\alpha_0^k + \\alpha_1^k$ in $\\mathscr{TL}_\\alpha$,\n\\begin{equation}\\label{eq:k dotted circle}\n\\begin{gathered}\n\\includegraphics{AD_TLa_circle_rel.pdf}\\, .\n\\end{gathered}\n\\end{equation}\nBy composing \\eqref{eq:spinning alpha} with the equivariant annular TQFT $\\mathcal{G}_{\\alpha}$, \n\\begin{equation}\\label{eq:TLa as linear maps}\n\\mathscr{TL}_\\alpha \\rar{S^1 \\times (-)} \\mathscr{BN}_\\alpha(\\mathbb{A}) \\rar{\\mathcal{G}_{\\alpha}} R_\\alpha{\\rm -}\\operatorname{ggmod},\n\\end{equation}\none can view dotted planar tangles as $R_\\alpha$-linear maps between tensor powers of $A_\\alpha$. \n\n\\begin{remark}\\label{rmk:U(2) vs U(1) times U(1)}\nAs is the case for $\\mathscr{BN}_\\alpha(\\mathbb{A})$, the relations in $\\mathscr{TL}_\\alpha$ involve only symmetric polynomials in $\\alpha_0$ and $\\alpha_1$, so one may consider the $U(2)$-equivariant analogue instead; see also Remark \\ref{rmk:cob cat BNa is induced from U(2)}. However, the functor \\eqref{eq:TLa as linear maps} is not present in the $U(2)$-equivariant setting. \n\\end{remark}\n\n\n\nThe functor \\eqref{eq:spinning alpha} is of course not faithful. For example, it factors through the local relations shown in Figure \\ref{fig:Russel relations}, which are equivariant analogues of the Russel relations, Figure \\ref{fig:Russel relations nonequiv}. Let\n\\[\n\\widetilde{\\mathscr{TL}}_{\\alpha}\n\\]\ndenote the category obtained from $\\mathscr{TL}_\\alpha$ by imposing the relations in Figure \\ref{fig:Russel relations} (it suffices to impose only the first). \n\n\\begin{figure}\n\\centering\n\\includegraphics{AD10.pdf}\n\n\\caption{Equivariant Russel relations.}\\label{fig:Russel relations}\n\\end{figure}\nWe end the section with two questions. The first is motivated by \\cite[Theorem 3.1]{Ru}. \n\\begin{question}{1}\nIs the induced spinning functor $S^1\\times (-)\\: \\widetilde{\\mathscr{TL}}_\\alpha \\to \\mathscr{BN}_\\alpha(\\mathbb{A})$ faithful? \n\\end{question}\n\n\n\nBy \\cite[Main Theorem]{Ru} and results in \\cite{Kh2}, the abelian group $\\widetilde{\\mathscr{TL}}_\\bullet(2n,0)$ is free of rank \n\\[\n\\binom{2n}{n}.\n\\]Note that for a symbol $\\star \\in \\{\\varnothing, \\bullet, \\alpha\\}$, the modules $\\mathscr{TL}_\\star(n,m)$ and $\\mathscr{TL}_\\star(k,\\l)$ are isomorphic whenever $n+m = k + \\l$. An isomorphism $\\mathscr{TL}_\\star(n,m) \\to \\mathscr{TL}_\\star(n+m,0)$ and its inverse are depicted in Figure \\ref{fig:tangle maps}. It clearly factors through the Russel relations, so $\\widetilde{\\mathscr{TL}}_\\bullet(n,m)$ is free of rank \n\\begin{equation}\\label{eq:rank of TLd}\n\\binom{n+m}{(n+m)\/2}\n\\end{equation}\nwhenever $n$ and $m$ have the same parity, and otherwise it is the zero module. \n\n\\begin{question}{2}\nIs $\\til{\\mathscr{TL}}_\\alpha(n,m)$ free over $R_\\alpha$, and, if so, of what rank?\n\\end{question}\n\n\\begin{figure}\n\\begin{subfigure}[b]{.4\\textwidth}\n\\begin{center}\n\\includegraphics{AD_tangle_iso1.pdf}\n\\end{center}\n\\caption{$\\mathscr{TL}_\\star(n,m) \\to \\mathscr{TL}_\\star(n+m,0)$}\\label{fig:tangle map1}\n\\end{subfigure}\n\\begin{subfigure}[b]{.4\\textwidth}\n\\begin{center}\n\\includegraphics{AD_tangle_iso2.pdf}\n\\end{center}\n\\caption{$\\mathscr{TL}_\\star(n+m,0) \\to \\mathscr{TL}_\\star(n,m)$}\\label{fig:tangle map2}\n\\end{subfigure}\n\\caption{An isomorphism $\\mathscr{TL}_\\star(n,m) \\cong \\mathscr{TL}_\\star(n+m,0)$.}\\label{fig:tangle maps}\n\\end{figure}\n\nNote that $\\mathscr{TL}_\\alpha(n,m) = 0$ if $n$ and $m$ have different parities, and otherwise $\\mathscr{TL}_\\alpha(n,m)$ is free of rank \n\\[\n2^{\\l} C(\\l),\n\\]\nwhere $\\l = \\frac{n+m}{2}$ and $C(\\l)$ is the $\\l$-th Catalan number. To see this, consider the collection of dotted planar $(n,m)$-tangles in which every component carries at most one dot and which has no closed components. They evidently form a basis for $\\mathscr{TL}_\\alpha(n,m)$. There are $C(\\l)$ undotted planar $(n,m)$-tangles with no closed components. A fixed such tangle has $\\l$ components, hence $2^\\l$ ways to put at most one dot on each component, which yields the count.\n\nWe can find bases for small values. A basis for $\\widetilde{\\mathscr{TL}}_{\\alpha}(1,1)$ is given by an undotted and once-dotted vertical strand, and a basis for $\\widetilde{\\mathscr{TL}}_{\\alpha}(2,2)$ is depicted in \\eqref{eq:TLa(2,2) basis}. That these elements generate the module follows from Figure \\ref{fig:Russel relations}, and linear independence can be verified using $S^1\\times (-)$ and the TQFT $\\mathcal{G}_{\\alpha}$. The ranks agree with \\eqref{eq:rank of TLd}, but it is not clear if this is a coincidence for small examples. For $\\widetilde{\\mathscr{TL}}_\\alpha(3,3)$, there are $2^3 C(3) = 40$ generators and many relations, making direct computation difficult. \n\n\\begin{equation}\\label{eq:TLa(2,2) basis}\n\\begin{gathered}\n\\includegraphics{AD_TLa2_basis.pdf}\n\\end{gathered}\n\\end{equation}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\subsection{Data Handling}\n\nBecause ``bad'' points frequently produce fake anomalies, \\cite{OB191053} aggressively removed all data points that have a sky background brighter than $5000$ ADU\/pixel\\footnote{For the KMTNet cameras, the gain is 1.0 photo-electrons per analog-to-digital unit (ADU).} or a seeing FWHM larger than 7 pixels ($0.4^{\\prime\\prime}$ per pixel) for the KMTA and KMTS data and 6.5 pixels for the KMTC data. However, many data points above these thresholds are of good quality and could contribute to the identification of anomalies (e.g., the planetary signal of KMT-2018-BLG-0029, \\citealt{KB180029}). We also find that there is no clear relation between fake anomalies and sky background. Thus, we group all the data points as ``good'' seeing (FWHM $< 7$ pixels for KMTA and KMTS, and FWHM $< 6.5$ pixels for KMTC), ``ok'' seeing ($7 \\leq$ FWHM $< 9$ pixels for KMTA and KMTS, and $6.5 \\leq$ FWHM $< 9$ pixels for KMTC), and ``bad'' seeing (FWHM $> 9$ pixels). For the PSPL fits, we only use the ``good'' points, but all points are shown to the operator. \n\nIn general, the errors derived from photometric pipelines, $\\sigma$, are underestimated and must be renormalized using the method of \\cite{MB11293}, which enables $\\chi^2\/{\\rm dof}$ for each data set to become unity. For each event, \\cite{OB191053} simply multiplied each error by $k$, and $k = 1.5$. For the optimized algorithm, we first follow \\cite{KMTeventfinder} in removing the 10\\% worst-$\\chi^2$ points and then calculate $k$ for each group of data. \n\n\n\\subsection{Manual Review}\n\nEach candidate identified by the automated algorithm is shown to the operator (W. Zang), and the operator then selects the plausible signals. See Figure \\ref{review} for an example. \\cite{OB191053} provided a four-panel display to the operator, including the light curves and residuals for the candidate signal and the data of the whole season. The optimized algorithm adds a panel that shows the seeing information for the candidate signals. This new panel shows the operator whether a candidate signal has a clear trend with seeing and whether ``ok'' and ``bad'' points dominate the detection. In addition, the optimized algorithm bins the residual of the whole season by 0.5 days, which enables the operator to see more clearly whether the data are variable and whether there is a buried long-duration trend for a short-timescale event (e.g., KMT-2018-BLG-0998, \\citealt{OB180383}). \n\n\\cite{OB191053} only tentatively classified the plausible signals by eye as ``possible planetary events'' and ``other anomalies''. The optimized algorithm allows them to be classified by eye into one of five categories, ``planet'', ``planet\/binary'' (more likely a planet), ``binary\/planet'' (more likely a binary), ``binary'', and ``finite source'' (finite-source point-lens \\citealt{1994ApJ...421L..75G,Shude1994,Nemiroff1994}). In principle, each of plausible signals should be fitted to determine its $q$, but the annual KMTNet data produces about 200 plausible signals, which is a heavy burden. Thus, the operator has to identify unambiguously non-planetary events (i.e., ``binary'') by eye, and we only fit the other four categories. There is a risk that the operator misidentifies a planetary event as ``binary''. However, co-author C. Han has personal modeling for $\\sim 75\\%$ of the ``binary''. We cross-checked his modeling for 2016--2019 ``binary'' and found that the operator only missed one massive ($q \\gtrsim 0.001$) planet on average per year. Thus, the potential buried planets in the remaining about $\\sim 25\\%$ of the ``binary'' can be ignored compared to the roughly 30--50 planets from the annual KMTNet data. In addition, for the other four categories for which we fit all events, we track the operator's by-eye sensitivity about planetary events.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Color Magnitude Diagram}\\label{CMD}\n\nTo obtain the angular Einstein radius by $\\theta_{\\rm E} = \\theta_*\/\\rho$, we estimate the angular source radius $\\theta_*$ by the de-reddened color and magnitude of the source. For each event, we place the source on a color magnitude diagram (CMD, \\citealt{Yoo2004}). For KMT-2019-BLG-1042\\ and KMT-2019-BLG-1552, the $V - I$ versus $I$ CMDs are constructed using the OGLE-III catalog \\citep{OGLEIII} of stars within a $180^{\\prime\\prime}$ square centered on the sources. For KMT-2019-BLG-2974, because the $V$-band detection limit of the OGLE-III survey affects the measurement of the red giant clump, the CMD is constructed by a combination of the OGLE-III $I$-band data and the VVV survey \\citep{VVV,VVV_cat} $K$-band data. See Figure \\ref{cmd} for the three CMDs. We calibrate the pyDIA photometry to the OGLE-III magnitudes. For KMT-2019-BLG-1552, to eliminate the effect from extinction, we compare the source position in the CMD, $(V - I, I)_{\\rm S}$, with the position of the centroid of the red giant clump, $(V - I, I)_{\\rm cl}$, whose de-reddened properties, $(V - I, I)_{\\rm cl,0}$, can be taken from \\cite{Bensby2013} and \\cite{Nataf2013}, respectively. Then, the source de-reddened color and magnitude can be derived by \n\\begin{equation}\n (V - I, I)_{\\rm S,0} = (V - I, I)_{\\rm S} - (V - I, I)_{\\rm cl} + (V - I, I)_{\\rm cl,0}.\n\\end{equation}\nFinally, using the color\/surface-brightness relation for dwarfs and subgiants of \\cite{Adams2018}, we obtain $\\theta_*$. For KMT-2019-BLG-1042\\ and KMT-2019-BLG-2974, due to the low S\/N of the $V$-band observations, we derive $I_{\\rm S, 0}$ from the CMDs and estimate $(V - I)_{\\rm S, 0}$ using the {\\it Hubble} Space Telescope ({\\it HST}) CMD of \\cite{HSTCMD}. We summarize the CMD values and ($\\theta_*$, $\\theta_{\\rm E}$, $\\mu_{\\rm rel}$) in Table \\ref{source}. \n\n\n\\subsection{Bayesian Analysis}\\label{Baye}\n\nIn principle, the lens mass $M_{\\rm L}$ and the lens distance $D_{\\rm L}$ can be determined if the two observables, $\\theta_{\\rm E}$ and $\\vec{\\pi}_{\\rm E}$, are both measured \\citep{Gould1992,Gould2000},\n\\begin{equation}\\label{eq:mass}\n M_{\\rm L} = \\frac{\\theta_{\\rm E}}{{\\kappa}|\\vec{\\pi}_{\\rm E}|};\\qquad D_{\\rm L} = \\frac{\\mathrm{au}}{|\\vec{\\pi}_{\\rm E}|\\theta_{\\rm E} + \\pi_{\\rm S}}.\n\\end{equation}\nHowever, neither the three planetary events have simultaneous measurements for the two observables. Thus, we estimate the physical parameters of the planetary systems from a Bayesian analysis using a Galactic model.\n\nWe apply the Galactic model and the procedures presented by \\cite{OB191053}. In each case, we create a sample of $10^8$ simulated events based on the Galactic model. For each simulated event, we weight it by\n\\begin{equation}\\label{eq:weight}\n \\omega_{{\\rm Gal}} = \\theta_{\\rm E}\\mu_{\\rm rel}\\mathcal{L}(t_{\\rm E})\\mathcal{L}(\\theta_{\\rm E}),\n\\end{equation}\nwhere $\\mathcal{L}(t_{\\rm E})$ and $\\mathcal{L}(\\theta_{\\rm E})$ are the likelihood of its inferred parameters $(t_{\\rm E}, \\theta_{\\rm E})$ respectively derived from the MCMC chain, and the minimum $\\chi^2$ for the lower envelope of the ($\\chi^2$ vs. $\\rho$) diagram and $\\theta_*$. For KMT-2019-BLG-1552\\ and KMT-2019-BLG-2974, we also weight the simulated events by the likelihood distribution of $\\vec{\\pi}_{\\rm E}$. For each event, we combine the posterior results of the two solutions by their Galactic-model likelihood and ${\\rm exp}(-\\Delta\\chi^2\/2)$, where $\\Delta\\chi^2$ is the $\\chi^2$ difference from the best-fit solution.\n\nThe resulting posterior distributions of the host mass $M_{\\rm host}$, the planet mass $M_{\\rm planet}$, the lens distance $D_{\\rm L}$, the projected planet-host separation $a_\\perp$ and the lens-source relative proper motion $\\mu_{\\rm rel}$ are listed in Table \\ref{phy}. The Bayesian analysis shows that all the lens systems consist of a cold giant planet beyond the snow line of its host star (assuming a snow line radius $a_{\\rm SL} = 2.7(M\/M_{\\odot})$~{\\rm au}, \\citealt{snowline}).\n\n\\subsection{Blended light and Future High-resolution Observations}\n\nThe brightness and the astrometric alignment of the blended light can sometimes provide additional constraints on the lens properties \\citep[e.g.,][]{OB181269, OB190960}. Thus, we check the baseline images taken from the 3.6m Canada-France-Hawaii Telescope (CFHT). For the astrometric and baseline brightness measurements, we adopt the CFHT images ($0.45'' <$ seeing FWHM $<0.55''$) that were taken from 2020 and 2021 in the SDSS-$i'$ band. For the baseline color measurements, we use the CFHT images obtained from the 2016 {\\it K2C9}-CFHT microlensing survey \\citep{CFHT} with the SDSS-$i'$ ($0.50'' <$ seeing FWHM $<0.60''$) and SDSS-$g'$ filters ($0.60'' <$ seeing FWHM $<0.75''$). We use DoPhot \\citep{dophot} to identify fields stars and do photometry and calibrate the photometry to the OGLE-III star catalog.\n\nFor KMT-2019-BLG-1042, the KMTNet Alert-Finder system identified an $I = 19.61$ catalog star as the event position. The CFHT images show that the catalog star is about $0.9''$ away from the true position of the source. There is no obvious object at the source position, and the nearest star is an $I \\sim 20.0$ star offset from the source position by about $0.33''$. The result from the CFHT images is consistent with the very faint lens ($I \\sim 24$) predicted by the Bayesian analysis.\n\nFor KMT-2019-BLG-1552, the CFHT images show significant blended light at the source position, with $(V - I, I)_{\\rm B} = (2.27 \\pm 0.04, 18.46 \\pm 0.03) $. We plot the blended light in the CMD (see the upper right panel of Figure \\ref{cmd}) and find that the blend belongs to the foreground main-sequence branch. However, the astrometric offset between the source and the blended light is\n\\begin{equation}\n \\Delta\\theta(N, E) = (73 \\pm 5, 99 \\pm 5) {\\rm mas},\n\\end{equation}\nindicating that the majority of the blended light is not the lens light. \n\nFor KMT-2019-BLG-2974, the CFHT images cannot identify any star within $1''$ around the source position, indicating that the lens is faint, which is also the prediction of the Bayesian analysis ($I \\sim 24.5$).\n\nFuture high-resolution observations can potentially unambiguously determine the mass and distance of the three planetary systems. When the source and the lens are resolved by adaptive optics (AO) imaging, the lens light can be measured, which combined with the angular Einstein radius $\\theta_{\\rm E}$ and\/or the microlensing parallax $\\vec{\\pi}_{\\rm E}$ can yield the mass and distance \\citep[e.g.,][]{Batista2015,Dave2015}. In addition, such observations can measure the lens-source relative proper motion $\\vec{\\mu}_{\\rm rel}$ and thus measure the angular Einstein radius by $\\theta_{\\rm E} = \\mu_{\\rm rel} \\times t_{\\rm E}$. We note that \\cite{OB120950} resolved the source and lens of the event OGLE-2012-BLG-0950 with Keck AO and the {\\it HST} when the lens-source separation is about 34 mas, for which the source and the lens have roughly equal brightness.\n\nFor KMT-2019-BLG-1042, the lens-source relative proper motion is about 10 mas\/yr. Considering the Bayesian analysis indicate that the lens is about 1--2 magnitudes fainter than the source in the near-infrared band, the source and the lens could be resolved using Keck AO in 2023. For KMT-2019-BLG-1552\\ and KMT-2019-BLG-2974, a measurement of the direction of proper motion, $\\hat{\\vec{\\mu}}_{\\rm rel} = \\hat{\\vec{\\pi}}_{\\rm E}$, combined with the 1-D measurement from annual parallax, would further constrain $\\vec{\\pi}_{\\rm E}$ \\citep{OB03175}. For KMT-2019-BLG-1552, the source and lens could have a similar brightness, but the predicted lens-source relative proper motion is low, i.e., about $3$ mas\/yr. It would probably require a 10-year wait for the current ground-based instruments to resolve the source and the lens, at which point AO on 30m class telescopes, with $\\sim 3$ times better resolution, is likely to be available. KMT-2019-BLG-2974\\ probably also requires a 10-year wait for high-resolution observations.\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\\label{intro}\n\\input{intro}\n\n\\section{The Optimized KMTNet AnomalyFinder Algorithm}\\label{anomaly}\n\\input{anomaly}\n\n\\section{AnomalyFinder Results for the 2019-Prime Fields and Protocols}\\label{2019_prime}\n\\input{2019_prime}\n\n\\section{Observations and Data Reduction}\\label{obser}\n\\input{obser}\n\n\\section{Light-curve Analysis}\\label{model}\n\\input{model}\n\n\\section{Physical Parameters}\\label{lens}\n\\input{lens}\n\n\\section{Discussion}\\label{dis}\n\\input{dis}\n\n\n\\subsection{Preamble}\n\nWe begin by introducing the processes common to light curve analysis for the three events. Their light curves all show deviations from the normal PSPL model characterized by three parameters: $(t_0, u_0, t_{\\rm E})$, i.e., the time of the closest lens-source alignment, the impact parameter scaled to the angular Einstein radius $\\theta_{\\rm E}$, and the timescale required to cross the unit Einstein radius,\n\\begin{equation}\\label{eqn:1}\nt_{\\rm E} = \\frac{\\theta_{\\rm E}}{|\\vec{\\mu}_{\\rm rel}|}; \\qquad \\theta_{\\rm E} = \\sqrt{\\kappa M_{\\rm L} \\pi_{\\rm rel}}; \\qquad \\kappa \\equiv \\frac{4G}{c^2\\mathrm{au}} \\simeq 8.144 \\frac{{\\rm mas}}{M_{\\odot}},\n\\end{equation} \nwhere $M_{\\rm L}$ is the mass of the lens system and $(\\pi_{\\rm rel},\\vec{\\mu}_{\\rm rel})$ are the lens-source relative (parallax, proper motion). \n\nWe first search for the 2L1S model of each event. The 2L1S model requires three additional parameters $(s, q, \\alpha)$ to describe the binary geometry, i.e., the projected separation between the binary components normalized to $\\theta_{\\rm E}$, the mass ratio between the binary components, and the angle between the source trajectory and the binary axis. We also introduce the parameter $\\rho$, the angular source radius $\\theta_*$ in units of $\\theta_{\\rm E}$ ($\\rho = \\theta_*\/\\theta_{\\rm E}$), for finite-source effects. We use the advanced contour integration code \\citep{Bozza2010,Bozza2018}, \\texttt{VBBinaryLensing}\\footnote{\\url{http:\/\/www.fisica.unisa.it\/GravitationAstrophysics\/VBBinaryLensing.htm}}, to compute the 2L1S magnification $A(t)$ at any given time $t$. In addition, for each data set $i$, there are two linear parameters ($f_{{\\rm S},i}$, $f_{{\\rm B},i}$) representing the flux of the source star and any blend flux, respectively. The observed flux, $f_{i}(t)$, is modeled as \n\\begin{equation}\n f_{i}(t) = f_{{\\rm S},i} A(t) + f_{{\\rm B},i}.\n\\end{equation}\nTo locate the local minima of the 2L1S model, we carry out grid searches for the parameters ($\\log s, \\log q, \\alpha, \\rho$). For each event, we first conduct a sparse grid, which consists of 21 values equally spaced between $-1.0 \\leq \\log s \\leq 1.0$, 20 values equally spaced between $0^{\\circ}\\leq \\alpha < 360^{\\circ}$, 61 values equally spaced between $-6.0 \\leq \\log q \\leq 0.0$ and five values equally spaced between $-3.5 \\leq \\log \\rho \\leq -1.5$. For each grid, we find the minimum $\\chi^2$ by Markov chain Monte Carlo (MCMC) $\\chi^2$ minimization using the \\texttt{emcee} ensemble sampler \\citep{emcee}, with fixed $\\log q$, $\\log s$ and free $t_0, u_0, t_{\\rm E}, \\rho, \\alpha$. For each local minimum, we then conduct a denser grid search (e.g., \\citealt{OB180799}). Finally, we find the best-fit models by MCMC with all parameters free. \n\n\\cite{Gaudi1998} pointed out that a 1L2S event can mimic a 2L1S event when the second source is much fainter than the primary source and moves much closer to the lens, and there have been several events with plausible 2L1S planetary solutions that proved to be 1L2S events \\citep{MB12486, OB160733, MB06074} or even a single-lens triple-source (1L3S) event \\citep{OB151459}. Thus, we check whether the observed data can be interpreted by the 1L2S model. The total magnification of a 1L2S model is the superposition of two single-lens events, \n\\begin{equation}\n A_\\lambda = \\frac{A_1f_{1,\\lambda}+A_2f_{2,\\lambda}}{f_{1,\\lambda}+f_{2,\\lambda}} = \\frac{A_{1}+q_{f,\\lambda}A_{2}}{1+q_{f,\\lambda}}; \\quad q_{f,\\lambda} \\equiv \\frac{f_{2,\\lambda}}{f_{1,\\lambda}}.\n\\end{equation}\nHere $A_\\lambda$ is total magnification, and $f_{{\\rm i},\\lambda}$ is the baseline flux at wavelength $\\lambda$ of each source, with $i=1$ and $2$ corresponding to the primary and the secondary sources, respectively. \n\nIn addition, we check whether higher-order effects can be constrained from the observed data. The first is the microlensing parallax effect \\citep{Gould1992,Gould2000,Gouldpies2004}, in which Earth's acceleration around the Sun introduces deviation from rectilinear motion between the lens and the source. We fit it by the inclusion of two parameters $\\pi_{\\rm E,N}$ and $\\pi_{\\rm E,E}$, the north and east components of the microlensing parallax vector $\\vec{\\pi}_{\\rm E}$ in equatorial coordinates, \n\\begin{equation}\n \\vec{\\pi}_{\\rm E} \\equiv \\frac{\\pi_{\\rm rel}}{\\theta_{\\rm E}} \\hat{\\vec{\\mu}}_{\\rm rel}.\n\\end{equation}\nWe also fit the $u_0 > 0$ and $u_0 < 0$ solutions for the ``ecliptic degeneracy'' \\citep{Jiang2004, Poindexter2005}. The second effect is the lens orbital motion \\citep{MB09387, OB09020}, and we fit it by introducing two parameters, $ds\/dt$ and $d\\alpha\/dt$, the instantaneous changes in the separation and orientation of the two lens components defined at $t_0$, for linearized orbital motion. In order to exclude unbound orbits, we restrict the MCMC trials to $\\beta < 1.0$ to remove unbound systems, where $\\beta$ is the absolute value of the ratio of projected kinetic to potential energy \\citep{An2002,OB050071D},\n\\begin{equation}\n \\beta \\equiv \\left| \\frac{{\\rm KE}_{\\perp}}{{\\rm PE}_{\\perp}} \\right| = \\frac{\\kappa M_{\\odot} {\\rm yr}^2}{8\\pi^2}\\frac{|\\vec{\\pi}_{\\rm E}|}{\\theta_{\\rm E}}\\gamma^2\\left(\\frac{s}{|\\vec{\\pi}_{\\rm E}| + \\pi_{\\rm S}\/\\theta_{\\rm E}}\\right)^3;\\quad \\vec{\\gamma} \\equiv \\left(\\frac{ds\/dt}{s}, \\frac{d\\alpha}{dt}\\right),\n\\end{equation}\nand where $\\pi_{\\rm S}$ is the source parallax. \n\n\\subsection{KMT-2019-BLG-1042}\n\nFigure \\ref{lc1} shows the observed data of KMT-2019-BLG-1042\\ together with the best-fit models. There is a $\\Delta I \\sim 0.9$ mag bump 1.2 days before the peak of an otherwise normal PSPL event, and both 2L1S and 1L2S models can produce such an anomaly. Table \\ref{parm1} presents the best-fit parameters and their $1\\sigma$ uncertainty for different models. The 2L1S modeling yields two degenerate solutions with $\\Delta\\chi^2 = 3.7$, and the top panels of Figure \\ref{cau} display the caustic structures and the source trajectories. The topology of the degeneracy is similar to the topology of the planetary event, OGLE-2019-BLG-0960 \\citep{OB190960}. The two solutions both have $s > 1$ and appear to be major-image perturbations. We label them by ``Inner'' and ``Outer''. We also find that this event does not suffer from the degeneracy in $\\rho$ which occurred in several known planetary events whose planetary signals are also bumps (e.g., \\citealt{MB11262,KB211391}). Because the 1L2S model is disfavored by $\\Delta\\chi^2 = 235$ and cannot fit the bump, we exclude the 1L2S model. \n\nWe find that the inclusion of higher-order effects only improves the fitting by $\\Delta\\chi^2 < 1$ and the $1\\sigma$ uncertainty of parallax at all directions is $> 5$. This is expected based on the relatively short event timescale ($t_{\\rm E} \\sim 11\\,$days) and the short duration of the planetary signal. Thus, we adopt the parameters of the 2L1S model without higher-order effects. This is a new microlensing planet with $q \\sim 6.3\\times10^{-4}$; i.e., about two times the Saturn\/Sun mass ratio. \n\n\\subsection{KMT-2019-BLG-1552}\n\nAs shown in Figure \\ref{lc2}, the light curve of KMT-2019-BLG-1552\\ exhibits two bumps and a trough between them. The 2L1S modeling yields a pair of inner\/outer solutions for the minor-image perturbations, for which the source passes on one side of the minor-image planetary caustics or the other \\citep{GG1997}. See Table \\ref{parm2} for their parameters and Figure \\ref{cau} for their geometries. However, the degeneracy is broken in this case, and the Inner solution is favored by $\\Delta\\chi^2 = 87$. \\cite{GG1997} argued that the degeneracy would be generally resolvable for the minor-image perturbations, because if the source trajectories approach the two different planetary caustics at different distances in the two solutions, then the bumps of the two solutions would be different. This argument is well-illustrated by the lower panels of Figure \\ref{lc2}, in which the main differences for the two solutions come from the two bumps and neither bump can be fitted by the Outer solution\\footnote{There are, however, many counter-examples, such as MOA-2016-BLG-319 \\citep{MB16319}, OGLE-2018-BLG-0677 \\citep{OB180677}, KMT-2019-BLG-0253, OGLE-2018-BLG-0506, OGLE-2018-BLG-0516, OGLE-2019-BLG-1492 \\citep{KB190253}, and KMT-2021-BLG-1253 \\citep{KB211391}.}. In addition, the Outer solution is still disfavored by $\\Delta\\chi^2 > 80$ with the inclusion of high-order effects, we thus exclude it. We also exclude the 1L2S model by $\\Delta\\chi^2 > 1500$. \n\nDue to the long Einstein timescale, $t_{\\rm E} \\sim 116$ days, it may be possible to either measure or place a strong constraint on the microlensing parallax vector $\\vec{\\pi}_{\\rm E}$. As presented in Table \\ref{parm2}, the addition of parallax improves the fit by $\\Delta\\chi^2 \\sim 97$. We find that all the seven data sets exhibit $\\chi^2$ improvement to the non-parallax model, so the parallax signal is reliable. The component of $\\vec{\\pi}_{\\rm E}$ that is parallel with the direction of Earth's acceleration, $\\pi_{\\rm E, \\parallel}$, is measured by $>6\\sigma$, and the perpendicular component, $\\pi_{\\rm E, \\bot}$, is constraint with $\\sigma(\\pi_{\\rm E, \\bot}) < 0.2$. For the lens orbital motion effect, we first fit it without the microlensing parallax and found $\\Delta\\chi^2 = 27$ worse than the parallax-only model. Then, the inclusion of both microlensing parallax and the lens orbital motion only improves the fit by $\\Delta\\chi^2 < 1$ compared with the parallax-only models, and $\\vec{\\gamma}$ has no correlation with $\\vec{\\pi}_{\\rm E}$. Thus, we conclude that only the microlensing parallax is measured and eliminate the lens orbital motion from the fit. This is a massive microlensing planet with about five times the Jupiter\/Sun mass ratio. \n\n\\subsection{KMT-2019-BLG-2974}\n\nFigure \\ref{lc3} shows that there is a $\\sim 4$-day dip followed by a $\\sim 2$-day bump around a step at ${\\rm HJD}^{\\prime} \\sim 8744 ({\\rm HJD}^{\\prime} = {\\rm HJD} - 2450000)$. Such an anomaly is likely due to a minor-image perturbation (e.g., \\citealt{MB14472}). That is, the source first passes on the relatively demagnified regions that are flanked by two minor-image planetary caustics and then crosses and\/or approaches one of the caustics. The 2L1S modeling yields only one solution, and its parameters and caustic geometry are respectively shown in Table \\ref{parm3} and Figure \\ref{cau}. The best-fit model exhibits two short-duration bumps during the anomaly, which are due to a cusp approach and a caustic crossing with one of the minor-image planetary caustics, respectively. Although the best-fit model shows a caustic crossing feature, due to the lack of data during the crossing, a point-source model is consistent within $1\\sigma$ level and the $3\\sigma$ upper limit is $\\rho < 0.010$. \n\nAlthough the inclusion of parallax improves the fit by only $\\Delta \\chi^2 = 1.7$ (see Table \\ref{parm3} for the parameters), the observed data provide a constraint with $\\sigma(\\pi_{\\rm E, \\parallel}) \\sim 0.2$. We find that the lens orbital motion effect is not detectable ($\\Delta\\chi^2 < 1$ for 2 degree-of-freedom) and not correlated with $\\vec{\\pi}_{\\rm E}$. Thus, we adopt the constraint on $\\vec{\\pi}_{\\rm E}$ in the Bayesian analysis in Section \\ref{Baye}. The binary mass ratio, $q \\sim 6.2 \\times 10^{-4}$, indicates that the companion is another microlensing planet with about two times the Saturn\/Sun mass ratio. \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzehbl b/data_all_eng_slimpj/shuffled/split2/finalzzehbl new file mode 100644 index 0000000000000000000000000000000000000000..358302c8ceaf7a0cb41ba150e7dff22a1b00ec4a --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzehbl @@ -0,0 +1,5 @@ +{"text":"\\section{Proofs of main results}\n\nThis appendix contains full proofs of the main results that are substantially novel (\\cref{NysInvBnd,NysPCGThm,ConvergenceCorollary}).\nThe supplement contains proofs of results that are similar to existing work, but do not appear explicitly in the literature.\n\n\\subsection{Proof \\cref{NysInvBnd}}\n\\label{section:NysSketchSolveProof}\n\nThis result contains the analysis of the Nystr{\\\"o}m sketch-and-solve method.\nWe begin with \\cref{eq-NysInvExpErr}, which provides an error bound that compares the regularized inverse of\na psd matrix $A$ with the regularized inverse of the randomized Nystr{\\\"o}m approximation $\\hat{A}_{\\mathrm{nys}}$.\nSince $0 \\preceq \\hat{A}_{\\mathrm{nys}} \\preceq A$, we can apply~\\cref{GenInvErrBnd} to obtain a deterministic\nbound for the discrepancy:\n\\[\n\\|(\\hat{A}_{\\textrm{nys}}+\\mu I)^{-1}-(A+\\mu I)^{-1}\\|\\leq \\frac{1}{\\mu}\\frac{\\|E\\|}{\\|E\\|+\\mu}\n\\quad\\text{where $E = A - \\hat{A}_{\\mathrm{nys}}$.}\n\\]\nThe function $f(t) = t\/(t+\\mu)$ is concave, so we can take expectations and invoke Jensen's inequality to obtain\n\\[\n\\E\\|(\\hat{A}_{\\textrm{nys}}+\\mu I)^{-1}-(A+\\mu I)^{-1}\\|\\leq \\frac{1}{\\mu}\\frac{\\E\\|E\\|}{\\E\\|E\\|+\\mu}.\n\\]\nInserting the bound \\cref{eq-RandNysAppxErrBnd} on $\\E \\| E \\|$ from \\cref{NysExpCorr} gives\n\\[\\E\\|(\\hat{A}_{\\textrm{nys}}+\\mu I)^{-1}-(A+\\mu I)^{-1}\\|\n \\leq \\frac{1}{\\mu} \\cdot \\frac{(3+ 4\\textup{e}^{2} \\textup{sr}_{p}(A) \/ p)\\lambda_{p}}\n {\\mu + (3+4\\textup{e}^{2}\\textup{sr}_{p}(A)\/p)\\lambda_{p} }.\\] \nTo conclude, observe that the denominator of the second fraction exceeds\n$\\mu + \\lambda_p$.\n\n\nNow, let us establish \\cref{eq-NysSSRelErr}, the error bound for Nystr{\\\"o}m sketch-and-solve.\nIntroduce the solution $\\hat{x}$ to the Nystr{\\\"o}m sketch-and-solve problem and the solution\n$x_{\\star}$ to the regularized linear system:\n\\begin{align*}\n (\\hat{A}_{\\textrm{nys}}+\\mu I)\\hat{x} = b \\quad\\text{and}\\quad\n (A+\\mu I)x_{\\star} = b.\n\\end{align*}\nWe may decompose the regularized matrix as $A+\\mu I = \\hat{A}_{\\textrm{nys}}+\\mu I+E$.\nSubtract the two equations in the last display to obtain\n\\[(\\hat{A}_{\\textrm{nys}}+\\mu I)(\\hat{x}-x_{\\star})-Ex_{\\star} = 0.\\]\nRearranging to isolate the error in the solution, we have\n\\[\\hat{x}-x_{\\star} = (\\hat{A}_{\\textrm{nys}}+\\mu I)^{-1}Ex_{\\star}.\\]\nTake the norm, apply the operator norm inequality, and use the elementary bound\n$\\| (\\hat{A}_{\\mathrm{nys}} + \\mu I)^{-1} \\| \\leq \\mu^{-1}$. We obtain\n\\[\\frac{\\|\\hat{x}-x_{\\star}\\|_{2}}{\\|x_{\\star}\\|_{2}}\\leq \\frac{\\|E\\|}{\\mu}.\\]\nFinally, take the expectation and repeat the argument used to control $\\E\\|E\\|\/\\mu$\nin the proof of \\cref{NysPCGEffDimThm}.\n\\qed\n\n\\subsubsection{Proof of \\cref{NysPCGThm}}\n\\label{section:NysPCGThmProof}\n\nLet $\\hat{A} = U \\hat{\\Lambda} U^T$ be an arbitrary rank-$\\ell$ Nystr{\\\"o}m approximation\nwhose $\\ell$th eigenvalue is $\\hat{\\lambda}_{\\ell}$.\n\\Cref{NysPCGThm} provides a deterministic bound on the condition number of\nthe regularized matrix $A_{\\mu}$ after preconditioning with\n\\[\nP = \\frac{1}{\\hat{\\lambda}_\\ell + \\mu} U (\\hat{\\Lambda} + \\mu I) U^T + (I - UU^T). \n\\]\nWe remind the reader that this argument is completely deterministic.\n\nFirst, note that the preconditioned matrix $P^{-1\/2}A_{\\mu}P^{-1\/2}$ is psd, so\n\\[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) = \\frac{\\lambda_1(P^{-1\/2}A_{\\mu}P^{-1\/2})}{\\lambda_n(P^{-1\/2}A_{\\mu}P^{-1\/2})}.\\]\nLet us begin with the upper bound on the condition number. We have the decomposition\n\\begin{equation}\n \\label{KeyDecomp}\n P^{-1\/2}A_{\\mu}P^{-1\/2} = P^{-1\/2}(\\hat{A}+\\mu I)P^{-1\/2} +P^{-1\/2}EP^{-1\/2},\n\\end{equation}\nowing to the relation $A_{\\mu} = \\hat{A} +\\mu I+E$.\nRecall that the error matrix $E$ is psd, so the matrix $P^{-1\/2}EP^{-1\/2}$ is also psd.\n\nFirst, we bound the maximum eigenvalue. Weyl's inequalities imply that\n\\[\\lambda_1(P^{-1\/2}A_{\\mu}P^{-1\/2})\\leq \\lambda_1(P^{-1\/2}(\\hat{A} +\\mu I)P^{-1\/2})+\\lambda_{1}(P^{-1\/2}EP^{-1\/2}).\\]\nA short calculation shows that $\\lambda_1(P^{-1\/2}(\\hat{A}+\\mu I)P^{-1\/2}) = \\hat{\\lambda}_{\\ell}+\\mu$.\nWhen $\\ell < n$, we have $\\lambda_{1}(P^{-1}) = 1$. Therefore,\n\\[\n\\lambda_{1}(P^{-1\/2}EP^{-1\/2}) = \\lambda_{1}(P^{-1}E) \\leq \\lambda_{1}(P^{-1})\\lambda_{1}(E) = \\lambda_{1}(E) = \\|E\\|.\n\\]\nIn summary,\n\\begin{equation}\n\\label{TopEigValBnd}\n \\lambda_1(P^{-1\/2}A_{\\mu}P^{-1\/2})\\leq \\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|.\n\\end{equation}\nFor the minimum eigenvalue, we first assume that $\\mu>0$. Apply Weyl's inequality to \\cref{KeyDecomp} to obtain\nto obtain\n\\begin{equation}\n\\label{SmallEigValBnd1}\n\\begin{aligned}\n\\lambda_{n}(P^{-1\/2}A_\\mu P^{-1\/2}) &\\geq \\lambda_{n}(P^{-1\/2}(\\hat{A}+\\mu I)P^{-1\/2})+\\lambda_{n}(P^{-1\/2}EP^{-1\/2}) \\\\\n &\\geq \\lambda_{n}(P^{-1\/2}(\\hat{A}+\\mu I)P^{-1\/2}) = \\mu.\n\\end{aligned}\n\\end{equation}\nCombining \\cref{TopEigValBnd} and \\cref{SmallEigValBnd1}, we reach\n\\[\n\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) \\leq \\frac{\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|}{\\mu}.\n\\]\nThis gives a bound for the maximum in case $\\mu > 0$.\n\nIf we only have $\\mu \\geq 0$, then a different argument is required for the smallest eigenvalue.\nAssume that $A$ is positive definite, in which case $\\hat{\\lambda}_{\\ell} > 0$.\nBy similarity,\n\\[\\lambda_n(P^{-1\/2}A_\\mu P^{-1\/2})\n = \\lambda_n(A_{\\mu}^{-1\/2}PA_{\\mu}^{-1\/2}))\n = \\frac{1}{\\lambda_{1}(A_{\\mu}^{-1\/2}PA_{\\mu}^{-1\/2})}.\n\\]\nIt suffices to produce an upper bound for $\\lambda_{1}(A_{\\mu}^{-1\/2}PA_{\\mu}^{-1\/2})$.\nTo that end, we expand\n\\begin{align*}\n\\lambda_{1}(A_{\\mu}^{-1\/2}PA_{\\mu}^{-1\/2})\n &= \\lambda_{1}\\bigg(A_{\\mu}^{-1\/2}\\left(\\frac{1}{\\hat{\\lambda}_{\\ell}+\\mu}(\\hat{A}+\\mu UU^{T})+(I-UU)^{T}\\right)A_{\\mu}^{-1\/2}\\bigg) \\\\\n &\\leq \\frac{1}{\\hat{\\lambda}_{\\ell}+\\mu}\\lambda_{1}\\left(A_{\\mu}^{-1\/2}(\\hat{A}+\\mu UU^{T})A_{\\mu}^{-1\/2}\\right)\\\\\n &+\\lambda_{1}\\bigg(A_{\\mu}^{-1\/2}\\left(I-UU^{T}\\right)A_{\\mu}^{-1\/2}\\bigg).\n\\end{align*}\nThe second inequality is Weyl's.\nSince $\\hat{A}\\preceq A$, we have $\\hat{A} +\\mu UU^{T} \\preceq A_{\\mu}$.\nThe last display simplifies to\n\\[\n\\lambda_{1}(A_{\\mu}^{-1\/2}PA_{\\mu}^{-1\/2}) \\leq \\frac{1}{\\hat{\\lambda}_{\\ell}+\\mu}+\\frac{1}{\\lambda_{n}+\\mu}.\n\\]\nPutting the pieces together with \\cref{TopEigValBnd}, we obtain\n\\[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) \\leq (\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|)\\bigg(\\frac{1}{\\hat{\\lambda}_{\\ell}+\\mu}+\\frac{1}{\\lambda_{n}+\\mu}\\bigg).\\]\nThus,\n\\[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})\\leq \\left(\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|\\right)\\min\\bigg\\{\\frac{1}{\\mu},~\n \\frac{\\hat{\\lambda}_{\\ell} + \\lambda_n + 2 \\mu}{(\\hat{\\lambda}_\\ell + \\mu)(\\lambda_n + \\mu)}\\bigg\\}.\n\\] \nThis formula is valid when $A$ is positive definite or when $\\mu > 0$.\n\nWe now prove the lower bound on $\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})$. Returning to \\cref{KeyDecomp} and invoking Weyl's inequalities yields\n\\[ \\lambda_1(P^{-1\/2}A_{\\mu}P^{-1\/2})\\geq \\lambda_1(P^{-1\/2}(\\hat{A}+\\mu I)P^{-1\/2})+\\lambda_{n}(P^{-1\/2}EP^{-1\/2})\\geq \\hat{\\lambda}_{\\ell}+\\mu.\\]\nFor the smallest eigenvalue we observe that\n\\[\\lambda_{n}(P^{-1\/2}A_{\\mu}P^{-1\/2}) = \\lambda_{n}(A_{\\mu}P^{-1}) \\leq \\lambda_{n}(A_{\\mu})\\lambda_{1}(P^{-1}) = \\lambda_{n}+\\mu.\\]\nCombining the last two displays, we obtain\n\\[\n\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\lambda_{n}+\\mu}\\leq \\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}).\n\\]\nCondition numbers always exceed one, so\n\\[\n\\max\\bigg\\{\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\lambda_{n}+\\mu},1\\bigg\\}\\leq \\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}).\n\\]\nThis point concludes the argument.\n\n\n\\subsubsection{Proof of \\cref{KeyLemma}}\n\\label{section:KeyLemmaProof}\n\n\\Cref{KeyLemma} establishes the central facts about the effective dimension.\nFirst, we prove \\cref{l-eff-max}.\nFix a parameter $\\gamma \\geq 1$, and set $j_{\\star} = \\max\\{1\\leq j\\leq n: \\lambda_{j}>\\gamma\\mu\\}$.\nWe can bound the effective dimension below by the following mechanism.\n\\[\nd_{\\textrm{eff}}(\\mu)\n = \\sum^{n}_{j=1}\\frac{\\lambda_{j}}{\\lambda_{j}+\\mu}\n \\geq \\sum^{j_{\\star}}_{j=1}\\frac{\\lambda_{j}}{\\lambda_{j}+\\mu}\\geq j_{\\star} \\cdot \\frac{\\lambda_{j_{\\star}}}{\\lambda_{j_{\\star}}+\\mu}.\n\\]\nWe have used the fact that $t \\mapsto t\/(1+t)$ is increasing for $t \\geq 0$,\nSolving for $j_{\\star}$, we determine that\n\\[\nj_{\\star}\\leq(1+\\mu\/\\lambda_{j_{\\star}})d_{\\textrm{eff}}(\\mu)<(1+\\gamma^{-1})d_{\\textrm{eff}}(\\mu).\n\\]\nThe last inequality depends on the definition of $j_{\\star}$. This is the required result.\n\n\\cref{l-eff-avg} follows from a short calculation:\n\\[\n\\begin{aligned}\n\\frac{1}{k}\\sum_{j>k}\\lambda_{j}\n &= \\frac{\\lambda_k+\\mu}{k}\\sum_{j>k}\\frac{\\lambda_{j}}{\\lambda_{k}+\\mu}\n \\leq \\frac{\\lambda_k+\\mu}{k}\\sum_{j>k}\\frac{\\lambda_{j}}{\\lambda_{j}+\\mu} \\\\\n &= \\frac{\\lambda_k+\\mu}{k}\\left(d_{\\textrm{eff}}(\\mu)-\\sum^{k}_{j=1}\\frac{\\lambda_{j}}{\\lambda_{j}+\\mu}\\right)\n \\leq\\frac{\\lambda_k+\\mu}{k}\\left(d_{\\textrm{eff}}(\\mu)-\\frac{k\\lambda_{k}}{\\lambda_{k}+\\mu}\\right) \\\\\n &= \\frac{\\mu \\,d_{\\textrm{eff}}(\\mu)}{k} +\\lambda_{k}\\left(\\frac{d_{\\textrm{eff}}(\\mu)}{k}-1\\right)\n \\leq \\frac{\\mu \\, d_{\\textrm{eff}}(\\mu)}{k}.\n\\end{aligned}\n\\]\nThe last inequality depends on the assumption that $k\\geq d_{\\textrm{eff}}(\\mu)$.\n\n\n\n\\subsection{Proof of \\cref{ConvergenceCorollary}}\n\\label{section:ConvergenceCorollaryPf}\n\nThis result gives a bound for the relative error $\\delta_t$ in the iterates of PCG.\nRecall the standard convergence bound for CG~\\cite[Theorem 38.5]{trefethen1997numerical}:\n\\[\n\\delta_{t}\\leq 2 \\, \\bigg(\\frac{\\sqrt{\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})}-1}{\\sqrt{\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})}+1}\\bigg)^{t}.\n\\]\nWe conditioned on the event that $\\{\\kappa(P^{-1\/2}A_{\\mu}P^{-1\/2}) \\leq 56\\}$. On this event,\nthe relative error must satisfy\n\\[\n\\delta_{t} < 2 \\, \\left( \\frac{\\sqrt{56} - 1}{\\sqrt{56} + 1}\\right)^t\n \\leq 2 \\cdot (0.77)^{t}.\n\\]\nSolving for $t$, we see that $\\delta_t < \\epsilon$ when $t\\geq \\lceil 3.8\\log\\left( 1\/\\epsilon \\right)\\rceil$. This concludes the proof.\n\\subsection{Proof of \\cref{AdaNysRankThm}}\n\\cref{AdaNysRankThm} establishes that with high probability \\cref{alg:AdaRandNysAppx} terminates in a logarithmic number of steps, the sketch size remains $O(d_{\\textrm{eff}}(\\delta\\tau\\mu))$, and PCG with the preconditioner constructed from the output converges fast. \n\\label{section:AdaNysRankThmPf}\n\n\\begin{proof}\nWe first recall that \\cref{alg:AdaRandNysAppx} terminates when the event\n\\[\\mathcal{E} = \\{\\|E\\|\\leq \\tau\\mu\\}\\cap\\{\\hat{\\lambda}_{\\ell}\\leq \\frac{\\tau\\mu}{11}\\}\\] \nholds. Observe that conditioned on $\\mathcal{E}$, \\cref{NysPCGThm} yields\n\\[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})\\leq \\frac{\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|}{\\mu} \\leq 1+(1+\\frac{1}{11})\\tau = 1+\\frac{12}{11}\\tau.\\]\nStatement 3 now follows from the above display and the standard convergence theorem for CG.\n\nNow, if \\cref{alg:AdaRandNysAppx} terminates with $N\\leq \\lceil\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)\\rceil-1$ steps of sketch size doubling, then $\\mathcal{E}$ holds with probability $1$.\nStatement 3 then follows by our initial observation, while statements 1 and 2 hold trivially. Hence statements $1$-$3$ all hold if the algorithm terminates in $N\\leq \\lceil\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)\\rceil-1$ steps.\n\nThus to conclude the proof, it suffices to show that if $N \\geq \\lceil\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)\\rceil$, then $\\mathcal{E}$ holds with probability at least $1-\\delta$, which implies that statements $1$-$3$ hold with probability at least $1-\\delta$, as above. \n\nWe now show that $\\mathcal{E}$ holds with probability at least $1-\\delta$ when $N = \\lceil\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)\\rceil$. To see this note that when $N = \\lceil\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)\\rceil$, we have $\\ell\\geq \\tilde{\\ell}$. Consequently, we may\ninvoke \\cref{NysExpBound} with $p = \\lceil 2d_{\\textup{eff}}\\left(\\frac{\\delta\\tau\\mu}{11}\\right)\\rceil+1 $ and \\cref{KeyLemma} to show\n\\begin{align*}\n \\mathbb{E}[\\|E\\|] &\\overset{(1)}{\\leq} 3\\lambda_{p}+\\frac{2\\textrm{e}^{2}}{p}\\left(\\sum^{n}_{j=p}\\lambda_{j}\\right)\\\\\n &\\overset{(2)}{\\leq} 3\\frac{\\delta \\tau \\mu}{11}+2\\textrm{e}^{2}\\frac{d_{\\textrm{eff}}(\\delta \\tau \\mu\/11)}{p}\\frac{\\delta \\tau \\mu}{11}\\\\\n &\\overset{(3)}{\\leq} \\frac{3}{11}\\delta \\tau \\mu+\\frac{2\\textrm{e}^{2}}{2}\\frac{\\delta \\tau \\mu}{11}= \\left(\\frac{3+\\textrm{e}^{2}}{11}\\right)\\delta \\tau \\mu \\leq \\delta \\tau \\mu.\n\\end{align*}\nWhere step (1) uses \\cref{NysExpBound}, step (2) uses items 1 and 2 of \\cref{KeyLemma} with $\\gamma = 1$, and step (3) follows from $p\\geq 2d_{\\textrm{eff}}(\\delta\\tau\\mu)$.\nThus,\n\\[\\mathbb{E}[\\|E\\|] \\leq \\delta\\tau \\mu.\\]\nBy Markov's inequality,\n\\[\\mathbb{P}\\{\\|E\\|>\\tau \\mu\\}\\leq \\delta.\\]\nHence $\\{\\|E\\|\\leq \\tau \\mu\\}$ holds with probability at least $1-\\delta$. Furthermore, by \\cref{KeyLemma} we have $\\{\\hat{\\lambda}_{\\ell}\\leq \\delta \\tau \\mu\/11\\}$ with probability $1$ as $\\hat{\\lambda}_{\\ell}\\leq \\lambda_{\\ell}\\leq \\lambda_{p}$. Thus when $N = \\lceil\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)\\rceil$, $\\mathcal{E}$ holds with probability at least $1-\\delta$, this immediately implies statements 1 and 3. Statement 2 follows as\n\\[\\ell = 2^{N}\\ell_0 \\leq 2^{\\log_{2}\\left(\\tilde{\\ell}\/\\ell_{0}\\right)+1}\\ell_0 = 2\\tilde{\\ell} = 4\\lceil2d_{\\textup{eff}}\\left(\\frac{\\delta\\tau\\mu}{11}\\right)\\rceil+2,\\] \nwhere in the first inequality we used $\\lceil x\\rceil\\leq x+1$, this completes the proof. \n\\end{proof}\n\n\\subsection{Proof of \\cref{AdaRankRatProp}}\n\\cref{AdaRankRatProp} shows once $\\ell = \\Omega\\left(d_{\\textrm{eff}}(\\tau \\mu)\\right)$, then with high probability $\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})$ differs from $(\\hat{\\lambda}_{\\ell}+\\mu)$ by at most a constant.\n\\begin{proof}\n\\cref{NysPCGThm} implies that\n\\[\\left(\\kappa_2(P^{-1\/2}A_{\\mu}P^{-1\/2})-\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\mu}\\right)_{+}\\leq \\frac{\\|E\\|}{\\mu}.\\]\n Combining the previous display with Markov's inequality yields \n\\[\\mathbb{P}\\left\\{\\left(\\kappa_2(P^{-1\/2}A_{\\mu}P^{-1\/2})-\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\mu}\\right)_{+}>\\frac{\\tau}{\\delta}\\right\\} \\leq \\frac{\\delta}{\\tau}\\frac{\\mathbb{E}[\\|E\\|]}{\\mu}.\\]\nNow, our choice of $\\ell$ combined with \\cref{NysExpBound} and \\cref{KeyLemma} implies that $\\mathbb{E}[\\|E\\|]\\leq \\tau\\mu$. \nHence we have\n\\[\\mathbb{P}\\left\\{\\left(\\kappa_2(P^{-1\/2}A_{\\mu}P^{-1\/2})-\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\mu}\\right)_{+}>\\frac{\\tau}{\\delta}\\right\\} \\leq \\delta,\\]\nwhich implies the desired claim.\n\\end{proof}\n\n\\section{Applications and experiments}\n\\label{section:NumericalExperiments}\nIn this section, we study the performance of Nystr{\\\"o}m PCG on real world data from three different applications: ridge regression, kernel ridge regression, and approximate cross-validation. \nThe experiments demonstrate the effectiveness of the preconditioner and our strategies for choosing the rank $\\ell$ compared to other algorithms in the literature:\non large datasets, we find that our method outperforms competitors by a factor of 5--10 (\\Cref{t-rfregress_results} and \\Cref{t-largekrr-results}).\n\n\\subsection{Preliminaries}\n\\label{section:ExperimentalPrelims}\nWe implemented all experiments in MATLAB R2019a and MATLAB R2021a on a server with 128 Intel Xeon E7-4850 v4 2.10GHz CPU cores and 1056 GB. Except for the very large scale datasets ($n\\geq 10^{5})$, every numerical experiment in this section was repeated twenty times;\ntables report the mean over the twenty runs, and the standard deviation (in parentheses) when it is non-zero. We highlight the best-performing method in a table in bold. \n\nWe select hyperparameters of competing methods by grid search to optimize performance.\nThis procedure tends to be very charitable to the competitors, and it may not be representative of their real-world performance.\nIndeed, grid search is computationally expensive, and it cannot be used as part of a practical implementation.\nA detailed overview of the experimental setup for each application may be found in the appropriate section of \\cref{section:Experimentaldetails}, and additional numerical results in \\cref{section:AddNumerics}.\n\n\\subsection{Ridge regression}\n\\label{section:RidgeRegression}\nIn this section, we solve the ridge regression problem \\eqref{eq-RidgeRegression} described in \\cref{IntroRidgeEx} on some standard machine learning data sets (\\cref{t-ridge-datasets}) from OpenML \\cite{vanschoren2014openml} and LIBSVM \\cite{chang2011libsvm}. \nWe compare Nystr{\\\"o}m PCG to standard CG and two state-of-the-art randomized preconditioning methods, \nthe sketch-and-precondition method of Rokhlin and Tygert~(R\\&T)~\\cite{rokhlin2008fast} and the Adaptive Iterative Hessian Sketch (AdaIHS)~\\cite{lacotte2020effective}.\n\n\\subsubsection{Experimental overview}\n\n \\begin{table}[t]\n \\caption{\\label{t-ridge-datasets} \\textbf{Ridge regression datasets.}}\n\\footnotesize\n\\begin{center}\n \\begin{tabular}{|c|c|c|}\n \\hline\n \\textbf{Dataset} & $\\mathbf{n}$ & $\\mathbf{d}$ \\\\\n \\hline\n CIFAR-10 & 50,000 & 3,072 \\\\\n \\hline\n Guillermo & 20,000 & 4,297 \\\\\n \\hline \n smallNorb-rf & 24,300 & 10,000 \\\\\n \\hline\n shuttle-rf & 43,300 & 10,000 \\\\\n \\hline\n Higgs-rf & 800,000 & 10,000 \\\\\n \\hline\n \n \n YearMSD-rf & 463,715 & 15,000\\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\nWe perform two sets of experiments: computing regularization paths on CIFAR-10 and Guillermo, \nand random features regression \\cite{rahimi2007random,rahimi2008uniform} \non shuttle, smallNORB, Higgs and YearMSD with specified values of $\\mu$. The values of $\\mu$ may be found in \\cref{section:RidgeRegressionDetails}.\nWe use Euclidean norm $\\|r\\|_{2}$ of the residual as our stopping criteria, with convergence declared when $\\|r\\|_2\\leq 10^{-10}$. \nFor both sets of experiments, we use Nystr{\\\"o}m PCG with adaptive rank selection, \\cref{alg:AdaRandNysAppx} in \\cref{section:AdaRankSelection}. For complete details of the experimental methodology, see \\cref{section:RidgeRegressionDetails}. \n\nThe regularization path experiments solve \\cref{eq-RidgeRegression} over a regularization path $\\mu = 10^{j}$ where $j = 3,\\cdots,-6$. \nWe begin by solving the problem for the largest $\\mu$ (initialized at zero), and solve for progressively smaller $\\mu$ with warm starting. For each value of $\\mu$, every method is allowed at most 500 iterations to reach the desired tolerance. \n\n\\subsubsection{Computing the regularization path}\n\\label{sec:RegPath}\n\\cref{fig:EffDimPlots} shows the evolution of $d_{\\textrm{eff}}(\\mu)$ along the regularization path. CIFAR-10 and Guillermo are both small, so we compute the exact effective dimension as a point of reference. \nWe see that we reach the sketch size cap of $\\ell_{\\textrm{max}} = 0.5d$ for CIFAR-10 and $\\ell_{\\textrm{max}} = 0.4d$ for Guillermo when $\\mu$ is small enough. For CIFAR-10, \nNystr{\\\"o}m PCG chooses a rank much smaller than the effective dimension for small values of $\\mu$. Nevertheless, the method still performs well (\\cref{fig:RidgeRegPathPlots}). \n\n\\begin{figure}[t]\n \\centering\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{lstsquaresfigs\/CIFAREffDimPlot.pdf}\n \\caption{CIFAR 10}\n \\label{fig:CifarEffDim}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{lstsquaresfigs\/GuillermoEffDimPlot.pdf}\n \\caption{Guillermo}\n \\label{fig:GuilEffDim}\n \\end{subfigure}\n\\caption{\\label{fig:EffDimPlots} \\textbf{Ridge regression: Adaptive sketch size selection.}\nNystr{\\\"o}m PCG with adaptive rank selection (\\cref{alg:AdaRandNysAppx}) selects a preconditioner whose rank is less than or equal to the effective dimension.We report error bars for the rank selected by the adaptive algorithm, however the variation is so small that the error bars aren't visible. Hence despite being inherently random, the adaptive algorithm's behavior is practically deterministic across runs. See \\cref{sec:RegPath}.} \n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{lstsquaresfigs\/CIFARRegPathPlot.pdf}\n \\caption{CIFAR 10: Runtime versus $\\mu$}\n \\label{fig:CifarCumTime}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{lstsquaresfigs\/GuillermoRegPathPlot.pdf}\n \\caption{Guillermo: Runtime versus $\\mu$}\n \\label{fig:GuilCumTime}\n \\end{subfigure}\n \\centering\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{lstsquaresfigs\/CIFARResidPlot.pdf}\n \\caption{CIFAR 10: Residual versus $\\mu$}\n \\label{fig:CifarResid}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{lstsquaresfigs\/GuillermoResidPlot.pdf}\n \\caption{Guillermo: Residual versus $\\mu$}\n \\label{fig:GuilResid}\n \\end{subfigure} \n\\caption{\\label{fig:RidgeRegPathPlots} \\textbf{Ridge regression: Runtime and residual.} Nystr{\\\"o}m PCG is either the fastest method, or it is competitive with the fastest method for all values of the regularization parameter $\\mu$. CG is generally the slowest method. All the methods reliably achieve the target residual along the entire regularization path, except for ordinary CG at small values of $\\mu$. See \\cref{sec:RegPath}.} \n\\end{figure}\n\n \\cref{fig:RidgeRegPathPlots} show the effectiveness of each method for computing the entire regularization path. Nystr{\\\"o}m PCG is the fastest of the methods along most of the regularization path. \n For CIFAR-10, Nystr{\\\"o}m PCG is faster than R\\&T until the very end of the regularization path, when $d_{\\textrm{eff}}(\\mu) \\approx d$. \n That is, the $O(d^{3})$ cost of forming the R\\&T preconditioner is not worthwhile unless $d_{\\textrm{eff}}(\\mu)\\approx d$.\n We expect Nystr{\\\"o}m PCG to perform better on ridge regression problems with non-vanishing regularization.\n\nAdaIHS is rather slow. It increases the sketch size parameter several times along the regularization path. Each time, AdaIHS must form a new sketch of the matrix, approximate the Hessian, and compute a Cholesky factorization. \n\n\\subsubsection{Random features regression}\n\n \\Cref{t-rfregress_results,t-rf-ErrCondEst} compare the performance of Nystr{\\\"o}m PCG, AdaIHS, and R\\&T PCG for random features regression. \\cref{t-rfregress_results} shows that Nystr{\\\"o}m PCG performs best on all datasets for all metrics.\nThe most striking feature is the difference between sketch sizes: \nAdaIHS and R\\&T require much larger sketch sizes than Nystr{\\\"o}m PCG, leading to greater computation time and higher storage costs.\n\\cref{t-rf-ErrCondEst} contains estimates for $\\|E\\|$ and the condition number of the preconditioned system,\nwhich explain the fast convergence.\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{\\label{t-rfregress_results} \\textbf{Ridge regression: Nystr{\\\"o}m PCG versus AdaIHS and R\\&T PCG.} Nystr{\\\"o}m PCG outperforms AdaIHS and R\\&T PCG in iteration and runtime complexity for both datasets. Additionally, Nystr{\\\"o}m PCG requires much less storage.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n\\thead{Dataset} & \\thead{Method} & \\thead{Final sketch size}& \\thead{Number of \\\\ iterations} & \\thead{Total \\\\\nruntime (s)}\\\\ \\hline\n\\multirow{3}{*}{shuttle-rf} & AdaIHS & $10,000$ & $66.9\\hspace{2pt}(0.933)$ & $66.9\\hspace{2pt}(5.27)$\\\\ \\cline{2-5}\n & R\\&T PCG & $20,000$ & $60.15$ & $242.6\\hspace{2pt} (12.24) $ \\\\ \\cline{2-5}\n & Nystr{\\\"o}m PCG & $800$ & $\\mathbf{13.1\\hspace{2pt}(1.47)}$ & $\\mathbf{9.78\\hspace{2pt}(0.943)}$ \\\\ \\hline\n\\multirow{3}{*}{smallNORB-rf} & AdaIHS & $12,800$ & $38.7\\hspace{2pt}(1.42)$ & $41.0\\hspace{2pt}(2.46)$\\\\ \\cline{2-5}\n & R\\&T PCG & $20,000$ & $34.5\\hspace{2pt}(1.31)$ & $181.5\\hspace{2pt}(6.53)$ \\\\ \\cline{2-5}\n & Nystr{\\\"o}m PCG & $800$ & $\\mathbf{31.5\\hspace{2pt}(0.489)}$ & $\\mathbf{6.67\\hspace{2pt}(0.372)}$ \\\\ \\hline\n\\multirow{3}{*}{YearMSD-rf} & AdaIHS & $30,000$ & $44$ & 1,327.3\\\\ \\cline{2-5}\n & R\\&T PCG & $30,000$ & $49$ & $766.5$ \\\\ \\cline{2-5}\n & Nystr{\\\"o}m PCG & $2,000$ & $\\mathbf{22}$ & $\\mathbf{209.7}$ \\\\ \\hline\n\\multirow{3}{*}{Higgs-rf} & AdaIHS & $6,400$ & $55$ & 1,052.7\\\\ \\cline{2-5}\n & R\\&T PCG & $20,000$ & $53$ & $607.4$ \\\\ \\cline{2-5}\n & Nystr{\\\"o}m PCG & $800$ & $\\mathbf{28}$ & $\\mathbf{91.26}$ \\\\ \\hline\n\\end{tabular}\n\n\\end{center}\n\\end{table}\n\n\\begin{table}[t]\n \\begin{center}\n\\footnotesize\n \\caption{\\label{t-rf-ErrCondEst} \\textbf{Ridge regression: Quality of Nystr{\\\"o}m preconditioner.} We use Nystr{\\\"o}m PCG as outlined in \\cref{section:LargeLStrategy}. The first column gives an estimate for the error $\\|E\\|$ in the randomized Nystr{\\\"o}m approximation. The second column gives an estimate for the condition number $\\kappa_2(P^{-1\/2}A_{\\mu}P^{-1\/2})$.}\n \\begin{tabular}{|c|c|c|}\n \\hline\n \\textbf{Dataset} & \\textbf{Estimate of $\\| E\\|$} & \\thead{Estimated condition number \\\\ of preconditioned system}\\\\\n \\hline\n shuttle-rf & $\\num{7.29e-13}$ & $4.17$ $(0.161)$ \\\\\n \\hline\n smallNorb-rf & $\\num{9.90e-3}$ & $18.5$ $(0.753)$ \\\\\n \\hline\n YearMSD-rf & $\\num{2.07e-4}$ & $22.7$\\\\\n \\hline\n Higgs-rf & $\\num{2.21e-3}$ & $23.8$\\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Approximate cross-validation}\nIn this subsection we use our preconditioner to compute approximate leave-one-out cross-validation (ALOOCV), which requires solving a large linear system with multiple right-hand sides.\n\n\\label{section:ALOOCV}\n\\subsubsection{Background}\nCross-validation is an important machine-learning technique\nto assess and select models and hyperparameters.\nGenerally, it requires re-fitting a model on many subsets of the data,\nso can take quite a long time.\nThe worst culprit is leave-one-out cross-validation (LOOCV), which requires running an expensive training algorithm $n$ times. \nRecent work has developed approximate leave-one-out cross-validation (ALOOCV), a faster alternative\nthat replaces model retraining by a linear system solve \\cite{giordano2019swiss,rad2020scalable,wilson2020approximate}. \nIn particular, these techniques yield accurate and computationally tractable approximations to LOOCV.\n\nTo present the approach, we consider the infinitesimal jacknife (IJ) approximation to LOOCV \\cite{giordano2019swiss,stephenson2020approximate}.\nThe IJ approximation computes\n\\begin{equation}\n \\label{eq-IJ} \n \\tilde{\\theta}_{\\textrm{IJ}}^{n\/j} = \\hat{\\theta}+\\frac{1}{n}H^{-1}(\\hat{\\theta})\\nabla_{\\theta} l(\\hat{\\theta},a_{j}),\n\\end{equation}\nwhere $H(\\hat{\\theta}) \\in \\R^{d \\times d}$ is the Hessian of the loss function at the solution $\\hat \\theta$, for each datapoint $a_j$.\nThe main computational challenge is computing the inverse Hessian vector product $H^{-1}(\\hat{\\theta})\\nabla_{\\theta} l(\\hat{\\theta},a_{j})$.\nWhen $n$ is very large, \nwe can also subsample the data and average \\cref{eq-IJ} over the subsample to estimate ALOOCV.\nSince ALOOCV solves the same problem with several right-hand sides,\nblocked PCG methods (here, Nystr{\\\"o}m blocked PCG) \nare the tool of choice\nto efficiently solve for multiple right-hand sides at once. \nTo demonstrate the idea, we perform numerical experiments on ALOOCV for logistic regression. \nThe datasets we use are all from LIBSVM \\cite{chang2011libsvm}; see \\cref{t-logistic-datasets}.\n\n\\begin{table}[t]\n\\footnotesize\n \\centering\n \\caption{\\label{t-logistic-datasets} \\textbf{ALOOCV datasets and experimental parameters.}} \n \\begin{tabular}{|c|c|c|c||c|c|}\n \\hline\n \\textbf{Dataset} & $\\mathbf{n}$ & $\\mathbf{d}$ & $\\mathbf{\\%} \\textbf{\\textrm{nz}}(\\mathbf{A})$\n & $\\boldsymbol \\mu$ & \\thead{Initial sketch size}\\\\\n \\hline\n Gisette & 6,000 & 5,000 & $99.1\\%$ & 1, 1e-4 & 850\\\\\n \\hline\n real-sim & 72,308 & 20,958 & $0.245\\%$ & 1e-4, 1e-8 & $500$ \\\\\n \\hline \n rcv1.binary & 20,242 & 47,236 & $0.157\\%$ & 1e-4, 1e-8 & $500$\\\\\n \\hline\n SVHN & 73,257 & 3,072 & $100\\%$ & 1, 1e-4 & 850\\\\\n \\hline\n \\end{tabular}\n\\end{table}\n\\subsubsection{Experimental overview}\nWe perform two sets of experiments in this section. The first set of experiments uses Gisette and SVHN to test the efficacy of Nystr{\\\"o}m sketch-and-solve. These datasets are small enough that we can factor $H(\\theta)$ using a direct method. We also compare to block CG and block PCG with the computed Nystr{\\\"o}m approximation as a preconditioner. \nTo assess the error due to an inexact solve for datapoint $a_j$, \nlet $x_\\star(a_j) = H^{-1}(\\theta)\\nabla_{\\theta} l(\\hat{\\theta},a_{j})$. \nFor any putative solution $\\hat{x}(a_j)$,\nwe compute the relative error $\\|\\hat{x}(a_j)-x_{\\star}(a_j) \\|_{2}\/\\|x_{\\star}(a_j)\\|_{2}$.\nWe average the relative error over 100 data-points $a_j$.\n\n The second set of experiments uses the larger datasets real-sim and rcv1.binary and small values of $\\mu$, the most challenging setting for ALOOCV. \n We restrict our comparison to block Nystr{\\\"o}m PCG versus the block CG algorithm, as Nystr{\\\"o}m sketch-and-solve is so inaccurate in this regime. We employ \\cref{alg:AdaRandNysAppx} to construct the preconditioner for block Nystr{\\\"o}m PCG.\n \n\\subsubsection{Nystr{\\\"o}m sketch-and-solve}\n\nAs predicted, Nystr{\\\"o}m sketch-and-solve works poorly (\\cref{t-gisette-svhn-results}).\nWhen $\\mu = 1$, the approximate solutions are modestly accurate,\nand the accuracy degrades as $\\mu$ decreases to $10^{-4}$. \nThe experimental results agree with the theoretical analysis presented in \\cref{Section:NystromSketchSolve}, which indicate that sketch-and-solve degrades as $\\mu$ decreases. In contrast, block CG and block Nystr{\\\"o}m PCG both provide high-quality solutions for each datapoint for both values of the regularization parameter. \n \n\\begin{table}[t]\n\\footnotesize\n \\caption{\\label{t-gisette-svhn-results} \\textbf{ALOOCV: Small datasets.} The error for a given value of $\\mu$ is the maximum relative error on $100$ randomly sampled datapoints, averaged over 20 trials.}\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\thead{Dataset} & $\\bm{\\mu}$ & \\thead{Nystr{\\\"o}m \\\\ sketch-and-solve \\\\ } & \\thead{Block CG } & \\thead{Block \\\\ Nystr{\\\"o}m PCG } \\\\\n \\hline\n Gisette & $1$ & $\\num{4.99e-2}$ & $\\num{2.68e-11}$ & $\\num{2.58e-12}$\\\\\n \\hline\n Gisette & $\\num{1e-4}$ & $\\num{1.22e-0}$ & $\\num{1.19e-11}$ & $\\num{6.59e-12}$ \\\\\n \\hline\n SVHN & $1$& $\\num{9.12e-5}$ & $\\num{2.80e-13}$ & $\\num{1.26e-13}$\\\\\n \\hline \n SVHN & $\\num{1e-4}$ & $\\num{3.42e-1}$ & $\\num{2.01e-10}$ & $\\num{1.41e-11}$\\\\\n \\hline\n \\end{tabular}\n\n\\end{center}\n\\end{table}\n\n\\subsection{Large scale ALOOCV experiments}\n \\cref{t-rcv1-realsim-results-part-i} summarizes results for block Nystr{\\\"o}m PCG and block CG on the larger datasets. \nWhen $\\mu = 10^{-4}$, block Nystr{\\\"o}m PCG offers little or no benefit over block CG\nbecause the data matrices are very sparse (see \\cref{t-logistic-datasets}) and the rcv1 problem is well-conditioned (see \\cref{t-rcv1-realsim-results-part-ii}).\n\nFor $\\mu = 10^{-8}$, block Nystr{\\\"o}m PCG reduces the number of iterations substantially,\nbut the speedup is negligible. \nThe data matrix $A$ is sparse, which reduces the benefit of the Nystr{\\\"o}m method.\nBlock CG also benefits from the presence of multiple right-hand sides just as block Nystr{\\\"o}m PCG.\nIndeed, O'Leary proved that the convergence of block CG depends on the ratio $(\\lambda_{s}+\\mu)\/(\\lambda_n+\\mu)$, where $s$ is is the number of right-hand sides \\cite{o1980block}. Consequently, multiple right-hand sides precondition block CG and accelerate convergence. \nWe expect bigger gains over block CG when $A$ is dense. \n\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n\\caption{\\label{t-rcv1-realsim-results-part-i} \\textbf{ALOOCV: Large datasets}. Block Nystr{\\\"o}m PCG outperforms block CG as $\\mu$ becomes small.}\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n\\thead{Dataset} & $\\bm{\\mu}$ & \\thead{Method} & \\thead{Number of \\\\ iterations} & \\thead{ \nRuntime (s)}\\\\ \\hline\n\\multirow{2}{*}{rcv1} & $\\num{1e-4}$ & Block CG & $12$ & $\\mathbf{11.06\\hspace{2pt}(0.874)}$\\\\ \\cline{2-5}\n & $\\num{1e-4}$ & Block Nystr{\\\"o}m PCG & $\\mathbf{10}$ & $11.87\\hspace{2pt}(0.767)$ \\\\ \\hline\n\\multirow{2}{*}{rcv1} & $\\num{1e-8}$ & Block CG & $52$ & $39.03\\hspace{2pt}(2.97)$\\\\ \\cline{2-5}\n & $\\num{1e-8}$ & Block Nystr{\\\"o}m PCG & $\\mathbf{15}$ & $\\mathbf{24.1\\hspace{2pt}(1.79)}$ \\\\ \\hline\n\\multirow{2}{*}{realsim} & $\\num{1e-4}$ & Block CG & $12$ & $23.04\\hspace{2pt}(2.04)$\\\\ \\cline{2-5}\n & $\\num{1e-4}$ & Block Nystr{\\\"o}m PCG & $\\mathbf{8}$ & $\\mathbf{19.05\\hspace{2pt}(1.10)}$ \\\\ \\hline\n\\multirow{2}{*}{realsim} & $\\num{1e-8}$ & Block CG & $90$ & $163.7\\hspace{2pt}(12.3)$\\\\ \\cline{2-5}\n & $\\num{1e-8}$ & Block Nystr{\\\"o}m PCG & $\\mathbf{32}$ & $\\mathbf{68.9\\hspace{2pt}(5.30)}$ \\\\ \\hline \n\\end{tabular}\n\n\n\\end{center}\n\\end{table}\n\n\\subsection{Kernel ridge regression}\nOur last application is kernel ridge regression (KRR),\na supervised learning technique that uses a kernel to \nmodel nonlinearity in the data. KRR leads to large dense linear systems that are challenging to solve. \n\\label{section:KernelRidgeRegression}\n\n\\subsubsection{Background}\nWe briefly review KRR \\cite{scholkopf2002learning}. Given a dataset of inputs $x_i \\in \\mathcal{D}$ and their corresponding outputs $b_i \\in \\R$ for $i = 1,\\dots, n$\nand a kernel function $\\mathcal{K}(x,y)$,\nKRR finds a function $f_{\\star}: \\mathcal{D} \\to \\R$ in the associated reproducing kernel Hilbert space $\\mathcal{H}$ that best predicts the outputs for the given inputs. \nThe solution $f_{\\star}\\in \\mathcal{H}$ minimizes the square error \nsubject to a complexity penalty:\n\\begin{equation}\n \\label{KRR}\n f_\\star = \\argmin_{f\\in \\Hc} \\quad \\frac{1}{2n}\\sum_{i=1}^{n}(f(x_{i})-b_{i})^{2}+\\frac{\\mu}{2}\\|f\\|^2_{\\Hc},\n\\end{equation}\nwhere $\\|\\cdot\\|_\\Hc$ denotes the norm on $\\Hc$. \nDefine the kernel matrix $K \\in \\R^{n \\times n}$ with entries $K_{ij} = \\mathcal{K}(x_i,x_j)$. \nThe representer theorem \\cite{steinwart2008support} states the solution to (\\ref{KRR}) is \n\\[\nf_\\star(x) = \\sum_{i=1}^{n}\\alpha_i \\mathcal{K}(x,x_{i}),\n\\]\nwhere $\\alpha = (\\alpha_1, \\ldots, \\alpha_n )$ solves the linear system\n\\begin{equation}\n\\label{KRRSys}\n (K+n\\mu I)\\alpha = b.\n\\end{equation}\nSolving the linear system (\\ref{KRRSys}) is the computational bottleneck of KRR. Direct factorization methods to solve \\eqref{KRRSys} are prohibitive for large $n$ as their costs grow as $n^3$; for $n > 10^{4}$ or so, iterative methods are generally preferred.\nHowever, $K$ is often extremely ill-conditioned, even with the regularization term $n\\mu I$. As a result, CG for Problem \\eqref{KRRSys} converges slowly.\n\n\\subsubsection{Experimental overview}\nWe use\nNystr{\\\"o}m PCG to solve several KRR problems derived from classification problems on real world datasets from \\cite{chang2011libsvm,vanschoren2014openml}.\nFor all experiments, we use the Gaussian kernel $\\mathcal{K}(x,y) = \\exp(-\\|x-y\\|^{2} \/ (2\\sigma^2))$. \nWe compare our method to random features PCG, proposed in \\cite{avron2017faster}. \nWe do not compare to vanilla CG as it is much slower than Nystr{\\\"o}m PCG \nand random features PCG. \n\nAll datasets either come with specified test sets, or we create one from a random $80$-$20$ split. \nThe PCG tolerance, $\\sigma$, and $\\mu$ were all chosen to achieve good performance \non the test sets (see \\cref{t-krr-ranks} below). \nBoth methods were allowed to run for a maximum of 500 iterations. \nThe statistics for each dataset and the experimental parameters are given in \n\\cref{t-KRR-datasets}. \n\nWe run two sets of experiments. For the datasets with $n<10^{5}$ we run oracle random features PCG against two versions of the Nystr{\\\"o}m PCG algorithm.\nThe first version uses the oracle best value of $\\ell$ found by grid search (from the same grid used to select $m_{\\textrm{rf}}$) to minimize the total runtime,\nand the second is the adaptive algorithm described in \\cref{section:RatioRankDoubling}. The grid for $\\ell$ and $m_{\\textrm{rf}}$ is restriced to less than $10,000$ to keep the preconditioners cheap to apply and store. \nThe adaptive algorithm for each dataset was initialized at $\\ell = 2,000$, which is smaller than $0.05n$ for all datasets.\nFor the datasets with $n\\geq 10^{5}$, we restricted both $\\ell$ and $m_{\\textrm{rf}}$ to $1,000$, which corresponds to less than $0.01n$. We then run both algorithms till they reach the desired tolerance or the maximum number of iterations are reached. \n\nWe use column sampling to construct the Nystr{\\\"o}m preconditioner for all KRR problems: on these problems, random projections take longer and yield similar performance (with somewhat lower variance).\n\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n \\caption{\\label{t-KRR-datasets} \\textbf{Kernel ridge regression datasets and experimental parameters.}}\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n \\textbf{Dataset} & $\\mathbf{n}$ & $\\mathbf{d}$ & $\\mathbf{n}_{\\textbf{\\textrm{classes}}}$ & $\\boldsymbol \\mu$ & $\\boldsymbol\\sigma$ & \\textbf{PCG tolerance}\\\\\n \\hline\n ijcnn1 & 49,990 & 49 & 2 & 1e-6 & 0.5 & 1e-3 \\\\\n \\hline\n MNIST & 60,000 & 784 & 10 & 1e-7 & 5 & 1e-4 \\\\\n \\hline\n Sensorless & 48,509 & 48 & 11 & 1e-8 & 0.8 & 1e-4 \\\\\n \\hline \n SensIT & 78,823 & 100 & 3 & 1e-8 & 3 & 1e-3 \\\\\n \\hline\n MiniBooNE & 104,052 & 50 & 2 & 1e-7& 5 & 1e-4 \\\\\n \\hline\n EMNIST-Balanced & 105,280 & 784 & 47 &1e-6 & 8 & 1e-3\\\\\n \\hline\n Santander & 160,000 & 200 & 2 & 1e-6 & 7 & 1e-3\\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsubsection{Experimental results}\n\n\\Cref{t-krr-results,t-largekrr-results,t-krr-ranks} summarize the results for the KRR experiments. \n\\Cref{t-krr-results} shows that both versions of Nystr{\\\"o}m PCG deliver better performance than random features preconditioning on all the datasets considered. Nystr{\\\"o}m PCG also uses less storage. \\Cref{t-largekrr-results} shows that Nystro{\\\"o}m PCG yields better performance than random features PCG on the larger scale datasets when both are restricted to ranks of $1,000$. \n\\cref{t-krr-ranks} shows the adaptive strategy proposed in \\cref{section:RatioRankDoubling} to select $\\ell$ works very well. \nIn contrast, it is difficult to choose $m_\\textrm{rf}$ for random features preconditioning: the authors of \\cite{avron2017faster} provide no guidance except for the polynomial kernel. \n \n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n\\caption{\\label{t-krr-results} \\textbf{Kernel ridge regression: Iteration count and runtime.} The adaptive and oracle Nystr{\\\"o}m PCG algorithms outperform oracle random features PCG in both time and iteration complexity.}\n\\begin{tabular}{|c|c|c|c|}\\hline\n\\thead{Dataset} & \\thead{Method} & \\thead{Number of \\\\ iterations} & \\thead{Total \\\\ runtime (s)}\\\\ \\hline\n\\multirow{3}{*}{icjnn1} & Oracle random features PCG & $63.8(2.66)$ & $38.3(2.33)$\\\\ \\cline{2-4}\n & Adaptive Nystr{\\\"o}m PCG & $43.7(1.77)$ & $\\mathbf{32.0(1.47)}$ \\\\\\cline{2-4}\n & Oracle Nystr{\\\"o}m PCG & $\\mathbf{31.8(0.835)}$ & $33.3(1.60)$ \\\\ \\hline\n\\multirow{3}{*}{MNIST} & Oracle random features PCG & $314.5(2.88)$ & $254.7(6.93)$\\\\ \\cline{2-4}\n & Adaptive Nystr{\\\"o}m PCG & $78.5(17.65)$ & $148.1(46.39)$ \\\\\\cline{2-4}\n & Oracle Nystr{\\\"o}m PCG & $\\mathbf{77.9(2.08)}$ & $\\mathbf{91.7(2.08)}$\\\\ \\hline\n\\multirow{3}{*}{Sensorless} & Oracle random features PCG & $55.4(2.35)$ & $39.9(3.96)$ \\\\ \\cline{2-4}\n & Adaptive Nystr{\\\"o}m PCG & $22.0(0.510)$ & $24.3(1.26)$\\\\\\cline{2-4}\n & Oracle Nystr{\\\"o}m PCG& $\\mathbf{21.7(0.571)}$ & $\\mathbf{22.7(1.63)}$ \\\\ \\hline\n\\multirow{3}{*}{SensIT} & Oracle random features PCG & $68.0(4.31)$ & $95.2(6.19)$ \\\\ \\cline{2-4}\n & Adaptive Nystr{\\\"o}m PCG & $\\mathbf{47.8(1.72)}$ & $70.1(2.43)$\\\\\\cline{2-4}\n & Oracle Nystr{\\\"o}m PCG & $48.7(3.40)$ & $\\mathbf{61.6(6.41)}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}[t]\n\\footnotesize\n\\caption{\\label{t-largekrr-results} \\textbf{Kernel ridge regression regression: Nystr{\\\"o}m PCG fixed rank versus random features PCG fixed rank.} Nystr{\\\"o}m PCG again outperforms random features PCG in iteration and time complexity.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\\hline\n\\thead{Dataset} & \\thead{Method} & \\thead{Number of \\\\ iterations} & \\thead{Total \\\\\nruntime (s)}\\\\ \\hline\n\\multirow{2}{*}{MiniBooNE} & Random Features PCG & $92$ & $154.2$\\\\ \\cline{2-4}\n & Nystr{\\\"o}m PCG & $\\mathbf{72}$ & $\\mathbf{137.4}$ \\\\ \\hline\n\\multirow{2}{*}{EMNIST-Balanced} & Random Features PCG & $154$ & $635.2$\\\\ \\cline{2-4}\n & Nystr{\\\"o}m PCG & $\\mathbf{32}$ & $\\mathbf{268.4}$ \\\\ \\hline \n\\multirow{2}{*}{Santander} & Random Features PCG & $160$ & $810.4$\\\\ \\cline{2-4}\n & Nystr{\\\"o}m PCG & $\\mathbf{31}$ & $\\mathbf{164.8}$ \\\\ \\hline \n\\end{tabular}\n\n\\end{center}\n\\end{table}\n\n\n\\begin{table}[ht]\n\\footnotesize\n\\begin{center}\n \\caption{\\label{t-krr-ranks} \\textbf{Kernel ridge regression: Oracle parameters and test errors.} The final ranks selected by the adaptive algorithm are almost in full agreement with the oracle ranks. Furthermore, the adaptive and oracle ranks for Nystr{\\\"o}m PCG are never larger than $m_{\\textrm{rf}}$.}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\textbf{Dataset} & \\thead{Adaptive \\\\Nystr{\\\"o}m \\\\ final rank} & \\thead{Oracle \\\\ Nystr{\\\"o}m \\\\ rank} & \\thead{Oracle} $\\mathbf{m_{\\textrm{rf}}}$ & \\thead{Test set error} \\\\\n \\hline\n ijcnn1 & $2,000$ & $3,000$ & $3,000$ & $1.25\\%$\\\\\n \\hline\n MNIST & $6,000\\hspace{2pt} (1,716)$ & $4,000$ & $9,000$ & $1.22\\%$ \\\\\n \\hline\n Sensorless & $2,000$ & $2,000$ & $5,000$ & $2.01\\%$ \\\\\n \\hline \n SensIT & $2,000$ & $2,000$ & $7,000$ & $12.83\\%$ \\\\\n \\hline\n MiniBooNE & NA & NA & NA & $7.93\\%$ \\\\\n \\hline\n EMNIST-Balanced &NA &NA &NA & $15\\%$ \\\\\n \\hline\n Santander & NA &NA &NA & $8.90\\%$\\\\\n \\hline\n \\end{tabular}\n\n\\end{center}\n\\end{table}\n\n\n\\section{Conclusion}\nWe have shown that Nystr{\\\"o}m PCG delivers a strong benefit \nover standard CG both in the theory and in practice,\nthanks to the ease of parameter selection,\non a range of interesting large-scale computational problems\nincluding ridge regression, kernel ridge regression, and ALOOCV.\nIn our experience, Nystr{\\\"o}m PCG outperforms all generic methods\nfor solving large-scale dense linear systems with spectral decay.\nIt is our hope that this paper motivates further research on randomized preconditioning for solving large scale linear systems\nand offers a useful speedup to practitioners.\n\n\\section{The Nystr{\\\"o}m approximation}\n\\label{section:NystromApproximation}\n\nLet us begin with a review of the Nystr{\\\"o}m approximation and the randomized Nystr{\\\"o}m approximation. \n\n\\subsection{Definition and basic properties}\n\nThe Nystr{\\\"o}m approximation is a natural way to construct a low-rank psd approximation of a psd matrix $A\\in \\mathbb{S}_{n}^+(\\R)$. Let $X\\in \\R^{n\\times \\ell}$ be an arbitrary test matrix. The \\textit{Nystr{\\\"o}m approximation} of $A$ with respect to the range of $X$ is defined by\n\\begin{equation}\n \\label{NysApprox}\n A\\langle X \\rangle = (AX)(X^{T}AX)^{\\dagger}(AX)^{T} \\in \\mathbb{S}_n^+(\\R).\n\\end{equation}\nThe Nystr{\\\"o}m approximation is the best psd approximation of $A$\nwhose range coincides with the range of $AX$. It has a deep relationship with the Schur complement and with Cholesky factorization~\\cite[Sec.~14]{MartTroppSurvey}.\n\nThe Nystr{\\\"o}m approximation enjoys several elementary properties that we record in the following lemma. \n\n\\begin{lemma}\n\\label{NysPropLemma}\nLet $A\\langle X\\rangle \\in \\mathbb{S}_n^+(\\R)$ be a Nystr{\\\"o}m approximation of the psd matrix $A \\in \\mathbb{S}_{n}^+(\\mathbb{R})$.\nThen \n\\begin{enumerate}\n \\item The approximation $A\\langle X\\rangle$ is psd and has rank at most $\\ell$.\n \\item The approximation $A\\langle X \\rangle$ depends only on $\\mathrm{range}(X)$.\n \\item\\label{it:nys-order} In the Loewner order, $A\\langle X\\rangle\\preceq A$.\n \\item\\label{it:nys-eigs} In particular, the eigenvalues satisfy $\\lambda_j(\\hat{A}) \\leq \\lambda_{j}(A)$ for each $1 \\leq j \\leq n$.\n\\end{enumerate}\n\\end{lemma}\nThe proof of \\cref{NysPropLemma}, \\cref{it:nys-order} is not completely obvious. It is a consequence of the fact that we may express $\\hat{A}_{\\textrm{nys}} = A^{1\/2}\\Pi A^{1\/2}$, where $\\Pi$ is an orthogonal projector.\n\n\n\\subsection{Randomized Nystr{\\\"o}m approximation}\n\\label{section:RandNysAppx}\n\\begin{algorithm}[t]\n\t\\caption{Randomized Nystr{\\\"o}m Approximation \\cite{li2017algorithm,tropp2017fixed}}\n\t\\label{alg:RandNysAppx}\n\t\\begin{algorithmic}[1]\n\t\t\\Require{Positive-semidefinite matrix $A \\in \\mathbb{S}_n^+(\\R)$, rank $\\ell$}\n\t\t\\Ensure{ Nystr{\\\"o}m approximation in factored form $\\hat{A}_{\\mathrm{nys}} = U \\hat{\\Lambda} U^T$}\n\t\t\n\t\t\\vspace{0.5pc}\n\n\t\t\\State $\\Omega = \\textrm{randn}(n, \\ell)$\n\t\t \\Comment{Gaussian test matrix}\n\t\t\\State $\\Omega = \\textrm{qr}(\\Omega,0)$\n\t\t \\Comment{Thin QR decomposition}\n\t\t\\State $Y = A\\Omega$\n\t\t \\Comment{$\\ell$ matvecs with $A$}\n\t\t\\State $\\nu = \\textrm{eps}(\\textrm{norm}(Y,\\textrm{'fro'}))$\n\t\t \\Comment{Compute shift}\n\t\t\\State $Y_{\\nu} = Y+\\nu\\Omega$\n\t\t \\Comment{Shift for stability}\n\t\t\\State $C = \\textrm{chol}(\\Omega^{T}Y_{\\nu})$\n\t\t\\State $B = Y_{\\nu}\/C$\n\t\t\\State $[U,\\Sigma,\\sim] = \\textrm{svd}(B,0)$\n\t\t \\Comment{Thin SVD}\n\t\t\\State $\\hat{\\Lambda} = \\max\\{0,\\Sigma^{2}-\\nu I\\}$ \\Comment{Remove shift, compute eigs}\n\t\\end{algorithmic}\n\\end{algorithm}\n\nHow should we choose the test matrix $X$ so that the Nystr{\\\"o}m approximation $A\\langle X\\rangle$ provides a good low-rank model for $A$? Surprisingly, we can obtain a good approximation simply by drawing the test matrix at random.\nSee~\\cite{tropp2017fixed} for theoretical justification of this claim.\n\nLet us outline the construction of the randomized Nystr{\\\"o}m approximation.\nDraw a standard normal test matrix $\\Omega \\in \\R^{n \\times \\ell}$ where $\\ell$\nis the sketch size, and compute the sketch $Y = A\\Omega$ By \\cref{NysPropLemma}, the sketch size $\\ell$ is equal to the rank of $\\hat{A}_{\\textrm{nys}}$ with probability 1,\nhence we use these terms interchangeably.\nThe Nystr{\\\"o}m approximation~\\cref{NysApprox} is constructed \ndirectly from the test matrix $\\Omega$ and the sketch $Y$:\n\\begin{equation} \\label{eqn:nys-unstable}\n \\hat{A}_{\\textrm{nys}} = A\\langle\\Omega\\rangle = Y(\\Omega^{T}Y)^{\\dagger}Y^{T}.\n\\end{equation}\nThe formula~\\cref{eqn:nys-unstable} is not numerically sound.\nWe refer the reader to~\\cref{alg:RandNysAppx}\nfor a stable and efficient implementation of the\nrandomized Nystr{\\\"o}m approximation~\\cite{li2017algorithm,tropp2017fixed}.\nConveniently, \\cref{alg:RandNysAppx} returns the truncated eigendecomposition $\\hat{A}_{\\textrm{nys}} = U\\hat{\\Lambda}U^{T}$, where $U \\in \\mathbb{R}^{n\\times \\ell}$ is an orthonormal matrix whose columns\nare eigenvectors and $\\hat{\\Lambda}\\in \\R^{\\ell\\times \\ell}$\nis a diagonal matrix listing the eigenvalues, which we often abbreviate\nas $\\hat{\\lambda}_1, \\dots, \\hat{\\lambda}_{\\ell}$.\n\nThe randomized Nystr{\\\"o}m approximation described in this section\nhas a key difference from the Nystr{\\\"o}m approximations that\nhave traditionally been used in the machine learning literature~\\cite{el2014fast,bach2013sharp,derezinski2020improved,gittens2011spectral,williams2001using}.\nIn machine learning settings, the Nystr{\\\"o}m approximation is usually constructed from a sketch $Y$ that samples random columns from the matrix (i.e., the random test matrix $\\Omega$ has 1-sparse columns). In contrast, \\cref{alg:RandNysAppx} computes a sketch $Y$ via random projection (i.e., the test matrix $\\Omega$ is standard normal). In most applications, we have strong reasons (\\cref{RandProjvsColSamp}) for preferring random projections to column sampling.\n\n\\subsubsection{Cost of randomized Nystr{\\\"o}m approximation} \n\\label{section:NystromAppxCost}\n Throughout the paper, we write $T_{\\textrm{mv}}$ for the time required to compute a matrix--vector product (matvec) with $A$. Forming the sketch $Y = A\\Omega$ with sketch size $\\ell$ requires $\\ell$ matvecs, which costs $T_{\\textrm{mv}}\\ell$. The other steps in the algorithm have arithmetic cost $O(n\\ell^{2})$. Hence, the total computational cost of \\cref{alg:RandNysAppx} is $O( T_{\\textrm{mv}}\\ell + \\ell^{2} n )$ operations. The storage cost is $O(\\ell n)$ floating-point numbers. \n\nFor \\cref{alg:RandNysAppx}, the worst-case performance occurs when $A$ is dense and unstructured. In this case, forming $Y$ costs $O(n^{2}\\ell)$ operations. However, if we have access to the columns of $A$ then we may reduce the cost of forming $Y$ to $O(n^{2}\\log \\ell)$ by using a structured test matrix $\\Omega$, such as a scrambled subsampled randomized Fourier transform (SSRFT) map or a sparse map~\\cite{MartTroppSurvey,tropp2017fixed}.\n\n\\subsubsection{A priori guarantees for the randomized Nystr{\\\"o}m approximation}\n\nIn this section, we present an a priori error bound for the randomized Nystr{\\\"o}m approximation.\nThe result improves over previous analyses~\\cite{gittens2011spectral,gittens2013revisiting,tropp2017fixed}\nby sharpening the error terms. This refinement is critical for the analysis of the preconditioner.\n\\begin{proposition}[Randomized Nystr{\\\"o}m approximation: Error] \n\\label{NysExpBound} \nConsider a psd matrix $A\\in \\mathbb{S}_{n}^+(\\R)$ with \neigenvalues $\\lambda_{1} \\geq \\cdots \\geq \\lambda_{n}$. \nChoose a sketch size $\\ell \\geq 4$, and draw a standard normal test matrix $\\Omega \\in \\R^{n\\times \\ell}$. \nThen the rank-$\\ell$ Nystr{\\\"o}m approximation $\\hat{A}_{\\textup{nys}}$ computed by \\cref{alg:RandNysAppx} satisfies\n\\begin{equation}\n\\label{eq-RandNysAppxErrBnd}\n \\mathbb{E}\\|A-\\hat{A}_{\\textup{nys}}\\| \\leq \\min_{2 \\leq p \\leq \\ell-2} \\left[ \\left(1+\\frac{2(\\ell-p)}{p-1}\\right)\\lambda_{\\ell-p+1}\n +\\frac{2\\textup{e}^{2}\\ell}{p^{2}-1}\\left(\\sum_{j>\\ell - p}\\lambda_{j}\\right) \\right].\n\\end{equation}\n\\end{proposition}\n\n\\noindent\nThe proof of \\cref{NysExpBound} may be found in \\cref{Section:NysExpBoundPf}. \n\n\\cref{NysExpBound} shows that, in expectation, the randomized Nystr{\\\"o}m approximation $\\hat{A}_{\\textrm{nys}}$ provides a good rank-$\\ell$ approximation to $A$. The first term in the bound is comparable with the spectral-norm error $\\lambda_{\\ell - p + 1}$ in the optimal rank-$(\\ell - p)$ approximation, $\\lfloor A \\rfloor_{\\ell - p}$. The second term in the bound is comparable with the trace-norm error $\\sum_{j > \\ell - p} \\lambda_j$ in the optimal rank-$(\\ell - p)$ approximation.\n\n\\Cref{NysExpBound} is better understood via the following simplification.\n\n\\begin{corollary}[Randomized Nystr{\\\"o}m approximation]\n \\label{NysExpCorr}\nInstate the assumptions of~\\cref{NysExpBound}. For $p \\geq 2$ and $\\ell = 2p - 1$, we have the bound \n\\[\\mathbb{E}\\|A-\\hat{A}_{\\textup{nys}}\\| \\leq \\left(3+\\frac{4\\textup{e}^{2}}{p}\\textup{sr}_{p}(A)\\right)\\lambda_{p}.\\]\nThe $p$-stable rank, $\\textup{sr}_{p}(A) = \\lambda_{p}^{-1}\\sum^{n}_{j = p}\\lambda_{j}$, reflects decay in the tail eigenvalues.\n\\end{corollary}\n\n\n\\cref{NysExpCorr} shows that the Nystr{\\\"o}m approximation error is on the order of $\\lambda_{p}$ when the rank parameter $\\ell = 2p - 1$. The constant depends on the $p$-stable rank $\\textrm{sr}_p(A)$, which is small when the tail eigenvalues decay quickly starting at $\\lambda_p$. This bound is critical for establishing our main results (\\cref{NysInvBnd,NysPCGEffDimThm}). \n\n\\subsubsection{Random projection versus column sampling}\n\\label{RandProjvsColSamp}\n\nMost papers in the machine learning literature~\\cite{el2014fast,bach2013sharp} construct Nystr{\\\"o}m approximations by sampling columns at random from an adaptive distribution.\nIn contrast, for most applications, we advocate using an oblivious random projection of the matrix to construct a Nystr{\\\"o}m approximation.\n\nRandom projection has several advantages over column sampling. \nFirst, column sampling may not be practical when we only have black-box matvec access to the matrix,\nwhile random projections are natural in this setting.\nSecond, it can be very expensive to obtain adaptive distributions for column sampling. Indeed, computing approximate ridge leverage scores costs just as much as solving the ridge regression problem directly using random projections \\cite[Theorem 2]{drineas2012fast}. \nThird, even with a good sampling distribution, column sampling\nproduces higher variance results than random projection,\nso it is far less reliable.\n\nOn the other hand, we have found that there are a few applications where it is more effective to compute a randomized Nystr{\\\"o}m preconditioner using column sampling in lieu of random projections. In particular, this seems to be the case for kernel ridge regression (\\cref{section:KernelRidgeRegression}). Indeed, the entries of the kernel matrix are given by an explicit formula, so we can extract full columns with ease. Sampling $\\ell$ columns may cost only $O(\\ell n)$ operations, whereas a single matvec generally costs $O(n^2)$. Furthermore, kernel matrices usually exhibit fast spectral decay, which limits the performance loss that results from using column sampling in lieu of random projection.\n\n\n\n\\section{Nystr{\\\"o}m Preconditioned Conjugate Gradients}\n\\label{section:NystromPCG}\nWe now present our main algorithm, Nystr{\\\"o}m PCG. This algorithm produces high accuracy solutions to a regularized linear system by using the Nystr{\\\"o}m approximation $\\hat{A}_{\\textrm{nys}}$ as a preconditioner. We provide a rigorous estimate for the condition number of the preconditioned system, and we prove that Nystr{\\\"o}m PCG leads to fast convergence for regularized linear systems. In contrast, we have shown that Nystr{\\\"o}m sketch-and-solve cannot be expected to yield accurate solutions.\n \n\\subsection{The preconditioner}\n\nIn this section, we introduce the optimal low-rank preconditioner,\nand we argue that the randomized Nystr{\\\"o}m preconditioner provides an approximation that is easy to compute.\n\n\\subsubsection{Motivation}\nAs a warmup, suppose we knew the eigenvalue decomposition of the best rank-$\\ell$ approximation of the matrix: $\\lfloor A \\rfloor_\\ell = V_{\\ell} \\Lambda_{\\ell} V_{\\ell}^T$.\nHow should we use this information to construct a good preconditioner for the regularized linear system~\\cref{eqn:reg-linsys-intro}?\n\nConsider the family of symmetric psd matrices that act as the identity on the orthogonal complement of $\\mathrm{range}(V_{\\ell})$. Within this class, we claim that the following matrix is the \\textit{optimal preconditioner}:\n\\begin{equation}\n\\label{eq-OptLowRankPre} \n P_\\star = \\frac{1}{\\lambda_{\\ell+1}+\\mu}V_{\\ell}(\\Lambda_{\\ell}+\\mu I)V_{\\ell}^{T}+(I-V_{\\ell}V_{\\ell}^{T}). \n\\end{equation}\nThe optimal preconditioner $P_{\\star}$ requires $O(n\\ell)$ storage, and we can solve linear systems in $P_{\\star}$ in $O(n\\ell)$ time. Whereas the regularized matrix $A_{\\mu}$ has condition number $\\kappa_2(A_{\\mu}) = (\\lambda_{1}+\\mu)\/(\\lambda_{n}+\\mu)$, the preconditioner yields\n\\begin{equation}\n\\label{eq-OptCondNumber}\n\\kappa_{2}(P_\\star^{-1\/2}A_{\\mu}P^{-1\/2}_{\\star}) = \\frac{\\lambda_{\\ell+1}+\\mu}{\\lambda_{n}+\\mu}.\n\\end{equation}\nThis is the minimum possible condition number attainable by a preconditioner from the class that we have delineated. It represents a significant improvement when $\\lambda_{\\ell + 1} \\ll \\lambda_1$.\nThe proofs of these claims are straightforward; for details, see \\cref{section:PrecondPfs}.\n\n\\subsubsection{Randomized Nystr{\\\"o}m preconditioner}\n\nIt is expensive to compute the best rank-$\\ell$ approximation $\\lfloor A \\rfloor_\\ell$ accurately. In contrast, we can compute the rank-$\\ell$ randomized Nystr{\\\"o}m approximation $\\hat{A}_{\\mathrm{nys}}$ efficiently (\\cref{alg:RandNysAppx}).\nFurthermore, we have seen that $\\hat{A}_{\\mathrm{nys}}$ approximates $A$ nearly as well as the optimal rank-$\\ell$ approximation (\\cref{NysExpCorr}).\nThese facts lead us to study the randomized Nystr{\\\"o}m preconditioner,\nproposed in~\\cite[Sec.~17]{MartTroppSurvey} without a complete justification.\n\nConsider the eigenvalue decomposition $\\hat{A}_{\\mathrm{nys}} = U \\hat{\\Lambda} U^T$, and write $\\hat{\\lambda}_{\\ell}$ for its $\\ell$th eigenvalue. The randomized Nystr{\\\"o}m preconditioner and its inverse take the form\n\\begin{equation}\n\\label{eq-NysPre}\n\\begin{aligned}\n P &= \\frac{1}{\\hat{\\lambda}_{\\ell}+\\mu}U(\\hat{\\Lambda}+\\mu I)U^{T}+(I-UU^{T}); \\\\\n P^{-1} &= (\\hat{\\lambda}_{\\ell}+\\mu)U(\\hat{\\Lambda}+\\mu I)^{-1}U^{T}+(I-UU^{T}).\n\\end{aligned}\n\\end{equation}\nLike the optimal preconditioner $P_\\star$, the randomized Nystr{\\\"o}m preconditioner \\eqref{eq-NysPre} is cheap to apply and to store.\nWe may hope that it damps the condition number of the preconditioned system $P^{-1\/2}A_{\\mu}P^{-1\/2}$ nearly as well as the optimal preconditioner $P_\\star$. We will support this intuition with a rigorous bound (\\cref{NysPCGThm}).\n\n\\subsection{Nystr{\\\"o}m PCG}\n\nWe can obviously use the randomized Nystr{\\\"o}m preconditioner within the framework of PCG.\nWe call this approach Nystr{\\\"o}m PCG, and we present a basic implementation in \\cref{alg:NysPCG}.\n\nMore precisely, \\cref{alg:NysPCG} uses left-preconditioned CG. This algorithm implicitly works with the unsymmetric matrix $P^{-1}A_{\\mu}$, rather than the symmetric matrix $P^{-1\/2}{A_\\mu}P^{-1\/2}$. The two methods yield identical sequences of iterates~\\cite{saad2003iterative}, but the former is more efficient. For ease of analysis, our theoretical results are presented in terms of the symmetrically preconditioned matrix.\n\n\\begin{algorithm}[t]\n\t\\caption{Nystr{\\\"o}m PCG} \n\t\\label{alg:NysPCG}\n\t\\begin{algorithmic}[1] \n\t\\Require{Psd matrix $A$, righthand side $b$, initial guess $x_{0}$, \n\tregularization parameter $\\mu$, sketch size $\\ell$,\n\tsolution tolerance $\\eta$}\n \\Ensure{Approximate solution $\\hat{x}$ to regularized system~\\cref{eqn:reg-linsys-intro}}\n\\vspace{0.5pc}\n \\State $[U, \\hat{\\Lambda}] =$ RandomizedNystr{\\\"o}mApproximation$(A, \\ell)$\n\t\\State $r_{0} = b-(A+ \\mu I) x_{0}$\n\t\\State $z_{0} = P^{-1}r_{0}$ \\Comment{using \\eqref{eq-NysPre}}\n\t\\State $p_{0} = z_{0}$\n\t\t\\While {$\\|r\\|_{2}>\\eta$}\n\t\t\\State $v = (A+ \\mu I) p_{0}$ \n\t\t\\State $\\alpha = (r_0^{T}z_{0}) \/ (p_{0}^{T}v_{0})$ \\Comment{compute step size} \n\t\t\\State $x = x_{0}+\\alpha p_{0}$ \\Comment{update solution}\n\t\t\\State $r = r_{0}-\\alpha v$ \\Comment{update residual}\n\t\t\\State $z = P^{-1}r$ \\Comment{find search direction via \\eqref{eq-NysPre}}\n\t\t\\State $\\beta = (r^{T}z)\/(r_{0}^{T}z_{0})$\n\t\t\\State $x_{0}\\leftarrow x$, $r_{0} \\leftarrow r$, $p_{0}\\leftarrow z+\\beta p_{0}$, $z_{0}\\leftarrow z$ \n\t\t\\EndWhile\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Complexity of Nystr{\\\"o}m PCG}\n\nNystr{\\\"o}m PCG has two steps. First, we construct the randomized Nystr{\\\"o}m approximation, and then we solve the regularized linear system using PCG.\nWe have already discussed the cost of constructing the Nystr{\\\"o}m approximation (\\cref{section:NystromAppxCost}).\nThe PCG stage costs $O(T_{\\max})$ operations per iteration, and it uses a total of $O(n)$ additional storage.\n\nFor the regularized linear system~\\cref{eqn:reg-linsys-intro}, \\cref{NysPCGEffDimThm,ConvergenceCorollary} demonstrate that it is enough to choose the sketch size $\\ell = 2 \\, \\lceil 1.5 d_{\\mathrm{eff}}(\\mu) \\rceil + 1$. In this case, the overall runtime of Nystr{\\\"o}m PCG is\n$$\nO\\big(d_{\\mathrm{eff}}(\\mu)^2 n + T_{\\mathrm{mv}} ( d_{\\mathrm{eff}}(\\mu) + \\log(1\/\\epsilon) ) \\big)\n\\quad\\text{operations.}\n$$\nWhen the effective dimension $d_{\\mathrm{eff}}(\\mu)$ is modest, Nystr{\\\"o}m PCG is very efficient.\n\nIn contrast, \\cref{NysSketchSolveAnalysis} shows that the running time for Nystr{\\\"o}m sketch-and-solve has the same form---with $d_{\\mathrm{eff}}(\\epsilon \\mu)$ in place of $d_{\\mathrm{eff}}(\\mu)$. This is a massive difference. Nystr{\\\"o}m PCG can produce solutions whose residual norm is close to machine precision; this type of successful computation is impossible with Nystr{\\\"o}m sketch-and-solve.\n\n\\subsubsection{Comparison to other randomized preconditioning methods}\n\\label{sec:preconditioner-comparison}\nIn this subsection we give a more comprehensive discussion of how Nystr{\\\"o}m PCG compares to prior work on randomized preconditioning \\cite{avron2010blendenpik,lacotte2020effective,meng2014lsrn,rokhlin2008fast} based on sketch-and-precondition and related ideas. All these prior methods were developed for least squares problems. We summarize the complexity of each method for regularized least-squares problems in \\cref{t-ComplexityComparison}. \n\n\\begin{table}[h]\n \\begin{center}\n\\footnotesize\n \\caption{\\label{t-ComplexityComparison} \\textbf{Regularized least-squares: complexity of prior randomized preconditoning methods vs. Nystr{\\\"o}m PCG}. The table compares the complexity of Nystr{\\\"o}m PCG and state-of-the-art randomized preconditioning methods in the overdetermined case $n\\geq d$, \n assuming we can access $A$ only via matrix-vector products. \n The sketch-and-precondition preconditioner is constructed from a sketch $SA$, where $S\\in \\mathbb{R}^{m\\times n}$ is a $(1\\pm \\gamma)$ Gaussian subspace embedding with sketch size $\\Omega(d\/\\gamma)$ and $\\gamma \\in (0,1).$ \n The time to compute the sketch is $O(T_{\\textrm{mv}}d \/ \\gamma)$ and the iteration complexity follows from the argument in ~\\cite[Sec ~2.6]{woodruff2014sketching}. \n For AdaIHS we use a sketch constructed from a Gaussian subspace embedding with sketch size $O(d_{\\textrm{eff}}(\\mu)\/\\rho)$ where $\\rho \\in (0,0.18)$. The complexity of AdaIHS follows from ~\\cite[Theorem 5]{lacotte2020effective}. \n The complexity of Nystr{\\\"o}m PCG is derived from \\cref{NysPCGEffDimThm,ConvergenceCorollary}.}\n \\begin{tabular}{|c|c|c|}\n \\hline\n \\textbf{Method} & \\thead{Complexity} & \\thead{References}\\\\\n \\hline\n Sketch-and-precondition & $O\\left(T_\\textrm{mv}d\/\\gamma + d^{3}\/\\gamma+T_{\\textrm{mv}}\\frac{\\log\\left(2\/\\epsilon\\right)}{\\log\\left(1\/\\gamma\\right)}\\right)$ & \\cite{avron2010blendenpik,meng2014lsrn,rokhlin2008fast} \\\\\n \\hline\n AdaIHS & $\\makecell{O\\left((T_{\\textrm{mv}}d_{\\textrm{eff}}\/\\rho+dd_{\\textrm{eff}}^{2}\/\\rho^{2})\\log(d_{\\textrm{eff}}\/\\rho)+T_{\\textrm{mv}}\\frac{\\log(1\/\\epsilon)}{\\log(1\/{\\rho})}\\right)}$ & \\cite{lacotte2020effective}\\\\\n \\hline\n Nystr{\\\"o}m PCG & $O\\left(T_\\textrm{mv}d_{\\textrm{eff}}+dd_{\\textrm{eff}}^{2}+T_{\\textrm{mv}}\\log(2\/\\epsilon)\\right)$ & This work \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\nThe time to construct the sketch-and-precondition preconditioner is always larger than the Nystr{\\\"o}m preconditioner,\nsince $d_\\text{eff} < d$ and $\\gamma < 1$.\nIndeed, constructing the preconditioner for sketch-and-precondition costs $\\Omega(d^{3})$, \nwhich is the same as a direct method when $d = \\Omega(n)$\nand is prohibitive for high-dimensional problems. \nHence Nystr{\\\"o}m PCG is amenable to problems with large $d$ and runs much faster than sketch-and-precondition whenever $d_{\\textrm{eff}}(\\mu)\\ll d$.\nThe Nystr{\\\"o}m preconditioner also enjoys wider applicability then sketch-precondition: it applies to square-ish systems, whereas the others only work for strongly overdetermined or underdetermined problems.\nIn addition to sketch-and-precondition, Nystr{\\\"o}m PCG also enjoys better complexity than AdaIHS. AdaIHS has linear dependence in $d$, but possesses additional logarithmic factors and scales in terms of $d_{\\textrm{eff}}(\\mu)\/\\rho$ where $\\rho<0.18$, leading to a worse runtime. \n\n\nIn the context of kernel ridge regression (KRR), \nthe random features method of \\cite{avron2017faster}\nmay be viewed as a randomized preconditioning technique.\n\\cite{avron2017faster} prove convergence guarantees for the polynomial kernel with a potentially prohibitive sketch size $\\ell = O(d_{\\textrm{eff}}(\\mu)^{2})$. \nIn contrast, Nystr{\\\"o}m PCG can be used for KRR with any kernel and requires smaller sketch size $\\ell = O(d_{\\textrm{eff}}(\\mu))$ to obtain fast convergence. \n\nIn summary, Nystr{\\\"o}m PCG applies to a wider class of problems than prior randomized preconditioners and enjoys stronger theoretical guarantees for regularized problems. \nNystr{\\\"o}m PCG also outperforms other randomized preconditioners numerically (\\cref{section:NumericalExperiments}). \n\n\\subsubsection{Block Nystr{\\\"o}m PCG}\n\nWe can also use the Nystr{\\\"o}m preconditioner with the block CG algorithm~\\cite{o1980block} to solve regularized linear systems with multiple right-hand sides. For this approach, we also use an orthogonalization pre-processing proposed in~\\cite{feng1995block} that ensures numerical stability without further orthogonalization steps during the iteration.\n\n\n\n\\subsection{Analysis of Nystr{\\\"o}m PCG}\n\nWe now turn to the analysis of the randomized Nystr{\\\"o}m preconditioner $P$.\n\\cref{NysPCGEffDimThm} provides a bound for the rank $\\ell$ of the Nystr{\\\"o}m preconditioner that reduces the condition number of $A_{\\mu}$ to a constant. In this case, we deduce that Nystr{\\\"o}m PCG converges rapidly (\\cref{ConvergenceCorollary}).\n\n\\begin{theorem}[Nystr{\\\"o}m preconditioning]\n\\label{NysPCGEffDimThm}\nSuppose we construct the Nystr{\\\"o}m preconditioner $P$ in \\cref{eq-NysPre} using \\cref{alg:RandNysAppx} with sketch size $\\ell = 2\\,\\lceil 1.5\\,d_{\\textup{eff}}(\\mu)\\rceil+1$.\nUsing $P$ to precondition the regularized matrix $A_{\\mu}$ results in the condition number bound\n\\[\n\\E\\big[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) \\big] < 28.\n\\]\n\\end{theorem}\n\n\\noindent\nThe proof of \\cref{NysPCGEffDimThm} may be found in \\cref{Section:NysPCGEffDimThmProof}.\n\n\\cref{NysPCGEffDimThm} has several appealing features. \nMany other authors have noticed that the effective dimension controls \nsample size requirements for particular applications\nsuch as discriminant analysis \\cite{chowdhury2018randomized},\nridge regression \\cite{lacotte2020effective}, and kernel ridge regression \\cite{el2014fast,bach2013sharp}. \nIn contrast, our result holds for any regularized linear system.\n\nOur argument makes the role of the effective dimension conceptually simpler,\nand it leads to explicit, practical parameter recommendations.\nIndeed, the effective dimension $d_{\\mathrm{eff}}(\\mu)$ is essentially\nthe same as the sketch size $\\ell$ that makes the approximation error\n$\\|A - \\hat{A}_{\\mathrm{nys}}\\|$ proportional to $\\mu$. \nIn previous arguments, such as those in \\cite{el2014fast,bach2013sharp,chowdhury2018randomized},\nthe effective dimension arises because the authors reduce the analysis to approximate matrix multiplication~\\cite{cohen2015optimal}, which produces inscrutable constant factors.\n\n\\Cref{NysPCGEffDimThm} ensures that Nystr{\\\"o}m PCG converges quickly.\n\\begin{corollary}[Nystr{\\\"o}m PCG: Convergence]\n\\label{ConvergenceCorollary}\nDefine $P$ as in \\cref{NysPCGEffDimThm}, and condition on the event \n$\\big\\{\\kappa_2\\left(P^{-1\/2}A_\\mu P^{-1\/2}\\right)\\leq 56\\big\\}$.\nLet $M = P^{-1\/2}A_{\\mu}P^{-1\/2}$. If we initialize \\cref{alg:NysPCG}\nwith initial iterate $x_0 = 0$, then the relative error $\\delta_t$ in the iterate $x_t$ satisfies\n\\[\n\\delta_t = \\frac{\\|x_{t}-x_{\\star}\\|_M}{\\|x_{\\star}\\|_{M}}\n < 2\\cdot \\left(0.77 \\right)^{t}\n \\quad\\text{where $A_{\\mu} x_{\\star} = b$.}\n\\] \nIn particular, after $t = \\lceil{3.8 \\log(2\/\\epsilon)\\rceil}$ iterations,\nwe have relative error $\\delta_t<\\epsilon$.\n\\end{corollary}\n\nThe proof of \\cref{ConvergenceCorollary} is an immediate consequence of the standard convergence result for CG \\cite[Theorem 38.5, p.~299]{trefethen1997numerical}. See~\\cref{section:ConvergenceCorollaryPf}. \n\n\\subsubsection{Analyzing the condition number}\n\nThe first step in the proof of \\cref{NysPCGEffDimThm} is a deterministic\nbound on how the preconditioner \\eqref{eq-NysPre} reduces the condition number\nof the regularized matrix $A_{\\mu}$. Let us emphasize that this bound is valid\nfor any rank-$\\ell$ Nystr{\\\"o}m approximation, regardless of the choice of test matrix. \n\n\\begin{proposition}[Nystr{\\\"o}m preconditioner: deterministic bound]\n\\label{NysPCGThm}\nLet $\\hat{A} = U \\hat{\\Lambda} U^T$ be any rank-$\\ell$ Nystr{\\\"o}m approximation, with $\\ell$th largest eigenvalue $\\hat{\\lambda}_{\\ell}$,\nand let $E = A-\\hat{A}$ be the approximation error. Construct the Nystr{\\\"o}m preconditioner $P$ as in \\eqref{eq-NysPre}.\nThen the condition number of the preconditioned matrix $P^{-1\/2}A_{\\mu}P^{-1\/2}$ satisfies\n\\begin{equation}\n \\label{eq-CondNumBnd}\n\\begin{aligned}\n \\max\\bigg\\{\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\lambda_{n}+\\mu},1\\bigg\\}\n &\\leq\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) \\\\\n &\\leq \\left(\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|\\right)\\min\\bigg\\{\\frac{1}{\\mu},~\n \\frac{\\hat{\\lambda}_{\\ell} + \\lambda_n + 2 \\mu}{(\\hat{\\lambda}_\\ell + \\mu)(\\lambda_n + \\mu)}\\bigg\\}.\n\\end{aligned}\n\\end{equation}\n\\end{proposition} \nFor the proof of \\cref{NysPCGThm} see \\cref{section:NysPCGThmProof}.\n\nTo interpret\nthe result, recall the expression~\\cref{eq-OptCondNumber} for the condition\nnumber induced by the optimal preconditioner. \\cref{NysPCGThm} shows that\nthe Nystr{\\\"o}m preconditioner may reduce the condition number\nalmost as well as the optimal preconditioner.\n\nIn particular, when $\\| E \\| = O(\\mu)$, \nthe condition number of the preconditioned system is bounded\nby a constant, independent of the spectrum of $A$. In this case,\nNystr{\\\"o}m PCG is guaranteed to converge quickly.\n\n\\subsubsection{The effective dimension and sketch size selection}\n\nHow should we choose the sketch size $\\ell$ to guarantee that $\\|E\\| = O( \\mu )$? \n\\cref{NysExpCorr} shows how the error in the rank-$\\ell$ randomized Nystr{\\\"o}m approximation\ndepends on the spectrum of $A$ through the eigenvalues of $A$ and the tail stable rank.\nIn this section, we present a lemma which demonstrates that the effective dimension\n$d_{\\mathrm{eff}}(\\mu)$ controls both quantities. As a consequence of this bound,\nwe will be able to choose the sketch size $\\ell$ proportional\nto the effective dimension $d_{\\mathrm{eff}}(\\mu)$.\n\nRecall from \\cref{EffDimDef} that the effective dimension of the matrix $A$\nis defined as\n\\begin{equation} \\label{eqn:deff-proof}\nd_{\\textrm{eff}}(\\mu) = \\textrm{tr}(A(A+\\mu I)^{-1}) = \\sum_{j=1}^{n}\\frac{\\lambda_j(A)}{\\lambda_j(A)+\\mu}.\n\\end{equation}\nAs previously mentioned, $d_{\\textrm{eff}}(\\mu)$ may be viewed as a smoothed count of the eigenvalues\nlarger than $\\mu$. Thus, one may expect that $\\lambda_{k}(A) \\lesssim \\mu$\nfor $k \\gtrsim d_{\\textrm{eff}}(\\mu)$. This intuition is correct, and it forms\nthe content of \\cref{KeyLemma}.\n\n\\begin{lemma}[Effective dimension]\n\\label{KeyLemma}\nLet $A\\in \\mathbb{S}_{n}^+(\\mathbb{R})$ with eigenvalues $\\lambda_{1}\\geq\\lambda_{2}\\geq\\cdots\\geq\\lambda_{n}$. Let $\\mu > 0$ be regularization parameter,\nand define the effective dimension as in~\\cref{eqn:deff-proof}. The following statements hold.\n\\begin{enumerate}\n \\item \\label{l-eff-max}\n \n Fix $\\gamma>0$. If $j\\geq (1+\\gamma^{-1})d_{\\textup{eff}}(\\mu)$, then $\\lambda_{j}\\leq \\gamma\\mu$.\n \\item \\label{l-eff-avg} If $k\\geq d_{\\textup{eff}}(\\mu)$, then\n $k^{-1} \\sum_{j>k}\\lambda_j \\leq (d_{\\textup{eff}}(\\mu)\/k) \\cdot \\mu$. \n\\end{enumerate}\n\\end{lemma}\nThe proof of \\cref{KeyLemma} may be found in \\cref{section:KeyLemmaProof}.\n\n \\cref{KeyLemma}, \\cref{l-eff-max} captures the intuitive fact that there are no more than $2 d_{\\mathrm{eff}}(\\mu)$ eigenvalues larger than $\\mu$. Similarly,\n \\cref{l-eff-avg} states that the effective dimension controls the sum\n of all the eigenvalues whose index exceeds the effective dimension.\n It is instructive to think about the meaning of these results\n when $d_{\\textrm{eff}}(\\mu)$ is small.\n\n\\subsubsection{Proof of \\cref{NysPCGEffDimThm}}\n\\label{Section:NysPCGEffDimThmProof}\n We are now prepared to prove \\cref{NysPCGEffDimThm}.\n The key ingredients in the proof are \\cref{NysExpBound}, \\cref{NysPCGThm}, and \\cref{KeyLemma}.\n \n\\begin{proof}[Proof of \\cref{NysPCGEffDimThm}]\nFix the sketch size $\\ell = 2 \\, \\lceil 1.5 \\, d_{\\mathrm{eff}}(\\mu) \\rceil + 1$.\nConstruct the rank-$\\ell$ randomized Nystr{\\\"o}m approximation $\\hat{A}_{\\mathrm{nys}}$\nwith eigenvalues $\\hat{\\lambda}_j$. Write $E = A - \\hat{A}_{\\mathrm{nys}}$ for the approximation error.\nForm the preconditioner $P$ via~\\cref{eq-NysPre}. We must bound the expected condition number\nof the preconditioned matrix $P^{-1\/2} A_{\\mu} P^{-1\/2}$\n \nFirst, we apply \\cref{NysPCGThm} to obtain a deterministic bound that is valid for any rank-$\\ell$\nNystr{\\\"o}m preconditioner:\n\\[\n\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})\\leq \\frac{\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|}{\\mu}\n \\leq 2 + \\frac{\\|E\\|}{\\mu}.\n\\]\nThe second inequality holds because $\\hat{\\lambda}_{\\ell} \\leq \\lambda_{\\ell} \\leq \\mu$.\nThis is a consequence of \\cref{NysPropLemma}, \\cref{it:nys-eigs}\nand \\cref{KeyLemma}, \\cref{l-eff-max} with $\\gamma = 1$.\nWe rely on the fact that $\\ell \\geq 2 \\, d_{\\mathrm{eff}}(\\mu)$.\n\nDecompose $\\ell = 2p-1$ where $p = \\lceil1.5\\,d_{\\textrm{eff}}(\\mu)\\rceil+1$.\nTake the expectation, and invoke \\cref{NysExpCorr} to obtain \n\\[\\E\\big[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})\\big]\n \\leq 2 + \\left(3+\\frac{4\\textrm{e}^{2}}{p}\\textrm{sr}_{p}(A)\\right) (\\lambda_{p} \/ \\mu).\\]\nBy definition, $\\textrm{sr}_{p}(A) \\cdot \\lambda_{p} = \\sum_{j \\geq p}\\lambda_{j}$.\nTo complete the bound, apply \\cref{KeyLemma} twice. We use \\cref{l-eff-max} with $\\gamma = 2$ and \\cref{l-eff-avg} with $k = p-1 = \\lceil 1.5\\,d_{\\textrm{eff}}(\\mu)\\rceil$ to reach\n\\[\n\\E\\big[\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})\\big]\\leq 2+\\frac{3\\cdot 2\\mu+4\\textrm{e}^{2}\\cdot 2\\mu\/3}{\\mu}<2+26 = 28,\n\\]\nwhich is the desired result.\n\\end{proof}\n\n \n\\subsection{Practical parameter selection}\n\\label{sec:adaptive-rank}\n\nIn practice, we may not know the regularization parameter $\\mu$ in advance,\nand we rarely know the effective dimension $d_{\\mathrm{eff}}(\\mu)$. As a consequence,\nwe cannot enact the theoretical recommendation for the rank of the Nystr{\\\"o}m\npreconditioner: $\\ell = 2 \\, \\lceil 1.5\\,d_{\\mathrm{eff}}(\\mu) \\rceil + 1$.\nInstead, we need an adaptive method for choosing the rank $\\ell$. Below,\nwe outline three strategies.\n\n\\subsubsection{Strategy 1: Adaptive rank selection by a posteriori error estimation}\n\\label{section:ErrorRankDoubling}\n\nThe first strategy uses the posterior condition number estimate adaptively\nin a procedure the repeatedly doubles the sketch size $\\ell$ as required.\n Recall that\n \\cref{NysPCGThm} controls the condition number\n of the preconditioned system:\n\\begin{equation}\n\\label{eq-kappa-reg-bnds}\n\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) \\leq \\frac{\\hat{\\lambda}_{\\ell}+\\mu+\\|E\\|}{\\mu}\n\\quad\\text{where $E = A - \\hat{A}_{\\mathrm{nys}}$.}\n\\end{equation} \nWe get $\\hat{\\lambda}_{\\ell}$ for free from \\cref{alg:RandNysAppx} and we can compute the error $\\|E\\|$ inexpensively with the randomized power method \\cite{kuczynski1992estimating};\nsee \\cref{alg:RandPowErrEst} in \\cref{section:AdaRankSelection}. \nThus, we can ensure the condition number is small by making $\\|E\\|$ and $\\hat{\\lambda}_{\\ell}$ fall below some desired tolerance.\nThe adaptive strategy proceeds to do this as follows.\nWe compute a randomized Nystr{\\\"o}m approximation with initial sketch size $\\ell_{0}$, and we estimate the error $\\| E \\|$ using randomized powering. \nIf $\\|E\\|$ is smaller than a prescribed tolerance, we accept the rank-$\\ell_0$ approximation. If the tolerance is not met, then we double the sketch size, update the approximation, and estimate $\\|E\\|$ again. \nThe process repeats until the estimate for $\\|E\\|$ falls below the tolerance or it breaches a threshold $\\ell_{\\textrm{max}}$ for the maximum sketch size. \\cref{alg:AdaRandNysAppx} usings the following stopping criterions $\\| E \\| \\leq \\tau\\mu$ and $\\hat{\\lambda}_{\\ell}\\leq \\tau\\mu\/11$ for a modest constant $\\tau$. The stopping criterion on $\\hat{\\lambda}_{\\ell}$ does not seem to be necessary in practice, as it is usually an order of magnitude small than $\\|E\\|$, but it is needed for \\cref{AdaNysRankThm}. \nBased on numerical experience, we recommend a choice $\\tau\\in [1,100]$. \nFor full algorithmic details of adaptive rank selection by estimating $\\|E\\|$,\n see \\ref{alg:AdaRandNysAppx} in \\ref{section:AdaRankSelection}. \n \n The following theorem shows that with high probability, \\cref{alg:AdaRandNysAppx} terminates with a modest sketch size in at most a logarithmic number of steps, and PCG with the resulting preconditioner converges rapidly.\n \n \\begin{theorem}\n \\label{AdaNysRankThm}\nRun \\cref{alg:AdaRandNysAppx} with initial sketch size $\\ell_0$ and tolerance $\\tau \\mu$ where $\\tau \\geq 1 $, and let $\\tilde{\\ell} = 2\\lceil 2d_{\\textup{eff}}\\left(\\frac{\\delta\\tau\\mu}{11}\\right)\\rceil+1$. Then with probability at least $1-\\delta$:\n\\begin{itemize}\n \\item[1.] \\cref{alg:AdaRandNysAppx} doubles the sketch size at most\n $\\lceil\\log_{2}\\left(\\frac{\\tilde{\\ell}}{\\ell_{0}}\\right)\\rceil$ times.\n \\item[2.] The final sketch size $\\ell$ satisfies\n \\[\\ell \\leq 4\\lceil2d_{\\textup{eff}}\\left(\\frac{\\delta\\tau\\mu}{11}\\right)\\rceil+2.\\]\n \\item[3.] With the preconditioner constructed from \\cref{alg:AdaRandNysAppx}, Nystr{\\\"o}m PCG converges in at most $\\lceil\\frac{\\log(2\/\\epsilon)}{\\log(1\/\\tau_{0})}\\rceil$ iterations, where $\\tau_{0} = \\frac{\\sqrt{1+12\\tau\/11}-1}{\\sqrt{1+12\\tau\/11}+1}$. \n\\end{itemize}\n\\end{theorem}\n\n\\cref{AdaNysRankThm} immediately implies the following concrete guarantee.\n\\begin{corollary}\nSet $\\tau = 44$ and $\\delta = 1\/4$ in \\cref{alg:AdaRandNysAppx} then with probability at least $3\/4$:\n\\item[1.] \\cref{alg:AdaRandNysAppx} doubles the sketch size at most\n $\\lceil\\log_{2}\\left(\\frac{\\tilde{\\ell}}{\\ell_{0}}\\right)\\rceil$ times.\n \\item[2.] The final sketch size $\\ell$ satisfies\n \\[\\ell \\leq 4\\lceil2d_{\\textup{eff}}(\\mu)\\rceil+2.\\]\n \\item[3.] With the preconditioner constructed from \\cref{alg:AdaRandNysAppx}, Nystr{\\\"o}m PCG converges in at most $\\lceil3.48\\log(2\/\\epsilon)\\rceil$ iterations. \n\\end{corollary}\n\n\\subsubsection{Strategy 2: Adaptive rank selection by monitoring $\\hat{\\lambda}_{\\ell}\/\\mu$}\n\\label{section:RatioRankDoubling}\n\nThe second strategy is almost identical to the first, except we monitor the ratio $\\hat{\\lambda}_{\\ell}\/\\mu$ instead of $\\|E\\|\/\\mu$. \nStrategy 2 doubles the approximation rank until $\\hat{\\lambda}_{\\ell}\/\\mu$ falls below some tolerance (say, 10) \nor the sample size reaches the threshold $\\ell_{\\max}$.\nThe approach is justified by the following proposition which shows that once the rank $\\ell$ is sufficiently large, with high probability, the exact condition number differs from the empirical condition number $(\\hat{\\lambda}_\\ell+\\mu)\/\\mu$ by at most a constant.\n\\begin{proposition}\n\\label{AdaRankRatProp}\nLet $\\tau\\geq 0$ denote the tolerance and $\\delta>0$ a given failure probability. Suppose the rank of the randomized Nystr{\\\"o}m approximation satisfies $\\ell \\geq 2\\lceil 2d_{\\textup{eff}}\\left(\\tau\\mu)\\right)\\rceil+1$. Then \n\\begin{equation}\n \\mathbb{P}\\left\\{\\left(\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})-\\frac{\\hat{\\lambda}_{\\ell}+\\mu}{\\mu}\\right)_{+}\\leq\\frac{\\tau}{\\delta}\\right\\}\\geq 1-\\delta,\n\\end{equation}\nwhere $X_{+} = \\max\\{X,0\\}.$\n\\end{proposition}\nThis strategy has the benefit of saving a bit of computation and is preferable when the target precision is not too important, eg, in machine learning problems where training error only loosely predicts test error.\n\n\\subsubsection{Strategy 3: Choose $\\ell$ as large as the user can afford}\n\\label{section:LargeLStrategy}\n\nThe third strategy is to choose the rank $\\ell$ as large as the user can afford.\nThis approach is coarse, and it does not yield any guarantees on the\ncost of the Nystr{\\\"o}m PCG method.\n\nNevertheless, once we have constructed a rank-$\\ell$ Nystr{\\\"o}m approximation we can combine the posterior estimate of the condition number used in strategy 1 with the standard convergence theory of PCG to obtain an upper bound for the iteration count of Nystr{\\\"o}m PCG.\nThis gives us advance warning\nabout how long it may take to solve the regularized linear system.\nAs in strategy 1 we compute the error $\\| E \\|$ in the condition number bound inexpensively with the randomized power method. \n\n\n\n\n\\section{Nystr{\\\"o}m sketch-and-solve}\n\nThe simplest mechanism for using the Nystr{\\\"o}m approximation\nis an algorithm called Nystr{\\\"o}m sketch-and-solve. This\nsection introduces the method, its implementation, and its history.\nWe also provide a general theoretical analysis that sheds light on its performance. In spite of its popularity, the Nystr{\\\"o}m sketch-and-solve method is rarely worth serious consideration.\n\n\\begin{algorithm}[t]\n\t\\caption{Nystr{\\\"o}m sketch-and-solve} \n\t\\label{alg:NysSketchSolve}\n\t\\begin{algorithmic}[1]\n\t\t\\Require{Psd matrix $A \\in \\mathbb{S}_n^+(\\R)$, right-hand side $b$, regularization $\\mu$, rank $\\ell$} \n\t\t\\Ensure{Approximate solution $\\hat{x}$ to~\\cref{eqn:reg-linsys-intro}}\n\\vspace{0.5pc}\n\t\t\\State $[U,\\hat{\\Lambda}]$ = RandomizedNystr{\\\"o}mApproximation($A,\\ell$)\n\t\t\\State Use~\\cref{eq-Woodbury} to compute $\\hat{x} = (\\hat{A}_{\\textrm{nys}}+\\mu I)^{-1}b$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\label{Section:NystromSketchSolve}\n\\subsection{Overview}\n\nGiven a rank-$\\ell$ Nystr{\\\"o}m approximation $\\hat A_\\textrm{nys}$ of the psd matrix $A$, it is tempting to replace the regularized linear system $(A + \\mu I) x = b$ with the proxy $(\\hat{A}_{\\textrm{nys}} + \\mu I) x = b$. Indeed, we can solve the proxy linear system in $O(\\ell n)$ time using the Sherman--Morrison--Woodbury formula~\\cite[Eqn. (2.1.4)]{golubvanloan2013}:\n\n\\begin{lemma}[Approximate regularized inversion]\n\\label{SketchInv}\nConsider any rank-$\\ell$ matrix $\\hat{A}$ with eigenvalue decomposition $\\hat{A} = U \\hat \\Lambda U^T$.\nThen\n\\begin{equation}\n\\label{eq-Woodbury}\n (\\hat{A}+\\mu I)^{-1} = U(\\hat{\\Lambda}+\\mu I)^{-1}U^{T}+\\frac{1}{\\mu}(I-UU^{T}).\n\\end{equation}\n\\end{lemma}\n\n\\noindent\nWe refer to the approach in this paragraph as the Nystr{\\\"o}m sketch-and-solve algorithm\nbecause it is modeled on the sketch-and-solve paradigm that originated in~\\cite{sarlos2006improved}.\n\nSee~\\cref{alg:NysSketchSolve} for a summary of the Nystr{\\\"o}m sketch-and-solve method.\nThe algorithm produces an approximate solution $\\hat x$ to the regularized linear system~\\cref{eqn:reg-linsys-intro} in time $O(T_{\\textrm{mv}}\\ell+\\ell^{2} n)$. The arithmetic cost is much faster than a direct method, which costs $O(n^3)$. It can also be faster than running CG for a long time at a cost of $O(T_{\\textrm{mv}})$ per iteration. The method looks attractive if we only consider the runtime, and yet...\n\nNystr{\\\"o}m sketch-and-solve only has one parameter, the rank $\\ell$ of the Nystr{\\\"o}m approximation, which controls the quality of the approximate solution $\\hat x$. When $\\ell \\ll n$, the method has an appealing computational profile. As $\\ell$ increases, the approximation quality increases but the computational burden becomes heavy. We will show that, alas, an accurate solution to the linear system actually requires $\\ell\\approx n$, at which point the computational benefits of Nystr{\\\"o}m sketch-and-solve evaporate completely.\n\nIn summary, Nystr{\\\"o}m sketch-and-solve is almost never the right algorithm to use. We will see that Nystr{\\\"o}m PCG generally produces much more accurate solutions with a similar computational cost.\n\n\n\\subsection{Guarantees and deficiencies}\n\\label{NysSketchSolveAnalysis}\n\nUsing \\cref{GenInvErrBnd} together with the a priori guarantee in \\cref{NysExpBound}, we quickly obtain a performance guarantee for \\cref{alg:NysSketchSolve}. \n\n\\begin{theorem}\n\\label{NysInvBnd}\nFix $p\\geq 2$, and set $\\ell = 2p - 1$. For a psd matrix $A \\in \\mathbb{S}_n^+(\\R)$, construct a randomized Nystr{\\\"o}m approximation $\\hat{A}_{\\textup{nys}}$ using \\cref{alg:RandNysAppx}. \nThen the approximation error for the inverse satisfies\n\\begin{equation}\n \\label{eq-NysInvExpErr}\n \\mathbb{E}\\big\\|(A+\\mu I)^{-1}-(\\hat{A}_{\\textup{nys}}+\\mu I)^{-1}\\big\\|\\leq \\bigg(3+\\frac{4\\textup{e}^{2}}{p}\\textup{sr}_{p}(A)\\bigg)\\frac{\\lambda_{p}}{\\mu \\cdot (\\lambda_{p}+\\mu)}.\n\\end{equation}\nDefine $x_{\\star} = (A+\\mu I)^{-1}b$, and select $\\ell = 2\\,\\lceil 1.5 \\, d_{\\textup{eff}}(\\epsilon\\mu)\\rceil+1$.\nThen the approximate solution $\\hat{x}$ computed by \\cref{alg:NysSketchSolve} satisfies\n\\begin{equation}\n \\label{eq-NysSSRelErr}\n \\E\\bigg[\\frac{\\|\\hat{x}-x_{\\star}\\|_{2}}{\\|x_{\\star}\\|_{2}}\\bigg] \\leq 26\\epsilon. \n\\end{equation}\n\\end{theorem}\nThe proof of \\cref{NysInvBnd} may be found in \\cref{section:NysSketchSolveProof}.\n\n\\cref{NysInvBnd} tells us how accurately we can hope to solve linear systems using Nystr{\\\"o}m sketch-and-solve (\\cref{alg:NysSketchSolve}). \nTo obtain relative error $\\epsilon$, we should choose $\\ell = O(d_{\\textrm{eff}}(\\epsilon\\mu))$. When $\\epsilon \\mu$ is small, we anticipate that $d_{\\textrm{eff}}(\\epsilon\\mu)\\approx n$. In this case, Nystr{\\\"o}m sketch-and-solve has no computational value. Our analysis is sharp in its essential respects, so the pessimistic assessment is irremediable.\n\n\\subsection{History}\n\nNystr{\\\"o}m sketch-and-solve has a long history in the machine learning literature. It was introduced in \\cite{williams2001using} to speed up kernel-based learning, and it plays a role in many subsequent papers on kernel methods. In this context, the Nystr{\\\"o}m approximation is typically obtained using column sampling \\cite{el2014fast,bach2013sharp,williams2001using}, which has its limitations (\\cref{RandProjvsColSamp}). More recently, Nystr{\\\"o}m sketch-and-solve has been applied to speed up approximate cross-validation \\cite{stephenson2020LowRank}.\n\nThe analysis of Nystr{\\\"o}m sketch-and-solve presented above differs from previous analysis. Prior works~\\cite{el2014fast,bach2013sharp} focus on the kernel setting, and they use properties of column sampling schemes to derive learning guarantees. In contrast, we bound the relative error for a Nystr{\\\"om} approximation based on a random projection.\nOur overall approach extends to column sampling if we replace \\cref{NysExpBound} by an appropriate analog, such as Gittens's results~\\cite{gittens2011spectral}.\n\n\n\\section{Approximating the regularized inverse}\n\\label{section:AppxRegInverse}\n\nLet us return to the regularized linear system~\\cref{eqn:reg-linsys-intro}.\nThe solution to the problem has the form $x_{\\star} = (A+\\mu I)^{-1}b$. \nGiven a good approximation $\\hat{A}$ to the matrix $A$, it is natural to ask whether $\\hat{x} = (\\hat{A} + \\mu I)^{-1} b$ is a good approximation to the desired solution $x_{\\star}$.\n\nThere are many reasons why we might prefer to use $\\hat{A}$ in place of $A$. In particular, we may be able to solve linear systems in the matrix $\\hat{A} + \\mu I$ more efficiently. On the other hand, the utility of this approach depends on how well the inverse $(\\hat{A} + \\mu I)^{-1}$ approximates the desired inverse $(A + \\mu I)^{-1}$. The next result addresses this question for a wide class of approximations that includes the Nystr{\\\"o}m approximation. \n\n\n\n\\begin{proposition}[Regularized inverses]\n\\label{GenInvErrBnd}\nConsider psd matrices $A$, $\\hat{A}\\in \\mathbb{S}_n^+(\\R)$, and assume that the difference $E = A-\\hat{A}$ is psd. Fix $\\mu > 0$. Then\n\\begin{equation}\n \\label{eq-InvInequality}\n \\|(\\hat{A}+\\mu I)^{-1}-(A+\\mu I)^{-1}\\|\\leq \\frac{1}{\\mu}\\frac{\\|E\\|}{\\|E\\|+\\mu}.\n\\end{equation}\nFurthermore, the bound~\\eqref{eq-InvInequality} is attained when $\\hat{A} = \\lfloor A\\rfloor_{\\ell}$ for $1\\leq \\ell \\leq n$.\n\\end{proposition}\nThe proof of \\cref{GenInvErrBnd} may be found in \\cref{section:GenInvErrBndPf}. It is based on~\\cite[Lemma~X.1.4]{bhatia2013matrix}.\n\n\n\\cref{GenInvErrBnd} has an appealing interpretation. When $\\|A - \\hat{A}\\|$ is small in comparison to the regularization parameter $\\mu$, then the approximate inverse $(\\hat{A} + \\mu I)^{-1}$ can serve in place of the inverse $(A + \\mu I)^{-1}$. Note that $\\|(A + \\mu I)^{-1}\\| \\leq 1\/\\mu$, so we can view~\\cref{eq-InvInequality} as a relative error bound.\n\n\n\n \n\n\\section*{Acknowledgments}\n\n\\bibliographystyle{siamplain}\n\n\\section{Proofs of additional propositions}\nIn this section we give the proofs for the propositions that are close to existing results but do not explicitly appear in the literature.\n\\subsubsection{Useful facts about Gaussian random matrices}\nIn this subsection we record some useful results about Gaussian random matrices that are necessary for the proof of \\cref{NysExpBound}. The proof of \\cref{NysExpBound} follows in \\cref{Section:NysExpBoundPf}.\n\\begin{proposition}[\\cite{halko2011finding,nakatsukasa2020fast}]\n\\label{GaussInvLemma}\nLet $G$ be $(\\ell-p)\\times \\ell$ standard Gaussian matrix with $\\ell\\geq 4$ and $2\\leq p \\leq \\ell-2$. Then\n\\begin{equation}\n \\left(\\E\\|G^{\\dagger}\\|^{2}_{F}\\right)^{1\/2} = \\sqrt{\\frac{\\ell-p}{p-1}},\n\\end{equation}\nand\n\\begin{equation}\n \\left(\\E\\|G^{\\dagger}\\|^{2}\\right)^{1\/2} \\leq \\textup{e}\\sqrt{\\frac{\\ell}{p^{2}-1}}.\n\\end{equation}\n\\end{proposition}\n\\begin{remark}\nThe first display in \\cref{GaussInvLemma} appears in \\cite{halko2011finding}, while the second display in \\cref{GaussInvLemma} is due to \\cite{nakatsukasa2020fast}.\n\\end{remark}\nWe also require one new result, which strengthens the improved version of Chevet's theorem due to Gordon \\cite{gordon1985some}.\n\\begin{proposition}[Squared Chevet]\n\\label{SquaredChevet}\nFix matrices $S\\in \\mathbb{R}^{r\\times m}$ and $T\\in \\mathbb{R}^{n\\times s}$ and let $G\\in\\mathbb{R}^{m\\times n}$ be a standard Gaussian matrix. Then\n\\[\\E\\|SGT\\|^{2} \\leq \\left(\\|S\\| \\|T\\|_{F}+\\|S\\|_{F}\\|T\\|\\right)^{2}.\\]\n\\end{proposition}\n\\iftutorial\nWe defer the proof of \\cref{SquaredChevet} to \\cref{section:SquaredChevetPf}.\n\\else\nAs the proof of Squared Chevet is rather technical and removed from the main ideas of this work, we have chosen to omit it. A full proof of the result may be found in \\cite{frangella2021randomized}.\n\\fi\n\\begin{remark}\n Chevet's theorem states that \\cite{halko2011finding}\n\\[\\E\\|SGT\\| \\leq \\|S\\| \\|T\\|_{F}+\\|S\\|_{F}\\|T\\|.\\]\n\\cref{SquaredChevet} immediately implies Chevet's theorem by H{\\\"o}lder's Inequality. \\end{remark}\n\n\\subsection{Proof of \\cref{NysExpBound}}\n\\label{Section:NysExpBoundPf}\n\\begin{proof}\nProposition 11.1 in \\cite[Sec. 11]{MartTroppSurvey} and the argument of Theorem 11.4 in \\cite[Sec. 11]{MartTroppSurvey} shows that\n\\[\\|A-\\hat{A}_{\\textrm{nys}}\\| \\leq \\|\\Sigma_{\\ell-p+1}\\|^{2}+\\|\\Sigma_{\\ell-p+1}\\Omega_{2}\\Omega_1^{\\dagger}\\|^{2}.\\]\nTaking expectations and using $\\|\\Sigma_{\\ell-p+1}\\|^{2} = \\lambda_{\\ell-p+1}$ gives\n\\[\\E\\|A-\\hat{A}_{\\textrm{nys}}\\| \\leq \\lambda_{\\ell-p+1}+\\E\\|\\Sigma_{\\ell-p+1}\\Omega_{2}\\Omega_1^{\\dagger}\\|^{2}.\\]\nUsing the law of total expectation, the second term may be bounded as follows\n\\begin{align*}\n \\E\\|\\Sigma_{\\ell-p+1}\\Omega_{2}\\Omega_1^{\\dagger}\\|^{2} &= \\E\\left(\\E_{\\Omega_{1}}\\left[\\|\\Sigma_{\\ell-p+1}\\Omega_{2}\\Omega_1^{\\dagger}\\|^{2}\\right] \\right) \\\\\n &\\overset{(a)}{\\leq}\\E\\left(\\|\\Sigma_{\\ell-p+1}\\|\\hspace{2pt}\\|\\Omega_{1}^{\\dagger}\\|_{F}+\\|\\Sigma_{\\ell-p+1}\\|_{F}\\hspace{2pt}\\|\\Omega_{1}^{\\dagger}\\| \\right)^{2} \\\\\n &\\overset{(b)}{\\leq} 2\\|\\Sigma_{\\ell-p+1}\\|^{2}\\E\\|\\Omega_{1}^{\\dagger}\\|^{2}_{F}+2\\|\\Sigma_{\\ell-p+1}\\|_{F}^{2}\\E\\|\\Omega_{1}^{\\dagger}\\|^{2}\\\\\n &\\overset{(c)}{\\leq} \\frac{2(\\ell-p)}{p-1}\\lambda_{\\ell-p+1}+\\frac{2\\textrm{e}^{2}\\ell}{p^{2}-1}\\left(\\sum_{j>\\ell-p}\\lambda_{j}\\right),\n\\end{align*}\n where in step $(a)$ we use Squared Chevet (\\cref{SquaredChevet}). In step $(b)$ we invoke the elementary identity $(a+b)^{2}\\leq 2a^{2}+2b^{2}$, and in step $(c)$ we apply the bounds from \\cref{GaussInvLemma}.\nInserting the above display into the bound for $\\E\\|A-\\hat{A}_{\\textrm{nys}}\\|$ yields\n\\[\\E\\|A-\\hat{A}_{\\textrm{nys}}\\| \\leq \\left(1+\\frac{2(\\ell-p)}{p-1}\\right)\\lambda_{\\ell-p+1}+\\frac{2\\textrm{e}^{2}\\ell}{p^{2}-1}\\left(\\sum_{j>\\ell-p}\\lambda_{j}\\right).\\]\nAs the bound above holds for any $2\\leq p\\leq \\ell-2$,\nwe may take the minimum over admissible $p$ to conclude the result.\n\\end{proof}\n\n\\subsubsection{Proof of \\cref{GenInvErrBnd}}\nWe require the following fact from \\cite[Chapter X]{bhatia2013matrix} ,\n\\begin{lemma}[\\cite{bhatia2013matrix} Lemma X.1.4.]\\label{fact-bhatia}\nLet $X,Y$ be psd matrices. Then\n\\[\\normiii{(X+I)^{-1}-(X+Y+I)^{-1}}\\leq \\normiii{Y(Y+I)^{-1}}\\]\nfor every unitarily invariant norm.\n\\end{lemma}\n\\label{section:GenInvErrBndPf}\n\\begin{proof}[Proof of \\cref{GenInvErrBnd}]\nWe first prove \\eqref{eq-InvInequality}.\nUnder the hypotheses of \\cref{GenInvErrBnd},\nwe may strengthen \\cref{fact-bhatia} by scaling the identity to deduce\n\\[\\|(\\hat{A}+\\mu I)^{-1}-(A+\\mu I)^{-1}\\| \\leq \\frac{1}{\\mu}\\|(E+\\mu I)^{-1}E\\|.\\]\n\nRecall that the function $f(t) = \\frac{t}{t+\\mu}$ is matrix monotone,\nso that $A\\preceq B$ implies $f(A)\\preceq f(B)$. As $E\\preceq \\|E\\|I$, it follows that\n\\[\\|(\\hat{A}+\\mu I)^{-1}-(A+\\mu I)^{-1}\\| \\leq \\frac{1}{\\mu}\\frac{\\|E\\|}{\\|E\\|+\\mu}.\\]\nHence we have established the desired inequality.\n\nNext we show the bound is attained when $\\hat{A} = [A]_{\\ell}$.\nApplying the Woodbury identity, we may write\n\\[([A]_{\\ell}+\\mu I)^{-1} = V_{\\ell}(\\hat{\\Lambda}_{\\ell}+\\mu I)^{-1}V_{\\ell}^{T}+\\frac{1}{\\mu}(I-V_{\\ell}V_{\\ell}^{T}).\\]\nUsing the eigendecomposition of $A = V_{\\ell}\\Lambda_{\\ell}V_{\\ell}^{T}+V_{n-\\ell}\\Lambda_{n-\\ell}V_{n-\\ell}^{T}$,\nwe obtain\n\\begin{align*}\n([A]_{\\ell}+\\mu I)^{-1} - (A+\\mu I)^{-1} &= \\frac{1}{\\mu}(I-V_{\\ell}V_{\\ell}^{T})-V_{n-\\ell}(\\Lambda_{n-\\ell}+\\mu I)^{-1}V_{n-\\ell}^{T}\\\\\n &= \\frac{1}{\\mu}V_{n-\\ell}(I-(\\Lambda_{n-\\ell}+\\mu I)^{-1})V_{n-\\ell}^{T} \\\\ &=\\frac{1}{\\mu}V_{n-\\ell}\\left(\\Lambda_{n-\\ell}(\\Lambda_{n-\\ell}+\\mu I)^{-1}\\right)V_{n-\\ell}^{T}.\n\\end{align*}\nHence\n\\[\\|(A+\\mu I)^{-1}-([A]_{\\ell}+\\mu I)^{-1}\\| = \\frac{\\lambda_{\\ell+1}}{\\mu(\\lambda_{\\ell+1}+\\mu)}.\\]\n\\end{proof}\n\\subsubsection{Proof of statements for the optimal low-rank preconditioner $P_{\\star}$}\n\\label{section:PrecondPfs}\nWe show that $P_{\\star}$ is the best symmetric positive definite preconditioner\nthat is constant off $V_{\\ell}$.\n\\begin{lemma}\nLet $\\mathcal{P} = \\{P: P = V_{\\ell}MV_{\\ell}^{T}+\\beta(I-V_{\\ell}V_{\\ell}^{T})$\\ \\textrm{where}\\ $\\beta>0$\\ \\textrm{and}\\ $M\\in \\mathbb{S}^{+}_{\\ell}(\\R)\\}$.\nWith this parametrization, define $P_\\star$ by setting $M= \\frac{1}{\\lambda_{\\ell+1}+\\mu}(\\Lambda_\\ell+\\mu I)$ and $\\beta = 1$.\nThen for any symmetric psd matrix $A$ and $\\mu \\geq 0$,\n\\begin{eqnarray}\n \\min_{P\\in\\mathcal{P}} \\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) = \\frac{\\lambda_{\\ell+1}+\\mu}{\\lambda_{n}+\\mu}, \\label{eqn-optimal-prec-value} \\\\\n P_\\star = \\argmin_{P\\in\\mathcal{P}} \\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe first prove the lefthand side of \\eqref{eqn-optimal-prec-value} is always at least as large as the righthand side,\nand then show the bound is attained by $P_{\\star}$.\nGiven $P \\in \\mathcal{P}$, we have\n\\[\nP^{-1\/2}A_{\\mu}P^{-1\/2} = V_{\\ell}M^{-1\/2}(\\Lambda_{\\ell}+\\mu I)M^{-1\/2}V_{\\ell}^{T}+\\frac{1}{\\beta}V_{n-\\ell}(\\Lambda_{n-\\ell}+\\mu I)V_{n-\\ell}^{T}.\n\\]\nFor any $1\\leq i,j\\leq n$,\n\\[\n\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2}) = \\frac{\\lambda_{1}(P^{-1\/2}A_{\\mu}P^{-1\/2})}{\\lambda_{n}(P^{-1\/2}A_{\\mu}P^{-1\/2})}\\geq \\frac{\\lambda_i(P^{-1\/2}A_{\\mu}P^{-1\/2})}{\\lambda_j(P^{-1\/2}A_{\\mu}P^{-1\/2})}.\n\\]\nFrom our expression for $P^{-1\/2}A_{\\mu}P^{-1\/2}$,\nwe see that $(\\lambda_{\\ell+1}+\\mu)\/\\gamma,(\\lambda_n+\\mu)\/\\gamma$ are eigenvalues of $P^{-1\/2}A_{\\mu}P^{-1\/2}$.\nHence for any $P\\in \\mathcal{P}$, the following inequality holds:\n\\[\n\\kappa_{2}(P^{-1\/2}A_{\\mu}P^{-1\/2})\\geq \\frac{\\lambda_{\\ell+1}+\\mu}{\\lambda_n+\\mu},\n\\]\nproving \\eqref{eqn-optimal-prec-value}.\nUsing the definition of $P_\\star$, we see\n\\begin{align*}\nP_{\\star}^{-1\/2}A_{\\mu}P^{-1\/2} &= (\\lambda_{\\ell+1}+\\mu)V_{\\ell}V_{\\ell}^{T}+V_{n-\\ell}(\\Lambda_{n-\\ell}+\\mu I)V_{n-\\ell}^{T}, \\\\\n\\kappa_2(P^{-1\/2}_{\\star}A_{\\mu}P^{-1\/2}_{\\star}) &= (\\lambda_{\\ell+1}+\\mu)\/(\\lambda_{n}+\\mu).\n\\end{align*}\n\\end{proof}\n\\iftutorial\n\\subsection{Proof of Squared Chevet}\n\\label{section:SquaredChevetPf}\nIn this subsection we provide a proof of \\cref{SquaredChevet}. The proof is based on a Gaussian comparison inequality argument, a standard technique in the high dimensional probability literature.\n\\begin{proof}\nLet \\[U = \\{S^{T}a: \\|a\\|_{2} = 1\\}\\subset \\R^{m}\\]\n\\[V = \\{Tb: \\|b\\|_{2} = 1\\}\\subset \\R^{n}\\]\nand for $u\\in U$, $v\\in V$ consider the Gaussian processes\n\\begin{equation*}\n Y_{uv} = \\langle u,Gv\\rangle +\\|S\\|\\hspace{2pt}\\|v\\|\\gamma\n\\qquad \\textup{and} \\qquad\n X_{uv} = \\|S\\|\\langle h,v \\rangle +\\|v\\|\\langle g,u\\rangle,\n\\end{equation*}\nwhere\n\\begin{itemize}\n \\item $G\\in \\R^{m\\times n}$ is a Gaussian random matrix,\n \\item $g,h$ are Gaussian random vectors in $\\R^m$ and $\\R^n$ respectively,\n \\item and $\\gamma$ is $N(0,1)$ in $\\R$.\n\\end{itemize}\nFurthermore, $G,g,h$ and $\\gamma$ are all independent.\n\nA standard calculation shows that the conditions of Slepian's lemma (\\cite[Corollary 3.12, p.~72]{ledoux2013probability}) are satisfied. Hence we conclude that\n\\begin{equation}\n\\label{eq-SlepIneq}\n \\P\\left(\\max_{u,v}Y_{uv}>t\\right)\\leq \\P\\left(\\max_{u,v}X_{uv}>t\\right).\n\\end{equation}\nWe are now ready to prove \\cref{SquaredChevet}.\nThroughout the argument below, we use the notation $X_{+} = \\max\\{X,0\\}$.\n\nWe first observe by Jensen's inequality with respect to $\\gamma$ and the variational characterization of the singular values that\n\\begin{align*}\n \\E\\max_{u,v} (Y_{uv})^{2}_{+}\n &= \\E\\max_{\\|a\\| = 1, \\|b\\| = 1} \\left( \\langle S^{T}a,GTb\\rangle+\\|S\\| \\|Tb\\|\\gamma\\right)^{2}_{+} \\\\\n &\\geq \\E_{G}\\max_{\\|a\\| = 1, \\|b\\| = 1}\\left( \\langle S^{T}a,GTb\\rangle^{2}\\right)_{+} = \\E_{G}\\|SGT\\|^{2}.\n\\end{align*}\nHence $\\E_{G}\\|SGT\\|^{2}$ is majorized by $\\E\\max_{u,v}\\hspace{2pt} (Y_{uv})^{2}_{+}.$\nFor $X_{uv}$, we note that\n\\begin{align*}\n \\E\\max_{u,v} (X_{uv})_{+}^{2}\n &\\leq\\E\\max_{u,v}X_{uv}^{2} = \\E\\max_{\\|a\\|=1,\\|b\\| = 1}\\left(\\|S\\|\\langle h,Tb \\rangle +\\|Tb\\|\\langle g,S^{T}a\\rangle\\right)^{2} \\\\\n &\\overset{(a)}{\\leq }\\E\\left(\\|S\\|^{2}\\|T^{T}h\\|^{2}+2\\|S\\|\\|T\\|\\|T^{T}h\\|\\|Sg\\|+\\|T\\|^{2}\\|Sg\\|^{2}\\right)\\\\\n &\\overset{(b)}{\\leq} \\|S\\|^{2}\\|T\\|^{2}_{F}+2\\|S\\|\\|T\\|\\|S\\|_{F}\\|T\\|_{F}+\\|T\\|^{2}\\|S\\|_{F}^{2} \\\\\n &= (\\|S\\|\\|T\\|_{F}+\\|T\\|\\|S\\|_{F})^{2},\n \\end{align*}\nwhere in step $(a)$ we expand the quadratic and use Cauchy-Schwarz. Step $(b)$ from a straightforward calculation and H{\\\"o}lder's inequality.\n\nTo conclude, we use integration by parts and \\eqref{eq-SlepIneq} to obtain\n\\begin{align*}\n\\E_{G}\\|SGT\\|^{2}\n&\\leq \\E\\max_{u,v}\\hspace{2pt} (Y_{uv})^{2}_{+} = \\int^{\\infty}_{0}t\\P\\left(\\max_{u,v}(Y_{uv})_{+}>t\\right)dt\\\\\n&= \\int^{\\infty}_{0}t\\P\\left(\\max_{u,v}Y_{uv}>t\\right)dt \\leq \\int^{\\infty}_{0}t\\P\\left(\\max_{u,v}X_{uv}>t\\right)dt \\\\\n&= \\int^{\\infty}_{0}t\\P\\left(\\max_{u,v}(X_{uv})_{+}>t\\right)dt = \\E\\max_{u,v} (X_{uv})^{2}_{+} \\\\ &\\leq (\\|S\\|\\|T\\|_{F}+\\|T\\|\\|S\\|_{F})^{2}\n\\end{align*}\ncompleting the proof.\n\\end{proof}\n\\else\n\n\\fi\n\n\\section{Additional experimental details}\n\\label{section:Experimentaldetails}\nHere we provide additional details on the experimental procedure and the methods we compared to.\n\\subsection{Ridge regression experiments}\n\\label{section:RidgeRegressionDetails}\n Most of the datasets used in our ridge regression experiments are classification datasets.\n We converted them to regression problems by using a one-hot vector encoding.\n The target vector $b$ was constructed by setting $b_i=1$ if example $i$ has the first label and 0 otherwise.\n We did no data pre-processing except on CIFAR-10,\n where we scaled the matrix by $255$ so that all entries lie in $[0,1]$.\n \nWe now give an overview of the hyperparameters of each method.\nThe R\\&T preconditioner has only one hyperparameter: the sketch size $\\ell_{\\textrm{RT}}$.\nAdaIHS has five hyperparameters: $\\rho,\\lambda_{\\rho},\\Lambda_{\\rho}, \\mu_{\\textrm{gd}}(\\rho)$, and $c_{\\textrm{gd}}(\\rho)$. The hyperparameter $\\rho\\in (0,1)$ controls the remaining four hyperparameters, which are set to the values recommended in \\cite{lacotte2020effective}.\nFor the regularization path experiments, $\\ell_{\\textrm{RT}}$ and $\\rho$ were chosen by grid search to minimize the time taken to solve the linear systems over the regularization path. We chose $m_{\\textrm{RT}}$ from the linear grid $jd$, where $j\\in\\{1,\\dots,8\\}$. Additionally, we restrict $j\\leq 4$ for Guillermo as $jd\\geq n$ when $j\\geq 5$, and hence no benefit is gained over a direct method. For AdaIHS, $\\rho$ was chosen from the linear grid $\\rho = j\\times 10^{-1}$ where $j \\in\\{1,\\dots,9\\}$. We set the initial sketch size for AdaIHS to $100$ for both sets of experiments.\n\nWe reused computation as much as possible for both R\\&T and AdaIHS, which we now detail. To construct the R\\&T preconditioner, we incur a $O(nd\\log(n)+\\ell_{\\textrm{RT}}d^{2})$ to cache the Gram matrix and pay an $O(d^{3})$ to update the preconditioner for each value of $\\mu$. In the case of AdaIHS, for each value of $\\mu$ we cache the sketch $SA$ and the corresponding Gram matrix. We then use them for the next value of $\\mu$ on the path until the adapativity criterion of the algorithm deems a new sketch necessary. For AdaIHS computing the sketch only costs $O(nd\\log(n)).$\n\nWe now give the details of the random features experiments. For Shuttle-rf we used random features corresponding to a Gaussian kernel with bandwidth parameter $\\sigma = 0.75$, we set $\\mu = 10^{-8}\/n$. For smallNORB-rf we used ReLU random features with $\\mu = 6\\times 10^{-4}$. For Higgs we normalized the features by their z-score and we used random features for a Gaussian Kernel with $\\sigma = 5$ and regularization $\\mu = 10^{-4}$. Similarly for YearMSD, we normalized the matrix by their z-score and used random features for a Gaussian kernel with $\\sigma = 8$ and $\\mu = 10^{-5}$. The sketch size for R\\&T was selected from $\\{d, 2d\\}$, to prevent the cost of the forming and applying preconditioner from becoming prohibitive. We selected the AdaIHS parameter $\\rho$ from the same grid used for the regularization path experiments. We also capped the sketch size for AdaIHS for each dataset by the sketch sized used for R\\&T. \n\nFinally, we give the details of our implementation of Nystr{\\\"o}m PCG. For both sets of experiments we used \\cref{alg:AdaRandNysAppx} initialized at $\\ell = 100$, with an error tolerance of $30\\mu$, and $q = 5$ power iterations. To avoid trivialities, the rank of the preconditioner is capped at $\\ell_{\\textrm{max}} = 0.5d$ for CIFAR-10 and $\\ell_{\\textrm{max}} = 0.4d$ for Guillermo. For the random features experiments we capped $\\ell$ at $\\ell_{\\textrm{max}} = 2000$. In the regularization path experiments, we keep track of the latest estimate $\\hat{E}$ of $\\|E\\|$, and do not compute a new Nystr{\\\"o}m approximation unless $\\hat{E}$ is larger than the error tolerance for the new regularization parameter. When we compute the new Nystr{\\\"o}m approximation, the adaptive algorithm is initialized with a target rank of twice the old one.\n\nThe values of hyperparameters used for all experiments are summarized in \\cref{t-parameters}.\n\\begin{table}[t]\n\\footnotesize\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\thead{Dataset} & \\thead{(R\\&T) sketch \\\\ size} & \\thead{AdaIHS rate} & \\thead{Initial AdaIHS \\\\sketch size} & \\thead{Initial \\\\ Nystr{\\\"o}m \\\\ sketch size} \\\\\n \\hline\n CIFAR-10 & $3d$ & $\\rho$ = 0.3 & $100$ & $100$\\\\\n \\hline\n Guillermo & $d$ & $\\rho$ = 0.3 & $100$ & $100$ \\\\\n \\hline\n shuttle-rf & $2d$ & $\\rho$ = 0.1 & $100$ & $100$ \\\\\n \\hline\n smallNORB-rf & $2d$ & $\\rho$ = 0.3 & $100$ & $100$ \\\\\n \\hline\n \\end{tabular}\n \\caption{\\label{t-parameters} Ridge regression experimental parameters.}\n\\end{center}\n\\end{table}\n\n\n\\subsection{ALOOCV}\nThe datasets were chosen so that $n$ and $d$ are both large, the challenging regime for ALOOCV. The first three datasets are binary classification problems, while SVHN has multiple classes. For SVHN we created a binary classification problem by looking at the first class vs. remaining classes.\n\nFor the large scale problems the adaptive algorithm for Nystr{\\\"o}m PCG was initialized at $\\ell_{0} = 500$ and is capped at $\\ell_{\\textrm{max}} = 4000$. We set the solve tolerances for both algorithms to $10^{-10}$. As before, we sample 100 points randomly from each dataset.\n\n\\subsection{Kernel Ridge Regression}\n\nWe converted the binary classification problem\nto a regression problem by constructing the target vector as follows:\nWe assign +1 to the first class and -1 to the second class.\nFor multi-class problems, we do one-vs-all classification; this formulation leads to multiple right hand sides, so we use block PCG for both methods. \nWe did no data pre-processing except for EMNIST, MiniBooNE, MNIST, and Santander. \nFor EMNIST and MNIST the data matrix was scaled by $255$ so that its entries lie in $[0,1]$, while for MinBooNE and Santander the features were normalized by their z-score.\nThe number of random features, $m_{\\textrm{rf}}$ from the linear grid $m_{\\textrm{rf}} = j\\times 10^{3}$ for $j = 1,\\dots,9$.\nFor adaptive Nystr{\\\"o}m PCG we capped the maximum rank for the preconditioner at $\\ell_{\\max} = \\lfloor0.1n\\rfloor$ and used a tolerance of $40$ for the ratio $\\hat{\\lambda}_{\\ell}\/{n\\mu}$ on all datasets.\n\n\\section{Additional numerical results}\n\\label{section:AddNumerics}\nHere we include some additional numerical results not appearing in the main paper.\n\\subsection{ALOOCV}\n\\cref{t-rcv1-realsim-results-part-ii} contains more details about the preconditioner and preconditioned system for the large scale ALOOCV experiments in \\cref{section:ALOOCV}. The original condition number in \\cref{t-rcv1-realsim-results-part-ii} below is estimated as follows. First we compute the top eigenvalue of the Hessian using Matlab's eigs() command, then we divide this by $\\mu$.\n\\begin{table}[H]\n\\footnotesize\n\\begin{center}\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\textbf{Dataset} & \\thead{Nystr{\\\"o}m \\\\ rank} & \\thead{Preconditioner\\\\ construction \\\\ time(s)} & \\thead{Condition \\\\ number\\\\ estimate} & \\thead{Preconditioned\\\\ condition \\\\ number\\\\ estimate} \\\\\n \\hline\n rcv1 $(\\mu = \\num{1e-4})$ & $1000$ & $19.5\\hspace{2pt}(0.523)$ & $21.6$ & $2.98\\hspace{2pt} (0.081)$\\\\\n \\hline\n rcv1 $(\\mu = \\num{1e-8})$ & $4000$ & $100.6\\hspace{2pt}(3.46)$ & $5.70\\mathrm{e}{+3}$ & $17.2\\hspace{2pt} (0.218)$ \\\\\n \\hline\n realsim $(\\mu = \\num{1e-4})$ & $3100\\hspace{2pt}(1.41\\mathrm{e}{+3})$ & $82.01 (2.04)$ & $10.0$ & $1.70 \\hspace{2pt}(0.2324)$\\\\\n \\hline\n realsim $(\\mu = \\num{1e-8})$ & $4000$ & $108.3\\hspace{2pt}(6.21)$ & $2.13\\mathrm{e}{+4}$ & $62.4\\hspace{2pt}(0.945)$ \\\\\n \\hline\n \\end{tabular}\n \\caption{\\label{t-rcv1-realsim-results-part-ii} For $\\mu = 10^{-4}$ the Hessian is well-conditioned for both datasets, so there is little value to preconditioning. For $\\mu = 10^{-8}$, the ill-conditioning of the Hessian increases significantly, making preconditioning more valuable. Furthermore, as ALOOCV uses Block PCG on at least several batches of data points, the cost of constructing the preconditioner is negligible compared to the cost of solving the linear systems (see \\cref{t-rcv1-realsim-results-part-i} in \\cref{section:ALOOCV}). }\n\\end{center}\n\\end{table}\n\n\\section{Adapative rank selection via a-posteriori error estimation}\nBelow we give the pseudocode for the algorithms used to perform adaptive rank selection using the strategy proposed in \\cref{section:ErrorRankDoubling}.\n\\label{section:AdaRankSelection}\n\\subsection{Randomized Powering algorithm}\nThe pseudo-code for estimating $\\|E\\|$ by the randomized power method is given in \\cref{alg:RandPowErrEst}\n\\begin{algorithm}[t]\n\t\\caption{Randomized Power method for estimating $\\|E\\|$}\n\t\\label{alg:RandPowErrEst}\n\t\\begin{algorithmic}[1]\n\t\t\\Require{symmetric PSD matrix $A\\in \\R^{n\\times n}$, approximate eigenvectors $U$, approximate eigenvalues $\\hat{\\Lambda}$, and number of power iterations $q$.}\n\t\t\\State $g = \\textrm{randn}(n,1)$\n\t\t\\State $v_0 = \\frac{g}{\\|g\\|_{2}}$\n\t\t \\For{$i = 1,\\dots,q$}\n\t\t \\State $v = Av_0-U(\\hat{\\Lambda}(U^{T}v_{0}))$\n\t\t \\State $\\hat{E} = v_{0}^{T}v$\n\t\t \\State $v = \\frac{v}{\\|v\\|_{2}}$\n\t\t \\State $v_{0}\\leftarrow v$\n\t\t \\EndFor\n\t\t\\Ensure{estimate $\\hat{E}$ of $\\|E\\|$}\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Adaptive rank selection algorithm}\nThe pseudocode for adaptive rank selection by a-priori error estimation is given in \\cref{alg:AdaRandNysAppx}. The code is structured to reuse use the previously computed $\\Omega$ and $Y$, resulting in significant computational savings. The error $\\|E\\|$ is estimated from $q$ iterations of the randomized power method on the error matrix $A-U\\hat{\\Lambda}U^{T}$.\n\n\\begin{algorithm}[t]\n\t\\caption{Adaptive Randomized Nystr{\\\"o}m Approximation}\n\t\\label{alg:AdaRandNysAppx}\n\t\\begin{algorithmic}[1]\n\t\t\\Require{symmetric psd matrix $A$, initial rank $\\ell_0$, maximum rank $\\ell_{\\textrm{max}}$, number of power iterations for estimating $E$, error tolerance $\\textrm{Tol}$ }\n\t\t\\State $Y = [\\hspace{5pt}], \\Omega = [\\hspace{5pt}],$ and $E = \\textrm{Inf}$\n\t\t\\State $m = \\ell_{0}$\n\t\t \\While {$E>\\textrm{Tol}$}\n\t\t \\State Generate Gaussian test matrix $\\Omega_{0}\\in \\mathbb{R}^{n\\times m}$\n\t\t \\State $[\\Omega_{0},\\sim] = \\textrm{qr}(\\Omega_{0},0)$\n\t\t \\State $Y_{0} = A\\Omega_{0}$\n\t\t \\State $\\Omega = [\\Omega\\hspace{5pt}\\Omega_{0}]$ and $Y = [Y\\hspace{5pt}Y_{0}]$\n\t\t \\State $\\nu = \\sqrt{n}\\hspace{2pt}\\textrm{eps}(\\textrm{norm}(Y,2))$\n\t\t \\State $Y_{\\nu} = Y+\\nu\\Omega,\\hspace{3pt}$\n\t\t \\State $C = \\textrm{chol}(\\Omega^{T}Y_{\\nu})$\n\t\t \\State $B = Y_{\\nu}\/C$\n\t\t \\State Compute $[U,\\Sigma,\\sim] = \\textrm{svd}(B,0)$\n\t\t \\State $\\hat{\\Lambda} = \\max\\{0,\\Sigma^{2}-\\nu I\\}$ \\Comment{remove shift}\n\t\t \\State $E = \\textrm{RandomizedPowerErrEst}(A,U,\\hat{\\Lambda},q)$\n \\Comment{estimate error}\n\t\t \\State $m\\leftarrow \\ell_{0}$, $\\ell_{0} \\leftarrow 2\\ell_{0}$\n \\Comment{double rank if tolerance is not met}\n\t\t \\If{$\\ell_{0}>\\ell_{\\textrm{max}}$}\n\t\t \\State $\\ell_{0} = \\ell_{0}-m$ \\Comment{when $\\ell_{0}>\\ell_{\\textrm{max}}$, reset to $\\ell_{0} = \\ell_{\\textrm{max}}$}\n\t\t \\State $m = \\ell_{\\textrm{max}}-\\ell_{0}$\n\t\t \\State Generate Gaussian test matrix $\\Omega_{0}\\in \\mathbb{R}^{n\\times m}$\n\t\t \\State $[\\Omega_{0},\\sim] = \\textrm{qr}(\\Omega_{0},0)$\n\t\t \\State $Y_{0} = A\\Omega_{0}$\n\t\t \\State $\\Omega = [\\Omega\\hspace{5pt}\\Omega_{0}]$ and $Y = [Y\\hspace{5pt}Y_{0}]$\n\t\t \\State $\\nu = \\sqrt{n}\\hspace{2pt}\\textrm{eps}(\\textrm{norm}(Y,2))$ \\Comment{compute final approximation and break}\n\t\t \\State $Y_{\\nu} = Y+\\nu\\Omega,\\hspace{3pt}$\n\t\t \\State $C = \\textrm{chol}(\\Omega^{T}Y_{\\nu})$\n\t\t \\State $B = Y_{\\nu}\/C$\n\t\t \\State Compute $[U,\\Sigma,\\sim] = \\textrm{svd}(B,0)$\n\t\t \\State $\\hat{\\Lambda} = \\max\\{0,\\Sigma^{2}-\\nu I\\}$\n\t\t \\State \\textbf{break}\n\t\t \\EndIf\n\t\t \\EndWhile\n\t\t\\Ensure{Nystr{\\\"o}m approximation $(U,\\hat{\\Lambda})$}\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Motivation}\n\nIn their elegant 1997 textbook on numerical linear algebra~\\cite{trefethen1997numerical},\nTrefethen and Bau write,\n\n\\begin{quotation}\n``In ending this book with the subject of preconditioners, we find ourselves at\nthe philosophical center of the scientific computing of the future...\nNothing will be more central to computational science in the next century\nthan the art of transforming a problem that appears intractable into another\nwhose solution can be approximated rapidly. For Krylov subspace matrix iterations,\nthis is preconditioning...\nwe can only guess where this idea will take us.''\n\\end{quotation}\n\nThe next century has since arrived, and one of the most fruitful developments\nin matrix computations has been the emergence of new algorithms that use randomness\nin an essential way. This paper explores a topic at the nexus of preconditioning\nand randomized numerical linear algebra. We will show how to use a\nrandomized matrix approximation algorithm to construct a preconditioner\nfor an important class of linear systems that arises throughout\ndata analysis and scientific computing.\n\n\\subsection{The Preconditioner}\n\nConsider the regularized linear system\n\\begin{equation} \\label{eqn:reg-linsys-intro}\n(A + \\mu I) x = b\n\\quad\\text{where $A \\in \\R^{n \\times n}$ is symmetric psd and $\\mu \\geq 0$.}\n\\end{equation}\nHere and elsewhere, psd abbreviates the term ``positive semidefinite.''\nThis type of linear system emerges whenever we solve a regularized\nleast-squares problem.\nWe will design a class of preconditioners for the problem~\\cref{eqn:reg-linsys-intro}.\n\nThroughout this paper, we assume that we can access the matrix\n$A$ through matrix--vector products $x \\mapsto Ax$, commonly\nknown as \\textit{matvecs}. The algorithms that we develop\nwill economize on the number of matvecs, and they may not be\nappropriate in settings where matvecs are very expensive or\nthere are cheaper ways to interact with the matrix.\n\n\nFor a rank parameter $\\ell \\in \\mathbb{N}$, the randomized Nystr{\\\"o}m approximation of $A$ takes the form\n\\begin{equation} \\label{eqn:randnys-intro}\n\\hat{A}_{\\textrm{nys}} = (A\\Omega)(\\Omega^{T}A\\Omega)^{\\dagger}(A\\Omega)^{T}\n\\quad\\text{where $\\Omega \\in \\R^{n \\times \\ell}$ is standard normal.}\n\\end{equation}\nThis matrix provides the best psd approximation of $A$ whose range coincides with the range of the sketch $A \\Omega$. The randomness in the construction ensures that $\\hat{A}_{\\textrm{nys}}$ is a good approximation to the original matrix $A$ with high probability~\\cite[Sec.~14]{MartTroppSurvey}.\n\nWe can form the Nystr{\\\"o}m approximation with sketch size $\\ell$, using $\\ell$ matvecs with $A$,\nplus some extra arithmetic. See \\cref{alg:RandNysAppx} for the implementation details. \n\nGiven the eigenvalue decomposition $\\hat{A}_{\\textrm{nys}} = U \\hat{\\Lambda} U^T$\nof the randomized Nystr{\\\"o}m approximation, we construct the Nystr{\\\"o}m preconditioner:\n\\begin{equation} \\label{eqn:precond-intro}\nP = \\frac{1}{\\hat{\\lambda}_{\\ell}+\\mu}U(\\hat{\\Lambda}+\\mu I)U^{T}+(I-UU^T).\n\\end{equation}\nIn a slight abuse of terminology, we refer to $\\ell$ as the rank of the Nystr{\\\"o}m preconditioner.\nThe key point is that we can solve the linear system $Py = c$ very efficiently,\nand the action of $P^{-1}$ dramatically reduces the condition number of the\nregularized matrix $A_{\\mu} = A + \\mu I$.\n\nWe propose to use~\\cref{eqn:precond-intro} in conjunction with the\npreconditioned conjugate gradient (PCG) algorithm. Each iteration of PCG\ninvolves a single matvec with $A$, and a single linear solve with $P$.\nWhen the preconditioned matrix $P^{-1} A_{\\mu}$ has a modest condition\nnumber, the algorithm converges to a solution of~\\cref{eqn:reg-linsys-intro}\nvery quickly. See~\\cref{alg:NysPCG} for pseudocode for Nystr{\\\"o}m PCG.\n\nThe randomized Nystr{\\\"o}m preconditioner~\\cref{eqn:precond-intro} was suggested by P.-G.~Martinsson in the survey~\\cite[Sec.~17]{MartTroppSurvey}, but it has not been implemented or analyzed.\n\n\\subsection{Guarantees}\n\n This paper contains the first comprehensive study of the preconditioner~\\cref{eqn:precond-intro}, including theoretical analysis and testing on prototypical problems from data analysis and machine learning. One of the main contributions is a rigorous method for choosing the rank $\\ell$ to guarantee good performance, along with an adaptive rank selection procedure that performs well in practice.\n\nA key quantity in our analysis is the \\textit{effective dimension} of the regularized matrix $A+\\mu I$. That is, \n\\begin{equation}\n\\label{EffDimDef}\n d_{\\textrm{eff}}(\\mu) = \\textrm{tr}\\left(A(A+\\mu I)^{-1}\\right) = \\sum_{j=1}^{n}\\frac{\\lambda_j(A)}{\\lambda_j(A)+\\mu}.\\\n\\end{equation}\nThe effective dimension measures the degrees of freedom of the problem after regularization. \nIt may be viewed as a (smoothed) count of the eigenvalues larger than $\\mu$.\nMany real-world matrices exhibit strong spectral decay, so the effective dimension is typically much smaller than the nominal dimension $n$.\nAs we will discuss, the effective dimension also plays a role in a number of machine learning papers~\\cite{el2014fast,avron2017faster,bach2013sharp,chowdhury2018randomized,lacotte2020effective} that consider randomized algorithms for solving regularized linear systems.\n\nOur theory tells us the randomized Nystr{\\\"o}m preconditioner $P$\nis successful when its rank $\\ell$ is proportional to the effective dimension.\n\n\\begin{theorem}[Randomized Nystr{\\\"o}m Preconditioner] \n\\label{MainThm}\nLet $A\\in \\mathbb{S}_{n}^+(\\mathbb{R})$ be a psd matrix, and write $A_\\mu = A+\\mu I$ where the regularization parameter $\\mu > 0$. Define the effective dimension $d_{\\textup{eff}}(\\mu)$ as in~\\cref{EffDimDef}. Construct the randomized preconditioner $P$ from~\\cref{eqn:randnys-intro,eqn:precond-intro} with rank parameter $\\ell = 2 \\,\\lceil 1.5 \\,d_{\\textup{eff}}(\\mu)\\rceil+1$. Then the condition number of the preconditioned system satisfies\n\\begin{equation} \\label{eqn:expect-cond-intro}\n\\E\\big[ \\kappa_{2}(P^{-1\/2}A_\\mu P^{-1\/2}) \\big] < 28.\n\\end{equation}\n\\end{theorem}\n\\noindent\n\\Cref{MainThm} is a restatement of \\cref{NysPCGEffDimThm}.\n\nSimple probability bounds follow from~\\cref{eqn:expect-cond-intro} via Markov's inequality. For example,\n\\[\n\\P\\big\\{ \\kappa_{2}(P^{-1\/2}A_\\mu P^{-1\/2}) \\leq 56 \\big\\} > 1\/2.\n\\]\nThe main consequence of \\cref{MainThm} is a convergence theorem for PCG with the randomized Nystr{\\\"o}m preconditioner.\n\n\\begin{corollary}[Nystr{\\\"o}m PCG: Convergence]\nConstruct the preconditioner $P$ as in \\cref{MainThm}, and condition on the event \n$\\{ \\kappa_2(P^{-1\/2}A_\\mu P^{-1\/2}) \\leq 56 \\}$.\nSolve the regularized linear system~\\cref{eqn:reg-linsys-intro} using Nystr{\\\"o}m PCG, starting with an initial iterate $x_0 = 0$. After $t$ iterations, the relative error $\\delta_t$ satisfies\n\\[\n\\delta_t \\coloneqq \\frac{\\|x_{t}-x_{\\star}\\|_{\\rm PCG}}{\\|x_{\\star}\\|_{\\rm PCG}}< 2 \\cdot \\left(0.77 \\right)^{t}\n\\quad\\text{where $A_{\\mu} x_{\\star} = b$.}\n\\]\nThe error norm is defined as $\\| u \\|_{\\rm PCG}^2 = u^T (P^{-1\/2} A_{\\mu} P^{-1\/2} ) u$.\nIn particular, $t\\geq \\lceil 3.9 \\log(2\/\\epsilon) \\rceil$ iterations\nsuffice to achieve relative error $\\epsilon$.\n\\end{corollary}\n\nAlthough \\cref{MainThm} gives an interpretable bound for the rank $\\ell$ of the preconditioner,\nwe cannot instantiate it without knowledge of the effective dimension. To address this shortcoming,\nwe have designed adaptive methods for selecting the rank in practice (\\cref{sec:adaptive-rank}).\n\nFinally, as part of our investigation, we will also develop a detailed understanding\nof Nystr{\\\"o}m sketch-and-solve, a popular algorithm in\nthe machine learning literature \\cite{el2014fast,bach2013sharp}.\nOur analysis highlights the deficiencies of Nystr{\\\"o}m sketch-and-solve relative to Nystr{\\\"o}m PCG.\n\n\\subsection{Example: Ridge Regression}\n\\label{IntroRidgeEx}\n\nAs a concrete example, we consider the $\\ell^2$ regularized least-squares problem, also known as ridge regression.\nThis problem takes the form\n\\begin{equation}\\label{eq:ridge-intro}\n\\mathrm{minimize}_{x\\in \\mathbb{R}^{d}} \\quad \\frac{1}{2n}\\|Gx-b\\|^{2}+\\frac{\\mu}{2}\\|x\\|^{2},\n\\end{equation}\nwhere $G\\in \\R^{n\\times d}$ and $b\\in \\R^{n}$ and $\\mu>0$.\nBy calculus, the solution to \\cref{eq:ridge-intro} also satisfies the regularized system of linear equations\n\\begin{equation} \n \\label{eq-RidgeRegression}\n \\left(\\frac{1}{n}G^{T}G+\\mu I\\right)x = \\frac{1}{n}G^{T}b. \n\\end{equation}\nA direct method to solve \\cref{eq-RidgeRegression} requires $O(nd^{2})$ flops,\nwhich is prohibitive when $n$ and $d$ are both large. \nInstead, when $n$ and $d$ are large, iterative algorithms, such as the conjugate gradient method (CG), become the tools of choice.\nUnfortunately, the ridge regression linear system~\\cref{eq-RidgeRegression} is often very ill-conditioned, and CG converges very slowly. \n\nNystr{\\\"o}m PCG can dramatically accelerate the solution of \\cref{eq-RidgeRegression}. \nAs an example, consider the {shuttle-rf} dataset (\\cref{section:RidgeRegression}). The matrix $G$ has dimension $43,300 \\times 10,000$, while the preconditioner is based on a Nystr{\\\"o}m approximation with rank $\\ell = 800$.\n\\Cref{fig:shuttlerfDemo} shows the progress of the residual as a function of the iteration count. Nystr{\\\"o}m PCG converges to machine precision in 13 iterations, while CG stalls.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.65\\textwidth]{lstsquaresfigs\/shuttle_rf_demo.pdf}\n \\caption{\\label{fig:shuttlerfDemo}\n \\textbf{Ridge regression: CG versus Nystr{\\\"o}m PCG.}\n For the {shuttle-rf} data set, Nystr{\\\"o}m PCG converges to machine precision in 13 iterations while CG stalls. See~\\cref{IntroRidgeEx,section:RidgeRegression}.}\n\\end{figure}\n\n\\subsection{Comparison to prior randomized preconditioners}\nPrior proposals for randomized preconditioners \\cite{avron2010blendenpik, meng2014lsrn, rokhlin2008fast}\naccelerate the solution of highly overdetermined or underdetermined least-squares problems using the sketch-and-precondition paradigm \\cite[Sec.~10]{MartTroppSurvey}.\nFor $n\\geq d$, these methods require $\\Omega(d^{3})$ \ncomputation to factor the preconditioner.\n\nIn contrast, the randomized Nystr{\\\"o}m preconditioner \napplies to any symmetric positive-definite linear system\nand can be significantly faster for regularized problems. \nSee \\cref{sec:preconditioner-comparison} more details.\n\n\\subsection{Roadmap}\n\\Cref{section:NystromApproximation} contains an overview of the Nystr{\\\"o}m approximation and its key properties. \\Cref{section:AppxRegInverse} studies the role of the Nystr{\\\"o}m approximation in estimating the inverse of the regularized matrix. We analyze the Nystr{\\\"o}m sketch-and-solve method in \\Cref{Section:NystromSketchSolve}, and we give a rigorous performance bound for this algorithm. \\Cref{section:NystromPCG} presents a full treatment of Nystr{\\\"o}m PCG, including theoretical results and guidance on numerical implementation. Computational experiments in \\Cref{section:NumericalExperiments} demonstrate the power of Nystr{\\\"o}m PCG for three different applications involving real data sets.\n\n\\subsection{Notation}\n\nWe write $\\mathbb{S}_{n}(\\R)$ for the linear space of $n\\times n$ real symmetric matrices, while $\\mathbb{S}_n^+(\\R)$ denotes the convex cone of real psd matrices. The symbol $\\preceq$ denotes the Loewner order on $\\mathbb{S}_{n}(\\R)$. That is, $A \\preceq B$ if and only if the eigenvalues of $B-A$ are all nonnegative.\nThe function $\\mathrm{tr}[\\cdot]$ returns the trace of a square matrix. The map $\\lambda_j(A)$ returns the $j$th largest eigenvalue of $A$; we may omit the matrix if it is clear. As usual, $\\kappa_2$ denotes the $\\ell^2$ condition number.\nWe write $\\|M\\|$ for the spectral norm of a matrix $M$. For a psd matrix $A$, we write $\\| u \\|_{A}^2 = u^T A u$ for the $A$-norm.\nGiven $A\\in \\mathbb{S}_{n}(\\R)$ and $1\\leq \\ell\\leq n$, the symbol $\\lfloor A\\rfloor_{\\ell}$ refers to any best rank-$\\ell$ approximation to $A$ relative to the spectral norm.\nFor $A \\in \\mathbb{S}_{n}^+(\\R)$ and $\\mu \\geq 0$, the regularized matrix is abbreviated $A_{\\mu} = A + \\mu I$. \nFor $A\\in \\mathbb{S}_{n}^{+}(\\R)$ and $\\mu>0$ effective dimension of $A_{\\mu}$ is defined as $d_{\\textrm{eff}}(\\mu) = \\textrm{tr}(A(A+\\mu I))^{-1}$.\nFor $A\\in \\mathbb{S}_{n}^{+}(\\R)$, the $p$-stable rank of $A$ is defined as $\\textrm{sr}_{p}(A) = \\lambda_{p}^{-1}\\sum^{n}_{j>p}\\lambda_{j}$.\nFor $A\\in \\mathbb{S}_{n}^{+}(\\R)$, we denote the time taken to compute a matvec with $A$ by $T_{\\textrm{mv}}$.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sect1}\n\n\n\n\\begin{figure}[htbp]\n\\input{pic3.tex}\n\\caption{Multiphase Flow and Phase Transition}\n\\label{Fig1}\n\\end{figure}\n\n\n\n\n\n\nWe are interested in the motion of the fluid particles near the boundary between the atmosphere and the ocean. We call the boundary the \\emph{air-water interface}. Since evaporation and condensation phenomena occur at the interface, we have to study multiphase flow with phase transition in order to understand air-sea interaction. This paper considers the governing equations for the motion of the fluid particles in two moving domains and the interface from an energetic point of view. We employ a generalized energetic variational approach to derive our multiphase flow systems with phase transition. Of course, this paper proposes our system as one of the models for phase transition.\n\n\nLet us first introduce fundamental notations. Let $t \\geq 0$ be the time variable, and $x( = { }^t (x_1 , x_2, x_3 ) ) \\in \\mathbb{R}^3$ the spatial variable. Fix $T >0$. Let $\\Omega \\subset \\mathbb{R}^3$ be a bounded domain with a smooth boundary $\\partial \\Omega$. The symbol $n_\\Omega = n_\\Omega (x) = { }^t (n^\\Omega_1 , n^\\Omega_2 , n^\\Omega_3 )$ denotes the unit outer normal vector at $x \\in \\partial \\Omega$. Let $\\Omega_A (t) (= \\{ \\Omega_A (t) \\}_{0 \\leq t < T} )$ be a bounded domain in $\\mathbb{R}^3$ with a moving boundary $\\Gamma (t)$. Assume that $\\Gamma (t) (= \\{ \\Gamma (t) \\}_{0 \\leq t < T})$ is a smoothly evolving surface and is a closed Riemannian 2-dimensional manifold. The symbol $n_\\Gamma = n_\\Gamma ( x , t ) = { }^t (n^\\Gamma_1 , n^\\Gamma_2 , n^\\Gamma_3)$ denotes the unit outer normal vector at $x \\in \\Gamma (t)$. For each $t \\in [0,T)$, assume that $\\Omega_A (t) \\Subset \\Omega$. Set $\\Omega_B (t) = \\Omega \\setminus \\overline{\\Omega_A (t)}$. It is clear that $\\Omega = \\Omega_A (t) \\cup \\Gamma (t) \\cup \\Omega_B (t)$ (see Figure \\ref{Fig1}). Set\n\\begin{multline}\\label{eq11}\n\\Omega_{A,T} = \\bigcup_{0< t < T} \\{ \\Omega_A (t) \\times \\{ t \\} \\},{ \\ } \\Omega_{B,T} = \\bigcup_{0< t < T} \\{ \\Omega_B (t) \\times \\{ t \\} \\},\\\\\n\\Gamma_T = \\bigcup_{0< t < T} \\{ \\Gamma (t) \\times \\{ t \\} \\},{ \\ }\\Omega_T = \\Omega \\times (0,T),{ \\ }\\partial \\Omega_T = \\partial \\Omega \\times (0,T).\n\\end{multline}\n\nIn this paper we assume that the fluid in $\\Omega_{A,T}$ is a compressible one, and that the fluid in $\\Omega_{B,T}$ is an incompressible one. Let us state physical notations. Let $\\rho_A = \\rho_A ( x , t)$, $v_A = v_A ( x , t) = { }^t (v^A_1 , v^A_2 , v^A_3 )$, $p_A = p_A (x,t)$, and $\\mu_A = \\mu_A (x,t)$, $\\lambda_A = \\lambda_A (x,t)$ be the density, the velocity, the pressure, and two viscosities of the fluid in $\\Omega_A (t)$, respectively. Let $\\rho_B = \\rho_B ( x , t)$, $v_B = v_B ( x , t) = { }^t (v^B_1 , v^B_2 , v^B_3 )$, $p_B = p_B (x,t)$, and $\\mu_B = \\mu_B (x,t)$ be the density, the velocity, the pressure, and the viscosity of the fluid in $\\Omega_B (t)$, respectively. Let $v_S = v_S (x , t) = { }^t (v^S_1 , v^S_2 , v^S_3 )$ be the motion velocity of the evolving surface $\\Gamma (t)$. The symbol $\\rho_0 >0$ denotes the density of the interface $\\Gamma (t)$, and $p_0 \\in \\mathbb{R} \\setminus \\{ 0 \\}$ denotes the surface tension (coefficient) at $x \\in \\Gamma (t)$. We assume that $\\rho_A$, $v_A$, $p_A$, $\\mu_A$, $\\lambda_A$, $\\rho_B$, $v_B$, $p_B$, $\\mu_B$, $v_S$ are smooth functions in $\\mathbb{R}^4$, and that $\\rho_0$, $p_0$ are constants. \n\n\nLet us explain the key assumptions of mathematical modeling of multiphase flow systems with phase transition. Let $r_A, r_B \\in C^\\infty (\\mathbb{R}^4)$. We assume that\n\\begin{equation}\\label{eq12}\n\\begin{cases}\n{\\rm{div}} v_B =0 & \\text{ in }\\Omega_{B,T},\\\\\nv_B \\cdot n_\\Omega =0 & \\text{ on }\\partial \\Omega_T,\\\\\nv_A \\cdot n_\\Gamma = r_A v_S \\cdot n_\\Gamma & \\text{ on } \\Gamma_T,\\\\\nv_B \\cdot n_\\Gamma = r_B v_S \\cdot n_\\Gamma & \\text{ on } \\Gamma_T.\n\\end{cases}\n\\end{equation}\nThe condition ${\\rm{div}} v_B =0$ means the incompressibility condition of the fluid in $\\Omega_{B,T}$, and $v_B \\cdot n_\\Omega =0$ means that fluid particles do not go out of the domain $\\Omega$. From $v_A \\cdot n_\\Gamma = r_A v_S \\cdot n_\\Gamma$ and $v_B \\cdot n_\\Gamma = r_B v_S \\cdot n_\\Gamma$, we have $(v_A - v_B) \\cdot n_\\Gamma = (r_A - r_B) v_S \\cdot n_\\Gamma$. This condition means \\emph{phase transition} in this paper. We call the condition a \\emph{jump condition} (see Figure \\ref{Fig1}, see \\eqref{eq31}, \\eqref{eq37} for $m_A,m_B$). In general, we assume several jump conditions when we make models for multiphase flow with phase transition. This paper uses only our jump condition to derive our systems. See Slattery-Sagis-Oh \\cite{SSO07} for several jump conditions on interfacial phenomena.\n\n\nThis paper has three purposes. The first one is to make the following inviscid model for multiphase flow with phase transition:\n\\begin{equation}\\label{eq13}\n\\begin{cases}\n{\\displaystyle{D_t^A \\rho_A + ({\\rm{div}} v_A) \\rho_A = {\\rm{div}}\\left\\{ \\frac{\\rho_0}{p_0}p_A v_A \\right\\} }}& \\text{ in } \\Omega_{A,T},\\\\\n{\\displaystyle{D_t^B \\rho_B = {\\rm{div}}\\left\\{ \\frac{\\rho_0}{p_0}p_B v_B \\right\\} }}& \\text{ in } \\Omega_{B,T},\\\\\n{\\displaystyle{\\rho_A D_t^A v_A + {\\rm{grad}}p_A = - {\\rm{div}}\\left\\{ \\frac{\\rho_0}{p_0}p_A v_A \\right\\} v_A }}& \\text{ in } \\Omega_{A,T},\\\\\n{\\displaystyle{\\rho_B D_t^B v_B + {\\rm{grad}}p_B = - {\\rm{div}}\\left\\{ \\frac{\\rho_0}{p_0}p_B v_B \\right\\} v_B }}& \\text{ in } \\Omega_{B,T},\\\\\np_0 H_\\Gamma n_\\Gamma - r_A p_A n_\\Gamma + r_B p_B n_\\Gamma ={ }^t(0,0,0) & \\text{ on } \\Gamma_T,\n\\end{cases}\n\\end{equation}\nwith \\eqref{eq12}. Here $D_t^A f = \\partial_t f + (v_A,\\nabla ) f $, $D_t^B f = \\partial_t f + (v_B, \\nabla ) f $, $(v_A , \\nabla ) f = v^A_1 \\partial_1 f + v^A_2 \\partial_2 f + v^A_3 \\partial_3 f$, $(v_B , \\nabla ) f = v^B_1 \\partial_1 f + v^B_2 \\partial_2 f + v^B_3 \\partial_3 f$, ${\\rm{div}}v_A = \\nabla \\cdot v_A$, ${\\rm{div}}v_B = \\nabla \\cdot v_B$, ${\\rm{grad}} f = \\nabla f$, $\\nabla = { }^t (\\partial_1 , \\partial_2 , \\partial_3)$, $\\partial_i = \\partial\/{\\partial x_i}$, and $\\partial_t = \\partial\/{\\partial t}$. The symbol $H_\\Gamma = H_\\Gamma (x , t )$ denotes the \\emph{mean curvature in the direction} $n_\\Gamma$ defined by $H_\\Gamma = - {\\rm{div}}_\\Gamma n_\\Gamma = - (\\partial_1^\\Gamma n_1^\\Gamma + \\partial_2^\\Gamma n^\\Gamma_2 + \\partial_3^\\Gamma n^\\Gamma_3)$, where $\\partial^\\Gamma_i f := \\sum_{j=1}^3(\\delta_{ij} - n^\\Gamma_i n^\\Gamma_j ) \\partial_j f$. More precisely, under the restriction \\eqref{eq12} we apply an energetic variational approach to derive system \\eqref{eq13}. See subsection \\ref{subsec41} for details.\n\n\nThe second one is to make the following viscous model for multiphase flow with phase transition:\n\\begin{equation}\\label{eq14}\n\\begin{cases}\n{\\displaystyle{D_t^A \\rho_A + ({\\rm{div}} v_A) \\rho_A = - \\frac{\\rho_0}{p_0} {\\rm{div}}( \\mathcal{T}_A v_A) }}& \\text{ in } \\Omega_{A,T},\\\\\n{\\displaystyle{D_t^B \\rho_B = - \\frac{\\rho_0}{p_0} {\\rm{div}} ( \\mathcal{T}_B v_B )}} & \\text{ in } \\Omega_{B,T},\\\\\n{\\displaystyle{\\rho_A D_t^A v_A = {\\rm{div}} \\mathcal{T}_A + \\frac{\\rho_0}{p_0} {\\rm{div}}( \\mathcal{T}_A v_A) v_A }}& \\text{ in } \\Omega_{A,T},\\\\\n{\\displaystyle{\\rho_B D_t^B v_B = {\\rm{div}} \\mathcal{T}_B + \\frac{\\rho_0}{p_0} {\\rm{div}}( \\mathcal{T}_B v_B)v_B }}& \\text{ in } \\Omega_{B,T},\\\\\np_0 H_\\Gamma n_\\Gamma + r_A \\widetilde{\\mathcal{T}}_A n_\\Gamma - r_B \\widetilde{\\mathcal{T}}_B n_\\Gamma= { }^t( 0,0,0) & \\text{ on } \\Gamma_T,\n\\end{cases}\n\\end{equation}\nwith\n\\begin{equation}\\label{eq15}\n\\begin{cases}\n{\\rm{div}} v_B =0 & \\text{ in }\\Omega_{B,T},\\\\\nv_B = { }^t (0,0,0) & \\text{ in }\\partial \\Omega_T,\\\\\nv_A \\cdot n_\\Gamma = r_A v_S \\cdot n_\\Gamma & \\text{ on } \\Gamma_T,\\\\\nv_B \\cdot n_\\Gamma = r_B v_S \\cdot n_\\Gamma & \\text{ on } \\Gamma_T,\\\\\nP_\\Gamma v_A = { }^t (0,0,0) & \\text{ on } \\Gamma_T,\\\\\nP_\\Gamma v_B = { }^t (0,0,0) & \\text{ on } \\Gamma_T,\n\\end{cases}\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq16}\n\\begin{cases}\n\\mathcal{T}_A =\\mathcal{T}_A (p_A , v_A) :=\\mu_A D (v_A) + \\lambda_A ({\\rm{div}} v_A)I_{3\\times 3} - p_A I_{3 \\times 3} ,\\\\\n\\mathcal{T}_B = \\mathcal{T}_B (p_B, v_B ) :=\\mu_B D (v_B) - p_B I_{3 \\times 3},\\\\ \n\\widetilde{\\mathcal{T}}_A =\\widetilde{\\mathcal{T}}_A (p_A , v_A) :=\\mu_A (n_\\Gamma \\cdot (n_\\Gamma , \\nabla ) v_A)+ \\lambda_A ({\\rm{div}} v_A) - p_A ,\\\\\n\\widetilde{\\mathcal{T}}_B = \\widetilde{\\mathcal{T}}_B (p_B, v_B ) :=\\mu_B (n_\\Gamma \\cdot (n_\\Gamma , \\nabla ) v_B) - p_B . \n\\end{cases}\n\\end{equation}\nHere $D (v_A) =\\{ { }^t(\\nabla v_A) +(\\nabla v_A) \\}\/2$ and $D (v_B) =\\{ { }^t(\\nabla v_B) +(\\nabla v_B) \\}\/2$. The symbol $I_{3\\times 3}$ denotes the $3 \\times 3$ identity matrix, and $P_\\Gamma = P_\\Gamma (x ,t)$ an \\emph{orthogonal projection to a tangent space} defined by $P_\\Gamma = I_{3 \\times 3} - n_\\Gamma \\otimes n_\\Gamma$, where $\\otimes$ is the tensor product. More precisely, under the restriction \\eqref{eq15} we apply an energetic variational approach to derive system \\eqref{eq14}. See subsection \\ref{subsec42} for details.\n\n\nThe third one is to investigate the conservative form, the conservation and energy laws of our systems \\eqref{eq13} and \\eqref{eq14}. In fact, any solution to system \\eqref{eq14} with \\eqref{eq15} satisfies that for $t_1 < t_2$,\n\\begin{multline}\\label{eq17}\n\\int_{\\Omega_A (t_2)} \\rho_A (x,t_2) { \\ } d x + \\int_{\\Omega_B (t_2)} \\rho_B (x,t_2) { \\ } d x + \\int_{\\Gamma (t_2)} \\rho_0 { \\ }d \\mathcal{H}_x^2 \\\\\n= \\int_{\\Omega_A (t_1)} \\rho_A (x,t_1) { \\ } d x + \\int_{\\Omega_B (t_1)} \\rho_B (x,t_1) { \\ } d x + \\int_{\\Gamma (t_1)} \\rho_0 { \\ }d \\mathcal{H}_x^2,\n\\end{multline}\nand\n\\begin{multline}\\label{eq18}\n\\int_{\\Omega_A (t_2)} \\frac{1}{2} \\rho_A \\vert v_A \\vert^2{ \\ }d x + \\int_{\\Omega_B (t_2)} \\frac{1}{2} \\rho_B \\vert v_B \\vert^2{ \\ }d x\\\\\n+ \\int_{t_1}^{t_2 } \\int_{\\Omega_A (t)}(\\mu_A \\vert D (v_A) \\vert^2+ \\lambda_A \\vert { \\rm{div} }v_A \\vert ^2) { \\ }d x d t + \\int_{t_1}^{t_2} \\int_{\\Omega_B (t)} \\mu_B \\vert D (v_B) \\vert^2{ \\ }d x dt\\\\\n= \\int_{\\Omega_A (t_1)} \\frac{1}{2} \\rho_A \\vert v_A \\vert^2{ \\ }d x + \\int_{\\Omega_B (t_1)} \\frac{1}{2} \\rho_B \\vert v_B \\vert^2{ \\ }d x\\\\\n+ \\int_{t_1}^{t_2} \\int_{\\Omega_B (t)} \\left( \\frac{\\rho_0}{2 p_0} {\\rm{div}}(\\mathcal{T}_B v_B) \\vert v_B \\vert^2\\right) { \\ }d x dt + \\int_{t_1}^{t_2} \\int_{\\Gamma (t)} ({\\rm{div}}_\\Gamma v_S )p_0 { \\ }d \\mathcal{H}^2_x d t\\\\\n+ \\int_{t_1}^{t_2 } \\int_{\\Omega_A (t)}\\left( ({\\rm{div}} v_A) p_A + \\frac{\\rho_0}{2 p_0} {\\rm{div}} ( \\mathcal{T}_A v_A) \\vert v_A \\vert^2\\right) { \\ }d x d t.\n\\end{multline}\nMoreover, any solution to system \\eqref{eq13} with \\eqref{eq12} satisfies \\eqref{eq17}, \\eqref{eq18} when $\\mu_A \\equiv \\lambda_A \\equiv \\mu_B \\equiv 0$, and\n\\begin{multline}\\label{eq19}\n\\int_{\\Omega_A (t_2)} \\rho_A v_A (x,t) { \\ }d x + \\int_{\\Omega_B (t_2)} \\rho_B v_B (x,t) { \\ }d x\\\\\n= \\int_{\\Omega_A (t_1)} \\rho_A v_A { \\ }d x + \\int_{\\Omega_B (t_1)} \\rho_B v_B { \\ }d x\\\\\n+ \\int_{t_1}^{t_2} \\int_{\\Gamma (t)} (p_B - p_A) n_\\Gamma { \\ }d \\mathcal{H}^2_x d t - \\int_{t_1}^{t_2} \\int_{\\partial \\Omega} p_B n_\\Omega { \\ } d \\mathcal{H}^2_x d t.\n\\end{multline}\nHere $ \\vert D (v_A) \\vert^2= D(v_A): D (v_A)$, $ \\vert D (v_B) \\vert^2= D(v_B): D (v_B)$, and $d \\mathcal{H}^2_x$ denotes \\emph{2-dimensional Hausdorff measure}. We often call \\eqref{eq17}, \\eqref{eq19}, and \\eqref{eq18}, the \\emph{law of conservation of mass}, the \\emph{law of conservation of momentum}, and the \\emph{energy equality}, respectively. We easily check that system \\eqref{eq14} satisfies the following conservative form:\n\\begin{equation*}\n\\begin{cases}\n{\\displaystyle{\\partial_t \\rho_A + {\\rm{div}}\\left( \\rho_A v_A + \\frac{\\rho_0}{p_0} \\mathcal{T}_A v_A \\right) = 0}}& \\text{ in } \\Omega_{A,T},\\\\\n{\\displaystyle{\\partial_t \\rho_B + {\\rm{div}}\\left( \\rho_B v_B + \\frac{\\rho_0}{p_0} \\mathcal{T}_B v_B \\right) = 0}}& \\text{ in } \\Omega_{B,T},\\\\\n{\\displaystyle{\\partial_t (\\rho_A v_A) + {\\rm{div}} \\left( \\rho_A v_A \\otimes v_A - \\mathcal{T}_A \\right) = 0}}& \\text{ in } \\Omega_{A,T},\\\\\n{\\displaystyle{\\partial_t (\\rho_B v_B) + {\\rm{div}}\\left( \\rho_B v_B \\otimes v_B - \\mathcal{T}_B \\right) = 0}}& \\text{ in } \\Omega_{B,T}.\n\\end{cases}\n\\end{equation*}\n\n\\begin{remark}\\label{rem11}$(\\rm{i})$ If we change \\eqref{eq12} or \\eqref{eq15} to another restriction, then we can derive another system by applying our approaches.\\\\\n$(\\rm{ii})$ If $r_A \\equiv r_B \\equiv 1 $, then we choose $\\rho_0 =0$ to derive the usual Euler and Navier-Stokes systems, or a two-phase flow system with surface tension by our approaches.\n\\end{remark}\n\nLet us explain three key ideas of deriving our multiphase flow systems with phase transition. The first point is to acknowledge the existence of the interface, that is, we assume that the density of the interface is a positive constant $\\rho_0$. The second point is to divide the jump condition $(v_A - v_B) \\cdot n_\\Gamma = (r_A - r_B) v_S \\cdot n_\\Gamma$ into $v_A \\cdot n_\\Gamma = r_A v_S \\cdot n_\\Gamma$ and $v_B \\cdot n_\\Gamma = r_B v_S \\cdot n_\\Gamma$. The third point is to make use of a generalized energetic variational approach. More precisely, we apply an energetic variational approach in order to look for functions $F_A$ and $F_B$ satisfying\n\\begin{equation*}\n\\begin{cases}\nD_t^A \\rho_A + ({\\rm{div}} v_A) \\rho_A = F_A & \\text{ in } \\Omega_{A,T},\\\\\nD_t^B \\rho_B = F_B & \\text{ in } \\Omega_{B,T},\n\\end{cases}\n\\end{equation*}\nand\n\\begin{equation*}\n\\frac{d }{d t} \\left( \\int_{\\Omega_A (t)} \\rho_A (x,t) { \\ }d x + \\int_{\\Omega_B(t)} \\rho_B (x,t) { \\ }d x + \\int_{\\Gamma (t)} \\rho_0 { \\ }d \\mathcal{H}^2_x \\right) =0 .\n\\end{equation*}\n\n\nAn energetic variational approach is a method for deriving PDEs by using the forces derived from a variation of energies. Gyarmati \\cite{Gya70} applied an energetic variational approach, which had been studied by Strutt \\cite{Str73} and Onsager \\cite{Ons31a, Ons31b}, to make several models for fluid dynamics in domains. Hyon-Kwak-Liu \\cite{HKL10} made use of their energetic variational approach to study complex fluid in domains. Koba-Sato \\cite{KS17} applied their energetic variational approach to make their non-Newtonian fluid systems in domains. Koba-Liu-Giga \\cite{KLG17} and Koba \\cite{K18} employed their energetic variational approaches to derive their fluid systems on an evolving closed surface. However, these papers \\cite{HKL10,KS17,KLG17,K18} did not consider multiphase flow. This paper improves and modifies their methods in \\cite{HKL10,KS17,KLG17,K18} to derive our multiphase flow systems. See Section \\ref{sect4} for details.\n\n\n\nFinally, we introduce the results related to this paper. Bothe-Pr\\\"{u}ss \\cite{BP12,BP17} considered multiphase flow with interface effects. In \\cite{BP12}, they made models for multiphase flow with surface tension and viscosities by applying the \\emph{Boussinesq-Scriven law}. In \\cite{BP17}, they made use of their jump conditions to make models for multi-component two-phase flow system with phase transition, and to study the individual mass densities of an isothermal mixture of $N$-species in a domain. Although this paper does not consider surface viscosity (surface flow), this paper considers interface effects such as surface tension and phase transition. Note that our models are different from the ones in \\cite{BP12, BP17}. See also \\cite{KLG17, K18} for models for surface flow.\n\n\n\nThe outline of this paper is as follows: In Section \\ref{sect2} we state the main results of this paper. In Section \\ref{sect3} we study the law of conservation of mass for multiphase flow with phase transition. In Section \\ref{sect4} we apply an energetic variational approach to make mathematical models for multiphase flow with phase transition. In Section \\ref{sect5} we investigate the conservation and energy laws of our systems. In Appendix, we provide two useful lemmas to derive our systems.\n\n\n\n\\section{Main Results}\\label{sect2}\nWe first introduce the transport theorems. Then we state the main results.\n\\begin{definition}[$\\Omega_T$ is flowed by the velocity fields $(v_A,v_B,v_S)$]\\label{def21}{ \\ }\\\\\n$(\\rm{i})$ We say that $\\Omega_{A,T}$ is \\emph{flowed by the velocity field} $v_A = v_A (x,t)$ if for each $0< t 4\\sigma$ lower than expectations from the MARLEY prediction.\n\nAs this result is discrepant with expectations, several checks were subsequently performed. First, to verify the transparency to neutrons of the lead, the lead used for the target was checked for impurities. The presence of lighter atoms in the surrounding lead may increase the neutron scattering cross section and would also decrease the density of the neutrino target. The lead is stamped as $99.99\\%$ natural lead. The density of lead used to cast the targets was measured and determined to be $11.48\\pm0.21$\\,$\\mbox{g\/cm}^3$, within uncertainty of the nominal value of $11.29$\\,$\\mbox{g\/cm}^3$. Second, the prompt neutron event rate was compared to the delivered proton beam power as a function of time. This was done to check the stability of the neutron selection efficiency during detector operations. The excess of neutron events in the prompt window was observed to follow the delivered POT exposure. Third, the PSD cut was extended to include lower energy recoils. This increases both the expected number of detected neutrons and the sensitivity to lower energy neutrons. Extending the PSD cut down to 50~keV increased the number of expected NINs by a factor of 1.43. However, these lower energy recoil events were not included in the original fit as the uncertainty on their selection efficiency is large. Fitting this low recoil energy sideband gave a NIN rate consistent with that observed in the analysis sample.\n\n\n\\begin{figure}[!bt]\n\\centering\n\\includegraphics[width=\\linewidth]{images\/combinedFit.png}\n\\caption{Likelihood profiles from the lead neutrino cube and Eljen cell detector updated analysis. A value of 1 corresponds to the predicted signal from the MARLEY event generator.}\n\\label{fig:LL}\n\\end{figure}\n\nThe analysis of the Eljen cell data in reference~\\cite{akimov2017} has been updated in three ways. The previous analysis assumed that the emitted NINs followed an evaporative neutron spectrum, and assumed the spectra of neutron emission events of all multiplicities were identical. The updated analysis has used MARLEY to generate a neutron energy distribution that depends on the multiplicity of the event along with new calculations of the number of neutrinos produced per incident POT from reference~\\cite{PhysRevD.106.032003}. Finally, time broadening of the beam-related neutron population was incorporated into the fit and allowed to float, as was done in the analysis of the lead neutrino cube, to incorporate dispersion effects due to time-of-flight varying with incident neutron energy. The influence of this additional uncertainty is detailed in supplementary materials~\\cite{nubesSupMat}.\n\nCombining the results yields a cross section scaling factor of $0.29^{+0.17}_{-0.16}$ relative to predictions from MARLEY (see Tab.~\\ref{marleyTable}) from a joint fit. For the neutrino cube sample, this corresponds to 100$^{+57}_{-54}$ events, consistent with the NIN fit using only data from this detector.\n\nThe lead neutrino cube was designed to measure the ${}^{nat}\\mbox{Pb}(\\nu_e,e^{-}+\\mbox{X}n)$ cross section using electron neutrinos produced at the SNS. The origin of the observed reduction in the signal expectations is unknown. It should be noted that while the suppression is observed in the measured results compared to the MARLEY model, the current result cannot provide meaningful data on charged-current events on lead without neutron emission, as the detector is not designed to measure the emitted electron.\n\nOne possible explanation could be that the inclusive charged-current cross section is lower than expected for lead, affecting both neutron emitting and non-neutron emitting interactions. This could be tested by a dedicated detector measuring the electromagnetic energy deposited in charged-current events on lead. This can be done within COHERENT, and there are also external plans (see reference~\\cite{pidar}). Additionally, a detector measuring the electromagnetic component of charged-current neutrino interactions on ${}^{127}$I has collected data for several years at the SNS, and may help to test theoretical predictions.\n\nAnother possibility is that the neutrons emitted by the charged-current interaction have energies lower than theoretically predicted in MARLEY. If this is the case, the sensitivity of the HALO experiment would be unaffected by the observed reduction in the charged-current cross section. However, there are no models predicting the emission of low energy neutrons from charged-current events on lead. A measurement of the NIN cross section with a capture-gated detector would test this possibility.\n\nMeasuring the NIN cross section on a lighter target may also provide understanding for the current result. In 2017, a similar detector to the lead neutrino cube was deployed to the SNS, seeking to measure the NIN cross section on iron with a $\\sim$700-kg target. The results described herein reduce the likelihood that the iron neutrino cube will successfully measure the NIN cross section, but analysis is still under way on the data it has collected. \n\nThere are several implications of a reduced lead NIN cross section for existing experiments. For CEvNS detectors at the SNS and ESS, this measurement impacts the expected backgrounds and design of detector shielding. This result has negative implications for HALO's potential to study supernova neutrinos. However, the HALO data continues to be vital for understanding the next galactic core-collapse supernova as HALO is sensitive to the $\\nu_e$ component of the supernova flux, while most large detectors are primarily sensitive to the $\\bar{\\nu}_e$ component through inverse beta decay.\n\nA separate analysis of the 2$n$ NIN emission cross section on lead is planned using the data collected by the lead neutrino cube. While the rate of observed events is expected to be lower, steady-state backgrounds are also significantly reduced.\n\n\\textit{Conclusion:} Five years of data have been recorded and analyzed to study NINs produced from electron neutrino charged-current interactions on lead at the SNS. Combining the results of this analysis with an updated analysis of the Eljen cell detector yields a cross section suppressed by a factor of $0.29^{+0.17}_{-0.16}$ compared to the MARLEY prediction. The cause of the apparent reduction in the NIN cross section is unknown, but future experiments will help to determine its origin. Within COHERENT, updated measurements of the neutrino flux with a heavy-water detector will improve systematic uncertainties on the existing measurements \\cite{COHERENT:2021xhx}, and measurements of charged-current cross sections on other targets may help to determine whether a similar suppression is seen for other nuclei.\n\n\\textit{Acknowledgements:} The COHERENT collaboration would like to thank the Duke Machine Shop for its help casting the lead targets used for this detector, Steven Gardiner for his assistance in adapting MARLEY for use with iron and lead, and the authors of reference~\\cite{wakasa2012} for providing the ${}^{208}\\mbox{Pb}(p,n)$ data used for generating predictions with MARLEY.\n\nThe COHERENT collaboration acknowledges the Kavli Institute at the University of Chicago for Eljen cell detector contributions. The COHERENT collaboration acknowledges the generous resources provided by the ORNL Spallation Neutron Source, a DOE Office of Science User Facility, and thanks Fermilab for the continuing loan of the CENNS-10 detector. We also acknowledge support from the Alfred~P. Sloan Foundation, the Consortium for Nonproliferation Enabling Capabilities, the National Science Foundation, the Korea National Research Foundation (NRF 2022R1A3B1078756), and the U.S. Department of Energy, Office of Science. Laboratory Directed Research and Development funds from ORNL also supported this project. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This research used the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility. The work was supported by the Ministry of Science and Higher Education of the Russian Federation, Project Fundamental properties of elementary particles and cosmology No. 0723-2020-0041.\n\n\\nocite{*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\\label{sec:intro}\n\t\t\n\n\tIn variational calculations of molecular rotation-vibration energies and wavefunctions, a matrix representation\n\tof the molecular rotation-vibration Hamiltonian, constructed in terms of suitable basis functions, is diagonalized\n\tnumerically. It is well known that this type of calculation can be facilitated by the introduction of symmetrized\n\tbasis functions, i.e., basis functions that generate irreducible representations of a symmetry group for the molecule\n\tin question (see, for example, Ref.~\\cite{98BuJexx}). Even though the calculation of energies and wavefunctions\n\tcan, in principle, be carried out without the use of symmetry, the subsequent simulation of molecular spectra, involving\n\tthe computation of transition intensities, would be impossible in practice without the consideration of symmetry~\\cite{18ChJeYu}.\n\tWe implement the symmetry information in the variational calculation by defining groups of matrices that are\n\tisomorphic or homomorphic to the original symmetry group of the molecule and that are associated with the various\n\tirreducible representations (irreps) of this symmetry group~\\cite{98BuJexx}. When suitable matrix group elements are known, projection\n\toperator techniques are employed to obtain the desired symmetrized basis functions for the variational calculation~\\cite{98BuJexx}.\n\tIn a recent paper~\\cite{18ChJeYu}, we described an example of such a symmetry adaption, in that we\n\tpresented character tables and irreducible representation transformation matrices for \\Dh{n} groups with arbitrary $n$-values.\n\tWith these results, we could practically utilize the linear-molecule symmetry properties described by the \\Dh{\\infty} point group\n\tin numerical calculations with the variational nuclear motion program {{\\sc TROVE}}~\\cite{07YuThJe.method}, the acetylene molecule $^{12}$C$_2$H$_2$\\ serving as an example.\n\tIn the present work, we report an analogous analysis for the molecular symmetry (MS) group~{\\itshape\\bfseries G}$_{36}$\\ and the extended molecular symmetry (EMS) group {\\itshape\\bfseries G}$_{36}$(EM) \\cite{98BuJexx} of the ethane molecule H$_3$$^{12}$C$^{12}$CH$_3$, shown in \\fig{fig:ethane_equilibrium}, with the aim of facilitating the solution of the rotation-vibration Schr\\\"{o}dinger equation for ethane and related molecules.\n\n\n\nThe MS group {\\itshape\\bfseries G}$_{36}$\\ has been the subject of a number of studies (see, for example, \\citep{81SmBexx,12CaAlSe}). The EMS group {\\itshape\\bfseries G}$_{36}$(EM) was studied in detail by Di Lauro and Lattanzi~\\citep{10LaDixx,05LaDixx}. Examples of the application of the {\\itshape\\bfseries G}$_{36}$(EM) group include studies of the rotation-torsional spectra of various ethane-type molecules \\citep{80Hougen,99LaLaxx,96LaLaVa,04LaDiHo,06LaDiHo}. {\\sc TROVE}\\ (Theoretical ROVibrational Energies) \\cite{07YuThJe.method,15YaYuxx.method} is a general variational program for computing ro-vibrational spectra and properties for small to medium-size polyatomic molecules of arbitrary structure. It has been applied to a large number of polyatomic species \\cite{09YuBaYa.NH3,11YaYuJe.H2CO,14SoHeYu.PH3,15SoAlTe.PH3,14UnYuTe.SO3,15AlYuTe.H2CO,14YuTexx.CH4,15YaYuxx.method,15AlOvPo.H2O2,15OwYuYa.CH3Cl,15OwYuYa.SiH4,15AdYaYuJe.CH3,15OwYuTh.NH3,16AlPoOv.HOOH,16UnTeYu.SO2,16OwYuYa,16OwYuYa1.CH4,18OwYaTh,18MaYaTe,15OwYuYa1.SiH4,19CoYuKo,19MaChYa}, most of which have considerable symmetry [e.g., with molecular symmetry groups~\\cite{98BuJexx} such as \\Cv{3}(M), \\Dh{n}(M) and {\\itshape\\bfseries T}$_{\\rm d}$(M)]. TROVE is an efficient computer program for simulating hot spectra of polyatomic molecules; it is one of the main tools of the ExoMol project \\cite{12TeYuxx.db,17TeYuxx}. The most recent updates of TROVE have been reported in Refs.~\\citenum{17TeYuxx,GAIN}. TROVE uses an automatic approach for constructing a symmetry-adapted basis set to be used in setting up a matrix representation of the molecular rotation-vibration Hamiltonian. As mentioned above, we require towards this end, for each irrep of the symmetry group in question~\\cite{98BuJexx}, a group of matrices constituting that irrep. However, for {\\itshape\\bfseries G}$_{36}$\\ or {\\itshape\\bfseries G}$_{36}$(EM), these matrices have not been available in the literature thus far. The present work aims at describing fully the groups {\\itshape\\bfseries G}$_{36}$\\ and {\\itshape\\bfseries G}$_{36}$(EM) and, in particular, determining matrix groups that define the irreps. The matrix groups obtained are used for symmetrizing the potential energy surface and the rotation-vibration basis functions for ethane.\n\n\n\n\t\\begin{figure}[htbp!]\n\t\t\\centering\n\t\t\\chemfig{{C}([:120]-{H})([:-100]<{H})([:-130]<:{H})-C([:-60]-H)([:80]<:H)([:50]] (0,0.2) -- (0.5,0.2) node[above] {$R$} -- (1,0.2);\n\\draw[<->] (-0.2,-0.1) -- ++ (120:0.5) node[below left] {$r$} -- ++(120:0.5);\n\\draw [domain=180:240] plot ({1 + 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[below] {$\\alpha$};\n\\draw [domain=240:300] plot ({1 + 0.3*cos(\\x)}, {0.3*sin(\\x)}) ;\n\\end{tikzpicture}\n\\caption{Representative members of three of the vibrational coordinate subsets. Here $R$ is the \\ce{C-C} bond length, $r$ is one of the six C--H$_k$ bond lengths $r_k$, and $\\alpha$ is one of the six $\\angle$({H$_k$-C-C}) bond angles $\\alpha_k$.}\n\\label{fig:three_vibrational_classes}\n\\end{figure}\nThe last vibrational subset is obtained form\nsix dihedral angles\n$\\theta_{12}$, $\\theta_{23}$, $\\theta_{31}$,\n$\\theta_{45}$, $\\theta_{56}$, $\\theta_{64}$,\none of which is labelled $\\theta$ in \\fig{fig:dihedrals}.\n$\\theta_{ij}$ is the angle between\nthe \\ce{H$_i$-C-C} and \\ce{H$_j$-C-C} planes, where protons $i$ and $j$ belong to the same \\ce{CH3} group. Only four of the six angles are linearly independent due to the constraints\n$\\theta_{12} + \\theta_{23} + \\theta_{31}$ $=$\n$\\theta_{45} + \\theta_{56} +\\theta_{64}$ $=$\n$2 \\pi$. The positive directions of rotation for the $\\theta_{ij}$ angles\nare from proton 1 $\\rightarrow$ 2 $\\rightarrow$ 3 for\nthe one \\ce{CH3} group, and from proton 4 $\\rightarrow$ 5 $\\rightarrow$ 6 for\nthe other.\nThe independent coordinates constructed from the six $\\theta_{ij}$ angles are\n\\begin{equation}\n\\begin{split}\n{\\gamma}_1 &= \\frac{1}{\\sqrt{6}}(2\\theta_{23}- \\theta_{31} - \\theta_{12}), \\\\\n{\\gamma}_2 &= \\frac{1}{\\sqrt{2}}(\\theta_{31} - \\theta_{12}), \\\\\n{\\delta}_1 &= \\frac{1}{\\sqrt{6}}(2\\theta_{56} - \\theta_{64} - \\theta_{45}), \\\\\n{\\delta}_2 &= \\frac{1}{\\sqrt{2}}(\\theta_{64} - \\theta_{45}),\n\\end{split}\n\\label{eq:gamma-delta}\n\\end{equation}\nwhich transform as the $G$ representation of {\\itshape\\bfseries G}$_{36}$.\n\nTo describe the orientation of the ethane molecule with Euler angles, one approach is to attach coordinate axes on each \\ce{CH3} group. Here, the $z$ axis is the same for both and points from \\ce{C_$b$} to \\ce{C_$a$}, while the $x_a$ and $x_b$ axes point in the direction of \\ce{C_$a$}-\\ce{H1} and \\ce{C_$b$}-\\ce{H4}, respectively, when the molecule is viewed in the Newman projection of \\fig{fig:coordinate_axes}. The $y$ axes ensure that the Cartesian axes are right handed. With this construction, the $\\theta$ and $\\phi$ Euler angles describing the direction of the $z$ axis for \\ce{CH3} group are the same while the $\\chi$ angles describing the rotation about the $z$ axis are different and are denoted by $\\chi_a$ and $\\chi_b$. These increase in the counterclockwise direction due to the right hand rule. This essentially follows Section 15.4.4 of \\cite{98BuJexx} in the definition of the Euler angles according to the convention described in Section 10.1.1 of \\cite{98BuJexx}.\n\n\\begin{figure}[htbp!]\n\\centering\n\\begin{tikzpicture}[scale=1.8]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 2};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {3};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {1};\n\n\t\\draw [dashed] (0,0) -- ++ (40+20:1) circle(2pt)[fill] node[above right] {5};\n\t\\draw [dashed] (0,0) -- ++(160+20:1) circle(2pt)[fill] node[above left] {4};\n\\draw [dashed] (0,0) -- ++(280+20:1) circle(2pt)[fill] ++ (280:0.15) circle(0pt) node {6};\n\\draw[->] (0,0) -- ++ (240:0.8) node[left] {$x_a$};\n\\draw[->] (0,0) -- ++ (240+90:0.8) node[right] {$y_a$};\n\\draw[->] (0,0) -- ++ (160+20:0.8) node[above] {$x_b$};\n\t\\draw[->] (0,0) -- ++ (160+90+20:0.8) node[below] {$y_b$};\n\n\n\\end{tikzpicture}\n\t\\caption{A Newman projection of ethane, with the CH$_3$ group containing protons 1, 2, and 3, indicated by solid \\ce{C-H} bonds, being closest to the viewer. The $x$ and $y$ components of the coordinate axes attached to each \\ce{CH3} group is shown, the subscript $a$ signifying that the coordinate axes are for the \\ce{C_$a$H3} group. To ensure the coordinate system is right handed, the $z$ axis (the same for both groups) points from \\ce{C_$b$} to \\ce{C_$a$}. With this construction, the $\\theta$ and $\\phi$ Euler describing the direction of the $z$ axis are the same for \\ce{CH3} group are the same while the $\\chi$ angle describing the rotation about the $z$ axis different and are denoted by $\\chi_a$ and $\\chi_b$. These increase in the counterclockwise direction due to the right hand rule.}\n\\label{fig:coordinate_axes}\n\\end{figure}\n\nAs described in \\cite{98BuJexx}, to achieve maximum separation of the torsional and rotational motion, it is expedient to define two new coordinates from $\\chi_a$ and $\\chi_b$. We set the rotational coordinate $\\chi$ to\n\\begin{equation}\\label{eq:chichiachib}\n\t\\chi = \\frac12 ( \\chi_a + \\chi_b)\n\\end{equation}\nand hence our $x$ axis shown in \\fig{fig:dihedrals} halves the angle between \\ce{H1-C-C} and \\ce{H4-C-C} and increases in the counterclockwise direction. The torsional angle $\\tau$ could be defined as\n\\begin{equation}\\label{eq:tau=chia-chib}\n\\tau = \\chi_a - \\chi_b\n\\end{equation}\nand hence, as indicated in \\fig{fig:dihedrals}, is the angle from \\ce{H4-C-C} to \\ce{H1-C-C} in the counterclockwise direction.\nIn the TROVE calculations, we use a different choice and define $\\tau$ as the average of three dihedral angles. To define these, we form three pairs of protons $(4,1)$, $(6,2)$, and $(5,3)$ with the two members belonging to different \\ce{CH3} groups. The pairs are chosen such that the protons $i$ and $j$ in each $(i,j)$ pair form a dihedral angle $\\tau_{ij}$ of $\\pi$ in the staggered equilibrium geometry of \\fig{fig:ethane_equilibrium} and $\\tau_{ij}=0$ for the eclipsed geometry (see \\fig{fig:ethane_eclipsed}), using the labelling of \\fig{fig:ethane_labelling}. In a general instantaneous geometry, each $(i,j)$ pair defines a dihedral angle $\\tau_{ij}$ (where the positive direction of rotation for the $\\tau_{ij}$ angles\nis from proton 1 $\\rightarrow$ 2 $\\rightarrow$ 3)\nand the torsional angle is then given by a symmetric combination\n\\begin{equation}\n\\tau =\n\\frac13 ( \\tau_{41} + \\tau_{62} + \\tau_{53}).\n\n\\label{eq:taudef}\n\\end{equation}\nWith this definition, $\\tau=0$, $2\\pi\/3$ and $4\\pi\/3$ correspond to eclipsed configurations, while at $\\tau = \\pi\/3$, $\\pi$, $5\\pi\/3$ the molecule is in one of its three equilibrium geometries. The torsional angle $\\tau$ has definite transformation\nproperties under the operations of {\\itshape\\bfseries G}$_{36}$\\ (see also \\ref{sect:appenix_b}). As discussed in Section~\\ref{sec:extenedGTS}, the two coordinate-pair values ($\\tau$, $\\chi$) and ($\\tau + 2\\pi$, $\\chi + \\pi$) describe the same physical situation. However, the coordinate which $\\tau$ is based on, $\\chi_a - \\chi_b$, has a range of $4\\pi$. Although a given geometry can be described by a value of $\\tau$ in the interval $[0, 2\\pi]$, we must allow $\\tau$ to range over $[0, 4\\pi]$ to obtain a correct correlation with $\\chi_a - \\chi_b$ (see Section~\\ref{sec:extenedGTS} below).\n\n\\begin{figure}[htbp!]\n\\centering\n\\begin{tikzpicture}[scale=1.8]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 2};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {3};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {1};\n\n\\draw [dashed] (0,0) -- ++ (40:1) circle(2pt)[fill] node[above right] {5};\n\\draw [dashed] (0,0) -- ++(160:1) circle(2pt)[fill] node[above left] {4};\n\\draw [dashed] (0,0) -- ++(280:1) circle(2pt)[fill] ++ (280:0.15) circle(0pt) node {6};\n\\draw[->] (0,0) -- ++ (200:0.8) node[left] {$x$};\n\\draw[->] (0,0) -- ++ (200+90:0.8) node[right] {$y$};\n\n\\draw [domain=0:60] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[above] {$\\theta$};\n\\draw [domain=60:120] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)});\n\n\\draw[-] [domain=160:200] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[above left] {$\\tau$};\n\t\\draw[->] [domain=200:240] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)});\n\\end{tikzpicture}\n\t\\caption{A Newman projection of ethane, with the CH$_3$ group containing protons 1, 2, 3, indicated by solid \\ce{C-H} bonds, being closest to the viewer. One of the dihedral angles used in the vibrational subsets is labelled by $\\theta$ and the torsional angle is labelled by $\\tau$ and is measured in the counterclockwise direction. The $x$ axis halves the dihedral angle between the \\ce{H$_1$-C-C} and \\ce{H$_4$-C-C} planes.\n}\n\\label{fig:dihedrals}\n\\end{figure}\n\n\\begin{figure}[htbp!]\n\t\t\\centering\n\t\t\\chemfig{{\\it a}([:120]-{1})([:-100]<{3})([:-130]<:{2})-{\\it b}([:60]-4)([:-50]<5)([:-80]<:6)}\n\t\t\\caption{Ethane in the eclipsed configuration.}\n\t\t\\label{fig:ethane_eclipsed}\n\t\\end{figure}\nIn conclusion, the coordinate subsets, for which we initially diagonalize reduced Hamiltonians, are\n\\begin{enumerate}\n\\item the \\ce{C-C} bond length $R$,\n\\item six \\ce{C-H} bond lengths $r_k$, $k$ $=$ 1, 2, \\ldots, 6,\n\\item six bond angles $\\angle$({H$_k$-C-C}) $=$ $\\alpha_k$, $k$ $=$ 1, 2, \\ldots, 6,\n\\item four dihedral-angle coordinates\n${\\gamma}_1$,\n${\\gamma}_2$,\n${\\delta}_1$, and\n${\\delta}_2$,\n\\item the torsional angle $\\tau$, and\n\\item the three rotational angles $(\\theta, \\phi, \\chi)$.\n\\end{enumerate}\nThe primitive basis functions for subsets 1, 2, 3, and 5 are obtained using the Numerov-Cooley approach \\citep{07YuThJe.method,24Numerov.method,61Coxxxx.method}, while we use harmonic-oscillator eigenfunctions~\\cite{98BuJexx} for subset 4.\nThe primitive basis set 5 is obtained by solving the 1D torsional Schr\\\"{o}dinger equation\n\\begin{equation}\n\\label{eq:torsion-schr-eq}\n \\left[-\\frac{1}{2}\\frac{\\partial}{\\partial \\tau} g_{\\tau,\\tau}^{\\rm 1D}(\\tau) \\frac{\\partial}{\\partial \\tau} + V^{\\rm 1D}(\\tau) \\right] | n \\rangle = E_n | n \\rangle\n\\end{equation}\nusing the basis set constructed from the normalised, $4\\pi$-periodic Fourier series functions $\\sqrt{1\/2\\pi} \\cos(k \\tau \/ 2 )$ and $\\sqrt{1\/2\\pi} \\sin(k \\tau \/ 2 )$. Here $g_{\\tau,\\tau}^{\\rm 1D}(\\tau)$ is the purely torsional element of the TROVE kinetic-energy $g$-matrix (see \\cite{07YuThJe.method}) and $V^{\\rm 1D}(\\tau)$ is the 1-dimensional, $2\\pi\/3$-periodic torsional part of the 18D potential energy function of ethane with the remaining 17 vibrational coordinates set to their equilibrium values. For both the kinetic-energy function $g_{\\tau,\\tau}^{\\rm 1D}(\\tau)$ and the potential function $V^{\\rm 1D}(\\tau)$ we must consider $\\tau$-values in the extended interval $[0, 4\\pi]$ as detailed in Section~\\ref{sec:extenedGTS} below.\n\n\n\\subsection{Transformation of the vibrational coordinates under {\\itshape\\bfseries G}$_{36}$\\ }\n\nAs described in \\sect{sect:irrep_g36}, one can construct every operation of {\\itshape\\bfseries G}$_{36}$\\ with only four generating operations. In the {\\sc TROVE}\\ calculations, we use the generators $R_2^{(-)} = (132)(456)$, $R_2^{(+)} = (123)(456)$, $R_3^{(+)} = (14)(26)(35)(ab)^*$ and $R_4^{(-)} = (14)(25)(36)(ab)$. Then, to describe the transformation properties of a coordinate for all elements in {\\itshape\\bfseries G}$_{36}$, we need only determine these properties for the four generating operations. We have implemented in {\\sc TROVE}\\ the procedure to generate the irreducible representations of {\\itshape\\bfseries G}$_{36}$\\ based on these four group generators and the multiplication rules from Table~\\ref{tab:multiplicationtable}. The latter can be conveniently represented as the following recursive rule:\n\\begin{equation}\nT_i = T_j \\, T_k,\\\n\\end{equation}\nwhere the operations $T_i$, $T_j$ and $T_k$ ($i,j,k = 1,\\ldots,36$) are as organised in Table~\\ref{tab:recurrence_prod}. \n\n\\begin{table}[hbtp!]\n\\caption{\\label{tab:recurrence_prod} The recursive rules to generate the elements of {\\itshape\\bfseries G}$_{36}$\\ using the four generators $T_2 = R_2^{(+)} = (123)(465)$, $T_{4} = R_4^{(-)}$ = $(14)(25)(36)(ab)$, $T_7 = R_2^{(-)}$ $=$ $(132)(456)$, and $ T_{19} = R_4^{(+)}$ $=$ $(14)(26)(35)(ab)^*$. See also Table~\\protect\\ref{tab:multiplicationtable} for the class structure of {\\itshape\\bfseries G}$_{36}$\\ and Fig.~\\protect\\ref{fig:4pi} for an illustration of the effects of the generators.}\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.35}\n\\begin{tabular}{lllclll}\n\\hline\\hline\n$T_{1\\phantom{0}}=$ & $E$ & \\phantom{\\huge l} && $T_{19}=$ &\n$(14)(25)(36)(ab)$ & $=R_4^{(-)}$ \\\\\n$T_{2\\phantom{0}}=$ & $(123)(456)$ & $=R_2^{(+)}$ && $T_{20}=$ & (16)(24)(34)$(ab)$ & $= T_{7} \\, T_{21}$ \\\\\n$T_{3\\phantom{0}}=$ & $(132)(465)$ & $={T_2}^2 $ && $T_{21}=$ & (15)(26)(34)$(ab)$ & $= T_{7}\\, T_{19}$ \\\\\n$T_{4\\phantom{0}}=$ & $(14)(26)(35)(ab)^*$ & $=R_4^{(+)}$ && $T_{22}=$ & (153426)$(ab)$ & $= T_{19} \\, T_2$ \\\\\n$T_{5\\phantom{0}}=$ & $(16)(25)(34)(ab)^*$ & $=T_2 \\, T_{6}$ && $T_{23}=$ & (162435)$(ab)$ & $=T_{19} \\, T_3 $ \\\\\n$T_{6\\phantom{0}}=$ & $(15)(24)(36)(ab)^*$ & $=T_{2} \\, T_{4}$ && $T_{24}=$ & (143624)$(ab)$ & $= T_{20} \\, T_2 $ \\\\\n$T_{7\\phantom{0}}=$ & $(132)(456)$ & $=R_2^{(-)}$ && $T_{25}=$ & (152634)$(ab)$ & $= T_{20} \\, T_3$ \\\\\n$T_{8\\phantom{0}}=$ & $(123)(465)$ & $={T_7}^2 $ && $T_{26}=$ & (163524)$(ab)$ & $= T_{21} \\,T_2 $ \\\\\n$T_{9\\phantom{0}}=$ & $(465)$ & $=T_7 \\, T_{2}$ && $T_{27}=$ & (142536)$(ab)$ & $= T_{21} \\, T_3 $ \\\\\n$T_{10}=$ & (123) & $=T_7 \\, T_{3}$ && $T_{28}=$ & (23)(56)$^*$ & $=T_{19} \\, T_4 $ \\\\\n$T_{11}=$ & $(132)$ & $=T_8 \\, T_{2}$ && $T_{29}=$ & (13)(46)$^*$ & $= T_{19} \\, T_{5} $ \\\\\n$T_{12}=$ & $(456)$ & $=T_8 \\, T_{3}$ && $T_{30}=$ & (12)(45)$^*$ & $= T_{19} \\, T_{6} $ \\\\\n$T_{13}=$ & (152436)$(ab)^*$ & $=T_7 \\, T_{4}$ && $T_{31}=$ & (12)(46)$^*$ & $= T_{20} \\, T_4 $ \\\\\n$T_{14}=$ & (142635)$(ab)^*$ & $=T_{7} \\, T_{5}$ && $T_{32}=$ & (23)(45)$^*$ & $=T_{20} \\, T_{5} $ \\\\\n$T_{15}=$ & (162534)$(ab)^*$ & $=T_{7} \\, T_{6}$ && $T_{33}=$ & (13)(56)$^*$ & $= T_{20} \\, T_{6} $ \\\\\n$T_{16}=$ & (163425)$(ab)^*$ & $=T_8 \\, T_{4}$ && $T_{34}=$ & (13)(45)$^*$ & $= T_{21} \\, T_4 $ \\\\\n$T_{17}=$ & (153624)$(ab)^*$ & $=T_{8} \\, T_{5}$ && $T_{35}=$ & (12)(56)$^*$ & $= T_{21} \\, T_{5} $ \\\\\n$T_{18}=$ & (143526)$(ab)^*$ & $=T_{8} \\, T_{6}$ && $T_{36}=$ & (23)(46)$^*$ & $=T_{21} \\, T_{6} $ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nEach of the vibrational coordinates and the torsional coordinate can be expressed as a function of the nuclear Cartesian coordinates. In the following, and in \\ref{sec:appendixB}, we describe the procedure in determining the transformation of these coordinates under the {\\itshape\\bfseries G}$_{36}$\\ operations in a systematic way, although in the majority of cases the result is intuitive.\n\nThe \\ce{C-C} bond length $R$ is invariant\nunder all {\\itshape\\bfseries G}$_{36}$\\ operations. We label the six \\ce{C-H_k} bond lengths $r_k$,\nand the six bond angles $\\angle${H$_k$-C-C} $=$ $\\alpha_k$, by the generic labels $\\beta_k$, as the two subsets transform identically. The transformation properties are determined by recognizing\n that after the operation $O_i$ $\\in$ {\\itshape\\bfseries G}$_{36}$\\ has been carried out, the proton 1, say, occupies the position\npreviously occupied by the proton $k$ (and the C nucleus to which it is bound has\nmoved with it), and therefore the transformed value of the \\ce{C-H$_1$} bond length is $r_{1}^{\\prime}$ $=$ $r_k$, the original value of the \\ce{C-H$_k$} bond length, with analogous considerations\n for the bond lengths $r_2$, \\dots, $r_6$, and the $\\alpha_i$ angles.\n\nFor ($\\beta_1$, $\\beta_2$, $\\beta_3$), we obtain the following transformation properties under the generating operations:\n\\begin{center}\n\\begin{tabbing}\n\\parbox[l]{7.4truecm}{\n$\n\\begin{pmatrix}\n\\beta_1^\\prime \\\\\n\\beta_2^\\prime \\\\\n\\beta_3^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n0 & 0 & 1 \\\\\n1 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_1 \\\\\n\\beta_2 \\\\\n\\beta_3\n\\end{pmatrix}\n\t$\nfor (123)(456),}\n\\parbox[l]{8.1truecm}{\n$\n\\begin{pmatrix}\n\\beta_1^\\prime \\\\\n\\beta_2^\\prime \\\\\n\\beta_3^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_4 \\\\\n\\beta_5 \\\\\n\\beta_6\n\\end{pmatrix}\n$\nfor $(14)(26)(35)(ab)^*,$}\n\\end{tabbing}\n\\begin{tabbing}\n\\parbox[l]{7.4truecm}{\n$\n\\begin{pmatrix}\n\\beta_1^\\prime \\\\\n\\beta_2^\\prime \\\\\n\\beta_3^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n0 & 1 & 0 \\\\\n0 & 0 & 1 \\\\\n1 & 0 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_1\\\\\n\\beta_2 \\\\\n\\beta_3\n\\end{pmatrix}\n$\nfor $(132)(456)$,}\n\\parbox[l]{8.0truecm}{\n$\n\\begin{pmatrix}\n\\beta_1^\\prime \\\\\n\\beta_2^\\prime \\\\\n\\beta_3^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_4 \\\\\n\\beta_5 \\\\\n\\beta_6\n\\end{pmatrix}\n$\nfor $(14)(25)(36)(ab)$}\n\\end{tabbing}\n\\end{center}\nwhile for $(\\beta_4, \\beta_5, \\beta_6)$ they are given by\n\\begin{center}\n\\begin{tabbing}\n\\parbox[l]{7.4truecm}{\n\t$\n\\begin{pmatrix}\n\\beta_4^\\prime \\\\\n\\beta_5^\\prime \\\\\n\\beta_6^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n0 & 0 & 1 \\\\\n1 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_4 \\\\\n\\beta_5 \\\\\n\\beta_6\n\\end{pmatrix}\n$\n\tfor $(123)(456)$,}\n\\parbox[l]{8.1truecm}{\n\t$\n\\begin{pmatrix}\n\\beta_4^\\prime \\\\\n\\beta_5^\\prime \\\\\n\\beta_6^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_1 \\\\\n\\beta_2 \\\\\n\\beta_3\n\\end{pmatrix}\n$\nfor $(14)(26)(35)(ab)^*,$}\n\\end{tabbing}\n\n\\begin{tabbing}\n\\parbox[l]{7.4truecm}{\n$\n\\begin{pmatrix}\n\\beta_4^\\prime \\\\\n\\beta_5^\\prime \\\\\n\\beta_6^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n0 & 0 & 1 \\\\\n1 & 0 & 0 \\\\\n0 & 1 & 0\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_4 \\\\\n\\beta_5 \\\\\n\\beta_6\n\\end{pmatrix}\n\t$\nfor $(132)(456)$,}\n\\parbox[l]{8.0truecm}{\n$\n\\begin{pmatrix}\n\\beta_4^\\prime \\\\\n\\beta_5^\\prime \\\\\n\\beta_6^\\prime\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n0 & 0 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n\\beta_1 \\\\\n\\beta_2 \\\\\n\\beta_3\n\\end{pmatrix}\n\t$\n\tfor $(14)(25)(36)(ab)$.}\n\\end{tabbing}\n\\end{center}\n\n\nThe dihedral-angle coordinates\n$({\\gamma}_1, {\\gamma}_2, {\\delta}_1,\n{\\delta}_2)$ are mixed by the {\\itshape\\bfseries G}$_{36}$\\ operations. The transformation properties are most easily determined by using the correspondence with the transformation of the\n$r_k$ and $\\alpha_k$ coordinates.\nFor $({\\gamma}_1, {\\gamma}_2)$ we write\n\\begin{equation}\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2 \\\\\n2\\pi \\\\\n\\end{pmatrix} =\n{\\mathbf Z} \\,\n\\begin{pmatrix}\n\\theta_{12} \\\\ \\theta_{23} \\\\ \\theta_{31} \\\\\n\\end{pmatrix}\n\\end{equation}\nwith\n\\begin{equation}\n{\\mathbf Z} =\n\\begin{pmatrix}\n-\\frac{1}{\\sqrt{6}}& \\frac{2}{\\sqrt{6}} &-\\frac{1}{\n \\sqrt{6}}\\\\ -{\\frac{1}{\\sqrt{2}}}&0&{\\frac{1}{\\sqrt{2}}}\n \\\\\n1 & 1 & 1 \\\\\n\\end{pmatrix}\n\\end{equation}\nwhere we have taken into account the constraint $\\theta_{12} + \\theta_{23} + \\theta_{31}$ $=$\n$2 \\pi$ in the third row of ${\\mathbf Z}$.\n\nAfter we have carried out the operation $(123)(456)$, the protons 1, 2, and 3 are found at the positions initially occupied by the protons 3, 1, and 2, respectively, and so\nthe angles\n $\\theta_{12}$,\n$\\theta_{23}$, $\\theta_{31}$\nare permuted as follows\n\\begin{equation}\n\\begin{pmatrix}\n\\theta_{12}' \\\\ \\theta_{23}' \\\\ \\theta_{31}' \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n\\theta_{31} \\\\ \\theta_{12} \\\\ \\theta_{23} \\\\\n\\end{pmatrix} =\n{\\mathbf S} \\,\n\\begin{pmatrix}\n\\theta_{12} \\\\ \\theta_{23} \\\\ \\theta_{31} \\\\\n\\end{pmatrix}\n= {\\mathbf S}\\, {\\mathbf Z}^{-1} \\,\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2 \\\\\n2\\pi \\\\\n\\end{pmatrix}\n\\end{equation}\nwith\n\\begin{equation}\n{\\mathbf S} =\n\\begin{pmatrix}\n0 & 0 & 1 \\\\\n1 & 0 & 0 \\\\\n0 & 1 & 0 \\\\\n\\end{pmatrix}\n\\end{equation}\nand we can finally calculate the transformed values of\n$({\\gamma}_1, {\\gamma}_2)$ as\n\\begin{equation}\n\\begin{pmatrix}\n{\\gamma}_1' \\\\\n{\\gamma}_2' \\\\\n2\\pi \\\\\n\\end{pmatrix} =\n{\\mathbf Z}\\,\n\\begin{pmatrix}\n\\theta_{12}' \\\\ \\theta_{23}' \\\\ \\theta_{31}' \\\\\n\\end{pmatrix} =\n{\\mathbf Z}\\, {\\mathbf S}\\, {\\mathbf Z}^{-1} \\,\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2 \\\\\n2\\pi \\\\\n\\end{pmatrix}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\mathbf Z}\\, {\\mathbf S}\\, {\\mathbf Z}^{-1} =\n\\begin{pmatrix}\n-\\frac12 & -\\frac{\\sqrt{3}}{2} & 0 \\\\\n\\frac{\\sqrt{3}}{2} & -\\frac12 & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{pmatrix}.\n\\end{equation}\nWith the upper 2~$\\times$~2 corner of this matrix we can express the transformed values $({\\gamma}_1', {\\gamma}_2')$ in terms of $({\\gamma}_1, {\\gamma}_2)$.\n The transformation matrix for $({\\gamma}_1, {\\gamma}_2)$ for $(123)(456)$, and the ones for the other generating operations, obtained in a similar manner, are\n\\begin{center}\n\\begin{tabbing}\n\\parbox[l]{7.3truecm}{\n$\n\\begin{pmatrix}\n{\\gamma}_1^\\prime \\\\\n{\\gamma}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n-\\frac12 & -\\frac{\\sqrt{3}}{2} \\\\\n\\frac{\\sqrt{3}}{2} & -\\frac12 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2\\\\\n\\end{pmatrix}\n$ for $(123)(456)$,}\n\\parbox[l]{8.2truecm}{\n$\n\\begin{pmatrix}\n{\\gamma}_1^\\prime \\\\\n{\\gamma}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\delta}_1 \\\\\n{\\delta}_2\\\\\n\\end{pmatrix}\n$ for $(14)(26)(35)(ab)^*,$}\n\\end{tabbing}\n\\begin{tabbing}\n\\parbox[l]{7.3truecm}{\n$\n\\begin{pmatrix}\n{\\gamma}_1^\\prime \\\\\n{\\gamma}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n-\\frac12 & \\frac{\\sqrt{3}}{2} \\\\\n-\\frac{\\sqrt{3}}{2} & -\\frac12 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2\\\\\n\\end{pmatrix}\n$ for $(132)(456)$,}\n\\parbox[l]{8.2truecm}{\n$\n\\begin{pmatrix}\n{\\gamma}_1^\\prime \\\\\n{\\gamma}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\delta}_1 \\\\\n{\\delta}_2\\\\\n\\end{pmatrix}\n$ for $(14)(25)(36)(ab),$}\n\\end{tabbing}\n\\end{center}\nand those for ($\\delta_1$, $\\delta_2$) are given by\n\\begin{center}\n\\begin{tabbing}\n\\parbox[l]{7.3truecm}{\n$\n\\begin{pmatrix}\n{\\delta}_1^\\prime \\\\\n{\\delta}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n-{\\frac{1}{2}}&{-\\frac{\\sqrt{3}}{2}}\\\\ {\\frac{\n \\sqrt{3}}{2}}&-{\\frac{1}{2}}\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\delta}_1 \\\\\n{\\delta}_2\\\\\n\\end{pmatrix}\n$ for $(123)(456)$,}\n\\parbox[l]{8.2truecm}{\n$\n\\begin{pmatrix}\n{\\delta}_1^\\prime \\\\\n{\\delta}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & -1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2\\\\\n\\end{pmatrix}\n$ for $(14)(26)(35)(ab)^*,$}\n\\end{tabbing}\n\n\\begin{tabbing}\n\\parbox[l]{7.3truecm}{\n$\n\\begin{pmatrix}\n{\\delta}_1^\\prime \\\\\n{\\delta}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n-{\\frac{1}{2}}&{-\\frac{\\sqrt{3}}{2}}\\\\ {\\frac{\n \\sqrt{3}}{2}}&-{\\frac{1}{2}}\\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\delta}_1 \\\\\n{\\delta}_2\\\\\n\\end{pmatrix}\n$ for $(132)(456)$,}\n\\parbox[l]{8.2truecm}{\n$\n\\begin{pmatrix}\n{\\delta}_1^\\prime \\\\\n{\\delta}_2^\\prime \\\\\n\\end{pmatrix} =\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 1 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n{\\gamma}_1 \\\\\n{\\gamma}_2\\\\\n\\end{pmatrix}\n$ for $(14)(25)(36)(ab).$}\n\\end{tabbing}\n\\end{center}\n\n\\subsection{The extended molecular symmetry group {\\itshape\\bfseries G}$_{36}$(EM) and the transformation of the torsional coordinate}\n\\label{sec:extenedGTS}\nAs explained in Section~{15.4.4} of \\cite{98BuJexx}, our separation of the rotational and torsional degrees of freedom has led $\\chi$ and $\\tau$ being double-valued. That is, there are two sets of\n$(\\chi,\\tau)$ values associated with the same physical situation.\nThis is most straightforwardly seen\nby considering Eq.~(\\ref{eq:chichiachib}) and the three angles\n$\\chi_a$, $\\chi_b$, and $\\chi$ appearing in it. The angle $\\chi_a$ is determined entirely by the positions in space of the protons 1, 2, 3 and their carbon nucleus C$_a$; $\\chi_b$ is determined by the positions of the protons 4, 5, 6 and their carbon nucleus C$_b$; and $\\chi$ is the average of the two. Due to the $2\\pi$ periodicity of $\\chi_a$, increasing it by $2\\pi$ does not change the positions in space of the nuclei in \\ce{C_$a$H3}, however in this case $\\chi$ $\\rightarrow$ $\\chi+\\pi$ and $\\tau \\rightarrow \\tau + 2\\pi$.\nThat is, the two coordinate pairs ($\\chi$,$\\tau$) and ($\\chi+\\pi$,$\\tau+2\\pi$) describe identical physical situations.\n\nOne way of avoiding the ambiguity described above would be to use a molecule-fixed axis system with, for example, $\\chi$ $=$ $\\chi_a$. This molecule-axis system is attached to the \\ce{CH3} group with the protons 1, 2, 3, and not influenced by the other \\ce{CH3} group. A $\\chi$ coordinate chosen in this manner has no ambiguity. However, as already mentioned it is advantageous to choose $\\chi$ according to Eq.~(\\ref{eq:chichiachib}) since this choice\n(called the Internal Axis Method (IAM), see Section~{15.2.2} of \\cite{98BuJexx}) yields a particularly useful form of the rotation-torsion-vibration Hamiltonian with minimized coupling between rotation and torsion. It is desirable to keep the definition of $\\chi$ from Eq.~(\\ref{eq:chichiachib}) and find a way of treating the ambiguity. We use here the procedure first proposed by Hougen~\\cite{64Hougen} in 1964, primarily for dimethylacetylene H$_3$CCCCH$_3$, whose internal rotation is essentially free, but generally applicable to other molecules such as ethane H$_3$CCH$_3$, hydrogen peroxide HOOH, and disulfane HSSH with two identical moieties carrying out internal rotation. Recently, these ideas have been extended to molecules with two different moieties and applied to oxadisulfane (a.k.a. hydrogen thio-peroxide) HSOH~\\cite{04YAMADA,09YAMADA}, and more recently they have been applied to computation of the rotation-torsion-vibrational spectra of hydrogen peroxide HOOH in Ref.~\\cite{15AlOvPo.H2O2,16AlPoOv.HOOH}, where the HOOH kinetic energy operator was built using the $x$-axis chosen to halve the dihedral angle between the \\ce{H$_1$-O-O} and \\ce{H$_4$-O-O} planes, as well as in Ref.~\\cite{18SzViRe}.\n\nWe consider the effect of {\\itshape\\bfseries G}$_{36}$\\ operations on $\\chi_a$, described thoroughly in Section 12.1.1 of \\cite{98BuJexx}, where the axes are rotated in such a way that the coordinates of the nuclei -- measured in a space-fixed axis system -- remain the same after the operation, assuming the \\ce{C_$a$H3} group is in the equilibrium configuration. Considering the operation (123), we see that, using our convention, $\\chi_a$ becomes $\\chi_a + 4\\pi\/3$ or, equivalently, $\\chi_a - 2\\pi\/3$. This would correspond to the pair $(\\tau, \\chi)$ becoming $(\\tau + 4\\pi\/3, \\chi + 2\\pi\/3)$ or $(\\tau - 2\\pi\/3, \\chi - \\pi\/3)$, which are no longer equivalent but correspond to the same physical situation as noted before. For $(123)^3 = E$, the changes become $(\\tau + 4\\pi, \\chi + 2\\pi) = (\\tau, \\chi)$ or $(\\tau -2\\pi, \\chi - \\pi) = (\\tau + 2\\pi, \\chi + \\pi)$. These are illustrated in \\fig{fig:4pi}(A) and \\fig{fig:4pi}(F), respectively. \n\n\n\n\n\n\n\nIn order to deal with the double-valuedness of ($\\tau$, $\\chi$), we extend the symmetry description in the manner first introduced by Hougen~\\cite{64Hougen}, by extending {\\itshape\\bfseries G}$_{36}$\\ to the extended molecular symmetry group {\\itshape\\bfseries G}$_{36}$(EM) as explained in Section~15.4.4 of \\cite{98BuJexx}.\nTo appreciate the definition of {\\itshape\\bfseries G}$_{36}$(EM), we note that the\ninternal coordinates $R$, $r_k$ and $\\alpha_k$ ($k$ $=$ 1, 2, \\ldots, 6),\n${\\gamma}_1$,\n${\\gamma}_2$,\n${\\delta}_1$,\n${\\delta}_2$,\n $\\theta$, and $ \\phi$ used for ethane in the present work are all ``space-fixed'' in the sense of Section~15.4.4 in~\\cite{98BuJexx}; the instantaneous values of these coordinates can be unambiguously obtained from the instantaneous coordinate values of the nuclei in a Cartesian, space-fixed axis system~\\cite{98BuJexx}, $XYZ$ say. \\ref{sec:appendixB} explains in detail how the internal coordinates values are determined from the Cartesian coordinate values.\n\n We present here the ideas of\nHougen~\\cite{64Hougen}, using the more modern notation of\nSection~15.4.4 in~\\cite{98BuJexx}. The extension of\n{\\itshape\\bfseries G}$_{36}$\\ to {\\itshape\\bfseries G}$_{36}$(EM) involves the introduction of\n a fictitious operation $E'$ (taken to be different from the identity $E$) which, for ethane, we can think of as letting the \\ce{C_$a$H3} do a full torsional revolution relative to the \\ce{C_$b$H3}. That is, $E'$ has the effect of transforming $\\chi$ $\\rightarrow$ $\\chi+\\pi$ and $\\tau$ $\\rightarrow$ $\\tau + 2\\pi$. After the application of $(E')^2$ the molecule-fixed axis system is back where it started, and so we take $(E')^2$ $=$ $E$. $E'$ does not affect the space-fixed nuclear coordinates; it has the same effect as the identity operation\non the complete rotation-torsion-vibration wavefunction of ethane.\n\nWe now define four generating operations $a$, $b$, $c$ and $d$\nthat transform $\\chi$ and $\\tau$ unambiguously and\n have the same effect on the space-fixed\ncoordinates [$R$, $r_k$ and $\\alpha_k$ ($k$ $=$ 1, 2, \\ldots, 6),\n${\\gamma}_1$,\n${\\gamma}_2$,\n${\\delta}_1$,\n${\\delta}_2$,\n $\\theta$, and $ \\phi$]\nas the {\\itshape\\bfseries G}$_{36}$\\ operations (123), (456), (14)(26)(35)($ab$)$^*$ and (23)(56)$^*$, respectively.\n Tables~A-28 and~A-33 of~\\cite{98BuJexx} are the character tables of {\\itshape\\bfseries G}$_{36}$\\ and {\\itshape\\bfseries G}$_{36}$(EM), respectively.\n Comparison of the two tables shows that\nthe four generating operations\n$R_2^{(-)} = (132)(456)$, $R_2^{(+)} = (123)(456)$, $R_4^{(+)} = (14)(26)(35)(ab)^*$ and $R_4^{(-)} = (14)(25)(36)(ab)$\nchosen for {\\itshape\\bfseries G}$_{36}$\\ in the present work correspond to $ab$, $a^5\\, b$, $c$, and $d\\, c$, respectively.\nFollowing Section~15.4.4 of~\\cite{98BuJexx},\nwe \\textit{define} in Table~\\ref{tab:torsion_transformations}\nthe effect on $\\chi$ and $\\tau$ of the {\\itshape\\bfseries G}$_{36}$(EM)\ngenerators. The approach of~\\cite{98BuJexx} to simply postulate, by definition, the\neffect of the {\\itshape\\bfseries G}$_{36}$(EM)\ngenerators on $\\chi$ and $\\tau$ may appear arbitrary. It is not, however.\nWe show in\nFig.~\\ref{fig:4pi} how, for each of the four {\\itshape\\bfseries G}$_{36}$(EM) generators\n$ab$, $a^5\\, b$, $c$, and $d\\, c$, the effect on\n$\\chi$ and $\\tau$ can be explained as the effect of the {\\itshape\\bfseries G}$_{36}$\\ partner\nof the {\\itshape\\bfseries G}$_{36}$(EM) generator in question (Table~\\ref{tab:torsion_transformations}).\n\n\n\\begin{table}[hbtp!]\n\\caption{\\label{tab:torsion_transformations} Transformation of the torsion angle $\\tau$ and the rotation angle $\\chi$ under the generators of {\\itshape\\bfseries G}$_{36}$(EM).}\n\\begin{center}\n\\begin{tabular}{llll}\n\\hline\\hline\nTransformed $\\tau$ &Transformed $\\chi$ &\n{\\itshape\\bfseries G}$_{36}$(EM) generator & {\\itshape\\bfseries G}$_{36}$\\ generator \\\\\n\\hline\n$\\tau - 4\\pi\/3$ & $\\chi$ & $ab$ & (123)(456) \\\\\n$\\tau$ & $\\chi + 2\\pi\/3$ & $a^5\\, b$ &(132)(456) \\\\\n$2\\pi- \\tau$ &$\\chi + \\pi$ & $c$ & (14)(26)(35)($ab$)$^*$ \\\\\n$\\tau$ & $\\chi$ & $d\\, c$ &(14)(25)(36)($ab$) \\\\\n$\\tau+2\\pi$ &$\\chi + \\pi$& $E'$ & \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe group generated by the five operations\n$a$, $b$, $c$, $d$, and $E'$ has 72 elements and, as mentioned above, it is denoted\n{\\itshape\\bfseries G}$_{36}$(EM), called an extended molecular symmetry (EMS)\ngroup, and its\ncharacter table is given as Table~A-33\nof~\\cite{98BuJexx}. This character table shows that we can think of {\\itshape\\bfseries G}$_{36}$(EM) as a direct product\n\\begin{equation}\n \\text{{\\itshape\\bfseries G}$_{36}$(EM)} = \\text{{\\itshape\\bfseries G}$_{36}$} \\times \\{ E,E' \\},\n\\end{equation}\nwhere the group $\\mathcal{G}_{36}$ is constructed from\nthe generating operations $a$, $b$, $c$ and $d$; it is isomorphic to {\\itshape\\bfseries G}$_{36}$, so that it has the irreducible representations given in Table~\\ref{tab:chartab}.\nThe two-element group $\\{ E,E' \\}$ is cyclic of order 2 and has two irreps $A'$ and $A''$, both one-dimensional, with the representation matrices $1$ or $-1$, respectively, under $E'$. The irreps of {\\itshape\\bfseries G}$_{36}$(EM) are straightforwardly constructed from those of {\\itshape\\bfseries G}$_{36}$.\n Each irrep $\\Gamma$ of {\\itshape\\bfseries G}$_{36}$\\ in Table~\\ref{tab:chartab} gives rise to two irreps of {\\itshape\\bfseries G}$_{36}$(EM),\n $\\Gamma_s$ $=$ $(\\Gamma,A')$ and\n $\\Gamma_d$ $=$ $(\\Gamma,A'')$ as given in Table~{A-33} of \\cite{98BuJexx}.\n An irrep $\\Gamma_s$ has identical characters for the operations $E$ and $E'$, $\\chi_s [E']$ $=$ $\\chi_s [E]$, whereas for the irrep\n $\\Gamma_d$, $\\chi_d [E']$ $=$ $-\\chi_d [E]$.\nAs long as we pretend that $E'$ $\\ne$ $E$, we must also pretend that the coordinate values $(\\tau,\\chi)$ and $E'\\, (\\tau,\\chi)$ $=$ $(\\tau+2\\pi,\\chi+\\pi)$ describe different physical situations. As a consequence, we must allow $\\tau$ to be periodic with a period of $4\\pi$ as already mentioned in connection with Eq.~(\\ref{eq:torsion-schr-eq}). The torsional potential energy function $V^{\\rm 1D}(\\tau)$ is periodic with period $2\\pi$, $V^{\\rm 1D}(\\tau)$ $=$ $V^{\\rm 1D}(\\tau+2\\pi)$ for $\\tau$ $\\in$ $[0, 2\\pi]$, and this symmetry causes the torsional wavefunctions $| n \\rangle$\nfrom Eq.~(\\ref{eq:torsion-schr-eq}) to be either symmetric (of $\\Gamma_s$ symmetry) or antisymmetric (of $\\Gamma_d$ symmetry) under $E'$.\n\n\nWe know that in reality, $E'$ $=$ $E$, and so only functions and coordinates with $\\Gamma_s$ symmetries occur in nature. Since we can form, for example, basis functions of an allowed $\\Gamma_s$ symmetry as products of an even number of factors, each with a forbidden symmetry $\\Gamma_d'$, say, we need to consider also the $\\Gamma_d$ symmetries initially for the torsional and rotational basis functions. The final wavefunctions resulting from our theoretical calculations should be subjected to a `reality check': They must necessarily have a $\\Gamma_s$ symmetry in {\\itshape\\bfseries G}$_{36}$(EM). In particular, torsional basis functions of $d$ symmetry must be combined with rotational basis functions of $d$ symmetry to produce a torsion-rotation basis function of an allowed $s$ symmetry.\n\n\n\\begin{figure}[htbp!]\n\\centering\n\n\\parbox{3.5cm}{(A)\n\\begin{tikzpicture}[scale=1.1]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 2};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {3};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {1};\n\n\\draw [dashed] (0,0) -- ++ (40:1) circle(2pt)[fill] node[above right] {5};\n\\draw [dashed] (0,0) -- ++(160:1) circle(2pt)[fill] node[above left] {4};\n\\draw [dashed] (0,0) -- ++(280:1) circle(2pt)[fill] node[below] {6};\n\\draw[->] (0,0) -- ++ (200:1) node[left] {$x$};\n\n\\draw [domain=0:60] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[above] {$\\theta$};\n\\draw [domain=60:120] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)});\n\n\\draw [domain=160:180] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[left] {$\\tau$};\n\\draw[->] [domain=180:240] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)});\n\\end{tikzpicture}\n}\n \\parbox{3.5cm}{(B)\n \\begin{tikzpicture}[scale=1.1]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 3};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {1};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {2};\n\n\\draw [dashed] (0,0) -- ++ (40:1) circle(2pt)[fill] node[above right] {6};\n\\draw [dashed] (0,0) -- ++(160:1) circle(2pt)[fill] node[above left] {5};\n\\draw [dashed] (0,0) -- ++(280:1) circle(2pt)[fill] node[below] {4};\n\\draw[->] (0,0) -- ++ (200:1) node[left] {$x$};\n\n\\draw [domain=280:400] plot ({ e^((\\x+200)\/(3000))*0.3*cos(\\x)}, {e^((\\x+200)\/(3000))*0.3*sin(\\x)}) node[above] {$\\tau$};\n\\draw[->] [domain=400:840] plot ({ e^((\\x+200)\/(3000))*0.3*cos(\\x)}, {e^((\\x+200)\/(3000))*0.3*sin(\\x)});\n\\end{tikzpicture}\n}\n\\parbox{3.5cm}{(C)\n\\begin{tikzpicture}[scale=1.1]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 1};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {2};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {3};\n\n\\draw [dashed] (0,0) -- ++ (40:1) circle(2pt)[fill] node[above right] {6};\n\\draw [dashed] (0,0) -- ++(160:1) circle(2pt)[fill] node[above left] {5};\n\\draw [dashed] (0,0) -- ++(280:1) circle(2pt)[fill] node[below] {4};\n\\draw[->] (0,0) -- ++ (320:1) node[right] {$x$};\n\n\\draw [domain=280:320] plot ({ e^((\\x+200)\/(3000))*0.3*cos(\\x)}, {e^((\\x+200)\/(3000))*0.3*sin(\\x)}) node[right] {$\\tau$};\n\\draw[->] [domain=320:360] plot ({ e^((\\x+200)\/(3000))*0.3*cos(\\x)}, {e^((\\x+200)\/(3000))*0.3*sin(\\x)});\n\\end{tikzpicture}\n}\n\\parbox{3.5cm}{(D)\n\\begin{tikzpicture}[scale=1.1]\n\\draw[dashed] (0,0) circle(2pt)[fill] -- ++ (0+180:1) circle(2pt)[fill] node[left] {\\ 6};\n\\draw[dashed] (0,0) -- ++(120+180:1) circle(2pt)[fill] node[below right] {5};\n\\draw[dashed] (0,0) -- ++(240+180:1) circle(2pt)[fill] node[above right] {4};\n\n\\draw [] (0,0) -- ++ (40+180:1) circle(2pt)[fill] node[below left] {3};\n\\draw [] (0,0) -- ++(160+180:1) circle(2pt)[fill] node[below right] {1};\n\\draw [] (0,0) -- ++(280+180:1) circle(2pt)[fill] node[above] {2};\n\\draw[->] (0,0) -- ++ (200+180:1) node[right] {$x$};\n\n\n\\draw [domain=60:200] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[left] {$\\tau$};\n\\draw[->] [domain=200:340] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) ;\n\\end{tikzpicture}\n}\n\n\\parbox{3.5cm}{(E)\n\\begin{tikzpicture}[scale=1.1]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 5};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {6};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {4};\n\n\\draw [dashed] (0,0) -- ++ (40:1) circle(2pt)[fill] node[above right] {2};\n\\draw [dashed] (0,0) -- ++(160:1) circle(2pt)[fill] node[above left] {1};\n\\draw [dashed] (0,0) -- ++(280:1) circle(2pt)[fill] node[below] {3};\n\\draw[->] (0,0) -- ++ (200:1) node[left] {$x$};\n\n\\draw[<-] [domain=160:180] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) node[left] {$\\tau$};\n\\draw [domain=180:240] plot ({ 0.3*cos(\\x)}, {0.3*sin(\\x)}) ;\n\\end{tikzpicture}\n}\n\\parbox{3.5cm}{(F)\n\\begin{tikzpicture}[scale=1.1]\n\\draw (0,0) circle(2pt)[fill] -- ++ (0:1) circle(2pt)[fill] node[right] {\\ 2};\n\\draw (0,0) -- ++(120:1) circle(2pt)[fill] node[above left] {3};\n\\draw (0,0) -- ++(240:1) circle(2pt)[fill] node[below left] {1};\n\n\\draw [dashed] (0,0) -- ++ (40:1) circle(2pt)[fill] node[above right] {5};\n\\draw [dashed] (0,0) -- ++(160:1) circle(2pt)[fill] node[above left] {4};\n\\draw [dashed] (0,0) -- ++(280:1) circle(2pt)[fill] node[below] {6};\n\\draw[->] (0,0) -- ++ (200+180:1) node[right] {$x$};\n\n\\draw [domain=160:330] plot ({ e^((\\x+200)\/(3000))*0.3*cos(\\x)}, {e^((\\x+200)\/(3000))*0.3*sin(\\x)}) node[right] {$\\tau$};Fa\n\\draw[->][domain=330:240+360] plot ({ e^((\\x+200)\/(3000))*0.3*cos(\\x)}, {e^((\\x+200)\/(3000))*0.3*sin(\\x)});\n\\end{tikzpicture}\n}\n\n\\caption{Newman projections of ethane showing the effects of the {\\itshape\\bfseries G}$_{36}$(EM) generators (and their {\\itshape\\bfseries G}$_{36}$\\ partners; see Table~\\protect\\ref{tab:torsion_transformations})\non $\\tau$ and $\\chi$; the change of $\\tau$ is represented by the curved arrow\nencircling the C--C axis and the change of $\\chi$ is illustrated by the change in $x$-axis orientation. (A): Starting configuration with $\\tau = 4\\pi\/9$ and the $x$-axis forming the clockwise angle $8\\pi\/9$ with the horizontal. (B): The effect of $ab$ $\\sim$ (123)(465) which causes $\\tau$ to decrease by $4\\pi\/3$ (equivalent to an increase of $8\\pi\/3$ ) and $\\chi$ to remain constant. (C): The effect of $a^5 \\, b$ $\\sim$ (132)(456) with $\\tau$ remaining constant and $\\chi$ changing by $+2\\pi\/3$. (D): The effect of $c$ $\\sim$ (14)(26)(35)$(ab)^*$ with $\\tau$ $\\rightarrow$ $2\\pi - \\tau$ and $\\chi$ $\\rightarrow$ $\\chi + \\pi$. (E): The effect of $d\\, c$ $\\sim$ (14)(25)(36)$(ab)$ under which $\\tau$ and $\\chi$ are both invariant. (F): The effect of $E'$ which has no {\\itshape\\bfseries G}$_{36}$\\ partner, $E'\\, \\tau$ $=$ $\\tau+2\\pi$ and $E'\\, \\chi$ $=$ $\\chi+\\pi$. }\n\\label{fig:4pi}\n\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Rigid-symmetric-rotor function representations}\nAs mentioned above, we use the rigid-symmetric-rotor eigenfunctions $\\vert J, k, m \\rangle$~\\cite{98BuJexx} as primitive rotational basis functions. For $K\\neq 0$ (where $K = \\abs{k}$), the corresponding symmetrized functions\n are defined as\n\\begin{equation}\n\\ket{JKm\\eta} = \\frac{i^{\\eta}(-1)^\\sigma}{\\sqrt{2}}(\\ket{J, k, m } + (-1)^{J+K}\\ket{J, -k, m})\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{cases}\n\\sigma = K \\ \\text{mod} \\ 3 & \\eta = 1 \\\\\n\\sigma = 0 & \\eta = 0, \\\\\n\\end{cases}\n\\end{equation}\nwhile for $K=0$ it is $\\ket{J 0 0 \\eta} = i^{\\eta}\\ket{J,0,0}$ where $\\eta= J \\ \\text{mod} \\ 2 $. Using the rigid rotor transformation properties given in Eqs. (12-46) and (12-47) of \\cite{98BuJexx}, along with Tables~{12-1} and~{15-6} of \\cite{98BuJexx}, we can determine the irreps of the rigid rotor basis functions; the results are summarised in \\tabl{tab:rotation_irreps}.\n\n\\begin{table}[hbtp!]\n\\caption{\\label{tab:rotation_irreps} The irreps of the rigid rotor wavefunctions. For $K > 0$, the two functions with $\\eta$ $=$ 0, 1 generate a two-dimensional irrep or the direct sum of two one-dimensional irreps. For a given $K$ value, we list first the $\\eta=0$ function and then the $\\eta=1$ one. $n$ is a positive integer.}\n\\begin{center}\n\\begin{tabular}{ll}\n\\hline\\hline\n$K$ & $\\Gamma$ \\\\\n\\hline\n0 ($J$ even) & $A_{1s}$ \\\\\n0 ($J$ odd) & $A_{2s}$ \\\\\n$K=3n$ ($K$ even) & $A_{1s} \\oplus A_{2s}$ \\\\\n$K=3n$ ($K$ odd) & $A_{4d} \\oplus A_{3d}$ \\\\\n$K=3n+1$ ($K$ even) & $E_{1s}$ \\\\\n$K=3n+1$ ($K$ odd) & $E_{2d}$ \\\\\n$K=3n+2$ ($K$ even) & $E_{1s}$ \\\\\n$K=3n+2$ ($K$ odd) & $E_{2d}$ \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\subsection{Symmetrisation of the basis set}\n\nOnce a set of basis functions has been constructed for one of the subsets mentioned above as eigenfunctions of a 1D Schr\\\"odinger equation, the degenerate eigenfunctions (which in general transform reducibly) are symmetrised by a sampling and projection procedure described in sections 4 and 5 of \\cite{17YuYaOv.methods}. Briefly, a set of geometries is sampled (typically around 50) and each MS operation is applied to the eigenfunctions to obtain an overdetermined set of linear equations which are solved for the transformation matrix effecting the operation on the degenerate eigenfunctions. Using the traces of these matrices and the irrep matrices, one can obtain the irreducible coefficients of the reducible matrix as well as projection operators onto the irreps. These operators are then applied to the eigenfunctions to generate the symmetrized basis functions.\n\nOne then produces the basis functions for the complete TROVE calculation as products of the symmetrised 1D basis functions, one 1D basis function from each of the sets defined above. In general, these total basis functions do not transform irreducibly, and so a further application of the symmetrisation procedure is required, although the sampling step can be omitted as the transformation properties of the `factor' functions are already known.\n\n\n\n\n\\section{Potential Energy Function of Ethane in a Symmetry Adapted Representation}\n\n\nTo represent the PES analytically, an on-the-fly symmetrization procedure has been implemented (see, e.g., \\citep{15OwYuYa.CH3Cl}). We first introduce the stretching coordinates\n\\begin{equation}\\label{eq:stretch1}\n\\xi_1=1-\\exp\\left(-a(R - R_{\\mathrm{e}})\\right)\n\\end{equation}\n\\begin{equation}\\label{eq:stretch2}\n\\xi_j=1-\\exp\\left(-b(r_i - r_{\\mathrm{e}})\\right){\\,};\\hspace{2mm}j=2,3,4,5,6,7{\\,}, \\hspace{2mm} i=j-1,\n\\end{equation}\n($a$ is used for the C--C internal coordinate $R$, and $b$ is used for the six C--H internal coordinates $r_1,r_2$, $r_3$, $r_4$, $r_5$ and $r_6$) the bending angular coordinates\n\\begin{equation}\\label{eq:angular1}\n\\xi_k = (\\alpha_i - \\alpha_{\\mathrm{e}}){\\,};\\hspace{2mm}k=8,9,10,11,12,13{\\,}, \\hspace{2mm} i=k-7,\n\\end{equation}\nthe dihedral coordinates from Eq.~(\\ref{eq:gamma-delta})\n\\begin{equation}\n\\left( \\xi_{14}, \\xi_{15}, \\xi_{16}, \\xi_{17} \\right) =\n\\left( {\\gamma}_1, {\\gamma}_2, {\\delta}_1, {\\delta}_2 \\right)\n\\end{equation}\nand, finally, the torsional term\n\\begin{equation}\\label{eq:angular3}\n\\xi_{18} = 1 + \\cos 3\\tau\n\\end{equation}\nwhere $\\tau$ is defined in Eq.~(\\ref{eq:taudef}).\nThe quantities\n $R_{\\mathrm{e}}$, $r_{\\mathrm{e}}$ and $\\alpha_{\\mathrm{e}}$ are the equilibrium structural parameter values of C$_2$H$_6$.\n\n\nTaking an initial potential term of the form\n\\begin{equation}\\label{eq:V_i}\nV_{\\bm k}^{\\mathrm{initial}}=\\prod_{i=1}^{18} \\xi_{i}^{\\,k_i}\n\\end{equation}\nwith maximum expansion order $\\sum_{i} k_i =6$, each symmetry operation $T_{X}$ of {\\itshape\\bfseries G}$_{36}$\\ (see Table~\\ref{tab:recurrence_prod}) is independently applied to $V_{\\bm k}^{\\mathrm{initial}}$, i.e.\n\\begin{equation}\\label{eq:V_op}\nV_{\\bm k}^{T_{X}} = T_{X}{\\,} V_{\\bm k}^{\\mathrm{initial}}(\\bm \\xi) = T_{X}\\left(\\prod_{i=1}^{18} \\xi_{i}^{\\,k_i}\\right)\n\\end{equation}\nwhere $T_{X}$ is one of the 36 operators of {\\itshape\\bfseries G}$_{36}$, to create 36 new terms. Here ${\\bm k}$ denotes the 18-dimensional hyper-index ${k_1, k_2, \\ldots , k_{18}}$ and $(\\bm \\xi)$ denotes $\\{\\xi_1,\\xi_2,\\ldots,\\xi_{18}\\}$. The results are summed up to produce a final term\n\\begin{equation}\\label{eq:V_f}\nV_{\\bm k}^{\\mathrm{final}}= \\sum_{T_{X}} V_{\\bm k}^{T_{X}},\n\\end{equation}\nwhich is itself subjected to the 36 {\\itshape\\bfseries G}$_{36}$\\ symmetry operations $T_{X}$ to check its invariance. The total potential function is then given by the expression\n\\begin{equation}\nV_{\\mathrm{total}}(\\bm \\xi)= \\sum_{\\bm k} {\\,} f_{\\bm k} V_{\\bm k}^{\\mathrm{final}}(\\bm \\xi)\n\\end{equation}\nwhere $f_{\\bm k}$ are the corresponding expansion coefficients determined through a least-squares fitting to the \\textit{ab initio}\\ data generated at the CCSD(T)-F12b\/aug-cc-pVTZ level of theory. The details of the PES will be given elsewhere. The symmetry operations are represented as matrices and are generated from the four chosen generators as described above (see also~\\ref{sec:appendixa}).\n\n\n\n\n\n\\section{Numerical Example}\n\nIn this section we will apply the symmetrisation procedure on a small test basis set for the ethane molecule. We will demonstrate how the method works for subsets 2 (six C$-$H$_i$ stretching modes), 4 (four dihedral-angle modes ${\\gamma}_1$, ${\\gamma}_2$, ${\\delta}_1$, ${\\delta}_2$), and 5 (the torsional mode $\\tau$). In subset 1 (C$-$C stretch), the symmetrised wavefunctions are of $A_{1s}$ symmetry only as each {\\itshape\\bfseries G}$_{36}$\\ operation leaves the bond length $R$ invariant; in subset 3 (six C$-$C$-$H$_i$ bending modes), each {\\itshape\\bfseries G}$_{36}$\\ element has the same effect on the coordinates as in subset 2, and so the irreps obtained for subset 3 are identical to those obtained for subset 2.\n\nIn TROVE calculations, the size of the primitive basis set is determined by the maximum polyad number $P_{\\text{max}}$, which occurs in the inequality\n\\begin{equation}\nP = a_1 n_1 + a_2 n_2 + a_3 n_3 + \\ldots \\leq P_{\\text{max}}\n\\end{equation}\nwhere $n_i$ is the excitation number for the primitive basis function of coordinate $i$ and $a_i$ is its polyad coefficient. In our example all $a_i$ $=$ 1.\n\n\\subsection{Subset 2 symmetrisation}\n\nFor the six-member subset 2, we set the maximum polyad number to be 1, so the space is seven-dimensional. We consider here the 7 unsymmetrised eigenfunctions at lowest energy, the latter 4 being degenerate. As the first three of these eigenfunctions eigenfunctions are non-degenerate, they are already symmetrised and no further work is required. The first one-dimensional energy solution (at $\\SI{0.0}{\\per\\centi\\meter}$) is essentially the primitive ground-state wavefunction with small contributions from the other primitives:\n\\begin{equation}\n\\label{eq:subset_1:no_1}\n\\Psi_1 = 0.9999374\\ket{000000} + \\ldots\n\\end{equation}\nwhere $\\ldots$ signifies small (of the order $10^{-14}$) contributions from other primitive functions (omitted in the following). For simplicity we give only 5--7 significant digits here, while the actual calculations are done in quadruple precision. The eigenfunction in Eq.~(\\ref{eq:subset_1:no_1}) has $A_{1s}$ symmetry.\n\nThe second one-dimensional solution (at $\\SI{2929.16}{\\per\\centi\\meter}$) is\n\\begin{equation}\n \\Psi_2 = \\frac{1}{\\sqrt{6}}(-\\ket{000001} -\\ket{000010} -\\ket{000100} +\\ket{001000} +\\ket{010000} +\\ket{100000})\n\\end{equation}\nand has $A_{4s}$ symmetry.\n\nThe final one-dimensional solution (at $\\SI{2940.44}{\\per\\centi\\meter}$) is\n\\begin{equation}\n \\Psi_3 = \\frac{1}{\\sqrt{6}}(\\ket{000001} +\\ket{000010} +\\ket{000100} +\\ket{001000} +\\ket{010000} + \\ket{100000})\n\\end{equation}\nwhich is totally symmetric and so of $A_{1s}$ symmetry.\n\nThe four unsymmetrised, degenerate solutions (at $\\SI{3007.21}{\\per\\centi\\meter}$) are\n\\begin{equation}\n\\begin{split}\n \\Psi_4 =& +0.565540\\ket{000001} -0.209498\\ket{000010} -0.356041\\ket{000100} -0.482230\\ket{001000} \\\\\n & -4.233691\\ket{010000} +0.524567\\ket{100000} \\\\\n \\Psi_5 =& -0.536788\\ket{000010} +0.446199\\ket{000100} +0.330258\\ket{001000} -0.578016\\ket{010000} \\\\\n &+0.247758\\ket{100000} \\\\\n \\Psi_6 =& +0.581276\\ket{000001} -0.308888\\ket{000010}-0.272388\\ket{000100} +0.398602\\ket{001000} \\\\\n &+0.157199\\ket{010000} -0.555801\\ket{100000} \\\\\n \\Psi_7 =& -0.489104\\ket{000010} +0.516345\\ket{000100} -0.407635\\ket{001000}+ 0.553226\\ket{010000} \\\\\n &-0.145592\\ket{100000}\n\\end{split}\n\\end{equation}\nwhich turns out to be of $G_s$ symmetry, and thus also already symmetrised. However,\n in this case there is ambiguity in the group matrices describing the transformation properties of the degenerate wavefunctions. We therefore find and apply a unitary transformation on the wavefunctions to produce functions with transformation properties described by our `standard' $G$ representation matrices collected in Section~\\ref{sec:finalmatrices}.\n To this end, we follow the sampling procedure from Ref.~\\cite{17YuYaOv.methods}, derive the transformation properties of the above wavefunctions and convert them to have our `standard' transformation properties. For this work, we modified the sampling procedure in TROVE by applying it to the five group generators only (out of 72). This simple modification led to a significant speedup of the sampling part of the code and is now the standard part of TROVE.\n The reduced symmetrised wavefunctions become\n\\begin{equation}\n\\begin{split}\n \\Psi_4 &= \\frac{1}{2\\sqrt{3}}( -\\ket{000001} -\\ket{000010} + 2\\ket{000100} -\\ket{001000} -\\ket{010000} + 2\\ket{100000}) \\\\\n \\Psi_5 &= \\frac{1}{2}(-\\ket{000001} + \\ket{000010} - \\ket{001000} + \\ket{010000})\\\\\n \\Psi_6 &= \\frac{1}{2\\sqrt{3}}(-\\ket{000001}-\\ket{000010} +2\\ket{000100} + \\ket{001000} +\\ket{010000} - 2\\ket{100000}) \\\\\n \\Psi_7 &= \\frac12 ( -\\ket{000001} +\\ket{000010} +\\ket{001000} - \\ket{010000})\n\\end{split}\n\\end{equation}\nwhich are recognised as the linear combinations (defined by the matrix ${\\mathbf V}$) of bond lengths $r_k$ that transform as the $G_s$ irrep [Eq.~\\eqref{eq:gtransform}].\n\n\\subsection{Subset 4 symmetrisation}\n\nFor subset 4, we use a maximum polyad number of 1 and thus we have 5 eigenfunctions, one non-degenerate and four degenerate ones.\n$\\ket{0000}$ is the first eigenfunction (at $\\SI{0.0}{\\per\\centi\\meter})$; it has $A_{1s}$ symmetry. The other four unsymmetrised eigenfunctions (at $\\SI{1469.30}{\\per\\centi\\meter})$ are\n\\begin{equation}\n\\begin{split}\n\\Psi_2 &= \\frac{1}{\\sqrt{2}}(\\ket{0001} -\\ket{0100}) \\\\\n\\Psi_3 &= -\\frac{1}{\\sqrt{2}}(\\ket{0001} + \\ket{0100}) \\\\\n\\Psi_4 &= -\\frac{1}{\\sqrt{2}}(\\ket{0010} + \\ket{1000}) \\\\\n\\Psi_5 &= \\frac{1}{\\sqrt{2}}(-\\ket{0010} + \\ket{1000})\n\\end{split}\n\\end{equation}\nwhich again turns out to be of $G_s$ symmetry. Here and in the following, $\\frac{1}{\\sqrt{2}}$ stands for the numerical value $0.707106781186545$ ($\\pm 10^{-15})$. The desired symmetrised functions are\n\\begin{equation}\n\\begin{split}\n\\Psi_2 &= \\frac{1}{\\sqrt{2}}(\n\\ket{0010} + \\ket{1000}) \\\\\n\\Psi_3 &=\n\\frac{1}{\\sqrt{2}}(\n\\ket{0001} + \\ket{0100}) \\\\\n\\Psi_4 &=\n\\frac{1}{\\sqrt{2}}(\n\\ket{0010} -\\ket{1000}\n) \\\\\n\\Psi_5 &= \\frac{1}{\\sqrt{2}}(\n\t\\ket{0001} -\\ket{0100}).\n\\end{split}\n\\end{equation}\n\\subsection{Subset 5 symmetrisation}\n\nFor the one dimensional subset 5 we use a maximum polyad number of 29 and show, for illustrative purposes, the symmetrisation of the first three eigenfunctions. The lowest energy solution (at $\\SI{0.0}{\\per\\centi\\meter}$ with $A_{1s}$ symmetry) is:\n\\begin{equation}\n\\Psi_1 = 0.999958\\ket{0} + \\ldots.\n\\end{equation}\n The second and third eigenfunctions (at $\\SI{0.0029}{\\per\\centi\\meter})$ are degenerate, and the unsymmetrised wavefunctions are\n\\begin{equation}\n\\begin{split}\n\\Psi_2 = & +0.0272058\\ket{1} + 0.999588\\ket{2} + \\ldots \\\\\n\\Psi_3 = &-0.999588\\ket{1} + 0.0272058\\ket{2} + \\ldots.\n\\end{split}\n\\end{equation}\nThese turn out to be of $E_{3d}$ symmetry, which we then convert to\n\\begin{equation}\n\\begin{split}\n\\Psi_2 = & -0.75926\\ket{1} - 0.65072\\ket{2} + \\ldots \\\\\n\\Psi_3 = &-0.65072\\ket{1} + 0.75926\\ket{2} + \\ldots\n\\end{split}\n\\end{equation}\nto obey our `standard' transformation properties imposed by the matrices in~\\ref{sec:appendixa}.\n\n\\subsection{Torsional basis function symmetries}\n\n\n\\fig{fig:irrep_energies} shows the torsional potential energy as a function of $\\tau$. In order for the torsional basis functions to transform according to the irreps of the extended MS group {\\itshape\\bfseries G}$_{36}$(EM), we obtain them as solutions of the 1D torsional Schr\\\"odinger equation in Eq.~\\eqref{eq:torsion-schr-eq}. In solving this Schr\\\"odinger equation in a Fourier-series basis [see the discussion of Eq.~\\eqref{eq:torsion-schr-eq}], we let $\\tau$ vary from 0 to $4\\pi$ so that the potential energy curve in \\fig{fig:irrep_energies} has six minima. In the figure, we indicate the lowest allowed torsional energies. Some of these energies are degenerate, i.e., associated with more than one eigenfunction of a given irreducible representation. In the limit of an infinite barrier height, we expect the energies to be six-fold degenerate. With the actual, finite height of the barrier, the energies form near-degenerate clusters with a total multiplicity of 6 as shown in \\fig{fig:zoomed_plot}. However, since our description of the torsional angle $\\tau$ $\\in$ $[0, 4 \\pi]$ is unphysical, only three of the states in the cluster (of $s$ symmetry) exist in nature if they are combined with rigid-rotor basis functions of $s$ symmetry. The other three states (of $d$ symmetry) must be combined with\nrigid-rotor basis functions of $d$ symmetry in order that the total rotation-torsion state can exist in nature.\nFor $J$ $=$ 0, only the $s$-type torsional states will exist since only $s$-type rigid-rotor basis functions are available.\n\n\\begin{figure}[htbp!]\n \\centering\n \\begin{tikzpicture}\n \\begin{axis}[\n xtick={\n0, 1.5708, 3.14159, 4.7123889, 6.28318,\n 7.85398, 9.42477, 10.9955689, 12.56636\n },\n xticklabels={\n $0$, $\\frac{\\pi}{2}$, $\\pi$, $\\frac{3\\pi}{2}$, $2\\pi$,\n $\\frac{5\\pi}{2}$, $3\\pi$, $\\frac{7\\pi}{2}$, $4\\pi$\n },\n xlabel={$\\tau$ (\\SI{}{\\radian})},\n ylabel={Energy\/$hc$ (\\SI{}{\\per\\centi\\meter})}]\n \\addplot[smooth] table[x=angle,y=energy] {\n angle energy\n 0\t1003.77951556\n0.069813170079778\t992.60301469816\n0.139626340159556\t959.590813837953\n0.209439510239333\t906.266645428049\n0.279252680319111\t835.078568736274\n0.349065850398889\t749.26978315875\n0.418879020478667\t652.712557344184\n0.488692190558445\t549.717316998068\n0.558505360638222\t444.8292039328\n0.628318530718\t342.623379896951\n0.698131700797778\t247.50848945125\n0.767944870877556\t163.545565412708\n0.837758040957333\t94.287710200816\n0.907571211037111\t42.6443703823134\n0.977384381116889\t10.7729746917228\n1.04719755119667\t4.10521118250522E-28\n1.11701072127644\t10.7729746917228\n1.18682389135622\t42.6443703823134\n1.256637061436\t94.287710200816\n1.32645023151578\t163.545565412708\n1.39626340159556\t247.50848945125\n1.46607657167533\t342.623379896951\n1.53588974175511\t444.8292039328\n1.60570291183489\t549.717316998068\n1.67551608191467\t652.712557344184\n1.74532925199444\t749.26978315875\n1.81514242207422\t835.078568736274\n1.884955592154\t906.266645428049\n1.95476876223378\t959.590813837953\n2.02458193231356\t992.60301469816\n2.09439510239333\t1003.77951556\n2.16420827247311\t992.60301469816\n2.23402144255289\t959.590813837953\n2.30383461263267\t906.266645428049\n2.37364778271244\t835.078568736274\n2.44346095279222\t749.26978315875\n2.513274122872\t652.712557344184\n2.58308729295178\t549.717316998069\n2.65290046303156\t444.829203932801\n2.72271363311133\t342.623379896951\n2.79252680319111\t247.50848945125\n2.86233997327089\t163.545565412708\n2.93215314335067\t94.2877102008161\n3.00196631343044\t42.6443703823134\n3.07177948351022\t10.7729746917228\n3.14159265359\t4.16074439846136E-28\n3.21140582366978\t10.7729746917228\n3.28121899374956\t42.6443703823133\n3.35103216382933\t94.287710200816\n3.42084533390911\t163.545565412708\n3.49065850398889\t247.50848945125\n3.56047167406867\t342.623379896951\n3.63028484414844\t444.8292039328\n3.70009801422822\t549.717316998068\n3.769911184308\t652.712557344184\n3.83972435438778\t749.26978315875\n3.90953752446756\t835.078568736274\n3.97935069454733\t906.266645428049\n4.04916386462711\t959.590813837953\n4.11897703470689\t992.60301469816\n4.18879020478667\t1003.77951556\n4.25860337486644\t992.60301469816\n4.32841654494622\t959.590813837953\n4.398229715026\t906.266645428049\n4.46804288510578\t835.078568736274\n4.53785605518556\t749.26978315875\n4.60766922526533\t652.712557344184\n4.67748239534511\t549.717316998069\n4.74729556542489\t444.829203932801\n4.81710873550467\t342.623379896951\n4.88692190558445\t247.50848945125\n4.95673507566422\t163.545565412708\n5.026548245744\t94.2877102008161\n5.09636141582378\t42.6443703823134\n5.16617458590356\t10.7729746917229\n5.23598775598333\t4.27181083037364E-28\n5.30580092606311\t10.7729746917228\n5.37561409614289\t42.6443703823133\n5.44542726622267\t94.2877102008159\n5.51524043630244\t163.545565412708\n5.58505360638222\t247.50848945125\n5.654866776462\t342.623379896951\n5.72467994654178\t444.8292039328\n5.79449311662156\t549.717316998068\n5.86430628670133\t652.712557344184\n5.93411945678111\t749.26978315875\n6.00393262686089\t835.078568736274\n6.07374579694067\t906.266645428049\n6.14355896702045\t959.590813837953\n6.21337213710022\t992.60301469816\n6.28318530718\t1003.77951556\n6.35299847725978\t992.60301469816\n6.42281164733956\t959.590813837953\n6.49262481741933\t906.266645428049\n6.56243798749911\t835.078568736274\n6.63225115757889\t749.26978315875\n6.70206432765867\t652.712557344184\n6.77187749773844\t549.717316998069\n6.84169066781822\t444.829203932801\n6.911503837898\t342.623379896951\n6.98131700797778\t247.50848945125\n7.05113017805756\t163.545565412708\n7.12094334813733\t94.2877102008161\n7.19075651821711\t42.6443703823134\n7.26056968829689\t10.7729746917229\n7.33038285837667\t4.43841047824206E-28\n7.40019602845644\t10.7729746917228\n7.47000919853622\t42.6443703823133\n7.539822368616\t94.2877102008159\n7.60963553869578\t163.545565412708\n7.67944870877556\t247.50848945125\n7.74926187885533\t342.623379896951\n7.81907504893511\t444.8292039328\n7.88888821901489\t549.717316998068\n7.95870138909467\t652.712557344184\n8.02851455917445\t749.26978315875\n8.09832772925422\t835.078568736274\n8.168140899334\t906.266645428049\n8.23795406941378\t959.590813837953\n8.30776723949356\t992.60301469816\n8.37758040957333\t1003.77951556\n8.44739357965311\t992.603014698161\n8.51720674973289\t959.590813837953\n8.58701991981267\t906.266645428049\n8.65683308989244\t835.078568736274\n8.72664625997222\t749.26978315875\n8.796459430052\t652.712557344184\n8.86627260013178\t549.717316998069\n8.93608577021156\t444.829203932801\n9.00589894029133\t342.623379896951\n9.07571211037111\t247.50848945125\n9.14552528045089\t163.545565412708\n9.21533845053067\t94.2877102008162\n9.28515162061044\t42.6443703823135\n9.35496479069022\t10.7729746917229\n9.42477796077\t4.66054334206662E-28\n9.49459113084978\t10.7729746917228\n9.56440430092956\t42.6443703823133\n9.63421747100933\t94.2877102008159\n9.70403064108911\t163.545565412708\n9.77384381116889\t247.50848945125\n9.84365698124867\t342.623379896951\n9.91347015132845\t444.8292039328\n9.98328332140822\t549.717316998068\n10.053096491488\t652.712557344184\n10.1229096615678\t749.26978315875\n10.1927228316476\t835.078568736274\n10.2625360017273\t906.266645428049\n10.3323491718071\t959.590813837953\n10.4021623418869\t992.60301469816\n10.4719755119667\t1003.77951556\n10.5417886820464\t992.603014698161\n10.6116018521262\t959.590813837953\n10.681415022206\t906.26664542805\n10.7512281922858\t835.078568736274\n10.8210413623656\t749.26978315875\n10.8908545324453\t652.712557344184\n10.9606677025251\t549.717316998069\n11.0304808726049\t444.829203932801\n11.1002940426847\t342.623379896951\n11.1701072127644\t247.50848945125\n11.2399203828442\t163.545565412708\n11.309733552924\t94.2877102008162\n11.3795467230038\t42.6443703823135\n11.4493598930836\t10.7729746917229\n11.5191730631633\t4.93820942184732E-28\n11.5889862332431\t10.7729746917228\n11.6587994033229\t42.6443703823132\n11.7286125734027\t94.2877102008159\n11.7984257434824\t163.545565412708\n11.8682389135622\t247.50848945125\n11.938052083642\t342.623379896951\n12.0078652537218\t444.8292039328\n12.0776784238016\t549.717316998068\n12.1474915938813\t652.712557344184\n12.2173047639611\t749.26978315875\n12.2871179340409\t835.078568736274\n12.3569311041207\t906.266645428049\n12.4267442742004\t959.590813837953\n12.4965574442802\t992.60301469816\n12.56637061436\t1003.77951556\n};\n\n \\addplot[mark=none,,blue,style=ultra thin] coordinates {(0.786555, 145.221) (1.30784, 145.221)};\n \\addplot[mark=none,blue,dashed, style=ultra thin] coordinates {(0.786553, 145.224) (1.30784, 145.224)};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.786547, 145.23) (1.30785, 145.23)};\n \\addplot[mark=none,blue, dashed, style=ultra thin] coordinates {(0.786544, 145.233) (1.30785, 145.233)};\n\n \\addplot[mark=none,blue,dashed, style=ultra thin] coordinates {(0.576294, 421.286) (1.5181, 421.286)};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.576223, 421.391) (1.51817, 421.391)};\n \\addplot[mark=none,blue, dashed, style=ultra thin] coordinates {(0.576073, 421.613) (1.51832, 421.613)};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.575998, 421.725) (1.5184, 421.725)};\n\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.41252, 663.551) (1.68187, 663.551)};\n \\addplot[mark=none,blue, dashed, style=ultra thin] coordinates {(0.411362, 665.193) (1.68303, 665.193)};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.408908, 668.661) (1.68549, 668.661)};\n \\addplot[mark=none,blue, dashed, style=ultra thin] coordinates {(0.407645, 670.445) (1.68675, 670.445)};\n\n \\addplot[mark=none,blue, dashed, style=ultra thin] coordinates {(0.265027, 850.114) (1.82937, 850.114)};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.256877, 858.736) (1.83752, 858.736)};\n \\addplot[mark=none,blue,dashed,style=ultra thin] coordinates {(0.235004, 880.801) (1.85939, 880.801)};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.217873, 896.944) (1.87652, 896.944)};\n\n\\end{axis}\n \\end{tikzpicture}\n \\caption{The torsional potential energy as a function of the torsion angle $\\tau$. The allowed energy values are marked by blue horizontal lines. Each energy may correspond to more than one eigenfunction of a given irreducible representation. Dashed lines indicate states of $d$-type symmetry.}\n \\label{fig:irrep_energies}\n\\end{figure}\n\n\\begin{figure}[htbp!]\n \\centering\n \\begin{tikzpicture}\n \\begin{axis}[\n ylabel style={yshift=0.4cm},\n xlabel={$\\tau$ (\\SI{}{\\radian})},\n ylabel={Energy\/$hc$ (\\SI{}{\\per\\centi\\meter})},\n yticklabel style={\/pgf\/number format\/.cd,fixed zerofill,precision=3}]\n \\addplot[smooth] table[x=angle,y=energy] {\n angle energy\n0.786557 145.22\n0.786538 145.24\n };\n \\addplot[smooth] table[x=angle,y=energy] {\n angle energy\n1.30784 145.22\n1.30786 145.24\n };\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.786555, 145.221) (1.30784, 145.221)} node[above left]{$A_{1s}$};\n \\addplot[mark=none,blue,dashed, style=ultra thin] coordinates {(0.786553, 145.224) (1.30784, 145.224)} node[above left]{$E_{3d}$};\n \\addplot[mark=none,blue,style=ultra thin] coordinates {(0.786547, 145.23) (1.30785, 145.23)} node[above left]{$E_{3s}$};\n \\addplot[mark=none,blue, dashed, style=ultra thin] coordinates {(0.786544, 145.233) (1.30785, 145.233)} node[above left]{$A_{1d}$};\n \\end{axis}\n \\end{tikzpicture}\n \\caption{A enlarged detail of \\protect\\fig{fig:irrep_energies}, showing the lowest energy cluster with the {\\itshape\\bfseries G}$_{36}$(EM) symmetry labels indicated.}\n \\label{fig:zoomed_plot}\n\\end{figure}\n\n\\fig{fig:torsion_ground} shows the ground state torsional wavefunction which, as always, is totally symmetric (of $A_{1s}$ symmetry) in {\\itshape\\bfseries G}$_{36}$(EM). The effect of the {\\itshape\\bfseries G}$_{36}$(EM) generators on $\\tau$ is shown in \\tabl{tab:torsion_transformations}. In \\fig{fig:torsion_first} (left display) we show the doubly-degenerate wavefunctions of the first excited state. They span the $E_{3d}$ irrep but must be transformed to obtain the transformation properties imposed by the $E_{3d}$ matrices described in \\sect{sec:Einalmatrices}. The transformed functions, with the desired transformation properties are shown in \\fig{fig:torsion_first} (right display).\n\n\n\\begin{figure}[htbp!]\n\t \\centering\n\t\t \\begin{tikzpicture}\n\t\t\t\t \\begin{axis}[\n\t\t\t\t\t\t\t xtick={\n\t\t\t\t\t\t\t\t 0, 1.5708, 3.14159, 4.7123889, 6.28318,\n\t\t\t\t\t\t\t\t 7.85398, 9.42477, 10.9955689, 12.56636\n\t\t\t\t\t\t\t\t },\n\t\t\t\t\t\t\t\t\t xticklabels={\n\t\t\t\t\t\t\t\t\t\t\t $0$, $\\frac{\\pi}{2}$, $\\pi$, $\\frac{3\\pi}{2}$, $2\\pi$,\n\t\t\t\t\t\t\t\t\t\t\t\t $\\frac{5\\pi}{2}$, $3\\pi$, $\\frac{7\\pi}{2}$, $4\\pi$\n\t\t\t\t\t\t\t\t\t\t\t\t\t },\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t xlabel={$\\tau$ (\\SI{}{\\radian})},\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t ylabel={Amplitude}]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\addplot[smooth] table[x=angle,y=energy] {\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t angle energy\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0\t-0.00277264\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.006981317\t-0.002778\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.013962634\t-0.00279411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.020943951\t-0.002821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.027925268\t-0.0028588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.034906585\t-0.00290762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.041887902\t-0.00296766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0488692191\t-0.00303911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0558505361\t-0.00312224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0628318531\t-0.00321734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0698131701\t-0.00332474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0767944871\t-0.00344484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0837758041\t-0.00357805\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0907571211\t-0.00372483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0977384381\t-0.00388569\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1047197551\t-0.0040612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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4.0910517667\t-0.00388569\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0980330837\t-0.00372483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1050144007\t-0.00357805\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1119957177\t-0.00344484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1189770347\t-0.00332474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1259583517\t-0.00321734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1329396687\t-0.00312224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1399209857\t-0.00303911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1469023027\t-0.00296766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1538836197\t-0.00290762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1608649368\t-0.0028588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1678462538\t-0.002821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1748275708\t-0.00279411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1818088878\t-0.002778\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1887902048\t-0.00277264\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1957715218\t-0.002778\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2027528388\t-0.00279411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2097341558\t-0.002821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2167154728\t-0.0028588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2236967898\t-0.00290762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2306781068\t-0.00296766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2376594238\t-0.00303911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2446407409\t-0.00312224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2516220579\t-0.00321734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2586033749\t-0.00332474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2655846919\t-0.00344484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2725660089\t-0.00357805\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2795473259\t-0.00372483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2865286429\t-0.00388569\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2935099599\t-0.0040612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3004912769\t-0.00425195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3074725939\t-0.00445859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3144539109\t-0.00468183\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3214352279\t-0.00492241\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3284165449\t-0.00518113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.335397862\t-0.00545886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.342379179\t-0.00575648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.349360496\t-0.00607498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.356341813\t-0.00641537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.36332313\t-0.00677873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.370304447\t-0.0071662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.377285764\t-0.00757898\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.384267081\t-0.00801835\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.391248398\t-0.00848562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.398229715\t-0.00898219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.405211032\t-0.00950953\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.412192349\t-0.01006918\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.419173666\t-0.01066272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4261549831\t-0.01129184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4331363001\t-0.01195829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4401176171\t-0.01266388\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4470989341\t-0.01341051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4540802511\t-0.01420016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4610615681\t-0.01503487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4680428851\t-0.01591677\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4750242021\t-0.01684806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4820055191\t-0.01783102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4889868361\t-0.01886802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4959681531\t-0.01996149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5029494701\t-0.02111394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5099307872\t-0.02232796\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5169121042\t-0.02360623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5238934212\t-0.02495147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5308747382\t-0.0263665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5378560552\t-0.0278542\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5448373722\t-0.02941753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5518186892\t-0.03105949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5588000062\t-0.03278316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5657813232\t-0.03459168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5727626402\t-0.03648824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5797439572\t-0.03847606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5867252742\t-0.04055844\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5937065912\t-0.04273869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6006879083\t-0.04502017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6076692253\t-0.04740625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6146505423\t-0.04990033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6216318593\t-0.05250581\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6286131763\t-0.0552261\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6355944933\t-0.05806461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6425758103\t-0.06102472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6495571273\t-0.06410978\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6565384443\t-0.06732312\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6635197613\t-0.070668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6705010783\t-0.07414764\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6774823953\t-0.07776517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6844637124\t-0.08152364\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6914450294\t-0.085426\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6984263464\t-0.08947509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7054076634\t-0.0936736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7123889804\t-0.09802411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7193702974\t-0.10252901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7263516144\t-0.10719052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7333329314\t-0.11201069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7403142484\t-0.11699133\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7472955654\t-0.12213405\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7542768824\t-0.12744021\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7612581994\t-0.13291091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7682395164\t-0.13854696\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7752208335\t-0.14434891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7822021505\t-0.15031697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7891834675\t-0.15645104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7961647845\t-0.16275067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8031461015\t-0.16921505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8101274185\t-0.17584299\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8171087355\t-0.18263294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8240900525\t-0.18958291\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8310713695\t-0.19669052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8380526865\t-0.20395293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8450340035\t-0.21136688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8520153205\t-0.21892865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8589966376\t-0.22663407\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8659779546\t-0.23447847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8729592716\t-0.24245672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8799405886\t-0.25056321\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8869219056\t-0.25879184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8939032226\t-0.26713601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9008845396\t-0.27558862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9078658566\t-0.2841421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9148471736\t-0.29278839\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9218284906\t-0.30151892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9288098076\t-0.31032467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9357911246\t-0.31919614\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9427724416\t-0.32812338\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9497537587\t-0.33709597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9567350757\t-0.34610308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9637163927\t-0.35513346\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9706977097\t-0.36417547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9776790267\t-0.37321708\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9846603437\t-0.38224592\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9916416607\t-0.39124929\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9986229777\t-0.40021419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0056042947\t-0.40912734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0125856117\t-0.41797525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0195669287\t-0.42674418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0265482457\t-0.43542024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0335295628\t-0.44398941\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0405108798\t-0.45243756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0474921968\t-0.46075048\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0544735138\t-0.46891397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0614548308\t-0.47691383\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0684361478\t-0.48473591\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0754174648\t-0.4923662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0823987818\t-0.49979079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0893800988\t-0.506996\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0963614158\t-0.51396833\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1033427328\t-0.52069461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1103240498\t-0.52716196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1173053668\t-0.53335785\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1242866839\t-0.53927018\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1312680009\t-0.54488727\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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6.9184851549\t-0.18958291\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9254664719\t-0.19669052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9324477889\t-0.20395293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9394291059\t-0.21136688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9464104229\t-0.21892865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9533917399\t-0.22663407\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.960373057\t-0.23447847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.967354374\t-0.24245672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.974335691\t-0.25056321\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.981317008\t-0.25879184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.988298325\t-0.26713601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.995279642\t-0.27558862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.002260959\t-0.2841421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.009242276\t-0.29278839\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.016223593\t-0.30151892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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10.0600778085\t-0.04502017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0670591255\t-0.04273869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0740404425\t-0.04055844\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0810217595\t-0.03847606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0880030765\t-0.03648824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0949843935\t-0.03459168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1019657105\t-0.03278316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1089470276\t-0.03105949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1159283446\t-0.02941753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1229096616\t-0.0278542\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1298909786\t-0.0263665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1368722956\t-0.02495147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1438536126\t-0.02360623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1508349296\t-0.02232796\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1578162466\t-0.02111394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1647975636\t-0.01996149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1717788806\t-0.01886802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1787601976\t-0.01783102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1857415146\t-0.01684806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1927228316\t-0.01591677\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1997041487\t-0.01503487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2066854657\t-0.01420016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2136667827\t-0.01341051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2206480997\t-0.01266388\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2276294167\t-0.01195829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2346107337\t-0.01129184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2415920507\t-0.01066272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2485733677\t-0.01006918\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2555546847\t-0.00950953\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2625360017\t-0.00898219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2695173187\t-0.00848562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2764986357\t-0.00801835\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2834799528\t-0.00757898\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2904612698\t-0.0071662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2974425868\t-0.00677873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3044239038\t-0.00641537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3114052208\t-0.00607498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3183865378\t-0.00575648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3253678548\t-0.00545886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3323491718\t-0.00518113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3393304888\t-0.00492241\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3463118058\t-0.00468183\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3532931228\t-0.00445859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3602744398\t-0.00425195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3672557568\t-0.0040612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3742370739\t-0.00388569\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3812183909\t-0.00372483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3881997079\t-0.00357805\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3951810249\t-0.00344484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4021623419\t-0.00332474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4091436589\t-0.00321734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4161249759\t-0.00312224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4231062929\t-0.00303911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4300876099\t-0.00296766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4370689269\t-0.00290762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4440502439\t-0.0028588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4510315609\t-0.002821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.458012878\t-0.00279411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.464994195\t-0.002778\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.471975512\t-0.00277264\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.478956829\t-0.002778\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.485938146\t-0.00279411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.492919463\t-0.002821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.49990078\t-0.0028588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.506882097\t-0.00290762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.513863414\t-0.00296766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.520844731\t-0.00303911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.527826048\t-0.00312224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.534807365\t-0.00321734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.541788682\t-0.00332474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5487699991\t-0.00344484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5557513161\t-0.00357805\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5627326331\t-0.00372483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5697139501\t-0.00388569\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5766952671\t-0.0040612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5836765841\t-0.00425195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5906579011\t-0.00445859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5976392181\t-0.00468183\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6046205351\t-0.00492241\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6116018521\t-0.00518113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6185831691\t-0.00545886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6255644861\t-0.00575648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6325458031\t-0.00607498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6395271202\t-0.00641537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6465084372\t-0.00677873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6534897542\t-0.0071662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6604710712\t-0.00757898\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6674523882\t-0.00801835\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6744337052\t-0.00848562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6814150222\t-0.00898219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6883963392\t-0.00950953\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6953776562\t-0.01006918\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7023589732\t-0.01066272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7093402902\t-0.01129184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7163216072\t-0.01195829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7233029243\t-0.01266388\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7302842413\t-0.01341051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7372655583\t-0.01420016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7442468753\t-0.01503487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7512281923\t-0.01591677\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7582095093\t-0.01684806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7651908263\t-0.01783102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7721721433\t-0.01886802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7791534603\t-0.01996149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7861347773\t-0.02111394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7931160943\t-0.02232796\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8000974113\t-0.02360623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8070787283\t-0.02495147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8140600454\t-0.0263665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8210413624\t-0.0278542\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8280226794\t-0.02941753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8350039964\t-0.03105949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8419853134\t-0.03278316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8489666304\t-0.03459168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8559479474\t-0.03648824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8629292644\t-0.03847606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8699105814\t-0.04055844\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8768918984\t-0.04273869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8838732154\t-0.04502017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8908545324\t-0.04740625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8978358495\t-0.04990033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9048171665\t-0.05250581\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9117984835\t-0.0552261\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9187798005\t-0.05806461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9257611175\t-0.06102472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9327424345\t-0.06410978\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9397237515\t-0.06732312\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9467050685\t-0.070668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9536863855\t-0.07414764\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9606677025\t-0.07776517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9676490195\t-0.08152364\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9746303365\t-0.085426\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9816116535\t-0.08947509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9885929706\t-0.0936736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9955742876\t-0.09802411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0025556046\t-0.10252901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0095369216\t-0.10719052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0165182386\t-0.11201069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0234995556\t-0.11699133\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0304808726\t-0.12213406\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0374621896\t-0.12744021\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0444435066\t-0.13291091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0514248236\t-0.13854697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0584061406\t-0.14434891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0653874576\t-0.15031697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0723687747\t-0.15645104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0793500917\t-0.16275067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0863314087\t-0.16921505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0933127257\t-0.175843\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1002940427\t-0.18263294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1072753597\t-0.18958292\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1142566767\t-0.19669052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1212379937\t-0.20395293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1282193107\t-0.21136688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1352006277\t-0.21892865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1421819447\t-0.22663407\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1491632617\t-0.23447847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1561445787\t-0.24245672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1631258958\t-0.25056322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1701072128\t-0.25879184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1770885298\t-0.26713601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1840698468\t-0.27558862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1910511638\t-0.2841421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1980324808\t-0.29278839\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2050137978\t-0.30151892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2119951148\t-0.31032467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2189764318\t-0.31919615\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2259577488\t-0.32812338\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2329390658\t-0.33709597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2399203828\t-0.34610308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2469016999\t-0.35513346\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2538830169\t-0.36417547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2608643339\t-0.37321708\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2678456509\t-0.38224592\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2748269679\t-0.39124929\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2818082849\t-0.40021419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2887896019\t-0.40912735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2957709189\t-0.41797525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3027522359\t-0.42674418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3097335529\t-0.43542025\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3167148699\t-0.44398942\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3236961869\t-0.45243756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3306775039\t-0.46075048\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.337658821\t-0.46891397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.344640138\t-0.47691383\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.351621455\t-0.48473592\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.358602772\t-0.4923662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.365584089\t-0.4997908\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.372565406\t-0.506996\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.379546723\t-0.51396834\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.38652804\t-0.52069462\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.393509357\t-0.52716196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.400490674\t-0.53335786\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.407471991\t-0.53927019\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.414453308\t-0.54488727\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4214346251\t-0.55019794\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4284159421\t-0.5551915\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4353972591\t-0.55985787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4423785761\t-0.56418752\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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11.6448367693\t-0.52716196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6518180863\t-0.52069462\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6587994033\t-0.51396834\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6657807203\t-0.506996\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6727620373\t-0.4997908\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6797433543\t-0.4923662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6867246714\t-0.48473592\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6937059884\t-0.47691383\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7006873054\t-0.46891397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7076686224\t-0.46075048\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7146499394\t-0.45243756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7216312564\t-0.44398942\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7286125734\t-0.43542025\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7355938904\t-0.42674418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7425752074\t-0.41797525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7495565244\t-0.40912735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7565378414\t-0.40021419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7635191584\t-0.39124929\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7705004754\t-0.38224592\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7774817925\t-0.37321708\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7844631095\t-0.36417547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7914444265\t-0.35513346\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7984257435\t-0.34610308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8054070605\t-0.33709597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8123883775\t-0.32812338\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8193696945\t-0.31919615\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8263510115\t-0.31032467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8333323285\t-0.30151892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8403136455\t-0.29278839\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8472949625\t-0.2841421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8542762795\t-0.27558862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8612575966\t-0.26713601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8682389136\t-0.25879184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8752202306\t-0.25056322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8822015476\t-0.24245672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8891828646\t-0.23447847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8961641816\t-0.22663407\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9031454986\t-0.21892865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9101268156\t-0.21136688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9171081326\t-0.20395293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9240894496\t-0.19669052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9310707666\t-0.18958292\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9380520836\t-0.18263294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9450334006\t-0.175843\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9520147177\t-0.16921505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9589960347\t-0.16275067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9659773517\t-0.15645104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9729586687\t-0.15031697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9799399857\t-0.14434891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9869213027\t-0.13854697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9939026197\t-0.13291091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0008839367\t-0.12744021\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0078652537\t-0.12213406\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0148465707\t-0.11699133\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0218278877\t-0.11201069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0288092047\t-0.10719052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0357905218\t-0.10252901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0427718388\t-0.09802411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0497531558\t-0.0936736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0567344728\t-0.08947509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0637157898\t-0.085426\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0706971068\t-0.08152364\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0776784238\t-0.07776517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0846597408\t-0.07414764\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0916410578\t-0.070668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0986223748\t-0.06732312\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1056036918\t-0.06410978\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1125850088\t-0.06102472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1195663258\t-0.05806461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1265476429\t-0.0552261\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1335289599\t-0.05250581\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1405102769\t-0.04990033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1474915939\t-0.04740625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1544729109\t-0.04502017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1614542279\t-0.04273869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1684355449\t-0.04055844\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1754168619\t-0.03847606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1823981789\t-0.03648824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1893794959\t-0.03459168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1963608129\t-0.03278316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.2033421299\t-0.03105949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.210323447\t-0.02941753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.217304764\t-0.0278542\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.224286081\t-0.0263665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.231267398\t-0.02495147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.238248715\t-0.02360623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.245230032\t-0.02232796\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.252211349\t-0.02111394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.259192666\t-0.01996149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.266173983\t-0.01886802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.2731553\t-0.01783102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.280136617\t-0.01684806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.287117934\t-0.01591677\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.294099251\t-0.01503487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3010805681\t-0.01420016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3080618851\t-0.01341051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3150432021\t-0.01266388\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3220245191\t-0.01195829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3290058361\t-0.01129184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3359871531\t-0.01066272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3429684701\t-0.01006918\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3499497871\t-0.00950953\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3569311041\t-0.00898219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3639124211\t-0.00848562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3708937381\t-0.00801835\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3778750551\t-0.00757898\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3848563722\t-0.0071662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3918376892\t-0.00677873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3988190062\t-0.00641537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4058003232\t-0.00607498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4127816402\t-0.00575648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4197629572\t-0.00545886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4267442742\t-0.00518113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4337255912\t-0.00492241\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4407069082\t-0.00468183\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4476882252\t-0.00445859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4546695422\t-0.00425195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4616508592\t-0.0040612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4686321762\t-0.00388569\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4756134933\t-0.00372483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4825948103\t-0.00357805\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4895761273\t-0.00344484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4965574443\t-0.00332474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5035387613\t-0.00321734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5105200783\t-0.00312224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5175013953\t-0.00303911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5244827123\t-0.00296766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5314640293\t-0.00290762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5384453463\t-0.0028588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5454266633\t-0.002821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5524079803\t-0.00279411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5593892974\t-0.002778\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5663706144\t-0.00277264\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{axis}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{tikzpicture}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\caption{The ground state torsional wavefunction.}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\label{fig:torsion_ground}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{figure}\n\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\begin{figure}[htbp!]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\centering\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\begin{tikzpicture}[scale=0.9]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\begin{axis}[\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t xtick={\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0, 1.5708, 3.14159, 4.7123889, 6.28318,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.85398, 9.42477, 10.9955689, 12.56636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t },\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t xticklabels={\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t $0$, $\\frac{\\pi}{2}$, $\\pi$, $\\frac{3\\pi}{2}$, $2\\pi$,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t $\\frac{5\\pi}{2}$, $3\\pi$, $\\frac{7\\pi}{2}$, $4\\pi$\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t },\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t xlabel={$\\tau$ (\\SI{}{\\radian})},\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t ylabel={Amplitude}]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\addplot[smooth, red] table[x=angle,y=energy] {\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t angle energy\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0\t0.00257847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.006981317\t0.0025048\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.013962634\t0.00244081\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.020943951\t0.00238625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.027925268\t0.00234091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.034906585\t0.00230461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.041887902\t0.00227718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0488692191\t0.00225852\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0558505361\t0.00224853\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0628318531\t0.00224717\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0698131701\t0.0022544\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0767944871\t0.00227024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0837758041\t0.00229472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0907571211\t0.0023279\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.0977384381\t0.00236989\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1047197551\t0.0024208\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1117010721\t0.0024808\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1186823891\t0.00255008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1256637061\t0.00262885\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1326450232\t0.00271735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1396263402\t0.00281587\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1466076572\t0.00292471\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1535889742\t0.00304422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1605702912\t0.00317477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1675516082\t0.00331676\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1745329252\t0.00347062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1815142422\t0.00363683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1884955592\t0.00381589\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.1954768762\t0.00400832\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2024581932\t0.00421471\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2094395102\t0.00443565\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2164208272\t0.00467179\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2234021443\t0.0049238\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2303834613\t0.00519238\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2373647783\t0.00547829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2443460953\t0.0057823\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2513274123\t0.00610525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2583087293\t0.00644798\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2652900463\t0.00681139\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2722713633\t0.00719642\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2792526803\t0.00760403\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2862339973\t0.00803525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.2932153143\t0.0084911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.3001966313\t0.00897269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.3071779484\t0.00948113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.3141592654\t0.01001758\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.3211405824\t0.01058325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.3281218994\t0.01117937\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t0.3351032164\t0.01180721\n\t\t\t\t\t\t\t\t\t\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};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\addplot[smooth, blue] table[x=angle,y=energy] {\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t angle energy\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1125850088\t-0.01592123\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1195663258\t-0.01515312\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1265476429\t-0.01441673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1335289599\t-0.01371121\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1405102769\t-0.01303567\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1474915939\t-0.01238923\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1544729109\t-0.01177102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1614542279\t-0.01118015\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1684355449\t-0.01061576\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1754168619\t-0.010077\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1823981789\t-0.00956299\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1893794959\t-0.00907291\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.1963608129\t-0.00860591\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.2033421299\t-0.00816119\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.210323447\t-0.00773794\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.217304764\t-0.00733538\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.224286081\t-0.00695273\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.231267398\t-0.00658924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.238248715\t-0.00624418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.245230032\t-0.00591684\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.252211349\t-0.00560653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.259192666\t-0.00531256\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.266173983\t-0.00503429\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.2731553\t-0.00477109\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.280136617\t-0.00452234\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.287117934\t-0.00428747\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.294099251\t-0.0040659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3010805681\t-0.00385709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3080618851\t-0.00366052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3150432021\t-0.0034757\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3220245191\t-0.00330213\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3290058361\t-0.00313938\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3359871531\t-0.00298701\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3429684701\t-0.0028446\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3499497871\t-0.00271177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3569311041\t-0.00258814\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3639124211\t-0.00247339\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3708937381\t-0.00236718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3778750551\t-0.0022692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3848563722\t-0.00217918\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3918376892\t-0.00209687\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.3988190062\t-0.00202201\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4058003232\t-0.00195439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4127816402\t-0.00189381\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4197629572\t-0.0018401\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4267442742\t-0.0017931\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4337255912\t-0.00175267\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4407069082\t-0.0017187\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4476882252\t-0.0016911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4546695422\t-0.0016698\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4616508592\t-0.00165473\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4686321762\t-0.00164588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4756134933\t-0.00164323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4825948103\t-0.00164679\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4895761273\t-0.00165661\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.4965574443\t-0.00167272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5035387613\t-0.00169522\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5105200783\t-0.0017242\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5175013953\t-0.0017598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5244827123\t-0.00180216\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5314640293\t-0.00185145\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5384453463\t-0.00190788\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5454266633\t-0.00197167\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5524079803\t-0.00204307\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5593892974\t-0.00212237\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t12.5663706144\t-0.00220988\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{axis}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{tikzpicture}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\begin{tikzpicture}[scale=0.9]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\begin{axis}[\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t xtick={\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0, 1.5708, 3.14159, 4.7123889, 6.28318,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.85398, 9.42477, 10.9955689, 12.56636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t },\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t xticklabels={\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t $0$, $\\frac{\\pi}{2}$, $\\pi$, $\\frac{3\\pi}{2}$, $2\\pi$,\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t $\\frac{5\\pi}{2}$, $3\\pi$, $\\frac{7\\pi}{2}$, $4\\pi$\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t },\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t xlabel={$\\tau$ (\\SI{}{\\radian})},\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t ylabel={Amplitude}]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\addplot[smooth, blue] table[x=angle,y=energy] {\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t angle energy\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0\t1.14903999997532E-008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.006981317\t0.000120877\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.013962634\t0.0002422161\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.020943951\t0.0003644876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.027925268\t0.0004881698\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.034906585\t0.0006137216\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.041887902\t0.000741656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0488692191\t0.0008724328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0558505361\t0.0010065727\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0628318531\t0.0011445648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0698131701\t0.0012869364\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0767944871\t0.001434221\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0837758041\t0.001586961\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0907571211\t0.0017457247\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0977384381\t0.0019110784\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1047197551\t0.0020836489\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1117010721\t0.0022640319\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1186823891\t0.002452878\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1256637061\t0.0026508479\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1326450232\t0.0028586433\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1396263402\t0.0030769791\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1466076572\t0.0033065939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1535889742\t0.0035482622\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1605702912\t0.0038027911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1675516082\t0.0040710286\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1745329252\t0.0043538492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1815142422\t0.0046521477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1884955592\t0.0049668743\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1954768762\t0.0052990269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2024581932\t0.0056496262\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2094395102\t0.0060197309\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2164208272\t0.0064104517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2234021443\t0.0068229395\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2303834613\t0.0072584091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2373647783\t0.0077180972\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2443460953\t0.0082032945\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2513274123\t0.008715345\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2583087293\t0.0092556458\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2652900463\t0.0098256308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2722713633\t0.0104267916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2792526803\t0.0110606825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2862339973\t0.0117288883\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2932153143\t0.0124330762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3001966313\t0.0131749382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3071779484\t0.0139562274\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3141592654\t0.0147787628\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3211405824\t0.0156443916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3281218994\t0.0165550316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3351032164\t0.0175126352\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3420845334\t0.0185192342\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.3490658504\t0.0195768656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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0.5235987756\t0.0692673668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5305800926\t0.0724541067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5375614096\t0.0757515121\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5445427266\t0.0791610462\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5515240436\t0.0826839857\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5585053606\t0.0863214674\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5654866776\t0.0900744719\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.5724679947\t0.0939437671\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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0.9704030641\t0.3989253208\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9773843811\t0.4017428481\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9843656981\t0.4043101825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9913470151\t0.4066220024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9983283321\t0.4086734821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0053096491\t0.4104603389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0122909662\t0.4119787895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0192722832\t0.4132256471\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0262536002\t0.4141982767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0332349172\t0.414894611\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0402162342\t0.4153131748\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0471975512\t0.415453084\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0541788682\t0.415314037\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0611601852\t0.414896329\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0681415022\t0.4142008439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0751228192\t0.4132290624\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0821041362\t0.411983053\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0890854532\t0.4104654451\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0960667703\t0.4086794439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1030480873\t0.4066287994\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1100294043\t0.4043178146\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1170107213\t0.4017513088\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1239920383\t0.3989346026\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1309733553\t0.3958735273\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1379546723\t0.3925743536\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1449359893\t0.3890438145\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1519173063\t0.3852890537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1588986233\t0.381317609\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1658799403\t0.3771374062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1728612573\t0.3727566806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1798425743\t0.3681840067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1868238914\t0.3634282446\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1938052084\t0.3584984937\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2007865254\t0.3534040686\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2077678424\t0.3481545158\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2147491594\t0.3427595163\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2217304764\t0.3372288747\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2287117934\t0.3315725381\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2356931104\t0.3258004985\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2426744274\t0.3199227811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2496557444\t0.313949442\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2566370614\t0.307890528\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2636183784\t0.3017560226\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2705996955\t0.295555851\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2775810125\t0.2892998266\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2845623295\t0.2829976597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2915436465\t0.2766589\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2985249635\t0.2702929478\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3055062805\t0.2639089644\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3124875975\t0.257515954\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3194689145\t0.2511226649\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3264502315\t0.244737599\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3334315485\t0.2383689729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3404128655\t0.2320247637\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3473941825\t0.2257126217\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3543754995\t0.2194399112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3613568166\t0.2132136697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3683381336\t0.2070406148\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3753194506\t0.20092713\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3823007676\t0.1948792681\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3892820846\t0.1889027335\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3962634016\t0.1830028858\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4032447186\t0.1771847406\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4102260356\t0.1714529612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4172073526\t0.1658118538\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4241886696\t0.1602654081\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4311699866\t0.1548172436\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4381513036\t0.1494706547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4451326207\t0.1442286048\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4521139377\t0.1390937059\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4590952547\t0.1340682868\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4660765717\t0.1291543369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4730578887\t0.1243535571\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4800392057\t0.1196673389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4870205227\t0.1150967944\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4940018397\t0.110642752\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5009831567\t0.1063058042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5079644737\t0.1020862492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5149457907\t0.0979841673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5219271077\t0.0939994174\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5289084247\t0.0901316157\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5358897418\t0.0863801806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5428710588\t0.0827443344\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5498523758\t0.0792231193\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5568336928\t0.0758153987\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5638150098\t0.0725199076\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5707963268\t0.0693351849\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5777776438\t0.0662596591\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5847589608\t0.0632916562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5917402778\t0.0604293389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5987215948\t0.0576708283\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6057029118\t0.0550140973\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6126842288\t0.052457075\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6196655459\t0.0499976061\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6266468629\t0.0476334812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6336281799\t0.0453624189\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6406094969\t0.0431821307\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6475908139\t0.0410902551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6545721309\t0.0390844306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6615534479\t0.0371622688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6685347649\t0.0353213692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6755160819\t0.0335593258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6824973989\t0.0318737382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6894787159\t0.0302622017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6964600329\t0.0287223461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.7034413499\t0.0272517915\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.710422667\t0.0258482178\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.717403984\t0.0245092928\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.724385301\t0.0232327452\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.731366618\t0.0220163252\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.738347935\t0.0208578266\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.745329252\t0.0197550799\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.752310569\t0.0187059699\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.759291886\t0.0177084152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.766273203\t0.0167603991\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.77325452\t0.0158599302\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.780235837\t0.0150050907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.787217154\t0.0141940171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.7941984711\t0.0134248816\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8011797881\t0.0126959205\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8081611051\t0.0120054275\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8151424221\t0.0113517658\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8221237391\t0.0107333213\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8291050561\t0.0101485787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8360863731\t0.0095960303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8430676901\t0.0090742659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8500490071\t0.0085819069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8570303241\t0.0081176352\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8640116411\t0.0076802047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8709929581\t0.0072683981\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8779742751\t0.0068810593\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8849555922\t0.0065171046\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8919369092\t0.0061754634\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8989182262\t0.0058551474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9058995432\t0.0055552196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9128808602\t0.005274761\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9198621772\t0.0050129377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9268434942\t0.004768949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9338248112\t0.0045420357\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9408061282\t0.0043314916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9477874452\t0.0041366463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9547687622\t0.0039568891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9617500792\t0.0037916376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9687313962\t0.0036403678\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9757127133\t0.0035025763\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9826940303\t0.0033778206\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9896753473\t0.0032656897\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9966566643\t0.0031658254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0036379813\t0.0030778828\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0106192983\t0.0030015757\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0176006153\t0.0029366706\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0245819323\t0.0028829236\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0315632493\t0.0028401936\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0385445663\t0.0028083212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0455258833\t0.0027872129\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0525072003\t0.0027768277\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0594885174\t0.0027771143\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0664698344\t0.0027881066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0734511514\t0.0028098575\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0804324684\t0.0028424636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0874137854\t0.0028860582\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0943951024\t0.0029407997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1013764194\t0.0030069256\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1083577364\t0.0030846668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1153390534\t0.0031743369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1223203704\t0.0032762623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1293016874\t0.0033908289\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1362830044\t0.0035184686\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1432643214\t0.003659639\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1502456385\t0.0038148802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1572269555\t0.0039847465\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1642082725\t0.0041698506\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1711895895\t0.0043708692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1781709065\t0.0045885238\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1851522235\t0.0048235864\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1921335405\t0.0050768922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1991148575\t0.0053493219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2060961745\t0.0056418343\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2130774915\t0.0059554305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2200588085\t0.0062911983\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2270401255\t0.0066502625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2340214426\t0.0070338399\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2410027596\t0.0074432104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2479840766\t0.007879721\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2549653936\t0.0083448024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2619467106\t0.0088399502\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2689280276\t0.0093667534\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2759093446\t0.0099268714\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2828906616\t0.0105220482\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2898719786\t0.0111541374\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2968532956\t0.0118250393\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3038346126\t0.0125367787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3108159296\t0.0132914697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3177972466\t0.0140912922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3247785637\t0.0149385683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3317598807\t0.0158356832\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3387411977\t0.0167851368\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3457225147\t0.0177895419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3527038317\t0.0188516047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3596851487\t0.0199741299\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3666664657\t0.0211600361\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3736477827\t0.0224123636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3806290997\t0.0237342371\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3876104167\t0.0251289005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3945917337\t0.0265997152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4015730507\t0.0281501388\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4085543678\t0.0297837505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4155356848\t0.0315042011\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4225170018\t0.033315303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4294983188\t0.0352209282\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4364796358\t0.0372250594\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4434609528\t0.0393318019\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4504422698\t0.0415453236\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4574235868\t0.0438699003\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4644049038\t0.0463098843\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4713862208\t0.0488697214\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4783675378\t0.0515539422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4853488548\t0.0543671127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4923301718\t0.0573138837\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4993114889\t0.060398957\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5062928059\t0.0636270886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5132741229\t0.0670030595\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5202554399\t0.0705316613\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5272367569\t0.0742177315\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5342180739\t0.07806609\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5411993909\t0.08208156\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5481807079\t0.0862689129\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5551620249\t0.0906329066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5621433419\t0.095178257\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5691246589\t0.0999095727\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5761059759\t0.1048314069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.583087293\t0.1099481835\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.59006861\t0.1152642432\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.597049927\t0.1207837325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.604031244\t0.1265106818\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.611012561\t0.1324489033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.617993878\t0.138602036\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.624975195\t0.1449734704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.631956512\t0.1515663529\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.638937829\t0.1583835706\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.645919146\t0.1654276986\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.652900463\t0.1727010111\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.65988178\t0.180205417\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.666863097\t0.1879424891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6738444141\t0.1959133847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6808257311\t0.2041188618\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6878070481\t0.2125592241\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6947883651\t0.2212343383\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7017696821\t0.2301435644\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7087509991\t0.239285755\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7157323161\t0.248659257\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7227136331\t0.2582618286\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7296949501\t0.2680906976\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7366762671\t0.2781424593\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7436575841\t0.2884131431\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7506389011\t0.2988981101\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7576202182\t0.309592127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7646015352\t0.3204892503\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7715828522\t0.3315829278\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7785641692\t0.342865884\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7855454862\t0.3543301883\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7925268032\t0.3659671975\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7995081202\t0.377767585\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8064894372\t0.3897213312\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8134707542\t0.4018177269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8204520712\t0.4140453467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8274333882\t0.4263921081\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8344147052\t0.4388452238\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8413960222\t0.4513912824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8483773393\t0.4640161875\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8553586563\t0.4767052296\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8623399733\t0.4894430833\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8693212903\t0.5022138427\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8763026073\t0.5150010405\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8832839243\t0.527787672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8902652413\t0.540556233\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8972465583\t0.5532887754\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9042278753\t0.5659669031\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9112091923\t0.578571859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9181905093\t0.5910845303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9251718263\t0.6034855155\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9321531434\t0.6157551616\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9391344604\t0.6278736419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9461157774\t0.6398209641\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9530970944\t0.651577052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9600784114\t0.6631218079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9670597284\t0.6744351548\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9740410454\t0.6854971011\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9810223624\t0.6962878051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9880036794\t0.7067876149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9949849964\t0.7169771637\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0019663134\t0.7268373962\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0089476304\t0.7363496539\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0159289474\t0.7454957252\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0229102645\t0.7542579104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0298915815\t0.7626190814\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0368728985\t0.770562723\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0438542155\t0.7780730206\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0508355325\t0.7851348818\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0578168495\t0.7917340192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0647981665\t0.7978569707\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0717794835\t0.8034911787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0787608005\t0.8086250043\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0857421175\t0.8132477891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0927234345\t0.8173499029\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0997047515\t0.8209227372\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1066860685\t0.8239587952\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1136673856\t0.8264516518\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1206487026\t0.8283960519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1276300196\t0.8297878687\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1346113366\t0.8306241379\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1415926536\t0.8309030939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1485739706\t0.83062414\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1555552876\t0.8297878653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1625366046\t0.8283960506\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1695179216\t0.8264516526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1764992386\t0.8239587917\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1834805556\t0.8209227369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1904618726\t0.8173498993\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1974431897\t0.8132477952\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2044245067\t0.8086250071\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2114058237\t0.8034911781\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2183871407\t0.7978569755\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2253684577\t0.7917340218\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2323497747\t0.7851348833\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2393310917\t0.7780730209\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2463124087\t0.7705627233\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2532937257\t0.7626190751\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2602750427\t0.7542579051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2672563597\t0.745495722\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2742376767\t0.7363496529\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2812189937\t0.7268373984\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2882003108\t0.7169771626\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2951816278\t0.7067876181\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3021629448\t0.6962877995\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3091442618\t0.6854971085\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3161255788\t0.6744351545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3231068958\t0.6631218076\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3300882128\t0.6515770528\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3370695298\t0.6398209594\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3440508468\t0.6278736394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3510321638\t0.6157551687\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3580134808\t0.6034855183\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3649947978\t0.5910845298\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3719761149\t0.5785718552\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3789574319\t0.5659669035\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3859387489\t0.5532887736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3929200659\t0.5405562356\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3999013829\t0.5277876659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4068826999\t0.5150010473\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4138640169\t0.5022138473\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4208453339\t0.4894430856\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4278266509\t0.4767052298\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4348079679\t0.464016192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4417892849\t0.4513912771\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4487706019\t0.4388452216\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4557519189\t0.4263921027\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.462733236\t0.4140453509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.469714553\t0.4018177246\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.47669587\t0.3897213365\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.483677187\t0.3777675826\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.490658504\t0.3659671929\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.497639821\t0.3543301804\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.504621138\t0.3428658847\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.511602455\t0.3315829296\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.518583772\t0.3204892498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.525565089\t0.3095921222\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.532546406\t0.2988981128\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.539527723\t0.2884131371\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5465090401\t0.278142463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5534903571\t0.2680906925\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5604716741\t0.2582618267\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5674529911\t0.2486592539\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5744343081\t0.2392857551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5814156251\t0.2301435634\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5883969421\t0.2212343372\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5953782591\t0.2125592273\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6023595761\t0.2041188627\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6093408931\t0.1959133866\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6163222101\t0.187942492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6233035271\t0.180205422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6302848441\t0.1727010051\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6372661612\t0.1654277022\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6442474782\t0.1583835709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6512287952\t0.1515663541\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6582101122\t0.1449734716\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6651914292\t0.1386020436\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6721727462\t0.1324489064\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6791540632\t0.1265106781\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6861353802\t0.1207837331\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6931166972\t0.1152642426\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7000980142\t0.1099481924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7070793312\t0.1048314047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7140606482\t0.0999095672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7210419653\t0.0951782545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7280232823\t0.090632906\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7350045993\t0.0862689121\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7419859163\t0.0820815493\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7489672333\t0.0780660931\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7559485503\t0.0742177289\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7629298673\t0.0705316617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7699111843\t0.067003053\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7768925013\t0.0636270895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7838738183\t0.0603989597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7908551353\t0.0573138774\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7978364523\t0.0543671114\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8048177693\t0.0515539383\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8117990864\t0.0488697248\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8187804034\t0.0463098828\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8257617204\t0.0438698929\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8327430374\t0.0415453234\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8397243544\t0.0393317957\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8467056714\t0.0372250614\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8536869884\t0.0352209209\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8606683054\t0.0333153016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8676496224\t0.0315042034\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8746309394\t0.0297837445\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8816122564\t0.0281501428\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8885935734\t0.0265997174\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8955748905\t0.025128904\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9025562075\t0.0237342353\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9095375245\t0.0224123598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9165188415\t0.0211600422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9235001585\t0.0199741305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9304814755\t0.018851601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9374627925\t0.0177895435\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9444441095\t0.0167851328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9514254265\t0.0158356767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9584067435\t0.0149385625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9653880605\t0.0140912892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9723693775\t0.013291464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9793506945\t0.0125367767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9863320116\t0.0118250397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9933133286\t0.0111541402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0002946456\t0.0105220533\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0072759626\t0.009926871\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0142572796\t0.0093667517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0212385966\t0.0088399514\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0282199136\t0.0083447966\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0352012306\t0.0078797199\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0421825476\t0.0074432094\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0491638646\t0.0070338325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0561451816\t0.0066502603\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0631264986\t0.0062911859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0701078157\t0.0059554294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0770891327\t0.0056418301\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0840704497\t0.005349322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0910517667\t0.0050768907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0980330837\t0.004823582\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1050144007\t0.0045885227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1119957177\t0.004370869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1189770347\t0.0041698464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1259583517\t0.0039847369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1329396687\t0.0038148692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1399209857\t0.0036596369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1469023027\t0.0035184597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1538836197\t0.0033908269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1608649368\t0.0032762545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1678462538\t0.003174323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1748275708\t0.0030846656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1818088878\t0.0030069223\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1887902048\t0.0029408023\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1957715218\t0.0028860486\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2027528388\t0.0028424584\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2097341558\t0.0028098559\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2167154728\t0.0027880924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2236967898\t0.0027771008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2306781068\t0.0027768127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2376594238\t0.0027872063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2446407409\t0.0028083076\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2516220579\t0.0028401817\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2586033749\t0.0028829131\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2655846919\t0.0029366483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2725660089\t0.0030015704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2795473259\t0.0030778617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2865286429\t0.0031657997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2935099599\t0.003265673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3004912769\t0.0033778014\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3074725939\t0.0035025525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3144539109\t0.0036403439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3214352279\t0.0037916192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3284165449\t0.0039568608\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.335397862\t0.0041366165\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.342379179\t0.0043314588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.349360496\t0.0045420038\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.356341813\t0.0047689151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.36332313\t0.0050129042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.370304447\t0.0052747237\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.377285764\t0.0055551739\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.384267081\t0.0058551095\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.391248398\t0.0061754207\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.398229715\t0.006517062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.405211032\t0.0068810115\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.412192349\t0.0072683462\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.419173666\t0.0076801451\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4261549831\t0.0081175795\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4331363001\t0.0085818423\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4401176171\t0.0090741969\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4470989341\t0.0095959589\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4540802511\t0.010148499\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4610615681\t0.0107332521\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4680428851\t0.0113516811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4750242021\t0.0120053412\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4820055191\t0.0126958244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4889868361\t0.013424777\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4959681531\t0.0141939114\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5029494701\t0.0150049877\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5099307872\t0.0158598096\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5169121042\t0.0167602791\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5238934212\t0.017708293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5308747382\t0.0187058328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5378560552\t0.0197549362\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5448373722\t0.0208576725\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5518186892\t0.0220161596\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5588000062\t0.0232325666\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5657813232\t0.0245091079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5727626402\t0.025848021\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5797439572\t0.0272515876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5867252742\t0.0287221285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5937065912\t0.0302619819\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6006879083\t0.0318734979\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6076692253\t0.0335590809\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6146505423\t0.0353211037\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6216318593\t0.0371619967\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6286131763\t0.0390841403\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6355944933\t0.0410899422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6425758103\t0.0431818002\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6495571273\t0.0453620773\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6565384443\t0.0476331195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6635197613\t0.0499972279\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6705010783\t0.0524566876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6774823953\t0.055013685\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6844637124\t0.0576703922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6914450294\t0.0604288862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6984263464\t0.0632911755\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7054076634\t0.0662591675\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7123889804\t0.0693346562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7193702974\t0.0725193572\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7263516144\t0.0758148343\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7333329314\t0.079222519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7403142484\t0.082743714\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7472955654\t0.0863795307\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7542768824\t0.0901309397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7612581994\t0.093998709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7682395164\t0.097983435\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7752208335\t0.1020854864\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7822021505\t0.1063050042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7891834675\t0.1106419211\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7961647845\t0.1150959239\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8031461015\t0.1196664353\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8101274185\t0.124352626\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8171087355\t0.1291533722\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8240900525\t0.1340672794\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8310713695\t0.1390926655\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8380526865\t0.1442275198\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8450340035\t0.1494695269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8520153205\t0.1548160862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8589966376\t0.1602642034\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8659779546\t0.1658106065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8729592716\t0.1714516669\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8799405886\t0.1771834044\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8869219056\t0.1830015146\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8939032226\t0.1889013147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9008845396\t0.1948777997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9078658566\t0.2009256208\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9148471736\t0.2070390574\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9218284906\t0.2132120633\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9288098076\t0.2194382585\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9357911246\t0.2257109212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9427724416\t0.2320230195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9497537587\t0.2383671789\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9567350757\t0.2447357492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9637163927\t0.2511207768\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9706977097\t0.2575140206\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9776790267\t0.2639069824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9846603437\t0.2702909157\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9916416607\t0.2766568269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9986229777\t0.2829955301\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0056042947\t0.2892976467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0125856117\t0.2955536287\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0195669287\t0.3017537558\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0265482457\t0.3078882111\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0335295628\t0.3139470843\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0405108798\t0.3199203755\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0474921968\t0.3257980464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0544735138\t0.3315700491\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0614548308\t0.3372263467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0684361478\t0.3427569324\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0754174648\t0.3481518962\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0823987818\t0.35340142\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0893800988\t0.3584957975\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0963614158\t0.3634255113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1033427328\t0.3681812465\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1103240498\t0.3727538772\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1173053668\t0.3771345624\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1242866839\t0.3813147433\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1312680009\t0.3852861527\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1382493179\t0.389040888\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1452306349\t0.3925713956\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1522119519\t0.3958705438\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1591932689\t0.3989316065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1661745859\t0.4017482862\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1731559029\t0.4043147709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1801372199\t0.4066257357\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1871185369\t0.4086763699\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1940998539\t0.4104623614\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.2010811709\t0.4119799547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.208062488\t0.4132259537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.215043805\t0.4141977307\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.222025122\t0.4148932067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.229006439\t0.4153109122\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.235987756\t0.4154499655\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.242969073\t0.4153100586\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.24995039\t0.414891493\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.256931707\t0.4141951634\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.263913024\t0.4132225469\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.270894341\t0.4119756877\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.277875658\t0.4104572484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.284856975\t0.4086704185\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.291838292\t0.4066189448\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.2988196091\t0.4043071415\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3058009261\t0.4017398325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3127822431\t0.398922322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3197635601\t0.3958604501\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3267448771\t0.3925604928\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3337261941\t0.3890291695\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3407075111\t0.3852736327\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3476888281\t0.3813014293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3546701451\t0.3771204685\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3616514621\t0.372738997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3686327791\t0.3681656017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3756140961\t0.3634091094\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3825954132\t0.358478632\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3895767302\t0.353383505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3965580472\t0.3481332458\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4035393642\t0.3427375543\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4105206812\t0.3372062418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4175019982\t0.3315492316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4244833152\t0.3257765163\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4314646322\t0.3198981404\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4384459492\t0.3139241594\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4454272662\t0.3078646028\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4524085832\t0.3017294577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4593899002\t0.2955286549\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4663712172\t0.2892720047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4733525343\t0.2829692135\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4803338513\t0.2766298421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4873151683\t0.2702632639\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4942964853\t0.2638786765\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5012778023\t0.2574850542\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5082591193\t0.2510911499\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5152404363\t0.2447054617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5222217533\t0.2383362308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5292030703\t0.2319914033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5361843873\t0.2256786304\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5431657043\t0.2194052854\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5501470213\t0.2131784145\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5571283383\t0.2070047101\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5641096554\t0.2008905696\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5710909724\t0.1948420499\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5780722894\t0.1888648306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5850536064\t0.1829642811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5920349234\t0.1771454279\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5990162404\t0.1714129116\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6059975574\t0.1657710497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6129788744\t0.1602238298\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6199601914\t0.1547748601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6269415084\t0.1494274461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6339228254\t0.1441845268\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6409041424\t0.1390487376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6478854595\t0.1340223938\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6548667765\t0.1291074703\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6618480935\t0.1243056773\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6688294105\t0.1196184107\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6758107275\t0.115046756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6827920445\t0.1105915732\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6897733615\t0.1062534176\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6967546785\t0.1020326004\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7037359955\t0.0979292062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7107173125\t0.0939430551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7176986295\t0.0900737933\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7246799465\t0.0863208191\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7316612635\t0.0826833624\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7386425806\t0.0791604451\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7456238976\t0.0757509425\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7526052146\t0.0724535586\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7595865316\t0.0692668456\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7665678486\t0.0661892419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7735491656\t0.063219023\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7805304826\t0.0603543812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7875117996\t0.0575934088\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7944931166\t0.0549340886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8014744336\t0.0523743233\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8084557506\t0.0499119545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8154370676\t0.0475447734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8224183847\t0.0452704882\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8293997017\t0.0430867749\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8363810187\t0.0409912953\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8433623357\t0.0389816559\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8503436527\t0.0370554536\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8573249697\t0.0352102849\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8643062867\t0.0334437207\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8712876037\t0.0317533512\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8782689207\t0.0301367515\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8852502377\t0.0285915389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8922315547\t0.0271153134\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8992128717\t0.0257057206\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9061941887\t0.0243604255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9131755058\t0.0230771376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9201568228\t0.0218535688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9271381398\t0.0206874968\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9341194568\t0.0195767158\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9411007738\t0.0185190849\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9480820908\t0.0175125057\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9550634078\t0.0165549036\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9620447248\t0.0156442731\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9690260418\t0.0147786537\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9760073588\t0.0139561257\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9829886758\t0.0131748358\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9899699928\t0.0124329836\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9969513099\t0.0117288079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0039326269\t0.0110605941\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0109139439\t0.0104267068\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0178952609\t0.0098255518\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0248765779\t0.0092555718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0318578949\t0.0087152776\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0388392119\t0.0082032335\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0458205289\t0.0077180415\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0528018459\t0.0072583503\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0597831629\t0.0068228845\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0667644799\t0.0064103981\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0737457969\t0.0060196797\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0807271139\t0.0056495764\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.087708431\t0.0052989852\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.094689748\t0.0049668402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.101671065\t0.0046521099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.108652382\t0.0043538064\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.115633699\t0.0040710024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.122615016\t0.0038027619\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.129596333\t0.0035482283\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.13657765\t0.0033065566\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.143558967\t0.0030769433\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.150540284\t0.0028586162\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.157521601\t0.0026508323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.164502918\t0.0024528596\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1714842351\t0.0022640095\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1784655521\t0.0020836258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1854468691\t0.0019110654\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1924281861\t0.0017457057\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1994095031\t0.0015869471\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2063908201\t0.0014341984\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2133721371\t0.0012869163\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2203534541\t0.0011445498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2273347711\t0.0010065556\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2343160881\t0.000872424\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2412974051\t0.0007416386\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2482787221\t0.0006137125\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2552600391\t0.000488148\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2622413562\t0.0003644655\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2692226732\t0.0002421955\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2762039902\t0.0001208641\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2831853072\t-1.14903999997532E-008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2901666242\t-0.000120877\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2971479412\t-0.0002422161\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3041292582\t-0.0003644876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3111105752\t-0.0004881698\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3180918922\t-0.0006137216\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3250732092\t-0.000741656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3320545262\t-0.0008724328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3390358432\t-0.0010065727\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3460171603\t-0.0011445648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3529984773\t-0.0012869364\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3599797943\t-0.001434221\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3669611113\t-0.001586961\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3739424283\t-0.0017457247\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3809237453\t-0.0019110784\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3879050623\t-0.0020836489\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3948863793\t-0.0022640319\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4018676963\t-0.002452878\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4088490133\t-0.0026508479\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4158303303\t-0.0028586433\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4228116473\t-0.0030769791\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4297929643\t-0.0033065939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4367742814\t-0.0035482622\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4437555984\t-0.0038027911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4507369154\t-0.0040710286\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4577182324\t-0.0043538492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4646995494\t-0.0046521477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4716808664\t-0.0049668743\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4786621834\t-0.0052990269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4856435004\t-0.0056496186\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4926248174\t-0.0060197309\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4996061344\t-0.0064104517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5065874514\t-0.006822946\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5135687684\t-0.0072584091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5205500855\t-0.0077180972\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5275314025\t-0.0082032945\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5345127195\t-0.008715345\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5414940365\t-0.0092556382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5484753535\t-0.0098256308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5554566705\t-0.0104267916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5624379875\t-0.0110606825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5694193045\t-0.0117288948\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5764006215\t-0.0124330762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5833819385\t-0.0131749382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5903632555\t-0.0139562274\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5973445725\t-0.0147787628\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6043258895\t-0.0156443916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6113072066\t-0.0165550316\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6182885236\t-0.0175126352\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6252698406\t-0.0185192266\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6322511576\t-0.0195768656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6392324746\t-0.0206876511\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6462137916\t-0.0218537341\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6531951086\t-0.0230773112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6601764256\t-0.0243606135\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6671577426\t-0.0257059147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6741390596\t-0.0271155186\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6811203766\t-0.0285917487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6881016936\t-0.0301369816\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6950830107\t-0.0317535907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7020643277\t-0.0334439721\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7090456447\t-0.0352105511\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7160269617\t-0.0370557348\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7230082787\t-0.0389819508\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7299895957\t-0.0409916038\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7369709127\t-0.0430871062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7439522297\t-0.0452708227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7509335467\t-0.0475451311\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7579148637\t-0.0499123384\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7648961807\t-0.0523747193\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7718774977\t-0.0549344996\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7788588147\t-0.0575938434\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7858401318\t-0.0603548398\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7928214488\t-0.0632194987\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7998027658\t-0.0661897434\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8067840828\t-0.0692673733\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8137653998\t-0.0724541056\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8207467168\t-0.0757515121\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8277280338\t-0.0791610451\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8347093508\t-0.0826839846\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8416906678\t-0.0863214739\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8486719848\t-0.0900744708\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8556533018\t-0.0939437736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8626346188\t-0.0979299432\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8696159358\t-0.1020333754\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8765972529\t-0.1062542199\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8835785699\t-0.1105924107\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8905598869\t-0.1150476276\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8975412039\t-0.1196193131\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9045225209\t-0.1243066245\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9115038379\t-0.1291084499\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9184851549\t-0.1340234052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9254664719\t-0.1390497949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9324477889\t-0.1441856168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9394291059\t-0.1494285702\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9464104229\t-0.1547760343\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9533917399\t-0.1602250415\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.960373057\t-0.1657723083\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.967354374\t-0.171414203\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.974335691\t-0.1771467633\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.981317008\t-0.1829656699\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.988298325\t-0.1888662573\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.995279642\t-0.194843525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.002260959\t-0.2008920995\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.009242276\t-0.2070062838\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.016223593\t-0.2131800242\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.02320491\t-0.2194069587\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.030186227\t-0.2256803482\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.037167544\t-0.231993155\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.044148861\t-0.2383380301\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0511301781\t-0.244707321\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0581114951\t-0.2510930529\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0650928121\t-0.2574870079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0720741291\t-0.263880669\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0790554461\t-0.2702653108\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0860367631\t-0.2766319355\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0930180801\t-0.2829713534\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0999993971\t-0.2892741928\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1069807141\t-0.2955308896\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1139620311\t-0.3017317467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1209433481\t-0.307866931\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1279246651\t-0.3139265381\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1349059822\t-0.3199005626\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1418872992\t-0.3257789861\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1488686162\t-0.3315517393\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1558499332\t-0.337208795\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1628312502\t-0.3427401491\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1698125672\t-0.3481358894\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1767938842\t-0.353386183\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1837752012\t-0.3584813467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1907565182\t-0.3634118654\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1977378352\t-0.3681683879\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2047191522\t-0.3727418296\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2117004692\t-0.3771233285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2186817862\t-0.3813043186\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2256631033\t-0.3852765508\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2326444203\t-0.3890321131\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2396257373\t-0.3925634602\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2466070543\t-0.3958634494\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2535883713\t-0.3989253458\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2605696883\t-0.401742873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2675510053\t-0.4043102074\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2745323223\t-0.4066220274\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2815136393\t-0.408673507\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2884949563\t-0.4104603639\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2954762733\t-0.4119788145\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3024575903\t-0.413225672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3094389074\t-0.4141983016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3164202244\t-0.4148946425\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3234015414\t-0.4153132062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3303828584\t-0.4154531089\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3373641754\t-0.4153140619\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3443454924\t-0.4148963539\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3513268094\t-0.4142008688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3583081264\t-0.4132290873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3652894434\t-0.4119830779\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3722707604\t-0.41046547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3792520774\t-0.4086794689\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3862333944\t-0.4066288244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3932147114\t-0.4043178395\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4001960285\t-0.4017513337\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4071773455\t-0.3989346275\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4141586625\t-0.3958735522\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4211399795\t-0.3925743786\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4281212965\t-0.3890438394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4351026135\t-0.3852890722\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4420839305\t-0.3813176339\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4490652475\t-0.3771374246\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4560465645\t-0.372756699\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4630278815\t-0.3681840317\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4700091985\t-0.3634282695\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4769905155\t-0.3584985122\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4839718326\t-0.3534040946\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4909531496\t-0.3481545342\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4979344666\t-0.3427595347\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5049157836\t-0.3372289008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5118971006\t-0.3315725641\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5188784176\t-0.325800517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5258597346\t-0.3199227995\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5328410516\t-0.3139494604\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5398223686\t-0.3078905464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5468036856\t-0.301756041\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5537850026\t-0.2955558694\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5607663196\t-0.289299845\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5677476366\t-0.2829976781\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5747289537\t-0.2766589261\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5817102707\t-0.2702929597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5886915877\t-0.2639089828\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5956729047\t-0.2575159735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6026542217\t-0.2511226844\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6096355387\t-0.2447376109\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6166168557\t-0.2383689924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6235981727\t-0.2320247756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6305794897\t-0.2257126337\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6375608067\t-0.2194399231\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6445421237\t-0.2132136751\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6515234407\t-0.2070406202\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6585047578\t-0.200927142\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6654860748\t-0.1948792811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6724673918\t-0.1889027465\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6794487088\t-0.1830028988\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6864300258\t-0.177184746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6934113428\t-0.1714529666\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7003926598\t-0.1658118668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7073739768\t-0.1602654211\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7143552938\t-0.154817249\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7213366108\t-0.1494706677\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7283179278\t-0.1442286037\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7352992448\t-0.1390937189\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7422805618\t-0.1340682998\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7492618789\t-0.1291543499\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7562431959\t-0.1243535701\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7632245129\t-0.1196673379\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7702058299\t-0.1150967933\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7771871469\t-0.1106427575\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7841684639\t-0.1063058031\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7911497809\t-0.1020862557\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7981310979\t-0.0979841803\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8051124149\t-0.0939994305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8120937319\t-0.0901316222\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8190750489\t-0.0863801795\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8260563659\t-0.0827443333\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.833037683\t-0.0792231258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.840019\t-0.0758154052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.847000317\t-0.0725199141\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.853981634\t-0.0693351839\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.860962951\t-0.0662596656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.867944268\t-0.0632916551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.874925585\t-0.0604293454\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.881906902\t-0.0576708348\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.888888219\t-0.0550140973\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.895869536\t-0.0524570815\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.902850853\t-0.0499976126\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.90983217\t-0.0476334812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.916813487\t-0.0453624255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9237948041\t-0.0431821307\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9307761211\t-0.0410902616\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9377574381\t-0.0390844306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9447387551\t-0.0371622753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9517200721\t-0.0353213692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9587013891\t-0.0335593258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9656827061\t-0.0318737382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9726640231\t-0.0302622082\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9796453401\t-0.0287223461\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9866266571\t-0.0272517915\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9936079741\t-0.0258482178\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0005892911\t-0.0245092993\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0075706081\t-0.0232327452\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0145519252\t-0.0220163252\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0215332422\t-0.0208578266\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0285145592\t-0.0197550799\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0354958762\t-0.0187059765\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0424771932\t-0.0177084217\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0494585102\t-0.0167604056\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0564398272\t-0.0158599302\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0634211442\t-0.0150050907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0704024612\t-0.0141940171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0773837782\t-0.013424874\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0843650952\t-0.0126959205\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0913464122\t-0.0120054275\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0983277293\t-0.0113517658\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1053090463\t-0.0107333213\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1122903633\t-0.0101485787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1192716803\t-0.0095960303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1262529973\t-0.0090742659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1332343143\t-0.0085819069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1402156313\t-0.0081176417\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1471969483\t-0.0076802047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1541782653\t-0.0072684046\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1611595823\t-0.0068810658\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1681408993\t-0.0065171046\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1751222163\t-0.0061754634\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1821035333\t-0.0058551474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1890848504\t-0.0055552196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1960661674\t-0.005274761\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2030474844\t-0.0050129377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2100288014\t-0.004768949\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2170101184\t-0.0045420357\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2239914354\t-0.0043314916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2309727524\t-0.0041366463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2379540694\t-0.0039568891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2449353864\t-0.0037916376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2519167034\t-0.0036403678\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2588980204\t-0.0035025763\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2658793374\t-0.0033778206\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2728606545\t-0.0032656897\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2798419715\t-0.0031658254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2868232885\t-0.0030778828\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2938046055\t-0.0030015757\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3007859225\t-0.0029366706\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3077672395\t-0.0028829236\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3147485565\t-0.0028401936\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3217298735\t-0.0028083212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3287111905\t-0.0027872129\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3356925075\t-0.0027768277\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3426738245\t-0.0027771143\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3496551415\t-0.0027881066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3566364585\t-0.0028098575\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3636177756\t-0.0028424701\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3705990926\t-0.0028860582\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3775804096\t-0.0029407997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3845617266\t-0.0030069256\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3915430436\t-0.0030846668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3985243606\t-0.0031743369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4055056776\t-0.0032762623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4124869946\t-0.0033908289\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4194683116\t-0.0035184686\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4264496286\t-0.003659639\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4334309456\t-0.0038148802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4404122626\t-0.0039847465\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4473935797\t-0.0041698506\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4543748967\t-0.0043708692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4613562137\t-0.0045885238\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4683375307\t-0.0048235864\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4753188477\t-0.0050768922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4823001647\t-0.0053493219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4892814817\t-0.0056418343\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4962627987\t-0.0059554305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5032441157\t-0.0062911983\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5102254327\t-0.0066502625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5172067497\t-0.0070338399\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5241880667\t-0.0074432104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5311693837\t-0.007879721\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5381507008\t-0.0083448024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5451320178\t-0.0088399502\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5521133348\t-0.0093667534\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5590946518\t-0.0099268714\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5660759688\t-0.0105220482\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5730572858\t-0.0111541374\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5800386028\t-0.0118250393\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5870199198\t-0.0125367787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5940012368\t-0.0132914697\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6009825538\t-0.0140912922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6079638708\t-0.0149385683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6149451878\t-0.0158356832\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6219265049\t-0.0167851368\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6289078219\t-0.0177895419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6358891389\t-0.0188516047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6428704559\t-0.0199741299\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6498517729\t-0.0211600361\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6568330899\t-0.0224123636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6638144069\t-0.0237342371\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6707957239\t-0.0251289081\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6777770409\t-0.0265997152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6847583579\t-0.0281501388\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6917396749\t-0.0297837505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6987209919\t-0.0315042011\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.7057023089\t-0.033315303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.712683626\t-0.0352209282\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.719664943\t-0.0372250594\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.72664626\t-0.0393318019\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.733627577\t-0.0415453236\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.740608894\t-0.0438699003\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.747590211\t-0.0463098843\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.754571528\t-0.048869729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.761552845\t-0.0515539422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.768534162\t-0.0543671127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.775515479\t-0.0573138837\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.782496796\t-0.0603989635\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.789478113\t-0.0636270962\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.7964594301\t-0.0670030595\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8034407471\t-0.0705316689\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8104220641\t-0.0742177315\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8174033811\t-0.0780660976\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8243846981\t-0.08208156\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8313660151\t-0.0862689129\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8383473321\t-0.0906329142\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8453286491\t-0.095178257\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8523099661\t-0.0999095803\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8592912831\t-0.1048314069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8662726001\t-0.1099481976\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8732539171\t-0.1152642497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8802352341\t-0.1207837401\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8872165512\t-0.1265106883\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8941978682\t-0.1324489174\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9011791852\t-0.1386020436\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9081605022\t-0.1449734769\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9151418192\t-0.1515663605\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9221231362\t-0.1583835782\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9291044532\t-0.1654277127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9360857702\t-0.1727010186\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9430670872\t-0.1802054311\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9500484042\t-0.1879424967\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9570297212\t-0.1959133988\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9640110382\t-0.2041188759\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9709923553\t-0.2125592382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9779736723\t-0.2212343524\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9849549893\t-0.230143572\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9919363063\t-0.2392857691\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9989176233\t-0.2486592711\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0058989403\t-0.2582618427\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0128802573\t-0.2680907052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0198615743\t-0.2781424734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0268428913\t-0.2884131572\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0338242083\t-0.2988981318\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0408055253\t-0.3095921411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0477868423\t-0.320489272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0547681593\t-0.331582943\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0617494764\t-0.3428658981\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0687307934\t-0.3543302024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0757121104\t-0.3659672116\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0826934274\t-0.3777676066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0896747444\t-0.3897213594\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0966560614\t-0.4018177486\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1036373784\t-0.4140453684\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1106186954\t-0.4263921298\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1176000124\t-0.438845252\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1245813294\t-0.4513913041\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1315626464\t-0.4640162092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1385439634\t-0.4767052578\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1455252805\t-0.4894431115\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1525065975\t-0.5022138785\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1594879145\t-0.5150010698\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1664692315\t-0.5277877002\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1734505485\t-0.5405562612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1804318655\t-0.5532888036\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1874131825\t-0.5659669389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1943944995\t-0.5785718948\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2013758165\t-0.5910845596\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2083571335\t-0.6034855437\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2153384505\t-0.6157551974\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2223197675\t-0.6278736777\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2293010845\t-0.6398209934\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2362824016\t-0.6515770878\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2432637186\t-0.6631218437\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2502450356\t-0.6744351906\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2572263526\t-0.6854971369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2642076696\t-0.6962878409\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2711889866\t-0.7067876507\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2781703036\t-0.7169771995\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2851516206\t-0.726837432\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2921329376\t-0.7363496897\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2991142546\t-0.745495761\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3060955716\t-0.7542579462\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3130768886\t-0.7626191172\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3200582056\t-0.7705627588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3270395227\t-0.7780730564\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3340208397\t-0.7851349252\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3410021567\t-0.791734055\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3479834737\t-0.7978570141\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3549647907\t-0.8034912145\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3619461077\t-0.8086250401\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3689274247\t-0.8132478325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3759087417\t-0.8173499463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3828900587\t-0.8209227806\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3898713757\t-0.823958831\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3968526927\t-0.8264517017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4038340097\t-0.8283960953\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4108153268\t-0.8297879121\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4177966438\t-0.8306241813\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4247779608\t-0.8309031362\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4317592778\t-0.8306241758\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4387405948\t-0.8297879087\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4457219118\t-0.828396094\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4527032288\t-0.826451696\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4596845458\t-0.8239588351\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4666658628\t-0.8209227803\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4736471798\t-0.8173499492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4806284968\t-0.8132478386\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4876098138\t-0.8086250429\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4945911308\t-0.8034912215\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5015724479\t-0.7978570113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5085537649\t-0.7917340576\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5155350819\t-0.7851349267\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5225163989\t-0.7780730567\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5294977159\t-0.7705627667\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5364790329\t-0.7626191185\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5434603499\t-0.7542579485\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5504416669\t-0.7454957654\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5574229839\t-0.7363496887\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5644043009\t-0.7268374342\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5713856179\t-0.7169771984\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5783669349\t-0.7067876473\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.585348252\t-0.6962878353\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.592329569\t-0.6854971378\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.599310886\t-0.6744351903\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.606292203\t-0.6631218434\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.61327352\t-0.651577082\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.620254837\t-0.6398209952\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.627236154\t-0.6278736751\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.634217471\t-0.6157551904\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.641198788\t-0.6034855399\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.648180105\t-0.5910845591\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.655161422\t-0.578571891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.662142739\t-0.5659669328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.669124056\t-0.5532888094\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6761053731\t-0.5405562648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6830866901\t-0.5277877017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6900680071\t-0.5150010689\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6970493241\t-0.5022138755\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7040306411\t-0.4894431073\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7110119581\t-0.476705258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7179932751\t-0.4640162136\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7249745921\t-0.4513913063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7319559091\t-0.4388452433\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7389372261\t-0.4263921244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7459185431\t-0.414045365\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7528998601\t-0.4018177463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7598811772\t-0.3897213582\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7668624942\t-0.3777676043\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7738438112\t-0.3659672146\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7808251282\t-0.3543302021\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7878064452\t-0.3428659064\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7947877622\t-0.3315829437\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8017690792\t-0.3204892715\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8087503962\t-0.3095921439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8157317132\t-0.298898128\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8227130302\t-0.2884131512\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8296943472\t-0.2781424771\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8366756642\t-0.2680907066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8436569812\t-0.2582618408\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8506382983\t-0.2486592615\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8576196153\t-0.2392857692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8646009323\t-0.2301435775\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8715822493\t-0.2212343513\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8785635663\t-0.2125592414\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8855448833\t-0.2041188703\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8925262003\t-0.1959134007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8995075173\t-0.1879424996\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9064888343\t-0.1802054296\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9134701513\t-0.1727010192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9204514683\t-0.1654277098\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9274327853\t-0.1583835775\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9344141024\t-0.1515663617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9413954194\t-0.1449734792\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9483767364\t-0.1386020436\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9553580534\t-0.1324489139\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9623393704\t-0.1265106922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9693206874\t-0.1207837407\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9763020044\t-0.1152642502\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9832833214\t-0.1099481924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9902646384\t-0.1048314047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9972459554\t-0.0999095813\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0042272724\t-0.0951782545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0112085894\t-0.0906329136\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0181899064\t-0.0862689121\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0251712235\t-0.0820815634\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0321525405\t-0.0780660931\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0391338575\t-0.0742177365\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0461151745\t-0.0705316682\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0530964915\t-0.0670030606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0600778085\t-0.063627096\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0670591255\t-0.0603989597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0740404425\t-0.057313885\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0810217595\t-0.0543671114\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0880030765\t-0.0515539448\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0949843935\t-0.0488697248\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1019657105\t-0.0463098828\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1089470276\t-0.0438699005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1159283446\t-0.0415453234\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1229096616\t-0.0393318033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1298909786\t-0.0372250614\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1368722956\t-0.0352209285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1438536126\t-0.0333153016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1508349296\t-0.0315042034\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1578162466\t-0.0297837445\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1647975636\t-0.0281501428\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1717788806\t-0.0265997174\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1787601976\t-0.025128904\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1857415146\t-0.0237342353\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1927228316\t-0.0224123598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1997041487\t-0.0211600422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2066854657\t-0.0199741305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2136667827\t-0.018851601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2206480997\t-0.0177895435\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2276294167\t-0.0167851328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2346107337\t-0.0158356767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2415920507\t-0.0149385625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2485733677\t-0.0140912892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2555546847\t-0.013291464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2625360017\t-0.0125367767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2695173187\t-0.0118250397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2764986357\t-0.0111541402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2834799528\t-0.0105220533\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2904612698\t-0.009926871\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2974425868\t-0.0093667517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3044239038\t-0.0088399514\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3114052208\t-0.0083447966\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3183865378\t-0.0078797199\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3253678548\t-0.0074432094\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3323491718\t-0.0070338325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3393304888\t-0.0066502603\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3463118058\t-0.0062911859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3532931228\t-0.0059554294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3602744398\t-0.0056418301\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3672557568\t-0.005349322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3742370739\t-0.0050768907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3812183909\t-0.004823582\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3881997079\t-0.0045885227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3951810249\t-0.004370869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4021623419\t-0.0041698464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4091436589\t-0.0039847369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4161249759\t-0.0038148692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4231062929\t-0.0036596369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4300876099\t-0.0035184597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4370689269\t-0.0033908269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4440502439\t-0.0032762545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4510315609\t-0.003174323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.458012878\t-0.0030846656\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.464994195\t-0.0030069223\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.471975512\t-0.0029408023\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.478956829\t-0.0028860486\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.485938146\t-0.0028424584\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.492919463\t-0.0028098559\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.49990078\t-0.0027880924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.506882097\t-0.0027771008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.513863414\t-0.0027768127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.520844731\t-0.0027872063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.527826048\t-0.0028083076\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.534807365\t-0.0028401817\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.541788682\t-0.0028829066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5487699991\t-0.0029366483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5557513161\t-0.0030015704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5627326331\t-0.0030778617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5697139501\t-0.0031657997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5766952671\t-0.003265673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5836765841\t-0.0033778014\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5906579011\t-0.0035025525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5976392181\t-0.0036403439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6046205351\t-0.0037916192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6116018521\t-0.0039568608\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6185831691\t-0.0041366165\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6255644861\t-0.0043314588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6325458031\t-0.0045420038\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6395271202\t-0.0047689151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6465084372\t-0.0050129042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6534897542\t-0.0052747237\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6604710712\t-0.0055551739\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6674523882\t-0.0058551029\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6744337052\t-0.0061754207\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6814150222\t-0.0065170555\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6883963392\t-0.0068810115\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6953776562\t-0.0072683462\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7023589732\t-0.0076801451\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7093402902\t-0.0081175795\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7163216072\t-0.0085818423\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7233029243\t-0.0090741904\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7302842413\t-0.0095959524\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7372655583\t-0.010148499\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7442468753\t-0.0107332456\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7512281923\t-0.0113516811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7582095093\t-0.0120053347\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7651908263\t-0.0126958244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7721721433\t-0.0134247705\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7791534603\t-0.0141939049\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7861347773\t-0.0150049736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7931160943\t-0.0158598096\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8000974113\t-0.016760265\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8070787283\t-0.0177082865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8140600454\t-0.0187058328\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8210413624\t-0.0197549297\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8280226794\t-0.0208576725\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8350039964\t-0.0220161596\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8419853134\t-0.0232325666\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8489666304\t-0.0245091014\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8559479474\t-0.025848021\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8629292644\t-0.0272515876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8699105814\t-0.0287221285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8768918984\t-0.0302619678\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8838732154\t-0.0318734914\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8908545324\t-0.0335590668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8978358495\t-0.0353211037\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9048171665\t-0.0371619826\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9117984835\t-0.0390841327\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9187798005\t-0.0410899357\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9257611175\t-0.0431817937\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9327424345\t-0.0453620708\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9397237515\t-0.047633113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9467050685\t-0.0499972214\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9536863855\t-0.0524566735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9606677025\t-0.0550136709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9676490195\t-0.0576703857\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9746303365\t-0.0604288797\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9816116535\t-0.063291169\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9885929706\t-0.0662591534\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9955742876\t-0.0693346497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0025556046\t-0.0725193507\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0095369216\t-0.0758148213\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0165182386\t-0.0792225125\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0234995556\t-0.0827436934\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0304808726\t-0.08637951\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0374621896\t-0.0901309191\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0444435066\t-0.093998696\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0514248236\t-0.097983422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0584061406\t-0.1020854658\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0653874576\t-0.1063049836\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0723687747\t-0.1106419005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0793500917\t-0.1150959032\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0863314087\t-0.1196664158\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0933127257\t-0.1243526054\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1002940427\t-0.1291533516\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1072753597\t-0.1340672588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1142566767\t-0.1390926384\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1212379937\t-0.1442274927\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1282193107\t-0.1494695063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1352006277\t-0.1548160526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1421819447\t-0.1602641687\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1491632617\t-0.1658105794\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1561445787\t-0.1714516333\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1631258958\t-0.1771833773\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1701072128\t-0.1830014734\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1770885298\t-0.1889012811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1840698468\t-0.194877766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1910511638\t-0.2009255872\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1980324808\t-0.2070390162\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2050137978\t-0.2132120232\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2119951148\t-0.2194382184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2189764318\t-0.2257108876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2259577488\t-0.2320229783\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2329390658\t-0.2383671387\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2399203828\t-0.2447357156\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2469016999\t-0.2511207367\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2538830169\t-0.2575139729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2608643339\t-0.2639069347\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2678456509\t-0.2702908615\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2748269679\t-0.2766567792\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2818082849\t-0.2829954824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2887896019\t-0.289297599\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2957709189\t-0.2955535745\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3027522359\t-0.3017537016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3097335529\t-0.3078881568\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3167148699\t-0.3139470301\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3236961869\t-0.3199203137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3306775039\t-0.3257979922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.337658821\t-0.3315699884\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.344640138\t-0.3372262784\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.351621455\t-0.3427568782\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.358602772\t-0.348151842\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.365584089\t-0.3534013451\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.372565406\t-0.3584957292\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.379546723\t-0.3634254506\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.38652804\t-0.3681811782\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.393509357\t-0.3727538088\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.400490674\t-0.3771345016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.407471991\t-0.3813146684\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.414453308\t-0.3852860779\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4214346251\t-0.3890408132\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4284159421\t-0.3925713348\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4353972591\t-0.3958704765\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4423785761\t-0.3989315317\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4493598931\t-0.4017482113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4563412101\t-0.404314696\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4633225271\t-0.4066256685\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4703038441\t-0.4086762886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4772851611\t-0.4104622866\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4842664781\t-0.4119798799\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4912477951\t-0.4132258789\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4982291121\t-0.4141976494\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5052104291\t-0.4148931254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5121917462\t-0.4153108308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5191730632\t-0.4154498841\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5261543802\t-0.4153099772\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5331356972\t-0.4148914181\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5401170142\t-0.4141950885\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5470983312\t-0.413222472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5540796482\t-0.4119756129\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5610609652\t-0.4104571736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5680422822\t-0.4086703436\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5750235992\t-0.4066188775\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5820049162\t-0.4043070667\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5889862332\t-0.4017397576\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5959675503\t-0.3989222472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6029488673\t-0.3958603753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6099301843\t-0.3925604179\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6169115013\t-0.3890290947\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6238928183\t-0.3852735579\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6308741353\t-0.3813013545\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6378554523\t-0.3771203937\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6448367693\t-0.3727389363\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6518180863\t-0.3681655269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6587994033\t-0.3634090345\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6657807203\t-0.3584785637\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6727620373\t-0.3533834366\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6797433543\t-0.3481331851\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6867246714\t-0.3427374935\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6937059884\t-0.3372061735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7006873054\t-0.3315491633\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7076686224\t-0.3257764555\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7146499394\t-0.3198980861\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7216312564\t-0.3139241052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7286125734\t-0.3078645486\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7355938904\t-0.3017294035\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7425752074\t-0.2955285931\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7495565244\t-0.2892719505\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7565378414\t-0.2829691658\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7635191584\t-0.2766297944\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7705004754\t-0.2702632162\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7774817925\t-0.2638786223\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7844631095\t-0.2574850065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7914444265\t-0.2510911022\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7984257435\t-0.2447054216\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8054070605\t-0.2383361831\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8123883775\t-0.2319913556\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8193696945\t-0.2256785968\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8263510115\t-0.2194052518\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8333323285\t-0.2131783733\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8403136455\t-0.2070046765\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8472949625\t-0.200890536\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8542762795\t-0.1948420087\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8612575966\t-0.1888647894\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8682389136\t-0.182964254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8752202306\t-0.1771453943\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8822015476\t-0.171412878\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8891828646\t-0.1657710226\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8961641816\t-0.1602238027\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9031454986\t-0.154774833\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9101268156\t-0.149427419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9171081326\t-0.1441844997\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9240894496\t-0.1390487105\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9310707666\t-0.1340223667\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9380520836\t-0.1291074497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9450334006\t-0.1243056643\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9520147177\t-0.1196183836\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9589960347\t-0.115046743\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9659773517\t-0.1105915526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9729586687\t-0.106253397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9799399857\t-0.1020325874\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9869213027\t-0.0979291856\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9939026197\t-0.093943042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0008839367\t-0.0900737803\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0078652537\t-0.0863207985\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0148465707\t-0.0826833418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0218278877\t-0.0791604321\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0288092047\t-0.0757509295\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0357905218\t-0.0724535445\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0427718388\t-0.0692668391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0497531558\t-0.0661892213\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0567344728\t-0.0632190089\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0637157898\t-0.0603543747\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0706971068\t-0.0575933947\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0776784238\t-0.0549340745\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0846597408\t-0.0523743103\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0916410578\t-0.049911948\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0986223748\t-0.0475447669\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1056036918\t-0.0452704741\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1125850088\t-0.0430867683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1195663258\t-0.0409912812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1265476429\t-0.0389816494\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1335289599\t-0.037055447\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1405102769\t-0.0352102784\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1474915939\t-0.0334437142\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1544729109\t-0.0317533372\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1614542279\t-0.030136745\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1684355449\t-0.0285915324\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1754168619\t-0.0271152993\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1823981789\t-0.0257057141\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1893794959\t-0.0243604255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1963608129\t-0.0230771376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.2033421299\t-0.0218535688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.210323447\t-0.0206874903\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.217304764\t-0.0195767093\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.224286081\t-0.0185190849\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.231267398\t-0.0175124992\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.238248715\t-0.0165549036\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.245230032\t-0.0156442666\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.252211349\t-0.0147786396\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.259192666\t-0.0139561192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.266173983\t-0.0131748358\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.2731553\t-0.0124329771\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.280136617\t-0.0117288014\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.287117934\t-0.0110605941\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.294099251\t-0.0104267068\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3010805681\t-0.0098255518\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3080618851\t-0.0092555718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3150432021\t-0.0087152776\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3220245191\t-0.0082032335\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3290058361\t-0.007718035\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3359871531\t-0.0072583503\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3429684701\t-0.0068228845\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3499497871\t-0.0064103905\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3569311041\t-0.0060196797\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3639124211\t-0.0056495764\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3708937381\t-0.0052989787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3778750551\t-0.0049668402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3848563722\t-0.0046521099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3918376892\t-0.0043538064\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3988190062\t-0.0040709948\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4058003232\t-0.0038027543\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4127816402\t-0.0035482283\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4197629572\t-0.0033065566\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4267442742\t-0.0030769433\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4337255912\t-0.0028586162\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4407069082\t-0.0026508323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4476882252\t-0.0024528531\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4546695422\t-0.0022640095\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4616508592\t-0.0020836258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4686321762\t-0.0019110654\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4756134933\t-0.0017456992\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4825948103\t-0.0015869471\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4895761273\t-0.0014341984\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4965574443\t-0.0012869163\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5035387613\t-0.0011445498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5105200783\t-0.0010065556\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5175013953\t-0.000872424\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5244827123\t-0.0007416386\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5314640293\t-0.0006137125\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5384453463\t-0.000488148\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5454266633\t-0.0003644655\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5524079803\t-0.0002421955\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5593892974\t-0.0001208641\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5663706144\t1.14903999997532E-008\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\addplot[smooth, red] table[x=angle,y=energy] {\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t angle energy\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0\t0.0033957422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.006981317\t0.0034023092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.013962634\t0.0034220302\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.020943951\t0.0034549693\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.027925268\t0.0035012598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.034906585\t0.0035610578\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.041887902\t0.0036345792\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0488692191\t0.0037220864\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0558505361\t0.0038238939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0628318531\t0.0039403683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0698131701\t0.0040719088\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0767944871\t0.0042189989\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0837758041\t0.004382143\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0907571211\t0.0045619073\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.0977384381\t0.0047589218\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1047197551\t0.0049738688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1117010721\t0.0052074825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1186823891\t0.0054605709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1256637061\t0.0057339766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1326450232\t0.0060286173\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1396263402\t0.0063454874\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1466076572\t0.0066856152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1535889742\t0.0070501253\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1605702912\t0.0074401966\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1675516082\t0.0078570829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1745329252\t0.0083020998\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1815142422\t0.0087766463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1884955592\t0.0092821954\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.1954768762\t0.009820287\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2024581932\t0.010392573\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2094395102\t0.0110007377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2164208272\t0.0116465891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2234021443\t0.0123319959\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2303834613\t0.0130589211\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2373647783\t0.0138294255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2443460953\t0.0146456293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2513274123\t0.0155097907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2583087293\t0.0164242132\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.2652900463\t0.0173913112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 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0.9424777961\t0.6673296917\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9494591131\t0.6738335647\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9564404301\t0.6799490751\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9634217471\t0.6856638588\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9704030641\t0.6909662437\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9773843811\t0.6958453591\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9843656981\t0.7002911435\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9913470151\t0.7042943477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 0.9983283321\t0.7078466345\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0053096491\t0.7109405541\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0122909662\t0.713569611\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0192722832\t0.7157282424\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0262536002\t0.7174118856\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0332349172\t0.7186169908\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0402162342\t0.7193409766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0471975512\t0.7195823085\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0541788682\t0.7193404776\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0611601852\t0.7186160004\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0681415022\t0.7174104115\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0751228192\t0.7157262704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0821041362\t0.7135671411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0890854532\t0.710937608\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.0960667703\t0.707843197\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1030480873\t0.7042904275\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1100294043\t0.7002867407\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1170107213\t0.6958404812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1239920383\t0.6909608841\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1309733553\t0.6856580177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1379546723\t0.679942774\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1449359893\t0.6738267972\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1519173063\t0.6673224653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1588986233\t0.6604428497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1658799403\t0.653201644\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1728612573\t0.6456131495\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1798425743\t0.6376922084\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1868238914\t0.6294541447\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.1938052084\t0.6209147247\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2007865254\t0.6120901095\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2077678424\t0.6029968042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2147491594\t0.5936515871\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2217304764\t0.5840714609\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2287117934\t0.5742736161\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2356931104\t0.5642753744\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2426744274\t0.5540941115\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2496557444\t0.5437472436\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2566370614\t0.5332521519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2636183784\t0.5226261375\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2705996955\t0.5118863854\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2775810125\t0.501049919\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2845623295\t0.4901335285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2915436465\t0.4791537612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.2985249635\t0.468126878\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3055062805\t0.4570688037\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3124875975\t0.445995078\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3194689145\t0.4349208631\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3264502315\t0.4238608865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3334315485\t0.4128293939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3404128655\t0.4018401887\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3473941825\t0.3909065177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3543754995\t0.3800411323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3613568166\t0.3692562274\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3683381336\t0.3585634339\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3753194506\t0.3479738199\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3823007676\t0.3374978405\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3892820846\t0.3271453892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.3962634016\t0.3169257477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4032447186\t0.3068476003\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4102260356\t0.2969190128\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4172073526\t0.2871474688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4241886696\t0.277539838\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4311699866\t0.2681024227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4381513036\t0.2588408909\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4451326207\t0.2497603901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4521139377\t0.2408654634\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4590952547\t0.2321601086\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4660765717\t0.2236477822\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4730578887\t0.2153314168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4800392057\t0.2072134301\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4870205227\t0.1992957362\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.4940018397\t0.1915797961\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5009831567\t0.1840665782\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5079644737\t0.1767566412\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5149457907\t0.1696501065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5219271077\t0.1627467084\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5289084247\t0.1560457886\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5358897418\t0.1495463351\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5428710588\t0.1432469966\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5498523758\t0.1371461093\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5568336928\t0.1312417113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5638150098\t0.1255315597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5707963268\t0.1200131643\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5777776438\t0.1146837952\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5847589608\t0.1095405035\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5917402778\t0.1045801473\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.5987215948\t0.0997994174\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6057029118\t0.0951948248\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6126842288\t0.0907627774\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6196655459\t0.0864995053\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6266468629\t0.0824011788\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6336281799\t0.0784638675\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6406094969\t0.0746835422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6475908139\t0.071056151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6545721309\t0.0675775496\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6615534479\t0.0642436103\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6685347649\t0.0610501422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6755160819\t0.0579929762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6824973989\t0.0550679217\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6894787159\t0.0522708236\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.6964600329\t0.0495975171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.7034413499\t0.047043895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.710422667\t0.044605888\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.717403984\t0.042279456\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.724385301\t0.0400606112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.731366618\t0.0379454631\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.738347935\t0.0359301322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.745329252\t0.0340108497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.752310569\t0.0321839063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.759291886\t0.0304456612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.766273203\t0.0287925819\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.77325452\t0.0272211836\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.780235837\t0.0257281108\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.787217154\t0.0243100673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.7941984711\t0.0229638538\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8011797881\t0.021686365\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8081611051\t0.0204745979\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8151424221\t0.0193256219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8221237391\t0.0182366182\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8291050561\t0.0172048526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8360863731\t0.0162276897\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8430676901\t0.0153025643\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8500490071\t0.0144270438\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8570303241\t0.0135987476\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8640116411\t0.012815396\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8709929581\t0.0120748131\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8779742751\t0.0113748756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8849555922\t0.0107135746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8919369092\t0.0100889787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.8989182262\t0.0094992137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9058995432\t0.008942515\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9128808602\t0.0084171732\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9198621772\t0.0079215648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9268434942\t0.0074541346\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9338248112\t0.007013402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9408061282\t0.0065979322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9477874452\t0.0062063868\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9547687622\t0.0058374653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9617500792\t0.0054899576\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9687313962\t0.0051626769\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9757127133\t0.0048545202\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9826940303\t0.0045644367\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9896753473\t0.0042914155\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 1.9966566643\t0.0040344913\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0036379813\t0.0037927758\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0106192983\t0.0035654047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0176006153\t0.003351559\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0245819323\t0.0031504633\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0315632493\t0.0029613909\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0385445663\t0.002783652\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0455258833\t0.0026165872\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0525072003\t0.0024595685\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0594885174\t0.0023120254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0664698344\t0.0021733808\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0734511514\t0.0020431272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0804324684\t0.0019207679\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0874137854\t0.0018058244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.0943951024\t0.0016978663\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1013764194\t0.0015964781\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1083577364\t0.0015012519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1153390534\t0.0014118375\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1223203704\t0.0013278692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1293016874\t0.0012490196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1362830044\t0.001175\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1432643214\t0.0011054917\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1502456385\t0.0010402331\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1572269555\t0.0009789618\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1642082725\t0.0009214391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1711895895\t0.0008674286\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1781709065\t0.0008167186\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1851522235\t0.0007691151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1921335405\t0.0007244118\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.1991148575\t0.0006824418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2060961745\t0.0006430389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2130774915\t0.000606034\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2200588085\t0.0005712796\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2270401255\t0.0005386463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2340214426\t0.0005080052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2410027596\t0.0004792151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2479840766\t0.0004521795\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2549653936\t0.0004267811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2619467106\t0.000402919\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2689280276\t0.0003805063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2759093446\t0.0003594508\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2828906616\t0.0003396535\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2898719786\t0.000321057\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.2968532956\t0.0003035646\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3038346126\t0.0002871339\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3108159296\t0.0002716802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3177972466\t0.0002571491\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3247785637\t0.0002434896\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3317598807\t0.0002306384\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3387411977\t0.0002185523\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3457225147\t0.0002071789\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3527038317\t0.0001964801\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3596851487\t0.0001864102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3666664657\t0.0001769289\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3736477827\t0.0001680079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3806290997\t0.0001596123\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3876104167\t0.0001516908\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.3945917337\t0.0001442402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4015730507\t0.000137221\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4085543678\t0.000130606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4155356848\t0.0001243767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4225170018\t0.0001184946\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4294983188\t0.0001129598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4364796358\t0.000107735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4434609528\t0.0001028092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4504422698\t9.81593874000014E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4574235868\t9.37755360000021E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4644049038\t8.96346342000032E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4713862208\t8.57277544000015E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4783675378\t8.20394565999999E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4853488548\t7.85586372000005E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4923301718\t7.52676802000009E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.4993114889\t7.21663204000002E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5062928059\t6.9221751399999E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5132741229\t6.64488880000036E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5202554399\t6.38192602000071E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5272367569\t6.13434411999986E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5342180739\t5.89907902000045E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5411993909\t0.000056773\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5481807079\t5.4670176000006E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5551620249\t5.26778140000073E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5621433419\t5.07945292000031E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5691246589\t0.000049006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.5761059759\t4.7316193400003E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.583087293\t4.57108746000009E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.59006861\t4.41962682000072E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.597049927\t4.27439405999985E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.604031244\t0.000041384\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.611012561\t4.00783704000013E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.617993878\t3.88375924000039E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.624975195\t3.7670185000005E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.631956512\t3.65466863999908E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.638937829\t3.54788644000015E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.645919146\t3.4474166000012E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.652900463\t3.35064324000073E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.65988178\t3.25918250000112E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.666863097\t3.17085629999991E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6738444141\t3.0873934800002E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6808257311\t3.00792208000056E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6878070481\t2.93081347999963E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.6947883651\t0.000028578\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7017696821\t2.78672744000175E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7087509991\t2.71932623999993E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7157323161\t2.65485030000179E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7227136331\t2.59211920000113E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7296949501\t2.53190779999912E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7366762671\t2.47347608000048E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7436575841\t2.41760565999893E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7506389011\t2.36312963999841E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7576202182\t2.31061984000136E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7646015352\t2.25902518000143E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7715828522\t2.20914184000109E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7785641692\t2.16166162000098E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7855454862\t2.11380928000215E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7925268032\t2.06758585999944E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.7995081202\t2.02174244000042E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8064894372\t1.97752860000289E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8134707542\t0.000019337\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8204520712\t1.88957152000291E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8274333882\t1.84737085999753E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8344147052\t1.80554645999864E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8413960222\t1.7630893600018E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8483773393\t1.72213982000124E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8553586563\t1.68104544000236E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8623399733\t1.63935036000262E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8693212903\t1.59822890000161E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8763026073\t0.000015568\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8832839243\t1.51688834000296E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8902652413\t1.47567082000033E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.8972465583\t1.434439539999E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9042278753\t1.39297288000439E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9112091923\t0.000013517\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9181905093\t1.30954170000641E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9251718263\t1.2683367200006E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9321531434\t1.22548642000342E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9391344604\t1.18327184000511E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9461157774\t1.14028730000548E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9530970944\t1.09718833999906E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9600784114\t1.05408832000164E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9670597284\t1.01023280000834E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9740410454\t9.65626080001725E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9810223624\t9.21247760005128E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9880036794\t8.75907160002853E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 2.9949849964\t8.30472740004362E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0019663134\t7.84835059997047E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0089476304\t7.38665520000348E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0159289474\t6.91741460001616E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0229102645\t6.45572700003205E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0298915815\t5.97978180000291E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0368728985\t5.50569500007425E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0438542155\t5.0311048000462E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0508355325\t4.54059339999313E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0578168495\t4.0631664001145E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0647981665\t3.57138340001395E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0717794835\t3.08872939996929E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0787608005\t2.59633799998271E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0857421175\t2.0991488000055E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0927234345\t0.000001601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.0997047515\t0.000001112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1066860685\t6.20809600004879E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1136673856\t1.2553580008845E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1206487026\t-3.73510000040156E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1276300196\t-8.70672000052863E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1346113366\t-1.36469840000819E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1415926536\t-1.85440500000134E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1485739706\t-2.35060420000766E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1555552876\t-0.000002849\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1625366046\t-3.33773459992859E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1695179216\t-3.82459459996598E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1764992386\t-4.30763400000034E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1834805556\t-4.78823460003941E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1904618726\t-5.26350420004817E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.1974431897\t-5.74471279995992E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2044245067\t-6.21173699999478E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2114058237\t-6.68355899996476E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2183871407\t-7.14451139999239E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2253684577\t-7.60284199990968E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2323497747\t-8.06142880005467E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2393310917\t-8.5069311999697E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2463124087\t-8.95425499991553E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2532937257\t-9.39448959996758E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2602750427\t-9.82635859997361E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2672563597\t-1.02583853999749E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2742376767\t-1.0682859199973E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2812189937\t-1.11007783999972E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2882003108\t-1.15164235999776E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.2951816278\t-1.19275941999897E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3021629448\t-1.23245177999776E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3091442618\t-1.27256424000133E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3161255788\t-1.31158049999081E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3231068958\t-0.000013507\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3300882128\t-1.38915949999263E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3370695298\t-1.42686355999455E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3440508468\t-1.46403131999762E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3510321638\t-1.50131831999634E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3580134808\t-1.53786105999698E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3649947978\t-1.57463913999512E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3719761149\t-1.6104618799917E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3789574319\t-1.64652335999937E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3859387489\t-1.68162977999819E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3929200659\t-1.71773084000071E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.3999013829\t-1.75232713999773E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4068826999\t-1.78768931999307E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4138640169\t-1.82283129999405E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4208453339\t-1.85828292000034E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4278266509\t-1.89316091999681E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4348079679\t-0.000019294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4417892849\t-1.96567314000029E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4487706019\t-2.00195835999839E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.4557519189\t-2.0381170199979E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.462733236\t-2.07541793999788E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.469714553\t-0.000021135\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.47669587\t-0.00002152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.483677187\t-0.000021914\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.490658504\t-2.23135333999869E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.497639821\t-2.27233211999855E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.504621138\t-2.31461048000037E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.511602455\t-2.35791168000099E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.518583772\t-2.40303969999966E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.525565089\t-2.44927636000036E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.532546406\t-2.49752637999923E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.539527723\t-2.5465215999998E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5465090401\t-2.59781017999949E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5534903571\t-2.65206884000058E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5604716741\t-2.70747631999912E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5674529911\t-0.00002766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5744343081\t-2.82636609999914E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5814156251\t-2.88944284000114E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5883969421\t-2.95665357999963E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.5953782591\t-3.02551021999775E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6023595761\t-3.09872782000009E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6093408931\t-3.17457181999853E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6163222101\t-3.25564353999969E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6233035271\t-3.33966103999994E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6302848441\t-3.42824461999902E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6372661612\t-0.000035216\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6442474782\t-3.6189645599996E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6512287952\t-3.72281259999885E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6582101122\t-3.83172209999921E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6651914292\t-3.94621912000043E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6721727462\t-4.06747887999881E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6791540632\t-4.19428924000043E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6861353802\t-4.32923279999886E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.6931166972\t-0.000044712\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7000980142\t-0.000046205\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7070793312\t-0.000047783\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7140606482\t-4.94544533999955E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7210419653\t-5.12190511999938E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7280232823\t-5.3087414200001E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7350045993\t-5.50592824000024E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7419859163\t-5.71398123999992E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7489672333\t-5.93482606000006E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7559485503\t-6.16756785999928E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7629298673\t-6.41358957999966E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7699111843\t-6.67535852000023E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7768925013\t-6.95056986000017E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7838738183\t-7.24331795999983E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7908551353\t-7.55292505999969E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.7978364523\t-7.88055753999997E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8048177693\t-8.22673105999941E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8117990864\t-8.5947817600001E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8187804034\t-8.98414145999969E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8257617204\t-0.000093976\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8327430374\t-9.83438707999988E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8397243544\t-0.0001029837\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8467056714\t-0.0001079018\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8536869884\t-0.0001131152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8606683054\t-0.0001186509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8676496224\t-0.000124513\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8746309394\t-0.000130736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8816122564\t-0.0001373478\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8885935734\t-0.0001443558\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.8955748905\t-0.0001518119\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9025562075\t-0.0001597116\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9095375245\t-0.0001681089\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9165188415\t-0.0001770242\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9235001585\t-0.0001864986\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9304814755\t-0.0001965687\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9374627925\t-0.0002072659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9444441095\t-0.000218628\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9514254265\t-0.0002307101\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9584067435\t-0.0002435522\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9653880605\t-0.0002572171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9723693775\t-0.0002717404\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9793506945\t-0.0002871924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9863320116\t-0.0003036177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 3.9933133286\t-0.0003211016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0002946456\t-0.0003396941\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0072759626\t-0.0003594892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0142572796\t-0.000380551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0212385966\t-0.0004029651\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0282199136\t-0.0004268223\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0352012306\t-0.0004522149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0421825476\t-0.0004792533\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0491638646\t-0.000508033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0561451816\t-0.0005386785\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0631264986\t-0.0005713092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0701078157\t-0.000606054\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0770891327\t-0.0006430647\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0840704497\t-0.0006824649\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0910517667\t-0.000724439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.0980330837\t-0.0007691332\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1050144007\t-0.0008167427\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1119957177\t-0.0008674509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1189770347\t-0.0009214605\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1259583517\t-0.0009789786\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1329396687\t-0.0010402513\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1399209857\t-0.0011055086\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1469023027\t-0.0011750108\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1538836197\t-0.0012490377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1608649368\t-0.0013278852\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1678462538\t-0.0014118474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1748275708\t-0.0015012647\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1818088878\t-0.0015964854\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1887902048\t-0.001697877\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.1957715218\t-0.0018058311\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2027528388\t-0.0019207718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2097341558\t-0.0020431394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2167154728\t-0.0021733906\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2236967898\t-0.0023120306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2306781068\t-0.0024595791\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2376594238\t-0.0026165947\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2446407409\t-0.0027836608\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2516220579\t-0.0029614065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2586033749\t-0.0031504838\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2655846919\t-0.0033515703\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2725660089\t-0.0035654179\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2795473259\t-0.0037927922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2865286429\t-0.0040345089\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.2935099599\t-0.0042914271\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3004912769\t-0.0045644577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3074725939\t-0.0048545368\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3144539109\t-0.0051626895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3214352279\t-0.005489971\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.3284165449\t-0.0058374822\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.335397862\t-0.0062064\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.342379179\t-0.0065979457\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.349360496\t-0.007013422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.356341813\t-0.0074541563\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.36332313\t-0.0079215935\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.370304447\t-0.0084171956\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.377285764\t-0.0089425419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.384267081\t-0.0094992441\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.391248398\t-0.0100890018\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.398229715\t-0.0107136048\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.405211032\t-0.0113749013\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.412192349\t-0.0120748391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.419173666\t-0.0128154305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4261549831\t-0.0135987794\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4331363001\t-0.014427077\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4401176171\t-0.0153026118\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4470989341\t-0.0162277331\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4540802511\t-0.017204901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4610615681\t-0.0182366653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4680428851\t-0.0193256704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4750242021\t-0.0204746529\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4820055191\t-0.021686423\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4889868361\t-0.0229639159\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.4959681531\t-0.0243101284\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5029494701\t-0.0257281744\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5099307872\t-0.0272212493\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5169121042\t-0.0287926483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5238934212\t-0.030445737\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5308747382\t-0.0321839797\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5378560552\t-0.0340109372\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5448373722\t-0.0359302227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5518186892\t-0.0379455519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5588000062\t-0.0400607157\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5657813232\t-0.0422795582\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5727626402\t-0.0446059958\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5797439572\t-0.0470440126\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5867252742\t-0.0495976379\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.5937065912\t-0.0522709521\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6006879083\t-0.055068065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6076692253\t-0.0579931188\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6146505423\t-0.0610502959\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6216318593\t-0.0642437688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6286131763\t-0.0675777135\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6355944933\t-0.0710563221\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6425758103\t-0.0746837291\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6495571273\t-0.0784640596\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6565384443\t-0.0824013874\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6635197613\t-0.0864997192\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6705010783\t-0.0907629965\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6774823953\t-0.0951950612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6844637124\t-0.0997996617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6914450294\t-0.1045804045\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.6984263464\t-0.1095407702\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7054076634\t-0.1146840757\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7123889804\t-0.1200134587\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7193702974\t-0.1255318739\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7263516144\t-0.1312420372\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7333329314\t-0.1371464487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7403142484\t-0.1432473503\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7472955654\t-0.1495466999\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7542768824\t-0.1560461676\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7612581994\t-0.1627471106\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7682395164\t-0.1696505319\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7752208335\t-0.1767570771\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7822021505\t-0.1840670335\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7891834675\t-0.1915802612\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.7961647845\t-0.1992962254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8031461015\t-0.2072139458\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8101274185\t-0.2153319471\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8171087355\t-0.2236483344\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8240900525\t-0.2321606815\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8310713695\t-0.2408660577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8380526865\t-0.2497610105\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8450340035\t-0.2588415322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8520153205\t-0.2681030825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8589966376\t-0.2775405234\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8659779546\t-0.287148178\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8729592716\t-0.2969197532\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8799405886\t-0.3068483614\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8869219056\t-0.3169265372\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.8939032226\t-0.3271462043\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9008845396\t-0.3374986801\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9078658566\t-0.3479746812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9148471736\t-0.3585643212\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9218284906\t-0.3692571495\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9288098076\t-0.3800420799\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9357911246\t-0.3909074896\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9427724416\t-0.4018411785\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9497537587\t-0.4128304237\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9567350757\t-0.4238619351\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9637163927\t-0.4349219455\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9706977097\t-0.4459961831\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9776790267\t-0.457069938\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9846603437\t-0.4681280416\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9916416607\t-0.4791549444\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 4.9986229777\t-0.4901347348\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0056042947\t-0.5010511529\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0125856117\t-0.5118876494\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0195669287\t-0.5226274274\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0265482457\t-0.53325347\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0335295628\t-0.5437485901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0405108798\t-0.55409548\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0474921968\t-0.5642767725\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0544735138\t-0.5742750429\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0614548308\t-0.5840729097\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0684361478\t-0.5936530578\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0754174648\t-0.6029983\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0823987818\t-0.6120916335\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0893800988\t-0.6209162626\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.0963614158\t-0.629455704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1033427328\t-0.637693788\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1103240498\t-0.6456147492\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1173053668\t-0.6532032629\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1242866839\t-0.6604444867\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1312680009\t-0.6673241187\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1382493179\t-0.6738284663\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1452306349\t-0.6799444597\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1522119519\t-0.6856597208\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1591932689\t-0.690962591\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1661745859\t-0.6958422031\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1731559029\t-0.7002884723\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1801372199\t-0.7042921737\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1871185369\t-0.7078449587\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.1940998539\t-0.7109393683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.2010811709\t-0.7135689154\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.208062488\t-0.7157280421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.215043805\t-0.7174121919\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.222025122\t-0.7186177866\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.229006439\t-0.7193422665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.235987756\t-0.7195840956\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.242969073\t-0.719342761\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.24995039\t-0.7186187681\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.256931707\t-0.7174136679\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.263913024\t-0.7157300137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.270894341\t-0.7135713739\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.277875658\t-0.7109423149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.284856975\t-0.7078483868\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.291838292\t-0.7042960974\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.2988196091\t-0.7002928776\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3058009261\t-0.695847091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3127822431\t-0.6909679538\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3197635601\t-0.6856655532\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3267448771\t-0.6799507617\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3337261941\t-0.6738352304\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3407075111\t-0.6673313458\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3476888281\t-0.6604521704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3546701451\t-0.6532114042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3616514621\t-0.6456233406\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3686327791\t-0.637702824\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3756140961\t-0.6294651782\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3825954132\t-0.6209261673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3895767302\t-0.6121019699\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.3965580472\t-0.6030090691\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4035393642\t-0.5936642531\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4105206812\t-0.5840845171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4175019982\t-0.5742870635\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4244833152\t-0.5642892063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4314646322\t-0.5541083205\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4384459492\t-0.5437618242\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4454272662\t-0.5332671054\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4524085832\t-0.5226414565\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4593899002\t-0.5119020732\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4663712172\t-0.501065965\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4733525343\t-0.4901499276\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4803338513\t-0.4791705254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4873151683\t-0.4681439967\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.4942964853\t-0.4570862825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5012778023\t-0.4460129093\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5082591193\t-0.4349390535\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5152404363\t-0.4238794248\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5222217533\t-0.4128482951\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5292030703\t-0.4018594382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5361843873\t-0.39092613\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5431657043\t-0.3800611075\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5501470213\t-0.3692765717\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5571283383\t-0.3585841578\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5641096554\t-0.3479949104\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5710909724\t-0.3375193235\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5780722894\t-0.3271672663\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5850536064\t-0.3169480309\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5920349234\t-0.3068702943\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.5990162404\t-0.2969421297\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6059975574\t-0.2871710176\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6129788744\t-0.2775638391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6199601914\t-0.2681268786\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6269415084\t-0.2588658369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6339228254\t-0.2497858336\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6409041424\t-0.2408914187\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6478854595\t-0.2321866\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6548667765\t-0.2236748342\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6618480935\t-0.2153590541\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6688294105\t-0.2072416721\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6758107275\t-0.1993246155\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6827920445\t-0.1916093333\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6897733615\t-0.1840968117\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.6967546785\t-0.1767876133\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7037359955\t-0.1696818371\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7107173125\t-0.1627792421\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7176986295\t-0.15607917\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7246799465\t-0.1495805959\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7316612635\t-0.1432821909\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7386425806\t-0.1371822914\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7456238976\t-0.1312789243\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7526052146\t-0.1255698552\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7595865316\t-0.1200526113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7665678486\t-0.1147244483\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7735491656\t-0.1095824323\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7805304826\t-0.1046234244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7875117996\t-0.0998441077\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.7944931166\t-0.0952410222\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8014744336\t-0.0908105528\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8084557506\t-0.0865489522\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8154370676\t-0.0824523979\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8224183847\t-0.0785169409\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8293997017\t-0.0747385939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8363810187\t-0.0711132767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8433623357\t-0.0676368903\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8503436527\t-0.0643052783\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8573249697\t-0.0611142846\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8643062867\t-0.0580597222\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8712876037\t-0.0551374274\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8782689207\t-0.0523432491\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8852502377\t-0.0496730398\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8922315547\t-0.047122693\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.8992128717\t-0.0446881553\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9061941887\t-0.0423653962\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9131755058\t-0.0401504526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9201568228\t-0.038039425\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9271381398\t-0.0360284761\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9341194568\t-0.0341138334\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9411007738\t-0.0322918005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9480820908\t-0.0305587764\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9550634078\t-0.0289112187\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9620447248\t-0.0273456966\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9690260418\t-0.0258588467\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9760073588\t-0.0244474151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9829886758\t-0.0231082096\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9899699928\t-0.021838177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 5.9969513099\t-0.0206343095\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0039326269\t-0.0194937308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0109139439\t-0.0184136413\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0178952609\t-0.0173913512\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0248765779\t-0.0164242443\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0318578949\t-0.0155098302\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0388392119\t-0.0146456653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0458205289\t-0.0138294577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0528018459\t-0.0130589547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0597831629\t-0.0123320226\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0667644799\t-0.0116466083\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0737457969\t-0.0110007606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.0807271139\t-0.0103925964\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.087708431\t-0.0098203219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.094689748\t-0.0092822166\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.101671065\t-0.0087766624\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.108652382\t-0.0083021224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.115633699\t-0.0078570996\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.122615016\t-0.0074402102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.129596333\t-0.007050147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.13657765\t-0.0066856324\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.143558967\t-0.0063454983\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.150540284\t-0.0060286285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.157521601\t-0.0057339872\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.164502918\t-0.0054605818\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1714842351\t-0.0052075015\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1784655521\t-0.0049738729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1854468691\t-0.0047589303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1924281861\t-0.0045619166\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.1994095031\t-0.0043821474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2063908201\t-0.0042190099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2133721371\t-0.0040719162\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2203534541\t-0.0039403771\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2273347711\t-0.0038238968\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2343160881\t-0.0037220959\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2412974051\t-0.0036345869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2482787221\t-0.003561063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2552600391\t-0.003501266\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2622413562\t-0.0034549746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2692226732\t-0.003422025\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2762039902\t-0.0034023098\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2831853072\t-0.0033957422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2901666242\t-0.0034023092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.2971479412\t-0.0034220302\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3041292582\t-0.0034549693\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3111105752\t-0.0035012598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3180918922\t-0.0035610578\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3250732092\t-0.0036345792\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3320545262\t-0.0037220864\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3390358432\t-0.0038238939\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3460171603\t-0.0039403683\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3529984773\t-0.0040719088\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3599797943\t-0.0042189989\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3669611113\t-0.004382143\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3739424283\t-0.0045619073\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3809237453\t-0.0047589218\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3879050623\t-0.0049738688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.3948863793\t-0.0052074825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4018676963\t-0.0054605709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4088490133\t-0.0057339766\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4158303303\t-0.0060286173\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4228116473\t-0.0063454874\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4297929643\t-0.0066856152\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4367742814\t-0.0070501253\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4437555984\t-0.0074401966\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4507369154\t-0.0078570829\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4577182324\t-0.0083020998\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4646995494\t-0.0087766463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4716808664\t-0.0092821954\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4786621834\t-0.009820287\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4856435004\t-0.0103925665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4926248174\t-0.0110007377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.4996061344\t-0.0116465891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5065874514\t-0.0123319883\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5135687684\t-0.0130589211\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5205500855\t-0.0138294255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5275314025\t-0.0146456293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5345127195\t-0.0155097907\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5414940365\t-0.0164242067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5484753535\t-0.0173913112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5554566705\t-0.0184136007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5624379875\t-0.0194936783\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5694193045\t-0.0206342511\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5764006215\t-0.0218381139\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5833819385\t-0.0231081561\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5903632555\t-0.0244473494\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.5973445725\t-0.0258587804\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6043258895\t-0.027345626\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6113072066\t-0.0289111494\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6182885236\t-0.0305586968\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6252698406\t-0.0322917147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6322511576\t-0.0341137464\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6392324746\t-0.036028382\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6462137916\t-0.038039335\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6531951086\t-0.040150358\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6601764256\t-0.0423652859\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6671577426\t-0.0446880429\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6741390596\t-0.0471225788\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6811203766\t-0.0496729131\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6881016936\t-0.0523431108\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.6950830107\t-0.0551372868\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7020643277\t-0.0580595753\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7090456447\t-0.0611141227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7160269617\t-0.0643051123\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7230082787\t-0.0676367184\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7299895957\t-0.0711130961\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7369709127\t-0.0747384069\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7439522297\t-0.078516745\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7509335467\t-0.0824521895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7579148637\t-0.0865487327\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7648961807\t-0.0908103127\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7718774977\t-0.0952407861\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7788588147\t-0.0998438514\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7858401318\t-0.104623156\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7928214488\t-0.1095821478\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.7998027658\t-0.1147241606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8067840828\t-0.1200523039\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8137653998\t-0.1255695376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8207467168\t-0.1312785915\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8277280338\t-0.1371819352\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8347093508\t-0.1432818245\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8416906678\t-0.1495802139\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8486719848\t-0.1560787671\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8556533018\t-0.1627788267\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8626346188\t-0.1696814135\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8696159358\t-0.1767871563\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8765972529\t-0.1840963416\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8835785699\t-0.1916088468\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8905598869\t-0.1993241079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.8975412039\t-0.2072411466\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9045225209\t-0.2153585032\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9115038379\t-0.2236742632\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9184851549\t-0.2321860158\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9254664719\t-0.2408908066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9324477889\t-0.2497851942\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9394291059\t-0.2588651787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9464104229\t-0.2681261963\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.9533917399\t-0.2775631354\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.960373057\t-0.2871702846\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.967354374\t-0.2969413709\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.974335691\t-0.3068695082\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.981317008\t-0.3169472283\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.988298325\t-0.3271664369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 6.995279642\t-0.337518463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.002260959\t-0.3479940259\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.009242276\t-0.3585832386\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.016223593\t-0.3692756318\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.02320491\t-0.380060143\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.030186227\t-0.3909251357\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.037167544\t-0.401858415\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.044148861\t-0.4128472428\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0511301781\t-0.4238783438\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0581114951\t-0.434937943\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0650928121\t-0.4460117804\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0720741291\t-0.4570851156\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0790554461\t-0.4681428059\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0860367631\t-0.479169304\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0930180801\t-0.4901486873\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.0999993971\t-0.5010646927\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1069807141\t-0.5119007741\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1139620311\t-0.5226401259\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1209433481\t-0.5332657477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1279246651\t-0.5437604456\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1349059822\t-0.5541069133\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1418872992\t-0.5642877672\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1488686162\t-0.5742856065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1558499332\t-0.5840830272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1628312502\t-0.5936627369\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1698125672\t-0.6030075353\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1767938842\t-0.6121004219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1837752012\t-0.6209245901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1907565182\t-0.6294635752\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.1977378352\t-0.6377011974\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2047191522\t-0.6456217035\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2117004692\t-0.6532097414\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2186817862\t-0.6604504893\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2256631033\t-0.6673296472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2326444203\t-0.6738335202\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2396257373\t-0.6799490306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2466070543\t-0.6856638144\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2535883713\t-0.6909661992\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2605696883\t-0.6958453146\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2675510053\t-0.700291099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2745323223\t-0.7042943032\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2815136393\t-0.70784659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2884949563\t-0.7109405096\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.2954762733\t-0.7135695665\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3024575903\t-0.715728198\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3094389074\t-0.7174118412\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3164202244\t-0.7186169387\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3234015414\t-0.7193409245\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3303828584\t-0.719582264\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3373641754\t-0.7193404332\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3443454924\t-0.718615956\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3513268094\t-0.717410367\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3583081264\t-0.715726226\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3652894434\t-0.7135670967\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3722707604\t-0.7109375636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3792520774\t-0.7078431526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3862333944\t-0.704290383\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.3932147114\t-0.7002866962\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4001960285\t-0.6958404367\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4071773455\t-0.6909608397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4141586625\t-0.6856579732\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4211399795\t-0.6799427295\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4281212965\t-0.6738267527\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4351026135\t-0.6673224284\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4420839305\t-0.6604428052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4490652475\t-0.6532016072\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4560465645\t-0.6456131126\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4630278815\t-0.637692164\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4700091985\t-0.6294541002\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4769905155\t-0.6209146878\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4839718326\t-0.6120900791\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4909531496\t-0.6029967674\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.4979344666\t-0.5936515502\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5049157836\t-0.5840714306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5118971006\t-0.5742735858\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5188784176\t-0.5642753375\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5258597346\t-0.5540940746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5328410516\t-0.5437472067\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5398223686\t-0.533252115\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5468036856\t-0.5226261006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5537850026\t-0.5118863485\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5607663196\t-0.5010498821\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5677476366\t-0.4901334916\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5747289537\t-0.4791537308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5817102707\t-0.4681268487\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5886915877\t-0.4570687668\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.5956729047\t-0.4459950552\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6026542217\t-0.4349208403\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6096355387\t-0.4238608572\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6166168557\t-0.4128293711\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6235981727\t-0.4018401594\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6305794897\t-0.3909064884\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6375608067\t-0.3800411031\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6445421237\t-0.3692562058\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6515234407\t-0.3585634123\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6585047578\t-0.3479737906\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6654860748\t-0.3374978253\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6724673918\t-0.327145374\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6794487088\t-0.3169257325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6864300258\t-0.3068475786\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.6934113428\t-0.2969189911\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7003926598\t-0.2871474536\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7073739768\t-0.2775398228\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7143552938\t-0.268102401\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7213366108\t-0.2588408758\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7283179278\t-0.249760376\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7352992448\t-0.2408654482\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7422805618\t-0.2321600934\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7492618789\t-0.223647767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7562431959\t-0.2153314017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7632245129\t-0.207213416\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7702058299\t-0.1992957221\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7771871469\t-0.1915797744\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7841684639\t-0.1840665641\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7911497809\t-0.1767566336\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.7981310979\t-0.1696500914\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8051124149\t-0.1627466933\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8120937319\t-0.156045781\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8190750489\t-0.149546321\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.8260563659\t-0.1432469825\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.833037683\t-0.1371461017\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.840019\t-0.1312417038\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.847000317\t-0.1255315521\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.853981634\t-0.1200131502\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.860962951\t-0.1146837876\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.867944268\t-0.1095404894\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.874925585\t-0.1045801397\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.881906902\t-0.0997994098\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.888888219\t-0.0951948248\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.895869536\t-0.0907627698\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.902850853\t-0.0864994977\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.90983217\t-0.0824011788\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.916813487\t-0.0784638599\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9237948041\t-0.0746835422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9307761211\t-0.0710561434\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9377574381\t-0.0675775496\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9447387551\t-0.0642436027\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9517200721\t-0.0610501422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9587013891\t-0.0579929762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9656827061\t-0.0550679217\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9726640231\t-0.052270816\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9796453401\t-0.0495975171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9866266571\t-0.047043895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 7.9936079741\t-0.044605888\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0005892911\t-0.0422794484\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0075706081\t-0.0400606112\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0145519252\t-0.0379454631\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0215332422\t-0.0359301322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0285145592\t-0.0340108497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0354958762\t-0.0321838988\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0424771932\t-0.0304456536\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0494585102\t-0.0287925743\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0564398272\t-0.0272211836\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0634211442\t-0.0257281108\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0704024612\t-0.0243100673\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0773837782\t-0.0229638473\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0843650952\t-0.021686365\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0913464122\t-0.0204745979\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.0983277293\t-0.0193256219\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1053090463\t-0.0182366182\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1122903633\t-0.0172048526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1192716803\t-0.0162276897\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1262529973\t-0.0153025643\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1332343143\t-0.0144270438\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1402156313\t-0.0135987401\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1471969483\t-0.012815396\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1541782653\t-0.0120748055\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1611595823\t-0.011374868\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1681408993\t-0.0107135746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1751222163\t-0.0100889787\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1821035333\t-0.0094992137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1890848504\t-0.008942515\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.1960661674\t-0.0084171732\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2030474844\t-0.0079215648\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2100288014\t-0.0074541346\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2170101184\t-0.007013402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2239914354\t-0.0065979322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2309727524\t-0.0062063868\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2379540694\t-0.0058374653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2449353864\t-0.0054899576\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2519167034\t-0.0051626769\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2588980204\t-0.0048545202\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2658793374\t-0.0045644367\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2728606545\t-0.0042914155\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2798419715\t-0.0040344913\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2868232885\t-0.0037927758\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.2938046055\t-0.0035654047\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3007859225\t-0.003351559\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3077672395\t-0.0031504633\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3147485565\t-0.0029613909\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3217298735\t-0.002783652\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3287111905\t-0.0026165872\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3356925075\t-0.0024595685\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3426738245\t-0.0023120254\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3496551415\t-0.0021733808\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3566364585\t-0.0020431272\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3636177756\t-0.0019207603\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3705990926\t-0.0018058244\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3775804096\t-0.0016978663\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3845617266\t-0.0015964781\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3915430436\t-0.0015012519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.3985243606\t-0.0014118375\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4055056776\t-0.0013278692\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4124869946\t-0.0012490196\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4194683116\t-0.001175\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4264496286\t-0.0011054917\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4334309456\t-0.0010402331\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4404122626\t-0.0009789618\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4473935797\t-0.0009214391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4543748967\t-0.0008674286\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4613562137\t-0.0008167186\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4683375307\t-0.0007691151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4753188477\t-0.0007244118\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4823001647\t-0.0006824418\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4892814817\t-0.0006430389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.4962627987\t-0.000606034\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5032441157\t-0.0005712796\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5102254327\t-0.0005386463\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5172067497\t-0.0005080052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5241880667\t-0.0004792151\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5311693837\t-0.0004521795\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5381507008\t-0.0004267811\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5451320178\t-0.000402919\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5521133348\t-0.0003805063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5590946518\t-0.0003594508\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5660759688\t-0.0003396535\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5730572858\t-0.000321057\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5800386028\t-0.0003035646\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5870199198\t-0.0002871339\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.5940012368\t-0.0002716802\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6009825538\t-0.0002571491\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6079638708\t-0.0002434896\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6149451878\t-0.0002306384\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6219265049\t-0.0002185523\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6289078219\t-0.0002071789\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6358891389\t-0.0001964801\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6428704559\t-0.0001864102\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6498517729\t-0.0001769289\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6568330899\t-0.0001680079\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6638144069\t-0.0001596123\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6707957239\t-0.0001516974\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6777770409\t-0.0001442402\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6847583579\t-0.000137221\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6917396749\t-0.000130606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.6987209919\t-0.0001243767\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.7057023089\t-0.0001184946\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.712683626\t-0.0001129598\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.719664943\t-0.000107735\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.72664626\t-0.0001028092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.733627577\t-9.81593874000014E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.740608894\t-9.37755360000021E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.747590211\t-8.96346342000032E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.754571528\t-8.57342616000026E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.761552845\t-8.20394565999999E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.768534162\t-7.85586372000005E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.775515479\t-7.52676802000009E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.782496796\t-7.21587277999966E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.789478113\t-6.92282586000001E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.7964594301\t-6.64488880000036E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8034407471\t-6.38257674000081E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8104220641\t-6.13434411999986E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8174033811\t-5.89972974000055E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8243846981\t-0.000056773\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8313660151\t-5.4670176000006E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8383473321\t-5.26843212000014E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8453286491\t-5.07945292000031E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8523099661\t-4.90124690000066E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8592912831\t-4.7316193400003E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8662726001\t-4.57097892000019E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8732539171\t-4.41886756000001E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8802352341\t-4.27504477999996E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8872165512\t-4.13764408000103E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.8941978682\t-4.00772849999953E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9011791852\t-3.8844099600005E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9081605022\t-3.76625924000118E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9151418192\t-3.65531936000057E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9221231362\t-3.54853716000025E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9291044532\t-3.4473080600006E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9360857702\t-3.35129396000083E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9430670872\t-3.25907396000052E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9500484042\t-3.17150702000002E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9570297212\t-3.0872849399996E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9640110382\t-3.00781354000135E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9709923553\t-0.000029307\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9779736723\t-0.000028577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9849549893\t-2.78737816000046E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9919363063\t-2.71921770000072E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 8.9989176233\t-2.65474176000258E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0058989403\t-2.59201066000192E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0128802573\t-2.53255851999923E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0198615743\t-2.47336754000127E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0268428913\t-0.000024175\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0338242083\t-2.36367182000208E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0408055253\t-2.31051129999937E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0477868423\t-2.25956735999955E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0547681593\t-2.21044328000131E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0617494764\t-2.16155307999899E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0687307934\t-0.000021137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0757121104\t-2.06747732000023E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0826934274\t-2.02228462000409E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0896747444\t-1.97731151999891E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.0966560614\t-1.9342453999982E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1036373784\t-1.89011370000103E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1106186954\t-1.8479130400012E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1176000124\t-1.80532938000022E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1245813294\t-1.76363154000547E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1315626464\t-1.72268200000214E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1385439634\t-1.68082836000116E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1455252805\t-1.63913328000143E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1525065975\t-1.59866254000329E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1594879145\t-1.55799070000739E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1664692315\t-1.51667125999899E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1734505485\t-1.47545374000191E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1804318655\t-1.43422246000058E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1874131825\t-1.39340652000608E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.1943944995\t-1.35213566000436E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2013758165\t-1.31073460000186E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2083571335\t-1.26811964000217E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2153384505\t-1.22592005999955E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2223197675\t-1.18370548000124E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2293010845\t-1.14148020000093E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2362824016\t-1.09762198000074E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2432637186\t-1.05452195999778E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2502450356\t-1.01066643999892E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2572263526\t-9.66059720003409E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2642076696\t-9.21681400001262E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2711889866\t-8.76340799998987E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2781703036\t-8.30906380006047E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2851516206\t-7.85268699998731E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2921329376\t-0.000007391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.2991142546\t-6.92175099997749E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3060955716\t-6.4600634000489E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3130768886\t-5.98411819996425E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3200582056\t-0.00000551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3270395227\t-5.03544120006305E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3340208397\t-4.55143700001104E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3410021567\t-4.06750280007584E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3479834737\t-3.58222700008737E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3549647907\t-3.09306580004165E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3619461077\t-2.60067440005507E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3689274247\t-0.00000211\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3759087417\t-1.61180239999226E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3828900587\t-1.12285860004446E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3898713757\t-6.25146000021726E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.3968526927\t-1.28786800057679E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4038340097\t3.62666400022249E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4108153268\t8.59828400034957E-007\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4177966438\t1.35385479999028E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4247779608\t1.85766119992214E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4317592778\t2.34626779999081E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4387405948\t2.83814719997943E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4457219118\t3.32689099996619E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4527032288\t3.81375100000358E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4596845458\t4.29679039998243E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4666658628\t4.7773910000215E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4736471798\t5.26025320002344E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4806284968\t5.73386919999752E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4876098138\t6.20740059997793E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.4945911308\t6.67271540000236E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5015724479\t7.14017499997555E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5085537649\t7.59850560000386E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5155350819\t8.05058520003676E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5225163989\t8.50259480000837E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5294977159\t8.94341139989763E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5364790329\t9.38364600000519E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5434603499\t9.81551500001121E-006\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5504416669\t1.0247541799957E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5574229839\t1.06785228000117E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5644043009\t1.10964419999249E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5713856179\t1.15120871999608E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.5783669349\t1.19156651999797E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.585348252\t1.23201813999607E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.592329569\t1.27137133999478E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.599310886\t1.31114685999467E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.606292203\t1.35026359999735E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.61327352\t1.38796659999718E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.620254837\t1.42642991999842E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.627236154\t0.000014636\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.634217471\t1.50077614000099E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.641198788\t1.53731887999609E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.648180105\t1.57344623999967E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.655161422\t1.61002823999556E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.662142739\t1.64533045999837E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.669124056\t0.000016812\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6761053731\t1.71653793999971E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6830866901\t1.7518935000016E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6900680071\t1.78714713999772E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.6970493241\t1.82304837999803E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7040306411\t1.85774073999945E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7110119581\t1.89337800000078E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7179932751\t1.92885669999809E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7249745921\t1.96448024000206E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7319559091\t2.00141617999472E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7389372261\t2.037574839997E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7459185431\t2.07552647999709E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7528998601\t2.11295705999648E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7598811772\t2.15145321999788E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7668624942\t2.19086235999644E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7738438112\t2.23081116000057E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7808251282\t2.27178993999766E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7878064452\t2.31406830000225E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.7947877622\t2.3580202200002E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8017690792\t0.000024025\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8087503962\t2.44873417999947E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8157317132\t2.49622493999901E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8227130302\t2.54663013999901E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8296943472\t2.5979187199987E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8366756642\t2.65217737999701E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8436569812\t2.70758486000111E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8506382983\t2.76535329999938E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8576196153\t2.82647463999974E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8646009323\t2.88955137999897E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8715822493\t2.95676211999885E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8785635663\t3.02561876000112E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8855448833\t3.09807709999998E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8925262003\t3.17468035999913E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.8995075173\t3.25499282000097E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9064888343\t3.33901032000122E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9134701513\t3.42835315999962E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9204514683\t3.52095179999845E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9274327853\t3.61972382000031E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9344141024\t3.72216188000013E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9413954194\t3.83107138000049E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9483767364\t3.94621912000043E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9553580534\t4.0668281599987E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9623393704\t0.000041944\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9693206874\t4.32858207999945E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9763020044\t4.4705526399999E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9832833214\t0.000046205\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9902646384\t0.000047783\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 9.9972459554\t4.94555387999945E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0042272724\t5.12190511999938E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0112085894\t5.3080906999993E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0181899064\t5.50592824000024E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0251712235\t5.71408977999913E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0321525405\t5.93482606000006E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0391338575\t6.16691713999917E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0461151745\t6.41434883999967E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0530964915\t6.67470780000012E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0600778085\t6.95132911999949E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0670591255\t7.24331795999983E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0740404425\t7.55227433999993E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0810217595\t7.88055753999997E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0880030765\t8.22749031999977E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.0949843935\t8.5947817600001E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1019657105\t8.98414145999969E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1089470276\t9.39695392000034E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1159283446\t9.83438707999988E-005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1229096616\t0.0001029772\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1298909786\t0.0001079018\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1368722956\t0.0001131087\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1438536126\t0.0001186509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1508349296\t0.000124513\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1578162466\t0.000130736\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1647975636\t0.0001373478\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1717788806\t0.0001443558\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1787601976\t0.0001518119\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1857415146\t0.0001597116\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1927228316\t0.0001681089\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.1997041487\t0.0001770242\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2066854657\t0.0001864986\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2136667827\t0.0001965687\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2206480997\t0.0002072659\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2276294167\t0.000218628\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2346107337\t0.0002307101\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2415920507\t0.0002435522\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2485733677\t0.0002572171\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2555546847\t0.0002717404\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2625360017\t0.0002871924\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2695173187\t0.0003036177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2764986357\t0.0003211016\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2834799528\t0.0003396941\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2904612698\t0.0003594892\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.2974425868\t0.000380551\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3044239038\t0.0004029651\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3114052208\t0.0004268223\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3183865378\t0.0004522149\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3253678548\t0.0004792533\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3323491718\t0.000508033\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3393304888\t0.0005386785\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3463118058\t0.0005713092\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3532931228\t0.000606054\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3602744398\t0.0006430647\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3672557568\t0.0006824649\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3742370739\t0.000724439\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3812183909\t0.0007691332\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3881997079\t0.0008167427\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.3951810249\t0.0008674509\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4021623419\t0.0009214605\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4091436589\t0.0009789786\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4161249759\t0.0010402513\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4231062929\t0.0011055086\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4300876099\t0.0011750108\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4370689269\t0.0012490377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4440502439\t0.0013278852\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.4510315609\t0.0014118474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.458012878\t0.0015012647\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.464994195\t0.0015964854\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.471975512\t0.001697877\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.478956829\t0.0018058311\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.485938146\t0.0019207718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.492919463\t0.0020431394\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.49990078\t0.0021733906\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.506882097\t0.0023120306\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.513863414\t0.0024595791\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.520844731\t0.0026165947\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.527826048\t0.0027836608\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.534807365\t0.0029614065\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.541788682\t0.0031504762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5487699991\t0.0033515703\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5557513161\t0.0035654179\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5627326331\t0.0037927922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5697139501\t0.0040345089\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5766952671\t0.0042914271\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5836765841\t0.0045644577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5906579011\t0.0048545368\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.5976392181\t0.0051626895\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6046205351\t0.005489971\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6116018521\t0.0058374822\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6185831691\t0.0062064\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6255644861\t0.0065979457\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6325458031\t0.007013422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6395271202\t0.0074541563\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6465084372\t0.0079215935\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6534897542\t0.0084171956\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6604710712\t0.0089425419\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6674523882\t0.0094992365\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6744337052\t0.0100890018\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6814150222\t0.0107135972\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6883963392\t0.0113749013\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.6953776562\t0.0120748391\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7023589732\t0.0128154305\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7093402902\t0.0135987794\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7163216072\t0.014427077\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7233029243\t0.0153026042\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7302842413\t0.0162277255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7372655583\t0.017204901\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7442468753\t0.0182366577\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7512281923\t0.0193256704\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7582095093\t0.0204746453\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7651908263\t0.021686423\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7721721433\t0.0229639083\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7791534603\t0.0243101208\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7861347773\t0.0257281733\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.7931160943\t0.0272212493\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8000974113\t0.0287926472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8070787283\t0.0304457294\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8140600454\t0.0321839797\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8210413624\t0.0340109296\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8280226794\t0.0359302227\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8350039964\t0.0379455519\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8419853134\t0.0400607157\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8489666304\t0.0422795506\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8559479474\t0.0446059958\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8629292644\t0.0470440126\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8699105814\t0.0495976379\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8768918984\t0.052270951\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8838732154\t0.0550680574\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8908545324\t0.0579931177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.8978358495\t0.0610502959\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9048171665\t0.0642437677\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9117984835\t0.06757772\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9187798005\t0.0710563145\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9257611175\t0.0746837215\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9327424345\t0.078464052\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9397237515\t0.0824013798\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9467050685\t0.0864997116\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9536863855\t0.0907629954\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9606677025\t0.0951950601\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9676490195\t0.0997996541\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9746303365\t0.1045803969\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9816116535\t0.1095407626\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9885929706\t0.1146840746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 10.9955742876\t0.1200134511\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0025556046\t0.1255318663\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0095369216\t0.131242022\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0165182386\t0.1371464411\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0234995556\t0.1432473416\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0304808726\t0.1495466912\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0374621896\t0.156046159\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0444435066\t0.1627470954\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0514248236\t0.1696505168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0584061406\t0.1767570684\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0653874576\t0.1840670248\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0723687747\t0.1915802525\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0793500917\t0.1992962168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0863314087\t0.207213923\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.0933127257\t0.2153319384\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1002940427\t0.2236483257\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1072753597\t0.2321606729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1142566767\t0.2408660414\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1212379937\t0.2497609942\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1282193107\t0.2588415235\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1352006277\t0.2681030586\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1421819447\t0.2775405137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1491632617\t0.2871481618\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1561445787\t0.2969197293\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1631258958\t0.3068483452\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1701072128\t0.3169265199\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1770885298\t0.3271461804\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1840698468\t0.3374986562\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1910511638\t0.3479746574\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.1980324808\t0.3585643038\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2050137978\t0.369257118\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2119951148\t0.3800420485\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2189764318\t0.3909074657\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2259577488\t0.4018411611\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2329390658\t0.4128303922\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2399203828\t0.4238619113\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2469016999\t0.434921914\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2538830169\t0.4459961581\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2608643339\t0.4570699131\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2678456509\t0.4681280091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2748269679\t0.4791549195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2818082849\t0.4901347099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2887896019\t0.501051128\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.2957709189\t0.5118876168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3027522359\t0.5226273948\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3097335529\t0.5332534375\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3167148699\t0.5437485576\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3236961869\t0.554095454\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.3306775039\t0.56427674\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.337658821\t0.5742750027\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.344640138\t0.5840728761\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.351621455\t0.5936530253\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.358602772\t0.6029982675\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.365584089\t0.6120915923\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.372565406\t0.620916229\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.379546723\t0.6294556639\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.38652804\t0.6376937544\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.393509357\t0.6456147156\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.400490674\t0.6532032228\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.407471991\t0.6604444455\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.414453308\t0.6673240775\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4214346251\t0.6738284251\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4284159421\t0.6799444195\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4353972591\t0.6856596731\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4423785761\t0.6909625498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4493598931\t0.6958421619\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4563412101\t0.7002884311\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4633225271\t0.7042921259\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4703038441\t0.7078449099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4772851611\t0.7109393271\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4842664781\t0.7135688741\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4912477951\t0.7157280009\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.4982291121\t0.7174121431\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5052104291\t0.7186177378\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5121917462\t0.7193422177\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5191730632\t0.7195840468\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5261543802\t0.7193427122\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5331356972\t0.7186187269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5401170142\t0.7174136267\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5470983312\t0.7157299725\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5540796482\t0.7135713327\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5610609652\t0.7109422737\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5680422822\t0.7078483456\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5750235992\t0.7042960497\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5820049162\t0.7002928363\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5889862332\t0.6958470498\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.5959675503\t0.6909679126\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6029488673\t0.685665512\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6099301843\t0.6799507205\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6169115013\t0.6738351891\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6238928183\t0.6673313046\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6308741353\t0.6604521292\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6378554523\t0.653211363\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6448367693\t0.6456233005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6518180863\t0.6377027828\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6587994033\t0.629465137\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6657807203\t0.6209261337\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6727620373\t0.6121019362\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6797433543\t0.603009029\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6867246714\t0.593664213\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.6937059884\t0.5840844835\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7006873054\t0.5742870298\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7076686224\t0.5642891662\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7146499394\t0.5541082879\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7216312564\t0.5437617917\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7286125734\t0.5332670729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7355938904\t0.5226414239\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7425752074\t0.5119020472\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7495565244\t0.5010659325\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7565378414\t0.4901499026\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7635191584\t0.4791705005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7705004754\t0.4681439718\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7774817925\t0.45708625\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7844631095\t0.4460128844\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7914444265\t0.4349390285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.7984257435\t0.4238793934\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8054070605\t0.4128482702\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8123883775\t0.4018594132\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8193696945\t0.3909261061\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8263510115\t0.3800610836\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8333323285\t0.3692765544\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8403136455\t0.3585841339\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8472949625\t0.3479948865\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8542762795\t0.3375193062\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8612575966\t0.327167249\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8682389136\t0.3169480146\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8752202306\t0.3068702705\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8822015476\t0.2969421058\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8891828646\t0.2871710013\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.8961641816\t0.2775638228\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9031454986\t0.2681268623\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9101268156\t0.2588658206\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9171081326\t0.2497858173\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9240894496\t0.2408914024\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9310707666\t0.2321865837\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9380520836\t0.2236748255\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9450334006\t0.2153590389\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9520147177\t0.2072416558\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9589960347\t0.1993246003\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9659773517\t0.1916093247\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9729586687\t0.184096803\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9799399857\t0.1767875981\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9869213027\t0.1696818284\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 11.9939026197\t0.1627792269\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0008839367\t0.1560791548\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0078652537\t0.1495805872\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0148465707\t0.1432821823\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0218278877\t0.1371822762\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0288092047\t0.1312789091\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0357905218\t0.1255698542\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0427718388\t0.1200526037\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0497531558\t0.1147244396\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0567344728\t0.1095824312\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0637157898\t0.1046234168\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0706971068\t0.0998441066\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0776784238\t0.0952410211\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0846597408\t0.0908105377\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0916410578\t0.0865489446\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.0986223748\t0.0824523903\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1056036918\t0.0785169398\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1125850088\t0.0747385863\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1195663258\t0.0711132756\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1265476429\t0.0676368827\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1335289599\t0.0643052708\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1405102769\t0.061114277\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1474915939\t0.0580597146\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1544729109\t0.0551374263\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1614542279\t0.0523432415\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1684355449\t0.0496730322\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1754168619\t0.0471226919\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1823981789\t0.0446881477\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1893794959\t0.0423653962\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.1963608129\t0.0401504526\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.2033421299\t0.038039425\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.210323447\t0.0360284685\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.217304764\t0.0341138258\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.224286081\t0.0322918005\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.231267398\t0.0305587688\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.238248715\t0.0289112187\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.245230032\t0.027345689\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.252211349\t0.0258588456\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.259192666\t0.0244474075\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.266173983\t0.0231082096\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.2731553\t0.0218381694\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.280136617\t0.0206343019\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.287117934\t0.0194937308\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.294099251\t0.0184136413\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3010805681\t0.0173913512\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3080618851\t0.0164242443\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3150432021\t0.0155098302\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3220245191\t0.0146456653\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3290058361\t0.0138294501\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3359871531\t0.0130589547\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3429684701\t0.0123320226\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3499497871\t0.0116466148\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3569311041\t0.0110007606\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3639124211\t0.0103925964\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3708937381\t0.0098203143\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3778750551\t0.0092822166\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3848563722\t0.0087766624\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3918376892\t0.0083021224\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.3988190062\t0.0078571061\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4058003232\t0.0074402167\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4127816402\t0.007050147\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4197629572\t0.0066856324\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4267442742\t0.0063454983\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4337255912\t0.0060286285\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4407069082\t0.0057339872\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4476882252\t0.0054605743\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4546695422\t0.0052075015\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4616508592\t0.0049738729\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4686321762\t0.0047589303\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4756134933\t0.004561909\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4825948103\t0.0043821474\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4895761273\t0.0042190099\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.4965574443\t0.0040719162\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5035387613\t0.0039403771\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5105200783\t0.0038238968\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5175013953\t0.0037220959\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5244827123\t0.0036345869\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5314640293\t0.003561063\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5384453463\t0.003501266\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5454266633\t0.0034549746\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5524079803\t0.003422025\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5593892974\t0.0034023098\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t 12.5663706144\t0.0033957422\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t };\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{axis}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t \\end{tikzpicture}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\caption{The degenerate first excited state torsional wavefunctions. Left display: The two wavefunctions which generate the $E_{3d}$ irrep, but which do not transform as given by the transformation matrices of \\protect\\sect{sec:Einalmatrices}, displayed as a red and a blue curve, respectively. Right display: The corresponding two $E_{3d}$ wavefunctions obtained as transforming according to the transformation matrices of \\protect\\sect{sec:Einalmatrices} and displayed as a red and a blue curve, respectively.}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{fig:torsion_first}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\\end{figure} \n\n\n\n\nThe symmetry adapted ro-vibrational basis set functions are constructed as direct products of the symmetrized component\nfunctions from the different subspaces as $ \\Psi_{\\lambda_0}^{(0),\\Gamma_0} \\otimes\n\\Psi_{\\lambda_1}^{(1),\\Gamma_1} \\otimes \\Psi_{\\lambda_2}^{(2),\\Gamma_2} \\cdots \\otimes\n\\Psi_{\\lambda_L}^{(L),\\Gamma_L}$, where $L$ is the number of vibrational subspaces. The symmetry of a product basis function is the direct product of the irreducible representations of the component functions. This symmetry must be further reduced, but this is rather straightforward when each component function generates one of the irreps of the group and has defined, `standard' transformation properties. In this case we use the same projection operator symmetrization technique described above without further sampling of the symmetry properties of the corresponding components. The $d$-symmetry components are unphysical and must be disregarded. Some $s$ symmetry ro-vibrational basis functions will necessarily contain products of the $d$ components, underpinning the importance of the {\\itshape\\bfseries G}$_{36}$(EM) extended symmetry group. The nuclear spin statistical weights of the ro-vibrational states can be deduced using the procedure from \\cite{98BuJexx}. We obtain nuclear spin statistical weights of 6, 10, 6, 10, 4, 4, 2, 6, 12 for the {\\itshape\\bfseries G}$_{36}$(EM) symmetries $A_{1s}, A_{2s}, A_{3s}, A_{4s}, E_{1s}, E_{2s}, E_{3s}, E_{4s}$, and $G_s$, respectively.\n\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe have presented a detailed description of the {\\itshape\\bfseries G}$_{36}$\\ and {\\itshape\\bfseries G}$_{36}$(EM) molecular symmetry groups. A full set of irreducible representation matrices have been derived and tested for constructing symmetry-adapted potential energy functions and basis functions of ethane C$_2$H$_6$. Both the construction of the transformation matrices and the symmetry adaption can be implemented as numerical procedures as part of computational approaches to the solution of the ro-vibrational Schr\\\"odinger equation. A self-consistent choice of the vibrational, torsional and rotational coordinates for ethane, satisfying the {\\itshape\\bfseries G}$_{36}$(EM) symmetry requirements have been introduced and analyzed in full detail. The irreducible representation matrices as well as the coordinate choice made for ethane in the present work have been implemented as part of\nthe program system TROVE, and we have discussed a few examples of symmetrized wavefunctions. These results of the present work will be important in our planned line-list calculations for ethane.\n\nThe results of the present work are, of course, not only applicable to\nethane H$_3$CCH$_3$, but also to other molecules with MS groups {\\itshape\\bfseries G}$_{36}$\\ and {\\itshape\\bfseries G}$_{36}$(EM).\nA prominent molecule of this kind is dimethylacetylene\nH$_3$CCCCH$_3$. Also, subgroups of the \n matrix groups constructed can be applied to molecules whose MS groups are subgroups of {\\itshape\\bfseries G}$_{36}$\\ and {\\itshape\\bfseries G}$_{36}$(EM), such as D$_3$CCH$_3$. The general ideas used for generating the matrix groups can be applied to other MS groups. In particular, generation of matrix groups in a manner similar to that of the present work will be required for MS groups with irreps of dimension 4 and higher.\n\n\n\n\n\\section{Acknowledgements}\n\n\nThis work was supported by the STFC Projects No. ST\/M001334\/1 and ST\/R000476\/1, and by the COST action MOLIM (CM1405). The authors acknowledge the use of the UCL Legion High Performance Computing Facility (Legion@UCL) and associated support services in the completion of this work, along with the Cambridge Service for Data Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component of CSD3 was funded by BIS capital funding via STFC capital grants ST\/P002307\/1 and ST\/R002452\/1 and STFC operations grant ST\/R00689X\/1. DiRAC is part of the National e-Infrastructure. The present work was supported in part by the Deutsche Forschungsgemeinschaft.\n\n\n\n\\setcounter{section}{0}\n\\setcounter{equation}{0}\n\\setcounter{table}{0}\n\\renewcommand\\thesection{Appendix \\Alph{section}}\n\\renewcommand\\thesubsection{\\Alph{section}.\\arabic{subsection}}\n\\renewcommand\\theequation{\\Alph{section}.\\arabic{equation}}\n\\renewcommand\\thetable{\\Alph{section}.\\arabic{table}}\n\\renewcommand\\thesubsubsection{\\Alph{section}.\\arabic{subsection}.\\arabic{subsubsection}}\n\n\\section{Character table of the isomorphic groups\n{\\itshape\\bfseries C}$_{3{\\rm v}}^{(-)}$ and {\\itshape\\bfseries C}$_{3{\\rm v}}^{(+)}$.}\\label{sec:appendixC3vgroups}\n\n{\\itshape\\bfseries G}$_{36}$ is the direct product of\n{\\itshape\\bfseries C}$_{3{\\rm v}}^{(-)}$ and {\\itshape\\bfseries C}$_{3{\\rm v}}^{(+)}$, \n{\\itshape\\bfseries G}$_{36}$ $=$\n{\\itshape\\bfseries C}$_{3{\\rm v}}^{(-)}$ $\\times$ {\\itshape\\bfseries C}$_{3{\\rm v}}^{(+)}$,\nand so the irreps of {\\itshape\\bfseries G}$_{36}$ are obtained from those of\n{\\itshape\\bfseries C}$_{3{\\rm v}}^{(-)}$ and {\\itshape\\bfseries C}$_{3{\\rm v}}^{(+)}$.\nThese latter irreps are given in Table~\\ref{tab:C3vchartab}. \n\n\\begin{table}[h]\n\\caption{\\label{tab:C3vchartab}Common character tables of \n{\\itshape\\bfseries C}$_{3{\\rm v}}^{(-)}$ and {\\itshape\\bfseries C}$_{3{\\rm v}}^{(+)}$. }\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{lrrr}\n\\hline\\hline\n$\\Gamma$ &\n${\\mathcal C}_1^{(\\pm)}$ & ${\\mathcal C}_2^{(\\pm)}$ & ${\\mathcal C}_3^{(\\pm)}$ \\\\\n& 1 & 2 & 3 \\\\\n\\hline\n$A_1$ & 1 & 1 & 1 \\\\\n$A_2$ & 1 & 1 & $-$1 \\\\\n$E$ & 2 & $-$1 & 0 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nWe label\n\tthe elements of {\\itshape\\bfseries C}$_{3{\\rm v}}^{(\\pm)}$ as\n\t$R_j^{(\\pm)}$, $j$ $=$ 1, 2, \\dots, 6 (see \\tabl{tab:multiplicationtable}). The nuclei are labelled as in \\fig{fig:ethane_labelling}.\n\t {\\itshape\\bfseries C}$_{3{\\rm v}}^{(\\pm)}$ has three\nclasses,\n${\\mathcal C}_1^{(\\pm)}$ $=$ $\\{ E \\}$ $=$ $\\{ R_1^{(\\pm)} \\}$,\n${\\mathcal C}_2^{(\\pm)}$ $=$ $\\{ R_2^{(\\pm)}, R_3^{(\\pm)} \\}$, and\n${\\mathcal C}_3^{(\\pm)}$ $=$ $\\{ R_4^{(\\pm)}, R_5^{(\\pm)}, R_6^{(\\pm)} \\}$.\n\n\\section{The {\\itshape\\bfseries G}$_{36}$\\ Transformation Matrices}\\label{sec:appendixa}\n\\setcounter{equation}{0}\n\nIn Sections~\\ref{sec:Einalmatrices}\nand~\\ref{sec:finalmatrices}, we have abbreviated some of the operations\n$R$ $\\in$ {\\itshape\\bfseries G}$_{36}$\\ as $R$ $=$ $R_j^{(-)} \\, R_k^{(+)}$ $=$ $R_k^{(+)} \\, R_j^{(-)}$, where $R_j^{(-)}$ $\\in$ {\\itshape\\bfseries C}$_{3{\\rm v}}^{(-)}$ and\n$R_k^{(+)}$ $\\in$ {\\itshape\\bfseries C}$_{3{\\rm v}}^{(+)}$ (since, if we used the complete labels such as (15)(24)(36)$(ab)^*$ and (143526)$(ab)^*$, the text would not fit on the page). All such products, and the complete classes of {\\itshape\\bfseries G}$_{36}$, are given in\n \\tabl{tab:multiplicationtable}.\n\n\n\\subsection{$E_i$ transformation matrices in {\\itshape\\bfseries G}$_{36}$}\\label{sec:Einalmatrices}\nFor each operation $R$ $\\in$ {\\itshape\\bfseries G}$_{36}$, we list here the four matrices\n$ {\\mathbf M}_{E_1}\\left[ R \\right]$,\n$ {\\mathbf M}_{E_2}\\left[ R \\right]$,\n$ {\\mathbf M}_{E_3}\\left[ R \\right]$, and\n$ {\\mathbf M}_{E_4}\\left[ R \\right]$ in that order.\n\\subsubsection{One-member class ${\\mathcal C}_1^{(-)} \\times {\\mathcal C}_1^{(+)}$ containing $E$}\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ E \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\\\[5pt]\n\\end{tabbing}}\n\\end{center}\n\\subsubsection{Two-member class ${\\mathcal C}_1^{(-)} \\times {\\mathcal C}_2^{(+)}$ containing (123)(456) and (132)(465)}\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_2^{(+)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}}} \\\\[5pt]\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(+)} \\right] $ \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\[5pt]\n\\end{tabbing}}\n\\end{center}\n\\subsubsection{Three-member class ${\\mathcal C}_1^{(-)} \\times {\\mathcal C}_3^{(+)}$ containing (14)(26)(35)($ab$)$^*$}\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_4^{(+)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(+)} \\right] $ \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{-1}{0}{0}{-1} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(+)} \\right] $ \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{-1}{0}{0}{-1} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n\\end{tabbing}}\n\\end{center}\n\\subsubsection{Two-member class ${\\mathcal C}_2^{(-)} \\times {\\mathcal C}_1^{(+)}$ containing (123)(456) and (132)(456)}\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_2^{(-)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} } \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(-)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{1}{0}{0}{1} \\\\\n\\end{tabbing}}\n\\end{center}\n\n\\subsubsection{Four-member class ${\\mathcal C}_2^{(-)} \\times {\\mathcal C}_2^{(+)}$ containing (123) and (456)}\n\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_2^{(-)}R_2^{(+)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} } \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} } \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_2^{(-)}R_3^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(-)}R_2^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(-)}R_3^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n\\end{tabbing}}\n\\end{center}\n\n\\subsubsection{Six-member class ${\\mathcal C}_2^{(-)} \\times {\\mathcal C}_3^{(+)}$ containing (142635)($ab$)$^*$ }\n\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_2^{(-)} R_4^{(+)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{ \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} } \\=\n\\parbox[b]{3.0truecm}{ \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1} } \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1} } \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_2^{(-)} R_5^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_2^{(-)} R_6^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(-)} R_4^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{1}{0}{0}{-1} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(-)} R_5^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_3^{(-)} R_6^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n\\end{tabbing}}\n\\end{center}\n\n\\subsubsection{Three-member class ${\\mathcal C}_3^{(-)} \\times {\\mathcal C}_1^{(+)}$ containing (14)(25)(36)($ab$) }\n\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_4^{(-)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-1}{0}{0}{-1}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(-)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{-1}{0}{0}{-1} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(-)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{1}{0}{0}{1} \\> \\twobytwo{-1}{0}{0}{-1} \\\\\n\\end{tabbing}}\n\\end{center}\n\n\\subsubsection{Six-member class ${\\mathcal C}_3^{(-)} \\times {\\mathcal C}_2^{(+)}$ containing (142536)($ab$) }\n\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_4^{(-)} R_2^{(+)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} } \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} } \\\\[5pt]\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(-)} R_2^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(-)} R_2^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_4^{(-)} R_3^{(+)} \\right] $ \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(-)} R_3^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(-)} R_3^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\\\\n\\end{tabbing}}\n\\end{center}\n\n\\subsubsection{Nine-member class ${\\mathcal C}_3^{(-)} \\times {\\mathcal C}_3^{(+)}$ containing (12)(45)$^*$}\n\n\\begin{center}\n{\\small\\baselineskip=32pt\n\\begin{tabbing}\n\\parbox[b]{3.5truecm}{$ {\\mathbf M}_{E_i}\\left[ R_4^{(-)} R_4^{(+)} \\right]$}\n\\= \\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-1}{0}{0}{1}} \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{1}{0}{0}{-1} } \\=\n\\parbox[b]{3.0truecm}{\\twobytwo{-1}{0}{0}{1} } \\\\[5pt]\n$ {\\mathbf M}_{E_i}\\left[ R_4^{(-)} R_5^{(+)} \\right] $ \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{-1}{0}{0}{1} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_4^{(-)} R_6^{(+)} \\right] $ \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{-1}{0}{0}{1} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(-)} R_4^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{-1}{0}{0}{1} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(-)} R_5^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_5^{(-)} R_6^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(-)} R_4^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{1}{0}{0}{-1} \\> \\twobytwo{-1}{0}{0}{1} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(-)} R_5^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n$ {\\mathbf M}_{E_i}\\left[ R_6^{(-)} R_6^{(+)} \\right] $ \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\> \\twobytwo{-\\frac{1}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{\\sqrt{3}}{2}}{\\frac{1}{2}} \\> \\twobytwo{\\frac{1}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} \\\\\n\\end{tabbing}}\n\\end{center}\n\n\\subsection{$G$ transformation matrices in {\\itshape\\bfseries G}$_{36}$}\n\\label{sec:finalmatrices}\n\n\\subsubsection{One-member class ${\\mathcal C}_1^{(-)} \\times {\\mathcal C}_1^{(+)}$ containing $E$}\n\\begin{displaymath}\n{\\mathbf M}_G\\left[ R_1^{(-)} R_1^{(+)} \\right] =\n\\left( \\begin{array}{rrrr} 1&0&0&0\\\\ 0&1&0&0\\\\ 0&0&1&0\\\\ 0&0&0&1\\end{array} \\right) \\end{displaymath}\n\n\\subsubsection{Two-member class ${\\mathcal C}_1^{(-)} \\times {\\mathcal C}_2^{(+)}$ containing (123)(456) and (132)(465)}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_1^{(-)} R_2^{(+)} \\right] =\n \\left( \\begin{array}{rrrr} -{\\frac{1}{2}}&-{\\frac{\\sqrt{3}}{2}}&0&0\\\\ {\\frac{\\sqrt{3}\n }{2}}&-{\\frac{1}{2}}&0&0\\\\ 0&0&-{\\frac{1}{2}}&-{\\frac{\\sqrt{3}\n }{2}}\\\\ 0&0&{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_1^{(-)} R_3^{(+)} \\right] =\n \\left( \\begin{array}{rrrr}-{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{2}}&0&0\\\\ -{\\frac{\\sqrt{3}\n }{2}}&-{\\frac{1}{2}}&0&0\\\\ 0&0&-{\\frac{1}{2}}&{\\frac{\\sqrt{3}\n }{2}}\\\\ 0&0&-{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\n\\subsubsection{Three-member class ${\\mathcal C}_1^{(-)} \\times {\\mathcal C}_3^{(+)}$ containing (14)(26)(35)($ab$)$^*$}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_1^{(-)} R_4^{(+)} \\right] =\n \\left( \\begin{array}{cccc}1&0&0&0\\\\ 0&-1&0&0\\\\ 0&0&-1&0\\\\ 0&0&0&1\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_1^{(-)} R_5^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&-{\\frac{\\sqrt{3}}{2}}&0&0\\\\ -{\\frac{\\sqrt{3}\n }{2}}&{\\frac{1}{2}}&0&0\\\\ 0&0&{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{2\n }}\\\\ 0&0&{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_1^{(-)} R_6^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{2}}&0&0\\\\ {\\frac{\\sqrt{3}\n }{2}}&{\\frac{1}{2}}&0&0\\\\ 0&0&{\\frac{1}{2}}&-{\\frac{\\sqrt{3}}{\n 2}}\\\\ 0&0&-{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\n\\subsubsection{Two-member class ${\\mathcal C}_2^{(-)} \\times {\\mathcal C}_1^{(+)}$ containing (123)(456) and (132)(456)}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_2^{(-)} R_1^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&0&0&-{\\frac{\\sqrt{3}}{2}}\\\\ 0&-{\\frac{1}{2\n }}&{\\frac{\\sqrt{3}}{2}}&0\\\\ 0&-{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}&\n 0\\\\ {\\frac{\\sqrt{3}}{2}}&0&0&-{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_3^{(-)} R_1^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&0&0&{\\frac{\\sqrt{3}}{2}}\\\\ 0&-{\\frac{1}{2\n }}&-{\\frac{\\sqrt{3}}{2}}&0\\\\ 0&{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}&\n 0\\\\ -{\\frac{\\sqrt{3}}{2}}&0&0&-{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\n\\subsubsection{Four-member class ${\\mathcal C}_2^{(-)} \\times {\\mathcal C}_2^{(+)}$ containing (123) and (456)}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_2^{(-)}R_2^{(+)} \\right] =\n\\left( \\begin{array}{cccc} {\\frac{1}{4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{4}}&{\\frac{\\sqrt{\n 3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&-{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3}{4}}\\\\ -{\\frac{3}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{1\n }{4}}&{\\frac{\\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{\n 4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_2^{(-)}R_3^{(+)} \\right] =\n\\left( \\begin{array}{cccc} {\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4}}&{\\frac{\\sqrt{\n 3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&-{\\frac{\\sqrt{3}\n }{4}}&{\\frac{3}{4}}\\\\ {\\frac{3}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{1\n }{4}}&-{\\frac{\\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{\n 4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_3^{(-)}R_2^{(+)} \\right] =\n\\left( \\begin{array}{cccc} {\\frac{1}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4}}&-{\\frac{\\sqrt{\n 3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&{\\frac{\\sqrt{3}\n }{4}}&{\\frac{3}{4}}\\\\ {\\frac{3}{4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{1\n }{4}}&{\\frac{\\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4\n }}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_3^{(-)}R_3^{(+)} \\right] =\n\\left( \\begin{array}{cccc} {\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{4}}&-{\\frac{\n \\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3}{4}}\\\\ -{\\frac{3}{4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{\n 1}{4}}&-{\\frac{\\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&-{\\frac{3\n }{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\n\n\n\n\\subsubsection{Six-member class ${\\mathcal C}_2^{(-)} \\times {\\mathcal C}_3^{(+)}$ containing (142635)($ab$)$^*$ }\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_2^{(-)} R_4^{(+)} \\right] =\n\\left( \\begin{array}{cccc} -{\\frac{1 }{2}}&0&0&-{\\frac{\\sqrt{3} }{2}}\\\\ 0&{\\frac{1 }{2\n }}&-{\\frac{\\sqrt{3} }{2}}&0\\\\ 0&{\\frac{\\sqrt{3} }{2}}&{\\frac{1 }{2}}&0\n \\\\ {\\frac{\\sqrt{3} }{2}}&0&0&-{\\frac{1 }{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_3^{(-)} R_4^{(+)} \\right] =\n\\left( \\begin{array}{cccc} -{\\frac{1 }{2}}&0&0&{\\frac{\\sqrt{3} }{2}}\\\\ 0&{\\frac{1 }{2}}\n &{\\frac{\\sqrt{3} }{2}}&0\\\\ 0&-{\\frac{\\sqrt{3} }{2}}&{\\frac{1 }{2}}&0\n \\\\ -{\\frac{\\sqrt{3} }{2}}&0&0&-{\\frac{1 }{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_2^{(-)} R_5^{(+)} \\right] =\n\\left( \\begin{array}{cccc }{\\frac{1 }{4}}&{\\frac{\\sqrt{3} }{4}}&-{\\frac{3 }{4}}&{\\frac{\\sqrt{\n 3} }{4}}\\\\ {\\frac{\\sqrt{3} }{4}}&-{\\frac{1 }{4}}&{\\frac{\\sqrt{3}\n }{4}}&{\\frac{3 }{4}}\\\\ {\\frac{3 }{4}}&-{\\frac{\\sqrt{3} }{4}}&-{\\frac{1\n }{4}}&-{\\frac{\\sqrt{3} }{4}}\\\\ -{\\frac{\\sqrt{3} }{4}}&-{\\frac{3\n }{4}}&-{\\frac{\\sqrt{3} }{4}}&{\\frac{1 }{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_3^{(-)} R_5^{(+)} \\right] =\n\\left( \\begin{array}{cccc }{\\frac{1 }{4}}&{\\frac{\\sqrt{3} }{4}}&{\\frac{3 }{4}}&-{\\frac{\\sqrt{\n 3} }{4}}\\\\ {\\frac{\\sqrt{3} }{4}}&-{\\frac{1 }{4}}&-{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3 }{4}}\\\\ -{\\frac{3 }{4}}&{\\frac{\\sqrt{3} }{4}}&-{\\frac{\n 1 }{4}}&-{\\frac{\\sqrt{3} }{4}}\\\\ {\\frac{\\sqrt{3} }{4}}&{\\frac{3 }{\n 4}}&-{\\frac{\\sqrt{3} }{4}}&{\\frac{1 }{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_2^{(-)} R_6^{(+)} \\right] =\n\\left( \\begin{array}{cccc }{\\frac{1 }{4}}&-{\\frac{\\sqrt{3} }{4}}&{\\frac{3 }{4}}&{\\frac{\\sqrt{\n 3} }{4}}\\\\ -{\\frac{\\sqrt{3} }{4}}&-{\\frac{1 }{4}}&{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3 }{4}}\\\\ -{\\frac{3 }{4}}&-{\\frac{\\sqrt{3} }{4}}&-\n {\\frac{1 }{4}}&{\\frac{\\sqrt{3} }{4}}\\\\ -{\\frac{\\sqrt{3} }{4}}&{\\frac{3\n }{4}}&{\\frac{\\sqrt{3} }{4}}&{\\frac{1 }{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_3^{(-)} R_6^{(+)} \\right] =\n\\left( \\begin{array}{cccc }{\\frac{1 }{4}}&-{\\frac{\\sqrt{3} }{4}}&-{\\frac{3 }{4}}&-{\\frac{\n \\sqrt{3} }{4}}\\\\ -{\\frac{\\sqrt{3} }{4}}&-{\\frac{1 }{4}}&-{\\frac{\\sqrt{\n 3} }{4}}&{\\frac{3 }{4}}\\\\ {\\frac{3 }{4}}&{\\frac{\\sqrt{3} }{4}}&-{\\frac{\n 1 }{4}}&{\\frac{\\sqrt{3} }{4}}\\\\ {\\frac{\\sqrt{3} }{4}}&-{\\frac{3 }{\n 4}}&{\\frac{\\sqrt{3} }{4}}&{\\frac{1 }{4}}\\end{array} \\right) \\end{displaymath}\n\n\\subsubsection{Three-member class ${\\mathcal C}_3^{(-)} \\times {\\mathcal C}_1^{(+)}$ containing (14)(25)(36)($ab$) }\n\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_4^{(-)} \\right] =\n\\left( \\begin{array}{cccc}1&0&0&0\\\\ 0&1&0&0\\\\ 0&0&-1&0\\\\ 0&0&0&-1\\end{array} \\right) \\end{displaymath}\n\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_5^{(-)} \\right] =\n\\left( \\begin{array}{cccc}-{\\frac{1}{2}}&0&0&-{\\frac{\\sqrt{3}}{2}}\\\\ 0&-{\\frac{1}{2\n }}&{\\frac{\\sqrt{3}}{2}}&0\\\\ 0&{\\frac{\\sqrt{3}}{2}}&{\\frac{1}{2}}&0\n \\\\ -{\\frac{\\sqrt{3}}{2}}&0&0&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_6^{(-)} \\right] =\n\\left( \\begin{array}{cccc}-{\\frac{1}{2}}&0&0&{\\frac{\\sqrt{3}}{2}}\\\\ 0&-{\\frac{1}{2\n }}&-{\\frac{\\sqrt{3}}{2}}&0\\\\ 0&-{\\frac{\\sqrt{3}}{2}}&{\\frac{1}{2}}&\n 0\\\\ {\\frac{\\sqrt{3}}{2}}&0&0&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\n\\subsubsection{Six-member class ${\\mathcal C}_3^{(-)} \\times {\\mathcal C}_2^{(+)}$ containing (142536)($ab$) }\n\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_4^{(-)} R_2^{(+)} \\right] =\n\\left( \\begin{array}{cccc}-{\\frac{1}{2}}&-{\\frac{\\sqrt{3}}{2}}&0&0\\\\ {\\frac{\\sqrt{3}\n }{2}}&-{\\frac{1}{2}}&0&0\\\\ 0&0&{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{\n 2}}\\\\ 0&0&-{\\frac{\\sqrt{3}}{2}}&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_5^{(-)} R_2^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{4}}&{\\frac{\\sqrt{\n 3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&-{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3}{4}}\\\\ {\\frac{3}{4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{\n 1}{4}}&-{\\frac{\\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{3}{\n 4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_6^{(-)} R_2^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4}}&-{\\frac{\\sqrt{\n 3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&{\\frac{\\sqrt{3}\n }{4}}&{\\frac{3}{4}}\\\\ -{\\frac{3}{4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{1\n }{4}}&-{\\frac{\\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&-{\\frac{3\n }{4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_4^{(-)} R_3^{(+)} \\right] =\n\\left( \\begin{array}{cccc}-{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{2}}&0&0\\\\ -{\\frac{\\sqrt{3}\n }{2}}&-{\\frac{1}{2}}&0&0\\\\ 0&0&{\\frac{1}{2}}&-{\\frac{\\sqrt{3}\n }{2}}\\\\ 0&0&{\\frac{\\sqrt{3}}{2}}&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_5^{(-)} R_3^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4}}&{\\frac{\\sqrt{\n 3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&-{\\frac{\\sqrt{3}\n }{4}}&{\\frac{3}{4}}\\\\ -{\\frac{3}{4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{\n 1}{4}}&{\\frac{\\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{\n 4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_6^{(-)} R_3^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{4}}&-{\\frac{\n \\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{1}{4}}&{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3}{4}}\\\\ {\\frac{3}{4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{1\n }{4}}&{\\frac{\\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4\n }}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\n\n\\subsubsection{Nine-member class ${\\mathcal C}_3^{(-)} \\times {\\mathcal C}_3^{(+)}$ containing (12)(45)$^*$}\n\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_4^{(-)} R_4^{(+)} \\right] =\n\\left( \\begin{array}{cccc}1&0&0&0\\\\ 0&-1&0&0\\\\ 0&0&1&0\\\\ 0&0&0&-1\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_4^{(-)} R_5^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&-{\\frac{\\sqrt{3}}{2}}&0&0\\\\ -{\\frac{\\sqrt{3}\n }{2}}&{\\frac{1}{2}}&0&0\\\\ 0&0&-{\\frac{1}{2}}&-{\\frac{\\sqrt{3}\n }{2}}\\\\ 0&0&-{\\frac{\\sqrt{3}}{2}}&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_4^{(-)} R_6^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{2}}&0&0\\\\ {\\frac{\\sqrt{3}\n }{2}}&{\\frac{1}{2}}&0&0\\\\ 0&0&-{\\frac{1}{2}}&{\\frac{\\sqrt{3}}{\n 2}}\\\\ 0&0&{\\frac{\\sqrt{3}}{2}}&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_5^{(-)} R_4^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&0&0&-{\\frac{\\sqrt{3}}{2}}\\\\ 0&{\\frac{1}{2\n }}&-{\\frac{\\sqrt{3}}{2}}&0\\\\ 0&-{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}\n &0\\\\ -{\\frac{\\sqrt{3}}{2}}&0&0&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_5^{(-)} R_5^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{4}}&{\\frac{\\sqrt{\n 3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}&{\\frac{\\sqrt{3}\n }{4}}&{\\frac{3}{4}}\\\\ -{\\frac{3}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{1\n }{4}}&{\\frac{\\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4\n }}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_5^{(-)} R_6^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4}}&{\\frac{\\sqrt{\n 3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}&{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3}{4}}\\\\ {\\frac{3}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{1\n }{4}}&-{\\frac{\\sqrt{3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{\n 4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_6^{(-)} R_4^{(+)} \\right] =\n \\left( \\begin{array}{cccc}-{\\frac{1}{2}}&0&0&{\\frac{\\sqrt{3}}{2}}\\\\ 0&{\\frac{1}{2}}\n &{\\frac{\\sqrt{3}}{2}}&0\\\\ 0&{\\frac{\\sqrt{3}}{2}}&-{\\frac{1}{2}}&0\n \\\\ {\\frac{\\sqrt{3}}{2}}&0&0&{\\frac{1}{2}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_6^{(-)} R_5^{(+)} \\right] =\n \\left( \\begin{array}{cccc}{\\frac{1}{4}}&{\\frac{\\sqrt{3}}{4}}&{\\frac{3}{4}}&-{\\frac{\\sqrt{\n 3}}{4}}\\\\ {\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}&-{\\frac{\\sqrt{3}\n }{4}}&-{\\frac{3}{4}}\\\\ {\\frac{3}{4}}&-{\\frac{\\sqrt{3}}{4}}&{\\frac{1\n }{4}}&{\\frac{\\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{\n 4}}&{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\\begin{displaymath} {\\mathbf M}_G\\left[ R_6^{(-)} R_6^{(+)} \\right] =\n\\left( \\begin{array}{cccc}{\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{3}{4}}&-{\\frac{\n \\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}&-{\\frac{\\sqrt{\n 3}}{4}}&{\\frac{3}{4}}\\\\ -{\\frac{3}{4}}&-{\\frac{\\sqrt{3}}{4}}&\n {\\frac{1}{4}}&-{\\frac{\\sqrt{3}}{4}}\\\\ -{\\frac{\\sqrt{3}}{4}}&{\\frac{3\n }{4}}&-{\\frac{\\sqrt{3}}{4}}&-{\\frac{1}{4}}\\end{array} \\right) \\end{displaymath}\n\n\n\n\t\\section{Derivation of the transformation of internal coordinates}\\label{sec:appendixB}\n\\setcounter{equation}{0}\n\n\tIn the following we will describe the procedure with which to calculate the transformation of the internal coordinates due to the molecular symmetry group operations. If $f$ is a function of the Cartesian coordinates $\\vect{X}_1$, $\\vect{X}_2,$ and $\\vect{X}_3$ of three nuclei 1, 2, and 3, respectively, then, after the operation (123), nucleus 1 is at the position nucleus 3 was previously, 2 is at 1, and 3 is at 2. We thus have $(123)f(\\vect{X}_1,\\vect{X}_2,\\vect{X}_3) = f(\\vect{X}_3,\\vect{X}_1,\\vect{X}_2)$. Also, $E^*f(\\vect{X}_1,\\vect{X}_2,\\vect{X}_3) = f(-\\vect{X}_1,-\\vect{X}_2,-\\vect{X}_3)$. Since any curvilinear coordinate can be written as a function of the Cartesian coordinates, this defines the coordinate change due to the operation. We will now go through the three types of internal coordinates that TROVE uses.\n\n\t\\subsection{Bond Lengths}\n\t\\renewcommand{\\vect}[1]{\\mathbf{#1}}\n\t\n\n\tThe bond length between nuclei 1 and 2 (with position vectors $\\vect{r}_1$ and $\\vect{r}_2$, respectively in the space-fixed axis system) is defined by $|\\vect{r}_1 - \\vect{r}_2|.$ We see that this value is unaffected by $E^*$ and the other operations simply result in the relabelling of the bond lengths. For example, the \\ce{H1-C_$a$} bond length is given by $r_1$ $=$ $|\\vect{r}_1 - \\vect{r}_a|$ and the operation (123) transforms $r_1$ to $r_1'$ $=$ $|\\vect{r}_3 - \\vect{r}_a|$ $=$ $r_3$, and we obtain, once again, an obvious result. The operation $(14)(26)(35)(ab)^*$ yields $r_1'$ $=$ $|-\\vect{r}_4 + \\vect{r}_b| = |\\vect{r}_4 - \\vect{r}_b| = r_4$\n\n\t\\subsection{Bond Angles}\n\n\tTo obtain the angle between nuclei 1 and $b$ via nucleus $a$ we define $\\vect{r}_{1a} = \\vect{r}_1 - \\vect{r}_a$ and likewise for $\\vect{r}_{ba}$. Then the sought angle is\n\t\\begin{equation}\n\t\\alpha_1 =\n\t\t\\arccos \\left(\\frac{\\vect{r}_{1a}\\cdot \\vect{r}_{ba}}{|\\vect{r}_{1a}|| \\vect{r}_{ba}|} \\right).\n\t\t\\end{equation}\n\t\tDespite the expression for $\\alpha_1$ being more complex than the expression for $r_1$,\n\t\tthe effect of MS group operations can be determined completely analogously.\n\t\t $E^*$ produces no effect and the other operations relabel. For $(123)$, we have\n\t\t\\begin{equation}\n\t\t\\alpha_1' =\n\t\t\t\\arccos \\left(\\frac{\\vect{r}_{3a}\\cdot \\vect{r}_{ba}}{|\\vect{r}_{3a}|| \\vect{r}_{ba}|} \\right) = \\alpha_3,\n\t\t\t\\end{equation}\n\t\t\twhile for $(14)(26)(35)(ab)^*$\n\t\t\t\\begin{equation}\n\t\t\t\\alpha_1' =\n\t\t\t\t\\arccos \\left(\\frac{(-\\vect{r}_{4b})\\cdot (-\\vect{r}_{ab})}{|-\\vect{r}_{4b}|| -\\vect{r}_{ab}|} \\right) =\\arccos \\left(\\frac{\\vect{r}_{4b}\\cdot \\vect{r}_{ab}}{|\\vect{r}_{4b}|| \\vect{r}_{ab}|} \\right) = \\alpha_4.\n\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\n\t\t\t\t\\subsection{Dihedral Angles}\n\\def\\vect{e}_{z}{\\vect{e}_{z}}\n\\def\\mathrm{arctan2}{\\mathrm{arctan2}}\n\t\t\t\tThe expressions involving the dihedral angles are the most complicated. Consider the four nuclei labelled 1, a, b, and 4 in \\fig{fig:dihedral-atoms}. The dihedral angle between the plane spanned by 1, $a$, and $b$ and that spanned by $a$, $b$, and 4 is shown in \\fig{fig:dihedral-atoms} and \\fig{fig:dihedral-angle}. We define\n\t\t\t\t$\\normvec{\\vect{v}}$ as the unit vector obtained by normalization of $\\vect{v}$,\n\t\t\t\t$\\normvec{\\vect{v}}$ $=$ $\\vect{v}\/\\vert \\vect{v} \\vert$. The orientation of the $z$ axis is defined by $\\vect{e}_{z}$ $=$ $ \\normvec{\\vect{r}_{ab}}$. We need two further axes $x$ and $y$ (whose orientations are defined by the unit vectors $\\vect{e}_{x}$ and $\\vect{e}_{y}$, respectively) which, together with $z$, form a right-handed axis system.\n\t\t\t\t\n\t\t\t\tTo simplify the discussion, we take the $xy$ plane to be horizontal and the $z$ axis to be vertical.\n\t\t\t\tWe aim at obtaining the $x$ and $y$ components of $\\vect{r}_{1a}$ in order to determine the ``horizontal'' angle $\\theta$ it makes with $\\vect{r}_{4b}$ as shown in \\fig{fig:dihedral-angle}. We require the $x$ axis to be directed along the horizontal component (i.e., the component perpendicular to $\\vect{e}_{z}$)\n\t\t\t\tof $\\vect{r}_{4b}$. Thus, the $y$ axis is perpendicular to the plane defined by $\\vect{e}_{z}$ and $\\vect{r}_{4b}$, so that we have $\\vect{e}_{y} = \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}$ and therefore $\\vect{e}_{x} = \\vect{e}_{z} \\times \\vect{e}_{y} = \\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}$ as shown in \\fig{fig:dihedral-cross}. The unit vector $\\vect{e}_{x'} = \\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}$ defines an $x'$ axis in the 1--$a$--$b$ plane; this axis is analogous to the $x$ axis in the 4--$a$--$b$ plane. The dihedral angle between the two planes is the angle between the $x$ and $x'$ axes.\n\t\t\t\t\n\t\t\t\t\n\t\t\t\tThe $x$ and $y$ components of $\\vect{e}_{x'}$ are $\\vect{e}_{x} \\cdot \\vect{e}_{x'}$ and $\\vect{e}_{y} \\cdot \\vect{e}_{x'}$, respectively. In order to obtain the dihedral angle in the range $[-\\pi,\\pi]$ with the correct sign, we use the standard trigonometric\n\t\t\t\tfunction\\footnote{For $(x,y)$ $=$ $(r\\,\\cos\\varphi, r\\,\\sin\\varphi)$, the function\n\t\t\t\t$\\mathrm{arctan2} (y,x) = \\varphi \\in [-\\pi,\\pi]$.}\n\t\t\t\t$\\mathrm{arctan2}$ to obtain\n\t\t\t\t\\begin{eqnarray}\n\t\t\t\\theta =\t\\tau_{41} & = &\n\t\t\t\t\\mathrm{arctan2}(\\vect{e}_{y} \\cdot \\vect{e}_{x'}, \\vect{e}_{x} \\cdot \\vect{e}_{x'})\n\t\t\t\t\\nonumber\\\\ & = &\n\t\t\t\t\\mathrm{arctan2}(\\vect{e}_{y} \\cdot \\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}},\\vect{e}_{x} \\cdot \\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}})\n\t\t\t\t\\end{eqnarray}\n\t\t\t\twhich, written more explicitly, is\n\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\begin{split}\n\t\t\t\t\\theta =\t\\tau_{41} =\n\t\t\t\t\t\t\\mathrm{arctan2} [& \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}} \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}), \\\\\n\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}) ] \\\\\n\t\t\t\t\t\t=\t\\mathrm{arctan2} [& \\vect{e}_{z} \\cdot ( \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}), \\\\\n\t\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}) ]\n\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\\label{eq:died-def}\n\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\twhere the last expression emphasizes the equivalence of nuclei 1 and 4.\n\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\begin{figure}[htbp!]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\centering\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\tdplotsetmaincoords{70}{110}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{tikzpicture}[tdplot_main_coords]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,o-o] (0,0,0) node[above left]{$b$} -- (0,0,1) node[left]{$a$};\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,-o] (0,0,1) -- (0.353553,0.612372,1.707107) node[right]{1};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,-o] (0,0,0) -- (-0.700629,-0.509037,-0.866025) node[left]{4};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\caption{Four nuclei 1, 4, $a$, and $b$. The dihedral angle\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t$\\theta$ is the angle between\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tthe 1--$a$--$b$ and 4--$a$--$b$ planes. We define the positive direction of the angle by the right hand rule with the thumb pointing in the $\\vect{r}_{ab}$ direction.}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{fig:dihedral-atoms}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{figure}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{figure}[htbp!]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\centering\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\tdplotsetmaincoords{0}{0}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{tikzpicture}[tdplot_main_coords]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick] (0,0,0) -- (0,0,1) ;\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,-o] (0,0,1) -- (0.353553,0.612372,1.707107) node[right]{1};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,-o] (0,0,0) -- (-0.700629,-0.509037,-0.866025) node[left]{4};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\tdplotdrawarc[->]{(0,0,0)}{0.3}{-145}{60}{anchor = west}{$\\theta$};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\caption{Top down view of \\fig{fig:dihedral-atoms} with the dihedral angle (with the defined direction) marked as $\\theta$.}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{fig:dihedral-angle}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{figure}\n\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{figure}[htbp!]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\centering\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\tdplotsetmaincoords{0}{0}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{tikzpicture}[tdplot_main_coords]\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick] (0,0,0) -- (0,0,1) ;\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,->] (0,0,1) -- (0.353553,0.612372,1.707107) node[right]{$\\vect{r}_{1a}$};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,->] (0,0,0) -- (-0.700629,-0.509037,-0.866025) node[left]{$\\vect{r}_{4b}$};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,->] (0,0,0) -- (-0.700629,-0.509037,-0.866025);\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\draw[thick,->] (0,0,0) -- (0.573576,-0.819152, 0) node[below right]{$\\vect{e}_{y} = \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}$};\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\caption{The $y$ axis.}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\label{fig:dihedral-cross}\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{figure}\n\nWe define the dihedral angles $\\theta_{ij}$ and $\\tau_{ij}$ in terms of Eq.~\\eqref{eq:died-def} and, by means of this equation,\nwe can determine their transformation properties under the generating operations of {\\itshape\\bfseries G}$_{36}$.\n\t\t\t\t\t\t We note that generally $E^*\\mathrm{arctan2}(y,x) = \\mathrm{arctan2}(-y,x) = 2\\pi - \\mathrm{arctan2}(y,x).$\n\t\t\t\t\t\t The dihedral angle $\\theta_{23}$, for example, is defined by\n\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\\theta_{23} = \\mathrm{arctan2} [& \\vect{e}_{z} \\cdot ( \\normvec{\\vect{e}_{z} \\times \\vect{r}_{3a}} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{2a}}), \\\\\n\t\t\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{3a}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{2a}}) ]\n\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tUnder (123) we obtain\n\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\\theta_{23}' =\n\t\t\t\t\t\t\t\t\t\\mathrm{arctan2} [& \\vect{e}_{z} \\cdot ( \\normvec{\\vect{e}_{z} \\times \\vect{r}_{2a}} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}), \\\\\n\t\t\t\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{2a}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}) ] = \\theta_{12}.\n\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\tTo obtain the effect of (14)(26)(35)$(ab)^*$, we initially apply (14)(26)(35)$(ab)$ with the result\n\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\\mathrm{arctan2} [& -\\vect{e}_{z} \\cdot ( \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{5b}} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{6b}}), \\\\\n\t\t\t\t\t\t\t\t\t\t&(-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{5b}}) \\cdot (-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{6b}}) ].\n\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\tApplying $E^*$ reverses $\\vect{e}_{z}$, and by swapping $\\vect{r}_{5b}$ and $\\vect{r}_{6b}$ we obtain the final result:\n\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\\theta_{23}' =\t\t\t\\mathrm{arctan2} [& -\\vect{e}_{z} \\cdot ( \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{6b}} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{5b}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t&(-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{6b}}) \\cdot (-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{5b}}) ] = \\theta_{56}.\n\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\t\twhere the positive direction of the dihedral angle is in the sense of the proton numbering.\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tFinally, applying (14)(25)(36)$(ab)$ to $\\theta_{23}$ gives\n\t\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\\theta_{23}' =\t\\mathrm{arctan2} [& -\\vect{e}_{z} \\cdot ( \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{6b}} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{5b}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t&(-\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{6b}}) \\cdot (-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{5b}}) ] = \\theta_{56}.\n\t\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\t\t\tFor $\\tau_{41}$, which is defined by going counterclockwise from 4 to 1, the equation is\n\t\t\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\mathrm{arctan2} [& \\vect{e}_{z} \\cdot ( \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}) ] = \\tau_{41}.\n\t\t\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\t\t\t\tUnder operation (123)(456), this becomes\n\t\t\t\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\\tau_{41}' =\t\t\\mathrm{arctan2} [& \\vect{e}_{z} \\cdot ( \\normvec{\\vect{e}_{z} \\times \\vect{r}_{3a}} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{6b}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{3a}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{6b}}) ] = \\tau_{63}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\tTo determine the effect of (14)(26)(35)$(ab)^*$, we first apply (14)(26)(35)$(ab)$ to $\\tau_{41}$\nand obtain\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mathrm{arctan2} [& -\\vect{e}_{z} \\cdot ( \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{4b}} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{1a}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t&(-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{4b}}) \\cdot (-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{1a}}) ].\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tApplying $E^*$ reverses $\\vect{e}_{z}$. After swapping the order of $\\vect{r}_{1a}$ and $\\vect{r}_{4b}$, we have\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\tau_{41}' =\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mathrm{arctan2} [& -\\vect{e}_{z} \\cdot ( \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t&(\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{1a}}) \\cdot (\\vect{e}_{z} \\times \\normvec{\\vect{e}_{z} \\times \\vect{r}_{4b}}) ] = 2\\pi - \\tau_{41}.\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tFinally, applying (14)(25)(36)$(ab)$ to $\\tau_{41}$ gives\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\tau_{41}' =\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\mathrm{arctan2} [& -\\vect{e}_{z} \\cdot ( \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{4b}} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{1a}}), \\\\\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t&(-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{4b}}) \\cdot (-\\vect{e}_{z} \\times \\normvec{-\\vect{e}_{z} \\times \\vect{r}_{1a}}) ] = \\tau_{41}.\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\end{equation}\n\t\n\t\t\t\t\n\\section{Transformation of $\\tau$ under the generating operations of {\\itshape\\bfseries G}$_{36}$}\n\\setcounter{equation}{0}\n\\label{sect:appenix_b}\n\nThe torsional coordinate $\\tau$ is defined by\nEq.~(\\ref{eq:taudef}):\n\\begin{equation}\n\\tau =\n\\frac13 ( \\tau_{41} + \\tau_{62} + \\tau_{53}),\n\\end{equation}\nand we will investigate how this coordinate transforms under the generating operations of {\\itshape\\bfseries G}$_{36}$, $R_2^{(+)} = (123)(456)$, $R_2^{(-)} = (132)(456)$, $R_4^{(+)} = (14)(26)(35)(ab)^*$, and $R_4^{(-)} = (14)(25)(36)(ab)$ used in the {\\sc TROVE}\\ calculations. The transformation properties of the dihedral angles $\\tau_{ij}$ are derived as outlined in \\ref{sec:appendixB}.\n\nThe operation (123)(456) permutes the protons 1, 2, 3, 4, 5, 6 to the positions previously occupied by the protons labelled\n3, 1, 2, 6, 4, 5, and the transformed value of $\\tau$ is given by\n\\begin{equation}\n\\tau^\\prime =\n\\frac13 ( \\tau_{63} +\\tau_{51} + \\tau_{42})\n\\end{equation}\nwhere\n\\begin{align}\n\\tau_{63} & = \\tau_{62} + \\theta_{23} \\\\\n\t\\tau_{51} & = \\tau_{53} + \\theta_{31} \\\\\n\\tau_{42} & = \\tau_{41} + \\theta_{12}. \\\\\n\\end{align}\nFor example, $\\tau_{51}$ $=$ $\\tau_{53}$ $+$ $\\theta_{31}$, the plus sign coming about because the positive direction of rotation for the $\\tau_{ij}$ dihedral angles (proton 1 $\\rightarrow$ 2 $\\rightarrow$ 3) is the same as that of\n$\\theta_{12}$, $\\theta_{23}$, and $\\theta_{31}$.\nHence\n\\begin{align}\n \\tau^\\prime\n & = \\frac13 \\left(\n \\tau_{41} + \\tau_{62} +\n\\tau_{53}\n+ \\left[ \\theta_{12} + \\theta_{23} + \\theta_{31} \\right] \\right) \\\\\n & = \\frac13 \\left(\n \\tau_{41} + \\tau_{62} + \\tau_{53}\n + 2\\pi \\right) = \\tau + 2\\pi\/3\n\\end{align}\nor, equivalently, $\\tau^\\prime$ $=$ $\\tau - 4\\pi\/3$ as given in \\tabl{tab:torsion_transformations}, to ensure that $(123)^3(456)^3 = E$.\n\n\nAfter carrying out the operation (132)(456), the protons 1, 2, 3, 4, 5, 6 are found at the positions previously occupied by the protons labelled\n2, 3, 1, 6, 4, 5, respectively. Consequently, the transformed value of $\\tau$ is given by\n\\begin{equation}\n\\tau^\\prime =\n\\frac13 ( \\tau_{62} + \\tau_{53} + \\tau_{41})\n= \\tau.\n\\end{equation}\n\n\nFor $(14)(26)(35)(ab)^*$,\n\\begin{align}\n\\tau^\\prime & =\n\\frac13 ( \\tau_{14} + \\tau_{26} + \\tau_{35}) \\\\\n & =\n\\frac13 ( -\\tau_{41} - \\tau_{62} - \\tau_{53}) = -\\tau\n\\end{align}\nor, equivalently, $2\\pi - \\tau$ as given in \\tabl{tab:torsion_transformations}.\n\nFinally, for (14)(25)(36)$(ab)$\n\\begin{align}\n\\tau^\\prime & =\n\\frac13 ( -\\tau_{14} -\\tau_{26} - \\tau_{35}) \\\\\n & =\n\\frac13 ( \\tau_{41} + \\tau_{62} + \\tau_{53}) = \\tau.\n\\end{align}\nThe transformation properties derived here are summarized in\n\\tabl{tab:torsion_transformations}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgcfk b/data_all_eng_slimpj/shuffled/split2/finalzzgcfk new file mode 100644 index 0000000000000000000000000000000000000000..cf4c86308fa91c6cff46e47e46a3c6b971b9c87e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgcfk @@ -0,0 +1,5 @@ +{"text":"\n\n\\section{Introduction}\n\\label{sec:introduction}\n\\IEEEPARstart{U}{}nderstanding the system model is essential for robotic applications, especially safety-critical autonomous systems. In particular, at high-speed driving, various dynamics elements such as chassis, tires, or engines become crucial to implement high-speed autonomy.\nModel-based optimal control\\cite{lewis2012optimal,jung2021game} is well-suited for handling those factors and is widely used to design dynamics system control.\nBy leveraging physics-based parametric dynamic models, it optimizes driving behavior with respect to a designed objective function and enables safe and reliable control system design.\n\nDespite the success of the model-based approach in robotics, the model-based algorithm has two fundamental challenges: model fidelity and tractability.\nThe performance of model-based approaches relies heavily on the accuracy of the model. However, identifying accurate models is often laborious or intractable because of their large search space and nonlinearity. Other than the model accuracy, models also need to be computationally feasible for the real-time control applications. High-fidelity but highly complex models are often difficult to integrate into real-time safety-critical driving systems.\n\nTo tackle these challenges, conventional approaches, including the Prediction Error Method, are used to identify model parameters\\cite{tangirala2018principles}.\nHowever, those methods often require the model structure to be linear or in specific mathematical forms, which might not be feasible to describe the nonlinear high-speed autonomous driving system.\nOn the other hand, in several recent works, data-driven approaches using neural networks, Gaussian processes, or Bayesian methods have been actively employed for nonlinear system dynamics modeling and have shown promising results\\cite{brunton2022data}. In \\cite{spielberg2019neural}, they proposed a simple neural network to replace a single-track vehicle model and used it to generate feedforward control signals. Similarly, in \\cite{hermansdorfer2021end}, they designed Deep Neural Networks (DNN) as a model approximator to identify the vehicle dynamics model in an end-to-end learning fashion.\nHowever, while DNN is an efficient way to approximate nonlinear systems, it is difficult to integrate with non-learning model-based methods, which are reliable in real-world applications. Furthermore, it is challenging to ensure the validity of the DNN model in unseen driving scenarios without large-scale field tests.\n\nIn this letter, we propose a data-driven model identification method via hyperparameter optimization (MIHO) for high-speed autonomous racing systems. Our key idea is to leverage a parameter optimization approach from machine learning to identify physics-based parametric models in a data-driven manner without any limitation on the form of the model equation.\nTo this end, we adopt a novel hyperparameter optimization (HPO) method that has an efficient exploration and exploitation strategy.\nUsing the proposed method, we estimate the parameters of the integrable parametric dynamics models for a full-scaled autonomous racecar platform, Dallara AV-21 (Fig. \\ref{fig:method_overview}), at the Indy Autonomous Challenge (IAC) \\cite{IAC}.\nWe validate our proposed approach by integrating identified models into the high-speed autonomous system and conducting extensive field experiments, including over $200 km\/h$ autonomous driving and obstacle avoidance scenarios in the Indianapolis Motor Speedway (IMS) and Las Vegas Motor Speedway (LVMS).\n\nIn summary, our technical contributions are as follows:\n\\begin{itemize}\n \\item We propose a data-driven model identification method via hyperparameter optimization.\n \\item We design model-based planning and control systems incorporating the learned vehicle dynamics models.\n \\item We integrate the systems with learned model parameters into the full-scaled autonomous race vehicle and extensively validate them during the IAC.\n\\end{itemize}\n\\section{Model Identification via Hyperparameter Optimization}\n\\label{sec:model_identification}\nVehicle dynamic models allow us to describe the race vehicle's motion accurately. However, dynamics models with high fidelity are often challenging to identify due to their high nonlinearity and a large number of model parameters. Therefore, an efficient parameter estimation approach is necessary to find the parameter configuration of such complex models.\nIn this letter, we propose a model identification method via hyperparameter optimization (MIHO) to learn the optimal model parameter configuration by a data-driven approach.\nHyperparameter optimization (HPO) is the problem of selecting an optimal hyperparameter configuration required for neural network training in the machine learning field \\cite{feurer2019hyperparameter}.\nA hyperparameter is a parameter that controls the training process.\nHPO optimizes the hyperparameter configuration by evaluating the performance of the configuration during the model training process.\nSince one course of neural network training requires a substantial time, HPO focuses on the balanced exploration and exploitation strategy for the efficient optimal hyperparameter selection.\\cite{feurer2019hyperparameter}.\nMotivated by the balanced strategy, we design MIHO by adopting the HPO to the model identification problem. First, we regard a parameter configuration $p \\in \\mathbb{R}^{n_p}$ with $n_p$ model parameters as a set of hyperparameters of a nonlinear dynamics model $f$. Then, we identify the parameter configuration by evaluating the following objective function inspired by the standard supervised learning problem:\n\\begin{equation}\n \\label{eq:objective}\n \\mathcal{L} = \\frac{1}{\\lvert D \\rvert} \\sum_{(x,y) \\in D} \\lVert y - {f}(x; p) \\rVert^2,\n\\end{equation}\nwhere $x, y$ denote the sampled input and output data of the model $f$ from a given dataset $D$. By minimizing this learning objective, we find an optimized model parameter configuration $p^*$ that has the minimum model error with the observed model output $y$.\n\\begin{algorithm}[b]\n\\caption{MIHO Algorithm based on Hyperband}\n\\textbf{Input: } $R, \\eta, D$\n\\begin{algorithmic}[1]\n\\State $ s_{max} \\leftarrow \\lfloor \\text{log}_{\\eta}(R) \\rfloor, B = (s_{max} + 1)R $\n \n\\For{$s \\in \\{s_{max},s_{max}-1, ..., 0 \\}$}\n \\State $n = \\lceil \\frac{B}{R} \\frac{\\eta^{s}}{(s+1)} \\rceil, r = R\\eta^{-s}$\n \\State $P = \\text{get\\_model\\_param\\_config}(n)$ \n \\For{$j \\in \\{ 0, ..., s \\}$}\n \\State $ n_j = \\lfloor n\\eta^{-j} \\rfloor, r_j = r\\eta^j $\n \\State $ L = \\{ \\text{eval\\_with\\_mutation}(p,r_j,D) : p \\in P \\} $\n \\State $ P = \\text{select\\_top\\_k\\_config}(P, L, \\lfloor n_j\/\\eta \\rfloor) $\n \\EndFor\n\\EndFor\n\\end{algorithmic}\n\\textbf{Output: } \\text{Optimized parameters $p^*$ with the smallest loss.}\n\\label{algo:MIHO}\n\\end{algorithm}\nThe model $f$ has no limitation on its form of the equation. Thus, our method can be used for arbitrary parametric models, such as a combination of polynomial or mathematical terms, as well as analytic physics-based models.\n\nWe implement MIHO incorporating a bandit-based HPO algorithm, Hyperband\\cite{li2017hyperband}, as summarized in Algorithm \\ref{algo:MIHO}.\nIt is a variation of a random search algorithm with explore-exploit theory to find the optimal hyperparameter configuration based on an evaluation loss.\nThe algorithm needs two arguments: $R$, the maximum amount of resource (e.g., the number of evaluation iterations) that can be allocated to a single configuration, and $\\eta$, a value that determines the proportion of the discarded configurations.\nThe two arguments derive $s_{max}+1$ combinations (called \"brackets\" in \\cite{li2017hyperband}) of the values $n$ and $r$, which enables various ratios of exploration and exploitation for finding the optimal parameter configuration.\nHyperband compares the evaluation loss of each sampled configuration and allocates more resources to the configurations with lower evaluation losses, excluding the configurations with higher losses. It repeats the sampling and exclusion processes until the last configuration remains to obtain the optimal set of hyperparameters.\nTo adjust the HPO algorithm to the model parameter optimization, we add the Gaussian mutation \\cite{mutation2019genetic} during the evaluation to explore the new neighbor parameters that might have less model loss. Unlike the original HPO, which only allocates more resources $r_i$, our approach, MIHO, adds noise perturbation with a noise random variable $\\epsilon \\in \\mathbb{R}^{n_p}$ at the selected configuration $p$ after the resource allocation as:\n\\begin{equation}\n \\label{eq:gaussian_mutation}\np_{mut} = p + \\sigma \\odot \\epsilon, \\quad \\epsilon \\sim N(0, I),\n\\end{equation}\nwhere $\\odot$ is the element-wise product and $\\sigma \\in \\mathbb{R}^{n_p}$ is the standard deviation of the exploration noise that is annealed over the course of the evaluation \\cite{kirkpatrick1983optimization}.\nWe define the following three functions for the HPO process in MIHO:\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.44\\textwidth]{figures\/method_overview.png}\n\\caption{\nOverview of our autonomous driving system in the AV-21. Our learned model parameters are embedded in the planning and control modules that are covered in this letter (highlighted in blue).\nSeveral input variables are omitted for clarity.\n}\n\\label{fig:method_overview}\n\\vspace{-1.5em}\n\\end{figure}\n\\begin{itemize}\n\\item \\textit{$\\text{get\\_model\\_param\\_config}(n)$}\\text{:} a function that returns a set of $n$ random parameter configurations from the normal distribution pre-defined over the configuration space.\n\n\\item \\textit{$\\text{eval\\_with\\_mutation}(p,r_j,D)$}\\text{:} a function that receives a parameter configuration $p$, an allocated resource $r_j$, and a dataset $D$ as arguments. Using the dataset, this function evaluates an initial configuration and mutates it for the allocated $r_j$ iterations by Eq. \\ref{eq:gaussian_mutation}.\nIf a mutated configuration $p_{mut} \\in \\mathbb{R}^{n_p}$ has a less loss than the initial one, the function replaces $p$ with $p_{mut}$. It returns the final loss after spending the allocated resources.\n \n\\item \\textit{$\\text{select\\_top\\_k\\_config}(P, L, k)$}\\text{:} a function that receives a set of hyperparameter configurations $P$ with their corresponding evaluation losses $L$ and returns the top $k$ high-performing configurations (here, $k = \\lfloor n_j\/\\eta \\rfloor$).\n\\end{itemize}\n\\section{Vehicle Dynamics Model}\n\\label{sec:vehicle_dynamics_model}\n\\subsection{Tire Dynamics Model}\nTire dynamics is one of the factors that significantly affect the nonlinearity of driving dynamics. Especially the lateral tire model is crucial to design stable path-tracking control in high-speed driving.\nThe tire model\\cite{bakker1987tyre} can be described as a function of the slip angle $\\alpha_i$, slip ratio $\\rho_{x,i}$, inclination angle $\\theta_i$, tire load $F_{z,i}$, and current velocity $v_{x,i}$, which has a lateral tire force $F^{*}_{y,i}$ of each tire ($i \\in \\{LF, LR, RF, RR\\}$) as,\n\\begin{equation}\n \\label{eq:pacejka_tire_function}\n \\begin{aligned}\n F^{*}_{y,i} = f_{tire}(\\alpha_i, \\rho_{x,i}, \\theta_i, F_{z,i}, v_{x,i}).\n \\end{aligned}\n\\end{equation}\nAlthough the model has high fidelity with various dynamics perspectives, it has low suitability for designing the controller of high-speed driving, which requires real-time performance.\nTherefore, we first define a tire model with dimension-reductio\nthat can be applied to model-based control design within an acceptable complexity. We then optimize the model's parameter configuration to represent the overall tire characteristic of a given dataset using our MIHO algorithm.\nWe follow the Pacejka tire model\\cite{kabzan2020amz} to define the tire dynamics.\nWhile the prior work neglect vertical and horizontal offsets of the model, we formulate a tire model $F_{y,i} = f_{t,i}(\\alpha_i; {p}_{t,i})$ containing the offset parameters $S_{x,i}, S_{y,i}$ to describe the asymmetric tire characteristic determined to maximize cornering performance on an oval track:\n\\begin{equation}\n \\label{eq:tire_model}\n \\begin{aligned}\n F_{y,i} &= D_i \\sin(C_i \\arctan(B_i (\\alpha_i + S_{x,i}))) + S_{y,i},\n \\end{aligned}\n\\end{equation}\nwhere the tire model parameter configuration ${p}_{t,i} = \\{B_i, C_i, D_i, S_{x,i}, S_{y,i}\\}$ is identified by minimizing the following tire model objective with a given dataset $D_{t,i}$ as\n\\begin{equation}\n \\label{eq:objective_tire}\n \\mathcal{L}_{t,i} = \\frac{1}{\\lvert D_{t,i} \\rvert} \\sum_{(\\alpha_i, F^{*}_{y,i}) \\in D_{t,i}} \\lVert F^{*}_{y,i} - f_{t,i}(\\alpha; {p}_{t,i}) \\rVert^2.\n\\end{equation}\n\n\\subsection{Engine Torque Model}\nThe powertrain system of our racecar consists of an internal combustion engine, transmission, and wheels. The AV-21 is a rear-wheel-drive vehicle whose traction force $F_{x,r}$ is generated by engine-based driveline dynamics. We model the equation of the longitudinal dynamics\\cite{rajamani2011vehicle} as follows:\n\\begin{equation}\n\\label{eq:longi_model}\n\\begin{aligned}\n m a_{x} = F_{x,r} - C_d v^{2}_{x} - C_r,\n\\end{aligned}\n\\end{equation}\nwhere $m$ is the vehicle mass, $v_x$ is the longitudinal velocity, $C_d$ denotes the drag coefficient, and $C_r$ denotes the rolling resistance.\nFollowing a prior work \\cite{engineSAGE16}, the traction force can be expressed as:\n\\begin{equation}\n\\label{eq:traction_force}\n\\begin{aligned}\n F_{x,r} = m a_{x,r} = \\frac{T_{e} \\eta_{t} i_{g} i_{0}}{R_w},\n\\end{aligned}\n\\end{equation}\nwhere $a_{x,r}$ denotes the traction acceleration, $\\eta_{t}$ denotes the efficiency of the transmission, $i_{g}, i_{0}$ denote the transmission ratio of the current gear and final reducer, and $R_w$ denotes the wheel radius. $T_{e} = f_e (w_e, \\tau_t)$ is the engine torque map in terms of the engine speed $w_e$ and throttle command $\\tau_t$. Due to the high complexity of the engine characteristic, the torque map is expressed as an experimental lookup table based on engine torque curves of specific throttle opening commands.\nAlthough the engine torque model can be obtained by the engine dynamometer testing\\cite{killedar2012dynamometer}, it could suffer from modeling errors because the dynamometer testing is done in a static environment. Therefore, we build the engine torque map that integrates the dyno data with learned torque curves based on our data-driven model identification approach. We express an engine torque curve $T_{e,\\tau_{t}} = f_{\\tau_{t}}(w_e; {p}_{\\tau_{t}})$ of a throttle command $\\tau_t$ as a third order polynomial function of the engine speed $w_e$ as:\n\\begin{equation}\n\\label{eq:engine_map}\n\\begin{aligned}\n T_{e,\\tau_{t}} = p_{\\tau_{t},0} + p_{\\tau_{t},1} w_e + p_{\\tau_{t},2} w^{2}_e + p_{\\tau_{t},3} w^{3}_e,\n\\end{aligned}\n\\end{equation}\nwhere $p_{\\tau_{t}} = \\{p_{\\tau_{t},0}, p_{\\tau_{t},1}, p_{\\tau_{t},2}, p_{\\tau_{t},3}\\}$ is the torque model parameter configuration. Using the resultant traction accelerations $a_{x,r}$ while driving, the engine torque output $T^{*}_{e,\\tau_{t}}$ is obtained by Eq. \\ref{eq:traction_force}.\nThen, $p_{\\tau_{t}}$ is learned by minimizing the following engine model objective with a given dataset $D_{\\tau_{t}}$:\n\\begin{equation}\n \\label{eq:objective_engine}\n \\mathcal{L}_{\\tau_{t}} = \\frac{1}{\\lvert D_{\\tau_{t}} \\rvert} \\sum_{(w_{e}, T^{*}_{e,\\tau_{t}}) \\in D_{\\tau_{t}}} \\lVert T^{*}_{e,\\tau_{t}} - f_{\\tau_{t}}(w_e; {p}_{\\tau_{t}}) \\rVert^2.\n\\end{equation}\nTo stabilize the learning process, we normalize the engine speed in the range [0,1] with the maximum engine speed.\n\n\n\\section{Model-based Planning and Control}\n\\label{sec:model_based_vehicle_control}\nIn this session, we introduce a model-based planning and control algorithm that uses the learned model parameters. We exploit the learned tire parameters to design a dynamics-aware velocity planning and model-based lateral controller (Fig. \\ref{fig:method_overview}). We also integrate the engine dyno data with the learned engine torque model to construct an engine lookup table.\n\\subsection{Dynamics-aware Velocity Planning}\nHigh-speed cornering during racing causes significant tire load transfer on each wheel due to a lateral acceleration at the roll axis.\nSince the tire load governs the maximum performance of the tire, a model-based velocity strategy accounting for the real-time wheel load is necessary to maximize the tire performance without losing tire grip.\nWe introduce a dynamics-aware velocity planning algorithm that derives the velocity plans with maximum tire performance based on the learned tire dynamics. We first compute the real-time vertical tire load $F_{z,i}$ affected by the lateral load transfer $\\Delta W_f$\\cite{seward2017race}. The diagram of the load transfer at the roll axis is illustrated in the left of Fig. \\ref{fig:method_vehicle_dynamics_control}. The load transfer is computed by the roll couple $C_{roll} = m_s \\dot{v}_y h_a$, where $m_s$ is the sprung mass, $\\dot{v}_y$ is the lateral acceleration, and $h_a$ is the roll height. As the learned tire model describes the characteristic for the nominal tire load $\\bar{F}_{z,i}$, we compute the maximum lateral force of each tire $F^{max}_{y,i}$ in terms of the tire load ratio with peak value of the tire model as:\n\\begin{equation}\n\\label{eq:max_lat_tire}\n\\begin{aligned}\n F^{max}_{y,i} &= \\mu \\frac{F_{z,i}}{\\bar{F}_{z,i}} F^{peak}_{y,i},\n\\end{aligned}\n\\end{equation}\nwhere $\\mu$ is a tire performance factor to control the confidence and maximum performance of the tire model, $\\frac{{F}_{z,i}}{\\bar{F}_{z,i}}$ is the tire load ratio.\nThe maximum lateral acceleration is determined by the following lateral motion dynamics \\cite{kabzan2020amz}:\n\\begin{equation}\n\\label{eq:lat_accel_limit}\n\\begin{aligned}\n a_{y,max} = \\frac{1}{m} (F^{max}_{y,r} + F^{max}_{y,f} cos(\\delta) - m v_x \\dot{\\psi}),\n\\end{aligned}\n\\end{equation}\nwhere $\\delta$ is the steering angle and $\\dot{\\psi}$ is the yaw rate.\nThen a desired maximum velocity $v_{x,des}$ is planned according to the curvature $\\kappa$ of a reference path from a planning module \\cite{lee2022resilient}:\n\\begin{equation}\n\\label{eq:velocity_limit}\n\\begin{aligned}\n v_{x,des} = \\sqrt{a_{y,max} \/ \\kappa}.\n\\end{aligned}\n\\end{equation}\n\n\\subsection{Throttle and Brake Control}\nThe planned desired velocity is fed to a feedback control module \\cite{doyle2013feedback} to compute the traction force. However, as shown in Fig \\ref{fig:method_overview}, another low-level controller to transform the traction force to the throttle command is required to control the racecar with nonlinear driveline dynamics. We design the throttle and brake control system following \\cite{hedrick1997brake}.\nFig. \\ref{fig:method_throttle_brake_control} shows the details of the low-level control system.\nWe exploit the integrated engine torque map to convert the desired engine torque $T_{e,des}$ to the desired throttle command $\\tau_{t,des}$.\nAs the torque map is built as a lookup table, we search the desired throttle with respect to a given engine speed and desired torque.\nThe inverse brake model is a module to convert the braking force to the brake pedal command, which is activated if $F_{x,r}$ is negative. The brake model is also attained by our proposed MIHO, but details are omitted to conserve space.\n\n\\subsection{Model-based Path Tracking Control}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.46\\textwidth]{figures\/method_vehicle_dynamics_for_control.png}\n\\caption{\n\\textbf{Left: } Lateral load transfer generation by the lateral acceleration at the roll axis. \n\\textbf{Right: } Overall diagram of the vehicle model.\n}\n\\label{fig:method_vehicle_dynamics_control}\n\\vspace{-1em}\n\\end{figure}\nWe follow the lateral vehicle dynamics of \\cite{rajamani2011vehicle} illustrated in the right of Fig \\ref{fig:method_vehicle_dynamics_control}. The lateral model is derived from the objective of tracking a reference trajectory. We implement path tracking control by stabilizing a velocity-dependent chassis model in terms of the error state variables $\\xi$ and control $u$.\n\\begin{equation}\n\\label{eq:state_control}\n\\begin{aligned}\n \\xi = [e_y, \\dot{e}_y, e_{\\psi}, \\dot{e}_{\\psi}]^T, \\quad u = \\delta,\n\\end{aligned}\n\\end{equation}\nwhere $e_y, e_{\\psi}$ denote the position and orientation error with respect to a given trajectory. The lateral model contains tire-related model parameters such as the cornering stiffnesses of the front and rear tires $C_{\\alpha,f}, C_{\\alpha,r}$. Since the lateral dynamics is obtained from the bicycle model, the front and rear tire models are optimized according to Eq \\ref{eq:tire_model}, but the sum of the left and right tire forces is used as the model output. The cornering stiffnesses then can be approximated as follows \\cite{rajamani2011vehicle}:\n\\begin{equation}\n\\label{eq:cornering_stiffness}\n\\begin{aligned}\n C_{\\alpha f} &\\approx B_{f} \\times C_{f} \\times D_{f}, \\quad C_{\\alpha r} \\approx B_{r} \\times C_{r} \\times D_{r}.\n\\end{aligned}\n\\end{equation}\n\nBased on the lateral vehicle model, we design the Linear Quadratic Regulator with the following optimization problem: \n\\begin{equation}\n\\label{eq:LQR}\n\\begin{aligned}\n \\underset{u}{\\text{min }} \\int_{0}^{\\infty} (\\xi^T Q \\xi + u^T R u ) dt,\n\\end{aligned}\n\\end{equation}\nwhere $Q, R$ denote gain matrices for LQR.\nFor the real-time control performance, we compute the state feedback optimal LQR gains over piecewise velocity intervals offline\\cite{spisak2022robust}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{figures\/method_throttle_brake_control.png}\n\\caption{\nThrottle and brake control system. \n}\n\\label{fig:method_throttle_brake_control}\n\\vspace{-1em}\n\\end{figure}\n\\section{Evaluation}\n\\label{sec:evaluation}\n\\subsection{Analysis for Model Identification}\n\\subsubsection{Tire Dynamics Model}\nFig. \\ref{fig:result_tire_model} illustrates the learned tire models with datasets provided as the tire property files (*.tir).\nThe files contain tire force and moment characteristics with high fidelity\\cite{schmeitz2013mf}.\nFor model identification, we sampled 3000 data for each tire using the property files in various tire state conditions such as tire load, camber angle, slip angle, and slip ratio.\nAs illustrated in the left and middle of Fig. \\ref{fig:result_tire_model}, the learned models show good fitness to the tire characteristic distribution.\nWe further investigated the tire model with the test drive data collected during track racing.\nThe right of Fig. \\ref{fig:result_tire_model} illustrates the front tire model of the single-track bicycle dynamics learned by the provided tire property data comparing it with the driving data that is not used for learning.\nThe learned model shows the generalization ability for the overall data distribution represented by blue dots.\nHowever, since we obtained the model by offline optimization and focused on the representativeness of data, the model needs higher accuracy in some edge cases near the peaks of the lateral force.\nTo handle these cases, an online parameter optimization can be used by parallelizing the HPO process in MIHO, and we will implement it in future work.\n\n\\subsubsection{Engine Torque Model}\nFig. \\ref{fig:result_engine_brake} illustrates the learned engine torque curves and integrated engine map. The data for the engine map was provided by engine dynamometer testing. \nFor higher reliability, we incorporated our data-driven engine torque models with the dyno data, especially for the throttle pedals 5, 15, and 20\\%, where the dynamometer showed insufficient accuracy in torque measurements.\nThe result shows that the learned torque curves are able to represent the change of the maximum torque according to the throttle commands. Moreover, the learned models also fit the torque curves that change nonlinearly as a function of engine speed.\nWe integrated the learned torque models with the provided dyno data and interpolated the torque data to construct an engine lookup table. The blue area on the right of Fig \\ref{fig:result_engine_brake} shows the interpolated region by the learned torque curves.\nOur vehicle utilized the learned region in racing scenarios such as pit-in\/out, obstacle avoidance, and driving within $100 km\/h$ (Fig. \\ref{fig:result_IMS_obstacle_only}).\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.86\\textwidth]{figures\/learned_tire_model.png}\n\\caption{\nLearned tire models with the provided tire data (\\textbf{Left:} left-front and right-front, \\textbf{Middle:} left-rear and right-rear). \\textbf{Right:} \nLearned front tire dynamics of the single-track bicycle model and the distribution of the collected data on the track.\n}\n\\label{fig:result_tire_model}\n\\vspace{-1em}\n\\end{figure*}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figures\/learned_engine_only.png}\n\\caption{\nLearned engine torque curves and the integrated engine map.\n}\n\\label{fig:result_engine_brake}\n\\vspace{-1em}\n\\end{figure}\n\\subsection{Control Performance in Indy Autonomous Challenge}\nOur model-based planning and control algorithms were deployed in the full-scale racecar platform. Moreover, we extensively validated our learned model parameter-based algorithms in the real-world race tracks, IMS and LVMS.\nThe algorithms successfully performed various race scenarios, such as obstacle avoidance and high-speed autonomous driving over $200 km\/h$ on the race tracks (Fig. \\ref{fig:result_IMS_velocity_perform}).\n\n\\subsubsection{Obstacle Avoidance at the IMS}\nFig. \\ref{fig:result_IMS_obstacle_only} shows the quantitative results of the obstacle avoidance mission.\nIn this mission, obstacles are located before the first-corner section, where the velocity plan is critical for avoiding collision while keeping close to the racing line. For the sake of safety, we set the tire performance factor $\\mu$ as 0.7.\nOur dynamics-aware velocity planner was able to allow the racecar to maximize the velocity while regulating the lateral acceleration within the learned maximum tire performance during the rapid avoidance maneuvers.\nThe obstacle avoidance was initiated in high-speed driving at $100 km\/h$, and steering commands were computed up to -10 degrees to follow the generated collision-free reference trajectory.\nThe sharp steering command could cause significant lateral acceleration higher than $9.0 m\/s^2$, which might cause critical tire grip loss. \nHowever, our tire model-based velocity planner inferred the allowable maximum lateral acceleration based on the real-time tire load. As a result, it was possible to plan for safe desired velocity within lateral acceleration limit capable of preserving the tire grip performance.\n\\subsubsection{High-Speed Autonomous Driving in LVMS}\nFurthermore, we extensively validated the control performance based on the optimized model parameters at LVMS.\nFig. \\ref{fig:result_LVMS_tracking_perform_only} illustrates the quantitative results of the lateral and longitudinal control while our vehicle raced more than nine laps ($23 km$). Our path-tracking algorithm shows robust control performance leveraging the learned tire parameters. The largest position and orientation errors were $0.6 m$ and $-2.2$ degrees, respectively.\nIn addition, the AV-21 succeeded high-speed autonomous driving at above $144 km\/h$ (with a top speed of $217.4 km\/h$), where the dynamic scenario had yet to be visited and adjusted before this track experiment.\nThese results demonstrate that MIHO can optimize and provide appropriate prior dynamics models offline for the design of model-based control before deployment.\nHowever, the characteristic of vehicle dynamics changed and affected the control performance over high-speed driving. As shown in the bottom of Fig. \\ref{fig:result_LVMS_tracking_perform_only}, the tire temperatures were increased after reaching the unseen velocity range. In addition, after visiting the range of over $144 km\/h$, our low-level controller computed throttle command of over 50\\% with the engine range consisting only of the provided dyno data. Those factors might have an effect on the velocity error in the velocity range $[42, 52] m\/s$ of Fig \\ref{fig:result_LVMS_tracking_perform_only}.\nNevertheless, the control system can be improved with an extended data-driven engine map for throttle control at high-speed.\nWe also point out that our method has the potential to be processed online by parallelizing the HPO process\\cite{li2017hyperband}, which enables the method to identify the model in real-time during deployment. The online MIHO could be incorporated with the offline model optimization introduced in this work, and we leave it as an important future work.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.80\\textwidth]{figures\/result_scenario.png}\n\\caption{\n\\textbf{Left: } Team KAIST's successful obstacle avoidance at IMS. The bottom illustrates the point cloud data and traveled trajectory during avoidance.\n\\textbf{Right: } Our AV-21 drove more than nine laps ($23 km$) at the Tri-Oval Superspeedway of LVMS.\n}\n\\label{fig:result_IMS_velocity_perform}\n\\vspace{-1.5em}\n\\end{figure*}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figures\/result_IMS_obstacle_only.png}\n\\caption{\nResults of the velocity control, lateral accelerations, and steering angles during the obstacle avoidance mission at the IMS.\n}\n\\label{fig:result_IMS_obstacle_only}\n\\vspace{-1.5em}\n\\end{figure}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{figures\/result_LVMS_tracking_perform_only.png}\n\\caption{\nResults of the errors, velocity control, throttle\/brake controls, and temperature of the right-rear and right-front tires in LVMS.\n}\n\\label{fig:result_LVMS_tracking_perform_only}\n\\vspace{-1.5em}\n\\end{figure}\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe present MIHO, a data-driven model identification method via hyperparameter optimization. Our approach showed the ability to optimize the parameters of the dynamics models, such as the tire models and engine torque curves. Furthermore, the model-based planning and control system with the learned model parameters demonstrated stable performance in the real-world track environments, IMS and LVMS. In future works, we will implement the online HPO method and integrate it with the offline method of this work to iteratively infer the changing parameters of the vehicle dynamics while on track.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAbout 30-40\\% of the Galactic WR stars have a visible OB-type companion (Vanbeveren and Conti, 1980; van der Hucht, 2001; Crowther, 2007). Detailed massive single star and massive binary population synthesis\\footnote {including a model to account for mergers and a formalism to account for the effects on binary parameters of asymmetrical supernova explosions, a process responsible for separating binaries and thus for making single stars which were born as binary components} makes it then possible to answer the question: `What must be the initial O-type close binary frequency to explain the observed WR+OB binary frequency?'. An answer was presented by Vanbeveren et al. (1998a, b, c) who concluded that the initial massive close binary frequency must be at least 70\\%, at that time a result that was hard to accept by the massive star scientific community. Fortunately, this predicted high O-type close binary percentage was observationally confirmed about 1.5 decades later by Sana et al. (2012) and it is now generally accepted that most of the massive stars are born in close binaries.\n\n`Close' means that at some point during the binary evolution Roche lobe overflow (RLOF) will happen. RLOF may be accompanied by mass transfer, may result in the formation of a common envelope (note that during a common envelope phase mass transfer is not expected to happen) and\/or may result in the merger of both binary components. The interested reader may find useful information regarding these processes and the uncertainties in extended reviews and references therein (van den Heuvel, 1993; Vanbeveren et al., 1998b, c; Langer, 2012; De Marco \\& Izzard, 2017). Massive close binary evolution and the study of the binary processes listed above have been the subject of numerous papers since the sixties but it is fair to state that the RLOF with all its facets is still not fully understood and it therefore requires further theoretical and observational research.\n\nExcept for about a quarter of Galactic massive WR stars, which are H-rich main-sequence stars of very high mass, most of the remaining Galactic WR-stars are hydrogen deficient core helium burning stars. Their progenitors have therefore experienced extensive mass loss. In the case of single stars this is possibly due to an LBV-type and\/or a red supergiant-type mass loss. When the WR-star is a binary component its progenitor may have lost its hydrogen rich layers by RLOF\/common envelope evolution. A comparison between the observations of WR+OB binaries and evolutionary prediction may therefore help to understand the RLOF\/common envelope evolution of their progenitors. Earlier attempts can be found in e.g., Vanbeveren et al., 1998 b, c; Petrovic et al., 2005a; Eldridge, 2009; Shenar et al., 2016.\n\nMass transfer is accompanied by angular momentum transfer and the mass gainer is expected to spin up. When mass transfer\/accretion goes via an accretion-disk Packet (1981) showed that this spin-up is very rapid and that soon after the onset of the RLOF the gainer will rotate at its critical break-up velocity. The possible effects of binary interaction on the rotation rates and the rotation velocity distribution of massive stars was investigated by De Mink et al. (2013). It was concluded that a significant percentage of all rapid rotators (maybe all of them) could have a binary origin. The authors note that a main uncertainty affecting this conclusion is the neglect of magnetic fields generated during mass accretion and stellar mergers (see also section 4). \n\nTo test these theoretical expectations the observed spin-rates of the O components in WR+O binaries may be most illuminating. Shara et al. (2017) measured and analyzed the spin rates of the O stars in eight WR+O binaries using the Southern African Large Telescope (SALT) increasing the Galactic sample size from 3 to 11. The 5 known WR+O binaries of the SMC were investigated by Shenar et al. (2016). The available data are further discussed in section 2 of the present paper. In sections 3, 4 and 5, we will address the following two questions:\n\n\\begin{itemize}\n\\item \tWhen the RLOF is accompanied by mass transfer do massive mass gainers always spin-up until they reach their critical rotation velocity?\\\\ \n\\item In anticipation, we will show that the answer to the foregoing question is `no' and that tidal effects are probably not the (only) cause. The second question then is: Is there a physical spin-down process different from tidal effects which is of the same order as, but still less than the spin-up process during RLOF?\n \n\\end{itemize}\n\n\\section{The WR+O binaries: the spin-rates of the O components}\n\nLet us start from Table 2 of Shara et al. (2017). The table lists the Galactic WR+O binaries (11 systems) for which the rotational velocity of the O-component has been determined\\footnote{Shara et al. (2017) determined and discussed the rotation speeds by analyzing two He-lines, He I 4922 and He II 4541; here we consider both.} or for which this rotational velocity was discussed in previous work (WR11 from Baade et al. 1990, WR127 from de la Chevrotiere et al. 2011, and WR139 from Marchenko et al. 1994). The table gives the observed mass functions and inclination angle-ranges from Lamontagne et al. (1996). For the WR binaries WR11 and WR139 the inclination angle is quite accurately known and therefore so are the orbital masses\\footnote{Note that North et al. (2007) reconsidered WR11 ($\\gamma$-Velorum) and proposed very similar mass estimates as those given in Table 2 of Shara et al., 2017}. For the other systems we have further restricted the inclination angle-ranges using the spectral type of the O-type components given by Crowther (2017) and applying the mass-spectral type\/luminosity class relation proposed by Vanbeveren et al. (1998c) for the O-type component \\footnote{A mass-spectral type-luminosity class relation relies on evolutionary tracks and therefore depends on parameters whose values are uncertain to some extent; in appendix A we discuss the relation.}. This yields the mass-range estimates of Table 1. Assuming alignment of orbital and spin vectors (not always obeyed: see e.g. Villar-Sbaffi et al. 2005 \\& 2006 for the WR+O binaries CQ Cep \\& CX Cep, resp.), this restriction then obviously also restricts the possible values of the equatorial velocity. To illustrate the procedure let us consider WR31. It is a WN4o+O8V binary with Msin$^3$i values 2.7 $M_{\\odot}$+ 6.3 $M_{\\odot}$. The inclination angle-range i = 40-62$^{\\circ}$. The mass-spectral type relation predicts a (generous) 24-34 $M_{\\odot}$ mass range for an O8V star; this means that an i-value close to 40$^{\\circ}$ is to be preferred and therefore an equatorial velocity close to the maximum value given in Shara et al. (2017) should be retained, i.e. 336\/274 km\/s. This exercise then yields a set of most probable WR+O star binary properties that are given in Table 2. \n\nAs was already concluded by Shara et al. (2017) the 11 WR+O binaries in our sample have O-type components that seem to be spun up, i.e. the 11 WR+O binaries all show the signature of mass transfer \\footnote{Although WR47 looks like an outlier our conclusions related to this binary also rely on the assumption that the WN6o component is hydrogen deficient, post-RLOF and core helium burning}. Since it is expected that a common envelope process is not accompanied by mass transfer we are inclined to conclude that {\\it common envelope evolution has not played a significant role in the formation of our WR+O sample}. This also means that the WR stars in our sample of WR+O binaries are not formed by a stellar wind mass loss process only because stellar wind mass loss is not expected to be accompanied by significant mass accretion either. Note that if the 11 WR+O binaries in our sample are representative for the whole Galactic WR+O population, the foregoing conclusions may apply for this whole population.\nOne can think of two reasons why common envelope evolution may be rare among the most massive binaries: \n\na. as was shown by Sana et al. (2012) the period distribution of the most massive binaries may be skewed towards values smaller than the typical periods that are needed for a common envelope phase to happen (of the order of years) and \n\nb. once a massive star reaches the red supergiant (RSG) phase it is subject to very large stellar wind mass loss (Vanbeveren et al., 2007; Meynet et al., 2015); when a star like that is a component of a binary that would evolve through a common envelope phase if the effects of a RSG wind are ignored, the inclusion of the RSG-wind may significantly reduce the effects of the common envelope process; it may even completely suppress the common envelope phase. Note that most binary population studies (in particular those that tend to predict the merger rates of double BHs by means of isolated close binary evolution) do not account properly for the effect of RSG stellar winds and this makes the predictions very uncertain.\n\n\\section{WR+O binaries: the spin-rates of the RLOF progenitors of the O-components}\n\nSome of the WR+O binaries in our sample are very close and they may have been subject to tidal effects which would slow the rotation of the O-type star down; this means that the rotation of the O-stars may have been faster in the past. To estimate the rotation speed of the progenitors of the O-components we proceed as follows. It is generally known that the overall structure of a post-RLOF mass loser of a case B (AB) binary (given its mass) is largely independent of the details of the progenitor evolution in general, the details of the RLOF in particular. Using the Brussels binary evolutionary code we computed the helium burning evolution of a grid of post-RLOF mass losers\\footnote {For a description of our evolutionary code and the basic ingredients, see Vanbeveren et al., 1998b, c and Mennekens and Vanbeveren, 2014.}. These computations reveal that Galactic hydrogen deficient WN stars which are mostly early type WN stars (WNE stars using the nomenclature introduced by Vanbeveren and Conti, 1980) are on average 2.10$^{5}$ yrs past their progenitor RLOF; for the WC stars this is on average 3.10$^{5}$ yrs. These timescales hold for WR stars of the binaries listed in Table 2. Given a WR+O system with the parameters of Table 2 we interpolate in the grid in order to estimate how the binary looked immediately after RLOF, using the timescales given above and assuming that during these short WR-timescales the structure of the O-component does not change (a very reasonable assumption accounting for the fact that the O-component is a core hydrogen burning star evolving on a timescale that is typically 10-times longer than the WR-timescale). It is then straightforward, using a scenario that treats tidal effects for stars with a radiative envelope (we used the formalism of Zahn, 1977) to calculate backwards the rotation speed of the O star progenitor at the end of the RLOF. The results are shown in Table 2 as well (v$_{rot}$ of the O-type star at the beginning of the WR phase of the binary is of course v$_{rot}$ of the O-type component at the end of the RLOF). Note that (as is well known) tides are effective mainly in the shortest period binaries. Accounting for the fact that the critical rotation velocity of the O-type stars in our sample ranges between 600 km\/s and 800 km\/s the results lead us to conclude that\n\n\\begin{itemize}\n\\item most of the mass gainers in the progenitors of WR+O binaries are spun up by mass transfer but their rotation velocity at the end of the RLOF may be significantly smaller than the critical value. The latter is a fortiori true when the rotational velocities are considered resulting from the He II lines.\n\\end{itemize}\n\nThe foregoing conclusion also applies for the longest period WR+O binaries where the effects of tides (and their uncertainties) are small.\n\nAs a note added in proof, in a recent paper (Shenar et al., 2016) the authors have investigated the 5 known WR+O binaries in the SMC. The O-type stars seem to rotate super synchronously but also here the rotation speed may be far below critical. In particular, the binary AB8 consists of a WO4 component with an O4V companion. The O4V star has an apparent age of $\\sim$10$^{6}$ yrs (see also Appendix A) which is similar to the ages of O4V-III stars in the Magellanic Clouds determined by Massey et al. (2000). One concludes that the O4V component is rejuvenated with respect to the age of the WO4 star (3.10$^{6}$ yrs; see Shenar et al., 2016) meaning that mass transfer happened. The rotation speed of the O-type star lies in the range 130-230 km\/s. Since the binary period = 16.6 days the effects of tidal interaction are small so that the present rotation speed of the O-type star almost equals the speed at the end of the RLOF of the progenitor. Compared to the critical rotation speed (which is of the order of 700 km\/s using the system parameters tabulated by Shenar et al., 2016) we conclude that also here it is very likely that the mass gainer has been spun-up by mass transfer but at the end of RLOF its super-synchronous rotation velocity was considerably smaller than the critical speed. \n\n\\section{The WR+O binaries: why the progenitors of the O-components spin up during RLOF but may not reach the critical speed}\n\nThe conclusions of the previous section lead us to suspect that during the RLOF of the progenitors of the WR+O binaries there must be an efficient spin-down process, otherwise the O components would all be observed to rotating at or near critical speed. Since our WR+O sample reveals that all O-type stars rotate super-synchronously the magnitude of the spin-down process should be of the same order as the spin-up but it should not completely suppress the spin-up. Note that this also applies for the longer period systems where the effects of tides (and their uncertainties) are expected to be small. Let us first remind the interested reader of a previous study (Dervisoglu, Tout and Ibanoglu, 2010) where the authors investigated the spins of the mass gainers of Algol binaries. In long-period Algols (P $>$ 5 days) mass transfer happens through a disc, implying a rapid spinning-up of the mass gainer. According to the analysis of Packet (1981) the rotation speed is expected to become critical soon after the onset of the mass accretion. However, the observations of longer period Algols reveal that the mass gainers have spins typically between 10 and 40\\% of the critical rate and therefore an efficient spin-down mechanism for the gainers is required. The authors argue that radiative tides are too weak to account for sufficient spin-down and they propose `{\\it the generation of magnetic fields in the radiative atmospheres of a differentially rotating star (the Spruit-Tayler type dynamo, Spruit, 2002 and Tayler, 1973) and the possibility of angular momentum loss driven by strong magnetic stellar winds (the model of Tout and Pringle, 1992)}'. Dervisoglu et al. (2010) presented a possible mathematical formalism for this process and concluded that the gainers in Algols with orbital periods longer than $\\sim$5 days rotate below breakup, when a small amount of mass, ($\\sim$10\\% of the transferred matter) is lost by the gainer which has a rotationally induced magnetic field of the order of one or a few kG.\n\nSince the Algol-situation shows some resemblance with the WR+O binary situation (mass transfer caused the spin-up of the mass gainer but the speed of the gainer at the end of RLOF is significantly below critical), we were tempted to test if the same process could work for the massive binaries as well. We follow the formalism described by Dervisoglu et al. and compare the following two processes.\n\nWhen during RLOF mass is transferred at a rate $\\dot{M}_{acc}$ from a disc to the star (with mass M and radius R) the rate of angular momentum transferred from the disc to the outer layers of the star is\n\n\\begin{equation}\n\\frac{dJ_{acc}}{dt} = \\dot{M}_{acc}\\sqrt{GMR}\n\\end{equation}\n\n\\noindent (G is the gravitational constant). \t \n\n\tThe angular momentum lost from a magnetic star due to a wind that is corotating up to the Alfv\u00e9n surface (with radius $R_{A}$) is\n\n\\begin{equation}\n\\frac{dJ_{w}}{dt} = \\dot{M}_{w}{R_{A}}^{2}\\Omega\n\\end{equation}\n\n\\noindent with $\\dot{M_{w}}$ the wind mass loss rate and $\\Omega$ the angular velocity of the outer layers of the star. If it is assumed that the generated magnetic field (with magnetic flux density at the stellar surface $B_s$ expressed in Gauss) is a dipole field, a straightforward calculation gives (using cgs units)\n\n\n\\begin{equation}\n\\frac{dJ_{w}}{dt} = -(-\\dot{M}_{w})^{3\/7}{B_{s}}^{8\/7}(2GM)^{-2\/7}R^{24\/7}\\Omega\n\\end{equation}\n\n\tDervisoglu et al. (2010) argued that a fully efficient Spruit-Tayler dynamo can generate a dipole field of the order of one or a few kG and it can support a wind with a rate up to 0.01 $M_{\\odot}$\/yr. This rate is comparable to the mass loss rate of the mass loser during the RLOF in intermediate-mass close binaries (forming the Algols) and in massive close binaries (forming the WR+O systems). If $\\dot{M}_{RLOF}$ is the mass loss rate of the mass loser during its RLOF it is useful to write $\\dot{M}_{acc}=-\\beta\\dot{M}_{RLOF}$ \nand \n$\\dot{M}_{w}=(1-\\beta)\\dot{M}_{RLOF}$\n\n\nNote that the Spruit-Tayler dynamo operates as long as the star is differentially rotating, i.e. as long as mass transfer happens. If mass transfer stops the star is expected to regain a state of uniform rotation and the magnetic field may vanish. Post-RLOF binaries or case A binaries \\footnote{Case A binaries are very short period binaries where RLOF starts while the mass loser is a core hydrogen burning star; case A RLOF starts with a rapid mass transfer phase on the thermal timescale of the loser, followed by a slow phase acting on the nuclear timescale of the loser, i.e. the slow phase lasts almost the entire remaining core hydrogen burning phase.} during the slow phase of RLOF may therefore not show the remains of such fields. We will come back to this later. \n\nTo illustrate whether or not the magnetically induced spin-down (formula 3) can compensate for the spin-up (formula 1) let us consider the evolution of a 30 $M_{\\odot}$ + 20 $M_{\\odot}$ zero age main sequence binary with a period assuring a case B type RLOF (we use the computations of de Loore and Vanbeveren, 1995). The computations reveal that during the rapid (resp. slow) phase of the RLOF the mass loss rate of the loser ($\\dot{M}_{RLOF}$) typically equals 5.10$^{-3}$ $M_{\\odot}$\/yr (resp. 10$^{-3}$ $M_{\\odot}$\/yr). Tables 3 and 4 show the ratio dJ+\/dJ- = $\\vert$expression (1)\/expression (3)$\\vert$ for the two $\\dot{M}_{RLOF}$ values, for different values of the magnetic field strength and different values of $\\beta$. Note that a ratio smaller than 1 corresponds to a situation where rapid spin-up is compensated. The results illustrate the following conclusion\n\n\\begin{itemize}\n\\item\nWhen during the mass transfer phase of a massive binary the Spruit-Taylor dynamo can generate a magnetic field of the order of a few kG then rapid spin-up of the mass gainer can be slowed down at the expense of a moderate mass loss from the binary.\n\\end{itemize}\n\nA more detailed calculation of this effect would require the implementation of rotation and magnetic fields in the stellar structure equations (as was done by Petrovic et al., 2005b, see also Potter et al., 2012) but this is beyond the scope of the present paper.\n\n\\section{Observational signature of magnetic fields in the O-type components of massive close binaries}\n\nPlaskett's star (HD 47129) is a massive binary (O8I+O7.5III) with a 14.4-day orbital period (Linder et al., 2008; Mahy et al., 2011). When accounting for the inclination angle range i = 69.3$^\\circ$-72.7$^\\circ$ proposed by Bagnuolo et al. (1992) both components have a mass between 50 $M_{\\odot}$ and 60 $M_{\\odot}$. The secondary is a fast rotator (v$_{rot}$sini $\\sim$300 km\/s) and it was argued by Linder et al. (2008) that the secondary may rather be an O7V star showing an apparent O7.5III spectrum because of its rapid rotation. Compared to the primary star the secondary is then rejuvenated indicating that mass transfer has happened and that the primary is on its way to become a WN star. It is tempting to conclude that the secondary was spun up by mass transfer but since the critical rotation speed of an O7V star is about 800 km\/s, the mass gainer did not reach this critical limit (note that tides in a binary with a 14.4-day period are not expected to be very effective so that the presently observed rotation speed almost equals the speed at the end of the previous RLOF), and all this is much like in many WR+O binaries in our sample. Most interestingly, a strong magnetic field was detected in the secondary (B $>$ 2.85 kG, Grunhut et al., 2013) and this makes Plaskett's star a strong candidate for the Spruit-Tayler dynamo scenario for the generation of kG-like magnetic fields in RLOF\/mass transferring binaries (see also Langer, 2014). The results of Tables 3 and 4 then illustrate that this magnetic field may be responsible for sufficient spin-down explaining why the mass gainer did not reach the critical speed.\n\nNaze et al. (2017) observed a sample of 15 interacting and post-interaction O-type binaries and, very interestingly, they found no indication of a magnetic field in any of the 15 stars and the authors conclude that binary interactions do not systematically trigger stable, strong magnetic fields in such systems. Most of the binaries in the sample of Naze et al. (2017) are very short period systems which makes them case A binary candidates during the slow phase of mass transfer. During the slow phase in a case A binary, rapid spin-up of the gainer is not expected to happen so that a Spruit-Tayler dynamo may not be operational. The results of Naze et al. support this view. Note that most of the binaries in the sample of Naze et al. (2017) are distinctly different from Plaskett's star which, according to its period, is most likely a case B binary with a mass transfer history that is quite different from that in case A binaries. \n\n\\section{Conclusions}\n\nThe rotational velocities of the O-type components in 8 WR+O binaries have been measured by Shara et al (2017) using SALT; 3 measurements of the O-type component of Galactic WR-binaries and at least 1 in the SMC were available in the literature. All 12 rotate super-synchronously and we consider this as strong evidence that they were spun up during the RLOF\/mass transfer of the progenitor binary. We conclude that common envelope evolution (meaning no mass transfer and thus no spin-up) has not played an important role in the formation of these 12 binaries, and generalizing, that common envelope evolution does not happen frequently in the massive binaries that produce WR+O systems. We implemented the formalism describing the effects of tides in stars with a radiative envelope in our binary evolutionary code and we computed the rotation speed of the O-type stars at the end of the RLOF of the progenitor binary. Many of them rotate significantly more slowly than critical. As an explanation we propose a model that has been worked out for long period Algols (period $>$ 5 days) where the mass gainers do not rotate at the critical speed either, i.e., - mass transfer causes the mass gainer to rotate differentially - differential rotation makes the Spruit-Tayler dynamo operational and a magnetic field is generated - when the magnetic field is of the order of a few kG a magnetic wind sets in which the mass loss rate is of the order of the mass transfer rate of the mass loser - the combination of wind and magnetic field spins the star down at a rate which is comparable to the mass accretion induced spin-up. In Plaskett's star the mass gainer has been spun-up but also here the critical speed was not reached. The star has an observed magnetic field B $>$ 2.85 kG and is therefore an observational test bed for the spin-up-spin-down scenario outlined above.\n\n\\begin{table*}\n\\caption{The mass-range of the O-type components in WR+O binaries resulting from the observed spectral type.}\n\\label{table1}\n\\centering\n\\begin{tabular}{c c c}\n\\hline\nSystem & Sp. Type & Mass estimate of O-star ($M_{\\odot}$) \\\\\n\\hline\nWR21 & WN5o+O4-5 & 37-60 \\\\\nWR30 & WC6+O6-8 & 24-50 \\\\\nWR31 & WN4o+O8V & 24-34 \\\\\nWR42 & WC7+O7V & 30-37 \\\\\nWR47 & WN6o+O5V & 37-70 \\\\\nWR79 & WC7+O5-8 & 24-60 \\\\\nWR97 & WN5b+O7 & $>$30 \\\\\nWR113 & WC8d+O8-9IV & 20-30 \\\\\nWR127 & WN5o+O8.5V & 17-24 \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{Most probable masses and rotation speed of the O-type component of the 11 WR+O binaries in our sample. Column 7 lists the rotation velocities at the end of the RLOF (= the beginning of the WR phase) of the progenitor binary calculated as explained in the text. The v$_{rot}$ column 6 lists the speed values resulting from the He I\/He II line measurements as explained in Shara et al. (2017). A question mark means that this particular measurement is not available.}\n\\label{table4}\n\\centering\n\\begin{tabular}{c c c c c c c}\n\\hline\nSystem & Sp. Type & P (days) & WR mass & O-mass & v$_{rot}$ (km\/s) & v$_{rot}$ (km\/s) \\\\\n & & & & & & begin WR \\\\\n\\hline\nWR21 & WN5o+O4-5 & 8.3 & $>$19 & $>$37 & 331\/138 & 415\/173 \\\\\nWR30 & WC6+O6-8 & 18.8 & 16.4 & 34 & ?\/$>$178 & ?\/$>$205 \\\\\nWR31 & WN4o+O8V & 4.8 & $>$11 & $>$24 & 336\/274 & 493\/402 \\\\\nWR42 & WC7+O7V & 7.9 & 16 & 27 & 500-511\/170-177 & 620-630\/211-218 \\\\\nWR47 & WN6o+O5V & 6.2 & 52 & 60 & ?\/$>$88 & ?\/$>$190 \\\\\nWR79 & WC7+O5-8 & 8.9 & $>$9.5 & $>$24 & ?\/$>$174 & ?\/$>$250 \\\\\nWR97 & WN5b+O7 & 12.6 & 18 & 30 & 470\/279 & 502\/298 \\\\\nWR113 & WC8d+O8-9IV & 29.7 & $>$11 & $>$22 & 310-330\/170-181 & 320-340\/175-186 \\\\\nWR11 & WC8+O7.5III-V & 78.5 & 9.6 & 30 & 220\/? & 232\/? \\\\\nWR127 & WN5o+O8.5V & 9.5 & $>$9 & $>$20 & 300-366\/? & 350-416\/? \\\\\nWR139 & WN5o+O6III-V & 4.2 & 9.4 & 28 & 215\/? & 365\/? \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\caption{dJ+\/dJ- as function of $\\beta$ and $B_s$ for $\\dot{M}_{RLOF}$ = 10$^{-3}$ $M_{\\odot}$\/yr}\n\\label{table5}\n\\centering\n\\begin{tabular}{c c c c c c}\n\\hline\n$\\beta$ & $B_s$ = 500 G & $B_s$ = 1 kG & $B_s$ = 2 kG & $B_s$ = 3 kG & $B_s$ = 4 kG \\\\\n\\hline\n0.9 & 9 & 4.1 & 1.8 & 1.2 & 0.8\\\\\n0.8 & 6 & 2.7 & 1.2 & 0.8 & 0.6\\\\\n0.6 & 3 & 1.5 & 0.7 & 0.4 & 0.3\\\\\n0.4 & 1.9 & 0.8 & 0.4 & 0.2 & 0.2\\\\\n0.2 & 0.8 & 0.4 & 0.2 & 0.1 & 0.1\\\\\n0.1 & 0.4 & 0.2 & 0.1 & 0.05 & 0.04\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\caption{dJ+\/dJ- as function of $\\beta$ and $B_s$ for $\\dot{M}_{RLOF}$ = 5.10$^{-3}$ $M_{\\odot}$\/yr}\n\\label{table4}\n\\centering\n\\begin{tabular}{c c c c c c}\n\\hline\n$\\beta$ & $B_s$ = 500 G & $B_s$ = 1 kG & $B_s$ = 2 kG & $B_s$ = 3 kG & $B_s$ = 4 kG \\\\\n\\hline\n0.9 & 22.5 & 10.2 & 4.6 & 2.9 & 2.1\\\\\n0.8 & 15 & 6.7 & 3 & 1.9 & 1.4\\\\\n0.6 & 8.3 & 3.8 & 1.7 & 1.1 & 0.8\\\\\n0.4 & 4.6 & 2.1 & 1 & 0.6 & 0.4\\\\\n0.2 & 2.1 & 0.9 & 0.4 & 0.3 & 0.2\\\\\n0.1 & 1 & 0.4 & 0.2 & 0.1 & 0.05\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Related Work}\n\n\tIn this section, we provide a brief overview of some related work in inertial navigation systems, pedestrian dead reckoning (PDR) and sequential deep learning. \n \n \\textbf{Strapdown Inertial Navigation System}: strapdown inertial navigation systems (SINS) have been studied for decades \\cite{Savage1998}. Previous inertial systems heavily relied on expensive, heavy, high-precision inertial measurement units, hence their main application had to be constrained on moving vehicles, such as automobiles, ships, aircraft, submarines and spacecraft. Recent advances of MEMS technology enable low-cost MEMS IMU to be deployed on robotics, UAV \\cite{Bloesch2015}, and mobile devices \\cite{Lymberopoulos2015}. However, restricted by size and cost, the accuracy of a MEMS IMU is extremely limited, and has to be integrated with other sensors, such as visual inertial odometry \\cite{Leutenegger2015}. Another solution is to attach an IMU on the user's foot in order to take advantage of heel strikes for zero-velocity update to compensate system error drifts \\cite{Skog2010}. These inconveniences prevent wide adoption of inertial solutions on consumer grade devices \\cite{Harle2013}. \n \n \\textbf{Pedestrian Dead Reckoning}: Unlike SINS's open-loop integration of inertial sensors, PDR uses inertial measurements to detect steps, estimate stride length and heading via an empirical formula \\cite{Shu2015a}. System errors still quickly accumulate, because of incorrect step displacement segmentation and inaccurate stride estimation. In addition, a large number of parameters have to be carefully tuned according to a user's walking habits. Recent research mainly focused on fusing PDR with external references, such as a floor plan \\cite{Xiao2014a}, WiFi fingerprinting \\cite{Hilsenbeck2014} or ambient magnetic fields \\cite{Wang2016a}, still leaving fundamental problem of rapid drift unsolved. Compared with prior work, we abandon step-based approach and present a new general framework for inertial odometry. This allows us to handle more general tracking problems, including trolley\/wheeled configurations, which step-based PDR cannot address.\n \n \\textbf{Sequential Deep Learning}: \n\tDeep learning approaches have recently shown excellent performance in handling sequential data, such as speech recognition \\cite{Graves2014}, machine translation \\cite{Dai2015}, visual tracking \\cite{Ondruska2016} and video description \\cite{Donahue2015}. To the best of our knowledge, our IONet is the first neural network framework to achieve inertial odometry using inertial data only. Previous learning-based work has tackled localization problems, by employing visual odometry \\cite{Zhou2017,Wang2017,Clark2017} and visual inertial odometry \\cite{Clark2017a}. Some other work has concentrated on learning intuitive physics \\cite{Hooman2017}, modeling state space models \\cite{Karl2016}, and supervising neural networks via physics knowledge \\cite{Stewart2017}. While most of them use visual observations, our work exploits real world sensor measurements to learn high-level motion trajectories.\n\n\\section{The Curse of Inertial Tracking}\n\nThe principles of inertial navigation are based on Newtonian mechanics. They allow tracking the position and orientation of an object in a navigation frame given an initial pose and measurements from accelerometers and gyroscopes.\n\n\t\\begin{figure}\n \t\\centering\n \\includegraphics[width=0.48\\textwidth]{existing_methods.pdf}\n \\caption{\\label{fig:existing_methods} Architecture of existing methods: SINS and PDR}\n \\end{figure} \n\nFigure \\ref{fig:existing_methods} illustrates the basic mechanism of inertial navigation algorithms. The three-axis gyroscope measures angular velocities of the body frame with respect to the navigation frame, which are integrated into pose attitudes in Equations (\\ref{eq:att_update1}-\\ref{eq:att_update3}). To represent the orientation, the direction cosine $\\mathbf{C}_b^n$ matrix is used to represent the transformation from the body (b) frame to the navigation (n) frame, and is updated with a relative rotation matrix $\\mathbf{\\Omega}(t)$. The 3-axis accelerometer measures proper acceleration vectors in the body frame, which are first transformed to the navigation frame and then integrated into velocity, discarding the contribution of gravity forces $\\mathbf{g}$ in Equation (\\ref{eq:vel_update}). The locations are updated by integrating velocity in Equation (\\ref{eq:loc_update}). Equations (\\ref{eq:att_update1}-\\ref{eq:loc_update}) describe the attitude, velocity and location update at any time stamp $t$. In our application scenarios, the effects of earth rotation and the Coriolis accelerations are ignored. \n\n\tAttitude Update:\n\t\\begin{equation}\t\n \t\\label{eq:att_update1}\n \t\\mathbf{C}_b^n(t)=\\mathbf{C}_b^n(t-1)*\\mathbf{\\Omega}(t) \\\\ \n \\end{equation}\n \\begin{equation}\n \t\\label{eq:att_update2}\n \t\\mathbf{\\sigma}=\\mathbf{w}(t)dt\n \\end{equation}\n\t\\begin{equation}\n \t\\label{eq:att_update3}\n\t\t\\mathbf{\\Omega}(t)=\\mathbf{C}_{b_t}^{b_{t-1}}=I+\\frac{sin(\\sigma)}{\\sigma}[\\mathbf{\\sigma}\\times]+\\frac{1-cos(\\sigma)}{\\sigma^2}[\\mathbf{\\sigma}\\times]^2\n\t\\end{equation}\n \n\tVelocity Update:\n\t\\begin{equation}\n \t\\label{eq:vel_update}\n \t\\mathbf{v}(t)=\\mathbf{v}(t-1)+((\\mathbf{C}_b^n(t-1))*\\mathbf{a}(t)-\\mathbf{g}_n)dt \n\t\\end{equation}\n\n\tLocation Update:\n\t\\begin{equation}\n \t\\label{eq:loc_update}\n \t\\mathbf{L}(t)=\\mathbf{L}(t-1)+\\mathbf{v}(t-1)dt \n \\end{equation}\nwhere $\\mathbf{a}$ and $\\mathbf{w}$ are accelerations and angular velocities in body frame measured by IMU, $\\mathbf{v}$ and $\\mathbf{L}$ are velocities and locations in navigation frame, and $\\mathbf{g}$ is gravity.\n \n \tUnder ideal condition, SINS sensors and algorithms can estimate system states for all future times. High-precision INS in military applications (aviation and marine\/submarine) uses highly accurate and costly sensors to keep measurement errors very small. Meanwhile, they require a time-consuming system initialization including sensor calibration and orientation initialization. However, these requirements are inappropriate for pedestrian tracking. Realizing a SINS mechanism on low-cost MEMS IMU platform suffers from the following two problems: \n \n\\begin{itemize}\n\\item The measurements from IMUs embedded in consumer phones are corrupted with various error sources, such as scale factor, axis misalignment, thermo-mechanical white noise and random walking noise \\cite{NaserEl-SheimyHaiyingHou2008}. From attitude update to location update, the INS algorithm sees a triple integration from raw data to locations. Even a tiny noise will be highly exaggerated through this open-loop integration, causing the whole system to collapse within seconds.\n\\item A time-consuming initialization process is not suitable for everyday usage, especially for orientation initialization. Even small orientation errors would lead to the incorrect projection of the gravity vector. For example, a 1 degree attitude error will cause an additional 0.1712 $m\/s^2$ acceleration on the horizontal plane, leading to 1.7 m\/s velocity error and 8.56 m location error within 10 seconds.\n\\end{itemize}\n\nEven if we view this physical model as a state-space-model, and apply a deep neural network directly to model inertial motion, the intrinsic problems would still `curse' the entire system.\n\n\\section{Tracking Down A Cure} \n\tTo address the problems of error propagation, our novel insight is to break the cycle of continuous integration, and segment inertial data into independent windows. This is analogous to resetting an integrator to prevent windup in classical control theory \\cite{Hippe2006}.\n \n However, windowed inertial data is not independent, as Equations (\\ref{eq:att_update1}-\\ref{eq:loc_update}) clearly demonstrate. This is because key states (namely attitude, velocity and location) are \\textit{unobservable} - they have to be derived from previous system states and inertial measurements, and propagated across time. Unfortunately, errors are also propagated across time, cursing inertial odometry. It is clearly impossible for windows to be truly independent. However, we can aim for pseudo-independence, where we estimate the \\textit{change} in navigation state over each window.\nOur problem then becomes how to constrain or estimate these unobservable states over a window. Following this idea, we derive a novel sequence-based physical model from basic Newtonian Laws of Motion, and reformulate it into a learning model.\n \n The unobservable or latent system states of an inertial system consist of orientation $\\mathbf{C}_b^n$, velocity $\\mathbf{v}$ and position $\\mathbf{L}$. In a traditional model, the transformation of system states could be expressed as a transfer function\/state space model between two time frames in Equation (\\ref{eq:transfer_function}), and the system states are directly coupled with each other. \n \\begin{equation}\n \t\\label{eq:transfer_function}\n \t\\begin{matrix}\n \t[\\mathbf{C}_b^n & \\mathbf{v} & \\mathbf{L}]\n \\end{matrix}_t = f(\\begin{matrix} [\\mathbf{C}_b^n & \\mathbf{v} & \\mathbf{L}] \\end{matrix}_{t-1}, \\begin{matrix} [\\mathbf{a} & \\mathbf{w}] \\end{matrix}_t)\n \\end{equation}\n \nWe first consider displacement. To separate the displacement of a window from the prior window, we compute the change in displacement $\\Delta \\mathbf{L}$ over an independent window of $n$ time samples, which is simply:\n\t\\begin{equation}\n \t\\label{eq:displacement1}\n\t\t\\Delta \\mathbf{L} =\\int_{t=0}^{n-1} \\mathbf{v}(t) dt\n\t\\end{equation}\n\nWe can separate this out into a contribution from the initial velocity state, and the accelerations in the navigation frame:\n\t\\begin{equation}\n \t\\label{eq:displacement2}\n\t\t\\Delta \\mathbf{L} =n\\mathbf{v}(0)dt+[(n-1)\\mathbf{s}_1+(n-2)\\mathbf{s}_2+\\dotsi+\\mathbf{s}_{n-1}]dt^2\n\t\\end{equation}\nwhere \\begin{equation}\n \t\\mathbf{s}(t)=\\mathbf{C}_b^n(t-1)\\mathbf{a}(t)-\\mathbf{g}\n \\end{equation} is the acceleration in the navigation frame, comprising a dynamic part and a constant part due to gravity. \n\nThen, Equation (\\ref{eq:displacement2}) is further formulated as:\n\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\\Delta \\mathbf{L} &=n\\mathbf{v}(0)dt+[(n-1)\\mathbf{C}_b^n(0)*\\mathbf{a}_1+(n-2)\\mathbf{C}_b^n(0)\\mathbf{\\Omega}(1)\\\\\n &*\\mathbf{a}_2+\\dotsi +\\mathbf{C}_b^n(0)\\displaystyle\\prod_{i=1}^{n-2} \\mathbf{\\Omega}(i)*\\mathbf{a}_{n-1}]dt^2-\\frac{n(n-1)}{2}\\mathbf{g}dt^2\\\\\n\t\t\\end{split}\n\t\\end{equation}\nand simplified to become:\n\t\\begin{equation}\n\t\t\\begin{split}\n\t\t\t\\Delta \\mathbf{L} =n\\mathbf{v}(0)dt+ \\mathbf{C}_b^n(0) \\mathbf{T} dt^2-\\frac{n(n-1)}{2}\\mathbf{g}dt^2 \\\\\n\t\t\\end{split}\n\t\\end{equation}\nwhere \\begin{equation}\n \t\\mathbf{T}=(n-1)\\mathbf{a}_1+(n-2)\\mathbf{\\Omega}(1)\\mathbf{a}_2+\\dotsi+\\displaystyle\\prod_{i=1}^{n-2} \\mathbf{\\Omega}(i)\\mathbf{a}_{n-1}\n \\end{equation}\n\nIn our work, we consider the problem of indoor positioning i.e. tracking objects and people on a horizontal plane. This introduces a key observation: in the navigation frame, there is no long-term change in height\\footnote{This assumption can be relaxed through the use of additional sensor modalities such as a barometer to detect changes in floor level due to stairs or elevator.}.\nThe mean displacement in the z axis over a window is assumed to be zero and thus can be removed from the formulation. We can compute the absolute change in distance over a window as the L-2 norm i.e. $\\Delta l=\\|\\Delta \\mathbf{L} \\|_2$, effectively decoupling the distance traveled from the orientation (e.g. heading angle) traveled, leading to:\n\t\\begin{equation}\n \t\\begin{split}\n \t&\\Delta l=\\| n\\mathbf{v}(0)dt+ \\mathbf{C}_b^n(0) \\mathbf{T} dt^2-\\frac{n(n-1)}{2}\\mathbf{g}dt^2\\|_2 \\\\\n &=\\|\\mathbf{C}_b^n(0)(n\\mathbf{v}^b(0)dt+ \\mathbf{T} dt^2-\\frac{n(n-1)}{2}\\mathbf{g}_0^b dt^2)\\|_2\n \\end{split}\n \\end{equation}\n\nBecause the rotation matrix $\\mathbf{C}_b^n(0)$ is an orthogonal matrix i.e. $\\mathbf{C}_b^n(0)^T\\mathbf{C}_b^n(0)=\\mathbf{I}$, the initial unknown orientation has been successfully removed from, giving us: \n \\begin{equation}\n \t\\begin{split}\n \t\t\\Delta l =\\|\\Delta \\mathbf{L} \\|_2&=\\| n\\mathbf{v}^b(0)dt+ \\mathbf{T} dt^2-\\frac{n(n-1)}{2}\\mathbf{g}_0^b dt^2\\|_2 \\\\\t\t\t\n \\end{split}\n \\end{equation} \nHence, over a window, the horizontal distance traveled can be expressed as a function of the initial velocity, the gravity, and the linear and angular acceleration, all in the body frame:\n\t\\begin{equation}\n\t\t\\Delta l=f(\\mathbf{v}^b(0), \\mathbf{g}_0^b, \\mathbf{a}_{1:n}, \\mathbf{w}_{1:n})\n\t\\end{equation}\n\nTo determine the change in the user's heading, we consider that a user's real accelerations and angular rates $(\\mathbf{a}_{1:n},\\mathbf{w}_{1:n})$ are also latent variables of IMU raw measurements $(\\mathbf{\\hat{a}}_{1:n},\\mathbf{\\hat{w}}_{1:n})$, and on the horizontal plane, only the heading attitude is essential in our system. From Equations (2-3) the change in the heading $\\Delta \\psi$ is expressed as a function of the raw data sequence. Therefore, we succeed in reformulating traditional model as a polar vector $(\\Delta l, \\Delta \\psi)$ based model, which is only dependent on inertial sensor data, the initial velocity and gravity in the body frame: \n \\begin{equation}\n \t\\label{eq:polar_vector}\n \t\t(\\Delta l, \\Delta \\psi)=f_{\\theta}(\\mathbf{v}^b(0), \\mathbf{g}_0^b, \\mathbf{\\hat{a}}_{1:n}, \\mathbf{\\hat{w}}_{1:n})\n \\end{equation}\n \nTo derive a global location, the starting location $(x_0, y_0)$ and heading $\\psi_0$ and the Cartesian projection of a number of windows can be written as, \n \\begin{equation}\n \t\\left\\{\n \t\\begin{aligned}\n \t\tx_n=x_0+\\Delta l cos(\\psi_0+\\Delta \\psi) \\\\\n \ty_n=y_0+\\Delta l sin(\\psi_0+\\Delta \\psi)\n \\end{aligned}\n \\right.\n \\end{equation}\n \n\nOur task now becomes how to implicitly estimate this initial velocity and the gravity in body frame, by casting each window as a sequence learning problem.\n\nThis section serves as an overview showing the transition from the traditional model-based method to the proposed neural-network-based method. It takes the traditional state-space-model described in Equations (1-5), which converts raw data to poses in a step-by-step manner, to a formulation where a window of raw inertial data is processed in a batch to estimate a displacement and an angle change. Note that in both formulations, the final output depends on the initial attitude and velocity. As a result, in both cases, the curse of error accumulation won't be avoided if using the model-based integration approach. However, our sequence based formulation paves the way to our proposed neural network approach. \n\n\n\\section{Deep Neural Network Framework}\nEstimating the initial velocity and the gravity in the body frame explicitly using traditional techniques is an extremely challenging problem. Rather than trying to determine the two terms, we instead treat Equation (\\ref{eq:polar_vector}) as a sequence, where the inputs are the observed sensor data and the output is the polar vector. The unobservable terms simply become latent states of the estimation. Intuitively, the motivation for this relies on the regular and constrained nature of pedestrian motion. Over a window, which could be a few seconds long, a person walking at a certain rate induces a roughly sinusoidal acceleration pattern. The frequency of this sinusoid relates to the walking speed. In addition, biomechanical measurements of human motion show that as people walk faster, their strides lengthen \\cite{Hausdorff2007}. Moreover, the gravity in body frame is related to the initial yaw and roll angle, determined by the attachment\/placement of the device, which can be estimated from the raw data \\cite{Xiao2015a}. In this paper, we propose the use of deep neural networks to learn the relationship between raw acceleration data and the polar delta vector, as illustrated in Figure \\ref{fig:overview}.\n\nInput data are independent windows of consecutive IMU measurements, strongly temporal dependent, representing body motions. To recover latent connections between motion characteristics and data features, a deep recurrent neural network (RNN) is capable of exploiting these temporal dependencies by maintaining hidden states over the duration of a window. Note however that latent states are not propagated between windows. Effectively, the neural network acts as a function $f_{\\theta}$ that maps sensor measurements to polar displacement over a window\n\t\\begin{equation}\n\t\t(\\mathbf{a}, \\mathbf{w})_{200*6} \\xrightarrow{f_{\\theta}} (\\Delta l, \\Delta \\psi)_{1*2},\n\t\\end{equation}\nwhere a window length of 200 frames (2~s) is used here\\footnote{We experimented with a window size of 50, 100, 200 and 400 frames, and selected 200 as an optimal parameter regarding the trade-off between accumulative location error and predicted loss.}.\n \\begin{figure}\n \t\\centering\n \\includegraphics[width=0.5\\textwidth]{overview_new.pdf}\n \\caption{\\label{fig:overview}Overview of IONet framework}\n \\end{figure}\n\n\t\\begin{figure*}\n \t\\centering\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{handheld_users.pdf}\n \t\\caption{\\label{fig:hand_users} Handheld}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{pocket_users.pdf}\n \t\\caption{\\label{fig:pocket_users} In Pocket}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{handbag_users.pdf}\n \t\\caption{\\label{fig:handbag_users} In Handbag}\n \\end{subfigure}\n \\caption{\\label{fig:multi_users} Performance in experiments involving different users.}\n \\end{figure*}\n\n\t\\begin{figure*}\n \t\\centering\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{handheld_phones.pdf}\n \t\\caption{\\label{fig:hand_phones} Handheld}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{pocket_phones.pdf}\n \t\\caption{\\label{fig:pocket_phones} In Pocket}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{handbag_phones.pdf}\n \t\\caption{\\label{fig:handbag_phones} Handbag}\n \\end{subfigure}\n \\caption{\\label{fig:multi_phones} Performance in experiments involving different devices.}\n \\end{figure*}\n\n\t\\begin{figure*}\n \t\\centering\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor1_handheld.pdf}\n \t\\caption{\\label{fig:floor1_hand_phones} Handheld}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor1_pocket.pdf}\n \t\\caption{\\label{fig:floor1_pocket_phones} In Pocket}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor1_handbag.pdf}\n \t\\caption{\\label{fig:floor1_handbag_phones} In Handbag}\n \\end{subfigure}\n \\caption{\\label{fig:floor1} Trajectories on Floor A}\n \\end{figure*}\n \n \\begin{figure*}\n \t\\centering\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor4_handheld.pdf}\n \t\\caption{\\label{fig:floor4_hand_phones} Handheld}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor4_pocket.pdf}\n \t\\caption{\\label{fig:floor4_pocket_phones} In Pocket}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.3\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor4_handbag.pdf}\n \t\\caption{\\label{fig:floor4_handbag_phones} In Handbag}\n \\end{subfigure}\n \\caption{\\label{fig:floor4} Trajectories on Floor B}\n \\end{figure*}\n\n\tIn the physical model, orientation transformations impact all subsequent outputs. We adopt Long Short-Term Memory (LSTM) to handle the exploding and vanishing problems of vanilla RNN, as it has a much better ability to exploit the long-term dependencies \\cite{Greff2016}. In addition, as both previous and future frames are crucial in updating the current frame a bidirectional architecture is adopted to exploit dynamic context.\n \n\tEquation (\\ref{eq:polar_vector}) shows that modeling the final polar vector requires modeling some intermediate latent variables, e.g. initial velocity and gravity. Therefore, to build up higher representation of IMU data, it is reasonable to stack 2-layer LSTMs on top of each other, with the output sequences of the first layer supplying the input sequences of the second layer. The second LSTM outputs one polar vector to represent the transformation relation in the processed sequence. Each layer has 96 hidden nodes. To increase the output data rate of polar vectors and locations, IMU measurements are divided into independent windows with a stride of 10 frames (0.1s).\n \n\tThe optimal parameter $\\mathbf{\\theta}^*$ inside the proposed deep RNN architecture can be recovered by minimizing a loss function on the training dataset $\\mathbf{D}=(\\mathbf{X},\\mathbf{Y})$. \n \\begin{equation}\n \t\\theta^*= \\operatorname*{arg\\, min}_{\\theta} \\ell(f_{\\theta}(\\mathbf{X}),\\mathbf{Y})\n \\end{equation}\n\n\tThe loss function is defined as the sum of Euclidean distances between the ground truth $(\\Delta \\tilde{l}, \\Delta \\tilde{\\psi})$ and estimated value $(\\Delta l, \\Delta \\psi)$. \n\t\\begin{equation}\n \t\\ell=\\sum \\|\\Delta \\tilde{l} - \\Delta l \\|_2^2+\\kappa \\|\\Delta \\tilde{\\psi} - \\Delta \\psi \\|_2^2\n \\end{equation}\n\twhere $\\kappa$ is a factor to regulate the weights of $\\Delta l$ and $\\Delta \\psi$.\n\n\t\\begin{figure}\n \t\\centering\n \\includegraphics[width=0.5\\textwidth]{loss.pdf}\n \\caption{\\label{fig:loss}Loss of adopting various frameworks}\n \\end{figure}\n\n\t \\begin{figure*}\n \t\\centering\n \\begin{subfigure}[t]{0.32\\textwidth}\n \t\\includegraphics[width=\\textwidth]{traj_vicon.pdf}\n \t\\caption{\\label{fig:traj_vicon} Ground Truth}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.32\\textwidth}\n \t\\includegraphics[width=\\textwidth]{traj_ionet.pdf}\n \t\\caption{\\label{fig:traj_ionet} IONet}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.32\\textwidth}\n \t\\includegraphics[width=\\textwidth]{traj_tango.pdf}\n \t\\caption{\\label{fig:traj_tango} Tango}\n \\end{subfigure}\n \\caption{\\label{fig:traj_trolley} Trolley tracking trajectories of (a) Ground Truth (b) IONet (c) Tango}\n \\end{figure*} \n \n\\section{Experiments}\n\n\\subsection{Training Details}\n\n\t\\textbf{Dataset}: There are no public datasets for indoor localization using phone-based IMU. We collected data ourselves with pedestrian walking inside a room installed with an optical motion capture system (Vicon) \\cite{Vicon2017}, providing very high-precise full pose reference (0.01m for location, 0.1 degree for orientation) for our experimental device. The training dataset consists of IMU data from the phone in different attachments, e.g. hand-held, in pocket, in handbag, on trolley each for 2 hours, collected by a very common consumer phone iPhone 7Plus. Note that all of our training data was collected by User 1 carrying iPhone 7. To test our model's ability to generalize, we invited 3 new participants and evaluated on another two phones, i.e. iPhone 6 and iPhone 5. \n \n \\textbf{Training}: We implemented our model on the public available TensorFlow framework, and ran training process on a NVIDIA TITAN X GPU. During training, we used Adam, a first-order gradient-based optimizer \\cite{Kingma2014} with a learning rate of 0.0015. The training converges typically after 100 iterations. To prevent our neural networks from overfitting, we gathered data with abundant moving characteristics, and adopted Dropout \\cite{Srivastava2014} in each LSTM layer, randomly dropping 25\\% units from neural networks during training. This method significantly reduces overfitting, and proves to perform well on new users, devices and environments. \n\n\t\\textbf{Comparison with Other DNN Frameworks}: To evaluate our assumption of adopting a 2-layer Bidirectional LSTM for polar vector regression, we compare its validation results with various other DNN frameworks, including frameworks using vanilla RNN, vanilla Convolution Neural Network, 1-layer LSTM and 2-layer LSTM without Bi-direction. The training data are from all attachments. Figure \\ref{fig:loss} shows their validation loss lines. Our proposed framework with 2-layer Bi-LSTM descends more steeply, and stays lower and more smoothly during the training than all other neural networks, supporting our assumption, while vanilla RNN is stuck in vanishing problems, and CNN doesn't seem to capture temporary dependencies well. \n \n \\textbf{Testing:} We also found that a separate training on every attachment shows better performance in prediction than training jointly, hence we implemented the prediction model of 2-layer Bi-LSTM trained on separate attachments in our following experiments. In a practical deployment, existing techniques can be adopted to recognize different attachments from pure IMU measurements \\cite{Brajdic2013}, providing the ability to dynamically switch between trained models.\n\n\t\\textbf{Baselines}: Two traditional methods are selected as baselines, pedestrian dead reckoning (PDR) and strapdown inertial navigation system (SINS) mechanism \\cite{Savage1998}, to compare with our prediction results. PDR algorithms are seldom made open-sourced, especially a robust PDR used in different attachments, so we implement code ourselves according to \\cite{Brajdic2013} for step detection and \\cite{Xiao2014} for heading and step length estimation.\n\n\\subsection{Tests Involving Multiple Users and Devices}\n\t\n A series of experiments were conducted inside a large room with new users and phones to show our neural network's ability to generalize. Vicon system provides highly accurate reference to measure the location errors.\n\n The first group of tests include four participants, walking randomly for two minutes with the phone in different attachments, e.g. in hand, pocket and handbag respectively, covering everyday behaviors. Our training dataset doesn't contain data from three of these participants. The performance of our model is measured as error cumulative distribution function (CDF) against Vicon ground truth and compared with conventional PDR and SINS. Figure \\ref{fig:multi_users} illustrates that our proposed approach outperforms the competing methods in every attachment. If raw data is directly triply integrated by SINS, its error propagates exponentially. The maximum error of our IOnet stayed around 2 meter within 90\\% testing time, seeing 30\\%- 40\\% improvement compared with traditional PDR in Figure \\ref{fig:error_users}.\n\n\tAnother group of experiments is to test the performance across different devices, shown in Figure \\ref{fig:multi_phones}. We choose another two very common consumer phones, iPhone 6 and iPhone 5, whose IMU sensors, InvenSense MP67B and ST L3G4200DH, are quite distinct from our training device, iPhone 7 (IMU: InvenSense ICM-20600). Although intrinsic properties of IMU influence the quality of inertial measurements, our neural network shows good robustness.\n\n\\subsection{Large-scale Indoor Localization}\n\n\tHere, we apply our model on more challenging indoor localization experiment to present its performance in a new environment. Our model \\textit{without training outside} Vicon room, is directly applied to six large-scale experiments conducted on two floors of an office building. The new scenarios contained long straight lines and slopes, which were not contained in the training dataset. Lacking the high precise reference from Vicon, we take Google Tango Tablet \\cite{Tango}, a famous visual-inertial device, as pseudo ground truth.\n \n \\begin{figure}\n \t\\centering\n \\begin{subfigure}[t]{0.23\\textwidth}\n \t\\includegraphics[width=\\textwidth]{error_users.png}\n \t\\caption{\\label{fig:error_users} Multiple Users}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.23\\textwidth}\n \t\\includegraphics[width=\\textwidth]{error_phones.png}\n \t\\caption{\\label{fig:error_phones} Multiple Phones}\n \\end{subfigure}\n \\caption{\\label{fig:small_error}Maximum position error within 90\\% test time}\n \\end{figure}\n\n \t\\begin{figure}\n \t\\centering\n \\begin{subfigure}[t]{0.23\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor1_errnew.pdf}\n \t\\caption{\\label{fig:floor1_err} Floor A}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.23\\textwidth}\n \t\\includegraphics[width=\\textwidth]{floor4_errnew.pdf}\n \t\\caption{\\label{fig:floor4_err} Floor B}\n \\end{subfigure}\n \\caption{\\label{fig:large_error} Position error in large-scale indoor localization}\n \\end{figure}\n\n The floor maps are illustrated in Figure \\ref{fig:floor1} (about 1650 $m^2$ size) and Figure \\ref{fig:floor4} (about 2475 $m^2$). Participants walked normally along corridors with the phone in three attachments respectively. The predicted trajectories from our proposal are closer to Tango trajectories, compared with the two other approaches in Figure \\ref{fig:floor1} and \\ref{fig:floor4}. The continuously propagating error of SINS mechanism caused trajectory drifts that grow exponentially with time. Impacted by wrong step detection or inaccurate step stride and heading estimation, PDR accuracy is limited. We calculate absolute position error against pseudo ground truth from Tango at a distance of 50m, 100m and the end point in Figure \\ref{fig:large_error}. Our IONet shows competitive performance over traditional PDR and has the advantage of generating continuous trajectory at 10 Hz, though its heading attitude deviates from true values occasionally.\n\n\\subsection{Trolley Tracking}\n\t\n We consider a more general problem without periodic motion, which is hard for traditional step-based PDR or SINS on limited quality IMU. Tracking wheel-based motion, such as a shopping trolley\/cart, robot or baby-stroller is highly challenging and hence under-explored. Current approaches to track wheeled objects are mainly based on visual odometry or visual-inertial odometry (VIO) \\cite{Li2013b,Bloesch2015}. They won't work when the device is occluded or operating in low light environments, such as placed in a bag. Moreover, their high energy- and computation-consumption also constrain further application. Here, we apply our model on a trolley tracking problem \\textit{using only inertial sensors}. Due to a lack of comparable technique, our proposal is compared with the state-of-art visual-inertial odometry Tango.\n \n \\begin{figure}\n \t\\centering\n \\includegraphics[width=0.45\\textwidth]{trolley_cdf.pdf}\n \\caption{\\label{fig:trolley_cdf}CDF of Trolley Tracking}\n \\end{figure} \n \n Our experiment devices, namely an iPhone 7 and the Google Tango are attached on a trolley, pushed by a participant. Detailed experiment setup and results could be found in supplementary video \\footnote{Video is available at: https:\/\/youtu.be\/pr5tR6Wz-zs}. High-precision motion reference was provided by Vicon. From the trajectories from Vicon, our IONet and Tango in Figure \\ref{fig:traj_trolley}, our proposed approach shows almost the same accuracy as Tango, and even better robustness, because our pure inertial approach suffers less from environmental factors. With the help of visual features, VIO (Tango) can constrain error drift by fusing visual transformations and the inertial system, but it will collapse when capturing wrong features or no features, especially in open spaces. This happened in our experiment, shown in Figure \\ref{fig:trolley_cdf}. Although VIO can recover from the collapse, it still left a large distance error.\n\n\\section{Conclusion and Future Work}\n\nWe presented a new neural network framework to learn inertial odometry directly from IMU raw data. Our model is derived from Newtonian mechanics and formulated as a sequential learning problem using deep recurrent neural networks to recover motion characteristics. The performance of IONet is evaluated through extensive experiments including tests involving multiple users\/devices, large-scale indoor localization and trolley tracking. It outperforms both traditional step-based PDR and SINS mechanisms. We believe our work lays foundations for a more accurate and reliable indoor navigation system fusing multiple sources. \n\nA number of research challenges lie ahead: 1) The performance of our proposed IONet degraded when the input IMU measurements are corrupted with a high bias value, because the training and testing data are not in the same domains. 2) Challenging motions and distinct walking habits of new users can influence the effectiveness of our proposed approach. 3) Our current model is trained and tested on different attachments separately. The challenge becomes how to jointly train all the attachments on one model and improve its robustness. 4) Combining the sequence-based physical model with the proposed neural network model would be an interesting point to investigate in future.\n\nTherefore, our future work includes extending our work to learn transferable features without the limitations of measurement units, users and environments, realized on all platforms including mobile devices and robots.\n\n\\section{ Acknowledgments}\nWe would like to thank the anonymous reviewers for their valuable comments. We also thank our colleagues, Dr. Sen Wang, Dr. Stepheno Rosa, Shuyu Lin, Linhai Xie, Bo Yang, Jiarui Gan, Zhihua Wang, Ronald Clark, Zihang Lai, for their help, and Prof. Hongkai Wen at University of Warwick for useful discussions and the usage of GPU computational resources.\n\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{section one}\nIn this paper, we are concerned with the bias voter model on regular\ntrees. First we introduce the definition of this process on general\ngraphs. The bias voter model $\\{\\eta_t\\}_{t\\geq 0}$ on a graph $S$\nis with state space $\\{0,1\\}^S$. That is to say, at each vertex\n$x\\in S$, there is a spin taking value from $\\{0,1\\}$. For any\nconfiguration $\\eta\\in \\{0,1\\}^S$ and $x\\in S$, we denote by\n$\\eta(x)$ the value of the spin at $x$.\n\nAt $t=0$, each spin takes a value from $\\{0,1\\}$ according to some\nprobability distribution. $\\lambda>\\theta>0$ are two constants. For\neach vertex $x$ in state $0$ (resp. $1$), it waits for an\nexponential time with rate $\\lambda$ (resp. $\\theta$) to choose a\nneighbor $y$ uniformly. Then, the value of the spin at $x$ flips to\nthat of the spin at $y$. Therefore, $\\{\\eta_t\\}_{t\\geq 0}$ is a spin\nsystem (see Chapter 3 of \\cite{LIG1985}) with flip rates function\ngiven by\n\\begin{equation}\\label{equ 1.1 flip rate}\nc(x,\\eta)=\n\\begin{cases}\n\\frac{\\lambda}{{\\rm deg}(x)}\\sum_{y:y\\sim x}\\eta(y)\n\\text{\\quad if~}\\eta(x)=0,\\\\\n\\frac{\\theta}{{\\rm deg}(x)}\\sum_{y:y\\sim x}[1-\\eta(y)] \\text{\\quad\nif~}\\eta(x)=1\n\\end{cases}\n\\end{equation}\nfor any $(x,\\eta)\\in S\\times \\{0,1\\}^S$, where we denote by $x\\sim\ny$ that $x$ and $y$ are neighbors and denote by ${\\rm deg}(x)$ the\ndegree of $x$.\n\nIntuitively, $0$ and $1$ are two opposite opinions of a topic.\nVertices in state $0$ (resp. $1$) are individuals holding the\nopinion $0$ (resp. $1$). Each individual waits for an exponential\ntime to choose a neighbor randomly and take the neighbor's opinion\nas its. The assumption that $\\lambda>\\theta$ can be considered as\nthat $0$ is a more controversial idea such that individuals holding\nit prefer to reconsider their opinions.\n\nIn this paper, we assume that the initial distribution of the\nprocess is the product measure with density $p$ for $p\\in (0,1)$,\nwhich is denoted by $\\mu_p$. In other words,\n\\[\n\\mu_p(\\eta:\\eta(x)=1,\\forall~x\\in A)=p^{|A|}\n\\]\nfor any $A\\subseteq S$.\n\nWe denote by $P_S^p$ the probability measure of the process\n$\\{\\eta_t\\}_{t\\geq 0}$ on $S$ with initial distribution $\\mu_p$. It\nis natural to consider the estimation of $P^p_S(\\eta_t(x)=1)$, which\nis the probability that $x$ takes $1$ at the moment $t$. When $S$ is\nthe complete graph with $N$ vertices, which we denote by $C_N$, it\nis easy to prove that $P^p_{C_N}(\\eta_t(x)=1)$ satisfies the\nfollowing limit theorem such that\n\\begin{equation}\\label{equ 1.2}\n\\lim_{N\\rightarrow+\\infty}P^p_{C_N}(\\eta_t(x)=1)=\\frac{pe^{(\\lambda-\\theta)t}}{1-p+pe^{(\\lambda-\\theta)t}}\n\\end{equation}\nfor any $t>0$.\n\nEquation \\eqref{equ 1.2} follows from a classic theory about density\ndependent population processes constructed by Ethier and Kurtz (see\nChapter 11 of \\cite{Ethier1986}). Let $N_t$ be the number of\nvertices in state $1$ at the moment $t$, then\n\\begin{equation}\\label{equ 1.3}\nN_t\\rightarrow\n\\begin{cases}\nN_t+1 \\text{~at rate~} \\frac{\\lambda}{N}(N-N_t)N_t, \\\\\nN_t-1 \\text{~at rate~} \\frac{\\theta}{N}(N-N_t)N_t.\n\\end{cases}\n\\end{equation}\n\nWhen the initial distribution of $\\{\\eta_t\\}_{t\\geq 0}$ is $\\mu_p$,\nthen according to Theorem 11.2.1 of \\cite{Ethier1986} and \\eqref{equ\n1.3}, $N_t\/N$ converges weakly to the solution $f(t,p)$ to the\nfollowing ODE\n\\[\n\\begin{cases}\n\\frac{d}{dt}f(t,p)=(\\lambda-\\theta)f(t,p)[1-f(t,p)],\\\\\nf(0,p)=p\n\\end{cases}\n\\]\nas $N$ grows to infinity. The mathematical expression of $f(t,p)$ is\nexactly the right side of \\eqref{equ 1.2}.\n\nIt is natural to ask whether \\eqref{equ 1.2} holds for other\nhomogeneous graph $S$. We manage to prove that the answer is\npositive when $S$ is a regular tree, which we denote by\n$\\mathbb{T}^N$. For mathematical details, see Section \\ref{section\ntwo}.\n\nSince\n\\[\\lim_{t\\rightarrow+\\infty}f(t,p)=1\\]\nwhen $\\lambda>\\theta$, it is natural to guess that $\\eta_t$\nconverges weakly to $\\delta_1$, the configuration where all vertices\nare in state $1$, as $t$ grows to infinity. For $S$ is a regular\ntree or a lattice, we can prove that this guess is correct when\n$\\lambda\/\\theta$ is sufficiently large. For mathematical details,\nsee Section \\ref{section two}.\n\nWhen $\\lambda=\\theta$, our model degenerates to the classic voter\nmodel introduced by Clifford and Sudbury in \\cite{Cli1973}. In\n\\cite{Holley1975}, Holley and Liggett give an important dual\nrelationship between the classic voter model and the coalescent\nrandom walks, which shows that any invariant measure of the classic\nvoter model is a convex combination of $\\delta_1$ and $\\delta_0$\nwhen and only when two independent simple random walks on $S$ will\nmeet with probability one. More details can be found in Section 3.4\nand Chapter 5 of \\cite{LIG1985}. The classic voter model is also a\nlinear system (see Chapter 9 of \\cite{LIG1985}), which makes the\nprocess has some good properties, such as $\\sum_{x\\in S}\\eta_t(x)$\nis a martingale. When $\\lambda>\\theta$, the bias voter model can not\nbe described via a linear system and has no good duality properties,\nwhich makes the classic approach to deal with voter models not be\nvalid.\n\n\\section{Main results}\\label{section two}\nIn this section we give the main results of this paper. We denote by\n$\\mathbb{T}^N$ the regular tree with degree $N+1$. We obtain that\n\\eqref{equ 1.2} holds for the bias voter model on $\\mathbb{T}^N$.\n\n\\begin{theorem}\\label{theorem 2.1 main}\nFor any $t>0$ and $p\\in (0,1)$,\n\\begin{equation}\\label{equ 2.1}\n\\lim_{N\\rightarrow+\\infty}P^p_{\\mathbb{T}^N}(\\eta_t(x)=1)=\\frac{pe^{(\\lambda-\\theta)t}}{1-p+pe^{(\\lambda-\\theta)t}}.\n\\end{equation}\n\\end{theorem}\n\nEquation \\eqref{equ 2.1} gives the limit of the probability that a\ngiven vertex is in state $1$ at the moment $t$ as the degree of the\ntree grows to infinity for any $t>0$. The limit function\n\\[\nf(t,p)=\\frac{pe^{(\\lambda-\\theta)t}}{1-p+pe^{(\\lambda-\\theta)t}}\n\\]\nis usually called the mean-field limit. Please note that\n$P_{\\mathbb{T}^N}^p(\\eta_t(x)=1)$ does not depend the choice of $x$\nsince $\\mu_p$ is a translation invariant measure on\n$\\{0,1\\}^{\\mathbb{T}^N}$ and the flip rate function of\n$\\{\\eta_t\\}_{t\\geq 0}$ given by \\eqref{equ 1.1 flip rate} is also\ntranslation invariant.\n\nThe proof of Theorem \\ref{theorem 2.1 main} is in Section\n\\ref{section 3}. The core step of the proof is to show that\n$\\eta_t(x)$ and $\\eta_t(y)$ are asymptotically independent for a\npair of neighbors $x$ and $y$ as the degree of the tree grows to\ninfinity.\n\nIt is natural to ask whether the counterpart of Theorem \\ref{theorem\n2.1 main} for the bias voter model on lattices $\\mathbb{Z}^d,\nd=1,2,\\ldots$ holds. We guess the answer is positive but we have not\nmanage to prove that.\n\nWe denote by $\\eta_t\\Rightarrow \\mu$ when the process\n$\\{\\eta_t\\}_{t\\geq 0}$ converges weakly to a probability measure\n$\\mu$. That is to say, $\\eta_t\\Rightarrow \\mu$ when and only when\n\\[\n\\lim_{t\\rightarrow+\\infty}Ef(\\eta_t)=\\int_{\\{0,1\\}^S}f(\\eta)~\\mu(d\\eta)\n\\]\nfor any $f\\in C(\\{0,1\\}^S)$. The mean-field limit $f(t,p)$ given by\n\\eqref{equ 2.1} satisfies that $f(t,p)\\rightarrow 1$ as\n$t\\rightarrow+\\infty$. So it is natural to guess that\n$P_{\\mathbb{T}^N}^p(\\eta_t(x)=1)\\rightarrow 1$ and therefore\n$\\eta_t\\Rightarrow \\delta_1$, the configuration where all the\nvertices are in state $1$. The following two theorems show that this\nguess holds for the bias voter model on trees and lattices when\n$\\lambda\/\\theta$ is sufficiently large.\n\n\\begin{theorem}\\label{theorem 2.2}\nFor each $N\\geq 2$, there is a constant $A(N)>0$ such that when\n$\\lambda\/\\theta>A(N)$, then\n\\[\n\\eta_t\\Rightarrow\\delta_1\n\\]\nfor the bias voter model $\\{\\eta_t\\}_{t\\geq 0}$ on $\\mathbb{T}^N$\nwith initial distribution $\\mu_p$ with $p\\in (0,1)$. The sequence\n$\\{A(N)\\}_{N\\geq 2}$ satisfies that\n\\begin{equation}\\label{equ 2.2}\n\\limsup_{N\\rightarrow+\\infty}\\frac{A(N)}{\\sqrt{N}}\\leq 1.\n\\end{equation}\n\\end{theorem}\n\nThe main approach to prove Theorem \\ref{theorem 2.2} is to compare\nthe bias voter model with a contact process on tree. The fact that\nthe strong survived contact process on tree satisfies the complete\nconvergence theorem is crucial for our proof. The limit theorem\n\\eqref{equ 2.2} of $A(N)$ follows from an important estimation of\nthe second critical value of the contact process on tree. For\nmathematical details, see Section \\ref{section 4}. For more about\nthe contact process on tree, see \\cite{Pem1992} and Part 1 of\n\\cite{LIG1999}.\n\nWe denote by $\\mathbb{Z}^d$ the lattice with degree $2d$. The\nfollowing theorem is a counterpart of Theorem \\ref{theorem 2.2} for\nthe bias voter model on $\\mathbb{Z}^d$.\n\n\\begin{theorem}\\label{theorem 2.3}\nFor each $d\\geq 1$ and the bias voter model $\\{\\eta_t\\}_{t\\geq 0}$\non $\\mathbb{Z}^d$ with initial distribution $\\mu_p$ with $p\\in\n(0,1)$, when $\\lambda\/\\theta>4$, then\n\\[\n\\eta_t\\Rightarrow\\delta_1.\n\\]\n\\end{theorem}\nThe proof of Theorem \\ref{theorem 2.3} is nearly the same analysis\nas that of Theorem \\ref{theorem 2.2}. The assumption\n$\\lambda\/\\theta>4$ relies on the fact that the critical value for\nthe contact process on $\\mathbb{Z}^d$ is at most $2\/d$. For more\ndetails, see Section \\ref{section 4}.\n\n\\section{Mean-field limit}\\label{section 3}\nIn this section, we will prove Theorem \\ref{theorem 2.1 main}. For\nany $t>0$ and $p\\in (0,1)$, we define\n\\[\nf(t,p)=\\frac{pe^{(\\lambda-\\theta)t}}{1-p+pe^{(\\lambda-\\theta)t}}.\n\\]\nSince we are focused on the case where $S=\\mathbb{T}^N$ in this\nsection, we rewrite $P^p_{\\mathbb{T}^N}$ as $P^p_N$. First it is\neasy to show that $f(t,p)$ is an upper bound of $P^p_N(\\eta_t(x)=1)$\nfor any $t\\geq 0$ and $N\\geq 1$.\n\n\\begin{lemma}\\label{lemma 3.1}\nFor any $t\\geq 0$ and $N\\geq 1$,\n\\begin{equation}\\label{equ 3.1}\nP_N^p(\\eta_t(x)=1)\\leq f(t,p).\n\\end{equation}\n\\end{lemma}\n\n\\proof\n\nAccording to the flip rate function $c(x,\\eta)$ of\n$\\{\\eta_t\\}_{t\\geq 0}$ given by \\eqref{equ 1.1 flip rate} and\nHille-Yosida Theorem,\n\\begin{align}\\label{equ 3.2}\n\\frac{d}{dt}P^p_N(\\eta_t(x)=1)=&\\frac{\\lambda}{N+1}\\sum_{y:y\\sim\nx}P_N^p(\\eta_t(x)=0,\\eta_t(y)=1)\\\\\n&-\\frac{\\theta}{N+1}\\sum_{y:y\\sim\nx}P_N^p(\\eta_t(x)=1,\\eta_t(y)=0)\\notag\n\\end{align}\nfor any $t>0$.\n\nSince $\\mu_p$ and $c(x,\\eta)$ are translation invariant,\n\\[\nP_N^p(\\eta_t(x)=0,\\eta_t(y)=1)=P_N^p(\\eta_t(x)=1,\\eta_t(y)=0)\n\\]\nand does not rely on the choose of the neighbor $y$.\n\nTherefore,\n\\begin{equation}\\label{equ 3.3}\n\\frac{d}{dt}P^p_N(\\eta_t(x)=1)=(\\lambda-\\theta)P_N^p(\\eta_t(x)=1,\\eta_t(y)=0),\n\\end{equation}\nwhere $y$ is a fixed neighbor of $x$.\n\nIt is easy to check that the bias voter model is an attractive spin\nsystem (see Section 3.2 of \\cite{LIG1985}). Therefore, the two\nevents $\\{\\eta_t(x)=1\\}$ and $\\{\\eta_t(y)=0\\}$ are negative\ncorrelated when the initial distribution is $\\mu_p$ according to\nTheorem 2.2.14 of \\cite{LIG1985}.\n\nAs a result,\n\\begin{align*}\nP_N^p(\\eta_t(x)=1,\\eta_t(y)=0)&\\leq\nP_N^p(\\eta_t(x)=1)P_N^p(\\eta_t(y)=0)\\\\\n&=P_N^p(\\eta_t(x)=1)[1-P_N^p(\\eta_t(x)=1)]\n\\end{align*}\nand hence\n\\begin{equation}\\label{equ 3.4}\n\\frac{d}{dt}\\Big[\\log\n\\frac{P_N^p(\\eta_t(x)=1)}{1-P_N^p(\\eta_t(x)=1)}\\Big]\\leq\n(\\lambda-\\theta)\n\\end{equation}\nby \\eqref{equ 3.3}.\n\nAccording to \\eqref{equ 3.4},\n\\begin{equation}\\label{equ 3.5}\n\\log \\frac{P_N^p(\\eta_t(x)=1)}{1-P_N^p(\\eta_t(x)=1)}-\\log\n\\frac{p}{1-p}\\leq (\\lambda-\\theta)t\n\\end{equation}\nfor any $t>0$.\n\nEquation \\eqref{equ 3.1} follows from \\eqref{equ 3.5} directly.\n\n\\qed\n\nTo give a lower bound of $P_N^p(\\eta_t(x)=1)$, we give another\ndescription of the bias voter model $\\{\\eta_t\\}_{t\\geq 0}$. We are\ninspired by the approach of graphical representation introduce by\nHarris in \\cite{Har1978} and the construction of stochastic\nprocesses of spin systems with exchange dynamics introduced by\nDurrett and Neuhauser in \\cite{Durrett1994}. For any $x,y\\in\n\\mathbb{T}^N, x\\sim y$, we assume that $\\{N_{(x,y)}(t):t\\geq 0\\}$ is\na Poisson process with rate $(\\lambda+\\theta)\/(N+1)$. Please note\nthat we care the order of $x$ and $y$, so $N_{(x,y)}\\neq N_{(y,x)}$.\nWe assume that all these Poisson processes are independent. At\n$t=0$, each spin takes a value from $\\{0,1\\}$ according to the\ndistribution $\\mu_p$. Then, the spin at $x$ may change its value\nonly at event times of $N_{(x,y)},y\\sim x$. For any $t>0$, we define\n\\[\n\\eta_{t-}(x)=\\lim_{s\\uparrow t,s0$, we define\n\\[\nA_{x,y}(T)=\\{N_{(x,y)}(T)=N_{(y,x)}(T)=0\\}\n\\]\nas the random event that the first event time of $N_{(x,y)}$ and\n$N_{(y,x)}$ does not come before $T$.\n\nThen,\n\\[\nP_N^p(A_{x,y}(T))=e^{-\\frac{2(\\lambda+\\theta)}{N+1}T}\n\\]\nand hence\n\\begin{equation}\\label{equ 3.6}\n\\lim_{N\\rightarrow+\\infty}P_N^p(A_{x,y}(T))=1\n\\end{equation}\nfor any $T>0$ and $p\\in (0,1)$.\n\nAfter all the prepared work, we can give the proof of Theorem\n\\ref{theorem 2.1 main} now.\n\n\\proof[Proof of Theorem \\ref{theorem 2.1 main}]\n\nAccording to \\eqref{equ 3.3}, $P^p_N(\\eta_t(x)=1)$ is increasing\nwith $t$, therefore by Lemma \\ref{lemma 3.1},\n\\begin{equation}\\label{equ 3.7}\np\\leq P^p_N(\\eta_t(x)=1)\\leq f(T,p)<1\n\\end{equation}\nfor any $t\\in [0,T]$ and $N\\geq 1$.\n\nFor any $t\\in [0,T]$, $x\\sim y\\in \\mathbb{T}^N$.\n\\begin{align}\\label{equ 3.8}\nP_N^p(\\eta_t(x)=1,\\eta_t(y)=0)&\\geq P_N^p\\Big(\\eta_t(x)=1,\\eta_t(y)=0,A_{x,y}(T)\\Big)\\\\\n&=P_N^p\\Big(\\eta_t(x)=1,\\eta_t(y)=0\\Big|A_{x,y}(T)\\Big)P_N^p\\Big(A_{x,y}(T)\\Big).\\notag\n\\end{align}\n\nFor $x\\sim y\\in \\mathbb{T}^N$, let\n\\[\nC_y(x)=\\{z\\in \\mathbb{T}^N: \\text{~there is a path avoiding $y$ from\n$x$ to $z$}\\}.\n\\]\nConditioned on $A_{x,y}(T)$, $\\{\\eta_t(z):z\\in C_y(x)\\}_{t\\leq T}$\nand $\\{\\eta_t(w):w\\in C_x(y)\\}_{t\\leq T}$ are independent when the\ninitial distribution is $\\mu_p$, since vertices in $C_x(y)$ can not\nexchange opinions with vertices in $C_y(x)$ before the moment $T$.\n\nAs a result,\n\\begin{align}\\label{equ 3.9}\n&P_N^p\\Big(\\eta_t(x)=1,\n\\eta_t(y)=0\\Big|A_{x,y}(T)\\Big)\\notag\\\\\n&=P_N^p\\Big(\\eta_t(x)=1\\Big|A_{x,y}(T)\\Big)P_N^p\\Big(\\eta_t(y)=0\\Big|A_{x,y}(T)\\Big)\\\\\n&\\geq\n\\frac{\\Big[P_N^p\\big(\\eta_t(x)=1\\big)-P_N^p\\big(A^c_{x,y}(T)\\big)\\Big]\n\\Big[P_N^p\\big(\\eta_t(y)=0\\big)-P_N^p\\big(A^c_{x,y}(T)\\big)\\Big]}{\\Big[P^p_N\\big(A_{x,y}(T)\\big)\\Big]^2}\\notag\n\\end{align}\nfor ant $t\\in [0,T]$, where $A^c_{x,y}(T)$ is the complementary set\nof $A_{x,y}(T)$.\n\nBy \\eqref{equ 3.3}, \\eqref{equ 3.8} and \\eqref{equ 3.9}, for $t\\in\n[0,T]$,\n\\begin{equation}\\label{equ 3.10}\n\\frac{d}{dt}P_N^p(\\eta_t(x)=1)\\geq(\\lambda-\\theta)P_N^p(\\eta_t(x)=1)\\big[1-P_N^p(\\eta_t(x)=1)\\big]G_t^p(x,y,N),\n\\end{equation}\nwhere\n\\begin{align}\\label{equ 3.11}\nG_t^p(x,y,N)&=\\frac{\\Big[1-\\frac{P_N^p\\big(A^c_{x,y}(T)\\big)}{P_N^p\\big(\\eta_t(x)=1\\big)}\\Big]\\Big[1-\\frac{P_N^p\\big(A^c_{x,y}(T)\\big)}\n{P_N^p\\big(\\eta_t(x)=0\\big)}\\Big]}{P_N^p\\big(A_{x,y}(T)\\big)}\\notag\\\\\n&\\geq\n\\frac{\\Big[1-\\frac{P_N^p\\big(A^c_{x,y}(T)\\big)}{p}\\Big]\\Big[1-\\frac{P_N^p\\big(A^c_{x,y}(T)\\big)}\n{1-f(T,p)}\\Big]}{P_N^p\\big(A_{x,y}(T)\\big)}\n\\end{align}\naccording to \\eqref{equ 3.7}.\n\nBy \\eqref{equ 3.6} and \\eqref{equ 3.11}, for any $\\epsilon>0$ and\n$T>0$, there exists $N(\\epsilon,T)>0$ such that\n\\begin{equation}\\label{equ 3.12}\nG_t^p(x,y,N)\\geq 1-\\epsilon\n\\end{equation}\nfor any $N\\geq N(\\epsilon,T)$ and $t\\in [0,T]$.\n\nBy \\eqref{equ 3.10} and \\eqref{equ 3.12},\n\\begin{equation}\\label{equ 3.13}\n\\frac{d}{dt}P_N^p(\\eta_t(x)=1)\\geq\n(\\lambda-\\theta)(1-\\epsilon)P_N^p(\\eta_t(x)=1)\\big[1-P_N^p(\\eta_t(x)=1)\\big]\n\\end{equation}\nfor $N\\geq N(\\epsilon,T)$ and $t\\in [0,T]$.\n\nBy \\eqref{equ 3.13},\n\\[\n\\frac{d}{dt}\\Big[\\log\n\\frac{P_N^p(\\eta_t(x)=1)}{1-P_N^p(\\eta_t(x)=1)}\\Big]\\geq\n(\\lambda-\\theta)(1-\\epsilon)\n\\]\nand hence\n\\begin{equation}\\label{equ 3.14}\nP_N^p(\\eta_t(x)=1)\\geq\n\\frac{pe^{(\\lambda-\\theta)(1-\\epsilon)t}}{1-p+pe^{(\\lambda-\\theta)(1-\\epsilon)t}}\n\\end{equation}\nfor $N\\geq N(\\epsilon, T)$ and $t\\in [0,T]$.\n\nBy \\eqref{equ 3.14},\n\\begin{equation}\\label{equ 3.15}\n\\liminf_{N\\rightarrow+\\infty}P_N^p(\\eta_t(x)=1)\\geq\n\\frac{pe^{(\\lambda-\\theta)(1-\\epsilon)t}}{1-p+pe^{(\\lambda-\\theta)(1-\\epsilon)t}}\n\\end{equation}\nfor any $t>0$ and $\\epsilon>0$.\n\nTheorem \\ref{theorem 2.1 main} follows from \\eqref{equ 3.1} and\n\\eqref{equ 3.15} directly.\n\n\\qed\n\nIn the proof above, we show that $\\eta_t(x)$ and $\\eta_t(y)$ are\nasymptotically independent as $N$ grows to infinity, since\n$\\eta_t(x)$ and $\\eta_t(y)$ are independent conditioned on\n$A_{x,y}(t)$ and the probability of $A_{x,y}(t)$ converges to $1$ as\n$N$ grows to infinity. If we could show that $\\eta_t(x)$ and\n$\\eta_t(y)$ are asymptotically independent for $x\\sim y, x,y\\in\n\\mathbb{Z}^d$ as $d$ grows to infinity, then we could extend Theorem\n\\ref{theorem 2.1 main} to the case where the bias voter model is on\nthe lattice. We will work on this problem as a further study.\n\n\\section{Weak convergence}\\label{section 4}\nIn this section we will give the proofs of Theorem \\ref{theorem 2.2}\nand Theorem \\ref{theorem 2.3}. After a scaling of the time, it is\neasy to see that the limit behavior of $\\{\\eta_t\\}_{t\\geq 0}$ only\ndepends on $\\lambda\/\\theta$, so in this section we assume that\n$\\theta=1$.\n\nThe proofs of Theorem \\ref{theorem 2.2} and \\ref{theorem 2.3} are\nvery similar, so we only give details of the proof of Theorem\n\\ref{theorem 2.2}. For Theorem \\ref{theorem 2.3}, we only give a\nsketch of the proof.\n\nFirst we introduce the definition of the contact process\n$\\{\\zeta_t\\}_{t\\geq 0}$ on $\\mathbb{T}^N$. $\\{\\zeta_t\\}_{t\\geq 0}$\nis a spin system with state space $\\{0,1\\}^{\\mathbb{T}^N}$ and flip\nrate function given by\n\\begin{equation}\\label{equ 4.1 rate function of contact process}\nc_1(x,\\zeta)=\n\\begin{cases}\n1 \\text{\\quad if~} \\zeta(x)=1,\\\\\n\\frac{\\lambda}{N+1}\\sum_{y:y\\sim x}\\zeta(y) \\text{\\quad if~}\n\\zeta(x)=0\n\\end{cases}\n\\end{equation}\nfor any $(x,\\zeta)\\in \\{0,1\\}^{\\mathbb{T}^N}$.\n\nThe contact process is first introduced by Harris in \\cite{Har1974}.\nChapter 6 of \\cite{LIG1985} and Part 1 of \\cite{LIG1999} give a\ndetailed summary of main properties of the contact process.\nIntuitively, the contact process describes the spread of an\ninfection disease. Vertices in state $1$ are infected while vertices\nin state $0$ are healthy. An infected vertex waits for an\nexponential time with rate $1$ to become healthy and a healthy\nvertex is infected at a rate proportional to the number of infected\nneighbors.\n\nAccording to the basic coupling of spin systems (see Section 3.1 of\n\\cite{LIG1985}), we can also use $P_N^p$ to denote the probability\nmeasure of the contact process $\\{\\zeta_t\\}_{t\\geq 0}$ on\n$\\mathbb{T}^N$ with initial distribution $\\mu_p$. We write $P_N^p$\nas $P_{N,\\lambda}^p$ when we need to distinguish $\\lambda$.\n\nThe following lemma shows that we can control the evolution of the\nbias voter model $\\{\\eta_t\\}_{t\\geq 0}$ from below by the contact\nprocess $\\{\\zeta_t\\}_{t\\geq 0}$, which is crucial for us to prove\nTheorem \\ref{theorem 2.2}.\n\n\\begin{lemma}\\label{lemma 4.1}\nAssume that $\\{\\eta_t\\}_{t\\geq 0}$ is the bias voter model with flip\nrate function $c(x,\\eta)$ given by \\eqref{equ 1.1 flip rate} with\n$\\theta=1$ and $\\{\\zeta_t\\}_{t\\geq 0}$ is the contact process with\nflip rate function $c_1(x,\\zeta)$ given by \\eqref{equ 4.1 rate\nfunction of contact process}, then\n\\begin{equation}\\label{equ 4.2}\nP_{N,\\lambda}^p(\\eta_t(x)=0,\\forall~x\\in A)\\leq\nP_{N,\\lambda}^p(\\zeta_t(x)=0,\\forall~x\\in A)\n\\end{equation}\nfor any $A\\subseteq \\mathbb{T}^N$ and any $t\\geq 0$.\n\\end{lemma}\n\n\\proof\n\nFor any $\\eta,\\zeta\\in \\{0,1\\}^{\\mathbb{T}^N}$, we write $\\eta\\geq\n\\zeta$ when and only when $\\eta(x)\\geq \\zeta(x)$ for any $x\\in\n\\mathbb{T}^N$.\n\nBy direct calculation, it is easy to check that\n\\begin{equation}\\label{equ 4.3}\n\\begin{cases}\nc(x,\\eta)\\geq c_1(x,\\zeta) \\text{\\quad if~} \\eta(x)=\\zeta(x)=0,\\\\\nc(x,\\eta)\\leq c_1(x,\\zeta) \\text{\\quad if~} \\eta(x)=\\zeta(x)=1\n\\end{cases}\n\\end{equation}\nfor any $\\eta\\geq \\zeta$.\n\nBy \\eqref{equ 4.3} and Theorem 3.1.5 of \\cite{LIG1985},\n\\begin{equation}\\label{equ 4.4}\n\\eta_t\\geq \\zeta_t\n\\end{equation}\nfor any $t>0$ in the sense of coupling when $\\eta_0$ and $\\zeta_0$\nhave the same distribution $\\mu_p$.\n\nEquation \\eqref{equ 4.2} follows from \\eqref{equ 4.4} directly.\n\n\\qed\n\nNow we introduce the second critical value of the contact process on\ntree, $\\lambda$ above which makes the complete convergence theorem\nhold.\n\nThe contact process $\\{\\zeta_t\\}_{t\\geq 0}$ is an attractive spin\nsystem (see Section 3.2 of \\cite{LIG1985}), therefore\n\\[\nP_{N,\\lambda_1}^1(\\exists~ t_n \\uparrow +\\infty,\n\\eta_{t_n}(x)=1,\\forall~n\\geq 1)\\geq P_{N,\\lambda_2}^1(\\exists~ t_n\n\\uparrow +\\infty, \\eta_{t_n}(x)=1,\\forall~n\\geq 1)\n\\]\nfor $\\lambda_1>\\lambda_2$. As a result, it is reasonable to define\nthe following critical value for each $N\\geq 2$,\n\\begin{equation}\\label{equ 4.5}\nA(N)=\\sup\\{\\lambda: P_{N,\\lambda}^1(\\exists~ t_n \\uparrow +\\infty,\n\\eta_{t_n}(x)=1,\\forall~n\\geq 1)=0\\}.\n\\end{equation}\n\n$A(N)$ is called the second critical value of the contact process on\n$\\mathbb{T}^N$. When $\\lambda>A(N)$, the contact process is called\nstrong survived. For more details, see Section 1.4 of\n\\cite{LIG1985}.\n\nAccording to Theorem 1.4.65 of \\cite{LIG1985},\n\\[\n\\limsup_{N\\rightarrow+\\infty}\\sqrt{N}\\frac{A(N)}{N+1}\\leq 1,\n\\]\nwhich is exactly equation \\eqref{equ 2.2}.\n\nThe following lemma is a corollary of the complete convergence\ntheorem of strong survived contact process on tree. Please note that\nwe denote by $\\delta_0$ the configuration in\n$\\{0,1\\}^{\\mathbb{T}^N}$ where all the vertices are in state $0$.\n\n\\begin{lemma}\\label{lemma 4.2}\nWhen $\\lambda>A(N)$, then there is a probability measure\n$\\nu_\\lambda$ on $\\{0,1\\}^{\\mathbb{T}^N}$ such that\n\\begin{equation}\\label{equ 4.6}\n\\nu_\\lambda(\\zeta:\\zeta=\\delta_0)=0\n\\end{equation}\nand\n\\begin{equation}\\label{equ 4.7}\n\\zeta_t\\Rightarrow \\nu_\\lambda\n\\end{equation}\nwhen $\\zeta_0$ has probability distribution $\\mu_p$ with $p\\in\n(0,1)$.\n\\end{lemma}\n\n\\proof\n\nWe denote by $\\zeta_t^1$ the contact process with\n$\\zeta_0=\\delta_1$. According to Theorem 3.2.3 and Theorem 6.1.6 of\n\\cite{LIG1985}, when $\\lambda>A(N)$, there exists probability\nmeasure $\\nu_\\lambda$ such that\n\\[\n\\zeta_t^1\\Rightarrow \\nu_\\lambda\n\\]\nand\n\\[\n\\nu_\\lambda(\\zeta:\\zeta=\\delta_0)=0.\n\\]\n\nLet\n\\[\n\\tau=\\inf\\{t:\\zeta_t=\\delta_0\\},\n\\]\nthen Theorem 1 of \\cite{Zhang1996} shows that for any probability\nmeasure $\\mu$ on $\\{0,1\\}^{\\mathbb{T}^N}$ and $\\{\\zeta_t\\}_{t\\geq\n0}$ with initial distribution $\\mu$,\n\\begin{equation}\\label{equ 4.8}\n\\zeta_t\\Rightarrow\nP_\\mu(\\tau<+\\infty)\\delta_0+P_\\mu(\\tau=+\\infty)\\nu_\\lambda\n\\end{equation}\nwhen $\\lambda>A(N)$.\n\nWhen $\\mu=\\mu_p$ for $p\\in (0,1)$, there are infinite many vertices\nin state $1$ at $t=0$ with probbaility one and hence\n\\begin{equation}\\label{equ 4.9}\nP_N^p(\\tau<+\\infty)=0.\n\\end{equation}\n\nLemma \\ref{lemma 4.2} follows from \\eqref{equ 4.8} and \\eqref{equ\n4.9} directly.\n\n\\qed\n\nEquation with form as \\eqref{equ 4.8} is called the complete\nconvergence theorem, which shows that the process with any initial\ndistribution converges weakly to a convex combination of invariant\nmeasures. In \\cite{Bez1990}, Bezuidenhout and Grimmett show that the\ncomplete convergence theorem holds for the contact process on\n$\\mathbb{Z}^d$. References \\cite{Zhang1996} authored by Zhang and\n\\cite{Sal1998} authored by Salzano and Schonmann give two different\nproofs of the complete convergence theorem of the strong survived\ncontact process on trees. In \\cite{ChenX2009} and \\cite{Yao2012},\nChen and Yao show that the complete convergence theorem holds for\ncontact process in a random environment on $Z^+\\times Z^d$. In\n\\cite{Handjani1999}, Handjani shows that the complete convergence\ntheorem holds for the threshold-one voter model on $\\mathbb{Z}^d$\nsuch that the process with any initial distribution converges weakly\nto a convex combination of three invariant measures.\n\nBy \\eqref{equ 3.3}, $P_N^p(\\eta_t(x)=1)$ is increasing with $t$, so\nit is reasonable to define\n\\[\nh(N,p)=\\lim_{t\\rightarrow+\\infty}P_N^p(\\eta_t(x)=1)\n\\]\nfor each $N\\geq 1$ and $p\\in (0,1)$. It is easy to see that\n$\\{\\eta_t\\}_{t\\geq 0}$ with initial distribution $\\mu_p$ converges\nweakly to $\\delta_1$ when and only when $h(N,p)=1$. The following\nlemma shows that there is a subsequence of $\\{\\eta_t\\}_{t\\geq 0}$\nconverges weakly to a convex combination of $\\delta_1$ and\n$\\delta_0$.\n\n\\begin{lemma}\\label{lemma 4.3}\nFor $\\{\\eta_t\\}_{t\\geq 0}$ with initial distribution $\\mu_p$ on\n$\\mathbb{T}^N$, there is a sequence $\\{t_n\\}_{n\\geq 1}$ increasing\nto infinity such that\n\\[\n\\eta_{t_n}\\Rightarrow h(N,p)\\delta_1+[1-h(N,p)]\\delta_0.\n\\]\n\\end{lemma}\n\n\\proof\n\nBy \\eqref{equ 3.3}, it is easy to see that\n\\[\n\\liminf_{t\\rightarrow+\\infty}P_N^p(\\eta_t(x)=1,\\eta_t(y)=0)=0\n\\]\nfor $x\\sim y$. Otherwise, there would be $\\alpha>0$ such that\n$P_N^p(\\eta_t(x)=1,\\eta_t(y)=0)\\geq \\alpha$ for any $t>T_0$, where\n$T_0$ is a sufficiently large number. Then, by \\eqref{equ 3.3},\n\\[\nP_N^p(\\eta_t(x)=1)-P_N^p(\\eta_{T_0}(x)=1)\\geq\n\\alpha(t-T_0)\\rightarrow +\\infty\n\\]\nas $t$ grows to infinity, which is contradictory.\n\nTherefore, there exists sequence $\\{t_n\\}_{n\\geq 1}$ increasing to\ninfinity such that\n\\begin{equation}\\label{equ 4.10}\n\\lim_{n\\rightarrow+\\infty}P_N^p(\\eta_{t_n}(x)=1,\\eta_{t_n}(y)=0)=0\n\\end{equation}\nfor $x\\sim y$.\n\nSince $\\{0,1\\}^{\\mathbb{T}^N}$ is a compact space, there is a\nsubsequence of $\\{\\eta_{t_n}\\}_{n\\geq 1}$ that converges weakly to a\nprobability measure on $\\{0,1\\}^{\\mathbb{T}^N}$ according to the\nHelly's selection theorem (see Theorem 3.2.6 of \\cite{Dur2010}).\nWithout loss of generality, we can assume that\n$\\{\\eta_{t_n}\\}_{n\\geq 1}$ is a convergent sequence itself.\n\nWe denote by $\\varphi$ the limit distribution of $\\eta_{t_n}$ as $n$\ngrows to infinity. Then, according to \\eqref{equ 4.10},\n\\begin{equation}\\label{equ 4.11}\n\\varphi(\\eta(x)=1,\\eta(y)=0)=0\n\\end{equation}\nfor any $x\\sim y$.\n\nBy \\eqref{equ 4.11},\n\\begin{align*}\n\\varphi(\\eta:\\eta\\neq \\delta_0,\\delta_1)&=\\varphi(\\exists~x\\sim y,\\eta(x)\\neq \\eta(y))\\\\\n&\\leq \\sum_{x\\sim y}\\varphi(\\eta(x)=1,\\eta(y)=0)+\\sum_{x\\sim\ny}\\varphi(\\eta(x)=0,\\eta(y)=1)=0.\n\\end{align*}\n\nAs a result, $\\varphi$ is a convex combination of $\\delta_1$ and\n$\\delta_0$. Since\n\\[\n\\varphi(\\eta(x)=1)=\\lim_{n\\rightarrow+\\infty}P_N^p(\\eta_{t_n}(x)=1)=h(N,p)\n\\]\naccording to definition of $h$,\n\\[\n\\varphi=h(N,p)\\delta_1+[1-h(N,p)]\\delta_0.\n\\]\n\n\\qed\n\nFinally we can give the proof of Theorem \\ref{theorem 2.2}.\n\n\\proof[Proof of Theorem \\ref{theorem 2.2}]\n\nWe only need to show that $h(N,p)=1$ when $\\lambda>A(N)$. When\n$\\lambda>A(N)$, for any $\\epsilon>0$, by \\eqref{equ 4.6} in Lemma\n\\ref{lemma 4.2}, there exists finite subset $D$ of $\\mathbb{T}^N$\nsuch that\n\\begin{equation}\\label{equ 4.12}\n\\nu_\\lambda(\\zeta(x)=0,\\forall~x\\in D)\\leq \\epsilon.\n\\end{equation}\n\nBy Lemma \\ref{lemma 4.3}, there exists sequence $\\{t_n\\}_{n\\geq 1}$\nincreasing to infinity such that\n\\begin{equation}\\label{equ 4.13}\n\\lim_{n\\rightarrow+\\infty}P_{N,\\lambda}^p(\\eta_{t_n}(x)=0,\\forall~x\\in\nD)=1-h(N,p).\n\\end{equation}\n\nBy Lemma \\ref{lemma 4.2} and \\eqref{equ 4.12},\n\\begin{equation}\\label{equ 4.14}\n\\lim_{n\\rightarrow+\\infty}P_{N,\\lambda}^p(\\zeta_{t_n}(x)=0,\\forall~x\\in\nD)=\\nu_\\lambda(\\zeta(x)=0,\\forall~x\\in D)\\leq \\epsilon.\n\\end{equation}\n\nBy \\eqref{equ 4.2}, \\eqref{equ 4.13} and \\eqref{equ 4.14},\n\\[\n1-h(N,p)\\leq \\epsilon\n\\]\nfor any $\\epsilon>0$.\n\nAs a result,\n\\[\nh(N,p)=1\n\\]\nwhen $\\lambda>A(N)$ and the proof is complete.\n\n\\qed\n\nAt the end of this section, we give a sketch of the proof of Theorem\n\\ref{theorem 2.3}.\n\n\\proof[Proof of Theorem \\ref{theorem 2.3}]\n\nLet $\\{\\xi_t\\}_{t\\geq 0}$ be contact process on $\\mathbb{Z}^d$ with\nflip rate function given by\n\\[\nc_2(x,\\xi)=\n\\begin{cases}\n1 \\text{\\quad if~}\\xi(x)=1,\\\\\n\\frac{\\lambda}{2d}\\sum_{y:y\\sim x}\\xi(y)\\text{\\quad if~}\\xi(x)=0\n\\end{cases}\n\\]\nfor any $(x,\\xi)\\in \\mathbb{Z}^d\\times \\{0,1\\}^{\\mathbb{Z}^d}$\n\nLet $\\lambda(d)$ be the first critical value of $\\{\\xi_t\\}_{t\\geq\n0}$, that is to say,\n\\[\n\\lambda(d)=\\sup\\{\\lambda:\\lim_{t\\rightarrow+\\infty}P_{\\mathbb{Z}^d,\\lambda}^1(\\xi_t(x)=1)=0\\}.\n\\]\nIt is shown in \\cite{Bez1990} that the complete convergence theorem\nholds for $\\{\\xi_t\\}_{t\\geq 0}$ when $\\lambda>\\lambda(d)$. Then,\naccording to a similar analysis with that in the proof of Theorem\n\\ref{theorem 2.2},\n\\[\n\\eta_t\\Rightarrow \\delta_1\n\\]\nfor the bias voter model $\\{\\eta_t\\}_{t\\geq 0}$ on $\\mathbb{Z}^d$\nwith initial distribution $\\mu_p$ with $p\\in (0,1)$ when\n$\\lambda>\\lambda(d)$.\n\nAccording to Corollary 6.4.4 of \\cite{LIG1985},\n\\[\n\\frac{\\lambda(d)}{2d}\\leq \\frac{2}{d}\n\\]\nand hence\n\\[\n\\lambda(d)\\leq 4.\n\\]\nTherefore, when $\\lambda>4$, the bias voter model on $\\mathbb{Z}^d$\nwith initial distribution $\\mu_p$ with $p\\in (0,1)$ converges weakly\nto $\\delta_1$.\n\n\\qed\n\n\\section{Two conjectures}\\label{section 5}\nIn this section we propose two conjectures. The first one is about\nthe mean field limit of the bias voter model on lattices.\n\n\\begin{conjecture}\\label{conjecture 5.1}\nFor $p\\in (0,1)$,\n\\[\n\\lim_{d\\rightarrow+\\infty}P_{\\mathbb{Z}^d}^p(\\eta_t(x)=1)=\\frac{pe^{(\\lambda-\\theta)t}}{1-p+pe^{(\\lambda-\\theta)t}}\n\\]\nfor any $t>0$.\n\\end{conjecture}\nAs we introduced in Section \\ref{section 3}, the main difficulty to\nprove Conjecture \\ref{conjecture 5.1} is to show that $\\eta_t(x)$\nand $\\eta_t(y)$ are asymptotically independent for $x\\sim y, x,y\\in\n\\mathbb{Z}^d$ as $d$ grows to infinity. Since there are infinite\nmany paths on the lattice from $x$ to $y$ avoiding the edge\nconnecting $x$ and $y$, our proof of Theorem \\ref{theorem 2.1 main}\nis not applicable for the case where the process is on the lattice.\n\nThe second conjecture is about the weak convergence of the process.\nWe guess that Theorem \\ref{theorem 2.2} and Theorem \\ref{theorem\n2.3} hold under a generalized condition.\n\n\\begin{conjecture}\\label{conjecture 5.2}\nFor any $\\lambda>\\theta$, $S=\\mathbb{T}^d$ or $\\mathbb{Z}^d$ with\n$d\\geq 1$,\n\\[\n\\eta_t\\Rightarrow \\delta_1\n\\]\nfor $\\{\\eta_t\\}_{t\\geq 0}$ on $S$ with initial distribution $\\mu_p$\nwith $p\\in (0,1)$.\n\\end{conjecture}\n\nAccording to the proof of Theorem \\ref{theorem 2.2}, the core step\nto prove Conjecture \\ref{conjecture 5.2} is to verify a claim that\nthe limit distribution of any convergent subsequence of\n$\\{\\eta_t\\}_{t\\geq 0}$ puts no mass on $\\delta_0$. However, for\n$\\lambda$ not large enough for the complete convergence theorem of\nthe contact process to hold, we have not find a way to prove this\nclaim yet.\n\nWe will work on this two conjectures as a further study and hope to\ndiscuss with readers who are interested in them.\n\n\\quad\n\n\\textbf{Acknowledgments.} We are grateful to the financial support\nfrom the National Natural Science Foundation of China with grant\nnumber 11171342 and China Postdoctoral Science Foundation (No.\n2015M571095).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{section:introduction}\n\nThe (reduced) \\emph{Burau representation} \n$$\\rho_n:\\ \\ B_n \\longrightarrow \\text{GL}\\left(n-1,\\mathbb Z[q^{\\pm 1}]\\right)$$ \nwas the first possible candidate for a faithful linear representation of the braid group on $n$ strands $B_n$ and it has been known for long to be faithful in the case of the 3-strand braid group~\\cite{magnuspeluso}. \nHowever, Moody \\cite{moody} showed that the Burau representation is not faithful for any braid index $n\\geqslant 9$. This was brought down to $n\\geqslant 6$ by Long and Paton \\cite{lp} and finally Bigelow showed the non-faithfulness of $\\rho_5$ \\cite{bigelow}. Despite these negative results, the linearity question of the braid groups was settled in the positive independently by Krammer \\cite{krammer} and Bigelow \\cite{bigelowlinear}. They showed that another linear representation \n$$\\mathcal L_n:\\ \\ B_n\\longrightarrow GL\\left(\\frac{n(n-1)}{2},\\mathbb Z[q^{\\pm 1},t^{\\pm 1}]\\right)$$ \nconstructed by Lawrence \\cite{lawrence} is faithful for all $n$. The representation $\\mathcal{L}_{n}$ is now known as the \\emph{Lawrence-Krammer-Bigelow representation}, or LKB representation for short.\n\nAt present, the question of the faithfulness of the Burau representation in the case $n=4$ remains open. \nThe linearity question itself was solved, nevertheless the problem to determine whether $\\rho_4$ is faithful or not remains of considerable importance:\na negative answer would be of great interest in \nquantum topology, since it is equivalent to the non-faithfulness of Jones and Temperley-Lieb representations of $B_{4}$, and would\nprovide a non-trivial knot with trivial Jones polynomial \\cite{bi3}.\nAnother interesting related problem is to study the image of the Burau representation -- an old question asks which $(n-1)\\times (n-1)$ matrices over $\\Z[q^{\\pm 1}]$ can appear as the image under the (reduced) Burau representation of some braid \\cite{birman}; it is widely open.\n\n\n\nThe present paper aims to establish relations between the Garside structures of the braid group and the Burau representation. Our motivation was to understand to what extent the Burau representation is close to be faithful, and when faithfulness property breaks down. This not only helps to attack the faithfulness problem of the 4-strand Burau representation, but also provides new insights for the image and the kernel of the Burau representation for arbitrary braid index, even for the simplest case $n=3$ (see Corollary \\ref{cor:3strands} below). \n\n\n\n\nThe \\emph{classical Garside structure}\nconsists in a lattice structure together with a special element $\\Delta$ satisfying some properties initially discovered by Garside in \\cite{garside}. \nA crucial output of this structure is the\n\\emph{classical (left) normal form} of a braid $x$, which is a unique decomposition of the form \n$$N_{\\sf c}(x) = \\Delta^p s_1\\cdots s_r$$\n in which the factors belong to the set of the so-called \\emph{simple elements}. \nThe \\emph{classical} \\emph{supremum} and \\emph{infimum} of $x$ are defined by $\\sup_{\\sf c}(x)= p+r$, $\\inf_{\\sf c}(x)=p$, respectively. The \\emph{classical canonical length} of $x$ is defined by $\\ell_{\\sf c}(x)=r$; \nthe \\emph{classical Garside length} $l_{\\sf c}(x)$ is the length of $x$ with respect to the simple elements. The latter satisfies\n $l_{\\sf c}(x)=\\max(\\sup_{\\sf c}(x),0)-\\min(\\inf_{\\sf c}(x),0)$.\n\nSlightly different but very close in spirit is the \\emph{dual Garside structure} (or BKL structure) discovered by Birman, Ko and Lee \\cite{bkldual} which leads to the \\emph{dual (left) normal form} of a braid $x$:\n$$N_{\\sf d}(x) = \\delta^p d_1\\cdots d_r$$\nwhere the factors belong to the set of the so-called \\emph{dual simple elements}.\nThe dual \\emph{supremum}, \\emph{infimum} and \\emph{canonical length} of a braid $x$ are defined similarly and denoted by $\\sup_{\\sf d}(x)$, $\\inf_{\\sf d}(x)$ and $\\ell_{\\sf d}(x)$ respectively. The \\emph{dual Garside length} $l_{\\sf d}(x)$ is the length of $x$ with respect to the dual simple elements; it satisfies $l_{\\sf d}(x)=\\max(\\sup_{\\sf d}(x),0)-\\min(\\inf_{\\sf d}(x),0)$. \nSee Section \\ref{section:garside} for more details on both classical and dual Garside structures of the braid group.\n\n\nOur first main result provides a non-vanishing criterion for the Burau representation $\\rho_4$ using the classical Garside structure.\n\\theoremstyle{plain}\n\\newtheorem*{thmMainclassical}{Theorem \\ref{theorem:main_classical}}\n\\begin{thmMainclassical}\nIf the classical left normal form of a 4-braid $x$\ndoes not contain a factor $(\\sigma_{2}\\sigma_{1}\\sigma_{3})$\nthen $\\rho_{4}(\\beta) \\neq 1$. \n\\end{thmMainclassical}\n\n\nIn the dual framework, we obtain more general and strong connections.\nFor a non-zero Laurent polynomial $\\Lambda$ in the variable $q$, \nlet us denote by $m(\\Lambda)$ and $M(\\Lambda)$ the minimal and maximal degrees of the variable, respectively. As a convention, we define $m(0)= +\\infty$ and $M(0)=-\\infty$.\n\n For a matrix $\\Lambda=(\\Lambda_{ij})\\in GL\\left(n-1,\\mathbb Z[q^{\\pm 1}]\\right)$, \nwe set $$m(\\Lambda)=\\min \\{m(\\Lambda_{ij}), 1\\leqslant i,j\\leqslant n-1\\},\\ \\textrm{and} \\ M(\\Lambda)=\\max \\{M(\\Lambda_{ij}, 1\\leqslant i,j\\leqslant n-1\\}.$$\n\n\n\nIn Section \\ref{section:dual} we will introduce a notion of \\emph{simply-nested} braid; roughly speaking, simply-nestedness is a local condition \non the factors of the dual left normal form of a braid. \nWe will show that the Burau representation completely determines the normal form of simply-nested braids.\n\n\n\n\\newtheorem*{ThmMainDual}{Theorem \\ref{theorem:main_dual}}\n\\begin{ThmMainDual}\nLet $x \\in B_n$ be a simply-nested braid. \n\\begin{enumerate}\n\\item[(i)] $\\sup_{\\sf d}(x) = M(\\rho_n(x))$.\n\\item[(ii)] One can compute the dual normal form from the matrix $\\rho_{n}(x)$, so the restriction of the Burau representation on the set of simply-nested braid $B_{n}^{\\sf sn}$ is injective.\n\\end{enumerate} \\end{ThmMainDual}\n\nThis provides several consequences for faithfulness questions in general.\nFirst of all, it follows that the Burau matrix of a 3-strand braid completely determines its dual normal form.\n\\newtheorem*{Corollary3strds}{Corollary \\ref{cor:3strands}}\n\\begin{Corollary3strds}\nLet $x\\in B_3$. Then \n\\begin{enumerate}\n\\item[(i)] $\\sup_{\\sf d}(x) = M(\\rho_{3}(x))$,\n\\item[(ii)] $\\inf_{\\sf d}(x) = m(\\rho_{3}(x))$.\n\\item[(iii)] One can compute the dual normal form of $x$ from the matrix $\\rho_{3}(x)$.\n\\end{enumerate}\\end{Corollary3strds}\n\n\nFor the 4-strand braid group, we will see:\n\n\\newtheorem*{Corollary4stds}{Corollary \\ref{cor:4strands}}\n\\begin{Corollary4stds}\nLet $x\\in B_4$ and $N_{\\sf d}(x)=\\delta^pd_1\\cdots d_r$. Assume that for all $i=1,\\ldots,r-1$,\n$(d_i,d_{i+1})$ is not in the following list:\n$$ \\left\\{ \\begin{array}{c}(a_{1,2}a_{3,4},a_{2,4}),(a_{1,2}a_{3,4},a_{3,4}a_{2,3}),(a_{1,2}a_{3,4},a_{1,2}a_{1,4}),\\\\\n\\ (a_{2,3}a_{1,4},a_{1,3}), (a_{2,3}a_{1,4},a_{1,3}a_{2,3}),(a_{2,3}a_{1,4},a_{1,3}a_{1,4}) \\end{array}\\right\\}$$\n\nThen \n\\begin{enumerate}\n\\item[(i)] $\\sup_{\\sf d}(x) = M_{q}(\\rho_{4}(x))$,\n\\item[(ii)] one can compute the dual normal form of $x$ from the matrix $\\rho_4(x)$. \n\\end{enumerate}\nIn particular, if the dual left normal form of a 4-braid $x$ \ndoes not contain a factor $(a_{1,2}a_{3,4})$ or $(a_{2,3}a_{1,4})$ then $\\rho_{4}(\\beta) \\neq 1$.\n\\end{Corollary4stds}\n\nFinally we give Garside-theoretical constraints for braids of arbitrary braid index to belong to the kernel of the Burau representation. \nLet $e:B_n \\rightarrow \\mathbb Z$ be the abelianization map. \n\\newtheorem*{Corollarynstds}{Corollary \\ref{cor:nstrands}}\n\\begin{Corollarynstds}\nLet $x\\in B_{n}$ be a non-trivial braid and $N_{\\sf d}(x)=\\delta^{p}d_{1}\\cdots d_{r}$. If there exists $r'\\leqslant r$ such that\n\\begin{enumerate}\n\\item[(i)] The subword $x_{r'}=\\delta^{p}d_{1}\\cdots d_{r'}$ is simply-nested,\n\\item[(ii)] $r' > e(d_{r'+1}\\cdots d_{r})$,\n\\end{enumerate}\nthen $\\rho_n(x)\\neq 1$.\\end{Corollarynstds}\n\nThus, we conclude that if a braid $x$ is sufficiently close to simply nested braids, then its Burau matrix is never trivial.\n\nNow we explain the organization of the paper. \nIn section \\ref{section:garside} we recall Garside theoretical no(ta)tions to be used later. \nSection \\ref{section:classical} shows Theorem \\ref{theorem:main_classical}.\nSections \\ref{section:diagrams}-\\ref{section:dual} are devoted to the proof of Theorem \\ref{theorem:main_dual}. This \ncan be sketched as follows. First we recall from \\cite{iw} the wall-crossing labeling of the curve diagram of a braid and how it is related to the dual Garside normal form (Section \\ref{section:diagrams}). Section \\ref{section:burau} reviews a homological interpretation of the reduced Burau representation; in this context we show how the Burau matrix is related to the wall-crossing labeling.\nWall-crossing labeling therefore serves as a bridge between Burau representation and the dual Garside structure. \nFinally, Section \\ref{section:dual} introduces the notion of simply-nestedness and proves Theorem \\ref{theorem:main_dual}\nand its above mentionned corollaries.\n\n\n\n\n\\section{Reminders on the Garside structures of braid groups}\\label{section:garside}\n\nLet $\\mathbb D^2$ be the closed disk in $\\mathbb C$ with diameter the real segment $[0,n+1]$ and $\\mathbb D_n$ be the $n$-times punctured disk: \n$\\mathbb D_{n}=\\mathbb D^2-\\{1,\\ldots,n \\}$. \nWe denote the $i$-th puncture point $i \\in \\C$ by $p_{i}$ and put $p_0=0\\in \\mathbb C$.\nAs is well-known, the braid group $B_{n}$ is identified with the mapping class group of $\\mathbb D_{n}$ (with boundary fixed pointwise).\nWe identify the standard Artin generator $\\sigma_{i}$ ($i=1,\\ldots,n-1$) with the \\emph{left-handed} (that is, \\emph{clockwise}) half Dehn twist along the real segment $[i,i+1]$. Throughout the paper we will consider braids acting on the \\emph{right}.\n\n\n\nFor $1\\leqslant i\\neq j\\leqslant n$, we denote by $a_{i,j}$ (or $a_{j,i}$ indifferently) the isotopy class of the left-handed half Dehn twist along an arc connecting the punctures $p_i$ and $p_j$ through the lower part of the disk $\\{z \\in \\mathbb D^{2}\\: | \\: \\textrm{Im}\\, z < 0\\}$. Using the Artin generators, $a_{i,j}$ $(iM_2$ and $M_1>M_3+1$, \n\\item if $S(s_1)=\\{2\\}$ then $M_2\\geqslant M_1$ and $M_2>M_3$, \n\\item if $S(s_1)=\\{3\\}$ then $M_3\\geqslant M_1$ and $M_3\\geqslant M_2$, \n\\item if $S(s_1)=\\{1,2\\}$ then either $M_1>M_2$ and $M_1>M_3+1$, or $M_2\\geqslant M_1$ and $M_2>M_3$,\n\\item if $S(s_1)=\\{2,3\\}$ then either $M_2\\geqslant M_1$ and $M_2>M_3$, or $M_3\\geqslant M_1$ and $M_3\\geqslant M_2$, \n\\item if $S(s_1)=\\{1,3\\}$ then either $M_1>M_2$ and $M_1>M_3$,\nor $M_3\\geqslant M_1$ and $M_3\\geqslant M_2$.\n\\end{itemize}\nMoreover, the following inequality holds: $\\sup_{\\sf c}(x)\\leqslant M(\\rho_4(x))\\leqslant 3\\sup_{\\sf c}(x)$.\n\\end{lemma}\n\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:classicalLemme}]\nThe proof is by induction on $r$.\nA direct calculation shows that all conclusions are correct for the case $r=2$. Here we remark that in the case $r=1$ and $s_{1}=\\sigma_{1}$ ($S(s_1)=\\{1\\}$), the conclusion does not hold since $M_1(\\sigma_1)=M_3(\\sigma_1)+1$.\n\nSuppose now $r>2$. Write $x=s_1x'$; by induction $x'$ satisfies the conclusions of the lemma. \nWe now distinguish 6 cases, according to the possible values of $S(s_1)$. \nIn each case, there are several possibilities for\n$s_1$. Each of them leads to conditions on the \nstarting set of $s_2$, the first factor of $x'$, because of the left-weightedness condition on the pair $(s_1,s_2)$.\nBy induction hypothesis this gives relations between the integers $M'_i:=M_i(x')$. \nIn each case, using the explicit computation of $\\rho_4(s_1)$, we express the integers $M_i=M_i(s_1x')$ in terms of the $M'_i$ and show that they satisfy the expected relations. In each case, the computations to be performed show that \n$M(\\rho_4(x'))+1\\leqslant M(\\rho_4(s_1x'))\\leqslant M(\\rho_4(x'))+3$; this shows the last claim in the lemma.\n\nWe present the cases $S(s_1)=\\{2\\}$ and $S(s_1)=\\{1,3\\}$; this will have the advantage to show the failure in the argument when a factor $\\sigma_2\\sigma_1\\sigma_3$ appears. Other cases are proven similarly.\n\n{\\bf Case $S(s_1)=\\{2\\}$.} The simple element $s_1$ is one of the following: $\\sigma_2$, $\\sigma_2\\sigma_1$, $\\sigma_2\\sigma_3$,\n$\\sigma_2\\sigma_1\\sigma_3\\sigma_2$ or $\\sigma_2\\sigma_1\\sigma_3$. We treat two examples; again the three others are dealt with similarly. \n\n\n{\\underline{Suppose $s_1=\\sigma_2$.}} Then $F(s_1)=\\{2\\}$ and by left-weightedness $S(s_2)=\\{2\\}$. By induction, \nwe have $M'_2\\geqslant M'_1$ and $M'_2>M'_3$. Multiplying $\\rho_4(x')$ on the left by $\\rho_4(\\sigma_2)=\\begin{pmatrix}\n1 & q & 0\\\\\n0 & -q & 0\\\\\n0 & 1 & 1\\\\\n\\end{pmatrix}$, the new degrees $M_i$ in the product satisfy $M_1=M'_2+1$, $M_2=M'_2+1$ and $M_3\\leqslant M'_2$ (possibly the terms of highest degrees in the second and third row of $\\rho_4(x')$ cancel with each other). \nTherefore we have $M_2=M_1$ and $M_2>M_3$, thus satisfying the expected conditions when $S(s_1)=\\{2\\}$. \n\n{\\underline{Suppose $s_1=\\sigma_2\\sigma_1\\sigma_3$}}. \nThen $F(s_1)=\\{1,3\\}$ and by left-weightedness, $S(s_2)=\\{1\\},\\{3\\}$ or $\\{1,3\\}$.\nBy induction $M'_3\\geqslant M'_1,M'_2$ or $M_1>M_2,M_3$ (with possibly $M_1>M_3+1$). \nComputing $\\rho_4(\\sigma_2\\sigma_1\\sigma_3)=\\begin{pmatrix} \n0 & q & q^2\\\\\n-q & -q & -q^2\\\\\n1 & 1 & 0\\\\\n\\end{pmatrix}$ we get in the first case $M_1=M'_3+2$, $M_2=M'_3+2$ and $M_3\\leqslant \\max(M'_1,M'_2)\\leqslant M'_3$; whence $M_2\\geqslant M_1$ and $M_2>M_3$. \nIn the second case, unless the strongest inequality $M'_1>M'_3+1$ holds, there is no reason why \na cancellation could not yield $M_3\\geqslant M_2$. Therefore the desired conclusion ($M_2>M_3$) possibly does not hold and we see that the argument fails when $\\sigma_2\\sigma_1\\sigma_3$ is a factor of $x$. \n\n{\\bf Case $S(s_1)=\\{1,3\\}$.} Then $s_1$ is $\\sigma_1\\sigma_3$, $\\sigma_1\\sigma_3\\sigma_2$, $\\sigma_1\\sigma_3\\sigma_2\\sigma_1$, $\\sigma_1\\sigma_3\\sigma_2\\sigma_3$ or $\\sigma_1\\sigma_3\\sigma_2\\sigma_3\\sigma_2$. \n\n{\\underline{Suppose $s_1=\\sigma_1\\sigma_3$.}} \nWe compute $\\rho_4(\\sigma_1\\sigma_3)=\\begin{pmatrix}\n-q & 0 &0 \\\\\n1 & 1 & q \\\\\n0 & 0 & -q\\\\\n\\end{pmatrix}.$\nBy left-weightedness, we have $S(s_2)\\subset \\{1,3\\}$ and therefore \nby induction the $M'_i$ satisfy \n$M'_1>M'_2,M'_3$ or $M'_3\\geqslant M'_1,M'_2$. \nIn the first case $x=\\sigma_1\\sigma_3x'$ satisfies \n$M_1=M'_1+1$, $M_2\\leqslant M'_1$ and $M_3=M'_3+1$: \nwe have, as expected, $M_1>M_2,M_3$. In the second case, we have $M_1=M'_1+1,M_2=M'_3+1,M_3=M'_3+1$ whence $M_3\\geqslant M_1,M_2$. \n\n{\\underline{Suppose $s_1=\\sigma_1\\sigma_3\\sigma_2$.}}\nThen we have to check the product of the matrix $\\rho_4(\\sigma_1\\sigma_3\\sigma_2)=\\begin{pmatrix}\n-q & -q^2 & 0 \\\\\n1 & q & q \\\\\n0 & -q & -q\\\\\n\\end{pmatrix}$ by $\\rho_4(x')$, where the $M'_i$ satisfy by induction $M'_2\\geqslant M'_1$ and $M'_2>M'_3$. \nThis gives $M_1=M'_2+2$, $M_2=M'_2+1$ and $M_3=M'_2+1$ whence $M_1>M_2,M_3$. \n\n{\\underline{Suppose $s_1=\\sigma_1\\sigma_3\\sigma_2\\sigma_1$.}}\nCompute $\\rho_4(\\sigma_1\\sigma_3\\sigma_2\\sigma_1)=\\begin{pmatrix}\n0 & -q^2 & 0 \\\\\n0 & q & q\\\\\n-q & -q & -q \\\\\n\\end{pmatrix}$. \nOn the other hand we have by induction one of the following set of conditions on $x'$: \n$M'_1>M'_2$ and $M'_1>M'_3+1$; or $M'_2\\geqslant M'_1$ and $M'_2>M'_3$.\nIn the first case we obtain $M_1=M'_2+2,M_2\\leqslant \\max(M'_2+1,M'_3+1)$ and $M_3=M'_1+1$ whence \n$M_3>M_1,M_2$. In the second case we get $M_1=M'_2+2,M_2=M'_2+1$ and $M_3\\leqslant M'_2+1$ whence \n$M_1>M_2,M_3$. \n\n{\\underline{Suppose $s_1=\\sigma_1\\sigma_3\\sigma_2\\sigma_3$.}}\nCompute $\\rho_4(\\sigma_1\\sigma_3\\sigma_2\\sigma_3)=\\begin{pmatrix}\n-q & -q^2 & -q^3\\\\\n1 & q & 0 \\\\\n0 & -q & 0\\\\\n\\end{pmatrix}.$\nBy induction hypothesis, as $S(S_2)\\subset F(s_1)=\\{2,3\\}$, we have $M'_2\\geqslant M'_1$ and $M'_2>M'_3$ or $M'_3\\geqslant M'_1,M'_2$. \nIn the first case: \n\\begin{itemize}\n\\item if $M'_2>M'_3+1$ then $M_1=M'_2+2$, $M_2=M'_2+1=M_3$ whence $M_1>M_2,M_3$,\n\\item if $M'_2=M'_3+1$ then $M_1\\leqslant M'_2+2$ and $M_2=M'_2+1=M_3$ whence we get $M_1>M_2,M_3$ if $M_1=M'_2+2$ and $M_3\\geqslant M_1,M_2$ if $M_1M_2,M_3$. \n\n{\\underline{Suppose $s_1=\\sigma_1\\sigma_3\\sigma_2\\sigma_1\\sigma_3$.}}\nThe reduced Burau matrix of $s_1$ is \n$\\begin{pmatrix}\n0 & -q^2 & -q^3\\\\\n0 & q & 0 \\\\\n-q & -q & 0\\\\\n\\end{pmatrix}$. \nOn the other hand $F(s_1)=\\{1,3\\}$ whence by induction $x'$ satisfies: \n$M'_1>M'_2,M'_3$ or $M'_3\\geqslant M'_1,M'_2$. \nIn the first case we get $M_1\\leqslant \\max(M'_2+2,M'_3+3)$, $M_2=M'_2+1$, $M_3=M'_1+1$. This implies $M_3\\geqslant M_1,M_2$ provided \n$M'_1>M'_3+1$ holds. If on the contrary $M'_1=M'_3+1$ we can say more about $M_1$ (actually there will be no cancellation there) because the inequality $M'_1>M'_2$ then implies $M'_3+1>M'_2$ whence $M_1=M'_3+3$. This finally shows $M_1>M_2,M_3$. \nIn the second case we obtain $M_1=M'_3+3$, $M_2=M'_2+1$ and $M_3\\leqslant \\max(M'_1+1,M'_2+1)$ whence \n$M_1>M_2,M_3$. \n\n\n\\end{proof}\n\n\n\\begin{example} \nWe show that the conclusion for $S(s_1)=\\{2\\}$ in Lemma \\ref{lem:classicalLemme} does not necessarily hold if \n$s_1=\\sigma_2\\sigma_1\\sigma_3$. \nIndeed, let\n$x=\\sigma_2\\sigma_1\\sigma_3\\cdot\\sigma_1\\sigma_3\\sigma_2\\sigma_1\\cdot\\sigma_1\\sigma_2\\sigma_1\\sigma_3\\cdot\\sigma_1\\sigma_3\\cdot\\sigma_1\\sigma_2\\cdot \\sigma_2$. This braid has infimum 0 and is in normal form as written; the degrees of the entries of its Burau matrix are indicated in the following matrix:\n$\\begin{pmatrix} 5 & 8 & 7\\\\\n6 & 7 & 7\\\\\n5 & 7 & 2\\\\\n\\end{pmatrix}$.\n\\end{example}\nLemma \\ref{lem:classicalLemme} leads to the following non-vanishing criterion for the reduced Burau representation of 4-braids. \n\n\\begin{theorem}\\label{theorem:main_classical}\nIf the classical left normal form of a 4-braid $x$\ndoes not contain a factor $(\\sigma_{2}\\sigma_{1}\\sigma_{3})$\nthen $\\rho_{4}(x) \\neq 1$. \n\\end{theorem} \n\n\\begin{proof}\nLet $N_{\\sf c}(x)=\\Delta^{p}s_{1}\\cdots s_{r}$.\nIt is easy to check that if $r\\leq 2$, then $\\rho_4(x) \\neq 1$ so we may assume $r > 2$.\n\nFirst we observe that \n$$ \\rho_4(\\Delta)=\n\\begin{pmatrix}\n0 & 0 & -q^{3}\\\\\n0 & -q^{2} & 0\\\\\n-q & 0 & 0\\\\\n\\end{pmatrix}$$\nhence $M_{1}(\\Delta x) = M_{3}(x)+3$, $M_{2}(\\Delta x) = M_{2}(x)+2$ and $M_{3}(\\Delta x) = M_{1}(x)+1$.\n\nAssume that $S(s_{1})\\neq \\{1,3\\}$.\nIf $p$ is even, then by conjugating by $\\Delta$ if necessary, we may assume that $S(s_{1})=\\{1\\}, \\{2\\}$, or $\\{1,2\\}$. By Lemma \\ref{lem:classicalLemme}, $\\rho_4(s_1\\cdots s_r)$ is not an homothety hence $\\rho_4(\\Delta^ps_1\\cdots s_r) \\neq 1$. If $p$ is odd, we may assume similarly that $S(s_{1})=\\{2\\}, \\{3\\}$, or $\\{2,3\\}$ hence by Lemma \\ref{lem:classicalLemme}, \n$M_2(s_1\\cdots s_r)\\geqslant M_1(s_1\\cdots s_r)$ or $M_3(s_1\\cdots s_r)\\geqslant M_1(s_1\\cdots s_r)$. \nOn the other hand, $\\rho_4(x)=1$ implies \n$$M_{3}(s_{1}\\cdots s_{r})+2 = M_{2}(s_{1}\\cdots s_{r})+1 = M_{1}(s_{1}\\cdots s_{r}),$$\nwhich is a contradiction.\n\nNow we consider the case $S(s_{1})=\\{1,3\\}$. Assume for a contradiction that $\\rho_4(x)=1$. This implies in particular $M_{i}(yxy')=M_{i}(yy')$ for any 4-braids $y$ and $y'$. We deduce a contradiction by finding appropriate braids $y$ and $y'$. \n\n\\textbf{Case 1:} $s_{1} = \\sigma_{1}\\sigma_{3}$.\nIf $p$ is even, put $y=\\Delta\\sigma_{2}\\sigma_{1}\\sigma_{3}\\sigma_{2}$: $N_{\\sf c}((\\Delta\\sigma_{2}\\sigma_{1}\\sigma_{3}\\sigma_{2})x) = \\Delta^{p+2} s_{2}\\cdots s_{r}$. By direct calculation \nand under our hypothesis that $\\rho_4(x)=Id$,\n$$\\rho_4(\\Delta^{p+2} s_{2}\\cdots s_{r})=\\rho_4(\\Delta\\sigma_{2}\\sigma_{1}\\sigma_{3}\\sigma_{2})=\n\\begin{pmatrix}\n-q^3 & 0 & 0\\\\\nq^3 & q^{4} & q^{4}\\\\\n0 & 0 & -q^3\\\\\n\\end{pmatrix}.$$\nIt follows that $M_2(s_2\\ldots s_r)=M_1(s_2\\ldots s_r)+1=M_3(s_2\\ldots s_r)+1$; contradicting Lemma \\ref{lem:classicalLemme}\napplied with $S(s_2)\\subset \\{1,3\\}$ (which implies in particular $M_20 \\: \\}$. The lines $W_i$ are called the {\\em walls}, and their union $\\bigcup_{i} W_{i}$ is denoted $W$. Let $U_i$ be a disk-neighborhood of the puncture $p_i$ and set $U = \\bigcup_{i}U_{i}$. See Figure \\ref{fig:curvediagram} (b), (c).\n\n\n\\begin{figure}[htbp]\n\\centerline{\\includegraphics[scale=1.2]{curvediagram_0_2.eps}}\n \\caption{Curve diagram and walls}\n \\label{fig:curvediagram}\n\\end{figure}\n\nThe (\\emph{total}) \\emph{curve diagram} of a braid $x$ is the respective image of $E$ (or $\\widebar E$) under a diffeomorphism $\\phi$ representing $x$ which satisfies:\n\\begin{enumerate}\n\\item $(\\widebar E)\\phi$ coincides with the real line on $U$,\n\\item $(\\widebar{E})\\phi$ is transverse to $W$ and the number of intersections of $(\\widebar{E})\\phi$ with $W$ is as small as possible (which is equivalent to saying that $(\\widebar E)\\phi$ and $W$ do not bound together any bigon \\cite{fgrrw}).\n\\end{enumerate}\n\n\nThe (total) curve diagram is uniquely defined up to isotopy of $\\mathbb D_n$ that fixes $\\partial \\mathbb D_n$. We denote by $D_x$ ($\\widebar{D_x}$ respectively) the (total) curve diagram of a braid $x$.\nFigure \\ref{fig:curvediagram} (c) shows the (total) curve diagram of the braid $\\sigma_1\\in B_4$; according \nto our previous convention, dashed line represents the image of the initial segment $E_0$. \n\nAn \\emph{arc segment} (or simply an arc) of the (total) curve diagram $D_x$ (or $\\widebar {D_x}$) is a connected component of $D_x-(W\\cup U)$ (or $\\widebar{D_x}-(W\\cup U)$). \nNotice that an arc segment of $\\widebar{D_x}$ is in one of the three following cases:\n\\begin{itemize}\n\\item it connects two walls $W_i$ and $W_j$,\n\\item it connects a wall $W_i$ and a puncture $p_j$ (more precisely the neighborhood $U_j$), \n\\item it connects two punctures $p_i$ and $p_j$ (more precisely the neighborhoods $U_i$ and $U_j$).\n\\end{itemize}\nIn all cases, $i\\neq j$ by construction of the curve diagram. We denote such an arc segment, in either case, by $\\wideparen{(ij)}$. \nUnless explicitly specified, we will not care about the orientation of an arc segment; this is reflected in our notation. \n\n\\subsection{Wall-crossing labeling and dual normal form}\n\nWe now describe the wall-crossing labeling. To that purpose, we need to introduce a modified version of the curve diagrams. \n\nLet $x\\in B_n$. Around each puncture $p_{i}$ distinct from the image of $p_n$ under $x$, we modify the total curve diagram $\\widebar {D_{x}}$ inside the neighborhood $U_i$ as shown in Figure \\ref{fig:modcurve} (a). \nWe denote the resulting (total) curve diagram by $MD_{x}$ $(\\widebar{MD_{x}})$, and call it the (\\emph{total}) \\emph{modified curve diagram} of $x$.\nFigure \\ref{fig:modcurve} (b) shows the (total) modified curve diagram of $\\sigma_{1}\\in B_4$.\n\n\\begin{figure}[htbp]\n \\begin{center}\n\\includegraphics*[scale=1]{modcurve_0_2.eps}\n \\caption{Modified curve diagrams}\n \\label{fig:modcurve}\n \\end{center}\n\\end{figure}\n\nTake a smooth parametrization of $\\widebar{MD_x}$, viewed as the image of a function $\\gamma\\co [0,1] \\rightarrow \\mathbb D^{2}$. For each connected component $\\alpha$ of $\\widebar{MD_{x}}- (W\\cup U)$, we assign the algebraic intersection number of $W$ and the arc $\\gamma([0,v])$, where $v \\in [0,1]$ is taken so that $\\gamma(v) \\in \\alpha$. \nNotice that a connected component of $\\widebar{MD_{x}}-(W\\cup U)$ naturally corresponds to an arc segment of $\\widebar {D_x}$, since \n$\\widebar{MD_{x}}$ and $\\widebar{D_{x}}$ are identical except on $U$. This allows to attribute a label to each arc segment of $\\widebar {D_x}$; this integer-valued labeling is called the \\emph{wall-crossing labeling} of $x$. We define\n$\\LWcr(x)$ and $\\SWcr(x)$ as the largest and smallest possible labels occuring in the wall-crossing labeling for arc segments \\emph{in the curve diagram} $D_x$, respectively. \n\nNotice that to define $\\LWcr$ and $\\SWcr$, we used the largest and smallest labels only of the curve diagram $D_{x}$, not the total curve diagram $\\widebar{D_{x}}$. However, in order to determine the wall crossing labelings we need to consider the total curve diagram. \n\nThe following relates the wall-crossing labeling with the dual length of a braid:\n\\begin{theorem}\\cite[Theorem 3.3]{iw}\n\\label{theorem:wall}\nFor a braid $x \\in B_{n}$, we have the following equalities: \n\\begin{enumerate}\n\\item $\\sup_{\\sf d}(x) = \\LWcr(x)$.\n\\item $\\inf_{\\sf d}(x) = \\SWcr(x)$.\n\\end{enumerate}\n\\end{theorem}\n\nHere we show a stronger result than Theorem \\ref{theorem:wall}, which is suggested by and is implicit in the proof of \\cite[Theorem 3.3]{iw}: one can read not only supremum, infimum, but also dual Garside normal form from the curve diagram.\nRecall from Section \\ref{section:garsidedual} the lattice ordering $\\preccurlyeq_{\\sf d}$ on $B_n$. \n\n\\begin{theorem}\n\\label{theorem:finalfactor}\nLet $x \\in B_{n} $ be a braid and put $\\ell=\\LWcr(x)-\\SWcr(x)$.\nFor $k=\\ell,\\ldots,1$, we define $d_{k}$ inductively as follows: \n\\begin{enumerate}\n\\item $d_{\\ell}$ is the least common multiple (with respect to $\\preccurlyeq_{\\sf d}$) of all letters $a_{i,j}$ such that the curve diagram $D_{x}$ contains an arc segment $\\wideparen{(i j)}$ with wall-crossing labeling $\\LWcr(x)$.\n\\item $d_{k}$ is the least common multiple (with respect to $\\preccurlyeq_{\\sf d}$) of all letters $a_{i,j}$ such that the curve diagram $D_{x d_{\\ell}^{-1}\\cdots d_{k+1}^{-1}}$ contains an arc segment $\\wideparen{(i j)}$ with wall-crossing labeling $(k+\\SWcr(x))$.\n\\end{enumerate}\nThen the dual normal form of $x$ is given by\n$$ N_{\\sf d}(x)=\\delta^{\\SWcr(x)}d_1\\ldots d_{\\ell}.$$.\n\\end{theorem}\n\n\nBefore proving Theorem \\ref{theorem:finalfactor}, we review from \\cite{iw} the description of how the action of a dual simple element affects the curve diagram of a braid and its wall-crossing labeling. This was the key of the proof of Theorem \\ref{theorem:wall}.\n\n\nDealing with the dual Garside structure, it will be convenient to work with the \nmodel of the punctured disk described in Section \\ref{section:garsidedual}; in that context the wall $W_i$ is the shortest straight segment connecting the puncture $p_i$ to the boundary, oriented outwards. Notice that the isotopy involved in the change of model for the punctured disk does not affect the wall-crossing labeling since the latter \nis defined in terms of algebraic intersection of arcs and walls. \n\n\nLet $x\\in B_n$; let $d$ be a dual simple element. Write $d=P_1\\cdots P_r$ the decomposition of $d$ into a product of disjoint polygons. For $i=1,\\ldots,r$, let $N_i$ be a regular neighborhood of the polygon $P_i$ in $\\mathbb D_n$. Let $A_i$ be an annulus which is a regular neighborhood of the boundary of $N_i$. Suppose moreover that $A_i$ is chosen so that none of its two boundary components forms a bigon together with the walls $W$ or the diagram $D_{x}$ and so that as many intersection points of $D_{x}$ and $W$ as possible lie in $A_i$.\n\nNow $D_{xd}$ and its wall-crossing labeling are obtained as follows. The respective actions of each of the polygons $P_i$ are independent; each of them acts non-trivially only on the inner complementary component of the corresponding annulus and on the annulus itself (where the diagram just describes a spiral). \nFor each $i=1,\\ldots,r$, $N_i$ is turned by one notch in the clockwise direction and all labels are increased by one; on the annulus $A_i$, $D_{xd}$ and the corresponding labels are interpolated linearly; see Figure \\ref{fig:Paction}. \nThe action of the inverse of a dual simple element can be described in a very similar way, the twisting on $N_i$ being in the opposite direction, and all labels being decreased by one. \n\n\\begin{figure}[hbtp] \n \\begin{center}\n\\includegraphics*[scale=0.5, width=110mm]{Paction_0_1.eps}\n\\caption{How to draw curve diagram of $xd$ from $D_{x}$}\\label{fig:Paction}\n \\end{center}\n\\end{figure}\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:finalfactor}]\nWe prove the theorem by induction on $\\ell=\\LWcr(x)-\\SWcr(x)$. \nWhen $\\ell \\leqslant 1$, the result is explicitly contained in the proof of Lemma 3.5 in \\cite{iw}. \nSuppose that $\\ell\\geqslant 2$. By induction, it is sufficient to show that $(d_{\\ell-1},d_{\\ell})$ is left-weighted. \n\nWe check the left-weightedness using Proposition \\ref{prop:left-weighted}.\nWrite the dual simple elements $d_{\\ell}$ and $d_{\\ell-1}$ as products of disjoint polygons: $d_{\\ell-1}=P_1\\cdots P_{r_{\\ell-1}}$ and $d_{\\ell}=Q_{1}\\cdots Q_{r_{\\ell}} $, respectively.\nLet $i,j$ be two vertices of some polygon $Q \\in \\{Q_1,\\ldots,Q_{r_{\\ell}}\\}$. We must show that there exists a polygon $P \\in \\{ P_1,\\ldots, P_{r_{\\ell}}\\}$ having vertices $k,l$ such that $a_{k,l}\\vdash a_{i,j}$. \nBy definition of $d_{\\ell-1}$, it is sufficient to show that $D_{xd_{\\ell}^{-1}}$ admits an arc segment $\\wideparen{(kl)}$ with label $\\LWcr(x)-1$ and such that $a_{k,l}\\vdash a_{i,j}$.\n\n\nAssume first that the diagram $D_x$ admits an arc segment $\\wideparen{(ij)}$ with label $\\LWcr(x)$. Then according to the description above of the action of the inverse of a polygon, the diagram $D_{xd_{\\ell}^{-1}}$ admits an arc segment $\\wideparen{(kl)}$ with label $\\LWcr(x)-1$ such that $k\\in (j,i-1)$ and $l\\in(i,j-1)$, as desired. Moreover, we notice that if $g_{i,j}$ and $h_{i,j}$ are the rightmost vertex of $Q$ in $(j,i-1)$ and $(i,j-1)$ respectively, then \n$k\\in (g_{i,j},i-1)$ and $l\\in (h_{i,j},j-1)$. See Figure \\ref{Fig:LW} (a).\n\nAssume now that $D_x$ does not have an arc segment $\\wideparen{(ij)}$ with label $\\LWcr(x)$.\nSince both $i$ and $j$ are vertices of $Q$, by definition of $Q$ there must exist arc segments $\\wideparen{(ib)},\\wideparen{(jc)}$ of $D_{x}$ with label $\\LWcr(x)$, for some punctures $b, c \\not \\in \\{i,j\\}$, possibly $b=c$.\n \nSuppose that such a puncture $b$ can be chosen so that \n$a_{b,i}a_{i,j}$ is a dual simple braid. This means that $b\\in (i+1,j-1)$. \nBut we have just seen that the action of $Q^{-1}$ produces an arc segment $\\wideparen{(kl)}$ labeled by $\\LWcr(x)-1$ in the diagram $D_{xd_{\\ell}^{-1}}$, such that $k\\in (j,i-1)$ (because the rightmost vertex of $Q$ in the arc $(b+1,i-1)$ certainly lies in the subarc $(j,i-1)$) and $l\\in (i,b-1)\\subset (i,j-1)$). Similarly, if $c$ can be chosen so that $a_{c,j}a_{i,j}$ is a dual simple braid, we get a pair of punctures $k,l$ with the expected property. See Figure \\ref{Fig:LW} (b).\n\n\n\\begin{figure}[hbtp] \n \\begin{center}\n\\includegraphics*[scale=0.5, width=120mm]{alphaMax.eps}\n\\caption{Proof of Theorem \\ref{theorem:finalfactor}; all arc segments represented are labeled $\\LWcr(x)$, crosses indicate vertices of the polygon $Q$.}\\label{Fig:LW}\n \\end{center}\n\\end{figure}\n\nFinally, suppose that no arc segment $\\wideparen{(bi)}$ nor $\\wideparen{(jc)}$ with labeling $\\LWcr(x)$ of $D_x$ has the above property. \nThen $b\\in (j+1,i-1)$, $c\\in (i+1,j-1)$ and $b\\neq c$. Among all $b$ so that $D_x$ admits an arc $\\wideparen{(bi)}$ labeled $\\LWcr(x)$, let $b_0$ be the leftmost one. Similarly, among all $c$ so that $D_x$ admits an arc $\\wideparen{(jc)}$ labeled $\\LWcr(x)$, let $c_0$ be the leftmost one. \nThe punctures $i,j,b_0$ and $c_0$ are all distinct and vertices of the polygon $Q$. By definition of $d_\\ell$, there must exist \nan arc segment $\\wideparen{(f_1f_2)}$ in $D_{x}$ with labeling $\\LWcr(x)$ such that $f_1\\in (j+1,b_0)$, $f_2\\in (i+1,c_0)$; the punctures $f_1,f_2$ are also vertices of $Q$. \nBut then $D_{xd_{\\ell}^{-1}}$ admits an arc $\\wideparen{(kl)}$ labeled $\\LWcr(x)-1$ with $k\\in(j,f_1-1)$ and $l\\in(i,f_2-1)$, thus with the required property. See Figure \\ref{Fig:LW} (c) (an example where $f_1=b_0$).\nThis completes the proof of Theorem \\ref{theorem:finalfactor}.\n\\end{proof}\n\n\n\n\\section{Burau representation}\\label{section:burau}\n\nIn this section we review a homological construction of the Burau representation; \nthis interpretation is used to relate the latter with the wall-crossing labeling.\n\n\\subsection{The Burau representation}\nFix the base point $\\ast=-(n+1) \\in \\partial \\mathbb D_{n}$ on the boundary of $\\mathbb D_n$. The fundamental group $\\pi_{1}(\\mathbb D_{n})$ is a free group of rank $n$ where the free generator $x_{i}$ is represented by a loop which rounds the $i$th puncture $p_i$ once clockwise.\nLet $\\epsilon: \\pi_{1}(\\mathbb D_{n}) \\rightarrow \\Z=\\langle q \\rangle$ be the homomorphism which sends all $x_{i}$ to the generator $q$. Geometrically, for a loop $\\gamma$, $\\epsilon([\\gamma])$ is the sum of the algebraic winding number of $\\gamma$ about the puncture points $\\{p_{i}\\}$ (in the clockwise direction).\n\nLet $\\pi: \\widetilde{\\mathbb D_{n}} \\rightarrow \\mathbb D_{n}$ be the infinite cyclic covering corresponding to $\\textrm{Ker}(\\epsilon)$, and fix a lift $\\widetilde{\\ast}$ of the base point.\nThe group of covering transformations of $\\widetilde{\\mathbb D_{n}}$ is identified with the cyclic group $\\left$.\nThen $H_{1}(\\widetilde{\\mathbb D_{n}};\\Z)$ can be endowed with a structure of $\\Z[q,q^{-1}]$-module, where multiplication by $q$ corresponds to the deck transformation. Moreover it turns out that $H_{1}(\\widetilde{\\mathbb D_{n}};\\Z)$ is free of rank $(n-1)$ as a $\\Z[q,q^{-1}]$-module. Since $\\epsilon$ is $B_{n}$-invariant, we have a linear representation \n\\[ \\rho: B_{n} \\rightarrow \\GL (H_{1}(\\widetilde{\\mathbb D_{n}};\\Z) ). \\]\nThis is called the {\\em (reduced) Burau representation}. In the rest of this section, we keep the same notation $\\epsilon$, $\\widetilde{\\mathbb D_n}$ and $\\widetilde{\\ast}$ for the above defined winding number evaluation morphism, covering space of $\\mathbb D_n$ and base point.\n\n\\subsection{Forks}\\label{section:forks}\nLet $Y$ be the $Y$-shaped graph \nconsisting of three external vertices: a distinguished one $r$, two others $v_{1}$ and $v_{2}$ and one internal vertex~$c$ and three edges relating \neach external vertex to the internal one (see Figure \\ref{fig:fork} (a)). \nWe orient the edges of $Y$ as shown in Figure \\ref{fig:fork} (a).\n\nA {\\em fork} is an embedded image of $Y$ into $\\mathbb D_n$ such that:\n\\begin{itemize}\n\\item All points of $Y- \\{r,v_1,v_2\\}$ are mapped to the interior of $\\mathbb D_n$.\n\\item The distinguished vertex $r$ is mapped to the base point $\\ast$.\n\\item The other two external vertices $v_{1}$ and $v_{2}$ are mapped to two different puncture points.\n\\end{itemize}\n\nGiven a fork $F$, the image of the edge $[r,c]$ is called the {\\em handle} of~$F$\nand the image of $[v_1,v_2]=[v_{1},c] \\cup [c,v_{2}]$, regarded as a single oriented arc, is called the {\\em tine} of $F$ and denoted by $T(F)$. The image of $c$ is called the \\emph{branch point} of~$F$.\nFigure \\ref{fig:fork} (b) shows a fork $F$ (with the handle depicted in grey line and the tine in black line).\n\n\n\n\\begin{figure}[htbp]\n \\begin{center}\n\\includegraphics*[scale=0.5, width=100mm]{fork_0_1.eps}\n \\caption{Fork and standard fork $F_i$}\n \\label{fig:fork}\n \\end{center}\n\\end{figure}\n\n\nFor a fork $F$, let $\\gamma \\co [0,1] \\rightarrow \\mathbb D_{n}$ be the handle of $F$, viewed as a path in $\\mathbb D_{n}$ and take a lift \n\\[ \\widetilde{\\gamma} \\co [0,1] \\rightarrow \\widetilde{\\mathbb D_{n}} \\]\nof $\\gamma$ so that $\\widetilde{\\gamma}(0)=\\widetilde{\\ast}$.\nLet $\\Sigma(F)$ be the connected component of $\\pi^{-1}(T(F))$ that contains the point $\\widetilde{\\gamma}(1)$. \nThe \\emph{homology class of $H_{1}(\\widetilde{\\mathbb D_n};\\Z)$ represented by} $F$ is then defined as the homology class represented by $\\Sigma(F)$. \nBy abuse of notation, we still denote this homology class by $F$.\nStrictly speaking, since $\\Sigma(F)$ is not compact we need to work with the homology of locally finite chains $H_{1}^{lf}(\\widetilde{\\mathbb D_n};\\Z)$ or $H_{1}(\\widetilde{\\mathbb D_n},\\widetilde{P};\\Z)$, where $\\widetilde{P}$ is the preimage of a small neighborhood of the punctures in $\\mathbb D_{n}$. Rigorous treatments are well-known and give rise to the same conclusions (see \\cite{bi3}, for example), so we do not take care of these subtle points.\n \n\nOf special importance is the following family of particularly simple forks: \nfor $i=1,\\ldots,n-1$, let $F_{i}$ be the fork whose tine is a straight arc connecting the $i$th and the $(i+1)$st punctures and whose handle is contained in the lower half of the disk $\\mathbb D_{n}$ (see Figure \\ref{fig:fork} (c)). \nThese are called {\\em standard forks}.\nStandard forks $F_{1}, \\ldots, F_{n-1}$ form a basis of $H_{1}(\\widetilde{\\mathbb D_{n}};\\Z)$. \nThe group $\\GL(H_{1}(\\widetilde{\\mathbb D_{n}};\\Z))$ can be identified with $\\GL(n-1;\\Z[q,q^{-1}])$ using the basis of standard forks. \nThis allows to get the familiar matrix description of the reduced Burau representation:\n\n\n\\[\n\\rho_n(\\sigma_{1}) = \\left(\n\\begin{array}{cc} \n-q & 0 \\\\\n1 & 1 \\\\\n\\end{array}\n\\right) \\oplus I_{n-3}, \\;\\;\\; \n\\rho_n(\\sigma_{n-1})=I_{n-3}\\oplus\n\\left(\\begin{array}{cc}\n1 & q\\\\\n0 &-q\\\\\n\\end{array}\n\\right),\n\\] \n\\[\n\\rho_n(\\sigma_{i}) = I_{i-2} \\oplus \n\\left(\n\\begin{array}{ccc} \n1 & q & 0 \\\\\n0 & -q & 0 \\\\\n0 & 1 & 1 \\\\\n\\end{array}\n\\right) \n\\oplus I_{n-i-2}, \\;\\;\\; (i=2,\\ldots,n-2)\n\\]\n\n\n\n\n\\subsection{The noodle-fork pairing}\nA {\\em noodle} is an embedded oriented arc in $\\mathbb D_n$ which begins at the base point $\\ast$ and ends at some point of the boundary $\\partial\\mathbb D_n$. \nNoodles represent relative homology classes in $H_1(\\widetilde{\\mathbb D_n}, \\partial \\widetilde{\\mathbb D_n};\\Z)$. \n\nThe \\emph{noodle-fork pairing} (in our notation, it should say fork-noodle pairing) is a homology intersection (algebraic intersection) pairing \n$$\\langle\\;,\\;\\rangle:H_1(\\widetilde{\\mathbb D_n} ;\\Z) \\times H_1(\\widetilde{\\mathbb D_n}, \\partial \\widetilde{\\mathbb D_n};\\Z) \\rightarrow \\Z[q,q^{-1}].$$\nGeometrically, it is computed in the following way (see \\cite{bi3} Section 4).\n\n\nGiven a fork $F$ and a noodle $N$, put $T(F)$ and $N$ transverse with minimal intersections.\nLet $z_1,\\ldots,z_r$ be the intersection points. Each intersection point $z_{i}$ then contributes a monomial $\\varepsilon_{i}q^{e_i}$ to $\\langle N,F\\rangle$, where $\\varepsilon_i$ is the sign of the intersection between $T(F)$ and $N$ at $z_{i}$ and $e_i$ is an integer.\nThe noodle-fork pairing is then given by\n\\[ \\langle F,N\\rangle = \\sum_{1 \\leq i \\leq r} \\varepsilon_{i}q^{e_i} \\in \\Z[q,q^{-1}]. \\]\n\nThe integer $e_i$ is computed as follows. Let $\\gamma_i$ be the loop which is the composition of three paths $A$, $B$ and $C$ in $\\mathbb D_{n}$:\n\\begin{itemize}\n\\item $A$ is a path from $\\ast$ to the branch point of~$F$ along the handle of~$F$.\n\\item $B$ is a path from the branch point of $F$ to $z_{i}$ along the tine $T(F)$.\n\\item $C$ is a path from $z_{i}$ to $\\ast$ along the noodle $N$.\n\\end{itemize}\nThen $e_{i} = \\epsilon([\\gamma_i])$: that is, $d_i$ is the sum of the winding numbers of the loop $\\gamma_i$ about the puncture points $p_1,\\ldots,p_n$.\n\nAs for forks, we define a distinguished family of noodles: for $i=1,\\ldots,n-1$, the \\emph{standard noodle} $N_i$ is the noodle which has empty intersection with the walls and ends at some boundary point between $W_{i}$ and $W_{i+1}$. \nGiven a braid $x$, the entries of its Burau matrix can be computed using the noodle-fork pairing in a fairly direct manner.\n\n\n\\begin{lemma}[Burau Matrix formula]\n \\label{lemma:Buraumatrix}\nLet $x\\in B_n$. \nThen for $1\\leqslant i,j\\leqslant n$, the entry $\\rho_n(x)_{ij}$ of its Burau matrix is given by $\\rho_n(x)_{ij}=\\langle (F_{i})x,N_{j} \\rangle$.\n\\end{lemma}\n\\begin{proof}\nBy definition, $(F_i)x = \\sum_{k=1}^{n-1} F_k \\rho_n(x)_{ik} \\in H_{1}(\\widetilde{\\mathbb D_n};\\Z)$, hence for $i,j \\in 1,\\ldots,n-1$ we have\n\\[ \\langle (F_i)x , N_j \\rangle = \\sum_{k=1}^{n-1} \\langle F_{k},N_{j} \\rangle \\rho_n(x)_{ik}\\]\nIt is directly checked that $\\langle F_{k},N_{j}\\rangle = \\delta_{kj}$ (Kronecker's delta) hence\n\\[ \\langle (F_{i})x,N_{j} \\rangle = \\rho_n(x)_{ij}. \\]\n\\end{proof}\n\n\\begin{example}\nAs an example of application of Lemma \\ref{lemma:Buraumatrix}, we can retrieve the Burau matrices associated to Artin generators $\\sigma_i$. First, we notice that for $k=1,\\ldots,n-1$, $i\\neq k-1,k,k+1$, $(F_i)\\sigma_k=F_i$, so that $\\langle (F_i)\\sigma_k,N_j\\rangle=\\delta_{i,j}$.\nFor the remaining values of $i$, Figure \\ref{fig:example} shows the images $(F_i)\\sigma_k$. \n\\begin{figure}[htbp]\n \\begin{center}\n\\includegraphics*[scale=0.5, width=100mm]{Example.eps}\n \\caption{On the left part, forks $F_{k-1}$, $F_k$ and $F_{k+1}$ and on the right part, their images under the action of the braid $\\sigma_k$; relevant noodles are depicted in dashed lines.}\n \\label{fig:example}\n \\end{center}\n\\end{figure}\n\nWith the help of Figure \\ref{fig:example} we can conclude:\n\n\\begin{tabular}{lr}\n{$\\langle (F_{k-1})\\sigma_k,N_j\\rangle=\\begin{cases} \n0 & {\\text{if}}\\ ji,\\\\\n1 & {\\text{if}}\\ j=k-1,\\\\\nq & {\\text{if}}\\ j=k.\\\\\n\\end{cases}$} & \n\n{$\\langle (F_{k+1})\\sigma_k,N_j\\rangle=\\begin{cases} \n0 & {\\text{if}}\\ jk+1,\\\\\n1 & {\\text{if}}\\ j=k,\\\\\n1 & {\\text{if}}\\ j=k+1.\\\\\n\\end{cases}$} \\\\\n\\end{tabular}\n\n$$\\langle (F_{k})\\sigma_k,N_j\\rangle=\n\\begin{cases} \n0 & {\\text{if}}\\ j\\neq k,\\\\\n-q & {\\text{if}}\\ j=k .\\\\\n\\end{cases}\n$$\nLemma \\ref{lemma:Buraumatrix} then allows to retrieve the matrices given at the end of Section \\ref{section:forks}.\n\n\n\\end{example}\n\n\\subsection{Noodle-fork pairing and wall-crossing labeling}\nWe finally review a connection between the integers $e_i$ in the computation of the noodle-fork pairing and the wall-crossing labeling. \nThis will yield the expected relation between the Burau representation and the wall-crossing labeling. \n\nLet $x\\in B_n$. First we recall how to assign wall-crossing labelings for points belonging to the image $(F_i)x$ of the standard fork $F_i$ under $x$. \nLet us consider the part of the curve diagram $D_{x}$ that is the image of $E_{i}$ (the line segment between the $i$-th and $(i+1)$-st punctures). We identify this part $(E_{i})x$ of the curve diagram with $(T(F_{i}))x$. Moreover, a part of the modified curve diagram can naturally be regarded as the handle of $(F_{i})x$, as shown in Figure~\\ref{fig:cdtofork}. This identification induces the wall crossing labeling on each connected component of $(F_{i})x- (W\\cup U)$. For a point $z \\in (F_i)x- (W\\cup U)$ we denote by $\\Wcr_x(z)$ the corresponding label. \n\n\\begin{figure}[htbp]\n \\begin{center}\n\\includegraphics*[scale=0.5, width=80mm]{cdtofork_0_1.eps}\n\\caption{Viewing a curve diagram as a union of tines of forks, and viewing initial segments of modified curve diagrams as handles.}\n\\label{fig:cdtofork}\n\\end{center}\n\\end{figure}\n\nLet $N$ be a noodle; we may assume that no intersection point in $(T(F_i))x \\cap N$ belongs to $W\\cup U$. \n\n\\begin{lemma}\n\\label{lemma:exponent}\nFix an intersection point $z\\in (T(F_i))x \\cap N$.\nLet $c(z)$ be the algebraic intersection number of $W$ and the path $C$ in the definition of the pairing $\\langle (F_i)x,N\\rangle$\n(i.e. $C$ is a path from $z$ to $\\ast$ along $N$). Let $e(z)$ be the degree of $q$ in the $z$-contribution to $\\langle (F_i)x,N\\rangle$.\nThen\n\\[ e(z) = \\Wcr_x(z) + c(z)\\]\n\\end{lemma}\n\\begin{proof}\nLet $A$ and $B$ be the paths in the definition of the pairing $\\langle (F_i)x,N\\rangle$. Then $\\Wcr(z)$ is nothing but the algebraic intersection number of $W$ and the composite path $BA$. Hence the algebraic intersection number of $W$ and the loop $\\gamma = CBA$ is $\\Wcr_x(z) + c(z)$, which is, by definition, equal to $e(z)= \\epsilon(\\gamma)$.\n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:ineqn}\nFor any braid $x\\in B_n$, the following inequality holds:\n$$M(\\rho_n(x))\\leqslant \\sup \\!{}_{\\sf d}(x).$$\n\\end{corollary}\n\\begin{proof}\nFor we have, by definition and thanks to Lemma \\ref{lemma:Buraumatrix},\n$$M(\\rho_n(x)) = \\max_{i,j}\\{ M(\\langle (F_{i})x,N_{j} \\rangle) \\}.$$\nFor a standard noodle $N_j$ and a point $z\\in (T(F_{i}))x \\cap N_j$ the integer $c(z)$ in Lemma \\ref{lemma:exponent} is always $0$ because standard noodles do not intersect with walls.\nTherefore we have \n$$\\max_{i,j}\\{ M (\\langle (F_{i})x,N_{j} \\rangle) \\}\\leq \\LWcr(x)$$\nand finally, as $\\LWcr(x) = \\sup_{\\sf d} (x)$ (Theorem \\ref{theorem:wall}) we are done.\n\\end{proof}\n\n\n\n\n\\section{Braids whose Burau Matrix detects the dual Garside normal forms}\\label{section:dual}\n\nIn view of Corollary \\ref{cor:ineqn}, a natural question is to ask when the converse inequality holds. Theorem \\ref{theorem:main_dual}\nwill give a sufficient condition for the maximal degree appearing in the Burau matrix of a braid to be equal to its dual supremum. Actually we will prove more: under the same condition, it is possible to determine the dual normal form from the Burau matrix.\n\nTo state Theorem \\ref{theorem:main_dual} we first introduce the notion of simply-nestedness as a refinement of the left-weightedness condition (Proposition \\ref{prop:left-weighted}), which will allow us to get a better control on the action of a braid in dual normal form. \n\nLet $d,d'$ be two dual simple elements, expressed as products of disjoint polygons $d=P_1\\ldots P_r$ and $d'=Q_1\\ldots Q_{r'}$ respectively. \nWe say that the ordered pair $(d,d')$ is \\emph{simply-nested} if for any polygon $Q$ among $Q_{1},\\ldots, Q_{r'}$, there exists a {\\em unique} polygon $P$ among $P_1,\\ldots, P_{r}$ such that for any two vertices $i,j$ of $Q$, the polygon $P$ has two vertices $k,l$ such that \n$a_{k,l}\\vdash a_{i,j}$. \nA braid $x$ will be said to be \\emph{simply-nested} if each pair of consecutive factors in its dual normal form is simply-nested.\n\nLet $B_{n}^{\\sf sn}$ be the set of simply nested $n$-braids.\nAlthough $B_{n}^{\\sf sn}$ does not form a group, $B_{n}^{\\sf sn}$ is a regular language over the alphabet $[1,\\delta]$.\nWe also remark that $B_{n}^{\\sf sn}$ is not symmetric: $x \\in B_{n}^{\\sf sn}$ does not imply $x^{-1} \\in B_{n}^{\\sf sn}$. A simple example is the 4-braid $x=(a_{3,4})(a_{2,4})$. Although $x$ is simply-nested, $N_{\\sf d}(x^{-1}) = \\delta^{-2}(a_{1,2}a_{3,4})(a_{1,2}a_{1,4})$ which is not simply nested.\n\n\nWe now can state our second main result:\n\n\\begin{theorem}\n\\label{theorem:main_dual}\nLet $x \\in B_n$ be a simply-nested braid.\n\\begin{enumerate}\n\\item[(i)] $\\sup_{\\sf d}(x) = M(\\rho_n(x))$.\n\\item[(ii)] One can compute the dual normal form from the matrix $\\rho_{n}(x)$, so the restriction of the Burau representation on the set of simply-nested braid $B_{n}^{\\sf sn}$ is injective.\n\\end{enumerate} \n\\end{theorem}\n\nFor a braid $x$ and $i=1,\\ldots,n-1$ let $\\mathcal M_{x}(E_i)$ be the set of the arc segments of $(E_i)x$ whose wall-crossing labeling attains the maximal value $\\LWcr(x)$ (possibly empty). We say that two arc segments in the curve diagram are \\emph{parallel} if both are described by $\\wideparen{(ij)}$ for some $i,j$. \nWe consider the following property \\textbf{(C)} (\\emph{Coherence property}) for a braid $x$:\n\\begin{definition}\nLet $x\\in B_n$ and $N_{\\sf d}(x) = \\delta^{p}d_1\\cdots d_r$. Express $d_{r}$ as a product of disjoint polygons: $d_{r}=Q_{1}\\cdots Q_{b}$.\nWe say that \\emph{$x$ has the property} \\textbf{(C)} if for each $i=1,\\ldots,n-1$, any two arc segments $\\alpha$ and $\\alpha'$ in $\\mathcal {M}_{x}(E_i)$ intersecting a common polygon $Q \\in \\{Q_{1},\\ldots,Q_{b}\\}$ are parallel and have the same orientation.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lemma:CondC}\nIf $x$ has the property \\textbf{(C)}, then $\\sup_{\\sf d}(x)= M(\\rho_n(x))$ holds.\n\\end{lemma}\n\n\n\\begin{proof\nLet $N_{\\sf d}(x) = \\delta^{p}d_1\\cdots d_r$. \nTake $i$ so that $\\mathcal{M}_{x}(E_i)$ is non-empty. Take the minimal number $k$ so that there exists an arc segment $\\alpha = \\wideparen{(kp)} \\in \\mathcal{M}_{x}(E_i)$ for some $p>k$. We look at the entry $\\rho_n(x)_{ik}$ in the Burau matrix of $x$, which is equal to $\\langle (F_i)x, N_{k}\\rangle$ by Lemma \\ref{lemma:Buraumatrix}. In view of Corollary \\ref{cor:ineqn} and Theorem \\ref{theorem:wall}, the desired equality will be shown provided $M_q(\\langle (F_i)x,N_{k}\\rangle)=\\LWcr(x)$. \n\nLet $\\alpha'$ be another arc segment in $\\mathcal{M}_{x}(E_i)$ which intersects the noodle $N_k$. By minimality of $k$, $\\alpha'=\\wideparen{(ku)}$ for some $u\\in (k+1,n)$. By Theorem \\ref{theorem:finalfactor},\nsome polygon $Q$ in the decomposition of $d_r$ \nhas vertices $k,p,u$; both arcs $\\alpha$ and $\\alpha'$ intersect $Q$. Hence by property \\textbf{(C)}, $\\alpha$ and $\\alpha'$ are parallel with the same orientation (notice that, in particular, $u=p$ holds). \nThis shows that all arcs in $\\mathcal M_{x}(E_i)$ intersecting the noodle $N_k$ have the same sign of intersection so $M_q (\\langle (F_i)x,N_{k}\\rangle)=\\LWcr(x)$. \n\\end{proof}\n\n\\begin{lemma}\\label{lemma:SNimplies_C}\nLet $x\\in B_n^{\\sf sn}$. Then $x$ has Property {\\rm{\\textbf{(C)}}}. \n\\end{lemma}\n\\begin{proof}\nThe proof is by induction on the number $r$ of non-$\\delta$ factors in the dual normal form of $x$. \nThe case $r=1$ is checked by direct calculation. Actually, in this case $(E_j)x$ has at most one maximal labeled arc for any $j$. \n\nSuppose $N_{\\sf d}(x)=\\delta^{p}d_1\\cdots d_r$ with $r>1$. Then $x'=\\delta^{p} d_1\\cdots d_{r-1}$ is also simply-nested and has the Property \\textbf{(C)} by induction hypothesis. Let us express\n$d_{r-1}$ and $d_{r}$ as products of disjoint polygons: $d_{r-1}=P_{1}\\cdots P_{b'}$ and $d_{r}=Q_{1}\\cdots Q_{b}$.\n\nFor $f=1,\\ldots,n-1$, suppose that $\\alpha=\\wideparen{(ij)}$ and $\\alpha'=\\wideparen{(i'j')}$ are two arcs in $\\mathcal{M}_{x}(E_f)$ that intersect a common polygon $Q \\in \\{Q_{1},\\ldots,Q_{b}\\}$. \nBy Theorem \\ref{theorem:finalfactor}, all of $i,i',j,j'$ are vertices of $Q$.\nFollowing the proof of Theorem \\ref{theorem:finalfactor} we can find arcs $\\beta=\\wideparen{(kl)},\\beta'=\\wideparen{(k'l')}$ in the diagram $D_{x'}$ with label $\\Wcr(x)-1$ ($\\beta,\\beta'\\in \\mathcal M_{x'}(E_f)$) and $a_{k,l}\\vdash a_{i,j}$ and $a_{k',l'}\\vdash a_{i',j'}$. Moreover we can choose $\\beta,\\beta'$ so that $\\alpha$ and $\\alpha'$ come from $\\beta$ and $\\beta'$ respectively under the action of $Q$ (see Figure \\ref{Fig:LW} (a)). \n\nBy simply-nestedness assumption, $k,l,k',l'$ must be vertices of a common polygon $P \\in \\{P_{1},\\ldots,P_{b'}\\}$. This implies that both $\\beta$ and $\\beta'$ intersect with the same polygon $P$, hence by Property \\textbf{(C)} for $x'$, the arc segments $\\beta$ and $\\beta'$ are parallel with the same direction. Therefore the same property holds true for $\\alpha$ and $\\alpha'$, as we wanted to show.\n\\end{proof} \n \n\\begin{remark}\nWe observe that, although it is a stronger property, simply-nestedness is fairly easy to check whereas checking Property {\\bf{(C)}} directly is often a hard task since we need to know both dual normal form and the curve diagram of braids.\n\\end{remark}\n\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:main_dual}]\nLemmas \\ref{lemma:CondC} and \\ref{lemma:SNimplies_C} show part (i).\n\n\nWe explain how to compute the final factor $d_{r}$ of the dual normal form of $x$, which gives an algorithm to compute the whole dual normal form of $x$ from its Burau matrix. Let $N_{\\sf d}(x)=\\delta^{p}d_{1}\\cdots d_{r}$ and write $d_{r}$ as a product of disjoint polygons: $d_{r}=Q_{1}\\cdots Q_{b}$.\n\n\nOur strategy to determine $d_{r}$ is as follows.\nWe show how to find some $a_{i,j}$ satisfying $a_{i,j} \\preccurlyeq_{\\sf d} d_{r}$ from $\\rho_{n}(x)$. Since $d_{r}$ is written as a product of at most $(n-2)$ letters $a_{i,j}$, by iterating this procedure at most $(n-2)$ times, we eventually determine $d_{r}$.\n\nFor $i=1,\\ldots,n-1$, let $M^{c}_{i}(x)=\\max\\{\\rho_n(x)_{ji}\\: | \\: j=1,\\ldots,n-1\\}$, namely, the maximal degrees of the variable $q$ in the $i$-th \\emph{column} of the Burau matrix of $x$ (do not confuse $M_{i}(x)$ in Section \\ref{section:classical}, where we used the maximal degrees of the $i$-th \\emph{row}. First we show that $M^{c}_{i}(x)$ gives candidates of $a_{i,j}$ satisfying $a_{i,j}\\preccurlyeq_{\\sf d} d_{r}$.\n\n\\begin{claim}\n\\label{claim:1p}\nWe have $$\\min\\{i\\in \\{1,\\ldots,n\\}\\: | \\: \\exists j,a_{i,j}\\preccurlyeq_{\\sf d}d_r\\}=\n\\min\\{i\\in \\{1,\\ldots,n-1\\}\\: | \\: M^{c}_i(x)=M(\\rho_n(x))\\}.$$ \n\\end{claim}\n\\begin{proof}\nLet $i_0=\\min\\{i\\in \\{1,\\ldots,n\\}\\: | \\: \\exists j,a_{i,j}\\preccurlyeq_{\\sf d}d_r\\}$.\nLet $k>i_0$ be such that $a_{i_0,k}\\preccurlyeq d_r$. \n\nFirst, we show that $M_{i_0}^{c}(x)=M(\\rho_n(x))$. \nSince $a_{i_0,k}\\preccurlyeq d_r$ and by Theorem \\ref{theorem:finalfactor} there must exist some $p\\in\\{1,\\ldots,n\\}$, $p>i_0$, such that $D_x$ admits an arc $\\alpha=\\wideparen{(i_0p)}$ labeled $\\LWcr(x)=M(\\rho_n(x))$. Let also $Q\\in\\{Q_1,\\ldots,Q_b\\}$ having vertices $i_0,p,k$ and let $j$ be such that $\\alpha\\in\\mathcal M_x(E_j)$.\nWe observe that $\\alpha$ intersects the noodle $N_{i_0}$. We will show that $M(\\rho_n(x)_{j,i_{0}})=M(\\rho_n(x))$.\nIndeed, let $\\alpha'\\in \\mathcal M_x(E_j)$ and suppose that $\\alpha'$ intersects $N_{i_0}$. By minimality of $i_0$, $\\alpha'$ must intersect with the polygon $Q$ and by Property {\\bf{(C)}}, $\\alpha'$ is parallel to $\\alpha$ with the same orientation. Hence the $\\alpha$ and $\\alpha'$ intersect with $N_{i_0}$ with the same sign. \nTherefore $M(\\langle (F_j)x,N_{i_0}\\rangle)=M(\\rho_n(x))$ as we wanted to show. \n\nSecond, we show that for $i e(d_{r'+1}\\cdots d_{r})$,\n\\end{enumerate}\nthen $\\rho_n(x)\\neq 1$.\nMoreover the condition (ii) is always satisfied if $r' > \\frac{n-2}{n-1}r$.\n\\end{corollary}\n\\begin{proof}\nPut $E=e(d_{r'+1}\\cdots d_{r})$. Assume contrary, $\\rho_n(x)=1$.\nSince $e(d_{i})\\leqslant(n-2)$, we have \n$ 0 = e(x) \\leqslant (n-1)p + (n-2)r' + E $ so $-p \\leqslant \\frac{1}{n-1}((n-2)r'+ E)$.\nOn the other hand, by (i)\n\\[ 0=M(\\rho_n(x)) = M(\\rho_{n}(\\delta^{p}d_{1}\\cdots d_{r'}) \\rho_n(d_{r'+1} \\cdots d_{r})) \\geqslant M(\\rho_{n}(\\delta^{p}d_{1}\\cdots d_{r'}) )= p+r' \\]\nhence $r' \\leqslant -p$. Therefore $r' \\leqslant \\frac{1}{n-1}((n-2)r'+ E)$, which is equivalent to $r' \\leqslant E$. This contradicts to (ii). The last assertion follows from the inequality $E \\leqslant (n-2)(r-r')$. \n\\end{proof}\n\n\n\nWe close this section by looking at some known examples of elements in the kernel of the Burau representations $\\rho_5$ and $\\rho_6$. \n\nConsider the braids $$x=[v_2^{-1}v_1\\sigma_3v_1^{-1}v_2,\\sigma_3] \\in B_6,$$ where $v_1=\\sigma_1\\sigma_2^{-1}\\sigma_5^{-1}\\sigma_4$ and \n$v_2=\\sigma_1^{-2}\\sigma_2\\sigma_5^{2}\\sigma_4^{-1}$ and \n$$y=[w_1^{-1}\\sigma_4w_1,w_2^{-1}\\sigma_4\\sigma_3\\sigma_2\\sigma_1\\sigma_1\\sigma_2\\sigma_3\\sigma_4w_2]\\in B_5,$$ where \n$w_1=\\sigma_3^{-1}\\sigma_2\\sigma_1^{2}\\sigma_2\\sigma_4^{3}\\sigma_3\\sigma_2$ and $w_2=\\sigma_4^{-1}\\sigma_3\\sigma_2\\sigma_1^{-2}\\sigma_2\\sigma_1^{2}\\sigma_2^{2}\\sigma_1\\sigma_4^{5}$. \n\nIt is known that $\\rho_5(y)=Id$ and $\\rho_6(x)=Id$. \nThe following are dual normal forms of a conjugate $x'$ and $y'$ of $x$ and $y$, respectively: \n\n$$N_{\\sf d}(x')=\\delta_6^{-6}(a_{1,6}a_{4,5})(a_{1,6}a_{2,5})(a_{1,6}a_{4,6}a_{2,3})(a_{1,5}a_{4,5}a_{2,3})(a_{3,6}a_{4,5})(a_{1,6}a_{2,5}a_{4,5})(a_{1,6}a_{3,5})(a_{1,6}a_{5,6}a_{2,4})$$\n$$(a_{1,3}a_{5,6})(a_{2,4}a_{5,6})(a_{1,3}a_{5,6}a_{4,5})(a_{2,6}a_{4,5})(a_{1,3}),$$\n\n\n$$N_{\\sf d}(y') =\\delta_5^{-23} (a_{2,5}a_{4,5})(a_{1,5}a_{3,5})(a_{1,4}a_{3,4})(a_{2,5})(a_{1,5}a_{2,3})(a_{1,5}a_{3,4})^{2}(a_{1,3})(a_{2,5}a_{3,4})$$\n$$(a_{1,4}a_{3,4})(a_{1,2}a_{1,4})(a_{1,2}a_{3,5})(a_{1,2}a_{1,5}a_{3,4})(a_{1,5})(a_{1,2})(a_{2,3})(a_{3,4})$$\n$$(a_{2,4})(a_{1,3}a_{4,5})(a_{1,2}a_{4,5})(a_{2,3}a_{4,5})(a_{1,3}a_{4,5})(a_{1,2}a_{3,5}a_{4,5})(a_{2,5}a_{3,5})$$\n$$(a_{1,3}a_{1,4})(a_{1,2}a_{1,4})(a_{1,2}a_{1,3}a_{4,5})(a_{1,2}a_{4,5})(a_{2,3}a_{4,5})^{2}(a_{2,5})(a_{1,4}a_{2,3})$$\n$$(a_{2,5}a_{3,5})(a_{1,5}a_{3,5})(a_{1,5}a_{2,4})(a_{1,5}a_{4,5}a_{2,3})(a_{1,5}a_{3,5}a_{2,3})^{4}(a_{2,4})(a_{1,3}a_{4,5})$$\n$$(a_{1,2}a_{4,5})(a_{2,3}a_{4,5})(a_{1,3}a_{4,5})(a_{1,2}a_{4,5}a_{3,4}).$$\n\n\nSee Figure \\ref{fig:NormalForm6} for pictorial (polygon) expression of $N_{\\sf d}(x')$. One notices that $N_{\\sf d}(x')$ contains many non-simply-nested pairs. Similarly, one observes that $N_{\\sf d}(y')$ also contains a lot of non-simply-nested pairs.\nThese examples and our results on simply-nested braid suggest the Burau matrix of a braid $x$ is close to be the identity matrix only when its dual normal form contains many non-simply nested pairs.\n\n\\begin{figure}[htbp]\n \\begin{center}\n\\includegraphics*[scale=1.4]{NormalForm.eps}\n\\caption{A pictorial way to represent the dual normal form of the braid $x'\\in B_6$ (the puncture rounded by a circle is the first); the symbol $\\ast$ represents non simply-nested pairs.}\n\\label{fig:NormalForm6}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgoim b/data_all_eng_slimpj/shuffled/split2/finalzzgoim new file mode 100644 index 0000000000000000000000000000000000000000..8a6696f401ffd5c9e2f82985486a6f1dfa984dcc --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgoim @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\nThe interplay of topology and interactions is at the heart of the condensed matter community's ongoing fascination with moir\\'e materials \\cite{macdonald2019bilayer,andrei2020graphene,balents2020superconductivity}. Not only do they provide a platform to engineer band structures of minimal bandwidth, dramatically increasing the importance of electron-electron interactions, but they can also be set up to host topologically non-trivial Chern bands --- paving the way for the emergence of novel ordered phases. \nTwisted double-bilayer graphene (TDBG) \\cite{Shen2020,Liu2020,Cao2020} is composed of four stacked layers of graphene, where the top and bottom pairs are in a Bernal configuration while the middle ones are twisted by a small angle $\\theta$, typically around ${\\theta \\simeq 1.3^{\\circ}}$, which results in a long-range moir\\'e pattern with a period of about ${10\\,\\mathrm{nm}}$. Similarly, twisted mono-bilayer graphene (TMBG) \\cite{Polshyn2020} is engineered by slightly twisting a Bernal stack relative to a single graphene sheet, totalling in three graphene layers. \nAs with other 2D moir\\'e systems, the density of electrons may be efficiently tuned via electrostatic doping.\nHowever, TDBG and TMBG stand out in the landscape of moir\\'e materials due to their lack of $C_2$ inversion symmetry, which leads to finite Berry curvature in a given valley. It also allows to induce a gap at charge neutrality via an external displacement field and stabilize an energetically isolated conduction band in each valley with Chern number ${\\abs{C} = 2}$.\nAt an integer number of filled moir\\'e bands, these systems were experimentally found to exhibit a tendency towards polarization of the spin and valley degrees of freedom \\cite{Shen2020,Liu2020,Cao2020,Burg2019, He2021a,Chen2021,Polshyn2020}. \nIn addition, TDBG was found to support a nesting induced correlated electron-hole state at a slightly higher twist angle, where the carrier concentration could be tuned individually in the top and bottom bilayers \\cite{Rickhaus2021}.\n\\begin{figure}[!ht]\n\t\\includegraphics[width=.95\\linewidth]{figs\/diagram.png}\n\t\\caption{(a) Phase diagram of twisted double-bilayer graphene at ${\\nu = 1\/2}$ as a function of the moir\\'e relaxation ratio ${w_{AA}\/w_{AB}}$ as well as $\\eta$ controlling the interaction strength, based on the combined data from exact diagonalization and Hartree-Fock. As illustrated in (b), the system realizes a tetrahedral antiferromagnetic (TAF) spin texture in a large parameter regime. At large values of ${w_{AA}\/w_{AB}}$, a ferromagnetic topological charge density wave (TCDW) with an approximate O$(3)$ symmetry emerges. Both types of order break time-reversal symmetry and exhibit a finite, quantized Hall conductance. \n\t(c) Illustration of the O$(3)$ TCDW manifold with exemplary anisotropies. \n\tMinor cubic and quartic anisotropies select between even (E), odd (O) and nematic (N) realizations of charge order.\n\t\\label{fig:phase_diagram}}\n\\end{figure}\nUpon doping away from integer filling, electronic nematic order was observed in TDBG \\cite{Rubio2022}. Moreover, signatures of symmetry broken Chern insulators (SBCI) with ${\\abs{C} = 1}$ at conduction band filling $\\nu=7\/2$ in both TDBG \\cite{He2021} and TMBG \\cite{Polshyn2022} highlight the intertwining of topology and translational symmetry breaking order at non-integer filling fractions in these moir\\'e systems.\nCorrelated states at half-integer band fillings are also seen in experiments on related graphene based moir\\'e materials \\cite{Bhowmik2022,Siriviboon2021,Xie2021a}.\n\n\nIn this paper we numerically study the emergence of correlated phases of strongly interacting fermions at half-integer fillings of a realistic model for the conduction band of TDBG and TMBG. More specifically, we focus on fillings ${\\nu=N_e\/N=1\/2}$ as well as ${\\nu=7\/2}$, where $N_e$ is the number of electrons living on the lattice with $N$ moir\\'e unit cells. For this purpose, we use extensive exact diagonalization (ED), fully accounting for quantum fluctuations, complemented by unrestricted Hartree-Fock (HF) calculations, where the ansatz for the HF order parameter is guided by the unbiased ED results. \nWe comprehensively demonstrate that the phase diagrams of both systems, exemplarily presented in \\csubfig{phase_diagram}{a} for TDBG, are dominated by a tetrahedral antiferromagnetic (TAF) phase, characterized by a non-coplanar spin texture and finite, periodic skyrmion density, which evolves into an almost O(3) degenerate manifold of ferromagnetic, topological charge density waves (TCDW). The real-space texture of these phases is illustrated in \\csubfig{phase_diagram}{b}.\nRelated TAF phases were the subject of previous numerical studies in the Mott limit of localized magnetic moments of extended Heisenberg models with a scalar chirality term \\cite{Hickey2016,Wietek2017} as well as investigations of weak-coupling nesting instabilities on the triangular lattice at the level of mean-field and perturbation theory \\cite{Martin2008,Akagi2010,Akagi2012,Hayami2014}. In a comparable scenario, the phase appeared close to the van Hove energy of a topologically trivial band at relatively weak interactions in an effective moir\\'e-scale Hubbard model for transition metal dichalcogenides \\cite{TAFHubbard}.\nConversely, we establish the importance of this non-coplanar spin texture to the strong-coupling regime of half-integer filled topological flat bands of the continuum model of TDBG and TMBG; we apply the full machinery of ED to directly verify its level structure corresponding to spontaneously broken SU(2) symmetry. \nNote that, while long-wavelength skyrmions have recently been studied \\cite{Khalaf2021,Christos2020,2020arXiv201001144C,Kwan2021,2021arXiv210602063C} as excitations of the correlated insulators in twisted graphene systems with $C_2$ symmetry, due their potential relevance for pairing in the presence of Dirac cones, we find a moir\\'e-scale skyrmion lattice as a ground state away from integer filling.\n\nIn addition, we uncover its potentially intimate relationship with the emergence of a continuous (approximate) symmetry leading to a degenerate manifold of distinct ferromagnetic charge density waves (CDWs). These CDWs can be described by a three-component real order parameter ${\\boldsymbol{\\phi}=(\\phi_1,\\phi_2,\\phi_3)^T}$, with $\\phi_j$ associated with charge-density modulation with wave vector given by the three moir\\'e \\textbf{M} points, and the approximate symmetry corresponds to an arbitrary O(3) rotation of these components [cf. \\csubfig{phase_diagram}{c}].\nThe identification of such \\emph{hidden symmetries} allows for a more profound understanding of the dominant physics in integer-filled flat bands of twisted bilayer graphene \\cite{Kang2019,Bultinck2020} and hence may be critical for a coherent picture of correlated phases at fractional fillings of moir\\'e Chern bands as well.\nBy assessing an exceptional level of agreement between the ED and HF methods, with a total weight of $> 96\\%$ of the HF state in the ED ground state manifold on accessible sizes, we deduce the many-body ground state to be surprisingly close to a Slater determinant. This correspondence is unexpected since, in contrast to previous findings on twisted bilayer graphene \\cite{Soejima2020,Xie2021}, we operate at fractional fillings of a $\\abs{C}=2$ Chern band with translational symmetry broken ground states and, to date, no exactly solvable limits are known which could explain their product state-nature.\nOur finding of an insulating ferromagnetic, topological charge density wave with Chern number $\\abs{C}=1$ readily explains the experimental observation of SBCI signatures in TMBG of \\cref{Polshyn2022}.\n\nThe formation of a tetrahedral antiferromagnet, which constitutes a crystallized form of magnetic skyrmions, holds promise for the future exploration of moir\\'e spintronics. \nIt further highlights that moir\\'e superlattices without $C_2$ symmetry and resultant Berry curvature are ideally suited to stabilize magnetic orders with finite scalar spin chirality. Due to the non-coplanar and likely strongly frustrated nature of the magnetic order, these systems could even host more exotic phases like chiral spin liquids. \nFurthermore, the emergent (approximate) continuous O(3) symmetry in the space of topological CDWs we uncover does not only shed new light on the ground state selection as a delicate perturbation to a more general set of charge ordered phases but also leads to an exotic form of (pseudo) Goldstone modes.\n\n\\begin{figure}[!tb]\n\t\\includegraphics[width=.95\\linewidth]{figs\/band_structure.png}\n\t\\caption{(a) Non-interacting band structure of the (${\\tau = -}$) valley TDBG continuum model near the charge neutral Fermi energy and (b) inhomogeneity of the conduction band Berry curvature [defined in \\ceqn{rmse_berry_curvature}] as a function of the moir\\'e relaxation ratio ${w_{AA}\/w_{AB}}$, see \\capp{moire} for parameters used. The red bands in (a) have Chern number $\\abs{C}=2$. Their energetic minimum is shifted to zero for illustration purposes.\n\t\\label{fig:band_structure}}\n\\end{figure}\n\n\\begin{figure*}[!htb]\n\t\\includegraphics[width=\\textwidth]{figs\/sp_ed_competition.png}\n\t\\caption{ED Results: Valley (a) and spin (b) polarization extracted from the lowest energy levels from ED provide evidence for the transition from a time-reversal symmetry breaking singlet (no SP) to a fully spin polarized (SP) state near ${w_{AA}\/w_{AB} \\simeq 0.7}$. (c) ED spectrum along the ${\\eta = 0}$ (flat band) slice in (b), corresponding to a pure interaction Hamiltonian. The low lying levels below {2\\,meV} encode the \"tower of states\", a clear spectral signature for the pattern of continuous symmetry breaking discussed in the main text. \n\t(d)--(e) In addition to ${\\nu=1}$ (orange shaded region), the prominent plateau at $\\nu=1\/2$ band filling (blue shaded region) as a function of the calculated chemical potential [defined in \\ceqn{chemical_potential}] points to an incompressible, insulating phase at $\\nu=1\/2$ for both (d) $w_{AA}\/w_{AB} = 0.0$ as well as (e) $w_{AA}\/w_{AB} = 0.9$. \n\tThe used simulation clusters are (a) C12, (b)--(c) C16 and (d)--(e) disclosed in the legend of (d). Square\/triangular\/diamond shaped markers in (d) and (e) indicate data from spin and valley resolved\/valley polarized\/spin and valley polarized simulations.\n\t\\label{fig:sp_ed_competition}}\n\\end{figure*}\n\nThe remainder of this paper is organized as follows: Section~\\ref{sec:model} introduces the band-projected interaction Hamiltonian and we briefly describe the applied numerical methods. In \\csec{ed_competition} we establish the emergence of spontaneous antiferromagnetic and ferromagnetic translational symmetry broken phases at ${\\nu=1\/2}$.\nTo draw the connection to the quantum Hall literature and previous studies of moir\\'e Chern bands, we perform a comprehensive analysis of ED spectra and structure factors measured on the exact finite size ground state of the normal ordered interaction Hamiltonian. We provide an intuitive picture of the states and describe the close connection to Hartree-Fock states in \\csec{hf} before presenting the phase diagram of the realistic models for TDBG and TMBG with particle-hole symmetric interactions at $\\nu=7\/2$ in \\csec{exp} and assessing its agreement with the experimental signatures reported in \\cref{Polshyn2022} and \\cref{He2021}.\n\n\n\\section{Model and methods}\n\\label{sec:model}\n\nOur numerical simulations of TDBG and TMBG are based on their respective continuum model description \\cite{Koshino2019,2019arXiv190500622M}, which we briefly recapitulate in \\capp{moire}, supplemented by a screened density-density Coulomb interaction. The introduction of the moir\\'e modulation creates an effective triangular lattice, where the new length scale is determined by the moir\\'e period ${L^{\\mathrm{M}} \\simeq 10\\,\\mathrm{nm}}$. As displayed in red in \\csubfig{band_structure}{a} for TDBG, near a twist angle of ${\\theta = 1.3^{\\circ}}$ a comparatively flat, spin-degenerate, isolated conduction band may be engineered in each valley by tuning an electric displacement field perpendicular to the graphene layers. For the parameters chosen, this band has a finite Chern number ${\\abs{C} = 2}$ and consequently a topologically non-trivial character. While other configurations of TDBG and TMBG with altered topological properties can be realized, we focus on the ${\\abs{C} = 2}$ case and comment on variants in \\csec{exp} as well as in \\capp{spectra} and \\capp{moire}. \nAt finite values of the displacement field, the only lattice symmetries unbroken are the moir\\'e translations, as well as $C_3$ rotational symmetry. The resulting Hamiltonian is time-reversal $\\mathcal{T}$ invariant since the bands from both valleys $\\tau = \\pm$ are degenerate but for an inversion of the coordinates. Therefore, the Chern number of the flat bands in each valley are opposite in sign and, hence, cancel out as long as $\\mathcal{T}$ is not spontaneously broken. \nFinally, the independence of spin ${\\sigma = \\updownarrows}$ and the absence of inter-valley scattering terms yield an overall ${\\mathrm{U}(2)_+\\times\\mathrm{U}(2)_-}$ symmetry in our model, corresponding to the separate conservation of spin and charge within each valley. \nSince a continuum model description technically has an infinite number of bands, we have to restrict our numerical analysis to a finite subset of them. For all parameters studied in this work, the approximately flat conduction band is energetically separated from the rest of the spectrum. This motivates projecting the model to the red bands in \\csubfig{band_structure}{a}, thus, keeping only one band per spin and valley flavor. \nDenoting the conduction band fermionic annihilation (creation) operators acting in spin and valley space by ${\\mathbf{c}^{(\\dagger)}_{\\mathbf{k}}}$, the projected, normal ordered many-body Hamiltonian is given by\n\\begin{equation}\n\t\\label{eq:H_NO}\n\tH = \\eta \\!\\!\\underbrace{\\sum_{\\mathbf{k} \\in \\mathrm{MBZ}}\\!\\! \\mathbf{c}\\dag_{\\mathbf{k}} \\epsilon^{\\phantom{\\dagger}}_{\\mathbf{k}} \\mathbf{c}^{\\phantom{\\dagger}}_{\\mathbf{k}}}_{H_{\\mathrm{kin}}} +\\, (1 - \\eta) \\underbrace{\\frac{1}{2 \\Omega} \\sum_{\\mathbf{q} \\in \\mathrm{BZ}} V(\\mathbf{q}) :\\!\\rho^{\\phantom{\\dagger}}_{-\\mathbf{q}} \\rho^{\\phantom{\\dagger}}_{\\mathbf{q}}\\!:}_{H_{\\mathrm{int}}^{\\mathrm{NO}}}\\;,\n\\end{equation}\nwhere the momenta $\\mathbf{k}$, $\\mathbf{q}$ are those of the moir\\'e Brillouin zone (MBZ) and graphene Brillouin zone (BZ) \\footnote{Strictly speaking this is only true when thinking in terms of the lattice model for graphene, since for the continuum model description $\\mathbf{q}$ is in $\\mathbb{R}^2$. For small twist angles and quickly decaying form factors they are practically equivalent.}, respectively. The density operators are defined as ${\\rho^{\\phantom{\\dagger}}_{\\mathbf{q}} = \\sum_{\\mathbf{k}} \\mathbf{c}\\dag_{\\mathbf{k}} \\Lambda(\\mathbf{k}, \\mathbf{q}) \\mathbf{c}^{\\phantom{\\dagger}}_{\\mathbf{k}+\\mathbf{q}}}$ via the (flavor diagonal) form factor matrix ${\\Lambda(\\mathbf{k}, \\mathbf{q})}$, with entries given by the matrix elements of the corresponding Bloch states of the continuum model; $\\Omega$ is the total area of the moir\\'e system and ${V(\\mathbf{q})}$ denotes the Fourier transform of a Yukawa potential, specified in \\capp{projection}. The moir\\'e relaxation ratio ${w_{AA}\/w_{AB}}$ enters the model through the dispersion ${\\epsilon_{\\mathbf{k}}}$ as well as the form factors ${\\Lambda(\\mathbf{k}, \\mathbf{q})}$. The convex parameter ${\\eta \\in [0,1]}$ allows us to tune the effective interaction strength across the range ${[0,\\infty]}$, while the overall energy scale is maintained. The effective relative permittivity is obtained by ${\\epsilon_{\\mathrm{eff}} = \\epsilon \\eta\/(1-\\eta)}$, where the material-specific relative permittivity $\\epsilon$ enters the Coulomb potential as $\\propto \\epsilon^{-1}$ (see \\capp{projection} for details).\n\nOur primary method of choice for the analysis of \\ceqn{H_NO} is exact diagonalization in momentum space on finite systems with periodic boundary conditions, which has previously been applied to study other graphene moir\\'e systems \\cite{Abouelkomsan2020,Repellin2020,Repellin2020a,Liu2021,Wilhelm2021,Xie2021,Abouelkomsan2022}.\nED provides us with an unbiased, numerically exact way of obtaining the many-body wave functions and full set of low-energy levels, which collectively constitute a spectral fingerprint for the prevalent ordered phases. In combination with group theoretical tools, this constitutes a powerful machinery for the identification of symmetry broken, but also topological phases \n\\cite{Bernu1992,Bernu1994,Misguich2007,Lauchli2013,Wietek2017}\nfor various system geometries. \nAlthough commonly applied in ED studies of spin Hamiltonians, the analysis of spontaneously broken continuous symmetries via the so called \\emph{tower of states} (TOS) method isn't equally well established in the context of fractionally filled bands. In this work, we corroborate its applicability to itinerant electron systems. \nThe challenge of exponentially increasing Hilbert spaces is mitigated by decomposing them into separate symmetry sectors, projecting to the most relevant electronic bands and exploiting the polarization tendencies of the studied systems. \nIn order to corroborate and complement the ED results for larger system sizes and further probe the nature of the investigated phases we additionally perform a comprehensive unrestricted Hartree-Fock treatment. Inspired by the patterns of symmetry breaking manifest in the ED spectrum we construct an effective single-particle Hamiltonian, where the original interactions enter as direct (Hartree) and exchange (Fock) contributions only and the exponential scaling of ED may be avoided at the cost of restricting the ground state to be a Slater determinant. As HF neglects some fluctuations, we directly compare with ED and explicitly show the reliability of the ansatz for specific parameters.\nAll of our numerical investigations are performed on finite simulation clusters labeled by a unique ID of the form C$N$, where $N$ is the system size. More information is compiled in \\capp{ED}.\n\n\\section{Non-coplanar antiferromagnet and ferromagnetic charge density waves}\n\\label{sec:ed_competition}\n\nWe start our numerical study by performing ED simulations of ABAB stacked TDBG at a filling fraction ${\\nu = N_e\/N = 1\/2}$, that is one electron per two moir\\'e unit cells above charge neutrality or, put differently, the red conduction bands in \\cfig{band_structure}(a) would be at $1\/8$ filling in the absence of interactions. The system has a strong tendency towards polarization of fermions into a single valley degree of freedom for a wide range of parameters as shown in \\csubfig{sp_ed_competition}{a}, which is consistent with previous HF studies \\cite{TDBGAshvin} at integer moir\\'e filling that have found spin or valley polarized states to dominate over intervalley coherence. Spontaneous valley polarization breaks time-reversal symmetry $\\mathcal{T}$ and consequently introduces a notion of chirality into the the system, as the original single-valley moir\\'e band is characterized by a finite Chern number ${\\abs{C} = 2}$. Here, we will focus on this valley polarized (VP), strongly interacting regime at ${\\eta = 0}$.\nMost interestingly, the remaining SU$(2)$ spin symmetry in a single valley is broken in different ways depending on $w_{AA}\/w_{AB}$, in the sense that a phase transition from an antiferromagnet to a maximally polarized ferromagnetic state occurs near ${w_{AA}\/w_{AB} \\simeq 0.7}$ in \\csubfigrange{sp_ed_competition}{b}{c}.\nPlotting the filling as a function of the chemical potential in \\csubfigrange{sp_ed_competition}{d}{e} for both parameter regimes reveals that, in addition to the expected spin polarized state at {$\\nu=1$} \\cite{Liu2020,Cao2020,Burg2019, He2021a}, ${\\nu=1\/2}$ stands out as a filling fraction with particularly robust insulating signatures. The clear gap in $\\mu$ when adding a single additional electron ${\\delta N_e = 1}$ strongly points to an incompressible phase while the classification at other $\\nu$ is more ambiguous. On this note, the comparatively small steps in $\\mu$ which do not coincide among multiple system sizes might correspond to a compressible, metallic state in multiple ranges of $\\nu$. The lack of jumps in $\\nu$ by ${\\delta N_e>1}$ indicates the likely absence of superconductivity, since, in the simplest scenario, such a phase would be compressible towards the introduction of Cooper pairs with ${\\delta N_e=2}$.\nIn the following sections we discuss what forms of spontaneous symmetry breaking give rise to the incompressible nature of this otherwise itinerant system.\n\n\\subsection{Tetrahedral order}\n\n\\begin{figure}[tb]\n\t\\includegraphics[width=\\linewidth]{figs\/ed_data_tetrahedral.png}\n\t\\caption{ED evidence for non-coplanar tetrahedral magnetic order at $w_{AA}\/w_{AB}=0$. (a) Multiplets located at center of mass momenta $\\mathbf{\\Gamma}$ and $\\mathbf{M}_j$ form a tower of states characteristic for the tetrahedral antiferromagnet. (b) The low-energy spectrum collapses ${\\propto\\, S(S+1)\/N}$ with increasing system size $N$, a clear manifestation of the spontaneously broken ${\\mathrm{SU}(2)}$ symmetry. (c)-(d) Measurements of the spin-$S_z$ structure factor $S_S(\\mathbf{q})$ point to the formation of long-range spin order. The peaks at ${\\mathbf{q} = \\mathbf{M}_j}$ in (c) extrapolate to finite values in the thermodynamic limit in (d), while other signals vanish. \\label{fig:ed_tetrahedral}}\n\\end{figure}\n\nThe type of the interaction induced symmetry broken state manifests in the structure of the energy levels as provided by ED. The low-energy spectrum in \\csubfig{sp_ed_competition}{c} near ${w_{AA}\/w_{AB} \\simeq 0}$ reveals a set of low lying energy levels forming exact SU(2) spin multiplets with ${S=0, \\ldots, 4=N_e \/ 2}$, which are well separated by a gap from higher excitations. The momentum-resolved low-energy manifold of states shown in \\csubfig{ed_tetrahedral}{a} reveals that all levels are located either at zero center of mass (COM) momentum ${\\mathbf{k}_{\\mathrm{COM}} = \\mathbf{\\Gamma}}$ or at the MBZ boundary at half a moir\\'e reciprocal lattice vector ${\\mathbf{k}_{\\mathrm{COM}} = \\mathbf{M}_j}$. Furthermore, the number of approximately degenerate spin multiplets increases itself as ${2 S +1}$, which strongly hints at a non-collinear magnetic symmetry breaking pattern. The energy fine structure of the so called tower of states has two characteristic features at display here. On the one hand the energy splitting has a ${S(S+1)}$ form for fixed size, and a common ${1\/N}$ scaling with the system size~\\cite{Bernu1992,Misguich2007}. Both of these hallmark features are clearly visible in \\csubfig{ed_tetrahedral}{b} --- a strong indicator for the spontaneous breaking of the continuous SU$(2)$ spin symmetry. Contributions from the distinct momenta $\\mathbf{M}_j$ as well as the collapse of the Anderson tower of states are key signatures also found in studies of the extended Heisenberg model on the triangular lattice \\cite{Wietek2017}. This and related work \\cite{Hickey2016} established that along with the chiral spin liquid phase, the TAF formed by localized magnetic moments is stabilized by the introduction of a scalar spin chirality term ${\\mathbf{S}_i \\cdot (\\mathbf{S}_j \\times \\mathbf{S}_k)}$, where the sites ${i,j,k}$ form a nearest-neighbor triangle in real-space. Such a term can be induced by an external orbital magnetic field, or by the non-trivial topology of the underlying Chern band, as was found by perturbative expansions of the Haldane model at Mott insulator filling \\cite{Hickey2016}. \n\nThis suggests that TDBG supports the formation of a non-coplanar TAF as depicted in \\csubfig{phase_diagram}{b}, \nalbeit its interpretation at the currently studied half-integer filling is less clear than for conventional strong-coupling spin-models.\nThe critical importance of band topology at fractional fillings is corroborated by the observation that the ${w_{AA}\/w_{AB}}$-region of TAF stability in \\csubfigrange{sp_ed_competition}{b}{c} correlates well with the regime where the Berry curvature ${\\mathcal{F}(\\mathbf{k})}$ of the conduction band is homogeneously distributed, see \\csubfig{band_structure}{b}.\nThe k-space homogeneity of the Berry curvature could be of similar importance to the formation of chiral phases in this fractionally filled system as was observed in numerical studies of fractional Chern insulators \\cite{Lauchli2013, Repellin2020,Wilhelm2021}.\n\nTo fortify our TOS analysis of the low-energy spectrum, we evaluate the signatures of translational symmetry breaking in the many-body ground state ${\\ket{\\Psi_0}}$ using the static spin structure factor\n\\begin{align}\n\tS_S(\\mathbf{q}) = \\frac{1}{N}\\norm{\\left(\\tilde{\\rho}_{\\uparrow, \\mathbf{q}} - \\tilde{\\rho}_{\\downarrow, \\mathbf{q}}\\right) \\ket{\\Psi_0}}^2 \\;, \\label{SpinStructurFactorDefinition}\n\\end{align}\non the topmost layer. Here, the spin density operators are defined as ${\\tilde{\\rho}_{\\sigma, \\mathbf{q}} = \\sum_{\\tau, \\mathbf{k}} \\lambda^{\\tau}(\\mathbf{k}, \\mathbf{q}) c\\dag_{\\sigma, \\tau, \\mathbf{k}} c^{\\phantom{\\dagger}}_{\\sigma, \\tau, \\mathbf{k}+\\mathbf{q}}}$, where ${\\lambda^{\\tau}(\\mathbf{k}, \\mathbf{q}) = \\braket{P_{\\mathrm{A}_1}u_{\\tau}(\\mathbf{k})}{P_{A_1} u_{\\tau}(\\mathbf{k}+\\mathbf{q})}}$ are the form-factors in valley $\\tau$ associated with the Bloch states projected onto the $A$ sublattice of the first graphene layer. While the projection to the $A$ sublattice does not affect the results qualitatively, since the weight of the Bloch states on the $B$ sublattice in layer one are vanishingly small (cf.~\\cfig{single_particle_properties}), we found the projection to the first layer to reveal the signatures of the different phases more clearly. This also underscores the relevance and reliability of scanning tunneling microscopy measurements \\cite{Rubio2022,TheorySTMNematic}, where primarily the topmost layer is probed, and establishes a direct connection to experiments. \n\nThe spin structure factor of the TAF state is shown in \\csubfig{ed_tetrahedral}{c}, which, as $S_S(\\mathbf{q})$ quickly vanishes beyond the first MBZ, reveals that the spin orientations are dominantly evolving over the moir\\'e length scale. \nThis fact is also rooted in the conduction band form factors, which also decay quickly with $\\mathbf{q}$. The dominant features at the three $\\mathbf{M}$ and the six inner k-points in \\csubfig{ed_tetrahedral}{c} might suggest a complex ordering pattern with a quite large unit cell. However, the finite size extrapolation in \\csubfig{ed_tetrahedral}{d} establishes that only the signals with ${\\mathbf{q} = \\mathbf{M}_j}$ survive in the thermodynamic limit (TDL), corresponding to long-range spin order with a ${2 \\times 2}$ unit cell.\n\nThese signatures are in good agreement with the properties of a TAF phase, which quadruples the moir\\'e unit cell in the spin channel with effective magnetic moments aligning towards the corners of a regular tetrahedron, as illustrated in \\csubfigrange{phase_diagram}{a}{b}. Furthermore, the pattern is invariant under the product of $C_3$ and an appropriately chosen spin rotation, such that the charge density remains invariant under moir\\'e translations. As illustrated in the topmost box of \\csubfig{phase_diagram}{b}, the real-space spin texture of the TAF state can be thought of as a skyrmion lattice.\nThe emergence of a phase with finite spin chirality at half-integer band filling is remarkably reminiscent of earlier works \\cite{Martin2008,Akagi2010,Akagi2012,Hayami2014} on nesting induced multiple-$\\mathbf{Q}$ instabilities in Kondo lattice models. However, in stark contrast, the instability in the current situation cannot be caused by nesting of the non-interacting Fermi surface since at ${\\eta = 0}$ the kinetic part of the Hamiltonian vanishes. What is more, in contrast to the Kondo model, the Hamiltonian in \\ceqn{H_NO} does not incorporate contributions from physically distinct localized magnetic moments but is exclusively composed of dynamic band fermions. \n\nAs demonstrated in \\csubfig{sp_ed_competition}{d} at $\\nu=1\/2$, the TAF is an insulator, characterized by a finite many-body Chern number ${\\abs{C} = 1}$, a value we explicitly calculate within HF in \\capp{hf_bc}, thus exhibiting an anomalous quantum Hall response. In a mean-field picture, the emergence of a finite Hall conductance can be thought of as the consequence of the coupling between itinerant electrons and spin textures with non-zero spin-chirality which breaks time-reversal symmetry in the charge channel \\cite{Ohgushi2000}.\n\n\\begin{figure}[!t]\n\t\\includegraphics[width=\\linewidth]{figs\/ed_data_cdw.png}\n\t\\caption{ED evidence for ferromagnetic charge order with an approximate ${\\mathrm{O}(3)}$ symmetry at $w_{AA}\/w_{AB}=0.9$. (a) The lowest energy states are completely spin polarized and have center of mass momenta $\\mathbf{\\Gamma}$ and $\\mathbf{M}_j$. (b) The ${2 J + 1}$ lowest levels are nearly degenerate, approximating an O$(3)$ symmetric manifold of states. The relatively minor splitting is well described by an effective ${\\mathrm{O}(3)}$ theory with cubic ${\\abs{b}>0}$ and quartic ${c_2<0}$ anisotropies. (c)-(d) Measurements of the charge structure factor strongly point to the formation of a ${\\mathbf{q} = \\mathbf{M}_j}$ charge density wave. The contributions at ${\\mathbf{q} = \\mathbf{0}}$ in (c) and (d) are discarded and the values of $b$ and $c_2$ in (b) are given in units of meV.\n\t \\label{fig:ed_cdw}}\n\\end{figure}\n\n\\subsection{Emergent, approximate O(3) charge density wave symmetry}\\label{EmergentSymmetry}\n\nIncreasing ${w_{AA}\/w_{AB}}$ beyond ${\\sim 0.7}$ in \\csubfig{sp_ed_competition}{c}, the order of the TOS levels of the TAF reverses and the ground state becomes ferromagnetic (FM); this can be clearly seen by inspection of the $S_z$ quantum number of the lowest-energy many-body states in \\csubfig{ed_cdw}{a}.\nWith both spin and valley polarized, the only remaining degree of freedom is charge and any additional symmetry breaking will be associated with charge density wave order. This is indeed what we find as discussed next. \n\nStarting from the TAF and increasing ${w_{AA}\/w_{AB}}$, the low-energy spectrum evolves continuously and all low lying levels are still found at the COM momenta $\\mathbf{\\Gamma}$ and $\\mathbf{M}_j$. This fits the presence of pronounced peaks in the charge structure factor ${S_C(\\mathbf{q})}$ at ${\\mathbf{q} = \\mathbf{M}_j}$, $j=1,2,3$, in \\csubfig{ed_cdw}{c}, defined as\n\\begin{align}\n\t\\label{eq:charge_structure_factor}\n\tS_C(\\mathbf{q}) = \\frac{1}{N}\\norm{\\tilde{\\rho}_{\\mathbf{q}} \\ket{\\Psi_0}}^2\\;,\n\\end{align}\nwhere ${\\tilde{\\rho}_{\\mathbf{q}} = \\sum_{\\sigma} \\tilde{\\rho}_{\\sigma, \\mathbf{q}}}$ denotes the charge density operator on the first layer. Together with the finite extrapolation to the TDL in \\csubfig{ed_cdw}{d}, these are hallmark signatures of CDW order. What is unconventional, however, is the amount of ferromagnetic levels that remain bundled up below ${\\Delta E \\lesssim 0.2\\,\\mathrm{meV}}$: a particular pattern of CDW order usually manifests in a fixed number of states corresponding to the ground state degeneracy of the translational symmetry broken states. In contrast, analyzing multiple system sizes, here we find that ${2 J + 1}$ levels are grouped exceptionally close in energy, where $J$ increases linearly with $N$ (or, equivalently, $N_e$ since we work at fixed $\\nu$) and equals the maximum total spin $S$ of the ferromagnetic ground state. The extensive nature of this suspected ground state manifold is incompatible with a single type of CDW,\nbut instead points at an approximate degeneracy of several density waves, corresponding to an approximate emergent continuous symmetry, as we explain in the following.\n\nTo develop intuition for the form and physical meaning of this emergent symmetry, we recall that the spin and valley polarization only leave the spatial modulation of the charge density, $\\delta \\varrho(\\mathbf{r})$, to construct order parameters. Furthermore, the behavior of the charge structure factor found above motivates only taking into account the momenta $\\mathbf{M}_j$ in its Fourier expansion,\n\\begin{equation}\n \\delta \\varrho(\\mathbf{r}) = \\sum_{j=1}^3 \\phi_j {\\rm e}^{{\\rm i} \\mathbf{M}_j \\mathbf{r}} + \\text{c.c.}\\;, \\quad \\phi_j \\in \\mathbb{C}. \\label{eq:ExpansionOfDensity}\n\\end{equation}\nUnder translation $T_j$ by a primitive vector $\\mathbf{L}_j$, ${j=1,2,3}$, of the triangular moir\\'e lattice, it holds ${\\phi_j\\rightarrow \\phi_j}$, ${\\phi_{j'\\neq j}\\rightarrow -\\phi_{j'\\neq j}}$, while $C_{3}$ acts as ${\\phi_j \\rightarrow \\phi_{(j+1 )\\text{mod}\\,3}}$. The change of the system's free energy $F_\\phi$ when turning on CDW order must then have the form ${F_\\phi \\sim \\sum_j (a_0 |\\phi_j|^2 + \\text{Re}[c\\, \\phi_j^2])}$ with ${a_0\\in\\mathbb{R}}$, ${c\\in\\mathbb{C}}$ up to quadratic order in $\\phi_j$. Consequently, the complex phase of all three $\\phi^2_j$ has to be the same, as long as this Ginzburg-Landau (GL) expansion is valid. Since the representation of $T_j$ and $C_{3}$ both commute with $\\phi_j\\rightarrow {\\rm e}^{{\\rm i}\\varphi}\\phi_j$, we can set $\\phi_j\\in\\mathbb{R}$ in \\ceqn{ExpansionOfDensity} without loss of generality. Physically, this means that although $\\phi_j\\rightarrow {\\rm e}^{{\\rm i}\\varphi}\\phi_j$ generally changes the spatial texture in $\\delta \\varrho(\\mathbf{r})$, its behavior under all symmetries of the system remains the same and, hence, belongs to the same phase. We note that this would be different for twisted bilayer or trilayer graphene as long as their $C_2\\mathcal{T}$ symmetry is preserved. \n\nUsing real $\\phi_j$ and defining ${\\boldsymbol{\\phi}=(\\phi_1,\\phi_2,\\phi_3)^T}$, the Ginzburg-Landau expansion becomes\n\\begin{equation}\n F_\\phi \\sim a\\,\\vec{\\phi}^2 + b \\,\\phi_1 \\phi_2\\phi_3 + c_1 \\left( \\vec{\\phi}^2\\right)^2 + c_2 \\sum_{j0$, respectively, since they are not related by symmetry. We refer to these states as TCDW$_{\\text{E}}$ and TCDW$_{\\text{O}}$. As can be seen in their respective boxes in \\csubfig{phase_diagram}{b}, both of these CDW states enlarge the moir\\'e unit cell by a factor of four.\n\n\\begin{figure*}[!htb]\n\t\\includegraphics[width=.95\\linewidth]{figs\/hf_correspondence.png}\n\t\\caption{Results from self-consistent unrestricted HF calculations. (a)-(b) The relative valley polarization $p_v$ and magnetization $m$ reproduce the respective quantities from \\csubfigrange{sp_ed_competition}{a}{b} to a high degree of accuracy. (c) The finite charge order parameters $\\phi_j$ suggest general charge density wave tendencies in the ferromagnetic regime while (f) the product ${\\phi_1 \\phi_2 \\phi_3}$ discriminates ${\\mathrm{TCDW}_{\\mathrm{E}}\\,(>0)}$, ${\\mathrm{TCDW}_{\\mathrm{O}}\\,(<0)}$ and the $C_3$ breaking ${\\mathrm{TCDW}_{\\mathrm{N}}\\,( \\simeq 0)}$ order. (g) Measurements of the scalar chirality $\\chi$ confirm the non-coplanar nature of the antiferromagnet. (h) The small difference in ground state energies per site relative to the non-symmetry-breaking state, attributes a surprising quality to the HF solution. {(d)-(e) (i)-(j)} The HF Slater determinants are almost completely composed of ED states contained in the ground state manifolds at COM orbitals $\\mathbf{\\Gamma}$ and $\\mathbf{M}_j$. In (a)--(c), (f)--(g) we used cluster C144, (h) is combined from C144 in HF and C12 in ED, (d) is computed on C16 and (e), (i)--(j) is C28. \\label{fig:hf_correspondence}}\n\\end{figure*}\n\nComing back to the emergent symmetry, we see that $F_{\\phi}$ in \\ceqn{FreeEnergyExp} only has discrete symmetries as long as $b$ or $c_2$ is non-zero. If, however, both $b$ and $c_2$ are zero (small compared to $\\sqrt{|ac_1|}$ and $c_1$, respectively), we obtain an emergent (approximate) O(3) symmetry; it acts as $\\mathbf{\\phi}\\rightarrow O \\mathbf{\\phi}$, $O\\in\\text{O}(3)$, i.e., it mixes the three distinct CDW states TCDW$_{\\text{N}}$, TCDW$_{\\text{E}}$, and TCDW$_{\\text{O}}$ defined above.\nOn this note, it should be emphasized that this continuous O(3) symmetry is independent of the ${\\mathrm{U}(2)_+\\times\\mathrm{U}(2)_-}$ symmetry of the model, but is an emergent feature in the spin and valley polarized regime of the studied many-body Hamiltonian.\nTo corroborate this Ginzburg-Landau picture and also resolve the finite breaking of the O(3) symmetry within our ED spectra, we note that the approximate symmetry implies that the low-energy space of the many-body Hamiltonian can be described as an effective \\emph{macroscopic} angular-momentum operator $\\hat{\\vec{J}}$. If the O(3) symmetry were exact, the ground state manifold would just be given by $2J+1$ exactly degenerate states, where ${\\hat{\\vec{J}}^2 = J(J+1)\\mathbb{1}}$. Finite ${b,c_2\\neq 0}$, however, induces anisotropy terms in the effective Hamiltonian, which lead to the contribution\n\\begin{align}\n\t\\label{eq:H_GL}\n\t\\Delta H_{J} =& \\frac{b}{6 J ^3} \\sum_{\\substack{(\\alpha,\\beta,\\gamma) \\in \\\\ \\pi{(x,y,z)}}} J_{\\alpha} J_{\\beta} J_{\\gamma} \\nonumber \\\\\n\t& + \\frac{c_2}{6 J ^4} \\sum_{\\substack{(i,j) \\in \\\\ \\{(x,y),(y,z),(x,z)\\} }}\\sum_{\\substack{(\\alpha,\\beta,\\gamma,\\delta) \\in \\\\ \\pi{(i,i,j,j)}}} J_{\\alpha} J_{\\beta} J_{\\gamma} J_{\\delta}\\;,\n\\end{align}\nin analogy to \\ceqn{FreeEnergyExp}. Here $\\pi$ denotes all possible permutations and the normalization by $J^n$ is added in order for the parameters $b$ and $c_2$ to be of similar magnitude across multiple system sizes. Diagonalization of $\\Delta H_{J}$ leads to ${2J+1}$ eigenvalues and fitting to the low-energy states of ED, allows to determine $b$ and $c_2$.\n\nIn \\csubfig{ed_cdw}{b} we directly compare the energy levels and their degeneracy of the effective model \\ceqn{H_GL} with the low-energy structure from ED of the full TDBG model for an optimal choice for the parameters $b$ and $c_2$. We find a remarkable agreement with the GL theory, which nearly perfectly reproduces the whole ${2 J +1}$-dimensional spectrum. Since the spectrum of \\ceqn{H_GL} is symmetric under ${b\\rightarrow -b}$, its sign cannot be inferred from the optimization procedure. In contrast, $c_2$ is found to be necessarily negative for all larger system sizes, and thus energetically favors TCDW$_{\\text{E}}$ or TCDW$_{\\text{O}}$ over TCDW$_{\\text{N}}$. \n\nThe analysis of the system's compressibility at ${w_{AA}\/w_{AB}=0.9}$ in \\csubfig{sp_ed_competition}{e} indicates insulating behavior of the TCDW phase at band filling ${\\nu=1\/2}$, which is in accordance with charge order.\nIn the following, we further substantiate the evidence for the approximate O$(3)$ character of the CDW phases by turning to a HF treatment. It will also allow us to reach larger system sizes, investigate how close the ground states are to product states, and develop a more intuitive picture for the phenomenology.\n\n\n\\section{Correspondence with Hartree-Fock Slater determinants}\n\\label{sec:hf}\n\nHF and generic mean-field calculations are routinely applied in studies of moir\\'e materials \\cite{TDBGAshvin,Bultinck2020,Xie_2020,Liao_2021,TAFHubbard,PhysRevX.11.041063,2021arXiv210602063C}. Despite making the quite restrictive assumption that the many-body ground state wave function is represented by a single Slater determinant, in certain cases of an integer number of filled bands the method was found to capture the physics of twisted bilayer graphene with surprising accuracy \\cite{Soejima2020, Xie2021}; for twisted bilayer \\cite{PhysRevLett.122.246401,Bultinck2020,PhysRevB.103.205414} and trilayer \\cite{2021arXiv210602063C} graphene, this can be understood analytically by constructing exactly solvable limits where the exact symmetry-breaking ground states are product states. For TDBG, TMBG and for fractional filling studied here, the accuracy of HF has not been tested with unbiased numerics and no realistic exactly solvable limits are known. In order to address this and complement our unbiased, but size-limited, ED results for the current situation at fractional filling, we perform self-consistent unrestricted HF calculations for the band-projected Hamiltonian of \\ceqn{H_NO}. \nBased on indications provided by the ED many-body spectrum, we allow the HF correlation matrix to simultaneously support translational symmetry breaking in all flavor channels for up to three independent momenta $\\mathbf{q}$\n\\begin{align}\n\t\\label{eq:density_matrix}\nP_{\\sigma, \\sigma^{\\prime}} ^{\\tau, \\tau^{\\prime}}(\\mathbf{k}, \\mathbf{q}) = \\langle c\\dag_{\\sigma, \\tau, \\mathbf{k}} c^{\\phantom{\\dagger}}_{\\sigma^{\\prime}, \\tau^{\\prime}, \\mathbf{k}+ \\mathbf{q}} \\rangle\\;,\n\\end{align}\nwith $\\ \\mathbf{q} \\in \\{\\mathbf{0},\\mathbf{M}_1,\\mathbf{M}_2,\\mathbf{M}_3\\}$.\nThe ordering tendencies of the self-consistent solutions are probed via the following set of observables: The valley polarization $p_v$ and magnetization $m$ are defined as \n\\begin{align}\n\t\\label{eq:HF_pv_m}\n\tp_v =& \\frac{1}{N_e}\\abs{\n\t\t\\sum_{\\mathbf{k}} \\langle \\mathbf{c}\\dag_{\\mathbf{k}} \\sigma_0 \\tau_z \\mathbf{c}^{\\phantom{\\dagger}}_{\\mathbf{k}} \\rangle\n\t}\\;, \\nonumber \\\\\n\tm =& \\frac{1}{N_e}\\norm{\\sum_{\\mathbf{k}} \\langle \\mathbf{c}\\dag_{\\mathbf{k}} (\\sigma_x, \\sigma_y, \\sigma_z) \\tau_0 \\mathbf{c}^{\\phantom{\\dagger}}_{\\mathbf{k}} \\rangle\n\t}\\;,\n\\end{align}\nwith $\\sigma_{\\alpha}$ ($\\tau_{\\alpha}$) denoting Pauli matrices in spin (valley) space. Translational symmetry breaking is captured by\n\\begin{align}\n\t\\mathcal{M}^{\\alpha}_j = \\frac{1}{2}\\sum_{\\tau, \\mathbf{k}}\\sum_{\\xi=\\pm}\\lambda^{\\tau}(\\mathbf{k}, \\xi \\mathbf{M}_j) \\langle \\mathbf{c}\\dag_{\\tau, \\mathbf{k}} \\sigma_{\\alpha} \\mathbf{c}^{\\phantom{\\dagger}}_{\\tau, \\mathbf{k}+\\mathbf{M}_j} \\rangle\\;,\n\\end{align}\nat the momenta ${\\mathbf{q} = \\mathbf{M}_j}$, $j=1,2,3$, which yields the spin density wave vector ${\\bm{\\mu}_{j} = \\left(\\mathcal{M}_j^x, \\mathcal{M}_j^y, \\mathcal{M}_j^z\\right)}$ and the CDW amplitude ${\\phi_j = \\mathcal{M}_j^0}$\n\\footnote{Although $-\\mathbf{q} = \\mathbf{q}+\\mathbf{G}$ for $\\mathbf{q} = \\mathbf{M}_j$, the form factors $\\lambda^{\\tau}(\\mathbf{k}, \\mathbf{q})$ (as well as $\\Lambda(\\mathbf{k}, \\mathbf{q})$) are in general not periodic in $\\mathbf{G}$ and hence both contributions are accounted for by an average over $\\mathbf{q} = \\pm\\mathbf{M}_j$.}.\nA characteristic quantity of the TAF phase, sensitive to its chiral, non-coplanar nature, is given by the momentum space scalar chirality ${\\chi = \\bm{\\mu}_1 \\cdot \\left(\\bm{\\mu}_2 \\times \\bm{\\mu}_3\\right)}$. Being odd under $\\mathcal{T}$, but invariant under $C_3$ and spin rotations, it is reminiscent of the scalar spin chirality term induced by an orbital magnetic field in lattice Heisenberg models \\cite{PhysRevB.51.1922}.\n\nIn agreement with the previous results from ED in \\cfig{sp_ed_competition}, the HF procedure yields full valley polarization ${p_v = 1}$ for most of the studied phase diagram in \\csubfig{hf_correspondence}{a}, except for $\\eta$ close to $1$, where the kinetic term in \\ceqn{H_NO} dominates. Near ${w_{AA}\/w_{AB} \\simeq 0.6}$ (at ${\\eta = 0}$) in \\csubfig{hf_correspondence}{b}, the system transitions from a state with zero magnetization (${m=0}$; labeled by ''no SP'') to a ferromagnetically ordered one (${m=1}$; indicated by ''SP'')---again in qualitative agreement with ED. In contrast to the ${m=0}$ phase, the completely polarized state exhibits pronounced CDW order ${\\sum_j \\phi_j^2 > 0}$ in its entire region of stability [cf.~\\csubfig{hf_correspondence}{c}]. A cubic combination of the real CDW amplitudes ${\\phi_1 \\phi_2 \\phi_3}$ in \\csubfig{hf_correspondence}{f} subdivides the SP regime into $C_3$ symmetric TCDW$_{\\mathrm{E}}$ and TCDW$_{\\mathrm{O}}$ domains partially separated by a distinct nematic TCDW$_{\\mathrm{N}}$ phase. The entire region with ${m=0}$ exhibits finite and non-coplanar $\\vec{\\mu}_j$ at wavevectors ${\\mathbf{q} = \\mathbf{M}_j}$, which results in finite values of $\\chi$ in \\csubfig{hf_correspondence}{g}, whose sign is governed by the spontaneously polarized valley $\\tau$. The same holds true for the composite Chern number of the HF state, which we explicitly calculate to be ${\\tau C = 1}$ for the TAF as well as all TCDWs.\nA direct comparison of the ground state energies per site ${\\Delta \\mathcal{E}_0^{\\mathrm{HF}-\\mathrm{ED}} = \\mathcal{E}_0^{\\mathrm{HF}} - \\mathcal{E}_0^{\\mathrm{ED}}}$ relative to the energy of the non-symmetry breaking state $\\mathcal{E}_{0,\\mathrm{NSB}}^{\\mathrm{HF}}$ in \\csubfig{hf_correspondence}{h} affirms the quality of the HF solution on a quantitative level. With no more than ${\\sim 4.3\\%}$ deviation from the per-site energy provided by ED, the optimized Slater determinant comes remarkably close to the true ground state energy on the finite lattice. \\csubfigrange{hf_correspondence}{d}{e} as well as \\csubfigrange{hf_correspondence}{i}{j} reiterate on the connection of the self-consistent Slater determinants with the ED states by explicitly representing them in the Fock state basis and subsequently performing a decomposition in the ED spectrum. Both the TAF as well as the TCDW phases reach overlaps of more than ${96\\%}$ upon accumulating the contributions from all symmetry sectors in the ground state manifolds.\nSymmetry related incarnations may feature an altered share of certain COM sectors, but accumulate to the same total value. \nWhat is more, the finite order parameter of the devised phases opens an interaction induced gap at the Fermi level, which perfectly agrees with the ED results in \\csubfigrange{sp_ed_competition}{d}{e} and underlines the bulk insulating nature of the obtained phases. \nThe combination of our ED and HF results culminates in the phase diagram presented in \\csubfig{phase_diagram}{a}.\n\n\\begin{figure}[tb]\n\t\\includegraphics[width=.8\\linewidth]{figs\/hf_subspace_energies.png}\n\t\\caption{Energies per site of the TAF and TCDW phases relative to the HF ground state as a function of $w_{AA}\/w_{AB}$. The different forms of TCDW order stay close in energy, especially for $w_{AA}\/w_{AB} {\\gtrsim} 0.7$, while there is a significant gap to the TAF phase. The crossover of the TAF and TCDW energy lines is reminiscent of the way the spin polarized levels and the singlet evolve through the phase transition in \\csubfig{sp_ed_competition}{c}. The applied discretization is C576.\n\t\\label{fig:hf_subspace_energies}}\n\\end{figure}\n\nRestricting the correlation matrix in \\ceqn{density_matrix} to each of the subspaces corresponding to the four discussed self-consistent solutions, TAF and TCDW$_{\\text{E,O,N}}$, we may track the evolution of their relative energies. \\cfig{hf_subspace_energies} clearly shows the crossover of the TAF and the set of ferromagnetic TCDW phases near ${w_{AA}\/w_{AB} \\simeq 0.6}$--${0.7}$. Most notably, whilst there is a clear energetic separation of the TAF and TCDW phases (except for the region of the phase transition) the TCDW phases remain very close in energy throughout the tuning range, especially for ${w_{AA}\/w_{AB} \\gtrsim 0.7}$. This is in accordance with the previously established spectral evidence for the emergence of an approximate O$(3)$ symmetry in the CDW channel in Sec.~\\ref{EmergentSymmetry}. In general, the overall shape of the crossover in \\cfig{hf_subspace_energies} is reminiscent of how the singlet and maximally polarized multiplets evolve through the spin phase transition in \\csubfig{sp_ed_competition}{c}.\n\nAs discussed in \\capp{hf_bc}, the states apparently inherit a lot of their quantum geometry from the original moir\\'e band. Despite the very different nature of the TAF and TCDW phases, their Berry curvatures are very similar.\nAlong with the evolution of the ${2 J + 1}$ FM levels containing the TCDWs from the TAF tower of states, while remaining separated from higher excited ones, [cf. \\csubfig{sp_ed_competition}{c}] this might point to a more intimate connection of antiferromagnetic skyrmion textures and generic topological charge density waves rooted in the topologically non-trivial nature of the moir\\'e Chern band. \n\n\n\\section{Phase diagrams at \n\\texorpdfstring{$\\bm{\\nu=7\/2}$}{nu=7\/2}\nand connection to experiments}\n\\label{sec:exp}\n\n\\begin{figure}[tb]\n\t\\includegraphics[width=\\linewidth]{figs\/phases_7_over_2_ph.png}\n\t\\caption{Phase diagram of (a) TDBG and (b) TMBG at ${\\nu = 7\/2}$ with particle-hole symmetric interactions. Except for the lack of stabilization of the nematic TCDW as the absolute ground state, the same phases as in \\csubfig{phase_diagram}{a} compete in the valley polarized parameter regime.\n\tBelow the solid black line, signatures of valley polarized TAF or TCDW phases are present in ED while the dashed line indicates the spin phase transition. The magnetic order in light-colored regions manifests only in one of the two methods. \\label{fig:hf_phases_7_over_2_ph}}\n\\end{figure}\n\nThe physics of TDBG as well as TMBG has been investigated in multiple experimental works, including with a focus on fractional fillings of the moir\\'e flat bands \\cite{Polshyn2022,He2021}. Both systems allow for gate-controllable, isolated conduction bands with Chern number ${\\abs{C} = 2}$. We hereby study these systems at the experimentally relevant filling of ${\\nu = 7\/2}$, which corresponds to a hole filling ${\\nu_h = 1\/2}$ of the spin and valley degenerate bands. \nMotivated by insights about the band-projected interaction Hamiltonian in the context of twisted bilayer graphene \\cite{Bultinck2020,PhysRevB.103.205414} and in order to counteract the large asymmetry of the electron and hole side in the interaction part of \\ceqn{H_NO}, we replace the density operators ${\\rho^{\\phantom{\\dagger}}_{\\mathbf{q}}}$ by ${\\delta\\rho^{\\phantom{\\dagger}}_{\\mathbf{q}} = \\rho_{\\mathbf{q}} {-} \\frac{1}{2} \\sum_{\\mathbf{k}} \\Tr\\left[ \\Lambda(\\mathbf{k},\\mathbf{q}) \\right] \\delta_{\\mathbf{q}\\in\\text{RL}}}$, where $\\delta_{\\mathbf{q}\\in\\text{RL}}$ is only non-zero if $\\vec{q}$ is a reciprocal moir\\'e lattice vector.\nWhilst omitting normal ordering, this results in a (up to a constant) particle-hole symmetric interaction Hamiltonian \n\\begin{align}\n\t\\label{eq:H_PH}\n\tH_{\\mathrm{int}}^{\\mathrm{PH}} = \\frac{1}{2 \\Omega} \\sum_{\\mathbf{q} \\in \\mathrm{BZ}} V(\\mathbf{q}) \\delta\\rho^{\\phantom{\\dagger}}_{-\\mathbf{q}} \\delta\\rho^{\\phantom{\\dagger}}_{\\mathbf{q}}\\;.\n\\end{align}\nThe Hamiltonian of \\ceqn{H_PH} differs from \\ceqn{H_NO} by a quadratic term which corresponds to a single-band version of the subtraction method applied in \\cref{Bultinck2020} and \\cref{PhysRevB.103.205414}.\n\nWe examine the nature of the ground state for both systems again by means of ED and HF. The findings are compiled in the phase diagrams of \\cfig{hf_phases_7_over_2_ph}. Remarkably, antiferromagnetic and ferromagnetic domains are located in approximately the same regions of the parameter space for both systems. The TAF is, as noted before, stable at low ${w_{AA}\/w_{AB}}$, whereas ${w_{AA}\/w_{AB} \\gtrsim 0.25}$ leads to a spin-polarized TCDW ground state at ${\\eta = 0}$.\nDue to the particle-hole symmetry of the Hamiltonian $H_{\\mathrm{int}}^{\\mathrm{PH}}$ at ${\\eta = 0}$, the phase diagram in the purely interacting limit is the same for ${\\nu = 7\/2}$ and ${\\nu = 1\/2}$. Hence, the shift of the phase transition from ${w_{AA}\/w_{AB} \\simeq 0.7}$ in \\csubfig{phase_diagram}{a} to ${w_{AA}\/w_{AB} \\simeq 0.2}$ in \\csubfig{hf_phases_7_over_2_ph}{a} must be caused by the additional dispersive term only. \nThis might indicate, that nesting processes related to the ones discussed in Refs.\\,\\cite{Martin2008,Akagi2010,Akagi2012,Hayami2014} could play a role in the stabilization of the TAF and TCDW phases, where the effective dispersion stems from the originally normal ordered and band-projected interactions in $H_{\\mathrm{int}}^{\\mathrm{NO}}$ in \\ceqn{H_NO} rather than the usual hopping of electrons.\nThis is in line with observations that the orbital occupation in such band-projected models strongly depends on the energetic landscape of this effective hole dispersion \\cite{Lauchli2013, Abouelkomsan2020, Wilhelm2021}, which recently has also been related to the quantum geometry of the band wave functions \\cite{Abouelkomsan2022}. In fact, in the single-band case, the dispersive term responsible for the particle-hole invariance of $H_{\\mathrm{int}}^{\\mathrm{PH}}$ coincides with half the effective dispersion felt by holes upon a particle-hole transformation of $H_{\\mathrm{int}}^{\\mathrm{NO}}$ (see \\capp{ED}).\n\nTuning ${\\eta > 0}$, the TAF regime exhibits a more or less pronounced cone-like front of stabilization until the lowest energy state in ED is no longer valley polarized near ${\\eta \\simeq 0.5}$. This tendency also applies to the TCDW phases, whose maximal valley polarizations appear to vanish at even slightly lower $\\eta$ according to ED. Compared to \\csubfig{phase_diagram}{a}, the discrepancy of ED and HF is slightly increased, in particular by HF overestimating the stability of spin and valley polarization. Despite the fact that both diagrams completely lack the previously present TCDW$_{\\mathrm{N}}$ phase and the absolute ground state for TDBG is the TCDW$_{\\mathrm{O}}$ while for TMBG it is the TCDW$_{\\mathrm{E}}$ phase, the crucial physical features are very similar to \\csubfig{phase_diagram}{a}. This is because the low-energy spectrum in the FM regime still exhibits the ${2 J + 1}$-fold quasi-degeneracy suspected to be rooted in an emergent O$(3)$ CDW symmetry. From that point of view, the selection of a specific ground state is exclusively due to the minor cubic and quartic anisotropies $b$ and $c_2$ in \\ceqn{FreeEnergyExp} resulting in the formation of local extrema in the energetic manifold. More so, the splitting of the lowest ferromagnetic levels is found to be significantly decreased in relation to the multiplet gap by switching to $H_{\\mathrm{int}}^{\\mathrm{PH}}$ and especially the spectrum of TMBG suggests extraordinarily low coefficients for the primary anisotropic terms in the GL theory of ${\\abs{b} = 0.06\\,\\mathrm{meV}}$ and ${c_2 = -0.05\\,\\mathrm{meV}}$ [see \\cfig{spec_20_spin} and \\cfig{spec_36_polarized}] . \n\nAn explicit calculation of the HF Chern number ${\\abs{C} = 1}$ again attributes a topologically non-trivial character to the wave function. As expected, for polarization to identical valleys, the sign of the Chern number as well as the momentum space scalar chirality $\\chi$ for the TAF are flipped compared to the $\\nu=1\/2$ case. In combination with the insulating, ferromagnetic nature of the TCDW in the realistic parameter range of ${w_{AA}\/w_{AB} \\simeq 0.6}$--${0.9}$, our findings are in excellent agreement with the experimental signatures of a ferromagnetic symmetry broken Chern insulator with ${\\abs{C} = 1}$ reported in \\cref{Polshyn2022} for TMBG. The proposed form of translational symmetry breaking along one moir\\'e lattice vector coincides with our definition of the TCDW$_{\\mathrm{N}}$ phase, where strain or boundary effects may play a crucial role in the sub-selection of the ground state from the quasi O(3) degenerate manifold of ordered phases.\nOn the other hand, \\cref{He2021} finds no signatures of an SBCI ground state for ABAB stacked TDBG, primarily studied in this work, but only for the flipped stacking chirality ABBA. \nThe absence of SBCI signatures for ABAB stacked TDBG in \\cref{He2021} may be rooted in the delicate interplay of band-projected interactions and kinetic dispersion contained in \\csubfig{hf_phases_7_over_2_ph}{a}. The exact location and shape of the valley polarization transition line may sensitively depend on various system parameters, potentially causing time-reversal symmetry to be restored and any quantum Hall signatures to vanish. Nevertheless, the extraordinary similarity of \\csubfig{hf_phases_7_over_2_ph}{a} and \\csubfig{hf_phases_7_over_2_ph}{b} in combination with the experimental finding that TMBG behaves similar to TDBG at integer fillings \\cite{Chen2021} holds potential that ABAB stacked TDBG may also host ferromagnetic topological charge density waves at half-integer filling. \n\n\n\\section{Conclusion and outlook}\n\nWe performed extensive numerical studies of the ground state order in a continuum model description of twisted double-bilayer graphene and twisted mono-bilayer graphene at half-integer filling of the flat Chern conduction band using a combination of exact diagonalization and unrestricted Hartree-Fock methods. \nWe provided conclusive evidence for the emergence of a non-coplanar magnetic state with the symmetries of the tetrahedral antiferromagnet of the underlying triangular lattice and finite skyrmion density in its 4-moir\\'e-site unit cell, see upper panel in \\csubfig{phase_diagram}{b}. While this state is found to be energetically favored for small $w_{AA}\/w_{AB}$, a set of almost degenerate spin-polarized charge density wave phases [lower three panels in \\csubfig{phase_diagram}{b}] dominates for larger $w_{AA}\/w_{AB}$, corresponding to an approximate emergent O$(3)$ symmetry. All of these phases are incompressible and exhibit a finite Chern number. The intimate connection between the phases on either side of this transition is clearly visible in our many-body spectra, where the tower of states of the tetrahedral state evolve into the set of ferromagnetic charge density levels.\nRemarkably, the explicit Fock space construction of the Hartree-Fock state and its decomposition in the ED ground state manifold establishes that all observed phases are exceptionally close to a fermionic product state.\n\nOur findings provide a natural explanation of the experimental signatures of a ferromagnetic Chern insulator observed in twisted mono-bilayer graphene \\cite{Polshyn2022} in an unbiased way and give perspective on the possibly delicate interplay of interactions and band dispersion responsible for the formation of such phases. Our work further illustrates that graphene moir\\'e systems where $C_2$ symmetry is explicitly broken by the lattice and can thus exhibit bands with finite Berry curvature and Chern numbers in each valley are promising platforms for stabilizing frustrated magnetic phases or potentially even spin-liquid states at fractional filling. \n\nAs the tetrahedral antiferromagnet at small $w_{AA}\/w_{AB}$ is invariant under a combination of spin rotation and translations, it will not give rise to a charge modulation on the triangular moir\\'e lattice. However, applying a magnetic field, would induce charge modulations that could for instance be observed in future scanning tunneling microscopy experiments \\cite{Rubio2022,CrommieSTM,SpectroscopyTDBG}. \nAnother perspective for future experiments \\cite{CollectiveModesTBG,YacobiMagnon} would be to identify the (almost) gapless collective modes associated with the (approximate) emergent O(3) symmetry and its explicit breaking by unintentional or controlled strain (e.g., via a piezoelectric substrate). \n\nThe work also poses many important open theoretical questions such as studying the physical consequences of charge-density fluctuations close to the O(3) symmetric limit \\cite{IsospinPomeranchuk}. We also believe that the emergent symmetry and the proximity of the ground states to Slater determinants we establish could pave the way for an analytical understanding of the strongly correlated low-temperature physics in twisted double- and mono-bilayer graphene. \n\n\\section*{Acknowledgments}\n\nWe thank Y. Kwan for valuable discussions on twisted bilayer graphene. Furthermore, we acknowledge support by the Austrian Science Fund FWF within the DK-ALM (W1259-N27). The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).\n\n\\emph{Note added:} In the final steps of preparation of this manuscript, experimental signatures of an anomalous Hall effect in ABAB stacked TDBG near $\\nu=7\/2$ and its stability under both in- and out-of-plane magnetic fields were reported in \\cref{Kuiri2022}. This is consistent with our results for TDBG and strengthens the established similarity of TDBG and TMBG in that it is in line with the observation of symmetry broken Chern insulators at $\\nu=7\/2$ in \\cref{Polshyn2022}.\n\n\\setcounter{equation}{0}\n\\setcounter{figure}{0}\n\\setcounter{table}{0}\n\\makeatletter\n\\renewcommand{\\theequation}{A\\arabic{equation}}\n\\renewcommand{\\thefigure}{A\\arabic{figure}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{Background and Motivation}\n In traditional networks, wireless nodes are powered by limited capacity batteries which should be regularly charged or replaced. Energy harvesting has been recognized as a promising solution to replenish batteries without using any physical connections for charging. In simultaneous wireless information and power transfer (SWIPT), the incoming RF signal is used for both energy harvesting and decoding of information bits. The inherent challenge of energy harvesting (EH) is the stochastic nature of the EH process, which dictates the amount and availability of harvested energy that is beyond the control of system designers. However, SWIPT may provide the network administrators a leverage on replenishing the remote devices for proper network operations.\n \n\nIn the seminal paper \\cite{varshney2008transporting}, the rates at which energy and reliable information can be transferred over a single point-to-point noisy link were characterized. This result was later extended to frequency-selective channels with additive white Gaussian noise (AWGN) in \\cite{grover2010shannon}. In \\cite{zhang2013mimo}, the authors examined separated and co-located information and energy receiver architectures in a multiple-input multiple-output (MIMO) wireless broadcast system. In separated architecture, both receivers have separate antennas, whereas in co-located architecture a single antenna is shared by both. In general, EH devices have small footprints necessitating a co-located architecture. This arises a resource allocation problem of sharing the RF signal among the two receivers. The incoming RF signal is fed to Information Decoding (ID) and Energy Harvesting (EH) circuitries by applying either time-switching (TS) or power splitting (PS) schemes. In TS, the RF signal is split over two different parts of the time slot, one for EH and the other for ID, whereas in PS the incoming RF signal is fed to both, proportional to a given factor. In this work, we consider the class of PS policies. In particular, we consider two types of PS policies: \\emph{splitting} and \\emph{no-splitting}. A splitting policy divides the RF signal into two parts with strictly non-zero power and feeds them to ID and EH circuitries, whereas no-splitting policy feeds the RF signal completely to either EH or ID.\n\n\nIn inherently error-prone wireless communications systems, re-transmissions triggered by decoding errors have a major impact on the energy consumption of wireless devices. Hybrid automatic repeat request (HARQ) schemes are frequently used in order to reduce the number of re-transmissions by employing various channel coding techniques \\cite{WHARQ}. Nevertheless, this comes at the expense of extra processing time and energy associated with the enhanced error-correction decoders. A receiver employing HARQ encounters two major energy consuming operations: (1) sampling or Analog-to-Digital Conversion (ADC), which includes all RF front-end processing, and (2) decoding. The energy consumption attributed to sampling, quantization and decoding plays a critical role in energy-constrained networks which makes their study a non-trivial problem. The authors in \\cite{de2014performance} investigated the performance of HARQ over an RF-energy harvesting point-to-point link, where the power transfer occurs over the downlink and the information transfer over the uplink. The authors studied the use of a TS policy when two HARQ mechanisms are used for information transfer; Simple HARQ (SH) and HARQ with Chase Combining (CC) \\cite{harq}. Also, the authors in \\cite{maha} studied the performance of HARQ in RF energy harvesting receivers, where heuristic TS policies are proposed to reduce the number of re-transmissions. \n\nIn this paper, we consider a point-to-point link where an energy-abundant transmitter employs HARQ to deliver a message reliably to an EH receiver. The receiver has no energy source, so it relies on harvesting energy from the information-bearing RF signal. The channel is time-varying where the amount of energy harvested and information collected varies depending on the quality of the channel. The receiver aims to split the incoming RF signal between EH and ID so that the expected number of re-transmissions is minimized. Unlike prior works, e.g., \\cite{zorzi}, in our work, we do not assume the availability of the channel state information (CSI) at the receiver\\footnote{Due to the time and energy cost, the acquisition of CSI in EH networks is challenging. Some interesting ideas along this line, such as limited CSI feedback, have been discussed in \\cite{Clerckx}.}. \n\\subsection{Contributions}\nOur main contributions in this paper are summarized as follows:\n\\begin{itemize}\n\\item We formulate the problem of minimizing the expected number of re-transmissions using a Markov decision process (MDP). \n\\item Due to the excessive number of states and actions in the MDP formulation, we use the special features of the EH HARQ framework to recast the MDP as a problem of minimizing the expected time to absorption in an absorbing Markov chain, significantly reducing the complexity associated with the MDP, when the wireless channel exhibits independent and identically distributed (i.i.d.), and time-correlated properties, respectively.\n\\item For i.i.d. channels, we prove that there is an optimal policy that does not split the incoming RF energy and uses it solely either for ID or EH. As a result, we convert the original problem whose states and actions take over continuous values into discrete ones, enabling a tractable solution.\n\\item The numerical solution of the MDP identifies multiple distinct policies that achieve the minimum expected number of re-transmissions, implying that the optimal policy is not unique. Hence, we later completely characterize a class of simple-to-implement optimal policies. Among those, harvest-first-store-later is an optimal policy lending itself for simple implementation on low complexity devices.\n\\item For a time-correlated channel, we once again show that there is an optimal policy that does not split the incoming RF energy. We develop a low complexity algorithm to determine the EH\/ID decision for each state of the receiver. Note that unlike the i.i.d. case, a simple policy such as harvest-first-store-later is no longer optimal for correlated channels as demonstrated in our numerical analysis.\n\\item We provide extensive numerical simulations to verify the analytical results established in the paper.\n\\end{itemize}\n\n\n\\subsection{Related Work}\nEarly works on wireless energy transfer \\cite{early2} considered a point-to-point single antenna communication system and studied its rate-energy trade-off. Single antenna systems are extended to single-input-multiple-output (SIMO) in \\cite{liangliu}, multiple-input-single-output (MISO) in \\cite{miso} and multiple-input-multiple-output (MIMO) system in \\cite{mimo}. \n\nNote that EH devices harvest energy only in minuscule amounts (orders of $\\mu W$s), so the energy consumption of the receiver circuitry to perform simple sampling and decoding can no longer be neglected. The authors in \\cite{doost1} addressed the energy consumption of sampling and decoding operations over a point-to-point link where the receiver harvests energy at a constant rate. In \\cite{retarq2}, a decision-theoretic approach is developed to optimally manage the transmit energy of an EH transmitter transmitting to an EH receiver, where both the transmitter and the receiver harvests energy independently from a Bernoulli energy source. The receiver uses selective sampling (SS) and informs the transmitter about the SS information and its delayed battery state by feedback. Based on this feedback, the transmitter adjusts its transmission policy to minimize the packet error probability. \n\nMeanwhile, in \\cite{doost2}, the performance of different HARQ schemes for an EH receiver harvesting energy from a deterministic energy source with a constant energy rate was studied.\nIn \\cite{resist}, the impact of the battery's internal resistance at the receiver was analyzed for an EH receiver with imperfect battery, with the aim of maximizing the amount\nof information decoded by the EH receiver.\nWhile ignoring the sampling energy cost at the receiver, \\cite{mehlul} investigates the performance of TS policies to maximize the amount of information decoded at the receiver operating over a binary symmetric channel (BSC), by optimizing the fraction of time used for harvesting energy and for extracting information.\nFor an EH transmitter and an EH receiver pair both harvesting ambient environmental energy with possible spatial correlation, \\cite{outage_zhou} addresses the problem of outage minimization over a fading wireless channel with ACK-based re-transmission scheme by optimizing the power allocation at the transmitter. \nIn \\cite{R3R1}, for a pair of EH transmitter-receiver employing ARQ and HARQ with binary EH process, packet drop probability over fading channels is minimized by optimally allocating power over different rounds of re-transmissions. In \\cite{adaptarq}, an adaptive feedback mechanism for an EH receiver is proposed by taking into account the energy cost of sampling and decoding is proposed. The receiver is allowed to transmit a delayed feedback with the aim of efficiently utilizing the harvested energy in order to minimize the packet drop probability in the long run. In \\cite{R3R4}, the outage probability for an EH receiver powered by RF transmissions is minimized by implementing HARQ. In particular, the transmitter optimally allocates two different power levels in charging and information transmission periods so that the probability of the event that information is not correctly received by the receiver due to either unsuccessful\nmessage decoding or lack of minimum energy at the receiver is minimized. Although \\cite{R3R4} is the most similar study to our work, it assumes that the channel stays constant during re-transmissions and it is known by the receiver. Differently, we assume that the wireless channel, with and without memory, varies over different instances of re-transmissions which calls for an online framework rather than an offline framework as in \\cite{R3R4}. The problem of throughput optimization for an EH receiver operating in a multi-access network was studied in \\cite{sufficient} where the receiver takes samples from the incoming RF signal to calculate the probability of a collision event and based on that decides to either utilize the incoming RF energy to replenish its battery or to extract information bits.\n\n\n\nIn \\cite{mehdigilbert}, an EH transmitter intelligently adapts its channel sensing strategy with respect to a belief parameter it has about the channel condition to maximize its long term discounted throughput over a time correlated channel. In \\cite{7492928}, maximization of long term weighted sum throughput, in an uplink scenario, for two RF EH transmitters is studied. The AP has the complete knowledge of the state of the network, i.e., battery levels, uplink and downlink CSI, and it calculates the optimal EH period, and the uplink durations of each transmitter at the beginning of each time slot. The finite horizon uplink throughput maximization for an EH transmitter with imperfect CSI and random EH process is studied in \\cite{7865904}, and the optimal power allocation problem at each time slot is formulated using dynamic programming (DP). \\cite{R1} studies the rate-energy (R-E) region of separated and co-located SWIPT architectures where R-E region characterizes all the achievable rate and harvested energy pairs under a given transmit power constraint. A strategy achieving the optimal R-E region is developed for the case of separated architecture. For the case of co-located architecture, two policies namely power splitting and time switching is investigated in terms of their achievable R-E region. In \\cite{R2}, for a network with a transmitter, a relay and a destination node, two relaying protocols namely power splitting based relaying (PSR) and time switching based relaying (TSR)protocols are proposed. Analytical expressions for outage probability of delay limited transmission mode and ergodic capacity of delay tolerant transmission mode are derived. In contrast to \\cite{R1,R2}, we show that there exists an optimal policy that does not split the incoming RF energy when HARQ mechanism is employed.\n\n\nDifferently from the available literature, we study the reliability of transmission by an HARQ mechanism in a SWIPT scenario, over time varying channels with unknown CSI and by considering an accurate model of energy consumption of the EH receiver. We develop a novel Markovian framework for the analysis which facilitates characterizing the optimal decision at any given time. A major contribution of this work is that we prove that there exists an optimal no-splitting policy that minimizes the number of re-transmissions. This finding enables a tractable optimal solution by reducing a two dimensional uncountable state MC into a countable state MC. In particular, for i.i.d. channels, we show that policies such as harvest-first-store-later are optimal enabling simple-to-implement optimal policies suitable for low power EH devices. However, for the case of correlated channels, we show that an intelligent algorithm that utilizes the correlation information of the channel states, can significantly outperform those simple-to-implement policies.\n\n\n\n\n\n\n\\section{System Model and Preliminaries}\\label{SystemModel}\n\\subsection{Channel Model and Receiver Architecture}\nConsider a point-to-point time varying wireless link between a transmitter-receiver pair. \nThe wireless channel is modeled according to a two-state block fading model where the states are GOOD and BAD\\footnote{Note that the two-state channel process is an approximation of a more general multi-state time varying channel, where each state of the channel supports a maximum transmission rate. Here, we employ two-state channel process due to its analytical tractability.}. Let $G_t\\in\\left\\{0,\\ 1\\right\\}$ be the state of the channel at time slot $t$ where BAD and GOOD states are denoted by $0$ and $1$, respectively. The CSI is neither available at the transmitter nor at the receiver due to the high computational and energy costs of transmitting and receiving a pilot signal necessary for measuring the CSI. We consider a communication scheme where the transmitter is connected to a power source with an unlimited energy supply. The receiver is equipped with a separate rectifier circuit for EH and a transceiver for ID, both connected to the same antenna. \n\n\n\n\nTime is slotted and each slot has a length of $N$ channel uses. We assume that $N$ is sufficiently large so that we can apply information theoretic arguments. The instantaneous achievable rate of the receiver is the maximum achievable mutual information between the output symbols of the transmitter and input symbols at the receiver. Let the achievable rate of the receiver be $R(t)$ at time $t$. As $N\\to \\infty$, $R(t)$ approaches the Shannon rate, and it can be computed as:\n\\small\n\\begin{align}\nR(t) = \\log_2(1 + P g(t)),\n\\end{align}\n\\normalsize\nwhere $g(t)\\in \\left\\{g_0, g_1\\right\\}$ is the channel power gain at time $t$ and $P$ is the noise-normalized transmit power of the transmitter. We assume that the transmitter power is fixed and known to the receiver. Let $R_1$ and $R_0$ be the achievable rates corresponding to the channel states GOOD and BAD, respectively:\n\\small\n\\begin{align}\nR_1 = \\log_2(1 + P g_1),\\label{R1R2a}\\\\\nR_0 = \\log_2(1 + P g_0).\\label{R1R2b}\n\\end{align}\n\\normalsize\nThe instantaneous channel states are not known a priori so we employ an HARQ scheme with incremental redundancy (IR) for providing reliability \\cite{wicker1995error}. In the following, we give a brief overview of HARQ-IR.\n\n\n\n\n\n\n\n \\subsection{Brief Overview of HARQ}\n \\label{sec:harq}\n HARQ is a well known method to provide reliable point to point communications \\cite{wicker1995error}. There are several types of HARQ implementations, e.g., simple HARQ, HARQ with Chase Combining (CC), repetition time diversity and incremental redundancy (IR). Note that in EH devices, CSI acquisition is cost prohibitive due to the energy and temporal cost of probing the channel. Hence, in this work, the transmitter is blind to the instantaneous channel conditions and it cannot adapt the code rates according to a particular channel gain. Thus, in our system, we consider HARQ-IR due to its superior throughput performance \\cite{930931} compared to other alternatives as well as its robustness against the absence of CSI \\cite{INRref}. Let us denote a message of the transmitter by $W\\in \\left\\{1,2,\\ldots,2^{NC}\\right\\}$, where $C$ denotes the rate of the information. Every incoming transport layer message into the transmitter is encoded by using a mother code of length $MN$ channel uses. The encoded message, $\\mathbf{x}$, is divided into $M$ blocks, each of length $N$ channel uses, with a variable redundancy and it is represented by $\\mathbf{x}=[x^1,\\ldots,x^M]$. Let us assume that $x^1$ is transmitted at $t_1$. If $x^1$ is successfully decoded, then the receiver sends a 1-bit, error-free, zero-delay, Acknowledgement (ACK) message, otherwise, the transmitter times out after waiting a certain time period. In case of no ACK received, the transmitter transmits $x^2$ at time slot $t_2$ and the receiver combines the previous block $x^1$ with $x^2$. This procedure is repeated until the receiver accumulates $C$ bits of mutual information or maximum blocks of information, $M$, is sent. We assume that, $M$ is chosen sufficiently large so that the probability of decoding failure, due to exceeding the maximum number of re-transmissions, is approximately equal to zero. With HARQ-IR scheme, after $r$ re-transmissions, the amount of accumulated mutual information at the receiver is $\\sum^{r}_{k=1} R(t_k)$. The receiver, given that it has sufficient energy, can perform a successful decoding attempt after $r$ re-transmissions, if the amount of accumulated mutual information exceeds the information rate of the transmitted message, i.e., $\\sum^{r}_{k=1} R(t_k)\\geq C$. We assume that each message is encoded at rate $R_1$ i.e., $C=R_1$ so that a transmission in a GOOD channel state carries all the information needed for decoding\\footnote{Note that this assumption is practically reasonable, since a time slot is typically defined as the duration of time necessary for transmission of a single information packet.}.\n \n\\subsection{Energy Harvesting and Consumption Model}\nIn the following, we assume that the receiver has a sufficiently large battery and memory, so there is no energy or information overflow. The receiver utilizes a PS policy, where $\\rho(t)\\in [0,1]$ denotes the power splitting parameter at the beginning of time slot $t$. Note that $\\rho(t)=0$ indicates that the received signal is used solely for mutual information accumulation, and $\\rho(t)=1$ indicates that the received signal is used solely for harvesting energy. Any value of $\\rho(t)$ between 0 and 1 refers to the case where the received signal is used for both harvesting energy and mutual information accumulation. \n\nWe incorporate a simplified energy harvesting model, which facilitates the formulation of a tractable optimization problem. In this model, the receiver harvests a maximum of $e\\geq 1$ energy units in the GOOD channel state and zero units during the BAD channel state\\footnote{The maximum energy is harvested if the received signal is completely directed to the energy harvester, i.e., $\\rho(t)=1$.}. Typically, an EH device has two stages in its energy harvesting circuitry \\cite{Talla}: a rectifier stage that converts the incoming alternating current (AC) radio signals into direct current (DC); and a DC-DC converter that boosts the converted DC signal to a higher DC voltage value to produce the voltage required to charge the battery. The main limitation in an energy harvester is that every DC-DC converter has a minimum input voltage threshold below which it cannot operate. Hence, when the channel is in a BAD state, the input voltage is below the threshold of the DC-DC converter and no energy is harvested. Even though the receiver cannot harvest any RF energy in a BAD channel state, it can still accumulate mutual information since ID circuit operates at a lower power sensitivity, e.g., $-10$ dBm for EH and $-60$ dBm for ID circuits \\cite{EHsurvey}.\n\n\n\n\n\n\n\nThe energy consumption of HARQ was recently investigated in \\cite{rosas2016optimizing}, and it was identified that the energy is consumed at the start up of the receiver, during decoding, for operating passband receiver elements (low-noise amplifiers, mixers, filters,\nfrequency synthesizers, etc.), and for providing feedback to the transmitter. In order to develop a tractable optimization frame work, we consider the model in \\cite{rosas2016optimizing}, and combine the individual costs of energy into two parameters only: the receiver consumes $E_d\\geq 1$ energy units for a decoding attempt and 1-energy unit for each mutual information accumulation event per time slot\\footnote{One energy unit is normalized to the energy cost of operating the RF transceiver circuit during one time slot.}, i.e., operating the passband receiver elements.\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Minimum Expected Number of Re-transmissions For I.I.D. Channels}\n\\label{MCF}\nIn this section, we calculate the minimum expected number of re-transmissions needed for successful decoding for time varying channels. We first consider an i.i.d. channel, and in Section VI, we will investigate the system under a time correlated channel model. Note that the receiver requires at least $E_d$ units of energy and $R_1$ bits of information before it can successfully decode the transmitted packet. Let the system states be $(b,\\ m)$, where $b$ is the total residual battery level and $m$ is the total accumulated mutual information normalized by $R_0$. For clarity of presentation, in the rest of the paper we assume that $R_0=1$.\nOur objective is to optimally determine a scheduling policy $\\rho(t)$ so that the transmission is successfully decoded with a minimum delay at the receiver. We formally define $\\rho(t)$ next.\n\n\n\\begin{definition}\nA scheduling policy $\\boldsymbol{\\pi}= (\\rho(1), \\rho(2), \\ldots,)$ is\na sequence of decision rules as such the $k$th element of $\\boldsymbol{\\pi}$\ndetermines the power splitting ratio at $k$th time slot based on the observed system state $(b,\\ m)$ at the beginning of this\ntime-slot for $t\\in \\{1,2,\\ldots\\}$. Similarly, a tail scheduling\npolicy $\\boldsymbol{\\pi}_t = (\\rho(t), \\rho(t+1), \\ldots)$ is a sequence of decision rules\nthat determines the power splitting ratios for the time slots from $t$ to $\\infty$.\n\\end{definition}\n\n\nLet the probability that the channel is in GOOD\nstate be $\\lambda$, i.e., $\\mathds{Pr}\\left[G_t=1\\right]=\\lambda$. The problem can be mathematically modeled as a two-state Markov chain (MC). Also, let the states of the MC be $(b,\\ m)$. It should be noted that the receiver is blind to the CSI before choosing the power splitting ratio. However, after it decides to sample the incoming RF signal for mutual information accumulation, the amount of the information in the sampled portion of the RF signal is revealed to the receiver. Because the scheduling policy is blind to the CSI, its decision only depends on $(b,\\ m)$.\n\n\\subsection{Markov Decision Process (MDP) Formulation}\n\\label{sec:MDP}\n\nAt any given time $t$, the next state of the system only depends on the current state, $(b,\\ m)$, and the power split ratio $\\rho(t)$. Hence, we can formulate the problem as an MDP. Let $f^{\\boldsymbol{\\pi}}(t)\\in\\{-1,0\\}$ be an indicator function taking a value of $0$ if the message can be decoded at the end of slot $t$ under policy $\\boldsymbol{\\pi}$, and a value of $-1$ otherwise. Then, the optimization problem we aim to solve is given as,\n\\small\n \\begin{align}\n \\max_{\\boldsymbol{\\pi}} \\sum_{t=0}^{\\infty} f^{\\boldsymbol{\\pi}}(t).\n \\end{align}\n\\normalsize\n Let $V^{\\boldsymbol{\\pi}}(b,0)$ be the expected discounted reward with initial state $S_0 =(b,0)$ under policy $\\boldsymbol{\\pi}$ with discount factor $\\beta \\in [0, 1)$. The expected discounted reward has the following expression\n\\small\n \\begin{align}\n V^{\\pi}(b,\\ 0) = \\mathds{E}^{\\boldsymbol{\\pi}}\\left[\\sum^{\\infty}_{t=0}\\beta^{t}U(S_{t},\\rho(t))|S_{0}=(b,\\ 0)\\right],\\label{Vdef}\n \\end{align}\n\\normalsize\n where $\\mathds{E}^{\\boldsymbol{\\pi}}$ is the expectation with respect to the policy $\\boldsymbol{\\pi}$, $t$ is the time index, $\\rho(t) \\in [0,1]$ is the action chosen at time $t$, and $U(S_{t},\\rho(t))$ is the instantaneous reward acquired when the current state is $S_t$.\nIn the rest of the paper, we use $\\rho(t)$ and $\\rho(b,m)$ interchangeably by assuming that at time slot $t$, the system is at state $(b,\\ m)$. The battery is recharged with incoming RF signal depending on the value of the power split ratio $\\rho(t)$. Meanwhile, one unit of energy is consumed in order to accumulate non-zero bits of mutual information. Hence, the evolution of the battery state is characterized as follows:\n\\small\n\\begin{align}\nB(t)=&\\left\\{\n\\begin{array}{ll}\nB(t-1)+\\rho(t) e-\\mathds{1}_{\\rho(t)\\neq 1},& \\text{if}\\ G_t=1\\\\\nB(t-1)-\\mathds{1}_{\\rho(t)\\neq 1},& \\text{if}\\ G_t=0\n\\end{array}\n\\right.,\\label{B}\n\\end{align}\n\\normalsize\nwhere $\\mathds{1}_{\\rho(t)\\neq 1}=0$, if $\\rho(t)=1$, and $\\mathds{1}_{\\rho(t)\\neq 1}=1$, otherwise.\n\nAccording to (\\ref{R1R2a}) and (\\ref{R1R2b}), the transmit power is equal to $P = \\frac{2^{R_1}-1}{g_1} = \\frac{2^{R_0}-1}{g_0}$. At the power splitter, $1-\\rho(t)$ portion of the received power is directed into the ID, so the achievable mutual information accumulation is:\n\\small\n\\begin{align}\nR(t) = \\log_2(1 + g(t)P(1-\\rho(t))).\\label{splittedR}\n\\end{align} \n\\normalsize\nNote that the maximum value of the mutual information is attained by setting $\\rho=0$. Inserting the value of $P$ in (\\ref{splittedR}) for GOOD and BAD channel states gives the mutual information accumulation in these states respectively for a given power splitting ratio $\\rho$ as\n\\small\n\\begin{align}\nR^H(\\rho)= \\log_2(\\rho+(1-\\rho) 2^{R_1}),\\\\\nR^L(\\rho)= \\log_2(\\rho+(1-\\rho) 2^{R_0}).\n\\end{align}\n\\normalsize\n Thus, the accumulated mutual information, $I(t)$, evolves as:\n\\small\n\\begin{align}\nI(t)=&\\left\\{\n\\begin{array}{ll}\n\\min(I(t-1)+R^H(\\rho(t)),R_1),& \\text{if}\\ G_t=1\\\\\n\\min(I(t-1)+R^L(\\rho(t)),R_1),& \\text{if}\\ G_t=0\n\\end{array}\n\\right..\\label{I}\n\\end{align}\n\\normalsize\nNote that (\\ref{I}) follows from the operation of HARQ-IR which is described in Section \\ref{sec:harq} where the received messages over different time slots are combined in such a way that the mutual information of the combined messages is the summation of the individual mutual information of the messages. The instantaneous reward is zero if the message can be correctly decoded, and it is minus one otherwise. Recall that the decoding operation is successful if and only if the accumulated mutual information is above a certain threshold, and the battery level is sufficient to decode the message. Hence, the instantaneous reward is given as follows:\n\\small\n \\begin{align}\nU(S_{t},\\rho(t))=&\\left\\{\n \\begin{array}{ll}\n 0,& \\text{if}\\ B_t\\geq E_d,\\ \\text{and}\\ I(t)\\geq R_1,\\\\\n -1,& \\text{if}\\ \\text{otherwise}.\n \\end{array}\n \\right..\n \\end{align}\n\\normalsize\n Define the value function $V(b,m)$ as\n\\small\n \\begin{align}\n V(b,\\ m) &= \\max_{\\pi}V^{\\pi}(b,\\ m),\\ \\forall b\\in[0,\\infty),\\ \\forall m\\in\\left[0,\\ R_1\\right]\\label{Vmax}.\n \\end{align}\n\\normalsize\n The value function $V(b, m)$ satisfies the Bellman equation\n\\small\n \\begin{align}\n V(b, m) = \\max_{0\\leq \\rho \\leq 1}V_\\rho(b,m), \\label{belman}\n \\end{align}\n\\normalsize\n where $V_\\rho(b,m)$ is the expected reward achieved by taking action $\\rho$ when\n the state is $(b, m)$ and is given by\n\\small\n \\begin{align}\n V_{\\rho}(b,m)=U((b,\\ m),\\rho)+\\beta\\mathds{E}\\left[V(\\acute{b},\\ \\acute{m})|S=(b,\\ m)\\right], \\label{actionvalue}\n \\end{align}\n\\normalsize\n where $(\\acute{b}, \\acute{m})$ is the next visited state and the expectation is over the distribution of the next state. The use of expected discounted reward allows us to obtain a tractable solution, and one can gain insights into the optimal policy when $\\beta$ is close to $1$. Value iteration algorithm (VIA) is a standard tool to solve Bellman equations such as the one in (\\ref{belman}). However, this problem suffers from the curse of dimensionality \\cite{sutton}. Note that from (6) and (10), the problem is a two dimensional uncountable state MDP with continuous actions at every state. Also, letting $\\beta \\rightarrow 1$, to approximate the average reward, slows down the algorithm to the point of infeasibility [30]. Hence, in the following, we take advantage of the special structure of our problem to derive an important characteristic of the optimal policy. The flow of the paper is depicted in Figure \\ref{fig:diagram}.\n \n \n \\begin{figure}[ht]\n \\centering\n \\includegraphics[scale=0.2]{diagram.eps}\n\t\t \\caption{A brief overview of the paper.}\n\t\t\t\\label{fig:diagram}\n\\end{figure}\n\n\\subsection{Absorbing Markov Chain Formulation}\nNote that the MC describing the operation of our system is an {\\em absorbing} MC, where all states except those $(b, m)$ where $b\\geq E_d$, and $m\\geq R_1$ are transient states. The absorbing states are those where the receiver has both sufficient energy and information accumulated to correctly decode. In an absorbing chain, starting from a transient state, the chain makes a finite number of visits to some transient states before its eventual absorption into one of the absorbing states. Hence, the mean time to absorption of the chain, starting from transient state $i$ initially, is the sum of the expected numbers of visits made to transient states. In an absorbing MC, the expected number of steps taken before being absorbed in an absorbing state characterizes the \\emph{mean time to absorption}. Hence, the mean time to absorption starting from a given transient state $(b,\\ m)$ provides the number of re-transmissions until successful decoding when the battery has $b$ units of energy and the memory contains $m$ bits of information. \n\n\n\n\n\n After establishing the $\\rho$ dependent state evolution of $B(t)$ and $I(t)$, we can formally introduce the state transition probabilities of the Markov chain as follows:\n\n \\begin{align}\n \\rho=1\\Rightarrow&\\left\\{\n \\begin{array}{ll}\n \\mathds{Pr}\\left((B,I),(B+l,I)\\right) = \\lambda\\\\\n \\mathds{Pr}\\left((B,I),(B,I)\\right) = 1-\\lambda\n \\end{array}\n \\right.,\n \\end{align}\n \\begin{align}\n \\rho=0\\Rightarrow&\\left\\{\n \\begin{array}{ll}\n \\mathds{Pr}\\left((B,I),(B-1,R_1)\\right) = \\lambda\\\\\n \\mathds{Pr}\\left((B,I),(B-1,I+1)\\right) = 1-\\lambda\n \\end{array}\n \\right.,\n \\end{align}\n\n \\begin{align}\n 0<\\rho<1\\Rightarrow&\\left\\{\n \\begin{array}{ll}\n \\mathds{Pr}\\left((B,I),(B-1+\\rho l,I+R^H(\\rho))\\right) = \\lambda\\\\\n \\mathds{Pr}\\left((B,I),(B-1,I+R^L(\\rho))\\right) = 1-\\lambda\n \\end{array}\n \\right.,\n \\end{align}\nwhere $\\mathds{Pr}(\\mathbf{x},\\mathbf{y})$ is the transition probability from state $\\mathbf{x}$ into state $\\mathbf{y}$, $B\\in [0,\\infty)$ and $I\\in [0,\\ R_1]$. The state transition probabilities of the Markov chain associated with $\\rho$ is depicted in Figure \\ref{fig:DTMC}.\n\n \n\n \\begin{figure}\n \\centering\n \\begin{subfigure}[t]{0.15\\textwidth}\n \\includegraphics[width=\\textwidth]{DTMC1.eps}\n \\caption{$\\rho = 0$.}\n \\label{fig:DTMC1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[t]{0.15\\textwidth}\n \\includegraphics[width=\\textwidth]{DTMC2.eps}\n \\caption{$\\rho = 1$. }\n \\label{fig:DTMC2}\n \\end{subfigure}\n \\begin{subfigure}[t]{0.15\\textwidth}\n \\includegraphics[width=\\textwidth]{DTMC3.eps}\n \\caption{$0<\\rho<1$.}\n \\label{fig:DTMC3}\n \\end{subfigure}\n \t \\caption{State transition probabilities of the Markov chain associated with $\\rho$.}\n \\label{fig:DTMC}\n \n \\end{figure}\nIn the following, we perform \\emph{first-step analysis}, by conditioning on the first step\nthe chain makes after moving away from a given initial state to obtain the mean time to absorption. Let $k_{b,m}$ be the expected number of transitions needed to hit\nan absorbing state when the MC starts from state $(b,\\ m)$. The analysis is performed by assuming that the MC is in steady-state.\n\nLet us first consider two trivial cases; when the battery has less than one unit of energy, i.e., $b<1$, in which case the receiver has no option but harvest the incoming RF signal, and when the amount of accumulated mutual information is $R_1$, in which case there is no point in further accumulating mutual information since the receiver has sufficient mutual information to decode the incoming packet. For these cases, the mean time to absorption starting from an initial state $(b,\\ m)$ is\n\\begin{equation} \\label{kBless1}\n\\begin{split}\nk_{b,m} &= 1+ \\lambda k_{b+e,m}+(1-\\lambda) k_{b,m} \\\\\n &=\\frac{1}{\\lambda} + k_{b+e,m},\\hspace{1cm} \\text{if}\\ b<1\\ \\text{or}\\ m = R_1.\n\\end{split}\n\\end{equation}\nNote that in (\\ref{kBless1}), one slot is needed to harvest energy, and depending on the channel state in that slot, the battery state either transitions to $b+e$ or remains the same. The following lemma plays an important role in establishing the structure of the optimal policy.\n\n\\begin{lemma}\n\\label{lemma1}\nFor any $E_d-i\\cdot e \\leq b < E_d-(i-1)\\cdot e$ such that $i=1,\\ldots,E_d$, given that $m=R_1$, the mean time to absorption is given by, $k_{b,R_1} = \\frac{i}{\\lambda}$.\n\\end{lemma}\n\\begin{proof}\nThe proof is given in Appendix \\ref{prooflemma1}.\n\\end{proof}\n\n\nWe will use Lemma \\ref{lemma1} to show that the optimal policy minimizing the mean time to absorption \\emph{does not} need to split the incoming RF signal. In order to show this, let us define two tail policies $\\boldsymbol{\\pi}^i_t=(a_i,\\boldsymbol{\\pi}_{t+1})$, $i=split,no-split$ taking different actions $a_i$, in the current slot, but following the same set of actions, $\\boldsymbol{\\pi}_{t+1}$ afterwards\\footnote{Note that $(a_i,\\boldsymbol{\\pi}_{t+1})$ defines a tail policy obtained by concatenating action $a_i$ in the current slot with tail policy $\\boldsymbol{\\pi}_{t+1}$.}. Let policy $\\boldsymbol{\\pi}^{split}_t=(\\rho, \\boldsymbol{\\pi}_{t+1})$ be a tail policy that always splits the incoming RF energy, i.e., $0<\\rho<1$, except when $B(t)<1$ or $I(t)=R_1$, when it only harvests energy.\nAssume that the state of the system is $(b,\\ m)$ at time slot $t$. Then, the mean time to absorption for tail policy $\\boldsymbol{\\pi}^{split}_t$ is:\n\\small\n\\begin{align}\nk^{\\boldsymbol{\\pi}^{split}}_{b,m} = 1 + \\lambda k_{b-1+\\rho e,m+R^H(\\rho)} + (1-\\lambda) k_{b-1,m+R^L(\\rho)},\\label{k_pi}\n\\end{align}\n\\normalsize\nwhere $k_{x,y}$ is the mean time to absorption of policy $\\boldsymbol{\\pi}_{t+1}$ beginning at state $(x,y)$. Note that with probability $\\lambda$ the channel is in GOOD state, and thus, $\\rho\\cdot e$ units of energy is harvested\\footnote{We assume that the energy harvesting circuit is generating energy linearly proportional to the energy of the incoming RF signal.}. However, one unit of energy is spent by operating the transceiver to accumulate $R^H(\\rho)$ bits of mutual information. Meanwhile, with probability $1-\\lambda$ the channel is in BAD state, and no energy is harvested, but the transceiver still consumes one unit of energy to accumulate $R^L(\\rho)$ bits of mutual information.\n\n\nUnder tail policy $\\boldsymbol{\\pi}^{no-split}_t$ the RF signal is never split at time slot $t$, but rather, it is completely used for mutual information accumulation except when $B(t)<1$ or $I(t) = R_1$ when it harvests energy only. In a similar way as before, we may calculate $k^{\\boldsymbol{\\pi}^{no-split}}_{b,m}$ as follows:\n\\small\n\\begin{align}\nk^{\\boldsymbol{\\pi}^{no-split}}_{b,m} = 1 + \\lambda k_{b-1,R_1} + (1-\\lambda)k_{b-1,m+ R_0}.\\label{kalpha}\n\\end{align}\n\\normalsize\n\\begin{theorem}\n\\label{TH_dis}\nPolicy $\\boldsymbol{\\pi}^{no-split}_t$ in \\eqref{kalpha} achieves an expected number of re-transmission that is never worse than that of policy $\\boldsymbol{\\pi}^{split}_t$ in (19), i.e., $k^{\\boldsymbol{\\pi}^{no-split}}_{b,m}\\leq k^{\\boldsymbol{\\pi}^{split}}_{b,m}$ for every $b=0,1,\\ldots$ and $m=0,1,\\ldots,R_1$.\n\\end{theorem}\n\\begin{proof}\nThe proof is given in Appendix \\ref{proofTH_dis}.\n\\end{proof}\nTheorem \\ref{TH_dis} proves that a no-splitting policy can achieve the minimum number of re-transmissions. Hence, in the latter part of the paper, we focus on characterizing the optimal no-splitting policy by determining the scheduling decision between EH or ID for each state of the MC. Therefore, the state space of the discrete MC associated with the optimal no-splitting policy is $b = 0,1,\\ldots,\\infty$, and $m = 0,1,\\ldots,R_1$\\footnote{Note that in the original problem the states of the MC are $[0,\\infty)\\times[0,R_1]$.}. \n \\begin{remark}\nTheorem \\ref{TH_dis} plays an important role in simplifying the original problem by reducing the two dimensional uncountable state MDP with continuous action space into a two dimensional countable state MDP with binary decision space. This significantly reduces the complexity of numerical methods such as VIA. However, as we shall see in Section \\ref{sec:iid}, the absorbing MC framework helps prove the optimality of a class of simple-to-implement algorithms that is more suitable for resource-deficient EH devices. \n \\end{remark}\n\n\n\nSince the class of policies that we are interested in does not observe the channel, but make a decision based only on $(b,\\ m)$, the time of the decision is irrelevant. Hence, given $(b,\\ m)$, time $t$ and $t+1$ are stochastically identical. Therefore, in the rest of the paper we will omit the time index and optimize the scheduling decisions for any given state $(b,\\ m)$. Define $\\boldsymbol{\\pi}^*$ as the optimal policy minimizing the mean time to absorption beginning at any given state $(b,\\ m)$. Let $k^{\\boldsymbol{\\pi}^*}_{b,m}$ be the minimum mean time to absorption obtained by policy $\\boldsymbol{\\pi}^*$\\footnote{Note that the mean time to absorption calculated in Lemma \\ref{lemma1} is the smallest possible value, i.e., $k^{\\boldsymbol{\\pi}^*}_{b,R_1}=k_{b,R_1}$ for $b=0,1,\\ldots, E_d-1$.}. Define the tail policy $\\boldsymbol{\\pi}^{i}(b,m) = (i,\\boldsymbol{\\pi}^*(\\acute{b},\\ \\acute{m}))$, $i=0,1$ such that it chooses $\\rho=i$ at state $(b,\\ m)$ but follows policy $\\boldsymbol{\\pi}^*$ after transitioning into the new state $(\\acute{b},\\ \\acute{m})$. Let $k^{\\boldsymbol{\\pi}^i}_{b,m}$ be the mean time to absorption of policy $\\boldsymbol{\\pi}^{i}(b,m)$, $i=0,1$. We can characterize $k^{\\boldsymbol{\\pi}^0}_{b,m}$ and $k^{\\boldsymbol{\\pi}^1}_{b,m}$ as follows:\n\\small\n\\begin{align}\nk^{\\boldsymbol{\\pi}^0}_{b,m} &= 1 + \\lambda k^{\\boldsymbol{\\pi}^*}_{b-1,R_1} + (1-\\lambda)k^{\\boldsymbol{\\pi}^*}_{b-1,m+1},\\label{k0}\\\\\nk^{\\boldsymbol{\\pi}^1}_{b,m} &= 1 + \\lambda k^{\\boldsymbol{\\pi}^*}_{b+e,m} + (1-\\lambda)k^{\\boldsymbol{\\pi}^1}_{b,m}\\nonumber\\\\\n&=\\frac{1}{\\lambda} + k^{\\boldsymbol{\\pi}^*}_{b+e,m}\\label{k1}.\n\\end{align}\n\\normalsize\nNote that by evaluating and then comparing the values of $k^{\\boldsymbol{\\pi}^0}_{b,m}$ and $k^{\\boldsymbol{\\pi}^1}_{b,m}$, at all possible states $(b,\\ m)$ for $b = 0,1,\\ldots,\\infty$, and $m = 0,1,\\ldots,R_1$, one can obtain the optimal policy $\\boldsymbol{\\pi}^*$ and its associated $k^{\\boldsymbol{\\pi}^*}_{b,m}$.\n\n\\begin{theorem}\n\\label{lemma2}\nFor states $(b,\\ m)=(E_d+j,\\ R_1-j)$ for $j = 1,2,\\ldots,R_1$, the minimum mean time to absorption, $k^{\\boldsymbol{\\pi}^*}_{b,m}$ is given by\n\\small\n\\begin{align}\nk^{\\boldsymbol{\\pi}^*}_{E_d+j,R_1-j} = k^{\\boldsymbol{\\pi}^0}_{E_d+j,R_1-j} = \\sum^{j}_{i=1} (1-\\lambda)^{i-1}.\n\\end{align}\n\\normalsize\nFurthermore, $k^{\\boldsymbol{\\pi}^*}_{b,R_1-j}=k^{\\boldsymbol{\\pi}^0}_{E_d+j,R_1-j}$ for $b=E_d+j+1, E_d+j+2,\\ldots$.\n\\end{theorem}\n\\begin{proof}\nThe proof is given in Appendix \\ref{prooflemma2}.\n\\end{proof}\nTheorem \\ref{lemma2} states that if the receiver has $R_1-n$ bits of mutual information accumulated and more than $E_d+n$ units of energy in its battery, then it should use the incoming RF signal for mutual information accumulation only. For any given state $(b,\\ m)$, we exploit Lemma \\ref{lemma1} and Theorem \\ref{lemma2} to develop Algorithm \\ref{alg1} for calculating the minimum mean time to absorption, $k^{\\boldsymbol{\\pi}^*}_{b,m}$, and the optimal scheduling decision at every state.\n\n\nThe idea of Algorithm \\ref{alg1} is to use Lemma \\ref{lemma1} and Theorem \\ref{lemma2} as boundary conditions and to recursively calculate the mean time to absorption $k^{\\boldsymbol{\\pi}^0}_{b,m}$ and $k^{\\boldsymbol{\\pi}^1}_{b,m}$ starting from $(b,\\ m)=(E_d,R_1 -1)$. Note that $k^{\\boldsymbol{\\pi}^0}_{E_d,R_1 -1}$ and $k^{\\boldsymbol{\\pi}^1}_{E_d,R_1 -1}$ depend on the values of $k^{\\boldsymbol{\\pi}^*}_{E_d-1,R_1}$ and $k^{\\boldsymbol{\\pi}^*}_{E_d+1,R_1 -1}$, which are obtained in the initialization step, and the optimal scheduling decision at state $(E_d, R_1-1)$ is given by $\\arg\\min_{i\\in{0,1}} k^{\\boldsymbol{\\pi}^i}_{b,m}$. The procedure in Algorithm \\ref{alg1} continues by decrementing the value of $b$ by $1$ at each iteration, until $b=0$ at which time the value of $m$ is decremented by $1$, $b$ is initialized to $E_d+n$ and the procedure is repeated. The aforementioned order of spanning the states of the MC ensures that at each iteration the mean time to absorption can be calculated from the values determined in the previous iterations. We have shown in Appendix \\ref{proofalg1}, that Algorithm \\ref{alg1} minimizes the expected number of re-transmissions starting from any state $(b,\\ m)$.\n\n\n\\begin{algorithm}\n\\caption{Calculating the minimum mean time to absorption for an i.i.d. channel}\\label{alg1}\n\\begin{algorithmic}[1]\n\\State Initialize $k^{\\boldsymbol{\\pi}^*}_{b,R_1}$ for $b = 0,\\ldots,E_d-1$ using Lemma \\ref{lemma1}.\n\\State Initialize $k^{\\boldsymbol{\\pi}^*}_{E_d+j,R_1-j}$ for $j = 1,\\ldots,R_1$ using Theorem \\ref{lemma2}.\n\\State $n\\gets 0$\n\\For{$m = R_1-1:0$}\n\\For{$b = E_d+n:0$}\n\\State Calculate $k^{\\boldsymbol{\\pi}^0}_{b,m}$, $k^{\\boldsymbol{\\pi}^1}_{b,m}$ from (\\ref{k0}) and (\\ref{k1}), respectively.\n\\State $k^{\\boldsymbol{\\pi}^*}_{b,m}=\\min\\left(k^{\\boldsymbol{\\pi}^0}_{b,m}, k^{\\boldsymbol{\\pi}^1}_{b,m}\\right)$.\n\\State $\\rho^*(b,m) = \\arg\\min_i k^{\\boldsymbol{\\pi}^i}_{b,m}$ for $i = 0,1$\n\\EndFor\n\\State $n\\gets n+1$\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\section{Optimal Class of Policies for i.i.d. Channels}\n\\label{sec:iid}\n\n\nIn the previous section, we have given a procedure to obtain the optimal scheduling decision of a no-splitting policy, once we established that there exists a no-splitting policy achieving the minimum number of re-transmissions. In this section, we formally determine the optimal class of scheduling policies minimizing the number of re-transmissions until successful decoding. In the following, we obtain our analytical results for $e=1$ and $R_0=1$. However, our analysis holds in general for different values of $e$ and $R_0$, as demonstrated by the numerical results presented in Section \\ref{Results}. Note that once the battery has sufficient charge to decode the packet, i.e., $b=E_d+1,E_d+2,\\ldots$, it is better to use the incoming RF signal only for information accumulation.\nFor the remaining states, i.e., $b=1,2,\\ldots, E_d$, and $m=0,1,\\ldots R_1-1$, any scheduling decision, either $\\rho=0$ or $\\rho=1$, is optimal. These two facts are proven formally in Appendix \\ref{proofTH_str1} and \\ref{proofTH_str2} respectively. This result, in essence, proves that there is no unique optimal policy. Instead, there exists a family of optimal policies achieving the same minimum mean time to absorption. We summarize our findings so far in the following theorem by formally characterizing the family of optimal policies.\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\n\\label{TH_str}\nOptimal policy, $\\boldsymbol{\\pi}^*$, satisfies the following properties.\n\\begin{enumerate}\n\\item If $b=0$ or $m=R_1$, it chooses $\\rho=1$.\n\\item If $b=E_d+1,E_d+2,\\ldots$, it chooses $\\rho=0$.\n\\item If $b=1,2,\\ldots, E_d$ and $m=0,1,\\ldots R_1-1$, chooses either $\\rho=0$ or $\\rho=1$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nThe proof is given in Appendix \\ref{proofTH_str}.\n\\end{proof}\n\nSimple examples of such optimal policies that belong to the optimal family of policies characterized in Theorem \\ref{TH_str}, are:\n\n\\begin{itemize}\n\\item Battery First (BF): the receiver harvests energy until it acquires $E_d$ units of energy and then starts accumulating the mutual information.\n\\item Information First (IF): the receiver always accumulates mutual information \nunless $b=0$ or $m=R_1$.\n\\item Coin Toss (CT): the receiver harvests energy when $b=0$ or $m=R_1$, while it accumulates mutual information when $b=E_d+1,E_d+2,\\ldots$. Otherwise, it tosses a fair coin to choose between harvesting energy or accumulating mutual information.\n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\\section{Expected Number of Re-Transmissions for a Correlated Channel}\n\\label{sec:cor}\nIn many wireless systems, the wireless channel cannot be modeled as an i.i.d. channel. In this section, we investigate optimal\nscheduling policies under a time-correlated channel model. Our analysis for a correlated channel follows a similar approach to our analysis for i.i.d. channels. However, due to correlation between the subsequent channel states, the receiver can improve its decision by incorporating its knowledge of the current state. \nLet the transition probabilities of the channel states be $\\mathds{Pr}\\left[G_t=1|G_{t-1}=1\\right]=\\lambda_1$ and $\\mathds{Pr}\\left[G_t=1|G_{t-1}=0\\right]=\\lambda_0$. Note that due to time correlation, the previous state of the channel provides information about the current channel state to the receiver. Hence, although once again we model the system as a MC, this time the state space of MC is extended where the states are $(b,\\ m,\\ G)$ with $G$ being the previous state of the channel\\footnote{Note that the receiver becomes aware of the channel state after it decides to sample the incoming RF signal.}. The resulting MC is still an absorbing MC, and the mean time to absorption is equivalent to the minimum expected number of re-transmissions until successful decoding. Define $\\boldsymbol{\\pi}^*$ as the optimal policy minimizing the mean time to absorption at any given state $(b,\\ m,\\ G)$. Let $k^{\\boldsymbol{\\pi}^*}_{b,m,G}$ be the mean time to absorption obtained by policy $\\boldsymbol{\\pi}^*$ at state $(b,\\ m,\\ G)$. \n\n\\begin{lemma}\n\\label{cor1}\nFor any $E_d-i\\cdot e \\leq b < E_d-(i-1)\\cdot e$ such that $i=1,\\ldots,E_d$, and given that $m=R_1$, the minimum mean time to absorption is given by \n\\small\n\\begin{align}\n&k^{\\boldsymbol{\\pi}^*}_{b,R_1,1} =i \\frac{1+\\lambda_0-\\lambda_1}{\\lambda_0}, &i=1,\\ldots,E_d,\\label{eq:alg1}\\\\\n&k^{\\boldsymbol{\\pi}^*}_{b,R_1,0} =\\frac{1}{\\lambda_0} + (i-1) \\frac{1+\\lambda_0-\\lambda_1}{\\lambda_0}, &i=1,\\ldots,E_d.\\label{eq:alg2}\n\\end{align}\n\\normalsize\n\\end{lemma}\n\\begin{proof}\nThe proof is similar to that of Lemma \\ref{lemma1}. The detailed proof is given in \\cite{arxivmehdi}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\nSimilar to Theorem \\ref{TH_dis}, by exploiting Lemma \\ref{cor1}, we can prove that the optimal policy should either choose energy harvesting or information accumulation at any given state $(b,\\ m,\\ G)$. Therefore, MC associated with the optimal strategy has discrete states in which $b = 0,1,\\ldots,\\infty$, $m = 0,1,\\ldots,R_1$ and $G = 0,1$. Define the tail policy $\\boldsymbol{\\pi}^{i}(b,m,G) = (i,\\boldsymbol{\\pi}^*(\\acute{b},\\ \\acute{m},\\ \\acute{G}))$, $i=0,1$ that chooses $\\rho=i$ at state $(b,\\ m,\\ G)$ but follows policy $\\boldsymbol{\\pi}^*$ after transitioning into the new state $(\\acute{b},\\ \\acute{m},\\ \\acute{G})$. Let $k^{\\boldsymbol{\\pi}^i}_{b,m,G}$ be the mean time to absorption of policy $\\boldsymbol{\\pi}^{i}(b,m,G)$, $i=0,1$. We can calculate $k^{\\boldsymbol{\\pi}^0}_{b,m,G}$ and $k^{\\boldsymbol{\\pi}^1}_{b,m,G}$ as follows:\n\\small\n\\begin{align}\n&k^{\\boldsymbol{\\pi}^0}_{b,m,0}= 1+ \\lambda_0 k^{\\boldsymbol{\\pi}^*}_{b-1,R_1,1} + (1-\\lambda_0)k^{\\boldsymbol{\\pi}^*}_{b-1,m+1,0}\\label{inf0},\\\\\n&k^{\\boldsymbol{\\pi}^0}_{b,m,1}= 1+ \\lambda_1 k^{\\boldsymbol{\\pi}^*}_{b-1,R_1,1} + (1-\\lambda_1)k^{\\boldsymbol{\\pi}^*}_{b-1,m+1,0}\\label{inf1},\\\\\n&k^{\\boldsymbol{\\pi}^1}_{b,m,0}= 1+ \\lambda_0 k^{\\boldsymbol{\\pi}^*}_{b+1,m,1} + (1-\\lambda_0)k^{\\boldsymbol{\\pi}^1}_{b,m,0}=\\frac{1}{\\lambda_0} + k^{\\boldsymbol{\\pi}^*}_{b+1,m,1},\\label{eh0}\\\\\n&k^{\\boldsymbol{\\pi}^1}_{b,m,1}= 1+ \\lambda_1 k^{\\boldsymbol{\\pi}^*}_{b+1,m,1} + (1-\\lambda_1)k^{\\boldsymbol{\\pi}^*}_{b,m,0}.\\label{eh1}\n\\end{align}\n\\normalsize\n\n\n\nSimilar to the outline of the Theorem \\ref{lemma2}, in the following, we consider states $(b,\\ m,\\ G)=(E_d+j,R_1-j,G)$ for $j=1,\\ldots,R_1$ and derive the optimal strategy for those states. \n\n\\begin{lemma}\n\\label{cor-lema}\nThe optimal strategy in states $(E_d+j,R_1-j,G)$ for $j=1,\\ldots,R_1$ and $G = 0,1$ is to accumulate mutual information ($\\rho^*(E_d+j,R_1-j,G) = 0$) and also $k^{\\boldsymbol{\\pi}^*}_{b,R_1-j,G} =k^{\\boldsymbol{\\pi}^0}_{E_d+j,R_1-j,G}$ for $b= E_d+j+1,E_d+j+2,\\ldots$.\n\n\\end{lemma}\n\\begin{proof}\nThe proof is similar to that of Theorem \\ref{lemma2}. The detailed proof is given in \\cite{arxivmehdi}.\n\\end{proof}\nNow that we know the optimal policy for states $(E_d+j,R_1-j,G)$, we can calculate the minimum mean time to absorption for those states as follows:\n\\small\n\\begin{align}\n&k^{\\boldsymbol{\\pi}^*}_{E_d+j,R_1-j,0} =k^{\\boldsymbol{\\pi}^0}_{E_d+j,R_1-j,0}= \\sum^{j}_{i=1}(1-\\lambda_0)^{i-1},\\hspace{0.5cm} j=1,\\ldots,R_1,\\label{eq:alg3}\\\\\n&k^{\\boldsymbol{\\pi}^*}_{E_d+j,R_1-j,1} =k^{\\boldsymbol{\\pi}^0}_{E_d+j,R_1-j,1}=1+ (1-\\lambda_1)\\sum^{j-1}_{i=1}(1-\\lambda_0)^{i-1},\\nonumber\\\\ &\\hspace{6cm} j=2,\\ldots,R_1,\\label{eq:alg4}\\\\\n&k^{\\boldsymbol{\\pi}^*}_{E_d+j,R_1-j,1} = 1,\\hspace{3.65cm} j=1.\\label{eq:alg5}\n\\end{align}\n\\normalsize\n\nAlgorithm \\ref{alg2} calculates the $k^{\\boldsymbol{\\pi}^*}_{b,m,G}$ and the corresponding $\\rho^*$ for any $b$, $m$, and $G$. Proving the optimality of Algorithm \\ref{alg2} is similar to the outline of the optimality proof of Algorithm \\ref{alg1} and hence it is omitted here. Note that the knowledge of the previous channel state, $G$, enables the receiver to be able to fully utilize the information yielded by the correlation. However, it also results in four coupled equations, (\\ref{inf0})-(\\ref{eh1}), over numerous states which makes the analysis extremely hard. For this reason, we omit the full characterization of the structure of the optimal policy. Nevertheless, note that Algorithm \\ref{alg2} provides a recursive method to determine the optimal scheduling decisions for each state $(b,m, G)$. In fact, we use these optimal decisions in the numerical experiments discussed in Section \\ref{Results} to calculate the minimum number of re-transmissions.\n\n\\begin{algorithm}\n\\caption{Calculating the minimum mean time to absorption for correlated channel}\\label{alg2}\n\\begin{algorithmic}[1]\n\\State Initialize $k^{\\boldsymbol{\\pi}^*}_{b,R_1,G}$ for $b = 0,\\ldots,E_d-1$ using (\\ref{eq:alg1}) and (\\ref{eq:alg2}).\n\\State Initialize $k^{\\boldsymbol{\\pi}^*}_{E_d+j,R_1-j,G}$ for $j = 1,\\ldots,R_1$ using (\\ref{eq:alg3}), (\\ref{eq:alg4}) and (\\ref{eq:alg5}).\n\\State $n\\gets 0$\n\\For{$m = R_1-1:0$}\n\\For{$b = E_d+n:0$}\n\\State Calculate $k^{\\boldsymbol{\\pi}^0}_{b,m,G}$ for $G=0,1$ using (\\ref{inf0}) and (\\ref{inf1}), respectively.\n\\State Calculate $k^{\\boldsymbol{\\pi}^1}_{b,m,G}$ for $G=0,1$ using (\\ref{eh0}) and (\\ref{eh1}), respectively.\n\\State $k^{\\boldsymbol{\\pi}^*}_{b,m,G}=\\min\\left(k^{\\boldsymbol{\\pi}^0}_{b,m,G}, k^{\\boldsymbol{\\pi}^1}_{b,m,G}\\right)$.\n\\State $\\rho^*(b,m,G) = \\arg\\min_i k^{\\boldsymbol{\\pi}^i}_{b,m,G}$ for $i = 0,1$\n\\EndFor\n\\State $n\\gets n+1$\n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\n\\section{Numerical Results}\\label{Results}\n\nIn this section, we provide numerical evidence to support the analytical results established in the paper. VIA is a standard tool for solving the bellman equations in (\\ref{actionvalue}). However, VIA iterates for numerous passes over each state, which is increasing in $\\beta$, before converging to a steady solution, whereas Algorithm \\ref{alg1} and \\ref{alg2} needs a single iteration. Moreover, VIA achieves exactly the same performance as Algorithm \\ref{alg1} and \\ref{alg2}. Thus, we omit the results obtained by VIA.\n\nWe will divide our attention to validate the optimal policy for i.i.d. and correlated channel models. Although the framework discussed is sufficiently general to determine the number of re-transmissions starting from any residual battery level, in this section for the clarity of presentation, we consider that the initial battery level is zero. We use a simple ARQ mechanism as a baseline for understanding the performance merits of the HARQ mechanism. In the following, we formally define the simple ARQ scheme for i.i.d. and correlated channels.\n\n\\subsection{Simple ARQ}\\label{simple_harq}\nIn simple ARQ, the packet is transmitted successfully whenever the channel is in a GOOD state and the receiver has sufficient energy to decode the packet. Otherwise, the receiver drops the packet and awaits re-transmissions. When the receiver employs simple ARQ, before any decoding attempt, it has to make sure that its battery has at least $E_d+1$ units of energy. Otherwise, after consuming 1 unit of energy for sampling, it will not have sufficient energy to decode the data packet and it will drop the packet. It is easy to prove that the optimal simple ARQ policy minimizing the mean time to absorption first harvests $E_d+1$ units of energy and then attempts decoding. If the decoding attempt is not successful, it harvests energy until its battery state reaches $E_d+1$ units again before attempting to decode. \n\n\n\n\n\\subsection{i.i.d. Channel States}\n\\label{num_iid}\nIn this section, we evaluate the minimum mean time to absorption obtained from Algorithm 1, and compare it to that of the following three simple policies. The studied policies are as follows: $i)$ Battery First (BF), $ii)$ Information First (IF), and $iii)$ Coin Toss (CT). Also, we compare the performance of the receiver equipped with HARQ mechanism with the case of a receiver equipped with simple ARQ mechanism. We determine the mean number of re-transmissions by Monte Carlo simulations, and compare them with that of analytical calculation described in Algorithm 1. Note that Monte Carlo simulations provide only \\emph{sample} mean time which is a random variable. The mean of this random variable is equal to the mean time to absorption and its variance decreases with the number of samples and becomes zero only if the number of iterations go to infinity. Hence, we expect to see small differences between the results obtained by the Monte Carlo simulations and analytical results, which is the reason why some policies have slightly smaller mean time to absorption than the optimal analytical value.\n\n\nTable \\ref{son} summarizes the mean time to absorption for $R_1 = 10$, $e=1$, $\\lambda=0.5$ and $E_d=5$ with respect to $R_0$ associated with different policies. For IF, BF, CT and simple ARQ policies, we run Monte Carlo simulations for $10^7$ iterations and evaluate the sample mean. It can be seen from Table \\ref{son} that all policies have almost the same performance. This observation confirms our major finding that the optimal policy achieving the minimum mean time to absorption is not unique. \n\nThe effect of quality of the channel on the mean time to absorption for $R_0 = 5$, $R_1=10$, $E_d=5$ and $e=2$ with respect to $\\lambda$ is summarized in Table \\ref{son_vs_lamda}. As expected, it can be seen that the mean time to absorption decreases as the channel quality improves. Also, the performance gap between the HARQ and simple ARQ mechanism becomes smaller as the channel quality improves. This is because as the channel quality improves, the probability of harvesting energy and accumulating $R_1$ bits of mutual information also increases. Finally, the mean time to absorption for $R_0 = 5$, $R_1=10$, $E_d=10$ and $\\lambda=0.3$ with respect to $e$ is summarized in Table \\ref{son_vs_l}. We observe that the mean time to absorption is approximately the same for all policies and it is decreasing with respect to the amount of harvested energy, $e$.\n\n\nThe results presented in Table \\ref{son}, \\ref{son_vs_lamda} and \\ref{son_vs_l} confirm our theoretical results that, indeed, the optimal policy harvests energy whenever $b = 0$ or $m=R_1$ and accumulates mutual information whenever $b>E_d$. For the rest of the states it does not matter what the receiver does, as long as, it does not split the received RF signal.\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Mean time to absorption for $R_1 = 10$, $e=1$ and $E_d=5$ vs. $R_0$}\n\\label{son}\n\\begin{tabular}{lllllllllll}\n & $R_0=1$ & $R_0=2$ & $R_0=3$ & $R_0=4$ & $R_0=5$ & $R_0=6$ & $R_0=7$ & $R_0=8$ & $R_0=9$ \\\\\nOptimal analytical & 15.9941 & 15.8125 & 15.6250 & 15.2500 & 14.5000 & 14.5000 & 14.5000 & 14.5000 & 14.5000 \\\\\nOptimal Monte-Carlo & 15.9910 & 15.8103 & 15.6235 & 15.2490 & 14.4992 & 14.5001 & 14.4998 & 14.5000 & 14.4983 \\\\\nBF & 15.9938 & 15.8116 & 15.6259 & 15.2504 & 14.4999 & 14.4995 & 14.4993 & 14.5012 & 14.5000 \\\\\nIF & 15.9941 & 15.8143 & 15.6245 & 15.2508 & 14.4987 & 14.4997 & 14.5017 & 14.4989 & 14.5003 \\\\\nCT & 15.9966 & 15.8140 & 15.6266 & 15.2491 & 14.5020 & 14.5007 & 14.5009 & 14.4984 & 14.5001 \\\\\nSimple ARQ & 15.9992 & 15.9992 & 15.9992 & 16.0006 & 16.0007 & 15.9995 & 15.9996 & 16.0008 & 16.0011 \n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{Mean time to absorption for $R_1 = 10$, $R_0=5$, $e=2$ and $E_d=5$ vs. $\\lambda$}\n\\label{son_vs_lamda}\n\\begin{tabular}{lllllllllll}\n & $\\lambda=0.1$ & $\\lambda=0.2$ & $\\lambda=0.3$ & $\\lambda=0.4$ & $\\lambda=0.5$ & $\\lambda=0.6$ & $\\lambda=0.7$ & $\\lambda=0.8$ & $\\lambda=0.9$ \\\\\nOptimal analytical & 40.9000 & 20.8000 & 14.0333 & 10.6000 & 8.5000 & 7.0667 & 6.0143 & 5.2000 & 4.5444 \\\\\nOptimal Monte-Carlo & 40.8904 & 20.7979 & 14.0320 & 10.5985 & 8.4989 & 7.0659 & 6.0140 & 5.1999 & 4.5443 \\\\\nBF & 40.8920 & 20.7962 & 14.0337 & 10.6002 & 8.4995 & 7.0666 & 6.0153 & 5.1998 & 4.5445 \\\\\nIF & 40.8978 & 20.7960 & 14.0331 & 10.5991 & 8.5002 & 7.0667 & 6.0132 & 5.1998 & 4.5443 \\\\\nCT & 40.8961 & 20.8006 & 14.0333 & 10.5973 & 8.4986 & 7.0665 & 6.0137 & 5.2001 & 4.5444 \\\\\nSimple ARQ & 87.3286 & 31.1145 & 17.9077 & 12.3428 & 9.3310 & 7.4591 & 6.1846 & 5.2607 & 4.5568 \n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}\n\\centering\n\\caption{Mean time to absorption for $R_1 = 10$, $R_0=5$, $\\lambda=0.3$ and $E_d=10$ vs. $e$}\n\\label{son_vs_l}\n\\begin{tabular}{lllllllllll}\n & $e=1$ & $e=2$ & $e=3$ & $e=4$ & $e=5$ & $e=6$ & $e=7$ & $e=8$ & $e=9$ \\\\\nOptimal analytical & 40.7000 & 21.7000 & 15.0333 & 11.7000 & 11.7000 & 8.3667 & 8.3667 & 8.3667 & 8.3667 \\\\\nOptimal Monte-Carlo & 40.6956 & 21.6999 & 15.0320 & 11.6995 & 11.7009 & 8.3648 & 8.3651 & 8.3675 & 8.3670 \\\\\nBF & 40.7015 & 21.6987 & 15.0381 & 11.6986 & 11.6995 & 8.3677 & 8.3653 & 8.3667 & 8.3672 \\\\\nIF & 40.7023 & 21.7020 & 15.0308 & 11.7030 & 11.7010 & 8.3667 & 8.3658 & 8.3671 & 8.3654 \\\\\nCT & 40.6980 & 21.7006 & 15.0345 & 11.6995 & 11.6992 & 8.3674 & 8.3663 & 8.3657 & 8.3670 \\\\\nSimple ARQ & 47.7832 & 26.5340 & 19.1515 & 15.4839 & 14.0076 & 11.8479 & 10.8730 & 10.4191 & 10.2021\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Correlated Channel}\nIn this section, we investigate the performance of the optimal policy presented in Algorithm \\ref{alg2} for the case of correlated channel and compare its performance to the three baseline policies that employ HARQ mechanism as well as a simple ARQ mechanism. We also consider a randomized policy, which we call \\emph{Bernoulli} policy which harvests energy with probability, $p$, unless its battery state is less than one unit or it has accumulated sufficient mutual information during when it solely harvests energy. In the following, we study the effects of the encoding rate, the time correlation, and the EH rate. Note that the mean time to absorption is determined by calculating $k_{b,m,0}$ and $k_{b,m,1}$ and then averaging them with respect to the steady-state distribution of the channel states, i.e., $k_{b,m} = \\phi(0)k_{b,m,0} + \\phi(0)k_{b,m,1}$, where $\\phi(0) =1-\\phi(1)= \\frac{1-\\lambda_1}{1+\\lambda_0-\\lambda_1}$.\n\n\\begin{remark}\nNote that, in this section, we do not calculate the mean time to absorption by Algorithm \\ref{alg2} (i.e., $k^{\\boldsymbol{\\pi}^*}_{b,m,0}$ and $k^{\\boldsymbol{\\pi}^*}_{b,m,1}$). Instead, we use the optimal scheduling decisions dictated by Algorithm \\ref{alg2} for each state $(b,m, G)$ to determine the mean time to absorption by Monte-Carlo simulations. This is because both methods yield the same mean time to absorption for the optimal policy and illustrating both on the same figure distinctly is not possible.\n\\end{remark}\n\n\nTo investigate the effect of the encoding rate on the mean time to absorption, we set the simulation parameters as $R_1 = 10$, $e=1$, $E_d=5$ and $p=0.1$. The mean time to absorption with respect to $R_0$, for negatively and positively correlated channel states, are depicted in Figures \\ref{cor_VS_R1} and \\ref{cor_VS_R2}, respectively. \nUnlike the i.i.d. case the knowledge of the channel state makes a significant difference in the performance of the proposed optimal policy as compared to the baseline policies. Hence, when the channel is correlated, a simple scheduling policy is not sufficient to achieve a low number of re-transmissions.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[t]{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{vsrate1.eps}\n \\caption{Negatively correlated channel, $\\lambda_0 =0.7$ and $\\lambda_1 =0.2$.}\n \\label{cor_VS_R1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[t]{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{vsrate2.eps}\n \\caption{Positively correlated channel $\\lambda_0 =0.2$ and $\\lambda_1 =0.7$. }\n \\label{cor_VS_R2}\n \\end{subfigure}\n\t\t\t\t\\caption{The effect of the encoding rate on the minimum expected number of re-transmissions for $R_1 = 10$, $e=1$, $E_d=5$ and $p=0.1$.}\n\\label{cor_VS_R}\n\\end{figure}\n\nNext, we study the effect of the channel quality and the correlation on the mean time to absorption. We set $R_1 = 10$, $e=1$, $E_d=5$ and $p=0.1$. We fix $\\lambda_1=0.2$ and by varying $\\lambda_0$, we calculate the mean time to absorption as illustrated in Figure \\ref{cor_VS_lamda1}. Similarly, we fix $\\lambda_0=0.2$ and by varying $\\lambda_1$, we calculate the mean time to absorption by the aforementioned baseline policies and illustrate the results in Figure \\ref{cor_VS_lamda2}. Note that when the channel is negatively correlated, as in Figure \\ref{cor_VS_lamda2}, the gap between the optimal policy and the baseline policies is high. However, when the channel is positively correlated, as in Figure \\ref{cor_VS_lamda2}, the gap disappears as $\\lambda_1$ increases. This is because, when the channel is positively correlated, the channel tends to stay in the same state for a longer time before changing its state. On the contrary, in negatively correlated channel states, the channel is more likely to change its state at any time. This rapid change in state transition in the case of negatively correlated channel states requires a more adaptive policy rather than the case of the positively correlated channel state which rarely changes its state. Thus, the performance gain of Algorithm \\ref{alg2} is more evident in negatively correlated channels.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[t]{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{vs_cor1.eps}\n \\caption{Negatively correlated channel, $\\lambda_1 =0.2$.}\n \\label{cor_VS_lamda1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[t]{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{vs_cor2.eps}\n \\caption{Positively correlated channel, $\\lambda_0 =0.2$. }\n \\label{cor_VS_lamda2}\n \\end{subfigure}\n\t\t\t\t\\caption{The effect of the channel quality and correlation on the minimum expected number of re-transmissions for $R_1 = 10$, $R_0 = 3$, $e=1$, $E_d=5$ and $p=0.1$.}\n\\label{cor_VS_lamda}\n\\end{figure}\n\n\nFinally the effect of EH rate, $e$, on the mean time to absorption for negatively and positively correlated channel states is depicted in Figure \\ref{cor_VS_l1} and \\ref{cor_VS_l2}, respectively. The results are obtained by setting $R_1 = 10$, $R_0=5$, $E_d=10$, $p=0.1$, $\\lambda_0 =0.7$ and $\\lambda_1 =0.2$ for negatively correlated channel states; and $\\lambda_0 =0.2$ and $\\lambda_1 =0.7$ for positively correlated channel states. We, again, observe that the optimal policy outperforms the baseline policies and the performance gain is more evident for negatively correlated channel states for the same reason we provided for the results in Figure \\ref{cor_VS_lamda}. \n\n\nIt should be noted that when the channel states are correlated, the knowledge about the future channel states plays a major role in making decision about the power splitting ratio. On the contrary, when the channel states evolve i.i.d. over time, there exist a class of optimal policies instead of a single optimal policy. \n\n\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[t]{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{vs_l1.eps}\n \\caption{$\\lambda_0 =0.7$ and $\\lambda_1 =0.2$.}\n \\label{cor_VS_l1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[t]{0.4\\textwidth}\n \\includegraphics[width=\\textwidth]{vs_l2.eps}\n \\caption{$\\lambda_0 =0.2$ and $\\lambda_1 =0.7$. }\n \\label{cor_VS_l2}\n \\end{subfigure}\n\t\t\t\t\\caption{The effect of the EH rate on the minimum expected number of re-transmissions for $R_1 = 10$, $R_0=5$, $E_d=10$ and $p=0.1$.}\n\\label{cor_VS_l}\n\\end{figure}\n\n\\section{Conclusion}\\label{Conclusions}\nWe analyzed a point-to-point wireless link employing HARQ for reliable transmission, where the receiver can only empower itself via the transmitter's RF signal. We modeled the problem of optimal power splitting using a Markovian framework, and developed an optimal algorithm achieving the minimum mean time to absorption for both time varying i.i.d. and correlated channels. We developed computationally inexpensive algorithms to calculate the minimum mean time to absorption and optimize the power splitting ratio starting at any arbitrary state. \n\nWe proved that the optimal policy in case of i.i.d. channel states is not unique, and indeed the optimal policy belongs to the optimal family of policies. For correlated channel, we observed that it is only possible to achieve the optimal performance by intelligently utilizing the information offered by channel's correlation information. Finally, we numerically validated the analytical results established in the paper by providing extensive number of simulations.\n\nIt is worth mentioning that the two-state model, adopted here, is an approximation of a more general multi-state wireless channel. As a future work, we aim to extend this work for a more general setting where we will consider multi-rate information transmission, multi-state EH process, and non-linear EH efficiency. Due to to analytical complexity, it is uncertain that the optimality result of no-split policy carries over to the more general setting. In this case, deep reinforcement learning techniques can be used as a promising approach to address the aforementioned extensions.\n\n\n\n\n\n\\begin{appendices}\n\\section{Proof of Lemma \\ref{lemma1}}\n\\label{prooflemma1}\nThe proof is by induction. \n\\begin{enumerate}\n\\item Base case: Let us consider the smallest possible value for $i$, i.e., $i=1$, such that $E_d-e\\leq b < E_d$. Note that since $m=R_1$, the optimal decision is to use incoming RF signal only for harvesting energy, i.e., $\\rho^*(b,\\ R_1)=1$. Thus, we get\n\\small\n\\begin{align}\nk_{b,R_1} = 1 + \\lambda k_{b+e,R_1}+ (1-\\lambda) k_{b,R_1}.\n\\end{align}\n\\normalsize\nFor $E_d-e\\leq b < E_d$, if the channel is GOOD then the MC transitions into state $(b+e,\\ R_1)$, which is an absorbing state, so $k_{b+e,R_1}=0$. Hence, $k_{b,R_1}=\\frac{1}{\\lambda}$ and thus, the lemma holds for $i = 1$.\n\\item Induction step: assume that the lemma is true for some $i=n$, i.e., $k_{b,R_1}=n\/\\lambda$ for $E_d-n\\cdot e\\leq b&k^{\\boldsymbol{\\pi}^0}_{E_d+1,R_1-1}.\n\\end{align}\n\\normalsize\nNote that when $b= E_d+1, E_d+2, \\ldots$, by choosing $\\rho =0$, regardless of the channel state, the next state, $(b-1,R_1)$, is an absorbing state so $k^{\\boldsymbol{\\pi}^0}_{b,R_1-1}=1$. Thus, the lemma holds for $j = 1$. In the induction step assume that the theorem holds for $j=n-1$, i.e., $k^{\\boldsymbol{\\pi}^*}_{b,R_1-n+1}=k^{\\boldsymbol{\\pi}^0}_{E_d+n-1,R_1-n+1}=\\sum^{n-1}_{i=1} (1-\\lambda)^{i-1}$ for $b= E_d+n-1, E_d+n, \\ldots$. Now, we prove that the claim is also true for $j=n$.\n\\small\n\\begin{align}\nk^{\\boldsymbol{\\pi}^0}_{E_d+n,R_1-n} =& 1 + (1-\\lambda)k^{\\boldsymbol{\\pi}^*}_{E_d+n-1,R_1-n+1}\\nonumber\\\\\n=& 1 + \\sum^{n-1}_{i=1} (1-\\lambda)^{i}\\nonumber\\\\\n=& \\sum^{n}_{i=1} (1-\\lambda)^{i-1},\\\\\nk^{\\boldsymbol{\\pi}^1}_{E_d+n,R_1-n}=& \\frac{1}{\\lambda} + k^{\\boldsymbol{\\pi}^*}_{E_d+n+e,R_1-n}\\nonumber\\\\\n>&\\frac{1}{\\lambda} + k^{\\boldsymbol{\\pi}^*}_{E_d+n+e,R_1-n+1}\\nonumber\\\\\n=& \\frac{1}{\\lambda} + k^{\\boldsymbol{\\pi}^*}_{E_d+n-1,R_1-n+1}\\nonumber\\\\\n=&\\frac{1}{\\lambda} + \\frac{1-(1-\\lambda)^{n-1}}{\\lambda}\n\\end{align}\n\\normalsize\nFurthermore,\n\\small\n\\begin{align}\nk^{\\boldsymbol{\\pi}^0}_{E_d+n,R_1-n} =&\\frac{1-(1-\\lambda)^n}{\\lambda}\\nonumber\\\\\n=& 1 + (1-\\lambda)\\frac{1-(1-\\lambda)^{n-1}}{\\lambda}E_d$}\n \\label{proofTH_str1}\n Due to space limitations, we only provide a sketch of the proof with complete details given in \\cite{arxivmehdi}. We need to show that $k^{\\boldsymbol{\\pi}^0}_{E_d+j-i,R_1-j}3$ odd,\n\\begin{displaymath}\nF^\\prime(x)=x^{2^i+1}+(x^{2^i}+x)tr(x^{2^i+1}+x), \\textbf{ } \\gcd(m,i)=1,\n\\end{displaymath}\nis an AB function which is EA-inequivalent to any power function.\n\\end{Theorem}\n\\emph{Proof.} It is easy (but lengthy) to prove that for \n$a\\in\\mathbb{F}_{2^m}^*$ the linear mapping\n$${\\cal L}(x,y)=(x+a\\ tr(a^{-1}x)+a\\ \ntr(a^{-(2^i+1)}y),y+a^{2^i+1}tr(a^{-(2^i+1)}y)+a^{2^i+1}tr(a^{-1}x))$$\nis an involution.\nWe have\n\\begin{displaymath}\n{\\cal L}(0,y)=\n\\left\\{ \\begin{array}{ll}\n(a,y+a^{2^i+1})& \\textrm{if } \\textrm{ } y\\in H_a\\\\\n(0,y)& \\textrm{if } \\textrm{ } y\\in \\mathbb{F}_{2^m}\\backslash H_a\n\\end{array}\\right..\n\\end{displaymath}\nSince $tr(a^{-(2^i+1)}a^{2^i+1})=1$, then $a^{2^i+1}\\in H_a$. The set $H_a$ is a \ncomplement of a linear hyperplane, hence the sum of any two elements from $H_a$ \nbelongs to $\\mathbb{F}_{2^m}\\backslash H_a$. Therefore, ${\\cal L}^{-1}(V)={\\cal \nL}(V)=V^\\prime$ and by Proposition \\ref{prop2} the function $F$ is \nCCZ-equivalent to $F_2\\circ F_1^{-1}$, where\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_1(x)&=&L_1(x,F(x))= x+a\\ tr(x\/a)+a\\ tr((x\/a)^{2^i+1}),\\\\\nF_2(x)&=&L_2(x,F(x))= x^{2^i+1}+a^{2^i+1}tr((x\/a)^{2^i+1})+a^{2^i+1}tr(x\/a).\n\\end{eqnarray*}}\nThe function $F_1$ is an involution (this proves again that it is bijective):\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_1\\circ F_1(x)&=&x+a\\ tr(x\/a)+a\\ tr((x\/a)^{2^i+1})+a\\ tr(a^{-1}( x+a\\ tr(x\/a)\\\\\n&+&a\\ tr((x\/a)^{2^i+1})))+a\\ tr(a^{-(2^i+1)}( x+a\\ tr(x\/a)+a\\ \ntr((x\/a)^{2^i+1}))^{2^i+1})\\\\\n&=&x+3a\\ tr(x\/a)+2a\\ tr((x\/a)^{2^i+1})+a\\ \ntr(a^{-(2^i+1)}(x^{2^i+1}+(ax^{2^i}+a^{2^i}x\\\\\n&+&a^{2^i+1})(tr(x\/a)+tr((x\/a)^{2^i+1}))))=x+a\\ tr(x\/a)+a\\ tr((x\/a)^{2^i+1})\\\\\n&+&a\\ tr((x\/a)^{2^i}+(x\/a)+1)(tr(x\/a)+tr((x\/a)^{2^i+1}))=x,\n\\end{eqnarray*}}\nsince\n$$( x+a\\ tr(x\/a)+a\\ tr((x\/a)^{2^i+1}))^{2^i+1}=x^{2^i+1}+(ax^{2^i}+a^{2^i}x\n+a^{2^i+1})(tr(x\/a)+tr((x\/a)^{2^i+1}))$$\n$$+2a^{2^i+1}tr(x\/a)tr((x\/a)^{2^i+1})=x^{2^i+1}+(ax^{2^i}+a^{2^i}x\n+a^{2^i+1})(tr(x\/a)+tr((x\/a)^{2^i+1})).$$\nTherefore,\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_2\\circ F_1^{-1}(x)&=&(x+a\\ tr(x\/a)+a\\ \ntr((x\/a)^{2^i+1}))^{2^i+1}+a^{2^i+1}tr(a^{-(2^i+1)}(x+a\\ tr(x\/a)\\\\\n&+&a\\ tr((x\/a)^{2^i+1}))^{2^i+1})+a^{2^i+1}tr(a^{-1}(x+a\\ tr(x\/a)+a\\ \ntr((x\/a)^{2^i+1})))\\\\\n&=&x^{2^i+1}+(ax^{2^i}+a^{2^i}x+a^{2^i+1})(tr(x\/a)+tr((x\/a)^{2^i+1}))+a^{2^i+1}tr((x\/a)^{2^i}\\\\\n&+&(x\/a)+1)(tr(x\/a)+tr((x\/a)^{2^i+1}))+2a^{2^i+1}tr(x\/a)+2a^{2^i+1}tr((x\/a)^{2^i+1})\\\\\n&=&a^{2^i+1}[(x\/a)^{2^i+1}+((x\/a)^{2^i}+(x\/a))tr((x\/a)+(x\/a)^{2^i+1})]=a^{2^i+1}F^\\prime(x\/a),\n\\end{eqnarray*}}\nwhere $F^{\\prime}(x)=x^{2^i+1}+(x^{2^i}+x)tr(x+x^{2^i+1})$.\nBy Proposition \\ref{prop1} the function $F^{\\prime}$ is AB.\\\\ \n It is not difficult to see that the algebraic degree of $F'$ is 3 for $m>3$.\n On the other hand\n$tr(F^\\prime(x))=tr(x^{2^i+1})$ and\n$d^{\\circ}(tr(F^\\prime(x)))=2$. It follows from Proposition \\ref{prop5} that \n$F^\\prime$\nis EA-inequivalent to any power function.\\hfill $\\Box$\\\\\n\n\\noindent \\textbf{Remark 4 } It was conjectured in \\cite{CC-carchzin} that any \nAB function is EA-equivalent to a permutation. The AB function from Theorem 1 \nserves as a counterexample for this conjecture.\nIt was checked by the help of a computer, that for no linear function $L$ on\n$\\mathbb{F}_{2^5}$ the sum $F^\\prime+L$ is a permutation for the AB\nfunction $F^\\prime(x)=x^{2^i+1}+(x^{2^i}+x)tr(x^{2^i+1}+x)$, $\\gcd(5,i)=1$. \nThus, $F^\\prime$ is EA-inequivalent to any permutation but it is CCZ-equivalent \nto the permutation $x^{2^i+1}$.\nFor $m$ even the existence of APN permutations on $\\mathbb{F}_{2^m}$ is an \nopen problem. It is shown by Nyberg that there exist no quadratic APN permutations when $m$ is even \\cite{nyb2}. But as we have seen the nonexistence of permutations EA-equivalent to the Gold functions does not mean yet that there exist no permutations CCZ-equivalent to them.\n\\hfill $\\Diamond$\n\\begin{Theorem}\\label{thm2}\nThe function $F^\\prime:\\mathbb{F}_{2^m}\\to \\mathbb{F}_{2^m}$, $m\\ge 4$ even,\n\\begin{displaymath}\nF^\\prime(x)=x^{2^i+1}+(x^{2^i}+x+1)tr(x^{2^i+1}), \\textbf{ } \\gcd(m,i)=1,\n\\end{displaymath}\nis an APN function which is EA-inequivalent to any power function.\n\\end{Theorem}\n\\emph{Proof.} For $a \\in \\mathbb{F}_{2^m}^{*},$ the linear mapping $${\\cal \nL}(x,y)=(L_1,L_2)(x,y)=(x+a\\ tr(a^{-(2^i+1)}y),y)$$ is an involution and, \nobviously, ${\\cal L}(V)=V^\\prime$. Thus, by Proposition \\ref{prop2} the function \n$F$ is CCZ-equivalent to $F_2\\circ F_1^{-1}$, where\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_1(x)&=&L_1(x,F(x))= x+a\\ tr((x\/a)^{2^i+1}),\\\\\nF_2(x)&=&L_2(x,F(x))= x^{2^i+1}.\n\\end{eqnarray*}}\nThe function $F_1$ is an involution:\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_1\\circ F_1(x)&=&x+a\\ tr((x\/a)^{2^i+1})+a\\ \ntr(a^{-(2^i+1)}(x^{2^i+1}+ax^{2^i}tr((x\/a)^{2^i+1})\\\\\n&+&a^{2^i}x\\ tr((x\/a)^{2^i+1})+a^{2^i+1}tr((x\/a)^{2^i+1})))\\\\\n&=&x+2a\\ tr((x\/a)^{2^i+1})+a\\ tr((x\/a)^{2^i}+(x\/a)+1)tr((x\/a)^{2^i+1})=x.\n\\end{eqnarray*}}\nWe have\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_2\\circ F_1^{-1}(x)&=&(x+a\\ \ntr((x\/a)^{2^i+1}))^{2^i+1}=x^{2^i+1}+ax^{2^i}tr((x\/a)^{2^i+1})\\\\\n&+&a^{2^i}x\\ tr((x\/a)^{2^i+1})+a^{2^i+1}tr((x\/a)^{2^i+1})\\\\\n&=&a^{2^i+1}[(x\/a)^{2^i+1}+((x\/a)^{2^i}+(x\/a)+1)tr((x\/a)^{2^i+1})].\n\\end{eqnarray*}}\nThe function $F_2\\circ F_1^{-1}$ is EA-equivalent to the function\n$$F^\\prime(x)=x^{2^i+1}+(x^{2^i}+x+1)tr(x^{2^i+1}).$$\nHence, $F^\\prime$ is CCZ-equivalent to $F$ and it is APN by Proposition \n\\ref{prop1}.\nThe algebraic degree of $F^\\prime$ is $3$ and\n$d^\\circ(tr(F^\\prime))=2$ as $tr(F^\\prime(x))=tr(x^{2^i+1})$. Therefore, \n$F^\\prime$ is EA-inequivalent to power functions\nby Proposition \\ref{prop5}. \\hfill $\\Box$\\\\\n\nNote that the proofs of Theorems \\ref{thm1} and \\ref{thm2} do not depend on the \ncondition $\\gcd (i,m)=1$. When $\\gcd (i,m)=s$ then the functions $F'$ have the \nsame differential and linear properties as $x^{2^i+1}$ and, therefore, if $m\/s$ \nis odd they can be considered as the first polynomials with three valued Walsh \nspectrum $\\{0,\\pm 2^\\frac{m+s}{2}\\}$, which are EA-inequivalent to power \nfunctions. \n\n\\begin{Theorem}\\label{thm3}\nThe function $F^\\prime:\\mathbb{F}_{2^m}\\to \\mathbb{F}_{2^m}$, $m$ divisible by \n$6$, \n$$F^\\prime(x)=[x+tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)})+tr(x)tr_{m\/3}(x^{2^i+1}+x^{2^{2i}(2^i+1)})]^{2^i+1},$$\nwith $\\gcd(m,i)=1$, is an APN function.\n\\end{Theorem}\n\\emph{Proof.} The linear function ${\\cal L}:\\mathbb{F}_{2^m}^2\\to \n\\mathbb{F}_{2^m}^2$, \n$${\\cal L}(x,y)=(L_1,L_2)(x,y)=(x+tr_{m\/3}(y^2+y^4),y)$$\nis a permutation since it has the kernel $\\{(0,0)\\}$. For $m$ divisible by $6$, \nthe function \n$$F_1(x)=L_1(x,F(x))=x+tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)})$$ \nis a permutation. Indeed, by Proposition \\ref{prop3} the function $F_1$ is a \npermutation \nif for every $v\\in \\mathbb{F}_{2^m}$ such that $tr(v)=0$ and every $u\\in \n\\mathbb{F}_{2^m}^*$ the condition $tr_{m\/3}((u^{2^i+1}v)^2+(u^{2^i+1}v)^4)\\ne u$ \nholds. Obviously, for any $u\\notin \\mathbb{F}_{2^3}$ the condition is \nsatisfied and when $u\\in \\mathbb{F}_{2^3}^*$ the condition \n$tr_{m\/3}((u^{2^i+1}v)^2+(u^{2^i+1}v)^4)\\ne u$ is equivalent to \n$(u^{2^i+1}tr_{m\/3}(v))^2+(u^{2^i+1}tr_{m\/3}(v))^4\\ne u$.\nTherefore, $F_1$ is a permutation if, for every $u,w\\in \\mathbb{F}_{2^3}^*$, \n$tr_3(w)=0$ the condition $(u^{2^i+1}w)^2+(u^{2^i+1}w)^4\\ne u$ is satisfied and \nthat was easily checked by a computer. \n\nWe show below that $F_{1}^{-1}=F_1\\circ F_1\\circ F_1\\circ F_1\\circ F_1$.\\\\\nWe denote \n$$T(x)=tr_{m\/3}(x^{2^i+1}),$$\nthen\n$$F_1(x)=x+T(x)^2+T(x)^4.$$\nSince every element of $\\mathbb{F}_8$ is equal to its $8$ power and the function \n$tr_{m\/3}(x)$ is $0$ on $\\mathbb{F}_8$, then\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\n&&T\\circ F_1(x)=tr_{m\/3}[(x+tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)}))^{2^i+1}]\\\\\n&=&tr_{m\/3}[(x+tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)}))(x^{2^i}+tr_{m\/3}(x^{2^{i+1}(2^i+1)}\\\\\n&+&x^{2^{i+2}(2^i+1)}))]=tr_{m\/3}[x^{2^i+1}+x\\ \ntr_{m\/3}(x^{2^{i+1}(2^i+1)}+x^{2^{i+2}(2^i+1)})\\\\\n&+&x^{2^i}tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)})]=tr_{m\/3}(x)tr_{m\/3}(x^{2^{s+1}(2^i+1)}+x^{2^{s+2}(2^i+1)})\\\\\n&+&tr_{m\/3}(x^{2^s})tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)})+tr_{m\/3}(x^{2^i+1}),\n\\end{eqnarray*}} \nwhere $s\\equiv i\\mod 3$.\\\\\nTherefore,\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\n&&F_1\\circ F_1(x)=F_1(x)+[T\\circ F_1(x))]^2+[T\\circ F_1(x))]^4\\\\\n&=&x+2tr_{m\/3}(x^{2(2^i+1)}+x^{4(2^i+1)})+tr_{m\/3}(x^2)tr_{m\/3}(x^{2^{s+2}(2^i+1)}\\\\\n&+&x^{2^{s}(2^i+1)})+tr_{m\/3}(x^{2^{s+1}})tr_{m\/3}(x^{4(2^i+1)}+x^{2^i+1})\\\\\n&+&tr_{m\/3}(x^4)tr_{m\/3}(x^{2^{s}(2^i+1)}+x^{2^{s+1}(2^i+1)})+tr_{m\/3}(x^{2^{s+2}})tr_{m\/3}(x^{2^i+1}+x^{2(2^i+1)})\n\\end{eqnarray*}} \nConsidering separately the cases $s=1$ and $s=2$ we get\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF_1\\circ F_1(x)&=&x+tr_{m\/3}(x+x^2+x^4)tr_{m\/3}(x^{2^i+1}+x^{2^s(2^i+1)})\\\\\n&=&x+tr(x)tr_{m\/3}(x^{2^i+1}+x^{2^s(2^i+1)})=x+(T(x)+T(x)^{2^s})tr(x).\n\\end{eqnarray*}} \nLike this we get\n$$F_1\\circ F_1\\circ F_1\\circ F_1\\circ F_1 \n(x)=x+T(x)^2+T(x)^4+tr(x)(T(x)+T(x)^{2^{2s}}),$$\n$$F_1\\circ F_1\\circ F_1\\circ F_1\\circ F_1\\circ F_1 (x)= x.$$\nThus,\n{\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*}\nF^\\prime(x)&=&F_2 \\circ \nF_1^{-1}(x)=[x+T(x)^2+T(x)^4+tr(x)(T(x)+T(x)^{2^{2s}})]^{2^i+1}=x^{2^i+1}+T(x)^{2^s+1}\\\\\n&+&tr(x^{2^i+1})T(x)^{2^{2s}}+tr(x)(T(x)+T(x)^4)+xtr(x)(T(x)+T(x)^{2^s})\\\\\n&+&x^{2^i}tr(x)(T(x)+T(x)^{2^{2s}})+x(T(x)+T(x)^{2^{2s}})+x^{2^i}(T(x)^2+T(x)^4),\n\\end{eqnarray*}} \nwhere $F_2(x)=L_2(x,F(x))=x^{2^i+1}.$\n\nThe function $F'$ has the algebraic degree 4, this can be shown by lengthy but \n uncomplicated calculations. Hence, $F^\\prime$ is EA-inequivalent to other known \n APN functions since for $m$ divisible by 6 we have no known APN functions of \n algebraic degree 4.\n\\hfill $\\Box$\\\\\n\n\nLet $F:\\mathbb{F}_{2^m}\\to \\mathbb{F}_{2^m}$, $F(x)=x^{2^i+1}$ with \n$\\gcd(m,i)=1$, $m$ odd and $n$ a divisor of $m$. Then $A_F\\cap \nV^\\prime=\\emptyset$ for the subgroup \n$$V^\\prime=\\{(b,x) : b\\in\\mathbb{F}_{2^n}, x\\in\\mathbb{F}_{2^m}, tr_{m\/n}(x)=0 \n\\},$$\nsince if $(b,x)\\in V^\\prime$ then \n$tr(b^{-(2^i+1)}x)=tr_n(b^{-(2^i+1)}tr_{m\/n}(x))=0$ and $(b,x)\\notin A_F$.\nHence, $G_F$ is transversal to $V^\\prime$. Using the subgroup $V^\\prime$ we \nconstruct an AB function $F^\\prime$ given in the following theorem.\n\\begin{Theorem}\\label{thm5}\nThe function $F^\\prime:\\mathbb{F}_{2^m}\\to \\mathbb{F}_{2^m}$, where $m$ is odd \nand divisible by $n$, $m\\ne n$ and $\\gcd(m,i)=1$,\n {\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*} \nF^\\prime(x)&=&x^{2^i+1}+tr_{m\/n}(x^{2^i+1})+x^{2^i}tr_{m\/n}(x)+x\\ \ntr_{m\/n}(x)^{2^i}\\\\\n&+&[tr_{m\/n}(x)^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)]^{\\frac{1}{2^i+1}}(x^{2^i}+tr_{m\/n}(x)^{2^i}+1)\\\\\n&+&[tr_{m\/n}(x)^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)]^{\\frac{2^i}{2^i+1}}(x+tr_{m\/n}(x)),\n\\end{eqnarray*}} \nis an AB function which is EA-inequivalent to any power function.\n\\end{Theorem}\n\\emph{Proof.} Let $m$ be odd and divisible by $n$. Obviously, the linear \nfunction \n$${\\cal L}(x,y)=(L_1,L_2)(x,y)=(x+tr_{m\/n}(x)+tr_{m\/n}(y),y+tr_{m\/n}(x))$$\nis a permutation on $\\mathbb{F}_{2^m}^2$ and\n$${\\cal L}^{-1}(x,y)=(x+tr_{m\/n}(y),y+tr_{m\/n}(x)+tr_{m\/n}(y)).$$\nWe have\n $$F_1(x)=x+tr_{m\/n}(x)+tr_{m\/n}(x^{2^i+1}),$$ \n $$F_2(x)=x^{2^i+1}+tr_{m\/n}(x). $$\nBy Proposition \\ref{prop2}, the function $F_1$ is a permutation since ${\\cal \nL}^{-1}(V)=V'$ and $G_F$ is transversal to $V^\\prime$.\nWe need the inverse of the function $F_1$ to construct $F^\\prime=F_2\\circ \nF_1^{-1}$.\\\\\nFor any fixed element $x\\in \\mathbb{F}_{2^m}$ we have\n$$y=x+tr_{m\/n}(x)+tr_{m\/n}(x^{2^i+1})=x+u,$$\nfor some $u\\in \\mathbb{F}_{2^n}$, and, therefore, $x=y+u$. Then\n$$y=(y+u)+tr_{m\/n}(y+u)+tr_{m\/n}((y+u)^{2^i+1})$$ which yields \n\\begin{equation} \\label{eq1}\nu^{2^i+1}+u^{2^i}tr_{m\/n}(y)+u(tr_{m\/n}(y))^{2^i}+tr_{m\/n}(y^{2^i+1})+tr_{m\/n}(y)=0.\n\\end{equation} \nIf $tr_{m\/n}(y)\\ne 0$ then we denote $v=u\/tr_{m\/n}(y)$ and we get\n$$v^{2^i+1}+v^{2^i}+v+\\frac{tr_{m\/n}(y^{2^i+1})+tr_{m\/n}(y)}{(tr_{m\/n}(y))^{2^i+1}}=0.$$\nSince $v^{2^i+1}+v^{2^i}+v=(v+1)^{2^i+1}+1$ then \n$$v+1=\\left[\\frac{tr_{m\/n}(y^{2^i+1})+tr_{m\/n}(y)}{(tr_{m\/n}(y))^{2^i+1}}+1\\right]^{\\frac{1}{2^i+1}}.$$\nReplacing $v$ by $u\/tr_{m\/n}(y)$ we have\n$$u=[(tr_{m\/n}(y))^{2^i+1}+tr_{m\/n}(y^{2^i+1})+tr_{m\/n}(y)]^{\\frac{1}{2^i+1}}+tr_{m\/n}(y).$$\nIf $tr_{m\/n}(y)=0$ then from the equality (\\ref{eq1}) we get \n$u=[tr_{m\/n}(y^{2^i+1})]^{\\frac{1}{2^i+1}}$ and we observe that $u$ equals again \n$[(tr_{m\/n}(y))^{2^i+1}+tr_{m\/n}(y^{2^i+1})+tr_{m\/n}(y)]^{\\frac{1}{2^i+1}}+tr_{m\/n}(y).$\nThus, in all cases, we have \n$$F_1^{-1}(y)=y+u=y+[(tr_{m\/n}(y))^{2^i+1}+tr_{m\/n}(y^{2^i+1})+tr_{m\/n}(y)]^{\\frac{1}{2^i+1}}+tr_{m\/n}(y)$$\nand\n {\\setlength\\arraycolsep{2pt}\n\\begin{eqnarray*} \nF^\\prime(x)&=&F_2\\circ \nF_1^{-1}(x)=[x+[(tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)]^{\\frac{1}{2^i+1}}+tr_{m\/n}(x)]^{2^i+1}\\\\\n&+&tr_{m\/n}[x+[(tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)]^{\\frac{1}{2^i+1}}+tr_{m\/n}(x)]\\\\\n&=& \nx^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)+x^{2^i}tr_{m\/n}(x)+x(tr_{m\/n}(x))^{2^i}\\\\\n&+&[(tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)]^{\\frac{1}{2^i+1}}(x^{2^i}+(tr_{m\/n}(x))^{2^i}+1)\\\\ \n&+&[(tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x)]^{\\frac{2^i}{2^i+1}}(x+tr_{m\/n}(x)).\n \\end{eqnarray*}}\n \nWe show below that \nthe function $F^\\prime$ has the algebraic degree $n+2$. \nIt means that the number of functions CCZ-equivalent to a Gold AB function and \nEA-inequivalent to it is not smaller than the number of divisors of $m$.\n\nThe inverse of $x^{2^i+1}$ on $F_{2^n}$ is $x^d$, where \n$$d=\\sum_{k=0}^{\\frac{n-1}{2}}2^{2ik},$$ \nand $x^d$ has the algebraic degree $\\frac{n+1}{2}$ \\cite{nyb}.\nObviously, \n$((tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})+tr_{m\/n}(x))^d$ \nhas the algebraic degree $n+1$ if and only if \n$((tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1}))^d$ has this algebraic degree.\n\nWe assume that $m\\ne n$ and $n\\ne 1$ since when $m=n$ we get \n$F^\\prime(x)=x^{\\frac{1}{2^i+1}}+x$ and Theorem \\ref{thm1} gives the case $n=1$.\n\nWe have\n$$tr_{m\/n}(x)=\\sum_{k=0}^{\\frac{m}{n}-1}x^{2^{kn}}$$\nand\n$$(tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})=\\sum_{k=0}^{\\frac{m}{n}-1}x^{2^{kn}}\\sum_{k=0}^{\\frac{m}{n}-1}x^{2^{kn+i}}+\\sum_{k=0}^{\\frac{m}{n}-1}(x^{2^i+1})^{2^{kn}}$$\n$$=\\sum_{k,j=0}^{\\frac{m}{n}-1}x^{2^{kn}+2^{jn+i}}+\\sum_{k=0}^{\\frac{m}{n}-1}x^{2^{kn}+2^{kn+i}}=\\sum_{\\begin{subarray}{c} \nk,j=0\\\\k\\ne j\\end{subarray}}^{\\frac{m}{n}-1}x^{2^{kn}+2^{jn+i}}.$$\n\nNote that we have\n$$[(tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1})]^{2^{2i}+1}=\\sum_{\\begin{subarray}{c} \nk,j=0\\\\k\\ne \nj\\end{subarray}}^{\\frac{m}{n}-1}x^{2^{kn}+2^{jn+i}}\\sum_{\\begin{subarray}{c} \nk,j=0\\\\k\\ne j\\end{subarray}}^{\\frac{m}{n}-1}x^{2^{kn+2i}+2^{jn+3i}}$$\n$$=\\sum_{\\begin{subarray}{c} k,j,s,t=0\\\\k\\ne j, s\\ne t \n\\end{subarray}}^{\\frac{m}{n}-1}x^{2^{kn}+2^{jn+i}+2^{sn+2i}+2^{tn+3i}}.$$\nSimilarly, we have\n\\begin{equation}\\label{1}\n((tr_{m\/n}(x))^{2^i+1}+tr_{m\/n}(x^{2^i+1}))^d=\\sum_{(k_0,...,k_n)\\in \nI}x^{\\sum_{s=0}^{n}2^{k_sn+si}}, \n\\end{equation}\nwhere $I=\\{(l_0,...,l_n):\\quad 0\\le l_t\\le \\frac{m}{n}-1,\\quad l_{2t}\\ne \nl_{2t+1}\\}$.\n\nThe equality $k_sn+si= k_tn+ti$ is possible for $0\\le s{\\columncolor[HTML]{EFEFEF}}c c\n>{\\columncolor[HTML]{EFEFEF}}c c|\n>{\\columncolor[HTML]{EFEFEF}}c c\n>{\\columncolor[HTML]{EFEFEF}}c c}\n\\hline\n\\multicolumn{4}{c|}{\\cellcolor[HTML]{EFEFEF}\\textbf{Real x LDPL 4}} & \\multicolumn{4}{c}{\\cellcolor[HTML]{EFEFEF}\\textbf{Regression x LDPL 4}} \\\\ \\hline\n\\textbf{P1} & 25\\% & \\textbf{P11} & 12\\% & \\textbf{P1} & 19\\% & \\textbf{P11} & 3\\% \\\\ \\hline\n\\textbf{P2} & 17\\% & \\textbf{P12} & 7\\% & \\textbf{P2} & 11\\% & \\textbf{P12} & 4\\% \\\\ \\hline\n\\textbf{P3} & 9\\% & \\textbf{P13} & 2\\% & \\textbf{P3} & 7\\% & \\textbf{P13} & 4\\% \\\\ \\hline\n\\textbf{P4} & 5\\% & \\textbf{P14} & 2\\% & \\textbf{P4} & 4\\% & \\textbf{P14} & 5\\% \\\\ \\hline\n\\textbf{P5} & 2\\% & \\textbf{P15} & 2\\% & \\textbf{P5} & 2\\% & \\textbf{P15} & 5\\% \\\\ \\hline\n\\textbf{P6} & 11\\% & \\textbf{P16} & 7\\% & \\textbf{P6} & 0\\% & \\textbf{P16} & 5\\% \\\\ \\hline\n\\textbf{P7} & 8\\% & \\textbf{P17} & 3\\% & \\textbf{P7} & 1\\% & \\textbf{P17} & 6\\% \\\\ \\hline\n\\textbf{P8} & 6\\% & \\textbf{P18} & 1\\% & \\textbf{P8} & 1\\% & \\textbf{P18} & 6\\% \\\\ \\hline\n\\textbf{P9} & 7\\% & \\textbf{P19} & 1\\% & \\textbf{P9} & 2\\% & \\textbf{P19} & 6\\% \\\\ \\hline\n\\textbf{P10} & 5\\% & \\textbf{P20} & 2\\% & \\textbf{P10} & 3\\% & \\textbf{P20} & 6\\% \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nWith the exception of some discrepancies for a few points (\\(P1\\) and \\(P2\\)), relative errors are all below 15\\%, and they are even lower when considering the logarithmic regression. We noticed that the engine room walls worked as a kind of shielding, helping to keep the signal quality, and attenuating the electromagnetic interferences effect on the signal strength. A possible explanation for the discrepancies is that, during the data collection, workers were accessing the engine room for performing their regular activities, and as the doors were opened the shielding effect appeared to be somehow broken, by allowing some signal leakage while impacting the overall signal quality. This makes the results more realistic, since the data collection was carried out in real working shifts, and being the least possible invasive.\n\nAs the employed model was not designed to take into account electromagnetic interferences, we are inducted to conclude that the 802.11g signal quality, while considering the power plant environment (characteristics and \"shielding\"), is not much affected by the interferences from the motors, since the estimated power computed with propagation exponent 4 (which is used to estimate the path loss in a regular environment with the presence of several obstacles) showed a good approximation with the measured power. In other words, it seems that the existing shielding in the engine room counterbalances the negative effect of electromagnetic interference, characterizing the network behavior as that of a common environment, in which the signal propagation is, many times, more reduced by distance and obstacles.\n\nThus, the main contributions of this work are: \n\\begin{itemize}\n\\item A practical evaluation in a real industrial environment (i.e., thermoelectric power plant), while similar studies\/works usually simulate the local;\n\\item The determination of the propagation exponent to be considered for 802.11g signal strength estimation in industrial environments (achieving an error bellow 15\\% for most of the spots in our evaluation).\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhqzp b/data_all_eng_slimpj/shuffled/split2/finalzzhqzp new file mode 100644 index 0000000000000000000000000000000000000000..a87f23c09630a63c47e558fdd2be7903740501dc --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhqzp @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nGeometrically branched molecules (small-molecule dendrimers\n\\refto{discovery} \n and \ndendritic polymers \\refto {dendrimer_polymer_review}) \nare seeing progressively sophisticated \napplications \\refto{dendrimer_review_2,dendrimer_review_1},\nparticularly as biomedical and drug-delivery agents\n\\refto{dendritic_polymer_medicine}.\nThe proliferation of free ends (possibly functionalized) and\na rich design space in the interior of these single molecules\nmakes them ideal as engineering platforms.\n\nThis work is particularly focused on determining the collective \nstructure of large collections of dendritic \nor hyper-branched \\refto{hyperbranched}, \npolymers end-grafted on a surface\n\\refto{brush}.\nEven ordinary, linear polymers gain interesting properties when they \nare crowded together onto a surface by one end.\nIn making an analogy between the Edwards single-chain free energy and\nan electrostatic system, Semenov \\refto{semenov}, and subsequently\nMilner, Witten, and Cates \\refto{mwc} (in making an analogy to the classical\nmechanics of a particle falling in an external potential), determined that\nthe monomer chemical potential took on a universal profile - the\n``parabolic'' potential (and density profile, when the polymer brush was\nsolvated).\nMonodispersity of the chains is a very powerful constraint producing this\nbehavior, as is the unique self-consistent potential thus determined.\nWhen the chains are grafted on a convex surface \\refto{marko}\nthe parabolic potential profile is no longer the self-consistent solution\nas such an {\\it ansatz} would require monomers to overfill space near the\ngrafting surface (so that the self-consistently determined distribution\nof free ends would be {\\it negative} in some regions).\nA considerably more complex analysis is required to determine the \nself-consistent potential in the presence of regions with zero end-density \n(so-called ``dead'' or ``exclusion'' zones).\n\nHowever, when the polymers in the brush\n are branched (star-like \\refto{katya} \nor dendritic \\refto{galen_forrest}-\\refto{galen_zook})\nthe architectural increase \nof free ends counteracts the ``overfilling'' effect, and it can \noccur that non-zero end-densities are achieved even for \nthe extreme case of a single dendritic polymer in a good solvent.\nHere \\refto{galen_zook}, the absence of dead-zones and the consequent \nparabolic potential and therefore monomer density causes \na single dendritic polymer to have a dense core of monomers that \ndecreases monotonically to its edge in good solvent.\nThis is in contrast to the original prediction \\refto{degennes_dendrimer}\nthat the tips of the dendritic polymer would all extend to the\nsame spherical surface producing a large, characteristic ``hollow core''\nwhich had eluded detection in early simulations \\refto{filled_core}.\nThe monodispersity of the hyper-branched polymer is again the culprit, as\na massive free-energy degeneracy is a consequence of the parabolic\npotential (and is the cause of the isochronous behavior of the harmonic\noscillator \\refto{goldstein}).\n\nIn this work, we generalize a previous theory \n\\refto{galen_forrest,dendrimer_copoly} as in Ref.~\\refto{galen_zook} in a\n``continuous branching'' model for the dendrimer brush. \nVastly different architectures (regularly branched, randomly branched,\nand various power-law schemes of branching) can all be handled in the\nsame theory with few assumptions and fitting parameters.\nThe parabolic shape of the insertion potential survives the\ncontinuous branching treatment, and is the exact analytic solution for\nan unreasonably large class of polymer brush architectures.\n\nThe paper is organized as follows. First, we introduce the strong \nstretching self-consistent theory for continuously branched brushes, \nthen we analyze the end-distributions for various architectures. Finally we\ndiscuss the implications for the near-universal stabilization of the\nparabolic profile by branching, and make our conclusions.\n\n\n\\omit{\nDendritic molecules have received much attention since their \ninvention \\refto{discovery},\npartly because of their intriguing architecture, but more importantly\nfor their promise in producing materials of decidedly interesting\nproperties \\refto{dendrimer_review_2,dendrimer_review_1}.\nSingle dendrimers can be synthesized with very extremely regular\narchitecture, and are capable of forming thermally tunable complexes\nas drug-delivery agents \\refto{thermal_tune}.\nTheir well-characterized size and relative stiffness makes them suited\nto forming complex two-dimensionally packed arrays that can then be\ndecorated or etched to produce nanoscopically patterned surfaces \n\\refto{patterns}.\nAnd, the geometric proliferation of chain tips, each capable of being\ntagged with a bio-specific functionality, makes them an ideal material\nfor engineering smart surfaces \\refto{darrel_tips}.\nIt should be pointed out that most naturally occurring biologically\nrelevant molecules are either lightly branched (the three-armed fatty-acids)\nor linear (phospholipids, DNA, and proteins), and it is therefore\nentirely possible that hyperbranching in and of itself can grant\nunusual single-molecules properties with no natural analog.\n\nRelated to these dendrimers are dendritic polymers, where the \nbranching points are connected by flexible spacer polymers of a controllable\nmolecular weight and composition.\nIn some respects, these dendritic polymers resemble \npolymer stars \\refto{daoud}, with\na general splaying of their many free arms away from a dense core.\nIndeed, miktoarm stars with an unequal number of $A$ and $B$ arms have been\npredicted to drastically reshape the $AB$ diblock copolymer \nphase diagram \\refto{milner_stars},\nand hyperbranched dendritic polymers are predicted to have the same\neffect \\refto{fredrickson,dendrimer_copoly}.\nThe microphases are consistently skewed toward having the component\nwith the most branches on the exterior of a curved surface,\nwith the most pronounced effects occurring for stars with, for example, \na single $A$ type arm and many $B$ arms.\nSimilarly, when the $A$ block is a linear flexible homopolymer, and the\n$B$ block is a $G$-generation dendritic polymer, the phase diagram\nis skewed considerably toward keeping the branched block on the\nexterior of the cylindrical and spherical domains.\nFor example, it was found that for compositionally symmetric ``tadpole''\ncopolymers \\refto{tadpole}, the lamellar \nphase is stable when the branched block \nis $G1-5$, while the cylindrical phase obtains for $G6-8$, and the spherical\nphase occurs for $G9$ and more branched \\refto{dendrimer_copoly}.\n\nThe purpose of the present work is to extend earlier treatments of this system\nwhich had been limited to this most dramatic case where\nthe $A$ and $B$ blocks of the copolymer are both dendrimers of independent\ngeneration.\nThus, we investigate the strong-segregation limit for $GA-GB$ block \ncopolymers, and determine the phase boundaries between the ``classical''\nblock copolymer phases.\nIn the lamellar phase (L), each of the branched blocks stretches\naway from the flat $AB$ interface, producing a layered material as in \nFigure~1.\nThere are two cylindrical phases, one (CA) with the $A$ material forming\na cylindrical core with the $B$ phase forming a continuous matrix in \nwhich the $A$ cores are hexagonally packed.\nThe second cylindrical phase (CB) has the $B$ material confined to the cores,\nwhile the $A$ material forms the continuous matrix.\nLikewise there are two spherical phases, (SA) and (SB) in which one material \nis confined to spherical domains packed on a bcc lattice, while the\nother material forms a continuous matrix.\nDenoting $\\phi_A$ and $\\phi_B$ the overall compositions of the single\ndendriblock copolymers, the phase boundaries will be controlled by the\noverall composition and the generations of the $A$ and $B$\nblocks, $\\phi_B(G_A,G_B)$.\nThe transitions between these phases will always follow the sequence\n(SA) - (CA) - (L) - (CB) - (SB) as $\\phi_B$ is increased.\nThe calculations we make are in the spherical\/cylindrical unit cell \napproximation, and rule out a priori exotic bicontinuous\nphases.\n\nThe first set of calculations we employ involve the Alexander-de Gennes\napproximation \\refto{fredrickson,alexander,degennes}.\nWe assume that all of the $A$ tips and the $B$ tips reside \non the same surfaces. \nIn the (L) phase, this results in the $A$ tips being segregated to a single\nsurface extending away from the $AB$ interface, and similarly for the $B$\ntips.\nIn the (C) phases, the outer block tips all reside on a cylindrical surface\nenclosing the core cylinder, while the inner tips\nare brought toward the center axis of the core,\nand similarly for the (S) phases.\nIt should be noted that the (L) phase thus resembles back-to-back brushes\nof dendritic molecules, while the (C) phase resembles the \nconformation of single dendrimer-comb copolymers, where dendrimers are\ngrafted to a single flexible backbone chain at regular lengths \n\\refto{dendrimer_review_2}.\nThe (S) phase resembles the single-molecule conformation of an individual\ndendrimer molecule.\nThus, the Alexander approximation can be seen as being similar to\nthe de Gennes and Hervet ansatz for a ``hollow-core'' \ndendrimer \\refto{degennes_dendrimer}.\nThe ``filled-core'' picture of Muthukumar and \nLescansec \\refto{filled_core}, however, seems\nto be theoretically and experimentally a more sound description.\nIndeed, allowing the tips of the internal and external dendrimer to occupy\nthe entire lamellar\/cylindrical\/spherical domain turns out to be a more\nrealistic assumption.\nIn the strong-segregation limit, relaxing the confinement of the\nchain free ends is achieved in the so-called ``classical limit'' \n\\refto{semenov,mwc} of\nthe Edwards self-consistent field.\n\n\n\nThe paper is organized as follows. We initially describe the\nAlexander-de Gennes calculation, and then the classical path approximation.\nThen, the phase diagrams for the dendrimer-dendrimer \ncopolymer will be developed.\nThe results herein rest on the stong-segregation limit, the robustness\nof which can be determined in a simple Random Phase Approximation\n(RPA) calculation.\nFinally, our conclusions will be offered.\n}\n\n\n\\section{Model}\nWe use $n$ as a ``chemical index'' marking the fewest monomers required from\na given monomer to arrive at a non-grafted free end.\nThus, each free tip has a chemical index of $0$, and the unique grafted monomer\nhas a chemical index of $N$ (analogous to the overall\nmolecular weight for a linear polymer).\nWe denote the branching profile, $f(n)$ to stand for the number of statistically\nequivalent monomers with the chemical index $n$.\nFor an unbranched polymer, $f(n)=1$, while for a regular \n(double tip-splitting) dendritic polymer \nof generation $G$, \n$f(n)$ is an exponentially decreasoing step function with $f(0) = 2^G$ and discontinuities at \n$n=1\/G, 2\/G ... (G-1)\/G$. This branching profile has been used to\nanalyze brushes of dendritic polymers \\refto{galen_forrest} and \ncopolymers of dendritic polymers \\refto{dendrimer_copoly} and brushes\nof star polymers \\refto{katya}.\nWe can model polymers, however, with continuous branching profiles, such as\n\\begin{equation}\n\\label{geometric}\nf(n) = 2^{-nG+G} \\equiv e^{-b n}\n\\end{equation}\nas in \\refto{galen_zook} where the conformation of a single dendrimer\nwas considered. \nThe ``branching index'', $b$ can be chosen so that we model polymers with\n$G$ generations and a junction functionality of 3 modeling tip-splitting\ndendrimers ... but any positive value of $b$ makes physical sense.\nThis ``continuously branched'' polymer model has significant analytic advantages\nand loses only some of the (admittedly interesting) self-similar details of\nthe exact free-end density.\nWe will consider another class of branching profiles:\n\\begin{equation}\n\\label{powerlaw}\nf(n) = \\left[\\frac{1 - \\alpha n\/N}{1-\\alpha}\\right]^b\n\\end{equation}\nmodeling polymers which are branched so that the average number of \nstatistically equivalent monomers at any chemical index \ndecreases as a power law from the free tips.\nHere, $\\alpha$ is a measure of how strongly branched the polymers are, with the degree of branching diverging at $\\alpha \\rightarrow 1$.\nThese two scenarios are depicted in Figure~\\ref{figure1} (top panel). \n\nAt any rate, we consider a set of polymers characterized by a maximal \nchemical index $N$, of monomers of volume $a^3$ with the $n=N$ monomer \nirreversibly grafted to a flat surface with a grafting density of $\\sigma$\nchains per unit area.\nWe consider a melt brush thus formed so that the overall height of the \ngrafted layer is consistent with the incompressibility of the monomers:\n\\begin{equation}\n\\frac{h}{\\sigma} = a^3 \\int_0^N dn f(n).\n\\end{equation}\nIf $\\sigma$ is large enough, then the chains in the brush will be considerably\nstretched from the grafting surface. \nAs in Figure~\\ref{figure1}, the $n=N$ monomer is located at $z=0$, and \nthe $f(0)$ equivalent chain tips are located at $z=z_o < h$.\nThe single-chain Edwards free energy of this chain is\n\\begin{equation}\nF_{single} = \\int_0^N dn f(n) \\left[ \\frac{1}{2 a^2} \\left|\\frac{dz}{dn}\\right|^2 +\n a^3 P(z(n)) \\right]\n\\label{single}\n\\end{equation}\nUnder conditions in which $F_{single} >> 1$ (in natural energy units of\n$k_B T$), fluctuations of the chain conformations around the\nconfiguration which minimizes Eq.~\\ref{single} are negligible, and all\nstatistical averages can be calculated from the configurations minimizing \nthe single chain free energy itself.\nHere, the first term counts the Gaussian elastic energy of $f(n)$ equivalent\nchains each consisting of $dn$ monomers and stretched a distance $dz$.\nThe second term counts the energy required to evacuate a volume $a^3$ for\neach monomer in these test chains, at a cost in free energy of $P(z)a^3$.\nThus, $P(z)$ is the monomer pressure in the layer, created by the crowding and \nthe conformations of all of the other chains.\n\nThe minimization of Eq.~\\ref{single} is effected through the usual calculus\nof variations, easily recognizable if the kernel of the integral is\ninterpreted as the Lagrangian of a particle with a time-dependent mass in \na gravitational field $-P(z)$:\n\\begin{equation}\nL[z;\\frac{dz}{dn}] = f(n) \\left[ \\frac{1}{2 a^2} |\\frac{dz}{dn}|^2 +\n a^3 P(z(n)) \\right],\n\\end{equation}\nwith the momentum conjugate to $z(n)$ given by\n\\begin{equation}\np = \\frac{\\partial}{\\partial dz\/dn} L = \\frac{f(n)}{a^2} \\frac{dz}{dn},\n\\end{equation}\nand the generalized force given by \n\\begin{equation}\nf = \\frac{\\partial L}{\\partial z} = f(n) a^2 \\frac{d}{dz} P(z(n)).\n\\end{equation}\nThus, the Euler-Lagrange equation governing the minimzation of $F_{single}$ is\n\\begin{equation}\n\\frac{d}{dn} p = f \n\\end{equation}\nor\n\\begin{equation}\n\\frac{d}{dn} f(n) \\frac{dz}{dn} = f(n) a^5 \\frac{dP}{dz}.\n\\label{eqmot}\n\\end{equation}\nThe physical initial condition on this equation of motion is that \n\\begin{equation}\n\\left. \\frac{dz}{dn}\\right|_{n=0} =0,\n\\end{equation} \nthus requiring that there be no tension on the free chain end.\n\nThe pressure field, $P$, can be calculated self-consistently from the \nsolutions to Eq.~\\ref{eqmot}.\nFirst, we need that $P$ satisfies an isochronous property.\nFollowing the time-dependent mass analogy, a particle of mass $f(t)$ released at rest \nfrom a position $z_o$ must hit the grafting surface $z=0$ when $n=N$.\nThis condition must obtain from every position with a non-zero density of \nfree ends in the layer:\n\\begin{equation}\n\\label{sc1}\nN = \\int_o^N dn = \\int_0^{z_o} dz \\frac{1}{dz\/dn}.\n\\end{equation}\nAlso, given such a $P$ it must be possible to create a layer completely\nfilled with monomers at every $z$ without overfilling space for the melt brush we consider here. The\ncommon formulation of this condition is that the height density of free\ntips $d \\sigma \/ dz$ be postive:\n\\begin{equation}\n\\label{sc2}\n\\Phi(z) \\equiv 1 = \\int_z^h dz_o \\frac{d \\sigma (z_o)}{dz_o} \\phi(z;z_o)\n\\end{equation}\nwith the volume fraction of monomers in the vicinity of $z$ for a chain with its \n$f(0)$ ends located at $z_o$:\n\\begin{equation}\n\\phi(z;z_o) = \\sigma a^3 f(n) \\frac{dn}{dz}.\n\\end{equation}\nWhen $f(n)\\equiv1$, (an ordinary polymer brush) self-consistency and the isochronous condition are satisfied \nwhen \\refto{mwc}\n\\begin{equation}\nP(z) = P_o (1-z^2\/h^2)\n\\end{equation}\nwith $P_o = \\pi^2\/8 a \\sigma^2$ and\n\\begin{equation}\n\\frac{d \\sigma}{dz} = \\sigma \\frac{z\/h}{\\sqrt{h^2 - z^2}},\n\\end{equation}\nremarkably compact and elegant expressions.\n\nOne appraoch to solving the self-consistency equations \nEqs~\\ref{sc1}-\\ref{sc2} when $f(n) \\ne 1$ is\nto solve this set of integral equations for the unknown fields $P(z)$ and\n$d \\sigma \/dz$ simultaneously and numerically. \nHowever, knowing that an arbitrary $f(n)$ gives the same physics as a time\ndependent mass in a gravitational field, we can be confident that the \n{\\em parabolic} pressure profile is a fruitful {\\it ansatz}.\nA time-dependent mass in a harmonic gravitational potential will have the\nsame falling time no matter where it is released from rest.\nThe equation of motion under these conditions becomes:\n\\begin{equation}\n\\label{parabolic_eq}\n\\frac{df}{dn} \\frac{dz}{dn} + f(t) \\frac{d^2 z}{dn^2} = -2 f(n) a^5 P_o z\n\\end{equation}\nwith $dz\/dn|_0=0$ where the pressure scale $P_o$ still has to be \ndetermined self-consistently by enforcing $z(N)=0$ numerically.\n\nTo make further progress, we need to specify $f(t)$, but there are two\ngeneral results that can be gleaned at once.\nFirst, if the distribution of ends is non-zero for all $0 0$ throughout the brush, so that the parabolic\n{\\it ansatz} is indeed the correct self-consistent inter-monomer interaction\npotential.\n\n\n\\section{Discussion}\nGiven how successful the parabolic {\\it ansatz} is, it is tempting to\njump to the erroneous conclusion that the parabolic potential is the\nself-consistent potential for {\\em any} choice of $f(n)$.\nThe two choices we have made result in nicely analytic minimum-action\ntrajectories, $z(n;z_o)$, but the critical feature required is that\n$f(n)$ be a decreasing function of chemical index.\nThat is, the chains are unbranched at the grafting surface, and become more\nand more branched as they continue away from the grafting surface.\nIf the chains started highly branched at the grafting surface, with\nsubchains combining into loops as one left the grafting surface so that the\nchain became less and less branched toward the free ends \nthen the brush will resemble a layer or Alexander-deGennes \n\\refto{alexander} \\refto{degennes} brushes \nof loops, with the free ends evacuated from a large region of the \nbrush \\refto{fredrickson}. The self-consistent potential is then known\nto be radically different from the parabolic potential here, although the\nreal potential still satisfies an isochronous, or monodisperse, condition.\n\nIndeed, if we start with an unbranched set of chains, $f(n)=1$,\nand alter the branching a bit by\n\\begin{equation}\nf(n) = 1 + \\epsilon(n)\n\\end{equation}\nwith $\\epsilon(n)$ a small function that is very strongly peaked at $n=0$, \nit is easy to show that the surface end-density must be negative, and\nspace is overfilled if the parabolic potential is followed.\nEssentially, we need to increase the number of chain free ends located with\n$z_o$ near the full brush height, $h$, above the unbranched limit. This causes \nthe number of monomers entrained into the layer to be slightly larger than they would \nhave been had $\\epsilon(n)=0$, with the result that building the brush from the\nouter edge to the surface maintaining $\\phi(z)=1$ will overfill space with \nmonomers at the grafting surface.\n\nSo, if $f(n)$ is a {\\em rapidly} decreasing function of $n$, each chain ``dumps''\nalmost all of its monomers right in the vicinity of the free end location,\n$z_o$.\nIn this case, the molten layer resembles a set of large, dense microgels \ntethered by much lower molecular weight net of branched chains. \nThe minimum-action chain trajectories become (essentially) particles at rest\nfor most of $n0.2$). Fainter than an absolute magnitude of $H=5$, deep water ice absorption is never seen.\n\nWhile such a correlation of absolute magnitude and presence of\nwater ice might be expected simply from the increased albedo\nof objects with more water ice on their surfaces, Spitzer radiometry\nhas shown that the $H>3$ KBOs are indeed smaller than the $H<3$ KBOs\n\\citep{2008ssbn.book..161S}.\nSomewhere between the $\\sim$650 km diameter of Ixion and \nthe $\\sim$900 km diameters\nof Quaoar and Orcus, KBOs appear\nto dramatically change in surface composition.\n\nThe change in surface composition is clearly not monotonic with\nsize. Ixion, 2002 UX25, 2002 AW197, 2004 GV9,\nand 2003 AZ84 have $\\sim$650 km diameters\nwithin their uncertainties \\citep{2008ssbn.book..161S}, but\nonly 2003 AZ84 has a large abundance of water ice on the surface. \nLikewise, the smaller 2005 RM43, which would have a diameter of \n$\\sim$520 km if it has the same albedo as 2003 AZ84, has water\nice absorption as deep as the larger objects Quaoar and 2007 OR10,\nwhile the similarly-sized objects Varuna and Huya have much\nsmaller absorption depths. \n\nSmaller than the $\\sim$520 km size of 2005 RM43, however, no (non-Haumea\nfamily member) KBO has been found with strong water ice absorption.\nFew objects in this range are bright enough for\nhigh quality spectroscopy (1996 TP66, with a diameter of $\\sim$180 km\nfor an assumed 0.1 albedo is the smallest KBO with a well measured\nspectrum), however. Nonetheless, spectroscopic evidence from\ncentaurs and photometric evidence from smaller KBOs (see below)\nsuggests that this trend continues, and that between 500 and 700 km\nin diameter KBOs transition from typical surfaces of small KBOs\nto those dominated by absorption due to water ice.\n\nAdditional evidence for a surface change in this diameter\nrange comes from measurements\nof albedos. \nThough the size and\nalbedo data available from Spitzer photometry have large\nuncertainties, a jump in albedo at this size range \nis apparent.\nThe same effect is see in the\ncumulative absolute magnitude distribution of the large\nobjects. While this distribution follows a power law from \nabsolute magnitudes of 3 until at least 4.5, for objects \nbrighter than $H=3$, the cumulative number distribution is\nlarger than would be expected from this power law, a clear\nsign of the increased albedos of these mid-sized\nobjects \\citep{2008ssbn.book..335B}. Implications for this surface\nchange are discussed below.\n\n\\subsection{Ammonia}\nOf these mid-sized objects with strong water ice absorption\nfeatures, two have evidence for a 2.25 $\\mu$m absorption\nfeature due to ammonia. Ammonia was first detected on Charon\n\\citep{2000Sci...287..107B,2007ApJ...663.1406C,2010Icar..210..930M} and later suggested to be present also on\nOrcus \\citep{2008A&A...479L..13B}. In both cases, the presence of ammonia\nwas postulated to be due to the flow of ammonia-rich interior\nliquid water on to the surface of the object at some point in\nthe past. Detailed models of the interior structure and evolution\nof these two bodies have suggested that such a scenario \nappears reasonable \\citep{2007ApJ...663.1406C, 2010A&A...520A..40D}.\n\nAmmonia has not been detected on the other water ice-rich objects,\nbut for Quaoar a bright object where high quality\nspectra can be obtained, the 2.3 $\\mu$m region of the spectrum\nwhere ammonia has its only detectable near-infrared feature is\ninstead dominated by the strongest absorption feature due to CH$_4$.\nAmmonia has additional absorption features beyond 3 $\\mu$m which\ncould be distinguished from CH$_4$ but spectral measurement\nat these wavelengths awaits larger ground or space-based telescopes.\nThe smaller water-ice-rich objects 2003 AZ84 and 2005 RM43 have spectra with\ninsufficient signal-to-noise to detect ammonia. Larger telescopes will\nagain be required.\n\nBased on the likely presence of ammonia on Charon and Orcus,\nand the sharp increase in water ice absorption with larger\nsize for these objects, \nwe hypothesize that on these largest objects the presence\nof water ice -- whether ammonia is detectable or not -- is a\nremnant of past liquid flows on the surface. We predict that\nfor these objects ammonia will always be detected when sufficient\nsignal-to-noise is available. The liquid flows\nneed not be recent -- the purity of the water ice on the surface of\nHaumea and its family shows that water ice can remain pristine through\nthe age of the solar system -- but the liquid flows must have \noccurred after the volatiles whose irradiation would cause coloration\n(see below) have all escaped. The increase in water ice absorption\nwith size would be a natural consequence of the larger \ninterior liquid reservoirs of larger objects. The sharp increase\nin water ice absorption starting at diameters around $\\sim$650 km\ngives an important clue into the physics of liquid interiors\nand surface water flows.\n\n\\section{Small objects: spectroscopic constraints}\n\\subsection{Kuiper belt objects}\n\nMany objects smaller than the size where strong water ice\nabsorption is present are statistically\nconsistent with having no water ice on their surface, but there is\na significant prevalence in the data for positive detection of water ice.\nRandom and known systematic errors would not produce a bias towards\nwater ice detection. Indeed, we regard the nearly complete lack of \nobjects with negative water ice fraction as an indication of the\nrobustness of our method. We conclude, therefore, that even the low\nlevel of water ice fraction detected in the majority of the\nobjects is a real indication that water ice at a low level\nis common even on the objects smaller than Ixion.\n\n2002 VE95, the smallest Kuiper belt object with a very high quality \nspectrum ($\\sim$330 km diameter assuming a 0.1 albedo), shows\nclear evidence of crystalline water ice on its surface\n\\citep{2008AJ....135...55B, 2006A&A...455..725B}.\nIndeed, the 1.65 $\\mu$m\nabsorption feature of crystalline water ice can always be detected when\nthe signal-to-noise is sufficiently high. \n\nBased on their smaller sizes, it\nseems likely that the water ice on these smaller objects is\nnot caused by liquid flows on the exterior, but rather by the\nexposure of crustal water ice. An important test of this expectation\nwould be that these smaller KBOs should not show the presence\nof ammonia on their surfaces. Currently feasible spectroscopy\ncannot achieve the signal-to-noise required to make this test, however.\n\n\n\\subsection{Centaurs}\nKBOs smaller than those discussed above are too faint for high quality\nspectroscopy even with the largest telescopes in the world. To understand\nthe compositions of these objects, we have to resort to\nobservational proxies. One important\nproxy has been spectroscopic observations of Centaurs. Centaurs are\nformer KBOs which have been perturbed onto short-lived planet-crossing\norbits. Being much closer to the sun than typical KBOs, they are \nbrighter and easier to study in detail.\n\nSpectroscopically, the Centaurs appear indistinguishable\nfrom the smallest KBOs whose spectra can be measured: objects contain\neither no detectable absorption features, a small amount of absorption \ndue to water ice, or (in one case) absorption due to water ice and \nmethanol. Figure 5 includes water ice absorption depth for a large\nsample of Centaurs, compared to KBOs. No discernible difference\ncan be seen between the largest of the centaurs and the smallest of\nthe measured KBOs. \nThe amount of water ice absorption seen on the surface of a centaur\ndoes not appear to correlate with anything, including\nperihelion, semimajor axis, optical color, dynamical lifetime, or \nactivity. Interestingly, while much speculation has occurred about \nCentaur surface evolution as objects move closer to the sun and \nbegin heating, infrared spectroscopy shows no such evidence of any\nchange in the distribution of water ice absorption depth.\n\n\\subsection{Methanol}\nAn absorption band at $\\sim$2.27 $\\mu$m\nwas first detected on the bright\ncentaur Pholus \\citep{1993Icar..102..166D}, and \\citet{1998Icar..135..389C} present \nthe case that this band is plausibly due to the presence of\nmethanol, though they point out that the identification is\nnot unique and that other low molecular weight hydrocarbons\nor photolytic products of methanol might fit the spectrum equally well.\nPholus is one of the reddest objects in the solar system, again\nfitting our picture of optical colors of irradiated hydrocarbons well.\n\nBased on lower signal-to-noise spectra with properties\nwhich resemble the water-ice-plus-methanol spectrum of Pholus, the presence of\nmethanol was suggested on the KBOs 1996 GQ21 and 2002 VE95\n\\citep{2008ssbn.book..143B}. \nTo examine this possibility more closely, we combine the Keck spectra\nof these two objects to increase signal-to-noise and consider\nthe presence of methanol. While the signal-to-noise\nremains low, the presence of absorption features similar\nto those on Pholus is certainly plausible.\nBoth objects, like Pholus, are red.\n\nA handful of other KBOs have recently been reported to also have\nabsorption features near 2.27 $\\mu$m, but, unlike\nPholus, 1996 GQ21 and 2002 VE95, to not have absorption\ndue to water ice \\citep{2011Icar..214..297B}. The signal-to-noise in\nthe spectral region are low,\nso it is difficult to determine\nif the absorption features are real. \nTo examine the possibility that\na $\\sim$2.27 $\\mu$m absorption feature occurs on faint KBOs,\nwe first examine all of the KBOs and centaurs in the Keck sample\n(which includes none of the potential methanol objects from\nthe VLT sample).\nWe find that a small number of objects have absorption features\nat or near the 2.27 $\\mu$m methanol absorption line, but\nno single spectrum is sufficiently reliable in this region to\nassert a positive detection. To increase the signal-to-noise,\nwe sum the spectra of all of the KBOs and centaurs in the Keck\nsample\nexcept for those with water ice absorption at the level of\nthat seen on 2003 AZ84 or deeper and those already suspected\nto contain methanol-like features (1996 GQ21 and 2002 VE95). This combined spectrum shows\nresidual absorptions due to water, as would be inferred from\nthe positive detections on most objects. The only major deviation\nfrom the water ice spectrum occurs at precisely the wavelength\nof the feature seen on Pholus and suspected on 1996 GQ21 and 2002 VE95.\nWe conclude that\nthe methanol-like feature is indeed present at a low-level on\nKBOs even though the feature cannot be reliably identified in\nindividual spectra\n(Figure 6).\n\\begin{figure}\n\\includegraphics[scale=.75]{fig6.eps}\n\\caption{The region of the near-infrared spectrum containing \nthe 2.27 $\\mu$m absorption feature attributed to methanol.\nThe feature can clearly be seen on Pholus, superimposed on a \nsmall amount of water ice absorption. The combined\n(to increase signal-to-noise) spectrum of 1996 GQ21 and 2002 VE95\nshows hints of a feature at the same location. The sum of 38\nKBO and centaur spectra, none of which individually show clear evidence\nfor the feature, clearly shows a feature at the same spot.\n}\n\\end{figure}\n\n\nNo hypothesis has ever been formulated for the \nsporadic presence of methanol on KBOs or centaurs, other\nthan to point out that methanol is common in cometary comae,\nso it is expected to be present in the interior of KBOs.\nIts presence is less expected on the surface of KBOs, however,\nas the absorption features of hydrocarbon ices quickly degrade\nunder irradiation while the remnants turn red \\citep{2006ApJ...644..646B}.\nVisible methanol absorption features suggest that the\nmethanol has only recently been exposed at the surface, perhaps\nas a result of a collision exposing the subsurface. \nOne prediction of this suggestion would be that the amount of\nmethanol observed would vary as different faces of the object were\nobserved. While such a test is possible in principle, in practice\nspectroscopy of these faint objects \nis sufficiently difficult that variation would be difficult to\nprove. If, however, irradiation preferentially turned the regions\nwith exposed methanol red, these objects should perhaps show color\nvariation with rotation, something which is otherwise rarely \nobserved in objects of this size \\citep{2008ssbn.book..129S}.\n\n\\subsection{Silicates}\nIn addition to ices and their irradiation products, we should\nexpect that KBOs that are small enough to be\nundifferentiated should expose some of their\nrock component on the surface. While silicates such as\nolivine or pyroxene\nare often included in detailed models of KBO spectra \n\\citep[i.e.][]{2010Icar..208..945M,2011Icar..214..297B}\n no\nspecific absorptions are easily observable: olivine, for example, has a \nbroad absorption centered at around 1 $\\mu$m, where observations\nare usually poor. These silicates are thus included in models\nto fit the overall spectral shapes over the $\\sim$0.6 - 1.2 $\\mu$m\nrange.\n\nMid-infrared observations have the possibility of positively\nidentifying silicate emission in the 10 and 20 $\\mu$m region \non KBOs. To date only the centaur Asbolus has had a \nspectrum measured with sufficient signal-to-noise to even\ndetect these spectral regions, and, in this case, emissivity\npeaks around 10 and 20 $\\mu$m are indeed seen, similar to\nthat observed in Trojan asteroids \\citep{2006Icar..182..496E,2008ssbn.book..143B},\nand interpreted to be due to fine-grained silicates.\nFuture observations will require improved space-based \nmid-infrared facilities, but characterization of \nactual silicate composition is indeed possible.\n\nWhile olivines or pyroxenes cannot be specifically identified\nin the visible to near-infrared range,\nthere have been a few reports of shallow broad absorption\nfeatures in the visible wavelength range\nsimilar to absorptions seen on some asteroids.\nOn asteroids these are generally attributed to aqueously \naltered silicates.\nConfirmation of these features has been difficult; the absorptions\nare subtle and have frequently appeared changed or absent upon reobservation\nof the same object \\citep{2004A&A...421..353F,2004A&A...416..791D, 2009A&A...508..457F, 2008A&A...487..741A, 2003AJ....125.1554L}\nWhile these changes are usually attributed to rotational\nvariability, it is worth noting that this speculation has never\nbeen verified. Indeed, for one object observations over half \nof a rotational period showed no signs of the visible absorption \nfeatures.\n\nWhile it is possible that these difficult-to-confirm features \nare a product of sporadic systematic error, the possibility\nof the existence of aqueously altered silicates is \nan interesting one to consider. The presence of\nliquid water on surfaces in the Kuiper belt might seem surprising, but\nhydrous materials seem to be present in small comets, interplanetary\ndust particles, and debris disks \\citep{2004A&A...416..791D}. \nIf these detections are indeed real, the most surprising thing\nabout them, perhaps, is that they are uncommon. \n\n\n\n\\section{Small objects: photometric constraints}\nSmall objects in the Kuiper belt are too faint to detect spectroscopically,\nbut photometric measurements can still give information -- albeit limited --\nabout the surfaces of these objects. To date, the single most robust\nconclusion based on photometry is that Kuiper belt surfaces are\ndiverse.\nLarge surveys of KBO optical\ncolors have found a wider range of\nsurface colors\nin the Kuiper belt than in any other small body population in the \nsolar system. The colors are uncorrelated with\nmost dynamical or physical properties \\citep[see review in][]{2008ssbn.book...91D}.\nA few systematic patterns\nhave been found, however, which are important clues to understanding the\nsurface compositions of these outer solar system objects.\n\n\\subsection{The bifurcated colors of centaurs} \nThe range of centaur optical colors generally covers the same wide\n range of colors\nfound in the Kuiper belt, but the centaurs are deficient in colors in\nthe middle part of the range, giving the distribution of centaur optical\ncolors\na bimodal appearance \\citep{2008ssbn.book..105T} with a neutral\nand a red clump of objects. Interestingly, this bifurcation is not\nseen only in the centaurs, but it appears to extend to all objects with\nlow perihelion distance whether the objects are dynamically unstable or\nnot. This result immediately suggests that the bifurcation in the \noptical colors is somehow formed through the increased heating or\nirradiation experienced by lower perihelion objects.\n\nThe H\/WTSOSS (Hubble\/WFC3 Test of Surfaces in the Outer Solar System) survey which used the Hubble Space Telescope to extend\n optical photometry of centaurs (and other low\nperihelion objects) into the near-infrared\n\\citep{hwtsoss} found that the neutral and red clumps do not consist\nof two groups with identical surfaces, \nbut rather are best described by two groups that fall along two\nseparate \nmixing lines.\nThe neutral clump of objects consists of a mixture of\na nearly neutrally reflecting material and a slightly red material,\nwhile the red clump of objects consists of a mixture of the same\nneutrally reflecting material and a much red material\n(Fig 7). \n\\begin{figure}\n\\includegraphics[scale=.65]{fig7.eps}\n\\caption{\nThree-color HST photometry\nof objects with perihelia inside of 30 AU from\nthe H\/WTSOSS survey. The filters are at 0.606, 0.814 1.39, and\n1.54 $\\mu$m, corresponding roughly to $R$, $I$, $H$-continuum, \nand a filter sampling the 1.6 $\\mu$m water ice absorption \nfeature. \nThe yellow point shows the\ncolors of the sun. \nThe objects appear to bifurcate into two clumps, one with near-solar optical colors\n(the ''neutral'' clump) and one with significantly redder optical colors (the ''red clump'').\nThe neutral clump (on the left in [F606W]-[F814W]\ncolors) shows a clear spread in the other colors, while the\nred clump shows a spread in all three colors. The color variation in\nboth clumps can be described by a mixing line (purple lines) where\nthe neutral and red clumps both have a single \ncommon end member but are mixed with either a more neutral or more \nred component, respectively.}\n\n\n\\end{figure}\n\n\nWhile three color photometry is generally incapable of identifying specific \nices or minerals, \\citet{hwtsoss} find that the neutral component \ncommon to all centaurs is consistent with some of the same\nhydrated silicates suggested from the optical spectroscopy\ndiscussed above. Significantly more spectral work is required,\nhowever, to further explore this possibility.\n\n \\subsection{The uniform colors of the cold classical KBOS.}\nWhile most of the rest of the \nKuiper belt appears to be composed of essentially the\nsame distribution\n of neutral to red objects \\citep{morbandbrown,2008ssbn.book...91D}\none dynamical region stands out for its\nhomogeneous composition. The cold classical Kuiper belt was first identified\nas a dynamically unique region of the Kuiper belt -- a difficult-to-explain \noverabundance of low inclination, dynamically cold objects beyond about\n41 AU \\citep{2001AJ....121.2804B}. \nSubsequent observations revealed that these objects\nshared a common red coloring \\citep{2002ApJ...566L.125T}. \nIn the H\/WTSOSS survey these objects do not fall along mixing\nlines, as the neutral and red clumps of centaurs do, but instead appear\nnearly uniform in color space \\citep{hwtsoss}.\nThese cold classical KBOs are\nalso know to be unique in their lack of large bodies \n\\citep{2001AJ....121.1730L}, their\nhigher abundance of satellites \\citep{2008Icar..194..758N}, \nand their different size distribution \\citep{2010Icar..210..944F}.\nAnd though our understanding of albedos in the Kuiper belt is\nstill poor, preliminary results suggest that the cold classical KBOs also appear to have higher albedos than\nthose of the remaining population\n\\citep{2009Icar..201..284B}. \nAll of these properties appear to signify a population with a \ndifferent -- and perhaps unique -- formation location or history. \nUnderstanding the surface compositions of these objects will be \na challenge given their distance and small sizes. In addition, these\nobjects are dynamically stable, so we likely never get samples\nof this population as centaurs or comets.\n\n\n\\subsection{The diverse colors of the remainder of the Kuiper belt}\nOther than the unique color properties of the centaurs and cold classical\nKBOs, the bulk of KBOs have no discernible pattern to their colors.\nThis finding in itself is significant for understanding the causes\nof the colors of KBOs.\nThis lack of\na connection between color and any dynamical property, \nparticularly with semi-major axis or perihelion\/aphelion \ndistance argues strongly that local heating, UV irradiation, \nand solar wind and cosmic ray\nbombardment \\citep{2003EM&P...92..261C} cannot be responsible for\nthe varying colors of the Kuiper belt. Local conditions appear to\nhave no primary influence on the colors of KBOs. \nFurthermore, careful measurement of the colors of the separate \ncomponents of binary KBOs has shown a tight correlation over the full\nrange of Kuiper belt colors \\citep{2009Icar..200..292B}. \nThe colors of two KBOs in orbit around\neach other are almost always nearly identical.\nThis fact immediately rules out any of the stochastic process such as\ncollisions for the causes of these Kuiper belt colors. Indeed, \ngiven the lack of correlation of color with local conditions, the\nnearly identical colors of binary KBOs argues that colors are simply primordial.\nIf binary KBOs were formed by early mutual capture in a \nquiescent disk \\citep{2002Natur.420..643G}, \nthe two component would likely have formed in very similar locations. If,\nalternatively, binary KBOs were formed in an initial gravitational\ncollapse \\citep{2010AJ....140..785N}, the objects would of necessity have formed at the \nsame location and of the same materials. \n\n\\subsection{The transition from KBOs to centaurs}\nThe manner in which primordial KBO surfaces evolve to become\nthe color-bifurcated centaur population could provide important \nclues to the compositions of both surfaces. While it appears\nthat a transition from a unimodal to bimodal color distribution\nmust occur as objects move to lower perihelia, the actual\nevidence for change is, in fact, weak.\nThe KBOs and centaurs with measured colors\nhave very different ranges of sizes, \nand, as demonstrated above, large KBOs have their\nsurfaces modified by the presence of volatiles and water ice. We\nthus must only compare like-sized objects. In addition, the cold\nclassical KBOs, with their unique surfaces, likely never enter the\ncentaur population, so these should be excluded from the \ncomparison. Finally,\nthe KBOs, being more distant, are likely to have higher uncertainty in\ntheir color measurements. \n\n\\begin{figure}\n\\plotone{fig8.eps}\n\\caption{\nA comparison of the colors of (non-cold classical)\nKBOs (thin line) and centaurs (thick line) in the $6P_z\/A$.\nAs a reasonable first approximation, we treat the nucleus as a\ncollection of $N=3A$ valence quarks, which, on the average, carry each\nlongitudinal momentum $x_{0}P_{z}\/A$ with $x_{0}=1\/3$.\nIn our approach the cumulative pion production proceeds in two steps.\nFirst a valence quark with a scaling variable $x>1$ is created.\nAfterwards\nit decays into the observed hadron with its scaling variable $x$ close\nto the initial cumulative quark's one. This second step is described by\nthe well-known quark fragmentation functions\n\\cite{CapellaT81} and will not be\ndiscussed here.\n\nThe produced cumulative (\"active\") quark acquires\nthe momentum much greater than $x_{0}P_{z}\/A$ only if this\nquark has interacted by means of gluon exchanges\nwith other $p$ quarks of flucton (\"donors\")\nand has taken some of their longitudinal momenta (see Fig.1).\nIf this active quark accumulates all\nlongitudinal momentum of these $p$ quarks then\n$K_z=(p+1)x_0P_z\/A$ and the donors become soft.\nIt is well-known that interactions which make the longitudinal momentum\nof one of the quark equal to zero may be treated by perturbation theory\n\\cite{Brodsky92}. This allows to calculate the part of Fig. 1\nresponsible for the creation of a cumulative quark explicitly.\nThis was done in \\cite{NPB94,YF97}, to which papers we refer the\nreader for all the details. As a result we were able to explain the\nexponential fall-off of the production rate in the cumulative region.\n\nSince with the rise of $x$ the active quark has to interact with\na greater number of donors, one expects that its average transverse momentum\nalso grows with $x$. Roughly one expects that $\\langle K_{\\perp}^2 \\rangle$\nis proportional to the number of interactions, that is, to $x$.\nIn \\cite{NPB94,YF97}\nthis point was not studied:\nwe have limited ourselves with the inclusive cross-section\nintegrated over the transverse momenta,\nwhich lead to some simplifications.\nThe aim of the present paper is to find\nthe pion production rate dependence on\nthe transverse momentum\nand the mean value of the latter\nas a function of $x$ in the cumulative region.\nThis dependence and also the magnitude of $\\langle K_{\\perp}^2 \\rangle$\nhave been studied experimentally. The comparison of our predictions\nwith the data allows to obtain further support for our model and\nfix one of the two its parameters (the infrared cutoff).\n\n\n\\section{The $K_{\\perp}$ dependence}\n\nRepeating the calculations of the diagram in Fig.1 described in\n\\cite{NPB94,YF97} but not\nlimiting ourselves with the inclusive cross-section\nintegrated over the transverse momentum,\nwe readily find that all dependence upon\nthe transverse momentum $K_{\\perp}$ of the produced particle\nis concentrated in a factor:\n\\begin{equation}\nJ (K_{\\perp})=\n\\int\n\\rho_A(\\underbrace{r,...,r}_{p+1}|\\underbrace{\\ol{r},...,\\ol{r}}_{p+1})\nG(c_1,...,c_p)\n\\prod_{i=1}^{p}\n\\lambda(c_i-r)\\lambda(c_i-\\ol{r})d^2 c_i\ne^{i(\\ol{r}-r)K_{\\perp}}\nd^2 r d^2 \\ol{r}\n\\end{equation}\nHere $\\rho_A$ is the (translationally invariant)\nquark density matrix of the nucleus:\n\\begin{equation}\n\\rho_A(r_i|\\ol{r}_i)\\equiv\n\\int\n\\psi_{\\perp A}(r_i,r_m)\n\\psi^*_{\\perp A}(\\ol{r}_i,r_m)\n\\prod_{m=p+2}^{N} d^2 r_m\n\\end{equation}\nwhere\n$\\psi_{\\perp A}$ is the transverse part of\nthe nuclear quark wave function. The propgation of soft donor quarks\nis decribed by\n\\begin{equation}\n\\lambda(c)=\\frac{K_0(m|c|)}{2\\pi}\n\\label{lam}\n\\end{equation}\nwhere $m$ is the constituent quark mass and $K_0$\nis the modified Bessel function (the Mac-Donald function).\nThe interaction with the projectile contributes a factor\n\\begin{equation}\nG(c_1,...,c_p)=\n\\int\n\\prod_{i=1}^{p}\n\\sigma_{qq}(c_i-b_i)\n\\eta_H(b_1,...,b_p) d^2 b_i\n\\end{equation}\nwhere $\\sigma_{qq}(c)$ is the quark-quark cross-section\nat a given value of impact parameter $c$ and\n\\begin{equation}\n\\eta_H(b_1,...,b_p)=\n\\sum_{L\\geq p}\\frac{L!}{(L-p)!}\n\\int|\\psi_{\\perp H}(b_i)|^2\n\\delta^{(2)}(\\frac{1}{L}\\sum_{i=1}^{L} b_i)\n\\ d^2 b_{p+1}...d^2 b_L\n\\end{equation}\nis a multiparton distribution in the projectile, expressed via\nthe transverse part of\nits partonic wave function $\\psi_{\\perp H}$ .\nIf we integrate $J (K_{\\perp})$ over $K_{\\perp}$ we come back\nto our old result (Eq. (33) in \\cite{NPB94}):\n$$\n\\int J (K_{\\perp})\n\\frac{d^2 K_{\\perp}}{(2\\pi)^2}=\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n\\int\nG(c_1,...,c_p)\n\\prod_{i=1}^{p} \\lambda^2(c_i-r)\nd^2 c_i d^2 r\n$$\n\nIf one assumes factorization of the multiparton distribution\n$\\eta_H(b_1,...,b_p)$ then\\\\\n$G(c_1,...,c_p)$ also factorizes:\n\\begin{equation}\nG(c_1,...,c_p)=\n\\prod_{i=1}^{p}\nG_0(c_i)\n\\label{facG}\n\\end{equation}\nFollowing \\cite{YF97} we use the quasi-eikonal approximation\nfor $\\eta_H$:\n$$\n\\eta_H(b_1,...,b_p)=\n\\xi^{(p-1)\/2}\\nu_H^{p}\\prod_{i=1}^{p}\\eta_H(b_i)\n$$\nwhere $\\xi$ is the quasi-eikonal diffraction factor,\n$\\nu_H^{}$ is the mean number of partons in the projectile hadron\nand the single parton distribution $\\eta_H(b)$\nis normalized to unity.\nIn a Gaussian approximation\nfor $\\sigma(c)$ and $\\eta_H(b)$ we find:\n$$\nG_0(c)=\n\\xi^{\\frac{1}{2}-\\frac{1}{2p}}\n\\frac{\\nu_H\\sigma_{qq}}{\\pi r_{0H}^2}\ne^{-\\frac{c^2}{r_{0H}^2}}\n$$\nwhere $\\sigma_{qq}$ is the total quark-quark cross-section,\n$r_{0H}^2=r_0^2+r_H^2$, $r_0$ and $r_H$ are the widths of\n$\\sigma(c)$ and $\\eta_H(b)$ respectively.\n\nWith the factorised $G(c_1,...,c_p)$ (\\ref{facG}) we have\n$$\nJ (K_{\\perp})=\n\\int\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{\\ol{r}-r,...,\\ol{r}-r}_{p+1})\nj^p(r,\\ol{r})\ne^{i(\\ol{r}-r)K_{\\perp}}\nd^2 r d^2 \\ol{r}\n$$\nwhere\n$$\nj(r,\\ol{r})=\n\\int d^2 c\nG_0(c)\n\\lambda(c-r)\n\\lambda(c-\\ol{r})\n$$\nWe also have used the translational invariance of the $\\rho$-matrix.\nNote that near the real threshold we have no spectators and\n$$\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{\\ol{r}-r,...,\\ol{r}-r}_{p+1})\n=\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n$$\nIn any case large $K_{\\perp}$ corresponds to small $\\ol{r}-r$ so\nwe factor $\\rho_A$ out of the integral in zero point.\nIn the rest integral we pass to the variables\n$$\nB=\\frac{r+\\ol{r}}{2}, \\hs 1 b=\\ol{r}-r\n$$\nand shift the integration variable $c$, then\n\\begin{equation}\nJ (K_{\\perp})=\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n\\int\nj^p(B,b)\ne^{ibK_{\\perp}}\nd^2 b d^2 B\n\\end{equation}\nwhere\n\\begin{equation}\nj(B,b)=\n\\int\nG_0(B+c)\n\\lambda(\\frac{b}{2}-c)\n\\lambda(\\frac{b}{2}+c)\nd^2 c\n\\label{j}\n\\end{equation}\n\n\n\\section{The calculation of $\\langle |K_{\\perp}|\\rangle $}\n\nNow we would like to find the width of the distribution on $K_{\\perp}$\nas a function of $p$ or what is the same of the cumulative\nnumber $x=(p+1)\/3$.\nFrom the mathematical point of view it is simpler to calculate\nthe mean squared width of the distribution $\\langle K_{\\perp}^2\\rangle $.\nUnfortunately in our case this quantity is logarithmically\ndivergent at large $K_{\\perp}$.\nThis divergency results from the behavior of $j(B,b)$\nat small $b$. This behavior is determined by the behavior of the\n$ \\lambda(b)=K_0(m|b|)\/(2\\pi) $ (\\ref{lam}),\nwhich has a logarithmical singularity\nat $|b|=0$. Smooth $G_0(B+c)$ does not affect this behavior.\n\nFor this reason we shall rather calculate $\\langle |K_{\\perp}|\\rangle $:\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\n=\\frac{1}{J_N}\n\\int\nj^p(B,b)\n|K_{\\perp}|e^{ibK_{\\perp}}\nd^2 b d^2 B\n\\frac{d^2 K_{\\perp}}{(2\\pi)^2}\n\\end{equation}\nwhere $J_N$ is the same integral as in the numerator\nbut without $|K_{\\perp}|$.\nPresenting $|K_{\\perp}|$ as $K_{\\perp}^2\/|K_{\\perp}|$ and\n$K_{\\perp}^2$ as the Laplacian $\\Delta_b$ applied to\nthe exponent we find\n$$\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{1}{J_N}\n\\int\nj^p(B,b)\n\\Delta_b e^{ibK_{\\perp}}\nd^2 b d^2 B\n\\frac{d^2 K_{\\perp}}{|K_{\\perp}|(2\\pi)^2}\n$$\nTwice integrating by parts\nand using the formula\n$$\n\\int\n\\frac{d^2 K_{\\perp}}{|K_{\\perp}|}e^{ibK_{\\perp}} =\n\\frac{2\\pi}{|b|}\n$$\nwe find\n$$\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{1}{2\\pi J_N}\n\\int\n\\frac{1}{|b|}\n\\Delta_b j^p(B,b)\nd^2 b d^2 B\n$$\nNow we again integrate by parts once to find\n$$\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{1}{2\\pi J_N}\n\\int\nd^2 B\n\\frac{d^2 b}{|b|^2}\n(n_b \\nabla_b) j^p(B,b)\n$$\nwhere $n_b=b\/|b|$. This leads to our final formula\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\n=-\\frac{p}{2\\pi J_N}\n\\int\nd^2 B\n\\frac{d^2 b}{|b|^2}\nj^{p-1}(B,b)\n(n_b \\nabla_b) j(B,b)\n\\label{Kperp}\n\\end{equation}\nwhere $j(B,b)$ is given by (\\ref{j}),\n$ \\lambda(b) $ is given by (\\ref{lam}) and\n$$\nJ_N=\\int d^2 B j^{p}(B,b=0)\n$$\n\n\n\\section{Approximations}\nTo simplify numerical calculations we make some additional\napproximations, which are not very essential but are well supported\nby the comparison with exact calulations at a few sample points.\n\nAs follows from the\nthe asymptotics of $K_0(z)$ at large $z$\n$$\nK_0(z)\\simeq\\sqrt{\\frac{\\pi}{2z}}e^{-z}\n$$\nthe width of $\\lambda(b)$ (\\ref{lam}) is of the order $m^{-1}$.\nThe function $G_0$ is smooth in the vicinity of the origin\nand its width $r_{0H}=\\sqrt{r_0^2+r_H^2}$ is substancially\nlarger than the width of $\\lambda$.\nFor this reason we factor $G_0(B+c)$ out of the integral (\\ref{j})\nover $c$ at the point $B$:\n\\begin{equation}\nj(B,b)=\nG_0(B)\\Lambda(b),\n\\hs 1\n\\Lambda(b) \\equiv\n\\int\n\\lambda(c)\n\\lambda(c-b)\nd^2 c=\n\\frac{|b|}{4\\pi m}K_1(m|b|)\n\\label{Lam}\n\\end{equation}\nThen we find that the integrals over $B$ and $b$ decouple\n\\begin{equation}\nJ (K_{\\perp})=\n\\rho_A(\\underbrace{0,...,0}_{p+1}|\\underbrace{0,...,0}_{p+1})\n\\int G^p_0(B) d^2 B\n\\int \\Lambda^p(b)\ne^{ibK_{\\perp}}\nd^2 b\n\\end{equation}\n\nIn this approximation we find that $\\langle |K_{\\perp}|\\rangle $\ndepends only\non one parameter - the constituent quark mass $m$,\nwhich in our approach plays the role of an infrared cutoff:\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\n=pm\n\\int_0^{\\infty} dz K_0(z) (zK_1(z))^{p-1}\n\\label{apprKperp}\n\\end{equation}\nThis allows to relate $m$ directly to the experimental data on\nthe transverse momentum dependence.\n\n\\section{Comparison with the data and discussion}\n\nThe integral in (\\ref{apprKperp}) can be easy calculated\nnumerically. For values of $p=1,...,12$ it is very well\napproximated by a power dependence (see Fig.2), so that we obtain\n\\begin{equation}\n\\langle |K_{\\perp}|\\rangle\/m\n=1.594\\, p^{0.625}\n\\end{equation}\nAs we observe, the rise of $\\langle |K_{\\perp}|\\rangle$ turns out to be\neven faster than\nexpected on naive physical grounds mentioned in the Introduction\n($\\sim\\sqrt{p}$).\nThe resulting plots for $\\langle |K_{\\perp}|\\rangle^2$\nas a function of the cumulative number $x=(p+1)\/3$\nat different values of parameter $m$ are shown in Fig.3\ntogether with avaiable experimental data from \\cite{Boyarinov94}\non $\\langle K_{\\perp}^2\\rangle$ for pion production obtained in\nexperiments\n\\cite{Boyarinov94}-\\cite{Boyarinov87}\nwith 10 $GeV$ protons\nand \\cite{Baldin82, Baldin83} with 8.94 $GeV$ protons.\n\nNote that earlier publications of the first group\n\\cite{Boyarinov92,Boyarinov87}\nreported a much stronger increase of $\\langle K_{\\perp}^2\\rangle$\nwith $x$, up to value 2 $(GeV\/c)^2$ at $x=3$ for pion production.\nIn our approach such an increase would require the quark mass to be as high\nas $m \\simeq 225 MeV$.\nIn a more recent publication \\cite{Boyarinov94}\nthe rise of $\\langle K_{\\perp}^2\\rangle$ is substancially weaker\n(it corresponds to $m \\simeq 175 MeV$ in our approach).\nThe authors of \\cite{Boyarinov94} explain this\nby new experimental data obtained and by a cutoff $K_{\\perp max}$\nintroduced in calculations of $\\langle K_{\\perp}^2\\rangle$ in \\cite{Boyarinov94}.\nThe introduction of this cutoff considerably\n(approximatly two times)\ndecreases the experimental value of $\\langle K_{\\perp}^2\\rangle$ at $x=3$.\nIn our opinion this is a confirmation that the\ncumulative pion production rate only weakly decreases with $K_{\\perp}$\nin the cumulative region so that the\nthe integral over $K_{\\perp}^2$ which enters the definition of\n$\\langle K_{\\perp}^2\\rangle$ is weakly convergent or even divergent,\nas in our approach.\nUndoubtedly presentation of the experimental data in terms of\nthe mean value\n$\\langle |K_{\\perp}| \\rangle^2$,\nrather than $\\langle K_{\\perp}^2\\rangle$ should reduce\nthe dependence on the cutoff\n$K_{\\perp max}$ and make the results more informative.\n\nOne of the ideas behind the investigations of the cumulative\nphenomena is that they may be a manifestation\nof a cold quark-gluon plasma formed when several nucleons overlap in the\nnuclear matter. In \\cite{NPB94} we pointed out that our model does not\ncorrespond to this picture. It implies coherent interactions of the\nactive quark with donors and, as a result, strong correlations between the\nlongitudinal and transverse motion. Predictions for the dependence of\n$\\langle |K_{\\perp}| \\rangle$ on $x$ are also different. From the cold\nquark-gluon plasma model one expects $\\langle |K_{\\perp}| \\rangle$ to behave\nas $x^{1\/3}$, since the Fermi momentum of the quarks inside the overlap\nvolume is proportional to the cubic root of the quark density. Our model\npredicts a much faster increase, with a power twice larger. The\nexperimental data seem to support our predictions.\n\n\\section{Acknowledgments}\n\nThe authors are greatly thankful to Prof. P.Hoyer who attracted\ntheir attention to the problem.\n\nThis work is supported by the Russian Foundation for Fundamental\nResearch, Grant No. 97-02-18123.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Acknowledgments}\nWe would like to thank De Meng, Maryam Fazel and Mehran Mesbahi for their help.\n\n\\section{Conclusions}\\label{conclusion}\nWe derived a class of $N$ player, weighted potential games under heterogeneous MDP dynamics and with application to multi-agent path coordination. For these games, we show equivalence between the unique Nash equilibrium and the global solution of a potential minimization problem, which we solve via gradient descent and single-player dynamic programming. Future work include deriving learning-based solutions for the games and \\change{integrating partially observable scenarios in which players have local observations only.} \n\n\n\n\n\\section{Multi-agent path coordination}\\label{sec:mapf}\n\\noindent We apply our game model to a multi-agent pick up and delivery scenario \\change{with stochastic package arrival times}. As shown in Figure~\\ref{fig:grid_world}, $N$ players navigate a 2D \\change{space}. \\change{Each player's goal is} to transport packages from the pick up chutes to the drop off chutes while \\change{avoiding collision with others}. Code for the simulation is available at \\url{https:\/\/github.com\/lisarah\/mdp_path_coordination}.\n\\begin{figure}\n \\centering\n \\vspace*{0.3cm} \n \\includegraphics[width=0.7\\columnwidth]{Figure\/plzwork.png}\n \\caption{Operation environment for multi-robot warehouse scenario.}\n \\label{fig:grid_world}\n\\end{figure}\n\\subsection{Stationary MDP Model}\n\\change{Players operate in a} two dimensional grid world with $5$ rows and $10$ columns. In addition to capturing location, each state also dictates whether the robot is in pick up or delivery mode. The state space is given by\n\\[[S] = \\Big\\{ (\\change{v, w}, m) \\ | \\ 1\\leq \\change{v} \\leq 5, \\ 1\\leq \\change{w} \\leq 10, \\ m \\in \\change{\\{1,2\\}}\\Big\\}.\\]\nAt each state, available actions are $[A] = \\{u, d, r, l, s\\}$, corresponding to up, down, right, left, stay. Player transition dynamics and rewards are \\emph{stationary} in time.\nThe transition probability of each state $(\\change{v, w}, m)$ extends the location-based transition probabilities $P^0$. \n\n\\noindent\\textbf{Location-based transition}. Let $u =\\change{(v, w)}$ denote the location component of the state. At each location, each action either points to a feasible target $u_{targ}(a)$ or is infeasible. The set of all feasible targets from $u$ is $\\mathcal{N}(u)$. \nWhen a target exists, players have $1 > q > 0$ chance of reaching it and $1 - q$ chance of reaching other states in $\\mathcal{N}(u)$. \n\\begin{equation}\\label{eqn:feasible_location_transitions}\nP^0_{u'ua} = \\begin{cases}\nq & u' = u_{targ}(a), \\\\\n\\frac{1 - q}{|\\mathcal{N}(u)|} & u' \\in \\mathcal{N}(u)\/\\{u_{targ}(a)\\},\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nWhen the target location is infeasible, the player transitions into a neighboring state $u' \\in \\mathcal{N}(u)$ at random.\n\\begin{equation}\\label{eqn:infeasible_location_transitions}\nP^0_{u'ua} = \\begin{cases}\n\\frac{1}{|\\mathcal{N}(u)|} & u' \\in \\mathcal{N}(u),\\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\n\\textbf{Full transition dynamics}. \\change{Within the same mode}, players transition between locations via dynamics $P^0$. Player modes \\change{transition at pick up chutes $\\mathcal{P}$ and drop off chutes $\\mathcal{D}$}.\n\\change{\n\\begin{enumerate}\n \\item When player $i$ is in mode $1$ (pick up) and about to transition into pick chute $p^i \\in \\mathcal{P}$, player $i$'s mode has $r^i$ probability of switching to mode $2$ (drop off). \n \\[\\begin{cases}\n P^i_{t(p^i, 2)sa} = r^iP^0_{tp^iua},\\\\\n P^i_{t(p^i, 1)sa} = (1 - r^i)P^0_{tp^iua},\n \\end{cases}\\ \\forall s = (u, 1), \\ s \\in [S].\\]\n \\item When player $i$ is in mode $2$ (drop off) and about to transition into drop chute $d^i \\in \\mathcal{D}$, player $i$ transitions to mode $1$ with probability $1$.\n \\[\\textstyle \\begin{cases}\n \\textstyle P^i_{t(d^i, 1)sa} = P^0_{td^iua},\\\\\n \\textstyle P^i_{t(p^i, 2)sa} = 0,\n \\end{cases}\\ \\forall s = (u, 2), \\ s \\in [S].\\]\n\\end{enumerate}\n}\nHere, $r^i \\in {\\mathbb{R}}$ denotes the probability of package arrival when player $i$ is in \\change{$p^i$}. Modeled as an independent Poisson process with rate $\\lambda_i$ and interval $\\Delta t = 1s$, $r^i = \\exp(-\\lambda_i \\Delta_t)$.\n\n\\subsection{Player Costs}\n\\noindent For all $(t,s, a) \\in \\mathcal{T}\\times[S]\\times[A]$ and congestion distribution $y$~\\eqref{eqn:congestion_distribution}, player $i$'s \\change{cost is given by\n\\[\\textstyle\\ell_{tsa}^i(y, x^i) = \\epsilon x^i_{tsa} - c^i_{tsa} + \\alpha_i f_{ts}(y).\\]}\n\\noindent The player-specific objective $c^i_{tsa}$ is defined as \n\\begin{equation}\nc^i_{t(\\change{v, w}, m) a} = \\begin{cases}\n1 & \\change{(v, w) = p^i}, \\ m = \\change{1}, \\\\\n1 & \\change{(v, w) = d^i}, \\ m = \\change{2}, \\\\\n0 & \\text{otherwise.}\n\\end{cases}\n\\end{equation}\nThe congestion function is strictly state-based and is an exponential function given by\n\\begin{equation}\\label{eqn:simulation_congestion_function}\nf_{t\\change{(v, w, m)}}(y) = -\\beta\\exp\\big(\\beta(\\sum_{m' \\in \\change{\\{1,2\\}}}\\sum_{a' \\in [A]}y_{t(\\change{v, w}, m')a'} - 1)\\big),\n\\end{equation}\nwhere $\\alpha_i> 0$ for all $(t,s,a) \\in \\mathcal{T}\\times[S]\\times[A]$. \\change{As opposed to~\\eqref{eqn:individual_costs}, function~\\eqref{eqn:simulation_congestion_function} calculates the congestion in $(v,w, \\cdot)$ using both $(v,w, 1)$'s and $(v, w, 2)$'s congestion level. }\n\\begin{figure}\n \\centering\n \\vspace*{0.3cm} \n \\includegraphics[trim={0 0 0 0.5cm},width=0.85\\columnwidth]{Figure\/polished_figures\/distribution_convergence.png}\n \\caption{$\\norm{\\cdot}_2$ of player $i$'s state-action distribution over Algorithm~\\ref{alg:frank_wolfe} iterations.}\n \\label{fig:frank_wolfe}\n\\end{figure}\n\\subsection{Simulation Results}\n\\noindent We simulate the path coordination game using parameters from Table~\\ref{tab:sim_params}. Player $i$'s pick up locations is the $i^{th}$ element of $ \\mathcal{P}= \\{(4, \\change{w^i}) \\ | \\ \\change{w^i} \\in [8, 7, 2]\\}$, \\change{and its drop-off location is the $i^{th}$ element of $\\mathcal{D} = \\{(0, w^i) \\ | \\ w^i \\in [4,5,8]\\}$}. At $t = 0$, players are initialized \\change{at their drop off location. }\n\\begin{table}[h!]\n\\begin{center}\n\\begin{tabular}{|ccccccccc|}\n\\hline\n$N$ & $q$ & $\\gamma_i$ & $\\lambda_i$ & \\change{$\\alpha_i$} & $\\Delta t$ & $T$ & \\change{$\\epsilon$} & \\change{$\\beta$} \\\\\n\\hline\n3 & 0.98 & 0.99& 0.5 & \\change{\\{0.5, 1, 1.5\\}} & 1s & \\change{120s} & \\change{1e-3} & \\change{40}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Parameters for simulation environment.}\\label{tab:sim_params}\n\\end{table}\n\n\\noindent We run Algorithm~\\ref{alg:frank_wolfe} for $100$ iterations, where line~\\ref{alg:mdp} is solved via value iteration~\\eqref{eqn:value_iteration}. \\change{The two norm of $x^i$ is shown in Figure~\\ref{fig:frank_wolfe} as a function of the algorithm iterations. We see that the state-action densities stabilize in about $20$ steps.} Performance is evaluated by: 1) expected number of collisions, 2) expected packages delivery time, 3) worst package delivery time. The results over $100$ random trials are visualized in Figures~\\ref{fig:collision_results} and~\\ref{fig:waiting_time_results}. \n\\begin{figure}\n \\centering\n \\vspace*{0.3cm} \n \\includegraphics[trim={0 0 0 0.5cm},width=0.8\\columnwidth]{Figure\/polished_figures\/collision_time_line.png}\n \\caption{Collisions per player as a function of MDP time step $t$.}\n \\label{fig:collision_results}\n\\end{figure}\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.95\\columnwidth]{Figure\/polished_figures\/collision_stats.png}\n \\caption{Average waiting time per package, worst case waiting time per package, and average number of collisions in $T$ for each player.}\n \\label{fig:waiting_time_results}\n\\end{figure}\n\n\\noindent\\change{\nWe compare the \\emph{jointly optimal congestion-free wait time} computed using Algorithm~\\ref{alg:frank_wolfe}, and compare them to the shortest \\change{wait time} available in the absence of opponents.\n}\nEach path is the number of steps to complete the drop off-pick up-drop off cycle. \\change{Based on players' pick-up and drop-off locations, their shortest \\change{wait time} in the absence of opponents is $16$, $12$, $20$ respectively.} We note that this matches well with the average wait time shown in Figure~\\ref{fig:waiting_time_results}.\n\nWe set the player impact factors as $\\{0.5, 1, 1.5\\}$ as in Table~\\ref{tab:sim_params}. From Figure~\\ref{fig:waiting_time_results}, the impact factors directly correlate with rate of collision players experience. Player $0$ impacts congestion the least and is the least sensitive to congestion. As a result, it encountered more collisions. Player $2$ impacts congestion the most and is the most sensitive to congestion. As a result, it encountered the least collisions. The collision rate is spread out evenly over $\\mathcal{T}$ (Figure~\\ref{fig:collision_results}).\n\\section{Individual Players}\n\nTo analyze how the penalty functions affect the individual player's decision making, we develop their perspective through the value function formulation, make connections to Q-Functions, and show that the penalties directly affect the value functions of individual players.\n\n\\subsection{Dual Formulation: Value Iteration}\nIt is well know that the dual of the linear programming MDP optimization problem is the dynamic programming formulation of the same problem. Here we show that in the non-linear, state dependent case, the dual constraints forms the dynamic programming problem new players joining the game would solve at time $t$.\n\nIndividual players try to optimize their value functions through solving the dual of the primal MDP problem. Given the MDP game primal problem, Eqn. \\ref{MDP game}, the dual can be written as: \n\\begin{equation}\n\\begin{aligned}\n\\max_{V_{ts}} & \\quad \\sum_s V_{0s}p_s \\\\\n\\text{s.t } & V_{ts} \\leq \\sum_{s'} P(s'| s, a)V_{t+1,s'} + \\ell_{tsa}(y_{tsa}) \\quad \\\\\n& \\forall \\, s \\in \\mathcal{S}, a\\in \\mathcal{A}, t \\in [T-1] \\\\\n& V_{Ts} \\leq \\ell_{tsa}(y_{tsa}) \\quad \\forall \\, s \\in \\mathcal{S},\\,\\, a\\in \\mathcal{A}\n\\end{aligned} \\label{unconstrained dual}\n\\end{equation}\nFrom this we can derive the value iteration method shown for latency functions $\\ell_{tsa}(y_{tsa}) $. Let the Q function be defined as following: \n\\begin{equation}\\label{q function}\nQ_{tsa}(y_{tsa}) = \\sum_{s'} P(s'| s, a)V_{t+1,s'} + \\ell_{tsa}(y_{tsa}) \\\\\n\\end{equation}\nThe difference $V^*_{ts} - Q_{tsa}(y^*_{tsa}) = \\mu^*_{tsa} \\geq 0$ is the dual variable of constraint $y_{tsa} \\geq 0 $, and is also the inefficiency of the action $a$. Any action $\\bar{a} $ whose inefficiency is zero is the optimal action at that particular state and time. \n\n\nWhich leads to the value iteration method Alg. \\ref{value iteration}. \n\\begin{algorithm}\n\\caption{Value iteration}\n\\begin{algorithmic}[h]\n\\Require \\(R\\), \\(P\\).\n\\Ensure \\(V^\\star\\), \\(\\pi^\\star\\).\n\\ForEach{\\(s,\\in\\mathcal{S}\\)}\n\\State{\\(V^\\star_{Ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{min}} \\,\\,\\ell_{Tsa}(y_{Tsa})\\)}\n\\State{\\(\\pi^\\star_{Ts}= \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}} \\,\\,\\ell_{Tsa}(y_{Tsa})\\)}\n\\EndFor\n\\For{\\(t=T-1, T-2, \\ldots, 0\\)}\n\\ForEach{\\(s\\in\\mathcal{S}\\)}\n\\State{\\(\\displaystyle V^\\star_{ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{min}}\\,\\, Q_{tsa}(y_{tsa})\\)}\n\\State{\\(\\displaystyle \\pi^\\star_{ts} = \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}}\\,\\, Q_{tsa}(y_{tsa})\\)}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\label{value iteration}\n\\end{algorithm}\n\n\\begin{theorem}\nAssuming existing players have reached nash equilibirum, and the rewards are state dependent $R_{tsa}(y_{tsa}) $, algorithm \\ref{value iteration} with optimal $y^*_{tsa} $ is the dynamic programming problem that any new player solves entering the game at time $t $.\n\nFurthermore, the best reward that the new player can gain is defined by \n\\[\\min_s V^*_{ts}\\]\n\\end{theorem}\n\n\\begin{proof}\nSince the algorithm depends on current state-action distribution, $y_{tsa}$. At Nash Equilibrium the constraints would be: \n\\[V_{ts} \\leq \\sum_{s'} P(s'| s, a)V_{t+1,s'} + \\ell_{tsa}(y^*_{tsa}) \\]\n\\[V_{Ts} \\leq \\ell_{tsa}(y^*_{tsa}) \\]\nAny player entering at time t would need to maximize its value function according to $V_{ts} $. Since the game is non-atomic, the new player would not affect the state depedent reward functions. Therefore, the new player's optimal policy can be found from algorithm \\ref{value iteration}.\n\n\\end{proof}\n\n\n\nSuppose we impose constraint $y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}\\bar{a}} $ for one state $\\bar{s} $ and all times t, and suppose it was violated during time step $\\bar{t}$. From an individual game player's perspective, the exact penalty function coefficients $k_t(\\bar{s},\\bar{a}) $, directly offsets the value functions at constrained optimal $y_c^*(s,a)$. \n\nUsing the exact penalty theorem, we find the minimum coefficient $k_{\\bar{t}\\bar{s}\\bar{a}} \\geq \\tau^*_{\\bar{t}\\bar{s}\\bar{a}} $ to ensure that the optimal solutions are between the penalized problem and the unconstrained problem are the same. However in order to ensure that the next player will also avoid the constraints, additional \"toll\" need to be charged. \n\n\n\\begin{theorem}\nIn addition to $\\tau^*_{\\bar{t}\\bar{s}}$ the dual variable's optimal value corresponding to the constraint $y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}} $ in problem \\eqref{MDP game}. Let \n\\[k_{\\bar{t}\\bar{s}\\bar{a}} > \\max\\{\\tau^*_{\\bar{t}\\bar{s}\\bar{a}}, \\mu_{\\bar{t}\\bar{s}a_{new}} \\}\\]\n\\[a_{new} = \\text{argmin}_{a \\neq \\bar{a}} Q_{\\bar{t}\\bar{s}a} \\]\nThen we can guarantee that the next player will not choose to violate the constraint $y_{\\bar{t}\\bar{s}\\bar{a}} \\leq d_{\\bar{t}\\bar{s}\\bar{a}} $\n\n\\end{theorem}\n\n\\begin{proof}\nFrom the exact penalty theorem, the solution to the unconstrained problem \\eqref{MDP Game} and the constrained problem \\eqref{cMDP game} share the same value and argument. \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\sum\\limits_{a\\in\\mathcal{A}} \\int_0^{y_{tsa}}\\ell_{tsa}(x)dx\\\\\n\\mbox{s.t.} &\\sum\\limits_{a\\in\\mathcal{A}} y_{t+1, sa} = \\sum\\limits_{s'\\in\\mathcal{S}}\\sum\\limits_{a\\in\\mathcal{A}}P_{s'as}y_{ts'a}, \\quad t\\in[T-1],\\\\\n&\\sum\\limits_{a\\in\\mathcal{A}}y_{0sa}=p_s,\\\\\n& y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}\\bar{a}}, \\\\\n&y_{tsa}\\geq 0,\\quad \\forall s\\in\\mathcal{S}, a\\in\\mathcal{A}, t\\in[T]\n\\end{aligned}\\label{cMDP game}\n\\end{equation}\n\nFor at the optimal solution of this constrained MDP game, the dual value iteration method occurs according to algorithm \\ref{constrained value iteration}. Here we see the value of $k_{t\\bar{s}\\bar{a}} $ directly offsets the Q-functions at the times when the unconstrained solution violated the state constraints.\n\n\\begin{algorithm}\n\\caption{Constrained Value iteration at constrained optimal $y_c^* $}\n\\begin{algorithmic}[h]\n\\Require \\(R\\), \\(P\\).\n\\Ensure \\(V^\\star\\), \\(\\pi^\\star\\).\n\\ForEach{\\(s,\\in\\mathcal{S}\\)}\n\\State{\\(\nV^\\star_{Ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{min}} \\,\\,Q_{Tsa} + k_T(s,a)\\)}\n\\State{\\(\n\\pi^\\star_{T}(s)= \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}} \\,\\,Q_{Tsa} + k_T(s,a)\\)}\n\\EndFor\n\\For{\\(t=T-1, T-2, \\ldots, 0\\)}\n\\ForEach{\\(s\\in\\mathcal{S}\\)}\n\\State{\\(\\displaystyle V^\\star_{t}(s)= \\underset{a\\in\\mathcal{A}}{\\mbox{min}}\\,\\, Q_{tsa} + k_t(s,a)\\)}\n\\State{\\(\\displaystyle \\pi^\\star_{t}(s) = \\underset{a\\in\\mathcal{A}}{\\mathop{\\rm argmin}}\\,\\, Q_{tsa}+ k_t(s,a)\\)}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\label{constrained value iteration}\n\\end{algorithm}\n\nSuppose the violated constraint occurs at $\\bar{t}, \\bar{s}, \\bar{a} $. Let $\\tilde{\\mu} $ be the smallest inefficiency of the set of actions available at $\\bar{s} $, excluding optimal action $\\bar{a} $. If $k_{\\bar{t}\\bar{s}\\bar{a}} > \\tilde{\\mu} $, then $\\bar{a} $ is no longer the optimal, and so the player will not choose $\\bar{a} $ as optimal policy. Therefore, the original constraint $y_{t\\bar{s}\\bar{a}} \\leq d_{t\\bar{s}\\bar{a}} $ cannot be violated at $\\bar{t} $ if the original solution was feasible. \n\nSince $k_{\\bar{t}\\bar{s}\\bar{a}} > \\tau^*_{\\bar{t}\\bar{s}\\bar{a}} $, then we still satisfy the exact penalty theorem, and so the optimal solution and arguments of the penalized problem still correspond to the constrained problem's solution and arguments. \n\\end{proof}\n\n\\begin{theorem}\nFor a new player entering the game at time $t $, their best expected reward is upperbounded by the optimal value function at time $t $. \n\\[\\max_{s} \\tilde{V}_{ts} \\]\nWhere $\\tilde{V}_{ts} $ is the result of solving algorithm \\ref{constrained value iteration}. Furthermore, the player can achieve this value by starting at state $s = \\text{argmax} V_{ts} $ and follow optimal policy $\\pi_{ts}^* $from algorithm \\ref{constrained value iteration}.\n\\end{theorem}\n\\begin{proof}\nThis follows from the fact that the value functions is equivalent to the expected sum of rewards over time.\n\\end{proof}\n\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}\nAs autonomous \\change{path} planning algorithms become widely-adapted by aeronautical, robotics, and operational sectors~\\cite{yun2020multi,ota2006multi}, the standard underlying assumption that the operating environment is stationary is no longer sufficient. More likely, autonomous players \\emph{share} the operating environment with other players who may have conflicting objectives. While the possibility for multi-agent conflicts has pushed single-agent \\change{path} planning towards greater emphasis on \\change{robust planning} and collision avoidance, we believe that the overarching goal should be to consider other players' trajectories and achieve optimality with respect to the \\emph{multi-agent dynamics}. \n\nWe focus on the scenario where a group of heterogeneous players collectively perform path planning in response to stochastic demands. We are inspired by fleets of robo-taxis fulfilling ride demands while avoiding congestion in traffic~\\cite{vosooghi2019robo} and warehouse robots retrieving packages under dynamic arrival rates~\\cite{kumar2018development,li2020mechanism} while avoiding collisions. The common feature in these applications is that the players must plan with respect to a forecasted demand distribution rather than a deterministic demand. We assume that the desirable outcome is a competitive equilibrium. Beyond competitive settings, a competitive equilibrium can be used in cooperative settings to ensure that each player achieves identical costs and each demand is \\emph{optimally} fulfilled with respect to other demands, thus ensuring a degree of fairness.\n\nWe propose MDP congestion games as \\change{a theoretical framework for analyzing the resulting path coordination problem}. By leveraging common congestion features in multi-agent \\change{path} planning, our key contribution is \\emph{reducing} the $N$-player coupled MDP problem to a \\emph{single} potential minimization problem. As a result, we can use optimization techniques to analyze the Nash equilibrium as well as apply gradient descent methods to compute it.\n\n\n\n\\textbf{Contributions}. To address the lack of \\change{game-theoretical models for} path \\change{coordination under} MDP dynamics, we propose an MDP congestion game with finite players and heterogeneous \\change{player} costs and dynamics. We define Bellman equation-type conditions for the Nash equilibrium, formulate a \\emph{potential function} and provide a necessary and sufficient condition for its existence. \\change{Under certain assumptions on the player costs}, we show equivalence between the \\change{Nash equilibrium} and the \\change{global solution of} the potential minimization problem, and provide sufficient conditions for a unique Nash equilibrium. Specifically \\change{for multi-player path coordination}, we \\change{formulate a class of cost functions that allows players to have different sensitivities to the total congestion and to find congestion-free paths that optimally achieve their individual objectives}. Finally, we provide a \\change{distributed} algorithm that converges to the Nash equilibrium and give rates of \\change{its} convergence. We demonstrate our model and algorithm on a 2D autonomous warehouse problem where robots retrieve and deliver packages \\change{with stochastic arrival times} while sharing a common navigation space. \n\n \n\n\n\n\n\n\n\n\n\n\\section{Unknown Initial Distribution}\n\n(I believe this is the the most general way to constrain the policy to maintain safety in the MDP.)\n\nI think we want to solve the following problem for all $t' = 0,\\dots,T$.\n\n\\begin{align}\n\\max_{y,p_{t'}} & \\quad \\sum_{s} \\sum_{a} \\Gamma_{t'}(s'|s,a) K_{t'}(s,a)p_{t'}(s) \\\\\n\\text{s.t.} & \\quad \\sum_s \\sum_a y_0(s,a) = 1 \\notag \\\\\n& \\quad \\sum_a y_{t+1}(s',a) = \\sum_{s \\in \\mathcal{S}} \\sum_{a \\in \\mathcal{A}} \\Gamma_t(s'|s,a) y_t(s,a) \\notag \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad t = 0,\\dots,t'-1 \\notag \\\\\n& \\quad \\sum_a y_{t'}(s,a) = p_{t'}(s) \\notag \\\\\n& \\quad 0 \\leq y_t(s,a) \\leq d_t(s,a) \\qquad t=0,\\dots,t' \\notag\n\\end{align}\n\nWe want to get a bound for the maximum amount of mass in state $s^*$. \n\\begin{align}\n\\min & \\quad \\lambda + \\sum_t\\sum_s\\sum_a d_t(s,a)\\tau_t(s,a) \\\\\n\\text{s.t.} & \\quad \\lambda = V_0(s)+\\tau_0(s,a) \\qquad \\qquad \\qquad \\forall s \\in \\mathcal{S} \\notag \\\\\n& \\quad V_t(s') = \\sum_s \\Gamma_t(s'|s,a)\\Big(V_{t+1}(s)+\\mu_t(s,a)\\Big)+\\tau_t(s,a) \\notag \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\forall a \\in \\mathcal{A} \\notag \\\\\n& \\quad V_{t'} = \\sum_a \\Gamma_{t'}(s^*|s,a)K_{t'}(s,a) + \\tau_{t'}(s,a) \\notag\n\\end{align}\n\n\n\\section{Useless Stuff}\n\nWe now consider the case where the initial distribution is not known a-priori but rather only constrained to be within some set. We assume $p_0 \\in \\mathcal{P}_0$. We first consider box capacity constraints on $p_0 \\leq d_0$ and defining the least conservative constant constraints on the policy that keep the set invariant. Set $\\bar{d}_0 = d_0$. Let $\\bar{d}_t$ be upper bounds on the state distribution at time $t$. We now derive bounds on the entries of $K_t(s,a)$. If the constraint at the next time step \n\\begin{align}\nd_{t+1}(s')= \\Gamma_t(s'|s,a)K_t(s,a)\\bar{d}_t(s)\n\\end{align}\n\\begin{align}\nK_t(s,a) = \n\\frac{d_{t+1}(s')}{\\Gamma_t(s'|s,a)\\bar{d}_t(s)}\n\\end{align}\n\\begin{align}\nD_t(s,a) = \\min \\left\\{\\min_{s'}\n\\frac{d_{t+1}(s')}{\\Gamma_t(s'|s,a)\\bar{d}_t(s)},1\n\\right\\}\n\\end{align}\n\\begin{align}\n\\bar{d}_{t+1}(s') = \\max_{y_t} & \\quad \\sum_s \\sum_a \\Gamma_t(s'|s,a)y_t(s,a) \\\\ \n\\text{s.t.}& \\quad \\sum_a y_t(s,a) \\leq \\bar{d}_t(s) \\notag \\\\\n& \\quad \\sum_{s,a} y_t(s,a) = 1 \\notag \\\\\n& \\quad y_t(s,a) \\leq \\text{diag}(y_t\\mathbf{1})D_t(s,a) \\notag\n\\end{align}\n\\section{Heterogeneous MDP Congestion Game}\\label{sec:notation}\nConsider a finite number of players $[N] = \\{1, \\ldots, N\\}$ with a \\emph{shared finite state-action space} given by $([S], [A])$ and common time interval $\\mathcal{T} = \\{0, 1, \\ldots, T\\}$. Each player $i$ has \\emph{individual} time-dependent transition probabilities given by $P^i\\in {\\mathbb{R}}_+^{TSSA}$, where at time $t$, $P^i_{ts'sa}$ is the transition probability from state $s$ to state $s'$ using action $a$ \\change{satisfying} the simplex constraints:\n\\begin{equation}\n \\textstyle\\sum_{s'} P^i_{t s' s a} = 1, \\ \\forall (i, t, s, a) \\in [N]\\times \\change{[T]}\\times [S] \\times [A].\n\\end{equation}\n\\textbf{State-action distribution}. \\change{At time $t$, let player $i$'s state be $s^i(t) \\in [S]$ and action taken be $a^i(t) \\in [A]$, then $x^i_{tsa} = \\mathbb{P}[s^i(t) = s, a^i(t) = a]$ is player $i$'s probability of being in state $s$ taking action $a$ at time $t$.}\n\\change{Player $i$'s state-action probability trajectory over time period $\\mathcal{T}$ is $x^i \\in{\\mathbb{R}}^{(T+1)SA}$, its \\emph{state-action distribution}.} \\change{We use $\\mathcal{X}(P^i, z^i_0)$ to denote the set of all feasible state-action distributions under transition dynamics $P^i$ and initial condition $z^i_0 \\in{\\mathbb{R}}^S_+$, where $z^i_{0s}= \\mathbb{P}[s^i(0) = s]$ is player $i$'s probability of starting in state $s$.}\n\\begin{multline}\\label{eqn:feasible_mdp_flows}\n\\textstyle\\mathcal{X}(P^i, \\change{z^i_{0}}):= \\Big\\{\\change{x^i} \\in {\\mathbb{R}}_+^{(T+1)SA} \\Bigg| \\sum_{a} \\change{x^i_{0sa} = z^i_{0s}}, \\forall s \\in [S], \\Big. \\\\ \n\\textstyle\\Big.\\sum_{s', a} P^i_{tss'a} \\change{x^i_{(t-1)s'a}} = \\sum_{a} \\change{x^i_{tsa}}, \\ \\forall (t,s) \\in \\change{[T]}\\times [S] \\Big\\}.\n\\end{multline}\nThe \\emph{joint state-action distribution} of all players is given by\n\\begin{equation}\\label{eqn:joint_state_action}\n x = (x^1, \\ldots, x^N) \\in {\\mathbb{R}}_+^{N(T+1)SA}.\n\\end{equation}\n\\change{We assume that $x$ is fully observable and may denote it as $x = (x^i, x^{-i})$ where $x^{-i} = (x^j)_{j \\in [N]\/\\{i\\}}$.}\n\n\\noindent\\textbf{\\change{Player} costs}. \\change{Similar to stochastic games, the player} costs are continuously differentiable \\emph{functions} of $x$: player $i$ incurs a cost $\\ell^i_{tsa}(x)$ for taking action $a$ at state $s$ and time $t$.\n\\begin{equation}\\label{eqn:player_cost_def}\n \\ell^i_{tsa}: {\\mathbb{R}}_+^{N(T+1)SA} \\mapsto {\\mathbb{R}}, \\ \\forall (i,t,s,a) \\in [N]\\times \\mathcal{T} \\times [S] \\times [A].\n\\end{equation}\n\\change{Compared to stochastic games where player costs are coupled to the opponent policies,~\\eqref{eqn:player_cost_def} is better suited to model collision events. For example, the expectation of the log-barrier function for players $i$ and $j$ at time $t$ can be modeled as $\\sum_{s, s' \\in [S]} (\\sum_a x^i_{tsa})(\\sum_a x^j_{ts'a})\\log(d_{s,s'})$, in which $d_{s,s'}$ denotes the distance between states $s, s'\\in [S]$. }\n\n\\noindent The \\emph{cost vector} of $(\\ell^1,\\ldots \\ell^N)$~\\eqref{eqn:player_cost_def} is given by $\\xi:{\\mathbb{R}}_+^{N(T+1)SA}\\mapsto {\\mathbb{R}}_+^{N(T+1)SA}$,\n\\begin{equation}\\label{eqn:cost_vector}\n \\xi(x) = [\\ell^1_{011}(x), \\ell^1_{012}(x), \\ldots, \\ell^N_{TSA}(x)] \\in {\\mathbb{R}}_+^{N(T+1)SA}.\n\\end{equation}\nWe assume that $\\xi$ has a positive definite gradient in $x$. \n\\begin{assumption}\\label{assum:positive_definite} The player cost vector $\\xi$~\\eqref{eqn:cost_vector} satisfies $\\nabla \\xi(x) \\succ 0$ for all $x$~\\eqref{eqn:joint_state_action} where $x^i \\in \\mathcal{X}(P^i, x_0^i), \\ \\forall i \\in [N]$.\n\\end{assumption}\n\\noindent For the class of player costs considered in Section~\\ref{sec:path_coordination_costs}, Assumption~\\ref{assum:positive_definite} implies that the player costs strictly increase as the number of players increases.\n\n\\noindent\\textbf{Coupled MDPs}. Given an initial distribution $z_0^i \\in {\\mathbb{R}}^S_+$ and fixed state-action distributions \\change{$x^{-i}$}~\\eqref{eqn:joint_state_action}, player $i$ solves the following optimization problem under MDP dynamics.\n\\begin{equation}\n\\begin{aligned}\\label{eqn:individual_player_mdp}\n \\underset{x^i}{\\mbox{min}} & \\change{\\sum_{t, s, a}\\int_0^{x^i_{tsa}}\\ell^i_{tsa}(u^i, x^{-i}) \\partial u^i_{tsa}} \\ \\mbox{s.t. } & x^i \\in \\mathcal{X}(P^i, z^i_{0}).\n\\end{aligned}\n\\end{equation}\n\\change{In~\\eqref{eqn:individual_player_mdp}, we note that each integral is taken over $u^i_{tsa}$, the $(t, s, a)^{th}$ element of $u^i$.} \nWhen $\\ell^i_{tsa}(x)$ is constant for all $(t,s,a) \\in \\mathcal{T} \\times [S]\\times [A]$, player $i$ solves a standard \\emph{linear program} MDP.\n\n\\noindent\\textbf{Dynamic programming}. At a joint state-action distribution $x$~\\eqref{eqn:joint_state_action}, player $i$'s cost-to-go in~\\eqref{eqn:individual_player_mdp} can be recursively defined via Q-value functions~\\cite{puterman2014markov} as\n\\begin{multline}\n Q^i_{Tsa}(x) := \\ell^i_{Tsa}(x), \\label{eqn:q_value} \\\\ \n Q^i_{(t-1)sa}(x) := \\ell^i_{(t-1)sa}(x) + \\textstyle\\sum_{s'} P^i_{ts'sa}\\underset{a'}{\\min}\\, Q^i_{t,s'a'}(x),\\\\\n\\forall \\ t \\in [T]\n\\end{multline}\n\\noindent\\change{\nThe optimal solution of~\\eqref{eqn:individual_player_mdp} can be stated using~\\eqref{eqn:q_value}.}\n\n\\begin{theorem}\n\\change{Under Assumption~\\ref{assum:positive_definite}, ${x}^i$~\\eqref{eqn:feasible_mdp_flows} uniquely minimizes~\\eqref{eqn:individual_player_mdp} with respect to the state-action distribution $x^{-i}$ if and only if its associated $Q^i({x}^i, x^{-i})$~\\eqref{eqn:q_value} satisfies\n\\begin{equation}\\label{eqn:individual_optimality}\n {x}^i_{tsa} > 0 \\Rightarrow \\textstyle Q^i_{tsa}({x}^i, x^{-i})= \\min_{a'} Q^i_{tsa'}({x}^i, x^{-i}),\n\\end{equation}\nfor all $(t, s, a) \\in \\mathcal{T}\\times[S]\\times[A]$.} I.e., ${x}^i$ is optimal for~\\eqref{eqn:individual_player_mdp} if and only if every action played with nonzero probability achieves the minimum cost-to-go~\\eqref{eqn:q_value} among available actions. \n\\end{theorem}\n\\begin{proof}\n\\change{\nLet $F({x}^i, x^{-i}) = \\sum_{t, s, a}\\int_0^{{x}^i_{tsa}}\\ell^i_{tsa}(u^i, x^{-i}) \\partial u^i_{tsa}$, then $\\partial F({x}^i, x^{-i}) \/\\partial x^i = \\ell({x}^i, x^{-i})$. We then apply Proposition A\\ref{prop:dp_to_minimizer} to~\\eqref{eqn:individual_player_mdp} and the theorem's results follow directly.}\n\\end{proof}\n\\noindent\\change{\nWhen all players jointly achieve the optimal cost-to-go~\\eqref{eqn:individual_optimality}, a stable equilibrium for unilateral optimality is achieved.\n}\n\\begin{definition}[Nash Equilibrium] \\label{def:wardrop} \nThe joint state-action distribution $\\hat{x}= [\\hat{x}^1, \\ldots, \\hat{x}^N]$~\\eqref{eqn:joint_state_action}\nis a \\emph{Nash equilibrium} if \\change{$\\big(\\hat{x}^i, Q^i(\\hat{x})\\big)$} satisfies~\\eqref{eqn:individual_optimality} for all $i \\in [N]$.\n\\end{definition}\n\\subsection{Potential optimization form}\n\\noindent We are interested in MDP congestion games that can be reduced from the coupled MDPs~\\eqref{eqn:individual_player_mdp} to a single minimization problem given by\n\\begin{equation}\n \\begin{aligned}\\label{eqn:convex_opt_eqn}\n \\min_{x^1, \\ldots, x^N} F(x), \\ \n \\text{s.t. } x^i \\in \\mathcal{X}(P^i, z^i_0) , \\ \\forall \\ i \\in [N],\n \n \\end{aligned}\n\\end{equation}\nwhere $F$ is the \\emph{potential function} of the corresponding game.\n\n\\begin{definition}[Potential Function]\\label{def:potential}\nWe say an MDP congestion game with {player costs} $\\{\\ell^{i}\\}_{i \\in [N]}$~\\eqref{eqn:player_cost_def} has a \\emph{potential function} $F: {\\mathbb{R}}^{N(T+1) S A}\\mapsto {\\mathbb{R}}$ if $F$ satisfies\n\\begin{equation}\\label{eqn:potential_first_order}\n \\frac{\\partial F(x)}{\\partial x^i_{tsa}} = \\ell^i_{tsa}(x), \\ \\forall \\ (i, t, s, a) \\in [N]\\times\\mathcal{T}\\times[S]\\times [A].\n \\end{equation}\n\\end{definition}\n\\noindent The following assumption on $\\{\\ell^{i}\\}_{i \\in [N]}$ is necessary and sufficient for the existence of $F$~\\cite[Eqn.2.44]{patriksson2015traffic}.\n\\begin{assumption}\\label{ass:potential_existence}\nFor all $(i, t, s, a) , (i', t', s', a') \\in [N]\\times\\mathcal{T}\\times[S]\\times [A]$, the player costs $\\{\\ell^{i}\\}_{i \\in [N]}$ satisfy\n \\begin{equation}\\label{eqn:potential_second_order}\n \n \\frac{\\partial \\ell^i_{tsa}(x)}{\\partial x^{i'}_{t's'a'}} = \\frac{\\partial \\ell^{i'}_{t's'a'}(x)}{\\partial x^i_{tsa}}.\n \\end{equation}\n\\end{assumption}\n\\begin{remark}\nAssumption~\\ref{ass:potential_existence} \\change{is equivalent to} $F$ \\change{being} conservative: $\\forall \\ x_1, x_2 \\in \\{x^i_{tsa}\\ | \\ (i, t, s, a) \\in [N] \\times \\mathcal{T} \\times [S] \\times [A]\\}$, \n\\begin{equation}\\label{eqn:potential_conservative}\n {\\partial^2 F(x)}\/{\\partial x_1\\partial x_2} = {\\partial^2 F(x)}\/{\\partial x_2\\partial x_1}.\n\\end{equation}\nIn other words, the Jacobian of $\\xi$~\\eqref{eqn:cost_vector}, ${\\partial\\xi(x)}\/{\\partial x}$, is symmetrical.\n\\end{remark}\n\\noindent Verifying the existence of $F$~\\eqref{eqn:potential_first_order} is non-trivial. However, if $F$ exists, the \\change{solution} of~\\eqref{eqn:convex_opt_eqn} is the Nash equilibrium~\\cite{calderone_finite}. \n\\begin{theorem}\\label{thm:kkt_pts} \n\\change{If the player costs $\\{\\ell^{i}\\}_{i \\in [N]}$~\\eqref{eqn:player_cost_def} satisfy Assumption~\\ref{assum:positive_definite}, \n\\begin{enumerate}\n \\item the potential function (Definition~\\ref{def:potential}) exists,\n \\item $\\hat{x}$~\\eqref{eqn:joint_state_action} is the global optimal solution of~\\eqref{eqn:convex_opt_eqn} if and only if $\\hat{x}$ is a Nash equilibrium (Definition~\\ref{def:wardrop}). \n\\end{enumerate}\n}\n\\end{theorem}\n\\begin{proof}\n\\change{We prove statement $1$ by showing that Assumption~\\ref{assum:positive_definite} implies Assumption~\\ref{ass:potential_existence}: if $\\nabla \\xi(x)\\succ 0$ for all feasible joint state-action distributions $x$~\\eqref{eqn:joint_state_action}, then $\\nabla \\xi(x)$ is symmetrical and satisfies~\\eqref{eqn:potential_second_order}.}\n\\change{Next, we show the forward direction of the statement $2$. If $(\\hat{x}^1,\\ldots \\hat{x}^N)$ minimizes~\\eqref{eqn:convex_opt_eqn}, then for each $i\\in [N]$, $\\hat{x}^i$ minimizes~\\eqref{eqn:mdp_variant} at $\\hat{x}^{-i}$. From Proposition A\\ref{prop:dp_to_minimizer}, $\\hat{x}^i$ satisfies~\\eqref{eqn:individual_optimality} for all $i\\in [N]$, therefore $\\hat{x}$ is a Nash equilibrium. To show the reverse direction of $2$, if~\\eqref{eqn:individual_optimality} is satisfied for all $i \\in [N]$, $\\hat{x}^i$ is coordinate-wise optimal for coordinate $i$ (Proposition A\\ref{prop:dp_to_minimizer}). Under Assumption~\\ref{assum:positive_definite},~\\eqref{eqn:convex_opt_eqn} has a strictly convex differentiable objective with separable convex constraints $\\mathcal{X}(P^i, z^i_0)$---each $x^i$ is constrained independently of $x^j$, $\\forall j \\in[N]\/\\{i\\}$, then the jointly coordinate-wise optimal $\\hat{x}$ is the global optimal solution of~\\eqref{eqn:convex_opt_eqn}~\\cite[Thm 4.1]{tseng2001convergence}. }\n\\end{proof}\n\\subsection{Path Coordination as an MDP Congestion Game}\\label{sec:path_coordination_costs}\n\\noindent We now model the path coordination problem as an MDP congestion game and demonstrate how players can achieve individual objectives while avoiding each other.\n\n\\noindent To reflect the congestion level of each state-action, we first define a \\textbf{congestion distribution} as the weighted sum of individual state-action distributions.\n\\begin{equation}\\label{eqn:congestion_distribution}\n \\textstyle y := \\sum_{i\\in[N]} \\alpha_ix^i \\in {\\mathbb{R}}^{(T+1)SA}, \\ \\alpha_i > 0, \\ \\forall i \\in [N],\n\\end{equation}\n\\noindent where $\\alpha_i$ is player $i$'s \\emph{impact factor}. If all players contribute to congestion equally, $\\alpha_i = 1 \\ \\forall i \\in [N]$. \n\n\\noindent\\textbf{Player costs}. \\change{We derive a class of player costs that satisfy Assumption~\\ref{assum:positive_definite}, incorporate congestion-based penalties, and enable players to pursue individual objectives. For all $(i,t,s,a) \\in [N]\\times\\mathcal{T}\\times[S]\\times[A]$, the player cost is given by }\n\\begin{equation}\\label{eqn:individual_costs}\n \\textstyle\\ell^i_{tsa}(y, x^i) = \\alpha_i f_{ts}\\big(\\sum_{a'} y_{tsa'}\\big) + \\alpha_i g_{tsa}\\big(y_{tsa}\\big) + h^i_{tsa}(x^i_{tsa}),\n\\end{equation}\nwhere \\change{$\\alpha_i$ is the same as in~\\eqref{eqn:congestion_distribution}, $f_{ts}:{\\mathbb{R}} \\mapsto{\\mathbb{R}}$ is the state-dependent congestion and takes the congestion level of $(t,s)$ as input, $g_{tsa}:{\\mathbb{R}} \\mapsto{\\mathbb{R}}$ is the state-action-dependent congestion and takes the congestion level of $(t,s,a)$ as input, and $h^i_{tsa}:{\\mathbb{R}}\\mapsto{\\mathbb{R}}$ is the player-specific objective and takes player $i$'s probability of being in $(t,s,a)$ as input. Player-specific objectives such as obstacle avoidance and target reachability can be incorporated as constant offsets in $h^i$}. \n\\begin{remark}[Effect of $\\alpha_i$]\nThe impact factor $\\alpha_i$ scales player $i$'s \\change{relative impact on the total congestion and the total congestion's impact on player $i$. When $\\alpha_i < \\alpha_j$, player $i$ impacts congestion less and cares about the congestion less than player $j$. When $\\alpha_i > \\alpha_j$, player $i$ impacts congestion more and cares about the congestion more than player $j$. }\n\\end{remark}\n\\noindent The potential function~\\eqref{eqn:potential_first_order} of the game with costs~\\eqref{eqn:individual_costs} is\n\\begin{equation}\n \\begin{aligned}\\label{eqn:state_state_action_potential}\n F(x)= & \n\\textstyle \\sum_{t,s} \\int_0^{\\sum_{a'}y_{tsa'}}f_{ts}(u) \\partial u + \\sum_{t,s,a} \\int_0^{y_{tsa}}g_{tsa}(u) \\partial u \\\\\n & \\textstyle \n + \\sum_{i, t, s, a} \\int_{0}^{x^i_{tsa}} h^i_{tsa} (u)\\partial u.\n\\end{aligned} \n\\end{equation}\n\\begin{remark}\n\\change{Congestion costs $f$ and $g$ must be identical for all players in order for a potential (Definition~\\ref{def:potential}) to exist.}\n\\end{remark}\n\\begin{example}[Road-sharing Vehicles]\n\\change{\nConsider a sedan (player $1$, $\\alpha_1=1$) and a trailer (player $2$, $\\alpha_2=2$) sharing a road network modeled by $[S]\\times [A]$. Player $i$ wants to reach state $s_i \\in [S]$. The player-specific objective is $h^i_{tsa}(x^i_{tsa}) = -\\mathbb{1}[s = s_i] + \\epsilon_i x^i_{tsa}$, where $\\mathbb{1}[w]$ is $1$ when $w$ is true and $0$ otherwise. The term $\\epsilon_i x^i_{tsa}$ where $\\epsilon_i > 0$ encourages player $i$ to randomize its policy over all optimal actions. Players experience state-based congestion as $f_{ts}(w) =\\exp(w)$. The player cost~\\eqref{eqn:individual_costs} is $\\ell^i_{tsa}(y, x^i) = \\alpha_i\\exp(\\sum_{a'} y_{tsa'}) + \\epsilon_ix^i_{tsa} -\\mathbb{1}[s = s_i]$. }\n\\end{example}\n\n\n\n\n\n\n\n\n\n\\begin{corollary}\\label{cor:unique_ne}\nPlayer costs of form~\\eqref{eqn:individual_costs} \\change{satisfy Assumption~\\ref{assum:positive_definite} if $h^i_{tsa}(\\cdot)$ is strictly increasing and $f_{ts}(\\cdot)$, $g_{tsa}(\\cdot)$ are non-decreasing} $\\forall (i,t,s,a)\\in [N]\\times\\mathcal{T}\\times[S]\\times[A]$.\n\\end{corollary}\n\\begin{proof}\n \\change{Let $I_Z$ be an identity matrix of size $Z\\times Z$, $\\mathbf 1_Z$ be a ones vector of size $Z\\times 1$, $\\vec{\\alpha} = [\\alpha_1,\\ldots,\\alpha_N] \\in {\\mathbb{R}}^{N\\times 1}$, $h(x) = [h^1(x),\\ldots,h^N(x)] \\in {\\mathbb{R}}^{N(T+1)SA}$, and $\\otimes$ be a kronecker product. We define the matrices $M = \\vec{\\alpha} \\otimes I_{(T+1)SA}$ and $J = (I_{(T+1)S}\\otimes \\mathbf 1_A^\\top)M$, and verify that $Mx = y$, $[Jx]_{ts} = \\sum_{a'}y_{tsa'}$ $\\forall (t,s) \\in \\mathcal{T}\\times[S]$, and $\\xi(x) = J^\\top f(Jx) + M^\\top g(Mx) + h(x)$. Let $w = Jx$, we can take $\\xi$'s gradient as\n$\n\\textstyle \\nabla \\xi(x) = \n J^\\top \\nabla f(w) J + \n M^\\top \\nabla g(y) M + \\nabla h(x).\n $\nUnder Corollary assumptions,\n$\\nabla f(w)$ and $\\nabla g(y)$ are non-negative diagonal matrices and $\\nabla h(x) $ is a strictly positive diagonal matrix. Therefore}, $\\nabla \\xi(x) \\succ 0$. \n\\end{proof}\n\\begin{remark}\nCorollary~\\ref{cor:unique_ne} implies that a strictly increasing $h^i$ is crucial to ensuring a unique Nash equilibrium. \nTherefore, $h^i$ can be interpreted as a regularization term.\n\\end{remark}\n\\subsection{Frank-Wolfe Learning Dynamics}\n\\noindent We find the Nash equilibrium of MDP congestion games by {leveraging} single-agent dynamic programming. \n\\begin{algorithm}[ht!]\n\\caption{Frank-Wolfe with dynamic programming}\n\\begin{algorithmic}[1]\n\\Require \\(\\{\\ell^i\\}_{i\\in[N]}\\), \\(\\{P^i\\}_{i\\in[N]}\\), \\(\\{z^i_0\\}_{i\\in[N]}\\), \\(N\\), \\([S], [A], \\mathcal{T}\\).\n\\Ensure \\(\\change{\\{\\hat{x}^{i}_{tsa}\\}_{t \\in \\mathcal{T}, s\\in[S], a \\in [A]}}\\).\n\\State{\\( x^{i0}\\in\\mathcal{X}(P^i, z^i_0) \\in {\\mathbb{R}}^{(T+1)SA}, \\quad \\forall \\ i \\in [N]. \\)}\n\n\\For{\\(k = 1, 2, \\ldots, \\)}\n\\For{\\(i = 1,\\ldots,N\\)}\n \\State{\\(\\change{C^{ik}} = \\ell^i([x^{1k},\\ldots, x^{Nk}])\\)}\\label{alg:cost_retrieval}\n\t\\State{\\(\\pi^i\\) = \\text{MDP}(\\change{\\(C^{ik}\\)}, \\(P^i\\), \\([S]\\), \\([A]\\), \\(T\\), \\change{\\(z^i_0\\)})}\\label{alg:mdp}\n\t\\State{\\(b^{ik} = \\) \\Call{RetrieveDensity}{\\(P\\), \\(z^i_0\\), \\(\\pi^i\\)}}\\Comment{Alg.~\\ref{alg:density}}\n\t\\State{\\(x^{i(k+1)}= (1 - \\frac{2}{k+1})x^{ik} + \\frac{2}{k+1} b^{ik}\\)}\n\\EndFor\n\\EndFor\n\\end{algorithmic}\n\\label{alg:frank_wolfe}\n\\end{algorithm}\n\n\\noindent In Algorithm~\\ref{alg:frank_wolfe}, each player can access an \\emph{oracle} \\change{that} returns the cost for a given joint state-action distribution. \n\\change{In line~\\ref{alg:mdp}, $\\pi^i \\in [A]^{(T+1)S}$ is any deterministic policy that solves the finite time MDP with cost $C^{ik}$, transition probability $P^i$, and initial distribution $z^i_0$. We use value iteration to recursively find $\\pi^i$ as}\n\\change{\\begin{equation}\n\\begin{aligned}\\label{eqn:value_iteration}\n V^i_{Ts} & = \\textstyle\\min_{a}C^{ik}_{Tsa}, \\ \\pi^i_{Ts} \\in \\mathop{\\rm argmin}_{a}C^{ik}_{Tsa}, \\\\\n V^i_{(t-1)s} & \\textstyle = \\min_{a} C^{ik}_{(t-1)sa} + {\\sum_{s'}}P^i_{ts'sa}V^i_{ts'} \\ \\forall t \\in [T] \\\\\n \\pi^i_{(t-1)s} & \\textstyle \\in \\mathop{\\rm argmin}_{a} C^{ik}_{(t-1)sa} + {\\sum_{s'}}P^i_{ts'sa}V^i_{ts'} \\ \\forall t \\in [T]\n\\end{aligned}\n\\end{equation}}\n\\noindent \\change{Algorithm~\\ref{alg:frank_wolfe}} then retrieves the corresponding state-action density $b^{ik}$ via Algorithm~\\ref{alg:density} and \\change{combines it} with the current state-action density $x^{ik}$ to derive the next joint state-action density. All steps within lines 4 to 7 are parallelizable.\n\n\\begin{algorithm}[ht!]\n\\caption{Retrieving \\change{state-action distribution} from \\change{$\\pi$} }\n\\begin{algorithmic}[1]\n\\Require \\(P\\), \\(z\\), \\(\\pi\\).\n\\Ensure \\(\\{d_{tsa}\\}_{t \\in\\mathcal{T}, s\\in[S], a\\in[A]}\\)\n\\State{\\(d_{0s\\pi_{0s}} = z_s, \\ \\forall s \\in [S]\\)}\n\\For{\\(t=1, \\ldots, T\\)}\n\t \n\t \n\t \n\t\t \\State{\\(d_{ts(\\pi_{ts})} = \\textstyle\\sum_a\\sum_{s'} P_{tss'a}d_{(t-1)s'a}, \\ \\forall\\ s \\in [S]\\)}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:density}\n\\end{algorithm}\n\n\\begin{theorem}\nUnder Assumption~\\ref{assum:positive_definite}, \nAlgorithm~\\ref{alg:frank_wolfe} converges towards the Nash equilibrium \\change{$\\hat{x} = (\\hat{x}^1,\\ldots, \\hat{x}^N)$} as \n\\begin{equation}\\label{eqn:alg_convergence}\n \\textstyle \\frac{\\alpha}{2}\\sum_{i\\in[N]} \\norm{x^{ik} - \\change{\\hat{x}^{i}}}^2_2 \\leq \\frac{2C_F}{k+2} \n\\end{equation}\nwhere $C_F$ is the potential function $F$'s~\\eqref{eqn:potential_first_order} \\emph{curvature constant} given by\n\\[ C_F = \\underset{\\substack{x^i, s^i \\in \\mathcal{X}(P^i, z^i_0) \\\\ \\gamma \\in [0, 1]\\\\\nw^i = x^i + \\gamma(s^i - x^i)}}{\\sup} \\frac{2}{\\gamma^2} \\Big(F(\\change{s}) - F(x) - \\sum_{i \\in [N]}(x^i - w^i)^\\top \\ell^i(x)\\change{\\Big)}.\\]\n\\end{theorem}\n\\begin{proof}\nAlgorithm~\\ref{alg:frank_wolfe} is a straight-forward implementation of~\\cite[Alg.2]{jaggi2013revisiting}. \n \\change{From Assumption~\\ref{assum:positive_definite}, $\\nabla \\xi(\\hat{x}) \\succ 0$. Therefore}, the potential function $F$ is \\change{strongly} convex and satisfies $ \\frac{\\alpha}{2}\\sum_{i\\in[N]} \\norm{x^{ik} - \\change{\\hat{x}^{i}}}^2_2 \\leq F(x^{k}) - F(\\hat{x})$. Equation~\\eqref{eqn:alg_convergence} then follows directly from~\\cite[Thm.1]{jaggi2013revisiting}.\n\\end{proof}\n\\begin{remark}[Scalability]\nAlgorithm~\\ref{alg:frank_wolfe} has linear complexity in the number of players.\n\\end{remark}\n\\subsection{Population Perspective: Numerical Method}\\label{sec:dualFormulation}\n\nAfter the social planner has offered incentives, the population plays the Wardrop equilibrium defined by modified rewards~\\eqref{eq:generalReward}; this equilibrium can be computed using the Frank Wolfe (FW) method~\\cite{freund2016new}, given in Algorithm~\\ref{alg:MSA}, with known optimal variables $\\{\\tau_i^{\\star}\\}$. \n\nFW is a numerical method for convex optimization problems with continuously differentiable objectives and compact feasible sets \\cite{powell1982convergence}, including routing games. One advantage of this learning paradigm is that the population does not need to know the function $r(\\cdot)$. Instead, they simply react to the realized rewards of previous game at each iteration. It also provides an interpretation for how a Wardrop equilibrium might be asymptotically reached by agents in MDPCG in an online fashion.\n\nAssume that we have a repeated game play, where players execute a fixed strategy determined at the start of each game. At the end of each game $k$, rewards of game $k$ based on $y^k_{tsa}$ are revealed to all players. FW models the population as having two sub-types: \\emph{adventurous} and \\emph{conservative}. Upon receiving reward information $\\ell_{tsa}(y^{k}_{tsa})$, the adventurous population decides to change its strategy while the conservative population does not. To determine its new strategy, the adventurous population uses value iteration on the latest reward information---i.e.~Algorithm \\ref{alg:valueIteration}---to compute a new optimal policy. Their resultant density trajectory is then computed using Algorithm \\ref{alg:policy iteration}. The step size at each iteration is equivalent to the fraction of total population who switches strategy. \nThe stopping criteria for the FW algorithm is determined by the Wardrop equilibrium notion---that is, as the population iteratively gets closer to an optimal strategy, the marginal increase in potential decreases to zero.\n\\begin{algorithm}\n\\caption{Value Iteration Method}\n\\begin{algorithmic}[h]\n\\Require \\(r\\), \\(P\\).\n\\Ensure \\( \\{\\pi^{\\star}_{ts}\\}_{ t \\in [T], \\, s\\in \\mathcal{S}} \\)\n\\For{\\(t=T, \\ldots, 1\\)}\n\t\t\\State{\\(\\displaystyle V_{ts}= \\underset{a\\in\\mathcal{A}}{\\mbox{max}}\\, Q_{tsa}\\), \\(\\displaystyle \\pi^{\\star}_{ts} = \\underset{a\\in\\mathcal{A}}{\\mbox{argmax}}\\, Q_{tsa},\\ \\forall \\ s \\in \\mathcal{S}\\)} \\Comment{Eqn.~\\eqref{eq:qvalue}}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:valueIteration}\n\\end{algorithm}\n\\begin{algorithm}\n\\caption{Retrieving density trajectory from a policy }\n\\begin{algorithmic}[h]\n\\Require \\(P\\), \\(p\\), \\(\\pi\\).\n\\Ensure \\(\\{d_{tsa}\\}_{t \\in[T], s\\in\\mathcal{S}, a\\in\\mathcal{A}}\\)\n\\State{\\(d_{tsa} = 0, \\ \\forall \\ t \\in [T], s \\in \\mathcal{S}, a \\in \\mathcal{A}\\)}\n\\State{\\(d_{1s\\pi_{1s}} = p_s, \\ \\forall s \\in \\mathcal{S}\\)}\n\\For{\\(t=2, \\ldots, T\\)}\n\t \n\t \n\t \n\t\t \\State{\\(d_{ts(\\pi_{ts})} = \\sum\\limits_{a \\in \\mathcal{A}}\\sum\\limits_{s' \\in \\mathcal{S}} P_{t-1,ss'a}d_{t-1,s'a}, \\ \\forall\\ s \\in \\mathcal{S} \\)}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:policy iteration}\n\\end{algorithm}\n\\begin{algorithm}\n\\caption{Frank Wolfe Method with Value Iteration}\n\\begin{algorithmic}[h]\n\\Require \\(\\bar{\\ell}\\), \\(P\\), \\(p_s\\), \\(N\\), \\(\\epsilon\\).\n\\Ensure \\(\\{y^{\\star}_{tsa}\\}_{t \\in [T], s\\in\\mathcal{S}, a \\in \\mathcal{A}}\\).\n\\State{\\( y^0 = 0 \\in {\\mathbb{R}}^{T \\times|\\mathcal{S}| \\times |\\mathcal{A}|} \\)}\n\\For{\\(k = 1, 2, \\ldots, N\\)}\n \\State{\\(c^k_{tsa} = \\bar{\\ell}_{tsa}(y^k)\\), \\(\\quad \\forall \\,\\, t \\in [T], s \\in \\mathcal{S}, a \\in \\mathcal{A}\\)}\n\t\\State{\\(\\pi_{ts} = \\) \\Call{ValueIteration}{\\(c^k\\), \\(P\\)}}\n\t\\Comment{Alg.~\\ref{alg:valueIteration}}\n\t\\State{\\(d^k\\) = \\Call{RetrieveDensity}{\\(P, p_s, \\pi_{ts}\\)}}\\Comment{Alg.~\\ref{alg:policy iteration}}\n\n\t\\State{\\( y^k = (1 - \\frac{2}{k+1})y^{k-1} + \\frac{2}{k+1} d^k\\)}\n\t\\State{Stop if }\n\t\\State{\\(\\quad \\sum\\limits_{t\\in[T]} \\sum\\limits_{s\\in\\mathcal{S}} \\sum\\limits_{a\\in\\mathcal{A}}\\Big(c^k_{tsa} - c^{k-1}_{tsa}\\Big)^2 \\leq \\epsilon\\)}\n\\EndFor\n\\end{algorithmic}\n\\label{alg:MSA}\n\\end{algorithm}\n\nIn contrast to implementations of FW in routing game literature, Algorithm \\ref{alg:MSA}'s descent direction is determined by solving an MDP \\cite[Section~4.5]{puterman2014markov} as opposed to a shortest path problem from origin to destination \\cite[Sec.4.1.3]{patriksson2015traffic}.\nAlgorithm~\\ref{alg:MSA} is guaranteed to converge to a Wardrop equilibirum if the predetermined step sizes decrease to zero as a harmonic series \\cite{freund2016new}---e.g., $\\frac{2}{k+1}$.\nFW with predetermined step sizes has been shown to have sub-linear worst case convergence in routing games \\cite{powell1982convergence}. \nOn the other hand, replacing fixed step sizes with optimal step sizes found by a line search method leads to a much better convergence rate.\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{Problem Formulation}\\label{problemFormulation}\n\n\\section{Problem Formulation}\\label{problemFormulation}\nConsider a finite horizon, Markov Decision Process (MDP) congestion game, where a group of players individually solves an MDP to maximize expected self gain while competing for global resources. We assume that all players have the same set of possible actions and stochastic transitions. This type of finite horizon, Markov Decision Process congestion game (MDPCG) is formulated as a convex optimization problem in \\cite{MDPCongestionGame}. \n\nIn this paper, we develop a dual interpretation for the role that state constraints play in MDPCG: through a social planner's perspective and individual players' perspective. Players will attempt to maximize their individual objectives by finding an optimal sequence of actions, while a social planner strives for game-level objectives, such as maximizing total gain or enforcing state constraints. To achieve these objectives, the planner cannot impose explicit constraints, but instead can adjust individual objectives to incentivize\/discourage certain actions. A toll\/reward method is developed for implicitly realizing global state dependent constraints .\n\nThe rest of the paper is organized as follows: in Section \\ref{notation}, we introduce the convex optimization model of state-constrained MDPCG. Section \\ref{convergence} connects the constrained optimal solution to Wardrop equilibrium with respect to a generalized cost function. In Section \\ref{dualFormulation}, we perform simulations of repeated game play using a fixed step-sized Frank Wolfe Method. The simulations show that when the cost functions are adjusted for tolls, players indeed converge at the optimal strategy that does not violate corresponding state constraints.\n\n\n\\section{Related work}\n\\label{sec:related work}\n\n\n\nAn MDP congestion game~\\cite{calderone2017infinite} is a stochastic population game and is related to potential mean field games \\cite{lasry2007mean,gueant2011infinity} in the discrete time and state-action space \\cite{gomes2010discrete} and mean field games on graphs \\cite{gueant2015existence}. \n\\change{In this paper, we extend our previous framework from continuous populations of identical MDP decision makers~\\cite{calderone2017infinite} to a finite number of heterogenous MDP decision makers.} In the continuous population case, MDP congestion games have been analyzed for constraint satisfaction in~\\cite{li2019tolling} and sensitivity to \\change{hyper}parameters in~\\cite{li2019sensitivity}. \n\nModel-based multi-agent \\change{path} planning is typically solved via graph-based searches~\\cite{cohen2019optimal} and mixed integer linear programming~\\cite{chen2020scalable}. Recently, reinforcement learning has been introduced as a viable method for solving multi-agent \\change{path} planning~\\cite{semnani2020multi,yun2020multi}. In most scenarios, the \\change{path} planning problem is modeled as an MDP~\\cite{bayerlein2021multi,lo2021towards}. In particular, \\cite{lo2021towards} adopts a stochastic game model for human-robot collision avoidance, but focuses more on algorithm development rather than game structure analysis. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Reward Uncertainty}\nHere we introduce slightly different notation: \n\\[y_{ts} = \\begin{pmatrix}\ny_{tsa_1} \\\\ y_{tsa_2}\\\\\\vdots\\\\y_{tsa_m}\\\\\n\\end{pmatrix} \\in {\\mathbb{R}}^{|\\mathcal{A}|} \\]\n\\[\n\\ell_{ts}(y_{ts}) = \\begin{pmatrix}\n\\ell_{tsa_1}(y_{tsa_1}) \\\\ \\ell_{tsa_2}(y_{tsa_2})\\\\\\vdots\\\\\\ell_{tsa_m}(y_{tsa_m})\\\\\n\\end{pmatrix} \\in {\\mathbb{R}}^{|\\mathcal{A}|}\n\\]\nThe optimization problem can be rewritten in the new vector variable form: \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\int_0^{y_{ts}}\\ell_{ts}(x)dx\\\\\n\\mbox{s.t.} & \\mathbf 1^T y_{t+1, s} = \\sum\\limits_{s'\\in\\mathcal{S}}P_{s\\cdot s'}^Ty_{ts'}, \\quad t\\in[T-1],\\\\\n&\\mathbf 1^Ty_{0s}=p_s,\\\\\n&y_{ts}\\geq 0,\\quad \\forall s\\in\\mathcal{S}, t\\in[T]\n\\end{aligned}\\label{mdp game vector}\n\\end{equation}\n\n\\subsection{Uncertainty Model}\nFor each individual solving the problem, their model of the reward may be entirely experience based and therefore prone to uncertainty. \n\nHere, we model their uncertainty as the following: for $(t,s,a)$, the reward can be anywhere within the following ellipse, which is also state dependent in its mean value\n\\begin{equation}\n\\begin{aligned}\n\\ell_{ts}(x) &\\in \\mathcal{E}_{ts} \\quad \\forall \\,\\, x \\in [0,1] \\\\\n& \\in \\{R_{ts}x + Q_{ts}u_{ts}\\, | \\quad \\norm{u}_2 \\leq 1\\} \\\\\n\\end{aligned}\n\\end{equation}\nWhere $R_{ts} \\in {\\mathbb{R}}^{|S| \\times |S| } $ is a time and state dependent matrix. \n\nSuppose we are considering worstcase scenario, the objective can be rewritten as: \n\\begin{equation}\n\\begin{aligned}\n&\\sup\\{\\mathbf 1^T\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\int_0^{y_{ts}}\\ell_{ts}(x)dx\\} \\\\\n= &\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}} \\mathbf 1^T\\int_0^{y_{ts}}R_{ts}xdx + \\sup\\{\\mathbf 1^T\\int_0^{y_{ts}}Q_{ts}u dx | \\norm{u}_2 \\leq 1\\} \\\\\n\\leq &\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 + \\sup\\{\\norm{u_{ts}^TQ_{ts}^Ty_{ts}}_2| \\norm{u}_2 \\leq 1\\} \\\\\n\\leq &\\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 +\\norm{Q_{ts}^Ty_{ts}}_2 \\\\\n\\end{aligned}\n\\end{equation} \nThen the optimization problem becomes: \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 +\\norm{Q_{ts}^Ty_{ts}}_2 \\\\\n\\mbox{s.t.} & \\mathbf 1^T y_{t+1, s} = \\sum\\limits_{s'\\in\\mathcal{S}}P_{s\\cdot s'}^Ty_{ts'}, \\quad t\\in[T-1],\\\\\n&\\mathbf 1^Ty_{0s}=p_s,\\\\\n&y_{ts}\\geq 0,\\quad \\forall s\\in\\mathcal{S}, t\\in[T]\n\\end{aligned}\\label{mdp game vector}\n\\end{equation}\n\nThe Lagangian becomes\n\\begin{equation}\n\\begin{aligned}\nL(y_{ts}, V_{ts}, \\mu_{ts}) & = \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}y_{ts}}^2_2 +\\norm{Q_{ts}^Ty_{ts}}_2 \\\\\n& + \\sum\\limits_{t \\in [1,T]}\\sum\\limits_{s\\in\\mathcal{S}} V_{ts}\\Big(\\sum\\limits_{s' \\in \\mathcal{S}}P^T_{s\\cdot s'} y_{t-1,s'} - \\mathbf 1^Ty_{ts}\\Big) \\\\\n& + \\sum\\limits_{s \\in \\mathcal{S}}V_{0s}\\Big(p_s - \\mathbf 1^Ty_{0s}\\Big) \\\\\n& - \\sum\\limits_{s \\in \\mathcal{S}}\\sum\\limits_{t \\in[T]}\\mu_{ts}^Ty_{ts} \\\\\n\\end{aligned}\n\\end{equation}\n\nThen the corresponding KKT conditions for the value iteration is: \n\n\\begin{equation}\n\\begin{aligned}\n&t = 0 \\cdots T-1: \\\\\n&\\quad V_{ts}\\mathbf 1 = -\\mu_{ts} + R_{ts}y_{ts} + \\frac{1}{\\norm{Q_{ts}^Ty_{ts}}_2}Q_{ts}Q_{ts}^Ty_{ts}\\\\\n& \\quad\\quad + \\sum_{s'}P_{s'\\cdot s}V_{t+1,s'} \\\\\n&t = T: \\\\\n&\\quad V_{Ts}\\mathbf 1 = -\\mu_{tsa} + R_{Ts}y_{Ts} + \\frac{1}{\\norm{Q_{Ts}^Ty_{Ts}}_2}Q_{Ts}Q_{Ts}^Ty_{Ts}\\\\\n\\end{aligned} \\label{robust kkt condition}\n\\end{equation} \n\nWe can estimate the worst case of $\\frac{1}{\\norm{Q_{ts}^Ty_{ts}}_2}Q_{ts}Q_{ts}^Ty_{ts} $ by the following inequality based on Cauchy-Schwarz inequality: \n\\[\\frac{1}{\\norm{Q_{ts}^Ty_{ts}}_2}Q_{ts}Q_{ts}^Ty_{ts} \\geq -\\norm{Q_{ts}}_2 \\]\n\nThen the worst case dual problem is the following: \n\\begin{equation}\n\\begin{aligned}\n\\underset{y}{\\mbox{min.}} & \\sum\\limits_{t\\in[T]}\\sum\\limits_{s\\in\\mathcal{S}}\\frac{1}{2}\\norm{R_{ts}{y_{ts}}}^2_2 - y^T_{ts}R_{ts}y_{ts}+ \\sum\\limits_{s \\in \\mathcal{S}}p_sV_{0s} \\\\\n\\mbox{s.t.} & V_{ts}\\mathbf 1 \\leq R_{ts}y_{ts} - \\norm{Q_{ts}}_2 + \\sum_{s'}P_{s'\\cdot s}V_{t+1,s'} \\\\\n&V_{Ts}\\mathbf 1 \\leq R_{Ts}y_{Ts} - \\norm{Q_{Ts}}_2 \\\\\n\\end{aligned}\\label{mdp game vector}\n\\end{equation}\nNotice that since $\\norm{Q_{ts}}_2 = \\lambda_{max} $, this is equivalent to offsetting the reward function by the worst possible variation, to obtain a robust reward function $R_{ts} - \\lambda_{max}(Q_{ts}) $\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjdbm b/data_all_eng_slimpj/shuffled/split2/finalzzjdbm new file mode 100644 index 0000000000000000000000000000000000000000..5e6089f426d734f732a1f4734d39ba4c4a766ce1 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzjdbm @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nWhen a cosmic ray interacts with particles in the Earth atmosphere, it produces\na shower of elementary particles propagating towards the grounds with almost\nthe speed of light. The first suggestion that these air showers can produce radio\nemission was given by Askaryan \\cite{askaryan} \nbased on a charge-excess mechanism. Recently, Falcke\\&Gorham \\cite{falcke-gorham}\nproposed that the mechanism for radio emission of air showers is coherent geosynchrotron\nradiation. Secondary electrons and positrons produced in the particle cascade are deflected\nin the Earth magnetic field and this produces dipole radiation that is relativistically\nbeamed in the forward direction.\nThe shower front emitting the radiation has a thickness which is comparable to a wavelength\nfor radio emission below 100MHz (around few meters). The emission is coherent which amplifies the \nsignal. \n\nRadio emission of cosmic ray air showers has been detected by LOPES (LOFAR Prototype Station) \n\\cite{falcke-nature}, a phased array of dipole antennas co-located with\nthe KASCADE (Karlsruhe Shower Core and Array Detector) experiment which provides \ncoincidence triggers for LOPES and well-calibrated informations about air-shower properties, like\nelectron number $N_e$, reconstructed muon number $N_{\\mu}$, azimuth and zenith angle of the event.\nThe LOPES experiment and data reduction are described in detail by Horneffer et al. \\cite{horneffer}.\n\nHighly inclined showers are expected to be very well detectable in the radio domain\n\\cite{huege-falcke},\\cite{gousset}. \nHowever, we have to mention that neither LOPES nor KASCADE are optimized for large\nzenith angles. For example, KASCADE reconstruction of electron and muon number can be\nnot accurate especially in cases when the shower core of specific event falls out\nof the KASCADE array. \n\nInclined cosmic ray air showers are specific, since they travel trough few times longer\ndistances in the Earth atmosphere compared to vertical showers, and due to this most of the \nelectromagnetic (particle) component of those showers has been \nabsorbed. So, inclined \nshowers that start high in the atmosphere (initiated by protons, iron nuclei or gamma-photons) will have\nlarge electron deficiency on the ground level compared to vertical showers. \nOn the other hand, neutrino induced showers may be generated at any distance from the \nground \\cite{capelle} so they could clearly\nbe distinguished from those whose primary particle initiated the shower\nhigh in the atmosphere by the number of electrons that reach the ground level.\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[width=0.3\\textwidth,angle=270,clip]{guipipeline-All-18.eps}\n\\includegraphics*[width=0.3\\textwidth,angle=270,clip]{guipipeline-X-18.eps}\n\\caption{\\label{fig1}Left: Electric field as a function of time for 10 LOPES antennas for an event \ndetected in March 2004 with zenith angle 53.3$^o$ and azimuth \n54.7$^o$. The angle between the shower core and the Earth magnetic field is 69.8$^o$.\n$N_e$=1.5$\\cdot$10$^6$, $N_{\\mu}$=1.5$\\cdot$10$^6$, reconstructed by KASCADE. Right:\nRadio emission as a function of time after beam forming for the same event.\nDark blue line represents X-beam, light blue line the total power and \npink line the Gaussian fit for X-beam.}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussion}\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[width=0.49\\textwidth,clip]{map18.ps}\n\\caption{\\label{fig3}Radio map of the cosmic ray shower with zenith angle 53.3$^o$ detected in March 2004. \nAzimuth (AZEL longitude) and elevation (AZEL latitude) of the event are given on the axes.}\n\\end{center}\n\\end{figure}\n\nWe made a selection of inclined events from data taken during year 2004 (also detected by the KASCADE\narray where the reconstructed shower cores were less than 100m away from the array center).\nWe found 2017 events with zenith angle larger than 50$^o$.\nThen we introduced an additional condition: $N_{\\mu}$$>$10$^5$.\nIn this way we narrowed the selection to 51 event and more than 40\\% of those are detected\nin the radio domain, many with very large field strengths, even though a threshold\non muon number is lower than the one used for bright events in \\cite{falcke-nature}.\nHowever, as we already mentioned, KASCADE is not optimized for large zenith angles, \nso the reconstruction of the electron and muon number failed for half of the detected events.\nThis leaves us with 10 events with strong radio signal and reliable shower properties\nreconstructed by KASCADE.\n\nAs an example, we show here one of those events, detected in March 2004 with zenith angle \n53.3$^o$ and azimuth 54.7$^o$ with roughly the same number of electrons and \nmuons $N_e$,$N_{\\mu}$$\\approx$1.5$\\cdot$10$^6$ \n(reconstructed by KASCADE). The angle between the shower axis and the Earth magnetic field \n(geomagnetic angle) is 69.8$^o$.\n \nIn Figure 1 (left) we show the electric field as a function of time for each antenna. The field is coherent \nat -1.825$\\mu$s which is the arrival time of the shower. The incoherent noise after is radio emission from\nphotomultipliers and in this case is very weak.\nFigure 1 (right) shows the radio emission as a function of time after X(excess)-beam forming.\nThis beam is formed in the following way. First squared signals of all antennas are summed which gives\nthe total power. Then signals of all two antenna combinations\nare multiplied and summed which gives the CC(cross correlation)-beam. Finally, CC-beam is multiplied \nwith the ratio \nbetween CC-beam and total power. In this way a suppression of incoherent noise is achieved.\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics*[width=0.35\\textwidth,clip]{r4776e150302eb1.eps}\n\\includegraphics*[width=0.35\\textwidth,clip]{r4776e150302eb2.eps}\n\\caption{\\label{fig3}\nLeft: Energy deposit of the \ncosmic ray shower with zenith angle 53.5$^o$ detected in March 2004 for e\/$\\gamma$ KASCADE detectors.\nDark blue color shows energy deposit of 100Mev\/m$^2$, red color 1000Mev\/m$^2$. Maximum energy deposit\nfor this event is $\\sim$4500Mev\/m$^2$.\nRight: Particle arrival time for the same event. Dark blue color represents $\\sim$300ns, red $\\sim$800ns.\n}\n\\end{center}\n\\end{figure}\n\nFigure 2 is a radio map of the example event. The air shower is the brightest point in the sky for\nseveral tens of nanoseconds. The resolution of the map is $\\sim$2$^o$ in azimuth and elevation towards the\nzenith. \n\nFigure 3 gives the energy deposit of the chosen cosmic ray shower over the KASCADE array with \ne\/$\\gamma$ detectors\n(left panel) and particle arrival time (right panel). \nWe can see that the shower core falls within the KASCADE array and that\nthe maximum energy is deposited in the north-western part, within the LOPES array. We can \nnotice elliptical shapes of isolines of energy deposit, which is typical for inclined events.\n\n\n\\section{Conclusions}\n\nEven though neither LOPES nor KASCADE are completely \noptimized for the detection of highly inclined events,\nwe find that in a selection of events with zenith angle larger than\n50$^o$ and $N_{\\mu}$$>$10$^5$ (51 event)\naround 40\\% of all events is detected in the radio domain, and some of them \nwith very high field strengths, like the example we have presented in this paper.\nThe most inclined cosmic ray air shower that we detected with LOPES has\na zenith angle of almost 70$^o$.\n\nAfter checking the resonctruction of $N_e$ and $N_{\\mu}$ different correlations \ncan be considered, for example between radio pulse height and muon number, electron number or \ngeomagnetic angle. This will also give more insight into the nature of primary particles that\ninitiate showers and the possibility that some of detected showers might have been triggered\nby neutrinos. \n\n\n\n\n \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this paper, $X$ denotes a smooth complex projective variety of dimension $n$, \n$D$ a (big) divisor on $X$ and $V \\subseteq X$ an irreducible subvariety of dimension $d$.\nThen the restricted volume of $D$ along $V$ is defined to be \n\\begin{equation*}\n\\V{V}(D) = \\limsup_{k \\to \\infty} \\dfrac\n{\\dim \\mathrm{Im} \\Big( H^{0}\\big(X,\\OX(kD) \\big) \\longrightarrow H^{0}\\big(V,\\OV(kD) \\big) \\Big) }{k^d\/d!}.\n\\end{equation*}\nRoughly speaking restricted volumes measure the number of sections of $\\OV(kD)$ \nwhich can be extended to $X$.\nRestricted volumes have many applications in various situations.\nThey are studied in [ELMNP], [BFJ09] and so on.\n\n\nOn the other hand, it is important problem when $D$ admits a Zariski decomposition.\nHere $D$ is said to admit a Zariski decomposition if \nthere exist a nef $\\RR$-divisor $P$ and an effective $\\RR$-divisor $N$ \nsuch that following map is an isomorphism for every integer $k>0$:\n\\begin{equation*}\nH^{0}(X, \\OX(\\lfloor kP \\rfloor)) \\longrightarrow H^{0}(X, \\OX(kD)).\n\\end{equation*}\nThe map is multipling the section $e_{k}$, \nwhere $e_{k}$ is the standard section of the effective divisor $\\lceil kN \\rceil$.\nHere $\\lfloor G \\rfloor$ (\\textit{resp}. $\\lceil G \\rceil$) denotes round down\n(\\textit{resp}. round up) of an $\\RR$-divisor $G$.\n\nAssume $V$ is not contained in $\\mathbb{B}_{+}(D)$.\nHere $\\mathbb{B}_{+}(D)$ denotes the augmented base locus of $D$.\nIt is the closed analytic set in $X$, which is the defect of $D$ to be ample.\nThen the restricted volume $\\V{V}(\\cdot)$ along $V$ depends only on the first Chern class (the numerical class) of $D$.\nOne ask whether the restricted volume $\\V{\\cdot}{D}$ of $D$ depends only on the numerical class of $V$.\n[BFJ09] shows the question is affirmative when the codimension of $V$ is one.\nIn this paper, we give a necessary and sufficient condition for $D$, that\nthe restricted volume of $D$ depends only on the numerical class of $V$.\nThe condition is related to the existence of a Zariski decomposition of $D$ as follows:\n\n\\begin{theo}\\label{MaB}\nAssume that $D$ is big. \nThe following two conditions are equivalent.\n\n$(1)$ $D$ admits a Zariski decomposition.\n\n$(2)$ $\\V{V}(D) = \\V{V^{'}}(D)$ holds for any pair of subvarieties $V$ and $V'$ of $X$\nsuch that $V\\sim V^{'}$ and $V, V^{'} \\not \\subseteq \\mathbb{B}_{+}(D)$.\n\\end{theo}\nWhen $V$ and $V^{'}$ are numerically equivalent, we write $V \\sim V^{'}$.\nRoughly speaking, the condition (2) claims that the restricted volume $\\V{\\cdot}(D)$\nof $D$ is determined only by the numerical class of $V$.\nTheorem \\ref{MaB} implies \nthat the restricted volumes along some numerically equivalent subvarieties $V$ and $V'$ can be different \nwhen $D$ does not admit a Zariski decomposition.\nSuch example is given by so-called Cutkosky construction.\n\n\nWhen $V$ is the ambient space $X$, the restricted volume of $D$ is the usual volume $\\mathrm{vol}_{X}(D)$ of $D$.\nThe usual volume has been studied by several authors.\nThe general theory is presented in detail and with full references in \\cite{La04}.\nBoucksom expressed the usual volume $\\mathrm{vol}_{X}(D)$ by current integration,\nusing the Calabi-Yau technique to solve Monge-Amp$\\rm{\\grave{e}}$re equations and the singular \nholomorphic Morse inequalities (see \\cite{Bou02}).\nIn other words, Boucksom expressed the volume of $D$ in terms of only the first Chern class $c_{1}(D)$.\n\nRestricted volumes along $V$ depend only on the first Chern class $c_{1}(D)$ \nif $V$ is not contained in the augmented base locus $ {\\mathbb{B}}_{+}(D)$ of $D$.\nThen Boucksom's formula can be extended to restricted volumes as follows:\n\n\\begin{theo}\\label{MaA}\nAssume that $V$ is not contained in the augmented base locus $ {\\mathbb{B}}_{+}(D)$.\nThen the restricted volume of $D$ along $V$ satisfies the following equality:\n\\begin{equation*}\n\\vol (D) = \\sup_{T\\in c_1(D)} \\int_{V_{\\reg}} { (T\\vert_{V_{\\reg}})_{\\ac}^d} ,\n\\end{equation*}\nfor $T$ ranging among a positive $(1,1)$-current with analytic singularities whose singular locus does not contain \n$V$.\n\\end{theo}\nHere $T\\vert_{V_{\\reg}}$ denotes the restriction of $T$ to the regular locus $V_{\\reg}$ of $V$ and $(T\\vert_{V_{\\reg}})_{\\ac}$ denotes the absolutely continuous part (see subsection 2.4).\nMoreover we can express the restricted volumes with non-pluripolar product, which \nis introduced in \\cite{BEGZ}.\nTheorem \\ref{MaA} implies that we can define the restricted volume of \na transcendental class on a compact K\\\"ahler manifold in natural way.\n\n\n\n\\begin{defi} \\label{defi}\nLet $V$ be an irreducible variety of dimension $d$ on a compact K\\\"ahler manifold $M$ \nand $\\A$ the class in $H^{1,1}(M , \\RR)$.\nAssume that $V$ is not contained in the non-K\\\"ahler locus $E_{nK}(\\A)$.\nThen \\textit{the restricted volume} of $\\A$ along $V$ is defined to be\n\\begin{equation*}\n\\mathrm{vol}_{M|V}(\\A) = \\sup_{T\\in \\A} \\int_{V_{\\reg}}{({T|_{V_{\\reg}}})_{ac}^d}, \n\\end{equation*}\nfor $T$ ranging among a positive $(1,1)$-current with analytic singularities and the singular locus of \n$T$ does not contain $V$.\n\\end{defi}\n\n\n\nHere the non-K\\\"ahler locus is analytic counter part of the augmented base locus. \nWhen $\\A$ is the Chern class of some divisor $D$, the non-K\\\"ahler locus $E_{nK}(\\A)$ coincides \nwith the augmented base locus $\\mathbb{B}_{+}(D)$.\nFor this extended definition, the properties of usual restricted volumes hold. \nFor example, the continuity, log concavity, Fujita's approximations and so on (see subsection 4.2).\nMoreover the analogue of Theorem 1.1 holds for this extended definition as follows.\nThe principle of the proof is essentially same as Theorem 1.1.\nThe proof gives another proof of Theorem 1.1 by using analytic methods (see subsection 4.3).\n\\begin{theo}\\label{MaBB}\nLet $\\A$ be a big class in $H^{1,1}(X, \\RR)$.\nThen the following two conditions are equivalent.\n\n$(1)$ $\\A$ admits a Zariski decomposition.\n\n$(2)$ $\\V{V}(\\A) = \\V{V^{'}}(\\A)$ holds for any pair of subvarieties $V$ and $V'$ of $X$\nsuch that $V$, $V^{'}$ defines the same cohomology class and $V, V^{'} \\not \\subseteq E_{nK}(\\A)$.\n\\end{theo}\nHere the Zariski decomposition of a class $\\A$ is defined by divisorial Zariski decompositions \nintroduced in \\cite{Bou04} (see subsection 2.7).\nWhen $\\A$ is the first Chern class of some divisor, the Zariski decomposition of $\\A$ \ncorrespond with that of the divisor.\n\\begin{rem}\nThe class $\\A$ is not always contained in the N\\'eron-Severi group.\nTherefore Theorem \\ref{MaBB} is essentially stronger statement than Theorem \\ref{MaB}. \n\\end{rem}\n\nIn the proof of Theorem 1.1 and 1.4, to consider the restricted base locus of a divisor is important.\nThe restricted base locus $\\mathbb{B}_{-}(D)$ of $D$ is a countable union of closed analytic set,\nwhich is the defect of $D$ to be nef.\nIn section 5, we accurize the simplest version of Kawamata-Viehweg vanishing theorem\nin response to the dimension of the restricted base locus. \nThis proof is essentially to use Kawamata-Viehweg vanishing theorem (see \\cite{Ka82}, \\cite{Vie82}).\nA divisor $D$ is said to be nef in codimension $k$ if the codimension of non-nef locus greater than $k$.\n\n\n\\begin{theo}\\label{KV} \nAssume that $D$ is nef in codimension $k$ and big.\nThen\n\\begin{equation*}\nH^{q}(X, \\OX(\\K + D))=0\\ \\ \\ for\\ any\\ q\\ \\geq \\ n-k .\n\\end{equation*} \n\\end{theo}\n\n\\begin{rem}\nNote $D$ is nef in codimension $n-1$ if and only if $D$ is nef.\nThen Theorem \\ref{KV} implies $H^{q}(X, \\OX(\\K + D))=0\\ \\ \\ (q \\geq 1)$.\nThis is the simplest version of Kawamata-Viehweg vanishing theorem.\n\\end{rem}\n\nThis theorem can be generalized by using the generalized numerical Kodaira dimension $\\nu(D)$ \nof $D$, which is introduced in \\cite{BDPP}.\nTheorem \\ref{KV} implies an asymptotic estimate of higher cohomology as follows:\n\n\\begin{cor}\\label{asy}\nLet $E$ be an arbitrary divisor on $X$.\nAssume that $D$ is nef in codimension $k$.\nThen the following estimate holds for $q \\geq n-k$ as $\\ell \\to \\infty$:\n\\begin{equation*}\n\\dim H^{q}(X, \\OX (E + \\ell D)) =O(\\ell ^{n-q}) .\n\\end{equation*}\n\\end{cor}\nThis estimate is similar to \\cite[Proposition 2.15]{Ku06}.\nCorollary \\ref{asy} and asymptotic Riemann-Roch formula shows that\nthe inequality $\\V{X}(D) \\geq D^{n}$ holds when $D$ nef in codimension $n-2$.\nIn particular, then $D$ is big when the self-intersection number $D^{n}$ is positive.\nIn the case that $X$ is surface, this fact is well-known.\n\n\n\n\\subsection*{Acknowledgment}\nThe author is very grateful to his supervisor Prof.\\ Shigeharu Takayama \nfor valuable comments and various suggestion.\nHe is indebted to Dr.\\ Tomoyuki Hisamoto for stimulating discussions.\n\n\n\\section{Terminology}\n\n\\subsection{Currents}\nLet $M$ be a compact K\\\"ahler manifold in this section.\nThen $H^{p,p}(M,\\CC)$ is identified with the quotient of the space of $d$-closed $(p,p)$-currents modulo the \n$\\ddbar$-exact currents.\nFor our purpose the case of $p=1$ is important.\nA $d$-closed almost positive $(1,1)$-current $T$ is said to have analytic (\\textit{resp}. algebraic) singularities \n(along a subscheme $V(\\I)$ defined by an ideal sheaf $\\I$), \nif its potential function $\\varphi$ can be locally written as \n\\begin{equation*}\n\\varphi =\\frac{c}{2}\\log(|f_1|^2+...+|f_k|^2) + v\n\\end{equation*}\nfor some $c \\in \\RR_{>0}$ (\\textit{resp}. $c \\in \\QQ_{>0}$), \nwhere $f_1,\\dots,f_k$ are local generators of $\\I$ and $v$ is a smooth function.\n(Refer to \\cite{Dem} for details.)\n\n\\subsection{Pull-back of $(1,1)$-current}\nWe can handle $(1,1)$-currents easier than high degree currents.\nFor example we can define the pull-back of a $d$-closed $(1,1)$-current.\nLet $f : Z\\longrightarrow M$ be a holomorphic map between complex manifolds.\nAssume that the image of $Z$ by $f$ is not contained in the polar set of $T$.\nLet $T= \\theta + \\ddbar \\varphi$ be a $d$-closed $(1,1)$-current in the class $\\A \\in H^{1,1}(M, \\RR)$.\nHere $\\theta$ is a smooth $(1,1)$-form in the class $\\A$ and $\\varphi$ is a $L^{1}$-function on $M$.\nThen the pullback of $T$ is defined to be $f^{*}T:= f^{*}\\theta + \\ddbar{f^{*}\\varphi }$ \nby using the potential $\\varphi $ of $T$. \nIn particular, we can restrict a $d$-closed $(1,1)$-current to any submanifold \nwhich is not contained in the polar set.\n\n\n\\subsection{Multiplier ideal sheaves and Skoda's lemma.}\nIn this paper, multiplier ideal sheaves is used.\nWe denote by $\\J(\\Vert kD \\Vert)$ the asymptotic multiplier ideal sheaves \nassociated to a divisor $kD$ and \nby $\\I(kT)$ the multiplier ideal sheaves associated to \na $d$-closed current $kT$.\nWe can refer to \\cite{DEL00}, \\cite{Dem} for more details.\nThe Skoda's Lemma implies that a multiplier ideal sheaf is estimated by the Lelong numbers (see \\cite[Lemme 6.6]{Dem}). \n\n\\begin{lemm}{\\rm [Skoda's Lemma] }\n$\\mathrm{(a)}$ If $\\nu(T,x) <1$, then $e^{-2 \\varphi}$ is integrable in a neighborhood of $x$.\nIn particular $\\I (T)_{M,x} = \\mathcal{O}_{M,x}$.\\\\\n$\\mathrm{(b)}$ If $\\nu(T,x) > n+s$ for some positive integer $s$, then $e^{-2 \\varphi}\\geq C |z-x|^{-2n-2s}$ in \na neighborhood of $x$. In particular $\\I (T)_{M,x} \\subseteq \\mathcal{M}_{M,x}^{s+1}$, \nwhere $\\mathcal{M}_{M,x}$ is the maximal ideal of $\\mathcal{O}_{M,x}$.\n\\end{lemm}\n\n\n\n\n\\subsection{Lebesgue decompositions}\nWe can always locally see a positive current $T$ as the differential form with positive measure coefficients.\nA positive measure admits a Lebesgue decomposition into the absolutely continuous part and \nthe singular part with respect to the Lesbegue measure. \nThis decomposition is local.\nHowever we can globally glue the absolutely continuous part and the singular \npart by the uniqueness of a Lebesgue decomposition. \nSo we obtain the docomposition $T = T_{\\ac} + T_{\\sing}$, where $T_{\\ac}$ is the form \nwith ${L^{1}_{loc}}$-function coefficients. \nThen we have $T_{\\ac} \\geq \\gamma $ when $T \\geq \\gamma $ for some smooth form $\\gamma$. \nIn particular the absolutely continuous part $T_{\\ac}$ is positive, when $T$ is a positive current.\nNote that $T_{\\ac}$ is not $d$-closed in general even if $T$ is $d$-closed.\nThen we can define the wedge $T_{\\ac}^{k}$ almost everywhere \nsince $T_{\\ac}$ is a positive form with ${L^{1}_{loc}}$-function coefficients (Refer to \\cite{Bou02} for more details). \n\n\\subsection{Approximations of currents}\nFix a K\\\"ahler form $\\omega$ on $M$. \nLet $T= \\theta + \\ddbar \\varphi$ be a $(1,1)$-current in the class $\\A \\in H^{1,1}(M, \\RR)$, \nwhere $\\theta$ is a smooth $(1,1)$-form in the class $\\A$.\nWe assume that $T\\geq \\gamma $ holds for a smooth form $\\gamma $.\nThen we can approximate $T$ by smooth forms in the following sense:\n\n\\begin{theo}\\label{ap-dem82} \\textrm { \\cite[TH\\'EOR\\`EME9.1]{Dem82} }\nThere exist a decreasing sequence of smooth functions $\\varphi _{k}$ converging to $\\varphi $ \nsuch that if we set $T_{k} = \\theta + \\ddbar \\varphi _{k} \\in \\A$, we have\n\n\\ \\ $\\mathrm{(a)}$\\ \\ $T_{k} \\longrightarrow T$ weakly and $T_{k} \\longrightarrow T_{\\ac} $ almost everywhere .\n\n\\ \\ $\\mathrm{(b)}$\\ \\ $T_{k} \\geq \\gamma - C\\lambda _{k} \\omega $, \nwhere $C$ is a positive constant depending on $(M,\\omega )$ only, \nand $\\lambda _{k}$ is a decreasing sequences of continuous functions such that \n$\\lambda _{k} \\searrow \\nu(T,x) $ for all $x \\in M$.\n\n\\end{theo}\nRoughly speaking Theorem \\ref{ap-dem82} says that it is possible to smooth a given current $T$ insides \nthe class $\\A$, but only with the loss of positivity controled by the Lelong numbers of $T$. \nBy the proof of Theorem \\ref{ap-dem82} in \\cite{Dem82}, we can add the following property $\\mathrm{(d)}$.\n\n\\textit{\n$\\mathrm{(c)}$\\ \\ When $T$ is smooth on some neighborhood of $x_{0} \\in M$, \n$T_{k} \\longrightarrow T_{\\ac}$ is a pointwise convergence on some neighborhood of $x_{0}$.\n}\\\\\nThe following theorem says that it is possible to approximate a given current $T$ with currents \nwith analytic singularities without the loss of positivity. \n\n\\begin{theo}\\label{ap-bou02} \\textrm{ \\cite{Dem92}},\\textrm{ \\cite[Theorem 2.4]{Bou02} }\nThere exists a decreasing sequence of functions $\\varphi _{k}$ with analytic singularities \nconverging to $\\varphi $ such that if we set $T_{k} = \\theta + \\ddbar \\varphi _{k} \\in \\A$, we have\n\n\\ \\ $\\mathrm{(a')}$\\ \\ $T_{k} \\longrightarrow T$ weakly and $T_{k,\\ac} \\longrightarrow T_{\\ac} $ almost everywhere.\n\n\\ \\ $\\mathrm{(b')}$\\ \\ $T_{k} \\geq \\gamma -\\e_{k}\\omega $, where $\\e_{k}$ is a positive number converging to zero.\n\n\\ \\ $\\mathrm{(c')}$\\ \\ The Lelong number $\\nu(T_{k},x)$ increase to $\\nu(T,x)$ uniformly with respect to $x \\in M$.\n\\end{theo}\n\nIn the proof of \\cite[Theorem 2.4]{Bou02}, \nthe convergence $T_{k,\\ac} \\longrightarrow T_{\\ac} $ in $\\mathrm{(a')}$ is proved only from the property \n$\\mathrm{(a)}$ in Theorem 2.2.\nTherefore we can add the following property $\\mathrm{(d')}$ from the property $\\mathrm{(c)}$.\n \n \n\\textit{\n\\ $\\mathrm{(d')}$\\ \\ When $T$ is smooth on some neighborhood of $x_{0} \\in M$, \n$T_{k} \\longrightarrow T_{\\ac}$ is pointwise convergence on some neighborhood of $x_{0}$.}\n\\\\\nThis consideration shows the following corollary.\n\\begin{cor}\nAssume that $T$ has analytic singularities whose polar set does not contain a closed analytic set $V$. \nThen $T_{k}$ in Theorem 2.3 satisfies the following:\n\\begin{equation*}\n\\big( T_{k} \\vert_{V_{\\reg}} \\big)_{\\ac} \\longrightarrow {\\big( T \\vert_{V_{\\reg}} \\big) }_{\\ac}\\ \\ \\ \\ \nfor\\ almost\\ point\\ in\\ V_{\\reg}.\n\\end{equation*}\n\\end{cor}\n\n\n\n\\subsection{Restricted volumes and properties.}\nFor simplicity, we denote by $H^{0}(X|V,\\OX (D))$ $$H^{0}(X|V,\\OX (D)):=\n\\mathrm{Im}\\Big(H^{0}(X,\\OX(D)) \\longrightarrow H^{0}(V,\\OV(D)) \\Big).$$\nThe restricted volume of $D$ along $V$ is defined to be \n\\begin{equation*}\n\\V{V}(D) = \\limsup_{k \\to \\infty} \\dfrac\n{\\dim H^{0}(X|V,\\OX (D)) }{k^d\/d!}.\n\\end{equation*}\nNote that when $V$ is the ambient space $X$, the restricted volume of $D$ is the usual volume.\nWhen $V$ is not contained in $\\mathbb{B}_{+}(D)$, \nmany results for usual volumes are extended to restricted volumes.\nIn this paper, we refer the basic properties of restricted volumes to [ELMNP].\nHere $\\mathbb{B}_{+}(D)$ (\\textit{resp}. $\\mathbb{B}_{-}(D)$) means the augmented base locus \n(\\textit{resp}. the restricted base locus) of $D$\n(see \\cite{ELMNP2}, \\cite{ELMNP3}).\nThe non-K\\\"ahler locus $E_{nK}(\\A)$ (\\textit{resp}. non-nef locus $E_{nn}(\\A)$) \nof a class $\\A$ in $H^{1,1}(X,\\RR)$ is the analytic counter part of the augmented base locus \n(\\textit{resp}. restricted base locus) (see \\cite{Bou04}).\nWhen $\\A $ is the first Chern class $c_{1}(D)$ of some divisor $D$, the the augmented base locus \n(\\textit{resp}.restricted base locus ) coincide with the non-K\\\"ahler locus (\\textit{resp}. the non-nef locus).\nIf $\\A$ is a big class, $E_{nn}(\\A)$ coincide with $E_{+}(T_{\\min}):=\\big\\{x \\in X \\big| \\nu(T_{\\mathrm{min}}, x) >0 \\big\\}$, where \n$T_{\\min}$ is a minimal singular current in $\\A$ (see \\cite[proposition 3.8]{Bou04}).\n\n\\subsection{Divisorial Zariski decomposition.}\n\nIn this subsection, we recall the definition of divisorial Zariski decompositions.\nDivisorial Zariski decomposition for a big divisor coincides with $\\sigma$-decompositions.\nDivisorial Zariski decompositions are studied in \\cite{Bou04} and $\\sigma$-decompositions are studied in \\cite{Nak}.\n\n\nLet $\\A$ be a pseudo-effective class in $H^{1,1}(M,\\RR)$.\nThen an effective $\\RR$-divisor $N$ is defined as follows: \n\\begin{equation*}\n N:=\\sum_{F:\\mathrm{prime\\ div}} \\nu(T_{\\min} , F) F,\n\\end{equation*}\nwhere $T_{\\min}$ is a minimal singular current in $\\A$.\nHere $\\nu(T_{\\min} , F)$ is the Lelong number along $F$, \nwhich is defined by $\\inf_{x \\in F} \\nu(T_{\\min}, x)$.\nThe class $P$ is defined by $P:= \\A -\\{ N \\}$.\nHere $\\{N\\}$ denotes the class of $N$.\nThen the decomposition $\\A = P + \\{ N \\}$ is said to be the divisorial Zariski decomposition of $\\A$.\n$P$ is called the positive part and $\\{ N \\}$ is called the negative part of $\\A$.\nIn general the positive part $P$ is nef in codimension 1.\n(that is, the codimension of the non-nef locus is greater than 1.)\nWe say that $\\A$ admits a Zariski decomposition if the positive part $P$ is nef.\nIf $\\A$ is the class of some divisor, this definition coincides with that of the divisor (see section 1).\nFor example if $M$ is surface, any big class admits a Zariski decompoition (see \\cite[section 4]{Bou04}).\nBy the construction of $N$, positive currents in $\\A$ and positive currents in $P$ are identified \nby the following correspondence:\n\\begin{equation*}\n\\A\\ni T \\longmapsto T-[N] \\in P.\n\\end{equation*}\n\n\n\\section{Restricted volumes and Zariski decompositions }\n\n\\subsection{Positive parts and restricted volumes }\nThe main purpose in this section is to prove Theorem \\ref{MaB}.\nIn this section, we assume $D$ is big.\nWe begin to prepare for the proof of Theorem \\ref{MaB}.\nConsider the divisorial Zariski decomposition $D = P+N$ of $D$.\nSince $D$ is big, the divisorial Zariski decomposition coincide with the $\\sigma$-decomposition.\nNote that $\\mathbb{B}_{+}(D) = \\mathbb{B}_{+}(P) $ and $\\mathrm{Supp}(N) \\subseteq \\mathbb{B}_{+}(D)$.\nThe following proposition shows that the restricted volume of $D$ can be computed by the positive part $P$.\n\\begin{prop}\\label{positive}\nLet $W\\subseteq X$ be an irreducible subvariety which is not contained in $\\mathbb{B}_{+}(D)$.\nThen the equality $\\V{W} (D) = \\V{W} (P)$ holds.\n\\end{prop}\n\n\\begin{rem}\nIn general $P$ is an $\\RR$-divisor. \nThen $\\V{W} (P)$ is defined by the limit of the restricted volumes of \n$\\QQ$-divisors which converges to $P$ in N\\'eron-Severi group.\nBy the continuity of restricted volumes (\\cite[Theorem 5.2]{ELMNP}), $\\V{W} (P)$ does not depend on the choice\n of $\\QQ$-divisors which converges to $P$.\n\\end{rem}\n\n\\begin{proof}\nBy the bigness of $D$, we can take an effective $\\QQ$-divisor $D^{'}$ with $D\\sim_{\\QQ} D^{'}$.\nBy multipling positive integer, we can assume that $D$ is effective divisor.\nSince $W$ is not contained in $\\mathbb{B}_{+}(D)$, we can assume \n$W \\not\\subseteq \\mathrm{Supp}(D) $. \nMoreover $W$ are not contained in $\\mathrm{Supp}(N \\cup P)$, \nsince $ \\mathrm{Supp}(N \\cup P)$ are contained in $\\mathrm{Supp}(D)$.\nFrom \\cite[Theorem 5.5]{Bou04}, there exists a canonical isomorphism \n$H^{0}(X, \\OX(\\lfloor kP \\rfloor)) \\cong H^{0}(X, \\OX(kD))$ by multipling the section $e_{k}$ for a positive intger $k$, \nwhere $e_{k}$ is the standard section of the effective divisor $\\lceil kN \\rceil$. \nWe consider the following commutative diagram:\n\\[\n\\begin{CD} H^0(X,\\OX (\\lfloor kP \\rfloor)) @>\\cdot e_{k}>>H^0(X,\\OX (kD))\\\\\n@V{f}VV @V{g}VV\\\\\nH^0(W,\\OW (\\lfloor kP \\rfloor)) @> \\cdot e_{k} \\vert_{W}>>H^0(W,\\OW (kD))\n\\end{CD}\n\\]\n\n$f$ and $g$ are the restriction maps.\nThis diagram induces the map $\\mathrm{Im}(f) \\longrightarrow \\mathrm{Im}(g)$.\nThis map is surjective since the vertical map is an isomorphism.\nMoreover the map is injective since ${e_{k}} \\vert_{W}$ is a nonzero section \nfrom $W\\not\\subseteq \\mathrm{Supp}(N)$.\nSo we have $$ \\V{W}(D) = \\limsup _{k \\to \\infty}\\dfrac{h^{0}(X|W,\\OX (\\lfloor kP \\rfloor))}{k^{d}\/d! }.$$\nWhen $P$ is a $\\QQ$-divisor, Proposition \\ref{positive} follows from this equation and \nthe homogeneity of restricted volumes (\\cite[Lemma 2.2]{ELMNP}).\nWhen $P$ is an $\\RR$-divisor, we can reduct to a $\\QQ$-divisor by the continuity of restricted volumes.\n\\big(Note that $\\lfloor kP \\rfloor \/k \\to P$ $(k \\to \\infty)$.\\big)\n\\end{proof}\n\n\\begin{cor}\\label{positive-co}\nLet $W \\subseteq X$ be an irreducible subvariety of dimension $d$ which is not contained in $\\mathbb{B}_{+}(D)$.\nWhen $D$ admits a Zariski decomposition (that is, the positive part $P$ is nef), \nthe equality $\\V{W} (D) = (W\\cdot P^{d})$ holds.\n\\end{cor}\n\n\n\\begin{proof}\nBy Theorem \\ref{positive}, we obtain $\\V{W} (D) = \\V{W}(P)$.\nSince $P$ is nef, there exists ample $\\QQ$-divisors $B_{k}$ such that \n$B_{k} \\longrightarrow P$ in the N\\'eron-Severi group.\nThe Ampleness of $B_{k}$ give $\\V{W}(B_k) = (V \\cdot B_{k}^{d})$.\nTherefore the continuity of restricted volumes shows \n\\begin{align*}\n\\V{W} (D) &= \\V{W}(P) \\\\\n&= \\lim_{k \\to \\infty} \\V{W} (B_k) \\\\\n&= \\lim_{k \\to \\infty} (W \\cdot B_{k}^{d}) = (W \\cdot P^{d}).\n\\end{align*}\n\\end{proof}\n\n\n\\subsection{Proof of the main Theorem}\n\nIn this subsection we complete the proof of Theorem \\ref{MaB}.\nWe take subvarieties $V$ and $V^{'}$ on $X$ such that $V \\sim V^{'}$ and $V, V^{'} \\not\\subseteq \\mathbb{B}_{+}(D)$. \nAssume that $D$ admits a Zariski decomposition.\nBy Corollary \\ref{positive-co} the restricted volume of $D$ can be computed by the intersection number \nof the positive part.\nThat is, $\\vol(D) =(V\\cdot P^{d}) $ and $\\vo(D) =(V^{'}\\cdot P^{d}) $ holds.\nSo $V \\sim V^{'}$ shows $(V\\cdot D^{d}) = (V^{'}\\cdot D^{d})$.\nHence $\\vol(D) = \\vo (D)$ holds if $D$ admits a Zariski decomposition.\n\n\nNext we prove the converse direction.\nAssume the condition (2).\nMoreover we assume that $P$ is not nef.\nThen the restricted base locus $\\mathbb{B}_{-}(P)$ is not empty.\nFix a very ample divisor $A$ on $X$ and an arbitrary point $x_{0}$ in $\\mathbb{B}_{-}(P)$.\nThen we can take irreducible smooth curves $C$ and $C^{'}$ with the following properties:\\\\\n\\ \\ \\ (1)\\ \\ \\ $C$ and $C^{'}$ are not contained in the augmented base locus $\\mathbb{B}_{+}(D)$. \\\\\n\\ \\ \\ (2)\\ \\ \\ $C$ intersects with the restricted base locus $\\mathbb{B}_{-}(P)$ at $x_{0} \\in X$.\\\\\n\\ \\ \\ (3)\\ \\ \\ $C^{'}$ does not intersect with the restricted base locus $\\mathbb{B}_{-}(P)$.\\\\\n\\ \\ \\ (4)\\ \\ \\ $C$ and $C^{'}$ are the complete intersections of the linear system of $A$.\n\n\nWe can take such curves. \nSince the codimension of the restricted base locus $\\mathbb{B}_{-}(P)$ is greater than one,\na very general complete intersection curve $C^{'}$ satisfies the properties (1), (3) by Bertini's Theorem. \n(We can not take $C^{'}$ which does not intersect $\\mathbb{B}_{-}(D)$ in general.)\n\\cite[Theorem 2.5]{Zha09} assure that a general hyperplane passing through $x_{0}$ \nis irreducible and smooth. \nTherefore we can take a smooth curve $C$ which satisfies the properties (1), (2).\n\n\nNow $C$ and $C^{'}$ are numerically equivalent \nsince $C$ and $C^{'}$ are the complete intersections of the same linear system.\nSo the equality $\\V{C}(P) = \\V{C^{'}}(P) $ holds by the assumption and Theorem \\ref {positive}.\nTherefore it is sufficient to prove the following claim for the contradiction to the assumption that $\\mathbb{B}_{-}(P)$ is not empty.\n\n\n\\begin{lemm}\n$\\mathrm{(A)}$\\ \\ \\ $\\V{C^{'}}(P) = (C^{'} \\cdot P) \\ \\big(=(C\\cdot P) \\big)$. \\\\\n\\ \\ \\ $\\mathrm{(B)}$\\ \\ \\ $ \\V{C}(P) < (C\\cdot P)$.\n\\end{lemm}\n\n\\begin{proof}\nFor simplicity, we assume $P$ is a $\\QQ$-divisor. \nWhen $P$ is an $\\RR$-divisor, the similar argument give the same results by using\ncontinuity of restricted volumes.\nFix a positive integer $a$ such that $aP$ is a $\\ZZ$-divisor.\n\\\\\n(A): Then from \\cite[Theorem 2.13]{ELMNP}, the restricted volume of $P$ can be computed as follows:\n\\begin{equation*}\n\\V{C^{'}}(P) = \\limsup_{\\ell \\to \\infty}\n\\dfrac{h^{0} \\big( C^{'}, \\mathcal{O}_{C^{'}} \\big( \\ell a P \\big) \\otimes \\J \n\\big(\\Vert \\ell aP \\Vert\\big)\\big\\vert_{C^{'}} \\big)}{\\ell a},\n\\end{equation*}\nSince $P$ is a big divisor, the restricted base locus of $P$ equals to the set \n$\\big\\{x \\in X \\ \\big| \\ \\nu(T_{\\mathrm{min}}, x) >0 \\big\\}$, \nwhere $T_{\\mathrm{min}}$ is a minimal singular current in $c_{1}(P)$.\nSo the Lelong number of every point $x$ on $C^{'}$ is zero by the property (3).\nThis implies $\\I_{+}(\\ell T_{\\mathrm{min}})$ and $\\I(\\ell T_{\\mathrm{min}})$ are \ntrivial along $C^{'}$ by Skoda's Lemma.\nThis shows $\\J(\\Vert\\ell P \\Vert)$ is trivial along $C^{'}$ by \\cite[Theorem 1.1]{DEL00}.\nHence we obtain \n\\begin{equation*}\n \\V{C^{'}}(P) = \\limsup_{\\ell \\to \\infty} \n\\dfrac{h^{0} \\big( C^{'}, \\mathcal{O}_{C^{'}} ( \\ell aP) \\big)}{\\ell a} .\n\\end{equation*}\n\nSince $C^{'}$ is not contained in $\\mathbb{B}(P)$, $P$ \nis an ample divisor on $C^{'}$.\nBy Riemann-Roch formula on compact Riemann surfaces, \nwe have $\\V{C^{'}} (P+A_{k}) = \\big((P+A_{k}) \\cdot C^{'} \\big)$.\nIn fact, Riemann-Roch formula shows \n\\begin{equation*}\nh^{0} \\big( C^{'}, \\mathcal{O}_{C^{'}} ( \\ell aP ) \\big) =\nh^{1} \\big( C^{'}, \\mathcal{O}_{C^{'}} ( \\ell aP) \\big) + \n\\big( \\ell a P \\cdot C^{'} \\big) -g +1,\n\\end{equation*}\nwhere $g$ is a genus of $C^{'}$.\nBy the ampleness of $P$, the first cohomology vanishes \nfor sufficietly large $\\ell$. \nSo we obtain $\\V{C^{'}} (P) = \\big( P \\cdot C^{'} \\big)$. \nTherefore (A) in Lemma 3.4 holds.\\\\\n\n(B):\\ By the same argument of (A), we get the following equality:\n\n\\begin{equation*}\n\\V{C}(P) = \\limsup_{\\ell \\to \\infty}\n\\dfrac{h^{0} \\big( C, \\mathcal{O}_{C} \\big( \\ell a P \\big) \\otimes \\J \\big(\\Vert \\ell aP \\Vert\\big)\n\\big\\vert_{C} \\big)}{\\ell a},\n\\end{equation*}\nNow we estimate the asymptotic multiplier ideal by using Skoda's Lemma. \nFirst of all, we obtain $\\J (\\Vert k P \\Vert) \\subseteq \\I( k T_{\\mathrm{min}})$ for every positive integer $k$\nfrom the minimal singularity of $T_{\\mathrm{min}}$, \nwhere $T_{\\mathrm{min}}$ is a minimal singular current in $c_{1}(P)$.\nSince $\\nu(T_{\\mathrm{min}},x_{0}) $ is positive by the property (2), \nwe can take a positive rational number $p\/q$ which is smaller than $\\nu(T_{\\mathrm{min}},x_{0})$.\nSo we obtain $p \\ell a < \\nu(q \\ell a T_{\\mathrm{min}}, x_{0})$.\nSkoda's Lemma implies $\\nu(k q T_{\\mathrm{min}})_{x_{0}} \\subseteq {\\mathcal{M}^{{p \\ell a -n+1}}_{x_{0}, X}}$, \nwhere $\\mathcal{M}_{x_{0},X}$ is the maximal ideal in $\\mathcal{O}_{x_{0,X}}$.\nSo we obtain \n\\begin{align*}\n\\V{C}(P) \n&\\leq \\limsup_{k \\to \\infty} \\dfrac{h^0 \\big( C,\\mathcal{O}_{C} (\\ell a P )\\otimes\n {\\mathcal{M}^{{p \\ell a -n+1}}_{x_{0}, X}} \\vert_{C} \\big) }{\\ell a} \\\\\n &\\leq \\limsup_{k \\to \\infty} \\dfrac{h^0 \\big( C,\\mathcal{O}_{C} (\\ell a P )\\otimes\n{\\mathcal{M}^{{p \\ell a -n+1}}_{x_{0}, C}}\\big) }{\\ell a} \\\\\n& = \\limsup_{k \\to \\infty} \\dfrac{h^0 \\big( C,\\mathcal{O}_{C} (\\lfloor \\ell a P \\rfloor - (p \\ell a -n+1)[x_{0}] )\\big) } {\\ell a} ,\n\\end{align*}\nwhere $[x_{0}]$ is a divisor on $C$ defined by $x_{0}$.\nBy the same argument of (A), we obtain $\\V{C}(P) < (C\\cdot P) -{p}\/{q}$.\n\\end{proof}\nBy the proof of Lemma 3.5 (B), the following corollary is proved.\n\\begin{cor}\nLet $C$ be an irreducible smooth curve. \nAssume that $C$ is not contained in the augmented base locus of $D$ \nThen the following inequality holds.\n\\begin{equation*}\n\\V{C}(D) \\leq (C\\cdot D) - \\sum_{x \\in C \\cap \\mathbb{B}_{-}(D)} \\nu(T_{\\mathrm{min}},x)\n\\end{equation*}\n\\end{cor}\n\n\n\n\n\\section{Expression of restricted volumes by current integration}\n\\subsection{Proof of the expression formula}\nThe main purpose of this subsection is to prove Theorem\\ref{MaA}.\nBefore the proof of Theorem \\ref{MaA}, we must show that the integral \n$\\int_{V_{\\reg}}{({T\\vert_{V_\\reg}})_{\\ac}^d}$ is always finite.\n\n\\begin{prop}\\label{F}\nLet $W$ be an irreducible subvariety of dimension $d$ on a compact K\\\"ahler manifold $Y$ \nand $S$ a $d$-closed positive $(1,1)$-current on $Y$.\nAssume that the polar set of $S$ does not contain $W$.\nThen the integral $\\int_{W_{\\reg}}({S\\vert_{W_{\\reg} } })_{\\ac}^d$ is finite.\n\\end{prop} \n\n\\begin{proof}\nNote the restriction $S\\vert_{W_{\\reg}}$ is $d$-closed positive.\nHence the integral $\\int_{W}({S\\vert_{W}})_{\\ac}^d$ is finite by \\cite[Lemma2.11]{Bou02} when $W$ is non-singular.\nTherefore it is enough to show the integral is finite when $W$ has singularities. \nLet $\\mu :\\widetilde {W} \\subseteq \\widetilde {X} \\longrightarrow W \\subseteq X$ be the embedded resolution \nof $W \\subseteq X$.\nThat is, $\\mu: \\widetilde {X} \\longrightarrow X$ is a modification and the restriction\n$\\mu : \\widetilde {W} \\longrightarrow W$ is the resolution of singularities of $W$.\nThen the following lemma assures Proposition \\ref{F} holds.\nIn fact, $\\int_{\\widetilde{W} } { \\big( (\\mu^{*} S)\\vert_{\\widetilde{W}} \\big)_{\\ac}^{d} }$ is finite, \nsince $\\widetilde{W}$ is non-singular.\n\\end{proof}\n\n\\begin{lemm}\\label{f}\n\\[ \\ \\ \\ \\ \\int_{W_{\\reg}}{({S\\vert_{W_{\\reg}}})_{\\ac}^d} = \n\\int_{\\widetilde{W} } { \\big( (\\mu^{*} S)\\vert_{\\widetilde{W}} \\big)_{\\ac}^{d} }\\]\n\\end{lemm}\n\n\\begin{proof}\nThe map $\\widetilde {W}\\xrightarrow{\\ \\ \\mu \\ \\ } W$ is an isomorphism except some subvarieties.\nTherefore $(\\mu ^{*}S) \\vert _{\\widetilde{W}}$ is identified with $S\\vert_{W_{\\reg}}$ by $\\mu$ on Zariski open set.\n$\\big( (\\mu ^{*}S) \\vert _{\\widetilde{W}} \\big)_{\\ac}$ and $\\big( S\\vert_{W_{\\reg}} \\big)_{\\ac}$\n are absolutely continuous with respect to the Lebesgue measure by the definition. \nHence we obtain $\\int_{W_{\\reg}}{({S\\vert_{W_{\\reg}}})_{\\ac}^d} = \n\\int_{\\widetilde{W} } { \\big( (\\mu^{*} S)\\vert_{\\widetilde{W}} \\big)_{\\ac}^{d} }$ \nsince a Zariski closed set is measure zero with respect to the Lebesgue measure .\n\\end{proof}\n\\textit{Proof of Theorem \\ref{MaA}.)}\n{\\bf{ (Step1)} } First of all we show the inequality $\\geq$ in Theorem \\ref{MaA}\nby using the singular holomorphic Morse inequalities (see [Bon93]) and Proposition \\ref{min}.\nProposition \\ref{min} is proved in the end of this subsection.\nAt first we consider the case when $V$ is non-singular. \nFor a positive $(1,1)$-current $T$ with analytic singularities in the Chern class $c_1(D)$ \nwhose singular locus does not contain $V$, we obtain the following inequality:\n\\begin{align*}\n\\vol (D) &= \\limsup_{k \\to \\infty} \\frac{\\dim\n H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT_{\\mathrm{min}})\\vert_{V} \\bigr)}{{ k^d}\/{d!} } \\\\\n& \\geq \\limsup_{k \\to \\infty} \\frac{\\dim \nH^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT)\\vert_{V} \\bigr)}{{ k^d}\/{d!} } \\\\\n& \\geq \\limsup_{k \\to \\infty} \\frac{\\dim \nH^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT\\vert_{V})\\bigr)}{{ k^d}\/{d!} }.\n\\end{align*}Here $T_{\\min}$ is a minimal singular current in $c_{1}(D)$.\nThe first equality follows from Proposition \\ref{min} and the second inequality follows from the restriction formula.\nBy using the singular holomorphic Morse inequality we have \n\\begin{align*}\n\\vol (D) &\\geq \\limsup_{k \\to \\infty} \\frac{\\dim \nH^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} (kT\\vert_{V})\\bigr)}{{ k^d}\/{d!} }\\\\\n& \\geq \\int_{V}{{(T\\vert_{V})}_{\\ac}^d}.\n\\end{align*}Therefore when $V$ is non-singular, Step 1 is proved.\n\nNext we consider the case when $V$ has singularities.\nThen we consider the embedded resolution \n$\\mu: \\widetilde V \\subseteq \\widetilde X \\longrightarrow V\\subseteq X$.\nNote the augmented base locus of the pull back $\\mu ^{*}D$ does not contain $\\widetilde V$ since \nthe restriction $\\mu : \\widetilde V \\longrightarrow V$ is isomorphism at a generic point.\nThen by applying the singular holomorphic Morse inequality and \nrestriction formula to $\\mu ^{*}D$, $\\widetilde V$ and $\\mu ^{*}T$ again, we obtain\n\\begin{equation*} \n\\voll (\\mu ^{*}D) \\geq \\int_{\\widetilde V} { ((\\mu ^{*}T)\\mid _{\\widetilde V})_{\\ac}^{d}}.\n\\end{equation*}\n\nBy Lemma \\ref{F} and \\cite[Lemma 6.7]{ELMNP} shows the following lemma.\nBy this lemma and the above inequalities, Step 1 is proved even if $V$ has singularities.\n\\begin{lemm}\\label{cc}\n\\[ (1)\\ \\ \\ \\ \\vol (D)=\\voll (\\mu ^{*}D) \\]\n\\[ \\ \\ (2)\\ \\ \\ \\ \\int_{V} \\T=\\int_{\\widetilde{V}} \\Tm \\]\n\\end{lemm}\n{\\bf(Step2)}\\ \\ \nWe show the converse inequality $\\leq$ to complete the proof of Theorem \\ref{MaA}.\nBy \\cite[Proposition 2.11]{ELMNP}, for an arbitrary number $\\e >0$, we can find \nthe modification $\\pi_{\\e}:X_{\\e} \\longrightarrow X$\n and the expression ${\\pi_{\\e}}^{*} (D)=A_{\\e}+E_{\\e}$ \nsuch that $({A_{\\e}}^{d}\\cdot V_{\\e}) \\geq \\vol(D) - \\e$.\nHere $A_{\\e}$ is an ample $\\QQ$-divisor and $E_{\\e}$ is an effective $\\QQ$-divisor such that\nthe proper transformation $V_{\\e}$ is not contained in $\\mathrm{Supp}(E_{\\e})$. \nWe express the intersection number $({A_{\\e}}^{d}\\cdot V_{\\e})$ by current integration.\nLet $\\omega _{\\e}$ be a smooth positive $(1,1)$-form on $X_{\\e}$ in the first Chern class $c_{1}(A_{\\e})$.\nSince $\\mathrm{Supp}(E_{\\e})$ does not contain $V_{\\e}$, \nwe can restrict $[E_{\\e}]$ to $V_{\\e}$.\nSince $[E_{\\e}]$ is a positive current, we obtain\n\\begin{align*}\n({A_{\\e}}^{d} \\cdot {V}_{\\e}) &= \\int_{V_{\\e}}{({\\omega _{\\e}}\\vert_{V_{\\e}})^{d}} \\\\\n&\\leq \\int_{V_{\\e}}{ \\big( { (\\omega _{\\e} + [E_{\\e}])\\vert_{V_{\\e} } } \\big) _{\\ac}^{d}} \\\\\n&= \\int_{V_{\\reg}} { \\big\\{ \\big ({\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])\\big)\\vert_{V_{\\reg} } \\big\\} _{\\ac}^{d} } . \\\\\n\\end{align*}\nThe third equality follows from the property that $\\pi_{\\e}$ is an isomorphism at a general point of $V$ and the same \nargument of Lemma \\ref{f}.\n\nSince $\\pi_{\\e}$ is a modification, the push-forward ${\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])$ is \nthe positive current in the Chern class $c_{1}(D)$.\nHowever the push-forward may not have analytic singularities.\nSo we want to approximate the push-forward by positive currents with analytic singularities. \nWhen we use Theorem \\ref{ap-bou02}, the positivity of a current can be lost.\nNote $(\\omega _{\\e} + [E_{\\e}])$ is a K\\\"ahler current but \nthe push-forward may not be a K\\\"ahler current.\nHence we can not always take the sequence of approximation by positive currents.\nNow we consider the approximations of the push-forward ${\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])$\nafter we change the current for the K\\\"ahler current.\n\n\nFor simplicity we write $T_{\\e} := {\\pi _{\\e}}_{*}(\\omega _{\\e} + [E_{\\e}])$. \nSince $V$ is not contained in the augmented base locus $ {\\mathbb{B}}_{+}(D)$, \nwe can take a K\\\"ahler current $S$ with analytic singularities such that the pole does not contain in $V$. \nFatou's Lemma assures\n\\begin{align*}\n\\vol(D) - \\e & \\leq ({A_{\\e}}^{d}\\cdot V_{\\e}) \\\\\n&= \\int_{\\Vr} \\liminf _{\\del \\to 0} \\big\\{ (1-\\del)({T_{\\e}}\\vert_{\\Vr}) + \\del (S\\vert_{V_{\\reg}}) \\big\\}_{\\ac}^{d} \\\\\n&\\leq \\liminf _{\\del \\to 0} \\int_{\\Vr} \\big\\{ (1-\\del)({T_{\\e}}\\vert_{\\Vr}) + \\del (S\\vert_{V_{\\reg}}) \\big\\}_{\\ac}^{d} .\n\\end{align*}\nHence there exists a sufficiently small $\\del_{0} >0$ with the following:\n\\begin{equation*}\n\\vol(D) - 2\\e \\leq \\int_{\\Vr} \\{ (1-\\del_{0})({T_{\\e}}\\vert_{\\Vr})_{\\ac} + \\del_{0} (S\\vert_{V})_{\\ac} \\}^{d}.\n\\end{equation*}\nNote that $(1-\\del_{0})T_{\\e} + \\del_{0} S$ is a K\\\"ahler current in $c_{1}(D)$.\nBy applying the approximation theorem (Theorem 2.3 and Corollary 2.4) to $(1-\\del_{0})T_{\\e} + \\del_{0} S$, \nwe can take positive currents $\\big\\{ U_{k} \\big\\}_{k=1}^{\\infty}$ in $c_1(D)$ with the following properties.\n\\\\\n\\ \\ \\ (1)\\ \\ \\ $U_{k}$ has an analytic singularities for every integer $k$. \\\\\n\\ \\ \\ (2)\\ \\ \\ $({U_{k}}\\vert_{\\Vr})_{\\ac} \\longrightarrow \n\\big\\{ (1-\\del_{0})T_{\\e} \\vert_{\\Vr}+ \\del_{0} S\\vert_{\\Vr} \\big\\}_{\\ac}$\\ \\ \\ for almost point in $V_{\\reg}$ \\\\\n\\ \\ \\ (3)\\ \\ \\ $U_{k}$ is a positive current for a sufficiently large $k$\n\n\nThe property (2) and Fatou's Lemma show \n\\begin{align*}\n\\vol(D) - 2\\e &\\leq \\int_{\\Vr} \\big\\{ (1-\\del_{0})({T_{\\e}}\\vert_{\\Vr})_{\\ac} + \\del_{0} (S\\vert_{V_{\\reg}})_{\\ac} \\big\\}^{d} \\\\\n&= \\int_{\\Vr} \\liminf_{k \\to \\infty}({U_{k}}\\vert_{\\Vr})_{\\ac}^{d}\\\\\n&\\leq \\liminf_{k \\to \\infty} \\int_{\\Vr} ({U_{k}}\\vert_{\\Vr})_{\\ac}^{d}.\\\\\n\\end{align*}\nTherefore we have $$\\vol(D)-3\\e\\leq \\int_{\\Vr} ({U_{k_{0}}}\\vert_{\\Vr})_{\\ac}^{d}$$ for a sufficiently large $k_{0}$.\nSince $\\e$ is an arbitrary positive number and $U_{k_{0}}$ is a positive current with analytic singularities \nin the Chern class $c_1(D)$ whose pole of $U_{k_{0}}$ does not contain $V$.\nHence Step 2 is finished.\nTheorem \\ref{MaA} is proved.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\nIn the end of this subsection, we prove the following proposition.\nIt is the variation of \\cite[Theorem 2.13]{ELMNP}.\n\n\\begin{prop}\\label{min}\nUnder the assumption that $V$ is not contained in $ {\\mathbb{B}}_{+}(D)$, the following equality holds.\n\n\\begin{equation*}\n\\vol (D) = \\limsup_{k \\to \\infty} \\frac{h^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{I} \n(kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } ,\n\\end{equation*}\nwhere $T_{\\mathrm{min}}$ is a minimal singular current in the class $c_1(D)$.\n\\end{prop}\n\n\\begin{proof}\n\\cite[Theorem 2.13]{ELMNP} assures the following equality under the assumption. \n\\begin{equation*}\n\\vol (D) = \\limsup_{k \\to \\infty} \\frac{h^{0}\\bigl(V,\\mathcal{O} \n(kD) \\otimes \\mathcal{J} ( \\Vert kD \\Vert)|_{V} \\bigr)}{{ k^d}\/{d!} } \n\\end{equation*}\nTo prove Proposition \\ref{min}, we compare the multiplier ideal sheaf \n$\\mathcal{I}(kT_{\\mathrm{min}})$ with the asymptotic multiplier ideal sheaf $\\mathcal{J}(\\Vert kD\\Vert)$.\nBy the definition of a minimal singular current, we obtain \n$\\mathcal{I}(kT _{\\mathrm{min}}) \\supseteq \\mathcal{J}(\\Vert kD\\Vert)$ for all positive integer $k$.\nHence we obtain the inequality $ \\leq $ in Proposition \\ref{min}.\nTo prove the converse inequality we show the following Lemma.\n\\begin{lemm}\\label{c}\nThere is an effective divisor $E$ (independent of $k$) which satisfies the following properties:\n\n\n\\ \\ \\ \\ $\\mathrm{(i)}$\\ \\ \\ \\ \\ $\\mathrm{Supp} (E)$ does not contain V.\n\n\\ \\ \\ \\ $\\mathrm{(ii)}$\\ \\ \\ \\ \\ $\\mathcal{I}(kT_{\\mathrm{min}})\\cdot \\mathcal{O}_{X}(-E) \n\\subseteq \\mathcal{J}(\\Vert kD\\Vert )$\n for all $k$ sufficiently large.\n\\end{lemm}\n\n\\begin{proof}\nThis proof is essentially based on the argument in \\cite{DEL00}.\nFix a very ample divisor $A$.\nFor an arbitrary point $x \\in X$, there exists a zero dimensional complete intersection $P_{x}$ \nof the linear system $\\mid A \\mid$ containing $x$. \nThe Ohsawa-Takegoshi-Manivel $L^2$-extension theorem shows \nfor every divisor $F$ with a singular hermitian metric $h$ with the nonnegative curvature $T_{h}$,\nthere exists an ample divisor $B$ ($B$ depends only $A$) and \nthe following restriction map is surjective (see \\cite{OT87}, \\cite{Man93}).\n\n\\begin{equation*}\nH^{0}\\bigl(X,\\mathcal{O}_{X} (F+B) \\otimes \\mathcal{I}(T_{h} )\\bigl) \n\\longrightarrow H^{0}\\bigl(P_{x},\\mathcal{O}_{P_{x}} (F+B)\\otimes \\mathcal{I}(T_{h} \\vert_{P_{x}})\\bigl )\n\\end{equation*}\nMoreover the Ohsawa-Takegoshi-Manivel $L^2$-extension claims that for every section on $P_{x}$, the extension is satisfied an $L^{2}$-estimate with a constant independent of $F$.\nNow since the dimension of $P_{x}$ is zero, the $L^{2}$-estimate does not depend on $P_{x}$.\nThat is, $L^{2}$-estimate of the extension of a section on $P_{x}$ depend only on $B$.\nSince $B$ depends only $A$, the $L^{2}$-estimate of the extension of a section on $P_{x}$ depend only on $A$.\n\n\nSince $D$ is big and $V$ is not contained in the augmented base locus of $D$, \nwe can take $E \\in \\big| k_{0}D - B \\big|$ with the property (i) by choosing a sufficiently large $k_{0}$.\nWe apply the Ohsawa-Takegoshi-Manivel $L^{2}$-extension theorem to \n$F_{k}:=(k-k_0)D + E$ equipped with the singular hermitian metric $h_{\\mathrm{min}}^{\\otimes k-k_{0}}\\otimes h_{E}$.\nHere $h_{\\mathrm{min}}$ is a minimal singular hermitian metric \nand $h_{E}$ is a singular hermitian metric defined by the standard section of the effective divisor $E$.\nThen for a sufficiently large $k$ and a point $x\\in X$, we obtain the global section $s_{x}$ of \n$F_{k}+B \\equiv _{\\mathrm{lin}} kD$ satisfied the following estimates:\n \n\\begin{equation*}\n \\int_{X}{ \\| s_{x} {\\|^{2}}_{{h_{\\mathrm{min}}^{\\otimes k-k_{0}}\\otimes h_{E}\\otimes h_{B} } } \\leq C\\ \\ \\ \\ \\ \\ and\\ \\ \\ \\ \\ \n| s_{x}(x) |^{2}}_{h_{\\mathrm{min}}^{\\otimes k-k_{0}}\\otimes h_{E}\\otimes h_{B} } =1,\n \\end{equation*}\nwhere $C$ is a constant depending only $A$ and $h_{B}$ is a smooth hermitian metric of B with the positive curvature.\nFrom the second equality, we infer the following equality:\n\\begin{equation*}\n|s_{x}(x)|^{2}e^{-2(k-k_{0}) \\varphi _{\\mathrm{min}} -2\\varphi _{E}-2\\varphi _{B} }=1,\n\\end{equation*} \nwhere $\\varphi _{\\mathrm{min}}$, $\\varphi _{E}$, $\\varphi _{B}$ is the weight of the hermitian metric \n$h_{\\mathrm{min}}$,$h_{E}$,$h_{B}$ respectively.\nSince $\\varphi _{B}$ is a smooth function and $X$ is compact, we obtain\n \\begin{equation*}\n \\varphi _{\\mathrm{min}}+\\frac{1}{k-k_0}\\varphi _{E} \\leq \\frac{1}{k-k_{0}} \\log|s_{x}(x)| +C.\n \\end{equation*}\nThe evaluation map $H^{0}\\bigl(X,\\mathcal{O}_{X} (kD))\\longrightarrow \\CC$ is \na bounded operator on the Hilbert space \n$H^{0}\\bigl(X,\\mathcal{O}_{X} (kD))$ with $L^{2}$-norm. \nMoreover the operation norm equals to the Bergman kernel\n ${\\log\\sum_{j}{|f_{j}}|}$, \nwhere $\\{f_{j}\\}$ is the orthonormal basis of $H^{0}\\bigl(X,\\mathcal{O}_{X} (kD))$.\n\nTherefore we get\n\\begin{equation*}\n \\frac{1}{k-k_{0}} \\log|s_{x}(x)|+C \\leq \\frac{1}{k-k_{0}} \\log\\sum_{j}{|f_{j}}| +C \n\\end{equation*}\nThis inequality implies that the function $\\frac{1}{k-k_{0}} \\log\\sum_{j}{|f_{j}}|$ has \nless singularities than $\\varphi _{\\mathrm{min}}+\\frac{1}{k-k_0}\\varphi _{E} $.\nConsidering the definition of the asymptotic multiplier ideal sheaf, we get the property (ii). \n\\end{proof}\n\n\nWe complete the proof of Proposition \\ref{min} by using Lemma \\ref{c}.\nFrom the property (i) in the previous Lemma, we can consider the following short exact sequence:\n\\begin{equation*}\n0 \\longrightarrow \\mathcal{O}_{V}(kD-E) \\longrightarrow \\mathcal{O}_{V}(kD) \\longrightarrow\n \\mathcal{O}_{V\\cap E}(D) \\longrightarrow\t0 \n\\end{equation*}\nSince the dimension of the intersection $V\\cap E$ is smaller than $\\dim{V}=d$, \nwe have\n\\begin{equation*}\n\\limsup_{k \\to \\infty} \\dfrac{\\dim \nH^{0}\\big(V \\cap E,\\mathcal{O}_{V \\cap E} (kD) \\big)}{k^{d}\/d!} =0.\n\\end{equation*}\nHence we obtain the following inequality by twisting the restriction to $V$ of the multiplier ideal \nsheaf $\\mathcal{I}(kT_{\\mathrm{min}})$.\n\\begin{equation*}\n\\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \n\\otimes \\mathcal{I} (kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } \\leq\n \\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD-E) \n\\otimes \\mathcal{I} (T_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } \n\\end{equation*}\nSince we chose the effective divisor $E$ satisfied the property (ii) of the previous Lemma, \nwe obtain\n \\begin{equation*}\n\\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD-E) \\otimes \\mathcal{I}\n (kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } \\leq \n\\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \\otimes \\mathcal{J}\n(\\Vert kD \\Vert )|_{V} \\bigr)}{{ k^d}\/{d!} }.\n\\end{equation*}\nThese inequalities assure\n\\begin{equation*}\n\\vol (D) \\geq \\limsup_{k \\to \\infty} \\frac{\\dim H^{0}\\bigl(V,\\mathcal{O}_{V} (kD) \n\\otimes \\mathcal{I} (kT_{\\mathrm{min}})|_{V} \\bigr)}{{ k^d}\/{d!} } .\n\\end{equation*}\nThe proof of Proposition \\ref{min} is finished.\n\\end{proof}\n\n\n\n\n\\subsection{Properties of restricted volumes.}\nTheorem \\ref{MaA} implies that the definition of restricted volumes can be extended to a big class on \na compact K\\\"ahler manifold (see Definition \\ref{defi}).\nLet $M$ be a compact K\\\"ahler manifold, $V\\subseteq M$ a irreducible subvariety of dimension $d$ and \n$\\A$ a big class in $H^{1,1}(M,\\RR)$ in this section. \nIn this subsection, we study the properties of the restricted volumes of a class on \ncompact K\\\"ahler manifold.\n.\n\n\n\\begin{prop}\\label{nef}\nAssume $\\A$ is a nef class and $V$ is not contained in the non K\\\"ahler locus $E_{nK}(\\A)$.\nThen the restricted volume $\\volm(\\A)$ is computed by the intersection number $(\\A^{d}\\cdot V)$.\nThat is, the equality $\\volm(\\A) = (\\A^{d}\\cdot V)$ holds.\n \\end{prop}\n\n\\begin{proof}\nWhen $V$ is non-singular, this proposition is proved by using the similar argument with \\cite[Theorem 4.1]{Bou02}.\nThe case when $V$ has singularities is reduced to the non-singular case by same \nthe argument in Lemma \\ref{f}.\n\\end{proof}\n\nProposition \\ref{div} is generalization of Lemma \\ref{positive} to a class on a compact K\\\"ahler manifold. \nMoreover the proof gives another proof of Lemma \\ref{positive}. \nLet $\\A = P + \\{N\\}$ be the divisorial Zariski decomposition of $\\A$.\n\n\\begin{prop}\\label{div}\nAssume that $V$ is not contained in the non-K\\\"ahler locus $E_{nK}(\\A)$. \nThen $V$ is not contained in $E_{nK}(P)$ and \nthe equality $\\volm(\\A) = \\volm(P) $ holds.\n\\end{prop}\n\n\n\n\\begin{proof}\nThis proposition is based on the following fact. \nPositive currents in $\\A$ and positive currents in $P$ are identified \nby the following correspondence.\n\\begin{align*}\n&\\A \\longrightarrow P \\\\\n&T \\longmapsto T-[N]\n\\end{align*}\nNote $T-[N]$ is a K\\\"ahler current if $T$ is a K\\\"ahler current in $\\A$.\nIn fact if $T \\geq \\delta \\omega $ holds, $T-\\delta \\omega -[E] $ is a positive current\nby considering the Siu decomposition for the positive current $T-\\delta \\omega$.\nHere $E$ is an effective divisor defined by $ E:=\\sum_{D} \\nu(T-\\delta \\omega, D)D$.\nSince $\\omega$ is a smooth form, we get $\\nu(T-\\delta \\omega, D) = \\nu(T, D)$.\nSince $E-N$ is an effective divisor, $T-[N] \\geq \\delta \\omega$ holds.\n\n\n\nFirst of all we show the following Lemma to prove that \n$V$ is not contained in $E_{nK}(P)$.\n\\begin{lemm}\\label{augment}\nThe equality $E_{nK}(\\A) = E_{nK}(P)$ holds.\n\\end{lemm}\n\\begin{proof}\nNote $\\mathrm{Supp}(N)$ is clearly contained in $E_{nK}(\\A)$ from the definition.\nFor a point $x \\not\\in E_{nK}(\\A) $, we can take a K\\\"ahler current $T$ in $\\A$ with analytic singularities \nsuch that $T$ is smooth at $x$.\nThen $T-[N]$ is a K\\\"ahler current in $P$.\nSince $x$ is not contained in $\\mathrm{Supp}(N)$, the K\\\"ahler current $T-[N]$ is smooth at $x$.\nSo $x \\not\\in E_{nK}(P)$ holds.\n\n\n\nConversely for a point $x \\not\\in E_{nK}(P)$, we take a K\\\"ahler current $S$ such that $S$ is smooth on $x$ .\nWe can assume that $S\\geq \\omega $.\nWe prove that $x$ is not contained in $\\mathrm{Supp}(N)$.\nTo prove this, we consider the surjective map:\n\\begin{equation*}\n\\big\\{ \\mathrm{smooth\\ real\\ \\textit{d}\\textit{-}closed\\ (1,1)\\textit{-}form} \\big\\} \\longrightarrow H^{1,1} (M,\\mathbb{R}),\n\\ \\ \\theta \\mapsto \\big\\{ \\theta \\big\\}.\n\\end{equation*}\nWe regard the space of smooth real $d$-closed $(1,1)$-forms as topological space \nwith Fr\\'echet topology. \nFor a smooth $(n-1,n-1)$-form $\\gamma $, the integral \n$\\int_{M}{\\theta _{k}\\wedge \\gamma } $ converges to $ \\int_{M}{\\theta \\wedge \\gamma} $ \nif $\\theta _{k} \\longrightarrow \\theta $ in Fr\\'echt topology.\nHence the above map $\\theta \\longmapsto \\big\\{ \\theta \\big\\}$ is continuous by Serre duality .\nFrom open mapping theorem the map is a open map.\n\n\nSince the map is open, for a positive number $\\e$ we can take a sufficietly small $\\delta >0$ such that \nthe Chern class $\\delta c_{1}(N)$ contains a smooth form $\\eta $ with $-\\e \\omega \\leq \\eta \\leq \\e\\omega $.\nSo $S+\\eta + (1-\\delta)[N]$ is a positive current in the class $\\A$ and \nthe Lelong number equals to $\\nu \\big((1-\\delta)[N] , x \\big)$ since $S$ is smooth at $x$.\nIf $\\nu([N], x)$ is positive, it is contradiction to minimality of $N$.\nSo $x$ is not contained in $\\mathrm{Supp}(N)$.\nThis implies that the positive current $S+[N]$ lies in $\\A$ and is smooth at $x$.\nHence $x$ is not contained in $E_{nK}(\\A)$.\n\\end{proof}\nBy this lemma we can define the restricted volume of $P$.\nNote that $\\mathrm{Supp} \\big( [E] \\vert_{\\Vr} \\big)$ is contained in $E\\cap V$. \nThis assures Lemma \\ref{N}.\nThe Lemma \\ref{N} implies the exceptional divisor part $[N]$ does\nnot effect the integration on $V$.\n\\begin{lemm}\\label{N}\n$( [E] \\vert_{\\Vr})_{\\mathrm{ac}}=0$\n\\end{lemm}\n\nLemma \\ref{N} shows $\\int_{\\Vr}(T\\vert_{\\Vr})_{\\mathrm{ac}}^{d} =\n \\int_{\\Vr} (T\\vert_{\\Vr}-[E] \\vert_{\\Vr})_{\\mathrm{ac}}^{d}$.\nTherefore Theorem \\ref{div} is proved by correspondence between positive currents in $\\A$ and in $P$. \n\\end{proof}\nThe following Proposition says that \nFujita's approximation theorem for the restricted volume holds for a class $\\A$.\nThis implies the continuity of a restricted volume.\n\n\\begin{theo}\\label{Fujita}\n\nThe restricted volume of the class $\\A$ along $V$ can be approximated by intersection numbers of semi-positive classes.\nThat is, the following equality holds.\n\\begin{equation*}\n\\volm (D) = \\sup_{\\pi^{*}T =B +[E]} \\big( \\{B \\} ^{d} \\cdot \\widetilde{V} \\big)\n\\end{equation*}\n, where the supremum is taken over all resolution $\\pi : \\widetilde{M}\\longrightarrow M$ of the positive current\n $T \\in \\A$ with analytic singularities such that $\\pi$ is an isomorphism at a generic point of $V$\nand $\\widetilde{V} \\not\\subseteq \\mathrm{Supp}(E)$.\n(Here $\\widetilde{V}$ denotes the proper transformation of $V$.)\n\\end{theo}\n\n\\begin{rem}\nBy subtracting a class of exceptional divisors from semi-positive class $\\{ B \\}$, \nwe can assume that $\\{ B \\}$ ranges among K\\\"ahler classes.\n\\end{rem}\n\n\\begin{proof}\nLet $T$ be a positive current with analytic singularities in the class $\\A$\n whose polar set is not contained $V$.\nThen We take the modification $\\mu$ such that $\\mu^{*}(T) = (B + [E])$.\n(We can assume that the map is an isomorphism for generic point on $V$.)\nBy Lemma \\ref{f} and Lemma \\ref{N} we obtain\n\\begin{align*}\n\\int_{V_{\\reg}}{\\T} &= \\int_{\\widetilde{V}}{ (\\mu^{*}T \\vert_{\\widetilde{V}})_{\\ac}^{d}} \\\\\n&= \\int_{\\widetilde{V}}{\\big( (B + [E]) \\vert_{\\widetilde{V}} \\big)_{\\ac}^{d}} \\\\\n&= \\int_{\\widetilde{V}}{ B _{\\ac}^{d}} = \\big( \\{B \\} ^{d} \\cdot \\widetilde{V} \\big).\n\\end{align*}\nTherefore we get $\\volm (\\A) =\\mathrm{sup} \\big(\\{B\\} ^{d} \\cdot \\widetilde{V} \\big)$ \nfrom the definition of a restricted volume of the class $\\A$ along $V$.\n\\end{proof}\nTo show the continuity of a restricted volume, we consider the domain of a restricted volume for\n a subvariety $V$.\nMoreover we prove the convexity of the domain and concavity of restricted volumes.\n\n\\begin{defi}\nFor a subvariety $V$, \\textit{the domain of a restricted volume} is defined to be \n$\\mathrm{Big}^{V}(M) := \\big\\{ \\B \\in H^{1,1} (M,\\mathbb{R}) \\ \\big|\\ V \\not\\subseteq E_{nK}(\\B) \\big\\}$.\n\\end{defi}\n\n\n\\begin{prop}\\label{concave}\n\n$(1)$\\ \\ $\\mathrm{Big}^{V}(M)$ is a open convex domain in $H^{1,1} (M,\\mathbb{R})$ \\\\\n$(2)$\\ \\ For $\\B_{1}, \\B_{2} \\in \\mathrm{Big}^{V}(M)$, the following inequality holds. \n$$ \\volm(\\B_{1}+\\B_{2})^{1\/d} \\geq \\volm(\\B_{1})^{1\/d} + \\volm(\\B_{2})^{1\/d}$$ \n\n\\end{prop}\n\\begin{proof}\n(1):\\ \\ \nThe concavity is directly followed from $E_{nK}(\\B + \\B^{'}) \\subseteq \nE_{nK}(\\B^{'}) \\bigcup E_{nK}(\\B)$ \nand $E_{nK}(\\B) = E_{nK}(k\\B)$ for $k \\geq 0$.\nFor given a class $\\B$, we can prove $E_{nK}(\\B^{'}) \\subseteq E_{nK}(\\B)$ for every class $\\B^{'}$ in a suitable open neighborhood of $\\B$\nby using the argument in Lemma \\ref{augment}.\nThis assures the domain is open.\n\n\n$(2)$\\ \\ Boucksom shows the log concavity for the volume of a transcendental class\n in \\cite{Bou02}.\nHence Proposition \\ref{nef} shows a restricted volume has log concavity\n for nef classes. \nWe can reduct to the case of nef classes by using the Proposition \\ref{Fujita}.\n \n\\end{proof}\n\n\n\\begin{cor}The following map is continuous.\n\\begin{align*}\n \\volm : \\mathrm{Big}^{V}(M) & \\longrightarrow \\mathbb{R} \\\\\n \\B & \\longmapsto \\volm(\\B)\n\\end{align*}\n\\end{cor}\n\\begin{proof}\nIt is well-known fact that a concave function on an open convex set $\\mathbb{R}^{N}$ is continuous.\nTherefore Corollary followed from Proposition \\ref{concave}.\n\\end{proof}\n\n\\subsection{Another proof of Theorem \\ref{MaB}}\nIn this subsection we prove Theorem \\ref{MaBB} by using the expression of a restricted volume with current integration. \nThis proof gives another proof of Theorem \\ref{MaB}.\nLet $\\A \\in H^{1,1}(X,\\RR)$ be a big class on $X$ and $\\A = P + \\{N \\}$ be the divisorial Zariski decomposition. \nNote $E_{nK} (\\A) = E_{nK} (P)$ holds by Lemma \\ref{augment}.\nHence we can consider the restricted volume of $P$ along $V$.\n\n\\textit{Proof of Theorem \\ref{MaBB}.)}\nThe principle of the proof is essentially one of Theorem \\ref{MaB}.\nBy Proposition \\ref{div} we get $\\vol(\\A) =\\vol(P) $ for an irreducible subvariety $V$ \nwith $V\\not\\subseteq E_{nK}(\\A)$. \nMoreover Proposition \\ref {nef} claim that $ \\vol (P) = (V\\cdot P ^{d} )$ holds when $P$ is nef.\nHence the restricted volume are unchangeable for cohomologous \nsubvarieties which is not contained the non-K\\\"ahler locus \nif $\\A$ admits a Zariski decomposition.\n\n\n \nWe prove the converse direction.\nAssume the non-nef locus $E_{nn}(P)$ is not empty. \nFix a very ample divisor $A$ on $X$. \nWe take smooth curves $C$ and $C^{'}$ with the following properties:\\\\\n\\ \\ \\ (1)\\ \\ \\ $C^{'}$ does not intersect with the non-nef locus $E_{nn}(P)$.\\\\\n\\ \\ \\ (2)\\ \\ \\ $C$ and $C^{'}$ are not contained in the non-K\\\"ahler locus $E_{nK}(\\A)$. \\\\\n\\ \\ \\ (3)\\ \\ \\ $C$ intersects with the non-nef locus $E_{nn}(P)$ at $x_{0} \\in X$.\\\\\n\\ \\ \\ (4)\\ \\ \\ $C$ and $C^{'}$ are the complete intersections of the linear system of $A$.\\\\\nThen we lead the contradictions by proving the following lemma. \n\n\\begin{lemm}\n$(A)$ \\ \\ $\\V{C^{'}}(\\A) = (C^{'}\\cdot P) \\big( = (C \\cdot P) \\big)$ \\\\\n$(B)$ \\ \\ $\\V{C}(\\A) < (C\\cdot P)$ \n\\end{lemm}\nBefore the proof of Lemma 4.15, we prove Lemma 4.16.\nLemma 4.16 assures that the restricted volume is computed by the Lelong number.\n\\begin{lemm}\nLet $S$ be a positive current with analytic singularities on a smooth curve $C$.\nThen the Siu decomposition coincide with the Lebesgue decomposition of $S$.\nThat is, $S_{\\ac} = S- \\displaystyle \\sum_{x \\in C}\\nu(S,x)[x] $ holds.\n\\end{lemm}\n\\begin{proof}\nA current $S$ is locally expressed by $S=\\dfrac{c}{2} \\log \\big( \\sum_{i=1}^N \\big| f_{i} \\big|^{2} \\big) + \\theta $, \nwhere $\\theta $ is a smooth $(1,1)$-form and $f_{i}$ is a holomorphic function.\nWe take the maximal effective divisor $D$ with $\\mathrm{div}(f_{i}) \\geq D \\geq 0$ for all $i=1,2,\\dots,N$ and \nthe holomorphic function $g$ with $\\mathrm{div}(g) = D$.\nWe denote by $g_{i}$ a holomorphic function defined by $g_{i} =f_{i} \/ g$.\nNote $\\frac{c}{2} \\ddbar \\log \\big| g \\big|^{2} =[cD]$ by Poincar\\`e-Lelong formula.\nHence the following decomposition is the Siu decomposition.\n\\begin{equation*}\nS =\\Big\\{ \\frac{c}{2} \\log (\\sum_{i=1}^N \\big| g_{i} \\big|^{2} \\big) + \\theta \\Big\\} + \n\\frac{c}{2} \\log \\big| g \\big|^{2}\n\\end{equation*}\nSince $g_{i}$ does not have common zero point, $\\log (\\sum_{i=1}^N \\big| g_{i} \\big|^{2} )$ is smooth.\nThe absolutely continuous part of $[cD]$ is zero. \nHence the decomposition is the Lebesgue decomposition.\n\\end{proof}\nBy using Lemma 4.16, we prove (B) in Lemma 4.15.\nBy the expressions of a restricted volume with current integration and Theorem \\ref {div} we obtain\n\\begin{equation*}\n \\V{C}(\\A) = \\V{C} (P) = \\sup_{T \\in P} \\int_{C}{( T\\vert_{C} ) _{\\ac}} \n\\end{equation*}\nWe take a positive current $T$ with analytic singularities in the class $P$ \nand whose singular locus does not contain $C$.\n$T \\vert_{C}$ is a positive current with analytic singularities on $C$.\nBy Lemma 4.16 we obtain ${ (T \\vert_{C})_{\\ac} = T \\vert_{C} -\\sum_{x \\in C} \\nu(T \\vert_{C} , x)[x] }$.\nNote that the restriction $T \\vert_{C}$ lies in $P \\vert_{C}$.\nThis implies that \n\\begin{equation*}\n\\int_{C} {T \\vert_{C}} = \\mathrm{deg}_{C} (P \\vert_{C}) = (C\\cdot P).\n\\end{equation*}In fact $T \\vert_{C}$ is expressed $T \\vert_{C} = \\theta \\vert_{C} + \\ddbar \\varphi \\vert_{C} $, \nwhere $\\theta$ is a smooth $(1,1)$-form in $P$ and $\\varphi $ is a $L^{1}$-function on $X$.\nHence we obtain \n\\begin{equation*}\n\\int_{C} {T \\vert_{C}} = (C\\cdot P) + \\int_{C} {\\ddbar \\varphi \\vert_{C}}.\n\\end{equation*}\nBy applying the approximation theorem (Theorem 2.2) to $\\varphi \\vert_{C}$, we get smooth functions $\\varphi _{k}$ on $C$ \nsuch that $\\ddbar \\varphi _{k} $ converges to $\\ddbar \\varphi \\vert_{C}$ weakly.\nSo $\\int_{C}{\\ddbar \\varphi _{k}} $ converges to $ \\int_{C} {\\ddbar \\varphi \\vert_{C}}$.\nHere $\\int_{C}{\\ddbar \\varphi _{k}}$ equals to zero by smoothness of $\\varphi _{k}$ and Stokes's theorem.\nHence we have\n\\begin{align*}\n\\V{C}(\\A) &= \\sup_{T \\in c_{1}(P)} \\{ (C \\cdot P) - \\sum_{x \\in C} \\nu(T \\vert_{C} , x)\\} \\\\\n& = (C \\cdot P) - \\inf_{T \\in P} \\sum_{x \\in C} \\nu(T \\vert_{C} , x).\n\\end{align*}The Lelong number of the restriction of a current is more than that of the current.\nMoreover $\\nu(T_{\\mathrm{min}}, x) \\leq \\nu(T , x) $ holds from the definition of a minimal singular current. \nTherefore we obtain\n\\begin{equation*}\n\\V{C}(\\A) \\leq (C \\cdot P) -\\sum_{x \\in C} \\nu(T_{\\mathrm{min}} , x). \n\\end{equation*}\nThe curve $C$ intersects with the non-nef locus $E_{nn}(P)$ at $x_{0}$ from the property (3). \nHence $\\nu(T_{\\mathrm{min}} , x_{0})$ is positive.\nThis implies $\\V{C}(\\A) \\leq (C \\cdot P) - \\nu(T_{\\mathrm{min}} , x_{0}) < (C \\cdot P) $.\nTherefore Lemma (B) holds. \n\nWe prove (A).\nBy the first half argument we get $\\V{C^{'}}(\\A) \\leq (C^{'} \\cdot P) $.\nTo show the converse inequality we take a K\\\"ahler current $S \\in \\A$ with analytic singularities.\nSo We can assume $S \\geq \\omega$, where $\\omega$ is a K\\\"ahler form on $X$. \nBy applying the approximation Theorem (Theorem 2.3) to a minimal singular current $T_{\\min}$ in $P$, \nWe get positive currents $T_{k}$ with analytic singularities with the following properties.\n\\\\\n\\ \\ \\ $\\mathrm{(b')}$\\ \\ \\ $T_{k} \\geq -\\e _{k} \\omega$ and $\\e_{k}$ converges to zero.\\\\\n\\ \\ \\ $\\mathrm{(c')}$\\ \\ \\ The Lelong number $\\nu(T_{k},x)$ increase to\n$\\nu(T_{\\mathrm{min}} , x)$ for every point $x \\in X$.\\\\\nFor every positive number $\\delta $, there is a $k(\\delta )$ such that \n$(1-\\delta ) T_{k(\\delta)} + \\delta S$ is a positive current. \nSince the current has an analytic singularities, \nthe inequality $$\\V{C^{'}} (\\A) \\geq \\int_{C^{'}} \\big( ((1-\\delta ) T_{k(\\delta)} + \\delta S )\\vert_{C^{'}} \\big)_{\\ac}$$ holds\nfor every $\\delta>0$ by the definition of restricted volumes. \nThe Lelong number of $T_{k}$ is zero for every point on $C$ by the property (3), (b). \nThis shows $T_{k}$ is smooth on $C$.\nSo we obtain \n\\begin{equation*}\n\\V{C^{'}} (\\A) \\geq (1-\\delta)(C^{'} \\cdot P) - \\delta\\int_{C^{'}} \\big( S \\vert_{C^{'}} \\big)_{\\ac}.\n\\end{equation*} holds for every $\\delta$. \nWhen $\\delta$ converges to zero, we get $\\V{C^{'}} (\\A) \\geq (C^{'} \\cdot P) $. \n\n\n\n\\section{Asymptotic estimates on higher cohomology}\n\n\\subsection{Kawamata-Viehweg type vanishing}\nThe purpose of this section is to prove the asymptotic estimates of higher cohomology of $\\ell D$.\nWe recall the definition. Let $k$ be an integer $k = 0,1,2,\\ldots, n-1$.\n\\begin{defi}\nA divisor $D$ is said to be \\textit{nef in codimension $k$} if the codimension of the restricted base locus $\\mathbb{B}_{-}(D)$\n is greater than $k$.\n\\end{defi}\n\\begin{rem}\n$D$ is nef in codimension $0$ if and only if the Chern class is a pseude-effective class.\n$D$ is nef in codimension $n-1$ if and only if the Chern class is a nef class.\n\\end{rem}\nTheorem \\ref{KV} is the accurization of the simplest version of Kawamata-Viehweg vanishing theorem.\nLemma \\ref{Key} is usefully worked in the proof of Theorem \\ref{KV} and Corollary \\ref{asy}.\nFirst of all we prove Lemma \\ref{Key}.\n\n\n\\begin{lemm}\\label{Key}\nAssume that $D$ is nef in codimension $k$ and $k$ is smaller than $(n-1)$.\nFix a very ample divisor $A$ on $X$.\nThen a very general hyperplane $H\\in |A|$ satisfies the following properties.\\\\\n\\ \\ \\ $\\mathrm{(i)}$\\ \\ \\ \\ $H \\cap X$ is a smooth projective variety of dimension $(n-1)$. \\\\\n\\ \\ \\ $\\mathrm{(ii)}$\\ \\ \\ \\ The restriction $D\\vert_{H}$ is nef in codimension $k$ on $H \\cap X$.\n\\end{lemm}\n\\begin{proof}\nSince $D$ is nef in codimension $k$, for an arbitrary number $\\e>0$, we can take\n a positive current $T_{\\e}$ in the class $c_{1}(D)$ with the following properties \nby the approximation theorem (Theorem 2.3).\\\\\n\\ \\ \\ \\ (1)\\ \\ \\ $T_{\\e}$ has analytic singularities. \\\\\n\\ \\ \\ \\ (2)\\ \\ \\ The codimension of the singular locus of $T_{\\e}$ is greater than $k$. \\\\\n\\ \\ \\ \\ (3)\\ \\ \\ $T_{\\e} \\geq -\\e \\omega $ holds, where $\\omega $ is a K\\\"ahler form on $X$.\n\\\\\nNow we define a countable union of closed analytic sets by \n\\begin{equation*}\n Q := \\bigcup _{\\QQ \\ni \\e >0} \\bigcup_{\\QQ\\ni c>0} E_{c}(T_{\\e}) .\n\\end{equation*}\nHere $E_{c}(T_{\\e})$ is a Lelong number upper level set, which is defined by \n$\\big\\{ x \\in X \\big| \\nu(T_{\\e}, x) \\geq c \\big\\}$.\nA theorem of \\cite{Siu74} asserts this is a closed analytic set.\n\\begin{lemm}\nFor a very general hyperplane $H\\in |A|$, $H$ does not contain analytic\n set with the codimension $(k+1)$ in $Q$ and $H\\cap X$ is smooth and irreducible.\n\\end{lemm}\n\\begin{proof}\nLet $\\{ C_{i}\\}_{i \\in I}$ be a family of analytic sets with codimension $(k+1)$ in $Q$.\nFor each $i \\in I$, we define $Q_{i}$ by $Q_{i} := \\big\\{ H \\in |A| \\ \\big|\\ C_{i} \\subseteq H \\big\\}$.\nThen $Q_{i}$ is a (proper) analytic set in $|A|$ for each $i$.\nNote $I$ is a countable set.\nEvery hyperplane $H \\in |A|-\\bigcup_{i \\in I}{Q_{i}} $ does not \ncontained an analytic set with the codimension $(k+1)$ in $Q$.\nMoreover Bertini's Theorem shows that $H\\cap X$ is smooth and irreducible for a general member \n$|A|$.\nLemma 5.4 is concluded. \n\\end{proof}\n\n\nFor a very general hyperplane $H\\in |A|$, \n$T_{\\e} \\vert_{H} \\in c_{1}(D \\vert_{H})$ and $T_{\\e} \\vert_{H} \\geq -\\e \\omega\\vert_{H} $ holds.\n$\\omega\\vert_{H}$ is a K\\\"ahler form on a smooth projective variety $X \\cap H$.\nHence following lemma concludes Lemma \\ref{Key}.\n\n\\begin{lemm}\nThe codimension in $X \\cap H$ of the singular locus of $T_{\\e}\\vert_{H}$ greater than $k$.\n\\end{lemm}\n\nLet $C$ be an analytic set with codimension $\\ell =0, 1,2,\\ldots,k$ on $X \\cap H$.\nThen $C$ is the analytic set with codimension $\\ell+1$ on $X$.\nWhen $\\ell+1$ is smaller than $k+1$, \n$\\nu (T_{\\e} , C) = 0$ for every positive number $\\e$ by the choice of $T_{\\e}$.\nWhen $\\ell+1$ equals to $k+1$, \n$\\nu (T_{\\e} , C) = 0$ by the choice of $H$.\nSince $T_{\\e}$ has analytic singularities, \nthe Lelong number is zero if and only if $T_{\\e}$ has smooth at $x \\in X$.\nThis shows that the singular locus of $T_{\\e}\\vert_{H}$ does not contain an analytic set $C$ with \ncodimension $\\ell =0, 1,2,\\ldots,k$ on $X \\cap H$.\nHence Lemma \\ref{Key} holds. \n\\end{proof}\n\n\nWe prove Theorem \\ref{KV} by using Lemma \\ref{Key}.\nThe principle of the proof is to reduct to the ordinary Kawamata-Viehweg vanishing theorem\nby using Lemma \\ref{Key} and the induction on $\\dim {X} =n$.\\\\\n\\textit{Proof of Theorem \\ref{KV}.)}\nWhen $n$ is one, bigness of $D$ implies that $D$ is ample.\nIn this case, Theorem \\ref{KV} follows from Kodaira vanishing theorem.\nHence we can assume that $n$ is greater than one.\nIf $k$ is $(n-1)$, $D$ is nef.\nThen Theorem \\ref{KV} follows from the ordinary Kawamata-Viehweg vanishing theorem.\nTherefore we can assume that $k$ is smaller than $(n-1)$.\nIt is sufficient to show $H^{p}(X, \\OX( -D))=0\\ \\ (0 \\leq p \\leq k)$ by Serre duality.\nWe obtain the following short exact sequence for a hyperplane $H \\in |A|$ with the properties in Lemma \\ref{Key}.\n\\[\n\\begin{CD} \n0 @>>> \\OX(-D-A) @>>> \\OX(-D) @>>> {\\mathcal{O}}_{H}(-D) @>>> 0\\\\\n\\end{CD}\n\\]\nWe consider the long exact sequence induced by the above exact sequence.\nBy choosing $A$ sufficiently positive, we can assume that $A+D$ is ample on $X$.\nThen Serre duality and Kodaira vanishing theorem show that $H^{\\ell}(X,\\OX(-D-A) )=0,\\ \\ (0\\leq \\ell \\leq n-1)$ .\n$D\\vert_{H} $ is nef in codimension $k$ on $X \\cap H$ by the choice of $H$.\nTherefore $H^{p}(H,\\mathcal{O}_{H}(-D) )=0\\ \\ (0\\leq p \\leq k)$ holds by the hypothesis of the induction.\nThese show that $H^{p}(X,\\mathcal{O}_{X} (-D))=0$ $\\ \\ (0\\leq p \\leq k)$.\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\\textit{Proof of Corollary \\ref{asy}.)}\nCorollary \\ref{asy} is also proved with the induction on $\\dim X =n$.\nWhen $n$ is one, pseudo-effective line bundle is nef.\nWhen the degree of $D$ is positive, Kodaira vanishing theorem shows that $H^{1}(X, \\OX (M + \\ell D)) =0$\n for a sufficietly large $\\ell$.\n When the degree of $D$ is zero, Riemann-Roch formula shows \n$\\dim H^{1}(X, \\OX (M + \\ell D)) =O(1)$.\n\nWe consider when the dimension of $X$ is greater than one.\nBy choosing a sufficietly positive line bundle $A$, we can assume that $\\OX (M+\\ell D-K_{X} +A)$\n is big for all $\\ell$, since $D$ is pseudo-effective. \nMoreover we can take a member $H$ of $|A|$ such that $D\\vert_{H}$ is nef in codimension $k$ on $X\\cap H$\nby Lemma \\ref{Key}.\nThen we obtain the following short exact sequence.\n\\[\n\\begin{CD} \n0 @>>> \\OX(M+\\ell D) @>>> \\OX(M+\\ell D +A) @>>> {\\mathcal{O}}_{H}(M+\\ell D + A) @>>> 0\\\\\n\\end{CD}\n\\]\nWe consider the long exact sequence induced by the above short exact sequence.\n\nNote that $(E+\\ell D-K_{X} +A)$ is big and nef in codimension $k$ for all $\\ell>0$ .\nSo Theorem \\ref{KV} shows $H^{q} \\big( X,\\OX(E+\\ell D +A) \\big) = 0$ for $q \\geq n-k$.\nTherefore we obtain from the long exact sequence\n\\begin{align*}\n&h^{q} \\big( X,\\OX(E+\\ell D) \\big)= h^{q-1} \\big( H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A) \\big)\\ \\ \\mathrm{for} \\ \\ q \\geq n-k +1 \\\\\n\\mathrm{and}\\ \\ \\ &h^{n-k} \\big( X,\\OX(E+\\ell D \\big) \\leq h^{n-k-1} \n\\big( H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A) \\big). \\\\\n\\end{align*}By hypothesis of induction we have \n\\begin{align*}\n&h^{q-1}(H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A))=O(\\ell^{(n-1)-(q-1)}) =O(\\ell^{n-q})\\ \\ \\ \\mathrm{for}\\ \\ q \\geq n-k +1 \\\\\n\\mathrm{and}\\ \\ \\ &h^{n-k-1}(H , \\mathcal{O}_{H}(\\OX(E+\\ell D+A))= O(\\ell^{(n-1)-(n-k-1)}) =O(\\ell^k)\n\\end{align*}\nTherefore Corollary \\ref{asy} holds.\n\n\\begin{flushright}\n$\\square$\n\\end{flushright}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDialogue state tracking (DST) modules, which aim to extract dialogue states during conversation \\cite{DBLP:journals\/pieee\/YoungGTW13}, is an important component for task-oriented dialog systems to understand users' goals and needs \\cite{DBLP:conf\/eacl\/Rojas-BarahonaG17,lei2018sequicity}. Dialogue states are sets of slots and their corresponding values. Collecting state labels can be costly \\cite{DBLP:conf\/emnlp\/BudzianowskiWTC18}, requiring experts to indicate all \\textit{(slot, value)} information for each turn in dialogues. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figs\/DST.pdf}\n\\caption{Dialogue state tracking (DST) task. U and A represent user's and system's utterances respectively. DST aims to extract dialogue states pairs (\\textcolor{orange}{slot}, {\\color{OliveGreen}{value}}), for each user's utterance. Values are usually the explicit needs expressed in the utterances.}\n\\label{DST}\n\\end{figure}\n\nTo reduce the dependency on large amounts of training data, some few-shot methods are proposed recently for low-resource DST. Most of them apply domain transfer-based methods \\cite{trade,DBLP:conf\/aaai\/LeeJ19,DBLP:conf\/aaai\/RastogiZSGK20} which rely on the assumption about the similarity among different domains. \nAnother series of approaches has tried to exploit external knowledge. \\citet{DBLP:conf\/asru\/ChenWR13} and \\citet{ddswws} consider slots and frames are similar semantic units and use the FrameNet semantic parsers to automatically induce slots. \\citet{todbert} fine-tune BERT with a task-oriented dialogue dataset and utilize it for the downstream DST task. Recently, a new paradigm, \"\\textit{Pre-train, Prompt and Predict}\" \\cite{prompt-survey}, which aims at utilizing PLM in a more effective way, has aroused the public's attention. It is supposed to be useful in few-shot scenarios as it can ``probe'' the task-related knowledge from PLM efficiently using a prompt.\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs\/train-arch.pdf}\n\\caption{Overview of the prompt-learning framework for few-shot DST. While training, prompt function $f(v)$ accepts {\\color{OliveGreen}value} candidate $v$ and get the {\\color{RoyalBlue}\\textbf{value-based prompt}}, which is concatenated to dialog history for PLM to generate the corresponding {\\color{orange}slot} $s$. Dashed arrows demonstrate the process of {\\color{RoyalBlue}\\textbf{inverse prompt}} mechanism which gets inverse prompt via the inverse prompt function $I(s)$ for the generated $s$ and aims to generate value $v'$ which is supposed to be close to the original input value $v$.}\n\\label{train-arch}\n\\end{figure*}\nIn 2021, \\citet{sup-dst-prompt} introduce prompt learning for DST task. This work aims to encode slot information into PLM via prompt and improve prediction performance when there are plenty of labeled data. However, the potential of prompt learning for low-resource DST is still under-explored. To further exploit this paradigm, we design a prompt learning framework for low-resource DST, which has value-based prompt module and inverse prompt module.\n\n\n\n\nFirst, we design \\textbf{value-based prompt functions} which use values as inputs to construct prompts. Slots are the target outputs. For few-shot scenarios, the slots that appear in the few labeled dataset may not include all possible needs a user will propose. Defining all possible slots in advance is also difficult because of the rapid application in different domains and users' continuous needs. \\%\\% Thus, we rethink about DST task and consider values as prompts to probe the corresponding slots. We design value-based prompt functions which equip the textual prompt with values. As shown in Figure \\ref{train-arch}, a prompt function is a textual template, e.g., ``\\textit{belief states: value = [v], slot = [s]}''. Given the dialogue history ``\\textit{...Plan a train to London}'', after extracting the value candidate \n\\textit{London}, the prompt becomes ``\\textit{belief states: value = London, slot = [s]}'' where \\textit{[s]} is supposed to be generated as \\textit{destination} by the PLM. Value-based prompting method can alleviate the dependency on pre-defined slot types and generate unseen slots.\n\n\nIn addition, values and slots are all semantic units that describe the users' needs during conversations. Prompting values via slots (value generation) can be seen as a dual-task of prompting slots with values (slot generation). Thus, we design \\textbf{inverse prompt} mechanism as shown in Figure \\ref{train-arch}. While training, after generating slots $s$ via value-based prompts, slots are presented to the inverse prompt function $I$. The inverse prompt process aims to generate the corresponding value $v'$ which is supposed to be close to the original input $v$. Naturally, there exists an internal correlation between these two types of prompt tasks and they can benefit each other, especially under the few shot settings. The inverse prompt mechanism can also help to self-check and restrict the output of the prompt process: if a generated slot can be used to prompt the original value, the value belongs to the slot with a larger probability. Finally, a simple but effective ensemble method is used to leverage the complementary advantages of different prompt functions while testing.\n\nThe main contributions of our work can be summarized as (1) We reformulate DST as a language modeling task for slots generation and design value-based prompt functions to probe DST-related knowledge from PLM for low-resource scenarios. The framework doesn't rely on the known ontology of target slots. (2) We propose an inverse prompt mechanism to further help the PLM understand the essence of DST with few labels and tune the generated slots. (3) Experimental results show that our model can generate unseen slots and significantly outperforms state-of-the-art few-shot approaches.\n\n\\section{Preliminary}\n\\subsection{Prompt Learning}\nPrompt learning, which aims to utilize pre-trained language models more effectively with the help of \\textit{prompt}, is a new NLP paradigm (``\\textit{Pre-train, Prompt and Predict}'') proposed recently. Usually, the original task input $x$ is used to construct a prompt that can reformulate the original task into a language modeling task. Take the emotion classification task as an example, when recognizing the emotion of a social media post, \\textit{\"I missed the bus today\"}, we may continue with a prompt \\textit{\"I felt so \\_\\_\"}, and ask the PLM to fill the blank with an emotion-bearing word. With the appropriate prompts, PLM can be pushed to generate the task-related output directly.\n\n\nGiven the \\textit{prompt function f} which maps original input x to the prompt, the goal is to learn:\n\\begin{equation}\nP\\left(y \\mid x, f(x)\\right)\n\\end{equation}\nwhere $y$ is the \\textit{answer} to be generated\/filled. In DST, $y$ can be a word in dialogue state $B$. \n\n\n\\subsection{Dialogue State Tracking}\nWe consider each conversation with $T$-turn utterances alternating between the user and system: $c=\\left\\{a_1, u_1, ...a_T, u_T\\right\\}$ where $u_t$ and $a_t$ represent the user's and system's utterance respectively. Given the dialog history $c_t$ (including current user utterance $u_t$ and the former utterances $h_t = \\left\\{a_1, u_1, ...u_{t-1}, a_t\\right\\}$), a DST model aims to extract the dialogue state (belief state) $B_t$ for $u_t$ which comprises multiple tuples of slots $s$ and their associated values $v$ ($B_t = \\left\\{(s_1, v_1),...(s_n, v_n)\\right\\}$). For example, given the dialog history $c_t$ (``\\textit{...Plan a train to London on this Tuesday}''), DST model is supposed to generate belief states $B_t = \\left\\{(destination, London), (day, this\\ Tuesday)\\right\\}$). The goal is to learn probability distribution $P$ \\cite{soloist} for $t$-th turn:\n\\begin{equation}\nP\\left(B_{t} \\mid c_t\\right)\n\\end{equation}\n\nIf $B_t$ is considered as a word sequence \\cite{simple-tod}, DST is essentially a language modeling task. Large-scale pre-trained language models (PLM) show outstanding language modeling and generation ability. Following the existing paradigm (\\textit{Pre-train and Fine-tune}), we need to fine-tune PLM with the task-related dataset. Fine-tuning with few labeled dataset may lead to over-fitting. Thus, an effective way to help PLM understand DST task in their familiar way (language modeling) and utilize the generation ability is important, inspiring the exploration of prompt learning for few-shot DST.\n\n\\section{Prompt Learning for Few-Shot DST}\n\n\n\nIn the following section, we will describe the main components of the designed prompt learning framework for few-shot DST, including \\textbf{value-based prompt} (Section \\ref{prompt function}), \\textbf{inverse prompt} (Section \\ref{inverse prompting}) and \\textbf{prompt ensemble} (Section \\ref{prompt ensemble}).\n\n\\subsection{Value-based Prompt}\\label{prompt function}\nPrevious work \\cite{DBLP:journals\/tacl\/JiangXAN20,DBLP:conf\/eacl\/SchickS21,template-ner} show that the design of prompting function $f$ is a key factor that influences the final performance. \nUsually, the function function contains a \\textit{template} which is a textual string and remains two blanks to be filled (the input and the answer). \n\n\n\nA natural idea is using slots as prompts to generate corresponding values, which is called slot-based prompt \\cite{sup-dst-prompt}. For example, given $c$ (``\\textit{...plan a train to London.}'') and slot (\\textit{destination}), the input of PLM becomes ``\\textit{...Plan a train to London. destination [y]}'' where \\textit{[y]} is supposed to be generated as \\textit{``London}''. This method relies on the known ontology of slot type. For few-shot DST, the slots that appear in the few labeled datasets may not include all possible needs. In addition, defining all possible slots are difficult as the rapid application in different new-rising domains and user's continuous need. In the real-world application, the candidate set of $s$ may be unknown and changeable.\n\nThus, we rethink about DST task and consider values as prompts to generate the corresponding slots as shown in Figure \\ref{train-arch}. Such a method doesn't require any knowledge about the target slot types and thus can generate unseen slots under the generative framework. Four intuitive templates for value-based prompts are shown in Table \\ref{prompts}. Take the first template for an example (\\textit{``belief states: value = [v], slot = [s]''}), given value candidate $v$ = \\textit{``London''}, $f(v)$ = \\textit{``belief states: value = London, slot = [s]''}. The goal is to learn the probability of slots in $t$-th turn given $c_t$ and the value $v$:\n\\begin{equation}\nP\\left(s \\mid c_t, f(v)\\right)\n\\end{equation}\nThe overall learning objective of this generation processing is maximizing the log-likelihood of slots in the training dataset $D$:\n\\begin{equation}\n\\mathcal L = \\sum_{t}^{|D|} log\\ P\\left(s_t \\mid c_t, f(v_t)\\right)\n\\label{loss_p}\n\\end{equation}\n\n\nAs a turn may contain multiple values and slots, each pair of (slot, value) constructs an instance for training and testing. While training, the values are known with annotations. Real values are utilized to construct prompts and train the slot-generation process. While testing, values are unknown. Referring to \\cite{DBLP:conf\/interspeech\/GoelPH19,latent}, candidate values are extracted from current user's utterance. We extract POS tags \\cite{DBLP:conf\/emnlp\/CuiZ19} and consider adjective and adverb as possible values as well as their previous negator (e.g., ``\\textit{not expensive}''). Named entities (e.g., name of place, time) and co-references are extracted via Stanford CoreNLP toolkit \\cite{corenlp}. Stop words and repeated candidates are filtered. Each value candidate is used to construct the value-based prompt and generate the slot. Actually, value generation can also be modeled as a summarization task and take advantage of few-shot summarization methods, which can be improved in future work but is not the focus of this paper.\n\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{cl}\n\\hline\n\\multicolumn{2}{c}{\\textbf{Prompt Functions}}\\\\\n\\hline\n$f_1$ & belief states: value = [v], slot = \\textbf{[s]} \\\\\n$f_2$ & belief states: [v] = \\textbf{[s]} \\\\\n$f_3$ & [v] is the value of \\textbf{[s]} \\\\\n$f_4$ & What is the slot type of [v] ? \\textbf{[s]} \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{prompts}\nDifferent prompt functions $f$. [v] is the input of value candidate and [s] is the slot to be generated.\n}\n\\end{table}\n\n\n\n\n\n\n\\subsection{Inverse Prompt}\\label{inverse prompting}\nValues and slots are both core semantic units in utterances that describe users' needs. Thus, prompting values with slots (value generation) can be seen as a dual-task of prompting slots with values (slot generation). Naturally, these two types of prompt tasks are supposed to hold an intrinsic correlation correlation and can benefit each other, especially in the few-shot settings.\n\n\nThus, we design \\textit{inverse prompt} as shown in Figure \\ref{train-arch}. While training, after generating the slot ($s$) via value-based prompts, $s$ is presented to inverse prompt function ($I$). The inverse prompt process aims to answer the corresponding value $v'$ which is supposed be close to the original input one $v$. We take ``\\textit{belief states: [s] = [v]}'' as the template in $I$. The loss function $\\widetilde\\mathcal{L}$ for inverse prompt mechanism is:\n\\begin{equation}\n\\widetilde\\mathcal{L} = \\sum_{t}^{|D|} log\\ P\\left(v_t \\mid c_t, I(s_t)\\right)\n\\label{loss_ip}\n\\end{equation}\nThe final loss function $\\mathcal L^*$ consists of loss functions in prompt learning $\\mathcal L$ and inverse prompt learning $\\widetilde\\mathcal{L}$:\n\\begin{equation}\n\\mathcal L^* = \\mathcal L + w * \\widetilde\\mathcal{L}\n\\label{loss}\n\\end{equation}\nwhere $w$ is a decimal in $(0,1)$ and used to adjust the influence of inverse prompt.\n\nThere are two main differences between our work and existing work using slots as prompts: (1) Inverse prompt is used as an auxiliary task for the original prompt learning. Our goal is to utilize it to help PLM understand the task and tune the output further: if a slot can be used to prompt the original value, it means there is a larger probability that the value belongs to the generated slot. (2) In our work, the slot types are not static. We generate the slots in the whole vocabulary space, making generating unseen slots is possible. \n\n\n\\subsection{Prompt Ensemble}\\label{prompt ensemble}\nIn the previous section, we described methods to generate a set of value-based prompt functions as shown in Table \\ref{prompts}. Each of these prompts may be more or less effective at eliciting knowledge from the PLM, and thus it is necessary to decide how to use these generated prompts at test time. Unfortunately, under few-shot settings, it's hard to get enough training and development set to automatically select or generate the best-performing prompt \\cite{DBLP:journals\/tacl\/JiangXAN20,DBLP:conf\/acl\/GaoFC20,ben2021pada,DBLP:conf\/emnlp\/DavisonFR19,DBLP:journals\/corr\/abs-2103-10385}. We introduce a multi-prompt learning method (\\textit{prompt ensemble}) for few-shot DST task in this section to effectively utilize different prompts.\n\n\nPrompt ensemble methods use multiple unanswered prompts for input at inference time to make predictions \\cite{prompt-survey}. It can leverage the complementary advantages of different prompts and alleviate the cost of choosing one best-performing prompt. There is relatively little work on prompt ensemble for generation tasks. A simple way for ensemble in this case it to train a separate model for each prompt and generate the output based on the vocabulary distribution learned by several models while testing. The probability of slot $s_t$ is calculated via:\n\n\n\\begin{equation}\nP\\left(s_t \\mid c_t \\right)= \\sum_{k}^{K} \\alpha_k * P\\left(s_t \\mid c_t, f_k(v_t)\\right)\n\\label{eq:ensemble}\n\\end{equation}\nwhere $f_k$ is the $k$-th prompt and $\\alpha_k$ is its weight. $K$ is the number of prompt functions.\n\n\n\\section{Experiments}\n\\subsection{Experimental Setup}\n\n\\paragraph{Datesets}\nWe evaluate our methods on MultiWOZ 2.1 \\cite{multiwoz2.1} dataset. It's a multi-domain task-oriented dialog dataset and contains 8438\/1000\/1000 dialogues for training\/validation\/testing, respectively. Following existing work~\\cite{trade}, we keep five domains (\\textit{Restaurant, Hotel, Attraction, Taxi, Train}) because the other two domains only appear in the training set. Each turn can include multiple slots. \n\n\\paragraph{Evaluation Metrics}\nWe adopt the standard metric in DST \\cite{trade}: joint goal accuracy (JGA). The metric compares the entire predicted belief states to the gold one at each dialog turn. The prediction is considered correct if and only if all the predicted states exactly match the ground truth states. Only when the values and slots are both correct, the prediction is correct. To omit the influence of the effect of value extraction, we also present the corresponding accuracy (JGA*) while the values are correctly identified for our methods.\n\n\\paragraph{Implement Details}\nWe choose SOLOIST \\cite{soloist} as our base model. SOLOIST is initialized with the 12-layer GPT-2 \\cite{gpt2} and further trained on multiple task-oriented dialog corpora (Schema \\cite{sgd} and Taskmaster \\cite{taskmaster} ) for two dialogue-related tasks (belief prediction and response generation). Specifically, the belief prediction task accepts utterance as input and generates the belief states as a word sequence (e.g., ``\\textit{Belief state: destination = London}''). Thus, we suppose that knowledge about DST may be learned via SOLOIST, and what we need to do is to find an effective way to ``probe'' the knowledge and apply it to few-shot scenarios. In addition, the moderate size of SOLOIST (117M parameters) makes fine-tuning for the task-related prompts computationally efficient. $\\alpha$ for each prompt function in Eq.\\ref{eq:ensemble} is set to the same value (1\/4). $w$ in Eq.\\ref{loss} is 0.1. Inverse prompt is used for training. While testing, value-based prompts are given to the models for target slot generation.\n\n\\paragraph{Baselines}\nWe compare our methods with several strong baselines capable of few-shot inference, which achieve SoTA on MultiWOZ 2.1 dataset. (1) \\textbf{TRADE} \\cite{trade} requires the embedding of slots as inputs and uses a soft copy mechanism to either copy the corresponding values from utterance pairs or generate them using RNN. (2) \\textbf{Self-Sup} \\cite{self-sup} adds two self-supervised objectives: preserving latent consistency and modeling conversational behavior for TRADE. (3) \\textbf{TOD-BERT} \\cite{todbert} trained BERT with several task-oriented dialogue relevant tasks: masked language modeling and response generation with large-scale corpora (100k dialogues across over 60 different domains). For DST, it learns a classifier to predict the value over the pre-defined possible value set for each known slot.\n(4) \\textbf{DSI} \\cite{latent} introduces latent variable models for DST which consider dialogue states and slots as latent variables. Although the original paper didn't focus on few-shot scenarios, we compare it here as both of us use the same ways of extracting value candidates as inputs and aim to generate slots. (5) \\textbf{SOLOIST}. It's the base model of our method. To investigate the effect of prompt learning, we fine-tune SOLOIST on the DST task which accepts dialog history and uses values as prompts directly to generate slots. \n\nIt's worthy to mention that except SOLOIST, most baselines need pre-defined slots while inference and some need the entire ontology including both slots and their all possible value sets. Although DSI doesn't need pre-defined slot types, the number of target slots should be given.\n\n\\subsection{Experiment Results}\nTo simulate the few-shot scenarios, we randomly select a limited quantity of labeled training data for training. Similar to the previous few-shot works \\cite{todbert,self-sup}, we conduct experiments on the data ratio of 1\\%, 5\\%, 10\\% and 25\\%. Specifically, to observe the performance in an extreme low-resource scenario, we also compare their performances while the training ratio is only 0.1\\% (8 dialogues and less than 60 users' utterances). \n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{l|c|c|c|c|c}\n\\hline\n& 0.1\\% & 1\\% & 5\\% & 10\\% & 25\\% \\\\\n\\hline\n& \\multicolumn{4}{c}{JGA}\\\\\n\\hline\nTRADE &1.4 & 10.4 & 27.7 & 32.6 & 38.5 \\\\\nSelf-Sup & 15.4 & 19.5 & 30.6 & 34.5 & 40.2 \\\\\nTOD-BERT & 6.3 & 9.9 & 28.6 & 37.0 & 44.3 \\\\\nDSI & 1.0 & 1.1 & 1.3 & 1.3 & 2.6 \\\\\nSOLOIST & 26.8 & 36.4 & 37.1 & 37.9 & 39.4 \\\\\n\\textbf{Ours} & \\textbf{36.1} & \\textbf{44.3} & \\textbf{44.7}& \\textbf{44.7} & \\textbf{45.4} \\\\\n\\hline\n& \\multicolumn{4}{c}{JGA*}\\\\\n\\hline\n\\textbf{Ours} & \\textbf{75.4} & \\textbf{92.4} & \\textbf{94.7}& \\textbf{94.7} & \\textbf{95.1} \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{fewshot}\nComparison of different models under the different ratios of training data (0.1\\% corresponds to less than 60 users' utterances).\n}\n\\end{table}\n\nFew-shot performances are shown in Table \\ref{fewshot}. We can observe that our model outperforms existing works by a large margin. From 0.1\\% to 10\\%, our model increases JGA by more than 7\\%, indicating the superiority of our model in low-resource scenarios. For 25\\%, the improvement is relatively small (1\\%). It may be owing to that using 1\\% data almost performs the same as using 25\\% data (1.2\\% difference) for our model. Using a few labeled data can help the PLM understand DST task. \n\nWe then analyze the results of value extracting and find it can only achieve an accuracy less than 61\\%. Table \\ref{errors} lists the three main types of wrong extraction: missed, partially extracted and totally wrong. However, our model still outperforms others as JGA in Table \\ref{fewshot} shows. It attributes to the high accuracy of slot generation while turn-level values are correctly extracted. When the ratio of training data is 1\\%, 92.4\\% turn-level states are correctly generated given correct value candidates as JGA* shown in Table \\ref{fewshot}.\nThe high values of JGA* indicate the potential of performance along with the improvement of value extraction in future work. \n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{p{0.95\\columnwidth}}\n\\hline\n\\textbf{Dialogue history:} ...[user]i need a taxi to pick me up at regency gallery and take me to {\\color{OliveGreen}{Don Pasquale Pizzeria}}.\\\\\n\\textbf{Gold value:} {\\color{OliveGreen}{Don Pasquale Pizzeria}} \\\\\n\\textbf{Our value:} (\\textbf{Missed})\\\\\n\\hline\n\\textbf{Dialogue history:} ...[user] i am looking for {\\color{OliveGreen}city centre north bed and breakfast}\\\\\n\\textbf{Gold value:} {\\color{OliveGreen}city centre north bed and breakfast}\\\\\n\\textbf{Our value:} {\\color{OliveGreen} north} (\\textbf{Partially extracted})\\\\\n\\hline\n\\textbf{Dialogue history:} ...[system] any preference on star rating [user] no , that s not important to me .\\\\\n\\textbf{Gold value:} {\\color{OliveGreen}don't care}\\\\\n\\textbf{Our value:} {\\color{OliveGreen}not important} (\\textbf{Totally wrong})\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{errors}\nError analyses for extracting {\\color{OliveGreen}value candidates}. Three main types of errors are listed (missed, partially extracted and totally wrong).\n}\n\\end{table}\n\n\n\\subsection{Unseen Slot Generation}\nFor MultiWOZ2.1, we present the slots of each domain in Table \\ref{slots}. We find that some domains share some slots with other domains. For example, all slots of \\textit{Attraction} can be found in \\textit{Hotel}. On the contrary, some domains hold some slots that are not seen in other domain. For \\textit{Hotel}, it has four unseen slots: \\textit{parking}, \\textit{book stay}, \\textit{stars} and \\textit{internet}. \\textit{Restaurant} has two unseen slots (\\textit{food} and \\textit{time}). Here, we consider ``unseen slots'' as both ``unseen'' in the labeled training data and ``unseen'' in the slot names of the source domains.\n\nTo observe the extension and generation ability for unseen slots, we design two zero-shot experiments: leave \\textit{Hotel} or \\textit{Restaurant} as held-out-domain respectively, and train on other four domains. We present slots accuracy which evaluates the slot-level accuracy of correctly generated slots while values are correctly extracted. From the results in Figure \\ref{hotel}, we find that: \n\n(1) For seen slots that have the same names as that of source domains, our model can generate them with high accuracy. For example, \\textit{area} in \\textit{Hotel} domain is a common slot for other two source domains (\\textit{Attraction} and \\textit{Restaurant}), which can be generated with 96.92\\% accuracy. It indicates the good transfer ability across domains.\n\n(2) For some unseen slots (\\textit{book stay} and \\textit{stars} in \\textit{Hotel} of Figure \\ref{hotel}, \\textit{book time} and \\textit{food} in \\textit{Restaurant} of Figure \\ref{restaurant}), our model can generate them with more than 87\\% accuracy. \nFor example, given the dialogue history ``\\textit{...yes, please book it for 1 person and for 5 nights starting Friday.}'' The model successfully generates ``\\textit{book stay}'' for ``\\textit{5}'' even it has never seen the instances of book stay while training. Without known slot types, our model can infer the hidden semantic from the value and contexts, which is supposed to be the slot.\n\n(3) Figure \\ref{hotel} misses two unseen slots (\\textit{parking} and \\textit{internet}). For these two slots, the value extraction module fails to extract its gold value (``\\textit{yes}'') from utterance. The accuracy of value extraction is 0. So, the slot accuracy is also 0. However, we extract the adjective ``\\textit{free}'' from the user utterance as shown in Table \\ref{a-example}. Then PLM model can infer that the word ``\\textit{free}'' is the property of ``\\textit{internet}''. Actually, we think ``\\textit{internet: free}'' can also clearly describe a user's need. The case indicates a possible drawback of the existing annotation system. \n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{p{0.95\\columnwidth}}\n\\hline\nslots\\\\\n\\hline\n$area^{123}$, $arrive\\ by^{45}$, $day^{235}$, $departure^{45}$, $destination^{45}$, \n$food^3$, $internet^2$, $leave^{45}$, $name^{123}$, $people^{235}$,\n$parking^2$, $price^{23}$, $stars^2$, $stay^2$, $time^3$, $type^{12}$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{slots}\nAll slots in MultiWOZ2.1. The upper script on slot indicates the domain it belongs to (1: \\textit{Attraction}, 2: \\textit{Hotel}, 3: \\textit{Restaurant}, 4: \\textit{Taxi}, 5: \\textit{Train}).\n}\n\\end{table}\n\n\\begin{figure*}\n\t\\subfigure[\\textit{Hotel}]{\n\t\t\\includegraphics[width=0.46\\textwidth]{figs\/hotel.pdf}\n\t\\label{hotel}\n\t}\n\t\\subfigure[\\textit{Restaurant}]{\n\t\t\\includegraphics[width=0.47\\textwidth]{figs\/restaurant.pdf}\n\t\\label{restaurant}\n\t}\n\t\\caption{Slot accuracy of each slot in \\textit{Hotel} and \\textit{Restaurant} domain under zero-shot settings. X-axis is the slot accuracy and y-axis is the slot. Pink bars mark {\\color{CarnationPink}unseen slots}.}\n\t\\label{slot-acc}\n\\end{figure*}\n\n\\begin{table}[t]\n\\small\n\\centering\n\\begin{tabular}{p{0.95\\columnwidth}}\n\\hline\n\\textbf{Dialogue history:} ... [system] could you please give me any preferences for internet and parking ? [user] I would like it to be a guesthouse that has {\\color{OliveGreen}free} WiFi. \\\\\n\\textbf{Gold states:} {\\color{OliveGreen}yes}, \\textcolor{orange}{internet} \\\\\n\\textbf{Our states:} {\\color{OliveGreen}free}, \\textcolor{orange}{internet}\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{a-example}\n A test instance that has a wrong value candidate (``{\\color{OliveGreen}free}'') and a correct generated slot (``\\textcolor{orange}{internet}''). However, we think ({\\color{OliveGreen}free}, \\textcolor{orange}{internet}) can also describe the user's need here.\n}\n\\end{table}\n\n\\subsection{Ablation Studies}\nWe further observe the performances of different components including different value-based prompt functions, inverse prompt mechanism and prompt ensemble. We first train separate models with each value-based prompt (``Orig'' for $f_1, ...f_4$), then add inverse prompt mechanism for each of them (``+Inv''). Finally, we apply prompt ensemble (``En'') for the trained models with and without inverse prompt respectively. Experiments with 0.1\\% training data are shown in Table \\ref{ablstudy}.\n\nWe observe that:\n(1) The first four numerals in the first row show the original performance with different prompt functions (no inverse prompt mechanism and prompt ensemble). Among the four prompts, $f_2$ performs best without inverse prompt which may attribute to the similar format of $f_2$ compared with the output sequences in a pre-training task of SOLOIST (Considering dialogue history as inputs and generate dialogue states in the format as ``\\textit{belief states: [s1] = [v1], [s2] = [v2]}''). It's also the reason why inverse prompt doesn't improve its performance much as the prompt (``\\textit{belief states: [v] = [s]}'') and inverse prompt (``\\textit{belief states: [s] = [v]}'') are too similar to learn complementary knowledge.\n(2) Inverse prompt brings improvement for all prompt functions, especially for the ``weakest'' prompt function $f_1$.\n(3) The prompt ensemble enables further improvement for both models with and without inverse prompt. Under few-shot settings, prompt ensemble is a simple but efficient way of utilizing different prompt functions.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{c|c|c|c|c||c}\n\\hline\n& $f_1$ & $f_2$ & $f_3$ & $f_4$ & $En$ \\\\\n\\hline\nOrig & 24.3 & 30.9 & 25.7 & 27.7 & 31.9 \\\\\n+ Inv & 32.4 & 31.4 & 29.6 & 33.6 & \\textbf{36.1} \\\\ \n\\hline\n\\end{tabular}\n\\caption{\\label{ablstudy}\nJGA results for our models trained with 0.1\\% data. ``+Inv'' means adding inverse prompt mechanism for the original models (``orig'') given different prompt functions (from $f_1$ to $f_4$). ``En'' means the ensemble of models trained on different prompt functions with and without inverse prompt.\n}\n\\end{table}\n\n\\subsection{Analyses about the Weight of Inverse Prompt}\nWe conduct experiments to observe the influence of weight $w$ in Eq.\\ref{loss}. $w$ is set to {0, 0.1, 0.3, 0.5}. Experiments using 0.1\\% training data and different value-based prompts are shown in Figure \\ref{weight}. We find that the JGA performance always increases with the value of $w$ first and then begins to decrease. It means that the inverse prompt is actually an auxiliary task and can provide useful knowledge when the weight is relatively small. All experiments for the four prompts perform best when the $w$ is 0.1. So we set it to 0.1 in the former experiments. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{figs\/weight.pdf}\n\\caption{The influence of weight $w$ for inverse prompt using different prompt functions $f$. Experiments with $w=0.1$ always perform best for all prompt functions.}\n\\label{weight}\n\\end{figure}\n\n\\section{Related Work}\n\n\\subsection{Few-Shot Dialogue State Tracking}\n\nSome few-shot methods used data augmentation to get more labeled data for training. \\citet{DBLP:conf\/acl\/CampagnaFML20} and \\citet{HouCCCL21} propose to synthesize dialogues for a new domain using the small number of domain templates derived from observing a small dataset and the ontology of the domain. These methods depend on the ontology about slots of the target domain.\n\nMost of the existing work focuses on transferring from other resource-rich DST domains. \\citet{DBLP:conf\/aaai\/LeeJ19} and \\citet{DBLP:conf\/aaai\/RastogiZSGK20} utilize the slot description for transferring reusable concepts across domains. \\citet{todbert} learn similarity functions between slots and values, and transfer them into unseen domains. \\citet{d-reptile} introduces meta-learning and uses source domains to meta-learn the parameters of the model used to initialize the fine-tuning process of the target domain. One constraint of such methods is that they rely on domain similarity for transfer, and therefore cannot be applied to general domains.\n\nAnother thread of approaches tries to exploit external knowledge. \\citet{DBLP:conf\/asru\/ChenWR13} and \\citet{ddswws} utilize FrameNet-style \\cite{framenet} semantic frames and named entity recognition (NER) as the weak supervision for slot candidates. \\citet{rc2},\\citet{rc1}, \\citet{zero-dst-qa} and \\citet{transferQA} reformulate DST into a Reading Comprehension (RC) task and make use of the abundant RC data and frameworks to overcome the data scarcity issue in the DST task. \\citet{self-sup} investigate two self-supervised objectives: preserving latent consistency and modeling conversational behavior. However, they have limited performance owing to the limited common knowledge. \n\n\n\n\n\\subsection{Prompt Learning}\nWith the rapid development of large-scale pre-trained language models (PLM), a new paradigm arise public's attention: ``\\textit{pre-train, prompt, and predict} \\cite{prompt-survey}''. Instead of adapting PLM to downstream tasks via objective engineering, prompt learning reformulates downstream tasks to look more like those solved during the original PLM training with the help of a textual prompt. GPT-3 model \\cite{gpt3} achieves remarkable few-shot performance solely by leveraging a few task demonstrations as input context (e.g., \\textit{``Translate English into French''}) and a natural-language prompt (e.g., \\textit{``cheese ==> ''}). However, training such a huge model (175B parameters) is difficult. A more usual prompt learning method is ``prompt-based fine-tune'': utilize a moderately-sized PLM for which fine-tuning is computationally efficient and fine-tune it with the task-related prompts. It shows good performance in many few-shot scenarios. \\citet{DBLP:conf\/acl\/GaoFC20} use RoBERT-large and design automatic prompt generation for text classification. \\citet{DBLP:conf\/acl\/LiL20} add continuous task-specific vector as prompt to each transformer layer and achieve improvements in low-resource text summarization. For DST task, \\citet{sup-dst-prompt} use slots as prompt directly and generate the corresponding values, which needs a lot of labeled training data for fine-tuning PLM. For few-shot DST, the prompt learning-based methods are still under-explored.\n\n\n\\section{Conclusion}\nFor the lack of labeled data in practical DST tasks, we design a prompt learning framework, which consists of two main components (value-based prompt and inverse prompt mechanism). Our model can effectively probe DST-related knowledge from pre-trained language models and utilize it for DST task. Experiments show that our model outperforms existing state-of-the-art methods under different levels of resources. It achieves an improvement of 7.3\\% joint goal accuracy under the extreme low-resource settings (only 0.1\\% training data). In addition, this framework doesn't rely on the known ontology of slot types. With extensive experiments, we find that it can generate slots that are not seen in source domains and are not pre-defined as well with high probabilities. In the future, we'll focus on improving the performance of extracting value candidates.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Acknowledgement}\nWe thank M. V. Berry and J. H. Eberly for discussions, and\nacknowledge funding by Projects No. GIU07\/40, No. FIS2009-12773-C02-01,\nNo. NSFC 60806041, No. 08QA14030, No. 2007CG52, No. S30105,\nNo. ANR-09-BLAN-0134-01, and Juan de la Cierva Program.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{s_intro}\n\nLet $\\mathcal{M}$ be a compact surface homeomorphic to $\\mathbb{S}^2$, embedded in\n$\\mathbb{R}^3.$ For $\\kappa, h>0$ and $\\mathbf{A}$ a vector field on\n$\\mathcal{M},$ we consider the Ginzburg-Landau functional\n$\\mathcal{G}_{\\mathcal{M},\\kappa}:H^1(\\mathcal{M};\\mathbb{C})\\to\\mathbb{R}_{+},$\n\\begin{equation}\\label{GLS}\n\\mathcal{G}_{\\mathcal{M},\\kappa}(\\psi)=\\int_\\mathcal{M}\\(\\abs{\\nabla_\\mathcal{M}-ih\\mathbf{A}\\psi}^2+\\frac{\\kappa^2}{2}\\(\\abs{\\psi}^2-1\\)^2\\)d\\mathcal{H}_{\\mathcal{M}}^2(x).\n\\end{equation}\nThe functional $\\mathcal{G}_{\\mathcal{M},\\kappa}$ arises as the $\\Gamma$-limit (see \\cite{CS}) of the full 3d Ginzburg-Landau energy\n\n\\begin{equation}\\label{GL3D}\n\\begin{split}\nG_{\\varepsilon,\\kappa}(\\psi,A)& =\\frac{1}{\\varepsilon}\\Bigg[\\int_{\\Omega_\\varepsilon}\\(\\abs{(\\nabla-iA)\\psi}^2+\\frac{\\kappa^2}{2}\\(\\abs{\\psi}^2-1\\)^2\\)dx \\\\\n&\\quad +\\int_{{\\mathbb R}^3}\\abs{\\nabla\\times A-\\mathbf{H}_{ext}}^2 dx\\Bigg].\n\\end{split}\n\\end{equation}\nwhere for all $\\varepsilon>0$ sufficiently small, $\\Omega_\\varepsilon$ corresponds to a uniform tubular neighborhood of $\\mathcal{M}.$\nIn \\eqref{GL3D} $\\mathbf{H}_{ext}$ is the external magnetic field. As $\\varepsilon \\to 0 ,$ the field completely penetrates the sample which then implies that in the $\\Gamma$-limit $A$ is prescribed to be equal to $\\mathbf{A},$ the tangential component of a divergence free vector field $\\mathbf{A}^e$ such that $\\nabla\\times h \\mathbf{A}^e=\\mathbf{H}_{ext}.$\n\nA central question in Ginzburg-Landau theory is the determination of the so-called {\\it critical fields}.\nThe first critical field corresponds to the appearance of zeros of $\\psi$ carrying non-trivial degree -- called vortices in this context -- in minimizers of the energy.\n\n\nThe analysis in \\cite{CS} includes the computation of the first critical field of a thin shell of a surface of revolution subject to a constant vertical field which turns out to be surprisingly simple and depending only on an intrinsic quantity, in the $ \\kappa\\to\\infty$ limit:\n$$\nH_{c_1}\\sim\\(\\frac{4\\pi}{\\mbox{Area of }\\mathcal{M}}\\)\\ln \\kappa.\n$$ \nThis result is extended in \\cite{C}, to general surfaces and magnetic fields. For a fixed field $\\mathbf{H}^e$, an external magnetic field of the form $\\mathbf{H}_{ext}=h( \\kappa)\\mathbf{H}^e=h( \\kappa)\\nabla\\times \\mathbf{A}^e$ is considered. Then the first critical field is\n$$\nH_{c_1}\\sim\\frac{1}{\\max_\\mathcal{M} *F-\\min_\\mathcal{M} *F}\\ln \\kappa,\n$$\nwhere $d^{*}F=*d*F=\\mathbf{A}$ and $*$ denotes the Hodge star-operator.\nIn fact, the study shows also that, somewhat remarkably, not all fields $\\mathbf{H}^e$ give rise to a first critical field. This phenomenon is related to the geometry and relative location of $\\mathcal{M}$ with respect to $\\mathbf{H}^e.$ For $\\mathbf{H}^e$ that yield a finite $H_{c_1},$ the topological obstruction imposed by $\\mathcal{M}$ implying that the total degree of $\\frac{\\psi}{\\abs{\\psi}}$ is zero is used in \\cite {C} to show that there is an even number of vortices in minimizers of $\\mathcal{G}_{\\mathcal{M},\\kappa},$ half with positive degree, half with negative degree concentrating respectively on the set where $*F$ achieves its minimum and maximum. The optimal number $2n$ and location of vortices and anti-vortices in $\\mathcal{M}$ is established in \\cite{C} for values of $h(k)$ slightly above $H_{c_1}$ and in addition it is shown that if the minimum and maximum of $*F$ is attained at finitely many points then the two sets of vortices minimize, independently, a renormalized energy.\n\n\nThe results in \\cite{C} and \\cite{CS} cover only a moderate regime; in these works the intensity of the applied field is $H_{c_1} +\\mathcal{O}(\\ln\\ln \\kappa )$ and thus the number of vortices remains bounded as $\\kappa$ goes to infinity.\n\nOnce the value of $h$ becomes much larger than $H_{c_1},$ that is there is a constant $C>0$ such that $h-H_{c_1}\\geq C\\ln\\kappa,$ then the number of vortices in minimizers diverges as $\\kappa\\to\\infty$. For even larger $h$, superconductivity persists only in a narrow region in the sample.\n\n\nIn the case of an infinite cylinder whose cross section is a domain $\\Omega\\subseteq{\\mathbb R}^2$ and for constant applied fields parallel to the axis of the cylinder a reduction to a two-dimensional problem is possible.\nIn this case it is known that as the intensity increases superconductivity is lost in the bulk and only a thin superconductivity region near $\\partial\\Omega$ persists (see Chapter 7 in \\cite{SS2}).\nFor much higher values still, superconductivity is completely lost: this value is known as $H_{c_3}$ and is estimated by a delicate spectral analysis of the magnetic Laplacian operator as in the monograph \\cite{fournaishelffer10}.\n\nIn our setting, corresponding to the above functional $\\mathcal{G}_{\\mathcal{M},\\kappa}$ \\eqref{GLS} on the compact surface $\\mathcal M$, there is no boundary, so what happens to the superconductivity region is not obvious. Another crucial difference lies in the behaviour of the (normalized) magnetic field $H$ induced on $\\mathcal M$, which is the normal component of $\\mathbf H^e$, or equivalently $H\\, d\\mathcal H^2_{\\mathcal M}=d\\mathbf A$ (viewing $\\mathbf A$ as a 1-form). Namely, in our case, $H$ vanishes and changes sign. The spectral analysis in \\cite{montgomery95} therefore suggests that superconductivity should persist near the set $\\lbrace H=0\\rbrace$, where the external magnetic field is tangent to the surface $\\mathcal M$. In \\cite{pankwek02} the authors study the case of a vanishing magnetic field in the infinite cylinder model, and observe indeed nucleation of superconductivity near the zero locus of the magnetic field, for very high values of the applied field (near the putative $H_{c_3}$) under the condition that the gradient of the magnetic field does not vanish on its zero locus. The problem of the determination of the upper critical field for vanishing fields remains largely open otherwise. Here, we are concerned with much lower values of the applied field: a main motivation of this work is to understand the transition from the vortexless to normal state regimes.\n\nAnother interesting difference is the fact that in the infinite cylinder model only positive vortices exist and so the location and growth of the vortex region is always ruled by the competing effects of mutual repulsion, and confinement provided by the external field. In the present setting, this is no longer the case. Vortices of positive and negative degree must coexist and so repulsion and attraction are common features of the relative placement of vortices in $\\mathcal{M}$, this without taking into account the external field.\n\n In this way, the shrinking of the superconductivity region is a multifaceted phenomenon. Moreover, the problems mentioned in the characterization of this region are present even in the most emblematic case of a constant external field $\\mathbf H^e$: the region of persistence of superconductivity does not only depend on the field and on the topology of $\\mathcal{M},$ but also on extrinsic geometric properties of the surface; the relative position of $\\mathcal{M}$ with respect to $\\mathbf H^e$ affects $H$ and therefore the zero locus of the induced field. \n\n\nIn the present work we address the question of identifying the region where superconductivity persists in the $\\kappa\\to\\infty$ limit, when \n$$\\frac{H_{c1}}{h}$$ is small; we show that as this quantity gets small superconductivity persists in a small neighborhood of the place where the applied field is tangential to the sample, provided the field satisfies a generic non-degeneracy condition(see \\eqref{Hnondegen} below). \nAnother thrust of this work is aimed at uncovering some new intermediate regimes only present in this setting, when the normal component of the external field changes sign multiples times. In the model problem of a surface of revolution and constant vertical field, we identify several structural transitions undergone by the superconductivity region. Furthermore, we observe a new phenomenon which we refer to as {\\it freezing of the boundary}, where a component of the vortex region stops growing even after increasing the intensity of the external field. This phenomenon holds in great generality (not only in the surface of revolution case), as is shown at the end of section \\ref{s_interm}.\n\n\n\nTo carry out our analysis we start by using a reduction to a mean field model, first derived rigorously in \\cite{SS1}. More precisely, if we write a critical point $\\psi$ of $\\mathcal{G}_{\\mathcal{M},\\kappa}$ in polar form $\\psi=\\rho e^{i\\phi},$ variations of the phase yield $d(\\rho^2(d\\phi-hd^{*}F))=0,$ and because $H_{dr}^1(\\mathcal{M})=0$ this implies there is a $V$ such that $*dV=\\rho^2(d\\phi-hd^{*}F).$\nTaking $V=hW$, the function $W$ is expected to minimize\n\\begin{equation}\\label{mf}\n\\int_\\mathcal{M}\\abs{\\nabla_\\mathcal{M} W}^2 d\\mathcal{H}_\\mathcal{M}^2\n+\\frac{\\ln \\kappa}{h}\\int_\\mathcal{M}\\abs{-\\Delta_\\mathcal{M} W+\\Delta_\\mathcal{M} *F}d\\mathcal{H}_\\mathcal{M}^2.\n\\end{equation}\n\nThe details of this mean field reduction can be found in \\cite{SS1} in the case of a positive external field applied in a bounded planar domain. However, the analysis in \\cite{SS1} does not handle the additional restriction of total zero mass which affects the construction of an upper bound in this setting. The steps needed to extend the proof to the present case are included in Appendix~\\ref{a_meanfield}. \n\n\nThe measure $-\\Delta_{\\mathcal{M}} V +\\Delta_{\\mathcal{M}} *F$ can be interpreted as the normalized measure generated by the vortices. \nOn the other hand,\nwe observe that \n\\begin{equation*}\n\\Delta_{\\mathcal{M}} *F \\, d\\mathcal H^2_{\\mathcal M} = d*d*F =d\\mathbf A= Hd\\mathcal H^2_{\\mathcal M},\n\\end{equation*}\nwhere the function $H$ is the normal component of the external magnetic field $\\mathbf H^e$ relative to $\\mathcal{M}$. In what follows we refer to $H$ simply as \\textit{the} magnetic field, and we assume that $H\\in C^1(\\mathcal M)$. Moreover, we drop the explicit dependence on $\\mathcal{M}$ in expressions like $\\Delta_{\\mathcal M}$, $\\nabla_{\\mathcal M}$. \n\n\nBefore we state our main result we make the following assumption: there exists $\\beta>0$ such that\n\\begin{equation} \\label{defbeta}\\lim_{\\kappa\\to \\infty }\\frac{\\ln \\kappa}{h}=\\beta.\\end{equation}\n Once the connection to the mean field problem \\eqref{mf} is established we proceed to locate very precisely the region of persistence of superconductivity, that is, the region $SC_\\beta$ where the vorticity measure $-\\Delta V + H$ vanishes. We find that this region corresponds to a $\\beta^{\\frac{1}{3}}$ neighborhood of the set where $H$ vanishes, in the $\\beta\\to 0$ limit. More precisely,\n\n\\begin{theorem}\\label{simplifiedmain} Under the nondegeneracy assumption that $\\nabla H$ is nowhere vanishing on $\\{H=0\\}, $ there exists $C>0$ independent of $\\beta$ such that the superconductivity region $SC_\\beta$ is contained in $\\{x\\in \\mathcal{M}: d(x, \\{H=0\\})\\beta_1^* \\geq \\beta_2^* >0$ such that\n\\begin{itemize}\n\\item for $\\beta\\in (\\beta^*_1,\\beta_c)$, $SC_\\beta$ has one connected component,\n\\item for $\\beta\\in (\\beta^*_2,\\beta^*_1)$, $SC_\\beta$ has two connected components,\n\\item for $\\beta\\in (0,\\beta^*_2)$, $SC_\\beta$ has three connected components.\n\\end{itemize}\nMoreover, for $\\beta\\in(\\beta^*_2,\\beta^*_1)$, one connected component of $SC_\\beta$ remains constant.\n\\end{prop}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{subfigure}{.32\\textwidth}\n\\includegraphics[width=\\textwidth]{reg1.pdf}\n\\caption{$\\beta\\in (\\beta_1^*,\\beta_c)$}\n\\label{figreg1}\n\\end{subfigure}\n\\hspace{.05\\textwidth}\n\\begin{subfigure}{.24\\textwidth}\n\\includegraphics[width=\\textwidth]{reg2.pdf}\n\\caption{$\\beta\\in (\\beta_2^*,\\beta_1^*)$}\n\\label{figreg2}\n\\end{subfigure}\n\\hspace{.05\\textwidth}\n\\begin{subfigure}{.24\\textwidth}\n\\includegraphics[width=\\textwidth]{reg3.pdf}\n\\caption{$\\beta\\in (0,\\beta_2^*)$}\n\\label{figreg3}\n\\end{subfigure}\n\\caption{The region $SC_\\beta$ in the three regimes of Proposition~\\ref{loose1d}}\n\\label{figreg}\n\\end{center}\n\\end{figure}\n\nThe most striking part of Proposition~\\ref{loose1d} is the appearance of an intermediate regime in which one connected component of $SC_\\beta$ remains constant: one part of the free boundary is \\textit{frozen}. In Section~\\ref{ss_freez} we identify the features responsible for such `freezing' of the boundary and prove a similar `freezing property' in a general (non-symmetric) setting (see Proposition~\\ref{prop_freez}).\n\n\nAn other interesting outcome of the precise version of Proposition~\\ref{loose1d} (Proposition~\\ref{prop_regimes1d} in Section~\\ref{ss_symcase}) are the expressions of the critical values $\\beta^*_1$ and $\\beta^*_2$, in terms of integral quantities involving $\\mathbf A$ and the parametrization of $\\mathcal M$. Transfering these conditions to a general non-symmetric setting seems far from obvious and constitutes an interesting challenge. \n\n\nThe plan of the paper is as follows.\nIn the next section we collect some basic properties of solutions to an obstacle problem that serves as the starting point in our analysis. In section \\ref{s_smallbeta} we identify the thin region of superconductivity when $\\beta$ is small. In section \\ref{s_interm} we turn to the symmetric situation and identify in Proposition~\\ref{prop_regimes1d} the further transitions as $\\beta$ decreases to zero from $\\beta_c=\\max(*F)-\\min(*F).$ We also prove the `freezing of the boundary' property at the end of section \\ref{s_interm}. \n\n\\subsubsection*{Acknowledgements}\n\nA.C. is\nsupported by the Fields postdoctoral fellowship. He wishes to thank\nRobert L. Jerrard for his incredible support. X.L. is supported by a Labex Milyon doctoral mobility scholarship for his stay at McMaster University. He wishes to thank his Ph.D. advisor Petru Mironescu and his McMaster supervisors Stanley Alama and Lia Bronsard for their great support.\n\n\\section{The obstacle problem}\\label{s_obstacle}\nThis preamble is devoted to the derivation of the obstacle problem dual to the mean field approximation. We also prove some basic results we will need later on. We think it is worthwhile recording these properties because in our setting, even in the by now classical application of the duality theorem which allows for the obstacle problem formulation, there is an inherent degeneracy we have to account for which is not present in other similar results in the literature. \n\nIn the first part of this section we show that -- as in \\cite[Chapter~7]{SS2} -- the minimizer of \n\\begin{equation}\\label{Ebeta}\nE_\\beta (V)=\\int_{\\mathcal M} \\abs{\\nabla V}^2 +\\beta \n \\int_{\\mathcal M} \\abs{ -\\Delta V + H} \n\\end{equation}\n is the solution of an obstacle problem, and then we study general properties of the contact set. There are two main differences with the obstacle problem arising in \\cite[Chapter~7]{SS2}.\n \\begin{itemize}\n \\item In our case there are no boundary conditions and the minimizer is well-defined only up to a constant. We need to deal with this degeneracy.\n \\item While in \\cite[Chapter~7]{SS2} the obstacle problem is one-sided, we have to consider a two-sided obstacle problem. This is due to the fact that, in our case, the magnetic field $H$ changes sign.\n \\end{itemize}\n \nThe functional $E_\\beta$ is, under assumption \\eqref{defbeta}, the limit of the sequence of energies considered in \\eqref{mf}. The link between $E_\\beta$ and the superconductivity region is, as mentioned in the introduction, proved in appendix A. \n \n\n\\subsection{Derivation of the obstacle problem}\\label{ss_derivobstacle}\n\n\\begin{prop}\\label{prop_obstacle}\nLet $\\beta>0$. A function $V_0\\in H^1(\\mathcal M)$ minimizes $E_\\beta$ \\eqref{Ebeta} if and only if $V_0$ minimizes\n\\begin{equation}\\label{F(V)}\n\\mathcal F (V) = \\int_{\\mathcal M}\\!\\left(\\abs{\\nabla V}^2 + 2 HV\\right)\n\\end{equation}\namong all $V\\in H^1(\\mathcal M)$ such that $(\\esssup V-\\essinf V)\\leq \\beta$.\n\\end{prop}\n\n\\begin{remark}\\label{rem_obstacle}\nSince the functional $\\mathcal F(V)$ is translation invariant, $V_0$ coincides, up to a constant, with any minimizer of the two-sided obstacle problem\n\\begin{equation*}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\!\\left(\\abs{\\nabla V}^2 + 2 HV\\right) \\colon V\\in H^1(\\mathcal M),\\: \\abs{V}\\leq\\beta \/2 \\right\\rbrace.\n\\end{equation*}\nMoreover, recalling that $H=\\Delta *F$, this obstacle problem can also be rephrased as\n\\begin{equation}\\label{obstacle*F}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\!\\abs{\\nabla(V-*F)}^2 \\colon V\\in H^1(\\mathcal M),\\: \\abs{V}\n \\leq \\beta\/2 \\right\\rbrace.\n\\end{equation}\nThe fact that minimizers coincide only up to a constant does not matter, since the physically relevant object is the vorticity measure $-\\Delta V +H$. Moreover, it is easy to check that, if the obstacle problem \\eqref{obstacle*F} admits a solution $V$ that `touches' the obstacles, i.e. satisfies $\\max V -\\min V=\\beta$, then this solution is unique because any other solution differs from it by a constant, which has to be zero. On the other hand, a solution satisfying $\\max V-\\min V <\\beta$ would have to be $V=*F+\\alpha$ for some constant $\\alpha$. Therefore, for $\\beta\\leq \\max *F-\\min *F$ the solution is unique.\n\\end{remark}\n\nThe proof of Proposition~\\ref{prop_obstacle} relies on the following classical result of convex analysis (easily deduced from \\cite{rockafellar66} or \\cite[Theorem~1.12]{B}). \n\n\\begin{lemma}\\label{lem_dual}\nLet $\\mathcal H$ be a Hilbert space and $\\varphi: \\mathcal H\\to{\\mathbb R}\\cup\\{+\\infty\\}$ be a convex lower semi-continuous function.\nThen the minimizers of the problems \n\\begin{equation*}\\min_{x\\in \\mathcal H}\\(\\frac{1}{2}\\norm{x}^2_{\\mathcal H}+\\varphi(x)\\)\\quad \\mbox{ and } \\quad\n\\min_{y\\in \\mathcal H}\\(\\frac{1}{2}\\norm{y}^2_{\\mathcal H}+\\varphi^{*}(-y)\\)\n\\end{equation*}\ncoincide, where $\\varphi^{*}$ denotes the Fenchel conjugate of $\\varphi$, \n\\begin{equation*}\n\\varphi^{*}(y):=\\sup_{z\\in \\mathcal H}\\langle y,z\\rangle_{\\mathcal H}-\\varphi(z).\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_obstacle}:]\nWe apply Lemma~\\ref{lem_dual} in the Hilbert space\n\\begin{equation*}\n\\mathcal H:=\\dot H^1(\\mathcal M)=\\left\\lbrace V\\in H^1(\\mathcal M)\\colon \\int_{\\mathcal M} V =0\\right\\rbrace,\n\\end{equation*}\nendowed with the norm $\\norm{V}^2=\\int \\abs{\\nabla V}^2$, to the function\n\\begin{equation}\\label{phibeta}\n\\varphi(V)=\\varphi_\\beta(V)=\\frac\\beta 2 \\int_{\\mathcal M} \\abs{-\\Delta V +H }.\n\\end{equation}\nIn formula \\eqref{phibeta}, it is implicit that $\\varphi(V)=+\\infty$ if $\\mu = -\\Delta V + H$ is not a Radon measure. Note that, when $\\mu$ \\textit{is} a Radon measure, it must have zero average $\\int\\mu=0$, since $\\mu=\\Delta (*F-V)$.\n\nWe compute the Fenchel conjugate of $\\varphi$. It holds\n\\begin{align*}\n\\varphi^*(V) & \n= \\sup_{U\\in \\mathcal H } \\left\\lbrace \\int_{\\mathcal M}\\! \\nabla V \\cdot \\nabla U -\\frac\\beta 2 \\int_{\\mathcal M}\\! \\abs{-\\Delta U +H} \\right\\rbrace \\\\\n& = -\\int_{\\mathcal M} HV \n + \\sup_{U\\in \\mathcal H} \\left\\lbrace \\int_{\\mathcal M}\\! (-\\Delta U + H)V -\\frac\\beta 2 \\int_{\\mathcal M}\\! \\abs{-\\Delta U +H} \\right\\rbrace \\\\\n& = -\\int_{\\mathcal M} H V + \\sup_{ \\int P =0} \\left\\lbrace \\int_{\\mathcal M}\\left( P V-\\frac\\beta 2 \\abs{P}\\right) \\right\\rbrace.\n\\end{align*}\nIn the last equality, the supremum may -- by a density argument -- be taken over all $L^2$ functions $P$ with zero average.\n\nIf $(\\esssup V-\\essinf V)\\leq\\beta$, then $\\abs{V+\\alpha}\\leq \\beta\/2$ for some $\\alpha\\in{\\mathbb R}$, so that\n\\begin{equation*}\n\\int_{\\mathcal M}\\!\\left( P V-\\frac\\beta 2 \\abs{P}\\right) = \\int_{\\mathcal M}\\! \\left( (V+\\alpha)P -\\frac\\beta 2 |P| \\right) \\leq 0,\n\\end{equation*}\nand in that case\n\\begin{equation*}\n\\varphi^*(V)=-\\int_{\\mathcal M}\\! HV.\n\\end{equation*}\nOn the other hand, if $(\\esssup V-\\essinf V)>\\beta$, then up to translating $V$ we may assume that $\\lbrace V>\\beta\/2\\rbrace$ and $\\lbrace V<-\\beta\/2\\rbrace$ have positive measures. It is then easy to construct a function $P$ supported in those sets, such that $\\int P=0$, $\\int |P|=1$, and $\\int PV>\\beta\/2$. Using $\\lambda P$ as a test function for arbitrary $\\lambda>0$, we deduce that $\\varphi^*(V)=+\\infty$.\n\nFrom Lemma~\\ref{lem_dual} it follows that $V_0\\in \\dot H^1(\\mathcal M)$ minimizes $E_\\beta$ if and only if $V_0$ minimizes \n\\begin{equation*}\n\\frac 12 \\int _{\\mathcal M}\\! \\abs{\\nabla V}^2 + \\int_{\\mathcal M}\\! HV \n\\end{equation*}\namong $V\\in \\dot H^1(\\mathcal M)$ such that $\\esssup V-\\essinf V\\leq \\beta$. Since both problems are invariant under addition of a constant, the restriction to the space $\\dot H^1(\\mathcal M)$ can be relaxed to obtain Proposition~\\ref{prop_obstacle}.\n\\end{proof}\n\n\\subsection{Basic properties}\\label{ss_basicprop}\n\nIn this section we concentrate on the obstacle problem\n\\begin{equation}\\label{obstacleH}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\!\\left( \\abs{\\nabla V}^2 + 2HV\\right)\\colon V\\in H^1(\\mathcal M),\\: |V|\\leq \\beta\/2 \\right\\rbrace.\n\\end{equation}\nWe recall the classical interpretation of \\eqref{obstacleH} as a free boundary problem, and establish a monotonicity property of the free boundary.\n\nThe first step to these basic properties is the reformulation of the obstacle problem \\eqref{obstacleH} as a variational inequality: a function $V\\in H^1(\\mathcal M)$ solves \\eqref{obstacleH} if and only if $\\abs{V}\\leq \\beta\/2$ and\n\\begin{equation}\\label{varineq}\n\\int_{\\mathcal M}\\! \\nabla V \\cdot \\nabla (W-V) \\geq -\\int_{\\mathcal M}\\! H(W-V) \\qquad\\forall W\\in H^1(\\mathcal M),\\: |W|\\leq \\beta\/2.\n\\end{equation}\nThe proof of this weak formulation is elementary and can be found in many textbooks on convex analysis. See for instance \\cite{rodrigues}.\n\nNext we recall the standard reformulation of \\eqref{varineq} as a free boundary problem.\n\\begin{lemma}\\label{lem_freebound}\nA function $V\\in H^1(\\mathcal M)$ with $\\abs{V}\\leq \\beta\/2$ solves \\eqref{obstacleH} or equivalently \\eqref{varineq} if and only if \n\\begin{equation}\\label{freebound}\n\\left\\lbrace\n\\begin{aligned}\n V&\\in W^{2,p}(\\mathcal M),\\quad 1 \\beta_c$, where\n\\begin{equation}\\label{beta_c}\n\\beta_c := \\max (*F) - \\min (*F),\n\\end{equation}\nthe function $*F + \\alpha$ solves the obstacle problem \\eqref{obstacleH}, as long as the constant $\\alpha$ satisfies $\\max (*F) -\\beta\/2 \\leq \\alpha \\leq \\min (*F) +\\beta\/2$, and the vorticity measure $-\\Delta V+H$ is identically zero.\n\nFor $\\beta\\leq \\beta_c$, the solution $V=V_\\beta$ of the obstacle problem~\\eqref{obstacleH} must satisfy\n\\begin{equation*}\n\\max V_\\beta -\\min V_\\beta =\\beta,\n\\end{equation*}\nand therefore is unique (see Remark~\\ref{rem_obstacle}). Recall that the superconductivity region $SC_\\beta$ is defined as the set where the vorticity measure $-\\Delta V +H$ vanishes. According to Lemma~\\ref{lem_freebound}, that region is exactly\n\\begin{equation}\\label{SCbeta}\nSC_\\beta =\\lbrace \\abs{V_\\beta}<\\beta\/2 \\rbrace.\n\\end{equation}\n\n\n\nA first basic property of the superconductivity region $SC_\\beta$ is its monotonicity.\n\\begin{prop}\\label{prop_monot}\nFor any $0<\\beta_1 < \\beta_2\\leq \\beta_c$, it holds\n\\begin{equation*}\nSC_{\\beta_1} \\subset SC_{\\beta_2}.\n\\end{equation*}\n\\end{prop}\nIn other words, increasing the intensity of the applied magnetic field shrinks the region of persisting superconductivity, which consistant with physical intuition.\nSince we have to deal with a two-sided obstacle problem, this monotonicity property is not as obvious as in \\cite[Chapter~7]{SS2}. To prove it, we use a comparison principle for two-sided obstacle problems \\cite[Lemma~2.1]{dalmasomoscovivaldi}. We state and prove here a particular form that will also be useful later on.\n\n\\begin{lemma}\\label{lem_comparison}\nLet $H_1\\geq H_2$ be bounded, real-valued functions on $\\mathcal M$. Let also $\\alpha_1\\leq\\alpha_2$ and $\\beta_1\\leq\\beta_2$ be real numbers. For $j=1,2$, let $V_j\\in H^1(\\mathcal M)$ solve respectively the obstacle problems\n\\begin{equation*}\n\\min \\left\\lbrace \\int_{\\mathcal M}\\! \\left(\\abs{\\nabla V}^2 + 2H_jV\\right)\\colon \\alpha_j\\leq V\\leq \\beta_j\\right\\rbrace.\n\\end{equation*}\nThen either $V_1-V_2$ is constant, or $V_1\\leq V_2$.\n\\end{lemma}\n\\begin{proof}\nFor the convenience of the reader, we provide here the elementary proof, which consists in remarking that\n\\begin{equation*}\nW_1 = \\min(V_1,V_2)\\quad\\text{and }\nW_2 = \\max(V_1,V_2)\n\\end{equation*}\nare admissible test functions in the variational inequalities \n\\begin{equation*}\n \\int_{\\mathcal M}\\! \\nabla V_j \\cdot \\nabla (W_j-V_j)\\geq -\\int_{\\mathcal M}\\! H_j(W_j-V_j),\\qquad\\forall W_j\\in H^1,\\: \\alpha_j\\leq W_j\\leq \\beta_j.\n\\end{equation*}\n Substracting the resulting inequalities, we obtain\n\\begin{equation*}\n\\int_{\\mathcal M}\\abs{\\nabla (V_1-V_2)_+}^2 \\leq \\int_{\\mathcal M}(H_2-H_1)(V_1-V_2)_+ \\leq 0,\n\\end{equation*}\nwhere $(V_1-V_2)_+=\\max(V_1-V_2,0)$. We conclude that $(V_1-V_2)_+$ is a constant function.\n\\end{proof}\n\nWith Lemma~\\ref{lem_comparison} at hand, we may prove the monotonicity of the superconductivity region.\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_monot}:]\nLet $V_1$ and $V_2$ denote the solution of the obstacle problem \\eqref{obstacleH} corresponding respectively to $\\beta=\\beta_1$ and $\\beta=\\beta_2$. Let \n\\begin{equation*}\n\\widetilde V_1 =V_1 +\\beta_1\/2,\\quad\\text{and}\\quad \\widetilde V_2=V_2+\\beta_2\/2,\n\\end{equation*}\nso that for $j=1,2$, $\\widetilde V_j$ solves the obstacle problem\n\\begin{equation*}\n\\min\\left\\lbrace \\int_{\\mathcal M}\\!\\left(\\abs{\\nabla V}^2 + 2HV\\right)\\colon 0\\leq V\\leq \\beta_j\\right\\rbrace.\n\\end{equation*}\nTherefore, applying Lemma~\\ref{lem_comparison} with $H_1=H_2=H$, $\\alpha_1=\\alpha_2=0$ and $\\beta_1\\leq \\beta_2$, we deduce that\n\\begin{equation*}\nV_1+\\beta_1\/2\\leq V_2 +\\beta_2\/2.\n\\end{equation*}\n(If $\\widetilde V_1- \\widetilde V_2$ is constant, then $\\beta_2 = \\max V_1-\\min V_1 = \\beta_1$.)\nIn particular, we obtain that\n\\begin{equation*}\n\\lbrace V_1 >-\\beta_1\/2 \\rbrace \\subset \\lbrace V_2 > -\\beta_2\/2\\rbrace.\n\\end{equation*}\nBy a similar argument, we show that\n\\begin{equation*}\n\\lbrace V_1 <\\beta_1\/2 \\rbrace \\subset \\lbrace V_2 < \\beta_2\/2\\rbrace,\n\\end{equation*}\nand conclude that $SC_{\\beta_1}\\subset SC_{\\beta_2}$.\n\\end{proof}\n\n\\begin{remark}\\label{rem_cont}\nIt follows from the above proof that\n\\begin{equation*}\n|V_1-V_2|\\leq (\\beta_2-\\beta_1)\/2,\n\\end{equation*}\nthus proving the continuity of $\\beta\\mapsto V_\\beta$ for $0\\leq \\beta\\leq\\beta_c$.\n\\end{remark}\n\n\\section{The small $\\beta$ limit}\\label{s_smallbeta}\nIn this section we study what happens to the superconductivity set when the intensity of the field is high enough to confine it in a narrow region. \nWe make the (generic) non degeneracy assumption that\n\\begin{equation}\\label{Hnondegen}\n|H|+|\\nabla H| > 0\\quad\\text{in }\\mathcal M.\n\\end{equation}\nIn other words, $\\nabla H\\neq 0$ in $\\lbrace H=0\\rbrace$. This implies in particular that the set $\\Sigma:=\\lbrace H=0\\rbrace$ where the magnetic field vanishes is a finite disjoint union of smooth closed curves. We also note that condition \\eqref{Hnondegen} also implies that we are not in the situation where not even the first critical field is defined(see \\cite{C}, Theorem 3.1).\n\nLet us say a few words here about the nondegeneracy assumption \\eqref{Hnondegen}. This is the same nondegeneracy assumption that has been considered in works on the spectral analysis of the magnetic Laplacian \\cite{montgomery95} and on higher applied magnetic fields in Ginzburg-Landau \\cite{pankwek02,attar14}.\nMoreover, we emphasize that \\eqref{Hnondegen} is a generic assumption, in the following sense.\n\\begin{lem}\\label{generic}\nThe set of $H$ satisfying \\eqref{Hnondegen} is open and dense in $C^1(\\mathcal M)$.\n\\end{lem}\n\\begin{proof}\nThe fact that \\eqref{Hnondegen} is an open condition in $C^1(\\mathcal M)$ is clear. The density follows from a transversality theorem by Quinn \\cite[Theorem~3]{quinn70}, applied to the $C^1$ map\n\\begin{equation*}\n\\Phi\\colon C^1(\\mathcal M) \\times \\mathcal M\\to\\mathbb R,\\quad (H,x)\\mapsto H(x).\n\\end{equation*}\nFor a function $H\\in C^1(\\mathcal M)$, \\eqref{Hnondegen} is equivalent to $\\Phi(H,\\cdot)$ being transverse to $\\lbrace 0\\rbrace$. Clearly, $D_H\\Phi(H,x)=I_{C^1(\\mathcal M)}$ is Fredholm, and $\\Phi$ is transverse to $\\lbrace 0\\rbrace$. Therefore, the set of $H$ such that $\\Phi(H,\\cdot)$ is transverse to $\\lbrace 0\\rbrace$ is dense in $C^1(\\mathcal M)$. \n\\end{proof}\n\n\n\nWe are interested in the behavior, as $\\beta\\to 0$, of the superconductivity region $SC_\\beta$ \\eqref{SCbeta}.\n\nWe let $d\\colon\\mathcal M\\to\\mathbb R_+$ denote the distance function to the set $\\Sigma=\\lbrace H=0\\rbrace$, that is\n\\begin{equation}\\label{defd}\nd(x)=\\mathrm{dist}(x,\\lbrace H=0\\rbrace).\n\\end{equation}\nIn this context we characterize the behavior of $SC_\\beta$ in terms of the function $d$, as follows(this is a more explicit version of Theorem \\ref{simplifiedmain}).\n\n\\begin{theorem}\\label{thm_scbeta}\nUnder the non-degeneracy assumption \\eqref{Hnondegen} on the magnetic field, there exists $\\beta_0>0$ and $C>0$ such that, for $\\beta\\in(0,\\beta_0)$, \\begin{equation}\\label{boundsscbeta}\n\\left\\lbrace d\\leq \\frac 1C \\beta^{1\/3}\\right\\rbrace \\subset SC_\\beta \\subset \\left\\lbrace d\\leq C\\beta^{1\/3} \\right\\rbrace,\n\\end{equation}\nwhere $SC_\\beta$ is the superconductivity region \\eqref{SCbeta}, and $d$ denotes the distance to the zero locus of the magnetic field \\eqref{defd}.\n\\end{theorem}\n\nIn the proof we construct explicit solutions to modified obstacle problems, in order to apply the comparison principle Lemma~\\ref{lem_comparison}. The comparison functions are constructed locally near each component $\\Gamma$ of $\\lbrace H=0\\rbrace$, and then we need to extend and paste these functions and the associated modified obstacle problem data. Although the construction looks local, it is worth noting that we really need to make it near \\textit{every} component $\\Gamma$ of $\\lbrace H=0\\rbrace$. Otherwise the pasting would not provide us with obstacle problems comparable to the original one, because a solution has to change sign near \\textit{every} curve $\\Gamma$.\n\n\n\n\\begin{remark}\\label{rem_outer+thick}\nAnother natural approach to proving Theorem~\\ref{thm_scbeta} would be to construct separate comparison functions in $\\lbrace H>0\\rbrace$ and $\\lbrace H<0\\rbrace$. In those regions, the obstacle problem becomes one-sided, so that more standard constructions with a classical comparison principle can be made. On the other hand, there is no boundary conditions in those regions, so that such a construction would only provide us with the outer bound \n\\begin{equation}\\label{outer}\nSC_\\beta \\subset \\lbrace d\\leq C \\beta^{1\/3}\\rbrace.\n\\end{equation}\n To obtain the bounds \\eqref{boundsscbeta} which show that\nthe superconductivity set extends to \\underline{both} sides of the zero locus of $H$ by a $\\beta^{\\frac{1}{3}}$ margin, it seems that we really have to appeal to the comparison principle for two-sided obstacle problems. \nHowever, if we would just content ourselves with showing that the superconductivity set had `thickness' proportional to $\\beta^{\\frac{1}{3}}$, namely\n\\begin{equation}\\label{thickness}\n\\mathrm{dist}(\\lbrace V=\\beta\/2\\rbrace,\\lbrace V=-\\beta\/2\\rbrace)\\geq c\\beta^{1\/3},\n\\end{equation}\nthere would be a simpler way. In fact \\eqref{thickness} can be directly inferred from \\eqref{outer}. This is a simple consequence of the interpolated elliptic estimate (see \\cite[Appendix~A]{BBH1})\n\\begin{equation}\\label{interpolestim}\n\\norm{\\nabla V}^2_\\infty \\leq C \\norm{\\Delta V}_\\infty \\norm{V}_\\infty,\n\\end{equation}\nwhich implies, since $|V|\\leq\\beta$ and \n$|\\Delta V|=|H\\ensuremath{\\mathonebb{1}}_{SC_\\beta}| \\leq C \\beta^{1\/3},$\nthat\n\\begin{equation}\\label{gradientbound}\n|\\nabla V|\\leq C \\beta^{2\/3} \\quad\\text{in }\\mathcal M.\n\\end{equation}\nHence, for any $x_\\pm\\in \\lbrace V=\\pm\\beta\/2\\rbrace$ and any arc-length parametrized curve $\\gamma(s)$, ($0\\leq s\\leq \\ell$) going from $x_-$ to $x_+$, it holds\n\\begin{equation*}\n\\beta =V(x_+)-V(x_-) = \\int_0^\\ell \\nabla V(\\gamma(s))\\cdot\\gamma'(s) \\, ds \\leq C \\beta^{2\/3}\\ell,\n\\end{equation*}\nso that the length of $\\gamma$ satisfies $\\ell\\geq c\\beta^{1\/3}$, which proves \\eqref{thickness}.\n\\end{remark}\n\nNext we turn to the proof of Theorem~\\ref{thm_scbeta}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm_scbeta}:]\nWe will construct, for small enough $\\beta$, bounded functions $H_1\\leq H\\leq H_2$, and comparison functions $V_1$ and $V_2$ of regularity $W^{2,\\infty}$, satisfying for $j=1,2$,\n\\begin{equation}\\label{eqVj}\n\\begin{gathered}\n\\Delta V_j=H_j \\ensuremath{\\mathonebb{1}}_{\\abs{V_j}<\\beta\/2},\\\\\n \\abs{V_j}\\leq \\beta\/2,\\quad H_j\\geq 0 \\text{ in }\\lbrace V_j=-\\beta\/2\\rbrace,\\quad H_j\\leq 0\\text{ in }\\lbrace V_j=\\beta\/2\\rbrace,\n \\end{gathered}\n\\end{equation}\nand the bounds\n\\begin{equation}\\label{boundsVj}\n\\left\\lbrace d\\leq \\frac 1C \\beta^{1\/3}\\right\\rbrace \\subset \\left\\lbrace \\abs{V_j}<\\beta\/2\\right\\rbrace \\subset \\left\\lbrace d\\leq C\\beta^{1\/3} \\right\\rbrace.\n\\end{equation}\nBy Lemma~\\ref{lem_freebound}, \\eqref{eqVj} implies that $V_j$ solves the obstacle problem \\eqref{obstacleH} with $H=H_j$. Therefore we may apply the comparison principle for two-sided obstacle problems (Lemma~\\ref{lem_comparison}) to conclude that $V_1\\geq V \\geq V_2$. In view of the bounds \\eqref{boundsVj} satisfied by $V_1$ and $V_2$, this obviously implies that the superconductivity region satisfies the bounds \\eqref{boundsscbeta}.\n\nThe rest of the proof is devoted to constructing $V_1$ and $V_2$. To this end we introduce good local coordinates in a neighborhood of $\\Sigma=\\lbrace H=0\\rbrace$. Recall that, thanks to the nondegeneracy assumption \\eqref{Hnondegen}, $\\Sigma$ is a finite union of closed smooth curves. Let us fix one of them, $\\Gamma$, together with an arc-length parametrization of it:\n\\begin{equation*}\n\\Gamma =\\left\\lbrace \\gamma(x) \\colon x\\in{\\mathbb R}\/\\ell\\mathbb Z\\right\\rbrace,\\quad \\abs{\\gamma'(x)}=1.\n\\end{equation*}\nLet us also fix a smooth normal vector $\\nu(x)$ to $\\Gamma$ on $\\mathcal M$, that is\n\\begin{equation*}\n\\nu(x)\\in T_{\\gamma(x)}\\mathcal M,\\quad \\abs{\\nu}=1,\\quad \\nu\\cdot\\gamma'=0,\n\\end{equation*}\nand impose that $\\nu(x)$ points in the direction of $\\lbrace H>0\\rbrace$ (since $H<0$ on one side of $\\Gamma$ and $H<0$ on the other side). We introduce Fermi coordinates along $\\Gamma$: for small enough $\\delta$, the map\n\\begin{equation*}\n{\\mathbb R}\/\\ell\\mathbb Z \\times (-\\delta,\\delta)\\to\\mathcal M,\\quad (x,y)\\mapsto \\exp_{\\gamma(x)}(y\\nu(x)),\n\\end{equation*}\nis a diffeomorphism. It defines local coordinates $(x,y)$ on $\\mathcal M$ in a neighborhood of $\\Gamma$, in which the Laplace operator has the form\n\\begin{equation}\\label{laplacefermi}\n\\Delta =\\frac 1f\\left(\\partial_y f\\partial_y +\\partial_x f^{-1} \\partial_x\\right),\n\\end{equation}\nwhere $f(x,y)=1-y\\kappa(x,y)$ for some smooth function $\\kappa$. Note that $y$ is nothing else than the signed distance to $\\Gamma$, and in particular $|y|=d$ in a neighborhood of $\\Gamma$. While this is a coordinate system that follows well the geometry of a neighborhood of $\\gamma , $ we actually need one where the Laplacian allows us to reduce our construction to a $1d$ problem. To that end let $(x,z)$ be the local coordinates where\n\\begin{equation}\\label{zcoord}\nz=y+\\frac 12 y^2\\kappa(x,y).\n\\end{equation}\nClearly the map $(x,y)\\mapsto (x,z)$ is a diffeomorphism for small enough $y$, so that $(x,z)$ define indeed local coordinates on $\\mathcal M$. The reason for using the coordinates $(x,z)$ is that the Laplace operator is then approximately\n\\begin{equation*}\n\\Delta \\approx \\partial_x^2 + \\partial_z^2,\n\\end{equation*}\nwhich will allow us to obtain nice bounds for functions depending only on $z$.\n\nNote that, since we choose the normal vector $\\nu$ to point in the direction of $\\lbrace H>0\\rbrace$, and since $\\abs{\\nabla H}\\geq c>0$ in a neighborhood of $\\Gamma$ thanks to the nondegeneracy assumption \\eqref{Hnondegen}, it holds\n\\begin{equation*}\n\\partial_z H \\geq c>0,\\quad \\abs{z}<\\delta.\n\\end{equation*}\nOn the other hand, $\\nabla H$ is bounded, so that there exist $C\\geq c>0$ such that\n\\begin{equation}\\label{Hz}\nCz\\ensuremath{\\mathonebb{1}}_{z<0} + cz\\ensuremath{\\mathonebb{1}}_{z>0} \\leq H \\leq cz\\ensuremath{\\mathonebb{1}}_{z<0} + Cz\\ensuremath{\\mathonebb{1}}_{z>0},\\qquad \\abs{z}<\\delta.\n\\end{equation}\n\nNext we concentrate on the construction of $V_1$ ($H_1$ will be defined accordingly). Away from the set $\\Sigma$, we simply define\n\\begin{equation}\\label{V1away}\nV_1=-\\mathrm{sign}(H)\\beta\/2 \\quad\\text{in }\\lbrace d>\\delta\/2\\rbrace.\n\\end{equation}\nThe interesting part is of course what happens near $\\Sigma$. Near each of the smooth curves $\\Gamma\\subset\\Sigma$, we will look for $V_1$ in the form $V_1=v(z)$, where $v$ is a $W^{2,\\infty}$ function satisfying\n\\begin{equation}\\label{V1near}\nv(z)=\\begin{cases}\n\\beta\/2 & \\text{ for }z<-\\eta_-,\\\\\n-\\beta\/2 & \\text{ for }z>\\eta_+,\n\\end{cases}\n\\end{equation}\nfor some parameters $\\eta_\\pm>0$ that will depend on $\\beta$. A straightforward computation using \\eqref{laplacefermi} and \\eqref{zcoord} shows that\n\\begin{equation}\\label{DeltaV1}\n\\Delta V_1 =v''(z) + z \\left(g_1(x,z) v''(z) + g_2(x,z) v'(z)\\right),\n\\end{equation}\nwhere $g_1$ and $g_2$ are bounded functions. We are going to define in $(-\\eta_-,\\eta_+)$ the function $v$ so that \n\\begin{equation}\\label{boundsv}\nv''\\leq 2Cz\\ensuremath{\\mathonebb{1}}_{z<0} + \\frac{c}{2}z\\ensuremath{\\mathonebb{1}}_{z>0},\\quad |v'|= o(\\beta),\\; |v''|= o(\\beta).\n\\end{equation}\nWe then define $H_1$ in $(-\\eta_{-} , \\eta_+)$ simply as $\\Delta V_1 .$\nThus, recalling \n\\eqref{Hz}, we will have, for small enough $\\beta>0$,\n\\begin{equation}\n\\Delta V_1 = H_1\\ensuremath{\\mathonebb{1}}_{|V_1|<\\beta\/2}\\qquad\\text{with }H_1\\leq H\\;\\text{in }\\lbrace -\\eta_-0},\n\\end{equation}\nso we impose\n\\begin{equation}\\label{condv''}\n6A_- = 2C,\\quad 6A_+ = \\frac c2,\\quad B_-+2\\eta_-A_- = B_+-2\\eta_+A_+ =0,\n\\end{equation}\nso that we even have an equality in \\eqref{v''}.\nPlugging \\eqref{condv''} into \\eqref{condv0}, we find\n\\begin{equation}\\label{condeta}\n\\frac c6 \\eta_+^3 + \\frac{2C}{3}\\eta_-^3 =\\beta,\\quad 4C\\eta_-^2 = c\\eta_+^2,\n\\end{equation}\nwhich leads us to choose\n\\begin{equation}\\label{eta}\n\\eta_\\pm = \\alpha_\\pm\\beta^{1\/3},\n\\end{equation}\nwhere $\\alpha_\\pm>0$ are the solutions of\n\\begin{equation*}\n4C\\alpha_-^2 = c\\alpha_+^2,\\quad \\frac c6 \\alpha_+^3 + \\frac{2C}{3}\\alpha_-^3=1.\n\\end{equation*}\nWith $A_\\pm$, $B_\\pm$ and $\\eta_\\pm$ chosen as in \\eqref{condv''}-\\eqref{eta}, the function $v$ is of class $W^{2,\\infty}$ and satisfies \\eqref{v''}. Moreover, it is straightforward to check that\n\\begin{equation*}\n|v'|+|v''|\\leq C\\beta^{1\/3} \\quad\\text{ in }(-\\eta_-,\\eta_+),\n\\end{equation*}\nso that \\eqref{boundsv} is satisfied, which concludes the construction of $V_1$ satisfying \\eqref{eqVj}. On the other hand $V_1$ obviously satisfies \\eqref{boundsVj} since\n\\begin{equation*}\n\\lbrace \\abs{V_1}<\\beta\/2\\rbrace = \\lbrace -\\eta_- < z<\\eta_+ \\rbrace.\n\\end{equation*}\n\nWe omit the construction of $V_2$, which is completely similar to the one just performed.\n\\end{proof}\n\n\n\n\n\n\\section{Intermediate regimes}\\label{s_interm}\n\nAs discussed in the introduction (Section~\\ref{s_intro}), in the present section we want to understand the transitions occurring as $\\beta$ decreases from $\\beta_c$ to 0, when the set $\\lbrace H=0\\rbrace$ has more than one connected component.\n\n\nIn Section~\\ref{ss_symcase} we study in detail a special case with rotational symmetry along a vertical axis, to provide some insight into the transition from the vortexless state to the zero solution. The reason to restrict to this setting is that it encapsules, what we believe are, the most interesting changes in the superconducting set that can occur. \n\nOn the one hand, once we drop the assumption of rotational symmetry, changes in $H$ inside the sample could lead to arbitrarily intricate solutions to the obstacle problem for different values of $\\beta,$ so a general theorem is not available. On the other hand the symmetries we consider highlight many model situations with remarkable properties. One of these is the striking phenomenon that some parts of the free boundary may freeze: that is, remain constant with respect to $\\beta$, for $\\beta$ in some interval. In Section~\\ref{ss_freez} we generalize this observation to the general, non-symmetric case. \n\nAs mentioned earlier, a generalization of the other properties is precluded due to the wide variety of solutions one could construct, having the freedom to choose both $H$ and $\\mathcal{M}.$ Nevertheless, we believe that under some more restrictive assumptions, in particular fixing the topology of the level sets of $H,$ one could extend the result on existence of the transitions observed in Proposition~\\ref{prop_regimes1d}, however the role of the integral conditions on $I_{\\pm}, J$ is not so easily transferable or even identifiable anymore. \n\n\n\n\n\\subsection{Detailed study of a symmetric case}\\label{ss_symcase}\n\nHere we consider a surface of revolution of the form\n\\begin{equation*}\n\\mathcal M =\\left\\lbrace (\\rho(\\phi)\\cos \\theta,\\rho(\\phi)\\sin\\theta,z(\\phi))\\colon \\phi\\in [0,\\pi],\\,\\theta\\in[0,2\\pi] \\right\\rbrace,\n\\end{equation*}\nwhere $\\rho$ and $z$ are smooth functions linked by the relation \n\\begin{equation*}\nz(\\phi)\\tan\\phi=\\rho(\\phi),\n\\end{equation*}\nand satisfying $\\rho(0)=\\rho(\\pi)=0$, $\\rho>0$ in $(0,\\pi)$, $z'(0)=z'(\\pi)=0$, and\n\\begin{equation*}\n\\gamma:=\\sqrt{(\\rho')^2+(z')^2}\\geq c>0.\n\\end{equation*}\nThe volume form on such $\\mathcal M$ is $d\\mathcal H^2_{\\mathcal M}=\\rho\\gamma d\\theta d\\phi$.\n\nThe induced magnetic potential $\\mathbf A$ on $\\mathcal M$ is also assumed to be symmetric, of the form\n\\begin{equation*}\n\\mathbf A = a(\\phi) d\\theta = \\frac{a(\\phi)}{u(\\phi)} \\hat e_\\theta,\n\\end{equation*}\nand we make the following assumptions on the functions $a$:\n\\begin{itemize}\n\\item[(a1):] $a(0)=a(\\pi)=0$, and $a>0$ in $(0,\\pi)$.\n\\item[(a2):] $a'>0$ in $(0,\\phi_1)$ and $(\\phi_2,\\phi_3)$ and $a'<0$ in $(\\phi_1,\\phi_2)$ and $(\\phi_3,\\pi)$, for some $0<\\phi_1<\\phi_2<\\phi_3<\\pi$.\n\\end{itemize}\nThe function $a(\\phi)$ hast two local maxima $a_1=a(\\phi_1)$ and $a_3=a(\\phi_3)$, and one local minimum $a_2=a(\\phi_2)$. To simplify notations to come, we assume in addition that $a_1\\beta^*_1 \\geq \\beta^*_2 >0$ be defined by\n\\begin{equation*}\n\\beta^*_1:=\\max(I_\\pm(\\alpha^*)),\\quad \\beta^*_2:=\\min (I_\\pm(\\alpha^*)).\n\\end{equation*}\nThen the conclusion of Propostion~\\ref{loose1d} holds:\n\\begin{itemize}\n\\item For $\\beta_c >\\beta >\\beta^*_1$, $SC_\\beta$ is an interval.\n\\item For $\\beta^*_1>\\beta>\\beta^*_2$, $SC_\\beta$ is the union of two disjoint intervals, one of them independent of $\\beta$. \n\\item For $\\beta^*_2>\\beta>0$, $SC_\\beta$ is the union of three disjoint intervals.\n\\end{itemize}\n\\end{prop}\n\n\\begin{remark}\nIt may happen that $I_-(\\alpha^*)=I_+(\\alpha^*)$. In that case, $\\beta^*_1=\\beta^*_2$ and the second regime predicted by Proposition~\\ref{prop_regimes1d} never happens.\n\\end{remark}\n\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_regimes1d}:]\nBy uniqueness (see Remark~\\ref{rem_obstacle}), it suffices to exhibit, for each regime listed in Proposition~\\ref{prop_regimes1d}, a solution of \\eqref{1dfreebound} satisfying the desired properties.\n\n\\textbf{Case 1:} $\\beta\\in (\\beta^*_1,\\beta_c)$. The function\n\\begin{equation*}\nI(\\alpha):=\\int_{\\phi_-}^{\\phi_+} (a-\\alpha)\\frac\\gamma\\rho\\, d\\phi,\\quad\\alpha\\in (0,a_1),\n\\end{equation*}\nis continuous, decreasing and satisfies $I(0)=\\beta_c$ and $I(\\alpha^*)=\\beta^*_1$. Therefore there exists a unique $\\alpha\\in (0,\\alpha^*)$ such that $I(\\alpha)=\\beta$. We define\n\\begin{equation*}\nv(\\phi)=\n\\begin{cases}\n-\\beta\/2 & \\text{ for }\\phi\\in (0,\\phi_-),\\\\\n-\\beta\/2 + \\int_{\\phi_-}^{\\phi} (a-\\alpha)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\phi_-,\\phi_+),\\\\\n\\beta\/2 & \\text{ for }\\phi\\in (\\phi_+,\\pi).\n\\end{cases}\n\\end{equation*}\nThe shape of the function $v$ is sketched in Figure~\\ref{figu1}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{u1.pdf}\n\\caption{The shape of $v$ for $\\beta\\in(\\beta^*_1,\\beta_c)$}\n\\label{figu1}\n\\end{center}\n\\end{figure}\n\nThe function $v$ is clearly continuous since $\\beta$ has been chosen accordingly. Moreover, it holds\n\\begin{equation*}\nv'(\\phi_+)=(a(\\phi_+)-\\alpha)\\frac\\gamma\\rho = v'(\\phi_-)=0,\n\\end{equation*}\nsince by definition $a(\\phi_+)=a(\\phi_-)=\\alpha$. Hence $v$ is in fact $C^1$ in $[0,\\pi]$. Also by definition, $a'\\geq 0$ in $(0,\\phi_-)$ and $a'\\leq 0$ in $(\\phi_+,\\pi)$. In addition, we clearly have $(\\rho\\gamma^{-1}v'-a)'=0$ in $(\\phi_-,\\phi_+)$. To prove that $v$ solves \\eqref{1dfreebound}, it only remains to show that $|v|<\\beta\/2$ in $(\\phi_+,\\phi_-)$. We consider two different cases, depending on whether $\\alpha\\in (0,a_2]$ or $\\alpha\\in (a_2,\\alpha^*)$.\n\nIf $\\alpha\\in (0,a_2)$, then (see Figure~\\ref{figa2})\n\\begin{equation*}\nv'=(a-\\alpha)\\frac\\gamma\\rho>0\\quad\\text{in }(\\phi_-,\\phi_+),\n\\end{equation*}\nso that $v$ is increasing on $(\\phi_-,\\phi_+)$ and it clearly holds $|v|<\\beta\/2$. For $\\alpha=a_2$ the derivative $v'$ only vanishes at one point and the same conclusion is valid.\n\nIf, on the other hand $\\alpha\\in (a_2,\\alpha^*)$, then (see Figure~\\ref{figu1})\n\\begin{equation*}\nv'=(a-\\alpha)\\frac\\gamma\\rho\\begin{cases}\n>0 &\\text{ in }(\\phi_-,\\psi_+),\\\\\n<0 & \\text{ in }(\\psi_+,\\psi_-),\\\\\n>0 & \\text{ in }(\\psi_-,\\phi_+).\n\\end{cases}\n\\end{equation*}\nTherefore it suffices to check that $v(\\psi_+)<\\beta\/2$ and $v(\\psi_-)>-\\beta\/2$. We have, since $I(\\alpha)=\\beta$ and by definition of $I_\\pm$ and $J$ (see Figure~\\ref{figIJ}),\n\\begin{equation*}\n\\begin{split}\nv(\\psi_+)-\\beta\/2 & = I_-(\\alpha)-\\beta = I_-(\\alpha)-I(\\alpha)=J(\\alpha)-I_+(\\alpha),\\\\\nv(\\psi_-)+\\beta\/2 & = I_-(\\alpha)-J(\\alpha).\n\\end{split}\\end{equation*}\nSince $\\alpha<\\alpha^*$ we find indeed (by definition of $\\alpha^*$) that $v(\\psi_+)<\\beta\/2$ and $v(\\psi_-)>-\\beta\/2$, and in that case also we conclude that $v$ solves the free boundary problem\n\\eqref{1dfreebound}.\n\n\\textbf{Case 2:} $\\beta\\in (\\beta^*_2,\\beta^*_1)$. We treat the case where $\\min (I_\\pm(\\alpha^*))=I_-(\\alpha^*)$. Thus $\\beta^*_1=I_+(\\alpha^*)$ and $\\beta^*_2=I_-(\\alpha^*)$. The other case can be dealt with similarly. \n\nThe function $I_+(\\alpha)$ is continuous and decreasing on $(a_2,a_3)$ and satisfies $I_+(\\alpha^*)=\\beta^*_1$ and $I_+(a_3)=0<\\beta^*_2$ (see Figure~\\ref{figIJ}). Therefore there exists $\\alpha>\\alpha^*$ such that $I_+(\\beta)=\\alpha$. We denote by $\\psi_-$ and $\\phi_+$ the two points of $\\lbrace a=\\alpha \\rbrace \\cap (\\phi_2,\\pi)$, and by $\\phi_-^*<\\psi_+^*<\\psi_-^*$ the three points of $\\lbrace a=\\alpha^*\\rbrace \\cap (0,\\phi_3)$ (as in Figure~\\ref{figu2} below). Note that, since $\\alpha>\\alpha^*$, $\\psi_-^*<\\psi_-$. Next we define\n\\begin{equation*}\nv(\\phi)=\n\\begin{cases}\n-\\beta\/2 &\\text{ for }\\phi\\in (0,\\phi_-^*),\\\\\n-\\beta\/2 +\\int_{\\phi_-^*}^{\\phi}(a-\\alpha^*)\\frac\\gamma\\rho\\, d\\tilde\\phi &\\text{ for }\\phi\\in (\\phi_-^*,\\psi_-^*),\\\\\n-\\beta\/2 & \\text{ for }\\phi\\in (\\psi_-^*,\\psi_-),\\\\\n-\\beta\/2 +\\int_{\\psi_-}^{\\phi}(a-\\alpha)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\psi_-,\\phi_+),\\\\\n\\beta\/2 & \\text{ for }\\phi\\in(\\phi_+,\\pi).\n\\end{cases}\n\\end{equation*}\nThe shape of the function $v$ is sketched in Figure~\\ref{figu2}.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{u2.pdf}\n\\caption{The shape of $v$ for $\\beta\\in(\\beta^*_2,\\beta^*_1)$}\n\\label{figu2}\n\\end{center}\n\\end{figure}\n\nContinuity of $v$ at $\\psi_-^*$ is ensured by the fact that $I_-(\\alpha^*)=J(\\alpha^*)$. Continuity at $\\phi_+$ by $I_+(\\alpha)=\\beta$. The function $v$ is $C^1$ because the facts that $a(\\phi_-^*)=a(\\psi_-^*)=\\alpha^*$ and $a(\\psi_-)=a(\\phi_+)=\\alpha$ guarantee that $v'(\\phi_-^*)=v'(\\psi_-^*)=v'(\\psi_-)=v'(\\phi_+)=0$. The sign of $a'$ is positive in $(0,\\phi_-^*)$ and $(\\psi_-^*,\\psi_-)$ and negative in $(\\phi_+,\\pi)$. In the two intervals $(\\phi_-^*,\\psi_-^*)$ and $(\\psi_-,\\phi_+)$, the equation $(\\rho\\gamma^{-1}v'-a)'=0$ is obviously satisfied, and it remains to check that $|v|<\\beta\/2$ in those intervals.\n\nSince $v'=(a-\\alpha)\\gamma\\rho^{-1} >0$ in $(\\psi_-,\\phi_+)$, it clearly holds $|v|<\\beta\/2$ in $(\\psi_-,\\phi_+)$.\n\nIn the interval $(\\phi_-^*,\\psi_-^*)$, the sign of $v'$ shows that $v$ attains its minimum at the boundary and its maximum at $\\psi_+^*$, and it holds\n\\begin{equation*}\nv(\\psi_+^*)-\\beta\/2=-\\beta + I_-(\\alpha^*) =-\\beta + \\beta^*_2 <0.\n\\end{equation*}\nWe conclude that $v$ solves the free boundary problem \\eqref{1dfreebound}. Moreover, the interval $(\\phi_-^*,\\psi_-^*)$ clearly does not depend on $\\beta$.\n\n\\textbf{Case 3:} $\\beta\\in (0,\\beta^*_2)$. \nSince $I_-$ is continuous and decreasing, $I_-(\\alpha^*)>\\beta^*_2$ and $I_-(a_1)=0$, there exists $\\alpha_1>\\alpha^*$ such that $I_-(\\alpha_1)=\\beta$. Similarly, there exist $\\alpha_2<\\alpha^*$ and $\\alpha_3>\\alpha^*$ such that $J(\\alpha_2)=I_+(\\alpha_3)=\\beta$. We denote by\n\\begin{equation*}\n0<\\phi_-^1<\\psi_+^1 <\\psi_+^2 <\\psi_-^2 <\\psi_-^3 <\\phi_+^3<\\pi\n\\end{equation*}\nthe points such that (see Figure~\\ref{figu3})\n\\begin{equation*}\n\\begin{gathered}\n\\lbrace a=\\alpha_1\\rbrace\\cap (0,\\phi_2)=\\lbrace \\phi_-^1,\\psi_+^1\\rbrace,\\\\\n\\lbrace a=\\alpha_2\\rbrace\\cap (\\phi_1,\\phi_3) =\\lbrace \\psi_+^2,\\psi_-^2\\rbrace,\\\\\n\\lbrace a=\\alpha_3\\rbrace\\cap (\\phi_2,\\pi)=\\lbrace \\psi_-^3,\\phi_+^3\\rbrace.\n\\end{gathered}\\end{equation*}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=8cm]{u3.pdf}\n\\caption{The shape of $v$ for $\\beta\\in(0,\\beta^*_2)$}\n\\label{figu3}\n\\end{center}\n\\end{figure}\n\nThen we define\n\\begin{equation*}\nv(\\phi)=\\begin{cases}\n-\\beta\/2 &\\text{ for }\\phi\\in (0,\\phi_-^1)\\text{ or }\\phi\\in (\\psi_-^2,\\psi_-^3)\\\\\n-\\beta\/2 +\\int_{\\phi_-^1}^{\\phi}(a-\\alpha_1)\\frac\\gamma\\rho\\, d\\tilde\\phi &\\text{ for }\\phi\\in (\\phi_-^1,\\psi_+^1),\\\\\n\\beta\/2 & \\text{ for }\\phi\\in (\\psi_+^1,\\psi_+^2)\\text{ or }\\phi\\in (\\phi_+^3,\\pi),\\\\\n\\beta\/2 +\\int_{\\psi_+^2}^{\\phi}(a-\\alpha_2)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\psi_+^2,\\psi_-^2)\\\\\n-\\beta\/2 +\\int_{\\psi_-^3}^{\\phi}(a-\\alpha_3)\\frac\\gamma\\rho\\, d\\tilde\\phi & \\text{ for }\\phi\\in (\\psi_-^3,\\phi_+^3).\n\\end{cases}\n\\end{equation*}\nThe shape of the function $v$ is sketched in Figure~\\ref{figu3}.\n\n\n\nAs above the $C^1$ regularity of $v$ follows from the definitions of $\\alpha_1$, $\\alpha_2$ and $\\alpha_3$. The sign of $a'$ is positive in $(0,\\phi_-^1)\\cup (\\psi_-^2,\\psi_-^3)$ and negative in $(\\psi_+^1,\\psi_+^2)\\cup (\\phi_+^3,\\pi) $. The equation $(\\rho\\gamma^{-1}v'-a)'=0$ is satisfied in the three intervals $(\\phi_-^1,\\psi_+^1)$, $(\\psi_+^2,\\psi_-^2)$ and $(\\psi_-^3,\\phi_+^3)$. Moreover in those intervals, the function $v$ is monotone, hence $|v|<\\beta\/2$. Therefore $v$ solves the free boundary problem \\eqref{1dfreebound}.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{`Freezing' of the free boundary}\\label{ss_freez}\n\n\\begin{prop}\\label{prop_freez}\nAssume that, for some $\\beta_0\\in (0,\\beta_c)$, one connected component $\\omega$ of the superconductivity set $SC_{\\beta_0}$ is such that $V_{\\beta_0}$ takes the same value on each connected component of $\\partial\\omega$. Then there exists $\\delta>0$ such that\n\\begin{equation}\\label{freez}\nSC_\\beta \\cap\\overline \\omega = SC_{\\beta_0}\\cap\\overline\\omega =\\omega,\n\\end{equation}\nfor all $\\beta\\in (\\beta_0-\\delta,\\beta_0]$.\n\\end{prop}\n\nIn Figure~\\ref{figfreez} we show a situation corresponding to Proposition~\\ref{prop_freez}, with $V=-\\beta\/2$ on every connected component of $\\partial\\omega$.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=4.5cm]{freez.pdf}\n\\caption{An example of the situation of Proposition~\\ref{prop_freez}}\n\\label{figfreez}\n\\end{center}\n\\end{figure}\n\n\\begin{remark}\nThe assumption on $\\beta_0$ in Proposition~\\ref{prop_freez} corresponds exactly to what happens in the symmetric case (Proposition~\\ref{prop_regimes1d}) in the regime $\\beta_1^*>\\beta>\\beta_2^*$, where $u(\\phi_-^*)=u(\\psi_-^*)=-\\beta\/2$ (Figure~\\ref{figu2}).\n\\end{remark}\n\n\\begin{proof}[Proof of Proposition~\\ref{prop_freez}:]\nWe present the proof in the case where $V=-\\beta_0\/2$ on every connected component of $\\partial\\omega$. The case $V=\\beta_0\/2$ on $\\partial\\omega$ can be dealt with similarly.\n\nSince $V<\\beta_0\/2$ in $\\omega$ and $V=-\\beta_0\/2$ on $\\partial\\omega$, it holds\n\\begin{equation*}\nm:=\\max_{\\overline\\omega} V <\\beta_0\/2,\n\\end{equation*}\nand we define\n\\begin{equation*}\n\\delta:= \\frac 12 {\\beta_0}-m >0.\n\\end{equation*}\n\nLet $\\beta\\in (\\beta_0-\\delta,\\beta_0]$, and define\n\\begin{equation}\\label{tildeV0}\n\\widetilde V_0 := V_{\\beta_0} + \\frac{1}{2}(\\beta_0-\\beta).\n\\end{equation} \nThe definitions of $m$ and $\\delta$ ensure that it holds\n\\begin{equation}\\label{ineqtildeV0}\n-\\beta\/2 \\leq \\widetilde V_0 \\leq \\frac 12 {\\beta_0} -\\delta + \\frac{1}{2}(\\beta_0-\\beta) < \\beta\/2\\quad\\text{in }\\overline\\omega.\n\\end{equation}\nWe claim that \n\\begin{equation}\\label{freezV}\nV_\\beta =\\widetilde V_0 \\quad\\text{in }\\overline\\omega,\n\\end{equation}\nwhich obviously implies \\eqref{freez}. \n\nNote that the proof of Proposition~\\ref{prop_monot} implies that it always holds\n\\begin{equation}\\label{VleqtildeV}\nV_\\beta \\leq \\widetilde V_0 \\quad\\text{in }\\mathcal M.\n\\end{equation}\nLet $\\omega_\\beta = SC_\\beta\\cap \\omega$, and \n\\begin{equation}\\label{Unonneg}\nU:=\\widetilde V_0 - V_\\beta \\geq 0.\n\\end{equation}\nNote that $U\\in C^{1,\\alpha}(\\overline\\omega)$, and $U=0$ on $\\partial\\omega$ (since, by definition of $\\omega$, $\\widetilde V_0=0$ on $\\partial\\omega$). \n\nLet $\\omega':=\\omega\\cap SC_\\beta$. It holds\n\\begin{equation}\\label{DeltaU}\n\\Delta U = H\\ensuremath{\\mathonebb{1}}_{\\omega\\setminus\\omega'}\\quad\\text{ in }\\omega.\n\\end{equation}\nFrom \\eqref{ineqtildeV0} and \\eqref{VleqtildeV} it follows that\n\\begin{equation*}\nV_\\beta <\\beta\/2 \\quad\\text{in }\\overline \\omega.\n\\end{equation*} \nTherefore, recalling the free boundary formulation \\eqref{freebound}, we have $H\\geq 0$ in $\\omega\\setminus\\omega'$. In particular \\eqref{DeltaU} implies that\n\\begin{equation*}\n\\Delta U\\geq 0 \\quad \\text{ in }\\omega.\n\\end{equation*}\nLet $\\varepsilon>0$ and consider\n\\begin{equation*}\n\\varphi:=\\max(U-\\varepsilon,0)\\in H^1(\\omega).\n\\end{equation*}\nRecalling that $U\\in C(\\overline \\omega)$ and $U=0$ on $\\partial\\omega$, we know that $\\varphi$ has compact support inside $\\omega$. Thus we may integrate by part (without knowing anything about the regularity of $\\partial\\omega$) to obtain\n\\begin{equation*}\n\\int_\\omega |\\nabla \\varphi|^2 =\\int_\\omega \\nabla\\varphi\\cdot\\nabla U = -\\int_\\omega \\varphi \\Delta U \\leq 0,\n\\end{equation*}\nand we deduce that $\\varphi\\equiv 0$ in $\\omega$, which implies that $U\\leq\\varepsilon$ in $\\omega$. Letting $\\varepsilon\\to 0$, we conclude that $U\\leq 0$ in $\\omega$, which, together with \\eqref{Unonneg}, shows that \\eqref{freezV} holds.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzjzcz b/data_all_eng_slimpj/shuffled/split2/finalzzjzcz new file mode 100644 index 0000000000000000000000000000000000000000..3f7d81f8c76fb81fdcb8521423b5c5ecf71c4e5e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzjzcz @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe intersection pairing between two divisors on a projective non-singular surface is the unique bilinear and symmetric pairing with values in $\\mathbb Z$ that satisfies some very natural properties: it counts the number of intersection points (when the divisor are normal crossing) and it is invariant if we ``move'' any of the two divisors in their linear class of equivalence. With the same philosophy, such a definition of intersection pairing between divisors can be extended naturally for projective non-singular varieties of any dimension: we list a number of natural properties and we find a unique multi-linear symmetric pairing satisfying them. It turns out that this unique intersection pairing on algebraic varieties can be expressed explicitly in terms of the Euler-Poincare characteristics of (invertible sheaves associated to the) divisors. For example, for a surface over a field $k$ we have the well known formula:\n\\begin{equation}\\label{eq_intro}\nC.D=\\chi_k(\\mathscr O_X)-\\chi_k(\\mathscr O_X(C)^{-1})-\\chi_k(\\mathscr O_X(D)^{-1})+\\chi_k(\\mathscr O_X(C)^{-1}\\otimes\\mathscr O_X(D)^{-1}) \n\\end{equation}\nwhich is involved in the proof of the Riemann-Roch theorem for surfaces (see for example \\cite{Beau}).\n\n\nFor a relative scheme $X\\to S$, if we don't appeal to any ``compactification arguments'' of $X$ and $S$, there is in general no hope for finding a reasonable intersection pairing for divisors which is invariant up linear equivalence. Let's see an example in the case of an arithmetic surface $X\\to \\spec \\mathbb Z$. Consider a prime $p\\in\\spec\\mathbb Z$, the fibre $X_{p}$ is a principal vertical divisor on $X$. Now let $D$ be an irreducible horizontal\ndivisor on $X$, then certainly $D$ meets $X_p$, which means $D.X_p> 0$ because our phantomatic intersection pairing should count the number of intersection points with multiplicity. Thus, for any other divisor $E\\in \\Div(X)$ we obtain\n$D.E < D.(E + X_p) $\nbut $E \\sim(E + X_p)$ since $X_p$ is principal by construction.\n\nThe closest object to an intersection pairing on a relative scheme $X\\to S$ of relative dimension $n$ is called the Deligne pairing: it is a map\n$$\\left<\\;,\\ldots,\\,\\right>_{X\/S}:\\operatorname{Vec}_1(X)^{n+1}\\to \\operatorname{Vec}_1(S)\\,,$$ \nwhere $\\operatorname{Vec}_1(\\cdot)$ denotes the set of invertible sheaves, which descends to a symmetric, multi-linear map at the level of Picard groups. This pairing is of crucial importance in arithmetic geometry, since it gives ``the schematic contribution'' to the Arakelov intersection number.\n\nThe Deligne pairing was constructed by Deligne in \\cite{Del} for arithmetic surfaces and then generalised to any dimension in \\cite{Elk}, \\cite{Zh} and \\cite{Gar}. Its definition was not built as the unique solution of a universal problem, but it was given a list of properties satisfied by the Deligne pairing. A set of axioms that uniquely identify the Deligne pairing were find recently in the preprint \\cite{Xia}. \n\n\nFor arithmetic surfaces one can show the following isomorphism of invertible sheaves which turns out to be very useful in the proof of Faltings-Riemann-Roch theorem (see for example \\cite{Del} and \\cite{MB} for more details):\n\\begin{equation}\\label{eq_intro1}\n\\left<\\mathscr L,\\mathscr M\\right>_{X\/S}\\cong \\det Rf_\\ast(\\mathscr O_X)\\otimes (\\det Rf_\\ast(\\mathscr L))^{-1}\\otimes(\\det Rf_\\ast(\\mathscr M))^{-1}\\otimes\\det Rf_\\ast(\\mathscr L\\otimes \\mathscr M)\\,.\n\\end{equation}\nOne can notice immediately the similarities between equations (\\ref{eq_intro}) and (\\ref{eq_intro1}). The only substantial difference is that for algebraic surfaces we use the Euler-Poincare characteristic whereas for arithmetic surfaces we use the determinant of the cohomology. Such a distinction makes perfectly sense because the determinant of the cohomology is constructed to be the arithmetic analogue of the Euler-Poincare characteristic.\n\nAt this point the natural question is the following one: is it possible to give an explicit definition of the Deligne pairing (in the most general case) in terms of the determinant of cohomology\\footnote{This was briefly conjectured already in \\cite{Elk}: `` (...) Dans le cas g\\'en\\'eral, c'est-\\`a-dire en dimension quelconque, et sans hypoth\\`ese de lissit\\' e, l'intersection doit aussi s'exprimer en termes de d\\'eterminants d'images directes (...) malheureusement, pour l'instant, des probl\\'e mes de signe obscurcissent s\\'e rieusement la situation.''}? In this paper we give an affirmative answer. By working in complete analogy of the theory of algebraic varieties, we write down a simple explicit formula for the Deligne pairing in terms of the determinant of cohomology. Let $f:X\\to S$ be a proper, flat morphism of reduced Noetherian schemes, and assume that $f$ has pure dimension $n$, then we put\n\n\\begin{equation}\\label{eq_intro2}\n\\left\\{\\begin{array} {lll}\n\\left<\\mathscr L_0,\\ldots, \\mathscr L_n\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)c_1(\\mathscr L^{-1}_1)\\ldots c_1(\\mathscr L^{-1}_n)\\mathscr O_X\\right)}\\, & \\textrm{if } n>0\\\\\n&\\\\\n\\left< \\mathscr L_0\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)\\mathscr O_X\\right)}^{-1}=N_{X\/S}(\\mathscr L_0)& \\textrm{if } n=0\\\\\n\\end{array}\\right.\n\\end{equation}\n\n\\noindent where $c_1(\\mathscr L)\\mathscr F:=\\mathscr F-\\mathscr L^{-1}\\otimes \\mathscr F\\in K_0(X)$ for any coherent sheaf $\\mathscr F$ and any invertible sheaf $\\mathscr L$ (here only the class of $\\mathscr F$ in the Grothendieck group matters). Moreover $N_{X\/S}$ is the norm, relative to $f$, of an invertible sheaf. We show that definition (\\ref{eq_intro2}) satisfies the axioms of \\cite{Xia}, and this implies that our definition is exactly the Deligne pairing.\n\n\nLet's mention some other papers that previously investigated in our direction: an explicit formula for the Deligne pairing when $X$ and $S$ are integral schemes over $\\mathbb C$ was announced in \\cite{BSW}, although a complete proof is not given. The approach of \\cite{BSW} is essentially different from ours, indeed the authors work on local trivializations of invertible sheaves. A more complicated expression of the Deligne pairing in terms of symmetric difference of the functor $\\det Rf_\\ast$ is proved in \\cite{Ducr} with heavy usage of category theory. An explicit calculation of the Deligne pairing $\\left<\\mathscr L,\\ldots,\\mathscr L\\right>$ (here $\\mathscr L$ is always the same) is given in \\cite{Phon}.\\\\\n\nThis paper is organized in the following way: in section \\ref{endo} we introduce the map $c_1(\\mathscr L):K_0(X)\\to K_0(X)$ with all its properties. Section \\ref{int_var} is a review of intersection theory for algebraic varieties and it gives to the reader the philosophical guidelines for the case of relative schemes. In section \\ref{rel_sch} we give the axioms of the Deligne pairing and we show with all details that if such pairing exists it must be unique (we follow \\cite{Xia}). Afterwards we show that the pairing (\\ref{eq_intro2}) satisfies all the axioms. appendix \\ref{det} is a review of the determinant of cohomology, this part is crucial in order to understand section \\ref{rel_sch}. Finally appendix \\ref{ori_constr} is a ``bonus section'' where we show with all details the original construction of Deligne pairing of \\cite{Del} (very often this construction is just sketched in the literature).\n\n\\paragraph{Acknowledgements.}\nThe author wants to express his gratitude to R.S. De Jong for his time spent in discussing the topic during summer 2019 in Nottingham and for his precious comments. A special thanks goes also to P. Corvaja, I. Fesenko, S. Urbinati and F. Zucconi.\\\\\n\n\\noindent This research was supported by the Italian national grant ``Ing. Giorgio Schirillo\" conferred by INdAM and partially by the EPSRC programme grant EP\/M024830\/1 (Symmetries and correspondences: intra-disciplinary developments and applications).\n\n\\section{An endomomorphism of the group $K_0(X)$}\\label{endo}\n Let's briefly recall the abstract construction of the Groethendieck group $K_0(\\catname C)$. Fix an abelian category $\\catname{C}$ and let $F(\\catname C)$ be the free abelian group over the set $\\Ob(\\catname C)\/\\cong$, where $\\cong$ is the isomorphism relation. If $C\\in \\Ob(\\catname C)$, then $(C)$ denotes isomorphism class in $\\Ob(\\catname C)\/\\cong$. To any short exact sequence in $\\catname C$:\n$$\\mathcal S\\colon \\quad 0\\to C'\\to C\\to C''\\to 0$$\nwe associate an element $ Q(\\mathcal S):=(C)- (C')-(C'')\\in F(\\catname C)$. Now $H(\\catname C)$ is the subgroup of $F(\\catname C)$ generated by all the elements $Q(\\mathcal S)$ for $\\mathcal S$ running over all short exact sequences. Then:\n$$K_0(\\catname C):= F(\\catname C)\/H(\\catname C)\\,,$$\nand $[C]\\in K_0(\\catname C)$ denotes the equivalence class associated to $C\\in \\Ob(\\catname C)$.\n\n\n\nLet's fix a reduced Noetherian scheme $X$, then $K_0(X):=K_0(\\catname {Coh}(X))$, where $\\catname {Coh}(X)$ is the category of coherent sheaves on $X$. From now on, by an abuse of notation we identify any coherent sheaf $\\mathscr F$ with its class in $K_0(X)$. In this paper, with the notation $\\catname {Coh}_r(X)$ we denote the category of coherent sheaves on $X$ whose support has dimension at most $r$, and we define $K_{0,r}(X):= K_0(\\catname {Coh}_r(X))$. Clearly when $0\\le i\\le j$, then $K_{0,i}(X)\\subseteq K_{0,j}(X)$.\n\nFor any invertible sheaf $\\mathscr L$ on $X$ we define a map:\n\n\\begin{equation}\n\\begin{aligned}\n c_1(\\mathscr L):K_0(X)&\\to K_0(X)\\\\\n\\mathscr F &\\mapsto c_1(\\mathscr L)\\mathscr F:= \\mathscr F-\\mathscr L^{-1}\\otimes \\mathscr F\\,.\n\\end{aligned}\n\\end{equation}\nNote that it is well defined because tensoring with an invertible sheaf is an exact functor, moreover it defines and endomorphism of the group $K_0(X)$. Since the notation for the function $c_1(\\mathscr L)$ is multiplicative, the symbol $c_1(\\mathscr L)c_1(\\mathscr L')$ denotes the composition of functions. The properties of the operator $c_1(\\mathscr L)$ are well described in \\cite[Appendix B]{Kl}, so here we just recall them.\n\n\\begin{proposition}\\label{propc_1}\nThe following properties hold for the operator $c_1(\\mathscr L)$:\n\\begin{itemize}\n\\item[$(i)$] $c_1(\\mathscr L)c_1(\\mathscr M)=c_1(\\mathscr L)+ c_1(\\mathscr M)-c_1(\\mathscr L\\otimes \\mathscr M)$, where clearly the sum is taken in $\\operatorname{End}(K_0(X))$.\n\\item[$(ii)$] $c_1(\\mathscr M) c_1(\\mathscr L)=c_1(\\mathscr L) c_1(\\mathscr M).$\n\\item[$(iii)$] If $Z\\subset X$ is a closed subscheme and $\\mathscr L_{|Z}=\\mathscr O_Z(D)$ where $D$ is an effective Cartier divisor on $Z$, then $c_1(\\mathscr L) \\mathscr O_Z=\\mathscr O_D$.\n\\end{itemize}\n\\end{proposition}\n\\proof\nBoth sides of the equality in $(i)$ expand to:\n\n\\begin{equation}\\label{e_comm}\n\\mathscr F-\\mathscr L^{-1}\\otimes \\mathscr F-\\mathscr M^{-1}\\otimes \\mathscr F\\mathscr +\\mathscr L^{-1}\\otimes\\mathscr M^{-1}\\otimes\\mathscr F\\,.\n\\end{equation}\n$(ii)$ Follows easily by looking at equation (\\ref{e_comm}). For $(iii)$ Consider the short exact sequence\n$$0\\to\\mathscr O_Z(-D)\\to\\mathscr O_Z\\to\\mathscr O_D\\to 0\\,.$$\n\\endproof\n\\begin{proposition}\nLet $\\mathscr F\\in K_{0,r}(X)$ and let $Z_1,\\ldots, Z_s$ the $r$-dimensional irreducible components of $\\supp(\\mathscr F)$ whose generic points are denoted respectively by $z_i$. Let $n_i=\\len \\mathscr F_{z_i}$. Then in $K_{0,r}(X)$ we have the equality:\n$$\\mathscr F\\equiv \\sum^s_{i=1} n_i\\mathscr O_{Z_i}\\mod K_{0,r-1}(X)\\,.$$\n\\end{proposition}\n\\proof\nSee \\cite[Lemma B4]{Kl}.\n\\endproof\n\n\\begin{proposition}\\label{decreasing_sup}\nlet $\\mathscr L$ an invertible sheaf on $X$, then $c_1(\\mathscr L)K_{0,r}(X)\\subset K_{0,r-1}(X)$ for any $r\\ge 0$.\n\\end{proposition}\n\\proof\nSee \\cite[Lemma B5]{Kl}.\n\\endproof\n\n\n\\begin{remark}\nThe operator $c_1(\\mathscr L)$ can be ``extended'' to bounded complexes of coherent sheaves on $X$. Let $\\mathscr F^\\bullet$ be a bounded complex of objects in $\\catname{Coh}(X)$ then we can define:\n$$c_1(\\mathscr L)\\mathscr F^\\bullet:=\\sum_i(-1)^ic_1(\\mathscr L)\\mathscr F^i\\in K_0(X)\\,.$$\nSuch a map is clearly zero on short exact sequences. \n\\end{remark}\n\n\\section{Intersection theory for algebraic varieties}\\label{int_var}\n\n\n\\begin{definition}\\label{inters_var}\nLet $X$ be a $n$-dimensional projective, non-singular algebraic variety over a field $k$. An \\emph{intersection pairing} on $X$ is a map:\n\\begin{equation}\n\\begin{aligned}\n\\Div(X)^n&\\to \\mathbb Z\\\\\n(D_1,\\ldots,D_n) &\\mapsto D_1.D_2.\\,\\ldots\\,.D_n\n\\end{aligned}\n\\end{equation}\nsatisfying the following properties:\n\\begin{itemize}\n\\item[$(1)$] It is symmetric and $\\mathbb Z$-multilinear.\n\\item[$(2)$] It descends to a pairing $\\Pic(X)^n\\to \\mathbb Z$.\n\\item[$(3)$] Let $D_i$ be an effective divisor for any $i$ and let $e_{i,x}\\in\\mathscr O_{X,x}$ be a local equation of $D_i$ at the point $x$. Assume that for all $x$ in the support of all divisors $D_i$, the $e_{i,x}$'s form a regular sequence in $\\mathscr O_{X,x}$ (i.e. the divisors are in general position), then: \n$$D_1.D_2.\\,\\ldots\\,.D_n=\\sum_{x\\in\\cap D_i}\\len\\frac{\\mathscr O_{X,x}}{(e_{1,x}, e_{2,x},\\ldots, e_{n,x})}$$ \n\\end{itemize}\n\\end{definition}\nNow we show that if an intersection pairing exists, it is uniquely defined by the three axioms of Definition \\ref{inters_var}.\n\\begin{proposition}\nIf an intersection pairing exists, then it is unique. \n\\end{proposition}\n\\proof\nLet $\\left<\\;\\right>_1$ and $\\left<\\;\\right>_2$ two pairings satisfying the axioms $(1)-(3)$ and fix $D_1,\\ldots,D_n\\in \\Div(X)^n$; by $(1)$ we can assume that all $D_i$ are effective. Thanks to Chow's moving lemma we can find some divisors $D'_i$ such that $D\\sim D_i'$ and $D'_1,\\ldots,D'_n$ are in general position. Therefore by using $(2)$ and $(3)$ we get:\n$$\\left_1=\\left_1=\\left_2=\\left_2\\,.$$\n\\endproof\nThe remaining part of this section is devoted to give the explicit expression of the intersection pairing on $X$ as in \\cite{Sn1}, \\cite{Sn2} and later \\cite{Ca}; then we see that the axioms of definition \\ref{inters_var} are satisfied. Such an intersection pairing uses the endomorphism defined in section \\ref{endo} and the Euler-Poincare characteristic for coherent sheaves. \n\nWe actually give a definition of the intersection pairing in a more general setting, in fact we will assume that $X$ is a relative scheme over a scheme $S$ and we define a ``partial'' intersection number of a particular subclass of divisors: roughly speaking, if we restrict to the case of arithmetic surfaces, we define an intersection number between two divisors provided that one of them is vertical. \n\nFrom now on in this section we assume that $X\\to S$ is a flat and proper morphism of irreducible Noetherian schemes. Let's denote with $\\catname{Coh}(X\/S)$ the category of coherent sheaves on $X$ whose schematic support is a proper over a $0$-dimensional subscheme of $S$. Moreover $\\catname{Coh}_r(X\/S)$ is the subcategory of $\\catname{Coh}(X\/S)$ made of sheaves whose support has dimension at most $r$. The motivation behind the restriction to sheaves with this kind of support is that for any $\\mathscr F\\in \\catname{Coh}(X\/S)$ we have a well defined notion of Euler-Poincare characteristic. In fact if $T$ is the schematic support of $\\mathscr F$ and $S_0=f(T)$, we know that $S_0$ is Noetherian of dimension $0$, so $S_0=\\spec A$ with $A$ artinian; at this point we can put\n\n$$\\chi_S(\\mathscr F):=\\sum_{i\\ge 0} (-1)^i\\len_A H^i(X, \\mathscr F)\\,.$$\nWhen $S=\\spec k$, then $\\chi_S$ is the usual Euler-Poicare characteristic (for coherent sheaves with proper support). Thanks to the ``additivity'' of $\\chi_S$ with respect to short exact sequences, it is immediate to notice that we have a naturally induced group homomorphism $\\chi_S:K_0(\\catname{Coh}_r(X\/S))\\to\\mathbb Z$.\n\\begin{definition}\nLet $X\\to S$ as above and let $\\mathscr F\\in\\catname{Coh}_r(X\/S)$. Then the \\emph{intersection number of the invertible sheaves $\\mathscr L_1,\\ldots\\mathscr L_r$ (with respect to $\\mathscr F$)} is defined as:\n\n$$(\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_r,\\mathscr F):=\\chi_S\\left(c_1(\\mathscr L_1)c_1(\\mathscr L_2)\\ldots c_1(\\mathscr L_r)\\mathscr F\\right)\\,.$$\nWhen $\\mathscr F=\\mathscr O_X$, which implies $r\\ge\\dim(X)$, we put for simplicity \n\n$$\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_r:=(\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_r,\\mathscr O_X)\\,.$$\nMoreover if $\\mathscr L_i=\\mathscr O_X(D_i)$ for a Cartier divisor $D_i$ on $X$, then:\n$$D_1.D_2.\\,\\ldots\\,.D_r:=\\mathscr O_X(D_1).\\mathscr O_X(D_2).\\,\\ldots\\,.\\mathscr O_X(D_r)\\,.$$ \n\n\\end{definition}\n\n\n\n\n\\begin{example}\nIf $X$ is a surface over $k$ and $C,D$ are two divisors, then:\n$$C.D=\\chi_k(c_1(\\mathscr O_X(C)) c_1(\\mathscr O_X(D))\\mathscr O_X)=\\chi_k(c_1(\\mathscr O_X(C))(\\mathscr O_X-\\mathscr O_X(D)^{-1}))=$$\n$$=\\chi_k(\\mathscr O_X)-\\chi_k(\\mathscr O_X(C)^{-1})-\\chi_k(\\mathscr O_X(D)^{-1})+\\chi_k(\\mathscr O_X(C)^{-1}\\otimes\\mathscr O_X(D)^{-1}) $$\n\\end{example}\n\nThe mere definitions tell us that we can intersect a number of divisors which is greater or equal to the dimension on $X$. On the other hand the next lemma shows that the intersection of a number of divisor which is strictly bigger than the dimension of $X$ is always $0$.\n\\begin{lemma}\\label{no_more_dim}\nIf $\\mathscr F\\in\\catname{Coh}_{r}(X\/S)$, then $(\\mathscr L_1.\\mathscr L_2.\\,\\ldots\\,.\\mathscr L_{r+1},\\mathscr F)=0$.\n\\end{lemma}\n\\proof\nIt follows directly from Proposition \\ref{decreasing_sup}.\n\\endproof\n\\begin{proposition}\\label{bilinear}\nThe intersection number of $\\mathscr L_1,\\ldots,\\mathscr L_m$ with respect to $\\mathscr F$ is a $\\mathbb Z$-multilinear map in the $\\mathscr L_j$'s (the operation is the tensor product).\n\\end{proposition}\n\\proof\nFollows by Proposition \\ref{propc_1}(i) and Lemma \\ref{no_more_dim}.\n\\endproof\n\\begin{proposition}\\label{mor_int}\nLet $g:X'\\to X$ be an morphism of $S$-schemes and let $\\mathscr F\\in\\catname{Coh}_r(X'\/S)$, then\n$$(g^\\ast\\mathscr L_1.g^\\ast\\mathscr L_1\\,.\\ldots g^\\ast\\mathscr L_n, \\mathscr F)=(\\mathscr L_1.\\mathscr L_1\\,.\\ldots \\mathscr L_n, g_\\ast\\mathscr F)\\,.$$ \n\\end{proposition}\n\\proof\nSee \\cite[Lemma B.15]{Kl}.\n\\endproof\n\nWe can give the explicit expression of the intersection number on varieties:\n\\begin{proposition}\nLet $X$ be an algebraic variety of dimension $r$ over a field $k$. The pairing $$(D_1,\\ldots,D_r)\\mapsto D_1.D_2.\\,\\ldots\\,.D_r$$ defines the intersection number on $X$.\n\\end{proposition}\n\\proof\nAxiom $(1)$ is satisfied thanks to Proposition \\ref{bilinear}. Axiom $(2)$ is obvious and axiom $(3)$ is \\cite[IV, Theorem 2.8]{Kol}.\n\\endproof\n\nFinally we state a proposition regarding the intersection along fibres:\n\n\\begin{proposition}\\label{gen_fiber}\nLet $s\\in S$ be a closed point and let $X_s$ be the fibre over $b$. Assume that $\\dim(X_s)=d$, then the map:\n$$s\\mapsto (\\mathscr L_1,\\ldots,\\mathscr L_d; \\mathscr O_{X_s})$$\nis locally constant on $S$.\n\\end{proposition}\n\\proof\nSee \\cite[VI, Proposition 2.10]{Kol}.\n\\endproof\n\n\\section{The case of schemes over a general base}\\label{rel_sch}\n\n\n\\subsection{Multi-monoidal and symmetric functors}\nThe Deligne pairing will be expressed as a collection of functors, so in this section we recall what is the functorial equivalent of a multi-linear homomorphism of abelian groups. \n\nWe assume that the reader is familiar with some basic notions of category theory and the concept of \\emph{Picard groupoid}. Roughly speaking a Picard groupoid is a category where morphisms are only isomorphisms and moreover there is a ``group-like'' operation between the object of the category. The simplest example is the \\emph{Picard category} $\\catname{Pic}(X)$ made of all invertible sheaves over a scheme $X$ where the morphisms are just the isomorphisms. The ``operation'' in $\\catname{Pic}(X)$ is clearly the tensor product of invertible sheaves and the identity element is the structure sheaf. The morphisms we want to consider between Picard groupoids are monoidal functors, i.e. functors that preserve the monoidal structure of the categories: \n\n\nFor the remaining part of this subsection we fix two Picard groupoids $(\\catname{C},\\otimes)$ and $(\\catname{D},\\otimes)$.\n\\begin{definition}\nA monoidal functor $\\catname{C}\\to\\catname{D}$ is a collection $(F,\\epsilon, \\mu)$ where $\\mu:=\\{\\mu_{X,Y}\\}_{X,Y\\in\\text{Obj(}\\catname C)}$, satisfying the following properties:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $F:\\catname{C}\\to\\catname{D}$ is a functor.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\epsilon\\colon F(1_{\\catname C})\\xrightarrow{\\cong} 1_{\\catname D}$ is an isomorphism.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\mu_{X,Y}\\colon F(X)\\otimes F(Y)\\xrightarrow{\\cong} F(X\\otimes Y)$ is an isomorphism functorial in $X$ and $Y$ which satisfies associativity and unitality in the obvious categorical sense.\n\\end{enumerate}\nFor simplicity we often omit $\\epsilon$ and $\\mu$ and we say that $F$ is a monoidal functor between $\\catname C$ and $\\catname{D}$. In symbols we write $F\\in L^1(\\catname C,\\catname D)$.\n\\end{definition}\n\n\\begin{definition}\\label{nat1}\nA \\emph{natural transformation} between monoidal functors $(F, \\epsilon,\\mu)$ and $(F',\\epsilon', \\mu')$ is a monoidal natural transformation $\\alpha:F\\to F'$ which maps $\\epsilon$ to $\\epsilon'$ and $\\mu$ to $\\mu'$.\n\\end{definition}\nIn order to give the next definition we need to introduce some notations. An object of the category $\\catname{C}^n$ (i.e. a $n$-uple of objects of $\\catname{C}$) is denoted by $ X=(X_1,\\ldots, X_n)$. Let $X, Y\\in \\catname{C}^n$ and let $i\\in\\{1,\\ldots,n\\}$ such that for any $j\\in\\{1,\\ldots,n\\}$ with $j\\neq i$ we have $X_j=Y_j$, then we define $ X\\otimes_i Y\\in\\catname{C}^n$ in the following way:\n\n$$\n(X\\otimes_i Y)_j=\\left\\{\\begin{array} {ll}\nX_j\\otimes Y_j & \\text{if } i=j\\\\\nX_j & \\text{if } i\\neq j\\\\\n\\end{array}\\right.\n$$\n\n\n\\begin{definition}\nA multi-monoidal\\footnote{Very often in literature one can find the term multi-additive.} functor $\\catname{C}^n\\to\\catname{D}$ is the datum of\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] A functor $F:\\catname{C}^n\\to \\catname{D}$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] For any functor $F':\\catname{C}\\to\\catname{D}$ obtained by fixing $n-1$ components in $\\catname C$, we have a collection $\\mu'$ such that $(F',\\mu')$ is a monoidal functor $\\catname{C}$ and $\\catname D$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] For every $i,j\\in\\{1,\\ldots, n\\} $ and $X,Y,Z,W\\in\\catname{C}^n$ such that $X_k=Y_k=Z_k=W_k$ for all $k\\neq i,j$, a commutative diagram:\n$$\n\\adjustbox{scale=0.8,center}{%\n\\begin{tikzcd}\n & F(X)\\otimes F(Y)\\otimes F(Z)\\otimes F(W) \\arrow{dl}\\arrow{dr} &\\\\\n(F(X)\\otimes_j F(Y))\\otimes (F(Z)\\otimes_j F(W))\\arrow{d}& & (F(X)\\otimes_i F(Y))\\otimes (F(Z)\\otimes_i F(W))\\arrow{d}\\\\\nF((X\\otimes_j Y)\\otimes_i (Z\\otimes_j W))\\arrow[\"=\"]{rr}&& (F(X)\\otimes_i F(Y))\\otimes_j (F(Z)\\otimes_i F(W))\n\\end{tikzcd}\n}\n$$\n\\end{enumerate}\n\\end{definition}\nThe notion of symmetry is what one expects:\n\\begin{definition}\nA multi-monoidal functor $\\catname{C}^n\\to\\catname{D}$ is \\emph{symmetric} if for any $c_i\\in\\catname C$ and any permutation $\\sigma\\in\\Sigma_n$ we have $F(c_1,\\ldots c_n)\\cong F(c_{\\sigma(1)},\\ldots c_{\\sigma(n)})$.\n\\end{definition}\n\nThe set of symmetric multi-monoidal morphisms from $\\catname{C}^n$ to $\\catname D$ is denoted by $L^n(\\catname C,\\catname D)$.\n\n\\begin{definition}\n\\emph{A natural transformation between two multi-monoidal functors} $F,F'\\in L^n(\\catname C,\\catname D)$ is a functorial isomorphism $\\alpha: F\\to F'$ which restricts to a natural transformation to each component in the sense of definition \\ref{nat1}.\n\\end{definition}\n\n\\subsection{Axiomatic Deligne pairing}\nThe Deligne pairing was introduced in \\cite{Del} as a bilinear and symmetric map $\\left<\\,,\\,\\right>:\\operatorname{Vec}_1(X)\\times\\operatorname{Vec}_1(X)\\to\\operatorname{Vec}_1(S)$, where $X\\to S$ is an arithmetic surface. Such a definition is quite implicit, in fact requires the choice of rational sections of invertible sheaves and some operations in the \\'etale topology on the base $S$. The Deligne pairing satisfies some compatibility conditions: it behaves well with respect to the base change and the pullback functor; moreover it is strictly related to the norm functor. In \\cite{Elk} Deligne's construction was extended straight away for proper flat morphisms of integral schemes of any dimension.\n\nLet $f:X\\to S$ a proper flat morphism between Noetherian reduced schemes, the guiding idea of this paper is that the Deligne pairing relative to $f$ should be the generalisation the intersection pairing described in section \\ref{int_var}. We want to work in complete analogy with the case of algebraic varieties, so in this section we give a set of ``natural axioms'' that uniquely define the Deligne pairing\\footnote{We follow \\cite{Xia}, but we prefer to give a self contained presentation with all details.}. The explicit construction of the Deligne pairing will be carried out in section \\ref{det_del}.\n\nLet $X$ and $S$ be two Noetherian reduced schemes, with the symbol $\\mathcal F^n(X,S)$ we denote the set of all proper flat morphisms $X\\to S$ of pure dimension $n$.\n\n\\begin{definition}\\label{axiom_deligne}\nA \\emph{Deligne pairing} consists of the following data for any two irreducible Noetherian schemes $X,S$, any $n\\in\\mathbb N$ and any $f\\in\\mathcal F^n(X,S)$: a functor \n$$\\left< \\,\\;,\\;,\\ldots,\\;\\right>_f=\\left< \\,\\;,\\;,\\ldots,\\;\\right>_{X\/S}\\in L^{n+1}(\\catname{Pic}(X), \\catname{Pic}(S))$$\nand a collection of natural transformations $\\alpha,\\beta,\\gamma,\\delta$ described below:\n\\begin{enumerate}\n\\item[(1)] For any commutative square given by a base change $g:S'\\to S$ which is proper, flat and with connected fibres \n\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\nX'=X\\times_S S'\\arrow[\"f'\"]{r}\\arrow[\"g'\"]{d} & S'\\arrow[\"g\"]{d}\\\\\nX\\arrow[\"f\"]{r} & S\n\\end{tikzcd}\n$$\na natural transformation between multi-monoidal functors $\\alpha_{f,g}:\\catname{Pic}(X)^{n+1}\\to \\catname{Pic}(S')$ such that\n $$\\alpha_{f,g}:g^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>_{X\/S}\\xrightarrow{\\cong}\\left_{X'\/S'}\\,.$$\n \n \\item[(2)] When $n>0$ and $D\\in\\Div(X)$ is an effective relative Cartier divisor, a natural transformation between multi-monoidal functors $\\beta_{f,D}:\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$ such that\n $$\\beta_{f,D}:\\left<\\mathscr L_1,\\ldots,\\mathscr L_n, \\mathscr O_X(D)\\right>_{X\/S}\\xrightarrow{\\cong}\\left<\\mathscr L_1|_D,\\ldots,\\mathscr L_n|_D\\right>_{D\/S}\\,.$$\nMoreover $\\beta_{f,D}$ is natural with respect to base change in the following sense: for a base change diagram as in axiom $(1)$ we have a commutative diagram: \n $$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\left_{X'\/S'}\\arrow[\"\\beta_{f',g'^\\ast D}\"]{r}\\arrow[\"\\cong\"]{d} & \\left_{{g'^\\ast D}\/S'}\\arrow[\"\\cong\"]{d}\\\\\ng^\\ast\\left<\\mathscr L_1,\\ldots,\\mathscr L_n, \\mathscr O_X(D)\\right>_{X\/S} & g^\\ast \\left<\\mathscr L_1|_D,\\ldots,\\mathscr L_n|_D\\right>_{D\/S}\\arrow[\"g^\\ast \\beta_{f,D}\"]{l} \n\\end{tikzcd}\n$$\nWhere the vertical isomorphisms are given by $\\alpha_{f,g}$ (remember that $g'^\\ast\\mathscr O_X(D)=\\mathscr O_{X'}(g'^\\ast D)$).\n \\item[(3)] When $n>0$, a natural transformation between multi-monoidal functors $\\gamma_{f}:\\catname{Pic}(S)\\times\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$\n such that\n $$\\gamma_{f}:\\left_{X\/S}\\xrightarrow{\\cong}\\mathscr L^{(\\mathscr L_1|_{X_s}.\\mathscr L_2|_{X_s}.\\ldots\\mathscr L_n|_{X_s};\\mathscr O_{X_s})}$$\nwhere $X_s$ is a generic fibre of $f$ (see Proposition \\ref{gen_fiber}). Moreover $\\gamma_{f}$ is natural with respect to base change in the following sense: for a base change diagram as in axiom $(1)$ we have a commutative diagram: \n $$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\left_{X'\/S'}\\arrow[\"\\gamma_{f'}\"]{r}\\arrow[\"\\cong\"]{d} & g^\\ast \\mathscr L^{(g'^\\ast \\mathscr L_1|_{g'^\\ast X_s}. g'^\\ast \\mathscr L_2|_{g'^\\ast X_s}.\\ldots g'^\\ast\\mathscr L_n|_{g'^\\ast X_s};\\mathscr O_{g'^\\ast X_s})}\\arrow[\"=\"]{d}\\\\\ng^\\ast\\left_{X\/S} & g^\\ast\\mathscr L^{(\\mathscr L_1|_{X_s}.\\mathscr L_2|_{X_s}.\\ldots\\mathscr L_n|_{X_s};\\mathscr O_{X_s})}\\arrow[\"g^\\ast \\gamma_{f}\"]{l} \n\\end{tikzcd}\n$$\nwhere the vertical isomorphism is given $\\alpha_{f,g}$ and the equality follows from proposition \\ref{mor_int} and the properties of $g$.\n\n\\item[(4)] When $n=0$, a natural transformation between monoidal functors $\\delta_{f}:\\catname{Pic}(X)\\to \\catname{Pic}(S)$\nsuch that\n $$\\delta_{f}:\\left<\\mathscr L\\right>_{X\/S}\\xrightarrow{\\cong} N_{X\/S}(\\mathscr L)$$\n where $N_{X\/S}$ is the norm of $f$ (see Definition \\ref{norm}). Moreover $\\delta_f$ is natural with respect to base change in the following sense: for a base change diagram as in axiom $(1)$ we have a commutative diagram:\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\left_{X'\/S'}\\arrow[\"\\delta_{f'}\"]{r}\\arrow[\"\\cong\"]{d} & N_{X'\/S'}(g'^\\ast\\mathscr L)\\arrow[\"\\cong\"]{d}\\\\\ng^\\ast\\left<\\mathscr L\\right>_{X\/S} & g^\\ast N_{X\/S}(\\mathscr L)\\arrow[\"g^\\ast \\delta_f\"]{l} \n\\end{tikzcd}\n$$\nwhere the vertical isomorphisms are given respectively by $\\alpha_{f,g}$ and thanks to the properties of the norm.\n\\end{enumerate} \n\\end{definition}\nWe have to show that if a Deligne pairing exists, then it is unique. Roughly speaking we will show that any two pairings $(\\left< \\,\\;,\\;,\\ldots,\\;\\right>^i,\\alpha^i,\\beta^i,\\gamma^i,\\delta^i)$, with $i=1,2$, satisfying the axioms of definition \\ref{axiom_deligne} are related by natural transformation of functors that respects all the data. We will work by induction on the relative dimension of the morphism $f$. Note that we cannot use straight away property $(2)$ to pass from relative dimension $n$ to $n-1$, since the whole construction will depend on the choice of the relative divisor $D$, but we want our constructions to be natural in a functorial way. Let's describe a general well known procedure to reduce the relative dimension of $f$ without any particular choice of divisors. It is called \\emph{universal extension}.\n\nLet $f\\in\\mathcal F^n(X,S)$ and let $\\mathscr L$ be an invertible sheaf on $X$. We assume that $\\mathscr L$ is \\emph{sufficiently ample (with respect to $f$)}, i.e. that the following properties are satisfied: $\\mathscr L$ is very ample (with respect to $f$) and $R^i f_\\ast\\mathscr L=0$ for $i>0$. \n\\begin{remark}\nThe following properties hold for sufficient ampleness:\n\\begin{itemize}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] It is preserved after base change.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] If $f\\in\\mathcal F^n(X,S)$ and $\\mathscr L$ is sufficiently ample on $X$, then $f_\\ast \\mathscr L$ is a locally free sheaf on $S$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] If $\\mathscr L_0$ is an invertible sheaf on $X$, then there exists a sufficiently ample $\\mathscr L$ such that $\\mathscr L_0\\otimes \\mathscr L$ is sufficiently ample. In particular we can always find on $X$ a sufficiently ample invertible sheaf.\n\\end{itemize}\n\\end{remark}\nPut $\\mathscr M=(f_\\ast\\mathscr L)^{\\vee}$ and let $\\mathbb P:=\\mathbb P_S(\\mathscr M)$ be the projective vector bundle associated to $\\mathscr M$, over $S$. Then we obtain the following base change diagram: \n\\begin{equation}\\label{diag_univ}\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\mathbb X:=\\mathbb P\\times_S X\\arrow[\"p_2\"]{r}\\arrow[\"p_1\"]{d} & X\\arrow[\"f\"]{d}\\\\\n\\mathbb P\\arrow[\"\\pi\"]{r} & S\n\\end{tikzcd}\n\\end{equation}\nConsider now the invertible sheaf $\\mathscr L_f:=p_1^\\ast\\mathscr O_\\mathbb P(1)\\otimes p_2^\\ast\\mathscr L$ on $\\mathbb X$. We want to construct a \\emph{canonical global section} $\\sigma$ of $\\mathscr L_f$. It is enough to find a canonical non-zero element in $\\mathscr L^{-1}_f$, because if $\\phi\\in\\Hom(\\mathscr O_X,\\mathscr L_f)=\\mathscr L^{-1}_f$ then we put $\\sigma:=\\phi_X(1)$.\nFirst of all we construct a surjective canonical morphism \n$$\\Psi:f^\\ast \\mathscr M\\to\\mathscr L \\,.$$ \nThanks to the properties of the pullback we have a canonical isomorphism $f^\\ast \\mathscr M \\cong (f^\\ast f_\\ast \\mathscr L)^\\vee$. Since $\\mathscr L$ is sufficiently ample, we have a canonical isomorphism $(f^\\ast f_\\ast \\mathscr L)^\\vee\\cong \\mathscr L^\\vee$. Moreover there is a surjective canonical map $\\mathscr L^\\vee\\to\\mathscr L$ given in the following way:\n\\begin{eqnarray*}\n\\Hom(\\mathscr O_X(U), \\mathscr L(U))& \\to& \\mathscr L(U)\\\\\n\\varphi_U &\\mapsto& \\varphi_U(1)\\,.\n\\end{eqnarray*}\nBy taking all compositions, we finally get our surjective $\\Psi$. We have to prove that $\\Psi$ induces a canonical element in $\\mathscr L^{-1}_f$ (in order to get $\\sigma$). Note that $\\mathscr L^{-1}_f$ is canonically isomorphic to $\\Hom(\\mathscr L^{-1}, {p_2}_\\ast p_1^\\ast \\mathscr O_{\\mathbb P}(1))$, but \n$${p_2}_\\ast p_1^\\ast \\mathscr O_{\\mathbb P}(1)=f^\\ast \\pi_\\ast\\mathscr O_{\\mathbb P}(1)=f^\\ast (\\mathscr M^\\vee)=(f^\\ast \\mathscr M)^\\vee\\,.$$\nWe conclude that the dual map of $\\Psi$ induces the non-zero element of $\\mathscr L^{-1}_f$ that we were searching for.\n\nFrom now on we will say that the section $\\sigma$ constructed above is the universal section relative to $\\mathscr L$. The following remark explains why we can use the universal section for our inductive step in the proof of uniqueness:\n\n\\begin{remark}\nIn \\cite[2.2]{Gar} it is shown that $\\sigma$ is a regular section, which is equivalent to say that the zero locus of $\\sigma$ (considered with its reduced scheme structure):\n$$Z(\\sigma):=\\{x\\in\\mathbb X\\colon 0=\\sigma(x)\\in \\mathscr L_f\/\\mathfrak m_x \\mathscr L_f\\}$$\nis a relative Cartier divisor on $\\mathbb X$. In this case we also have that $\\mathscr L_f$ is canonically isomorphic to $\\mathscr O_{\\mathbb X}(Z(\\sigma))$. Now consider the restriction \n$$p:=(p_1)|_{Z(\\sigma)}: Z(\\sigma )\\to \\mathbb P\\,;$$\nLet $U$ be the flat locus of $p$ and put $V:=p(U)$. Then $V$ is open in $\\mathbb P$, and we denote its closed complementar with $W$, then we conclude that \n\\begin{equation}\\label{section_map}\np: Z(\\sigma)-p^{-1}(W)\\to V\n\\end{equation}\nis flat of relative dimension $n-1$.\n\\end{remark}\n\n\n\n\n\n\n\nThe following theorem ensures the unicity of the Deligne pairing:\n\\begin{theorem}\nThe Deligne pairing is unique: given two sets of data $(\\left< \\,\\;,\\;,\\ldots,\\;\\right>^i,\\alpha^i,\\beta^i,\\gamma^i,\\delta^i)$, with $i=1,2$, satisfying the conditions of Definition \\ref{axiom_deligne}, there is a unique multi-additive morphism $\\left< \\,\\;,\\;,\\ldots,\\;\\right>^1\\to \\left< \\,\\;,\\;,\\ldots,\\;\\right>^2$ that transforms $\\alpha^i,\\beta^i,\\gamma^i,\\delta^i$ accordingly.\n\\end{theorem}\n\\proof\nWe proceed by induction on $n$. When $n=0$, the claim follows directly from axiom $(4)$. Let's work now with $n>0$; first of all we want a functorial isomorphism:\n\\begin{equation}\\label{iso_to_show}\n\\Psi(\\mathscr L_0,\\ldots, \\mathscr L_n):\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^1_{X\/S}\\xrightarrow{\\cong}\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^2_{X\/S}\\,.\n\\end{equation}\nLet's first construct it by assuming that one invertible sheaf $\\mathscr L=\\mathscr L_0$, is chosen sufficiently ample; we will denote it as $\\Psi'(\\mathscr L_0,\\ldots, \\mathscr L_n)$. Let's construct for $\\mathscr L$ the base change diagram (\\ref{diag_univ}), with the same notations. Then $\\sigma$ is the universal section of $\\mathscr L_f$ and we also have the map $p$ described in equation (\\ref{section_map}). Thanks to \\cite[Lemme 21.13.2]{EGAIV}, in order to give isomorphism (\\ref{iso_to_show}), it is enough to give a functorial isomorphism:\n\\begin{equation}\\label{iso_to_show1}\n(\\pi|_{V})^\\ast\\left<\\mathscr L, \\mathscr L_1,,\\ldots,\\mathscr L_n\\right>^1_{(\\mathbb X-p_1^{-1}(W))\/V}\\xrightarrow{\\cong}(\\pi|_{V})^\\ast\\left<\\mathscr L,\\mathscr L_1,\\ldots,\\mathscr L_n\\right>^2_{(\\mathbb X-p_1^{-1}(W))\/V}\\,.\n\\end{equation}\nwhere $V\\subset \\mathbb P$ is the image of the flat locus of $p$ (remember that $V$ is open) and $W=\\mathbb P-V$. Let's now put $q:=p_2|_{\\mathbb X-p_1^{-1}(W)}$. By applying axiom $(1)$, it is enough to get a functorial isomorphism:\n\\begin{equation}\\label{iso_to_show2}\n\\left^1_{(\\mathbb X-p_1^{-1}(W))\/V}\\xrightarrow{\\cong}\\left^2_{(\\mathbb X-p_1^{-1}(W))\/V}\\,.\n\\end{equation}\nNow remember that by definition of $\\mathscr L_f$ we have:\n$$q^\\ast\\mathscr L=\\mathscr L_f|_{(\\mathbb X-p_1^{-1}(W))}\\otimes (p_1^\\ast\\mathscr O_\\mathbb P(1))^{-1}$$ \nLet's put for simplicity of notations $\\mathscr M:=\\mathscr L_f|_{(\\mathbb X-p_1^{-1}(W))}$; by multi-additivity and axiom $(3)$ we only need to find a functorial isomorphism\n\\begin{equation}\\label{iso_to_show3}\n\\left<\\mathscr M,q^\\ast\\mathscr L_1\\ldots,q^\\ast\\mathscr L_n\\right>^1_{(\\mathbb X-p_1^{-1}(W))\/V}\\xrightarrow{\\cong}\\left<\\mathscr M,q^\\ast\\mathscr L_1\\ldots,q^\\ast\\mathscr L_n\\right>^2_{(\\mathbb X-p_1^{-1}(W))\/V}\\,.\n\\end{equation}\nAt this point put $Z'(\\sigma):=Z(\\sigma)-p_1^{-1}(W)$; thanks to axiom $(2)$, it is enough to find a functorial isomorphism: \n\\begin{equation}\\label{iso_to_show4}\n\\left<(q^\\ast\\mathscr L_1)|_{Z'(\\sigma)}\\ldots,q^\\ast(\\mathscr L_n)|_{Z'(\\sigma)}\\right>^1_{Z'(\\sigma)\/V}\\xrightarrow{\\cong}\\left<(q^\\ast\\mathscr L_1)|_{Z'(\\sigma)}\\ldots,q^\\ast(\\mathscr L_n)|_{Z'(\\sigma)}\\right>^2_{Z'(\\sigma)\/V}\\,.\n\\end{equation}\nThe relative dimension of the map $p:Z'(\\sigma)\\to V$ is now $n-1$ and we can apply the inductive hypothesis.\n\nWe still have to prove the the existence of $\\Psi(\\mathscr L_0,\\ldots, \\mathscr L_n)$ for a general $\\mathscr L_0$. For any invertible sheaf $\\mathscr L_0$ there exists a sufficiently ample one $\\mathscr M$ such that $\\mathscr L_0\\otimes\\mathscr M$ is again sufficiently ample. So we can put:\n$$\\Psi(\\mathscr L_0,\\ldots, \\mathscr L_n):=\\Psi'(\\mathscr L_0\\otimes \\mathscr M,\\ldots, \\mathscr L_n)\\otimes \\Psi'(\\mathscr M,\\ldots, \\mathscr L_n)^{-1}$$\nprovided that the construction doesn't depend on the choice of $\\mathscr M$. Such a claim is equivalent to show that $\\Psi'(\\cdot\\,,\\mathscr L_1,\\ldots, \\mathscr L_n)$ is additive with respect to sufficiently ample invertible sheaves.\n\nNow we consider two sufficiently ample invertible sheaves $\\mathscr L^{(i)}$ for $i=1,2$ and the associated diagrams:\n\\begin{equation}\\label{diag_univ1}\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\mathbb X^{(i)}:=\\mathbb P^{(i)}\\times_S X\\arrow[\"p^{(i)}_2\"]{r}\\arrow[\"p^{(i)}_1\"]{d} & X\\arrow[\"f\"]{d}\\\\\n\\mathbb P^{(i)}\\arrow[\"\\pi^{(i)}\"]{r} & S\n\\end{tikzcd}\n\\end{equation}\nwhere clearly $\\mathbb P^{(i)}:=\\mathbb P_S(\\mathscr M^{(i)})$ for $\\mathscr M^{(i)}:=(f_\\ast(\\mathscr L^{(i)}))^\\vee$. On the other hand if we put $\\mathscr L:=\\mathscr L^{(1)}\\otimes \\mathscr L^{(2)}$ and $\\mathbb P:=\\mathbb P_S((f_\\ast\\mathscr L^{(1)})^\\vee \\otimes (f_\\ast\\mathscr L^{(2)})^\\vee)$ we end up with the diagram (\\ref{diag_univ1}). There is a natural map \n$$\\iota: \\mathbb X^{(1)}\\times_X \\mathbb X^{(2)}\\to\\mathbb X$$\nLet $q_i$ the projections of $\\mathbb X^{(1)}\\times_X \\mathbb X^{(2)}$ on the factors $\\mathbb X^{(i)}$, then one can show that:\n$$\\iota^\\ast\\mathscr L_f=q_1^\\ast\\mathscr L^{(1)}_f\\otimes q_2^\\ast\\mathscr L^{(2)}_f\\,,$$\n$$\\iota^\\ast \\sigma= q_1^\\ast\\sigma^{(1)}\\otimes q_2^\\ast\\sigma^{(2)}\\,.$$\nFrom the properties of the universal extension discussed in \\cite[I.2]{Elk} the claim follows.\n\nIt remains to show that $\\Psi$ transforms $\\alpha^{1},\\beta^{1},\\gamma^{1},\\delta^{1}$ to $\\alpha^{2},\\beta^{2},\\gamma^{2},\\delta^{2}$. Let's do it for $\\alpha^{i}$, the other cases are similar. In particular we have to prove that, given a base change diagram as in axiom (1), we get a commutative diagram:\n\\begin{equation}\\label{diag_alpha}\n\\begin{tikzcd}[row sep=large, column sep = huge]\ng^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^{1}_{X\/S}\\arrow[\"\\cong\"]{r}\\arrow[\"\\cong\"]{d} & \\left^1_{X'\/S'}\\arrow[\"\\cong\"]{d}\\\\\ng^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>^{2}_{X\/S}\\arrow[\"\\cong\"]{r} & \\left^2_{X'\/S'}\\,.\n\\end{tikzcd}\n\\end{equation}\nIn order to construct (\\ref{diag_alpha}) it is enough to proceed similarly as we did above: we work by induction on $n$. If $n=0$ the claim follows from the property of $\\delta^i$ with respect to base change. For the generic $n$ we can use the universal extension procedure described above and the properties of $\\beta^i$ with respect to base change to reduce to $n-1$.\n\\endproof\n\n\\subsection{Deligne pairing in terms of determinant of cohomology}\\label{det_del}\nIn this section we heavily use the proprieties of the determinant functor presented in appendix \\ref{det} in order to give an explicit expression of the Deligne pairing in terms of the determinant of cohomology.\n\n \nLet $f:X\\to S$ be a flat morphism between Noetherian irreducible schemes. If $\\catname{Vec}(X)$ is the category of locally free sheaves of finite rank on $X$, the first think to notice is that $\\det Rf_\\ast $ descends to a map on $K_0(\\catname{Vec}(X))$. Now we put:\n$$\n\\left\\{\\begin{array} {lll}\n\\left<\\mathscr L_0,\\ldots, \\mathscr L_n\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)c_1(\\mathscr L^{-1}_1)\\ldots c_1(\\mathscr L^{-1}_n)\\mathscr O_X\\right)}\\, & \\textrm{if } n>0\\\\\n&\\\\\n\\left< \\mathscr L_0\\right>_{X\/S}:={\\det Rf_\\ast\\left(c_1(\\mathscr L^{-1}_0)\\mathscr O_X\\right)}^{-1}=N_{X\/S}(\\mathscr L_0)& \\textrm{if } n=0\\\\\n\\end{array}\\right.\n$$\nWe want to show that this defines the Deligne pairing i.e. that there are some ``canonical'' natural transformations associated to $\\left<\\;,\\ldots, \\;\\right>_{X\/S}$ satisfying all axioms of definition \\ref{axiom_deligne}. \n\\begin{remark}\nWhen $n=1$, after some simple algebraic manipulations we obtain the expected result:\n$$\\left<\\mathscr L,\\mathscr M\\right>=\\det Rf_\\ast(c_1(\\mathscr L^{-1}) c_1(\\mathscr M^{-1})\\mathscr O_X)=\\det Rf_\\ast(c_1(\\mathscr L^{-1})(\\mathscr O_X-\\mathscr M))=$$\n$$=\\det Rf_\\ast(\\mathscr O_X)\\otimes\\det Rf_\\ast(\\mathscr L)^{-1}\\otimes\\det Rf_\\ast(\\mathscr M)^{-1}\\otimes\\det Rf_\\ast(\\mathscr L\\otimes\\mathscr M) $$\n\\end{remark}\n\nLike in the case of algebraic varieties, Proposition \\ref{propc_1} ensures that $\\left<\\;,\\ldots, \\;\\right>_{X\/S}$ is multi-monoidal and symmetric. Moreover axiom (4) of Definition \\ref{axiom_deligne} is trivially satisfied by definition (see Definition \\ref{norm}). So it remains to show that axioms (1)-(3) are satisfied. \n\n\n\\begin{proposition}[Axiom (1) holds]\nFor any commutative square given by a base change $g:S'\\to S$ which is proper, flat and with connected fibres \n\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\nX'=X\\times_S S'\\arrow[\"f'\"]{r}\\arrow[\"g'\"]{d} & S'\\arrow[\"g\"]{d}\\\\\nX\\arrow[\"f\"]{r} & S\n\\end{tikzcd}\n$$\nthere is a natural transformation between multi-additive morphisms $\\alpha_{f,g}:\\catname{Pic}(X)^{n+1}\\to \\catname{Pic}(S')$ such that\n $$\\alpha_{f,g}:g^\\ast\\left<\\mathscr L_0,\\ldots,\\mathscr L_n\\right>_{X\/S}\\xrightarrow{\\cong}\\left_{X'\/S'}\\,.$$\n\\end{proposition}\n\\proof\nFirst of all we have that for any invertible sheaf $\\mathscr L$ on $X$ and any coherent sheaf $\\mathscr F'$ on $X'$:\n$$c_1(\\mathscr L)Rg'_\\ast\\mathscr F'=Rg'_\\ast (c_1(g'^\\ast\\mathscr L)\\mathscr F')$$\n(see for example the proof of \\cite[Lemma B.15]{Kl} for a detailed explanation of the above equality). Therefore\n$$c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_n)Rg'_\\ast(\\mathscr O_{X'})=c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_{n-1})Rg'_\\ast (c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'})=\\ldots$$\n$$\\ldots=Rg'_\\ast(c_1(g'^\\ast\\mathscr L_0)\\ldots c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'})\\,.$$\nIt means that:\n\n\\begin{equation}\\label{functor_mess}\ng^\\ast\\det Rf_\\ast\\left(c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_n)Rg'_\\ast(\\mathscr O_{X'})\\right)=g^\\ast\\det Rf_\\ast \\left(Rg'_\\ast(c_1(g'^\\ast\\mathscr L_0)\\ldots c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'})\\right)\\,.\n\\end{equation}\nBut thanks to the properties of the morphism $g:S'\\to S$ we have that $g'_\\ast\\mathscr O_{X'}=\\mathscr O_X$ (see for example \\cite[Exercise 3.11]{Kol}, so it follows that the left hand side of equation (\\ref{functor_mess}) is \n$$g^\\ast\\det Rf_\\ast\\left(c_1(\\mathscr L_0)\\ldots c_1(\\mathscr L_n)\\mathscr O_X\\right)\\,.$$\n On the right hand side of equation (\\ref{functor_mess}) note that we have the composition of the following functors:\n\\begin{equation}\\label{functor_mess1}\ng^\\ast\\circ\\det{\\!}_{S}\\circ Rf_\\ast\\circ Rg'_\\ast\\,.\n\\end{equation}\nBy the properties of the determinant functor equation (\\ref{functor_mess1}) is naturally isomorphic to\n$$\n\\det{\\!}_{S'}\\circ Lg^\\ast \\circ Rf_\\ast\\circ Rg'_\\ast\\cong\\det{\\!}_S'\\circ Rf'_\\ast=\\det Rf'_\\ast\\,.\n$$\nIn other words we obtained that the right hand side of equation (\\ref{functor_mess}) is naturally isomorphic to \n$$\\det Rf'_\\ast\\left(c_1(g'^\\ast\\mathscr L_0)\\ldots c_1(g'^\\ast\\mathscr L_n)\\mathscr O_{X'}\\right)\\,.$$\n\\endproof\n\n\\begin{remark}\nFor axioms (2) and (3), we only have to show that the natural transformations exist, since their ``good behaviour'' with respect to base change is ensured by the properties of the determinant of cohomology with respect to base change i.e. equation (\\ref{det_bchange}).\n\\end{remark}\n\n\\begin{proposition}[Axiom (2) holds]\nWhen $n>0$ and $D\\in\\Div(X)$ is an effective relative Cartier divisor, there is a natural transformation between multi-additive morphisms $\\beta_{f,D}:\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$ such that\n $$\\beta_{f,D}:\\left<\\mathscr O_X(D),\\ldots,\\mathscr L_n\\right>_{X\/S}\\xrightarrow{\\cong}\\left<\\mathscr L_1|_D,\\ldots,\\mathscr L_n|_D\\right>_{D\/S}\\,.$$\nMoreover such a transformation is natural with respect to base change.\n\\end{proposition}\n\\proof\nThis follows by the simple fact that $c_1(\\mathscr O_X(D))\\mathscr O_X=\\mathscr O_D$ (see Proposition \\ref{propc_1}(iii)).\n\\endproof\n\n\n\\begin{proposition}[Axiom (3) holds]\nWhen $n>0$ there is a natural transformation between multi-additive morphisms $\\gamma_{f}:\\catname{Pic}(S)\\times\\catname{Pic}(X)^{n}\\to \\catname{Pic}(S)$\n such that\n $$\\gamma_{f}:\\left_{X\/S}\\xrightarrow{\\cong}\\mathscr L^{(\\mathscr L_1|_{X_s}.\\mathscr L_2|_{X_s}.\\ldots\\mathscr L_n|_{X_s};\\mathscr O_{X_s})}$$\nwhere $X_s$ is a generic fibre of $f$. Moreover such a transformation is natural with respect to base change.\n\\end{proposition}\n\\proof\nLet's put $\\mathscr E:= c_1(\\mathscr L_1)\\ldots c_1(\\mathscr L_n)\\mathscr O_X$, then:\n$$\n\\det Rf_\\ast (c_1(f^\\ast \\mathscr L)c_1(\\mathscr L_1)\\ldots c_1(\\mathscr L_n)\\mathscr O_X)=\\det Rf_\\ast(c_1(f^\\ast \\mathscr L)\\mathscr E)=\\det Rf_\\ast(\\mathscr E-(f^\\ast \\mathscr L)^{-1}\\otimes \\mathscr E)=\n$$\n$$\n=\\det Rf_\\ast(\\mathscr E)\\otimes \\det Rf_\\ast((f^\\ast \\mathscr L)^{-1}\\otimes \\mathscr E)^{-1}=(\\ast)\n$$\nNow, thanks to Proposition \\ref{det_chi} the above chain of equalities can be continued in the following way through a canonical isomorphism:\n$$\n(\\ast)\\cong\\det Rf_\\ast(\\mathscr E)\\otimes \\det Rf_\\ast(\\mathscr E)^{-1}\\otimes \\mathscr L^{\\chi_S(\\mathscr E|_{X_s})}=\\mathscr L^{\\chi_S(\\mathscr E|_{X_s})}\n$$\nwhere $X_s$ is a generic fibre. In order to conclude, it is enough to notice that $\\mathscr E|_{X_s}=c_1(\\mathscr L_1|_{X_s})\\ldots c_1(\\mathscr L_n|_{X_s})\\mathscr O_{X_s}$. \n\n\\endproof\n\n\n\\begin{appendices}\n\n\n\n\n\\section{Determinant functor and determinant of cohomology}\\label{det}\n\nIn this section we briefly discuss, without proofs, the determinant functor by following \\cite{KM}. First we define the determinant for locally free sheaves, then we extend it to complexes of locally free sheaves and then we extend it further for perfect complexes. We will use some basic notions from the theory of derived category (see \\cite{Hoche} for a concise introduction.)\n\n\nWe fix a Noetherian irreducible scheme $X$. A \\emph{graded invertible sheaf} on $X$ is a couple $(\\mathscr L, \\alpha)$ where $\\mathscr L$ is an invertible sheaf on $X$ and $\\alpha: X\\to\\mathbb Z$ is a continuous function. A morphism of graded invertible sheaves $\\phi: (\\mathscr L, \\alpha)\\to(\\mathscr M,\\beta)$ is a morphism of invertible sheaves such that the following condition hold: for any $x\\in X$, if $\\alpha(x)\\neq \\beta(x)$, then $\\phi_x=0$. We denote with $\\catname{Gr}(X)$ the category of graded invertible sheaves, and $\\catname{isGr}(X)$ is the category whose objects are graded invertible sheaves and the morphisms are just the isomorphisms; note that $\\catname{isGr}(X)$ is a Picard groupoid. The tensor product (i.e. the group operation) between graded invertible sheaves is defined as $(\\mathscr L, \\alpha)\\otimes(\\mathscr M, \\beta)=(\\mathscr L\\otimes\\mathscr M,\\alpha+\\beta)$. The unit graded invertible sheaf is $(\\mathscr O_X,0)$. Furthermore we can define the isomorphism $\\tau:\\mathscr L\\otimes\\mathscr M\\to\\mathscr M\\otimes\\mathscr L$ such that locally and on pure tensors is given by:\n$$\\tau(l\\otimes m)=(-1)^{\\alpha\\beta}m\\otimes l\\,.$$\nLet $\\catname{Vec}(X)$ be the category of locally free sheaves on $X$ (of finite rank) and let $\\catname{isVec}(X)$ its subcategory where the morphisms are only the isomorphisms. \n\n For a locally free sheaf $\\mathscr E$ of rank $r$, we denote with the symbol $\\bigwedge^r\\mathscr E$ the sheafification of the following presheaf:\n$$U\\mapsto\\bigwedge^r\\mathscr E(U)\\,.$$\nThen we can construct graded invertible sheaf $\\det{\\!}^\\star\\mathscr E$ in the following way:\n$$\\det{\\!}^\\star\\mathscr E:=\\left(\\bigwedge^r\\mathscr E, r\\right)\\,.$$\nThen we have a functor:\n$$\\det{\\!}_X^\\star\\colon \\catname{isVec}(X)\\to\\catname{isGr}(X)\\,.$$\nFor any short exact sequence of locally free sheaves:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H\\arrow[r,\"\\alpha\"]&\\mathscr F\\arrow[r,\"\\beta\"]&\\mathscr G\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*} \nthere is an isomorphism of graded invertible sheaves\n\\begin{equation*}\n\\begin{aligned}\ni_X^\\star(\\alpha,\\beta):\\det{\\!}_X^\\star(\\mathscr H)\\otimes\\det{\\!}_X^\\star(\\mathscr G)\\xrightarrow{\\cong} \\det{\\!}_X^\\star(\\mathscr F)\n\\end{aligned}\n\\end{equation*}\nthat locally is given in the following way: assume that $\\mathscr H$ has rank $r$ and $\\mathscr G$ has rank $s$, then for any local sections $h_i\\in\\mathscr H(U)$ and $\\beta(f_i)\\in\\mathscr G(U)$, for $f_i\\in\\mathscr F(U)$ we have:\n\n$$i_X^\\star(\\alpha,\\beta)((h_1\\wedge\\ldots\\wedge h_r)\\otimes(\\beta(f_1)\\wedge\\ldots\\wedge\\beta(f_s)))=\\alpha(h_1)\\wedge\\ldots\\wedge \\alpha(h_r)\\wedge f_1\\wedge\\ldots\\wedge f_n\\,.$$\nWe are ready to give the definition of determinant for bounded complexes of locally free sheaves.\n\\begin{definition}\\label{def_det}\nLet $\\catname{isVec}_b^\\bullet(X)$ be the category of bounded complexes in $\\catname{Vec}(X)$ where the morphism are just the quasi-isomorphisms between complexes. Then a \\emph{determinant functor on $X$} consists of the following data:\n\\begin{enumerate}\n\\item[$(1)$] A functor $\\mathfrak f_X:\\catname{isVec}_b^\\bullet(X)\\to\\catname{isGr(X)}$\n\\item[$(2)$] For any short exact sequence in $\\catname{isVec}_b^\\bullet(X)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r,\"\\alpha\"]&\\mathscr F^\\bullet\\arrow[r,\"\\beta\"]&\\mathscr G^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nan isomorphism:\n\n\\begin{equation*}\n\\begin{aligned}\ni_X(\\alpha,\\beta):\\mathfrak f_X(\\mathscr H^\\bullet)\\otimes\\mathfrak f_X(\\mathscr G^\\bullet)\\xrightarrow{\\cong} \\mathfrak f_X(\\mathscr F^\\bullet)\n\\end{aligned}\n\\end{equation*}\n\n\\end{enumerate} \n\nMoreover $(\\mathfrak f_X, i_X)$ satisfies the following conditions:\n\\begin{enumerate}\n\\item[$(i)$] Given a commutative diagram in $\\catname{isVec}_b^\\bullet(X)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r, \"\\alpha\"]\\arrow[d,\"\\lambda'\"]&\\mathscr F^\\bullet\\arrow[r, \"\\beta\"]\\arrow[d, \"\\lambda\"]&\\mathscr G^\\bullet\\arrow[r]\\arrow[d,\"\\lambda''\"]& 0\\\\\n0\\arrow[r]&\\mathscr H_1^\\bullet\\arrow[r, \"\\alpha_1\"]&\\mathscr F_1^\\bullet\\arrow[r, \"\\beta_1\"]&\\mathscr G_1^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nsuch that the rows are short exact sequences, then the following diagram commutes\n$$\n\\begin{tikzcd}[row sep=large, column sep = large]\n\\mathfrak f_X({\\mathscr H}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G}^\\bullet)\\arrow[\"{i_X(\\alpha,\\beta)}\"]{r}\\arrow[\"\\mathfrak f_X(\\lambda')\\otimes \\mathfrak f_X(\\lambda'')\"]{d} & \\mathfrak f_X({\\mathscr F}^\\bullet)\\arrow[\"\\mathfrak f_X(\\lambda)\"]{d}\\\\\n\\mathfrak f_X({\\mathscr H_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G_1}^\\bullet)\\arrow[\"{i_X(\\alpha_1,\\beta_1)}\"]{r} & \\mathfrak f_X({\\mathscr F_1}^\\bullet)\n\\end{tikzcd}\n$$\n\n\\item[$(ii)$] Given a commutative diagram in $\\catname{isVec}_b^\\bullet(X)$:\n$$\n\\begin{tikzcd}\n & 0\\arrow{d} & 0\\arrow{d} & 0\\arrow{d} &\\\\\n 0 \\arrow{r} & \\mathscr H_1^\\bullet \\arrow[r,\"\\alpha_1\"]\\arrow[d,\"\\gamma_1\"] & \\mathscr F_1^\\bullet \\arrow[r,\"\\beta_1\"]\\arrow[d,\"\\gamma\"] & \\mathscr G_1^\\bullet \\arrow{r}\\arrow[d,\"\\gamma_2\"] & 0\\\\\n 0 \\arrow{r} & {\\mathscr H}^\\bullet \\arrow[r,\"\\alpha\"]\\arrow[d,\"\\delta_1\"] & {\\mathscr F}^\\bullet \\arrow[r,\"\\beta\"]\\arrow[d, \"\\delta\"] & {\\mathscr G}^\\bullet \\arrow{r}\\arrow[d,\"\\delta_2\"] & 0\n\\\\\n 0 \\arrow{r} & {\\mathscr H_2}^\\bullet \\arrow[r,\"\\alpha_2\"]\\arrow[d] & {\\mathscr F_2}^\\bullet \\arrow[r,\"\\beta_2\"]\\arrow[d] & {\\mathscr G_2}^\\bullet \\arrow{r}\\arrow[d] & 0\\\\\n & 0 & 0 & 0 &\n\\end{tikzcd}\n$$\nsuch that rows and columns are a short exact sequence, then the following diagram commutes\n$$\n\\begin{tikzcd}[row sep=huge, column sep = 4.5cm]\n\\mathfrak f_X({\\mathscr H_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr H_2}^\\bullet)\\otimes\\mathfrak f_X({\\mathscr G_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G_2}^\\bullet)\\arrow[\"{i_X(\\gamma_1,\\gamma_2)\\otimes i_X(\\delta_1,\\delta_2)}\"]{r}\\arrow[\"{i_X(\\alpha_1,\\beta_1)\\otimes i_X(\\alpha_2,\\beta_2)}\"]{d} & \\mathfrak f_X({\\mathscr H}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr G}^\\bullet)\\arrow[\"{i_X(\\alpha,\\beta)}\"]{d}\\\\\n\\mathfrak f_X({\\mathscr F_1}^\\bullet)\\otimes \\mathfrak f_X({\\mathscr F_2}^\\bullet)\\arrow[\"{i_X(\\gamma,\\delta)}\"]{r} & \\mathfrak f_X({\\mathscr F}^\\bullet)\n\\end{tikzcd}\n$$\n\\item[$(iii)$] $\\mathfrak f_X$ and $i_X(\\,,\\,)$ both commute with the base change of $X$. The explicit expression of such a property is the following: fix a morphism of schemes $\\psi:Y\\to X$ and let $L\\psi^\\ast:\\catname D_-(\\catname{ QCoh}(X))\\to\\catname D_-(\\catname {QCoh}(Y))$ be the left derived functor of to the pullback $\\psi^\\ast$; then the following properties hold:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] There is a natural transformation between functors: $\\eta(\\psi)\\colon\\mathfrak f_Y\\circ L\\psi^\\ast\\xrightarrow{\\cong}\\psi^\\ast\\circ\\mathfrak f_X$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] For any short exact sequence in $\\catname{isVec}_b^\\bullet(Y)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r,\"\\alpha\"]&\\mathscr F^\\bullet\\arrow[r,\"\\beta\"]&\\mathscr G^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nthen the following diagram commutes\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\n\\mathfrak f_Y(L\\psi^\\ast{\\mathscr H}^\\bullet)\\otimes\\mathfrak f_Y(L\\psi^\\ast{\\mathscr G}^\\bullet)\\arrow[\"{i_Y(L\\psi^\\ast(\\alpha,\\beta))}\"]{r}\\arrow[\"\\eta(\\psi)(\\mathscr H^\\bullet)\\otimes \\eta(\\psi)(\\mathscr G^\\bullet)\"]{d} & \\mathfrak f_Y(L\\psi^\\ast{\\mathscr F}^\\bullet)\\arrow[\"\\eta(\\psi)(\\mathscr F^\\bullet)\"]{d}\\\\\n\\psi^\\ast\\mathfrak f_X({\\mathscr H}^\\bullet)\\otimes \\psi^\\ast\\mathfrak f_X({\\mathscr G}^\\bullet)\\arrow[\"{\\psi^\\ast i_X(\\alpha,\\beta)}\"]{r} & \\psi^\\ast\\mathfrak f_X({\\mathscr F}^\\bullet)\n\\end{tikzcd}\n$$\n\\end{enumerate}\n\n\\item[$(iv)$] $\\mathfrak f_X(0)=(\\mathscr O_X, 0)$. Moreover for the short exact sequence:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr F^\\bullet\\arrow[r,\"\\id\"]&\\mathscr F^\\bullet\\arrow[r,\"0\"]&0\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nwe have that $i_X(\\id,0)\\colon \\mathfrak f_X(\\mathscr F^\\bullet)\\otimes(\\mathscr O_X,0)\\to \\mathfrak f_X(\\mathscr F^\\bullet)$ is the canonical map (i.e. the ``projection on the first component'').\n\\item[$(v)$] If we canonically identify $\\catname{isVec}(X)$ as a subcategory of $\\catname{isVec}_b^\\bullet(X)$, then $\\mathfrak f_X$ restricts to $\\det{\\!}_X^\\star$ and $i_X(\\,,\\,)$ restricts to $i^\\star_X(\\,,\\,)$.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\n\\begin{theorem}\nUp to natural transformation of functors there exists a unique determinant $(\\mathfrak f_X, i_X)$ on $X$. It will be denoted as $(\\det_X, i_X)$.\n\\end{theorem}\n\\proof\nSee \\cite[Theorem 1]{KM} for a complete proof. Here we write down just the explicit expressions for $\\det_X$:\n\\begin{equation}\n\\det{\\!}_X(\\mathscr F^\\bullet)=\\bigotimes_i\\det{\\!}^\\star_X(\\mathscr F^i)^{(-1)^i}\\,.\n\\end{equation}\n\n\\endproof\n\nThe category $\\catname{isVec}_b^\\bullet(\\cdot)$ is quite restrictive, for example it doesn't behave well with respect to the pushforward functor. Therefore, we would like to have a determinant functor for a more general category.\n\\begin{definition}\nA complex $\\mathscr F^\\bullet $ of $\\mathscr O_X$-modules is said \\emph{perfect} if for any $x\\in X$ there exist an open neighbourhood $U\\ni x$, a complex $\\mathscr G^\\bullet $ in $\\catname {Vec}^\\bullet_b(U)$ and a quasi isomorphism of complexes of $\\mathscr O_U$-modules $\\mathscr G^\\bullet\\to\\mathscr F^\\bullet_{|U}$. The category of perfect complexes on $X$ is denoted by $\\catname{Perf}(X)$, whereas $\\catname{isPerf}(X)$ denotes the category of perfect complexes where the morphisms are just the quasi-isomorphisms.\n\\end{definition}\n\\begin{theorem}\nThe determinant $(\\det_X,i_X)$ can be extended uniquely, up to natural transformation, to a determinant on $\\catname{isPerf}(X)$. We will denote this extension again with the symbol $(\\det_X,i_X)$ and formally it is the datum of:\n\\begin{enumerate}\n\\item[$(1)$] A functor $\\det_X\\colon\\catname{isPerf}(X)\\to\\catname{isGr(X)}$\n\\item[$(2)$] For any short exact sequence in $\\catname{isPerf}(X)$:\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H^\\bullet\\arrow[r,\"\\alpha\"]&\\mathscr F^\\bullet\\arrow[r,\"\\beta\"]&\\mathscr G^\\bullet\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*}\nan isomorphism:\n\\begin{equation*}\n\\begin{aligned}\ni_X(\\alpha,\\beta):\\det{\\!}_X(\\mathscr H^\\bullet)\\otimes\\det{\\!}_X(\\mathscr G^\\bullet)\\xrightarrow{\\cong} \\det{\\!}_X (\\mathscr F^\\bullet)\n\\end{aligned}\n\\end{equation*}\n\\end{enumerate} \nMoreover the properties $(i)-(v)$ listed in definition \\ref{def_det} are satisfied in $\\catname{isPerf}(X)$. \n\\end{theorem}\n\\proof\nSee \\cite[Theorem 2]{KM}.\n\\endproof\nOne of the most important applications of the determinant functor appears in arithmetic geometry if we consider its interaction with the usual pushfoward functor.\n\n\\begin{definition}\nLet $f:X\\to S$ be a flat morphism between irreducible Noetherian schemes and let $\\mathscr E$ be a locally free sheaf of finite rank on $X$. It is well known (\\cite[Exp. 3, Proposition 4.8]{SGA}) that the complex $Rf_\\ast\\mathscr E$ induced by the right derived functor of $f_\\ast$ is a perfect complex on $S$. Then, just by composing $Rf_\\ast$ with $\\det_S$ it is possible to define the functor: \n$$\\det Rf_\\ast:=\\det{\\!}_S\\circ Rf_\\ast : \\catname{isVec}(X)\\to \\catname{isGr}(S)$$\nwhich is called the \\emph{determinant of cohomology (relative to $f$)}. Very often, for simplicity we want to forget about the graduation on the target of the determinant of cohomology, so it becomes a functor $\\catname{isVec}(X)\\to \\catname{Pic}(S)$.\n\\end{definition}\n\nSince the right derived functor $Rf_\\ast$ is exact in the derived sense (see \\cite{Hoche}) it is not hard to show that for any short exact sequence of locally free sheaves on $X$\n\\begin{equation*}\n\\begin{tikzcd}\n0\\arrow[r]&\\mathscr H\\arrow[r,\"\\alpha\"]&\\mathscr F\\arrow[r,\"\\beta\"]&\\mathscr G\\arrow[r]& 0\\\n\\end{tikzcd}\n\\end{equation*} \nthere is an isomorphism of graded invertible sheaves\n\\begin{equation*}\n\\begin{aligned}\ni_f(\\alpha,\\beta):\\det Rf_\\ast\\mathscr H\\otimes\\det Rf_\\ast\\mathscr G\\xrightarrow{\\cong} \\det Rf_\\ast\\mathscr F\n\\end{aligned}\n\\end{equation*}\nMoreover the whole construction behaves well with respect to flat base change in the following sense: assume that the following commutative square is given by a flat base change from $S$ to $S'$\n\n$$\n\\begin{tikzcd}[row sep=large, column sep = huge]\nX'=X\\times_S S'\\arrow[\"f'\"]{r}\\arrow[\"g'\"]{d} & S'\\arrow[\"g\"]{d}\\\\\nX\\arrow[\"f\"]{r} & S\n\\end{tikzcd}\n$$\nthen for any locally free sheaf $\\mathscr E$ on $X$ we have\n\n\\begin{equation}\\label{det_bchange}\ng^\\ast(\\det Rf_\\ast \\mathscr E)\\cong \\det Rf'_\\ast(g'^\\ast\\mathscr E)\\,.\n\\end{equation}\nWe will also need an important property of the determinant of cohomology:\n\n\\begin{proposition}\\label{det_chi}\nLet $f:X\\to S$ be a flat morphism between irreducible Noetherian schemes and let $\\mathscr E$ be a locally free sheaf of finite rank on $X$. Moreover let $\\mathscr L$ be an invertible sheaf on $S$. Then there is a canonical isomorphism between invertible sheaves on $S$:\n$$\\det Rf_\\ast(f^\\ast \\mathscr L\\otimes \\mathscr E)\\xrightarrow{\\cong} \\mathscr L^{\\otimes \\chi_S(\\mathscr E|_{X_s})}\\otimes \\det Rf_\\ast(\\mathscr E)$$\nwhere $X_s$ is a generic fibre of $f$ .\n\\end{proposition}\n\\proof\nSee \\cite{KM}.\n\\endproof\n\n\\begin{definition}\\label{norm}\nLet $\\varphi:X\\to S$ be a finite morphism between irreducible Noetherian schemes, then \\emph{the norm of $\\varphi$} is defined as the functor:\n\\begin{eqnarray*}\nN_{\\varphi}=N_{X\/S}:\\catname{Pic}(X) &\\to & \\catname{Pic}(S)\\\\\n\\mathscr L &\\mapsto & \\det R\\varphi_\\ast \\mathscr L\\otimes (\\det R\\varphi_\\ast \\mathscr O_X)^{-1}\\,.\n\\end{eqnarray*} \n\\end{definition}\n\n\n\\section{Original construction of Deligne pairing}\\label{ori_constr}\nIn this section we give all details of the construction of the Deligne pairing described in \\cite{Del}.\n\n\n$S$ is an irreducible Dedekind scheme and we put $K:=K(S)$. $\\varphi:X\\to S$ is a $S$-scheme satisfying the following properties:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $X$ is two dimensional, integral, and regular. The generic point of $X$ is $\\eta$ and the function field of $X$ is denoted by $K(X)$.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\varphi$ is proper and flat.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] The generic fibre, denoted by $X_K$, is a geometrically integral, smooth, projective curve over $K$. \n\\end{enumerate} \nWe say that $X$ is an \\emph{arithmetic surface over $S$}. \n\nIn this section we will also need recall the norm operator $\\mathcal N$ in dimension $1$ and $2$ (it is formally different from the norm of an invertible sheaf defined above). \n\\begin{definition}\nlet $C$ be a projective, non-singular curve over a field $k$, then for a closed point $x\\in C$ and any non-zero rational function $f\\in K(C)^\\times$ we put\n\\begin{equation}\\label{N_oncurves}\n\\mathcal N_x(f):=N_{k(x)|k} \\left(f(x)\\right)\\,,\n\\end{equation}\nwhere $f(x)$ is the obvious element of $k(x)$ associated to $f$. So if $D=\\sum_{x\\in X}v_x(D)[x]\\in \\Div(X)$ and $f\\in k(X)^\\times$ is a non-zero rational function such that $(f)$ and $D$ have no common components, then it is well defined the following element:\n$$ \\mathcal N_D(f):=\\prod_{x\\in X} \\mathcal N_x(f)^{v_x(D)}\\quad\\in k^\\times$$\n\\end{definition}\nThe well known Weil reciprocity law says that:\n$$\\mathcal N_{(g)}(f)=\\mathcal N_{(f)}(g)\\rlap.$$\n\n\\noindent Coming back to our arithmetic surface $\\varphi:X\\to S$, consider \n $$\\Upsilon:=\\Set{(D,E)\\in\\Div(X)\\times\\Div(X)\\colon\\textrm{$D$ and $E$ have no common components}}\\rlap,$$\nand note that if $(D_j,E_j)\\in\\Upsilon$ with $j=1,2$, then $(D_1+D_2,E_1+E_2)\\in\\Upsilon$. \n\\begin{definition}\\label{def1}\nLet $(D,E)\\in \\Upsilon$ such that $D$ and $E$ are both effective, then for any closed point $x\\in X$ we put:\n\n$$i_x(D,E):= \\len_{\\mathscr O_{X,x}}\\mathscr O_{X,x}\/\\left (\\mathscr O_X(-D)_x+\\mathscr O_X(-E)_x\\right)\\,.$$\nThis is called the \\emph{local intersection number} of $D$ and $E$ at $x$. \n\\end{definition}\nThe local intersection number assigns the multiplicity of the intersection at each point of $X$, and the following basic result summarizes its naive properties. \n\\begin{proposition}\\label{prop_loc_int}\nLet $(E,D)\\in \\Upsilon$ and $(E_j,D_j)\\in\\Upsilon$ with $j=1,2$ such that all the divisors are effective, then\n\\begin{enumerate} \n\\item[$(1)$]$i_x(D,E)=i_x(E,D)$.\n\\item[$(2)$]$\\displaystyle i_x(D_1+D_2,E_1+E_2)=\\sum_{j,k=1}^2 i_x(D_j,E_k)$.\n\\item[$(3)$] $i_x(D,E)\\neq 0$ if and only if $x\\in\\supp(D)\\cap\\supp(E)$.\n\\item[$(4)$] If $x\\in E$, $i_x(D,E)=\\mult_x(D|_E)$.\n\\end{enumerate}\n\\end{proposition}\n\\proof\n$(1)$ and $(3)$ are obvious. For $(2)$ and $(4)$ see \\cite[lemma 9.1.4]{Liu}.\n\\endproof\nAny divisor $D\\in\\Div(X)$ can be written in a unique way as $D=D_+-D_-$ where both $D_+$ and $D_-$ are effective and if $(D,E)\\in\\Upsilon$, then $(D_\\pm,E_\\pm)\\in\\Upsilon$. We can use definition \\ref{def1} in order to have the local intersection at $x$ of $D$ and $E$ when $(D,E)$ is any element of $\\Upsilon$ (so not necessarily effective):\n$$i_x(D,E):=i_x(D_+,E_+)-i_x(D_+,E_-)-i_x(D_-,E_+)+i_x(D_-,E_-)\\,.$$\n\\begin{definition}\\label{def2}\nLet $(D,E)$ be an element of $\\Upsilon$, then we define the $0$-cycle on $X$ given by:\n$$i(D,E):=\\sum_{x\\in X^{(0)}}i_x(D,E)[x]\\rlap,$$\nwhere here $[x]$ is a shorthand of $[\\overline{\\{x\\}}].$\n\\end{definition}\n\\begin{remark}\nThe sum in definition \\ref{def2} is finite because if $D$ and $E$ are effective without common components, then $i_x(D,E)=\\mult_x(D|_E)$ (proposition \\ref{prop_loc_int}(4)) and there is only a finite number of points on $E$ at which the divisor $D|_E$ has non-zero multiplicity. \n\\end{remark}\n\\begin{proposition}\nIf $(D,E),(D_j,E_j)\\in\\Upsilon$ with $j=1,2$, then the following properties hold for $i(D,E)$:\n\\begin{enumerate}\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $i(D,E)=i(E,D)$ (symmetry)\\,.\n\\item[\\,\\begin{picture}(-1,1)(-1,-3)\\circle*{3}\\end{picture}\\ ] $\\displaystyle i(D_1+D_2,E_1+E_2)=\\sum_{j,k=1}^2 i(D_j,E_k)$ (bilinearity)\\,.\n\\end{enumerate}\n\\end{proposition}\n\\proof\nIt follows immediately from proposition \\ref{prop_loc_int}.\n\\endproof\n\\begin{definition}\\label{finite_int_numb}\nWe have the symmetric and bilinear pairing on $\\Upsilon$:\n\\begin{eqnarray*}\n\\Upsilon &\\to& \\Div(S)\\\\\n(D,E) &\\mapsto& \\left\n\\end{eqnarray*}\nwhere\n$$\\left:=\\varphi_\\ast i(D,E)=\\sum_{x\\in X} [k(x):k(\\varphi(x))]\\,i_{x}(D,E)\\, [\\varphi(x)]\\,.$$\n\\end{definition}\n\nLet $\\Gamma$ be a prime divisor of $X$ with generic point $\\gamma$ and consider a non-zero rational function $f\\in K(X)^\\times$ such that $(f)$ and $\\Gamma$ have no common components, then define $\\mathcal N_{\\Gamma}(f)\\in K^\\times$ in the following way:\n$$\\mathcal N_{\\Gamma}(f):=\\left\\{\\begin{array} {cc}\nN_{K(\\Gamma)|K}(f|_{\\Gamma}) & \\textrm{if $\\Gamma$ is horizontal}\\\\\n1 & \\textrm{if $\\Gamma$ is vertical}\\\\\n\\end{array}\\right.\n$$\nwhere $N_{K(\\Gamma)|K}$ is the usual field norm and $f|_{\\Gamma}$ is defined as follows: since $(f)$ and $\\Gamma$ have no common components it follows that $v_{\\gamma}(f)=0$, that is $f\\in\\mathscr O_{X,\\gamma}^\\times$. So $f|_{\\Gamma}$, is the natural image of $f$ in $k(\\gamma)=K(\\Gamma)$. At this point for any $D=\\sum_i n_i\\Gamma_i\\in \\Div(X)$ such that $D$ and $(f)$ have no common components we have:\n$$\\mathcal N_{D}(f):=\\prod_i\\mathcal N_{\\Gamma_i}(f)^{n_i}\\quad \\in K^\\times$$\nSince $K(X)$ is the function field of any open subscheme $U\\subseteq X$ and of $X_K$ we can restrict the operator $\\mathcal N_\\ast(\\cdot)$ to $U$ and to $X_K$.\n\n\\begin{proposition}\\label{restr_prop}\nLet $f\\in K(X)^\\times$ and let $D\\in Div(X)$ such that $(f)$ and $D$ have no common components, then the following claims hold:\n\\begin{itemize}\n\\item[$(1)$] Let $U\\subseteq X$ be an open subscheme, then $\\mathcal N_{D|_U}(f)=\\mathcal N_D(f)$.\n\n\\item[$(2)$] $\\mathcal N_{D|_{X_K}}(f)=\\mathcal N_D(f)$, where the left hand side is the one-dimensional operator defined in equation (\\ref{N_oncurves}).\n\n\\end{itemize}\n\\end{proposition}\n\\proof\nIn both items we can restrict to the case when $D=\\Gamma$ is an irreducible horizontal divisor.\\\\\n$(1)$ The function fields and the generic points of $\\Gamma$ and $\\Gamma|_U$ coincide, so the claim follows trivially.\\\\\n$(2)$ Let $\\gamma\\in X_K$ be the generic point of $\\Gamma$, it is a closed point of $X_K$ such that $k(\\gamma)=K(\\Gamma)$. By the bare definitions we can check the required equality. \n\\endproof\n\n\\begin{proposition}\\label{intwithrat}\nLet $f\\in K(X)^\\times$ and let $D\\in\\Div(X)$ a divisor such that $D$ and $(f)$ have no common components, then\n$$\\left=\\left(\\mathcal N_D(f)\\right)\\;\\in \\Princ(S)\\,.$$ \n\\end{proposition}\n\\proof\nSee \\cite[Proposition 4.3]{Mor}.\n\\endproof\n\n\n\n\nNow we will construct the Deligne pairing and see the relation with the pairing $\\left$ for divisors. We divide the construction in two steps:\n\n\\emph{Step 1.} Definition of the $K$-vector space $\\left<\\mathscr L,\\mathscr M\\right>_K$.\\\\\n\nConsider the sets:\n\n\\begin{alignat*}{2}\n \\Upsilon_K &:= \\biggl\\{(D,E)\\in \\Div(X)\\times\\Div(X) &&\\;\\colon\\; \\pctext{2in}{$D|_{X_K}$ and $E|_{X_K}$ have no common components (as divisors on $X_K$)}\\biggr\\}\\rlap, \n\\end{alignat*}\n\\begin{alignat*}{2}\n \\Sigma_K &:= \\biggl\\{(l,m) &&\\;\\colon\\; \\pctext{3.5in}{ $l$ and $m$ are non-zero meromorphic sections of $\\mathscr L$ and $\\mathscr M$ such that $(\\divi(l),\\divi(m))\\in \\Upsilon_K$}\\biggr\\}\\rlap. \n\\end{alignat*}\nNote that $\\Upsilon_K$ is just the set of couple of divisors with no common horizontal components. Now we define some vector spaces over $K$.\n$$V:= K^{(\\Sigma_K)}\\,,$$\nnamely $V$ is the free $K$-vector space over $\\Sigma_K$. \n\\begin{equation}\\label{rel1}\nW':=\\left\\{(fl,m)-\\mathcal N_{\\divi(m)|_{X_K}}(f)\\cdot(l,m)\\colon f\\in K(X)^\\times, (l,m),(fl,m)\\in \\Sigma_K\\right\\}\\,,\n\\end{equation} \n\n\\begin{equation}\\label{rel2}\nT':=\\left\\{(l,gm)-\\mathcal N_{\\divi(l)|_{X_K}}(g)\\cdot(l,m)\\colon g\\in K(X)^\\times, (l,m),(l,gm)\\in \\Sigma_K \\right\\}\\,.\n\\end{equation} \nNote that the above ``$\\mathcal N_{\\ast}(\\cdot)$'' is the one-dimensional operator of definition \\ref{N_oncurves} considered on the curve $X_K$.\n\n\\begin{remark}\n$\\mathcal N_{\\divi(m)|_{X_K}}(f)$ and $\\mathcal N_{\\divi(l)|_{X_K}}(g)$ are well defined since $(l,m)$, $(fl,m)$, $(l,gm) \\in \\Sigma_K$, so $\\divi(m)|_{X_K}$ and $(f)$ have no common components. The same holds for $\\divi(l)|_{X_K}$ and $(g)$.\n\\end{remark}\n\nDefine the free vector spaces $W:= K^{(W')}$ and $T:=K^{(T')} $; moreover put \n$$\\left<\\mathscr L,\\mathscr M\\right>_K:=V\/(W+T)\\,\\rlap,$$\nwhich is considered as a constant sheaf (of $K$-vector spaces) over $X$. The natural image of any element $(l,m)\\in \\Sigma_K\\subset V$ in $\\left <\\mathscr L,\\mathscr M\\right>_K$ is denoted as $\\left _K$.\\\\\n\\begin{proposition}\\label{sstep1}\n $\\left<\\mathscr L,\\mathscr M\\right>_K$ is a one-dimensional vector space over $K$.\n\\end{proposition}\n\\proof\nFix $(l_0,m_0)\\in\\Sigma_K$, then for any $(l,m)\\in\\Sigma_K$ there are two elements $f_0,g_0\\in K(X)^\\times$ such that $l=f_0l_0$, $m=g_0m_0$ and moreover:\n$$((f_0),(g_0)),\\;((f_0),\\;\\divi(m_0)),\\;((g_0),\\divi(l_0))\\in\\Upsilon_K\\,.$$\nBy equations (\\ref{rel1}) and (\\ref{rel2}), in $\\left<\\mathscr L,\\mathscr M\\right>_K$ we can write:\n\\begin{equation}\\label{rel3}\n\\left_K= \\left_K=[f_0,g_0]\\,\\mathcal N_{\\divi(m_0)|_{X_K}}(f_0)\\,\\mathcal N_{\\divi(l_0)|_{X_K}}(g_0)\\left_K\\rlap.\n\\end{equation}\nwhere, in order to simplify the notations, we put $[f_0,g_0]:=\\mathcal N_{(f_0)}(g_0)$ intended as operation on the curve $X_K$. This shows that $\\left<\\mathscr L,\\mathscr M\\right>_K$ has dimension at most $1$ over $K$.\nDefine the homomorphism of $K$-vector spaces:\n$$\\theta:V\\to K$$\nsuch that\n$$\\theta(l,m):=[f_0,g_0] \\mathcal N_{\\divi(m_0)|_{X_K}}(f_0) N_{\\divi(l_0)|_{X_K}}(g_0)\\,.$$\n Note that $\\theta$ is non-trivial, so surjective, since $\\theta(l_0,m_0)=1$. Now by using the Weil reciprocity law we prove that $\\theta$ descends to a non-trivial morphism $\\overline\\theta:\\left<\\mathscr L,\\mathscr M\\right>_K\\to K$, indeed for $f,g\\in K(X)^\\times$:\n\\begin{eqnarray*}\n\\theta(fl,m)&=&[ff_0,g_0]\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(ff_0)\\, \\mathcal N_{\\divi(l_0)|_{X_K}}(g_0)=\\\\\n&=& [f,g_0]\\,[f_0,g_0]\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(f)\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(f_0)\\, \\mathcal N_{\\divi(l_0)|_{X_K}}(g_0)=\\\\\n&=&[g_0,f]\\, \\mathcal N_{\\divi(m_0)|_{X_K}}(f)\\, \\theta(l,m)=\\\\\n&=& \\mathcal N_{\\divi(m)|_{X_K}}(f)\\, \\theta(l,m)\\rlap.\n\\end{eqnarray*}\nSimilarly it holds that\n$$\\theta(l,gm)=\\mathcal N_{\\divi(l)|_{X_K}}(g)\\, \\theta(l,m)\\rlap.$$\nIn other words equation \\ref{rel3} can we written as:\n$$\n\\left_K=\\overline{\\theta}(\\left_K)\\left_K\n$$\nhence, by the non triviality of $\\overline{\\theta}$ we conclude that $\\left<\\mathscr L,\\mathscr M\\right>_K$ has dimension $1$.\n\\endproof\n\\emph{Step 2.} Definition of $\\left<\\mathscr L,\\mathscr M\\right>$.\\\\\n Let $U\\subseteq S$ be a non-empty open subset and denote with $X_U$ the schematic inverse image of $U$ with respect to $\\varphi$. We clearly have a flat map $X_U\\to U$, so we define:\n\\begin{alignat*}{2}\n \\Upsilon_U &:= \\biggl\\{(D,E)\\in \\Div(X)\\times\\Div(X) &&\\;\\colon\\; \\pctext{2in}{$D|_{X_U}$ and $E|_{X_U}$ have no common components (as divisors on $X_U$)}\\biggr\\}\\rlap, \n\\end{alignat*}\n\\begin{alignat*}{2}\n \\Sigma_U &:=\\Bigg\\{(l,m) &&\\;\\colon\\; \\pctext{3in}{$l$ and $m$ are non-zero meromorphic sections of $\\mathscr L$ and $\\mathscr M$ such that $(\\divi(l),\\divi(m))\\in\\Upsilon_U$ and $\\left<\\divi(l)|_{X_U}, \\divi(m)|_{X_U} \\right>$ is effective on $U$}\\Bigg\\}\\rlap. \n\\end{alignat*}\nMoreover notice that $\\Sigma_U\\subset \\Sigma_K$. We define a sheaf of $\\mathscr O_S$-modules $\\mathscr A$ on $X$ given by:\n$$\\mathscr A|_{U}:={\\mathscr O_{S}|_{U}}^{(\\Sigma_{U})}\\,.$$\nFinally consider the morphism of sheaves: $\\Phi:\\mathscr A\\to\\left<\\mathscr L,\\mathscr M\\right>_K$ which sends $(l,m)\\in\\Sigma_U$ to $\\left_K$ and define\n$$\\left<\\mathscr L,\\mathscr M \\right>:=\\faktor{\\mathscr A}{\\ker(\\Phi)}\\rlap.$$\nThe canonical image of $(l,m)\\in\\Sigma_U$ in $\\mathscr A(U)$ is denoted as $\\left_U$.\n\n\\begin{proposition}\\label{sstep2}\nLet $(l,m)\\in \\Sigma_{U}$ such that $\\left<\\divi(l)|_{X_{U}}, \\divi(m)|_{X_{U}}\\right>=0\\in \\Div(U)$. Then for any $(l',m')\\in\\Sigma_{U}$ there exists an element $a\\in\\mathscr O_S(U)$ such that $\\left_{U}=a\\left_{U}$.\n\\end{proposition}\n\\proof\n\nThere are two elements $f,g\\in K(X)^\\times$ such that $l'=fl$, $m'=gm$ and moreover:\n$$((f),(g)),\\;((f),\\;\\divi(m)),\\;((g),\\divi(l))\\in\\Upsilon_K\\,.$$\nHence by using proposition \\ref{intwithrat}:\n$$\\left<\\divi(l')|_{X_{U}}, \\divi(m')|_{X_{U}}\\right>=\\left<(f)|_{X_{U}}\\divi(l)|_{X_{U}}, (g)|_{X_{U}}\\divi(m)|_{X_{U}}\\right>=$$\n$$=\\left< (f)|_{X_{U}},(g)|_{X_{U}}\\right>+\\left< (f)|_{X_{U}},\\divi(m)|_{X_{U}}\\right>+\\left<\\divi(l)|_{X_{U}}, (g)|_{X_{U_i}}\\right>+0=$$\n$$= \\left(\\mathcal N_{(f)|_{X_{U}}}(g)\\right)+\\left(\\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\right)+\\left(\\mathcal N_{\\divi(l)|_{X_{U}}}(g)\\right)=$$\n$$=\\left(\\mathcal N_{(f)|_{X_{U}}}(g)\\, \\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\, \\mathcal N_{\\divi(l)|_{X_{U}}}(g) \\right)\\rlap.$$\nSince $\\left<\\divi(l')|_{X_{U}}, \\divi(m')|_{X_{U}}\\right>$ is effective, then \n$$a:=\\mathcal N_{(f)|_{X_{U}}}(g)\\,\\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\, \\mathcal N_{\\divi(l)|_{X_{U}}}(g)\\in \\mathscr O_S(U)\\,.$$\nOn the other hand\n$$\\left_K=[f,g] \\mathcal N_{\\divi(m)|_{X_K}}(f)\\mathcal N_{\\divi(l)|_{X_K}}(g)\\left_K$$\ntherefore by proposition \\ref{restr_prop} we can conclude that:\n$$\\left_{U}=\\mathcal N_{(f)|_{X_{U}}}(g) \\mathcal N_{\\divi(m)|_{X_{U}}}(f)\\mathcal N_{\\divi(l)|_{X_{U}}}(g)\\left_U=a\\left_{U}\\,.$$\n\\endproof\n\nWe are ready to show that $\\left<\\mathscr L,\\mathscr M\\right>$ is an invertible sheaf on $S$. By proposition \\ref{sstep1} $\\left<\\mathscr L,\\mathscr M\\right>$ is non-zero; now assume $\\mathscr L=\\mathscr O_X(D)$, $\\mathscr M=\\mathscr O_X(E)$ and fix a point $s_0\\in S$. By the moving lemma we can find a divisor $D'$ such that $D'\\sim D$ and $D'$ doesn't have components in $X_{s_0}$. Suppose that $x_1,\\ldots x_m$ are the intersection points of $D'$ and $X_{s_0}$, by applying again the moving lemma we can find a divisor $E'$ such that: $E'\\sim E$, $E'$ and $D'+X_{s_0}$ have no common components, and $E$ doesn't pass by $x_1,\\ldots, x_m$. Consider the finite subset of $S$ \n$$C:=\\{s\\in S'\\colon D'\\cap E'\\cap X_s\\neq \\emptyset\\}\\,$$\nand note that its complement $U:=S\\setminus C$ has the following properties: $s_0\\in U$ and $\\left=0$. At this point any two meromorphic sections of $\\mathscr L$ and $\\mathscr M$ corresponding respectively to the divisors $D'$ and $E'$ will satisfy the hypothesis of proposition \\ref{sstep2} on $U$. This implies that $\\left<\\mathscr L,\\mathscr M\\right>$ is an invertible sheaf.\n\nOne can show that the pairing constructed above satisfies the axioms (1)-(4) of definition \\ref{axiom_deligne} for $n=1$, moreover we have the following additional properties:\n\n\\begin{theorem}\\label{del_thm}\nThe Deligne pairing $(\\mathscr L,\\mathscr M)\\to \\left<\\mathscr L,\\mathscr M\\right>$ satisfies the properties listed below. We assume that $\\mathscr L$ and $\\mathscr M$ are two invertible sheaves on $X$.\n\\begin{itemize}\n\\item[$(1)$] The induced map $\\Pic(X)\\times\\Pic(X)\\to\\Pic(S)$ is bilinear and symmetric. \n\\item[$(2)$] Let $l$ and $m$ be two non-zero meromorphic sections of $\\mathscr L$ and $\\mathscr M$, respectively, such that $\\divi(l)$ and $\\divi(m)$ have no common components. Then, there exists a non-zero meromorphic section $\\left$ with the following properties: \n\\begin{itemize}\n\\item[$(i)$] If $f,g\\in K(X)^\\times$ such that $(\\divi(fl),\\divi(m)),(\\divi(l),\\divi(gm))\\in \\Upsilon$, then:\n$$\\left=\\mathcal N_{\\divi(m)}(f)\\left$$\n$$\\left=\\mathcal N_{\\divi(l)}(g)\\left$$\n\\item[$(ii)$] There is an isomorphism of invertible sheaves \n$$\\left<\\mathscr L,\\mathscr M\\right>\\cong\\mathscr O_S(\\left<\\divi(l),\\divi(m)\\right>)$$\nMoreover, under the above isomorphism $\\left$ corresponds to $1_{\\left<\\divi(l),\\divi(m)\\right>}$. In particular:\n$$\\divi(\\left)=\\left<\\divi(l),\\divi(m)\\right>\\,.$$\n\\end{itemize}\n\\end{itemize}\n\\end{theorem}\n\\proof\nSee \\cite[Theorem 4.7]{Mor}.\n\\endproof\n\\begin{remark}\\label{del_vectspace}\nNote that when $S=\\spec k$ for any field $k$ (in other words $X$ is an algebraic curve), then $\\left<\\mathscr L,\\mathscr M\\right>$ is just a one dimensional $k$-vector space.\n\\end{remark}\n \n\\begin{remark}\nIf $S$ is a non-singular projective curve over a field $k$, and $\\varphi: X\\to S$ is a morphism over $\\spec k$ (i.e. $X$ is a fibred surface over $S$), then it is evident that $(D,E)\\mapsto \\deg\\left<\\mathscr O_X(D),\\mathscr O_X(D)\\right>$ satisfies all the axioms of definition \\ref{inters_var}, so it is the intersection pairing on $X$. The same argument holds also when $X$ has generic dimension $n$.\n\\end{remark}\n\n\n\\end{appendices}\n\n\\bibliographystyle{hplain.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{INTRODUCTION AND MOTIVATION}\nObservation of rare processes demands a careful study and implementation of signal processing and detector design to reach new levels of background suppression. Both the potential backgrounds and detector response to backgrounds must be thoroughly understood. Radon (Rn) gas provides a problematic class of backgrounds due to its terrestrial presence and its long-lived progeny. Exposure to radon at any stage of assembly of an experiment can result in surface contamination by progeny supported by the long half-life (22 yr) of \\nuc{210}{Pb} on sensitive surfaces of a detector \\cite{ams07, leu05,gui11}. The radon progeny surface contaminants can continue to produce unwanted background even after the detector is moved to its final laboratory or configuration with a lower radon level. In the case of neutrinoless double-beta decay experiments, energetic $\\alpha$ and $\\beta$ decays of \\nuc{210}{Po} and \\nuc{210}{Bi}, respectively, produce a background near or above the region of interest. Particularly problematic is the degraded alpha deposit that can fall under the region of interest requiring pulse shape or fiducial cuts to be rejected. In the case of a dark matter detector, the $\\alpha$ decay of \\nuc{210}{Po} produces a nuclear recoil similar to that of a WIMP-nucleon scatter. Because of the importance of this class of backgrounds, all low background rare event searches pay close attention to Rn exposure and are exploring ways to mitigate and control Rn-generated backgrounds. \n\nWhile the Rn progeny deposit directly on a material surface, the nuclear recoils of the alpha decay of \\nuc{218}{Po} and \\nuc{214}{Po} cause the daughters to be implanted deeper into the surface. Modeling and surface alpha measurements \nindicate that the Rn-borne \\nuc{210}{Pb} is implanted down to 0.05 -- 0.1 $\\mu$m in metals due to the deposition of progeny earlier in the chain. Further, exposure to Rn gas invites diffusion of Rn that can carry progeny deeper into the surface layers of material. The diffused Rn progeny and any possible bulk contamination of \\nuc{210}{Pb} creates an additional path of material for the degradation of an emitted alpha background. Though the majority of alpha emission may come from deposited daughters close to the surface, the surface roughness of a material adds an additional layer of material an alpha must traverse \\cite{per13}. \nThe net effect is an emitted alpha with a peak near its full energy, but with a low energy tail extended down to low energies as if the alpha originated deeper than the implantation depth of 0.05 - 0.1 $\\mu$m. For a typical metal, the range of a 5 MeV alpha is 20 $\\mu$m. One method to mitigate surface alpha contamination is to chemically remove the Rn progeny that exist down to a surface depth that they are imbedded.\n\nThe deposition of Rn progeny continues to be thoroughly modeled and studied by low background experiments. The findings are helping establish the infrastructure and facility designs needed for future experiments to control the exposure to Rn gas. Naturally, the deposition studies are complimented by separate studies evaluating the cleaning and surface removal of Rn progeny. Since the start of the Low Radioactivity Techniques workshop series, there have been over 10 papers focusing on the modeling, deposition, or removal of surface Rn progeny \\cite{leu05,woj07,gui11,woj11,jil13,pat13,per13,sch13,bru15,kob15,zuz15}, not to mention the papers where Rn studies are described in a more general experimental program overview paper or studies reported elsewhere. The findings of surface progeny removal studies have shown mixed results. Generally, the Pb and Bi progeny has been found to be easily removed using a variety of standard cleaning methods. However, in some materials, Po has been more difficult to remove and more aggressive chemical cleaning methods are often recommended. \n\nThe mixed results for the removal of surface \\nuc{210}{Po} are worrisome given the needs of next generation low background, rare event searches. With larger experiments, there will be a greater number of parts with a more stringent surface contamination control required. When deciding on a cleaning procedure, next generation experiments must consider: 1) efficiencies of removing Rn progeny, 2) quantities and purities of chemicals needed, 3) chemical waste, 4) generation and mitigation of chemical fumes, 5) underground cleaning limitations and logistics, 6) number of parts and process automation compatibility, and 6) maintaining dimensional and mechanical tolerances.\n\n\\section{Po REMOVAL TECHNIQUES}\n\nSurface cleaning techniques have been evaluated with varying results. In this section, we describe some of the methods used to highlight the generally accepted findings. The work of \\citet{hop07} looked at the cleaning of Cu and evaluated two methods. The Cu samples used in the study were initially loaded with $^{209}$Po by electrodeposition and the removal of the Po atoms was used to evaluate the various surface cleaning methods. The most effective treatment of removing Po surface contamination was a concentrated nitric acid etch. Lowering the concentration of nitric also lowered the effectiveness of Po removal. In the paper, the authors also introduce an alternate acidified peroxide cleaning method containing a dilute piranha solution of 1\\% H$_2$SO$_4$ and 3\\% H$_2$O$_2$ (hereinafter referred to the PNNL solution for cleaning Cu). This PNNL solution can effectively etch away surface layers in a more controlled fashion than concentrated nitric solution. Though it is shown to be effective at removing Cu atoms from the surface, it has limited effect of removing Po surface contamination. The authors stated some concern with Po solubility in the solution given they found improved Po removal when starting with Cu samples of lower initial Po activity. \n\nSimilar tests were conducted by other groups and two studies are worth mentioning here. \\citet{zuz15} looked at the removal of all three long-lived Rn progeny ($^{210}$Pb, $^{210}$Bi, $^{210}$Po) from samples of Cu, stainless steel and Ge. Using the standard PNNL acidified peroxide solution, the authors found that cleaning Cu samples showed better reduction of Bi and Pb than of Po, which showed very little, if any, reduction. Explanations for the poor removal of Po, including redeposition in solution, are given in Ref. \\cite{zuz12}. The removal of Rn progeny from contaminated samples of steel and Ge faired a bit better than Cu indicating the substrate material has a significant effect on contamination removal efficiency. The authors additionally tried a separate electropolishing technique on Cu and stainless steel. The electropolishing treatment significantly improved the removal efficiency of all three progeny, including a significant improvement for Po. The next study given in Ref. \\cite{sch13} also found effective reduction of Po through an electropolishing procedure on stainless steel. Interestingly, this study reduced Po to detection sensitivity levels after a depth of 0.6 $\\mu$m steel was removed, while the implantation depth of Rn progeny is modeled to be 0.05 - 0.1 $\\mu$m in most metals. \n\nThe fact that methods to remove Po have had mixed results indicates the importance of understanding the chemical behavior of Po in solution and the substrate being cleaned. \nMost methods tried using some form of an acidic solution, but given the behavior of Po in the Pourbaix diagram (see Ref. \\cite{ans12}), neutral Po can exist in solution over the entire pH range. In the neutral state, Po will likely redeposit onto a surface. However at low pH, Po can be driven into the Po(IV) oxidized state. When in a ion state, Po should favor staying in solution rather than redepositing based on the aqueous solution chemistry. The Po can be converted into a stable ion state by applying an oxidation potential or an oxidizing agent. The standard electrode potentials for Po in an acidic solution suggests that a potential of 0.73 V separates neutral Po from Po$^{4+}$, which is expected to be the most stable Po ion in solution \\cite{ans12}.\n\nWorking off the scenario that the ion state of Po is a significant factor in its removal from a contaminated sample, the following requirements are needed. First, the Po must be converted into an ion state through oxidation and favorably into the Po$^{4+}$ state. Solubility plays a role so a sufficient amount of surface area should be given within a cleaning solution for the amount of Po present. Finally, the stability of the ion state must be maintained to avoid reduction back to a neutral atom. The substrate atoms (e.g Cu) will likely play a role as they compete for the oxidizer (either an applied potential or an oxidizing agent). Given the separate oxidation potentials, the kinetics of the Po and the substrate atom removal may occur at different rates. That is, the depth of the substrate that is removed may not necessarily mean all other species located within that depth have had time to be oxidized before redepositing back onto the surface. Instead, sufficient exposure to the oxidizer may be the determining factor of effective Po removal, rather than the depth of the substrate removed. Greater exposure can be accomplished several ways: the contaminated sample can be agitated to make use of the volume of solution exposure, greater concentrations of an oxidizing agent, longer exposure times, larger volumes of solution surrounding the contaminated sample, etc. \n\nReturning to the mixed results for Po noted earlier, we can now put the results in the context of the Po ion state scenario. The tests with nitric acid \\cite{hop07} worked very well for all progeny. Nitric acid is an excellent oxidizer and is aggressive to almost all metals. A problem with concentrated nitric is that it will remove a large amount of material in the process in an uncontrolled way due to thermodynamic changes during dissolution. Electropolishing was found to be very efficient in removing Po and the other progeny \\cite{zuz15,sch13}. Electropolishing recipes typically call for concentrated acids to keep the pH low and an applied electric potential provides the necessary oxidation. This method drives off a large portion of the surface layers, which could compromise mechanical properties if used on small parts. The PNNL acidified peroxide solution uses H$_2$0$_2$ as the oxidizing agent. With this method, there is a modest and controlled material removal, which is an important consideration for tight dimensional tolerances on small parts. Though this solution meets some of the the requirements for removal of Po, the previous tests had found this method does not effectively remove Po from Cu. It is possible the method can be effective with increased exposure to the oxidizing agent, which is what we explore and test in the next section. \n\n\\section{Po REMOVAL STUDY}\n\nFor our study, we wish to explore the hypothesis of the role played by oxidation in removing Po from a sample. Given previous studies have shown the poor removal of Po from Cu, we will focus only on quantifying the removal of Po from Cu samples. Cu foil samples 50 mm in diameter and 0.5-mm thick are exposed to a 100 kBq radon source for about 1 month. The samples are left for over 2 years to allow the $^{210}$Po activity to grow-in to equilibrium with the $^{210}$Pb. The net surface alpha rate achieved is around 300 cts\/day (not corrected for detection efficiency). An alpha spectrometer using an ion-implanted silicon detector is used to count the Cu samples before and after chemical treatment. The alpha detector has a background count rate of 6 counts\/day. The goals of this study are to explore certain features of chemical treatment to assess the role played by oxidizers on the removal of Po from the contaminated surface. Based on the ratio of the pre-treatment and post-treatment alpha rates, the fraction of Po removed is determined from the background-subtracted alpha rates, which is not dependent on the detector counting efficiency since the geometry remains unchanged. \n\n\\section{RESULTS AND DISCUSSION}\n\nThe first test was to simply look at the effect of increasing the H$_2$0$_2$ concentration from 3\\% to 9\\% but otherwise following the PNNL acidified peroxide solution process. Each sample was exposed to a fresh solution for a varying amount of time to achieve a range of Cu thickness removed from each face of the sample. The samples were agitated equally during all tests and all but one test was performed with a 100 ml solution. The results in Fig. \\ref{fig:1}(a) show a general trend of improved Po removal with greater Cu removal up to the maximum 100\\% Po removal (within counting uncertainties consistent with the background rate of the alpha detector). However, one should not generally conclude that the depth of Cu removed down to the expected Po implantation depth should determine when all of the Po is removed. Recall, the implanted Po should exist down to 0.05 $\\mu$m. Instead, the amount of Cu removed is likely a proxy for quantifying the exposure to the oxidizing agent converting Po into the Po$^{4+}$ ion state. The conditions present may allow the Cu removal to indicate when sufficient exposure for removal of the Po has occurred. Further, the greater concentration of H$_2$0$_2$ does not have a strong effect on the Po removal; but it does give sufficient exposure in a shorter time as confirmed by the amount of Cu removed. \n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalStd.eps} \\\\\n \\small (a)\n\\end{tabular}\n \\quad\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalWithVolt.eps}\\\\\n \\small (b) \n \\end{tabular}\n \\caption{(a) The fraction of Po removed from a cleaning solutions plotted against the depth of Cu removed. The standard PNNL acidified peroxide solution is used for a varying amount of time, though the H$_2$0$_2$ is increased for one set of samples. (b) The same data points from (a), but with an additional set where a cell potential is applied.}\n \\label{fig:1}\n\\end{figure}\n\nTo test if a cell potential could aid in the oxidation of Po in the same acidified peroxide solution, three additional samples were cleaned with an applied potential greater than 0.73 V. The results of the three tests are given in Fig. \\ref{fig:1}(b) and plotted with the same non-potential tests from Fig. \\ref{fig:1}(a), where there is no longer a distinction made for the concentration of H$_2$0$_2$. The results show no added benefit of adding a cell potential, which suggests the oxidizing agent present is the dominant oxidizing mechanism under these conditions. Instead, the additional samples follow the same general trend of increasing Po removal with greater exposure, as indicated by the increased Cu removal. With the additional samples present, the variability among samples is more visible when plotted against exposure. This observation indicates another variable may be controlling the effectiveness of Po removal.\n\n\n\nThe data from Fig. \\ref{fig:1}(b) with the most effective Po removal (where $>2\\, \\mu$m of Cu removed) is plotted in Fig. \\ref{fig:2}(a) as a function of initial $^{210}$Po surface activity. The fraction of Po removed appears to generally increase as the initial $^{210}$Po activity decreases, with one notable exception. The sample with a starting activity of 400 cts\/day had an acidified peroxide exposure sufficient to remove 8 $\\mu$m of Cu while the other two samples with starting activity $>$300 cts\/day had exposures removing $<6\\,\\mu$m of Cu. The results suggest the initial Po activity dictates the exposure to the solution required. When the exposure to the oxidizing agent is increased, commensurate with increased initial $^{210}$Po activity, the Po removal can be effective. \n\nIn a final series of tests, some alternative cleaning methods are explored to further investigate the role of oxidizers. \nGiven there was no added benefit to adding a cell potential to the standard acidified peroxide solution, we attempted to increase the acidity to see if the cell potential could be made to play a larger role. The standard PNNL solution is adjusted to 1M H$_2$SO$_4$ and a slight increase of H$_2$0$_2$ to 6\\%. Two separate cleanings were made with this solution while one has an applied cell potential. The results in Fig. \\ref{fig:2}(b) show that both cleanings were effective at removing Cu ($\\sim 8\\,\\mu$m of depth) while the very little Po removal occurred, though the sample with the cell potential present hints at slightly more removal of Po. The chemical conditions of this solution provide a competition between the Po and Cu with an advantage to Cu with its lower oxidation potential. The test further confirms that, for this type of solution, there is no advantage to be gained by applying an oxidizing potential when a strong oxidizing agent is present.\n\nTo seek some benefit of an oxidizing potential, a test is performed without H$_2$0$_2$ but with a 6M H$_2$SO$_4$ solution for an even lower pH solution. Again, two samples are cleaned in separate solutions while one sample has an applied cell potential. From the results in Fig. \\ref{fig:2}(b), the acidic solution alone was not effective at removing either Cu or Po, while the applied cell potential was sufficient to effectively remove both. This observation suggests that a cell potential alone can oxidize Po in the presence of Cu in an acidic environment, which is consistent with the previous finding of the effectiveness of electropolishing.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalByActivity.eps} \\\\\n \\small (a)\n\\end{tabular}\n \\quad\n\\begin{tabular}[b]{c}\n \\includegraphics[width=0.49\\textwidth]{PoRemovalSpecials.eps}\\\\\n \\small (b) \n \\end{tabular}\n \\caption{(a) The fraction of Po removed from a cleaning solutions plotted against the initial $^{210}$Po surface activity. for the samples in Fig. \\ref{fig:1} where $>2 \\mu$m of Cu was removed. See the discussion in the text regarding the sample with an activity of 400 cts\/day. \n (b) The fraction of Po removed from a cleaning solutions plotted against the depth of Cu removed for two alternate cleaning methods. In one set, an acidified peroxide solution of lower pH is used with and without an applied cell potential resulting a small fraction of Po removal. In the second set, only a stronger acidic solution is used with and without an applied cell potential. Removal of Cu and Po is found with the application of a cell potential.\n }\n \\label{fig:2}\n\\end{figure}\n\n\\section{OUTLOOK}\nThere is a rich history of studying Rn progeny deposition and plate-out on surfaces to help guide the design of low background rare event searches. Likewise, successful studies demonstrate methods to remove Rn progeny from metal surfaces, including the problematic $^{210}$Po. Next-generation experiments will have even more stringent demands for the cleaning and removal of Rn progeny surface contamination. Given the need, it is desirable to use methods that provide both efficient progeny removal and ease of process implementation. \n\nSeveral factors are found that determine if the problematic $^{210}$Po progeny will stay in solution during cleaning to be fully removed from a contaminated surface. Oxidizing the Po atoms should keep them in solution and prevent redeposition. As shown here, oxidation of Po can be achieved by an oxidizing agent or an applied potential in the right environment. So the ultimate method to achieve full progeny removal will be dependent on the ideal chemical conditions for the material being cleaned and the chemistry required to achieve and maintain oxidation of the progeny. In the case of Cu samples, the controlled PNNL acidified peroxide solution is capable of fully removing Po atoms rather than more aggressive methods. However, the cleaning must be performed in a way to give sufficient oxidizing agent exposure commensurate wth the initial $^{210}$Po surface activity. Increased exposure can be achieved through agitation of the samples, greater solution volume in contact with the samples, greater concentration of H$_2$0$_2$, or a longer duration time in solution.\n\nFurther studies are needed to fully explore the process variables tested here. The effects of Po solubility, as noted by previous studies \\cite{hop07}, will aid in understanding the correlation between $^{210}$Po activity and Po removal. The role of the kinetics should be explored to understand why a greater exposure is required to oxidize the Po atoms and prevent redeposition than that needed for the substrate atoms. In studies to determine acceptable cleaning solutions, it is desirable to seek out chemical conditions and processes that favor keeping the progeny in solution.\n\n\n\\section{ACKNOWLEDGMENTS}\nWe thank our {\\sc Majorana} collaborators for fruitful discussions. \nThis material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-SC0012612. This material is based upon cooperation with S.R. Elliott and Los Alamos National Laboratory.\n\n\\bibliographystyle{aipnum-cp}%\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nSubluminous B stars (subdwarf B stars or sdBs) are stars with thin hydrogen envelopes, currently undergoing helium-core burning, which are found on the extreme horizontal branch (EHB). Their masses were determined to be around $0.47$\\,M$_{\\rm \\odot}$ \\citep{heber09, heber16}. About half of the known single-lined sdB stars are found to be members of short-period binaries \\citep[P $\\lesssim$ 30 d, most even with P $\\lesssim$ 10 d,][]{maxted01,napiwotzki04a,kupfer15}. A large mass loss on the red giant branch (RGB) is required to form these stars, which can be caused by mass transfer to the companion, either via stable Roche lobe overflow or the formation and eventual ejection of a common envelope \\citep{han02,han03}. \nFor the existence of apparently single sdB stars binary evolution might play an important role as well, as such stars could be remnants of helium white dwarf mergers \\citep{webbink84,ibentutukov84} or from engulfing a substellar object, which might get destroyed in the process \\citep{soker98,nelemans98}.\n\nEclipsing sdB+dM binaries (HW\\,Vir systems) having short orbital periods ($0.05-1\\,{\\rm d}$) and low companion masses between $0.06$\\,M$_{\\rm \\odot}$ and $0.2$\\,M$_{\\rm \\odot}$ \\citep[see][for a summary of all known HW Vir systems]{schaffenroth18,schaffenroth19} have been known for decades \\citep{menzies} and illustrate that objects close to the nuclear burning limit of $\\sim 0.070-0.076$\\,M$_{\\rm \\odot}$ for an object of solar metallicity and up to $0.09\\,\\rm M_{\\rm \\odot}$ for metal-poor objects \\citep[see][for a review]{2014AJ....147...94D} can eject a common envelope and lead to the formation of an sdB. The light travel-time technique was used to detect substellar companion candidates to sdB stars \\citep[e.g.][and references therein]{beuermann12, kilkenny12}. However, in these systems the substellar companions\nhave wide orbits and therefore cannot have influenced the evolution of the host star.\n\n\nThe short-period eclipsing HW~Vir type binary SDSS J082053.53+000843.4, hereafter J08205+0008, was discovered as part of the MUCHFUSS project \\citep{geier11a, geier11b}. \\citet{geier11c} derived an orbital solution based on time resolved medium resolution spectra from SDSS \\citep{abazajian09} and ESO-NTT\/EFOSC2. The best fit orbital period was $P_{\\rm orb}=P=0.096\\pm0.001\\,{\\rm d}$ and the radial velocity (RV) semi-amplitude $K=47.4\\pm1.9\\,{\\rm km\\,s^{-1}}$ of the sdB. An analysis of a light curve taken with Merope on the Mercator telescope allowed them to constrain the inclination of the system to $85.8^{\\rm \\circ}\\pm0.16$. \n\nThe analysis resulted in two different possible solutions for the fundamental parameters of the sdB and the companion. As the sdB sits on the EHB the most likely solution is a core-He burning object with a mass close to the canonical mass for the He flash of $0.47 \\,\\rm M_\\odot$. Population synthesis models \\citep{han02,han03} predict a mass range of $M_{\\rm sdB}=0.37-0.48\\,\\rm M_\\odot$, which is confirmed by asteroseismological measurements \\citep{fontaine12}. A more massive ($2-3\\,\\rm M_\\odot$) progenitor star would ignite the He core under non-degenerate conditions and lower masses down to $0.3\\,\\rm M_\\odot$ are possible. Due to the shorter lifetime of the progenitors such lower mass hot subdwarfs would also be younger. Higher masses for the sdB were ruled out as contemporary theory did not predict that. By a combined analysis of the spectrum and the light curve the companion was derived to have a mass of $0.068\\pm0.003\\,\\rm M_\\odot$. However, the derived companion radius for this solution was significantly larger than predicted by theory.\n\nThe second solution that was consistent with the atmospheric parameters was a post-RGB star with an even lower mass of only $0.25\\,\\rm M_\\odot$. Such an object can be formed whenever the evolution of the star on the RGB is interrupted due to the ejection of a common envelope before the stellar core mass reaches the mass, which is required for helium ignition. Those post-RGB stars, also called pre-helium white dwarfs, cross the EHB and evolve directly to white dwarfs.\nIn this case the companion was determined to have a mass of $0.045\\pm0.003\\,\\rm M_\\odot$ and the radius was perfectly consistent with theoretical predictions.\n\n\nThe discovery of J08205+0008 was followed by the discovery of two more eclipsing systems with brown dwarf (BD) companions, J162256+473051 \\citep{schaffenroth14} and V2008-1753 \\citep{schaffenroth15}, both with periods of less than 2 hours. Two non-eclipsing systems were also discovered by \\citet{schaffenroth14a}, and a subsequent analysis of a larger population of 26 candidate binary systems by \\citet{schaffenroth18} suggests that the fraction of sdB stars with close substellar companions is as high as 3 per cent, much higher than the $0.5\\pm0.3$ per cent that is estimated for brown dwarf companions to white dwarfs (e.g. \\citealt{steele11}). Seven of the nine known white dwarf-brown dwarf systems have primary masses within the mass range for a He-core burning hot subdwarf and might therefore have evolved through this phase before. \n\nIn this paper, we present new phase-resolved spectra of J08205+0008 obtained with ESO-VLT\/UVES and XSHOOTER and high cadence light curves with ESO-NTT\/ULTRACAM. Combining these datasets, we have refined the radial velocity solution and light curve fit. We performed an in-depth analysis of the sdB atmosphere and a fit of the spectral energy distribution using the ULTRACAM secondary eclipse measurements to better constrain the radius and mass of the sdB primary and the companion. We also present our photometric campaign using the SAAO\/1m-telescope and BUSCA mounted at the Calar Alto\/2.2m telescope which has been underway for more than 10 years now, and which has allowed us to derive variations of the orbital period.\n\n\\section{Spectroscopic and photometric data}\n\n\\subsection{UVES spectroscopy}\nWe obtained time-resolved, high resolution ($R\\simeq40\\,000$) spectroscopy of J08205+0008 with ESO-VLT\/UVES \\citep{dekker04} on the night of 2011-04-05 as part of program 087.D-0185(A). In total 33 single spectra with exposure times of $300\\,{\\rm s}$ were taken consecutively to cover the whole orbit of the binary. We used the 1\" slit in seeing of $\\sim$ 1\" and airmass ranging from 1.1. to 1.5. The spectra were taken using cross dispersers CD\\#2 and CD\\#3 on the blue and red chips respectively to cover a wavelength range from \\SI{3300}{\\angstrom} to \\SI{6600}{\\angstrom} with two small gaps ($\\simeq$ \\SI{100}{\\angstrom}) at \\SI{4600}{\\angstrom} and \\SI{5600}{\\angstrom}. \n\nThe data reduction was done with the UVES reduction pipeline in the \\textsc{midas} package \\citep{midas}. In order to ensure an accurate normalisation of the spectra, two spectra of the DQ type white dwarf WD\\,0806$-$661 were also taken \\citep{subasavage09}. Since the optical spectrum of this carbon-rich white dwarf is featureless, we divided our data by the co-added and smoothed spectrum of this star.\n\nThe individual spectra of J08205+0008 were then radial velocity corrected using the derived radial velocity of the individual spectra as described in Sect. \\ref{rvs} and co-added for the atmospheric analysis.\nIn this way, we increased the signal-to-noise ratio to S\/N\\,$\\sim$\\,90, which was essential for the subsequent quantitative analysis.\n\n \n\n\\subsection{XSHOOTER spectroscopy}\nWe obtained time resolved spectra of J08205+0008 with ESO-VLT\/XSHOOTER \\citep{vernet11} as part of programme 098.C-0754(A). The data were observed on the night of 2017-02-17 with 300~s exposure times in nod mode and in seeing of $0.5-0.8$\". We obtained 24 spectra covering the whole orbital phase (see Fig. \\ref{change of atmospheric parameters vs. orbital phase}) in each of the UVB ($R\\sim$\\,5400), VIS ($R\\sim$\\,8900) and NIR ($R\\sim$\\,5600) arms with the $0.9-1.0$\" slits. The spectra were reduced using the ESO \\textsc{reflex} package \\citep{reflex} and the specific XSHOOTER routines in nod mode for the NIR arm, and in stare mode for the UVB and VIS arms.\n\nTo correct the astronomical observations for atmospheric absorption features in the VIS and NIR arms, we did not require any observations of telluric standard stars, as we used the \\texttt{molecfit} software, which is based on fitting synthetic transmission spectra calculated by a radiative transfer code to the astronomical data \\citep{2015A&A...576A..77S, 2015A&A...576A..78K}. The parameter set-up (fitted molecules, relative molecular column densities, degree of polynomial for the continuum fit, etc.) for the telluric absorption correction evaluation of the NIR-arm spectra were used according to Table 3 of \\citet{2015A&A...576A..78K}. Unfortunately, the NIR arm spectra could not be used after the telluric corrections since the signal-to-noise (S\/N) ratio and the fluxes are too low. Figure \\ref{VIS X-Shooter arm before and after telluric correction with molecfit} shows an example comparison between the original and the telluric absorption corrected XSHOOTER VIS arm spectra. The quality of the telluric correction is sufficient to allow us to make use of the hydrogen Paschen series for the quantitative spectral analysis.\n\nAccurate radial velocity measurements for the single XSHOOTER spectra were performed within the analysis program SPAS \\citep{2009PhDT.......273H}, whereby selected sharp metal lines listed in Table \\ref{list of lines detected} were used. We used a combination of Lorentzian, Gaussian and straight line (in order to model the slope of the continuum) function to fit the line profiles of the selected absorption lines.\nAfter having corrected all single spectra by the averaged radial velocities, a co-added spectrum was created in order to achieve S\/N\\,$\\sim 460\/260$ in the UVB and VIS channels, respectively.\n\nThe co-added spectrum then was normalized also within SPAS. Numerous anchor points were set where the stellar continuum to be normalized was assumed. In this way, the continuum was approximated by a spline function. To obtain the normalized spectrum, the original spectrum was divided by the spline. \n\n\\subsection{ULTRACAM photometry}\\label{ULTRACAM}\nLight curves in the SDSS $u'g'r'$ filters were obtained simultaneously using the ULTRACAM instrument \\citep{dhillon} on the 3.5m-ESO-NTT at La Silla. The photometry was taken on the night of 2017-03-19 with airmass $1.15-1.28$ as part of programme 098.D-679 (PI; Schaffenroth). The data were taken in full frame mode with 1$\\times$1 binning and the slow readout speed with exposure times of 5.75~s resulting in 1755 frames obtained over the full orbit of the system. The dead-time between each exposure was only 25 msec. We reduced the data using the HiperCam pipeline (\\url{http:\/\/deneb.astro.warwick.ac.uk\/phsaap\/hipercam\/docs\/html}). The flux of the sources was determined using aperture photometry with an aperture scaled variably according to the full width at half-maximum. The flux relative to a comparison star within the field of view (08:20:51.941 +00:08:21.64) was determined to account for any variations in observing conditions. This reference star has SDSS magnitudes of $u'$=15.014$\\pm$0.004, $g'$=13.868$\\pm$0.003, $r'$=13.552$\\pm$0.003 which were used to provide an absolute calibration for the light curve. \n\n\n\\subsection{SAAO photometry}\\label{saao}\n\n\nAll the photometry was obtained on the 1m (Elizabeth) telescope at the Sutherland\nsite of the South African Astronomical Observatory (SAAO). Nearly all observations\nwere made with the STE3 CCD, except for the last two (Table H1), which were made\nwith the STE4 camera. The two cameras are very similar with the only difference being the pixel size as the STE3 is $512\\times512$ pixels in size and the STE4 is $1024\\times1024$. We used a $2\\times2$ pre-binned mode for each CCD resulting in a read-out time of around 5 and 20s, respectively, so that\nwith typical exposure times around 10-12s, the time resolution of STE4 is only about\nhalf as good as STE3. Data reduction and eclipse analysis were carried out as outlined\nin \\citep{kilkenny11}; in the case of J08205+008, there are several useful comparison \nstars, even in the STE3 field, and - given that efforts were made to observe eclipses near the meridian - usually there were no obvious \"drifts\" caused by differential\nextinction effects. In the few cases where such trends were seen, these were removed\nwith a linear fit to the data from just before ingress and just after egress. The stability of the\nprocedures (and the SAAO time system over a long time base) is demonstrated by the\nconstant-period system AA Dor \\citep[Fig.1 of ][]{kilkenny14} and by the intercomparisons in Fig. 8 of \\citet{baran18}, for example. \n\n\n\\subsection{BUSCA photometry}\\label{calaralto}\nPhotometric follow-up data were also taken with the Bonn University\nSimultaneous CAmera \\cite[BUSCA; see][]{busca}, which is mounted to the\n2.2 m-telescope located at the Calar Alto Observatory in Spain.\nThis instrument observes in four bands simultaneously giving a very accurate eclipse measurement and good estimate of the errors. The four different bands we used in our observation are given solely by the intrinsic transmission curve given by the\nbeam splitters (UB, BB, RB, IB, \\url{http:\/\/www.caha.es\/CAHA\/Instruments\/BUSCA\/bands.txt}) and the efficiency of the CCDs, as no filters where used to ensure that all the visible light is used most efficiently. \n\nThe data were taken during one run on 25 Feb 2011 and 1 Mar 2011. We used an exposure time of 30~s. Small windows were defined around the target and four comparison stars to decrease the read-out time from 2 min to 15 s. As comparison stars we used stars with similar magnitudes ($\\Delta m< 2$mag) in all SDSS bands from $u$ to $z$, which have been pre-selected using the SDSS DR 9 skyserver (\\url{http:\/\/skyserver.sdss.org\/dr9\/en\/}). The data were reduced using IRAF\\footnote{http:\/\/iraf.noao.edu\/}; a standard CCD reduction was performed using the IRAF tools for bias- and flatfield-correction. Then the light curves of the target and the comparison stars were extracted using the aperture photometry\npackage of DAOPHOT. The final light was constructed by dividing the light curve of the target by the light curves of the comparison stars.\n\n\n\n\n\\section{Analysis}\n\n\\subsection{The hybrid LTE\/NLTE approach and spectroscopic analysis}\\label{The spectroscopic analysis technique}\nBoth the co-added UVES and XSHOOTER (UVB and VIS arm) spectra were analyzed using the same hybrid local thermodynamic equilibrium (LTE)\/non-LTE (NLTE) model atmospheric approach. This approach has been successfully used to analyze B-type stars (see, for instance, \\citealt{2006BaltA..15..107P, 2006A&A...445.1099P, 2011JPhCS.328a2015P}; \\citealt{2007A&A...467..295N, Nieva_2008}) and is based on the three generic codes \\textsc{atlas12} \\citep{1996ASPC..108..160K}, \\textsc{detail}, and \\textsc{surface} (\\citealt{1981PhDT.......113G}; \\citealt{Butler_1985}, extended and updated). \n\nBased on the mean metallicity for hot subdwarf B stars according to \\citet{naslim13}, metal-rich and line-blanketed, plane-parallel and chemically homogeneous model atmospheres in hydrostatic and radiative equilibrium were computed in LTE within \\textsc{atlas12}. \nOccupation number densities in NLTE for hydrogen, helium, and for selected metals (see Table \\ref{summary of model atoms used for the hybrid LTE\/NLTE approach}) were computed with \\textsc{detail} by solving the coupled radiative transfer and statistical equilibrium equations. The emergent flux spectrum was synthesized afterwards within \\textsc{surface}, making use of realistic line-broadening data.\nRecent improvements to all three codes \\citep[see][for details]{2018A&A...615L...5I} with regard to NLTE effects on the atmospheric structure as well as the implementation of the occupation probability formalism \\citep{1994A&A...282..151H} for H\\,{\\sc i} and He\\,{\\sc ii} and new Stark broadening tables for H \\citep{2009ApJ...696.1755T} and He\\,{\\sc i} \\citep{1997ApJS..108..559B} are considered as well. For applications of these models to sdB stars see \\citet{schneider18}. \n\n\n\nWe included spectral lines of H and \\ion{He}{i}, and in addition, various metals in order to precisely measure the projected rotational velocity ($v\\sin i$), radial velocity ($v_{\\rm rad}$), and chemical abundances of J08205+0008. The calculation of the individual model spectra is presented in detail in \\citet{2014A&A...565A..63I}.\nIn Table \\ref{Hybrid LTE\/NLTE model grid used for the quantitative spectral analysis of SDSS J08205+0008}, the covered effective temperatures, surface gravities, helium and metal abundances for the hybrid LTE\/NLTE model grid used are listed.\n\n\n\n\nThe quantitative spectral analysis followed the methodology outlined in detail in \\citet{2014A&A...565A..63I}, that is, the entire useful spectrum and all 15 free parameters ($T_{\\rm eff}$, $\\log g$, $v_{\\rm rad}$, $v\\sin i$, $\\log{n(\\text{He})}:=\\log{\\left[\\frac{\\text{N(He)}}{\\text{N(all elements)}}\\right]}$, plus abundances of all metals listed in Table \\ref{summary of model atoms used for the hybrid LTE\/NLTE approach}) were simultaneously fitted using standard $\\chi^2$ minimization techniques. Macroturbulence $\\zeta$ and microturbulence $\\xi$ were fixed to zero because there is no indication for additional line-broadening due to these effects in sdB stars \\citep[see, for instance,][]{geier:2012,schneider18}.\n\n\n\\begin{table}\n\\caption{Metal abundances of J08205+0008 derived from XSHOOTER and UVES.$^\\dagger$}\\label{abundance table}\n\\centering\n\\begin{tabular}{lll}\n\\hline\\hline\nParameter & XSHOOTER & UVES\\\\\n\\hline\n$\\log{n(\\text{C})}$ & $-4.38\\pm0.05$ & $-4.39^{+0.04}_{-0.03}$\\\\\n$\\log{n(\\text{N})}$ & $-4.00^{+0.03}_{-0.02}$ & $-3.98\\pm0.03$\\\\\n$\\log{n(\\text{O})}$ & $-4.01^{+0.05}_{-0.06}$ & $-3.86^{+0.07}_{-0.06}$\\\\\n$\\log{n(\\text{Ne})}$ & $\\leq -6.00$ & $\\leq -6.00$\\\\\n$\\log{n(\\text{Mg})}$ & $-4.98^{+0.05}_{-0.04}$ & $-5.03\\pm0.05$\\\\\n$\\log{n(\\text{Al})}$ & $-6.20\\pm0.03$ & $\\leq -6.00$\\\\\n$\\log{n(\\text{Si})}$ & $-5.13\\pm0.04$ & $-5.17^{+0.07}_{-0.08}$\\\\\n$\\log{n(\\text{S})}$ & $-5.31^{+0.11}_{-0.10}$ & $-5.12^{+0.06}_{-0.08}$\\\\\n$\\log{n(\\text{Ar})}$ & $-5.54^{+0.15}_{-0.27}$ & $-5.32^{+0.19}_{-0.23}$\\\\\n$\\log{n(\\text{Fe})}$ & $-4.39\\pm0.04$ & $-4.41^{+0.04}_{-0.05}$\\\\\n\\hline\n\\multicolumn{3}{l}{$\\dagger$: Including 1$\\sigma$ statistical and systematic errors.}\\\\\n\\multicolumn{3}{l}{$\\log{n(\\text{X})}:=\\log{\\left[\\frac{\\text{N(X)}}{\\text{N(all elements)}}\\right]}$}\n\\end{tabular}\n\\end{table}\\noindent\n\n\\subsection{Effective temperature, surface gravity, helium content and metal abundances} \\label{Effective temperature, surface gravity, and helium content}\nThe excellent match of the global best fit model spectrum to the observed one is demonstrated in Fig. \\ref{XSHOOTER hydrogen and helium lines 1} for selected spectral ranges in the co-added XSHOOTER spectrum of J08205+0008 (UVB + VIS arm).\n\nThe wide spectral range covered by the XSHOOTER spectra allowed, besides the typical hydrogen Balmer series and prominent $\\ion{He}{i}$ lines in the optical, Paschen lines to be included in the fit, which provides additional information that previously could not be used in sdB spectral analysis, but provides important consistency checks.\\\\\n\nIn the framework of our spectral analysis, we also tested for variations of the atmospheric parameters over the orbital phase as seen in other reflection effect systems \\citep[e.g.][]{heber04,schaffenroth13}. As expected, due to the relatively weak reflection effect of less than 5\\%, the variations were within the total uncertainties given in the following and can therefore be neglected (see also Fig. \\ref{change of atmospheric parameters vs. orbital phase} for details).\n\n \n \\begin{figure*}\n \\begin{center}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_1_v2.pdf}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_2_v2.pdf}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.49]{SDSS0820_hydrogen_and_helium_line_profiles_3_v2.pdf}\n \\caption{Comparison between observation (solid black line) and global best fit (solid red line) for selected spectral ranges in the co-added XSHOOTER spectrum of J08205+0008. Prominent hydrogen and $\\ion{He}{i}$ lines are marked by blue labels and the residuals for each spectral range are shown in the bottom panels, whereby the dashed horizontal lines mark mark deviations in terms of $\\pm1\\sigma$, i.e., values of $\\chi=\\pm1$ (0.2\\% in UVB and 0.4\\% in VIS, respectively).} Additional absorption lines are caused by metals (see Fig. \\ref{SDSS0820_metal_line_profiles}). Spectral regions, which have been excluded from the fit, are marked in grey (observation) and dark red (model), respectively. Since the range between $\\ion{H}{i}$ \\SI{9230}{\\angstrom} and $\\ion{H}{i}$ \\SI{9546}{\\angstrom} strongly suffers from telluric lines (even after the telluric correction with \\texttt{molecfit}), it is excluded from the figure.\n \\label{XSHOOTER hydrogen and helium lines 1}\n \\end{center}\n \\end{figure*}\n\n\n\n\n \n \n \n \n\n \n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=\\linewidth]{J0820_kiel.pdf}\n \\caption\n $T_{\\text{eff}}-\\log{(g)}$ diagram for J08205+0008. While the blue square represents the UVES solution, the red square results from XSHOOTER. The grey square marks the LTE solution of \\citet{geier11c}. The zero-age (ZAEHB) and terminal-age horizontal branch (TAEHB) for a canonical mass sdB are shown in grey as well as evolutionary tracks for a canonical mass sdB with different envelope masses from \\citet{1993ApJ...419..596D} with black dotted lines. Additionally we show evolutionary tracks with solar metallicity for different sdB masses with hydrogen layers of $0.005\\rm\\,M_\\odot$, according to \\citet{han02} to show the mass dependence of the EHB. The error bars include 1$\\sigma$ statistical and systematic uncertainties as presented in the text (see Sect. \\ref{Effective temperature, surface gravity, and helium content} for details).}\n \\label{Kiel diagram}\n \\end{center}\n \\end{figure}\n \n \\begin{figure}\n \\begin{center}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.45]{NLTE_abundances_plot_v2.pdf}\n \\caption{The chemical abundance pattern of J08205+0008 (red: XSHOOTER, blue: UVES) relative to solar abundances of \\citet{asplund09}, represented by the black horizontal line. The orange solid line represents the mean abundances for hot subdwarf B stars according to \\citet{naslim13} used as the metallicity for our quantitative spectral analysis. Upper limits are marked with downward arrows and $\\left[\\frac{\\text{N}(\\text{X})}{\\text{N}(\\text{total})}\\right]:=\\log_{10}{\n \\left\\{\\frac{\\text{N}(\\text{X})}{\\text{N}(\\text{total})}\\right\\}}-\\log_{10}{\\left\\{\\frac{\\text{N}(\\text{X(solar)})}{\\text{N}(\\text{total})}\\right\\}}$.}\n \n \\label{abundance}\n \\end{center}\n \\end{figure}\n \n \n \n \\begin{figure*}\n \\begin{center}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.5]{SDSS0820_metal_line_profiles_1.pdf}\n \\includegraphics[trim = 0cm 0cm 0cm 0cm, clip, scale=0.5]{SDSS0820_metal_line_profiles_2.pdf}\n \\caption{Selected metal lines in the co-added XSHOOTER spectrum of J08205+0008. The observed spectrum (solid black line) and the best fit (solid red line) are shown. Solid blue vertical lines mark the central wavelength positions and the ionization stages of the individual metal lines according to Table \\ref{list of lines detected}.}\n \\label{SDSS0820_metal_line_profiles}\n \\end{center}\n \\end{figure*}\n \nThe resulting effective temperatures, surface gravities, and helium abundances derived from XSHOOTER and UVES are listed in Table \\ref{tab:par}. The results include 1$\\sigma$ statistical errors and systematic uncertainties according to the detailed study of \\citet{2005A&A...430..223L}, which has been conducted in the framework of the ESO Supernova Ia Progenitor Survey. For stars with two exposures or more, \\citet{2005A&A...430..223L} determined a systematic uncertainty of $\\pm$374\\,K for $T_{\\text{eff}}$, $\\pm$\\,0.049\\,dex for $\\log{(g)}$, and $\\pm$\\,0.044\\,dex for $\\log{n(\\text{He})}$ (see Table 2 in \\citealt{2005A&A...430..223L} for details).\n\nFigure \\ref{Kiel diagram} shows the $T_{\\text{eff}}-\\log{(g)}$ diagram, where we compare the UVES and XSHOOTER results to predictions of evolutionary models for the horizontal branch for a canonical mass sdB with different envelope masses from \\citet{1993ApJ...419..596D}, as well as evolutionary tracks\nassuming solar metallicity and masses of $0.50$\\,M$_{\\rm \\odot}$ and $0.55$\\,M$_{\\rm \\odot}$ \\citep{han02}. With $T_{\\text{eff}}=26\\,000\\pm400$\\,\\si{\\kelvin} and $\\log{(g)}=5.54\\pm0.05$ (XSHOOTER, statistical and systematic errors) and $T_{\\text{eff}}=25\\,600\\pm400$\\,\\si{\\kelvin} and $\\log{(g)}=5.51\\pm0.05$ (UVES, statistical and systematic errors), J08205+0008 lies within the EHB, as expected. Our final result ($T_{\\text{eff}}=25\\,800\\pm290$\\,\\si{\\kelvin}, $\\log{(g)}=5.52\\pm0.04$), the weighted average of the XSHOOTER and UVES parameters, is also in good agreement with the LTE results of \\citet{geier11c}, which are $T_{\\text{eff}}=26\\,700\\pm1000$\\,\\si{\\kelvin} and $\\log{(g)}=5.48\\pm0.10$, respectively.\n\nThe determined helium content of J08205+0008 is $\\log{n(\\text{He})}=-2.06\\pm0.05$ (XSHOOTER, statistical and systematic errors) and $\\log{n(\\text{He})}=-2.07\\pm0.05$ (UVES, statistical and systematic errors), hence clearly subsolar (see \\citealt{asplund09} for details).\nThe final helium abundance ($\\log{n(\\text{He})}=-2.07\\pm0.04$), the weighted average of XSHOOTER and UVES, therefore is comparable with \\citet{geier11c}, who measured $\\log{n(\\text{He})}=-2.00\\pm0.07$, and with the mean helium abundance for sdB stars from \\citet{naslim13}, which is $\\log{n(\\text{He})}=-2.34$ (see also Fig. \\ref{abundance}).\n\n \n\nMoreover, it was possible to identify metals of various different ionization stages within the spectra (see Table \\ref{list of lines detected} and Fig. \\ref{SDSS0820_metal_line_profiles}) and to measure their abundances. Elements found in more than one ionization stage are oxygen ($\\ion{O}{i\/ii}$), silicon ($\\ion{Si}{ii\/iii}$), and sulfur ($\\ion{S}{ii\/iii}$), whereas carbon ($\\ion{C}{ii}$), nitrogen ($\\ion{N}{ii}$), magnesium ($\\ion{Mg}{ii}$), aluminum ($\\ion{Al}{iii}$), argon ($\\ion{Ar}{ii}$), and iron ($\\ion{Fe}{iii}$) are only detected in a single stage. We used the model grid in Table \\ref{Hybrid LTE\/NLTE model grid used for the quantitative spectral analysis of SDSS J08205+0008} to measure the individual metal abundances in both the co-added XSHOOTER and the UVES spectrum. We were able to fit the metal lines belonging to different ionization stages of the same elements similarly well (see Fig. \\ref{SDSS0820_metal_line_profiles}). The corresponding ionization equilibria additionally constrained the effective temperature.\\\\\nAll metal abundances together with their total uncertainties are listed in Table 1. Systematic uncertainties were derived according to the methodology presented in detail in \\citet{2014A&A...565A..63I} and cover the systematic uncertainties in effective temperature and surface gravity as described earlier.\n\nThe results of XSHOOTER and UVES are in good agreement, except for the abundances of oxygen, sulfur, and argon, where differences of $0.15$\\,dex, $0.19$\\,dex, and $0.22$\\,dex, respectively, are measured. However, on average these metals also have the largest uncertainties, in particular argon, such that the abundances nearly overlap if the corresponding uncertainties are taken into account. According to Fig. \\ref{abundance}, J08205+0008 is underabundant in carbon and oxygen, but overabundant in nitrogen compared to solar (\\citealt{asplund09}), showing the prominent CNO signature as a remnant of the star's hydrogen core-burning through the CNO cycle. Aluminum and the alpha elements (neon, magnesium, silicon, and sulfur) are underabundant compared to solar. With the exception of neon, which is not present, the chemical abundance pattern of J08205+0008 generally follows the metallicity trend of hot subdwarf B stars (\\citealt{naslim13}), even leading to a slight enrichment in argon and iron compared to solar. The latter may be explained by radiative levitation, which occurs in the context of atomic transport, that is, diffusion processes in the stellar atmosphere of hot subdwarf stars (\\citealt{Greenstein_1967}; see \\citealt{Michaud_Atomic_Diffusion_in_Stars_2015} for a detailed review). \n\n \n\nDue to the high resolution of the UVES (and XSHOOTER) spectra, we were also able to measure the projected rotational velocity of J08205+0008 from the broadening of the spectral lines, in particular from the sharp metal lines, to $v\\sin{i}=66.0\\pm0.1\\,{\\rm km\\,s^{-1}}$ (UVES, 1$\\sigma$ statistical errors only) and $v\\sin{i}=65.8\\pm0.1\\,{\\rm km\\,s^{-1}}$ (XSHOOTER, 1$\\sigma$ statistical errors only)\n\n\n\n \n\n\\subsection{Search for chemical signatures of the companion} \nAlthough HW\\,Vir type systems are known to be single-lined, traces of the irradiated and heated hemisphere of the cool companion have been found in some cases. \\citet{wood99} discovered the H$\\alpha$ absorption component of the companion in the prototype system HW\\,Vir \\citep[see also][]{edelmann08}. \n\nMetal lines in emission were found in the spectra of the hot sdOB star AA\\,Dor by \\citet{vuckovic16} moving in antiphase to the spectrum of the hot sdOB star indicating an origin near the surface of the companion. After the removal of the contribution of the hot subdwarf primary, which is dominating the spectrum, the residual\nspectra showed more than 100 shallow emission lines originating from the heated side of the secondary, which show their maximum intensity close to the phases\naround the secondary eclipse. They analysed the residual spectrum in order to model the irradiation of the low-mass companion by the hot subdwarf star.\nThe emission lines of the heated side of the secondary star allowed them to determine the radial velocity semi-amplitude of the centre-of-light. After the correction to the centre-of-mass of the secondary they could derive accurate masses of both components of the AA Dor system, which is consistent with a canonical sdB mass of $0.46\\,\\rm M_\\odot$ and a companion of $0.079\\pm0.002\\rm\\,M_\\odot$ very close to the hydrogen burning limit. They also computed a first generation atmosphere model\nof the low mass secondary including irradiation effects. \n\nJ08205+0008 is significantly fainter and cooler than AA Dor but with a much shorter period.\nWe searched the XSHOOTER spectra for signs of the low-mass companion of J08205+0008.\nThis was done by subtracting the spectrum in the secondary minimum where the companion is eclipsed from the spectra before and after the secondary eclipse where most of the heated atmosphere of the companion is visible. \nHowever, no emission or absorption lines from the companion were detected (see Fig. \\ref{balmer_companion_uvb} and \\ref{balmer_companion_vis}). Also, in the XSHOOTER NIR arm spectra, no emission lines could be found.\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\\subsection{Photometry: Angular diameter and interstellar reddening}\\label{Photometric analysis}\\label{SED fitting}\n\nThe angular diameter of a star is an important quantity, because it\nallows the stellar radius to be determined, if the distance is known e.g. from\ntrigonometric parallax. The angular diameter can be determined by\ncomparing observed photometric magnitudes to those calculated from\nmodel atmospheres for the stellar surface. \nBecause of contamination by the reflection effect the apparent magnitudes of the hot subdwarf can be measured only during \nthe secondary eclipse, where the companion is completely eclipsed by the larger \nsubdwarf. We performed a least squares fit to the flat bottom of the secondary eclipse in the ULTRACAM light curves to \ndetermine the apparent magnitudes and \nderived $u'=14.926\\pm$ 0.009mag, $g'=15.025\\pm$ 0.004mag, and $r'=15.450\\pm$ 0.011mag (1$\\sigma$ statistical errors).\n\nBecause the star lies at low Galactic latitude (b=19$^\\circ$) interstellar reddening is expected to be significant.\nTherefore, both the angular diameter and the interstellar colour\nexcess have to be determined simultaneously. \nWe used the reddening law of \\citet{2019ApJ...886..108F} and matched a synthetic flux distribution calculated from the same grid of model \natmospheres that where also used in the quantitative spectral analysis (see Sect. \\ref{The spectroscopic analysis technique}) to the observed magnitudes as \ndescribed in \\citet{2018OAst...27...35H}. The $\\chi^2$ based fitting routine uses two free parameters: the angular diameter $\\theta$, \nwhich shifts the fluxes up and down according to $f(\\lambda)=\\theta^2 F(\\lambda)\/4$, where f($\\lambda$) is the observed flux at the detector \nposition and $F(\\lambda)$ is the synthetic model flux at the stellar surface, and the color excess \n\\footnote{\\citet{2019ApJ...886..108F} use $E(44-55)$, the monochromatic equivalent of the usual $E(B-V)$ in the Johnson system, \n using the wavelengths $\\lambda$ =4400\\,\\AA and 5000\\,\\AA, respectively. In fact, $E(44-55)$ is identical to $E(B-V)$ \n for high effective temperatures as determined for J08205+0008.}.\nThe final atmospheric parameters and their respective uncertainties derived from the quantitative spectral analysis (see Sect. \\ref{Effective temperature, surface gravity, and helium content}) \nresult in an angular diameter of $\\theta= 6.22\\, (\\pm 0.15) \\cdot 10^{-12}\\,\\rm rad$ and an interstellar reddening of \n$E(B-V)=0.041 \\pm 0.013$ mag.\nThe latter is consistent with values from reddening maps of \\citet{1998ApJ...500..525S} and \\citet{2011ApJ...737..103S}: 0.039 mag and 0.034 mag, \nrespectively. \n\nIn addition, ample photometric measurements of J08205+0008 are available in different filter systems, covering the spectral range all the way \nfrom the ultraviolet (GALEX) through the optical (e.g. SDSS) to the infrared (2MASS; UKIDDS; WISE). However, those measurements are mostly averages of \nobservations taken at multiple epochs or single epoch measurements at unknown orbital phase. \nTherefore, those measurements do not allow us to determine the angular diameter of the sdB because of the contamination by light from the heated hemisphere of the companion. However, an average spectral energy distribution of the system\ncan be derived. This allows us to redetermine the interstellar reddening and to search for an infrared excess caused by light from the \ncool companion.\n\nThe same fitting technique is used in the analysis of the SED as\ndescribed above for the analysis of the ULTRACAM magnitudes. Besides\nthe sdB grid, a grid of synthetic spectra of cool stars \\citep[$2300\\,{\\rm K} \\le T_{\\rm eff} \\le 15 000\\,\\rm K$,][]{2013A&A...553A...6H} is used.\nIn addition to the angular diameter and reddening parameter, the temperature of the cool companion as well as the surface ratio are \nfree parameters in the fit. The fit results in $E(B-V) = 0.040 \\pm\n0.010$ mag, which is fully consistent with the one derived from the ULTRACAM photometry as well as with the reddening map. \nThe apparent angular diameter is larger than that from ULTRACAM photometry by 2.8\\%, which is caused by the contamination by \nlight from the companion's heated hemisphere. The effective temperature of the companion is unconstrained and the best match is achieved for the surface ratio of zero, which means there is no signature from the \ncool companion.\nIn a final step we allow the effective temperature of the sdB to vary and determine it along with the angular diameter and the interstellar reddening, which results in $T_{\\rm eff}$= 26900$^{+1400}_{-1500}$\\,K in agreement with the spectroscopic result.\n\n\n \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.05\\columnwidth]{photometry_SED.pdf}\n \\caption{\\label{fig:photometry_sed}Comparison of synthetic and\n observed photometry: \\textit{Top panel:} Spectral\n energy distribution: Filter-averaged fluxes converted from\n observed magnitudes are shown in different colours. \n The respective full width at tenth maximum are shown as dashed\n horizontal lines. The best-fitting model, degraded to a\n spectral resolution of 6\\,{\\tiny\\AA} is plotted in gray. \n In order to reduce the steep SED slope the\n flux is multiplied by the wavelength cubed.\n \\textit{Bottom panel:} Difference between synthetic and observed magnitudes\n divided by the corresponding uncertainties (residual $\\chi$).\n The following color code is used for the different photometric\n systems: GALEX\n \\citep[violet,][]{2017yCat.2335....0B}; SDSS\n \\citep[golden,][]{2015ApJS..219...12A}; Pan-STARRS1\n \\citep[dark red,][]{2017yCat.2349....0C}; Johnson\n \\citep[blue,][]{2015AAS...22533616H};\n {\\it Gaia}\n \\citep[cyan,][with corrections and\n calibrations from \\citet{2018A&A...619A.180M}]{2018A&A...616A...4E}; 2MASS\n \\citep[red,][]{2003yCat.2246....0C}; UKIDSS \n \\citep[pink,][]{2007MNRAS.379.1599L}; \n WISE \\citep[magenta,][]{2014yCat.2328....0C,2019ApJS..240...30S}.}\n \\end{figure}\n \n\\subsection{Stellar radius, mass and luminosity}\\label{Stellar radius, mass and luminosity}\n \nSince Gaia data release 2 \\citep[DR2;][]{2018A&A...616A...1G}, trigonometric parallaxes are available for a\nlarge sample of hot subdwarf stars, including J08205+0008\nfor which 10\\% precision has been reached. We corrected for the Gaia DR2 parallax zero point offset of $-$0.029 mas \\citep{2018A&A...616A...2L}.\n\n Combining the parallax measurement\nwith the results from our quantitative spectral analysis ($\\log{g}$ and $T_{\\rm eff}$) and with the angular diameter $\\theta$ derived\nfrom ULTRACAM photometry, allows for the determination of the mass of the sdB primary in J08205+0008 via:\n\\begin{equation}\n M = \\frac{g \\theta^2}{4 G \\varpi^2}\\label{mass}\n\\end{equation}\n\nThe respective uncertainties of the stellar parameters are derived by Monte Carlo error propagation. The uncertainties are dominated by the error of the parallax measurement. Results are summarized in Table \\ref{tab:par}. \n Using the gravity and effective temperature derived by the spectroscopic analysis, the mass for the\nsdB is $M= 0.48^{+0.12}_{-0.09}$ M$_\\odot$ and its luminosity is $L=16^{+3.6}_{-2.8}$ L$_\\odot$ in agreement with canonical models for EHB stars \\citep[see Fig. 13][]{1993ApJ...419..596D}.\nThe radius of the sdB is calculated by the angular diameter and the parallax to\n$R = 0.200^{+0.021}_{-0.018}$ R$_\\odot$.\n \n \n \\subsection{Radial velocity curve and orbital parameters}\\label{rvs}\n\n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=\\linewidth]{rv_j0820.pdf}\n \\caption{Radial velocity of J08205+0008 folded on the orbital period.\n The residuals are shown together with a prediction of the Rossiter-McLaughlin effect using the parameters derived in this paper in blue and a model with a higher rotational velocity assuming bound rotation in green. The radial velocities were determined from spectra obtained with XSHOOTER (red circles), UVES (black triangles), EFOSC2 (black circles), and SDSS (black rectangles). The EFOSC2 and SDSS RVs have been corrected by a systematic shift (see text for details).}\n \\label{rv}\n \\end{center}\n \\end{figure}\n \n\nThe radial velocities of the individual XSHOOTER spectra were measured by fitting all spectral features simultaneously to synthetic models as described in Sect.~\\ref{The spectroscopic analysis technique}. \n\nDue to lower S\/N of the individual UVES spectra, which were observed in poor conditions, only the most prominent features in the spectra are suitable for measuring the Doppler shifts. After excluding very poor quality spectra, radial velocities of the remaining 28 spectra were measured using the {\\sc fitsb2} routine \\citep{napiwotzki04b} by fitting a set of different mathematical functions to the hydrogen Balmer lines as well as He\\,{\\sc i} lines. The continuum is fitted by a polynomial, and the line wings and line core by a Lorentzian and a Gaussian function, respectively. The barycentrically corrected RVs together with formal $1\\sigma$-errors are summarized in Table~\\ref{RVs}. \n\nThe orbital parameters $T_{\\rm 0}$, period $P$, system velocity $\\gamma$, and RV-semiamplitude $K$ as well as their uncertainties were derived with the same method described in \\citet{geier11b}. To estimate\nthe contribution of systematic effects to the total error budget additional to the statistic errors determined by the {\\sc fitsb2} routine, we normalised the $\\chi^{2}$ of the most probable solution by adding systematic errors to each data point $e_{\\rm norm}$ until the reduced $\\chi^{2}$ reached $\\simeq1.0$.\n\nCombining the UVES and XSHOOTER RVs we derived $T_{0}=57801.54954\\pm0.00024\\,{\\rm d}$, $P=0.096241\\pm 0.000003\\,{\\rm d}$, $K=47.9\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second} and the system velocity $\\gamma=26.5\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second}. No significant systematic shift was detected between the two datasets and the systematic error added in quadrature was therefore very small $e_{\\rm norm}=2.0$\\,\\si{\\kilo\\metre\\per\\second}. The gravitational redshift is significant at $1.6_{+0.05}^{-0.02}$\\,\\si{\\kilo\\metre\\per\\second} and might be important if the orbit of the companion could be measured by future high resolution measurements \\citep[see, e.g.,][]{vos13}.\n\nTo improve the accuracy of the orbital parameters even more we then tried to combine them with the RV dataset from \\citet{geier11c}, medium-resolution spectra taken with ESO-NTT\/EFOSC2 and SDSS. A significant, but constant systematic shift of $+17.4$\\,\\si{\\kilo\\metre\\per\\second} was detected between the UVES+XSHOOTER and the SDSS+EFOSC2 datasets. Such zero-point shifts are common between low- or medium-resolution spectrographs. It is quite remarkable that both medium-resolution datasets behave in the same way. However, since the shift is of the same order as the statistical uncertainties of the EFOSC2 and SDSS individual RVs we refrain from interpreting it as real. \n\nAdopting a systematic correction of $+17.4$\\,\\si{\\kilo\\metre\\per\\second} to the SDSS+EFOSC2 dataset, we combined it with the UVES+XSHOOTER dataset and derived $T_{0}\\,(\\rm BJD_{TDB})=2457801.59769\\pm0.00023\\,{\\rm d}$, $P=0.09624077\\pm 0.00000001\\,{\\rm d}$, which is in perfect agreement with the photometric ephemeris, $K=47.8\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second} and $\\gamma=26.6\\pm0.4$\\,\\si{\\kilo\\metre\\per\\second}. This orbital solution is consistent with the solution from the XSHOOTER+UVES datasets alone. Due to the larger uncertainties of the SDSS+EFOSC2 RVs, the uncertainties of $\\gamma$ and $K$ did not become smaller. The uncertainty of the orbital period on the other hand improved by two orders of magnitude due to the long timebase of 11 years between the individual epochs. Although this is still two orders of magnitude larger than the uncertainty derived from the light curve (see Sect.~\\ref{timing}), the consistency with the light curve solution is remarkable. The RV curve for the combined solution phased to the orbital period is given in Fig.~\\ref{rv}. Around phase 0 the Rossiter-McLaughin effect \\citep{rossiter,mclaughlin} is visible. This effect is a RV deviation that occurs as parts of a rotating star are blocked out during the transit of the companion. The effect depends on the radius ratio and the rotational velocity of the primary. We can derive both parameters much more precisely with the spectroscopic and photometric analysis, but we plotted a model of this effect using our system parameters on the residuals of the radial velocity curve to show that is consistent. \n\nExcept for the corrected system velocity, the revised orbital parameters of J08205+0008 are consistent with those determined by \\citet{geier11c} ($P=0.096\\pm 0.001\\,{\\rm d}$, $K=47.4\\pm1.9$\\,\\si{\\kilo\\metre\\per\\second}), but much more precise.\n\n \n \\subsection{Eclipse timing}\\label{timing}\n \\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{o-c_J0820_res}\n \\caption{(O--C) diagram for J08205+0008 using eclipse times observed with Merope (red squares), BUSCA (blue diamonds), ULTRACAM (green triangles) and the SAAO-1m\/1.9m telescope (black circles). The solid line represents a fit of a parabola to account for the period change of the orbital period. The derived quadratic term is given in the legend. The parameters of the fit are provided in the legend. In the lower panel the residuals between the observations and the best fit are shown.}\n \\label{o-c}\n\\end{figure}\n \n Since the discovery that J08205+00008 is an eclipsing binary in November 2009, we have monitored the system regularly using BUSCA mounted at the 2.2m-telescope in Calar Alto, Spain, ULTRACAM and the 1m in Sutherland Observatory (SAAO), South Africa. Such studies have been performed for several post-common envelope systems with sdB or white dwarf (WD) primaries and M dwarf companions \\citep[see][for a summary]{lohr14}. In many of those systems period changes have been found\n \nThe most convenient way to reveal period changes is to construct an observed minus calculated (O--C) diagram. Thereby we compare the observed mid-eclipse times (O) with the expected mid-eclipse times (C) assuming a fixed orbital period $P_0$ and using the mid-eclipse time for the first epoch $T_0$. Following \\citet{kepler91}, if we expand the observed mid-eclipse of the Eth eclipse ($T_E$ with $E=t\/P$) in a Taylor serie\n, we get the (O--C) equation:\n\\begin{equation}\n \\mathrm{O-C}=\\Delta T_0+\\frac{\\Delta P_0}{P_0}t+\\frac{1}{2}\\frac{\\dot{P}}{P_0}t^2+...\n\\end{equation}\nThis means that with a quadratic fit to the O--C data we can derive the ephemeris $T_0$, $P$, and $\\dot{P}$ in $BJD_{TDB}$.\n\nTogether with the discovery data observed with Merope at the Mercator telescope on La Palma \\citep{geier11c} it was possible to determine timings of the primary eclipse over more than 10 years, as described in Sect. \\ref{saao} and \\ref{calaralto}. \nAll measured mid-eclipse times can be found in Table \\ref{ecl_time}.\n\n\n\n We used all eclipse timings to construct an O--C diagram, which is shown in Fig. \\ref{o-c}. We used the ephemeris given in \\citet{geier11c} as a starting value to find the eclipse numbers of each measured eclipse time and detrended the O-C diagram by varying the orbital period until no linear trend was visible to improve the determination of the orbital period. During the first $7-8$ years of observations, the ephemeris appeared to be linear. This was also found by \\citet{pulley:18}. As their data show a large scatter, we do not use it in our analysis. However, in the last two years a strong quadratic effect was revealed. The most plausible explanation is a decrease in the orbital period of the system.\n This enabled us to derive an improved ephemeris for J08205+0008:\n \\begin{eqnarray*}\n T_0&=&2455165.709211(1)\\\\\n P&=&0.09624073885(5)\\rm\\,d\\\\\n \\dot{P}&=&-3.2(8)\\cdot 10^{-12}\\,\\rm dd^{-1}\n \\end{eqnarray*}\n \n \n \n \\subsection{Light curve modeling}\n \n \n\n With the new very high quality ULTRACAM $u'g'r'$ light curves we repeated the light curve analysis of \\citep{geier11c} obtaining a solution with much smaller errors.\n\n For the modeling of the light curve we used {\\sc lcurve}, a code written to model detached and accreting binaries containing a white dwarf \\citep[for details, see][]{copperwheat10}. It has been used to analyse several detached white dwarf-M dwarf binaries \\citep[e.g.,][]{parsons_nnser}. Those systems show very similar light curves with very deep, narrow eclipses and a prominent reflection effect, if the primary is a hot white dwarf. Therefore, {\\sc lcurve} is ideally suited for our purpose.\n \n The code calculates monochromatic light curves by subdividing each star into small elements with a geometry fixed by its radius as measured along the line from the center of one star towards the center of the companion. The flux of the visible elements is always summed up to get the flux at a certain phase. A number of different effects that are observed in compact and normal stars are considered, e.g. Roche distortions observed when a star is distorted from the tidal influence of a massive, close companion, as well as limb-darkening and gravitational darkening. Moreover, lensing and Doppler beaming, which are important for very compact objects with close companions, can be included. The Roemer delay, which is a light travel-time effect leading to a shift between primary and secondary eclipse times due to stars of different mass orbiting each other and changing their distance to us, and asynchronous orbits can be considered. The latter effects are not visible in our light curves and can hence be neglected in our case. \n \n \\begin{figure}\n \\centering\n \\includegraphics[width=1.05\\linewidth]{lc.pdf}\n \\caption{ULTRACAM $u'g'r'$ light curves of J08205+0008 together with the best fit of the most consistent solution. The light curves in the different filters have been shifted for better visualisation. The lower panel shows the residuals. The deviation of the light curves from the best fit is probably due to the fact that the comparison stars cannot completely correct for atmospheric effects due to the different colour and the crude reflection effect model used in the analysis is insufficient to correctly describe the shape of the reflection effect.}\n \\label{lc_0820}\n \\end{figure}\n \n \\begin{figure*}\n \\centering\n \\includegraphics[width=1.0\\linewidth]{mcmc_g_newer.pdf}\n \\caption{MCMC calculations showing the distributions of the parameter of the analysis of the ULTRACAM g'-band light curve.}\n \\label{mcmc_r}\n\\end{figure*}\n\n\n \n As we have a prominent reflection effect it is very important to model this effect as accurately as possible. The reflection effect, better called the irradiation effect, results from the huge difference in temperature between the two stars, together with their small separation.\nThe (most likely) tidally locked companion is heated up on the side facing the hot subdwarf because of the strong irradiation by the hot primary. Therefore, the contribution of the companion to the total flux of the system varies with phase and increases as more of the heated side is visible to the observer. We use a quite simple model, which calculates the fluxes from the temperatures of both companions using a black body approximation. The irradiation is approximated by assigning a new temperature to the heated side of the companion \n \\begin{equation}\n \\sigma T'^4_{\\rm sec}=\\sigma T^4_{\\rm sec}+F_{\\rm irr}=\\sigma T^4_{\\rm sec}\\left[1+\\alpha\\left(\\frac{T_{\\rm prim}}{T^{\\rm sec}}\\right)^4\\left(\\frac{R_{\\rm prim}}{a}\\right)^2\\right],\n \\end{equation}\n with $\\alpha$ being the albedo of the companion and $F_{\\rm irr}$ the irradiating flux, accounting for the angle of incidence and distance from the hot subdwarf. The irradiated side is heated up to a temperature of $13\\,000-15\\,000$ K similar to HW Vir \\citep{kiss00}, which is slightly hotter but has a longer period. Hence, the amplitude of the effect is increasing from blue to red as can be seen in Fig. \\ref{lc_0820}, as the sdB is getting fainter compared to the companion in the red. If the irradiation effect is very strong, the description given above might not be sufficient, as the back of the irradiated star is completely unaffected in this description, but heat transport could heat it up, increasing the luminosity of unirradiated parts as well. This is not considered in our simple model. \n\nAs the light curve model contains many parameters, not all of them independent, we fixed as many parameters as possible (see Table \\ref{param}). The temperature of the sdB was fixed to the temperature determined from the spectroscopic fit. We used the values determined by the coadded XSHOOTER spectra, as they have higher signal-to-noise.\nThe gravitational limb darkening coefficients were fixed to the values expected for a radiative atmosphere for the primary \\citep{von_zeipel} and a convective atmosphere for the secondary \\citep{lucy} using a blackbody approximation to calculate the resulting intensities. For the limb darkening of the primary we adopted a quadratic limb darkening law using the tables by \\citet{claret}. As the tables include only surface gravities up to $\\log{g}=5$ we used the values closest to the parameters derived by the spectroscopic analysis. \n\nAs it is a well-separated binary, the two stars are approximately spherical, which means the light curve is not sensitive to the mass ratio. Therefore, we computed solutions with different, fixed mass ratios. To localize the best set of parameters we used a \\textsc{simplex} algorithm \\citep{press92} varying the inclination, the radii, the temperature of the companion, the albedo of the companion (absorb), the limb darkening of the companion, and the time of the primary eclipse to derive additional mid-eclipse times. Moreover, we also allowed for corrections of a linear trend, which is often seen in the observations of hot stars, as the comparison stars are often redder and so the correction for the air mass is often insufficient. This is given by the parameter \"slope\". The model of the best fit is shown in Fig. \\ref{lc_0820} together with the observations and the residuals.\n\nTo get an idea about the degeneracy of parameters used in the light curve solutions, as well as an estimation of the errors of the parameters we performed Markov-Chain Monte-\nCarlo (MCMC) computations with \\textsc{emcee} \\citep{emcee} using the best solution we obtained with the \\textsc{simplex} algorithm as a starting value varying the radii, the inclination, the temperature of the companion as well as the albedo of the companion. As a prior we constrained the temperature of the cool side of the companion to $3000\\pm500$ K. Due to the large luminosity difference between the stars the temperature of the companion is not significantly constrained by the light curve. The computations were done for all three light curves separately.\n\nFor the visualisation we used the python package \\textsc{corner} \\citep[see Fig. \\ref{mcmc_r}]{corner}. The results of the MCMC computations of the light curves of all three filters agree within the error (see Table \\ref{param}). A clear correlation between both radii and the inclination is visible as well as a weak correlation of the albedo of the companion (absorb) and the inclination. This results from the fact that the companion is only visible in the combined flux due to the reflection effect and the eclipses and the amplitude of the reflection effect depends on the inclination, the radii, the separation, the albedo and the temperatures. Looking at the $\\chi^2$ of the temperature of the companion we see that all temperatures give equally good solutions showing that the temperature can indeed not be derived from the light curve fit. The albedo we derived has, moreover, a value > 1, which has been found in other HW Vir systems as well and is due to the simplistic modeling of the reflection effect. The reason for the different distribution in the inclination is not clear to us. However, it is not seen in the other bands. It might be related to the insufficient correction of atmospheric effects by the comparison stars. \n\n\\begin{table}\\caption{Parameters of the light curve fit of the ULTRACAM u'g'r' band light curves}\\label{param}\n\\begin{tabular}{llll}\n\t\\hline\\hline\n\tband & u'&g'&r'\\\\\\hline\n\t\\multicolumn{4}{c}{Fixed Parameters}\\\\\\hline\n\tq&\\multicolumn{3}{c}{0.147}\\\\\n\t$P$&\\multicolumn{3}{c}{0.09624073885}\\\\\n\t$T_{\\rm eff,sdB}$&\\multicolumn{3}{c}{25800}\\\\\n\t$x_{1,1}$&0.1305&0.1004&0.0788\\\\\n\t$x_{1,2}$&0.2608&0.2734&0.2281\\\\\n\t$g_1$&\\multicolumn{3}{c}{0.25}\\\\\n\t$g_2$&\\multicolumn{3}{c}{0.08}\\\\\n\t\\hline\n\t\\multicolumn{4}{c}{Fitted parameters}\\\\\\hline\n\t$i$&$85.3\\pm0.6$&$85.6\\pm0.2$&$85.4\\pm0.3$\\\\\n\t$r_1\/a$&$0.2772\\pm0.0029$&$0.2734\\pm0.0010$&$0.2748\\pm0.0014$\\\\\n\t$r_2\/a$&$0.1322\\pm0.0018$&$0.1297\\pm0.0006$&$0.1304\\pm0.0008$\\\\\n\t$T_{\\rm eff,comp}$&$3000\\pm500$&$2900\\pm500$&$3200\\pm560$\\\\\n\tabsorb&$1.54\\pm0.08$&$1.58\\pm0.03$&$2.08\\pm0.05$\\\\\n\t$x_2$&0.70&0.78&0.84\\\\\n\t$T_0$ [MJD]&57832.0355&57832.0354&57832.0354\\\\\n\tslope&-0.000968&-0.002377&0.00013417\\\\\n\t$\\frac{L_1}{L_1+L_2}$&0.992578&0.98735&0.97592\\\\\n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Absolute parameters of J08205+00008}\\label{Absolute parameters of J08205+00008}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{sdB_python.pdf}\n \\caption{Mass of the sdB versus the photometric $\\log g$ for J08205+0008 for different mass ratios from $0.11-0.20$ in steps of 0.01 (red solid line). The parameters were derived by combining the results from the analysis of the light curves and radial velocity curve. The grey area marks the spectroscopic $\\log{g}$ that was derived from the spectroscopic analysis.\n The blue dashed lines indicate the $\\log g$ derived by the radius from the SED fitting and the {\\it Gaia} distance for different sdB masses.\n The red area marks the mass range for the sdB for which we get a consistent solution by combining all different methods. The red vertical line represents the solution for a canonical mass sdB.} \n \\label{sdb}\n\\end{figure}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{brown_dwarf_python.pdf}\n \\caption{Comparison of theoretical mass-radius relations of low-mass stars \\citep{baraffe,baraffe_2} to results from the light curve analysis of J08205+0008. We used tracks for different ages of 1 Gyr (dashed), 5 Gyr (dotted-dashed) and 10 Gyr (dotted). Each red square together with the errors represents a solution from the light curve analysis for a different mass ratio ($q = 0.11-0.20$ in steps of 0.01). The red vertical line represents the solution for a canonical mass sdB. The red area marks the mass range of the companion corresponding to the mass range we derived for the sdB.}\n \\label{bd}\n\\end{figure}\nAs explained before, we calculated solutions for different mass ratios ($q=0.11-0.20$). We obtain equally good $\\chi^2$ for all solutions, showing that the mass ratio cannot be constrained by the light curve fit as expected. Hence, the mass ratio needs to be constrained differently.\nHowever, the separation, which can be calculated from the mass ratio, period, semi-amplitude of the radial-velocity curve and the inclination, is different for each mass ratio.\nThe masses of both companions can then be calculated from the mass function. From the relative radii derived from the light curve fit together with the separation, the absolute radii can be calculated. This results in different radii and masses for each mass ratio. \n\nAs stated before, the previous analysis of \\citet{geier11c} resulted in two possible solutions: A post-RGB star with a mass of 0.25 $\\rm M_\\odot$ and a core helium-burning star on the extreme horizontal branch with a mass of $\\rm 0.47\\,M_\\odot$.\nFrom the analysis of the photometry together with the Gaia magnitudes (see Sect. \\ref{Stellar radius, mass and luminosity}) we get an additional good constraint on the radius of the sdB. Moreover, the surface gravity was derived from the fit to the spectrum. This can be compared to the mass and radius of the sdB (and a photometric $\\log g$: $g=GM\/R^2$) derived in the combined analysis of radial velocity curve and light curve. This is shown in Fig. \\ref{sdb}. We obtain a good agreement for of all three methods (spectroscopic, photometric, parallax-based) for an sdB mass between 0.39-0.60 $\\rm M_\\odot$. This means that we can exclude the post-RGB solution. The position of J0820 in the $T_{\\rm eff}-\\log{g}$ diagram, which is shown in Fig. \\ref{Kiel diagram}, gives us another constraint on the sdB mass. By comparing the atmospheric parameters of J08205+0008 to theoretical evolutionary tracks calculated by \\citet{han02} it is evident that the position is not consistent with sdB masses larger than $\\sim 0.50\\,\\rm M_\\odot$, which we, therefore, assume as the maximum possible mass for the sdB.\n\nAccordingly, we conclude that the solution that is most consistent with all different analysis methods is an sdB mass close to the canonical mass ($0.39-0.50\\,\\rm M_\\odot$). For this solution we have an excellent agreement of the parallax radius with the photometric radius only, if the parallax offset of $-0.029$ mas suggested by \\citet{2018A&A...616A...2L} is used. Otherwise the parallax-based radius is too large. The companion has a mass of $0.061-0.71\\,\\rm M_\\odot$, which is just below the limit for hydrogen-burning. Our final results can be found in Table \\ref{tab:par}. The mass of the companion is below the hydrogen burning limit and the companion is hence most likely a massive brown dwarf.\n\nWe also investigated the mass and radius of the companion and compared it to theoretical calculations by \\citet{baraffe} and \\citet{baraffe_2} as shown in Fig.~\\ref{bd}. It is usually assumed that the progenitor of the sdB was a star with about $1-2\\,\\rm M_\\odot$ \\citep{heber09,heber16}. Therefore, we expect that the system is already quite old (5-10 Gyrs). For the solutions in our allowed mass range the measured radius of the companion is about 20\\% larger than expected from theoretical calculations. Such an effect, called inflation, has been observed in different binaries and also planetary systems with very close Jupiter-like planets. A detailed discussion will be given later. This effect has already been observed in other hot subdwarf close binary systems \\citep[e.g.][]{schaffenroth15}. \n\nHowever, if the system would still be quite young with an age of about 1 Gyr, the companion would not be inflated. We performed a kinematic analysis to determine the Galactic population of J08205+0008. As seen in Fig. \\ref{toomre} the sdB binary belongs to the thin disk where star formation is still ongoing and could therefore indeed be as young as 1 Gyr, if the progenitor was a $2\\,\\rm M_\\odot$ star. About half of the sdO\/Bs at larger distances from the Galactic plane (0.5 kpc) are found in the thin disk \\citep{martin17}. However, it is unclear whether a brown dwarf companion can eject the evelope from such a massive 2 $\\rm M_{\\odot}$ star. Hydrodynamical simulations performed by \\citet{kramer20} indicate that a BD companion of $\\sim0.05-0.08\\,\\rm M_\\odot$ might just be able to eject the CE of a lower mass ($1\\,\\rm M_\\odot$) red giant.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{{SDSSJ082053.53+000843.4_Toomre_AS}.pdf}\n \\caption{Toomre diagram of J08205+0008: the quantity $V$ is the velocity in direction of Galactic rotation, $U$ towards the Galactic center, and $W$ perpendicular to the Galactic plane. The two dashed ellipses mark boundaries for the thin (85\\,km\\,s$^{-1}$) and thick disk (180\\,km\\,s$^{-1}$) following Fuhrmann (2004). The red cross marks J08205+0008, the yellow circled dot the Sun, and the black plus the local standard of rest. The location of J08205+0008 in this diagram clearly hints at a thin disk membership. }\n \\label{toomre}\n\\end{figure}\n\n\\begin{table}\n\\caption{Parameters of J08205+0008.\n}\\label{tab:par}\n\\vspace{0.5cm}\n\\begin{tabular}{lll}\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{SPECTROSCOPIC PARAMETERS}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$\\gamma$& [${\\rm km\\,s^{-1}}$] & {$26.5\\pm0.4$} \\\\\n$K_1$ & [${\\rm km\\,s^{-1}}$] & {$47.8\\pm0.4$} \\\\\n$f(M)$& [$M_{\\rm \\odot}$] & {$0.0011\\pm0.0001$} \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$T_{\\rm eff,sdB}$ & [K] & {$25800\\pm290^\\ast$} \\\\ \n$\\log{g,sdB}$ & & {$5.52\\pm0.04^\\ast$} \\\\\n$\\log{n(\\text{He})}$ & & {$-2.07\\pm0.04^\\ast$} \\\\\n$v\\sin{i}$ & [${\\rm km\\,s^{-1}}$] & $65.9\\pm0.1^\\dagger$\\\\\n\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$a$ &[$\\rm R_\\odot$]&$0.71\\pm0.02$\\\\\n$M_{\\rm 1}$&[$M_{\\rm \\odot}$]& $0.39 - 0.50$ \\\\\n$M_{\\rm 2}$ & [$M_{\\rm \\odot}$] & $0.061-0.071$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{PHOTOMETRIC PARAMETERS}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$T_0$ & [BJD$_{TDB}$]&2455165.709211(1)\\\\\n$P$&[d]&$0.09624073885(5)$\\\\\n$\\dot{P}$&dd$^{-1}$&$-3.2(8)\\cdot 10^{-12}$\\\\\n$i$&[$^\\circ$]&$85.6\\pm0.3$\\\\\n$R_{\\rm 1}$ & [$R_{\\rm \\odot}$] & $0.194\\pm0.008$ \\\\\n$R_{\\rm 2}$ & [$R_{\\rm \\odot}$] & $0.092\\pm 0.005$ \\\\\n$\\log g$ & &$5.52\\pm0.03$\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\multicolumn{3}{l}{SED FITTING}\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n$\\varpi_{\\text{Gaia}}$ & [mas] & $0.6899\\pm0.0632^\\dagger$ \\\\\n$E(B-V)$ & [mag] & $0.040\\pm0.010^\\dagger$ \\\\\n$\\theta$ & [$10^{-12}$\\,rad] & $6.22\\pm0.15^\\ast$ \\\\\n$R_{\\text{Gaia}}$ & [$R_{\\odot}$] & 0.200$^{+0.021\\ast}_{-0.018}$ \\\\\n$M_{\\text{Gaia}}$ & [$M_{\\odot}$] & 0.48$^{+0.12\\ast}_{-0.09}$ \\\\\n$\\log{(L_{\\text{Gaia}}\/L_{\\odot})}$ & & 16$^{+3.6\\ast}_{-2.8}$ \\\\\n\\noalign{\\smallskip}\n\\hline\n\\multicolumn{3}{l}{Gaia: Based on measured \\textit{Gaia} parallax, but applying a zero}\\\\ \\multicolumn{3}{l}{point offset of $-0.029$\\,mas (see Sect. \\ref{Stellar radius, mass and luminosity} for details).}\\\\\n\\multicolumn{3}{l}{$\\dagger$: 1$\\sigma$ statistical errors only.}\\\\\n\\multicolumn{3}{l}{$\\ast$: Listed uncertainties result from statistical and systematic}\\\\ \\multicolumn{3}{l}{errors (see Sects. \\ref{Effective temperature, surface gravity, and helium content} and \\ref{SED fitting} for details).}\\\\\n\\end{tabular}\n\\end{table}\n\n\n\\section{Discussion}\n\n\\subsection{Tidal synchronisation of sdB+dM binaries} \\label{syncro}\n\n\nIn close binaries, the rotation of the components is often assumed to be synchronised to their orbital motion. In this case the projected rotational velocity can be used to put tighter constraints on the companion mass. \\citet{geier10b} found that assuming tidal synchronisation of the subdwarf primaries in sdB binaries with orbital periods of less than $\\simeq1.2\\,{\\rm d}$ leads to consistent results in most cases. In particular, all the HW\\,Vir type systems analysed in the \\citet{geier10b} study turned out to be synchronised. \n\nIn contrast to this, the projected rotational velocity of J08205+0008 is much smaller than is required for tidal synchronisation. We can calculate the expected rotational velocity ($v_{\\rm rot}$) using the inclination ($i$), rotational period ($P_{\\rm rot}$) and the radius of the primary ($R_1$) from the light curve analysis if we assume the system is synchronised:\n\t\\begin{equation}\n\tP_{\\rm rot, 1}=\\frac{2\\pi R_1}{v_{\\rm rot}}\\equiv P_{\\rm orb} \\rightarrow v_{\\rm synchro} \\sin i=\\frac{2\\pi R_1\\sin i}{P_{\\rm orb}}.\n\t\\end{equation}\n\nDue to the short period of this binary, the sdB should spin with $v_{\\rm syncro}\\simeq102\\,{\\rm km\\,s^{-1}}$ similar to the other known systems \\citep[see][and references therein]{geier10b}. \n\n\nOther observational results in recent years also indicate that tidal synchronisation of the sdB primary in close sdB+dM binaries is not always established in contrast to the assumption made by \\citet{geier10b}. New theoretical models for tidal synchronisation \\citep{preece:18,preece:19} even predict that none of the hot subdwarfs in close binaries should rotate synchronously with the orbital period. \n\nFrom the observational point of view, the situation appears to be rather complicated.\n\\citet{geier10b} found the projected rotational velocities of the two short-period ($P=0.1-0.12\\,{\\rm d}$) HW\\,Vir systems HS\\,0705+6700 and the prototype HW\\,Vir to be consistent with synchronisation. \\citet{charpinet08} used the splitting of the pulsation modes to derive the rotation period of the pulsating sdB in the HW\\,Vir-type binary PG\\,1336$-$018 and found it to be consistent with synchronised rotation. This was later confirmed by the measurement of the rotational broadening \\citep{geier10b}. \n\nHowever, the other two sdBs with brown dwarf companions J162256+473051 and V2008-1753 \\citep{schaffenroth14,schaffenroth15} have even shorter periods of only 0.07 d and both show sub-synchronous rotation with 0.6 and 0.75 of the orbital period, respectively, just like J0820+0008. AA Dor on the other hand, which has a companion very close to the hydrogen burning limit and a longer period of 0.25 d, seems to be synchronised \\citep[and references therein]{vuckovic16}, but it has already evolved beyond the EHB and is therefore older and has had more time to synchronise.\n\n\\citet{pablo11} and \\citet{pablo12} studied three pulsating sdBs in reflection effect sdB+dM binaries with longer periods and again used the splitting of the pulsation modes to derive their rotation periods ($P\\simeq0.39-0.44\\,{\\rm d}$). All three sdBs rotate much slower than synchronised. But also in this period range the situation is not clear, since a full asteroseismic analysis of the sdB+dM binary Feige\\,48 ($P\\simeq0.38\\,{\\rm d}$) is consistent with synchronised rotation. \n\nSince synchronisation timescales of any kind \\citep{geier10b} scale dominantely with the orbital period of the close binary, these results seem puzzling. Especially since the other relevant parameters such as mass and structure of the primary or companion mass are all very similar in sdB+dM binaries. They all consist of core-helium burning stars with masses of $\\sim0.5\\,M_{\\odot}$ and low-mass companions with masses of $\\sim0.1\\,M_{\\odot}$. And yet 5 of the analysed systems appear to be synchronised, while 6 rotate slower than synchronised without any significant dependence on companion mass or orbital period. \nThis fraction, which is of course biased by complicated selection effects, might be an observational indication that the synchronisation timescales of such binaries are of the same order as the evolutionary timescales\n\nIt has to be pointed out that although evolutionary tracks of EHB stars exist, the accuracy of the derived observational parameters (usually $T_{\\rm eff}$ and $\\log{g}$) is not high enough to determine their evolutionary age on the EHB by comparison with those tracks as accurate as it can be done for other types of stars (see Fig.~\\ref{Kiel diagram}). As shown in Fig. \\ref{Kiel diagram}, the position of the EHB is also dependent on the core and envelope mass and so it is not possible to find a unique track to a certain position in the $T_{\\rm eff}-\\log{g}$ diagram and in most sdB systems the mass of the sdB is not constrained accurately enough.\n\n\\citet{2005A&A...430..223L} showed that sdB stars move at linear speed over the EHB and so the distance from the zero-age extreme horizontal branch (ZAEHB) represents how much time the star already spent on the EHB.\nIf we look at the position of the non-synchronised against the position of the synchronised systems in the $T_{\\rm eff}-\\log{g}$ diagram (Fig. \\ref{Kiel diagram synchro}), it is obvious that all the systems, which are known to be synchronised, appear to be older. There also seems to be a trend that systems with a higher ratio of rotational to orbital velocity are further away from the ZAEHB. This means that the fraction of rotational to orbital period might even allow an age estimate of the sdB. \n\nThe fact that the only post-EHB HW\\,Vir system with a candidate substellar companion in our small sample (AA\\,Dor) appears to be synchronised, while all the other HW\\,Vir stars with very low-mass companions and shorter periods are not, fits quite well in this scenario. \nThis could be a hint to the fact that for sdB+dM systems the synchronisation timescales are comparable to or even smaller than the lifetime on the EHB.\nHot subdwarfs spend $\\sim100\\,{\\rm Myrs}$ on the EHB before they evolve to the post-EHB stage lasting $\\sim10\\,{\\rm Myrs}$. So we would expect typical synchronisation timescales to be of the order of a few tens of millions of years, as we see both synchronised and unsynchronised systems.\n\n\n\n\n\n \\begin{figure}\n \\begin{center}\n \\includegraphics[width=\\linewidth]{J0820_synchro.pdf}\n \\caption{$T_{\\text{eff}}-\\log{(g)}$ diagram for the sdB+dM systems with known rotational periods mentioned in Sect. \\ref{syncro}. The filled symbols represent synchronized systems, the open symbols, systems which are known to be non-synchronised. The square marks the position of J08205+0008. The sizes of the symbols scale with the orbital period, with longer periods having larger symbols. Plotted error bars are the estimated parameter variations due to the reflection effect, as found e.g. in \\citet{schaffenroth13}. The zero-age (ZAEHB) and terminal-age extreme horizontal branch (TAEHB) for a canonical mass sdB as well as evolutionary tracks for a canonical mass sdB with different envelope masses from \\citet{1993ApJ...419..596D} are also shown.}\n \\label{Kiel diagram synchro}\n \\end{center}\n \\end{figure}\n\n \\subsection{A new explanation for the period decrease}\n\n There are different mechanisms of angular momentum loss in close binaries leading to a period decrease: gravitational waves, mass transfer (which can be excluded in a detached binary), or magnetic braking \\citep[see][]{quian:08}.\n Here, we propose that tidal synchronisation can also be an additional mechanism to decrease the orbital period of a binary.\n \n From the rotational broadening of the stellar lines (see Sect. \\ref{syncro}) we derived the rotational velocity of the subdwarf to be about half of what would be expected from the sdB being synchronised to the orbital period of the system. This means that the sdB is currently spun up by tidal forces until synchronisation is reached causing an increase in the rotational velocity.\n As the mass of the companion is much smaller than the mass of the sdB, we assume synchronisation for the companion. \n \n The total angular momentum of the binary system is given by the orbital angular momentum $J_{\\rm orb}$ and the sum of the rotational angular momentum of the primary and secondary star $I_{\\rm spin,1\/2}$, with $\\omega$ being the orbital angular velocity and $\\Omega_i$ the rotational, angular velocity\n \\begin{align}\n\tJ_{\\rm tot}&=J_{\\rm orb}+\\sum_{i=1}^{2}I_{\\rm spin,i} \\\\\n\tJ_{\\rm orb}&=(m_1a_1^2+m_2a_2^2)\\,\\omega = \\frac{m_1m_2}{m_1+m_2}a^2\\omega\\\\%\\frac{2\\pi}{P_{\\rm orb}} \\\\\n\ta^2&=\\left(\\frac{G(m1+m2)}{\\omega^2}\\right)^{2\/3}\\\\%\\left(\\frac{G(m_1+m_2)}{4\\pi^2}\\right)^{2\/3}P_{\\rm orb}^{4\/3} \\\\\n I_{\\rm spin,i}&=k_r^2M_iR_i^2\\Omega_\n \\end{align}\n with $k_r^2$ the radius of gyration of the star. It refers to the distribution of the components of an object around its rotational axis. It is defined as $k_r^2=I\/MR^2$, where $I$ is the moment of inertia of the star. \\citet{geier10b} used a value of 0.04 derived from sdB models, which we adopt.\n\nFor now we neglect angular momentum loss due to gravitational waves and magnetic braking.\nIf we assume that the companion is already synchronised and its rotational velocity stays constant ($\\frac{\\rm d\\Omega_{2}}{\\mathrm{d}t}=0$) and that the masses and radii do not change, as we do not expect any mass transfer after the common envelope phase, we obtain \n\t\\begin{equation}\n\t\\frac{dJ_{\\rm tot}}{dt}=p_1\\frac{\\mathrm{d}\\omega^{-1\/3}}{\\mathrm{d}t}+p_2\\frac{\\rm d\\Omega_1}{\\mathrm{d}t}=-p_1\\frac{\\dot{\\omega}}{3\\omega^{4\/3}}+p_2\\dot{\\Omega}=0\n\t\\end{equation}\n\twith \n\t\\begin{equation}\n\tp_1=\\frac{m_1m_2G^{2\/3}}{(m_1+m_2)^{1\/3}}\n\t\\end{equation}\n\tand\n\t\t\\begin{equation}\n\tp_2=k_r^2m_1R_1^2\n\t\\end{equation}\nThis shows that from an increase in the rotational velocity of the primary, which is expected from tidal synchronisation, we expect an increase of the orbital velocity, which we observe in the case of J08205+0008.\nWe can now calculate the current change of orbital velocity:\n\\begin{equation}\n\t\\dot{\\Omega}_1=\\frac{p_1}{3p_2}\\frac{\\dot{\\omega}}{\\omega^{4\/3}}=\\frac{m_2G^{2\/3}}{3k_r^2R_1^2(m_1+m_2)^{1\/3}}\\frac{\\dot{\\omega}}{\\omega^{4\/3}}\n\t\\end{equation}\n\nFrom this equation we can clearly see that rotational velocity change depends on the masses of both stars, the radius of the primary, the orbital velocity change and the current orbital velocity. An increasing rotational velocity causes an increasing orbital velocity and hence a period decrease.\n\n\\subsection{Synchronisation timescale}\n\nIf we assume that the observed period decrease is only due to the rotational velocity change, we can calculate the rate of the rotational velocity change and the timescale until synchronisation is reached.\n According to \\citet{preece:18}, the change of rotational angular velocity is given by\n\\begin{equation}\n \\frac{\\mathrm{d}\\Omega}{\\mathrm{d}t}=\\frac{\\omega}{\\tau_{\\rm tide}}\\left(1-\\frac{\\Omega}{\\omega}\\right)\\frac{M_2}{M_1+M_2}\\frac{a^2}{R^2k_r^2}\\propto \\left(1-\\frac{\\Omega}{\\omega}\\right)\\label{eq}\n\\end{equation}\nwhere $\\tau_{\\rm tide}$ is the tidal time-scale depending on the density, radius and mass of the star and the viscous time-scale of the convective region. The current position of J08205+0008 on the $T_{\\rm eff}-\\log g$ diagram and the mass we derived from our analysis suggest that the sdB is currently in the evolutionary phase of helium-burning. The lifetime of this phase is approximately 100 Myrs. So we do not expect the structure of the star to change significantly in the next few Myr. Because the moment of inertia of an sdB star is small compared to that of the binary orbit, the change in separation and angular velocity can be neglected.\n\nTherefore, we can calculate the timescale until synchronisation is reached using the equation given in \\citet{zahn:89}:\n\\begin{equation}\n\\frac{1}{T_{\\rm sync}}=-\\frac{1}{\\Omega_1-\\omega} \\frac{\\mathrm{d}\\Omega_1}{\\mathrm{d}t}\n\\end{equation}\nUsing our equation (\\ref{eq}) and calculating and substituting the angular velocities by the periods we derive an expression for the synchronisation time scale:\n\\begin{equation}\n T_{\\rm sync}=\\left(1-\\frac{2\\pi R_1\\sin i}{P_{\\rm orb}v\\sin i}\\right)\\frac{P_{\\rm orb}^{2\/3}v\\sin i}{\\dot{P}_{\\rm orb}\\sin i}\\frac{3(2\\pi)^{1\/3}k_r^2R_1(m_1+m_2)^{1\/3}}{m_2G^{2\/3}}\n\\end{equation}\nUsing the orbital period, the masses, radii and inclination from our analysis, we calculate a synchronisation time $T_{\\rm sync}$ of $2.1\\pm0.1$ Myrs, well within the lifetime of a helium burning object on the extreme horizontal branch. The orbital period will change by about 200 s (3.5\\%) in this 2 Myrs, which means a change in the separation of only 0.01 $\\rm R_\\odot$, which shows that our assumption of a negligible change in separation is valid. If we assume that the rotation after the common envelope phase was close to zero, the total timescale until the system reaches synchronisation is about 4 Myrs. This assumption is plausible as most red giant progenitors rotate slowly and the common envelope phase is very short-lived and so no change of the rotation is expected.\n\nThis means that this effect could significantly add to the observed period decrease. The fact that the synchronised systems appear to be older than the non-synchronised ones confirms that the synchronisation timescale is of the expected order of magnitude and it is possible that we might indeed measure the synchronisation timescale.\n\nAs mentioned before \\citet{preece:18} predict that the synchronisation timescales are much longer than the lifetime on the EHB and that none of the HW Vir systems should be synchronised.\n\\citet{preece:19} investigated also the special case of NY Vir, which was determined to be synchronised from spectroscopy and asteroseismolgy, and came to the conclusion that they cannot explain, why it is synchronised. They proposed that maybe the outer layers of the sdB were synchronised during the common envelope phase. However, observations show that synchronised sdB+dM systems are not rare, but that synchronisation occurs most likely during the phase of helium-burning, \nwhich shows that synchronisation theory is not yet able to predict accurate synchronization time scales\n\n\n\\subsection{Orbital period variations in HW Vir systems}\n\nAs mentioned before, there are several mechanisms that can explain period changes in HW Vir systems. \nThe period change due to gravitational waves is usually very small in HW Vir systems and would only be observable after observations for many decades \\citep[e.g.][]{kilkenny14}. Using the equation given in \\citet{kupfer20} with the system parameters derived in this paper, we predict an orbital period decay due to gravitational waves of $\\dot{P}=4.5\\cdot10^{-14}\\,\\rm ss^{-1}$.\nThe observed change in orbital period is hence about 100 times higher than expected by an orbital decay due to gravitational waves\n \n\nHW Vir and NY Vir have also been observed to show a period decrease of the same order of magnitude \\citep{quian:08,kilkenny14} but have been found to rotate (nearly) synchronously.\nBoth also show additionally to the period decrease a long-period sinusoidal signal \\citep{Lee:09,Lee:14}. These additional variations in the O--C diagram have been interpreted as caused by circumbinary planets in both cases, however the solutions were not confirmed with observations of longer baselines. Observations of more than one orbital period of the planet would be necessary to confirm it. The period decrease was explained to be caused by angular momentum loss due to magnetic stellar wind braking.\n\nFollowing the approach of \\citet{qian:07} we calculated the relation between the mass-loss rate and the Alfv\\'{e}n radius that would be required to account for the period decrease in J08205+0008 due to magnetic braking. This is shown in Fig. \\ref{mag_braking}. Using the tidally enhanced mass-loss rate of \\citet{tout:88} we derive that an Alfv\\'{e}n radius of $75\\,\\rm R_\\odot$ would be required to cause the period decrease we measure, much larger than the Alfv\\'{e}n radius of the Sun. This shows that, as expected, the effect of magnetic braking in a late M dwarf or massive brown dwarf is very small at best and cannot explain the period decrease we derive.\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{mag_braking.pdf}\n \\caption{Correlation between the Alfv\\'{e}n radius and the mass loss rate\nfor the companion of J08205+0008. The red dashed line marks the Alfv\\'{e}n radius for the Sun, the blue dotted line indicates the tidally enhanced mass-loss rate determined using the parameters of the sdB using the formula of \\citet{tout:88}.}\n \\label{mag_braking}\n\\end{figure}\n\n\\citet{bours:16} made a study of close white dwarf binaries and observed that the amplitude of eclipse arrival time variations in K dwarf and early M dwarf companions is much larger than in late M dwarf, brown dwarf or white dwarf companions, which do not show significant orbital period variations. They concluded that these findings are in agreement with the so-called Applegate mechanism, which proposes that variability in the binary orbits can be driven by magnetic cycles in the secondary stars. In all published HW Vir systems with a longer observational baseline of several years quite large period variations on the order of minutes have been detected \\citep[see][for an overview]{zorotovic:13, pulley:18}, with the exception of AA Dor \\citep{kilkenny14}, which still shows no sign of period variations after a baseline of about 40 years. Also the orbital period decrease in J08205+0008 is on the order of seconds and has only been found after 10 years of observation and no additional sinusoidal signals have been found as seen in many of the other systems. This confirms that the findings of \\citet{bours:16} apply to close hot subdwarf binaries with cool companions. The fact that the synchronised HW Vir system AA Dor does not show any period variations also confirms our theory that the period variations in HW Vir systems with companions close to the hydrogen-burning limit might be caused by tidal synchronisation. In higher-mass M dwarf companions the larger period variations are likely caused by the Applegate mechanism and the period decrease can be caused dominantly by magnetic braking and additionally tidal synchronisation.\n\nIt seems that orbital period changes in HW Vir systems are still poorly understood and have also not been studied observationally to the full extent. More observations over long time spans of synchronised and non-synchronised short-period sdB binaries with companions of different masses will be necessary to understand synchronisation and orbital period changes of hot subdwarf binaries. Most likely it cannot be explained with just one effect and is likely an interplay of different effects.\n\n\\subsection{Inflation of brown dwarfs and low-mass M dwarfs in eclipsing WD or sdB binaries}\nClose brown dwarf companions that eclipse main sequence stars are rare, with only 23 known to date \\citep{carmichael20}. Consequently, brown dwarf companions to the evolved form of these systems are much rarer with only three (including J08205+0008) known to eclipse hot subdwarfs, and three known to eclipse white dwarfs. These evolved systems are old ($>$ 1 Gyr), and the brown dwarfs are massive, and hence not expected to be inflated \\citep{thorngren18}.\n\nSurprisingly, of the three hot subdwarfs with brown dwarf companions, J08205+0008 is the one that receives the least irradiation - almost half that received by V2008-1753 and SDSSJ162256.66+473051.1, both of which have hotter primaries (32000~K, 29000~K) and shorter periods ($\\sim$1.6~hr) than J08205+0008. This suggests that more irradiation, and more irradiation at shorter wavelengths does not equate to a higher level of inflation of a brown dwarf. Indeed this finding is consistent with that for brown dwarfs irradiated by white dwarfs, where the most irradiated object with a measured radius is SDSS J1205-0242B, in a 71.2 min orbit around a 23681~K white dwarf and yet the brown dwarf is not inflated \\citep{parsons17}. The brown dwarf in this system only receives a hundredth of the irradiation that J08205+0008 does. However, WD1032+011, an old white dwarf ($T_{\\rm eff} \\sim$ 10000~K) with a high mass brown dwarf companion (0.0665 M$_{\\odot}$) does appear to be inflated \\citep{casewell:2020}. As can be seen from Figure \\ref{MR}, the majority of the low mass brown dwarfs (M$<$35 M$_{\\rm Jup}$) are inflated, irrelevant of how much irradiation they receive. For the few old (5-10 Gyr), higher mass inflated brown dwarfs, the mechanism leading to the observed inflation is not yet understood.\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=1.05\\linewidth]{sds_mr_jul31.pdf}\n \\caption{All known eclipsing binary white dwarfs with detached brown dwarf (triangles: \\citealt{parsons17, littlefair14}) and late M dwarf companions (triangles) from \\citet{parsons18}, hot subdwarfs with eclipsing brown dwarf companions (circles: \\citealt{schaffenroth14a, schaffenroth15}) and all known eclipsing brown dwarf companions to main sequence stars (+: \\citealt{carmichael20}). J08205+0008 is plotted as the filled square. The colour is proportional to the effective temperature of the primary in each system and the coloured circle size is proportional to the amount of total incident radiation the secondary receives. Also shown are the Sonora Bobcat brown dwarf evolutionary models of \\citet{bobcat} for solar and sub-solar metallicity and the NextGen models \\citep{baraffe97}.}\n \\label{MR}\n\\end{figure*}\n\n\n\\subsection{Previous and future evolution of the system}\nAs stated before, stars with a cool, low-mass companion sitting on the EHB are thought to have formed by a common-envelope phase from a progenitor of up to two solar mass on the RGB. Due to the large mass ratio only unstable mass transfer is possible. If the mass transfer happened at the tip of the RGB, a core-helium burning object with about $0.5\\,\\rm M_\\odot$ will be formed. If the mass transfer happened earlier then the core of the progenitor has not enough mass to start He-core burning and the pre-He WD will move to the WD cooling track crossing the EHB. Our analysis of J08205+0008 showed that a low-mass solution ($0.25\\,M_\\odot$, as discussed previously) can be excluded and that the primary star is indeed currently a core He-burning object. \n\n\\citet{kupfer15} calculated the evolution of J08205+0008 and considering only angular momentum loss due to gravitational waves and found that the companion will fill its Roche lobe in about 2.2 Gyrs and mass transfer is expected to start forming a cataclysmic variable.\nWe detected a significantly higher orbital period decrease in this system than expected from gravitational waves. Up to now, we could not detect any change in the rate of this period decrease. If we assume that the orbital period change is due to rotational period change\nuntil synchronisation is reached and afterwards the period decrease will be solely due to gravitational waves, \nwe can calculate when the companion will fill its Roche lobe and accretion to the primary will start. To calculate the Roche radius the equation derived in \\citet{eggleton83} was used:\n\\begin{equation}\n R_L=\\frac{0.49q^{2\/3}}{0.6q^{2\/3}+\\ln(1+q^{1\/3})}a\n\\end{equation}\nUsing the values derived in our analysis we calculate that the Roche lobe of the companion will be filled at a system separation of 0.410 $\\rm R_\\odot$, 56\\% of the current separation, which is reached at a period of 3525 s.\nFrom this we calculate a time scale of 1.8 Gyrs until the Roche lobe will be filled.\n\nSystems with a mass ratio $q=M_2\/M_1<2\/3$, with $M_1$ being the mass of the accretor, are assumed to be able to undergo stable mass transfer. Our system has a mass ratio of $0.147\\ll2\/3$.\nThe subdwarf will already have evolved to a white dwarf and a cataclysmic variable will be formed. It is expected that the period of an accreting binary with a hydrogen-rich donor star will decrease until a minimum period of $\\simeq70$ min is reached at a companion mass around $0.06\\,M_\\odot$ and the period will increase again afterwards \\citep{nelson18}. Such systems are called period bouncers. Our system comes into contact already close to the minimum period and should hence increase the period when the mass transfer starts.\n\nThe future of the system depends completely on the period evolution. A longer baseline of observations of this system is necessary to confirm that the period decrease is indeed stable and caused by the tidal synchronisation. \n\n\n\\section{Conclusion and summary}\n\nThe analysis of J08205+0008 with higher quality data from ESO-VLT\/XSHOOTER, ESO-VLT\/UVES and ESO-NTT\/ULTRACAM allowed us to constrain the masses of the sdB and the companion much better by combining the analysis of the radial velocity curve and the light curve. We determine an sdB mass of $0.39-0.50\\,\\rm M_\\odot$ consistent with the canonical mass and a companion mass of $0.061-0.071\\,\\rm M_\\odot$ close to the hydrogen burning limit. Therefore, we confirm that the companion is likely be a massive brown dwarf. \n\nThe atmospheric parameters and abundances show that J08205+0008 is a typical sdB and comparison with stellar evolution tracks suggest that the mass has to be less than $0.50\\,M_\\odot$ consistent with our solution and also the mass derived by a spectrophotometric method using Gaia parallaxes and the SED derived in the secondary eclipse, where the companion is not visible. \n\nIf the sdB evolved from a $1\\,\\rm M_\\odot$ star, the age of the system is expected to be around 10\\, Gyrs. In this case the radius of the brown dwarf companion is about 20\\% inflated compared to theoretical calculations. Such an inflation is observed in several sdB\/WD+dM\/BD systems but not understood yet. However, the inflation seems not to be caused by the strong irradiation.\nThe sdB binary belongs to the thin disk, as do about half of the sdB at this distance from the Galacic plane. This means that they also could be young, if they have evolved from a more massive progenitor. Then we get a consistent solution without requiring inflation of the companion. However, a brown dwarf companion might not be able to remove the envelope of a more massive progenitor.\n\nWe detected a significant period decrease in J0820+0008. This can be explained by the spin-up of the sdB due to tidal sychronisation. We calculated the synchronisation timescale to 4 Myrs well within the lifetime on the EHB. The investigation of the parameters of all known Vir systems with rotational periods (see Sect. \\ref{syncro}) shows that the synchronised systems tend to be older, showing that the synchronisation timescale seems to be comparable but smaller than the lifetime on the EHB in contrast to current synchronisation theories.\n\nBy investigating the known orbital period variations in HW Vir systems, we can confirm the findings by \\citet{bours:16} that period variations in systems with higher mass M dwarf companions seem to be larger. Hence, we conclude that the large period variations in those systems are likely caused by the Applegate mechanism and the observed period decreases dominantly by magnetic braking. In lower-mass companions close to the hydrogen-burning limit, on the other hand, tidal synchronisation spinning up the sdB could be responsible for the period decrease, allowing us to derive a synchronisation timescale. \n\n\nThe results of our analysis are limited by the precision of the available trigonometric parallax. As the Gaia mission proceeds, the precision and accuracy of the trigonometric parallax will improve, which will narrow down the uncertainties of the stellar parameters.\nA very important goal is to detect spectral signatures from the companion and to measure the radial velocity curve of the companion. We failed to do so, because the infrared spectra at hand are of insufficient quality. The future IR instrumentation on larger telescopes, such as the ESO-ELT, will be needed. A high precision measurement of the radial velocity curves of both components will then allow us to derive an additional constraint on mass and radius from the difference of the stars' gravitational redshifts \\citep{vos13}. \nSuch measurements will give an independent determination of the nature of the companion and will help to test evolutionary models for low mass star near the hydrogen burning limit via the mass-radius relation.\n\nThe combination of many different methods allowed us to constrain the masses of both components much better without having to assume a canonical mass for the sdB. This is only the fourth HW Vir system for which this is possible.\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nD.S. is supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HE 1356\/70-1 and IR190\/1-1.\nV.S. is supported by the Deutsche Forschungsgemeinschaft, DFG through grant GE 2506\/9-1. S.L.C. is supported by an STFC Ernest Rutherford Fellowship ST\/R003726\/1.\nDK thanks the SAAO for generous allocations of telescope time and the National Research Foundation of South Africa and the University of the Western Cape for financial support. VSD, SPL and ULTRACAM are supported by the STFC.\n We thank J. E. Davis for the development of the \\texttt{slxfig} module, which has been used to prepare figures in this work. \\texttt{matplotlib} \\citep{2007CSE.....9...90H} and \\texttt{NumPy} \\citep{2011CSE....13b..22V} were used in order to prepare figures in this work. This work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia}\nData Processing and Analysis Consortium (DPAC, \\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement. Based on observations at the Cerro Paranal Observatory of the European Southern Observatory (ESO) in Chile under the program IDs 087.D-0185(A), and 098.C-0754(A). Based on observations at the La Silla Observatory of the European Southern Observatory (ESO) in Chile under the program IDs 082.D-0649(A), 084.D-0348(A), and 098.D-679. \nThis paper uses observations made at the South African Astronomical Observatory.\nWe made extensive use of NASAs Astrophysics Data System Abstract Service (ADS) and the SIMBAD and VizieR database, operated at CDS, Strasbourg, France.\n\n\\section*{Data availability statement}\nMost data are incorporated into the article and its online supplementary material. All other data are available on request.\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe Apache Point Observatory Galactic Evolution Experiment 2 (APOGEE-2; \\inprep{S.~Majewski et al.~in prep}) is one of the programs in the Sloan Digital Sky Survey IV \\citep[SDSS-IV;][]{blanton_2017} and the successor survey to APOGEE \\citep[referred to as APOGEE-1, hereafter, for clarity;][]{majewski_2017} in SDSS-III \\citep{eisenstein_2011}. \nAPOGEE-2 operates in both the Northern and Southern hemispheres, with the original APOGEE spectrograph on the Sloan Foundation Telescope at Apache Point Observatory \\citep[APO;][]{gunn_2006} and an almost clone spectrograph on the Ir\\'en\\'ee DuPont telescope at Las Campanas Observatory \\citep[LCO;][]{Bowen_1973}; both instruments are described in \\citet{wilson_2019}.\nWhen referring to the SDSS-III program, we will use ``APOGEE-1'' and, for the combination of APOGEE-1 and APOGEE-2, we will use ``APOGEE'' to encompass the joint dataset.\\footnote{Because SDSS data release are cumulative, the user will find the distinction between APOGEE-1 and APOGEE-2 to be academic. Yet, in terms of targeting, targeting-flags, and some elements of operations, there are many notable distinctions.}\nWhen necessary to specify a specific APOGEE-2 survey component, ``APOGEE-2N'' (for APOGEE-2 North) refers to the survey, data, or instrument associated with APO, and ``APOGEE-2S'' (for APOGEE-2 South) to those at LCO. \nIf we need to refer to the spectrograph specifically, APOGEE-N is the spectrograph at APO and APOGEE-S is the spectrograph at LCO.\n\n\n\\defcitealias{zasowski_2017}{Z17}\n\\defcitealias{zasowski_2013}{Z13}\n\nThe scientific goals of the APOGEE-1 survey \\revise{in SDSS-III \\citep{eisenstein_2011}}, and how those scientific goals were mapped into hardware, software, and data processing requirements, are given in \\citet{majewski_2017}. \nThe overall targeting strategy for APOGEE-1, including a description of APOGEE-1's ancillary programs, was given in \\citet[][hereafter Z13]{zasowski_2013}. \nThe cornerstone targeting strategy for APOGEE-1 was the use of a simple set of color and magnitude criteria in de-reddened color magnitude diagrams (CMDs) that permits precise modelling of the survey selection function; this targeting strategy forms the ``main red star sample'' that aims to target late-type giants based on their intrinsic colors in the near-infrared (NIR) and mid-infrared (MIR). \nThere were, however, deviations from this strategy when deemed necessary to best achieve specific scientific goals---for example, the use of Washington$+DDO51$ photometry to pre-select likely red giant stars in the Milky Way halo among the dominant disk foreground of dwarf type stars \\citep[the technique is described in][]{majewski_2000}. \nWhile some of these deviations require only a modification to the selection function, other programs demanded the selection of individual stars for explicit inclusion in the survey, such as confirmed member-stars in open clusters \\citep[a detailed description is given in][]{frinchaboy_2013}, or the targets required for ancillary programs \\citep[such as the M\\,31 star clusters presented in][]{sakari_2016}, reflecting the specificity of their focused scientific goals.\nThus, \\citetalias{zasowski_2013} not only included the methodology for target selection, but also how to identify the methodology for a given source using a set of targeting flags. \n\\revise{APOGEE-1 observations occurred from September 2011 until July 2014, with Data Releases in 2013 \\citep[DR10;][]{dr10_sdss,dr10_apogee}, 2015 \\citep[DR12][]{dr12_sdss,holtzman_2015}, and the final final data release from SDSS-III in 2016 \\citep[DR13][]{dr13_sdss,holtzman_2015}.}\n\n\\revise{In SDSS-IV \\citep{blanton_2017}}, APOGEE-2 continues the large-scale goal of chemo-dynamical mapping of the key structural components of the Milky Way and its environment in both the Northern Hemisphere and the Southern Hemisphere, where it expands to new scientific areas due to its access to the full sky \\revise{and a six-year operational timescale for APOGEE-2N} (\\inprep{S.~Majewski et al.~in prep}).\nAPOGEE-2 largely adopted the same underlying targeting strategies described above for APOGEE, including its foundation ``main red star sample.''\n\\revise{However,} APOGEE-2 also elevated several ancillary projects from APOGEE-1 to key ``core'' programs, such that the science goals of these programs became part of APOGEE-2's primary \\revise{scientific goals}, while, at the same time, granted time to new ancillary programs that further expand the impact and legacy of APOGEE \\revise{as a scientific project}.\n\\revise{APOGEE-2 also included new targeting classes, spanning from RR Lyrae in the inner Galaxy to red giants in the dwarf Spheroidal companions to the Milky Way. \nWith its broader goals, longer timeline, and dual-instruments, APOGEE-2 presented a significant change from the overall targeting strategies of APOGEE-1.}\n\nThe initial targeting plans for APOGEE-2 were presented in \\citet[][hereafter Z17]{zasowski_2017} and focused on the overall strategy for the \\revise{joint APOGEE-2N and APOGEE-2S observing programs.}\nThe publication of \\citetalias{zasowski_2017} was timed to accompany the first \\revise{APOGEE-2} data release from SDSS-IV \\citep[Data Release 14;][DR14]{dr14} \\revise{that contained new observations with the APOGEE-N spectrograph from July 2014 to July 2016 \\citep{holtzman_2018}.}\n\\revise{A subsequent APOGEE-2 Data Release occurred in 2019 with Data Release 16 \\citep[DR16;][]{dr16,Jonsson_2020} that included observations from 2016 to 2018, including $\\sim$1 year of observations from APOGEE-2S.\nThe final Data Release will occur in December 2021 (DR17) and will include the complete observational program from APOGEE-2 alongside new processing of APOGEE-1 observations.} \n\nWhen \\citetalias{zasowski_2017} was written, modifications to this \\revise{base plan} were anticipated, \\revise{at a minimum} due to the then-incomplete commissioning of the APOGEE-2S instrument \\citep[see][]{wilson_2019} and the on-going APOGEE-2N ancillary program allocation process \\revise{(there were application cycles in 2015 and 2017, with implementation of programs often taking months, and observations taken over several years)}.\n\\revise{Since \\citetalias{zasowski_2017}, additional changes in the targeting plan have also occurred through the Contributed Programs in APOGEE-2S and the Bright Time Extension (BTX) for APOGEE-2N, the latter served as an effective 1.5 year extension due to unanticipated gains in operational efficiency at APO.}\n\nThe objective of this paper is to present the summation of modifications to the field plan and targeting strategy for APOGEE-2N that was presented in \\citepalias[][]{zasowski_2017} as it applies to the \\revise{now complete} APOGEE-2N survey from APO (with observations from 2014 to 2020). \nA companion paper, F.~Santana et al. (submitted), provides a complementary presentation of the changes made in the APOGEE-2S observing program at LCO \\revise{that began in February 2017 and completed in January 2021 as were required by on-sky performance and time allocations.} \n\\revise{These papers are intended to be a formal presentation of survey strategy and motivations.}\n\\revise{The material is presented separately for APOGEE-2N and APOGEE-2S to make apparent the different contexts through which the survey plans from \\citetalias{zasowski_2017} were modified in terms of nights available, target visibility, instrumental throughput, and operational procedures that required distinct planning and implementation strategies.}\n\\revise{The web documentation accompanying Data Release 17 contains a streamlined and, perhaps more practical, presentation of the total observational programs spanning APOGEE-1 and APOGEE-2.\\footnote{The DR16 Documentation for Targeting is here: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/targets\/}, for Special Programs here: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/targets\/special-programs\/}, and a discussion of Selection Biases and the Survey Selection Function is here: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/targets\/selection-biases\/}. The URLs can be updated for ``dr17'' to point to the DR17 versions when DR17 becomes public in December 2021.} }\n\n\n\\revise{This paper for APOGEE-2N and its companion for APOGEE-2S (F.~Santana et al. submitted)\\ are} intended to serve in supplement to the existing APOGEE-2 targeting paper, \\citetalias{zasowski_2017}, \\revise{and are intended to accompany Data Release 17 (DR17) planned for December 2021 that will release the cumulative APOGEE-1 and APOGEE-2 observations (\\inprep{J.~Holtzman et al.~in prep.}).}\nThus, readers new to APOGEE-2 are strongly advised to review \\citetalias{zasowski_2017} as well as \\revise{the APOGEE-1 Targeting Paper \\citetalias{zasowski_2013} that provides the base-line strategies}.\nOnly those programs \\revise{in APOGEE-2N} that were modified relative to the descriptions in \\citetalias{zasowski_2017} or programs that are completely new are discussed in this paper. \nHowever, certain keystone-information regarding targeting is repeated from \\citetalias{zasowski_2017} to provide enough context that elements of this paper can stand alone \\revise{(predominantly in \\autoref{sec:prelims})}.\nBecause of the broad scientific scope of the programs in APOGEE-2N, descriptions of prior work from APOGEE-1 along with other scientific background is required to explain how that prior work has influenced the targeting and implementation of programs within APOGEE-2N. \n\nThis paper is organized as follows. \nA summary of general information regarding APOGEE-2N targeting and the general motivation for the modifications of its strategy are given in \\autoref{sec:prelims}.\nThe paper is organized by the \\revise{significance of the} modification from \\revise{the original survey strategies and plans for APOGEE-2N described in} \\citetalias{zasowski_2017} from the most significant to the least significant.\n\\autoref{sec:strategy} details \\revise{changes to the targeting strategies or selection algorithms described in \\citetalias{zasowski_2017}.}\n\\autoref{sec:new_programs} describes the newly added programs that were not described in \\citetalias{zasowski_2017} \\revise{because they were added after that publication because survey observations beyond the initial six year plan became feasible}.\n\\autoref{sec:expansion} describes the extensions of some programs in \\citetalias{zasowski_2017} during the BTX \\revise{to include additional fields or span longer time baselines}.\nFinally, \\autoref{sec:wrapup} provides a summary that reflects on the overall process of modifying the combined APOGEE-1 and APOGEE-2 survey over its decade of operations \\revise{at APO}, with a focus on highlighting intentional choices in our survey-planning \\revise{specific to its targeting operations} that enabled the project to grow and adapt. \n\\revise{As discussed in F.~Santana et al. (submitted), these strategies were pivotal in the successful completion of the APOGEE-2S scientific program.}\n\n\n\\section{Preliminaries} \\label{sec:prelims}\n\nTo aid the reader, in the subsections that follow, we briefly review observing and targeting concepts from \\citetalias{zasowski_2013} that continue to be used in APOGEE-2 (\\autoref{sec:obsoverview} and \\autoref{sec:targoverview}).\nA number of SDSS- and APOGEE-specific terms and acronyms will be introduced and, as an additional aid, a glossary of these terms is given in \\autoref{sec:glossary} (following that of \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017}, but also including new terms used for this present paper). \nAfter these summaries, we provide a broad overview of the updates to the APOGEE-2N survey that motivated modifications to its targeting (\\autoref{ssec:ancprograms_overview} and \\autoref{ssec:btx_overview}).\nWe close by reviewing the final field plan for the APOGEE-2 Survey (\\autoref{ssec:field_plan}) and summarizing the datasets used in this paper (\\autoref{ssec:datasets}).\n\n\\subsection{Observational Framework} \\label{sec:obsoverview}\n\nAs described in depth by \\citet{wilson_2019}, both APOGEE spectrographs are fed by fibers that are held to a target position using plug-plates; thus, the specifics of the plate and fiber positioning places fundamental constraints on the targeting for the survey.\nPlates designed for the APOGEE-N spectrograph have a 3$^{\\circ}$ diameter field-of-view; each fiber is approximately 3\\arcsec\\ in diameter on the sky, and two adjacent fibers cannot be placed closer than $\\sim$72\\arcsec\\ apart, the ``fiber collision radius''. \n\\revise{In addition, no targets can be placed within 96$''$ of the field center due to the central post that supports the plate \\citep{Owen_1994}.}\nEach unique APOGEE pointing on the sky is referred to as a field, and each unique set of targets selected for that field is called a design; even a difference of a single target will create a unique design that will be indicated by a unique design ID (an integer assigned to each design).\nDesigns are then drilled onto plates, but the exact locations of target holes on given plate is set by the intended hour angle at observation, to minimize the impacts of differential refraction across the field-of-view and during an integration \\citep[see discussion in][]{majewski_2017,wilson_2019}. \nThus, a given field can have multiple designs, and any given design can have multiple associated plates. \nThe final APOGEE-2 field plan is given in \\autoref{fig:fieldmap}, where each circle represents a single field, but the number of targets per field depends on the number of designs and the number of stars common to those designs. \n\n\nEach observation of a given plate is known as a ``visit''; a typical visit consists of about one hour of integration that is broken into eight exposures separated into two sets of spectral dithering sequences (each an ``ABBA'' sequence).\\footnote{Half pixel dithering in the spectral dimension is employed by APOGEE to recover Nyquist sampling of the intrinsic spectrograph resolution \\citep[see][]{majewski_2017,wilson_2019}.}\nDeviations from this procedure occur when a plate has been designed specifically to focus on faint targets, and the changes are in two forms, intended to improve the overall signal-to-noise ($S\/N$) in the visit spectra: \n\\begin{enumerate} \\itemsep -2pt\n \\item beginning in late 2017, when a given plate has more than five faint targets (defined as $H$\\textgreater~13.5), only a single dither sequence is used over the same time frame, with each individual exposure being doubled in length \n (such a visit is referred to as ``DAB''); and \n \\item beginning in late 2019, when a given plate is dominated by faint stars, the plate receives an extra dither pair in a single DAB-style visit, such that its visit would sum to an hour and a half of exposure time (referred to as ``TDAB''). \n\\end{enumerate}\nA given field is planned to have a specific number of visits determined by its target magnitude depth, and these visits are implemented according to specific temporal spacing requirements, referred to as the ``cadence rule''.\n\\revise{Cadence rules vary by the scientific program. \nIn the main red star sample, the cadence rules were designed with a spacing optimized to the anticipated radial velocity variation from close binaries on the red giant branch with the goal of, at a minimum, removing such stars as a potential source of uncertainty in studies of detailed Galactic dynamics and, more optimally, to determine the true systematic velocity of the binary system for use in such studies.\nWe note that programs specifically aiming to characterize and not just detect such variations have more complicated cadence rules.}\n\nGenerally, APOGEE aims for spectra with a minimum $S\/N$ of 100 per pixel to ensure the highest quality stellar parameters and chemical abundances. \nThis $S\/N$ target is a fundamental constraint on the targeting and, along with the intended magnitude limit, determines the number of visits a plate will receive. \nBecause the APOGEE reduction pipeline performance has proven to be similarly reliable from $S\/N$ = 70 to 100 \\citep[for a detailed performance assessment, see][]{Jonsson_2020}, for some programs the targeting-imposed magnitude limits have been altered; when this is the case, it will is noted.\\footnote{For the purposes of this paper, ``reliable'' from $S\/N$ = 70 to 100 refers to the observation that the stellar parameter and chemical abundance uncertainties, generally, over this $S\/N$-range are similar. In contrast, for $S\/N<70$, the uncertainties increase rapidly. \nBy this reasoning, the \\texttt{STARFLAG} bit for \\texttt{SN\\_WARN} is automatically set for all stars with $S\/N<70$ \\citep{holtzman_2015,holtzman_2018,Jonsson_2020}.} \n \n \n \n\\autoref{tab:maglims} summarizes the approximate $S\/N$ attained for a typical star at the intended faint $H$ magnitude limit on some common design configurations (we note that the brightest target allowed is $H\\sim7$ due to instrument detector saturation concerns). \nThe visit number needed to reach $S\/N = 100$ for a given plate magnitude limit is shown in \\autoref{tab:maglims} and provides the guideline for survey planning and scheduling. \nPlates are generally categorized as ``3-visit plates'', ``6-visit plates'', etc., according to the prescriptions shown. \nObviously, the actual on-sky $S\/N$ performance achieved during visits varies due to observing conditions.\n\nHowever, as can be seen, any $H=11$ star that might happen to be in a 24-visit plate, would obtain an estimated $S\/N\\sim490$, greatly exceeding what is required to achieve our scientific goals. \nFor this reason, stars in targeted fields are often grouped by magnitude bins to form ``cohorts''; cohorts are normally only observed for the number of visits required to obtain $S\/N\\sim$100 for the faintest star in the cohort. \nMany fields have more than one cohort per design and this strategy combines each faint cohort with several brighter cohorts, so that the same faint stars are included in multiple designs while bright targets are switched out; this strategy increases the overall number of stars in a field that are included in the survey.\nAs a rule, a design will have no more than three cohorts, generally referred to as the `short,' `medium', and `long' cohorts, where the short cohorts require the fewest visits, and the long cohorts are included on the full complement of field designs to achieve the maximum exposure time.\nThe visits per cohort and relative number of stars in each cohorts are specific to a scientific program \\citepalias{zasowski_2013,zasowski_2017}. \n\n\\input{Tables\/table1_signaltonoise}\n\n\\subsection{General Targeting Overview} \\label{sec:targoverview}\n\n`Targeting' is our term for the implementation of the observational strategies of APOGEE to achieve its scientific goals through the assignment of fibers to targets. \nBoth APOGEE spectrographs have 300 total fibers. \nFor all plates designed in APOGEE-2, 15 fibers are assigned to hot (more ``featureless'') stars for the derivation of telluric absorption corrections, 35 fibers are assigned to ``blank sky'' positions distributed across the plate \\citepalias[see][]{zasowski_2013}, and 250 fibers are available for science programs.\n\nBecause our data are only as good as the calibrations, the selection of suitable calibration fibers, both tellurics and blank sky, occur at high priority.\nTelluric stars are selected to be the bluest stars in a given field that can also achieve $S\/N>100$ in a single visit; because of their importance, telluric stars are selected and assigned first.\nCandidate ``blank sky'' positions are selected as regions with no 2MASS point source \\citep{Skrutskie_06_2mass} within 6$\\arcsec$ of the position.\nThough the selection of a large number of candidate sky regions occurs early in the design process, their fibers are not assigned until the science targets have been selected and \\revise{then the 35 blank sky fibers are selected from the candidate positions and distributed uniformly across the plate. \nThe design will not be drilled unless the 15 tellurics and 35 blank sky fibers are successfully allocated.}\n\nAllocation of science fibers occurs in a two phase process. \nIn the first phase, `special targets' are assigned following star-by-star priorities.\nSpecial targets are generally stars that are unlikely to be picked from our standard algorithms; specific cases for the main survey include \n (i) extremely rare or sparse targets (for example, a specific type of photometric variable star), or \n (ii) known member stars of a substructure (for example, a star cluster or dwarf galaxy).\nAll targets for scientifically-focused Survey programs (for example, programs specifically targeting stars with {\\it Kepler}\\ observations) and targets from Ancillary Science Programs (\\autoref{ssec:ancprograms_overview}) are considered special targets. \nTypically, special target lists are prepared and prioritized field-by-field by the relevant science working group or Ancillary Science Program principal investigator (PI) and submitted to the targeting team for plate design. \nIn all cases, special targets are identified by special targeting flags (\\autoref{tab:targeting_bits}).\nIf multiple sets of special targets exist for a given field, the special programs themselves are given a priority schema such that the special program with the smallest number of special targets is given the highest priority; the one exception to this rule is a field assigned to a specific program, like an open cluster, in which case the targets for that program come at highest priority. \n\nOnce special targets are assigned, the remaining fibers are allocated following specific selection algorithms.\nThe core of APOGEE is the ``main red star sample'' \\citepalias{zasowski_2013,zasowski_2017}, which is selected by fixed and simple color-magnitude criteria.\nThe color-limit and color-selection criteria adopted are set by the primary Galactic component targeted by the plate (effectively, disk, bulge, halo) and the magnitude range is set in accordance with the number of visits and cohorting scheme for a given field.\nThe main red star sample is well described in \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017}, with modifications for the APOGEE-2S survey given in F.~Santana et al. (submitted). \nThe definitions of the plates designed to target specific Galactic components and their associated color cuts are given in \\autoref{tab:colorcuts}. \n\nFor fields that are devoted to specific science cases, the majority of fibers are assigned to that science case as special targets and the remaining targets are drawn from the appropriate ``main red star sample'' selection function for that part of the sky. \nFor example, fibers not assigned to dwarf spheroidal galaxy members or candidate members would be assigned following the halo selection function and cohort scheme for a 24-visit field \\citepalias[see \\autoref{sec:dsph} in this work and also][]{zasowski_2017}.\n\n\\input{Tables\/table2_colorcuts}\n\n\\subsubsection{Targeting Bits} \\label{sec:targbits}\nAPOGEE-2 uses bit flags to convey the targeting schema. \nThe flags are not a comprehensive way of identifying stars that meet particular scientific criteria. \nRather the flags serve to identify why a particular set of stars was targeted to enable study of (and correction for) the selection function of the survey. \n\\autoref{tab:targeting_bits} provides a summary and description of the APOGEE-2 targeting bits that span four bit flags; three are used currently, \\texttt{APOGEE2\\_TARGET1}, \\texttt{APOGEE2\\_TARGET2}, and \\texttt{APOGEE2\\_TARGET3}, and the fourth, \\texttt{APOGEE2\\_TARGET4}, was added to the DR17 data model but is not currently in use. \nThe specific bits that have been put into use since \\citetalias{zasowski_2017} are shown in \\autoref{tab:targeting_bits} in bold for ease of identification. \nThe newly allocated bits largely indicate those targets selected under a specific schema, which will be described in the sections that follow. \nF.~Santana et al. (submitted)\\ provides a more comprehensive discussion of modifications to the targeting bits relative to that described in \\citetalias{zasowski_2017}. \n\n\\input{.\/Tables\/table3_bitflags}\n\n\\subsection{Ancillary Science Programs} \\label{ssec:ancprograms_overview}\n\nIn APOGEE-1 and in APOGEE-2N, a fiber reserve was intentionally budgeted into the survey plan for Ancillary Science Programs. \nThis aspect of the survey design has been exceptionally beneficial, as Ancillary Science Programs from APOGEE-1 (e.g., the APOKASC and KOI programs, see \\citetalias{zasowski_2013}) eventually became core components of APOGEE-2N \\citepalias{zasowski_2017}.\n\nIn APOGEE-2N, approximately 5\\% of the fiber hours \\revise{for a six years of bright time operations} were reserved for Ancillary Science Programs. \nA fiber hour is defined as one visit for a single fiber, such that a single plate represents 265 fiber hours allocated for stars; \\revise{allocation by fiber hour} allows for more flexibility in the implementation of Ancillary Science Programs. \nThese were awarded by way of a competitive, internal review process that resulted in the selection of 23 programs \\revise{over two application cycles}. \nThe corresponding allocations could be through sparse fibers across many plates, a concentration of targets in dedicated (often new) fields or APOGEE-N observations via the fiber link to the NMSU 1-meter telescope \\citep{holtzman_2010,holtzman_2015}.\n\\citetalias{zasowski_2017} described the general process of selecting and implementing Ancillary Science Programs, but, because the full implementation and even allocation of some of these programs was then still underway, could not include detailed descriptions of the programs, as had been done for APOGEE-1 Ancillary Science Programs in \\citetalias{zasowski_2013}.\n\nA \\revise{description} of each program is given in \\autoref{sec:ancillary2015} for the 2015 programs and \\autoref{sec:ancillary2017} for the 2017 programs. \n\\revise{These descriptions include the scientific motivations for the observations, the observations undertaken, and the specific goals of the program. \nThe scope of the programs vary a great deal. \nAll Ancillary Targets are input into the targeting procedure as ``special targets'' and are then drilled onto plates at high priority. \nOnly in rare cases would the targets be modified (e.g., if the star was too bright and could compromise other observations). \n}\n\n\nFor ease, \\autoref{tab:anc_sum} summarizes the 23 programs with their title and subsection reference, contact scientists, and appropriate targeting flag.\nDedicated fields for ancillary programs have \\texttt{PROGRAMNAME} given as ``ancillary,'' \\revise{such that whole fields could be identified and all} individual targets are flagged as summarized in \\autoref{tab:anc_sum}.\nWe note that the timing of the 2017 call for Ancillary Science Programs relative to the Bright Time Extension (BTX; \\autoref{ssec:btx_overview}) resulted in some programs from the former being absorbed into the latter; if this occurred for a particular program, it is noted both in the main text and in the appendices. \n\n\\input{.\/Tables\/table4_ancprograms.tex} \n\n\\subsection{The Bright Time Extension} \\label{ssec:btx_overview}\nThe \\revise{Bright Time Extension} (BTX) was an expansion of APOGEE-2N programs to fill an excess of bright time anticipated toward the end of the SDSS-IV survey as a result of improved observational efficiencies and better than average weather at APO during the first few years of the survey. \nPlanning for the BTX began in mid-2017 and fiber-hours were awarded through an open call across SDSS-IV that anticipated approximately $\\sim$1200 hours would be available (corresponding to the equivalent of $\\sim$1.5 years, \\revise{or 20\\%}, of APOGEE-2 bright time observations). \nThis time was divided between a mix of programs initiated by the APOGEE-2 team and jointly with the ``After SDSS-IV (AS4)'' scientific collaboration. \nThe latter has since been formally established as SDSS-V\\footnote{\\url{https:\/\/www.sdss5.org\/}}, but the implementation of these programs used the acronym ``as4'' and throughout this paper we will use both AS4 and SDSS-V when referring to these programs. \n\n\nThe general APOGEE-2N BTX strategy had three primary components:\n\\begin{enumerate}\n \\item expanding ``core'' programs and modifying their target selection to better meet the strategic science objectives of the APOGEE-2 survey (\\autoref{sec:strategy}), \n \\item the construction of new programs as a reaction to developments in the scientific community, to secure a more comprehensive legacy of the APOGEE survey (\\autoref{sec:new_programs}), or to build synergy with the After Sloan-IV collaboration, and\n \\item the expansion of some programs to better meet their overall scientific goals (\\autoref{sec:expansion}).\n\\end{enumerate}\nAll observations for the BTX have `\\_btx' appended to their \\texttt{FIELD} and \\texttt{PROGRAMNAME} tag; one exception to this policy was the ``odisk'' program.\\footnote{We note that the Outer Disk program was a completely new program and did not need `\\_btx' appended to differentiate it and its targeting strategy from a similar non-BTX program.}\n\n\n\\subsection{The Final Field Plan} \\label{ssec:field_plan}\n\n\\autoref{fig:fieldmap} shows the final field plan for the APOGEE-2 survey overlaid on the \\citet{sfd98} all sky infrared dust map in Galactic coordinates, with colored circles representing APOGEE-2N (this paper and \\citetalias{zasowski_2017}) and grey circles representing APOGEE-1 \\citepalias{zasowski_2013} and APOGEE-2S F.~Santana et al. (submitted). \nThe color-coding in \\autoref{fig:fieldmap} is used to indicate the primary scientific program for a given field, which, in some cases, combine programs core to APOGEE-2 (e.g., disk, halo) and BTX expansions. \nThe labels roughly correspond to terms used in the \\texttt{programname} tag in the summary files produced for the SDSS-IV data releases \\citep[e.g.,][]{Jonsson_2020}\\footnote{See also the DR16 documentation: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/dr_synopsis\/}}; the tag(s) specific to a program will be given with its description. \n\n\\begin{figure*}\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=\\textwidth]{f01_fieldplan.pdf}\n \\caption{The complete APOGEE-1 and APOGEE-2 field plan overlaid on the \\citet{sfd98} all sky infrared dust map and shown in Galactic coordinates. \n Fields designed and observed for APOGEE-2N are color-coded by program and fields designed and those observed in APOGEE-1 or APOGEE-2S are shown in gray.\n The FOV for APOGEE-S is smaller than APOGEE-N, such that APOGEE-S pointings can be distinguished from APOGEE-N pointings by the point size. \n The general motivations for this field plan are given in \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017}, whereas this paper describes the Ancillary Science Programs and the BTX. \n }\n \\label{fig:fieldmap}\n \\end{mdframed}\n\\end{figure*}\n\n\\subsection{APOGEE Datasets Used in this Paper} \\label{ssec:datasets}\n\n\\revise{Throughout this paper, we will show results from APOGEE-1 and APOGEE-2 using the final sample and pipeline to be released as Data Release 17 (DR17) planned for December 2021 (J.~Holtzman et al., in prep.). \nEach of the image reduction, radial velocity measurements, spectral combination, and the derivation of stellar parameters and chemical abundances have been modified from that of Data Release 16 described in \\citet{Jonsson_2020}. \nFor the purposes of this paper, we typically show targets using targeting-related planes (typically \\emph{Gaia} color-magnitude diagrams), but for the evaluation of some targeting strategies, we will use the stellar parameters ($\\log{g}$\\ and $T_{\\rm eff}$).\nFrom the perspective of what is used in this paper, however, the impacts are overall small and the reader may use intuition from \\citet{Jonsson_2020} to understand these limited ASPCAP-results when presented.\n}\n\nWe will often refer to ``tags'' or ``fields'' that occur in various APOGEE-2 data products using their official names.\nSuch ``tags'' are found in multiple APOGEE data products, as in the headers affiliated with APOGEE spectra, as well as in the more commonly used summary files, \\texttt{allStar} and \\texttt{allVisit}. \nGenerally the ``tags'' described here will refer to those in the summary files, unless otherwise noted and tags will be referred to in true-type fonts, e.g., {\\tt APOGEE\\_ID}. \nA full description of the data products and their affiliated data models are given in the online documentation for the Data Release.\\footnote{For DR16: \\url{https:\/\/www.sdss.org\/dr16\/irspec\/spectro_data\/}}\n\nTo estimate how successful our targeting methods were, throughout the paper we will classify stars using their DR17 ASPCAP $\\log{g}$\\ into dwarfs, sub-giants, and giants. \nTo determine the numbers of stars observed in each of these stellar classifications, we will only compare the numbers of those stars with calibrated measurements \\citep[see][(J.~Holtzman et al., in prep.)]{Jonsson_2020}, e.g., those with the \\texttt{LOGG} tag populated. \nWe will use the following definitions:\n a dwarf is a star with $\\log{g}$ \\textgreater\\ 4.1,\n a giant has -1 \\textless $\\log{g}$ \\textless 3.5, and \n subgiants have 3.5 \\textless $\\log{g}$ \\textless 4.1. \n\n\\revise{Because spectro-photometric can be determined for stars that are well beyond the current reach of trigonometric parallaxes from {\\it Gaia}, will also use spectro-photometric distances} following the methods of \\citet{Rojas-Arriagada_2017,Rojas-Arriagada_2019,Rojas-Arriagada_2020}, but applied to the \\sout{internal, incremental data release mentioned above that uses the DR16 pipeline}\n\\revise{final DR17 dataset described above (e.g. not the specific datasets described in those works, but using the same spectro-photometric distance method)}. \n\\revise{As described in \\citet[][their Section 2.2]{Rojas-Arriagada_2020}, the APOGEE spectroscopic parameters ($T_{\\rm eff}$, $\\log{g}$, and [M\/H]) are used to match an observed star to potential absolute magnitudes on PARSEC stellar evolution tracks \\citep{Bressan_2012,Marigo_2017}.\nUsing the observed 2MASS photometry ($JHK_s$), the distance and extinction can be determined.}\nA detailed comparison of these distances to those derived by other studies \\revise{including {\\it Gaia}~trigonometric parallaxes} is given in \\citet[][their appendix A]{Rojas-Arriagada_2020}.\n\n\\section{Targeting Strategy Changes in the Bright Time Extension} \\label{sec:strategy}\n\nThis section discusses changes to the general targeting strategy established in \\citetalias{zasowski_2017} as they apply to observations planned in the BTX. Fields that are subject to these changes in strategy have `\\_btx' appended to their field name (and given in the \\texttt{FIELD} tag) and also have `btx' appearing in the \\texttt{PROGRAMNAME} tag.\\footnote{There is one exception to this rule in the case of the new Outer Disk BTX program which can be identified according to it's `odisk' \\texttt{PROGRAMNAME} tag (\\autoref{sec:odisk}).} \n\\revise{Two strategy modifications are described: (1) a change in priority star selection (\\autoref{sec:mastar}) and (2) the remaining sections discuss a major targeting change aimed at bolstering the sample of distant halo stars.}\n\n\\subsection{Telluric Selection and MaStar Co-Targeting} \\label{sec:mastar}\n\nThe SDSS-IV MaNGA Stellar program \\citep[MaStar;][]{yan_2019} uses the MaNGA fiber bundles \\revise{\\citep{bundy_2015}} to collect stellar spectra; this is distinct from APOGEE-2N co-targeting with MaNGA galaxy observations that is described in \\citetalias{zasowski_2017}. \n\\revise{The aim of these observations is to build a \\emph{fully empirical} stellar library with a broad span in stellar type and abundance for use in the MaNGA project \\citep[][]{yan_2016,yan_2019}.}\nBecause the observations for MaStar occur in tandem with the APOGEE-2N observations using the same plates, the plate design process takes into account the locations of MaStar targets, and this can influence APOGEE-2N targeting.\n\n\nPrior to the BTX, the MaStar targets were included on the plate at the lowest priority, with the intent of having the smallest impact on the APOGEE-2 targeting. \nHowever, this resulted in the under-sampling of some of some of MaStar's target classes due to targeting collisions or conflicts with APOGEE-2N targets. \nOne example deficiency in the MaStar sampling occurred for very luminous B and A type stars with low foreground extinction; this is a natural consequence of APOGEE's reliance on these stars as telluric calibrators, which are the highest priority targets.\n\nTo remedy the lack of such stars in the MaStar sample, the priority for plate design within the BTX fields was altered so that the highest priority MaStar targets were selected first. \nThe revised default priority scheme for the BTX is as follows: \n\\begin{enumerate} \\itemsep -2pt\n \\item MaStar high priority targets (but see description of reconciliation process below),\n \\item APOGEE telluric calibration stars,\n \\item APOGEE targets (but see description of reconciliation process below),\n \\item MaStar low priority targets,\n \\item MaStar standard stars,\n \\item MaStar sky fibers,\n \\item APOGEE sky fibers.\n\\end{enumerate}\nThough this scheme solves the MaStar under-sampling issue, it poses a potential problem for APOGEE-2N science goals.\n\nThere are two impacts: the selection of the same target (a conflict) and the selection of a neighboring target the precludes another target (a collision).\nFor the latter, the MaStar fiber-bundle collision radius (102\\arcsec) is larger than that for the APOGEE-N fibers (72\\arcsec) and, as a result, MaStar targets are not a one-to-one replacement of an APOGEE-N target in a given plate design; this is particularly challenging for plates with spatially clustered targets. \nSome of the more complicated cases include targeting for star clusters, photometric objects of interest (e.g., {\\it Kepler}, {K2}, and {\\it TESS}), and confirmed distant halo stars; in most of these cases, the scientific motivation to collect data for a specific target is similar for MaStar and APOGEE-2N. \nThis competition for targets created a logistical and managerial challenge.\n\nThus, a target reconciliation process was constructed that compared the highest priority MaStar and APOGEE-2N targets for a given plate to determine conflicts (same target desired by both surveys) and collisions (where the MaStar fiber bundle precludes an APOGEE-2N target).\nA team of MaStar and APOGEE-2N scientists carefully evaluated these issues (at a typical rate of only a 1-2 incidents per plate, but potentially dozens for the ensemble of plates being designed at a given time) with the aim that both programs were ensured success of their science goals.\nSolutions to these conflicts and collisions included: \n (i) one survey ceding the target, \n (ii) for a multi-visit field, splitting the visits between MaStar and APOGEE-2N data collection, \n (iii) inclusion of an additional design or visit to satisfy the needs of both surveys. \nThis process was time-consuming, but assured mutual success of the respective goals for each survey. \n\nWhile we anticipate the net impact of these priority changes to be small, those science investigations requiring a detailed selection function analysis may want to exclude all plates designed with MaStar modifications --- i.e., all plates that have ``\\_btx'' appended to their field name.\n\n\\begin{figure*}[h]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{f02_halo_mainsurvey}\n \\caption{Evaluation of the Washington+$DDO51$ giant pre-selection in the APOGEE-2N Main Survey halo program for distant halo stars (e.g., \\texttt{PROGRAMNAME}=``halo'' and \\texttt{TELESCOPE}=``apo25m''). \n \n (a) Spectroscopic $\\log{g}$\\ versus $T_{\\rm eff}$\\ for APOGEE-observed giant candidates (grey) and dwarf candidates (black) demonstrating the overall effectiveness of the Washginton$+DDO51$ pre-selection technique. \n Stars with distances between 15 and 25 kpc are shown as green circles and those with distances greater than 25 kpc as yellow diamonds.\n \n (b) Spectro-photometric distance against apparent $H$ magnitude for giant candidates (grey) and the distant star samples (green and yellow). \n Horizontal dashed lines indicate the magnitude limits for targeting at the specified number of visits (labeled to the right).\n \n (c) After separating the giant candidate sample into distance bins ($d$~\\textless~15~kpc, 15 \\textless~$d$~\\textless~25~kpc, 25~\\textless~$d$), the fraction of the sample identified in the long (24-visit), medium (12-visit) and short (3- or 6-visit) ``cohorts'' on the plate designs. \n \n (d) The yield of distant stars within each ``cohort'' defined as $N_{\\rm distant}\/N_{\\rm total}$, with $N_{\\rm total}$ being the number of giant candidates.\n }\n \\label{fig:halo_bad}\n \\end{mdframed}\n\\end{figure*}\n\n\n\\subsection{Sampling Stars in the Distant Halo} \\label{sec:halo}\n\nA high-level scientific goal of APOGEE is to define the chemo-dynamical fingerprint for stars in all of the structural components of the Milky Way. \nOne particularly difficult component to sample is the distant halo, both because it is sparsely populated and because the stars are fainter due to their distance. \nTo sample this distant and diffuse component of the Milky Way, the APOGEE-2 Science Requirements Document (SRD) set a benchmark APOGEE-2N goal of 1000 stars at distances beyond 15 kpc, such that at least 100 were ``distant'' stars beyond 25~kpc and the remaining $\\sim$900 were at ``intermediate'' distances between 15 and 25 kpc.\nThe SRD was specific that these stars were to be identified from ``halo'' targeting (e.g., not counting stars in dwarf satellite galaxies, but including deliberately targeted streams or serendipitous targets in the background or foreground of dwarf satellite galaxies). \n \n \n \n \n \n \n \n \n \n \nBecause only giants can be seen at these distances (given APOGEE's magnitude limits), giants are the targets of interest to probe the Milky Way halo, however, local dwarf stars in the disk of the Milky Way provide significant levels of contamination, providing a challenge to identify giants of interest. \nTo mitigate this foreground contamination, APOGEE-1 used the Washington+$DDO51$ dwarf-giant separation technique \\citep[as defined by][]{majewski_2000} to pre-select likely giant stars.\nThis technique combines the $DDO51$ intermediate-band filter \\citep{McClure_1973}, which is centered on the surface gravity sensitive Mgb triplet at 5051~\\AA, with the Washington $M$ and $T_2$ filters \\citep{Canterna_1976}, which are optimized for temperature separation in late-spectral types.\nAs demonstrated in \\citetalias{zasowski_2013} for the APOGEE-specific use of Washington+$DDO51$, a star is classified as a likely dwarf or giant based on its location in the $M-DDO51$ versus $M-T_2$ color-color diagram.\n\nAs discussed in detail in \\citetalias{zasowski_2017}, the APOGEE-1 Washington+$DDO51$ dwarf-giant separation methodology was adopted for APOGEE-2N. \nThose stars meeting our NIR color-magnitude criteria {\\it and} classified as giants using the Washington$+DDO51$ criterion were targeted at the highest priority for halo fields, with the likely dwarf stars targeted at lowest priority \\citepalias[see Section~7.1 of][]{zasowski_2013}.\nThe APOGEE-1 strategy was continued in APOGEE-2 \\citepalias{zasowski_2017} in new 3-visit ``short'' fields and additional visits to APOGEE-1 fields (so-called ``deep-drill fields'') for a total of 24 visits per field (to reach $H\\sim13.8$).\nThe ``short'' fields had a single short cohort of 3-visits, while the ``deep-drill fields'' had four short cohorts observed for 6 visits each, two medium cohorts of 12 visits, and a single long cohort of 24 visits. \n\n\nAs discussed in \\autoref{sec:badhalo}, at early targeting reviews, it was clear that the halo program was deficient in its number of observed giants in both the intermediate and distant samples defined in the SRD. \nThe subsections that follow first explain the halo-targeting problem in more detail (\\autoref{sec:badhalo}), then describe our BTX targeting scheme where we attempted to remedy this sampling problem (\\autoref{sec:known_giants} and \\autoref{sec:new_halo_targeting}), evaluate the new targeting scheme (\\autoref{sec:how_did_we_do}), and summarize our investigation into halo targeting (\\autoref{sec:halo_sum}). \n\n\\subsection{Detailed Evaluation of Halo Targeting} \\label{sec:badhalo}\n\nThe spectroscopic $T_{\\rm eff}$--$\\log{g}$\\ diagram for the stars in the APOGEE-2N halo program (selected using the \\texttt{PROGRAMNAME} `halo' and \\texttt{TELESCOPE} `apo25m'), for which target selection relied on Washington$+DDO51$ photometry, is shown in \\autoref{fig:halo_bad}a.\nThe data points in \\autoref{fig:halo_bad}a are color-coded by their Washington$+DDO51$ photometric classification as a dwarf (black filled circle; \\texttt{APOGEE2\\_TARGET1} bit 8) or a giant (grey filled circle; \\texttt{APOGEE2\\_TARGET1} bit 7).\n\nFor our initial, targeting review evaluations of the efficacy of our halo targeting and whether we were meeting our SRD goal of probing the more distant halo, we computed spectro-photometric distances of our observed halo stars. \nHere we reproduce that initial assessment from our targeting reviews, but using updated distances \\citep[using the methods of][]{Rojas-Arriagada_2020} and sub-divide the stars into the three SRD-relevant distance bins as shown in \\autoref{fig:halo_bad}a:\n (i) stars with $d$~\\textless~15 kpc (black for dwarfs and grey for giants), \n (ii) stars with 15~\\textless~$d$~\\textless~25 kpc (green circles), and \n (iii) stars with 25~\\textless~$d$ (orange diamonds). \nAs visible in \\autoref{fig:halo_bad}a, the distant stars (green and orange) are among the most intrinsically luminous in this sample.\n\nOverall, the Washington$+DDO51$ pre-selection has proven very effective at identifying giant stars, since, of the stars with spectroscopically measured parameters, only 12\\% of those pre-selected to be giant candidates turned out to be dwarf stars. \nMany of the pre-selected giants do not fall into these intermediate and distant halo bins, and instead appear to be at distances associated with the thin or thick disk. \n\\autoref{fig:halo_bad}b compares spectro-photometric distances, $d$, of stars to their apparent $H$ magnitude (note that extinction is negligible in these fields) for the giant candidate sample having reliable ASPCAP results and distances \\revise{(1214 stars)} with the same color-coding as \\autoref{fig:halo_bad}a.\nThe dashed horizontal lines in \\autoref{fig:halo_bad}b indicate the magnitude limits from \\citetalias{zasowski_2013} and \\citetalias{zasowski_2017} to obtain $S\/N\\sim$100 in the labeled number of visits. \nThe majority of stars with $d > $1~kpc are fainter than $H$=12.2 (the 3-visit depth), but the $d$\\textgreater~25~kpc stars are largely fainter than $H$=12.8 (the 6-visit depth). \n\n\n\\autoref{fig:halo_bad}c dissects the sample into the three distance bins by the targeting cohort; specifically, there were \\revise{195, 395, and 624 stars} in the short (3- or 6-visit), medium (12-visit) and long (24-visit) cohorts, respectively.\n\\revise{The fractions of the stars in the nearest distance-bin were targeted using the magnitude selection for the the short, medium, and long cohorts were 17\\%, 32\\%, 50\\%, respectively}. \nFor the intermediate and distant distance-bins, \\revise{just over} half of the sample was targeted using the long cohort strategy \\revise{(57\\% and 61\\% )} with the other half coming from the sum of the medium and short cohorts \\revise{(43\\% and 39\\%)}.\nDespite the greater 0.7 mag greater $H$ limit and, especially, the commitment of $\\sim$12 hours more observing time, \\autoref{fig:halo_bad}c demonstrates that the long, 24-visit cohort is not more effective at yielding distant halo stars than the combination of medium and short cohorts.\n\n\n\\autoref{fig:halo_bad}d, which shows the intermediate and distant star targeting efficiency for each targeting cohort, further clarifies the optimal observing strategy to yield such stars. \nWhile the efficiency in the medium cohort is double (triple) that of the short for intermediate (distant) stars, there is no commensurate gain in efficiency from the long cohort. \nThus, despite requiring two to four times more fiber hours per target, the long cohort did not provide a commensurate gain in the harvest of intermediate and distant star samples. \nFurthermore, while the medium cohorts yield a higher percentage of distant stars than the short cohorts, these also required more visits to reach their requisite $S\/N$, meaning that for the same amount of time, the short cohorts were nearly as efficient at accumulating distant halo stars.\nThe ultimate reason for this is difficult to diagnose, but it could be reflective of a number of operational reasons (specific fields targeted, the depth of the Washington$+DDO51$ photometry, etc.) or astrophysical reasons (halo giant luminosity function, global density law, the presence of halo substructure, variations of chemistry and luminosity functions in accreted systems, etc.).\n\n\n\nOf the $\\sim$900 intermediate distance stars and the 100$+$ distant stars required by the SRD, there are only \\revise{91 stars in the intermediate bin} and \\revise{59 in the distant bin} for the data shown in \\autoref{fig:halo_bad} (\\revise{this is only for the Washington-selected sample and} most of this data was in hand by the time of BTX planning). \nThat means that around the time of the BTX planning, only \\revise{$\\sim$15\\% of the goal} was met for stars beyond 15~kpc and \\revise{59\\% of the goal} for beyond 25~kpc.\nFortunately, the BTX program offered us an opportunity not only to add to, but also to course correct, our halo targeting to help meet our SRD goal. \n\nBecause of the diminishing returns of the medium and long cohorts (corresponding to the deep-drill fields in the original APOGEE-2N halo plan), we opted to take a ``broad and shallow'' approach to the BTX halo targeting program (corresponding roughly 1\/3 of the APOGEE-2N BTX allocation). \nWe used 6-visit, single cohort halo fields rather than deeper 12- or 24-visit fields, which, despite their comparable effective yields, are more challenging to schedule and complete. \nCombined with this alteration in how observing hours were distributed by field, we also modified our target selection strategy, with the goal of a higher yield of more distant stars collected with a higher efficiency.\nThis modified target selection strategy employed a two-tiered prioritization scheme, with one tier focusing on fields containing confirmed distant halo stars previously identified by the SEGUE survey (\\autoref{sec:known_giants}), and the second tier relying on new giant star candidate selection criteria that exploited \\revise{the (then) newly available {\\it Gaia}~DR1 astrometry} (\\autoref{sec:new_halo_targeting}). \n\n\\subsection{BTX Halo Targeting Method 1: Known Halo Giants in SEGUE} \\label{sec:known_giants}\nCoincident with the BTX planning was the submission of an Ancillary Science Program on the distant halo, described in \\autoref{anc:distanthalo}, that intended to use spectro-photometric distances derived from SEGUE and SEGUE-II observations in SDSS-II and SDSS-III, respectively, to expand upon the APOGEE-2 halo targeting at large heliocentric distances \\citep[][\\inprep{C.~Rockosi et al.~in prep}]{yanny_2009, eisenstein_2011}. \nWe opted to fold the Ancillary Science Program into the BTX halo program, making the former much larger in scope and implemented at a high priority.\n\n\\citet{xue_2014} identified over 6000 giants in the SEGUE sample, many of which were both metal poor and distant; exploiting this dataset is an ideal way to ensure that APOGEE-2 adequately samples the chemical fingerprint of the distant halo.\nAPOGEE-2N fields were selected to contain 2-3 of the \\citeauthor{xue_2014} K-giants. \nWe did not place constraints on the distance or metallicity of the \\citeauthor{xue_2014} stars and, instead, opted to target any stars from the \\citeauthor{xue_2014}, as any overlap with SEGUE expands upon our inter-survey cross-targeting (\\autoref{sec:cross}). \n\nIn the figures developed in the discussion to follow, the targets from \\citeauthor{xue_2014} are always shown as filled black symbols.\nBecause we consider these targets part of our ``BTX Halo'' strategy, they will be included in efficiency metrics, in part because all of the \\citet{xue_2014} targets would have been selected from our algorithmic targeting strategy (\\autoref{sec:new_halo_targeting}) and, in effect, we have just prioritized their selection by including them as special targets. \n\nStars targeted from \\citet{xue_2014} have \\texttt{APOGEE2\\_TARGET2} bit 20 set and, because a different scheme was used for the APOGEE-2S Halo Program, will have a \\texttt{TELESCOPE} tag of ``apo25m''. \n\n\\begin{figure*}[h]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{f03_halo_newstrategy}\n \\caption{Giant star pre-selection using the total proper motion ($\\mu_{\\rm tot}$) and fractional proper motion uncertainty ($\\sigma_{\\mu_{tot}}\/ \\mu_{tot}$) from HSOY \\citep{altmann_2017,altmann_2017_catalog}. \n The original halo targeting, (a) and (b), was evaluated in all ``deep'' fields to demonstrate that distant giants almost always have $\\mu_{\\rm tot}$ \\textless~5~mas yr$^{-1}$ and $\\sigma_{\\mu_{tot}}\/ \\mu_{tot}$ \\textgreater~0.4 (indicated by dotted lines in these panels). \n Similar to \\autoref{fig:halo_bad}, confirmed giant candidates are in grey, confirmed dwarf candidates in black, and the distant samples are in green for 15~\\textless~$d$~\\textless~25~kpc and yellow for 25~\\textless~$d$~kpc. \n Open symbols are colored by distance bin and represent stars from the ``main red star sample'' that were targeted without Washington$+DDO51$ classification. \n The result of using this proper motion-based strategy in the BTX is shown in panels (c) and (d), with spectroscopically-confirmed giants in grey, dwarfs in black, and the distant samples as in (a) and (b).\n }\n \\label{fig:halo_pm}\n \\end{mdframed}\n\\end{figure*}\n\n\\subsection{BTX Halo Targeting Method 2: Exploiting Astrometry} \\label{sec:new_halo_targeting}\nDue to the short time between the BTX planning and the beginning of its observations, it was impossible to acquire and process Washington$+DDO51$ pre-imaging for plate design. \nThus, we used the existing APOGEE observations, spectro-photometric distances, and, at that time, the recently released Hot Stuff for One Year \\citep[HSOY;][]{altmann_2017,altmann_2017_catalog} catalog of proper motions, combining URAT1 \\citep{urat1} and {\\it Gaia}~DR1 \\citep{gaia_dr1} astrometry, to see if an alternative strategy for increasing our yield of distant halo stars could be constructed.\nOur original investigations of how to identify distant giants effectively explored any fields in APOGEE reaching sufficient depth (e.g., 6, 12, or 24 visits) and included tests with a number of different spectro-photometric distance codes; but for simplicity, we will demonstrate some of the test results using the original APOGEE-2 halo sample and the spectro-photometric distances from \\autoref{fig:halo_bad}. \n\nFor these stars, \\autoref{fig:halo_pm}a compares the spectro-photometric distance to the total proper motion from HSOY, where $\\mu_{\\rm tot}$ is taken to be the quadrature sum of $\\mu_{\\alpha}$ and $\\mu_{\\delta}$.\n\\autoref{fig:halo_pm}a demonstrates that the vast majority of the Washington$+DDO51$-selected distant stars (green circles and orange diamonds as in \\autoref{fig:halo_bad}) have $\\mu_{\\rm tot}$ \\textless 10 mas year$^{-1}$. \n\\autoref{fig:halo_pm}b compares the derived distance to the fractional proper motion uncertainty from the HSOY measurements. \nWe found that all of our distant stars not only had small $\\mu_{\\rm tot}$ but also had large uncertainties relative to those motions. \nThese studies on data in-hand led to an algorithmic strategy that had three criteria to select halo candidates:\n\\begin{enumerate} \\itemsep -2pt\n \\item $J-K_{\\rm s}$ \\textgreater\\ 0.5 \n \\item $\\mu_{\\rm tot}$ \\textless\\ 10 mas year$^{-1}$ \n \\item $\\sigma_{\\mu_{tot}}\/ \\mu_{tot}$ \\textgreater\\ 0.4\n\\end{enumerate}\nThe latter two, proper motion criteria are illustrated in \\autoref{fig:halo_pm}c and \\ref{fig:halo_pm}d, along with the targets observed in BTX halo fields, again color-coding distant halo stars (green circles and orange diamonds as in \\autoref{fig:halo_bad}).\n \nTo further increase our yield and acknowledging that ASPCAP produces reliable stellar parameters and chemical abundances for spectra with $S\/N$ as low as 70 \\citep[][]{holtzman_2015,holtzman_2018,Jonsson_2020}, we also increased the $H$-magnitude limits to $H$=13.5 in 6-visit fields (i.e., that corresponding to $S\/N\\sim$70 per pixel for the faintest stars; \\autoref{tab:maglims}) for any halo star candidates; stars drawn from the main red star sample were still restricted to $H$=12.8 for a 6-visit field (\\autoref{tab:maglims}).\nTargets selected following this scheme have \\texttt{APOGEE2\\_TARGET2} bit 21 set and, because a different method was used for APOGEE-2S, will have \\texttt{TELESCOPE} of ``apo25m''.\n\n\\begin{figure*}[ht]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=1.0\\textwidth]{f04_halo_fulleval} \n \\caption{\n Evaluation of the new halo targeting using HSOY proper motions described in \\autoref{sec:new_halo_targeting} \\citep{altmann_2017}.\n (a) Kiel diagram for BTX halo targeting. \n Giant candidates are shown in grey, SEGUE targets are shown in black, and the distant samples are in green for the intermediate bin and yellow for the distant bin.\n (b) Spectro-photometric distance versus apparent $H$ magnitude with color-coding as in (a). \n The depth of various targeting strategies is labeled.\n These panels can be compared with their counterparts for the original halo targeting \\autoref{fig:halo_bad}a and \\ref{fig:halo_bad}b.\n (c) Total number of stars in distance bins for three targeting methods: the original halo selection scheme in blue circles, the BTX scheme in purple triangles, and stars collected via MaNGA co-targeting in red down-triangles. \n As discussed in the text, the strategies are quite different, but produce similar numbers of distant stars.\n (d) The number of stars in a given bin normalized by the total fiber hours applied to a given targeting scheme, following the color-coding in (c). \n Here the increased yield from the BTX halo targeting is evident; indeed, the BTX targeting improved over the original halo targeting \\revise{by a factor of $\\sim$3$\\times$ to $\\sim$5$\\times$ for all distance bins and by 5$\\times$ to 18$\\times$ over the MaNGA Co-Targeting strategy in the more distant bins.}\n Broadly we interpret the efficiency panels to indicate that dampening the dwarf-star-foreground has led to our gains.\n The data used for panels c and d is given in \\autoref{tab:halo_distant_stars}.\n }\n \\label{fig:halo_newtarg}\n \\end{mdframed}\n\\end{figure*}\n\n\\input{Tables\/table5_halonumbers}\n\n\\subsection{Effectiveness of the BTX Halo Targeting Strategies} \\label{sec:how_did_we_do}\n\nAfter several years of observing, it is now possible to evaluate the effectiveness of the strategies employed for the BTX observing.\n\\autoref{fig:halo_newtarg} illustrates the current status of the BTX program by way of the metrics used to understand the original halo targeting (e.g., \\autoref{fig:halo_bad}a and \\autoref{fig:halo_bad}b).\n\\autoref{fig:halo_newtarg}a shows the Kiel diagram for all targets in halo fields (grey), SEGUE distant halo stars (black), and distant stars identified based on our altered BTX targeting scheme described in Section 3.5 (green and orange). \nAs may be seen in \\autoref{fig:halo_newtarg}a and \\autoref{fig:halo_bad}a, the distant stars in the BTX targeting span a larger range in $\\log{g}$, meaning that they provide a more broadly representative sample of giant stars, not just the most intrinsically luminous ones, as in the original scheme. \n\\autoref{fig:halo_newtarg}b and \\autoref{fig:halo_bad}b also show that the BTX targeting has identified stars over a wide range of apparent magnitudes than in the original halo targeting scheme.\nBased on \\autoref{fig:halo_newtarg}, we believe our targeting criteria are actually building an overall less biased sample of stars at large distances.\n\nThe remaining panels of \\autoref{fig:halo_newtarg} show metrics to quantify the relative success of our BTX halo targeting strategy against those of the other targeting scenarios used in APOGEE-2. \nHere we focus on three in particular:\n\\begin{itemize} \\itemsep -2pt\n \\item {\\it MaNGA Co-Targeting:} As described in \\citetalias{zasowski_2017}, APOGEE-2 observations were obtained in tandem with MaNGA observations of their galaxy sample. \n These fields are in the North Galactic Cap, which is in our halo ($\\ell$,$b$) range. \n The APOGEE-2N targeting strategy for these ``free fibers'' adopted the high-latitude halo color-cut (\\autoref{tab:colorcuts}), but a faint limit of $H\\sim$11.5.\\footnote{The pointings for MaNGA co-targeting are not shown in \\autoref{fig:fieldmap} because their positions are not determined by APOGEE-2; these fields and their data are shown in the data release papers, see e.g., \\citet[][see their Figure 1]{Jonsson_2020} and \\inprep{S.~Majewski et al. (in prep.)}.} These pointings can be identified using the \\texttt{PROGRAMNAME} of ``manga'' and the program is comprised of 3-visit fields. \n While not formally part of the APOGEE-2N Halo Program, the MaNGA Co-Targeting sample is a good comparison set as it is a true \\revise{near-infrared-based} magnitude-color selection without any other interventions or modifications aiming to isolate giants. \\revise{There were 606 individual, but sometimes overlapping, fields in the MaNGA Co-Targeting program all to equal depth and using identical selection.}\n %\n \\item {\\it Original Halo:} As described both in \\citetalias{zasowski_2017} and in this paper, the original halo strategy is a combination of the Washington+$DDO51$ strategy and the color-magnitude criteria (\\autoref{tab:colorcuts}). The results of this strategy are summarized in \\autoref{fig:halo_bad}. This program consists of 52 pointings with a range of depths. \n %\n \\item {\\it BTX Halo:} Halo giant candidates selected according to the criteria given in \\autoref{sec:known_giants} and \\autoref{sec:new_halo_targeting} or as filler according to typical color-magnitude criteria (\\autoref{tab:colorcuts}). This program is comprised of 32 pointings of 6-visits each.\n\\end{itemize}\n\\autoref{tab:halo_distant_stars} summarizes statistics for these three strategies, including the total number of targets and total number of fiber hours. \nThe row ``All Targets'' is the number of targets with calibrated stellar parameters (e.g., {\\tt LOGG} and {\\tt TEFF} tags in the summary file), but excluding duplicates ({\\tt EXTRATARG} bit 4) and telluric observations ({\\tt EXTRATARG} bit 2) \\citep{Jonsson_2020}. \nWe use the same spectro-photometric distances to separate targets into four distance bins: \n (1) 5 \\textless~$d$~\\textless 10 kpc, \n (2) 10 \\textless~$d$~\\textless 15 kpc,\n (3) 15 \\textless~$d$~\\textless 25 kpc, and \n (4) $d$ \\textgreater 25kpc. \nIn \\autoref{fig:halo_newtarg}c, the total number of targets in each of these distance bins is plotted for each of the three strategies; here we see that the MaNGA Co-Targeting strategy actually produces a large sample of distant stars in all distance bins, larger even than the focused halo plates, with the exception of the most distant bin.\nHowever, the raw counts of stars is not necessarily the best metric of success and we should take into account the ``resource cost'' for different means of targeting.\n\nIn \\autoref{fig:halo_newtarg}d, the number of targets in a given distance bin is normalized by the total fiber hours in the program; this normalization takes into account the different spatial areas (e.g., number of distinct fields) and magnitude-depths of the strategies. \nTo compute fiber hours, we sum the number of visits contributing to the final spectrum ({\\tt NVISITS}) for all targets meeting the criteria of the sample; the total fiber hours is the total {\\tt NVISITS} for all targets (not just those in a given distance bin). \nWe note that the MaNGA Co-targeting and the Original Halo programs used approximately the same number of total fiber hours (see \\autoref{tab:halo_distant_stars}), but used them differently, with MaNGA Co-targeting accumulating \\revise{606 unique, but sometimes overlapping, fields} to a depth of $H\\sim$11.5 and the Original Halo only targeting 52 fields, but employing the wedding-cake targeting strategy of deeper cohorts going as faint as $H\\sim$13.8 (see \\autoref{fig:halo_bad}b, and the star counts enumerated in \\autoref{tab:halo_distant_stars}).\nHere, the overall inefficiency of the MaNGA Co-Targeting for distances beyond 10~kpc is evident, however, these fields were not designed specifically for reaching distant halo stars, and, remarkably, the Original Halo strategy does not perform distinctly better. \nIn the end, these performance metrics indicate that distant stars are sufficiently sparse on the sky that that a shallow-depth, but wide-area sampling strategy has benefits in terms of overall sample size.\n\nIn contrast, the BTX Halo program used \\revise{about $\\sim$30\\% of the fiber hours} as employed in the original halo targeting (\\revise{in contrast, MaNGa Co-Targing used 2x the fiber hours}), and employed them to target only 32 fields (61\\% of fields in original halo \\revise{and 5\\% of MaNGA Co-Targeting}). \nThe efficiency of the BTX halo program, as shown in the \\autoref{fig:halo_newtarg}d, is \\revise{$\\sim$3$\\times$ to $\\sim$5$\\times$} more efficient than the original halo targeting and \\revise{from $\\sim$2$\\times$ to $\\sim$18$\\times$} more efficient than the MaNGA co-targeting strategy (the exact numbers are given in \\autoref{tab:halo_distant_stars}). \nThe greatest BTX gains are at the largest distances and, using \\autoref{fig:halo_newtarg}b, we can see that the bulk of the distant stars have apparent magnitudes fainter than the depth probed by the MaNGA co-targeting.\n\n\\subsection{Summary of the Efficacy of APOGEE-2N Halo Targeting} \\label{sec:halo_sum}\n\nThanks to routine evaluations of its targeting strategies, the APOGEE-2 targeting team was able to identify the shortcomings of the original halo targeting strategy, and then determine a set of new strategies to reduce the effect of the dominant foreground contamination of disk stars, and increase both the total number of distant halo stars in the sample as well as the efficiency at which they were accrued. \nAs a result, the BTX program netted a comparable number of distant stars as the original strategy, \\revise{but with only $\\sim$30\\% of the total fiber hours.}\nIt is worth noting that the strategy was successful not so much through more efficient, deliberate targeting of distant halo stars, but rather by more effectively limiting the amount of foreground disk contamination.\nWe also compromised between depth and area, choosing to increase to an $H\\sim$13.5 limit using 6-visits, rather than using past strategies that prioritized depth with $H\\sim$13.8 limit in 24-visit fields.\nIn the end, the BTX Strategy of dampening the foreground improved our yield per fiber hour \\revise{by factors of $\\sim$3 to $\\sim$5} over that of the original halo targeting strategy for stars with distances greater than 5~kpc.\n\nAPOGEE-2 also sampled the halo using a simple magnitude-color criterion with the MaNGA co-targeting. \nThis sample represents a wide-area, but shall depth ($H\\sim$11.5) targeting strategy that differs from the original APOGEE-2 strategy and that adopted in the BTX using proper motions. \nWe find that the MaNGA co-targeting produced comparable, or even substantially larger, samples of stars at heliocentric distances greater than 5~kpc. \nHowever, this was due to the sheer number of fibers allocated in this way, and the halo stars accumulated were collected very inefficiently; \\revise{the BTX strategy is more efficient by factors of $\\sim$2 to $\\sim$18} over the MaNGA co-targeting program in terms of yield per fiber hour (\\autoref{tab:halo_distant_stars}). \n\nThe counts of the MaNGA sample suggest that even relatively shallow but wide-area strategies can accumulate large numbers of stars at halo-relevant distances; however, the stars are accumulated highly inefficiently and adding in a proper-motion based criteria would improve the efficiency at reaching distant halo stars.\nWe note that in many cases the highest-latitude fields are so sparse in stars of any type that cohorted targeting strategies like the Original Halo will ``run-out'' of suitable bright targets.\nThis too motivates the sample building potential of a wide-and-shallow approach, like those taken in the MaNGA co-targeting or BTX halo.\n\nFuture surveys aiming to target the halo will need to weigh their observing strategies carefully, as we did here, but also take into account the impacts on the selection function.\nFor APOGEE-2, we deliberately altered our halo targeting strategy during the course of the survey to explicitly build a sufficiently large sample (e.g., hundreds of stars) at large distances so that we might better probe the chemical distributions of stars in the outer halo.\nWe made this choice while fully recognizing the impact it would have in complicating our selection function and other scientific investigations.\nFor example, the proper motion-prior imposed in the BTX halo targeting could impact dynamics-focused studies of the halo using APOGEE-2N data. On the other hand, the halo star sample that resulted is far less-biased in $\\log{g}$\\ and $T_{\\rm eff}$, and therefore better samples the luminosity function. \nAnd for all of the subsamples of halo stars within the APOGEE dataset, there are complex distance-luminosity biases imposed by the varying magnitude limits employed.\nSampling the halo is difficult and compromises have to be made that are driven by the scientific aims. \n\n\n\\section{New Programs in the Bright Time Extension} \\label{sec:new_programs}\nThis section discusses programs undertaken in the BTX that add new scientific objectives to the APOGEE-2N program.\nFor each, a scientific motivation is provided to place the targeting needs and constraints in context.\n\\revise{There are four new programs:\n (1) mapping of the California Giant Molecular Cloud (\\autoref{sec:calicloud}), \n (2) coverage in the TESS Continous Viewing Zone (\\autoref{fig:tess}),\n (3) probing the outer Galacitic disk (\\autoref{sec:odisk}), \n (4) main sequence calibrations (\\autoref{sec:mainsequence}).\n }\n\n\\subsection{Mapping the Interstellar Medium in the California Giant Molecular Cloud}\\label{sec:calicloud}\n\nThis program is a pathfinder to understand the observational limits for a larger program to map Giant Molecular Clouds (GMCs) planned for SDSS-V \\citep{sdssv}.\nTo understand GMC evolution, one needs to understand better the relative importance of colliding flows, gravitational contraction, magnetic support, and turbulence over the range of physical scales and internal conditions spanned by GMCs \\citep[e.g.,][]{VazquezSemadeni_2007_GMCformation,Clark_2012_GMCformation,Heitsch_2013_GMCformation,Fujimoto_2014_GMCformation}.\nThe velocity field of the GMC's environment may hold the key: for a GMC to contract and form stars, material must flow together.\nA promising way to probe these flows around the GMC formation regime is to use the 1.5272 \\micron\\ Diffuse Interstellar Band (DIB) present in the APOGEE spectral range \\citep{Zasowski_2015a}, in combination with line-of-sight dust column measurements \\citep{Zasowski_2019_highDdust}.\n\nThe California GMC\\footnote{As noted in \\citet{Harvey_2013}, this cloud is called ``Auriga'' in the {Spitzer}\\ Legacy survey. \\citet{Harvey_2013} use the name ``Auriga-California Cloud.'' We stay consistent with the naming convention within APOGEE-2N of ``California Cloud'' following \\citet{lada_2009}.} is a nearby ($d_{\\rm los} = 400$~pc), massive (10$^{5}$ $M_{\\odot}$), isolated, quiescent GMC with a large reservoir of surrounding dust and gas \\citep{lada_2009,Harvey_2013}. \n{\\it Gaia}~DR2 \\citep{gaia_dr2} provides distances accurate to 10~pc for a large number of stars within about 50~pc of the California GMC. \nThus, APOGEE spectra can be used to measure the mean line-of-sight velocity of the 1.5272~$\\mu$m DIB, map the flow of material onto or away from the molecular cloud, and thus probe the dynamical environment of the California GMC. \nMany of the stars in the vicinity of the California GMC are either F-type stars or red giants with $H$ \\textless\\ 11.5 that can be targeted with single, one-hour visits to provide $S\/N\\sim$100 spectra suitable for measuring the DIB in $A_{\\rm V}\\sim1$ material \\citep[a discussion of observational requirements for DIB measurements can be found in][]{Zasowski_2015a}. \nA target density of at least 1 star per 10 pc$^{3}$ is estimated to meet these scientific aims, which requires $\\sim$1200 stars within 50 pc of the California GMC.\nLater catalogs from {\\it Gaia}\\ will be released during the operation of SDSS-V and are likely to provide comparable stellar distance accuracy to GMCs that are $\\sim$1 kpc away, enabling an expansion of this program to other spiral arms and star formation environments \\citep[SDSS-V Science Drivers are explained in][]{sdssv}.\nThis pilot program was designed to determine what quantities can be recovered from different observing strategies and thus optimize the upcoming SDSS-V observations. \n\nThis program was implemented with seven dedicated 1-visit fields with field names of the form `CA\\_$lll$-$bb$\\_btx', where $lll$-$bb$ are the Galactic coordinates of the field center. \nThe \\texttt{PROGRAMNAME} is `as4\\_btx,' where `as4' stands for After Sloan-IV, the project now established as SDSS-V. \nIndividual stars observed as part of this program have \\texttt{APOGEE2\\_TARGET2} bit 1 set. \n\n\n\\subsection{{\\it TESS}\\ Northern Continuous Viewing Zone} \\label{sec:tessncvz}\n\n\n\\begin{figure*}\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=0.95\\textwidth]{f05_tess}\n \\caption{Overview of the {\\it TESS}~N-CVZ program in APOGEE-2N. \n (a) CMD from {\\it Gaia}~DR2 photometry and using {\\it Gaia}~DR2 parallaxes to compute a distance modulus ($\\mu_{\\rm {\\it Gaia}~DR2}$). \n The color-coding indicates the four targeting categories of \n (1) OBAF stars (red), \n (2) stars from the {\\it TESS}~Guest Investigator programs (orange), \n (3) stars identified as dwarf-type from the TESS Candidate Target List (CTL; green),and \n (4) stars identified as giants in the CTL (blue). \n (b) Sky distribution of targets. \n (c) Total $m_{\\rm H}$ distribution of targets (black), as well as histograms for each of the the four targeting categories separately (colored lines).\n }\n \\label{fig:tess}\n \\end{mdframed}\n\\end{figure*}\n\nAs described in \\citet{ricker2015}, the {\\it TESS}\\ mission was approved to enter Phase B implementation in 2013 with a launch no earlier than March 2018.\\footnote{ \\url{https:\/\/tess.mit.edu\/science\/}}.\nThis timeline precluded large-scale consideration of the {\\it TESS}\\ mission for APOGEE-2 planning given that APOGEE-2N operations began in 2014 (APOGEE-2S began in 2017), and with the APOGEE-2 science requirements, field plan, and targeting schema largely in place prior to even the {\\it TESS}\\ Phase B approval.\nThus, no specific effort to coordinate with {\\it TESS}\\ observations was included in the original APOGEE-2 targeting plan \\citepalias{zasowski_2017}.\n\nThe BTX thus provided a key opportunity to capitalize on the scientific opportunities feasible from a joint-analysis of {\\it TESS}\\ and APOGEE data products.\nSuch opportunities include, but are not limited to: \n characterization of planet-hosting stars \\citep[e.g.,][]{canas_2019}, \n radial velocity monitoring \\citep[e.g.,][]{troup_2016}, \n binary star identification \\citep[e.g.,][]{El-Badry_2018}, \n and stellar astrophysics \\citep[e.g.,][]{apokasc,apokasc2}. \nMany of these science goals are fundamental rationales for SDSS-V \\citep{sdssv}, and indeed SDSS-V planned to synergize with the {\\it TESS}\\ mission to help address them. \nThe BTX represented a timely opportunity to test several strategies for SDSS-V, and this motivated a joint effort from APOGEE-2 and the AS4 teams. \nBecause SDSS-V will operate with robotic fiber positioners that lend it a greater ability to survey sparsely-spaced {\\it TESS}\\ targets across the full sky, the APOGEE-2 observations of {\\it TESS}\\ targets have focused on the {\\it TESS}\\ Continuous Viewing Zones (CVZ, hereafter) in the Northern and Southern Hemispheres, which correspond to a circular area 15$^{\\circ}$ in diameter around the ecliptic poles. \nThe latter regions of sky have received 365 day coverage at 30-minute cadence by {\\it TESS}\\ in the original two year mission, with additional observations occurring in the on-going {\\it TESS}\\ extended mission.\nWhile similar in concept and yielding complementary data, the Southern (F.~Santana et al. submitted)\\ and Northern {\\it TESS}\\ CVZ programs differ in their logistical implementation.\n\nA {\\it TESS}-focused APOGEE-2N program is challenging because, despite spanning a large area on the sky, the CVZ is only accessible for observations over a limited range of local sidereal time (LST) for ground-based observations (with a bulk of the N-CVZ accessible to the APOGEE-N spectrograph from roughly at LSTs of 17 to 19 hours).\nMoreover, this LST range is already over-subscribed in APOGEE-2N owing to the location of the {\\it Kepler}\\ field (R.A.,Dec = 19:22:40,+44:30:00) and the cadence requirements for observing programs in the {\\it Kepler}\\ field (see \\autoref{sec:koi} and discussion in \\citetalias{zasowski_2017}). \nThus, only a limited number (75) of hours were available and allocated to this program over the 1.5 years of the BTX, and these observations had to be spread out over several years because of these LST constraints.\n\n\nPlanning how to implement the 75 1-visit fields was a joint effort between the scientific teams within SDSS-IV and SDSS-V, who identified four classes of targets to observe, and which are, in priority order (see Fig. \\ref{fig:tess}a): \n\\begin{enumerate} \\itemsep -2pt\n \\item hot stars of OBAF spectral type (\\texttt{APOGEE2\\_TARGET2} bit 27), \n \\item stars on {\\it TESS}\\ 2-min cadence, largely those either from the {\\it TESS}\\ Guest Investigator programs or candidates for such programs (\\texttt{APOGEE2\\_TARGET2} bit 28), \n \\item dwarf stars in the Asterosiesmic Target List \\citep[ATL;][]{tess_atl} or Candidate Target List \\citep[CTL;][]{stassun_2019} (\\texttt{APOGEE2\\_TARGET2} bit 29),\n \\item giant type stars generally meeting the specifications of the APOGEE `main red star sample' \\revise{and drawn from the TESS Input Catalog (TIC;} \\texttt{APOGEE2\\_TARGET2} bit 30). \n\\end{enumerate}\nAn effort was made to have roughly 50\\% dwarfs and 50\\% giants on a given plate to ensure broad coverage in the Hertzsprung-Russell diagram.\nThese four classes of science targets are briefly described below, before discussion of the detailed implementation of the {\\it TESS}\\ N-CVZ program (\\autoref{sec:tessncvz_implement}). \n\n\\subsubsection{OBAF Stars} \\label{sec:tessncvz_obaf}\nHigh-resolution spectra for oscillating OBAF stars are the foundation for an goal of ``dynamical asteroseismology'' for stars with convective cores; this program was designed in synergy with the program planned for SDSS-V. \nThese stars contribute to the dynamical and chemical evolution of galaxies, but the models of their stellar structure and evolution are known to be inadequate. \nBy comparing seismically determined parameters with spectroscopic parameters and dynamical masses from modelling multi-epoch radial velocities, we will infer precise constraints, for example on the size of the convective core.\nSuch constraints are necessary for the calibration and improvement of the present-day models of stellar structure and evolution for these stellar types \\citep[see e.g.,][]{Pedersen_2018}. \nThe SDSS-IV observations for the OBAF stars provide a first epoch radial velocity measurement to aid in the orbital determinations as well as allow for a first pass on their stellar parameters.\nThese stars have \\texttt{APOGEE2\\_TARGET2} bit 27 set. \n\n\\subsubsection{Stars with 2-min Cadence Observations} \\label{sec:tessncvz_gi}\n\n{\\it TESS}\\ Cycle 1 and Cycle 2 observations produced two data products: full-frame images (FFIs) at 30 minute cadence and ``postage stamp'' images at 2 minute cadence. \nThe former category ensures that every bright star has some data, whereas the latter category was reserved for Guaranteed Time Observations (GTO) from the {\\it TESS}\\ team and Guest Investigator (GI) observations from the broader astronomical community, with the latter awarded through competitive proposal cycles. \nThus, a portion of our program was reserved for stars with 2-minute cadence observations from Guest Investigator samples and given high priority to ensure these rare-target classes would be sufficiently sampled. \nSuch targets include known planet hosts, cool dwarfs, and subgiants. \nThese targets are all identified by \\texttt{APOGEE2\\_TARGET2} bit 28. \n\n\\subsubsection{Dwarf Stars} \\label{sec:tessncvz_ctl}\n\nAfter the rare target classes were selected, roughly half of the fibers ($\\sim$125) were reserved for dwarf-type stars drawn from two {\\it TESS}\\ scientific target lists: \n (i) first, stars from the \\revise{Asterosiesmic Target List \\citep[ATL;][]{tess_atl}} and then \n (ii) stars from the \\revise{Candidate Target List \\citep[CTL;][]{stassun_2019}}.\n\nStars with solar-like oscillations were selected from the {\\it TESS}\\ ATL produced by the {\\it TESS}\\ Asteroseismic Science Consortium (TASC)\\footnote{\\url{http:\/\/tasoc.dk}} as of Version 4.\\footnote{This is Version 4 of the ATL produced in $\\sim$October 2017 (see TASOC website \\url{https:\/\/tasoc.dk\/wg1\/Targetselection}) that implemented discussion from the TASC3\/KASC10 workshop; \\url{https:\/\/www.tasc3kasc10.com\/}} \nThe final ATL sample and its detailed derivation is described in \\cite{tess_atl}, but we provide a brief summary of the intermediate catalog and priority scheme that was available for our plate design.\nStars were selected based on their assigned priority in the ATL. \nThe ATL priorities that we used were assessed based on a term known as $P_{\\rm mix}$ that is a linear combination of the likelihood of the detection of seismic modes and the likelihood that such modes are falling-off beyond detectability based on the stellar parameters relative to the known space in the Hertzsprung-Russell diagram with seismic modes.\\footnote{$P_{\\rm mix} = (1-\\alpha)~P_{vary} + \\alpha~P_{\\rm fix}$, where $\\alpha$=0.5 in the version of the catalog used for APOGEE-2 targeting.}\nThese parameters were selected to balance competing scientific objectives in the ATL.\nThis is similar to what is described in \\citet[][their Subsubsection 4.3.3]{tess_atl} and the APOGEE-2 sample should mimic the overall distribution of the larger ATL. \nAt the time of plate design, $\\sim$900 ATL stars with solar-like oscillations were in the {\\it TESS}~N-CVZ footprint. \nThe entire list was folded into the priority schema and the ATL stars will have \\texttt{APOGEE2\\_TARGET2} bit 28 and \\texttt{APOGEE2\\_TARGET2} bit 29 set because they were targeted as part of a GI program {\\it and} counted toward the dwarf-stars quota for each plate.\n\nAfter selecting candidates from the ATL, the remaining fibers for dwarf-type targets were assigned from the CTL version 8 (CTL8) \\citep[][]{stassun_2019}.\nAs described in \\citet{stassun_2018, stassun_2019}, the CTL is a set of stars selected from the \\revise{{\\it TESS}~Input Catalog (TIC)} that are ideal candidates for the {\\it TESS}\\ planet-finding mission;\nCTL stars are high-likelihood main-sequence or sub-giant type stars with their likelihood determined from the broad range of photometric and astrometric measurements collated into the TIC.\n\nA major component of the classification of stars from the TIC and into the CTL was the likelihood that a star was a dwarf. \nAccordingly, a key component of this determination was the use of the reduced proper motion diagram (RPM), more specifically the NIR version of RPM developed by \\citet{colliercameron_2007} known as RPMJ. \nThis means that astrometric information (proper motions) are important for the CTL and the classification of sources may have some dependence on the astrometric catalog being adopted \\citep[for evaluations pre- and post- adoption of {\\it Gaia}\\ astrometry see][]{stassun_2019}. \nWe will return to this in \\autoref{sec:tessncvz_implement} for the {\\it TESS}~N-CVZ program and, because the RPMJ technique is used in the Outer Disk Program, more discussion is included in \\autoref{sec:outerdist_assess}.\nAPOGEE-2N targets selected from the CTL will have \\texttt{APOGEE2\\_TARGET2} bit 29 set. \n\nTwo sets of fiber assignment were performed from the CTL: \n a ``bright'' set limited to targets with $H<$ 12 and \n a ``faint'' set of targets than with $12 < H < 14$; \nthe latter ``faint'' selection was performed after selecting ``bright'' giants but before selecting ``faint'' giants as described in the next subsection.\nIn either selection round, the stars in the CTL were ranked by their CTL-priority \\citep[see][their Section 3.4]{stassun_2018}, which, briefly, is the probability of detecting a transit signal from a small, rocky planet from a typical {\\it TESS}\\ 2-min postage stamp observation.\n\n\\subsubsection{Giant Stars} \\label{sec:tessncvz_giants}\n\nAfter the rare target classes were selected, roughly half of the fibers ($\\sim$125) were reserved for candidate giant-type stars drawn from the TIC \\citep{stassun_2018,stassun_2019}. \nTwo rounds of fiber assignment were performed, a ``bright'' with $H<$13 and a ``faint'' set with $13 < H < 14$. \nThese two rounds occurred subsequent to the``bright'' and ``faint'' dwarf samples, respectively, to both prioritize dwarfs over giants while prioritizing bright stars over faint stars. \nIn both cases, the giant-candidates were selected following the dwarf-giant classification criteria given in \\citet{stassun_2018} using reduced-proper motion diagrams \\citep[the criterion of][]{colliercameron_2007}. \nGiants selected from the TIC have \\texttt{APOGEE2\\_TARGET2} bit 30 set. \n\nAfter all of the above {\\it TESS}\\ target selections were completed, any remaining fibers were assigned following the standard ``main red star sample'' criteria in APOGEE-2 \\citepalias[][]{zasowski_2013,zasowski_2017} and are flagged accordingly.\n\n\\subsubsection{Implementation of the N-CVZ Program} \\label{sec:tessncvz_implement}\nThe first year of APOGEE-2N observations of the {\\it TESS}~N-CVZ was planned in advance of the {\\it TESS}~Guest Investigator cycles and the release of {\\it Gaia}~DR2 \\citep{gaia_dr2}, both of which were deemed to likely have a significant impact on how to optimize our limited number of fiber hours to apply to the {\\it TESS}\\ science cases.\nThus, the first set of APOGEE-2N observations of the {\\it TESS}~N-CVZ were drilled around the locations of rare, O and B type stars (priority 1), with the remaining fibers assigned following our general schema but using the Tycho-{\\it Gaia}\\ Astrometric Solution from {\\it Gaia}~DR1 \\citep[TGAS][]{gaia_dr1} for the RPMJ computations;\n21 such plates were designed around these OB stars (\\texttt{FIELD} names of CVZ\\_OB\\#\\#\\_btx) plus three additional plates (\\texttt{FIELD} names of CVZ\\_FILL\\#\\#\\_btx).\n\nA second set of plates were designed to attain more-or-less uniform spatial coverage of the N-CVZ; \n51 plates were required and these have \\texttt{FIELD} names CVZTILE\\_$lll\\pm bb$\\_btx, with $lll\\pm bb$ representing the Galactic coordinates of the field center. \nThese plates were designed simultaneously from a consistent list of input targets and relied on {\\it Gaia}~DR2 \\citep{gaia_dr2} for the RPMJ computation; the plates, themselves, were drilled over time to optimize our use of this LST range (e.g., selecting Hour Angles of observation that optimized our observing schedule).\nAll plates from this program have \\texttt{PROGRAMNAME} `cvz\\_btx' with individual targets flagged as described above.\n\n\n\\autoref{fig:tess} gives a summary of the targeting for the {\\it TESS}~N-CVZ program.\n\\autoref{fig:tess}a shows a {\\it Gaia}~DR2 color-absolute magnitude diagram with the distinct targeting classes identified using colors (OBAF stars are red, GI are orange, dwarfs are green, and giants are blue). \n\\autoref{fig:tess}b provides the sky distribution of the targets.\nDuring the first year of observations, an error in target lists led to spatial distributions for science targets that did not fill the full plate footprint. These plates are visible as oblong footprints in \\autoref{fig:tess}b, surrounded by telluric standards that did utilize the full circular footprint of the plate. \n\\autoref{fig:tess}c provides histograms of the magnitude range spanned by each of the four targeting classes; these plates were only intended to have single visits, but span a larger range of magnitudes than typical 1-visit plates in APOGEE-2, as a test of strategies planned for SDSS-V.\n\n\\subsection{Probing the Outer Disk} \\label{sec:odisk}\n\nThe advent of photometric and spectroscopic surveys over large areas of the sky have led to a revolution in our understanding of the structure of the outer disk.\nFar from there being a slow ramp down of the inner disk properties, the outer disk is abundant with star clusters \\citep[e.g.,][]{Zasowski_2013clusters}, apparent substructure \\citep[e.g.,][]{Slater_2014}, and perhaps additional, yet undiscovered features. \nWith the advent of large-area imaging surveys, such as SDSS and 2MASS, star-count maps revealed apparent stellar overdensities in the outer disk; these include the \n Monoceros Ring \\citep[sometimes included into the ``Galactic Anticenter Stellar Structure,'' or GASS;][]{Newberg_2002,crane_2003,yanny2003}, \n Canis Major \\citep{martin_2004}, \n Triangulum-Andromeda \\citep[TriAnd;][]{majewski_2004,rocha-pinto_2004}, \n and A13 \\citep{Sharma_2010,li_2017}, \n among other, smaller features \\citep[for a more exhaustive list see the overview from][]{grillmair_carlin_2016}. \n\nDue to their discovery at a time when clearly identifiable stellar streams from dwarf galaxies were also being uncovered, such as the Sagittarius stream \\citep{ivezic_2000,Newberg_2002,majewski_2003}, these overdensities were broadly interpreted to be debris from dwarf satellite mergers \\citep{rocha-pinto_2003,penarrubia_2005,chou_2010,chou_2011,sollima_2011,sheffield_2014}. \nSpectroscopic follow-up of such features largely seemed to justify these interpretations \\citep{crane_2003,chou_2011,sheffield_2014, deason_2014}, but typically limited themselves to only sampling the most likely member stars and not broadly examining the larger scale behavior of the stars in these areas of the Milky Way.\n\nWhile the dwarf galaxy debris origin was originally favored, evidence grew that these Galactic Anticenter overdensities could also be related to the Milky Way disk, in particular as perturbations to the disk excited by orbiting dwarf galaxies \\citep{kazantzidis_2008,purcell_2011,gomez_2013,gomez_2016,price-whelan_2015,xu_2015,li_2017,newberg_2017,laporte_2018}. \nIn this picture, density waves in the Milky Way disk would be expressed as vertical oscillations of the disk midplane and and the crests and troughs of these waves would appear as apparent overdensities \\citep[although in reality, because of the low density in the outer disk, these overdensities may instead appear to be more feathery, spiral arm-like features rather than a strictly rippled disk; for more realistic examples see][]{laporte_2018}. \n\nVarious studies argued for this revised picture of the Galactic Anticenter overdensities, given that\n (1) the ratio of RR Lyrae to M-giant type stars implied a lack of old stars, which are normally seen in dwarf spheroidal galaxies \\citep{price-whelan_2015}, \n (2) star counts suggest a connection between the apparent substructures that could be interpreted as one continuous feature originating from the Galactic disk \\citep{xu_2015}, and \n (3) we may expect to see these kinds of corrugations across the outer disk excited by known dwarf galaxy satellites like the Sagittarius dSph \\citep[e.g.,]{laporte_2018}. \nHowever, the few existing spectroscopic analyses of the Galactic Anticenter overdensities largely focused on small numbers of stars, were of moderate spectral resolution, and lacked strong sampling of the chemo-dynamics of the outer disk, where these overdensities are located, so that chemistry could not decisively be brought to bear on the origin of these overdensities.\n\nThe original targeting plan for APOGEE-2N contained five pointings on one of these overdensities, Triangulum-Andromeda \\citepalias[referred to as TriAnd in][]{zasowski_2017}, each of which contains only a handful of confirmed members of the TriAnd feature \\citep[drawn from][]{chou_2011,sheffield_2014}, but designed with the hope that the strategy for the main star red sample would naturally identify additional members \\revise{over the extent of TriAnd on the sky (from 100$^{\\circ}$\\textless$\\ell$\\textless150$^{\\circ}$ and -50$^{\\circ}$\\textless$b$\\textless-15$^{\\circ}$).}\nDespite the relatively small sample of TriAnd stars as well as the limited sample of APOGEE targets in the outer disk available at that time, \\citet{hayes_2018} used samples of TriAnd and available outer disk stars to show convincingly that the TriAnd chemistry in [$\\alpha$\/Fe]-[Fe\/H] space is a natural extension of disk chemical patterns to lower metallicity, a finding in agreement with the concurrent study by \\citet{bergemann_2018}.\nThese high resolution spectroscopic studies revealed the potential for chemistry to be used to help clarify the origin of these other Galactic Anticenter overdensities and better understand their evolution, and motivated a more thorough probe of the Milky Way's outer disk with available time in the BTX.\n\n\\begin{figure*}\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=0.95\\textwidth]{f06_odisk_skymap.pdf}\n \\caption{\n PanStarrs DR1 star-count map \\citep[for stars at distances between 7.6 $-$ 11.0 kpc, adapted from][]{Slater_2014} showing also the main survey APOGEE-2 fields (grey) and the outer disk fields for the BTX (red). \n Previously identified member stars in the outer disk substructures are indicated as well; more specifically, \n GASS \\citep{crane_2003} in yellow diamonds, \n A13 \\citep{li_2017} as blue triangles, and\n TriAnd \\citep{rocha-pinto_2004,sheffield_2014} as green circles.\n }\n \\label{fig:outerdisk_map}\n \\end{mdframed}\n\\end{figure*} \n\n\\subsubsection{Outer Disk Program Implementation} \\label{sec:outerdist_implement}\n\nThe \\citet{hayes_2018} study motivated a larger-scale effort to use the APOGEE-2N BTX program to trace the outer disk using a more deliberate targeting strategy; this was essentially accomplished by systematically extending the APOGEE disk field grid out to larger Galactic latitudes ($|b|$~\\textgreater~20$^{\\circ}$), while also pursuing a focused targeting of previously studied stars in GASS, TriAnd, and A13 to define robustly the multi-abundance ``chemical fingerprint'' of these systems relative to the disk \\citep[with targets drawn from][]{crane_2003,chou_2011,sheffield_2014,li_2017}.\nStars targeted from these prior surveys have bit 7 set in \\texttt{APOGEE2\\_TARGET2}.\n\nThe BTX Galactic Anticenter plan encompasses 50 fields, with 26 having $|b|$~\\textless~$24^{\\circ}$ but that expand the disk grid out of the Galactic plane to probe lower latitude features like the Monoceros Ring \\citep{Newberg_2002,yanny2003}, another 19 fields with $|b|$ $\\gtrsim 24^{\\circ}$ that target known members of these features (described below), and two fields, 162$+$34\\_btx and 186$+$31\\_btx, designed to fall on regions of the sky along the AntiCenter Stream \\citep[ACS;][]{grillmair_2006}.\nThe three remaining fields, 108$-$31\\_btx, 114$-$25\\_btx, and 129$-$21\\_btx, were intended to target known TriAnd members but due to an error in the design of these fields, instead probe the lower latitude disk closer to the disk midplane. \n\nEach of these pointings is comprised of two short (3-visit) cohorts (7 \\textless\\ $H$ \\textless\\ 12.2) and 1 medium (6-visit) cohort (12.2 \\textless\\ $H$ \\textless\\ 13.3). \nKnown members of anticenter structures were always given the highest priority.\nNext, likely dwarf stars were identified and removed using a reduced proper motion (RPM) diagram in the NIR; we followed the the example of \\citet{colliercameron_2007}, who adapted a \\emph{Tycho}-based algorithm established by \\citet[][]{gould_2003} to the 2MASS $J$ filter \\citep{Skrutskie_06_2mass}. \nWe used the best fit from \\citet{colliercameron_2007} to separate dwarfs from giants as follows.\nFirst, the RPMJ for a star is defined as follows, \n \\begin{equation}\n \\label{eq:rpmj_def}\n RPMJ = m_{\\rm J} + 5\\log(\\mu) \\\\\n \\end{equation}\n\\noindent and the empirical division between giant- and dwarf-like RPMJ, is \n \\begin{equation}\n \\label{eq:rpmj_division}\n RPMJ \\ {\\rm Division} = -141.25 (J-H)^{4} + 473.18 (J-H)^{3} - 583.6 (J-H)^{2} + 313.42 (J-H) - 58.\n \\end{equation}\nThus, for source $i$, if RPMJ$_{i}$ is brighter than \\autoref{eq:rpmj_division}, then the star is a RPMJ giant candidate and, conversely, if its RPMJ$_{i}$ is fainter, then the star is a RPMJ dwarf candidate; we will refer to \\autoref{eq:rpmj_division} this as the ``RPMJ Division'' in the text that follows.\nAny source defined as an RPMJ dwarf candidate using the URAT1 proper motions \\citep{urat1} was removed from initial consideration and only the remaining, giant candidates were available for the first phase of targeting; however, the dwarf candidates would still be eligible for selection in``main red star sample.'' \nStars targeted as RPMJ giant candidates have bit 8 set in \\texttt{APOGEE2\\_TARGET2}. \n\nFor any remaining fibers, targets were selected largely following the normal red star sample color-cuts (without consideration its RPMJ; \\autoref{eq:rpmj_def}). \nFields with $|b|$\\textless $24^{\\circ}$ followed a disk-like cut with $(J-K)_{0}$ \\textgreater\\ 0.5 and a requirement that 50\\% of the stars were assigned to each the short and medium samples per design.\nFields with $|b|$\\textgreater $24^{\\circ}$ followed a halo-like cut with $(J-K)_{0}$ \\textgreater\\ 0.3, but also had a 50:50 short\/medium cohort requirement.\nThe targeting flags for these selections are as given in \\autoref{tab:colorcuts}. \n\n\n\\subsubsection{Outer Disk Program Assessment} \\label{sec:outerdist_assess}\n\nThe implementation of the foreground removal using the RPMJ division (\\autoref{eq:rpmj_division}) represents a different targeting strategy than that typically employed in APOGEE \\citepalias{zasowski_2013,zasowski_2017}. \nThus, we provide evaluation of the targeting strategy in three ways:\n (i) by the fraction of spectroscopic giants and dwarfs obtained using the RPMJ strategy compared to that obtained in the Main Survey,\n (ii) by the reliability of the RPMJ giant selection using newer, and more precise, proper motions, and \n (iii) by the fraction of targets at the distances of the overdensities of interest.\n\\autoref{fig:pm_comp} is used to illustrate these evaluations, with the columns in this figure representing the BTX targeting (\\autoref{fig:pm_comp}a and \\ref{fig:pm_comp}c) and a selection of stars from the ``main red star sample'' at a similar location in the sky, i.e., in original APOGEE-1 and -2 fields, which we refer to as the ``Main Survey'' sample below (\\autoref{fig:pm_comp}b and \\ref{fig:pm_comp}d). \n\nThe BTX Sample is selected using \\texttt{PROGRAMNAME} of ``odisk''.\nThe ``Main Survey'' sample covers the same area on the sky (see \\autoref{fig:outerdisk_map}; $90^{\\circ} < \\ell < 220^{\\circ}$ and $5^{\\circ}<|b|<40^{\\circ}$), but we have eliminated fields dominated by special targeting (e.g., using \\texttt{PROGRAMNAME} to remove stars in the young stellar clusters, radial velocity monitoring, and contributed programs, among others) and removed the mid-plane to avoid extremely dusty sightlines ($|b| > 5^{\\circ}$). \nThe Main Survey and BTX Samples used to compare the Outer Disk targeting each have $\\sim$18,000 stars.\nBecause the BTX targeting (\\autoref{fig:outerdisk_map}) was designed to target high-latitude fields, the samples are not perfectly matched in terms of the underlying stellar density, but the samples are comparable enough for present purposes.\n\nThe rows in \\autoref{fig:pm_comp} show the differences in the RPMJ (\\autoref{eq:rpmj_def}) distribution obtained using the URAT1 proper motions adopted for the BTX design \\citep{urat1} (a and b) and what would be obtained using the \\revise{{\\it Gaia}~eDR3 proper motions \\citep{gaia_edr3}} (c and d).\nThe \\citet{colliercameron_2007} RPMJ division between dwarf- and giant- candidates (\\autoref{eq:rpmj_division}) is shown in each panel as the dashed line.\nThe color-coding is based on spectroscopic luminosity class using the calibrated $\\log{g}$, such that dwarfs are blue, giants are red, and subgiants are green (see \\autoref{ssec:datasets} for the $\\log{g}$\\ limits). \nGiant-type stars in the spectro-photometric distance catalog with heliocentric distances larger than 12~kpc, placing them at the distance of the TriAnd overdensity (or beyond), are shown as the larger, yellow symbols. \n\n\n\\subsubsection{RPMJ Efficacy for Identifying of Giants}\n\n\\autoref{fig:pm_comp}a shows the BTX Sample with the RPMJ determined from the URAT1 proper motions; as designed, 98\\% of the targets are giant-candidates using the RPMJ division (above the dashed line; \\autoref{eq:rpmj_division}) and the 2\\% of targets that are classified as RPMJ dwarf candidates are all from the ``main red star sample'' (using the targeting flags).\nIn \\autoref{fig:pm_comp}b, the Main Survey sample (that only followed the ``main red star sample'' color-magnitude criteria) RPMJ-color distribution using URAT1 is shown. \nFor the Main Survey sample, only \\revise{81\\% of the targets} would have been classified as giants and \\revise{19\\% being classified} as dwarfs following the RPMJ division (\\autoref{eq:rpmj_division}).\nThus, \\revise{19\\%} of the targets in the Main Survey sample could have been excluded were the RPMJ criterion employed, and opened up fibers for more giant candidates to be observed.\n\nIn the end, the efficacy of the RPMJ criterion can be evaluated using the spectroscopic parameters. \nOf the BTX Sample, \\revise{69\\% of the stars} are spectroscopic giants comprise whereas 72\\% of the targets from the Main Survey sample were giants; thus, the end yield of giant stars were similar from either program.\nHowever, it is important to note that the bulk of the BTX Sample were targeting regions of dramatically lower stellar density than the Main Survey. \nThe initial targeting simulations of APOGEE-1 illustrated that the main red star sample would achieve a $\\sim 50-75$\\% giant fraction in traditional disk fields, but at higher latitudes such as the BTX outer disk fields, the respective giant fraction was significantly lower, around $\\sim 25-50$\\% \\citep[see Appendix D. of][]{majewski_2017}. \nTherefore, achieving the same fraction of giants in the BTX and Main Survey samples is impressive.\nPerhaps most importantly, 75\\% of the spectroscopic dwarfs from the Main Survey sample would have been identified as dwarf-candidates from using the RPMJ division.\n\n\\begin{figure}[h]\n \\begin{mdframed}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{f07_odisk_rpmj}\n \\caption{ \n RPMJ selection criteria applied in the outer disk (left column) and the same visualized for the main survey sample in the same region of the sky (right column) for different proper motion catalogs, at top URAT1 and middle \\revise{{\\it Gaia}~eDR3 \\citep{gaia_edr3}}. \n The dashed line shows the RPMJ division given in \\autoref{eq:rpmj_division}.\n The points are color-coded by spectroscopic giants (red), spectroscopic sub-giants (green), and spectroscopic dwarfs (blue) with the larger symbols indicating spectroscopic giants at heliocentric distances greater than 12 kpc. \n The BTX and Main Survey samples both have $\\sim$18,000 stars. \n The BTX sample has systematically bluer colors than the main survey, though the main survey sample does have higher-extinction sight-lines. \n (e) Fraction of targets at a given heliocentric distance for the main survey (thin blue) and the BTX (thick purple). \n }\n \\label{fig:pm_comp}\n \\end{mdframed}\n\\end{figure} \n\n\\subsubsection{Reliability of URAT1 RPMJ}\nWhile scientifically interesting, the BTX Outer Disk program was designed to fill an imminent need for observations in specific regions of the sky and it was one of the first programs planned in the BTX.\nAt that time, the URAT1 proper motions were the best available and, for the purposes of screening out nearby stars, should have been more than sufficient given our goals.\nSince that time, {\\it Gaia}~eDR3 proper motions were released that attain significantly higher precision \\citep{gaia_edr3}.\n\\autoref{fig:pm_comp}c and \\ref{fig:pm_comp}d investigate \\revise{if the more precise, {\\it Gaia}~eDR3 proper motions could have impacted our selection}.\\footnote{\\revise{We also performed this exercise with {\\it Gaia}~DR2 \\citep{gaia_dr2} and drew identical conclusions.}}\n\nIn the BTX Sample, the number of targets classified as RPMJ giant candidates changes from 98\\% using URAT1 to 92\\% using {\\it Gaia}~eDR3 (\\revise{note $\\sim$1\\%} of the targets did not have proper motions in {\\it Gaia}~eDR3, so if we only consider the sample of stars with measured proper motions, the number of giant candidates \\revise{increases to 93\\%}).\nThus, using the same RPMJ division and the more precise {\\it Gaia}~DR2 proper motions would reclassify $\\sim$6\\% of the targets. \nHowever, or the BTX Sample, 99.8\\% of the spectroscopic giants were RPMJ giants \\revise{in {\\it Gaia}~eDR3} (with a similar fraction from the Main Survey), so therefore, this $\\sim$6\\% reclassification is predominantly changing spectroscopic dwarfs from giant-candidates to dwarf-candidates and all coming from the ``main red star sample''. \n\nBecause the majority of the fields in the BTX Sample still had open fibers after selecting all of the available RPMJ giant candidates and the {\\it Gaia}~eDR3 proper motions tended to only reclassify ``main red star sample'' dwarfs, it is unclear that {\\it Gaia}~eDR3 proper motions would have had a strong impact on the BTX targeting.\nThus, we see no significant impact to our targeting by having used the less precise URAT1 motions over those of {\\it Gaia}~eDR3, largely because we use these not to select the giants, but to suppress the dwarf foreground and the dwarf foreground has sufficiently large proper motions that the precision of the underlying astrometric catalog has less of an impact.\n\nWe further note that the RPMJ division defined by \\citet{colliercameron_2007} was designed to construct a ``pure'' sample of dwarf stars and our comparisons reinforce the reliability of this tool for that purpose. \nHowever, inspection of \\autoref{fig:pm_comp}c and \\ref{fig:pm_comp}d suggests that the {\\it Gaia}~eDR3 proper motions are sufficiently precise that the RPMJ division could be refined.\nThe refinements could act to build a more complete dwarf sample by including the ``cloud'' of spectroscopic dwarfs just above the RPMJ division and, in turn, to make a more pure giant sample with their exclusion. \n\n\n\\subsubsection{Using RPMJ to Access Distant Stars}\n\nLastly, \\autoref{fig:pm_comp}e is a histogram of fraction of targets at a given heliocentric distance for the Main Survey (thin, light blue) and BTX outer disk sampling (thick, purple). \nOverall, the BTX targeting scheme has a higher fraction of stars at larger distances ($d>15$~kpc) and a similar fraction of stars from $10$ 4-5 $M_{\\odot}$) AGB stars form very different isotopes (such as $^{87}$Rb, $^{7}$Li, $^{14}$N) from the isotopes formed by lower mass AGB stars and supernova explosions, as a consequence of different dominant nuclear reaction mechanisms \\citep{abia_2001,garcia-hernandez_2007}. \nStars evolving from the AGB phase to the PNe stage also form complex organic molecules (such as polycyciclic aromatic hydrocarbons or PAHs, fullerenes, and graphene) and inorganic solid-state compounds. \nThus, the $\\sim$10$^{2}$ to $\\sim$10$^{4}$ years of evolution following the end of the AGB phase represent a most fascinating laboratory for astrochemistry. \n\nIn AGB stars, the CNO elemental and isotopic abundances, aluminum as well as the abundances of several $\\alpha$-elements together with s-process abundances (Rb, Sr, Y, Cs, Nd, Ce) can be measured with APOGEE spectra.\nComplementary, circumstellar effects on the chemical abundances in AGB stars will be also investigated and we will attempt the identification of high excitation s-process lines for bright PNe (e.g., [Kr {\\sc III}], [Se {\\sc IV}], [Rb {\\sc IV}]).\n\nThe targets were selected using four different sources:\nThe General Catalogue of Galactic Carbon stars \\citep{alksnis_2001}, \nThe Strasbourg-ESO Catalogue of Galactic Planetary Nebulae \\citep{acker_1992},\nThe Torun catalogue of Galactic post-AGB and related objects \\citep{szczerba_2007} and the PhD thesis of Pedro Garcia-Lario (1992).\nTargets were selected with 7.0 \\textless\\ $H$ \\textless\\ 13.5.\n\\section{2017 Ancillary Programs} \\label{sec:ancillary2017}\nAncillary programs were solicited via an open call across the SDSS-IV collaboration (i.e., not limited to just APOGEE) and were awarded based on a review that focused on technical and scientific feasibility.\nThe 2017 proposals were due in February 2017, selected by peer review during March\/April 2017, and implemented starting in August\/September 2017. \nBecause the bulk of the Main Survey observations had already been drilled when implementation began, the 2017 Ancillary Science Programs had more limited access to certain parts of the sky.\n\nThe timing of the 2017 Ancillary Call and the formulation phases of the Bright Time Extension, while distinct processes, meant that programs were merged during implementation for those cases where the scientific or targeting goals were aligned. \nSuch instances have been noted in the main text, but given that these projects were approved based on their scientific merits independent of the Bright Time Extension, we include them here.\n\\subsection{M33 Globular Clusters} \\label{anc:m33}\nM33 has both young and old globular clusters (GCs) that span from 1 to 12 Gyr in age \\citep{chandar_2006,beasley_2015}. We might expect both old and young GCs to show the CNO and Na\/O anomalies -- extragalactic GCs have been inferred to host Na\/O anti-correlations since many have high [Na\/Fe] as measured from integrated light \\citep{colucci_2014, sakari_2015}.\nSurprisingly, abundance variations have not been observed in young to intermediate-aged LMC GCs \\citep[e.g.,][]{sakari_2017}. \nThis motivated a program to collect integrated-light spectra on young and old GCs in the Triangulum Galaxy (M\\,33) to test this hypothesis. \n\nM\\,33 is the third most massive galaxy in the Local Group and it is is much less studied than the Milky Way (MW), Andromeda (M31) and the Magellanic Clouds (MCs).\nThe Pan-Andromeda Archaeological Survey (PANDAS) revealed large-scale substructures of low surface brightness, including arcs, stream and globular clusters, connecting the M\\,31 and M\\,33 galaxies \\citep{huxor_2011}. Substructure in our own Galactic halo reveals its merger history, and certain globular clusters (GCs) appear to be associated with specific accreted satellites (e.g., the Sagittarius dwarf spheroidal galaxy).\nHence, Globular Cluster Systems (GCSs) can be used as tracers of the formation processes and the assembly history of a galaxy. \n\nTo date, integrated-light spectroscopy in the optical has been obtained for only twelve M33 GCs \\citep{beasley_2015}; no such data exist in the NIR.\nYet the APOGEE $H$-band wavelength coverage confers some significant advantages for integrated-light spectroscopy: insensitivity to hot stars, but high sensitivity to red giant branch (RGB) and asymptotic giant branch (AGB) stars, a feature that simplifies integrated-light analysis \\citep{schiavon_2004,sakari_2014}. \nThe $H$ also offers access to some chemical features not easily available to optical spectroscopy, in particular, the presence of strong molecular lines of CN, CO and OH, which enable determinations of C, N and O abundances \\citep{smith_2013,sakari_2016}Other useful lines for this work are those for Mg, Al, Si, Ca and Ti. \nAPOGEE spectra also bring the opportunity to probe multiple populations in GCs, the ability to detect [O\/Fe], and to probe directly the Na\/O anti-correlation \\citep{sakari_2016}. \n\nWe selected our targets from the catalogue with homogeneous $UBVRI$ photometry of 708 M\\,33 star clusters and cluster candidates based on archival images from the Local Group Galaxies Survey \\citep{fan_2014}.\nWe use the photometry for the M\\,31 GCs \\citep[][their table 1]{sakari_2016} to convert from the $V$ to $H$ in the \\citet{fan_2014} catalogue. \nWe select all the clusters with 12.5 \\textless\\ $H$ \\textless\\ 15 for a total of 132~clusters; these also have initial estimates of metallicity, age and mass from \\citet{fan_2014}.\n\\subsection{Cepheids Calibrators} \\label{anc:cephcalib}\nThe goal of this program is to provide homogeneous chemical characterization of Galactic Cepheids that have multi-wavelength photometric characterization and will have sub-percent precision trigonometric parallaxes from {\\it Gaia}. \nMultiple epochs of APOGEE spectra were obtained for each star using the NMSU 1-meter fiber feed. Chemical abundances will be used to calibrate the metallicity effects in the period-luminosity relationship for ten photometric bands and will provide high signal-to-noise templates over multiple phase points for other Cepheid-based programs in APOGEE-2.\n\nFor nearly a century, Cepheid variables have been the de facto standard candle buttressing the extra-galactic distance scale. While nearby dwarf galaxies provide excellent probes of the Leavitt law at low metallicity, the overall metallicity sensitivity of the Leavitt Law remains relatively poorly unconstrained due to the lack of high metallicity calibrators (the Galactic field Cepheids) with independent distances. \nThe trigonometric parallaxes delivered by {\\it Gaia}\\ are the first to provide these independent distances for a large sample of Cepheids in the Galaxy. \nThis NMSU 1-meter fiber extension ancillary project targets a set of the most well characterized Cepheids in the Galaxy that are poised to be the best calibrators in {\\it Gaia}.\nThese stars serve not only an important scientific role in themselves, but will support other Cepheid projects in APOGEE-2 and SDSS-V.\n\nThe first goal of the program is to calibrate the metallicity dependency of the Leavitt Law and utilize the \\citet{fouque_2007} sample of Cepheids with magnitudes in eight optical and near-infrared bandpasses, to which we add newer observations in the near-infrared and mid-infrared for a total of ten bands. \nThis sample includes the most nearby Cepheids that will, in turn, have the best {\\it Gaia}\\ parallax measurements. \nThe stars will be sampled over a number of epochs to study the stability of the metallicity measurements over phase, which helps to place potential single phase measurements in context. \nA second goal of the project is to calibrate the \\citet{scowcroft_2016b} mid-infrared color-metallicity relationship using a homogeneous metallicity characterization spanning 2 dex in [Fe\/H] (in combination with other APOGEE-2 Cepheid programs).\nThis Galactic sample will have the best mid-infrared light curves for Cepheids \\citep[][]{monson_2012}. \n\nWe selected the sample from \\citet{fouque_2007}, which has homogeneously derived magnitudes in eight bands, to which we add 3.6\\micron\\ and 4.5\\micron\\ measurements from the Carnegie Hubble Program \\citep{monson_2012}, better NIR sampling \\citep{monson_2011}, and revised line-of-sight extinction measurements \\citep{madore_2017}. \n\nThe \\citet{fouque_2007} multi-wavelength sample contains 60 stars with 2MASS apparent magnitudes in the range 1.8 \\textless\\ $H$ \\textless\\ 8.2 (note that for bright stars the 2MASS magnitude uncertainties are 50\\% of the pulsation amplitude of the star). Restricting to those objects with $\\delta$ \\textgreater\\ $-30^{\\circ}$, there are 34 remaining sources that range in magnitude over 1.8 \\textless\\ $H$ \\textless\\ 7.7. We prioritized the sources by putting those sources with TGAS parallaxes as top priority and those without TGAS parallaxes as lower priority \\citep{gaia_dr1}. \nDue to saturation limits with {\\it Gaia}, this distinction amounts to down-weighting the brightest sources (in the range 1.8 \\textless\\ $H$ \\textless\\ 5). \n\n\\subsection{Brown Dwarfs} \\label{anc:browndwarf}\nThis program aims to build a library of late-M and L dwarf APOGEE spectra for the purpose of (1) measuring spatial and rotational kinematics for the nearby population and (2) extending spectral modeling for abundance analysis to $T_{\\rm eff}$\\ \\textless\\ 2700~K. Sources with spectral types later than M7 and 11 \\textless\\ $H$ \\textless\\ 14.5 were selected. The desired scientific outcome is improved spectral models across the hydrogen-burning limit, with an eye toward improving characterization of potential low-mass terrestrial exoplanet host systems.\n\nThe transition between the M dwarf and L dwarf spectral classes for very low mass, $M_{*}$ $\\leq$ 0.1 $M_{\\odot}$ \\citep[VLM;][]{kirkpatrick_2005}, stars and brown dwarfs is a critical benchmark in studies of Galactic populations, substellar evolution, stellar magnetic field generation, angular momentum evolution, star formation processes and history, and exoplanet atmospheric chemistry and habitability. \nThis transition spans the temperature range for atmospheric condensate formation, a focus of current star and exoplanet spectral modeling work \\citep{helling_2008,marley_2010}; and the decoupling of atmospheres from internally-generated magnetic fields \\citep{mohanty_2002}, which results in sharp declines in the incidence and strength of H$\\alpha$ and X-ray non-thermal emission \\citep{west_2011} -- albeit with dramatic exceptions \\citep[e.g.,][]{schmidt_2014}, and reduced angular momentum transport resulting in rotation periods as short as 1-3 hr \\citep{konopacky_2010,irwin_2011}. \nLate-M and L dwarfs also span the hydrogen-burning mass limit ($M_{*}=0.07~M_{\\odot}$) and are the densest hydrogen-rich bodies known, probing a minimum in the mass-radius relationship and potentially exotic states of matter \\citep{burrows_2001}. \nThe long lifetimes of these sources (trillions of years) and their limited fusion and full convective mixing make them ideal time capsules for Galactic star formation and chemical evolution history.\nFinally, their small radii ($R_{*} \\sim$0.1 $R_{\\odot}$), close-in habitable zones ($d R_{*}^{-1} \\sim$ 10-30), and apparent preference for forming Earth-sized planets \\citep{dressing_2013} make VLM dwarfs ideal targets for probing Galactic habitability through the transit method \\citep[e.g., Trappist-1,][]{gillon_2017}.\n\nCharacterizing the physical and populative properties of local VLM dwarfs is optimally accomplished with high-resolution infrared spectroscopy. \nAPOGEE's resolution and sensitivity are well matched to the infrared magnitudes and rapid rotation of these objects (up to 80 km s$^{-1}$), while its broad spectral coverage is critical for measuring atmospheric abundances, probing both bulk composition and atmospheric chemical dynamics. \nHowever, late-M and L dwarfs fall below current APOGEE stellar modeling temperature limits \\citep[][]{garciaperez_2016,schmidt_2016,souto_2017}. \nDR14 included $\\sim$25 VLM dwarfs \\citep{holtzman_2018} and this provided only a narrow scope for examining population properties. \nThis program aims to increase the observe sample 20-fold by targeting up to 448 M6-L6 dwarfs. \nThese data will provide radial velocities, which, combined with PanSTARRS\/{\\it Gaia}\\ proper motions, will: yield precise kinematics (\\textless 1-2 km s$^{-1}$); measure rotational velocities down to 5 km s$^{-1}$, to examine angular momentum evolution and magnetic activity trends in conjunction with ancillary optical spectral observations; enable searches for close-separation companions down to planetary masses through RV variability measurements; improve spectral modeling in the low-temperature regime; and characterize planet-host candidates targeted by MEarth \\citep{nutzman_2008}, SPECULOOS \\citep{gillon_2013}, and {\\it TESS}\\ \\citep{ricker2015}, among others.\n\nTargets were selected from compilations of known VLM dwarfs, including \n Dwarf Archives\\footnote{\\url{DwarfArchives.org}}; \n Pan-STARRS DR1 \\citep{best_2017}; \n BOSS Ultracool Dwarfs \\citep{schmidt_2015}, \n BASS \\citep{gagne_2015}, \n LaTE-MoVeRS \\citep{theissen_2017}, \n MEarth \\citep{newton_2017}, \n SPECULOOS \\citep{gillon_2013}, and \n the {\\it TESS}\\ Input Catalog \\citep{stassun_2018}. \nTargets were selected to have reported spectral types later than M7 ($T_{\\rm eff}$\\ \\textless 2700~K) and 11 \\textless $H$ \\textless 14.5. \n\\subsection{Distant Halo Giants} \\label{anc:distanthalo}\nAs discussed in Section \\autoref{sec:halo}, the initial targeting strategy of APOGEE unfortunately yielded relatively few spectra for distant halo stars ($D$\\textgreater10~kpc).\nOn the other hand, the SEGUE\ncatalog consists of 6000 confirmed K~giants \\citep[][]{xue_2014}, many of which are at large heliocentric distances.\nThe 0.5 \\textless\\ ($g$-$r$) \\textless\\ 1.3 color range is within the temperature range of the APOGEE pipeline. \nObservations that deliberately target known SEGUE giants will dramatically increase the number of distant stars observed by APOGEE and, in particular, those in the distant halo. \nBy comparison there were approximately 60 halo giants observed by APOGEE in DR14 \\citep[based on metallicity, velocity and distance;][]{holtzman_2018,dr14} from approximately 100 halo or stream plates observed by APOGEE to that point in the survey \\citepalias[][]{zasowski_2013,zasowski_2017}.\nThe halo is lumpy and so the number of giants within the APOGEE magnitude varies greatly, but by targeting known stars we will confidently boost this sample of stars and ensure that APOGEE obtains a chemical fingerprint of this Galactic component.\nThis program was subsumed, fully, into the BTX Halo targeting described in \\autoref{sec:new_halo_targeting}. \n\\subsection{The Young Galaxy} \\label{anc:younggal}\n{\\it Gaia}\\ DR2 has increased our understanding of the Galaxy. \nHowever, a complete description of the Galactic thin disk remains challenging, because even {\\it Gaia}\\ has severe limitations in the Galactic plane, where dust extinction dims stars below the detection limit at distances greater than 5~kpc. \nWe target faint\/distant young Cepheids that were identified in the PanSTARRs \\citep[PS1;][]{panstars_dr1} multi-epoch catalog to sample the farthest obscured reaches of the Galactic disk.\n\nCepheids are powerful probes of both the structure and the recent history of the Milky Way: \nthey are luminous and can be seen to great distances, even through substantial dust extinction;\ntheir individual ages and distances can be precisely determined from their periods; and, with ages of 20-150 Myrs, they are young stars but they are relatively cool and hence their spectra show a rich metal-line absorption spectrum, from which many element abundances can be determined. \nThe new PanSTARRs (PS1) catalog of variable stars is currently still under construction.\n\nWe selected a list of targets determined on the basis of the following steps:\n[1] identification of Cepheids-like variables in the entire PS1 database, by using the variability parameters by \\citet{Hernitschek_2016} and colors defined on the basis of magnitudes in the $grizy$ (PS1) and near-infrared bands from 2MASS \\citep{Skrutskie_06_2mass} and ALLWISE \\citep{cutri_2014}; \n[2] multi-band template fitting of the sparsely sampled PS1 light curves for all the 200,000 sources selected and determination of pulsation period and other parameters (including distance modulus and extinction);\n[3] use of machine-learning techniques to extract the variables classified as Cepheids with highly significant probability (i.e., higher classification score).\nThe targets existing in APOGEE fields will be targeted for multi-epoch observations as is feasible.\n\\subsection{Kapteyn Selected Area 57} \\label{anc:sa57}\n\nSelected Area 57 (SA57) is the nearest of Kapteyn's Selected Areas (an ambitious program to systmatically study the Milky Way first organized by Jacobus Kapteyn in the year 1906) to the North Galactic Pole. \nSA57, therefore, lies in a direction relatively free of reddening and where ``in situ'' halo stars can most easily be accessed. \nSA57 has traditionally played a significant role for a variety of astronomical studies, from probing the vertical density laws of the Galactic stellar populations to deep studies of galaxies and quasars, the latter because of the minimal stellar foreground. \nAs a result of more than a century of interest in this direction of the sky, SA57 has received extensive attention by surveys of photometry, astrometry and spectroscopy --- but never, to this point, high-resolution spectroscopy. \n\nThe APOGEE observations here are intended to tie state-of-the-art chemistry of giant stars in SA57 to the rich legacy of previous observations in this field. \nStars were selected following the criteria of main red star sample \\citepalias{zasowski_2013,zasowski_2017}.\nFour overlapping plate centers (N, S, E, W) were targeted yielding three cohorts in a ``wedding-cake'' arrangement that follows the magnitude limits of the main red star sample, but also takes into account the overlap between the plates to reach the full $S\/N$.\n\\subsection{Tidally Synchronized Binaries} \\label{anc:tlockbinary}\nIt is now common knowledge that more stars have binary companions than are truly isolated stars \\citep{Raghavan_2010}. \nWhile the formation of single-star systems is still under active study, the formation and evolution of binaries demonstrates rich behavior and, yet, is even more poorly understood. \nBinary formation is governed by the interplay between fragmentation and the circumprotobinary disk\n\\citep{Artymowicz_1991,Artymowicz_1994,Bonnell1994,Bate_2000,gunther_2002}, while the subsequent evolution occurs due to dynamical interactions in the cluster \\citep{Bodenheimer_2001,clarke_2001}.\nUnfortunately, obtaining data on the field binary population to compare to evolutionary models is notoriously difficult.\nThe selection function for field binaries is so heterogeneous and biased that statistical analyses do not provide strong constraints. \nBinaries in clusters are more easily interpretable; however, binaries currently in clusters may not be representative of the escaped field population.\n\nThe aim of this project is to collect a verified sample of tidally-synchronized binaries selected from the rapid rotators in the {\\it Kepler}\\ field. \nThe modulation amplitude in tidally-synchronized binaries should be easily measured because rapid rotation enhances starspot activity \\citep{Basri_1987}.\nApplying solar neighborhood binary properties \\citep{Raghavan_2010} to the {\\it Kepler}\\ field predicts around 300 non-eclipsing tidally-synchronized binaries showing detectable rotational modulation, compatible with the actual number of rapid-rotators found.\n\\subsection{SubStellar Companions} \\label{anc:substellar}\nWith the ever-growing sample of companions (both stellar and substellar) being discovered around stars from ever more diverse environments, population studies comparing host environments are now possible.\nFor example, recent work suggests the hot Jupiter occurrence rate in M67 is higher than in the field \\citep{Brucalassi_2016}, although it is not clear whether this is the norm for open clusters generally.\nWe aim to expand upon the substellar companion science program in APOGEE-2 \\citepalias{zasowski_2017} to include stars in a variety of cluster environments with a wide range of ages, metallicities, and densities.\nThese observations will help constrain the role environment has to play in the formation and evolution of stellar systems containing companions of various masses. \nCritically, this program will both extend the time baseline for targets observed in APOGEE-1 and increase the number of RV measurements to enable robust orbital fits in the final dataset. \nThe \\texttt{FIELD} names from APOGEE-1 are: IC348\\_RV, M67\\_RV, and M3\\_RV. \n\n\\subsection{M dwarfs in {K2}} \\label{anc:mdwarfk2}\nM-dwarf stars are increasingly important objects in both the fields of exoplanet searches, due to the emphasis on cool dwarfs in the target lists of {K2}\\ and {\\it TESS}, as well as studies of Milky Way stellar populations, as these are the most numerous stars in the Galaxy. \nM-dwarfs are notoriously difficult to analyze via optical spectroscopy, due to intense molecular line absorption; this difficulty is alleviated greatly at infrared wavelengths. \nOur team is using APOGEE spectra of M dwarfs to pioneer the derivation of fundamental stellar parameters and detailed chemical compositions for these types of stars \\citep{souto_2017,Souto_2020}. \nM dwarfs are a particularly important component in both transit and radial-velocity searches for exoplanets, thanks to the enhanced detectability of small planets due to low stellar masses, low luminosities, and small stellar radii. \nThis work opens a new window into a class of planet-hosting stars that will play an increasingly important role in ongoing and future planet surveys and missions, such as {K2}\\ and {\\it TESS}. \nTargets from this program are shown in \\autoref{fig:k2}a.\n\\subsection{Local Group Stellar Populations in Integrated Light} \\label{anc:m31}\n\nThe MW provides us with a unique opportunity to constrain galaxy evolution using individual interstellar clouds and single stars, both of which serve as the fundamental building blocks of galaxies. \nThe results from these stellar measurements in the Galaxy have guided our understanding of the chemo-dynamical processes that impact galaxies on all spatial scales.\nThis understanding, then, relies on the assumption that the processes required to produce these spatial and chemical distributions are universally available and commonplace for galaxies generally. M\\,31 provides one opportunity for testing this assumption.\n\nWith 3\\arcsec\\ fibers at APO, APOGEE-2 cannot resolve individual stars in M\\,31, so integrated light observations are required to identify multiple stellar populations with distinct chemistry and dynamics.\nAPOGEE collected integrated light spectra of the inner disk of M\\,31, spaced in a grid of points within $R_{\\rm M31} < 5$~kpc, and of the centers of M\\,32 and M\\,110. \nAll fiber positions were visually checked against optical and near-IR imaging and shifted by up to 10\\arcsec\\ to ensure that no bright stars, clusters, or emission line regions fell within the aperture. \nFibers that could not be fit onto these targets due to fiber packing and plugging limitations were placed on positions observed by the MaNGA survey and on Luminous Blue Variable stars in the field of view.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction \\label{section_intro} }\n\n\nNGC~2419 is a globular cluster (GC) in the outer halo of the Galaxy at\na distance of 84~kpc\\footnote{The distance and other parameters for NGC~2419\n are from the 2010 version of the on-line GC database of\n \\cite{harris96}, based on the photometry of \\cite{harris97}.}.\n\\cite{sigmav09} determined a velocity dispersion for this cluster\nbased on 40 stars which leads to $M\/L = 2.05\\pm0.50~M_{\\odot}\/L_{\\odot}$, a normal\nvalue for an old stellar system, with no evidence for dark\nmatter at the present time. \\citet*{cls10} further asserted that this cluster could not\nhave formed in a now-defunct dark matter halo. \\cite{harris97}\nobtained a deep CMD of NGC~2419 with HST\/WFPC2 and demonstrated that\nthere is no detectable difference in age between it and M92, an\nancient, well-studied inner halo GC of comparable metallicity.\nFurthermore, there is no evidence from CMDs of multiple stellar\npopulations in NGC~2419.\n\n\nHowever, some of the characteristics of NGC~2419 are anomalous for a\nGC. It has the highest luminosity ($M_V \\sim -9.6$~mag) of any GC\nwith Galactocentric radius $R > 20$~kpc, higher than all other GCs\nwith $R > 15$~kpc with the exception of M54, which is believed to represent\nthe nuclear region of the Sgr dSph galaxy now being accreted by the Milky Way\n\\citep{ibata95,sarajedini95}. NGC~2419 also has an unusually large\nhalf-light radius ($r_h = 19$~pc) and core radius ($r_c = 7$~pc)\nfor a massive Galactic GC and a relaxation time which exceeds the Hubble\ntime, also unusual for a GC. Every other luminous GC in the outer\nhalo is considerably more compact. The 2010 version of the on-line\ndatabase of \\cite{harris96} lists only three Galactic GCs that lie\nanomalously above the bulk of the Galactic GCs in the plane $r_h$ vs\n$M_V$, one of which is NGC~2419. The other two are M54 and $\\omega$\nCen, the most luminous Galactic GC, and one with a large internal\nspread in metallicity, also believed to be the stripped core of a\ndwarf galaxy accreted by the Milky Way. For these reasons,\n\\cite{vdb04} and \\cite{mackey05}, among others, suggested that\nNGC~2419 is also the remnant of an accreted dwarf galaxy. In fact,\n\\citet{irwin99} and \\citet{newberg03} suggested that NGC~2419 is one\nof many GCs associated with the disrupted Sgr dSph, but \\citet{law10}\nfirmly concluded that it is not. Therefore, the cluster is potentially\na piece of an unidentified, tidally disrupted dSph.\n\n\n\nTo follow up on these earlier suggestions, \\cite{deimos} obtained\nmoderate resolution spectra of a large sample of luminous giants in\nNGC~2419 and analyzed line strengths in the region of the\nnear-infrared Ca triplet. We found that there is a small but measurable\nspread in Ca abundance of $\\sim$0.3~dex within this massive metal-poor\nglobular cluster. We present here an analysis of high dispersion\nspectra of seven luminous giant members of NGC~2419, whose chemical\ninventory we compare\nto that inferred from a similar sized sample of luminous RGB members of the nearby\ninner halo globular cluster M30 (NGC~7099), observed and analyzed\nin the same way as the NGC~2419 stars. Note that\nthese two globular clusters\nhave the same metallicity in the 2003 version of the\non-line database of \\cite{harris96}.\nWe explore two issues:\n\n\\begin{enumerate}\n\n\\item Is the chemical inventory in such a distant cluster identical to\n that of a globular cluster of similar [Fe\/H] in the inner halo?\n\n\\item What star-to-star variations are revealed by our high-dispersion\nspectra?\n\n\n\\end{enumerate}\n\n\n\\section{Observations and Analysis\\label{section_obs} }\n\n\nWe obtained high-spectral-resolution and reasonably high-SNR spectra\nof 7 luminous giants in NGC~2419 using HIRESr \\citep{vogt94} on\nthe Keck~I 10~m Telescope. \nThe stars were selected from both the early spectroscopic study of\n\\cite{suntzeff88} and the current version of the\n on-line photometric database described by \\cite{stetson05}.\nThe sample stars range in $V$ from 17.2 to\n17.6~mag. The spectral range is 3900 to 8350~\\AA\\ with gaps\nbetween the reddest echelle orders. We used a 1.1~arcsec slit,\nequivalent to 6.3 pixels at 15$\\mu$\/pixel, to achieve a spectral\nresolution of 34,000. The maximum total exposure time for one star\nwas 2.5 hours, split into shorter segments to improve cosmic ray\nremoval. The code MAKEE\\footnote{MAKEE was developed\nby T.A. Barlow specifically for reduction of Keck HIRES data. It is\nfreely available on the world wide web at the\nKeck Observatory home page, \nhttp:\\\\www2.keck.hawaii.edu\/inst\/hires\/data\\_reduction.html.}\nwas used to reduce the HIRES spectra, which\nwere obtained in various runs over the past 5\nyears. The majority of the seven spectra have a SNR exceeding 90 per\nspectral resolution element at 5500~\\AA\\ in the continuum. The SNR\ndegrades towards the blue, and we often eliminated lines bluer than\n5000~\\AA\\ for species with many detected features. The radial\nvelocities for these seven stars have a mean of $-20.6$~km\/sec with\n$\\sigma$ of 3.7~km\/sec. \\cite{deimos}\ndiscussed membership issues for NGC~2419 in detail. \nOur sample of 5 RGB\nmembers in the sparse nearby GC M30 (NGC~7099) were observed\nwith the same HIRES configuration; all the resulting spectra\nare of high SNR. Table~\\ref{table_sample} gives details of the samples and\nspectra. Figs.~\\ref{figure_7099_cmd} and \\ref{figure_2419_cmd}\nshow CMDs for the full Stetson on-line database \\citep{stetson05}\nas well as our HIRES sample for each of these two GCs\nwith 12~Gyr isochrones from the Y2 grid \\citep{yale03} superposed.\n\n\nThe determination of stellar parameters followed our earlier papers,\n\\cite[see, e.g.][~and references therein]{cohen05a}\nand is based on $V-I$, $V-J$, and $V-K_s$ where the optical colors are from\n\\cite{stetson05} and the infrared colors are from 2MASS\n\\citep{2mass1,2mass2}. The uncertainties in 2MASS $K_s$ are\nrather large for the faint NGC~2419 giants, and only two of the\nNGC~7099 giants have an $I$~mag in the \\cite{stetson05} database. \nWe use the predicted color grid as a function of $T_{\\rm eff}$,\nlog($g$), and [Fe\/H]\\footnote{The \nstandard nomenclature is adopted; the abundance of\nelement $X$ is given by $\\epsilon(X) = N(X)\/N(H)$ on a scale where\n$N(H) = 10^{12}$ H atoms. Then\n[X\/H] = log[N(X)\/N(H)] $-$ log[N(X)\/N(H)]$_{\\odot}$, and similarly\nfor [X\/Fe].} \nfrom \\cite{houdashelt00}. If we use the recent $T_{\\rm eff}$ --\ncolor relatoins of \\cite{irfm_teff}, which are not calibrated\nfor such luminous giants, we obtain $T_{\\rm eff}$ $\\sim$35~K\nlower for the cooler luminosity stars in our sample for\nNGC~2419, ranging up to 105~K lower for the most luminous\nstar in this GC.\nThe surface gravities\nwere calculated assuming a mass of 0.8~$M_{\\odot}$, the known\ndistance, and the (low) interstellar extinction to each GC. The isochrone of\nthe upper RGB for a very metal-poor old stellar system\nis such that an error of 100~K in $T_{\\rm eff}$ produces\nan error of 0.2~dex in the inferred surface gravity using\nthis procedure.\n\nThe red giants in our sample in NGC~2419 span 0.36~mag\nin $V$ and a range of less than 200~K in $T_{\\rm eff}$.\nThe RGB stars in our sample for NGC~7099 are somewhat hotter\nin the mean\ndue both to the somewhat lower metallicity of this GC \nand to the scarcity of upper RGB stars in this rather sparse\ncluster.\n\n\nThe detailed abundance analysis for each of the cluster giants follows\nJ.~Cohen's previous work \\citep[see, e.g.,][]{cohen05a} using\n\\cite{kurucz93} LTE plane-parallel \nmodel stellar atmospheres and the analysis code MOOG\n\\citep{moog}. The adopted transition probabilities are largely\nfrom NIST version 3. Note that the $gf$ values adopted here for\nthe observed Mg~I and Ca~I lines have\nbeen updated from those we consistently used\nup to the present \\citep[see, e.g.][]{cohen05a}\nto match the current on-line values in\nthe NIST version 4.0 \\citep{nist} database. \nThe changes \nin abundance resulting from these updates are small, about\n$-0.05$~dex for Mg for a typical set of observed lines,\n and $+0.03$~dex for Ca~I\nwhere more lines are usually observed,\nonly some of which have $gf$ values that have been updated.\n\n\nHyper-fine energy level splitting was used where\nappropriate, with many such patterns taken from\n\\cite{prochaska}. The adopted damping constants are those\ndescribed in \\cite{cohen05a}.\n\n\nAlthough it appears that small non-LTE corrections\nshould be applied for several species, for example Na~I \\citep{takeda03}\nand Ba~II \\citep{nonlte_ba2}, for consistency with\nour earlier work,\nthe only non-LTE correction we have\nimplemented is a fixed value of $+0.60$~dex for\nthe resonance Al~I doublet at 3944 and 3961~\\AA\\ following\n\\cite{al1_nonlte}, \\citep[see also][]{andrievsky08}.\nNon-LTE for K~I is discussed in \\S\\ref{section_s1131}.\n\n\nEquivalent widths were measured in some cases by the script\ndescribed in {\\S}3 of \\cite{cohen05a},\nwhich automatically\nsearches for absorption features, fits a Gaussian,\nthen matches those found\nto a master line list given the radial velocity of the star.\nFor other sample stars,\nW.~Huang used an IDL script to determine $W_{\\lambda}$\nfor the set of lines in the master list.\nThere was extensive hand checking\nof weak features, of most rare earth lines, and of many\nof the strongest accepted lines. Lines with $W_{\\lambda} > 175$~m\\AA\\\nwere rejected unless the species had only a few detected features.\nThe resulting values, together with the\natomic parameters for each line used, are given\nin Tables~\\ref{table_2419_eqw} and \\ref{table_7099_eqw}.\n\nThe $W_{\\lambda}$ for the 3961~\\AA\\ Al~I line, when given, \nare particularly uncertain for the NGC~2419 giants; the\n3944~\\AA\\ line was never used to due contamination by CH.\nBoth lines are uncomfortably far in the blue,\nwhere the SNR of the HIRES spectra for such faint stars\nis degraded, but the 3961~\\AA\\ line is a key feature for this element as \nthe 6690~\\AA\\ Al~I doublet is very weak and difficult to\ndetect at the low metallicity of this GC.\n\nBecause of the uncertainty of the photometry for the NGC~2419 giants, \nwe felt free to make\nsmall adjustments to the photometrically derived stellar parameters,\nprimarily changing $T_{\\rm eff}$. These adjustments,\ndetailed in Table~\\ref{table_temp}, improved the\nexcitation equilibrium for \\ion{Fe}{1} lines and the ionization balance\nbetween neutral and singly ionized lines of Fe and Ti. No such\nadjustment exceeded 100~K, and was less than\n50~K for four of the NGC~2419 giants. In the case of M30,\nonly one star had $T_{\\rm eff}$ modified by more than 20~K.\nFor each star in NGC~2419, we measured 67 to 108 \\ion{Fe}{1}\nlines, which allowed us to determine the microturbulent velocity\ndirectly from the spectra; every star in our sample in NGC~7099\nhad 99 or more detected Fe~I lines.\n\nTable~\\ref{table_fe_slopes} gives the slope of a linear\nfit to the Fe abundance deduced from each of the Fe~I lines\ndetected in a specific sample star\nas a function of excitation potential, reduced equivalent\nwidth ($W_{\\lambda}\/{\\lambda}$), and $\\lambda$, where the\nfirst is most sensitive to the adopted $T_{\\rm eff}$, the\nsecond to $v_t$, and the third indicates any systematic\nproblems in continuum opacity as a function of wavelength.\nValues are given for both GCs. In the ideal case, these slopes\nare zero. For NGC~2419 the typical\nrange in $\\chi$ for the Fe~I lines is 0.9 to 4.2~eV;\nthe stars with the best spectra span a larger\nrange from 0.0 to 4.8~eV. The wavelength range for Fe~I lines\nin most of the NGC~2419 (NGC~7099) spectra is $\\sim$2400~\\AA\\\n($\\sim$3000~\\AA),\nand a typical range in reduced equivalent width for\nsuch spectra is $-5.4$ to $-4.5$~dex for NGC~2419\nand $-6.3$ to $-4.4$~dex for NGC~7099. \nThe crucial slope with $\\chi$ ranged from $-0.083$\nto $-0.038$~dex\/eV for the NGC~2419 sample, and is\n$-0.048\\pm{0.007}$~dex\/eV for the M30 RGB sample.\n \n \nThe resulting abundances for the 7 RGB stars in NGC~2419\nand the 5 in M30 are given in\nTables~\\ref{table_ngc2419_abund} and \\ref{table_7099_abund},\nwhere all abundances are relative to Fe~I.\nThe adopted Solar absolute abundances for each element can be\ninferred from the entries in Table~\\ref{table_7099_abund}.\nWith regard to the ionization equilibrium, that for Fe~I\nvs Fe~II is generally quite good, within 0.1~dex, for all\nstars in both NGC~2419 and in M30. \nThe ionization equilibrium of Ti~I vs Ti~II is somewhat worse, typically\nwithin $\\sim$0.2~dex, with the Ti~II lines giving a somewhat\nhigher Ti abundance.\n\nTables~\\ref{table_abund_sens_abs} and \\ref{table_abund_sens_rel}\ngive the sensitivity of the derived\nabsolute abundances [X\/H] and those relative to Fe [X\/Fe]\nfor small variations in the various relevant stellar and\nanalysis parameters.\n\n\n\n\n\n\\section{Alternative Choices of Stellar \n Parameters \\label{section_caspread} }\n\n\nThe choice of stellar parameters is crucial for determining\nabsolute stellar abundances, although less critical for abundance\nratios [X\/Fe] as many of the systematic effects cancel,\nas is shown by comparing the entries for each species in\nTable~\\ref{table_abund_sens_abs} with those in \nTable~\\ref{table_abund_sens_rel}. We have\ntherefore explored several methods of determining $T_{\\rm eff}$.\nWe call the $T_{\\rm eff}$ determined by the mean of that\ninferred from each available\ncolor ($V-I,~V-J,$ and $V-K_s$), the photometric $T_{\\rm eff}$. \nThe value of $T_{\\rm eff}$ adopted for the abundance analysis,\n $T_{\\rm eff}$(adopt), is initially\nset to $T_{\\rm eff}$(phot), with small adjustments permitted to improve\nthe Fe~I slopes and the ionization equilibrium as described above.\n\nAnother\n$T_{\\rm eff}$ can be determined by assuming that all the stars are\nRGB members of their cluster, and that all the cluster stars\nhave the same initial chemical inventory and age. Hence they must \nlie along a single old very metal-poor isochrone.\nHere we are using a single magnitude,\nchosen on the basis of its high measurement accuracy\nand discrimination along the isochrone,\nto determine $T_{\\rm eff}$(iso).\nThis removes the issue of random errors in colors,\nwhich is of particular concern for NGC~2419, with its large distance and\nhence faint RGB stars near the limit for 2MASS photometry.\nWith a large enough sample, the ``isochrone''\nitself can be defined from the mean locus of the stars in the cluster CMD,\nas was done in \\cite{cohen05a}, see also the large VLT\nstudy of \\cite{carretta10} and references therein.\nIn the present case, even if the HIRES sample in NGC~2419\nwere much larger, \nthe concern\nthat this particular GC may not be chemically uniform would \nlimit the applicability of such an approach.\nIf the isochrone chosen is appropriate\nfor the cluster,\nthen the mean ${\\Delta}[T_{\\rm eff}(phot) - T_{\\rm eff}(iso)]$ will be zero.\nIf the isochrone is close, but not exactly that of the cluster, then\nthe mean difference will be a constant.\n\nTable~\\ref{table_temp} gives the values of the adopted, photometric,\nand isochrone $T_{\\rm eff}$ for the sample of RGB stars in NGC~2419.\nThe rms $\\sigma$ about the mean for the set of available colors\nfrom the three we consider is\ngiven for the $T_{\\rm eff}$(phot) in parentheses. \nThe photometric $T_{\\rm eff}$ was adopted for the detailed\nabundance analysis for 3 of the 5\nstars in NGC~7099. For the other two, the differences are 18~K (0.8 $\\sigma$)\nfor S31 and 79~K (1.2 $\\sigma$) for S34. The isochrone and photometric\nvalues of $T_{\\rm eff}$ for this nearby GC are essentially identical.\nFor NGC~2419 that is not the case. The brightest star (S223)\nat the RGB tip is somewhat redder than would be expected from the isochrone.\n\n\n\\section{Inner vs Outer Halo }\n\nA comparison of the mean abundance ratios in the two globular\nclusters discussed here\nM30 (NGC~7099), at a distance of only 8~kpc, with the very distant\nouter halo\nglobular cluster NGC~2419 is given in Table~\\ref{table_abund_mean}\nand illustrated in detail in Fig.~\\ref{figure_abund}.\nThere is no substantive difference between the elemental abundance\nratios for luminous red giants in the outer halo cluster NGC~2419 and\nthe ratios of inner halo clusters. The inner halo abundance pattern \nis also\nobserved in other distant clusters: NGC~7492\n\\citep[$R_G \\sim 25$~kpc,][]{cohen05b}, the low luminosity GC Pal~3\n\\citep[$R_G \\sim 90$~kpc,][]{pal3}, and in coadditions of\nhigh-resolution, low-SNR spectra of 19 luminous giants in the very\ndistant, low-luminosity GC Pal~4 \\citep[$R_G \\sim 109$~kpc,][]{pal4}.\nThe low [C\/Fe] ratios seen among the NGC~2419 and M30 \nluminous giants are common\namong such stars in inner halo GCs \\citep[see, e.g.][]{m13c}, \npresumably due to depletion via deep mixing.\n\nWhen comparing Galactic GC abundances to those of dSph\nsatellites of the Galaxy, it has become apparent\nvia extensive surveys in recent years \n\\citep{cohen_draco, cohen_umi} and at moderate resolution\nthe extensive work of \\cite{kirby4} that\nthe metal-poor end of all known systems seems to\nhave, at least to first order, an identical chemical \ninventory. That should not be a surprise, since differences\nproduced through different star formation rates\nor gas flow history (accretion and\/or outflows) between the dSph satellites and\nthe Galactic GCs will only show up at late times. Initially\nonly the metals produced in and ejected by SNII\nenrich the system's gas, and hence,\nexcept for small metallicity dependent effects\non nucleosynthetic yields, all\nvery metal-poor stellar systems should have identical abundance\nratios. Those GCs with peculiar abundance ratios,\nincluding no enhancement of the $\\alpha$-elements, have\nintermediate [Fe\/H] and most are\nknown to be associated with the ongoing accretion of the Sgr dSph;\nexamples include\nPal~12 \\citep{cohen_pal12} with [Fe\/H] $\\sim -0.7$~dex, \nTerzan~7 \\citep{terzan7} with\n[Fe\/H] $-0.6$~dex, as well as\nRup~106 \\citep{rup106} with [Fe\/H] $-1.45$~dex, whose age is several\nGyr less than the bulk of the halo, suggesting an accretion origin.\n\n\n\n\\section{Comparison with Previously Published Abundance Analyses}\n\nThe only previously published high-dispersion analysis of any star in\nNGC~2419 is for the star S1305, which was observed by\n\\cite{shetrone01}\\footnote{\\cite{shetrone01} call this star RH~10.\nThe coordinates of this star, privately communicated from\n M.~Shetrone, match those of S1305.}. Our measurement of [Fe\/H] is\n0.13~dex higher than the value they obtained, which we ascribe\nlargely to our adopted $T_{\\rm eff}$ being 125~K\nhigher than theirs. The two sets of [X\/Fe] for all elements in \ncommon through atomic number 30 (Zn) are in good agreement\nexcept for Na. Na is a difficult case as the NaD lines are somewhat\ncorrupted by interstellar features at the cluster's $v_r$,\nwhile the 5682, 5688~\\AA\\ doublet used by \\cite{shetrone01}\nis very weak. Among the heavier elements detected in\nboth studies, only Y (in the form of Y~II)\nhas a large discrepancy. There are 3 detected lines from\nour spectra, and a claim of 4 from theirs, but the line \nat 4900.1~\\AA\\ is quite crowded, and perhaps should be\nignored. For the two Y~II lines in common, our $W_{\\lambda}$\nare $\\sim$20\\% smaller. Considering how faint this star is \n($V = 17.61$~mag) we regard the overall agreement\nin abundances between the two analyses to be very good.\n\n\\cite{shetrone03} observed one star in NGC~7099. Once\nthe difference in adopted solar Fe abundance is taken into\naccount, their derived [Fe\/H] for this star differs from\nthe mean for our sample of five stars by only 0.05~dex.\nThe abundance ratios agree satisfactorily in most cases,\nwith 7 elements differing in [X\/Fe] by less than 0.10~dex.\nTheir UVES observations cover only 4800 to 6800~\\AA,\nand so are missing many key blue lines for the rare earths,\nwhere the differences in relative abundances between\nthe two studies tend to be the largest. The choice of\nadopted transition probabilities, particularly for the rare earths, also contributes;\nwe adopted them from the recent studies of Lawler\nand collaborators, \n\\citep[see, e.g.][for Eu as an example]{lawler01} \nwhen available, while \\cite{shetrone03} tended to use older values.\n\nThe Padua group (see, e.g. Carretta et al 2009abc and\nCarretta et al 2010) has completed a large VLT study\nof the Na\/O anti-correlation in GCs. Although their survey does not\ninclude NGC~2419, they have a observed a\nsample of 10 RGB stars in M30 with UVES. Thus far they have published\nabundances for only the light elements. \nTable~\\ref{table_7099_comp} presents a comparison of our\nresults from HIRES at Keck with their UVES\/VLT study\nand highlights the good agreement between\nthese two completely independent detailed abundance\nanalyses for RGB stars in M30.\n\n\n\n\n\n\n\n\n\n\\section{Consistency with Our Previous Deimos Range in Ca Abundance \n\\label{section_ca_2419} }\n\n\n\n\nOur previous medium-resolution study of RGB stars in NGC~2419\n\\citep{deimos} uncovered a range in [Ca\/H] of a factor of \ntwo or three among the sample of 43 stars found to be definite\nmembers of this GC. The range in $V$ of these stars is from 17.4 to 19.3~mag.\nFig.~\\ref{figure_cah_hist} shows a histogram of [Ca\/H] for these\nstars, on which is superposed the distribution of the\ngiants in NGC~2419 with high quality HIRES spectra analyzed \nhere.\nThe two additional probable members discussed in \n\\cite{deimos}, each of which has\nsome remaining concern about membership, are indicated in this figure\nwith cross hatching. \n\n\nThe first point to note is that the histogram of [Ca\/H] for an unbiased\nsample of members of NGC~2419, the moderate resolution Deimos survey\nof \\cite{deimos}, is sharply peaked at [Ca\/H] $-1.95$~dex.\nWhile there is a rapidly declining tail extending towards higher metallicities,\nthe low metallicity peak dominates, and the fraction of stars\nin the extended tail is small. The minimum [Ca\/H] is $-2.0$~dex.\nOnly 13 (30\\%) of the 43 definite members lie outside the range\n$-2.0$ to $-1.85$~dex.\n\nOur sample of RGB stars in NGC~2419 with HIRES\nspectra good enough for a detailed abundance analysis \ncontains only 7 stars.\nThis sample was, to a large extent, selected by brightness before\nthe full analysis of the moderate resolution spectra was completed.\nOne of these 7 stars (S1131) has\n[Ca\/H] $-1.72$~dex from its HIRES spectrum ($-1.76$~dex from\nits Deimos Ca triplet region analysis) and a second star (S973) is\nmarginally above the $-1.85$~dex [Ca\/H] cutoff at\n$-1.81$~dex from our detailed abundance analysis\n($-1.93$~dex from its moderate resolution Deimos spectrum).\nThe others all are more metal-poor than [Ca\/H] $= -1.85$~dex as deduced\nfrom both their HIRES and Deimos spectra.\n\nThe total range in Ca and Fe abundance within the NGC~2419 sample\nis small (see Table~\\ref{table_ngc2419_abund}.\nWe used the three different choices of $T_{\\rm eff}$\ndiscussed in \\S\\ref{section_obs}\nand given in Table~\\ref{table_temp} as an indication of the maximum\nrange in this parameter that might be appropriate for each of the NGC~2419 \nstars\\footnote{We adjusted log($g$) appropriately in each case to maintain the star\non the RGB.}.\nIn effect, we carried out the abundance analysis three times for each star\nusing different sets of adopted stellar parameters.\nWe find that for each of these three choices, S1131 {\\it{always}} has\nthe highest value of [Ca\/H] of the 7 stars in the HIRES sample for NGC~2419,\n and it is ${\\sim}0.2$~dex (2 to 3 $\\sigma$) higher than\nthe mean value for the other 6 stars in this cluster.\nThe Fe abundance for S1131, as derived from either Fe~I or Fe~II,\nis always the highest or second highest as well, but is\nwithin 1~$\\sigma$ of the mean of the other cluster stars. \n\nThe uncertainties\nin absolute abundances (see Table~\\ref{table_abund_sens_abs})\nare daunting for such a small range in Ca abundance in such\na distant cluster with spectra and near-IR photometry\nthat are less than ideal. Lowering \nthe $T_{\\rm eff}$ of S1131 in NGC~2419 by $\\sim$100~K\nrelative to the other RGB stars in this cluster in our HIRES\nsample would eliminate its high Ca and Fe abundance.\nBut this requires a change \nof this one star relative to all the others which\nis about half of the\nentire range in $T_{\\rm eff}$ spanned by the 7 cluster giants,\nfor which there is no justification.\nGiven the small size of our HIRES sample,\nall we can say is that the Ca\/H distribution of the HIRES\nsample in NGC~2419 is consistent with that of the Deimos sample, which\ndoes appear to show a small but measurable range in Ca\/H among\nthe cluster members \\citep{deimos}.\n\n\nBoth of two probable additional members of NGC~2419 discussed\nin \\cite{deimos} from the Deimos sample, shown by cross hatching in\nFig.~\\ref{figure_cah_hist},\nhave been observed with HIRES. However observations were\nterminated after the first 30~min exposure revealed\nconcerns about cluster membership. Hence\nthe resulting HIRES spectra for these two stars are not good enough for a detailed\nabundance analysis.\nOne of these (S951) has a spectrum which appears to be that of a RGB\nstar in this very metal-poor GC, but with $v_r$ off from the cluster mean\nby $\\sim$15~km s$^{-1}$. This star has a value from an analysis of\nthe Ca triplet lines in its Deimos spectrum of\n[Ca\/H] of $-1.91$~dex, consistent with the bulk of the cluster\npopulation. The concern with the second star, S1673, is twofold; it lies\nslightly bluer than the main RGB locus of NGC~2419 in $V-I$\n(see Fig.~\\ref{figure_2419_cmd}) and its HIRES spectrum\nappears metal-rich. The [Ca\/H] inferred from its Deimos \nspectrum ( $-1.68$~dex)\nputs this star, if it is a member, firmly in the metal-rich\ntail of the histogram shown in Fig.~\\ref{figure_cah_hist}. \nSee \\cite{deimos} for further discussion\nregarding membership in NGC~2419 of these two stars.\n\nThere are 13 stars in the Deimos sample that may\nbe members of NGC~2419 with $V < 17.7$, which is slightly fainter\nthan the faintest star in the present HIRES sample analyzed here.\nIn addition to all 7 RGB stars in our present HIRES sample,\nboth\nof the two probable cluster members discussed above\nare brighter than that cutoff.\nThis leaves 4 stars brighter than $V = 17.7$~mag without\nHIRES observations at present,\n3 of which have [Ca\/H] $\\sim -1.9$~dex. Only one is in the high\nmetallicity tail for NGC~2419. \n\nOur present\nHIRES sample, which was selected primarily on the\nbasis of brightness given the 84~kpc distance to NGC~2419, \nis not ideal for exploring the star-to-star variation.\nThe small variations of [Ca\/H] or [Fe\/H] from the 7 stars with\nhigh dispersion spectra are only marginally larger than the\nuncertainties, but are broadly consistent with our\nDeimos results \\citep{deimos} which found a range\nin [Ca\/H] within NGC~2419. In the future we hope to provide a sample that better\nprobes the metal-rich tail of NGC~2419 as\ndefined by the larger Deimos sample of \\cite{deimos}.\n\n\n\n\\section{One Star with a Peculiar Abundance Pattern \\label{section_s1131} }\n\nOne star, NGC~2419 S1131,deviates from the\narchetypal GC abundance pattern. \n S1131 is\na definite member of NGC~2419 based on its \nHIRES radial velocity (its heliocentric $v_r$ is $-17.2$~km s$^{-1}$)\nand spectrum; see also \\cite{deimos}. As discussed above\nin \\S\\ref{section_ca_2419}\nour detailed abundance analysis\nsuggests that this star is slightly more metal-rich \nthan the other NGC~2419 giants\nin Fe and Ca as well as \nseveral other elements with many detected lines.\nIt has an\nenhancement of $\\sim 0.2$~dex in [Ca\/H] \n(with slight variation depending on the choice of\n$T_{\\rm eff}$ discussed in \\S\\ref{section_caspread}), corresponding\nto Ca enhanced by a factor of 1.6, compared to the mean \nof the other 6 NGC~2419 giants.\n\nIt is interesting to note that S1131 is the only one of the six\nNGC~2419 sample giants included in the moderate resolution study of\n\\cite{deimos} found to be Ca-rich. The remaining 6 NGC~2419 giants in\nthe HIRES sample all lie at the low Ca\nabundance end of the distribution inferred by \\cite{deimos}. \nHowever, given the small abundance\nrange involved, observations of additional giants believed\nto be more metal-rich than the bulk of the cluster stars\nare required for a definitive confirmation for star-to-star\nvariation of Ca, Fe, and other heavy elements within\nNGC~2419.\n\n\nFurthermore, NGC~2419 S1131 has some anomalous abundance ratios,\nirrespective of whether its $T_{\\rm eff}$ has been overestimated. It\nhas [Mg\/Fe] = $-0.47$~dex from five \\ion{Mg}{1} lines, in contrast\nto the mean of the other six stars,\n$+0.44\\pm0.06$~dex. In deriving the Mg abundance for S1131, the upper limit\nto the strength of the 5711~\\AA\\ line was considered as a detection,\nand the two unblended Mg triplet lines were retained. These two lines\ngive Mg abundances slightly higher than the mean, but the highest\nMg abundance is from the 5711~\\AA\\ line, 0.18~dex above the mean\nof the 5 lines used. Note that the first ionization potential of\nMg is only 0.25~eV lower than that of Fe. The [Mg\/Fe]\nmeasurement would increase by only 0.08~dex if $T_{\\rm eff}$ were\nreduced by 100~K (see Table~\\ref{table_abund_sens_rel}). \nThe regions of the spectra of two \\ion{Mg}{1} lines\nin this star and in S1209, also known as Suntzeff~16\n\\citep{suntzeff88}, with $T_{\\rm eff}$ lower than that of S1131 by\nonly 85~K, are shown in Fig.~\\ref{figure_spectra}. Although most GCs\nshow a small spread in [Mg\/Fe] \\citep{gratton_araa}, \ngenerally interpreted as the result of proton-burning\nof Mg into Al in\nintermediate mass AGB stars,the expected enhancement of Al does not \nappear to be present in S1131. The abundance pattern of this\nspecific star cannot be reproduced by proton-burning.\nFurthermore $-0.47$~dex is an\nunprecedentedly low [Mg\/Fe] abundance for a globular cluster star; the\nonly stars that have such low [Mg\/Fe] ratios are the halo star\nCS22966-043 \\citep{ivans03}, a hot SX~Phe variable, \nand stars in dSph satellite galaxies, such\nas COS~171 in Ursa Minor \\citep{cohen_umi} and S58 in Sextans\n\\citep{shetrone01}. NGC~2419\nS1131 is unique in that no other elements appear depleted, including\nSi. A very low value of Mg and super-solar values of other alpha\nelements are inconsistent with any Type~II SN yields\n\\citep[e.g.,][]{woosleyweaver,nomoto06,heger10}.\n\n\nIn addition, NGC~2419 S1131 also has an unusually high [K\/Fe] of\n$+1.13$~dex, also\nillustrated in Fig.~\\ref{figure_spectra}. This figure includes\nthe spectrum of a rapidly rotating B star to demonstrate that\nany telluric absorption at the wavelength of the K~I line,\ntaking into account the $v_r$ of this GC, is negligible.\nFurthermore, two other NGC~2419 sample stars were observed\non the same run (see Table~\\ref{table_sample}) and do not\nshow this abnormality.\nThis abundance ratio would\ndecrease by less than 0.10~dex if $T_{\\rm eff}$ were reduced by 200~K. \nThis is an internal comparison within our NGC~2419\n sample, which has a total range in $T_{\\rm eff}$ of less than 200~K.\nStudies of non-LTE corrections for the 7699~\\AA\\ \nresonance line of K~I\n(the only line used here) by \\cite{andrievsky10} and by \\cite{takeda_k_nonlte}\n suggest that the appropriate non-LTE corrections\nchange by only 0.15~dex over such a small range in $T_{\\rm eff}$,\nranging from about $-0.35$ to $-0.2$~dex.\nThus differential non-LTE corrections for K~I within our HIRES \nsample of luminous RGB stars are much too small to explain the\ndeviant potassium abundance of S1131.\n\n\\cite{takeda_k_nonlte} very recently found a red giant in M13 and another\nin M4 with similarly high [K\/Fe]. They ascribe the unexpectedly\nstrong K~I resonance lines to an increase in activity or turbulent\nvelocity fields high in the stellar atmosphere where \nthe core of such a strong line is formed.\nHowever, while S1131 does show emission on both the red and blue wings\nof H$\\alpha$, the emission wings of H$\\alpha$ \nin the spectrum of NGC~2419 S223 are considerably stronger,\nyet this star does not show an anomalously strong K~I line.\nFurthermore if this explanation is correct, \nother strong resonance lines arising from neutral species\nof elements with (low) first ionization potentials comparable \nto that of K should be similarly affected. Unfortunately\nthe NaD lines, the most likely candidates to check, are\nbadly corrupted\nby interstellar absorption due to the radial velocity of NGC~2419.\n\n\nA careful inspection of Fig.~\\ref{figure_spectra} \nstrongly suggests, by comparison with the spectrum of NGC~2419\nS1209, which has a $V$ magnitude only 0.2~mag brighter than S1131, \nthat S1131 is indeed\nslightly more metal-rich in other species as well as K. \n\nThe Li~I blend at 6707~\\AA\\ was too weak to be detected\nin any of the NGC~2419 stars. The upper limit for the equivalent\nwidth of this feature of 7~m\\AA\\ in\nS1131 corresponds to log[$\\epsilon$(Li)]\nof $-0.01$~dex, indicates extensive depletion from the Li production \nin the Big Bang, typical of red giants, which have extensive\nsurface convection zones.\n\n\nThe stellar parameters, spectra, and\nanalyses have been carefully checked and these specific differences in\nabundance ratios discussed here (i.e. for Mg and for K) \nwithin the NGC~2419 sample are real. \n\n\n\n\\section{Summary}\n\nOur key result from the analysis of 7 luminous giant members of the\ndistant outer halo globular cluster NGC~2419 is that one of these\nstars is extremely peculiar, having a very low [Mg\/Fe] ratio, but\nbeing normal in all other element ratios except for a high [K\/Fe].\nSimilarly low [Mg\/Fe] ratios are seen in stars in dSph satellites of the\nGalaxy, specifically in Ursa Minor \\citep{cohen_umi} and in Sextans\n\\citep{shetrone01}. But in both cases these stars have low values of\nother $\\alpha$ elements, which is not the case for NGC~2419 S1131.\nJ.~Cohen has examined and analyzed spectra of $\\gtrsim$100 stars in\nGalactic GCs, and has never seen one similar to S1131 in NGC~2419.\n\nFurthermore, the HIRES spectra support the small range in\nmetallicity within the giants in NGC~2419 found by \\cite{deimos},\nwhich might suggest that this distant outer halo GC is\nthe remnant of an accreted dwarf galaxy. A definitive result\nwill require further observations of the faint upper RGB\nstars in this GC, concentrating on those whose moderate\nresolution Deimos spectra from \\cite{deimos}\nsuggest they are more metal-rich than\nthe bulk of the stars in this cluster, which will be carried out\nin a future campaign.\n \n\nIgnoring the limited peculiarities of S1131, we find that there is\nno substantive difference between the mean behavior of the abundance\nratios for six other red giants in NGC~2419 and those ratios characteristic of\nthe inner halo despite the structural peculiarities of NGC~2419. \nThe influence of star formation rates\nand other aspects of the detailed chemical evolution of a stellar\nsystem only affect such ratios once star formation\nhas been underway long enough for sources other than SNII to\nbecome important contributors, which does not seem to be the case\nin very metal-poor systems such as NGC~2419, nor in the\nmost metal-poor stars in the dSph satellites of the Milky Way\n(Cohen \\& Huang 2009, 2010; Kirby et al 2011).\n\nThus, like the previous studies based on detailed\nabundance analyses of individual or co-added stellar spectra of stars\nin distant outer halo clusters, the chemical inventory of the globular\ncluster system appears to be independent of location. With the\nexception of those clusters known to be associated with the\ncurrently disrupting Sgr dSph galaxy, every globular cluster\nexhibits the same trends as a function of overall cluster\nmetallicity---parameterized by [Fe\/H]---irrespective of kinematics or\nlocation within the halo. \n\nThe existence of a metallicity gradient in the Galactic halo is\ncontroversial. \\citet{carollo07} found evidence for a break in the\nmetallicity distribution of halo field stars in the sense that the\nmore distant stars are more metal-poor; this conclusion is\nsupported by \\cite{rix10}. Furthermore, \\cite{carollo07} showed that\nthe metallicity within the outer halo population displays a gradient\ntoward even lower metallicities for the most distant stars or stars\nshowing the highest magnitude of rotation retrograde to the Milky Way\ndisk. However, both \\citet{sesar11} and \\citet{schoenrich10} improved\nupon \\citeauthor{carollo07}'s study. These new works,\nwhich are disputed by \\cite{beers11}, found no\nevidence for a dichotomous halo, particularly in regard to the\nkinematics and density profile. Our measurements for NGC~2419 \nadditionally provide\nevidence against a dichotomy in detailed abundances. The outer halo\nglobular clusters show the same bulk abundance\nproperties---and, by extension, (short) star formation timescales---as the\ninner halo globular clusters.\n\n\n\n\\acknowledgements\n\n\nWe are grateful to the many people who have worked to make the Keck\nTelescope and its instruments a reality and to operate and maintain\nthe Keck Observatory. The authors wish to extend special thanks to\nthose of Hawaiian ancestry on whose sacred mountain we are privileged\nto be guests. Without their generous hospitality, none of the\nobservations presented herein would have been possible. We thank \nAlex Heger and Ken Nomoto for helpful conversations. J.G.C.\nand W.H. thank NSF grants AST-0507219 and\nAST-0908139 for partial support. Work by E.N.K. was supported by NASA\nthrough Hubble Fellowship grand HST-HF-01233.01 awarded to ENK by the\nSpace Telescope Science institute, which is operated by the\nAssociation of Universities for Research in Astronomy, Inc., for NASA,\nunder contract NAS 5-26555. \n\n\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzlrpo b/data_all_eng_slimpj/shuffled/split2/finalzzlrpo new file mode 100644 index 0000000000000000000000000000000000000000..a1eb7a2cd140edc36edb9585ffe426baab026afa --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzlrpo @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nSome people believe that Twitter and Blogs are enthusiastically posted, but most people are supposed to search on the Internet. Therefore, analysis of the search behavior of people of society is very important in grasping social movements. In this research, we analyze theoretically by the idea of social physics, how television information, net information, or Blog, Twitter etc. influences people's search behavior.\n\n\nThe subjects of the analysis are seasonally limited dates, and many people participate in the event. Specifically, Valentine's Day, Halloween, Christmas, and New Year's Countdown. In addition, in the habit peculiar to Japan, we also analyzed Eho-maki, which eats scroll sushi on February 3, and an event to eat eel on the Midsummer Occasion Day of Summer (the day of ox in midsummer). In particular, Specifically, Valentine's Day, Halloween, Christmas, and New Year's Countdown last for only one day and thus, it is easier to analyze the rising excitement and subsequent wane in interest. In many European countries, people return to their homelands for Christmas and enjoy a long Christmas break. In constast, in Japan, however, Christmas is a one-day event in which young people have Christmas parties with friends but do not return home. In Japan, therefore, it is not until the New Year's vacation of the following week that people return to their hometowns. Thus, in Japan, Christmas is very similar to Halloween.\n\n\nIn Japan, Feb.3 is the traditional day to eat \"Eho-maki\". Eho-maki are thick sushi rolls \nshown in fig.\\ref{eho-maki} which is believed to bring good fortune if eaten while facing the year's \"Eho\" (good luck direction of the year) like fig.\\ref{eho-maki2}. Not all, but many Japanese people have Eho-maki at this day.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6cm]{eho-maki.eps}\n\\caption{Eho-maki in Japan.}\n\\label{eho-maki}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6cm]{eho-maki2.eps}\n\\caption{How to eat eho-maki in Japan.}\n\\label{eho-maki2}\n\\end{center}\n\\end{figure}\n\n\n\nOn the day of the ox in midsummer Japanese have a custom to eat eel which started in the Edo period, 18th century. Eel is a popular food for Japanese people, and it is expensive, so eating eel on the day of the ox in midsummer once a year is a big concern. On the day of the ox in midsummer Japanese have a custom to eat eel which started in the Edo period. The day of the ox, which is named after one of the twelve animals of the Chinese zodiac. According to one legend, long before the scientific reasons were established, in the 1700s well-known scholar Gennai Hiraga came up with the custom as part of a marketing ploy to boost limp summer sales when the owner of a struggling eel store asked the wise man for some business advice. In fig.\\ref{unagi}, we show the typical cooking of eel in Japan for the day of the ox in midsummer.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=6cm]{unagi.eps}\n\\caption{Typical cooking of eel in Japan.}\n\\label{unagi}\n\\end{center}\n\\end{figure}\n\n\nThus, this study examines the peaks and falls in interest in these time-limited events, focusing on the medium used to perform searches and on what cohort of the population perform those searches. The mathematical model of search behavior is used for the analysis\\cite{Ishii2018,Okano} .\n\n\n\\section{Theory}\n\n\nIn the theory of search behavior \\cite{Ishii2018,Okano} , the interest and concern on a certain topic can be calculated using a mathematical model of differential equations. Here, we introduce I(t) as the interest or concern on a certain topic. We construct a mathematical model based on the mathematical model for the hit phenomenon within a society presented as a stochastic process of interactions of human dynamics in the sense of many body theory in physics \\cite{Ishii2012a,Ishii2017}. As in the model in \\cite{Ishii2012a,Ishii2017}, we assume that the intention of humans in a society is affected by the three factors: advertisement, communication with friends, and rumors. Advertisements act as external forces; communications with friends are a form of direct communication and its effect is considered as interaction with the intention of friends. The rumor effect is considered as the interaction among three persons and a form of indirect communication as described. In the model, we use only the time distribution of advertisement budget as an input, and word-of-mouth (WOM) represented by posts on social network systems is the observed data for comparison with the calculated results. The parameters in the model are adjusted by the comparison with the calculated and observed social media posting data. \n\nAccording to \\cite{Ishii2012a,Ishii2017}, we write down the equation of purchase intention at the individual level $I_i(t)$ as\n\n \n \\begin{equation}\n\\frac{dI_i(t)}{dt} = \\sum_{\\xi} c_{\\xi}A_{\\xi}(t) - a I_i(t) + \\sum_j d_{ij} I_j(t) + \\sum_j \\sum_k p_{ijk} I_j(t) I_k(t)\n \\end{equation}\n\n\nwhere t is the time, $d_{ij}$, $p_{ijk}$, and $f_i(t)$ are the coefficient of the direct communication, the coefficient of the indirect communication, and the random effect for person i, respectively\\cite{Ishii2012a}. The advertisement and publicity effects are include in $A_{\\xi}(t)$ which is treated as an external force. The index $\\xi$ means sum up of the multi media exposures. \nWord-of-mouth (WOM) represented by posts on social network systems like blog or twitter is used as observed data which can be compared with the calculated results of the model. The unit of time is a day. \n\nHere, it is assumed that the height of interest $I(t)$ of people attenuates exponentially. Although it is known that this is known to occur in movies and the like \\cite{Ishii2012a}, attention such as events and anniversaries is known to attenuate by a power function. \\cite{Sano2013a, Sano2013b} In the case of social interest, we attenuate the intermediate between the exponential function and the power function \\cite{Ishii-Koyabu}, but here we simply adopt exponential decay.\n\nWe consider the above equation for every consumers in the society. Taking the effect of direct communication, indirect communication, and the decline of audience into account, we obtain the above equation for the mathematical model for the hit phenomenon. Using the mean field approximation, we obtain the following equation as equation for averaged intention in the society. The derivation of the equation is explained in detail in ref.\\cite{Ishii2012a}. \n\n\\begin{equation}\n\\label{eq:eq13}\n\\frac{dI(t)}{dt} = \\sum_{\\xi} c_{\\xi}A_{\\xi}(t) + (D-a) I(t) + P I^2(t)\n \\end{equation}\n\nThis equation is the macroscopic equation for the intention of whole society. Using this equation, our calculations for the Japanese motion picture market have agreed very well with the actual residue distribution in time \\cite{Ishii2012a}. The advertisement and publicity effects are obtained from the dataset of M Data and the WOM represented by posts on social network systems are observed using the system of Hottolink. We found that the indirect communication effect is very significant for huge hit movies. \n\n\\subsection{Extension to include Twitter and Blog} \n\nIn the new mathematical model for search behavior, we use daily blog and Twitter postings as the external force. Therefore, we extend the above mathematical model for hit phenomena to include the effects of Twitter and blog as external field as follows.\n\n\\begin{eqnarray}\n\\label{eq:eq14}\n\\frac{dI(t)}{dt} &=& C_{TV}A_{TV}(t) \\nonumber \\\\\n&+& C_{NetNews}A_{NetNews}(t) + C_{Twitter}A_{Twitter}(t) \\nonumber \\\\\n&+& C_{blog}A_{blog}(t) + (D-a) I(t) + P I^2(t)\n\\end{eqnarray}\n\nIn the above equation(\\ref{eq:eq14}), $I(t)$ is the intention to search a certain topic using Google Trend and $C_{TV}$, $C_{NetNews}$, $C_{Twitter}$ and $C_{blog}$ correspond to the strength to influence willingness to search the certain topic. $(D-a) I(t) + P I^2(t)$ correspond to the direct and indirect communications in the previous mathematical model for hit phenomena. In the model of the present paper, these terms correspond to the direct and indirect effect of searching the topic by other people. \n\nOn the real calculation, we use advertisement time data on TV from M Data co. ltd. and the internet news site data, daily Twitter posting data and daily blog posting data on the certain topic from Hottolink co.ltd. The daily search data comes from Google Trend as the reference of our calculation. \n\nThe parameters $C_{TV}$, $C_{NetNews}$, $C_{Twitter}$, $C_{blog}$, $D$ and $P$ are determined in similar way as the previous model by using the metropolis-like Mote Carlo method as noted in the previous paper\\cite{Ishii2012a, Ishii2017}. We define here \"R-factor\" to check the correctness of the adjustment of parameters.\n\n\\begin{equation}\nR = \\frac{\\sum_i (f(i)-g(i))^2}{\\sum_i (f(i)^2 + g(i)^2)},\n\\end{equation}\n\nwhere $f(i)$ and $g(i)$ correspond to the calculated $I(t)$ and the observed number of Google Trend data. The R-factor is originally defined by J B Pendry\\cite{Pendry} to adjust the positions of atoms on surface in the calculation of low energy electron diffraction experiment where measured electric current - voltage curve compared with the corresponding calculation. The smaller the value of R, the better the functions $f$ and $g$. Thus, we use a random number to search for the parameter set that minimizes R. This random number technique is similar to the Metropolis method\\cite{Metropolis}, which we have used previously\\cite{Ishii2012a}. We use this R-factor as a guide to obtain the best parameters for each calculation in this paper.\n\nWe employ the Monte Carlo method like Metropolis method\\cite{Metropolis} to fine the minimum of R. This is very similar for finding the minimum of total energy in the first principle calculation. In the real calculation to adjust the parameters $C_x$, $D$, and $P$, we should take care of the local minimum trapping like the first principle calculation in material physics. It is well-known that there are several ways to find the minimum condition like the steepest descent, the equation of motion method and the conjugate gradient method. Even in the actual calculation of the first principle calculation or the density functional theory, we should be careful of the danger of local minimum trapping. In this paper, the way we employ is just do the calculation using the several initial value in the Metropolis-like method to avoid the local minimum trapping. To check the accuracy of the parameters adjusting, we use the R-factor value. For every calculation which we show in this paper, the R-factor is below 0.01. \n\nActually, the parameters $C_{TV}$, $C_{NetNews}$, $C_{Twitter}$, $C_{blog}$, $D$ and $P$ in equation (\\ref{eq:eq14}) can be considered as functions of time, because people's attention changes over time. However, if we introduce the functions $C_{TV}(t)$, $C_{NetNews}(t)$, $C_{Twitter}(t)$, $C_{blog}(t)$, $D(t)$ and $P(t)$, we can tune any phenomena by adjusting these functions. Thus, we retain $C_{TV}$, $C_{NetNews}$, $C_{Twitter}$, $C_{blog}$, $D$ and $P$ as constant values to examine whether equation (\\ref{eq:eq14}) can be applied to any social phenomena.\n\n\\section{Data}\n\nFor the investigation of this article, we should use Google Trends data for the target data of our calculation. The data of Google Trends can be obtained on the Google Trends page. \n\nThe daily posting number to Twitter and Blog are obtained from the service \"Kuchikomi Kakaricyo\" of Hottolink co.ltd. The mass media advertisement data can be obtained from M Data co.ltd. via the \"Kuchikomi Kakaricyo\" service. \n\n\n\n\\section{Result}\n\nAnalyze the events 1 month before and 1 month after the events, and calculate each coefficients $C_{TV}$, $C_{NetNews}$, $C_{Twitter}$, $C_{blog}$, $D$ and $P$ so that the calculated $I(t)$ is combined with the measured value by Google Trend to best match.\n\nThe calculation results of all coefficients for Christmas are shown in the fig.\\ref{christmas}. For Christmas, the calculation results of all coefficients for the first month before Christmas and one month after Christmas are shown in the fig.\\ref{christmas}.\n\nIn particular, here we examine the difference whether search behavior is affected by Blog or influenced by Twitter. Shown in the fig.\\ref{christmas-blogtwitter} are $C_{blog}$ and $C_{twitter}$ values before and after Christmas. Looking at the results, Twitter's influence is big before Christmas and blog influence is big after Christmas.\n\nIn the same way, the results calculated for Halloween, New Year's countdown, Valentine's Day are shown in the fig.\\ref{halloween-blogtwitter}, fig.\\ref{countdown-blogtwitter} and fig.\\ref{valentine-blogtwitter}, respectively. For Holloween and Valentine's Day, we analyze the events 1 month before and 1 month after the events. For New Year's countdown, we analyze the event 1week before and 1 week after the event in order to avoid the effect of the Christmas before Dec.24. From these, it is possible to read the common tendency that Twitter influences strongly before the event, and after the event the effect of Blog is strong.\n\nAs mentioned above, qualitatively, Halloween, Christmas, Countdown and Valentine's Day are in agreement that it is influenced by Twitter before the event and after the event is affected by Blog.\n\nNext, we introduce the calculation results of Eho-maki and Eel of the Midsummer's Day eating foods decided on a fixed date in fig.\\ref{eho-maki-blogtwitter} and \\ref{unagi-blogtwitter}. As you can see from the results, the influence of Blog and Twitter on people's search behavior is opposite to the previous examples. In the case of this foods, it is from Blog that is influenced before the event and it is influenced by Twitter after the event. The qualitative behavior is same for the two event foods.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{christmas.eps}\n\\caption{Result of $D$, $P$, $C_{TV}$, $C_{NetNews}$, $C_{blog}$, $C_{twitter}$ in calulation for Christmas.}\n\\label{christmas}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{christmas-blogtwitter.eps}\n\\caption{Result of calulation of $C_{Twitter}$ and $C_{blog}$ before and after Christmas.}\n\\label{christmas-blogtwitter}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{halloween-blogtwitter.eps}\n\\caption{Result of calulation of $C_{Twitter}$ and $C_{blog}$ before and after Halloween.}\n\\label{halloween-blogtwitter}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{countdown-blogtwitter.eps}\n\\caption{Result of calulation of $C_{Twitter}$ and $C_{blog}$ before and after New Year Countdown. The analysis is done for a week before the Christmas, a week before the New Year's Day, a week after the New Year's Day and a week after January 8. }\n\\label{countdown-blogtwitter}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{valentine-blogtwitter.eps}\n\\caption{Result of calulation of $C_{Twitter}$ and $C_{blog}$ before and after Valentine's Day.}\n\\label{valentine-blogtwitter}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{Eho-maki-blogtwitter.eps}\n\\caption{Result of calulation of $C_{Twitter}$ and $C_{blog}$ before and after the Eho-maki.}\n\\label{eho-maki-blogtwitter}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{unagi-blogtwitter.eps}\n\\caption{Result of calulation of $C_{Twitter}$ and $C_{blog}$ before and after the day of the ox in midsummer.}\n\\label{unagi-blogtwitter}\n\\end{center}\n\\end{figure}\n\n\\section{Discussion}\n\nWe analyzed Christmas, Halloween, New Year Countdown, Valentine's Day, Ewaki Roll, and Eel of the Midsummer Day of the ox as events with limited time. Looking at the analysis results, the results were divided for four of Christmas, Halloween, Countdown, Valentine's Day, and the two of Eho-maki and Eel. \n\nThese two groups are thought to depend on surprises or happening for the event and its preparation. In Halloween, there are many matters to investigate beforehand, such as what kind of costumes they themselves, what kind of costumes they are going to do, what kinds of costumes will be popular this year. At Christmas in Japan, there are not many things to consider beforehand, such as what kind of surprise there are lovers to book a Christmas dinner with. In the New Year 's countdown, since the taste of the countdown is different for each gathering, it is necessary to gather information in advance according to which place to go to, which event to go to and which bar to go to. At Japanese Valentine's Day, women collect information in advance, whether women make their own chocolate for her lover or they purchase high-end chocolate at some famous brand shop. \n\nMeanwhile, there are few kinds of Eho-maki to eat as an event, and there is no element to look into in advance as there are also decided how to eat. Also, there is no surprise when eating Eho-maki. As for the Eel to eat on the Midsummer Day of the ox, as shown in the fig.\\ref{unagi}, the method of cooking has been decided traditionally. It is impossible to make home made, so Japanese people have to eat Eel at a restaurant. Therefore, the information to be checked in advance is the only restaurant to eat. Besides, there is not much difference in Eel's cuisine for each restaurant.\n\nIn this way, events that need to check information sufficiently in advance are affected by Twitter. On the contrary, in the case of an event where there is no surprise and there is no need to check in advance, the effect of Twitter at the prior stage is small, and the influence of Blog is expected to be relatively high. \n\nTherefore, if search actions that are strongly influenced by Twitter are observed beforehand, those who come to the event are collecting information. For those people, event-related marketing will be effective. In this way, the method of this research can be applied to marketing.\n\n\\section{Conclusion}\n\nPosting on Blog and Twitter is an act performed by some people in society. On the other hand, search behavior is used by most people who use the Internet. Therefore, analysis of search behavior on the Internet is very important in social analysis in that it can also target people without voice.\n\nIn this research, we have analyzed such search behavior on the Internet using mathematical model of search behavior. As an object, we observed preliminary excitement and post cool down at the event to be held in a short time. The events analyzed are Halloween, Christmas, Countdown, Valentine's Day, Eho-maki, Eel of the Midsummer Day of the ox in Japan. According to the analysis, it turned out that these are divided into events requiring preparatory preparations for surprises, and events not being prepared. Twitter has a strong influence on events requiring advance preparations for surprises. In other events the effect of Twitter is after the incident.\n\nThis research result is expected to be applicable to marketing.\n\n\\section*{Acknowlegement}\n\nThe authors are grateful for helpful discussion to Yasuko Kawahata of Gunma University, Japan.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Background and Scope}\nAndroid applications are developed in Java and then compiled to a custom bytecode format called \\emph{Dalvik}, which is run by the Dalvik Virtual Machine (DVM). Unlike Java VMs, which are stack machines, the DVM adopts a register-based architecture.\nAndroid applications are different from standard Java programs, since\nthey are structured in \\emph{components} of four different types:\nactivities, services, content providers and broadcast\nreceivers~\\cite{Android}. These components represent distinct entry\npoints of the Android framework into the application. Hence, the\noperational behaviour of an Android application does not simply amount\nto the sequential execution of its bytecode implementation, but it\nheavily relies on callbacks from the Android framework, as a reaction\nto user inputs, system events, or inter-component\ncommunication. Different Android components, either in the same\napplication or from different applications, can communicate by\nexchanging \\emph{intents}, i.e., dictionary-like messaging\nobjects. Intents may be sent either to a specific component\n(\\emph{explicit} intents) or to any component which declares the will\nof providing a given functionality (\\emph{implicit} intents). \n\nIn our formal model we consider Android applications consisting of\nactivities only. We focus on activities, since a tested semantics is\navailable for them and because they exhibit the most complicated\nlife-cycle among all the component types~\\cite{PayetS14}. Also, we\nonly model intra-application communication based on explicit\nintents: implicit intents are mostly, if not only, used for inter-application\nmessages. As we discuss in Section~\\ref{sec:experiments}, {$\\mu\\text{-Dalvik}_{A}$}\\xspace does not\ncover all the Android features supported by {HornDroid}\\xspace{}: the purpose of {$\\mu\\text{-Dalvik}_{A}$}\\xspace is\nensuring that the design principles at the core of {HornDroid}\\xspace{} are sound and\nthat most of the Android-specific subtleties have been taken into due account. \n\n\\subsection{Syntax}\nWe write $(r_i)^{i \\leq n}$ for the sequence $r_1,\\ldots,r_n$. If the length of the sequence is immaterial, we just write $r^*$ and we still let $r_j$ stand for its $j$-th element. We represent the empty sequence with a dot ($\\cdot$). We let $r^*[j \\mapsto r']$ be the sequence obtained from $r^*$ by replacing its $j$-th element with $r'$. A \\emph{partial map} is a sequence of key-value bindings $(k_i \\mapsto v_i)^*$, where all the keys $k_i$ are pairwise distinct. Given a partial map $M$, let $\\textit{dom}(M)$ stand for the set of its keys and let $M(k) = v$ whenever the binding $k \\mapsto v$ occurs in $M$. We identify partial maps which are identical up to the order of their key-value bindings.\n\nTable~\\ref{tab:dalvik} provides the syntax of {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} programs. It is an extension of the original $\\mu$-Dalvik syntax~\\cite{JeonMF12} with a few additional statements modelling method calls to Android APIs used for inter-component communication.\n\n\\begin{table}[t]\n\\[\\begin{array}{lll}\nP & ::= & \\mathit{cls}^* \\\\\n\\mathit{cls} & ::= & \\cls{c}{c'}{c^*}{\\mathit{fld}^*}{\\mathit{mtd}^*} \\\\\n\\mathit{\\tau_{prim}} & ::= & \\type{bool} ~|~ \\type{int} ~|~ \\dots \\\\\n\\tau & ::= & c ~|~ \\mathit{\\tau_{prim}} ~|~ \\arrtype{\\tau} \\\\ \n\\mathit{fld} & ::= & f: \\tau \\\\\n\\mathit{mtd} & ::= & \\meth{m}{\\methsign{\\tau^*}{\\tau}{n}}{\\mathit{st}^*} \\\\\\\\\n\\mathit{st} & ::= & \\goto{\\mathit{pc}} \\\\\n & | & \\move{\\mathit{lhs}}{\\mathit{rhs}} \\\\\n & | & \\ifbr{r_1}{r_2}{\\mathit{pc}} \\\\\n & | & \\unop{r_d}{r_s} \\\\\n & | & \\binop{r_d}{r_1}{r_2} \\\\\n & | & \\new{r_d}{c} \\\\\n & | & \\newarray{r_d}{r_l}{\\tau} \\\\\n & | & \\checkcast{r_s}{\\tau} \\\\\n & | & \\instanceof{r_d}{r_s}{\\tau} \\\\\n & | & \\invoke{r_o}{m}{r^*} \\\\\n & | & \\sinvoke{c}{m}{r^*} \\\\\n & | & \\texttt{return} \\\\\n & | & \\newintent{r_i}{c} \\\\\n & | & \\putextra{r_i}{r_k}{r_v} \\\\\n & | & \\getextra{r_i}{r_k}{\\tau} \\\\\n & | & \\startact{r_i}\n\\\\\\\\\nr & \\in & \\textit{Registers} \\\\\n\\mathit{pc} & \\in & \\mathbb{N} \\\\\n\\oplus & ::= & + ~|~ - ~|~ \\dots \\\\\n\\odot & ::= & - ~|~ \\neg ~|~ \\dots \\\\\n\\varolessthan & ::= & < ~|~ > ~|~ \\dots \\\\\n\\mathit{prim} & ::= & \\mathtt{true} ~|~ \\mathtt{false} ~|~ \\dots \\\\\n\\mathit{lhs} & ::= & r \\\\\n& | & r[r] \\\\\n& | & r.f \\\\\n& | & c.f \\\\\n\\mathit{rhs} & ::= & \\mathit{lhs} \\\\\n& | & \\mathit{prim}\n\\end{array}\n\\]\n\\caption{{$\\mu\\text{-Dalvik}_{A}$}\\xspace{} Syntax}\n\\label{tab:dalvik}\n\\end{table}\n\nA {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} program $P$ is a sequence of classes $\\mathit{cls}^*$, which in turn are defined by a class name $c$, a direct super-class $c'$, some implemented interfaces $c^*$, and a number of fields $\\mathit{fld}^*$ and methods $\\mathit{mtd}^*$. Field declarations $f: \\tau$ include the field name $f$ and its type $\\tau$, while method declarations $\\meth{m}{\\methsign{\\tau^*}{\\tau}{n}}{\\mathit{st}^*}$ include the method name $m$, the argument types $\\tau^*$, the return type $\\tau$, and the method body $\\mathit{st}^*$. The annotation $n$ on top of the arrow tracks the number of local registers used by the method, which is statically known in Dalvik.\n\nWe briefly discuss below the statements of the language. An\nunconditional branch $\\goto{\\mathit{pc}}$ sets the program counter to\n$\\mathit{pc}$. The statement $\\move{\\mathit{lhs}}{\\mathit{rhs}}$ moves the right-hand side\n$\\mathit{rhs}$ into the left-hand side $\\mathit{lhs}$: here, $\\mathit{lhs}$ may be a register\n$r$, an array cell $r_1[r_2]$, an object field $r.f$, or a static field\n$c.f$; $\\mathit{rhs}$ may be any of these elements or a constant. A\nconditional branch $\\ifbr{r_1}{r_2}{\\mathit{pc}}$ compares the content of two\nregisters $r_1$ and $r_2$ using the comparison operator $\\varolessthan$ and\nsets the program counter to $\\mathit{pc}$ if the check is successful,\notherwise it moves to the next instruction. We then have unary and\nbinary operations, represented by $\\unop{r_d}{r_s}$ and\n$\\binop{r_d}{r_1}{r_2}$ respectively, where $r_d$ is the destination\nregister where the result of the operation must be stored and the\nother registers contain the operands. Object creation is modelled by\n$\\new{r_d}{c}$, which creates an object of class $c$ and stores a\npointer to it in $r_d$; array creation is similarly handled by\n$\\newarray{r_d}{r_l}{\\tau}$, where $r_d$ is the destination register\nwhere the pointer to the new array must be stored, $r_l$ contains the\narray length and $\\tau$ specifies the type of the array cells. The\ntype cast statement $\\checkcast{r_s}{\\tau}$ checks whether the\nregister $r_s$ contains a pointer to an object of type $\\tau$ and it\nmoves to the next instruction if this is the case, otherwise it stops\nthe execution\\footnote{The corresponding Dalvik opcodes would raise an\nexception, but we do not model exceptions in our formalism.}. The\nstatement $\\instanceof{r_d}{r_s}{\\tau}$ stores $\\mathtt{true}$ in $r_d$ if\n$r_s$ points to an object of type $\\tau$, otherwise it stores\n$\\mathtt{false}$. A method invocation $\\invoke{r_o}{m}{r^*}$ calls the method\n$m$ on the receiver object pointed by $r_o$, passing the values in the\nregisters $r^*$ as actual arguments. The invocation of static methods\nis modelled by $\\sinvoke{c}{m}{r^*}$. The $\\texttt{return}$ statement has no\nargument, rather there is a special register $r_{\\mathit{ret}}$ for holding\nreturn values: the return value must be moved to $r_{\\mathit{ret}}$ by the\ncallee before calling $\\texttt{return}$. \n\nThe last four statements are used to model inter-component\ncommunication. Intent creation is modelled by $\\newintent{r_i}{c}$,\nwhich creates an intent for the activity $c$ and stores a pointer to\nit in $r_i$. The statement $\\putextra{r_i}{r_k}{r_v}$ adds to the\nintent pointed by $r_i$ a new key-value binding $k \\mapsto v$, where\n$k$ and $v$ are the contents of $r_k$ and $r_v$ respectively. The\nstatement $\\getextra{r_i}{r_k}{\\tau}$ retrieves from the intent\npointed by $r_i$ the value bound to key $k$, where $k$ is the content\nof $r_k$, provided that this value has type $\\tau$. Finally,\n$\\startact{r_i}$ sends the intent pointed by $r_i$, thus starting a\nnew activity. Throughout the paper, we only consider \\emph{well-formed} programs.\n\n\\begin{definition}\nA program $P$ is \\emph{well-formed} iff: (1) all its class names are pairwise distinct, (2) for each of its classes, all the field names are pairwise distinct, and (3) for each of its classes, all the method names are pairwise distinct.\n\\end{definition}\n\nNotice that the last condition of the definition above is not restrictive, since overloading resolution is performed at compile time in Java~\\cite{JavaSpec} and Dalvik bytecode thus identifies methods through their signature, rather than their name. In our formalism, we then suppose that method names are tagged with some distinctive information drawn from their signature, so that we can identify each method of a given class just by its name. Notice that two different classes can still define two methods with the same name, which is important to model dynamic dispatching.\n\nFrom now on, we focus our attention on some well-formed program $P = \\mathit{cls}^*$. Most of the definitions we present in the paper depend on $P$, but we do not make this dependence explicit in the notation to keep it lighter.\n\n\\subsection{Dalvik Semantics}\n\\label{sec:dalvik}\nTable~\\ref{tab:dalvik-domains} defines the semantic domains employed by the operational semantics of {$\\mu\\text{-Dalvik}_{A}$}\\xspace. Values include primitive values and \\emph{locations}, i.e., pointers to heap elements extended with an annotation $\\lambda$. Annotations have no semantic import and are only needed for our static analysis: we will discuss their role in Section~\\ref{sec:statics}. \n\n\\begin{table*}[htb]\n\\begin{mathpar}\n\\begin{array}{llcl}\n\\text{Pointers} & p & \\in & \\textit{Pointers} \\\\\n\\text{Program points} & \\mathit{pp} & ::= & c,m,\\mathit{pc} \\\\\n\\text{Annotations} & \\lambda & ::= & \\mathit{pp} ~|~ c ~|~ \\astart{c} \\\\\n\\text{Locations} & \\ell & ::= & \\pointer{p}{\\lambda} \\\\\n\\text{Values} & u,v & ::= & \\mathit{prim} ~|~ \\ell \\\\\n\\text{Registers} & R & ::= & (r \\mapsto v)^* \\\\\n\\text{Local states} & L & ::= & \\slocstate{\\mathit{pp}}{\\mathit{st}^*}{R} \\\\\n\\text{Call stacks} & \\alpha & ::= & \\varepsilon ~|~ L :: \\alpha \\\\\n\\text{Pending activity stacks} & \\pi & ::= & \\varepsilon ~|~ i :: \\pi \\\\\n\\end{array}\n\\begin{array}{llll}\n\\text{Objects} & o & ::= & \\obj{c}{(f_{\\tau} \\mapsto v)^*} \\\\\n\\text{Arrays} & a & ::= & \\arr{\\tau}{v^*} \\\\\n\\text{Intents} & i & ::= & \\intent{c}{(k \\mapsto v)^*} \\\\\n\\text{Memory blocks} & b & ::= & o ~|~ a ~|~ i \\\\\n\\text{Heaps} & H & ::= & (\\ell \\mapsto b)^* \\\\\n\\text{Static heaps} & S & ::= & (c.f \\mapsto v)^* \\\\\n\\text{Local configurations} & \\Sigma & ::= & \\smethconf{\\alpha}{\\pi}{H}{S}\n\\end{array}\n\\end{mathpar}\n\\caption{{$\\mu\\text{-Dalvik}_{A}$}\\xspace{} Semantic Domains}\n\\label{tab:dalvik-domains}\n\\end{table*}\n\nA \\emph{local configuration} $\\Sigma =\n\\smethconf{\\alpha}{\\pi}{H}{S}$ represents the state of a\nspecific activity. It includes a call stack $\\alpha$, a pending\nactivity stack $\\pi$, a heap $H$, and a static heap $S$. A\ncall stack $\\alpha$ is a list of \\emph{local states}, which is\npopulated upon method invocation. Each local state includes: (1) a\nprogram point $\\mathit{pp} = c,m,\\mathit{pc}$, where $c$ and $m$ identify the invoked\nmethod, while $\\mathit{pc}$ points to the next instruction to execute; (2) a list\nof statements $\\mathit{st}^*$, modelling the method body; and (3) a map $R$ \nbinding local registers to their current value. \n\nA pending activity stack $\\pi$ is a list of intents, which are treated\nas (untyped) dictionaries in our formalism. As anticipated, for the\nsake of simplicity, we only consider\nexplicit intents in the formalization, i.e., intents which are meant to be delivered\nto an activity of a given class $c$: this class is specified after the\n\\lq at\\rq\\ symbol (@) in the intent syntax\\footnote{Extending the formalism\nto include implicit intents would not be difficult, but this would introduce\nnon-determinism on the choice of the receiving activity, thus making\nthe presentation harder to follow.}. We use $\\pi$ to keep track of\nwhich activities have been started by the activity modelled by the\nlocal configuration.\n\nFinally, a heap $H$ is a mapping between locations and memory\nblocks, where each block is either an object, an array or an\nintent. Object fields are annotated with their static type, though we\ntypically omit this annotation when it is unimportant. The static heap\n$S$ simply binds static fields to their corresponding value. \n\nThe small-step operational semantics of {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} is defined by a reduction relation $\\Sigma \\rightsquigarrow \\Sigma'$. Reduction takes place by fetching the next statement to execute, based on the program counter of the top-most local state of the call stack in $\\Sigma$, and by running it to produce $\\Sigma'$. The definition of the reduction relation is lengthy, but unsurprising, and it is given in \\iffull Appendix~\\ref{sec:bytecode}. \\else the full version~\\cite{full}. \\fi The only point worth noticing here is that, when a new memory block is created, e.g., by \\texttt{new}, the corresponding pointer to the heap is annotated with the program point $c,m,\\mathit{pc}$ where creation takes place.\n\n\\subsection{Activity Semantics}\n\nThe operational behaviour of an activity does not depend only on its bytecode implementation, but also on external events, like user inputs and system callbacks. The event-driven nature of Android applications gives rise to highly non-deterministic executions, which are not trivial to approximate correctly by static analysis.\n\n\\subsubsection{Formalizing Activities}\n\nWe start by introducing a formal notion of activity.\n\n\\begin{definition}\nA class $\\mathit{cls}$ is an \\emph{activity class} if and only if $\\mathit{cls} = \\cls{c}{c'}{c^*}{\\mathit{fld}^*}{\\mathit{mtd}^*}$ for some $c' \\leq \\texttt{Activity}$. An \\emph{activity} is an instance of an activity class. We stipulate that each activity has the following fields: (1) $\\textit{finished}$: a boolean flag stating whether the activity has finished or not; (2) $\\textit{intent}$: a pointer to the intent which started the activity; (3) $\\textit{result}$: a pointer to an intent storing the result of the activity computation; and (4) $\\textit{parent}$: a pointer to the parent activity, i.e., the activity which started the present one.\n\\end{definition}\n\nWe require that each activity has a (possibly empty) set of \\emph{event handlers} for user inputs: given an activity class $c$, we let $\\textit{handlers}(c) = \\{m_1,\\ldots,m_n\\}$ be the set of the names of the methods of $c$ which may be dispatched when some user input event occurs. We assume a set of activity states $\\textit{ActStates}$ and a relation $\\textit{Lifecycle} \\subseteq \\textit{ActStates} \\times \\textit{ActStates}$ defining the state transitions admitted by the activity lifecycle~\\cite{PayetS14}. We assume that each activity class $c$ has a set of callbacks for each activity state $s$, whose names are returned by a function $\\mathit{cb}(c,s)$; for the $\\actstate{running}$ state we let $\\mathit{cb}(c,\\actstate{running}) = \\textit{handlers}(c)$, i.e., when an activity is running, any callback set for user inputs may be dispatched.\n\nWe then extend the syntax of {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} with the elements in Table~\\ref{tab:ext-dalvik}. A \\emph{frame} $\\varphi$ includes a location $\\ell$ pointing to an activity, a corresponding activity state $s$, a pending activity stack $\\pi$ and a call stack $\\alpha$. Frames are organized in an \\emph{activity stack} $\\Omega$, modelling different activities executing in the same application: a single frame in $\\Omega$ has priority of execution and is underlined. A \\emph{configuration} $\\Psi$ includes an activity stack $\\Omega$, a heap $H$ and a static heap $S$.\n\n\\begin{table}[ht]\n\\[\n\\begin{array}{llcl}\n\\text{Activity states} & s & \\in & \\textit{ActStates} \\\\\n\\text{Frames} & \\varphi & ::= & \\actframe{\\ell}{s}{\\pi}{\\alpha} ~|~ \\uactframe{\\ell}{s}{\\pi}{\\alpha} \\\\\n\\text{Activity stacks} & \\Omega & ::= & \\varphi ~|~ \\varphi :: \\Omega \\\\\n\\text{Configurations} & \\Psi & ::= & \\actconf{\\Omega}{H}{S}\n\\end{array}\n\\]\n\\textbf{Convention:} each activity stack $\\Omega$ contains at most one active (underlined) frame.\n\\caption{Extensions to the Syntax of {$\\mu\\text{-Dalvik}_{A}$}\\xspace}\n\\label{tab:ext-dalvik}\n\\end{table}\n\n\\subsubsection{Reduction Rules}\nBefore presenting the formal semantics, we need to introduce some additional definitions. We start with the notion of \\emph{callback stack}, identifying the admissible format of a call stack for new frames pushed on the activity stack upon the invocation of a callback from the Android system. Let $\\textit{sign}(c,m) = \\methsign{\\tau^*}{\\tau}{n}$ iff there exists a class $\\mathit{cls}_i$ such that $\\mathit{cls}_i = \\cls{c}{c'}{c^*}{\\mathit{fld}^*}{\\mathit{mtd}^*, \\meth{m}{\\methsign{\\tau^*}{\\tau}{n}}{\\mathit{st}^*}}$. Let then $\\textit{lookup}$ stand for a \\emph{method lookup} function such that $\\textit{lookup}(c,m) = (c',\\mathit{st}^*)$ iff: (1) $c'$ is the class defining the method which is dispatched when $m$ is invoked on an object of type $c$, and (2) $\\mathit{st}^*$ is the method body. \n\n\\begin{definition}\nGiven a location $\\ell$ pointing to an activity of class $c$, we let $\\getcb{\\ell}{s}$ stand for an arbitrary \\emph{callback stack} for state $s$, i.e., any call stack $\\slocstate{c',m,0}{\\mathit{st}^*}{R} :: \\varepsilon$, where $(c',\\mathit{st}^*) = \\textit{lookup}(c,m)$ for some $m \\in \\mathit{cb}(c,s)$, $\\textit{sign}(c',m) = \\methsign{\\tau_1,\\ldots,\\tau_n}{\\tau}{\\mathit{loc}}$ and:\n\\[ \nR = ((r_i \\mapsto \\mathbf{0})^{i \\leq \\mathit{loc}}, r_{\\mathit{loc}+1} \\mapsto \\ell, (r_{\\mathit{loc}+1+j} \\mapsto v_j)^{j \\leq n}),\n\\]\nfor some values $v_1,\\ldots,v_n$ of the correct type $\\tau_1,\\ldots,\\tau_n$.\n\\end{definition}\n\nIn the definition, we let $\\mathbf{0}$ be the default value for local registers. There is just one default value for registers in the model, since registers are untyped in Dalvik. In the following, it is also convenient to presuppose for each type $\\tau$ the existence of a a default value $\\mathbf{0}_{\\tau}$, used to initialize fields of type $\\tau$ upon object creation.\n\nA tricky aspect of the operational semantics of activities, which has never been formalized before, is the \\emph{serialization} of objects upon inter-component communication. Different activities may exchange objects using intents, but these objects are never passed by reference: rather, they are serialized at the sender side and a copy of them is created at the receiver side. The intent itself is serialized upon communication. We formalize this serialization routine by two mutually recursive functions $\\serval{H}(v) = (v',H')$ and $\\serblock{H}(b) = (b',H')$, returning a serialized copy of their argument and a new heap where all the pointers created in the serialization process have been instantiated correctly. We refer to Table~\\ref{tab:activity-semantics} below for the definition of the two functions. Their definition uses a set of pointers $\\Gamma$ to keep track of which pointers have already been followed in the serialization process, so as to allow the serialization of memory blocks including self-references.\n\nFinally, the operational semantics requires the next definition of \\emph{successful} call stack. A successful call stack is the call stack of an activity which has completed its computation.\n\n\\begin{definition}\nA call stack $\\alpha$ is \\emph{successful} if and only if $\\alpha = \\slocstate{\\mathit{pp}}{\\texttt{return}}{R} :: \\varepsilon$ for some $\\mathit{pp}$ and $R$. We let $\\overline{\\callstack}$ range over successful call stacks.\n\\end{definition}\n\nNow we have all the ingredients to define the formal semantics of\nactivities, which is given by the reduction rules in\nTable~\\ref{tab:activity-semantics}. As anticipated, the rules closely\nfollow previous work by Payet and Spoto~\\cite{PayetS14}, which we\nextend to provide a more accurate account of inter-component\ncommunication by modelling value-passing based on a serialization routine. \nWe give a short explanation of all the rules, we refer\nto~\\cite{PayetS14} for a longer description. \n\n\\begin{table*}[p]\\small\n\\begin{mathpar}\n\\inferrule[(A-Active)]\n{\\smethconf{\\alpha}{\\pi}{H}{S} \\rightsquigarrow \\smethconf{\\alpha'}{\\pi'}{H'}{S'}}\n{\\actconf{\\Omega :: \\uactframe{\\ell}{s}{\\pi}{\\alpha} :: \\Omega'}{H}{S} \\Rightarrow \\actconf{\\Omega :: \\uactframe{\\ell}{s}{\\pi'}{\\alpha'} :: \\Omega'}{H'}{S'}}\n\n\\inferrule[(A-Deactivate)]\n{}\n{\\actconf{\\Omega :: \\uactframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} :: \\Omega'}{H}{S} \\Rightarrow \\actconf{\\Omega :: \\actframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} :: \\Omega'}{H}{S}}\n\n\\inferrule[(A-Step)]\n{(s,s') \\in \\textit{Lifecycle} \\\\\n\\pi \\neq \\varepsilon \\Rightarrow (s,s') = (\\actstate{running},\\actstate{onPause}) \\\\\nH(\\ell).\\textit{finished} = \\mathtt{true} \\Rightarrow (s,s') \\in \\{(\\actstate{running},\\actstate{onPause}),(\\actstate{onPause},\\actstate{onStop}),(\\actstate{onStop},\\actstate{onDestroy})\\}}\n{\\actconf{\\actframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} :: \\Omega}{H}{S} \\Rightarrow \\actconf{\\uactframe{\\ell}{s'}{\\pi}{\\getcb{\\ell}{s'}} :: \\Omega}{H}{S}}\n\n\\inferrule[(A-Destroy)]\n{H(\\ell).\\textit{finished} = \\mathtt{true}}\n{\\actconf{\\Omega :: \\actframe{\\ell}{\\actstate{onDestroy}}{\\pi}{\\overline{\\callstack}} :: \\Omega'}{H}{S} \\Rightarrow \\actconf{\\Omega :: \\Omega'}{H}{S}}\n\n\\inferrule[(A-Back)]\n{H' = H[\\ell \\mapsto H(\\ell)[\\textit{finished} \\mapsto \\mathtt{true}]]}\n{\\actconf{\\actframe{\\ell}{\\actstate{running}}{\\varepsilon}{\\overline{\\callstack}} :: \\Omega}{H}{S} \\Rightarrow \\actconf{\\actframe{\\ell}{\\actstate{running}}{\\varepsilon}{\\overline{\\callstack}} :: \\Omega}{H'}{S}}\n\n\\inferrule[(A-Replace)]\n{H(\\ell) = \\obj{c}{(f_{\\tau} \\mapsto v)^*,\\textit{finished} \\mapsto u} \\\\ \no = \\obj{c}{(f_{\\tau} \\mapsto \\mathbf{0}_{\\tau})^*,\\textit{finished} \\mapsto \\mathtt{false}} \\\\ \nH' = H, \\pointer{p}{c} \\mapsto o}\n{\\actconf{\\actframe{\\ell}{\\actstate{onDestroy}}{\\pi}{\\overline{\\callstack}} :: \\Omega}{H}{S} \\Rightarrow \\actconf{\\uactframe{\\pointer{p}{c}}{\\actstate{constructor}}{\\pi}{\\getcb{\\pointer{p}{c}}{\\actstate{constructor}}} :: \\Omega}{H'}{S}}\n\n\\inferrule[(A-Hidden)]\n{\\varphi = \\actframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} \\\\\ns \\in \\{\\actstate{onResume},\\actstate{onPause}\\} \\\\\n(s',s'') \\in \\{(\\actstate{onPause},\\actstate{onStop}),(\\actstate{onStop},\\actstate{onDestroy})\\} }\n{\\actconf{\\varphi :: \\Omega :: \\actframe{\\ell'}{s'}{\\pi'}{\\overline{\\callstack}'} :: \\Omega'}{H}{S} \\Rightarrow \\actconf{\\varphi :: \\Omega :: \\uactframe{\\ell'}{s''}{\\pi'}{\\getcb{\\ell'}{s''}} :: \\Omega'}{H}{S}}\n\n\\inferrule[(A-Start)]\n{s \\in \\{\\actstate{onPause},\\actstate{onStop}\\} \\\\\ni = \\intent{c}{(k \\mapsto v)^*} \\\\\n\\emptyset \\vdash \\serblock{H}(i) = (i',H') \\\\\n\\pointer{p}{c},\\pointer{p'}{\\astart{c}} \\not\\in \\textit{dom}(H,H') \\\\\no = \\obj{c}{(f_{\\tau} \\mapsto \\mathbf{0}_{\\tau})^*,\\textit{finished} \\mapsto \\mathtt{false}, \\textit{intent} \\mapsto \\pointer{p'}{\\astart{c}}, \\textit{parent} \\mapsto \\ell} \\\\\nH'' = H,H',\\pointer{p}{c} \\mapsto o, \\pointer{p'}{\\astart{c}} \\mapsto i'}\n{\\actconf{\\actframe{\\ell}{s}{i :: \\pi}{\\overline{\\callstack}} :: \\Omega}{H}{S} \\Rightarrow \\actconf{\\uactframe{\\pointer{p}{c}}{\\actstate{constructor}}{\\varepsilon}{\\getcb{\\pointer{p}{c}}{\\actstate{constructor}}} :: \\actframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} :: \\Omega}{H''}{S}}\n\n\\inferrule*[width=30em,lab=(A-Swap)]\n{\\varphi' = \\actframe{\\ell'}{\\actstate{onPause}}{\\varepsilon}{\\overline{\\callstack}'} \\\\\nH(\\ell').\\textit{finished} = \\mathtt{true} \\\\\n\\varphi = \\actframe{\\ell}{s}{i :: \\pi}{\\overline{\\callstack}} \\\\\ns \\in \\{\\actstate{onPause},\\actstate{onStop}\\} \\\\\nH(\\ell').\\textit{parent} = \\ell}\n{\\actconf{\\varphi' :: \\varphi :: \\Omega}{H}{S} \\Rightarrow \\actconf{\\varphi :: \\varphi' :: \\Omega}{H}{S}}\n\n\\inferrule*[width=50em,lab=(A-Result)]\n{\\varphi' = \\actframe{\\ell'}{\\actstate{onPause}}{\\varepsilon}{\\overline{\\callstack}'} \\\\\nH(\\ell').\\textit{finished} = \\mathtt{true} \\\\\n\\varphi = \\actframe{\\ell}{s}{\\varepsilon}{\\overline{\\callstack}} \\\\\ns \\in \\{\\actstate{onPause},\\actstate{onStop}\\} \\\\\nH(\\ell').\\textit{parent} = \\ell \\\\\n\\emptyset \\vdash \\serval{H}(H(\\ell').\\textit{result}) = (\\ell'',H') \\\\\nH'' = (H,H')[\\ell \\mapsto H(\\ell)[\\textit{result} \\mapsto \\ell'']]}\n{\\actconf{\\varphi' :: \\varphi :: \\Omega}{H}{S} \\Rightarrow \\actconf{\\uactframe{\\ell}{s}{\\varepsilon}{\\getcb{\\ell}{\\actstate{onActivityResult}}} :: \\varphi' :: \\Omega}{H''}{S}}\n\\end{mathpar}\nwhere:\n\\begin{mathpar}\n\\inferrule\n{}\n{\\Gamma \\vdash \\serval{H}({\\mathit{prim}}) = (\\mathit{prim}, \\cdot)}\n\n\n\\inferrule\n{\\pointer{p}{\\lambda} \\in \\Gamma}\n{\\Gamma \\vdash \\serval{H}(\\pointer{p}{\\lambda}) = (\\newpointer{\\pointer{p}{\\lambda}}, \\cdot)}\n\n\\inferrule\n{\\pointer{p}{\\lambda} \\notin \\Gamma \\\\ \n\\Gamma \\cup \\{\\pointer{p}{\\lambda}\\} \\vdash\n\\serblock{H}(H(\\pointer{p}{\\lambda})) = (b, H'') \\\\\nH' = H'',\\newpointer{\\pointer{p}{\\lambda}} \\mapsto b }\n{\\Gamma \\vdash \\serval{H}(\\pointer{p}{\\lambda}) = (\\newpointer{\\pointer{p}{\\lambda}}, H')}\n\n\\inferrule\n{\\forall i \\in [1,n]: \\Gamma \\vdash \\serval{H}(v_i) = {(u_i,H_i)} \\\\\nH' = H_1, \\ldots, H_n}\n{\\Gamma \\vdash \\serblock{H}(\\arr{\\tau}{(v_i)^{i \\leq n}}) = (\\arr{\\tau}{(u_i)^{i \\leq n}}, H')}\n\n\\inferrule\n{\\forall i \\in [1,n]: \\Gamma \\vdash \\serval{H}(v_i) = {(u_i,H_i)} \\\\\nH' = H_1, \\ldots, H_n}\n{\\Gamma \\vdash \\serblock{H}(\\obj{c'}{(f_i \\mapsto v_i)^{i \\leq n}}) = (\\obj{c'}{(f_i \\mapsto u_i)^{i \\leq n}}, H')}\n\n\\inferrule\n{\\forall i \\in [1,n]: \\Gamma \\vdash \\serval{H}(v_i) = {(u_i,H_i)} \\\\\nH' = H_1, \\ldots, H_n }\n{\\Gamma \\vdash \\serblock{H}(\\intent{c'}{(k_i \\mapsto v_i)^{i \\leq n}}) = (\\intent{c'}{(k_i \\mapsto u_i)^{i \\leq n}}, H')}\n\\end{mathpar}\n\\textbf{Conventions:} the activity stack on the left-hand side does\nnot contain underlined frames, but for the first two rules. In the\nserialization rules we assume the existence of a function $\\newpointer{\\_}$\nassigning to each pointer a fresh pointer with the same annotation, used to \nstore the result of the serialization.\n\\caption{Reduction Relation for Configurations ($\\actconf{\\Omega}{H}{S} \\Rightarrow \\actconf{\\Omega'}{H'}{S'}$)}\n\\label{tab:activity-semantics}\n\\end{table*}\n\nRule \\irule{A-Active}\nallows the execution of the statements in the active frame, using the\nreduction relation for local configurations described in\nSection~\\ref{sec:dalvik}. Rule \\irule{A-Deactivate} models the\nsituation where the active frame has run up to completion: the frame\nloses priority and one of the other rules can be applied. Rule\n\\irule{A-Step} models the transition of the top-level activity from\nstate $s$ to one of its successors $s'$ in the activity lifecycle:\ncorrespondingly, a new callback method is executed. Two\nside-conditions constrain the possible state transitions, based on the\npresence of pending activities to start and on whether the activity has\nfinished or not. \n\nRule \\irule{A-Destroy} models the removal of a finished activity from the activity stack. Rule \\irule{A-Back} models the scenario where the user hits the back button on the Android device and the top-most activity gets finished by the system. Rule \\irule{A-Replace} corresponds to screen orientation changes: the foreground activity is destroyed and gets replaced by a fresh activity instance; notice that the new pointer to the heap is annotated with the class of the activity. Rule \\irule{A-Hidden} models the scenario where a new activity (the frame $\\varphi$) has come to the foreground and hides a previously running activity, which gets stopped or destroyed by the system.\n\nThe starting of a new activity is modelled by rule \\irule{A-Start}. The top-most activity is paused or stopped and there is some intent $i$ to be sent to $c$: the intent is serialized and a new instance of $c$ is pushed on the activity stack, setting its $\\textit{intent}$ field to a pointer to the serialized copy of $i$ and setting its $\\textit{parent}$ field to a pointer to the activity which sent the intent. The pointer to the new activity is annotated with the class $c$, while the pointer to the serialized copy of the intent gets the annotation $\\astart{c}$: again, this is needed just for the static analysis and will be discussed later. Notice that, if multiple activities need to be started, rule \\irule{A-Swap} allows a parent activity to substitute itself to a child activity on the top of the activity stack, so that rule \\irule{A-Start} can be applied again to fire the remaining intents. Finally, rule \\irule{A-Result} allows a finished activity in the foreground to return the result of its computation to the parent activity: the parent activity gets a serialized copy of the result and becomes active by executing a corresponding callback, bound to the \\actstate{onActivityResult} state.\n\n\\subsection{Examples}\n\\label{sec:examples}\nOne reason why it is useful to have a formal semantics before devising\na static analysis technique is to pinpoint corner cases which may\npotentially lead\nto unsound analysis results. We discuss two examples below. \n\n\\subsubsection{Static Fields}\nEven though inter-component communication does not allow for the\nexchange of references, activities in the same application can still\nshare memory by using static fields. This is apparent in the formal\nsemantics, since the syntax of configurations $\\Psi$ contains a global\nstatic heap $S$, which can be accessed by using publicly known\nnames of static fields. We then observe that the order of execution of\ndifferent activities, or even different callbacks inside the same\nactivity, is very hard to predict: for instance, the rules in\nTable~\\ref{tab:activity-semantics} highlight that even activities\nwhich are not on the top of the activity stack may become active and\nexecute callbacks by rule \\irule{A-Hidden}. Also, the same callback\nmay be executed multiple times, since an activity may be routinely\nrecreated by the Android system due to user activities (e.g., screen\norientation changes), which cannot be known statically, \nas modelled by rule \\irule{A-Replace}. \n\nThe implication on static analysis is that it is extremely challenging to implement flow-sensitivity on accesses to static fields without producing unsound results. Furthermore, given that static fields may be used to share pointers to heap locations, flow-sensitivity for heap accesses is also hard to achieve. Since we target soundness in this work, the static analysis we devise in the next section is flow-insensitive on both static fields and heap locations.\n\n\\subsubsection{Serialization}\nRule \\irule{A-Start} of the operational semantics highlights that\nintents are serialized upon inter-component communication. This means\nthat, when a parent activity starts a child activity, the latter\noperates on a copy of the intent sent by the former and not on the\nsame intent. \n\nThe implication on static analysis is that, although the callback bound to the \\actstate{onActivityResult} state of the parent activity is always executed after the construction of the child activity, no change to the intent done by the child activity should overwrite the original over-approximation of the intent computed for the parent\nactivity when a result is returned to it. This applies to any object which is serialized with the intent. The static analysis in the next section provides a conservative over-approximation of this behaviour.\n\n\n\n\n\\subsection{Value-sensitivity}\nValue-sensitivity is the ability of a static analysis to approximate runtime values and use this information to improve precision, e.g., by skipping unreachable program branches~\\cite{NielsonNH99}. Concretely, consider the following code:\n\\begin{verbatim}\nint x = 0;\nfor (int y = 0; y <= 10; y++) { x++; }\nTelephonyManager tm = ...\nString imei = tm.getDeviceId();\nif (x == 0) { leak(imei); }\n\\end{verbatim}\nThough this code is perfectly safe, all the existing tools (IccTA, AmanDroid and DroidSafe) will identify it as leaky. IccTA and DroidSafe conservatively assume all the program points to be potentially reachable. Even AmanDroid raises a false alarm for this code, though it internally implements a dedicated data-flow analysis~\\cite{WeiROR14}. \n\nBesides this simple example, there are many reasons why real-world static analysis tools for Android applications should be value-sensitive to be practically useful. First, several features of Java and the Android APIs, most notably \\emph{reflection} and \\emph{dictionary-like} containers, e.g., intents and bundles, need value-sensitivity to be analysed precisely. Second, the loss of precision entailed by value-insensitivity may creep and interact badly with other desirable features of the static analysis, e.g., \\emph{context-sensitivity}, which has been deemed as crucial by previous studies~\\cite{ArztRFBBKTOM14,GordonKPGNR15}. \n\nContext-sensitivity is the ability of the analysis to compute\ndifferent static approximations upon different method calls. To\nunderstand why the benefits of context-sensitivity can be voided by\nvalue-insensitivity, consider the following method, where we assume to\nknow a valid upper bound for the GPS location values:\n\n\\begin{verbatim}\nvoid m (double x, double y) {\n if (x <= MAX_X && y <= MAX_Y)\n ...\n else\n leak(\"Invalid location:\" + x + y);\n}\n\\end{verbatim} \nContext-insensitive static analyses would detect a dangerous information flow whenever the method \\texttt{m} is invoked at two different program points and one of these invocations provides the location of the device in the actual parameters, while the other one provides an invalid location. The reason is that the method \\texttt{m} would be analysed only once, hence the static analysis would detect that both public and confidential values may reach a sink. Conversely, a context-sensitive analysis potentially has the ability to discriminate between the two methods invocations and be precise, but the lack of value-sensitivity would necessarily lead to the detection of a non-existent information flow.\n\nFinally, it is worth noticing that value-sensitivity is crucial to support \nsecurity-relevant, value-dependent security queries (e.g., ``Is the credit card \nnumber sent on HTTP rather than on HTTPS?'' or ``Is the picture actually \nuploaded on Facebook, as opposed to some other untrusted website?'').\n\n\\subsection{Flow-sensitivity}\nFlow-sensitivity is the ability of a static analysis to take the order of statements into account and compute different approximations at different program points~\\cite{NielsonNH99}. To understand its importance, consider the following code:\n\\begin{verbatim}\nTelephonyManager tm = ...\nString imei = tm.getDeviceId();\nimei = new String(\"empty\");\nleak(imei);\n\\end{verbatim}\nThough the code above is safe, the flow-insensitive analysis implemented in DroidSafe will identify it as leaky, since the variable \\texttt{imei} does contain a secret information at some program point. Conversely, both FlowDroid and AmanDroid will correctly deem the program as safe.\n\nClearly, it is tempting to target a flow-sensitive information flow analysis tool to achieve a higher level of precision, but, as pointed out by the authors of DroidSafe~\\cite{GordonKPGNR15}, flow-sensitivity is very hard to get right for Android applications, due to their massive use of asynchronous callbacks. Both FlowDroid and AmanDroid suggest to tackle this problem by introducing a \\emph{dummy main method} emulating each possible interleaving of the callbacks defining the application life-cycle. Unfortunately, it is difficult to ensure that the dummy main method construction is accurate and comprehensive, which leads to missing malicious information flows~\\cite{GordonKPGNR15}. \n\n\\subsection{HornDroid}\nOur tool, {HornDroid}\\xspace, targets a \\emph{sound} and \\emph{practical} information flow analysis for Android applications. We report on the design choices we made to hit the sweet spot between these two potentially conflicting requirements.\n\n{HornDroid}\\xspace{} implements a \\emph{value-sensitive} information flow analysis. As anticipated, value-sensitivity is crucial to support a practically useful analysis of real-world applications. The analysis implemented in {HornDroid}\\xspace{} is reminiscent of \\emph{abstract interpretation}, whereby the operational semantics of a program is over-approximated by a computable abstract semantics. As it is customary for abstract interpretation, the design of the analysis is parametric with respect to the choice of a set of \\emph{abstract domains}, defining how runtime values are statically approximated: one can then fine-tune the precision of the analysis by testing different abstract domains. To ensure the scalability of our value-sensitive analysis, the abstract semantics implemented in {HornDroid}\\xspace{} is based on \\emph{Horn clauses}, whose efficient resolution is supported by state-of-the-art SMT solvers~\\cite{BjornerMR12}.\n\n{HornDroid}\\xspace{} performs a \\emph{flow-sensitive} information flow analysis on the registers employed by the Dalvik Virtual Machine, while implementing a \\emph{flow-insensitive} analysis for callback methods and heap locations. This is crucial to preserve the precision of the analysis, without sacrificing soundness. We already mentioned that previous studies highlighted that flow-sensitive analyses may easily produce unsound results, due to the challenges of predicting all the possible orderings of the Android callbacks~\\cite{GordonKPGNR15}. Moreover, while carrying out the soundness proof for {HornDroid}\\xspace, we realized that \\emph{static fields} are particularly delicate to treat in a flow-sensitive fashion. The reason is that static fields provide a way to implement a shared memory between otherwise memory-isolated components running in the same application. Given that the execution order of different Android components is extremely hard to predict, due to their callback-driven nature, it turns out that flow-insensitivity for static fields is in practice needed for soundness. Indeed, since static fields can be used to exchange pointers to heap locations, a sound flow-sensitive analysis for heap locations is in general hard to achieve. Our soundness proof, instead, confirms that flow-sensitivity can be implemented for the registers employed by the Dalvik Virtual Machine without missing any malicious information flow.\n\n\n\\subsection{Evaluation on DroidBench}\nDroidBench~\\cite{ArztRFBBKTOM14} is a set of small applications which\nhas been proposed by the research community as a testing ground for\nstatic information flow analysis tools for Android. The current\nversion of the benchmark (2.0) includes 120 test cases, featuring both\nleaky (positive) and benign (negative) examples. We tested IccTA, AmanDroid, DroidSafe and {HornDroid}\\xspace{} on this benchmark, the results are summarized in the confusion matrix in Table~\\ref{tab:droidbench}, reporting the number of true positives ($tp$), true negatives ($tn$), false positives ($fp$) and false negatives ($fn$) produced by the tools.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{c|c|c|}\n\\cline{2-3} & \\multicolumn{2}{ c| }{Output} \\\\ \n\\cline{2-3} & \\emph{leaky} & \\emph{benign} \\\\ \n\\cline{2-3}\n\\cline{2-3} & \\emph{IccTA}\/\\emph{AD}\/\\emph{DS}\/\\emph{HD} & \\emph{IccTA}\/\\emph{AD}\/\\emph{DS}\/\\emph{HD} \\\\\n\\cline{1-3}\n\\multicolumn{1}{ |c| }{\\emph{leaky}} & $tp:$ 64 \/ 70 \/ 89 \/ 96 & $fn:$ 36 \/ 30 \/ 11 \/ 4 \\\\ \n\\cline{1-3}\n\\multicolumn{1}{ |c| }{\\emph{benign}} & $fp:$ 8 \/ 5 \/ 10 \/ 6 & $tn:$ 11 \/ 14 \/ 9 \/ 13 \\\\\n\\cline{1-3}\n\\end{tabular}\n\\end{center}\n\\caption{Confusion Matrix on DroidBench}\n\\label{tab:droidbench}\n\\end{table}\n\nIccTA does not detect 36 out of 100 leaky applications, AmanDroid misses 30 and DroidSafe still misses 11. Most of the leaks missed by IccTA and AmanDroid are due to flow-sensitivity and some callbacks which are not correctly detected by the analysis; as to DroidSafe, we do not have definite answers on the unsound results, given the sheer size of the project and the lack of complete documentation. {HornDroid}\\xspace{} performs much better than all its competitors on DroidBench, since it only misses 4 leaky applications: all these cases are related to \\emph{implicit flows}, which are not covered by standard taint analyses (and our formal proof). \n\nBut even better, despite the strong security guarantees it provides,\nthe analysis performed by {HornDroid}\\xspace{} is not overly conservative, since it\ndetects as potentially leaky only 6 out of 19 benign applications. We\nnotice that 3 of these false alarms are due to flow insensitivity of\nthe heap abstraction, one to an over-approximation of exceptions, and\n2 to an over-approximated treatment of inter-app communication. Only\nAmanDroid is more precise, since it produces one less false positive;\non the other hand, it misses many more malicious information flows\nthan {HornDroid}\\xspace{} (30 vs 4). For the sake of completeness, we report in Table~\\ref{tab:droidbench-res}\na full breakdown of the experiments on DroidBench, omitting the cases where all the tools agree with the ground truth.\n\nThe experimental results on DroidBench are summarized by a few standard statistical measures in Table~\\ref{tab:spec-sens}, which highlight that soundness in HornDroid does not come at the\ncost of precision.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c|}\n\\cline{2-5}\n& \\emph{IccTA} & \\emph{AD} & \\emph{DS} & \\emph{HD} \\\\\n\\cline{1-5}\n\\multicolumn{1}{|c|}{\\emph{Sensitivity}} & 0.64 & 0.70 & 0.89 & 0.96 \\\\ \n\\cline{1-5}\n\\multicolumn{1}{|c|}{\\emph{Specificity}} & 0.58 & 0.74 & 0.47 & 0.68 \\\\\n\\cline{1-5}\n\\multicolumn{1}{|c|}{\\emph{F-Measure}} & 0.61 & 0.72 & 0.62 & 0.80 \\\\\n\\cline{1-5}\n\\end{tabular}\n\\end{center}\n\\emph{Sensitivity} = $tp \/ (tp + fn)$ $\\sim$ Soundness \\\\\n\\emph{Specificity} = $tn \/ (tn + fp)$ $\\sim$ Precision \\\\\n\\emph{F-Measure} = $2 * (sens * spec) \/ (sens + spec)$ $\\sim$ Aggregate\n\\caption{Performance Measures on DroidBench}\n\\label{tab:spec-sens}\n\\end{table}\n\n\\iffull\n\\input{results}\n\\fi\n\nBesides the quality of the results, also performances are important. Table~\\ref{tab:performances} reports the mean and the median of the analysis times for the applications in DroidBench. As it turns out, {HornDroid}\\xspace{} is one order of magnitude faster than both IccTA and AmanDroid, which in turn are one order of magnitude faster than DroidSafe. The extremely good performances of {HornDroid}\\xspace{} are due to both design choices, like flow insensitivity on the activity life-cycle, and excellent support by Z3 in Horn clauses resolution. \n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c|}\n\\cline{2-5}\n& \\emph{IccTA} & \\emph{AD} & \\emph{DS} & \\emph{HD} \\\\\n\\cline{1-5}\n\\multicolumn{1}{|c|}{\\emph{Average Analysis Time}} & 19 & 11 & 176 & 1 \\\\ \n\\cline{1-5}\n\\multicolumn{1}{|c|}{\\emph{Median Analysis Time}} & 15 & 10 & 186 & 1 \\\\\n\\cline{1-5}\n\\end{tabular}\n\\end{center}\n\\caption{Analysis Time for DroidBench (Seconds)}\n\\label{tab:performances}\n\\end{table}\n\n\n\\subsection{Evaluation on Real Applications}\nIn order to evaluate the practicality of our analysis, we performed a\ntest on the two largest applications available in the Google Play Top\n30: the game Candy Crash Soda Saga (51.7 Mb) and the Facebook\napplication (46.5 Mb). We ran the experiments on a server with 64\nmulti-thread cores and 758 Gb of memory, however the highest memory\nconsumption by {HornDroid}\\xspace{} was around 10 Gb, so it is possible to\nreproduce our results even on a modern commercial machine. \n\n{HornDroid}\\xspace{} found an information leak in Facebook, while Candy Crash Soda Saga\nappears to be secure. The analysis took around 30 minutes and 60\nminutes respectively. We tested all the existing competitors on both\napplications, to check whether they could confirm the analysis\nresults. Unfortunately, AmanDroid crashed just after the beginning of\nthe analysis of Facebook, while both DroidSafe and IccTA failed to\nterminate within the timeout we set (2 hours). We were able instead to\nanalyse Candy Crash Soda Saga using AmanDroid in around 50 minutes,\ngetting an information flow. After a manual inspection, we realized\nthis is a false positive due to the incorrect inclusion of the\n\\texttt{onHandleIntent} method of the class \\texttt{IntentService}\namong the possible sources of sensitive information: this is not\nincluded in more recent proposals~\\cite{GordonKPGNR15,LiEtAl15}. Both\nIccTA and DroidSafe were not able to analyse the application within 2\nhours. Due to space constraints, we refer to~\\cite{full} for a more\ncomprehensive experimental evaluation on real applications. \n\n\\subsection{Features and Limitations}\n\\label{sec:features}\nAs anticipated, the formalization in the previous sections only captures the \\emph{core} of the analysis implemented in {HornDroid}\\xspace{} and establishes the soundness of its principles. The tool, however, supports more features which are needed to make the analysis scale to real applications. We detail here some important aspects of {HornDroid}\\xspace{} which are not covered by our formal model and we comment on current limitations.\n\n\\subsubsection*{Android Components}\nAlthough the {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} model only represents activities and their life-cycle, {HornDroid}\\xspace{} supports all the component types available on the Android platform, including services, broadcast receivers and content providers~\\cite{Android}. The implementation of the analysis for these components does not significantly differ from the one for activities we presented in the paper, though it requires a correct modelling of their specific life-cycle.\n\n\\subsubsection*{Fragments}\nFragments are used to separate the functionality of an activity among different independent sub-components~\\cite{Fragments}. In order to support a sound analysis of fragments, {HornDroid}\\xspace{} over-approximates their life-cycle by executing all the fragments along with the containing activity in a flow-insensitive way. This might lead to precision problems on real applications, but this is the simplest of the sound options, which follows the philosophy we adopted for activity analysis.\n\n\\subsubsection*{Arrays} \nThough the static analysis we formalized is field-insensitive on\narrays, {HornDroid}\\xspace{} supports a more precise treatment of array\nindexes. Being value-sensitive, {HornDroid}\\xspace{} statically approximates which\nindexes of an array may be accessed at runtime: if a secret value is\nstored in the first position of the array, but only the second element\nof the array is leaked, the tool does not raise an alarm, contrarily\nto all the other existing tools (cf. the breakdown on the experiments\nin~\\cite{full}).\n\n\\subsubsection*{Exceptions} \n{HornDroid}\\xspace{} implements a conservative solution to handle exceptions, i.e., exceptions are always assumed to be thrown. A similar coarse over-approximation is implemented in FlowDroid~\\cite{ArztRFBBKTOM14}. We leave a more precise treatment of exceptions to future work: we believe that the value-sensitivity of the analysis implemented in our tool will be crucial to limit the number of false alarms for exception handling. For instance, a value-sensitive analysis can ensure that a null pointer exception is never raised at runtime, since it over-approximates the set of the possible runtime values.\n\n\\subsubsection*{Inter-app Communication} \n{HornDroid}\\xspace{} has limited support for inter-application communication, i.e., it conservatively detects an information leak whenever an intent storing secret data is sent to another application. More precise results could be achieved by analysing all the communicating applications simultaneously, but the current implementation of {HornDroid}\\xspace{} only supports the analysis of a single application. We plan to leverage existing state-of-the-art solutions to overcome this limitation~\\cite{LiEtAl15}. \n\n\\subsubsection*{Threading} \n{HornDroid}\\xspace{} handles multithreading by assuming that threads are executed in a sequential,\nbut arbitrary order, much in the same spirit of the callbacks defining\nthe activity life-cycle. This is the same strategy used in FlowDroid. We conjecture, but did not prove yet, that\nthis strategy is sound in our case, since the analysis is flow\ninsensitive on everything except for registers, which are not shared. For flow-sensitive analysis techniques (e.g.,\nFlowDroid), instead, this strategy is in general unsound, since it\nmay miss potential interleavings arising due to synchronization on\nshared memory (e.g., static heaps). The only aspect that should be added to our\nstatic analysis is a thread pool simulation. In Java, every time the\nmethod \\texttt{execute} is called on a thread, this is placed in a\npool and then executed by the system by calling the runnable method\n\\texttt{run}. Our static analysis similarly binds each invocation of\n\\texttt{execute} to a corresponding \\texttt{run} method. \n\n\\subsubsection*{Reflection} \nThough supporting reflection soundly is an open research problem~\\cite{SmaragdakisKB14}, {HornDroid}\\xspace{} still covers a significant fraction of common reflection cases by implementing a simple string analysis. The solution we propose is in the same spirit of DroidSafe, i.e., reflective calls which can be statically resolved are replaced by direct calls to the appropriate method. Pragmatically, however, we observed that we are able to achieve much better results than DroidSafe for the reflection cases in DroidBench.\n\n\\subsubsection*{Limitations}\nA comprehensive implementation of analysis stubs for method calls to the Android APIs is still lacking: we only implemented some selected stubs for our experiments, to show that our approach is feasible and practical. When a stub to an external library is missing, the tool tries to be conservative: the return value of the call is over-approximated to the top element of the corresponding abstract domain, and it is tainted whenever at least one of the arguments is tainted. \nOther important limitations of {HornDroid}\\xspace{} are shared with existing\nsolutions~\\cite{ArztRFBBKTOM14,GordonKPGNR15}. First, the analysis\ndoes not capture \\emph{implicit} information flows at\npresent. Second, the analysis does not consider \\emph{native\ncode}: this is a point we leave as a future work, observing that SMT solving has \nbeen successfully applied in the past to C code (see, e.g., the SLAM \nproject~\\cite{Ball_adecade}). Third, the analysis is oblivious to the \\emph{semantics}\nof the information flows, i.e., it lacks any built-in declassification mechanism\nto qualify legitimate data flows. Since our analysis approximates data\ninformation rather than just tracking taints, however, it is in principle \npossible to encode expressive data-dependent declassification policies, e.g.,\none could define the result of an encryption as untainted\nonly if the encryption is performed with the right key.\n\n\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\\input{intro}\n\n\\section{Design and Motivations}\n\\label{sec:design}\n\\input{design}\n\n\\section{Operational Semantics}\n\\label{sec:activity}\n\\input{activity}\n\n\\section{Static Analysis}\n\\label{sec:statics}\n\\input{statics}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{experiments}\n\n\\section{Additional Related Work}\n\\label{sec:related}\n\\input{related}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{conclusion}\n\n\n\n\\bibliographystyle{IEEEtranS}\n\n\\subsection{Representation Functions}\nWe presuppose the existence of a representation function $\\beta_{\\textit{Prim}}$ which associates to each primitive value $\\mathit{prim}$ a corresponding abstract value $\\{\\widehat{\\prim}\\}$. For a location $\\ell = \\pointer{p}{\\lambda}$, we let $\\beta_{\\textit{Loc}}(\\ell) = \\{\\lambda\\}$. Based on this, we define $\\beta_{\\textit{Val}}(v)$ as follows:\n\\[\n\\beta_{\\textit{Val}}(v) =\n\\begin{cases}\n\\beta_{\\textit{Prim}}(v) & \\text{if } v = \\mathit{prim} \\\\\n\\beta_{\\textit{Loc}}(v) & \\text{if } v = \\ell\n\\end{cases}\n\\]\nWe typically omit brackets around singleton abstract values. We then define $\\beta_{\\textit{Blk}}(b)$ as follows:\n\\[\n\\beta_{\\textit{Blk}}(b) =\n\\begin{cases}\n\\absobj{c}{(f \\mapsto \\hat{v})^*} & \\text{if } b = \\obj{c}{(f \\mapsto v)^*} \\text{ and } \\forall i: \\beta_{\\textit{Val}}(v_i) = \\hat{v}_i \\\\\n\\absintent{c}{\\hat{v}} & \\text{if } b = \\intent{c}{(f \\mapsto v)^*} \\text{ and } \\hat{v} = \\sqcup_i\\, \\beta_{\\textit{Val}}(v_i) \\\\\n\\absarray{\\tau}{\\hat{v}} & \\text{if } b = \\arr{\\tau}{v^*} \\text{ and } \\hat{v} = \\sqcup_i\\, \\beta_{\\textit{Val}}(v_i)\n\\end{cases}\n\\]\nUsing these definitions, we can define how configurations are translated into facts by a corresponding representation function. This requires one to define a number of clauses, summarized below:\n\n\\[\n\\begin{array}{lcl}\n\\beta_{\\textit{Lst}}(\\locstate{c,m,\\mathit{pc}}{\\mathit{st}^*}{R}{u^*}) & = & \\{\\absreg{c,m,\\mathit{pc}}{\\hat{u}^*}{\\hat{v}^*} ~|~ \\forall j: \\hat{u}_j = \\beta_{\\textit{Val}}(u_j) \\wedge \\forall k: \\hat{v}_k = \\beta_{\\textit{Val}}(R(r_k))\\} \\cup \\bigcup_i\\, \\ainstfull{\\mathit{st}_i}{c,m,i} \\\\\n\n\\beta_{\\textit{Call}}(\\alpha) & = & \\bigcup_{i \\in [1,n]} \\beta_{\\textit{Lst}}(L_i) \\text{ whenever } \\alpha = L_1 :: \\ldots :: L_n \\\\\n\n\\beta_{\\textit{Heap}}(H) & = & \\{\\mathsf{H}(\\lambda,\\hat{b}) ~|~ H = H',\\ell \\mapsto b \\wedge \\lambda = \\beta_{\\textit{Loc}}(\\ell) \\wedge \\hat{b} = \\beta_{\\textit{Blk}}(b)\\} \\\\\n\n\\beta_{\\textit{Stat}}(S) & = & \\{\\mathsf{S}(c,f,\\hat{v}) ~|~ S = S',c.f \\mapsto v \\wedge \\hat{v} = \\beta_{\\textit{Val}}(v)\\} \\\\\n\n\\rfdispatch{\\ell}(\\pi) & = & \\{\\mathsf{I}(c,\\hat{b}) ~|~ c = \\beta_{\\textit{Loc}}(\\ell) \\wedge \\pi = \\pi_0 :: i :: \\pi_1 \\wedge \\hat{b} = \\beta_{\\textit{Blk}}(i)\\} \\\\\n\n\\beta_{\\textit{Lcnf}}(\\methconf{\\alpha}{\\pi}{H}{S}{\\ell}) & = & \\beta_{\\textit{Call}}(\\alpha) \\cup \\rfdispatch{\\ell}(\\pi) \\cup \\beta_{\\textit{Heap}}(H) \\cup \\beta_{\\textit{Stat}}(S) \\\\\n\n\\beta_{\\textit{Frm}}(\\actframe{\\ell}{s}{\\pi}{\\alpha}) & = & \\beta_{\\textit{Frm}}(\\uactframe{\\ell}{s}{\\pi}{\\alpha}) = \\rfdispatch{\\ell}(\\pi) \\cup \\beta_{\\textit{Call}}(\\alpha) \\\\\n\n\\beta_{\\textit{Stk}}(\\Omega) & = & \\bigcup_{i \\in [1,n]} \\beta_{\\textit{Frm}}(\\varphi_i) \\text{ whenever } \\Omega = \\varphi_1 :: \\ldots :: \\varphi_n \\\\\n\n\\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S}) & = & \\beta_{\\textit{Stk}}(\\Omega) \\cup \\beta_{\\textit{Heap}}(H) \\cup \\beta_{\\textit{Stat}}(S)\n\\end{array}\n\\]\n\n\\subsection{Ordering Abstract Values and Facts}\nWe presuppose the existence of a pre-order $\\sqsubseteq_{\\textit{Prim}}$ on primitive singleton abstract values. Based on this, we define a pre-order $\\sqsubseteq_{\\textit{Val}}$ on abstract values by having $\\hat{u} \\sqsubseteq_{\\textit{Val}} \\hat{v}$ iff:\n\\begin{itemize}\n\\item $\\forall \\widehat{\\prim} \\in \\hat{u}: \\exists \\widehat{\\prim}' \\in \\hat{v}: \\widehat{\\prim} \\sqsubseteq_{\\textit{Prim}} \\widehat{\\prim}'$;\n\\item $\\forall \\lambda \\in \\hat{u}: \\lambda \\in \\hat{v}$.\n\\end{itemize}\nWe then build a pre-order $\\sqsubseteq_{\\textit{Seq}}$ on sequences of abstract values by having $\\hat{u}^* \\sqsubseteq_{\\textit{Seq}} \\hat{v}^*$ iff $\\hat{u}^*$ and $\\hat{v}^*$ have the same length and:\n\\[\n\\forall i: \\hat{u}_i \\sqsubseteq_{\\textit{Val}} \\hat{v}_i.\n\\]\nWe can then define a pre-order $\\sqsubseteq_{\\textit{Blk}}$ on abstract blocks as follows:\n\\begin{itemize}\n\\item if $\\hat{b} = \\absobj{c}{(f \\mapsto \\hat{u})^*}$ and $\\hat{b}' = \\absobj{c}{(f \\mapsto \\hat{v})^*}$ and $\\hat{u}^* \\sqsubseteq_{\\textit{Seq}} \\hat{v}^*$, then $\\hat{b} \\sqsubseteq_{\\textit{Blk}} \\hat{b}'$;\n\\item if $\\hat{b} = \\absintent{c}{\\hat{u}}$ and $\\hat{b}' = \\absintent{c}{\\hat{v}}$ and $\\hat{u} \\sqsubseteq_{\\textit{Val}} \\hat{v}$, then $\\hat{b} \\sqsubseteq_{\\textit{Blk}} \\hat{b}'$;\n\\item if $\\hat{b} = \\absarray{\\tau}{\\hat{u}}$ and $\\hat{b}' = \\absarray{\\tau} {\\hat{v}}$ and $\\hat{u} \\sqsubseteq_{\\textit{Val}} \\hat{v}$, then $\\hat{b} \\sqsubseteq_{\\textit{Blk}} \\hat{b}'$.\n\\end{itemize}\n\nFinally, we let $\\mathsf{f} \\sqsubseteq \\mathsf{f}'$ be the least pre-order on facts such that:\n\\begin{itemize}\n\\item $\\absreg{c,m,\\mathit{pc}}{\\hat{u}_{call}^*}{\\hat{u}^*} \\sqsubseteq \\absreg{c,m,\\mathit{pc}}{\\hat{v}_{call}^*}{\\hat{v}^*}$ whenever $\\hat{u}_{call}^* \\sqsubseteq_{\\textit{Seq}} \\hat{v}_{call}^*$ and $\\hat{u}^* \\sqsubseteq_{\\textit{Seq}} \\hat{v}^*$;\n\\item $\\mathsf{H}(\\lambda,\\hat{b}) \\sqsubseteq \\mathsf{H}(\\lambda,\\hat{b}')$ whenever $\\hat{b} \\sqsubseteq_{\\textit{Blk}} \\hat{b}'$;\n\\item $\\mathsf{S}(c,f,\\hat{u}) \\sqsubseteq \\mathsf{S}(c,f,\\hat{v})$ whenever $\\hat{u} \\sqsubseteq_{\\textit{Val}} \\hat{v}$;\n\\item $\\prhs{\\hat{u}} \\sqsubseteq \\prhs{\\hat{v}}$ whenever $\\hat{u} \\sqsubseteq_{\\textit{Val}} \\hat{v}$;\n\\item $\\absresult{c,m}{\\hat{u}_{call}^*}{\\hat{u}^*} \\sqsubseteq \\absresult{c,m}{\\hat{v}_{call}^*}{\\hat{v}^*}$ whenever $\\hat{u}_{call}^* \\sqsubseteq_{\\textit{Seq}} \\hat{v}_{call}^*$ and $\\hat{u}^* \\sqsubseteq_{\\textit{Seq}} \\hat{v}^*$;\n\\item $\\mathsf{I}(c,\\hat{b}) \\sqsubseteq \\mathsf{I}(c,\\hat{b}')$ whenever $\\hat{b} \\sqsubseteq_{\\textit{Blk}} \\hat{b}'$.\n\\end{itemize}\n\n\\subsection{Formal Results}\n\n\\subsubsection{Preliminaries}\n\n\\begin{definition}\nA local configuration $\\Sigma = \\methconf{\\alpha}{\\pi}{H}{S}{\\ell}$ is \\emph{well-formed} if and only if, whenever $\\alpha = L_1 :: \\ldots :: L_n$, we have:\n\\begin{itemize}\n\\item either $n \\in \\{0,1\\}$, i.e., $\\alpha$ is either empty or it contains just a single local state;\n\\item or $n \\geq 2$ and for each $i \\in [2,n]$, either of the\n following conditions hold true:\n\\MESSAGEIfor{MS}{151202}{We replace $\\mathit{st}_{\\mathit{pc}-1}$ with $\\mathit{st}_{\\mathit{pc}}$, without\n that analysis cannot over-approximate representation function (i.e.,\n we\n will have function calls treated as both nops and real calls in the\n representation function).}\n\\begin{itemize}\n\\item $L_i = \\locstate{c',m',\\mathit{pc}'}{\\mathit{st}'^*}{R'}{v^*}$ and $L_{i-1} = \\locstate{c,m,\\mathit{pc}}{\\mathit{st}^*}{R}{\\_}$ with $\\mathit{st}_{\\mathit{pc}} = \\invoke{r_o}{m'}{r_1',\\ldots,r_n'}$, \\\\ $\\textit{lookup}(\\gettype{H}{\\regval{r_o}},m') = (c',\\mathit{st}'^*)$, $\\textit{sign}(c',m') = \\methsign{\\tau_1,\\ldots,\\tau_n}{\\tau}{\\mathit{loc}}$ and $v^* = (\\regval{r_k'})^{k \\leq n}$\n\\item $L_i = \\locstate{c',m',\\mathit{pc}'}{\\mathit{st}'^*}{R'}{v^*}$ and $L_{i-1} = \\locstate{c,m,\\mathit{pc}}{\\mathit{st}^*}{R}{\\_}$ with $\\mathit{st}_{\\mathit{pc}} = \\sinvoke{c'}{m'}{r_1',\\ldots,r_n'}$, \\\\ $\\textit{lookup}(c',m') = (c',\\mathit{st}'^*)$, $\\textit{sign}(c',m') = \\methsign{\\tau_1,\\ldots,\\tau_n}{\\tau}{\\mathit{loc}}$ and $v^* = (\\regval{r_k'})^{k \\leq n}$.\n\\end{itemize}\n\\end{itemize} \n\\end{definition}\n\n\\begin{lemma}[Preserving Local Well-formation]\n\\label{lem:preserve-local}\nIf $\\Sigma$ is well-formed and $\\Sigma \\rightsquigarrow^* \\Sigma'$, then $\\Sigma'$ is well-formed.\n\\end{lemma}\n\\begin{IEEEproof}\nBy induction on the length of the reduction sequence and a case analysis on the last rule applied.\n\\end{IEEEproof}\n\n\\begin{definition}\nA heap $H$ is \\emph{well-typed} if and only if, whenever $H(\\ell) = \\obj{c}{(f_i \\mapsto v_i)^{i \\leq n}}$, for all $i \\in [1,n]$ we have $\\gettype{H}{v_i} \\leq \\tau_i$, where $\\tau_i$ is the declared type of field $f_i$ for an object of type $c$ according to the underlying program.\n\\end{definition}\n\n\\begin{assumption}[Java Type Soundness]\n\\label{asm:java-sound}\nIf $\\methconf{\\alpha}{\\pi}{H}{S}{\\ell} \\rightsquigarrow \\methconf{\\alpha'}{\\pi'}{H'}{S'}{\\ell}$, then for any value $v$ we have $\\gettype{H'}{v} \\leq \\gettype{H}{v}$. Moreover, if $H$ is well-typed, then also $H'$ is well-typed.\n\\end{assumption}\n\n\\begin{definition}\nA configuration $\\Psi = \\actconf{\\Omega}{H}{S}$ is \\emph{well-formed} if and only if:\n\\begin{itemize}\n\\item whenever $\\Omega = \\Omega_0 :: \\varphi :: \\Omega_1$ with $\\varphi \\in \\{\\actframe{\\ell}{s}{\\pi}{\\alpha},\\uactframe{\\ell}{s}{\\pi}{\\alpha}\\}$, we have $H(\\ell) = \\obj{c}{(f \\mapsto v)^*}$ for some activity class $c$ and $\\ell = \\pointer{p}{c}$ for some pointer $p$;\n\n\\item whenever $\\Omega = \\Omega_0 :: \\varphi :: \\Omega_1$ with $\\varphi \\in \\{\\actframe{\\ell}{s}{\\pi}{\\alpha},\\uactframe{\\ell}{s}{\\pi}{\\alpha}\\}$, we have that $\\Sigma = \\methconf{\\alpha}{\\pi}{H}{S}{\\ell}$ is a well-formed local configuration;\n\n\\item $H$ is a well-typed heap.\n\\end{itemize}\n\\end{definition}\n\n\\begin{lemma}[Preserving Well-formation]\n\\label{lem:preserve-well}\nIf $\\Psi$ is well-formed and $\\Psi \\Rightarrow^* \\Psi'$, then $\\Psi'$ is well-formed.\n\\end{lemma}\n\\begin{IEEEproof}\nBy induction on the length of the reduction sequence and a case analysis on the last rule applied, using Lemma~\\ref{lem:preserve-local} and Assumption~\\ref{asm:java-sound} to deal with case \\irule{A-Active}.\n\\end{IEEEproof}\nFrom now on, we tacitly focus only on well-formed configurations. All the formal results only apply to them: notice that well-formed configurations always reduce to well-formed configurations by Lemma~\\ref{lem:preserve-well}.\n\n\\subsubsection{Main Results}\n\n\\begin{lemma}\n\\label{lem:sub-order}\nIf $\\Delta \\subseteq \\Delta'$, then $\\Delta <: \\Delta'$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem:trans-order}\nIf $\\Delta <: \\Delta'$ and $\\Delta' <: \\Delta''$, then $\\Delta <: \\Delta''$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lem:join-order}\nIf $\\Delta_1 <: \\Delta_2$ and $\\Delta_3 <: \\Delta_4$, then $\\Delta_1 \\cup \\Delta_3 <: \\Delta_2 \\cup \\Delta_4$.\n\\end{lemma}\n\n\\begin{assumption}[Soundness of the Abstract Operations]\n\\label{asm:sound-op}\nWe assume all the following properties:\n\\begin{itemize}\n\\item if $u \\varolessthan v$, then $\\hat{u}\\ \\hat{\\comp}\\ \\hat{v}$ for any $\\hat{u},\\hat{v}$ such that $\\hat{u} :> \\beta_{\\textit{Val}}(u)$ and $\\hat{v} :> \\beta_{\\textit{Val}}(v)$\n\\item for any $\\hat{v} :> \\beta_{\\textit{Val}}(v)$, we have $\\hat{\\odot} \\hat{v} :> \\beta_{\\textit{Val}}(\\odot v)$\n\\item for any $\\hat{u},\\hat{v}$ such that $\\hat{u} :> \\beta_{\\textit{Val}}(u)$ and $\\hat{v} :> \\beta_{\\textit{Val}}(v)$, we have $\\hat{u}\\ \\hat{\\oplus}\\ \\hat{v} :> \\beta_{\\textit{Val}}(u \\oplus v)$\n\\end{itemize}\n\\end{assumption}\n\n\\begin{assumption}[Overriding]\n\\label{asm:overriding}\nIf $\\textit{lookup}(c,m) = (c',\\mathit{st}^*)$, then $c \\leq c'$.\n\\end{assumption}\n\nIn the next results, let $\\Delta \\vdash \\Delta'$ whenever $\\Delta \\vdash \\mathsf{f}$ for each $\\mathsf{f} \\in \\Delta'$.\n\n\\begin{lemma}[Right-hand Sides]\n\\label{lem:rhs}\nLet $\\Sigma = \\methconf{\\alpha}{\\pi}{H}{S}{\\ell}$ with $\\alpha = \\locstate{\\mathit{pp}}{\\mathit{st}^*}{R}{u^*}$ and let $\\regval{\\mathit{rhs}} = v$, then for any $\\Delta :> \\beta_{\\textit{Lcnf}}(\\Sigma)$ there exists $\\hat{v}$ such that $\\beta_{\\textit{Val}}(v) \\sqsubseteq_{\\textit{Val}} \\hat{v}$ and $\\Delta \\cup \\arhs{\\mathit{rhs}} \\vdash \\prhs{\\hat{v}}$.\n\\end{lemma}\n\\begin{IEEEproof}\nBy a case analysis on the structure of $\\mathit{rhs}$.\n\\end{IEEEproof}\n\n\\begin{lemma}[Local Preservation]\n\\label{lem:local}\nIf $\\Sigma \\rightsquigarrow \\Sigma'$ under a given program $P$, then for any $\\Delta :> \\beta_{\\textit{Lcnf}}(\\Sigma)$ there exists $\\Delta' :> \\beta_{\\textit{Lcnf}}(\\Sigma')$ such that $\\translate{P} \\cup \\Delta \\vdash \\Delta'$.\n\\end{lemma}\n\\begin{IEEEproof}\n(Sketch) By a case analysis on the rule applied in the reduction step. The cases for the $\\texttt{move}$ instruction use Lemma~\\ref{lem:rhs}. The case for the $\\texttt{return}$ instruction exploits the (implicit) well-formation assumption of the local configuration $\\Sigma$. The case for the $\\texttt{invoke}$ instruction uses Assumption~\\ref{asm:overriding}. The cases for comparison operators and primitive operations exploit Assumption~\\ref{asm:sound-op}.\n\\end{IEEEproof}\n\n\\begin{lemma}[Serialization]\n\\label{lem:serialization}\nBoth the following statements hold true:\n\\begin{itemize}\n\\item if $\\serval{H}(v) = (v',H')$, then $\\beta_{\\textit{Val}}(v) = \\beta_{\\textit{Val}}(v')$\n\\item if $\\serblock{H}(b) = (b',H')$, then $\\beta_{\\textit{Blk}}(b) = \\beta_{\\textit{Blk}}(b')$\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nIf $v = \\mathit{prim}$, then $v' = \\mathit{prim}$ and $\\beta_{\\textit{Val}}(v) = \\beta_{\\textit{Val}}(v') = \\beta_{\\textit{Prim}}(\\mathit{prim})$. If $v = \\pointer{p}{\\lambda}$, then $v' = \\pointer{p'}{\\lambda}$ for some pointer $p'$ and $\\beta_{\\textit{Val}}(v) = \\beta_{\\textit{Loc}}(\\pointer{p}{\\lambda}) = \\lambda = \\beta_{\\textit{Loc}}(\\pointer{p'}{\\lambda}) = \\beta_{\\textit{Val}}(v')$. The second point is a direct consequence of the first one.\n\\end{IEEEproof}\n\n\n\\MESSAGEIfor{MS}{151202}{New definition for the size function.}\n\\begin{definition}\nWe define a function $\\bvsize{H}$ which assigns to values and blocks a natural number as follows:\n\\begin{itemize}\n\\item $\\Gamma \\vdash \\bvsize{H}(\\mathit{prim}) = 1$\n\\item $\\ell \\notin \\Gamma; \\Gamma,\\ell \\vdash\n \\bvsize{H}(\\ell) = 1 + \\bvsize{H}(H(\\ell))$\n\\item $\\ell \\in \\Gamma; \\Gamma,\\ell \\vdash \\bvsize{H}(\\ell) = 0$\n\\item $\\Gamma \\vdash \\bvsize{H}(\\obj{c}{(f_i \\mapsto v_i)^*}) = 1 + \\sum_i \\bvsize{H}(v_i)$\n\\item $\\Gamma \\vdash \\bvsize{H}(\\intent{c}{(k_i \\mapsto v_i)^*}) = 1 + \\sum_i \\bvsize{H}(v_i)$\n\\item $\\Gamma \\vdash \\bvsize{H}(\\arr{\\tau}{v^*}) = 1 + \\sum_i \\bvsize{H}(v_i)$\n\\end{itemize}\n\\end{definition}\n\n\\begin{lemma}[Heap Serialization]\n\\label{lem:heap-serialization}\nIf $\\Delta :> \\beta_{\\textit{Heap}}(H)$, then:\n\\begin{itemize}\n\\item $\\serval{H}(v) = (v',H')$ implies $\\Delta :> \\beta_{\\textit{Heap}}(H')$\n\\item $\\serblock{H}(b) = (b',H')$ implies $\\Delta :> \\beta_{\\textit{Heap}}(H')$\n\\end{itemize}\n\\end{lemma}\n\\begin{IEEEproof}\nBy simultaneous induction on the size of the syntactic element in the\nantecedent. If $v = \\mathit{prim}$, then $H'$ is empty, hence\n$\\beta_{\\textit{Heap}}(H') = \\emptyset$ and we are done. If $v =\n\\pointer{p}{\\lambda}$, then $H' = H'',\\pointer{p'}{\\lambda} \\mapsto\nb$ with $\\serblock{H}(H(\\pointer{p}{\\lambda})) = (b, H'')$\nand $v' = \\pointer{p'}{\\lambda}$. By induction hypothesis $\\Delta :> \\beta_{\\textit{Heap}}(H'')$, so to conclude we just need to show that:\n\\[\n\\begin{array}{lcll}\n\\Delta & :> & \\beta_{\\textit{Heap}}(\\pointer{p'}{\\lambda} \\mapsto b) \\\\\n & = & \\{\\mathsf{H}(\\lambda,\\beta_{\\textit{Blk}}(b))\\} & \\text{by definition} \\\\\n & = & \\{\\mathsf{H}(\\lambda,\\beta_{\\textit{Blk}}(H(\\pointer{p}{\\lambda})))\\} & \\text{by Lemma~\\ref{lem:serialization}} \\\\\n & = & \\beta_{\\textit{Heap}}(\\pointer{p}{\\lambda} \\mapsto H(\\pointer{p}{\\lambda})) & \\text{by definition}\n\\end{array}\n\\]\nbut this follows from the hypothesis $\\Delta :> \\beta_{\\textit{Heap}}(H)$. The remaining cases for blocks follow by inductive hypothesis.\n\\end{IEEEproof}\n\n\\begin{theorem}[Preservation]\n\\label{trm:preservation}\nIf $\\Psi \\Rightarrow^* \\Psi'$ under a given program $P$, then there exists $\\Delta :> \\beta_{\\textit{Cnf}}(\\Psi')$ such that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta$.\n\\end{theorem}\n\\begin{IEEEproof}\nBy induction on the length of the reduction sequence. If the reduction sequence is empty, we have $\\Psi' = \\Psi$ and the result follows by picking $\\Delta = \\beta_{\\textit{Cnf}}(\\Psi)$. Otherwise, assume that $\\Psi \\Rightarrow^* \\actconf{\\Omega}{H}{S}$ in $n \\geq 0$ reduction steps and let $\\actconf{\\Omega}{H}{S} \\Rightarrow \\actconf{\\Omega'}{H'}{S'}$. By induction hypothesis there exists $\\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$ such that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta'$, we show that there exists $\\Delta$ such that $\\Delta :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'})$ and $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta$. The proof is by a case analysis on the rule applied in the last reduction step:\n\\begin{itemize}\n\\item[\\irule{A-Active}]: let $\\Omega = \\Omega_0 :: \\uactframe{\\ell}{s}{\\pi}{\\alpha} :: \\Omega_1$ and $\\Omega' = \\Omega_0 :: \\uactframe{\\ell}{s}{\\pi'}{\\alpha'} :: \\Omega_1$ with $\\methconf{\\alpha}{\\pi}{H}{S}{\\ell} \\rightsquigarrow \\methconf{\\alpha'}{\\pi'}{H'}{S'}{\\ell}$. Since $\\beta_{\\textit{Lcnf}}(\\methconf{\\alpha}{\\pi}{H}{S}{\\ell}) \\subseteq \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have $\\beta_{\\textit{Lcnf}}(\\methconf{\\alpha}{\\pi}{H}{S}{\\ell}) <: \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$ by Lemma~\\ref{lem:sub-order}. Since $\\beta_{\\textit{Lcnf}}(\\methconf{\\alpha}{\\pi}{H}{S}{\\ell}) <: \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$ and $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S}) <: \\Delta'$, we get $\\beta_{\\textit{Lcnf}}(\\methconf{\\alpha}{\\pi}{H}{S}{\\ell}) <: \\Delta'$ by Lemma~\\ref{lem:trans-order}. \nHence, by Lemma~\\ref{lem:local} there exists $\\Delta'' :> \\beta_{\\textit{Lcnf}}(\\methconf{\\alpha'}{\\pi'}{H'}{S'}{\\ell})$ such that $\\translate{P} \\cup \\Delta' \\vdash \\Delta''$. By the weakening property of the logic, the latter implies $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\cup \\Delta' \\vdash \\Delta''$. Since we have $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta'$ and $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\cup \\Delta' \\vdash \\Delta''$, we get $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta''$ by the admissibility of the cut rule. Recall now that $\\Delta'' :> \\beta_{\\textit{Lcnf}}(\\methconf{\\alpha'}{\\pi'}{H'}{S'}{\\ell}) = \\beta_{\\textit{Call}}(\\alpha') \\cup \\rfdispatch{\\ell}(\\pi') \\cup \\beta_{\\textit{Heap}}(H') \\cup \\beta_{\\textit{Stat}}(S')$, so we have:\n\\begin{itemize}\n\\item[(1)] $\\Delta'' :> \\beta_{\\textit{Call}}(\\alpha')$\n\\item[(2)] $\\Delta'' :> \\rfdispatch{\\ell}(\\pi')$\n\\item[(3)] $\\Delta'' :> \\beta_{\\textit{Heap}}(H')$\n\\item[(4)] $\\Delta'' :> \\beta_{\\textit{Stat}}(S')$\n\\end{itemize}\nWe then observe that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, which similarly implies:\n\\begin{itemize}\n\\item[(5)] $\\Delta' :> \\beta_{\\textit{Stk}}(\\Omega_0)$\n\\item[(6)] $\\Delta' :> \\beta_{\\textit{Stk}}(\\Omega_1)$\n\\end{itemize}\nCombining all these facts, we get $\\Delta' \\cup \\Delta'' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'})$ by Lemma~\\ref{lem:join-order}. Given that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' \\cup \\Delta''$, we conclude the case;\n\n\\item[\\irule{A-Deactivate}]: in this case $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S}) = \\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'})$, hence the conclusion immediately follows by the induction hypothesis;\n\n\\item[\\irule{A-Step}]: let $\\Omega = \\actframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} :: \\Omega_0$ and $\\Omega' = \\uactframe{\\ell}{s'}{\\pi}{\\getcb{\\ell}{s'}} :: \\Omega_0$ for some $(s,s') \\in \\textit{Lifecycle}$, $H' = H$ and $S' = S$. Since $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have:\n\\begin{itemize}\n\\item[(1)] $\\Delta' :> \\beta_{\\textit{Stk}}(\\Omega_0)$\n\\item[(2)] $\\Delta' :> \\rfdispatch{\\ell}(\\pi)$\n\\end{itemize}\nSince we only focus on well-formed configurations, we have $H(\\ell) = \\obj{c}{(f \\mapsto u)^*}$ for some activity class $c$ and $\\ell = \\pointer{p}{c}$ for some pointer $p$. We then observe that $\\getcb{\\ell}{s'} = \\locstate{c',m,0}{\\mathit{st}^*}{R}{v^*} :: \\varepsilon$, where $(c',\\mathit{st}^*) = \\textit{lookup}(c,m)$ for some $m \\in \\mathit{cb}(c,s)$, $\\textit{sign}(c',m) = \\methsign{\\tau_1,\\ldots,\\tau_n}{\\tau}{\\mathit{loc}}$ and:\n\\[ \nR = ((r_i \\mapsto \\mathbf{0})^{i \\leq \\mathit{loc}}, r_{\\mathit{loc}+1} \\mapsto \\ell, (r_{\\mathit{loc}+1+j} \\mapsto v_j)^{j \\leq n}),\n\\]\nfor some values $v_1,\\ldots,v_n$ of the correct type $\\tau_1,\\ldots,\\tau_n$. By Assumption~\\ref{asm:overriding}, we also have $c \\leq c'$.\n\nGiven that $\\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have $\\Delta' :> \\beta_{\\textit{Heap}}(H)$, which implies that there exists $\\mathsf{H}(\\lambda,\\hat{b}) \\in \\Delta'$ such that $\\lambda = \\beta_{\\textit{Loc}}(\\ell) = c$ and $\\hat{b} \\sqsupseteq \\beta_{\\textit{Blk}}(\\obj{c}{(f \\mapsto u)^*})$. This implies that $\\hat{b} = \\absobj{c}{(f \\mapsto \\hat{v})^*}$ for some $v^*$ such that $\\forall i: \\hat{v}_i \\sqsupseteq \\beta_{\\textit{Val}}(u_i)$. Since $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta'$ and $\\mathsf{H}(\\lambda,\\hat{b}) = \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{v})^*}) \\in \\Delta'$, we have in particular $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{v})^*})$, hence:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}},\n\\]\nby using the implications $\\rulename{Cbk}$ included in $\\translate{P}$. We then observe that:\n\\[\n\\{\\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}}\\} :> \\beta_{\\textit{Call}}(\\getcb{\\ell}{s'})\n\\]\nBy combining (1), (2) and the last observation through Lemma~\\ref{lem:join-order} we then get:\n\\[\n\\{\\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}}\\} \\cup \\Delta' :> \\beta_{\\textit{Call}}(\\getcb{\\ell}{s'}) \\cup \\beta_{\\textit{Stk}}(\\Omega_0) \\cup \\rfdispatch{\\ell}(\\pi) = \\beta_{\\textit{Stk}}(\\Omega')\n\\]\nSince $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\{\\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}}\\} \\cup \\Delta'$, we conclude the case;\n\n\\item[\\irule{A-Destroy}]: in this case $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'}) \\subseteq \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, hence $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'}) <: \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$ by Lemma~\\ref{lem:sub-order}. Since $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'}) <: \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$ and $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S}) <: \\Delta'$, we have $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'}) <: \\Delta'$ by Lemma~\\ref{lem:trans-order}. Given that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta'$, we conclude the case;\n\n\\item[\\irule{A-Back}]: let $\\Omega' = \\Omega = \\actframe{\\ell}{\\actstate{running}}{\\varepsilon}{\\overline{\\callstack}} :: \\Omega_0$, $H' = H[\\ell \\mapsto H(\\ell) [\\textit{finished} \\mapsto \\mathtt{true}]]$ and $S' = S$. Let $b = H(\\ell)$. Since we only focus on well-formed configurations, we have $b = \\obj{c}{(f \\mapsto u)^*,\\textit{finished} \\mapsto v}$ for some activity class $c$ and some boolean value $v$. Let then $b' = H'(\\ell) = \\obj{c}{(f \\mapsto u)^*,\\textit{finished} \\mapsto \\mathtt{true}}$ according to the reduction rule. \n\nGiven that $\\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have $\\Delta' :> \\beta_{\\textit{Heap}}(H)$, which implies that there exists $\\mathsf{H}(\\lambda,\\hat{b}) \\in \\Delta'$ such that $\\lambda = \\beta_{\\textit{Loc}}(\\ell)$ and $\\hat{b} \\sqsupseteq \\beta_{\\textit{Blk}}(b)$. This means that $\\hat{b} = \\absobj{c}{(f \\mapsto \\hat{u})^*,\\textit{finished} \\mapsto \\hat{v}}$ for some $u^*,v$ such that $\\forall i: \\hat{u}_i \\sqsupseteq \\beta_{\\textit{Val}}(u)$ and $\\hat{v} \\sqsupseteq \\beta_{\\textit{Val}}(v)$. We then observe that:\n\\[\n\\beta_{\\textit{Blk}}(b') = \\absobj{c}{(f \\mapsto \\beta_{\\textit{Val}}(u))^*, \\textit{finished} \\mapsto \\widehat{\\mathtt{true}}}\n\\]\n\nSince $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta'$ and $\\mathsf{H}(\\lambda,\\hat{b}) \\in \\Delta'$, we have in particular $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(\\lambda,\\hat{b})$, hence:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(\\lambda,\\absobj{c}{(f \\mapsto \\hat{u})^*,\\textit{finished} \\mapsto \\top_{\\type{bool}}}),\n\\]\nby using the implication $\\rulename{Fin}$ included in $\\translate{P}$. We then observe that:\n\\[\n\\begin{array}{lcll}\n\\mathsf{H}(\\lambda,\\absobj{c}{(f \\mapsto \\hat{u})^*,\\textit{finished} \\mapsto \\top_{\\type{bool}}}) & \\sqsupseteq & \\mathsf{H}(\\lambda,\\absobj{c}{(f \\mapsto \\hat{u})^*,\\textit{finished} \\mapsto \\widehat{\\mathtt{true}}}) \\\\\n& = & \\mathsf{H}(\\beta_{\\textit{Loc}}(\\ell),\\absobj{c}{(f \\mapsto \\hat{u})^*,\\textit{finished} \\mapsto \\widehat{\\mathtt{true}}}) \\\\\n& \\sqsupseteq & \\mathsf{H}(\\beta_{\\textit{Loc}}(\\ell), \\beta_{\\textit{Blk}}(b'))\n\\end{array}\n\\]\nHence, $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' \\cup \\{\\mathsf{H}(\\lambda,\\absobj{c}{(f \\mapsto \\hat{u})^*,\\textit{finished} \\mapsto \\top_{\\type{bool}}})\\} :> \\beta_{\\textit{Heap}}(H')$, which is enough to conclude the case;\n\n\\item[\\irule{A-Replace}]: let $\\Omega = \\actframe{\\ell}{\\actstate{onDestroy}}{\\pi}{\\overline{\\callstack}} :: \\Omega_0$ and $\\Omega' = \\uactframe{\\pointer{p}{c}}{\\actstate{constructor}}{\\pi}{\\getcb{\\pointer{p}{c}}{\\actstate{constructor}}} :: \\Omega_0$ with $H(\\ell) = \\obj{c}{(f \\mapsto v)^*,\\textit{finished} \\mapsto u}$, $H' = H, \\pointer{p}{c} \\mapsto o$ with $o = \\obj{c}{(f \\mapsto \\mathbf{0}_{\\tau})^*,\\textit{finished} \\mapsto \\mathtt{false}}$, and $S' = S$. Since we only focus on well-formed configurations, we know that $c$ is an activity class and $\\ell = \\pointer{p'}{c}$ for some pointer $p'$.\n\nGiven that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta':> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have:\n\\begin{itemize}\n\\item[(1)] $\\Delta' :> \\rfdispatch{\\ell}(\\pi)$\n\\item[(2)] $\\Delta' :> \\beta_{\\textit{Stk}}(\\Omega_0)$\n\\end{itemize}\nSince $\\beta_{\\textit{Loc}}(\\ell) = \\beta_{\\textit{Loc}}(\\pointer{p'}{c}) = \\beta_{\\textit{Loc}}(\\pointer{p}{c})$, from (1) we get:\n\\begin{itemize}\n\\item[(3)] $\\Delta' :> \\rfdispatch{\\pointer{p}{c}}(\\pi)$\n\\end{itemize}\nWe then observe that $\\getcb{\\pointer{p}{c}}{\\actstate{constructor}} = \\locstate{c',m,0}{\\mathit{st}^*}{R}{v^*} :: \\varepsilon$, where $(c',\\mathit{st}^*) = \\textit{lookup}(c,\\actstate{constructor})$, $\\textit{sign}(c',\\actstate{constructor}) = \\methsign{\\tau_1,\\ldots,\\tau_n}{\\tau}{\\mathit{loc}}$ and:\n\\[ \nR = ((r_i \\mapsto \\mathbf{0})^{i \\leq \\mathit{loc}}, r_{\\mathit{loc}+1} \\mapsto \\pointer{p}{c}, (r_{\\mathit{loc}+1+j} \\mapsto v_j')^{j \\leq n}),\n\\]\nfor some values $v_1',\\ldots,v_n'$ of the correct type $\\tau_1,\\ldots,\\tau_n$. By Assumption~\\ref{asm:overriding}, we also have $c \\leq c'$.\n\nGiven that $\\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have $\\Delta' :> \\beta_{\\textit{Heap}}(H)$, which implies that there exists $\\mathsf{H}(\\lambda,\\hat{b}) \\in \\Delta'$ such that $\\lambda = \\beta_{\\textit{Loc}}(\\ell) = c$ and $\\hat{b} \\sqsupseteq \\beta_{\\textit{Blk}}(H(\\ell))$. This implies that $\\hat{b} = \\absobj{c}{(f \\mapsto \\hat{v})^*,\\textit{finished} \\mapsto \\hat{u}}$ for some $\\hat{v}^*,\\hat{u}$ such that $\\forall i: \\hat{v}_i \\sqsupseteq \\beta_{\\textit{Val}}(v_i)$ and $\\hat{u} \\sqsupseteq \\beta_{\\textit{Val}}(u)$. Since $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta'$ and $\\mathsf{H}(\\lambda,\\hat{b}) \\in \\Delta'$, we have in particular $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(\\lambda,\\hat{b}) = \\mathsf{H}(c, \\absobj{c}{(f \\mapsto \\hat{v})^*,\\textit{finished} \\mapsto \\hat{u}})$, hence:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}},\n\\]\nby using the implications $\\rulename{Cbk}$ included in $\\translate{P}$. We then observe that:\n\\[\n\\{\\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}}\\} :> \\beta_{\\textit{Call}}(\\getcb{\\pointer{p}{c}}{\\actstate{constructor}})\n\\]\nBy combining (2), (3) and the last observation through Lemma~\\ref{lem:join-order} we then get:\n\\[\n\\{\\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}}\\} \\cup \\Delta' :> \\beta_{\\textit{Call}}(\\getcb{\\pointer{p}{c}}{\\actstate{constructor}}) \\cup \\beta_{\\textit{Stk}}(\\Omega_0) \\cup \\rfdispatch{\\pointer{p}{c}}(\\pi) = \\beta_{\\textit{Stk}}(\\Omega')\n\\]\nSince $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\{\\absreg{c',m,0}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}})^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}}\\} \\cup \\Delta'$, we proved that the change to the activity stack is correctly over-approximated.\n\nTo conclude, we need to deal with the change to the heap. We first observe that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$ and $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S}) :> \\beta_{\\textit{Heap}}(H)$, hence:\n\\begin{itemize}\n\\item[(4)] $\\Delta':> \\beta_{\\textit{Heap}}(H)$\n\\end{itemize}\nSince $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(\\lambda,\\hat{b}) = \\mathsf{H}(c, \\absobj{c}{(f \\mapsto \\hat{v})^*,\\textit{finished} \\mapsto \\hat{u}})$, we have\\footnote{We assume here that boolean fields are initialized to $\\mathtt{false}$. The proof can be adapted to the case where they are initialized to $\\mathtt{true}$ by using the implication in rule $\\rulename{Fin}$.}:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*,\\textit{finished} \\mapsto \\widehat{\\mathtt{false}}})),\n\\]\nby using the implication $\\rulename{Rep}$. We then observe that:\n\\[\n\\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*,\\textit{finished} \\mapsto \\widehat{\\mathtt{false}}}))\\} :> \\beta_{\\textit{Heap}}(\\pointer{p}{c} \\mapsto o).\n\\]\nBy combining (4) with the latter observation by Lemma~\\ref{lem:join-order}, we get:\n\\[\n\\Delta' \\cup \\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*,\\textit{finished} \\mapsto \\widehat{\\mathtt{false}}}))\\} :> \\beta_{\\textit{Heap}}(H')\n\\]\nSince $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' \\cup \\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*,\\textit{finished} \\mapsto \\widehat{\\mathtt{false}}}))\\}$, we proved that also the change to the heap is over-approximated correctly;\n\n\\item[\\irule{A-Hidden}]: analogous to case \\irule{A-Step};\n\n\\item[\\irule{A-Start}]: let $\\Omega = \\actframe{\\ell}{s}{i :: \\pi}{\\overline{\\callstack}} :: \\Omega_0$ and $\\Omega' = \\uactframe{\\pointer{p}{c}}{\\actstate{constructor}}{\\varepsilon}{\\getcb{\\pointer{p}{c}}{\\actstate{constructor}}} :: \\actframe{\\ell}{s}{\\pi}{\\overline{\\callstack}} :: \\Omega_0$ with $i = \\intent{c}{(k \\mapsto v)^*}$. Also, let $S' = S$ and $H' = H,H'',\\pointer{p}{c} \\mapsto o, \\pointer{p'}{\\astart{c}} \\mapsto i'$ with $\\serblock{H}(i) = (i',H'')$ and $o = \\obj{c}{(f \\mapsto \\mathbf{0}_{\\tau})^*,\\textit{finished} \\mapsto \\mathtt{false}, \\textit{intent} \\mapsto \\pointer{p'}{\\astart{c}}, \\textit{parent} \\mapsto \\ell}$. Since we only focus on well-formed configurations, we know that $\\ell = \\pointer{p'}{c'}$ for some pointer $p'$ and some activity class $c'$.\n\nGiven that $\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta':> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have $\\Delta' :> \\rfdispatch{\\ell}(i :: \\pi)$, which implies that there exists $\\mathsf{I}(\\lambda,\\hat{b}) \\in \\Delta'$ such that $\\lambda = \\beta_{\\textit{Loc}}(\\ell) = c'$ and $\\hat{b} \\sqsupseteq \\beta_{\\textit{Blk}}(i)$. This implies that $\\hat{b} = \\absintent{c}{\\hat{v}}$ for some $\\hat{v}$ such that $\\hat{v} \\sqsupseteq \\sqcup_i\\, \\beta_{\\textit{Val}}(v_i)$. We then have:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(\\astart{c},\\absintent{c}{\\hat{v}}),\n\\]\nand:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(c, \\absobj{c}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*, \\textit{finished} \\mapsto \\widehat{\\mathtt{false}}, \\textit{parent} \\mapsto c', \\textit{intent} \\mapsto \\astart{c}}),\n\\]\nby using the implications $\\rulename{Act}$ included in $\\translate{P}$. Using the latter fact and the implications $\\rulename{Cbk}$, we can prove that the change to the activity stack is over-approximated correctly, similarly to what we did in case \\irule{A-Replace}: we omit details.\n\nWe focus instead on the changes to the heap. Since $\\Delta' :> \\beta_{\\textit{Heap}}(H)$ and $\\serblock{H}(i) = (i',H'')$, we know that $\\Delta' :> \\beta_{\\textit{Heap}}(H'')$ by Lemma~\\ref{lem:heap-serialization}. We then observe that:\n\\[\n\\{\\mathsf{H}(c, \\absobj{c}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*, \\textit{finished} \\mapsto \\widehat{\\mathtt{false}}, \\textit{parent} \\mapsto c', \\textit{intent} \\mapsto \\astart{c}})\\} = \\beta_{\\textit{Heap}}(\\pointer{p}{c} \\mapsto o)\n\\]\nFinally, we notice that:\n\\[\n\\begin{array}{lcll}\n\\{\\mathsf{H}(\\astart{c},\\absintent{c}{\\hat{v}})\\} & :> & \\{\\mathsf{H}(\\astart{c},\\beta_{\\textit{Blk}}(i)\\} & \\text{since } \\hat{b} = \\absintent{c}{\\hat{v}}) \\sqsupseteq \\beta_{\\textit{Blk}}(i) \\\\\n& = & \\beta_{\\textit{Heap}}(\\pointer{p'}{\\astart{c}} \\mapsto i) & \\text{by definition} \\\\\n& = & \\beta_{\\textit{Heap}}(\\pointer{p'}{\\astart{c}} \\mapsto i') & \\text{by Lemma~\\ref{lem:serialization}}\n\\end{array}\n\\]\nBy combining all these observations, we prove that the new heap is over-approximated correctly;\n\n\\item[\\irule{A-Swap}]: in this case $\\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S}) = \\beta_{\\textit{Cnf}}(\\actconf{\\Omega'}{H'}{S'})$, hence the conclusion immediately follows by the induction hypothesis;\n\n\\item[\\irule{A-Result}]: let:\n\\[\n\\Omega = \\actframe{\\ell'}{\\actstate{onPause}}{\\varepsilon}{\\overline{\\callstack}'} :: \\actframe{\\ell}{s}{\\varepsilon}{\\overline{\\callstack}} :: \\Omega_0,\n\\]\nand: \n\\[\n\\Omega' = \\uactframe{\\ell}{s}{\\varepsilon}{\\getcb{\\ell}{\\actstate{onActivityResult}}} :: \\actframe{\\ell'}{\\actstate{onPause}}{\\varepsilon}{\\overline{\\callstack}'} :: \\Omega_0,\n\\]\nwith $H(\\ell').\\textit{parent} = \\ell$. Also, let $S' = S$ and $H' = (H,H'')[\\ell \\mapsto H(\\ell)[\\textit{result} \\mapsto \\ell'']]$ with: \n\\[\n\\serval{H}(H(\\ell').\\textit{result}) = (\\ell'',H'').\n\\]\nSince we focus only on well-formed configurations, we have $\\ell = \\pointer{p}{c}$ and $\\ell' = \\pointer{p'}{c'}$ for some pointers $p,p'$ and some activity classes $c,c'$. Also, let $H(\\ell) = \\obj{c}{(f \\mapsto \\hat{v})^*}$ and $H(\\ell') = \\obj{c'}{(f' \\mapsto \\hat{v}')^*, \\textit{parent} \\mapsto \\ell}$. Since $H(\\ell) = \\obj{c}{(f \\mapsto \\hat{v})^*}$, to prove that the changes to the activity stack are correctly over-approximated we can proceed like in case \\irule{A-Step}, using the implications in \\rulename{Cbk}: we omit details.\n\nWe focus instead on the changes to the heap. Since $\\Delta' :> \\beta_{\\textit{Cnf}}(\\actconf{\\Omega}{H}{S})$, we have in particular:\n\\begin{itemize}\n\\item[(1)] $\\Delta' :> \\beta_{\\textit{Heap}}(H)$\n\\end{itemize}\nBy (1) and $\\serval{H}(H(\\ell').\\textit{result}) = (\\ell'',H'')$, using Lemma~\\ref{lem:heap-serialization}, we prove:\n\\begin{itemize}\n\\item[(2)] $\\Delta' :> \\beta_{\\textit{Heap}}(H'')$ \n\\end{itemize}\nAgain by (1), there exists $\\mathsf{H}(\\lambda,\\hat{b}) \\in \\Delta'$ such that $\\lambda = \\beta_{\\textit{Loc}}(\\ell) = c$ and $\\hat{b} \\sqsupseteq \\beta_{\\textit{Blk}}(H(\\ell))$. This implies that $\\hat{b} = \\absobj{c}{(f \\mapsto \\hat{v})^*}$ for some $\\hat{v}^*$ s.t. $\\forall i: \\hat{v}_i \\sqsupseteq \\beta_{\\textit{Val}}(v_i)$. Similarly, we show that there exists $\\mathsf{H}(\\lambda',\\hat{b}') \\in \\Delta'$ s.t. $\\lambda' = \\beta_{\\textit{Loc}}(\\ell') = c'$ and $\\hat{b}' \\sqsupseteq \\beta_{\\textit{Blk}}(H(\\ell'))$, and $\\hat{b}' = \\absobj{c'}{(f' \\mapsto \\hat{v}')^*,\\textit{parent} \\mapsto c,\\textit{result} \\mapsto \\lambda''}$ for some $\\hat{v}'^*,\\lambda''$ such that $\\forall i: \\hat{v}_i' \\sqsupseteq \\beta_{\\textit{Val}}(v_i')$ and $\\lambda'' = \\beta_{\\textit{Loc}}(H(\\ell').\\textit{result})$. Hence, we have:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{v})^*}) \\wedge \\mathsf{H}(c',\\absobj{c'}{(f' \\mapsto \\hat{v}')^*,\\textit{parent} \\mapsto c}),\n\\]\nwhich allows us to prove:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{v})^*[\\textit{result} \\mapsto \\lambda'']}),\n\\]\nby using the implication $\\rulename{Res}$. We then observe that:\n\\[\n\\begin{array}{lcll}\n\\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{v})^*[\\textit{result} \\mapsto \\lambda'']})\\} & :> & \\beta_{\\textit{Heap}}(\\ell \\mapsto H(\\ell)[\\textit{result} \\mapsto H(\\ell').\\textit{result}]) & \\text{by definition} \\\\\n& = & \\beta_{\\textit{Heap}}(\\ell \\mapsto H(\\ell)[\\textit{result} \\mapsto \\ell'']) & \\text{by Lemma~\\ref{lem:serialization}}\n\\end{array}\n\\]\nSince $H' = (H,H'')[\\ell \\mapsto H(\\ell)[\\textit{result}\n\\mapsto \\ell'']] = H[\\ell \\mapsto H(\\ell)[\\textit{result} \\mapsto\n\\ell'']],H''$, by combining (1), (2) and the last observation\nusing Lemma~\\ref{lem:join-order}, we conclude as follows:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta' \\cup \\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\hat{v})^*[\\textit{result} \\mapsto \\lambda'']})\\} :> \\beta_{\\textit{Heap}}(H')\n\\]\n\\end{itemize}\n\\end{IEEEproof}\n\n\n\n\n\\subsection{Overview}\nThe analysis is based on the syntactic categories in Table~\\ref{tab:absdoms}. We start by discussing how values are approximated. We presuppose the existence of an arbitrary set of abstract domains used to approximate primitive values: for each primitive value $\\mathit{prim}$, we assume that there exists a corresponding abstraction $\\widehat{\\prim}$, e.g., integer numbers could be approximated by their sign. Locations of the form $\\ell = \\pointer{p}{\\lambda}$, instead, are abstracted into their annotation $\\lambda$. An \\emph{abstract value} $\\hat{v}$ is a set of elements drawn from either the abstract domains or the set of annotations.\n\n\\begin{table*}[htb]\n\\begin{mathpar}\n\\begin{array}{llcl}\n\\text{Facts} & \\mathsf{f} & ::= & \\\\\n\\text{Abs. registers} & & | & \\absreg{\\mathsf{pp}}{t^*}{t^*} \\\\\n\\text{Abs. heap entries} & & | & \\mathsf{H}(t,t') \\\\\n\\text{Abs. static fields} & & | & \\mathsf{S}_{\\mathsf{c},\\mathsf{f}}(t) \\\\\n\\text{Abs. right-hand sides} & & | & \\prhs{t} \\\\\n\\text{Abs. results} & & | & \\absresult{\\mathsf{c},\\mathsf{m}}{t^*}{t} \\\\\n\\text{Abs. pending activities} & & | & \\mathsf{I}(t,t') \\\\\n\\text{Set membership} & & | & t \\in t' \\\\\n\\text{Subtyping} & & | & t \\leq t' \\\\\n\\text{Horn clauses} & & | & \\forall x^*.\\bigwedge_i \\mathsf{f}_i \\implies \\mathsf{f} \\\\\n\\text{Abs. programs} & \\Delta & ::= & \\{\\mathsf{f}_1,\\ldots,\\mathsf{f}_n\\}\n\\end{array}\n\\begin{array}{llcl}\n\\text{Abs. values} & \\hat{u},\\hat{v} & ::= & \\emptyset ~|~ \\{\\widehat{\\prim}\\} ~|~ \\{\\lambda\\} ~|~ \\hat{v} \\cup \\hat{v} \\\\\n\\text{Abs. objects} & \\hat{o} & ::= & \\absobj{c}{(f_{\\tau} \\mapsto \\hat{v})^*} \\\\\n\\text{Abs. arrays} & \\hat{a} & ::= & \\absarray{\\tau}{\\hat{v}} \\\\\n\\text{Abs. intents} & \\hat{i} & ::= & \\absintent{c}{\\hat{v}} \\\\\n\\text{Abs. mem. blocks} & \\hat{b} & ::= & \\hat{o} ~|~ \\hat{a} ~|~ \\hat{i} \\\\\n\\\\\n\\text{Variables} & x,y & \\in & \\textit{Vars} \\\\\n\\text{Constants} & \\const{k} & ::= & \\hat{v} ~|~ \\hat{b} ~|~ \\tau ~|~ \\lambda \\\\\n\\text{Terms} & t & ::= & \\const{k} ~|~ x ~|~ \\astart{t}\n\\end{array}\n\\end{mathpar}\n\\caption{Abstract Domains and Analysis Facts}\n\\label{tab:absdoms}\n\\end{table*}\n\nThe different forms of annotations $\\lambda$ provide insight on different\naspects of the static analysis. Program point annotations $\\mathit{pp} =\nc,m,\\mathit{pc}$ are used to represent pointers to memory blocks instantiated\nusing the statements \\texttt{new}, \\texttt{newarray} and\n\\texttt{newintent}: by abstracting these elements with the program\npoint where they are created, we implement a\n\\emph{plain-object-sensitive} static\nanalysis~\\cite{Smaragdakis:2011:PYC:1926385.1926390}. {We chose it because it is \nwell-understood and convenient to both formalize and\npresent: we plan to integrate more advanced analyses like 2full+1H in\nfuture releases. Class name \n annotations $c$, instead, are used to represent activities in an \n object-insensitive way: different activities of the same class $c$ are all \n abstracted by the annotation $c$, since it is generally hard to statically \n discriminate between different activity instances. Finally, we use the annotation \n $\\astart{c}$ to abstract all the intents which are used to start an activity of class \n $c$.\n\nComing to memory blocks, our analysis is field-sensitive on objects, but field-insensitive on both arrays and intents. It is easier to implement field-sensitivity for objects, since field names are statically known in Java. Implementing field-sensitivity for arrays would require precise information on array bounds and indexes; intents, instead, would need an accurate string analysis, to deal with their dictionary-like programming patterns. It would be possible to leverage existing proposals~\\cite{DilligDA11} to implement a more precise analysis in terms of field-sensitivity, but we propose a simpler framework here to focus on the Android-specific aspects of the analysis. Notice that, just like the objects they approximate, abstract objects $\\hat{o}$ feature type annotations on their fields, which are omitted when unimportant. \n\nAbstract values and abstract memory blocks, plus all the types available in the analysed program and the annotations, determine a universe of \\emph{constants}, ranged over by $\\const{k}$. A \\emph{term} $t$ is either a constant $\\const{k}$, a variable $x$ drawn from a denumerable set $\\textit{Vars}$ disjoint from the set of constants, or an expression of the form $\\astart{t'}$ for some term $t'$. The set of terms is used to define the syntax of \\emph{facts} $\\mathsf{f}$, logical formulas built on selected predicate symbols used by the analysis.\n\nThe fact $\\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc}}{\\hat{u}^*}{\\hat{v}^*}$ states that, whenever the method $m$ of class $c$ is invoked with some arguments over-approximated by $\\hat{u}^*$, the state of the local registers at the $\\mathit{pc}$-th statement is over-approximated by $\\hat{v}^*$. The syntax of the fact highlights that: (1) the analysis is flow-sensitive for register values, since it computes different static approximations at different program points, and (2) method invocations are handled in a context-sensitive way, where the notion of context coincides with the (abstraction of) the actual arguments supplied to the method upon invocation. The fact $\\mathsf{H}(\\lambda,\\hat{b})$ states that some location $\\pointer{p}{\\lambda}$ refers to a heap element storing a memory block over-approximated by $\\hat{b}$ at some point of the program execution. Notice that the fact does not contain any program point information, i.e., the analysis is flow-insensitive for heap locations, which is important for soundness (see Section~\\ref{sec:examples}). Similarly, the fact $\\mathsf{S}_{\\mathsf{c},\\mathsf{f}}(\\hat{v})$ states that the static field $f$ of class $c$ contains a value which is over-approximated by $\\hat{v}$ at some point of the program execution. The fact $\\prhs{\\hat{v}}$ states that the right-hand side of the \\texttt{move} statement at program point $\\mathit{pp}$ evaluates to a value over-approximated by $\\hat{v}$. The fact $\\absresult{\\mathsf{c},\\mathsf{m}}{\\hat{u}^*}{\\hat{v}}$ states that, whenever the method $m$ of class $c$ is invoked with some arguments over-approximated by $\\hat{u}^*$, its return value is over-approximated by $\\hat{v}$. The fact $\\mathsf{I}(c,\\hat{i})$ tracks that an activity of class $c$ has sent an intent which is over-approximated by $\\hat{i}$. We then have set membership facts $t \\in t'$ and subtyping facts $\\tau \\leq \\tau'$ with the obvious meaning. \n\nFinally, Horn clauses define the abstract semantics of programs. A Horn clause has the form:\n\\[ \n\\forall x_1,\\ldots,\\forall x_m.\\mathsf{f}_1 \\wedge \\ldots \\wedge \\mathsf{f}_n \\implies \\mathsf{f},\n\\]\nwhere all the variables of $\\mathsf{f}_1,\\ldots,\\mathsf{f}_n,\\mathsf{f}$ belong to $\\{x_1,\\ldots,x_m\\}$ and each variable of $\\mathsf{f}$ occurs among the variables of $\\mathsf{f}_1,\\ldots,\\mathsf{f}_n$. Since most of the Horn clauses we present do not make use of constants, to improve readability we omit the universal quantifiers in front of Horn clauses and we just represent each variable occurring therein with a constant of the expected type. The few exceptions where constants are actually used are disambiguated using a $\\mathsf{sans\\ serif}$ font, e.g., we use $\\const{c}$ to denote the constant corresponding to the activity class $c$ specifically, rather than some universally quantified variable standing for an arbitrary activity class. We let an underscore (\\_) stand for any syntactic element occurring in a Horn clause which is not significant to understanding.\n\n\\subsection{Analysis Specification}\n\n\\subsubsection{Abstract Semantics of Dalvik}\nWe start by presenting the abstract evaluation rules for right-hand sides, which are simple and provide a good intuition on how the static analysis works. These rules are given in Table~\\ref{tab:abs-rhs}.\n\n\\begin{table*}[htb]\n\\begin{mathpar}\n\\arhs{\\mathit{prim}} = \\{\\prhs{\\{\\widehat{\\prim}\\}}\\}\n\n\\arhs{r_i} = \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\prhs{\\hat{v}_i }\\}\n\n\\arhs{c.f} = \\{\\mathsf{S}_{\\mathsf{c},\\mathsf{f}}(\\hat{v}) \\implies \\prhs{\\hat{v}}\\}\n\n\\arhs{r_i.f} = \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_i \\wedge \\mathsf{H}(\\lambda,\\absobj{c}{(f' \\mapsto \\hat{v}')^*, f \\mapsto \\hat{u}}) \\implies \\prhs{\\hat{u}}\\}\n\n\\arhs{r_i[r_j]} = \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_i \\wedge \\mathsf{H}(\\lambda, \\absarray{\\tau}{\\hat{u}}) \\implies \\prhs{\\hat{u}}\\}\n\\end{mathpar}\n\\caption{Abstract Evaluation of Right-hand Sides}\n\\label{tab:abs-rhs}\n\\end{table*}\n\nTo abstract a primitive value $\\mathit{prim}$ at any program point $\\mathit{pp}$, we just pick the corresponding element $\\widehat{\\prim}$ from the underlying abstract domain. To abstract the content of the register $r_i$ at program point $\\mathit{pp}$, we take the fact $\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*}$ and we return the $i$-th abstract value $\\hat{v}_i$. To abstract the content of a static field $c.f$ at any program point, we take any fact $\\mathsf{S}_{\\mathsf{c},\\mathsf{f}}(\\hat{v})$ and we return the abstract value $\\hat{v}$. Abstracting the content of the field $f$ of an object at program point $\\mathit{pp}$ is slightly more complicated: if the pointer to the object is stored in the register $r_i$, we pick the $i$-th abstract value $\\hat{v}_i$ from the fact $\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*}$ modelling the state of the registers at $\\mathit{pp}$; then, if $\\hat{v}_i$ contains any pointer abstraction $\\lambda$, we use it to match a corresponding abstract heap entry $\\mathsf{H}(\\lambda,\\hat{o})$ and we return the value of the field $f$ of the abstract object $\\hat{o}$ contained therein. We similarly abstract the content of array cells: just notice that, since the representation of arrays is field-insensitive, the index of the cell does not play any role in the static analysis.\n\nThe rules for abstracting a right-hand side are useful to define the abstract semantics of the \\texttt{move} statement. Other statements require some additional definitions. First, for each comparison operator $\\varolessthan$ and each primitive operation $\\odot,\\oplus$ of the concrete semantics, we presuppose the existence of a corresponding abstract operation $\\hat{\\comp}$, $\\hat{\\odot}$ and $\\hat{\\oplus}$ defined over the elements of the appropriate abstract domain. Then, given an abstract memory block $\\hat{b}$, we define a function $\\widehat{\\textit{get-type}}(\\hat{b})$ as follows:\n\\[\n\\widehat{\\textit{get-type}}(\\hat{b}) =\n\\begin{cases}\nc & \\text{if } \\hat{b} = \\absobj{c}{(f \\mapsto \\hat{v})^*} \\\\\n\\arrtype{\\tau} & \\text{if } \\hat{b} = \\absarray{\\tau}{\\hat{v}} \\\\\n\\texttt{Intent} & \\text{if } \\hat{b} = \\absintent{c}{\\hat{v}}\n\\end{cases}\n\\]\nFinally, we assume a function $\\widehat{\\textit{lookup}}(m)$, which returns the set of classes which define (or inherit) a method called $m$.\n\nWith these definitions, we are ready to introduce the abstract semantics of statements. The idea is to define, for each possible form of statement $\\mathit{st}$, a translation $\\ainst{\\mathit{st}}$ into a set of Horn clauses, which over-approximate the semantics of $\\mathit{st}$ at program point $\\mathit{pp}$. The full formal semantics of the translation is given in Table~\\ref{tab:abs-statements} and explained below. \n\n\\begin{table*}[htb] \\small\n\\begin{mathpar}\n\\begin{array}{lcl}\n\\ainst{\\goto \\mathit{pc}'} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc'}}{ \\_ }{\\hat{v}^*}\\} \\\\\n\n\\ainst{\\ifbr {r_i}{r_j} {\\mathit{pc}'}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\hat{v}_i\\ \\hat{\\comp}\\ \\hat{v}_j \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc'}}{ \\_ }{\\hat{v}^*}\\}\\, \\cup \\\\\n& & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\neg(\\hat{v}_i\\ \\hat{\\comp}\\ \\hat{v}_j) \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*}\\} \\\\\n\n\\ainst{\\binop{r_d}{r_i}{r_j}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies\n\\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*[d \\mapsto \\hat{v}_i\\ \\hat{\\oplus}\\ \\hat{v}_j]}\\} \\\\\n\n\\ainst{\\unop{r_d}{r_i}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies\n\\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*[d \\mapsto \\hat{\\odot}\\, \\hat{v}_i]}\\} \\\\\n\n\\ainst{\\move{r_d}{\\mathit{rhs}}} & = & \\{\\prhs{\\hat{v}'} \\wedge \\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_} {\\hat{v}^*[d \\mapsto \\hat{v}']}\\} \\cup \\arhs{\\mathit{rhs}} \\\\\n\n\\ainst{\\move{r_a[r_\\mathit{idx}]}{\\mathit{rhs}}} & = & \\{\\prhs{\\hat{v}''} \\wedge \\absreg{\\mathsf{pp}}{\\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_a \\wedge \\mathsf{H}(\\lambda,\\absarray{\\tau}{\\hat{v}'}) \\implies \\mathsf{H}(\\lambda,\\absarray{\\tau}{\\hat{v}' \\cup \\hat{v}''})\\}\\, \\cup \\\\\n& & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*}\\} \\cup \\arhs{\\mathit{rhs}} \\\\\n\n\\ainst{\\move{r_o.f}{\\mathit{rhs}}} & = & \\{\\prhs{\\hat{v}''} \\wedge \\absreg{\\mathsf{pp}}{ \\_} {\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_o \\wedge \\mathsf{H}(\\lambda,\\absobj{c'}{(f' \\mapsto \\hat{u}')^*, f \\mapsto \\hat{v}'}) \\implies \\\\\n& & \\mathsf{H}(\\lambda,\\absobj{c'}{(f' \\mapsto \\hat{u}')^*, f \\mapsto \\hat{v}'')})\\} \\cup \\{\\absreg{\\mathsf{pp}}{ \\_} {\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*}\\} \\cup \\arhs{\\mathit{rhs}} \\\\\n\n\\ainst{\\move{c'.f}{\\mathit{rhs}}} & = & \\{\\prhs{\\hat{v}'} \\implies \\mathsf{S}_{\\mathsf{c'},\\mathsf{f}}(\\hat{v}')\\} \\cup \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_}{\\hat{v}^*}\\} \\cup \\arhs{\\mathit{rhs}} \\\\\n\n\\ainst{\\instanceof{r_d}{r_s}{\\tau}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_}{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_s \\wedge \\mathsf{H}(\\lambda, \\hat{b}) \\wedge \\widehat{\\textit{get-type}}(\\hat{b}) \\leq \\tau \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*[d \\mapsto \\widehat{\\mathtt{true}}]}\\}\\, \\cup \\\\\n& & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_s \\wedge \\mathsf{H}(\\lambda,\\hat{b}) \\wedge \\widehat{\\textit{get-type}}(\\hat{b}) \\not\\leq \\tau \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_ }{\\hat{v}^*[d \\mapsto \\widehat{\\mathtt{false}}]}\\} \\\\\n\n\\ainst{\\checkcast{r_s}{\\tau}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_s \\wedge \\mathsf{H}(\\lambda,\\hat{b}) \\wedge \\widehat{\\textit{get-type}}(\\hat{b}) \\leq \\tau \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_}{\\hat{v}^*}\\} \\\\\n\n\\ainst{\\invoke{r_o}{m'}{(r_{i_j})^{j \\leq n}}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_o \\wedge \\mathsf{H}(\\lambda,\\absobj{c'}{(f \\mapsto \\hat{u})^*}) \\wedge c' \\leq \\const{c''} \\implies \\\\\n& & \\absreg{\\mathsf{c''},\\mathsf{m'},\\mathsf{0}}{(\\hat{v}_{i_j})^{j \\leq n}}{(\\hat{\\mathbf{0}}_k)^{k \\leq \\mathit{loc}}, (\\hat{v}_{i_j})^{j \\leq n}} ~|~ c'' \\in \\widehat{\\textit{lookup}}(m') \\wedge \\textit{sign}(c'',m') = \\methsign{(\\tau_j)^{j \\leq n}}{\\tau}{\\mathit{loc}}\\}\\, \\cup \\\\\n& & \\{ \\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_o \\wedge \\mathsf{H}(\\lambda,\\absobj{c'}{(f \\mapsto \\hat{u})^*}) \\wedge c' \\leq \\const{c''} \\wedge \\absresult{\\mathsf{c''},\\mathsf{m'}}{(\\hat{v}_{i_j})^{j \\leq n}}{\\hat{v}'_{\\mathit{ret}}} \\implies \\\\\n& & \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_} {\\hat{v}^*[\\mathit{ret} \\mapsto \\hat{v}'_{\\mathit{ret}}]} ~|~ c'' \\in \\widehat{\\textit{lookup}}(m')\\} \\\\\n\n\\ainst{\\sinvoke{c'}{m'}{(r_{i_j})^{j \\leq n}}} & = & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c'},\\mathsf{m'},\\mathsf{0}}{(\\hat{v}_{i_j})^{j \\leq n}}{(\\hat{\\mathbf{0}}_k)^{k \\leq \\mathit{loc}},(\\hat{v}_{i_j})^{j \\leq n}} ~|~ \\textit{sign}(c', m') = \\methsign{(\\tau_j)^{j \\leq n}}{\\tau}{\\mathit{loc}} \\}\\, \\cup \\\\\n& & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\wedge \\absresult{\\mathsf{c'},\\mathsf{m'}}{(\\hat{v}_{i_j})^{j \\leq n}}{\\hat{v}'_{\\mathit{ret}}} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{ \\_} {\\hat{v}^*[\\mathit{ret} \\mapsto \\hat{v}'_{\\mathit{ret}}]}\\} \\\\\n\n\\ainst{\\new{r_d}{c'}} & = & \\{ \\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\implies \\mathsf{H}(\\mathsf{pp}, \\absobj{c'}{(f \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*}\\} \\cup \\{ \\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{\\_}{\\hat{v}^*[d \\mapsto \\mathsf{pp}]}\\} \\\\\n\n\\ainst{\\newarray{r_d}{r_l}{\\tau}} & = & \\{ \\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\implies \\mathsf{H}(\\mathsf{pp}, \\absarray{\\tau}{\\hat{\\mathbf{0}}_{\\tau}})\\} \\cup \\{ \\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{\\_}{\\hat{v}^*[d \\mapsto \\mathsf{pp}]}\\} \\\\\n\n\\ainst{\\texttt{return}} & = & \\{\\absreg{\\mathsf{pp}}{\\hat{v}^*_{call}}{\\hat{v}^*} \\implies \\absresult{\\mathsf{c},\\mathsf{m}}{\\hat{v}^*_{call}}{\\hat{v}_{\\mathit{ret}}}\\} \\\\\n\n\\ainst{\\startact{r_i}} & = & \\{\\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_i \\wedge \\mathsf{H}(\\lambda,\\absintent{c'}{\\hat{u}}) \\implies \\mathsf{I}(\\const{c},\\absintent{c'}{\\hat{u}})\\}\\, \\cup \\\\ \n& & \\{ \\absreg{\\mathsf{pp}}{\\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{\\_}{\\hat{v}^*}\\} \\\\\n\n\\ainst{\\newintent{r_d}{c'}} & = & \\{\\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\implies \\mathsf{H}(\\mathsf{pp}, \\absintent{c'}{\\emptyset})\\} \\cup \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{\\_}{\\hat{v}^*[d \\mapsto \\mathsf{pp}]}\\} \\\\\n\n\\ainst{\\putextra{r_i}{r_k}{r_j}} & = & \\{\\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_i \\wedge \\mathsf{H}(\\lambda, \\absintent{c'}{\\hat{v}'}) \\implies \\mathsf{H}(\\lambda, \\absintent{c'}{\\hat{v}' \\cup \\hat{v}_j})\\}\\, \\cup \\\\\n& & \\{\\absreg{\\mathsf{pp}}{ \\_ }{\\hat{v}^*} \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{\\_}{\\hat{v}^*}\\} \\\\\n\n\\ainst{\\getextra{r_i}{r_k}{\\tau}} & = & \\{\\absreg{\\mathsf{pp}}{\\_}{\\hat{v}^*} \\wedge \\lambda \\in \\hat{v}_i \\wedge \\mathsf{H}(\\lambda, \\absintent{c'}{\\hat{v}'}) \\implies \\absreg{\\mathsf{c},\\mathsf{m},\\mathsf{pc+1}}{\\_}{\\hat{v}^*[\\mathit{ret} \\mapsto \\hat{v}']}\\}\n\\end{array}\n\\end{mathpar}\n\\caption{Abstract Semantics of {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} - Statements (let $\\mathit{pp} = c,m,\\mathit{pc}$)}\n\\label{tab:abs-statements}\n\\end{table*}\n\nThe rule for $\\goto{\\mathit{pc}'}$ propagates the state of the registers at the current program counter $\\mathit{pc}$ to $\\mathit{pc}'$. The rule for $\\ifbr{r_i}{r_j}{\\mathit{pc}'}$ propagates the state of the registers at the current program counter $\\mathit{pc}$ either to $\\mathit{pc}'$ or to $\\mathit{pc}+1$, based on the outcome of a comparison $\\hat{\\comp}$ between the abstract values $\\hat{v}_i$ and $\\hat{v}_j$ approximating the content of registers $r_i$ and $r_j$ respectively: both branches may be enabled, as the result of an over-approximation of the contents of the registers. The two rules for unary and binary operations just employ the appropriate abstract operation to update the approximation of the content of the destination register $r_d$. The four rules for the \\texttt{move} statement rely on the auxiliary rules for abstracting a right-hand side we introduced before: these rules store their result in a $\\mathsf{RHS}$ fact, which occurs in the premises of the Horn clause used to update the abstraction of the left-hand side. The most interesting point to notice here is that field-sensitivity or its absence has an import on how fields are updated: for objects, we replace the old value of the field with the new one; for arrays and intents, instead, we add the new value to the old approximation, since their abstraction over-approximates the content of the entire data structure, rather than just the single element which is updated. The rules for \\texttt{instof} and \\texttt{checkcast} use the $\\widehat{\\textit{get-type}}$ function previously defined.\n\nThe rule for \\texttt{invoke} is the most complicated one, since it has to deal with dynamic dispatching. The challenge here is that the name of the invoked method is statically known from the syntax of the statement, but the method implementation is not, since it depends on the runtime type of the receiver object, an information which is only over-approximated when solving the Horn clauses, rather than when generating them. We then use the method name and the number of arguments passed upon invocation to narrow the set of possible classes of the receiver object, using the functions $\\widehat{\\textit{lookup}}$ and $\\textit{sign}$, and we generate one Horn clause for each of them. We then rely on subtyping to make the analysis precise, by imposing that a Horn clause generated for class $c''$ can only be fired if the class $c'$ of (the abstraction of) the receiver object is a subtype of $c''$. Besides implementing a sound approximation of the dynamic dispatching mechanism, the rule for \\texttt{invoke} generates additional Horn clauses used to propagate the abstraction of the method return value from the callee to the caller: this is done by using a $\\mathsf{Res}$ fact, which is introduced by a \\texttt{return} statement in the implementation of the callee, as we discuss below. The rule for static method invocation follows a similar logic, but it is significantly simpler, due to the lack of dynamic dispatching on static calls.\n\nThe rules for object and array creation create a new abstract heap entry $\\mathsf{H}(\\lambda,\\hat{b})$, where $\\lambda$ is the current program point and $\\hat{b}$ is the abstraction of a freshly initialized object\/array. The rule for \\texttt{return} introduces a $\\mathsf{Res}$ fact, storing an over-approximation of the method return value; notice that the arguments $\\hat{v}_{call}^*$ supplied upon method invocation are propagated in the $\\mathsf{Res}$ fact, which is important to implement context-sensitivity, i.e., to propagate the result to the right caller. The rule for \\texttt{start-activity} tracks that the present activity $c$ has sent an intent: an over-approximation of the intent is propagated from the corresponding abstract heap entry into the $\\mathsf{I}$ fact modelling the presence of a pending activity which is about to start. The last rules for managing intents should be easy to understand, based on the intuitions given for the other rules.\n\n\\subsubsection{Abstract Semantics of Activities}\nWe can finally introduce the abstract semantics of activities. Intuitively, it is defined by: (1) the Horn clauses produced by translating each statement in the bytecode, and (2) a small set of bytecode-independent Horn clauses, abstracting the event-driven behaviour of activities. This is formalized next.\n\n\\begin{definition}\nLet $P = (\\mathit{cls}_i)^{i \\leq n}$ be a program where $\\mathit{cls}_i = \\cls{c_i}{c'}{c^*}{\\mathit{fld}^*}{(\\mathit{mtd}_j)^{j \\leq h_i}}$ and $\\mathit{mtd}_j = \\meth{m_j}{\\methsign{\\tau^*}{\\tau}{\\mathit{loc}}}{(\\mathit{st}_k)^{k \\leq s_{ij}}}$, we let $\\translate{P}$ be defined as follows:\n\\[\n\\translate{P} = \\bigcup_{i \\leq n, j \\leq h_i, k \\leq s_{ij}} \\ainstfull{\\mathit{st}_k}{c_i,m_j,k} \\cup \\mathcal{R},\n\\]\nwhere $\\mathcal{R}$ stands for the union of all the rules in Table~\\ref{tab:abs-activity}.\n\\end{definition}\n\n\\begin{table*}[htb]\n\\begin{mathpar}\n\\begin{array}{lcl}\n\\rulename{Cbk} & = & \\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\_)^*}) \\wedge c \\leq \\const{c'} \\implies \\absreg{\\mathsf{c'},\\mathsf{m},\\mathsf{0}}{(\\top_{\\tau_j})^{j \\leq n}}{(\\hat{\\mathbf{0}}_k)^{k \\leq \\mathit{loc}},c,(\\top_{\\tau_j})^{j \\leq n}} ~|~ \\\\\n& & c' \\text{ is an activity class} \\wedge \\exists s: m \\in \\mathit{cb}(c',s) \\wedge \\textit{sign}(c',m) = \\methsign{\\tau_1,\\ldots,\\tau_n}{\\tau}{\\mathit{loc}}\\} \\\\\n\n\\rulename{Fin} & = & \\{\\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\_)^*,\\textit{finished} \\mapsto \\_}) \\implies \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\_)^*,\\textit{finished} \\mapsto \\top_{\\type{bool}}})\\} \\\\\n\n\\rulename{Rep} & = & \\{\\mathsf{H}(c,\\absobj{c}{(f_{\\tau} \\mapsto \\_)^*}) \\implies \\mathsf{H}(c,\\absobj{c}{(f_{\\tau} \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*})\\} \\\\\n\n\\rulename{Act} & = & \\{\\mathsf{I}(c',\\absintent{c}{\\hat{v}})) \\implies \\mathsf{H}(\\astart{c},\\absintent{c}{\\hat{v}})\\}\\, \\cup \\\\\n& & \\{\\mathsf{I}(c',\\absintent{c}{\\hat{v}})) \\implies \\mathsf{H}(c, \\absobj{c}{(f_{\\tau} \\mapsto \\hat{\\mathbf{0}}_{\\tau})^*, \\textit{finished} \\mapsto \\widehat{\\mathtt{false}}, \\textit{parent} \\mapsto c', \\textit{intent} \\mapsto \\astart{c}})\\} \\\\\n\n\\rulename{Res} & = & \\{\\mathsf{H}(c',\\absobj{c'}{(f' \\mapsto \\_)^*,\\textit{parent} \\mapsto c,\\textit{result} \\mapsto \\lambda} \\wedge \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\_)^*,\\textit{result} \\mapsto \\_} \\implies \\\\\n& & \\mathsf{H}(c,\\absobj{c}{(f \\mapsto \\_)^*,\\textit{result} \\mapsto \\lambda}\\} \\\\\n\n\\rulename{Sub} & = & \\{\\tau \\leq \\tau' ~|~ \\tau \\leq \\tau' \\text{ is a valid subtyping judgement} \\}\n\\end{array}\n\\end{mathpar}\n\\caption{Abstract Semantics of {$\\mu\\text{-Dalvik}_{A}$}\\xspace{} - Activity Rules}\n\\label{tab:abs-activity}\n\\end{table*}\n\nWe explain the rules from Table~\\ref{tab:abs-activity}. Rule \\rulename{Cbk} simulates the invocation of a callback: since we do not approximate the activity state in the abstract semantics, any callback method bound to a state $s$ of the activity lifecycle may be non-deterministically dispatched; the statically unknown arguments supplied to the callback are abstracted by the top element ($\\top$) of the abstract domain associated to their type, which is a sound over-approximation of any value of that type. Rule \\rulename{Fin} tracks updates to the $\\textit{finished}$ field of an activity in the abstract semantics: since it is hard to statically track whether an activity has finished or not, the rule sets the field to the top element of the abstract domain used to represent boolean value ($\\top_{\\type{bool}}$). Rule \\rulename{Rep} approximates the behaviour of rule \\irule{A-Replace} of the concrete semantics: the activity fields may be reset to their default abstract value as the result of a screen orientation change. \n\nRule \\rulename{Act} represents the starting of a new activity. If an intent has been sent by an activity of class $c'$ to start an activity of class $c$, we introduce: (1) a new abstract heap entry to bind an abstraction of the intent to $\\astart{c}$, and (2) a new abstract heap entry to bind an abstraction of the started activity to $c$. No serialization happens in the abstract semantics: if an intent is used to send an object in the concrete semantics, a reference to the corresponding abstract object is sent in our abstraction. This is sound, since our analysis is flow-insensitive on heap values, hence no over-approximation of the original object is ever lost as the result of an update to the heap at the receiver side. We then have rule \\rulename{Res}, which is used to communicate a result from a child activity to its parent, thus simulating the behaviour of rule \\irule{A-Result} in the concrete semantics; again, no serialization happens in the process, rather a pointer to the result is passed. Finally, rule \\rulename{Sub} corresponds to an axiomatization of the subtyping relationships for the analysed program.\n\n\\subsection{Formal Results}\nThe soundness of the analysis is proved using \\emph{representation functions}, a standard approach in program analysis~\\cite{NielsonNH99}.\nThe representation function $\\beta_{\\textit{Cnf}}$ maps an arbitrary configuration $\\Psi$ into a corresponding set of facts $\\Delta$, modelling an over-approximation of $\\Psi$. Its definition is lengthy, but unsurprising, e.g., each element $\\ell \\mapsto b$ of the heap is converted into an abstract heap entry $\\mathsf{H}(\\lambda,\\hat{b})$, where $\\lambda$ is the annotation on $\\ell$ and $\\hat{b}$ is an abstraction of $b$. After defining $\\beta_{\\textit{Cnf}}$, we introduce a partial order $\\sqsubseteq$ on analysis facts, with the intuitive understanding that $\\mathsf{f} \\sqsubseteq \\mathsf{f}'$ whenever $\\mathsf{f}$ is a more precise abstraction than $\\mathsf{f}'$. The partial order is then lifted to abstract programs by having\n$\\Delta <: \\Delta'$ if and only if $\\forall \\mathsf{f} \\in \\Delta: \\exists \\mathsf{f}' \\in \\Delta': \\mathsf{f} \\sqsubseteq \\mathsf{f}'$.\n\nOur main theorem states that any reachable configuration in the concrete semantics is over-approximated by some set of facts which is provable using the abstract semantics of the program and an abstraction of the initial configuration. The proof is parametric with respect to the choice of the abstract domains\/operations used for primitive values, provided they offer some minimal soundness guarantees. This allows for choosing different trade-off between efficiency and precision of the analysis.\n\n\\begin{theorem}[Preservation]\nIf $\\Psi \\Rightarrow^* \\Psi'$ under a program $P$, there exists $\\Delta :> \\beta_{\\textit{Cnf}}(\\Psi')$ such that:\n\\[\n\\translate{P} \\cup \\beta_{\\textit{Cnf}}(\\Psi) \\vdash \\Delta.\n\\]\n\\end{theorem}\n\nBy providing an over-approximation of any reachable configuration of the concrete semantics in terms of a corresponding set of facts, the theorem can be used to prove the absence of undesired information flows of sensitive data into local registers of selected sink methods. In particular, we leverage the theorem to develop a provably sound taint analysis, based on standard ideas. Due to space constraints, we refer to \\iffull Appendix~\\ref{sec:proofs} \\else the online version~\\cite{full} \\fi for full details.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nOne of the most well-studied mechanisms for setting the observed dark matter (DM) abundance is thermal freeze-out, where DM is in equilibrium with the Standard Model (SM) thermal bath at very early times. The DM abundance is then depleted through annihilations at later times until the DM drops out of chemical equilibrium. The appeal of this mechanism is that the final relic abundance is generally independent of the high-temperature initial conditions at reheating. Furthermore, producing the observed relic abundance requires a particular thermally averaged annihilation cross section in most thermal freeze-out scenarios, $\\langle \\sigma v\\rangle \\sim 10^{-26}$ cm$^3$\/s. This weak-scale cross section provides a target that can be probed by direct and indirect detection experiments. Assuming the relic abundance is set by annihilations to SM particles, then consistency with Big Bang Nucleosynthesis (BBN) generally requires that thermal freeze-out candidates have masses $m_\\chi \\gtrsim 1$~MeV~\\cite{Boehm:2013jpa, Nollett:2013pwa,Nollett:2014lwa}. \nThe appealing simplicity of this scenario has led to an enormous number of DM searches targeting the thermal freeze-out mechanism, with a particular emphasis on weakly interacting massive particle (WIMP) candidates in the $m_\\chi\\sim$~GeV$-$TeV mass range. More recently, there has been a growing interest in $m_\\chi\\sim$~MeV$-$GeV thermal candidates where interactions with the SM or within a hidden sector deplete the DM density to the observed value \\cite{Boehm:2003hm,Pospelov:2007mp,Feng:2008ya,Hochberg:2014dra,Hochberg:2014kqa,Hochberg:2018rjs,Choi:2017zww,DAgnolo:2018wcn,DAgnolo:2017dbv,DAgnolo:2015ujb,Pappadopulo:2016pkp,Cline:2017tka,Kopp:2016yji}. \n\nThe freeze-in mechanism for DM production is a compelling alternative to thermal freeze-out, where DM is instead produced by feeble, sub-Hubble interactions of SM particles~\\cite{Asaka:2005cn,Asaka:2006fs,Gopalakrishna:2006kr,Page:2007sh,Hall:2009bx,Bernal:2017kxu}. If the dominant freeze-in process is annihilation of SM particles into DM via a light mediator, then many of the appealing features of thermal freeze-out are maintained. For annihilation through a mediator lighter than the DM, the thermal cross section typically scales as $\\langle \\sigma v \\rangle \\sim g_\\chi^2 g_{\\rm SM}^2\/(4 \\pi T)^2$ where $g_\\chi$ is the mediator-DM coupling, $g_{\\rm SM}$ is the mediator-SM coupling, and $T$ is the SM temperature. With this scaling, DM freeze-in dominantly occurs at the lowest temperature where the process is kinematically accessible, and thus the mechanism is not sensitive to the reheat scale.\\footnote{We assume the minimal scenario where the dark sector is not populated in abundance at reheating.}\n \n\nFreeze-in through a light vector mediator has emerged as a key benchmark for sub-GeV direct detection experiments. Producing the observed DM relic abundance implies a tiny value for the coupling constants, which is difficult to target with accelerator searches. However, sufficiently light mediators give rise to scattering cross sections that scale as $\\sigma \\propto 1\/v^4$ for relative velocity $v$, implying that the kinematics of the Milky Way (where $v\\sim 10^{-3}$) can enhance the detectability of DM coupling to a light mediator. If the mediator also couples to charged SM fermions, then the DM can scatter off of electrons or nuclei and may be detectable with the next generations of direct detection experiments~\\cite{Essig:2011nj,Essig:2012yx,Essig:2015cda,Hochberg:2016ntt,Derenzo:2016fse,Hochberg:2017wce,Knapen:2017ekk,Griffin:2018bjn,Schutz:2016tid,Knapen:2016cue,Hochberg:2015pha,Hochberg:2015fth} (see also Ref.~\\cite{Battaglieri:2017aum} for a recent review). Indeed, recent experimental results by SENSEI~\\cite{Crisler:2018gci,sensei}, SuperCDMS~\\cite{Agnese:2018col}, and DarkSide~\\cite{Agnes:2018oej} are demonstrating significant progress towards achieving the sensitivity needed in the MeV-GeV mass range. It was also shown recently that Xenon1T~\\cite{Aprile:2018dbl} is for the first time constraining freeze-in in the GeV-TeV mass range~\\cite{Hambye:2018dpi}. \n\nIn the keV$-$MeV DM mass range, freeze-in is the leading scenario that could be tested by proposed low-threshold direct detection experiments. Refs.~\\cite{Knapen:2017xzo,Green:2017ybv} studied the possible direct detection cross sections in models of sub-MeV DM, finding that it would be difficult to observe thermal freeze-out scenarios (even purely within a dark sector) due to a combination of BBN, CMB, fifth force, and stellar emission constraints. Obtaining accurate predictions of freeze-in is thus an important step in the program to search for low-mass DM. While freeze-in from electron-positron annihilations via a light vector mediator has been studied in the past~\\cite{Chu:2011be,Essig:2011nj}, in this work we thoroughly explore a previously overlooked production mechanism: freeze-in through plasma effects. The contribution of plasma effects to dark sector thermalization was estimated earlier in Refs.~\\cite{Davidson:2000hf,Vogel:2013raa} and the effect on freeze-in via a heavy mediator was recently considered in Ref.~\\cite{An:2018nvz} as we were in the late stages of completing this work, but it was not included in previous studies of freeze-in through a light vector mediator. We find that the plasma production of DM is a dominant channel for sub-MeV DM masses, and will therefore restrict our discussion to this mass range. The additional contribution to the relic abundance implies that the target cross section for direct detection is lower by roughly an order of magnitude for the lowest experimentally accessible DM masses.\n\nThe rest of this paper is organized as follows. We begin in Section~\\ref{sec:model} by reviewing the arguments for the simplest viable freeze-in models in the keV-MeV mass range: either pure millicharged DM arising from a DM hypercharge or \\emph{effectively} millicharged DM that is coupled to an ultralight dark photon mediator. These two scenarios are almost phenomenologically identical, with the key difference being that DM-DM scattering can be parametrically larger when dark photon interactions are present. These DM candidates have recently received considerable attention in the context of the anomalous 21~cm global signal~\\cite{Bowman:2018yin,barkana2018possible,Barkana:2018cct,Berlin:2018sjs,Munoz:2018pzp}. In Section~\\ref{sec:prod} we compute the DM relic abundance from freeze-in via a light mediator. We include the effects of plasmon decays for the first time, and show the impact for direct detection. We then present the calculation of the phase space distribution for freeze-in DM in Section~\\ref{sec:phasespace}. A summary of our results can be found in Section~\\ref{summary}.\nIn a companion paper~\\cite{inprep}, we will apply the calculations of the phase-space distribution to cosmological observables, showing that the cosmic microwave background (CMB) and probes of large-scale structure (LSS) provide a strong complementary test of DM freeze-in for $m_\\chi\\sim$~keV$-$MeV. In particular, we find that existing cosmological constraints restrict $m_\\chi \\gtrsim$ tens of keV for freeze-in via a light mediator, and it will be possible to probe even higher masses with planned experiments. \n\n\\section{Models for sub-MeV freeze-in \\label{sec:model}}\n\n\\subsection{The case for light vector mediators}\n\\label{vectors}\nThe simplest observationally viable models for sub-MeV freeze-in through a light mediator can be divided into two classes, where (1) the DM only has interactions mediated by the SM photon or (2) the DM has interactions with an ultralight kinetically mixed dark photon. We note that models of millicharged DM~\\cite{Davidson:2000hf,Dubovsky:2003yn} can fall under either category: they can arise as a limit of the dark-photon model where the dark photon is nearly massless, or they could be present as Dirac fermions with a tiny hypercharge.\\footnote{Other models that have been considered in the past require giving neutrinos small charges as well~\\cite{Foot:1992ui}, which we do not consider further due to strong experimental bounds on neutrino charge~\\cite{Chen:2014dsa}.}\n\nFor sub-MeV freeze-in to be relevant for direct detection, vector mediators are the only observationally viable option due to stringent constraints on other light mediators with the requisite couplings to the SM, as outlined below. For direct detection of freeze-in, the mediator masses must be sufficiently small compared to the typical momentum transfer for scattering processes. If the mediators are heavier, then they do not give rise to the $v^{-4}$ enhancement that would render extremely feeble DM-SM interactions detectable on Earth. For nuclear recoils the relevant momentum scale is set by galactic kinematics $q \\sim m_\\chi v \\sim 10^{-3} m_\\chi$, while for electron recoils the typical electron momentum in the target material is most relevant $q \\sim \\alpha m_e \\approx 4$~keV, where $m_e$ is the electron mass and $\\alpha$ is the electromagnetic fine structure constant. Thus for sub-MeV DM, the experimentally relevant mediators have masses below $\\mathcal{O}(1)$~keV. \n\nAssuming an annihilation cross section of SM fermions into DM with the form $\\langle \\sigma v\\rangle \\sim g_\\chi^2 g_{\\rm SM}^2\/(4 \\pi T)^2$, the relic abundance can be estimated as \n\\begin{align}\n Y_\\chi = \\frac{n_\\chi}{s} \\sim \\frac{n_{\\rm SM}^2 \\langle \\sigma v\\rangle}{s H} \\sim 2\\times10^{-4} \\, \\frac{g_\\chi^2 g_{\\rm SM}^2 M_{\\rm Pl} }{ T},\n\\end{align}\nwhere $M_{\\rm Pl}=1\/\\sqrt{8 \\pi G}$ is the reduced Planck mass and we assumed $T \\sim $ MeV. Then for $m_\\chi\\sim$ MeV, we find that $g_\\chi g_{\\rm SM} \\simeq 10^{-12}$ to saturate the relic abundance. This order-of-magnitude estimate is in agreement with more detailed calculations below. Since obtaining the relic abundance from freeze-in requires $g_\\chi g_{\\rm SM} \\sim 10^{-12}$, $g_{\\rm SM}$ must be greater than $10^{-12}$ if we require the dark sector to be perturbative (i.e. $g_\\chi~\\lesssim~1$). Weakly coupled, sub-keV mediators can be emitted in stars, affecting their luminosity and lifetime. The observed properties of stars lead to strong bounds on such mediators, which we summarize here (see also Refs.~\\cite{Knapen:2017xzo,Green:2017ybv} where these bounds are collected and discussed in the context of sub-MeV DM models): \n\\begin{itemize}\n\\item {\\emph{Scalars and pseudoscalars coupled to electrons}} $-$ The strongest bound on a light scalar with interaction $g_{\\phi ee} \\phi \\bar e e$ comes from helium ignition in red giants, with $g_{\\phi e e} \\lesssim 7 \\times 10^{-16}$ for sub-keV masses~\\cite{Hardy:2016kme}. For a sub-keV pseudoscalar, observations of white dwarfs give typical constraints of $g_{a e e} \\lesssim 2 \\times 10^{-13}$~\\cite{Raffelt:2006cw,Viaux:2013lha,Bertolami:2014wua}. A caveat for most stellar emission bounds is that when the coupling is increased, the new particle may be trapped within the star and would not lead to anomalous energy loss. However, this would still affect energy transport in the star, which can be constrained for the range of couplings relevant for freeze-in through this mediator~\\cite{Carlson:1988jg,Raffelt:1988rx}.\n\\item {\\emph{Scalars and pseudoscalars coupled to nucleons}} $-$ Similar to the case of mediators coupling to electrons, red giants constrain $g_{\\phi n n} \\lesssim 10^{-12}$ for a scalar~\\cite{Hardy:2016kme} and $g_{a n n} \\lesssim {\\rm few} \\times 10^{-10}$ for a pseudoscalar~\\cite{Raffelt:2006cw,Giannotti:2017hny}. While the latter coupling appears at face value to be sufficiently large, freeze-in through baryons is largely suppressed after the QCD phase transition due to the low baryon number density. Therefore, in this case our estimate for the minimum $g_{\\rm SM}$ with $T\\sim 1$~MeV is much too low and freeze-in would have to occur with a larger value of $g_{\\rm SM}$ that is in tension with stellar bounds. \n\\item {\\emph{Scalar mixing with the Higgs}} $-$ The bounds here are similar to those in the two previous cases, and it has been shown in Ref.~\\cite{Krnjaic:2017tio} that freeze-in through this portal is only a viable mechanism for producing all of the DM for DM masses above a few hundred MeV. \n\\item {\\emph{Kinetically mixed dark photon}} $-$ In this case, the stellar constraints on $g_{\\rm SM}$ decrease linearly with the mediator mass for masses below $\\sim 100$ eV~\\cite{An:2013yfc,An:2013yua} because of the in-medium plasma mass suppression of producing dark photons from SM interactions, as detailed in Eq.~\\eqref{eq:inmedium_L} and the surrounding discussion in Section~\\ref{sec:darkphoton}. From the collected bounds on dark photons from Refs.~\\cite{Jaeckel:2013ija}, a dark photon can have $g_{\\rm SM} > 10^{-12}$ when its mass is well below $1$~eV. At even lower masses, the coupling could be $\\sim 10^{-3}$ for masses below $\\lesssim 10^{-14}$~eV. \n\\item {\\emph{$B-L$ vector}} $-$ Stellar constraints on a $B-L$ vector are similar to that for the dark photon. However, for eV-scale and lighter mediator masses, a $B-L$ vector is also strongly constrained by fifth force searches (e.g.~\\cite{Murata:2014nra,Adelberger:2003zx}), which limits the mediator-SM coupling to below $10^{-12}$. \n\\end{itemize} \nSince we are focusing on the simplest benchmarks for direct detection, we do not consider more exotic possibilities with additional particles and interactions. From the bounds on new particles with the couplings described above, we conclude that freeze-in through a light mediator is viable either when the mediator is (1) the SM photon, and the DM has a tiny electric charge, or (2) when the mediator is an ultralight kinetically mixed dark photon. \n\nWe discuss these two closely related scenarios in the rest of the section. In both cases, DM has an effective charge $Q e$ (or millicharge $Q$) with respect to the SM photon. This parameter determines the relic abundance, irrespective of which of the two models is under consideration. Both models allow for heat and momentum transfer between SM particles and DM during epochs when the typical relative velocities are low (as discussed in Section~\\ref{sec:scattering}), which is relevant to observations of the CMB~\\cite{Dvorkin:2013cea,Xu:2018efh,Slatyer:2018aqg,Kovetz:2018zan,Boddy:2018wzy,Dubovsky:2001tr,Boddy:2018kfv} and the cosmological 21~cm global signal~\\cite{Bowman:2018yin,barkana2018possible,Barkana:2018cct,Berlin:2018sjs,Munoz:2018pzp}. The main phenomenological difference between these two possibilities is that DM-DM scattering via a dark photon can be parametrically larger than DM-DM scattering mediated by the SM photon, as discussed below. If present at a sufficient level, the DM self-scattering can play an important role in determining the DM phase space distribution at late times, well after freeze-in.\n\n\n\\subsection{DM with photon-mediated interactions \\label{sec:millicharge}}\n \nIf the DM is a Dirac fermion $\\chi$ with a tiny hypercharge $Q_Y$ (the only gauge-invariant, renormalizable operator leading to a bare millicharge), then it can interact via the SM photon. After electroweak symmetry breaking, the DM obtains an electric charge given by $e Q_Y \\equiv eQ$ (taking the convention where the Gell-Mann Nishijima formula reads $Q = I_3 +Y$). Although there are also $Z$-mediated DM interactions, they are negligible for the relevant epochs where $T \\ll m_Z$. This gives the simplest model of millicharged DM. It is difficult to incorporate such matter content into a Grand Unified Theory (GUT)~\\cite{Okun:1983vw}; however, this scenario is economical in that it requires that no additional particles be introduced to the SM aside from the DM itself. \n\nThe possibility that this DM candidate obtains its relic abundance by thermal freeze-out has been considered before in Ref.~\\cite{McDermott:2010pa}, where it was shown to be excluded by structure formation when all of the DM is produced this way. Thus, freeze-in is the simplest remaining possibility for producing this DM candidate, with $g_\\chi = eQ$ and $g_{\\rm SM} = e$ in the language of the previous subsections.\n\nThere are stellar emission bounds on this DM candidate because the DM can be pair produced by the decay of plasmons in stars, leading to additional energy loss. These bounds are shown as the shaded region in our summary plot, Fig.~\\ref{fig:summaryplot}. Constraints on DM pair produced in SN1987a were derived in Refs.~\\cite{Davidson:2000hf,Chang:2018rso} and require $Q\\lesssim 10^{-9}$ for $m_\\chi$ up to a few MeV, which does not impact freeze-in. However, there are constraints for $m_\\chi$ below $\\mathcal{O}(10)$~keV from emission in white dwarfs, horizontal branch stars, and red giants (see Appendix of Ref.~\\cite{Vogel:2013raa}). Note that the range of $m_\\chi$ where stellar emission can constrain freeze-in is exponentially sensitive to assumptions about temperatures within the stars. In addition, the bounds derived are applicable in the weak coupling limit where the DM escapes cleanly from the star. For sufficiently large $Q$, DM emission could contribute to energy transport within the star and the effects have not been carefully studied in this regime. The couplings for freeze-in are large enough that they could be in this regime and stellar bounds on freeze-in should be regarded with care. \n\nThe relevant interactions for the relic abundance and phase space distribution in this model are SM annihilations and plasma decay into the DM. DM-SM scattering can become important at late times but, as we discuss in Section~\\ref{sec:scattering}, the effect must be small to be consistent with limits from the CMB. The DM self-scattering cross section is proportional to $Q^4$, and we find it to be irrelevant for the phase space. Finally DM-photon scattering is also proportional to $Q^4$ and is not enhanced in the low-velocity limit, so it is also irrelevant.\n\n\\subsection{DM with dark photon interactions \\label{sec:darkphoton}}\n\nWe next consider Dirac fermion DM coupled to a kinetically mixed dark photon $A'$, with the vacuum Lagrangian given by\n\\begin{align}\n\t{\\cal L} \\supset & -\\frac{1}{4} F_{\\mu \\nu} F^{\\mu \\nu} +\\frac{\\kappa}{2} F_{\\mu \\nu} F^{\\prime \\mu \\nu} -\\frac{1}{4} F^\\prime_{\\mu \\nu} F^{\\prime \\mu \\nu} + \\frac{1}{2} m_{A'}^2 A_\\mu^\\prime A^{\\prime \\mu} \\nonumber \\\\\n &+ \\, e J^\\mu_{\\rm EM} A_\\mu + g_\\chi\\bar \\chi \\gamma^\\mu \\chi A^\\prime_{\\mu} + \\bar \\chi (i \\partial - m_\\chi) \\chi, \\label{eq:vacuumL} \n\\end{align}\nwhere $A$ is the SM photon, $\\kappa$ is the kinetic mixing parameter and $\\chi$ is Dirac fermion DM. For the purposes of this discussion, we consider Abelian kinetic mixing, noting that non-Abelian kinetic mixing is also possible~\\cite{Barello:2015bhq,Arguelles:2016ney}. The mixing parameter $\\kappa$ could have any number of origins; for instance, it could be generated as a result of loop diagrams with heavy matter fields charged under both $A$ and $A'$ \\cite{Dienes:1996zr} or from certain compactifications of type IIB strings \\cite{Abel:2008ai,Goodsell:2009xc}. Since the kinetic mixing term is a marginal operator, we take the point of view of a bottom-up effective field theory and we will treat it here as a small free parameter without specifying its origin. \nIn this model, the combination of couplings relevant for the relic abundance is $g_\\chi g_\\text{SM} = g_\\chi \\kappa e$.\n\n\n\nAs discussed in Section~\\ref{vectors}, the dark photon mass must satisfy $m_{A'} \\lesssim 1$~eV in order to give a sufficient coupling for freeze-in while also evading existing bounds on stellar energy loss~\\cite{Jaeckel:2013ija}. However, the requirements are even more stringent because unlike the model presented in Section~\\ref{sec:millicharge} there could be large $A'$-mediated DM self interaction. For $m_{A'} <$~eV, the mediator would be light enough to give rise to $v^{-4}$ enhanced DM self-scattering in astrophysical environments, with a rate proportional to $g_\\chi^4$. Furthermore, as mentioned before, the freeze-in relic abundance is determined by the product $g_\\chi \\kappa e$, meaning that large $g_\\chi$ can be compensated by reducing $\\kappa$ to give the same observed relic abundance. Thus a sizable DM self-interaction is possible, and could be relevant to astrophysical probes of self-interacting DM (SIDM). The effects of SIDM are typically parameterized by the momentum-transfer self-scattering cross section, which in the limit of a very light vector mediator is given by~\\cite{Feng:2009hw} \n\\begin{align}\n\\sigma_{T,\\, \\chi \\chi} &= \\int d\\cos \\theta_\\text{CM} \\, \\frac{d\\sigma_{\\chi \\chi}}{d\\cos \\theta_\\text{CM}} (1 - \\cos \\theta_\\text{CM}) \\approx \\frac{8\\pi\\alpha_\\chi^2}{m_\\chi^2 v^4} \\ln \\frac{(m_\\chi v)^2}{m_{A'}^2},\n \\label{eq:transfer}\n\\end{align}\nwhere $\\theta_\\text{CM}$ is the scattering angle in the center-of-mass (CM) frame, $\\sigma_{\\chi\\chi}$ is the self-interaction cross section, and $\\alpha_\\chi$ is the dark equivalent of the electromagnetic fine structure constant, $\\alpha_\\chi \\equiv g_\\chi^2\/4 \\pi$.\nTypical bounds on SIDM require $\\sigma_{\\chi \\chi}\/m_\\chi < 1-10$ cm$^2$\/g for systems ranging from dwarf galaxies where $v\\sim 10^{-4}$ to merging clusters where $v\\sim 10^{-2}$ (for a recent review, see Ref.~\\cite{Tulin:2017ara}). While few simulation-based studies of self-interactions have been done in the ultralight mediator limit (see for instance Ref.~\\cite{Kummer:2019yrb}), we can estimate the expected bound. \nTaking the more restrictive limit of $\\sigma_{\\chi \\chi}\/m_\\chi\\sim$1~cm$^2\/$g, the bound is \n\\begin{equation} g_\\chi \\lesssim 4 \\times 10^{-5}\\times \\left( \\frac{v}{10^{-3}}\\right) \\times \\left(\\frac{m_\\chi}{1\\,\\text{MeV}}\\right)^{3\/4}\\times \\left(\\frac{10}{\\ln \\left(m_\\chi^2 v^2\/m_{A'}^2\\right) }\\right)^{1\/4}. \\label{SIDMbound}\n\\end{equation}\nSince $\\kappa e g_\\chi \\gtrsim 10^{-12}$ is needed for sub-MeV freeze-in, the SIDM bounds imply that the kinetic mixing is $\\kappa \\gtrsim 10^{-7}$ for MeV-scale DM. For sub-eV dark photons, such large kinetic mixing is only possible when $m_{A'} \\lesssim 10^{-10}$ eV~\\cite{Jaeckel:2013ija}. For even lighter DM, $g_\\chi$ is even more restricted so $\\kappa \\gtrsim 10^{-5}$ is required for freeze-in, which is possible when $m_{A'}\\lesssim 10^{-14}$~eV. Therefore, we are required to consider an ``ultralight'' dark photon~\\cite{Knapen:2017xzo}. Note that black hole superradiance constrains dark photons being present in the mass spectrum (in the small-coupling limit) between $\\sim 10^{-14} - 10^{-11}$~eV and preliminarily between $\\sim 10^{-19} - 10^{-17}$~eV~\\cite{Baryakhtar:2017ngi}.\n\nSuch a light dark photon is phenomenologically equivalent to the massless dark photon limit for all processes considered in this paper because the $m_{A'}$ is much lower than the effective in-medium photon mass $m_A$ in the early universe. Then, following Appendix D of Ref.~\\cite{Knapen:2017xzo}, the vacuum Lagrangian in Eq.~\\eqref{eq:vacuumL} is modified with an additional term $m_{A}^2 A^\\mu A_\\mu \/2$.\\footnote{For simplicity we consider a constant $m_A^2$ for the schematic purposes of this discussion, although the photon polarization tensor $\\Pi^{\\mu \\nu}(\\vec{q}, \\omega)$ (which gives rise to the in-medium effective mass) depends on the photon momentum $\\vec{q}$, energy $\\omega$, polarization, and thermal properties of the medium. For an on-shell mode with $\\omega \\sim |\\vec{q}|$, $m_A^2$ would correspond to the plasma mass, as discussed in Section~\\ref{sec:plasmon}. For scattering processes with a highly off-shell mode, $|\\vec{q}| \\gg \\omega$, $m_A^2$ is given by the Debye mass~\\cite{Blaizot:1995kg}.} Rotating away the mixing term in the presence of $m_{A}$ and $m_{A'}$ and rewriting in terms of the mass eigenstates $\\tilde A$ and $\\tilde A'$, the in-medium Lagrangian is given by\n\\begin{align}\n\t{\\cal L}_{\\rm IM} \\supset & -\\frac{1}{4} \\tilde F_{\\mu \\nu} \\tilde F^{\\mu \\nu} \n\t\t\t- \\frac{1}{4} \\tilde F'_{\\mu \\nu} \\tilde F'^{\\mu \\nu} \n\t\t\t+ \\frac{m_{A}^2}{2} \\tilde A^\\mu \\tilde A_\\mu\n\t\t\t+ \\frac{m_{A'}^2}{2} \\tilde A'^\\mu \\tilde {A'}_\\mu \\nonumber \\\\\n\t &+ J_{\\rm EM}^\\mu \\left(e \\tilde A_\\mu + \\frac{e \\kappa \\, m_{A'}^2}{m_{A'}^2 - m_{A}^2} \\tilde {A'}_\\mu\\right) + g_\\chi \\bar \\chi \\gamma^\\mu \\chi \\left( \\tilde {A^\\prime}_\\mu - \\frac{\\kappa m_{A}^2}{m_{A'}^2 - m_{A}^2} \\tilde A_\\mu \\right).\n\t\\label{eq:inmedium_L}\n\\end{align}\nFrom this, we see that when $m_{A} \\gg m_{A'}$, the interaction terms above reduce to\n\\begin{align}\n\t{\\cal L}_{\\rm IM} \\supset J_{\\rm EM}^\\mu \\left(e \\tilde A_\\mu \\right) + g_\\chi \\bar \\chi \\gamma^\\mu \\chi \\left(\\tilde {A^\\prime}_\\mu + \\kappa \\tilde A_\\mu \\right),\n\t\\label{eq:inmedium_interactions}\n\\end{align}\nmeaning that DM has an effective millicharge parameter $Q = \\kappa g_\\chi\/e$, and the interactions are identical to those for a massless dark photon. Note that this suppression of the $A'$-SM coupling in the $m_{A'} \\ll m_A$ limit is the source of the in-medium (plasma mass) suppression of the stellar constraints on dark photons~\\cite{An:2013yfc,An:2013yua} discussed in Section~\\ref{vectors}. Also note that this suppression means that the dark photon is not abundantly produced by SM interactions in the early universe and does not contribute to the effective number of relativistic species, $N_\\text{eff}$.\n\nIn the exactly massless $A'$ limit, we are free to perform a field redefinition on $A' \\to A' + \\kappa A$ in the vacuum Lagrangian, Eq.~\\eqref{eq:vacuumL}, which eliminates the kinetic mixing term and generates a DM interaction term $g_\\chi \\bar \\chi \\gamma^\\mu \\chi (A^\\prime_\\mu + \\kappa A_\\mu)$, which is again identical to having a millicharge $Q = \\kappa g_\\chi\/e$ under $U(1)_{EM}$. \n\nThe model considered here thus provides another realization of millicharged DM, and all of the stellar constraints discussed in the previous section apply. The only difference is the additional DM self-interaction via the $A'$, which potentially leads to sizeable self-interactions. \n\n\n\\section{Relic abundance from freeze-in \\label{sec:prod}}\n\nHere we compute the relic abundance of DM from freeze-in. We begin by reproducing the contribution from annihilation of SM fermions $f \\bar f \\to \\chi \\bar \\chi$ that was previously calculated in Refs.~\\cite{Essig:2011nj,Chu:2011be}. Because freeze-in is peaked at low temperatures and this paper concerns sub-MeV DM, electrons are the primary source of DM for this channel; in the rest of this section we explicitly refer to freeze-in off electrons, noting we have numerically checked that adding heavier fermions (for instance muons) to the calculation changes the results by less than 1\\%. In addition to freeze-in off electrons, there is a contribution from plasmon decays, $\\gamma^* \\to \\chi \\bar \\chi$, which we calculate for the first time. Photon annihilation into DM $\\gamma \\gamma \\to \\chi \\bar \\chi$ is suppressed by an additional factor of $Q^2$ and can be safely neglected. \n\nIn what follows, we take the observed present-day relic DM abundance to be $\\omega_{c} \\equiv \\Omega_{c} h^2 = 0.12$~\\cite{Aghanim:2018eyx}. After freeze-in, the DM density should scale like $a^{-3}$ and it is common practice to compare this to another quantity that has the same scaling irrespective of changes to the SM bath temperature. In this work we choose to compare the number density to the entropy density. Taking the present-day CMB temperature to be 2.73~K, the observed yield is then \\begin{equation} Y\\equiv n_\\chi\/s = 4.35\\times 10^{-7} \\times \\left(\\frac{1\\, \\mathrm{ MeV}}{m_\\chi}\\right) .\\end{equation}\nFor $m_\\chi\\gtrsim1$~keV, the DM yield is much lower than the order unity yield for relativistic species, such that DM contributes negligibly to $N_\\text{eff}$. This is in contrast to other DM models, such as thermal freeze-out, where sub-MeV DM would generically inject a considerable amount of entropy to the photon or neutrino sectors and would violate observational bounds on $N_\\text{eff}$.\\footnote{An exception for thermal, sub-MeV DM was pointed out in Ref.~\\cite{Berlin:2017ftj}, where the DM thermalizes with the SM thermal bath {\\emph{after}} neutrino-photon decoupling, reducing the contribution to $N_\\text{eff}$. Furthermore, in this model changes to $N_\\text{eff}$ that occur after DM thermalization are compensated by decoupling at a later time.}\n\nThe low DM occupation number also implies that it is possible to self-consistently ignore back-reactions that would reduce the DM number density, namely DM annihilation to electrons and inverse decays to plasmons. For instance, if we ignore the back-reaction, the solution for the number density of DM is significantly lower than the electron number density during the entirety of freeze-in in spite of the fact that the latter is becoming Boltzmann suppressed. Depletion of the DM number density through annihilation to dark photons $\\chi \\bar \\chi \\to \\gamma^{'} \\gamma^{'}$ is negligible for the same reason. In what follows, we solve the $0^\\text{th}$ moment of the Boltzmann equation ignoring back-reactions, noting that we have numerically checked that they are negligible. The relevant equation is then\n \\begin{equation} \n \\frac{d n_\\text{DM}}{d a} + \\frac{3 n_\\text{DM}}{a} = \\frac{2}{a H}\\left( \\avg{\\sigma v}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} n_e^2\n + \\avg{\\Gamma}_{\\gamma^{*} \\rightarrow \\chi \\bar{\\chi}} n_{\\gamma^*}\\right).\n \\label{0thboltz}\n \\end{equation}\nHere we are using $a$ as our time variable. The relationship between $a$ and the SM temperature $T$ (which determines the DM production rate) is not adiabatic during freeze-in because the electrons are leaving the thermal bath at this time; this is discussed further in Appendix~\\ref{clock}. Note that we are solving for the total DM density which includes both $\\chi$ and $\\bar{\\chi}$ in the matter budget; assuming zero DM chemical potential, $n_\\text{DM} = 2 n_\\chi = 2 n_{\\bar{\\chi}}$, which accounts for the factor of two in Eq.~\\eqref{0thboltz}.\\footnote{This factor is related to the usual factor of $1\/2$ that appears in the Boltzmann equation for Dirac fermions~\\cite{Gondolo:1990dk, Srednicki:1988ce}; however, unlike the ordinary case of thermal DM, the change in the comoving DM density for freeze-in is independent of the DM number density (i.e. there is no factor of $n_\\text{DM}^2$ appearing in Eq.~\\eqref{0thboltz}) which accounts for the factor of four difference.}\n\n\n\\subsection{Annihilations}\n\n\nIn computing the DM relic abundance from annhilations of electron-positron pairs, we treat the two scenarios discussed in Section~\\ref{sec:model} as indistinguishable in the limit that $m_{A'}\\rightarrow 0$. We also ignore the in-medium photon mass for this process, which we find to be a percent level effect for $s$-channel annihilations happening at the relevant range of temperatures. In this limit, the matrix element squared is\n \\begin{align*} \n \\sum_\\mathrm{d.o.f.}\\abs{\\mathcal{M}}^2_{e^+ e^- \\leftrightarrow \\chi \\bar{\\chi}} =\\frac{32 \n Q^2 e^4}{ (p_{e^+}+p_{e^-})^4 \n} \\Big(& (p_{e^+}\\cdot p_\\chi) (p_{e^-}\\cdot p_{\\bar{\\chi}}) + (p_{e^+}\\cdot p_{\\bar{\\chi}}) (p_{e^-}\\cdot p_{\\chi})\\\\ &\n + m_e^2 (p_\\chi\\cdot p_{\\bar{\\chi}}) + m_\\chi^2(p_{e^+}\\cdot p_{e^-}) + 2 m_e^2 m_\\chi^2 \\Big), \\numberthis\n \\end{align*} \n where we sum over both initial \\emph{and} final spin degrees of freedom (d.o.f.) without averaging and where $Q$ is the effective millicharge in the dark photon case, $Q = \\kappa g_\\chi \/e$.\nThe thermally averaged cross section appearing in Eq.~\\eqref{0thboltz} for this process is given by\n\\begin{align*} \\avg{\\sigma v}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} n_e^2 = \\int \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{e^+}}{2 E_{e^+}} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{e^-}}{2 E_{e^-}}& \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_\\chi}{2 E_\\chi} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{\\bar{\\chi}}}{2 E_{\\bar{\\chi}}} ~e^{-(E_{e^+} + E_{e^-})\/T}\\numberthis \\label{sigmavann}\\\\& \\times \\sum_\\mathrm{d.o.f.} \\abs{\\mathcal{M}}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}}^2 (2 \\pi)^4 \\delta^{(4)}(p_{e^+} + p_{e^-} - p_\\chi - p_{\\bar{\\chi}}) \\quad \\quad \\end{align*}\nwhere $\\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p \\equiv d^3 p\/(2\\pi)^3$. \nWe assume that from the onset of freeze-in, the electrons have entered the non-relativistic regime where their phase space is given by a Maxwell-Boltzmann distribution with temperature $T$ and zero chemical potential. As we will show, sub-MeV DM freeze-in through the annihilation channel is most effective at temperatures $T\\lesssim m_e$ where the effects of Fermi-Dirac statistics can be neglected. We also ignore Pauli blocking of the DM due to its low occupation number.\n\n\n\n \n To evaluate the thermal cross section, we note that the primordial plasma has a preferred rest frame (where bulk motions average to zero), which breaks Lorentz invariance. The phase space factors of Eq.~\\eqref{sigmavann} are evaluated in a frame that is comoving with the plasma. Practically, we can perform the integration by inserting factors of unity, \n\\begin{equation} \n \\int \\frac{ d^3 q_{12} d s_{12} }{2 E_{12}} \\delta^{(4)}(q_{12} - p_1 - p_2)=1, \n\\label{eq:unity}\n\\end{equation} \n where $q_{12}$ is the effective bulk 4-momentum of the particles labelled 1 and 2 and $s_{12}$ can be thought of as the effective (Lorentz invariant) mass-squared of a single particle with that bulk 3-momentum and energy (\\emph{i.e.} here $E_{12} = \\sqrt{s_{12} + \\vec{q}_{12}^{\\,2}}$). Inserting such a factor into Eq.~\\eqref{sigmavann} gives\n \\begin{align*} \\avg{\\sigma v}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} n_e^2 &= \n \\int \\frac{ d^3 q_{\\chi \\bar{\\chi}} d s_{\\chi \\bar{\\chi}} }{2 E_{\\chi \\bar{\\chi}}} \\int \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{e^+}}{2 E_{e^+}} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{e^-}}{2 E_{e^-}} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_\\chi}{2 E_\\chi} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{\\bar{\\chi}}}{2 E_{\\bar{\\chi}}} ~e^{-(E_{e^+} + E_{e^-})\/T}\\numberthis \\\\& \\times \\sum_\\mathrm{d.o.f.} \\abs{\\mathcal{M}}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}}^2 (2 \\pi)^4 \\delta^{(4)}(p_{e^+} + p_{e^-} - p_\\chi - p_{\\bar{\\chi}}) \\delta^{(4)}(q_{\\chi \\bar{\\chi}} - p_\\chi - p_{\\bar{\\chi}}).\\quad \\quad \\end{align*}\nThe integral over $p_\\chi$ and $p_{\\bar{\\chi}}$ does not depend on the frame of $q_{\\chi \\bar{\\chi}}$, so the two-body phase space of $p_\\chi$ and $p_{\\bar{\\chi}}$ can be evaluated in the CM frame of $q_{\\chi \\bar{\\chi}}$. We define\n \\begin{align*} &\\Phi_{\\chi \\bar{\\chi}}(s_{\\chi \\bar{\\chi}}) \\abs{\\mathcal{M}}^2_\\text{CM}(s_{\\chi \\bar{\\chi}}) \\equiv \\int \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_\\chi}{2 E_\\chi} \\int \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{\\bar{\\chi}}}{2 E_{\\bar{\\chi}}} (2 \\pi)^4 \\delta^{(4)}(q_{\\chi \\bar{\\chi}} - p_\\chi - p_{\\bar{\\chi}}) \\sum_\\mathrm{d.o.f.} \\abs{\\mathcal{M}}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}}^2 \\\\\n& = \\frac{ Q^2 e^4}{2 \\pi s_{\\chi \\bar{\\chi}}^2 } \\sqrt{1 -\\frac{4 m_\\chi^2}{s_{\\chi \\bar{\\chi}}}} \\left( s_{\\chi \\bar{\\chi}}^2 + \\frac{1}{3}(s_{\\chi \\bar{\\chi}} - 4 m_e^2) (s_{\\chi \\bar{\\chi}} - 4 m_\\chi^2)+ 4 s_{\\chi \\bar{\\chi}} (m_\\chi^2 +m_e^2) \\right),\\quad \\quad \\quad \\numberthis \\end{align*}\n and insert this into the expression for the thermally averaged cross section\n \\begin{align*} \\avg{\\sigma v}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} n_e^2 = \\int \\frac{ d^3 q_{\\chi \\bar{\\chi}} d s_{\\chi \\bar{\\chi}} }{2 E_{\\chi \\bar{\\chi}}}~ e^{-E_{\\chi \\bar{\\chi}}\/T} & \\Phi_{\\chi \\bar{\\chi}}(s_{\\chi \\bar{\\chi}})\\abs{\\mathcal{M}}^2_\\text{CM}(s_{\\chi \\bar{\\chi}})\\\\\n \\times & \\int \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{e^+}}{2 E_{e^+}} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{e^-}}{2 E_{e^-}} \\delta^{(4)}(p_{e^+}+p_{e^-}-q_{\\chi \\bar{\\chi}} ) . \\numberthis \\end{align*}\nAgain, we can evaluate the integral over $p_{e^+}$ and $p_{e^-}$ in the center-of-mass frame. Defining\n\\begin{equation} \\Phi_{e^+ e^- }(s_{\\chi \\bar{\\chi}}) \\equiv \\frac{1}{8 \\pi} \\sqrt{1 - \\frac{4 m_e^2}{s_{\\chi \\bar{\\chi}}}},\\end{equation}\nthe thermally averaged cross section becomes\n \\begin{align}\n\\avg{\\sigma v}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} n_e^2 &= \\frac{1 }{ (2 \\pi)^4} \\int \\frac{ d^3 q_{\\chi \\bar{\\chi}} d s_{\\chi \\bar{\\chi}} }{2 E_{\\chi \\bar{\\chi}}} e^{-E_{\\chi \\bar{\\chi}}\/T} \\Phi_{e^+ e^- }(s_{\\chi \\bar{\\chi}})\\Phi_{\\chi \\bar{\\chi}}(s_{\\chi \\bar{\\chi}})\\abs{\\mathcal{M}}^2_\\text{CM}(s_{\\chi \\bar{\\chi}}).\n\\end{align}\nWe can write this result in terms of the first order modified Bessel function of the second kind $K_1(z) = z \\int_1^\\infty d u \\,e^{- z u} \\sqrt{u^2 - 1} $ with $u = \\sqrt{1+ q_{\\chi \\bar{\\chi}}^2 \/s_{\\chi \\bar{\\chi}}} \\,$ :\n\\begin{equation} \\avg{\\sigma v}_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} n_e^2 = \\frac{ T}{(2 \\pi)^3} \\int ds\\,\\sqrt{s}~ \\Phi_{e^+ e^- }(s)\\,\\Phi_{\\chi \\bar{\\chi}}(s) \\abs{\\mathcal{M}}^2_\\text{CM}(s)\\, K_1(\\sqrt{s}\/T)\\end{equation}\nwhere we have dropped the subscript on the integration variable $s$. Note that $s$ is restricted to $s > 4\\max\\left(m_e^2, m_\\chi^2\\right)$. The procedure above provides an alternate derivation of the well-known results from Ref.~\\cite{Gondolo:1990dk}, and we have validated this method here because we use similar techniques to derive the full collision term for annihilation in Section~\\ref{sec:ann_phasespace}.\n\n\n\n\\subsection{Plasmon decay \\label{sec:plasmon}}\n\nThe early Universe is an optically thick plasma where photons acquire an in-medium mass; this can be understood classically as arising from the electrons' oscillatory response to a propagating electric field and the dynamical shielding of that electric field. This effective mass is also manifest in the photon propagator and the polarization vectors of external photon legs in the medium; in other words, the photon mass and wavefunction are renormalized in the plasma. The effective masses and dressed polarization functions for the transverse and longitudinal ``plasmon'' modes are shown in Fig.~\\ref{fig:plasmon} and explicit formulae are provided in Appendix~\\ref{plasma}. The effective mass for plasmons is closely related to the plasma frequency. For a relativistic plasma at zero chemical potential, the plasma frequency is $\\omega_p = eT\/3 \\approx 0.1 T$ where $e$ is electric charge. \n\nPlasmons can undergo decay provided that it is kinematically allowed. For instance, plasmons can decay to neutrino pairs through mixing with the $Z$ boson \\cite{Braaten:1993jw}. Plasmons cannot decay to charged particles in the SM because their effective mass is also renormalized in the medium and it is always kinematically forbidden. However, this is not the case for millicharged DM where corrections to the mass are suppressed by powers of $Q$. \n\n\\onecolumngrid\n\\begin{figure*}[t!]\n\\includegraphics[width=\\textwidth]{m_eff_Zk.pdf}\n\\caption{The effective in-medium mass (left) and wavefunction renormalization (right) for photons, as computed in Coulomb gauge for a plasma with $T=1$~MeV and zero chemical potential (see Appendix~\\ref{plasma} for relevent formulae). The transverse mode is relevant at all wavelengths while the longitudinal mode crosses the lightcone at high $k$ and can thus only propagate at low $k$. Also shown are the low-$k$, low-$T$ and high-$k$, high-$T$ limits for the effective transverse mass, $m_t = \\omega_p$ and $m_t = \\sqrt{3\/2}\\omega_p$, respectively.}\n\\label{fig:plasmon}\n\\end{figure*}\n \n\nThe effective matrix element that captures plasmons decaying to DM is \n\\begin{equation} \ni \\mathcal{M}_{\\gamma^* \\rightarrow \\chi \\bar{\\chi}} = i Q e\\, \\tilde{\\epsilon}_\\mu(k) \\bar{u}(p_\\chi) \\gamma^\\mu v(p_{\\bar{\\chi}}), \\end{equation} \nwhere $\\tilde{\\epsilon}_\\mu(k)$ is the dressed polarization vector for the longitudinal and transverse plasmon modes as detailed in Appendix~\\ref{plasma}, where we work in Coulomb gauge. We express this process in terms of the DM effective millicharge $Q$ and in Appendix~\\ref{basis} we show explicitly that decaying through a dark photon gives the same effective matrix element in the limit $m_{A'}\\rightarrow 0$. In squaring and summing over polarizations, only the diagonal terms ($LL$, $++$, and $--$) contribute,\n\\begin{equation} \\sum_\\mathrm{d.o.f.} \\abs{\\mathcal{M}}^2_{\\gamma^* \\rightarrow \\chi \\bar{\\chi}} = 4 Q^2 e^2 \\times \\begin{cases}\n2 Z_t(k) ( p_\\chi^2 \\sin^2 \\theta + \\omega_t(k) E_\\chi - k p_\\chi \\cos \\theta) & {++ \\& --} \\\\\nZ_\\ell(k) \\frac{\\omega_\\ell(k)^2}{k^2} (\\omega_\\ell(k) E_\\chi - 2 E_\\chi^2 + k p_\\chi \\cos \\theta) & \\text{LL},\n\\end{cases}\n\\end{equation}\nwhere the photon four-momentum is given by $K^\\mu = \\big(\\omega(k), \\vec{k}\\big)^\\mu$ with appropriate dispersion relations for transverse and longitudinal modes $\\omega_t(k)$ and $\\omega_\\ell(k)$ (see Appendix~\\ref{plasma}), the DM four-momentum is given by $\\left(E_\\chi, \\vec{p}_\\chi\\right)^\\mu$, $\\theta$ is the angle between $\\vec{k}$ and $\\vec{p}_\\chi$, and $Z_t(k)$ and $Z_\\ell(k)$ are wavefunction renormalization factors (shown in Fig.~\\ref{fig:plasmon}) that are related to the dressed polarization vectors for the transverse and longitudinal modes.\n\nThe thermally averaged decay rate is \\begin{equation} \\left< \\Gamma\\right>_{\\gamma^* \\rightarrow \\chi \\bar{\\chi}} n_{\\gamma^*} = \\int \\frac{\\ensuremath{\\mathchar'26\\mkern-12mu d}^3 k}{2 \\omega(k)} \\frac{\\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_\\chi}{2 E_\\chi}\\frac{\\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{\\bar{\\chi}}}{2 E_{\\bar{\\chi}}} ~f\\left(\\omega(k)\\right) \n (2\\pi)^4 \\delta^{(4)}\\left(K - p_\\chi - p_{\\bar{\\chi}}\\right) \\sum_\\mathrm{d.o.f.} \\abs{\\mathcal{M}}^2_{\\gamma^* \\rightarrow \\chi \\bar{\\chi}},\\end{equation}\nand can be evaluated directly. Taking the plasmons to be Bose-Einstein distributed, the longitudinal and transverse contributions to this rate are\n\\begin{equation} \\left< \\Gamma\\right>_{\\gamma^*_\\ell \\rightarrow \\chi \\bar{\\chi}} n_{\\gamma^*_\\ell} =\\frac{ Q^2 e^2}{(2 \\pi)^3} \\int k^2 \\,d k \\, \\frac{Z_\\ell(k) \\omega_\\ell (k)(m_\\ell(k)^2 + 2 m_\\chi^2 )\\sqrt{m_\\ell(k)^2(m_\\ell(k)^2 -4 m_\\chi^2)}}{3 m_\\ell(k)^4 \\left(e^{\\omega_\\ell(k)\/T}-1 \\right)} \\end{equation}\n\\begin{equation} \\left< \\Gamma\\right>_{\\gamma^*_t \\rightarrow \\chi \\bar{\\chi}} n_{\\gamma^*_t}= \\frac{ 4Q^2 e^2 }{(2 \\pi)^3} \\int k^2 \\,d k \\, \\frac{ Z_t(k)(m_t(k)^2 -m_\\chi^2)\\sqrt{m_t(k)^2 (m_t(k)^2 - 4 m_\\chi^2)}}{3 \\omega_t (k)\\,m_t(k)^2 \\left(e^{\\omega_t(k)\/T}-1 \\right)}, \\end{equation}\nwhere the effective plasmon masses are $m_\\ell(k)^2 = \\omega_\\ell(k)^2 - k^2$ for the longitudinal modes and $m_t(k)^2 = \\omega_t(k)^2 - k^2$ for the tranverse ones.\nThe final integrals over $k$ can be computed numerically and the total plasmon contribution to decay is dominated by the transverse modes (note that we are working in Coulomb gauge). This is because the longitudinal mode has a finite range of $k$ over which it can propagate, meaning that it has less available phase space than the transverse mode which has no restriction in $k$. Furthermore, the longitudinal mass and renormalization factors fall steeply within the range of $k$ where this mode can propagate. \n\n\n\\subsection{Couplings for freeze-in}\n\nIn solving the zeroth moment of the Boltzmann equation for the DM relic abundance, we find that the relative contributions from $e^+ e^-$ annihilation and plasmon decays are starkly different in different mass ranges, as illustrated in Fig.~\\ref{masscomp_abundance}. \n\\begin{figure*}[t]\n\\includegraphics[width=0.5\\textwidth]{abundance_40.pdf}\\includegraphics[width=0.5\\textwidth]{abundance_400.pdf}\n\\caption{Evolution of the comoving DM number density for $m_\\chi = 40$~keV (left) and $m_\\chi = 400$~keV (right) as compared to the relic abundance of DM with that mass. Also shown are the relative contributions from electron-positron annihilations and plasmon decays, as discussed in the text.}\n\\label{masscomp_abundance}\n\\end{figure*}\nThis can be understood by considering the fact that freeze-in is dominant at low temperatures, provided that it is kinematically allowed and that the population the DM is freezing in from has a sufficient abundance. For sub-MeV DM, freeze-in from $e^+e^-$ annihilation is always kinematically allowed and this process only ends when the electron number density becomes Boltzmann suppressed, namely $T\\lesssim m_e$. Meanwhile, the plasmon abundance is not Boltzmann suppressed but the mass runs with temperature, so freeze-in through plasmon decay ends when it is no longer kinematically allowed, namely when $m_{\\gamma^*}\\sim \\omega_p = 2 m_\\chi$. Since $\\omega_p \\approx 0.1 T$ in the relativistic limit, plasmon decay to millicharged DM shuts off at an earlier time compared to annihiliation. These two criteria are shown in Fig.~\\ref{masscomp_abundance} and indeed we see that plasmon decays are more dominant in determining the relic abundance for lower mass DM because the decays are active for a longer period of time.\n\nIn terms of the effective millicharge needed to produce the observed DM relic abundance, we find that including plasmon decays leads to a significant reduction in coupling for keV-mass DM while the effect is small once $m_\\chi$~=~MeV. \nThe change to the freeze-in benchmark for direct detection is shown in Fig.~\\ref{DDshift},\n\\begin{figure}[t]\n\\includegraphics[width=0.7\\textwidth]{DD_comparison.pdf} \n\\caption{The effect of plasmon decays on the freeze-in benchmark for direct detection via electron recoils. Also shown are the projected sensitivities of low-threshold experiments with kg-day exposure, including a SuperCDMS G2 experiment~\\cite{Battaglieri:2017aum} and proposals using polar materials (GaAs and Al$_2$O$_3$)~\\cite{Griffin:2018bjn, Knapen:2017ekk}, Dirac materials (ZrTe$_5$)~\\cite{Hochberg:2017wce}, or superconductors (Al SC)~\\cite{Hochberg:2015fth}.\n\\label{DDshift}}\n\\end{figure} \nwhere the cross section for electron recoils is \n\\begin{equation} \\sigma_e = \\frac{16 \\pi Q^2 \\alpha^2 \\mu_{\\chi e}^2}{ (\\alpha m_e)^4} . \\end{equation}\nHere $\\mu_{\\chi e}$ is the electron-DM reduced mass, $\\mu_{\\chi e} = m_e m_\\chi \/ (m_e + m_\\chi)$. At the lowest mass where proposed low-threshold direct detection experiments are sensitive, the plasmon decay channel for DM production lowers the expected signal strength by roughly an order of magnitude.\n\nIt has been noted in the literature~\\cite{Chuzhoy:2008zy,Hu:2016xas,Dunsky:2018mqs} that millicharged DM could be efficiently accelerated in supernova remnants, which would lead to an accelerated component of dark cosmic rays and eject DM from the disk. Both of these effects can lead to substantial changes to the predicted direct detection rates and sensitivities of proposed experiments shown above. However, the conclusions are highly sensitive to aspects of cosmic ray physics which are not fully understood, such as the injection of particles into the diffusive shock acceleration process. The predictions would also be sensitive to whether the DM obtains its effective millicharge through a kinetic mixing portal; in this case, the dark photon mass and couplings can affect the acceleration, and an exploration of these effects is beyond the scope of this work.\n\n\n\n\\section{Dark matter phase space distribution \\label{sec:phasespace}}\n\nSince freeze-in DM is so weakly coupled to the SM, it does not thermalize with the SM during freeze-in and the phase space distribution can deviate substantially from a thermal distribution. While this has no clear impact on direct detection, since galaxy assembly is expected to significantly alter the DM velocity distribution, it does affect DM free-streaming and DM-SM scattering in the early universe. Here we compute the full phase space distributions needed to determine the cosmological observables; the signatures, constraints, and detection prospects will be presented in a companion paper~\\cite{inprep}.\n\nWe must solve the full Boltzmann equation in an expanding background, given by \n\\begin{equation} \n \\frac{\\partial f_\\chi}{\\partial t} - H \\frac{p_\\chi^2}{E_\\chi} \\frac{\\partial f_\\chi}{\\partial E_\\chi} = \\frac{C(p_\\chi, t)}{E_\\chi}, \n\\end{equation}\nwhere $C(p_\\chi, t)$ is the collision term, which encapsulates all interactions that affect the phase space. At early times, the interactions that determine the phase space evolution are $e^+ e^-$ annihilation and plasmon decay. We have checked numerically that heavier fermion annihilation processes (for instance the annihilation of muon-antimuon pairs) affect the phase space by a negligible amount because they occur only at early times when freeze-in is less efficient. Scattering has a negligible impact on the phase space during freeze-in since the DM occupation number is much smaller than that of electrons or plasmons. Neglecting the small effect of scattering during freeze-in, the collision term is independent of $f_\\chi$ to leading order and the Boltzmann equation can be solved by direct integration~\\cite{Bae:2017dpt}, \n\\begin{equation} \n f_\\chi(p_\\chi, t) = \\int_{t_i}^{t} dt'\\, \\frac{C\\left(\\frac{a(t)}{a(t')}\\, p_\\chi, t'\\right)}{\\sqrt{\\frac{a(t)^2}{a(t')^2}\\, p_\\chi^2+m_\\chi^2} } = \\int_{a_i}^{a(t)} \\frac{da'}{a' H(a')}\\, \\frac{C\\left(\\frac{a(t)}{a'}\\, p_\\chi, a'\\right)}{\\sqrt{\\frac{a(t)^2}{a'^2}\\, p_\\chi^2+m_\\chi^2} }. \\label{axinoapprox}\n\\end{equation}\nHere the factors of $a$ in the integrand keep track of redshifting of momentum due to expansion. We use the scale factor $a$ as our time variable rather than the common choice of using the SM temperature because it is not evolving adiabatically as the electron-positron pairs leave the bath during freeze-in. The temperature evolution and the evolution of the Hubble parameter are detailed in Appendix~\\ref{clock}. \n\nAfter freeze-in ends, the DM momenta redshift and the phase space distribution is constant in comoving momentum. However, at late times DM-SM and DM-DM scattering eventually can become important since the scattering cross sections are peaked at low relative velocities. The effects of DM-SM scattering on the phase space are generally negligible for the allowed parameter space, but DM self-scattering can lead to thermalization of the DM phase space distribution. Whether this occurs is model-dependent, and we discuss the conditions for this to occur in Section~\\ref{sec:DMscattering}.\n \n\\subsection{Phase space from annihilation \\label{sec:ann_phasespace} }\nThe computation of the full collision term from annihilation proceeds similarly to the computation of its zeroth moment. Once again, inserting a factor of unity as defined in Eq.~\\eqref{eq:unity}, we find \n\\begin{align*} \n C(p_\\chi, t)_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} \n = \\frac{1}{2(2 \\pi)^3} \\int \\frac{ d^3 q_{e^+ e^-} d s_{e^+ e^- } }{2 E_{\\bar{\\chi}} 2 E_{e^+ e^-}} & \\delta(E_{e^+ e^-} - E_\\chi - E_{\\bar{\\chi}}) ~e^{-E_{e^+ e^-}\/T}\\\\\n &\\times \\Phi_{e^+ e^-}(s_{e^+ e^-}) \\abs{\\mathcal{M}}^2_\\text{CM}(s_{e^+ e^-}), \\numberthis\n\\end{align*}\nwhere $E_{\\bar{\\chi}} = \\sqrt{m_\\chi^2 + p_{{\\chi}}^2 + q_{e^+ e^-}^2 - 2 p_\\chi q_{e^+ e^-} \\cos \\theta}$, $E_{e^+ e^-} = \\sqrt{s_{e^+e^-}+q^2_{e^+ e^-}}$ and $\\theta$ is the angle that $\\vec{q}_{e^+ e^-}$ makes with the unconstrained, unintegrated $\\vec{p}_\\chi$. \nDefining $x \\equiv \\cos\\theta$ and dropping the subscript on the bulk electron momentum, we find \n\\begin{align}\n& C(p_\\chi, t)_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} = \\frac{1}{2 (2 \\pi)^2 p_\\chi} \\int \\frac{ dx \\,q d q\\, d s }{ 4 E} \\delta \\left(x-\\frac{2 E_\\chi E -s}{2 p_\\chi q}\\right) e^{-E\/T} \\Phi_{e^+ e^-}(s) \\abs{\\mathcal{M}}^2_\\text{CM}(s).\n\\end{align}\nRequiring that $x \\in [-1, 1]$ and switching integration variables,\n\\begin{align*} &C(p_\\chi, t)_{e^+ e^- \\rightarrow \\chi\\bar{\\chi}} = \\frac{1}{8 p_\\chi (2 \\pi)^2} \\int\nd s\\int_{\\frac{E_\\chi s - p_\\chi \\sqrt{s(s-4 m_\\chi^2)}}{2 m_\\chi^2}}^{\\frac{E_\\chi s + p_\\chi \\sqrt{s(s-4 m_\\chi^2)}}{2 m_\\chi^2}} d E \\, e^{-E\/T} \\Phi_{e^+ e^-}(s) \\abs{\\mathcal{M}}^2_\\text{CM}(s) \\\\\n&= \\frac{ T }{ 4 p_\\chi (2 \\pi)^2} \\int\nd s \\, e^{-\\frac{ E_\\chi s}{2 m_\\chi^2 T}} \\sinh \\left( \\frac{ p_\\chi \\sqrt{s(s-4 m_\\chi^2)}}{2 m_\\chi^2 T}\\right) \\Phi_{e^+ e^-}(s) \\abs{\\mathcal{M}}^2_\\text{CM}(s). \\numberthis \\label{colltermann} \\end{align*}\nThen, to solve for the final phase space from annihilation, we can combine Eqs.~\\eqref{axinoapprox} and \\eqref{colltermann}. Note that because $p_\\chi$ is fixed (rather than an integration variable), $s$ in the above integral is restricted to $s > \\max \\left(4 m_e^2, 2 m_\\chi (E_\\chi + m_\\chi)\\right)$ unlike in the integral for determining the thermally averaged cross section. The resulting evolution of the phase space distribution is shown in the left panel of Fig.~\\ref{freezephase}.\n\n\n\\onecolumngrid\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{freezein_temp_new.pdf}\n\\caption{A comparison of the phase space evolution of DM being produced by $e^+ e^-$ annihilation (left) and $\\gamma^*$ decay (right) at $m_\\chi = 40$~keV. The momenta shown here are comoving, $P_\\chi \\equiv a p_\\chi$ where $a=1$ corresponds to $T= 1$~MeV. The phase space is normalized arbitrarily for the purposes of comparing the $P_\\chi$-dependence side by side. Over time, the comoving phase space converges to its final frozen-in shape. The phase space from annihilation is similar to that of the thermal electrons from which they inherit their kinematics. Meanwhile, the phase space from plasmon decay is highly peaked at low $P_\\chi$ because freeze-in through this channel occurs predominantly at threshold when $\\omega_p \\sim 2 m_\\chi$ and the decay is peaked when the plasmon is ``at rest,'' $k\\rightarrow0$. }\n\\label{freezephase}\n\\end{figure*}\n \n\n\\subsection{Phase space from plasmon decay \\label{sec:plasmon_phasespace} }\n\nThe collision term from plasmon decay,\n\\begin{equation} C(p_\\chi, t)_{\\gamma^* \\rightarrow \\chi \\bar{\\chi}} = \\frac{1}{2}\\int \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 k}{2 \\omega(k)} \\frac{ \\ensuremath{\\mathchar'26\\mkern-12mu d}^3 p_{\\bar{\\chi}}}{2 E_{\\bar{\\chi}}} \\frac{1}{e^{\\omega(k)\/T}-1} (2 \\pi)^4 \\delta^{(4)}(K - p_\\chi - p_{\\bar{\\chi}}) \\sum_\\mathrm{d.o.f.} \\abs{\\mathcal{M}}^2_{\\gamma^* \\rightarrow \\chi \\bar{\\chi}}\\end{equation} \nproceeds through direct computation. We find\n\\begin{align} C(p_\\chi, t)_{\\gamma^*_\\ell \\rightarrow \\chi \\bar{\\chi}} &= \\frac{ Q^2 e^2 }{4 \\pi p_\\chi} \\int \\frac{dk\\, \\omega_\\ell (k) Z_\\ell(k) }{k\\, (e^{\\omega_\\ell(k)\/T}-1)} \\left(2 E_\\chi (\\omega_\\ell(k) - E_\\chi) - m_\\ell(k)^2\/2\\right)\n \\\\ C(p_\\chi, t)_{\\gamma^*_t \\rightarrow \\chi \\bar{\\chi}} &= \\frac{ Q^2 e^2 }{4 \\pi p_\\chi} \\int \\frac{dk\\,k Z_t(k)}{\\omega_t(k) (e^{\\omega_t(k)\/T}-1)} \\left(2 p_\\chi^2 - \\frac{( 2 E_\\chi \\omega_t(k) - m_t(k)^2)^2}{2 k^2} +m_t(k)^2\\right)\n\\end{align}\nwhere the limits of the $k$ integral are determined by the requirement that $x_0 = (2 E_\\chi \\omega_{\\ell, t}(k) - m_{\\ell, t}(k)^2)\/2 k p_\\chi$ lies in the range $[-1,1]$. The limits of integration cannot be solved for in closed form because of the nontrivial dispersion relations, so the phase space must be determined numerically.\n\nThe evolution of the phase space from plasmon decays is shown in the right panel of Fig.~\\ref{freezephase}, and our results for the combined phase space can be found in Fig.~\\ref{masscomp_phase}. The distributions are noticeably nonthermal due to plasmon decays. Fig.~\\ref{Teff} compares the average momentum and momentum-squared of the DM to the SM photons, which serves as a useful metric to determine the DM free-streaming and suppression of the growth of structure.\n\n\\onecolumngrid\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{phase_comp_new.pdf}\n\\caption{A comparison of the contributions to the phase space for $m_\\chi = 40$~keV (left) and $m_\\chi = 400$~keV (right). The momenta shown here are comoving, $P_\\chi \\equiv a p_\\chi$ where $a=1$ corresponds to $T= 1$~MeV. The phase space is normalized to the comoving DM relic abundance for each mass depicted. The plasmon contribution dominates more at low masses than at high masses because freeze-in through this channel persists for longer at lower masses, ending when the plasmon mass is at threshold, $\\omega_p \\sim 2 m_\\chi$. Also shown (dashed lines) are the phase space distributions that would arise if the DM could thermalize within its own sector, conserving $\\left$ for non-relativistic DM.}\n\\label{masscomp_phase}\n\\end{figure*}\n \n\n\\begin{figure}[t]\n\\includegraphics[width=0.7\\textwidth]{Teff_new.pdf}\n\\caption{A comparison between moments of the DM phase space and the SM photon phase space as a function of DM mass. For reference, the moments for the SM photon are $\\langle p_\\gamma \\rangle = 2.7\\, T_\\gamma$ and $\\langle p_\\gamma^2 \\rangle = 10.35\\, T_\\gamma^2$. While the DM phase space is not thermal, these moments can be thought of as relating to the DM effective temperature, which will have ramifications for the subsequent cosmology. As the DM mass rises, the effective temperature increases because $e^+ e^-$ annihilations become more important than plasmon decays and have a comparatively fatter high-$p_\\chi$ tail. At even larger masses where $m_\\chi$ is comparable to $m_e$, that high-$p_\\chi$ tail is suppressed because the DM mass becomes relevant to the kinematics of annihilation, causing the effective temperature to drop. }\n\\label{Teff}\n\\end{figure}\n\n\\subsection{Effect of DM-SM scattering \\label{sec:scattering}}\n\nWe argue here that the effect of DM-SM scattering on the DM phase-space distribution is small from freeze-in until the onset of recombination. The relevant quantity is the momentum-transfer rate, which we estimate in the limits where the DM is relativistic and non-relativistic. We do not consider scattering by relativistic, charged SM particles because this is only relevant for electrons during freeze-in; during freeze-in, the number density of DM is many orders of magnitude smaller than the number density of electrons and the effect of electron-DM scattering is suppressed by $n_\\chi\/n_e$ relative to the dominant effect of electron-positron annihilations on the phase space. \nAs outlined below, DM-SM scattering becomes more important at low velocities, corresponding to later cosmological times. This can affect CMB anisotropies and the cosmological 21~cm signal, and we provide more detailed calculations in that context in our companion paper~\\cite{inprep}.\n\nIn the limit of relativistic DM scattering with non-relativistic SM particles (the case after freeze-in until $T_\\gamma \\sim m_\\chi$), the differential cross section with respect to the center-of-mass scattering angle $\\theta_\\text{CM}$ is given by\n\\begin{align}\n \\frac{d\\sigma_{\\chi b}}{d\\cos \\theta_\\text{CM}} = \\frac{ \\pi Q^2 \\alpha^2 }{ p_\\text{CM}^2} \\frac{ (1+ \\cos \\theta_\\text{CM})}{(1 - \\cos \\theta_\\text{CM} +m_D^2\/2p_{\\text{CM}}^2)^2}, \n \\label{eq:scattering_relativistic}\n\\end{align}\nwhere $p_{\\text{CM}}\\equiv | \\vec{p}_{\\text{CM}}|$ is the momentum in the CM frame. Here we have taken $p_\\chi \\ll m_e$, which is a good approximation after freeze-in has ended. In this approximation, the dependence on the SM particle mass drops out, making scattering with electrons and protons equally important (we refer to them collectively as ``baryons,'' in the remainder of this discussion, hence the subscript $b$ in the cross section). The dependence on the Debye mass $m_D$ comes from the photon propagator for electric scattering in a medium~\\cite{Blaizot:1995kg}. The usual $t$-channel divergence is thus regulated in the forward-scattering limit by the Debye angle, defined as $\\theta_D \\equiv m_D\/p_{\\text{CM}}$. Once the plasma has become non-relativistic with $T_\\gamma \\lesssim m_e$, the Debye mass is given by \\begin{equation} m_D = \\sqrt{4 \\pi \\alpha n_e\/T_\\gamma} = 3.7\\times 10^{-6}\\, T_\\gamma \\end{equation}\nin natural units, assuming $\\Omega_b h^2 = 0.022$ \\cite{Aghanim:2018eyx} and that the ionization fraction is unity. The momentum transfer cross section is defined for DM self-scattering in Eq.~\\eqref{eq:transfer} and the analogous definition applies for scattering between DM and SM particles. For relativistic DM, we find that in the limit of the Debye angle $\\theta_D \\ll 1$\n\\begin{align}\n \\sigma_{T,\\, \\chi b} = \\frac{4 \\pi Q^2 \\alpha^2}{p_\\chi^2} \\log \\frac{2}{\\theta_{D}} .\n\\end{align}\nSince $m_b \\gg m_\\chi$ and the baryons are non-relativistic, the DM momentum in the CM frame can be approximated by the DM momentum in the comoving frame, $p_\\chi$. As illustrated in Fig.~\\ref{Teff}, the typical DM momentum is comparable to the SM photon temperature, with both quantities redshifting after freeze-in. Therefore, we can estimate the momentum transfer rate per DM particle and per Hubble time as\n\\begin{align}\n \\frac{ n_p \\sigma_{T, \\,\\chi b}}{H} \\approx 5.3 \\times 10^{-11} \\, \\left( \\frac{ Q}{10^{-10}} \\right)^2 \\left( \\frac{ \\textrm{MeV} }{T_\\gamma} \\right) ,\n\\end{align}\nwhere $n_p \\approx 1.5 \\times 10^{-10}\\, T_\\gamma^3$ and $p_\\chi \\approx 0.4 \\, p_\\gamma \\approx T_\\gamma$. For $T_\\gamma$ in the keV-MeV range and $Q < 10^{-10}$ for freeze-in, this rate is tiny and thus scattering in this regime has a negligible effect on the DM phase space. \n\nFor scattering of non-relativistic DM and charged SM particles, the differential cross section is instead given by \\begin{align}\n \\frac{d\\sigma_{\\chi b}}{d\\cos \\theta_\\text{CM}} = \\frac{2 \\pi Q^2 \\alpha^2 }{ \\mu_{\\chi b}^2 v^4} \\frac{ 1}{(1 - \\cos \\theta_\\text{CM} +m_D^2\/2p_{\\rm CM}^2)^2}, \n \\label{eq:scattering_NR}\n\\end{align}\nwhere $\\mu_{\\chi b}$ is the reduced mass of the DM and baryon, $\\mu_{\\chi b} = m_\\chi m_b\/(m_\\chi+m_b)$, $v$ is the relative velocity between DM and SM particles, and $ p_{\\rm CM} = \\mu_{\\chi b} v$. The momentum transfer cross section is\n\\begin{align}\n\t\\sigma_{T,\\, \\chi b} = \\frac{4 \\pi \\, Q^2 \\alpha^2 }{ \\mu_{\\chi b}^2 v^4} \\log \\frac{2}{\\theta_{D}}\n \\label{eq:NRmomentumtransfer} \\, ,\n\\end{align}\nwhere again we take the $\\theta_D \\ll 1$ limit.\nNote that the Coulomb logarithm appearing here differs from the one that appears in the often-quoted Ref.~\\cite{McDermott:2010pa}; however, that reference did not include the Debye mass in the photon propagator, as discussed in Appendix~\\ref{appendix:debeye}. Compared to the Coulomb logarithm in Ref.~\\cite{McDermott:2010pa}, our treatment of the Debye mass results in a factor of $2.5-3$ smaller momentum transfer rate at recombination; this will translate to a weaker CMB bound on generic millicharged DM than has been reported previously~\\cite{Dvorkin:2013cea,Xu:2018efh,Slatyer:2018aqg,Kovetz:2018zan,Boddy:2018wzy}, which we explore in more detail in our companion paper~\\cite{inprep}.\n\nGiven the velocity scaling in Eq.~\\eqref{eq:NRmomentumtransfer}, momentum transfer is most important at late times. For freeze-in couplings, there may be a substantial effect at the recombination epoch. In particular, momentum transfer during this epoch leads to a drag force between the DM and baryon fluids, which can affect CMB anisotropies~\\cite{Dubovsky:2001tr,Dvorkin:2013cea,Boddy:2018kfv,Boddy:2018wzy}. The CMB bounds {\\emph{require}} that the momentum transfer rate is slow compared to the rate of Hubble expansion at $z \\approx 1100$, thus limiting the possible effect on the DM phase space. We calculate the bounds in detail in the companion paper~\\cite{inprep}, properly accounting for the velocity distribution for freeze-in DM with the updated Coulomb logarithm.\n\nIn addition to DM-baryon scattering as discussed above, DM-photon scattering is possible. However, these processes do not have the low-velocity $v^{-4}$ enhancement in the rate and the cross section scales as $Q^4$, so the effects are negligible. In the model with a dark photon $A'$, scattering processes such as $ e^-+ \\gamma \\to e^- + A'$ are also possible and scale only as kinetic mixing squared $\\kappa^2$. However, these processes are still negligible compared to DM-baryon scattering since they lack the low-$v$ enhancement and have an additional large suppression due to the in-medium kinetic mixing effects, as discussed in Section~\\ref{sec:darkphoton}. Processes like $\\chi + \\gamma\\to \\chi + A'$ scale as $Q^2 g_\\chi^2$; these also lack the $v^{-4}$ enhancement and any enhancement (relative to DM-baryon scattering) from the large photon-to-baryon ratio is more than compensated by the factor of $g_\\chi^2$, even at the largest values of $g_\\chi$ that saturate SIDM bounds. \n\n\\subsection{Effect of DM-DM scattering \\label{sec:DMscattering}}\n\nIn the absence of a dark photon, DM self scattering is proportional to $Q^4$, rendering it entirely negligible. However, self-interactions of the DM can effectively thermalize the phase space distribution in the model with a dark photon. The rate for dark photon mediated DM scattering is proportional to $g_\\chi^4$, and thus may be important if $g_\\chi$ is sufficiently large compared to $\\kappa$. Similar to DM-baryon scattering, the cross section scales as $1\/v^4$ and so these effects are most important at later times when the DM is cooler. Sufficient levels of self-scattering will convert a free-streaming phase space distribution into a Maxwell-Boltzmann or Gaussian velocity distribution. In the non-relativistic limit, the quantity $\\langle a(t)^2 p_\\chi^2 \\rangle$ will remain the same after this process (by conservation of comoving energy), although other moments of the phase space differ.\n\n\n\n\nTo determine when self-scattering becomes important, we estimate the redshift $z_{\\rm therm}$ when the momentum transfer rate per DM particle and per Hubble time is order unity:\n\\begin{align}\n \\frac{ n_\\chi \\sigma_{T,\\,\\chi \\chi} v}{ H(z_{\\rm therm})} = 1\n\\end{align}\nwhere $v$ is the relative velocity between DM particles and $\\sigma_{T,\\, \\chi \\chi}$ is the self-scattering momentum transfer cross section given in Eq.~\\eqref{eq:transfer}, with the dark photon mass regulating the forward scattering instead of the Debye mass that is present for DM-baryon scattering. Using the ratio of the average DM momentum to the photon momentum in Fig.~\\ref{Teff}, we approximate the relative velocity as $v \\approx p_\\chi\/m_\\chi \\approx T_\\gamma(z)\/m_\\chi$. In this estimate, we have assumed that DM is non-relativistic at the time self-interactions become important.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{DMselftherm_new.pdf}\n\\caption{The approximate redshift when DM self-scattering becomes important, $z_{\\rm therm}$, as a function of DM mass in the model with dark photon mediated interactions. The freeze-in relic abundance is determined by $Q= g_\\chi \\kappa \/e$ and we show $z_\\text{therm}$ assuming two values of $\\kappa$ (where $g_\\chi$ is fixed to obtain the DM relic abundance). The epoch when DM self-thermalization becomes relevant is highly sensitive to the choice of couplings, which can yield different results for CMB observables depending on whether thermalization occurs before recombination. Note that DM halo formation is neglected in this estimate. Also shown are bounds on DM self-thermalization which come from the SIDM limits on $g_\\chi$ in Eq.~\\eqref{SIDMbound}. For illustration, we assume $\\sigma_{T,\\,\\chi \\chi} \\lesssim 1$~cm$^2\/$g for scattering via an ultralight mediator and show both $v\\sim10^{-3}$ and $v\\sim10^{-4}$, speeds relevant to a halo the size of the Milky Way and to a dwarf galaxy. In this figure we have taken $m_{A'} = 10^{-14}$~eV, which is sufficiently light that the constraints on the kinetic mixing parameter $\\kappa$ are rather weak. \\label{fig:DMtherm} }\n\\end{figure} \n\n\nThe self scattering randomizes the DM velocities while preserving the average kinetic energy $ \\tfrac{3}{2} T^{\\rm eff}_\\chi(z) \\equiv \\langle p_\\chi^2 \\rangle\/(2 m_\\chi)$, where $p_\\chi$ is physical momentum and the average momentum-squared is given in Fig.~\\ref{Teff}. After self-scattering becomes significant, the DM phase space is described by a thermal Maxwell-Boltzmann distribution,\n\\begin{align}\n\tf_{\\rm DM}(p_\\chi,z) = n_{\\rm DM}(z) \\, \\left( \\frac{2\\pi }{ m_\\chi T_\\chi^{\\rm eff}(z)} \\right)^{3\/2} 4\\pi p_\\chi^2 \\exp \\left( - \\frac{p_\\chi^2 }{2 m_\\chi T_\\chi^{\\rm eff}(z) } \\right) ,\n \\label{eq:fdm_PSD_Gaussian}\n\\end{align}\nwhere $n_{\\rm DM}(z)$ is the DM number density. \n\n\n\nFig.~\\ref{fig:DMtherm} shows the redshift of thermalization for two representative choices of $\\kappa$ (thus fixing $g_\\chi$ to yield the observed relic abundance), where we see the assumption of non-relativistic DM is a reasonably good approximation in our estimates. Since the phase space calculations here will be an input to determining CMB constraints on freeze-in DM, we compare $z_{\\rm therm}$ with the redshift of recombination $z \\approx 1100$. For constraints from structure formation, a range of redshifts will be relevant. We also show some fiducial limits from SIDM, which give upper bounds on $g_\\chi$. Fig.~\\ref{fig:DMtherm} illustrates that the DM phase space at the time of recombination depends sensitively on the model parameters and on the robustness of SIDM limits in different astrophysical systems. For the largest values of $g_\\chi$ consistent with the weaker assumed SIDM bounds, the DM phase space is described by a Maxwell-Boltzmann distribution at the time of recombination for all the DM masses we consider. However, for $\\kappa = 10^{-3}$ (which is consistent with bounds on ultralight dark photons), $g_\\chi$ is small enough that DM self-interactions are not important at recombination and the phase space is described by the results of Sections~\\ref{sec:ann_phasespace}-\\ref{sec:plasmon_phasespace}. The comparison of the free-streaming and thermalized phase space can be seen in Fig.~\\ref{masscomp_phase}.\n\n\n\\section{Results and Discussion}\n\\label{summary}\nIn this paper, we have shown that DM freeze-in through a light vector mediator is substantially affected by plasmon decay, which constitutes a new production channel. This is an efficient way of producing sub-MeV DM and is dominant over SM fermion annihilation for masses below a few hundred keV. To account for this extra production channel, the couplings between the DM and the SM must be reduced in order to obtain the observed relic abundance of DM. For the lightest DM masses that are accessible to low-threshold direct detection experiments, the predicted cross section is lowered by roughly an order of magnitude. Updated predictions for freeze-in through a light vector mediator are shown in Fig.~\\ref{fig:summaryplot}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Q_new.pdf} \n\\caption{Summary plot including early-universe plasma effects for the parameter space of sub-MeV freeze-in DM. The correct DM relic abundance is obtained for couplings on the freeze-in line. We show constraints coming from emission of DM pairs in white dwarf, horizontal branch and red giant stars~\\cite{Vogel:2013raa}, while bounds from emission of DM pairs in supernovae apply for $Q \\gtrsim 10^{-9}$~\\cite{Chang:2018rso}. Dotted lines are projected sensitivities of proposed direct detection experiments as in Fig.~\\ref{DDshift}. \n\\label{fig:summaryplot}}\n\\end{figure} \n\nThe presence of this channel also affects the DM phase space. In the absence of plasmon decays, the DM is never technically thermal but it acquires a distribution that appears thermal by inheriting the electron phase space distribution at the time of production. At early times $f_{\\chi, \\, e^+e^-}(p_\\chi)\\sim e^{-p_\\chi\/T_{\\chi, \\, e^+e^-}}$, where $T_{\\chi,\\, e^+e^-}$ is an effective DM temperature inherited from the electrons; at late times, this exponential distribution persists because the DM does \\emph{not} thermalize to give the Maxwell-Boltzmann distribution that would be expected for non-relativistic matter in equilibrium. On the other hand, the plasmon decay channel yields a DM phase space distribution that never appears thermal, which can be attributed to the running of the plasmon mass with temperature and the fact that plasmon decays occur dominantly as the plasmon wavenumber $k\\rightarrow 0$. For DM masses where plasmon decays are the dominant production mode, the phase space is peaked at low momentum and has a long tail; for DM masses where contributions from both channels are important, the phase space distribution is bimodal.\n\nThough the DM is born with a highly non-thermal distribution, it may be possible for the DM to thermalize with itself under the right circumstances. For DM that is only charged under the SM $U(1)_{EM}$ with millicharge $Q$, the thermalization rate is suppressed by a factor of $Q^4$ where the requisite $Q$ to produce the DM relic abundance is $Q \\sim \\mathcal{O}\\left(10^{-11}\\right)$. If the DM is also charged under a dark $U(1)$ gauge group that kinetically mixes with the SM $U(1)_{EM}$ (with mixing parameter $\\kappa$), it may be possible for DM self-scattering to thermalize the DM phase space distribution. In this case, $Q = \\kappa g_\\chi\/e$ (where $\\kappa$ can take on a wide range of values) and DM self-scattering via the dark photon scales as $g_\\chi^4$, meaning that with the appropriate choice of $\\kappa$ and $g_\\chi$ it is possible to efficiently self-scatter while still producing the observed relic abundance. The coupling $g_\\chi$ cannot be arbitrarily large due to observational limits on SIDM in astrophysical systems; however, there is a range of $g_\\chi$ where self-scattering thermalizes the DM before recombination and where the SIDM bounds are simultaneously satisfied. Energy is conserved within the DM fluid, so for non-relativistic DM $\\left$ will be conserved and the resulting distribution has a well-defined notion of temperature.\n\nAlthough the freeze-in DM phase space distribution may not be thermal, it is still informative to take moments of the distribution. When comparing the first and second moments of $f_\\chi(p_\\chi)$ to the equivalent quantities for the SM photon bath, we find that the typical DM momentum is similar to the typical photon momentum, $\\langle p_\\chi \\rangle \\approx (0.4-0.7)\\times \\langle p_\\gamma \\rangle$ depending on the DM mass. In other words, the DM is born considerably warmer than what is typically assumed for cold DM initial conditions. This will have ramifications for cosmology in two key ways:\n\\begin{itemize}\n \\item Freeze-in DM will behave like warm DM, leading to suppression of the matter power spectrum below some physical scale roughly corresponding to the free-streaming length. This effect is not already captured by existing limits on warm DM, where different DM phase space distributions are assumed. To understand this suppression quantitatively, a Boltzmann code is necessary that accounts for the potentially nonthermal phase space from freeze-in. Having understood this, it will be possible to constrain DM freeze-in via a light vector mediator using probes of the matter power spectrum and the halo mass function.\n \\item Existing CMB limits on DM with an effective millicharge do not straightforwardly apply to the case of freeze-in. These limits stem from a DM-baryon drag; because the drag is highly sensitive to the relative DM-baryon velocity (the cross section scales like $\\sim v^{-4}$), modifications to the DM phase space can substantially alter the size of the effect. Existing limits have made the assumption of cold dark matter, and the larger DM velocities for freeze-in will lead to reduced drag force. Taking into account the updated Debye logarithm (which may weaken existing limits by a factor of $\\sim 2-3$), the limit on freeze-in will be further reduced compared to previously reported results.\n\\end{itemize}\nBoth of these effects will be thoroughly explored in our companion paper~\\cite{inprep}, which will place restrictions on the range of masses where DM freeze-in via a light mediator is observationally viable.\n\n\n \n\n\n\n\\section*{Acknowledgments}\nWe thank Masha Baryakhtar, Asher Berlin, Simon Knapen, Jung-Tsung Li, Adrian Liu, Aneesh Manohar, Sam McDermott, Julian Mu\\~noz, and Tomer Volansky for helpful discussions. We acknowledge the importance of equity and inclusion in this work and are committed to advancing such principles in our scientific communities. CD was supported by NSF grant AST-1813694, Department of Energy (DOE) grant DE-SC0019018, and the Dean's Competitive Fund for Promising Scholarship at Harvard University. TL is supported by an Alfred P. Sloan Research Fellowship and Department of Energy (DOE) grant DE-SC0019195. Parts of this paper were prepared while at the KITP, supported by the National Science Foundation under Grant No. NSF PHY-1748958. KS is supported by a National Science Foundation Graduate Research Fellowship and a Hertz Foundation Fellowship. KS is grateful for the hospitality of the Lisanti group at Princeton University and of the Center for Cosmology and Particle Physics at New York University, where part of this work was completed.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe studied an ensemble of 42 MSPs from the EPTA, combining multifrequency\ndatasets from four different observatories, with data spanning more than 15\nyears for almost half of our sample. The analysis was performed with TempoNest\nallowing the simultaneous determination of the white noise parameters and\nmodeling of the stochastic DM and red noise signals. We achieved the detection\nof several new parameters: seven parallaxes, nine proper motions and six apparent\nchanges in the orbital semi-major axis. We also measured Shapiro delay in\ntwo systems, PSRs J1600$-$3053 and J1918$-$0642, with low-mass Helium white dwarf\ncompanions.\nFurther observations of PSR J1600$-$3053 will likely yield the detection of the advance of\nperiastron, dramatically improving the mass measurement of this system and\nimproving the constraints on the geometry of the system. We presented an\nupdated mass measurement for PSR J0751+1807,\nroughly consistent with the previous work by \\citet{nsk08}.\nWe searched for the presence of annual-orbital parallax in three systems, PSRs J1600$-$3053, J1857$+$0943 and J1909$-$3744.\nHowever we could only set constraints on the\nlongitude of ascending node in PSRs J1600$-$3053 and\nJ1909$-$3744 with marginal evidence of annual-orbital parallax in PSR J1600$-$3053.\n\nWith an improved set of parallax distances, we investigated the difference\nbetween the predictions from the NE2001 Galactic electron density model and the\nLK-corrected parallax distances. On average we found an error of $\\sim$ 80\\% in\nthe NE2001 distances, this error increasing further at high Galactic latitudes.\nDespite its flaws for high galactic latitude lines-of-sight, we find NE2001 to\nstill predict more accurate distances than two recent models, M2 and M3, proposed by\n\\citet{sch12}, based respectively on the TC93 and NE2001 models with an extended thick disk. \nWe showed that a change in the scale height of the thick disk of\n the current electron density models also dramatically affects the pulsars that\nare located in the Galactic plane. Our updated set of parallaxes presented here\nwill likely contribute to improving\non any future model of the Galactic electron density model\n\nA comparison of the 2-D velocity distribution between isolated and binary MSPs\nwith a sample two times larger than the last published study \\citep{gsf+11}\nstill shows no statistical difference, arguing that both populations originate from the same underlying population.\nThrough precision measurement of the orbital period derivative, we\nachieved better constraints on the distance to two pulsars, PSRs J1012+5307 and\nJ1909$-$3744, than is possible via the detection of the annual parallax.\n\nBased on the timing results presented in this paper and the red noise\nproperties of the pulsars discussed in \\citet{cll+15}, we will\nrevisit and potentially remove some MSPs from the EPTA observing list.\nThe EPTA is also continuously adding more sources to its observing list, especially in\nthe last five years, as more MSPs are discovered through the targeted survey of\n\\textit{Fermi} sources \\citep{rap+12} and large-scale pulsar surveys\n\\citep[e.g. the PALFA, HTRU and GBNCC collaborations;\n][]{lbh+15,bck+13,nbb+14,slr+14}. Over 60 MSPs\nare now being regularly monitored as part of the EPTA effort.\n\nRecent progress in digital processing, leading in some cases to an increase of\nthe processed bandwidth by a factor of $2-4\\times$, allowed new wide-band coherent\ndedispersion backends to be commissioned at all EPTA sites in the last few\nyears \\citep[see e.g.][]{kss08,dbc+11}. These new instruments provide TOAs with\nimproved precision that will be included in a future release of the EPTA\ndataset.\n The long baselines of MSPs timing data presented here, especially when\nrecorded with a single\nbackend, are of great value, not only for the detection of the GWB but also to\na wide range of astrophysics as shown in this paper.\n\n\\section{Discussion}\n\\label{sec:discussions}\n\n\n\\subsection{Distances}\n\\label{sec:dis_distances}\n\nIn Table~\\ref{tab:distances}, we present the parallaxes measured from our\ndata, based on the distance-dependent curvature of the wave-front\ncoming from the pulsar. This curvature causes an arrival-time delay\n$\\tau$ (in seconds) with a periodicity of six months and a maximal amplitude of\n\\citep{lk04}:\n\\begin{equation}\\label{eq:px}\n\\tau = \\frac{d_{\\odot}^2 \\cos^2 \\beta}{2 c d}\n\\end{equation}\nwhere $d_{\\odot}$ is the distance of the Earth to the Sun, $d$ is the\ndistance of the SSB to the pulsar, $c$ is the\nspeed of light and $\\beta$ is the ecliptic latitude of the pulsar.\n\nBecause of the asymmetric error-volume, parallax measurements with\nsignificance less than $\\sim 4\\sigma$, are unreliable as the\nLutz-Kelker bias dominates the measurement \\citep{lk73,vlm10}. The\nLutz-Kelker-corrected parallax values as well as the derived\ndistances\\footnote{We remind the reader that the most likely distance\nis not necessarily equal to the inverse of the most likely parallax,\ngiven the non-linearity of the inversion.} are also given in\nTable~\\ref{tab:distances}, based on the analytical corrections proposed by\n\\citet{vwc+12} and the flux density values shown in Table~\\ref{tab:summary}.\n\nIn total, we present 22 new parallax measurements. Seven of these new\nmeasurements are for MSPs that had no previous distance measurement,\nbut all of these are still strongly biased since their significance is\nat best $3\\sigma$. For five pulsars (specifically for PSRs J0030+0451,\nJ1012+5307, J1022+1001, J1643$-$1224 and J1857+0943) our parallax\nmeasurement is of comparable significance than the previously\npublished value, but with the exception of PSR J1857+0943, our\nmeasurement precision is better than those published previously; and\nthe lower significance is a consequence of the smaller parallax value\nmeasured (as predicted by the bias-correction). Our measurement for\nPSR J1857+0943 is slightly less precise than the value published by\n\\citet{vbc+09}, but consistent within $1\\sigma$.\n\nFinally, we present improved parallax measurements for ten pulsars:\nPSRs J0613$-$0200, J0751+1807, J1024$-$0719, J1600$-$3053, J1713+0747,\nJ1744$-$1134, J1909$-$3744, J1939+2134, J2124$-$3358 and J2145$-$0750.\nFor seven of these the previous measurement was already free of bias,\nfor the remaining three (PSRs J0613$-$0200, J0751+1807 and\nJ2124$-$3358) our update reduces the bias to below the $1\\sigma$\nuncertainty level (with two out of three moving in the direction predicted\nby the bias-correction code). For three pulsars with previously\npublished parallax measurements we only derive upper limits, but two\nof these previous measurements (for PSRs J0218+4232 and J1853+1303)\nwere of low significance and highly biased. Only PSR J1738+0333's\nparallax was reliably measured with GBT and Arecibo data\n\\citep{fwe+12} and not confirmed by us. Four pulsars had a known\nparallax before the creation of the NE2001 model, namely PSRs J1713$+$0747\n\\citep{cfw94}, J1744$-$1134 \\citep{tbm+99}, J1857$+$0943 and J1939$+$2134\n\\citep{ktr94}. These pulsars are therefore not included in our analysis of the\nNE2001 distance (see below), leaving us with a total of 21 parallaxes.\n\n\\subsubsection{Distance comparison with NE2001 predictions}\nWhen comparing the bias-corrected distances presented in\nTable~\\ref{tab:distances} with those predicted by the widely used NE2001\n electron-density model for the Milky Way \\citep{cl02},\n we find that the model performs reasonably well overall.\nHowever, significant offsets exist, primarily at high positive\nlatitudes and large distance ($d>2$ kpc) into the Galactic plane.\nIn Fig~\\ref{fig:dist1}, we plot this comparison for three ranges of Galactic\n latitude $b$ (defined as low: $\\left|b\\right| < 20^\\circ$, intermediate:\n $20^\\circ < \\left|b\\right| < 40^\\circ$ and high: $\\left|b\\right| > 40^\\circ$)\n highlighting the weakness of NE2001 at high latitude.\nWe find a mean uncertainty of 64\\%, 55\\% and 117\\% respectively for the NE2001\ndistances to be consistent with our measurement.\n On average, the NE2001 distances would require an uncertainty of 80\\%. This\nvalue is significantly higher than the 25\\% uncertainty typically assumed in\nthe literature for this model; or than the fractional uncertainties displayed\nin Figure~12 of \\citet{cl02}.\n\n\\begin{figure}\n\\includegraphics[height=85mm,angle=-90]{plots\/comparison_Px_NE2001_M2_M3.ps}\n\\caption{Comparison between the Lutz-Kelker bias corrected parallax distances\n(in ordinates) and the DM distances (in abscissa) for different Galactic\nlatitudes $b$ on logarithmic scales. The DM distances in the left, middle and\nright panels are derived from the NE2001, M2 and M3 models respectively. Top\npanels: the stars show pulsars with $b>40^{\\circ}$ and the crosses pulsars with\n$b<-40^{\\circ}$. Middle panels: the stars show pulsars with\n$40^{\\circ}>b>20^{\\circ}$ and the crosses pulsars with\n$-40^{\\circ}b>0^{\\circ}$ and the crosses pulsars with\n$-20^{\\circ} D_\\pi \\\\\n {D_\\pi}\/{D_\\text{model}}, & \\text{otherwise}\n\\end{cases}\n\\end{equation}\nwith $D_\\pi$ and $D_\\text{model}$ being the parallax distance and distance\nfrom a given Galactic electron density model (NE2001, M2 or M3), respectively.\n As can be seen, the NE2001 model \nprovides on average slightly better distance estimates (lower $N$)\nthan the M2 or M3 models. M3 gives more accurate distance than M2 for the first\nhalf of lines-of-sight (when the prediction of both models is the best) but\ngets superseded by M2 when N increases.\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=90mm]{plots\/cum_distrib_NE2001_M2_M3.ps}\n\\caption{Cumulative distribution of the $N$ factor between \nthe DM distance and the parallax distance (see Eq.~\\ref{eq:Nfactor}). These distributions include\nthe 21 pulsars with measured parallaxes in Table~\\ref{tab:distances}. The DM distances are derived \nfrom the NE2001, M2 and M3 models and represented in black, red and blue respectively.}\n\\label{fig:dist2}\n\\end{figure}\n\n\\input{table-distances}\n\n\n\n\n\n\n\\subsection{Proper motions and 2-D spatial velocities}\n\\label{sec:dis_vel}\n\n\nStellar evolution modeling by \\citet{tb96} and \\citet{cc97} predicted that the\nrecycled MSP population would have a smaller spatial velocity than the normal\npulsar population. A study by \\citet{tsb+99} found a mean transverse velocity\n\\vt for MSPs of $85\\pm13$~km~s$^{-1}$ based on a sample of 23 objects. They\nnoted that this value is four times lower than the ordinary young pulsar\nvelocity. The authors also observed isolated MSPs to have a velocity two-thirds\nsmaller than the binary MSPs.\nWith an ever increasing number of MSPs, further studies by \\citet{hll+05} and\n\\citet{gsf+11} found no statistical evidence for a difference in the velocity\ndistribution of isolated and binary MSPs. \\citet{hll+05} reported on \\vt\n$=76\\pm16$ and \\vt $=89\\pm15$ km~s$^{-1}$ for isolated and binary MSPs\nrespectively while \\citet{gsf+11} found \\vt $=68\\pm16$ and \\vt $=96\\pm15$\nkm~s$^{-1}$ for isolated and binary MSPs. \nAll these results are in agreement with other work by \\citet{lkn+06}.\n\nWithin our sample of 42 MSPs, we measured seven new proper motions, of which\nthree are for \nisolated MSPs (PSRs J1843$-$1113, J1911$+$1347 and J2010$-$1323) and 4 are\nfor binary MSPs (PSRs J0034$-$0534, J0900$-$3144, J1751$-$2857 and J1804$-$2717).\nIn addition, we improved the precision of the proper-motion measurement by a\nfactor of ten for seven other MSPs (PSRs J0610$-$2100, J0613$-$0200, J1455$-$3330,\nJ1801$-$1417, J1911$-$1114, J2229$+$2643 and J2317$+$1439). \n\nThese improvements in the proper motion as well as the distance estimates\npresented in Section~\\ref{sec:dis_distances} and recent discoveries of MSPs\npublished elsewhere led us to re-examine the distribution of $V_T$, the transverse\nvelocity of MSPs in km~s$^{-1}$, where\n\n\\begin{equation}\n V_T =\\text{ 4.74 km s}^{-1} \\times \\mu \\times d.\n\\end{equation}\nAgain, $\\mu$ is the proper motion in mas yr$^{-1}$ and $d$ the distance to the\npulsar in kpc. In this\nanalysis we considered all known MSPs listed in the ATNF pulsar catalogue, but\ndiscarding pulsars in globular clusters, double neutron stars or pulsars with\n$P>20$ ms. This represents 19 isolated and 57 binary pulsars for a total of 76\nMSPs. In comparison, the last published MSP velocity study by \\citet{gsf+11}\nmade use of 10 isolated and 27 binary MSPs with $P$ below 10 ms. If we choose\nto restrict our sample to pulsars with $P$ below 10 ms, only 6 binary pulsars\nwould not pass our criteria. The selected isolated and binary pulsars are\nlisted in Tables \\ref{tab:iso_velocities} and \\ref{tab:bin_velocities}\nrespectively. The distances used in the calculation of $V_T$ and reported in\nthe third column of Tables \\ref{tab:iso_velocities} and\n\\ref{tab:bin_velocities} are the best distance estimates available, either\ncoming from the Lutz-Kelker-corrected parallax or the NE2001 model.\n\nWe find an average velocity of $88\\pm17$ km~s$^{-1}$ and $93\\pm13$ km~s$^{-1}$\nfor the isolated and binary MSPs, respectively. For the entire MSP dataset, we\nget an average velocity of $92\\pm10$ km~s$^{-1}$. Our results are consistent\nwith the work by \\citet{hll+05} and \\citet{gsf+11}.\n\nWhen we keep only the pulsars with a more reliable distance estimate (i.e.\npulsars with a parallax measurement), 8 isolated and 20\nbinary MSPs are left in our sample. In this case, we find an average velocity\nof $75\\pm10$ km~s$^{-1}$ and \n$56\\pm3$ km~s$^{-1}$ for the isolated and binary MSPs respectively. \nConversely, we get an average velocity of $98\\pm29$ km~s$^{-1}$ and $113\\pm20$\nkm~s$^{-1}$ for the pulsars with a distance coming from the Galactic electron\ndensity models. The explanations for this discrepancy are twofold: the\nNE2001 model is overestimating the distances for low Galactic latitude as shown\nin Fig.~\\ref{fig:dist1} and our sample of 2-D velocities is biased against\ndistant low-velocity MSPs. Nearby\npulsars are likely to have a parallax and a proper-motion measurement whereas\ndistant pulsars would most likely have a distance estimate from the NE2001 model and\na proper-motion measurement for the high-velocity pulsars only.\n\nFig.~\\ref{fig:2D_hist} shows the histogram of the velocities for both the\nisolated and binary MSPs populations. A two-sample Kolgomorov-Smirnov (KS) test\nbetween the full isolated and binary MSPs velocity distributions results in a\nKS-statistic of 0.14 and a p-value of 0.92. If we perform the same test on the\npulsars with a parallax distance, we get a KS-statistic of 0.25 and a p-value\nof 0.81. For both cases, we therefore cannot reject the null hypothesis and we argue that there is\nno statistical evidence for the measurements to be drawn from different\ndistributions. This supports the scenario that both isolated and binary MSPs\nevolve from the same population of binary pulsars.\n\n\n\n\n\n\n\n\\begin{figure}\n\\includegraphics[height=80mm,angle=-90]{plots\/PM_histo.ps}\n\\caption{Histogram of the 2-D velocity distribution for a sample of 19 isolated\nMSPs (top panel) and 57 binary MSPs (bottom panel). The respective average\nvelocities are $88\\pm17$ km s$^{-1}$ and $96\\pm12$ km s$^{-1}$. The hatched\npart of the histogram shows the pulsars with a distance estimate from the\nparallax measurement (8 isolated and 21 binary MSPs).}\n\\label{fig:2D_hist}\n\\end{figure}\n\n\\input{table-velocities-isolated}\n\n\\input{table-velocities-binaries}\n\n\n\n\n\\subsection{Shklovskii and Galactic acceleration contributions}\n\\label{sec:dis_shk}\n\nThe observed pulse period derivatives, $\\dot{P}$, reported in Tables \\ref{tab:param1}\n to \\ref{tab:param11} are different from their intrinsic values\n$\\dot{P}_{\\text{int}}$. This is because it includes the 'Shklovskii' contribution due\nto the transverse velocity of the pulsar \\citep[$\\dot{P}_{\\text{shk}}$,][]{shk70}, the\nacceleration from the differential Galactic rotation ($\\dot{P}_{\\text{dgr}}$) and the\nacceleration towards the Galactic disk ($\\dot{P}_{\\text{kz}}$) \\citep{dt91,nt95}.\nHence $\\dot{P}_{\\text{int}}$ can be written as\n\n\\begin{equation}\n\\label{eq:pdot}\n \\dot{P}_{\\text{int}}=\\dot{P}-\\dot{P}_{\\text{shk}}-\\dot{P}_{\\text{dgr}}-\\dot{P}_{\\text{kz}},\n\\end{equation}\nwhere the Shklovskii contribution $\\dot{P}_{\\text{shk}}$ is given by\n\\begin{equation}\n \\frac{\\dot{P}_{\\text{shk}}}{P} = \\frac{\\mu^2 d}{c}.\n\\end{equation}\nAgain $d$ is our best distance estimate for the pulsar and $\\mu$ our measured\ncomposite proper motion. The equation for $\\dot{P}_{\\text{dgr}}$\nis taken from \\citet{nt95} with updated values for the distance to the Galactic center\n$R_0=8.34\\pm0.16$ kpc and the Galactic rotation speed at the Sun\n$\\Theta=240\\pm8$~km~s$^{-1}$ \\citep{rmb+14}. $\\dot{P}_{\\text{kz}}$ is taken from the\nlinear interpolation of the $K_z$ model in \\citet[see Fig.~8]{hf04}.\n\n\nTo compute these contributions with full error propagation we use the distances from\nTable~\\ref{tab:distances} and the proper motions shown in\nTables~\\ref{tab:iso_velocities} and \\ref{tab:bin_velocities}. These values are\nreported for each pulsar at the bottom of Tables~\\ref{tab:param1} to\n\\ref{tab:param11}. The magnitudes of all three corrective\nterms to $\\dot{P}$ depend on the distance $d$ to the pulsar. \nAlternatively, as the pulsar braking torque causes the spin period to increase\n(i.e. $\\dot{P}$ to be positive) in systems where no mass transfer is taking\nplace, we used this constraint to set an upper limit, $D_{\\dot{P}}$, on the\ndistance to the pulsar by assuming all the observed $\\dot{P}$ is a result of\nkinematic and Galactic acceleration effects. This upper limit $D_{\\dot{P}}$ is\nshown in column 5 of Table~\\ref{tab:distances} for 19 pulsars, where this upper\nlimit is below 15 kpc.\n\n\nFor all pulsars except PSRs J0610$-$2100, J1024$-$0719 and J1721$-$2457, the\nupper limits $D_{\\dot{P}}$ are consistent with both the NE2001 and M3\ndistances, $D_{\\text{NE2001}}$ and $D_{\\text{M3}}$ respectively. For PSR\nJ0610$-$2100, $D_{\\text{M3}}=8.94$ kpc is ruled out by\n$D_{\\dot{P}} < 3.89$ kpc. We note that for this pulsar, $D_{\\text{M3}}$ is 2.5 times higher than\n$D_{\\text{NE2001}}$. For PSR J1721$-$2457, both $D_{\\text{NE2001}}$ and\n$D_{\\text{M3}}$ are ruled out by $D_{\\dot{P}} < 0.96$ kpc. The case of PSR J1024$-$0719 is discussed below.\n\n\n\\input{table-pbdot} \n\n For nine pulsars, an independent estimate of\nthe distance from the parallax measurement is available. For all nine pulsars\nbut PSR J1024$-$0719, the parallax distance is consistent with the upper limit\n$D_{\\dot{P}}$.\nPSR J1024$-$0719 has $D_{\\dot{P}} < 0.42$ kpc but a reported Lutz-Kelker-corrected\ndistance $D_{\\pi} = 1.08_{-0.16}^{+0.28}$ kpc, $\\sim4\\sigma$ away above the\nupper limit $D_{\\dot{P}}$.\nTo explain this discrepancy (also discussed in \\citet{egc+13,aaa+13}), we argue that PSR J1024$-$0719 must be subject to\na minimum relative acceleration $a$ along the line of sight,\n\\begin{equation} \n a = \\frac{ \\left| \\dot{P} - \\dot{P}_{\\text{int}} \\right| }{P} \\times c = 1.7 \\times 10^{-9} \\text{m s}^{-2}.\n\\end{equation}\nA possible explanation for this acceleration is the presence of a nearby star,\norbiting PSR J1024$-$0719 in a very long period. A possible companion has been\nidentified by \\citet{srr+03}.\n\nThe same reasoning behind the corrections of Eq.~\\ref{eq:pdot} also apply to the\nobserved orbital period derivative $\\dot{P_b}$. In addition to the previous\nterms, we also consider the contribution due to gravitational radiation\nassuming GR, $\\dot{P_b}_{\\_GR}$ but neglect the contributions from mass loss in the\nbinary, tidal interactions or changes in the gravitational constant $G$.\n $\\dot{P_b}_{\\_GR}$ is therefore the only\ncontribution independent of the distance to the pulsar system but requires an\nestimate of the masses of the binary.\n\n As we measured the orbital period derivative for four pulsars\n(PSRs J0613$-$0200, J0751$+$1807, J1012$+$5307 and J1909$-$3744), we investigate here\nthe possible bias in those measurements assuming the parallax distances from\nTable~\\ref{tab:distances}. Conversely, \\citet{bb96} (hereafter BB96) pointed out that the\nmeasurement of $\\dot{P_b}$ would potentially lead to more accurate distance\nthan the annual parallax. Hence, we also present a new distance estimate,\n$D_{\\dot{P_b}}$, assuming the observed $\\dot{P_b}$ is the sum of all four\ncontributions described above. These results are shown in Table~\\ref{tab:pbdot}. \n\n To estimate the gravitational radiation contribution to $\\dot{P_b}$ for PSR\nJ0613$-$0200 without a mass measurement, we assumed $m_p=1.4$ M$_\\odot$\nand $i=60^\\circ$. The resulting distance estimate is $D_{\\dot{P_b}} = 1.68\\pm0.33$ kpc. This result is\n2.2$\\sigma$ consistent with the parallax distance and currently limited by the precision\non the measured $\\dot{P_b}$. Continued timing of this pulsar will greatly\nimprove this test as the uncertainty on $\\dot{P_b}$ decrease as $t^{-2.5}$.\nFor PSR J0751+1807, we measure a negative orbital period derivative,\n$\\dot{P_b}=(-3.50\\pm0.25) \\times 10^{-14}$, meaning the Shklovskii effect is not\nthe dominant contribution to $\\dot{P_b}$ in this system. We also note that our\nmeasured composite proper motion is 3.3$\\sigma$ higher than the value in \\citet{nss+05}\nresulting in a Shklovskii contribution to $\\dot{P_b}$ that is five times larger than the one quoted\nin \\citet{nss+05}.\nIn the next section, we will combine the corrected orbital period derivative from\nacceleration bias $\\dot{P_b}_{\\text{corr}} = \\dot{P_b} -\n\\dot{P_b}_{\\_\\text{kin}} -\n\\dot{P_b}_{\\_\\text{kz}} - \\dot{P_b}_{\\_\\text{dgr}} = (-4.6\\pm0.4) \\times 10^{-14} $ with the measurement of the Shapiro delay to constrain the masses of the two stars.\n\n\nFor PSR J1012$+$5307, we measured the orbital period derivative\n$\\dot{P_b}=(6.1\\pm0.4) \\times 10^{-14}$, a value similar to the one reported by \\citet{lwj+09}. We\nalso find the contributions to $\\dot{P_b}$ to be consistent with their work.\n After taking into account the companion mass and\ninclination angle from \\citet{kbk96,cgk98} to compute $\\dot{P_b}_{\\_\\text{GR}}$, we find\n$D_{\\dot{P_b}} = 940\\pm30$ pc, in very good agreement with the optical\n\\citep{kbk96,cgk98} and parallax distance, but more precise by a factor three and\neight, respectively.\n\nThe bias in the orbital period derivative measured for PSR J1909$-$3744 is\nalmost solely due to the Shklovskii effect. We get $D_{\\dot{P_b}} = 1140\\pm11$\npc. This result with a fractional uncertainty of only 1\\% is also in very good\nagreement with the parallax distance.\n\nTwenty years ago, BB96 predicted that after only 10 years, several of the MSPs included in\nthis paper would have a better determination of the distance through the measurement of\nthe Shklovskii contribution to $\\dot{P_b}$ compared to the annual parallax.\nHowever we achieved a better distance estimate from\n$\\dot{P_b}$ than the parallax for only two pulsars so far.\n\nWe investigate here the pulsars for which we should have detected $\\dot{P_b}$\nbased on the work by BB96 (i.e. PSRs J1455$-$3330, J2019+2425\nJ2145$-$0750 and J2317+1439). \nIn their paper, BB96 assumed a transverse velocity of 69 km~s$^{-1}$\nfor pulsars where the proper motion was not measured and adopted the distance to the\npulsar based on the \\citet{tc93} Galactic electron density model.\n\n\nIn the case of PSR J2019+2425, our measured proper motion is similar to the\nvalue used by BB96 and the time span of our data is nine years.\n The peak-to-peak timing signature of the Shklovskii contribution\nto $\\dot{P_b}$ (see Eq.~1 of BB96) is $\\Delta T_{pm} = 6\\pm5~\\mu s$,\nwith the large uncertainty coming from the NE2001 distance assumed.\nFor the three remaining pulsars, no proper-motion measurement was available at\nthe time and BB96 assumed in those cases a transverse velocity of 69\nkm~s$^{-1}$. However our new results \nreported in Table~\\ref{tab:bin_velocities} show much smaller transverse\nvelocities for PSRs J1455$-$3330, J2145$-$0750 and J2317+1439, with\n$31\\pm12$ km s$^{-1}$, $40\\pm3$ km s$^{-1}$,\n$17\\pm6$ km s$^{-1}$ respectively, resulting in a much lower Shklovskii\ncontribution to $\\dot{P_b}$ than predicted by BB96, explaining the non-detection of this parameter after 10 to 17 years\nof data with our current timing precision. \n\n\n\n\\subsection{Shapiro delay and mass measurement}\n\\label{sec:dis_mass}\n\n\\input{table-mass}\n\nIn Figures \\ref{fig:0751_mass} to \\ref{fig:1918_mass}, we plot, assuming GR, the\njoint \\mbox{2-D}\nprobability density function of the Shapiro delay that comes directly out of the TempoNest\nanalysis for the three pulsars we achieve greatly improved mass measurements, PSRs J0751+1807, J1600$-$3053 and J1918$-$0642,\nrespectively. For PSR J0751+1807, we use the corrected orbital period\nderivative, $\\dot{P_b}_{corr} = (-4.6\\pm0.4) \\times 10^{-14} $, derived in the previous section to further constrain the\nmasses of the system. The projection of the parameters $\\dot{P_b}$ and\n$\\varsigma$ gives the following 68.3\\%\nconfidence levels:\n$m_p=1.64\\pm0.15$~M$_{\\odot}$ and\n$m_c=0.16\\pm0.01$~M$_{\\odot}$. The inclination angle is constrained with $\\cos\ni < 0.64$ (2$\\sigma$). Our new pulsar mass\nmeasurement is 1.3$\\sigma$ larger from the latest mass value published in \\cite{nsk08}. \n\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}{150mm}\n\\includegraphics[width=150mm]{plots\/J0751_mass_mass.eps}\n\\caption{Constraints on PSR J0751$+$1807 parameters from the\nmeasurement of Shapiro delay and orbital period derivative $\\dot{P_b}$. The bottom left plot shows the $\\cos i - m_c$\nplane. The bottom right plot shows the $m_p - m_c$ plane. The continuous black\nline, the dashed line and the dotted line represent, respectively, the 68.3\\%,\n95.4\\% and 99.7\\% confidence levels of the 2-D probability density function.\nThe grey area is\nexcluded by the mass function with the condition $\\sin i \\leq 1$. The red\ncurves indicate the 1-$\\sigma$ constraint required by $\\dot{P_b}$ assuming GR. The other\nthree panels show the projected 1-D distributions based on $\\dot{P_b}$ and the\ninclination angle (given by $\\varsigma$). The dashed lines\nindicate the median value and the continuous lines the 1-$\\sigma$ contours.}\n\\label{fig:0751_mass}\n\\end{minipage}\n\\end{figure*}\n\nIn the case of PSR J1600$-$3053, the posterior results from TempoNest give\n $\\cos i=0.36\\pm0.06$, $m_p=1.22_{-0.35}^{+0.50}$~M$_{\\odot}$ and\n$m_c=0.21_{-0.04}^{0.06}$~M$_{\\odot}$. We now have\nan accurate mass of the companion compared to the marginal detection\nby \\citet{vbc+09}. Given the eccentricity $e \\sim 1.7\\times10^{-4}$ of this system, a detection of\nthe precession of periastron is likely to happen in the near future and would\ngreatly improve the pulsar mass measurement.\n\nThe results for PSR J1918$-$0642 translate into a pulsar mass $m_p =\n1.25_{-0.4}^{+0.6}$~M$_{\\odot} $ and a companion mass $m_c =\n0.23\\pm0.07$~M$_{\\odot}$. The cosine of the inclination angle\nis $0.09_{-0.04}^{+0.05}$. Based on the mass estimates for the companions to\nPSRs J1600$-$3053 and J1918$-$0642, it is expected that these are low-mass\nHelium white dwarfs.\n\nIn Table~\\ref{tab:mass}, we summarize all our mass measurements and compare them to the values previously published in the literature. We find that PSRs J1713+0747,\nJ1802$-$2124, J1857+0943 and J1909$-$3744 have a mass measurement that is in very good\nagreement to the values reported in the literature \\citep{zsd+15,\nfsk+10, vbc+09}.\n\n\n\\begin{figure*}\n\\centering\n\\begin{minipage}{150mm}\n\\includegraphics[width=150mm]{plots\/J1600_mass_mass.eps}\n\\caption{Constraints on PSR J1600$-$3053 parameters from the\nmeasurement of Shapiro delay. The bottom left plot shows the $\\cos i - m_c$\nplane. The bottom right plot shows the $m_p - m_c$ plane. The continuous black\nline, the dashed line and the dotted line represent, respectively, the 68.3\\%,\n95.4\\% and 99.7\\% confidence levels of the 2-D probability density function.\nThe grey area is\nexcluded by the mass function with the condition $\\sin i \\leq 1$. The other\nthree panels show the projected 1-D distributions with the dashed line\nindicating the median value and the continuous lines the 1-$\\sigma$ contours.}\n\\label{fig:1600_mass}\n\\end{minipage}\n\\end{figure*}\n\n\n\n\n\n\n\\subsection{Search for annual-orbital parallax}\n\\label{sec:dis_kom}\nFor pulsars in binary systems, any change in the direction to the orbit\nnaturally leads to apparent variations in two of the Keplerian parameters, the intrinsic\nprojected semi-major axis $x_{\\text{int}}$ and longitude of periastron\n$\\omega_{\\text{int}}$.\nIn the case of nearby binary pulsars in wide orbits, a small periodic variation\nof $x$ and $\\omega$ due to the annual motion of the Earth around the Sun as\nwell as the orbital motion of the pulsar itself can be measured. This effect,\nknown as the annual-orbital parallax, can be expressed as \\citep{kop95}: \n\n\\begin{equation}\n x = x_{\\text{int}} \\left\\lbrace 1 + \\frac{\\cot i }{d} (\\Delta _{I_0} \\sin \\Omega - \\Delta _{J_0} \\cos \\Omega) \\right\\rbrace \n\\end{equation}\nand \n\\begin{equation}\n \\omega = \\omega_{\\text{int}} - \\frac{\\csc i }{d} (\\Delta _{I_0} \\cos \\Omega + \\Delta _{J_0} \\sin \\Omega), \n\\end{equation}\nwhere $\\Omega$ is the longitude of the ascending node. $\\Delta_{I_0}$ and\n$\\Delta_{J_0}$ are defined in \\citet{kop95} as:\n\\begin{equation}\n\\Delta_{I_0} = -X \\sin \\alpha + Y \\cos \\alpha\n\\end{equation}\nand\n\\begin{equation}\n\\Delta_{J_0} = -X \\sin \\delta \\cos \\alpha - Y \\sin \\delta \\sin \\alpha + Z \\cos \\delta,\n\\end{equation}\nwhere $\\mathbf{r} = (X,Y,Z)$ is the position vector of the Earth in the SSB\ncoordinate system.\n\nThe proper motion of the binary system also changes the apparent viewing\ngeometry of the orbit by \\citep{ajr+96,kop96}:\n\\begin{equation}\n\\label{eq:hot}\n x = x_{\\text{int}} \\left\\lbrace 1 + \\frac{1}{\\tan i} (-\\mu_{\\alpha} \\sin \\Omega +\n\\mu_{\\delta} \\cos \\Omega ) ( t - t_0) \\right\\rbrace ,\n\\end{equation}\n\n\\begin{equation}\n \\omega = \\omega_{\\text{int}} + \\frac{1}{\\sin i} (\\mu_{\\alpha} \\cos \\Omega +\n\\mu_{\\delta} \\sin \\Omega ) ( t - t_0) . \n\\end{equation}\nThe time derivative of Eq.~\\ref{eq:hot} can be expressed as\n\\begin{equation}\n\\label{eq:xdot}\n\\frac{\\dot{x}}{x} = \\mu \\cot i \\sin (\\theta_\\mu - \\Omega),\n\\end{equation}\nwhere $\\theta_\\mu$ is the position angle of the proper motion on the sky. If the inclination\nangle, $i$, can be measured through, e.g., the detection of Shapiro delay, then a\nmeasurement of $\\dot{x}$ can constrain the longitude of ascending node $\\Omega$.\nThese apparent variations in $x$ and $\\omega$ are taken into account in Tempo2's\nT2 binary model with the KOM and KIN parameters, corresponding to the\nposition angle\nof the ascending node $\\Omega$ and inclination angle $i$ (without the 90$^{\\circ}$\nambiguity inherent to the Shapiro delay measurement). Therefore the parameter\n$s \\equiv \\sin i$ of the Shapiro delay has to become a function of KIN.\n\nEven a null $\\dot{x}$ can, if measured precisely enough, be useful.\nAccording to Eq.~\\ref{eq:xdot}, the maximum value for $\\left|\\dot{x}\\right|$ is\n$\\dot{x}_{\\max} = |x \\mu \\cot i|$ (obtained using the inequality $| \\sin\n(\\theta_{\\mu} - \\Omega) | \\leq 1$). Thus whenever the observed value and\nuncertainty represent a small fraction of the interval from $- \\dot{x}_{\\max}$\nto $\\dot{x}_{\\max}$, they are placing a direct constraint on $\\sin\n(\\theta_{\\mu} - \\Omega)$\n\n\nIn our dataset, we measured the apparent variation of $\\dot{x}$ for\n13 pulsars, among which six are new measurements (PSRs J0751+1807,\nJ1455$-$3330, J1640+2224, J1751$-$2857, J1857+0943 and J1955+2908). \n\n\\begin{figure*}\n\\begin{minipage}{150mm}\n\\includegraphics[width=150mm]{plots\/J1918_mass_mass.eps}\n\\caption{Same caption as Fig.~\\ref{fig:1600_mass} for PSR J1918$-$0642.}\n\\label{fig:1918_mass}\n\\end{minipage}\n\\end{figure*}\n\n\nFor the three pulsars where we measured both the Shapiro delay and the variation\nof the semi-major axis (i.e. PSRs J0751+1807, J1600$-$3053 and\nJ1857+0943) and PSR J1909$-$3744 (where $\\dot{x} =0.6\\pm1.7 \\times 10^{-16}$\n and $x \\mu \\cot i = 1.08\\times 10^{-14}$), we map the KOM-KIN space with TempoNest\nusing the following procedure. First, we reduce the dimensionality of the Bayesian\nanalysis by fixing the set of white noise parameters to their maximum\nlikelihood values from the\ntiming analysis. We also choose to marginalize analytically over the astrometric and spin\nparameters. Then we manually set the priors on KOM, KIN and M2 to encompass any\nphysical range of solution. Finally we perform the\nsampling with TempoNest with the constant efficiency option turned off, in\norder to more carefully explore the complex multi-modal parameter space. Because of the strong\ncorrelation between the companion mass and the inclination angle in the\ncase of PSR J0751+1807, (see Fig.~\\ref{fig:0751_mass}), we do not report our\nmeasurements as they were not constrained enough. The results are shown in\nFigures \\ref{fig:1600_komkin} to \\ref{fig:1909_komkin} for the other three pulsars. \n\nFor PSR J1600$-$3053, the 1-$\\sigma$ contours of the 2-D posterior distribution\n(Fig.~\\ref{fig:1600_komkin}) give three solutions for\n($\\Omega, i$): $219^\\circ < \\Omega < 244^\\circ$ and $63^\\circ < i < 71^\\circ$ or\n$303^\\circ < \\Omega < 337^\\circ$ and $61^\\circ < i< 72^\\circ$ and the preferred\nsolution, $37^\\circ < \\Omega < 163^\\circ$ and $105^\\circ < i < 122^\\circ$.\nThe 2.5$\\sigma$ detection of $\\dot{x}$ in the PSR J1857+0943 binary system\nstill limit the constraints that can be set on $\\Omega$ (see\nFig.~\\ref{fig:1857_komkin}). Even though we do not detect $\\dot{x}$ for PSR\nJ1909$-$3744, we can constrain $\\Omega$ (see Fig.~\\ref{fig:1909_komkin}) to $-2^\\circ < \\Omega < 33 ^\\circ$ or $181^\\circ <\n\\Omega < 206 ^\\circ$. The preferred solution is $-2^\\circ < \\Omega < 33 ^\\circ$\nand $93.78^\\circ < i < 93.95^\\circ$.\nHowever, with this EPTA dataset, we still have no statistical evidence for the\ndetection of annual-orbital parallax as we cannot distinguish between the symmetric solutions of\nthe pulsar orbits in these three pulsars.\n\n\n\n\\begin{figure}\n\\includegraphics[width=84mm]{plots\/J1600-3053_komkin.eps}\n\\caption{One and two-dimensional marginalized posterior distributions of the\nlongitude of ascending node $\\Omega $ and inclination angle $i$ for PSR\nJ1600$-$3053. The continuous black line, the dashed line and the dotted line\nrepresent, respectively, the 68.3\\%, 95.4\\% and 99.7\\% confidence levels of the\n2-D probability density function. The red cross indicates the maximum\nlikelihood location. The continuous lines in the panels of the projected 1-D\ndistributions of KOM and KIN show the 68.3\\% confidence levels for each\nparameter.}\n\\label{fig:1600_komkin}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=84mm]{plots\/J1857+0943_komkin.eps}\n\\caption{Same caption as Fig.~\\ref{fig:1600_komkin} for PSR J1857$+$0943.}\n\\label{fig:1857_komkin}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=84mm]{plots\/J1909-3744_komkin.eps}\n\\caption{Same caption as Fig.~\\ref{fig:1600_komkin} for PSR J1909$-$3744.}\n\\label{fig:1909_komkin}\n\\end{figure}\n\n\n\\subsection{Comparison with the latest NANOGrav and PPTA results}\nWhile this work was under review, similar analysis by NANOGrav and the PPTA were\npublished elsewhere \\citep[][hereafter \\citetalias{abb+15} and\n\\citetalias{rhc+16}, respectively]{abb+15,rhc+16}. \\citetalias{abb+15} presents a thorough\ndescription of their data analysis and \\citet[][hereafter\n\\citetalias{mnf+16}]{mnf+16} report on the study of astrometric parameters.\nOther timing results and their interpretations (e.g. pulsar mass measurements) will be presented in a\nseries of upcoming papers. Hence, we briefly summarize here the similarities and\ndifferences between our work and the ones by \\citetalias{rhc+16} and \\citetalias{mnf+16}.\n\n\n\\citetalias{rhc+16} used Tempo2 linearized, least-squares fitting methods to present timing models for a set of 20 MSPs. White noise, DM variations and red\nnoise are included in the timing analysis and modeled with completely independent techniques from the ones\ndescribed in Section~\\ref{sec:timing}. \nFor all 13 pulsars observed commonly by the EPTA and the PPTA, both PTAs\nachieve the detection of the parallax with consistent results\n(within 1.5$\\sigma$). We note here that the parallax value of PSR J1909$-$3744 should\nread $\\pi=0.81\\pm0.03$~mas in \\citetalias{rhc+16} (Reardon, private communication).\nAlso, the seven new proper motions values reported in this paper are for pulsars that are not\nobserved by \\citetalias{rhc+16}.\nWe obtain similar results for the pulsar and companion masses to the\nvalues reported in \\citetalias{rhc+16}, albeit with much greater precision\nin the case of PSRs J1600$-$3053. Furthermore, all our measurements of $\\dot{x}$ agree with \\citetalias{rhc+16}.\nWhile \\citetalias{rhc+16} measure $\\dot{P_b}$ in PSR J1022$+$1001 for the\nfirst time, the EPTA achieve the detection of $\\dot{P_b}$ for another MSP\n(PSR J0613$-$0200), allowing us to get an independent distance estimate for\nthese systems.\n\n\\citetalias{mnf+16} report on the astrometric results for a set of 37 MSPs analyzed\nwith the linearized least-squares fitting package\nTempo\\footnote{http:\/\/tempo.sourceforge.net\/}. More details on the DM and red\nnoise models included in their analysis can be found in \\citetalias{abb+15}.\nAll 14 parallax measurements for the pulsars presented commonly in this work\nand in \\citetalias{mnf+16} are consistent at the 2-$\\sigma$ level. \nIn addition, \\citetalias{mnf+16} show a new parallax measurement for PSR J1918$-$0642\nthat was not detected with our dataset.\n\\citetalias{mnf+16} also present updated proper motions for 35 MSPs and derived the pulsar\nvelocities in galactocentric coordinates.\nThe new proper-motion measurement for PSR J2010$-$1323 reported in our work is\nconsistent at the 2-$\\sigma$ level with the independent measurement from\n\\citetalias{mnf+16}. \n\n\nFinally, \\citetalias{mnf+16} discuss in detail the same discrepancy reported\nin Section~\\ref{sec:dis_shk} between\ntheir measured parallax distance for PSR J1024$-$0719 and its constraint from $D_{\\dot{P}}$.\n\n\n\n\\section{Timing results}\n\\label{sec:results}\n\nIn this section we summarize the timing results of the 42 MSPs obtained from\nTempoNest. Among these sources, six pulsars, namely PSRs J0613$-$0200, J1012+5307,\nJ1600$-$3053, J1713+0747, J1744$-$1134 and J1909$-$3744, have been selected by\n\\citet{bps+15} to form the basis of the work presented by\n\\citet{ltm+15,tmg+15,bps+15}. The quoted uncertainties represent the $68.3\\%$\nBayesian credible interval of the one-dimensional marginalized posterior\ndistribution of each parameter. The timing models are shown in \nTables~\\ref{tab:param1} to \\ref{tab:param11}. These models, including the\nstochastic parameters, are made publicly available on the\nEPTA website\\footnote{http:\/\/www.epta.eu.org\/aom.html}. The reference\nprofiles at L-Band can be found in Fig.~\\ref{plot:templates-1} and \\ref{plot:templates-2}.\nThroughout the paper, we refer to RMS as the weighted Root Mean Square timing\nresiduals. The details on the data sets used in this paper can be found in\nTable~\\ref{tab:data}.\n\n\\input{table_summary}\n\n\\subsection{PSR J0030$+$0451}\nA timing ephemeris for this isolated pulsar has been published by\n\\citet{aaa+09a} with a joint analysis of gamma-ray data from the \\textit{Fermi} Gamma-ray\nSpace Telescope. Because the authors used the older DE200 version of the Solar\nSystem ephemeris model, we report here updated astrometric measurements.\nWhile our measured proper motion is consistent with the \\citet{aaa+09a} value,\nwe get a significantly lower parallax value $\\pi=2.79\\pm0.23 $ mas that we\nattribute partly to the errors in the DE200 ephemeris. Indeed reverting back to\nthe DE200 in our analysis yields an increased value of the parallax by 0.3 mas\nbut still below the parallax $\\pi=4.1\\pm0.3$ mas determined by \\citet{aaa+09a}.\n\n\\subsection{PSR J0034$-$0534}\nPSR J0034$-$0534 is a very faint MSP when observed at L-Band with a flux\ndensity $S_{1400}=0.01$ mJy leading to profiles with very low S\/N compared to\nmost other MSPs considered here. Helped by the better timing precision at\n 350 MHz, we were able to improve on the previously\npublished composite proper motion $\\mu=31\\pm9$\nmas yr$^{-1}$ by \\citet{hll+05} to $\\mu=12.1\\pm0.5$ mas yr$^{-1}$. We also\nmeasure the eccentricity $e=(4.3\\pm0.7) \\times 10^{-6}$ of this system for the\nfirst time. Even with our improved timing precision characterized by a timing\nresiduals RMS of 4 \\us, the detection of the parallax signature (at most 2.4\n\\us~according to \\citet{aaa+10a}) is still out of reach.\n \n \\input{table_param1}\n\n\\subsection{PSR J0218$+$4232}\nThe broad shape of the pulse profile of this pulsar (with a duty cycle of about\n50\\%, see Figure~\\ref{plot:templates-1}) and its low flux density limit\nour timing precision to about 7~$\\mu$s and, therefore, its use for GWB\ndetection. \\citet{dyc+14} recently published the pulsar composite proper\nmotion $\\mu = 6.53\\pm0.08$ mas~yr$^{-1}$ from very long baseline interferometry\n(VLBI). With EPTA data, we find $\\mu=6.14\\pm0.09$ mas~yr$^{-1}$. This value is\nin disagreement with the VLBI result. A possible explanation for this\ndiscrepancy is\nthat \\citet{dyc+14} overfitted their model with five parameters for five\nobserving epochs. \\citet{dyc+14} also\nreported a distance $d=6.3_{-2.3}^{+8.0}$ kpc from VLBI parallax measurement.\n\\citet{vl14} later argued that the \\citet{dyc+14} parallax suffers from the Lutz-Kelker bias\nand corrected the distance to be $d=3.2_{-0.6}^{+0.9}$ kpc. This\ndistance is consistent with the 2.5 to 4 kpc range estimated from the\nproperties of the white dwarf companion to PSR J0218+4232 \\citep{bkk02}. Even with the\n\\citet{vl14} $3\\sigma$ lowest distance estimate, the parallax would induce a\nsignature on the timing residuals of less than 800 ns \\citep{lk04}, which is far from\nour current timing precision. We therefore cannot further constrain the\ndistance with our current dataset. Our measurement of the system's eccentricity\n$e=(6.8\\pm0.4)\\times10 ^{-6} $ is significantly lower than the previously\nreported value $e = (22 \\pm 2) \\times 10^{-6}$ by \\citet{hlk+04}.\n\n\n\\subsection{PSR J0610$-$2100}\nWith a very low-mass companion ($0.02~\\text{M}_{\\odot} <\n\\text{M}_c<0.05~\\text{M}_{\\odot}$), PSR J0610$-$2100 is a member of the\n`black widow' family, which are a group of (often) eclipsing binary MSPs believed\nto be ablating their companions. Here we report on a newly measured\neccentricity, $e=(2.9\\pm0.8) \\times 10^{-5}$, and an improved proper motion\n($\\mu_{\\alpha}=9.0\\pm0.1$ mas~yr$^{-1}$ and $\\mu_{\\delta}=16.78\\pm0.12$\nmas~yr$^{-1}$) compared to the previous values ($\\mu_{\\alpha}=7\\pm3$\nmas~yr$^{-1}$ and $\\mu_{\\delta}=11\\pm3$ mas~yr$^{-1}$) from \\citet{bjd+06}\nderived with slightly more than two years of data. It is interesting to note\nthat, in contrast to another well studied black widow pulsar, PSR J2051$-$0827 \\citep{lvt+11}, no secular\nvariations of the orbital parameters are detected in this system.\nThere is also no evidence for eclipses of the radio signal in our data.\n\nWe checked our data for possible orbital-phase dependent DM-variation that could\naccount for the new measurement of the eccentricity. We found no evidence for\nthis within our DM precision. We also obtained consistent results for the eccentricity and longitude of\nperiastron after removing TOAs for given orbital phase ranges.\n\n\\subsection{PSR J0613$-$0200}\nFor PSR J0613$-$0200, we measure a parallax $\\pi=1.25\\pm 0.13$ mas that is\n consistent with the value published in \\citet{vbc+09} ($\\pi=0.8\\pm\n0.35$ mas). In addition,\nwe report on the first detection of the orbital period derivative $\\dot{P_b} =\n(4.8 \\pm 1.1) \\times\n10^{-14}$ thanks to our 16-yr baseline. This result will be discussed further\nin Section \\ref{sec:dis_shk}. Finally, we improve on the precision of the\nproper motion with $\\mu_{\\alpha}=-1.822\\pm0.008$ mas~yr$^{-1}$ and\n$\\mu_{\\delta}=-10.355\\pm0.017$ mas~yr$^{-1}$.\n\n\\subsection{PSR J0621$+$1002}\nDespite being the slowest rotating MSP of this dataset with a period of almost 30 ms,\nPSR J0621$+$1002 has a profile with a narrow peak feature of width $\\sim500$\n\\us. We are able to measure the precession of the periastron\n$\\dot{\\omega}=0.0113\\pm0.0006$ deg yr$^{-1}$ and find it\nto be within 1 $\\sigma$ of the value reported by \\citet{nsk08} using\n Arecibo data. We also find a similar value of the proper motion\nto \\citet{sna+02}.\n\n\\subsection{PSR J0751$+$1807}\nPSR J0751$+$1807 is a 3.5-ms pulsar in an approximately 6-h orbit. \\citet{nss+05}\noriginally reported a parallax $\\pi=1.6\\pm0.8$~mas and a measurement of the\norbital period derivative $\\dot{P_b}=(-6.4 \\pm 0.9) \\times 10^{-14}$. Together\nwith their detection of the Shapiro delay, they initially derived a large\npulsar mass $m_p=2.1\\pm0.2 \\text{M}_{\\odot}$. \\citet{nsk08} later corrected\nthe orbital period derivative measurement to $\\dot{P}_b=(-3.1\\pm 0.5) \\times 10^{-14}$,\ngiving a much lower pulsar mass $m_p=1.26 \\pm 0.14 \\text{M}_{\\odot}$. Here we\nreport on a parallax $\\pi=0.82 \\pm 0.17$ mas and $\\dot{P}_{b} = (-3.5 \\pm\n0.25) \\times 10^{-14}$ that is similar to the value in \\citet{nsk08}. \nHowever, we measured a precise composite proper motion of\n$13.7\\pm0.3$ mas yr$^{-1}$, inconsistent with the result ($6\\pm2$ mas yr$^{-1}$) from \\citet{nss+05}.\n\\citet{nsk08} explained the issue found with the timing solution presented\nin \\citet{nss+05} but did not provide an update of the proper motion for\ncomparison with our value.\nWe are also able to measure an apparent change in the semi-major axis\n$\\dot{x}=(-4.9\\pm0.9) \\times 10^{-15}$. Finally, we applied the orthometric\nparametrization of the Shapiro delay to get $h_3=(3.0 \\pm 0.6) \\times 10^{-7}$ and $\\varsigma=0.81\\pm0.17$.\n The interpretation of these results will be discussed in Section \\ref{sec:dis_mass}.\n\n\n\\subsection{PSR J0900$-$3144}\nWith about seven years of timing data available for PSR J0900$-$3144 (discovered\nby \\citep{bjd+06}) we detect the proper motion for the first time,\nrevealing it to be one of the lowest composite proper-motion objects among our\ndata set with $\\mu=2.26\\pm0.07$ mas yr$^{-1}$. We also uncover a marginal\nsignature of the parallax $\\pi=0.77\\pm 0.44$ mas. However, we do not detect the\nsignature of the Shapiro delay despite the improvement in timing precision\ncompared to \\citet{bjd+06}.\nFollowing the criterion introduced in Section~\\ref{sec:criteria}, we\n get $h_{3o}=0.4 \\mu$s. With $\\delta_{\\text{TOAs}} = 4.27\\mu$s and N$_\\text{TOAs} =\n875$, we find $\\xi =0.14$~$\\mu$s. Hence, given $ \\xi < h_{3o}$, we argue for $i \\lesssim 60^\\circ$ to explain\nthe lack of Shapiro delay detection in this system.\n\n\\input{table_param2}\n\n\\subsection{PSR J1012$+$5307}\n\\citet{lwj+09} previously presented a timing solution using a subset of these\nEPTA data to perform a test on gravitational dipole radiation and variation of the\ngravitational constant, $\\dot{G}$. The $\\dot{x}$ and $\\dot{P_b}$ parameters we\npresent here are consistent with the values from \\citet{lwj+09} but we\nimprove on the uncertainties of these parameters by factors of two and three,\nrespectively. Nonetheless, we note that our value for the parallax $\\pi=0.71\\pm 0.17$\nmas differs by less than 2$\\sigma$ from the value measured by \\citet{lwj+09} using\nthe DE405 ephemeris. \n\n\n\\subsection{PSR J1022$+$1001}\nAs recently pointed out by \\citet{van13}, this source requires a high level of\npolarimetric calibration in order to reach the best timing precision. Indeed,\nby carefully calibrating their data, \\citet{van13} greatly improved on the\ntiming model of \\citet{vbc+09} and successfully unveiled the precession of the\nperiastron $\\dot{\\omega }=0.0097\\pm0.0023$ deg yr$^{-1}$, the presence of \nShapiro delay and the secular variation of $\\dot{x}$. Here we find similar\nresults with $\\dot{\\omega }=0.010 \\pm 0.002$ deg yr$^{-1}$ and a 2-$\\sigma$ \nconsistent $\\dot{x}$ with a completely independent dataset.\n Nonetheless, we can not confirm the measurement of Shapiro delay\nwith our dataset. For this pulsar, we get $h_{3o}=0.62 \\mu$s. With $\\xi =\n0.14$ $\\mu$s, our constraint implies that the inclination angle $i \\lesssim 60^\\circ$, in\nagreement with the result presented by \\citet{van13}.\n\n\\subsection{PSR J1024$-$0719}\n\\citet{hbo06} were the first to announce a parallax $\\pi=1.9\\pm 0.4$ mas for\nthis nearby and isolated MSP that shows a large amount of red noise\n\\citep{cll+15}. More recently, \\citet{egc+13} used a subset of this\nEPTA dataset to produce an\nephemeris and detected gamma-ray pulsations from this pulsar. The authors assumed\nthe LK bias corrected distance \\citep{vwc+12} from the \\citet{hbo06} parallax\nvalue to estimate its gamma-ray efficiency. However, it should be noted that\n\\citet{vbc+09} did not report on the measurement of the parallax using an\nextended version of the \\citet{hbo06} dataset. \nWith this independent dataset we detect a parallax $\\pi=0.80\\pm0.17$ mas, a\nvalue inconsistent with the early measurement reported by \\citet{hbo06}. A\npossible explanation for this discrepancy could be that \\citet{hbo06} did not\ninclude a red noise model in their analysis.\n\n\\subsection{PSR J1455$-$3330}\nThe last timing solution for this pulsar was published by \\citet{hlk+04} and\ncharacterized by an RMS of 67 \\us. Thanks to our 9 years of data\nwith an RMS of less than 3 \\us, we successfully detect the signature of\nthe proper motion $\\mu_{\\alpha}=7.88\\pm0.08$ mas~yr$^{-1}$ and\n$\\mu_{\\delta}=-2.23\\pm0.19$ mas~yr$^{-1}$, the parallax $\\pi=1.04\\pm 0.35$ mas\nand the secular variation of the semi-major axis,\n$\\dot{x}=(-1.7\\pm0.4)\\times10^{-14}$ for the first time.\n\n\\subsection{PSR J1600$-$3053}\nThis 3.6-ms pulsar can be timed at very high precision thanks to the $\\sim 45$\n\\us~wide peak on the right edge of its profile (see\nFig.~\\ref{plot:templates-1}). We present here a precise measurement of the\nparallax $\\pi = 0.64 \\pm 0.07$ mas, a value marginally consistent with the\n$\\pi = 0.2 \\pm 0.15$ mas from \\citet{vbc+09}. We also show a large improvement\non the Shapiro delay detection through the use of the orthometric\nparametrization \\citep{fw10} with $h_3= (3.3\\pm0.2) \\times 10^{-7}$ and\n$\\varsigma=0.68\\pm0.05$. The resulting mass measurement of this system\nis discussed in Section~\\ref{sec:dis_mass}.\n\n\\input{table_param3}\n\n\\subsection{PSR J1640$+$2224}\n\\citet{llw+05} used early Arecibo and Effelsberg data to report on the tentative detection of Shapiro delay for\nthis wide binary system in a 6-month orbit. From this measurement they deduced\nthe orientation of the system to be nearly edge-on ($78^{\\circ} < i <\n88^{\\circ}$) and a companion mass for the white dwarf $m_p=0.15_{-0.05}^{+0.08}\n\\text{M}_{\\odot}$. We cannot constrain the Shapiro delay with the\ncurrent EPTA data, even though our data comprise almost twice the number of\nTOAs with a similar overall timing precision. The parallax signature in the\nresiduals also remains undetected (based on Bayesian evidence\\footnote{A\ndifference of 3 in the log evidence between two models is usually required to\njustify the introduction of an additional parameter \\citep{kr95}.}) but we find a significant\n$\\dot{x}=(1.07\\pm0.16)\\times 10^{-14}$, consistent with the upper limit set by \\citet{llw+05}.\n\n\\subsection{PSR J1643$-$1224}\nUsing PPTA data, \\citet{vbc+09} previously announced a parallax value $\\pi =\n2.2 \\pm 0.4$ mas that is marginally consistent with our value of $\\pi = 1.17\n\\pm 0.26$ mas. We get a similar proper motion and $\\dot{x}=(-4.79\\pm0.15) \\times\n10^{-14}$, albeit measured with a greater precision.\n\n\\subsection{PSR J1713$+$0747}\n\\label{sec:1713}\nPSR J1713$+$0747 is one of the most precisely timed pulsars over two decades\n\\citep{vbc+09, zsd+15}.\nOur proper motion and parallax values are consistent with the ones from \\citet{vbc+09} and\n\\citet{zsd+15}. Nonetheless we can not detect any hint of the orbital period derivative $\\dot{P_{b}}$.\nThe measurement of the Shapiro delay yields the following masses of the system,\n$m_p=1.33_{-0.08}^{+0.09} \\text{M}_{\\odot}$ and $m_c=0.289_{-0.011}^{+0.013}\n\\text{M}_{\\odot}$, in very good agreement with \\citet{zsd+15}.\n\n\nWhen inspecting the residuals of PSR J1713$+$0747 we noticed successive TOAs\ntowards the end of 2008 that arrived significantly earlier ($\\sim3$ \\us) than\npredicted by our ephemeris (see top panel of Figure~\\ref{plot:1713_event}).\nAfter inspection of the original archives and comparison with other high\nprecision datasets like those on PSRs J1744$-$1134 and J1909$-$3744, we ruled out any\ninstrumental or clock issue as an explanation for this shift. We therefore\nattribute this effect to a deficiency of the electron content towards the line\nof sight of the pulsar. This event has also been observed by the other PTAs\n\\citep{zsd+15,cks+15} and interpreted as possibly a kinetic shell propagating\nthrough the interstellar medium \\citep{cks+15} followed by a rarefaction of the electron\ncontent.\n\nTo model this DM event we used shapelet basis functions. A thorough\ndescription of the shapelet formalism can be found in \\cite{ref03}, with\nastronomical uses being described in e.g., \\cite{rb03,km04,lah15}. Shapelets\nare a complete ortho-normal set of basis functions that allow us to recreate\nthe effect of non-time-stationary DM variations in a statistically robust\nmanner, simultaneously with the rest of the analysis. We used the Bayesian\nevidence to determine the number of shapelet coefficients to include in the\nmodel (only one coefficient was necessary in this study, i.e. the shapelet is\ngiven by a Gaussian). Our priors on the\nlocation of the event span the entire dataset, while we assume an event width\nof between five days and one year. The maximum likelihood results indicate\nan event centered around MJD 54761 with a width of 10 days. The resulting DM signal\n(including the shapelet functions) and the residuals corrected from it are\nplotted in the middle and bottom panels of Fig.~\\ref{plot:1713_event}\nrespectively. The DM model hence predicts a drop of $(1.3\\pm0.4) \\times 10^{-3}$\npc~cm$^{-3}$.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[height=80mm,angle=-90]{plots\/1713-event.ps}\n\\caption{Top panel: zoom-in on the PSR J1713$+$0747 residuals (black dots and red\ntriangles are L-Band and S-Band data respectively). Middle panel: DM signal\nfrom the maximum likelihood DM model incorporating the shapelet basis functions\n(see Section~\\ref{sec:1713} for details). The bottom panel shows the residuals after\nsubtraction of the DM signal. The uncertainties on the DM signal come directly\nfrom the 1-$\\sigma$ uncertainties on the shapelet amplitudes used to model the\nevent, obtained from the full Bayesian analysis.}\n\\label{plot:1713_event}\n\\end{figure}\n\n\\input{table_param4}\n\n\\subsection{PSR J1721$-$2457}\nThanks to an additional five years of data compared to \\citet{jsb+10}, the\nproper motion of this isolated MSP is now better constrained. Our current\ntiming precision is most likely limited by the pulsar's large duty cycle (see\nFig.~\\ref{plot:templates-1}) and the apparent absence of sharp features in the\nprofile. The flux density of this pulsar is also quite low with a value of 1 mJy at 1400MHz.\n\n\\subsection{PSR J1730$-$2304}\nThis low-DM and isolated MSP has a profile with multiple pulse components (see\nFig.~\\ref{plot:templates-1}). As this pulsar lies very near to the ecliptic\nplane ($\\beta= 0.19^{\\circ}$), we are unable to constrain its proper motion in\ndeclination, similar to the previous study \\citep{vbc+09}. Assuming the NE2001 distance, the\nexpected parallax timing signature would be as large as 2.3 \\us. We report here on a\ntentative detection of the parallax, $\\pi = 0.86\\pm0.32$ mas.\n\n\\subsection{PSR J1738$+$0333}\nAfter the determination of the masses in this system from optical observations\n\\citep{akk+12}, \\citet{fwe+12} used the precise measurements of the proper motion, parallax\n and $\\dot{P_b}$ in this binary system to put constraints on\nscalar-tensor theories of gravity. Our measured proper motion remains\nconsistent with their measurements. With a longer baseline and more\nobservations recorded with the sensitive Arecibo Telescope, \\citet{fwe+12}\nwere able to detect the parallax and the orbital period derivative of the\nsystem. However, we do not yet reach the sensitivity to detect these two\nparameters with our dataset.\n\n\n\\subsection{PSR J1744$-$1134}\nThis isolated MSP was thought to show long-term timing noise by \\citet{hbo06}\neven with a dataset shorter than 3 years.\nIn our data set we detect a (red) timing noise component \\citep[see][]{cll+15}.\nThe RMS of the time-domain noise signal is\n$\\sim\\,0.4\\mu$s, but has a peak-to-peak variation of $\\sim\\,2\\mu$s. The\nhigher latter value, however, is due to a bump which appears localized\nin time (MJD $\\sim$ 54000 to 56000). As discussed in \\citet{cll+15},\nnon-stationary noise\nfrom instrumental instabilities may cause such effects, but data with\nbetter multi-telescope coverage are necessary to verify such a\npossibility. This is further investigated in Lentati et al. (submitted)\nusing a more extended dataset from the International Pulsar Timing Array (IPTA) \n\\citep{vlh+16}.\n\n\n\\input{table_param5}\n\n\\subsection{PSR J1751$-$2857}\n\\citet{sfl+05} announced this wide ($P_b = 111$~days) binary MSP after\ntiming it for 4 years with an RMS of 28 $\\mu s$ without a detection of the proper\nmotion. With 6 years of data at a much lower RMS, we are able to constrain its\nproper motion ($\\mu_{\\alpha}=-7.4\\pm0.1$ mas~yr$^{-1}$ and\n$\\mu_{\\delta}=-4.3\\pm1.2$ mas~yr$^{-1}$) and detect $\\dot{x}=(4.6\\pm0.8) \\times\n10^{-14}$.\n\n\n\\subsection{PSR J1801$-$1417}\nThis isolated MSP was discovered by \\citet{lfl+06}. With increased timing\nprecision, we measure a new composite proper motion $ \\mu = 11.3\\pm0.3$ mas\nyr$^{-1}$. As our dataset for this pulsar does not include multifrequency\ninformation; we can not rule out DM variations.\n\n\\subsection{PSR J1802$-$2124}\n\\citet{fsk+10} recently reported on the mass measurement of this system by\ncombining TOAs from the Green Bank, Parkes and Nan\\c cay radio telescopes.\nTherefore, our dataset shows no improvement in the determination of the system\nparameters but gives consistent results to \\citet{fsk+10}.\n\n\\subsection{PSR J1804$-$2717}\nWith an RMS timing residual improved by a factor 25 compared to the last\nresults published by \\citet{hlk+04}, we obtain a reliable measurement of the\nproper motion of this system. Assuming the distance based on the NE2001 model\n$d_{\\text{NE2001}} = 780$ pc, the parallax timing signature can amount to 1.5\n\\us, still below our current timing precision.\n\n\\input{table_param6}\n\n\\subsection{PSR J1843$-$1113}\nThis isolated pulsar discovered by \\citet{hfs+04} is the second fastest-spinning\nMSP in our dataset. Its mean flux density (S$_{1400} = 0.6$ mJy)\nis among the lowest, limiting our current timing precision to $\\sim$ 1 \\us. For\nthe first time, we report the detection of the proper motion\n$\\mu_{\\alpha}=-1.91\\pm0.07$ mas~yr$^{-1}$ and $\\mu_{\\delta}=-3.2\\pm0.3$\nmas~yr$^{-1}$ and still low-precision parallax $\\pi = 0.69 \\pm 0.33$ mas.\n\n\\subsection{PSR J1853$+$1303}\nOur values of proper motion and semi-major axis change are consistent with the\nrecent work by \\citet{gsf+11} using high-sensitivity Arecibo and Parkes data,\nthough there is no evidence for the signature of the parallax in our data\n , most likely due to our less precise dataset.\n\n\\subsection{PSR J1857$+$0943 (B1855+09)}\nOur measured parallax $\\pi=0.7\\pm0.26$ mas is lower than, but still compatible\nwith, the value reported by \\citet{vbc+09}. We also report a marginal detection\nof $\\dot{x} = (-2.7 \\pm 1.1) \\times 10^{-15}$. Our measurement of the Shapiro\ndelay is also similar to the previous result from \\citet{vbc+09}.\n\n\\subsection{PSR J1909$-$3744}\nPSR J1909$-$3744 \\citep{jbv+03} is the most precisely timed source with a RMS\ntiming residual of about 100 ns. As these authors pointed out, this pulsar's\nprofile has a narrow peak with a pulse duty cycle of 1.5\\% (43$\\mu$s) at FWHM (see\nFig.~\\ref{plot:templates-2}). Unfortunately its declination makes it only\nvisible with the NRT but it will be part of the SRT timing campaign. We improved\nthe precision of the measurement of the orbital period derivative $\\dot{P_b}$ by a factor\nof six compared to \\citet{vbc+09} and our constraint on $\\dot{x}$ is\nconsistent with their tentative detection.\n\n\n\\subsection{PSR J1910$+$1256}\nWe get similar results as recently published by \\citet{gsf+11} with Arecibo and\nParkes data. In addition, we uncover a marginal signature of the parallax\n$\\pi = 1.44 \\pm 0.74$ mas, consistent with the upper limit set by \\citet{gsf+11}.\n\n\\input{table_param7}\n\n\\subsection{PSR J1911$+$1347}\n With a pulse width at 50\\% of the main peak amplitude (see\nFig.~\\ref{plot:templates-2}), $W_{50} = 89 $ $\\mu$s (only twice the width of\nJ1909$-$3744), this isolated MSP is potentially a good candidate for PTAs.\nUnfortunately it has so far been observed at the JBO and NRT observatories only and\nno multifrequency observations are available. Based on this work, this pulsar\nhas now been included in the observing list at the other EPTA telescopes.\nDespite the good timing precision we did not detect the parallax but we did\nmeasure the proper motion for the first time with $\\mu_{\\alpha}=-2.90\\pm0.04$\nmas~yr$^{-1}$ and $\\mu_{\\delta}=-3.74\\pm0.06$ mas~yr$^{-1}$.\n\n\\subsection{PSR J1911$-$1114}\nThe last ephemeris for this pulsar was published by \\citet{tsb+99} 16 years\nago using the DE200 planetary ephemeris. Our EPTA dataset spans three times\nlonger than the one from \\citet{tsb+99}. We hence report here on a greatly\nimproved position, proper motion ($\\mu_{\\alpha}=-13.75\\pm0.16$ mas~yr$^{-1}$\nand $\\mu_{\\delta}=-9.1\\pm1.0$ mas~yr$^{-1}$) and a new eccentricity\n$e=(1.6\\pm1.0) \\times 10^{-6}$, lower by a factor of 10 than the previous\nmeasurement.\n\n\\subsection{PSR J1918$-$0642}\nPSR J1918$-$0642 is another MSP studied by \\citet{jsb+10} with EPTA data.\nCompared to \\citet{jsb+10} we extended the baseline with an additional five\nyears of data. We unveil the signature of Shapiro delay in this system\nwith $h_3=(8.6\\pm1.2)\\times 10^{-7}$ and $\\varsigma =0.91\\pm0.04$.\nThe masses of the system are discussed in Section~\\ref{sec:dis_mass}.\n\n\n\\input{table_param8}\n\n\\subsection{PSR J1939$+$2134 (B1937+21)}\nThanks to the addition of early Nan\\c cay DDS TOAs, our dataset span over 24\nyears for this pulsar. This pulsar has been long known to show significant DM\nvariations as well as a high level of timing noise \\citep{ktr94}; see\nresiduals in Fig.~\\ref{plot:residuals-2}. A\npossible interpretation of this red noise is the presence of an asteroid belt\naround the pulsar \\citep{scm+13}. Despite this red noise, the timing signature\nof the parallax has successfully been extracted to get $\\pi=0.22 \\pm 0.08$ mas,\na value consistent with \\citet{ktr94} and \\citet{vbc+09}.\n\n\\subsection{PSR J1955$+$2908 (B1953+29)}\nPSR J1955$+$2908 is another MSP recently analyzed by \\citet{gsf+11}. With an\nindependent dataset, we get similar results to \\citet{gsf+11}. We report here on\nthe tentative detection of $\\dot{x}=(4.0\\pm1.4) \\times 10^{-14}$.\n\n\\subsection{PSR J2010$-$1323}\nThis isolated MSP was discovered a decade ago \\citep{jbo+07} and no update on\nthe pulsar's parameters has been published since then. Hence we announce here\nthe detection of the proper motion $\\mu_{\\alpha}=-2.53\\pm0.09$ mas~yr$^{-1}$\nand $\\mu_{\\delta}=-5.7\\pm0.4$ mas~yr$^{-1}$. Assuming the NE2001 distance of 1\nkpc, the parallactic timing signature would amount to 1.17 $\\mu s$ but was not\ndetected in our data.\n\n\\subsection{PSR J2019$+$2425}\nCompared to the Arecibo 430-MHz dataset used by \\citet{nss01}, the EPTA timing\nprecision for this pulsar is limited due to its low flux density at 1400 MHz. Because\nof this we are not able to measure the secular change of the projected\nsemi-major axis $\\dot{x}$.\n\n\\input{table_param9}\n \n\\subsection{PSR J2033$+$1734}\nIn spite of a narrow peak of width $\\sim 160$ $\\mu$s this MSP has a very large\ntiming RMS of 14 $\\mu$s. With the absence of obvious systematics in the\nresiduals, we attribute the poor timing precision to the extremely low flux\ndensity of this pulsar at 1400 MHz, $S_{1400} = 0.1$ mJy where all of our\nobservations were performed. Indeed this pulsar was discovered by\n\\citet{rtj+96} with the Arecibo telescope at 430 MHz and later followed up by\n\\citet{spl04} still at 430 and 820 MHz with the Green Bank 140-ft telescope.\nHere we report with an independent dataset at 1400 MHz a similar proper motion\nresult to \\citet{spl04}. \n\n\\subsection{PSR J2124$-$3358}\nFor the isolated PSR J2124$-$3358, our measured proper motion is consistent\nwith the already precise value published by \\citet{vbc+09}. Our parallax\n$\\pi=2.50\\pm0.36$ mas is also consistent with their results but with a better\nprecision.\n\n\\subsection{PSR J2145$-$0750}\nDespite its rotational period of 16 ms PSR J2145$-$0750 is characterized by a\ntiming RMS of 1.8 $\\mu$s thanks to its narrow leading peak and large average\nflux density, $S_{1400}=7.2$~mJy. The EPTA dataset does not show any\nevidence for a variation of the orbital period of PSR J2145$-$0750 or a\nprecession of periastron, even though \\citet{vbc+09} reported a marginal\ndetection with a slightly shorter data span characterized by a higher RMS\ntiming residual. On the other hand, we detect a significant\n$\\dot{x}=(8.2\\pm0.7)\\times10^{-15}$, which is not consistent with the\nmarginal detection, $\\dot{x}=(-3\\pm1.5)\\times10^{-15}$, reported by \\citet{vbc+09}. \n\n\\subsection{PSR J2229$+$2643}\nWith eight years of data on PSR J2229$+$2643, we measure\n$\\mu_{\\alpha}=-1.73\\pm0.12$ mas~yr$^{-1}$ and $\\mu_{\\delta}=-5.82\\pm0.15$\nmas~yr$^{-1}$. Our measured $\\mu_{\\delta}$ is inconsistent with the last timing solution by\n\\citet{wdk+00} using the DE200 ephemeris ($\\mu_{\\alpha}=1\\pm4$ mas~yr$^{-1}$ and $\\mu_{\\delta}=-17\\pm4$\nmas~yr$^{-1}$). Given our much smaller timing residual RMS, our use of the\nsuperior DE421 model and longer baseline, we are confident our value is\nmore reliable. The expected timing signature of the parallax (0.7 \\us) is too small\nto be detected with the current dataset. Note that the early Effelsberg data\nrecorded with the EPOS backend included in \\citet{wdk+00} are not part of this\ndataset.\n\n\\input{table_param10}\n\n\\subsection{PSR J2317$+$1439}\nCompared to \\citet{cnt96} we are able to constrain the proper motion\n($\\mu_{\\alpha}=-1.19\\pm0.07$ mas~yr$^{-1}$ and $\\mu_{\\delta}=3.33\\pm0.13$\nmas~yr$^{-1}$) and eccentricity $e=(5.7\\pm1.6) \\times 10^{-7}$ of the system\nthrough the use of the ELL1 parametrization. We also detect a marginal\nsignature of the parallax $\\pi=0.7\\pm0.3$ mas. \n\n\\subsection{PSR J2322$+$2057}\nPSR J2322$+$2057 is an isolated MSP with a pulse profile consisting of two\npeaks separated by $\\simeq$ 200\\dg (see Fig.~\\ref{plot:templates-2}).\n\\citet{nt95} were the last to publish a timing solution for this last source in\nour dataset. We measure a proper motion consistent with their results albeit\nwith much greater precision, $\\mu=24.0\\pm0.4$ mas yr$^{-1}$.\n\n\\input{table_param11}\n\n\\section{Introduction}\n\nThree decades ago \\citet{bkh+82} discovered the first millisecond pulsar (MSP), \nspinning at 642 Hz. Now over 300 MSPs have been found; see the Australia\nTelescope National Facility (ATNF) pulsar\ncatalog\\footnote{http:\/\/www.atnf.csiro.au\/people\/pulsar\/psrcat\/} \\citep{mht+05}. MSPs are\nthought to be neutron stars spun-up to rotation periods (generally) shorter than 30 ms via the\ntransfer of mass and angular momentum from a binary companion\n\\citep{acr+82,rs82}. We know that the vast majority of the MSP population ($ \n\\simeq 80$\\%) still\nreside in binary systems and these objects have been shown to be incredible\nprobes for testing physical theories. Their applications range from\nhigh-precision tests of general relativity (GR) in the quasi-stationary strong-field regime\n\\citep{ksm+06,fwe+12} to constraints on the equation of state of matter at supra-nuclear densities\n\\citep{dpr+10,afw+13}. Binary systems with a MSP and a white dwarf in wide orbits\noffer the most stringent tests of the strong equivalence principle\n\\citep[e.g.][]{sfl+05,fkw12,rsa+14}.\n\n Most\nof these applications and associated results mentioned above arise from the use of the pulsar\ntiming technique that relies on two properties of the radio MSPs: their\nextraordinary rotational and average pulse profile stability. The pulsar\ntiming technique tracks the times of arrival (TOAs) of the pulses recorded at\nthe observatory and compares them to the prediction of a best-fit model. This\nmodel, which is continuously improved as more observations are made available,\ninitially contains the pulsar's astrometric parameters, the rotational parameters\nand the parameters describing the binary orbit, if applicable. With the recent\nincrease in timing precision due to e.g. improved receivers, larger available \nbandwidth and the use of coherent dedispersion \\citep{hr75},\nparameters that have a smaller effect on the TOAs have become measurable.\n\nThe first binary pulsar found, PSR B1913+16 \\citep{ht75}, yielded the first evidence\nfor gravitational waves (GWs) emission. Since then, several ground-based detectors have been built\naround the globe, e.g. Advanced LIGO \\citep{lig15} and Advanced Virgo\n\\citep{vir15}, to\ndirectly detect GWs in the frequency range of 10-7000 Hz.\nAlso a space mission, eLISA \\citep{lis+13}, is being designed to study GWs in the mHz regime.\nPulsars, on the other hand, provide a complementary probe for GWs\n by opening a new window in the nHz regime \\citep{saz78,det79}.\nPrevious limits on the amplitude of the stochastic GW background (GWB) have been set by studying individual MSPs\n\\citep[e.g.][]{ktr94}.\nHowever, an ensemble of pulsars spread over the sky (known as Pulsar Timing\nArray; PTA) is required to ascertain the presence of a GWB and discriminate between\npossible errors in the Solar System ephemeris or in the reference time\nstandards \\citep{hd83,fb90}.\n\nA decade ago, \\citet{jhl+05} claimed that timing a set of a least 20 MSPs with a precision of\n100 ns for five years would allow a direct detection of the GWB.\nSuch high timing precision has not yet been reached \\citep{abb+15}.\nNonetheless, \\citet{sej+13} recently argued that when a PTA enters a new signal\nregime where the GWB signal starts to prevail over the low frequency pulsar\ntiming noise, the sensitivity of this PTA\ndepends more strongly on the number of pulsars than the cadence of the\nobservations or the timing precision. Hence, datasets consisting of many pulsars\nwith long observing baselines, even with timing precision of $\\sim 1 \\mu$s, constitute\nan important step towards the detection of the GWB. In addition to the GWB\nstudies, such long and precise datasets allow additional timing parameters, and\ntherefore science, to be extracted from the same data.\n\nParallax measurements can contribute to the construction of Galactic electron\ndensity models \\citep{tc93,cl02}. Once built, these models can provide distance\n estimates for pulsars along generic lines-of-sight. New parallax measurements hence\nallow a comparison and improvement of the current free electron distribution\nmodels \\citep{sch12}.\nAn accurate distance is also crucial to correct the spin-down rate of the\npulsar from the bias introduced by its proper motion \\citep{shk70}. This same\ncorrection has to be applied to the observed orbital period derivative before\nany test of GR can be done with this parameter \\citep{dt91}.\n\nIn binary systems, once the Keplerian parameters are known, it may be possible\nto detect post-Keplerian (PK) parameters. These theory-independent parameters \n describe the relativistic\ndeformation of a Keplerian orbit as a function of the Keplerian parameters and\nthe {\\it a priori} unknown pulsar\nmass ($m_p$), companion mass ($m_c$) and inclination angle ($i$).\nMeasurement of the Shapiro delay, an extra propagation delay of the radio waves\ndue to the gravitational potential of the companion, gives 2 PK parameters\n(range $r$ and shape $s\\equiv \\sin i$).\nOther relativistic effects such as the advance of periastron $\\dot{\\omega}$\nand the orbital decay $\\dot{P_b}$ provide one extra PK parameter each.\nIn GR, any PK parameter can be described by the Keplerian parameters plus the\ntwo masses of the system. Measuring three or more PK parameters therefore\noverconstrains the masses, allowing one to perform tests of GR\n\\citep{tw89,ksm+06}.\n\nThe robustness of the detections of these parameters can be hindered\nby the presence of stochastic influences like dispersion measure (DM)\nvariations and red (low-frequency) spin noise in the timing residuals \\citep{chc+11,lah+14}.\nRecent work by \\citet{kcs+13} and \\citet{lbj+14} discussed the modeling of the\nDM variations while \\citet{chc+11} used Cholesky decomposition of the covariance\nmatrix to properly estimate the parameters in the presence of red noise.\n Correcting for the DM\nvariations and the effects of red noise has often been done through an iterative\nprocess. However, TempoNest, a Bayesian pulsar timing analysis software\n\\citep{lah+14} used in this work allows one to model these stochastic\ninfluences\nsimultaneously while performing a non-linear timing analysis. \n\n\nIn this paper we report on the timing solutions of 42 MSPs observed by the\nEuropean Pulsar Timing Array (EPTA). The EPTA is a collaboration of European\nresearch institutes and radio observatories that was established in 2006\n\\citep{kc13}. The EPTA makes use of the five largest (at decimetric\nwavelengths) radio telescopes in Europe: the Effelsberg Radio Telescope in\nGermany (EFF), the Lovell Radio\nTelescope at the Jodrell Bank Observatory (JBO) in England, the Nan\\c cay Radio\nTelescope (NRT) in France, the Westerbork Synthesis Radio Telescope (WSRT) in\nthe Netherlands and the Sardinia Radio Telescope (SRT) in Italy. As the SRT is\ncurrently being commissioned, no data from this telescope are included in this paper.\nThe EPTA also operates the Large European Array for Pulsars (LEAP), where\ndata from the EPTA telescopes are coherently combined to form a tied-array telescope with\nan equivalent diameter of 195 meters, providing a significant improvement in\nthe sensitivity of pulsar timing observations \\citep{bjk+15}.\n\nThis collaboration has already led to previous publications. Using multi-telescope\ndata on PSR J1012$+$5307, \\citet{lwj+09} put a limit on the gravitational\ndipole radiation and the variation of the gravitational constant $G$.\n\\citet{jsb+10} presented long-term timing results of four MSPs, two of which are\nupdated in this work. More recently, \\citet{hlj+11} set the first EPTA upper\nlimit on the putative GWB. Specifically for a GWB formed by circular, GW-driven\nsupermassive black-hole binaries, they measured the amplitude $A$ of the\ncharacteristic strain level at a frequency of 1\/yr, $A < 6 \\times\n10^{-15}$, using a subset of the EPTA data from only 5 pulsars.\n\n\nSimilar PTA efforts are ongoing around the globe with the Parkes Pulsar\nTiming Array (PPTA; \\citet{mhb+13}) and the NANOGrav collaboration\n\\citep{mac13}, also setting limits on the GWB \\citep{dfg+13, src+13}.\n\n\n\n\nThe EPTA dataset introduced here, referred to as the EPTA Data\nRelease 1.0, serves as the reference dataset for the\nfollowing studies: an analysis of the DM variations (Janssen et al., in prep.),\n a modeling of the red noise in each pulsar \\citep{cll+15}, a limit on the stochastic GWB\n\\citep{ltm+15} and the anisotropic background \\citep{tmg+15} as well as a\nsearch for continuous GWs originating from single sources\n\\citep{bps+15}. The organization of this paper is as follows. The\ninstruments and methods to extract the TOAs at each observatory are described\nin Section~\\ref{sec:obs}. The combination and timing procedures are detailed\nin Section~\\ref{sec:timing}. The timing results and new parameters are\npresented in Section~\\ref{sec:results} and discussed in Section ~\\ref{sec:discussions}.\n Finally, we summarize and present some prospects about the EPTA in Section~\\ref{sec:conclusions}.\n\n\\section{Observations and data processing}\n\\label{sec:obs}\n\nThis paper presents the EPTA dataset, up to mid-2014, that was gathered\nfrom the `historical' pulsar instrumentations at\nEFF, JBO, NRT and WSRT with, respectively, the EBPP (Effelsberg-Berkeley Pulsar Processor), \nDFB (Digital FilterBank), BON (Berkeley-Orl\\'eans-Nan\\c cay) and PuMa (Pulsar\nMachine) backends.\nThe data recorded with the newest generation of instrumentations, e.g. PSRIX\n at EFF \\citep{lkg+16} and PuMaII at WSRT \\citep{kss08}, will be part of a future EPTA data release.\n\n Compared to the dataset presented in\n\\citet{hlj+11}, in which timing of only five pulsars was presented, this\nrelease includes 42 MSPs (listed in Table\n\\ref{tab:summary} with their distribution on the sky shown in\nFig.~\\ref{fig:aitoff}). Among those 42 MSPs, 32 are members of binary systems. The timing\nsolutions presented here span at least seven years, and for 16 of the MSPs the\nbaseline extends back $\\sim 15$ years. For the five pulsars included in\n\\citet{hlj+11}, the baseline is extended by a factor 1.7-4.\nWhen comparing our set of pulsars with the NANOGrav Nine-year Data Set\n\\citep{abb+15} (consisting of 37 MSPs) and the PPTA dataset \\citep{mhb+13,rhc+16}\n(consisting of 20 MSPs), we find\nan overlap of 21 and 12 pulsars, respectively. However, we note that the\nNANOGrav dataset contains data for 7 MSPs with a baseline less than two years.\n\n\n In this paper, we define an observing system as a specific combination\nof observatory, backend and frequency band. The radio telescopes and pulsar\nbackends used for the observations are described below.\n\n\n\n\\begin{figure}\n\\includegraphics[height=80mm,angle=-90]{plots\/MSPs_aitoff.ps}\n\\caption{Distribution of the 42 MSPs, represented with a star, in Galactic\ncoordinates (longitude $l$ and latitude $b$). The center of the plot is\noriented towards the Galactic Center. The hatched area is the part of the sky\n(declination $\\delta < -39^\\circ$) that is not accessible to the EPTA.}\n\\label{fig:aitoff}\n\\end{figure}\n\n\\subsection{Effelsberg Radio Telescope}\n\nThe data from the 100-m Effelsberg Radio Telescope presented in this paper were\nacquired using the EBPP, an online\ncoherent dedispersion backend described in detail by \\citet{bdz+97}. This\ninstrument can process a bandwidth (BW) up to 112 MHz depending on the DM value.\nThe signals from the two circular polarizations are split into 32 channels each\nand sent to the dedisperser boards. After the dedispersion takes place, the\noutput signals are folded (i.e. individual pulses are phase-aligned and summed)\nusing the topocentric pulse period.\n\n \nEPTA timing observations at Effelsberg were made at a central frequency of 1410\nMHz until April 2009 then moved to 1360 MHz afterwards due to a change in the\nreceiver. Additional observations at S-Band (2639 MHz) began in November 2005\nwith observations at both frequencies taken during the same two-day observing\nrun. Typically, the observations occur on a monthly basis with an integration time per source\nof about 30 minutes. The subintegration times range from 8 to 12 mins before\n2009 and 2 mins thereafter. For 4 pulsars, namely PSRs J0030$+$0451,\nJ1024$-$0719, J1730$-$2304 and J2317$+$1439, there is a gap in the data from\n1999 to 2005 as these sources were temporarily removed from the observing list. Data reduction was\nperformed with the PSRCHIVE package \\citep{hvm04}. The profiles were cleaned\nof radio frequency interference (RFI) using the PSRCHIVE {\\tt paz} tool but\nalso examined and excised manually with the {\\tt pazi} tool. No standard\npolarization calibration using a pulsed and linearly polarized noise diode was\nperformed. However the EBPP automatically adjusts the power levels of both \npolarizations prior to each observation. The TOAs were calculated by\ncross-correlating the time-integrated,\nfrequency-scrunched, total intensity profile, with an analytic and noise free\ntemplate. This template was generated using the {\\tt paas} tool to fit a set\nof von Mises functions to a profile formed from high signal-to-noise ratio\n(S\/N) observations. In general, we used the standard `Fourier phase gradient'\nalgorithm \\citep{tay92} implemented in PSRCHIVE to estimate the TOAs and their\nuncertainties. We used a different template for each observing frequency,\nincluding different templates for the 1410 and 1360 MHz observations. Local\ntime is kept by the on-site H-maser clock, which is corrected to Coordinated\nUniversal Time (UTC) using recorded offsets between the maser and the Global\nPositioning System (GPS) satellites.\n\n\\subsection{Lovell Radio Telescope}\n\nAt Jodrell bank, the 76-m Lovell telescope is used in a regular monitoring\nprogram to observe most of the pulsars presented in this paper. All TOAs used\nhere were generated by using the DFB, a clone of the Parkes Digital FilterBank.\n Each pulsar was observed with a typical cadence\nof once every 10 days for 30 mins with a subintegration time of 10~s. The DFB\ncame into operation in January 2009 observing at a central frequency of 1400\nMHz with a BW of 128 MHz split into 512 channels. From September 2009, the center frequency was\nchanged to 1520 MHz and the BW increased to 512 MHz (split into 1024 channels) of which\napproximately 380 MHz was usable, depending on RFI conditions. As this is a\nsignificant change, and to account for possible profile evolution with\nobserving frequency, both setups are considered as distinct observing systems\nand different templates were used. \nData cleaning and TOA generation were done in a similar way to the\nEffelsberg data. There is no standard polarization calibration\n(through observations of a noise diode) applied\nto the DFB data. However the power levels of both polarizations are\nregularly and manually adjusted via a set of attenuators. Local time is kept by the on-site H-maser clock, which is\ncorrected to UTC using recorded offsets between\nthe maser and the GPS satellites.\n\n\\subsection{Nan\\c cay Radio Telescope}\n\\label{sec:obs_nrt}\nThe Nan\\c cay Radio Telescope is a meridian telescope with a collecting area\nequivalent to a 94-m dish. The moving focal carriage that allows an observing\ntime of about one hour per source hosts the Low Frequency (LF) and High\nFrequency (HF) receivers covering 1.1 to 1.8 GHz and 1.7 to 3.5 GHz, respectively.\n A large timing program of MSPs started in late 2004 with the commissioning of\nthe BON instrumentation, a member of the\nASP-GASP coherent dedispersion backend family \\citep{d07}. A 128~MHz BW\nis split into 32 channels by a CASPER\\footnote{https:\/\/casper.berkeley.edu}\nSerendip V board and then sent to servers to be coherently dedispersed and\nfolded to form 2-min subintegrations.\n\nFrom 2004 to 2008 the BW was limited to 64 MHz and then extended to 128~MHz.\n At the same time, the NRT started to regularly observe a pulsed noise\ndiode prior to each observation in order to properly correct for the\ndifference in gain and phase between the two polarizations. In August 2011, the\nL-Band central frequency of the BON backend shifted from 1.4 GHz to 1.6 GHz to\naccommodate the new wide-band NUPPI dedispersion backend \\citep{ldc+14}. Due to\nknown instrumental issues between November 2012 and April 2013\n(i.e. loss of one of the polarization channels, mirroring of the spectrum),\nthese data have not been included in the analysis.\n\nThe flux density values at 1.4 GHz reported in Table \\ref{tab:summary} are\nderived from observations recorded with the NUPPI instrument between MJD 55900\nand 56700. The quasar 3C48 was chosen to be the reference source for the\nabsolute flux calibration. These flux density values have been corrected for\nthe declination-dependent illumination of the mirrors of the NRT. \nAlthough the NUPPI timing data are not included in this work, we used these\nobservations to estimate the median flux densities as no other EPTA data were\nflux-calibrated. The NUPPI timing data will be part of a future EPTA\ndata release along with the data from other telescopes recorded with\nnew-generation instrumentations.\n\nThe data were reduced with the PSRCHIVE package and automatically cleaned\nfor RFI. Except for pulsars with short orbital periods, all daily observations\nare fully scrunched in time and frequency to form one single profile. For PSRs\nJ0610$-$2100, J0751$+$1807, J1738$+$0333, J1802$-$2124 the data were integrated\nto form 6, 12, 16\nand 8 min profiles respectively. The templates for the three observing\nfrequencies are constructed by phase-aligning the $\\sim$10\\% profiles with the\nbest S\/N. The resulting integrated profiles are made noise free with the same\nwavelet noise removal program as in \\citet{dfg+13}. As stated above, we used\nthe standard `Fourier phase gradient' from PSRCHIVE to estimate the TOAs and\ntheir uncertainties. However, we noticed that in the case of very low S\/N\nprofiles, the reported uncertainties were underestimated. \\citet{abb+15} also\nobserved that TOAs extracted from low S\/N profiles deviate from a Gaussian\ndistribution and therefore excluded all TOAs where S\/N <8 (see Appendix B of their paper\nfor more details). Here,\nwe made use of the Fourier domain Markov Chain Monte Carlo TOA estimator\n(hereafter FDM) to properly estimate the error bars in this low S\/N regime. We applied the FDM method\nto PSRs J0034$-$0534, J0218$+$4232, J1455$-$3330, J2019$+$2425, J2033$+$1734.\nAll the BON data are time-stamped with a GPS-disciplined clock.\n\nFor PSR J1939$+$2134, archival data from 1990 to 1999 recorded with a swept-frequency\n local oscillator (hereafter referred to as DDS) at a frequency of 1410\nMHz \\citep{cbl+95} were added to the dataset. These data are time-stamped with\nan on-site Rubidium clock, which is corrected to UTC using recorded offsets between\n the Rubidium clock and the Paris Observatory Universal Time.\n\n\\subsection{Westerbork Synthesis Radio Telescope}\n\nThe Westerbork Synthesis Radio Telescope is an East-West array\nconsisting of fourteen 25-m dishes, adding up to the equivalent size of a 94-m\ndish when combined as a tied-array. From 1999 to 2010, an increasing\n number of MSPs were observed once a\nmonth using the PuMa pulsar machine (a digital filterbank) at WSRT \\citep{vkh+02}. In each observing\nsession, the pulsars were observed for 25 minutes each at one or more\nfrequencies centered at 350 MHz (10 MHz BW), 840 MHz (80 MHz BW) and 1380 MHz\n(80 MHz spread across a total of 160 MHz BW). Up to 512 channels were used to\nsplit the BW for the observations at 350 MHz. At 840 MHz and 1380 MHz, 64 channels were used per 10 MHz subband. For a more detailed description of\nthis instrumentation, see e.g. \\citet{jsk+08}. Since 2007, the 840 MHz band was no longer used for\nregular timing observations, however, an additional observing frequency\ncentered at 2273 MHz using 160 MHz BW was used for a selected set of the\nobserved pulsars. The data were dedispersed and folded offline using custom\nsoftware, and then\nintegrated over frequency and time to obtain a single profile for each\nobservation. \nGain and phase difference between the two polarizations are adjusted during the phased-array calibration of the dishes.\nTo generate the TOAs, a high-S\/N template based on the observations was used\nfor each observing frequency separately.\n Local time is kept by the on-site H-maser\nclock, which is corrected to UTC using recorded\noffsets between the maser and the GPS satellites.\n\n\n\\section{Data combination and timing}\n\\label{sec:timing}\n\nThe topocentric TOAs recorded at each observatory are first converted to the\nSolar System barycenter (SSB) using the DE421 planetary ephemeris \\citep{fwb09} with\nreference to the latest Terrestrial Time standard from the Bureau International\ndes Poids et Mesures (BIPM) \\citep{pet10}. The DE421 model is a major\nimprovement on the DE200 ephemeris that was used for older\npublished ephemerides and later found to suffer from inaccurate values of planetary\nmasses \\citep{sns+05,hbo06,vbs+08}.\n\nWe used TempoNest \\citep{lah+14}, a Bayesian analysis software that uses the\nTempo2 pulsar timing package \\citep{hem06, ehm06} and MULTINEST \\citep{fhb09}, a\nBayesian inference tool, to evaluate and explore the parameter\nspace of the non-linear pulsar timing model. \nAll pulsar timing parameters are sampled in\nTempoNest with uniform priors. The timing model includes the\nastrometric (right ascension, $\\alpha$, declination, $\\delta$, proper motion in\n$\\alpha$ and $\\delta$, $\\mu_{\\alpha}$ and $\\mu_{\\delta}$) and rotational parameters (period $P$\nand period derivative $\\dot{P}$). If the pulsar is part of a binary system,\nfive additional parameters are incorporated to describe the Keplerian binary\nmotion: the orbital period $P_b$, the projected semi-major axis $x$ of the\npulsar orbit, the longitude of periastron $\\omega$, the epoch $T_0$ of the periastron\npassage and the eccentricity $e$. For some pulsars in our set, we require theory-independent PK parameters\n\\citep{dd85, dd86} to account for deviations from a Keplerian motion, or\nparameters to describe changes in the viewing geometry of the systems. The\nparameters we used include the precession of periastron\n$\\dot{\\omega}$, the orbital period derivative $\\dot{P_b}$, the Shapiro delay\n(`range' $r$ and `shape' $s$; $s$ has a uniform prior in $\\cos i$ space) and the apparent\nderivative of the projected semi-major axis $\\dot{x}$. These parameters are implemented\nin Tempo2 under the `DD' binary model. In the case of quasi-circular orbits, the `ELL1'\nmodel is preferred and replaces $\\omega$, $T_0$ and $e$ with the two\nLaplace-Lagrange parameters $\\kappa$ and $\\eta$ and the time of ascending node\n$T_{\\text{asc}}$ \\citep{lcw+01}. For the description of the Shapiro delay in PSRs\nJ0751$+$1807, J1600$-$3053 and J1918$-$0642 we adopted the orthometric\nparametrization of the Shapiro delay introduced by \\citet{fw10} with the\namplitude of the third harmonic of the Shapiro delay $h_3$ and the ratio of\nsuccessives harmonics $\\varsigma$.\n \nTo combine the TOAs coming from the different observing systems described in\nSection~\\ref{sec:obs}, we first corrected them for the phase difference between\nthe templates by cross-correlation of the reference template with the\nother templates. We then fit for the arbitrary time offsets, known as JUMPs, between\nthe reference observing system and the remaining systems. These\nJUMPs encompass, among other things: the difference in instrumental delays,\nthe use of different templates and the choice for the fiducial point on the\ntemplate. The JUMPs are analytically marginalized over during the\nTempoNest Bayesian analysis. In\norder to properly weight the TOAs from each system, the\ntiming model includes a further two {\\it ad~hoc} white noise parameters per observing\nsystem. These parameters known as the error factor `EFAC', $E_f$, and the error\nadded in quadrature `EQUAD', $E_q$ (in units of seconds), relate to a TOA with uncertainty\n$\\sigma_p$ in seconds as:\n\\begin{equation}\n\\sigma = \\sqrt{E_q^2 + E_f^2 {\\sigma_p}^2}.\n\\end{equation}\nNote that this definition of EFAC and EQUAD in TempoNest is different from the\ndefinition employed in Tempo2 and the earlier timing software Tempo, where\n$E_q$ was added in quadrature to $\\sigma_p$ before applying $E_f$. The $E_f$\nand $E_q$ parameters are set with\nuniform {priors in the \nlogarithmic space (log-uniform priors)} in the $\\log_{10}$-range $[-0.5,1.5], [-10, -3]$, respectively.\nThese prior ranges are chosen to be wide enough to include any value of EFAC\nand EQUAD seen in our dataset.\n\nEach pulsar timing model also includes two stochastic models to describe the\nDM variations and an additional {achromatic} red noise process.\n{Both processes are modeled as stationary, stochastic signals with \na power-law spectrum of the form $S(f)\\propto{}A^2f^{-\\gamma}$,\nwhere $S(f)$, $A$, and $\\gamma$ are the power spectral density as function of\nfrequency $f$, the\namplitude and \nthe spectral index, respectively. The power laws \nhave a cutoff frequency at the lowest frequency, equal to the inverse of the data span, \nwhich is mathematically necessary for the subsequent calculation of the\ncovariance matrix \\citep{hlm+09}.\nIt has been shown that this cutoff rises naturally for the achromatic red noise power law in pulsar timing data\nbecause any low-frequency signal's power below the cutoff frequency is absorbed by the fitting of the pulsar's \nrotational frequency and frequency derivative \\citep{hlm+09,lbj+12}. \nIt is possible to do the same for the DM variations model, by fitting a first\nand a second DM derivative (parameters DM1 and DM2)\nin the timing model \\citep{lbj+14}. \nImplementation of the models is made using the time-frequency method of\n\\citet{lah+13}. \nDetails on this process and applications can be found in \\citet{ltm+15} and \\citet{cll+15}.\nIn brief, denoting matrices with boldface letters, the red noise process \ntime-domain signal, is expressed as a Fourier series,\n${\\rm \\bf t}_{\\rm TN}=\\mathbfss{F}_{\\rm TN}\\textrm{{\\bf a}}$, where $\\mathbfss{F}_{\\rm TN}$ \nis the sum of sines and cosines with coefficients given by the matrix $\\textrm{{\\bf a}}$. \nFourier frequencies are sampled with integer multiples of the lowest frequency, \nand are sampled up to $1\/14$\\,days$^{-1}$. The \nFourier coefficients are free parameters.} \n\n{The DM variations component \nis modeled similarly, with the only difference being that the time-domain signal \nis dependent on the observing frequency. According to the \ndispersion law from interstellar plasma, the delay in the arrival time \nof the pulse depends on the inverse square of the \nobserving frequency, see e.g. \\citet{lg12}. As such, the Fourier \ntransform components are \n${\\mathbfss{F}^{DM}_{ij}={\\mathbfss{F}_{ij}}D_{i} D_{j}}$, \nwhere the i,j indices denote the residual index number, $D_i=1\/(k\\nu^2_i)$, \nand $k=2.41\\times 10^{-16}$~Hz$^{-2}$cm$^{-3}$pc~s$^{-1}$, is the dispersion constant. \nThis stochastic DM variations component is additional to the deterministic \nlinear and quadratic components\nimplemented as part of the Tempo2 timing model.} \nIn addition, we used the standard electron density model for the solar wind\nincluded in Tempo2 with a value of 4~cm$^{-3}$ at 1 AU. This solar wind model\ncan be covariant with the measured astrometric parameters of the pulsar.\n\n{The covariance matrix of each of these two components \nis then calculated with a function of the form \\citep{ltm+15}:\n\\begin{equation}\n\\label{eq:BayesCovRed}\n{\\bf C} = {\\bf C^{-1}_{\\textrm{w}}} - {\\bf \nC^{-1}_{\\textrm{w}}}\\mathbfss{F} \\left[(\\mathbfss{F})^{\\textrm{{\\bf T}}}{\\bf \nC^{-1}_{\\textrm{w}}}\\mathbfss{F} + (\\Psi)^{-1}\\right]^{-1} (\\mathbfss{F})^{\\textrm{{\\bf T}}}{\\bf C^{-1}_{\\textrm{w}}} .\n\\end{equation}\nThe equation is valid for both the DM variations and achromatic \nred noise process, by using the corresponding Fourier transform \n$\\mathbfss{F}$ and covariance matrix of the Fourier coefficients $\\Psi=\\langle\n\\textrm{a}_i\\textrm{a}_j\\rangle$. \nThe ${\\bf C_{\\textrm{w}}}$ term is the white noise covariance matrix \nand is a diagonal matrix \nwith the main diagonal formed by the residual uncertainties squared. \nThe superscript ${\\textrm{{\\bf T}}}$ denotes the transpose of the matrix.}\n\n{The power-law parameterization of the DM variations and red noise \nspectra means that the parameters we need to sample are \nthe amplitudes and spectral indices of the power law. We do so by using \nuniform priors in the range $[0,7]$ for the spectral index and\nlog-uniform priors for the amplitudes, in the $\\log_{10}$-range $[-20,-8]$. \nFor discussion on the impact of our prior type selection, \nsee \\citet{lah+14} and \\citet{cll+15}.\nHere, we have used the least informative priors on the noise parameters. \nThis means that the Bayesian inference will assign equal probability to \nthese parameters if the data are insufficient to break the degeneracy between\nthem. This \napproach is adequate to derive a total noise covariance matrix (addition of \nwhite noise, red noise and DM variations covariance matrices) that allows \nrobust estimation of the timing parameters. \nThe prior ranges are set\nto be wide enough to encompass any DM or red noise signal seen in the data. The\nlower bound on the spectral index of the red noise process is set to zero as we\nassume there is no blue process in the data. Together with the EFAC and EQUAD\nvalues, the DM and red noise spectral indices and amplitudes are used by the timing \nsoftware to form the timing residuals.}\n\n \n\n\\subsection{Criterion for Shapiro delay detectability}\n\\label{sec:criteria}\nTo assess the potential detectability of Shapiro delay, we used the following\ncriterion. With the orthometric parametrization of Shapiro delay, we can compute the\n amplitude $h_3$ (in seconds) in the timing residuals \\citep{fw10},\n\\begin{equation}\nh_3 = \\left( \\frac{\\sin i}{1+\\cos i} \\right)^3 m_c T_\\odot.\n\\end{equation}\nHere, $c$ is the speed of light, $ T_\\odot = 4.925~490~947$ $\\mu$s is the mass\nof the Sun in units of time. By assuming a median companion mass, $m_c$, given\nby the mass function with $m_p=1.35$ M$_\\odot$ and an inclination angle\n$i=60^\\circ$, we can predict an observable $h_{3o}$. We can then compare this\n$h_{3o}$ value to the expected precision given by $\\xi = \\delta_{\\text{TOAs}}\n{\\text{N}_{\\text{TOAs}}}^{-1\/2}$ where $ \\delta_{\\text{TOAs}}$ is the median\nuncertainty of the TOAs and ${\\text{N}_{\\text{TOAs}}}$ the number of TOAs in\nthe dataset. The criterion $h_{3o} \\gtrsim \\xi$ associated with a non detection\nof Shapiro delay would likely mean an unfavorable inclination angle, i.e. $i\n\\lesssim 60^\\circ$.\n\n \n\n\\input{paper-results}\n\n\\input{paper-discussion}\n\n\\input{paper-conclusion}\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank D.~Schnitzeler for providing us with the M2 and M3\ndistances used in this work, P.~Freire, M.~Bailes, T.~Tauris and N.~Wex for\nuseful discussions, P.~Demorest for his contribution to the pulsar\ninstrumentation at the NRT.\n\nPart of this work is based on observations with the 100-m telescope of the\nMax-Planck-Institut f\\\"ur Radioastronomie (MPIfR) at Effelsberg in Germany. Pulsar\nresearch at the Jodrell Bank Centre for Astrophysics and the observations using\nthe Lovell Telescope are supported by a consolidated grant from the STFC in the\nUK. The Nan{\\c c}ay radio observatory is operated by the Paris Observatory,\nassociated to the French Centre National de la Recherche Scientifique (CNRS).\nWe acknowledge financial support from `Programme National de Cosmologie et\nGalaxies' (PNCG) of CNRS\/INSU, France. The Westerbork Synthesis Radio Telescope\nis operated by the Netherlands Institute for Radio Astronomy (ASTRON) with\nsupport from the Netherlands Foundation for Scientific Research (NWO).\n\nCGB, GHJ, RK, KL, KJL, DP acknowledge the support from the `LEAP' ERC Advanced\nGrant (337062).\nRNC acknowledges the support of the International Max Planck Research School\nBonn\/Cologne and the Bonn-Cologne Graduate School.\nJG and AS are supported by the Royal Society.\nJWTH acknowledges funding from an NWO Vidi fellowship and CGB, JWTH acknowledge\nthe support from the ERC Starting Grant `DRAGNET' (337062).\nKJL is supported by the National Natural Science Foundation of China (Grant No.11373011).\nPL acknowledges the support of the International Max Planck Research School Bonn\/Cologne.\nCMFM was supported by a Marie Curie International Outgoing Fellowship within\nthe 7th European Community Framework Programme.\nSO is supported by the Alexander von Humboldt Foundation.\nThis research was in part supported by ST's appointment to the NASA\nPostdoctoral Program at the Jet Propulsion Laboratory, administered by Oak\nRidge Associated Universities through a contract with NASA.\nRvH is supported by NASA Einstein Fellowship grant PF3-140116.\n\nThe authors acknowledge the use of the Hydra and Hercules computing cluster from\nRechenzentrum Garching. This research has made extensive use of NASA's Astrophysics Data\nSystem, the ATNF Pulsar Catalogue and the Python Uncertainties package,\nhttp:\/\/pythonhosted.org\/uncertainties\/.\n\n\n\n\\bibliographystyle{mnras}\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}\n\nLight can exist in a chiral state, and this property makes it possible to study chiral objects, such as molecules and nanostructures, in an optical manner.\\cite{tang2011enhanced,zhao2016enantioselective,gorkunov2017enhanced,govorov2010theory,govorov2011plasmon,amabilino2009chirality} Optical detection of chirality is important in a myriad of scientific fields, with key applications in molecular biology and pharmacology.\\cite{behr2008lock,denmark2006topics,tverdislov2017periodic,kasprzyk2010pharmacologically} In common applications, the specimen is illuminated sequentially with right-handed and left-handed circularly polarized light (RCP and LCP), and the chiral information is inferred from the differential response of the material. The chiral state of circularly polarized light is well understood, which facilitates the analysis of such measurements. However, performing similar measurements at the nanoscale can be challenging~\\cite{ho2017enhancing,hanifeh2020helicity,hanifeh2020helicitymax}, because the chiral state of light in the near zone, close to a nano-object, is generally unknown and cannot be directly assessed experimentally. Without experimental tools for extracting information about the chirality of the electromagnetic field, optical examination of chirality at the nanoscale is severely complicated. \n\nThe chiral state of light is related to the curled character of the electric and magnetic fields. The \nquantity commonly used for quantifying this `degree of curliness' is the time-averaged helicity density \\cite{bliokh2013dual,cameron2012optical}, which is defined as {$h=\\frac{1}{2\\omega c}$}\\rm{Im}($\\mathbf{E}\\cdot\\mathbf{H^{\\ast}}$), where $\\mathbf{E}$ and $\\mathbf{H}$ are the phasor electric and magnetic field at angular frequency $\\omega$, and $c$ is the speed of light ($^\\ast$ denotes complex conjugation) \\cite{trueba1996electromagnetic,hanifeh2020optimally}. The helicity density is a time-even, pseudo-scalar conserved quantity that corresponds to the difference between the numbers of RCP and LCP photons \\cite{bliokh2013dual,cameron2012optical,bliokh2011characterizing}. The flux of the helicity density, i.e. the chiral momentum density, is related to the spin angular momentum of the beam \\cite{bliokh2013dual,bliokh2011characterizing}. The latter quantity is a time-odd, pseudo-vector quantity \\cite{cameron2012optical} and its density is defined as ${\\boldsymbol\\sigma}=-\\frac{\\varepsilon_0}{4\\omega}$\\rm{Im}($\\mathbf{E}\\times\\mathbf{E^{\\ast}})-\\frac{\\mu_0}{4\\omega}$\\rm{Im}($\\mathbf{H}\\times\\mathbf{H^{\\ast}}$), where $\\varepsilon_0$ and $\\mu_0$ are the absolute permittivity and permeability of free space \\cite{cameron2012optical,barnett2010rotation}, respectively. Although both quantities describe angular momentum associated with the polarization state of light\\cite{bliokh2013dual}, only the time-averaged helicity density is classified as a conserved property of the electromagnetic field \\cite{tang2010optical,tang2011enhanced}, and, therefore, considered to be the proper descriptor of optical chirality.\n\nThe question arises whether a quantity like the helicity density can be properly measured at the nanoscale. Unlike quantities such as energy, mechanical force and torque carried or performed by the electromagnetic field, properties like linear and angular momentum of light are more difficult to measure and are commonly deduced from other measurable quantities \\cite{emile2018energy}. Helicity density of light falls into the latter category, and measuring it requires a connection to a quantity that can be experimentally assessed. One possibility is to detect helicity density through a photo-induced force. For instance, the technique of photo-induced force microscopy (PiFM) has been used to map electric and magnetic field distributions through the optical force exerted on a metallic tip \\cite{zeng2018sharply} or a specially designed magnetic nanoprobe \\cite{zeng2021photoinduced}. The photo-induced force has also been used for in sorting and trapping of chiral nanoparticles\\cite{hou2021separating,kamenetskii2021chirality,hayat2015lateral,tkachenko2014optofluidic,li2019optical}. In addition, PiFM has been employed to determine enantioselective chirality of nanosamples \\cite{kamandi2017enantiospecific}, and for measuring the geometric chirality of broken symmetry structures with differential force measurements under RCP\/LCP illumination\\cite{rajaei2019giant} .\n\nIn this work, we theoretically investigate the connection between the photo-induced forces felt by a chiral tip under differential RCP\/LCP illumination and the chiral state of light at the nanoscale. Using image dipole theory, we have found a direct relation between the measured differential force and the chiral properties of the incident electromagnetic field in terms of the time-averaged helicity density and spin-angular momentum density. We further model the chiral tip as an isotropic chiral sphere to examine several design considerations and validate our dipole model with full-wave, finite element method (FEM) simulations.\n\n\n\\section{Theoretical Analysis}\n\nWe are interested in detecting the mechanical force that acts on the tip due to the presence of the electromagnetic field. We can model the tip as particle and write the time-averaged force exerted on this particle as:\\cite{novotny2012principles, yang2016resonant}\n\\begin{equation}\\label{eq:Force_general}\n \\langle\\mathbf{F}\\rangle=\\frac{1}{2}\n\\mathrm{Re}\\Big\\{\\int\\limits_{\\mathit{S}}\n{[\\varepsilon(\\mathbf{E}\\cdot\\mathbf{n})\\mathbf{E}\n+\\mu^{-1}(\\mathbf{B}\\cdot\\mathbf{n})\\mathbf{B}\n-\\frac{1}{2}(\\varepsilon|\\mathbf{E}|^2+\\mu^{-1}|\\mathbf{B}|^2)\\mathbf{n}}]\n\\mathit{dS}\\Big\\}\n\\end{equation}\nwhere the integration is over the arbitrary surface $S$ that encloses the particle, $\\mathbf{n}$ is the unit vector normal to this surface, and $\\mathbf{E}$ and $\\mathbf{B}$ are the total electric field and magnetic flux density, which include both the incident light and the scattered light contributions. It is assumed that the particle is embedded in a non-dissipative medium with permittivity {$\\varepsilon$} and permeability $\\mu$. To simplify this expression, we next assume that the tip can be described as a dipolar particle with an electric and magnetic dipole moment in free space, written as $\\mathbf{p}_{tip}~[\\rm{Cm}]$ and $\\mathbf{m}_{tip}~[\\rm{Am}^2]$, respectively. Adopting the $e^{-i\\omega t}$ time convention for the time-harmonic field, the time-averaged electromagnetic force becomes \\cite{nieto2010optical}:\n\\begin{equation}\\label{eq:dipolar_force}\n\\begin{split}\n\\langle\\mathbf{F}\\rangle=\\frac{1}{2}\\mathrm{Re}\\Big\\{\n\\mathbf{p}_{tip}\\cdot\\big(\\nabla\\mathbf{E}^{loc}(\\mathbf{r}_{tip})\\big)^\\ast\n+\\mu_0\\mathbf{m}_{tip}\\cdot\\big(\\nabla\\mathbf{H}^{loc}(\\mathbf{r}_{tip})\\big)^\\ast \\\\\n-\\frac{c\\mu_0 k^4}{6\\pi}(\\mathbf{p}_{tip}\\times\\mathbf{m}^{\\ast}_{tip})\\Big\\}\n\\end{split}\n\\end{equation}\nwhere $\\mathbf{E}^{loc}(\\mathbf{r}_{tip})$ and $\\mathbf{H}^{loc}(\\mathbf{r}_{tip})$ are the local electric and magnetic field at the tip dipole position. The time-averaged force has three distinct contributions: the first term on the right hand side of equation (\\ref{eq:dipolar_force}) is recognized as the electric dipolar force $\\langle\\mathbf{F}_{e}\\rangle$, the second term is known as the magnetic dipolar force $\\langle\\mathbf{F}_{m}\\rangle$, and the last term is called the interaction force $\\langle\\mathbf{F}_{int}\\rangle$. \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=10cm]{Fig1.eps}\n\\caption{(a) Schematic of the PiFM system with a chiral with illumination from the bottom. (b) Photo-induced chiral tip dipole and image dipole according to image dipole theory. }\n\\label{fig:fig1}\n\\end{figure}\n\nTo calculate the force, we require expressions for the induced electric and magnetic dipole moments. Assuming that the tip is an isotropic and reciprocal chiral object, the dipole moments can be related to the local fields through polarizabilities as \\cite{A}:\n\\begin{align}\n \\mathbf{p}_{tip}&=\\alpha_{ee}\\mathbf{E}^{loc}(\\mathbf{r}_{tip})+\\alpha_{em}\\mathbf{H}^{loc}(\\mathbf{r}_{tip})\\label{eq:electric_dipole_tip}\\\\\n \\mathbf{m}_{tip}&=-\\mu^{-1}_0\\alpha_{em}\\mathbf{E}^{loc}(\\mathbf{r}_{tip})+\\alpha_{mm}\\mathbf{H}^{loc}(\\mathbf{r}_{tip})\\label{eq:magnetic_dipole_tip}\n\\end{align}\nwhere $\\alpha_{ee}$, $\\alpha_{mm}$ and $\\alpha_{em}$ are the electric, magnetic and electro-magnetic polarizabilities, respectively. The local fields at the tip dipole position can be obtained from image dipole theory \\cite{novotny2012principles, jackson2009classical,rajapaksa2010image}. We model the effect of the substrate by substituting the substrate response by the image of the photo-induced (chiral) dipole in the tip, written as $\\mathbf{p}_{img}$ and $\\mathbf{m}_{img}$ and shown in Figure \\ref{fig:fig1}(b). The local electric and magnetic fields at the tip position can then be expressed as:\n\\begin{align}\n \\mathbf{E}^{loc}(\\mathbf{r}_{tip})&=\\mathbf{E}^{inc}(\\mathbf{r}_{tip})+\\mathbf{E}^{sca}_{img\\rightarrow tip}(\\mathbf{r}_{tip})\\label{eq:E_loc}\\\\\n \\mathbf{H}^{loc}(\\mathbf{r}_{tip})&=\\mathbf{H}^{inc}(\\mathbf{r}_{tip})+\\mathbf{H}^{sca}_{img\\rightarrow tip}(\\mathbf{r}_{tip})\\label{eq:H_loc}\n\\end{align}\nwhere $\\mathbf{E}^{inc}(\\mathbf{r}_{tip})$, $\\mathbf{H}^{inc}(\\mathbf{r}_{tip})$ are the incident electric and magnetic field at the tip dipole location, and $\\mathbf{E}^{sca}_{img\\rightarrow tip}(\\mathbf{r}_{tip})$, $\\mathbf{H}^{sca}_{img\\rightarrow tip}(\\mathbf{r}_{tip})$ are the scattered electric and magnetic field by the image dipoles at the tip dipole location.\n\nThe scattered fields are defined through the image dipole moments and their Green's functions as: \\cite{novotny2012principles,jackson2009classical} \n\\begin{align}\n \\mathbf{E}^{sca}_{img\\rightarrow tip}(\\mathbf{r}_{tip})&=\\mathbf{\\underline{G}}^{ee}(\\mathbf{r}_{tip}-\\mathbf{r}_{img})\\cdot\\mathbf{p}_{img}+\\mathbf{\\underline{G}}^{em}(\\mathbf{r}_{tip}-\\mathbf{r}_{img})\\cdot\\mathbf{m}_{img}\\\\\n \\mathbf{H}^{sca}_{img\\rightarrow tip}(\\mathbf{r}_{tip})&=\\mathbf{\\underline{G}}^{me}(\\mathbf{r}_{tip}-\\mathbf{r}_{img})\\cdot\\mathbf{p}_{img}+\\mathbf{\\underline{G}}^{mm}(\\mathbf{r}_{tip}-\\mathbf{r}_{img})\\cdot\\mathbf{m}_{img}\n\\end{align}\nwhere $\\mathbf{\\underline{G}}(\\mathbf{r})=G_x\\mathbf{\\hat{x}\\hat{x}}+G_y\\mathbf{\\hat{y}\\hat{y}}+G_z\\mathbf{\\hat{z}\\hat{z}}$ is the dyadic Green's function in Cartesian coordinates and the four different electric [$\\mathbf{\\underline{G}}^{ee}(\\mathbf{r})$, $\\mathbf{\\underline{G}}^{em}(\\mathbf{r})$] and magnetic [$\\mathbf{\\underline{G}}^{me}(\\mathbf{r})$, $\\mathbf{\\underline{G}}^{mm}(\\mathbf{r})$] dyadic Green's functions for an electric and magnetic dipole are defined as:\n\\cite{campione2013effective} \n\\begin{align}\n \\mathbf{\\underline{G}}^{ee}(\\mathbf{r})&=\\frac{1}{4\\pi\\varepsilon_0|\\mathbf{r}|^{3}}(3\\mathbf{\\hat{r}}\\mathbf{\\hat{r}}-\\mathbf{\\underline{I}})\\\\\n \\mathbf{\\underline{G}}^{mm}(\\mathbf{r})&=\\frac{1}{4\\pi|\\mathbf{r}|^{3}}(3\\mathbf{\\hat{r}}\\mathbf{\\hat{r}}-\\mathbf{\\underline{I}})\\\\\n \\mathbf{\\underline{G}}^{em}(\\mathbf{r})&=-\\frac{1}{i\\omega\\varepsilon_0}\\nabla\\times\\mathbf{\\underline{G}}^{mm}(\\mathbf{r})\\\\\n \\mathbf{\\underline{G}}^{me}(\\mathbf{r})&=\\frac{1}{i\\omega\\mu_0}\\nabla\\times\\mathbf{\\underline{G}}^{ee}(\\mathbf{r})\n\\end{align}\nHere $\\mathbf{\\underline{I}}$ is a unit tensor of rank 2 and $\\mathbf{\\hat{r}}$ is the unit vector from the source point to the point of observation.\n\nIn the same fashion, we can also define the image dipole moments and the local fields at the image location. Our goal is to obtain self-consistent expressions for the local fields at tip location and for the dipole moments at the time in terms of the incident light components. To achieve this, we first ignore any phase retardation effects in the fields between the tip and image dipole position. Second, we will only consider terms in the polarizability up to second order. With these assumptions, we may use equations (\\ref{eq:electric_dipole_tip})-(\\ref{eq:H_loc}) to determine the longitudinal ($z$) component of the time-averaged force in the dipole approximation as:\n\\begin{equation}\n\\begin{split}\n\\langle{F_{z,e}}\\rangle+\\langle{F_{z,m}}\\rangle=~&\\frac{3}{2\\pi|\\mathit{z}|^4}\\Bigg[\\frac{1}{2}\\Big(\\frac{|\\alpha_{ee}|^2}{\\varepsilon_0}+\\mu_0|\\mu^{-1}_0\\alpha_{em}|^2\\Big)\\Big(\\frac{1}{2}|\\mathbf{E}^{inc}_\\parallel|^2-|\\mathbf{E}^{inc}_z|^2\\Big)\\\\\n&+\\frac{1}{2}\\Big(\\frac{|\\alpha_{em}|^2}{\\varepsilon_0}+\\mu_0|\\alpha_{mm}|^2\\Big)\\Big(\\frac{1}{2}|\\mathbf{H}^{inc}_\\parallel|^2-|\\mathbf{H}^{inc}_z|^2\\Big)\\\\\n&+\\mathrm{Re}\\Big\\{\\frac{\\alpha_{ee}\\alpha^{\\ast}_{em}}{\\varepsilon_0}-\\alpha_{mm}\\alpha^{\\ast}_{em}\\Big\\}\n\\mathrm{Re}\\Big\\{\\frac{1}{2}\\mathbf{E}^{inc}_{\\parallel}\\cdot\\mathbf{H}^{inc\\ast}_{\\parallel}-\\mathbf{E}^{inc}_{z}\\cdot\\mathbf{H}^{inc\\ast}_{z}\\Big\\}\\\\\n&-\\mathrm{Im}\\Big\\{\\frac{\\alpha_{ee}\\alpha^{\\ast}_{em}}{\\varepsilon_0}-\\alpha_{mm}\\alpha^{\\ast}_{em}\\Big\\}\n\\mathrm{Im}\\Big\\{\\frac{1}{2}\\mathbf{E}^{inc}_{\\parallel}\\cdot\\mathbf{H}^{inc\\ast}_{\\parallel}-\\mathbf{E}^{inc}_{z}\\cdot\\mathbf{H}^{inc\\ast}_{z}\\Big\\}\\Bigg]\n\\end{split}\n\\label{eq:force_em}\n\\end{equation}\n\n\\begin{equation}\n\\begin{split}\n\\langle{F_{z,int}}\\rangle=~&-\\frac{ck^{4}}{12\\pi}\\Bigg[\\mathrm{Re}\\Big\\{-\\alpha^{\\ast}_{em}\\alpha_{ee}\\Big\\}\\mathrm{Re}\\Big\\{[\\mathbf{E}^{inc}_{\\parallel}\\times\\mathbf{E}^{inc\\ast}_{\\parallel}]_{z}\\Big\\}\\\\\n&+\\mathrm{Re}\\Big\\{\\mu_0\\alpha^{\\ast}_{mm}\\alpha_{em}\\Big\\}\\mathrm{Re}\\Big\\{[\\mathbf{H}^{inc}_{\\parallel}\\times\\mathbf{H}^{inc\\ast}_{\\parallel}]_{z}\\Big\\}\\\\\n&+4\\omega\\mathrm{Im}\\Big\\{-\\frac{\\alpha^{\\ast}_{em}\\alpha_{ee}}{\\varepsilon_0}\\Big\\}[\\boldsymbol{\\sigma}^{inc}_{E}]_{z}+4\\omega\\mathrm{Im}\\Big\\{-\\alpha^{\\ast}_{mm}\\alpha_{em}\\Big\\}[\\boldsymbol{\\sigma}^{inc}_{H}]_{z}\\\\\n&+\\mathrm{Re}\\Big\\{\\alpha^{\\ast}_{em}\\alpha_{em}[\\mathbf{S}^{inc\\ast}]_{z}\\Big\\}+\\mathrm{Re}\\Big\\{\\mu_o\\alpha^{\\ast}_{mm}\\alpha_{ee}[\\mathbf{S}^{inc}]_{z}\\Big\\}\\Bigg]\n\\end{split}\n\\label{eq:force_int}\n\\end{equation}\nwhere $\\mathbf{E}^{inc}_{\\parallel}=\\mathit{E}^{inc}_x\\mathbf{\\hat{x}}+\\mathit{E}^{inc}_y\\mathbf{\\hat{y}}$,~$\\mathbf{H}^{inc}_{\\parallel}=\\mathit{H}^{inc}_x\\mathbf{\\hat{x}}+\\mathit{H}^{inc}_y\\mathbf{\\hat{y}}$ and $\\mathbf{E}^{inc}_{z}=\\mathit{E}^{inc}_z\\mathbf{\\hat{z}}$, $\\mathbf{H}^{inc}_{z}=\\mathit{H}^{inc}_z\\mathbf{\\hat{z}}$ are the transverse and longitudinal components of the incident fields at the tip dipole location, the notation $(\\mathbf{r}_{tip})$ has been avoided here for brevity; $\\mathbf{S}^{inc}=\\mathbf{E}^{inc}\\times\\mathbf{H}^{inc\\ast}$ is the time-averaged Poynting vector of the incident field; $\\boldsymbol{\\sigma}^{inc}_{E}$ and $\\boldsymbol{\\sigma}^{inc}_{H}$ are the electric and magnetic parts of the time-averaged total spin angular momentum density of the incident light; and $z$ is the vertical distance between the tip dipole and its image. See the Supporting Information for a detailed derivation of equations (\\ref{eq:force_em}) and (\\ref{eq:force_int}).\n\nEquation (\\ref{eq:force_em}) describes the force due to the combined electric and magnetic dipolar response of the tip. This force shows a distance dependence on the tip-sample distance which scales as $\\mathit{z}^{-4}$, similar to the distance dependence of the gradient force that is typically measured in PiFM \\cite{novotny2012principles,jahng2014gradient}. The first two lines of equation (\\ref{eq:force_em}) are recognized as the purely electric and purely magnetic dipolar force contributions to the gradient force, whereas the latter two lines describe the forces that arise from a nonzero scalar product of the electric and magnetic fields, i.e. the helicity density.\n\nEquation (\\ref{eq:force_int}) accounts for the interaction force, which lacks a direct dependence on the tip-sample distance. This contribution to the force resembles the scattering force \\cite{novotny2012principles,jahng2014gradient}. In addition to the purely electric and magnetic contributions to the scattering force, described by the first two lines in equation (\\ref{eq:force_int}), the latter two lines add force contributions that scale with the longitudinal component the spin angular momentum density and with the time-averaged Poynting vector. \n\nWe are interested in finding an expression for the differential force, i.e the force difference between sequential measurements where the tip-sample junction is illuminated with incident light of opposite handedness. For this purpose, we assume two states for the incident light, indicated by $\\mathbf{E}^{inc+}$, $\\mathbf{H}^{inc+}$ and $\\mathbf{E}^{inc-}$, $\\mathbf{H}^{inc-}$, describing the input fields of different handedness. The two illumination states have the same energy densities, $|\\mathbf{E}^{inc+}|^2=|\\mathbf{E}^{inc-}|^2$ and $|\\mathbf{H}^{inc+}|^2=|\\mathbf{H}^{inc-}|^2$, but exhibit opposite helicity densities, $\\mathrm{Im}(\\mathbf{E}^{inc+}\\cdot\\mathbf{H}^{inc+\\ast})=-\\mathrm{Im}(\\mathbf{E}^{inc-}\\cdot\\mathbf{H}^{inc-\\ast})$. Note that the longitudinal spin angular momentum density components are related as $[\\boldsymbol{\\sigma}^{inc+}_{E}]_z=-[\\boldsymbol{\\sigma}^{inc-}_{E}]_z$ and $[\\boldsymbol{\\sigma}^{inc+}_{H}]_z=-[\\boldsymbol{\\sigma}^{inc-}_{H}]_z$. We also find that $\\mathbf{S}^{inc+}=\\mathbf{S}^{inc-}$, and that the following relations hold:\n\\begin{equation}\n \\begin{split}\n \\mathrm{Re}\\Big\\{[\\mathbf{E}^{inc+}_{\\parallel}\\times\\mathbf{E}^{inc+\\ast}_{\\parallel}]_{z}\\Big\\}&=\\mathrm{Re}\\Big\\{[\\mathbf{E}^{inc-}_{\\parallel}\\times\\mathbf{E}^{inc-\\ast}_{\\parallel}]_{z}\\Big\\}\\\\\n \\mathrm{Re}\\Big\\{[\\mathbf{H}^{inc+}_{\\parallel}\\times\\mathbf{H}^{inc+\\ast}_{\\parallel}]_{z}\\Big\\}&=\\mathrm{Re}\\Big\\{[\\mathbf{H}^{inc-}_{\\parallel}\\times\\mathbf{H}^{inc-\\ast}_{\\parallel}]_{z}\\Big\\}\n \\end{split}\n\\end{equation}\nBy measuring the force under illumination with incident light of $(+)$ and $(-)$ handedness, and taking the difference, the differential gradient force $\\Delta\\langle{F_{z,grad}}\\rangle$ and the differential scattering force $\\Delta\\langle{F_{z,scat}}\\rangle$ can be obtained from equations (\\ref{eq:force_em}) and (\\ref{eq:force_int}) as:\n\\begin{align}\n\\Delta\\langle{F_{z,grad}}\\rangle&=-\\frac{6\\omega c}{\\pi|\\mathit{z}|^4}\n\\mathrm{Im}\\Big\\{\\frac{\\alpha_{ee}\\alpha^{\\ast}_{em}}{\\varepsilon_o}-\\alpha_{mm}\\alpha^{\\ast}_{em}\\Big\\}\n\\Big(\\frac{1}{2}\\mathit{h}^{inc}_{\\parallel}-\\mathit{h}^{inc}_{z}\\Big)\\label{eq:force_grad}\\\\\n\\Delta\\langle{F_{z,scat}}\\rangle&=-\\frac{2\\omega ck^4}{3\\pi}\\Big[\n\\mathrm{Im}\\Big\\{-\\frac{\\alpha^{\\ast}_{em}\\alpha_{ee}}{\\varepsilon_o}\\Big\\}[\\boldsymbol{\\sigma}^{inc}_{E}]_{z}+\\mathrm{Im}\\Big\\{-\\alpha^{\\ast}_{mm}\\alpha_{em}\\Big\\}[\\boldsymbol{\\sigma}^{inc}_{H}]_{z}\\Big]\n\\label{eq:force_scat}\n\\end{align} \nwhere $\\mathit{h}^{inc}_{\\parallel}=\\frac{1}{2\\omega c}\\mathrm{Im}\\left\\{\\mathbf{E}^{inc}_{\\parallel}\\cdot\\mathbf{H}^{inc\\ast}_{\\parallel}\\right\\}$ and $\\mathit{h}^{inc}_{z}=\\frac{1}{2\\omega c}\\mathrm{Im}\\left\\{\\mathbf{E}^{inc}_{z}\\cdot\\mathbf{H}^{inc\\ast}_{z}\\right\\}$ are the tangential and longitudinal helicity density of the incident chiral light. We see that the differential gradient force depends on the helicity density of the incident beam, whereas the differential scattering force contains qualitative information about the spin angular momentum. For small tip-sample distances, the differential gradient force is expected to form the dominant contribution to the measured force, thus offering a means of experimentally extracting information about the helicity density. Note that the differential force in equation (\\ref{eq:force_grad}) depends on the difference between the helicity density of the transverse and longitudinal components of the incident field. \n\n\n\\section{Mie Scattering Formalism}\n\nWe next study the characteristics of the differential gradient force. We see from equation (\\ref{eq:force_grad}) that $\\Delta\\langle{F_{z,grad}}\\rangle$ depends on the polarizability of the tip material, including the electro-magnetic polarizability. To model the tip polarizability, we assume that the tip can be described as a spherical chiral nanoparticle (NP). We also assume that the following constitutive relations hold: $\\mathbf{D}=\\varepsilon_o\\varepsilon\\mathbf{E}+i\\sqrt{\\varepsilon_0\\mu_0}\\kappa\\mathbf{H}$ and $\\mathbf{B}=\\mu_0\\mu\\mathbf{H}-i\\sqrt{\\varepsilon_0\\mu_0}\\kappa\\mathbf{E}$, where $\\varepsilon$ and $\\mu$ are the relative permittivity and permeability, respectively.\\cite{A} The chirality parameter $\\kappa$ is an empirical quantity that provides the chiral strength of the material under consideration. We can next relate the electric, magnetic and electro-magnetic polarizabilities through the material parameters by using Mie scattering theory as:\\cite{kamandi2017enantiospecific,bohren2012univ,hanifeh2020optimally} $\\alpha_{ee}=-{6\\pi i \\varepsilon_0 b_1}\/{k^3_0}$, $\\alpha_{mm}=-{6\\pi i a_1}\/{k^3_0}$ and $\\alpha_{em}={6\\pi i c_1}\/{(c k^3_0)}$ where $c$ is the free space speed of light, $k_0$ is the wavenumber in free space and $b_1$, $a_1$, $c_1$ are the Mie coefficients. In our calculations, the material parameters of the isotropic chiral NP such as $\\varepsilon$ and $\\mu$ are those of silicon\\cite{aspnes1983dielectric} with $\\mu=1$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=6cm]{Fig2.eps}\n\\caption{(a) The total difference force spectrum in pN. (b) The normalized magnitude spectrum of $\\alpha_{ee}-\\varepsilon_0\\alpha_{mm}$. In all calculations, the tip-image dipole distance is $|z|=10~\\rm{nm}$. Black lines indicate resonance condition of $c_1$ and white lines indicate minima due to satisfying the Kerker condition. }\n\\label{fig:fig2}\n\\end{figure}\n\nWe next place the sphere above a flat and transparent dielectric surface at a tip-surface distance of 5~nm. The system is subsequently illuminated with a plane wave of field strength $1.5\\times10^6 ~\\rm{Vm}^{-1}$ that is either in the RCP or LCP state, and which propagates through the transparent dielectric material toward the sphere. The wave induces a polarization in the sphere, which in turn produces an image dipole in the dielectric material. For simplicity, it is assumed that the image dipole strength is identical to the induced dipole in the sphere. In the current configuration, the image dipole is found at a distance $z=10~\\rm{nm}$ from the outer diameter of the sphere. We calculate the total difference force spectrum with the aid of equations (\\ref{eq:force_grad}) and (\\ref{eq:force_scat}), defined as $\\Delta\\langle{F_{z,total}}\\rangle=\\Delta\\langle{F_{z,grad}}\\rangle+\\Delta\\langle{F_{z,scat}}\\rangle$, and plotted in \\ref{fig:fig2}(a) as a function of sphere radius and excitation wavelength. In all calculations, the value of $\\kappa$ is set to 0.1.\\cite{ali2020enantioselective,ali2020probing} It is clear that nonzero $\\Delta\\langle{F_{z,total}}\\rangle$ is achieved under certain experimental conditions. In particular, the maxima and minima are seen to co-localize with the resonant spectral position of the electro-magnetic Mie coefficient $c_1$ of the chiral NP. The black dotted lines show the location of the peak values of $\\mathrm{Im}(c_1)$. Changing $\\kappa$ does not alter the spectral resonances, but instead changes the sign and magnitude of the total differential force proportionally. \n\nA careful inspection of equation \\ref{eq:force_grad} suggests that, for any chosen $r$ and $\\kappa$ of the tip, $\\Delta\\langle{F_{z,total}}\\rangle$ may approach zero when the first Kerker condition of the chiral NP is met, irrespective of helicity density of the incident light. The Kerker condition states that $\\alpha_{ee}=\\varepsilon_0\\alpha_{mm}$~\\cite{kerker1983electromagnetic,lee2017reexamination,hanifeh2020helicity}, and this condition is plotted in Figure \\ref{fig:fig2}(b) as a function of radius and excitation wavelength, while $\\kappa$ is fixed at $0.1$. The white dotted lines depict the conditions where the relation $\\alpha_{ee}=\\varepsilon_0\\alpha_{mm}$ is satisfied whereas the black dotted lines show the resonant position of electro-magnetic Mie coefficient $c_1$ as found in panel (a). The finite separation between the two curves assures that the above relation is less likely to have any impact on helicity density measurements if the force difference measurement is properly maximized. \n\n\\section{Full Wave Simulations}\nTo further validate our analytical findings, we perform 3D full wave simulations to determine the forces between the dipole induced in the chiral sphere and its image in the dielectric material below. We use the finite element method in COMSOL Multiphysics for this purpose. In the simulation, the tip and image dipoles are modeled as isotropic chiral spheres \\cite{yang2016resonant, nieto2004near,PhysRevLett.99.127401} with similar parameters as used above in Figure \\ref{fig:fig2}(a). The system is sequentially illuminated by RCP and LCP plane waves of field strength $1.5\\times10^6~\\rm{Vm}^{-1}$ in the bottom illumination scheme. The force exerted on the tip is determined using Maxwell's stress tensor formalism. \n\nFigures \\ref{fig:fig3}(b), (c) and (d) show the time averaged-force difference spectrum for spheres of radius of 50~nm, 60~nm and 70~nm. The black curve shows the analytical result whereas the full wave FEM result is indicated by the green solid line. For $r=50~\\rm{nm}$, the FEM simulation closely corroborates the analytical result obtained in the dipole approximation, where the dip corresponds to the Mie resonance of the $c_1$ coefficient. The spectra of Figure \\ref{fig:fig4}(c) and (d) display some dissimilarities between the FEM and analytical calculations, which can be attributed to higher order multipole contributions in the lower wavelength range. Yet, in the range examined the dipolar contribution remains dominant and provides a maximum force difference value $-0.32~\\rm{pN}$ for a tip radius of $r_{NP}=70~\\rm{nm}$ when $\\kappa=0.1$. Near resonance, the differential force is of the order of 0.1 pN. Although such forces are near the noise floor of a typical PiFM microscope, sensitive experiments are likely able to resolve the targeted differental force under optimized conditions.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=10cm]{Fig3.eps}\n\\caption{(a) Schematic of chiral tip sphere and image sphere with radius $r_{NP}$ and tip-image dipole distance of $|z|=10~\\rm{nm}$ used in FEM simulation. Time-averaged differential force from analytical calculation (black solid curve) and from full wave simulation (green solid curve) for a chiral NP with radius (b) $r_{NP}=50~\\rm{nm}$, (c) $r_{NP}=60~\\rm{nm}$, (d) $r_{NP}=70~\\rm{nm}$ and $\\kappa=0.1$ under RCP\/LCP illumination. }\n\\label{fig:fig3}\n\\end{figure}\n\nFinally, we present the differential force map of the helicity density for a focused chiral beam, a scenario of direct relevance to microscopy applications. We assume a circularly polarized beam of $2~\\rm{mW}$ average power at a wavelength of $\\lambda=520\\rm{nm}$ as the incident light source, which is focused by a 1.4 NA oil~($n=1.518$) objective lens. The focal plane electric and magnetic field field components used in the simulation are obtained from \\cite{novotny2012principles}. We first calculate the normalized helicity density distribution at the focal plane for a focused LCP beam, which is shown in \\ref{fig:fig4}(a). As shown in \\ref{fig:fig4}(b) and (c), spatial distribution of the the transverse and longitudinal components of helicity density resemble a circularly and doughnut-shaped profile, respectively. We next calculate the differential force. In Figure \\ref{fig:fig4}(d), $\\Delta\\langle{F_{z,total}}\\rangle$ is shown for the case of a tip of radius $r_{NP}=60~\\rm{nm}$. The central region of the beam is characterized by a negative differential force, surrounded by region where the differential force is positive. We can understand this by realizing that the transverse and longitudinal helicity density contributions have different spatial profiles with opposite signs. We can further separate the transverse and longitudinal contributions by proper design of the tip's chiral properties. For example, if the chiral tip is designed such that it exhibits only longitudinal chirality, i.e. $\\alpha_{em,xx}=\\alpha_{em,yy}=0$ and $\\alpha_{em,zz}=\\alpha_{em}$, then the differential force map will track the doughnut-shaped longitudinal helicity density as shown in \\ref{fig:fig4}(e). Conversely, a chiral tip having solely transverse chirality with $\\alpha_{em,xx}=\\alpha_{em,yy}=\\alpha_{em}$ and $\\alpha_{em,zz}=0$ produces a map that reports on the central transverse helicity density, as shown in \\ref{fig:fig4}(f). In the supporting information, we also present the helicity density maps determined with the differential force of a focused azimuthally radially polarized beam (ARPB)\\cite{kamandi2018unscrambling,hanifeh2020optimally,novotny2012principles}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=10cm]{Fig4.eps}\n\\caption{(a) The focal plane distribution of normalized helicity density of a circularly polarized(LCP) light focused by a 1.4 NA oil objective. Distributions of (b) longitudinal and (c) transverse component of the normalized helicity density in (a).(d)The total force difference map at the focal plane for a chiral isotropic tip of radius $r_{NP}=60~\\rm{nm}$ under the same focused illumination of LCP and RCP. Force difference maps for a (b)longitudinally chiral($\\alpha_{em,xx}=\\alpha_{em,yy}=0$ and $\\alpha_{em,zz}=\\alpha_{em}$) and (c)transversely chiral($\\alpha_{em,xx}=\\alpha_{em,yy}=\\alpha_{em}$ and $\\alpha_{em,zz}=0$) tip tracking the respective helicity densities with opposite sign. }\n\\label{fig:fig4}\n\\end{figure}\n\n\\section{Conclusion}\nIn this work, we have studied the information contained in the photo-induced force when illuminated by chiral light. Our theoretical analysis reveals that the differential force is directly sensitive to the chiral properties of light. In particular, the differential gradient force is proportional to the helicity density of the incident light field, whereas the differential scattering force is sensitive to the spin angular momentum of the applied light. Using realistic values for the illumination intensity, tip dimension, and the chirality parameter of the tip, we find that the differential force can reach detectable values of several hundreds of fN, just above the noise floor of common scan probe microscopy systems. These findings are significant because a direct characterization of optical chirality at the nanoscale has hitherto been extremely challenging. The observation that the differential gradient force can map out the local helicity density of the light is relevant for numerous applications where knowledge of the chiral state of light at sub-diffraction-limited dimensions is important.\n\n\n\\begin{acknowledgements}\n\nThe authors thank the Keck Foundation and the National Science Foundation, grant CMMI-1905582.\n\n\\end{acknowledgements}\n\n\\section*{Supplementary Information}\n\nDetailed derivations are available in the Supporting Information. Please contact the corresponding author for a copy.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzltps b/data_all_eng_slimpj/shuffled/split2/finalzzltps new file mode 100644 index 0000000000000000000000000000000000000000..834b554bce7527568c7a1769f9fb7c6a2c09c257 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzltps @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\subsection{Statement of the problem}\n Our objective is to study a fluid structure interaction problem in a 2d channel. The fluid flow here is modeled by the compressible Navier-Stokes equations. Concerning the structure we will consider an Euler-Bernoulli damped beam located on a portion of the boundary. As remarked in \\cite{gavalos1}, such dynamical models arise in the study of many engineering systems (e.g., aircraft, bridges etc). In the present article we establish a result on the local in time existence of strong solutions of such a fluid structure interaction problem. To the best of our knowledge, this is the first article dealing with the existence of local in time strong solutions for the complete non-linear model considered here.\\\\\n We consider data and solutions which are periodic in the \\textquoteleft channel direction\\textquoteright\\, (with period $L,$ where $L>0$ is a constant). Here $L$-periodicity of a function $f$ (defined on $\\mathbb{R}$) means that $f(x+L)=f(x)$ for all $x\\in \\mathbb{R}.$\\\\ \nWe now define a few notations. Let $\\Omega$ be the domain $\\mathbb{T}_{L}\\times (0,1)\\subset {\\mathbb{R}}^{2},$ where $\\mathbb{T}_{L}$ is the one dimensional torus identified with $(0,L)$ with periodic conditions. The boundary of $\\Omega$ is denoted by $\\Gamma$. We set \n \\begin{equation}\\nonumber\n \\begin{array}{l}\n \\Gamma_{s}=\\mathbb{T}_{L}\\times \\{1\\},\n \\quad\n \\Gamma_{\\ell}=\\mathbb{T}_{L}\\times\\{0\\},\\quad\\Gamma=\\Gamma_{s}\\cup\\Gamma_{\\ell}.\n \\end{array}\n \\end{equation} \nNow for a given function $$\\eta : \\Gamma_{s}\\times (0,\\infty)\\rightarrow (-1,\\infty),$$ which will correspond to the displacement of the one dimensional beam, let us denote by $\\Omega_{t}$ and $\\Gamma_{s,t}$ the following sets\n\\begin{equation}\\nonumber\n\\begin{array}{ll}\n\\Omega_{t}=\\{(x,y) \\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; x\\in (0,L),\\quad 0, color=red, thick] (3.2,5) -- (3.2,6);\n \t\\end{tikzpicture}\n \t\\caption{Domain $\\Omega_{t}$.}\n \\end{figure} \\\\ \n We consider a fluid with density $\\rho$ and velocity ${\\bf u}.$ The fluid structure interaction system coupling the compressible Navier-Stokes and the Euler-Bernoulli damped beam equation is modeled by\n \\begin{equation}\\label{1.1}\n \\left\\{\n \\begin{array}{ll}\n \\rho_{t}+\\mbox{div}(\\rho {\\bf u})=0\\quad &\\mbox{in} \\quad \\widetilde{Q_{T}},\n \\vspace{1.mm}\\\\\n (\\rho {\\bf u}_{t}+\\rho({\\bf u}.\\nabla){\\bf u})-(2\\mu \\mbox{div} (D({\\bf u}))+\\mu{'}\\nabla\\mbox{div}{\\bf u}) +\\nabla p(\\rho) =0\\quad &\\mbox{in} \\quad \\widetilde{Q_T},\n \\vspace{1.mm}\\\\\n {\\bf u}(\\cdot,t)=(0,\\eta_{t})\\quad & \\mbox{on}\\quad \\widetilde{\\Sigma^{s}_{T}},\n \\vspace{1.mm}\\\\\n {\\bf u}(\\cdot,t)=(0,0)\\quad &\\mbox{on}\\quad \\Sigma^{\\ell}_{T},\n \\vspace{1.mm}\\\\\n {\\bf u}(\\cdot,0)={\\bf u}_{0}\\quad& \\mbox{in} \\quad \\Omega_{\\eta(0)}=\\Omega,\n \\vspace{1.mm}\\\\\n \\rho(\\cdot,0)=\\rho_{0}\\quad &\\mbox{in}\\quad \\Omega_{\\eta(0)}=\\Omega,\n \\vspace{1.mm}\\\\\n \\eta_{tt}-\\beta \\eta_{xx}- \\delta\\eta_{txx}+\\alpha\\eta_{xxxx}=(T_{f})_{2} \\quad& \\mbox{on}\\quad \\Sigma^{s}_{T},\n \\vspace{1.mm}\\\\\n \\eta(\\cdot,0)=0\\quad \\mbox{and}\\quad \\eta_{t}(\\cdot,0)=\\eta_{1}\\quad &\\mbox{in}\\quad \\Gamma_{s}.\n \\end{array} \\right.\n \\end{equation} \n The initial condition for the density is assumed to be positive and bounded. We fix the positive constants $m$ and $M$ such that\n \\begin{equation}\\label{cor0}\n \\begin{array}{l}\n 00,\\quad {\\mu}{'}\\geqslant 0.$$ \n In our case the fluid is isentropic i.e. the pressure $p(\\rho)$ is only a function of the fluid density $\\rho$ and is given by\n $$p(\\rho)=a\\rho^{\\gamma},$$\n where $a>0$ and $\\gamma>1$ are positive constants.\\\\ \n We assume that there exists a constant external force ${p_{ext}}>0$ which acts on the beam. The external force ${p_{ext}}$ can be written as follows\n $${p_{ext}}=a\\overline{\\rho}^{\\gamma},$$\n for some positive constant $\\overline{\\rho}.$\\\\\n To incorporate this external forcing term ${p_{ext}}$ into the system of equations \\eqref{1.1}, we introduce the following \n \\begin{equation}\\label{1.2}\n P(\\rho)=p(\\rho)-{p_{ext}}=a\\rho^{\\gamma}-a\\overline{\\rho}^{\\gamma}.\n \\end{equation}\n Since $\\nabla p(\\rho)=\\nabla P(\\rho),$ from now onwards we will use $\\nabla P(\\rho)$ instead of $\\nabla p(\\rho)$ in the equation \\eqref{1.1}$_{2}.$\\\\\n In the beam equation the constants, $\\alpha>0,$ $\\beta\\geqslant0$ and $\\delta>0$ are respectively the adimensional rigidity, stretching and friction coefficients of the beam. The non-homogeneous source term of the beam equation $(T_{f})_{2}$ is the net surface force on the structure which is the resultant of force exerted by the fluid on the structure and the external force ${p_{ext}}$ and it is assumed to be of the following form\n\\begin{equation}\\label{1.3}\n(T_{f})_{2}=([-2\\mu D({\\bf u})-\\mu{'}(\\mbox{div}{\\bf u}){\\bf I}_{d}]\\cdot {{\\bf n}_{t}}+P{{\\bf n}_{t}})\\mid_{\\Gamma_{s,t}}\\sqrt{1+\\eta^{2}_{x}}\\cdot \\vec{e}_{2}\\quad\\mbox{on}\\quad \\Sigma^{s}_{T},\n\\end{equation}\nwhere ${\\bf I}_{d}$ is the identity matrix, ${\\bf n}_{t}$ is the outward unit normal to $\\Gamma_{s,t}$ given by\n$${\\bf n}_{t}=-\\frac{\\eta_{x}}{\\sqrt{1+\\eta^{2}_{x}}}\\vec{e}_{1}+\\frac{1}{\\sqrt{1+\\eta^{2}_{x}}}\\vec{e}_{2}$$\n($\\vec{e}_{1}=(1,0)$ and $\\vec{e}_{2}=(0,1)$).\\\\\nObserve that $(\\rho,{\\bf u},\\eta)=(\\overline{\\rho},0,0)$ is a stationary solution to \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3}.\n\\begin{remark}\n\tNow we can formally derive a priori estimates for the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} and show the following energy equality \n\t\\begin{equation}\\label{1.15l}\n\t\\begin{split}\n\t&\\frac{1}{2}\\frac{d}{dt}\\left(\\int\\limits_{\\Omega_{t}}\\rho|{\\bf u}|^{2}\\,dx\\right)+\\frac{d}{dt}\\left(\\int\\limits_{\\Omega_{t}}\\frac{a}{(\\gamma-1)}\\rho^{\\gamma}\\,dx \\right) +\\frac{1}{2}\\frac{d}{dt}\\left(\\int\\limits_{0}^{L}|\\eta_{t}|^{2}\\,dx\\right)\n\t+\\frac{\\beta}{2}\\frac{d}{dt}\\left(\\int\\limits_{0}^{L}|\\eta_{x}|^{2}\\,dx\\right)\\\\[1.mm]\n\t& +\\frac{\\alpha}{2}\\frac{d}{dt}\\left(\\int\\limits_{0}^{L}|\\eta_{xx}|^{2}\\,dx\\right)\n\t+2\\mu\\int\\limits_{\\Omega_{t}}| D{\\bf u}|^{2}\\,dx+\\mu'\\int\\limits_{\\Omega_{t}}|\\mathrm{div}{\\bf u}|^{2}\\,dx+\\delta\\int\\limits_{0}^{L}|\\eta_{tx}|^{2}\\,dx =-{p_{ext}}\\int\\limits_{\\Gamma_{s}}\\eta_{t}.\n\t\\end{split}\n\t\\end{equation}\nThe equality \\eqref{1.15l} underlines the physical interpretation of each coefficient and in particular of the viscosity coefficients, $\\mu,$ $\\mu'$ and $\\delta$.\n\\end{remark}\n \\begin{remark}\\label{eta0}\n \tObserve that in \\eqref{1.1} we have considered the initial displacement $\\eta(0)$ of the beam to be zero. This is because we prove the local existence of strong solution of the system \\eqref{1.1} with the beam displacement $\\eta$ close to the steady state zero. There are several examples in the literature where the authors consider the initial displacement of the structure (in a fluid-structure interaction problem) to be equal to zero. For instance the readers can look into the articles \\cite{kukavica} and \\cite{boukir}. We also refer to the article \\cite{veiga} where the initial displacement of the structure is non zero but is considered to be suitably small. The issues involving the existence of strong solution for the model \\eqref{1.1} but with a non zero initial displacement $\\eta(0)$ of the beam is open. The case of a system coupling the incompressible Navier-Stokes equations and an Euler-Bernoulli damped beam with a non zero initial beam displacement is addressed in \\cite{casanova}. \n \\end{remark}\n Our interest is to prove the local in time existence of a strong solution to system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} i.e we prove that given a prescribed initial datum $(\\rho_{0},{\\bf u}_{0},\\eta_{1}),$ there exists a solution of system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} with a certain Sobolev regularity in some time interval $(0,T),$ provided that the time $T$ is small enough.\\\\ \n We study the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} by transforming it into the reference cylindrical domain $Q_{T}.$ This is done by defining a diffeomorphism from $\\Omega_{t}$ onto $\\Omega.$ We adapt the diffeomorphism used in \\cite{veiga} in the study of an incompressible fluid-structure interaction model. The reader can also look at \\cite{raymondbeam}, \\cite{grand} where the authors use a similar map in the context of a coupled fluid-structure model comprising an incompressible fluid. \n \\subsection{Transformation of the problem to a fixed domain}\\label{transfixdm}\n To transform the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} in the reference configuration, for $\\eta$ satisfying\n $\n 1+\\eta(x,t)>0$ for all $(x,t)\\in\\Sigma^{s}_{T},\n$\n we introduce the following change of variables\n \\begin{equation}\\label{1.14}\n \\begin{array}{l}\n \\displaystyle\n {\\Phi}_{\\eta(t)}:\\Omega_{t}\\longrightarrow \\Omega\\quad\\mbox{defined by}\\quad {\\Phi}_{\\eta(t)}(x,y)=(x,z)=\\left(x,\\frac{y}{1+\\eta(x,t)}\\right),\\\\\n \\displaystyle\n {\\Phi}_{\\eta}:\\widetilde{{Q}_{T}}\\longrightarrow Q_{T}\\quad\\mbox{defined by}\\quad {\\Phi}_{\\eta}(x,y,t)=(x,z,t)=\\left(x,\\frac{y}{1+\\eta(x,t)},t\\right).\n \\end{array} \n \\end{equation}\n \\begin{remark}\n \tIt is easy to prove that for each $t\\in[0,T),$ the map ${\\Phi}_{\\eta(t)}$ is a $C^{1}-$ diffeomophism from $\\Omega_{t}$ onto $\\Omega$ provided that $(1+\\eta(x,t))>0$ for all $x\\in \\mathbb{T}_L$ and that $\\eta(\\cdot,t)\\in C^{1}(\\Gamma_{s}).$\n \\end{remark}\n Notice that since $\\eta(\\cdot,0)=0,$ ${\\Phi}_{\\eta(0)}$ is just the identity map. We set the following notations\n \\begin{equation}\\label{1.15}\n \\begin{array}{l}\n \\widehat{\\rho}(x,z,t)=\\rho(\\Phi^{-1}_{\\eta}(x,z,t)),\\,\\,\\widehat{{\\bf u}}(x,z,t)=(\\widehat{u}_{1},\\widehat{u}_{2})={\\bf u}(\\Phi^{-1}_{\\eta}(x,z,t)).\\\\\n\n \\end{array}\n \\end{equation}\n After transformation and using the fact that $\\widehat{{ u}}_{1,x}=0$ on $\\Sigma^{s}_{T}$ (since $\\widehat {\\bf u}=\\eta_{t}\\vec{e_{2}}$ on $\\Sigma^{s}_{T}$) the nonlinear system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} is rewritten in the following form\n \\begin{equation}\\label{1.16}\n \\left\\{\n \\begin{array}{ll}\n \\widehat{\\rho_{t}}+\\begin{bmatrix}\n \\widehat {u}_{1}\\\\\n \\frac{1}{(1+\\eta)}(\\widehat {u}_{2}-\\eta_{t}z-\\widehat{u}_{1}z\\eta_{x})\n \\end{bmatrix}\n \\cdot \\nabla\\widehat{\\rho}+\\widehat{\\rho}\\mbox{div}{\\widehat{\\bf u}}={F}_{1}(\\widehat{\\rho},\\widehat{\\bf u},\\eta)\\quad& \\mbox{in}\\quad Q_{T},\n \\vspace{1.mm}\\\\\n \\widehat\\rho \\widehat {\\bf u}_{t}-\\mu\\Delta \\widehat {\\bf u}-(\\mu'+\\mu)\\nabla(\\mbox{div}\\widehat {\\bf u})+ \\nabla P(\\widehat{\\rho}) ={F}_{2}(\\widehat \\rho,\\widehat {\\bf u},\\eta)\\quad &\\mbox{in} \\quad {Q}_{T},\n \\vspace{1.mm}\\\\\n \\widehat{\\bf u}=\\eta_{t}\\vec{e_{2}}\\quad& \\mbox{on}\\quad \\Sigma^{s}_{T},\\\\[1.mm]\n \\widehat {\\bf u}(\\cdot,t)=0\\quad& \\mbox{on}\\quad \\Sigma^{\\ell}_{T},\n \\vspace{1.mm}\\\\\n \\widehat {\\bf u}(\\cdot,0)={\\bf u}_{0}\\quad& \\mbox{in} \\quad \\Omega,\n \\vspace{1.mm}\\\\\n \\widehat{\\rho}(\\cdot,0)={\\rho_{0}}\\quad& \\mbox{in}\\quad \\Omega,\n \\vspace{1.mm}\\\\\n \\eta_{tt}-\\beta \\eta_{xx}- \\delta\\eta_{txx}+\\alpha\\eta_{xxxx}=F_{3}(\\widehat{\\rho},\\widehat{\\bf u},{\\eta}) \\quad& \\mbox{on}\\quad \\Sigma^{s}_{T},\n \\vspace{1.mm}\\\\\n \\eta(0)=0\\quad \\mbox{and}\\quad \\eta_{t}(0)=\\eta_{1}\\quad &\\mbox{in}\\quad \\Gamma_{s},\n \\end{array} \\right.\n \\end{equation}\n\n where \n \\begin{equation}\\label{F123}\n \\begin{aligned}\n \\displaystyle\n {F}_{1}(\\widehat {\\rho},\\widehat {\\bf u},\\eta)= & \\frac{1}{(1+\\eta)}(\\widehat { u}_{1,z}z\\eta_{x}\\widehat\\rho+\\eta\\widehat{\\rho}\\widehat{u}_{2,z}),\\\\\n {F}_{2}(\\widehat \\rho,\\widehat {\\bf u},\\eta)= & -\\eta\\widehat{\\rho}\\widehat {\\bf u}_{t}+z\\widehat\\rho\\widehat {\\bf u}_{z}\\eta_{t}-\\eta\\widehat \\rho\\widehat {u}_{1}\\widehat {\\bf u}_{x}+\\widehat { u}_{1}\\widehat {\\bf u}_{z}\\eta_{x}\\widehat \\rho z+\\mu \\big(\\eta\\widehat {\\bf u}_{xx}-\\frac{\\eta \\widehat {\\bf u}_{zz}}{(1+\\eta)}-2\\eta_{x}z\\widehat {\\bf u}_{zx}+\\frac{\\widehat {\\bf u}_{zz}z^{2}\\eta^{2}_{x}}{(1+\\eta)}\\\\\n & -\\widehat {\\bf u}_{z}\\big( \\frac{(1+\\eta)z\\eta_{xx}-2\\eta_{x}^{2}z}{(1+\\eta)}\\big)\\big)-\\widehat\\rho(\\widehat {\\bf u}.\\nabla)\\widehat {\\bf u}+(\\mu+\\mu')\\\\\n\n\n\n & \\displaystyle \\cdot\\begin{bmatrix}\n \\eta\\widehat {u}_{1,xx}-\\widehat {u}_{1,xz}z\\eta_{x}-\\eta_{x}z\\big(\\widehat { u}_{1,zx}-\\frac{\\widehat {u}_{1,zz}z\\eta_{x}}{(1+\\eta)}\\big)+\\widehat { u}_{1,z}\\big(\\frac{(1+\\eta)z\\eta_{xx}-2\\eta^{2}_{x}z}{(1+\\eta)}\\big) -\\frac{\\eta_{x}\\widehat {u}_{2,z}}{(1+\\eta)}-\\frac{\\eta_{x}z\\widehat { u}_{2,zz}}{(1+\\eta)}\\\\[2.mm]\n -\\frac{\\eta_{x}\\widehat {u}_{1,z}}{(1+\\eta)}-\\frac{\\eta_{x}z\\widehat { u}_{1,zz}}{(1+\\eta)}-\\frac{\\eta\\widehat {u}_{2,zz}}{(1+\\eta)}\n \\end{bmatrix}\\\\\n & -(\\eta P_{x}(\\widehat{\\rho})-P_{z}(\\widehat{\\rho})z\\eta_{x})\\vec{e_{1}},\\\\\n \\pagebreak\n F_{3}(\\widehat{\\rho},\\widehat {\\bf u},\\eta)= & -\\mu\\big(-{\\widehat{ u}_{2,z}}+\\eta_{x}\\widehat {u}_{2,x}+\\frac{\\widehat { u}_{2,z}}{(1+\\eta)}\\eta^{2}_{x}z-\\frac{2\\eta\\widehat { u}_{2,z}}{(1+\\eta)}-\\frac{\\eta_{x}\\widehat {u}_{1,z}}{(1+\\eta)}\\big)\n -\\mu'\\big(-2\\widehat{u}_{2,z}+\\frac{\\widehat { u}_{1,z}}{(1+\\eta)}\\eta_{x}z\\\\\n &-\\frac{\\eta\\widehat {u}_{2,z}}{(1+\\eta)}\\big)+P(\\widehat{\\rho}).&\n \\end{aligned}\n \\end{equation}\n The transport equation for density \\eqref{1.16}$_{1}$-\\eqref{1.16}$_{6}$ is of the form\n \\begin{equation}\\label{1.17}\n \\left\\{\n \\begin{array}{ll}\n \\widehat{\\rho_{t}}+\\begin{bmatrix}\n \\widehat {u}_{1}\\\\\n \\frac{1}{(1+\\eta)}(\\widehat {u}_{2}-\\eta_{t}z-\\widehat{u}_{1}z\\eta_{x})\n \\end{bmatrix}\n \\cdot \\nabla\\widehat{\\rho}+\\widehat{\\rho}\\mbox{div}{\\widehat{\\bf u}}={F}_{1}\\quad& \\mbox{in}\\quad Q_{T},\n \\vspace{1.mm}\\\\\n \\widehat{\\rho}(\\cdot,0)={\\rho_{0}}\\quad& \\mbox{in}\\quad \\Omega.\n \\end{array} \\right.\n \\end{equation}\n Due to the interface condition, $\\widehat {\\bf u}=\\eta_{t}\\vec{e_{2}}$ on $\\Sigma^{s}_{T},$ we get that the velocity field $(\n \\widehat {u}_{1},\n \\frac{1}{(1+\\eta)}(\\widehat {u}_{2}-\\eta_{t}z-\\widehat{u}_{1}z\\eta_{x})\n )$ satisfies\n $$\\begin{bmatrix}\n \\widehat {u}_{1}\\\\\n \\frac{1}{(1+\\eta)}(\\widehat {u}_{2}-\\eta_{t}z-\\widehat{u}_{1}z\\eta_{x})\n \\end{bmatrix}\\cdot {\\bf n}=0\\quad \\mbox{on}\\quad \\Sigma^{s}_{T},$$\n where ${\\bf n}$ is the unit outward normal to $\\Omega.$ \n Hence we shall not prescribe any boundary condition on the density for the system \\eqref{1.17} to be well posed.\\\\\nTo avoid working in domains which deform when time evolves, the meaning of solutions for \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} will be understood as follows: The triplet $(\\rho,{\\bf u},\\eta)$ solves \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} if and only if $(\\widehat{\\rho},\\widehat{\\bf u},\\eta)$ solves \\eqref{1.16}. This notion will be detailed in the next section. \n\\subsection{Functional settings and the main result}\nIn the fixed domain $\\Omega$ we have the following spaces of functions with values in $\\mathbb{R}^{2},$\n$${\\bf H}^{s}(\\Omega)=H^{s}(\\Omega;\\mathbb{R}^{2})\\quad\\mbox{for all}\\quad s\\geqslant 0.$$\nWe also introduce the following spaces of vector valued functions \n\\begin{equation}\\label{fntlsp}\n\\begin{array}{l}\n{\\bf H}^{1}_{0}(\\Omega)=\\{{\\bf z}\\in{\\bf H}^{1}(\\Omega)\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; {\\bf z}=0\\,\\,\\mbox{on}\\,\\, \\Gamma \\},\\\\[1.mm]\n{\\bf H}^{2,1}(Q_{T})=L^{2}(0,T;{\\bf H}^{2}(\\Omega))\\cap H^{1}(0,T;{\\bf L}^{2}(\\Omega)),\\\\[1.mm]\n{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})=\\{{\\bf z}\\in {\\bf H}^{2,1}(Q_{T})\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; {\\bf z}=0\\,\\,\\mbox{on}\\,\\,\\Sigma_{T} \\}.\n\\end{array}\n\\end{equation}\nSimilarly for $s\\geqslant 0,$ we can define $H^{s}(\\Omega),$ the Sobolev space for the scalar valued functions defined on $\\Omega.$ Now for $\\theta,\\tau\\geqslant 0,$ we introduce the following spaces which we use to analyze the beam equation\n\\begin{equation}\\nonumber\n\\begin{array}{l}\n H^{\\theta,\\tau}(\\Sigma^{s}_{T})=L^{2}(0,T;H^{\\theta}(\\Gamma_{s}))\\cap H^{\\tau}(0,T;L^{2}(\\Gamma_{s})).\n\\end{array}\n\\end{equation} \n\\begin{remark}\n\tSince $\\Omega=\\mathbb{T}_{L}\\times (0,1)$ and $\\Gamma_{s}=\\mathbb{T}_{L}\\times \\{1\\},$ the above definitions of the functional spaces implicitly assert that the functions are $L-$ periodic in the $x$ variable. \n\\end{remark}\n\\begin{prop}\\label{pr1}\nLet $T>0.$ If $\\eta$ is regular enough in the space variable, say $\\eta(\\cdot,t)\\in H^{m}(\\Gamma_{s})$ for $m\\geqslant 2$ and the following holds\n\\begin{equation}\\label{coet}\n\\begin{array}{l}\n1+\\eta(x,t)\\geqslant\\delta_{0}>0\\quad\\mbox{on}\\quad \\Sigma^{s}_{T},\n\\end{array}\n\\end{equation}\nfor some constant $\\delta_{0},$ the map ${{g}}\\mapsto \\widehat{{g}}=g(\\Phi^{-1}_{\\eta(t)}(x,z))$ is a homeomorphism from ${ H}^{s}(\\Omega_{t})$ to ${ H}^{s}(\\Omega)$ for any $s\\leqslant m.$\n\\end{prop}\nThe proposition stated above can be proved in the same spirit of \\cite[Proposition 2, Section 3]{grand}.\\\\\nNow in view of Proposition \\ref{pr1}, we define the notion of strong solution of the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} in terms of the strong solution of the system \\eqref{1.16}.\n\\begin{mydef}\\label{doss}\n\t The triplet $(\\rho,{\\bf u},\\eta)$ is a strong solution of the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} if\n\t\t\\begin{equation}\\label{conoeta}\n\t\t\\begin{array}{ll}\n\t\t\\eta\\in C^{0}\\big([0,T]; H^{9\/2}(\\Gamma_{s})\\big),&\n\t\t\\eta_{t}\\in L^{2}\\big(0,T;{ H}^{4}(\\Gamma_{s})\\big)\\cap C^{0}\\big([0,T];{ H}^{3}(\\Gamma_{s}) \\big),\\\\[1.mm]\n\t\t\\eta_{tt}\\in L^{2}\\big(0,T;{ H}^{2}(\\Gamma_{s})\\big)\\cap C^{0}\\big([0,T];{ H}^{1}(\\Gamma_{s}) \\big),&\n\t\t\\eta_{ttt}\\in L^{2}\\big(0,T;{ L}^{2}(\\Gamma_{s})\\big),\n\t\t\\end{array}\n\t\t\\end{equation}\n\t \\eqref{coet} holds for every $(x,t)\\in \\Sigma^{s}_{T}$ and the triplet $(\\widehat{\\rho},\\widehat{\\bf u},\\eta)=({\\rho}\\circ\\Phi_{\\eta}^{-1},{\\bf u}\\circ\\Phi_{\\eta}^{-1},\\eta )$ solves \\eqref{1.16} in the following Sobolev spaces\n\t\\begin{equation}\\label{de2}\n\t\\begin{array}{l}\n\t\\widehat\\rho\\in C^{0}\\big([0,T]; H^{2}(\\Omega) \\big),\\,\n\t\\widehat\\rho_{t}\\in C^{0}\\big([0,T]; H^{1}(\\Omega)\\big),\\\\[1.mm]\n\t\\widehat{\\bf u}\\in L^{2}\\big(0,T;{\\bf H}^{3}(\\Omega)\\big)\\cap C^{0}\\big([0,T]; {\\bf H}^{5\/2}(\\Omega) \\big),\n\t\\vspace{1.mm}\\\\\n\t\\widehat{\\bf u}_{t}\\in L^{2}\\big(0,T;{\\bf H}^{2}(\\Omega)\\big)\\cap C^{0}\\big([0,T];{\\bf H}^{1}(\\Omega) \\big),\n\t\\vspace{1.mm}\\\\\n\t\\widehat{\\bf u}_{tt}\\in L^{2}\\big(0,T;{\\bf L}^{2}(\\Omega)\\big).\n\t\\end{array}\n\t\\end{equation}\n\t($\\eta$ is in the space mentioned in \\eqref{conoeta}). Note that $(\\rho,{\\bf u})$ can then be obtained from $(\\widehat{\\rho},\\widehat{\\bf u})$ by $(\\rho,{\\bf u})=(\\widehat{\\rho}\\circ\\Phi_{\\eta},\\widehat{\\bf u}\\circ\\Phi_{\\eta}).$\n\\end{mydef}\nIn relation with Definition \\ref{doss}, we introduce the following functional spaces\n\\begin{equation}\\label{dofYi}\n\\begin{split}\nY_{1}^{T}=&\\{{\\rho}\\in C^{0}([0,T];H^{2}(\\Omega))\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; \\rho_{t}\\in C^{0}([0,T]; H^{1}(\\Omega))\\},\\\\[1.mm]\nY_{2}^{T}=&\\{{\\bf u}\\in L^{2}(0,T;{\\bf H}^{3}(\\Omega))\\cap C^{0}([0,T];{\\bf H}^{5\/2}(\\Omega))\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; {\\bf u}_{t}\\in L^{2}(0,T;{\\bf H}^{2}(\\Omega))\n\\cap C^{0}([0,T];{\\bf H}^{1}(\\Omega)),\\\\[1.mm]\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad{\\bf u}_{tt}\\in L^{2}(0,T;{\\bf L}^{2}(\\Omega)) \\},\\\\[1.mm]\nY_{3}^{T}=&\\{\\eta\\in C^{0}([0,T]; H^{9\/2}(\\Gamma_{s})),\\,\\eta(x,0)=0\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; \\eta_{t}\\in L^{2}(0,T;{ H}^{4}(\\Gamma_{s}))\\cap C^{0}([0,T];{ H}^{3}(\\Gamma_{s})), \\\\[1.mm]\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\eta_{tt}\\in L^{2}(0,T;{ H}^{2}(\\Gamma_{s}))\\cap C^{0}([0,T];{ H}^{1}(\\Gamma_{s})),\n\\eta_{ttt}\\in L^{2}(0,T;{ L}^{2}(\\Gamma_{s}))\\}.\n\\end{split}\n\\end{equation}\nThe spaces $Y_{1}^{T},$ $Y_{2}^{T}$ and $Y_{3}^{T}$ correspond to the spaces in which the unknowns $\\widehat{\\rho},$ $\\widehat{\\bf u}$ and $\\eta$ respectively.\\\\\n Now we precisely state the main result of the article.\n\\begin{thm}\\label{main}\n\tAssume that \n\t\\begin{equation}\\label{cit}\n\t\\left\\{ \\begin{array}{ll}\n\t(i)\\,&(a)\\,\\,{Regularity\\,of\\,initial\\,conditions:}\\,\\rho_{0}\\in {H}^{2}(\\Omega),\\,\\, \\eta_{1}\\in {H}^{3}(\\Gamma_{s}),\\,\\, {\\bf u}_{0}\\in {\\bf H}^{3}(\\Omega).\\\\[2.mm]\n &(b)\\,\\,{Compatibility\\,between\\,initial\\, and\\,boundary\\,conditions}:\\\\\n\t&\\quad(b)_{1}\\,\\,\\left( {\\bf u}_{0}-\\begin{bmatrix}\n\t0\\\\\n\tz\\eta_{1}\n\t\\end{bmatrix}\\right)=0\\,\\mbox{on}\\,\\Gamma,\\\\[2.mm]\n\t &\\quad(b)_{2}\\,\\,-P'(\\rho_{0})\\nabla\\rho_{0}-(\\delta\\eta_{1,xx}-(\\mu+2\\mu')({ u}_{0})_{2,z}+P(\\rho_{0}))z\\rho_{0}\\vec{e}_{2}+z\\rho_{0}({\\bf u}_{0})_{z}\\eta_{1}\\\\\n\t &\\quad\\,\\,\\quad\\quad-\\rho_{0}({\\bf u}_{0}\\cdot\\nabla){\\bf u}_{0}-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div}){\\bf u}_{0}=0\\,\\mbox{on}\\,\\Gamma,\\\\\n\t(ii)\\,&\\eqref{cor0}\\,\\mbox{holds},\n\t\\end{array}\\right.\n\t\\end{equation}\n\t where we use the notations $P'({\\rho}_{0})=\\nabla P({\\rho}_{0}),$ $P(\\rho_{0})=(a\\rho_{0}^{\\gamma}-a\\overline{\\rho}^{\\gamma})$ and ${\\bf u}_{0}=((u_{0})_{1},(u_{0})_{2}).$\n\tThen there exists $T>0$ such that the system \\eqref{1.16} admits a solution $(\\widehat{\\rho},\\widehat{\\bf u},\\eta)\\in Y_{1}^{T}\\times Y_{2}^{T}\\times Y_{3}^{T}.$ \n\tConsequently in the sense of Definition \\ref{doss} the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} admits a strong solution $({\\rho},{\\bf u},\\eta).$ \n\\end{thm} \n \\begin{remark}\n \tOur analysis throughout the article can be suitably adapted to consider any pressure law $p(\\cdot)\\in C^{2}(\\mathbb{R}^{+})$ (in this article we present the proofs with the pressure law given by $p(\\rho)=a\\rho^{\\gamma},$ with $\\gamma>1$) such that there exists a positive constant $\\overline{\\rho}$ satisfying $p(\\overline{\\rho})={p_{ext}},$ where ${p_{ext}}(>0)$ is the external force acting on the beam. The adaptation is possible since we only consider the case where the fluid density $\\rho$ has a positive lower and upper bound. \n \\end{remark} \nNow let us sketch the strategy towards the proof of Theorem \\ref{main}.\n\\subsection{Strategy}\\label{Chap3strategy}\n(i) $\\mathit{Changing\\,\\,\\eqref{1.16}\\,\\, to\\,\\, a\\,\\, homogeneous\\,\\, boundary\\,\\, value\\,\\, problem}$: Recall that (see Remark \\ref{eta0}) we will prove the existence of local in time strong solution of the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} only when the beam displacement $\\eta$ is close to zero. Again observe that $(\\widehat{\\rho}=\\overline{\\rho},\\widehat{\\bf u}=0,\\widehat{\\eta}=0)$ is a steady state solution of the system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} and hence of the system \\eqref{1.16}. So to work in a neighborhood of $\\eta=0,$ we make the following change of unknowns in \\eqref{1.16},\n\\begin{equation}\\label{1.20}\n\\begin{array}{l}\n\\sigma=\\widehat\\rho-\\overline{\\rho},\n\\quad\n{\\bf v}=(v_{1},v_{2})=\\widehat {\\bf u}-0,\n\\quad\n\\eta=\\eta-0.\n\\end{array}\n\\end{equation} \nIn view of the change of unknowns \\eqref{1.20} one obtains\n\\begin{equation}\\label{1.21}\n\\left\\{\n\\begin{array}{ll}\n{\\sigma_{t}}+\\begin{bmatrix}\n{v}_{1}\\\\\n\\frac{1}{(1+\\eta)}({v}_{2}-\\eta_{t}z-{v}_{1}z\\eta_{x})\n\\end{bmatrix}\n\\cdot \\nabla\\sigma+(\\sigma+\\overline{\\rho})\\mbox{div}({\\bf v})={F}_{1}(\\sigma+\\overline{\\rho},{\\bf v},\\eta)\\quad &\\mbox{in}\\quad Q_{T},\n\\vspace{1.mm}\\\\\n(\\sigma+\\overline{\\rho}){\\bf v}_{t}-\\mu \\Delta {\\bf v}-(\\mu+\\mu{'})\\nabla\\mathrm{div}{\\bf v}=-{P}'(\\sigma+\\overline{\\rho}) \\nabla \\sigma+{F}_{2}(\\sigma+\\overline{\\rho},{\\bf v},\\eta)\\quad &\\mbox{in} \\quad {Q}_{T},\n\\vspace{1.mm}\\\\\n{\\bf v}=\\eta_{t}\\vec{e_{2}}\\quad& \\mbox{on}\\quad \\Sigma^{s}_{T},\\\\[1.mm]\n{\\bf v}=0\\quad &\\mbox{on}\\quad\\Sigma^{\\ell}_{T},\n\\vspace{1.mm}\\\\\n{\\bf v}(\\cdot,0)={\\bf u}_{0}\\quad& \\mbox{in} \\quad \\Omega,\n\\vspace{1.mm}\\\\\n\\sigma(\\cdot,0)=\\sigma_{0}={\\rho_{0}}-{\\overline\\rho}\\quad &\\mbox{in}\\quad \\Omega,\n\\vspace{1.mm}\\\\\n\\eta_{tt}-\\beta \\eta_{xx}- \\delta\\eta_{txx}+\\alpha\\eta_{xxxx}={F}_{3}(\\sigma+\\overline{\\rho},{\\bf v},\\eta)\\quad &\\mbox{on}\\quad \\Sigma^{s}_{T},\n\\vspace{1.mm}\\\\\n\\eta(0)=0\\quad \\mbox{and}\\quad \\eta_{t}(0)=\\eta_{1}\\quad &\\mbox{in}\\quad \\Gamma_{s}.\n\\end{array} \\right.\n\\end{equation}\nWe transform the system \\eqref{1.21} into a homogeneous Dirichlet boundary value problem by performing further the following change of unknown\n\\begin{equation}\\label{cvhd}\n\\begin{array}{l}\n{\\bf w}=(w_{1},w_{2})={\\bf v}-z\\eta_{t}\\vec{e_{2}}.\n\\end{array}\n\\end{equation}\nSince ${\\bf v}$ and ${\\eta}_{t}$ both are $L$-periodic in the $x-$direction, the new unknown ${\\bf w}$ is also $L$-periodic in the $x-$direction. With the new unknown ${\\bf w},$ we write the transformed system in the following form\n\\begin{equation}\\label{chdb}\n\\left\\{\n\\begin{array}{ll}\n{\\sigma_{t}}+\\begin{bmatrix}\n{w}_{1}\\\\\n\\frac{1}{(1+\\eta)}({w}_{2}-{w}_{1}z\\eta_{x})\n\\end{bmatrix}\n\\cdot \\nabla\\sigma=G_{1}(\\sigma,{\\bf w},\\eta)\\quad& \\mbox{in}\\quad Q_{T}, \n\\vspace{1.mm}\\\\\n(\\sigma+\\overline{\\rho}){\\bf w}_{t}-\\mu \\Delta {\\bf w} -(\\mu+\\mu{'})\\nabla\\mathrm{div}{\\bf w}=G_{2}(\\sigma,{\\bf w},\\eta)\\quad& \\mbox{in}\\quad Q_{T},\n\\vspace{1.mm}\\\\\n{\\bf w}=0\\quad &\\mbox{on}\\quad \\Sigma_{T},\n\\vspace{1.mm}\\\\\n{\\bf w}(\\cdot,0)={\\bf w}_{0}={\\bf u}_{0}-z\\eta_{1}\\vec{e}_{2}\\quad &\\mbox{in} \\quad \\Omega,\n\\vspace{1.mm}\\\\\n\\sigma(\\cdot,0)=\\sigma_{0}={\\rho_{0}}-{\\overline\\rho}\\quad &\\mbox{in}\\quad \\Omega,\n\\vspace{1.mm}\\\\\n\\eta_{tt}-\\beta \\eta_{xx}- \\delta\\eta_{txx}+\\alpha\\eta_{xxxx}=G_{3}(\\sigma,{\\bf w},\\eta)\\quad & \\mbox{on}\\quad \\Sigma^{s}_{T},\n\\vspace{1.mm}\\\\\n\\eta(0)=0\\quad \\mbox{and}\\quad \\eta_{t}(0)=\\eta_{1}\\quad& \\mbox{in}\\quad \\Gamma_{s},\n\\end{array} \\right.\n\\end{equation}\nwhere\n\\begin{equation}\\label{1.22}\n\\begin{split}\n& G_{1}(\\sigma,{\\bf w},\\eta)=-(\\sigma+\\overline{\\rho})\\mbox{div}({\\bf w}+z\\eta_{t}\\vec{e_{2}})+{F}_{1}(\\sigma+\\overline{\\rho},{{\\bf w}+z\\eta_{t}\\vec{e_{2}}},\\eta),\\\\[1.mm]\n& G_{2}(\\sigma,{\\bf w},\\eta)=-{P}'(\\sigma+\\overline{\\rho}) \\nabla \\sigma-\tz\\eta_{tt}(\\sigma+\\overline{\\rho})\\vec{e_{2}}-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\eta_{t}\\vec{e_{2}})\\\\[1.mm]\n& \\qquad\\qquad\\qquad+{F}_{2}(\\sigma+\\overline{\\rho},{\\bf w}+z\\eta_{t}\\vec{e_{2}},\\eta),\n\\vspace{1.mm}\\\\\n& G_{3}(\\sigma,{\\bf w},\\eta)={F}_{3}(\\sigma+\\overline{\\rho},{\\bf w}+\\eta_{t}\\vec{e_{2}},\\eta).\n\\end{split}\n\\end{equation}\n(ii) $\\textit{Study\\,\\,of\\,\\,some\\,\\,decoupled\\,\\,linear\\,\\,problems}$: Observe that in the new system \\eqref{chdb} the coupling between the velocity of the fluid and the elastic structure appears only as source terms. In order to solve the system \\eqref{chdb} we first study some linear equations in Section \\ref{lineqs}. In order to analyze the local in time existence of strong solution the difficulty is to track the dependence of the constants (appearing in the inequalities) with respect to the time parameter \\textquoteleft T\\textquoteright.\\ \\,In this direction we first obtain a priori estimates for the linear density and velocity equations with non homogeneous source terms in the spirit of \\cite{vallizak}. Then we prove the existence of strong solutions for a linear beam equation. The proof strongly relies on the analyticity of the corresponding beam semigroup (see \\cite{chen} for details). At this point we refer the readers to the articles \\cite{denk} (maximal $L^{p}-L^{q}$ regularity of structurally damped beam equation), \\cite{fanli} (analyticity and exponential stability of beam semigroup), \\cite{raymondbeam} (study of beam equation in the context of an incompressible fluid structure interaction problem) and the references therein for the existence and regularity issues of the damped beam equation. In our case to obtain estimates with the constants independent of \\textquoteleft T\\textquoteright\\ for the beam equation we first fix a constant $\\overline{T}>0$ and restrict ourselves to work in the time interval $(0,T)$ where\n\\begin{equation}\\label{Tbar}\n\\begin{array}{l}\nT<\\overline{T}.\n\\end{array}\n\\end{equation}\nThis technique is inspired from \\cite{rayvan}.\\\\ \n(iii) $\\mathit{Fixed\\,\\,point\\,\\,argument}$: In Section \\ref{fixdpt} we prove the existence of a strong solution of \\eqref{chdb} by using the Schauder's fixed point theorem based on \\eqref{chdb}-\\eqref{1.22}.\n\\begin{remark}\n\tSince $\\eta(0)=0$ the regularity \\eqref{conoeta} of $\\eta$ guarantees that \n\t\\begin{equation}\\label{smalleta}\n\t\\begin{array}{l}\n\t\\|\\eta\\|_{L^{\\infty}(\\Sigma^{s}_{T})}\\leqslant CT\\|\\eta_{t}\\|_{L^{\\infty}(0,T;H^{3}(\\Gamma_{s}))},\n\t\\end{array}\n\t\\end{equation}\n\tfor a constant $C$ independent of $T.$ For small enough time $T,$ \\eqref{smalleta} furnishes $\\eta\\approx 0$ and hence during small times, the beam stays close to the steady state zero.\n\\end{remark}\n\\subsection{Comments on initial and compatibility conditions}\\label{incom}\n(i)\nRecall from \\eqref{cit}$(i)$$(a)$ that we assume ${\\bf u}_{0}\\in {\\bf H}^{3}(\\Omega).$ Also observe that in our solution (see \\eqref{de2}) the vector field $\\widehat{\\bf u}\\in C^{0}([0,T];{\\bf H}^{5\/2}(\\Omega))$ i.e for the velocity field there is a loss of $\\frac{1}{2}$ space regularity as the time evolves. One can find such instances of a loss of space regularity in many other articles in the literature, for instance we refer the readers to \\cite{bougue3}, \\cite{kukavica} (for the coupling of fluid-elastic structure comprising a compressible fluid) and \\cite{cout1}, \\cite{cout2}, \\cite{rayvan} (for incompressible fluid structure interaction models).\\\\[2.mm] \n(ii) We use \\eqref{1.22}$_{3}$ to obtain the following expression of $G_{3}\\mid_{t=0}$ (the value of $G_{3}(\\sigma,{\\bf w},\\eta)$ at time $t=0$)\n\\begin{equation}\\label{G30}\n\\begin{array}{l}\nG_{3}\\mid_{t=0}=-(\\mu+2\\mu')({u}_{0})_{2,z}+P(\\rho_{0}).\n\\end{array}\n\\end{equation} \nUsing $\\rho_{0}\\in H^{2}(\\Omega),$ ${\\bf u}_{0}\\in {\\bf H}^{3}(\\Omega)$ (see \\eqref{cit}$(i)$$(a)$) and standard trace theorems one easily checks that \n\\begin{equation}\\label{regG3t0}\n\\begin{array}{l}\nG_{3}\\mid_{t=0}\\in H^{3\/2}(\\Gamma_{s}).\n\\end{array}\n\\end{equation}\nWe will use the regularity of $G_{3}\\mid_{t=0}$ (in fact we will only use $G_{3}\\mid_{t=0}\\in H^{1}(\\Gamma_{s})$) to prove the regularity of $\\eta.$ This will be detailed in Theorem \\ref{t23}.\\\\[2.mm]\n (iii) We use \\eqref{G30} and the equation \\eqref{chdb}$_{6}$ to check that\n $$\\eta_{tt}(\\cdot,0)=\\delta\\eta_{1,xx}-(\\mu+2\\mu')(u_{0})_{2,z}+P(\\rho_{0}).$$\n Hence using \\eqref{1.22}$_{2}$ one obtains the following expression of $G_{2}\\mid_{t=0}$ (the value of $G_{2}(\\sigma,{\\bf w},\\eta)$ at time $t=0$)\n\\begin{equation}\\label{iG0}\n\\begin{split}\nG_{2}\\mid_{t=0}&=-P'(\\rho_{0})\\nabla\\rho_{0}-(\\delta\\eta_{1,xx}-(\\mu+2\\mu')({ u}_{0})_{2,z}\n+P(\\rho_{0}))z\\rho_{0}\\vec{e}_{2}\n+z\\rho_{0}({\\bf u}_{0})_{z}\\eta_{1}\\\\\n&-\\rho_{0}({\\bf u}_{0}\\cdot\\nabla){\\bf u}_{0}-(-\\mu\\Delta-(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\eta_{1}\\vec{e_{2}}).\n\\end{split}\n\\end{equation}\n This gives\n\\begin{equation}\\label{compat}\n\\begin{split}\nG_{2}\\mid_{t=0}&-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div})\\left( {\\bf u}_{0}-\\begin{bmatrix}\n0\\\\\nz\\eta_{1}\n\\end{bmatrix}\\right)\n=-P'(\\rho_{0})\\nabla\\rho_{0}-(\\delta\\eta_{1,xx}-(\\mu+2\\mu')({ u}_{0})_{2,z}\\\\\n&+P(\\rho_{0}))z\\rho_{0}\\vec{e}_{2}\n+z\\rho_{0}({\\bf u}_{0})_{z}\\eta_{1}-\\rho_{0}({\\bf u}_{0}\\cdot\\nabla){\\bf u}_{0}-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div}){\\bf u}_{0}.\n\\end{split}\n\\end{equation}\nThe regularity assumptions \\eqref{cit}$(i)$$(a)$ and \\eqref{compat} furnish the following\n\\begin{equation}\\label{regcom}\n\\begin{array}{l}\nG_{2}\\mid_{t=0}-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div})\\left( {\\bf u}_{0}-\\begin{bmatrix}\n0\\\\\nz\\eta_{1}\n\\end{bmatrix}\\right)\\in {\\bf H}^{1}(\\Omega).\n\\end{array}\n\\end{equation}\nHence one obtains (recalling that ${\\bf w}_{0}={\\bf u}_{0}-z\\eta_{1}\\vec{e}_{2}$)\n\\begin{equation}\\label{asm2}\n\\begin{array}{ll}\n&\\mbox{the assumption}\\, \\eqref{cit}(i)(a)\\,\\mbox{and}\\, \\eqref{cit}(i)(b)_{2}\\\\\n&\\Longrightarrow G_{2}\\mid_{t=0}-(-\\mu \\Delta{\\bf w}_{0} -(\\mu+\\mu{'})\\nabla\\mathrm{div}{\\bf w}_{0})\\in {\\bf H}^{1}_{0}(\\Omega).\n\\end{array}\n\\end{equation}\nWe need this to prove some regularity of ${\\bf w}$ and hence of ${\\widehat{\\bf u}}.$ This will be detailed in Theorem \\ref{t21}.\\\\[2.mm] \n\\subsection{Bibliographical comments}\\label{bibcomlclext} Here we mainly focus on the existing literature devoted to the study of fluid structure interaction problems.\\\\ \nTo begin with we quote a few articles dedicated to the mathematical study of compressible Navier-Stokes equations. The existence of local in time classical solutions for the compressible Navier-Stokes equations in a time independent domain was first proved in \\cite{nash} and the uniqueness was established in \\cite{serrin}. The global existence of strong solutions for a small perturbation of a stable constant state was established in the celebrated work \\cite{matnis}. In the article \\cite{vallizak} the authors established the local in time existence of strong solutions in the presence of inflow and outflow of the fluid through the boundary. In the same article they also present the proof of global in time existence for small data in the absence of the inflow. P.-L. Lions proved (in \\cite{lions}) the global existence of renormalized weak solution with bounded energy for an isentropic fluid (i.e $p(\\rho)=\\rho^{\\gamma}$) with the adiabatic constant $\\gamma>3d\/(d+2),$ where $d$ is the space dimension. E. Feireisl $\\mathit{et\\, al.}$ generalized the approach to cover the range $\\gamma>3\/2$ in dimension $3$ and $\\gamma>1,$ in dimension $2$ in \\cite{feireisl}. The well-posedness issues of the compressible Navier-Stokes equations for critical regularity data can be found in \\cite{danchin}, \\cite{danchin1}. For further references and a very detailed development of the mathematical theory of compressible flow we refer the reader into the books \\cite{novotny1} and \\cite{bresch1}.\\\\\nIn the last decades the fluid-structure interaction problems have been an area of active research. There is a rich literature concerning the motion of a structure inside or at the boundary of a domain containing a viscous incompressible Newtonian fluid, whose behavior is described by Navier-Stokes equations. For instance local existence and uniqueness of strong solutions of incompressible fluid-structure models with the structure immersed inside the fluid are studied in \\cite{cout1} (the elastic structure is modeled by linear Kirchhoff equations) and \\cite{cout2} (the elastic structure is governed by quasilinear elastodynamics). There also exist articles dealing with incompressible fluid-structure interaction problems where the structure appears on the fluid boundary and is modeled by Euler-Bernoulli damped beam equations \\eqref{1.1}$_{7}$-\\eqref{1.1}$_{8}.$ For example we refer the readers to \\cite{veiga} (local in time existence of strong solutions), \\cite{esteban} (existence of weak solutions), \\cite{raymondbeam} (feedback stabilization), \\cite{grand} (global in time existence) and the references therein for a very detailed discussion of such problems.\\\\\nDespite of the growing literature on incompressible fluids the number of articles addressing the compressible fluid-structure interaction problems is relatively limited and the literature has been rather recently developed. One of the fundamental differences between the incompressible and compressible Navier-Stokes equations is that the pressure of the fluid in incompressible Navier-Stokes equations is interpreted as the Lagrange multiplier whereas in the case of compressible Navier-Stokes equations the pressure is given as a function of density with the density modeled by a transport equation of hyperbolic nature. The strong coupling between the parabolic and hyperbolic dynamics is one of the intricacies in dealing with the compressible Navier-Stokes equations and this results in the regularity incompatibilities between the fluid and the solid structure. However in the past few years there have been works exploring the fluid-structure interaction problems comprising the compressible Navier-Stokes equations with an elastic body immersed in the fluid domain. For instance in the article \\cite{bougue2} the authors prove the existence and uniqueness of strong solutions of a fluid structure interaction problem for a compressible fluid and a rigid structure immersed in a regular bounded domain in dimension 3. The result is proved in any time interval $(0,T),$ where $T>0$ and for a small perturbation of a stable constant state provided there is no collision between the rigid body and the boundary $\\partial\\Omega$ of the fluid domain. In \\cite{boulakia1} the existence of weak solution is obtained in three dimension for an elastic structure immersed in a compressible fluid. The structure equation considered in \\cite{boulakia1} is strongly regularized in order to obtain suitable estimates on the elastic deformations. A result concerning the local in time existence and uniqueness of strong solutions for a problem coupling compressible fluid and an elastic structure (immersed inside the fluid) can be found in \\cite{bougue3}. In the article \\cite{bougue3} the equation of the structure does not contain any extra regularizing term. The flow corresponding to a Lagrangian velocity is used in \\cite{bougue3} in order to transform the fluid structure interaction problem in a reference fluid domain $\\Omega_{F}(0),$ whereas in the present article we use the non physical change of variables \\eqref{1.14} for the similar purpose of writing the entire system in a reference configuration. A similar Navier-Stokes-Lam\\'{e} system as that of \\cite{bougue3} is analyzed in \\cite{kukavica} to prove the existence of local in time strong solutions but in a different Sobolev regularity framework. In the article \\cite{kukavica} the authors deal with less regular initial data. We also quote a very recent work \\cite{boukir} where the authors prove the local in time existence of a unique strong solution of a compressible fluid structure interaction model where the structure immersed inside the fluid is governed by the Saint-Venant Kirchhoff equations.\\\\\n On the other hand there is a very limited number of works on the compressible fluid-structure interaction problems with the structure appearing on the boundary of the fluid domain. The article \\cite{flori2} deals with a 1-D structure governed by plate equations coupled with a bi-dimensional compressible fluid where the structure is located at a part of the boundary. Here the authors consider the velocity field as a potential and in their case the non linearity occurs only in the equation modeling the density. Instead of writing the system in a reference configuration in \\cite{flori2} the authors proved the existence and uniqueness of solution in Sobolev-like spaces defined on time dependent domains. The existence of weak solution for a different compressible fluid structure interaction model (with the structure appearing on the boundary) is studied in dimension three by the same authors in \\cite{flori1}. In the model considered in \\cite{flori1}, the fluid velocity $v$ satisfies $\\mbox{curl}v\\wedge n=0$ on the entire fluid boundary and the plate is clamped everywhere on the structural boundary. In a recent article \\cite{gavalos1} the authors prove the Hadamard well posedness of a linear compressible fluid structure interaction problem (three dimensional compressible fluid interacting with a bi-dimensional elastic structure) defined in a fixed domain and considering the Navier-slip boundary condition at the interactive boundary. They write the coupled system in the form\n $$\\frac{d}{dt}\\begin{pmatrix}\n \\rho\\\\\n {\\bf u}\\\\\n \\eta\\\\\n \\eta_{t}\n \\end{pmatrix}=A\\begin{pmatrix}\n \\rho\\\\\n {\\bf u}\\\\\n \\eta\\\\\n \\eta_{t}\n \\end{pmatrix}\\,\\,\\mbox{in}\\,\\,(0,T),\\quad \\mbox{and}\\quad\\begin{pmatrix}\n \\rho(0)\\\\\n {\\bf u}(0)\\\\\n \\eta(0)\\\\\n \\eta_{t}(0)\n \\end{pmatrix}=\\begin{pmatrix}\n \\rho_{0}\\\\\n {\\bf u}_{0}\\\\\n \\eta_{1}\\\\\n \\eta_{2}\n \\end{pmatrix},$$\n and prove the existence of mild solution $(\\rho,{\\bf u},\\eta,\\eta_{t})$ in the space $C^{0}([0,T];\\mathcal{D}(A))$ where $\\mathcal{D}(A)$ is the domain of the operator $A.$ Their approach is based on using the Lumer-Phillips theorem to prove that $A$ generates a strongly continuous semigroup. In yet another recent article \\cite{breit} the authors consider a three dimensional compressible fluid structure interaction model where the structure located at the boundary is a shell of Koiter-type with some prescribed thickness. In the spirit of \\cite{lions} and \\cite{feireisl} the authors prove the existence of a weak solution for their model with the adiabatic constant restricted to $\\gamma>\\frac{12}{7}.$ They show that a weak solution exists until the structure touches the boundary of the fluid domain.\\\\\n To the best of our knowledge there is no existing work (neither in dimension 2 nor in 3) proving the existence of strong solutions for the non-linear compressible fluid-structure interaction problems (defined in a time dependent domain) considering the structure at the boundary of the fluid domain. In the present article we address this problem in the case of a fluid contained in a 2d channel and interacting with a 1d structure at the boundary. Our approach is different from that of \\cite{gavalos1} and \\cite{breit}. In \\cite{gavalos1}, since the problem itself is linearized in a fixed domain, the authors can directly use a semigroup formulation to study the existence of mild solution, whereas \\cite{breit} considers weak solutions and a $4$ level approximation process (using artificial pressure, artificial viscosity, regularization of the boundary and Galerkin approximation for the momentum equation). In the study of weak solutions (in \\cite{lions}, \\cite{feireisl}, \\cite{breit}) one of the major difficulties is to pass to the limit in the non-linear pressure term which is handled by introducing a new unknown called the effective viscous flux. In our case of strong regularity framework we do not need to introduce the effective viscous flux and for small enough time $T,$ the term $\\nabla P(\\sigma+\\overline{\\rho})$ can be treated as a non homogeneous source term. Our approach is based on studying the regularity properties of a decoupled parabolic equation, continuity equation and a beam equation. This is done by obtaining some apriori estimates and exploiting the analyticity of the semigroup corresponding to the beam equation. Then the existence result for the non-linear coupled problem is proved by using the Schauder's fixed point argument. We prove the existence of the fixed point in a suitable convex set, which is constructed very carefully based on the estimates of the decoupled problems and the estimates of the non-homogeneous source terms. This led us to choose this convex set as a product of balls (in various functional spaces) of different radius. In the present article we prove a local in time existence result of strong solutions whose incompressible counterpart was proved in \\cite{veiga}.\\\\ \n Let us also mention the very recent article \\cite{shibata} where the global existence for the compressible viscous fluids (without any structure on the boundary) in a bounded domain is proved in the maximal $L^{p}-L^{q}$ regularity class. In this article the authors consider a slip type boundary condition. More precisely the fluid velocity ${\\bf u}$ satisfies the following on the boundary\n $$\nD({\\bf u}){\\bf n}-\\langle D({\\bf u}){\\bf n},{\\bf n}\\rangle {\\bf n}=0,\\quad\\mbox{and}\\quad {\\bf u}\\cdot {\\bf n}=0\\quad\\mbox{on}\\quad \\partial\\Omega\\times (0,T).$$\n In a similar note one can consider a fluid structure interaction problem with slip type boundary condition. In that case the velocity field ${\\bf u}$ solves the following\n \\begin{equation}\\label{fslip}\n \\begin{array}{l}\n D({\\bf u}){\\bf n}-\\langle D({\\bf u}){\\bf n},{\\bf n}\\rangle {\\bf n}=0,\\quad\\mbox{and}\\quad {\\bf u}\\cdot {\\bf n}=\\eta_{t}\\quad\\mbox{on}\\quad \\Gamma_{s}\\times (0,T),\n \\end{array}\n\\end{equation}\n where $\\eta_{t}$ is the structural velocity at the interactive boundary $\\Gamma_{s}\\times(0,T).$ To the best of our knowledge for a compressible fluid structure interaction problem the condition \\eqref{fslip} is treated only in \\cite{gavalos1}, proving the existence of mild solution. Of course the boundary condition \\eqref{fslip} is different from the one we consider in the present article since in our case we do not allow the fluid to slip tangentially through the fluid structure interface (i.e recall in our case ${ u}_{1}=0$ on $\\Sigma^{s}_{T}$).\\\\\n A more generalized slip boundary condition is considered in \\cite{muhacan} in the context of an incompressible fluid structure interaction problem. In the model examined in \\cite{muhacan} the structural displacement has both tangential and normal components with respect to the reference configuration. At the interface the fluid and the structural velocities are coupled via a kinematic coupling condition and a dynamic coupling condition (stating that the structural dynamics is governed by the jump of the normal stress at the interface). The kinematic coupling conditions at the interface treated in \\cite{muhacan} consists of continuity of the normal velocities and a second condition stating that the slip between the tangential components of the fluid and structural velocities is proportional to the fluid normal stress. The authors in \\cite{muhacan} prove the existence of a weak solution for their model. \n \\subsection{Outline} Section \\ref{lineqs} contains results involving the existence and uniqueness of some decoupled linear equations. We state the existence and uniqueness result for a parabolic equation in Section \\ref{veleq}, continuity equation in Section \\ref{deneq}, linear beam equation in Section \\ref{beameq}. In Section \\ref{fixdpt} we prove Theorem \\ref{main} by using the Schauder fixed point theorem. \n\n\\section{Analysis of some linear equations}\\label{lineqs}\nWe will prove the existence and uniqueness of strong solutions of a parabolic equation, a continuity equation and a damped beam equation with prescribed initial data and source terms in appropriate Sobolev spaces.\\\\\nFrom now onwards all the constants appearing in the inequalities will be independent of the final time $T,$ unless specified. We also comment that we will denote many of the constants in the inequalities using the same notation although they might vary from line to line. \n\\subsection{ Study of a parabolic equation}\\label{veleq}\nAt first we consider the following linear problem\n\\begin{equation}\\label{2.1.1}\n\\left\\{ \\begin{array}{ll}\n\\overline{\\sigma}{\\bf w}_{t}-\\mu \\Delta {\\bf w} -(\\mu+\\mu{'})\\nabla\\mathrm{div}{\\bf w}=G_{2}&\\quad \\mbox{in}\\quad Q_{T},\n\\vspace{1.mm}\\\\\n{\\bf w}=0&\\quad\\mbox{on}\\quad \\Sigma_{T},\\\\[1.mm]\n{\\bf w}(0)={\\bf w}_{0}&\\quad \\mbox{in}\\quad \\Omega,\n\\end{array}\\right.\n\\end{equation}\nwhere $\\overline{\\sigma},$ ${\\bf w}_{0}$ and $G_{2}$ are known functions which are $L$-periodic in the $x$ direction.\\\\\nLet $m$ and $M$ be positive constants such that $m0$ small enough and hence we will conclude Theorem \\ref{main}.\\\\\n\t\t\t Now we sketch the steps towards the proof of Theorem \\ref{main}:\\\\\n\t\t\t\t(i) First in Section \\ref{dfp} we define a suitable map for which a fixed point gives a solution of the system \\eqref{chdb}.\\\\\n\t\t\t\t(ii) Next we design a suitable convex set such that the map defined in step (i) maps this set into itself. This is done in Section \\ref{Enht}. \\\\\n\t\t\t\t(iii) In Section \\ref{comcon} we show that the convex set defined in step (ii) is compact in some appropriate topology. We further prove that the fixed point map from step (i), is continuous in that topology.\\\\\n\t\t\t\t(iv) At the end in Section \\ref{conc} we draw the final conclusion to prove Theorem \\ref{main}.\\\\\n\t\t\t\tIn what follows all the constants appearing in the inequalities may vary from line to line but will never depend on $T.$\n\t\t\t\t \\subsection{Definition of the fixed point map}\\label{dfp}\n\t\t\t\t For $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ satisfying\n\t\t\t\t \\begin{equation}\\label{regswe}\n\t\t\t\t \\left\\{ \\begin{array}{l}\n\t\t\t\t \\widetilde{\\sigma}\\in L^{\\infty}(0,T;H^{2}(\\Omega))\\cap W^{1,\\infty}(0,T;H^{1}(\\Omega)),\\\\\n\t\t\t\t \\widetilde{\\bf w}\\in L^{\\infty}(0,T;{\\bf H}^{5\/2}(\\Omega))\\cap L^{2}(0,T;{\\bf H}^{3}(\\Omega))\\cap W^{1,\\infty}(0,T;{\\bf H}^{1}(\\Omega))\\cap H^{1}(0,T;{\\bf H}^{2}(\\Omega))\\\\\n\t\t\t\t \\qquad\\cap H^{2}(0,T;{\\bf L}^{2}(\\Omega)),\\\\\n\t\t\t\t \\widetilde{\\eta}\\in L^{\\infty}(0,T;H^{9\/2}(\\Gamma_{s}))\\cap W^{1,\\infty}(0,T;H^{3}(\\Gamma_{s}))\\cap H^{1}(0,T;H^{4}(\\Gamma_{s}))\\cap W^{2,\\infty}(0,T;H^{1}(\\Gamma_{s}))\\\\\n\t\t\t\t \\qquad \\cap H^{2}(0,T;H^{2}(\\Gamma_{s}))\\cap H^{3}(0,T;L^{2}(\\Gamma_{s})),\n\t\t\t\t \\end{array}\\right.\n\t\t\t\t \\end{equation}\n\t\twe consider the following problem:\n\t\t\t\t \\begin{equation}\\label{3.2}\n\t\t\t\t \\left\\{\n\t\t\t\t \\begin{array}{lll}\n\t\t\t\t &{\\sigma_{t}}+\\widetilde{W}(\\widetilde{\\bf w},\\widetilde{\\eta})\n\t\t\t\t \\cdot \\nabla\\sigma=G_{1}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})&\\quad \\mbox{in}\\quad Q_{T},\n\t\t\t\t \\vspace{1.mm}\\\\\n\t\t\t\t &(\\widetilde\\sigma+\\overline{\\rho}){\\bf w}_{t}-\\mu \\Delta {\\bf w} -(\\mu+\\mu{'})\\nabla\\mathrm{div}{\\bf w}=G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})&\\quad \\mbox{in} \\quad {Q}_{T},\n\t\t\t\t \\vspace{1.mm}\\\\\n\t\t\t\t &{\\bf w}=0&\\quad \\mbox{on}\\quad \\Sigma_{T},\n\t\t\t\t \\vspace{1.mm}\\\\\n\t\t\t\t &{\\bf w}(\\cdot,0)={\\bf w}_{0}={\\bf u}_{0}-z\\eta_{1}\\vec{e}_{2}&\\quad \\mbox{in} \\quad \\Omega,\n\t\t\t\t \\vspace{1.mm}\\\\\n\t\t\t\t &\\sigma(\\cdot,0)=\\sigma_{0}={\\rho_{0}}-{\\overline\\rho}&\\quad\\mbox{in}\\quad \\Omega,\n\t\t\t\t \\vspace{1.mm}\\\\\n\t\t\t\t &\\eta_{tt}-\\beta \\eta_{xx}- \\delta\\eta_{txx}+\\alpha\\eta_{xxxx}=G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta}) &\\quad\\mbox{on}\\quad \\Sigma^{s}_{T},\n\t\t\t\t \\vspace{1.mm}\\\\\n\t\t\t\t &\\eta(0)=0\\quad \\mbox{and}\\quad \\eta_{t}(0)=\\eta_{1}&\\quad\\mbox{in}\\quad \\Gamma_{s},\n\t\t\t\t \\end{array} \\right.\n\t\t\t\t \\end{equation}\n\t\t\t\t where $G_{1},$ $G_{2},$ $G_{3}$ are as defined in \\eqref{1.22} and $\\widetilde{W}(\\widetilde{\\bf w},\\widetilde{\\eta})$ is defined as follows\n\t\t\t\t \\begin{equation}\\label{dotW}\n\t\t\t\t \\begin{array}{l}\n\t\t\t\t \\displaystyle\n\t\t\t\t \\widetilde{W}(\\widetilde{\\bf w},\\widetilde{\\eta})=\\begin{bmatrix}\n\t\t\t\t \\widetilde{w}_{1}\\\\\n\t\t\t\t \\frac{1}{(1+\\widetilde{\\eta})}(\\widetilde{w}_{2}-\\widetilde{w}_{1}z\\widetilde{\\eta}_{x})\n\t\t\t\t \\end{bmatrix},\\qquad \\left(\\widetilde{\\bf w}=\\begin{pmatrix}\n\t\t\t\t \\widetilde{w}_{1}\\\\\n\t\t\t\t \\widetilde{w}_{2}\n\t\t\t\t \\end{pmatrix}\\right).\n\t\t\t\t \\end{array}\n\t\t\t\t \\end{equation}\n\t\t\t\t \n\t\t\t\t It turns out that it will be important for us to check that $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ and $ G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ respectively coincide at time $t = 0$ with the values $G_2^0$ and $G_3^0$ computed in \\eqref{iG0} and \\eqref{G30}, and given as follows:\n\t\t\t\t \\begin{equation}\\label{iG0-bis}\n\\begin{split}\nG_{2}^0&=-P'(\\rho_{0})\\nabla\\rho_{0}-(\\delta\\eta_{1,xx}-(\\mu+2\\mu')({ u}_{0})_{2,z}\n+P(\\rho_{0}))z\\rho_{0}\\vec{e}_{2}\n+z\\rho_{0}({\\bf u}_{0})_{z}\\eta_{1}\\\\\n&-\\rho_{0}({\\bf u}_{0}\\cdot\\nabla){\\bf u}_{0}-(-\\mu\\Delta-(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\eta_{1}\\vec{e_{2}}),\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{G30-bis}\n\\begin{array}{l}\nG_{3}^0=-(\\mu+2\\mu')({u}_{0})_{2,z}+P(\\rho_{0}).\n\\end{array}\n\\end{equation} \n\t\t\t\t This will be imposed by assuming $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta},\\widetilde{\\eta}_{t})(\\cdot,0)=(\\sigma_{0},{\\bf w}_{0},0,\\eta_{1})$ and \n\t\t\t\t \\begin{equation}\\label{tetatt0}\n\t\t\t\t \\begin{array}{l}\n\t\t\t\t\\displaystyle \\widetilde{\\eta}_{tt}(\\cdot,0)=\\delta\\eta_{1,xx}-(\\mu+2\\mu')(u_{0})_{2,z}+P(\\rho_{0})\\,\\,\\mbox{in}\\,\\,\\Omega, \n\t\t\t\t \\\\\n\t \t\t \t\\displaystyle \\widetilde{\\bf w}_{t}(\\cdot,0)=\\frac{1}{\\rho_{0}}\\big(G_{2}^0-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div})({\\bf u}_{0}-z\\eta_{1}\\vec{e_{2}})\\big)\\,\\,\\mbox{in}\\,\\,\\Omega,\n\t\t\t\t \\end{array}\n\t\t\t\t \\end{equation}\n\t\t\t\tIndeed, under the above conditions, one can check from the expressions of $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ and $G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ that $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\mid_{t=0} = G_2^0$ and $G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\mid_{t=0} = G_3^0$.\n\t\t\t\t \\begin{lem}\\label{wellpos}\n\t\t\t\t\tLet the constant $\\delta_{0}$ be fixed by \\eqref{fixd0}.\n\t\t\t\t \tFor $T < \\overline{T}$, let us assume the following \n\t\t\t\t \t\\begin{equation}\\label{tsweY}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\,\\,\\mbox{satisfies}\\,\\,\\eqref{regswe},\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{tweq0*}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t\\widetilde{\\bf w}=0\\,\\,\\mathrm{on}\\,\\,\\Sigma_{T},\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{tswe0*}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t(\\widetilde{\\sigma}(\\cdot,0),\\widetilde{\\bf w}(\\cdot,0),\\widetilde{\\eta}(\\cdot,0),\\widetilde{\\eta}_{t}(\\cdot,0))=(\\rho_{0}-\\overline{\\rho},{\\bf u}_{0}-z\\eta_{1}\\vec{e}_{2},0,\\eta_{1})\\,\\,\\mbox{in}\\,\\,\\Omega,\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{etatt0*}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t\\eqref{tetatt0}\\,\\,\\mbox{holds},\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{1etad0*}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t1+\\widetilde{\\eta}(x,t)\\geqslant\\delta_{0}>0\\,\\,\\mathrm{on}\\,\\,\\Sigma^{s}_{T},\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{tsgmM*}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t\\displaystyle 0<\\frac{m}{2}\\leqslant\\widetilde{\\sigma}+\\overline{\\rho}\\leqslant 2M\\,\\,\\mathrm{in}\\,\\, Q_{T},\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \twhere $m$ and $M$ were fixed in \\eqref{cor0}.\\\\\n\t\t\t\t Then $G_{1}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta}),$ $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ and $G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ satisfy the following\n\t\t\t\t \\begin{equation}\\label{G123}\n\t\t\t\t \\begin{array}{ll}\n\t\t\t\t & G_{1}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\in L^{1}(0,T;H^{2}(\\Omega))\\cap L^{\\infty}(0,T;H^{1}(\\Omega)),\\\\\n\t\t\t\t & G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\in L^{2}(0,T;{\\bf H}^{1}(\\Omega))\\cap H^{1}(0,T;{\\bf L}^{2}(\\Omega))\\cap L^{\\infty}(0,T;{\\bf L}^{2}(\\Omega)),\\\\\n\t\t\t\t & G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\in L^{\\infty}(0,T;H^{1\/2}(\\Gamma_{s}))\\cap H^{1}(0,T;L^{2}(\\Gamma_{s})),\\\\\n\t\t\t\t & \\widetilde{W}(\\widetilde{\\bf w},\\widetilde{\\eta})\\in L^{1}(0,T;{\\bf H}^{3}(\\Omega))\\cap L^{\\infty}(0,T;{\\bf H}^{2}(\\Omega)), \\\\\n\t\t\t\t& G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\mid_{t=0} = G_2^0 \\text{ and } G_{3}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\mid_{t=0} = G_3^0.\n\t\t\t\t \\end{array}\n\t\t\t\t \\end{equation}\n\t\t\t\t \\end{lem}\n\t\t\t\t \\begin{proof}\n\t\t\t\t \tThe detailed computations to verify \\eqref{G123} follows from Lemma \\ref{eog1} (for estimates of $G_{1}$), Lemma \\ref{eog2} (for estimates of $G_{2}$), Lemma \\ref{eog3} (for estimates of $G_{3}$) and Lemma \\ref{eoWs} (for estimates of $\\widetilde{W}$) in the Section \\ref{estimates}.\n\t\t\t\t \\end{proof}\n\t\t\t\t Observe that the condition \\eqref{tweq0*} implies that $\\widetilde{W}(\\widetilde{\\bf w},\\widetilde{\\eta})\\cdot {\\bf n}=0$ (where $\\widetilde{W}$ is as defined in \\eqref{dotW}) on $\\Sigma_{T}.$ Hence in view of Lemma \\ref{wellpos}, for all $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ satisfying the conditions \\eqref{tsweY}-\\eqref{tweq0*}-\\eqref{tswe0*}-\\eqref{etatt0*}-\\eqref{1etad0*}\n\t\t\t\t -\\eqref{tsgmM*}, the system \\eqref{3.2} admits a unique solution as a consequence of Theorem \\ref{t21}, Theorem \\ref{t22} and Theorem \\ref{t23} in the space $Z^{T}_{1}\\times Y^{T}_{2}\\times Z^{T}_{3},$ where $Y^{T}_{2}$ is defined in \\eqref{dofYi}, $Z^{T}_{1}$ and $Z^{T}_{3}$ are defined as follows\n\t\t\t\t \\begin{equation}\\label{Z23}\n\t\t\t\t \\begin{array}{ll}\n\t\t\t\t & Z^{T}_{1}=\\{{\\rho}\\in C^{0}([0,T];H^{2}(\\Omega))\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; \\rho_{t}\\in L^{\\infty}(0,T; H^{1}(\\Omega))\\},\\\\\n\t\t\t\t & Z^{T}_{3}=\\{\\eta\\in L^{\\infty}(0,T; H^{9\/2}(\\Gamma_{s})),\\,\\eta(x,0)=0\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\; \\eta_{t}\\in L^{2}(0,T;{ H}^{4}(\\Gamma_{s}))\\cap C^{0}([0,T];{ H}^{3}(\\Gamma_{s})), \\\\[1.mm]\n\t\t\t\t &\\qquad\\qquad\\qquad\\qquad\\qquad\\eta_{tt}\\in L^{2}(0,T;{ H}^{2}(\\Gamma_{s}))\\cap C^{0}([0,T];{ H}^{1}(\\Gamma_{s})),\n\t\t\t\t \\eta_{ttt}\\in L^{2}(0,T;{ L}^{2}(\\Gamma_{s}))\\}.\n\t\t\t\t \\end{array}\n\t\t\t\t \\end{equation}\n\t\t\t\t Observe that the only difference between $Y^{T}_{1}$ (defined in \\eqref{dofYi}) and $Z^{T}_{1}$ is that the elements of $Y^{T}_{1}$ belongs to $C^{1}([0,T];H^{1}(\\Omega))$ while the elements of $Z^{T}_{1}$ are in $W^{1,\\infty}(0,T;H^{1}(\\Omega)).$ Also one observes that the elements of $Y^{T}_{3}$ (defined in \\eqref{dofYi}) are in $C^{0}([0,T];H^{9\/2}(\\Gamma_{s}))$ while $Z^{T}_{3}$ is only a subset of $L^{\\infty}(0,T;H^{9\/2}(\\Gamma_{s})).$\\\\\n\t\t\t\t Before defining a suitable fixed point map (in order to solve the non-linear problem \\eqref{chdb}), we will introduce a convex set $\\mathscr{C}_{T}$ (where we will show the existence of a fixed point). The set $\\mathscr{C}_{T}$ will be defined as a subset of $L^{2}(0,T;L^{2}(\\Omega))\\times L^{2}(0,T;{\\bf L}^{2}(\\Omega))\\times L^{2}(0,T;L^{2}(\\Gamma_{s}))$ such that the elements of $\\mathscr{C}_{T}$ satisfy some norm bounds and some conditions at initial time $t=0.$\\\\\n\t\t\t\t Let us make precise the assumptions which will be used to define the set $\\mathscr{C}_{T}.$\\\\ \n\t\t\t\t $Regularity\\,\\,assumptions\\,\\,and\\,\\,norm\\,\\,bounds\\,\\,of\\,\\, (\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$:\n\t\t\t\t\\begin{subequations}\\label{normbnd}\n\t\t\t\t \t\\begin{equation}\\label{tsB12}\n\t\t\t\t \t\\begin{array}{l}\n\t\t\t\t \t\\|\\widetilde{\\sigma}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\leqslant B_{1},\\quad\\|\\widetilde\\sigma_{t}\\|_{L^{\\infty}(0,T; H^{1}(\\Omega))}\\leqslant B_{2},\n\t\t\t\t \t\\end{array}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{twB3}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t\\|\\widetilde{\\bf w}\\|_{L^{\\infty}(0,T;{\\bf H}^{2}(\\Omega))}+\\|\\widetilde{\\bf w}\\|_{{L^{2}}(0,T;{\\bf H}^{3}(\\Omega))}\n\t\t\t\t \t+\\|\\widetilde{\\bf w}_{t}\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\n\t\t\t\t \t&+\\|\\widetilde{\\bf w}_{t}\\|_{L^{2}(0,T;{\\bf H}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t \t&+\\|\\widetilde{\\bf w}_{tt}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant B_{3},\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{teB4}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t\\|\\widetilde\\eta\\|_{L^{\\infty}(0,T;H^{9\/2}(\\Gamma_{s}))}+\t\\|\\widetilde\\eta_{t}\\|_{L^{\\infty}(0,T;H^{3}(\\Gamma_{s}))}&+\t\\|\\widetilde\\eta_{t}\\|_{L^{2}(0,T;H^{4}(\\Gamma_{s}))}+\t\\|\\widetilde\\eta_{tt}\\|_{L^{\\infty}(0,T; H^{1}(\\Gamma_{s}))}\\\\[1.mm]\n\t\t\t\t \t&+\t\\|\\widetilde\\eta_{tt}\\|_{L^{2}(0,T;H^{2}(\\Gamma_{s}))}\n\t\t\t\t \t+\\|\\widetilde\\eta_{ttt}\\|_{L^{2}(0,T;L^{2}(\\Gamma_{s}))}\\leqslant B_{4},\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation} \n\t\t\t\t \t\\vspace{1.mm}\\\\\n\t\t\t\t \t\\begin{equation}\\label{1etad0}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t1+\\widetilde{\\eta}(x,t)\\geqslant\\delta_{0}>0\\,\\,\\mathrm{on}\\,\\,\\Sigma^{s}_{T},\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{tsgmM}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t0<\\frac{m}{2}\\leqslant\\widetilde{\\sigma}+\\overline{\\rho}\\leqslant 2M\\,\\,\\mathrm{in}\\,\\, Q_{T},\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \\end{subequations} \n\t\t\t\t where $B_{i}$'s ($1\\leqslant i\\leqslant 4$) are positive constants and will be chosen in the sequel. The norm bound \\eqref{twB3} implicitly asserts (by interpolation) that $\\widetilde{\\bf w}$ is in $C^{0}([0,T];{\\bf H}^{5\/2}(\\Omega)).$\\\\\n\t\t\t\t $Assumptions\\,\\,on\\,\\,initial\\,\\,and\\,\\,boundary\\,\\,conditions$:\\\\\n\t\t\t\t \\begin{subequations}\\label{inbndc}\n\t\t\t\t \t\\begin{equation}\\label{tweq0}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t\\widetilde{\\bf w}=0\\,\\,\\mbox{on}\\,\\,\\Sigma_{T},\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{tswe0}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t(\\widetilde{\\sigma}(\\cdot,0),\\widetilde{\\bf w}(\\cdot,0),\\widetilde{\\eta}(\\cdot,0),\\widetilde{\\eta}_{t}(\\cdot,0))=(\\rho_{0}-\\overline{\\rho},{\\bf u}_{0}-z\\eta_{1}\\vec{e}_{2},0,\\eta_{1})\\,\\,\\mbox{in}\\,\\,\\Omega,\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{etatt0}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t\\widetilde{\\eta}_{tt}(\\cdot,0)=\\delta\\eta_{1,xx}-(\\mu+2\\mu')(u_{0})_{2,z}+P(\\rho_{0})\\,\\,\\mbox{in}\\,\\,\\Omega,\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \t\\begin{equation}\\label{twt0}\n\t\t\t\t \t\\begin{split}\n\t\t\t\t \t\\widetilde{\\bf w}_{t}(\\cdot,0)=\\frac{1}{\\rho_{0}}\\big(G_{2}^0-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div})({\\bf u}_{0}-z\\eta_{1}\\vec{e_{2}})\\big)\\,\\,\\mbox{in}\\,\\,\\Omega.\n\t\t\t\t \t\\end{split}\n\t\t\t\t \t\\end{equation}\n\t\t\t\t \\end{subequations}\n\t\t\t\t For $T<\\overline{T},$ let us define the following set\n\t\t\t\t \\begin{equation}\\label{defRT}\n\t\t\t\t \\begin{split}\n\t\t\t\t \\mathscr{C}_{T}(B_{1},B_{2},B_{3},B_{4})= \\{(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})&\\in L^{2}(0,T;L^{2}(\\Omega))\\times L^{2}(0,T;{\\bf L}^{2}(\\Omega))\\times L^{2}(0,T;L^{2}(\\Gamma_{s}))\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\;\\\\\n\t\t\t\t & \\mbox{the relations}\\, \\eqref{normbnd}-\\eqref{inbndc}\\,\\,\\mbox{are true}\\}.\n\t\t\t\t \\end{split}\n\t\t\t\t \\end{equation} \n\t\t\t\t Now for $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\in\\mathscr{C}_{T}(B_{1},B_{2},B_{3},B_{4}),$ let $(\\sigma,{\\bf w},\\eta)\\in Z_{1}^{T}\\times Y_{2}^{T}\\times Z_{3}^{T}$ (recall the definition of $Y^{T}_{2},$ from \\eqref{dofYi} and $Z^{T}_{1},$ $Z^{T}_{3}$ are defined in \\eqref{Z23}) be the solution of the problem \\eqref{3.2} corresponding to $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta}).$ This defines the map\n\t\t\t\t \\begin{equation}\\label{welposL}\n\t\t\t\t \\begin{matrix}\n\t\t\t\t L: \\mathscr{C}_{T}(B_{1},B_{2},B_{3},B_{4})&\\longrightarrow& Z_{1}^{T}\\times Y_{2}^{T}\\times Z_{3}^{T}\\\\\n\t\t\t\t (\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})&\\mapsto& ({\\sigma},{\\bf w},{\\eta}).\n\t\t\t\t \\end{matrix}\n\t\t\t\t \\end{equation}\n\t\t\t\t Now observe that if the map $L$ admits a fixed point $(\\sigma_{f},{\\bf w}_{f},\\eta_{f})$ on the set $\\mathscr{C}_{T}(B_{1},B_{2},B_{3},B_{4}),$ then the triplet $(\\sigma_{f},{\\bf w}_{f},\\eta_{f})$ is a solution to the system \\eqref{chdb}. \n\t\t\t\t Thus, our goal from now is to prove the existence of a fixed point to the map $L.$ In that direction we first show that for suitable parameters $B_{i}$ ($1\\leqslant i\\leqslant 4$) and $T,$ the set $\\mathscr{C}_{T}(B_{1},B_{2},B_{3},B_{4})$ is non-empty.\n\t\t\t\t \\begin{lem}\\label{nonempty}\n\t\t\t\t \tThere exists a constant $B_{0}^{*}>0$ such that for all $B_{i}\\geqslant B^{*}_{0}$ ($1\\leqslant i \\leqslant 4$) there exists $T^{*}_{0}(B_{1},B_{2},B_{3},B_{4} )\\in (0, \\min\\{1,\\overline{T}\\})$ such that for all $00\\quad\\mbox{on}\\quad \\Sigma^{s}_{T},$$\n\t\t\t\t \n\t\t\t\t \n\t\t\t\t \t$i.e$ $\\eta^{*}$ satisfies \\eqref{1etad0}.\\\\\n\t\t\t\t \t$Step\\, 3.$\\, One easily checks that $\\sigma^{*}={\\rho}_{0}-\\overline{\\rho}$ verifies \\eqref{tsB12} and \\eqref{tsgmM}.\\\\\n\t\t\t\t \tWe further observe that $(\\sigma^{*},{\\bf w}^{*},\\eta^{*})$ satisfies \\eqref{tweq0} and \\eqref{tswe0} automatically by construction.\\\\ \n\t\t\t\t \t So we have shown that if we choose $B^{*}_{0}$ (and hence $B_{i}\\geqslant B^{*}_{0},$ for all $1\\leqslant i\\leqslant 4$) as in \\eqref{coBi} and $0\\frac{d}{2},$ $0\\leqslant s\\leqslant r.$ If $v\\in H^{r}(\\Omega_{0})$ and $w\\in H^{s}(\\Omega_{0})$ then $vw\\in H^{s}(\\Omega_{0})$ with\n\t\t\t\t\t$$\\|vw\\|_{H^{s}(\\Omega_{0})}\\leqslant K(\\Omega_{0})\\|v\\|_{H^{r}(\\Omega_{0})}\\|w\\|_{H^{s}(\\Omega_{0})}.$$ \n\t\t\t\t\tSimilar estimates hold when $v$ and $w$ are vector valued functions i.e for ${\\bf v}\\in {\\bf H}^{r}(\\Omega_{0})$ and ${\\bf w}\\in {\\bf H}^{s}(\\Omega_{0}).$\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{lem}\\label{leoin}\n\t\t\t\t\tLet $T<\\overline{T}$ (recall that we have fixed $\\overline{T}$ in \\eqref{Tbar}). We assume that ${\\bf f}\\in {\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T}).$ As usual we use the notation ${\\bf f}_{z}$ to denote the directional derivative $\\partial_{z}{\\bf f}$ of ${\\bf f}$ with respect to $z.$ Also suppose that $\\Gamma_{s}$ is a smooth subset of $\\Gamma.$ Then the trace ${\\bf f}_{z}\\mid_{\\Sigma_{T}}$ on $\\Gamma_{s}$ (i.e the normal derivative of ${\\bf f}$ on $\\Gamma_{s}$) belongs to $H^{1\/6}(0,T;{\\bf L}^{2}(\\Gamma_{s})).$ In particular there exists a constant $K>0$ such that for all ${\\bf f}\\in {\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})$ we have the following\n\t\t\t\t\t\\begin{equation}\\label{ibin}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}_{z}\\mid_{\\Sigma_{T}}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Gamma_{s}))}\\leqslant T^{1\/6}K(\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}+\\|{\\bf f}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}),\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\twhere ${\\bf f}(0)$ denotes the function ${\\bf f}$ at time $t=0.$ We specify that in our case the space ${\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})$ is endowed with the following norm\n\t\t\t\t\t$$\\|{\\bf f}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}=\\|{\\bf f}\\|_{L^{2}(0,T;{\\bf H}^{2}(\\Omega))}+\\|{\\bf f}_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}.$$\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{remark}\n\t\t\t\t\tThe appearance of ${\\bf f}(0)$ in the inequality \\eqref{ibin} might seem redundant since for all, ${\\bf f}\\in {\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})$\n\t\t\t\t\t$$\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}\\leqslant K_{T}\\|{\\bf f}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}.$$ \n\t\t\t\t\tBut the constant $K_{T}$ there may depend on $T$ while the constant $K$ in \\eqref{ibin} is independent of $T.$ This is the reason why we prefer working with \\eqref{ibin}.\n\t\t\t\t\\end{remark}\n\t\t\t\t\\begin{proof}[Proof of Lemma \\ref{leoin}]\n\t\t\t\t\tWe have to estimate $\\|{\\bf f}_{z}\\mid_{\\Sigma_{T}}\\|_{L^{2}(0,T;L^{2}(\\Gamma_\n\t\t\t\t\t\t{s}))}.$ Using H\\\"{o}lder's inequality we get the following\n\t\t\t\t\t\\begin{equation}\\label{il32}\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\\left(\\int\\limits_{0}^{T}\\|{\\bf f}_{z}\\mid_{\\Sigma_{T}}\\|^{2}_{L^{2}(\\Gamma_{s})}\\right)^{1\/2}\\leqslant \\left(\\int\\limits_{0}^{{T}}\\|{\\bf f}_{z}\\mid_{\\Sigma_{T}}\\|^{3}_{L^{2}(\\Gamma_{s})}\\right)^{1\/3}T^{1\/6}\\leqslant K(\\Omega)\\left(\\int\\limits_{0}^{{T}}\\|{\\bf f}\\|^{3}_{{\\bf H}^{5\/3}{(\\Omega)}}\\right)^{1\/3}T^{1\/6}.\n\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tTo prove \\eqref{ibin}, in view of \\eqref{il32} it is enough to show the following inequality\n\t\t\t\t\t\t\\begin{equation}\\label{triain3}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\\|{\\bf f}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\Omega,\\overline{T})(\\|{\\bf f}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}+\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}).\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\tIn order to prove \\eqref{triain3}, first let us consider the solution ${\\bf{f}}^{*}$ of\n\t\t\t\t\t\\begin{equation}\\label{ellstf}\n\t\t\t\t\t\t\\left\\{ \\begin{array}{lll}\n\t\t\t\t\t\t\t{\\bf f}^{*}_{t}-\\Delta{\\bf f}^{*}=0&\\quad\\mbox{in}\\quad& Q_{T},\\\\[1.mm]\n\t\t\t\t\t\t\t{\\bf f}^{*}=0&\\quad\\mbox{on}\\quad &\\Sigma_{T},\\\\[1.mm]\n\t\t\t\t\t\t\t{\\bf f}^{*}(.,0)={\\bf f}(0)&\\quad\\mbox{in}\\quad&\\Omega.\n\t\t\t\t\t\t\\end{array}\\right.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tAs ${\\bf f}(0)\\in {\\bf H}^{1}_{0}(\\Omega),$ ${\\bf f}^{*}\\in {\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T}).$ It is also well known that there exists a constant $K(\\Omega)$ such that ${\\bf f}^{*}$ satisfies the following inequalities \n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\t\\begin{equation}\\label{inell}\n\t\t\t\t\t\t\\begin{array}{ll}\n\t\t\t\t\t\t\t(i)&\\|{\\bf f}^{*}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}\\leqslant K(\\Omega)\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)},\\\\[1.mm]\n\t\t\t\t\t\t\t(ii)& \\|{\\bf f}^{*}\\|_{L^{\\infty}(0,T;{\\bf H}^{1}_{0}(\\Omega))}+\\|{\\bf f}^{*}\\|_{L^{2}(0,T;{\\bf H}^{2}(\\Omega))}\\leqslant K(\\Omega)\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}.\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tNow we will estimate the norm of ${\\bf f}^{*}$ in $L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega)).$\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\tUsing interpolation we have for $a.e$ $t$\n\t\t\t\t\t$$\\|{\\bf f}^{*}(t)\\|_{{\\bf H}^{5\/3}(\\Omega)}\\leqslant K(\\Omega)\\|{\\bf f}^{*}(t)\\|^{2\/3}_{{\\bf H}^{2}(\\Omega)}\\|{\\bf f}^{*}(t)\\|^{1\/3}_{{\\bf H}^{1}_{0}(\\Omega)}.$$\n\t\t\t\t\tFrom the last inequality one obtains the following\n\t\t\t\t\t\\begin{equation}\\label{iafin}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\|{\\bf f}^{*}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}=\\left(\\int\\limits_{0}^{T}\\|{\\bf f}^{*}(t)\\|^{3}_{{\\bf H}^{5\/3}(\\Omega)}\\right)^{1\/3}\\leqslant K(\\Omega)\\|{\\bf f}^{*}\\|^{1\/3}_{L^{\\infty}(0,T;{\\bf H}^{1}_{0}(\\Omega))}\\|{\\bf f}^{*}\\|^{2\/3}_{L^{2}(0,T;{\\bf H}^{2}(\\Omega))}.\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tHence using inequality $(ii)$ of \\eqref{inell} in \\eqref{iafin} we obtain\n\t\t\t\t\t\\begin{equation}\\label{iafin1}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}^{*}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\Omega)\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}.\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tNow let us observe that $({\\bf f}-{\\bf f}^{*})(0)=0.$ Extend the function $({\\bf f}-{\\bf f}^{*})$ by defining it zero in the time interval $(T-\\overline{T},0)$ (the extended function is also denoted by $({\\bf f}-{\\bf f}^{*})$). In what follows we will use the notation \n\t\t\t\t\t\\begin{equation}\\nonumber\n\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t{Q_{T-\\overline{T},T}}=\\Omega\\times(T-\\overline{T},T).\n\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tWe also introduce the space ${\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})$ which is defined as in \\eqref{fntlsp} with $Q_{T}$ replaced by ${Q_{T-\\overline{T},T}}.$\\\\\n\t\t\t\t\tOne can check that the extended function $({\\bf f}-{\\bf f}^{*})\\in {\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})$ and\n\t\t\t\t\t\\begin{equation}\\label{noeq}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|({\\bf f}-{\\bf f}^{*})\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})}=\\|({\\bf f}-{\\bf f}^{*})\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}.\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tAgain due to the embedding ${\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})\\hookrightarrow H^{1\/6}(T-\\overline{T},{T};{\\bf H}^{5\/3}(\\Omega))$ we have the following \n\t\t\t\t\t\\begin{equation}\\label{il321}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}-{\\bf f}^{*}\\|_{{H}^{1\/6}(T-\\overline{T},T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\overline{T},\\Omega)\\|{\\bf f}-{\\bf f}^{*}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})}.\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tSince $H^{1\/6}(T-\\overline{T},T)$ is continuously embedded into $L^{3}(T-\\overline{T},T)$, hence from \\eqref{il321}\n\t\t\t\t\t\\begin{equation}\\label{il32ne}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}-{\\bf f}^{*}\\|_{{L}^{3}(T-\\overline{T},T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\overline{T},\\Omega)\\|{\\bf f}-{\\bf f}^{*}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})}.\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tUse of triangle inequality furnishes the following\n\t\t\t\t\t\\begin{equation}\\label{triain}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\|{\\bf f}-{\\bf f}^{*}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}+\\|{\\bf f}^{*}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}).\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tIncorporate inequalities \\eqref{iafin1} and \\eqref{il32ne} in \\eqref{triain} in order to obtain\n\t\t\t\t\t\\begin{equation}\\label{triain1}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\Omega,\\overline{T})(\\|{\\bf f}-{\\bf f}^{*}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}({Q_{T-\\overline{T},T}})}+\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}).\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tIn view of the equality \\eqref{noeq} we can obtain the following from \\eqref{triain1},\n\t\t\t\t\t\\begin{equation}\\label{triain2}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{\\bf f}\\|_{L^{3}(0,T;{\\bf H}^{5\/3}(\\Omega))}\\leqslant K(\\Omega,\\overline{T})(\\|{\\bf f}-{\\bf f}^{*}\\|_{{\\bf H}^{2,1}_{\\Sigma_{T}}(Q_{T})}+\\|{\\bf f}(0)\\|_{{\\bf H}^{1}_{0}(\\Omega)}).\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tOnce again use triangle inequality and \\eqref{inell} $(i)$, in order to prove \\eqref{triain3}.\\\\\n\t\t\t\t\tFinally use \\eqref{triain3} in \\eqref{il32} to show \\eqref{ibin}. This completes the proof.\n\t\t\t\t\\end{proof}\n\t\t\t\n\t\t\t\tThe following lemma is a simple consequence of the fundamental theorem of calculus, whose proof is left to the reader.\n\t\t\t\t\\begin{lem}\\label{futcl}\n\t\t\t\t\tFix $i\\geqslant 0$ and a domain $\\Omega_{0}$ in $\\mathbb{R}^{d}$ ($d$ is either $1$ or $2$). Then there exists a constant $K>0$ such that for all $\\psi\\in H^{1}(0,T;H^{i}(\\Omega_{0})),$ the following holds \n\t\t\t\t\t\\begin{equation}\\label{eoLit}\n\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\|\\psi\\|_{L^{\\infty}(0,T;H^{i}(\\Omega_{0}))}\\leqslant K(\\|\\psi(0)\\|_{H^{i}(\\Omega_{0})}+T^{1\/2}\\|\\psi_{t}\\|_{L^{2}(0,T;H^{i}(\\Omega_{0}))}),\n\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\\end{equation} \n\t\t\t\t\twhere $\\psi(0)$ denotes $\\psi$ at time $t=0.$ The inequality \\eqref{eoLit} is true even for a vector valued function ${\\Psi}\\in H^{1}(0,T;{\\bf H}^{i}(\\Omega_{0})).$\n\t\t\t\t\\end{lem}\n\t\t\t\t\\subsubsection{Estimates of $G_{1},$ $G_{2},$ $G_{3}$ and $\\widetilde{W}$} \\label{estimates} \n\t\t\t\t\\begin{lem}\\label{eog1}\n\t\t\t\t\tLet $B^{*}_{0}$ and $T^{*}_{0}$ are as in Lemma \\ref{nonempty} and $B_{i}\\geqslant B^{*}_{0}$ ($\\forall\\, 1\\leqslant i\\leqslant 4$). Then there exist $K_{1}=K_{1}(B_{1},B_{2},B_{3},B_{4})>0$ and $K_{2}>0$ such that for all $00,$ $K_{4}=K_{4}(B_{1},B_{4})>0$ and $K_{5}>0$ such that for all $0 1,$\n\t\t\t\t\t\t$$(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\in C^{0}([0,T];H^{2}(\\Omega))\\,\\,\\mbox{and}\\,\\,(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-2}\\in C^{0}([0,T];H^{2}(\\Omega))$$ \n\t\t\t\t\t\tand\n\t\t\t\t\t\t\\begin{equation}\\label{eog2*}\n\t\t\t\t\t\t\\left\\{ \\begin{split}\n\t\t\t\t\t\t\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\leqslant K\\|\\widetilde{\\sigma}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\leqslant K(B_{1}),\\\\\n\t\t\t\t\t\t\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-2}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\leqslant K\\|\\widetilde{\\sigma}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\leqslant K(B_{1}).\n\t\t\t\t\t\t\\end{split}\\right.\n\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t (i)\tWe first estimate $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in $L^{2}(0,T;{\\bf H}^{1}(\\Omega)).$\\\\ \n\t\t\t\t\t\t{Estimate of $P'(\\widetilde{\\sigma}+\\overline{\\rho})\\nabla\\widetilde{\\sigma}$ in $L^{2}(0,T;{\\bf H}^{1}(\\Omega))$}:\\\\\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog21}\n\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t&\\|P'(\\widetilde{\\sigma}+\\overline{\\rho})\\nabla\\widetilde{\\sigma}\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t& \\leqslant T^{1\/2}\\|P'(\\widetilde{\\sigma}+\\overline{\\rho})\\nabla\\widetilde{\\sigma}\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant T^{1\/2}K(\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\|\\nabla{\\widetilde{\\sigma}}\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad(\\mbox{using the definition of}\\,P\\,\\mbox{and}\\,\\mbox{Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t\t& \\leqslant K(B_{1})T^{1\/2},\\quad(\\mbox{using}\\,\\eqref{tsB12}).\n\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t{Estimate of $z\\widetilde{\\eta}_{tt}(\\widetilde{\\sigma}+\\overline{\\rho})\\vec{e}_{2}-(\\mu \\Delta +(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\widetilde{\\eta}_{t}\\vec{e}_{2})$ in $L^{2}(0,T;{\\bf H}^{1}(\\Omega))$:}\\\\\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog22}\n\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t&\\|z\\widetilde{\\eta}_{tt}(\\widetilde{\\sigma}+\\overline{\\rho})\\vec{e}_{2}-(\\mu \\Delta +(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\widetilde{\\eta}_{t}\\vec{e}_{2})\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant T^{1\/2}\\|z\\widetilde{\\eta}_{tt}(\\widetilde{\\sigma}+\\overline{\\rho})\\vec{e}_{2}-(\\mu \\Delta +(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\widetilde{\\eta}_{t}\\vec{e}_{2})\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant T^{1\/2}K(\\|\\widetilde{\\eta}_{tt}\\|_{L^{\\infty}(0,T; H^{1}(\\Omega))}\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}+\\|\\widetilde{\\eta}_{t}\\|_{L^{\\infty}(0,T; {H}^{3}(\\Omega))})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad(\\mbox{using Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant K(B_{1},B_{4})T^{1\/2},\\quad(\\mbox{using}\\,\\eqref{tsB12},\\eqref{teB4}).\n\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t{Estimate of $F_{2}(\\widetilde{\\sigma}+\\overline{\\rho},\\widetilde{\\bf w}+z\\widetilde{\\eta}_{t}\\vec{e}_{2},\\widetilde{\\eta})$ (defined in \\eqref{F123}) in $L^{2}(0,T;{\\bf H}^{1}(\\Omega))$:}\n\t\t\t\t\t\t We will only estimate the terms of $F_{2}(\\widetilde{\\sigma}+\\overline{\\rho},\\widetilde{\\bf w}+z\\widetilde{\\eta}_{t}\\vec{e}_{2},\\widetilde{\\eta})$ which are the most intricate to deal with. The others are left to the reader.\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog23}\n\t\t\t\t\t\t\t\\begin{array}{ll}\n\t\t\t\t\t\t\t(a)&\\quad\\|\\widetilde{\\eta}(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}})\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t& \\leqslant T^{1\/2}\\|\\widetilde{\\eta}(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}})\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant T^{1\/2}K(\\|\\widetilde{\\eta}\\|_{L^{\\infty}(0,T;{H}^{2}(\\Gamma_{s}))}\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;{H}^{2}(\\Omega))}\n\t\t\t\t\t\t\t\\|(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}})\\|_{{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad(\\mbox{using Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant K(B_{1},B_{3},B_{4})T^{1\/2},\\quad(\\mbox{using}\\,\\eqref{tsB12},\\eqref{twB3},\\eqref{teB4}).\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog24}\n\t\t\t\t\t\t\t\\begin{array}{ll}\n\t\t\t\t\t\t\t(b)&\\quad\\|z(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{z}+\\widetilde{\\eta}_{t}\\vec{e_{2}})\\widetilde{\\eta}_{t}\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t& \\leqslant T^{1\/2}\\|z(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{z}+\\widetilde{\\eta}_{t}\\vec{e_{2}})\\widetilde{\\eta}_{t}\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t &\\leqslant T^{1\/2} K(\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\|(\\widetilde{\\bf w}_{z}+\\widetilde{\\eta}_{t}\\vec{e_{2}})\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\\|\\widetilde{\\eta}_{t}\\|_{L^{\\infty}(0,T;H^{2}(\\Gamma_{s}))})\\\\[1.mm]\n\t &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad(\\mbox{using Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant K(B_{1},B_{3},B_{4})T^{1\/2},\\quad(\\mbox{using}\\,\\eqref{tsB12},\\eqref{twB3},\\eqref{teB4}).\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog25}\n\t\t\t\t\t\t\t\\begin{array}{ll}\n\t\t\t\t\t\t\t(c) &\\displaystyle\\quad\\left\\|\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\eta^{2}_{x}}{(1+\\widetilde\\eta)}\\right\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\displaystyle\\leqslant K(\\|\\widetilde{\\bf w}_{zz}\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\|\\widetilde{\\eta}^{2}_{x}\\|_{L^{\\infty}(0,T;H^{2}(\\Gamma_{s}))}\\left\\|\\frac{1}{(1+\\widetilde{\\eta})}\\right\\|_{L^{\\infty}(0,T;H^{9\/2}(\\Gamma_{s}))})\\\\\n\t\t\t\t\t\t\t&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\,\\,(\\mbox{using Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\displaystyle\\leqslant T^{1\/2}K(\\|\\widetilde{\\bf w}_{zz}\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\|\\widetilde{\\eta}^{2}_{xt}\\|_{L^{2}(0,T;H^{2}(\\Gamma_{s}))}\\left\\|\\frac{1}{(1+\\widetilde{\\eta})}\\right\\|_{L^{\\infty}(0,T;H^{9\/2}(\\Gamma_{s}))})\\\\[1.mm]\n\t\t\t\t\t\t\t&\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad(\\mbox{using}\\,\\eqref{eoLit}\\,\\mbox{with}\\,\\psi=\\widetilde{\\eta}^{2}_{x}\\,\\mbox{and the fact}\\,\\widetilde{\\eta}_{x}(,0)=0)\\\\[1.mm]\n\t\t\t\t\t\t\t&\\displaystyle\\leqslant K(B_{3},B_{4})T^{1\/2},\\quad(\\mbox{using}\\,\\eqref{tsB12},\\eqref{twB3},\\eqref{teB4}\\,\\mbox{and}\\,\\eqref{eog1*}).\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t \\,\\,\\,\\,\\,\\,\\quad$(d)$\\quad Using arguments similar to that in the computation \\eqref{eog21} we show the following\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog26}\n\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t\\|(\\widetilde{\\eta} P'\\widetilde{\\sigma}_{x}- P'\\widetilde{\\sigma}_{z}z\\widetilde{\\eta}_{x})\\vec{e_{1}}\\|_{L^{2}(0,T; {\\bf H}^{1}(\\Omega))}\n\t\t\t\t\t\t\t\\leqslant K(B_{1},B_{4})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tNow the reader can deal with the other terms using similar arguments in order to prove\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog27}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|{F}_{2}(\\widetilde{\\sigma}+\\overline{\\rho},(\\widetilde{\\bf{w}}+z\\widetilde\\eta_{t}\\vec{e_{2}}),\\widetilde{\\eta})\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}\\leqslant K(B_{1},B_{3},B_{4})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tCombining the estimates \\eqref{eog21}, \\eqref{eog22} and \\eqref{eog27} we conclude the proof of the inequality \\eqref{eg2}$(i)$.\\\\[2.mm]\n\t\t\t\t\t\t\t(ii)\n\t\t\t\t\t\t\tWe now estimate $\\|(G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta}))_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}.$\\\\\n\t\t\t\t\t\t\t{Estimate of $(P'(\\widetilde{\\sigma}+\\overline{\\rho})\\nabla\\widetilde{\\sigma})_{t}$ in $L^{2}(0,T;{\\bf L}^{2}(\\Omega))$:}\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog28}\n\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t &\\|(P'(\\widetilde{\\sigma}+\\overline{\\rho})\\nabla\\widetilde{\\sigma})_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t &\\leqslant T^{1\/2}\\|(P'(\\widetilde{\\sigma}+\\overline{\\rho})\\nabla\\widetilde{\\sigma})_{t}\\|_{L^{\\infty}(0,T;{\\bf L}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t &\\leqslant T^{1\/2}K(\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{(\\gamma-2)}\\widetilde{\\sigma}_{t}\\nabla{\\widetilde\\sigma}\\|_{L^{\\infty}(0,T;{\\bf L}^{2}(\\Omega))}+\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\nabla\\widetilde{\\sigma}_{t}\\|_{L^{\\infty}(0,T;{\\bf L}^{2}(\\Omega))})\\\\[1.mm]\n\t\t\t\t\t\t &\\leqslant T^{1\/2}K(\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-2}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\|\\widetilde{\\sigma}_{t}\\|_{L^{\\infty}(0,T; H^{1}(\\Omega))}\\|\\nabla\\widetilde{\\sigma}\\|_{L^{\\infty}(0,T; {\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t &\\qquad+\\|(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\|\\nabla\\widetilde{\\sigma}_{t}\\|_{L^{\\infty}(0,T;{\\bf L}^{2}(\\Omega))})\n\t\t\t\t\t\t \\qquad(\\mbox{using Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t &\\leqslant K(B_{1},B_{2})T^{1\/2},\\quad(\\mbox{using}\\,\\eqref{tsB12}\\,\\mbox{and}\\,\\eqref{eog2*}).\n\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t{Estimate of $(z\\widetilde{\\eta}_{tt}(\\widetilde{\\sigma}+\\overline{\\rho})\\vec{e_{2}}-(\\mu \\Delta +(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\widetilde{\\eta}_{t}\\vec{e}_{2}))_{t}$ in $L^{2}(0,T;{\\bf L}^{2}(\\Omega))$:}\\\\\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog29}\n\t\t\t\t\t\t\t\\begin{split}\n\t\t\t\t\t\t\t&\\|(z\\widetilde{\\eta}_{tt}(\\widetilde{\\sigma}+\\overline{\\rho})\\vec{e_{2}}-(\\mu \\Delta +(\\mu+\\mu{'})\\nabla\\mathrm{div})(z\\widetilde{\\eta}_{t}\\vec{e}_{2}))_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant T^{1\/2}K(\\|\\widetilde{\\eta}_{tt}\\|_{L^{\\infty}(0,T; H^{1}(\\Gamma_{s}))}\\|\\widetilde{\\sigma}_{t}\\|_{L^{\\infty}(0,T; H^{1}(\\Omega))})\n\t\t\t\t\t\t\t +K(\\|\\widetilde{\\eta}_{ttt}\\|_{L^{2}(0,T;L^{2}(\\Gamma_{s}))}\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t & \\qquad+\\|\\widetilde{\\eta}_{tt}\\|_{L^{2}(0,T;H^{2}(\\Gamma_{s}))})\n\t\t\t\t\t\t\t \\qquad(\\mbox{using Lemma}\\,\\ref{lfpss})\\\\[1.mm]\n\t\t\t\t\t\t\t & \\leqslant K(B_{2},B_{4})T^{1\/2}+K(B_{1},B_{4}).\n\t\t\t\t\t\t\t\\end{split}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t{Estimate of $({F}_{2}(\\widetilde{\\sigma}+\\overline{\\rho},(\\widetilde{\\bf w}+z\\widetilde\\eta_{t}\\vec{e_{2}}),\\widetilde{\\eta}))_{t}$ in $L^{2}(0,T;{\\bf L}^{2}(\\Omega))$:}\\\\\n\t\t\t\t\t\t\\begin{equation}\\label{eog210}\n\t\t\t\t\t\t\t\\begin{array}{ll}\n\t\t\t\t\t\t\t(a)\\quad&\\|(\\widetilde{\\eta}(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}}))_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t&\\leqslant K(\\|(\\widetilde{\\eta}_{t}(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}}))\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\n\t\t\t\t\t\t\t+\\|\\widetilde{\\eta}\\widetilde{\\sigma}_{t}(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}})\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t &\\qquad+\\|(\\widetilde{\\eta}(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{tt}+z\\widetilde{\\eta}_{ttt}\\vec{e_{2}}))\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))})\\\\[1.mm]\n\t\t\t\t\t\t\t &\\leqslant T^{1\/2}K(\\|\\widetilde{\\eta}_{t}\\|_{L^{\\infty}(0,T;H^{2}(\\Gamma_{s}))}\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\|(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}})\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t &\\qquad+\\|\\widetilde{\\eta}\\|_{L^{\\infty}(0,T;H^{2}(\\Gamma_{s}))}\\|\\widetilde{\\sigma}_{t}\\|_{L^{\\infty}(0,T; H^{1}(\\Omega))}\\|(\\widetilde{\\bf w}_{t}+z\\widetilde{\\eta}_{tt}\\vec{e_{2}})\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))})\\\\[1.mm]\n\t\t\t\t\t\t\t &\\qquad+\\|\\widetilde{\\eta}\\|_{L^{\\infty}(0,T;H^{2}(\\Gamma_{s}))}\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\|(\\widetilde{\\bf w}_{tt}+z\\widetilde{\\eta}_{ttt}\\vec{e_{2}})\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\\\[1.mm]\n\t\t\t\t\t\t\t & \\leqslant T^{1\/2}K(B_{1},B_{2},B_{3},B_{4})+T^{1\/2}\\|\\widetilde{\\eta}_{t}\\|_{L^{2}(0,T;H^{2}(\\Gamma_{s}))}\\|(\\widetilde{\\sigma}+\\overline{\\rho})\\|_{L^{\\infty}(0,T;H^{2}(\\Omega))}\\\\\n\t\t\t\t\t\t\t &\\qquad\\|(\\widetilde{\\bf w}_{tt}+z\\widetilde{\\eta}_{ttt}\\vec{e_{2}})\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\n\t\t\t\t\t\t\t \\qquad(\\mbox{using }\\,\\eqref{eoLit}\\,\\mbox{with}\\,\\psi=\\widetilde{\\eta}\\,\\mbox{and the fact}\\,\\widetilde{\\eta}(,0)=0)\\\\[1.mm]\n\t\t\t\t\t\t\t& \\leqslant K(B_{1},B_{2},B_{3},B_{4})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\\,\\,\\,\\,\\,\\,\\,$(b)$\\quad Using similar estimates we can have the following\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog211}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|(z(\\widetilde{\\sigma}+\\overline{\\rho})(\\widetilde{\\bf w}_{z}+\\widetilde{\\eta}_{t}\\vec{e_{2}})\\eta_{t})_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{1},B_{2},B_{3},B_{4})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\t\\,\\,\\,\\,\\,\\,\\,$(c)$\\quad Now we estimate \n\t\t\t\t\t\t\t$$\t\\left\\|\\left(\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\widetilde{\\eta}^{2}_{x}}{(1+\\widetilde{\\eta})}\\right)_{t}\\right\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}.$$\n\t\t\t\t\t\t\tTo start with, we have the following identity of distributional derivatives\n\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\\label{HugeTerms}\n\t\t\t\t\t\t\t\t\\left(\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\widetilde{\\eta}^{2}_{x}}{(1+\\widetilde{\\eta})}\\right)_{t}=\\frac{z^{2}\\widetilde{\\bf w}_{tzz}\\widetilde{\\eta}^{2}_{x}}{(1+\\widetilde{\\eta})}+\\frac{2\\widetilde{\\eta}_{x}\\widetilde{\\eta}_{xt}\\widetilde{\\bf w}_{zz}}{(1+\\widetilde{\\eta})}-\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\widetilde{\\eta}^{2}_{x}\\widetilde{\\eta}_{t}}{(1+\\widetilde{\\eta})^{2}}.\n\t\t\t\t\t\t\t\\end{equation}\nWe now estimate the first term of the summands. Using \\eqref{eog1*} one obtains\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog212}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\left\\|\\frac{z^{2}\\widetilde{\\bf w}_{tzz}\\widetilde{\\eta}^{2}_{x}}{(1+\\widetilde{\\eta})}\\right\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{4})(\\|\\widetilde{\\bf w}_{tzz}\\|_{{L^{2}(0,T;{\\bf L}^2(\\Omega))}}\\|\\widetilde{\\eta}_{x}\\|_{{{L^{\\infty}(\\Sigma^{s}_{T})}}}^2).\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tNow we use inequality \\eqref{eoLit} and $\\widetilde{\\eta}_{x}(\\cdot,0)=0$ to get\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog213}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\|\\widetilde{\\eta}_{x}\\|_{{{L^{\\infty}(\\Sigma^{s}_{T})}}}\n\t\t\t\t\t\t\t\\leq\n\t\t\t\t\t\t\tC \\|\\widetilde{\\eta}_{x}\\|_{L^{\\infty}(0,T;{ H}^{2}(\\Gamma_{s}))}\\leqslant K(B_{3})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tHence we use \\eqref{eog213} in \\eqref{eog212} to obtain\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog214}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\left\\|\\frac{z^{2}\\widetilde{\\bf w}_{tzz}\\widetilde{\\eta}^{2}_{x}}{(1+\\widetilde{\\eta})}\\right\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{3},B_{4})T.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation} \n\t\t\t\t\t\t\tFor the second and third summands of \\eqref{HugeTerms}, we similarly obtain:\n\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\left\\|\\frac{2\\widetilde{\\eta}_{x}\\widetilde{\\eta}_{xt}\\widetilde{\\bf w}_{zz}}{(1+\\widetilde{\\eta})}\\right\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{3},B_{4})T^{1\/2}, \\text{ and } \n\t\t\t\t\t\t\t%\n\t\t\t\t\t\t\t\\left\\|-\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\widetilde{\\eta}^{2}_{x}\\widetilde{\\eta}_{t}}{(1+\\widetilde{\\eta})^{2}} \\right\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{3},B_{4})T.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation} \n\n\t\t\t\t\t\t\tSo altogether we get\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog215}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\displaystyle\n\t\t\t\t\t\t\t\\left\\|\\left(\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\widetilde\\eta^{2}_{x}}{(1+\\widetilde\\eta)}\\right)_{t}\\right\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{3},B_{4})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tThe remaining terms in the expression of ${F}_{2}$ are relatively easier to deal with and hence we leave the details to the reader to show\n\t\t\t\t\t\t\t\\begin{equation}\\label{eog216}\n\t\t\t\t\t\t\t\\begin{array}{l}\n\t\t\t\t\t\t\t\\|({F}_{2}(\\widetilde{\\sigma}+\\overline{\\rho},\\widetilde{\\bf w}+z\\widetilde\\eta_{t}\\vec{e_{2}},\\widetilde{\\eta}))_{t}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}\\leqslant K(B_{1},B_{2},B_{3},B_{4})T^{1\/2}.\n\t\t\t\t\t\t\t\\end{array}\n\t\t\t\t\t\t\t\\end{equation}\n\t\t\t\t\t\t\tHence combining the estimates \\eqref{eog28}, \\eqref{eog29} and \\eqref{eog216} one gets \\eqref{eg2}$(ii).$\\\\[1.mm]\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t(iii) In \\eqref{eoLit} replace $\\psi$ by $G_{2}(\\widetilde{\\bf w},\\widetilde{\\sigma},\\widetilde{\\eta})$ and use the estimate \\eqref{eg2}$(ii)$ to prove \\eqref{eg2}$(iii).$\n\t\t\t\t\t\\end{proof}\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{lem}\\label{eog3}\n\t\t\t\t\t\t\tLet $B^{*}_{0}$ and $T^{*}_{0}$ are as in Lemma \\ref{nonempty} and $B_{i}\\geqslant B^{*}_{0}$ ($\\forall\\, 1\\leqslant i\\leqslant 4$). Then there exist $K_{6}>0$ and $K_{7}=K_{7}(B_{1},B_{2},B_{3},B_{4})>0$ such that for all $00,$ and $K_{9}=K_{9}(B_{3},B_{4})>0$ such that for all $00,&\\quad \\mbox{on}\\quad \\Sigma^{s}_{T}\\\\[1.mm]\n\t\t\t \t\\displaystyle \\frac{m}{2}\\leqslant \\sigma(x,z,t)+\\overline{\\rho}\\leqslant 2M,&\\quad \\mbox{in}\\quad Q_{T}.\n\t\t\t \t\\end{array}\n\t\t\t \t\\end{equation}\n\t\t\t \tAgain it follows from the equation \\eqref{3.2}$_{2}$ that ${\\bf w}_{t}(0)$ satisfies the condition \\eqref{twt0}. Similarly one uses \\eqref{3.2}$_{6}$ to show that $\\eta_{tt}(\\cdot,0)$ satisfies \\eqref{etatt0}. Now we set\n\t\t\t \t$$T^{*}=T^{*}(B_{1},B_{2},B_{3},B_{4})=T^{*}_{5}.$$\n\t\t\t \tHence if $B_{i}$ ($\\forall\\,1\\leqslant i\\leqslant 4$) is chosen as in \\eqref{chooseBi1}, \\eqref{chooseBi2} and \\eqref{chooseBi3} and $00\\quad \\mbox{on}\\quad \\Sigma^{s}_{T}.\n\t\t\t \n\t\t\t \t\\end{array}\n\t\t\t \t\\end{equation}\n\t\t\t \tObserve that the weak* convergence of $\\widetilde{\\sigma}_{n}$ to $\\overline{\\sigma}$ in $L^{\\infty}(0,T;L^{2}(\\Omega))$ is enough to conclude that (since $\\widetilde{\\sigma}_{n}$ satisfies \\eqref{tsgmM})\n\t\t\t \t\\begin{equation}\\label{bndsigma}\n\t\t\t \t\\begin{array}{l}\n\t\t\t \t\\displaystyle \\frac{m}{2}\\leqslant \\overline\\sigma(x,z,t)+\\overline{\\rho}\\leqslant 2M\\quad \\mbox{in}\\quad Q_{T}.\n\t\t\t \t\\end{array}\n\t\t\t \t\\end{equation}\n\t\t\t \tUsing the strong convergence of $(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ to $(\\overline{\\sigma},\\overline{\\bf w},\\overline{\\eta})$ in $\\mathcal{X}$ furnishes\n\t\t\t \t \\begin{equation}\\label{3.31}\n\t\t\t \t \\begin{array}{ll}\n\t\t\t \t \\overline{\\bf w}(\\cdot,0)=({\\bf u}_{0}-z\\eta_{1}\\vec{e_{2}})&\\quad \\mbox{in}\\quad \\Omega,\\\\\n\t\t\t \t \\overline{\\sigma}(\\cdot,0)=\\sigma_{0}&\\quad\\mbox{in}\\quad \\Omega.\n\t\t\t \t \\end{array}\n\t\t\t \t \\end{equation}\n\t\t\t \t Now we can use the uniform bounds of $\\|\\widetilde{\\bf w}_{n,t}\\|_{L^{\\infty}(0,T;{\\bf H}^{1}(\\Omega))}$ and $\\|\\widetilde{\\bf w}_{n,tt}\\|_{L^{2}(0,T;{\\bf L}^{2}(\\Omega))}$ and the Aubin Lions lemma to have the convergence $\\widetilde{\\bf w}_{n,t}\\rightarrow \\overline{\\bf w}_{t}$ in $C^{0}([0,T];{\\bf L}^{2}(\\Omega)).$ Consequently\n\t\t\t \t \\begin{equation}\\label{icbwt}\n\t\t\t \t \\begin{split}\n\t\t\t \t \\overline{\\bf w}_{t}(.,0)=\\frac{1}{\\rho_{0}}(G_{2}^0-(-\\mu \\Delta -(\\mu+\\mu{'})\\nabla\\mathrm{div})({\\bf u}_{0}-z\\eta_{1}\\vec{e}_{2})).\n\t\t\t \t \\end{split}\n\t\t\t \t \\end{equation}\n\t\t\t \tSo combining \\eqref{3.27}-\\eqref{3.28}-\\eqref{inoeta}-\\eqref{3.29}-\\eqref{3.30}-\\eqref{3.32}-\\eqref{bndsigma}-\\eqref{3.31}-\\eqref{icbwt} we conclude that the limit point $(\\overline{\\sigma},\\overline{\\bf w},\\overline{\\eta})\\in \\mathscr{C}_{T}$ and hence $\\mathscr{C}_{T}$ is closed in $\\mathcal{X}$.\n\t\t\t \t\\newline\n\t\t\t \tOnce again using Aubin Lions lemma we get that $\\mathscr{C}_{T}$ is a compact subset of $\\mathcal{X}.$ \n\t\t\t \\end{proof}\n\t\t\t \tNow to apply Schauder's fixed point theorem one only needs to prove that $L$ is continuous on $\\mathscr{C}_{T}.$ \n\t\t\t \t\\begin{lem}\\label{cont}\n\t\t\t \t\tLet $\\mathscr{C}_{T}$ be the set in \\eqref{TCT}. The map $L$ is continuous from $\\mathscr{C}_{T}$ into itself for the topology of $\\mathcal{X}.$\n\t\t\t \t\\end{lem}\n\t\t\t \t\\begin{proof}\n\t\t\t \t Suppose that $(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})\\in \\mathscr{C}_{T},$ converges to $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ strongly in $\\mathcal{X}.$ Then, according to Lemma \\ref{com}, $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})\\in \\mathscr{C}_{T}.$ We thus set $(\\widehat{\\sigma}_{n},\\widehat{\\bf w}_{n},\\widehat{\\eta}_{n})= L(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n}),$ $(\\widehat{\\sigma},\\widehat{\\bf w},\\widehat{\\eta})= L(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta}).$ Our goal is to show that $(\\widehat{\\sigma}_{n},\\widehat{\\bf w}_{n},\\widehat{\\eta}_{n})$ strongly converges to $(\\widehat{\\sigma},\\widehat{\\bf w},\\widehat{\\eta})$ in $\\mathcal{X}.$ Using that $(\\widehat{\\sigma}_{n},\\widehat{\\bf w}_{n},\\widehat{\\eta}_{n})$ belongs to $\\mathscr{C}_{T}$ (see Lemma \\ref{inclusion}) we get that there exists a triplet $(\\overline{\\sigma},\\overline{\\bf w},\\overline{\\eta})$ such that up to a subsequence \n\t\t\t \t \\begin{equation}\\label{weakconv}\n\t\t\t \t \\begin{array}{lll}\n\t\t\t \t & \\widehat{\\sigma}_{n}\\stackrel{\\ast}{\\rightharpoonup}\\overline{\\sigma}&\\,\\,\\mbox{in}\\,\\, L^{\\infty}(0,T;H^{2}(\\Omega))\\cap W^{1,\\infty}(0,T;H^{1}(\\Omega)),\\\\\n\t\t\t \t & \\widehat{\\bf w}_{n}\\rightharpoonup \\overline{\\bf w}&\\,\\,\\mbox{in}\\,\\, L^{2}(0,T;{\\bf H}^{3}(\\Omega))\\cap H^{1}(0,T;{\\bf H}^{2}(\\Omega))\\cap H^{2}(0,T;{\\bf L}^{2}(\\Omega)),\\\\\n\t\t\t \t & \\widehat{\\bf w}_{n}\\stackrel{\\ast}{\\rightharpoonup} \\overline{\\bf w}&\\,\\,\\mbox{in}\\,\\, L^{\\infty}(0,T;{\\bf H}^{2}(\\Omega))\\cap W^{1,\\infty}(0,T;{\\bf H}^{1}(\\Omega)),\\\\\n\t\t\t \t & \\widehat{\\eta}_{n}\\rightharpoonup \\overline{\\eta}&\\,\\,\\mbox{in}\\,\\, H^{1}(0,T;H^{4}(\\Gamma_{s}))\\cap H^{2}(0,T;H^{2}(\\Gamma_{s}))\\cap H^{3}(0,T;L^{2}(\\Gamma_{s})),\\\\\n\t\t\t \t & \\widehat{\\eta}_{n}\\stackrel{\\ast}{\\rightharpoonup} \\overline{\\eta}&\\,\\,\\mbox{in}\\,\\, L^{\\infty}(0,T;H^{9\/2}(\\Gamma_{s}))\\cap W^{1,\\infty}(0,T;H^{3}(\\Gamma_{s}))\\cap W^{2,\\infty}(0,T;H^{1}(\\Gamma_{s})).\n\t\t\t \t \\end{array}\n\t\t\t \t \\end{equation}\n\t\t\t \t The compactness result proved in Lemma \\ref{com} provides the strong convergence in $\\mathcal{X}$ i.e, up to a subsequence, $(\\widehat{\\sigma}_{n},\\widehat{\\bf w}_{n},\\widehat{\\eta}_{n})$ converges strongly in $\\mathcal{X}$ to $(\\overline{\\sigma},\\overline{\\bf w},\\overline{\\eta}).$ It is clear that in order to prove that the map $L$ is continuous it is enough to show that $(\\overline{\\sigma},\\overline{\\bf w},\\overline{\\eta})=(\\widehat{\\sigma},\\widehat{\\bf w},\\widehat{\\eta}).$ This will be verified in the following steps.\\\\\n\t\t\t \t$(i)$ We first claim that $G_{2}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ converges weakly to $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in $L^{2}(0,T;{\\bf L}^{2}(\\Omega)).$\\\\\n\t\t\t \t Since $(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ belongs to $\\mathscr{C}_{T}$ and we have fixed $B_{i}$ (for all $1\\leqslant i\\leqslant 4$) and $T,$ one can use Lemma \\ref{eog2} to show that $\\|G_{2}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})\\|_{L^{2}(0,T;L^{2}(\\Omega))}$ is uniformly bounded. Hence, to prove our claim it is enough to show that $G_{2}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ converges to $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in $\\mathcal{D}'(Q_{T})$ ($\\mathcal{D}'(Q_{T})$ is the space of distributions on $Q_{T}$).\\\\\n\t\t\t \t \n\t\t\t \n\t\t\t \n\t\t\t \tLet us consider the term $\\displaystyle\\frac{\\widetilde{\\bf w}_{n,zz}z^{2}\\widetilde{\\eta}^{2}_{n,x}}{(1+\\widetilde{\\eta}_{n})}.$\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t \n\t\t\t \tFrom the uniform norm bound over $\\|\\widetilde{\\bf w}_{n,zz}\\|_{L^{2}(0,T;{\\bf H}^{1}(\\Omega))}$ we get that $\\widetilde{\\bf w}_{n,zz}$ converges weakly in $L^{2}(0,T;{\\bf H}^{1}(\\Omega))$ to $\\widetilde{\\bf w}_{zz}.$ Since $\\widetilde{\\eta}_{n}$ strongly converges to $\\widetilde{\\eta}$ in $C^{0}([0,T];H^{2}(\\Gamma_{s}))$ and both $\\widetilde{\\eta}_{n}$ and $\\widetilde{\\eta}$ satisfy \\eqref{1etad0}, $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta}_{n})}$ and $\\widetilde{\\eta}_{n,x}$ converge strongly to $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta})}$ and $\\widetilde{\\eta}_{x}$ respectively in the spaces $C^{0}([0,T];H^{2}(\\Gamma_{s}))$ and $C^{0}([0,T];H^{1}(\\Gamma_{s})).$\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \tHence one gets in particular the strong convergence of $\\widetilde{\\eta}^{2}_{n,x}$ to $\\widetilde{\\eta}^{2}_{x}$ in the space $C^{0}([0,T];L^{2}(\\Gamma_{s})).$ This implies that $\\displaystyle\\frac{\\widetilde{\\bf w}_{n,zz}z^{2}\\widetilde{\\eta}^{2}_{n,x}}{(1+\\widetilde{\\eta}_{n})}$ converges to $\\displaystyle\\frac{\\widetilde{\\bf w}_{zz}z^{2}\\widetilde{\\eta}^{2}_{x}}{(1+\\widetilde{\\eta})}$ weakly in $L^{2}(0,T;L^{1}(\\Omega))$ and hence particularly in the space $\\mathcal{D}'(Q_{T}).$\\\\\n\t\t\t \tNow we consider the term $P'\\widetilde{\\sigma}_{n,z}z\\widetilde{\\eta}_{n,x}\\vec{e}_{1}=(\\widetilde{\\sigma}_{n}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{n,z}z\\widetilde{\\eta}_{n,x}\\vec{e}_{1}.$ Since $\\|(\\widetilde{\\sigma}_{n}+\\overline{\\rho})\\|_{C^{0}(0,T;H^{2}(\\Omega))}$ is uniformly bounded so is $\\|(\\widetilde{\\sigma}_{n}+\\overline{\\rho})^{\\gamma-1}\\|_{C^{0}(0,T;H^{2}(\\Omega))}$ and hence $(\\widetilde{\\sigma}_{n}+\\overline{\\rho})^{\\gamma-1}$ converges weakly to $(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}$ in $L^{2}(0,T;H^{2}(\\Omega)).$ We also have that $\\widetilde{\\sigma}_{n,z}$ converges strongly to $\\widetilde{\\sigma}_{z}$ in $C^{0}([0,T];L^{2}(\\Omega)).$ Hence $(\\widetilde{\\sigma}_{n}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{n,z}$ converges weakly to $(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{z}$ in $L^{2}(0,T;L^{2}(\\Omega)).$ Now the strong convergence of $\\widetilde{\\eta}_{n,x}$ to $\\widetilde{\\eta}_{x}$ in $C^{0}([0,T];H^{1}(\\Gamma_{s}))$ furnish that $(\\widetilde{\\sigma}_{n}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{n,z}z\\widetilde{\\eta}_{n,x}$ weakly converges to $(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{z}z\\widetilde{\\eta}_{x}$ in $L^{2}(0,T;L^{1}(\\Omega)).$ Hence $(\\widetilde{\\sigma}_{n}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{n,z}z\\widetilde{\\eta}_{n,x}\\vec{e}_{1}$ converges to $(\\widetilde{\\sigma}+\\overline{\\rho})^{\\gamma-1}\\widetilde{\\sigma}_{z}z\\widetilde{\\eta}_{x}\\vec{e}_{1}$ in the space $\\mathcal{D}'(Q_{T}).$\\\\\n\t\t\t \tWe can apply similar line of arguments to prove that $G_{2}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ converges to $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in $\\mathcal{D}'(Q_{T}).$ Hence we have proved that $G_{2}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ converges to $G_{2}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ weakly in $L^{2}(0,T;{\\bf L}^{2}(\\Omega)).$\n\t\t\t \t\\newline\n\t\t\t \tAlso observe that $(\\widetilde{\\sigma}_{n}+\\overline{\\rho})$ converges strongly to $(\\widetilde{\\sigma}+\\overline{\\rho})$ in $C^{0}([0,T];H^{1}(\\Omega))$ and $\\widehat{\\bf w}_{n,t},$ $(-\\mu\\Delta-(\\mu'+\\mu)\\nabla(\\mbox{div}))\\widehat{\\bf w}_{n}$ converge up to a subsequence weakly to $\\displaystyle\\overline{\\bf w}_{t}$ and $\\displaystyle(-\\mu\\Delta-(\\mu'+\\mu)\\nabla(\\mbox{div}))\\overline{\\bf w}$ respectively in the spaces $L^{2}(0,T;{\\bf H}^{2}(\\Omega))$ and $L^{2}(0,T;{\\bf H}^{1}(\\Omega)).$ Hence up to a subsequence one obtains in particular the following convergence\n\t\t\t \t$$(\\widetilde\\sigma_{n}+\\overline{\\rho})\\widehat{\\bf w}_{n,t}-\\mu\\Delta\\widehat{\\bf w}_{n}-(\\mu'+\\mu)\\nabla(\\mbox{div}\\widehat {\\bf w}_{n})\\rightharpoonup (\\widetilde\\sigma+\\overline{\\rho})\\overline{\\bf w}_{t}-\\mu\\Delta\\overline{\\bf w}-(\\mu'+\\mu)\\nabla(\\mbox{div}\\overline{\\bf w})\\quad\\mbox{in}\\quad L^{2}(0,T;{\\bf L}^{2}(\\Omega)).$$ \n\t\t\t \tNow consider \\eqref{3.2}$_{2}$ with $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ and ${\\bf w}$ replaced respectively by $(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ and $\\widehat{\\bf w}_{n}.$ The weak convergences discussed so far allow to pass to the limits in both sides of this equation. So using the uniqueness of weak solution for the linear problem \\eqref{2.1.1} we conclude that $\\overline{\\bf w}=\\widehat{\\bf w}.$\\\\\n\t\t\t \t$(ii)$ Now we claim that $G_{1}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ converges weakly to $G_{1}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in $L^{2}(0,T;L^{2}(\\Omega)).$\\\\ \n\t\t\t \tLet us consider the term $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta}_{n})}(\\widetilde{ w}_{n})_{1,z}z\\widetilde{\\eta}_{n,x}(\\widetilde{\\sigma}_{n}+\\overline{\\rho}).$ We already know that $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta}_{n})}$ and $\\widetilde{\\eta}_{n,x}$ converge strongly to $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta})}$ and $\\widetilde{\\eta}_{x}$ respectively in the spaces $C^{0}([0,T];H^{2}(\\Gamma_{s})$ and $C^{0}([0,T];H^{1}(\\Gamma_{s})).$ One also observes that $(\\widetilde{w}_{n})_{1,z}$ weakly converges to $\\widetilde{w}_{1,z}$ in $L^{2}(0,T;{H}^{2}(\\Omega))$ (since $\\widetilde{\\bf w}_{n}\\rightharpoonup \\widetilde{\\bf w}$ in $L^{2}(0,T;{\\bf H}^{3}(\\Omega))$). Finally the strong convergence of $(\\widetilde{\\sigma}_{n}+\\overline{\\rho})$ to $(\\widetilde{\\sigma}+\\overline{\\rho})$ in $C^{0}([0,T];H^{1}(\\Omega))$ furnish the weak convergence of $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta}_{n})}(\\widetilde{ w}_{n})_{1,z}z\\widetilde{\\eta}_{n,x}(\\widetilde{\\sigma}_{n}+\\overline{\\rho})$ to $\\displaystyle\\frac{1}{(1+\\widetilde{\\eta})}(\\widetilde{ w})_{1,z}z\\widetilde{\\eta}_{x}(\\widetilde{\\sigma}+\\overline{\\rho})$ in $L^{2}(0,T;L^{2}(\\Omega)).$ We can apply similar arguments for other terms in the expression of $G_{1}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in order to prove the weak convergence of $G_{1}(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ to $G_{1}(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ in $L^{2}(0,T;L^{2}(\\Omega)).$\\\\\n\t\t\t \t We further observe that $\\nabla\\widehat{\\sigma}_{n}$ strongly converges to $\\nabla\\overline{\\sigma}$ in $C^{0}([0,T];L^{2}(\\Omega)).$ Since $(\\widetilde{ w}_{n})_{1}$ weakly converges to $\\widetilde{w}_{1}$ in $L^{2}(0,T;H^{3}(\\Omega)),$ $(\\widetilde{\\eta}_{n})_{x}$ strongly converges to $\\widetilde{\\eta}_{x}$ in $L^{\\infty}(\\Sigma^{s}_{T})$ (because $(\\widetilde{\\eta}_{n})_{x}$ strongly converges to $\\widetilde{\\eta}_{x}$ in $C^{0}([0,T];H^{1}(\\Gamma_{s}))$ and the embedding $H^{1}(\\Gamma_{s})\\hookrightarrow L^{\\infty}(\\Gamma_{s})$ is continuous) and $\\frac{1}{(1+\\widetilde{\\eta}_{n})}$ strongly converges to $\\frac{1}{(1+\\widetilde{\\eta})}$ in $C^{0}([0,T];H^{2}(\\Gamma_{s}))$, $\\frac{1}{(1+\\widetilde{\\eta}_{n})}(\\widetilde{w}_{n})_{1}z(\\widetilde{\\eta}_{n})_{x}(\\widehat{\\sigma}_{n})_{z}$ weakly converges to $\\frac{1}{(1+\\widetilde{\\eta})}\\widetilde{w}_{1}z\\widetilde{\\eta}_{x}\\widehat{\\sigma}_{z}$ in $L^{2}(0,T;L^{2}(\\Omega)).$ Besides, up to a subsequence $(\\widehat{\\sigma}_{n})_{t}$ weakly converges to $\\overline{\\sigma}_{t}$ in $L^{2}(0,T;L^{2}(\\Omega)).$ Hence up to a subsequence we have \n\t\t\t $$(\\widehat{\\sigma}_{n})_{t}+\\begin{bmatrix}\n\t\t\t (\\widetilde{w}_{n})_{1}\\\\\n\t\t\t \\frac{1}{(1+\\widetilde{\\eta}_{n})}((\\widetilde{w}_{n})_{2}-(\\widetilde{ w}_{n})_{1}z(\\widetilde{\\eta}_{n})_{x})\n\t\t\t \\end{bmatrix}\\cdot\\nabla\\widehat{\\sigma}_{n}\\rightharpoonup {\\overline{\\sigma}_{t}}+\\begin{bmatrix}\n\t\t\t \\widetilde{ w}_{1}\\\\\n\t\t\t \\frac{1}{(1+\\widetilde{\\eta})}(\\widetilde{w}_{2}-\\widetilde{ w}_{1}z\\widetilde\\eta_{x})\n\t\t\t \\end{bmatrix}\n\t\t\t \\cdot \\nabla\\overline{\\sigma}\\quad\\mbox{in}\\quad L^{2}(0,T;L^{2}(\\Omega)).$$\n\t\t\t \tNow consider \\eqref{3.2}$_{1}$ with $(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})$ and ${\\sigma}$ replaced respectively by $(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})$ and $\\widehat{\\sigma}_{n}.$ The weak type convergences discussed so far allow to pass to the limits in both sides of this equation. Hence from uniqueness of weak solution of the linear problem \\eqref{2.2.1} we conclude that $\\overline{\\sigma}=\\widehat{\\sigma}.$\\\\\n\t\t\t \t$(iii)$ One can use similar line of arguments as used so far to show that $G_{3}{(\\widetilde{\\sigma}_{n},\\widetilde{\\bf w}_{n},\\widetilde{\\eta}_{n})}$ converges weakly to $G_{3}{(\\widetilde{\\sigma},\\widetilde{\\bf w},\\widetilde{\\eta})}$ in $L^{2}(0,T;L^{2}(\\Gamma_{s})).$ Using the norm bounds of $\\widehat{\\eta}_{n}$ (since $(\\widehat{\\sigma}_{n},\\widehat{\\bf w}_{n},\\widehat{\\eta}_{n})\\in \\mathscr{C}_{T}$) we can prove that up to a subsequence the left hand side of \\eqref{3.2}$_{6}$ with $\\eta$ replaced by $\\widehat{\\eta}_{n}$ converges weakly to $$\\displaystyle\\overline{\\eta}_{tt}-\\beta \\overline{\\eta}_{xx}- \\delta\\overline{\\eta}_{txx}+\\alpha\\overline{\\eta}_{xxxx}$$\n\t\t\t \tin $L^{2}(0,T;L^{2}(\\Gamma_{s})).$ Now the uniqueness of weak solution to the problem \\eqref{2.3.1} furnishes $\\overline{\\eta}=\\widehat{\\eta}.$\n\t\t\t \tHence the proof of Lemma \\ref{cont} is complete.\n\t\t\t \t\\end{proof}\n\t\t\t \t\\subsection{Conclusion}\\label{conc}\n\t\t\t The following properties hold\\\\\n\t\t\t \t(i) The convex set $\\mathscr{C}_{T}$ is non-empty (Lemma \\ref{nonempty}) and is a compact subset of $\\mathcal{X}$ (Lemma \\ref{com}).\\\\\n\t\t\t \t(ii) The map $L$, defined in \\eqref{welposL}, is continuous on $\\mathscr{C}_{T}$ in the topology of $\\mathcal{X}$ (Lemma \\ref{cont}).\n\t\t\t\t\\\\\n\t\t\t\t(iii) The map $L$ maps $\\mathscr{C}_{T}$ to itself (Lemma \\ref{inclusion}).\\\\ \n\t\t\t \tThus, all the assumptions of Schauder fixed point theorem are satisfied by the map $L$ on $\\mathscr{C}_{T},$ endowed with the topology of $\\mathcal{X}.$ Therefore, Schauder fixed point theorem yields a fixed point $(\\sigma_{f},{\\bf w}_{f},\\eta_{f})$ of the map $L$ in $\\mathscr{C}_{T}.$ From the definition of the map $L$, one has $(\\sigma_{f},{\\bf w}_{f},\\eta_{f})\\in Z^{T}_{1}\\times Y^{T}_{2}\\times Z^{T}_{3}.$ Hence we have the following time continuities (since still now one only has the regularities \\eqref{eg1} of $G_{1}(\\sigma_{f},{\\bf w}_{f},\\eta_{f}),$ \\eqref{eg2} of $G_{2}(\\sigma_{f},{\\bf w}_{f},\\eta_{f})$ and \\eqref{eg3} of $G_{3}(\\sigma_{f},{\\bf w}_{f},\\eta_{f})$)\n\t\t\t \t\\begin{equation}\\label{timecont}\n\t\t\t \t\\begin{array}{l}\n\t\t\t \t{\\sigma}_{f}\\in C^{0}([0,T];H^{2}(\\Omega)),\\\\\n\t\t\t \t{\\bf w}_{f}\\in C^{0}([0,T];{\\bf H}^{5\/2}(\\Omega))\\cap C^{1}([0,T];{\\bf H}^{1}(\\Omega)),\\\\\n\t\t\t \t\\eta_{f} \\in C^{0}([0,T];H^{4}(\\Gamma_{s}))\\cap C^{1}([0,T];H^{3}(\\Gamma_{s}))\\cap C^{2}([0,T];H^{1}(\\Gamma_{s})).\n\t\t\t \t\\end{array}\n\t\t\t \t\\end{equation}\n\t\t\t \tThe regularities \\eqref{timecont} can be used to further check that $G_{1}(\\sigma_{f},{\\bf w}_{f},\\eta_{f})\\in C^{0}([0,T];H^{1}(\\Omega))$ and $G_{3}(\\sigma_{f},{\\bf w}_{f},\\eta_{f})\\in C^{0}([0,T];H^{1\/2}(\\Gamma_{s})).$ Hence we use Corollary \\ref{dencor} and the Corollary \\ref{timebeam} to obtain the following\n\t\t\t \t\\begin{equation}\\nonumber\n\t\t\t \t\\begin{array}{l}\n\t\t\t \t(\\sigma_{f})_{t}\\in C^{0}([0,T];H^{1}(\\Omega))\\,\\,\\mbox{and}\\,\\, \\eta_{f}\\in C^{0}([0,T];H^{9\/2}(\\Gamma_{s})).\n\t\t\t \t\\end{array}\n\t\t\t \t\\end{equation} \n\t\t\t \t Hence, $(\\sigma_{f},{\\bf w}_{f},\\eta_{f})\\in Y^{T}_{1}\\times Y^{T}_{2}\\times Y^{T}_{3}.$ The trajectory $(\\sigma_{f},{\\bf w}_{f},\\eta_{f})$ solves the nonlinear problem \\eqref{chdb} in $Y^{T}_{1}\\times Y^{T}_{2}\\times Y^{T}_{3}.$ Consequently the system \\eqref{1.21} admits a solution. This further implies that the original system \\eqref{1.1}-\\eqref{1.2}-\\eqref{1.3} admits a strong solution in sense of the Definition \\ref{doss}. Finally the proof of Theorem \\ref{main} is complete.\n\t\t\t \t\\end{proof}\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t \n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\n\n\\bibliographystyle{plain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\\subsection{Physical motivation} The study of quantized vortex dynamics in Bose-Einstein condensates (BECs)\nis a topic of intense experimental and theoretical investigations. A particular interesting situation is created when the BEC is stirred through an\nexternal {\\it rotating} confinement potential. Indeed, if the rotation speed exceeds some critical value \n{\\it vortices} and, more generally, {\\it vortex lattices} are being created, see, e.g., \\cite{A, CPRY} for a broader introduction. \n\nFrom a mathematical point of view, rotating BECs can be described within the realm of a mean-field model, the so-called \n{\\it Gross-Pitaevskii equation} \\cite{PiSt}. In the following, we shall assume, without loss of generality, that the system rotates around the $z$-axis with a \ngiven speed $\\Omega \\in \\R$. Placing ourselves in the associated rotating reference frame, \nthe corresponding mathematical model is a {\\it nonlinear \nSchr\\\"odinger equation} (NLS) given by\n\\begin{equation}\n\\label{NLS_rot}\ni \\partial_t \\psi = -\\frac{1}{2} \\Delta \\psi + \\lambda |\\psi|^{2} \\psi + V(x) \\psi - \\Omega L \\psi .\n\\end{equation}\nHere, $t\\in \\R$, $x\\in \\R^d$ with $d=3$, or $d=2$, respectively. The latter corresponds to the assumption of homogeneity of the BEC \nalong the $z$-axis (see, e.g., \\cite{BMSW, MeSp}, for a rigorous scaling limit from three to effective two-dimensional models for BEC). The parameter $\\lambda \\ge 0$ describes \nthe strength of the inter-particle interaction, which in this work is assumed to be repulsive.\nThe potential $V$ describes the magnetic trap and is usually taken in the form of a harmonic oscillator, i.e.\n\\begin{equation}\n\\label{Vquadr}\nV(x)=\\frac{1}{2} \\omega^2 |x|^2, \\quad \\omega \\in \\R.\n\\end{equation}\nHere, and in the following, we choose $V$ to be rotationally symmetric for simplicity. All our results can be easily generalized to the case of \nan anisotropic harmonic oscillator. \nFinally, $ \\Omega L\\psi$ describes the rotation around the $z$-axis, where \n\\begin{equation}\\label{eq:angular_momentum}\n L \\psi:= - i (x_1 \\partial_{x_2} \\psi - x_2 \\partial_{x_1}\\psi)\\equiv -i x^\\perp \\cdot \\nabla \\psi,\n\\end{equation}\ndenotes the corresponding quantum mechanical rotation operator. \n\nMost rigorous mathematical results on vortex creation are based on standing wave solutions of \\eqref{NLS_rot}, i.e. \nsolutions of the form $\\psi(t,x) = \\varphi(x) e^{-i \\mu t}$, $\\mu\\in \\R$, which leads to the following nonlinear elliptic equation\n\\begin{equation}\n\\label{stat_NLS}\n-\\frac{1}{2} \\Delta \\varphi + \\lambda |\\varphi|^{2} \\varphi + V(x) \\varphi - \\Omega L \\varphi - \\mu \\varphi =0.\n\\end{equation}\nEquation \\eqref{stat_NLS} can be interpreted as the \nEuler-Lagrange equation of the associated Gross-Pitaevskii energy functional \\cite{PiSt, S}:\n\\begin{equation}\\label{GPenergy}\nE_{\\rm GP}(\\varphi):= \\int_{\\R^d} \\bigg( \\frac{1}{2} |\\nabla \\varphi|^2 + V(x) |\\psi|^2 + \\frac{\\lambda}{2}|\\varphi |^{4} - \\Omega \\overline{\\varphi}\nL \\varphi \\bigg) \\;dx,\n\\end{equation}\nOne possible way of constructing solutions to \\eqref{stat_NLS} is thus to minimize \\eqref{GPenergy} under the constraint $\\|\\varphi \\|^2_{L^2}=M$, \nwhere $M>0$ denotes a given mass. This consequently yields a {\\it chemical potential} $\\mu=\\mu(M)\\ge 0$ playing the role of a Lagrange multiplier. \nIn order to do so, one requires $\\omega>|\\Omega| $ which ensures that $E_{\\rm GP}$ is bounded from below. \nPhysically speaking, this condition means that the confinement potential $V(x)$ is stronger than the rotational forces, \nensuring that the BEC stays trapped.\nWithin this framework, it was proved in \\cite{S} \nthat the hereby obtained physical {\\it ground states}, i.e. energy minimizing solutions of \\eqref{stat_NLS}, undergo a symmetry breaking (of the rotational symmetry) \nfor sufficiently strong $\\Omega$ and\/or $\\lambda \\ge 0$. The latter is interpreted as the onset of vortex-lattice creation. \n\nOn the other hand, it is often argued in the physics literature that a small amount of dissipation must be present for the \nexperimental realization of stable vortex lattices, cf. \\cite{CMK, KT, KNKM}. \nIn order to describe such dissipative effects, not present in the original Gross-Pitaevskii equation \\eqref{NLS_rot}, \nthe following phenomenological model has been proposed in \\cite{TKU} and subsequently been studied in, e.g., \\cite{CGKT, CP, GAF, KT, KNKM}:\n\\begin{equation}\\label{dissGP}\n(i\\beta-\\gamma) \\partial_t \\psi = -\\frac{1}{2} \\Delta \\psi + \\lambda |\\psi|^{2} \\psi + V(x) \\psi - \\Omega L \\psi - \\mu \\psi.\n\\end{equation}\nHere $\\beta\\in\\R$ and $\\gamma>0$ are physical parameters whose ratio describes the strength of the dissipation. (In \\cite{GAF} the authors use \nformal arguments based on quantum kinetic theory to obtain $\\frac{\\gamma}{\\beta} \\approx\n0.03$.) Note that any {\\it time-independent solution} $\\psi = \\varphi(x)$ of \\eqref{dissGP} solves the stationary NLS \\eqref{stat_NLS}. \nIn contrast to \\eqref{NLS_rot}, equation \\eqref{dissGP} is no longer Hamiltonian and only makes sense for $t\\in \\R_+$.\n\n\\subsection{Mathematical setting and main result} \nThis work is devoted to a rigorous mathematical analysis of \\eqref{dissGP}. In particular, we shall be interested in the long time behavior of its solutions as $t \\to +\\infty$. \nTo this end, it is convenient to re-scale time such that $\\beta^2+\\gamma^2=1$. \nThen we can write \\[\ni\\beta-\\gamma = -e^{i\\vartheta}, \\quad \\text{for some $\\vartheta\\in\\Big(-\\frac \\pi 2,\\frac \\pi 2\\Big)$.}\n\\] \nNote that by doing so, the real part of $e^{i\\vartheta}$ has the same (positive) sign as $\\gamma > 0$. We shall thus be concerned with the following initial value problem for $(t,x)\\in \\R_+\\times \\R^d$ and $d=2,3$:\n\\begin{equation}\n\\label{NLS_diss}\n-e^{i\\vartheta}\\partial_t \\psi = -\\frac{1}{2} \\Delta \\psi + \\lambda |\\psi|^{2\\sigma} \\psi + V(x) \\psi - \\Omega L \\psi - \\mu \\psi, \\quad \n\\psi _{\\mid t =0} =\\psi_0(x),\n\\end{equation}\nwhere $\\psi_0$ will be chosen in some appropriate function space (see below), and $\\sigma > 0$ a generalized nonlinearity. \nFormally, the usual Gross-Pitaevskii equation \\eqref{NLS_rot} is obtained from \\eqref{NLS_diss} in the limit $\\vartheta \\to \\pm \\frac{\\pi}{2}$. On the other hand, if $\\vartheta =0$ the Hamiltonian character \nof the model is completely lost and \\eqref{NLS_diss} instead resembles a nonlinear parabolic \nequation of {\\it complex Ginzburg-Landau} (GL) type, cf. \\cite{AK} for a review on this type of models.\n\n\nEquation \\eqref{NLS_diss} can thus be seen as a hybrid between the Gross-Pitaevskii\/Non-linear Schr\\\"odinger equation and \nthe complex Ginzburg-Landau equation. Both kind of models have been extensively studied in the mathematical literature:\nFor local and global well-posedness results on NLS, \nwith or without quadratic potentials $V$, we refer to \\cite{Caze, Car05, Car09}. Allowing for the inclusion of a rotation term, the initial value problem for \\eqref{NLS_rot} \nhas been analyzed in \\cite{AMS}.\nSimilarly, well-posedness results for the complex GL equation in various spaces can be found in \\cite{DGL, GV1,GV2}. The existence and basic properties of \na global attractor for solutions to GL (on bounded domains $D\\subset \\R^d$) are studied in \\cite{Te} and \\cite{MM}. \nMoreover, the so-called {\\it inviscid limit} which links solutions of \nGL to solutions of NLS has been established in \\cite{Wu}. However, none of the aforementioned results directly apply to the model \n\\eqref{NLS_diss}, which involves an unbounded (quadratic) potential $V$ and a rotation term, \nneither of which have been included in the studies on GL cited above. \nOne should also note that the GL equation in its most general form allows for \ndifferent complex pre-factors in front of the Laplacian and the nonlinearity. In our case those\npre-factors coincide, allowing for a closer connection to NLS.\nVery recently, a similar type of such restricted GL models with $\\lambda <0$ (and without potential and rotation terms) has \nbeen studied in \\cite{CDW1, CDW2} as an ``intermediate step\"\nbetween the NLS and the nonlinear heat equation. \nFinally, we also mention that equation \\eqref{dissGP} with $\\beta=0$ is used to \nnumerically obtain the Gross-Pitaevskii ground states, cf. \\cite{BaoDu, CST}. \n\n\nAs announced before, we shall mainly be interested in the long time behavior of solutions to \\eqref{NLS_diss}. In view of this the main result \nof our paper can be stated in the following form:\n\n\\begin{theorem}\\label{thmmain} Let $d\\in \\{2,3\\}$, $\\omega>|\\Omega|$, $\\vartheta \\in (-\\frac \\pi 2 , \\frac \\pi 2 )$, $\\lambda \\ge 0$, and $0< \\sigma < \\frac{d}{2(d-2)}$. \nThen for any \n\\[\n\\psi_0 \\in \\Sigma:=\\big\\{f\\in H^1(\\R^d)\\; : \\; |x|f\\in L^2(\\R^d)\\big\\}\n\\]\nthere exits a unique strong solution $\\psi \\in C([0,\\infty), \\Sigma)$ to \\eqref{NLS_diss}. The associated mass and energy thereby satisfy the identities \\eqref{eq:mass} and \\eqref{eq:energy} below. \nIf, in addition, $\\lambda >0$, the evolutionary system \\eqref{NLS_diss} possesses a global attractor $\\mathcal A\\subset \\Sigma$, i.e., $\\mathcal A$ is\nis invariant under the time-evolution associated to \\eqref{NLS_diss} and such that\n\\[\n\\inf_{\\phi \\in \\mathcal A} \\| \\psi(t) - \\phi \\|_{L^2(\\R^d)} \\stackrel{t\\to +\\infty}{\\longrightarrow} 0. \n\\]\nMore precisely, \n\\[\n\\mathcal A=\\big \\{\\psi_0 : \\psi_0 = \\psi(0) \\mbox{ for some } \\psi \\in C({(-\\infty}, \\infty);\\Sigma) \\, \\text{solution to \\eqref{NLS_diss}} \\big \\} \n\\]\nis a connected compact set in $L^2(\\R^d)$ and uniformly attracts bounded sets in $L^2(\\R^d)$. \nFurthermore, $\\mathcal A$ has finite Hausdorff and fractal dimensions which depend on the given parameters as described in Proposition \\ref{prop:dimension}.\nFinally, if $\\mu < \\frac{\\omega d}{2}$ it holds $\\mathcal A = \\{ 0 \\}$.\n\\end{theorem}\n\nHere, $\\Sigma$ is the physical energy space ensuring that $E_{\\rm GP}(\\psi(t))$ is finite. \nThe assumption on $\\sigma>0$ is thereby slightly more restrictive than the one for the usual $H^1$-subcritical nonlinearities (see Remark \\ref{remH1} below). \nNote however, that we may always take $\\sigma = 1$ in the above theorem which corresponds to the usual cubic nonlinearity. In addition, the condition $\\omega>|\\Omega|$ \nensures that the confinement is stronger than the rotation, and thus, the system remains trapped for all times $t\\ge 0$. \n\nAs we shall see, neither the mass nor the (total) energy are conserved quantities of the time-evolution, but for $\\lambda >0$, there are {\\it absorbing balls} for $\\psi$ in both \nthe mass and the energy space, see Section \\ref{sec:uniform} for a precise definition. The existence of a global attractor $\\mathcal A$ therefore requires the presence of the nonlinearity \nand, of course, the presence of the confining potential $V$. Clearly, all stationary solutions $\\varphi \\in \\Sigma$ of \\eqref{stat_NLS} \nare members of $\\mathcal A$. However, since for $\\mu$ sufficiently large \nthere are always at least two such solutions (namely, zero and the nontrivial energy minimizer) and since $\\mathcal A$ is connected, it is unclear what the precise \nlong-time behavior of \\eqref{NLS_diss} is. \nIndeed, in the case of the GL equation for superconducting materials it is known \\cite{TW} \nthat the global attractor contains not only all possible steady state solutions, but also the heteroclinic orbits joining these steady states, and we \nconsequently expect a similar behavior to also hold also in our model. \n\nExcept in the case $\\mu<\\frac{\\omega d}{2}$, the precise dependence of the dimension of $\\mathcal A$ on the given physical parameters is not known. \nIn Section \\ref{sec:dimension} we shall prove that the Hausdorff dimension ${\\rm dim}_{\\rm H}(\\mathcal A)\\le m$, where $m$ depends in a \nrather complicated way on all the involved parameters. It is interesting, however, to check that $m\\to +\\infty$, as $|\\Omega|\\to \\omega$. \nIn other words, the influence of the rotation term potentially increases the dimension of the attractor. This is consistent with \nnumerical and physical experiments on the creation of vortex lattices in rotating BEC. For a recent (non-rigorous) study which employs numerical simulations and asymptotic analysis \nto investigate the corresponding pattern formation mechanism, we refer to \\cite{CGKT}. In fact, one easily observes that in the linear case ($\\lambda =0$) the dynamics \nadmits exponentially growing modes, cf. Section \\ref{sec:linear} below for more details. It is argued in \\cite{CGKT} that this type of instability mechanism is responsible for the \nnucleation of a large number of vortices at the periphery of the atomic cloud, as can be seen in physical experiments.\\\\\n\n\n\nThe proof of Theorem \\ref{thmmain} will be done in several steps: First, we shall establish local (in-time) well-posedness of \\eqref{NLS_diss} in Section \\ref{sec:local} below.\nThen, we will show how to extend this result to global in-time solutions in Section \\ref{sec:global}, where we also prove that for $\\mu < \\frac{\\omega d}{2}$ solutions decay to zero as $t\\to +\\infty$. \nThe main technical step for the existence of an attractor is then to prove certain uniform bounds on the total mass and energy as done in Section \\ref{sec:uniform}.\nThis will allow us to conclude the existence of an absorbing ball and of a global attractor in Section \\ref{sec:attractor}, where we shall also prove the announced estimates on the dimension. \nFinally, we collect some basic computations regarding the kernel of the linear semigroup in the appendix.\n\n\n\\section{Mathematical preliminaries}\n\nIn this section we shall collect several preliminary results to be used later on.\n\n\\subsection{Spectral properties of the linear Hamiltonian}\\label{sec:linear}\nIn the following, we denote by\n\\begin{equation}\\label{H}\nH_\\Omega := -\\frac{1}{2} \\Delta + V(x) - \\Omega L ,\\quad x\\in \\R^d, \n\\end{equation}\nthe linear Hamiltonian operator, with $V(x)$ given in \\eqref{Vquadr}. Note that in the case without rotation, i.e. $\\Omega=0$, the operator \n\\begin{equation}\\label{H_0}\nH_0=\\frac12 \\left(- \\Delta+\\omega ^2 {|x|^2}\\right),\n\\end{equation} is nothing but the \n(isotropic) quantum mechanical harmonic oscillator in, respectively, $d=2$, or $3$ spatial dimensions. The spectral properties of $H_0$ are well known \\cite{Fl, T}: \n\\begin{lemma}\\label{lem:H}\n$H_0$ is essentially self-adjoint on $C_0^\\infty(\\R^d)\\subset L^2(\\R^d)$ with compact resolvent. The \nspectrum of $H_0$ is given by $\\sigma(H_0) = \\{ E_{0,n}\\}_{n\\in \\N}$, where\n\\[\nE_{0,n} = \\omega \\Big( \\frac{d}{2} +n-1 \\Big),\\quad n =1,2, \\dots.\n\\]\nIn addition, the eigenvalue $E_{0,n}$ is $\\left(\\begin{matrix} d+n -2 \\\\ n-1 \\end{matrix}\\right)-$fold degenerate.\n\\end{lemma}\nIn particular, $E_{0,n} \\ge E_{0,1}\\equiv \\frac{\\omega d}{2}>0$, for all $n \\in\\N$. \nThe associated eigenfunctions form a complete orthonormal basis of $L^2(\\R^d)$. In $d=2$, they are explicitly given by \\cite{Fl}:\n\\[\n\\chi^0_{n_1, n_2}(x_1, x_2) = f_{n_1}(x_1) f_{n_2}(x_2),\\quad n_j\\in \\N,\n\\]\nwhere $n_1 + n_2 = n$ and the $f_{n_j}\\in \\mathcal S(\\R)$ are the eigenfunctions of the one-dimensional harmonic oscillator, i.e., an appropriately normalized Gaussians times a Hermite polynomial of order $n_j-1$. \nAn analogous formula holds in $d=3$ dimensions.\n\nIn the case with $\\Omega\\not =0$, we first note that the commutator $[H_\\Omega, L]=0$, due to the rotational symmetry of the potential $V$. \nThis implies that $H_\\Omega$ and $L$ have a common orthonormal basis of eigenfunctions $\\{\\chi_n\\}_{n\\in \\N_0}$, which can be obtained by taking appropriate linear combinations of the \neigenvalues of $H_0$, see \\cite{Fl}. An important assumption throughout this work, will \nbe that $\\omega>|\\Omega|$, ensuring confinement of the BEC. \nIn mathematical terms, this condition implies that the rotational term can be seen as a perturbation of the positive definite operator\n$H_0$, such that $H_\\Omega$ is still positive definite. In other words, we have that \n\\begin{equation}\\label{eigenvalue}\nH_\\Omega \\chi_n = E_{\\Omega,n} \\chi_n,\n\\end{equation}\nwhere the new eigenvalues $E_{\\Omega, n}\\in \\R$ (indexed in increasing order) are related to the unperturbed $E_{0,n}$ via \n\\[\n\\{E_{\\Omega, n}, \\ n \\in \\N\\}= \\{ E_{0,\\ell} + m \\Omega, \\ -\\ell+1 \\le m\\le \\ell-1, \\, \\text{for}\\, \\ell \\in\\N\\}.\n\\] \nIn particular, under the assumption that $\\omega > \\Omega$, we still have: $E_{\\Omega, n}\\ge \\frac{\\omega d}{2}$, for all $n\\in \\N$. Thus, the ground state energy \neigenvalue stays the same with and without rotation. \\\\\n\nWith these spectral data at hand, we can now look at the {\\it linear} time-evolution ($\\lambda =0$) associated to \\eqref{dissGP}, i.e.\n\\begin{equation}\\label{linGP}\n(i\\beta-\\gamma) \\partial_t \\psi =H_\\Omega \\psi - \\mu \\psi.\n\\end{equation}\nUsing the fact that $\\{\\chi_n\\}_{n\\in \\N}$ comprises an orthonormal basis, we can decompose the solution to this equation via\n\\begin{equation}\\label{decomp}\n\\psi(t,x) = \\sum_{n \\in \\N} c_n(t) \\chi_n (x),\n\\end{equation}\nwhere $ \\{ c_n (t) \\}_{n\\in \\N} \\in \\ell^2$, i.e. $\\sum |c_n(t)|^2 < +\\infty$. In view of \\eqref{eigenvalue}, \\eqref{linGP} we find\n\\[\nc_n(t) = c_n(0) \\exp (-(i\\beta + \\gamma)(E_{\\Omega, n}-\\mu) t),\n\\]\nIn particular, the normalization $\\beta^2+\\gamma^2=1$ yields \n\\[\n\\| \\psi_n (t) \\|^2_{L^2} \\equiv \\sum_{n=1}^\\infty |c_n(t)|^2 = \\sum_{n=1}^\\infty |c_n(0)|^2 e^{- 2\\cos\\vartheta (E_{\\Omega, n} - \\mu) t},\n\\]\nwhere we identify $\\gamma =\\cos \\vartheta$.\nFor $ \\vartheta \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ \nthe right hand side exponentially decays to zero as $t\\to+\\infty$, provided $\\mu < E_{\\Omega, n}$, for all $n\\in \\N$. This is equivalent to saying that $\\mu 0$. Hence, given a $\\mu > E_{\\Omega, 1}$ the solution is exponentially decaying as long as \nthe initial data is such that $c_n(0)=0$ for all $n\\in \\N$ for which $E_{\\Omega, n} < \\mu$. Otherwise, we have, in general, exponential growth of the \n$L^2$-norm of $\\psi(t)$. \n\n\\begin{remark}\nIn the case where we choose $\\mu = E_{\\Omega, m}$ for some fixed $m\\in \\N_0$, we see that the $|c_m(t)|^2 = |c_m(0)|^2$ is a conserved quantity of the linear time evolution. \nAll higher modes exponentially decay towards zero, whereas all lower modes will exponentially increase. We consequently expect linear instability of stationary states of the nonlinear system.\n\\end{remark}\n\n\n\n\n\n\\subsection{Dispersive properties of the linear semi-group} \nIn order to set up a well-posedness result for the nonlinear equation \\eqref{dissGP}, we need to study the regularizing properties of the \nlinear semigroup associated to $H_\\Omega$, i.e.\n\\[\nS_\\Omega(t):= \\exp\\left(-e^{-i\\vartheta} t H_\\Omega\\right), \\quad t\\in \\R_+,\n\\]\nAs usual we identify $S_\\Omega(t)$ with its associated integral kernel via\n\\[\nS_\\Omega(t) f(x) = \\int_{\\R^d} S_\\Omega(t,x,y) f(y) \\, dy,\\quad f \\in L^2(\\R^d).\n\\]\nThe following lemma states some basic properties of $S_\\Omega(t)$ to be used later on.\n\n\\begin{lemma}\\label{lem:kernel}\nLet $\\vartheta \\in (-\\frac{\\pi}{2}, \\frac{\\pi}{2})$ and $t>0$. Then \n\\begin{align}\\label{SG_diss}\n S_\\Omega(t,x,y) = \\bigg(\\frac{\\omega}{2 \\pi \\sinh(e^{-i\\vartheta} \\omega t)}\\bigg)^\\frac{d}{2} \\exp\\left({\\Phi(t,x,y)}\\right),\n\\end{align}\nwhere the pre-factor in front of the exponent is understood in terms of the principal value of the complex logarithm, and the phase function $F$ is given by\n\\begin{align*}\n \\Phi(t,x,y) &= -\\frac{\\omega}{\\sinh(e^{-i\\vartheta} \\omega t)}\\bigg(\\frac{1}{2}(x^2+y^2)\\cosh(e^{-i\\vartheta}\n\\omega t) - \\cosh(e^{-i\\vartheta}\\Omega t) (x_1 y_1 + x_2 y_2) \\\\&\\qquad\\qquad\\qquad\\qquad\\qquad + i \\sinh(e^{-i\\vartheta} \\Omega t) (x_2y_1 - x_1y_2) \\bigg).\n\\end{align*}\nMoreover, for $\\omega>|\\Omega|$, there exists $\\delta>0 $ such that \n\\begin{equation}\\label{kernel_Lp}\n \\|S_\\Omega(t) f \\|_{L^r} \\le C\\ t^{\\frac{d}{2}(\\frac{1}{r}-\\frac{1}{q})} \\| f \\|_{L^q}\n\\end{equation}\nand\n\\begin{equation}\\label{kernel_Sigma}\n \\|\\nabla S_\\Omega(t) f \\|_{L^r} + \\|x S_\\Omega(t) f \\|_{L^r},\n\\le C\\ t^{-\\frac{1}{2}+\\frac{d}{2}(\\frac{1}{r}-\\frac{1}{q})} \\| f \\|_{L^q},\n\\end{equation}\nfor all $1\\le q\\le r \\le \\infty$ and all $0|\\Omega|$, and $d\\in \\{2,3\\}$. \n\\begin{itemize}\n\\item[(i)] Let $p > \\max (\\sigma d, 2\\sigma +1)$ and $\\psi_0\\in L^p(\\R^d)$. Then there exists a time $T>0$ and a unique solution $\\psi \\in C([0,T];L^p(\\R^d))$ to \\eqref{NLS_diss}, depending continuously on the initial data.\n\\item[(ii)] If, in addition, $0 < \\sigma < \\frac{d}{2(d-2)}$ and $\\psi_0 \\in \\Sigma $, then there exists a $T^*>0$ such that the solution from ${\\rm (i)}$ satisfies \n\\[\n \\psi \\in C([0,T^*]; \\Sigma).\n\\]\nMoreover, the solution is maximal in the sense that either $T^*=+\\infty$, or the following blow-up alternative holds:\n\\begin{equation*}\\label{blowup_alt}\n\\lim_{t\\to T_-^*} \\| \\psi(t)\\|_{\\Sigma} = \\infty.\n\\end{equation*}\n\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nThe proof is based on a fixed point argument using Duhamel's formula and the properties of the semigroup $S_\\Omega(t)$. To this end, we first note that the term $\\mu\\psi$ is of no importance here, as it can always be added in a subsequent step (in fact,\nwe could have included it in the kernel of $S_\\Omega(t)$). Hence let us assume that $\\mu=0$ for notational convenience. \n\nTo prove (i), we will show that the mapping\n\\[\n\\psi \\mapsto \\Xi(\\psi)(t) := S_\\Omega(t)\\psi_0 - e^{-i\\vartheta} \\int_0^t S_\\Omega(t-\\tau) \\big(\\lambda |\\psi(\\tau)|^{2\\sigma}\\psi(\\tau)\\big) \\;d\\tau\n\\]\nis a contraction in the space\n\\[\n X_T:=\\big\\{ \\psi\\in C([0,T];L^p(\\R^d)) \\ : \\ \\|\\psi\\|_{L^\\infty(0,T;L^p)} \\le 2 \\|\\psi_0\\|_{L^p} \\big\\}\n\\]\nfor small enough $T>0$. To do so, we can use the kernel estimate \\eqref{kernel_Lp} with the following choice of parameters:\n\\[\n\\begin{split}\nr=p\\ge 2\\sigma+1, \\quad q=\\frac{p}{2\\sigma+1},\\qquad \\text{when} \\qquad d&=2,\\\\\nr=p > \\max (\\sigma d, 2\\sigma +1), \\quad q=\\frac{p}{2\\sigma+1},\\qquad \\text{when} \\qquad d&=3.\n\\end{split} \n\\]\nNote that any such a choice of $p$ implies that $d\\sigma 0$. Thus, for $T>0$ sufficiently small, we conclude that $\\Xi$ indeed maps $X_T$ onto itself.\nLikewise it holds that for two solutions $\\psi$ and $\\tilde \\psi$\n\\begin{align*}\n& \\|\\Xi(\\tilde\\psi)(t)-\\Xi(\\psi)(t)\\|_{L^p}\\\\ &\\le \\lambda \\int_0^t\n\\big\\|S_\\Omega(t-\\tau)\\big(|\\tilde\\psi(\\tau)|^{2\\sigma}\\tilde\\psi(\\tau)-|\\psi(\\tau)|^{2\\sigma}\\psi(\\tau)\\big)\\big\\|_{L^p} \\;d\\tau\\\\\n &\\le C \\int_0^t (t-\\tau)^{-{d\\sigma}\/{p}} \\big(\\|\\tilde\\psi(\\tau)\\|_{L^p}^{2\\sigma} + \\|\\psi(\\tau)\\|_{L^p}^{2\\sigma}\\big) \\|\n\\tilde\\psi(\\tau)-\\psi(\\tau) \\|_{L^p} \\;d\\tau\\\\\n &\\le C T^{1-\\frac{d\\sigma}{p}} \\|\\psi\\|_{L^\\infty(0,T;L^p)}^{2\\sigma} \\| \\tilde\\psi-\\psi\n\\|_{L^\\infty(0,T;L^p)},\n\\end{align*}\nwhich shows that $\\Xi$ is a contraction for $T>0$ sufficiently small. \n\n\\medskip\n\nTo prove (ii), we first note that by Sobolev imbedding $\\Sigma \\hookrightarrow L^p(\\R^d)$, for $p< p^*=\\frac{2d}{d-2}$\nwhen $d=3$ and $p<\\infty$ when $d=2$, respectively. Thus $\\Sigma\\cap L^p(\\R^d) = \\Sigma$ for $p0$ sufficiently small (depending on the size of $\\|\\psi\\|_{L^\\infty L^p}$). Similar arguments for $\\psi$ and $x\\psi$ imply\n\\begin{align*}\n\\| \\psi \\|_{L^\\infty(0,T;\\Sigma)} \\le \\| \\psi_0 \\|_{\\Sigma}+ C T^{1-\\frac{d\\sigma}{p}} \\|\\psi\\|_{L^\\infty(0,T;L^p)}^{2\\sigma} \\| \\psi\\|_{L^\\infty(0,T;\\Sigma)} .\n\\end{align*}\nChoosing $T>0$ even smaller, if necessary, the second term on the right hand side can be absorbed on the left hand side and we are done. \nAs before, this inequality also applies to the differences of two solutions $\\psi, \\tilde \\psi$, which yields the continuity of $\\psi$ in $\\Sigma$. \n\nWe denote by $T^*>0$ the maximal time of existence in $\\Sigma$. This is always less than or equal to $T>0$, the maximal time of existence in $L^p(\\R^d)$. \nTo prove the blow-up alternative, assume by contradiction that $T^*<\\infty$, and $\\| \\psi(t, \\cdot)\\|_{\\Sigma}$ remains bounded for $t\\in [0, T^*]$. Then, by Sobolev imbedding \n$\\| \\psi(t, \\cdot)\\|_{L^p}$ also remains bounded and thus, we can restart the local existence argument in $\\Sigma$ leading to a contradiction. \n\\end{proof}\n\n\n\\begin{remark}\\label{remH1}\nUnfortunately, our method of proof does not yield existence of solutions for the full $H^1$-subcritical regime, i.e., $\\sigma < \\frac{2}{d-2}$. We expect that \nthis is only a technical issue that can be overcome using a different approach (for example, by using ideas from \\cite{GV1}, or by \ngeneralizing the space-time estimates of \\cite{Bax} to $S_\\Omega$). Note, however, that our slightly more restrictive condition $\\sigma < \\frac{d}{2(d-2)}$ still allows to take $\\sigma =1$ in \n$d=3$. Hence, the physically most relevant case of a cubic nonlinearity is covered.\n\\end{remark}\n\n\\section{Global existence and asymptotic vanishing of solutions} \\label{sec:global}\n\nIn this section, we shall first prove the global existence of solutions in the energy space before showing that for any choice of $\\mu < E_{\\Omega,1}$, the \nsolutions asymptotically vanish as $t\\to +\\infty$.\n\n\n\\subsection{Global existence}\n\nIn order to prove global well-posedness of \\eqref{NLS_diss}, we \nwill need to collect some useful a-priori estimates. To this end, we denote for $\\psi\\in \\Sigma$ the {\\it total mass} by\n\\begin{equation}\\label{mass}\nM(\\psi):= \\| \\psi \\|_{L^2}^2,\n\\end{equation}\nand the {\\it total energy} by\n\\begin{equation}\\label{energy}\nE(\\psi):= \\int_{\\R^d} \\bigg( \\frac{1}{2} |\\nabla \\psi|^2 + V(x) |\\psi|^2 + \\frac{\\lambda}{\\sigma +1}|\\psi |^{2\\sigma+2} - \\Omega \\overline{\\psi}\nL \\psi \\bigg) \\;dx.\n\\end{equation}\nThe latter is nothing but the sum of the kinetic, potential, nonlinear potential, and rotational energy. Clearly, for $\\psi \\in \\Sigma$, \nSobolev's imbedding implies that all the terms in $E(\\psi)$ are finite, provided $\\sigma < \\frac{2}{d-2}$ (and hence also for our range of $\\sigma$). \nFor simplicity of notation, we will write $E(t) \\equiv E(\\psi(t,\\cdot))$ and likewise for $M(t)$, whenever we compute the \nmass and energy of the time-dependent solution $\\psi(t,x)$ to \\eqref{NLS_diss}.\nIn addition, the {\\it free energy} is \ngiven by\n\\begin{equation}\\label{free_energy}\nF(\\psi):=E(\\psi) - \\mu M(\\psi).\n\\end{equation}\n\nIn the case of the usual Gross-Pitaevskii equation, i.e. $\\vartheta = \\pm \\frac \\pi 2$, one finds, that both $M(t)=M(0)$ and $E(t)=E(0)$ are conserved in time \\cite{AMS}. \nIn our dissipative model this is no longer the case. Instead we have the following result, which can be seen as an extension of \nsome well-known identities proved for the classical GL equation, cf. \\cite{DGL, GV2, Te, Wu}.\n\n\\begin{lemma}\\label{lem:Lyapunov}\nLet $\\sigma <\\frac{d}{2(d-2)}$ and $\\psi \\in C([0,T]; \\Sigma)$ be a solution to \\eqref{NLS_diss}. Then the following identities hold:\n\\begin{equation}\\label{eq:mass}\nM(t) + 2\\cos \\vartheta \\int_0^t \\left( E(s) + \\frac{\\lambda\\sigma}{\\sigma+1} \\| \\psi(s,\\cdot) \\|_{L^{2\\sigma+2}}^{2\\sigma+2} - \\mu M(s)\\right) ds =M(0),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:energy}\nF(t) + 2\\cos \\vartheta \\int_0^t \\int_{\\R^d} | \\partial_t \\psi(s,x)|^2\\, dx \\, ds= F(0).\n\\end{equation}\nIn particular, for $\\vartheta \\in (-\\frac \\pi 2 ,\\frac \\pi 2 )$, the free energy $F(\\psi)$ is a non-increasing functional along solutions of \\eqref{NLS_diss}.\n\n\\end{lemma}\n\\begin{proof} In a first step, let us assume sufficient regularity (and spatial decay) of $\\psi$, such that all the following calculations are justified. Then, \nas in the case of the usual NLS, \nidentity \\eqref{eq:mass} is obtained by multiplying \\eqref{NLS_diss} by $\\bar \\psi$, integrating with respect to $x\\in \\R^d$ and taking the real part of the resulting expression (see, e.g., \\cite{AMS, Caze}). \nThis yields\n\\begin{equation}\\label{mass-derivative}\n\\frac{d}{dt} M(t) = -2\\cos \\vartheta \\left( E(t) + \\frac{\\lambda\\sigma}{\\sigma+1} \\| \\psi(t) \\|_{L^{2\\sigma+2}}^{2\\sigma+2} - \\mu M(t)\n\\right)\n\\end{equation}\nwhich directly implies \\eqref{eq:mass} after an integration in time. Similarly, after multiplying \\eqref{NLS_diss} by $\\partial_t \\bar \\psi$, \nintegrating with respect to $x$, and taking the real part, we obtain\n\\begin{equation}\\label{energy-derivative}\n \\frac{d}{dt} \\big(E(t) - \\mu M(t)\\big) = -2\\cos \\vartheta \\int_{\\R^d} | \\partial_t \\psi(t,x)|^2\\, dx,\n\\end{equation}\nwhich yields \\eqref{eq:energy} after integration w.r.t. time. \n\nThe second step then consists of a classical density argument (cf. \\cite{CDW1}), which, together with the fact that \n$\\psi(t)$ depends continuously on the initial data $\\psi_0\\in \\Sigma$, \nallows us to extend \\eqref{eq:mass} and \\eqref{eq:energy} to the case of general solutions $\\psi \\in C([0,T;\\Sigma)$. \nFinally, we note that for $\\vartheta \\in (-\\frac \\pi 2 ,\\frac \\pi 2 )$ we have $\\cos \\vartheta >0$, and thus \n\\eqref{eq:energy} directly implies that $F(t) \\le F(0)$, for all $t\\ge 0$.\n\\end{proof}\n\n\n\nHaving in mind that $\\psi \\in C([0,T], \\Sigma)$ the assumption on $\\sigma$ implies (via Sobolev imbedding) that the integrand \nappearing in identity \\eqref{eq:mass} is a continuous function of time. \nThe fundamental theorem of calculus therefore allows us to differentiate \\eqref{eq:mass} w.r.t. $t$ and consequently use the \ndifferential inequality \\eqref{mass-derivative}. However, the same is not true for \\eqref{eq:energy}, i.e., we cannot use \\eqref{energy-derivative}, \nsince at this point we do not know wether $\\partial_t \\psi \\in C([0,T; L^2(\\R^d))$ holds true. This fact will play a role in some of the proofs given below. \\\\\n\n\n\nAnother preliminary result, to be used several times in the following, is the fact that under our assumptions on the parameters $\\omega, \\Omega, \\lambda, \\sigma$, \nthe energy is indeed non-negative.\n\n\\begin{lemma} \\label{lem:Ebound}\nLet $\\omega>|\\Omega|$, $\\lambda \\ge 0$, and $\\sigma < \\frac{2}{d-2}$. Then for any \n$u \\in \\Sigma$ there exists a constant $c=c(\\omega, \\Omega, \\lambda , \\sigma)>0$, such that\nsuch that \n\\[\n\\|\\nabla u \\|_{L^2}^2 + \\| x u \\|_{L^2}^2 + \\| u \\|_{L^{2\\sigma +2}}^{2\\sigma+2} \\le cE(u).\n\\]\n\\end{lemma}\n\\begin{proof}\nSince $\\lambda\\ge0$, the only possibly negative term within $E(u)$ is given by the rotational energy. However, since \n$\\Omega^2\/\\omega^2=:\\epsilon < 1$, Young's inequality applied to~\\eqref{eq:angular_momentum} yields the pointwise interpolation estimate\n\\[\n\\big| \\Omega \\overline{u} L u \\big| \\le \\frac{\\omega^2}{2} |x^\\bot|^2 | u |^2 \\, dx +\\frac{\\epsilon}{2} |\\nabla^\\bot u|^2 \\le V(x) |u|^2 + \\frac{\\epsilon}{2} |\\nabla u |^2.\n\\]\nWe therefore can bound the energy from below via\n\\begin{equation*}\\label{grad_by_en}\n 0 \\le \\frac{1-\\epsilon}{2} \\|\\nabla u \\|_{L^2}^2 + \\frac{\\lambda\\sigma}{\\sigma+1} \\| u \\|_{L^{2\\sigma+2}}^{2\\sigma+2} \\le E(u).\n\\end{equation*}\nAnalogously, we have\n\\[\n 0 \\le \\frac{1-\\epsilon}{2} \\|x u \\|_{L^2}^2+ \\frac{\\lambda\\sigma}{\\sigma+1} \\| u \\|_{L^{2\\sigma+2}}^{2\\sigma+2} \\le E(u).\n\\]\nCombining these two estimates then yields the desired result with a constant\n\\[\nc=\\frac{4}{\\min\\{1-\\epsilon, \\frac{2\\lambda \\sigma}{\\sigma +1}\\}}.\n\\]\nNote that $c\\to +\\infty$ as $|\\Omega|\\to \\omega$.\n\\end{proof}\n\n\nThe mass\/energy-relations stated in Lemma \\ref{lem:Lyapunov} can now be used to infer global existence of solutions in the \ncase of {\\it defocusing} case $\\lambda > 0$.\n\n\\begin{proposition} \\label{prop:global} Let $\\omega>|\\Omega|$, $\\vartheta \\in (-\\frac \\pi 2 , \\frac \\pi 2 )$, $\\lambda \\ge 0$, and $\\sigma < \\frac{d}{2(d-2)}$. \nThen, for any $\\psi_0 \\in\n\\Sigma$\nthere exists a unique global-in-time solution $ \\psi \\in C([0,\\infty); \\Sigma)$ to \\eqref{NLS_diss}.\n\\end{proposition}\n\\begin{proof}\nIn view of the blow-up alternative stated in Proposition \\ref{prop:loc_ex}, all we need to show is that the $\\Sigma$-norm remains bounded for all $t\\ge 0$. \nLemma \\ref{lem:Ebound} implies that this is the case, as soon as we we can show that both $M(t)$ and $E(t)$ are bounded. \nIn order to do so, we first consider the case $\\mu <0$ and recall that $\\cos \\vartheta >0$ for $\\vartheta \\in (-\\frac \\pi 2 , \\frac \\pi 2 )$. In this case \nidentity \\eqref{eq:energy} implies\n\\[\nE(t) + |\\mu| M(t) \\le F(0) <+\\infty,\n\\]\nand since both $E(t)$ and $M(t)$ are non-negative, we directly infer the required bound on the mass and energy.\n\nOn the other hand, for $\\mu \\ge 0$, identity \\eqref{eq:mass} yields (since $\\lambda \\ge 0$)\n\\[\nM(t) \\le M(0) + 2 \\mu \\cos \\vartheta \\int_0^t M(s) \\, ds,\n\\]\nand hence, Grownwall's lemma implies\n\\begin{equation}\\label{Mbound}\nM(t) \\le M(0)\\left( 1+ 2 \\mu t \\cos \\vartheta \\, e^{2\\mu t \\cos \\vartheta } \\right).\n\\end{equation}\nUsing this estimate in identity \\eqref{eq:energy} we obtain\n\\[\n E(t) \\le F(0) + \\mu M(t) \\le E(0) + 2 \\mu^2 t \\cos \\vartheta M(0) e^{2\\mu t \\cos \\vartheta } .\n\\]\nThe right hand side is finite, for all $t\\ge0$ and thus, the assertion is proved.\n\\end{proof}\n\n\\begin{remark}\nThe global in-time strong solutions constructed above are of the same type as the corresponding solutions for NLS with quadratic potentials, cf. \\cite{AMS, Car05}. \nIt is certainly possible to, alternatively, construct global weak solutions to \\eqref{NLS_diss} as has been done for the usual GL model in, e.g., \\cite{DGL, GV1}. \nBut since we consider the equation \\eqref{NLS_diss} as a \ntoy model describing possible relaxation phenomena in the mean-field dynamics of BEC, we have decided to remain as close as possible to the corresponding NLS theory. \nIn particular, we do not make any use of the strong smoothing property of the linear (heat type) semigroup $S_\\Omega(t)$ for $\\vartheta \\in (-\\frac{\\pi}{2}, \\frac \\pi 2)$. \nWe finally note that our set-up makes it possible to directly generalize the inviscid limit results of \\cite{Wu} to our model.\n\\end{remark}\n\n\\subsection{Asymptotically vanishing solutions}\n\nThe discussion in Section \\ref{sec:linear} shows that solutions to the linear time evolution $\\lambda =0$ asymptotically vanish, provided \n$\\mu < E_0$, i.e., the lowest (positive) energy eigenvalue of $H_\\Omega$. \nWe shall prove that the same is true for in the nonlinear case $\\lambda >0$.\n\n\\begin{proposition} Let $\\vartheta \\in (-\\frac \\pi 2 , \\frac \\pi 2 )$, $\\lambda \\ge 0$, $\\omega>|\\Omega|$, and $\\psi \\in C([0,\\infty), \\Sigma)$ be a solution of \\eqref{NLS_diss} with \n$\\mu0$, and $M(t) = \\sum_{n=0}^\\infty |c_n(t)|^2$.\nThe inequality above can thus be rewritten as\n\\[\n\\frac{d}{dt} \\left(e^{+2 t\\cos \\vartheta (E_{\\Omega,0} - \\mu) } M(t) \\right) \\le 0,\n\\]\nwhich after an integration in time implies \n\\[\nM(t)\\le M(0) e^{-2 t\\cos \\vartheta (E_{\\Omega,0} - \\mu) } \\xrightarrow{t\\to +\\infty}0,\n\\]\nsince $\\vartheta \\in (-\\frac \\pi 2 , \\frac \\pi 2 )$. \n\\end{proof}\n\nAt this point, it is unclear if the decay rate given above is indeed sharp.\n\n\\begin{remark} In the case where $\\mu <0$, one does not need to use the decomposition of $\\psi$ via the spectral subspaces of $H_\\Omega$, at the expense of a slightly worse decay rate. \nIndeed, for $\\mu<0$, the inequality \\eqref{mass-derivative} directly yields\n\\[\n\\frac{d}{dt} M(t) \\le - 2 |\\mu| \\cos \\vartheta M(t),\n\\]\nand thus\n\\[\n M(t) \\le M(0) e^{-2 t|\\mu| \\cos \\vartheta} , \\quad \\forall t \\ge 0.\n\\]\nNote that for $\\mu <0$ there are no nontrivial steady states $\\varphi(x)\\not =0$, satisfying \\eqref{stat_NLS}. This can be seen by \nmultiplying equation \\eqref{stat_NLS} with $\\bar \\varphi$, integrating in $x\\in \\R^d$, \nand recalling the restriction $\\omega>\\Omega\\ge0$, which implies that $\\mu$ has to be non-negative.\n\\end{remark}\n\n\\section{Bounds on the mass and energy}\\label{sec:uniform}\n\nIn this section we shall prove the existence of absorbing balls in both $L^2(\\R^d)$ and $\\Sigma$ for solutions to \\eqref{NLS_diss}.\nIn view of the discussion on the linear model, cf. Section \\ref{sec:linear}, this might seem surprising, given that for general $\\mu>0$ we can expect exponentially growing modes. \nHowever, we shall see that for $\\lambda>0$, the nonlinearity, combined with the confining potential, mixes the dynamics in a way that \nmakes it possible to infer a uniform bound on the mass and energy, and hence on the $\\Sigma$--norm of the solution.\nTo this end, the following lemma is the key technical step.\n\n\\begin{lemma}\\label{lem:glob_bound}\nLet $\\lambda > 0$, $\\omega > |\\Omega|$ and $0 < \\sigma < \\frac{d}{2(d-2)}$. Then there exists a constant $C=C(\\omega, \\Omega, \\lambda, \\sigma)>0$, such that\n\\[\n M(\\psi) \\le C E(\\psi)^{\\frac{\\sigma\\theta + 1}{\\sigma+1}} ,\\quad \\text{with $\\theta = \\frac{d\\sigma}{2\\sigma + 2 + d\\sigma}$.}\n \\]\n\\end{lemma}\n\\begin{proof}\nThe proof of this result relies on the following localization property: \n For all $d\\ge1$ and all $p\\ge2$ and any $f\\in C^\\infty_0(\\R^d)$:\n\\begin{equation}\\label{localize}\n \\| f \\|_{L^2(\\R^d)} \\le { 2} \\| x f \\|_{L^2(\\R^d)}^{\\theta} \\| f \\|_{L^p(\\R^d)}^{1-\\theta},\n\\end{equation}\nwith \n\\[\n\\theta = \\frac{d(\\frac{1}{2}-\\frac{1}{p})}{1+d(\\frac{1}{2}-\\frac{1}{p})} = \\frac{d(p-2)}{2p + d(p-2)}.\n\\]\nIn order to show this, let $B_R$ denote the ball around the origin of radius $r>0$. We rewrite \n\\begin{align*}\n \\|f\\|_{L^2(\\R^d)} &= \\|f\\|_{L^2(B_r)} + \\|f\\|_{L^2(\\R^d\\setminus B_r)} \\le r^{d(\\frac{1}{2} - \\frac{1}{p})} \\|f\\|_{L^p(B_r)} + \\frac{1}{r} \\|x f\\|_{L^2(\\R^d\\setminus B_r)}\\\\\n &\\le r^{d(\\frac{1}{2} - \\frac{1}{p})} \\|f\\|_{L^p(\\R^d)} + \\frac{1}{r} \\|x f\\|_{L^2(\\R^d)}.\n\\end{align*}\nThe right-hand side is minimal if both summands are of the same order of magnitude, i.e.\n\\[\n r^{1 + d(\\frac{1}{2} - \\frac{1}{p})} = \\frac{\\|x f\\|_{L^2(\\R^d)}}{\\|f\\|_{L^p(\\R^d)}}.\n\\]\nWith this choice of $r$, the estimate \\eqref{localize} follows and a density argument allows to extend it to any $f\\in \\Sigma$. \nSpecifying $p=2\\sigma+2$, consequently yields\n\\begin{equation}\\label{proof_glob_bound}\n\\|\\psi \\|_{L^2}^2 \\le { 2} \\bigg( \\int_{\\R^d} |x|^2 |\\psi(x)|^2 \\;dx \\bigg)^{\\theta} \\bigg( \\int_{\\R^d} |\\psi(x)|^{2\\sigma+2} \\;dx \\bigg)^{\\frac{1-\\theta}{\\sigma+1}},\n\\end{equation}\nwhere $\\theta = \\frac{d\\sigma}{2\\sigma + 2 + d\\sigma}.$ \nIn view of Lemma \\ref{lem:Ebound}, both factors on the right hand side of \\eqref{proof_glob_bound} are bounded by the energy. \nMore precisely, \n\\[\n M(\\psi) \\le 2(c E(\\psi))^{\\theta + \\frac{1-\\theta}{\\sigma+1}} = C E(\\psi)^{\\frac{\\sigma\\theta+1}{\\sigma+1}},\n\\]\nwhere $C=2c^{\\frac{\\sigma\\theta+1}{\\sigma+1}}$ and $c=c(\\omega, \\Omega, \\lambda, \\sigma)>0$ is the constant from Lemma~\\ref{lem:Ebound}. \n\\end{proof}\n\\begin{remark}\nNote that in order to infer this bound one needs \nthe presence of {\\it both} the confinement and the nonlinearity, since the proof requires $\\sigma>0$, $\\lambda>0$ and $\\omega>0$. Moreover, one checks that $C\\to +\\infty$, as $|\\Omega| \\to \\omega$.\n\\end{remark}\n\n\nWith this result in hand, we can deduce global bounds on $M(t)$ and $E(t)$ along solutions of \\eqref{NLS_diss}.\n\\begin{proposition}\\label{prop:Ebound}\n Let $\\psi \\in C([0,\\infty), \\Sigma)$ be a solution to \\eqref{NLS_diss} with $\\vartheta \\in (-\\frac \\pi 2 ,\\frac \\pi 2 )$.\n Under the assumptions of Lemma \\ref{lem:glob_bound}, if additionally $\\mu>0$, there exists a constant $K=K(\\omega, \\Omega, \\sigma, \\lambda, \\mu)>0$, independent of time, \n such that\n\\[\n E(t) \\le K + e^{- t \\mu \\cos \\vartheta} E(0) , \\quad \\forall \\, t\\ge 0.\n\\]\n\\end{proposition}\n\\begin{proof}\nWe first note that Lemma \\ref{lem:glob_bound} and the differential inequality \\eqref{mass-derivative} imply\n\\[\n\\frac{d}{dt}M(t) \\le -2\\cos \\vartheta\\, E(t) + C \\mu E(t)^{\\frac{\\sigma\\theta + 1}{\\sigma+1}} .\n\\]\nNow, for any $\\vartheta \\in (-\\frac \\pi 2 ,\\frac \\pi 2 )$ and $\\tilde\\theta= \\frac{\\sigma\\theta + 1}{\\sigma+1}$, by Young's inequality, we obtain\n\\[\n E(t)^{\\tilde\\theta} \\le \\frac{\\cos\\vartheta}{C\\mu} E(t) + (1-\\tilde \\theta) \\left(\\frac{ C \\mu \\tilde \\theta}{\\cos \\vartheta}\\right)^{\\frac{\\tilde \\theta} {1-\\tilde \\theta}} = \n \\frac{\\cos \\vartheta}{C\\mu} \\, E(t)+\\tilde C,\n\\]\nwhere $\\tilde C>0$, depends on all the parameters involved, but not on time. Thus, we have\n\\[\n\\frac{d}{dt}M(t) \\le - \\cos\\vartheta E(t) + \\mu C \\tilde C.\n\\]\nOn the other hand, identity \\eqref{eq:energy} implies \n\\[\nE(t) -E(t_0) \\leq \\mu M(t)-\\mu M(t_0), \\qquad 0 \\leq t_0 \\leq t,\n\\]\nand hence\n\\[\nE(t)-E (s) \\le \\int_{s}^t (- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C )\\, d\\tau, \\qquad 0 \\leq s \\leq t,\n\\]\nas well as\n\\[\nE(t)-E (s) \\ge \\int_{s}^t (- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C )\\, d\\tau, \\qquad 0 \\leq t \\leq s.\n\\]\n\nNow, given any positive bump function $\\chi \\in C_0^{\\infty}((t-\\epsilon,t+\\epsilon))$, such that\n$\\chi' \\ge 0$ on $(t-\\epsilon, t)$ and $\\chi' \\le 0$ on $(t,t+\\epsilon)$, we multiply by $\\chi'(s)$ and integrate in $s$, to obtain\n\\[\n\\begin{split}\n\\int_{t-\\epsilon}^{t+\\epsilon} [E(t)-E(s)]\\chi'(s)\\, ds \\le& \\int_{t-\\epsilon}^{t+\\epsilon} \\int_s^t \n(- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C ) \\chi'(s) \\, d\\tau \\, ds\\\\\n =& \\int_{t-\\epsilon}^{t} \\int_{t-\\epsilon}^\\tau \n(- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C ) \\chi'(s) \\, ds \\, d\\tau \\\\\n&-\\int_{t}^{t+\\epsilon} \\int_\\tau^{t+\\epsilon} \n(- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C ) \\chi'(s) \\, ds \\, d\\tau \\\\\n =& \\int_{t-\\epsilon}^{t+\\epsilon} \n(- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C ) \\chi(\\tau) \\, d\\tau.\\\\\n\\end{split}\n\\]\nA similar computation gives the same inequality for a negative bump function function $\\chi \\in C_0^{\\infty}((t-\\epsilon,t+\\epsilon))$, such that\n$\\chi' \\le 0$ on $(t-\\epsilon, t)$ and $\\chi' \\ge 0$ on $(t,t+\\epsilon)$. Since an arbitrary test function can be written as a linear\ncombination of positive and negative bump functions, we have \n\\[\n-\\int_{t_0}^t E(\\tau) \\chi'(\\tau) \\, d \\tau \\le \\int_{t_0}^t \\left(- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C \\right) \\, \\chi(\\tau) \\, d\\tau,\n\\]\nfor any $\\chi \\in C_0^{\\infty}((t_0,t))$. Here, we have also used the fact that $\\chi$ has compact support on $(t_0,t)$. \nChoosing $\\chi(\\tau)= e^{\\mu \\tau \\cos \\vartheta } \\phi(\\tau)$ with $\\phi \\in C_0^{\\infty}((t_0,t))$, we obtain\n\\[\n-\\int_{t_0}^t E(\\tau) \\left(e^{\\mu \\tau \\cos \\vartheta } \\phi(\\tau)\\right)' \\, d \\tau \\le \\int_{t_0}^t \\left(- \\mu \\cos \\vartheta \\, E(\\tau) + \\mu^2 C \\tilde C \\right) \\, e^{\\mu \\tau \\cos \\vartheta } \\phi(\\tau) \\, d\\tau,\n\\]\nand thus \n\\[\n\\begin{split}\n-\\int_{t_0}^t E(\\tau) e^{\\mu \\tau \\cos \\vartheta } \\phi'(\\tau) \\, d \\tau &\\le\n\\int_{t_0}^t \\mu^2 C \\tilde C e^{\\mu \\tau \\cos \\vartheta } \\phi(\\tau) \\, d\\tau\\\\\n&\\le \\int_{t_0}^t \\frac{\\mu^2 C \\tilde C}{\\mu \\cos \\vartheta}(1- e^{\\mu \\tau \\cos \\vartheta }) \\phi'(\\tau) \\, d\\tau.\n\\end{split}\n\\]\nHence\n\\[\nE(t) e^{\\mu t \\cos \\vartheta } + \\frac{\\mu^2 C \\tilde C}{\\mu \\cos \\vartheta}(1- e^{\\mu t \\cos \\vartheta })\n\\le E(t_0) e^{\\mu t_0 \\cos \\vartheta } + \\frac{\\mu^2 C \\tilde C}{\\mu \\cos \\vartheta}(1- e^{\\mu t_0 \\cos \\vartheta }),\n\\]\nfor almost all $0\\leq t_0 \\leq t$. In summary, for almost all $t\\ge 0$ we have\n\\[\nE(t) \\le E(0) e^{-\\mu t \\cos \\vartheta } +K(1- e^{-\\mu t \\cos \\vartheta }),\n\\]\nwhere\n\\[\nK= \\frac{\\mu C \\tilde C}{\\cos \\vartheta}.\n\\]\nHowever, since $\\psi\\in C([0,\\infty;\\Sigma))$ implies that $E (t)$ is continuous in time, we \nconsequently infer the inequality for all $t\\ge 0$.\\end{proof} \n\n\\begin{remark}\nThe proof above is slightly complicated due to the fact that we cannot use the energy identity \\eqref{eq:energy} in its differentiated form \\eqref{energy-derivative}, see \nthe discussion below the proof of Lemma \\ref{lem:Lyapunov}. \nIf we ignore this problem for the moment, then we have\n\\[\n\\frac{d}{dt} E(t) \\le\\mu\\frac{d}{dt}M(t)\\le - \\mu \\cos \\vartheta E(t) + \\mu^2 C \\tilde C,\n\\]\nwhich directly allows us to conclude the assertion proved above.\n\\end{remark}\n\nIn view of Lemma \\ref{lem:glob_bound} the bound on $E(t)$ obtained above implies a similar bound on $M(t)$. In particular, \nthere is some constant $\\rho_M >0$ and a function $t_M(\\cdot)$, such that for all $\\psi\\in C([0,\\infty);\\Sigma)$ solutions to \\eqref{NLS_diss}, it holds\n\\[\n\\| \\psi (t, \\cdot)\\|_{L^2} \\le \\rho_M, \\quad \\forall t \\ge t_M(M(0)).\n\\]\nTherefore $$\\{ \\psi \\in L^2(\\R^d): \\|\\psi\\|_{L^2} \\le \\rho_M \\}\\subset L^2(\\R^d)$$\nis an absorbing ball for trajectories $t \\mapsto \\psi(t, \\cdot)$. Similarly, we know, that there exists a \n$\\rho_\\Sigma\\ge \\rho_M$ and a function $t_\\Sigma(\\cdot)$, such that\n\\[\n\\|\\psi (t, \\cdot)\\|_\\Sigma \\le \\rho_\\Sigma, \\quad \\forall t \\ge t_\\Sigma(\\|\\psi(0)\\|_\\Sigma).\n\\]\nIn other words, \n\\begin{equation}\\label{X}\nX:=\\{ \\psi \\in \\Sigma: \\|\\psi\\|_{\\Sigma} \\leq \\rho_\\Sigma \\}\n\\end{equation}\nis an absorbing ball in $\\Sigma$ for trajectories $t\\mapsto \\psi(t, \\cdot)$.\nIn our study of long time dynamics of \\eqref{NLS_diss}, the set $X$ will play the role of a {\\it phase space}.\n\n\\section{The global attractor and its properties} \\label{sec:attractor}\nIn the previous section we proved that solutions $\\psi(t)$ exist globally in $\\Sigma$, and, moreover, all such solutions\nremain within an absorbing ball $X \\subset \\Sigma$ for $t>0$ large enough. It is therefore natural to ask whether\nthere exists an $\\mathcal{A} \\subset \\Sigma$ that {\\it attracts} all trajectories $t\\mapsto \\psi(t, \\cdot)\\in \\Sigma$. Unfortunately,\nclassical theories of global attractors (see, e.g., \\cite{CV, Te}) do not apply to our situation as they typically require asymptotic\ncompactness, which is unknown in $\\Sigma$. However, the trajectories might still converge to the global attractor $\\mathcal{A}$\nin some weaker metric, say $L^2$. To prove this we revisit the rather general framework of evolutionary systems introduced in \\cite{C5} and adapt it to our situation. \n\n\n\\subsection{Existence of a global attractor}\n\n\nFirst, recall that our phase space is the metric space $(X,\\mathrm{d}_{L^2}(\\cdot,\\cdot))$ where $X\\subset \\Sigma$ is given by \\eqref{X} and \n$\\mathrm{d}_{L^2}({\\psi,\\phi}) = \\| {\\psi - \\phi} \\|_{L^2}$. We note that $X$ is $\\mathrm{d}_{L^2}$-compact.\nIn addition, we also have the stronger $\\Sigma$-metric $\\mathrm{d}_\\Sigma({\\psi,\\phi}):= \\| {\\psi-\\phi}\\|_{\\Sigma}$ on $X$, which satisfies: If $\\mathrm{d}_\\Sigma(\\psi_n, \\phi_n) \\to 0$ as $n \\to \\infty$ for some\n$\\psi_n, \\phi_n \\in X$, then $\\mathrm{d}_{L^2}(\\psi_n, \\phi_n) \\to 0$ as $n \\to \\infty$.\nNote that any $\\Sigma$-compact set is $L^2$-compact, and any $L^2$-closed set is $\\Sigma$-closed.\n\nNow, let $C([a, b];X_\\bullet)$, where $\\bullet = \\Sigma$ or $L^2$, be the space of $\\mathrm{d}_{\\bullet}$-continuous $X$-valued\nfunctions on $[a, b]$ endowed with the metric\n\\[\n\\,d_{C([a, b];X_\\bullet)}(\\psi,\\phi) := \\sup_{t\\in[a,b]}\\mathrm{d}_{\\bullet}(\\psi(t),\\phi(t)). \n\\]\nAlso, let $C([a, \\infty);X_\\bullet)$ be the space of $\\mathrm{d}_{\\bullet}$-continuous\n$X$-valued functions on $[a, \\infty)$ endowed with the metric\n\\[\n\\,d_{C([a, \\infty);X_\\bullet)}(\\psi,\\phi) := \\sum_{T\\in \\mathbb{N}} \\frac{1}{2^T} \\, \\frac{\\sup\\{\\mathrm{d}_{\\bullet}(\\psi(t),\\phi(t)):a\\leq t\\leq a+T\\}}\n{1+\\sup\\{\\mathrm{d}_{\\bullet}(\\psi(t),\\phi(t)):a\\leq t\\leq a+T\\}}.\n\\]\nIn order to define a general evolutionary system, we introduce\n\\[\n\\mathcal{T} := \\{ I: \\ I=[T,\\infty) \\subset \\mathbb{R}, \\mbox{ or } \nI=(-\\infty, \\infty) \\},\n\\]\nand for each $I \\subset \\mathcal{T}$, we denote \nthe set of all $X$-valued functions on $I$ by $\\mathcal X(I)$.\n\n\\begin{definition} \\label{Dc}\nA map $\\mathcal{E}$ that associates to each $I\\in \\mathcal{T}$ a subset\n$\\mathcal{E}(I) \\subset \\mathcal X(I)$ will be called an {\\it evolutionary system} if\nthe following conditions are satisfied:\n\n\\begin{itemize}\n\\item[(i)] $\\mathcal{E}([0,\\infty)) \\ne \\emptyset$.\n\\item[(ii)]\n$\\mathcal{E}(I+s)=\\{\\psi(\\cdot): \\ \\psi(\\cdot +s) \\in \\mathcal{E}(I) \\}$ for\nall $s \\in \\mathbb{R}$.\n\n\\item[(iii)] For all pairs $I_2\\subset I_1 \\in \\mathcal{T}$: $\\{\\psi(\\cdot)|_{I_2} : \\psi(\\cdot) \\in \\mathcal{E}(I_1)\\}\n\\subset \\mathcal{E}(I_2)$.\n\n\\item[(iv)]\n$\\mathcal{E}((-\\infty , \\infty)) = \\{\\psi(\\cdot) : \\ \\psi(\\cdot)|_{[T,\\infty)}\n\\in \\mathcal{E}([T, \\infty)) \\ \\forall T \\in \\mathbb{R} \\}.$\n\\end{itemize}\nIn general, $\\mathcal{E}(I)$ will be referred to as {\\it set of trajectories} on the time interval $I$, and trajectories in $\\mathcal{E}((-\\infty,\\infty))$ will be called {\\it complete}.\n\\end{definition}\n\nWe now consider the specific evolutionary system induced by the family of trajectories of \\eqref{NLS_diss}\nin $X$. More precisely, we set\n\\begin{equation}\n\\begin{split}\n\\mathcal{E}([T,\\infty)) := \\Big \\{ &\\, \\psi \\in C([T, \\infty);X) \\ \\text{a solution to \\eqref{NLS_diss}, with $\\vartheta \\in \\big (-\\frac \\pi 2 ,\\frac \\pi 2 \\big)$,} \\\\ \n&\\, \\text{$\\lambda, \\mu > 0$, $\\omega > |\\Omega| $, and $0 < \\sigma < \\frac{d}{2(d-2)}$} \\Big \\} .\n\\end{split}\n\\label{eq:syst}\n\\end{equation}\nClearly, the properties (i)--(iv) above hold for the evolutionary system associated to \\eqref{NLS_diss}. In addition, due to Proposition \\ref{prop:global}, for\nany $\\psi_0 \\in X$ there exists $\\psi \\in \\mathcal{E}([{T},\\infty))$ with $\\psi({T})=\\psi_0$. Standard techniques then imply the following lemma:\n\\begin{lemma} \\label{l:convergenceofLH}\nLet $(\\psi_n)_{n\\in \\N}$ be a sequence of functions, such that $\\psi_n \\in \\mathcal{E}([T_1, \\infty))$ for all $n\\in \\N$. Then for any $T_2>T_1$ there exists a sub-sequence $(\\psi_{n_j})_{j\\in \\N}$ which converges in \n$C([T_1, T_2]; X_{L^2})$ to $\\psi\\in \\mathcal{E}([T_1, \\infty))$.\n\\end{lemma}\n\n\\begin{proof}\nSince $X$ is compact in $L^2(\\R^d)$, there exists a sequence $(\\psi_{n_j})_{j\\in \\N}$ such that $\\psi_{n_j}(T_1) \\to \\tilde \\psi$ for some $\\tilde \\psi \\in L^2(\\R^d)$. \nHowever, since lower-semicontinuity and the definition of $X$ yield\n\\[\n\\| \\tilde \\psi \\|_\\Sigma \\le \\liminf_{j \\to \\infty} \\| \\psi_{n_j}(T_1, \\cdot) \\|_\\Sigma \\le \\rho_\\Sigma,\n\\]\nwe have that $\\tilde \\psi \\in X$. In view of proposition \\ref{prop:global} there exists $\\psi \\in \\mathcal{E}([T_1, \\infty))$ with $\\psi(T_1) = \\tilde \\psi$. \nContinuous dependence on the initial data, then gives the desired result.\n\\end{proof}\n\n\nUsing this, we can prove one of the main structural properties of the set of trajectories induced by \\eqref{NLS_diss}:\n\\begin{proposition} \\label{prop:compact} $\\mathcal{E}([0,\\infty))$ is a compact set in $C([0,\\infty); X_{L^2})$.\n\\end{proposition}\n\\begin{proof}\nFirst note that $\\mathcal{E}([0,\\infty)) \\subset C([0,\\infty);X_{L^2})$. Now take any sequence\n$(\\psi_n)_{n\\in \\N} \\in \\mathcal{E}([0,\\infty))$.\nThanks to Lemma~\\ref{l:convergenceofLH}, there exists\na subsequence, still denoted by $\\psi_n$, that converges\nto some $\\psi^{1} \\in \\mathcal{E}([0,\\infty))$ in $C([0, 1];X_{L^2})$ as $n \\to \\infty$.\nPassing to a subsequence and dropping a subindex once more, we obtain that\n$\\psi_n \\to \\psi^2$ in $C([0, 2];X_{L^2})$ as $n \\to \\infty$ for some \n$\\psi^{2} \\in \\mathcal{E}([0,\\infty))$.\nNote that $\\psi^1(t)=\\psi^2(t)$ on $[0, 1]$.\nContinuing and picking a diagonal sequence, we obtain a subsequence $\\psi_{n_j}$\nof $\\psi_n$ that converges\nto some $\\psi \\in \\mathcal{E}([0,\\infty))$ in $C([0, \\infty);X_{L^2})$ as $n_j \\to \\infty$.\n\\end{proof}\n\n\nIn order to proceed further, we denote, as usual, the set of all subsets of $X$ by $P(X)$.\nFor every $t \\ge 0$, we can then define a map $R(t):P(X) \\to P(X)$, by\n\\[\nR(t)A := \\{\\psi(t): \\psi(0) \\in A, \\ \\text{such that} \\ \\psi \\in \\mathcal{E}([0,\\infty))\\}, \\quad\n\\text{for any $A \\subset X.$}\n\\]\nNote that the assumptions on $\\mathcal{E}$ imply that $R(s)$ enjoys\nthe following property:\n\\begin{equation} \\label{eq:propR(T)}\nR(t+s)A \\subset R(t)R(s)A, \\qquad A \\subset X,\\quad t,s \\ge 0.\n\\end{equation}\n\n\\begin{definition}\nA set $A$ is called {\\it invariant} under the dynamics, if $R(t)A = A$ for all $t\\ge 0$.\n\\end{definition}\n\nWe also recall the standard notion of and $\\omega$-limit associated to an evolutionary system (see also \\cite{Te}).\n\n\\begin{definition}The ${\\omega}_{\\bullet}${\\it-limit} ($\\bullet= \\Sigma, \\, L^2$) of a set $A\\subset X$ is\n\\[{\\omega}_{\\bullet}(A):=\\bigcap_{T\\ge0}\\overline{\\bigcup_{t\\ge T}R(t)A}^{\\bullet}.\\]\\end{definition}\n\nWe also note that an equivalent definition of the ${\\omega}_{\\bullet}$-limit set is given by\n\\[\n\\begin{split}\n{\\omega}_{\\bullet}(A)=\\big\\{&{\\psi}\\in X: \\mbox{ there exist sequences }\nt_n \\xrightarrow{n\\to \\infty}\\infty \\mbox { and } {\\psi_n} \\in R(t_n)A,\\\\\n& \\mbox{such that } {\\psi_n(t_n)}\\xrightarrow{n\\to \\infty}\\psi \\mbox{ in the } \\mathrm{d}_{\\bullet}\\mbox{-metric} \\big \\}.\n\\end{split}\n\\]\n\nFinally, we will give a precise definition of what we mean by an attractor.\n\\begin{definition}\nA set $A \\subset X$ is a $\\mathrm{d}_{\\bullet}${\\it -attracting set}, if it uniformly\nattracts $X$ in $\\mathrm{d}_{\\bullet}$-metric, i.e. \n\\[\n\\liminf_{{\\phi} \\in A}\\mathrm{d}_{\\bullet}(R(t) X, {\\phi}) \\xrightarrow{t\\to +\\infty}0.\n\\]\nA set\n$\\mathcal{A} \\subset X$ is a\n$\\mathrm{d}_{\\bullet}$-{\\it global attractor} if\n$\\mathcal{A}$ is a minimal $\\mathrm{d}_{\\bullet}$-closed\n$\\mathrm{d}_{\\bullet}$-attracting set.\n\\end{definition}\n\nAfter these preparations, we are able to prove the main result of this section:\n\\begin{corollary} \\label{thm:Attractor}\nThe evolutionary system \\eqref{eq:syst} possesses a unique $d_{L^2}$-global attractor $\\mathcal{A}=\\omega_{L^2}(X)$, which has the following structure\n\\[\n\\mathcal{A}=\\{\\psi_0: \\psi_0 =\\psi(0) \\mbox { for some } \\psi\\in \\mathcal{E}((-\\infty,\\infty))\\}\\\\\n\\]\nFurthermore, it holds:\n\\begin{enumerate}\n\\item For any $\\epsilon > 0$ and $T > 0$, there exists a $t_0\\in \\R$, such that for any $t^* > t_0$, every trajectory\n $\\psi\\in\\mathcal{E}([0,\\infty))$ satisfies $\\mathrm{d}_{L^2}(\\psi(t), \\phi(t)) < \\varepsilon$, for all $t \\in [t^*, t^* +T ]$, where $\\phi \\in \\mathcal{E} ((-\\infty, \\infty))$ is some complete trajectory, i.e., \n the uniform tracking property holds.\n\\item If the $\\Sigma$ global attractor exists, then it coincides with $\\mathcal{A}$.\n\\item $\\mathcal{A}$ is connected in $L^2$.\n\\item $\\mathcal A$ is the maximal invariant set.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nAssertion (1) and (2) follow from the results proved in \\cite{C5}. To this end, one first shows that the $\\omega_{L^2}$-limit of $X$ is an attracting set, which by definition is closed \nand the minimal set satisfying these two properties. Then, using Proposition \\ref{prop:compact} and a diagonalization process, one can prove the structural properties of $\\mathcal A$, cf. \n\\cite[Theorem 5.6]{C5}.\nThe fact that $\\mathcal A$ is connected then follows from Lemma~\\ref{l:convergenceofLH} and uniqueness: We argue by contradiction and hence assume \nthat $\\mathcal{A}$ is not $L^2$-connected. Then there exist disjoint $\\mathrm{d}_{L^2}$-open sets $U_1, U_2 \\in X$\nsuch that $\\mathcal{A} \\subset U_1 \\cup U_2$ and $\\mathcal{A} \\cap \\, U_1$, $\\mathcal{A} \\cap U_2$ are nonempty. Define\n\\[\nX_j= \\{ {\\psi}\\in X: \\omega_{L^2}({\\psi}) \\in U_j\\}, \\qquad j=1,2.\n\\] \nSince $U_1$ and $U_2$ are disjoint, we also have that $X_1$, $X_2$ are disjoint. Continuity of trajectories implies that\n$X_1 \\cup X_2 =X$. Since $\\mathcal{A}$ is $\\mathrm{d}_{L^2}$-attracting, there exists $T> 0$ such that\n\\[\nR(t)X \\in U_1 \\cup U_2, \\qquad \\forall t > T.\n\\]\nBy continuity of trajectories we have that for each $\\psi \\in \\mathcal{E}([0,\\infty))$, either $\\psi(t) \\in U_1$ for all $t >T$, or $\\psi(t) \\in U_2$ for all $t >T$. \nThis implies that both $X_1$ and $X_2$ are nonempty. Moreover, Lemma~\\ref{l:convergenceofLH} implies that $X_1$ and\n$X_2$ are $\\mathrm{d}_{L^2}$-open. This contradicts the fact that $X$ is $\\mathrm{d}_{L^2}$-connected. \nFinally we note that the structure of $\\mathcal A$, together with uniqueness of solutions, imply that $\\mathcal A$ is an invariant set. \nClearly, only complete trajectories are invariant, hence $\\mathcal A$ is the maximal invariant set.\n\\end{proof}\n\n\\begin{remark}\nIn the case of the usual GL equation (posed on bounded domains $D\\subset \\R^d$) many more details concerning the global attractor are known, see, e.g., \\cite{MM, Te, TW}. \nIt is an interesting open problem to check which of these results can be extended to our situation and what the main structural differences between \\eqref{NLS_diss} and the usual GL equation are.\n\\end{remark}\n\n\\subsection{Dimension of the attractor} \\label{sec:dimension} We hereby follow the, by now, classical theory of estimating the Lyapunov numbers associated to $\\mathcal{E}([0,\\infty))$ by studying \nthe evolution of an $m$-dimensional volume element of our phase space $X$, cf. \\cite[Chapter V]{Te} for a general introduction. Using this technique, \nthe case of the usual GL equation on bounded domains $D\\subset \\R^n$, \nwith $n=1,2$ is studied, e.g., in \\cite[Chapter VI, Section 7]{Te}. In our case, the same idea works, but requires several adaptions on a technical level. \n\nTo this end, we first rewrite \\eqref{NLS_diss} as\n\\[\n\\partial_t \\psi = - e^{-i \\vartheta } G(\\psi),\\quad \\psi_{\\mid t =0} = \\psi_0.\n\\]\nand, for any $\\psi_0 \\in \\mathcal A$, consider the linearization around a given orbit $\\psi(t) = R(t) \\psi_0$, i.e.,\n\\begin{equation}\\label{eq:linear}\n\\partial_t \\phi = - e^{-i \\vartheta } G'(\\psi) \\phi, \\quad \\psi_{\\mid t =0} = \\xi.\n\\end{equation}\nHere, $\\xi \\in X$ and $G'$ denotes the Frechet derivative\n\\[\nG'(\\psi) \\phi = H_{\\Omega} \\phi - \\mu \\phi + \\lambda \\left(|{\\psi}|^{2\\sigma} \\phi + \\sigma \\psi |\\psi|^{2\\sigma -2} \\text{Re}\\, (\\overline \\psi \\phi) \\right),\n\\]\nwhere $H_\\Omega$ is the linear Hamiltonian (with rotation) defined \nin \\eqref{H}. It is easy to see, that the linearized equation \\eqref{eq:linear} admits a unique strong solution for any given $\\xi \\in X$ and $\\psi\\in \\mathcal A$. \nWe now consider $\\phi_1(t), \\dots, \\phi_m(t)$ solutions to \\eqref{eq:linear}, corresponding to initial data $\\xi_1, \\dots, \\xi_m$, $m\\in \\N$, and choose an \n$L^2$-orthonormal basis $\\chi_1(t), \\dots, \\chi_m(t)$ of \n\\[\nP_m(t) X:= \\text{span}\\{ \\phi_1(t), \\dots, \\phi_m(t)\\},\n\\]\nwhere $P_m$ denotes the corresponding orthogonal projection. Then, it is easy to see (cf. \\cite{Te}), that the evolution of the $m$-dimensional volume element in $X$ is given by\n\\[\n|\\phi_1(t)\\wedge \\dots \\wedge \\phi_m(t)| = | \\xi_1 \\wedge \\dots \\wedge \\xi_m| \\exp \\left(- \\int_0^t \\text{Re} \\, \\text{Tr}\\, e^{-i \\vartheta } G'(\\psi(s)) \\circ P_m(s) \\, ds \\right).\n\\]\nIn order to proceed, we first note that:\n\n\\begin{lemma}\\label{lem:collective} Let $H_0$ be given by \\eqref{H_0}. Then, for any orthonormal family $\\{ \\chi_j\\}_{j=1}^m \\subset L^2(\\R^d)$ there exists a constant $c=c(\\omega, d)>0$, such that\n\\[\n\\sum_{j=1}^m \\langle H_0 \\chi_j , \\chi_j \\rangle_{L^2} \\ge c \\, m^{1+1\/d}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nHaving in mind the form and the degeneracy of the eigenvalues stated Lemma \\ref{lem:H}, one checks that when counted with multiplicity $E_{0,m}\\sim m^{1\/d}$, as $m\\to \\infty$.\nThe desired result then follows directly from \\cite[Chapter VI, Lemma 2.1]{Te}. \\end{proof}\n\nUsing this, we can prove the following result for the dimension of $\\mathcal A$:\n\n\\begin{proposition}\\label{prop:dimension}\nConsider the dynamical system \\eqref{eq:syst} and let $m$ be defined by\n\\[\nm-1 < \\left( \\frac{2 \\kappa_2 }{\\kappa_1} \\right)^{d\/(d+1)}\\le m,\n\\]\nwhere\n\\[\n\\kappa_1 = \\frac{\\gamma c}{4}\\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big), \\quad \\kappa_2 = c' \\gamma \\mu^{1+d} \\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)^{-d}+\n\\frac{c''( \\lambda |\\beta| )^{1+\\alpha} }{\\gamma^\\alpha}\\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)^{-\\alpha} \\delta,\n\\]\nwith $c, c', c'', \\alpha, \\tilde \\alpha$ positive constants depending only on $\\omega, d, \\sigma$, and\n\\[\n\\delta = \\limsup_{t\\to \\infty} \\sup_{\\psi_0 \\in \\mathcal A} \\left( \\frac{1}{t} \\int_0^t \\| R(s) \\psi_0\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}} \\, ds \\right) \\le \\left(K\\frac{\\sigma+1}{\\lambda}\\right)^{\\frac{2\\sigma}{2\\sigma+2-d\\sigma}}.\n\\]\nHere, $K$ is the constant from Proposition~\\ref{prop:Ebound}.\n\nThen, as $t\\to +\\infty$, the $m$-dimensional volume element in $X$ is exponentially decaying.\nMoreover, the Hausdorff dimension of $\\mathcal A$ is less than or equal to $m$ and its fractal dimension is less than or equal to $2m$.\n\\end{proposition}\n\n\\begin{proof}\nHaving in mind the representation formula for the $m$-dimensional volume element as given above, we introduce\n\\[\nq_m:= \\limsup_{t\\to \\infty} \\sup_{ \\| \\xi_j \\|_{L^2} \\le 1} \\left(- \\frac{1}{t} \\int_0^t \\text{Re} \\, \\text{Tr}\\, e^{-i \\vartheta } G'(\\psi(s)) \\circ P_m(s) \\, ds \\right).\n\\]\nand quote the following result from \\cite[Chapter VI, Lemma 2.2]{Te}: If there are constants $\\kappa_{1,2}\\ge 0$, such that\n\\[\nq_j \\le - \\kappa_1 j^{\\theta} + \\kappa_2, \\quad \\forall j=1, \\dots, m,\n\\]\nthen the Hausdorff dimension of $\\mathcal A $ is less than or equal to $m$ and its fractal dimension is less than or equal to $2m$, where\n\\[\nm-1< \\left( \\frac{2 \\kappa_2 }{\\kappa_1} \\right)^{1\/\\theta}\\le m.\n\\]\nIn order to obtain the required estimate on $q_j$, we first note that\n\\[\n\\text{Re}\\, \\text{Tr} \\, e^{-i \\vartheta } G'(\\psi(t)) \\circ P_m(t) = \\sum_{j=1}^m \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi(t)) \\chi_j(t), \\chi_j(t)\\rangle_{L^2}.\n\\]\nNext, we recall that $e^{-i \\vartheta } = \\gamma + i \\beta$, with $\\gamma >0$, and compute (suppressing all the $t$-dependence for a moment)\n\\begin{align*}\n&\\, - \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi) \\chi_j, \\chi_j\\rangle_{L^2} = - \\frac{\\gamma}{2}\\left( \\| \\nabla \\chi_j \\|^2_{L^2} + \\omega^2 \n\\| x \\chi_j \\|_{L^2}^2\\right) + \\gamma \\Omega \\int_{\\R^d} \\overline {\\chi_j}L\\chi_j \\, dx + \\gamma \\mu \\\\\n&\\ - \\lambda \\gamma \\int_{\\R^d} |\\psi|^{2\\sigma} |\\chi_j|^2 \\, dx + \\sigma \\lambda \\int_{\\R^d} |\\psi|^{2\\sigma -2} \n\\text{Re}\\, (\\overline \\psi \\chi_j)\\big( \\beta \\text{Im}\\, (\\psi \\overline{\\chi_j})- \\gamma \\text{Re}\\, (\\psi \\overline {\\chi_j}) \\big)\\, dx,\n\\end{align*}\nwhere we have also used the fact that $\\| \\chi(t) \\|_{L^2}=1$. Next, we estimate the term {proportional to} $\\Omega$ as we did in the proof of Lemma \\ref{lem:Ebound} and we also \nuse that fact that \n\\[\n\\int_{\\R^d} \\beta \\text{Re}\\, (\\overline \\psi \\chi_j) \\beta \\text{Im}\\, (\\psi \\overline{\\chi_j})\\,dx \\le |\\beta| \\int_{\\R^d} |\\psi|^2 |\\chi_j|^2\\, dx.\n\\]\nIn summary, this yields\n\\begin{align*}\n - \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi) \\chi_j, \\chi_j\\rangle_{L^2} \\le &\\, - \\frac{\\gamma}{2}\\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)\\left( \\| \\nabla \\chi_j \\|^2_{L^2} + \\omega^2 \\| x \\chi_j \\|_{L^2}^2\\right) \n + \\gamma \\mu \\\\\n&\\ - \\lambda (\\gamma- \\sigma |\\beta| ) \\int_{\\R^d} |\\psi|^{2\\sigma} |\\chi_j|^2 \\, dx .\n\\end{align*}\nThus,\n\\begin{equation}\\label{mest1}\n\\begin{split}\n- \\sum_{j=1}^m \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi) \\chi_j, \\chi_j\\rangle_{L^2} \\le &\\, - \\gamma \\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)\\sum_{j=1}^m \\langle H_0 \\chi_j , \\chi_j \\rangle_{L^2} + \\gamma \\mu m \\\\\n&\\, + \\sigma \\lambda |\\beta| \\sum_{j=1}^m \\int_{\\R^d} |\\psi|^{2\\sigma} |\\chi_j|^2 \\, dx,\n\\end{split}\n\\end{equation}\nin view of definition \\eqref{H_0}. To further estimate the right hand side of \\eqref{mest1}, we use H\\\"older's inequality and Gagliardo-Nirenberg to obtain\n\\[\n\\int_{\\R^d} |\\psi|^{2\\sigma} |\\chi_j|^2 \\, dx \\le \\| \\psi\\|^{2\\sigma}_{L^{2\\sigma +2}} \\| \\chi_j \\|_{L^{2\\sigma+2}}^2 \\le c_1 \\| \\psi\\|^{2\\sigma}_{L^{2\\sigma +2}} \\| \\nabla \\chi_j\\|_{L^2}^{d\\sigma \/(\\sigma +1)},\n\\]\nwhere $c_1=c_1(d,\\sigma)>0$ some absolute constant. Young's inequality then implies that for any $\\varepsilon >0$, there exists a $c_2 = c_2(d, \\sigma)>0$, such that\n\\begin{align*}\n \\| \\psi\\|^{2\\sigma}_{L^{2\\sigma +2}} \\| \\nabla \\chi_j\\|_{L^2}^{d\\sigma \/(\\sigma +1)} \\le &\\, \\frac{c_2}{\\varepsilon^{\\alpha}} \\| \\psi\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}} + \\varepsilon \\| \\nabla \\chi_j\\|^2_{L^2} \\\\\n \\le & \\, \\frac{c_2}{\\varepsilon^{\\alpha}} \\| \\psi\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}} + \\varepsilon \\langle H_0 \\chi_j , \\chi_j \\rangle_{L^2},\n \\end{align*}\n where $\\alpha = \\frac{d\\sigma}{2\\sigma +2-d\\sigma}$, and $\\tilde \\alpha = \\frac{2\\sigma+2}{2\\sigma +2-d\\sigma}$. Note that both of these exponents are positive for $\\sigma <\\frac{d}{2(d-2)}$.\nThus, we an appropriate choice of $\\varepsilon$, we obtain from \\eqref{mest1}, that\n\\begin{equation*}\\label{mest2}\n\\begin{split}\n- \\sum_{j=1}^m \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi) \\chi_j, \\chi_j\\rangle_{L^2} \\le &\\, - \\frac{\\gamma}{2} \\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)\\sum_{j=1}^m \\langle H_0 \\chi_j , \\chi_j \\rangle_{L^2} + \\gamma \\mu m \\\\\n&\\, + \\frac{c_3( \\lambda |\\beta| )^{1+\\alpha} }{\\gamma^\\alpha}\\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)^{-\\alpha} \\| \\psi\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}}.\n\\end{split}\n\\end{equation*}\nNow, using the estimate from Lemma \\ref{lem:collective} above, we have\n\\begin{equation*}\\label{mest3}\n\\begin{split}\n- \\sum_{j=1}^m \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi) \\chi_j, \\chi_j\\rangle_{L^2} \\le &\\, - \\frac{\\gamma c}{2} \\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)m^{1+1\/d} + \\gamma \\mu m \\\\\n&\\, + \\frac{c_3( \\lambda |\\beta| )^{1+\\alpha} }{\\gamma^\\alpha}\\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)^{-\\alpha} \\| \\psi\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}}.\n\\end{split}\n\\end{equation*}\nThis can be estimated further by\n\\[\n- \\sum_{j=1}^m \\text{Re}\\, \\langle e^{-i \\vartheta } G'(\\psi(t)) \\chi_j, \\chi_j\\rangle_{L^2} \\le - \\kappa_1 m^{1+1\/d} + \\rho(t) ,\n\\]\nwhere $\\kappa_1$ is as defined above and\n\\[\n\\rho(t) = c_4 \\gamma \\mu^{1+d} \\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)^{-d}+\n\\frac{c_3( \\lambda |\\beta| )^{1+\\alpha} }{\\gamma^\\alpha}\\Big(1- \\frac{\\Omega^2}{\\omega^2}\\Big)^{-\\alpha} \\| \\psi (t)\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}}, \n\\]\nwith $c_4=c_4(\\omega, d)>0$. \n\nNow, for $\\psi(t) = R(t)\\psi_0 \\in \\mathcal A$, we have that\n\\[\n\\delta = \\limsup_{t\\to \\infty} \\sup_{\\psi_0 \\in \\mathcal A} \\left( \\frac{1}{t} \\int_0^t \\| R(s) \\psi_0\\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}} \\, ds \\right) <\\infty,\n\\]\ndue to Lemma \\ref{lem:Ebound} and Proposition \\ref{prop:Ebound}, which imply that for $\\psi(t)\\in \\mathcal A$:\n\\[\n\\| \\psi(t) \\|^{2\\sigma \\tilde \\alpha}_{L^{2\\sigma +2}} \\lesssim \\| \\psi(t) \\|^{2\\sigma \\tilde \\alpha\/(2\\sigma +2)}_{\\Sigma} \\lesssim \\rho_\\Sigma ^{2\\sigma \\tilde \\alpha\/(2\\sigma +2)}.\n\\]\nThis consequently yields\n\\[\nq_m \\le - \\kappa_1 m^{1+1\/d} + \\kappa_2, \\quad \\text{ for all $m\\ge 1$,}\n\\] \nwhich finishes the proof.\n\\end{proof}\n\n\\begin{remark}\nIn comparison to many \nclassical results on the dimensions of global attractors (cf. \\cite{Te}), the proof above avoids the use of a Lieb-Thirring type inequality to control the \nterm proportional to $ \\lambda$. \n\\end{remark}\n\nWe expect that a similar analysis can be done to estimate the box dimension of the attractor, cf. \\cite{CV} for more details.\nWe finally note that a careful analysis of all the involved constants in $\\kappa_1, \\kappa_2$ shows that for a given, fixed $\\omega>0$, \nthe fraction\n\\[\n\\left( \\frac{ \\kappa_2 }{\\kappa_1} \\right) \\to +\\infty, \\quad \\text{as $|\\Omega| \\to \\omega$.}\n\\]\nThe estimate on the dimension of $\\mathcal A$ thus becomes larger the larger the rotation speed.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLet $\\Omega \\subset \\mathbb{R}^{n}$ ($n \\in \\{2,3\\}$) be a bounded and convex domain with boundary $\\Gamma := \\partial \\Omega$ and $s \\in (0,1)$. \nFor $u_{d} : \\Omega \\rightarrow \\mathbb{R}$ we\ndefine the objective functional\n\\begin{equation}\\label{eq:Functional}\nJ(u, z) := \\frac{1}{2}\\|u - u_d\\|_{L^{2}(\\Omega)}^{2} + \\frac{\\mu}{2} \\|z\\|_{L^{2}(\\Omega)}^{2},\n\\end{equation}\nwhere $\\mu > 0$ denotes a regularization parameter. \nIn this work we consider the optimal control problem of finding\n\\begin{equation}\n\\label{eq:min_J}\n\\argmin_z J(u,z),\n\\end{equation}\nsubject to the \\textit{fractional state equation}\n\\begin{equation}\n\\label{eq:state_equation}\n\\left(-\\Delta\\right)^{s}u = z \\textrm{ in } \\Omega, \\quad u=0 \\textrm{ on } \\Gamma,\n\\end{equation}\nand the \\textit{control constraints}\n\\begin{equation}\n\\label{eq:control_constraints}\na \\leq z(x) \\leq b \\textrm{ a.e. in } \\Omega,\n\\end{equation}\nwith constants $a, b \\in \\mathbb{R}$ satisfying $a \\leq 0 \\leq b$.\nHere, we understand the operator $(-\\Delta)^{s}$ in the sense of its spectral\ndefinition, compare e.g.~\\cite{Cabre, Capella, Nochetto}.\n \nThe main difficulty in studying this problem is the nonlocality of the fractional Laplace operator \n\\cite{Cabre}. \nOne way to overcome this issue is based on the Cafarelli--Silvestre extension~\\cite{Cafarelli} on unbounded domains\nand its extension to bounded domains~\\cite{Cabre, Capella, Stinga}.\n In this approach, an auxiliary problem in an extended domain \n $\\mathcal{C} := \\Omega \\times (0,\\infty)$ is introduced and the solution of\n the state equation~\\eqref{eq:state_equation} is then given as the Dirichlet trace on $\\Omega \\times \\left\\{0\\right\\}$ of the solution to the extended problem.\nExponential decay of the solution in the artificial dimension allows construction of different numerical methods, \nsee e.g. \\cite{Bonito, MeidPSV_2017_hpFE_fracDiff, Nochetto}. \nIn these publications, the problem is discretized by introducing a tensor product mesh of the domain \n$\\mathcal{C}_{\\mathcal{Y}} = \\Omega \\times (0,Y)$, \nwhich is constructed by a conformal triangulation of $\\Omega$ \nand a graded mesh in the artificial direction, \nsee e.g. \\cite[Section 5.1]{Nochetto}.\nA convergence rate of $h^{1 + s}$ (up to some logarithmic term) \nin the $L^2(\\Omega)$-norm can be obtained~\\cite{Antil, Nochetto2}, \nprovided that $z \\in \\mathbb{H}^{1-s}(\\Omega)$, where $h$ denotes the global mesh parameter.\nHowever, numerical experiments show that this convergence rate is not optimal in a specific range of fractional powers~$s$.\nThe cost of solving the problem is related to the number of elements in $\\mathcal{C}_{\\mathcal{Y}}$,\nand not only to the number of elements in $\\Omega$, \nresulting in an increased computational complexity.\nThis issue is first overcome in \\cite{MeidPSV_2017_hpFE_fracDiff} by \nexploiting $p$-finite elements in the extended direction.\n\n\nAn alternative approach for solving~\\textup {(\\ref {eq:state_equation})} uses the \nBalakrishnan representation formula \\cite[IX. 11.]{Yosida_FuncAna}, namely for\n$s \\in (0,1)$ and $z \\in \\mathbb{H}^{-s}(\\Omega)$\n\\begin{equation}\n\\label{eq:balakrishnan_formula}\n(-\\Delta)^{-s}z = \\frac{\\sin{(s\\pi)}}{\\pi} \\int_{0}^{\\infty} \\nu^{-s} (\\nu I - \\Delta)^{-1}z d \\nu.\n\\end{equation}\nNumerical approximation of \\textup {(\\ref {eq:balakrishnan_formula})} is then based on a\nsuitable quadrature formula for \\textup {(\\ref {eq:balakrishnan_formula})} with respect to\n$\\nu$ and a discretization of the operator $\\nu I-\\Delta$ using the finite element method,\nsee \\cite{Bonito}. \n\n\nWhile the numerical analysis of the optimal control problem\n\\eqref{eq:min_J}--\\eqref{eq:control_constraints} using an equivalent formulation with the Cafarelli--Silvestre extension is well established \\cite{Antil}, the numerical analysis using the Balakrishnan formula is still open.\n\n\nIn this article we propose and analyze two discrete schemes for the approximation of the solution to the\n optimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints} using\n the Balakrishnan representation of the solution $u$ of the state equation\n \\eqref{eq:state_equation}.\n Both schemes rely on a finite element discretization of the operator $\\nu I - \\Delta$ in \\eqref{eq:balakrishnan_formula} \n and a sinc quadrature approximation \\cite{2018_BonitoLeiPasciak} of the integral in \\eqref{eq:balakrishnan_formula}. \n The first method is the variational discretization approach \\cite{Hinze}, \n where the set of controls is not discretized a priori. However, it inherits its approximation properties from the approximation of the adjoint state. \n The second one uses a fully discrete setting, where the set of controls is discretized by piecewise constant functions \\cite{Arada2002, Casas2005, Roesch2006}. \n We derive $L^2(\\Omega)$-error estimates for the state and control for both types of the FE discretization of the optimal control problem.\n\nRegarding the variational approach for the discretization of the optimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints} \nwe show an optimal convergence rate of $h^{\\min{(2, 3\/2+2s - \\varepsilon)}}$ for the control and the state in the $L^{2}(\\Omega)$-norm, \nwhereas using the extension approach \\cite{Antil} yields a convergence rate of $h^{1+s}$ (up to some logarithmic term). \nIn the case of the fully discrete scheme we show the expected linear convergence for the control in the $L^{2}(\\Omega)$-norm \nand for the state in the $\\mathbb{H}^{s}(\\Omega)$-norm. \nNumerically we also consider the post-processing approach \\cite{MeyerRoesch2004}\nfor the optimal control and measure again the same rate of $h^{\\min{(2, 3\/2+2s - \\varepsilon)}}$ for the \npost-processed optimal control.\nSimilar results are shown for the extension approach in \\cite{Antil}.\nWhile the convergence rate for the optimal control is optimal, as confirmed by numerical experiments, \nthere is still a gap between the theoretical and practical rates for the state, \nwhich will be addressed in future work. \n\nThe outline of this paper is as follows.\n In Section~\\ref{sec:fractional_optimal_control} we review existence and uniqueness results for the \n fractional optimal control problem based on \\cite{Antil} as well as regularity properties of the optimal control problem. \n The numerical analysis of the mentioned discretization methods is conducted in Section~\\ref{sec:error_estimates}, \n starting with the derivation of the error estimates for the discretization of the state equation \\eqref{eq:state_equation} \n using the Balakrishnan formula. \n In Section~\\ref{subsec:semidiscrete_scheme} we study the convergence properties of the optimal control \n and state using the semidiscrete approach, \n while Section~\\ref{subsec:full_discrete_scheme} is devoted to the numerical analysis of the fully discrete scheme. \n In Section~\\ref{sec:Implementation} we introduce a solver for the finite element approximation of the problem. \n Numerical results validating the theoretical convergence results for the proposed discretization techniques are presented in Section~\\ref{sec:Numerics}.\n \n \n \n \n \n \n\\section{Existence and regularity of optimal controls}\n\\label{sec:fractional_optimal_control}\nIn this section we review existence and uniqueness as well as the regularity results for the \noptimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints} based on \\cite[Sec. 3]{Antil}.\nWe start this section with a brief introduction of the spectral definition of the fractional operator\n$(-\\Delta)^s$ following~\\cite{Cabre, Capella}.\n\nThe eigenfunctions $\\left\\{\\varphi_{k}\\right\\}_{k\\in \\mathbb{N}}$ with eigenvalues $\\left\\{\\lambda_{k}\\right\\}_{k \\in \\mathbb{N}}$ of the Laplace operator, i.e.,\n\\begin{equation*}\n-\\Delta \\varphi_{k} = \\lambda_{k} \\varphi_{k} \\textrm{ in } \\Omega, \\quad \\varphi_{k} = 0 \\textrm{ on } \\Gamma, \\quad k \\in \\mathbb{N}\n\\end{equation*}\nform an orthonormal basis of $L^{2}(\\Omega)$. The spectral fractional Laplace operator for ${w \\in C_{0}^{\\infty}(\\Omega)}$ is then defined as\n\\begin{equation*}\n(-\\Delta)^{s}w:= \\sum_{k = 1}^{\\infty} \\lambda_{k}^{s} w_{k} \\varphi_{k}, \\quad w_{k} := \\int_{\\Omega} w \\varphi_{k} dx, \\quad k \\in \\mathbb{N}.\n\\end{equation*}\nThis definition can be extended by density to the space $\\mathbb{H}^{s}(\\Omega)$ \\cite{Antil, Nochetto} defined as\n\\begin{equation}\n\\label{eq:fractional_sobolev_spaces}\n\\mathbb{H}^{s}(\\Omega) := \\left\\{w = \\sum_{k=1}^{\\infty} w_{k} \\varphi_{k} : \\sum_{k=1}^{\\infty} \\lambda_{k}^{s}w_{k}^{2} < \\infty \\right\\} = \n\\begin{cases}\nH^{s}(\\Omega) \\equiv H_{0}^{s}(\\Omega) & \\textrm{if } s \\in (0,\\frac{1}{2}),\\\\\nH_{00}^{1\/2}(\\Omega) & \\textrm{if } s = \\frac{1}{2}, \\\\\nH_{0}^{s}(\\Omega) & \\textrm{if } s \\in (\\frac{1}{2}, 1).\n\\end{cases}\n\\end{equation}\nThe characterization of the fractional Sobolev spaces on the right hand side in \\eqref{eq:fractional_sobolev_spaces} can be found, e.g., in \\cite{mclean2000strongly}. For $s \\in [1,2]$ we set $\\mathbb{H}^{s}(\\Omega) := H^{s}(\\Omega) \\cap H^{1}_{0}(\\Omega)$, whereas $\\mathbb{H}^{0}(\\Omega) := L^{2}(\\Omega)$. The dual space of $\\mathbb{H}^{s}(\\Omega)$ we denote by $\\mathbb{H}^{-s}(\\Omega)$.\nWe stress that this definition of $(-\\Delta)^s$ inherently assumes homogeneous Dirchlet boundary data (in a suitable sense). \nFor a generalization to inhomogeneous Dirichlet boundary data we refer to \\cite{Antil_Pfefferer_Rogovs} and the references therein.\n\n\n\\bigskip\n\n\nLet $u_{d} \\in L^{2}(\\Omega)$ and $a, b \\in \\mathbb{R}$ with $a \\leq 0 \\leq b$ be given.\nWe define the set of admissible controls $Z_{ad}$ by\n\\begin{equation*}\nZ_{ad} := \\left\\{w \\in L^{2}(\\Omega): a \\leq w(x) \\leq b \\textrm{ a.e. in } \\Omega \\right\\}.\n\\end{equation*}\nLet ${S: \\mathbb{H}^{-s}(\\Omega) \\rightarrow \\mathbb{H}^{s}(\\Omega)}$ denote the control to state operator defined as $Sz := u$, \nwhere $u \\in \\mathbb{H}^{s}(\\Omega)$ is the unique solution of the state equation \\eqref{eq:state_equation}. \nIt holds, that for any $z \\in \\mathbb{H}^{-s}(\\Omega)$, \nthe boundary value problem \\eqref{eq:state_equation} has a unique solution $u \\in \\mathbb{H}^{s}(\\Omega)$, see e.g. \\cite{MeidPSV_2017_hpFE_fracDiff}.\nWe may also consider the operator $S$ acting on $L^{2}(\\Omega)$ with range in $L^{2}(\\Omega)$. Note also that $S$ is self-adjoint, since the operator $(-\\Delta)^{s}$ is self-adjoint. The adjoint state $p \\in \\mathbb{H}^{s}(\\Omega)$ for $z \\in \\mathbb{H}^{-s}(\\Omega)$ is then given by $p = S(Sz - u_{d})$. \nIn \\cite[Section 3.1]{Antil} the existence and uniqueness of a solution to the optimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints} is shown. Let us recall the main result from that reference.\n\n\\begin{theorem}[existence, uniqueness, and optimality conditions, {\\cite[Section 3.1]{Antil}}]\nThe fractional optimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints} has a unique optimal solution \n$(\\bar{u}, \\bar{z}) \\in \\mathbb{H}^{s}(\\Omega) \\times Z_{ad}$. \nThese fulfill the necessary and sufficient optimality conditions\n\\begin{align}\n \\bar{u} &= S \\bar{z} \\in \\mathbb{H}^{s}(\\Omega),\\\\\n \\bar{p} &= S(\\bar{u}-u_{d}) \\in \\mathbb{H}^{s}(\\Omega),\\\\\n \\bar{z} \\in Z_{ad}, &\\quad (\\mu \\bar{z} + \\bar{p}, z - \\bar{z})_{L^{2}(\\Omega)} \\geq 0 \\quad \\textrm{for all } z \\in Z_{ad}.\n \\label{eq:variational_inequality}\n\\end{align}\n\\end{theorem}\n\n\n\nFor $\\mu > 0$ and $\\bar{p} = S(\\bar{u}-u_{d})$ the variational inequality \\eqref{eq:variational_inequality} is equivalent to the projection formula \\cite{Fredi}\n\\begin{equation*}\n\\bar{z}(x) = \\textrm{proj}_{[a, b]} \\left(-\\frac{1}{\\mu} \\bar{p}(x)\\right)\n\\end{equation*}\nwhere $\\textrm{proj}_{[a, b]}(v) := \\min{\\left\\{b, \\max{\\left\\{a,v\\right\\}}\\right\\}}$.\nSince we assume that $\\Omega$ is a convex domain and that $a \\leq 0 \\leq b$ we can prove the following regularity results for the control.\n\\begin{lemma}[$H^{1}$-regularity of the optimal control, {\\cite[Lemma 3.5]{Antil}}]\nLet $\\bar{z} \\in Z_{ad}$ be the optimal control and $u_{d} \\in \\mathbb{H}^{1-s}(\\Omega)$. \nThen $\\bar{z} \\in H_0^{1}(\\Omega)$.\n\\end{lemma}\n\\begin{proof}\nThe proof is based on bootstrapping. \nWe only comment on the case $s \\in (0,\\frac{1}{4})$. In this case,\nan intermediate regularity result is $\\bar{z} \\in \\mathbb{H}^{s}(\\Omega)$. \nAs $\\bar z = \\textrm{proj}_{[a,b]}\\left( -\\frac{1}{\\mu}\\bar p\\right)$ this,\nin turn, requires $a \\leq 0 \\leq b$.\n\\end{proof}\n\\begin{lemma}\n \\label{lem:regularity_control}\nLet $\\bar{z} \\in Z_{ad}$ be the solution of the optimal control problem \\textup {(\\ref {eq:min_J})}--\\textup {(\\ref {eq:control_constraints})} \nwith $u_{d} \\in \\mathbb{H}^{3\/2}(\\Omega)$. \nThen $\\bar{z} \\in \\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)$, where $\\varepsilon$ is a positive, arbitrary small number.\n\\end{lemma}\n\\begin{proof}\nThe proof follows from the standard bootstraping argument.\n\\end{proof}\n\\section{A priori error estimates}\n\\label{sec:error_estimates}\n\nIn this section, we analyse two finite element approximations of \nthe fractional optimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints}. \nFirst, we investigate the variational approach \\cite{Hinze}, \nwhere the control set is not discretized, and then move to a fully discrete scheme. \nBoth techniques are based on a finite element discretization of the state \nequation \\eqref{eq:state_equation} using the Balakrishnan formula \\eqref{eq:balakrishnan_formula}. \nIn the following subsection, we review the resulting FE error estimates, based on \\cite[Sec. 4]{Bonito}.\n \n \n\\begin{assumption} \nThroughout this and the following sections we assume that $\\Omega$ is a polygonal or polyhedral domain and that $u_{d} \\in \\mathbb{H}^{3\/2}(\\Omega)$, hence the regularity result from Lemma \\ref{lem:regularity_control} holds.\n\\end{assumption} \n\n\n\\subsection{A finite element method for the state equation}\n\\label{subsec:fem_state_equation}\n\nLet $\\mathbb{U}(\\mathcal{T}_{h})$ be the space of piecewise linear and globally continuous functions vanishing on the boundary $\\partial \\Omega$, defined with respect to a conforming quasi-uniform triangulation $\\mathcal{T}_{h}$ of the domain $\\Omega$. \nA FE approximation of problem \\eqref{eq:state_equation} for $z \\in L^{2}(\\Omega)$ is given by\n\\begin{equation}\n\\label{eq:dunford_taylor_FE}\nu_{h} = \\frac{\\sin{(s \\pi)}}{\\pi} \\int_{0}^{\\infty} \\nu^{-s}(\\nu I - \\Delta_{h})^{-1} z\\,d\\nu\n= \\frac{\\sin{(s \\pi)}}{\\pi}\\int_{-\\infty}^{\\infty} e^{(1-s)t}\\left(e^tI-\\Delta_h\\right)^{-1}z\\,dt,\n\\end{equation}\nwhere $\\Delta_{h}$ denotes the discrete Laplace operator.\n\nFor $k > 0$ we define the numbers\n$N_+ := \\Bigg\\lceil \\frac{\\pi^2}{4 s k^2} \\Bigg\\rceil$ and $N_- := \\Bigg\\lceil \\frac{\\pi^2}{4 (1-s) k^2} \\Bigg\\rceil$.\nThe sinc quadrature approximation of $u_{h}$ is then given by\n\\begin{equation}\n\\label{eq:sinc_quadrature}\nu_{h}^{k} := \\frac{\\sin{(s \\pi)}}{\\pi} k \\sum_{l = -N_{-}}^{N_{+}} e^{(1-s)kl}(e^{kl}I - \\Delta_{h})^{-1} z.\n\\end{equation}\n\nPractical aspects of the numerical implementation of this method are discussed in Section\\nobreakspace \\ref {sec:Implementation}.\n\n\nIn our problem set-up the following error estimates hold.\n\n\n\\begin{theorem}[finite element approximation,{\\cite[Theorem 4.2]{Bonito}}]\n\\label{thm:fe_error_dunford_talyor} \nGiven $r \\in [0,1]$ with $r \\leq 2s$, set $\\gamma := \\max{(r+2\\alpha_{\\star} - 2s, 0)}$ \nand $\\alpha_{\\star} := \\frac{1}{2}(\\alpha + \\min{(1-r, \\alpha)})$ with ${\\alpha \\in (0,1]}$. \nIf $z \\in \\mathbb{H}^{\\delta}(\\Omega)$ for $\\delta \\geq \\gamma$, then \n\\begin{equation*}\n\\norm{u - u_{h}}_{\\mathbb{H}^{r}(\\Omega)} \\leq C_{h}\\, h^{2\\alpha_{\\star}} \\norm{z}_{\\mathbb{H}^{\\delta}(\\Omega)}\n\\end{equation*}\nwhere $C_{h} \\leq c \\log{(2\/h)}$ if $\\delta = \\gamma$ and $r+2\\alpha_{\\star} \\geq 2s$, and $C_{h} \\leq c$ otherwise. \n\\end{theorem}\nNote, that we get a convergence rate of $h^{2 - r}$ if we set $\\alpha = 1$ and if $z$ is regular enough. However, in order to obtain the convergence rates depending on $s \\in (0,1)$ and on the regularity of $z$, we have to choose $\\alpha$ in Theorem \\ref{thm:fe_error_dunford_talyor} appropriately.\nFor $r=0$ and $r=s$ respectively in Theorem \\ref{thm:fe_error_dunford_talyor}, we conclude the following error estimates.\n\\begin{corollary}\nFor $z \\in \\mathbb{H}^{\\delta + \\varepsilon'}(\\Omega)$ with $\\varepsilon' > 0$ arbitrary small and $\\delta \\geq - \\varepsilon'$ there holds\n\\begin{equation}\n\\label{eq:fe_error_l2_hs}\n\\begin{aligned}\n\\norm{u - u_{h}}_{L^{2}(\\Omega)} &\\leq c \\, h^{\\min{(2, \\delta + 2s)}} \\norm{z}_{\\mathbb{H}^{\\delta + \\varepsilon'}(\\Omega)},\\\\\n\\norm{u - u_{h}}_{\\mathbb{H}^{s}(\\Omega)} &\\leq c \\, h^{\\min{(2-s, \\delta + s)}} \\norm{z}_{\\mathbb{H}^{\\delta + \\varepsilon'}(\\Omega)}.\n\\end{aligned}\n\\end{equation}\n\\end{corollary}\n\nThe quadrature formula \\eqref{eq:sinc_quadrature} possesses the following approximation property.\n\\begin{theorem}[sinc quadrature approximation, {\\cite[Theorem 4.3]{2018_BonitoLeiPasciak}}]\nFor $r \\in [0,1]$ and $z \\in \\mathbb{H}^{r}(\\Omega)$ there holds\n\\begin{equation*}\n\\norm{u_{h} - u_{h}^{k}}_{\\mathbb{H}^{r}(\\Omega)} \n\\leq c \\, e^{-\\pi^{2}\/(2k)} \\norm{z}_{\\mathbb{H}^{\\max(0,r-2s+\\epsilon)}(\\Omega)}\n\\leq c \\, e^{-\\pi^{2}\/(2k)} \\norm{z}_{\\mathbb{H}^{r}(\\Omega)}\n\\end{equation*}\n\\end{theorem}\n\nHence, if we choose $k$ appropriately, we can balance the sinc quadrature and the finite element errors.\n\\begin{lemma}\n\\label{lem:fe_approximation_error}\nAssume that the number of integration points in the\n sinc quadrature~\\textup {(\\ref {eq:sinc_quadrature})} is balanced with the FE errors\n \\textup {(\\ref {eq:fe_error_l2_hs})}, i.e., $k \\in \\mathcal O(\\big|\\ln{h}\\big|^{-1})$. For\n $z \\in \\mathbb{H}^{\\delta+\\varepsilon'}(\\Omega)$ with $\\varepsilon' > 0$ and $\\delta \\geq - \\varepsilon'$ we obtain\n\\begin{equation}\n\\label{eq:fe_approximation_error}\n\\begin{aligned}\n\\norm{u - u_{h}^{k}}_{L^{2}(\\Omega)} &\\leq c \\, h^{\\min{(2, \\delta+2s)}} \\norm{z}_{\\mathbb{H}^{\\delta+\\varepsilon'}(\\Omega)},\\\\\n\\norm{u - u_{h}^{k}}_{\\mathbb{H}^{s}(\\Omega)} &\\leq c \\, h^{\\min{(2-s, \\delta+s)}} \\norm{z}_{\\mathbb{H}^{\\delta+\\varepsilon'}(\\Omega)}.\n\\end{aligned}\n\\end{equation}\n\\end{lemma}\n\nGiven the regularity results of ~Lemma\\nobreakspace \\ref {lem:regularity_control} for $u_{d} \\in \\mathbb{H}^{3\/2}(\\Omega)$ we conclude the following error estimates.\n\\begin{corollary}\n\\label{cor:fe_error_estimates_regular}\nFor $z \\in \\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)$ there holds\n\\begin{equation*}\n\\norm{u - u_{h}^{k}}_{L^{2}(\\Omega)} \\leq c \\, h^{\\min{(2, 3\/2+2s-\\varepsilon')}} \\norm{z}_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)}\n\\end{equation*} \nand\n\\begin{equation*}\n\\norm{u - u_{h}^{k}}_{\\mathbb{H}^{s}(\\Omega)} \\leq c \\, h^{\\min{(2-s, 3\/2 + s - \\varepsilon')}} \\norm{z}_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)}\n\\end{equation*}\nwith $\\varepsilon' > 0$ and $\\varepsilon > 0$ arbitrary small and $\\varepsilon < \\varepsilon'$. Note, that the approximation $u^k_{h}$ converges quadratically in the\n$L^{2}(\\Omega)$-norm, provided that $s > \\frac{1}{4}$.\n\\end{corollary}\n\nIn the following we drop the superscript $k$ and write $z_h$, $u_h$, and $p_h$ for the discrete approximations of $z$, $u$ and $p$.\n\n\n\\subsection{Variational discretization}\n\\label{subsec:semidiscrete_scheme}\n\nWe define the variational discretization of the optimal control problem\n\\eqref{eq:min_J}--\\eqref{eq:control_constraints}\nas finding $\\bar{z}_h \\in Z_{\\text{ad}}$ such that\n\\begin{align}\\label{eq:var_dis}\n\\bar{z}_h := \\argmin_{z \\in Z_{\\text{ad}}}J_h(z) = \\argmin_{z \\in Z_{\\text{ad}}}\\frac{1}{2} \\lVert S_h z -u_d \\rVert_{L^2{(\\Omega})}^{2} + \\frac{\\mu}{2}\\lVert z\\rVert_{L^2{(\\Omega})}^{2}.\n\\end{align}\n\nSimilarly, as in the continuous setting, the discrete optimal control problem \\eqref{eq:var_dis} has a unique solution $\\bar{z}_h \\in Z_{\\text{ad}}$.\nWe denote by $\\bar{u}_h:= S_h \\bar{z}_h$ the optimal discrete state and by \n$\\bar{p}_h := S_h(S_h \\bar{z}_h-u_d)$ the optimal discrete adjoint state. In this case the variational inequality reads as\n\\begin{equation}\\label{eq:var_ineq_var}\n (\\bar{p}_h + \\mu \\bar{z}_h,z-\\bar{z}_h)_{L^2(\\Omega)} \\geq 0\\quad \\forall z \\in Z_{ad},\n\\end{equation}\nwhich implies \n \\begin{equation}\n \\label{eq:disc_proj}\n \\bar{z}_h = \\textrm{proj}_{[a,b]}\\left( -\\frac{1}{\\mu}\\bar{p}_h \\right).\n \\end{equation}\n Here and in the following, we denote by $S_h$ the discrete, self-adjoint solution operator defined by \\textup {(\\ref {eq:sinc_quadrature})}.\n \n\\begin{lemma}\\label{lemma:stab}\nThe following stability estimates hold\n\\begin{align}\n\\lVert Sv \\rVert_{L^2(\\Omega)} &\\leq c \\lVert v \\rVert_{L^2(\\Omega)}, \\quad \\lVert S_h v \\rVert_{L^2(\\Omega)} \\leq c \\lVert v \\rVert_{L^2(\\Omega)}, \\quad \\lVert S_h v \\rVert_{\\mathbb{H}^s(\\Omega)} \\leq c \\lVert v \\rVert_{L^2(\\Omega)}.\n \\label{eq:Sv}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nThe first estimate follows from a trivial embedding and the $2s$-shift of the fractional Laplace operator. \n\\[\n\\lVert Sv \\rVert_{L^2(\\Omega)} \\leq c \\lVert Sv \\rVert_{\\mathbb{H}^{2s}(\\Omega)} \\leq c \\lVert v \\rVert_{L^2(\\Omega)} .\n\\]\nTo prove the second estimate we introduce the intermediate function $Sv$ to obtain\n\\[\n\\norm{S_{h}v}_{L^{2}(\\Omega)} \\leq \\lVert (S_h - S) v \\rVert_{L^2(\\Omega)} + \\lVert Sv \\rVert_{L^2(\\Omega)},\n\\]\nand apply the a priori error estimate \\eqref{eq:fe_approximation_error} \nwith $\\delta = -\\varepsilon'$ \nas well as the first estimate in \\eqref{eq:Sv}. \nThe proof of the third estimate follows the same path, using the stability of the operator $S: L^{2}(\\Omega) \\rightarrow \\mathbb{H}^{s}(\\Omega)$.\n\\end{proof}\n \n\\begin{theorem}\nLet the pairs $(\\bar{u}(\\bar{z}),\\bar{z})$ and $(\\bar{u}_h(\\bar{z}_h),\\bar{z}_h)$ be the solutions to problems \\eqref{eq:min_J} and \\eqref{eq:var_dis}, respectively. Then the estimates\n\\begin{align}\\label{eq:var_a_priori_z}\n\\lVert \\bar{z} - \\bar{z}_h \\rVert_{L^2(\\Omega)} &\\leq c \\, h^{\\min(2,3\/2+2s -\\varepsilon')} \\left(\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} \\right),\\\\\\label{eq:var_a_priori_u}\n\\lVert \\bar{u} - \\bar{u}_h \\rVert_{L^2(\\Omega)} &\\leq c \\, h^{\\min(2,3\/2+2s -\\varepsilon')} \\left(\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} \\right),\\\\\\label{eq:var_a_priori_u_s}\n\\lVert \\bar{u} - \\bar{u}_h \\rVert_{\\mathbb{H}^s(\\Omega)} &\\leq c \\, h^{\\min(2-s,3\/2+s -\\varepsilon')} \\left(\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} \\right)\n\\end{align}\nhold, provided $\\varepsilon < \\varepsilon'$.\n\\end{theorem}\n\\begin{proof}\nWe begin by showing the first estimate.\nThe proof is similar to the proof of \\cite[Theorem 5.10]{Antil} based on ideas introduced in \\cite{Hinze}.\nTesting variational inequalities \\eqref{eq:variational_inequality} and \\eqref{eq:var_ineq_var} with $\\bar{z}_h \\in Z_{\\text{ad}}$ and $\\bar{z} \\in Z_{\\text{ad}}$, respectively, and adding both expressions, we arrive at\n\\begin{align}\n\\mu \\lVert \\bar{z}-\\bar{z}_h \\rVert_{L^2(\\Omega)}^2& \\leq ( \\bar{p}-\\bar{p}_h, \\bar{z}_h-\\bar{z})\\nonumber\\\\\n &\\leq ((S-S_h)S\\bar{z}, \\bar{z}_h-\\bar{z})+ (S_{h}(S-S_h)\\bar{z}, \\bar{z}_h-\\bar{z})\\nonumber\\\\\n &\\quad +((S_h-S) u_d, \\bar{z}_h-\\bar{z}) + (S_h^2(\\bar{z}-\\bar{z}_h), \\bar{z}_h-\\bar{z}).\\label{eq:4_terms_var}\n\\end{align}\nThe first two terms can be estimated using the Cauchy--Schwarz inequality, Lemma \\ref{lemma:stab} and the a priori estimate \\eqref{eq:fe_approximation_error}\n\\begin{align}\n((S-S_h)S\\bar{z}, \\bar{z}_h-\\bar{z}) &\\leq c \\, h^{\\min(2,3\/2+2s -\\varepsilon')}\n\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)}\\lVert\n\\bar{z}-\\bar{z}_h \\rVert_{L^2(\\Omega)},\\label{eq:var_proof_1}\\\\\n(S_h(S-S_h)\\bar{z}, \\bar{z}_h-\\bar{z}) &\\leq c \\, h^{\\min(2,3\/2+2s -\\varepsilon')} \\lVert \\bar{z} \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)}\\lVert \\bar{z}-\\bar{z}_h \\rVert_{L^2(\\Omega)}.\n\\end{align}\nThe estimate of the third term follows from the Cauchy--Schwarz inequality and the estimate \\eqref{eq:fe_approximation_error}\n\\begin{equation}\\label{eq:var_proof_3}\n((S_h-S) u_d, \\bar{z}_h-\\bar{z}) \\leq c \\, h^{\\min(2,3\/2+2s -\\varepsilon')} \\lVert u_d \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} \\lVert \\bar{z}-\\bar{z}_h \\rVert_{L^2(\\Omega)},\n\\end{equation}\n and the last term is non-positive, since $S_h$ is self-adjoint and therefore\n\\begin{equation}\\label{eq:var_proof_4}\n(S_h^2(\\bar{z}-\\bar{z}_h), \\bar{z}_h-\\bar{z}) \\leq -\\lVert S_h(\\bar{z}-\\bar{z}_h) \\rVert_{L^2(\\Omega)}^2 \\leq 0.\n\\end{equation}\nThe desired estimate follows from estimates \\eqref{eq:var_proof_1}--\\eqref{eq:var_proof_4}.\n\nApplication of Lemma\\nobreakspace \\ref {lem:fe_approximation_error}, Lemma\\nobreakspace \\ref {lemma:stab} and \\textup {(\\ref {eq:var_a_priori_z})} leads to \n\\begin{align*}\n\\lVert \\bar{u} - \\bar{u}_h \\rVert_{L^2(\\Omega)} &\\leq \\lVert S \\bar{z} -S_h \\bar{z} \\rVert_{L^2(\\Omega)} + \\lVert S_h \\bar{z} -S_h \\bar{z}_h \\rVert_{L^2(\\Omega)} \\\\\n&\\leq c \\, h^{\\min(2,3\/2+2s -\\varepsilon')} \\left(\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)} \\right),\n\\end{align*}\nand this proves \\textup {(\\ref {eq:var_a_priori_u})}. The proof of\n\\textup {(\\ref {eq:var_a_priori_u_s})} follows the same path.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{A fully discrete scheme}\n\\label{subsec:full_discrete_scheme}\n\nIn this section we consider a fully discrete \nscheme for the optimal control problem \\eqref{eq:min_J}--\\eqref{eq:control_constraints}. \nWe discretize the set of admissible controls with piecewise constant functions\n\\begin{align*}\n Z_h &:= \\{ z_h \\in L^{\\infty}(\\Omega) : z_h \\vert_T \\in \\mathcal{P}_0 \\text{ for all } T \\in \\mathcal{T}_h \\}, \n \\quad \\text{and }Z_h^{\\text{ad}} := Z_h \\cap Z_{\\text{ad}} .\n\\end{align*}\nThe discretized optimal control problem reads as: find $\\bar z_h \\in Z_h^{\\text{ad}}$ such that\n\\begin{align}\\label{eq:opt_cont_pb_full}\n\\bar{z}_h &= \\argmin_{z_h \\in Z^{\\text{ad}}_h}J_h (z_h) = \\argmin_{z_h \\in Z^{\\text{ad}}_h} \\frac{1}{2} \\lVert S_h z_h - u_d \\rVert^2_{L^2(\\Omega)} + \\frac{\\mu}{2} \\lVert z_h \\rVert^2_{L^2(\\Omega)}.\n\\end{align}\nUsing the same argumentation as in the continuous case, \nit can be shown that the optimal control problem \\eqref{eq:opt_cont_pb_full} \nhas a unique solution $\\bar{z}_h \\in Z_{h}^{\\text{ad}}$.\n Let $\\bar{u}_h = S_h\\bar{z}_h$ and $\\bar{p}_h = S_h(S_h \\bar{z}_h - u_d)$ \n be the optimal discrete state and optimal discrete adjoint state,\n respectively, associated with $\\bar{z}_h $. \n Then the discrete optimality condition reads as \n \\begin{equation}\\label{eq:full_ineq_var}\n (\\bar{p}_h + \\mu \\bar{z}_h, z_h - \\bar{z}_h)_{L^2(\\Omega)} \\geq 0\\quad \\forall z_h \\in Z_h^{\\text{ad}}.\n \\end{equation}\nBefore we state the main result of this section, \nwe define the $L^2(\\Omega)$-projection operator $Q_h: L^2(\\Omega) \\rightarrow Z_h$ by\n\\[\n\\int_{\\Omega} (z - Q_h z) v_h = 0 \\quad \\forall v_h \\in Z_h,\n\\]\nwhich has the following properties\n\\begin{enumerate}\n\\item[(L1)] $\\lVert Q_h v \\rVert_{L^2(\\Omega)} \\leq c\\lVert v \\rVert_{L^2(\\Omega)} \\quad \\forall v \\in L^2(\\Omega)$,\n\\item[(L2)] $\\lVert v - Q_h v \\rVert_{L^2(\\Omega)} \\leq c \\, h \\norm{v}_{\\mathbb{H}^{1}(\\Omega)} \\quad \\forall v \\in {H}^1(\\Omega)$.\n\\end{enumerate}\n\n\\begin{theorem}\n\\label{thm:FD:rates}\nLet the pairs $(\\bar{u}(\\bar{z}),\\bar{z})$ and $(\\bar{u}_h(\\bar{z}_h),\\bar{z}_h)$ \nbe the solutions to problems \\eqref{eq:min_J} and \\eqref{eq:opt_cont_pb_full}, respectively. Then the estimates\n\\begin{align}\\label{eq:full_a_priori_z}\n\\lVert \\bar{z} - \\bar{z}_h \\rVert_{L^2(\\Omega)} &\\leq c \\, h \\left(\\norm{\\bar{z}}_{\\mathbb{H}^{1}(\\Omega)} \n+ \\lVert u_d \\rVert_{\\mathbb{H}^{\\max{(0, 1-2s+\\varepsilon)}}(\\Omega)} \\right),\\\\\n\\label{eq:full_a_priori_u}\n\\lVert \\bar{u} - \\bar{u}_{h} \\rVert_{\\mathbb{H}^{s}(\\Omega)} &\\leq c \\, h \\left(\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)} \n+ \\lVert u_d \\rVert_{\\mathbb{H}^{\\max{(0, 1-2s+\\varepsilon)}}(\\Omega)} \\right),\\\\\n\\lVert \\bar{u} - \\bar{u}_{h} \\rVert_{L^{2}(\\Omega)} &\\leq c \\, h \\left(\\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)} \n+ \\lVert u_d \\rVert_{\\mathbb{H}^{\\max{(0, 1-2s+\\varepsilon)}}(\\Omega)} \\right)\n\\end{align}\nhold.\n\\end{theorem}\n\\begin{proof}\nThe proof is similar to the proof of \\cite[Theorem 5.16]{Antil}. \nFirst, we use $z = \\bar{z}_h \\in Z_{\\textrm{ad}}$ in the continous optimality condition \\eqref{eq:variational_inequality} \nto get\n\\[\n(\\bar{p} + \\mu \\bar{z},\\bar{z}_h - \\bar{z}) \\geq 0. \n\\]\nSecond, using $z_h = Q_h \\bar{z} \\in Z_h^{\\text{ad}}$ in the discrete optimality condition\n\\eqref{eq:full_ineq_var}\nand introducing $\\bar{z}$, we arrive at\n\\[\n(\\bar{p}_h + \\mu \\bar{z}_h , Q_h \\bar{z}- \\bar{z}) + (\\bar{p}_h + \\mu \\bar{z}_h , \\bar{z}- \\bar{z}_h) \\geq 0.\n\\]\nConsequently, adding the previous two inequalities together we get\n\\[\n(\\bar{p} -\\bar{p}_h + \\mu (\\bar{z} - \\bar{z}_h) ,\\bar{z}_h - \\bar{z}) + (\\bar{p}_h + \\mu \\bar{z}_h , Q_h \\bar{z}- \\bar{z}) \\geq 0.\n\\]\nHence, we can conclude\n\\begin{equation}\\label{eq:full_2_terms}\n\\mu \\lVert \\bar{z} - \\bar{z}_h \\rVert_{L^2(\\Omega)}^2 \\leq (\\bar{p} -\\bar{p}_h ,\\bar{z}_h - \\bar{z}) + (\\bar{p}_h + \\mu \\bar{z}_h , Q_h \\bar{z}- \\bar{z}).\n\\end{equation}\nThe estimate for the first term on the right hand side of \\eqref{eq:full_2_terms} \nfollows from the estimate for \\eqref{eq:4_terms_var} \nwith an appropriate application of estimate \\eqref{eq:fe_approximation_error}\n\\begin{equation}\\label{eq:full_2_1_terms}\n(\\bar{p} -\\bar{p}_h ,\\bar{z}_h - \\bar{z}) \\leq c \\, h \\left( \\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{\\max{(0, 1-2s+\\varepsilon)}}(\\Omega)} \\right) \\lVert \\bar{z}-\\bar{z}_h \\rVert_{L^2(\\Omega)}.\n\\end{equation}\nTo estimate the second term we add and substract $\\bar{p}$ and $\\mu \\bar{z}$ and get \n\\begin{equation}\\label{eq:full_3_terms}\n(\\bar{p}_h + \\mu \\bar{z}_h , Q_h \\bar{z}- \\bar{z}) \n= (\\bar{p} + \\mu \\bar{z} , Q_h \\bar{z}- \\bar{z}) + \\mu (\\bar{z}_h - \\bar{z} , Q_h \\bar{z}- \\bar{z}) + (\\bar{p}_h - \\bar{p} , Q_h \\bar{z}- \\bar{z}).\n\\end{equation}\nTo estimate the first term on the right hand side of \\eqref{eq:full_3_terms} we use the definition of the operator $Q_h$ and obtain\n\\[\n(\\bar{p} + \\mu \\bar{z} , Q_h \\bar{z}- \\bar{z}) = (\\bar{p} + \\mu \\bar{z} - Q_h(\\bar{p} + \\mu \\bar{z}) , Q_h \\bar{z}- \\bar{z}) \\leq c h^2 \\lVert \\bar{p} + \\mu \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)} \\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)},\n\\]\nwhere the last inequality follows from property (L2) of the $L^2$-projection. \nThe application of the Cauchy-Schwarz inequality yields the desired estimate of the second term \n\\begin{equation}\\label{eq:full_3_1_terms}\n\\mu (\\bar{z}_h - \\bar{z} , Q_h \\bar{z}- \\bar{z}) \\leq c \\, h \\lVert \\bar{z} - \\bar{z}_h \\rVert_{L^2(\\Omega)} \\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)}.\n\\end{equation}\nThe estimate of the third term can be shown analoguous to \\eqref{eq:full_2_1_terms} with an application of (L2)\nand yields\n\\begin{align}\n (\\bar p_h- \\bar p,Q_h \\bar z - \\bar z) \n &\\leq c \\, h \\left( \\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{\\max{(0, 1-2s+\\varepsilon)}}(\\Omega)} \\right)\n \\lVert Q_h \\bar z - \\bar z \\rVert_{L^2(\\Omega)} \\nonumber \\\\\n &\\leq c \\, h ^2\\left( \\lVert \\bar{z} \\rVert_{\\mathbb{H}^{1}(\\Omega)} + \\lVert u_d \\rVert_{\\mathbb{H}^{\\max{(0, 1-2s+\\varepsilon)}}(\\Omega)} \\right)\\lVert \\bar z \\rVert_{\\mathbb{H}^{1}(\\Omega)}\n \\label{eq:full_3_2_terms}\n\\end{align}\nEstimates \\eqref{eq:full_2_1_terms} -- \\eqref{eq:full_3_2_terms} together with an \nappropriate application of H\\\"older's and Young's inequality yield the desired estimate \\eqref{eq:full_a_priori_z}.\n\nIn order to prove estimate \\eqref{eq:full_a_priori_u} we proceed as follows. Introducing intermediate functions, applying the triangle inequality and using the stability results from Lemma \\ref{lemma:stab} yields\n\\begin{equation}\n\\label{eq:full_a_priori_u_1}\n\\begin{aligned}\n\\norm{\\bar{u} - \\bar{u}_{h}}_{\\mathbb{H}^{s}(\\Omega)} &\\leq \\norm{(S-S_{h})\\bar{z}}_{\\mathbb{H}^{s}(\\Omega)} + \\norm{S_{h}(\\bar{z} - \\bar{z}_{h})}_{\\mathbb{H}^{s}(\\Omega)}\\\\\n&\\leq \\norm{(S-S_{h})\\bar{z}}_{\\mathbb{H}^{s}(\\Omega)} + c \\norm{\\bar{z} - \\bar{z}_{h}}_{L^{2}(\\Omega)}.\n\\end{aligned}\n\\end{equation}\nHence an application of the a priori error estimate \\eqref{eq:fe_approximation_error} with $\\delta = 1 - \\varepsilon'$ and estimate \\eqref{eq:full_a_priori_z} proves \\eqref{eq:full_a_priori_u}.\nThe third estimate is obtained in the same way.\n\\end{proof}\n\n\n\\begin{remark}\n We see, that the rates, that are obtained by Theorem\\nobreakspace \\ref {thm:FD:rates} are not optimal with respect to the state\n and that they are dictated by the optimal linear rate, that we obtain for the control.\n In Section\\nobreakspace \\ref {sec:Numerics} we numerically measure higher rates for the optimal state, namely the same\n as for the variational discretization. The proof of higher rates will be addressed in future work and is mainly based on supercloseness results \\cite{MeyerRoesch2004} for the control. These convergence rates carry over to the rates for the discrete control computed by the so-called post-processing step, i.e. using the projection formula \\textup {(\\ref {eq:disc_proj})} to obtain a new, piecewise linear approximation of the control. Numerical experiments for this post-processing approach are also contained in Section\\nobreakspace \\ref {sec:Numerics}. The theoretical analysis is left for future work.\n\\end{remark}\n\\section{Implementation}\n\\label{sec:Implementation}\n\nIn this section, we introduce a solver for the finite element approximation\n\\textup {(\\ref {eq:dunford_taylor_FE})}.\nThe use of the Balakrishnan formula for inverting the fractional operator leads to the necessity of solving a large number of independent linear systems of equations to obtain an accurate solution. \nHowever, these systems carry a lot of structure that can be used to design efficient \niterative schemes based on tailored Krylov subspace methods.\n\nFollowing \\cite{Bonito}, application of the sinc quadrature to the\nBalakrishnan representation \\eqref{eq:balakrishnan_formula}\ngives rise to the discretization of the state\nequation \\eqref{eq:state_equation}.\nFor convenience, we repeat the resulting approximation here.\n\\begin{equation}\n \\label{eq:im:uhk}\nu_{h}^{k} = \\frac{\\sin{(s \\pi)}}{\\pi} k \\sum_{l = -N_{-}}^{N_{+}} e^{(1-s)kl}v_{h}^{l}\n\\end{equation}\nwhere \n$v_{h}^{l} \\in \\mathbb{U}(\\mathcal{T}_{h})$ is the unique solution of the Galerkin variational problem\n\\begin{equation}\n\\label{eq:DT_FEM_systems}\n\\int_{\\Omega} \\nabla v_{h}^l \\cdot \\nabla w_{h} dx \n+ e^{kl} \\int_{\\Omega} v_{h}^l w_{h} dx \n= \\int_{\\Omega} z w_{h} dx \\quad \\forall w_{h} \\in \\mathbb{U}(\\mathcal{T}_{h}).\n\\end{equation}\nThe evaluation of \\textup {(\\ref {eq:im:uhk})} requires the solution of $N_{+} + N_{-} + 1$ linear\nsystems of the form\n\\begin{equation}\n \\label{eq:im:DiscreteDTSystem}\n\\left( A + \\alpha_l M \\right) V^l = Z, \\quad -N_{-} \\leq l \\leq N_{+}.\n\\end{equation}\nHere, $\\alpha_l = e^{kl}$ and $A$, $M$ denote respectively the corresponding stiffness and mass\nmatrices of the system, $Z$ denotes the load vector, while $V^l$ denotes the node vector for\n$v^l_h$.\nNotice that the $N_{-}~+~N_{+}~+~1$ linear systems in~\\eqref{eq:im:DiscreteDTSystem} are independent \nfor different values of~$l$.\nHence, a first approach for solving systems \\textup {(\\ref {eq:im:DiscreteDTSystem})} might be the use of massive\nparallelization.\nHowever, we shall follow a more efficient approach that exploits the structure of the\nlinear systems and uses tailored conjugated gradients solvers.\n \nWe start by normalizing the systems.\nApplication of a standard mass-lumping strategy results in a diagonal mass matrix~$M_h$. \nWe define $\\rho := \\|M_h^{-1\/2} A M_h^{-1\/2}\\|_{\\infty}$,\n$\\tilde{A} = \\frac{1}{\\rho}{M_h}^{-1\/2}A{M_h}^{-1\/2}$, \n$\\tilde{\\alpha}_l = \\frac{1}{\\rho}\\alpha_l$, \n$\\tilde{V}^l = {M_h}^{1\/2} V^l$ \nand $\\tilde{Z} = \\frac{1}{\\rho}{M_h}^{-1\/2}Z$.\nThen, the linear systems~\\textup {(\\ref {eq:im:DiscreteDTSystem})} can be reformulated as\n\\begin{equation}\n \\label{eq:im:DiscreteDTSystem_modified}\n\\Big( \\tilde{A} + \\tilde{\\alpha}_l I \\Big) \\tilde{V}^l = \\tilde{Z}, \\quad -N_{-} \\leq l \\leq N_{+}.\n\\end{equation}\nWe can estimate the $2-$condition number of system $l$ in \\textup {(\\ref {eq:im:DiscreteDTSystem_modified})}\nby\n\\begin{align}\n \\label{eq:im:estimateCond}\n \\kappa\\left(\\tilde{A} + \\tilde{\\alpha}_l I\\right) = \n \\frac{\\lambda_{\\max}(\\tilde{A} + \\tilde{\\alpha}_l I)}\n {\\lambda_{\\min}(\\tilde{A} + \\tilde{\\alpha}_l I)}\n = \n \\frac{\\lambda_{\\max}(\\tilde{A}) + \\tilde{\\alpha}_l}\n {\\lambda_{\\min}(\\tilde{A}) + \\tilde{\\alpha}_l} \n \\leq 1 + \\min\\left(\n\\frac{\\lambda_{\\max}(\\tilde{A})}{\\tilde{\\alpha}_l},\n \\kappa(\\tilde{A})\n\\right)\n\\end{align}\nwhere $\\lambda_{\\max}(\\tilde A)$ and $\\lambda_{\\min}(\\tilde A)$ denote the largest and smallest\neigenvalue of the symmetric positive definit matrix $\\tilde A$, respectively.\nFrom \\textup {(\\ref {eq:im:estimateCond})} we observe, that for small $\\tilde \\alpha_l$ the condition number of\n$\\tilde A + \\tilde \\alpha_l I$ is close to the condition number of $\\tilde A$, which is a scaled\nstiffness matrix, while for large $\\tilde \\alpha_l$ the condition number converges to 1.\nBy introducing the scaling with $\\rho$, we fix $\\lambda_{\\max}(\\tilde A) \\leq 1$.\n\n\n\nThus for $l$ decreasing from $N_+$ to $N_-$ the condition number of the linear system $\\tilde A + \\tilde \\alpha_l I$\nis increasing. While for $l \\equiv N_+$ conjugated gradients without preconditioning is a well suited solver, \nfor $l\\equiv N_-$ preconditioning in general is required. \nDue to this observation, we consider two adapted linear solvers.\n\n\\begin{itemize}\n\\item Linear problems, for which $l$ is sufficiently large, \nare considered to be well-conditioned, and no further preconditioning is needed to obtain fast convergence of the conjugate gradient solver.\nThanks to the shift-invariance property of Krylov subspace methods \nthe Krylov spaces that are generated during the conjugate gradients method are independent of $l$.\nAs the build-up of the Krylov space contains the only matrix-vector multiplication in the conjugated gradients method,\nthe dimension of the space is equal to the number of matrix-vector multiplications. We fix a number $N_{\\max}$ of multiplications and proceed as follows.\nStarting with $l\\equiv N_+$ we solve linear systems for decreasing $l$, where we reuse the Krylov spaces from previous solutions.\nWe stop at $l\\equiv N_0$ as soon as the required Krylov space has reached the dimension of $N_{\\max}$. \n\nFor the implementation, we use a variant of \nthe conjugate gradient method proposed in~\\cite{Frommer1999}.\nIn Algorithm\\nobreakspace \\ref {alg:im:ShiftInvariant} we summarize the\npseudo code.\n\\item For the resulting systems $N_-,\\ldots, N_0$ preconditioning is necessary which conflicts\nwith the shift invariant property of Krylov methods.\nNote that subsequent systems are still similar, and thus, that a preconditioner for system\n$l$ is also a (worse, but not necessarily bad) preconditioner for system $l+1$. Therefore, we use the standard\napproach for solving system \\textup {(\\ref {eq:im:DiscreteDTSystem_modified})} sequentially and recalculating a\nnew preconditioner whenever the old one is no longer good enough, i.e. as soon as a given maximum number of iterations is exceeded in a conjugated gradients method, see Algorithm\\nobreakspace \\ref {alg:im:sequential}.\n\n\\end{itemize}\n\n\nIn Section\\nobreakspace \\ref {ssec:num:DunfordTaylor} we report on the behaviour of the proposed solver.\n\n\n\\begin{algorithm}\n\\SetAlgoLined\n\\KwIn{$\\tilde A,\\tilde \\alpha,\\tilde Z, N_{\\max}$}\n\\KwData{Set: $\\mathcal K=\\emptyset$, $l=N_+$}\n\\KwOut{$N_0$}\n\\While{$dim (\\mathcal K) < N_{\\max}$}\n{ \n\\For(\\tcp*[h]{cg-iteration for system $l$}){$k=1,\\ldots$}\n{\n\\If{$k>\\mbox{dim}(\\mathcal K)$}\n{ \nCalculate basis vector for $k$-th Krylov space and store into $\\mathcal K$\\;\n\\label{alg:im:ShiftCG_increaseKrylovSpace}\n\n{}\nSolve linear system $l$ in space $\\mathcal K_k$\nusing \\cite[Alg.~4]{Frommer1999}\\;\n\\label{alg:im:ShiftCG_solveInKrylovSpace}\n\n$l:=l-1$\\;\n\n$N_0 := l$\n\\caption{Pseudo code for solving the well-conditioned systems. The only matrix-vector\nmultiplication appears in line\\nobreakspace \\ref {alg:im:ShiftCG_increaseKrylovSpace}.\nHere the space $\\mathcal K_k$ in line\\nobreakspace \\ref {alg:im:ShiftCG_solveInKrylovSpace} is\nthe span over the first $k$ basis elements of $\\mathcal K$.}\n\\label{alg:im:ShiftInvariant}\n\\end{algorithm} \n\n\\begin{algorithm}\n\\SetAlgoLined\n\\KwIn{$\\tilde A,\\tilde \\alpha,\\tilde Z, N_0$}\n\\KwData{Set: $N_{\\max}>0$, $N_{iter} = N_{\\max}+1$, }\n\\For{$l=-N_-\\ldots N_0$}\n{ \n\\If{$N_{iter} >N_{\\max}$ }\n{\nBuild up amg preconditioner $P$ for system $l$\\;\n}\nSolve systems $l$ with preconditioned conjugate gradients method using preconditioner $P$ with\n$N_{iter}$ iterations\\;\n\n\\caption{Pseudo code for solving the not well-conditioned systems.}\n\\label{alg:im:sequential}\n\\end{algorithm}\n\n\\begin{remark}[An alternative solver]\n In \\cite{Chan1999} a conjugate gradients method is proposed that uses Krylov spaces generated for one of the linear systems \\textup {(\\ref {eq:im:DiscreteDTSystem_modified})}, called seed\n system, to generate good initial values, or even solutions, for the other systems.\n Thanks to the particular structure of\n the systems \\textup {(\\ref {eq:im:DiscreteDTSystem_modified})} this can be done without additional\n matrix-vector multiplications, see \\cite[Sec.~3.1]{Chan1999}. \n Upon convergence, another system is chosen as a seed system, for which the Krylov spaces are\n generated.\n In our implementation, we combine this approach with algebraic multigrid (amg) preconditioning, \n and choose that system as the next seed system, that\n currently has the largest residuum.\n \n This approach requires storing the solution to all systems in memory to apply the Krylov spaces and to find the next seed system. Unfortunately, this turned out to be not feasible for\n fine meshes in 3D, but we obtained very fast convergence of the method, when applicable.\n \n A combination of the proposed sequential solver as in Algorithm\\nobreakspace \\ref {alg:im:sequential} and the solver\n proposed in \\cite{Chan1999} seems possible (at least with some restrictions) and will be subject\n to future work.\n\\end{remark}\nFinally we note that a large number of tailored Krylov methods is proposed to deal with shifted\nsystems that require preconditioning and we only refer to\n\\cite{2003_Benzi_ApproximateInversePrecond_for_Shift,\n2003_Frommer_bicgstab_shiftedSystems,\n2016_Soodhalter_recursiveGMRES_generalPrecond,\n2014_SoodhalterSzydXue_KrylovRecycling_shiftesSystems,\n2017_ZhongGu_FlexibleAdaptiveSGMRES_shiftedSystems}.\n\n\\section{Numerical results}\n\\label{sec:Numerics}\n\nIn this section, we validate the theoretical rates of convergence derived in Section\\nobreakspace \\ref {sec:error_estimates}. \nIn Section\\nobreakspace \\ref {ssec:num:DunfordTaylor}, we investigate the solver \n for the fractional Laplace proposed in Section\\nobreakspace \\ref {sec:Implementation}.\nIn Section\\nobreakspace \\ref {ssec:OptControl}, we present the convergence rates of the fully discrete finite element scheme for the approximation of the optimal control problem. \nAll numerical experiments are conducted for a range of values of the fractional exponent~$s$.\n\n \nWe implement the solver proposed in Section\\nobreakspace \\ref {sec:Implementation} in C++ using the\nPETSc linear algebra package \\cite{petsc-web-page} and solve the optimal control problem\nusing the TAO package of PETSc using the bound-constrained limited-memory variable-metric method\n(tao\\_blmvm), which is a limited memory BFGS method.\n We generate meshes, finite element functions and assemble matrices\nusing FEniCS \\cite{LoggMardalEtAl2012a} through the C++ interface.\n \n\n For the solver of the fractional operator proposed in Section\\nobreakspace \\ref {sec:Implementation} we fix $N_{\\max} = 500$ for 2D simulations and \n $N_{\\max} = 250$ for 3D simulations. \n The individual linear systems are solved up to a relative accuracy of $10^{-8}$. \n For Algorithm\\nobreakspace \\ref {alg:im:sequential}, we calculate a \n new preconditioner as soon as more than $N_{\\max}=20$ iterations are taken in the preconditioned conjugate gradients method.\n As amg preconditioner we use 2 V-cycles of Hypre \\cite{hypre-web-page} that is accessed through the PETSc interface. We stop the optimization as soon as the $l^2$-norm of the projected gradient is smaller or equal to $10^{-5} \\sqrt{h^n}$, where $h$ is the length of the longest edge in the finite element mesh. Here the scaling with $h$ mimics the different scaling of the $l^2$-norm and the\n $L^2(\\Omega)$-norm.\n\n\n\\subsection{The solver for the fractional operator}\n\\label{ssec:num:DunfordTaylor}\nLet us first report on the performance of the proposed solver for the systems\n\\textup {(\\ref {eq:im:DiscreteDTSystem_modified})}.\nAs a test example we use $\\Omega = (0,1)^n$, $n \\in \\{2,3\\}$ and\nset $f = \\min(0.25,f_0)$, where $f_0(m) = 0.5$, with $m$ the center of $\\Omega$, \n$f_0|_{\\partial\\Omega} = 0$ and $f_0$ is linearly interpolated between these values.\nNote that no analytical solution is known for this\nright-hand side $f$, and that $f$ enjoys $\\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)$ regularity, which is the maximal regularity of the optimal control $z$.\nWe solve the equation $(-\\Delta)^s u_h^k = f$ on a sequence of homogeneously refined meshes and\nuse the solution on the finest mesh (with $N_\\Omega = 4198401$ nodes for $n=2$) as the reference solution. \nThese meshes are chosen, such that all\nkinks in $f$ are resolved, and the integration of $f$ is done with no numerical error.\n\nIn Table\\nobreakspace \\ref {tab:num:DunfordTaylor_2D} and Table\\nobreakspace \\ref {tab:num:DunfordTaylor_3D} we report on the solver for the cases $n=2$ and $n=3$, respectively, and for $s=0.05$ and $s=0.5$.\n \n\\begin{table}\n\\centering\n\\input{DF_generic_2D.tab.tex}\n\n\\caption{The behavior of the proposed solver for the fractional operator for 2D simulation.\n$N_\\Omega$ denotes the number of degrees of freedom in $\\Omega$, $N_\\alpha = N_- + N_+ + 1$ denotes the number of linear systems to solve. In brackets, we show how many systems are solved by\nAlgorithm\\nobreakspace \\ref {alg:im:ShiftInvariant} and Algorithm\\nobreakspace \\ref {alg:im:sequential} respectively.\nWe give results for $s=0.05$, $s=0.5$, and $s=0.95$.\nThe number of amg setups in Algorithm\\nobreakspace \\ref {alg:im:sequential} are equal in all cases.\n}\n\\label{tab:num:DunfordTaylor_2D}\n\\end{table}\n \n \\begin{table}\n \\centering\n\\input{DF_generic_3D.tab.tex}\n\\caption{Behavior of the proposed solver for the fractional operator for 3D simulation.\nFor an explanation of the abbreviations, see Table\\nobreakspace \\ref {tab:num:DunfordTaylor_2D}.\n}\n\\label{tab:num:DunfordTaylor_3D}\n\\end{table} \n\nWe observe that in fact, the number of amg setups is very small or a set up is not even necessary, which indicates, how closely\nrelated the systems are.\n\nFinally, for small $s$ the operator is closer to identity, and thus, more systems are\nwell-conditioned, which can be seen by the number of systems that are solved by\nAlgorithm\\nobreakspace \\ref {alg:im:ShiftInvariant} in comparison to the number of systems solved by\nAlgorithm\\nobreakspace \\ref {alg:im:sequential}.\n\nLet us briefly comment on the convergence rate from Corollary\\nobreakspace \\ref {cor:fe_error_estimates_regular} for $u_h^k$.\nAs the above defined right hand side enjoys $f \\in \\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)$,\nwe expect a rate of $h^{\\min(2,3\/2+2s-\\varepsilon')}$.\nIn Table\\nobreakspace \\ref {tab:num:DF_ratesL2} we show the observed convergence rates for $n=2$ and $s\\in \\{0.05,0.1,0.25\\}$, \nwhich indeed confirm the theoretical predictions. For $n=3$ memory consumption\nrestricts the quality of the reference solution, such that we do not measure a rate for $n=3$.\n \n\\begin{table}\n \n \\input{DF_generic_2D_rates.tab.tex}\n \n \\caption{Experimental convergence rates for the numerical solution of the fractional Laplace\n equation with $f \\in \\mathbb{H}^{3\/2-\\varepsilon}(\\Omega)$. \n We observe, that the expected rate $r^s_{L^2(\\Omega)} = \\min(2,3\/2+2s-\\varepsilon')$ is fulfilled for all examples.\n } \n \\label{tab:num:DF_ratesL2} \n\\end{table}\n\n\\subsection{Optimal Control Problem}\\label{ssec:OptControl}\nTo verify the theoretical convergence rates of the finite element discretization of the optimal control problem, \nwe perform numerical experiments without a known optimal solution.\nWe use the domain $\\Omega = (0, 1)^2$ and set the desired state to be equal to an eigenfunction of the Laplacian on the square, \nnamely $u_d = \\sin (2 \\pi x) \\sin (2 \\pi y)$ and $f \\equiv 0$. \nWe consider three different values of the fractional parameter, \nnamely $s \\in \\lbrace 0.05, 0.25, 0.5\\rbrace$ and choose $a = -0.8, b = 0.8$, \nsuch that the box-constraints are attained in some subdomain of~$\\Omega$.\nThe optimal solution for $h=0.0014$ is considered as reference solution.\n\nResults of the numerical tests are summarized in Figure\\nobreakspace \\ref {fig:OPT_Plots}. \nFirst order convergence of the approximation of the control is obtained, \nwhich is in line with~\\textup {(\\ref {eq:full_a_priori_z})}. \n\nWe also report on results using the post-processing approach \\cite{MeyerRoesch2004}.\nHere the projection formula \\textup {(\\ref {eq:disc_proj})} is used to obtain a higher order approximation for the optimal control.\nHigher order means, that instead of an approximation with piecewise constant functions, \na piecewise linear approximation is obtained that has the same structure as the optimal control obtained with variational discretization. \nWe expect thus the same optimal rate of convergence for this post-processed optimal control $\\bar{z}_h^{PP}$ \nas for the variational discretization approach, namely \n$\\|\\bar{z}-\\bar{z}_h^{PP}\\|_{L^2(\\Omega)} \\leq c \\, h^{\\min(2,3\/2+2s-\\varepsilon')}$.\nIn Figure\\nobreakspace \\ref {fig:OPT_Plots} we observe the expected higher rates for the optimal controls with post-processing approach for $n=2$.\n\nWe also investigate the finite element approximation of the state in the $L^2(\\Omega)$- and $H^s(\\Omega)$-norms, \nthe latter being estimated using Gagliardo-Nirenberg interpolation inequality \n$\\| \\bar{u} - \\bar{u}_h\\|_{H^s(\\Omega)} \\lesssim \\| \\bar{u} - \\bar{u}_h\\|^{1 - s}_{L^2(\\Omega)} \\|\\bar{u} - \\bar{u}_h \\|^s_{H^1(\\Omega)}$. We observe $h^{\\min(2,3\/2 + 2s)}$ order of convergence in the $L^2(\\Omega)$-norm and $h^{\\min(2-s,3\/2 + s)}$ order of convergence in the $H^s(\\Omega)$-norm. \nThe theoretical justification as well as the analysis of the post-processing approach is left for future work.\n\\begin{figure}[!]\n \n\n\\input{Plot_2D_control.new.tex}\n\\input{Plot_2D_state.new.tex}\n\n\n\\caption{Convergence rates of the discretization of the optimal control problem in 2D. \nFigure on the left-hand side presents approximations of the control~$z$. \nFirst order of convergence of the piecewise constant approximation can be observed. \nApplication of the additional post-processing significantly improves the convergence properties, \nand we observe $h^{\\min(2,3\/2 + 2s)}$ convergence. \nOn the right-hand side convergence of the approximation of the state~$u$ is shown. \nConvergence order of the piecewise linear finite element method measured in the $L^2(\\Omega)$-norm depends \non the choice of~$s$ and varies between $3\/2$ and $2$, which can be attained for sufficiently large~$s$. \nConvergence order in $H^s(\\Omega)$-norm is included for completeness.}\n\\label{fig:OPT_Plots}\n\\end{figure}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Approach}\n\nIn this section, we describe our classification model that incorporates the attention mechanism. It can be applied to both text and tabular data and is inspired by works in attention-based models in text-classification~\\cite{zhou-etal-2016-attention}. We incorporate attention over the input features. Next, we describe how this attention over features can attribute the model's unfairness to certain features. Finally, using this attribution framework, we propose a post-processing approach for mitigating unfairness.\n\nIn this work, we focus on binary classification tasks. We assume access to a dataset of triplets $\\mathcal D = \\{x_i , y_i , a_i \\}^N_{i=1}$, where $x_i,y_i,a_i$ are i.i.d. samples from data distribution $p(\\mathbf x, \\mathbf y, \\mathbf a)$. $\\mathbf a \\in \\{a_1,\\ldots a_l\\}$ is a discrete variable with $l$ possible values and denotes the sensitive or protected attributes with respect to which we want to be fair, $\\mathbf y\\in \\{0,1\\}$ is the true label, $\\mathbf x\\in \\mathbb{R}^m $ are features of the sample which may include sensitive attributes. We use $\\hat y_o$ to denote the binary outcome of the original model, and $\\hat y_z^k$ will represent the binary outcome of a model in which the attention weights corresponding to $k^\\text{th}$ feature are zeroed out. Our framework is flexible and general that it can be used to find attribution for any fairness notion. More particularly, we work with the group fairness measures like \\textit{Statistical Parity}~\\cite{Dwork:2012:FTA:2090236.2090255}, \\textit{Equalized Odds}~\\cite{NIPS2016_9d268236}, and \\textit{Equality of Opportunity}~\\cite{NIPS2016_9d268236}, which are defined as:\\footnote{We describe and use the definition of these fairness measures as implemented in \\textit {Fairlearn }package~\\cite{bird2020fairlearn}. } \n\n\\noindent\n\\textbf{Statistical Parity Difference (SPD)}:\n$$\n\\text{SPD}(\\hat {\\mathbf y}, \\mathbf a ) =\n \\max_{a_i, a_j}|P(\\hat{\\mathbf y}=1\\mid \\mathbf a=a_i)-P(\\hat{\\mathbf y}=1\\mid \\mathbf a =a_j)|\n$$\n\\textbf{Equality of Opportunity Difference (EqOpp)}:\n\\begin{align*}\n\\text{EqOpp}(\\hat {\\mathbf y}, \\mathbf a, \\mathbf y) =\n \\max_{a_i, a_j} |P(\\hat{\\mathbf y}=1\\mid\\mathbf a = a_i,\\mathbf y=1)&\\\\-P(\\hat{\\mathbf y}=1\\mid\\mathbf a = a_j,\\mathbf y=1)|\n\\end{align*}\n\\textbf{Equalized Odds Difference (EqOdd)}:\n\\begin{align*}\n\\text{EqOdd}(\\hat {\\mathbf y}, \\mathbf a, \\mathbf y) = \n\\max_{a_i, a_j} \\max_{y\\in\\{0,1\\}} |P(\\hat{\\mathbf y}=1\\mid\\mathbf a = a_i,\\mathbf y=y)&\\\\- P(\\hat{\\mathbf y}=1\\mid\\mathbf a = a_j,\\mathbf y=y)|\n\\end{align*}\n\n\n\n\\begin{figure}[h]\n \\centering\n \\begin{subfigure}[b]{0.21\\textwidth}\n \\includegraphics[width=0.96\\textwidth,trim=0cm 0cm 0cm 0cm,clip=true]{images\/general_model.pdf}\n \\caption{Classification model.}\n \\label{fig:original_model}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\includegraphics[width=\\textwidth,trim=0cm 0cm 0cm 0cm,clip=true]{images\/attr_framework.pdf}\n \\caption{Attribution framework.}\n \\label{fig:intervention_model}\n \\end{subfigure}\n \\caption{(a) In general classification model, for each feature $f_k$ a vector representation $e_k$ of length $d^e$ is learned. This is passed to the attention layer which produces a $d^e$-dimensional vector representation for the sample instance $i$ which is passed to two dense layers to get the final classification output. (b) The Attribution framework has the same architecture as the general model. One outcome is obtained through the original model and another through the model that has some attention weights zeroed. The observed difference in accuracy and fairness measures will indicate the effect of the zeroed out features on accuracy and fairness.}\n \\label{fig:model}\n \n\\end{figure}\n\n\n\n\\begin{algorithm}[h]\n\\SetAlgoLined\nInput: decay rate $d_r$ ($0 \\leq d_r <1$), $n$ test samples indexed by variable $i$. \\\\\nOutput: final predictions, unfair features. \\\\\nCalculate the attention weights $\\alpha_{ki}$ for $k^{\\text{th}}$ feature in sample $i$ using the attention layer as in Eq.~\\ref{eq:attention_op1}.\\\\\n{unfair\\_feature\\_set} = \\{\\} \\\\\n\\For{each feature (index) $k$}{\n \\If{$\\text{SPD}(\\hat {\\mathbf y}_o, \\mathbf {a})- \\text{SPD}(\\hat {\\mathbf y}_z^k, \\mathbf {a}) \\geq 0$}{\n {unfair\\_feature\\_set} = {unfair\\_feature\\_set} $\\cup\\ \\{k\\}$\n }\n}\n\\For{each feature (index) $k$}{\n\\If{$k$ in \\text{unfair\\_feature\\_set}}{\nSet $\\alpha_{ki} \\leftarrow (d_r \\times \\alpha_{ki})$ for all $n$ samples\n}\n}\nUse new attention weights \nto obtain the final predictions $\\hat Y$. \\\\\n\\Return $\\hat Y$, {unfair\\_feature\\_set}\n \\caption{Bias Mitigation with Attention}\n \\label{mitig_alg}\n\\end{algorithm}\n\n\\subsection{General Model: Incorporating Attention over Inputs in Classifiers}\n\nWe consider each feature value as an individual entity (like the words are considered in text-classification) and learn a fixed-size embedding $\\{e_k\\}_{k=1}^m,\\ e_k \\in \\mathbb{R}^{d^e}$ for each feature, $\\{f_k\\}_{k=1}^m$. These vectors are passed to the attention layer. The Computation of attention weights and the final representation for a sample is described in Eq.~\\ref{eq:attention_op1}. $E = [e_1 \\ldots e_m], \\ E \\in \\mathbb R ^{d^e\\times m} $ is the concatenation of all the embeddings, $w \\in \\mathbb{R}^{d^e} $ is a learnable parameter, $r \\in \\mathbb{R}^{d^e}$ denotes the overall sample representation, and $\\alpha \\in \\mathbb{R}^{m}$ denotes the attention weights.\n\\begin{align}\nH = \\tanh(E) \\quad \\label{eq:attention_op1}\n\\alpha = \\text{softmax}(w^TH) \\quad\nr = \\tanh(E\\alpha^T)\n\\end{align}\nThe resulting representation, $r$, is passed to the feed-forward layers for classification. In this work, we have used \ntwo feed-forward layers (See Fig.~\\ref{fig:model} for overall architecture).\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Fairness Attribution with Attention Weights}\nThe aforementioned classification model with the attention mechanism combines input feature embeddings by taking a weighted combination. By manipulating the weights, we can intuitively capture the effects of specific features on the output. To this end, we observe the effect of each attribute on the fairness of outcomes by zeroing out or reducing its attention weights and recording the change. Other works have used similar ideas to understand the effect of attention weights on accuracy and evaluate interpretability of the attention weights by comparing the difference in outcomes in terms of measures such as Jensen-Shannon Divergence~\\cite{serrano-smith-2019-attention} but not for fairness. We are interested in the effect of features on fairness measures. Thus, we measure the difference in fairness of the outcomes based on the desired fairness measure. A large change in fairness measure and a small change in performance of the model would indicate that this feature is mostly responsible for unfairness, and it can be dropped without causing large impacts on performance. The overall framework is shown in Fig.~\\ref{fig:model}. First, the outcomes are recorded with the original attention weights intact (Fig.~\\ref{fig:original_model}). Next, attention weights corresponding to a particular feature are zeroed out, and the difference in performance and fairness measures is recorded (Fig.~\\ref{fig:intervention_model}). Based on the observed differences, one may conclude how incorporating this feature contributes to fairness\/unfairness.\n\nTo measure the effect of the $k^{th}$ feature on different fairness measures, we consider the difference in the fairness of outcomes of the original model and model with $k^{th}$ feature's effect removed. For example, for statistical parity difference, we will consider $\\text{SPD}(\\hat {\\mathbf y}_o, \\mathbf {a})- \\text{SPD}(\\hat {\\mathbf y}_z^k, \\mathbf {a})$. A negative value will indicate that the $k^{th}$ feature helps mitigate unfairness, and a positive value will indicate that the $k^{th}$ feature contributes to unfairness. This is because $\\hat{y}_z^k$ captures the exclusion of the $k^{th}$ feature (zeroed out attention weight for that feature) from the decision-making process. If the value is positive, it indicates that not having this feature makes the bias lower than when we include it. Notice here, we focus on global attribution, so we measure this over all the samples; however, this can also be turned into local attribution by focusing on individual sample $i$ only.\n\\subsection{Mitigating Bias by Removing Unfair Features} \\label{mitig_sec}\nAs discussed in the previous section, we can identify features that contribute to unfair outcomes according to different fairness measures. A simple technique to mitigate or reduce bias is to reduce the attention weights of these features. This mitigation technique is outlined in Algorithm~\\ref{mitig_alg}. In this algorithm, we first individually set attention weights for each of the features in all the samples to zero and monitor the effect on the desired fairness measure. We have demonstrated the algorithm for SPD, but other measures, such as EqOdd, EqOpp, and even accuracy can be used (in which case the ``unfair\\_feature\\_set'' can be re-named to feature set which harms accuracy instead of fairness). If the $k^{th}$ feature contributes to unfairness, we reduce its attention weight using decay rate value. This is because $\\hat {\\mathbf y}_z^k$ captures the exclusion of the $k^{th}$ feature (zeroed attention weight for that feature) compared to the original outcome $\\hat {\\mathbf y}_o$ for when all the feature weights are intact; otherwise, we use the original attention weight. We can also control the fairness-accuracy trade-off by putting more attention weight on features that boost accuracy while keeping the fairness of the model the same and down-weighting features that hurt accuracy, fairness, or both. \n\nThis post-processing technique has a couple of advantages over previous works in bias mitigation or fair classification approaches. First, the post-processing approach is computationally efficient as it does not require model retraining to ensure fairness for each sensitive attribute separately. Instead, the model is trained once by incorporating all the attributes, and then one manipulates attention weights during test time according to particular needs and use-cases. Second, the proposed mitigation method provides an explanation and can control the fairness-accuracy trade-off. \nThis is because manipulating the attention weights reveals which features are important for getting the desired outcome, and by how much.\nThis provides an explanation for the outcome and also a mechanism to control the fairness-accuracy trade-off by the amount of the manipulation.\n\n\n\n\n\n\n\n\n\\section{Experiments}\n\\vspace{-0.2cm}\nWe perform a suite of experiments on synthetic and real-world datasets to evaluate our attention based interpretable fairness framework. First, we elucidate\ninterpretability on the synthetic dataset on which we can control the relations between input and output features. We verify that our interpretability framework recovers expected attributions and behaviour. We also demonstrate interpretability on the real-world dataset and identify features that cause unfairness. Exploiting these attributions, we test our proposed post-processing bias mitigation framework and compare it with various recent mitigation approaches. Finally, we show the effectiveness of our mitigation approach on non-tabular data format, namely text.\n\n\\subsection{Datasets}\n\n\\newcommand{\\text{Ber}}{\\text{Ber}}\n\n\\subsubsection{Synthetic Data} \\label{syn_data_sec}\nTo validate the interpretability framework, we created two synthetic datasets in which we control how features interact with each other and contribute to the accuracy and fairness of the outcome variable.\nThese datasets capture some of the common scenarios arising in decision or classification problems.\n\n\\paragraph{Scenario 1:} First, we create a simple scenario to demonstrate that our framework identifies correct feature attributions for fairness and accuracy. To this end, we create a feature that is correlated with the outcome (responsible for accuracy), a discrete feature that would cause the prediction outcomes to be biased (responsible for fairness), and a continuous feature that is independent of the label or the task; thus, irrelevant for the task. Intuitively, suppose the attention-based interpretability framework works correctly. In that case, we expect to see a reduction in accuracy upon removing (i.e., making the attention weight zero) the feature responsible for the accuracy, reduction in bias upon removing the feature responsible for bias, and very little or no change upon removing the irrelevant feature. With this objective, we generated a synthetic dataset with three features, i.e., $x = [f_1, f_2, f_3]$ as follows~\\footnote{We use $x\\sim \\text{Ber}(p)$ to denote that $x$ is a Bernoulli random variable with $P(x=1)=p$.}.\n\\begin{equation*}\n f_1 ~\\sim \\text{Ber}(0.9)\n \n \n \n \n \\quad f_2\\sim \\text{Ber}(0.5)\n \\quad f_3\\sim \\mathcal N (0,1)\n \\quad y\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.1) & \\text{if } f_2=0\n \\end{cases}\n\\end{equation*}\nClearly, $f_2$ has the most predictive information for the task and responsible for accuracy. Here, we consider $f_1$ as the sensitive attribute. $f_1$ is an imbalanced feature that can bias the outcome and is generated such that there is no intentional correlation between $f_1$ and the outcome, $y$ or $f_2$. $f_3$ is sampled from a normal distribution independent of the outcome $y$, or the other features, making it irrelevant to the task.\nThus, an ideal classifier would be fair if it captures the correct outcome without being affected by the imbalance in $f_1$. However, due to limited data and skew in $f_1$, there will be some undesired bias --- few errors when $f_1=0$ can lead to large statistical parity. \n\n \n\n\\paragraph{Scenario 2:} Using features that are not identified as sensitive attributes can result in unfair decisions due to their implicit relations or correlations with the sensitive attributes. This phenomenon is called indirect discrimination~\\cite{zliobaite2015survey,6175897,ijcai2017-549}. We designed this synthetic dataset to demonstrate and characterize the behavior of our framework under indirect discrimination. Similar to the previous scenario, we consider three features. Here, $f_1$ is considered as the sensitive attribute, and $f_2$ is correlated with $f_1$ and the outcome, $y$. The generative process is as follows:\n\\begin{equation*}\n f_1\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.1) & \\text{if } f_2=0\n \\end{cases}\n \\quad f_2\\sim\\text{Ber}(0.5)\n \\quad f_3\\sim \\mathcal N (0,1)\n \\quad y\\sim \\begin{cases}\n \\text{Ber}(0.7) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.3) & \\text{if } f_2=0\n \\end{cases}\n\\end{equation*}\n\nNote that, in this case $f_1$ and $y$ are correlated with \n$f_2$. The model should mostly rely on $f_2$ for its decisions. \nHowever, due to the correlation between $f_1$ and $f_2$, we expect $f_2$ to affect both the accuracy and fairness of the model. Thus, in this case, indirect discrimination is possible. Using such a synthetic dataset, we demonstrate a) indirect discrimination and b) the need to have an interpretable framework to reason about unfairness and not blindly focus on the sensitive attributes for bias mitigation.\n\n\n\n\\subsubsection{Real-world Datasets}\nIn addition to synthetic datasets, we demonstrate our approach on various real-world datasets which we discuss next.\n\n\\paragraph{Tabular Datasets:}\nWe conduct our experiments on two real-world tabular datasets often used to benchmark fair classification techniques --- \\textit{UCI Adult}~\\cite{Dua:2019} and \\textit{Heritage Health}~\\footnote{https:\/\/www.kaggle.com\/c\/hhp} datasets. The \\textit{UCI Adult} dataset contains census information about individuals, with the prediction task being whether the income of the individual is higher than \\$50k or not. The sensitive attribute, in this case, is gender (male\/female). The \\textit{Heritage Health} dataset contains patient information, and the task is to predict the Charleson Index (comorbidity index, which is a patient survival indicator). Each patient is grouped into one of the 9 possible age groups, and we consider this as the sensitive attribute. We used the same pre-processing and train-test splits as in~\\cite{gupta2021controllable}.\n\n\\paragraph{Non-Tabular or Text Dataset:}\nTo demonstrate the flexibility of our approach, we also experiment with a non-tabular text dataset. We used the \\textit{biosbias} dataset~\\cite{de2019bias}. The dataset contains short bios of individuals. The task is to predict the occupation of the individual from their bio.\nWe utilized the bios from the year 2018 from the \\texttt{2018\\_34} archive and considered two occupations for our experiments, namely, nurse and dentist. The dataset was split into $70-15-15$ train, validation, and test splits. \\cite{de2019bias} has demonstrated the existence of gender bias in this prediction task and showed that certain gender words are associated with certain job types. For example, for the bios that we consider, we expect female associated words, such as \\textit{she}, to cause skew towards the prediction of job nurse and male associated words, such as \\textit{he}, to skew the prediction towards the job dentist.\n\n\n\n\n\n\n\n\\subsection{Interpreting Fairness with Attention}\n\\label{sec:interpret_fairness}\n\\begin{figure*}[t]\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_newscen1.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_newscen2.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_real_data_adult_subset2.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.4cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_real_data_health_subset2.pdf}\n\\end{subfigure}\n\\caption{Results from the synthetic datasets on top and real-world datasets in the bottom. Labels on the points represent the feature name that was removed. The results show how the accuracy and fairness of the model (in terms of statistical parity difference) change by exclusion of each feature. The red original point represents the results of the model including all the features.}\n\\label{fig:interpret_results}\n\\end{figure*}\n\n\nFirst, we validate that our interpretability framework can capture correct attributions by controlled experiments with synthetic data. We also experiment with \\textit{UCI Adult} and \\textit{Heritage Health} and argue that the captured attributions are reasonable. Fig.~\\ref{fig:interpret_results} summarizes our results. The interpretability framework correctly captures the intentional behavior created in the synthetic datasets.\n\nIn \\textit{Scenario 1}, as expected, $f_2$ is correctly attributed to being responsible for the accuracy, and removing it hurts the accuracy drastically. Similarly, $f_1$ is correctly shown to be responsible for unfairness, and removing it creates a fairer outcome. Ideally, the model should not be using any information about $f_1$ as it is independent of the task, but it does. Therefore, by removing $f_1$, we can be sure that information is not used and hence outcomes are fair. Lastly, as expected, $f_3$ was the irrelevant feature, and its effects on accuracy and fairness are negligible. Another interesting observation is that $f_2$ is helping the model achieve fairness since its exclusion means that the model should rely on $f_1$ for decision making, resulting in more bias; thus, removing $f_2$ harms accuracy and fairness as expected.\n\nIn \\textit{Scenario 2}, our framework captures the effect of indirect discrimination. We can see that removing $f_2$ reduces bias as well as accuracy drastically. This is because $f_2$ is the predictive feature, but due to its correlation with $f_1$, it can also indirectly affect the model's fairness. More interestingly, although $f_1$ is the sensitive feature, removing it does not play a drastic role in fairness or the accuracy. This is an important finding as it shows why removing $f_1$ on its own can not give us a fairer model due to the existence of correlations to other features and indirect discrimination. \n\nLastly, Fig.~\\ref{fig:interpret_results}, also shows results on a subset of the features from the \\textit{UCI Adult} and \\textit{Heritage Health} datasets (To keep the plots uncluttered and readable, we incorporated the most interesting features in the plot). These can provide us with some intuition about how different features in these datasets contribute to the fairness and accuracy of the model. While features such as capital gain and capital loss in the \\textit{UCI Adult} dataset are responsible for improving accuracy and reducing bias, we can observe from the results in Fig.~\\ref{fig:interpret_results} that features such as relationship or marital status that can be directly or indirectly correlated with sex are causing unfairness. These results are intuitive and once again confirm the reliability of our framework.\n\nThe above results validate that attention weights capture the true or the expected behavior and that this framework can provide reliable attributions for the fairness and accuracy of the model.\n\n\\vspace{-0.3cm}\n\\subsection{Attention as a Mitigation Technique}\n\\vspace{-0.3cm}\n\\begin{figure*}[t]\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1.0cm,clip=true]{.\/images\/mitig_imgs\/adult_results_err.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1cm,clip=true]{.\/images\/mitig_imgs\/health_results_err.pdf}\n \\end{subfigure}\n \\caption{Accuracy vs parity curves for UCI Adult and Heritage Health datasets.}\n \\label{fig:mitig_results}\n\\end{figure*}\n\nAs we have highlighted earlier, understanding how the information within features interact and contribute to the decision making can be used to design effective bias mitigation strategies. One such example was shown in Sec. ~\\ref{sec:interpret_fairness}. Often real-world datasets have features which cause indirect discrimination, due to which fairness can not be achieved by simply eliminating the sensitive feature from the decision process. \nUsing the attributions derived from our attention-based interpretability framework, we propose a post-processing mitigation strategy. Our strategy is to intervene on attention weights as discussed in Sec.~\\ref{mitig_sec}. We first attribute and identify the features responsible for the unfairness of the outcomes, i.e., all the features whose exclusion will decrease the bias compared to the original model's outcomes and gradually decrease their attention weights to zero. We show that this procedure provides competitive trade-offs compared to several recently fair machine learning approaches. \n\n\\subsubsection{Experiments with Tabular Datasets}\n\\paragraph{Datasets, Baselines, and Evaluation Procedure:}\nMost of the fair decision-making algorithms are benchmarked on tabular datasets. To this end, we evaluate our approach on two real-world datasets, \\textit{UCI Adult} and \\textit{Heritage Health}. In this section, we use statistical parity as our notion of fairness. However, due to the flexibility of our interpretability framework, it can easily be extended to other fairness measures (discussed in Appendix Section A). We compare our method against several recent baselines that are specifically optimized to achieve statistical parity.\nSpecifically, we consider methods that use information-theoretic objectives to learn representations of data so that information about sensitive attributes is eliminated.\n\\textit{\\textbf{CVIB}}~\\cite{NEURIPS2018_415185ea} realize this objective through a conditional variational autoencoder, whereas \\textit{\\textbf{MIFR}}~\\cite{pmlr-v89-song19a} used a combination of information bottleneck term and adversarial learning to optimize the fairness objective. \n\\textit{\\textbf{FCRL}}~\\cite{gupta2021controllable} showed that optimizing information theoretic objectives can be used to achieve good trade-offs between fairness and accuracy by using specialized contrastive information estimators~\\cite{gupta2021controllable}. \nIn addition to information-theoretic approaches, we also considered baselines that use adversarial learning such as \\textit{\\textbf{MaxEnt-ARL}}~\\cite{roy2019mitigating}, \\textit{\\textbf{LAFTR}}~\\cite{pmlr-v80-madras18a}, and \\textit{\\textbf{Adversarial Forgetting}}~\\cite{advforg}. \n\nFor all these fair representation learning baselines, we used the approach outlined in~\\cite{gupta2021controllable} for training a downstream classifier and evaluating the accuracy\/fairness trade-offs. The downstream classifier was a 1-hidden-layer MLP with 50 neurons along with ReLU activation function. Our experiments were performed on Nvidia GeForce RTX 2080. Each method was trained with five different seeds, and we report the average accuracy and fairness measure as statistical parity difference (SPD). \\textit{\\textbf{CVIB}}, \\textit{\\textbf{MaxEnt-ARL}}, \\textit{\\textbf{Adversarial Forgetting}} and \\textit{\\textbf{FCRL}} are designed for statistical parity notion of fairness and are not applicable for other measures like Equalized Odds and Equality of Opportunity. \\textit{\\textbf{LAFTR}} can only deal with binary sensitive attributes and thus not applicable for Heritage Health dataset. For our approach, we vary the attention weights and report the resulting fairness-accuracy trade offs. \n\nNote that in contrast our approach, the baselines described above are not interpretable as they are incapable of directly attributing features to fairness outcomes.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\paragraph{Results:}\nFig.~\\ref{fig:mitig_results} compares fairness-accuracy trade-offs of different bias mitigation approaches. We desire outcomes to be fairer, i.e., lower values of \\texttt{SPD} and to be more accurate, i.e., towards the right. \nOur mitigation framework based on the manipulation of the attention weights is competitive with state-of-the-art mitigation strategies. However, most of these approaches are specifically designed and optimized to achieve parity and do not provide any interpretability. \nOur model can not only achieve comparable and competitive results, but it is also able to provide explanation such that the users exactly know what feature and by how much it was manipulated to get the corresponding outcome. Another advantage of our model is that it needs only one round of training. The adjustments to attention weights are made post-training, and thus it is possible to achieve different trade-offs. Moreover, our approach does not need to know sensitive attributes while training; thus, it could work with other sensitive attributes not known beforehand or during the training. Lastly, here we merely focused on mitigating bias (as our goal was to show that the interpretability framework can identify problematic features and their removal would result in bias mitigation) and did not focus too much on improving accuracy and achieving the best trade-off curve. We manipulated attention weights of all the features that contributed to unfairness irrespective of if they helped maintaining high accuracy or not. However, the trade-off results can be improved by carefully considering the trade-off each feature contributes to with regards to both accuracy and fairness (e.g., using results from Fig.~\\ref{fig:interpret_results}) to achieve even better trade-off results which can be investigated as a future direction (e.g., removing problematic features that contribute to unfairness only if their contribution to accuracy is below a certain threshold value). Picking the right threshold value is challenging and needs further investigation.\n\n\n\n\n\n\n\\vspace{-0.3cm}\n\\subsubsection{Experiments with Non-Tabular Data}\n\\vspace{-0.1cm}\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{ p{3.5cm} c c c}\n \\toprule\n\\textbf{Method}&\\textbf{Dentist TPRD (stdev)}&\\textbf{Nurse TPRD (stdev)}&\\textbf{Accuracy (stdev)}\\\\\n \\midrule\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Post-Processing (Ours)}}}\n&\\textbf{0.0202 (0.010)}&\\textbf{0.0251 (0.020)}&0.951 (0.013)\\\\[0.5pt]\n\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Pre-Processing}}}\n&0.0380 (0.016)&0.0616 (0.025)&0.946 (0.011)\\\\[0.5pt]\n\\midrule\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Not Debiased Model}}}\n&0.0474 (0.025)&0.1905 (0.059)&\\textbf{0.958 (0.011)}\\\\[0.5pt]\n\n\n \\bottomrule\n\\end{tabular}\n \\caption{Difference of the True Positive Rates (TPRD) amongst different genders for the dentist and nurse occupations on the biosbias dataset. Our introduced post-processing method is shown to be the most effective in reducing the disparity for both occupations compared to the previously used pre-processing technique.}\n\\label{bios_results}\n\\end{table}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth,trim=0cm 8cm 0cm 3cm,clip=true]{.\/images\/NLP_qual\/ex1.pdf}\n \\includegraphics[width=\\textwidth,trim=0.24cm 7cm 1.4cm 3cm,clip=true]{.\/images\/NLP_qual\/ex2.pdf}\n \\caption{Qualitative results from the non-tabular data experiment on the job classification task based on the bio texts. Green regions represent top three words that the model used for its prediction based on the attention weights. While the Not Debiased Model mostly focused on gendered words, our method focused on more useful words, such as R.N. (Registered Nurse), to predict the nurse label for the corresponding bio.}\n \\label{NLP_qualitative}\n\\end{figure}\n\nIn addition to providing interpretability, our approach is flexible and useful for controlling fairness in modalities other than tabular datasets. To put this to the test, we applied our model to mitigate bias in text-based data. We consider the \\textit{biosbias} dataset~\\cite{de2019bias}, and use our mitigation technique to reduce observed biases in the classification task performed on this dataset. We compare our approach with the debiasing technique proposed in the original paper~\\cite{de2019bias}, which works by masking the gender-related words and then training the model on this masked data. As discussed earlier, such a method is computationally inefficient. It requires re-training the model or creating a new masked dataset, each time it is required to debias the model against different attributes, such as gender vs. race. \nFor the baseline pre-processing method, we masked the gender-related words, such as names and gender words, as provided in the biosbias dataset and trained the model on the filtered dataset. On the other hand, we trained the model on the raw bios for our post-processing method and only manipulated attention weights of the gender words during the testing process. Notice, we eliminated the same features for both of the methods. Only the time of the elimination was different (one before training and one after training by manipulating the attention weights).\n\n\nIn order to measure the bias, we used the same measure as in~\\cite{de2019bias} which is based on the equality of opportunity notion of fairness~\\cite{NIPS2016_9d268236} and reported the True Positive Rate Difference (TPRD) of each occupation amongst different genders. As shown in Table~\\ref{bios_results}, our post-processing mitigation technique provides lower TRPD while being more accurate, followed by the technique that masks out the gendered words before training. Although both methods reduce the bias compared to a model trained on raw bios without applying any mask or invariance to gendered words, the post-processing method is more effective. \nFig.~\\ref{NLP_qualitative} also shows some qualitative results and highlights the differences between models in terms of their most attentive features for the prediction task. As shown in the results, our post-processing technique is able to use more meaningful words, such as R.N. which stands for registered nurse, to predict the outcome label nurse, while the not debiased model focuses on gendered words.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion}\nIn this work, we analyzed how attention weights contribute to fairness and accuracy of a predictive model. To do so, we proposed an attribution method that leverages the attention mechanism and showed the effectiveness of this approach on both tabular and text data. Using this interpretable attribution framework we then introduced a post-processing bias mitigation strategy based on attention weight manipulation. We validated the proposed framework by conducting experiments with different baselines, as well as fairness metrics, and different data modalities.\n\nAlthough our work can have a positive impact in allowing to reason about fairness and accuracy of models and reduce their bias, it can also have negative societal consequences if used unethically. For instance, it has been previously shown that interpretability frameworks can be used as a means for fairwashing which is when malicious users generate fake explanations for their unfair decisions to justify them \\cite{pmlr-v119-anders20a}. In addition, previously it has been shown that interpratability frameworks are vulnerable against adversarial attacks \\cite{slack2020fooling}. We acknowledge that our framework may also be targeted by malicious users for malicious intent that can manipulate attention weights to either generate fake explanations or unfair outcomes. An important future direction can be to analyze and improve robustness of our framework along with others. \n\n\n\n\n\\section{Experiments}\nWe perform a suite of experiments on synthetic and real-world datasets to evaluate our attention based interpretable fairness framework. First, we elucidate\ninterpretability on synthetic dataset for which we have the ground truth attributions. We verify that our interpretability framework recovers correct attribution. We also demonstrate our interpretability on real-world dataset and identify features that cause unfairness. Exploiting these attributions, we test our proposed post-processing bias mitigation framework and compare it with various different existing mitigation approaches. Finally, we show the effectiveness of our approach on non-tabular data format, namely text.\n\n\n\\subsection{Interpreting Fairness with Attention}\n\nIn this section, we demonstrate the effectiveness of our interpretability framework in being able to capture responsible features by controlled experiments with synthetic data as well as results on well-known real-world datasets.\n\n\\subsubsection{Synthetic Data}\nIn order to test our interpretability framework, we created some synthetic datasets in which we could control how features interact with each other and contribute to accuracy and fairness of the outcome variable and performed our first set of experiments.\n\n\\paragraph{Scenario 1:} First, we create a simple scenario to demonstrate that our framework identifies correct feature attributions for fairness and accuracy. To this end, we create a feature that is correlated with the outcome (responsible for accuracy), a feature that would cause the data to be biased (responsible for fairness), and a feature that is independent of the label or the task; thus, irrelevant for the task. Intuitively, if the attention based interpretability framework works correctly, we expect to see a reduction in accuracy upon removing (i.e., making the attention weight zero) the feature responsible for accuracy, reduction in bias upon removing the feature responsible for bias, and very little or no change upon removing the irrelevant feature. With this objective, we generated a synthetic dataset with three features, i.e., $x = [f_1, f_2, f_3]$ as follows~\\footnote{We use $x\\sim \\text{Ber}(p)$ to denote that $x$ is a Bernoulli random variable with $P(x=1)=p$.}.\n\\begin{equation*}\n f_1\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } y=1\\\\\n \\text{Ber}(0.9) & \\text{if } y=0\n \\end{cases}\n \\quad f_2\\sim \\text{Ber}(0.5)\n \\quad f_3\\sim \\mathcal N (0,1)\n \\quad y\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.1) & \\text{if } f_2=0\n \\end{cases}\n\\end{equation*}\n\nDue to the existing correlation between $f_2$ and the outcome variable $y$, $f_2$ can be considered as the feature with the most predictive information and responsible for accuracy. We consider $f_1$ as the sensitive attribute. It is generated such that there is no intentional correlation between $f_1$ and the outcome, $y$ or $f_2$. However, since $f_1$ is skewed towards $1$, there will be some undesired bias. Lastly, we sample $f_3$ from normal distribution that it is independent of the outcome $y$, or the other features; thus, making it irrelevant to the task.\n\n\\paragraph{Scenario 2:} Using features that are not identified as sensitive attributes can result in unfair decisions as a result of their implicit relations or correlations with the sensitive attributes. This phenomenon is called indirect discrimination~\\cite{zliobaite2015survey,6175897,ijcai2017-549}. We designed this synthetic dataset to demonstrate and characterize behaviour of our framework under indirect discrimination. This data has three features same as the previous scenario. Here, $f_1$ is considered as the sensitive attribute, and $f_2$ is correlated with $f_1$ and the outcome, $y$. The generative process is as follows:\n\\begin{equation*}\n f_1\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.1) & \\text{if } f_2=0\n \\end{cases}\n \\quad f_2\\sim\\text{Ber}(0.5)\n \\quad f_3\\sim \\mathcal N (0,1)\n \\quad y\\sim \\begin{cases}\n \\text{Ber}(0.7) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.3) & \\text{if } f_2=0\n \\end{cases}\n\\end{equation*}\n\nNote that $f_1$ is made to be correlated with $f_2$, and similarly the outcome is made to be correlated with $f_2$. Thus, the model mostly will rely on $f_2$ for its decisions as $f_2$ is the feature that is making the outcome to be correlated with itself. Therefore, despite the fact that $f_1$ is the sensitive attribute, we expect $f_2$ to affect both accuracy and fairness of the model more than $f_1$ as it is directly correlated with both the outcome and $f_1$ showing how indirect discrimination is possible. Using such synthetic dataset, we demonstrate a) indirect discrimination, and b) the need for having an interpretable framework to reason about unfairness and not blindly focus on the sensitive attributes for bias mitigation.\n\n\n\n\n\\begin{figure*}[t]\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_newscen1.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_newscen2.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_real_data_adult_subset2.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.5\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.4cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/no_err_real_data_health_subset2.pdf}\n\\end{subfigure}\n\\caption{Results from the synthetic datasets on top and real-world datasets in the bottom. Labels on the points represent the feature name that was removed. The results show how the accuracy and fairness of the model (in terms of statistical parity difference) change by exclusion of each feature. The red original point represents the results of the model including all the features.}\n\\label{interpret_results}\n\\end{figure*}\n\\subsubsection{Real Data}\nIn addition to synthetic datasets, we also used the UCI Adult and Heritage Health datasets and observed how different features contribute to accuracy and fairness of the model in these real-world datasets.\n\\subsubsection{Results}\nFor \\textit{Scenario 1}, as expected $f_2$ is correctly attributed to be responsible for accuracy and removing it hurts the accuracy drastically. Similarly $f_1$ is correctly shown to be responsible for unfairness as removing it creates a fairer outcome ($f_1$ was created to increase the bias thus its exclusion will result in more fair outcome). Lastly, as expected $f_3$ was the irrelevant feature and its effects on accuracy and fairness are negligible. Another interesting observation is that $f_2$ is actually helping the model to achieve fairness in this case since its exclusion means that the model should rely on $f_1$ for decision making, resulting in more bias; thus, removing $f_2$ will both harm accuracy and fairness as expected.\n\nFor \\textit{Scenario 2}, we were also able to capture the effect of indirect discrimination and show that $f_2$ indeed reduces bias and accuracy drastically by its removal. As explained previously, this is because $f_2$ is the predictive feature and due to its correlation with $f_1$, it can also affect fairness of the model indirectly. More interestingly, although $f_1$ is the sensitive feature, removing it does not play a drastic role in fairness nor the accuracy. This is an important finding as it shows why removing $f_1$ by its own can not give us a fairer model due to the existence of correlations of other features and indirect discrimination. Thus, our framework recognize such behavior and help design better mitigation strategies targeted towards responsible features which will lead us to our mitigation strategy discussed in the next section.\n\nLastly, Fig. \\ref{interpret_results}, also contains results on a subset of the features from the UCI Adult and Heritage Health datasets (To make the plots visually appealing and readable, we incorporated the interesting features in the plot.) which can give the reader some intuition on how different features in these datasets contribute to fairness and accuracy of the model. Gender was used as the sensitive feature in the Adult dataset and Age in the Heritage Health dataset.\n\nFrom the results, we conclude that attention weights capture the true and expected behavior and that this framework can be a reliable framework for explaining what contributes to fairness and accuracy of the model.\n\n\\subsection{Attention as a Mitigation Technique}\nAs discussed previously, it is important to know how features interact and contribute to the decision making outcome specifically if we want to design mitigation strategies. One obvious example was shown previously: When there is the effect of the indirect discrimination, it is not enough to just blindly put the focus on the sensitive feature; however, knowing how features interact and contribute to the outcome can help in designing such mitigation strategies. In light of this, we propose a mitigation strategy that relies on the observations from the interpretability framework that was introduced in the previous section. This mitigation strategy operates based on manipulation of the attention weights for the problematic features to achieve the desired outcome of the user. The proposed mitigation strategy is a post-processing technique that manipulates attention weights as discussed in Section \\ref{mitig_sec}. We first use our interpretability framework to identify the features that are responsible for fairness of the model (all the features whose exclusion will decrease the bias compared to the original model) and gradually decrease their attention weights to zero.\n\nFor our experiments, we used two real-world datasets and compared our mitigation technique with different mitigation strategies. We used statistical parity as our notion of fairness; however, due to the felxibility of our interpretability framework, our mitigation strategy can be extended to other fairness measures as well. Here, to compare our method against baselines that are mostly specifically optimized to achieve statistical parity, we decided to perform our main experiments with this notion of fairness. In our later experiments on text data, we utilized another fairness notion to highlight this flexibility. Additional results on other fairness measures and baseline methods can be found in our supplementary material. Each method was ran over five different seeds, and we reported the average of the accuracy and statistical parity difference (SPD). The downstream classifier was a 1-hidden-layer MLP with 50 neurons along with ReLU activation function. Our experiments were performed on Nvidia GeForce RTX 2080.\n\n\\begin{figure*}[t]\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1.0cm,clip=true]{.\/images\/mitig_imgs\/adult_results_err.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1cm,clip=true]{.\/images\/mitig_imgs\/health_results_err.pdf}\n\\end{subfigure}\n\\caption{Accuracy vs parity curves for UCI Adult and Heritage Health datasets.}\n\\label{mitig_results}\n\\end{figure*}\n\n\\subsubsection{Data}\nWe conduct our experiments on two real-world datasets-- UCI Adult~\\cite{Dua:2019} and Heritage Health \\footnote{https:\/\/www.kaggle.com\/c\/hhp} datasets. The UCI Adult dataset contains information about individuals with the prediction task of whether the income of the individual is greater than \\$50k or not. The Heritage Health dataset contains patient information and the task is to predict the Charleson Index (comorbidity index which is a patient survival indicator). We used the same pre-processing and train-test splits as in~\\cite{gupta2021controllable}. For the UCI Adult dataset we considered gender as the sensitive attribute and age as the sensitive attribute for the Herigate Health dataset.\n\\subsubsection{Baselines}\nTo measure the effectiveness of our interpretability framework and the manipulation of the attention weights as a bias mitigation strategy, we compared our mitigation results with different baselines. The results can also help us to notice that attention weights capture important information utilizing which in a smart manner can achieve us fairness. For our baselines, we included both information-theoretic as well as adversarial approaches that try to achieve fairness. \\textit{\\textbf{FCLR}} is a recent approach that learns representations to control parity through mutual information based on contrastive information estimators~\\cite{gupta2021controllable}. \\textit{\\textbf{CVIB}} also tries to learn fair representations through an information-theoretic approach that optimizes an objective which would minimize the mutual information between the encoding and the sensitive attribute without adversarial training~\\cite{NEURIPS2018_415185ea}. \\textit{\\textbf{MIFR}} also uses an information-theoretic objective to learn representations subject to fairness constraints~\\cite{pmlr-v89-song19a}. In addition to information theoretic approaches, we also compared our results with adversarial approaches that try to achieve invariance and fairness, such as \\textit{\\textbf{Adversarial Forgetting}}~\\cite{advforg} and \\textit{\\textbf{MaxEnt-ARL}}~\\cite{roy2019mitigating}. We also utilized \\textit{\\textbf{LAFTR}}~\\cite{pmlr-v80-madras18a} on the Adult dataset as another baseline and reported its results; however, since this method is feasible for binary sensitive attributes only, we were unable to use it on the Heritage Health dataset in which we have non-binary sensitive features.\n\n\\subsubsection{Results}\nFrom the results demonstrated in Fig.~\\ref{mitig_results}, one can observe that our mitigation framework based on the manipulation of the attention weights is comparable to state of the art mitigation strategies that are specifically targeted and optimized to achieve parity without providing interpretable outcomes. This sheds light on the importance of the interpretability framework and that it correctly captures how features interact and contribute to fairness of the system as knowing these types of information one can design strategies that can benefit the outcome. The results show the averaged accuracy vs parity curves for five different runs over various random seeds. Having results in the bottom-right corner is the ideal case as it represents the region with the highest accuracy and lowest bias. Our model is not only able to achieve comparable and competitive results, but it is also able to provide explanation such that we exactly know what feature and by how much it was manipulated to get the corresponding outcome. It addition, our approach needs only one round of training to get all the trade-off results for every possible sensitive attribute that might exist. Lastly, here we merely focused on mitigating bias (as our goal was to show that the interpretability framework is able to identify problematic features; thus, their removal would result in bias mitigation) and did not focus too much on improving accuracy and achieving the best trade-off curve. Therefore, we manipulated attention weights of all the features which contributed to unfairness irrespective of if they helped maintaining high accuracy or not. However, the trade-off results can be improved by considering the trade-off each feature contributes to with regards to both accuracy and fairness (e.g. using results from Fig.~\\ref{interpret_results}) for a stronger trade-off results which can be investigated as a future direction (e.g. removing problematic features that contribute to unfairness only if their contribution to accuracy is below a certain threshold value). Picking the right threshold value can be a challenge on its own that would need further investigation.\n\n\\subsection{Non-Tabular Experiments}\nIn addition to providing interpretability, an advantage of our proposed model over non-neural based models is its flexibility to be applied to large scale datasets as well as large text-based data. To put this into test, we applied our model over a text-based data and used our mitigation technique based on attention weight manipulation to reduce observed biases in the classification task performed on the dataset. The results confirm the effectiveness of our framework.\n\\subsubsection{Data}\nFor our experiments, we used the biosbias dataset~\\cite{de2019bias}. We utilized the bios from the year 2018 from the 2018\\_34 archive. We only considered two occupations to form our experiments namely, nurse and dentist. The dataset was split into 70-15-15 train, validation, and test splits. The dataset contains short bios about individuals with different occupations with the prediction task of the occupation for the given individual. Previous work~\\cite{de2019bias} has shown the existence of gender bias in such prediction tasks in which certain gender words are associated with certain job types. The goal of our study is to manipulate attention weights associated with the problematic features such as gender words, improve the performance of the model, and reduce the previously observed gender biases in this dataset. In our case, we expect female associated words, such as \\textit{she}, to cause skew towards the prediction of job nurse and male associated words, such as \\textit{he}, to skew the prediction towards the job dentist.\n\\subsubsection{Results}\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{ p{3.5cm} c c c}\n \\toprule\n\\textbf{Method}&\\textbf{Dentist TPRD (stdev)}&\\textbf{Nurse TPRD (stdev)}&\\textbf{Accuracy (stdev)}\\\\\n \\midrule\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Post-Processing (Ours)}}}\n&\\textbf{0.0202 (0.010)}&\\textbf{0.0251 (0.020)}&0.951 (0.013)\\\\[0.5pt]\n\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Pre-Processing}}}\n&0.0380 (0.016)&0.0616 (0.025)&0.946 (0.011)\\\\[0.5pt]\n\\midrule\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Not Debiased Model}}}\n&0.0474 (0.025)&0.1905 (0.059)&\\textbf{0.958 (0.011)}\\\\[0.5pt]\n\n\n \\bottomrule\n\\end{tabular}\n \\caption{Difference of the True Positive Rates (TPRD) amongst different genders for the dentist and nurse occupations on the biosbias dataset. Our introduced post-processing method is shown to be the most effective in reducing the disparity for both occupations compared to the previously used pre-processing technique.}\n\\label{bios_results}\n\\end{table}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth,trim=0cm 8cm 0cm 3cm,clip=true]{.\/images\/NLP_qual\/ex1.pdf}\n \\includegraphics[width=\\textwidth,trim=0.24cm 7cm 1.4cm 3cm,clip=true]{.\/images\/NLP_qual\/ex2.pdf}\n \\caption{Qualitative results from the non-tabular data experiment on the job classification task based on the bio texts. Green regions represent top three words that the model used for its prediction based on the attention weights. While the Not Debiased Model mostly focused on gendered words, our method focused on more useful words, such as R.N. (Registered Nurse), to predict the nurse label for the corresponding bio.}\n \\label{NLP_qualitative}\n\\end{figure}\nFor our results, we considered two approaches for bias mitigation. The first approach, is what is used in some previous work~\\cite{de2019bias} to debias the results. This method is based on pre-processing and works by masking the gender words and training the model. As explained previously, these types of pre-processing methods are not efficient since they require re-training the model each time we require to debias the model against different attributes, such as gender vs race. The second approach, however, is our proposed mitigation technique based on post-processing that trains the model on the original corpus without masking, but masks out (zeros out the attention weights) the effect of undesired features during test time. This method is more efficient since it does not require retraining for debiasing with regards to different attributes as debiasing happens during test time.\n\nFor the baseline pre-processing method, we masked out the gender related words, such as names and gender words, as also provided in the biosbias dataset and trained the model on the filtered dataset. On the other hand, for our post-processing method, we trained the model on the raw bios and only manipulated attention weights of the gender words during the testing process. Notice, we eliminated the same features for both of the methods. Only the time of the elimination was different (one before training and one after training by manipulating the attention weights).\n\nIn order to measure the bias, we used the same measure as in ~\\cite{de2019bias} which is based on the equality of opportunity notion of fairness~\\cite{NIPS2016_9d268236} that reports the True Positive Rate Difference (TPRD) of each occupation amongst different genders. As shown in Table~\\ref{bios_results}, our introduced post-processing mitigation technique that works by manipulating the attention weights, works the best in minimizing the bias for both occupations followed by the in-processing technique that masks out the gendered words before training. Although both methods reduce the bias compared to a model that is trained on raw bios without applying any mask or invariance to gendered words, the post-processing method achieves the best results. It also has the advantage of not needing retraining for debiasing with regards to different aspects aside from gender such as race. Fig.~\\ref{NLP_qualitative} also shows some qualitative results and highlights the differences between models in terms of their most attentive features for the prediction task. As shown in the results, our post-processing technique is able to use more meaningful words, such as R.N. which stands for registered nurse, to predict the outcome label nurse, while the not debiased model focuses on gendered words.\n\n\n\n\n\\section{Related Work}\n\\para{Fairness.} The research in fairness concerns itself with various topics, such as defining fairness metrics, proposing solutions for bias mitigation, and analyzing existing harms in various systems~\\cite{10.1145\/3457607}. In this work, we utilized different metrics that were introduced previously, such as statistical parity~\\cite{Dwork:2012:FTA:2090236.2090255}, equality of opportunity and equalized odds~\\cite{NIPS2016_9d268236}, to measure the amount of bias. We also used different bias mitigation strategies to compare against our mitigation strategy, such as FCRL~\\cite{gupta2021controllable}, CVIB~\\cite{NEURIPS2018_415185ea}, MIFR~\\cite{pmlr-v89-song19a}, adversarial forgetting~\\cite{advforg}, MaxEnt-ARL~\\cite{roy2019mitigating}, and LAFTR~\\cite{pmlr-v80-madras18a}. We also utilized concepts and datasets that were analyzing existing biases in NLP systems, such as~\\cite{de2019bias} which studied the existing biases in NLP systems on the occupation classification task on the bios dataset.\n\n\\para{Interpretability.} In this work, we introduced an attribution framework based on the attention weights that can analyze fairness and accuracy of the models at the same time and reason about the importance of each feature on fairness and accuracy. There is a body of work in NLP literature that tried to analyze the effect of the attention weights on interpretability of the model~\\cite{wiegreffe-pinter-2019-attention,jain-wallace-2019-attention,serrano-smith-2019-attention}. Other work also utilized attention weights to define an attribution score to be able to reason about how transformer models such as BERT work~\\cite{hao2021self}. Notice that although \\citet{jain-wallace-2019-attention} claim that attention might not be explanation, a body of work has proved otherwise including~\\cite{wiegreffe-pinter-2019-attention} in which authors directly target the work in \\citet{jain-wallace-2019-attention} and analyze in detail the problems associated with this study. In our work, we also find that attention can be useful and can extract meaningful information which can be beneficial in many aspects. In addition,~\\citet{NEURIPS2020_92650b2e} analyze the effect of the attention weights in transformer models for bias analysis in language models. However, their approach is different and has a more causal take on investigating the bias. Their study is specific to language models and does not necessarily apply to broader tasks and existing fairness definitions. Aside from interpretability and fairness, we utilized concepts from the NLP literature for designing our attention-based model that can be applicable to tabular data~\\cite{NIPS2017_3f5ee243,zhou-etal-2016-attention}.\n \n\\section{Experimental Setup}\nWe perform a suite of experiments on synthetic and real-world datasets to evaluate our attention based interpretable fairness framework. The experiments on synthetic data are intended to elucidate interpretability in controlled settings, where we can manipulate the relations between input and output feature. The experiments on real-world data aim to validate the effectiveness of the proposed approach on both tabular and non-tabular (textual) data. Below we describe the experiments, datasets, and respective baselines. \n\n\\subsection{Types of Experiments}\nWe enumerate the experiments and their goals as follows: \\\\\n\\noindent{\\bf Experiment 1: Attributing Fairness with Attention} The purpose of this experiment is to demonstrate that our attribution framework can capture correct attributions of features to fairness outcomes. We present our results for tabular data in Sec.~\\ref{sec:interpret_fairness}.\\\\\n\\noindent {\\bf Experiment 2: Bias Mitigation via Attention Weight Manipulation} In this experiment, we seek to validate the proposed post-processing bias mitigation framework and compare it with various recent mitigation approaches. The results for real-world tabular data are presented in Sec.~\\ref{sec:bias_mitigation}.\\\\\n\\noindent{\\bf Experiment 3: Validation on Textual Data} The goal of this experiment is to demonstrate the flexibility of the proposed attention-based method by conducting experiments on non-tabular, textual data. The results are presented in Sec.~\\ref{sec:textual_data}.\n\n\n\\subsection{Datasets}\n\\newcommand{\\text{Ber}}{\\text{Ber}}\n\n\\subsubsection{Synthetic Data} \\label{syn_data_sec}\nTo validate the attribution framework, we created two synthetic datasets in which we control how features interact with each other and contribute to the accuracy and fairness of the outcome variable.\nThese datasets capture some of the common scenarios, namely the data imbalance (skewness) and indirect discrimination issues, arising in fair decision or classification problems.\n\n\\paragraph{Scenario 1:} First, we create a simple scenario to demonstrate that our framework identifies correct feature attributions for fairness and accuracy. We create a feature that is correlated with the outcome (responsible for accuracy), a discrete feature that causes the prediction outcomes to be biased (responsible for fairness), and a continuous feature that is independent of the label or the task (irrelevant for the task). For intuition, suppose the attention-based attribution framework works correctly. In this case, we expect to see a reduction in accuracy upon removing (i.e., making the attention weight zero) the feature responsible for the accuracy, reduction in bias upon removing the feature responsible for bias, and very little or no change upon removing the irrelevant feature. With this objective, we generated a synthetic dataset with three features, i.e., $x = [f_1, f_2, f_3]$ as follows\\footnote{We use $x\\sim \\text{Ber}(p)$ to denote that $x$ is a Bernoulli random variable with $P(x=1)=p$.}:\n\\begin{align*}\n f_1 ~\\sim \\text{Ber}(0.9)\n \n \n \n \n \\quad f_2\\sim \\text{Ber}(0.5) \n \\quad f_3\\sim &\\mathcal N (0,1) \\\\\n \\quad y\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.1) & \\text{if } f_2=0\n \\end{cases}\n\\end{align*}\nClearly, $f_2$ has the most predictive information for the task and is responsible for accuracy. Here, we consider $f_1$ as the sensitive attribute. $f_1$ is an imbalanced feature that can bias the outcome and is generated such that there is no intentional correlation between $f_1$ and the outcome, $y$ or $f_2$. $f_3$ is sampled from a normal distribution independent of the outcome $y$, or the other features, making it irrelevant for the task.\nThus, an ideal classifier would be fair if it captures the correct outcome without being affected by the imbalance in $f_1$. However, due to limited data and skew in $f_1$, there will be some undesired bias --- few errors when $f_1=0$ can lead to large statistical parity. \n\n\n\\paragraph{Scenario 2:} Using features that are not identified as sensitive attributes can result in unfair decisions due to their implicit relations or correlations with the sensitive attributes. This phenomenon is called indirect discrimination~\\cite{zliobaite2015survey,6175897,ijcai2017-549}. We designed this synthetic dataset to demonstrate and characterize the behavior of our framework under indirect discrimination. Similar to the previous scenario, we consider three features. Here, $f_1$ is considered as the sensitive attribute, and $f_2$ is correlated with $f_1$ and the outcome, $y$. The generative process is as follows:\n\\begin{align*}\n f_1\\sim \\begin{cases}\n \\text{Ber}(0.9) & \\text{if } f_2=1\\\\\n \\text{Ber}(0.1) & \\text{if } f_2=0\n \\end{cases} \n \\quad f_2\\sim\\text{Ber}(0.5) &\n \\quad f_3\\sim \\mathcal N (0,1) \\\\ \n \\quad y\\sim \\begin{cases}\n \\text{Ber}(0.7) & \\text{if } f_2=1\\\\ \n \\text{Ber}(0.3) & \\text{if } f_2=0\n \\end{cases} \n\\end{align*}\n\nIn this case $f_1$ and $y$ are correlated with \n$f_2$. The model should mostly rely on $f_2$ for its decisions. \nHowever, due to the correlation between $f_1$ and $f_2$, we expect $f_2$ to affect both the accuracy and fairness of the model. Thus, in this case, indirect discrimination is possible. Using such a synthetic dataset, we demonstrate a) indirect discrimination and b) the need to have an attribution framework to reason about unfairness and not blindly focus on the sensitive attributes for bias mitigation.\n\n\n\n\\subsubsection{Real-world Datasets}\nWe demonstrate our approach on the following real-world datasets:\\\\\n\\noindent{\\bf Tabular Datasets:} We conduct our experiments on two real-world tabular datasets often used to benchmark fair classification techniques --- \\textit{UCI Adult}~\\cite{Dua:2019} and \\textit{Heritage Health}\\footnote{https:\/\/www.kaggle.com\/c\/hhp} datasets. The \\textit{UCI Adult} dataset contains census information about individuals, with the prediction task being whether the income of the individual is higher than \\$50k or not. The sensitive attribute, in this case, is gender (male\/female). The \\textit{Heritage Health} dataset contains patient information, and the task is to predict the Charleson Index (comorbidity index, which is a patient survival indicator). Each patient is grouped into one of the 9 possible age groups, and we consider this as the sensitive attribute. We used the same pre-processing and train-test splits as in~\\citet{gupta2021controllable}.\\\\\n\\noindent{\\bf Non-Tabular or Text Dataset:}\nTo demonstrate the flexibility of our approach, we also experiment with a non-tabular, text dataset. We used the \\textit{biosbias} dataset~\\cite{de2019bias}. The dataset contains short bios of individuals. The task is to predict the occupation of the individual from their bio.\nWe utilized the bios from the year 2018 from the \\texttt{2018\\_34} archive and considered two occupations for our experiments, namely, nurse and dentist. The dataset was split into 70-15-15 train, validation, and test splits. \\citet{de2019bias} has demonstrated the existence of gender bias in this prediction task and showed that certain gender words are associated with certain job types (e.g., \\textit{she} to nurse and \\textit{he} to dentist). \n\n\\subsection{Bias Mitigation Baselines} \n\\label{sec:baselines}\nWe compared our bias mitigation approach to a number of recent state of the art methods. For our experiments with tabular data, we focus on methods that are specifically optimized to achieve statistical parity. Results for other fairness notions can be found in the appendix. For our baselines, we consider methods that learn representations of data so that information about sensitive attributes is eliminated.\n\\textit{\\textbf{CVIB}}~\\cite{NEURIPS2018_415185ea} realizes this objective through a conditional variational autoencoder, whereas \\textit{\\textbf{MIFR}}~\\cite{pmlr-v89-song19a} uses a combination of information bottleneck term and adversarial learning to optimize the fairness objective. \n\\textit{\\textbf{FCRL}}~\\cite{gupta2021controllable} optimizes information theoretic objectives that can be used to achieve good trade-offs between fairness and accuracy by using specialized contrastive information estimators. \nIn addition to information-theoretic approaches, we also considered baselines that use adversarial learning such as \\textit{\\textbf{MaxEnt-ARL}}~\\cite{roy2019mitigating}, \\textit{\\textbf{LAFTR}}~\\cite{pmlr-v80-madras18a}, and \\textit{\\textbf{Adversarial Forgetting}}~\\cite{advforg}. Note that in contrast to our approach, the baselines described above are not interpretable as they are incapable of directly attributing features to fairness outcomes. For the textual data, we compare our approach with the debiasing technique proposed in~\\citet{de2019bias}, which works by masking the gender-related words and then training the model on this masked data. \n\n\n\n\n\\section{Results}\n\\subsection{Attributing Fairness with Attention} \n\\label{sec:interpret_fairness}\n\nFirst, we test our method's ability to capture correct attributions in controlled experiments with synthetic data (described in Sec.~\\ref{syn_data_sec}). We also conduct a similar experiment with \\textit{UCI Adult} and \\textit{Heritage Health} datasets which can be found in the appendix. Fig.~\\ref{fig:interpret_results} summarizes our results by visualizing the attributions, which we now discuss. \n\nIn \\textit{Scenario 1}, as expected, $f_2$ is correctly attributed to being responsible for the accuracy and removing it hurts the accuracy drastically. Similarly, $f_1$ is correctly shown to be responsible for unfairness and removing it creates a fairer outcome. Ideally, the model should not be using any information about $f_1$ as it is independent of the task, but it does. Therefore, by removing $f_1$, we can ensure that information is not used and hence outcomes are fair. Lastly, as expected, $f_3$ was the irrelevant feature, and its effects on accuracy and fairness are negligible. Another interesting observation is that $f_2$ is helping the model achieve fairness since its exclusion means that the model should rely on $f_1$ for decision making, resulting in more bias; thus, removing $f_2$ harms accuracy and fairness as expected.\n\nIn \\textit{Scenario 2}, our framework captures the effect of indirect discrimination. We can see that removing $f_2$ reduces bias as well as accuracy drastically. This is because $f_2$ is the predictive feature, but due to its correlation with $f_1$, it can also indirectly affect the model's fairness. More interestingly, although $f_1$ is the sensitive feature, removing it does not play a drastic role in fairness or the accuracy. This is an important finding as it shows why removing $f_1$ on its own can not give us a fairer model due to the existence of correlations to other features and indirect discrimination. Overall, our results are intuitive and thus validate our assumption that attention-based framework can provide reliable feature attributions for the fairness and accuracy of the model. \n\n\n\n\n\\subsection{Attention as a Mitigation Technique}\n\\label{sec:bias_mitigation}\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.35\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/attr_vis_sen1.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.35\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/attr_vis_sen2.pdf}\n\\end{subfigure}\n\\caption{Results from the synthetic datasets. Following the $\\hat{y}_o$ and $\\hat{y}_z$ notations, $\\hat{y}_o$ represents the original model outcome with all the attention weights intact, while $\\hat{y}_z^k$ represents the outcome of the model in which the attention weights corresponding to $k^{th}$ feature are zeroed out (e.g. $\\hat{y}_z^1$ represents when attention weights of feature $f_1$ are zeroed out). The results show how the accuracy and fairness of the model (in terms of statistical parity difference) change by exclusion of each feature.}\n\\label{fig:interpret_results}\n\\end{figure}\nAs we have highlighted earlier, understanding how the information within features interact and contribute to the decision making can be used to design effective bias mitigation strategies. One such example was shown in Sec.~\\ref{sec:interpret_fairness}. Often real-world datasets have features which cause indirect discrimination, due to which fairness can not be achieved by simply eliminating the sensitive feature from the decision process. \nUsing the attributions derived from our attention-based attribution framework, we propose a post-processing mitigation strategy. Our strategy is to intervene on attention weights as discussed in Sec.~\\ref{mitig_sec}. We first attribute and identify the features responsible for the unfairness of the outcomes, i.e., all the features whose exclusion will decrease the bias compared to the original model's outcomes and gradually decrease their attention weights to zero as also outlined in Algorithm~\\ref{mitig_alg}. We do this by first using the whole fraction of the attention weights learned and gradually use less fraction of the weights until the weights are completely zeroed out. \n\n\nFor all the baselines described in Sec.~\\ref{sec:baselines}, we used the approach outlined in~\\citet{gupta2021controllable} for training a downstream classifier and evaluating the accuracy\/fairness trade-offs. The downstream classifier was a 1-hidden-layer MLP with 50 neurons along with ReLU activation function. Our experiments were performed on Nvidia GeForce RTX 2080. Each method was trained with five different seeds, and we report the average accuracy and fairness measure as statistical parity difference (SPD). Results for other fairness notions can be found in the appendix. \\textit{\\textbf{CVIB}}, \\textit{\\textbf{MaxEnt-ARL}}, \\textit{\\textbf{Adversarial Forgetting}} and \\textit{\\textbf{FCRL}} are designed for statistical parity notion of fairness and are not applicable for other measures like Equalized Odds and Equality of Opportunity. \\textit{\\textbf{LAFTR}} can only deal with binary sensitive attributes and thus not applicable for Heritage Health dataset. Notice that our approach does not have these limitations. For our approach, we vary the attention weights and report the resulting fairness-accuracy trade offs. \n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[t]\n\\centering\n \\begin{subfigure}[b]{0.43\\textwidth}\n \\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1.0cm,clip=true]{.\/images\/mitig_imgs\/adult_results_err.pdf}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.43\\textwidth}\n \\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1cm,clip=true]{.\/images\/mitig_imgs\/health_results_err.pdf}\n \\end{subfigure}\n \\caption{Accuracy vs parity curves for UCI Adult and Heritage Health datasets.}\n \\label{fig:mitig_results}\n\\end{figure*}\n\nFig.~\\ref{fig:mitig_results} compares fairness-accuracy trade-offs of different bias mitigation approaches. We desire outcomes to be fairer, i.e., lower values of \\texttt{SPD} and to be more accurate, i.e., towards the right. The results show that using attention attributions can indeed be beneficial for reducing bias. Moreover, our mitigation framework based on the manipulation of the attention weights is competitive with state-of-the-art mitigation strategies. However, most of these approaches are specifically designed and optimized to achieve parity and do not provide any interpretability. \nOur model can not only achieve comparable and competitive results, but it is also able to provide explanation such that the users exactly know what feature and by how much it was manipulated to get the corresponding outcome. Another advantage of our model is that it needs only one round of training. The adjustments to attention weights are made post-training; thus, it is possible to achieve different trade-offs. Moreover, our approach does not need to know sensitive attributes while training; thus, it could work with other sensitive attributes not known beforehand or during training. Lastly, here we merely focused on mitigating bias (as our goal was to show that the attribution framework can identify problematic features and their removal would result in bias mitigation) and did not focus too much on improving accuracy and achieving the best trade-off curve which can be considered as the current limitation of our work. We manipulated attention weights of all the features that contributed to unfairness irrespective of if they helped maintaining high accuracy or not. However, the trade-off results can be improved by carefully considering the trade-off each feature contributes to with regards to both accuracy and fairness (e.g., using results from Fig.~\\ref{fig:interpret_results}) to achieve better trade-off results which can be investigated as a future direction (e.g., removing problematic features that contribute to unfairness only if their contribution to accuracy is below a certain threshold value). The advantage of our work is that this trade-off curve can be controlled by controlling how many features and by how much to be manipulated which is not the case for most existing work.\n\n\n\n\n\n\n\n\\subsection{Experiments with Non-Tabular Data}\n\\label{sec:textual_data}\n\n\nIn addition to providing interpretability, our approach is flexible and useful for controlling fairness in modalities other than tabular datasets. To put this to the test, we applied our model to mitigate bias in text-based data. We consider the \\textit{biosbias} dataset~\\cite{de2019bias}, and use our mitigation technique to reduce observed biases in the classification task performed on this dataset. We compare our approach with the debiasing technique proposed in the original paper~\\cite{de2019bias}, which works by masking the gender-related words and then training the model on this masked data. As discussed earlier, such a method is computationally inefficient. It requires re-training the model or creating a new masked dataset, each time it is required to debias the model against different attributes, such as gender vs. race. \nFor the baseline pre-processing method, we masked the gender-related words, such as names and gender words, as provided in the \\textit{biosbias} dataset and trained the model on the filtered dataset. On the other hand, we trained the model on the raw bios for our post-processing method and only manipulated attention weights of the gender words during the testing process as also provided in the \\textit{biosbias} dataset. \n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{ p{3.5cm} c c c}\n \\toprule\n\\textbf{Method}&\\textbf{Dentist TPRD (stdev)}&\\textbf{Nurse TPRD (stdev)}&\\textbf{Accuracy (stdev)}\\\\\n \\midrule\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Post-Processing (Ours)}}}\n&\\textbf{0.0202 (0.010)}&\\textbf{0.0251 (0.020)}&0.951 (0.013)\\\\[0.5pt]\n\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Pre-Processing}}}\n&0.0380 (0.016)&0.0616 (0.025)&0.946 (0.011)\\\\[0.5pt]\n\\midrule\n \\parbox[t]{2mm}{\\multirow{1}{*}{\\shortstack[l]{Not Debiased Model}}}\n&0.0474 (0.025)&0.1905 (0.059)&\\textbf{0.958 (0.011)}\\\\[0.5pt]\n\n\n \\bottomrule\n\\end{tabular}\n \\caption{Difference of the True Positive Rates (TPRD) amongst different genders for the dentist and nurse occupations on the biosbias dataset. Our introduced post-processing method is the most effective in reducing the disparity for both occupations compared to the pre-processing technique.}\n\\label{bios_results}\n\\end{table*}\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[width=0.95\\textwidth,trim=0cm 8cm 0cm 3.49cm,clip=true]{.\/images\/NLP_qual\/ex1.pdf}\n \\includegraphics[width=0.95\\textwidth,trim=0.24cm 7.8cm 1.4cm 3.49cm,clip=true]{.\/images\/NLP_qual\/ex2.pdf}\n \\caption{Qualitative results from the non-tabular data experiment on the job classification task based on bio texts. Green regions are the top three words used by the model for its prediction based on the attention weights. While the Not Debiased Model mostly focuses on gendered words, our method focused on profession-based words, such as R.N. (Registered Nurse), to correctly predict ``nurse.''}\n \\label{NLP_qualitative}\n\\end{figure*}\nIn order to measure the bias, we used the same measure as in~\\cite{de2019bias} which is based on the equality of opportunity notion of fairness~\\cite{NIPS2016_9d268236} and reported the True Positive Rate Difference (TPRD) for each occupation amongst different genders. As shown in Table~\\ref{bios_results}, our post-processing mitigation technique provides lower TRPD while being more accurate, followed by the technique that masks the gendered words before training. Although both methods reduce the bias compared to a model trained on raw bios without applying any mask or invariance to gendered words, our post-processing method is more effective. Fig.~\\ref{NLP_qualitative} also highlights qualitative differences between models in terms of their most attentive features for the prediction task. As shown in the results, our post-processing technique is able to use more meaningful words, such as R.N. (registered nurse) to predict the outcome label nurse compared to both baselines, while the non-debiased model focuses on gendered words.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n\nMachine learning algorithms that optimize for performance (e.g., accuracy) often result in unfair outcomes~\\cite{10.1145\/3457607}. These algorithms capture biases present in the training datasets causing discrimination toward different groups. As machine learning continues to be adopted into fields where discriminatory treatments can lead to legal penalties, fairness and interpretability have become a necessity and a legal incentive in addition to an ethical responsibility~\\cite{barocas2016big,hacker2020explainable}. Existing methods for fair machine learning include applying complex transformations to the data so that resulting representations are fair~\\cite{gupta2021controllable,NEURIPS2018_415185ea,roy2019mitigating,advforg, pmlr-v89-song19a}, adding regularizers to incorporate fairness~\\cite{pmlr-v54-zafar17a,kamishima2012fairness,mehrabi2020statistical}, or modifying the outcomes of unfair machine learning algorithms to ensure fairness~\\cite{NIPS2016_9d268236}, among others. Here we present an alternative approach \\footnote{Code can be found at: \\url{https:\/\/github.com\/Ninarehm\/Attribution}}, which works by identifying the significance of different features in causing unfairness and reducing their effect on the outcomes using an attention-based mechanism.\n\n\nWith the advancement of transformer models and the attention mechanism \\cite{NIPS2017_3f5ee243}, recent research in Natural Language Processing (NLP) has tried to analyze the effects and the interpretability of the attention weights on the decision making process~\\cite{wiegreffe-pinter-2019-attention,jain-wallace-2019-attention,serrano-smith-2019-attention,hao2021self}. Taking inspiration from these works, we propose to use an attention-based mechanism to study the fairness of a model. The attention mechanism provides an intuitive way to capture the effect of each attribute on the outcomes. Thus, by introducing the attention mechanism, we can analyze the effect of specific input features on the model's fairness. We form visualizations that explain model outcomes and help us decide which attributes contribute to accuracy vs.\\ fairness. We also show and confirm the observed effect of indirect discrimination in previous work \\cite{zliobaite2015survey,6175897,ijcai2017-549} in which even with the absence of the sensitive attribute, we can still have an unfair model due to the existence of proxy attributes. Furthermore, we show that in certain scenarios those proxy attributes contribute more to the model unfairness than the sensitive attribute itself.\n\nBased on the above observations, we propose a post-processing bias mitigation technique by diminishing the weights of features most responsible for causing unfairness. We perform studies on datasets with different modalities and show the flexibility of our framework on both tabular and large-scale text data, which is an advantage over existing interpretable non-neural and non-attention-based models. Furthermore, our approach provides a competitive and interpretable baseline compared to several recent fair learning techniques.\n\nTo summarize, the contributions of this work are as follows: (1) We propose a new framework for attention-based classification in tabular data, which is interpretable in the sense that it allows to quantify the effect of each attribute on the outcomes; (2) We then use these attributions to study the effect of different input features on the fairness and accuracy of the models; (3) Using this attribution framework, we propose a post-processing bias mitigation technique that can reduce unfairness and provide competitive accuracy vs. fairness trade-offs; (4) Lastly, we show the versatility of our framework by applying it to large-scale non-tabular data such as text.\n\n\n\n\n\n\n\n\n\n\n\\section{Appendix}\nWe included additional bias mitigation results using other fairness metrics, such as equality of opportunity and equalized odds on both of the Adult and Heritage Health datasets in this supplementary material. We also included additional post-processing results along with additional qualitative results both for the tabular and non-tabular dataset experiments. More details can be found under each sub-section.\n\\subsection{Results on Tabular Data}\nHere, we show the results of our mitigation framework considering equality of opportunity and equalized odds notions of fairness. We included baselines that were applicable for these notions. Notice not all the baselines we used in our previous analysis for statistical parity were applicable for equality of opportunity and equalized odds notions of fairness; thus, we only included the applicable ones. In addition, LAFTR is only applicable when the sensitive attribute is a binary variable, so it was not applicable to be included in the analysis for the heritage health data where the sensitive attribute is non-binary. Results of these analysis is shown in Figures~\\ref{mitig_results_eqop} and \\ref{mitig_results_eqod}. We once again show competitive and comparable results to other baseline methods, while having the advantage of being interpretable and not requiring multiple trainings to satisfy different fairness notions or fairness on different sensitive attributes. Our framework is also flexible for different fairness measures and can be applied to binary or non-binary sensitive features.\n\n\nIn addition, we show how different features contribute differently under different fairness notions. Fig.~\\ref{features} demonstrates the top three features that contribute to unfairness the most along with the percentages of the fairness improvement upon their removal for each of the fairness notions. As observed from the results, while equality of opportunity and equalized odds are similar in terms of their problematic features, statistical parity has different trends. This is also expected as equality of opportunity and equalized odds are similar fairness notions in nature compared to statistical parity.\n\nWe also compared our mitigation strategy with the Hardt etl al. post-processing approach~\\cite{NIPS2016_9d268236}. Using this post-processing implementation~\\footnote{\\url{https:\/\/fairlearn.org}}, we obtained the optimal solution that tries to satisfy different fairness notions subject to accuracy constraints. For our results, we put the results from zeroing out all the attention weights corresponding to the problematic features that were detected from our interpretability framework. However, notice that since our mitigation strategy can control different trade-offs we can have different results depending on the scenario. Here, we reported the results from zeroing out the problematic attention weights that is targeting fairness mostly. From the results demonstrated in Tables~\\ref{adult_post_proc} and~\\ref{health_post_proc}, we can see comparable numbers to those obtained from~\\cite{NIPS2016_9d268236}. This again shows that our interpretability framework yet again captures the correct responsible features and that the mitigation strategy works as expected.\n\n\\subsection{Results on non-tabular Data}\nWe also included some additional qualitative results from the experiments on non-tabular data in Fig.~\\ref{fig:NLP_qualitative_appendix}.\n\\begin{figure*}[h]\n \\centering\n \\includegraphics[width=\\textwidth,trim=0cm 6cm 0cm 1cm,clip=true]{.\/images\/NLP_qual\/ex3.pdf}\n \\caption{Additional qualitative results from the non-tabular data experiment on the job classification task based on the bio texts. Green regions represent top three words that the model used for its prediction based on the attention weights.}\n \\label{fig:NLP_qualitative_appendix}\n\\end{figure*}\n\n\\subsection{Interpreting Fairness with Attention}\nFig.~\\ref{fig:interpret_results_2} shows results on a subset of the features from the \\textit{UCI Adult} and \\textit{Heritage Health} datasets (to keep the plots uncluttered and readable, we incorporated the most interesting features in the plot), and provide some intuition about how different features in these datasets contribute to the model fairness and accuracy. While features such as {\\em capital gain} and {\\em capital loss} in the \\textit{UCI Adult} dataset are responsible for improving accuracy and reducing bias, we can observe that features such as {\\em relationship} or {\\em marital status}, which can be indirectly correlated with the feature {\\em sex}, have a negative impact on fairness. For the \\textit{Heritage Health} dataset, including the features {\\em drugCount ave} and {\\em dsfs max} provide accuracy gains but at the expense of fairness, while including {\\em no Claims} and {\\em no Specialities} negatively impact both accuracy and fairness. \n\n\n\\subsection{Information on Datasets and Features}\nMore details about each of the datasets along with the descriptions of each feature for the Adult dataset can be found at\\footnote{https:\/\/archive.ics.uci.edu\/ml\/datasets\/adult} and for the Heritage Health dataset can be found at\n\\footnote{https:\/\/www.kaggle.com\/c\/hhp}. In our qualitative results, we used the feature names as marked in these datasets. If the names or acronyms are unclear kindly reference to the references mentioned for more detailed description for each of the features. Although most of the features in the Adult datasets are self-descriptive, Heritage Health dataset includes some abbreviations that we list in Table~\\ref{feature_abbriv} for the ease of interpreting each feature's meaning.\n\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\n\\textbf{Abbreviation} & \\textbf{Meaning} \\\\ \\hline\n\\multirow{1}{*}{PlaceSvcs} & Place where the member was treated. \n \\\\ \\hline\n\\multirow{1}{*}{LOS} & Length of stay. \n \\\\ \\hline\n\\multirow{1}{*}{dsfs} & Days since first service that year. \n \\\\ \\hline\n\\end{tabular}\n\\caption{Some abbreviations used in Heritage Health dataset's feature names. These abbreviations are listed for clarity of interpreting each feature's meaning specifically in our qualitative analysis or attribution visualizations.}\n\\label{feature_abbriv}\n\\end{table}\n\n\\begin{figure*}[h]\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1.0cm,clip=true]{.\/images\/appendix_imgs\/adult_results_err_eqop.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1cm,clip=true]{.\/images\/appendix_imgs\/health_results_err_eqop.pdf}\n\\end{subfigure}\n\\caption{Accuracy vs equality of opportunity curves for UCI Adult and Heritage Health datasets.}\n\\label{mitig_results_eqop}\n\\end{figure*}\n\n\n\\begin{figure*}[h]\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1.0cm,clip=true]{.\/images\/appendix_imgs\/adult_results_err_eqod.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.49\\textwidth}\n\\includegraphics[width=\\textwidth,trim=0cm 0cm 1.8cm 1cm,clip=true]{.\/images\/appendix_imgs\/health_results_err_eqod.pdf}\n\\end{subfigure}\n\\caption{Accuracy vs equalized odds curves for UCI Adult and Heritage Health datasets.}\n\\label{mitig_results_eqod}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\\centering\n\\begin{subfigure}[b]{0.44\\textwidth}\n\\includegraphics[width=\\textwidth,trim=7cm 2cm 5cm 4cm,clip=true]{.\/images\/appendix_imgs\/adult_features.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.50\\textwidth}\n\\includegraphics[width=\\textwidth,trim=4cm 3cm 4cm 3cm,clip=true]{.\/images\/appendix_imgs\/health_features.pdf}\n\\end{subfigure}\n\\caption{Top three features for each fairness definition removing which caused the most benefit in improving the corresponding fairness definition. The percentage of improvement upon removal is marked on the $y$-axis for adult and heritage health datasets.}\n\\label{features}\n\\end{figure*}\n\n\\begin{table*}[h]\n\\centering\n\\scalebox{0.9}{\n \\begin{tabular}{c | c c | c c |cc}\n \\toprule\n & Accuracy & SPD & Accuracy & EQOP & Accuracy & EQOD\\\\\n \\midrule\n Attention (Ours)&\\textbf{0.77 (0.006)}&\\textbf{0.012 (0.003)}&0.81 (0.013)&\\textbf{0.020 (0.019)}&\\textbf{0.81 (0.021)}&\\textbf{0.027 (0.023)}\\\\\n \\midrule\n Hardt et al.&\\textbf{0.77 (0.012)}&0.013 (0.005)&\\textbf{0.83 (0.005)}&0.064 (0.016)&\\textbf{0.81 (0.007)}&0.047 (0.014)\\\\\n \\bottomrule\n \\end{tabular}}\n \\caption{Adult results on post-processing approach from Hardt et al. vs our attention method when all problematic features are zeroed out.}\n \\label{adult_post_proc}\n \\vspace{-1em}\n\\end{table*}\n\n\n\\begin{table*}[h]\n\\centering\n\\scalebox{0.9}{\n \\begin{tabular}{c | c c | c c |cc}\n \\toprule\n & Accuracy & SPD & Accuracy & EQOP & Accuracy & EQOD\\\\\n \\midrule\n Attention (Ours)&\\textbf{0.68 (0.004)}&\\textbf{0.04 (0.015)}&0.68 (0.015)&\\textbf{0.15 (0.085)}&0.68 (0.015)&\\textbf{0.10 (0.085)}\\\\\n \\midrule\n Hardt et al.&\\textbf{0.68 (0.005)}&0.05 (0.018)&\\textbf{0.75 (0.001)}&0.20 (0.033)&\\textbf{0.69 (0.012)}&0.19 (0.031)\\\\\n \\bottomrule\n \\end{tabular}\n }\n \\caption{Heritage Health results on post-processing approach from Hardt et al. vs our attention method when all problematic features are zeroed out.}\n \\label{health_post_proc}\n \\vspace{-1em}\n\\end{table*}\n\n\\begin{figure*}[h]\n\\centering\n\\begin{subfigure}[b]{0.45\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.5cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/attr_vis_adult.pdf}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.45\\textwidth}\n\\includegraphics[width=\\textwidth,trim=2.4cm 0cm 4.5cm 1.0cm,clip=true]{.\/images\/inter_imgs\/attr_vis_health.pdf}\n\\end{subfigure}\n\\caption{Results from the real-world datasets. Note that in our $\\hat{y}_z$ notation we replaced indexes with actual feature names for clarity in these results on real-world datasets as there is not one universal indexing schema, but the feature names are more universal and discriptive for this case. Labels on the points represent the feature name that was removed (zeroed out) according to our $\\hat{y}_z$ notation. The results show how the accuracy and fairness of the model (in terms of statistical parity difference) change by exclusion of each feature.}\n\\label{fig:interpret_results_2}\n\\end{figure*}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nWe consider a map $T \\colon [0,1] \\to [0,1]$. Let $r =\n(r_n)_{n=1}^\\infty$ be a sequence of decreasing positive numbers. In\nthis paper we shall investigate the size of the set\n\\begin{align*}\n E (x, r) &= \\{\\, y \\in [0,1] : d (T^n x, y) < r_n\n \\text{ for infinitely many } n \\,\\} \\\\ &= \\limsup_{n \\to \\infty}\n B(T^n x, r_n).\n\\end{align*}\n\nSets of this form with $T \\colon x \\mapsto 2x \\mod 1$ were studied by\nFan, Schmeling and Troubetzkoy in \\cite{Fanetal}. Li, Wang, Wu and Xu\nstudied in \\cite{Lietal} a related but different set in the case when\n$T$ is the Gau\\ss{} map.\n\nIn the paper \\cite{LiaoSeuret}, Liao and Seuret studied the case when\n$T$ is an expanding Markov map with a Gibbs measure $\\mu$. They proved\nthat if $r_n = n^{-\\alpha}$, then for $\\mu$-almost all $x$, the set $E\n(x, r)$ has Hausdorff dimension $1\/\\alpha$ provided that $1\/\\alpha$\nis not larger than the dimension of the measure $\\mu$.\n\nIn this paper we will consider more general maps than those studied by\nLiao and Seuret and prove results similar to those of the three papers\nmentioned above. We will use a method of statistical nature very\nsimilar to the one used in \\cite{Persson}. The maps we will work with\nare mostly piecewise expanding interval maps, but some of our results\nare valid for more abstract maps with certain statistical properties.\n\nWe will not assume that the maps have a Markov partition. In the case\nthat $\\mu$ is a measure that is absolutely continuous with respect to\nLebesgue measure, we can consider the sets $E(x,r)$ with $r_n =\nn^{-\\alpha}$ for any $\\alpha > 1$. However, for other measures $\\mu$\nwe have to impose extra restrictions on $\\alpha$ and our results are\nonly valid for sufficiently large $\\alpha$. This extra restriction is\nnot present in the works of Fan, Schmeling and Troubetzkoy; Li, Wang,\nWu and Xu; and Liao and Seuret.\n\nThe results of this paper are presented in two main theorems, found in\nSections~\\ref{sec:acim} and \\ref{sec:gibbs}. The first theorem treats\nthe case when $\\mu$ is absolutely continuous with respect to the\nLebesgue measure and no extra restriction is imposed on $\\alpha$. In\nthis case our result is a generalisations of the corresponding result\nby Liao and Seuret and we also prove that for almost all $x$ the set\n$E(x, r)$ has large intersections. This means that the set $E(x,r)$\nbelongs for some $0 < s <1$ to the class $\\mathscr{G}^s$ of\n$G_\\delta$-sets, with the property that any countable intersection of\nbi-Lipschitz images of sets in $\\mathscr{G}^s$ has Hausdorff dimension\nat least $s$. See Falconer's paper \\cite{Falconer} for more details\nabout those classes of sets. The large intersection property was not\nproved in any of the papers \\cite{Fanetal}, \\cite{Lietal} and\n\\cite{LiaoSeuret}.\n\nThe second theorem treats more general measures and is only valid for\nsufficiently large $\\alpha$. Restriction of this type are not present\nin the papers \\cite{Fanetal}, \\cite{Lietal} and \\cite{LiaoSeuret}. We\nhave not been able to prove the large intersection property in this\ncase in the general setting. However, we prove that if the map is a\nMarkov map, then the large intersection property holds.\n\nIn Section~\\ref{sec:examples} we provide explicit examples of maps\nthat satisfy the assumptions of the two main theorems. Most examples\nare for uniformly expanding maps, but we also give some examples with\nnon-uniformly expanding maps.\n\nOne can also study the Hausdorff dimension of the complement of $E\n(x,r)$. That was done both in the paper by Liao and Seuret as well as\nthat by Fan, Schmeling and Troubetzkoy, but we shall not do so in this\npaper.\n\n\n\\section{Maps with Absolutely Continuous Invariant Measures} \\label{sec:acim}\n\nWe will first work with maps $T \\colon [0,1] \\to [0,1]$ satisfying the\nfollowing assumptions.\n\\begin{assumption} \\label{as:acim}\n There exists an invariant measure $\\mu$ that is\n absolutely continuous with respect to Lebesgue measure, with density\n $h$ such that $c_h^{-1} < h < c_h$ holds Lebesgue almost everywhere\n for some constant $c_h > 0$.\n\\end{assumption}\n\\begin{assumption} \\label{as:decay1}\n Correlations decay with summable speed for functions of bounded\n variation: There is a function $p \\colon \\mathbb{N} \\to (0,\\infty)$\n such that if $f \\in L^1$ and $g$ is of bounded variation, then\n \\[\n \\biggl| \\int f \\circ T^n \\cdot g \\, \\mathrm{d} \\mu - \\int f \\,\n \\mathrm{d} \\mu \\int g \\, \\mathrm{d} \\mu \\biggr| \\leq \\lVert f\n \\rVert_1 \\lVert g \\rVert p(n),\n \\]\n where $\\lVert \\psi \\rVert = \\lVert \\psi \\rVert_1 + \\var \\psi$, and\n we assume that the correlations are summable in the sense that\n \\[\n C := \\sum_{n=0}^\\infty p(n) < \\infty.\n \\]\n\\end{assumption}\n\nWe prove the following theorem. The proof is in\nSection~\\ref{sec:absctsproof}.\n\n\\begin{theorem} \\label{the:shrinkingtarget}\n Under the Assumptions~\\ref{as:acim} and \\ref{as:decay1} above,\n $\\dimH E (x, r) \\geq s$ for Lebesgue almost all $x \\in [0,1]$, where\n \\[\n s = \\sup \\{\\, t : \\exists c, \\forall n : n^{-2} \\sum_{j=1}^n\n r_j^{-t} < c \\,\\}.\n \\]\n Moreover, the set $E(x,r)$ belongs to the class $\\mathscr{G}^s$ of\n $G_\\delta$-sets with large intersections for Lebesgue almost all\n $x$.\n\n In particular, if $r_n = n^{-\\alpha}$ then $\\dimH E (x, r) =\n 1\/\\alpha$ for Lebesgue almost all $x \\in [0,1]$.\n\\end{theorem}\n\nIn Section~\\ref{sec:examples} we provide some examples of maps\nsatisfying the assumptions of Theorem~\\ref{the:shrinkingtarget}.\n\n\n\\section{Maps with Gibbs Measures} \\label{sec:gibbs}\n\nWe will now consider a map $T \\colon [0,1] \\to [0,1]$ with a Gibbs\nmeasure $\\mu_\\phi$. Our assumptions are as follows.\n\n\\begin{assumption} \\label{as:piecewise}\n $T$ is piecewise monotone and expanding with respect to a finite\n partition, and there is bounded distortion for the derivative $T'$.\n\\end{assumption}\n\n\\begin{assumption} \\label{as:gibbs}\n The potential $\\phi \\colon [0,1] \\to \\mathbb{R}$ is of bounded\n distortion, and there is a Gibbs measure $\\mu_\\phi$ to the potential\n $\\phi$, with $\\mu_\\phi = h_\\phi \\nu_\\phi$ where $h_\\phi$ is a\n bounded function that is bounded away from zero, and $\\nu_\\phi$ is a\n conformal measure, that is, for any subset $A$ of a partition\n element holds\n \\[\n \\nu_\\phi (T(A)) = \\int_A e^{P(\\phi)-\\phi} \\, \\mathrm{d} \\nu_\\phi,\n \\]\n where $P (\\phi)$ denotes the topological pressure of $\\phi$.\n\\end{assumption}\n\n\\begin{assumption} \\label{as:decay2}\n We have summable decay of correlations for functions of bounded\n variation. That is we assume that there is a function $p \\colon\n \\mathbb{N} \\to (0,\\infty)$ such that if $f \\in L^1 (\\mu_\\phi)$ and\n $g$ is of bounded variation, then\n \\[\n \\biggl| \\int f \\circ T^n \\cdot g \\, \\mathrm{d} \\mu_\\phi - \\int f\n \\mathrm{d} \\mu_\\phi \\int g \\, \\mathrm{d} \\mu_\\phi \\biggr| \\leq\n \\lVert f \\rVert_1 \\lVert g \\rVert p(n),\n \\]\n holds for all $n$, and we assume that\n \\[\n C := \\sum_{n=0}^\\infty p(n) < \\infty.\n \\]\n\\end{assumption}\n\n\\begin{assumption} \\label{as:measureofball}\n There is a number $s_0 > 0$ such that for any $s < s_0$ there is a\n constant $c_s$ such that $\\mu_\\phi (I) \\leq c_s |I|^s$ holds for any\n interval $I \\subset [0,1]$.\n\\end{assumption}\n\n\\begin{remark} \\label{rem:dimension}\n We note that Assumption~\\ref{as:measureofball} implies that\n \\begin{equation} \\label{eq:localenergy}\n \\iint | x - y |^{-t} \\, \\mathrm{d} \\mu_\\phi (x) \\mathrm{d}\n \\mu_\\phi (y) \\leq \\frac{t c_s}{s-t}.\n \\end{equation}\n for any $t < s < s_0$. This follows since, for any $x$, we have\n \\begin{multline} \\label{eq:energyatx}\n \\int |x - y|^{-t} \\, \\mathrm{d} \\mu_\\phi (y) = \\int_1^\\infty\n \\mu_\\phi (B (x, u^{-1\/t})) \\, \\mathrm{d}u \\\\ \\leq \\int_1^\\infty\n c_s u^{-s\/t} \\, \\mathrm{d} u = \\frac{t c_s}{s-t},\n \\end{multline}\n which implies \\eqref{eq:localenergy}.\n\n Note also that \\eqref{eq:localenergy} implies that the lower pointwise\n dimension of $\\mu_\\phi$ is at least $s_0\/2$ at any point in\n $[0,1]$. Indeed, since $|I|^{-s} \\leq |x - y|^{-s}$ holds whenever\n $x, y \\in I$, we have together with \\eqref{eq:localenergy} that\n \\begin{align*}\n |I|^{-s} \\mu_\\phi (I)^2 &\\leq \\iint_{I \\times I} | x - y |^{-s} \\,\n \\mathrm{d} \\mu_\\phi (x) \\mathrm{d} \\mu_\\phi (y) \\\\ & \\leq \\iint |\n x - y |^{-s} \\, \\mathrm{d} \\mu_\\phi (x) \\mathrm{d} \\mu_\\phi (y) =\n c\n \\end{align*}\n holds whenever $s < s_0$. Hence $\\mu_\\phi (I) \\leq \\sqrt{c}\n |I|^{s\/2}$ and the claim follows.\n\\end{remark}\n\nIn this setting we can prove a similar result to\nTheorem~\\ref{the:shrinkingtarget}. The proof of the following theorem\nis in Section~\\ref{sec:gibbsproof}.\n\n\\begin{theorem} \\label{the:gibbs}\n Assume that $T \\colon [0,1] \\to [0,1]$ satisfies the\n Assumptions~\\ref{as:piecewise}, \\ref{as:gibbs}, \\ref{as:decay2} and\n \\ref{as:measureofball}. Then, we have that $\\dimH E(x,r) \\geq s$ for\n $\\mu_\\phi$-almost all $x$, where\n \\[\n s = \\sup \\{\\, t < s_0 : \\exists c, \\forall n : n^{-2} \\sum_{j=1}^n\n r_j^{-t} < c \\,\\}.\n \\]\n In particular, if $r_n = n^{-\\alpha}$ and $\\alpha > 1 \/ s_0$, then\n $\\dimH E (x,r) = 1\/\\alpha$ for $\\mu_\\phi$-almost every $x$.\n\\end{theorem}\n\n\\begin{remark}\n Note that if $\\alpha \\leq 1\/s_0$ then Theorem~\\ref{the:gibbs} gives us\n the result that $\\dimH E (x,r) \\geq s_0$. However, one would expect\n that $\\dimH E(x,r) = 1\/\\alpha$ as long as $1\/\\alpha$ is not larger\n than the dimension of $\\mu_\\phi$, which is the result proved by Liao\n and Seuret in their setting.\n\n As is clear from Remark~\\ref{rem:dimension}, our method cannot work\n for the full range of $\\alpha$, since we rely on\n Assumption~\\ref{as:measureofball}, so that we cannot consider\n $\\alpha$ such that $(2\\alpha)^{-1}$ is larger than the lower\n pointwise dimension of $\\mu_\\phi$ at any point.\n\\end{remark}\n\nIf we also assume that the map is Markov, then we can prove the large\nintersection property of the set $E (x,r)$.\n\n\\begin{theorem} \\label{the:markov}\n Assume that $T \\colon [0,1] \\to [0,1]$ is a Markov map that\n satisfies the Assumptions~\\ref{as:piecewise}, \\ref{as:gibbs},\n \\ref{as:decay2} and \\ref{as:measureofball}. Then, we have that\n $E(x,r) \\in \\mathscr{G}^s$ for $\\mu_\\phi$-almost all $x$, where\n \\[\n s = \\sup \\{\\, t < s_0 : \\exists c, \\forall n : n^{-2} \\sum_{j=1}^n\n r_j^{-t} < c \\,\\}.\n \\]\n\\end{theorem}\n\nIn the next section we give examples of maps satisfying the\nassumptions of Theorem~\\ref{the:gibbs}.\n\n\n\\section{Examples} \\label{sec:examples}\n\n\n\\subsection{Examples for Theorem~\\ref{the:shrinkingtarget}} \\label{ssec:example1}\nThere exist some dynamical systems that obviously satisfy the\nassumptions of Theorem \\ref{the:shrinkingtarget}, for example $n-1$\nexpanding diffeomorphisms of the circle. We are going to present less\nobvious examples of application of our results.\n\nFor instance, the maps studied by Liverani in \\cite{Liverani} satisfy\nthe assumptions of Theorem~\\ref{the:shrinkingtarget}. These maps are\ndefined as follows. Assume that there is a finite partition\n$\\mathscr{P}$ of $[0,1]$ into intervals, such that on every interval\n$I \\in \\mathscr{P}$, the map $T$ can be extended to a $C^2$ map on a\nneighbourhood of the closure of $I$, and assume that there is a\n$\\lambda > 1$ such that $|T'| \\geq \\lambda$ holds everywhere. To put\nit shortly, $T$ is piecewise $C^2$ with respect to a finite partition,\nand uniformly expanding. We assume also that $T$ is weakly covering,\nas defined by Liverani: The map $T$ is said to be weakly covering if\nthere exists an $N_0 \\in \\mathbb{N}$ such that if $I \\in \\mathscr{P}$,\nthen\n\\[\n \\bigcup_{k=0}^{N_0} T^k (I) \\supset [0,1] \\setminus W,\n\\]\nwhere $W$ is the set of points that never hit the discontinuities of\n$T$. Under the assumptions mentioned above, it is shown in\n\\cite{Liverani} that $T$ has an invariant measure $\\mu$ satisfying the\nAssumption~\\ref{as:acim} above, and the correlations decay\nexponentially. Hence they are summable and Assumption~\\ref{as:decay1}\nholds. We therefore have the following corollary.\n\n\\begin{corollary}\n If $T \\colon [0,1] \\to [0,1]$ is piecewise $C^2$ with respect to a\n finite partition, uniformly expanding, and weakly covering, then\n with $r_n = n^{-\\alpha}$, $\\alpha \\geq 1$ we have\n \\[\n \\dimH E (x,r) = \\frac{1}{\\alpha}\n \\]\n and $E (x,r) \\in \\mathscr{G}^{1\/\\alpha}$ for Lebesgue almost every\n $x \\in [0,1]$.\n\\end{corollary}\n\nIn fact, it is not necessary to assume that the map is piecewise\n$C^2$. It is sufficient that the derivative is of bounded variation,\nsince then one can combine the estimates by Rychlik \\cite{Rychlik}\nwith the method of Liverani \\cite{Liverani} to get the same result.\n\nIf the map is piecewise expanding with an indifferent fixed point,\nthen Assumption~\\ref{as:decay1} does not hold. However, as we will see\nbelow, we can still use Theorem~\\ref{the:shrinkingtarget} to get the\nfollowing result.\n\n\\begin{corollary}\n Let $T_\\beta:[0,1)\\to [0,1)$ with $\\beta > 1$ be the\n Manneville--Pomeau map\n \\[\n x \\mapsto \\left\\{ \\begin{array}{ll} x + 2^{\\beta-1} x^{\\beta} & x <\n 1\/2 \\\\ 2x - 1 & x \\geq 1\/2 \\end{array} \\right.\n \\]\n and $r_j = j^{-\\alpha}, \\alpha \\geq 1$. If\\\/ $1 < \\beta < 2$, then\n for Lebesgue almost every $x$ we have that $\\dim_H E(x,r) =\n 1\/\\alpha$ and $E (x,r) \\in \\mathscr{G}^{1\/\\alpha}$. If $\\beta \\geq\n 2$, then for Lebesgue almost every $x$ we have that $\\dimH E(x,r) =\n \\frac{1}{\\alpha (\\beta - 1)}$ and $E (x,r) \\in \\mathscr{G}^{1\/\\alpha\n \/(\\beta - 1)}$.\n\\end{corollary}\n\n\\begin{proof}\n Let $S_\\beta$ be the first return map on the interval $[1\/2, 1)$.\n Then there exists an $S_\\beta$-invariant measure $\\nu$ that is\n absolutely continuous with respect to Lebesgue measure, and $\\nu$\n is ergodic.\n\n Let $R(x)$ be the return time of $x$ to $[1\/2, 1)$, that is, we have\n $T_\\beta^{R(x)} = S_\\beta (x)$.\n\n It the case $1 < \\beta < 2$ we will do as follows. In this case $R$\n is integrable and so, for almost all $x$ there is a constant $c > 0$\n such that\n \\begin{equation} \\label{eq:returntimeestimate}\n n \\leq \\sum_{k=1}^n R_k (x) \\leq c n\n \\end{equation}\n for all sufficiently large $n$. (The lower bound always holds, since\n $R \\geq 1$.) We put\n \\[\n r_j' = (cj)^{-\\alpha} \\qquad \\text{and} \\qquad r_j'' = j^{-\\alpha}.\n \\]\n Then for almost all $x$ we will have that\n \\[\n B (S_\\beta^j (x), r_j') \\subset B (T_\\beta^{\\sum_{k=1}^j R_k (x)}\n (x), r_{\\sum_{k=1}^j R_k (x)}) \\subset B(S_\\beta^j (x), r_j'')\n \\]\n for sufficiently large $j$. Hence, with\n \\begin{align*}\n E' (x,r') &:= \\limsup_{j \\to \\infty} B (S_\\beta^j (x), r_j'), \\\\\n E'' (x,r'') &:= \\limsup_{j \\to \\infty} B (S_\\beta^j (x), r_j''),\n \\end{align*}\n we have\n \\[\n E' (x,r') \\subset E (x,r) \\cap [1\/2,1] \\subset E''(x,r'')\n \\]\n for almost all $x$.\n\n Now, Theorem~\\ref{the:shrinkingtarget} implies that $E' (x,r') \\cap\n [1\/2, 1) \\in \\mathscr{G}^{1\/\\alpha}$ for almost all $x$ and $\\dimH\n E'' (x,r'') = 1\/\\alpha$ for almost all $x$. This implies the\n desired result for $E(x,r) \\cap [1\/2, 1)$.\n\n In the same way we can get the result for $E (x,r) \\cap I_n$ where\n $I_n = [x_n, 1)$, where $x_n$ is the $n$-th pre-image of $1\/2$ with\n respect to the left branch of $T_\\beta$. This concludes the proof\n for the case $1 \\leq \\beta < 2$.\n\n The method above does not quite work when $\\beta \\geq 2$, since then\n $\\int R\\, \\mathrm{d} \\nu = \\infty$, and the upper bound of\n \\eqref{eq:returntimeestimate} fails. However, whenever $\\varepsilon\n > 0$, we have for almost all $x$ that\n \\[\n n^{\\beta - 1 - \\varepsilon} \\leq \\sum_{k = 1}^n R_k (x) \\leq\n n^{\\beta - 1 + \\varepsilon}.\n \\]\n holds for large $n$. The upper bound above follows from\n Theorem~2.3.1 of \\cite{Aaronson}. The lower bound follows using\n Theorem~1 in \\cite{AaronsonDenker}.\n\n We now proceed as in the case $1 < \\beta < 2$. Put\n \\[\n r_j' = (c_2 j)^{-\\alpha(\\beta - 1 + \\varepsilon)} \\qquad \\text{and}\n \\qquad r_j'' = (c_1 j)^{-\\alpha (\\beta - 1 - \\varepsilon)}.\n \\]\n With the same notation as previously we then have that\n \\[\n E' (x,r') \\subset E (x,r) \\cap [1\/2,1] \\subset E''(x,r'')\n \\]\n for almost all $x$.\n\n Theorem~\\ref{the:shrinkingtarget} implies that for almost all $x$\n $E'(x, r') \\cap [1\/2, 1) \\in \\mathscr{G}^{1\/\\alpha\/(\\beta - 1 +\n \\varepsilon)}$ and $\\dimH E''(x, r'') \\cap [1\/2,1) = (\\alpha\n (\\beta - 1 - \\varepsilon))^{-1}$. Since $\\varepsilon > 0$ can be\n chosen arbitrarily small, this implies the result for $E (x,r)\n \\cap [1\/2,1)$. As before, we get the result stated in the\n corollary by considering $E (x,r \\cap I_n)$ in the same way.\n\\end{proof}\n\n\n\\subsection{Examples for Theorem~\\ref{the:gibbs}}\n\nHere we will show that the Assumptions~\\ref{as:decay2} and\n\\ref{as:measureofball} are satisfied for a natural class of\nsystems. Consider a map $T$ which is piecewise $C^2$ with respect to a\nfinite partition, and uniformly expanding, as defined in\nSection~\\ref{ssec:example1}. Then Assumption~\\ref{as:piecewise} is\nsatisfied.\n\nSuppose that $\\phi$ satisfies the assumptions of Liverani, Saussol and\nVaienti in \\cite{Liveranietal}, that is, $e^\\phi$ is of bounded\nvariation and that there exists an $n_0$ such that\n\\begin{equation} \\label{eq:contractingpotential}\n \\sup e^{S_{n_0} \\phi} < \\inf L_\\phi^{n_0} 1,\n\\end{equation}\nwhere $S_{n_0} \\phi = \\phi + \\phi \\circ T + \\cdots + \\phi \\circ\nT^{n_0-1}$ and\n\\[\nL_\\phi f (x) = \\sum_{T(y) = x} e^{\\phi (y)} f(y)\n\\]\nis the transfer operator with respect to the potential $\\phi$. We\nassume moreover that $\\phi$ is piecewise $C^2$ with respect to the\npartition of the map, so that the bounded distortion part of\nAssumption~\\ref{as:gibbs} is satisfied.\n\nFinally, we assume that $T$ is covering, in the sense that for any non\ntrivial interval $I$ there is an $n$ such that $T^n (I) \\supset [0,1]\n\\setminus W$, where $W$ is the set of points that never hit the\ndiscontinuities of $T$. Under these assumptions, there exists a\nunique Gibbs measure $\\mu_\\phi$ and the Assumptions~\\ref{as:gibbs} and\n\\ref{as:decay2} hold, see Theorem 3.1 in \\cite{Liveranietal}. In this\nsetting, Assumption~\\ref{as:measureofball} will also be satisfied.\n\n\\begin{corollary} \n Assume that $T \\colon [0,1] \\to [0,1]$ is piecewise $C^2$ with\n respect to a finite partition, uniformly expanding and covering. If\n $\\phi$ satisfies the assumptions above, then the\n Assumption~\\ref{as:measureofball} is satisfied with\n \\[\n s_0 =\n \\limsup_{m \\to \\infty} \\inf \\frac{S_m \\phi - m P(\\phi)}{-\\log\n |(T^m)'|}.\n \\]\n Hence, if $r_n = n^{-\\alpha}$, $\\alpha > 1\/s_0$, then\n \\[\n \\dimH E(x,r) = \\frac{1}{\\alpha}\n \\]\n for $\\mu_\\phi$-almost every $x \\in [0,1]$.\n\\end{corollary}\n\n\\begin{proof}\n We will rely on the part of Assumption~\\ref{as:gibbs} that says that\n if $A$ is a subset of one of the partition elements, then\n \\begin{equation} \\label{eq:conformal}\n \\nu_\\phi (T(A)) = \\int_A e^{P(\\phi)-\\phi} \\, \\mathrm{d} \\nu_\\phi,\n \\end{equation}\n where $P (\\phi) = \\lim_{n \\to \\infty} n^{-1} \\log \\inf L_\\phi^n 1$\n denotes the topological pressure of $\\phi$. Since there are\n constants $c_1$ and $c_2$ such that $0 < c_1 < h < c_2$, it suffices\n to prove Assumption~\\ref{as:measureofball} for the measure\n $\\nu_\\phi$.\n\n Let $r_0 > 0$ be such that any interval of length $r_0$ intersects\n at most two partition elements. If $r < r_0$ and $I$ is an interval\n of length $r$, then $I$ intersects at most two different partition\n elements and therefore $T(I)$ consists of at most two intervals of\n length at most $r \\sup |T'|$. By \\eqref{eq:conformal}, it follows\n that\n \\[\n \\nu_\\phi (I) \\inf_I e^{P(\\phi) - \\phi} \\leq 2 \\sup_{|I_1| = r \\sup_I\n |T'|} \\nu (I_1).\n \\]\n Hence\n \\[\n \\nu_\\phi (I) \\leq 2 \\sup_I e^{\\phi - P(\\phi)} \\sup_{|I_1| = r \\sup_I\n |T'|} \\nu (I_1).\n \\]\n By induction, we conclude that\n \\[\n \\nu_\\phi (I) \\leq \\bigl(2 \\sup_I e^{\\phi - P(\\phi)} \\bigr)^n,\n \\]\n where $n$ is the largest integer such that $r (\\sup_I |T'|)^n \\leq\n r_0$. Hence we have that there is a constant $C_1$, that does not\n depend on $I$, such that\n \\[\n \\nu_\\phi (I) \\leq C_1 r^{\\theta_1} = C_1 |I|^{\\theta_1}, \\quad\n \\theta_1 = \\frac{\\log 2 + \\log \\sup_I e^{\\phi - P(\\phi)}}{- \\log\n \\sup_I |T'|}\n \\]\n By making the constant $C_1$ sufficiently large, we can ensure that\n the estimate above holds for all intervals $I$, not only those that\n are sufficiently small.\n\n By considering $T^m$ instead of $T$, where $m$ is a positive\n integer, the same argument gives us the existence of a constant $C_m$\n such that\n \\[\n \\nu_\\phi (I) \\leq C_m |I|^{\\theta_m}, \\quad \\theta_m = \\frac{\\log 2\n + \\log \\sup_I e^{S_m \\phi - m P(\\phi)}}{- \\log \\sup_I |(T^m)'|}\n \\]\n holds for any interval $I$.\n\n This shows that we may take $s_0 = \\limsup_{m \\to \\infty} \\inf\n \\frac{S_m \\phi - m P(\\phi)}{-\\log |(T^m)'|}$. The assumption\n \\eqref{eq:contractingpotential} guaranties that $s_0 > 0$.\n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{the:shrinkingtarget}} \\label{sec:absctsproof}\n\nThe proof of Theorem~\\ref{the:shrinkingtarget} will be based on the\nfollowing lemma. It is a special case of Theorem~1 in\n\\cite{PerssonReeve}. We refer to \\cite{PerssonReeve} for a proof.\n\n\\begin{lemma} \\label{lem:frostman}\n Let $E_n$ be open subsets of $[0,1]$, and $\\mu_n$ Borel probability\n measures with support in $E_n$, that converge weakly to a measure\n $\\mu$ that is absolutely continuous with respect to Lebesgue measure\n and with density that is bounded and bounded away for zero. Suppose\n there exists a constant $C$ such that\n \\[\n \\iint |x-y|^{-s} \\, \\mathrm{d} \\mu_n (x) \\mathrm{d} \\mu_n (y) < C\n \\]\n holds for all $n$. Then the set $\\displaystyle \\limsup_{n \\to\n \\infty} E_n$ belongs to the class $\\mathscr{G}^s$ and has\n Hausdorff dimension at least $s$.\n\\end{lemma}\n\nWe will also make use of the following two lemmata.\n\n\\begin{lemma} \\label{lem:energy}\n Let $0 < s < 1$. There is a constant $c_s > 0$ such that if $B_1 =\n B(x_1, r_1)$ and $B_2 = B(x_2,r_2)$ are two balls, then\n \\[\n \\frac{1}{r_1 r_2} \\int_{B_1} \\int_{B_2} |x - y|^{-s} \\, \\mathrm{d}x\n \\mathrm{d}y \\leq c_s \\min \\{|x_1 - x_2|^{-s}, r_1^{-s}, r_2^{-s} \\},\n \\]\n and for any fixed $x_2$, the variation of the function\n \\[\n x_1 \\mapsto \\frac{1}{r_1 r_2} \\int_{B_1} \\int_{B_2} |x - y|^{-s} \\,\n \\mathrm{d}x \\mathrm{d}y,\n \\]\n is less than $2 c_s \\min \\{r_1^{-s}, r_2^{-s}\\}$.\n\\end{lemma}\n\n\\begin{proof}\n This is intuitively clear, but we provide a proof.\n\n We suppose that $r_1 \\geq r_2$. Let\n \\[\n I (x_1,x_2) = \\frac{1}{r_1 r_2} \\int_{B_1} \\int_{B_2} |x - y|^{-s}\n \\, \\mathrm{d}x \\mathrm{d}y.\n \\]\n It is clear that $I$ achieves it's maximal value when $x_1 = x_2$,\n for instance when $x_1 = x_2 = 1\/2$. Then a direct calculation shows\n that there is a constant $c_1$ such that\n \\[\n I (1\/2, 1\/2) \\leq c_1 r_1^{-s}.\n \\]\n Hence $I (x_1,x_2) \\leq c_1 r_1^{-s} = c_1 \\min \\{r_1^{-s}, r_2^{-s}\n \\}$.\n\n Suppose that $|x_1 - x_2| > r_1$. It suffices to show that\n $I (x_1,x_2) \\leq c_2 |x_1 - x_2|^{-s}$ holds for some constant\n $c_2$. By a change of variables, we have that\n \\begin{align*}\n I (x_1, x_2) &= |x_1 - x_2|^{-s} \\int_{-1}^1 \\int_{-1}^1 \\Bigl| 1\n - \\frac{r_1}{|x_1-x_2|} u - \\frac{r_2}{|x_1 - x_2|} v \\Bigr|^{-s}\n \\, \\mathrm{d}u \\mathrm{d}v \\\\ &\\leq 4 |x_1 - x_2|^{-s} \\int_0^1\n \\int_0^1 \\Bigl| 1 - \\frac{r_1}{|x_1-x_2|} u - \\frac{r_2}{|x_1 -\n x_2|} v \\Bigr|^{-s} \\, \\mathrm{d}u \\mathrm{d}v.\n \\end{align*}\n Since $r_1 \/ |x_1 - x_2|$ and $r_2 \/ |x_1 - x_2|$ are not larger\n than $1$, we have that\n \\[\n I (x_1, x_2) \\leq 4 |x_1 - x_2|^{-s} \\int_0^1 \\int_0^1 | 1 - u - v\n |^{-s} \\, \\mathrm{d}u \\mathrm{d}v = c_2 |x_1 - x_2|^{-s}.\n \\]\n We can now conclude that $I (x_1, x_2) \\leq c_s \\min \\{|x_1 -\n x_2|^{-s}, r_1^{-s}, r_2^{-s} \\}$, with $c_s = \\max \\{c_1, c_2 \\}$.\n\n The statement about the variation is now a direct consequence since\n the function\n \\[\n x_1 \\mapsto \\frac{1}{r_1 r_2} \\int_{B_1} \\int_{B_2} |x - y|^{-s} \\,\n \\mathrm{d}x \\mathrm{d}y,\n \\]\n is positive, unimodal and with maximal value at most $c_s \\min\n \\{r_1^{-s}, r_2^{-s}\\}$.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:decay}\n Suppose that $F \\colon [0,1]^2 \\to \\mathbb{R}$ is a continuous and\n non-negative function, and that $D$ and $E$ are constants such that\n for each fixed $x$ the function $f \\colon y \\mapsto F(x,y)$\n satisfies $\\var f \\leq D$ and $\\int f \\, \\mathrm{d} \\mu \\leq\n E$. Then\n \\[\n \\int F(T^n x, x) \\, \\mathrm{d} \\mu (x) \\leq E + (D + E) p(n).\n \\]\n\\end{lemma}\n\n\\begin{proof}\n Let $\\varepsilon > 0$. Let $I_k = [k\/m, (k+1)\/m)$. There is an $m$\n such that if\n \\[\n G (x, y) = \\sum_{k = 0}^{m-1} F (k\/m, y) {1}_{I_k} (x),\n \\]\n where ${1}_{I_k}$ denotes the indicator function on $I_k$, then\n \\[\n | F (x,y) - G(x,y) | < \\varepsilon.\n \\]\n Hence we have\n \\[\n \\biggl| \\int F (T^n x, x) \\, \\mathrm{d} \\mu (x) - \\int G (T^n x, x)\n \\, \\mathrm{d} \\mu (x) \\biggr| < \\varepsilon.\n \\]\n\n For each term $F (k\/m, y) {1}_{I_k} (x)$ in the sum defining $G$, we\n have\n \\begin{multline*}\n \\biggl| \\int F (k\/m, x) {1}_{I_k} (T^n x) \\, \\mathrm{d} \\mu (x) -\n \\int F (k\/m, x) \\, \\mathrm{d} \\mu (x) \\int 1_{I_k} \\, \\mathrm{d}\n \\mu \\biggr| \\\\ \\leq \\mu (I_k) (D + E) p(n).\n \\end{multline*}\n by the decay of correlations. As a consequence, we have\n \\[\n \\int F (k\/m, x) {1}_{I_k} (T^n x) \\, \\mathrm{d} \\mu (x) \\leq E \\mu\n (I_k) + \\mu (I_k) (D + E) p(n).\n \\]\n and so\n \\[\n \\int F (T^n x, x) \\, \\mathrm{d} \\mu (x) \\leq \\varepsilon + \\int G\n (T^n x, x) \\, \\mathrm{d} \\mu (x) \\leq \\varepsilon + E + (D + E)\n p(n).\n \\]\n Let $\\varepsilon \\to 0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{the:shrinkingtarget}]\n Let $B_n (x) = B(T^n x, r_n)$. We consider the sets\n \\[\n V_n (x) = \\bigcup_{k = m(n)}^n B_k (x)\n \\]\n where $m(n)$ is a slowly increasing sequence such that $m(n) < n$\n and $m(n) \\to \\infty$ as $n \\to \\infty$. It then holds that $\\limsup\n V_n (x) = \\limsup B_n (x)$.\n\n We define probability measures $\\mu_{n, x}$ with support in $V_n\n (x)$ by\n \\[\n \\mu_{n,x} = \\frac{1}{n - m(n) + 1} \\sum_{k = m(n)}^n \\lambda_{B_k\n (x)},\n \\]\n where $\\lambda_A$ denotes the Lebesgue measure restricted to the set\n $A$ and normalised so that $\\lambda_A (A) = 1$. It is clear that\n $\\mu_{n,x}$ converges weakly to $\\mu$ as $n \\to \\infty$ for almost\n every $x$.\n\n We shall consider the quantities\n \\[\n I_s (\\mu_{n,x}) = \\iint |y - z|^{-s} \\, \\mathrm{d} \\mu_{n,x} (y)\n \\mathrm{d} \\mu_{n,x} (z).\n \\]\n From the definition of the measure $\\mu_{n,x}$ it follows that\n \\[\n I_s (\\mu_{n,x}) = \\frac{1}{(n - m(n) + 1)^2} \\sum_{i = m(n)}^n\n \\sum_{j = m(n)}^n \\frac{1}{4 r_i r_j} \\int_{B_i} \\int_{B_j} |y -\n z|^{-s} \\, \\mathrm{d}y \\mathrm{d}z,\n \\]\n We now assume that $m(n) < n \/ 2$. Together with\n Lemma~\\ref{lem:energy} we then get that\n \\[\n I_s (\\mu_{n,x}) \\leq \\frac{4 c_s}{n^2} \\sum_{m(n) \\leq i \\leq j \\leq\n n} \\min \\{|T^i x - T^j x|^{-s}, r_i^{-s} \\}.\n \\]\n\n Using that $\\mu$ is $T$-invariant, we can write\n \\[\n \\int I_s (\\mu_{n,x}) \\, \\mathrm{d}\\mu (x) \\leq \\frac{4 c_s}{n^2}\n \\sum_{m(n) \\leq i \\leq j \\leq n} \\int \\min \\{|T^{j -i} x - x|^{-s},\n r_i^{-s} \\wedge r_j^{-s} \\} \\, \\mathrm{d} \\mu (x),\n \\]\n where $a \\wedge b$ denotes the minimum of $a$ and $b$.\n\n An application of Lemma~\\ref{lem:decay} gives that\n \\begin{align*}\n \\int I_s (\\mu_{n,x}) \\, \\mathrm{d}\\mu (x) & \\leq \\frac{1}{n^2}\n \\sum_{m(n) \\leq i \\leq j \\leq n} \\bigl( C_1 + (C_1 + 2 (r_i^{-s}\n \\wedge r_j^{-s}) ) p(j-i) \\bigr) \\\\ & \\leq \\frac{1}{n^2}\n \\sum_{m(n) \\leq i \\leq j \\leq n} C_2 (1 + (r_i^{-s} \\wedge\n r_j^{-s}) p(j-i)) \\\\ & \\leq C_2 + \\frac{C_2}{n^2} \\sum_{j = 1}^n\n \\sum_{i=1}^j (r_i^{-s} \\wedge r_j^{-s}) p (j-i).\n \\end{align*}\n Since $p$ is summable, we can estimate that\n \\[\n \\sum_{j = 1}^n \\sum_{i=1}^j (r_i^{-s} \\wedge r_j^{-s}) p (j-i) \\leq\n \\sum_{j = 1}^n \\sum_{i=1}^j r_j^{-s} p (j-i) = \\sum_{j = 1}^n\n \\sum_{i=0}^{j-1} r_j^{-s} p (i) \\leq C \\sum_{j=1}^n r_j^{-s}.\n \\]\n (This estimate is actually not too rough, since\n \\[\n \\sum_{j = 1}^n \\sum_{i=1}^j (r_i^{-s} \\wedge r_j^{-s}) p (j-i) =\n \\sum_{j = 1}^n \\sum_{i=0}^{j-1} (r_{j-i}^{-s} \\wedge r_j^{-s}) p (i)\n \\geq \\sum_{j = 1}^n r_j^{-s} p(0),\n \\]\n which is of the same order of magnitude if $r_i \\to 0$ as $i \\to \\infty$.)\n\n We conclude that\n \\[\n \\int I_s (\\mu_{n,x}) \\, \\mathrm{d} \\mu (x) \\leq C_2 + \\frac{C\n C_2}{n^2} \\sum_{j=1}^n r_j^{-s},\n \\]\n and this is uniformly bounded for all $n$ if\n \\[\n s < \\sup \\{\\, t : \\exists c, \\forall n : n^{-2} \\sum_{j=1}^n\n r_j^{-t} < c \\,\\}.\n \\]\n\n Suppose $s$ satisfies the inequality above. Then, by Birkhoff's\n ergodic theorem, for $\\mu$-almost all $x$ the measures $\\mu_{n,x}$\n converges weakly to the measure $\\mu$, and, as follows from the\n considerations above, for $\\mu$-almost all $x$, there is a sequence\n $n_k$, with $n_k \\to \\infty$, such that the sequence $(I_s\n (\\mu_{n_k,x}))_{k=1}^\\infty$ is bounded. We can now apply\n Lemma~\\ref{lem:frostman} and conclude that for $\\mu$-almost all $x$\n the set $E(x, r)$ belongs to the class $\\mathscr{G}^s$. This proves\n the first part of Theorem~\\ref{the:shrinkingtarget}.\n\n If $r_n = n^{-\\alpha}$, then it is easy to check that the result\n above gives us that the set $E (x,r)$ belongs to\n $\\mathscr{G}^{1\/\\alpha}$ for almost all $x$. A simple covering\n argument shows that in fact the dimension is not larger than\n $1\/\\alpha$.\n\\end{proof}\n\n\n\\section{Proof of Theorems~\\ref{the:gibbs} and \\ref{the:markov}} \\label{sec:gibbsproof}\n\nAssume that we have a sequence of open sets $E_n$, such that each\n$E_n$ is a finite union of disjoint intervals, and that the diameters\nof these intervals go to zero as $n$ grows. We are first going to\nstudy the Hausdorff dimension of the set $\\limsup E_n$ in the\nfollowing lemmata. The proof of Theorem~\\ref{the:gibbs} will then be\nsimilar to that of Theorem~\\ref{the:shrinkingtarget}, but will instead\nbe based on the lemmata below.\n\n\\begin{lemma} \\label{lem:localfrostman}\n Let $E_n$ be open subsets of $[0,1]$. Suppose there are Borel\n probability measures $\\mu_n$ with support in $E_n$, that converge\n weakly to a measure $\\mu$ that satisfies \\eqref{eq:localenergy}. If\n for some $t < s < s_0$ there is a constant $C$ such that\n \\[\n \\iint |x - y|^{-s} \\, \\mathrm{d} \\mu_n (x) \\mathrm{d} \\mu_n (y) < C\n \\]\n for all $n$, then, whenever $I$ is an interval with\n \\[\n \\iint_{I \\times I} |x - y|^{-t} \\, \\mathrm{d} \\mu (x) \\mathrm{d}\n \\mu (y) < c |I|^{-t} \\mu (I)^2,\n \\]\n there is an $n_I$ such that\n \\[\n \\sum |U_k|^t \\geq \\frac{1}{2c} |I|^t\n \\]\n holds for any cover $\\{U_k\\}$ of $E_n \\cap I$, $n > n_I$.\n\\end{lemma}\n\n\\begin{proof}\n The assumptions implies that for any $t < s$\n \\[\n \\iint_{I \\times I} |x - y|^{-t} \\, \\mathrm{d} \\mu_n (x) \\mathrm{d}\n \\mu_n (y) \\to \\iint_{I \\times I} |x - y|^{-t} \\, \\mathrm{d} \\mu (x)\n \\mathrm{d} \\mu (y),\n \\]\n as $n \\to \\infty$. (See Corollary 2.3 of \\cite{PerssonReeve}.)\n\n For a measure $\\nu$ on $I$ we write $R_t \\nu (x) = \\int |x-y|^{-t}\n \\, \\mathrm{d} \\nu (y)$.\n\n Take an interval $I \\subset [0,1]$ satisfying the assumption of the\n lemma, and define the measure $\\nu_n$ on $I$ by\n \\begin{equation} \\label{eq:nudefinition}\n \\nu_n (A) = \\frac{\\int_A (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d}\n \\mu_n}{\\int_I (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d} \\mu_n},\n \\end{equation}\n where $\\mu_n |_I$ denotes the restriction of $\\mu_n$ to $I$.\n\n There is an $n_I$ such that if $n > n_I$ then\n \\begin{equation} \\label{eq:nuestimate}\n \\nu_n (U) \\leq 2 c \\frac{|U|^t}{|I|^t}\n \\end{equation}\n holds for all intervals $U \\subset I$. This is proved as follows. By\n the definition of $\\nu_n$ the estimate \\eqref{eq:nuestimate} is\n equivalent to\n \\[\n \\frac{1}{|U|^t} \\int_U (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d} \\mu_n \\leq\n \\frac{2 c}{|I|^t} \\int_I (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d} \\mu_n.\n \\]\n We prove the stronger statement that\n \\begin{equation} \\label{eq:Rtintegrals}\n \\frac{1}{|U|^t} \\int_U (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d} \\mu_n \\leq\n 1 \\leq \\frac{2 c}{|I|^t} \\int_I (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d}\n \\mu_n.\n \\end{equation}\n\n The first inequality in \\eqref{eq:Rtintegrals} is proved in\n \\cite{PerssonReeve}. (Use Lemma~2.4 of \\cite{PerssonReeve} and\n approximate with measures that are absolutely continuous with\n respect to Lebesgue.) To prove the second inequality we use Jensen's\n inequality and Assumption~\\ref{as:measureofball} (in particular\n \\eqref{eq:localenergy}) to conclude that\n \\begin{align*}\n \\int_I (R_t \\mu_n |_I)^{-1} \\, \\frac{\\mathrm{d} \\mu_n}{\\mu_n (I)}\n & \\geq \\biggl(\\int_I (R_t \\mu_n |_I) \\, \\frac{\\mathrm{d}\n \\mu_n}{\\mu_n (I)} \\biggr)^{-1} \\\\ &= \\biggl( \\frac{1}{\\mu_n (I)}\n \\iint_{I \\times I} |x-y|^{-t} \\mathrm{d} \\mu_n (x) \\mathrm{d}\n \\mu_n (y) \\biggr)^{-1} \\\\ &\\geq \\frac{1}{\\sqrt 2} \\biggl(\n \\frac{1}{\\mu (I)} \\iint_{I \\times I} |x-y|^{-t} \\mathrm{d} \\mu (x)\n \\mathrm{d} \\mu (y) \\biggr)^{-1} \\\\ &\\geq \\frac{1}{2c}\n |I|^t \\mu_n (I)^{-1},\n \\end{align*}\n provided $n > n_I$ for some $n_I$. Hence\n \\[\n \\frac{1}{|I|^t} \\int_I (R_t \\mu_n |_I)^{-1} \\, \\mathrm{d} \\mu_n\\geq\n \\frac{1}{2c}\n \\]\n and \\eqref{eq:Rtintegrals} follows.\n\n We have now proved \\eqref{eq:nuestimate}, and will use it as\n follows. Suppose that $\\{U_k\\}$ is a cover of $E_n \\cap I$, and $n >\n n_I$. Then\n \\[\n 1 = \\nu_n (\\bigcup_k U_k) \\leq \\sum_k \\nu_n (E_k) \\leq\n \\frac{2 c}{|I|^t} \\sum_k |U_k|^t.\n \\]\n This shows that $\\sum_k |U_k|^t \\geq \\frac{1}{2c} |I|^t$ for\n any cover $\\{U_k\\}$ of $E_n \\cap I$.\n\\end{proof}\n\nIf we would have known that for some constant $c$, the estimate\n\\[\n\\iint_{I \\times I} |x - y|^{-t} \\, \\mathrm{d} \\mu_\\phi (x) \\mathrm{d}\n\\mu_\\phi (y) < c |I|^{-t} \\mu_\\phi (I)^2,\n\\]\nholds for any $I$, then we could have used this to prove that the set\n$E (x, r)$ has a large intersection property, see the proof of\nTheorem~\\ref{the:gibbs}. However, we are unable to prove that such a\nconstant exists, and our strategy is instead to prove that we have\nsuch an estimate for sufficiently many intervals to get the dimension\nresult. The lemma below is what we need.\n\nIf $\\mathscr{Z}$ is the partition with respect to which $T$ is\npiecewise expanding, then the elements of the partition $\\mathscr{Z}\n\\vee T^{-1} \\mathscr{Z} \\vee \\cdots \\vee T^{-n+1} \\mathscr{Z}$\nare called cylinders of generation $n$.\n\n\\begin{lemma} \\label{lem:energyanddistorsion}\n Let $d_0 > 0$ be given and suppose that \\eqref{eq:localenergy} holds\n and that $s < s_0$. Then there is a constant $K = K (d_0)$ such\n that if $I$ is an interval that is a subset of a cylinder of\n generation $n$ and $|T^n (I)| > d_0$, then\n \\begin{equation} \\label{eq:goodinterval}\n \\iint_{I \\times I} |x - y|^{-s} \\, \\mathrm{d} \\mu_\\phi (x)\n \\mathrm{d} \\mu_\\phi (y) < K |I|^{-s} \\mu_\\phi (I)^2.\n \\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n Let\n \\[\n K_0 = \\sup_{|I| > d_0} \\frac{|I|^s}{\\mu_\\phi (I)^2} \\iint_{I \\times\n I} |x - y|^{-s} \\, \\mathrm{d} \\mu_\\phi (x) \\mathrm{d} \\mu_\\phi (y)\n < \\infty.\n \\]\n By the bounded distortion, there exists a constant $K_1$ such that\n \\begin{multline*}\n \\frac{|I|^s}{\\mu_\\phi (I)^2} \\iint_{I \\times I} |x - y|^{-s} \\,\n \\mathrm{d} \\nu_\\phi (x) \\mathrm{d} \\nu_\\phi (y)\\\\ < K_1\n \\frac{|T^n(I)|^s}{\\nu_\\phi (T^n (I))^2} \\iint_{T^n (I) \\times\n T^n (I)} |x - y|^{-s} \\, \\mathrm{d} \\nu_\\phi (x) \\mathrm{d}\n \\nu_\\phi (y),\n \\end{multline*}\n whenever $I$ is an interval contained in a cylinder of generation\n $n$. Since $\\mu_\\phi = h_\\phi \\nu_\\phi$, where $h_\\phi$ is bounded\n and bounded away from zero, the combination of these two estimates\n gives us the desired result.\n\\end{proof}\n\nBy Lemma~\\ref{lem:energyanddistorsion} we know that some particular\nintervals are good, in the sense that we have the estimate\n\\eqref{eq:goodinterval}. We will now use these intervals to construct\na Cantor set $N = \\cap N_n \\subset \\limsup E_n$ with large\ndimension. The following lemma describes the important properties of\nthis construction.\n\n\\begin{lemma} \\label{lem:net}\n Suppose that the assumptions of Lemma~\\ref{lem:localfrostman} hold\n with $\\mu = \\mu_\\phi$, and that \\eqref{eq:localenergy} is satisfied.\n Then, for any $\\varepsilon > 0$, there is a sequence of sets $N_n$\n with the following properties.\n \\begin{enumerate}\n \\addtocounter{enumi}{1}\n \\item[(\\roman{enumi})] All $N_n$ are compact, each $N_n = \\cup\n N_{n,i}$ is a finite and disjoint union of intervals $N_{n,i}$,\n and $N_{n+1} \\subset N_n$. \\addtocounter{enumi}{1}\n \\item[(\\roman{enumi})] There is an increasing sequence $m_n$ such\n that $N_n \\subset E_{m_n}$. \\addtocounter{enumi}{1}\n \\item[(\\roman{enumi})] For any $N_{n,i}$ we have\n \\[\n \\sum |U_k|^t \\geq \\frac{1}{4K} |N_{n,i}|^t,\n \\]\n for any cover $\\{U_k\\}$ of $N_{n,i} \\cap N_{n+1}$.\n \\addtocounter{enumi}{1}\n \\item[(\\roman{enumi})] For any $N_{n,i}$ and $N_{n+1,j}$ we have\n \\[\n \\frac{|N_{n,i}|}{|N_{n+1,j}|} > (4K)^{1\/\\varepsilon}.\n \\]\n \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n By Hofbauer \\cite{Hofbauer}, Lemma 13, we have that if we choose\n $d_0$ sufficiently small, then the Hausdorff dimension of the set of\n points, for which $|T^n I_n (x)| > d_0$ does not hold for infinitely\n many different $n$, is arbitrarily close to $0$. In particular, if\n we choose $d_0$ sufficiently small, then there is a set $A$ of full\n measure such that for any $x \\in A$ there are infinitely many $n$\n with $|T^n I_n (x)| > d_0$.\n\n If $x \\in A$ and $I_n (x)$ has the property that $|T^n I_n (x)| >\n r_0$, then we let $J_{x,n} = I_n (x)$. We denote by $\\mathscr{J}$\n the set of all $J_{x,n}$, that is\n \\[\n \\mathscr{J} = \\{\\, J_{x,n} : x \\in A \\, \\}.\n \\]\n\n We will define the sets $N_n$ inductively as follows. We set $N_0 =\n [0,1]$. Clearly $[0,1]$ satisfies the assumptions of\n Lemma~\\ref{lem:energyanddistorsion}. We let $m_0 = n_{[0,1]}$, where\n $n_{[0,1]}$ is by Lemma~\\ref{lem:localfrostman}.\n\n Suppose that $N_n$ has been defined together with a number $m_n$\n such that for any $N_{n,i}$, Lemma~\\ref{lem:localfrostman} is\n satisfied with $n_{N_{n,i}} \\leq m_n$.\n\n We wish to define $N_{n+1}$. The set $A \\cap N_n$ has full measure\n in $N_n$. Hence, for any $\\varepsilon_n > 0$, we can find a finite\n and disjoint collection $\\mathscr{J}_n \\subset \\mathscr{J}$ such\n that for all $J_{x,n} \\in \\mathscr{J}_n$ we have $|J_{x,n}| <\n \\varepsilon_n$ and $J_{x,n} \\subset E_{m_n}$. Moreover, we can\n choose the collection $\\mathscr{J}_n$ such that for any $N_{n,i}$,\n if $\\mathscr{J}_n'$ denotes the elements of $\\mathscr{J}_n$ that are\n subsets of $N_{n,i}$, then\n \\[\n \\nu_n (\\cup \\mathscr{J}_n') > \\frac{1}{2},\n \\]\n where the measure $\\nu_n$ is defined by \\eqref{eq:nudefinition}. As\n in the proof of Lemma~\\ref{lem:localfrostman}, we can then conclude\n that for any $N_{n,i}$ we have\n \\[\n \\sum |U_k|^t \\geq \\frac{1}{4}\n K^{-1} |N_{n,i}|^t,\n \\]\n for any cover $\\{U_k\\}$ of $N_{n,i}$.\n\n We put $N_{n+1} = \\cup \\mathscr{J}_n$ and $\\{N_{n+1,i}\\} =\n \\mathscr{J}_n$. The number $m_{n+1}$ is taken to be an upper bound\n of $\\{\\, n_I : I \\in J_n \\,\\}$. By taking $\\varepsilon_n$\n sufficiently small we can achieve that\n \\[\n \\frac{|N_{n,i}|}{|N_{n+1,j}|} > (4K)^{1\/\\varepsilon}\n \\]\n holds for any $N_{n,i}$ and $N_{n+1,j}$.\n\n By induction, we now get the sets $N_n$ with the desired properties.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:dimofcantorset}\n With the assumptions and notation of Lemma~\\ref{lem:net} we have\n that $\\dimH N \\geq t - \\varepsilon$, where $N = \\cap N_n$.\n\\end{lemma}\n\n\\begin{proof}\n Consider any countable cover $\\mathscr{U}=\\{U_k\\}$ of the set\n $N$. Since $N$ is compact, we can assume that $\\mathscr{U}$ is a\n finite cover. We will consider the sum\n \\[\n Z_{t-\\varepsilon}(\\mathscr{U}) = \\sum_k |U_k|^{t-\\varepsilon},\n \\]\n trying to prove that it is uniformly bounded away from 0.\n\n Step 1. There exists $n_0$ such that there is a finite cover\n $\\mathscr{U}' = \\{U_k'\\}$ of $N$ such that each intersection $U_k'\n \\cap N$ is a finite union of $N \\cap N_{n_0,i_\\ell}$ and\n \\begin{equation} \\label{eqn:ind1}\n Z_{t- \\varepsilon}(\\mathscr{U}) \\geq \\frac 12 Z_{t-\\varepsilon}\n (\\mathscr{U}').\n \\end{equation}\n This can be done by taking $n_0$ so large that the intervals\n $N_{n_0,i}$ are much smaller than all the (finitely many) elements\n of the cover $\\mathscr{U}$, and then perturb each $U_k$ so that it\n is aligned with the intervals $N_{n_0,i}$.\n\n Step 2. Consider a new cover $\\mathscr{U}''$, obtained in the\n following way. For any $U_k'$, the set $U_k'\\cap N$ must be\n contained in some $N_{n,i}$. There are at most two sets $N_{n+1, j}$\n that intersect $U_k'$ but are not contained in $U_k'$. We replace\n $U_k'$ by at most three open sets: $U_k' \\cap N_{n,i} \\cap N_{n+1,\n j_1}$, $U_k' \\cap N_{n,i} \\cap N_{n+1, j_2}$, and $U_k' \\cap\n N_{n,i} \\setminus \\overline{(N_{n+1, j_1} \\cup N_{n+1, j_2})}$. The\n latter we leave as is, with the former two we repeat the\n procedure. The end result of this procedure: instead of $U_k'$ we\n have a finite family of open sets $U_\\ell''$, each of which contains\n a finite union of $N_{n',i}$ for some $n'$ and does not intersect\n other $N_{n',j}$ (we will call this the {\\it wholeness\n property}). We will call such $U_\\ell''$ a $n'$-th level element.\n\n Note that in this subfamily there will be at most one element of\n level $n$ and at most two elements of each level $n'$, $n0$ for species $i$ it follows that all the\nclassical energy conditions hold \\cite{energy} for that species as\nlong as\n\\begin{equation}\n-\\frac{1}{3} \\leq w_{i} \\leq 1.\\label{energy}\n\\end{equation}\nThe situation is summarized for convenience in Fig \\ref{fig1}.\n\n\\begin{figure}[ht]\n\\epsfig{file=fig1.eps,height=0.5in,width=3in,angle=0}\n\\caption{\\label{fig1}The range in $w_{i}$ is shown (in the box)\nfor which the species satisfies the classical energy conditions.\nHere $M$ stands for dust ($w_{M}=0$) and $R$ for radiation\n($w_{R}=1\/3$).}\n\\end{figure}\n\nIt follows from (\\ref{Cconstant}) that\n\\begin{equation}\n2q-1=3\\sum_{i}w_{i}\\Omega_{i}\\label{q}\n\\end{equation}\nand writing $1+z \\equiv \\frac{a_{0}}{a}$ \\cite{nophotons}, it\nalso follows that\n\\begin{equation}\n\\Omega_{i}=\\frac{(1+z)^{3(1+w_{i})}\\Omega_{io}}{\\sum_{j}\\Omega_{jo}(1+z)^{3(1+w_{j})}}.\\label{OmegaZ}\n\\end{equation}\nAs $1+z \\rightarrow \\infty$ (Big Bang or Big Crunch) it follows\nthat $\\Omega_{i} \\rightarrow 0$ for all species except that with\nthe largest $w_{i}>-1$. Then $\\Omega_{i} \\rightarrow 1$. As $1+z\n\\rightarrow 0$, $\\Omega_{i} \\rightarrow 0$ for all species except\nthat with the smallest $w_{i}\\leq-1$. Then $\\Omega_{i} \\rightarrow\n1$. From (\\ref{q}) and (\\ref{OmegaZ}) it follows that the\nevolution of the universes considered are governed by the\nautonomous (but non-linear) system \\cite{dynamical}\n\\begin{equation}\n\\Omega_{i}^{'}=\\Omega_{i}((2q-1)-3w_{i})\\label{system}\n\\end{equation}\nwhere $^{'}\\equiv -(1+z)\\frac{d}{d(1+z)}$.\n\n\\section{Solving the Friedmann equation}\n\nWe solve the system (\\ref{system}) algorithmically by way of the\nconstruction of a complete set of constants of the motion.\nConsider all species or any subset thereof. To distinguish this\nlatter possibility we replace $i$ by $\\alpha$. A constant of the\nmotion, defined globally over the history of a universe, and not\nwith respect to any particular observer, is defined here to be a\nfunction\n\\begin{equation}\n\\mathcal{F}=\\mathcal{F}(\\Omega_{\\alpha})\\label{function}\n\\end{equation}\nfor which\n\\begin{equation}\n\\mathcal{F}^{\\;'}=0. \\label{constant}\n\\end{equation}\nConsider the product\n\\begin{equation}\n\\mathcal{F}=\n\\prod_{\\alpha}\\Omega_{\\alpha}^{v_{\\alpha}}\\label{product}\n\\end{equation}\nwhere the exponents $v_{\\alpha}$ are constants. Note that we do\nnot make use of (\\ref{omegasum}) at this stage. For\n(\\ref{constant}) to hold for (\\ref{product}), it follows from\n(\\ref{system}) that\n\\begin{equation}\n(2q-1)V=3W\\label{constantmotion}\n\\end{equation}\nwhere\n\\begin{equation}\nV\\equiv\\sum_{\\alpha}v_{\\alpha},\\;\\;\\;\nW\\equiv\\sum_{\\alpha}w_{\\alpha}v_{\\alpha}.\\label{VW}\n\\end{equation}\nIf $V \\neq 0$ then according to (\\ref{constantmotion}) $q$ itself\nis a constant and so from (\\ref{system}) for each species the\nevolution is given by\n\\begin{equation}\n\\Omega_{i}= \\Omega_{io} (1+z)^{1-2q+3w_{i}}. \\label{qconst}\n\\end{equation}\nWe do not consider this case further here. If $q$ is not constant\nthen for $\\mathcal{F}$ of the form (\\ref{product}) to be a\nconstant of the motion it follows from (\\ref{constantmotion}) that\n\\begin{equation}\nV=W=0 \\label{vwconst}\n\\end{equation}\nand the constant itself then reduces to\n\\begin{equation}\n\\mathcal{F}=\\prod_{\\alpha} \\mathcal{C}_{\\alpha}^{v_{\\alpha}}.\n\\label{constantC}\n\\end{equation}\n\nSums, differences, products and quotients of constants of the\nmotion are of course constants as well and so the range in\n$\\alpha$ must be examined. First consider two species, say\n$\\alpha=a, b$. It then follows form (\\ref{vwconst}) that\n$w_{a}=w_{b}$ which is not possible as we require that the species\nbe distinguished by $w$. Next suppose that $\\alpha$ ranges over\nmore than three species. It now follows from (\\ref{vwconst}) that\nany such constant can be reduced to products and quotients of\nconstants involving only three distinct species. As a result we\nneed consider only those constants $\\mathcal{F}$ constructed from\nthree distinct species which we now do.\n\nThe trivial case is that of a universe for which there are only\nthree species, say $a, b,$ and $c$. Then\n$\\mathcal{F}=\\Omega_{a}^{v_{a}}\\Omega_{b}^{v_{b}}\\Omega_{c}^{v_{c}}$\nand since we can always choose $v_{a}$, (\\ref{vwconst}) reduces to\ntwo equations in two unknowns. The constant of motion is then\ndetermined up to the choice in $v_{a}$. For example, with dust we\nhave the species $\\Lambda, M$ and $k$ (assumed $\\neq 0$\n\\cite{two}, the $\\Lambda$CDM model) and it follows immediately\nthat \\cite{alpha}\n\\begin{equation}\n\\mathcal{F}=\\frac{\\Omega_{\\Lambda}\\Omega_{M}^2}{\\Omega_{k}^3}.\n\\label{dust}\n\\end{equation}\nWe can now substitute for $\\Omega_{k}$ from (\\ref{omegasum}).\n\nMore generally, consider $n$ species, $n>3$. If, say, species $a$\nand $b$ are chosen to construct $\\mathcal{F}$ then there are $n-2$\nchoices for the remaining species, the number of independent\nconstants of the motion. Further, for example, there are\n$(n-2)(n-3)\/2$ dependent constants of motion that follow which\ninvolve $a$ but not $b$. In any event, as with the case of only\nthree species, since we can always specify one exponent,\n(\\ref{vwconst}) always reduces to two equations in two unknowns.\n\nAs the foregoing algorithm makes clear, if species are added to a\nmodel, but none taken away, the constants of motion are\n\\textit{inherited} from the simpler model. For example,\nirrespective of what we add to the $\\Lambda$CDM model,\n$\\mathcal{F}$, given by (\\ref{dust}), remains a constant of the\nmotion.\n\nNow consider evolution in a three dimensional subspace, say\n$\\Omega_{a}, \\Omega_{b}, \\Omega_{c}$. We can construct the\nconstant $\\mathcal{F}_{1}=\\Omega_{a}\\Omega_{b}^{v_{b}}\n\\Omega_{c}^{v_{c}}$, and the constant\n$\\mathcal{F}_{2}=\\Omega_{a}^{\\bar{v}_{a}}\\Omega_{b}^{\\bar{v}_{b}}\n\\Omega_{d}$ where $d$ is any species other than $a, b$ or $c$.\nThen with (\\ref{omegasum}) we replace $\\Omega_{d}$. All species\nother than $a,b$ or $c$ that now enter are replace with the aide\nof constants constructed from that species and any two of $a,b$\nand $c$. In the subspace then we have two independent relations\nthat relate $\\Omega_{a}, \\Omega_{b}, \\Omega_{c}$. The intersection\nof these surfaces in the subspace defines the evolution trajectory\nof the associated universe in that subspace.\n\nTo amplify the foregoing, consider, for example, dust and\nradiation. We have the species $\\Lambda, M, R$ and $k$ (assumed\n$\\neq 0$). Following the algorithm outlined above we immediately\nobtain the constants \\cite{rindler}\n\\begin{equation}\n\\mathcal{F}_{1}=\\frac{\\Omega_{\\Lambda}\\Omega_{M}^2}{\\Omega_{k}^3},\\;\\;\\;\\mathcal{F}_{2}=\\frac{\\Omega_{\\Lambda}\\Omega_{R}^3}{\\Omega_{M}^4},\n\\label{rad12}\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{F}_{3}=\\frac{\\Omega_{\\Lambda}\\Omega_{R}}{\\Omega_{k}^2},\\;\\;\\;\\mathcal{F}_{4}=\\frac{\\Omega_{k}\\Omega_{R}}{\\Omega_{M}^2}.\n\\label{rad34}\n\\end{equation}\n\nFirst note that the constant $\\mathcal{F}_{1}$ is\n\\textit{inherited} from the $\\Lambda$CDM model as discussed above.\nMoreover, only two of the above constants are independent. For\nexample, $\\mathcal{F}_{3}=\\mathcal{F}_{1}\\mathcal{F}_{4}$ and\n$\\mathcal{F}_{2}=\\mathcal{F}_{1}\\mathcal{F}_{4}^3$. In, say, the\n$\\Lambda, M, R$ subspace the history of a universe is obtained by\nthe trajectory defined by the intersection of $\\mathcal{F}_{2}$\nwith one of the other constants and with $\\Omega_{k}$ replaced by\n$1-\\Omega_{\\Lambda}-\\Omega_{M}-\\Omega_{R}$ in that constant\n\\cite{further}.\n\nWe end our discussion of constants of the motion here by noting\nthat there are an infinite number of universes, of arbitrary\ncomplexity (but containing the species $\\Lambda, M$ and $k$), for\nwhich\n\\begin{equation}\n\\frac{\\Omega_{k}^3}{\\Omega_{\\Lambda}\\Omega_{M}^2} \\sim 0\n\\label{flatness}\n\\end{equation}\nthroughout the entire history of the universe. The fact that we\nwould appear to live in such a universe is a problem for some\ncosmologists - the ``flatness problem\". We take the view that this\n``problem\" arises only when too few species are considered, in the\nlimit a two species model with only $M$ and $k$, in order to put\nthe issue in perspective.\n\nCentral to the discussion given above is the system\n(\\ref{system}). We now discuss how this system can be used to make\na fundamental distinction between model universes.\n\n\\section{The gravitational epoch function}\n\nIn rigorous texts on general relativity, for example\n\\cite{O'Neill} and \\cite{Sachs}, spacetime is defined (in part) as\na time-oriented manifold. As stated in \\cite{O'Neill},\ntime-oriented is often weakened simply to time-orientable, and as\npointed out in \\cite{Sachs}, the local time orientation is, with\nguesswork, extrapolated to the universe as a whole. Here we take a\ngravitational epoch function to be defined as a scalar field\nconstructed from dimensionless ratios of invariants, each\nderivable from the Riemann tensor without differentiation, such\nthat the function is monotone throughout the history of the\nuniverse. The purpose of an epoch function is to allow the\nordering of events without reference to any specific class of\nobservers or, of course, coordinates. The existence of an epoch\nfunction, when unique, allows the rigorous time-orientation of an\nentire manifold without guesswork. This is why the existence of an\nepoch function is interesting.\n\nIn a conformally flat four dimensional spacetime the maximum\nnumber of independent scalar invariants derivable from the Riemann\ntensor without differentiation is four. These are usually taken to\nbe the Ricci scalar $R$ at degree $d = 1$ and the Ricci invariants\n$r(d-1)$ for $d=2,3,4$. (Degree ($d$) here means the number of\ntensors that go into the construction of the scalar. Let $R_{a b c\nd}$ signify the Riemann tensor, $R_{a}^{b}$ the Ricci tensor, $R$\nthe Ricci scalar ($\\equiv R_{a}^{a}$) and $S_{a}^{b}$ the\ntrace-free Ricci tensor ($ \\equiv R_{a}^{b}-\\frac{R}{4\n}\\delta_{a}^{b} $). The Ricci invariants are defined by (the\ncoefficients are of no physical consequence as they derrive from\nthe spinor forms of the invariants): $r(1) \\equiv \\frac{1}{2^2}\nS_{a}^{b} S^{a}_{b}$, $r(2) \\equiv -\\frac{1}{2^3} S_{a}^{b}\nS_{b}^{c} S_{c}^{a}$, and $r(3) \\equiv \\frac{1}{2^4} S_{a}^{b}\nS_{b}^{c} S_{c}^{d}S_{d}^{a}$. The Kretschmann invariant, for\nexample, is not used since $6R_{a b c d}R^{a b c d} = R^2+48 r(1)$\nin the conformally flat case.) The dimension of these invariants\nis $L^{-2d}$. In a Robertson - Walker spacetime only two of these\ninvariants are independent and in this spacetime it follows that\n$r(1)^2\/r(3)=12\/7, \\,r(1)^3\/r(2)^2=3, \\, R^4\/r(3)=12f^2\/7$ and\n$(R^3\/r(2))^2=3f^3$ where\n\\begin{equation}\nf \\equiv \\frac{R^2}{r(1)}. \\label{ratio}\n\\end{equation}\n It is clear that $f$ given by (\\ref{ratio}), or any\nmonotone function thereof, is the only epoch function that can be\nconstructed in a Robertson - Walker spacetime. Universes can then\nbe fundamentally categorized as to whether or not they admit\nmonotone $f$.\n\nFor convenience consider $\\mathcal{T} \\equiv f\/48$. It follows\ndirectly that in any Robertson - Walker spacetime\n\\begin{equation}\n\\mathcal{T} = (\\frac{\\Omega_{k}+q-1}{\\Omega_{k}-q-1})^2.\n\\label{epoch}\n\\end{equation}\nWhereas $\\mathcal{T}$ is defined through a bounce, its\nrepresentation in the $\\Omega$ notation is not. This is of no\nconcern here since we are concerned with the behaviour of\n$\\mathcal{T}$ strictly prior to any bounce should one exist. With\nthe aide of (\\ref{system}), applied only to the species $k$, it\nalso follows that\n\\begin{equation}\n\\mathcal{T}^{\\;'} = \\frac{4(\\Omega_{k}+q-1)(-2q^2\n\\Omega_{k}+q^{'}(\\Omega_{k}-1) )}{(\\Omega_{k}-q-1)^3}.\n\\label{epochprime}\n\\end{equation}\nIf we now apply the restriction (\\ref{barotropic}) and use the\nfull set of equations (\\ref{system}) it follows that in addition\nto (\\ref{q})\n\\begin{equation}\nq^{\\;'}=\\frac{9}{2}((\\sum_{i}w_{i}\\Omega_{i})^2-\\sum_{i}w_{i}^2\\Omega_{i}).\n\\label{qprime}\n\\end{equation}\n\nLet us again consider the $\\Lambda$CDM model \\cite{rad}. It\nfollows from (\\ref{epoch}) that\n\\begin{equation}\n\\mathcal{T} = (\\frac{\\Omega_{M}+4\n\\Omega_{\\Lambda}}{3\\Omega_{M}})^2 \\label{epochdust}\n\\end{equation}\nand from (\\ref{epochprime}) and (\\ref{qprime}) that\n\\begin{equation}\n\\mathcal{T}^{\\;'} = 8 \\Omega_{\\Lambda}(\\frac{\\Omega_{M}+4\n\\Omega_{\\Lambda}}{3\\Omega_{M}^2}) \\label{epochdustprime}\n\\end{equation}\nand finally from (\\ref{system}) applied to $M$ and $\\Lambda$ that\n\\begin{equation}\n\\mathcal{T}^{\\;''} = 8 \\Omega_{\\Lambda}(\\frac{\\Omega_{M}+8\n\\Omega_{\\Lambda}}{\\Omega_{M}^2}). \\label{epochdustprimeprime}\n\\end{equation}\nAs a result, $\\mathcal{T}$ has a global minimum value of $0$ when\n\\begin{equation}\n\\Omega_{\\Lambda}=-\\frac{\\Omega_{M}}{4}. \\label{epochdustminimum}\n\\end{equation}\nThe question then is, what $\\Lambda$CDM models intersect the locus\n(\\ref{epochdustminimum}) and therefore fail to have $\\mathcal{T}$\nmonotone \\cite{degenerate}?\n\nSince $\\Omega_{M} > 0$ clearly all $\\Lambda$CDM models with\n$\\Lambda > 0$ admit the epoch function (\\ref{epochdust}). Now\nconsider $\\Lambda <0$. For $k=0$ condition\n(\\ref{epochdustminimum}) holds at\n$(\\Omega_{M},\\Omega_{\\Lambda})=(4\/3,-1\/3)$. For $k \\neq 0$ use\n(\\ref{epochdustminimum}) in (\\ref{dust}) to define\n\\begin{equation}\n\\mathcal{F}_{int}=\\frac{16 \\Omega_{M}^3}{(3\\Omega_{M}-4)^3}.\n\\label{dustintersection}\n\\end{equation}\nWith $k=-1$ choose a value of $\\Omega_{M}$ in the range $0 <\n\\Omega_{M} < 4\/3$ and insert this value into\n(\\ref{dustintersection}). The resultant $\\mathcal{F}_{int}$ (which\nis negative) is the constant of motion associated with the\nintersection of the associated integral curve with the locus\n(\\ref{epochdustminimum}) with the intersection taking place at the\nchosen value of $\\Omega_{M}$. For $k=1$ choose a value of\n$\\Omega_{M}$ in the range $4\/3 < \\Omega_{M} < \\infty$ and follow\nthe foregoing procedure ($\\mathcal{F}_{int}$ is now positive).\nHowever, with $k=1$, as $\\Omega_{M} \\rightarrow \\infty$ the\nconstant $\\mathcal{F}_{int}$ reaches the lower bound of $16\/27$.\nThere is then the range $0 < \\mathcal{F} < 16\/27$ for which the\nassociated integral curves do not intersect the locus\n(\\ref{epochdustminimum}). We conclude then that for the\n$\\Lambda$CDM models, all models with $\\Lambda > 0$ admit the epoch\nfunction (\\ref{epochdust}), but for $\\Lambda < 0$ only those with\n$k=1$ and $0 < \\mathcal{F} < 16\/27$ do. There is no evidence that\nour universe lies in this latter category. Indeed, all the current\nevidence would suggest that we are very far away from it.\n\nWe now make some general observations on the monotonicity of\n$\\mathcal{T}$, but restricted to the case $\\Omega_{k}=0$\n\\cite{furthert}. In this case it follows from (\\ref{epochprime})\nthat the monotonicity of $\\mathcal{T}$ is closely associated with\nthe monotonicity of $q$. For example, if we assume that $q$ is\nmonotone decreasing, then we can put a limit on the maximum $w$\nallowed for a monotone $\\mathcal{T}$. To do this note that for\nmodels with a Big Bang the initial value of $q$ is $(3\n\\tilde{w}+1)\/2$ where $\\tilde{w}$ signifies the largest value of\n$w$ (assumed $>-1$) for all species. It follows from\n(\\ref{epochprime}) then that $\\mathcal{T}$ can be monotone only\nfor $\\tilde{w}\\leq 1\/3$. In view of (\\ref{energy}) this is a\nrather surprising restriction, but one which is in accord with all\nobservations.\n\n\\section{Summary}\n\nA complete set of constants of the motion has been constructed for\nall FLRW models consisting of an arbitrary number of separately\nconserved species, each with a constant ratio of pressure to\ndensity. These constants of the motion are to be considered as the\nfundamental characteristics of the model universes. The unique\ncandidate for a gravitational epoch function has been constructed\nfor all FLRW models. In the simplest of all models, the\n$\\Lambda$CDM model, it has been shown that the epoch function\nexists for all models with $\\Lambda > 0$, for no models with\n$\\Lambda = 0$, and for almost no models with $\\Lambda < 0$. This\nfunction allows the global ordering of events without reference to\nany particular class of observers or, of course, coordinates.\n\n\\bigskip\n\n\\begin{acknowledgments}\nIt is a pleasure to thank Nicos Pelavas for comments. This work\nwas supported by a grant from the Natural Sciences and Engineering\nResearch Council of Canada.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe photons that Penzias \\& Wilson (1965) detected coming from all\ndirections with a temperature of about $3\\,$K have traveled freely since their\nlast Thomson scattering, when the Universe became cool enough for the\nions and electrons to form neutral atoms.\nDuring this recombination epoch, the\nopacity dropped precipitously, matter and radiation were decoupled,\nand the anisotropies of the Cosmic Microwave Background radiation (CMB) were\nessentially\nfrozen in. These anisotropies of the CMB have now been detected \non a range of scales (e.g.~White, Scott, \\& Silk 1994;\nSmoot \\& Scott 1998) and developments in the field have been very rapid.\nRecently, two post-{\\sl COBE\\\/}\\ missions, the Microwave Anisotropy Probe\n({\\sl MAP\\\/}) and the {\\sl Planck\\\/}\\ satellite, were approved with the major science goal of\ndetermining\nthe shape of the power spectrum of anisotropies with experimental\nprecision at a level similar to current theoretical predictions.\n\nDetailed understanding of the recombination process is crucial for\nmodeling the power spectrum of CMB anisotropies. Since the seminal work\nof the late 1960s (Peebles 1968; Zel'dovich et al.~1968), several\nrefinements have been introduced by, for example,\nMatsuda, Sato, \\& Takeda (1971), Zabotin \\& Nasel'skii (1982),\nLyubarsky \\& Sunyaev (1983), Jones \\& Wyse (1985), Krolik (1990),\nand others, but in fact little has changed (a fairly comprehensive overview\nof earlier work on recombination is to be found in Section~IIIC of\nHu et al.~1995, hereafter HSSW).\nMore recently, refinements have been made independently in\nthe radiative transfer to calculate secondary spectral distortions\n(Dell'Antonio \\& Rybicki 1993; Rybicki \\& Dell'Antonio 1994),\nand in the chemistry (Stancil, Lepp, \\& Dalgarno 1996b). These\nimprovements may have noticeable effects (at the 1\\% level) on the\ncalculated shapes of the power spectrum of anisotropies. Given the\npotential to measure important cosmological parameters\nwith {\\sl MAP\\\/}\\ and {\\sl Planck\\\/}\\\n(e.g.~Jungman et al.~1996; Bond, Efstathiou, \\& Tegmark~1997;\nZaldarriaga, Spergel, \\& Seljak~1997; Eisenstein, Hu, \\& Tegmark~1998;\nBond \\& Efstathiou~1998) it is of\ngreat interest to make a complete and detailed calculation of the\nprocess of recombination. Our view is that this is in principle such a\nsimple process that it should be so well understood that it could never\naffect the parameter estimation endeavor.\n\nOur motivation is to carry out a `modern' calculation of the cosmic\nrecombination process. All of the physics is well understood, and so it\nis surprising that cosmologists have not moved much beyond the solution of a\nsingle ODE, as introduced in the late 1960s. With today's computing power,\nthere is no need to make the sweeping approximations that were implemented\n30 years ago. We therefore attempt to calculate to as great an extent\nas possible the {\\it full\\\/} recombination problem. The other difference compared with\nthree decades ago is that we are now concerned with high precision\ncalculations, because of the imminent prospect of high fidelity data.\n\nIt is our intention to present here a coupled treatment of the\nnon-equilibrium radiative transfer and the detailed chemistry.\nThe present investigation was motivated by indications (HSSW) that multi-level\nnon-equilibrium effects in H and He, as well as in some molecular\nspecies, may have measurable effects on the power spectrum of CMB\nanisotropies by affecting the low-$z$ and high-$z$ tails\nof the visibility function \n${\\rm e}^{-\\tau}\\slantfrac{{d\\tau}}{{dz}}$ (where $\\tau$ is the optical depth).\n\nTo that effect this paper presents a study of the recombination era by\nevolving neutral and ionized species of H and He, and molecular species of H\nsimultaneously with the matter temperature. We believe our work represents\nthe most accurate picture to date of how exactly the Universe as a whole\nbecame neutral.\n\n\\section{Basic Theory}\n\\subsection{The Cosmological Picture}\n\nWe will assume that we live in a homogeneous, expanding Universe within\nthe context of the canonical Hot Big Bang paradigm. The general picture\nis that at some sufficiently early time the Universe can be regarded as\nan expanding plasma of hydrogen plus some helium, with\naround $10^9$ photons per\nbaryon and perhaps some non-baryonic matter. As it expanded\nand cooled there came a time when the protons were able to keep hold of the\nelectrons and the Universe became neutral. This is the period of cosmic\nrecombination.\n\nIn cosmology it is standard to use redshift $z$ as a time coordinate, so\nthat high redshift represents earlier times. Explicitly, $a(t)=1\/(1+z)$\nis the scale factor of the Universe, normalized to be unity today, and\nwith the relationship between scale factor and time depending on the\nparticular cosmological model. It is sufficient to consider only hydrogen\nand helium recombination, since the other elements exist in minute amounts.\nThe relevant range of redshift is then $\\lesssim10{,}000$, during which the\ndensities are typical of those astrophysicists deal with every day, and\nthe temperatures are low enough that there are no relativistic effects.\n\n\\subsection{The Radiation Field}\n\\label{sec-radfield}\nIn describing the radiation field and its interaction with matter,\nwe must use a specific form of the radiative transfer equation. It\nshould describe how radiation is absorbed, emitted and scattered as it\npasses through matter in a medium which is homogeneous, isotropic,\ninfinite, and expanding. The basic time-dependent form\nof the equation of transfer is\n\\begin{eqnarray} \n\\lefteqn{\\frac{1}{c}\\frac{\\partial{I(\\vec{r},\\hat{n},{\\nu},t)}}{\\partial{t}}\n + \\frac{\\partial{I(\\vec{r},\\hat{n},{\\nu},t)}}{\\partial{l}}=}\\nonumber \\\\\n & & j(\\vec{r},\\hat{n},{\\nu},t) - \\kappa(\\vec{r},\\hat{n},{\\nu},t)\n I(\\vec{r},\\hat{n},{\\nu},t).\n\\label{eq:radtrans}\n\\end{eqnarray} \nHere the symbols are:\n$I(\\vec{r},\\hat{n},{\\nu},t)$ is the specific intensity of\nradiation at\nposition $\\vec{r}$, traveling in direction $\\hat{n}$ (the unit\ndirection vector), with\nfrequency $\\nu$\nat a time $t$ (in units of ergs s$^{-1}$cm$^{-2}$Hz$^{-1}$sr$^{-1}$);\n$l$ is the path length along the ray (and is a coordinate-independent\npath length); $j$ is the emissivity,\nwhich is calculated by summing products of upper excitation state\npopulations and transition probabilities over all relevant processes\nthat can release a photon at frequency $\\nu$, including electron\nscattering; and $\\kappa$ is the absorption coefficient, which is the\nproduct of an atomic absorption cross section and the number density\nof absorbers summed over all states that can interact with photons of\nfrequency $\\nu$.\n \nIn the homogeneous and isotropic medium of the early Universe, we\ncan integrate equation~(\\ref{eq:radtrans}) over all solid angles\n$\\omega$ (i.e.~integrating over the unit direction vector $\\hat{n}$):\n\\begin{eqnarray} \n\\lefteqn{\\frac{4\\pi}{c}\\frac{\\partial{J(\\vec{r},\\nu,t)}}{\\partial{t}} +\n\\vec{\\nabla}{\\cdot}\\vec{F}(\\vec{r},\\nu,t) =} \\nonumber\\\\\n& & \\mbox{} - \\oint\\Big[j(\\vec{r},\\hat{n},{\\nu},t) -\n\\kappa(\\vec{r},\\hat{n},{\\nu},t)I(\\vec{r},\\hat{n},{\\nu},t)\\Big]\\, d\\omega.\n\\label{eq:angleav}\n\\end{eqnarray}\nHere $J(\\nu,t)$ is the mean intensity, the zeroth order moment of\nthe specific intensity over all angles (in units of\nergs ${\\rm s}^{-1}{\\rm cm}^{-2}{\\rm Hz}^{-1}$), $\\vec{F}$ is the flux of\nradiation, which is the net rate of radiant energy flow across an\narbitrarily oriented surface per unit time and frequency interval, and\n$c$ is the speed of light. If\nthe radiation field is isotropic there is a ray-by-ray cancellation\nin the net energy transport across a surface and the net flux is\nzero.\nAlso, because of the isotropy of the radiation field and the medium\nbeing static, we can drop the dependence upon angle of $j$ and\n$\\kappa$ in equation~(\\ref{eq:angleav}).\nWith the definition of $J$,\n$\\oint Idw = 4{\\pi}J$, this simplifies equation~(\\ref{eq:angleav}) to:\n\\begin{equation}\n\\label{eq:rt-2}\n\\frac{1}{c}\\frac{\\partial{J(\\nu,t)}}{\\partial{t}} =\nj(\\nu,t) - \\kappa(\\nu,t)J(\\nu,t).\n\\end{equation}\n\nThe above equation is for a static medium. An isotropically expanding\nmedium would reduce the number\ndensity of photons due to the expanding volume, and reduce their\nfrequencies due to redshifting. The term due to the density change\nwill be simply a $3\\frac{\\dot{a}(t)}{a(t)}$ factor, while the redshifting\nterm will involve the frequency derivative of $J(\\nu,t)$ and hence a \n$\\nu\\frac{\\dot{a}(t)}{a(t)}$ factor.\n\nThen the equation for the evolution\nof the radiation field as affected by the expansion and the sources and\nsinks of radiation becomes\n\\begin{eqnarray}\n\\lefteqn{\\frac{dJ(\\nu,t)}{dt} =\n \\frac{\\partial{J(\\nu,t)}}{\\partial{t}} -\n \\nu H(t)\\frac{\\partial{J(\\nu,t)}}{\\partial{\\nu}}}\\nonumber \\\\\n & & \\quad = -3H(t)J(\\nu,t) + c\\left[j(\\nu,t) - \\kappa(\\nu,t)J(\\nu,t)\\right],\n\\label{eq:radfield}\n\\end{eqnarray}\nwhere $H(t)\\equiv {\\dot{a}}\/a$.\n\nThis equation is in its most general form and difficult to solve; \nfortunately we can make two significant\nsimplifications, because the primary spectral\ndistortions{\\footnote{Not to be confused here with the power spectrum of\nspatial anisotropies.}} are of\nnegligible intensity (Dell'Antonio \\& Rybicki 1993) and the quasi-static\nsolution for spectral line profiles is valid (Rybicki \\& Dell'Antonio 1994).\nThe first simplification is that \nfor the purposes of this paper (in which we do not study secondary spectral\ndistortions),\nwe set $J(\\nu,t)= B(\\nu,t)$, the Planck function, which is observed to\napproximate $J(\\nu)$ to at least 1 part in 10$^4$ (Fixsen et\nal.~1996). Thus\nwe eliminate explicit frequency integration from the simultaneous integration\nof all equations (\\S2.6). The validity of this assumption is shown in\n\\S\\ref{sec-Distortions} where we follow the dominant secondary\ndistortions of H Ly$\\,\\alpha$ and H\n2-photon by including their feedback on the recombination process, and find\nthat the secondary spectral distortions from the other Lyman lines and He are\nnot strong enough to feed back on the recombination process. \n\nThe second simplification is in the treatment of the evolution of the\nresonance lines\n(Ly$\\,\\alpha$, etc.) which must still be treated explicitly -- because\nof cosmological redshifting they cause $J(\\nu,t) \\neq B(\\nu,t)$ in the\nlines. These we call the primary distortions. We use escape \nprobability methods for moving (expanding) media (\\S\\ref{sec-Sobolev} and\n\\ref{sec-3level}).\nThis simplification is not an approximation, but is an exact\ntreatment -- a simple solution to the multi-level radiative transfer\nproblem afforded by the physics of the expanding early Universe.\n\nNot only does using $B(\\nu,t)$ with the escape probability method\ninstead of $J(\\nu,t)$ simplify the \ncalculation and reduce computing time\nenormously, but the effects from following the actual radiation\nfield will be small compared to the main improvements of our\nrecombination calculation which are the level-by-level treatment of H,\n\\ion{He}{1} and \\ion{He}{2},\ncalculating recombination directly, and the correct treatment of\n\\ion{He}{1} triplet and singlet states.\n\n\\subsection{The Rate Equations}\n\\label{sec-species}\nThe species we evolve in the expanding Universe are \\ion{H}{1}, \\ion{H}{2}, \n \\ion{He}{1} , \\ion{He}{2}, \\ion{He}{3}, e${^-}$, H${^-}$, H$_{2}$,\nand H$_{2}{^+}$. \nThe chemistry of the early\nUniverse involves the reactions of association and dissociation among\nthese species, facilitated by interactions with the radiation field,\n$J(\\nu,t)$.\nThe rate equations\nfor an atomic system with $N$ energy levels can be described as\n\\begin{eqnarray}\na(t)^{-3}\\frac{d\\left(n_{i}(t)a(t)^{3}\\right)}{dt}\n \\,{=}&\\Big[n_{\\rm e}(t) n_{\\rm c}(t) P_{{\\rm c}i}\n - n_i(t) P_{i{\\rm c}}\\Big] \\nonumber \\\\\n {+}&\\!\\! \\sum_{j=1}^N \\Big[n_j(t) P_{ji} - n_i(t) P_{ij}\\Big],\n\\end{eqnarray}\nwhere the $P_{ij}$ are the rate coefficients between bound levels $i$ and\n$j$, and the $P_{i{\\rm c}}$ are the rate coefficients between bound levels and\nthe continuum ${\\rm c}$:\n$P_{ij} = R_{ij} + n_{\\rm e} C_{ij}$ and $P_{i{\\rm c}}\n = R_{i{\\rm c}} + n_{\\rm e} C_{i{\\rm c}}$,\nwhere $R$ refers to radiative rates and $C$ to collisional rates.\nHere the $n$s are physical (as opposed to comoving) number\ndensities: $n_i$ refers to number density of the $i$th excited atomic\nstate, $n_{\\rm e}$ to the number density of electrons, and\n$n_{\\rm c}$ to the number density of a continuum particle such as a proton,\n\\ion{He}{2} or \\ion{He}{3}.\n$a(t)$ is the cosmological scale factor. \nThe rate equations for molecules take\na slightly different form because their formation and destruction\ndepends on the rate coefficients for the reactions discussed in\n\\S\\ref{sec-Chem}, \nand molecular bound states are not included.\n\n\\subsubsection{Photoionization and Photorecombination}\n\\label{sec-Recomb}\nBy calculating photorecombination rates $R_{{\\rm c}i}$ directly to each level\nfor multi-level H,\n\\ion{He}{1}, and \\ion{He}{2} atoms, we avoid the problem of finding an accurate\nrecombination coefficient, the choice of which has a large effect on\nthe power spectra (HSSW).\n\nPhotoionization rates are calculated by\nintegrals of the incident radiation field $J(\\nu,t)$ and the bound-free cross\nsection $\\sigma_{i{\\rm c}}(\\nu)$. The photoionization rate \nin s$^{-1}$ is\n\\begin{equation}\n\\label{eq:defRic}\n R_{i{\\rm c}} = 4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}J(\\nu,t)d\\nu.\n\\end{equation}\nHere $i$ refers to the $i$th excited state and\n$c$ refers to the continuum. $\\nu_0$ is the threshold frequency for\nionization from the $i$th excited state. The radiation field\n$J(\\nu,t)$ depends on frequency $\\nu$ and time $t$. With $n_i$ as the\nnumber density of the $i$th excited state, the number of\nphotoionizations per unit volume per unit time (hereafter\nphotoionization rate) is $n_i R_{i{\\rm c}}$. \n\nBy using the principle of detailed balance\nin the case of local thermodynamic equilibrium (LTE), the radiative\nrecombination rate\ncan be calculated from the photoionization rate.\nThen, as described below, the photorecombination\nrate can be generalized to the non-LTE case by scaling the LTE\npopulations with the actual populations and substituting the actual\nradiation field for the LTE radiation field.\nIn LTE the radiation field $J(\\nu,t)$ is the Planck\nfunction $B(\\nu,t)$.\n$B(\\nu,t)$ is a function of time $t$ during recombination because\n$T_{\\rm R} = 2.728 (1 + z(t))\\,$K. We\nwill call the LTE temperature $T$ ($T = T_{\\rm R} = T_{\\rm M}$ at\nearly times), where $T_{\\rm R}$ is the radiation temperature, and\n$T_{\\rm M}$ the matter temperature.\nTo emphasize the Planck function's dependence on temperature,\nwe will use $B(\\nu,T)$ where $T$ is a function of time.\n\nBy detailed balance in LTE we have \n\\begin{displaymath}\n\\label{eq:db1}\n \\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}\\right]^{\\rm LTE} =\n \\left[n_i R_{i{\\rm c}}\\right]^{\\rm LTE}\\!.\n\\end{displaymath}\n\nRadiative recombination includes spontaneous and stimulated\nrecombination, so we must rewrite the above equation as\n\\begin{eqnarray}\n\\label{eq:db2}\n\\lefteqn{\\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}\\right]^{\\rm LTE} =\n \\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}^{\\rm spon}\\right]^{\\rm LTE}\n + \\left[n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}^{\\rm stim}\\right]^{\\rm LTE}} \\\\\n & & = \\left( \\left[n_i R_{i{\\rm c}}\\right]^{\\rm LTE}\n - \\left[n_i R_{i{\\rm c}}^{\\rm stim}\\right]^{\\rm LTE}\\right)\n + \\left[n_i R_{i{\\rm c}}^{\\rm stim}\\right]^{\\rm LTE}. \\nonumber\n\\end{eqnarray}\nUsing the definition of $R_{i{\\rm c}}$ in equation~(\\ref{eq:defRic}), \n\\begin{displaymath}\n \\left[n_{\\rm e} n_{c}R_{{\\rm c}i}\\right]^{\\rm LTE}\\! =\n 4\\pi n_{i}^{\\rm LTE}\\!\\!\\int_{\\nu_{0}}^{\\infty}\\!\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T)\\!\n \\left(1 - {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T}\\right)\\!d\\nu \\nonumber\n\\end{displaymath}\n\\vspace{-3mm}\n\\begin{equation}\n \\qquad\\qquad {+}\\ 4\\pi n_{i}^{\\rm LTE}\\!\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T)\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T}d\\nu.\n\\label{eq:recombLTE}\n\\end{equation}\nThe first term on the right hand side is the spontaneous recombination rate and the second term on the right hand side is the stimulated recombination rate.\nHere $h_{\\rm P}$ is Planck's constant and $k_{\\rm B}$ is Boltzmann's constant.\nThe factor $(1 - {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T})$ is the correction for\nstimulated recombination (see Mihalas 1978, $\\S$4--3 for a derivation of this\nfactor). \nStimulated recombination can be treated as either negative ionization or as\npositive recombination; the physics is the same (see Seager \\& Sasselov,\nin preparation, for some subtleties). \nWith the LTE expression for recombination (equation~(\\ref{eq:recombLTE})),\nit is easy\nto generalize to the non-LTE case, considering spontaneous\nand stimulated recombination separately. Because the matter\ntemperature $T_{\\rm M}$ and the radiation temperature $T_{\\rm R}$\ndiffer at low $z$, it is important to understand how recombination\ndepends on each of these separately.\n\nSpontaneous recombination involves a free electron but its calculation\nrequires no knowledge of the local radiation field,\nbecause the photon energy is derived from the electron's kinetic energy. In\nother words, whether or not LTE is valid, the LTE spontaneous recombination\nrate holds per ion, as long as the velocity distribution is Maxwellian. The\nlocal Planck function (as representing the Maxwell distribution) depends on\n$T_{\\rm M}$, because the Maxwell distribution describes a collisional process.\nFurthermore, since the \nMaxwellian distribution depends on $T_{\\rm M}$, so does the\nspontaneous rate. To get the non-LTE rate, we only have\nto rescale the LTE ion density to the actual ion density:\n\\begin{eqnarray}\n\\lefteqn{n_{\\rm e} n_{\\rm c}R_{{\\rm c}i}^{\\rm spon} =\n 4\\pi \\frac{n_{\\rm e} n_{\\rm c}}{(n_{\\rm e} n_{\\rm c})^{\\rm LTE}}n_{i}^{\\rm LTE}\n\\ \\times}\\nonumber \\\\\n & & \\qquad\\int_{\\nu_{0}}^{\\infty}\\frac{\\sigma_{i{\\rm c}}(\\nu)}\n {h_{\\rm P}\\nu}B(\\nu,T_{\\rm M})\n \\left(1 - {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}\\right)d\\nu\n\\end{eqnarray}\n\\vspace{-3mm}\n\\begin{equation}\n{\\!{=}\\, 4\\pi n_{\\rm e} n_{\\rm c}\n \\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE}\\!\\!\n \\int_{\\nu_{0}}^{\\infty}\\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}\n \\frac{2h_{\\rm P}\\nu^{3}}{c^{2}}\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\label{eq:spon}\n\\end{equation}\n\nTo generalize the stimulated recombination rate from the LTE rate to the\nnon-LTE rate, we rescale the LTE ion density to the\nactual ion density, and replace the LTE\nradiation field by the actual radiation field\n$J(\\nu,t)$ because that is what is `stimulating' the recombination.\nThe correction for stimulated recombination depends on $T_{\\rm M}$,\nbecause the recombination process is collisional; the term always remains\nin the LTE form because equation~(\\ref{eq:recombLTE}) was derived from\ndetailed balance. So we have\n\\begin{eqnarray}\n\\lefteqn{n_{\\rm e} n_{\\rm c} R_{{\\rm c}i}^{\\rm stim} =\n 4\\pi \\frac{n_{\\rm e} n_{\\rm c}}\n{(n_{\\rm e} n_{\\rm c})^{\\rm LTE}}n_{i}^{\\rm LTE} \\times} \\nonumber \\\\\n & & {\\displaystyle \\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}J(\\nu,t)\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\end{eqnarray}\nTherefore, the total non-LTE recombination rate $(R_{{\\rm c}i}^{\\rm spon} +\nR_{{\\rm c}i}^{\\rm stim})$ is\n\\begin{eqnarray}\n\\lefteqn{n_{\\rm e} n_{\\rm c} R_{{\\rm c}i} = n_{\\rm e} n_{\\rm c}\n \\left(\\frac{n{_i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} \\times}\\nonumber \\\\\n & {\\displaystyle 4\\pi\\int_{\\nu_{i}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}}\n + J(\\nu,t)\\right]{\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\end{eqnarray}\nThe LTE population ratios\n$\\left( n_i \/ {n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE}$\ndepend only on $T_{\\rm M}$ through the Saha relation:\n\\begin{equation}\n\\label{eq:saha}\n\\left( \\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} =\n \\left( \\frac{h^{2}}{2\\pi m_{\\rm e}k_{\\rm B}T_{\\rm M}} \\right)^{3\/2}\n \\frac{g_i}{2g_{\\rm c}} {\\rm e}^{E_{i}\/k_{\\rm B}T_{\\rm M}}.\n\\end{equation}\nHere $m_{\\rm e}$ is the electron mass,\nthe atomic parameter $g$ is the degeneracy of the\nenergy level, and $E_{i}$ is the ionization energy of level $i$.\nIn the recombination calculation presented in this paper we \nuse the Planck function $B(\\nu,T_{\\rm R})$ instead of the radiation field\n$J(\\nu,T)$ as described earlier.\n\nIn the early Universe\nCase B recombination is used. This excludes recombinations to the\nground state and considers the Lyman lines to be\noptically thick. An implied assumption necessary to compute the\nphotoionization rate is that the excited states (n~$\\ge 2$) are in equilibrium\nwith the radiation. Our approach is more general than Case B, because\nwe don't consider the Lyman lines to be optically thick and\ndon't assume equilibrium among the excited states. For more details on the\nvalidity of Case B recombination see \\S\\ref{sec-CaseB}.\nTo get the total recombination coefficient, we sum\nover captures to all\nexcited levels above the ground state.\n\nTo summarize, the form of the total photoionization rate is\n\\begin{equation}\n\\label{eq:integral1}\n\\sum_{i>1}^N n_{i}R_{i{\\rm c}} = \\sum_{i>1}^N n_{i}4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T_{\\rm R})d\\nu,\n\\end{equation}\nand the total recombination rate is\n\\begin{eqnarray}\n\\lefteqn{\\sum_{i>1}^N n_{\\rm e} n_{\\rm c} R_{{\\rm c}i}\n = n_{\\rm e}n_{\\rm c}\\sum_{i>1}^N\n \\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} \\times}\\nonumber \\\\\n & \\!\\!\\!{\\displaystyle 4\\pi\\!\\int_{\\nu_{i}}^{\\infty}\n \\frac{\\sigma_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}} + B(\\nu,T_{\\rm R})\\right]\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu.}\n\\label{eq:integral2}\n\\end{eqnarray}\n\n\\subsubsection{Comparison With the `Standard' Recombination\n\\label{sec-StandardRecomb}\nCalculation of Hydrogen}\nThe `standard' recombination calculation refers to the calculation\nwidely used today and first derived by Peebles (1968,\n1993) and Zel'dovich and collaborators (1968, 1983),\nupdated with the most recent parameters and recombination\ncoefficient (HSSW). See also \\S\\ref{sec-3level}.\n\nFor a 300-level H atom in our new recombination calculation, the\nexpressions (\\ref{eq:integral1}) and (\\ref{eq:integral2}) include\n300 integrals at each redshift step.\nThe standard recombination calculation does not go through this\ntime-consuming task but avoids it entirely by using a\npre-calculated recombination coefficient that is a single expression dependent on $T_{\\rm M}$ only.\nThe recombination coefficient to each excited state $i$ is defined by\n\\begin{equation}\n\\alpha_{i}(T_{\\rm M}) = R_{{\\rm c}i}^{\\rm spon}.\n\\end{equation}\nHere $\\alpha$ is a function of $T_{\\rm M}$, because spontaneous recombination\nis a collisional process, as described previously.\nThe total Case B recombination coefficient ($\\alpha_{\\rm B}$) is obtained from\n\\begin{equation}\n\\alpha_{\\rm B}(T_{\\rm M}) = \\sum_{i>1}^{N} \\alpha_{i}(T_{\\rm M}).\n\\end{equation}\nWe will refer to $\\alpha_{\\rm B}(T_{\\rm M})$ as the `pre-calculated\nrecombination coefficient' because the recombination to each atomic level $i$\nand the summation over $i$ are pre-calculated for LTE conditions.\nSee Hummer (1994) for an example of how these recombination\ncoefficients are calculated. \nSome more elaborate derivations of $\\alpha(T_{\\rm M}) = f(T_{\\rm M}, n)$\nhave also been tried (e.g.~Boschan \\& Biltzinger 1998).\n\nThe standard recombination calculation uses a photoionization\ncoefficient $\\beta_{\\rm B}(T_{\\rm M})$ which is derived\nfrom detailed balance using the recombination rate: \n\\begin{equation}\n{(n_i \\beta_i)}^{\\rm LTE} = ({n_{\\rm e} n_{\\rm p} \\alpha_i})^{\\rm LTE}.\n\\end{equation}\nTo get the non-LTE rate, one uses the actual populations $n_i$,\n\\begin{equation}\nn_i \\beta_i(T_{\\rm M}) =\n n_i{\\left(\\frac{n_{\\rm e} n_{\\rm p}}{n_i}\\right)}^{\\rm LTE}\n \\alpha_i(T_{\\rm M}),\n\\end{equation}\nor with the Saha relation (equation~(\\ref{eq:saha})),\n\\begin{equation}\n\\label{eq:beta_saha}\nn_i \\beta_i(T_{\\rm M}) = n_i \\left( \\frac{2\\pi m_{\\rm e}k_{\\rm\nB}T_{\\rm M}}{h_{\\rm P}^2} \\right)^{3\/2}\n\\frac{2}{g_{i}} {\\rm e}^{-E_{i}\/k_{\\rm B}T_{\\rm M}} \\alpha_i(T_{\\rm M}).\n\\end{equation}\nConstants and variables are as described before.\nTo get the total photoionization rate, $\\beta_i$ is summed over all\nexcited levels. Because \nthe `standard' calculation avoids use of all levels $i$\nexplicitly, the $n_i$ are assumed to be in equilibrium with the\nradiation, and thus can be related to the first excited state number\ndensity $n_{2s}$ by the Boltzmann relation,\n\\begin{equation}\n\\label{eq:boltz}\nn_i = n_{2s} \\frac{g_i}{g_{2s}} {\\rm e}^{-(E_2-E_i)\/k_{\\rm B}T_{\\rm M}}.\n\\end{equation}\nWith this relation, the total photoionization rate is\n\\begin{equation}\n\\sum_{i>1}^N n_i \\beta_i = n_{2s} \\alpha_{\\rm B}\n {\\rm e}^{-E_{2s}\/k_{\\rm B}T_{\\rm M}}\n \\left( \\frac{2\\pi m_{\\rm e}k_{\\rm B}T_{\\rm M}}{h_{\\rm P}^2} \\right)^{3\/2} \n \\equiv n_{2s} \\beta_{\\rm B}.\n\\end{equation}\nIn this expression for the total photoionization rate, the excited\nstates are populated according to a Boltzmann distribution. $T_{\\rm M}$\nis used instead of $T_{\\rm R}$, because the Saha and Boltzmann\nequilibrium used in the derivation are collisional processes. The\nexpression says nothing about the excited levels being in equilibrium \nwith the continuum, because the actual values of $n_{\\rm e}$, $n_1$ and\n$n_{2s}$ are used, and the $n_i$ are proportional to $n_{2s}$.\n\nTo summarize, the standard calculation uses a single expression for \neach of the total recombination rate and the total photoionization rate\nthat is dependent on\n$T_{\\rm M}$ only. The\ntotal photoionization rate is\n\\begin{eqnarray}\n\\lefteqn{\\sum_{i>1}^N n_i R_{i{\\rm c}} = n_{2s} \\beta_{\\rm B}(T_{\\rm M})}\n \\nonumber \\\\\n & = n_{2s} \\alpha_{\\rm B}(T_{\\rm M}) {\\rm e}^{-E_{2s}\/k_{\\rm B}T_{\\rm M}}\n \\left(2\\pi m_{\\rm e} k_{\\rm B} T_{\\rm M} \\right)^{3\/2}\/h_{\\rm P}^3,\n\\label{eq:Betatot}\n\\end{eqnarray}\nand the total recombination rate is\n\\begin{equation}\n\\sum_{i>1}^N n_{\\rm e} n_{\\rm p} R_{{\\rm c}i} = n_{\\rm e} n_{\\rm p}\n \\alpha_{\\rm B}(T_{\\rm M}).\n\\end{equation}\nComparing the right hand side of equation~(\\ref{eq:Betatot}) to our\nlevel-by-level total photoionization rate (equation~(\\ref{eq:integral1})),\nthe main improvement in our method over the standard one is clear:\nwe use the actual excited level populations $n_i$,\n{\\it assuming no equilibrium distribution among them}. In this way\nwe can test the validity of the equilibrium assumption.\nFar less important is that the standard recombination treatment cannot\ndistinguish\nbetween $T_{\\rm R}$ and $T_{\\rm M}$, even though photoionization and\nstimulated recombination are functions of \n$T_{\\rm R}$ while spontaneous recombination is a function of $T_{\\rm\nM}$, as shown in\nequations~(\\ref{eq:integral1}) and~(\\ref{eq:integral2}). The\nnon-equilibrium of excited states is important at the \n$10\\%$ level in the residual ionization fraction for $z\\lesssim800$, while using $T_{\\rm R}$ in\nphotoionization and photoexcitation is only important at the few percent\nlevel for $z\\lesssim300$ (for typical cosmological models). \nNote that although the pre-calculated recombination coefficient\nincludes spontaneous recombination only,\nstimulated recombination (as a function of $T_{\\rm M}$)\nis still included as negative\nphotoionization via detailed balance (see\nequation~(\\ref{eq:db2})). \n\n\\subsubsection{Photoexcitation}\n\\label{sec-Sobolev}\n\nIn the expanding Universe, redshifting of the photons must be taken\ninto account (see equation~(\\ref{eq:radfield})). Line photons emitted\nat one position may be redshifted \nout of\ninteraction frequency (redshifted more than the width of the line) by\nthe time they reach another position in the \nflow.\nWe use the Sobolev escape probability to account for this, a method\nwhich was first used for the expanding Universe by Dell'Antonio and\nRybicki (1993). \nThe Sobolev escape probability (Sobolev 1946), also sometimes called the\nlarge-velocity gradient approximation, is not an approximation but is\nan exact, simple solution to the multi-level radiative transfer in the\ncase of a large velocity gradient. It is this solution which allows\nthe explicit inclusion of the line distortions to the radiation field\n-- without it our detailed approach to the recombination problem would\nbe intractable. \nWe will call the net bound-bound rate for each line transition $\\Delta\nR_{ji}$, where $j$ is the upper level and $i$ is the lower level: \n\\begin{equation}\n\\label{eq:delrp}\n\\Delta R_{ji} = p_{ij} \\left\\{n_j\\left[A_{ji} + B_{ji} B(\\nu_{ij}, t)\\right]\n - n_i B_{ij} B(\\nu_{ij},t)\\right\\}.\n\\end{equation}\nHere the terms $A_{ji}, B_{ji}, B_{ij}$ are the Einstein coefficients;\nthe escape probability $p_{ij}$ is the probability that photons associated\nwith this transition will `escape' without being further scattered or absorbed.\nIf $p_{ij} = 1$ the photons produced in the line transition escape to\ninfinity -- they contribute no\ndistortion to the radiation field. If $p_{ij} = 0$ no photons\nescape to infinity; all of them get reabsorbed and the line is\noptically thick. This is the case of primary distortions to the radiation\nfield, and the Planck function cannot be used for the line radiation.\nIn general $p_{ij} \\ll 1$ for the Lyman lines and\n$p_{ij} = 1$ for all other line transitions. With this method we have\ndescribed the redshifting of photons through the resonance lines and\nfound a simple solution to the radiative transfer problem for all\nbound-bound transitions. The rest of this section is devoted to deriving\n$p_{ij}$. \n\nFor the case of no cosmological redshifting, the radiative rates per\ncm$^{3}$ for transitions between\nexcited states of an atom are\n\\begin{eqnarray}\n\\label{eq:bbup}\nn_{i}R_{ij} &=& n_{i}B_{ij} \\overline{J} \\\\\n\\label{eq:bbdown}\n{\\rm and}\\quad n_{j}R_{ji} &=& n_{j}A_{ji} + n_{j}B_{ji}\\overline{J},\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{eq:Jbar}\n\\overline{J} = \\int_{0}^{\\infty}J(\\nu,t)\\phi(\\nu)d\\nu,\n\\end{equation}\nand $\\phi(\\nu)$ is the line profile function with its area normalized by\n\\begin{equation}\n\\label{eq:normalize}\n \\int_{0}^{\\infty}\\phi(\\nu)d\\nu = 1.\n\\end{equation}\nThe line profile $\\phi(\\nu)$ is taken to be a Voigt function that includes\nnatural and Doppler broadening. In principle, equation~(\\ref{eq:Jbar})\nis the correct approach. In practise we take $\\phi(\\nu)$ as a delta\nfunction, and use $J(\\nu,t)$ instead of $\\overline{J}$. The smooth\nradiation field is essentially constant over the width of the line and \nso the line shape is not important; we get the same results using\n$\\overline{J}$ or $J(\\nu,t).$\n\nThe Sobolev escape probability considers\nthe distance over which the expansion of the medium\ninduces a velocity difference equal to the thermal velocity (for the\ncase of a Doppler width): $L = v_{\\rm th}\/|{v^{\\prime}}|$,\nwhere $v_{\\rm th}$ is the\nthermal velocity width and $v^{\\prime}$ the velocity gradient. The\ntheory is valid when this distance $L$ is much smaller than typical\nscales of macroscopic variation of other quantities.\n\nWe follow Rybicki (1984) in the derivation of the\nSobolev escape probability. The general definition of escape\nprobability is given by the exponential extinction law,\n\\begin{equation}\np_{ij} = \\exp[-\\tau(\\nu_{ij})],\n\\end{equation}\nwhere $\\nu_{ij}$ is the frequency for a given line transition, and\n$\\tau(\\nu_{ij})$ is the monochromatic optical depth forward along a\nray from a given point to the boundary of the medium. Here $\\tau(\\nu_{ij})$\nis defined by\n\\begin{equation}\nd\\tau(\\nu_{ij}) = -{\\tilde k} \\phi(\\nu_{ij})dl,\n\\end{equation}\nwhere ${\\tilde k}$ is the integrated line absorption coefficient, so that the\nmonochromatic absorption coefficient or opacity is\n$\\kappa={\\tilde k}\\phi(\\nu_{ij})$, and $l$ is the distance along the ray\nfrom the emission point ($l=0$).\nRewriting the optical depth for a line profile function\n(which has units of inverse frequency) of the dimensionless\nfrequency variable $x = (\\nu - \\nu_{ij}) \/ \\Delta $,\nwith $\\Delta$ the width of the line in Doppler units,\nand $\\nu_{ij}$ the central line frequency,\nwe have\n\\begin{equation}\nd\\tau(\\nu_{ij}) = -\\frac{\\tilde k}{\\Delta} \\phi(x)dl.\n\\end{equation}\nHere $\\kappa$ is the absorption coefficient (defined in\nequation~(\\ref{eq:radtrans})),\nwith ${\\tilde k}$ just dividing out the line profile\nfunction, and\n\\begin{equation}\n{\\tilde k} = \\frac{h_{\\rm P}\\nu}{4\\pi}(n_iB_{ij} - n_jB_{ji}).\n\\end{equation}\nUsing the Einstein relations\n$g_i B_{ij} = g_j B_{ji}$ and \n$A_{ji} = (2 h \\nu^3\/c^2) B_{ji}$\nthe absorption coefficient can also be written as\n\\begin{equation}\n{\\tilde k} = \\frac{A_{ji}\\lambda_{ij}^2}{8\\pi}\n \\left(n_i\\frac{g_j}{g_i} - n_j\\right).\n\\end{equation}\n\nNote that distances along a ray $l$ correspond to shifts in\nfrequency $x$. This is due to the Doppler effect induced by the velocity\ngradient and is the essence of the Sobolev escape probability approach.\nFor example, the Ly$\\,\\alpha$ photons cannot be reabsorbed in the\nLy$\\,\\alpha$ line if they redshift out of the frequency interaction range.\nThis case will happen at some frequency $x$, or at some distance $l$\nfrom the photon emission point, where because of the expansion the\nphotons have redshifted out of the frequency interaction range.\nFor an expanding medium with a constant velocity gradient $v^\\prime = dv\/dl$,\nthe escape probability along a ray is then\n\\begin{equation}\np_{ij} = \\exp\\left[-\\frac{\\tilde k}{\\Delta} \\int_{0}^{\\infty}\\!\n \\phi(x - l\/L)dl \\right] \\equiv \\exp\n \\left[-\\tau_{\\rm S}\\! \\int_{-\\infty}^{x}\\!\\phi({\\tilde x})d{\\tilde x} \\right].\n\\end{equation}\nThe velocity field has in effect introduced an intrinsic escape mechanism\nfor photons; beyond the interaction limit with a given atomic transition, the photons can no longer be absorbed\nby the material, even if it is of infinite extent, but escape\nfreely to infinity (Mihalas 1978).\nHere the Sobolev optical thickness along the ray is defined by\n\\begin{equation}\n\\label{eq:Tau}\n\\tau_{\\rm S} \\equiv \\frac{\\tilde k}{\\Delta}L,\n\\end{equation}\nwhere L is the Sobolev length,\n\\begin{equation}\nL = v_{\\rm th}\/|{v^{\\prime}}| =\n \\left.\\sqrt{\\frac{3k_{\\rm B} T_{\\rm M}}{m_{\\rm atom}}}\\right\/|{v^{\\prime}}|,\n\\end{equation}\nand $\\Delta$ is the width of the line, which in the case of Doppler\nbroadening is\n\\begin{equation}\n\\Delta = \\frac{\\nu_0}{c}\\sqrt{\\frac{3k_{\\rm B} T_{\\rm M}}{m_{\\rm atom}}}.\n\\end{equation}\nWith these definitions, equation~(\\ref{eq:Tau}) becomes\n\\begin{equation}\n\\tau_{\\rm S} = \\frac{\\lambda_{ij} {\\tilde k}}{|{v^{\\prime}}|}.\n\\end{equation}\nIn the expanding Universe, the velocity gradient $v^{\\prime}$ is given by the\nHubble expansion rate $H(z)$, and using the above definition for ${\\tilde k}$,\n\\begin{equation}\n\\label{eq:sobtau}\n\\tau_{\\rm S} = \\frac{A_{ji} {\\lambda_{ij}}^3 \\left[n_i (g_j\/g_i) - n_j\\right]}\n {8\\pi H(z)}.\n\\end{equation}\n \nTo find the Sobolev escape probability for the ray, we average over the\ninitial frequencies $x$, using the line profile function $\\phi(x)$\nfrom equation~(\\ref{eq:normalize}),\n\\begin{displaymath}\np_{ij} =\n\\int_{-\\infty}^{\\infty}\\!dx\\phi(x)\\exp\\left[-\\tau_{\\rm S}\\!\n\\int_{-\\infty}^{x}\\!\\phi({\\tilde x})d{\\tilde x} \\right]\\! =\\!\n\\int_{0}^{1}\\!d\\zeta\\,\\exp(-\\tau_{\\rm S}\\zeta), \\\\\n\\end{displaymath}\n\\begin{equation}\n\\label{eq:indofphi}\n{\\rm i.e.}\\quad p_{ij} = \\frac{1 - \\exp(-\\tau_{\\rm S})}{\\tau_{\\rm S}}.\n\\end{equation}\nNote that this expression is independent of the line profile shape $\\phi(x)$.\nThe escape probability $p_{ij}$ is defined as a frequency average at a\nsingle point.\nFinally, we must average over angle, but in the case of the isotropically \nexpanding Universe, the angle-averaged Sobolev escape probability\ntakes the same form as $p_{ij}$ above.\nFor further details on the Sobolev escape probability see Rybicki\n(1984) or Mihalas (1978), \\S14.2. \n \nHow does the Sobolev optical depth relate to the usual meaning of\noptical depth? The optical depth for a specific\nline at a specific redshift point is equivalent to the Sobolev optical\ndepth,\n\\begin{equation}\nd\\tau(\\nu_{ij}) = -\\tau_{\\rm S} \\phi(x)\\frac{dl}{L}.\n\\end{equation}\nIf no other line or continua photons are redshifted into that\nfrequency range before or after the redshift point, then the optical\ndepth at a given frequency \ntoday (i.e.~summed over all redshift points) will be equivalent to the\nSobolev optical depth at that past point. Generally, behaviors in frequency\nand space are interchangeable in a medium with a velocity gradient.\n \nIn order to derive an expression for the bound-bound rate\nequations, we must consider the mean radiation field $\\overline{J}$ in\nthe line. For the case of spectral distortions $\\overline{J}$ does not\nequal the Planck function at the line frequency. We \nuse the core saturation method (Rybicki 1984) to get $\\overline{J}$ using\n$p_{ij}$; from this we get\nthe net rate of deexcitations in that transition ($j\\rightarrow i$), \ngiven in equation~(\\ref{eq:delrp}).\nIn general, only the Lyman lines of H and \\ion{He}{2}, and the\n\\ion{He}{1} n$^1$p--1$^1$s lines have $p_{ij} < 1$.\nWith this solution we have accomplished two things: described the\nredshifting of photons through the resonance lines, and found a simple\nsolution to the radiative transfer problem for all bound-bound lines.\n\nPeculiar velocities during the recombination era may cause line\nbroadening of the same order of magnitude as thermal broadening over\ncertain scales (A.~Loeb, private communication).\nBecause we use $J(\\nu) = B(\\nu,T)$, the\nradiation field is essentially constant over the width of the line and\nso the line shape is \nnot important. Similarly, the peculiar velocities will not affect the\nSobolev escape probability because it is independent of line shape\n(equation~(\\ref{eq:indofphi})). If \npeculiar velocities were angle dependent there would be an effect on the\nescape probability which is an angle-averaged function.\nOnly for computing spectral distortions to the CMB, where the line\nshape is important, should\nline broadening from peculiar velocities be included.\n\n\\subsubsection{Chemistry}\n\\label{sec-Chem}\nHydrogen molecular chemistry has been included because\nit may affect the residual electron densities at low redshift ($z<200$).\nDuring the recombination epoch, the H$_2$ formation reactions include\nthe H$^-$ processes\n\\begin{equation}\n{\\rm H} + {\\rm e}^{-} \\longleftrightarrow {\\rm H}{^-} + \\gamma,\n\\end{equation}\n\\begin{equation}\n{\\rm H}^{-} + {\\rm H} \\longleftrightarrow {\\rm H}_{2} + {\\rm e}^{-},\n\\end{equation}\nand the $\\mathrm{H_2}^+$ processes\n\\begin{equation}\n{\\rm H} + {\\rm H}^{+} \\longleftrightarrow {{\\rm H}_2}^{+} + \\gamma,\n\\end{equation}\n\\begin{equation}\n{{\\rm H}_2}^{+} + {\\rm H} \\longleftrightarrow {\\rm H}_2 + {\\rm H}^{+},\n\\end{equation}\ntogether with\n\\begin{equation}\n{\\rm H} + {\\rm H}_2 \\longleftrightarrow {\\rm H} + {\\rm H} + {\\rm H},\n\\end{equation}\nand\n\\begin{equation}\n{\\rm H}_2 + {\\rm e}^{-} \\longleftrightarrow {\\rm H} + {\\rm H} + {\\rm e}^{-}.\n\\end{equation}\n(see~Lepp \\& Shull 1984). The direct three-body process for H$_2$ formation\nis significant only at much higher densities.\nThe most recent rate coefficients and cross sections are listed with\ntheir references in Appendix A (see also Cen~1992; Puy et al.~1993;\nTegmark et al.~1997; Abel et al.~1997; Galli \\& Palla~1998). \n\nWe have not included the molecular chemistry of Li, He, or D. \nIn general, molecular chemistry only becomes important\nat values of $z < 200$. Recent detailed analyses of H, D, and He chemistry\n(Stancil et al.~1996a) and of Li and H chemistry (Stancil et al.~1996b;\nStancil \\& Dalgarno~1997)\npresented improved relative abundances\nof all atomic, ionic and molecular species. The values are certainly too small\nto have any significant effect on the CMB power spectrum.\n\nThere are two reasons for this. The visibility function is\n${\\rm e}^{-\\tau}\\slantfrac{{d\\tau}}{{dz}}$, where $\\tau$ is the optical depth.\nThe main component of the optical depth is Thomson scattering by\nfree electrons. Because the populations of Li, LiH, HeH$^+$, HD, and\nthe other species are so small relative to the electron density,\nthey do not affect the\ncontributions from Thomson scattering. Secondly, these atomic species\nand their molecules themselves make no contribution to the visibility\nfunction because they have no strong opacities. HD has\nonly a weak dipole moment. And while LiH has a very strong dipole moment,\nits opacity during this epoch is expected to be\nnegligible because of its tiny ($< 10^{-18}$) fractional abundance (Stancil,\net al.~1996b). Similarly, the fractional abundance of\nH$_2\\mathrm{D}^+$ ($< 10^{-22}$) is too small to have an effect on\nthe CMB spectrum (Stancil et al.~1996a). For more details,\nsee Palla et al.~(1995) and Galli \\& Palla (1998).\nAn interesting additional point is that \nbecause of the smaller energy gap between n=2 and the continuum\nin \\ion{Li}{1}, it actually recombines at a slightly lower redshift than\nhydrogen does (see e.g.~Galli \\& Palla 1998). Of course this has\nno significant cosmological effects.\n\nWe have also excluded atomic D from the calculation. D, like H,\nhas an atomic opacity much lower\nthan Thomson scattering for the recombination era conditions. D parallels H\nin its reactions with electrons and protons, and recombines in the same\nway and at the same time as H (see Stancil et al.~1996a). Although\nthe abundance of D is small ([D\/H] $\\simeq 10^{-5}$), its Lyman\nphotons are still trapped because\nthey are shared with hydrogen. This is seen, for example, by the\nratio of the isotopic shift of D Ly$\\,\\alpha$ to the width of the H\nLy$\\,\\alpha$ line, on\nthe order of $10^{-2}$. Therefore by excluding D we expect no change in the\nionization fraction, and hence none in the visibility function.\n\nWhile the non-hydrogen chemistry is still extremely important for\ncooling and triggering the collapse of primordial gas clouds, it is\nnot relevant for CMB power spectrum observations at the level measurable by\n{\\sl MAP\\\/}\\ and {\\sl Planck\\\/}.\n\n\\subsection{Expansion of the Universe}\nThe differential equations in time for the number densities and matter\ntemperature must be\nconverted to differential equations in redshift by multiplying by a\nfactor of $dz\/dt$ . The redshift $z$ is related to time by the\nexpression\n\\begin{equation}\n \\frac{dz}{dt} = -(1+z) H(z),\n\\end{equation}\nwith scale factor\n\\begin{equation}\n a(t) = \\frac{1}{1 + z}.\n\\end{equation}\nHere $H(z)={\\dot a}\/a$ is the Hubble factor\n\\begin{eqnarray}\n\\lefteqn{H(z)^2 = H_0^2 \\Big[\\Omega_0(1+z)^4\/(1+z_{\\rm eq}) }\n \\nonumber \\\\\n & & \\qquad \\mbox{} + \\Omega_0(1+z)^3 + \\Omega_K(1+z)^2 + \\Omega_{\\Lambda}\\Big],\n\\end{eqnarray}\nwhere $\\Omega_0$ is the density contribution, $\\Omega_K$ is the\ncurvature contribution and $\\Omega_{\\Lambda}$ is\nthe contribution associated with the cosmological constant,\nwith $\\Omega_0 + \\Omega_{\\rm R} + \\Omega_K + \\Omega_{\\Lambda} = 1$,\nand $\\Omega_{\\rm R}=\\Omega_0\/(1+z_{\\rm eq})$.\nHere $z_{\\rm eq}$ is the redshift of matter-radiation equality,\n\\begin{equation}\n 1+z_{\\rm eq} = \\Omega_0\\frac{3(cH_{0}){^2}}{8\\pi G(1+f_{\\nu})U},\n\\end{equation}\nwith $f_{\\nu}$ the\nneutrino contribution to the energy density in relativistic species\n($f_\\nu\\simeq0.68$ for three massless neutrino types),\n$G$ the gravitational constant, $U$ the photon energy density, and\n$H_{0}$ the Hubble constant today, which will be written as\n$100\\,h\\,{\\rm km}\\,{\\rm s}^{-1}{\\rm Mpc}^{-1}$.\nSince we are interested in redshifts $z\\sim z_{\\rm eq}$,\nit is crucial to include the radiation contribution explicitly (see HSSW).\n \n\\subsection{Matter Temperature}\nThe important processes that are considered in following the matter\ntemperature are Compton cooling, adiabatic cooling, and Bremsstrahlung\ncooling. Less important but also included are photoionization\nheating, photorecombination cooling, radiative and collisional line\ncooling, collisional ionization cooling, and collisional recombination\ncooling.\nNote that throughout the relevant time period, collisions and Coulomb\nscattering hold all the matter species at very nearly the same temperature.\n`Matter' here means protons (and other nuclei), plus electrons, plus\nneutral atoms; dark matter is assumed to be decoupled.\n\nCompton cooling is a major source of energy\ntransfer between electrons and photons. It is described by\nthe rate of transfer of energy per unit volume between photons and\nfree electrons when the electrons are near thermal equilibrium with\nthe photons:\n\\begin{equation}\n\\label{eq:E1}\n \\frac{dE_{{\\rm e},\\gamma}}{dt} = \n \\frac{4{\\sigma_{\\rm T}}Un_{\\rm e}k_{\\rm B}}{m_{\\rm e}{c}}\n {(T_{\\rm R} - T_{\\rm M})},\n\\end{equation}\n\\begin{equation}\n\\label{eq:cool1}\n{\\rm or}\\qquad \\frac{dT_{\\rm M}}{dt} =\n \\frac{8}{3}\\frac{\\sigma_{\\rm T} U n_{\\rm e}}{m_{\\rm e}{c}n_{\\rm tot}}\n (T_{\\rm R} - T_{\\rm M}),\n\\end{equation}\n(Weymann 1965), \nwhere $E_{{\\rm e},\\gamma}$ is the electron energy density, $k_{\\rm\nB}$, $m_{\\rm e}$ and $c$ are constants as before,\n$\\sigma_{\\rm T}$ is the Thomson scattering cross section, $T_{\\rm R}$ is the\nradiation temperature and $T_{\\rm M}$ is the electron or matter temperature.\nTo get from equation~(\\ref{eq:E1}) to equation~(\\ref{eq:cool1}) we use\nthe energy of all particles; collisions among all particles keep them\nat the same temperature.\nHere $n_{\\rm tot}$ represents the total\nnumber density of particles, which includes all of the species mentioned in\n\\S\\ref{sec-species}, while $U$ represents the radiation energy density\n(integrated over all frequencies) in units of ${\\rm ergs}\\,{\\rm cm}^{-3}$:\n\\begin{equation}\n U = \\int_{0}^{\\infty}u(\\nu)d\\nu,\n\\end{equation}\nwhere $u(\\nu,t)= 4\\pi{J(\\nu,t)}\/{c}$.\nIn thermal equilibrium the radiation field has a frequency distribution\ngiven by the Planck function, $J(\\nu,t)= B(\\nu,T_{\\rm R})$, and\nthus, in thermal equilibrium the energy density is\n\\begin{equation}\n u(\\nu,t) = \\frac{4\\pi}{c}B_{\\nu}(T_{\\rm R}),\n\\end{equation}\nand the total energy density $U$ is given by Stefan's law\n\\begin{equation}\n U = \\frac{8\\pi{h_{\\rm P}}}{c^3}\\int_{0}^{\\infty}\n ({\\rm e}^{h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm R}} - 1)^{-1}\n \\nu^{3}d\\nu = a_{\\rm R}T_{\\rm R}^{4}.\n\\end{equation}\nThe spectrum of the CMB remains close to blackbody because the heat\ncapacity of the radiation is very much larger than that of the matter\n(Peebles 1993), i.e.~there are vastly more photons than baryons.\n\nAdiabatic cooling due to the expansion of the Universe is described by\n\\begin{equation} \n\\label{eq:adiabaticcooling}\n\\frac{dT_{\\rm M}}{dt} = -2H(t)T_{\\rm M},\n\\end{equation}\nsince $\\gamma=\\slantfrac{5}{3}$ for an ideal gas implies\n$T_{\\rm M}\\propto(1+z)^2$. \\newline\nThe following cooling and heating\nprocesses are often represented by approximate expressions.\nWe used the exact forms, with the exception of Bremsstrahlung\ncooling, and the negligible collisional cooling. \\newline\nBremsstrahlung, or free-free cooling:\n\\begin{equation}\n\\Lambda_{\\rm brem} = \\frac{2^5\\pi e^6 Z^2}\n {3^{3\/2}h_{\\rm P}m_{\\rm e} c^3}\n \\left(\\frac{2\\pi k_{\\rm B}T}{m_{\\rm e}}\\right)^{1\/2}\n \\!\\!\\!\\!g_{\\rm ff}n_{\\rm e}(n_{\\rm p}+n_{\\rm He II} + 4 n_{\\rm He III}),\n\\end{equation}\nwhere $g_{\\rm ff}$ is the free-free Gaunt factor (Seaton 1960), $n_{\\rm\np}$ is the number density of protons, $n_{\\rm He II}$ and $n_{\\rm He\nIII}$ the number density of singly and double ionized helium\nrespectively, and other\nsymbols are as previously described. \\newline\nPhotoionization heating:\n\\begin{equation}\n\\Pi_{\\rm p\\!-\\!i} = \\sum_{i=1}^N n_{i}4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\alpha_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T_{\\rm R})\n h_{\\rm P}(\\nu - \\nu_0)d\\nu.\n\\end{equation}\nPhotorecombination cooling:\n\\begin{eqnarray}\n\\lefteqn{\\Lambda_{\\rm p\\!-\\!r} = \\sum_{i=1}^N 4\\pi n_{\\rm e} n_{\\rm c}\n \\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm c}}\\right)^{\\rm LTE} \\times}\\nonumber \\\\\n & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\\displaystyle\n \\int_{\\nu_{i}}^{\\infty}\\!\\!\\frac{\\alpha_{i}(\\nu)}{h_{\\rm P}\\nu}\\!\\!\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}}\\,{+}\\,B(\\nu,T_{\\rm R})\\!\\right]\\!\\!\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}h_{\\rm P}(\\nu\\,{-}\\,\\nu_0)d\\nu}.\n\\end{eqnarray}\nLine cooling:\n\\begin{equation}\n\\Lambda_{\\rm line} = h_{\\rm P}\\nu_0[n_jR_{ji} - n_iR_{ij}].\n\\end{equation}\nCollisional ionization cooling:\n\\begin{equation}\n\\Lambda_{\\rm c\\!-\\!i} = h_{\\rm P}\\nu_0C_{i{\\rm c}}.\n\\end{equation}\nCollisional recombination heating:\n\\begin{equation}\n\\Lambda_{\\rm c\\!-\\!r} = h_{\\rm P}\\nu_0C_{{\\rm c}i}.\n\\end{equation}\nHere $\\nu_0$ is the frequency at the ionization edge.\nWe used approximations for collisional ionization and recombination\ncooling because these collisional processes are essentially\nnegligible during the recombination\nera. $C_{i{\\rm c}}$ and $C_{{\\rm c}i}$ are the collisional ionization and\nrecombination rates respectively, computed as in e.g.~Mihalas (1978),~\\S5.4.\n\nThus, with \n\\begin{equation}\n \\frac{dT_{\\rm M}}{dz} = \\frac{dt}{dz}\\frac{dT_{\\rm M}}{dt},\n\\end{equation}\nthe total rate of change of matter temperature\nwith respect to redshift becomes\n\\begin{eqnarray}\n\\lefteqn{(1+z)\\frac{dT_{\\rm M}}{dz} = \\frac{8\\sigma_{\\rm T}U}{3H(z)m_{\\rm e}c}\\,\n \\frac{n_{\\rm e}}{n_{\\rm e}+n_{\\rm H}+n_{\\rm He}}\\,(T_{\\rm M} - T_{\\rm R})}\n \\nonumber \\\\\n & \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\mbox{}+ 2T_{\\rm M} + {\\displaystyle\n \\frac{2\\!\\left(\\Lambda_{\\rm brem}\\,{-}\\,\\Pi_{\\rm p\\!-\\!i}\n \\,{+}\\,\\Lambda_{\\rm p\\!-\\!r}\\,{+}\\,\\Lambda_{\\rm c\\!-\\!i}\n \\,{+}\\,\\Lambda_{\\rm c\\!-\\!r}\n \\,{+}\\,\\Lambda_{\\rm line}\\right)}{3 k_{\\rm B} n_{\\rm tot} H(z)}.}\n\\label{eq:cooling}\n\\end{eqnarray}\nHere $n_{\\rm He}$ is the total number density of helium, and \nthe denominator $n_{\\rm e}+n_{\\rm H}+n_{\\rm He}$ ($n_{\\rm tot}$ from\nequation~(\\ref{eq:cool1}))\ntakes into account the fact that the energy is shared among all the\navailable matter particles. All the terms except adiabatic cooling\nin equation~(\\ref{eq:cooling}) involve matter energy conversion into\nphotons. In particular, Compton and Bremsstrahlung cooling are the\nmost important, and they\ncan be thought of as keeping $T_{\\rm M}$ very close to $T_{\\rm R}$\nuntil their time scales\nbecome long compared with the Hubble time, and\nthereafter the matter cools as $T_{\\rm M}\\propto(1+z)^2$.\nPrevious recombination calculations only included Compton and adiabatic\ncooling, however the additional terms add improvements only at the\n$10^{-3}\\%$ level in the ionization fraction.\nThe reason for the negligible improvement is that it \nmakes little difference which mechanism keeps $T_{\\rm M}$ close to\n$T_{\\rm R}$ early on,\nand adiabatic cooling still becomes important at the same time.\n\n\\subsection{Summary of Equations}\nThe system of equations to be simultaneously integrated in redshift are:\n\\begin{eqnarray}\n(1+z)\\frac{dn_{i}(z)}{dz} = \n-&\\!\\!\\! {\\displaystyle\\frac{1}{H(z)}}\n\\Big\\{\\left[n_{\\rm e}(z) n_{\\rm c}(z) P_{{\\rm c}i} - n_i(z) P_{ic}\\right]\n \\nonumber \\\\\n & \\quad \\mbox{} + {\\rm \\sum_{{\\it j}=1}^{N}} \\Delta R_{ji}\\Big\\} + 3 n_i(z),\n\\label{eq:sum1}\n\\end{eqnarray}\n\\begin{displaymath}\n{\\displaystyle (1+z)\\frac{dT_{\\rm M}}{dz} =\n \\frac{8\\sigma_{\\rm T}U(J_{\\nu,z})}{3H(z)m_{\\rm e}c}\\,\n \\frac{n_{\\rm e}}{n_{\\rm e} + n_{\\rm H} + n_{\\rm He}}\\,(T_{\\rm M} - T_{\\rm R})}\n\\end{displaymath}\n\\vspace{-3mm}\n\\begin{equation}\n\\!\\!\\mbox{}+ 2T_{\\rm M}\n - \\frac{2\\left(\\Lambda_{\\rm brem} +\n \\Pi_{\\rm p\\!-\\!i} + \\Lambda_{\\rm p\\!-\\!r} + \\Lambda_{\\rm c\\!-\\!i}\n + \\Lambda_{\\rm c\\!-\\!r} + \\Lambda_{\\rm line}\\right)}\n {3 k_{\\rm B}n_{\\rm tot}H(z)},\n\\label{eq:sum2}\n\\end{equation}\nand\n\\begin{equation}\n (1+z)\\frac{dJ(\\nu,z)}{dz} =\n 3J(\\nu,z)\n -\\frac{c}{H(z)}\\Big[j(\\nu,z) - \\kappa(\\nu,z)J(\\nu,z)\\Big].\n\\label{eq:sum3}\n\\end{equation}\n\nFor $J(\\nu,z) = B(\\nu,z) = B(\\nu,T_{\\rm R})$ (see~\\S\\ref{sec-radfield}),\nequation~(\\ref{eq:sum3}) can\nbe omitted because the expansion of the Universe preserves the thermal\nspectrum of non-interacting radiation, and we can use the Sobolev escape\nprobability method for the primary spectral line distortions.\nThe system of coupled equations~(\\ref{eq:sum1}) that we use contains up to 609\nseparate equations, 300 for H (one for each of a maximum of 300 levels we\nconsidered), 200 for \\ion{He}{1}, 100 for \\ion{He}{2}, 1 for \\ion{He}{3},\n1 for electrons, 1 for protons, and 1 for each of the 5 molecular or ionic H\nspecies.\nThis system of equations, along with (\\ref{eq:sum2}), is extremely stiff,\nthat is\nthe dependent variables are changing on very different time scales.\nWe used the Bader-Deuflhard semi-implicit numerical integration scheme,\nwhich is described in Press et al.~(1992).\nTo test the numerical integration we checked at each time step that the\ntotal charge and total number of particles are conserved to one part\nin $10^{7}$.\n\n\\section{Results and Discussion}\nBy an `effective 3-level' H atom we mean a hydrogen atom that includes\nthe ground state,\nfirst excited state, and continuum. In an effective 3-level atom, the\nenergy levels between n=2 and \nthe continuum are accounted for by a recombination coefficient which\nincludes recombinations to those levels.\nThis should be distinguished from an {\\it actual\\\/} 3-level atom, which would\ncompletely neglect all levels above n=2, and would be a hopeless\napproximation. Good accuracy is obtained by considering an n-level atom,\nwhere n is large enough. In practice we find that a 300-level atom is\nmore than adequate.\nWe do not explicitly include angular momentum states $\\ell$, whose\neffect we expect to be negligible.\nIn contrast to the effective 3-level H atom, the 300-level H\natom has no\nrecombination coefficient with `extra' levels.\nThe `standard' recombination calculation refers to the calculation\nwith the effective 3-level atom that is\nwidely used today and first derived by Peebles (1968) and Zel'dovich\net al.~(1968), updated with the most recent parameters and recombination\ncoefficient (HSSW). \n\nThe primordial He abundance was taken to be $Y_{\\rm P}=0.24$ by mass\n(Schramm \\& Turner 1998). The present-day CMB temperature $T_{0}$ was\ntaken to be $2.728\\,$K, the central value determined by the FIRAS\nexperiment (Fixsen et al.~1996).\n\n\\subsection{The `Effective 3-level' Hydrogen Atom}\n\\label{sec-3level}\nFor comparison with the standard recombination calculation that only\nincludes hydrogen (see~Peebles 1968, 1993; Scott 1988), we reduce our chemical\nreaction network to an effective 3-level atom, i.e.~a two-level\nhydrogen atom plus continuum. \nThe higher atomic energy levels are included by way of the recombination\ncoefficient, which can effectively include recombination to hundreds of levels.\nThe following reactions are included:\n\\begin{eqnarray}\n{\\rm H}_{\\rm n=2,\\ell=2s} + \\gamma &\\longleftrightarrow&\n {\\rm e}{^-} + {\\rm H}{^+} \\nonumber\\\\\n{\\rm H}_{\\rm n=1} + \\gamma &\\longleftrightarrow&\n {\\rm H}_{\\rm n=2,\\ell=2p} \\nonumber\\\\\n{\\rm H}_{\\rm n=1} + 2\\gamma &\\longleftrightarrow&\n {\\rm H}_{\\rm n=2,\\ell=2s}.\\nonumber\n\\end{eqnarray}\n\nAs described in Peebles (1993), we omit the recombinations and\nphotoionizations to the ground state because any recombination directly\nto the ground state will emit a photon with energy $>13.6\\,$eV, where\nthere are few blackbody photons, and this will immediately re-ionize a\nneighboring H atom.\nWe include the two-photon rate from the $2s$ state with the rate\n$\\Lambda_{2s{-}1s} = 8.22458\\,$s$^{-1}$ (Goldman 1989).\nThe most accurate total Case B recombination coefficient is\nby Hummer (1994) and is fitted by the function\n\\begin{equation}\n\\alpha_{\\rm B} = 10^{-13}\\frac{at^{b}}{1 + ct^{d}} \\, \\mathrm{cm^{3}s^{-1}},\n\\end{equation}\nwhere $a=4.309$, $b=-0.6166$, $c=0.6703$, $d=0.5300$ and\n$t= T_{\\rm M}\/10^{4}\\,$K (P{\\'e}quignot et al.~1991; see also Verner\n\\& Ferland~1996).\n\nConsideration of detailed balance in the effective 3-level atom leads to\na single ordinary differential equation for the ionization fraction:\n\\begin{eqnarray}\n\\lefteqn{{dx_{\\rm e}\\over dz} =\n \\frac{\\big[x_{\\rm e}^2 n_{\\rm H} \\alpha_{\\rm B}\n - \\beta_{\\rm B} (1-x_{\\rm e})\n {\\rm e}^{-h_{\\rm P}\\nu_{2s}\/k_{\\rm B}T_{\\rm M}}\\big]} {H(z)(1+z)}\\,\\times}\n \\nonumber \\\\\n & & \\frac{\\big[1 + K \\Lambda_{2s{-}1s} n_{\\rm H}(1-x_{\\rm e})\\big]}\n {\\big[1+K \\Lambda_{2s{-}1s} n_{\\rm H} (1-x_{\\rm e})\n + K \\beta_{\\rm B} n_{\\rm H}(1-x_{\\rm e})\\big]}\n\\label{eq:standard_xe}\n\\end{eqnarray}\n(see e.g.~Peebles~1968; extra terms included in Jones \\& Wyse 1985,\nfor example, are negligible). Here $x_{\\rm e}$ is the residual\nionization fraction, that is the\nnumber of electrons compared to the total number of hydrogen nuclei\n($n_{\\rm H}$). Here the Case B recombination coefficient\n$\\alpha_{\\rm B}=\\alpha_{\\rm B}(T_{\\rm M})$, the total photoionization rate\n$\\beta_{\\rm B}=\\alpha_{\\rm B} (2\\pi m_{\\rm e} k_{\\rm B}\nT_{\\rm M}\/h_{\\rm P}^2)^{3\/2} \\exp(-E_{2s}\/k_{\\rm B}T_{\\rm M})$ as described in\n\\S\\ref{sec-StandardRecomb}, $\\nu_{2s}$ is the frequency of the $2s$ level from\nthe ground state,\nand the redshifting rate $K\\equiv \\lambda_\\alpha^3\/(8\\pi H(z))$, where\n$\\lambda_\\alpha$ is the Ly$\\,\\alpha$ rest wavelength.\nNote that $T_{\\rm M}$ is used in equation~(\\ref{eq:standard_xe}) and\nin $\\beta_{\\rm B}$, because the temperature terms come from detailed balance\nderivations that use Boltzmann and Saha equilibrium distributions, which are\ncollisional descriptions.\nIn the past, this equation has been solved (for $x_{\\rm e}(z)$)\nsimultaneously with\na form of equation~(\\ref{eq:cooling}) containing only adiabatic and\nCompton cooling. We refer to the approach of\nequation~(\\ref{eq:standard_xe}) as the `standard calculation.'\n\nFor the comparison test with the standard recombination calculation,\nwe also use an effective recombination coefficient, but three equations to\ndescribe the three reactions listed above. That is, we\nsimplified equation~(\\ref{eq:sum1}) to three equations, one for the\nground state population ($n_1$), one for the first excited state\npopulation ($n_2$), and one for the\nelectrons (for H recombination $n_e = n_p$).\n\\begin{equation}\n(1+z)\\frac{dn_{1}(z)}{dz} =\n-\\frac{1}{H(z)}\n\\left[\\Delta R_{2p{-}1s} + \\Delta R_{2s{-}1s}\\right] + 3 n_{1}\n\\end{equation}\n\\vspace{-3mm}\n\\begin{eqnarray}\n(1+z)\\frac{dn_{2}(z)}{dz} =& \\!\\!\\!\\!\\mbox{}-{\\displaystyle\\frac{1}{H(z)}}\n\\Big[(n_{\\rm e}(z) n_{\\rm p}(z) \\alpha_{\\rm B}\n - n_{2s}(z) \\beta_{\\rm B} \\nonumber \\\\\n & \\ \\mbox{} - \\Delta R_{2p{-}1s} - \\Delta R_{2s{-}1s}\\Big] + 3 n_2\n\\end{eqnarray}\n\\vspace{-3mm}\n\\begin{equation}\n(1+z)\\frac{dn_{\\rm e}(z)}{dz} =\n-\\frac{1}{H(z)}\n\\big[(n_{2s}(z) \\beta_{\\rm B} - n_{\\rm e}(z) n_{\\rm p}(z) \\alpha_{\\rm B}\n\\big] + 3 n_{\\rm e}\n\\end{equation}\nThe remaining\nphysical difference between our effective 3-level atom approach and that of\nthe standard calculation is the treatment of the redshifting of H Ly$\\,\\alpha$\nphotons (included in the $\\Delta R_{2p{-}1s}$ terms). In our calculation the redshifting is accounted for by \nthe Sobolev escape probability (see~\\S\\ref{sec-Sobolev}).\nFollowing Peebles (1968, 1993), the standard calculation accounts for the\nredshifting by\napproximating the intensity distribution as a step, and in effect takes the\nratio of the redshifting of the photons through the line to the expansion\nscale that produces the same amount of redshifting.\nIt can be shown that Peebles' step method considered as an escape\nprobability scales as $1\/ \\tau_{\\rm S}$, where $\\tau_{\\rm S}$ is the\nSobolev optical depth.\nFor high Sobolev optical depth, which holds for H Ly$\\,\\alpha$ during\nrecombination for any cosmological model (see\nFig.~7),\nthe Sobolev escape probability also scales as $1\/\\tau_{\\rm S}$:\n\\begin{equation} \n \\lim_{\\tau_{\\rm S} \\gg 1} p_{ij} = \\lim_{\\tau_{\\rm S} \\gg 1}\n \\frac{1}{\\tau_{\\rm S}}(1 - {\\rm e}^{-\\tau_{\\rm S}}) = \\frac{1}{\\tau_{\\rm S}}.\n\\end{equation}\nTherefore the two approximations are equivalent for Ly$\\,\\alpha$,\nalthough we would expect differences for lines with $\\tau_{\\rm S}\\lesssim1$,\nwhere $p\\to1$.\nBecause we treat recombination in the same way as Peebles, \nno individual\ntreatment of other lines is permitted, and therefore there are no other\ndifferences between the two calculations for this simple case. \nNote that with Peebles' step method to compute $\\Delta R_{2p{-}1s}$,\nand the assumption that $n_1 = n_{\\rm H} - n_{\\rm p}$, the above equations will\nreduce to the single ODE equation~(\\ref{eq:standard_xe}).\n\nThe results from our effective 3-level recombination calculation are\nshown in\nFig.~1,\nplotted along with values from a\nseparate code as used in HSSW, which represents the standard\nrecombination calculation updated with the most recent parameters.\nThe resulting ionization fractions are equal, which shows that our new approach\ngives exactly the standard result when reduced to an effective 3-level atom.\nTwo other results are plotted for comparison, namely values of $x_{\\rm e}$\ntaken from\nPeebles (1968) and Jones \\& Wyse (1985). Their differences can be largely\naccounted for by\nthe use of an inaccurate recombination coefficient with\n$\\alpha_{\\rm B}(T_{\\rm M}) \\propto T_{\\rm M}^{-1\/2}$.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig1.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~1.---}\nComparison of effective three-level hydrogen recombination for\nthe parameters\n$\\Omega_{\\rm tot} = 1.0, \\Omega_{\\rm B} = 1.0, h=1.0$.\nNote that the Jones \\& Wyse (1985) and Peebles (1968) curves overlap as does\nour curve with the HSSW one.\n}\n\\label{fig:Hrec3level}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nAs an aside, we note the behavior for $z\\lesssim50$ in our curve and the HSSW\none. This is caused by inaccuracy in the recombination coefficient for very\nlow temperatures. The down-turn is entirely artificial and could be removed\nby using an expression for $\\alpha_{\\rm B}(T)$ which is more physical\nat small temperatures. The results of our detailed calculations are\nnot believable\nat these redshifts either, since accurate modeling becomes increasingly\ndifficult due to numerical precision as $T$ approaches zero.\nBut in any case the optical\ndepth back to such redshifts is negligible, and the real Universe is\nreionized at a similar epoch (between $z=5$ and 50 certainly).\n\n\\subsection{Multi-level Hydrogen Atom}\n\\label{sec-Multi}\nThe purpose of a multi-level hydrogen atom is to improve the\nrecombination calculation, by following the population of each atomic\nenergy level with redshift and by including all bound-bound and bound-free\ntransitions. This includes recombination to, and photoionization from,\nall levels {\\it directly\\\/} as a function of time,\nin place of a parameterized recombination and photoionization coefficient. The individual\ntreatment of all levels in a coupled manner allows for the\ndevelopment of departures from equilibrium among the states with time,\nand feedback on the rate of recombination. Since the accuracy of the\nrecombination coefficient is probably the single most important effect\nin obtaining accurate power spectra (HSSW), it makes sense to\nfollow the level populations as accurately as possible.\n\nIn the multi-level H atom recombination calculation, we do not consider\nindividual $\\ell$\nstates (with the exception of $2s$ and $2p$), but assume the $\\ell$ sublevels have populations proportional to\n(2$\\ell$ + 1). The $\\ell$ sublevels only deviate from this distribution\nin extreme non-equilibrium conditions (such as planetary nebulae).\nIn their H recombination calculation, Dell'Antonio \\& Rybicki (1993)\nlooked for such $\\ell$ level deviations for n$\\,{\\leq}\\,10$ and found none.\nFor n$\\,{>}\\,10$, the $\\ell$ states are even less likely to differ from an\nequilibrium distribution, because the energy gaps\nbetween the $\\ell$ sublevels are increasingly smaller as n increases.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig2.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~2.---}\nMulti-level hydrogen recombination for the standard CDM\nparameters $\\Omega_{\\rm tot} = 1.0, \\Omega_{\\rm B} = 0.05, h = 0.5$\n(top), and for $\\Omega_{\\rm tot} = 1.0, \\Omega_{\\rm B} = 1.0, h = 1.0$\n(bottom), both with \n$Y_{\\rm P} = 0.24, T_0 = 2.728\\,$K. The `effective 3-level' calculation\nis essentially the same as in HSSW, and uses a recombination coefficient\nwhich attempts to account for the net effect of all relevant levels. We find\nthat we require a model which considers close to 300 levels for full\naccuracy. Note also that although we plot all the way to $z=0$, we know\nthat the Universe becomes reionized at $z>5$, and that our calculations\n(due to numerical precision at low $T$) become unreliable\nfor $z\\lesssim50$ in the upper two curves, and for $z\\lesssim20$ in\nthe other curves.\n\\label{fig:HrecMulti}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsubsection{Results From a Multi-level H Atom}\n\\label{sec-resultsmulti}\nFig.~2\nshows the ionization fraction $x_{\\rm e}$ from\nrecombination of a 2-, 10-, 50-, 100- and 300-level H atom, compared with\nthe standard effective 3-level results. The $x_{\\rm e}$ converges\nfor the highest n-level atom calculations. The effective\n3-level atom actually includes about 800 energy levels via the recombination\ncoefficient (e.g.~Hummer 1994).\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig3.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~3.---}\nEnergy separations between various hydrogen atomic levels and the\ncontinuum. The solid curve shows the energy width of the same energy\nlevels due to thermal\nbroadening. Thermal broadening was calculated using\n$\\nu (2 k_{\\rm B} T_{\\rm M} \/ m_{\\rm H} c^2)^{1\/2}$. The\nfrequency of the highest atomic energy level (n=300, with an energy\nfrom the ground state of 109676.547 cm$^{-1}$; the continuum energy\nlevel is 109677.766 cm$^{-1}$) was used for $\\nu$, but a\nthermal broadening value for any atomic energy level would overlap on\nthis graph.\n\\label{fig:thermbroad}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nFig.~2\nshows that the more levels that are included in the\nhydrogen atom, the lower\nthe residual $x_{\\rm e}$. The simple explanation is that the\nprobability for electron capture increases with more energy levels per\natom. Once captured, the electron can cascade\ndownwards before being reionized. Together this means \nadding more higher energy levels per atom\nincreases the rate of recombination.\nEventually $x_{\\rm e}$\nconverges as the atom becomes complete in terms of electron energy\nlevels, i.e.~when there is no gap between the highest energy level and\nthe continuum (see\nFig.~3).\nUltimately the uppermost levels\nwill have gaps to the continuum which are smaller than the thermal\nbroadening of those levels, and so\nenergy levels higher than about n=300 do not need to be considered,\nexcept perhaps at the very lowest redshifts.\n \nFor other reasons entirely, our complete (300-level) H atom recombination\ncalculation gives an $x_{\\rm e}$ lower than that of the\neffective 3-level atom calculation. The faster production of\nhydrogen atoms is due to non-equilibrium processes in the excited states\nof H, made obvious by our new, level-by-level treatment of recombination. The\ndetails are described in \\S\\ref{sec-noneq} below.\n\n\\subsubsection{Faster H Recombination in our Level-by-level\nRecombination Calculation}\n\\label{sec-noneq}\n\nThe lower $x_{\\rm e}$ in our calculation compared to the standard calculation\nis caused by the strong but cool radiation field. Specifically, \nboth a faster downward cascade rate and a lower total photoionization rate\ncontribute to a faster net recombination rate.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig4.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~4.---}\nHow the bound-bound rates of large energy separation\n(e.g. n=70 to n=20) go out of\nequilibrium at low $T$, illustrated using an upper level of\nn=70 for definiteness. The case of equilibrium corresponds to net\nbound-bound rates of zero. Each upward and downward bound-bound rate for\na given transition is represented by the same curve; non-equilibrium occurs\nwhere a single curve separates into two as redshift decreases.\nThe rates shown are for the sCDM model.\n\\label{fig:bbrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nBy following the population of each\natomic energy level with redshift, we relax the assumption used in the standard\ncalculation that the excited states are in equilibrium. In addition, we\ncalculate all bound-bound rates which control equilibrium among\nthe bound states.\nIn the standard calculation, equilibrium among the excited states\nn~$\\geq 2$ is\nassumed, meaning that the net bound-bound rates are\nzero.\nFig.~4\nshows that the net bound-bound\nrates are actually different from zero at $z\\lesssim1000$.\nThe reason for this is that at low temperatures, the strong but cool\nradiation field means that high energy transitions are rare due to few high\nenergy photons. More specifically,\nphotoexcitation and stimulated photodeexcitation for high energy transitions\nbecome rare (e.g.~70--10, 50--4 etc.). \nIn this case\nspontaneous deexcitation dominates, causing a\nfaster downward cascade to the n=2 state. \nIn addition, the faster downward cascade rate is faster than the\nphotoionization rate from the upper state, and one might view this as\nradiative decay stealing some of the depopulation `flux' from\nphotoionization. Both the faster downward cascade and the lower\nphotoionization rate contribute to the faster net recombination rate.\n\nThe cool radiation field is strong, so photoexcitations and \nphotodeexcitations are rapid among nearby energy\nlevels (e.g.~70--65, etc.;\nsee Fig.~4). What we see in Fig.~4 is that with time, after $z\\lesssim1000$,\nthe n=70 energy level becomes progressively decoupled from the distant\nlower energy levels (n=2,3,...20,...), but remains tightly coupled to its\nnearby `neighbors' (n=60,65, etc.). This explains the departures from an\nequilibrium Boltzmann distribution (in the excited states) as seen in the shape of the curves in\nFig.~5.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig5a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig5b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~5.---}\nRatio of the actual number densities of the excited\nstates ($n_i$) and\nnumber densities for a Boltzmann distribution of excited states\n($n_i^*$) at\ndifferent redshifts for recombination within the standard CDM model.\nSee section~\\ref{sec-noneq} for details. We give two different plots with\nlinear and logarithmic y-axes and different redshift values, to show\na wider range of out-of-equilibrium conditions. The ratio approaches\na constant for high n at a given $z$, because the high energy (Rydberg)\nstates are all very close in energy, and thus have similar behavior\n(i.e. remain coupled to the radiation field and each other).\n\\label{fig:Boltz}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nFig.~5\nillustrates the non-equilibrium of the excited\nstates by showing the ratio of populations of the excited states\ncompared to a Boltzmann equilibrium distribution with respect to n=2. \nWe find that the\nupper levels of the hydrogen atom are not in thermal equilibrium with\nthe radiation, i.e.~the excited levels are not populated according to\na Boltzmann distribution. The excited states are in fact overpopulated\nrelative to a\nBoltzmann distribution. This is not a surprise for the population of the\nn=2 state, which is strongly overpopulated compared to the n=1 ground\nstate, and so should be all n~$\\geq 2$ states because all Lyman lines remain\noptically thick during recombination. What {\\em is\\\/} surprising is that\nall excited states develop a further overpopulation {\\em with respect to\\\/}\nn=2 and each other. Note that this is not a population inversion.\nThe recombination rate to a given high level is faster than the downward\ncascade rate, and this causes a `bottleneck' creating the\noverpopulation.\nFig.~5\nshows that all states\nare in equilibrium at high redshifts, with the highest states going\nout of equilibrium first, followed by lower and lower states as the\nredshift decreases. The factor by which the excited states are\nover-populated approaches a constant at high n for a given redshift,\nwith this factor increasing as $z$ decreases. The ratio is constant\nbecause the high energy level Rydberg states have very similar energy\nlevels to each other, with a relatively large energy separation from the n=2\nstate (i.e.~the exponential term in equation~(\\ref{eq:boltz}) dominates\nover the $g_i$ ratios, and the exponential term is similar for all of\nthe Rydberg states.) \n\nFig.~5\nalso shows an enormous ratio at low\nredshift ($z < 500$) for number densities\nof the actual excited states to the number densities of a Boltzmann\ndistribution of excited states, on the order of $10^6$. At such a low\nredshift, there are almost no electrons in\nthe excited states ($\\sim 10^{-20}$ cm$^{-3}$), and so\nunlike at higher redshifts, the ratio is only an\nillustration of the strong departure from an equilibrium distribution;\nthe actual populations are very low in any case.\n\nIn comparison with the standard\nequilibrium capture-cascade calculation for $\\alpha_{\\rm B}$, the unusual\nsituation described above (caused by the strong but cool radiation field)\nleads to higher effective recombination\nrates for the majority of excited states without increasing photoionization\nproportionally. This results in a higher net rate of production of neutral\nhydrogen atoms, i.e.~a lower $x_{\\rm e}$.\n\n\\subsubsection{Accurate Recombination vs Recombination Coefficient}\nTo demonstrate why the non-equilibrium in the excited states of H affects\nthe recombination rate, we must consider the\ndifference in our new treatment of recombination compared to the\nstandard treatment.\nAn important new benefit of our level-by-level calculation\nlies in replacing the recombination coefficient with\na direct calculation of recombination to and photoionization from each level at each redshift step.\nIn other words, we calculate the recombination rate and the\nphotoionization rate using\nindividual level populations and parameters of the excited states $i$,\n\\begin{eqnarray}\n\\lefteqn{\\sum_{i=1}^N n_{\\rm e} n_{\\rm p} R_{{\\rm c}i}\n = n_{\\rm e} n_{\\rm p} \\sum_{i=1}^N\n {\\left(\\frac{n_{i}}{n_{\\rm e} n_{\\rm p}}\\right)}^{\\rm LTE}\\times}\\nonumber \\\\\n& \\!\\!{\\displaystyle 4\\pi\\!\\int_{\\nu_{i}}^{\\infty}\n \\frac{\\alpha_{i}(\\nu)}{h_{\\rm P}\\nu}\n \\left[\\frac{2h_{\\rm P}\\nu^{3}}{c^{2}} + B(\\nu,T_{\\rm R})\\right]\n {\\rm e}^{-h_{\\rm P}\\nu\/k_{\\rm B}T_{\\rm M}}d\\nu},\n\\label{eq:feedback2}\n\\end{eqnarray}\nand similarly for the photoionization rate,\n\\begin{equation}\n\\label{eq:feedback}\n\\sum_{i=1}^N n_{i}R_{i{\\rm c}} = \\sum_{i=1}^N\n n_{i}4\\pi\\int_{\\nu_{0}}^{\\infty}\n \\frac{\\alpha_{i{\\rm c}}(\\nu)}{h_{\\rm P}\\nu}B(\\nu,T_{\\rm R})d\\nu.\n\\end{equation}\nFor the standard recombination calculation, the recombination rate is\n\\begin{equation}\n\\label{eq:standardrate}\n\\sum_{i=1}^N n_{\\rm e} n_{\\rm p} R_{{\\rm c}i} = n_{\\rm e} n_{\\rm p}\n \\alpha_{\\rm B}(T_{\\rm M}),\n\\end{equation}\nand the photoionization rate is\n\\begin{eqnarray}\n\\lefteqn{\\Sigma_i n_i R_{i{\\rm c}} \\equiv n_{2s} \\beta_{\\rm B}(T_{\\rm M})}\n \\nonumber \\\\\n& = n_{2s} \\alpha_{\\rm B}(T_{\\rm M}) {\\rm e}^{-E_{2s}\/k_{\\rm B}T_{\\rm M}}\n \\left(2\\pi m_{\\rm e} k_{\\rm B} T_{\\rm M} \\right)^{3\/2}\/h_{\\rm P}^3.\n\\label{eq:nofeedback}\n\\end{eqnarray}\nIn this last equation, the excited state populations are hidden by the\nBoltzmann relation with $n_{2s}$ (see~\\S\\ref{sec-StandardRecomb}).\nThe important point here is that our method allows redistribution of the\nH level populations over all 300 levels at each redshift step, which feeds\nback on the recombination process via\nequation~(\\ref{eq:feedback}), and leads to the lower $x_{\\rm e}$ shown in\nFig.~2.\nThis\nredistribution of the level populations is not possible in the standard \ncalculation's equation~(\\ref{eq:nofeedback}) which only considers the\npopulations $n_{\\rm e}$, $n_{1}$, and $n_{2s}$, and considers the\nexcited level populations $n\\,{>}\\,2s$\nto be proportional to $n_{2s}$ in an equilibrium distribution.\n\nA small improvement in our new recombination treatment over\nthe standard treatment is in our distinguishing the various temperature\ndependencies of recombination. Photoionization and\nstimulated recombination are radiative, so they should depend on\n$T_{\\rm R}$. Spontaneous recombination is collisional and depends on\n$T_{\\rm M}$ (see~\\S\\ref{sec-Recomb}). In the standard calculation, the radiative nature of \nrecombination and photoionization is\noverlooked because both the recombination\ncoefficient and photoionization coefficient are a function $T_{\\rm M}$ only\n(equations~(\\ref{eq:standardrate}) and (\\ref{eq:nofeedback})).\nAlthough adiabatic cooling (equation~(\\ref{eq:adiabaticcooling}))\ndoes not dominate until quite low redshifts ($z\\lesssim100$), it still\ncontributes partially to matter cooling\nthroughout recombination. The resulting difference between\n$T_{\\rm M}$ and\n$T_{\\rm R}$ in the net recombination rate affects $x_{\\rm e}$ at the\nfew percent level at $z\\lesssim300$ for the popular cosmologies,\nand has an even larger effect for high $\\Omega_{\\rm B}$ models.\n\n\\subsubsection{Collisions}\nThe standard recombination calculation omits collisional excitation\nand ionization because at the relevant temperatures and densities they\nare negligible for a two-level hydrogen\natom (Matsuda et al.~1971).\nWe have found that the collisional processes are also not important for \nthe higher levels, even though those electrons are bound with little\nenergy. In high $\\Omega_{\\rm B}$ models collisional ionization and\ncollisional recombination rates for \nthe highest energy levels are of the same order of\nmagnitude as the photoionization and recombination rates, though not\ngreater than them (see\nFig.~6).\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig6a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig6b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~6.---}\nComparison of ionization rates. The upper curves are\nphotoionization rates, the lower curves are collisional ionization rates.\nThe left panel shows a model with the standard CDM parameters, while\nthe right panel shows an extreme baryon cosmology.\n\\label{fig:Collrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsubsection{Departures From Case B}\n\\label{sec-CaseB}\nCase B recombination excludes recombination to the ground state and considers\nthe Lyman lines to be optically thick (i.e.~photons associated with all\npermitted radiative transitions to\nn=1 are assumed to be instantly reabsorbed).\nAn implied assumption necessary to compute the\nphotoionization rate is that the excited states (n~$\\ge 2$) are in\nequilibrium with the radiation.\nUnlike the standard recombination calculation, our method allows departures\nfrom Case B because the Lyman lines are\ntreated by the Sobolev escape probability method which is valid\nfor any optical thickness, and our\nmethod allows departures from equilibrium of the excited states\n(\\S\\ref{sec-resultsmulti}).\nWe find that the excited states depart from equilibrium at redshifts\n$\\lesssim800$, so Case B does not hold then.\nHowever, our calculations show that for hydrogen all Lyman lines\nare indeed\noptically thick during all of hydrogen recombination, so\nCase B holds for H recombination above redshifts $\\simeq 800$.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig7.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~7.---}\nSobolev optical depth ($\\tau_{\\rm S}$) for the H Lyman lines for\ntwo different\ncosmological models. From upper to lower the curves represent the\noptical depth in the Lyman transitions from n=2 (i.e.~Ly$\\,\\alpha$), n=10,\nn=50, n=100, and n=270. The curves show that all the Lyman transitions are\noptically thick during H recombination ($z\\lesssim2000$)\nbut some are optically thin ($\\tau_{\\rm S} < 1$)\nearlier, during He recombination ($z \\gtrsim$ 2000).\n\\label{fig:SobolevTau}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nFig.~7\nshows that the Lyman lines are not\noptically thick\nat earlier times, e.g.~during helium recombination, where we\nfind some optically thin H\nLyman lines. The Sobolev escape probability treats this consistently,\nwhich is necessary because we evolve H, \\ion{He}{1}, and \\ion{He}{2}\nsimultaneously.\nwhich is\nnecessary because we evolve H, \\ion{He}{1}, and \\ion{He}{2} simultaneously.\n\n\\subsubsection{Other Recent Studies}\n\nThe previous study that was closest in approach to our own was that of\nDell'Antonio \\& Rybicki (1993), who calculated recombination\nfor a ten-level hydrogen atom in order to estimate the spectral distortions\nto the CMB blackbody radiation spectrum. Ten levels are insufficient to\ncalculate recombination accurately, because the higher energy levels of the\natom are completely ignored (see\nFig.~3).\nHowever the accuracy of the ionization\nfraction ($x_{\\rm e}$) was sufficient to determine the magnitude\nof the spectral distortions. Their recombination model treated individual\nlevels, but used a recombination coefficient to each level of the form\n${T_{\\rm M}}^{-1\/2}$. Because the form of the recombination\ncoefficient dominates the H recombination process, our models are not\nequivalent, and so there is little use in comparing the results.\n\nMore recently, Boschan \\& Biltzinger (1998) derived a new\nparameterized recombination coefficient to solve the recombination\nequation of the standard calculation, and to\ngenerate spectral distortions in the CMB. \nTheir calculation differs from ours in that their\nrecombination coefficient is pre-calculated. Hence it is not an interactive\npart of the calculation, and does not allow the advantages that our\ncalculation does, mainly the feedback of the non-equilibrium in the excited\nstates on the net recombination rate. While they include pressure\nbroadening for a cutoff in the partition function, they neglect\nthermal broadening. A more serious problem is their method\nof inclusion of stimulated recombination, as originally suggested by\nSasaki \\& Takahara (1993), who included stimulated recombination as\npositive recombination instead of negative ionization.\nThe physics (as described in \\S\\ref{sec-Recomb}) and our\ncomputational results are the same regardless of whether stimulated\nrecombination is treated as positive recombination or negative ionization.\nHowever, this may not be the case computationally for the standard\ncalculation, if it is\nnot treated with care. We defer a full discussion of these matters to a\nseparate paper (Seager \\& Sasselov, in preparation).\n\nWe have also investigated how we can approximate our calculations, so\nthat other researchers can obtain approximately accurate results without\nthe need to follow 300 levels in a hydrogen atom. Because the net effect \nof our new H calculation is a faster recombination (a lower freeze-out\nionization fraction), our results can be reproduced by artificially\nspeeding up recombination in the standard calculation. Further\ndetails are described in Seager, Sasselov, \\& Scott (1999).\n\n\\subsection{Helium}\n\\label{sec-Helium}\nWe compute helium and hydrogen recombination simultaneously.\nThe recombination of \\ion{He}{3} into \\ion{He}{2} and \\ion{He}{2} \ninto \\ion{He}{1}\nis calculated in much the same way as hydrogen, with recombination,\nphotoionization,\nredshifting of the n$^1p$--$1^1s$ lines (in H these are the Lyman\nlines), inclusion of the $2^1s$--$1^1s$ two-photon rates, collisional\nexcitation, collisional deexcitation, collisional ionization, and\ncollisional recombination, as described in \n\\S\\ref{sec-Recomb}--\\ref{sec-Sobolev}. The multi-level \\ion{He}{1} atom\nincludes the first 4 angular momentum states up to the level n=20, above which\nonly the principal quantum number energy levels and transitions are used.\nFig.~8\n(which shows the levels up to n=4 only)\nindicates how much more complicated the\n\\ion{He}{1} atom is compared with H or the hydrogenic \\ion{He}{2}.\nOur multi-level \\ion{He}{2} atom includes the first 4 angular momentum states\nup to the level n=4, above which only the principal quantum number energy\nlevels and transitions are used.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig8.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~8.---}\nGrotrian diagram for \\ion{He}{1}, showing the states with\nn~$\\le4$ and the continuum. In practice our model atom explicitly\ncontains the first 4 angular momentum states up to n=20, and\n120 principal quantum number energy levels beyond.\n\\label{fig:Hegrotrian}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nPhotoionizations from\nany \\ion{He}{1} excited state are allowed only into the ground state\nof \\ion{He}{2}, because there are few photons energetic enough ($>40\\,$eV)\nto do more than that. Two electron transitions in \\ion{He}{1} are\nnegligible at recombination era temperatures.\n\nCosmological helium recombination was discussed explicitly in\nMatsuda et al.~(1969, 1971, hereafter MST), and Sato, Matsuda \\& Takeda (1971),\nand to a lesser extent in Lyubarsky \\& Sunyaev (1983),\nwhile several\nother papers give results, but no details (e.g.~Lepp \\& Shull~1984;\nFahr \\& Loch~1991; Galli \\& Palla~1998).\nThe main improvement in our calculation over previous treatments of helium is\nthat we use a multi-level \\ion{He}{2} atom, a multi-level \\ion{He}{1}\natom with triplets and singlets treated correctly, and evolve the\npopulation of each energy level with redshift by including all\nbound-bound and bound-free transitions. This is not possible for the\nstandard recombination calculation method\n(equation~(\\ref{eq:standard_xe})) extended to \\ion{He}{1}, using an\neffective three-level \\ion{He}{1} atom with only a singlet ground\nstate, singlet first excited state and continuum.\n\n\\subsubsection{Results From \\ion{He}{1} Recombination}\n\nFig.~9,\nshows the ionization fraction $x_{\\rm e}$\nthrough \\ion{He}{2}, \\ion{He}{1} and H recombination, plotted against the\nstandard H calculation that includes \\ion{He}{2} and \\ion{He}{1}\nrecombination via the Saha equation. For completeness we give the\nhelium Saha equations here: \\\\\nfor \\ion{He}{1}$\\,\\leftrightarrow\\,$\\ion{He}{2}\n\\begin{equation}\n\\label{eq:heonesaha}\n{(x_{\\rm e}-1)x_{\\rm e}\\over 1+f_{\\rm He}-x_{\\rm e}} = \n 4 {(2\\pi m_{\\rm e} k_{\\rm B} T)^{3\/2}\\over h_{\\rm P}^3 n_{\\rm H}}\n {\\rm e}^{-\\chi_{\\rm He I}\/k_{\\rm B}T},\n\\end{equation}\nand for \\ion{He}{2}$\\,\\leftrightarrow\\,$\\ion{He}{3}\n\\begin{equation}\n\\label{eq:hetwosaha}\n{(x_{\\rm e}-1-f_{\\rm He})x_{\\rm e}\\over 1+2f_{\\rm He}-x_{\\rm e}} = \n {(2\\pi m_{\\rm e} k_{\\rm B} T)^{3\/2}\\over h_{\\rm P}^3 n_{\\rm H}}\n {\\rm e}^{-\\chi_{\\rm He II}\/k_{\\rm B}T}.\n\\end{equation}\nHere the $\\chi$s are ionization potentials, $n_{\\rm H}$ is the total\nnumber density of hydrogen, $f_{\\rm He}$ is the total number fraction of\nhelium to hydrogen $f_{\\rm He}=n_{\\rm He}\/n_{\\rm H}=Y_{\\rm\nP}\/4(1-Y_{\\rm P})$, and our definition of $x_{\\rm e}\\equiv n_{\\rm e}\/n_{\\rm H}$\nresults in the complicated-looking left hand sides. The extra factor of\n4 on the right hand side for \\ion{He}{1}$\\,\\leftrightarrow\\,$\\ion{He}{2}\narises from the statistical weights factor. \n \n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig9.eps}}}}\n\\baselineskip=8pt\n\\vspace{6mm}\n{\\footnotesize {\\sc Fig.~9.---}\nHelium and hydrogen recombination for two cosmological models with\n$Y_{\\rm P} = 0.24$ and $T_0 = 2.728\\,$K. The first step from\nright to left is recombination of \\ion{He}{3} to \\ion{He}{2}, the second\nstep is \\ion{He}{2} to \\ion{He}{1}, and the third step is H recombination.\n\\label{fig:Herec}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig10.eps}}}}\n\\baselineskip=8pt\n\\vspace{3mm}\n{\\footnotesize {\\sc Fig.~10.---}\nDetails of helium recombination for the standard CDM\ncosmology (top figure) and the high $\\Omega_{\\rm B}$ cosmology (bottom\nfigure). The dashed lines show our new results, and the dotted lines \nshow the results assuming the `standard calculation' (equivalent to\nSaha equilibrium).\n\\label{fig:Herecdetails}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nWhile our improved $x_{\\rm e}$ agrees fairly closely with Saha recombination\nfor \\ion{He}{2} (see~\\S\\ref{sec-HeII}), the \ndifference in $x_{\\rm e}$ from Saha recombination during \\ion{He}{1}\nrecombination is dramatic.\nOur new detailed treatment of \\ion{He}{1} shows \\ion{He}{1}\nrecombination finishing just after the start of H recombination (see\nFig.~9),\ni.e.~significantly delayed compared with the\nSaha equilibrium case. This\nis different from the earlier calculations (e.g.~MST), in which\n\\ion{He}{1} recombination is\nfinished well before H recombination begins. In this previous case,\n\\ion{He}{1} recombination still affected the CMB anisotropy power\nspectrum on small angular scales because the diffusion damping length\ngrows continuously and is\nsensitive to the full thermal history (HSSW). In our new case,\nparticularly for our low\n$\\Omega_{\\rm B}$ models \\ion{He}{1} recombination is still finishing at the\nvery beginning of H recombination, which further affects the power spectrum\nat large angular scales (see~\\S\\ref{sec-Powerspectrum}).\nWe show a `blow-up' of the two helium recombination epochs in\nFig.~10.\n\n\\subsubsection{Physics of \\ion{He}{1} Recombination}\nThe physics of \\ion{He}{1} recombination can be summarized as follows. There\nare three major aspects to it: (1) the \\ion{He}{1} has excited states which\nare able to retain charge; but (2) being very close to the continuum, the\nhighly excited states are easily photoionized by the radiation field at\n$z \\simeq 3000$; then (3) we have a standard hydrogenic-like Case B\nrecombination, which is unaffected by neutral H removing \\ion{He}{1}\n$2^1p$--$1^1s$ (resonance line) photons.\n\n\\vspace{2mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig11.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~11.---}:\nThis figure shows why the Saha equilibrium recombination rate\n($R_{\\rm Saha}$) for\n\\ion{He}{1} is not valid. Comparing the dotted and dashed lines,\nthe photoexcitation rate (i.e.~photoabsorption rate) for He 2$^1p\\rm{-}1^1s$\n($R_{\\rm He}$) is orders\nof magnitude greater than the photoionization rate for H ($R_{\\rm H}$)\nfrom the same \\ion{He}{1} 2$^1p \\rm{-} 1^1s$\nphoton pool; there is no possibility for H to `steal' the\nphotons to speed up \\ion{He}{1} recombination. For\nSaha recombination to be valid, $R_{\\rm H} > R_{\\rm He}$, as well as\n$R_{\\rm H} \\geq R_{\\rm Saha}$. For the sCDM model shown here, \\ion{He}{1}\nrecombination begins around $z=3000$.\n\\label{fig:fracnu}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nThe \\ion{He}{1} atom has a metastable, i.e.~very slow, set of\nstates -- the triplets (e.g.~n$^3p$--n$^1s$). Therefore, overall the\nexcited states of \\ion{He}{1} can naturally retain more charge than a simple\nhydrogenic system under Boltzmann equilibrium. The situation would resemble\nwhat we found for H recombination with the enhanced populations of the\nhigher states, and would lead to faster reduction of $x_{\\rm e}$. However, the\nhigh excited states of \\ion{He}{1} are much more strongly `packed' towards the\ncontinuum compared to those of H;\nthe energy difference between the $3p$ levels and the continuum is\n$1.6\\,$eV for \\ion{He}{1} versus $1.5\\,$eV for H, \ncompared to $24.6\\,$eV versus $13.6\\,$eV for the ground state--continuum energy difference.\nThis is\nenough to depopulate the triplets (whose `ground state' is n=$2^3s$),\ngiven the much higher radiation temperature during \\ion{He}{1} recombination.\nLeft on its own under these circumstances, \\ion{He}{1} would recombine much\nlike the standard Case~B effective 3-level H atom, i.e.~slower than Saha\nrecombination. There is one possible obstacle -- it is the existence of some\nneutral H, which could `steal' \\ion{He}{1} resonance line photons,\ninvalidate the effective Case~B and make it a Saha recombination instead.\nHowever, our detailed calculation \nshows that neutral H during \\ion{He}{1} recombination is not able to\naccomplish that, and the process is {\\it not\\\/} described by Saha equilibrium.\n\nFig.~11\nshows that Saha equilibrium recombination is\ninvalid for \\ion{He}{1}, by comparing the three possible destruction\nprocesses of the \\ion{He}{1} $2^1p$--$1^1s$ (in H this is Ly$\\,\\alpha$)\nphotons: (1) cosmological redshift; (2) the $2^1s$--$1^1s$ two-photon rate;\nand (3) the photoionization rate of the ground state of H by the same\n\\ion{He}{1} $2^1p$--$1^1s$ photons.\nFig.~11\nclearly shows that process (3) is negligible\n(in contradiction to the discussion in HSSW).\nTo be doubly sure that\nabsorption of these photons by hydrogen is negligible we explicitly\nincluded the relevant rate in our models and found no discernible effects.\nIn order for \\ion{He}{1} recombination to be approximated by Saha equilibrium,\none of the 3 processes described above would have to be \nfaster than or equal to the Saha equilibrium rate, which we do not find to be\nthe case.\n\nThe recombination of \\ion{He}{1} is slow for the same reasons that H\nrecombination is,\nnamely because of the optically thick n$^1p$--$1^1s$ transitions which\nslow cascades to the ground state, and the \nexclusion of recombinations to the ground state.\nIn other words \\ion{He}{1} follows a Case B recombination. Because the\n`bottleneck' at n=2 controls recombination, it is not surprising\nthat \\ion{He}{1} and H recombination occur at a similar redshift; the\nionization energy of n=2 is similar in both. \\ion{He}{1}\nrecombination is slower than H recombination because of its different\natomic structure. The excited states of \\ion{He}{1} are more tightly packed,\nand the $2^1p$--$1^1s$ energy difference greater than that of H.\nThe strong radiation field keeps the ratio of photoionization rate\/downward\ncascade rate higher than in the H case, resulting in a slower recombination.\n\nWe find the strong radiation field also causes the triplet states to\nbe virtually unpopulated. \nThe lack of electrons in triplet states is easily understood by\nconsidering the blackbody radiation spectrum.\nAt \\ion{He}{1} recombination (z $\\simeq$ 3000), the blackbody\nradiation\npeak is around $2\\,$eV, so there are around 11 orders of magnitude\nmore photons that can\nionize the lowest triplet state 2$^3$s ($4.8\\,$eV),\nthan the singlet ground state ($24.6\\,$eV), since both are on the\nsteeply decreasing Wien tail.\nIt is interesting to note that in planetary nebulae where the young,\nhot ionizing star produces most of its energy in the UV, the opposite\noccurs: the \\ion{He}{1} atoms have few electrons in the singlet\nstates; instead most of them are in the triplet states.\n \nThere is one more possible method to speed up \\ion{He}{1}\nrecombination, and that is collisional rates between the triplets and\nsinglet states. If fast enough, the collisional rates would provide\nanother channel to keep hold of captured electrons -- by pumping them into the\ntriplet states faster than they can be reionized. The triplets are 3\ntimes as populated as the singlets due to the statistical weight factors.\nBy forcing the collisional\nrates to be greater than the recombination rates and the bound-bound\nradiative rates, we find an extremely fast He I recombination --\napproximated by the Saha equilibrium. Essentially we force electrons\nfrom the singlets into the triplets faster than they can cascade\ndownwards, and faster than they can be photoionized out of the\ntriplets. In reality, the collisions are\nnegligible, a few orders of magnitude less than the radiative rates.\nIt is important to note that\napart from collisions, the singlet and triplet states are {\\it only\\\/}\nconnected via the n$^3p$--n$^1s$ transitions, which are orders of\nmagnitude slower than the $2^3s$--$1^1s$ rate.\nWe note here that MST stated that the collisional rates were high\nenough to cause equilibrium between the triplet and singlet states.\nOne must be careful\nto compare all relevant rates, and we keep all of them in our code.\nWe find the allowed radiative rates\n(e.g.~photoexcitation and photodeexcitation) are greater than the\ncollisional rates. Therefore the allowed radiative rates control the\nexcited states' population distribution, {\\it not\\\/} the\ncollisional rates. In other words, electrons in the singlet states\nare jumping between bound singlet states faster than\nthe collisional rates can send them into the triplet states.\n\n\\subsubsection{Effective 3-level calculation for \\ion{He}{1}}\n\nWe note here that MST used an effective 3-level \\ion{He}{1} singlet atom\nand calculated\n\\ion{He}{1} recombination in the same way as the standard H calculation\n(equation~(\\ref{eq:standard_xe})) \nwith the appropriate \\ion{He}{1} parameters. When we follow their\ntreatment, we get essentially the same result as our multi-level \\ion{He}{1} calculation.\nWe are not sure why MST obtained such a fast \\ion{He}{1} recombination. \n\nAs with hydrogen, we have also investigated what is required to\nachieve an accurate solution for helium, without modeling the full\nsuite of atomic processes. We have found that the use of the\n`effective 3-level' equations for helium (as described in MST),\ntogether with an appropriate recombination coefficient for singlets only\n(equation~(\\ref{eq:hecoefficient})), results in a very accurate\ntreatment of $x_{\\rm e}(z)$ during the time of helium recombination.\nIn detail it is necessary to follow hydrogen and\nhelium recombination simultaneously, increasing the number of\ndifferential equations to solve. However, little accuracy is in fact lost\nby treating them independently -- since recombination is governed by dramatic\nchanges in time scales through Boltzmann factors and the like, and is\naffected little by small changes in the number of free electrons at a given\ntime. Further details are discussed in Seager, Sasselov, \\& Scott (1999).\n\nAlthough our model does not explicitly use a recombination coefficient,\nit does allow us to calculate one easily. \nTo aid other researchers it\nis worth presenting a fit for the singlet-only Case~B recombination\ncoefficient for \\ion{He}{1} (including recombinations to all states except the ground state) from the data in Hummer \\& Storey (1998). \nHummer \\& Storey (1998) compute photoionization cross sections that are more\naccurate than the ones we use (Hofsaess 1979), but are not publicly available. Following the functional forms used in the\nfits of Verner \\& Ferland (1996) we find\n\\begin{equation}\n\\label{eq:hecoefficient}\n\\alpha_{\\rm He}=a\\!\\left[\\sqrt{T_{\\rm M}\\over T_2}\n \\left(1+\\sqrt{T_{\\rm M}\\over T_2}\\right)^{1-b}\\!\\!\n \\left(1+\\sqrt{T_{\\rm M}\\over T_1}\\right)^{1+b}\\right]^{-1}\\!\\!\\! m^3 s^{-1},\n\\end{equation}\nwith\n$a=10^{-16.744}$m$^3$s$^{-1}$, $b=0.711$, $T_1=10^{5.114}\\,$K, and $T_2$ fixed\narbitrarily at $3\\,$K. This fit is good to $<0.1\\%$\nover the relevant temperature range (4{,}000--10{,}000\\,K), and still\nfairly accurate over a much wider range of temperatures.\n\n\\subsubsection{\\ion{He}{2} Recombination}\n\\label{sec-HeII}\n\\ion{He}{2} recombination occurs too early to affect the power spectrum\nof CMB anisotropies. For completeness, we mention it briefly\nhere. \\ion{He}{2} recombination is fast because of the very fast two\nphoton rate.\nFig.~14\nshows that for most cosmologies the two-photon\nrate is faster than the net recombination rate, meaning that as fast\nas electrons are captured from the continuum they can cascade down to\nthe ground state. Because of this, there is essentially no `bottleneck' at the\nn=2 level. In high baryon models, \\ion{He}{2} recombination can be\napproximated using the Saha recombination. As shown in\nFig.~9,\n\\ion{He}{2} recombination is slightly slower than the Saha recombination for\nlow baryon models. \n\n\\vspace{0.2in}\n\\centerline{{\\vbox{\\epsfxsize=8.25cm\\epsfbox{recombination_fig12.eps}}}}\n\\baselineskip=8pt\n\\vspace{0.3in}\n{\\footnotesize {\\sc Fig.~12.---}\nWhat controls H recombination? The net $2p$--$1s$ rate\n(dashed) compared to the $2s$--$1s$ two-photon rate (dotted) and the\nnet recombination rate (solid) for 4 different cosmologies. Except for\nlow $\\Omega_{\\rm B}$ and low $h$ models (e.g.~the sCDM model),\nthe $2s$--$1s$ rate dominates. The solid vertical line represents where\n5\\% of the atoms have recombined.\n\\label{fig:Hrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsubsection{What Controls Recombination?}\nH recombination is largely controlled by the $2s$--$1s$\ntwo-photon rate, which except for low-baryon cases, is much faster than\nthe H Ly$\\,\\alpha$ rate. The net recombination rate, net $2s$--$1s$\nrate, and net Ly$\\,\\alpha$ rate are compared for different cosmologies\nin\nFig.~12.\nFigs.~13\nand~14\nshow the same rate\ncomparison for \\ion{He}{1} and \\ion{He}{2}.\nThe three figures all have the same scale on the $x$ and $y$ axes, for\neasy comparison. \n\\ion{He}{1} recombination is controlled by the $2^1p$--$1^1s$ rate\nrather than the $2^1s$--$1^1s$ rate as previously stated (e.g.~MST).\nFig.~13\n(for \\ion{He}{1}) also illustrates the slow net\nrecombination rate, which is the primary factor in the slow Case B\n\\ion{He}{1} recombination.\nFig.~14\nalso illustrates that \\ion{He}{2} in the\nhigh $\\Omega_{\\rm B}$ and $h$ models has a $2s$--$1s$ rate faster\nthan the net recombination rate, meaning that there is no slowdown\nof recombination due to n$=2$, and the Saha equilibrium approximation\nis valid. \n\nThe rates change with cosmological model. Physically this is because\nall of the rates are very sensitive to the baryon density. The\n$2^1p$--$1^1s$ rates are further affected by the Hubble factor\nbecause the Sobolev approximation\n(equations~(\\ref{eq:indofphi}) and~(\\ref{eq:sobtau})), depends on the\nvelocity gradient. Whether most of the atoms in the Universe recombined\nvia a $2p$--$1s$ or a $2s$--$1s$ two-photon transition depends on the\nprecise values of the cosmological parameters. A confident answer to that\nquestion is still not known, given today's parameter uncertainties.\n\n\\vspace{0.2in}\n\\centerline{{\\vbox{\\epsfxsize=8.25cm\\epsfbox{recombination_fig13.eps}}}}\n\\baselineskip=8pt\n\\vspace{0.3in}\n{\\footnotesize {\\sc Fig.~13.---}\nWhat controls \\ion{He}{1} recombination? The net\n$2^1p$--$1^1s$ rate (dashed) compared to $2^1s$--$1^1s$\ntwo-photon rate (dotted)\nand the net recombination rate (solid) for 4\ndifferent cosmologies. Except for high $\\Omega_{\\rm B}$ and high $h$\nmodels, the $2^1p$--$1^1s$ rate dominates, in contrast to H. The solid\nvertical line represents where 5\\% of the atoms have recombined.\n\\label{fig:HeIrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\vspace{0.2in}\n\\centerline{{\\vbox{\\epsfxsize=8.5cm\\epsfbox{recombination_fig14.eps}}}}\n\\baselineskip=8pt\n\\vspace{0.25in}\n{\\footnotesize {\\sc Fig.~14.---}\nWhat controls \\ion{He}{2} recombination? The net $2p$--$1s$\nrate (dashed) compared to the $2s$--$1s$ two-photon rate (dotted) and\nthe net recombination rate (solid) for 4 different cosmologies. Except\nfor low $\\Omega_{\\rm B}$ and low $h$ \nmodels, the $2s$--$1s$ rate dominates during recombination, and the\n$2p$--$1s$ at the start of recombination. The solid vertical line\nrepresents where 5\\% of the atoms have recombined.\n\\label{fig:HeIIrates}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsection{Atomic Data and Estimate of Uncertainties}\n\\label{sec-AtomicData} \nOur approach in this work has been to include all relevant degrees of freedom\nof the recombining matter in a consistent and coupled manner. This requires\nspecial attention to the quality of the atomic data used. The challenge\nlies in building a consistent model for $all$ energy levels and transitions,\nnot just for the low-lying ones, which are often better known experimentally\nand theoretically.\n\n\\subsubsection{H and \\ion{He}{2}}\nHydrogen (and the hydrogenic ion of helium) have exactly known rate\ncoefficients for radiative processes from precise quantum-mechanical\ncalculations (uncertainties below 1\\%). We use exact values for the\nbound-bound radiative transitions and for radiative recombination,\nas in e.g.~Hummer (1994). For more details see Hummer \\& Storey (1987),\nbut also Brocklehurst (1970) and Johnson (1972). In particular, the\nrate of radiative recombination to level n of a hydrogenic ion can \nbe evaluated from the photoionization (bound-free) cross section for\nlevel n, ${\\sigma}_{{\\rm nc}}(\\nu)$, with the standard assumption of detailed\nbalance (see~\\S\\ref{sec-Recomb}).\nFor hydrogenic bound-free cross sections, we follow in essence\nSeaton's work (Seaton 1959) with its \nasymptotic expansion for the Gaunt factor (see Brocklehurst 1970).\nNote that the weak dependence of the Gaunt factor on wavelength has\na noticeable effect in our final recombination rate calculation.\nGiven our application, we do not require the resolution of resonances,\nas achieved for a few transitions by the Opacity Project (TOPbase,\nCanto et al.~1993).\nLike Hummer (1994), we work with the n-levels assuming that the\n$\\ell$-sublevels have populations proportional to (2$\\ell$+1). The\nresulting uncertainties for hydrogenic radiative rates at low\ntemperatures (T~$\\leq~$10$^5$K) certainly do not exceed the 1\\% level.\n\nCollisional rate coefficients cannot be calculated exactly.\nSo, compared to the hydrogenic radiative rate coefficients, the\nsituation for the bound-bound collisional rates\nand collisional ionization is poor, with errors typically about 6\\%,\nand as high as 20\\% in some cases (Percival \\& Richards 1978).\nA number of methods are used to evaluate\nelectron-impact excitation cross sections of hydrogen-like ions (Fisher\net al.~1997). These most recent values compare well to the older\nsources (Johnson 1972; Percival \\& Richards 1978). The helium ion, \\ion{He}{2},\nis hydrogenic and was treated accordingly. We basically\nfollowed Hummer \\& Storey (1987) and Hummer (1994) in building the model\natom. For our application,\ncollisional processes are negligible, so that the large\nuncertainties that still persist for the collisional rates have no\nimpact on our results.\n\n\\subsubsection{\\ion{He}{1}}\nHelium, in its neutral state, poses a challenge for building a multi-level\natomic model of high precision. Unlike atomic hydrogen, no exact solutions\nto the Schr\\\"odinger equation are available for helium. However, very high\nprecision approximations are now available (Drake 1992, 1994)\nwhich we have used. These approximations are essentially exact for all\npractical purposes. The largest \nrelativistic correction comes from singlet-triplet mixing between states\nwith the same n, $L$, and $J$, but is still small.\nTransition rates were calculated following\nthe recent comprehensive \\ion{He}{1} model built by Smits (1996) and some\nvalues in Theodosiou (1987). The source of our photoionization cross\nsections was TOPbase (Cunto et al.~1993) and Hofsaess (1979) for small n;\nabove n=10 we used scaled hydrogenic values.\nNew detailed calculations (Hummer \\& Storey 1998)\nshow that the \\ion{He}{1} \nphotoionization cross sections become strictly hydrogenic at about n~$>20$.\nThe uncertainties in the \\ion{He}{1} radiative rates are at the 5\\%\nlevel and below.\n\nThe situation with the collisional rates for \\ion{He}{1} is predictably much\nworse than for \\ion{He}{1} radiative rates, with good R-matrix\ncalculations existing only for n~$\\leq 5$ (Sawey \\&\nBerrington 1993). The collisional rates at large n are a crucial ingredient\nin determining the amount of singlet-triplet mixing, but fortunately\ncollisions are not very important for the low density conditions in the early\nUniverse, so the large uncertainty in these rates does not effect\nour calculation. The Born approximation,\nwhich assumes proportionality to the radiative transition rates, is used\n(see Smits 1996) to calculate the collisional cross sections for large n.\n\nFor the $2^1s$--$1^1s$ two-photon rate for \\ion{He}{1} we used the value\n$\\Lambda_{\\rm He I}=51.3\\,{\\rm s}^{-1}$ (Drake et al. 1969)\nwhich differs from a previously used value (Dalgarno~1966) by\n$\\sim10\\%$. An uncertainty even of this magnitude\nwould still make little difference in the final results.\nFor the \\ion{He}{2} $2s$--$1s$ two-photon rate we used the value\n$\\Lambda_{\\rm He I}=526.5\\,{\\rm s}^{-1}$ (for hydrogenic ions this is\nessentially $Z^6$ times the value for H) from Lipeles et al.~(1965).\nDielectronic recombination for \\ion{He}{1} is not at all\nimportant during \\ion{He}{1} recombination. While dielectronic\nrecombination dominates at temperatures above\n$6\\times10^4$K, for the range of temperatures relevant here it\nis at least 10 orders of magnitude below the radiative recombination\nrate (using the fit referred to in Abel et al.~1997).\n\n\\subsubsection{Combined Error From Atomic Data}\nWe have gathered together the uncertainties in the atomic data in\norder to estimate the resulting uncertainty in our derivation of $x_{\\rm e}$.\nThe atomic data with the dominant effect on our calculation are the set\nof bound-free cross sections for all H and \\ion{He}{1} levels -- not so much \nany individual values, but the overall consistency of the sets (which\nare taken from different sources). The differences between our\nmodel atom and Hummer's (1994) reflect the uncertainty in the atomic\ndata. To test the effect on our hydrogenic results, we compared the $x_{\\rm e}$\nresults of an effective 3-level atom using Hummer's (1994)\nrecombination coefficient with the\nresults using a recombination coefficient calculated with our own model H\natom. We find maximum\ndifferences of 1\\% at $z=300$, which corresponds to measurable effects on\nCMB anisotropies of much less than 1\\%.\n\nThe error in $x_{\\rm e}$ arising from \\ion{He}{1} is more difficult to\ncalculate. We estimate it to be considerably less than 1\\%, because the low\nlevel (n~$\\leq$~4)\nbound-bound and bound-free radiative rates dominate \\ion{He}{1}\nrecombination, and as described above, those data are accurate.\n\n\\subsection{Secondary Distortions in the Radiation Field}\n\\label{sec-Distortions}\nIn our recombination calculation we follow `secondary' distortions in the radiation\nfield that could affect the recombination process at a later time. The\nsecondary distortions are caused by the primary distortions that are\nfrozen into the radiation field. At a later time they are redshifted\ninto interaction frequency with other atomic transitions.\nExplicitly, we follow: \\newline\n(1) H Ly$\\,\\alpha$ photons; \\newline\n(2) H $2s$--$1s$ photons. \\newline\nBy the time of H recombination these photons have been redshifted\ninto an energy range where they\ncould photoionize H(n=2). In addition we follow: \\newline\n(3) \\ion{He}{1} $2^1p$--$1^1s$; \\newline \n(4) \\ion{He}{1} $2^1s$--$1^1s$. \\newline\nBy the time of H recombination these photons have been\nredshifted into an energy range which could photoionize H(n=1).\nAnd finally we also follow: \\newline\n(5) \\ion{He}{2} Ly$\\,\\alpha$ photons; \\newline\n(6) \\ion{He}{2} $2s$--$1s$ photons. \\newline\nBy the time of H recombination these photons have been redshifted\ninto an energy range\nwhich could photoionize H(n=1). These \\ion{He}{2} photons bypass\n\\ion{He}{1} because the photons have not been\nredshifted into a suitable energy range for interaction.\n\nHere we only attempt to investigate the maximum effects of secondary spectral\ndistortions. To that end we do not include additional distortions\nwhich are smaller. For example,\nLyman lines other than (1), (3), and (5), whose distortions are smaller \nthan Ly$\\,\\alpha$,\nwill produce a comparably smaller feedback on photoionization.\nThe \\ion{He}{1} singlet recombination photons could theoretically\nphotoionize \\ion{He}{1} triplet states, but as previously discussed\nthere are virtually no electrons in the triplet states, so this\nprocess is also negligible. Another possible effect is due to the\nsimilar energy levels of H and \\ion{He}{2}: $\\Delta E_{\\rm He II} =\n4\\Delta E_{\\rm H}$. For example, the transition from\n\\ion{He}{2} (n=4) to (n=2) produces the same frequency photons as the\ntransition from H~(n=2) to (n=1). These transitions are\ntheoretically competing for photons, and this effect can be important for other\nastrophysical situations (e.g.~planetary nebulae) where H and \\ion{He}{2}\nsimultaneously exist. However, any such effect is negligible for primeval\nrecombination because during \\ion{He}{2} recombination the amount of\nneutral H is very small ([H\/\\ion{He}{2}] $< 10^{-8}$), and during\nH recombination, there is almost no \\ion{He}{2}\n([\\ion{He}{2}\/H]$< 10^{-10}$).\n\nBecause we are only investigating maximum effects we assume the photons\nwere emitted at line center and are redshifted undisturbed until their\ninteraction with H(n=1) or H(n=2) as described above. We also \nassume two photons at half the energy for the $2s$--$1s$ transitions,\ncompared to the $2p$--$1s$ transitions.\nThe distorting photons emitted at a time $z_{\\rm em}$ are\nabsorbed at a later time $z$, where\n\\begin{equation}\nz = z_{\\rm em} \\nu_{\\rm edge}\/\\nu_{\\rm em}.\n\\end{equation}\nHere $\\nu_{\\rm edge}$ is the photoionization edge frequency where the\nphotons are being absorbed, and $\\nu_{\\rm em}$ is the photon's\nfrequency at emission.\nThe distortions are calculated as\n\\begin{eqnarray}\n\\lefteqn{J(\\nu,z) = {h \\nu(z) c}\\,\np_{ij}(z_{\\rm em})\\,\\times}\\nonumber \\\\\n& & \\qquad \\bigg\\{ n_j(z_{\\rm em})\\Big[A_{ji}\n+ B_{ji}B\\big(\\nu_{\\rm em},T_{\\rm R}(z_{\\rm em})\\big)\\Big]\\nonumber \\\\\n& & \\qquad\\qquad\\quad \\mbox{}- n_i(z_{\\rm em})B_{ij}\n B\\big(\\nu_{\\rm em},T_{\\rm R}(z_{\\rm em})\\big)\\bigg\\} ,\n\\end{eqnarray}\nwhere: $B(\\nu_{\\rm em},T_{\\rm R}(z_{\\rm em}))$ is the Planck function at the\ntime of emission; $A_{ji}, B_{ji}$ and $B_{ij}$ are the Einstein\ncoefficients; $p_{ij}$ is the Sobolev escape probability for the\nline; and the other variables are as described previously.\n\nThe distortions (1) and (2) were previously discussed by Rybicki\n\\& Dell'Antonio (1992). They pointed out that the effect from (1) should\nbe small, because the Ly$\\,\\alpha$ distortion must be redshifted by at\nleast a factor of 3 to have any effect. This means that the Ly$\\,\\alpha$\nphotons produced at $z\\lesssim2500$\nwill only affect the Balmer continuum at $z\\lesssim800$ when the\nrecombination process (and any possibility of photoionization) is\nalmost entirely over.\n\nWe find that including the distortions (1) through (6) improves\n$x_{\\rm e}$ during H\nrecombination at\nless than the $0.01\\%$ level.\nThis difference is far too small to make a\nsignificant change in the power spectrum, and it is negligible\ncompared to the major improvements in this paper, which are \nthe level-by-level treatment of H, \\ion{He}{1}, and \\ion{He}{2},\nallowing departures of the excited state populations from\nan equilibrium distribution, calculating recombination directly to\neach excited state, and the correct treatment of\n\\ion{He}{1} triplet and singlet states.\nHowever, the removal of these distorting photons by photoionization\nmust be taken into account when calculating spectral distortions to\nthe CMB blackbody, which we plan to study in a later paper.\n\n\\subsection{Chemistry}\nIncluding the detailed hydrogen chemistry (see~\\S\\ref{sec-Chem})\nmarginally affects the fractional abundances of protons and electrons at\nlow $z$. However, the correction is of the order\n$10^{-2}x_{\\rm e}$ at $z<150$.\nThis change in the electron density\nwould change the Thomson scattering optical depth by $\\sim 10^{-5}$,\ntoo little to make a difference in the CMB power spectrum.\n\n\\vspace{3mm}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig15a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig15b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~15.---}\nThe effect of the improved treatment of recombination on H\nchemistry. Shown is the standard CDM model. Solid lines are values\nfrom the standard calculation, dashed and dotted lines from our\nimproved results.\n\\label{fig:chem}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nOn the other hand, as shown in\nFig.~15,\nthe different\n$x_{\\rm e}(z)$ that we find will lead to \nfractional changes of similar size in molecular abundances at low $z$, since\nH$_2$ for example is formed via H$^{-}$ which is affected by the residual\nfree electron density. The delay in\n\\ion{He}{1} recombination compared to previous studies causes a\nsimilar delay in formation of He molecules (P.~Stancil, private\ncommunication). However, with the exception of He$_2^+$, no\nchanges are greater than those caused by the residual\nfree electron density at freeze-out. Since molecules can be important\nfor the cooling of primordial gas clouds and the formation of the first\nobjects in the Universe, the precise determination of molecular abundances is\nan important issue (e.g.~Lepp \\& Shull 1984, Tegmark et al.~1997, Abel\net al.~1997, Galli \\& Palla~1998). However, the roughly 10-20\\% change\nin the abundance of some chemical species is probably less than other\nuncertainties in the reaction rates (A.~Dalgarno, private communication).\nWith this in mind, we suspect no drastic implications for theories of\nstructure formation. \n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig16a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig16b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~16.---}\nDifferences in CMB power spectra arising from the improved treatment\nof hydrogen for: (a) the standard CDM parameters\n$\\Omega_{\\rm tot} = 1.0$, $\\Omega_{\\rm B} = 0.05$, $h =0.5$,\n$Y_{\\rm P} = 0.24$, $T_0 = 2.728\\,$K; and (b) an extreme baryon model\nwith $\\Omega_{\\rm B}=1$, $h=1.0$ (for which there is relatively little\neffect). The fractional\ndifference plotted is between our new hydrogen recombination calculation and\nthe standard hydrogen recombination calculation (e.g.~in HSSW), with the\nsense of $C_\\ell^{\\rm new}-C_\\ell^{\\rm old}$, and with the two calculations\nnormalized to have the same amplitude for the initial conditions.\nThe solid lines are for\ntemperature, while the dashed lines are for the (`E'-mode)\npolarization power spectrum.\n\\label{fig:h_powerspectrum}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\n\\subsection{Power Spectrum}\n\\label{sec-Powerspectrum}\nEven relatively small differences in the recombination history of the\nUniverse can have potentially measurable effects on the CMB\nanisotropies. And so we might expect our two main changes (one in H\nand one in He) to be noticeable in the power spectrum.\nAs a first example\nFig.~16\ncompares the difference in the anisotropy power spectrum derived\nfrom our new $x_{\\rm e}(z)$ to that derived from the standard\nrecombination $x_{\\rm e}$ (essentially identical to that described in\nHSSW), for hydrogen recombination only. Here the $C_{\\ell}$s are\nsquares of the amplitudes in a spherical \nharmonic decomposition of anisotropies on the sky (the azimuthal index\n$m$ depends on the choice of axis, and so is irrelevant for an\nisotropic Universe). They represent the power and angular scale of\nthe CMB anisotropies by describing the rms temperatures at fixed\nangular separations averaged over the whole sky (see e.g.~White, Scott,\n\\& Silk~1994). These $C_{\\ell}$s\ndepend on the ionization fraction $x_{\\rm e}$\nthrough the precise shape of the thickness of the photon last\nscattering surface (i.e.~the visibility function).\nSince the detailed shape of the power spectrum may allow\ndetermination of fundamental cosmological parameters,\nthe significance of the change in $x_{\\rm e}$ is evident.\nTo determine the effect of the change in $x_{\\rm e}$ we have\nused the code {\\tt cmbfast} written and made available by Seljak\n\\& Zaldarriaga (1996), with a slight modification to allow for the input\nof an arbitrary recombination history.\n\nThe dominant physical affect arising from the new H calculation comes\nfrom the change in $x_{\\rm e}$ at low $z$. A process seldom mentioned in\ndiscussions of CMB anisotropy physics (which are otherwise quite\ncomprehensive, e.g.~Hu, Silk, \\& Sugiyama~1997) is that the low-$z$\ntail of the visibility function results in {\\it partial erasure\\\/} of the\nanisotropies produced at $z\\sim1000$. The optical depth in Thomson\nscattering back to, for example, $z=800$\n($\\tau=c\\sigma_{\\rm T}\\int n_{\\rm e}(dt\/dz)\\,dz$)\ncan be several percent. This partial rescattering of the photons\nleads to partial erasure of the $C_\\ell$s by an amount ${\\rm e}^{-2\\tau}$. Let\nus look at the standard CDM calculation first\n(Fig.~16(a)).\nOur change in the optical depth back to $z\\simeq800$ (see\nFig.~2)\nis around 1\\% less than that obtained\nusing the standard\ncalculation, and so we find that the anisotropies suffer less partial erasure\nby about 2\\%.\nThere is no effect on angular scales larger than the horizon at the\nscattering epoch (here redshifts of several hundred), so that all multipoles\nare effected except for the lowest hundred or so $\\ell$s. Hence this\neffect is largely a change in the overall normalization of the power spectrum,\nwith some additional differences at low $\\ell$ which will be masked by the\n`cosmic variance'. In addition there are smaller effects due to changes in\nthe {\\it generation\\\/} of anisotropies in the low-$z$ tail, giving small\nchanges in the acoustic peaks, which can be seen as wiggles in the figure.\nSince the partial erasing effect is essentially unchanged in the case of the\n$\\Omega_{\\rm B}=h=1.0$ model, these otherwise sub-dominant effects are more\nobvious in\nFig.~16(b).\n\nDifferences in the power spectra are rather small in absolute terms,\nso\nFig.~16\nplots the relative difference. We\nhave shown this for our two chosen models, one being standard Cold Dark Matter\n(a), which we will refer to as sCDM,\nand the other being an extreme baryon-only model (b). These models are\nmeant to be representative only, and changes in cosmological parameters will\nresult in curves which differ in detail. We describe how to calculate an\napproximately correct recombination history for arbitrary models in a\nseparate paper (Seager, Sasselov, \\& Scott~1999). Since the main effect is\nsimilar to an overall amplitude change, we normalized our CMB power spectra\nto have\nthe same large-scale matter power spectrum, which is equivalent to normalizing\nto the same amplitude for the initial conditions. The amplitude of the\neffect of our new H calculation clearly depends on the cosmology. For the high\n$\\Omega_{\\rm B}$ and $h$ case, the freeze-out value of $x_{\\rm e}$ is\nmuch smaller (around $10^{-5}$),\nand since the fractional change in $x_{\\rm e}$ is similarly $\\sim10$\\%,\nthe absolute change in ionization fraction is much lower than for the sCDM\nmodel. The integrated optical depth is directly proportional to\n$\\Omega_{\\rm B}h\\Delta x_{\\rm e}$, which is small, despite the increase in\n$\\Omega_{\\rm B}$ and $h$. Hence we see a much smaller increase from our\nhydrogen improvement in\nFig.~16(b).\nThe normalization\nchange is rather difficult to see, since it is masked by relatively small\nchanges around the power spectrum peaks, giving wiggles in the difference\nspectrum.\n\nThe dashed lines in\nFig.~16\nshow the effect\non the power spectrum for CMB polarization. In standard models polarization\nis typically at the level of a few percent of the anisotropy signal, and\nso will be difficult to measure in detail (see Hu \\& White~1997b for a\ndiscussion of CMB polarization).\nWe show the results here to indicate that there\nare further observational consequences of our improved recombination\ncalculation (explicitly we have plotted the `E' mode of polarization, see\ne.g.~Seljak 1997). The effect of our improvements on the polarization can be\nunderstood similarly through the visibility function. Since the polarization\npower spectrum tends to have sharper acoustic peaks, the wiggles in the\ndifference spectrum are more pronounced than for the temperature anisotropies.\nNote that the large relative differences at low $\\ell$\nare actually very small in absolute terms, since the polarization\nsignal is so small there. The polarization-temperature correlation power\nspectrum and the `B' mode of polarization (for models with gravity\nwaves) could also be plotted, but little extra insight is gained, and\nso we avoid this for the sake of clarity.\n\nThe other major difference we find compared with previous treatments\nis in the delayed recombination of \\ion{He}{1}.\nIn\nFig.~17\nwe show the effect of our new \\ion{He}{1} calculation, again as a fractional\nchange in the CMB anisotropy power spectrum versus multipole\n$\\ell$. The change in the recombination of \\ion{He}{1} affects the density\nof free electrons just before\nhydrogen recombination, which in turn affects the diffusion of the\nphotons and baryons, and hence the damping scale for the acoustic\noscillations which give rise to the peaks in the power spectrum.\nThe phases of the acoustic oscillations will also be affected\nsomewhat, which shows up in the wiggles in the difference spectrum.\nFor CDM-like models the\nmain effect is the change in the damping scale, since we now think there are\nmore free electrons at $z\\sim1500$--2000. The resulting change in the\n$C_\\ell$s is essentially the same as assuming the wrong angular scale for the\ndamping of the anisotropies (see Hu \\& White~1997a), which is\nthe same physical effect that HSSW found in arguing\nfor the need to include \\ion{He}{1} recombination {\\it at all\\\/} for obtaining\npercent accuracy in the $C_\\ell$s. The effect of this improved \\ion{He}{1} on\nthe power spectrum will depend on the background cosmology through the\nbaryon density ($\\propto\\Omega_{\\rm B}h^2$) and the horizon size at\nlast scattering through $\\Omega_0h^2$ -- hence there is no simple\nfitting formula, and it is necessary to calculate the effect on the\nanisotropy damping tail for each cosmological model considered.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig17a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig17b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~17.---}\nThe effect of the improved treatment of helium on the CMB power\nspectra for: (a) the standard CDM model; and (b) the extreme baryonic model\n(for which there is essentially no change). The fractional\ndifference plotted is between our new helium recombination calculation and\nthe assumption that helium follows Saha equilibrium (as in HSSW), with\nthe same `effective 3-level' hydrogen recombination used in both cases.\nAgain the sense is $C_\\ell^{\\rm new}-C_\\ell^{\\rm old}$, solid lines are\ntemperature, and dashed lines are polarization.\n\\label{fig:he_powerspectrum}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nThere are really two parts to the \\ion{He}{1} effect. Firstly the extra\n$x_{\\rm e}$ makes the tight coupling regime tighter, so that the\nphoton mean free path is shorter, and the length scale for diffusion\nis smaller. Secondly, the effective damping scale comes from an\naverage over the visibility function, so an increase in the high-$z$\ntail also leads to a smaller damping scale. The CMB anisotropies can\nbe thought of as a series of acoustic peaks multiplied by a roughly\nexponential damping envelope, with the characteristic multipole of the\ncut-off being determined by the damping length scale. As a result of\nthis smaller damping scale, the high $\\ell$ part of the power\nspectrum is {\\it less\\\/} suppressed, and so we see an increase in\nFig.~17(a)\ntowards high $\\ell$. For the\n$\\Omega_{\\rm B}=h=1$ model\n(Fig.~17(b))\nwe\nsee only a very small effect at the highest $\\ell$s.\nThis is easily understood by examining\nFig.~10,\nwhere we see that \\ion{He}{1} recombination is\npushed back to higher redshifts than for the sCDM case, and also that H\nrecombination happens earlier, which, together with the higher\n$\\Omega_{\\rm B}$ and $h$,\nshifts the peak of the visibility function to lower redshifts relative\nto the recombination curve. Hence the the high-$z$ tail of the\nvisibility function is much less affected in this case, and our\nimproved \\ion{He}{1} calculation has essentially negligible effect. However,\nfor less extreme models we find that the \\ion{He}{1} effect is always at least\nmarginally significant.\n\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig18a.eps}}}}\n\\centerline{{\\vbox{\\epsfxsize=8cm\\epsfbox{recombination_fig18b.eps}}}}\n\\baselineskip=8pt\n{\\footnotesize {\\sc Fig.~18.---}\nThe total effect of our improvements on the CMB power\nspectra for: (a) the standard CDM model; and (b) an extreme baryonic model.\nThese plots are essentially the sum of the separate effects of hydrogen\nand helium.\n\\label{fig:tot_powerspectrum}\n}\n\\vspace{2mm}\n\n\\baselineskip=11pt\n\nTaking the two main effects together, for the sCDM model,\nwe find that they have essentially the\nsame sign, so that the total effect of our new calculation is more\ndramatic (see\nFig.~18(a)).\nBoth effects lead to a slight increase in the anisotropies,\nparticularly at small angular scales (high $\\ell$). Although the\nexact details will depend on the underlying cosmological model, we see\nthe same general trend for most parameters we have looked at. The\nchange can be\nhigher than 5\\% for the smallest scales, although it should be\nremembered that the absolute amplitude of the anisotropies at such scales is\nactually quite low. Nevertheless, changes of this size are well above the\nlevel which is relevant for future determinations of the power\nspectrum. As a rough measure, the cosmic variance at $\\ell\\sim1000$\nis about 3\\%. Hence\na 1\\% change over a range of say 1000 multipoles is something like a\n$10\\sigma$ effect for a cosmic variance limited experiment.\nWhat we can see is that although the effects are far from astonishing,\nthey are at a level which is potentially measurable. Hence our improvements\nare significant in terms of using future CMB data-sets to infer the values\nof cosmological parameters; if not properly taken into account\nthese subtle effects in the\natomic physics of hydrogen and helium might introduce biases in the\ndetermination of fundamental parameters.\n\n\\subsection{Spectral Distortions to the CMB}\n\\label{sec-Future}\nWith the model described in this paper we plan to calculate spectral\ndistortions to the blackbody radiation of the CMB today (see also\nDubrovich~1975; Lyubarsky \\& Sunyaev 1983;\nFahr \\& Loch 1991; Dell'Antonio \\& Rybicki~1993; Burdyuzha \\& Chekmezov 1994;\nDubrovich \\& Stolyarov 1995, 1997; Boschan \\& Biltzinger~1998).\nThe main emission from Ly$\\,\\alpha$ and the two-photon process will\nbe in the far-infrared part of the spectrum.\nTransitions among the very high energy levels, which have very\nsmall energy separations, may produce spectral distortions in the\nradio. Although far weaker than distortions by the lower Lyman lines,\nthey will be in a spectral region less contaminated by background\nsources.\n\nDetecting such distortions will not be easy, since they are generally swamped\nby Galactic infra-red or radio emission and other foregrounds.\nOur new calculation\ndoes not yield any vast improvement in the prospects for detection. However,\nconfirmation of the presence of these recombination lines would be a\ndefinitive piece of supporting evidence for the whole Big Bang paradigm.\nMoreover, detailed measurement of the lines, if ever possible to carry out,\nwould be a direct diagnostic of the recombination process. For these\nreasons we will present spectral results elsewhere.\n\n\\section{Conclusions}\n\nOne point we would like to stress is that\nour detailed calculation agrees very well\nwith the results of the effective 3-level atom. This underscores the\ntremendous achievement of Peebles, Zel'dovich and colleagues in so fully\nunderstanding cosmic recombination 30 years ago. However, the great goal\nof modern cosmology is to determine the cosmological parameters to an\nunprecedented level of precision, and in order to do so it is now\nnecessary to understand very basic things, like recombination, much more\naccurately. \n\nWe have shown that improvements upon previous recombination calculations\nresult in a roughly 10\\% change in $x_{\\rm e}$ at low redshift for most\ncosmological models, plus a substantial delay in \\ion{He}{1}\nrecombination, resulting in a few percent change in the CMB\npower spectrum at small angular scales.\nSpecifically, the low redshift difference in $x_{\\rm e}$ is due to the\nH excited states' departure from an equilibrium distribution. This in turn\ncomes from the level-by-level treatment of a 300-level H atom, which\nincludes all bound-bound radiative rates, and which allows\nfeedback of the disequilibrium of the excited states on\nthe recombination process. The large improvement in\n$x_{\\rm e}$ during \\ion{He}{1} recombination comes from the\ncorrect treatment of the atomic levels, including\ntriplet and singlet states. While\nit was already understood that \\ion{He}{1} recombination would\naffect the power spectrum at high multipoles (HSSW), our improved\n\\ion{He}{1} recombination affects even the start of H recombination for\ntraditional low $\\Omega_{\\rm B}$ models. There is thus a substantially\nbigger change in the $C_\\ell$s, reaching to larger angular scales.\n\nCareful use of $T_{\\rm M}$ rather than $T_{\\rm R}$ can also have\nnoticeable consequences, as to a lesser extent can the treatment of\nLy$\\,\\alpha$ redshifting using the Sobolev escape probability.\nOur other new contributions to the recombination calculation produce negligible \ndifferences in $x_{\\rm e}$. Collisional excitation and ionization for\nH, \\ion{He}{1} and for \\ion{He}{2} are of little importance. Inclusion\nof additional cooling and heating terms in the evolution of $T_{\\rm M}$ also\nproduce little change in $x_{\\rm e}$. The largest spectral distortions do not\nfeed back on the recombination process to a level greater than\n$0.01\\%$ in $x_{\\rm e}$. Finally, the H chemistry occurs too low in\nredshift to make any noticeable difference in the CMB power spectrum. \n\nAlthough we have tried to be careful to consider every process we can\nthink of, it is certainly possible that other subtle effects remain to\nbe uncovered. We hope that we do not have to wait another 30\nyears for the next piece of substantial progress in understanding how\nthe Universe became neutral.\n\n\\acknowledgments\nThe program {\\tt recfast}, which performs approximate calculations of the\nrecombination history is available at\n{\\tt http:\/\/www.astro.ubc.ca\/people\/scott\/recfast.html} (FORTRAN version) and\nat {\\tt http:\/\/cfa-www.harvard.edu\/ ${\\sim}$sasselov\/rec\/} (C version).\nWe would like to thank George Rybicki, Ian Dell'Antonio, Avi Loeb, and\nHan Uitenbroek for many useful conversations. Also David Hummer and\nAlex Dalgarno for discussions on the atomic physics, Alex Dalgarno and\nPhil Stancil for discussions on the chemistry, Martin White, Wayne Hu\nand Uro{\\v s} Seljak for discussion on the cosmology, and Jim Peebles\nfor discussions on several aspects of this work. We thank the referee\nfor a careful reading of the manuscript. Our study of effects\non CMB anisotropies was made much easier through the availability of\nMatias Zaldarriaga and Uro{\\v s} Seljak's code {\\tt cmbfast}. DS is\nsupported by the Canadian Natural Sciences and Engineering Research\nCouncil.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe study of centrality in networks goes back to the late forties. Since then,\nseveral measures of centrality with different properties have been published (see~\\cite{BoVAC} for a survey).\nTo sort out which measures are more apt for a specific application, one can try\nto classify them by means of some axiom that they might satisfy or not. \n\nIn a previous paper~\\cite{BLVRMCM}, two of the authors have studied in particular\n\\emph{score monotonicity}~\\cite{BoVAC} and \\emph{rank monotonicity} on directed graphs.\nThe first property says that when an arc\n$x\\to y$ is added to the graph, the score of $y$ strictly increases.\nRank monotonicity~\\cite{CDKLEAA}\nstates that after adding an arc $x\\to y$, all nodes with a score\nsmaller than or equal to $y$ have still a score smaller than or equal to $y$.\nScore and rank monotonicity complement themselves. Score\nmonotonicity tells us that ``something good happens''. Rank monotonicity\nthat ``nothing bad happens''. \n\nOnce we move to undirected graphs, however, previous definitions and results are no longer\napplicable. Note that adding a single edge to an undirected graph is equivalent to adding \\emph{two} opposite arcs\nin a directed graph, which may suggest why the situation is so different. In this paper, we\npropose definitions that are natural extensions of the directed case, and prove\nresults about classical types of spectral ranking~\\cite{VigSR}---eigenvector\ncentrality~\\cite{LanZRWT,BerTGA}, Seeley's\nindex~\\cite{SeeNRI}, and PageRank~\\cite{PBMPCR}. With minor restrictions,\nall these measures of centrality have been proven to be score and rank monotone in the directed case~\\cite{BLVRMCM}.\nHowever, we will prove that, surprisingly, this is no longer true in the undirected\ncase: in the case of eigenvector centrality and PageRank, at least one of the extremes of the edge might\nlower both its score and its rank.\n\nTo prove general results in the case of PageRank, we use the theory of graph fibrations~\\cite{BoVGF},\nwhich makes us able to reduce the computation of a spectral ranking of a graph of variable size to a similar\ncomputation on a finite graph.\nThis approach to proofs, which we believe is of independent interest,\nmakes it possible to use analytic techniques to control the PageRank values.\n\nWe conclude the paper with some anecdotal evidence from a medium-sized real-world\nnetwork, showing that violations of rank monotonicity do happen.\n\n\\section{Graph-theoretical preliminaries}\n\\label{sec:defs}\n\nWhile we will focus on simple undirected graphs, we are going to make use of\nsome proof techniques that require handling more general types of graphs.\n\nA \\emph{(directed multi)graph} $G$ is defined by a set $N_G$ of nodes, a set\n$A_G$ of arcs, and by two functions $s_G,t_G:A_G\\to N_G$ that specify the\nsource and the target of each arc (we shall drop the subscripts whenever no\nconfusion is possible); a \\emph{loop} is an arc with the same source and target.\nWe use $G(i,j)$ for denoting the set of\narcs from node $i$ to node $j$, that is, the set of arcs $a\\in A_G$ such that $s(a)=i$ and\n$t(a)=j$; the arcs in $G(i,j)$ are said to be \\emph{parallel} to one\nanother. \nSimilarly, we denote with $G(-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:,i)$ the set of arcs coming\ninto $i$, that is, the set of arcs $a\\in A_G$ such that $t(a)=i$, and analogously with\n$G(i,-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:)$ the set of arcs going out of $i$. \nFinally, we write $d_G^+(i)=|G(i,-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:)|$ for the \\emph{outdegree}\nof $i$ in $G$ and $d_G^-(i)=|G(-} % in alternative to... \\newcommand{\\noarg}{\\:\\cdot\\:,i)|$ for the \\emph{indegree} of $i$ in\n$G$.\n\nThe main difference between this definition and the standard definition of a\ndirected graph is that we allow for the presence of multiple arcs between any\npair of nodes.\nSince we do not need to distinguish between graphs that only differ because of node names, we will\nalways assume that $N_G=\\{\\,0,1,\\dots,n_G-1\\,\\}$ where $n_G$ is the number of nodes of $G$.\nEvery graph $G$ \nhas an associated $n \\times n$ \\emph{adjacency matrix}, also denoted by $G$, where $G_{ij}=|G(i,j)|$. \n\nA \\emph{(simple) undirected graph} is a loopless\\footnote{Note that our negative results are \\emph{a fortiori} true if we consider\nundirected graphs with loops. Our positive results are still valid in the same case\nusing the standard convention that loops increase the degree by two.} graph $G$ such that for all $i,j \\in N$,\n$|G(i,j)|=|G(j,i)|\\leq 1$. In other words, there are no parallel arcs and if there is an\narc from $i$ to $j$ there is also an arc in the opposite direction.\nIn an undirected graph, an \\emph{edge} is an unordered set of nodes $\\{\\,i,j\\,\\}$ (simply denoted by $i \\scalebox{0.5}[1.0]{-} j$) such that $|G(i,j)|=1$; \nthe set of all edges will be denoted by $E_G$; obviously, the number of edges is exactly half of the number of arcs. \nFor undirected graphs,\nwe prefer to use the word ``vertex'' instead of ``node'', and use $V$ (instead of $N$) for the set of vertices\nand $d(x)$ for the degree of a vertex $x$.\n\n\n\n\\section{Score and rank monotonicity axioms on undirected graphs}\n\nOne of the most important notions that researchers have been trying to capture\nin various types of graphs is ``node centrality'':\nideally, every node (often representing an\nindividual) has some degree of influence or importance within the social domain\nunder consideration, and one expects such importance to be reflected in the\nstructure of the social network; centrality is a quantitative measure that\naims at revealing the importance of a node.\n\nFormally, a \\emph{centrality} (measure or index) is any function $c$ that, given a graph $G$, assigns a \nreal number $c_G(x)$ to every node $x$ \tof $G$; countless notions of centrality have been proposed over time, for\ndifferent purposes and with different aims; each of them was originally defined only for a specific category of graphs. Later some of\nthese notions of centrality have been extended to more general classes; in this paper, we shall only consider centralities \nthat can be defined\nproperly on all undirected graphs (even disconnected ones).\n\nAxioms are useful to isolate properties of different centrality measures and make it possible to compare them. One\nof the oldest papers to propose this approach is Sabidussi's paper~\\cite{SabCIG}, and many other proposals have appeared in the\nlast two decades.\n\nIn this paper we will be dealing with two properties of centrality measures:\n\n\\begin{definition}[Score monotonicity]\nGiven an undirected graph $G$, \na centrality $c$ is said to be \\emph{score monotone on $G$} iff for every pair of non-adjacent vertices $x$ and $y$ we have that\n\\[\n\tc_{G'}(x) > c_G(x) \\text{\\ and\\ } c_{G'}(y) > c_G(y),\n\\]\nwhere $G'$ is the graph obtained adding the new edge $x \\scalebox{0.5}[1.0]{-} y$ to $G$.\nIt is said to be \\emph{weakly score monotone on $G$} iff the same property holds, with $\\geq$ instead of $>$.\nWe say that $c$ is \\emph{(weakly) score monotone on undirected graphs} iff it is (weakly) score monotone on all \nundirected graphs $G$.\n\\end{definition}\n\n\\begin{definition}[Rank monotonicity]\nGiven an undirected graph $G$, \na centrality $c$ is said to be \\emph{rank monotone on $G$} iff for every pair of non-adjacent vertices $x$ and $y$ we have that for all vertices $z\\neq x,y$\n\\[\n\tc_{G}(x) \\geq c_G(z) \\Rightarrow c_{G'}(x) \\geq c_{G'}(z) \\text{\\ and \\ }\n\tc_{G}(y) \\geq c_G(z) \\Rightarrow c_{G'}(y) \\geq c_{G'}(z),\\\\\n\\]\nwhere $G'$ is the graph obtained adding the new edge $x \\scalebox{0.5}[1.0]{-} y$ to $G$.\nIt is said to be \\emph{strictly rank monotone\\footnote{Note that the published version of this paper~\\cite{BFVSRMUN}\ncontains a slightly different (and mistaken) definition which does not extend correctly the definition given in~\\cite{BLVRMCM}.} on $G$} if instead\n\\[\n\tc_{G}(x) \\geq c_G(z) \\Rightarrow c_{G'}(x) > c_{G'}(z) \\text{\\ and \\ }\n\tc_{G}(y) \\geq c_G(z) \\Rightarrow c_{G'}(y) > c_{G'}(z).\\\\\n\\]\nWe say that $c$ is \\emph{(strictly) rank monotone on undirected graphs} iff it is (strictly) rank monotone on all \nundirected graphs $G$.\n\\end{definition}\n\n\nThese four properties\\footnote{The asymmetric use of strict\/weak in the two definitions is for consistency with the previous literature on this topic.} can be studied on the class of all undirected graphs or only on the connected class, giving rise to eight possible ``degrees\nof monotonicity'' that every given centrality may satisfy or not. This paper studies these different degrees of monotonicity for three popular \nspectral centrality measures, also comparing the result obtained with the corresponding properties in the directed case. As we shall see, the\nundirected situation is quite different.\n\n\n\\section{Eigenvector centrality}\n\nEigenvector centrality is probably the oldest attempt at deriving a centrality from matrix information: a first\nversion was proposed by Landau in 1895 for matrices representing the results of chess tournaments~\\cite{LanZRWT}, and\nit was stated in full generality in 1958 by Berge~\\cite{BerG}; it has been rediscovered many times since then.\nOne considers the adjacency matrix of the graph and computes its left or right dominant eigenvector, which in our case coincide: the result\nis thus defined modulo a scaling factor, and if the graph is strongly\nconnected, the result is unique (modulo the scaling factor).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics{albiro-203-mps.eps}\n\\caption{\\label{fig:ec}A counterexample to score monotonicity for eigenvector centrality. After adding the edge between $0$ and $1$, the score of $0$ decreases:\nin norm $\\ell_1$, from $0.30656$ to $0.29914$; in norm $\\ell_2$, from $0.65328$ to $0.63586$; and when projecting the\nconstant vector $\\mathbf1$ onto the dominant eigenspace, from $1.39213$ to $1.35159$.}\n\\end{figure}\n\nDiscussing score monotonicity requires some form of normalization, due to the presence of the scaling factor. In Figure~\\ref{fig:ec} we show a very simple graph violating the property.\nIn particular, node $0$ score decreases after adding the arc $0\\adj1$ both\nin norm $\\ell_1$ and norm $\\ell_2$, and when projecting the constant vector $\\mathbf1$ onto the dominant eigenspace, which is an alternative way of circumventing the\nscaling factor~\\cite{VigSR}. The intuition \nis that initially node $0$ has a high score because of its largest degree (three). However, once we close the triangle\nwe create a loop that absorbs a large amount of rank, effectively decreasing the score of $0$. We conclude that\n\\begin{theorem}\nEigenvector centrality does not satisfy weak score monotonicity, even on connected undirected graphs (using norm $\\ell_1$, $\\ell_2$, or projection onto the dominant eigenspace).\n\\end{theorem}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics{albiro-204-mps.eps}\n\\caption{\\label{fig:ec2}A counterexample to rank monotonicity for eigenvector centrality.\nBefore adding the edge between $0$ and $1$, the score of $1$ is greater than the score of $3$; after, it is smaller.}\n\\end{figure}\n\n\nA similar counterexample, shown in Figure~\\ref{fig:ec2}, shows that eigenvector centrality does not satisfy rank monotonicity.\nThe scores of nodes $3$ and $1$ go from strictly increasing (without the edge $0\\adj1$) to strictly decreasing (with the edge $0\\adj1$);\nthus, $1$ loses rank.\nNote that in this case we do not have to choose a normalization, as the order of the two values does not change upon normalization. We conclude that \n\\begin{theorem}\nEigenvector centrality does not satisfy weak rank monotonicity, even on connected undirected graphs.\n\\end{theorem}\n\n\n\\section{Seeley's index}\n\nSeeley's index~\\cite{SeeNRI} is simply the steady state of the natural (uniform)\nrandom walk on the graph (for more details, see~\\cite{BoVAC}). It is a\nwell-known fact that if the graph is connected the steady-state probability of\nnode $x$ is simply $d(x)\/2m$---essentially, the index is just the $\\ell_1$-normalized degree. We\nwill thus use this definition for all graphs. As a consequence:\n\n\\begin{theorem}\nSeeley's index ($\\ell_1$-normalized degree) is strictly rank monotone on undirected graphs.\n\\end{theorem}\n\nThe situation is slightly different for score monotonicity:\n\\begin{theorem}\nSeeley's index ($\\ell_1$-normalized degree) is score monotone on undirected graphs, except in the case of a disconnected graph formed by a star graph and by one or more additional isolated vertices,\nin which case it is just weakly score monotone.\n\\end{theorem} \n\\begin{proof}\nWhen we add an edge between $x$ and $y$ in a graph with $m$ edges, the score of $x$ changes from $d(x)\/2m$ to $(d(x)+1)\/(2m+2)$. If we require\n\\[\n\\frac{d(x)+1}{2m+2}> \\frac{d(x)}{2m}\n\\]\nwe obtain $d(x)< m$. Since obviously $d(x)\\leq m$, the condition is always true except when $d(x)=m$, which corresponds to\nthe case of a disconnected graph formed by a star graph and by additional isolated vertices.\nIndeed, in that case adding an edge between an isolated vertex and the center of the star will not change the center's Seeley's index. \\ifmmode\\mbox{ }\\fi\\rule[-.05em]{.3em}{.7em}\\setcounter{noqed}{0}\n\\end{proof}\n\n\n\n\\section{Graph fibrations and spectral ranking}\n\\label{sec:fib}\n\nIt is known since seminal works from the '50s in the theory of \\emph{graph divisors}~\\cite{CDSSG} that fibrations~\\cite{BoVGF}, defined below,\nhave an important relationship with eigenvalues and eigenvectors: if there is a fibration $f:G\\to B$, the eigenvalues\nof $G$ and $B$ are the same, modulo multiplicity, and eigenvectors of $G$ can be obtained from the eigenvectors of $B$. The\nresults extend to weighted graphs, too. In this section, we are going to extend such results to \\emph{damped spectral rankings}~\\cite{VigSR} of the form\n\\[\n\\bm v\\sum_{i=0}^\\infty \\beta^iM^i = \\bm v( 1 - \\beta M)^{-1},\n\\]\nwhere $M$ is the weighted adjacency matrix of a graph, $\\beta$ is a parameter satisfying the condition\n$0\\leq\\beta<1\/\\rho(M)$, $\\rho(M)$ is the spectral radius of $M$, and $\\bm v$ is a \\emph{preference vector}: \nKatz's index~\\cite{KatNSIDSA}, Hubbell's index~\\cite{HubIOACI} and PageRank~\\cite{PBMPCR} are all examples of damped spectral rankings.\n\nWhile determining a damped spectral ranking for a \\emph{specific} graph essentially requires solving a system of linear equations,\npossibly approximating its solution with an iterative method,\ndoing that for \\emph{parametric families} of graphs is tricky and often requires\n\\emph{ad hoc} approaches. Nonetheless, when the graphs under consideration are sufficiently symmetric, one can try to reduce\nthe computation using a technique based on fibrations. The idea was introduced in~\\cite{BLSGFGIP} for random walks\nwith restart, and in this section we will extend it to general damped spectral rankings, providing\nthus a self-contained (and, in fact, simpler) proof.\n\nLet us start with some additional definitions.\nA \\emph{path} (of length $n\\geq 0$) is a sequence $\\pi=\\langle i_0 a_1 i_1 \\cdots\ni_{n-1} a_n i_n\\rangle$, where $i_k\\in N_G$, $a_k\\in A_G$, $s(a_k)=i_{k-1}$ and\n$t(a_k)=i_k$. \nWe define $s(\\pi)=i_0$ (the \\emph{source} of $\\pi$), $t(\\pi)=i_n$ (the \\emph{target} of $\\pi$), \n$|\\pi|=n$ (the \\emph{length} of $\\pi$) and let $G^*(i,j) = \\{\\, \\pi\n\\mid s(\\pi)=i, t(\\pi)=j\\,\\}$ (the\nset of paths from $i$ to $j$). \n\n\nA \\emph{(graph) morphism} $f:G\\to H$ is given by a pair of functions\n$f_N:N_G\\to N_H$ and $f_A:A_G\\to A_H$ commuting with the source and\ntarget maps, that is, $s_H(f_A(a))=f_N(s_G(a))$ and $t_H(f_A(a))=f_N(t_G(a))$ for all\n$a \\in A_G$ (again, we shall drop the subscripts whenever no confusion is possible). In other\nwords, a morphism maps nodes to nodes and arcs to arcs in such a way to\npreserve the incidence relation. \nThe definition of morphism we give here is the obvious extension to the case of multigraphs of the standard notion the\nreader may have met elsewhere.\nAn \\emph{epimorphism} is a morphism $f$ such that both $f_N$ and $f_A$ are surjective.\n\nA \\emph{fibration}~\\cite{BoVGF} between the graphs $G$ and $B$ is a morphism $f: G\\to B$ such\nthat for each arc $a\\in A_B$ and each node $i\\in N_G$ satisfying\n$f(i)=t(a)$ there is a unique arc $\\lift ai\\in A_G$ (called the \\emph{lifting of\n$a$ at $i$}) such that $f(\\lift ai)=a$ and $t(\\lift ai)=i$.\nIf $f:G\\to B$ is a fibration, $G$\nis called the \\emph{total graph} and $B$ the \\emph{base} of $f$. \nWe shall also say that $G$ is \\emph{fibered (over $B$)}. The \\emph{fiber over a\nnode $h\\in N_B$} is the set of nodes of $G$ that are mapped to $h$, and shall\nbe denoted by $f^{-1}(h)$. \n\nA more geometric way of interpreting the definition of\nfibration is that given a node $h$ of $B$ and a path $\\pi$ terminating at $h$,\nfor each node $i$ of $G$ in the fiber of $h$ there is a unique path terminating\nat $i$ that is mapped to $\\pi$ by the fibration; this path is called the\n\\emph{lifting of $\\pi$ at $i$}. \n\nIn Figure~\\ref{fig:exfib}, we show two graph morphisms; the morphisms are\nimplicitly described by the colors on the nodes. The morphism displayed on the\nleft is not a fibration, as the loop\non the base has no counterimage ending at the lower gray node, and\nmoreover the other arc has two counterimages with the same target. The\nmorphism displayed on the\nright, on the contrary, is a fibration. Observe that loops are not necessarily\nlifted to\nloops.\n\n\\begin{figure}[htbp]\n \\begin{center}\n\t\\includegraphics{albiro-5-mps.eps}\\qquad\\qquad\\qquad\\qquad\\includegraphics{albiro-6-mps.eps}\n \\end{center}\n \\caption{\\label{fig:exfib}On the left, an example of graph morphism that is\nnot a fibration; on the right, a fibration. Colors on the nodes are used to\nimplicitly specify the morphisms.}\n\\end{figure}\n\nWe will now show how fibrations can be of help in the computation of a damped spectral ranking.\nFirst of all, we are now going to consider \\emph{weighted} graphs, in which each arc is assigned\na real weight, given by a weighting function $w:A_G\\to \\mathbf R$: the adjacency matrix associated to\na weighted graph $G$ is defined by letting\n\\[\n\tG_{ij}=\\sum_{a \\in G(i,j)} w(a)\n\\]\nand we obtain the unweighted case when $w$ is the constant $a \\mapsto 1$ function.\nAll the morphisms (especially, fibrations) between weighted graphs are assumed to preserve weights. \n\nNote that every morphism $f: G \\to B$ extends to a mapping $f^*$ between paths of $G$ and paths of $B$\nin an obvious way. This map $f^*$ preserves not only path lengths, but also weight sequences if $f$ does.\n\nFor fibrations we can say more; using the lifting property, one can prove by induction that:\n\\begin{theorem}\n\\label{thm:bij}\nIf $f: G \\to B$ is an epimorphic fibration between weighted graphs,\nthen for every two nodes $j \\in N_G$ and $k \\in N_B$ the map $f^*$ is a bijection \nbetween $\\cup_{i \\in f^{-1}(k)} G^*(i,j)$ and $B^*(k,f(j))$.\n\\end{theorem}\n\n\nNow, for every $t\\geq 0$, $G^t$ is the matrix whose $ij$ entry contains a summation of contributions, one for each path $\\pi \\in G^*(i,j)$, \nand the contribution is given by the product of the arc weights found along the way;\nhence, by Theorem~\\ref{thm:bij}, under convergence assumptions we have that for all $\\beta$ and all $i \\in N_G$ and $k \\in N_B$\n\\[\n\t\\sum_{i \\in f^{-1}(k)}\\left(\\sum_{t \\geq 0} \\beta^t G^t\\right)_{ij} = \\left(\\sum_{t \\geq 0} \\beta^t B^t\\right)_{kf(j)},\n\\]\nor equivalently\n\\[\t\n\\sum_{i \\in f^{-1}(k)}\\left((1-\\beta G)^{-1}\\right)_{ij} = \\left((1-\\beta B)^{-1}\\right)_{kf(j)}.\n\\]\nNow, for every vector\\footnote{All vectors in this paper are row vectors.} $\\bm u$ of size $n_B$, define its \\emph{lifting along $f$} as\nthe vector $\\bm u^f$ of size $n_G$ given by\n\\[\n\t\\left(u^f\\right)_i=u_{f(i)}.\n\\]\nFor every $j$, we have\n\\begin{multline*}\n\t\\left(\\bm u^f (1-\\beta G)^{-1}\\right)_j\n\t=\\sum_{i\\in N_G} u^f_i \\left((1-\\beta G)^{-1}\\right)_{ij}\n\t=\\\\\n\t=\\sum_{k\\in N_B}\\sum_{i \\in f^{-1}(k)} u_{f(i)} \\left((1-\\beta G)^{-1}\\right)_{ij}\n\t=\\sum_{k\\in N_B}u_k\\left(\\sum_{i \\in f^{-1}(k)} \\left((1-\\beta G)^{-1}\\right)\\right)_{ij}\n\t=\\\\\n\t=\\sum_{k\\in N_B}u_k\\left((1-\\beta B)^{-1}\\right)_{kf(j)}\n\t=\\left(\\bm u (1-\\beta B)^{-1}\\right)_{f(j)}\n\\end{multline*}\nwhich can be more compactly written as\n\\begin{equation}\n\\label{eqn:resumedsp}\n\t\\bm u^f (1-\\beta G)^{-1}=\\left(\\bm u (1-\\beta B)^{-1}\\right)^f.\n\\end{equation}\n\n\nEquation (\\ref{eqn:resumedsp}) essentially states that if we want to compute the damped spectral ranking of the weighted graph $G$,\nfor a preference vector that is\nconstant along the fibers of an epimorphic fibration $f: G \\to B$, and thus of the form $\\bm u^f$, we can compute the damped spectral ranking of the weighted base $B$ \nusing $\\bm u$ as preference vector, and then lift along $f$ the result. For example, in the\ncase of Katz's index a simple fibration between graphs is sufficient, as in that case there are no weights to deal with.\n\n\n\\paragraph{Implications for PageRank.}\nPageRank~\\cite{PBMPCR} can be defined as\n\\[\n(1-\\alpha)\\bm v\\sum_{i=0}^\\infty \\alpha^i\\bar G^i = (1-\\alpha) \\bm v(1-\\alpha \\bar G)^{-1},\n\\]\nwhere $\\alpha\\in[0\\..1)$ is the damping factor, \n$\\bm v$ is a non-negative preference vector with unit $\\ell_1$-norm, \nand $\\bar G$ is the row-normalized version\\footnote{Here we are assuming that $G$ has no \\emph{dangling nodes} (i.e., nodes with outdegree $0$).\nIf dangling nodes are present, you can still use this definition (null rows are left untouched in $\\bar G$), but then to obtain PageRank you \nneed to normalize the resulting vector~\\cite{BSVPFD,DCGRFPCSLS}. So all our discussion can also be applied\nto graphs with dangling nodes, up to $\\ell_1$-normalization.} of $G$.\n\nThen $\\bar G$ is just the (adjacency matrix of the) weighted version of $G$ defined by letting $w(a)=1\/d^+_G(s_G(a))$.\nHence, if you have a weighted graph $B$,\nan epimorphic weight-preserving fibration $f: \\bar G \\to B$, and a vector $\\bm u$ of size $n_B$ such that \n$\\bm u^f$ has unit $\\ell_1$-norm, you can deduce from (\\ref{eqn:resumedsp}) that\n\\begin{equation}\n\\label{eqn:resumepr}\n (1-\\alpha) \\bm u^f (1-\\alpha \\bar G)^{-1}=\\left(\\bm (1-\\alpha) \\bm u (1-\\alpha B)^{-1}\\right)^f.\n\\end{equation}\n\nOn the left-hand side you have the actual PageRank of $G$ for a preference vector that is fiberwise constant;\non the right-hand side you have a spectral ranking of $B$ for the projected preference vector.\nNote that $B$ is not row-stochastic, and $\\bm u$ has not unit $\\ell_1$-norm, \nso technically the right-hand side of equation (\\ref{eqn:resumepr}) is\nnot PageRank anymore, but it is still a damped spectral ranking.\n\n\\section{PageRank}\n\nArmed with the results of the previous section, we attack the case of PageRank, which is the most interesting. The first observation\nis that\n\\begin{theorem}\nGiven an undirected graph $G$ there is a value of $\\alpha$ for which PageRank is strictly rank monotone on $G$.\nThe same is true for score monotonicity, except when $G$ is formed by a star graph and by one or more additional isolated vertices.\n\\end{theorem}\n\\begin{proof}\nWe know that for $\\alpha\\to 1$, PageRank tends to Seeley's index~\\cite{BSVPFDF}. Since Seeley's index is strictly rank monotone, for each non-adjacent pair \nof vertices $x$ and $y$ there is a value $\\alpha_{xy}$ such that for $\\alpha\\geq\\alpha_{xy}$ adding the edge $x\\scalebox{0.5}[1.0]{-} y$ is strictly\nrank monotone. The proof is completed by taking $\\alpha$ larger than all $\\alpha_{xy}$'s. \nThe result for score monotonicity is similar. \\ifmmode\\mbox{ }\\fi\\rule[-.05em]{.3em}{.7em}\\setcounter{noqed}{0} \n\\end{proof}\n\nOn the other hand, we will now show that \n\\emph{for every possible value of the damping factor $\\alpha$} there is a graph on which PageRank violates rank and score monotonicity.\n\nThe basic intuition of our proof is that when you connect a high-degree node $x$ with a low-degree node $y$, $y$ will pass to $x$\na much greater fraction of its score than in the opposite direction. This phenomenon is caused by the stochastic normalization of the\nadjacency matrix: the arc from $x$ to $y$ will have a low coefficient, due to the high degree of $x$, whereas the arc from $y$ to $x$\nwill have a high coefficient, due to the low degree of $y$.\n\nWe are interested in a parametric example, so that we can tune it for different values of $\\alpha$. At the same time, we want to \nmake the example analytic, and avoid resorting to numerical computations, as that approach would make it impossible to prove a result\nvalid for every $\\alpha$---we would just, for example, prove it for a set of samples in the unit interval.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\raisebox{.5cm}{$G_k$\\qquad}&\\includegraphics{albiro-201-mps.eps}\\\\\n\\raisebox{.5cm}{$B_k$\\qquad}&\\includegraphics{albiro-202-mps.eps}\n\\end{tabular}\n\\caption{\\label{fig:pr}The parametric counterexample graph for PageRank. The two $k$-cliques are represented here as $5$-cliques for simplicity. Arc\nlabels represent multiplicity; weights are induced by the uniform distribution on the upper graph.}\n\\end{figure}\n\nWe thus resort to fibrations, using equation (\\ref{eqn:resumepr}). In Figure~\\ref{fig:pr} we show a parametric\ngraph $G_k$ comprising two $k$-cliques (in the figure, $k=5$). \nBelow, we show the graph $B_k$ onto which $G_k$ can be fibred by mapping\nnodes following their labels. The dashed edge is the addition that we will study:\nthe fibration exists whether the edge exists or not (in both graphs).\n\nWhile $G_k$ has $2k+4$ vertices, $B_k$ has $9$ vertices, independently of $k$, and\nthus its PageRank can be computed analytically as rational functions of $\\alpha$ whose coefficients are rational functions\nin $k$ (as the number of arcs of each $B_k$ is different). The adjacency matrix of $B_k$ without the dashed arc, considering multiplicities, is\\footnote{Note that in the published version of this paper~\\cite{BFVSRMUN}\nthe denominators of the second row are $k-1$, mistakenly, instead of $k+1$.}\n\\[\n\\left(\\begin{matrix}\n \\frac{k-2}{k-1}& \\frac{k-1}{k-1}& 0& 0& 0& 0& 0& 0& 0\\\\\n \\frac1{k+1}& 0& \\frac1{k+1}& 0& \\frac1{k+1}& 0& 0& 0& 0\\\\\n 0& \\frac12& 0& \\frac12& 0& 0& 0& 0& 0\\\\\n 0& 0& \\frac12& 0& 0& 0& \\frac12& 0& 0\\\\\n 0& \\frac12& 0& 0& 0& \\frac12& 0& 0& 0\\\\\n 0& 0& 0& 0& 1& 0& 0 &\\fcolorbox{gray}{gray}{0}& 0\\\\\n 0& 0& 0& \\frac1k& 0& 0& 0& \\frac1k& \\frac1k\\\\\n 0& 0& 0& 0& 0& \\fcolorbox{gray}{gray}{0}& \\frac1{k-1}& 0& \\frac1{k-1}\\\\\n 0& 0& 0& 0& 0& 0& \\frac{k-2}{k-1}& \\frac{k-2}{k-1}& \\frac{k-3}{k-1}\\\\\n\\end{matrix}\\right)\n\\]\nAfter adding the edge between $5$ and $7$ we must modify the matrix by setting the two grayed \nentries to one and fix normalization accordingly. We will denote with $\\operatorname{pre}_\\alpha(x)$ the rational function returning the PageRank of\nnode $x$ with damping factor $\\alpha$ before the addition of the dashed arc, and \nwith $\\operatorname{post}_\\alpha(x)$ the rational function returning the PageRank of\nnode $x$ with damping factor $\\alpha$ after the addition of the dashed arc. \n\nWe use the Sage computational engine~\\cite{Sage} to perform all computations, as\nthe resulting rational functions are quite formidable.\\footnote{The Sage worksheet can be found at \\url{https:\/\/vigna.di.unimi.it\/pagerank.ipynb}.} \nWe start by considering\nnode $5$: evaluating $\\operatorname{post}_\\alpha(5)-\\operatorname{pre}_\\alpha(5)$ in $\\alpha=2\/3$ we obtain\na negative value for all $k\\geq 12$, showing\nthere is always a value of $\\alpha$ for which node $5$ violates weak score monotonicity, as long as $k\\geq 12$. \n\nTo strengthen our results, we are now going to show that for \\emph{every} $\\alpha$ there is a $k$ such that\nweak score monotonicity is violated. We use Sturm polynomials~\\cite{RaSATP} to compute the number of sign changes of the \nnumerator $p(\\alpha)$ of $\\operatorname{post}_\\alpha(5)-\\operatorname{pre}_\\alpha(5)$ \nfor $\\alpha\\in[0\\..1]$, as the denominator cannot have zeros. Sage\nreports that there are two sign changes for $k\\geq 12$, which means that $p(\\alpha)$ is initially positive; then, somewhere before $2\/3$\nit becomes negative; and finally it returns positive again somewhere after $2\/3$.\n\nDetermining the behavior of the points at which $p(\\alpha)$ changes sign is impossible due to the high degree of the polynomials\ninvolved. However, we can take two suitable parametric points in the unit interval that sandwich $2\/3$, such as \n\\[\na = \\frac34 - \\frac{3k}{4k+ 1000} \\leq \\frac23 \\leq \\frac12+\\frac{k}{2k+1000} = b,\n\\]\nand use again Sturm polynomials to count the number of sign changes in $[0\\..a]$ and $[b\\..1]$. In both cases, if $k\\geq 15$\nthere is exactly one sign change in the interval, and since $a\\to0$, $b\\to 1$ as $k\\to \\infty$, we conclude that as $k$ grows\nthe size of the interval of $\\alpha$'s in which $p(\\alpha)<0$ grows, approaching $[0\\..1]$ in the limit. Thus,\n\\begin{theorem}\nFor every value of $\\alpha\\in[0\\..1)$, there is an undirected graph for which PageRank violates weak score monotonicity\nwhen $\\alpha$ is chosen as damping factor.\n\\end{theorem}\n\nWe now use the same example to prove the lack of rank monotonicity. In this case, we study in a similar way\n$\\operatorname{pre}_\\alpha(5)-\\operatorname{pre}_\\alpha(2)$, which is positive in $\\alpha=2\/3$\nif $k\\geq 14$. Its numerator has two sign changes in the unit interval,\nwhich means that initially $5$ has a smaller PageRank than $2$; then, somewhere before $2\/3$\n$5$ starts having a larger PageRank than $2$; finally, we return to the initial condition.\n\n Once again, we sandwich $2\/3$ using\n\\[\n\\frac34 - \\frac{3k}{4k+200}\\leq \\frac23\\leq \\frac12 + \\frac{k}{2k+200},\n\\]\nand with an argument analogous to the case of score monotonicity we conclude that as $k$ grows the subinterval\nof values of $\\alpha$ in $[0\\..1)$ for which the score of $5$ is greater than the score of $2$ grows up to the whole interval.\n\nFinally, we study $r(\\alpha)=\\operatorname{post}_\\alpha(5)-\\operatorname{post}_\\alpha(2)$ which is negative in $\\alpha=2\/3$\nif $k\\geq 6$, and whose numerator has three sign changes in the unit interval. Once again, we sandwich $2\/3$ using\n\\[\na = \\frac1{10} - \\frac{k}{10k+2000}\\leq \\frac23\\leq \\frac12 + \\frac{k}{2k+200} = b.\n\\]\nIn this case, there are always two sign changes in $[0\\..a]$ and one sign change in $[b\\..1]$ for $k\\geq 25$, so \nthere is a subinterval of values of $\\alpha$ in $[0\\..1)$ for which the score of $5$ is smaller than the score of $2$ after adding the\nedge $5\\scalebox{0.5}[1.0]{-} 2$, and this subinterval grows in size up to the whole unit interval as $k$ grows. All in all, we proved that:\n\\begin{theorem}\n\\label{th:prrank}\nFor every value of $\\alpha\\in[0\\..1)$, there is an undirected graph for which PageRank violates rank monotonicity\nwhen $\\alpha$ is chosen as damping factor.\n\\end{theorem}\n\n\\section{Experiments on IMDB}\n\n\n\\begin{table}[t]\n\\renewcommand{\\arraystretch}{1.2}\n\\renewcommand{\\tabcolsep}{1ex}\n\\begin{tabular}{lll}\nScore increase & Score decrease & Violations of rank monotonicity\\\\\n\\hline\nMeryl Streep & Yasuhiro Tsushima & Anne--Mary Brown, Jill Corso,~\\ldots\\\\\nDenzel Washington & Corrie Glass & Patrice Fombelle, John Neiderhauser,~\\ldots\\\\\nSharon Stone & Mary Margaret (V) & Dolores Edwards, Colette Hamilton,~\\ldots\\\\\nJohn Newcomb & Robert Kirkham & Brandon Matsui, Evis Trebicka,~\\ldots \n\\end{tabular}\n\\vspace*{.5em}\n\\caption{\\label{tab:rank}A few examples of violations of score monotonicity and rank monotonicity in the Hollywood co-starship graph\n\\texttt{hollywood-2011}. If we add an edge between the actors in the first and second column, the first actor has a score increase, the second actor has a score decrease,\nand the actors in the third column, which were less important than the second actor, become more important after the edge addition.}\n\\end{table}\n\n\nTo show that our results are not only theoretical, we provide a few interesting anecdotal examples from\nthe PageRank scores ($\\alpha=0.85$) of the Hollywood co-starship graph,\nwhose vertices are actors\/actresses in the Internet Movie Database, with an edge connecting them if played in the same movie.\nIn particular, we used the \\texttt{hollywood-2011} dataset from the Laboratory for Web Algorithmics,\\footnote{\\url{http:\/\/law.di.unimi.it\/}}\nwhich contains approximately two million vertices and $230$ million edges.\n\nTo generate our examples, we picked two actors either at random, or considering\nthe top $1\/10000$ of the actors of the graph in PageRank order and the bottom\nquartile, looking for a collaboration that would hurt either actor (or\nboth).\\footnote{Note that for this to happen, the collaboration should be a\ntwo-person production. A production with more people would actually add more\nedges.} About $4$\\% of our samples yielded a violation of monotonicity, and in\nTable~\\ref{tab:rank} we report a few funny examples.\n\nIt is interesting to observe that in the first three cases it is the less-known actor that loses score (and rank) by the collaboration\nwith the star, and not the other way round, which is counterintuitive.\nIn the last case, instead, a collaboration would damage the most important vertex, and\nit is an open problem to prove a result analogous to Theorem~\\ref{th:prrank} for this case.\nWe found no case in which both actors would be hurt by the collaboration.\n\n\n\\section{Conclusions}\n\nWe have studied score and rank monotonicity for three fundamental kinds of spectral ranking---eigenvector\ncentrality, Seeley's index, and PageRank. Our results show that except for\nSeeley's index on connected graphs, there are always cases in which score and rank monotonicity fail,\ncontrarily to the directed case, and these failures can be found in real-world graphs.\nIn particular, for PageRank we can find a counterexample for every value of the damping factor.\nFinding such a class of counterexamples for Katz's index~\\cite{KatNSIDSA} is an interesting open problem.\nAnother valuable contribution would be to find another class of counterexamples for PageRank that is amenable to a simpler analytic proof without having to rely on computer algebra. \n\nOur results suggest that common knowledge about the behavior of PageRank in the directed case cannot\nbe applied automatically to the undirected case.\n\n\n\\vspace*{-.7em}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzznumk b/data_all_eng_slimpj/shuffled/split2/finalzznumk new file mode 100644 index 0000000000000000000000000000000000000000..46e536e853e2a5eaa1bd5dd876247cf924d0a15b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzznumk @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and Main Result}\nThe property of exact solvability or even integrability of a non-trivial quantum field theory (QFT) in four dimensions is a very active and ongoing discipline in theoretical and mathematical physics. Usually, QFT's turned out to be accessible only by approximate methods, this is why one paid attention to simplified models. An established approach being of interest for us is the Euclidean one \\cite{Schwinger:1959} having a proximity to statistical physics, giving the chance of exchanging methods and concepts of both disciplines. New hope arose from the formulation of such models on a suitable noncommutative space. In particular, the fact that this novel geometry developed by A. Connes \\cite{Connes94noncommutativegeometry} also bridged the significant gap of how general relativity can be unified with the fundamental laws of quantum mechanics \\cite{Doplicher:1994tu} fuelled the debate around this type of toy models. At the turn of the millennium, two models celebrated noteworthy successes towards exact solvability and renormalisability in four dimensions. The quartic model of hermitian fields by Grosse and Wulkenhaar \\cite{Grosse:2004yu,Grosse:2012uv} and the model of complex-valued fields with arbitrary interaction by Langmann, Szabo and Zarembo \\cite{Langmann:2003if}, both formulated on the noncommutative Moyal space \\cite{rieffel1989}. An important fact is that quantum fields on the Moyal space admit an expansion in a matrix base (Moyal algebra).\n\nA suitable approach to the property of exact solvability was given in each case by the finite-dimensional approximation of the quantum fields by matrices. In this finite limit at the so-called self-duality point of the action functional \\cite{Langmann:2002cc}, the quantum field theories are reduced to hermitian or complex matrix models with an external field (representing the covariance), an area of research that has its own powerful approaches to solving. This reduction process to an ordinary matrix model starting from an abstract noncommutative algebra can be retraced in great detail in e.g. \\cite{SurveyNCG} which is also accessible for beginners in this area of research. By exact solvability we mean the expressibility, at least in principle, of the model's correlation functions by \"known\" functions. With new methods, it was recently possible to replace the principle expressibility for the hermitian model with concrete expressions that satisfy a remarkable algebraic structure whose investigation is still an ongoing process \\cite{Grosse:2019jnv,Branahl:2020yru,Hock:2021tbl}. The key to this is the universal structure of topological recursion (TR) of Chekhov, Eynard and Orantin \\cite{Chekhov:2006vd,Eynard:2007kz}. \n\nIn a similar way to the Grosse-Wulkenhaar model, which shrinks in the finite limit to a quartic analogue of the Kontsevich model (QKM) \\cite{Kontsevich:1992ti}, we turn to the complex-valued LSZ model, restricted to a quartic interaction. Due to its proximity to the action functional of the Higgs field and the possibility of treating charged quantum fields, this model is quite a bit closer to realistic physical QFTs. Although the complex nature of the fields imposes a more general structure on the partition function, the character of the loop equations for the correlation functions as well as the algebraic structure behind their solution will turn out to be much simpler. In the original work by Langmann, Szabo and Zarembo \\cite{Langmann:2003if}, it was proved to be useful to extend the partition function with a second external field. With this following expression of the \\textit{partition function}, a more concrete reformulation of exact solvability as well as a more precise notion of integrability based on topological recursion will be possible in this article. The partition function of the LSZ model reads:\n\\begin{align}\n\\label{partfunLSZ}\n&\\mathcal{Z}(J,J^\\dagger)\\\\\n&= \\int_{C_N} d \\Phi d \\Phi^\\dagger \\exp \\Big (- N \\mathrm{Tr}\\bigg [E \\Phi\\Phi^\\dagger+\\tilde{E} \\Phi^{\\dagger}\\Phi + \\frac{\\lambda}{2} (\\Phi^\\dagger \\Phi)^2\\bigg]+N\\mathrm{Tr} (\\Phi^\\dagger J + J^\\dagger \\Phi) \\Big ) \\nonumber\n\\end{align}\nHere, the integral is carried out over the space $C_N$ of complex $N \\times N$ matrices. $E$ and $\\tilde{E}$ are distinct $N \\times N$ hermitian matrices with positive eigenvalues that we call \\textit{external fields} responsible for the covariance. Their positive and distinct eigenvalues $(E_1,...,E_N)$ and $(\\tilde{E}_1,...,\\tilde{E}_N)$ are not related. Without loss of generality (see proof of Proposition \\ref{prop:ward}), we can assume that both matrices are diagonal due to the invariance of $\\mathcal{Z}$ under unitary tranformations. The quantum fields themselves are given by $\\Phi$, their interaction strength is encoded in the scalar $\\lambda$, the \\textit{coupling constant}. We also need to introduce besides the \\textit{source} $J\\in C_N$ together with its adjoint $J^\\dagger$ building together the \\textit{source term} that will give rise to the correlation functions by derivatives of $\\mathcal{Z}(J,J^\\dagger)$ at $J,J^\\dagger=0$. \n\nNote that Langmann, Szabo and Zarembo were in \\cite{Langmann:2003if} in particular interested in the case $\\tilde{E}=0$, but the same strong properties of exact solvability and integrability (to be precise: integrable in the Kadomtsev-Petviashvili hierarchy) were shown first for this regulated partition function \\eqref{partfunLSZ} with $\\tilde{E}\\neq 0$. Some correlation functions in the case $\\tilde{E}=0$ were explicitly solved by the authors, we generalise it to any $\\tilde{E}$ and all correlation functions. This extension to any $\\tilde{E}$ also became interesting for other applications: In \\cite{Santilli:2018} it was shown that the LSZ model computes the probability of atypically large fluctuations in the Stieltjes\u2013Wigert matrix model, which is a $q$-ensemble describing $\\mathcal{U}(N)$ Chern\u2013Simons theory on the three-sphere. In their work it is also stated that despite similarities between the LSZ model and the QKM, it \\textit{leads to a different analysis and very different set of results}. Using topological recursion, we will show that the two models are indeed deeply related and can be treated with the same techniques. More concretely, the LSZ model is solved by TR and the QKM by blobbed topological recursion (BTR), at least in genus $g=0$, with the same spectral curve $(\\overline{\\mathbb{C}},x,y,B)$ for $E=\\tilde{E}$ \\cite{Hock:2021tbl}.\n\nTo understand this in detail, we need to recapitulate the framework of topological recursion. This mathematical machinery is \\textit{topological} in the sense that it computes an infinite sequence of symmetric meromorphic $n$-forms $\\omega_{g,n}$ on $\\Sigma^n$, where $g$ can be understood as genus and $n$ to the number of marked points. $\\Sigma$ is a Riemann surface independent of $g$ and $n$. It is \\textit{recursive} in the sense that the computation is performed recursively in the negative Euler characteristic $-\\chi=2g+n-2$ and builds on a set of initial data $(\\Sigma,x,y,B)$, where $x,y:\\Sigma\\to \\Sigma_0$ are ramified covering of Riemann surfaces, $\\omega_{0,1}=y\\, dx$ is a meromorphic differential 1-form on $\\Sigma$, regular at the ramification points of $x$, and $\\omega_{0,2}=B$ a symmetric bilinear differential form on $\\Sigma\\times \\Sigma$ with double pole on the diagonal and no residue. \n\nFor particular spectral curves, the family of differential forms $\\omega_{g,n}$ is directly related for instance to correlation functions (expectation values) of some statistical models. Therefore, the spectral curve recursively encodes the whole information about the model by itself. In the following section we will introduce some mathematical details behind the recursion and a visualisation. \n\nAfter its discovery 15 years ago it enabled great success in various, seemingly disconnected fields of mathematical research - as enumerative geometry \\cite{Eynard:2016yaa}, intersection numbers of tautological classes of the moduli space of complex curves \\cite{Witten:1990hr,Eynard:2011} and integrable hierarchies \\cite{Eynard:2017}, and much more. Today, it is almost impossible to give a complete list of applications of this powerful universal procedure. Topological recursion entered the stage of (noncommutative) quantum field theory a few years ago during the attempt to understand the solution structure of the correlation functions of the QKM, formulated by the partition function:\n\\begin{align}\n\\label{partfunQKM}\n\\mathcal{Z}^{(QKM)}(J)= \\int_{H_N} d \\Phi \\exp \\bigg (- N \\mathrm{Tr}\\bigg [E \\Phi^2+ \\frac{\\lambda}{4} \\Phi^4\\bigg]+N\\mathrm{Tr} (\\Phi J ) \\bigg )\n\\end{align}\nHere, we integrate over the space $H_N$ of hermitian $N \\times N$ matrices where $E \\in H_N$ has positive eigenvalues $(E_1,...,E_N)$. The exact and concrete solution of the 2-point function of \\eqref{partfunQKM}, obtained after complexification and a crucial variable transform \\cite{Schurmann:2019mzu,Grosse:2019jnv,Panzer:2018tvy}, gave the possibility to read off the spectral curve for topological recursion \\cite{Branahl:2020yru}. For the complex LSZ model, now with two external fields, we will follow a very similar strategy. With established methods as a system of Ward identities (Sec. \\ref{ch:setup}) we derive a closed nonlinear equation for the 2-point function of the LSZ model in Sec. \\ref{ch:dse}, which is strikingly similar to the QKM \\eqref{partfunQKM} in the planar sector. Two variable transforms, occurring in the proof of its solution, yield the corresponding spectral curve. The introduction of a boundary creation operator, creating two families $T$ and $\\Omega$ of generalised correlators only building on the knowledge of the 2-point function, is the key step towards topological recursion. Their Dyson-Schwinger equations have exactly the same structure as the loop equations of the hermitian 2-matrix model \\cite{Chekhov:2006vd} and the generalised Kontsevich model \\cite{Belliard:2021jtj} such that a proof in Sec. \\ref{ch:tr} is straightforward. In summary, we will prove the following result:\n\\begin{theorem}\n\\label{th:main}\nLet $(\\overline{\\mathbb{C}},x,y,B= \\frac{dz_1\\, dz_2}{(z_1-z_2)^2})$ be a spectral curve with\n\\begin{align*}\n x(z) = z-\\frac{\\lambda}{N} \\sum_{k=1}^N\\frac{1}{y'(\\tilde{\\varepsilon}_k)(z-\\tilde{\\varepsilon}_k)} \\qquad y(z) =- z + \\frac{\\lambda}{N} \\sum_{k=1}^{N} \\frac{1}{x'(\\varepsilon_k)(z-\\varepsilon_k)}\n\\end{align*}\nwhere $x(\\varepsilon_k)=E_k$ and $y(\\tilde{\\varepsilon}_k)=\\tilde{E}_k$. Then, topological recursion generates meromorphic forms $\\omega_{g,n}(z_1,...,z_n)$ according to eq. (\\ref{eq:TR-intro}), that solve the Langmann-Szabo-Zarembo model for quartic field interaction. In particular, $\\omega_{g,n}$ relates to the family of correlators $\\Omega_n^{(g)}$ via $\\Omega_n^{(g)}(z_1,...,z_n)dx(z_1)...dx(z_n)=\\omega_{g,n}(z_1,...,z_n)$ where\n\\begin{align*}\n \\Omega_n^{(g)}(\\varepsilon_{p_1},...,\\varepsilon_{p_n}) = &\\bigg [ N^{2-2g-n} \\bigg ] (-1)^n \\frac{\\partial^n}{\\partial E_{p_1}...\\partial E_{p_n}} \\log[\\mathcal{Z}(0)] \\\\\n &+ \\frac{\\delta_{n,2}\\delta_{g,0}}{(E_{p_1}-E_{p_2})^2}\n +\\frac{\\delta_{n,1}\\delta_{g,0}}{\\lambda}V'(E_{p_1})\n\\end{align*}\nFrom the family $\\Omega_n^{(g)}$ of meromorphic functions, the exact and explicit solution of all correlation functions $G$ of the LSZ model can be computed recursively via Corollary \\ref{cor2+} and Theorem \\ref{thm:complete}.\n\\end{theorem}\n\\begin{remark}\n\tThe theorems and propositions throughout the article are slightly more general and hold even if we admit multiplicities for the eigenvalues $E_k,\\tilde{E}_l$. To avoid technicalities, we stated Theorem \\ref{th:main} in this simpler version.\n\\end{remark}\n\nInterchanging the role of $E$ and $\\tilde{E}$ in the LSZ model leads to an interchange between $x$ and $y$ in Theorem \\ref{th:main}, which is in general a meaningful phenomenon in TR \\cite{Borot:2021thu,Hock:2022wer}, but has in this specific model a trivial consequence (similar to the 2-matrix model).\n\nIn summary, topological recursion provides a direct access to the characterisation of the integrable hierarchy behind exact solvability, renewing the discussion of \\cite{Langmann:2003if}. Moreover, topological recursion is somehow a necessary starting point towards the representation of all correlation functions in full generality that we derive in Sec. \\ref{sec:compl}. We will also compare with results obtained in \\cite{Langmann:2003if} for $\\tilde{E}=0$ and point out the aforementioned drastic simplifications that occur in comparison to the QKM in Sec. \\ref{ch:cross}. The occurrence to $\\omega_{g,n}$ of holomorphic additions in the solutions of QKM, which generate poles away from the ramification points will be explained by the nature of hermitian instead of complex fields. With this comes the necessity to make use of the extended framework of \\textit{blobbed topological recursion}, initially developed in \\cite{Borot:2015hna} and proved in the planar sector for the QKM in \\cite{Hock:2021tbl}. In Sec. \\ref{ch:cross}, we also explain TR in in the context of perturbation theory, in particular in the combinatorial limit in which all eigenvalues of the external fields coincide and providing generating functions for the enumeration of bipartite quadrangulations. We will clarify the connection to matrix models tailored for such enumerative problems. In the final discussion and outlook, Sec. \\ref{ch:concl}, we give an overview of further connections and possible investigations such as the triviality problem, the integrable hierarchy behind the model and Hurwitz numbers. The technical proofs of the initial data of the recursion as well as of the extended Dyson-Schwinger equations for the correlation functions in full generality are shifted for readability to the appendix.\n\n\n\n\n\n\n\\section*{ Acknowledgements}\n 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. The work of JB at the University of Oxford was additionally financed by the \\emph{RTG 2149 Strong and Weak\nInteractions \u2013 from Hadrons to Dark Matter}. AH is supported by\nthe Walter-Benjamin fellowship\\footnote{``Funded by\n the Deutsche Forschungsgemeinschaft (DFG, German Research\n Foundation) -- Project-ID 465029630}. The authors are grateful to Harald Grosse and Raimar Wulkenhaar, since many of the used techniques are results of our collaboration over the last years.\n \n\\newpage \n\n{\\footnotesize\\tableofcontents}\n\n\n\\section{The Setup}\n\\label{ch:setup}\n\n\nOur path towards topological recursion of the LSZ model consists of establishing different families of correlators, becoming more and more simple in their algebraic structure. From the partition function, derivatives with respect to the source terms create the usually considered correlation functions $G_{...}$ from other common quantum field theoretical investigations. \nOne remarkable message of this article will be the fact that although they appear as the most natural objects seen from the field theoretical, especially perturbative perspective, another family $\\Omega_{...}$ is much more suitable to concretely formulate the exact solvability. \nIn this manner, it is almost impossible to unravel with usual techniques the common correlation functions $G$, their recursive structure is fairly complicated even after applying TR.\n\nIn fact, only the 2-point function will be needed for our proof of TR. We derive its Dyson-Schwinger equation (DSE), which are equations between correlation functions, using a system of Ward identities that are able to decouple the 2-point function in the large $N$-limit from higher correlation function. \n \\newpage\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width= 0.9\\textwidth]{flow.png} \n\t\\caption{A guide through the main steps in this article towards the complete solution of the LSZ model. In the sense of matrix models, we arrive there in the fourth step by reconstructing the partition function with the free energies of topological recursion. TR will be however only an intermediate step to formulate the complete solution in the quantum field theoretical sense: Exact solutions as rational functions of all arbitrarily complicated correlation functions.\n\t\t\\label{fig:flow}}\n\\end{figure}\n\n However, this yields for the planar 2-point function to a closed nonlinear equation which is known to be solvable. In this equation we recognise the appearance of derivatives with respect to the eigenvalues of one of the two external fields $E$, $\\partial_{E_p}$ (first observed in \\cite{Hock:2020rje}). They play the important role of boundary creation operators in TR, in principle enabling the transition from $\\omega_{g,n}$ to $\\omega_{g,n+1}$. As intermediate steps, we first need to introduce the family of generalised 2-point functions $T_{...}$ and their (finite) Hilbert transforms. Those are the objects to successfully carry out the final proof of TR via abstract loop equations. Afterwards, TR computes the complete family of $\\Omega_{...}$ giving eventually rise to the solution of all generalised correlation functions and thus all objects of quantum field theoretical interest with arbitrarily complicated boundary structure and topology. All these information is visualised in Fig. \\ref{fig:flow} also guiding through the main definitions and theorems. But now we will start from scratch in the following subsections, briefly introducing topological recursion and defining the fundamental objects of interest.\n\n\n \n\n\\subsection{Topological Recursion}\nIn the introductory section we have already mentioned the initial data $(\\Sigma,x,y,B)$ of topological recursion. It contains a Riemann surface $\\Sigma$ (which will be the Riemann sphere $\\overline{\\mathbb{C}}$ for the LSZ model) and the topologically unstable meromorphic forms $\\omega_{0,1}(z)=y(z)\\,dx(z)$ and $\\omega_{0,2}(z_1,z_2)=B(z_1,z_2)=\\frac{dz_1\\, dz_2}{(z_1-z_2)^2}$ on $\\overline{\\mathbb{C}}$ and $\\overline{\\mathbb{C}} \\times\\overline{\\mathbb{C}}$, respectively. They are unstable in the sense that the initial data contains the only $\\omega_{g,n}$ with positive Euler characteristic $\\chi=2-2g-n$. The recursion is then carried out with techniques of complex analysis \\cite{Eynard:2007kz} and yields meromorphic forms $\\omega_{g,n}$ on \n\\begin{align}\n \\label{eq:TR-intro}\n& \\omega_{g,n+1}(I,z)\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 +\\!\\!\\!\\!\\!\\!\\!\\!\n \\sum_{\\substack{g_1+g_2=g\\\\ I_1\\uplus I_2=I\\\\\n (g_i,I_i)\\neq (0,\\emptyset)}}\\!\\!\\!\\!\\!\n \\omega_{g_1,|I_1|+1}(I_1,q)\n \\omega_{g_2,|I_2|+1}(I_2,\\sigma_i(q))\\!\\bigg). \\nonumber\n\\end{align}\nThe notation includes:\n\\begin{itemize}\n\\item $I=\\{z_1,\\dots,z_n\\}$ as a collection of $n$ variables $z_j$\n\\item the ramification points\n$\\beta_i$ of $x$ defined by $dx(\\beta_i)=0$\n\\item the local Galois involution $\\sigma_i\\neq \\mathrm{id}$ with\n$x(q)=x(\\sigma_i(q))$ defined in the vicinity of $\\beta_i$ with fixed point $\\beta_i$\n\\item the recursion kernel $K_i(z,q)$ is also locally \ndefined in the vicinity of $\\beta_i$ by\n\\begin{align*}\nK_i(z,q)=\\frac{\\frac{1}{2}\\int^{q}_{\\sigma_i(q)}\n B(z,\\bullet)}{\\omega_{0,1}(q)-\\omega_{0,1}(\\sigma_i(q))}.\n\\end{align*}\n\\end{itemize}\nIt turns out that $\\omega_{g,n}$ are symmetric in its variables and for $\\chi<0$ all\n $\\omega_{g,n}$ have poles at the ramification points only, with vanishing residues. A pictorial access to this formula is given in Fig. \\ref{fig:tr}.\n\\begin{figure}[h!t]\n\\centering\n\\includegraphics[width= 1\\textwidth]{toprec.pdf} \n\\caption{This is a graphical description of the topological recursion formula eq. (\\ref{eq:TR-intro}). There are two different ways to obtain the left hand side\n $\\omega_{g,n}$ by gluing the recursion kernel $K$ (a pair of pants, built by the initial data) with something of lower topology: Either one glues\n one object with one genus less and one boundary more $(g-1,n+2)$ along\n two boundaries of the kernel creating the missing genus, or one glues\n two objects with the kernel along its boundaries (causing no genus change): Then one has to\n account any $(g_1,n_1),(g_2,n_2)$ conform with the left hand side --\n the sum over all possible partitions in the master formula\n eq. (\\ref{eq:TR-intro}) arises.}\n\t\\label{fig:tr}\n\\end{figure}\nFinally, we emphasise the existence of quantities belonging to the special case $n=0$. Instead of $\\omega_{g,0}$, one usually denotes these quantities by $\\mathcal{F}^{(g)}$, the so-called \\textit{free energy} of genus $g$. They can be obtained as follows: Define the loop annihilation operator $\\Phi(z)$ as a primitive $d_z \\Phi(z)=\\omega_{0,1}(z)$. Then, for negative Euler characteristic $\\chi$, the \\textit{Dilaton equation} holds:\n\\begin{align}\n\\label{dilaton2}\n\\sum_{\\beta_i} \\Res_{z \\to \\beta_i} \\Phi(z)\\omega_{g,1}(z)=(2-2g)\\mathcal{F}^{(g)} \\qquad g>1\n\\end{align}\nThe unstable free energies $\\mathcal{F}^{(0)}$ and $\\mathcal{F}^{(1)}$ obey completely different and much more complicated formulae which will not be of greater interest in this article, see \\cite{Eynard:2007kz}. Altogether, the formal series of $\\mathcal{F}^{(g)}$ encodes the complete information of the model due to\n\\begin{align}\n\\log \\mathcal{Z} = -\\sum_{g=0}^\\infty N^{2-2g} \\mathcal{F}^{(g)}\n\\end{align}\nwhere $\\mathcal{Z}$ is the partition function. Knowing the spectral curve for the LSZ model, we would in principal have its complete solution by topological recursion since the partition function can completely reconstructed from the genus-summed free energies. As announced, the overall goal will be the complete solution in the quantum field theoretical sence (regarding all correlation functions $G$ defined in Defintion \\ref{def:FullExpectation} together with \\eqref{defG}), where however TR will be the starting point. There is a clear strategy how to reach this: Astonishingly, there is also a universal way to prove that a model is governed by TR (or its extensions). Two relatively mild conditions for the $\\omega_{g,n}$ must be fulfilled - they were worked out in \\cite{Borot:2013lpa} and read concretely:\n\\begin{definition}[Abstract loop equations]\n\\label{def:absloopeq}\n A family of meromorphic differential forms $\\omega_{g,n}$ on $\\Sigma^n$,\nwith $g \\geq 0$ and $n >0$, fulfils the \\textbf{linear loop equation}\nif\n\\begin{align}\n\\omega_{g,n+1}(u_1,...,u_n,z)+\n\\omega_{g,n+1}(u_1,...,u_n,\\sigma_i(z))=\n\\mathcal{O}(z-\\beta_i)dz\n\\end{align}\nis a holomorphic linear form in $z$ with (at least) a simple zero\nat $z \\to \\beta_i$. The family of $\\omega_{g,n}$ fulfils the\n\\textbf{quadratic loop equation} if\n\\begin{align}\n&\\omega_{g-1,n+2}(u_1,...,u_n,z,\\sigma_i(z))+ \\hspace*{-0.5cm}\n\\sum_{\\substack{g_1+g_2=g \\\\ I_1\\uplus I_2=\\{u_1,...,u_n\\}}}\n\\hspace*{-0.8cm}\n \\omega_{g_1,|I_1|+1}(I_1,z)\n \\omega_{g_2,|I_2|+1}(I_2,\\sigma_i(z))\n\\nonumber \\\\ \n&=\\mathcal{O}((z-\\beta_i)^2)(dz)^2\n\\label{qle}\n\\end{align}\nis a holomorphic quadratic form in $z$ with at least a double zero at $z \\to \\beta_i$.\n\\end{definition}\nAn important subclass of solutions satisfying the abstract loop equations is given by differentials governed by TR\n\\cite{Borot:2013lpa}, whereas the entirety of solutions is provided\nby the so-called blobbed topological recursion that will become of interest when we compare the LSZ model with the case of hermitian matrices. More concretely, we cite:\n\\begin{theorem}[\\cite{Borot:2013lpa}]\n\\label{thm:ale} Let $\\chi=2-2g-n<0$. \nIf $\\omega_{g,n}$ satisfy the linear and quadratic loop equations and their poles only occur at the ramification points $\\beta_i$ of $x(z)$, they are uniquely given by topological recursion eq. (\\ref{eq:TR-intro}).\n\\end{theorem}\nIt thus solely remains to manipulate the Dyson-Schwinger equations (or loop equations) of suitable quantities in the LSZ model such that it will be possible to show that the two abstract loop equations are fulfilled. To do that, we start with the definition of the natural correlators, seen from the QFT point of view.\n \n\\subsection{Correlation Functions}\nFirst of all, we bring the partition function \\eqref{partfunLSZ} into a convenient form, in order to carry out future calculations simpler. To do that, note that the \\textit{fields} $\\Phi$, $\\Phi^\\dagger$ can be related to derivatives of the corresponding adjoint source term: $\\Phi^\\dagger_{ab}\\leftrightarrow \\frac{1}{N} \\frac{\\partial}{\\partial J_{ba}}$, $\\Phi_{ab} \\leftrightarrow \\frac{1}{N} \\frac{\\partial}{\\partial J^\\dagger_{ba}}$. With this duality, the \\textit{interaction term} can be thus written in a twofold manner:\n\\begin{align*}\n& S_{int}[ \\Phi,\\Phi^\\dagger ] = N\\frac{\\lambda}{2} \\sum_{n_i} \\Phi_{n_1n_2}\\Phi^\\dagger_{n_2n_3}\\Phi_{n_3n_4}\\Phi^\\dagger_{n_4n_1}\\\\\n& S_{int} \\bigg [ \\frac{1}{N} \\frac{\\partial}{\\partial J^\\dagger}, \\frac{1}{N} \\frac{\\partial}{\\partial J} \\bigg ] = N\\frac{\\lambda}{2N^4} \\sum_{n_i} \\frac{\\partial^4}{\\partial J^\\dagger_{n_2n_1} \\partial J_{n_3n_2}\\partial J^\\dagger_{n_4n_3} \\partial J_{n_1n_4}}\n\\end{align*}\nsuch that it can be pulled out of the integral.\n By symmetrisation and a simple remaining Gaussian integration, which is allowed after the upper rewriting, we can then prove the following rearrangement of the partition function:\n\\begin{proposition}\n\\label{prop:partnew}\n$\\mathcal{Z}(J,J^\\dagger)$ can be rewritten as follows:\n\\begin{align*}\n&\\mathcal{Z}(J,J^\\dagger)= C \\exp \\biggl ( - S_{int} \\bigg [ \\frac{1}{N} \\frac{\\partial}{\\partial J^\\dagger}, \\frac{1}{N} \\frac{\\partial}{\\partial J} \\bigg ] \\biggl ) \\mathcal{Z}_{free}(J,J^\\dagger)\n\\end{align*}\nwhere\n\\begin{align*}\nC=\\prod_{m,n=1}^N \\frac{\\pi }{N(E_m+\\tilde{E}_n)} \\qquad \\mathrm{and} \\qquad \\mathcal{Z}_{free}(J,J^\\dagger)=\\exp \\biggl ( N \\sum_{m,n=1}^N \\frac{J_{nm}J^\\dagger_{mn}}{E_n+\\tilde{E}_m} \\biggl ) \n\\end{align*}\n\\begin{proof}\n\tFirst, we symmetrise the kinetic term. The external fields can be diagonalised, $E= \\delta_{mn}E_m$ and $\\tilde{E}= \\delta_{mn}\\tilde{E}_m$, as the partition function is invariant under unitary transformations $U \\in \\mathcal{U}(N)$ (see later in the discussion of the Ward identity, Proposition \\ref{prop:ward}). Therefore, the trace can be rewritten to\n\t\\begin{align*}\n\t\t\\mathrm{Tr}[E \\Phi\\Phi^\\dagger+\\tilde{E} \\Phi^{\\dagger}\\Phi ] = \\sum_{m,n=1}^N (E_n \\Phi_{nm} \\Phi^\\dagger_{mn}+ \\tilde{E}_n \\Phi^\\dagger_{nm} \\Phi_{mn}) = \\sum_{m,n=1}^N (E_n+\\tilde{E}_m) \\Phi_{nm} \\Phi^\\dagger_{mn}\n\t\\end{align*}\n\tNext, we rewrite also the kinetic term combined with the source term:\n\t\\begin{align*}\n\t\t& (E_n+\\tilde{E}_m) \\Phi_{nm} \\Phi^\\dagger_{mn} -J_{nm}\\Phi^\\dagger_{mn}-\\Phi_{nm}J^\\dagger_{mn}\\\\\n\t\t&= (E_n+\\tilde{E}_m) \\bigg ( \\Phi_{nm}- \\frac{J_{nm}}{ (E_n+\\tilde{E}_m)} \\bigg )\\bigg (\\Phi^\\dagger_{mn}- \\frac{J^\\dagger_{mn}}{ (E_n+\\tilde{E}_m)} \\bigg) - \\frac{J_{nm}J^\\dagger_{mn}}{ (E_n+\\tilde{E}_m)}\n\t\\end{align*}\n\tNow, we use the invariance of the measure under the transformations $\\Phi_{nm} \\to \\Phi_{nm}' =\\Phi_{nm}- \\frac{J_{nm}}{ (E_n+\\tilde{E}_m)}$, implying $\\Phi_{mn}^\\dagger \\to (\\Phi_{mn}^\\dagger)'=\\Phi_{mn}^\\dagger- \\frac{J^\\dagger_{mn}}{ (E_n+\\tilde{E}_m)}$.\n\tFinally, one carries out the $2N^2$ Gaussian integrations.\n\\end{proof}\n\\end{proposition}\n\nIn order to define the correlation functions, let us first define the expectation value:\n\\begin{definition}\\label{def:FullExpectation}\nWe define moments\n\\begin{align}\\label{eq:FullExpectation}\n \\langle \\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}..\\Phi_{p_Kq_K}\\rangle :=\\frac{\\int d\\Phi d\\Phi^\\dagger\n \\, \\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}..\\Phi_{p_Kq_K} e^{-S[\\Phi,\\Phi^\\dagger]}}{\\int d\\Phi d\\Phi^\\dagger\n \\, e^{-S[\\Phi,\\Phi^\\dagger]}}.\n\\end{align}\nwhere $K$ even and the action functional reads $S[\\Phi,\\Phi^\\dagger]=- N \\mathrm{Tr}\\big [E \\Phi \\Phi^\\dagger+\\tilde{E} \\Phi^{\\dagger} \\Phi+ \\frac{\\lambda}{2} (\\Phi^\\dagger \\Phi)^2\\big]$. The connected part of this expression is denoted by $ \\langle \\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}..\\Phi_{p_Kq_K}\\rangle_c$. The full expectation value is given in terms of the connected ones by\n\\begin{align*}\n \\langle \\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}..\\Phi_{p_Kq_K}\\rangle=\\sum_{\\text{partitions}}\n \\langle \\Phi^\\dagger_{q_{i^1_1}p_{i^1_1}}..\\Phi_{p_{i^1_{k^1}}q_{i^1_{k^1}}}\\rangle_c\n ..\\langle \\Phi^\\dagger_{q_{i^j_1}p_{i^j_1}}..\\Phi_{p_{i^j_{k^j}}q_{{i^j_{k^j}}}}\\rangle_c,\n\\end{align*}\nwhere the sum over partitions is understood as a sum over all possible \ndecompositions.\n\\end{definition}\n\\begin{remark}\n\tIt is quite easy to see that if the number $K$ in Defintion \\ref{def:FullExpectation} is odd or the number of $\\Phi$ and $\\Phi^\\dagger$ is different that the expectation value vanishes.\n\\end{remark}\n As usual in quantum field theory and probability theory, the connected expectation value is equivalently obtained by \nderivatives with respect to to the source terms, applied on the logarithm of the partition function. In this article we will follow on this connected part of the expectation values\n\\begin{align*}\n \\frac{1}{N^K}\\frac{\\partial^K}{\\partial J_{p_1q_1}\\partial J^\\dagger_{q_2p_2}..\\partial J^\\dagger_{q_Kp_K}}\\log \\frac{\\mathcal{Z}(J,J^\\dagger)}{\\mathcal{Z}(0)}\\bigg\\vert_{J,J^\\dagger=0}=\n \\langle \\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}..\\Phi_{p_Kq_K}\\rangle_c.\n\\end{align*}\nThe indices $\\{p_i,q_i\\} \\in \\{1,N\\}$ may build cycles, otherwise the expectation value vanishes (see \\cite{Schurmann:2019mzu}). A correlation function $G$ built of $b$ cycles of indices $\\{p^\\beta_{1},q_1^\\beta,p^\\beta_{2},q_2^\\beta,...,p_{N_\\beta}^\\beta, q_{N_\\beta}^\\beta\\}$ with $\\beta\\in \\{1,...,b\\}$ having each cycle length $2N_\\beta$ will correspond to the following connected expectation value as follows:\n\\begin{align}\\label{defG}\n G_{|p_1^1q_1^1...p_{N_1}^1q_{N_1}^1|...|p_1^bq_1^b...p_{N_b}^bq_{N_b}^b|} :=N^{b-2} \\frac{\\partial^{2N_1+...+2N_b} \\log [\\mathcal{Z}(J,J^\\dagger)]}{\\partial J_{p_1^1q_1^1}...\\partial J^\\dagger_{q^1_{N_1}p^1_{1}}...\\, \\partial J_{p_1^bq_1^b}...\\partial J^\\dagger_{q^b_{N_b}p^b_{1}}} |_{J,J^\\dagger =0}.\n\\end{align}\nWe call this type of correlation function a $2N_1+...+2N_b$-\\textit{point functions}. This will be taken as definition for the correlation function which we are interested in from QFT perspective.\nThus, the logarithm of the partition function can be expressed formally in terms of correlation functions:\n\\begin{align}\\label{loginG}\n& \\log [\\mathcal{Z}(J,J^\\dagger)] \\\\\\nonumber\n&= \\log[\\mathcal{Z}(0)] +\\sum_{b=1}^\\infty \\sum_{N_1,...,N_b=1}^\\infty \\sum_{p_1^1...q_{N_b}^b=1}^N \\sum_{g=0}^\\infty \\frac{N^{2-b-2g}}{b!} G^{(g)}_{|p_1^1q_1^1...p_{N_1}^1q_{N_1}^1|...|p_1^bq_1^b...p_{N_b}^bq_{N_b}^b|} \\prod_{\\beta=1} ^b \\frac{\\mathbb{J}_\\beta}{N_\\beta}\n\\end{align}\nwith $\\mathbb{J}_\\beta= J_{p_1^\\beta q_1^\\beta}...J^\\dagger_{q^\\beta_{N_\\beta}p^\\beta_{1}}$. $J$ and $J^\\dagger$ show always up alternating, giving rise to pairs $\\{p_i^j,q_i^j\\}$. Furthermore, the common genus expansion $G = \\sum_{g=0}^\\infty N^{-2g} G^{(g)}$ is inserted. For the purpose of this article, it is sufficient for concrete calculations to focus on the 2-point function $G_{|pq|}$. \n\\begin{example}\n\\label{ex:2p}\nThe naive derivatives with respect to the (adjoint) source terms gives, making use of Proposition \\ref{prop:partnew}:\n\\begin{align}\nG_{|pq|}& =\\frac{1}{N} \\frac{\\partial^2 \\log [\\mathcal{Z}(J,J^\\dagger)]}{\\partial J_{pq} \\partial J^\\dagger_{qp}}|_{J,J^\\dagger =0} \\nonumber \\\\\n&= C \\frac{\\partial}{\\partial J_{qp}^\\dagger} \\frac{1}{\\mathcal{Z}(J,J^\\dagger)} e^{-S_{int}[\\partial_J,\\partial_{J^\\dagger}]} \\frac{J^\\dagger_{qp}}{E_p+\\tilde{E}_q} \\mathcal{Z}_{free}(J,J^\\dagger)_{J,J^\\dagger =0} \\nonumber \\\\\\nonumber\n& = \\frac{1}{E_p+\\tilde{E}_q} \\bigg [1-\\frac{\\partial}{\\partial J_{qp}^\\dagger} \\frac{1}{\\mathcal{Z}(J,J^\\dagger)} \\frac{\\partial S_{int}(\\partial_J,\\partial_{J^\\dagger})}{\\partial \\Phi_{qp}^\\dagger} \\mathcal{Z}(J,J^\\dagger) \\bigg ]_{J,J^\\dagger =0}\\\\\\nonumber\n&=\\frac{1}{E_p+\\tilde{E}_q} \\bigg [1-\\lambda \\bigg(\\frac{1}{N^2}\\sum_{n,m=1}^NG_{|pnmq|}+G_{|pq|}\\frac{1}{N}\\sum_{n=1}^N(G_{|pn|}+G_{|nq|})\\\\\n&\\qquad\\qquad\\qquad +\\frac{1}{N^3}\\sum_{n=1}^N(G_{|pq|pn|}+G_{|pq|nq|})\\bigg)\\bigg]\n\\label{eq:2p}\n\\end{align}\nwhere we used the general chain rule:\n\\begin{align*}\ne^{f(\\partial_x)}(x \\cdot g(x)) = xe^{f(\\partial_x)}g(x)+f'(\\partial_x)e^{f(\\partial_x)}g(x) \\qquad e^{f(\\partial_x)} := \\sum_{k=0} ^\\infty \\frac{[f(\\partial_x)]^k}{k!},\n\\end{align*}\nwhere $f(x)$ denotes some polynomial in $x$, $g(x)$ is smooth. We have applied in the last step the expansion \\eqref{loginG}. We observe that we need further relations between correlation functions to turn this ansatz into a closed Dyson-Schwinger equation, since the 2-point function depends on the 4-point function and on the $2+2$-point function which are a priori not known. In this way, Dyson-Schwinger equations are building a tower of coupled equations iteratively depending on higher topologies. \n\\end{example}\n\n\n\n\\subsection{Ward Identities}\nAn exceptional phenomenon in the treated class of QFT toy models is the decoupling of the correlation function of interest from all higher topologies. It was already observed in the early 2000's that a system of \\textit{Ward identities} offers the possibility to obtain closed Dyson-Schwinger equations. In this class of models the identities rely on the fact of unitary invariance of the partition function, first developed in \\cite{Disertori:2006nq} for complex quantum fields, but one external field. This procedure can be adapted to the LSZ model, we are going to prove:\n\\begin{proposition}\n\\label{prop:ward}\nLet $E_l\\neq E_k$. \nThe partition function of the LSZ model satisfies the following system of Ward-identities:\n\\begin{align*}\n\\sum_{n=1}^N\\bigg ( \\frac{E_k-E_l}{N} \\frac{\\partial^2}{\\partial J_{ln} \\partial J^\\dagger_{nk}} - J^\\dagger_{nl} \\frac{\\partial}{\\partial J^\\dagger_{nk}}+J_{kn} \\frac{\\partial}{\\partial J_{ln}}\\bigg )\\mathcal{Z}(J,J^\\dagger)=0\n\\end{align*}\n\\begin{proof}\n\tWe transform the fields under $U\\in \\mathcal{U}(N)$ as $\\Phi^{(U)} =U\\Phi $ and thus $(\\Phi^\\dagger)^{(U)} = \\Phi^\\dagger U^\\dagger$. First of all, we recognise that there is a $U \\in \\mathcal{U}(N)$ such that $ (\\Phi^\\dagger)^{(U)} E (\\Phi)^{(U)} $ diagonalises $E$, $E_{mn} = E_m\\delta_{mn}$. \\footnote{Introducing another unitary transformation $\\Phi^{(U)} = \\Phi U$ and thus $(\\Phi^\\dagger)^{(U)} = U^\\dagger \\Phi^\\dagger $ guarantees the claimed diagonalisability of $\\tilde{E}$ in the same manner.} Let now an other $U=e^{i A}$ a unitary transform on the space of complex $N \\times N$ matrices with $i A$ in the $\\mathfrak{u}(N)$ algebra. Assume $A$ is a small deviation from the unit element such that we expand the expansion around the identity $U = \\mathbbm{1} + i A + \\mathcal{O}(A^2)$. Noticing that the partition function is invariant under unitary transformations, we have the condition $\\mathcal{Z}(\\Phi^{(U)},(\\Phi^\\dagger)^{(U)})-\\mathcal{Z}(\\Phi,\\Phi^\\dagger)=0$.\n\tInsert the field transformations into the partition function and consider the small variation in $A$. The interaction term is untouched by $U$ and the source term behaves at order $A^1$ like $i\\, \\mathrm{Tr}(A(\\Phi J^\\dagger -J \\Phi^\\dagger ))$. We finally discuss the kinetic term. Despite the occurrence of an additional matrix $\\tilde{E}$, the Ward identity does not lose its original shape of \\cite{Disertori:2006nq}, as inserting the unitary transforms gives:\n\t\\begin{align*}\n\t\t& \\mathrm{Tr}(\\Phi^\\dagger E \\Phi + \\Phi \\tilde{E} \\Phi^\\dagger) - \\mathrm{Tr}( (\\Phi^\\dagger)^{(U)} E\\Phi^{(U)} + \\Phi^{(U)} \\tilde{E}(\\Phi^\\dagger)^{(U)}) \\\\\n\t\t& = \\mathrm{Tr}(\\Phi^\\dagger E \\Phi) - \\mathrm{Tr}( \\Phi^\\dagger U^\\dagger E U \\Phi ) \\\\\n\t\t& = \\mathrm{Tr}(\\Phi^\\dagger E \\Phi) - \\mathrm{Tr}( \\Phi^\\dagger ( \\mathbbm{1} - i A) E ( \\mathbbm{1} + i A) \\Phi ) + \\mathcal{O}(A^2) \\\\\n\t\t& = - i \\mathrm{Tr}( A\\Phi \\Phi^\\dagger E - A E\\Phi \\Phi^\\dagger)+ \\mathcal{O}(A^2)\n\t\\end{align*}\n\tExploiting $U^\\dagger U = \\mathbbm{1}$, the additional external field $\\tilde{E}$ cancels and does not show up in the identity. We also used invariance of the trace under cyclic permutations. Substituting $\\Phi^\\dagger_{ab}\\leftrightarrow \\frac{1}{N} \\frac{\\partial}{\\partial J_{ba}}$, $\\Phi_{ab} \\leftrightarrow \\frac{1}{N} \\frac{\\partial}{\\partial J^\\dagger_{ba}}$ and taking the derivative $\\frac{\\partial}{\\partial A_{lk}}$ at $A=0$ finishes the proof.\n\t\n\tA second way to deduce the Ward identity is by partial integration similar to the derivation of the Dyson-Schwinger equation (see \\cite{Hock:2018wup} for details).\n\\end{proof}\n\\end{proposition}\nThe most important tool to deduce closed Dyson-Schwinger equations shows up during the discussion of coinciding matrix indices. It is a fact that the cycle decomposition persists for such a coincidence, meaning that a 2+2-point function always stays topologically different from a 4-point function. As the correlation functions possess a perturbative expansion into rational functions of the eigenvalues $E_m,\\tilde{E}_n$ being continuously differentiable, one may interpret the correlation function as $\\mathcal{C}^1$-functions of all their indices. With this continuity argument, we prove:\n\\begin{proposition}\n\\label{prop:ward2}\nFor distinct $E_m$, the partition function $\\mathcal{Z}(J,J^\\dagger)$ \\eqref{partfunLSZ} of the LSZ model satisfies the system of Ward identities\n \\begin{align}\n\\label{eq:Ward}\n\\sum_{n=1}^N \\frac{\\partial^2 \\mathcal{Z}(J,J^\\dagger)}{\\partial J_{ln} \\partial J^\\dagger_{nk}} = \\delta_{k,l} W_k[J,J^\\dagger]\\mathcal{Z}(J,J^\\dagger) + \\frac{N}{E_k-E_l}\\sum_{n=1}^N \\biggl ( J^\\dagger_{nl} \\frac{\\partial}{\\partial J^\\dagger_{nk}}-J_{kn} \\frac{\\partial}{\\partial J_{ln}}\\bigg )\\mathcal{Z}(J,J^\\dagger)\n \\end{align}\nwhere\n \\begin{align*}\n W_k[J,J^\\dagger]:=&\\sum_{b=1}^\\infty \\sum_{N_1,..,N_b=1}^\\infty \\sum_{p_1^1,q_1^1..,q_{N_b}^b=1}^N\n \\sum_{g=0}^\\infty \\frac{N^{2-b-2g}}{b!}\\prod_{\\beta=1}^b\\frac{\\mathbb{J}_{\\beta}}{N_\\beta}\\times\\\\\n &\\bigg( \\frac{1}{N}\\sum_{n=1}^N G^{(g)}_{|kn|\n p_1^1q_1^1...p_{N_1}^1q_{N_1}^1|...|p_1^bq_1^b...p_{N_b}^bq_{N_b}^b|}\\\\\n &+\\sum_{M=1}^\\infty\\sum_{n,k_1,..,k_{2M-1}=1}^N G^{(g)}_{|knk_1..k_{2M-1}n|\np_1^1q_1^1...p_{N_1}^1q_{N_1}^1|...|p_1^bq_1^b...p_{N_b}^bq_{N_b}^b|}J_{nk_1}J^\\dagger_{k_1k_2}..J^\\dagger_{k_{2M-1}n}\\bigg) \n \\end{align*}\nand $\\mathbb{J}_\\beta= J_{p_1^\\beta q_1^\\beta}...J^\\dagger_{q^\\beta_{N_\\beta}p^\\beta_{1}}$.\n\\begin{proof}\n\tWe rewrite the second derivative as\n\t\\begin{align*}\n\t\t&\\frac{\\partial^2 \\mathcal{Z}(J,J^\\dagger)}{\\partial J_{ln}\\partial J^\\dagger_{nk}} = \\mathcal{Z}(J,J^\\dagger) \\bigg [ \\frac{\\partial^2 \\log[\\mathcal{Z}(J,J^\\dagger)]}{\\partial J_{ln}\\partial J^\\dagger_{nk}} + \\frac{\\partial \\log[\\mathcal{Z}(J,J^\\dagger)]}{\\partial J^\\dagger_{nk}} \\frac{\\partial \\log[\\mathcal{Z}(J,J^\\dagger)]}{\\partial J_{ln}} \\bigg ]\\\\\n\t\t&= (\\delta_{kl} W_k[J,J^\\dagger]+ W^{reg}_{k,l}[J,J^\\dagger])\\mathcal{Z}(J,J^\\dagger) \n\t\\end{align*}\n\tNote that the product of logarithms will not contribute. $W^{reg}_{k,l}[J,J^\\dagger]$ is regular and comes from the Ward identity of Proposition \\ref{prop:ward}, which is safely divide by $\\frac{E_k-E_l}{N}$ yielding\n\t\\begin{align*}\n\t\t W^{reg}_{k,l}[J,J^\\dagger] \\mathcal{Z}(J,J^\\dagger) = \\frac{N}{E_k-E_l}\\sum_{n=1}^N \\biggl ( J^\\dagger_{nl} \\frac{\\partial}{\\partial J^\\dagger_{nk}}-J_{kn} \\frac{\\partial}{\\partial J_{ln}}\\bigg )\\mathcal{Z}(J,J^\\dagger).\n\t\\end{align*}\n\tBy l'Hospital's rule, the coincidence of indices $k \\to l$ yields a continuous transition and a finite (regular) result.\n\tFor the term proportional to $\\delta_{k,l} $, we have to look at the action of\n\t $\\frac{\\partial^2}{\\partial J_{kn}\\partial J^\\dagger_{nk}}$ in a way which can not produced by the Ward identity. There are two cases, first $\\frac{\\partial^2}{\\partial J_{kn}\\partial J^\\dagger_{nk}}$ can act on a cycle of length 2 and produces a closed cycle. The second possibility is a cycle of length $2M+2 \\geq 4$, partially hit by the these derivatives. Expanding the $\\log \\mathcal{Z}$ as in \\eqref{loginG} leads to all terms appearing in $W_k$.\n\\end{proof}\n\\end{proposition}\n\n\\section{Dyson-Schwinger Equations}\n\\label{ch:dse}\n\\subsection{The 2-Point Function}\nWith the results of the previous sections we are prepared to easily derive the closed Dyson-Schwinger equation for the 2-point function.\n\\begin{proposition}\n\\label{thm:2p}\nThe 2-point function of the LSZ model satisfies\n \\begin{align*}\n G_{|pq|}&=\n\\frac{1}{E_p+\\tilde{E}_q}-\\frac{\\lambda}{E_p+\\tilde{E}_q} \n\\bigg\\{ G_{|pq|}\\cdot \\frac{1}{N}\\sum_{n=1}^N G_{|pn|} \n \\\\\n&+\\frac{1}{N^3}\\sum_{n=1}^N G_{|pn|pq|}\n+\\frac{1}{N^2} G_{|pqpq|}\n+\\frac{1}{N}\\sum_{n=1}^N\\frac{G_{|pq|}- G_{|nq|}}{E_n-E_p} \n\\bigg\\}\\;.\n \\end{align*}\n\\begin{proof}\n\tEq. \\ref{eq:2p} turns with the upper Ward identity eq. \\ref{eq:Ward} into\n\t\\begin{align*}\n\t\tG_{|pq|}& =\\frac{1}{E_p+\\tilde{E}_q} \\bigg [1- \\frac{\\lambda}{\\mathcal{Z}(0)} \\sum_{m,n=1}^N\\frac{\\partial^4}{\\partial J_{pm}\\partial J_{mn}^\\dagger\\partial J_{nq}\\partial J_{qp}^\\dagger} \\mathcal{Z}(J,J^\\dagger) \\bigg ]_{J,J^\\dagger =0}\\\\\n\t\t& = \\frac{1}{E_p+\\tilde{E}_q} \\bigg [1-\\frac{\\lambda}{N^3\\mathcal{Z}(0)} \\frac{\\partial^2 W_p[J,J^\\dagger] \\mathcal{Z}(J,J^\\dagger)}{\\partial J_{pq} \\partial J^\\dagger_{qp}} \\\\\n\t\t&- \\frac{\\lambda}{N^2\\mathcal{Z}(0)} \\sum_{m,n=1}^N \\frac{1}{E_n-E_p} \\frac{\\partial^2}{\\partial J_{nq} \\partial J^\\dagger_{qp}} \\bigg ( J_{nm} \\frac{\\partial}{\\partial J_{pm}}-J^\\dagger_{mp} \\frac{\\partial}{\\partial^\\dagger J_{mn}} \\bigg ) \\mathcal{Z}(J,J^\\dagger)\\bigg ]_{J,J^\\dagger =0}\n\t\\end{align*}\n\tNow, all surviving correlation functions after setting $J,J^\\dagger =0$ have to be identified. From $W_k$ we easily read off all the terms in Theorem \\ref{thm:2p} except $\\frac{1}{N}\\sum_{n=1}^N\\frac{G_{|pq|}- G_{|nq|}}{E_n-E_p}$. This term comes from the last line if $J_{nm}$ is hit by $\\frac{\\partial}{\\partial J_{nq}}$ fixing $m=q$, and if $J_{mp}^\\dagger$ is hit by $\\frac{\\partial}{\\partial J^\\dagger_{qp}}$ fixing $m=p$.\n\\end{proof}\n\\end{proposition} \nA formal genus expansion $G = \\sum_{g=0}^\\infty N^{-2g} G^{(g)}$ yields at order $N^{-2g}$:\n\\begin{align}\n\\label{genusexp}\n G^{(g)}_{|pq|}&=\n\\frac{\\delta_{0,g}}{E_p+\\tilde{E}_q}-\\frac{\\lambda}{E_p+\\tilde{E}_q} \n\\bigg\\{ \\sum_{h=0}^{g} G^{(h)}_{|pq|}\n\\frac{1}{N}\\sum_{n=1}^N G^{(g-h)}_{|pn|}\n\\\\\n&+\\frac{1}{N}\\sum_{n=1}^N G^{(g-1)}_{|pn|pq|}\n+G^{(g-1)}_{|pqpq|}\n+\\frac{1}{N}\\sum_{n=1}^N \\frac{G^{(g)}_{|pq|}- G^{(g)}_{|nq|}}{E_n-E_p} \n \\bigg\\}\\nonumber.\n\\end{align}\n Note that in contrast to hermitian fields, we do not have symmetry under exchanging indices, $G_{|pq|}\\neq G_{|qp|}$. For the planar 2-point function we thus have structurally the same nonlinear equation as in the QKM with hermitian fields (compare with \\cite[eq. (9)]{Schurmann:2019mzu}) except for $\\tilde{E}_q$. We will see that also their exact solutions are very similar.\n\n\\subsection{Generalised Correlation Functions}\n So far, we carried out derivatives with respect to the source term. A crucial step towards TR, however, is keeping the eyes on derivatives with respect to the eigenvalues of the external field $E$ - for example:\n\\begin{align*}\n&-N\\frac{\\partial}{\\partial E_{p}} \\int_{C_N} d \\Phi d \\Phi^\\dagger \\exp \\bigg (- N \\mathrm{Tr}\\bigg [E \\Phi \\Phi^\\dagger+\\tilde{E} \\Phi^{\\dagger} \\Phi+ \\frac{\\lambda}{2} (\\Phi^\\dagger \\Phi)^2 - (\\Phi^\\dagger J+J^\\dagger \\Phi)\\bigg] \\bigg )\\bigg\\vert_{J=0} \\\\\n=&N^2\\sum_{n=1}^N\\int_{C_N} d \\Phi d \\Phi^\\dagger\\, \\Phi_{pn}\\Phi^\\dagger_{np} \\exp \\bigg (- N \\mathrm{Tr}\\bigg [E \\Phi \\Phi^\\dagger+\\tilde{E} \\Phi^{\\dagger} \\Phi+ \\frac{\\lambda}{2} (\\Phi^\\dagger \\Phi)^2 - (\\Phi^\\dagger J+J^\\dagger \\Phi)\\bigg] \\bigg )\\bigg\\vert_{J=0}\n\\\\\n =& \\sum_n \\frac{\\partial^2}{\\partial J_{pn}\\partial J^\\dagger_{np}} \\int_{C_N} d \\Phi d \\Phi^\\dagger \\exp \\bigg (- N \\mathrm{Tr}\\bigg [E \\Phi^\\dagger \\Phi+\\tilde{E} \\Phi \\Phi^{\\dagger}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad + \\frac{\\lambda}{2} (\\Phi^\\dagger \\Phi)^2 - (\\Phi^\\dagger J+J^\\dagger \\Phi) \\bigg] \\bigg )\\bigg\\vert_{J=0} \\\\\n=& \\frac{1}{N} \\sum_{n=1}^N G_{|pn|}=:\\Omega_{p}\n\\end{align*}\nWith this simple calculation we defined the first member of the family $\\Omega$, namely $\\Omega_{p}$, via derivative with respect to $E_p$ that will be later related to topological recursion. In particular, we read off that in some sense the initial data of this TR must be encoded in the 2-point function. This appearance of derivatives also allows for a convenient new form of the Dyson-Schwinger equation for the 2-point function:\n\\begin{align}\n\\label{2pnew}\n G_{|pq|}&=\n\\frac{1}{E_p+\\tilde{E}_q} \n\\bigg (1- \\lambda G_{|pq|}\\cdot \\Omega_p\n +\\frac{\\lambda}{N} \\frac{\\partial G_{|pq|}}{\\partial E_p}\n-\\frac{\\lambda}{N}\\sum_{\\substack{{n=1}\\\\{n\\neq p}}}^N\\frac{G_{|pq|}- G_{|nq|}}{E_n-E_p} \n\\bigg )\n\\end{align}\n\\begin{remark}\n\\label{rem:xy}\nBy exchanging the role of the two external fields $E, \\tilde{E}$, it is possible to rewrite this equation in an equivalent form:\n\\begin{align}\n\\label{2pnew2}\n G_{|pq|}&=\n\\frac{1}{E_p+\\tilde{E}_q} \n\\bigg (1- \\lambda G_{|pq|}\\cdot \\frac{1}{N}\\sum_{n=1}^NG_{|nq|}\n +\\frac{\\lambda}{N} \\frac{\\partial G_{|pq|}}{\\partial \\tilde{E}_q}\n-\\frac{\\lambda}{N}\\sum_{\\substack{{n=1}\\\\{n\\neq q}}}^N\\frac{G_{|pq|}- G_{|pn|}}{\\tilde{E}_n-\\tilde{E}_q} \n\\bigg )\n\\end{align}\n\\end{remark}\nThe repeated appearance of the derivative suggests to introduce the following objects:\n\\begin{definition}[Generalised correlation functions]\n\\label{def:gDSE}\nFor pairwise different $ E_{p_i}$ we define the \\textit{generalised correlation functions} by repetitive action of the boundary creation operator $-N\\frac{\\partial}{\\partial E_{p_i}}$:\n\\begin{align*}\n T_{p_1,...,p_m\\|pq}\n &:=\\frac{(-N)^m\\partial^m}{\\partial E_{p_1}...\\partial \n E_{p_m}}G_{|pq|} \n\\end{align*}\nIn the same manner, we introduce a family $\\Omega$ defined as\n\\begin{align*}\n\\Omega_{p_1,...,p_m} &:= \n\\frac{(-N)^{m-1}\\partial^{m-1}\\Omega_{p_1}}{\n\\partial E_{p_2}...\\partial \nE_{p_{m}}} +\\frac{\\delta_{m,2}}{(E_{p_1}-E_{p_2})^2}\\;,\\qquad\nm\\geq 2\\;,\n\\end{align*}\nwhere $ \\Omega_{p_1}=\\frac{1}{N} \\sum_{k=1}^N G_{|p_1k|}$. \n\\end{definition}\nFor these objects we will establish similar Dyson-Schwinger equations. For coinciding indices a derivative with respect to one of the indices has to be taken into account, e.g.\n\\begin{align}\n\t-N\\frac{\\partial}{\\partial E_p}G_{pq}=T_{p\\|pq}-N\\lim_{n\\to p}\\frac{G_{nq}-G_{pq}}{E_n-E_q}.\n\\end{align}\nThe latter term is well-defined since, as argued earlier, any correlation function has a unique perturbative expansion in $\\lambda$ which is a rational function in $E_i$ (and $\\tilde{E}_i$) at each order in $\\lambda$. Using the representation \\eqref{2pnew} for the DSE of the 2-point function, it is straightforward to show:\n\\begin{corollary}\n\tLet $I=\\{p_1,...,p_m\\}$ be a index set and we abbreviate $T_{p_1,...,p_m\\|pq}=T_{I\\|pq}$ and $\\Omega_{p_1,...,p_m}=\\Omega_{I}$. Then the generalised correlation function $T_{I\\|pq}$ and $\\Omega_{I}$ satisfy the DSE's\n\\begin{align}\n&\\Big(\\tilde{E}_q+E_p+\\frac{\\lambda}{N}\n\\sum_{\\substack{l=1\\\\l\\neq p}}^N \\frac{1}{E_l-E_p}\\Big)T_{I\\|pq}\n- \\frac{\\lambda}{N}\\sum_{\\substack{l=1 \\\\ l\\notin I,p}}^N\n \\frac{T_{I\\|lq}}{E_l-E_p}\n\\label{DSE-T2}\n\\\\\n&= \\delta_{0,|I|}\n-\n\\lambda\\bigg\\{\n\\sum_{I'\\uplus I''=I}\n\\Omega_{I',p} T_{I''\\|pq}\n-\\frac{1}{N}\\frac{\\partial T_{I\\|pq}}{\\partial E_p}\n\\nonumber\n+ \n\\sum_{j=1}^m \\frac{\\partial}{\\partial E_{q_j}}\n\\Big(\\frac{T_{I\\setminus q_j\\|q_jq}}{E_{q_j}-E_p}\\Big)\n\\bigg\\}\\;.\n\\nonumber\n\\end{align}\nand \n\\begin{align}\n&\\Omega_{I,p}\n= \\frac{\\delta_{|I|,1}}{\n(E_{p_1}-E_p)^2}\n+\n\\frac{1}{N}\\sum_{k=1}^NT_{I\\|pk}\n\\label{DSE-Om}\n\\end{align}\n\\begin{proof}\n\tFollows immediately by careful application of Definition \\ref{def:gDSE} on eq. (\\ref{2pnew}). It is in particular noteworthy that the second matrix index of the 2-point function is never hit by the action of the derivative $\\partial_{E_{p_i}}$ - this is responsible for the major differences to the QKM (hermitian fields).\n\\end{proof}\n\\end{corollary}\nAs a final step, we have to compute one further DSE resulting from \\eqref{DSE-T2} by multiplying with $\\frac{\\lambda}{N}\\frac{1}{v-\\tilde{E}_q}$ for some complex variable $v$ and summing over $q$, which yields\n\\begin{align}\\nonumber\n\t&\\big(v+V'(E_p)\\big)\\bigg(-\\delta_{|I|,0}+\\frac{\\lambda}{N}\\sum_q \\frac{T_{I\\|pq}}{v-\\tilde{E}_q}\\bigg)+\\frac{\\lambda^2}{N^2}\\sum_q\\sum_{n\\notin I,p}\\frac{T_{I\\|nq}}{(E_p-E_n)(v-\\tilde{E}_q)}\\\\\\label{H2P}\n\t&=-\\delta_{|I|,0}(\\tilde{V}'(v)+V'(E_p))-\\lambda \\sum_{I_1\\uplus I_2=I} \\bigg(-\\delta_{|I_1|,0}+\\frac{\\lambda}{N}\\sum_q \\frac{T_{I_1\\|pq}}{v-\\tilde{E}_q}\\bigg) \\Omega_{I_2, p}\\\\\\nonumber\n\t&+\\lambda \\sum_{p_i\\in I}\\frac{\\partial}{\\partial E_{p_i}}\\bigg(\\frac{\\frac{\\lambda }{N}\\sum_{q}\\frac{T_{I\\setminus p_i\\| p_iq}}{v-\\tilde{E}_q}}{E_p-E_{p_i}}\\bigg)+\\frac{\\lambda }{N}\\frac{\\partial }{\\partial E_p}\\bigg(\\frac{\\lambda }{N}\\sum_{q} \\frac{T_{I\\|pq}}{v-\\tilde{E}_q}\\bigg),\n\\end{align}\nwhere $V'(E_p)=E_p-\\frac{\\lambda}{N}\\sum_{n\\neq p}\\frac{1}{E_p-E_n}$ and $\\tilde{V}'(x)=x-\\frac{\\lambda}{N}\\sum_{k}\\frac{1}{x-\\tilde{E}_k}$.\n\n\\begin{remark}\n\tNotice that the DSE \\eqref{DSE-T2} of $T_{I\\|pq}$ and \\eqref{H2P} of $\\big(-\\delta_{|I|,0}+\\frac{\\lambda}{N}\\sum_q \\frac{T_{I\\|pq}}{v-\\tilde{E}_q}\\big)$ are of very similar form. However, by construction the solution of latter one is necessarily a rational function in $v$ with poles only at $\\tilde{E_q}$, or equivalenty by multiply with $\\prod_{q} (v-\\tilde{E}_q)$ this would be a polynomial of degree $N$.\n\\end{remark}\n\nIn the next section, we will solve the DSE \\eqref{H2P} and start with the first one $I=\\emptyset$, which will provide the initial data. It is well-known that solutions of this type of models are achieved by complex analytic techniques. Therefore, we have to choose an analytic continuation of the correlations functions in the neighbourhood of $E_i$ (and $\\tilde{E}_j$, respectively). \n\n\n\n\n\\section{Topological Recursion of the LSZ Model}\n\\label{ch:tr} \n\n\n\\subsection{Analytic Continuation and Initial Data}\n\nTo determine the solution of a correlation function, we will choose an analytic continuation imposed by the DSE's. The correlation functions are not uniquely continued in this way, but very canonical. Evaluating the analytic functions at the points $E_i$ (and\/or $\\tilde{E}_j$) yields however the unique result of the DSE's (\\ref{DSE-T2}) and (\\ref{DSE-Om}). The same kind of continuation is performed in the Kontsevich model (see e.g. \\cite{Eynard:2016yaa}) or in the quartic analogue of the Kontsevich model \\cite{Branahl:2020yru}.\n\n\\begin{definition}[Analytic continuation]\\label{def:complexT}\n\tDSE \\ref{DSE-T2} (or equivalently DSE \\eqref{H2P}) suggests the following extension:\n\t\\begin{itemize}\n\t\t\n\t\t\\item[\\textup{(a)}] Introduce holomorphic functions \n\t\t$G,T,\\tilde{\\Omega}$ in several complex variables, \n\t\tdefined on Cartesian products of a connected neighbourhood \n\t\t$\\mathcal{V}$ of $\\{E_1,...,E_N\\}$ (and $\\tilde{\\mathcal{V}}$ of $\\{\\tilde{E}_1,...,\\tilde{E}_N\\}$) in $\\mathbb{C}$ , \n\t\twhich at $E_1,\\dots,E_N$ (and $\\tilde{E}_1,...,\\tilde{E}_N$) agree with the previous correlation functions:\n\t\t\\begin{align*}\n\t\t\tG(E_p,\\tilde{E}_q)&\\equiv G_{|pq|}\\;,\n\t\t\t\\\\\n\t\t\tT(E_{p_1},...,E_{p_n}\\| E_p,\\tilde{E}_q|)\n\t\t\t&\\equiv T_{p_1,\\dots,p_n\\|pq}\\;,\\qquad\n\t\t\t\\\\\n\t\t\t\\tilde{\\Omega}(E_{p_1},...,E_{p_n})\n\t\t\t&\\equiv \\Omega_{p_1,\\dots,p_n}\\;.\n\t\t\\end{align*}\n\t\t\n\t\t\\item[\\textup{(b)}] Write eq. \\eqref{DSE-T2} (or equivalently \\eqref{H2P})\n\t\tin terms of $G,T,\\tilde{\\Omega}$ and postulate that they extend to\n\t\tpairwise different points\n\t\t$\\{E_p\\mapsto\\zeta,\\tilde{E}_q\\mapsto \\eta,E_{p_j} \\mapsto \\zeta_j\\}$ of\n\t\t$\\mathcal{V}$ and $\\tilde{\\mathcal{V}}$.\n\t\t\n\t\t\n\t\t\\item[\\textup{(c)}] Complexify the derivative by \n\t\t\\begin{align*}\n\t\t\t\\frac{\\partial }{\\partial E_p} f(E_p)\\mapsto\n\t\t\t\\frac{f(\\zeta)-f(E_p)}{\\zeta-E_p}\n\t\t\t+\\frac{\\partial }{\\partial E_p}\\Big\\vert_{E_p\\mapsto \\zeta} f(\\zeta)\n\t\t\\end{align*}\n\t\tsuch that the\n\t\t$\\frac{\\partial }{\\partial E_p}\\big\\vert_{E_p\\mapsto \\zeta}$-derivative\n\t\tacts in the sense of Definition \\ref{def:gDSE} with extension to\n\t\t$E_p\\mapsto \\zeta$, and a difference quotient which tends for\n\t\t$\\zeta\\to E_p$ to the derivative on the argument of $f$.\n\t\t\n\t\t\\item[\\textup{(d)}] Keep the $E_n$ and $\\tilde{E}_k$ in summations over $k,n \\in \\{1,\\dots,N\\}$ and \n\t\tcomplete the $n$-summation of $E_n$ with the difference quotient term \n\t\tof \\textup{(c)}.\n\t\tConsider the equations for\n\t\t$\\zeta,\\zeta_i \\in \\mathcal{V}\\setminus\n\t\t\\{E_1,\\dots,E_N\\}$ and $\\eta\\in \\tilde{\\mathcal{V}}\\setminus \\{\\tilde{E}_1,...,\\tilde{E}_N \\} $.\n\t\t\n\t\t\\item[\\textup{(e)}] Define the values of $G,T,\\tilde{\\Omega}$ at \n\t\t$\\zeta=E_p,\\eta=\\tilde{E}_q,\\zeta_i =E_{p_i}$ and \n\t\tat coinciding points by a limit procedure.\n\t\\end{itemize}\n\\end{definition}\n\nThe holomorphic extension gives us more freedom for the correlation functions. The property of having pairwise distinct eigenvalues $E_1,...,E_N$ (and $\\tilde{E}_1,...,\\tilde{E}_N$) can be relaxed to have coinciding eigenvalues. We will deal in the following with the more general situation that $e_1,...,e_d$ are the $d$ pairwise distinct eigenvalues of $E$ with multiplicities $r_1,...,r_d$, i.e. $\\sum_{n=1}^dr_n=N$ and $\\tilde{e}_1,...,\\tilde{e}_{\\tilde{d}}$ are the $\\tilde{d}$ pairwise distinct eigenvalues of $\\tilde{E}$ with multiplicities $\\tilde{r}_1,...,\\tilde{r}_{\\tilde{d}}$, i.e. $\\sum_{k=1}^{\\tilde{d}}\\tilde{r}_k=N$). \n\n\nTogether with the complexification defined and described in Definition \\ref{def:complexT}, the analytically continued DSE's \\eqref{DSE-T2} and \\eqref{H2P} spell out in a straightforward way. We illustrate the procedure with the following example of $I=\\emptyset$:\n\\begin{example}\\label{ex:Gen2P}\n\tThe analytically continued DSE's \\eqref{DSE-T2} and \\eqref{H2P} read for $I=\\emptyset$:\n\t\\begin{align}\\label{Gen2Pcompl1}\n\t\t&\\bigg(\\eta+V'(\\zeta)+\\lambda \\tilde{\\Omega}(\\zeta)\\bigg)T(\\|\\zeta,\\eta|)+\\lambda\\frac{1}{N^2}T(\\zeta\\|\\zeta,\\eta|)\n\t\t=1 -\\frac{\\lambda}{N}\\sum_{n=1}^dr_n\\frac{T(\\|e_n,\\eta|)}{\\zeta-e_n}\n\t\\end{align}\nand\n\\begin{align}\\label{H2Pcompl1}\n\t&\\big(v+V'(\\zeta)+\\lambda \\tilde{\\Omega}(\\zeta)\\big)\\bigg(-1+\\frac{\\lambda}{N}\\sum_{q=1}^{\\tilde{d}}\\tilde{r}_q \\frac{T(\\|\\zeta,\\tilde{e}_q|)}{v-\\tilde{e}_q}\\bigg)+\\frac{\\lambda^2 }{N^3}\\sum_{q=1}^{\\tilde{d}}\\tilde{r}_q \\frac{T(\\zeta\\|\\zeta,\\tilde{e}_q|)}{v-\\tilde{e}_q}\\\\\\nonumber\n\t=&-\\tilde{V}'(v)-V'(\\zeta)-\\frac{\\lambda^2}{N^2}\\sum_{q,n=1}^{\\tilde{d},d}\\frac{T(\\|e_n,\\tilde{e}_q|)}{(\\zeta-e_n)(v-\\tilde{e}_q)}.\n\\end{align}\nThe potentials are given by $V'(\\zeta)=\\zeta-\\frac{\\lambda}{N}\\sum_{n=1}^d\\frac{r_n}{\\zeta-e_n}$ and $\\tilde{V}'(x)=x-\\frac{\\lambda}{N}\\sum_{k=1}^{\\tilde{d}}\\frac{\\tilde{r}_k}{x-\\tilde{e}_k}$ and the $n$-summation got unrestricted due to \\textup{(c)} of Definition \\ref{def:complexT}. We emphasise the crucial fact that the rhs (and therefore the lhs as well) of \\eqref{Gen2Pcompl1} is a rational function in $\\zeta$ with simple poles at the points $e_n$. Multiplying by $\\prod_{n=1}^d (\\zeta-e_n)$ generates a polynomial of degree $d$. Furthermore, the rhs (and therefore the lhs) of \\eqref{H2Pcompl1} is a rational function in $\\zeta$ and $v$ with simple poles at the points $e_n$ and $\\tilde{e}_q$. Multiplying by $\\prod_{n=1}^d (\\zeta-e_n)\\prod_{q=1}^{\\tilde{d}} (v-\\tilde{e}_q)$ generates a polynomial of degree $d+1$ in $\\zeta$ and $\\tilde{d}+1$ in $v$.\n\\end{example}\n\nAs mentioned earlier, the correlation functions are graded by the genus $g$ and have a formal genus expansion in $N$. In case of $g=0$, this decouples \\eqref{Gen2Pcompl1} and \\eqref{H2Pcompl1} from $T(\\zeta\\|\\zeta,.|)$ and yields a nonlinear equation for $T(\\|\\zeta,\\eta|)$. Together with the rationality property underlined in Example \\ref{ex:Gen2P}, these two equations can be solved simultaneously. The solution is based on some natural, weak assumptions.\n\\begin{theorem}[Initial data for topological recursion]\\label{Thm:InitialData}\n\t\tLet $\\lambda,e_n,\\tilde{e}_k>0$. Assume two rational functions $x,y:\\overline{\\mathbb{C}}\\to \\overline{\\mathbb{C}}$ exist with $x$ of rational degree $\\tilde{d}+1$ and $y$ of rational degree $d+1$, both normalised such that $ x(z)=z+\\mathcal{O}(1)$ and $y(z)=-z+\\mathcal{O}(1)$. \n\tLet further $G(x(z),y(w))=T(\\|x(z),y(w)|)=:\\mathcal{G}(z,w)$, and let $x,y$ satisfy the functional relations\n\t\\begin{align}\\label{yGsum}\n\t\ty(z)=&-V'(x(z))-\\frac{\\lambda}{N}\\sum_{k=1}^{\\tilde{d}}\\tilde{r}_k\\mathcal{G}(z,\\tilde{\\varepsilon}_k)\\\\\n\t\tx(z)=&\\tilde{V}'(y(z))+\\frac{\\lambda}{N}\\sum_{n=1}^{d}r_n\\mathcal{G}(\\varepsilon_n,z),\n\t\\end{align}\n\twhere the points $\\varepsilon_n$ and $\\tilde{\\varepsilon}_k$ are implicitly defined by $e_n=:x(\\varepsilon_n)$ and $\\tilde{e}_k=y(\\tilde{\\varepsilon}_k)$. We denote the $\\tilde{d}+1$ solutions of $x(v)=x(z)$ by $v\\in\\{z,\\hat{z}^1,...,\\hat{z}^{\\tilde{d}}\\}$ and the $d+1$ solutions of $y(v)=y(w)$ by $v\\in\\{w,\\hat{\\tilde{w}}^1,...,\\hat{\\tilde{w}}^{d}\\}$. Assume that $y(\\hat{z}^{l})$ and $x(\\hat{\\tilde{w}}^{l})$ as well as $\\mathcal{G}(z,\\hat{z}^{l})$ and $\\mathcal{G}(\\hat{\\tilde{w}}^{l},w)$ are finite $\\forall l$. Then, $x,y$ and $\\mathcal{G}$ are uniquely determined by\n\t\\begin{align}\\label{xy}\n\t\tx(z)=z-\\frac{\\lambda}{N}\\sum_{k=1}^{\\tilde{d}}&\\frac{\\tilde{r}_k}{y'(\\tilde{\\varepsilon}_k)(z-\\tilde{\\varepsilon}_k)},\\qquad y(z)=-z+\\frac{\\lambda}{N}\\sum_{n=1}^{d}\\frac{r_n}{x'(\\varepsilon_n)(z-\\varepsilon_n)}\\\\\n\t\t\\mathcal{G}(z,w)=&\\frac{E(x(z),y(w))}{(x(z)-x(w))(y(z)-y(w))},\\\\ \\text{where}\\qquad \n\t\t E(x(z),y(w))=&(y(w)-y(z))\\prod_{k=1}^{\\tilde{d}}\\frac{y(w)-y(\\hat{z}^k)}{y(w)-y(\\tilde{\\varepsilon}_k)}\\\\\n\t\t =&(x(z)-x(w))\\prod_{n=1}^{d}\\frac{x(z)-x(\\hat{\\tilde{w}}^n)}{x(z)-x(\\varepsilon_n)}.\n\t\\end{align}\n \\begin{proof}\n \tSee Appendix \\ref{sec:2p}\n \\end{proof}\n\\end{theorem}\nNote again that $\\mathcal{G}(z,w) \\neq \\mathcal{G}(w,z)$ for unrelated $E$ and $\\tilde{E}$ in contrast to the hermitian model.\nHere we encounter a general phenomenon during the solution strategy of a certain matrix model using TR. The correlator of the most basic topology is a sophisticated (nonlinear) problem, often related to a Riemann-Hilbert problem. Its solution already reveals the geometry of the spectral curve, implicitly containing all the information about higher topological sectors. Indeed, the same holds for the LSZ model: The solution given in Theorem \\ref{Thm:InitialData} provides most of the initial data for topological recursion $(\\Sigma,x,y,B)$ as we will show in the rest of this section, where $\\Sigma=\\overline{\\mathbb{C}}$ and $x,y$ given by \\eqref{xy}. The rational parametrisation of $x,y$ implies to a genus 0 spectral curve, which uniquely determines $B(z,w)=\\frac{dz\\,dw}{(z-w)^2}$. We will show that the Bergman kernel is directly related to $\\Omega$ with topology $(g,n)=(0,2)$ in Appendix \\ref{sec:om02}. \n\nIt still remains to show how the correlators $\\omega_{g,n}$ computed by TR, eq. (\\ref{eq:TR-intro}), are related to the correlation functions of the LSZ model. It turns out that $\\omega_{g,n}$ are in direct relation to $\\Omega_{q_1,...,q_n}$ defined in Definition \\ref{def:gDSE}. To prove this, we will need to perform a decisive variable transformation implied by the solution of $x(z)$ and $y(z)$. \n\n\n\\subsection{Complexified Dyson-Schwinger Equations}\nThe hidden algebraic structure behind the LSZ model becomes visible after a final variable transformation imposed by $x(z)$ and $y(z)$, which was achieved by the solution of the 2-point function in Theorem \\ref{Thm:InitialData}. The variable transformation via $x(z)$ and $y(z)$ is a transformation depending on the coupling constant $\\lambda$. We define the correlation functions on the (usually called) $z$-plane:\n\\begin{definition}[Correlation functions on the $z$-plane]\\label{def:compl}\n\tLet $G,T,\\tilde{\\Omega}$ be the functions in several \n\tcomplex variables obtained by the complexification of\n\tDefinition~\\ref{def:complexT} and by \n\tadmitting multiplicities $r_n,\\tilde{r}_k$ of $e_n,\\tilde{e}_k$. Then functions \n\t$\\mathcal{G},\\mathcal{T},\\Omega_n$ of several \n\tcomplex variables are introduced by\n\t\\begin{align*}\n\t\t\\mathcal{G}(u,w)&:=G(x(u),y(w))\\;,\n\t\t\\\\\n\t\t\\mathcal{T}(z_1,...,z_n\\|u,w|)\n\t\t&:=\n\t\tT(x(z_1),...,x(z_n)\\|x(u),y(w)|)\\;,\n\t\t\\\\\n\t\t\\Omega_{n}(z_1,....,z_n)&:=\n\t\t\\tilde{\\Omega}(x(z_1),...,x(z_n))+\\frac{\\delta_{n,1}}{\\lambda}V'(x(z_1)) \\;.\n\t\\end{align*}\n\tWe let $\\mathcal{T}(\\emptyset\\|u,w|):=\n\t\\mathcal{G}(u,w)$.\n\\end{definition}\nWe write the genus-expanded correlation functions as\n\\begin{align}\n\t\\mathcal{T}(z_1,...,z_n\\|u,w|)=&:\\sum_{g=0}^\\infty N^{-2g} \\mathcal{T}^{(g)}(z_1,...,z_n\\|u,w|)\\\\\n\t\\Omega_{n}(z_1,....,z_n)=&:\\sum_{g=0}^\\infty N^{-2g}\\Omega_{g,n}(z_1,....,z_n).\n\\end{align}\nCombining all of the previous constructions and definitions, we finally get a complexified DSE on the $z$-plane, which takes a well-known form.\n\\begin{corollary}[DSE of topological recursion type]\\label{Cor:DSE}\n\tLet $I=\\{z_2,...,z_n\\}$ and \n\t\\begin{align*}\n\t\tH_{g,n}(v;z;I):=&-\\delta_{|I|,0}\\delta_{g,0}+\\frac{\\lambda}{N}\\sum_{k=1}^{\\tilde{d}} \\tilde{r}_k\\frac{\\mathcal{T}^{(g)}(I\\|z,\\tilde{\\varepsilon}_k)}{v-\\tilde{e}_k},\\\\\n\t\tP_{g,n}(v;x;I):=&\\delta_{|I|,0}\\delta_{g,0}(\\tilde{V}'(v)+V'(x))+\\frac{\\lambda^2}{N^2}\\sum_{k,n=1}^{\\tilde{d},d} \\tilde{r}_kr_n\\frac{\\mathcal{T}^{(g)}(I\\|\\varepsilon_n,\\tilde{\\varepsilon}_k)}{(v-\\tilde{e}_k)(x-e_n))}\\\\\n\t\t&-\\sum_{z_i\\in I}\\frac{\\partial}{\\partial x(z_i)}\\frac{\\lambda^2}{N}\\sum_{k=1}^{\\tilde{d}} \\tilde{r}_k\\frac{\\mathcal{T}^{(g)}(I\\setminus z_i\\|z_i,\\tilde{\\varepsilon}_k)}{(v-\\tilde{e}_k)(x-x(z_i))}+\\frac{\\lambda \\delta_{g,0}\\delta_{n,2}}{(x-x(z_2))^2},\n\t\\end{align*}\n\twhere $H$ is by definition rational in $v$ and $P$ rational in $v$ and $x$. Then, $H$ and $P$ satisfy the following DSE\n\t\\begin{align}\\label{DSEHP}\n\t\t&(v-y(z))H_{g,n}(v;z;I)+P_{g,n}(v;x(z);I)\\\\\\nonumber\n\t\t=&-\\lambda \\sum_{\\substack{g_1+g_2=g\\\\ I_1\\uplus I_2=I}}^\\prime H_{g_1,|I_1|+1}(v;z;I_1)\\Omega_{g_2,|I_2|+1}(I_2,z)-\\lambda H_{g-1,n+1}(v;z;z,I),\n\t\\end{align}\n\twhere the primed sum excludes $(g_2,I_2)=(0,\\emptyset)$.\n\t\\begin{proof}\n\t\tThe DSE of $H$ and $P$ arises from DSE \\eqref{H2P}. First, the complexification of Definition \\ref{def:complexT} has to be considered to formulate the DSE in terms of $T$ and $\\tilde{\\Omega}$ in the variables $\\zeta_j, \\zeta$ and $\\eta$ similar to Example \\ref{ex:Gen2P}. Then, Theorem \\ref{Thm:InitialData} is applied with \\eqref{yGsum} to get $y(u)$ on the lhs of the equation. Following the variable transformation given in Definition \\ref{def:compl} via $x(z)$ and $y(z)$, and the definition of $H$ and $P$ yields the assertion.\n\t\\end{proof}\n\\end{corollary}\n\nExactly the same structure of the DSE of Corollary \\ref{Cor:DSE} appears in the 2-matrix model \\cite[eq. (2-18)]{Chekhov:2006vd} and the generalised Kontsevich model \\cite[eq. (3.16)]{Belliard:2021jtj}. We will follow the same steps to proof TR. From the definitions, we have for the asymptotic expansion in $v$ the following leading order terms: \n\\begin{align}\\label{HOmega}\n\tH_{g,n}(v;z;I)=-\\delta_{|I|,0}\\delta_{g,0}+&\\frac{1}{v}\\bigg(\\lambda\\Omega_{g,n}(z,I)\n\t-\\delta_{|I|,0}\\delta_{g,0}V'(x(z))-\\frac{\\lambda \\delta_{|I|,1}\\delta_{g,0}}{(x(z)-x(z_2))^2}\\bigg)\\\\\\nonumber\n\t&+\\mathcal{O}(v^{-2})\n\\end{align}\n\nWe recall that $H_{g,n}(v;z;I)$ is a rational function in $v$ with simple poles located at $\\tilde{e}_k$. Consequently, after multiplying by $\\prod_{k=1}^{\\tilde{d}}(v-y(\\tilde{\\varepsilon}_k))$ we obtain a polynomial in $v$ of degree $\\tilde{d}$ for $(g,n)=(0,0)$ and degree $\\tilde{d}-1$ for any other $(g,n)$. Furthermore, $P_{g,n}(v;x(z);I)$ of \\eqref{DSEHP} is a rational function in $v$ and $x(z)$, where all poles are known as well. After multiplying with $\\prod_{k=1}^{\\tilde{d}}(v-y(\\tilde{\\varepsilon}_k))$ it turns (for $2g+n-2\\geq 0$) into a polynomial of degree $\\tilde{d}$ in $v$. A polynomial of degree $\\tilde{d}$ is uniquely determined by $\\tilde{d}+1$ points.\nIn particular, we already know the lowest topology in $H$ and $P$:\n\\begin{align}\\label{H01P01}\tH_{0,1}(v;z;\\emptyset)=-\\prod_{k=1}^{\\tilde{d}}\\frac{v-y(\\hat{z}^k)}{v-\\tilde{e}_k},\n\\qquad P_{0,1}(v;x(z);\\emptyset)=(v-y(z))\\prod_{k=1}^{\\tilde{d}}\\frac{v-y(\\hat{z}^k)}{v-\\tilde{e}_k}.\n\\end{align}\nThese expressions were used in the proof of Theorem \\ref{Thm:InitialData}, see App. \\ref{sec:2p}. Additionally, the fact that $P_{g,n}(v;x(z);I)$ depends on $x(z)$ rather than on $z$ yields several identities for DSE \\eqref{DSEHP} by\n\\begin{align*}\n\tP_{g,n}(v;x(z);I)=P_{g,n}(v;x(\\hat{z}^k);I),\n\\end{align*}\nwhere the $\\hat{z}^k$'s were defined as the preimages under $x$, i.e. $x(z)=x(\\hat{z}^k)$ for $k=1,...,\\tilde{d}$.\nThese properties of the functions $H$ and $P$ will be crucial in the proof of TR. Before going into more details, we will finish this subsection with the solution for the second ingredient of the initial data, $\\Omega_{0,2}$.\n\\begin{proposition}\\label{pro:om02}\n\tThe cylinder amplitude of the LSZ model reads\n\t\\begin{align}\n\t\t\\Omega_{0,2}(z_1,z_2)=\\frac{1}{x'(z_1)x'(z_2)(z_1-z_2)^2}.\n\t\\end{align}\n\t\\begin{proof}\n\t\tSee Appendix \\ref{sec:om02}\n\t\\end{proof}\n\\end{proposition}\n\n\n\n\n\n\\subsection{Proof of Topological Recursion}\nThe proof will proceed in three steps, in complete analogy to the proof that the generalised Kontsevich model is governed by topological recursion \\cite{Belliard:2021jtj}:\n\\begin{itemize}\n\t\\item[\\textbf{I.}] The solution of $H_{g,n}$ and $P_{g,n}$ is derived in terms of sums of the $\\Omega_{g',n'}$ where the arguments are the preimages $\\hat{z}^k$ with $2g+n-2\\geq 2g'+n'-2$\n\t\\item[\\textbf{II.}] The linear and quadratic loop equations are extracted from the exact solution obtained in \\textbf{I.}\n\t\\item[\\textbf{III.}] TR is concluded from the linear and quadratic loop equations of \\textbf{II.}, see Theorem \\ref{thm:ale}\n\\end{itemize}\n\\textbf{I.}\nThe solutions of $H$ and $P$ are constructed by the following functions (first defined in \\cite{Bouchard:2012yg}, but we follow the notation of \\cite{Bouchard:2016obz,Belliard:2021jtj})\n\\begin{definition}\n\tLet $\\underline{t}:=\\{t_1,...,t_k\\}$ and $I=\\{z_2,...,z_n\\}$, then we define\n\t\\begin{align}\n\t\t\\mathcal{E}^{(k)}\\Omega_{g,n}(\\underline{t};I):=\\sum_{\\mu \\vdash \\underline{t}}\\sum_{I_1\\uplus ...\\uplus I_{l(\\mu)}=I}\\sum_{\\substack{h_1+...h_{l(\\mu)}\\\\\n\t\t=g+l(\\mu)-k}}\\bigg(\\prod_{i=1}^{l(\\mu)}\\Omega_{h_i,|\\mu_i\\uplus I_i|}(\\mu_i,I_i)\\bigg),\n\t\\end{align}\nwhere $\\mu \\vdash \\underline{t}$ means that $\\mu$ is a set partition of $\\underline{t}$, which consists of $l(\\mu)$ parts $\\mu_1,...,\\mu_{l(\\mu)}$. In particular, we set $\\mathcal{E}^{(0)}\\Omega_{g,n}(\\emptyset;I)=\\delta_{g,0}\\delta_{n,1}$.\n\\end{definition}\nIn the next step, we introduce the functions $\\check{H}_{g,n}$ and $\\check{P}_{g,n}$, which are designed to coincide with $H_{g,n}$ and $P_{g,n}$:\n\\begin{definition}\\label{Def:Hcheck}\n\tLet $\\tau(z)=\\{z,\\hat{z}^1,...,\\hat{z}^{\\tilde{d}}\\}$ be the set of solutions of the equation $x(\\bullet)=x(z)$, and let $\\tau_0(z)=\\tau(z)\\setminus \\{z\\}$. For $I=\\{z_2,...,z_n\\}$, define\n\t\\begin{align}\n\t\t\\check{H}_{g,n}(v;z;I)=&-\\frac{1}{\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)}\\sum_{i=0}^{\\tilde{d}}\\lambda^iv^{\\tilde{d}-i}\\sum_{\\underline{t}\\subset_i\\tau_0(z) }\\mathcal{E}^{(i)}\\Omega_{g,n}(\\underline{t};I)\\\\\n\t\t\\check{P}_{g,n}(v;x(z);I)=&\\frac{1}{\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)}\\sum_{i=0}^{\\tilde{d}+1}\\lambda^iv^{\\tilde{d}+1-i}\\sum_{\\underline{t}\\subset_i\\tau(z) }\\mathcal{E}^{(i)}\\Omega_{g,n}(\\underline{t};I),\n\t\\end{align}\n\twhere $\\subset_i$ takes a subset of $i$ elements. The main difference in the definition of $\\check{H}$ and $\\check{P}$ is the sum of $\\underline{t}$ which either lies in $\\tau_0(z)$ or $\\tau(z)=\\tau_0(z)\\cup \\{z\\}$. By definition, $\\check{P}$ is a rational function in $x(z)$ due to the symmetric representation in all preimages $\\hat{z}^i$, including $z$.\n\\end{definition}\nFor $g=0$ and $I=\\emptyset$ the expressions considerably simplify.\nIt is straightforward to show their lowest topology $(g,n)=(0,1)$ with $y(z)=-\\lambda \\Omega_{0,1}(z)$ by their definitions: \n\\begin{align}\\label{Hcheck01}\n\t\\check{H}_{0,1}(v;z;\\emptyset)=-\\prod_{k=1}^{\\tilde{d}}\\frac{v-y(\\hat{z}^k)}{v-\\tilde{e}_k}\\quad \\text{and}\\quad \\check{P}_{0,1}(v;x(z);\\emptyset)=(v-y(z))\\prod_{k=1}^{\\tilde{d}}\\frac{v-y(\\hat{z}^k)}{v-\\tilde{e}_k}\n\\end{align}\nThese expressions coincide with \\eqref{H01P01}. The coincidence holds for any $(g,n)$:\n\\begin{proposition}\\label{Prop:HHcheck}\n\tLet $I=\\{z_2,...,z_n\\}$. Then,\n\tthe following holds:\n\t\\begin{align*}\n\t\t\\check{H}_{g,n}(v;z;I)=H_{g,n}(v;z;I)\\qquad \\text{and}\\qquad \\check{P}_{g,n}(v;x;I)=P_{g,n}(v;x;I).\n\t\\end{align*}\n\\begin{proof}\n\tDecompose $\\check{P}_{g,n}$ of Definition \\ref{Def:Hcheck} in terms of $\\check{H}_{g',n'}$ by separating $z\\in \\tau(z)$ yields an equation between $\\check{P}_{g,n}$ and $\\check{H}_{g',n'}$ with $2g+n-2\\geq 2g'+n'-2$. Applying \\cite[Lemma 3.17]{Bouchard:2016obz} \n\t\\begin{align}\n\t\t\\mathcal{E}^{(k+1)}\\Omega_{g,n}(z,\\underline{t};I)=\\mathcal{E}^{(k)}\\Omega_{g-1,n+1}(\\underline{t};z,I)+\\sum_{\\substack{g_1+g_2=g\\\\I_1\\uplus I_2=I}}\\Omega_{g_1,|I_1|+1}(z,I_1)\\mathcal{E}^{(k)}\\Omega_{g_2,|I_2|+1}(\\underline{t};I_2),\n\t\\end{align}\nreduces the previous equation to \n\\begin{align}\\label{DSEHPcheck}\n\t&(v-y(z))\\check{H}_{g,n}(v;z;I)+\\check{P}_{g,n}(v;x(z);I)\\\\\\nonumber\n\t=&-\\lambda \\sum_{\\substack{g_1+g_2=g\\\\ I_1\\uplus I_2=I}}^\\prime \\check{H}_{g_1,|I_1|+1}(v;z;I_1)\\Omega_{g_2,|I_2|+1}(I_2,z)-\\lambda \\check{H}_{g-1,n+1}(v;z;z,I),\n\\end{align}\nwhich coincides with \\eqref{DSEHP} of Corollary \\ref{Cor:DSE} (we refer to \\cite{Belliard:2021jtj} for more details). Now, we proceed by induction in the negative Euler characteristic $-\\chi=2g+n-2$, where the initial step was already shown in \\eqref{Hcheck01}. Assume that Proposition \\ref{Prop:HHcheck} holds for all $2g'+n'-2<2g+n-2$, then we obtain by subtracting \\eqref{DSEHP} from \\eqref{DSEHPcheck}\n\\begin{align}\n\tP_{g,n}(v;x(z);I)-\\check{P}_{g,n}(v;x(z);I)=(v-y(z))(\\check{H}_{g,n}(v;z;I)-H_{g,n}(v;z;I)).\n\\end{align}\nSince the lhs is a rational function of $x(z)$, it does not change by $z\\mapsto \\hat{z}^j$\n\\begin{align}\n\tP_{g,n}(v;x(z);I)-\\check{P}_{g,n}(v;x(z);I)=(v-y(\\hat{z}^j))(\\check{H}_{g,n}(v;\\hat{z}^j;I)-H_{g,n}(v;\\hat{z}^j;I)).\n\\end{align}\nSetting $v=y(\\hat{z}^j)$ for $j=0,...,\\tilde{d}$, the rhs vanishes at $\\tilde{d}+1$ points, where $P$ and $\\check{P}$ coincide:\n\\begin{align}\\label{Pd+1}\n\tP_{g,n}(y(\\hat{z}^j);x(z);I)=\\check{P}_{g,n}(y(\\hat{z}^j);x(z);I),\\qquad \\forall j=0,...,\\tilde{d}\n\\end{align}\nRecall that after multiplying $\\check{P}$ by $\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)$ it becomes a polynomial of degree $\\tilde{d}$, which is uniquely determined by the $\\tilde{d}+1$ points in \\eqref{Pd+1}. This gives rise to the equality between $P$ and $\\check{P}$. Furthermore, the DSE \\eqref{DSEHPcheck} directly leads to the equality between $H$ and $\\check{H}$.\n\\end{proof}\n\\end{proposition}\n\n\\textbf{II.}\nThe equality given in Proposition \\ref{Prop:HHcheck} is more involved than it seems. Expanding it around $v$, we obtain equations for $\\Omega_{g,n}$ which are equivalent to the linear and quadratic loop equations (see Definition \\ref{def:absloopeq}):\n\\begin{proposition}\\label{Prop:linquad}\n\tLet $I=\\{z_2,...,z_n\\}$ and $2g+n-2 > 0$, then\n\t\\begin{align}\\label{linearloop}\n\t\t\\sum_{k=0}^{\\tilde{d}}\\Omega_{g,n}(\\hat{z}^k,I)=0\n\t\\end{align}\nand \n\\begin{align}\\label{quadloop}\n\t&\\frac{1}{2}\\sum_{k=0}^{\\tilde{d}}\\bigg(\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}\\Omega_{g_1,|I_1|+1}(\\hat{z}^k,I_1)\\Omega_{g_2,|I_2|+1}(\\hat{z}^k,I_2)+\\Omega_{g-1,n+1}(\\hat{z}^k,\\hat{z}^k,I)\\bigg)\\\\\\nonumber\n\t=&\\sum_{z_i\\in I}\\frac{\\partial}{\\partial x(z_i)}\\frac{\\Omega_{g,n-1}(I)}{x(z)-x(z_i)}-\\frac{1}{N}\\sum_{k=1}^{d}r_k\\frac{\\Omega_{g,n}(\\varepsilon_k,I)}{x(z)-e_k}\n\\end{align}\n\n\n\\begin{proof}\n\tThe first equation is obtained by identifying (due to Proposition \\ref{Prop:HHcheck}) the expressions\n\t\\begin{align*}\n\t\t[v^{\\tilde{d}-1}]\\bigg(\\check{H}_{g,n}(v;z;I)\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)\\bigg)=&-\\lambda \\sum_{k=1}^{\\tilde{d}}\\Omega_{g,n}(\\hat{z}^k,I)\\\\\n\t\t[v^{\\tilde{d}-1}]\\bigg(H_{g,n}(v;z;I)\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)\\bigg)=&\\lambda\\Omega_{g,n}(z,I),\n\t\\end{align*}\n\twhich follows directly by their definitions for $2g+n-2\\geq 0$. Going to the next order in $v$, we obtain\n\t\\begin{align*}\n\t\t&[v^{\\tilde{d}-2}]\\bigg(\\check{H}_{g,n}(v;z;I)\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)\\bigg)\\\\\n\t\t=&-\\frac{\\lambda^2}{2}\\bigg(\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}\\Omega_{g_1,|I_1|+1}(z,I_1)\\Omega_{g_2,|I_2|+1}(z,I_2)+\\Omega_{g-1,n+1}(z,z,I)\\bigg)\\\\\n\t\t&+\\frac{\\lambda^2}{2}\\sum_{k=1}^{\\tilde{d}}\\bigg(\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}\\Omega_{g_1,|I_1|+1}(\\hat{z}^k,I_1)\\Omega_{g_2,|I_2|+1}(\\hat{z}^k,I_2)+\\Omega_{g-1,n+1}(\\hat{z}^k,\\hat{z}^k,I)\\bigg)\\\\\n\t\t&+\\lambda^2\\sum_{z_i\\in I}\\frac{\\partial}{\\partial x(z_i)}\\frac{\\Omega_{g,n-1}(I\\setminus z_i,z)}{x(z)-x(z_i)}-\\frac{\\lambda^2\\delta_{g,0} \\delta_{|I|,2}}{(x(z)-x(z_2))^2(x(z)-x(z_3))^2}\\\\\n\t\t&+V'(x(z))\\bigg(\\Omega_{g,n}(z,I)-\\frac{\\lambda \\delta_{g,0}\\delta_{|I|,1}}{(x(z)-x(z_2))^2}\\bigg)\n\t\\end{align*}\n\twhere $\\mathcal{E}^{(2)}$ was written out explicitly, the sum over the preimages was anti-symmetrised and the linear loop equation \\eqref{linearloop} applied.\n\t\n\tOn the other hand, the expansion of $H_{g,n}(v;z;I)$ is achieved by expanding the DSE \\eqref{DSEHP} of Corollary \\ref{Cor:DSE}\n\t\\begin{align*}\n\t\t&[v^{\\tilde{d}-2}]\\bigg(H_{g,n}(v;z;I)\\prod_{k=1}^{\\tilde{d}}(v-\\tilde{e}_k)\\bigg)\\\\\n\t\t=&-\\lambda^2\\bigg(\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}\\Omega_{g_1,|I_1|+1}(z,I_1)\\Omega_{g_2,|I_2|+1}(z,I_2)+\\Omega_{g-1,n+1}(z,z,I)\\bigg)\n\t\t\\\\\n\t\t&-\\frac{\\lambda^2}{N}\\sum_{k=1}^{d}r_k\\frac{\\Omega_{g,n}(\\varepsilon_k,I)}{x(z)-e_k}+\\lambda^2\\sum_{z_i\\in I}\\frac{\\partial}{\\partial x(z_i)}\\frac{\\Omega_{g,n-1}(I\\setminus z_i,z)+\\Omega_{g,n-1}(I)}{x(z)-x(z_i)}\\\\\n\t\t&-\\frac{\\lambda^2\\delta_{g,0} \\delta_{|I|,2}}{(x(z)-x(z_2))^2(x(z)-x(z_3))^2}\n\t\t+V'(x(z))\\bigg(\\Omega_{g,n}(z,I)-\\frac{\\lambda \\delta_{g,0}\\delta_{|I|,1}}{(x(z)-x(z_2))^2}\\bigg).\n\t\\end{align*}\n\tComparing both expressions, we confirm the assertion.\n\\end{proof}\n\\end{proposition}\n\n\\textbf{III.}\nFirst, we comment on the pole structure of $\\Omega_{g,n}$. From the exact solutions\n\\begin{align*}\n\t\\Omega_{0,1}(z)=-\\frac{1}{\\lambda}y(z),\\qquad \\Omega_{0,2}(z_1,z_2)=\\frac{1}{x'(z_1)x'(z_2)(z_1-z_2)^2},\n\\end{align*}\nwe can deduce inductively that all $\\Omega_{g,n}(z,...)$ have for $2g+n-2>0$ poles only located at the ramification point $z$ of $x$, i.e. $x'(z)=0$. This is observed by expanding the DSE \\eqref{DSEHP} in $v$ with \n\\begin{align*}\n\tH_{g,n}(v;z;I)=\\frac{\\lambda \\Omega_{g,n}(z,I)}{v}+\\mathcal{O}(v^{-2}).\n\\end{align*}\nComparing the poles on both sides inductively, we see that\nsome additional possible poles could arise at $z=z_i$ or $z=\\varepsilon_n$. However, these poles cancel exactly due to the exact solution of $\\Omega_{0,1}$ and $\\Omega_{0,2}$ having also poles at the diagonal $z=z_i$ and $z=\\varepsilon_n$. Therefore, $\\Omega_{g,n}(z,I)$ has for $2g+n-2>0$ only poles at the ramification points of $x(z)$.\n \nLet the simple ramification points of $x(z)$, where $x(z)$ is given by Theorem \\ref{Thm:InitialData}, be denoted by $\\{\\beta_1,...,\\beta_{2\\tilde{d}}\\}$ with $x'(\\beta_i)=0$. At a ramification point two branches meet, i.e. for $z=\\beta_j$ exists an $i\\in \\{1,...,\\tilde{d}\\}$ with $\\hat{z}^i=\\beta_j$. We will denote in the vicinity of $\\beta_j$ this specific preimages $\\hat{z}^i$ around $\\beta_j$ by $\\sigma_j(z):=\\hat{z}^i$ being the local Galois involution introduced in Sec. \\ref{ch:setup}. \n\n\\begin{theorem}\\label{thm:TROm}\n\tLet $I=\\{z_2,...,z_n\\}$ and $2g+n-2>0$. Then, one can compute recursively in $2g+n-2$ all $\\Omega_{g,n}$ by topological recursion\n\t\\begin{align}\\label{TRTheorem}\n\t\t\\Omega_{g,n}(z,I)&x'(z)=\\lambda \\sum_{i=1}^{2 \\tilde{d}}\\Res\\displaylimits_{q\\to \\beta_i} \\frac{\\frac{1}{z-q}-\\frac{1}{z-\\sigma_i(q)}}{2(y(q)-y(\\sigma_i(q)))}dx(q)\\\\\\nonumber\n\t\t&\\times\\bigg\\{\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}^\\prime \\Omega_{g_1,|I_1|+1}(q,I_1)\\Omega_{g_2,|I_2|+1}(\\sigma_i(q),I_2)+\\Omega_{g-1,n+1}(q,\\sigma_i(q),I)\\bigg\\},\n\t\\end{align}\nwhere the sum is taken over all the residues are at the ramification points $\\beta_i$ of $x(z)$, i.e. $x'(\\beta_i)=0$. The primed sum $\\sum^\\prime$ excludes the cases $(g_i,I_i)=(0,\\emptyset)$.\n\n\\begin{proof}\n\tWe deduce from the linear loop equation \\eqref{linearloop} of Proposition \\ref{Prop:linquad} for the behavior around $\\beta_i$\n\t\\begin{align}\n\t\t\\Omega_{g,n}(z,I)+\\Omega_{g,n}(\\sigma_i(z),I)=\\mathcal{O}((z-\\beta_i)^0).\n\t\\end{align}\n\tCombining the quadratic loop equation \\eqref{quadloop} with the linear, we obtain in the vicinity of $\\beta_i$\n\t\\begin{align}\\label{quadproof}\n\t\t\\frac{1}{\\lambda}(y(z)-y(\\sigma_i(z)))\\Omega_{g,n}(z,I)+&\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}^\\prime\\Omega_{g_1,|I_1|+1}(z,I_1)\\Omega_{g_2,|I_2|+1}(\\sigma_i(z),I_2)\\\\\n\t\t+&\\Omega_{g-1,n+1}(z,\\sigma_i(z),I)=\\mathcal{O}((z-\\beta_i)^0).\n\t\\end{align}\n\tFinally, we compute by Cauchy's formula and by shifting the contour $\\Omega_{g,n}$ as\n\t\\begin{align*}\n\t\t\\Omega_{g,n}(z,I)x'(z)=& \\sum_{i=1}^{2 \\tilde{d}} \\Res\\displaylimits_{q\\to z}\\frac{\\Omega_{g,n}(q,I)dx(q)}{z-q}=-\\sum_{i=1}^{2 \\tilde{d}}\\Res\\displaylimits_{q\\to \\beta_i}\\frac{\\Omega_{g,n}(q,I)dx(q)}{z-q}\\\\\n\t\t=&-\\lambda \\sum_{i=1}^{2 \\tilde{d}} \\Res\\displaylimits_{q\\to \\beta_i}\\frac{\\sum_{\\substack{I_1\\uplus I_2=I\\\\g_1+g_2=g}}^\\prime\\Omega_{g_1,|I_1|+1}(q,I_1)\\Omega_{g_2,|I_2|+1}(\\sigma_i(q),I_2)}{(y(q)-y(\\sigma_i(q)))(z-q)}dx(q),\n\t\\end{align*}\n\twhere we took into account that $\\Omega_{g,n}$ has only poles at the ramification points $\\beta_i$ and the quadratic loop equation \\eqref{quadproof} after dividing by $\\frac{1}{\\lambda}(y(z)-y(\\sigma_i(z)))$. Since the residue is invariant under the local Galois involution $q\\mapsto \\sigma_i(q)$ around $\\beta_i$, we derived \\eqref{TRTheorem}.\n\\end{proof}\n\\end{theorem}\nRelating the correlation function $\\Omega_{g,n}$ to the differential forms $\\omega_{g,n}$ of TR by \n\\begin{align*}\n\t\\Omega_{g,n}(z_1,...,z_n)\\prod_{i=1}^ndx(z_i)=\\omega_{g,n}(z_1,...,z_n)\n\\end{align*}\ngives the most common representation of TR already given in \\eqref{eq:TR-intro}. This finishes the proof that the Langmann-Szabo-Zarembo model is governed by topological recursion.\n\n\n\n\n\n\n\n\\section{Complete Solution of the LSZ Model}\n\\label{sec:compl}\nIn some sense, we have already written down a complete solution of the LSZ model: Knowing all meromorphic forms $\\omega_{g,n}$ by the universal topological recursion formula, it is a feasible task to reconstruct the partition function eq. (\\ref{partfunLSZ}) using the genus-summed free energies $\\mathcal{F}^{(g)}$ as described in Sec. \\ref{ch:setup}. From the matrix model perspective, this is a satisfying answer. However, the LSZ model was designed as a quantum field theoretical toy model and thus we are eventually interested in the explicit solutions of the $2N_1+...+2N_b$-point function - in other words the shape of all correlation functions $G$ of arbitrary boundary structures. \n\nIn this section we will give a simple recursion formula for these most general correlation functions. There basic building blocks are, as indicated already in Fig. \\ref{fig:flow}, the generalised 2+...+2-point functions. We thus have to extend the procedure to obtain the Dyson-Schwinger equation for the (generalised) 2-point function - which is straightforward, but lengthy. The complexification also works in exactly the same manner and gives rise to a relatively simple residue formula for their computation that has a clear pictorial interpretation (see \\cite{Eynard:2007gw,Branahl:2020yru}). With this formula, all building blocks of aforementioned recursion can be calculated. This recursion will be the main result of this section as it extends the knowledge about the particular form of loop equations also showing up in the 2-matrix model and the generalised Kontsevich model. Due to a plethora of indices and little learning effects, the laborious proofs are shifted into Appendix \\ref{sec:appproof} for the sake of readability.\n\nAs announced, we start with a more general Dyson-Schwinger equation:\n\n\\begin{proposition}\n\\label{prop:GpqJ}\nLet $\\mathcal{J}=\\{J^2,...,J^b\\}$ for $J^s=[p^s,q^s]$.\nThen for pairwise different $p,q,p^s,q^s$ one has \nthe Dyson-Schwinger equation for the $2+...+2$-point function:\n\\begin{align}\nG_{|pq|\\mathcal{J}|}\n&= \\frac{\\delta_{|\\mathcal{J}|,0}}{E_p+\\tilde{E}_q}\n+\n\\frac{\\lambda}{E_p+\\tilde{E}_q}\\Big\\{\n-\\Omega_p G_{|pq|\\mathcal{J}|}\n+\\frac{1}{N}\\frac{\\partial G_{|pq|\\mathcal{J}|}}{\\partial E_p} \n\\nonumber\n\\\\\n& \n-\n\\sum_{\\substack{\\mathcal{J}'\\uplus \\mathcal{J}''=\\mathcal{J}\\\\\n\\mathcal{J}'\\neq \\emptyset}}\nT_{p\\|\\mathcal{J}'|} G_{|pq|\\mathcal{J}''|}+ \\frac{1}{N}\\sum_{\\substack{l=1 \\\\ l\\neq p}}^N \n\\frac{G_{|lq|\\mathcal{J}|}-G_{|pq|\\mathcal{J}|}}{E_l-E_p}\n\\nonumber\n\\\\\n&+ \\sum_{s=2}^b \n\\frac{G_{|p^sq^sp^sq|\\mathcal{J}\\setminus J^s|}\n-G_{|pq^sp^sq|\\mathcal{J}\\setminus J^s|}\n}{E_{p^s}-E_p} \n\\Big\\}\\;.\n\\label{GpqJ}\n\\end{align}\n \\end{proposition}\nIn the same manner as for the 2-point function, the repeated application of the boundary creation operator $-N\\frac{\\partial }{\\partial E_{p_i}}$ gives without further effort the following corollary.\n\\begin{corollary}\n\tLet $\\mathcal{J}=\\{J^2,...,J^b\\}$ for $J^s=[p^s,q^s]$ and $I=\\{p_1,...,p_m\\}$.\n\tThen for pairwise different $p,q,p^s,q^s$ one has \n\tthe Dyson-Schwinger equation for the generalised $2+...+2$-point function:\n\\begin{align}\n&\\Big(\\tilde{E}_q+E_p+\\frac{\\lambda}{N}\n\\sum_{\\substack{l=1\\\\l\\neq p}}^N \\frac{1}{E_l-E_p}\\Big)\nT_{I\\|pq|\\mathcal{J}|}\n- \\frac{\\lambda}{N}\n\\sum_{\\substack{l=1 \\\\ l\\notin I,p}}^N\n \\frac{T_{I\\|lq|\\mathcal{J}|}}{E_l-E_p}\n\\nonumber\n\\\\\n&= \\delta_{0,|\\mathcal{J}|}\\delta_{0,|I|}\n-\n\\lambda\\bigg\\{\n\\sum_{I'\\uplus I''=I}\n\\Omega_{I',p} T_{I''\\|pq|\\mathcal{J}|}\n-\\frac{1}{N}\\frac{\\partial T_{I\\|pq|\\mathcal{J}|}}{\\partial E_p}\n\\nonumber\n\\\\\n&+\n\\sum_{\\substack{I'\\uplus I''=I\\\\\n\\mathcal{J}'\\uplus \\mathcal{J}''=\\mathcal{J},~\\mathcal{J}'\\neq \\emptyset}}\n\\hspace*{-1.5em}\nT_{I',p\\|\\mathcal{J}'|} T_{I''\\|pq|\\mathcal{J}''|}\n+ \n\\sum_{i=1}^m \\frac{\\partial}{\\partial E_{p_i}}\n\\Big(\\frac{T_{I\\setminus p_i\\|p_iq|\\mathcal{J}|}}{E_{p_i}-E_p}\\Big) \\nonumber\n\\\\\n&+ \\sum_{s=2}^b \n\\frac{T_{I\\|p^sq^sp^sq|\\mathcal{J}\\setminus J^s|}\n-T_{I\\|pq^sp^sq|\\mathcal{J}\\setminus J^s|}\n}{E_{p^s}-E_p} \\Big\\}.\n\\label{DSE-T2long}\n\\end{align}\n\\end{corollary}\n\nIn analogy to the previous strategy of analytic continuation, we rewrite the (generalised) correlation functions for arbitrary boundary components as\n\\begin{align}\n\\label{eq:complexi}\nG(E_{p_1^1},\\tilde{E}_{q_{1}^{1}},...,\\tilde{E}_{q_{N_1}^{1}}|\\dots|E_{p_1^b},\\tilde{E}_{q_{1}^{b}},...,\\tilde{E}_{q^b_{N_b}}) \n&\\equiv G_{|p_1^1q_1^1...q^1_{N_1}|\\dots|p_1^bq_1^b...q^b_{N_b}|}\\;,\n\\\\\nT(E_{p_1},...,E_{p_m}\\| \nE_{p_1^1},...,\\tilde{E}_{q^1_{N_1}}|\\dots|E_{p_1^b},...,\\tilde{E}_{q^b_{N_b}}) \n&\\equiv T_{p_1,\\dots,p_m\\|p_1^1...q^1_{N_1}|\\dots|p_1^b...q^b_{N_b}|}\\;. \\nonumber\n\\end{align}\nMoreover, we use again the change of variables suggested by the solution of the 2-point function in Theorem~\\ref{Thm:InitialData}, giving us again meromorphic functions $\\mathcal{G}$ and $\\mathcal{T}$ of several complex variables. We thus define:\n\\begin{definition}\n\\label{def:complexification} \nWe define functions $\\mathcal{G}$ and $\\mathcal{T}$ in several complex variables by the variable transforms $x(z)$ and $y(w)$ and the functions $G$ and $T$ complexified in eq. (\\ref{eq:complexi}). They have the known formal genus expansion as well:\n\\begin{align*}\n&\\mathcal{G}^{(g)}(z_1^1,w_1^1,...,z^1_{N_1},w^1_{N_1}|\\dots|\nz_1^b,w_1^b,...,z^b_{N_b},w^b_{N_b})\\\\\n&:=\nG^{(g)}(x(z_1^1),y(w^1_{1}),...,x(z^1_{N_1}),y(w^1_{N_1})|\\dots|\nx(z_1^b),y(w^b_{1}),...,x(z^b_{N_b}),y(w^b_{N_b}))\\;,\n\\end{align*}\nand\n\\begin{align*}\n&\\mathcal{T}^{(g)}(u_1,...,u_m\\|z_1^1,...,w^1_{N_1}|\\dots|\nz_1^b,...,w^b_{N_b}|) \\\\\n&:=\nT^{(g)}(x(u_1),...,x(u_m)\\|x(z_1^1),...,y(w^1_{N_1})|\\dots|\nx(z_1^b),...,y(w^b_{N_b}))\n\\end{align*}\nWe let\n\\begin{align*}\n&\\mathcal{T}^{(g)}(\\emptyset\\|z_1^1,...,w^1_{N_1}|\\dots|\nz_1^b,...,w^b_{N_b}|)\n:=\n\\mathcal{G}^{(g)}(z_1^1,...,w^1_{N_1}|\\dots|\nz_1^b,...,w^b_{N_b})\n\\end{align*}\n \\end{definition}\nWith this preparation, we can formulate the complexified version of eq. (\\ref{DSE-T2long}):\n\\begin{corollary}\n\\label{corr:DSEcomplex}\nLet $\\mathcal{J}=\\{J^2,...,J^b\\}$ for $J^s=[z^s,w^s]$ \nand $I=\\{u_1,...,u_m\\}$.\nThe complexification of Definition~\\ref{def:complexification} \nturns after formal genus expansion, inclusion \nof multiplicities $r_i$ of $e_i$ as well as $\\tilde{r}_j$ of $\\tilde{e}_j$ and the change of variables, which involves \nthe rational functions $x$ and $y$ of Theorem~\\ref{Thm:InitialData},\nthe Dyson-Schwinger equation for the generalised $2+...+2$-point function eq. (\\ref{DSE-T2long}) into the following equation:\n\\begin{align}\n&(y(w)-y(z)) \\mathcal{T}^{(g)}(I\\|z,w|\\mathcal{J}|)\n- \\frac{\\lambda}{N}\n\\sum_{k=1}^d\n \\frac{r_k \\mathcal{T}^{(g)}(I\\|\\varepsilon_k,w|\\mathcal{J}|)}{\nx(\\varepsilon_k)-x(z)}\n\\label{DSE-cT2}\n\\\\\n&= \\delta_{0,|\\mathcal{J}|}\\delta_{0,|I|}\\delta_{g,0}\n-\n\\lambda\\bigg\\{\n\\sum_{\\substack{I_1\\uplus I_2=I\\\\\ng_1+g_2=g,~(g_1,I_1)\\neq (0,\\emptyset)}} \\hspace*{-2em}\n\\Omega^{(g_1)}_{|I_1|+1}(I_1,z) \n\\mathcal{T}^{(g_2)}(I_2\\|z,w|\\mathcal{J}|)\n\\nonumber\n\\\\\n&+\\mathcal{T}^{(g-1)}(I,z\\|z,w|\\mathcal{J}|)\n+\n\\sum_{\\substack{I_1\\uplus I_2=I\\\\\n\\mathcal{J}_1\\uplus \\mathcal{J}_2=\\mathcal{J},~\n\\mathcal{J}_1\\neq \\emptyset\\\\\ng_1+g_2=g}}\n\\hspace*{-1.5em}\n\\mathcal{T}^{(g_1)}(I_1,z\\|\\mathcal{J}_1|) \n\\mathcal{T}^{(g_2)}(I_2\\|z,w|\\mathcal{J}_2|)\n\\nonumber\n\\\\\n& \n+ \n\\sum_{i=1}^m \\frac{\\partial}{\\partial x(u_i)}\n\\Big(\\frac{\\mathcal{T}^{(g)}(I{\\setminus} u_i\\|u_i,w|\\mathcal{J}|)}{\nx(u_i)-x(z)}\\Big)\n\\nonumber\n\\\\\n&- \\sum_{s=2}^b\\frac{\n\\mathcal{T}^{(g)}(I\\|z^s,w^s,z^s,w|\\mathcal{J}{\\setminus} J^s|)\n-\n\\mathcal{T}^{(g)}(I\\|z,w^s,z^s,w|\\mathcal{J}{\\setminus} J^s|)\n}{x(z^s)-x(z)}\n\\nonumber\n\\bigg\\},\n\\nonumber\n\\end{align}\n\\end{corollary}\n\nThe Lagrange interpolation formula (see Lemma \\ref{lem:interpol}) together with the solution of the 2-point function allows to solve this equation by a residue formula:\n\n\n \\begin{proposition} \\label{prop2}\n \tLet $\\big\\{\\hat{\\tilde{w}}^1,...,\\hat{\\tilde{w}}^d\\big\\}$ be the solutions of $y(\\bullet)=y(w)$. Then, the Dyson-Schwinger equation of Corollary \\ref{corr:DSEcomplex} is solved by\n \\begin{align*}\n \\mathcal{T}^{(g)}&(I\\|z,w|\\mathcal{J}|)\n =\\lambda\\mathcal{G}^{(0)}(z,w)\\sum_{j=1}^d \\Res\\displaylimits_{t\\to z,\\hat{\\tilde{w}}^j}\\frac{x'(t)\\, dt}{(x(z)-x(t))(y(w)-y(t))\\mathcal{G}^{(0)}(t,w)} \\\\\n&\\times\\bigg[\\sum_{\\substack{I_1\\uplus I_2=I\\\\\ng_1+g_2=g,~(g_1,I_1)\\neq (0,\\emptyset)}} \\hspace*{-2em}\n\\Omega^{(g_1)}_{|I_1|+1}(I_1,t) \n\\mathcal{T}^{(g_2)}(I_2\\|t,w|\\mathcal{J}|)\n\\nonumber\n\\\\\n&+\\mathcal{T}^{(g-1)}(I,t\\|t,w|\\mathcal{J}|)\n+\n\\sum_{\\substack{I_1\\uplus I_2=I\\\\\n\\mathcal{J}_1\\uplus \\mathcal{J}_2=\\mathcal{J},~\n\\mathcal{J}_1\\neq \\emptyset\\\\\ng_1+g_2=g}}\n\\hspace*{-1.5em}\n\\mathcal{T}^{(g_1)}(I_1,t\\|\\mathcal{J}_1|) \n\\mathcal{T}^{(g_2)}(I_2\\|t,w|\\mathcal{J}_2|)\n\\nonumber\n\\\\\n& \n+ \n\\sum_{i=1}^m \\frac{\\partial}{\\partial x(u_i)}\n\\Big(\\frac{\\mathcal{T}^{(g)}(I{\\setminus} u_i\\|u_i,w|\\mathcal{J}|)}{\nx(u_i)-x(t)}\\Big)\n\\nonumber\n\\\\\n&- \\sum_{s=2}^b\\frac{\n\\mathcal{T}^{(g)}(I\\|z^s,w^s,z^s,w|\\mathcal{J}{\\setminus} J^s|)\n-\n\\mathcal{T}^{(g)}(I\\|t,w^s,z^s,w|\\mathcal{J}{\\setminus} J^s|)\n}{x(z^s)-x(t)}\\bigg]\n\\nonumber\n \\end{align*}\n \\end{proposition}\n\n\nThe result of Proposition \\ref{prop2} can be grasped in a more compact way by rewriting two terms:\n\\begin{corollary}\\label{cor2+} \t \nProposition \\ref{prop2} is equivalent to\n\\begin{align*}\n\\mathcal{T}^{(g)}(I\\|z,w|\\mathcal{J})\n&= \\sum_{s=2}^b \\sum_{j=1}^d \\sum_{i=1}^m\\Res\\displaylimits_{t\\to z,\\hat{\\tilde{w}}^j,u_i,z^s}\n \\frac{\\lambda\\mathcal{G}^{(0)}(z,w) x'(t)\\, dt}{(x(z)-x(t))(y(w)-y(t))\\mathcal{G}^{(0)}(t,w)} \n\\\\\n&\\times\\bigg[\\sum_{\\substack{I_1\\uplus I_2=I\\\\\n\\mathcal{J}_1\\uplus \\mathcal{J}_2=\\mathcal{J}\\\\\ng_1+g_2=g}}^\\prime\n\\mathcal{T}^{(g_1)}(I_1,t\\|\\mathcal{J}_1|) \n\\mathcal{T}^{(g_2)}(I_2\\|t,w|\\mathcal{J}_2|)\n\\nonumber\n\\\\\n&\\qquad +\\mathcal{T}^{(g-1)}(I,t\\|t,w|\\mathcal{J}|)\n+\\sum_{s=2}^b\\frac{\n\\mathcal{T}^{(g)}(I\\|t,w^s,z^s,w|\\mathcal{J}{\\setminus} J^s|)\n}{x(z^s)-x(t)}\n\t\t\\bigg],\n\t\\end{align*}\nwhere the primed sum excludes $(g_1,I_1,\\mathcal{J}_1)=(0,\\emptyset,\\emptyset)$.\n\\end{corollary}\n\\begin{remark}\\label{rmk:2MMH}\n\tThis residue formula has an intuitive geometric interpretation that the interested reader can find in \\cite[eq. (4-1) and (4-2)]{Eynard:2007gw}. We have a one-to-one correspondence to the 2-matrix model. Due to our restricitions to boundary lengths of 2, the first two visualised summands in \\cite[eq. (4-2)]{Eynard:2007gw} do not show up here, but have to be added when going to arbitrary $N_s$. The generalised correlation functions $\\mathcal{T}^{(g)}(I\\|z,w|\\mathcal{J})$ correspond to the $H^{(g)}_{k_L;m;n}(S_1,...,S_l;p_1,...,p_m;q_1,...,q_n)$, where the boundaries in $\\mathcal{J}$ correspond to $\\{S_2,...,S_l\\}$ and the set of marked points $u_i \\in I$ to $p_i$. Another set of marked points $q_j$ could be generated by acting with $\\frac{\\partial}{\\partial \\tilde{E}_q}$on $G_{...}$. We especially refer to eq. (3-9) in \\cite{Eynard:2007gw} to compare the complete structural equivalence to our 2-point function: $\\mathcal{G}^{(0)}(z,w) \\equiv H^{(0)}_{1;0;0}(z,w)$.\n\\end{remark}\nWe now have collected all components necessary to write the $2N_1+...+2N_b$-point function in terms of polynomials in (2+...+2)-point functions as well as denominators $\\frac{1}{E_k-E_l}$. The complete solution of the LSZ model in the quantum field theoretical perspective reads:\n\\begin{theorem}\n\\label{thm:complete}\nLet $\\mathcal{J}=\\{J^2,...,J^b\\}$, $J^\\beta=\\{p_1^\\beta, q_1^\\beta,...p_{N_\\beta}^\\beta, q_{N_\\beta}^\\beta \\}$ and $\\beta \\in \\{2,...,b\\}$. The generalised $2N_1+...+2N_b$-point function satisfies the following recursive equation:\n \\begin{align*}\n &T^{(g)}_{I\\|p_1^1 q^1_1..q^1_{N_1}|\\mathcal{J}|}=-\\frac{\\lambda}{\\tilde{E}_{q^1_1}-\\tilde{E}_{q^1_{N_1}}}\\bigg\\{\n \\sum_{k=2}^{N_1}\\frac{T^{(g-1)}_{I\\|p^1_kq^1_{1}..q^1_{k-1}|p^1_1q^1_k..q^1_{N_1}|\\mathcal{J}|}\n -T^{(g-1)}_{I\\|p^1_1q^1_1p^1_2..q^1_{k-1}|p^1_{k}..q^1_{N_1}|\\mathcal{J}|}}{E_{p^1_k}-E_{p^1_1}}\\\\ \n +&\\sum_{\\beta=2}^{b}\\sum_{k=1}^{N_\\beta}\\frac{T^{(g)}_{I\\|p_1^\\beta q_1^\\beta..p_k^\\beta q^1_1p^1_2..q^1_{N_1}\n p^1_1 q_{k}^\\beta..q^\\beta_{N_\\beta}|\\mathcal{J}\\backslash \\{J^\\beta\\}|}-\n T^{(g)}_{I\\|p_1^\\beta q_1^\\beta..q_{k-1}^\\beta p^1_1 q^1_1..q^1_{N_1}\n p_{k}^\\beta..q^\\beta_{N_\\beta}|\\mathcal{J}\\backslash \\{J^\\beta\\}|}}{E_{p^\\beta_k}-E_{p^1_1}}\\\\\n +&\\sum_{k=2}^{N_1}\n \\sum_{\\substack{\\mathcal{J}'\\uplus \\mathcal{J}''=\\mathcal{J}\\\\I_1\\uplus I_2=I\\\\h+h'=g}} \\frac{T^{(h)}_{I_1\\|p^1_kq^1_{1}..q^1_{k-1}|\\mathcal{J}'|} \n T^{(h')}_{I_2\\|p^1_1q^1_k..q^1_{N_1}|\\mathcal{J}''|}-\n T^{(h)}_{I_1\\|p^1_1 p^1_2..q^1_{k-1}|\\mathcal{J}'|} T^{(h')}_{I_2\\|p^1_{k}..q^1_{N_1}|\\mathcal{J}''|}}{E_{p^1_k}-E_{p^1_1}}\\bigg\\}\n \\end{align*}\n\\end{theorem}\n\n\\begin{remark}\n\tAs mentioned before, we have a direct correspondence of all correlation functions for the LSZ model in terms of $\\mathcal{T}$ to the hermitian 2-matrix model (see Remark \\ref{rmk:2MMH}). However, the algebraic recursive equation of Theorem \\ref{thm:complete} was never proved for the 2-matrix model. It is straightforward to write the theorem in terms of meromorphic functions $\\mathcal{T}$ with $x(z), y(w)$. Therefore, this algebraic formula should also hold in the 2-matrix model. Furthermore, cyclic symmetry within a boundary, i.e. \n\t\\begin{align*}\n\t\tT^{(g)}_{I\\|p_1^1 q^1_1p_2^1q_2^1..q^1_{N_1}|\\mathcal{J}|}=T^{(g)}_{I\\|p_2^1q_2^1..q^1_{N_1}p_1^1 q^1_1|\\mathcal{J}|}\n\t\\end{align*}\n\tis a direct consequence of the equation of Theorem \\ref{thm:complete}, which was also never proved for general correlation functions in the 2-matrix model, but should obviously hold by definition.\n\\end{remark}\n\n\n\nThus, we have a concrete algorithm to derive all correlation functions building on topological recursion. The algorithm is recursive in the negative Euler characteristic $-\\chi=2g+n-2$. For any $2N_1+...+2N_b$-point function, Theorem \\ref{thm:complete} has to be applied until one ends up just with $2+...+2$-point functions. For these, Corollary \\ref{cor2+} applies, which includes recursively further $2+...+2$-point functions and $\\Omega_{g,n}$ of less topology or equal Euler characteristic. The $\\Omega_{g,n}$ are derived by topological recursion, Theorem \\ref{thm:TROm}, where the initial data is provided by Theorem \\ref{Thm:InitialData}.\n\n\nWe end with an example of the above recursion formula being in particular interesting from combinatorial perspective:\n\\begin{example}\nThere is a particular example in the planar sector which analogously occurs for hermitian fields \\cite{DeJong} or in the 2-matrix model \\cite{Eynard:2005iq}: The recursion boils down for the planar $2N$-point function to:\n\\begin{align*}\n G^{(0)}_{|p_1 q_1..q_{N}|}=&-\\lambda\n \\sum_{k=1}^{N-2}\n \\frac{G^{(0)}_{|p_1q_{k+1}..q_{N}|} \n G^{(0)}_{|p_{k+1}q_1..q_k|}-\n G^{(0)}_{|p_{k+1}..q_{N}|} G^{(0)}_{|p_1 q_1..q_{k}|}}{(E_{p_{k+1}}-E_{p_1})(\\tilde{E}_{q_1}-\\tilde{E}_{q_{N}})},\n\t\\end{align*}\nThe explicit structure of the produced polynomials follow a remarkable combinatorial pattern based on nested Catalan tuples that was unraveled in \\cite{DeJong}. This combinatorial pattern might by generalised to higher genus and more boundaries by finding an explicit structure for the recursion of Theorem \\ref{thm:complete}.\n\\end{example}\nWe have now reached all solutions of correlators in the LSZ model in their full complexity. Some simple topologies shall now be compared with known limits of this model.\n\n\n\n\\section{Cross Checks}\n\\label{ch:cross}\n\nIn this section we take the obtained results from the LSZ model and compare them with known results from other models that can be reached by certain limits. Setting $\\tilde{E}=0$, we get back the original model the authors of \\cite{Langmann:2003if} were interested in - since they introduced the additional external field $\\tilde{E}$ only as an auxiliary quantity. Setting $\\tilde{E}=E$, the Quartic Kontsevich Model (QKM) is obtained for the first simple topologies (for higher topologies the combinatorics changes drastically). Finally, we perform a perturbative analysis for both complex and hermitian fields $\\Phi$ and compare the different classes of ribbon graphs that are generated. In particular, we relate them to known results from enumerative geometry for the complex and hermitian 1-matrix model, arising in the \\textit{combinatorial limit} of a single $N$-fold degenerate eigenvalue $e$ and $\\tilde{e}$, respectively (in other words $d=\\tilde{d}=1$).\n\n\\subsection{Recovering the Results of Langmann-Szabo-Zarembo \\cite{Langmann:2003if}}\nAs already mentioned, the original work of Langmann, Szabo and Zarembo used the external matrix $\\tilde{E}$ as a regulator, which was switched off for evaluating the exact solution of the correlators. In \\cite{Langmann:2003if}, Langmann, Szabo and Zarembo have computed the solution of $\\Omega^{(0)}_p$ with classical techniques like the Harish-Chandra-Itzykson-Zuber formula \\cite{Harish57,Itz1980} together with the Riemann-Hilbert equation. An important observation was that their ''master equation'' was of the same type as in the Kontsevich-Penner model, which had a universal procedure to solve it. This procedure relies essentially on the fact of having a one-cut solution. \n\nWe will see that this is just possible if all eigenvalues of $\\tilde{E}$ coincide. For the specific case of $\\tilde{E}=0$, the function $x(z)$ of Theorem \\ref{Thm:InitialData} breaks down to a degree-two function in $z$. Its inverse has just two branches so that only one cut occurs. In particular, we have in this case from our theorem with $\\tilde{d}=1$\n\\begin{align*}\n\tx(z)=z-\\frac{\\lambda}{y'(\\tilde{\\varepsilon})(z-\\tilde{\\varepsilon})},\\qquad y(z)=-z+\\frac{\\lambda}{N}\\sum_{n=1}^{d}\\frac{r_n}{x'(\\varepsilon_n)(z-\\varepsilon_n)},\n\\end{align*}\nwhere $0=y(\\tilde{\\varepsilon})$ and $e_n=x(\\varepsilon_n)$. Let now \n\\begin{align*}\n\tb_1+b_2=2\\tilde{\\varepsilon},\\qquad b_1-b_2=4\\sqrt{-\\frac{\\lambda}{ y'(\\tilde{\\varepsilon})}},\n\\end{align*}\nthen we can invert $x(z)$ to\n\\begin{align*}\n\tz(x)=\\frac{x+\\frac{b_1+b_2}{2}\\pm \\sqrt{(x-b_1)(x-b_2)}}{2}.\n\\end{align*}\nIt is straightforward to check the following relations\n\\begin{align*}\n\tx'(z)^2(\\tilde{\\varepsilon}-z)^2=&(x(z)-b_1)(x(z)-b_2) \\\\ \\bigg(\\frac{2 (x(z)-e_n)}{x'(\\varepsilon_n)(z-\\varepsilon_n)}-\\frac{x(z)-e_n}{x'(\\varepsilon_n)(\\tilde{\\varepsilon}-\\varepsilon_n)}-1\\bigg)^2=&\\frac{(x(z)-b_1)(x(z)-b_2)}{(e_n-b_1)(e_n-b_2)}\n\\end{align*}\nwhich implies (by taking the right square root branch) for $\\Omega^{(0)}_1$\n\\begin{align*}\n\t&-\\frac{y(z)+V'(x(z))}{\\lambda}=\\frac{z-x(z)}{\\lambda}+\\frac{1}{N}\\sum_{n=1}^d r_n\\bigg(\\frac{1}{x(z)-e_n}-\\frac{1}{x'(\\varepsilon_n)(z-\\varepsilon_n)}\\bigg)\\\\\n\t=&\\frac{\\sqrt{(x-b_1)(x-b_2)}-x}{2\\lambda}+\\frac{1}{2N}\\sum_{n=1}^d r_n\\bigg(\\frac{1}{x-e_n}-\\frac{\\sqrt{(x-b_1)(x-b_2)}}{(x-e_n)\\sqrt{(e_n-b_1)(e_n-b_2)}}\\bigg).\n\\end{align*}\nThis result is in perfect coincidence with the exact solution provided by \\cite[eq. (D.14)]{Langmann:2003if} (up to a global sign). The two implicit constraints on $b_1$ and $b_2$ in \\cite[eq. (D.15)]{Langmann:2003if} are in one-to-one correspondence to $0=y(\\tilde{\\varepsilon})$ and $b_1-b_2=4\\sqrt{-\\frac{\\lambda}{ y'(\\tilde{\\varepsilon})}}$.\nFrom this analysis, we see that the solution of Langmann, Szabo and Zarembo is a very specific case of the result provided by Theorem \\ref{Thm:InitialData}.\n\n\n\n\n\n\n\\subsection{Analytical Comparison with the Hermitian Model (\\ref{partfunQKM})}\nThis subsection is dedicated to a structural comparison of the different types of topological recursion solving the models defined by eq. (\\ref{partfunLSZ}) and eq. (\\ref{partfunQKM}) with complex and hermitian fields, respectively. Although looking at hermitian matrices $\\Phi=\\Phi^\\dagger$ at first sight seems to reduce the complexity of the model, the contrary is the case. Performing very similar steps as for the solution of the LSZ model, we end up in the hermitian model (QKM) of eq. (\\ref{partfunQKM}) with meromorphic forms which has additional properties. These properties can be described by three facts:\n\\begin{itemize}\n\\item having only one external field $E$, $x(z)$ and $y(z)$ boil down to \\cite[Thm. 4.1]{Branahl:2021}:\n\\begin{align*}\nx(z)=R(z)=z-\\sum_{k=1}^d \\frac{r_k}{x'(\\varepsilon_k)(z+\\varepsilon_k)} \\, , \\qquad y(z)=-R(-z)\\; .\n\\end{align*}\nThis limit is reached by $\\varepsilon_k = -\\tilde{\\varepsilon}_k$ implied by $E=\\tilde{E}$. It is decisive that both $x$ and $y$ can be expressed by the same change of variables $R(z)$, such that we have a global reflection symmetry $x(z)=-y(-z)$. In particular, this implies e.g. $\\mathcal{G}^{(0)}(z,w)=\\mathcal{G}^{(0)}(w,z)$\n\\item $\\omega_{0,2}$ is enlarged by a \"reflected\" Bergman kernel: $\\omega_{0,2}(z_1,z_2)=B(z_1,z_2)-B(z_1,-z_2)$\n\\item $\\omega_{g,n+1}(z,z_1,...,z_n)$ has poles on the antidiagonals $z=-z_i$, for $g>0$, $\\omega_{g,n+1}(z,z_1,...,z_n)$ has additionally poles at $z=0$\n\\end{itemize}\nAltogether, these three facts suggest to turn to the more general framework of \\textit{blobbed topological recursion} (BTR), developed in \\cite{Borot:2015hna}. This framework finds application in the solution of models in which the abstract loop equations are fulfilled, but additional terms with poles away from the ramification points show up. This suggests to decompose every $\\omega_{g,n}$ into a holomorphic (at the ramification points $\\beta_i$) part $\\mathcal{H}\\omega_{g,n}$ and a polar part $\\mathcal{P}\\omega_{g,n}$. The holomorphic parts may be interpreted as an infinite tower of surplus initial data, that cannot be handled with usual topological recursion. Instead, they successively contribute at each recursion step in the Euler characteristic and mix with polar contributions in the recursion formula. A visualisation of this phenomenon is depicted in Fig. \\ref{diagrams}.\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= 0.99\\textwidth]{btr.png} \n \\caption{Graphical interpretation of blobbed topological recursion: The usual recursion formula is enriched by a holomorphic (at ramification points) term $\\mathcal{H}_z\\omega_{g,n+1}$ (coloured) that appears as a surplus structure in the solution of the loop equations. It has to be seen as additional data that has to be taken into account at each further recursion step. \n \\label{diagrams}}\n\\end{figure}\n\nHistorically, the treatment of so-called \\textit{stuffed maps} first stressed the necessity of this enlarged framework \\cite{Borot:2013fla}. The detailed analysis of the loop equations of (so far) the planar sector of the QKM allowed for a concrete model-specific recursion, having astonishing similarities to the known formula generating the polar part \\cite{Hock:2021tbl}. The much more complicated non-planar sector is still work in progress, only some first examples were explicitly worked out. We do not want to review all these accomplishments and rather refer to the article \\cite{Branahl:2021} being readable also for beginners in both noncommutative quantum field theory and topological recursion. Instead, we want to analyse the differences in the Dyson-Schwinger equations for the $\\Omega_{q_1,...,q_n}^{(g)}$ and the reasons for it. The three above-mentioned facts correspond to the following phenomenological differences:\n\\begin{itemize}\n\\item Having only one external field $E$, the boundary creation operator $-N \\frac{\\partial}{\\partial E_p}$ additionally hits the second argument of the 2-point function $G_{|ak|}$ in $\\Omega_{a}=\\frac{1}{N}\\sum_{k} G_{|ak|}$. This has obviously an impact on the exact solution given by the ramified cover $y(z)_{\\tilde{E}=E}=-R(-z)$ - adding a second term to eq. (\\ref{DSE-Om}) and thus giving rise to the extended Bergman kernel. This does not happen in the LSZ model, where we have distinguished dependencies $E_k,\\tilde{E}_l$.\n\\item The hermitian model allows objects like a 1+1-point function (first solved in \\cite{Schurmann:2019mzu}), and in general any partition function with odd boundary lengths, as long as their sum is even, again - adding a third term to eq. (\\ref{DSE-Om}). We will see that also perturbatively, these objects are forbidden in the complex model.\n\\end{itemize}\nThese two observations lead to the following Dyson-Schwinger equation \\cite{Branahl:2020yru}:\n\\begin{align}\n&\\Omega^{(QKM)}_{I,q}\n= \\frac{\\delta_{|I|,1}}{\n(E_{q_1}-E_q)^2}\n+\n\\frac{1}{N}\\sum_{\\substack{l=1\\\\l\\notin I}}^NT_{I\\|ql|}\n\\textcolor{red}{- \\sum_{j=1}^m \n\\frac{\\partial T_{I\\setminus q_j\\|qq_j|}}{\\partial E_{q_j}} \n+\\frac{1}{N^2} T_{I\\|q|q|}\\;,}\n\\label{DSE-OmQKM}\n\\end{align}\nComparing with the proof of the Bergman kernel $B(z_1,z_2)$ in the LSZ model in App. \\ref{sec:om02}, the first new term compared to eq. (\\ref{DSE-Om}) gives rise to the pole on the antidiagonal being responsible for the additional $-B(z_1,-z_2)$, the first blob in blobbed topological recursion of the QKM:\n\\begin{align}\n&\\Omega^{(QKM)}_{0,2}(z_1,z_2)\n=\\frac{1}{x'(z_1)x'(z_2)}\\bigg( \\frac{1}{(z_1-z_2)^2}+\\frac{1}{(z_1+z_2)^2}\\bigg).\n\\end{align}\n This structure inherits to the whole planar sector with more holomorphic parts with poles of $\\omega_{0,n+1}(z,z_1,...,z_n)$ on the antidiagonals $z=-z_i$. We see that the second new term is suppressed by a factor $\\frac{1}{N^2}$, thus it only contributes for $g>0$. The evaluation of the 1+1-point function on the diagonal then gives rise to poles at $z=0$, the fixed point of the global reflection symmetry in the spectral curve, $x(z)=R(z)=-y(-z)$ (see above). Note that these two kinds of poles emerge from two important differences between the LSZ model and the QKM: The coexistence of one single external field together with the property of the quantum fields to be hermitian leads to the recursive structure of the QKM in its full complexity. We will see in the coming subsection that these consequences have clear interpretation in perturbation theory.\n\n \n\n\n\\subsection{Perturbation Theory}\nWhere in physical quantum field theory, perturbative expansions are often the only way to produce results, they become in toy models a powerful tool to justify the calculations for the exact solutions. Comparing hermitian and complex fields seen from the perturbative perspective is worth an investigation as well. \n\n\nMatrix models of the considered type are perturbatively (or combinatorially) expanded into ribbon graphs, which is known since the 70's \\cite{tHooft:1973alw,Brezin:1977sv}. The external matrix $E$ (and $\\tilde{E}$) associates an additional weight for the free propagator. We refer to \\cite{Branahl:2020uxs} for some more rigorous details on the hermitian model and try to summarize the most important facts. \n\nPerturbation theory is the formal expansion of the initially introduced cumulants (see eq. \\ref{eq:FullExpectation}) by interchanging the integration with the series in the coupling constant $\\lambda$:\n\\begin{align}\n\t&\\langle \\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}... \\Phi^\\dagger_{q_{n-1}p_{n-1}}\\Phi_{p_nq_n}\\rangle_c \n\t\\nonumber\n\t\\\\\\label{cumulants-1}\n\t&= \\sum_{v=0}^\\infty \\frac{N^v(-\\lambda)^v}{2^vv!}\n\t\\Big[\\int_{C_N}\\,d\\Phi\\, d\\Phi^\\dagger\\, e^{-N\\mathrm{Tr}(E\\Phi^\\dagger\\Phi +\\tilde{E}\\Phi\\Phi^\\dagger)}\\\\\\nonumber\n\t&\\qquad \\qquad\\qquad \\times\n\t\\Phi^\\dagger_{q_1p_1}\\Phi_{p_2q_2}... \\Phi^\\dagger_{q_{n-1}p_{n-1}}\\Phi_{p_nq_n} \\!\\!\\!\\!\\!\n\t\\sum_{j_1,...,m_v=1}^N\n\t\\prod_{i=1}^v\\big(\\Phi_{j_ik_i}\\Phi^\\dagger_{k_il_i}\\Phi_{l_im_i}\\Phi^\\dagger_{m_ij_i}\\big)\\Big]_c,\n\\end{align}\nPerforming the integration gives a graphical interpretation in terms of ribbon graphs with labelled strands.\nThe translation into graphs is performed by substituting a pair of fields by a ribbon (Wick contraction), and the cycle $\\Phi_{j_ik_i}\\Phi^\\dagger_{k_il_i}\\Phi_{l_im_i}\\Phi^\\dagger_{m_ij_i}$ (coming from the expanded interaction $S_{int}$) forming a 4-valent vertex. First recall the free propagator for \\textit{hermitian matrix} $\\Phi_{ij}\\Phi_{ji}$ is a ribbon with strands labelled by $i,j$ ,and the four-valent vertex $\\lambda \\Phi_{ij}\\Phi_{jk}\\Phi_{kl}\\Phi_{li}$ has four strands labelled by $i,j,k,l$ drawn as (see\\cite{Branahl:2020uxs}):\n\\begin{align*}\n\t\\begin{picture}(30,15)\n\t\\put(0,5){\\line(1,0){30}}\n\t\\put(0,9){\\line(1,0){30}}\n\t\\put(2,-3){\\mbox{\\scriptsize$j$}}\n\t\\put(2,11){\\mbox{\\scriptsize$i$}}\n\t\\put(28,-3){\\mbox{\\scriptsize$j$}}\n\t\\put(28,11){\\mbox{\\scriptsize$i$}}\n\t\\end{picture} \\qquad \\qquad \\qquad \\begin{picture}(24,19)\n\t\\put(0,6){\\line(1,0){10}}\n\t\\put(0,10){\\line(1,0){10}}\n\t\\put(24,6){\\line(-1,0){10}}\n\t\\put(24,10){\\line(-1,0){10}}\n\t\\put(10,-4){\\line(0,1){10}}\n\t\\put(14,-4){\\line(0,1){10}}\n\t\\put(10,20){\\line(0,-1){10}}\n\t\\put(14,20){\\line(0,-1){10}}\n\t\\put(5,-9.5){\\mbox{\\scriptsize$j$}}\n\t\\put(15,-9.5){\\mbox{\\scriptsize$k$}}\n\t\\put(14,21.5){\\mbox{\\scriptsize$l$}}\n\t\\put(3,21.5){\\mbox{\\scriptsize$i$}}\n\t\\put(-10,0){\\mbox{\\scriptsize$j$}}\n\t\\put(-10,8){\\mbox{\\scriptsize$i$}}\n\t\\put(26,0){\\mbox{\\scriptsize$k$}}\n\t\\put(26,8){\\mbox{\\scriptsize$l$}}\n\t\\end{picture} \n\\end{align*}\nLooking instead at \\textit{complex matrices} with $\\Phi, \\Phi^\\dagger$, the ribbons have additional structure in terms of an direction (depicted by arrows giving an direction to the strands). We will call these graphs \\textit{directed ribbon graphs}. $\\Phi^\\dagger$ will represent an ingoing arrow, whereas the adjoint matrix $\\Phi$ will represent an outgoing arrow. Thus, they always have to appear in pairs (otherwise the integation will give zero) like the free propagator $\\Phi^\\dagger_{ji}\\Phi_{ij}$ and four-valent vertex $\\lambda \\Phi_{ij}\\Phi_{jk}^\\dagger\\Phi_{kl}\\Phi_{li}^\\dagger$:\n\\begin{align*}\n\t\\begin{picture}(30,15)\n\t\\put(0,5){\\line(1,0){30}}\n\t\\put(0,9){\\line(1,0){30}}\n\t\\put(8,7){\\vector(1,0){15}}\n\t\\put(2,-3){\\mbox{\\scriptsize$j$}}\n\t\\put(2,11){\\mbox{\\scriptsize$i$}}\n\t\\put(28,-3){\\mbox{\\scriptsize$j$}}\n\t\\put(28,11){\\mbox{\\scriptsize$i$}}\n\t\\end{picture} \\qquad \\qquad \\qquad \\begin{picture}(24,19)\n\t\\put(0,6){\\line(1,0){10}}\n\t\\put(0,10){\\line(1,0){10}}\n\t\\put(24,6){\\line(-1,0){10}}\n\t\\put(24,10){\\line(-1,0){10}}\n\t\\put(10,-4){\\line(0,1){10}}\n\t\\put(14,-4){\\line(0,1){10}}\n\t\\put(10,20){\\line(0,-1){10}}\n\t\\put(14,20){\\line(0,-1){10}}\n\t\\put(5,-9.5){\\mbox{\\scriptsize$j$}}\n\t\\put(15,-9.5){\\mbox{\\scriptsize$k$}}\n\t\\put(14,21.5){\\mbox{\\scriptsize$l$}}\n\t\\put(3,21.5){\\mbox{\\scriptsize$i$}}\n\t\\put(-10,0){\\mbox{\\scriptsize$j$}}\n\t\\put(-10,8){\\mbox{\\scriptsize$i$}}\n\t\\put(26,0){\\mbox{\\scriptsize$k$}}\n\t\\put(26,8){\\mbox{\\scriptsize$l$}}\n\t\\put(25,8){\\vector(-1,0){9}}\n\t\\put(0,8){\\vector(1,0){9}}\n\t\\put(12,12){\\vector(0,1){9}}\n\t\\put(12,4){\\vector(0,-1){9}}\n\t\\end{picture} \n\\end{align*}\nAll directed ribbon graphs of the LSZ model have either 1- or 4-valent vertices. For the cumulant \\eqref{cumulants-1}, we would have $n$ 1-valent vertices associated to the $\\Phi_{p_iq_i},\\Phi^\\dagger_{q_ip_i}$, where the structure and the orientation is fixed by $\\Phi^\\dagger_{q_{2i+1}p_{2i+1}}$ giving ingoing arrows and $\\Phi_{p_{2i}q_{2i}}$ giving outgoing arrows. Consequently, all strands are either clockwise or counter-clockwise directed due to the arrows. We associate to a ribbon with counter-clockwise directed strand labelled by $i$ and clockwise directed strand labelled by $j$ the free propagator $\\frac{1}{E_i+\\tilde{E}_j}$ arising from the integration of a pair $\\Phi^{\\dagger}_{ji}\\Phi_{ij}$. Letting now $\\pi\\in \\mathcal{S}_n$ be a permutation $p_{2i+1}=\\pi(p_{2i+1})$ and $q_{2i}=\\pi(q_{2i})$, we state the fairly well-known (Feynman-)expansion with the additional structure entailed by the LSZ model (including multiplicities $r_i,\\tilde{r}_j$ of eigenvalues $e_i,\\tilde{e}_j$)\n\\begin{proposition}\\label{prop:weight}\n\tLet $p_1,q_1,...,p_n,q_n$ be pairwise different and\n\t$\\mathfrak{G}^{v,\\pi}_{p_1,q_1,...,p_n,q_n}$ be the set of labelled\n\tconnected and directed ribbon graphs with $v$ four-valent vertices and $n$\n\tone-valent vertices labelled $(q_1,\\pi(p_1)), (p_1,\\pi(q_2)) \\dots,\n\t(q_{n-1},\\pi(p_{n-1})),(p_{n},\\pi(q_{n}))$, where $p_i$ corresponds to the counter-clockwise directed strands and $q_i$ to clockwise. Then for $n$ even, the integral \\eqref{cumulants-1} evaluates to\n\t\\begin{align}\n\tN^n \\langle \\Phi^\\dagger_{q_1\\pi(p_1)}\\Phi_{p_2\\pi(q_2)}... \\Phi^\\dagger_{q_{n-1}\\pi(p_{n-1})}\\Phi_{p_n\\pi(q_n)}\\rangle_c \n\t=\\sum_{v=0}^\\infty \\sum_{\\Gamma \\in \\mathfrak{G}^{v,\\pi}_{p_1,q_1,...,p_n,q_n}}\n\tN^{v-r+n+s(\\Gamma)} \\varpi(\\Gamma)\\;,\n\t\\label{cumulants-2}\n\t\\end{align}\n\twhere $r=2v+n\/2$ is the number of ribbons,\n\t$s_1(\\Gamma)$ the number of counter-clockwise directed loops\n\tand $s_2(\\Gamma)$ the number of clockwise directed loops in $\\Gamma$\n\t and\n\tthe weight $\\varpi(\\Gamma)$ is derived from the following Feynman rules:\n\t\\begin{itemize} \n\t\t\\item label the $s_1=s_1(\\Gamma)$ counter-clockwise directed loops by $k_1,...,k_{s_1}$, and the $s_2=s_2(\\Gamma)$ clockwise directed loops by $l_1,...,l_{s_2}$;\n\t\t\\item associate a factor $-\\lambda$ to a 4-valent ribbon-vertex;\n\t\t\\item associate the factor $\\frac{1}{e_p+\\tilde{e}_q}$ to a ribbon with counter-clockwise directed\n\t\tstrand labelled by $p$ and clockwise directed strand labelled by $q$;\n\t\t\\item multiply all factors and apply the summation \n\t\toperator\n\t\t\\begin{align*}\n\t\t\t\\frac{1}{N^{s_1+s_2}}\\sum_{k_1,..,k_{s_1},l_1,...,l_{s_2}=1}^{d,\\tilde{d}}r_{k_1}..r_{k_{s_1}}\\tilde{r}_{l_1}...\\tilde{r}_{l_{s_2}}.\n\t\t\\end{align*}\n\t\\end{itemize}\n\\end{proposition}\nThe permutation $\\pi\\in \\mathcal{S}_n$ defines uniquely a cycle-type (see \\cite{Branahl:2020uxs} for the cycle-type reconstruction from $\\pi$). Thus, Proposition \\ref{prop:weight} gives via equation \\eqref{defG} the graphical expansion of $G_{|...|}$ as a generating function of directed ribbon graphs. \n\\begin{example}\\label{ex:2ppert}\n\tThe first ribbon graphs contributing to the 2-point and 4-point function are shown in Fig. \\ref{2p} and \\ref{4p}. Label the counter-clockwise directed strands by $p_i$ and clockwise directed by $q_i$. Up to order $\\lambda^1$, we get with Proposition \\ref{prop:weight}\n\t\\begin{align}\\nonumber\n\t\tG_{|pq|}= & \\frac{1}{N}\\frac{\\partial^2}{\\partial J_{pq}\\partial J^\\dagger_{qp}}\\log \\mathcal{Z}(J,J^\\dagger)\\vert_{J=0}=N\\langle \\Phi^\\dagger_{qp}\\Phi_{pq}\\rangle\\\\\n\t\t=&\\frac{1}{e_p+\\tilde{e}_q}- \\frac{\\lambda}{(e_p+\\tilde{e}_q)^2}\\frac{1}{N}\\bigg(\\sum_{l=1}^{\\tilde{d}}\\frac{\\tilde{r}_l}{e_p+\\tilde{e}_l}+\\sum_{k=1}^{d}\\frac{r_k}{\\tilde{e}_q+e_k}\\bigg)+\\mathcal{O}(\\lambda^2),\\\\\n\t\tG_{|p_1q_1p_2q_2|}=&\\frac{-\\lambda}{(e_{p_1}+\\tilde{e}_{q_1})(e_{p_2}+\\tilde{e}_{q_1})(e_{p_2}+\\tilde{e}_{q_2})(e_{p_1}+\\tilde{e}_{q_2})}+\\mathcal{O}(\\lambda^2).\n\t\\end{align}\n\tThe next order can easily read off the figures.\n\\end{example}\n\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width= 0.99\\textwidth]{2p_graphs.png} \n\t\\caption{Ribbon graphs contributing to the planar 2-point function of the quartic LSZ model up to order $\\mathcal{O}(\\lambda^2)$. At order $\\lambda^k$, a total amount of $\\frac{2 \\cdot 3^k(2k)!}{k!(k+2)!}$ ribbon graphs contributes.\n\t\t\\label{2p}}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width= 0.75\\textwidth]{4p_graphs.png} \n\t\\caption{Ribbon graphs contributing to the planar four-point function of the quartic LSZ model up to order $\\mathcal{O}(\\lambda^2)$, where the first two\n\t\tgraphs of order $\\lambda^2$ contribute with four different labellings and the last graph with two different labellings (after probably inverting the orientation).\n\t\t\\label{4p}}\n\\end{figure}\n\nThroughout this article, the main insight of the algebraic structure of the exact solutions is that the correlators $\\Omega_{p_1,...,p_n}$ are much simpler from analytic perspective. However, from physics perspective the correlation functions $G_{|...|}$ have the most natural significations and the most common perturbative expansion. We focus on the difference between $\\Omega$ and $G$ on the perturbative level and demonstrate the perturbative interpretion of $\\Omega$.\nIn doing so, we illustrate the action of the boundary creation operator (see Definition \\ref{def:gDSE}), which is by considering multiplicities $r_i$\n\\begin{align*}\n-\\frac{N}{ r_{p_{n+1}} }\\frac{\\partial}{\\partial e_{p_{n+1}}} \\Omega_{p_1,...,p_n}= \\Omega_{p_1,...,p_n,p_{n+1}}+\\frac{\\delta_{n,1}}{(e_{p_1}-e_{p_2})^2}.\n\\end{align*}\nTo show that the boundary creation operator for the LSZ model was properly chosen, we first need to prove the following correspondence:\n\n\\begin{proposition}\n\\label{prop:GOM}\n$\\Omega_{p_1,...,p_n}$ has for $n={1,2,3}$ and distinct $p_i$ the following representations in terms of correlation functions:\n\\begin{align*}\n&\\Omega^{(g)}_{p_1}=\\frac{1}{N} \\sum_l \\tilde{r}_l G^{(g)}_{|p_1l|} \\\\\n&\\Omega^{(g)}_{p_1,p_2}= \\frac{\\delta_{g,0}}{(e_{p_1}-e_{p_2})^2} + \\frac{1}{N^2} \\sum_{l_1,l_2} \\tilde{r}_{l_1}\\tilde{r}_{l_2} G^{(g)}_{|p_1l_1|p_2l_2|} + \\frac{1}{N} \\sum_{l}\\tilde{r}_l G^{(g)}_{|q_1lq_2l|} \\\\\n&\\Omega^{(g)}_{p_1,p_2,p_3}= \\frac{1}{N^3} \\sum_{l_1,l_2,l_3}\\tilde{r}_{l_1}\\tilde{r}_{l_2}\\tilde{r}_{l_3} G^{(g)}_{|p_1l_1|p_2l_2|p_3l_3|} + \\frac{1}{N^2} \\sum_{l_1,l_2}\\tilde{r}_{l_1}\\tilde{r}_{l_2} \\bigg (G^{(g)}_{|p_1l_1p_2l_1|p_3l_2|} + \\mathrm{cycl.} \\biggl )\\\\\n&\\qquad \\qquad + \\frac{1}{N} \\sum_{l}\\tilde{r}_{l} G^{(g)}_{|p_1lp_2lp_3l|} \n\\end{align*}\n\\begin{proof}\n\tThe expression for $\\Omega^{(g)}_{p_1}$ is already known. We write\n\t\\begin{align*}\n\t\t&\\Omega^{(g)}_{p_1,p_2}= [N^{2-2g}]\\sum_{l_1,l_2=1}^N \\frac{\\partial^4}{\\partial J_{q_1k}\\partial J^\\dagger_{kq_1}\\partial J_{q_2l}\\partial J^\\dagger_{lq_2}} \\log [\\mathcal{Z}(J,J^\\dagger)]_{J,J^\\dagger =0}\n\t\\end{align*}\n\tFor generic $l_1,l_2$, the derivatives produce two boundaries, yielding $ \\frac{1}{N^2} \\sum_{l_1,l_2} G^{(g)}_{|p_1l_1|p_2l_2|} $. For $l_1=l_2=l$, we additionally produce $\\frac{1}{N} \\sum_{l} G^{(g)}_{|p_1lp_2l|} $ as the derivatives commute and we rearrange them to a cycle of four indices, yielding a 4-point function. Allowing multiplicities leads to expression of $\\Omega^{(g)}_{p_1,p_2}$\n\t Analogous techniques give the expression for $\\Omega^{(g)}_{p_1,p_2,p_3}$.\n\\end{proof}\n\\end{proposition}\n\n\n\n\\begin{remark}\nA comparison to the Quartic Kontsevich model (hermitian matrices) shows the enormous simplification. For $\\Omega^{(g)}_{p_1,p_2}$ e.g., the following analogue to Proposition \\ref{prop:GOM} was proved in \\cite[Prop. 4.7]{Branahl:2020uxs}:\n \\begin{align}\n\\Omega^{(g)}_{p_1,p_2} &=\n\\textcolor{red}{\\frac{\\delta_{g,0}}{(E_{p_1}-E_{p_2})^2}}\n+\\sum_{g_1+g_2=g} G^{(g_1)}_{|p_1p_2|} G^{(g_2)}_{|p_1p_2|}\n\\nonumber\n\\\\\n&\n+ \\textcolor{red}{ \\frac{1}{N^2}\\sum_{l_1,l_2=1}^dr_{l_1} r_{l_2} G^{(g)}_{|p_1l_1|p_2l_2|}}\n+\\frac{1}{N}\\sum_{l=1}^d r_l \\Big(G^{(g)}_{|p_1lp_1p_2|}+G^{(g)}_{|p_2lp_2p_1|}+\\textcolor{red}{G^{(g)}_{|p_1lp_2l|}}\n\\Big)\n\\nonumber\n\\\\\n& +\\frac{1}{N} \\sum_{l=1}^d r_l \\Big(G^{(g-1)}_{|p_1l|p_2|p_2|}+G^{(g-1)}_{|p_2l|p_1|p_1|}\\Big)\n+G^{(g-1)}_{|p_1p_2p_2|p_2|}+G^{(g-1)}_{|p_2p_1p_1|p_1|}\n\\nonumber\n\\\\\n&+\\sum_{g_1+g_2=g-1} G^{(g_1)}_{|p_1|p_2|} G^{(g_2)}_{|p_1|p_2|}\n+G^{(g-2)}_{|p_1|p_1|p_2|p_2|}\\;.\n\\end{align} \nBesides odd boundary lengths, also products of correlation functions leading to disconnected ribbon graphs that contribute to $\\Omega^{(g)}$ are allowed, as well as chains of external indices $q_i$ within one boundary. This is possible since much more derivatives with respect to the source $J$ can be commuted and rearranged to different chains of indices. Hence, it is not surprising that several surplus structures (in terms of blobs) arise from the analytic perspective forming bobbed topological recursion. \n\\end{remark}\n\nCombining the perturbative expansion given by Proposition \\ref{prop:weight} together with Proposition \\ref{prop:GOM}, we get the perturbative expansion of $\\Omega^{(g)}_{p_1,...,p_n}$ in general, which can be compared with the exact results achieved for instance from Theorem \\ref{th:main}. \n\nWe illustrate this at the first orders for $\\Omega^{(0)}_p$ and $\\Omega^{(0)}_{p_1,p_2}$:\n\\begin{example}\n\tFrom the perturbative expansion of the 2- and 4-point function from Example \\ref{ex:2ppert}, we find \n\t\\begin{align}\\label{OM1}\n\t\t\\Omega^{(0)}_{p}=&\\frac{1}{N}\\sum_{q=1}^{\\tilde{d}}\\tilde{r}_q\\bigg[\\frac{1}{e_p+\\tilde{e}_q}- \\frac{\\lambda}{(e_p+\\tilde{e}_q)^2}\\frac{1}{N}\\bigg(\\sum_{l=1}^{\\tilde{d}}\\frac{\\tilde{r}_l}{e_p+\\tilde{e}_l}+\\sum_{k=1}^{d}\\frac{r_k}{\\tilde{e}_q+e_k}\\bigg)\\bigg]+\\mathcal{O}(\\lambda^2)\\\\\\label{OM2}\n\t\t\\Omega^{(0)}_{p_1,p_2}=&\\frac{1}{(e_{p_1}-e_{p_2})^2}-\\frac{\\lambda}{N}\\sum_{q=1}^{\\tilde{d}}\\frac{ \\tilde{r}_q}{(e_{p_1}+\\tilde{e}_{q})^2(e_{p_2}+\\tilde{e}_{q})^2}+\\mathcal{O}(\\lambda^2),\n\t\\end{align}\n\tsince the $2+2$-point function starts at order $\\mathcal{O}(\\lambda^2)$. Note that \\eqref{OM2} is also achieved by the boundary creation operator $-\\frac{N}{r_{p_2}}\\frac{\\partial}{\\partial e_{p_2}}$ on \\eqref{OM1}, as it should by Definition \\ref{def:gDSE}. \n\t\n\tOn the other hand, the initial data for $x$ and $y$ provided in Theorem \\ref{Thm:InitialData} expands $\\varepsilon_p,\\tilde{\\varepsilon}_q$ with $e_p=x(\\varepsilon_p)$ and $\\tilde{e}_q=y(\\tilde{\\varepsilon}_q)$ into (note that both expansions are highly coupled)\n\t\\begin{align*}\n\t\\varepsilon_p =& e_p + \\frac{\\lambda}{N} \\sum_{k=1}^{\\tilde{d}} \\frac{\\tilde{r}_k}{e_p+\\tilde{e}_k} -\\frac{\\lambda^2}{N^2}\\sum_{k=1}^{\\tilde{d}}\\frac{\\tilde{r}_k}{e_p+\\tilde{e}_k}\\bigg(\\sum_{n=1}^d\\frac{r_n}{(\\tilde{e}_k+e_n)^2}+\\frac{r_n}{(e_p+\\tilde{e}_k)(\\tilde{e}_k+e_n)}\\\\\n\t&\\qquad \\qquad\\qquad \\qquad\\qquad \\qquad\\qquad \\qquad +\\sum_{l=1}^{\\tilde{d}}\\frac{\\tilde{r}_l}{(e_p+\\tilde{e}_k)(\\tilde{e}_l+e_p)} \\bigg)+ \\mathcal{O}(\\lambda^3),\\\\\n\t x'(\\varepsilon_p) =& 1 + \\frac{\\lambda}{N} \\sum_{k=1}^{\\tilde{d}} \\frac{\\tilde{r}_k}{(e_p+\\tilde{e}_k)^2} + \\mathcal{O}(\\lambda^2),\\\\\n\t \\tilde{\\varepsilon}_q =& -\\tilde{e}_q - \\frac{\\lambda}{N} \\sum_{n=1}^{d} \\frac{r_n}{\\tilde{e}_q+e_n} + \\mathcal{O}(\\lambda^2), \\quad y'(\\tilde{\\varepsilon}_q) = -1 + \\frac{\\lambda}{N} \\sum_{n=1}^{d} \\frac{r_n}{(\\tilde{e}_q+e_n)^2} + \\mathcal{O}(\\lambda^2).\n\t\\end{align*}\n\tInserting this into the solution of $\\Omega^{(0)}_p$ yields\n\t\\begin{align*}\n\t\t\\Omega^{(0)}_p=&-\\frac{y(\\varepsilon_p)+V'(e_p)}{\\lambda}\\\\\n\t\t=&\\frac{\\varepsilon_p-e_p}{\\lambda}+\\frac{1}{N}\\sum_{n=1}^dr_n\\bigg(\\frac{1}{e_p-e_n}-\\frac{1}{x'(\\varepsilon_n)(\\varepsilon_p-\\varepsilon_n)}\\bigg)\\\\\n\t\t=&\\frac{1}{N}\\sum_{k=1}^{\\tilde{d}}\\frac{\\tilde{r}_k}{e_p+\\tilde{e}_k}-\\frac{\\lambda}{N^2}\\sum_{k=1}^{\\tilde{d}}\\sum_{n=1}^dr_n\\tilde{r}_k\\bigg(\\frac{1}{(\\tilde{e}_l+e_n)^2(e_n-e_p)}+\\frac{\\frac{1}{(\\tilde{e}_l+e_n) }-\\frac{1}{(\\tilde{e}_l+e_p)}}{(e_n-e_p)^2}\\bigg)\\\\\n\t\t-&\\frac{\\lambda}{N^2}\\sum_{k=1}^{\\tilde{d}}\\tilde{r}_k\\bigg[\\sum_{n=1}^d\\frac{r_n}{(e_p+\\tilde{e}_k)(\\tilde{e}_k+e_n)^2}+\\frac{r_n}{(e_p+\\tilde{e}_k)^2(\\tilde{e}_k+e_n)}\n\t\t+\\sum_{l=1}^{\\tilde{d}}\\frac{\\tilde{r}_l}{(e_p+\\tilde{e}_k)^2(\\tilde{e}_l+e_p)}\\bigg]\\\\\n\t\t&+ \\mathcal{O}(\\lambda^2),\n\t\\end{align*}\n\twhere the double sum in second last line cancels the first term in the last line, and therefore it is in full compliance with the perturbative result \\eqref{OM1}. Similarly, we find from Proposition \\ref{pro:om02}\n\t\\begin{align*}\n\t&\\Omega^{(0)}_{p_1,p_2}-\\frac{1}{(e_{p_1}-e_{p_2})^2}\n\t=\\frac{1}{x'(\\varepsilon_{p_1})x'(\\varepsilon_{p_2})(\\varepsilon_{p_1}-\\varepsilon_{p_2})^2}-\\frac{1}{(e_{p_1}-e_{p_2})^2}\n\t\\\\\n\t&=-\\frac{\\lambda}{N}\\sum_{k=1}^{\\tilde{d}} \\tilde{r}_k\\bigg(2\\frac{\\frac{1}{\\tilde{e}_k+e_{p_1}}\n\t\t-\\frac{1}{\\tilde{e}_k+e_{p_2}}}{(e_{p_1}-e_{p_2})^3}+\\frac{\\frac{1}{(e_{p_1}+\\tilde{e}_k)^2}+\\frac{1}{(e_{p_2}+\\tilde{e}_k)^2}}{(e_{p_1}-e_{p_2})^2}\n\t\\bigg)+\\mathcal{O}(\\lambda^2),\n\t\\end{align*}\n\twhich is also after trivial rearrangements in full compliance with the perturbative result \\eqref{OM2}.\n\t\n\tAn interested reader may also check the next orders, where surprising cancellations show up. The perturbative and the analytic result is represented in completely different partial franction decompositions very similar to \\cite{Branahl:2020uxs}, but with more structure due to the distinction between $e_n$ and $\\tilde{e}_k$.\n\\end{example}\n \n \n\n\n\n\\subsection{The Combinatorial Limit}\nFinally, we want to focus on an extreme limit for the LSZ model: The case where the external fields become a multiple of the identity matrix, meaning $d=\\tilde{d}=1$ or $N$-fold degenerate single eigenvalues of $E$ and $\\tilde{E}$. For a quartic interaction, this particular model aroused interest already in the 90's, but in the field of combinatorics\/enumeration of certain objects. For this reason, the aforementioned limit case shall be called the \\textit{combinatorial limit}. In particular, Morris aimed to count ways of checkering a genus $g$-surface like a chess board in \\cite{Morris:1990cq} whereas Zinn-Justin and Zuber used the same partition function for the enumeration of virtual tangles and links \\cite{Zinn-Justin:2003ecd}. Moreover, the complex 1-matrix model is known to give rise to generating functions of \\textit{bipartite maps}, whereas hermitian matrices yield in the same manner generating functions for the enumeration of \\textit{ordinary maps} (as they are today often called) - one of the first successes of TR \\cite{Eynard:2016yaa}. The spectral curve generating bipartite maps was recently worked out in \\cite{Branahl:2022}. \n\nInterestingly, these two kinds of maps show up again in the comparison between the QKM and the LSZ model: The evaluation of the corresponding $\\Omega_{g,1}(z)$ at $z=\\varepsilon$ of the QKM (LSZ model) gives rise to generating functions of the number of rooted (bipartite) quadrangulations of a genus-$g$ surface, where the boundary length is 2. This access to generating functions of the enumeration of maps is fairly different compared with earlier techniques - although in the combinatorial limit the same partition functions occur. Within this framework of the complex\/hermitian 1-matrix model, topological recursion gives rise to resolvents containing an infinite sum all possible boundary lengths. To evaluate the meromorphic forms $\\omega_{g.n}(z_1,...,z_n)$ of these 1-matrix models, one takes residues at $z_i = \\infty$ to filter the generating function of interest out of the Laurent series\/resolvent in powers of $x(z)$ (see e.g. again \\cite{Eynard:2016yaa,Branahl:2022}). Below, we give some exemplary numbers up to $\\mathcal{O}(\\lambda^5)$ in the LSZ model in the combinatorial limit that coincide with \\cite{Branahl:2022}:\n\\begin{table}[h]\n\\centerline{\\begin{tabular}[h!b]{|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nOrder &$\\Omega_{0,1}$&$\\Omega_{0,2}$ &$\\Omega_{0,3}$ &$\\Omega_{1,1}$&$\\Omega_{1,2}$ &$\\Omega_{2,1}$\\\\\n\\hline\n$\\lambda^0$ & 1 & 0 & 0 & 0 &0 & 0 \\\\\n\\hline\n$\\lambda^1$ & 2 &1 & 0 & 0 &0 &0 \\\\\n\\hline\n$\\lambda^2$ & 9 & 13 & 6 & 1 & 0&0 \\\\\n\\hline\n$\\lambda^3$ & 54 &144 &172 & 20 & 0 &0\\\\\n\\hline\n$\\lambda^4$ & 378 &1539 & 3294 & 307 & 21 & 21 \\\\\n\\hline\n$\\lambda^5$ & 2916 & 16335 &53136 & 4280 & 734 & 966 \\\\\n\\hline\n\\end{tabular}}\n\\hspace*{1ex}\n\\caption{These numbers are generated by Theorem \\ref{th:main} for $d=\\tilde{d}=1$, where $\\Omega^{(g)}_n(z_1,...,z_n)$ is evaluated at $z_i=\\varepsilon$ and then expanded in the coupling constant $\\lambda$. $\\Omega_{g,1}$ coincides with OEIS no. A000168 (g=0), no. A006300 (g = 1) and no. A006301 (g = 2) }\n\\label{tab1}\n\\end{table}\n\nMoreover, we can give concrete formulae for all planar correlation functions emerging from the LSZ model by exploiting two facts. On the one hand, we know that the correlation functions $G_{...}$ of the QKM count in the combinatorial limit \\textit{fully simple maps} in the sense of \\cite{Borot:2017agy}. Their generating functions were already worked out in \\cite{bernardi2017bijections}. On the other hand, taking only the bipartite ones is an easy task in the planar sector: Since the following relation \n \\begin{align*}\n (G^{(0)}_{|p_1^1q_1^1...p_{N_1}^1q_{N_1}^1|...|p_1^bq_1^b...p_{N_b}^bq_{N_b}^b|} )^{LSZ} = 2^{1-b} (G^{(0)}_{|p_1^1q_1^1...p_{N_1}^1q_{N_1}^1|...|p_1^bq_1^b...p_{N_b}^bq_{N_b}^b|} )^{QKM}\n\\end{align*}\nholds for $e=\\tilde{e}$, in analogy to the discussion in \\cite[Sec. 1]{Branahl:2022}, we conclude in the combinatrial limit using \\cite{bernardi2017bijections}:\n\\begin{corollary}\n \\begin{align*}\n \\bigg ( G^{(0)}_{|k_1^1...k_{n_1}^1|...|k_1^b...k_{n_b}^b|} \\bigg )^{LSZ}\n \\Big|_{d=1}\n =\\sum_{n=0}^{\\infty} \\frac{3^{b+n-2}(\\#n_e-1)!}{n!2^{b-1}(3l_h+b+n-2)!}\n \\prod_{i=1}^b n_i \\binom{\\frac{3n_i}{2}}{\\frac{n_i}{2}}\n \\cdot \\frac{(-\\lambda)^{n+l_h+b-2}}{(e+\\tilde{e})^{2(n+l_h+b-1)}}\\;.\n\\end{align*}\nHere $l_h:=\\frac{1}{2}\\sum_i n_i$ is the half boundary length and\n$\\#n_e:=3l_h+2b+2n-4$ the number of edges. For $b=1$, $n_1=2$ and thus $l_h= 1$, one recovers the famous\nresult of Tutte \\cite{Tuttbij} for the number of rooted planar quadrangulations\n$\\frac{2 \\cdot 3^n(2n)!}{n!(n+2)!}$, obtained as coefficient of\n$\\frac{(-\\lambda)^n}{(e+\\tilde{e})^{2n+2}}$ in the 2-point\nfunction. \n\\end{corollary}\n\n\nIn the combinatorial limit it is possible to write the solution of the implicitly defined variable transforms $x(z)$ and $y(z)$. With the notation $\\frac{\\tilde{r}_k}{y'(\\tilde{\\varepsilon}_k)} = \\tilde{\\rho}_k$ and $\\frac{r_k}{x'(\\varepsilon_k)} = \\rho_k$, the combinatorial limit gives rise to an exact solution of the set $\\{\\varepsilon,\\tilde{\\varepsilon},\\rho,\\tilde{\\rho} \\}$: The system of equations\n\\begin{align*}\n & \\biggl \\{ e= \\varepsilon-\\frac{\\lambda \\tilde{\\rho}}{N(\\varepsilon-\\tilde{\\varepsilon})} \\,\\, , \\,\\,\\, \\tilde{e}= -\\tilde{\\varepsilon}-\\frac{\\lambda \\rho}{N(\\varepsilon-\\tilde{\\varepsilon})} \\,\\, , \\,\\,\\,\\frac{N}{\\tilde{\\rho}} - \\frac{\\lambda \\rho}{N(\\varepsilon-\\tilde{\\varepsilon})^2}=\\frac{N}{\\rho} - \\frac{\\lambda \\tilde{\\rho}}{N(\\varepsilon-\\tilde{\\varepsilon})^2} \\biggl \\}\n\\end{align*}\nhas the solutions\n\\begin{align*}\n & \\varepsilon= \\frac{5e-\\tilde{e}+\\sqrt{(e+\\tilde{e})^2+12 \\lambda}}{6} \\qquad \\tilde{\\varepsilon} = \\frac{e-5\\tilde{e}-\\sqrt{(e+\\tilde{e})^2+12 \\lambda}}{6} \\\\\n& \\rho= \\tilde{\\rho} =N\\cdot \\frac{-(e+\\tilde{e})^2+12\\lambda +(\\tilde{e}+e)\\sqrt{(e+\\tilde{e})^2+12 \\lambda} }{18 \\lambda} \n\\end{align*}\nwith $ \\varepsilon+\\tilde{\\varepsilon}=e-\\tilde{e}$ and $\\lim_{\\lambda \\to 0} \\varepsilon(\\lambda)=e$ and $\\lim_{\\lambda \\to 0} \\tilde{\\varepsilon}(\\lambda)=-\\tilde{e}$. Setting $e=\\tilde{e}$, we get back the solutions of $\\{\\varepsilon,\\rho\\}$ in the combinatorial limit of the QKM. \n\n\n\n\\section{Discussion and Outlook}\n\\label{ch:concl}\nIn this article we have generalised the exact solvability of the Langmann-Szabo-Zarembo model with a quartic potential by identifying suitable meromorphic forms $\\omega_{g,n}$ that satisfy topological recursion and shed new light on the discussion of its solutions. It was a long way to these objects: Starting with the naturally occurring correlation functions in quantum field theory, that satisfy closed Dyson-Schwinger equations after applying a system of Ward identities, we identified a differential operator later creating boundaries in $\\omega_{g,n}$. This, together with the previous success of this method in the quartic analogue of the Kontsevich model, suggested the introduction of the first new set of correlators - the generalised correlation functions $T$. The creation operator gave birth to a second family of correlators $\\Omega$ which is eventually governed by topological recursion itself. From the solution of the 2-point function after analytic continuation and two important variable transforms we could directly read off the initial data of the recursion. Complexification of the Dyson-Schwinger equation luckily led to known loop equation for which the recipe to show the fulfilment of the two abstract loop equations was well-known already and applied in a couple of examples. \n\nThe journey to this solution started when finding an answer to the question, for which reasons the analogue of the LSZ model taking hermitian quantum fields into consideration is so much more complicated than the complex case. We analysed the exact and perturbative differences in these two models. \n\nAfter a concrete proof of the exact solution of the LSZ model, we are left with the insight that the 2-point function of quantum field theory is undoubtedly a necessary step towards topological recursion to read off the initial data and plays an important role in perturbation theory, but only a smart rearrangement of the correlation functions $G$ to the family of $\\omega_{g,n}$ allows for compact and concrete solutions with a simple recursive pattern. This insight might be crucial when turning to exact solutions of more complicated, physical quantum field theories.\n\nFinally, this discussion and outlook shall be dedicated to three interesting perspectives for the future. We refresh the discussion of integrability of the LSZ model, the triviality problem on the four-dimensional Moyal space and explain how topological recursion builds a bridge to Hurwitz numbers.\n\\begin{enumerate}\n\\item \\textit{The triviality problem.} In this article we did not pay much attention to the noncommutative Moyal space, where the actual LSZ model lives. Instead, we directly approximated the quantum fields by finite $N\\times N$ matrices and obtained a partition function being a matrix model (this procedure can be retraced in detail e.g. in \\cite[Ch. 2.5]{Hock:2020rje}) that could be treated with established methods. However, this is far away from quantum field theory in its original sense. It would be worth a detailed investigation to perform the $N \\to \\infty$ limit and to see which parts of topological recursion inherit to the Moyal plane in two dimensions or, most importantly, the four-dimensional Moyal space. This limit contains several technical procedures, as a regularisation of the integral equations (occurring by sending $N \\to \\infty$ in the Dyson-Schwinger equations) by a cut-off $\\Lambda^2$ (ratio of the largest eigenvalue and $N$ in the limit $N\\to \\infty$) and renormalisation of some physical quantities like mass, field and coupling constant (see \\cite{Grosse:2019jnv} for the hermitian model). \n\nThere is a severe danger for any quantum field theory: It has to be consistent with the scaling limit $\\Lambda^2 \\to \\infty$ (see e.g. \\cite{Landau:1954??} for the Landau ghost problem, a serious hindrance towards renormalisability in the childhood of QFT). The authors of \\cite{Langmann:2003if} remark that their scaling limit leads to only trivial Green's functions\/connected correlation functions, but also admit that the subtle choice of a proper scaling limit in noncommutative QFT may have meaningful consequences for the triviality problem. On commutative geometries, triviality of the $\\lambda \\phi^4$ model in $D=4 + \\varepsilon$ dimensions is known since the 80's \\cite{Aizenman:1981du,Frohlich:1982tw}, whereas in exactly $D=4$ dimensions, (marginal) triviality was recently proved \\cite{Aizenman:2019yuo}. But there is hope coming from the investigation of the noncommutative $\\lambda \\phi^4$ theory with hermitian fields on the four-dimensional Moyal space. This is exactly the $N \\to \\infty$ limit of the previously explained QKM, obtained by a suitable deformed spectral measure. As shown in \\cite{Grosse:2019jnv,Grosse:2019qps}, the spectral measure has a dimension drop in the planar case of $D=4-2 \\frac{\\arcsin(\\lambda \\pi)}{\\pi}$ implying \\textit{non-triviality} in four dimensions in the planar sector! We are convinced that a thorough future investigation of the $N \\to \\infty$ limit of the LSZ model will reveal many similarities to the hermitian model. There might be an adequate scaling limit to rescue the non-triviality of this noncommutative QFT with complex fields.\n\n\n\\item \\textit{Integrability.} Integrable systems in Nature are often those with the richest algebraic beauty in their mathematical description. The Kyoto school method (Sato, Hirota, Miwa, Jimbo et al. \\cite{Hirota:1971,Sato:1983}) associates a $\\tau$ function to the integrable system\nthat encodes the majority of its properties and strongly corresponds to the partition function. It obeys\nHirota's equation. The characterisation of the integrable system can be performed by finding the integrable PDE class the $\\tau$ function obeys. On the other hand, topological recursion provides a simple method to characterise the integrable hierarchy by the geometry of the spectral curve. A first success was the to figure out the bridge between the spectral curve for the Kontsevich model to intersection theory, enumeration of trivalent ribbon graphs and the $\\tau$ function of the Korteweg-de Vries (KdV) hierarchy \\cite{Witten:1990hr}.\n\n Today, it is known \\cite{Eynard:2017} that the entire moduli space of hyperelliptical spectral curves with $\\mathrm{deg}(x)=2$ and a polynomial relation $y^2=P(x)$ yields the space of KdV systems. This is a subspace of all Kadomtsev-Petviashvili (KP) systems corresponding to the moduli space of algebraic curves where $x$ maps to the Riemann sphere and $x$ and $y$ are meromorphic such that there is a polynomial $P(x,y\/dx)=0$. This is exactly the type of spectral curve we dealt with in the LSZ model. The authors of \\cite{Langmann:2003if} stated that $\\mathcal{Z}(J,J^\\dagger)$ from eq. (\\ref{partfunLSZ}) is a $\\tau$ function of the Toda lattice hierarchy. This is indeed true, but with the knowledge from topological recursion we can restrict this statement to the subspace $\\mathcal{M}_{KP} \\subset \\mathcal{M}_{Toda}$ as long as we are dealing with finite matrix approximation. The moduli space $\\mathcal{M}_{Toda}$ namely additionally allows for logarithmic singularities in $x$ and $y$. This kind of singularities will occur in the limit $N\\to\\infty$, changing the matrix model to a model on the $2D$-dimensional Moyal space - comparing with \\cite{Grosse:2019jnv}, where e.g. in 2 dimensions, the spectral curve turns into a logarithm. This kind of \\textit{Lambert curve} gives the connection to simple Hurwitz numbers.\n\n\\item \\textit{A matrix model for simple Hurwitz numbers.} In the early days of topological recursion, prime examples were mostly found in the realm of map enumeration. Bouchard and Mari\u00f1o then formulated a conjecture \\cite{Bouchard:2007hi} that the same recursive structure shows up in the calculation of (simple) Hurwitz numbers. These numbers count $n$-fold coverings over $\\overline{\\mathbb{C}}$ that are connected genus-$g$ curves with one branch point of arbitrary ramification profile and trivial profiles (only transpositions) for $m$ further points. This conjecture was proved in a twofold way: By so-called cut-and-join equations \\cite{Eynard:2011xy} and by a quite artificially constructed matrix model with external field \\cite{Borot:2009ix}. We will focus in this discussion on the latter one. \n\nThe partition function of this matrix model for simple Hurwitz numbers looks quite technical and is designed exactly for this purpose. However, spectral curves solving matrix models with external fields are always of the same algebraic structure. We cite their result \\cite[eq. (13)]{Borot:2009ix}\n\\begin{align*}\n\\quad x(z)=z+C-g_s \\sum_{i=1}^N \\frac{1}{y'(z_i)(z-z_i)} \\qquad y(z)= -z + \\sum_{j=1}^{\\hat{N}} \\frac{u_j}{x'(\\hat z_j)(z-\\hat z_j)}\n\\end{align*}\nand compare with the curve solving the LSZ model ($z_i \\leftrightarrow \\tilde{\\varepsilon}_i$, $\\hat z_j \\leftrightarrow \\varepsilon_j$, $g_s \\leftrightarrow \\lambda$, the occurrence of $u_j$ is not important). There is in fact one decisive difference: The coupling constant $\\lambda$ in the LSZ model is a global prefactor for both $x(z)$ and $y(z)$. The authors of \\cite{Borot:2009ix}, however, aim to reconstruct in the limit $N \\to \\infty$ the \\textit{Lambert curve} $e^x=ye^{-y}$ from the Bouchard-Mari\u00f1o conjecture that can only be obtained if $y(z)$ reduces to $y(z)=z$. For this purpose, they set the coupling $g_s$ to 0, which is needed before for regularisation. An equivalent apprach in the LSZ model might be to send the deformed eigenvalues either $\\varepsilon_k \\to \\infty$ or $\\tilde{\\varepsilon}_l \\to \\infty$ for $k \\in \\{1,..,d\\}$ and $l \\in \\{1,...,\\tilde{d}\\}$. Investigating this more thoroughly may be combined with the analysis of the four-dimensional case with respect to the triviality problem in some future.\n\\end{enumerate}\nWe have seen that using topological recursion to show the exact solvability of the LSZ model with quartic interaction bridges quantum field theory to several interesting mathematical field. As a summary, Fig. \\ref{mm} shall illustrate these connections for different limiting cases. \nHaving the last three points of (non-)triviality, integrability and Hurwitz numbers in mind, there remain fascinating questions for future investigations. Up to now, we have added another model to the wide range of models and theories obeying topological recursion in its original sense. This work strengthens furthermore the connection of noncommutative QFT toy models to topological recursion and thus renews and enriches discussions about QFT-related problems that were debated about two decades ago. \\newpage\n\n\\begin{landscape}\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width= 1.45\\textwidth]{mm.png} \n \\caption{There exist various connections of the LSZ model to other matrix models in different limits.\n\\label{mm}}\n\\end{figure}\n\\end{landscape}\n\\newpage\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Statistical models}\n\\label{sec:statistical-models}\n\nLet us start with a (very rough) scheme. Imagine an experiment that produces some bit string $x$. We know nothing about the device that produced this data, and cannot repeat the experiment. Still we want to suggest some statistical model that fits the data (``explains'' $x$ in a plausible way). This model is a probability distribution on some finite set of binary strings containing $x$. What do we expect from a reasonable model?\n\nThere are, informally speaking, two main properties of a good model. First, the model should be ``simple''. If a model contains so many parameters that it is more complicated than the data itself, we would not consider it seriously. To make this requirement more formal, one can use the notion of Kolmogorov complexity.\\footnote{We assume that the reader is familiar with basic notions of algorithmic information theory (complexity, a priory probability). See~\\cite{shen-vovk} for a concise introduction, and \\cite{LiVi97,usv} for more details.} Let us assume that measure $P$ (used as a model) has finite support and rational values. Then $P$ can be considered as a finite (constructive) object, so we can speak about Kolmogorov complexity of $P$. The requirement then says that complexity of $P$ should be much smaller than the complexity of the data string $x$ itself.\n\nFor example, if a data string $x$ contains $n$ bits, we may consider a model that corresponds to $n$ independent fair coin tosses, i.e., the uniform distribution $P$ on the set of all $n$-bit strings. Such a distribution is a constructive object that is completely determined by the value of $n$, so its complexity is $O(\\log n)$, while the complexity of most $n$-bit strings is close to $n$ (and therefore is much larger than the complexity of $P$, if $n$ is large enough).\n\nStill this simple model looks unacceptable if, for example, the sequence $x$ consists of $n$ zeros, or, more generally, if the frequency of ones in $x$ deviates significantly from $1\/2$, or if zeros and ones alternate. This feeling was one of the motivations for the development of algorithmic randomness notions: why some bit sequences of length $n$ look plausible as outcomes of $n$ fair coin tosses while other do not, while all the $n$-bit sequences have the same probability $2^{-n}$ according to the model? This question does not have a clear answer in the classical probability theory, but the algorithmic approach to randomness says that plausible strings should be incompressible: the complexity of such a string (the minimal length of a program producing it) should be close to its length.\n\nThis answer works for a uniform distribution on $n$-bit strings; for arbitrary $P$ it should be modified. It turns out that for arbitrary $P$ we should compare the complexity of $x$ not with its length but with the value $(-\\log P(x))$ (all logarithms are binary); if $P$ is the uniform distribution on $n$-bit strings, the value of $(-\\log P(x))$ is $n$ for all $n$-bit strings $x$. Namely, we consider the difference between $(-\\log P(x))$ and complexity of $x$ as \\emph{randomness deficiency} of $x$ with respect to $P$. We discuss the exact definition in the next section, but let us note here that this approach looks natural: different data strings require different models.\n\n\\emph{Disclaimer}. The scheme above is oversimplified in many aspects. First, it rarely happens that we have no a priori information about the experiment that produced the data. Second, in many cases the experiment can be repeated (the same experimental device can be used again, or a similar device can be constructed). Also we often deal with a data stream: we are more interested, say, in a good prediction of oil prices for the next month than in a construction of model that fits well the prices in the past. All these aspects are ignored in our simplistic model; still it may serve as an example for more complicated cases. One should stress also that algorithmic statistics is more theoretical than practical: one of the reasons is that complexity is a non-computable function and is defined only asymptotically, up to a bounded additive term. Still the notions and results from this theory can be useful not only as philosophical foundations of statistics but as a guidelines when comparing statistical models in practice (see, for example,~\\cite{deRV}).\n\nMore practical approach to the same question is provided by machine learning that deals with the same problem (finding a good model for some data set) in the ``real world''. Unfortunately, currently there is a big gap between the algorithmic statistics and machine learning: the first one provides nice results about mathematical models that are quite far from practice (see the discussion about ``standard models'' below), while machine learning is a tool that sometimes works well without any theoretical reasons. There are some attempts to close this gap (by considering models from some class or resource-bounded versions of the notions), but much more remains to be done.\n\n\\emph{A historical remark}. The principles of algorithmic statistics are often traced back to Occam's razor principle often stated as ``Don't multiply postulations beyond necessity'' or in a similar way. Poincare writes in his \\emph{Science and Method'} (Chapter 1, \\emph{The choice of facts}) that ``this economy of thought, this economy of effort, which is, according to Mach, the constant tendency of science, is at the same time a source of beauty and a practical advantage''. Still the mathematical analysis of these ideas became possible only after a definition of algorithmic complexity was given in 1960s (by Solomonoff, Kolmogorov and then Chaitin): after that the connection between randomness and incompressibility (high complexity) became clear. The formal definition of $(\\alpha,\\beta)$-stochasticity (see the next section) was given by Kolmogorov (the authors learned it from his talk given in 1981~\\cite{kolmogorov81}, but most probably it was formulated earlier in 1970s; the definition appeared in print in~\\cite{shen83}). For the other related approaches (the notions of logical depth and sophistication, minimal description length principle) see the discussion in the corresponding sections (see also~\\cite[Chapter 5]{LiVi97}.)\n\n\\section{$(\\alpha,\\beta)$-stochasticity}\\label{sec:stoch}\n\n\\subsection[Prefix complexity, a priori probability and randomness deficiency]{Prefix complexity, a priori probability and\\\\ randomness deficiency}\n\nPreparing for the precise definition of $(\\alpha,\\beta)$-stochasticity, we need to fix the version of complexity used in this definition. There are several versions (plain and prefix complexities, different types of conditions), see \\cite[Chapter 6]{usv}. For most of the results the choice between these versions is not important, since the difference between the different versions is small (at most $O(\\log n)$ for strings of length $n$), and we usually allow errors of logarithmic size in the statements.\n\nWe will use the notion of \\emph{conditional prefix complexity}, usually denoted by $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu c)$. Here $x$ and $c$ are finite objects; we measure the complexity of $x$ when $c$ is given. This complexity is defined as the length of the minimal prefix-free program that, given $c$, computes $x$.\\footnote{We do not go into details here, but let us mention one common misunderstanding: the set of programs should be prefix-free for each $c$, but these sets may differ for different $c$ and the union is not required to be prefix-free.} The advantage of this definition is that it has an equivalent formulation in terms of a priori probability~\\cite[Chapter 4]{usv}: if $\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu c)$ is the conditional a priori probability, i.e., the maximal lower semicomputable function of two arguments $x$ and $c$ such that $\\sum_x \\mm(x\\mskip 1.5mu | \\mskip 1.5 mu c)\\le 1$ for every $c$, then\n$$\n\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu c)=-\\log\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu c)+O(1).\n$$\nIn particular, if a probability distribution $P$ with finite support and rational values (we consider only distributions of this type) is considered as a condition, we may compare $\\mm$ with function $(x,P)\\mapsto P(x)$ and conclude that $\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\ge P(x)$ up to an $O(1)$-factor, so $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le -\\log P(x)$. So if we define the randomness deficiency as\n$$\nd(x\\mskip 1.5mu | \\mskip 1.5 mu P)=-\\log P(x)-\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P),\n$$\nwe get a non-negative (up to $O(1)$ additive term) function. One may also explain in a different way why $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le - \\log P(x)$: this inequality is a reformulation of a standard result from information theory (Shannon--Fano code, Kraft inequality).\n\nWhy do we define the deficiency in this way? The following proposition provides some additional motivation.\n\n\\begin{proposition} The function $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ is (up to $O(1)$-additive term) the maximal lower semicomputable function of two arguments $x$ and $P$ such that\n$$\n \\sum_x 2^{d(x\\mskip 1.5mu | \\mskip 1.5 mu P)}\\cdot P(x)\\le 1 \\eqno(*)\n$$\nfor every $P$.\n\\end{proposition}\n\nHere $x$ is a binary string, and $P$ is a probability distribution on binary strings with finite support and rational values. By lower semicomputable functions we mean functions that can be approximated from below by some algorithm (given $x$ and $P$, the algorithm produces an increasing sequence of rational numbers that converges to $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)$; no bounds for the convergence speed are required). Then, for a given $P$, the function $x\\mapsto 2^{d(x\\mskip 1.5mu | \\mskip 1.5 mu P)}$ can be considered as a random variable on the probability space with distribution $P$. The requirement $(*)$ says that its expectation is at most $1$. In this way we guarantee (by Markov inequality) that only a $P$-small fraction of strings have large deficiency: the $P$-probability of the event $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)>c$ is at most $2^{-c}$. It turns out that there exists a maximal function $d$ satisfying $(*)$ up to $O(1)$ additive term, and our formula gives the expression for this function in terms of prefix complexity.\n\n\\begin{proof}\nThe proof uses standard arguments from Kolmogorov complexity theory. The function $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ is upper semicomputable, so $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ is lower semicomputable. We can also note that\n$$\n \\sum_x 2^{d(x\\mskip 1.5mu | \\mskip 1.5 mu P)}\\cdot P(x) = \\sum_x \\frac{\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P)}{P(x)}\\cdot P(x)=\\sum_x \\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le1,\n$$\nso the deficiency function satisfies~$(*)$.\n\nTo prove the maximality, consider an arbitrary function $d'(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ that is lower semicomputable and satisfies $(*)$. Then consider a function $m(x\\mskip 1.5mu | \\mskip 1.5 mu P)=2^{d'(x\\mskip 1.5mu | \\mskip 1.5 mu P)}\\cdot P(x)$ (the function equals $0$ if $x$ is not in the support of $P$). Then $m$ is lower semicomputable, $\\sum_x m(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le 1$ for every $P$, so $m(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le \\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ up to $O(1)$-factor; this implies that $d'(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le d(x\\mskip 1.5mu | \\mskip 1.5 mu P)+O(1)$.\n\\end{proof}\n\n\nFor the case where $P$ is the uniform distribution on $n$-bit strings, using $P$ as a condition is equivalent to using $n$ as the condition, so\n$$\nd(x\\mskip 1.5mu | \\mskip 1.5 mu P) = n -\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu n)\n$$\nin this case, and small deficiency means that complexity $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu n)$ is close to the length $n$, so $x$ is incompressible.\\footnote{Initially Kolmogorov suggested to consider $n -\\KS(x)$ as ``randomness deficiency'' in this case, where $\\KS$ stands for the plain (not prefix) complexity. One may also consider $n-\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu n)$. But all three deficiency functions mentioned are close to each other for strings $x$ of length $n$; one can show that the difference between them is bounded by $O(\\log d)$ where $d$ is any of these three functions. The proof works by comparing the expectation and probability-bounded characterizations as explained in~\\cite{bienvenu-gacs-et-al}.}\n\n\\subsection{Definition of stochasticity}\n\n\\begin{definition}\nA string $x$ is called $(\\alpha,\\beta)$-stochastic if there exists some probability distribution $P$ (with rational values and finite support) such that $\\KP(P)\\le \\alpha$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le\\beta$.\n\\end{definition}\n\nBy definition every $(\\alpha,\\beta)$-stochastic string is $(\\alpha',\\beta')$-stochastic for $\\alpha'\\ge\\alpha$, $\\beta'\\ge\\beta$. Sometimes we say informally that a string is ``stochastic'' meaning that it is $(\\alpha,\\beta)$-stochastic for some reasonably small thresholds $\\alpha$ and $\\beta$ (for example, one can consider $\\alpha,\\beta=O(\\log n)$ for $n$-bit strings).\n\\smallskip\n\nLet us start with some simple remarks.\n\n\\begin{itemize}\n\n\\item Every simple string is stochastic. Indeed, if $P$ is concentrated on $x$ (singleton support), then $\\KP(P)\\le \\KP(x)$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)=0$ (in both cases with $O(1)$-precision), so $x$ is always $(\\KP(x)+O(1),O(1))$-stochastic.\n\n\\item On the other end of the spectrum: if $P$ is a uniform distribution on $n$-bit strings, then $\\KP(P)=O(\\log n)$, and most strings of length $n$ have $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)=O(1)$, so most strings of length $n$ are $(O(\\log n), O(1))$-stochastic. The same distribution also witnesses that every $n$-bit string is $(O(\\log n), n+O(1))$-stochastic.\n\n\\item It is easy to construct stochastic strings that are between these two extreme cases. Let $x$ be an incompressible string of length $n$. Consider the string $x0^n$ (the first half is $x$, the second half is zero string). It is $(O(\\log n), O(1))$-stochastic: let $P$ be the uniform distribution on all the strings of length $2n$ whose second half contains only zeros.\n\n\\item For every distribution $P$ (with finite support and rational values, as usual) a random sampling according to $P$ gives us a $(\\KP(P), c)$-stochastic string with probability at least $1-2^{-c}$. Indeed, the probability to get a string with deficiency greater than $c$ is at most $2^{-c}$ (Markov inequality, see above).\n\n\\end{itemize}\n\nAfter these observations one may ask whether non-stochastic strings exists at all --- and how they can be constructed? A non-stochastic string should have non-negligible complexity (our first observation), but a standard way to get strings of high complexity, by coin tossing or other random experiment, can give only stochastic strings (our last observation).\n\nWe will see that non-stochastic strings do exist in the mathematical sense; however, the question whether they appear in the ``real world'', is philosophical. We will discuss both questions soon, but let us start with some mathematical results.\n\nFirst of all let us note that with logarithmic precision we may restrict ourselves to uniform distributions on finite sets.\n\n\\begin{proposition}\\label{prop:models-to-sets-1}\nLet $x$ be an $(\\alpha,\\beta)$-stochastic string of length $n$. Then there exist a finite set $A$ containing $x$ such that $\\KP(A)\\le\\alpha+O(\\log n)$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu U_A)\\le \\beta+O(\\log n)$, where $U_A$ is the uniform distribution on $A$.\n\\end{proposition}\n\nSince $\\KP(A)=\\KP(U_A)$ (with $O(1)$-precision, as usual), this proposition means that we may consider only uniform distributions in the definition of stochasticity, and get an equivalent (up to logarithmic change in the parameters) definition. According to this modified definition, a string $x$ in $(\\alpha,\\beta)$-stochastic if there exists a finite set $A$ such that $\\KP(A)\\le \\alpha$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le \\beta$, where $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$ is now defined as $\\log \\#A - \\KP(x\\mskip 1.5mu | \\mskip 1.5 mu A)$. Kolmogorov originally proposed the definition in this form (but used plain complexity).\n\n\\begin{proof}\nLet $P$ be the (finite) distribution that exists due to the definition of $(\\alpha,\\beta)$-stochas\\-tic\\-i\\-ty of $x$. We may assume without loss of generality that $\\beta\\le n$ (as we have seen, all strings of length $n$ are $(O(\\log n), n+O(1))$-stochastic, so for $\\beta>n$ the statement is trivial). Consider the set $A$ formed by all strings that have sufficiently large $P$-probability. Namely, let us choose minimal $k$ such that $2^{-k}\\le P(x)$ and consider the set $A$ of all strings such that $P(x)\\ge 2^{-k}$. By construction $A$ contains $x$. The size of $A$ is at most $2^k$, and $-\\log P(x)=k$ with $O(1)$-precision. According to our assumption, $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)=k-\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le n$, so $k=d(x\\mskip 1.5mu | \\mskip 1.5 mu P)+\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le O(n)$. Then $$\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\ge \\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P,k)\\ge \\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)-O(\\log n),$$ since $A$ is determined by $P,k$, and the additional information in $k$ is $O(\\log k)=O(\\log n)$ since $k=O(n)$ by our assumption. So the deficiency may increase only by $O(\\log n)$ when we replace $P$ by $U_A$, and $$\\KP(A)\\le\\KP(P,k)\\le\\KP(P)+O(\\log n)$$ for the same reasons.\n\\end{proof}\n\n\\begin{remark}\nSimilar argument can be applied if $P$ is a computable distribution (may be, with infinite support) computed by some program $p$, and we require $\\KP(p)\\le \\alpha$ and $-\\log P(x)-\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu p)\\le \\beta$. So in this way we also get the same notion (with logarithmic precision). It is important, however, that program $p$ \\emph{computes} the distribution $P$ (given some point $x$ and some precision $\\varepsilon>0$, it computes the probability of $x$ with error at most $\\varepsilon$). It is \\emph{not} enough for $P$ to be an output distribution for a randomized algorithm $p$ (in this case $P$ is called the semimeasure lower \\emph{semicomputed} by $p$; note that the sum of probabilities may be strictly less than $1$ since the computation may diverge with positive probability). Similarly, it is very important in the version with finite sets $A$ (and uniform distributions on them) that the set $A$ is considered as a finite object: $A$ is simple if there is a short program that prints the list of all elements of $A$. If we allowed the set $A$ to be presented by an algorithm that enumerates $A$ (but never says explicitly that no more elements will appear), then situation would change drastically: for every string of complexity $k$ the finite set $S_k$ of strings that have complexity at most $k$, would be a good explanation for $x$, so all objects would become stochastic.\n\\end{remark}\n\n\\subsection{Stochasticity conservation}\\label{subsec:stoch-cons}\n\nWe have defined stochasticity for binary strings. However, the same definition can be used for arbitrary finite (constructive) objects: pairs of strings, tuples of strings, finite sets of strings, graphs, etc. Indeed, complexity can be defined for all these objects as the complexity of their encodings; note that the difference in complexities for different encodings is at most $O(1)$. The same can be done for finite sets of these objects (or probability distributions), so the definition of $(\\alpha,\\beta)$-stochasticity makes sense.\n\nOne can also note that computable bijection preserves stochasticity (up to a constant that depends on the bijection, but not on the object). In fact, a stronger statement is true: every total computable mapping preserves stochasticity. For example, consider a stochastic pair of strings $(x,y)$. Does it imply that $x$ (or $y$) is stochastic? It is indeed the case: if $P$ is a distribution on pairs that is a reasonable model for $(x,y)$, then its projection (marginal distribution on the first components) should be a reasonable model for $x$. In fact, projection can be replaced by any \\emph{total} computable mapping.\n\n\\begin{proposition}\\label{prop:stoch-cons}\nLet $F$ be a total computable mapping whose arguments and values are strings. If $x$ is $(\\alpha,\\beta)$-stochastic, then $F(x)$ is $(\\alpha+O(1),\\beta+O(1))$-stochastic. Here the constant in $O(1)$ depends on $F$ but not on $x,\\alpha,\\beta$.\n\\end{proposition}\n\n\\begin{proof}\nLet $P$ be the distribution such that $\\KP(P)\\le\\alpha$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le \\beta$; it exists according to the definition of stochasticity. Let $Q=F(P)$ be the image distribution. In other words, if $\\xi$ is a random variable with distribution $P$, then $F(\\xi)$ has distribution $Q$. It is easy to see that $\\KP(Q)\\le\\KP(P)+O(1)$, where the constant depends only on $F$. Indeed, $Q$ is determined by $P$ and $F$ in a computable way. It remains to show that $d(F(x)\\mskip 1.5mu | \\mskip 1.5 mu Q)\\le d(x\\mskip 1.5mu | \\mskip 1.5 mu P)+O(1)$.\n\nThe easiest way to show this is to recall the characterization of deficiency as the maximal lower semicomputable function such that\n$$\n\\sum_u 2^{d(u\\mskip 1.5mu | \\mskip 1.5 mu S)}S(u)\\le 1\n$$\nfor every distribution $S$. We may consider another function $d'$ defined as\n$$\n d'(u\\mskip 1.5mu | \\mskip 1.5 mu S) = d(F(u)\\mskip 1.5mu | \\mskip 1.5 mu F(S))\n$$\nIt is easy to see that\n$$\n\\sum_u 2^{d'(u\\mskip 1.5mu | \\mskip 1.5 mu S)} S(u)=\\sum_u 2^{d(F(u)\\mskip 1.5mu | \\mskip 1.5 mu F(S))}S(u)=\\sum_v 2^{d(v\\mskip 1.5mu | \\mskip 1.5 mu F(S))}{}\\cdot{}[F(S)](v)\\le 1\n$$\n(in the second equality we group all the values of $u$ with the same $v=F(u)$). Therefore the maximality of $d$ guarantees that $d'(u\\mskip 1.5mu | \\mskip 1.5 mu S)\\le d(u\\mskip 1.5mu | \\mskip 1.5 mu S)+O(1)$, so we get the required inequality.\n\nThis proof can be also rephrased using the definition of stochasticity with a priori probability. We need to show that for $y=P(x)$ and $Q=F(P)$ we have\n$$\n\\frac{\\mm(y\\mskip 1.5mu | \\mskip 1.5 mu Q)}{Q(y)}\\le O(1)\\cdot\\frac{\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P)}{P(x)}\n$$\nor\n$$\n\\frac{\\mm(F(x)\\mskip 1.5mu | \\mskip 1.5 mu F(P))\\cdot P(x)}{Q(F(x))}\\le O(\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P)).\n$$\nIt remains to note that the left hand side is a lower semicomputable function of $x$ and $P$ whose sum over all $x$ (for every $P$) is at most $1$. Indeed, if we group all terms with the same $F(x)$, we get the sum $\\sum_y \\mm(y\\mskip 1.5mu | \\mskip 1.5 mu F(P))\\le 1$, since the sum of $P(x)$ over all $x$ with $F(x)=y$ equals $Q(y)$.\n\\end{proof}\n\n\\begin{remark}\nIn this proof it is important that we use the definition with distributions. If we replace is with the definition with finite sets, the results remains true with logarithmic precision, but the argument becomes more complicated, since the image of the uniform distribution may not be a uniform distribution. So if a set $A$ is a good model for $x$, we should not use $F(A)$ as a model for $F(x)$. Instead, we should look at the maximal $k$ such that $2^k\\le \\#F^{-1}(y)$, and consider the set of all $y'$ that have at least $2^k$ preimages in $A$.\n\\end{remark}\n\n\\begin{remark}\\label{rem:non-total}\nIt is important in Proposition~\\ref{prop:stoch-cons} that $F$ is a total function. If $x$ is some non-stochastic object and $x^*$ is the shortest program for $x$, then $x^*$ is incompressible and therefore stochastic. Still the interpreter (decompressor) maps $x^*$ to $x$. We discuss the case of non-total $F$ below, see Section~\\ref{subsec:depth-appl}.\n\\end{remark}\n\n\\begin{remark}\\label{rem:cons-f}\nA similar argument shows that $d(F(x)\\mskip 1.5mu | \\mskip 1.5 mu F(P))\\le d(x\\mskip 1.5mu | \\mskip 1.5 mu P)+\\KP(F)+O(1)$ (for total $F$), so both $O(1)$-bounds in Proposition~\\ref{prop:stoch-cons} may be replaced by $\\KP(F)+O(1)$ where $O(1)$-constant does not depend on $F$ anymore.\n\\end{remark}\n\n\\subsection{Non-stochastic objects}\n\nNote that up to now we have not shown that non-stochastic objects exist at all. It is easy to show that they exist for rather large values of $\\alpha$ and $\\beta$ (linearly growing with $n$).\n\n\\begin{proposition}[\\cite{shen83}]\\label{prop:existence-nonstochastic}\nFor some $c$ and all $n$:\n\n(1)~if $\\alpha+2\\betan+c\\log n$, then every $n$-bit string is $(\\alpha,\\beta)$-stochastic.\n\\end{proposition}\n\nNote that the term $c\\log n$ allows us to use the definition with finite sets (i.e., uniform distributions on finite sets) instead of arbitrary finite distributions, since both versions are equivalent with $O(\\log n)$-precision.\n\n\\begin{proof}\nThe second part is obvious (and is added just for comparison): if $\\alpha+\\beta=n$, then all $n$-bit strings can be split into $2^\\alpha$ groups of size $2^\\beta$ each. Then the complexity of each group is $\\alpha+O(\\log n)$, and the randomness deficiency of every string in the corresponding group is at most $\\beta+O(1)$. It is slightly bigger than the bounds we need, but we have reserve $c\\log n$, and $\\alpha$ and $\\beta$ can be decreased, say, by $(c\/2)\\log n$ before using this argument.\n\n\\emph{The first part}: Consider all finite sets $A$ of strings that have complexity at most $\\alpha$ and size at most $2^{\\alpha+\\beta}$. Since $\\alpha+(\\alpha+\\beta)\\alpha+ c \\log n$ is bounded by $O(2^{-\\alpha})$, if $c$ is large enough. Indeed, for those strings we have\n $$\n -\\log \\tilde P(x) - \\KP(x\\mskip 1.5mu | \\mskip 1.5 mu \\tilde P) > \\alpha+c \\log n.\n $$\nThe complexity of $\\tilde P$ is bounded by $\\alpha+O(\\log n)$ and therefore $\\KP(x)$ exceeds $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu \\tilde P)$ at most by $\\alpha+O(\\log n)$, so\n $\n-\\log \\tilde P(x) - \\KP(x) > 1\n $\n(or $\\tilde P(x)< \\mm(x)\/2$)\nfor those strings, if $c$ is large enough (it should exceed the constants hidden in $O(\\log n)$ notation). The difference $1$ is enough for the estimate below, but we could have arbitrary constant or even logarithmic difference by choosing larger value of~$c$.\n\nPrefix complexity can be defined in terms of a priori probability, so we get\n $$\n \\log(\\mm(x)\/\\tilde P(x))>1\n $$\nfor all $x$ that have deficiency exceeding $\\alpha+c\\log n$ with respect to $\\tilde P$. The same inequality is true for $P$ instead of $\\tilde P$, since $P$ is smaller. So for all those $x$ we have $P(x)<\\mm(x)\/2$, or $(\\mm(x)-P(x))>\\mm(x)\/2$. Recalling that the sum of $\\mathbf{m}(x)-P(x)$ over all $x$ of length $n$ does not exceed $O(2^{-\\alpha})$ by construction of $\\tilde\\omega$, we conclude that the sum of $\\mathbf{m}(x)$ over all strings of randomness deficiency (with respect to $\\tilde P$) exceeding $\\alpha+c\\log n$ is at most $O(2^{-\\alpha})$.\n\nSo we have shown that the sum of $\\mm(x)$ for all $x$ of length $n$ that are not $(\\alpha+O(\\log n),\\alpha+O(\\log n))$-stochastic, does not exceed $O(2^{-\\alpha})$. This differs from our claim only by $O(\\log n)$-change in $\\alpha$.\n\\end{proof}\n\nBruno Bauwens noted that this argument can be modified to obtain a stronger result where $(\\alpha,\\alpha)$-stochasticity is replaced by $(\\alpha+O(\\log n),O(\\log n))$-stochasticity. Instead of one measure $P$, one should consider a family of measures. Let us approximate $\\omega$ and look when the approximations cross the thresholds corresponding to $k$ first bits of the binary expansion of $\\omega$. In this way we get $P=P_1+P_2+\\ldots+P_\\alpha$, where $P_i$ has total weight at most $2^{-i}$, and complexity at most $i+O(\\log n)$. Let us show that all strings $x$ where $P(x)$ is close to $\\mm(x)$ (say, $P(x)\\ge \\mm(x)\/2$) are $(\\alpha+O(\\log n), O(\\log n))$-stochastic, namely, one of the measures $P_i$ multiplied by $2^i$ is a good explanations for them. Indeed, for such $x$ and some $i$ the value of $P_i(x)$ coincides with $\\mm(x)$ up to polynomial (in $n$) factor, since the sum of all $P_i$ is at least $\\mm(x)\/2$. On the other hand, $\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu 2^iP_i)\\le 2^i \\mm(x) \\approx 2^i P_i(x)$, since the complexity of $2^iP_i$ is at most $i+O(\\log n)$. Therefore the ratio $\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu P_i)\/(2^iP_i(x))$ is polynomially bounded, and the model $2^iP_i$ has deficiency $O(\\log n)$. This better bound also follows from the Levin's explanation, see below.\n\nThis result shows that non-stochastic objects rarely appear as outputs of randomized algorithms. There is an explanation of this phenomenon (that goes back to Levin): non-stochastic objects provide a lot of information about halting problem, and the probability of appearance of an object that has a lot of information about some sequence $\\alpha$, is small (for any fixed $\\alpha$). We discuss this argument below, see Section~\\ref{subsec:halting-information}.\n\\smallskip\n\nIt is natural to ask the following general question. For a given string $x$, we may consider the set of all pairs $(\\alpha,\\beta)$ such that $x$ is $(\\alpha,\\beta)$-stochastic. By definition, this set is upwards-closed: a point in this set remains in it if we increase $\\alpha$ or $\\beta$, so there is some boundary curve that describes the trade-off between $\\alpha$ and $\\beta$. What curves could appear in this way? To get an answer (to characterizes all these curves with $O(\\log n)$-precision), we need some other technique, explained in the next section.\n\n\\section{Two-part descriptions}\\label{sec:two-part}\n\nNow we switch to another measure of the quality of a statistical model. It is important both for philosophical and technical reasons. The philosophical reason is that it corresponds to the so-called ``minimal description length principle''. The technical reason is that it is easier to deal with; in particular, we will use it to answer the question asked at the end of the previous section.\n\n\\subsection{Optimality deficiency}\n\nConsider again some statistical model. Let $P$ be a probability distribution (with finite support and rational values) on strings. Then we have\n$$\n\\KP(x)\\le \\KP(P)+\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le \\KP(P)+(-\\log P(x))\n$$\nfor arbitrary string $x$ (with $O(1)$-precision). Here we use that (with $O(1)$-precision):\n\\begin{itemize}\n\\item $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le -\\log P(x)$, as we have mentioned;\n\n\\item the complexity of the pair is bounded by the sum of complexities: $\\KP(u,v)\\le \\KP(u)+\\KP(v)$;\n\n\\item $\\KP(v)\\le \\KP(u,v)$ (in our case, $\\KP(x)\\le KP(x,P)$).\n\\end{itemize}\n\nIf $P$ is a uniform distribution on some finite set $A$, this inequality can be explained as follows. We can specify $x$ in two steps:\n %\n\\begin{itemize}\n %\n\\item first, we specify $A$;\n %\n\\item then we specify the ordinal number of $x$ in $A$ (in some natural ordering, say, the lexicographic one).\n\\end{itemize}\nIn this way we get $\\KP(x)\\le \\KP(A)+\\log \\#A$ for every element $x$ of arbitrary finite set $A$. This inequality holds with $O(1)$-precision. If we replace the prefix complexity by the plain version, we can say that $\\KS(x)\\le\\KS(A)+\\log\\#A$ with precision $O(\\log n)$ for every string $x$ of length at most $n$: we may assume without loss of generality that both terms in the right hand side are at most $n$, otherwise the inequality is trivial.\n\nThe ``quality'' of a statistical model $P$ for a string $x$ can be measured by the difference between sides of this inequality: for a good model the ``two-part description'' should be almost minimal. We come to the following definition:\n\n\\begin{definition}\nThe \\emph{optimality deficiency} of a distribution $P$ considered as the model for a string $x$ is the difference\n$$\n\\delta(x,P)=(\\KP(P)+(-\\log P(x)))-\\KP(x).\n$$\n\\end{definition}\nAs we have seen, $\\delta(x,P)\\ge 0$ with $O(1)$-precision.\n\nIf $P$ is a uniform distribution on a set $A$, the optimality deficiency $\\delta(x,P)$ will also be denoted by $\\delta(x,A)$, and\n$$\n\\delta(x,A)=(\\KP(A)+\\log\\#A) - \\KP(x).\n$$\nThe following proposition shows that we may restrict our attention to finite sets as models (with $O(\\log n)$-precision):\n\n\\begin{proposition}\\label{prop:models-to-sets-2}\nLet $P$ be a distribution considered as a model for some string $x$ of length~$n$. Then there exists a finite set $A$ such that\n$$\n\\KP(A)\\le \\KP(P)+O(\\log n);\\quad \\log\\#A\\le -\\log P(x)+O(1) \\eqno(*)\n$$\n\\end{proposition}\n\nThis proposition will be used in many arguments, since it is often easier to deal with sets as statistical models (instead of distributions). Note that the inequalities~$(*)$ evidently imply that\n$$\n\\delta(x,A)\\le \\delta(x,P)+O(\\log n),\n$$\nso arbitrary distribution $P$ may be replaced by a uniform one ($U_A$) with a logarithmic-only change in the optimality deficiency.\n\n\\begin{proof}\nWe use the same construction as in Proposition~\\ref{prop:models-to-sets-1}. Let $2^{-k}$ be the maximal power of $2$ such that $2^{-k}\\le P(x)$, and let $A=\\{x\\mid P(x)\\ge 2^{-k}\\}$. Then $k=-\\log P(x)+O(1)$. We may assume that $k=O(n)$: if $k$ is much bigger than $n$, then $\\delta(x,P)$ is also bigger than $n$ (since the complexity of $x$ is bounded by $n+O(\\log n)$), and in this case the statement is trivial (let $A$ be the set of all $n$-bit strings).\n\nNow we see that that $A$ is determined by $P$ and $k$, so $\\KP(A)\\le \\KP(P)+\\KP(k)\\le \\KP(P)+O(\\log n)$. Note also that $\\#A \\le 2^k$, so $\\log\\# A\\le -\\log P(x)+O(1)$.\n\\end{proof}\n\nLet us note that in a more general setting~\\cite{milovanov-stacs} where we consider several strings as outcomes of the repeated experiment (with independent trials) and look for a model that explains all of them, a similar result is not true: not every probability distribution can be transformed into a uniform one.\n\n\\subsection{Optimality and randomness deficiencies}\n\nNow we have two ``quality measures'' for a statistical model $P$: the randomness deficiency $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ and the optimality deficiency $\\delta(x,P)$. They are related:\n\n\\begin{proposition}\\label{prop:randomness-optimality}\n$$d(x\\mskip 1.5mu | \\mskip 1.5 mu P)\\le \\delta(x,P)$$\nwith $O(1)$-precision.\n\\end{proposition}\n\n\\begin{proof}\nBy definition\n\\begin{align*}\nd(x\\mskip 1.5mu | \\mskip 1.5 mu P) &= - \\log P(x) - \\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P);\\\\\n\\delta(x,P)&= -\\log P(x) + \\KP(P) - \\KP(x).\n\\end{align*}\nIt remains to note that $\\KP(x)\\le \\KP(x,P)\\le \\KP(P)+\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ with $O(1)$-precision.\n\\end{proof}\n\nCould $\\delta(x,P)$ be significantly larger than $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)$? Look at the proof above: the second inequality $\\KP(x,P)=\\KP(P)+\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P)$ is an equality with logarithmic precision. Indeed, the exact formula (Levin--G\\'acs formula for the complexity of a pair with $O(1)$-precision) is\n$$\n\\KP(x,P)=\\KP(P)+\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu P,\\KP(P)).\n$$\nHere the term $\\KP(P)$ in the condition changes the complexity by $O(\\log \\KP(P))$, and we may ignore models $P$ whose complexity is much greater than the complexity of $x$.\n\nOn the other hand, in the first inequality the difference between $\\KP(x,P)$ and $\\KP(x)$ may be significant. This difference equals $\\KP(P\\mskip 1.5mu | \\mskip 1.5 mu x)$ with logarithmic accuracy and, if it is large, then $\\delta(x,P)$ is much bigger than $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)$. The following example shows that this is possible. In this example we deal with sets as models.\n\n\\begin{example}\\label{ex:stochasticity-optimality}\nConsider an incompressible string $x$ of length $n$, so $\\KP(x)=n$ (all equalities with logarithmic precision). A good model for this string is the set $A$ of all $n$-bit strings. For this model we have $\\#A=2^n$, $\\KP(A)=0$ and $\\delta(x,A)=n+0-n=0$ (all equalities have logarithmic precision). So $d(x\\mskip 1.5mu | \\mskip 1.5 mu P)=0$, too. Now we can change the model by excluding some other $n$-bit string. Consider a $n$-bit string $y$ that is incompressible and independent of $x$: this means that $\\KP(x,y)=2n$. Let $A'$ be $A\\setminus\\{y\\}$.\n\nThe set $A'$ contains $x$ (since $x$ and $y$ are independent, $y$ differs from $x$). Its complexity is $n$ (since it determines $y$). The optimality deficiency is then $n + n - n =n$, but the randomness deficiency is still small: $d(x\\mskip 1.5mu | \\mskip 1.5 mu A')=\\log \\#A'-\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu A')= n - n = 0$ (with logarithmic precision). To see why $\\KP(A'\\mskip 1.5mu | \\mskip 1.5 mu x)=n$, note that $x$ and $y$ are independent, and the set $A'$ has the same information as $(n,y)$.\n\\end{example}\n\nOne of the main results of this section (Theorem~\\ref{th:deficiencies}) clarifies the situation: it implies that if optimality deficiency of a model is significantly larger than its randomness deficiency, then this model can be improved and another model with better parameters can be found. More specifically, the complexity of the new model is smaller than the complexity of the original one while both the randomness deficiency and optimality deficiency of the new model are not worse than the randomness deficiency of the original one. This is one of the main results of algorithmic statistics, but first let us explore systematically the properties of two-part descriptions.\n\n\\subsection{Trade-off between complexity and size of a model}\n\\label{sub:trade-off}\n\nIt is convenient to consider only models that are sets (=uniform distribution on sets). We will call them \\emph{descriptions}. Note that by Propositions~\\ref{prop:models-to-sets-1} and~\\ref{prop:models-to-sets-2} this restriction does not matter much since we ignore logarithmic terms. For a given string $x$ there are many different descriptions: we can have a simple large set containing $x$, and at the same time some more complicated, but smaller one. In this section we study the trade-off between these two parameters (complexity and size).\n\n\\begin{definition}\\label{def:px}\nA finite set $A$ is an $(i*j)$-description\\footnote{This notation may look strange; however, we speak so often about finite sets of complexity at most $i$ and cardinality at most $2^j$ that we decided to introduce some short name and notation for them.} of $x$ if $x\\in A$, complexity $\\KP(A)$ is at most $i$, and $\\log\\#A\\le j$. For a given $x$ we consider the set $P_x$ of all pairs $(i,j)$ such that $x$ has some $(i*j)$-description; this set will be called \\emph{the profile} of $x$.\n\\end{definition}\n\nInformally speaking, an $(i*j)$-description for $x$ consists of two parts: first we spend $i$ bits to specify some finite set $A$ and then $j$ bits to specify $x$ as an element of $A$.\n\nWhat can be said about $P_x$ for a string $x$ of length $n$ and complexity $k=\\KP(x)$? By definition, $P_x$ is closed upwards and contains the points $(0,n)$ and $(k,0)$. Here we omit terms $O(\\log n)$: more precisely, we have a $(O(\\log n)*n)$-description that consists of all strings of length $n$, and a $((k+O(1))*0)$-description $\\{x\\}$. Moreover, the following proposition shows that we can move the information from the second part of the description into its first part (leaving the total length almost unchanged). In this way we make the set smaller (the price we pay is that its complexity increases).\n\n\\begin{proposition}[\\cite{kolm,gtv,shen99}]\n \\label{prop:description-shift}\nLet $x$ be a string and $A$ be a finite set that contains $x$. Let $s$ be a non-negative integer such that $s\\le \\log\\# A$. Then there exists a finite set $A'$ containing $x$ such that $\\#A' \\le \\# A\/ 2^s$ and $\\KP(A')\\le \\KP(A)+s + O(\\log s)$.\n\\end{proposition}\n\n\\begin{proof}\nList all the elements of $A$ in some (say, lexicographic) order. Then we split the list into $2^s$ parts (first $\\#A\/2^s$ elements, next $\\#A\/2^s$ elements etc.; we omit evident precautions for the case when $\\#A$ is not a multiple of $2^s$). Then let $A'$ be the part that contains $x$. It has the required size. To specify $A'$, it is enough to specify $A$ and the part number; the latter takes at most $s$ bits. (The logarithmic term is needed to make the encoding of the part number self-delimiting.)\n\\end{proof}\n\nThis statement can be illustrated graphically. As we have said, the set $P_x$ is ``closed upwards'' and contains with each point $(i,j)$ all points on the right (with bigger $i$) and on the top (with bigger $j$). It contains points $(0,n)$ and $(\\KP(x),0)$; Proposition~\\ref{prop:description-shift} says that we can also move down-right adding $(s,-s)$ (with logarithmic precision). We will see that movement in the opposite direction is not always possible. So, having two-part descriptions with the same total length, we should prefer the one with bigger set (since it always can be converted into others, but not vice versa).\n\nThe boundary of $P_x$ is some curve connecting the points $(0,n)$ and $(k,0)$. This curve (introduced by Kolmogorov in 1970s, see~\\cite{kolmmmo}) never gets into the triangle $i+j< \\KP(x)$ and always goes down (when moving from left to right) with slope at least $-1$ or more.\n\n\\begin{figure}[h]\n\\begin{center}\\includegraphics{mdl-e-1.pdf}\\end{center}\n\\caption{The set $P_x$ and its boundary curve}\\label{mdl.1.eps}\n\\end{figure}\n\nThis picture raises a natural question: which boundary curves are possible and which are not? Is it possible, for example, that the boundary goes along the dotted line on Figure~\\ref{mdl.1.eps}? The answer is positive: take a random string of desired complexity and add trailing zeros to achieve desired length. Then the point $( 0,\\KP(x))$ (the left end of the dotted line) corresponds to the set $A$ of all strings of the same length having the same trailing zeros. We know that the boundary curve cannot go down slower than with slope $-1$ and that it lies above the line $i+j=\\KP(x)$, therefore it follows the dotted line (with logarithmic precision).\n\nA more difficult question: is it possible that the boundary curve starts from $( 0,n)$, goes with the slope $-1$ to the very end and then goes down rapidly to $(\\KP(x),0)$ (Figure~\\ref{mdl.2}, the solid line)? Such a string $x$, informally speaking, would have essentially only two types of statistical explanations: a set of all strings of length $n$ (and its parts obtained by Proposition~\\ref{prop:description-shift}) and the exact description, the singleton $\\{x\\}$.\n\n\\begin{figure}[h]\n\\begin{center}\\includegraphics{mdl-e-2.pdf}\\end{center}\n\\caption{Two opposite possibilities for a boundary curve}\\label{mdl.2}\n\\end{figure}\n\nIt turns out that not only these two opposite cases are possible, but also all intermediate curves (provided they decrease with slope $-1$ or faster, and are simple enough), at least with logarithmic precision. More precisely, the following statement holds:\n\n\\begin{thm}[\\cite{vv}]\\label{stat-any-curve}\nLet $k\\le n$ be two integers and let $t_0 > t_1 > \\ldots > t_k$ be a strictly decreasing sequence of integers such that $t_0\\le n$ and $t_k=0$; let $m$ be the complexity of this sequence. Then there exists a string $x$ of complexity $k+O(\\log n)+O(m)$ and length $n+O(\\log n)+O(m)$ for which the boundary curve of $P_x$ coincides with the line $(0,t_0)$--$(1,t_1)$--\\ldots--$(k,t_k)$ with $O(\\log n)+O(m)$ precision: the distance between the set $P_x$ and the set $T=\\{( i,j)\\mid (it_i)\\}$ is bounded by $O(\\log n)+O(m)$.\n\\end{thm}\n\n(We say that the distance between two subsets $P,Q\\subset\\mathbb{Z}^2$ is at most $\\varepsilon$ if $P$ is contained in the $\\varepsilon$-neighborhood of $Q$ and vice versa.)\n\n\\begin{proof}\nFor every $i$ in the range $0\\ldots k$ we list all the sets of complexity at most $i$ and size at most $2^{t_i}$. For a given $i$ the union of all these sets is denoted by $S_i$. It contains at most $2^{i+t_i}$ elements. (Here and later we omit constant factors and factors polynomial in $n$ when estimating cardinalities, since they correspond to $O(\\log n)$ additive terms for lengths and complexities.) Since the sequence $t_i$ strictly decreases (this corresponds to slope $-1$ in the picture), the sums $i+t_i$ do not increase, therefore each $S_i$ has at most $2^{t_0}\\le 2^n$ elements. The union of all $S_i$ therefore also has at most $2^n$ elements (up to a polynomial factor, see above). Therefore, we can find a string of length $n$ (actually $n+O(\\log n)$) that does not belong to any $S_i$. Let $x$ be a first such string in some order (e.g., in the lexicographic order).\n\nBy construction, the set $P_x$ lies above the curve determined by $t_i$. So we need to estimate the complexity of $x$ and prove that $P_x$ follows the curve (i.e., that $T$ is contained in the neighborhood of $P_x$).\n\nLet us start with the upper bound for the complexity of $x$. The list of all objects of complexity at most $k$ plus the full table of their complexities have complexity $k+O(\\log k)$, since it is enough to know $k$ and the number of terminating programs of length at most~$k$. Except for this list, to specify $x$ we need to know $n$ and the sequence $t_0,\\ldots,t_k$, whose complexity is $m$.\n\nThe lower bound: the complexity of $x$ cannot be less than $k$ since all the singletons of this complexity were excluded (via $S_k$).\n\nIt remains to show that for every $i\\le k$ we can put $x$ into a set $A$ of complexity $i$ (or slightly bigger) and size $2^{t_i}$ (or slightly bigger). For this we enumerate a sequence of sets of correct size and show that one of the sets will have the required properties; if this sequence of sets is not very long, the complexity of its elements is bounded. Here are the details.\n\nWe start by taking the first $2^{t_i}$ strings of length $n$ as our first set $A$. Then we start enumerating all finite sets of complexity at most $j$ and of size at most $2^{t_j}$ for all $j=0,\\ldots,k$, and get an enumeration of all sets $S_j$. Recall that all elements of all $S_j$ should be deleted (and the minimal remaining element should eventually be $x$). So, when a new set of complexity at most $j$ and of size at most $2^{t_j}$ appears, all its elements are included in $S_j$ and deleted. Until all elements of $A$ are deleted, we have nothing to worry about, since $A$ is covering the minimal remaining element. If (and when) all elements of $A$ are deleted, we replace $A$ by a new set that consists of first $2^{t_i}$ undeleted (yet) strings of length $n$. Then we wait again until all the elements of this new $A$ are deleted, if (and when) this happens, we take $2^{t_i}$ first undeleted elements as new $A$, etc.\n\nThe construction guarantees the correct size of the sets and that one of them covers $x$ (the minimal non-deleted element). It remains to estimate the complexity of the sets we construct in this way.\n\nFirst, to start the process that generates these sets, we need to know the length $n$ (actually something logarithmically close to $n$) and the sequence $t_0,\\ldots,t_k$. In total we need $m+O(\\log n)$ bits. To specify each version of $A$, we need to add its version number. So we need to show that the number of different $A$'s that appear in the process is at most $2^i$ or slightly bigger.\n\nA new set $A$ is created when all the elements of the old $A$ are deleted. These changes can be split into two groups. Sometimes a new set of complexity $j$ appears with $j\\le i$. This can happen only $O(2^i)$ times since there are at most $O(2^i)$ sets of complexity at most $i$. So we may consider the other changes (excluding the first changes after each new large set was added). For those changes all the elements of $A$ are gone due to elements of $S_j$ with $j>i$. We have at most $2^{j+t_j}$ elements in $S_j$. Since $t_j+j\\le t_i+i$, the total number of deleted elements only slightly exceeds $2^{t_i+i}$, and each set $A$ consists of $2^{t_i}$ elements, so we get about $2^i$ changes of $A$.\n %\n\\end{proof}\n\n\\begin{remark}\nIt is easy to modify the proof to get a string $x$ of length exactly $n$. Indeed, we may consider slightly smaller bad sets: decreasing the logarithms of their sizes by $O(\\log n)$, we can guarantee that the total number of elements in all bad sets is less than $2^n$. Then there exists a string of length $n$ that does not belong to bad sets. In this way the distance between $T$ and $P_x$ may increase by $O(\\log n)$, and this is acceptable.\n\\end{remark}\n\nTheorem~\\ref{stat-any-curve} shows that the value of the complexity of $x$ does not describe the properties of $x$ fully; different strings of the same complexity $x$ can have different boundary curves of $P_x$. This curve can be considered as an ``infinite-dimensional'' characterization of $x$.\n\nStrings $x$ with minimal possible $P_x$ (Figure~\\ref{mdl.2}, the upper curve) may be called \\emph{antistochastic}. They have quite unexpected properties. For example, if we replace some bits of an antistochastic string $x$ by stars (or some other symbols indicating erasures) leaving only $\\KP(x)$ non-erased bits, then the string $x$ can be reconstructed from the resulting string $x'$ with logarithmic advice, i.e., $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu x')=O(\\log n)$. This and other properties of antistochastic strings were discovered in~\\cite{milovanov-antistochastic}.\n\n\\subsection{Optimality and randomness deficiency}\\label{subsec:opt-rand}\n\nIn this section we establish the connection between optimality and randomness deficiency. As we have seen, the optimality deficiency can be bigger than the randomness deficiency (for the same description), and the difference is\n$\n\\delta(x,A)-d(x\\mskip 1.5mu | \\mskip 1.5 mu A)=\\KP(A)+\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu A) - \\KP(x).\n$\nThe Levin--G\\'acs formula for the complexity of pair ($\\KP(u,v)=\\KP(u)+\\KP(v\\mskip 1.5mu | \\mskip 1.5 mu u)$ with logarithmic precision, for $O(1)$-precision one needs to add $\\KP(u)$ in the condition, but we ignore logarithmic size terms anyway) shows that the difference in question can be rewritten as\n$$\n\\delta(x,A)-d(x\\mskip 1.5mu | \\mskip 1.5 mu A)=\\KP(A,x)-\\KP(x)=\\KP(A\\mskip 1.5mu | \\mskip 1.5 mu x).\n$$\nSo if the difference between deficiencies for some $(i*j)$-description $A$ of $x$ is big, then $\\KP(A\\mskip 1.5mu | \\mskip 1.5 mu x)$ is big. All the $(i*j)$-descriptions of $x$ can be enumerated if $x$, $i$, and $j$ are given. So the large value of $\\KP(A\\mskip 1.5mu | \\mskip 1.5 mu x)$ for some $(i*j)$-description $A$ means that there are many $(i*j)$-descriptions of $x$, otherwise $A$ can be reconstructed from $x$ by specifying $i,j$ (requires $O(\\log n)$ bits) and the ordinal number of $A$ in the enumeration. We will prove that if there are many $(i*j)$-descriptions for some $x$, then there exist a description with better parameters.\n\nNow we explain this in more detail. Let us start with the following remark. Consider all strings that have $(i*j)$-descriptions for some fixed $i$ and $j$. They can be enumerated in the following way: we enumerate all finite sets of complexity at most $i$, select those sets that have size at most $2^j$, and include all elements of these sets into the enumeration. In this construction\n\\begin{itemize}\n\\item the complexity of the enumerating algorithm is logarithmic (it is enough to know $i$ and $j$);\n\\item we enumerate at most $2^{i+j}$ elements;\n\\item the enumeration is divided into at most $2^i$ ``portions'' of size at most $2^j$.\n\\end{itemize}\nIt is easy to see that any other enumeration process with these properties enumerates only objects that have $(i*j)$-descriptions (again with logarithmic precision). Indeed, each portion is a finite set that can be specified by its ordinal number and the enumeration algorithm, the first part requires $i+O(\\log i)$ bits, the second is of logarithmic size according to our assumption.\n\n\\begin{remark}\\label{rem:portion}\nThe requirement about the portion size is redundant. Indeed, we can change the algorithm by splitting large portions into pieces of size $2^j$ (the last piece may be incomplete). This, of course, increases the number of portions, but if the total number of enumerated elements is at most $2^{i+j}$, then this splitting adds at most $2^i$ pieces. This observation looks (and is) trivial, still it plays an important role in the proof of the following proposition.\n\\end{remark}\n\n\\begin{proposition}\\label{prop:improving-descriptions}\nIf a string $x$ of length $n$ has at least $2^k$ different $(i,j)$-descriptions, then $x$ has some $(i*(j-k))$-description and even some $((i-k)*j)$-description.\n\\end{proposition}\n\nAgain we omit logarithmic term: in fact one should write $((i+O(\\log n))*(j-k+O(\\log n)))$, etc. The word ``even'' in the statement refers to Proposition~\\ref{prop:description-shift} that shows that indeed the second claim is stronger.\n\n\\begin{proof}\nConsider the enumeration of all objects having $(i*j)$-descriptions in $2^i$ portions of size $2^j$ (we ignore logarithmic additive terms and respective polynomial factors) as explained above. After each portion (i.e., new $(i*j)$-description) appears, we count the number of descriptions for each enumerated object and select objects that have at least $2^k$ descriptions. Consider a new enumeration process that enumerates only these ``rich'' objects (rich = having many descriptions). We have at most $2^{i+j-k}$ rich objects (since they appear in the list of size $2^{i+j}$ with multiplicity $2^k$), enumerated in $2^i$ portions (new portion of rich objects may appear only when a new portion appears in the original enumeration). So we apply the observation above to conclude that all rich objects have $(i*(j-k))$-descriptions.\n\nTo get the second (stronger) statement we need to decrease the number of portions (while not increasing too much the number of enumerated objects). This can be done using the following trick: when a new rich object (having $2^k$ descriptions) appears, we enumerate not only rich objects, but also ``half-rich'' objects, i.e., objects that currently have at least $2^k\/2$ descriptions. In this way we enumerate more objects --- but only twice more. At the same time, after we dumped all half-rich objects, we are sure that next $2^k\/2$ new $(i*j)$-descriptions will not create new rich objects, so the number of portions is divided by $2^k\/2$, as required.\n\\end{proof}\n\nLet us say more accurately how we deal with logarithmic terms. We may assume that $i,j=O(n)$, otherwise the claim is trivial. Then we allow polynomial (in $n$) factors and $O(\\log n)$ additive terms in all our considerations.\n\n\\begin{remark}\nIf we unfold this construction, we see that new descriptions (of smaller complexity) are not selected from the original sequence of descriptions but constructed from scratch. In Section~\\ref{sec:restricted-type} we deal with much more complicated case where we restrict ourselves to descriptions from some class (say, Hamming balls). Then the proof given above does not work, since the description we construct is not a ball even if we start with ball descriptions. Still some other (much more ingenious) argument can be used to prove a similar result for the restricted case.\n\\end{remark}\n\nNow we are ready to prove the promised results (see the discussion after Example~\\ref{ex:stochasticity-optimality}).\n\n\\begin{thm}~\\label{thm:improving-descriptions}\nIf a string $x$ of length $n$ is $(\\alpha,\\beta)$-stochastic, then there exists some finite set $B$ containing $x$ such that $\\KP(B)\\le \\alpha+O(\\log n)$ and $\\delta(x,B)\\le \\beta+O(\\log n)$.\n\\end{thm}\n\n\\begin{proof}\nSince $x$ is $(\\alpha,\\beta)$-stochastic, there exists some finite set $A$ such that $\\KP(A)\\le\\alpha$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le \\beta$. Let $i=\\KP(A)$ and $j=\\log\\# A$, so $A$ is an $(i*j)$-description of $x$. We may assume without loss of generality that both $\\alpha$ and $\\beta$ (and therefore $i$ and $j$) are $O(n)$, otherwise the statement is trivial. The value $\\delta(x,A)$ may exceed $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$, as we have discussed at the beginning of this section. So we assume that\n$$\nk=\\delta(x,A)-d(x\\mskip 1.5mu | \\mskip 1.5 mu A)>0;\n$$\nif not, we can let $B=A$. Then, as we have seen, $\\KP(A\\mskip 1.5mu | \\mskip 1.5 mu x)\\ge k-O(\\log n)$, and there are at least $2^{k-O(\\log n)}$ different $(i*j)$-descriptions of $x$. According to Proposition~\\ref{prop:improving-descriptions}, there exists some finite set $B$ that is an $(i*(j-k+O(\\log n)))$-description of $x$. Its optimality deficiency $\\delta(x,B)$ is $(k-O(\\log n))$-smaller (compared to $A$) and therefore $O(\\log n)$-close to $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$.\t\n\\end{proof}\n\nIn this argument we used the simple part of Proposition~\\ref{prop:improving-descriptions}. Using the stronger statement about complexity decrease, we get the following result:\n\n\\begin{thm}[\\cite{vv}]\\label{th:deficiencies}\nLet $A$ be a finite set containing a string $x$ of length $n$ and let $k=\\delta(x,A)-d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$. Then there is a finite set $B$ containing $x$ such that $\\KP(B)\\le \\KP(A)-k+O(\\log n)$ and $\\delta(x,B)\\le d(x\\mskip 1.5mu | \\mskip 1.5 mu A)+O(\\log n)$.\n\\end{thm}\n\n\\begin{proof}\nIndeed, if $B$ is an $((i-k)*j)$-description of $x$ (up to logarithmic terms, as usual), then its optimality deficiency is again $(k-O(\\log n))$-smaller (compared to $A$) and therefore $O(\\log n)$-close to $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$.\n\\end{proof}\n\nNote that the statement of the theorem implies that $d(x\\mskip 1.5mu | \\mskip 1.5 mu B)\\le d(x\\mskip 1.5mu | \\mskip 1.5 mu A)+O(\\log n)$.\n\nTheorem~\\ref{thm:improving-descriptions} and Proposition~\\ref{prop:randomness-optimality} show that we can replace the randomness deficiency in the definition of $(\\alpha,\\beta)$-stochastic strings by the optimality deficiency (with logarithmic precision). More specifically, for every string $x$ of length $n$ consider the sets\\label{def:qx}\n$$\nQ_x=\\{( \\alpha,\\beta)\\mid \\text{$x$ is\n$(\\alpha,\\beta)$-stochastic}\\},\n$$\nand\n$$\n\\tilde Q_x=\\{( \\alpha,\\beta)\\mid \\text{there exists $A\\ni x$ with }\n\\KP(A)\\le \\alpha,\\ \\delta(x,A)\\le\\beta)\\}.\n$$\nThen these sets are at most $O(\\log n)$ apart (each is contained in the $O(\\log n)$-neighborhood of the other one).\n\nThis remark, together with the existence of antistochastic strings of given complexity and length, allows us to improve the result about the existence of non-stochastic objects (Proposition~\\ref{prop:existence-nonstochastic}).\n\n\\begin{proposition}[\\protect{\\cite[Theorem IV.2]{gtv}}]\\label{prop:existence-nonstochastic-strong}\nFor some $c$ and for all $n$: if $\\alpha+\\beta\\KP(x)$ both sets contain all pairs with first component $\\alpha$.}\n\\end{thm}\n\n\\begin{figure}[h]\n\\begin{center}\\includegraphics{mdl-e-8.pdf}\\end{center}\n\\caption{The set $P_x$ and the boundary of the set $Q_x$ (bold dotted line); on every vertical line two intervals have the same length.}\\label{mdl.8}\n\\end{figure}\n\nAs usual, this statement is true with logarithmic accuracy: the distance between the image of the set $Q_x$ under this transformation and the set $P_x$ is claimed to be $O(\\log n)$ for string $x$ of length $n$.\n\n\\begin{proof}\nAs we have seen, we may use the optimality deficiency instead of randomness deficiency, i.e., use the set $\\tilde Q_x$ in place of $Q_x$. The preimage of the pair $(i,j)$ under our affine transformation is the pair $(i,i+j-\\KP(x))$. Hence we have to prove that a pair $(i,j)$ is in $P_x$ if and only if the pair $(i,i+j-\\KP(x))$ is in $\\tilde Q_x$. Note that $\\KP(A)=i$ and $\\log\\#A=j$ is equivalent to $\\KP(A)=i$ and $\\delta(x,A)=i+j-\\KP(x)$ just by definition of $\\delta(x,A)$. (See Figure~\\ref{mdl.8}: the optimality deficiency of a description $A$ with $\\KP(A)=i$ and $\\log\\#A =j$ is the vertical distance between $(i,j)$ and the dotted line.)\n\nBut there is some technical problem: in the definition of $P_x$ we used inequalities $\\KP(A)\\le i$ and $\\log\\#A\\le j$, not the equalities $\\KP(A)=i$ and $\\log\\#A=j$. The same applies to the definition of $\\tilde Q_x$. So we have two sets that correspond to each other, but their $\\le$-closures could be different. Obviously, $\\KP(A)\\le i$ and $\\log\\#A\\le j$ imply $\\KP(A)\\le i$ and $\\KP(A)+\\log\\#A -\\KP(x)\\le i+j-\\KP(x)$, but not vice versa.\n\nIn other words, the set of pairs $(\\KP(A),\\log\\#A)$ satisfying the latter inequalities (see the right set on Figure~\\ref{mdl-10-11}) is bigger than the set of pairs $(\\KP(A),\\log\\#A)$ satisfying the former inequalities (see the left set on Figure~\\ref{mdl-10-11}).\n %\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=1]{mdl-e-10.pdf} \\qquad\\qquad\n\\includegraphics[scale=1]{mdl-e-11.pdf}\n\\end{center}\n\\caption{The left picture shows (for given $i$ and $j$) the set of all pairs $(\\KP(A),\\log\\#A)$ such that $\\KP(A)\\le i$ and $\\log\\#A\\le j$; the right picture shows the pairs $(\\KP(A),\\log\\#A)$ such that $\\KP(A)\\le i$ and $\\delta(x,A)\\le i+j-\\KP(x)$.}\\label{mdl-10-11}\n\\end{figure}\n %\nNow Proposition~\\ref{prop:description-shift} helps: we may use it to convert any set with parameters from the right region into a set with parameters from the left region.\n\\end{proof}\n\n\n\n\\begin{remark}\nLet us stress again that Theorem~\\ref{thm:improving-descriptions} claims only that the \\emph{existence} of a set $A\\ni x$ with $\\KP(A)\\le\\alpha$ and $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le\\beta$ is equivalent to the existence of a set $B\\ni x$ with $\\KP(B)\\le\\alpha$ and $\\delta(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le\\beta$ (with logarithmic accuracy). The theorem does \\emph{not} claim that for \\emph{every} set $A\\ni x$ with complexity at most $\\alpha$ the inequalities $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le \\beta$ and $\\delta(x,A)\\le \\beta$ are equivalent (with logarithmic accuracy). Indeed, the Example~\\ref{ex:stochasticity-optimality} shows that this is not true: the first inequality does not imply the second one in general case. However, Theorems~\\ref{thm:improving-descriptions} and~\\ref{th:deficiencies} show that this can happen only for non-minimal descriptions (for which the description with smaller complexity and the same optimality deficiency) exists. Later we will see that all the minimal descriptions of the same (or almost the same) complexity have almost the same information. Moreover, if $A$ and $B$ are minimal descriptions and the complexity of $A$ is less than that of $B$ then $\\KS(A\\mskip 1.5mu | \\mskip 1.5 mu B)$ is small.\n\\end{remark}\n\nFor the people with taste for philosophical speculations the meaning of Theorems~\\ref{thm:improving-descriptions} and~\\ref{th:deficiencies} can be advertised as follows. Imagine several scientists that compete in providing a good explanation for some data $x$. Each explanation is a finite set $A$ containing $x$ together with a program $p$ that computes $A$.\n\nHow should we compare different explanations? We want the randomness deficiency $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$ of $x$ in $A$ to be negligible (no features of $x$ remain unexplained). Among these descriptions we want to find the simplest one (with the shortest $p$). That is, we look for a set $A$ corresponding to the point where the bold dotted line on Fig.~\\ref{mdl.8} touches the horizontal axis. (In fact, there is always some trade-off between the parameters, not the specific exact point where the curve touches the horizontal axis, but we want to keep the discussion simple though imprecise.)\n\nHowever, this approach meets the following obstacle: we are unable to compute randomness deficiency $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$. Moreover, the inventor of the model $A$ has no ways to convince us that the deficiency is indeed negligible if it is the case (the function $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)$ is not even upper semicomputable). What could be done? Instead, we may look for an explanation with (almost) minimal sum $\\log\\#A+|p|$ (minimum description length principle). Note that this quantity is known for competing explanation proposals. Theorems~\\ref{thm:improving-descriptions} and~\\ref{th:deficiencies} provide the connection between these two approaches.\n\nReturning to mathematical language, we have seen in this section that two approaches (based on $(i*j)$-descriptions and $(\\alpha,\\beta)$-stochasticity) produce essentially the same curve, though in different coordinates. The other ways to get the same curve will be discussed in Sections~\\ref{sec:bcl} and~\\ref{sec:depth}.\n\n\n\\subsection{Historical remarks}\nThe idea to consider $(i*j)$-descriptions with optimal parameters can be traced back to Kolmogorov. There is a short record for his talk given in 1974~\\cite{kolmmmo}. Here is the (full) translation of this note:\n\\begin{quote}\nFor every constructive object $x$ we may consider a function $\\Phi_x(k)$ of an integer argument $k\\ge 0$ defined as a logarithm of the minimal cardinality of a set of complexity at most $k$ containing $x$. If $x$ itself has a simple definition, then $\\Phi_x(1)$ is equal to one [a typo: cardinality equals $1$, and logarithm equals $0$] already for small $k$. If such a simple definition does not exist, $x$ is ``random'' in the negative sense of the word ``random''. But $x$ is positively ``probabilistically random'' only if the function $\\Phi$ has a value $\\Phi_0$ for some relatively small $k$ and then decreases approximately as $\\Phi(k)=\\Phi_0 -(k-k_0)$. [This corresponds to approximate $(k_0,0)$-stochasticity.]\n\\end{quote}\n\nKolmogorov also gave a talk in 1974~\\cite{kolm}; the content of this talk was reported by Cover~\\cite[Section 4, page 31]{cover1985}. Here $l(p)$ stands for the length of a binary string $p$ and $|S|$ stands for the cardinality of a set $S$.\n %\n\\begin{quote}\n4. \\textbf{Kolmogorov's $H_k$ Function}\n\nConsider the function $H_k\\colon \\{0,1\\}^k\\to N$, $H_k(x)=\\min_{p\\colon l(p)\\le k} \\log |S|$, where the minimum is taken over all subsets $S\\subseteq\\{0,1\\}^n$, such that $x\\in S$, $U(p)=S$, $l(p)\\le k$. This definition was introduces by Kolmogorov in a talk at the Information Symposium, Tallinn, Estonia, in 1974. Thus $H_k(x)$ is the log of the size of the smallest set containing $x$ over all sets specifiable by a program of $k$ or fewer bits. Of special interest is the value\n$$\nk^*(x)=\\min\\{k\\colon H_k(x)+k=K(x)\\}.\n$$\nNote that $\\log |S|$ is the maximal number of bits necessary to describe an arbitrary element $x\\in S$. Thus a program for $x$ can be written in two stages: ``Use $p$ to print the indicator function for $S$; the desired sequence is the $i$th sequence in a lexicographic ordering of the elements of this set''. This program has length $l(p)+\\log |S|$, and $k^*(x)$ is the length of the shortest program $p$ for which this $2$-stage description is as short as the best $1$-stage description $p^*$. We observe that $x$ must be maximally random with respect to $S$ --- otherwise the $2$-stage description could be improved, contradicting the minimality of $K(x)$. Thus $k^*(x)$ and its associated program $p$ constitute a minimal sufficient description for $x$. $\\langle\\ldots\\rangle$\n\nArguments can be provided to establish that $k^*(x)$ and its associated set $S^*$ describe all of the ``structure'' of $x$. The remaining details about $x$ are conditionally maximally complex. Thus $pp^{**}$, the program for $S^*$, plays the role of a sufficient statistic.\n\n\\end{quote}\n %\nIn both places Kolmogorov speaks about the place when the boundary curve of $P_x$ reaches its lower bound determined by the complexity of $x$.\n\nLater the same ideas were rediscovered and popularized by many people. Koppel in~\\cite{koppel87} reformulates the definition using total algorithms. Instead of a finite set $A$ he considered a total program $P$ that terminates on all strings of some length. The two-part description of some $x$ is then formed by this program $P$ and the input $D$ for this program that is mapped to $x$. In our terminology this corresponds to the set $A$ of all values of $P$ on the strings of the same length as $D$. He writes then~\\cite[p.~1089]{koppel87}\n %\n\\begin{quote}\n\\textbf{Definition 3.} The $c$-sophistication of a finite string $S$ [is defined as] $$\\text{SOPH}_c(S)=\\min\\{|P|\\mid \\exists D\\text{ s. t. } (P,D)\\text{ is a $c$-minimal description of $\\alpha$}\\}.$$\n %\n\\end{quote}\n %\nThere is a typo in this paper: $S$ should be replaced by $\\alpha$ (two times). Before in Definition 1 the description is called $c$-minimal if $|P|+|D|\\le H(\\alpha)+c$ (here $P$ and $D$ are the program and and its input, respectively, $H$ stands for complexity).\n\nThough this paper (as well as the subsequent papers~\\cite{koppel,koppel-atlan}) is not technically clear (e.g., it does not say what are the requirements for the algorithm $U$ used in the definition, and in~\\cite{koppel,koppel-atlan} only universality is required, which is not enough: if $U$ is not optimal, the definition does not make sense), the philosophic motivation for this notion is explained clearly~\\cite[p.~1087]{koppel87}:\n %\n\\begin{quote}\nThe total complexity of an object is defined as the size of its most concise description. The total complexity of an object can be large while its ``meaningful'' complexity is low; for example, a random object is by definition maximally complex but completely lacking in structure.\n\n$\\langle\\ldots\\rangle$ The ``static'' approach to the formalization of meaningful complexity is ``sophistication'' defined and discussed by Koppel and Atlan [reference to unpublished paper ``Program-length complexity, sophistication, and induction'' is given, but later a paper of same authors~\\cite{koppel-atlan} with a similar title appeared]. Sophistication is a generalization of the ``H-function'' or ``minimal sufficient statistic'' by Cover and Kolmogorov~$\\langle\\ldots\\rangle$ The sophistication of an object in the size of that part of that object which describes its structure, i.e. the aggregate of its projectible properties.\n\\end{quote}\n\nOne can also mention the formulation of ``minimal description length'' principle by Rissanen~\\cite{rissanen78}; the abstract of this paper says: ``Estimates of both integer-valued structure parameters and real-valued system parameters may be obtained from a model based on the shortest data description principle''; here ``integer-valued structure parameters'' may correspond to the choice of a statistical hypothesis (description set) while ``real-valued system parameters'' may correspond to the choice of a specific element in this set. The author then says that ``by finding the model which minimizes the description length one obtains estimates of both the integer-valued structure parameters and the real-valued system parameters''.\n\nWe do not try here to follow the development of these and similar ideas. Let us mention only that the traces of the same ideas (though even more vague) could be found in 1960s in the classical papers of Solomonoff~\\cite{sol1,sol2} who tried to use shortest descriptions for inductive inference (and, as a side product, gave the definition of complexity later rediscovered by Kolmogorov~\\cite{kolm65}). One may also mention a ``minimum message length principle'' that goes back to~\\cite{wallace-boulton68}; the idea of two-part description is explained in~\\cite{wallace-boulton68} as follows:\n\\begin{quote}\nIf the things are now classified then the measurements can be recorded by listing the following:\n\n1. The class to which each thing belongs.\n\n2. The average properties of each class.\n\n3. The deviations of each thing from the average properties of its parent class.\n\nIf the things are found to be concentrated in a small area of the region of each class in the measurement space then the deviations will be small, and with reference to the average class properties most of the information about a thing is given by naming the class to which it belongs. In this case the information may be recorded much more briefly than if a classification had not been used. We suggest that the best classification is that which results in the briefest recording of all the attribute information.\n\\end{quote}\nHere the ``class to which thing belongs'' corresponds to a set (statistical model, description in our terminology); the authors say that if this set is small, then only few bits need to be added to the description of this set to get a full description of the thing in question.\n\nThe main technical results of this sections (Theorems~\\ref{stat-any-curve},~\\ref{thm:improving-descriptions}, and~\\ref{th:deficiencies}) are taken from~\\cite{vv} (where some historical account is provided).\n\n\\section{Bounded complexity lists}\\label{sec:bcl}\n\nIn this section we show one more classification of strings that turns out to be equivalent (up to coordinate change) to the previous ones: for a given string $x$ and $m\\ge \\KS(x)$ we look how close $x$ is to the end in the enumeration of all strings of complexity at most $m$. For technical reasons it is more convenient to use plain complexity $\\KS(x)$ instead of the prefix version $\\KP(x)$. As we have mentioned, the difference between them is only logarithmic, and we mainly ignore terms of that size.\n\n\\subsection{Enumerating strings of complexity at most $m$}\n\nConsider some integer $m$, and all strings $x$ of (plain) complexity at most $m$. Let $\\Omega_m$ be the number of those strings. The following properties of $\\Omega_m$ are well known and often used (see, e.g., \\cite{bienvenu-desfontaines-shen}).\n\n\\begin{proposition}\\label{prop:omegas}\n\\leavevmode\n\\begin{itemize}\n\\item $\\Omega_m=\\Theta(2^m)$ (i.e., $c_12^m\\le \\Omega_m\\le c_2 2^m$ for some positive constants $c_1,c_2$ and for all~$m$;\n\\item $\\KS(\\Omega_m)=m+O(1)$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nThe number of strings of complexity at most $m$ is bounded by the total number of programs of length at most $m$, which is $O(2^m)$. On the other hand, if $\\Omega_m$ is an $(m-d)$-bit number, we can specify a string of complexity greater than $m$ using $m-d+O(\\log d)$ bits: first we specify $d$ in a self-delimiting manner using $O(\\log d)$ bits, and then append $\\Omega_m$ in binary. This information allows us to reconstruct $d$, then $m$ and $\\Omega_m$, then enumerate strings of complexity at most $m$ until we have $\\Omega_m$ of them (so all strings of complexity at most $m$ are enumerated), and then take the first string $x_m$ that has not been enumerated. As $m<\\KS(x_m)\\le m-d+O(\\log d)$, the value of $d$ is bounded by a constant and hence $\\Omega_m$ is an $(m-O(1))$-bit number.\n\nIn this argument the binary representation of $\\Omega_m$ can be replaced by its program, so $\\KS(\\Omega_m)\\ge m-O(1)$. The upper bound $m+O(1)$ is obvious, since $\\Omega_m=O(2^m)$.\n\\end{proof}\n\nGiven $m$, we can enumerate all strings of complexity at most $m$. How many steps needs the enumeration algorithm to produce all of them? The answer is provided by the so-called \\emph{busy beaver numbers}; let us recall their definition in terms of Kolmogorov complexity (see~\\cite[section 1.2.2]{usv} for details).\n\nBy definition, the number $B(m)$ is the maximal integer of complexity at most $m$. It is not hard to see that $\\KS(B(m))= m+O(1)$. Indeed, $\\KS(B(m))\\le m$ by definition. On the other hand, the complexity of the next number $B(m)+1$ is greater than $m$ and at the same time is bounded by $\\KS(B(m))+O(1)$.\n\nNote that $B(m)$ can be undefined for small $m$ (if there are no integers of complexity at most $m$) and that $B(m+1)\\ge B(m)$ for all $m$. For some $m$ this inequality may not be strict. This happen, for example, if the optimal algorithm used to define Kolmogorov complexity is defined only on strings of, say, even lengths; this restriction does not prevent it from being optimal, but then $B(2n)=B(2n+1)$ for all $n$, since there are no objects of complexity exactly $2n+1$. However, for some constant $c$ we have $B(m+c)>B(m)$ for all $m$. Indeed, consider a program $p$ of length at most $m$ that prints $B(m)$. Transform it to a program $p'$ that runs $p$ and then adds $1$ to the result. This program witnesses that $\\KS(B(m)+1)\\le m+c$ for some constant $c$. Hence $B(m+c)\\ge B(m)+1$.\n\nNow we define $B'(m)$ as follows. As we have said, the set of all strings of complexity at most $m$ can be enumerated given $m$. Fix some enumeration algorithm $A$ (with input $m$) and some computation model. Then let $B'(m)$ be the number of steps used by this algorithm to enumerate all the strings of complexity at most $m$.\n\n\\begin{proposition}\\label{prop:busy-beavers}\nThe numbers $B(m)$ and $B'(m)$ coincide up to $O(1)$-change in $m$. More precisely, we have\n$$\nB'(m)\\le B(m+c), \\qquad\nB(m)\\le B'(m+c)\n$$\nfor some $c$ and for all $m$.\n\\end{proposition}\n\n\\begin{proof}\nTo find $B'(m)$, it is enough to know $m$-bit binary string that represents $\\Omega_m$ (this string also determines $m$). Therefore $\\KS(B'(m))\\le m+c$ for some constant $c$. As $B(m+c)$ is the largest number of complexity $m+c$ or less, we have $B'(m)\\le B(m+c)$.\n\nOn the other hand, if some integer $N$ exceeding both $m$ and $B'(m)$ is given, we can run the enumeration algorithm $A$ within $N$ steps for each input smaller than $N$. Consider the first string that has not been enumerated. Its complexity is greater than $m$, so $\\KS(N)>m-c$ for some constant $c$. Thus the complexity of every number $N$ starting from $\\max\\{m,B'(m)\\}$ is greater than $m-c$, which means that $\\max\\{m,B'(m)\\}>B(m-c)$. It remains to note that for all large enough $m$ we have $m\\le B(m-c)$, as the complexity of $m$ is $O(\\log m)$. Thus for all large enough $m$ the number $B'(m)$ (and not $m$) must be bigger than $B(m-c)$. Replacing here $m$ by $m+c$ and increasing the constant $c$ if needed, we conclude that $B'(m+c)>B(m)$ for all $m$.\n %\n\\end{proof}\n\nA similar argument shows that $B(n)$ coincides (up to $O(1)$-change in the argument) with the maximal computation time of the universal decompressor (from the definition of plain Kolmogorov complexity) on inputs of size at most $m$, see~\\cite[section~1.2.2]{usv}\n\nThe next result says how many strings require long time to be enumerated.\n\n\\begin{proposition}\\label{prop:enumeration-tail}\nAfter $B'(m-s)$ steps of the enumeration algorithm on input $m$ there are $2^{s+O(\\log m)}$ strings that are not yet enumerated.\n\\end{proposition}\n\nWe assume that the algorithm enumerates strings (for every input $m$) without repetitions. Note also that here $B'$ can be replaced by $B$, since they differ at most by a constant change in the argument.\n\n\\begin{proof} To make the notation simpler\nwe omit $O(1)$- and $O(\\log m)$-terms in this argument. Given $\\Omega_{m-s}$, we can determine $B'(m-s)$. If we also know how many strings of complexity at most $m$ appear after $B'(m-s)$ steps, we can wait until that many strings appear and then find a string of complexity greater than $m$. If the number of remaining strings is smaller than $2^{s-O(\\log m)}$, we get a prohibitively short description of this high complexity string.\n\nOn the other hand, let $x$ be the last element that has been enumerated in $B'(m-s)$ steps. If there are significantly more than $2^s$ elements after $x$, say, at least $2^{s+d}$ for some $d$, we can split the enumeration in portions of size $2^{s+d}$ and wait until the portion containing $x$ appears. By assumption this portion is full. The number $N$ of steps needed to finish this portion is at least $B'(m-s)$ . This number $N$ and its successor $N+1$ can be reconstructed from the portion number that contains about $m-s-d$ bits. Thus the complexity of $N+1$ is at most $m-s-d+O(\\log m)$. Hence we have\n$$\nB(m-s-d+O(\\log m)) > N \\ge B'(m-s).\n$$\nBy Proposition~\\ref{prop:busy-beavers}\nwe can replace $B'$ by $B$ here:\n$$\nB(m-s-d+O(\\log m))> B(m-s).\n$$\n(with some other constant in $O$-notation). Since $B$ is a non-decreasing function, we get $d=O(\\log m)$.\n %\n\\end{proof}\n\n\\subsection{$\\Omega$-like numbers}\n\nG.~Chaitin introduced the ``Chaitin $\\Omega$-number''\n$\n\\Omega=\\sum_k \\mm(k);\n$\nit can also be defined as the probability of termination if the optimal prefix decompressor is applied to a random bit sequence~(see~\\cite[section 5.7]{usv}).%\n\\footnote{This number depends on the choice of the prefix decompressor, so it is not a specific number but a class of numbers. The elements of this class can be equivalently characterized as random lower semicomputable reals in $[0,1]$, see~\\cite[section~5.7]{usv}.}\n %\nThe numbers $\\Omega_n$ are finite versions of Chaitin's $\\Omega$-number. The information contained in $\\Omega_n$ increases as $n$ increases; moreover, the following proposition is true. In this proposition we consider $\\Omega_n$ as a bit string (of length $n+O(1)$) identifying the number $\\Omega_n$ and its binary representation.\n\n\\begin{proposition}\\label{prop:omega-equivalence}\n %\nAssume that $k\\le m$. Consider the string $(\\Omega_m)_k$ consisting of the first $k$ bits of $\\Omega_m$. It is $O(\\log m)$-equivalent to $\\Omega_k$: both conditional complexities $\\KS(\\Omega_k\\mskip 1.5mu | \\mskip 1.5 mu(\\Omega_m)_k)$ and $\\KS((\\Omega_m)_k\\mskip 1.5mu | \\mskip 1.5 mu\\Omega_k)$ are $O(\\log m)$.\n %\n\\end{proposition}\n\n\\begin{proof}\nThis is essentially the reformulation of the previous statement (Proposition~\\ref{prop:enumeration-tail}).\n\nRun the algorithm that enumerates strings of complexity at most $m$. Knowing $(\\Omega_m)_k$, we can wait until less than $2^{m-k}$ strings are left in the enumeration of strings of complexity at most $m$; we know that this happens after more than $B(k)$ steps, and in this time we can enumerate all strings of complexity at most $k$ and compute $\\Omega_k$. (In this argument we ignore $O(\\log m)$-terms, as usual.)\n\nNow the second inequality follows by the symmetry of information property. Indeed, since $\\KS(\\Omega_k)=k+O(1)$ and $\\KS((\\Omega_m)_k)\\le k+O(1)$, the inequality $\\KS(\\Omega_k\\mskip 1.5mu | \\mskip 1.5 mu(\\Omega_m)_k)=O(\\log m)$ implies the inequality $\\KS((\\Omega_m)_k\\mskip 1.5mu | \\mskip 1.5 mu\\Omega_k)=O(\\log m)$.\n\nA direct argument is also easy. Knowing $\\Omega_k$ and $k$, we can find the list of all the strings of complexity at most $k$ and the number $B'(k)$. Then we make $B'(k)$ steps in the enumeration of the list of strings of complexity at most $m$. Proposition~\\ref{prop:enumeration-tail} then guarantees that at that moment $\\Omega_m$ is known with error about $2^{m-k}$, so the first $k$ bits of $\\Omega_m$ can be reconstructed with small advice (of logarithmic size; we omit terms of that size in the argument).\n %\n\\end{proof}\n\nThere is a more direct connection with Chaitin's $\\Omega$-number: one can show that the number $\\Omega_m$ is $O(\\log m)$-equivalent to the $m$-bit prefix of Chaitin's $\\Omega$-number. Since in this survey we restrict ourselves to finite objects, we do not go into details of the proof here, see~\\cite[section 5.7.7]{usv}.\n\n\\subsection{Position in the list is well defined}\n\nWe discussed how much time is needed to enumerate all strings of complexity at most $m$ and how many strings remain not enumerated before this time. Now we want to study \\emph{which} strings remain not enumerated.\n\nMore precisely, let $x$ be some string of complexity at most $m$, so $x$ appears in the enumeration of all strings of complexity at most $m$. How close $x$ is to the end, that is, how many strings are enumerated after $x$? The answer depends on the enumeration, but only slightly, as the following proposition shows.\n\n\\begin{proposition}\\label{prop:pos-def}\n Let $A$ and $B$ be algorithms that both for any given $m$ enumerate (without repetitions) the set of strings of complexity at most $m$. Let $x$ be some string and let $a_x$ and $b_x$ the number of strings that appear after $x$ in $A$- and $B$-enumerations. Then $|\\log a_x - \\log b_x|=O(\\log m)$.\n\\end{proposition}\n\nWe may also assume that $A$ and $B$ are algorithms of complexity $O(\\log m)$ without input that enumerate strings of complexity at most $m$.\n\n\\begin{proof}\nAssume that $a_x$ is small: $\\log a_x \\le k$. Why $\\log b_x$ cannot be much larger than $k$? Given the first $m-\\log b_x$ bits of $\\Omega_m$ and $B$, we can compute a finite set of strings $B'$ that contains $x$ and consists only of strings of complexity at most $m$. Then we can wait until all strings from $B'$ appear in $A$-enumeration. After then at most $2^k$ strings are left, and we need $k$ bits to count them. In this way we can describe $\\Omega_m$ by $m-\\log b_x+k+O(\\log m)$ bits; however, Proposition~\\ref{prop:omegas} says that $\\KS(\\Omega_m)=m+O(1)$. Hence\n$\\log b_x \\le k+O(\\log m)$.\n\nThe other inequality is proven by a symmetric argument.\n\\end{proof}\n\nIn this theorem $A$ and $B$ enumerate exactly the same strings (though in different order). However, the complexity function is essentially defined with $O(1)$-precision only: different optimal programming languages lead to different versions. Let $\\KS$ and $\\tilde{\\KS}$ be two (plain) complexity functions; then $\\tilde{\\KS}(x)\\le\\KS(x)+c$ for some $c$ and for all $x$. Then the list of all $x$ with $\\KS(x)\\le m$ is contained in the list of all $x$ with $\\tilde{\\KS}(x)\\le m+c$. The same argument shows that the number of elements after $x$ in the first list cannot be much larger than the number of elements after $x$ in the second list. The reverse inequality is not guaranteed, however, even for the same version of complexity (small increase in the complexity bound may significantly increase the number of strings after $x$ in the list). We will return to this question in Section~\\ref{subsec:rel-px}, but let us note first that some increase is guaranteed.\n\n\\begin{proposition}\\label{prop:tail-monotonicity}\nIf for a string $x$ there are at least $2^s$ elements after $x$ in the enumeration of all strings of complexity at most $m$, then for every $d\\ge 0$ there are at least $2^{s+d-O(\\log m)}$ strings after $x$ in the enumeration of all strings of complexity at most $m+d$.\n\\end{proposition}\n\n\\begin{proof}\nEssentially the same argument works here: if there are much less than $2^{s+d}$ strings after $x$ in the bigger list, then this bigger list can be determined by $2^{m-s}$ bits needed to cover $x$ in the smaller list and less than $s+d$ bits needed to count the elements in the bigger list that follow the last covered element.\n\\end{proof}\n\nThe last proposition can be restated in the following way. Let us fix some complexity function and and some algorithm that, given $m$, enumerates all strings of complexity at most $m$. Then, for a given string $x$, consider the function that maps every $m\\ge \\KS(x)$ to the logarithm of the number of strings after $x$ in the enumeration with input $m$. Proposition~\\ref{prop:tail-monotonicity} says that $d$-increase in the argument leads at least to $(d-O(\\log m))$-increase of this function (but the latter increase could be much bigger). As we will see, this function is closely related to the set $P_x$ (and therefore $Q_x$): it is one more representation of the same boundary curve.\n\n\\subsection{The relation to $P_x$}\\label{subsec:rel-px}\n\nTo explain the relation, consider the following procedure for a given binary string $x$. For every $m\\ge \\KS(x)$ draw the line $i+j=m$ on $(i,j)$-plane. Then draw the point on this line with second coordinate $s$ where $s$ is the logarithm of the number of elements after $x$ in the enumeration of all strings of complexity at most $m$. Mark also all points on this line on the right of (=below) this point. Doing this for different $m$, we get a set (Figure~\\ref{mdl-e-12}).\n %\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=1]{mdl-e-12.pdf}\n\\caption{For each $m$ between $\\KP(x)$ and $n$ (length of $x$) we count elements after $x$ in the list of strings having complexity at most $m$; assuming there is about $2^s$ of them, we draw point $(m-s,s)$ and get a point on some curve. This curve turns out to be the boundary of $P_x$ (with logarithmic precision).}\n\\label{mdl-e-12}\n\\end{center}\n\\end{figure}\n %\nProposition~\\ref{prop:tail-monotonicity} guarantees that this set is upward closed with logarithmic precision: if some point $(i,j)$ belongs to this set, then the point $(i,j+d)$ is in $O(\\log(i+j))$-neighborhood of this set. This implies that the point $(i+d,j)$ is also in the neighborhood, since our set is closed by construction in the direction $(1,-1)$.\n\nIt turns out that this set coincides with $P_x$ \\textup(Definition~\\ref{def:px}\\textup) with $O(\\log n)$-precision for a string $x$ of length $n$ (this means, as usual, that each of the two sets is contained in the $O(\\log n)$-neighborhood of the other one):\n\n\\begin{thm}\\label{thm:tail-characterization}\nLet $x$ be a string of length $n$. If $x$ has a $(i*j)$-description then $x$ is at least $2^{j-O(\\log n)}$-far from the end of $(i+j+O(\\log n))$-list. Conversely, if there are at least $2^j$ elements that follow $x$ in the $(i+j)$-list then $x$ has a $((i+O(\\log n))*j)$-description.\n\\end{thm}\n\n\\begin{proof}\nWe need to verify two things. First, assuming that $x$ has a $(i*j)$-description, we need to show that it is at least $2^j$-far from the end of $(i+j)$-list. (With error terms: in $(i+j+O(\\log n))$-list there are at least $2^{j-O(\\log n)}$ elements after $x$.) Indeed, knowing some $(i*j)$-description $A$ for $x$, we can wait until all the elements of $A$ appear in $(i+j)$-list (as usual, we omit $O(\\log n)$-term: all elements of $A$ have complexity at most $i+j+O(\\log n)$, so we should consider $(i+j+O(\\log n))$-list to be sure that it contains all elements of $A$). In particular, $x$ has appeared at that moment. If there are (significantly) less than $2^j$ elements after $x$, then we can encode the number of remaining elements by (significantly) less than $j$ bits, and together with the description of $A$ we get less than $i+j$ bits to describe $\\Omega_{i+j}$, which is impossible.\n\nSecond, assume that there are at least $2^j$ elements that follow $x$ in the $(i+j)$-list. Then, splitting this list into $2^j$-portions, we get at most $2^i$ full portions, and $x$ is covered by one of them. Each portion has complexity at most $i$ and log-size at most $j$, so we get an $(i*j)$-description for $x$. (As usual, logarithmic terms are omitted.)\n %\n\\end{proof}\n\nNow we can reformulate the properties of stochastic and antistochastic objects. Every object of complexity $k$ appears in the list of objects of complexity at most $k'$ for all $k'>k$. Each stochastic object is far from the end of these lists (except, may be, for some $k'$-lists with $k'$ very close to $k$). Each antistochastic object of length $n$ is maximally close to the end of all $k'$-lists with $k'b>\\ldots$. The list itself then can be split into pieces of size $2^a$, $2^b$,\\ldots, and these pieces can be considered as description of corresponding objects. In this way for each string $x$ and for each $m\\ge\\KS(x)$ we get some description on $x$, a piece than contains $x$. Descriptions obtained in this way will be called \\emph{standard} descriptions. Note that for a given $x$ we have many standard descriptions (depending on the choice of $m$). One should have in mind also that the class of standard descriptions depends on the choice of the complexity function and the enumeration algorithm, and we assume in the sequel that they are fixed.\n\nThe following results show that standard descriptions are in a sense universal. First let us note that the standard descriptions have parameters close to the boundary curve of $P_x$ (more precisely, to the boundary curve of the set constructed in the previous section that is close to $P_x$).\\footnote{In general, if two sets $X$ and $Y$ in $\\mathbb{N}^2$ are close to each other (each is contained in the small neighborhood of the other one), this does not imply that their boundaries are close. It may happen that one set has a small ``hole'' and the other does not, so the boundary of the first set has points that are far from the boundary of the second one. However, in our case both sets are closed by construction in two different directions, and this implies that the boundaries are also close.}\n\n\n\n\\begin{proposition}\\label{prop:std-pos}\nConsider the standard description $A$ of size $2^j$ obtained from the list of all strings of complexity at most $m$. Then $\\KS(A)=m-j+O(\\log m)$, and the number of elements in the list that follow the elements of $A$ is $2^{j+O(\\log m)}$.\n\\end{proposition}\n\nThis statement says that parameters of $A$ are close to the point on the line $i+j=m$ considered in the previous section (Figure~\\ref{mdl-e-12}).\n\n\\begin{proof}\nTo specify $A$, it is enough to know the first $m-j$ bits of $\\Omega_m$ (and $m$ itself). The complexity of $A$ cannot be much smaller, since knowing $A$ and the $j$ least significant bits of $\\Omega_m$ we can reconstruct $\\Omega_m$.\n\nThe number of elements that follow $A$ cannot exceed $2^j$ (it is a sum of smaller powers of $2$); it cannot be significantly less since it determines $\\Omega_m$ together with the first $m-j$ bits of $\\Omega$. (In other words, since $\\Omega_m$ is an incompressible string of length $m$, it cannot have more that $O(\\log m)$ zeros in a row.)\n %\n\\end{proof}\n\n\n\nThis result does \\emph{not} imply that every point on the boundary of $P_x$ is close to parameters of some standard description. If some part of the boundary has slope $-1$, we cannot guarantee that there are standard descriptions along this part. For example, consider the list of strings of complexity at most $m$; the maximal complexity of strings in this list is $m-c$ for some $c=O(1)$; if we take first string of this complexity, there are $2^{m+O(1)}$ strings after it, so the corresponding point is close to the vertical axis, and due to Proposition~\\ref{prop:tail-monotonicity} all other standard descriptions of $x$ are also close to the vertical axis. However, descriptions with parameters close to arbitrary points on the boundary of $P_x$ can be obtained from standard descriptions by chopping them into smaller parts, as in Proposition~\\ref{prop:description-shift}. In that shopping it is natural to use the order in which the strings were enumerated. In other words, chop the list of strings of complexity at most $m$ into portions of size $2^j$. Consider all the full portions (of size exactly $2^j$) obtained in this way (they are parts of standard descriptions of bigger size). Descriptions obtained in this way are ``universal'' in the following sense: if a pair $(i,j)$ is on the boundary of $P_x$ then there is a set $A\\ni x$ of this type of complexity $i+O(\\log(i+j))$ and log-cardinality $j+O(\\log(i+j))$.\n\n\\smallskip\n\\ver{\n The following result says more: for every description $A$ for $x$ there is a ``better'' standard description that is simple given $A$.\n \n\n\n\\begin{proposition}\\label{prop:better-std}\nLet $A$ be an $(i*j)$-description of a string $x$ of length $n$. Then there is \n$$\nm\\le \\min\\{n,i+j\\}+O(\\log n)\n$$ \nsuch that the parameters of the standard description $B$ \nfor $x$ obtained from the list of strings of complexity at most \n$m$ satisfy the inequalities \n$$\n\\KS(B)\\le i+O(\\log m),\\qquad \\KS(B)+\\log\\#B =m+O(\\log m)\n$$\nMoreover, \n$B$ is simple given $A$, i.e., $\\KS(B\\mskip 1.5mu | \\mskip 1.5 mu A)=O(\\log m)$. \n\\end{proposition}\n\n\\begin{proof}\nIf $i+j\\le n$ then the complexity of every element from $A$ is at most\n$i+j+O(\\log j)=\\min\\{n,i+j\\}+O(\\log n)$. Otherwise remove from\n$A$ all strings of length different from $n$. In this way $A$ becomes $(i*j)$-description for $x$ with slightly larger $i$ than before the removal and the same or smaller $j$. Now all the elements of $A$ have complexity at most \n$n+O(1)=\\min\\{n,i+j\\}+O(1)$. Thus w.l.o.g. we may assume that the\ncomplexity of all strings from $A$ does not exceed some \n$m=\\min\\{n,i+j\\}+O(\\log n)$.\n\nConsider the list of all strings of complexity at most $m$ and the standard description $B$ of $x$ obtained from this list. As we know from Proposition~\\ref{prop:std-pos}, the sum of the parameters of this description is \n$m+O(\\log m)$. \nWe need to show that the size of $B$ is at least $2^{m-i-O(\\log m)}$ and hence\nthe complexity of $B$ is at most $i+O(\\log m)$. Why is this the case? Consider elements that appear after the last element of $A$ in the list. There are at least $2^{m-i-O(\\log m)}$ of them, otherwise the total number of elements in the list could be described in much less than $m$ bits. \nTherefore there are at least $2^{m-i-O(\\log m)}$ elements in the list that appear after $x$.\nAs $x\\in B$ the number of elements that appear after $x$ is less than $2\\# B$ therefore $\\# B\\ge 2^{m-i-O(\\log m)}$ and $\\K(B)\\le i+O(\\log m)$ .\n\n\nWhy $B$ is simple given $A$? Denote the size of $B$ by $2^{j'}$. Given $A$ and $m$, we can find the last element of $A$, call it $x'$, in the list of strings of complexity at most $m$. Chop the list into portions of size $2^{j'}$. Then $B$ is the last complete portion. If $B$ contains $x'$, we can find $B$ from $m$, $j'$, and $x'$ as the complete portion containing $x'$. Otherwise, $x'$ appears in the list after all the elements from $B$. In this case we can find $B$ from $m$ and $x'$ as the last complete portion before $x'$. Thus in any case we are able to find $B$ from $m$, $j'$, and $x'$ plus one extra bit.\n\\end{proof}\n}\n\nFor the same reason every standard description $B$ of some $x$ is simple given $x$ (and this is not a surprise, since we know that all optimal descriptions of $x$ are simple given $x$, see Proposition~\\ref{prop:improving-descriptions}).\n\nProposition~\\ref{prop:better-std} has the following corollary which we formulate in an informal way. Let $A$ be some $(i*j)$-description with parameters on the boundary of $P_x$. Assume that on the left of this point the boundary curve decreases fast (with slope less than $-1$). Then in Proposition~\\ref{prop:better-std} the value of $d$ is small, otherwise the point $(i-d,j+d)$ would be far from $P_x$. So the complexities of $A$ and the standard description $B$ are close to each other. We know also that $A$ is simple given $B$, therefore $B$ is also simple given $A$, and $A$ and $B$ have the same information (have small conditional complexities in both directions).\n\nIf \\label{discussion-minimal}\nwe have two different descriptions $A,A'$ with approximately the same parameters on the boundary of $P_x$, and the curve decreases fast on the left of the corresponding boundary point, the same argument shows that $A$ and $A'$ have the same information. Note that the condition about the slope is important: if the point is on the segment with slope $-1$, the situation changes. For example, consider a random $n$-bit string $x$ and two its descriptions. The first one consists of all $n$-bit strings that have the same left half as $x$, the second one consists of all $n$-bit strings that have the same right half. Both have the same parameters: complexity $n\/2$ and log-size $n\/2$, so they both correspond to the same point on the boundary of $P_x$. Still the information in these two descriptions is different (left and right halves of a random string are independent).\n\nThese results sound as good news. Let us recall our original goal: to formalize what is a good statistical model. It seems that we are making some progress. Indeed, for a given $x$ we consider the boundary curve $P_x$ and look at the place when it first touches the lower bound $i+j=\\KS(x)$; after that it stays near this bound. In other terms, we consider models with negligible optimality deficiency, and select among them the model with minimal complexity. Giving a formal definitions, we need to fix some threshold $\\varepsilon$. Then we say that a set $A$ is a \\emph{$\\varepsilon$-sufficient statistic} if $\\delta(x, A)<\\varepsilon$, and may choose the simplest one among them and call it the \\emph{minimal $\\varepsilon$-sufficient statistic}. If the curve goes down fast on the left of this point, we see that all the descriptions with parameters corresponding to minimal sufficient statistic are equivalent to each other.\n\nTrying to relate these notion to practice, we may consider the following example. Imagine that we have digitized some very old recording and got some bit string $x$. There is a lot of dust and scratches on the recording, so the originally recorded signal is distorted by some random noise. Then our string $x$ has a two-part description: the first part specifies the original recording and the noise parameters (intensity, spectrum, etc.) and the second part specifies the noise exactly. May be, the first part is the minimal sufficient statistic --- and therefore sound restoration (and lossy compression in general) is a special case of the problem of finding a minimal sufficient statistic? The uniqueness result above (saying that all the minimal sufficient statistics contain the same information under some conditions) seem to support this view: different good models for the same object contain the same explanation.\n\nStill the following observation (that easily follows from what we know) destroys this impression completely.\n\n\\begin{proposition}\\label{prop:std-omega}\nLet $B$ be some standard description of complexity $i$ obtained from the list of all strings of complexity at most $m$. Then $B$ is $O(\\log m)$-equivalent to $\\Omega_i$.\n\\end{proposition}\n\nThis looks like a failure. Imagine that we wanted to understand the nature of some data string $x$; finally we succeed and find a description for $x$ of reasonable complexity and negligible randomness and optimality deficiencies (and all the good properties we dreamed of). But Proposition~\\ref{prop:std-omega} says that the information contained in this description is more related to the computability theory than to specific properties of $x$. Recalling the construction, we see that the corresponding standard description is determined by some prefix of some $\\Omega$-number, and is an interval in the enumeration of objects of bounded complexity. So if we start with two old recordings, we may get the same information, which is not what we expect from a restoration procedure. Of course, there is still a chance that some $\\Omega$-number was recorded and therefore the restoration process indeed should provide the information about it, but this looks like a very special case that hardly should happen for any practical situation.\n\nWhat could we do with this? First, we could just relax and be satisfied that we now understand much better the situation with possible descriptions for $x$. We know that every $x$ is characterized by some curve that has several equivalent definitions (in terms of stochasticity, randomness deficiency, position in the enumeration --- as well as time-bounded complexity, see Section~\\ref{sec:depth} below). We know that standard descriptions cover the parts of the curve where it goes down fast, and to cover the parts where the slope is $-1$ one may use standard descriptions and their pieces; all these descriptions are simple given $x$. When curve goes down fast, the description is essentially unique (all the descriptions with the same parameters contain the same information, equivalent to the corresponding $\\Omega$-number); this is not true on parts with slope $-1$. So, even if this curve is of no philosophical importance, we have a lot of technical information about possible models.\n\nThe other approach is to go farther and consider only models from some class (Section~\\ref{sec:restricted-type}), or add some additional conditions and look for ``strong models'' (Section~\\ref{sec:strong-models}).\n\n\\subsection{Non-stochastic objects revisited}\\label{subsec:halting-information}\n\nNow we can explain in a different way why the probability of obtaining a non-stochastic object in a random process is negligible (Proposition~\\ref{prop:nonstochastic-counting}). This explanation uses the notion of mutual information from algorithmic information theory. The mutual information in two strings $x$ and $y$ is defined as\n$$ I(x : y)=\\KS(x)-\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu y) =\\KS(y)-\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)=\\KS(x)+\\KS(y)-\\KS(x,y);$$\nall three expressions are $O(\\log n)$-close if $x$ and $y$ are strings of length $n$ (see, e.g., \\cite[Chapter 2]{usv}).\n\nConsider an arbitrary string $x$ of length $n$; let $k$ be the complexity of $x$. Consider the list of all objects of complexity at most $k$, and the standard description $A$ for $x$ obtained from this list. If $A$ is large, then $x$ is stochastic; if $A$ is small, then $x$ contains a lot of information about $\\Omega_k$ and $\\Omega_n$.\n\nMore precisely, let us assume that $A$ has size $2^{k-s}$ (i.e., is $2^s$ times smaller than it could be). Then (recall Proposition~\\ref{prop:std-pos}) the complexity of $A$ is $s+O(\\log k)$, since we can construct $A$ knowing $k$ and the first $s$ bits of $\\Omega_k$ (before the bit that corresponds to $A$). So we get $(s+O(\\log k))*(k-s)$-description with optimality deficiency $O(\\log k)$.\n\nOn the other hand, knowing $x$ and $k$, we can find the ordinal number of $x$ in the enumeration, so we know $\\Omega_k$ with error at most $2^{k-s}$, so $\\KS(\\Omega_k\\mskip 1.5mu | \\mskip 1.5 mu x)\\le k-s+O(\\log k)$, and $I(x:\\Omega_k)\\ge s-O(\\log k)$ (recall that $\\KS(\\Omega_k)=k+O(1)$). In the last statement we may replace $\\Omega_k$ by $\\Omega_n$ (where $n$ is the length of $x$): we know from Proposition~\\ref{prop:omega-equivalence} that $\\Omega_k$ is simple given $\\Omega_n$, so if condition $\\Omega_k$ decreases complexity of $x$ by almost $s$ bits, the same is true for condition $\\Omega_n$.\n\nComparing arbitrary $i\\le n$ with this $s$ (it can be larger than $s$ or smaller than $s$), we get the following result:\n\n\\begin{proposition}\\label{prop:dilemma}\nLet $x$ be a string of length $n$. For every $i\\le n$\n\\begin{itemize}\n\\item either $x$ is $(i+O(\\log n),O(\\log n))$-stochastic,\n\\item or $I(x:\\Omega_n)\\ge i-O(\\log n)$.\n\\end{itemize}\n\\end{proposition}\n\nNow we may use the following (simple and general) observation: for every string $u$ the probability to generate (by a randomized algorithm) an object that contains a lot of information about $u$ is negligible:\n\n\\begin{proposition}\\label{prop:information-rare}\nFor every string $u$ and for every number $d$, we have\n$$\n\\sum\\{ \\mm(x)\\mid \\KP(x)-\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu u)\\ge d\\} \\le 2^{-d}.\n$$\n\\end{proposition}\nIn this proposition the sum is taken over all strings $x$ that have the given property (have a large mutual information with $u$). Note that we have chosen the representation of mutual information that makes the proposition easy (in particular, we have used prefix complexity). As we mentioned, other definitions differ only by $O(\\log n)$ if we consider strings $x$ and $u$ of length at most $n$, and logarithmic accuracy is enough for our purposes.\n\n\\begin{proof}\nRecall the definition of prefix complexity: $\\KP(x)=-\\log \\mm(x)$, and $\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu u)=-\\log \\mm(x\\mskip 1.5mu | \\mskip 1.5 mu u)$. So $\\KP(x)-\\KP(x\\mskip 1.5mu | \\mskip 1.5 mu u)\\ge d$ implies $\\mm(x)\\le 2^{-d}\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu u)$, and it remains to note that $\\sum_x\\mm(x\\mskip 1.5mu | \\mskip 1.5 mu u)\\le 1$ for every $u$.\n\\end{proof}\n\nPropositions~\\ref{prop:dilemma} and~\\ref{prop:information-rare} immediately imply the following improved version of Proposition~\\ref{prop:nonstochastic-counting} (page~\\pageref{prop:nonstochastic-counting}):\n\n\\begin{proposition}\\label{prop:nonstochastic-counting-improved}\n$$\n\\sum \\{\\,\\mm(x)\\mid \\text{$x$ is a $n$-bit string that is not $(\\alpha,O(\\log n))$-stochastic}\\ \\} \\le 2^{-\\alpha+O(\\log n)}\n$$\nfor every $\\alpha$.\n\\end{proposition}\nThe improvement here is the better upper bound for the randomness deficiency: $O(\\log n)$ instead of $\\alpha+O(\\log n)$.\n\n\\subsection{Historical comments}\n\nThe relation between busy beaver numbers and Kolmogorov complexity was pointed out in~\\cite{gacs1984} (see Section 2.1). The enumerations of all objects of bounded complexity and their relation to stochasticity were studied in~\\cite{gtv} (see Section III, E).\n\n\\section{Computational and logical depth}\\label{sec:depth}\n\nIn this section we reformulate the results of the previous one in terms of bounded-time Kolmogorov complexity and discuss the various notions of computational and logical depth that appeared in the literature. (The impatient reader may skip this section; it is not technically used in the sequel).\n\n\\subsection{Bounded-time Kolmogorov complexity}\n\nThe usual definition of Kolmogorov complexity of $x$ as the minimal length $l(p)$ of a program $p$ that produces $x$ does not take into account the running time of the program $p$: it may happen that the minimal program for $x$ requires a lot of time to produce $x$ while other programs produce $x$ faster but are longer (for example, program ``print $x$'' is rather fast). To analyze this trade-off, the following definition is used.\n\n\\begin{definition}\nLet $D$ be some algorithm; its input and output are binary strings. For a string $x$ and integer $t$, define\n$$\n\\KS^t_D=\\min\\{l(p)\\colon \\text{$D$ produces $x$ on input $p$ in at most $t$ steps}\\},\n$$\nthe time-bounded Kolmogorov complexity of $x$ with time bound $t$ with respect to $D$.\n\\end{definition}\n %\nThis definition was mentioned already in the first paper by Kolmogorov~\\cite{kolm65}:\n %\n\\begin{quote}\nOur approach has one important drawback: it does not take into account the efforts needed to transform the program $p$ and object $x$ [the description and the condition] to the object $y$ [whose complexity is defined]. With appropriate definitions, one may prove mathematical results that could be interpreted as the existence of an object $x$ that has simple programs (has very small complexity $K(x)$) but all short programs that produce $x$ require an unrealistically long computation. In another paper I plan to study the dependence of the program complexity $K^t(x)$ on the difficulty $t$ of its transformation into~$x$. Then the complexity $K(x)$ (as defined earlier) reappears as the minimum value of $K^t(x)$ if we remove restrictions on~$t$.\n %\n\\end{quote}\n %\nKolmogorov never published a paper he speaks about, and this definition is less studied than the definition without time bounds, for several reasons.\n\nFirst, the definition is machine-dependent: we need to decide what computation model is used to count the number of steps. For example, we may consider one-tape Turing machines, or multi-tape Turing machine, or some other computational model. The computation time depends on this choice, though not drastically (e.g., a multi-tape machine can be replaced with a one-tape machine with quadratic increase in time, and most popular models are polynomially related --- this observation is used when we argue that the class P of polynomial-time computable functions is well defined).\n\nSecond, the basic result that makes the Kolmogorov complexity theory possible is the Solomonoff--Kolmogorov theorem saying that there exists an optimal algorithm $D$ that makes the complexity function minimal up to $O(1)$ additive term. Now we need to take into account the time bound, and get the following (not so nice) result.\n\n\\begin{proposition}\\label{prop:time-bounded-optimal}\nThere exists an optimal algorithm $D$ for time-bounded complexity in the following sense: for every other algorithm $D'$ there exists a constant $c$ and a polynomial $q$ such that\n$$\n \\KS_{D'}^t(x)\\le \\KS_D^{q(t)}(x)+c\n$$\nfor all strings $x$ and integers $t$.\n\\end{proposition}\n\nIn this result, by ``algorithm'' we may mean a $k$-tape Turing machine, where $k$ is an arbitrary fixed number. However, the claim remains true even when $k$ is not fixed, i.e., we may allow $D'$ to have more tapes than $D$ has.\n\nThe proof remains essentially the same: we choose some simple self-delimiting encoding of binary strings $p\\mapsto\\hat p$ and some universal algorithm $U(\\cdot,\\cdot)$ and then let\n$$\nD(\\hat p x)=U(p,x)\n$$\nThen the proof follows the standard scheme; the only thing we need to note is that the decoding of $\\hat p$ runs in polynomial time (which is true for most natural ways of self-delimiting encoding) and that the universal algorithm simulation overhead is polynomial (which is also true for most natural constructions of universal algorithms).\n\nA similar result is true for conditional decompressors, so the conditional time-bounded complexity can be defined as well.\n\nFor Turing machines with fixed number of tapes the statement is true for some linear polynomial $q(n)=O(n)$. For the proof we need to consider a universal machine $U$ that simulates other machines efficiently: it should move the program along the tape, so the overhead is bounded by a factor that depends on the size of the program and not on the size of the input or computation time.\\footnote{%\n %\nThis observation motivates Levin's version of complexity ($Kt$, see~\\cite[Section 1.3, p.~21]{levin-conservation}) where the program size and logarithm of the computation time are added: linear overhead in computation time matches the constant overhead in the program size. However, this is a different approach and we do not use the Levin's notion of time bounded complexity in this survey.}\n\nLet $t(n)$ be an arbitrary total computable function with integer arguments and values; then the function\n$$\nx \\mapsto \\KS^{t(l(x))}_{D} (x)\n$$\nis a computable upper bound for the complexity $\\KS(x)$ (defined with the same $D$; recall that $l(x)$ stands for the length of $x$). Replacing the function $t(\\cdot)$ by a bigger function, we get a smaller computable upper bound. An easy observation: in this way we can match every computable upper bound for Kolmogorov complexity.\n\n\\begin{proposition}\\label{prop:any-upper-bound}\nLet $\\tilde{\\KS}(x)$ be some total computable upper bound for Kolmogorov complexity function based on the optimal algorithm $D$ from Proposition~\\ref{prop:time-bounded-optimal}. Then there exists a computable function $t$ such that $\\KS^{t(l(x))}_D(x)\\le \\tilde{\\KS}(x)$ for every $x$.\n\\end{proposition}\n\n\\begin{proof}\nGiven a number $n$, we wait until every string $x$ of length at most $n$ gets a program that has complexity at most $\\tilde{\\KS}(x)$, and let $t(n)$ be the maximal number of steps used by these programs.\n\\end{proof}\n\nSo the choice of a computable time bound is essentially equivalent to the choice of a computable total upper bound for Kolmogorov complexity.\n\nIn the sequel we assume that some optimal (in the sense of Proposition~\\ref{prop:time-bounded-optimal}) $D$ is fixed and omit the subscript $D$ in $\\KS^t_D(\\cdot)$. Similar notation $\\KS^t(\\cdot\\mskip 1.5mu | \\mskip 1.5 mu\\cdot)$ is used for conditional time-bounded complexity.\n\n\\subsection{Trade-off between time and complexity}\n\nWe use the extremely fast growing sequence $B(0),B(1),\\ldots$ as a scale for measuring time. This sequence grows faster than any computable function (since the complexity of $t(n)$ for any computable $t$ is at most $\\log n+O(1)$, we have $B(\\log n+O(1))\\ge t(n)$). In this scale it does not matter whether we use time or space as the resource measure: they differ at most by an exponential function, and $2^{B(n)}\\le B(n+O(1))$ (in general, $f(B(n))\\le B(n+O(1))$ for every computable $f$). So we are in the realm of general computability theory even if we technically speak about computational complexity, and the problems related to the unsolved P=NP question disappear.\n\nLet $x$ be a string of length $n$ and complexity $k$. Consider the time-bounded complexity $\\KS^t(x)$ as a function of $t$. (The optimal algorithm from Proposition~\\ref{prop:time-bounded-optimal} is fixed, so we do not mention it in the notation.) It is a decreasing function of $t$. For small values of $t$ the complexity $\\KS^t(x)$ is bounded by $n+O(1)$ where $n$ stands for the length of $x$. Indeed, the program that prints $x$ has size $n+O(1)$ and works rather fast. Formally speaking, $\\KS^t(x)\\le n+O(1)$ for $t=B(O(\\log n))$. As $t$ increases, the value of $\\KS^t(x)$ decreases and reaches $k=\\KS(x)$ as $t\\to\\infty$. It is guaranteed to happen for $t=B(k+O(1))$, since the computation time for the shortest program for $x$ is determined by this program.\n\nWe can draw a curve that reflects this trade-off using $B$-scale for the time axis. Namely, consider the graph of the function\n$$\n i \\mapsto \\KS^{B(i)}(x)-\\KS(x)\n$$\nand the set of points above this graph, i.e., the set\n$$\nD_x=\\{(i,j)\\mid \\KS^{B(i)}(x)-\\KS(x)\\le j\\}.\n$$\n\n\\begin{thm}[\\cite{bauwens,ABST}]\\label{thm:depth}\nThe set $D_x$ coincides with the set $Q_x$ with $O(\\log n)$-precision for a string $x$ of length~$n$.\n\\end{thm}\n\nRecall that the set $Q_x$ consists of pairs $(\\alpha,\\beta)$ such that $x$ is $(\\alpha,\\beta)$-stochastic (see p.~\\pageref{def:qx}).\n\n\\begin{proof}\nAs we know from Theorem~\\ref{thm:def-opt}, the sets $P_x$ and $Q_x$ are related by an affine transformation (see Figure~\\ref{mdl.8}). Taking this transformation into account, we need to prove two statements:\n\\begin{itemize}\n\\item if there exists an $(i*j)$-description $A$ for $x$, then $$\\KS^{B(i+O(\\log n))}(x)\\le i+j+O(\\log n);$$\n\\item if $\\KS^{B(i)}(x)\\le i+j$, then\n$$\\text{there exist an }((i+O(\\log n))*(j+O(\\log n)))\\text{-description for $x$.}$$\n\\end{itemize}\nBoth statements are easy to prove using the tools from the previous section. Indeed, assume that $x$ has an $(i*j)$-description $A$. All elements of $A$ have complexity at most $i+j+O(\\log n)$. Knowing $A$ and this complexity, we can find the minimal $t$ such that $C^t(x')\\le i+j+O(\\log n)$ for all $x'$ from $A$. This $t$ can be computed from $A$, which has complexity $i$, and an $O(\\log n)$-bit advice (the value of complexity). Hence $t\\le B(i+O(\\log n))$ and $C^t(x)\\le i+j+O(\\log n)$, as required.\n\nThe converse: assume that $C^{B(i)}(x)\\le i+j$. Consider all the strings $x'$ that satisfy this inequality. There are at most $O(2^{i+j})$ such strings. Thus we only need to show that given $i$ and $j$ we are able to enumerate all those strings in at most $O(2^i)$ portions.\n\nOne can get a list of all those strings $x'$ if $B(i)$ is given, but we cannot compute $B(i)$ given~$i$. Recall that $B(i)$ is the maximal integer that has complexity at most $i$; new candidates for $B(i)$ may appear at most $2^i$ times. The candidates increase with time; when this happens, we get a new portion of strings that satisfy the inequality $C^{B(i)}(x)\\le i+j$. So we have at most $O(2^{i+j})$ objects including $x$ that are enumerated in at most $2^i$ portions, and this implies that $x$ has an $((i+O(\\log n))*j)$-description. Indeed, we make all portions of size at most $2^j$ by splitting larger portions into pieces. The number of portions increases at most by $O(2^i)$, so it remains $O(2^i)$. Each portion (including the one that contains $x$) has then complexity at most $i+O(\\log n)$ since it can be computed with logarithmic advice from its ordinal number.\n\\end{proof}\n\nThis theorem shows that the results about the existence of non-stochastic objects can be considered as the ``mathematical results that could be interpreted as the existence of an object $x$ that has simple programs (has very small complexity $K(x)$) but all short programs that produce $x$ require an unrealistically long computation'' mentioned by Kolmogorov (see the quotation above), and the algorithmic statistics can be interpreted as an implementation of Kolmogorov's plan ``to study the dependence of the program complexity $K^t(x)$ on the difficulty $t$ of its transformation into~$x$'', at least for the simple case of (unrealistically) large values of $t$.\n\n\\subsection{Historical comments}\n\nSection~\\ref{sec:depth} has title ``logical and computational depth'' but we have not defined these notions yet. The name ``logical depth'' was introduced by C.~Bennett in~\\cite{bennett}. He explains the motivation as follows:\n\\begin{quote}\nSome mathematical and natural objects (a random sequence, a sequence of zeros, a perfect crystal, a gas) are intuitively trivial, while others (e.g., the human body, the digits of $\\pi$) contain internal evidence of a nontrivial causal history. $\\langle\\ldots\\rangle$\n\nWe propose depth as a formal measure of value. From the earliest days of information theory it has been appreciated that information per se is not a good measure of message value. For example, a typical sequence of coin tosses has high information content but little value; an ephemeris, giving the positions of the moon and the planets every day for a hundred years, has no more information than the equations of motion and initial conditions from which it was calculated, but saves its owner the effort of recalculating these positions. The value of a message thus appears to reside not in its information (its absolutely unpredictable parts), nor in its obvious redundancy (verbatim repetitions, unequal digit frequencies), but rather is what might be called its buried redundancy --- parts predictable only with difficulty, things the receiver could in principle have figured out without being told, but only at considerable cost in money, time, or computation. In other words, the value of a message is the amount of mathematical or other work plausibly done by its originator, which its receiver is saved from having to repeat.\n\\end{quote}\n %\nTrying to formalize this intuition, Bennett suggests the following possible definitions:\n\\begin{quote}\n\\textbf{Tentative Definition 0.1}: A string's depth might be defined as the execution time of its minimal program.\n\\end{quote}\n %\nThis notion is not robust (it depends on the specific choice of the optimal machine used in the definition of complexity). So Bennett considers another version:\n %\n\\begin{quote}\n\\textbf{Tentative Definition 0.2}: A string's depth at significance level $s$ [might] be defined as the time required to compute the string by a program no more than $s$ bits larger than the minimal program.\n\\end{quote}\n %\n We see that Definition 0.2 consider the same trade-off as in Theorem~\\ref{thm:depth}, but in reversed coordinates (time as a function of difference between time-bounded and limit complexities). Bennett is still not satisfied by this definition, for the following reason:\n %\n\\begin{quote}\nThis proposed definition solves the stability problem, but is unsatisfactory in the way it treats multiple programs of the same length. Intuitively, $2^k$ distinct $(n+k)$-bit programs that compute same output ought to be accorded the same weight as one $n$-bit program $\\langle\\ldots\\rangle$\n\\end{quote}\n %\nIn other language, he suggests to consider a priori probability instead of complexity:\n %\n\\begin{quote}\n\\textbf{Tentative Definition 0.3}: A string's depth at significance level $s$ might be defined as the time $t$ required for the string's time-bounded algorithmic probability $P_t(x)$ to rise to within a factor $2^{-s}$ of its asymptotic time-unbounded value $P(x)$.\n\\end{quote}\nHere $P_t (x)$ is understood as a total weight of all self-delimiting programs that produce $x$ in time at most $t$ (each program of length $s$ has weight $2^{-s}$). For our case (when we consider busy beaver numbers as time scale) the exponential time increase needed to switch from a priori probability to prefix complexity does not matter. Still Bennett is interested in more reasonable time bounds (recall that in his informal explanation a polynomially computable sequence of $\\pi$-digits was an example of a deep sequence!), and prefers a priori probability approach. Moreover, he finds a nice reformulation of this definition (almost equivalent one) in terms of complexity:\n\n\\begin{quote}\nAlthough Definition 0.3 satisfactorily captures the informal notion of depth, we propose a slightly stronger definition for the technical reason that it appears to yield a stronger slow growth property $\\langle\\ldots\\rangle$\n\n\\textbf{Definition 1} (Depth of Finite Strings): Let $x$ and $w$ be strings [probably $w$ is a typo: it is not mentioned later] and $s$ a significance parameter. A string's \\emph{depth} at significance level $s$, denoted $D_s(x)$, will be defined as $$\\min\\{T(p)\\colon (|p|-|p^*|\\K(x),\\ i+j\\ge \\K(x,y)\\}\n $$\nMore accurately, the sets $P_{(x,y)}$ and $\\phi(P_x)$ are in \nan $O(\\log l(x)+\\log l(y)+\\varepsilon)$-neighborhood of each other.\n\\end{thm}\nThe transformation of $P_x$ to $P_{(x,y)}$\nis shown on Fig.~\\ref{ver-1}.\n\\begin{figure}[h]\n\\begin{center}\\includegraphics{ver-1.pdf}\\end{center}\n\\caption\n \\ver{The boundary of $P_{xy}$\n is obtained by shifting the boundary of $P_x$\n vertically by $l(y)\\approx \\K(x,y)-\\K(x)$ and adding the sloping segment\n with the endpoints $(\\K(x),\\K(x,y)-\\K(x))$ and\n $(\\K(x,y),0)$.}}\\label{ver-1}\n\\end{figure}\n\\end{remark}\n\n\n\n\n\\begin{remark}\n An interesting special case of this theorem is obtained if we consider a string $u$ and its description $X$ with small randomness deficiency: $d(u\\mskip 1.5mu | \\mskip 1.5 mu X)\\approx 0$. Let $y$ be the ordinal number of $u$ in $X$. Then the small randomness deficiency guarantees that $y$ is conditionally random with respect to $X$. Therefore the pair $(X,y)$ has the same stochasticity profile as $u$. Since this pair is mapped to $u$ by a simple total computable function, we conclude (Proposition~\\ref{prop:stoch-cons}) that the stochasticity profile of $X$ is contained in the stochasticity profile of $u$ (more precisely, in its $O(\\log n + d(u\\mskip 1.5mu | \\mskip 1.5 mu X))$-neighborhood).\n For profiles, there is a more simple and direct proof of the\n inclusion of $\\phi(P_X)$ into a small neighborhood of $P_{u}$:\n if $\\cal{U}$ is an $(i*j)$-description for $X$, we consider the ``lifting'' of\n $\\mathcal U$, i.e. the union of all elements of $\\cal{U}$ that have approximately the same cardinality as $X$; in this way we obtain a $(i*(j+l(y))$-description for $u$. This shows\n that the set\n $\\{(i,j+l(y))\\mid i\\le\\K(x), (i,j)\\in P_x\\}$\n is included in $P_u$.\n For the set $\\{(i,j)\\mid i>\\K(x),\\ i+j\\ge \\K(x,y)\\}$\n the inclusion is obvious, as for all $i>\\K(x)$,\n $j\\ge \\K(x,y)-i$\n the set of all strings in $X$ whose index has the same\n $l(y)-j$ leading\n bits, as $u$, is a $(i*j)$-description of $u$. \n\\end{remark}\n}\n\n\\smallskip\n\n\n\\begin{proof}[A proof sketch of Theorem~\\ref{prop:add-noise}]\nUsing the depth characterization of stochasticity profile, we need to show that\n$$\n\\KS^{B(i)}(x,y) -\\KS(x,y) \\approx \\KS^{B(i)}(x)-\\KS(x).\n$$\nHere ``approximately'' means that these two quantities may differ by a logarithmic term, and also we are allowed to add logarithmic terms to $i$ (see below what does it mean). The natural idea is to rewrite this equality as\n$$\n\\KS^{B(i)}(x,y)-\\KS^{B(i)}(x) \\approx \\KS(x,y)-\\KS(x).\n$$\nThe right hand side is equal to $\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)$ (with logarithmic precision) due to Kol\\-mo\\-go\\-rov--Le\\-vin formula for the complexity of a pair (see, e.g.,~\\cite[Chapter 2]{usv}), and $\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)$ equals $l(y)$, as $y$ is random and independent of $x$. Thus it suffices to show that the left hand side also equals $l(y)$. To this end we can prove a version of Kolmogorov--Levin formula for bounded complexity and show that the left hand side equals to $\\KS^{B(i)}(y\\mskip 1.5mu | \\mskip 1.5 mu x)$. Again, since $y$ is random and independent of $x$, $\\KS^{BB(i)}(y\\mskip 1.5mu | \\mskip 1.5 mu x)$ equals $l(y)$.\n\nThis plan needs clarification. First of all, let us explain which version of Kol\\-mo\\-go\\-rov--Le\\-vin formula for bounded complexity we need. (Essentially it was published by Longpr\\'e in~\\cite{longpre-thesis} though the statement was obscured by considering time bound as a function of the input length.)\n\nThe equality $\\KS(x,y)=\\KS(x)+\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)$ should be considered as two inequalities, and each one should be treated separately.\n\n\\begin{lemma}\n1. There exist some constant $c$ and some polynomial $p(\\cdot,\\cdot)$ such that\n$$\n\\KS^{p(n,t)}(x,y)\\le \\KS^t(x)+\\KS^t(y\\mskip 1.5mu | \\mskip 1.5 mu x)+c\\log n\n$$\nfor all $n$ and $t$ and for all strings $x$ and $y$ of length at most $n$.\n\n2. There exist some constant $c$ and some polynomial $p(\\cdot,\\cdot)$ such that\n$$\n\\KS^{p(2^n,t)}(x)+\\KS^{p(2^n,t)}(y\\mskip 1.5mu | \\mskip 1.5 mu x) \\le \\KS^t(x,y)+c \\log n\n$$\nfor all $n$ and $t$ and for all strings $x$ and $y$ of length at most $n$.\n\\end{lemma}\n\n\\begin{proof}[Proof of the lemma]\nThe proof of this time-bounded version is obtained by a straightforward analysis of the time requirements in the standard proof. The first part says that if there is some program $p$ that produces $x$ in time $t$, and some program $q$ that produces $y$ from $x$ in time $t$, then the pair $(p,q)$ can be considered as a program that produces $(x,y)$ in time $\\poly(t,n)$ and has length $l(p)+l(q)+O(\\log n)$ (we may assume without loss of generality that $p$ and $q$ have length $O(n)$, otherwise we replace them by shorter fast programs).\n\nThe other direction is more complicated. Assume that $\\KS^t(x,y)=m$. We have to count for a given $x$ the number of strings $y'$ such that $\\KS^{t}(x,y')\\le m$. These strings ($y$ is one of them) can be enumerated in time $\\poly(2^n,t)$, so if there are $2^s$ of them, then $\\KS^{\\poly(2^n,t)}(y\\mskip 1.5mu | \\mskip 1.5 mu x)\\le s+O(\\log n)$ (the program witnessing this inequality is the ordinal number of $y$ in the enumeration plus $O(\\log n)$ bits of auxiliary information. Note that we do not need to specify $t$ in advance, we enumerate $y'$ in order of increasing time, and $y$ is among first $2^s$ enumerated strings.\n\nOn the other hand, there are at most $2^{m-s+O(1)}$ strings $x'$ for which this number (of different $y'$ such that $\\KS^t(x',y')\\le m$) is at least $2^{s-1}$, and these strings also could be enumerated in time $\\poly(2^n,t)$, so $\\KS^{\\poly(2^n,t)}(x)\\le m-s+O(\\log n)$ (again we do not need to specify $t$, we just increase gradually the time bound). When these two inequalities are added, $s$ disappears and we get the desired inequality.\n\\end{proof}\n\nOf course, the exponent in the lemma is disappointing (for space bound it is not needed, by the way), but since we measure time in busy beaver units, it is not a problem for us: indeed, $\\poly(2^n,B(i))\\le B(i+O(\\log n))$, and we allow logarithmic change in the argument anyway.\n\nNow we should apply this lemma, but first we need to give a full statement of what we want to prove. There are two parts (as in the lemma):\n\\begin{itemize}\n\\item for every $i$ there exists $j\\le i+O(\\log n)$ such that\n $$\n \\KS^{B(j)}(x,y)-\\KS(x,y) \\le \\KS^{B(i)}(x)-\\KS(x)+\\varepsilon+O(\\log n)\n $$\nfor all strings $x$ and $y$ of length at most $n$ such that $\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)\\le l(y)-\\varepsilon$;\n \\item for every $i$ there exists $j\\le i+O(\\log n)$ such that\n $$\n \\KS^{B(j)}(x)-\\KS(x) \\le \\KS^{B(i)}(x,y)-\\KS(x,y)+O(\\log n)\n $$\nfor all strings $x$ and $y$ of length at most $n$;\n\\end{itemize}\nBoth statements easily follow from the lemma. Let us start with the second statement where the hard direction of the lemma is used. As planned, we rewrite the inequality as\n$$\n \\KS^{B(j)}(x)+\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)\\le \\KS^{B(i)}(x,y)+O(\\log n)\n$$\nusing the unbounded formula. Our lemma guarantees that\n$$\n \\KS^{B(j)}(x)+\\KS^{B(j)}(y\\mskip 1.5mu | \\mskip 1.5 mu x)\\le \\KS^{B(i)}(x,y)+O(\\log n)\n$$\nfor some $j\\le i+O(\\log n)$, and it remains to note that\n$\n\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x) \\le \\KS^{B(j)}(y\\mskip 1.5mu | \\mskip 1.5 mu x).\n$\nFor the other direction the argument is similar: we rewrite the inequality as\n$$\n \\KS^{B(j)}(x,y)\\le \\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)+\\KS^{B(i)}(x)+O(\\log n)\n$$\nand note that\n$\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)\\ge l(y)-\\varepsilon \\ge C^{B(i)}(y\\mskip 1.5mu | \\mskip 1.5 mu x)-\\varepsilon$, assuming that $B(i)$ is greater than the time needed to print $y$ from its literary description (otherwise the statement is trivial). So the lemma again can be used (in the simple direction).\n\\end{proof}\n\nThis proof used the depth representation of the stochasticity curve; in other cases some other representation are more convenient. Our second example is the change in stochasticity profile when a simple algorithmic transformation is applied. We have seen (Section~\\ref{subsec:stoch-cons}) that a total mapping with a short program preserves stochasticity, and noted that for non-total mapping it is not the case (Remark~\\ref{rem:non-total}, p.~\\pageref{rem:non-total}). However, if the time needed to perform the transformation is bounded, we can get some bound (first proven by A.~Milovanov in a different way):\n\n\\ver{\n\\begin{thm}\\label{prop:stoch-nontotal}\nLet $F$ be a partial computable mapping whose arguments and values are strings. If some $n$-bit string $x$ is $(\\alpha,\\beta)$-stochastic, and $F(x)$ is computed in time $B(i)$ for some $i$, then $F(x)$ is $(\\max(\\alpha,i)+O(\\log n),\\beta+O(\\log n))$-stochastic. (The constant in $O(\\log n)$-notation depends on $F$ but not on $n,x,\\alpha,\\beta$.)\n\\end{thm}\n}\n\n\\begin{proof}[Proof sketch]\nLet us denote $F(x)$ by $y$. By assumption there exist a $(\\alpha*(\\KS(x)-\\alpha+\\beta))$-description of $x$ (recall the definition with optimality deficiency; we omit logarithmic terms as usual). So there exists a simple enumeration of at most $2^{\\KS(x)+\\beta}$ objects $x'$ in at most $2^\\alpha$ portions that includes $x$. Let us count $x'$ in this enumeration such that $F(x')=y$ and the computation uses time at most $B(i)$; assume there are $2^s$ of them. Then we can enumerate all $y$'s that have at least $2^s$ preimages in time $B(i)$, in $2^\\alpha + 2^i$ portions. Indeed, new portions appear in two cases: (1)~a new portion appears in the original enumeration; (2)~candidate for $B(i)$ increases. The first event happens at most $2^\\alpha$ times, the second at most $2^i$ times. The total number of $y$'s enumerated is $2^{\\KS(x)+\\beta-s}$; it remains to note that $\\KS(x)-s \\le \\KS(y)$. Indeed, $\\KS(x)\\le \\KS(y)+\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu y)$, and $\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu y)\\le s$, since we can enumerate all the preimages of $y$ in the order of increasing time, and $x$ is determined by $s$-bit ordinal number of $x$ in this enumeration.\n %\n\\end{proof}\n\nA special case of this proposition is Bennett's observation: if some $d$-incompressible program $p$ produces $x$ in time $B(i)$, then $p$ is $(0,d)$-stochastic, and $p$ is mapped to $x$ by the interpreter (decompressor) in time $B(i)$, so $x$ is $(0+i,d)$-stochastic. (For simplicity we omit all the logarithmic terms in this argument, as well as in the previous proof sketch.)\n\n\\begin{remark}\nOne can combine Remark~\\ref{rem:cons-f} (page~\\pageref{rem:cons-f}) with Proposition~\\ref{prop:stoch-nontotal} and show that if a program $F$ of complexity at most $j$ is applied to an $(\\alpha,\\beta)$-stochastic string $x$ of length $n$ and the computation terminates in time $B(i)$, then $F(x)$ is $(\\max(i,\\alpha)+j+O(\\log n),\\beta+j+O(\\log n))$-stochastic, where the constant in $O(\\log n)$ notation is absolute (does not depend on $F$). To show this, one may consider the pair $(x,F)$; it is easy to show (this can be done in different ways using different characterizations of the stochasticity curve) that this pair is $(\\alpha+j+O(\\log n), \\beta+j+O(\\log n))$-stochastic.\n\\end{remark}\n\nLet us note also that there are some results in algorithmic information theory that are true for stochastic objects but are false or unknown without this assumption. We will discuss (without proofs) two examples of this type. The first is Epstein--Levin theorem saying that for a stochastic set $A$ its total a priori probability is close to the maximum a priori probability of $A$'s elements; see~\\cite{shen-survey} for details. Here the result is (obviously) false without stochasticity assumption.\n\nIn the next example~\\cite{muchnik-romashchenko} the stochasticity assumption is used in the proof, and it is not known whether the statement remains true without it: \\emph{for every triple of strings $(x,y,z)$ of length at most $n$ there exists a string $z'$ such that\n\\begin{itemize}\n\\item $\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu z) = \\KS(x\\mskip 1.5mu | \\mskip 1.5 mu z')+O(\\log n)$,\n\\item $\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu z) = \\KS(y\\mskip 1.5mu | \\mskip 1.5 mu z')+O(\\log n)$,\n\\item $\\KS(x,y\\mskip 1.5mu | \\mskip 1.5 mu z) = \\KS(x,y\\mskip 1.5mu | \\mskip 1.5 mu z') + O(\\log n)$;\n\\item $\\KS(z') \\le I((x,y):z)+O(\\log n)$,\n\\end{itemize}\nassuming that $(x,y)$ is $(O(\\log n),O(\\log n))$-stochastic}.\n\nThis proposition is related to the following open question on ``irrelevant oracles'': assume that the mutual information between $(x,y)$ and some $z$ is negligible. Can an oracle $z$ (an ``irrelevant oracle'') change substantially natural properties of the pair $(x,y)$ formulated in terms of Kolmogorov complexity? For instance, can such an oracle $z$ allow us to extract some common information of $x$ and $y$? In~\\cite{muchnik-romashchenko} a negative answer to the latter question is given, but only for stochastic pairs $(x,y)$.\n\n\\section{Descriptions of restricted type}\\label{sec:restricted-type}\n\n\\subsection{Families of descriptions}\n\nIn this section we consider the restricted case: the sets (considered as descriptions, or statistical hypotheses) are taken from some family $\\mathcal{A}$ that is fixed in advance.\\footnote{One can also consider some class of probability distributions, but we restrict our attention to sets (uniform distributions).} (Elements of $\\mathcal{A}$ are finite sets of binary strings.) Informally speaking, this means that we have some \\emph{a priori} information about the black box that produces a given string: this string is obtained by a random choice in one of the $\\mathcal{A}$-sets, but we do not know in which one.\n\nBefore we had no restrictions (the family $\\mathcal{A}$ was the family of all finite sets). It turns out that the results obtained so far can be extended (sometimes with weaker bounds) to other families that satisfy some natural conditions. Let us formulate these conditions.\n\n(1)~The family $\\mathcal{A}$ is enumerable. This means that there exists an algorithm that prints elements of $\\mathcal{A}$ as lists, with some separators (saying where one element of $\\mathcal{A}$ ends and another one begins).\n\n(2)~For every $n$ the family $\\mathcal{A}$ contains the set $\\mathbb{B}^n$ of all $n$-bit strings.\n\n(3)~There exists some polynomial $p$ with the following property: for every $A\\in\\mathcal{A}$, for every natural $n$ and for every natural $c<\\#A$ the set of all $n$-bit strings in $A$ can be covered by at most $p(n)\\cdot\\#A\/c$ sets of cardinality at most $c$ from $\\mathcal{A}$.\n\nThe last condition is a replacement for splitting: in general, we cannot split a set $A\\in\\mathcal{A}$ into pieces from $A$, but at least we can cover a set $A\\in\\mathcal{A}$ by smaller elements of $\\mathcal{A}$ (of size at most $c$) with polynomial overhead in the number of pieces, compared to the required minimum $\\#A\/c$ (more precisely, we have to cover only $n$-bit elements of $A$).\n\nWe assume that some family $\\mathcal{A}$ that has properties (1)--(3) is fixed. For a string $x$ we denote by $P_x^\\mathcal{A}$ the set of pairs $( i,j)$ such that $x$ has $(i*j)$-description \\emph{that belongs to $\\mathcal{A}$}. The set $P_x^\\mathcal{A}$ is a subset of $P_x$ defined earlier; the bigger $\\mathcal{A}$ is, the bigger is $P_x^\\mathcal{A}$. The full set $P_x$ is $P_x^\\mathcal{A}$ for the family $\\mathcal{A}$ that contains all finite sets.\n\nFor every string $x$ the set $P_x^\\mathcal{A}$ has properties close to the properties of $P_x$ proved earlier.\n\n\\begin{proposition}\\label{prop:a-family}\nFor every string $x$ of length $n$ the following is true:\n\n\\begin{enumerate}\n \\item\\label{a1} The set $P_x^\\mathcal{A}$ contains a pair that is $O(\\log n)$-close to $( 0,n)$.\n\n \\item\\label{a2} The set $P_x^\\mathcal{A}$ contains a pair that is $O(1)$-close to $( \\KS(x),0)$.\n\n \\item\\label{a3} The adaptation of Proposition~\\ref{prop:description-shift} is true: if $( i,j)\\in P_x^\\mathcal{A}$, then $( i+k+O(\\log n),j-k)$ also belongs to $P_x^\\mathcal{A}$ for every $k\\le j$. (Recall that $n$ is the length of $x$.)\n %\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\ref{a1}. The property (2) guarantees that the family $\\mathcal{A}$ contains the set $\\mathbb{B}^n$ that is an $(O(\\log n)*n)$-description of $x$.\n\n\\ref{a2}. The property (3) applied to $c=1$ and $A=\\mathbb{B}^n$ says that every singleton belongs to $A$, therefore each string has $((\\KS(x)+O(1))*0)$-description.\n\n\\ref{a3}. Assume that $x$ has $(i*j)$-description $A\\in\\mathcal{A}$. For a given $k$ we enumerate $\\mathcal{A}$ until we find a family of $p(n)2^k$ sets of size $2^{-k}\\#A$ (or less) in $\\mathcal{A}$ that covers all strings of length $n$ in $A$. Such a family exists due to (3), and $p$ is the polynomial from~(3). The complexity of the set that covers $x$ does not exceed $i+k+O(\\log n+\\log k)$, since this set is determined by $A$, $n$, $k$ and the ordinal number of the set in the cover. We may assume without loss of generality that $k\\le n$, otherwise $\\{x\\}$ can be used as $((i+k+O(\\log n))*(j-k))$-description of $x$. So the term $O(\\log k)$ can be omitted.\n\\end{proof}\n\nFor example, we may consider the family that consists of all ``cylinders'': for every $n$ and for every string $u$ of length at most $n$ we consider the set of all $n$-bit strings that have prefix $u$. Obviously the family of all such sets (for all $n$ and $u$) satisfies the conditions (1)--(3).\n\nWe may also fix some bits of a string (not necessarily forming a prefix). That is, for every string $z$ in ternary alphabet $\\{0,1,*\\}$ we consider the set of all bit strings that can be obtained from $z$ by replacing stars with some bits. This set contains $2^k$ strings, if $u$ has $k$ stars. The conditions (1)--(3) are fulfilled for this larger family, too.\n\nA more interesting example is the family $\\mathcal{A}$ formed by all balls in Hamming sense, i.e., the sets $B_{y,r}=\\{x\\mid l(x)=l(y), d(x,y)\\le r\\}$. Here $l(u)$ is the length of binary string $u$, and $d(x,y)$ is the Hamming distance between two strings $x$ and $y$ of the same length. The parameter $r$ is called the \\emph{radius} of the ball, and $y$ is its \\emph{center}. Informally speaking, this means that the experimental data were obtained by changing at most $r$ bits in some string $y$ (and all possible changes are equally probable). This assumption could be reasonable if some string $y$ is sent via an unreliable channel. Both parameters $y$ and $r$ are not known to us in advance.\n\nIt turns out that the family of Hamming balls satisfies the conditions (1)--(3). This is not completely obvious. For example, these conditions imply that for every $n$ and for every $r\\le n$ the set $\\mathbb{B}^n$ of $n$-bit strings can be covered by $\\poly(n)2^n\/V$ Hamming balls of radius $r$, where $V$ stands for the cardinality of such a ball (i.e., $V=\\binom{n}{0}+\\ldots+\\binom{n}{r}$), and $p$ is some polynomial. This can be shown by a probabilistic argument: take $N$ balls of radius $r$ whose centers are randomly chosen in $\\mathbb{B}^n$. For a given $x\\in\\mathbb{B}^n$ the probability that $x$ is not covered by any of these balls equals $(1-V\/2^n)^N < e^{-VN\/2^n}$. For $N=n\\ln 2\\cdot 2^n\/V$ this upper bound is $2^{-n}$, so for this $N$ the probability to leave some $x$ uncovered is less than~$1$. A similar argument can be used to prove (1)--(3) in the general case.\n\n\\begin{proposition}[\\cite{vv10}]\\label{prop:hamming-balls}\nThe family of all Hamming balls satisfies conditions (1)--(3) above.\n\\end{proposition}\n\n\\begin{proof}[Proof sketch]\nLet $A$ be a ball of radius $a$ and let $c$ be a number less than $\\#A$. We need to cover $A$ by balls of cardinality $c$ or less, using almost minimal number of balls, close to the lower bound $\\#A\/c$ up to a polynomial factor. Let us make some observations.\n\n(1)~The set of all $n$-bit strings can be covered by two balls of radius $n\/2$. So we can assume without loss of generality that $a\\le n\/2$, otherwise we can apply the probabilistic argument above.\n\n(2)~Clearly the radius of covering balls should be maximal possible (to keep cardinality less than $c$); for this radius the cardinality of the ball equals $c$ up to polynomial factors, since the size of the ball increases at most by factor $n+1$ when its radius increases by $1$.\n\n(3)~It is enough to cover spheres instead of balls (since every ball is a union of polynomially many spheres); it is also enough to consider the case when the radius of the sphere that we want to cover ($a$) is bigger than the radius of the covering ball ($b$), otherwise one ball is enough.\n\n(4)~We will cover $a$-sphere by randomly chosen $b$-balls whose centers are uniformly taken at some distance $f$ from the center of $a$-sphere. (See below about the choice of $f$.) We use the same probabilistic argument as before (for the set of all strings). It is enough to show that for a $b$-ball whose center is at that distance, the polynomial fraction of points belong to $a$-sphere. Instead of $b$-balls we may consider $b$-spheres, the cardinality ratio is polynomial.\n\n(5)~It remains to choose some $f$ with the following property: if the center of a $b$-sphere $S$ is at a distance $f$ from the center of $a$-sphere $T$, then the polynomial fraction of $S$-points belong to $T$. One can compute a suitable $f$ explicitly. In probabilistic terms we just change $f\/n$-fraction of bits and then change random $b\/n$ fraction of bits. The expected fraction of twice changed bits is, therefore, about $(f\/n)(b\/n)$, and the total fraction of changed bits is about $f\/n+b\/n-2(f\/n)(b\/n)$. So we need to write an equation saying that this expression is $a\/n$ and the find the solution $f$. (Then one can perform the required estimate for binomial coefficients.)\n\nHowever, one can avoid computations with the following probabilistic argument: start with $b$ changed bits, and then change all the bits one by one in a random order. At the end we hat $n-b$ changed bits, and $a$ is somewhere in between, so there is a moment where the number of changed bits is exactly $a$. And if the union of $n$ events covers the entire probability space, one of these events has probability at least $1\/n$.\n\\end{proof}\n\nWhen a family $\\mathcal{A}$ is fixed, a natural question arises: does the restriction on models (when we consider only models in $\\mathcal{A}$) changes the set $P_x$? Is it possible that a string has good models in general, but not in the restricted class? The answer is positive for the class of Hamming balls, as the following proposition shows.\n\n\\begin{proposition}\\label{prop:hamming-gap}\nConsider the family $\\mathcal{A}$ that consists of all Hamming balls. For some positive $\\varepsilon$ and for all sufficiently large $n$ there exists a string $x$ of length $n$ such that the distance between $P_x^\\mathcal{A}$ and $P_x$ exceeds $\\varepsilon n$.\n\\end{proposition}\n\n\\begin{proof}[Proof sketch]\nFix some $\\alpha$ in $(0,1\/2)$ and let $V$ be the cardinality of the Hamming ball of radius $\\alpha n$. Find a set $E$ of cardinality $N=2^n\/V$ such that every Hamming ball of radius $\\alpha n$ contains at most $n$ points from $E$. This property is related to \\emph{list decoding} in the coding theory. The existence of such a set can be proved by a probabilistic argument: $N$ randomly chosen $n$-bit strings have this property with positive probability. Indeed, the probability of a random point to be in $E$ is an inverse of the number of points, so the distribution is close to Poisson distribution with parameter~$1$, and tails decrease much faster that $2^{-n}$ needed.\n\nSince $E$ with this property can be found by an exhaustive search, we can assume that $\\KS(E)=O(\\log n)$ and ignore the complexity of $E$ (as well as other $O(\\log n)$ terms) in the sequel. Let $x$ be a random element in $E$, i.e., a string $x\\in E$ of complexity about $\\log\\#E$. The complexity of a ball $A$ of radius $\\alpha n$ that contains $x$ is at least $\\KS(x)$, since knowing such a ball and an ordinal number of $x$ in $A\\cap E$, we can find $x$. Therefore $x$ does not have $(\\log\\#E,\\log V)$-descriptions in $\\mathcal A$. On the other hand, $x$ does have $(0,\\log\\#E)$-description if we do not require the description to be in $\\mathcal A$; the set $E$ is such a description. The point $(\\log\\#E, \\log V)$ is above the line $\\KS(A)+\\log\\#A=\\log\\#E$, so $P_x^\\mathcal A$ is significantly smaller than $P_x$.\n\\end{proof}\n\nThis construction gives a stochastic $x$ ($E$ is the corresponding model) that becomes maximally non-stochastic if we restrict ourselves to Hamming balls as descriptions (Figure~\\ref{mdl-e-13}).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=1]{mdl-e-13.pdf}\n\\end{center}\n\\caption{Theorem~\\ref{thm:improving-descriptions-1-gen} can be used (together with the argument above) to show that the border of the set $P_x^\\mathcal{A}$ (shown in gray) consists of a vertical segment $\\KS(A)=n-\\log V$, $\\log\\#A\\le \\log V$, and the segment of slope $-1$ defined by $\\KS(A)+\\log\\#A = n$, $\\log V \\le \\log\\#A$. The set $P_x$ contains also the hatched part.}\\label{mdl-e-13}\n\\end{figure}\n\n\\subsection{Possible shapes of boundary curve}\n\nOur next goal is to extend some results proven for non-restricted descriptions to the restricted case. Let $\\mathcal{A}$ be a family that has properties (1)--(3). We prove a version of Theorem~\\ref{stat-any-curve} where the precision (unfortunately) is significantly worse: $O(\\sqrt{n\\log n})$ instead of $O(\\log n)$. Note that with this precision the term $O(m)$ (proportional to the complexity of the curve) that appeared in Theorem~\\ref{stat-any-curve} is not needed. Indeed, if we draw the curve on the cell paper with cell size $\\sqrt{n}$ or larger, then it touches only $O(\\sqrt{n})$ cells, so it is determined by $O(\\sqrt{n})$ bits with $O(\\sqrt{n})$-precision, and we may assume without loss of generality that the complexity of the curve is $O(\\sqrt{n})$.\n\n\\begin{thm}[\\cite{vv10}]\\label{thm:family-curve}\nLet $k\\le n$ be two integers and let $t_0>t_1>\\ldots>t_k$ be a strictly decreasing sequence of integers such that $t_0\\le n$ and $t_k=0$.. Then there exists a string $x$ of complexity $k+O(\\sqrt{n\\log n})$ and length $n+O(\\log n)$ for which the distance between $P_x^\\mathcal{A}$ and $T=\\{( i,j) \\mid (i\\le k)\\Rightarrow (j\\ge t_i)\\}$ is at most $O(\\sqrt{n\\log n})$.\n\\end{thm}\n\nWe will see later (Theorem~\\ref{thm:improving-descriptions-1-gen}) that for every $x$ the boundary curve of $P_x^{\\mathcal{A}}$ goes down at least with slope $-1$, as for the unrestricted case, so this theorem describes all possible shapes of the boundary curve.\n\n\\begin{proof}\n %\nThe proof is similar to the proof of Theorem~\\ref{stat-any-curve}. Let us recall this proof first. We consider the string $x$ that is the lexicographically first string (of suitable length $n'$) that is not covered by any ``bad'' set, i.e., by any set of complexity at most $i$ and size at most $2^j$, where the pair $(i,j)$ is at the boundary of the set $T$. The length $n'$ is chosen in such a way that the total number of strings in all bad sets is strictly less than $2^{n'}$. On the other hand, we need ``good sets'' that cover $x$. For every boundary point $(i,j)$ we construct a set $A_{i,j}$ that contains $x$, has complexity close to $i$ and size $2^j$. The set $A_{i,j}$ is constructed in several attempts. Initially $A_{i,j}$ is the set of lexicographically first $2^j$ strings of length $n'$. Then we enumerate bad sets and delete all their elements from $A_{i,j}$. At some step $A_{i,j}$ may become empty; then we refill it with $2^j$ lexicographically first strings that are not in the bad sets (at the moment). By construction the final $A_{i,j}$ contains the first $x$ that is not in bad sets (since it is the case all the time). And the set $A_{i,j}$ can be described by the number of changes (plus some small information describing the process as a whole and the value of $j$). So it is crucial to have an upper bound for the number of changes. How do we get this bound? We note that when $A_{i,j}$ becomes empty, it is refilled again, and all the new elements should be covered by bad sets before the new change could happen. Two types of bad sets may appear: ``small'' ones (of size less than $2^j$) and ``large ones'' (of size at least $2^j$). The slope of the boundary line for $T$ guarantees that the total number of elements in all small bad sets does not exceed $2^{i+j}$ (up to a $\\poly(n)$-factor), so they may make $A_{i,j}$ empty only $2^i$ times. And the number of large bad sets is $O(2^i)$, since the complexity of each is bounded by $i$. (More precisely, we count separately the number of changes for $A_{i,j}$ that are first changes after a large bad set appears, and the number of other changes.)\n\nCan we use the same argument in our new situation? We can generate bad sets as before and have the same bounds for their sizes and the total number of their elements. So the length $n'$ of $x$ can be the same (in fact, almost the same, as we will need now that the union of all bad sets is less than half of all strings of length $n'$, see below). Note that we now may enumerate only bad sets in $\\mathcal{A}$, since $\\mathcal{A}$ is enumerable, but we do not even need this restriction. What we cannot do is to let $A_{i,j}$ to be the set of the first non-deleted elements: we need $A_{i,j}$ to be a set from $\\mathcal{A}$.\n\nSo we now go in the other direction. Instead of choosing $x$ first and then finding suitable ``good'' $A_{i,j}$ that contain $x$, we construct the sets $A_{i,j}\\in\\mathcal{A}$ that change in time in such a way that (1)~their intersection always contains some non-deleted element (an element that is not yet covered by bad sets); (2) each $A_{i,j}$ has not too many versions. The non-deleted element in their intersection (in the final state) is then chosen as $x$.\n\nUnfortunately, we cannot do this for all points $(i,j)$ along the boundary curve. (This explains the loss of precision in the statement of the theorem.) Instead, we construct ``good'' sets only for some values of $j$. These values go down from $n$ to $0$ with step $\\sqrt{n\\log n}$. We select $N=\\sqrt{n\/\\log n}$ points $(i_1,j_1),\\ldots,(i_N,j_N)$ on the boundary of $T$; the first coordinates $i_1,\\ldots,i_N$ form a non-decreasing sequence, and the second coordinates $j_1,\\ldots,j_N$ split the range $n\\ldots 0$ into (almost) equal intervals ($j_1=n$, $j_N=0$). Then we construct good sets of sizes at most $2^{j_1},\\ldots,2^{j_N}$, and denote them by $A_1,\\ldots,A_N$. All these sets belong to the family $\\mathcal{A}$. We also let $A_0$ to be the set of all strings of length $n'=n+O(\\log n)$; the choice of the constant in $O(\\log n)$ will be discussed later.\n\nLet us first describe the construction of $A_1,\\ldots,A_N$ assuming that the set of deleted elements is fixed. (Then we discuss what to do when more elements are deleted.) We construct $A'$ inductively (first $A_1$, then $A_2$ etc.). As we have said, $\\#A'\\le 2^{j_s}$ (in particular, $A_N$ is a singleton), and we keep track of the ratio\n $$(\\text{the number of non-deleted strings in $A_0\\cap A_1\\cap\\ldots\\cap A'$})\/2^{j_s}.$$\nFor $s=0$ this ratio is at least $1\/2$; this is obtained by a suitable choice of $n'$ (the union of all bad sets should cover at most half of all $n'$-bit strings). When constructing the next $A'$, we ensure that this ratio decreases only by $\\poly(n)$-factor. How? Assume that $A_{s-1}$ is already constructed; its size is at most $2^{j_{s-1}}$. The condition $(3)$ for $\\mathcal{A}$ guarantees that $A_{s-1}$ can be covered by $\\mathcal{A}$-sets of size at most $2^{j_s}$, and we need about $2^{j_{s-1}-j_s}$ covering sets (up to $\\poly(n)$-factor). Now we let $A'$ be the covering set that contains maximal number of non-deleted elements in $A_0\\cap\\ldots\\cap A_{s-1}$. The ratio can decrease only by the same $\\poly(n)$-factor. In this way we get\n $$(\\text{the number of non-deleted strings in $A_0\\cap A_1\\cap\\ldots\\cap A'$})\\ge \\alpha^{-s}2^{j_s}\/2,$$\nwhere $\\alpha$ stands for the $\\poly(n)$-factor mentioned above.\\footnote{Note that for the values of $s$ close to $N$ the right-hand side can be less than $1$; the inequality then claims just the existence of non-deleted elements. The induction step is still possible: non-deleted element is contained in one of the covering sets.}\n\nUp to now we assumed that the set of deleted elements is fixed. What happens when more strings are deleted? The number of the non-deleted in $A_0\\cap\\ldots\\cap A_{s}$ can decrease, and at some point and for some $s$ can become less than the declared threshold $\\nu_s=\\alpha^{-s} 2^{j_s}\/2$. Then we can find minimal $s$ where this happens, and rebuild all the sets $A',A_{s+1},\\ldots$ (for $A'$ the threshold is not crossed due to the minimality of $s$). In this way we update the sets $A'$ from time to time, replacing them (and all the consequent ones) by new versions when needed.\n\nThe problem with this construction is that the number of updates (different versions of each $A'$) can be too big. Imagine that after an update some element is deleted, and the threshold is crossed again. Then a new update is necessary, and after this update next deletion can trigger a new update, etc. To keep the number of updates reasonable, we agree that after the update \\emph{for all the new sets $A_l$} (starting from $A'$) \\emph{the number of non-deleted elements in $A_0\\cap\\ldots\\cap A_l$ is twice bigger than the threshold $\\nu_l=\\alpha^{-l}2^{j_l}\/2$}. This can be achieved if we make the factor $\\alpha$ twice bigger: since for $A_{s-1}$ we have not crossed the threshold, for $A'$ we can guarantee the inequality with additional factor $2$.\n\nNow let us prove the bound for the number of updates for some $A'$. These updates can be of two types: first, when $A'$ itself starts the update (being the minimal $s$ where the threshold is crossed); second, when the update is induced by one of the previous sets. Let us estimate the number of the updates of the first type. This update happens when the number of non-deleted elements (that was at least $2\\nu_s$ immediately after the previous update of any kind) becomes less than $\\nu_s$. This means that at least $\\nu_s$ elements were deleted. How can this happen? One possibility is that a new bad set of complexity at most $i_s$ (``large bad set'') appears after the last update. This can happen at most $O(2^{i_s})$ times, since there is at most $O(2^i)$ objects of complexity at most $i$. The other possibility is the accumulation of elements deleted due to ``small'' bad sets, of complexity at least $i_s$ and of size at most $2^{j_s}$. The total number of such elements is bounded by $nO(2^{i_s+j_s})$, since the sum $i_l+j_l$ may only decrease as $l$, increases. So the number of updates of $A'$ not caused by large bad sets is bounded by\n $$ n O(2^{i_s+j_s}) \/\\nu_s =\\frac{O(n2^{i_s+j_s})}{\\alpha^{-s}2^{j_s}} = O(n\\alpha^s 2^{i_s})=2^{i_s+NO(\\log n)}=2^{i_s+O(\\sqrt{n\\log n})}$$\n(recall that $s\\le N$, $\\alpha=\\poly(n)$, and $N\\approx \\sqrt{n\/\\log n}$). This bound remains valid if we take into account the induced updates (when the threshold is crossed for the preceding sets: there are at most $N\\le n$ these sets, and additional factor $n$ is absorbed by $O$-notation).\n\nWe conclude that all the versions of $A'$ have complexity at most $i_s+O(\\sqrt{n\\log n})$, since each of them can be described by the version number plus the parameters of the generating process (we need to know $n$ and the boundary curve, whose complexity is $O(\\sqrt{n})$ according to our assumption, see the discussion before the statement of the theorem). The same is true for the final version. It remains to take $x$ in the intersection of the final sets $A'$. (Recall that $A_N$ is a singleton, so final $A_N$ is $\\{x\\}$.) Indeed, by construction this $x$ has no bad $(i*j)$-descriptions where $(i,j)$ is on the boundary of $T$. On the other hand, $x$ has good descriptions that are $O(\\sqrt{n\\log n})$-close to this boundary and whose vertical coordinates are $\\sqrt{n\\log n}$-apart. (Recall that the slope of the boundary guarantees that horizontal distance is less than the vertical distance.) Therefore the position of the boundary curve for $P_x^\\mathcal{A}$ is determined with precision $O(\\sqrt{n\\log n})$, as required.\\footnote{Now we see why $N$ was chosen to be $\\sqrt{n\/\\log n}$: the bigger $N$ is, the more points on the curve we have, but then the number of versions of the good sets and their complexity increases, so we have some trade-off. The chosen value of $n$ balances these two sources of errors.}\n %\n \\end{proof}\n\n\\begin{remark}\\label{rem:family}\nIn this proof we may use bad sets not only from $\\mathcal{A}$. Therefore, the set $P_x$ is also close to $T$ (and the same is true for for every family $\\mathcal{B}$ that contains $\\mathcal{A}$). It would be interesting to find out what are the possible combinations of $P_x$ and $P_x^\\mathcal{A}$; as we have seen, it may happen that $P_x$ is maximal and $P_x^\\mathcal{A}$ is minimal, but this does not say anything about other possible combinations. \\end{remark}\n\nFor the case of Hamming balls the statement of Theorem~\\ref{thm:family-curve} has a natural interpretation. To find a simple ball of radius $r$ that contains a given string $x$ is the same as to find a simple string in a radius $r$ ball centered at $x$. So this theorem show the possible behavior of the ``approximation complexity'' function\n$$\n r\\mapsto \\min \\{\\KS(x')\\mid d(x,x')\\le r\\}\n$$\nwhere $d$ is Hamming distance. One should only rescale the vertical axis replacing the log-sizes of Hamming balls by their radii. The connection is described by the Shannon entropy function: a ball in $\\mathbb{B}^n$ of radius $r$ has log-size about $nH(r\/n)$ for $r\\le n\/2$, and has almost full size for $r\\ge n\/2$. For example, error correcting codes (in classical sense, or with list decoding) are example of strings where this function is almost a constant for small values of $r$: it is almost as easy to approximate a codeword as give it precisely (due to the possibility of error correction).\n\n\\subsection{Randomness and optimality deficiencies: restricted case}\n\nNot all the results proved for unrestricted descriptions have natural counterparts in the restricted case. For example, one hardly can relate the set $P_x^\\mathcal{A}$ with bounded-time complexity (is completely unclear how $\\mathcal{A}$ could enter the picture). Still some results remain valid (but new and much more complicated proofs are needed). This is the case for Proposition~\\ref{prop:description-shift} and~\\ref{prop:improving-descriptions}.\n\nLet again $\\mathcal{A}$ be the class of descriptions that satisfies requirements (1)--(3).\n\n\\begin{thm}[\\cite{vv10}]\\label{thm:improving-descriptions-1-gen}\n\\leavevmode\n\\begin{itemize}\n\\item If a string $x$ of length $n$ has an $(i*j)$-description in $\\mathcal{A}$, then it has $((i+d+O(\\log n))*(j-d+O(\\log n)))$-description in $\\mathcal{A}$ for every $d\\le j$.\n\n\\item Assume that $x$ is a string of length $n$ that has at least $2^k$ different $(i*j)$-descriptions in $\\mathcal{A}$. Then it has $((i-k+O(\\log n))*(j+O(\\log n))$-description in $\\mathcal{A}$.\n\\end{itemize}\n\\end{thm}\n\nIn fact, the second part uses only condition (1); it says that $\\mathcal{A}$ is enumerable. The first part uses also (3). It can be combined with the second part to show that $x$ has also $((i+O(\\log n))*(j-k+O(\\log n))$-description in $\\mathcal{A}$.\n\\smallskip\n\nThough theorem \\ref{thm:improving-descriptions-1-gen} looks like a technical statement, it has important consequences; it implies that the two approaches based on randomness and optimality deficiencies remain equivalent in the case of bounded class of descriptions. The proof technique can be also used to prove Epstein--Levin theorem~\\cite{epstein-levin}, as explained in~\\cite{shen-survey}; similar technique was used by A.~Milovanov in \\cite{milovanov-stacs} where a common model for several strings is considered.\n\n\\begin{proof}\nThe first part is easy: having some $(i*j)$-description for $x$, we can search for a covering by the sets of right size that exists due to condition~(3); since $\\mathcal{A}$ is enumerable, we can do it algorithmically until we find this covering. Then we select the first set in the covering that contains $x$; the bound for the complexity of this set is guaranteed by the size of the covering.\n\nThe proof of the second statement is much more interesting. In fact, there are two different proofs: one uses a probabilistic existence argument and the second is more explicit. But both of them start in the same way.\n\nLet us enumerate all $(i*j)$-descriptions from $\\mathcal A$, i.e., all finite sets that belong to $\\mathcal A$, have cardinality at most $2^j$ and complexity at most $i$. For a fixed $n$, we start a selection process: some of the generated descriptions are marked (=selected) immediately after their generation. This process should satisfy the following requirements: (1)~at any moment every $n$-bit string $x$ that has at least $2^k$ descriptions (among enumerated ones) belongs to one of the marked descriptions; (2)~the total number of marked sets does not exceed $2^{i-k}p(n)$ for some polynomial~$p$. Note that for $i\\ge n$ or $j\\ge n$ the statement is trivial, so we may assume that $i$, $j$ (and therefore $k$) do not exceed $n$; this explains why the polynomial depends only on $n$.\n\nIf we have such a strategy (of logarithmic complexity), then the marked set containing $x$ will be the required description of complexity $i-k+O(\\log n)$ and log-size $j$. Indeed, this marked set can be specified by its ordinal number in the list of marked sets, and this ordinal number has $i-k+O(\\log n)$ bits.\n\nSo we need to construct a selection strategy of logarithmic complexity. We present two proofs: a probabilistic one and an explicit construction.\n\n\\textsc{Probabilistic proof}. First we consider a finite game that corresponds to our situation. Two players alternate, each makes $2^i$ moves. At each move the first player presents some set of $n$-bit strings, and the second player replies saying whether it \\emph{marks} this set or not. The second player loses if after some moves the number of marked sets exceeds $2^{i-k+1}(n+1)\\ln 2$ (this specific value follows from the argument below) or if there exists a string $x$ that belongs to $2^k$ sets of the first player but does not belong to any marked set.\n\nSince this is a finite game with full information, one of the players has a winning strategy. We claim that the second player can win. If it is not the case, the first player has a winning strategy. We get a contradiction by showing that the second player has a \\emph{probabilistic} strategy that wins with positive probability against any strategy of the first player. So we assume that some (deterministic) strategy of the first player is fixed, and consider the following simple probabilistic strategy: every set $A$ presented by the first player is marked with probability $p=2^{-k}(n+1)\\ln 2$.\n\nThe expected number of marked sets is $p2^i=2^{i-k}(n+1)\\ln 2$. By Chebyshev's inequality, the number of marked set exceeds the expectation by a factor $2$ with probability less than $1\/2$. So it is enough to show that the second bad case (after some move there exists $x$ that belongs to $2^k$ sets of the first player but does not belong to any marked set) happens with probability at most $1\/2$.\n\nFor that, it is enough to show that for every fixed $x$ the probability of this bad event is at most $2^{-(n+1)}$, and then use the union bound. The intuitive explanation is simple: if $x$ belongs to $2^k$ sets, the second player had (at least) $2^k$ chances to mark a set containing $x$ (when these $2^k$ sets were presented by the first player), and the probability to miss all these chances is at most $(1-p)^{2^k}$; the choice of $p$ guarantees that this probability is less than $1\/2^{-(n+1)}$. Indeed, using the bound $(1-1\/x)^x < 1\/e$, it is easy to show that $(1-p)^{2^k} < e^{-(n+1)\\ln 2}=2^{-(n+1)}$.\n\nThe pedantic reader would say that this argument is not formally correct, since the behavior of the first player (and the moment when next set containing $x$ is produced) depends on the moves of the second player, so we do not have independent events with probability $1-p$ each (as it is assumed in the computation).\\footnote{The same problem appears if we observe a sequence of independent coin tossings with probability of success $p$, select some trials (before they are actually performed, based on the information obtained so far), and ask for the probability of the event ``$t$ first selected trials were all unsuccessful''. This probability does not exceed $(1-p)^t$; it can be smaller if the total number of selected trials is less than $t$ with positive probability. This scheme was considered by von Mises when he defined random sequences using selection rules, so it should be familiar to algorithmic randomness people.} The formal argument considers for each $t$ the event $R_t$: ``after some move of the second player the string $x$ belongs to at least $t$ sets provided by the first player, but does not belong to any marked set''. Then we prove by induction (over $t$) that the probability of $R_t$ does not exceed $(1-p)^t$. Indeed, it is easy to see that $R_t$ in a union of several disjoint subsets (depending on the events happening until the first player provides $t+1$ sets containing $x$), and $R_{t+1}$ is obtained by taking a $(1-p)$-fraction in each of them.\n\n\\textsc{Constructive proof}. We consider the same game, but now allow more sets to be marked (replacing the bound $2^{i-k+1}(n+1)\\ln 2$ by a bigger bound $2^{i-k}i^2\\ln 2$) and also allow the second player to mark sets that were produced earlier (not necessarily at the current move of the first player). The explicit winning strategy for the second player performs in parallel $i-k+\\log i$ substrategies (indexed by the numbers $\\log (2^k\/i),\\ldots,i$).\n\nThe substrategy number $s$ wakes up once in $2^s$ moves (when the number of moves made by the first player is a multiple of $2^s$). It considers a family $S$ that consists of $2^s$ last sets produced by the first player, and the set $T$ that consists of all strings $x$ covered by at least $2^k\/i$ sets from $S$. Then it selects and marks some elements in $S$ in such a way that all $x\\in T$ are covered by one of the selected sets. It is done by a greedy algorithm: first take a set from $S$ that covers maximal part of $T$, then the set that covers maximal number of non-covered elements, etc. How many steps do we need to cover the entire $T$? Let us show that\n $$ (i\/2^k)n 2^{s} \\ln 2 $$\nsteps are enough. Indeed, every element of $T$ is covered by at least $2^k\/i$ sets from $S$. Therefore, some set from $S$ covers at least $\\#T2^k\/(i2^s)$ elements, i.e., $2^{k-s}\/i$-fraction of $T$. At the next step the non-covered part is multiplied by $(1-2^{k-s}\/i)$ again, and after $in2^{s-k}\\ln 2$ steps the number of non-covered elements is bounded by\n $$ \\#T (1-2^{k-s}\/i)^{in2^{s-k}\\ln 2} < 2^n (1\/e)^{n\\ln 2} = 1,$$\ntherefore all elements of $T$ are covered. (Instead of a greedy algorithm one may use a probabilistic argument and show that randomly chosen $in2^{s-k}\\ln 2$ sets from $S$ cover $T$ with positive probability; however, our goal is to construct an explicit strategy.)\n\nAnyway, the number of sets selected by a substrategy number $s$, does not exceed\n $$ in2^{s-k}(\\ln 2)2^{i-s} = in2^{i-k}\\ln 2, $$\nand we get at most $i^2 n 2^{i-k} \\ln 2$ for all substrategies.\n\nIt remains to prove that after each move of the second player every string $x$ that belongs to $2^k$ or more sets of the first player, also belongs to some selected set. For $t$th move we consider the binary representation of $t$:\n $$t=2^{s_1}+2^{s_2}+\\ldots, \\text{ where } s_1>s_2>\\ldots $$\nSince $x$ does not belong to the sets selected by substrategies with numbers $s_1,s_2,\\ldots$, the multiplicity of $x$ among the first $2^{s_1}$ sets is less than $2^k\/i$, the multiplicity of $x$ among the next $2^{s_2}$ sets is also less than $2^k\/i$, etc. For those $j$ with $2^{s_j}<2^k\/i$ the multiplicity of $x$ among the respective portion of $2^{s_j}$ sets is obviously less than $2^k\/i$. Therefore, we conclude that the total multiplicity of $x$ is less that $i\\cdot 2^k\/i=2^k$ sets of the first player and the second player does not need to care about~$x$. This finishes the explicit construction of the winning strategy.\n\nNow we can assume without loss of generality that the winning strategy has complexity at most $O(\\log(n+k+i+j))$. (In the probabilistic argument we have proved the existence of a winning strategy, but then we can perform the exhaustive search until we find one; the first strategy found will have small complexity.) Then we use this simple strategy to play with the enumeration of all $\\mathcal{A}$-sets of complexity less than $i$ and size $2^j$ (or less). The selected sets can be described by their ordinal number (among the selected sets), so their complexity is bounded by $i-k$ (with logarithmic precision). Every string that has $2^k$ different $(i*j)$-descriptions in $\\mathcal{A}$, will also have one among the selected sets, and that is what we need.\n %\n\\end{proof}\n\nAs before (for the unrestricted case), this result implies that descriptions with minimal parameters are simple with respect to the data string:\n %\n\\begin{thm}[\\cite{vv10}]\\label{thm:improving-descriptions-2-gen}\nLet $\\mathcal A$ be an enumerable family of finite sets. If a string $x$ of length $n$ has $(i*j)$-description $A\\in\\mathcal{A}$ such that $\\KS(A\\mskip 1.5mu | \\mskip 1.5 mu x)\\ge k$, then $x$ has a $((i-k+O(\\log n))*(j+O(\\log n)))$-description in $\\mathcal{A}$. If the family $\\mathcal{A}$ satisfies the condition $(3)$, then $x$ has also a $((i+O(\\log n))*(j-k+O(\\log n)))$-description in~$\\mathcal{A}$.\n\\end{thm}\n\nThis gives us the same corollaries as in the unrestricted case:\n\n\\begin{corollary}\nLet $\\mathcal{A}$ be a family of finite sets that satisfies the conditions (1)--(3). Then for every string $x$ of length $n$ three statements\n\\begin{itemize}\n\\item there exists a set $A\\in\\mathcal{A}$ of complexity at most $\\alpha$ with $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le\\beta$;\n\\item there exists a set $A\\in\\mathcal{A}$ of complexity at most $\\alpha$ with $\\delta(x,A)\\le\\beta$;\n\\item the point $(\\alpha,\\KS(x)-\\alpha+\\beta)$ belongs to $P_x^\\mathcal{A}$\n\\end{itemize}\nare equivalent with logarithmic precision (the constants before the logarithms depend on the choice of the set $\\mathcal{A}$).\n\\end{corollary}\n\nIf we are interested in the uniform statements true for every enumerable family $\\mathcal{A}$, the same arguments prove the following result:\n\n\\begin{proposition}\nLet $\\mathcal{A}$ be an arbitrary family of finite sets enumerated by some program $p$. Then for every $x$ of length $n$ the statements\n\\begin{itemize}\n\\item there exists a set $A\\in\\mathcal{A}$ such that $d(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\le \\beta$;\n\\item there exists a set $A\\in\\mathcal{A}$ such that $\\delta(x,A)\\le\\beta$\n\\end{itemize}\nare equivalent up to $O(\\KS(p)+\\log\\KS(A)+\\log n+\\log\\log\\#A)$-change in the parameters.\n\\end{proposition}\n\n\\ver{\n\\section{Strong models}\\label{sec:strong-models}\n\n\\subsection{Information in minimal descriptions}\n\nA possible way to bring the theory in accordance to our intuition is to change the definition of ``having the same information''. Although we have not given that definition explicitly, we have adopted so far the following viewpoint: $x$ and $y$ have the same (or almost the same) information if both conditional complexities $\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu y),\\KS(y\\mskip 1.5mu | \\mskip 1.5 mu x)$ are small. If only one complexity, say $\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu y)$, is small, we said that all (or almost all) information contained in $x$ is present in $y$.\n\nNow we will adopt a more restricted viewpoint and say that $x$ and $y$ have the same information if there are short \\emph{total} (everywhere defined) programs mapping $x$ to $y$ and vice versa. From this viewpoint we cannot say anymore that a string $x$ and its shortest program $x^*$ have the same information: for example, $x$ may be non-stochastic while $x^*$ is always stochastic, so there is no short total program that maps $x^*$ to $x$ because of Proposition~\\ref{prop:stoch-cons}.\\footnote{It is worth to mention that on the other hand, for every string $x$ there is an almost minimal program for $x$ that can be obtained from $x$ by a simple total algorithm~\\cite[Theorem 17]{ver2015}.} Let us mention that if $x$ and $y$ have the same information in this new sense, then there exists a simple computable \\emph{bijection} that maps $x$ to $y$ (so they have the same properties if the property is defined in the computability language), see~\\cite{mezhirov} for the proof.\n\nFormally, let us define the total conditional complexity with respect to a computable function $D$ of two arguments, as\n $$\n\\KT_D(x\\mskip 1.5mu | \\mskip 1.5 mu y)=\\min\\{l(p)\\mid D(p,y)=x,\\text{ and $D(p,y')$ is defined for all } y'\\}.\n $$\n(Note that $D$ is not required to be total, but we consider only $p$ such that $D(p,y')$ is defined for all $y'$.)\n\nThere is a computable function $D$ such that $\\KT_D$ is minimal up to an additive constant. Fixing any such $D$ we obtain the \\emph{total conditional complexity} $\\KT(x\\mskip 1.5mu | \\mskip 1.5 mu y)$. In other way, we may define $\\KT(x\\mskip 1.5mu | \\mskip 1.5 mu y)$ as the minimal plain complexity of a total program that maps $y$ to $x$.\n\nWe will think that $y$ has all (or almost all) the information from $x$ if $\\KT(x\\mskip 1.5mu | \\mskip 1.5 mu y)$ is negligible. Formally, we write $x\\stackrel{\\varepsilon}{\\rightarrow}y$ if $\\KT(y\\mskip 1.5mu | \\mskip 1.5 mu x)\\le\\varepsilon$ and we call $x$ and $y$ \\emph{$\\varepsilon$-equivalent} and write $x\\stackrel{\\varepsilon}{\\leftrightarrow}y$, if both $\\KT(y\\mskip 1.5mu | \\mskip 1.5 mu x)$ and $\\KT(x\\mskip 1.5mu | \\mskip 1.5 mu y)$ are at most $\\varepsilon$.\n\n\\begin{proposition}\\label{prop:equivalence}\nIf $x\\stackrel{\\varepsilon}{\\leftrightarrow}y$ then the sets $P_x$ and $P_y$ are in $O(\\varepsilon)$ neighborhood of each other.\n\\end{proposition}\n\n\\begin{proof}\nIndeed, if $A$ is an $(i*j)$-description of $x$ and $p$ is a total program witnessing $x\\stackrel{\\varepsilon}{\\rightarrow}y$, then the set $B=\\{D(p,x')\\mid x'\\in A\\}$ is an $((i+O(\\varepsilon))*j)$-description for $y$. (We need $p$ to be total, as otherwise we cannot produce the list of $B$-elements from the list of $A$-elements and $p$.)\n\\end{proof}\n\n\\subsection{An attempt to separate ``good'' models from ``bad'' ones}\n\nNow we have a more fine-grained classification of descriptions and can try to distinguish between descriptions that were equivalent in the former sense. For example, consider a string $xy$ where $y$ is random conditionally to $x$. Let $A$ be a model for $xy$ consisting of all extensions of $x$ (of the same length). This model looks good (in particular, it has negligible optimality deficiency). On the other hand, we may consider a standard model $B$ for $xy$ of the same (or smaller) complexity. It also has negligible optimality deficiency but looks unnatural. In this section we are interested in the following question: how can we formally distinguish good models like $A$ from bad models like $B$? We will see that at least for some strings $u$ the value $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu u)$ can be used to distinguish between good and bad models for $u$. (Indeed, in our example $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu xy)$ is small, while $\\KT(B\\mskip 1.5mu | \\mskip 1.5 mu xy)$ can be large.)\n\n\\begin{definition}\nA set $A\\ni x$ is an \\emph{$\\varepsilon$-strong model} (or \\emph{statistic}) for a string $x$ if $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu x)\\le\\varepsilon$.\n\\end{definition}\n\nFor instance, the model $A$ discussed above is an $O(\\log n)$-strong model for $x$. On the other hand, we will see later that, if $y$ is chosen appropriately, then no standard description $B$ of the same complexity and log-cardinality as $A$ is an $\\varepsilon$-strong model for $x$, even for $\\varepsilon=\\Omega(n)$.\n\nThe following proposition explains the meaning of a strong model by providing an equivalent definition.\nA finite family of sets $\\mathcal{A}$ is called \\emph{partition} if \nfor every $A_1, A_2 \\in \\mathcal{A}$ \nwe have $A_1 \\cap A_2 \\not= \\varnothing \\Rightarrow A_1 = A_2$. \nFor any partition we can define its complexity. \nThe lemma states that strong statistics are those that belong \nto simple partitions.\n\n\n\\begin{proposition}\\label{part}\nAssume that $A$ is a model for $x$ that belongs to a partition of complexity $\\varepsilon$. Then \n$A$ is an $\\varepsilon+O(1)$-strong model for $x$.\nConversely, assume that $A$ is an $\\varepsilon$-strong statistic for \u0430 string $x$ of length $n$. Then there is a partition $\\mathcal{A}$ \nof complexity at most $\\varepsilon + O(\\log n)$ and a model $A'\\in\\mathcal A$ for $x$ such that: $\\# A' \\le \\# A$ and \nboth $\\CT(A \\mskip 1.5mu | \\mskip 1.5 mu A')$ and $\\CT(A' \\mskip 1.5mu | \\mskip 1.5 mu A)$ are at most $\\varepsilon + O(\\log n)$. \n\\end{proposition}\n\\begin{proof}\nAssume that $A$ is a model for $x$ that belongs to a partition $\\mathcal A$ of complexity $\\varepsilon$. \nThe program that maps every given string $x'$ to the set $A'\\in \\mathcal A$ which $x$ belongs to (if $x'$ belongs to no set in $\\mathcal A$,\nthen the program maps it, say, to the empty set) is total and has length at most $\\varepsilon+O(1)$.\n\nConversely, assume that $A$ is an $\\varepsilon$-strong statistic for a string $x$ of length $n$.\nThen there is a total program $p$ such that \n$p(x) = A$ and $|p| \\le \\varepsilon$. \n\nConsider the set $X$ of all strings $x'$ with $x'\\in p(x')$.\nObviously, $x\\in X$.\nPartition $X$ according to the value of $p(x')$:\nstrings $x'$ and $x''$ are in the same set of the partition if\n$p(x')=p(x'')$. The constructed partition $\\mathcal A$ has complexity at most $\\varepsilon + O(\\log n)$. \nIt includes the set $A'=\\{x'\\in X\\mid p(x')=A\\}$, which \nincludes $x$ and can be obtained from $x$ by a total program\nof length at most $\\varepsilon+O(\\log n)$: that program maps a given string $x'$ to the set $\\{x''\\in X\\mid p(x'')=p(x')\\}$.\n\nSince $A'\\subset A$, we have $ \\# A' \\le \\# A$. It remains to show that\nthat both $\\CT(A \\mskip 1.5mu | \\mskip 1.5 mu A')$ and \n$\\CT(A' \\mskip 1.5mu | \\mskip 1.5 mu A )$ are less than \n$|p| + O(\\log n) = \\varepsilon + O(\\log n)$.\nIndeed, $A'$ can be obtained from $A$ by a total program of length $|p| + O(\\log n)$ that maps \na given set $B$ to $\\{x'\\in X\\mid p(x')=B\\}$. On the other hand, $A$ can be obtained from $A'$\nby a total program of length $|p| + O(1)$ that for a given set $B$ picks any element $x'$ from $B$ and computes $p(x')$.\nIf $D$ is empty then that program outputs, say, the empty set. Recall that $A'$ is\nnon-empty, as it includes $x$. \n\\end{proof}\n\n\n\n\nStrong models satisfy an analog of Proposition~\\ref{prop:description-shift} (the same proof works):\n\n\\begin{proposition}\n \\label{prop:description-shift-1}\nLet $x$ be a string and $A$ be an $\\varepsilon$-strong model for $x$. Let $i$ be a non-negative integer such that $i\\le \\log\\# A$. Then there exists an $\\varepsilon+O(\\log i)$-strong model $A'$ for $x$ such that $\\#A' \\le \\# A\/ 2^i$ and $\\KS(A')\\le \\KS(A)+i + O(\\log i)$.\n\\end{proposition}\n\nTo take into account the strength of models, we may consider the set\n$$\nP_x(\\varepsilon)=\\{(i,j)\\mid x\\text{ has an $\\varepsilon$-strong $(i*j)$-description}\\}.\n$$\nObviously, we have\n$$\nP_x(\\varepsilon)\\subset P_x = P_x(n+O(1))\n$$\nfor all strings $x$ of length $n$ and for all $\\varepsilon$.\n\nIf the set $P_x(\\varepsilon)$ is not much smaller than $P_x$ for a reasonably small $\\varepsilon$, we will say that $x$ is a ``normal'' string and otherwise we call $x$ ``strange''. More precisely, a string $x$ is called $(\\varepsilon,\\delta)$-\\emph{normal} if $P_x$ is in $\\delta$-neighborhood of $P_x(\\varepsilon)$. Otherwise, $x$ is called \\emph{$(\\varepsilon,\\delta)$-strange}.\n\nIt turns out that there are $\\sqrt{n\\log n},O(\\log n)$-normal strings with any given set $P_x$ that satisfies the conditions of Theorem~\\ref{stat-any-curve}. On the other hand, there are $\\Omega(n),\\Omega(n)$-strange strings of length $n$. We are going to state these facts accurately.\n\n\\begin{thm}[\\cite{milovanov-csr}]\\label{stat-any-curve-1}\nLet $k\\le n$ be two integers and let $t_0 > t_1 > \\ldots > t_k$ be a strictly decreasing sequence of integers such that $t_0\\le n$ and $t_k=0$. Then there exists a string $x$ of complexity $k+O(\\sqrt{n\\log n})$ and length $n+O(\\log n)$ for which the distance between both sets $P_x$ and $P_x(O(\\log n))$ and the set $T=\\{( i,j)\\mid (i\\le k)\\Rightarrow (j\\ge t_i)\\}$ is at most $O(\\sqrt{n\\log n})$.\n\\end{thm}\n\n\\begin{proof}\nConsider the family $\\mathcal A$ of all cylinders, i.e., the family of all the sets of the form\n$\\{ur\\mid l(r)=m\\}$ for different strings $u$ and natural numbers $m$. Sets from this family have the following feature: if $A\\ni x$ then $A$ is an $O(\\log n)$-strong model for $x$. Hence for all strings $x$ we have $P_x^{\\mathcal A}=P_x^{\\mathcal A}(O(\\log n))$.\n\nBy Theorem~\\ref{thm:family-curve} and Remark~\\ref{rem:family} there is a string $x$ of length $n+O(\\log n)$ and complexity $k+O(\\sqrt{n\\log n})$ such that all sets $P_x, P_x^{\\mathcal A},T$ are $O(\\sqrt{n\\log n})$-close to each other. Hence all the three sets are close to the set $P_x^{\\mathcal A}(O(\\log n))$ as well. As the set $P_x(O(\\log n))$ includes the latter set and is included in $P_x$, all the three sets are close to the set $P_x(O(\\log n))$ as well.\n\\end{proof}\n\nThe next theorem from~\\cite{ver2015} shows that ``strange'' strings do exist.\n\n\\begin{thm}\\label{t1}\nAssume that natural numbers $k,n,\\varepsilon$ satisfy the inequalities\n$$\nO(1)\\le\\varepsilon\\le k\\le n.\n$$\nThen there is a string $x$ of length $n$ and complexity $k+O(\\log n)$ such that the sets $P_x$ and $P_x(\\varepsilon)$ are $O(\\log n)$-close to the sets shown on Fig.~\\textup{\\ref{f6}}. (The set $P_x$ is to the right of the dashed line. The set $P_x(\\varepsilon)$ is to the right of the solid line. The difference between the sets\nhas the shape of a parallelogram.)\n\\end{thm}\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=1]{mdl-e-14.pdf}\n\\caption{The sets $P_x$ and $P_x(\\varepsilon)$ for the strange string from Theorem~\\ref{t1}, with $O(\\log n)$-precision. The set $P_x$ is to the right of the dashed line. The set $P_x(k)$ is to the right of the solid line.}\\label{f6}\n\\end{center}\n\\end{figure}\n\n\nWe will prove this theorem later in Section~\\ref{s:strange}. To illustrate its statement\nlet $k=2n\/3$ and $\\varepsilon=n\/3$ in Theorem~\\ref{t1}. Then the sets $P_x$ and $P_x(n\/3)$ are almost $n\/3$-apart, since the point $(n\/3,n\/3)$ is in the $O(\\log n)$-neighborhood of $P_x$ while all points from $P_x(n\/3)$ are $(n\/3-O(\\log n))$-apart from $(n\/3,n\/3)$ (in $l_\\infty$-norm). Thus the string $x$ is $(n\/3,n\/3-O(\\log n))$-strange.\n\n\n\n\\medskip\nRecall that we have introduced the notion of a strong model to separate good models from bad ones. We will now present some results that justify this approach. The following theorem by Milovanov states, roughly speaking, that there exists a string $x$ of length $n$ and a strong model $A$ for $x$ such that the parameters (complexity, log-cardinality) of every strong \\emph{standard} model $B$ for $x$ are $\\Omega(n)$-far from those of $A$.\n\n\\begin{thm}\\label{thm:separation}\nFor some positive $c$ for almost all $k$ there is a string $x$ of length $n=4k$ whose profile $P_x$ is $O(\\log n)$-close to the gray set shown on Fig.~\\textup{\\ref{f4}}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=1]{m41.pdf}\n\\end{center}\n\\caption{The profile $P_x$ of a string $x$ from Theorem \\ref{thm:separation}.}\n\\label{f4}\n\\end{figure}\nsuch that\n\\begin{itemize}\n\\item there is an $O(\\log n)$-strong model $A$ for $x$ with complexity $k+O(\\log n)$ and log-cardinality $2k$ \\textup(that model witnesses the point $(k,2k)$ on the border of $P_x$\\textup), but\n\\item for every $m\\ge \\KS(x)$ and for every simple enumeration of strings of complexity at most $m$ the standard model $B$ for $x$ obtained from that enumeration is either not strong for $x$ or its parameters are far from the point $(k,2k)$. More specifically, if $B$ is a\nmodel for $x$ obtained from an enumeration provided by some program $q$, then \nat least one of the values \n$\\KT(B\\mskip 1.5mu | \\mskip 1.5 mu x),\\KS(q),|\\KS(B)-k|,|\\log\\#B-2k|$ is larger than $ck$.\n\\end{itemize}\n\\end{thm}\n\nWe will prove this theorem in Section~\\ref{s:normal}.\n\n\\subsection{Properties of strong models}\n\nOnce we have decided that non-strong descriptions are bad, it is natural to restrict ourselves to strong descriptions with negligible randomness deficiency (and hence negligible optimality deficiency). \n\nConsider some $n$-bit string $x$. Assume that $A$ is an $\\varepsilon$-strong description for $x$. Let $u$ be the ordinal number of $x$ in $A$ with respect to some fixed order. Then $\\KT(x\\mskip 1.5mu | \\mskip 1.5 mu A,u)=O(1)$ and $\\KT(A,u\\mskip 1.5mu | \\mskip 1.5 mu x)\\le \\varepsilon+O(1)$ (the latter inequality holds since $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu x)\\le\\varepsilon$ and $u$ can be easily found when $x$ and $A$ are known). \nThat is, $x\\stackrel{\\varepsilon+O(1)}{\\longleftrightarrow}(A,u)$. As the pairs $(A,u)$ are naturally partitioned into classes, in this way we obtain and alternative proof\nof Proposition~\\ref{part}: the image of that partition under the total mapping witnessing $(A,u)\\stackrel{O(1)}{\\rightarrow}x$ satisfies the second claim of Proposition~\\ref{part}.\n\nAssume further that the randomness deficiency of $x$ in $A$ is at most $\\varepsilon$. \nAs $u$ is random and independent of $A$ (with precision $\\varepsilon$; note that $\\KS(u\\mskip 1.5mu | \\mskip 1.5 mu A)\\approx \\KS(x\\mskip 1.5mu | \\mskip 1.5 mu A)\\ge \\log\\#A-\\varepsilon$), the sets $Q_{A,i}$ and $Q_{A}$ are $\\varepsilon+O(\\log n)$-close (Theorem~\\ref{prop:add-noise}). On the other hand, the sets $Q_{A,u}$ and $Q_x$ are $\\varepsilon+O(1)$-close by Proposition~\\ref{prop:equivalence}. Thus we obtain the first property of strong models:\n\n\\begin{proposition}\\label{prop:upward}\nIf both $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu x)$ and $\\log\\# A-\\KS(x\\mskip 1.5mu | \\mskip 1.5 mu A)$ are at most $\\varepsilon$, then the sets $Q_x$ and $Q_{A}$ are $O(\\varepsilon+\\log l(x))$-close.\n\\end{proposition}\n\n\nAssume that $A$ is an $\\varepsilon$-strong model for $x$ with negligible randomness deficiency and~$\\varepsilon$; for simplicity we ignore these negligible quantities in the sequel. Assume that $A$ is normal in the sense described above. Then the string $x$ is normal as well. Indeed, for every pair $(i,j)\\in P_x$ with $i\\le \\KS(A)$ the pair $(i,j-\\log\\# A)$ is in $P_{A}$ (Theorem~\\ref{rem:add-noise}; note that $x$ is equivalent to $(A,u)$ and $u$ is random with condition $A$) and hence there is a strong $(i*(j-\\log\\# A))$-description $\\mathcal{B}$ for $A$. Consider the ``lifting'' of $\\mathcal{B}$, that is, the union of all sets from $\\mathcal{B}$ that have approximately the same size as $A$. It is a strong $(i*j)$-description for $x$.\n\nIt remains to consider pairs $(i,j)\\in P_x$ where $i\\ge \\KS(A)$. Then $i+j\\ge \\KS(A)+\\log\\# A=\\KS(x)$. Hence the subset of $A$ consisting of all strings $x'$ whose ordinal number in $A$ has the same $i-\\KS(A)$ leading bits as the ordinal number of $x$, is a strong $(i*j)$-description for $x$.\n\nIt turns out that for minimal models the converse is true as well. A model $A$ for $x$ is called \\emph{$(\\delta,\\varkappa)$-minimal} if there is no model $B$ for $x$ with $\\KS(B)\\le \\KS(A)-\\delta$ and $\\delta(x,B)\\le \\delta(x,A)+\\varkappa$. The smaller $\\delta$ and the larger $\\varkappa$ are the stronger the property of $(\\delta,\\varkappa)$-minimality is.\n\n\nRecall that an $\\varepsilon$-sufficient statistic is a model whose optimality deficiency is smaller than~$\\varepsilon$.\n\n\\begin{thm}[\\cite{milovanov-csr}]\n\\label{thm:hereditary}\nFor some value $\\varkappa=O(\\log n)$ the following holds. Assume that $A$ is an $\\varepsilon$-strong \nand $\\varepsilon$-sufficient \nstatistic for an $(\\varepsilon,\\varepsilon)$-normal string $x$ of length $n$. Assume also that $A$ is a $(\\delta,\\varkappa)$-minimal model for $x$. Then $A$ is $(O(\\delta+(\\varepsilon+\\log n)\\sqrt n),O(\\delta+(\\varepsilon+\\log n)\\sqrt n))$-normal.\n\\end{thm}\n\n\\begin{remark}\nIn the original theorem from~\\cite{milovanov-csr} it is claimed only that $A$ is $(O((\\delta+\\varepsilon+\\log n)\\sqrt n),O((\\delta+\\varepsilon+\\log n)\\sqrt n))$-normal.\nHowever, the arguments from~\\cite{milovanov-csr} prove the theorem as stated here.\n\\end{remark}\n\nThis theorem will be proved in Section~\\ref{s:hereditary}.\nThe next theorem states that the total conditional complexity of any strong, sufficient and minimal statistic for $x$ conditioned by any other sufficient statistic for $x$ is negligible.\n\n\\begin{thm}[\\cite{ver}]\\label{thm:step-wise}\nFor some value $\\varkappa=O(\\log n)$ the following holds. Assume that $A,B$ are $\\varepsilon$-sufficient statistics for a string $x$ of length $n$. Assume also that $A$ is a $(\\delta,\\varepsilon+\\varkappa)$-minimal statistic for $x$.\nThen $\\K(A\\mskip 1.5mu | \\mskip 1.5 mu B) =O(\\delta+\\log n)$. Moreover, if, additionally, $A$ is an $\\varepsilon$-strong model for $x$,\nthen $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu B) =O(\\varepsilon+\\delta+\\log n)$. \n\\end{thm}\n\nThis theorem can be interpreted as follows: assume that we have removed some noise from a given data string $x$ by finding its description $B$ with negligible optimality deficiency. Let $A$ be any ``ultimately denoised'' model for $x$, i.e., a minimal model for $x$ with negligible optimality deficiency. Then $\\KS(A\\mskip 1.5mu | \\mskip 1.5 mu B)$ is negligible (the first part of Theorem~\\ref{thm:step-wise}). \nHence to obtain the ``ultimately denoised'' model for $x$ we do not need $x$: any such model can be obtained from $B$ by a short program. The second part of \nTheorem~\\ref{thm:step-wise} shows that any such \\emph{strong} model $A$ can be obtained from $B$ by a short \\emph{total} program.\n\n\n\\subsection{Strange strings}\\label{s:strange}\nIn this section we prove the existence of strange strings (Theorem \\ref{t1}). Then we prove that there are many such strings (Theorem \\ref{t3}).\n\nLet $A\\mapsto [A]$ denote a computable bijection from the family of finite sets to the set of binary strings. This bijection will be used \nto encode finite sets of strings. \n\\begin{lemma}\\label{l1}\nAssume that $A$ is an $\\varepsilon$-strong statistic for a string $x$\nof length $n$. Let $y=[A]$ be the code of $A$.\nThen $y$ has an $(\\varepsilon+O(\\log n)),n$-description.\n\\end{lemma}\n\\begin{proof}\nLet $p$ be a string of length at most $\\varepsilon$ \nsuch that $p(x)=y$ and $p(x')$ is defined for all strings $x'$.\nConsider the set $\\{p(x') \\mid x'\\in \\{0,1\\}^n \\}$.\nIts cardinality is at most $2^n$ and complexity at most $\\varepsilon + O(\\log n)$.\n\\end{proof} \n\\begin{proof}[Proof of Theorem \\ref{t1}]\nTo prove the theorem\nit suffices to find a set $A\\subset \\{0,1\\}^n$ \nwith\\\\\n(a) $\\K(A)\\le \\varepsilon+O(\\log n)$, $\\log \\# A\\le k-\\varepsilon$\\\\\nwhich is not covered by the union of sets from the union of the following\nthree families:\\\\\n(b) the family $\\mathcal B$ consisting of all sets \n$B\\subset\\{0,1\\}^*$ with with $\\K(B)\\le \\varepsilon$, $\\log \\#B\\le n-\\varepsilon-4$,\\\\\n(c) the family $\\mathcal C$ consisting of all sets $M$ with \n$\\K(M)\\le k$, $\\log \\# M\\le n-k-4$\nwhose code $[M]$ has a $(\\varepsilon+O(\\log n)),n$-description, and\\\\\n(d) the family $\\mathcal D$ consisting of all singletons sets $\\{x\\}$ where $\\K(x)< k$.\\\\\n\nIndeed, assume that we have such set $A$.\nAs $x$ we can take any non-covered string in $A$.\nNotice that item (a) implies that \nthe complexity of $x$ is at most $k+O(\\log n)$, and item (d)\nimplies that it is at least $k$. \nThus the membership of the pair $(k+O(\\log n), O(\\log n))$ in $P_x(\\varepsilon)$\nis witnessed by the singleton $\\{x\\}$, provided $\\varepsilon$ is greater than\nthe constant from the inequality $\\CT(\\{x\\}\\mskip 1.5mu | \\mskip 1.5 mu x)=O(1)$. \nThe membership of the pair $(O(\\log n), n)$ in $P_x(\\varepsilon)$ and \n$P_x$ will be witnessed by the set \nof all strings of length $n$, provided $\\varepsilon$ is greater than\nthen constant from the inequality $\\CT(\\{0,1\\}^n\\mskip 1.5mu | \\mskip 1.5 mu x)=O(1)$. \nThe membership of the pair $(\\varepsilon+O(\\log n), k-\\varepsilon)$ in $P_x$ will be witnessed by the set $A$. \n\nThe upper bounds for $P_x(\\varepsilon)$ and $P_x$ follow from (b), (c), and Lemma~\\ref{l1}.\nIndeed, item (b) implies that the pair $(\\varepsilon,n-\\varepsilon-O(1))$ is not in $P_x$ (hence by\nProposition~\\ref{prop:description-shift-1} for any $i\\le \\varepsilon$ the pair $(i-O(\\log n),n-i)$ is not in $P_x$ either).\nItem (c) implies that the pair $(k-O(\\log n), n-k-O(1))$ is not in $P_x(\\varepsilon)$.\n\nLet us show that there is a set $A$ of $n$-bit strings satisfying (a), (b), (c) and (d).\nA direct counting reveals\nthat the family $\\mathcal B\\cup\\mathcal C\\cup\\mathcal D$ covers at most\n$$\n2^{\\varepsilon+1}2^{n-\\varepsilon-4}+2^{k+1}2^{n-k-4}+2^{k}\\le2^{n-3}+2^{n-3}+2^{n-4}<2^{n-1}\n$$\nstrings\nand hence at least half of all $n$-bit strings are non-covered.\nHowever we cannot let $A$ be any $2^{k-\\varepsilon}$-element \nnon-covered set of $n$-bit strings, as in that case \n$\\K(A)$ could be large.\n\n\nWe first show how to find $A$, as in (a), that is not covered\nby $\\mathcal B\\cup\\mathcal D$ (but may be covered by $\\mathcal C$). This may be done using the same technique as in the proof of Theorem~\\ref{stat-any-curve}. \nTo construct $A$ notice that both the families \n$\\mathcal B$ and $\\mathcal D$ can be enumerated given $k,\\varepsilon,n$ \nby running the universal machine $U$ in parallel on all inputs.\nWe start such an enumeration and construct $A$ ``in several attempts''.\nDuring the construction \nwe maintain the list of all strings covered by sets\nfrom $\\mathcal B\\cup\\mathcal D$ \nenumerated so far.\nSuch strings are called \\emph{marked}.\nInitially, no strings are marked and\n$A$ contains the lexicographic first $2^{k-\\varepsilon}$ strings of length $n$.\nEach time a new set $B\\in\\mathcal B$ appears, all its elements \nreceive a b-mark and we replace $A$ by any set \nconsisting of $2^{k-\\varepsilon}$ yet non-marked $n$-bit strings.\nEach time a new set $\\{x\\}$ in \n$D$ appears, the string $x$ receives a\nd-mark, but we do not immediately replace $A$.\nHowever we do that when all strings in $A$ receive a d-mark,\nreplacing it by any set \nconsisting of $2^{k-\\varepsilon}$ yet non-marked $n$-bit strings.\nThe above counting shows that such replacements are always possible.\n\nThe last version of $A$ \n(i.e. the version obtained after the last set \nin $\\mathcal B\\cup\\mathcal D$ have appeared)\nis the sought set. \nIndeed, by construction $\\# A = 2^{k-\\varepsilon}$ \nand $A$ is not covered by sets in $\\mathcal B\\cup\\mathcal D$. \nIt remains to verify that $\\K(A)\\le \\varepsilon+O(\\log n)$. This follows \nfrom the fact that $A$ \nis replaced at most $O(2^{\\varepsilon})$ times, and hence \ncan be identified by the number of its replacements and $\\varepsilon,k,n$\n(we run the above construction of $A$\nand wait until the given number of replacements are made).\n\nWhy is $A$ \nreplaced at most $O(2^{\\varepsilon})$ times? The number of replacements\ncaused by appearance of a new set $B\\in\\mathcal B$ is \nat most $2^{\\varepsilon+1}$. The \nnumber of strings with a d-mark is at most $2^{k}$\nand hence $A$ can be replaced at most $2^{k}\/2^{k-\\varepsilon}=2^{\\varepsilon}$ times \ndue to receiving d-marks.\n\nNow we have to take into account \nstrings covered by sets from the family $\\mathcal C$. \nTo this end modify the construction as follows: put a c-mark\non all strings from each set $C$ enumerated into \n$\\mathcal C$ and replace $A$ each time when all its elements\nhave received c or d marks (or when a new set is enumerated into $\\mathcal B$). \n\nHowever this modification alone is not enough.\nIndeed, up to $\\Omega(2^{n})$ strings may receive a c-mark,\nand hence $A$ might be replaced up to $\\Omega(2^{n-(k-\\varepsilon)})$ times due to c-marks.\nThe crucial modification is the following: each time $A$ is replaced, its new version \nis not just any set of $2^{k-\\varepsilon}$ non-marked $n$-bit strings but a carefully\nchosen such set. \n\nTo explain how to choose $A$ we \nfirst represent $\\mathcal C$ as an intersection of two families,\n$\\mathcal C'$ and $\\mathcal C''$.\nThe first family $\\mathcal C'$ consists of all sets $M$ with\n$\\K(M)\\le k$ and the second family $\\mathcal C''$\nof all sets $C$ with $\\log \\# C \\le n-k-4$\nwhose code $[C]$ has a $(\\varepsilon+O(\\log n),n)$-description.\nThe first family is small (less than $2^{k+1}$ sets)\nand the second family has only small sets (at most $2^{n-k-4}$-element sets)\nand is not very large ($\\# \\mathcal C'' =2^{O(n)}$). \nBoth families can be enumerated given $\\varepsilon,k,n$\nand, moreover, the sets from $\\mathcal C''$ appear in the enumeration in \nat most $2^{\\varepsilon+O(\\log n)}$ portions. Due to this property of \n$\\mathcal C''$ we can update $A$ each time a new portion of sets in \n$\\mathcal C''$ appears---this will increase the number of replacements of $A$\nby $2^{\\varepsilon+O(\\log n)}$, which is OK. \n\nThe crucial change in construction\nis the following: each time $A$ is replaced, its new version \nis \\emph{a set \nof $2^{k-\\varepsilon}$ non-marked $n$-bit strings that has at most $O(n)$ common strings \nwith every set from the part of $\\mathcal C''$ \nenumerated so far}. (We will show later that such a set always exists.)\n\nWhy does this solve the problem? \nThere are two types \nof replacements of $A$: \nthose caused by enumerating a new set in $\\mathcal B$ or a new bunch of sets in \n$\\mathcal C''$\nand those caused by that all elements in $A$ have received c- or d-marks.\nThe number of replacement of \nthe first type is at most $2^{\\varepsilon+O(\\log n)}$. \nReplacements of the second type \nare caused by enumerating new singleton sets \nin $\\mathcal D$ and by enumerating new sets $C$ in \n$\\mathcal C'$ which were enumerated into $\\mathcal C''$\non earlier steps. Due to the careful choice of $A$,\nwhen each such set $C$ appears in the enumeration of $\\mathcal C'$\nit can mark only \n$O(n)$ strings in the current version of $A$.\nThe total number of sets in $\\mathcal C'$ is at most $2^{k+1}$.\nTherefore the total number of events ``a string in the current version of \n$A$ receives a c-mark'' is at most $O(n2^{k})$. \nThe total number of d-marks is at most $2^{k}$.\nHence the number of replacements of the second type is at most \n$$\n(O(n2^{k})+2^{k})\/2^{k-\\varepsilon}=O(n2^{\\varepsilon}).\n$$\n\n\nThus it remains to show that we indeed can always \nchoose $A$, as described above. This will follow\nfrom a lemma that says that in a large universe\none can always choose a large set \nthat has a small intersection with every set from\na given small family of small sets.\n\n\\begin{lemma}\nAssume that a finite family $\\mathcal C$ of \nsubsets of a finite universe $U$ is given\nand each set in $\\mathcal C$ has at most $s$ elements. \nIf \n$$\n\\# \\mathcal C \\binom{N}{t+1}\\left(\\frac{s}{|U|-t}\\right)^{t+1}<1\n$$\nthen there is an \n$N$-element set $A\\subset U$ \nthat has at most $t$ common elements with each set in $\\mathcal C$. \n\\end{lemma}\n\\begin{proof}\nTo prove the lemma we use probabilistic method.\nThe first element $a_1$ of $A$ is chosen at random \namong all elements in $U$ \nwith uniform distribution,\nthe second element $a_2$ is chosen with uniform distribution\namong the remaining elements and so forth.\n\nWe have to show that the statement of the theorem \nholds with positive probability.\nTo this end note that for every fixed $C$ in $\\mathcal C$ \nand for every fixed set of indexes \n$\\{i_1,\\dots,i_{t+1}\\}\\subset\\{1,2,\\dots,N\\}$\nthe probability that \\emph{all} $a_{i_1},\\dots,a_{i_{t+1}}$ \nfall in $C$ is at most \n$\\left(\\frac{s}{|U|-t}\\right)^{t+1}$.\nThe number of sets of indexes as above is $\\binom{N}{t+1}$.\nBy union bound the probability that a random set $A$ \ndoes not satisfy the lemma is \nupper bounded by the left hand side of the displayed inequality.\n\\end{proof}\n\nWe apply the lemma for \n$U$ consisting of all non-marked $n$-bit strings,\nfor $N=2^{k-\\varepsilon}$ and for $\\mathcal C$ consisting of\nall sets in $\\mathcal C''$ appeared so far. Thus $s=2^{n-k-4}$, $\\# U \\ge 2^{n-1}$, $\\# \\mathcal C =2^{O(n)}$, and we need to show that\nfor some $t=O(n)$ it holds\n$$\n2^{O(n)}\\binom{2^{k-\\varepsilon}}{t+1}\n\\left(\\frac{2^{n-k-4}}{2^{n-1}-t}\\right)^{t+1}<1,\n$$\nwhich easily\nfollows from the inequality $\\binom{2^{k-\\varepsilon}}{t+1}\\le 2^{(k-\\varepsilon)(t+1)}$.\n\\end{proof}\nTheorem~\\ref{t1} does not say anything about how rare are strange strings.\nSuch strings are rare, as \nfor majority of strings $x$ of length $n$ the set $\\{0,1\\}^n$ is a strong MSS for $x$.\nA more meaningful question is whether such strings might \nappear with high probability in a statistical experiment. More specifically, \nassume that we sample a string $x$ in a given set $A\\subset\\{0,1\\}^n$, where all elements \nare equiprobable. Might it happen that with high probability \n(say with probability $99\\%$) \n$x$ is strange? \nAn affirmative answer to this question is given in the following\n\n\\begin{thm}\\label{t3}\nAssume that natural \n$k,n,\\varepsilon,\\delta$ satisfy the inequalities \n$$\nO(1)\\le\\varepsilon\\le k\\le n,\\quad \\delta \\le k-\\varepsilon.\n$$\nThen there is set $A\\subset\\{0,1\\}^n$ of cardinality $2^{k-\\varepsilon}$ and complexity at most $\\varepsilon+O(\\delta+\\log n)$\nsuch that all but $2^{k-\\varepsilon-\\delta}$ its elements $x$ have complexity $k+O(\\delta+\\log n)$ and \nthe sets $P_x$ and $P_x(\\varepsilon)$ are $O(\\delta+\\log n)$-close to the sets shown on Fig.~\\ref{f6}.\n\\end{thm}\n\nTheorem~\\ref{t3}\nis proved similarly to Theorem~\\ref{t1}. The only difference is that we change $A$ each time\nwhen at least $2^{k-\\varepsilon-\\delta}$ strings in $A$ receive \nc- or d-marks. \nAs the result, the number of changes of $A$ will increase $2^\\delta$ times and the complexity of $A$ will increase by $\\delta$.\n\n\\subsection{Strong sufficient statistics}\nIn this section we prove Theorem \\ref{thm:step-wise}.\n\n\\emph{The proof of the first claim $\\K(A\\mskip 1.5mu | \\mskip 1.5 mu B)\\le O(\\delta+\\log n)$}.\nRecall the notion of a standard description and Proposition \\ref{prop:better-std}. By that \nproposition there exist standard descriptions $A',B' \\ni x$ that are ``better'' \nthan $A,B$, respectively, i.e. \n\\begin{align*}\n\\delta(x,A') &\\le \\delta (x, A) + O(\\log n),\\quad \\K(A' \\mskip 1.5mu | \\mskip 1.5 mu A) = O(\\log n),\\\\\n\\delta(x,B') &\\le \\delta (x, B) + O(\\log n),\\quad \\K(B' \\mskip 1.5mu | \\mskip 1.5 mu B) = O(\\log n).\n\\end{align*} \n Let $i,j$ denote the complexities of $A',B'$, respectively.\n Ignoring $O(\\log n)$ terms we have\n $$\n \\K(A\\mskip 1.5mu | \\mskip 1.5 mu B)\\le \\K(A\\mskip 1.5mu | \\mskip 1.5 mu A')+\\K(A'\\mskip 1.5mu | \\mskip 1.5 mu \\Omega_i)+\\K(\\Omega_i\\mskip 1.5mu | \\mskip 1.5 mu \\Omega_j)+\\K(\\Omega_j\\mskip 1.5mu | \\mskip 1.5 mu B')+\\K(B'\\mskip 1.5mu | \\mskip 1.5 mu B).\n $$\n We will show that in the right hand side of this inequality, each term is at most $\\delta+O(\\log n)$.\n\n For the last term this directly follows from the construction of $B'$.\n The second and forth terms can be estimated as follows.\nBy Proposition \\ref{prop:std-omega}, $A',B'$ are $O(\\log n)$-equivalent to $\\Omega_i,\\Omega_j$, respectively,\nthus the second and forth terms are $O(\\log n)$.\n\nTo upper bound the third term, we first show that \nthat $i\\le j+\\delta+O(\\log n)$. By construction of $B'$ we have $\\delta(x,B')\\le \\delta(x,B) + O(\\log n)$. Since $B$ is\n$\\varepsilon$-sufficient, this inequality implies that $\\delta(x,B')\\le \\varepsilon+O(\\log n)$.\nOn the other hand, $\\delta(x,A)\\ge -O(\\log n)$ and thus $\\delta(x,B')\\le \\delta(x,A) + \\varepsilon+O(\\log n)$.\nChoose the constant $d$ in the statement of the theorem so that this inequality implies that \n$\\delta(x,B')\\le \\delta(x,A) + \\varepsilon+\\varkappa$. We assume that \n $A$ is a $(\\delta,\\varepsilon+\\varkappa)$-minimal statistic for $x$ and hence $\\K(B')\\ge \\K(A)- \\delta$,\n that is, $j\\ge \\K(A)-\\delta$. On the other hand, by construction of $A'$ we have $i=\\K(A')\\le \\K(A)+O(\\log n)$\n and hence $i\\le j+\\delta+O(\\log n)$. \n\nBy Proposition\n\\ref{prop:omega-equivalence} \nwe have \n$$\n\\K(\\Omega_i\\mskip 1.5mu | \\mskip 1.5 mu \\Omega_j)\\le \\K(\\Omega_i\\mskip 1.5mu | \\mskip 1.5 mu \\Omega_{j+\\delta+O(\\log n)})+\\delta+O(\\log n).\n$$\nOn the other hand, \nthe inequality $i\\le j+\\delta+O(\\log n)$ and Proposition\n\\ref{prop:omega-equivalence} imply that \n$$\n\\K(\\Omega_i\\mskip 1.5mu | \\mskip 1.5 mu \\Omega_{j+\\delta+O(\\log n)})=O(\\log n).\n$$\n\n \n It remains to estimate the first term.\n Repeating the argument from the beginning of the last but one paragraph we can show that $\\K(A')\\ge \\K(A)- \\delta$,\nOn the other hand, by construction of $A'$ we have $\\K(A'\\mskip 1.5mu | \\mskip 1.5 mu A)=O(\\log n)$.\nBy Symmetry of information this implies that $\\K(A\\mskip 1.5mu | \\mskip 1.5 mu A')=\\delta+O(\\log n)$. \n Indeed, ignoring $O(\\log n)$ terms, we have:\n $$\n \\K(A\\mskip 1.5mu | \\mskip 1.5 mu A')=\\K(A)+\\K(A'\\mskip 1.5mu | \\mskip 1.5 mu A)-\\K(A')=\\K(A)-\\K(A')\\le\\delta.\n $$\n\n\\medskip\n\n\\emph{Proof of the second claim $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu B)\\le O(\\varepsilon+\\delta+\\log n)$} (provided $A$ is a strong models for $x$).\nIt suffices to we show that $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu B)$ is close to $\\K(A\\mskip 1.5mu | \\mskip 1.5 mu B)$ provided $A$ is a strong model for $x$.\n\nRecall that $A\\mapsto [A]$ denotes a computable bijection from the family of finite sets to the set of binary strings which is used \nto encode finite sets of strings. \n\nOn the top level the argument is as follows. Let $p$ be a program witnessing $\\KT(A\\mskip 1.5mu | \\mskip 1.5 mu x)\\le\\varepsilon$.\nWe first show that $A$ has the following feature: there are many strings $x'\\in B$ \nwith $p(x')=[A]$. More specifically, at least $2^{-\\K(A\\mskip 1.5mu | \\mskip 1.5 mu B)}$ fraction of $x'$\nfrom $B$ have this property. \nAt most $2^{\\K(A\\mskip 1.5mu | \\mskip 1.5 mu B)}$ sets $A'$ can have this feature, as each such $A'$ can be identified\nby the portion of $x'\\in B$ with $p(x')=[A']$.\nGiven $B$ and $p$ we are able to find a list \nof all such $A'$ by means of a short \ntotal program. Given $B$, the set $A$ can be identified \nby $p$ and its index in that list. \n\nLet us proceed to the detailed proof. In the proof, we will ignore terms of order $O(\\varepsilon+\\log k)$.\nFirst we show that there are many \n$x'\\in B$ with $p(x')=[A]$ (otherwise\n$B$ could not be a sufficient statistic for $x$). \nLet \n$$\nD=\\{x'\\in B\\mid p(x')=[A]\\}.\n$$\nWe have \n$$\n\\K(D \\mskip 1.5mu | \\mskip 1.5 mu B)\\le \\K(A \\mskip 1.5mu | \\mskip 1.5 mu B),\n$$\nas \n$$\n\\K(D \\mskip 1.5mu | \\mskip 1.5 mu B)\\le \\K(D\\mskip 1.5mu | \\mskip 1.5 mu A)+\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B)\n$$\nand $\\K(D\\mskip 1.5mu | \\mskip 1.5 mu A)=\\K(D\\mskip 1.5mu | \\mskip 1.5 mu [A])\\le \\K(p)+O(1)\\le\\varepsilon+O(1)$. \n\nObviously, $D$ includes $x$, thus given $B$ and $p$ the string $x$ can be identified \nby its index in $D$. Therefore \n$$\n\\K(x \\mskip 1.5mu | \\mskip 1.5 mu B)\\le\\K(D \\mskip 1.5mu | \\mskip 1.5 mu B)+\\log \\# D\\le\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B)+ \\log \\# D.\n$$\nOn the other hand, $\\K(x \\mskip 1.5mu | \\mskip 1.5 mu B)=\\log \\# B$, as\n$B$ is $\\varepsilon$-sufficient. Hence\n$$\n\\log \\# D\\ge\\log \\# B-\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B).\n$$ \n\nRecall that we ignored terms of order $O(\\varepsilon+\\log k)$.\nActually, we have shown that \n$\\log \\# D\\ge l$ for some\n$$\nl=\\log \\# B-\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B)-O(\\varepsilon+\\log k).\n$$\nConsider now all $A'$ for which the set $D'$ defined in a similar way has the same lower bound for its cardinality.\nThat is, consider sets $A'$ with \n$$\n\\log \\# \\{x'\\in B\\mid p(x')=[A']\\} \\ge l.\n$$\nEach such $A'$ can be identified\nby the portion of $x'\\in B$ with $p(x')=[A']$. \nThus there are at most $2^{\\K(A\\mskip 1.5mu | \\mskip 1.5 mu B)+O(\\varepsilon+\\log k)}$ \ndifferent such $A'$s.\nGiven $B$ and $\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B),p,\\varepsilon$ we are able to find\nthe list of all such $A'$s. The program that maps $B$ \nto the list of such $A'$s is obviously total. \nTherefore there is a $\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B)+O(\\varepsilon+\\log k)$-bit total program \nthat maps $B$ to $A$ and\n$\\KT(A \\mskip 1.5mu | \\mskip 1.5 mu B)=\\K(A \\mskip 1.5mu | \\mskip 1.5 mu B)+O(\\varepsilon+\\log k)$. \n\n\n\n\n\\subsection{Normal strings and standard descriptions}\\label{s:normal}\nHere we prove Theorem \\ref{thm:separation}, i.e. we \nexhibit an example of a normal string $x$ such that every standard description is not a strong sufficient minimal statistic for $x$.\nOur string $x$ will be obtained from an antistochastic string in the sense of Section \\ref{sub:trade-off}.\n\n\\begin{definition}\nA string $x$ of length $n$ and complexity\n$k$ is called \\emph{$\\varepsilon$-antistochastic}\nif for all $(m,l)\\in P_x$ either\n$m>k-\\varepsilon$, or $m+l>n-\\varepsilon$. (The profile of an antistochastic string is shown on Fig.~\\ref{f22}.)\n\\end{definition}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale=1]{curve22.pdf}\n\\end{center}\n\\caption{The profile of an $\\varepsilon$-antistochastic string $x$ of length $n$ and complexity $k$ for a \nsmall $\\varepsilon$.}\n\\label{f22}\n\\end{figure}\nBy Theorem~\\ref{stat-any-curve} for every $k< n$ there exists an $O(\\log n)$-antistochastic string of length $n$ and complexity $k + O(\\log n)$.\nIt is easy to see that all antistochastic strings are normal.\nRecall that a string $x$ is called $(\\varepsilon,\\delta)$-normal if $P_x$ is in $\\delta$-neighborhood of $P_x(\\varepsilon)$.\n\n\\begin{proposition} \nLet $x$ be an $\\varepsilon$-antistochastic string of length $n$ and complexity $k$. Then $x$ is $(O(\\log n),O(\\log n) + \\varepsilon)$-normal. \n\\end{proposition}\n\\begin{proof}\nTo prove the claim, it suffices to construct for every point $(i,j)$ on the boundary of the set shown on Figure~\\ref{f22}\nan $O(\\log n)$-strong $(i+O(\\log n)*j)$-description $A$ for $x$. \nIf $i\\ge k$ then\nlet $A=\\{x\\}$. \nOtherwise let \n$A$ be the set of all strings of length $n$ whose\nfirst $i$ bits are the same as those of $x$. \nBy construction $\\K(A) \\le i + O(\\log n)$ and \n$\\log \\# A = n - i=j$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:separation}]\nLet $y$ be an \n$O(\\log k)$-antistochastic string \nof complexity $k+O(\\log k)$ and length $2k$, which exists by Theorem~\\ref{stat-any-curve}.\nLet $z$ be a string of \nlength $2k$ such that $\\K(z \\mskip 1.5mu | \\mskip 1.5 mu y) \\ge 2k$. Finally, let $x=yz$\nbe the concatenation of $y$ and $z$. Obviously, $\\K(x) = 3k$ with accuracy $O(\\log k)$.\n\nBy Theorem~\\ref{rem:add-noise}\nthe set $P_x$ is $O(\\log k)$-close to the gray set on Fig. \\ref{f4}. From normality of $y$ \nit is not difficult to derive that $x$ \nis $O(\\log k),O(\\log k)$-normal.\n\n\n\nLet $A=\\{yz'\\mid l(z')=2k\\}$. Then $A$ is an $O(\\log k)$-strong statistic for $x$. \nNow we need to show that every standard description whose parameters (complexity, log-cardinality) are close to those of $A$\nis not strong.\nLet $q$ be a program enumerating all strings of complexity at most $m$. Let $B$ be the standard model for $x$ obtaining from this enumeration. \nWe have to show that for some positive $c$ for almost all $k$ we have\n\\begin{equation}\n\\label{req}\n\\min\\{\\KT(B\\mskip 1.5mu | \\mskip 1.5 mu x),\\KS(q),|\\KS(B)-k|,|\\log\\#B-2k|\\}\\ge c k.\n\\end{equation}\n\nFix some small positive $c$. For the sake of contradiction \nassume that for infinitely many $k$ there are $m,q,B$ such that \nthe inequality (\\ref{req}) does not hold. \n\nOn the top level, the argument is as follows. We have assumed \n$\\K(B)$ and $\\log\\#B$ are close to $k$ and $2k$, respectively. The shape of $P_x$\nguarantees that in this case $B$ is a sufficient minimal statistic for $x$. \nWe have assumed also that $\\KT(B\\mskip 1.5mu | \\mskip 1.5 mu x)$ is small, that is $B$ is a strong statistic for\n$x$. Hence we can apply Theorem \\ref{thm:step-wise} to $B$ and $A$ and\nconclude that $\\CT(B\\mskip 1.5mu | \\mskip 1.5 mu A)$ is small. As $A$ is strongly equivalent to $y$,\nthe total conditional complexity $\\CT(B\\mskip 1.5mu | \\mskip 1.5 mu y)$ is small as well.\nOn the other hand we will show that the total \ncomplexity of any standard statistic $B$ obtained from an enumeration\nof strings of complexity at most $m$ conditional to any string $y$\nis larger than $\\min\\{\\K(B),m-l(y)\\}$. In our case $m$ is at least $\\K(x)\\approx 3k$,\n$l(y)=2k$ and $\\K(B)\\approx k$ thus $\\CT(B\\mskip 1.5mu | \\mskip 1.5 mu y)\\ge k$ with accuracy $O(\\log k)$,\nwhich is a contradiction, if $c$ is small enough.\n\nLet us proceed to the detailed proof.\nLet us show first that $m=O(k)$. Recall that \n$B$ is a standard description obtained from a list of complexity at most $m$\nenumerated by program $q$. We have seen that in this case \n$$\n\\KS(B)+\\log\\#B=m+O(\\K(q)+\\log m). \n$$\nHence $m\\le (k+c k)+(2k+c k)=O(k)$ with accuracy $O(\\K(q)+\\log m)$.\nWe have assumed that $\\K(q)

0 \\; &\\Rightarrow& \\; -p^T\\nabla f(x^*) = \\alpha \\; \\gamma'(c_p) \\quad \\text{(no wiggle room)},\\\\\nc_p = 0 \\; &\\Rightarrow& \\; -p^T\\nabla f(x^*) \\in \\alpha\\; [-\\infty, \\gamma_{\\max}] \\quad \\text{(wiggle room exists)}.\n\\end{eqnarray*}\nHere, the wiggle room condition looks asymmetric, but note that if $p$ and $-p$ is in $\\mathcal P_0$, then $c_p = c_{-p} = 0$ implies $-p \\nabla f(x^*) \\in \\alpha [-\\gamma_{\\max},\\gamma_{\\max}]$, recovering the symmetric condition from secion \\ref{sec:simple}.\nAs before, since $\\gamma'$ is a decreasing function, a nonzero coefficient for $x^*$ does not mean a maximal gradient inner product. \n\n\n\n\n\\subsection{RP-CGM}\nIn the case that $\\mathcal P_0$ includes directions of recession, we again treat them separately; given assumption \\ref{asspt:atoms}, $\\mathcal P_0 = \\mathcal P_0' \\cup \\mathcal K$ and $\\mathcal P_0'$ is finite. \nWe define the \\emph{reweighted atomic set} for a given reference point $x$ as \n\\[\n\\mathcal P_0(x) = \\left\\{\\frac{1}{\\gamma'(\\mathbf{coeff}_\\mathcal P(x,p))}p : p\\in \\mathcal P'_0\\right\\},\\quad \\mathcal P(x) = \\mathbf{conv}(\\mathcal P_0(x)).\n\\]\nThen\n$r_\\mathcal P(s;x) = \\kappa_{\\mathcal P(x)}(s)$, \nwith corresponding reweighted support function\n\\begin{equation}\n\\sigma_{\\mathcal P(x)}(z) = \\max_{p\\in \\mathcal P_0}\\; \\frac{p^Tz}{\\gamma'(\\mathbf{coeff}_\\mathcal P(x,p))}.\n\\label{eq:reweighted-support}\n\\end{equation}\nAt each iteration, we take a penalized conditional gradient step toward solving the reweighted gauge optimization problem with dual\n\\begin{eqnarray*}\n \\text{(P-general)} \n &&\n \\minimize{x,y\\in \\mathcal K} \\quad f(x+y) + \\phi(r_0 + \\kappa_{\\mathcal P(\\bar x)}(x))\\\\\n \\text{(D-general)}\n &&\n \\maximize{z\\in \\mathcal K^\\circ} \\quad -f^*(-z) - \\phi^*(\\sigma_{\\mathcal P(\\bar x)}(z))+r_0\\cdot \\sigma_{\\mathcal P(\\bar x)}(z)\n\\end{eqnarray*}\nA description of the most generalized version of the reweighted method is given in Algorithm \\ref{alg:general-nonconvex}.\n\n\\begin{algorithm}\n\\caption{RP-CGM on general nonconvex sparse optimization}\n\\begin{algorithmic}[1]\n\n\\Procedure{RP-CGM}{$f$, $\\phi$, $\\gamma$, $\\mathcal P_0 = \\mathcal P_0' \\cup \\mathcal K$, max iter $T$} \n \\State Initialize with any $x^{(0)}\\in \\mathbf{cone}(\\mathcal P)$ where $\\mathcal P$ is the convex hull of $\\mathcal P_0'$, $y^{(0)} \\in \\mathcal K$\n \\State \n \\For{$t = 1,...T$}\n \\State Compute the projected negative gradient $z = -\\nabla f(x^{(t)}+y^{(t)})$\n \\State Compute the reweighted atomic set $\\mathcal P(x)$\n \n \\State Compute next atom $s =\\xi p$ where \\Comment{Pick next atom}\n \\[\n p = \\mathbf{LMO}_{\\mathcal P(x)}(z), \\quad \\xi = (\\phi^*)'(\\sigma_\\mathcal P(z))\n \\]\n \n \\State Update $x^{(t+1)}$ \\Comment{Merge}\n \\[\n x^{(t+1)} = (1-\\theta^{(t)})x^{(t)} + \\theta^{(t)} s, \\quad \\theta^{(t)} = 2\/(1+t)\n \\]\n \n \\State Update $y^{(t+1)}$ using exact minimization \\Comment{Recession component}\n \\[\n y^{(t+1)} = \\argmin{y\\in \\mathcal K}\\; f(x^{(t+1)} + y)\n \\]\n \\EndFor\n\\Return $x^{(T)} + y^{(T)}$\n\\EndProcedure\n\n\\end{algorithmic}\n\\label{alg:general-nonconvex}\n\\end{algorithm}\n\n\\subsection{Convergence}\n\n\n\n\\begin{property}[Residual]\nDenoting $\\mathbf{res}_\\mathcal P(x) = \\mathbf{gap}_\\mathcal P(x,-\\nabla f(x);x)$ the gap at $x$ with reference $x$, then\n \\[\n \\mathbf{res}_\\mathcal P(x) \\geq 0 \\; \\forall x, \\qquad \\mathbf{res}_\\mathcal P(x) =0 \\iff \\text{ $x$ is a stationary point of \\ref{eq:main-1norm}.}\n \\]\n \\label{prop:gapres-general}\n\\end{property}\nThe proof follows closely that of Property \\ref{prop:gapres-l1}; see appendix \\ref{app:convergence} for full details.\n\n\n\n\n\n\n\n\\begin{lemma}[One step descent]\n Suppose $f$ is $L$-smooth w.r.t. $\\mathcal P$ (unweighted). Take \n\\[\nx^+ = (1-\\theta) x + \\theta s, \\qquad s = \\argmin{\\tilde s} \\; \\nabla f(x+y)^T\\tilde s + \\bar h(s;x)\n\\]\nfor some $\\theta \\in (0,1)$. Define $y = \\argmin{y\\in \\mathcal K} \\; f(x+y)$,\\; $y^+ = \\argmin{y\\in \\mathcal K}\\; f(x+y)$.\nThen \n\\[\nf(x^++y^+) + h(x^+) - f(x+y)-h(x) \\leq -\\theta \\mathbf{res}(x) + \\frac{L\\theta^2}{2} \\kappa_\\mathcal P(s-x)^2.\n\\]\n\\label{lem:onestepdescend1}\n\n\\end{lemma}\n\\begin{proof} \nFrom $L$-smoothness we have \n\\begin{eqnarray}\nf(x^++y^+)-f(x+y) &\\leq& f(x^++y)-f(x+y)\\nonumber\\\\\n&\\leq&\\nabla f(x+y)^T(x^+-x) + \\frac{L}{2} \\kappa_\\mathcal P(x^+-x) \\nonumber\\\\\n&=& \\theta \\nabla f(x+y)^T(s-x) + \\frac{L\\theta^2}{2}\\kappa_\\mathcal P(s-x)^2\n\\label{eq:stephelper1}\n\\end{eqnarray}\nDenote $ \\nu = \\sigma_{\\widetilde\\mathcal P(x)} (-\\nabla f(x+y))$.\nSince $s = \\xi \\phi'(\\nu)$, then \n\\begin{eqnarray}\n\\nabla f(x+y)^Ts + \\phi(r_0 + \\bar r_\\mathcal P(s;x)) &=& \\min_{\\tilde s} \\underbrace{\\nabla f(x+y)^T \\tilde s}_{=-\\xi\\cdot\\nu} + \\phi\\bigg(r_0 + \\underbrace{\\bar r_\\mathcal P(s;x)}_{=\\xi}\\bigg) \\nonumber\\\\\n&=& \\nu r_0 - \\phi^* (\\nu).\n\\label{eq:stephelper2}\n\\end{eqnarray}\nAlso, by definition of residual,\n\\begin{eqnarray}\n\\mathbf{res}_\\mathcal P(x) &=& f(x+y) + f^*(\\nabla f(x+y)) +\\phi(r_\\mathcal P(x+y)) + \\phi^*(\\nu) - r_0\\cdot \\nu\\nonumber\\\\\n&= & \\underbrace{\\nabla f(x+y)^T(x+y)}_{\\nabla f(x+y)^Ty\\geq 0} +\\phi(r(x)) + \\phi^*(\\nu) - r_0\\cdot \\nu\\nonumber\\\\\n&\\geq & \\nabla f(x+y)^Tx +\\phi(r(x)) + \\phi^*(\\nu) - r_0\\cdot \\nu.\n\\label{eq:stephelper3}\n\\end{eqnarray}\n\n\nTherefore taking $F(x+y) = f(x+y) + \\phi(r(x))$ and combining \\eqref{eq:stephelper1}, \\eqref{eq:stephelper2}, and \\eqref{eq:stephelper3},\n\\begin{eqnarray*}\n F(x^++y^+)- F(x+y) \n\n&=&-\\theta \\mathbf{res}(x) + \\theta \\left(\\phi(r_\\mathcal P(x)) - \\phi(r_0 + \\bar r_\\mathcal P(s;x)) \\right) \\\\\n&& \\qquad + \\frac{L\\theta^2}{2}\\kappa_\\mathcal P(s-x)^2 + \\phi( r_\\mathcal P(x^+)) - \\phi( r_\\mathcal P(x)) \\\\\n\\end{eqnarray*}\nNext, by convexity of $\\phi$,\n\\begin{eqnarray*}\n (1-\\theta)\\phi( r_\\mathcal P(x)) + \\theta \\phi(r_0+ \\bar r_\\mathcal P(s;x)) &\\geq&\n\n \\phi(r_\\mathcal P(x) +\\bar r_\\mathcal P(x^+;x)-\\bar r_\\mathcal P(x;x) )\\\\\n &\\overset{\\text{majorant}}{\\geq}& \\phi(r_\\mathcal P(x^+;x))\n\\end{eqnarray*}\nwhich leaves the desired result.\n\\end{proof}\n\n\n\n\n\\begin{lemma}[Iterate gauge control.]\nSuppose additionally $\\theta^{(t)} = 2\/(t+1)$. Then\n\\begin{eqnarray*}\n \\kappa_\\mathcal P(s^{(t)}-x^{(t)}) \n &\\leq& \\frac{\\gamma_{\\max}}{\\gamma_{\\min}\\mu}\\bigg(2\\sigma_{\\widetilde\\mathcal P}(\\nabla f(x^*+y^*)) +\n \\sqrt{2L\\Delta^{(t)}} +\n \\frac{2}{t(t-1)}\\sum_{u=1}^{t-1} \\sqrt{2L\\Delta^{(u)}}\\bigg) \\\\\n &&\\qquad + 2\\nu_0\\gamma_{\\max}+\\kappa_\\mathcal P(x^{(0)}) .\n\\end{eqnarray*}\n\n\\label{lem:onestepdescend2}\n\\end{lemma}\n\n\\begin{proof}\n\\begin{eqnarray*}\n \\kappa_\\mathcal P(s^{(t)}-x^{(t)}) &\\overset{\\substack{\\text{subadditive}\\\\\\text{gauge}}}{\\leq} & \\kappa_\\mathcal P(s^{(t)})+\\kappa_\\mathcal P(x^{(t)})\\\\\n&\\overset{\\text{convexity}}{\\leq}& \\kappa_\\mathcal P(s^{(t)}) + \\theta^{(t-1)} \\kappa_\\mathcal P(s^{(t-1)} ) +(1-\\theta^{(t-1)})\\kappa_\\mathcal P(x^{(t-1)}) \\\\\n&\\overset{\\text{recursion}}{\\leq}& \\kappa_\\mathcal P(s^{(t)}) + \\kappa_\\mathcal P(x^{(0)}) + \\sum_{u=1}^{t-1} \\theta^{(u)} \\underbrace{\\prod_{u'=u+1}^{t-1}(1-\\theta^{(u')})}_{=\\frac{(u+1)u}{t(t-1)}}\\kappa_\\mathcal P(s^{(u)} ) \\\\\n &\\leq& \\kappa_\\mathcal P(s^{(t)}) +\\kappa_\\mathcal P(x^{(0)}) + \\frac{2}{t(t-1)}\\sum_{u=1}^{t-1} u \\kappa_\\mathcal P(s^{(u)}).\n \\end{eqnarray*}\nIn general, for any $x$, $z$, $\\bar x$, \n\\[\n\\kappa_{\\mathcal P}(x) \\leq \\gamma_{\\max}\\kappa_{\\mathcal P(\\bar x)}(x), \\qquad \n\\sigma_{\\mathcal P}(z) \\geq \\frac{1}{\\gamma_{\\min}}\\sigma_{\\mathcal P(\\bar x)}(z).\n\\]\nTaking $y^{(u)} = \\argmin{y\\in \\mathcal K}\\;f(x^{(u)} + y)$, $z^{(u)} = -\\nabla f(x^{(u)}+y^{(u)})$, ${z^* = -\\nabla f(x^*+y^*)}$:\n\\begin{eqnarray*}\n\\kappa_{\\mathcal P(\\bar x)}(s^{(u)}) \n=\n(\\phi^*)'\\left( \\sigma_{\\mathcal P(\\bar x)}(z^{(u)}) \\right)\n&\\overset{\\text{Asspt \\ref{asspt:phi2}}}{\\leq} & \\mu^{-1} \\cdot \\sigma_{\\mathcal P(\\bar x)}(z^{(u)}) + \\nu_0\\\\\n&\\overset{\\text{Bound on $\\gamma'$}}{\\leq} & \\frac{1}{\\mu r_{\\min}} \\sigma_{\\mathcal P}(z^{(u)}) + \\nu_0\\\\\n&\\overset{\\substack{\\text{$\\Delta$-ineq +}\\\\\\text{ Prop. \\ref{prop:res-bound-graderr}}}}{\\leq} & \\frac{1}{\\mu r_{\\min}}\\left( \\sigma_{\\widetilde \\mathcal P}(z^*) + \\sqrt{2L\\Delta^{(u)}}\\right) + \\nu_0.\n\\end{eqnarray*}\nPutting it all together gives the desired result.\n\\qed\n\\end{proof}\nFrom Lemmas \\ref{lem:onestepdescend1} and \\ref{lem:onestepdescend2}, we arrive at\n\\[\n\\Delta^{(t+1)}-\\Delta^{(t)} \\leq -\\theta^{(t)} \\mathbf{res}_\\mathcal P(x^{(t)}) + (\\theta^{(t)})^2\\left( B\\Delta^{(t)} + B \\bar \\Delta^{(t-1)} \n + A \\right) \n\\]\nfor constants \n\\[\nA = \\left(\\frac{6L \\gamma_{\\max}^2}{\\mu^2\\gamma_{\\min}^2 }\\sigma_{\\widetilde\\mathcal P}(-\\nabla f(x^*+y^*)+6\\gamma_{\\max}\\nu_0+3\\kappa_\\mathcal P(x^{(0)}) \\right)^2, \\qquad B = \\frac{3L^2\\gamma_{\\max}^2}{\\mu^{2}\\gamma_{\\min}^2}\n\\]\nand where $\\bar \\Delta^{(t)}$ is defined as an averaging over square roots, e.g. \n\\[\n\\sqrt{\\bar\\Delta^{(t)}} =\\frac{2}{t(t+1)} \\sum_{u=1}^{t} u \\sqrt{\\Delta^{(u)}}. \n\\]\n\n\\begin{thm}[Convergence]\nConsider $G$ large enough such that for all $t < 6B$, $\\Delta^{(t)} t \\leq G$ and $G > 24A$. \nGiven assumptions \\ref{asspt:phigamma1}, \\ref{asspt:phi2}, \\ref{asspt:atoms}, \\ref{asspt:gensmooth}, with iterates $x^{(t)}+y^{(t)}$ from algorithm \\ref{alg:general-nonconvex}, using $\\theta^{(t)} = 2\/(t+1)$, then\n\\[\n\\Delta^{(t)} \\leq \\frac{G}{t+1} \\qquad \\text{ and } \\qquad\n\\min_{i\\leq t} \\mathbf{res}(x^{(i)}) \\leq \\frac{3G}{2\\log(2) (t+1)}.\n\\]\n\\label{th:convergence}\n\\end{thm}\nGiven Lemmas \\ref{lem:onestepdescend1} and \\ref{lem:onestepdescend2}, the details of the proof closely mirror steps in previous works, and thus we give the explicit details in appendix \\ref{app:convergence}.\n\n\n\\paragraph{Comparison with CGM} In \\cite{jaggi2013revisiting}, the primal convergence rate for vanilla CGM (with noiseless gradients) is given as $\\Delta^{(t)} \\leq \\frac{2 C_f}{t+2}$ where $C_f$ is a curvature constant that depends on the conditioning of $f$ and the size of $P$. These players appear here in the form of the conditioning of $f$ (quadratic in $L\/\\mu$), and implicitly $\\sigma_{\\widetilde\\mathcal P}$ (which grows proportionally with $\\mathbf{diam}~\\mathcal P$). The new players $\\nu_0$, $\\gamma_{\\min}$, and $\\gamma_{\\max}$ account for the penalty and nonconvex generalizations.\n\n\n\n\n\n\n\\subsection{Invariance}\nConsider $\\mathcal Q = A\\mathcal P$, $f(x) = g(Ax)$, $w = Ax$, $\\bar w = A \\bar x$, where $A$ has full column rank. Additionally, assume $x, \\bar x \\in \\mathbf{cone}(\\mathcal P)$. Then the following hold.\n\\begin{itemize}\n\\item \\textbf{Penalty.} $r_\\mathcal P(x) = r_\\mathcal Q(w)$.\nThis follows from noting that \n\\[\nx = \\sum_{p\\in \\mathcal P} c_p p \\iff w = \\sum_{p\\in \\mathcal P} c_p (Ap) = \\sum_{q\\in \\mathcal Q} c'_q q\n\\]\nand in fact noting that the coefficients are equal ($c'_q = c_{Ap}$).\n\n\n\n\\item \\textbf{Stationarity.} \nWe construct $P$ with columns containing the atoms in $\\mathcal P_0'$, and $c$ such that\n$x = Pc$, $w = Ax = APc$. \n\\[\n P^T\\partial r_\\mathcal P(x)=\\partial r_\\mathcal P(c) \\overset{r_\\mathcal P(c) = r_\\mathcal Q(c)}{=} \\partial r_\\mathcal Q(c) = P^TA^T\\partial r_\\mathcal Q(w).\n\\]\nAdditionally, for any stationary point $x^*$, if $\\nabla f(x^*)\\not\\in \\mathbf{cone}(\\mathcal P)$ then there exists a descent direction that is uneffected by the penalty $r_\\mathcal P(x)$, and thus it must be that $\\nabla f(x^*)\\in \\mathbf{cone}(\\mathcal P)$. By the same token, $A^T\\nabla g(w^*)\\in \\mathbf{cone}(\\mathcal P)$. Therefore, the stationary conditions are equivalent: for $x^* = Aw^*$,\n\\[\n0\\in \\nabla f(x^*) + P^T\\partial r_\\mathcal P(x^*) \\iff 0\\in A^T\\nabla g(w^*) + A^T\\partial r_{\\mathcal Q}(w^*). \n\\]\n\n\n\n\\end{itemize}\nAdditionally, it can be shown through the chain rule that $A\\mathcal P(x) = \\mathcal Q(w)$ and $\\mathbf{res}_\\mathcal P(x) = \\mathbf{res}_\\mathcal Q(w)$.\nOverall, this shows that the steps and analysis of RP-CGM are all invariant to linear transformations on $x$.\n\n\\subsection{Screening}\n\n\nWe now describe the gradient error measured in terms of this ``dual gauge\"\n$\\sigma_{\\widetilde\\mathcal P}(\\nabla f(x)-\\nabla f(x^*))$ where the symmetrization $\\widetilde\\mathcal P := \\mathcal P \\cup -\\mathcal P$ ensures that ${\\sigma_{\\widetilde\\mathcal P}(z-z^*)=\\sigma_{\\widetilde\\mathcal P}(z^*-z)}$, bounding errors in both directions. \n\n\n\\begin{property}[Residual bound on gradient error.]\nDenote $D(x) = r_\\mathcal P(x)-r_\\mathcal P(x^*) + \\bar r_\\mathcal P(x;x) - \\bar r_\\mathcal P(x^*;x)$\nthe linearization error at $x$. \nDenoting $x^*$ a stationary point of \\eqref{eq:main-nonconvex} and $y(x) = \\argmin{y'\\in \\mathcal K}\\; f(x+y')$, then\n\\begin{multline*}\n\\sigma_{\\widetilde\\mathcal P}(\\nabla f(x+y(x))-\\nabla f(x^*+y(x^*))) \\leq\\\\ \\frac{LD(x)}{2\\gamma_{\\min}} + \\sqrt{\\frac{L^2D(x)^2}{4\\gamma^2_{\\min}} + L\\mathbf{res}(x) + LD(x) \\frac{\\sigma_{\\widetilde\\mathcal P}(\\nabla f(x+y(x)))}{\\gamma_{\\min}}}.\n\\end{multline*}\n\\label{prop:res-bound-graderr}\n\\end{property}\nThe linearization error $D(x) = 0$ when the regularizer is convex.\nThe proof is similar to that for Prop. \\ref{prop:resboundsgrad-l1}, and is detailed in appendix \\ref{app:convergence}.\n\n\n\\begin{thm}[Dual screening]\nFor any $x$ and some choice of $\\epsilon > 0$, define the screened set as \n\\begin{equation}\n\\mathcal I_\\epsilon(x) = \\{p\\in \\mathcal P_0 : \\sigma_{\\widetilde\\mathcal P}(\\nabla f(x)) + p^T\\nabla f(x) > \\epsilon + 2\\sqrt{L\\mathbf{res}(x)+\\epsilon}\\}.\n\\label{eq:gaprule}\n\\end{equation}\nThen given assumptions \\ref{asspt:phigamma1}, \\ref{asspt:phi2}, and \\ref{asspt:gensmooth},\nif \n\\[\n\\epsilon \\geq \\frac{L D(x)}{\\gamma_{\\min}} \\; \\max\\left\\{\\frac{1}{2},\\frac{L D(x)}{4\\gamma_{\\min}} + \\sigma_{\\widetilde \\mathcal P}(\\nabla f(x))\\right\\}\n\\]\nthen $p\\not\\in \\mathbf{supp}_\\mathcal P(x^*)$,\nwhere $x^*$ is the optimal variable in \\eqref{eq:main-convex}.\n\\label{th:screening2}\n\\end{thm}\n\n\n\n\n\n\nIn the convex case, $D(x) = 0$, and thus we pick $\\epsilon = 0$ in our screening rule. In this scenario, not only does this screening rule achieve finite-iteration support identification, but the finite time $\\bar t$ depends directly on $\\delta_{\\min}$.\n\\begin{thm}[Support identification of screened P-CGM]\n\\label{th:supportid}\nGiven assumptions \\ref{asspt:phigamma1}, \\ref{asspt:phi2}, \\ref{asspt:atoms}, \\ref{asspt:gensmooth}, then the screening rule for convex penalties \n\\[\n\\mathcal I^{(0)} = \\mathcal P_0, \\qquad \\mathcal I^{(t)} = \\mathcal I^{(t-1)}\\setminus \\{p \\in \\mathcal P_0 : p\\in \\mathcal I_0(x) \\mathrm{\\; for\\; } x=x^{(t)}\\},\n\\]\nis safe and convergent:\n\\[\n \\mathcal I^{(t)}\\supseteq \\mathbf{supp}_\\mathcal P(x^*),\\; \\forall t, \\quad \\text{ and } \\quad \\mathcal I^{(t)} = \\mathbf{supp}_{\\mathcal P(x^*)}(x^*), \\; t \\geq t'\n\\]\nwhere $t'$ is such that\n\\begin{equation}\n \\sqrt{L\\min_{i\\leq t'}\\mathbf{res}(x^{(i)})} < \\delta_{\\min}\/3\n\\label{eq:suppID:bound}\n\\end{equation}\nwhich happens at a rate $t' = O(1\/(\\delta_{\\min}^2))$.\n\\end{thm}\n\\begin{proof}\nThis is a direct consequence to theorems \\ref{th:convergence} and \\ref{th:screening2}.\n\\qed\n\\end{proof}\n\nNote that Theorem \\ref{th:supportid} imposes no conditions on the sequence $\\theta^{(k)}$, or choice of $\\phi$, $f$, etc., except $L$-smoothness of $f$. In other words, for any method where the gap is easily computable and its convergence rate known, then a corresponding screening rule and support identification rate automatically follow.\nAdditionally, computing $L$ may be challenging, depending on $\\kappa_\\mathcal P$; as shown previously, at the very least it may require a full pass over the data. However, this is a one-time calculation per dataset, and can be estimated if data are assumed to be drawn from specific distributions (as in sensing applications). \n\n\n \n\n\\section{Reweighted Penalized CGM for simple sparse recovery}\n\n\n\\label{sec:simple}\nWe begin by considering the optimization problem \n\\begin{equation}\n \\minimize{x\\in \\mathbb R^d} \\quad F(x) := f(x) + \\underbrace{\\phi(r(x))}_{h(x)}, \\qquad r(x) = \\sum_{i=1}^d \\gamma(|x_i|).\n \\label{eq:main-1norm}\n\\end{equation}\nThis is the simplification of \\eqref{eq:main0} with $r := r_\\mathcal P$ and $\\mathcal P_0 = \\{\\pm e_1,...,\\pm e_d\\}$ the signed unit basis. \nThe more general case of the $r_\\mathcal P$ gauge-like penalty follows a similar analysis to what is presented in this section, and can be viewed intuitively as sparsity in a preimage space.\nWe define the \\emph{support of $x$} as the indices of the nonzeros as\n$\\mathbf{supp}(x) = \\{i: x_i \\neq 0\\}$.\nFor a method producing iterates $x^{(1)}, x^{(2)}, \\to x^*$, we say that this method has \\emph{recovered the support at iteration $\\bar t$} if for all $t' \\geq \\bar t$, $\\mathbf{supp}(x^{(t)}) = \\mathbf{supp}(x^*)$. \n\n\n\n\\subsection{Stationary points}\n\nFor a continuous function $h:\\mathbb R^d\\to\\mathbb R$, the point $x^*$ is a \\emph{Clarke stationary point} of \\eqref{eq:main-1norm} if \n$0\\in \\nabla f(x^*) + \\partial h(x^*)$ where $\\partial h(x) = \\mathbf{conv}\\; \\{\\lim_{x'\\to x} \\nabla h(x')\\}$\nis the \\emph{Clarke subdifferential} of $h$ at $x$ \\citep{clarke1975generalized, clarke1983nonsmooth}.\n\n\\begin{assumption}[$\\phi$ and $\\gamma$]\nAssume that \n\\begin{itemize}\n \\item $\\phi:\\mathbb R_+\\to\\mathbb R_+$ is convex, monotonically increasing, and differentiable everywhere on its domain,\n \\item $\\gamma:\\mathbb R_+\\to\\mathbb R_+$ is concave, monotonically increasing, and differentiable everywhere on its domain, \n \\item the derivative $\\gamma'(\\xi)$ is lower and upper bounded by \n \\[\n 0 < \\gamma_{\\min} := \\lim_{\\xi\\to _\\infty} \\gamma'(\\xi) \\leq \\gamma'(\\xi) \\leq \\lim_{\\xi\\to 0^+} \\gamma'(\\xi) =:\\gamma_{\\max} < +\\infty, \\quad \\forall \\xi \\geq 0.\n \\]\n\\end{itemize}\n\\label{asspt:phigamma1}\n\\end{assumption}\nFrom these assumptions, the Clarke subdifferential for $h(x)$ is \n\\[\n(\\partial h(x))_i=\n\\begin{cases}\n\\phi'(r(x))\\cdot \\{\\mathbf{sign}(x_i) \\, \\gamma'(|x_i|)\\}, & x_i \\neq 0,\\\\\n\\phi'(r(x))\\cdot [-\\gamma_{\\max}, \\gamma_{\\max}], & x_i = 0.\n\\end{cases}\n\\]\nIn other words, the optimality conditions can be summarized as follows: $x^*$ is a stationary point of \\eqref{eq:main-1norm} if \n\\begin{eqnarray*}\nx^*_i \\neq 0 \\; &\\Rightarrow& \\; -\\nabla f(x^*)_i = \\phi'(r(x)) \\; \\gamma'(|x_i|) \\\\\nx^*_i = 0\\; &\\Rightarrow& \\; -\\nabla f(x^*)_i \\in \\phi'(r(x))\\; [-\\gamma_{\\max}, \\gamma_{\\max}].\n\\end{eqnarray*}\n\n\n\n\\paragraph{Example.}\nConsider the concave regularizer\n$h(x) := \\bigg(\\sum_i \\sqrt{|x_i|+ \\xi_0}\\bigg)^2$. \nThis construction arises from $\\phi(\\xi) = \\xi^2$ and $r(\\xi) = \\sqrt{|\\xi|+\\xi_0}$. \nIts Clarke-subdifferential can be expressed element-wise\n\\[\n(\\partial h(x))_i = \\left(\\sum_j \\sqrt{|x_j|+\\xi_0}\\right) \\cdot\n\\begin{cases}\n[-\\xi_0^{-1\/2},\\xi_0^{-1\/2}], & x_i = 0,\\\\\n\\left\\{ \\frac{\\mathbf{sign}(x_i)}{\\sqrt{|x_i|+\\xi_0}} \\right\\}, & |x_i| > 0.\n\\end{cases}\n\\]\nThe optimality conditions can also be summarized in terms of ``wiggle room\"; that is, whenever $x_i = 0$, then $\\nabla f(x)_i$ lies in an interval. But when $x_i \\neq 0$, $\\nabla f(x)_i$ must take a specific value. When $\\gamma$ is concave, however, that value could lie anywhere in the interval $[-\\xi_0^{-1\/2},\\xi_0^{-1\/2}]$. In this case, the gradient $\\nabla f(x)$ cannot reveal any hints as to the support of $x^*$, even as $x \\to x^*$.\n\n\n\\paragraph{Concave penalties} Popular concave penalties are listed in Table \\ref{tab:concavepenalties}. To satisfy assumption \\ref{asspt:phigamma1},\nwe propose a piecewise extension given a concave penalty $\\gamma_0$ and a ``boundary point\" $\\bar \\xi$:\n\\begin{equation}\n\\gamma(\\xi) = \n\\begin{cases}\n\\gamma_0(\\xi), & 0 \\leq \\xi \\leq \\bar\\xi,\\\\\n\\gamma_0'(\\bar\\xi)(\\xi - \\bar\\xi ) + \\gamma_0(\\bar\\xi) , & \\xi < \\bar\\xi.\n\\end{cases}\n\\label{eq:locallyconvex}\n\\end{equation}\nThis ensures that $\\gamma_{\\max} = \\gamma_0'(\\bar \\xi) < +\\infty$. \nA more complete table of commonly used concave penalties is given by \\citet{gong2013general, rakotomamonjy2019screening}.\nSee also figure \\ref{fig:phigamma}.\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{|r|l|c|c|}\n\\hline\n& $\\gamma(c)$ & $\\displaystyle\\lim_{c\\to 0} \\gamma'(c)$ & $\\displaystyle\\lim_{c\\to +\\infty} \\gamma'(c)$\\\\\\hline\nFractional fns & $q^{-1} c^q$, $0 < q < 1$ & $+\\infty$ & 0\\\\&&&\\\\\nLSP & $ \\log(1+|c|\/\\theta)$ for $\\theta > 0$ & $\\theta^{-1}$ & 0\\\\&&&\\\\\nSCAD & $\\begin{cases}\n\\lambda |c| & |c| \\leq \\lambda, \\\\\n\\frac{-c^2 + 2\\theta\\lambda |c|-\\lambda^2}{2(\\theta-1)}& \\lambda < |c| \\leq \\theta\\lambda, \\\\\n(\\theta+1)\\lambda^2\/2& |c| \\geq \\theta\\lambda, \n\\end{cases}$ for $\\theta > 2$ & $\\lambda$ & 0 \\\\&&&\\\\\nMCP & $\\begin{cases}\n\\lambda |c| - c^2\/(2\\theta) & |c| \\leq \\theta \\lambda,\\\\\n\\theta \\lambda^2\/2 & |c| > \\theta\\lambda,\n\\end{cases}$ for $\\theta > 0$ & $\\lambda$ & 0\\\\&&&\\\\\nLocally convex & \\eqref{eq:locallyconvex}, given $\\gamma_0$ and $\\bar \\xi$ & $\\displaystyle\\lim_{c\\to 0}\\gamma_0'(c)$ & $\\gamma_0'(\\bar \\xi)$\\\\\\hline\n\\end{tabular}\n\\label{tab:concavepenalties}\n\\caption{A list of several popular concave penalties, and their slope behavior at extremities. The last entry shows the effect of the piecewise construction, which becomes linear with non-zero slope at large values of $\\xi$.}\n\\end{table}\n\n\n\\subsection{Convex majorant}\nInspired by methods in majorization-minimization and difference-of-convex literature, we propose the RP-CGM, which at each iteration takes a conditional gradient step over the following convex proxy problem\n\\begin{equation}\n \\mini{x\\in \\mathbb R^d} \\quad \\bar F(x;x^{(t)}) := f(x) + \\phi(r(x^{(t)}) - \\bar r(x^{(t)};x^{(t)}) + \\bar r(x; x^{(t)})).\n\\label{eq:lin_primal_onenorm}\n\\end{equation}\nwhere $\\bar r(x;\\bar x) := \\sum_i \\gamma'(|\\bar x_i|) |x_i|$ is the linearized function of $x$ with reference point $\\bar x$.\nExpanding the terms, given $\\gamma$ concave,\n\\begin{equation}\n\\sum_{i=1}^d \\underbrace{\\gamma(|x^{(t)}_i|) }_{r(x^{(t)}) } + \\underbrace{\\gamma'(|x^{(t)}_i|)(|x_i| - |x^{(t)}_i|)}_{\\bar r(x; x^{(t)}) - \\bar r(x^{(t)}; x^{(t)})} \\geq \\sum_{i=1}^d \\gamma(|x_i|) = r(x)\n\\end{equation}\nshows that $\\bar F^{(t)}(x) \\geq F(x)$ for all $x^{(t)}$, and is indeed a majorant; additionally, when $x = x^{(t)}$, we have $\\bar F^{(t)}(x) = F(x)$.\n\n\n\n\\begin{figure}\n \\centering\n \\begin{tabular}[t]{ccc}\n \\begin{minipage}{3in}\n \\includegraphics[width=3in]{levelsets.eps}\n \\end{minipage}\n & & \n \\begin{minipage}{1.15in}\n \\includegraphics[width=1.25in]{example_gamma.eps}\n \\end{minipage}\n\\end{tabular}\n \\caption{\\textbf{Transformations $\\phi$ and $\\gamma$.} Left: Level sets for the penalty $h(x) = \\phi(\\sum_i \\gamma(|x_i|))$. The concave penalty $\\gamma$ increases the ``spike-ness\"; the convex penalty $\\phi$ increases the effect of the aggregate value.\n Right: Three example functions of $\\gamma$.\n RP-CGM will behave erratically when $\\gamma_{\\min} = 0$ (red and blue) and $\\gamma_{\\max}$ is unbounded (red), so we use a penalty that is bounded on both ends (green).\n }\n \\label{fig:phigamma}\n\\end{figure}\n\n\n\n\n\n\n\\paragraph{Primal-dual pair.} Given a reference point $\\bar x$, we define $r_0 := r(\\bar x) - \\bar r(\\bar x; \\bar x)$ and $w_i = \\gamma'(|\\bar x_i|)$. We can now ignore the reference point, given that $r_0$ and $w$ are fixed.\nThe penalty of the linearized problem is then \n\\[\n\\bar h(x) = \\phi(r_0+\\bar r(x)), \\qquad \\bar r(x) = \\sum_i w_i|x_i|.\n\\]\nFrom assumption \\ref{asspt:phigamma1}, for all $i$ $w_i \\geq \\gamma_{\\min} > 0$. Then the convex conjugate of the penalty function is \n\\begin{eqnarray*}\n\\bar h^*(y) &=& \\sup_{x}\\; x^Ty - \\phi(r_0+\\sum_iw_i|x_i|)\\\\\n &=& \\sup_{\\hat x, \\xi}\\;\\{\\xi \\hat x^Ty - \\phi(r_0+\\xi) : \\sum_i w_i |\\hat x_i| = 1\\}\\\\\n&\\overset{u_i = w_i\\hat x_i, \\; v_i = y_i\/w_i}{=}& \\sup_{u,\\xi}\\;\\{ \\xi u^Tv - \\phi(r_0 + \\xi) : v_i = \\frac{y_i}{w_i}, \\|u\\|_1 = 1\\}.\n\\end{eqnarray*}\nIn particular, note that $\\bar r(x)$ is positive homogeneous in $x$, which is why we can decompose $x = \\xi \\hat x$ where $\\bar r(x) = \\xi$ and $\\hat x = \\xi^{-1} x$, optimizing the two independently. \nThe maximization over $u$ does not involve $\\phi$, and is achieved at $u = \\mathbf{sign}(v_k)e_k$ where $|v_k| = \\|v\\|_\\infty$. Then \n\\begin{eqnarray*}\n\\bar h^*(y) &=& \\sup_{\\xi}\\;\\{ \\xi \\|v\\|_\\infty- \\phi(r_0 + \\xi) + \\underbrace{r_0\\|v\\|_\\infty-r_0\\|v\\|_\\infty}_{=0}: v_i = \\frac{y_i}{w_i}\\}\\\\\n&=& \\phi^*(\\|v\\|_\\infty) - r_0 \\|v\\|_\\infty.\n\\end{eqnarray*}\nWe can therefore we rewrite \\eqref{eq:lin_primal_onenorm} and its Fenchel dual as\n\\begin{eqnarray*}\n\\text{(P-simple)}&&\n \\mini{x\\in \\mathbb R^d} \\quad \\bar F(x; \\bar x) := f(x) + \\underbrace{\\phi\\left(r_0 + \\sum_i w_i |x_i|\\right)}_{=:\\bar h(x;\\bar x)}\\\\\n\\text{(D-simple)}&&\n\\maxi{z} \\quad \\bar F_D(z; \\bar x) := -f^*(-z) - \\phi^*\\left(\\max_i \\frac{|z_i|}{w_i}\\right) + r_0 \\cdot \\left(\\max_i \\frac{|z_i|}{w_i}\\right),\n\\end{eqnarray*}\nwhere if $\\bar x^*$ optimizes \\text{(P-simple)}~then $-\\nabla f(\\bar x^*)$ optimizes \\text{(D-simple)}. \n\n\n\n\n\n\\paragraph{Optimality conditions.} $\\bar x^*$ minimizes \\text{(P-simple)}~if and only if\n\\[\n0\\in \\nabla f(\\bar x^*) + \\partial \\bar h(\\bar x^*;\\bar x).\n\\]\n\n\\begin{assumption}\nWe assume that $f$ is convex and $L$-smooth w.r.t. $\\|\\cdot\\|_1$:\n\\begin{equation}\nf(y) - f(x) \\leq \\nabla f(x)^T(y-x) + \\frac{L}{2}\\|y-x\\|_1^2,\\quad \\forall x,y.\n\\label{eq:lsmooth-l1}\n\\end{equation}\n\\label{asspt:Lsmooth-l1}\n\\end{assumption}\nAn important consequence of \\eqref{eq:lsmooth-l1} is that, while the set of minimizers of \\text{(P-simple)}~may not necessarily be unique, their gradient $\\nabla f(\\bar x^*)$ will be unique. Specifically, \n\\eqref{eq:lsmooth-l1} implies that \n\\begin{equation}\nf(x)-f(y) \\geq \\nabla f(y)^T(x-y) + \\frac{1}{2L}\\|\\nabla f(x)-\\nabla f(y)\\|_\\infty^2, \\quad \\forall x,y\n\\label{eq:stronglyconvex:1norm}\n\\end{equation}\nand in particular taking $y = \\bar x^*$ where $-\\nabla f(\\bar x^*) \\in \\partial \\bar h(\\bar x^*;\\bar x)$, we have \n\\[\n f(x)+ \\bar h(x;\\bar x)-f(\\bar x^*) - \\bar h(\\bar x^*;\\bar x) \\geq \\frac{1}{2L}\\|\\nabla f(x)-\\nabla f(\\bar x^*)\\|_\\infty^2, \\quad \\forall x\n\\]\nand thus $x$ is optimal only if $\\nabla f(x) = \\nabla f(\\bar x^*)$.\n\n\n\\subsection{Reweighted penalized CGM}\nThe RP-CGM on \\eqref{eq:main-1norm} solves at each iteration the \\text{(Min-Maj)}~step\n\\begin{equation}\ns = \\argmin{s\\in \\mathbb R^d} \\; \\nabla f(x)^Ts + \\phi\\left(r_0 + \\sum_i w_i |s_i|\\right).\n\\label{eq:lmo-l1}\n\\end{equation}\n Following the derivation of the penalty conjugate, the optimization can be written as a composition of two steps, using a decomposition \n$s = \\xi \\hat s$, $\\xi = \\sum_i w_i |s_i|$,\n and noting that $\\sum_i |w_i| |s_i|$ is positive homogeneous w.r.t. $s$, the two variables can be treated separately. \n\n Specifically, \n \\[\n\\hat s = \\frac{\\mathbf{sign}(v_k)}{|w_k|}e_k, \\quad k = \\argmax{k}\\; |v_k| , \\quad v_i = \\begin{cases}\n-\\nabla f(x)_i\/|w_i|, & w_i \\neq 0,\\\\\n0, & w_i = 0,\n\\end{cases}\n\\]\nand\n\\[\n\\xi = \\argmin{\\xi \\geq 0}\\; -\\xi \\cdot \\|v\\|_\\infty + \\phi(r_0 + \\xi).\n\\]\n\n\nThis second step does not appear in the usual CGM, but is required for dealing with sparse penalties rather than constraints. It involves a one-dimensional optimization problem that at worst can be solved efficiently through bisection; more commonly, explicit low-cost expressions exist.\nIn particular, \n\\begin{eqnarray*}\n\\phi^*(\\|v\\|_\\infty) := \\sup_{\\xi} \\; \\xi\\cdot \\|v\\|_\\infty - \\phi(\\xi)\n=\\sup_{r_0 + \\xi} \\; (\\xi+r_0)\\cdot \\|v\\|_\\infty - \\phi(r_0 + \\xi),\n\\end{eqnarray*}\nand in fact \n\\[\n(\\phi^*)'(\\|v\\|_\\infty) = \\argsup{r_0 + \\xi'} \\; (\\xi'+r_0)\\cdot \\|v\\|_\\infty - \\phi(r_0 + \\xi') = \\xi + r_0\\cdot \\|v\\|_\\infty.\n\\]\nOften, closed-form solutions exist for this mapping $(\\phi^*)'$.\nTo relate to the vanilla CGM, where $\\phi(\\xi) = \\iota_{\\cdot \\leq 1}(\\xi)$, the convex conjugate $\\phi^*(\\nu) = \\nu$ and is always optimized at $\\xi = 1$.\n\n\n\\paragraph{Slope requirement on $\\gamma$.}\n To see why the bounds $0 < \\gamma_{\\min}<\\gamma_{\\max}<+\\infty$ are important, suppose that at any point $\\gamma'(|x_i|)= 0$. Then $v_i = \\|v\\|_\\infty \\to +\\infty$, and $\\xi = (\\phi^*)'(\\|v\\|_\\infty) \\to +\\infty$. Conversely if $\\gamma'(|x_i|) = +\\infty$ then $u_i = +\\infty$. In either case, the bounds on the slope of $\\gamma$ are required to control the norm of the iterate $x$. (See also fig. \\ref{fig:phigamma}.)\n\n \n\n\n\\paragraph{Minimum curvature on $\\phi$.}\nA key issue with changing the constrained formulation used in CGM to a penalized form is that without a norm bound on the iterates, it is not clear if the method will diverge. As an example, consider the minimization of $f(x) + g(x)$ where $f(x) = \\tfrac{1}{2} x^2$, $g(x) = |x|$. \nThen for any $|x| > 1$, the \\text{(Min-Maj)}~step \n$\\min_s s\\cdot x - |s| = -\\infty$ is undefined.\nWe therefore require a sufficient amount of curvature in $\\phi$ for convergent iterates.\n\n\n\n\n\n\n\n\n\\begin{assumption}[Lower quadratic bound]\nWe assume $\\phi$ is lower-bounded by a quadratic function\n$\\phi(\\xi) \\geq \\mu_\\phi \\xi^2-\\phi_0$,\nfor some $\\mu_\\phi > 0$ and $\\phi_0$. \n\\label{asspt:phi2}\n\\end{assumption}\n\\begin{property}\nIf \\eqref{asspt:phi2} holds, then\nthe derivative of $\\phi^*$ is asymptotically nonexpansive; e.g., for some finite-valued $\\xi_0$,\n$(\\phi^*)'(\\nu) \\leq \\frac{\\nu}{\\mu_\\phi}+\\xi_0$.\n\\end{property}\n\\begin{proof}\nAssume that $\\phi_0$ is as large as possible; e.g., there exists some finite $\\xi_0$ where $\\phi(\\xi_0) = \\mu\\xi_0^2 - \\phi_0$. Then for all $\\xi \\geq \\xi_0$, for all $\\nu\\in \\partial \\phi(\\xi)$,\n\\[\n\\mu (\\xi^2-\\xi_0^2 )\\leq \\phi(\\xi)-\\phi(\\xi_0) \\leq \\nu(\\xi-\\xi_0),\n\\]\nand therefore\n\\[\n\\nu\\geq\\mu \\frac{(\\xi+\\xi_0)(\\xi-\\xi_0)}{\\xi-\\xi_0} = \\mu \\xi+\\mu \\xi_0 \\; \\iff \\; \\xi \\leq \\mu^{-1} \\nu - \\xi_0.\n\\]\nThus, for any $\\xi$, $\\nu\\in \\partial \\phi( \\xi)$ must satisfy \n$\\xi \\leq \\max\\{\\xi_0, \\mu^{-1} \\nu - \\xi_0\\} \\leq \\mu^{-1} \\nu +\\xi_0$.\nBy Fenchel Young, this must apply to all $\\xi \\in \\partial \\phi^*(\\nu)$.\n\\qed\n\\end{proof}\n\n\n\n\n\n\\paragraph{Example: Monomials.}\nFor $1 \\leq \\alpha,\\beta \\leq +\\infty$, the following $\\phi:\\mathbb R_+\\to\\mathbb R_+$ and $\\phi^*:\\mathbb R_+\\to\\mathbb R_+$ form a conjugate pair:\n\\[\n\\phi(\\xi) = \\frac{1}{\\alpha} \\xi^{\\alpha}, \\qquad \\phi^*(\\nu) = \\frac{1}{\\beta}\\nu^\\beta, \\qquad \\frac{1}{\\alpha} + \\frac{1}{\\beta} = 1.\n\\]\nIn particular, in the case that $\\alpha = 1$, then $\\beta \\to +\\infty$, and the function \n\\[\n\\phi^*(\\nu) = \\lim_{\\beta\\to+\\infty} \\frac{1}{\\beta}\\nu^\\beta = \n\\begin{cases}\n0, & \\nu \\leq 1\\\\\n+\\infty, & \\nu > 1.\n\\end{cases}\n\\]\nIn this case, whenever $\\nu > 1$ then $\\phi^*(\\nu) = +\\infty$; we exclude this case as P-CGM will not converge in this case. \nWhen $\\alpha \\geq 2$, $\\phi$ is strongly convex and we can show $O(1\/t)$ convergence of P-CGM.\nWhen $1 < \\alpha < 2$, $\\phi^*(\\nu)$ is finite and the iterates are well-defined, but the method may converge or diverge.\n\n\n\n\n\n\n\n\n\n\n \\paragraph{Example: Barrier functions.}\nConsider \n\\begin{equation}\n\\phi(\\xi) = -\\frac{1}{\\beta}\\log(C-\\xi) - \\frac{\\xi}{C\\beta} + \\frac{\\log(C)}{\\beta},\n\\label{eq:logbarrier}\n\\end{equation}\nwhich is a log-barrier penalization function for $\\xi \\leq C$; as $\\beta \\to +\\infty$, $\\phi(\\xi)$ approaches the indicator function for this constraint. Its conjugate is \n\\[\n\\phi^*(\\nu) = C\\nu - \\beta^{-1} \\log(C\\beta\\nu+1),\n\\]\nachieved at $\\xi =C^2\\beta\\nu\/(C\\beta \\nu+1)$. For all $C > 0, \\beta > 0$, and $\\nu\\neq -(C\\beta)^{-1}$, both $\\phi^*$ and $\\xi^*$ exist and are finite. Note also the implicit constraint, as $\\phi(\\kappa_\\mathcal P(x))$ is finite only if $x\\in C\\mathcal P$.\n\n\n\n\n\nThe entire RP-CGM algorithm for the simple sparse optimization problem is given in Alg. \\ref{alg:simple}\n\n\\begin{algorithm}\n\\caption{RP-CGM on simple sparse optimization}\n\\begin{algorithmic}[1]\n\n\\Procedure{RP-CGM}{$f$, $\\phi$, $\\gamma$, $T$} \n \\State Initialize with any $x^{(0)}\\in \\mathbb R^d$\n \\For{$t = 1,...T$}\n \\State Compute negative gradient $z = -\\nabla f(x^{(t)})$\n\n \\State Compute weights $w$ and offset $r_0$ \\Comment{Reweight}\n \\[\n w_i = \\gamma'(|x^{(t)}_i|), \\quad i = 1,...,d, \\qquad r_0 = r(x^{(t)}) - \\bar r(x^{(t)};x^{(t)})\n \\]\n \\State Compute next atom $s =\\xi \\cdot \\mathbf{sign}(v_k) e_k $ where $v_i = z_i \/ w_i$, and\\Comment{Min-maj}\n \\[\n k = \\argmax{i}\\; |v_i|, \\quad \\xi = (\\phi^*)'(\\|v\\|_\\infty)-r_0\n \\]\n \n \\State Update $x^{(t+1)}$ \\Comment{Merge}\n \\[\n x^{(t+1)} = (1-\\theta^{(t)})x^{(t)} + \\theta^{(t)} s, \\quad \\theta^{(t)} = 2\/(1+t)\n \\]\n \\EndFor\n\\Return $x^{(T)}$\n\\EndProcedure\n\\end{algorithmic}\n\\label{alg:simple}\n\\end{algorithm}\n\n\n\\subsection{Convergence of RP-CGM}\nThe duality gap of the nonconvex original problem \\eqref{eq:main-1norm} is (as expected) bounded away from 0, and is thus an inadequate measure of suboptimality. \n\\begin{property}[Duality gap of nonconvex regularizer] \nFor $h(x) = \\phi(r(x))$,\n\\begin{eqnarray*}\nh(x) - h^{**}(x) \\geq \\phi(r(x)) - \\phi(\\gamma_{\\min} \\|x\\|_1).\n\\end{eqnarray*}\n\\end{property}\n\n\\begin{proof}\nFirst,\n\\[\nh^*(z) = \\sup_x \\; x^Tz - \\phi(r(x))\n= \\sup_x \\phi^*\\left(\\frac{x^Tz}{r(x)}\\right).\n\\]\nPicking \n\\[\n\\begin{cases}\nx_i\\to \\mathbf{sign}(z_i) \\cdot \\infty &\\text{ if } |z_i| = \\|z\\|_\\infty\\\\\nx_i=0 & \\text{ otherwise } \n\\end{cases}\n\\quad \\Rightarrow \\quad \n\\frac{x^Tz}{r(x)} = \\frac{\\|z\\|_\\infty}{\\gamma_{\\min}}\n\\]\ngives\n$h^*(z) \\geq \\phi^*\\left(\\frac{\\|z\\|_\\infty}{\\gamma_{\\min}}\\right)$, and so \n\\[\nh^{**}(x)\\leq \\sup_z \\; x^Tz - \\phi^*\\left(\\frac{\\|z\\|_\\infty}{\\gamma_{\\min}}\\right)\n= \\phi(\\|x\\|_1 \\gamma_{\\min}).\n\\]\n\\qed\n\\end{proof}\nInstead, we measure convergence via the gap of the linearized problem at $\\bar x = x$.\n\\begin{property}[Residual]\nThe gap of the linearized problem at $x$ with reference point $x$, denoted as $\\mathbf{res}(x) :=\\mathbf{gap}(x,-\\nabla f(x);x)$, satisfies\n\\[\n \\mathbf{res}(x) \\geq 0\\;\\forall x, \\qquad \\mathbf{res}(x) =0 \\iff \\text{ $x$ is a stationary point of \n (\\ref{eq:main-1norm}).}\n\\]\n\\label{prop:gapres-l1}\n\\end{property}\n\n\\begin{proof}\nSince $\\mathbf{res}(x)$ is a duality gap, it is always nonnegative. Explicitly,\ndenote $\\nu = \\max_i \\left(\\frac{|\\nabla f(x)_i|}{\\gamma'(|x_i|)}\\right)$. \n Then, since $f(x) + f^*(\\nabla f(x)) = \\nabla f(x)^Tx$,\n\\begin{eqnarray*}\n\\mathbf{res}(x) \n&=& x^T\\nabla f(x) + \\phi( r(x)) + \\phi^*(\\nu)+ (\\bar r(x;x)- r(x))\\cdot \\nu\\\\\n&\\overset{(a)}{\\geq}& x^T\\nabla f(x) + r(x)\\nu+ (\\bar r(x;x)- r(x))\\cdot \\nu\\\\\n&= & x^T\\nabla f(x) +\\sum_i \\gamma'(|x_i|)|x_i| \\cdot \\max_j \\left(\\frac{|\\nabla f(x))_j|}{\\gamma'(|x_j|)}\\right)\\\\\n&\\overset{(b)}{\\geq} & x^T\\nabla f(x) -x^T\\nabla f(x) = 0.\n\\end{eqnarray*}\nTightness of (a) occurs iff Fenchel-Young is satisfied with equality, e.g. $\\phi'(r(x)) = \\nu$.\nTightness of (b) occurs iff \n\\begin{equation}\n\\max_j \\;\\frac{|\\nabla f(x)_j|}{\\gamma'(|x_j|)} = \\frac{-\\nabla f(x)_i \\cdot \\mathbf{sign}(x_i)}{\\gamma'(|x_i|)}, \\quad \\forall x_i \\neq 0.\n\\label{eq:optcond-helper2}\n\\end{equation}\nCombining these two observations, then $\\mathbf{res}(x) = 0$ if and only if\n\\[\n-\\nabla f(x)_i \\in \\phi'(r(x)) \\cdot \\begin{cases}\n\\{\\gamma'(|x_i|) \\mathbf{sign}(x_i)\\} & x_i \\neq 0\\\\\n [-\\gamma'(0),\\gamma'(0)] & x_i = 0,\n\\end{cases}\n\\]\nwhich is the condition for $x = x^*$ a stationary point of \\eqref{eq:main-1norm}.\n\\qed\n\\end{proof}\n\n\n\n\\begin{thm}[Convergence of RP-CGM, simple case]\nPick any $x^{(0)}\\in \\mathbb R^d$ where $h(x^{(0)})$ is finite. Define the sequence $x^{(t)}$, $t = 1,...$ by the steps dictated in \\text{(Min-Maj)}~and \\text{(Merge)}, using the step size sequence $\\theta^{(t)} = 2\/(1+t)$. Given assumptions \\ref{asspt:phigamma1}, \\ref{asspt:Lsmooth-l1}, \\ref{asspt:phi2}, \nthen \n\\[\nF(x^{(t)}) - F(x^*) = O(1\/t), \\qquad \\min_{t' \\leq t}\\,\\mathbf{res}(x^{(t')}) = O(1\/t).\n\\]\n\\label{th:convergence-l1}\n\\end{thm}\nThis is a special case of Theorem \\ref{th:convergence}, which is proven in section \\ref{sec:nonconvex} and appendix \\ref{app:convergence}. The proof is inductive, and shows that $O(1\/t)$ behavior ``kicks in\" at a large enough $t$; explicit constants are given in section \\ref{sec:nonconvex}.\n\n\n\n\n\n\\subsection{Convex support recovery and screening}\nFrom optimality conditions,\n$\\bar x^*$ minimizes \\text{(P-simple)}~if \n\\begin{equation}\n- \\frac{\\nabla f(\\bar x^*)_i}{\\alpha w_i} \\in \n\\begin{cases}\n\\{\\mathbf{sign}(\\bar x^*_i)\\}, & \\bar x^*_i \\neq 0\\\\\n[-1,1], &\\bar x^*_i = 0,\n\\end{cases}\n\\label{eq:main-lin-onenorm-opt}\n\\end{equation}\nfor some $\\alpha \\in \\partial_{\\xi} \\phi(r_0 + \\xi)$ at $\\xi = \\sum_i w_i\\bar x_i^*$.\nIn other words, for this convex reweighting problem, the sparsity pattern of $\\bar x^*$ can be partially ascertained from $\\nabla f(\\bar x^*)$, in that the set of nonzeros of $\\bar x^*$ must be contained in the set of maximal indices of the reweighted $\\nabla f(\\bar x^*)$. Formally, define \n\\begin{equation}\n\\mathbf{dsupp}(x; \\bar x) :=\\left\\{i : \\frac{|\\nabla f(x)_i|}{\\gamma'(|\\bar x_i|)} =\\max_j \\frac{|\\nabla f(x)_j|}{\\gamma'(|\\bar x_j|)} \\right\\}.\n\\label{eq:dsupp-l1}\n\\end{equation}\nThen the optimality condition \\eqref{eq:main-lin-onenorm-opt} states that $\\mathbf{supp}(\\bar x^*) \\subseteq \\mathbf{dsupp}(\\bar x^*;\\bar x)$, where $\\bar x^*$ minimizes \\text{(P-simple)}. We are in particular interested in $\\bar x^* = x^*$ the stationary point of \\eqref{eq:main-1norm}.\nFrom this observation, we have our first screening property\n\\begin{property}[Screening for simple sparsity.]\nIf $\\|\\nabla f(x)-\\nabla f( x^*)\\|_\\infty \\leq \\epsilon$, then \n\\begin{equation}\n\\|\\nabla f(x)\\|_\\infty - |\\nabla f(x)_i| > 2\\epsilon\\gamma_{\\max}\\; \\Rightarrow \\; x_i^* = 0.\n\\label{eq:screen-l1-prop}\n\\end{equation}\n\\end{property}\n\\begin{proof}\nFirst, define $v_i = \\frac{|\\nabla f(x)_i|}{w_i}$ and $\\bar v_i^* = \\frac{|\\nabla f( x^*)_i|}{w_i} $. \nThen \n\\[\n\\frac{\\|\\nabla f(x^*)-\\nabla f(x)\\|_\\infty }{\\gamma_{\\max}}\\leq \\frac{\\|\\nabla f(x^*)-\\nabla f(x)\\|_\\infty }{\\max_i w_i} \\leq \\|v-\\bar v^*\\|_\\infty.\n\\]\nBy optimality conditions \n$\\bar v_i^* < \\|\\bar v^*\\|_\\infty \\Rightarrow \\bar x_i^* = 0$.\nThus, \\eqref{eq:screen-l1-prop} implies\n\\[\n2\\epsilon > \\frac{\\|\\nabla f(x)\\|_\\infty - |\\nabla f(x)_i|)}{\\gamma_{\\max}} \\geq \n\\|\\bar v\\|_\\infty-\\bar v_i\n\\]\nand therefore\n$\\|\\bar v^*\\|_\\infty-\\bar v^*_i \\leq \\|\\bar v\\|_\\infty-\\bar v_i + 2\\epsilon < 0$.\n\\qed\n\\end{proof}\n\n\n\nWhile this fundamental concept is very intuitive, it is not practical, as one does not have access to $\\nabla f(x^*)$ until the algorithm has terminated. \nIn the convex setting, this is overcome by showing that the duality gap between \\text{(P-simple)}~and \\text{(D-simple)}~upper bounds the gradient error. In the nonconvex setting, this bound is more complicated, but still leads to a useful heuristic screening rule.\n\\begin{property}[Residual bound on gradient error.]\nDefine $D(x) = \\sum_{i=1}^d \\gamma(|x_i|)-\\gamma(|x_i^*|) - \\gamma'(|x_i|)(|x_i-x_i^*|)$\nthe linearization error at $x$. \nDenoting $x^*$ a stationary point of \\eqref{eq:main-1norm}, then\n\\[\n\\|\\nabla f(x^*)-\\nabla f(x)\\|_\\infty \\leq \\frac{LD(x)}{2\\gamma_{\\min}} + \\sqrt{\\frac{L^2D(x)^2}{4\\gamma^2_{\\min}} + L\\mathbf{res}(x) + LD(x) \\frac{\\|\\nabla f(x)\\|_\\infty}{\\gamma_{\\min}}}.\n\\]\n\\label{prop:resboundsgrad-l1}\n\\end{property}\nNote that if $r(x) = \\|x\\|_1$ then $D(x) = 0$.\n\\begin{proof}\nFirst, note that \n\\begin{eqnarray}\n\\phi^*\\left(\\max_i \\frac{|z_i|}{w_i}\\right) + r_0\\cdot \\left(\\max_i \\frac{|z_i|}{w_i}\\right) &=& \\sup_y\\; y^Tz - \\phi(r_0 + \\sum_i w_i |y_i|) \\nonumber\\\\\n&\\geq & z^Tx^* - \\phi(r_0 + \\sum_i w_i |x^*_i|). \\label{eq:phistar-ell1-proofhelper1}\n\\end{eqnarray}\nAs before, $\\mathbf{res}(x) = \\bar F(x;x)-\\bar F_D(-\\nabla f(x);x)$.\nTaking $(x,-\\nabla f(x))$ as a feasible primal-dual pair and reference point $\\bar x=x$, and denoting $w_i =\\gamma'(|\\bar x_i|)$, $\\epsilon(x) = \\phi(r_0+\\sum_i w_i |x_i^*|) - \\phi(r(x^*))$, and $\\nu = \\max_i \\frac{|\\nabla f(x)_i|}{w_i}$,\n\\begin{eqnarray*}\n\\mathbf{res}( x) &=&\\underbrace{f(x) + f^*(\\nabla f(x))}_{\\text{use Fenchel-Young}}+\\phi(r(x)) + \\underbrace{\\phi^*(\\nu) -r_0 \\cdot (\\nu) }_{\\text{use \\eqref{eq:phistar-ell1-proofhelper1}}} \\\\\n&\\geq & \\nabla f(x)^T(x- x^*) + \\phi(r(x)) - \\phi\\left(r_0 + \\sum_i w_i | x^*_i|\\right)\\\\\n&\\overset{+\\epsilon(x)-\\epsilon(x)}{\\geq} & \\nabla f(x)^T(x- x^*) + \\underbrace{\\phi(r(x)) -\\phi(r(x^*))}_{\\text{convex in $x$}}- \\epsilon(x)\\\\\n&\\overset{g\\in \\partial h(x^*)}{\\geq} & \\nabla f(x)^T(x- x^*) + g^T(x- x^*) - \\epsilon(x).\n\\end{eqnarray*}\nPicking in particular $g = -\\nabla f( x^*)$,\n\\[\n\\mathbf{res}(x) +\\epsilon(x) \\geq (x- x^*)^T( \\nabla f(x) -\\nabla f( x^*)) \n\\overset{(\\star)}{\\geq} \\frac{1}{L}\\|\\nabla f( x^*)-\\nabla f(x)\\|_\\infty^2,\n\\]\nwhere $(\\star)$ follows from assumption \\ref{asspt:Lsmooth-l1}.\n\n\nNext, note that\n\\[\n\\epsilon(x) = \\phi(r(x)-r(x;x)+r(x^*;x))-\\phi(r(x^*)) \n\\leq \\phi'(r(x^*))D(x),\n\\]\nwhere in general, $D(x) \\leq (\\gamma_{\\max}-\\gamma_{\\min})\\|x-x^*\\|_1$ and $D(x) = 0$ if $\\gamma(\\xi) = \\xi$ (convex case).\nNoting that, at optimality, \n\\[\n\\phi'(r(x^*)) = \\max_i\\frac{|\\nabla f(x^*)_i|}{\\gamma'(|x_i^*|)}\\leq \\frac{\\|\\nabla f(x^*)\\|_\\infty}{\\gamma_{\\min}},\n\\]\nthen\n\\[\n\\gamma_{\\min}\\phi'(r(x^*)) \\leq \\|\\nabla f(x^*)\\|_\\infty \\leq \\|\\nabla f(x)\\|_\\infty + \\|\\nabla f(x) - \\nabla f(x^*)\\|_\\infty \n\\]\nand overall,\n\\begin{eqnarray*}\n\\|\\nabla f(x^*)-\\nabla f(x)\\|_\\infty^2 \n&\\leq& L \\mathbf{res}(x) + LD(x)\\frac{ \\|\\nabla f(x)\\|_\\infty + \\|\\nabla f(x) - \\nabla f(x^*)\\|_\\infty }{\\gamma_{\\min}}.\n\\end{eqnarray*}\nThis inquality is quadratic in $\\|\\nabla f(x)-\\nabla f(x^*)\\|_\\infty$, which leads to the stated bound. \n\\qed\n\\end{proof}\n\n\nFrom these two properties, we immediately get a screening rule for \\eqref{eq:main-1norm}:\n\\begin{thm} [Heuristic screening rule.]\n For any $x$ and some choice of $\\epsilon > 0$, define \n\\begin{equation}\n\\mathcal I_\\epsilon^{(t)} = \\left\\{i : \\|\\nabla f(x)\\|_\\infty - |\\nabla f(x)_i| \\geq \\epsilon + 2\\sqrt{L\\mathbf{res}(x)+\\epsilon } \\right\\}.\n\\label{eq:screening-l1}\n\\end{equation}\nIf \n\\[\n\\epsilon \\geq \\frac{L D(x)}{\\gamma_{\\min}} \\; \\max\\left\\{\\frac{1}{2},\\frac{L D(x)}{4\\gamma_{\\min}} + \\|\\nabla f(x)\\|_\\infty\\right\\}\n\\]\nthen \n $ x_i^* = 0$ for all $i\\in \\mathcal I_\\epsilon^{(t)}$, where $ x^*$ any minimizer of \\eqref{eq:main-1norm}. \n\\label{th:screen-l1}\n\\end{thm}\nNote that in the convex case ($\\gamma(|x_i|) = |x_i|$) then $D(x) = 0$ and $\\epsilon = 0$ is a safe choice, for all $x$. In the general case, since we do not know $D(x)$, we cannot guarantee the safety of an intermediate iterate; however, since $D(x^*) = 0$ by definition of stationary point, then $x^{(t)}\\to x^*$ implies $D(x^{(t)}) \\to 0$. Picking any decaying sequence $\\epsilon^{(t)}\\to 0$, therefore, forms a rule that converges to the true support.\n\n\n\n\\paragraph{Degeneracy and support recovery guarantee.}\nFollowing the terminology introduced in \\cite{hare2011identifying}, we say that $x^*$ is a \\emph{degenerate solution} if $\\mathbf{supp}(x^*) \\neq \\mathbf{dsupp}(x^*;x^*)$; that is, there exists $i$ where \n\\[\nx_i^* = 0 \\quad \\text{ and } \\quad \\frac{|\\nabla f(x^*)_i|}{\\gamma'(|x^*_i|)} = \\max_j\\, \\frac{|\\nabla f(x^*)_j|}{\\gamma'(|x^*_j|)}.\n\\]\nFinite-time support recovery is not possible for degenerate solutions \\citep{lewis2011identifying,hare2011identifying,burke1988identification}. We define\n\\[\n\\delta_i(x) = \\max_j\\, \\frac{|\\nabla f(x)_j|}{\\gamma'(|x_j|)} - \\frac{|\\nabla f(x)_i|}{\\gamma'(|x_i|)},\\quad \\delta_{\\min}(x) = \\min_{i : x_i = 0} \\delta_i(x)\n\\]\nand\nthe quantity $\\delta_{\\min}(x^*)$ expresses the distance to degeneracy for this solution. \n\n\\begin{corollary}\nIf $\\delta_{\\min} > 0$, then for a method $x^{(t)}\\to x^*$,\nthen the screening rule \\eqref{eq:screening-l1} with $\\epsilon = 0$\nidentifies $\\mathbf{supp}(x^*)$ after a finite number of iterations $\\bar t$; that is, for all $t \\geq \\bar t$, $\\mathcal I^{(t)}_0 = \\mathbf{supp}(x^*)$. In the convex case ($\\gamma(|x_i|) = |x_i|)$, this occurs when $\\|\\nabla f(x^*)-\\nabla f(x)\\|_\\infty\\leq \\delta_{\\min}\/3 $, which occurs at $\\bar t = O(1\/\\delta_{\\min}^2)$.\n\\end{corollary}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nTwo important characteristics of surfactants in binary fluid is that they can move towards the fluid interface due to their amphiphilic structure and they can reduce the interfacial tension and system energy \\cite{Myers2005Surfactant}. Therefore, surfactants have various applications in many fields such as biotechnology and industry because of their features~\\cite{Branger2002Accelerated,Liu2010Phase}. In the past two decades, there have been a number of excellent studies related to models with surfactants~\\cite{Fonseca2007Surfactants, Komura1997Two, Laradji1992The, Sman2006Diffuse, Xiao2018Numerical, Yang2017Linear}. Often, there are two different ideas to model the interfacial dynamics with surfactants. One is the sharp interface model which has a long history dated back to one century ago~\\cite{Gibbs1874Transactions,Stefan1970Uber}, and this kind of model has been adopted in \\cite{James2004A,Khatri2014An}. In fact, sharp interface models have made great progresses in explaining kinetics of diffusional phase transformations and simulating multiphase systems with surfactants at one time. However, there are some difficulties stemming from the interface interactions with various complex processes during the course of phase transformations \\cite{Liu2010Phase}. The other approach method is the known phase-field method~\\cite{Fix1983Phase, Langer1986Models, Rayleigh1892On, Waals1979The}. This approach makes use of an appropriate free energy functional to character the interfacial dynamics, and it has been adopted to investigate the interfacial dynamics with surfactants in~\\cite{Fonseca2007Surfactants, Laradji1992The, Teigen2011A, Teramoto2001Droplet, Theissen1999Lattice}. In particular, phase-field method was used to in \\cite{Laradji1992The} to study the phase transition behaviors of the monolayer microemulsion system, formed by surfactant molecules. Generally, the free energy of binary fluid-surfactant model consists of the following two parts: the first part is the classical Ginzburg-Landau double well potential, which is used to describe a binary mixture, and the other part, called nonlinear coupling entropy term, has a historical evolution process, and is used to account for the influence of the surfactant in boosting the formation of interfaces. Laradji et al. in the pioneering work \\cite{Laradji1992The} introduced two phase field variables to represent the local densities of the fluids, as well as the local concentration of the surfactant, respectively. As mentioned in~\\cite{Komura1997Two}, an extra diffusion term was added to prevent the model from becoming unbounded and a Ginzburg-Landau type potential for the concentration variable to allow the coexistence of the two bulk states. In order to restrict the range of the concentration variable, the authors added the logarithmic Flory-Huggins potential in \\cite{Sman2006Diffuse}, based on the nonlinear coupling entropy similar to~\\cite{Komura1997Two, Laradji1992The}. In consideration of penalizing the concentration to accumulate along the fluid interface, the authors changed the nonlinear coupled entropy slightly in~\\cite{Fonseca2007Surfactants}. In addition, a further modified model was considered in~\\cite{Teng2013Simulating} by adding the Flory-Huggins potential for the local concentration variable as well, in comparison with the model in~\\cite{Fonseca2007Surfactants}. \n\nIn this paper, we focus on constructing unconditionally energy stable numerical schemes for the binary fluid-surfactant model in \\cite{Teng2013Simulating}. \nThere have been some works about numerical approximation to multi-phase models~\\cite{Diegel2017A,Ji2019A}. \nOwing to the stiff nonlinear terms originated from the thin interface thickness parameter, \nthere are a lot of subtle difficulties to construct numerical schemes with unconditional energy stability, \nespecially for the second order accurate (in time) scheme. Lots of efforts have been made to solve these problems~\\cite{Diegel2017A, Yan2018A, Zhang2019Linear}, etc. \nSince a simple fully implicit or explicit type discretization brings extremely severe time step size constraint on the interfacial width~\n\\cite{Boyer2009Numerical, Feng2003Numerical, Shen2010Numerical}, a semi-implicit method was adopted in \\cite{Teng2013Simulating}. \nHowever, the author mentioned that it suffers from a small CFL conditional number. \nRecently, Gu et al. in \\cite{Gu2014An} constructed an energy stable finite difference scheme for the binary fluid-surfactant system, \nwhich is based on the convex splitting approach~\\cite{Eyre1998Un, Jie2012Second, Wang2011Energy, Wu2017Stabilized}: \nimplicit treatment for the convex part and explicit treatment for the concave part. \nMeanwhile, it is observed that, the convexity analysis for one mixed term has not been theoretically justified in \\cite{Gu2014An}, due to the multi variables involved in the system. \nIn addition, the positivity-preserving property has not been theoretically proved, so that the well-defined nature of the numerical scheme is not available. \nMore recently, Yang et al. constructed the linear and stable schemes for the binary fluid-surfactant system with constant mobility in~\\cite{Yang2017Linear}, \nusing the invariant energy quadratization (IEQ) technique~\n\\cite{Cheng2017Efficient, Guill2013On, Han2017Numerical, Yang2016Linear, Yang2017Efficient, Yang2017Numerical, Yang2017Numerical2, Jia2017Numerical}. \nIn this approach, the free energy is transformed into an equivalent quadratic form by introducing appropriate auxiliary variables, \nand all nonlinear terms in this system are treated semi-explicitly~\\cite{Yang2017Linear}. \nThe energy stability has been derived for the IEQ method, while such a stability has to be based on an alternate energy functional (involved with auxiliary variables), \nnot for the original energy functional, as always in the IEQ approach. Moreover, the positivity-preserving property is not available to the IEQ-based numerical method, \nbecause of the explicit treatment for the nonlinear logarithmic term. \nIn addition to the IEQ idea, Zhu et al. proposed the scalar auxiliary variable (SAV) method to the surfactant model in \\cite{Zhu2018Decoupled}, \nfollowing similar ideas in~\\cite{Jie2017A, Jie2018The}. The SAV approach introduces a constant-coefficient linear equation to solve at each time step, \nand the energy stability could be derived for an alternate energy involved with a scalar variable. The convergence and error estimate for the SAV approach, \nfor the typical Cahn-Hilliard equation with double-well potential, has also been established in recent works~\\cite{Xiao2018Energy, Jie2018Convergence}. \nHowever, an application of the SAV approach to the surfactant model could not overcome the difficulty to theoretically justify the positivity-preserving property, \ndue to the explicit treatment of the logarithmic term. In turn, the Flory-Huggins energy potential has to be re-defined and extended around and beyond the singular phase variable values. \nAlso see a more recent work~\\cite{Qin2019a} of SAV-based numerical algorithm for the surfactant model. \n\nAmong the existing numerical methods, different approaches have different advantages. Here, we pay attention to the the convex splitting approach, originated from the pioneering work of Eyre \\cite{Eyre1998Un}. The idea is that the energy admits a splitting into purely convex and concave parts, that is, $E=E_{c}-E_{e}$, where $E_{c}$ and $E_{e}$ are both convex. Such an idea has had wide applications in various gradient flow models, including either first or second order accurate schemes. See the related works for the phase field crystal (PFC) and the modified PFC (MPFC) equation \\cite{Wise2009An, Wang2010Global}, the epitaxial thin film growth models~\\cite{Chen2014A, Feng2018A}, and the Cahn-Hilliard flow coupled with fluid motion~\\cite{ChenConvergence, DiegelConvergence}, etc. \n\nMeanwhile, there have been extensive works of linear numerical schemes for the Cahn-Hilliard and epitaxial thin film equations~\\cite{LiD2017, LiD2017b, LiD2016a}, in which stabilized implicit-explicit approach has been applied, and energy stability has been theoretically provided. In the case of a Flory-Huggins energy potential with singular logarithmic terms, the positivity-preserving property has been recently established in~\\cite{Du2021a} for the corresponding Allen-Cahn equation, based on the maximum principle arguments. The advantage of such a linear scheme (corresponding to an explicit treatment of the nonlinear logarithmic term) is associated with the computational efficiency, so that a nonlinear Newton iteration is not required. On the other hand, this approach works very well for the positivity preserving analysis for the Allen-Cahn gradient flow, due to the availability of maximum principle, while its direct extension to the Cahn-Hilliard gradient flow would face a serious theoretical difficulty. In this paper, we design a uniquely solvable, positivity-preserving, unconditionally energy stable, and first order in time convergent scheme for the binary fluid-surfactant system, based on the convex-splitting idea, combined with the centered difference spatial approximation. For the theoretical analysis of the positivity-preserving property, we make use of the singular nature of the logarithmic function, and prove that such a singular nature prevents the numerical solution approaches the singular limit values, following similar ideas of in the analysis for the Cahn-Hilliard model~\\cite{Chen2019Pos, dong20b, Wang2019A, DOng2019A}, as well as the one for the Poisson-Nernst-Planck system~\\cite{LiuC20a, Qian20a}, droplet liquid film model~\\cite{ZhangJ2021}, etc. In addition, an optimal rate convergence analysis is provided, which is the first such work for the surfactant model. The key difficulty in such an analysis is associated with the logarithmic potential term and the coupled term. In this article, we can make full use of the convexity of energy associated with the nonlinear terms to directly deal with all logarithmic terms and coupled terms, because the convexity of energy indicates the corresponding nonlinear error inner product is always non-negative. \n\nThe rest of the paper is organized as follows. In Section~\\ref{sec: model}, we give a brief introduction to the binary fluid-surfactant phase field model and state its energy law. \nIn Section~\\ref{sec: numerical scheme}, the numerical scheme is proposed and analyzed, and we prove the unique solvability, positivity-preserving property, \nas well as the energy stability. An optimal rate convergence estimate is also provided. \nSome numerical experiments are presented in Section~\\ref{sec: numerical results}. \nFinally, some conclusions are made in Section~\\ref{sec: conclusion} .\n\\section{The mathematical model: binary fluid-surfactant system} \\label{sec: model}\nIn this paper, we consider the two-dimensional (2-D) binary fluid-surfactant system. \nWith the domain given by $\\Omega=\\left(0,L_{x}\\right)\\times\\left(0,L_{y}\\right)$, the binary fluid-surfactant system is formulated as \n\\begin{subequations}\\label{bfs}\n\\begin{align}\n\\phi_{t} &=M_{1}\\Delta \\mu_{\\phi},\\label{eqn_phi}\\\\\n\\rho_{t} &= M_{2}\\nabla\\cdot \\left(M\\left(\\rho\\right)\\nabla \\mu_{\\rho}\\right),\\label{eqn_rho}\\\\\n\\mu_{\\phi} &= \\frac{\\delta G}{\\delta \\phi} = \\frac{f' (\\phi)}{\\varepsilon} - \\varepsilon \\Delta \\phi \n+ \\alpha \\nabla \\cdot \\Big( \\frac{ \\rho \\nabla \\phi }{| \\nabla \\phi |} \\Big) ,\\label{eqn_mu_phi}\\\\\n\\mu_{\\rho} &= \\frac{\\delta G}{\\delta \\rho} = - \\alpha | \\nabla \\phi | + \\beta H' (\\rho) ,\\label{eqn_mu_rho}\n\\end{align}\n\\end{subequations}\nwith the periodic boundary condition and $M\\left(\\rho\\right)=\\rho\\left(1-\\rho\\right)$.\nThe PDE system~\\eqref{bfs} corresponds to the following free energy functional\n\\begin{equation} \\label{bfs-energy-1} \nG\\left(\\phi,\\rho\\right) = \\int_{\\Omega}\\Big(\\frac{f\\left( \\phi\\right)}{\\varepsilon}\n+\\frac{\\varepsilon}{2}|\\nabla \\phi|^{2}+\\frac{\\alpha}{2}\\left(\\rho-|\\nabla \\phi|\\right)^{2}\n+\\beta H\\left(\\rho\\right)\\Big)\\mathrm{d}\\bm{x},\n\\end{equation}\nwhere \n\\begin{equation*}\nf\\left(\\phi\\right) = \\frac{1}{4}\\phi^{2}\\left(1-\\phi\\right)^{2},\\quad \nH\\left(\\rho\\right) = \\rho\\ln\\rho+\\left(1-\\rho\\right)\\ln\\left(1-\\rho\\right) , \n\\end{equation*}\nand $\\alpha,\\beta, \\varepsilon$ are all small positive parameters. In this paper we assume $M\\left(\\rho\\right)=1$ and $M_{1}=M_{2}=\\mathcal{M}$ for simplicity. \nFurthermore, to avoid the singularity in calculating the coupled energy $\\left(\\rho-|\\nabla \\phi|\\right)^{2}$, \nwe use $\\sqrt{\\phi_{x}^{2}+\\phi_{y}^{2}+\\delta^{2}}$ to approximate $|\\nabla \\phi|$. \nAnd also, we add diffuse terms $\\frac{\\eta^{2}}{2}|\\Delta\\phi|^{2}$ and $\\frac{\\xi}{2}|\\nabla\\rho|^{2}$ in the energy density, so that the new free energy functional becomes \n\\begin{equation}\\label{energy}\nG_{new}\\left(\\phi,\\rho\\right) = \\int_{\\Omega}\n\\Big(\\frac{f\\left( \\phi\\right)}{\\varepsilon}\n+\\frac{\\varepsilon}{2}|\\nabla \\phi|^{2}\n+\\frac{\\eta^{2}}{2}|\\Delta\\phi|^{2}\n+\\frac{\\xi}{2}|\\nabla\\rho|^{2}\n+\\frac{\\alpha}{2}\\left(\\rho-|\\nabla \\phi|\\right)^{2}\n+\\beta H\\left(\\rho\\right)\\Big)\\mathrm{d}\\bm{x}.\n\\end{equation}\nFor simplicity, we still use $G$ to express $G_{new}$. In turn, the corresponding chemical potentials become \n\\begin{subequations}\\label{bfs-new}\n\\begin{align}\n\\mu_{\\phi} &= \\frac{\\delta G}{\\delta \\phi} = \\frac{f' (\\phi)}{\\varepsilon} - \\varepsilon \\Delta \\phi \n+ \\eta^2 \\Delta^2 \\phi + \\alpha \\nabla \\cdot \\Big( \\frac{ \\rho \\nabla \\phi }{| \\nabla \\phi |} \\Big) ,\\label{eqn_mu_phi-new}\\\\\n\\mu_{\\rho} &= \\frac{\\delta G}{\\delta \\rho} = - \\alpha | \\nabla \\phi | - \\xi \\Delta \\rho + \\beta H' (\\rho) .\\label{eqn_mu_rho-bew}\n\\end{align}\n\\end{subequations}\n\nAs always in the gradient system, the energy dissipation property is always valid: \n\\begin{equation*}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}G\\left(\\phi\\left(t\\right),\\rho\\left(t\\right)\\right)\\leq 0,\\quad t>0.\n\\end{equation*}\nBesides, the appearance of the Flory-Huggins energy indicates a positivity property for the density variable, $0 < \\rho <1$ at a point-wise level. Our primary aim is to develop a numerical scheme inheriting these properties at a theoretical level.\n\n\n\\begin{remark} \nTypically, in the context of physical models, the Dirichlet energy $\\| \\nabla u \\|^2$ represents surface tension, whereas the higher order term $\\| \\Delta u \\|^2$ represents bending rigidity. In general, it may be assumed that all orders of the energy density are represented in the expansion of the energy \n$$\n E (u) = f(u) + a_0 u^2 + a_2 \\| \\nabla u \\|^2 + a_4 \\| \\Delta u \\|^2 + ... , \n$$\nwhere f may be non-quadratic, and the coefficients $a_k$ may depend upon $u$, or derivatives of $u$ but are usually constants. On the other hand, it is typical to drop all higher order terms beyond those that are physically\/mathematically necessary to make the PDE well posed. One would assume that the corresponding coefficients are sufficiently small so as to justify this. In most cases, it may be more reasonable to assume that $a_2 = 0$ (or is negligible) and only keep terms of order higher than two. \n\nFor the binary fluid-surfactant system~\\eqref{bfs}, combined with the physical energy~\\eqref{bfs-energy-1}, we are able to construct a numerical scheme with an energy stability, while the optimal rate convergence analysis will face essential theoretical difficulties, due to the highly nonlinear and singular 1-Laplacian term involved for the variable $\\phi$. To overcome this subtle difficulty, we add an additional bi-harmonic diffusion term for $\\phi$, as well as a regular diffusion term for the $\\rho$, in the energy representation~\\eqref{energy}. As a result, both the energy stability and optimal rate convergence analysis could be theoretically justified, as will be demonstrated in the later section. In particular, an optimal rate convergence analysis will be the first such result for the binary fluid-surfactant system. Such an approach of adding higher order bi-harmonic diffusion process has been reported in many related nonlinear physical systems, in which the diffusion terms have played essential roles in the stability analysis, such as the planetary geostrophic equations of oceanic geophysical fluid model~\\cite{LiuJ2006, STWW07, STW98, STW00}, etc. \n\\end{remark} \n\n\n\\section{The numerical scheme} \\label{sec: numerical scheme}\n\nIn this section, we present a convex-concave decomposition of the energy $\\eqref{energy}$, and propose a convex splitting scheme based on such a decomposition. \nThe unique solvability, energy stability, positivity-preserving property will be analyzed afterward. \n\n\\subsection{The convex-concave decomposition of the energy}\n\\begin{lemma}\nSuppose that $\\Omega=\\left(0,L_{x}\\right)\\times\\left(0,L_{y}\\right)$ and $\\phi,\\rho: \\Omega \\to \\mathbb{R}$ are periodic and sufficiently regular. \nDefine the following energy functionals \n\\begin{align*}\nG_{c}\\left(\\phi,\\rho\\right)\n=& \\int_{\\Omega}\\frac{1}{4\\varepsilon} (\\phi-\\frac{1}{2} )^{4}+\\frac{1}{64\\varepsilon}\n +\\frac{\\varepsilon}{2}|\\nabla \\phi|^{2}+\\frac{\\eta^{2}}{2}|\\Delta\\phi|^{2}\n +\\beta H\\left(\\rho\\right)+\\frac{\\xi}{2}|\\nabla\\rho|^{2}\\\\\n &+\\frac{\\alpha}{2}\\left\\{\\left(\\rho-|\\nabla^\\delta \\phi|\\right)^{2}\n + (\\sqrt{2}-1 )\\rho^{2}+\\frac{1}{\\delta}|\\nabla \\phi|^{2}\\right\\}\\mathrm{d}\\bm{x},\\\\\nG_{e}\\left(\\phi,\\rho\\right)\n=& \\int_{\\Omega}\\frac{1}{8\\varepsilon} (\\phi-\\frac{1}{2} )^{2}\n +\\frac{\\alpha}{2}\\left\\{ (\\sqrt{2}-1 )\\rho^{2}\n +\\frac{1}{\\delta}|\\nabla \\phi |^{2}\\right\\}\\mathrm{d}\\bm{x} , \n\\end{align*}\nwith $|\\nabla^\\delta \\phi| := ( | \\nabla \\phi |^2 + \\delta^2 )^\\frac12$. \nThen $G_{c}\\left(\\phi,\\rho\\right)$ and $G_{e}\\left(\\phi,\\rho\\right)$ are both convex \nwith respect to $\\phi$ and $\\rho$, with $G\\left(\\phi,\\rho\\right)=G_{c}\\left(\\phi,\\rho\\right)-G_{e}\\left(\\phi,\\rho\\right)$.\n\\end{lemma}\n\n\\begin{proof}\nWe focus on the convexity analysis of $G_{c}\\left(\\phi,\\rho\\right)$ \nand $G_{e}\\left(\\phi,\\rho\\right)$. Let \n\\begin{align*}\ne_{c}\\left(\\phi,\\phi_{x},\\phi_{y},\\Delta\\phi,\\rho,\\rho_{x},\\rho_{y}\\right)\n=& \\frac{1}{4\\varepsilon} (\\phi-\\frac{1}{2} )^{4}+\\frac{1}{64\\varepsilon}\n +\\frac{\\varepsilon}{2}|\\nabla \\phi|^{2}+\\frac{\\eta^{2}}{2}|\\Delta\\phi|^{2}\n +\\beta H\\left(\\rho\\right)+\\frac{\\xi}{2}|\\nabla\\rho|^{2}\\\\\n &+\\frac{\\alpha}{2}\\left\\{\\left(\\rho-|\\nabla \\phi|\\right)^{2}\n + (\\sqrt{2}-1 )\\rho^{2}+\\frac{1}{\\delta}|\\nabla \\phi|^{2}\\right\\},\\\\\ne_{e}\\left(\\phi,\\phi_{x},\\phi_{y},\\Delta\\phi,\\rho,\\rho_{x},\\rho_{y}\\right)\n=& \\frac{1}{8\\varepsilon} (\\phi-\\frac{1}{2} )^{2}\n +\\frac{\\alpha}{2}\\left\\{ (\\sqrt{2}-1 )\\rho^{2}+\\frac{1}{\\delta}|\\nabla \\phi|^{2}\\right\\} . \n\\end{align*}\nWe also denote \n\\begin{subequations}\n\\begin{align*}\ne_{c_{1}}\\left(\\bm{v}\\right)\n& \\triangleq e_{c1}\\left(\\phi,\\phi_{x},\\phi_{y},\\Delta\\phi,\\rho,\\rho_{x},\\rho_{y}\\right)\n =\\frac{1}{4\\varepsilon} (\\phi-\\frac{1}{2} )^{4}+\\frac{1}{64\\varepsilon}\n +\\frac{\\varepsilon}{2}|\\nabla \\phi|^{2}+\\frac{\\eta^{2}}{2}|\\Delta\\phi|^{2}\n +\\beta H\\left(\\rho\\right)+\\frac{\\xi}{2}|\\nabla\\rho|^{2},\\\\\ne_{c_{2}}\\left(\\bm{v}\\right)\n& \\triangleq e_{c2}\\left(\\phi,\\phi_{x},\\phi_{y},\\Delta\\phi,\\rho,\\rho_{x},\\rho_{y}\\right)\n =\\frac{\\alpha}{2}\\left\\{\\left(\\rho-|\\nabla^\\delta \\phi|\\right)^{2}\n + (\\sqrt{2}-1 )\\rho^{2}+\\frac{1}{\\delta}|\\nabla \\phi|^{2}\\right\\},\\\\\ne_{e}\\left(\\bm{v}\\right)\n&\\triangleq e_{e}\\left(\\phi,\\phi_{x},\\phi_{y},\\Delta\\phi,\\rho,\\rho_{x},\\rho_{y}\\right)\n =\\frac{1}{8\\varepsilon} (\\phi-\\frac{1}{2} )^{2}\n +\\frac{\\alpha}{2}\\left\\{ (\\sqrt{2}-1 )\\rho^{2}+\\frac{1}{\\delta}|\\nabla \\phi|^{2}\\right\\},\n\\end{align*}\n\\end{subequations}\nwhere \n\\begin{equation*}\n{\\bm{v}}=\\left(v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)\n\\triangleq\\left(\\phi,\\phi_{x},\\phi_{y},\\Delta\\phi,\\rho,\\rho_{x},\\rho_{y}\\right) . \n\\end{equation*} \nThen we have \n\\begin{equation*}\ne_{c}\\left(\\bm{v}\\right)=e_{c_{1}}\\left(\\bm{v}\\right)+e_{c_{2}}\\left(\\bm{v}\\right), \\quad \nG_{c}\\left(\\phi,\\rho\\right)=\\int_{\\Omega}e_{c}\\left(\\bm{v}\\right)\\mathrm{d}\\bm{x}, \\quad \nG_{e}\\left(\\phi,\\rho\\right)=\\int_{\\Omega}e_{e}\\left(\\bm{v}\\right)\\mathrm{d}\\bm{x},\n\\end{equation*}\nand the following inequalities are derived: \n\\begin{subequations}\n\\begin{align*}\n\\partial_{v_{1}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)\n&=\\frac{3}{\\varepsilon} (\\phi-\\frac{1}{2} )^{2}\\geq 0,\\\\\n\\partial_{v_{2}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\varepsilon>0,\\\\\n\\partial_{v_{3}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\varepsilon>0,\\\\\n\\partial_{v_{4}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\eta^{2}> 0,\\\\\n\\partial_{v_{5}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)\n&=\\frac{\\beta}{\\rho\\left(1-\\rho\\right)}>0,\\\\\n\\partial_{v_{6}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\xi> 0,\\\\\n\\partial_{v_{7}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\xi> 0,\\\\\n\\partial_{v_{1}}^{2}e_{e}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\frac{1}{4\\varepsilon}>0,\\\\\n\\partial_{v_{2}}^{2}e_{e}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\frac{\\alpha}{\\delta}>0,\\\\\n\\partial_{v_{3}}^{2}e_{a}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=\\frac{\\alpha}{\\delta}>0,\\\\\n\\partial_{v_{4}}^{2}e_{c_{1}}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&=0,\\\\\n\\partial_{v_{5}}^{2}e_{e}\\left(v_1,v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}\\right)&= (\\sqrt{2}-1 )\\alpha>0.\n\\end{align*}\n\\end{subequations}\nThese facts imply that both $e_{c_{1}}\\left(\\bm{v}\\right)$ and $e_{e}\\left(\\bm{v}\\right)$ are convex. \nTo get the convexity of $e_{c_{2}}\\left(\\bm{v}\\right)$, we have to analyze the Hessian matrix of $e_{c_{2}}\\left(\\hat{\\bm{v}}\\right)$ as follows, \nwhere $\\hat{\\bm{v}}=\\left(v_{2},v_{3},v_{5}\\right)$: \n\\begin{equation}\nH\\left(\\hat{\\bm{v}}\\right)=\\alpha\\left[\n\\begin{matrix}\n\\displaystyle\n\\sqrt{2}& -\\frac{v_{2}}{\\sqrt{v_{2}^{2}+v_{3}^{2}+\\delta^2}} \n& -\\frac{v_{3}}{\\sqrt{v_{2}^{2}+v_{3}^{2}+\\delta^2}}\\\\\n-\\frac{v_{2}}{\\sqrt{v_{2}^{2}+v_{3}^{2}+\\delta^2}} \n& -\\frac{v_{5}\\left(v_3^{2}+\\delta^2\\right)}{\\left(v_{2}^{2}+v_{3}^{2}+\\delta^{2}\\right)^{\\frac{3}{2}}}\n+\\frac{1}{\\delta}+1 &\\frac{v_{2}v_{3}v_{5}}{\\left(v_{2}^{2}+v_{3}^{2}+\\delta^{2}\\right)^{\\frac{3}{2}}}\\\\\n-\\frac{v_{3}}{\\sqrt{v_{2}^{2}+v_{3}^{2}+\\delta^2}} \n&\\frac{v_{2}v_{3}v_{5}}{\\left(v_{2}^{2}+v_{3}^{2}+\\delta^{2}\\right)^{\\frac{3}{2}}}\n&-\\frac{v_{5}\\left(v_2^{2}+\\delta^2\\right)}{\\left(v_{2}^{2}+v_{3}^{2}+\\delta^{2}\\right)^{\\frac{3}{2}}}\n+\\frac{1}{\\delta}+1\n\\end{matrix}\n\\right] . \\label{Hessian-H-1} \n\\end{equation}\nSince $0 < \\rho < 1$, i.e. $v_{5}\\in\\left(0,1\\right)$, a careful application of calculus reveals that $H\\left(\\hat{\\bm{v}}\\right)$ is diagonally dominated, so that it is non-negative definite. This in turn indicates the convexity of $e_{c_{2}}$. Therefore, we obtain the following inequality, according to the definition of convex function: \n\\begin{subequations}\n\\begin{align}\n&e_{c}\\left(\\lambda\\bm{w}+\\left(1-\\lambda\\right)\\bm{v}\\right)\\leq \n\\lambda e_{c}\\left(\\bm{w}\\right)+\\left(1-\\lambda\\right)e_{c}\\left(\\bm{v}\\right),\\label{convex_c}\\\\\n&e_{e}\\left(\\lambda\\bm{w}+\\left(1-\\lambda\\right)\\bm{v}\\right)\\leq \n\\lambda e_{e}\\left(\\bm{w}\\right)+\\left(1-\\lambda\\right)e_{e}\\left(\\bm{v}\\right),\\label{convex_a}\n\\end{align}\n\\end{subequations}\nwhere $\\lambda\\in\\left(0,1\\right), \\bm{w}, \\bm{v}\\in\\mathbb{R}^{7}.$\nIntegrating both sides of $\\eqref{convex_c}$ and $\\eqref{convex_a}$ leads to \n\\begin{equation*}\nG_{c}\\left(\\lambda \\phi_{1}+\\left(1-\\lambda\\right) \\phi_{2}, \n\\lambda \\rho_{1}+\\left(1-\\lambda\\right) \\rho_{2}\\right)\\leq \n\\lambda G_{c}\\left(\\phi_{1},\\rho_{1}\\right)\n+\\left(1-\\lambda\\right)G_{c}\\left(\\phi_{2},\\rho_{2}\\right),\n\\end{equation*}\nand \n\\begin{equation*}\nG_{e}\\left(\\lambda\\phi_{1}+\\left(1-\\lambda\\right) \\phi_{2}, \n\\lambda \\rho_{1}+\\left(1-\\lambda\\right) \\rho_{2}\\right)\\leq \n\\lambda G_{e}\\left(\\phi_{1},\\rho_{1}\\right)\n+\\left(1-\\lambda\\right)G_{e}\\left(\\phi_{2},\\rho_{2}\\right),\n\\end{equation*}\nwhich indicates that both $G_{c}\\left(\\phi,\\rho\\right)$ and $G_{e}\\left(\\phi,\\rho\\right)$ \nare convex with respect to $\\phi$ and $\\rho$.\n\\end{proof}\n\nAs a generalization of the theorem presented in~\\cite{Wise2009An}, the following lemma is the foundation of energy stability for binary fluid-surfactant, \nor more generally, for two variable functional. The proof is similar to \\cite{Wise2009An}, so we skip it for the sake of brevity.\n \n\\begin{lemma}\nAssume $\\phi,\\varphi,\\rho,\\psi:\\Omega\\to\\mathbb{R}$ are periodic and smooth enough.\nIf $G=G_{c}-G_{e}$ gives a convex-concave decomposition, then we have \n\\begin{equation}\\label{une}\nG\\left(\\phi,\\rho\\right)-G\\left(\\varphi,\\psi\\right)\\leq\n\\left(\\delta_{\\phi}G_{c}\\left(\\phi,\\rho\\right)-\\delta_{\\phi}G_{e}\\left(\\varphi,\\psi\\right),\n\\phi-\\varphi\\right)_{L^{2}}+\n\\left(\\delta_{\\rho}G_{c}\\left(\\phi,\\rho\\right)-\\delta_{\\rho}G_{e}\\left(\\varphi,\\psi\\right),\n\\rho-\\psi\\right)_{L^{2}},\n\\end{equation}\nwhere $\\delta$ denotes the variational derivative.\n\\end{lemma}\n\nGiven a time step $\\Delta t>0$, we construct the discrete-time, continuous-space scheme of the binary fluid-surfactant system~$\\eqref{bfs}$ as follows\n\\begin{subequations}\\label{CS}\n\\begin{align}\n\\frac{\\phi^{n+1}-\\phi^{n}}{\\Delta t} &=\\mathcal{M}\\Delta \\mu_{\\phi}^{n+1},\\\\\n\\frac{\\rho^{n+1}-\\rho^{n}}{\\Delta t} &= \\mathcal{M}\\Delta \\mu_{\\rho}^{n+1},\\\\\n\\mu_{\\phi}^{n+1} \n&= \\delta_{\\phi} G_{c}\\left(\\phi^{n+1},\\rho^{n+1}\\right)-\\delta_{\\phi}G_{e}\\left(\\phi^{n},\\rho^{n}\\right),\\\\\n\\mu_{\\rho}^{n+1} \n&= \\delta_{\\rho} G_{c}\\left(\\phi^{n+1},\\rho^{n+1}\\right)-\\delta_{\\rho}G_{e}\\left(\\phi^{n},\\rho^{n}\\right).\n\\end{align}\n\\end{subequations}\n\n\n\\subsection{The spatial discretization and the fully discrete numerical scheme}\nThe centered difference approximation is applied to discretize the space. \nHere we first recall some basic notations of this methodology, and we use the similar notations and results for some discrete functions and operators introduced in~\n\\cite{Guo2016An, Wise2009An}. Let $\\Omega=(0,L_x)\\times(0,L_y)$, where for simplicity, we assume $L_x=L_y:=L>0$. Let $N\\in \\mathbb{N}$ be given, \nand define the grid mesh size $h:=L\/N$. Such a uniform mesh size assumption is only for simplicity of presentation.\nWe define the following two uniform, infinite node sets with grid spacing $h>0$:\n\\begin{equation}\nE:=\\{p_{i+\\frac{1}{2}}~|~i\\in\\mathbb{Z}\\},\\quad C:=\\{p_{i}~|~i\\in\\mathbb{Z}\\},\n\\end{equation}\nwhere $p_i=p(i):=(i-\\frac{1}{2})\\cdot h$. \nConsider the following 2-D discrete $N^{2}$-peroidic function spaces:\n\\begin{align*}\n&\\mathcal{C}_{per}:=\\{\\nu: C\\times C\\to\\mathbb{R}|\\nu_{i,j}\n=\\nu_{i+\\alpha N,j+\\beta N},~\\forall i,j,\\alpha,\\beta \\in \\mathbb{Z}\\},\\\\\n&\\mathcal{E}_{per}^{x}:=\\{\\nu: E\\times C\\to\\mathbb{R}|\\nu_{i+\\frac{1}{2},j}\n=\\nu_{i+\\frac{1}{2}+\\alpha N,j+\\beta N},~\\forall i,j,\\alpha,\\beta \\in \\mathbb{Z}\\}.\n\\end{align*}\nThe spaces $\\mathcal{E}_{per}^{y}$ can be analogously defined. Here we use the Greek symbols $\\nu_{i,j}=\\nu(p_{i},p_{j})$, et cetera. \nThe functions of $\\mathcal{C}_{per}$ are called $cell-centered$ functions, while the functions of $\\mathcal{E}_{per}^{x}$ and $\\mathcal{E}_{per}^{y}$ are called \n$x-direction$ and $y-direction$ $edge-centered$ functions, respectively. \nWe also define the mean zero space\n\\begin{equation*}\n\\mathring{\\mathcal{C}}_{per}:=\\left\\{\\nu \\in \\mathcal{C}_{per}~|~\n0=\\bar{\\nu}:=\\frac{h^{2}}{|\\Omega|}\\sum_{i,j=1}^{m}\\nu_{i,j}\\right\\}.\n\\end{equation*}\nAdditionally, we denote $\\vec{\\mathcal{E}}_{per}:=\\mathcal{E}_{per}^{x} \\times \\mathcal{E}_{per}^{y}$.\n\nNext, the important difference and average operators are introduced on the function spaces:\n\\begin{align*}\n&A_{x}\\nu_{i+\\frac{1}{2},j}:=\\frac{1}{2}(\\nu_{i+1,j}+\\nu_{i,j}),\\quad \n D_{x}\\nu_{i+\\frac{1}{2},j}:=\\frac{1}{h}(\\nu_{i+1,j}-\\nu_{i,j}),\\\\\n&A_{y}\\nu_{i,j+\\frac{1}{2}}:=\\frac{1}{2}(\\nu_{i,j+1}+\\nu_{i,j}),\\quad\n D_{y}\\nu_{i,j+\\frac{1}{2}}:=\\frac{1}{h}(\\nu_{i,j+1}-\\nu_{i,j}),\n\\end{align*}\nwith \n$A_{x},D_{x}:\\mathcal{C}_{per}\\to \\mathcal{E}_{per}^{x}$ and \n$A_{y},D_{y}:\\mathcal{C}_{per}\\to \\mathcal{E}_{per}^{y}$. Likewise,\n\\begin{align*}\n&a_{x}\\nu_{i,j}:=\\frac{1}{2}(\\nu_{i+\\frac{1}{2},j}+\\nu_{i-\\frac{1}{2},j}),\\quad \n d_{x}\\nu_{i,j}:=\\frac{1}{h}(\\nu_{i+\\frac{1}{2},j}-\\nu_{i-\\frac{1}{2},j}),\\\\\n&a_{y}\\nu_{i,j}:=\\frac{1}{2}(\\nu_{i,j+\\frac{1}{2}}+\\nu_{i,j-\\frac{1}{2}}),\\quad\n d_{y}\\nu_{i,j}:=\\frac{1}{h}(\\nu_{i,j+\\frac{1}{2}}-\\nu_{i,j-\\frac{1}{2}}),\n\\end{align*}\nwith $a_{x},d_{x}:\\mathcal{C}_{per}\\to \\mathcal{C}_{per}^{x}$ and $a_{y},d_{y}:\\mathcal{C}_{per}\\to \\mathcal{C}_{per}^{y}$. \nThe discrete gradient $\\nabla_{h}:\\mathcal{C}_{per}\\to\\vec{\\mathcal{E}}_{per}$ is defined via \n\\begin{equation*}\n\\nabla_{h}\\nu_{i,j}=(D_{x}\\nu_{i+\\frac{1}{2},j},D_{y}\\nu_{i,j+\\frac{1}{2}}),\n\\end{equation*}\nand the discrete divergence $\\nabla_{h}\\cdot:\\mathcal{E}_{per}\\to\\vec{\\mathcal{C}}_{per}$ becomes \n\\begin{equation*}\n\\nabla_{h}\\cdot \\vec{f}_{i,j}=d_{x}f_{i,j}^x +d_{y }f_{i,j}^y , \\quad \n\\mbox{for} \\, \\, \\, \\vec{f}=(f^{x},f^{y}) \\in \\vec{\\mathcal{E}}_{per} . \n\\end{equation*}\nThe standard 2-D discrete Laplacian, $\\Delta_{h}:\\mathcal{C}_{per}\\to\\mathcal{C}_{per}$, is given by\n\\begin{align*}\n\\Delta_{h}\\nu_{i,j}\n:&=\\nabla_{h}\\cdot(\\nabla_{h} \\nu)_{i,j}\n =d_{x}(D_{x}\\nu)_{i,j}+d_{y}(D_{y}\\nu)_{i,j}\\\\\n &=\\frac{1}{h^{2}}(\\nu_{i+1,j}+\\nu_{i-1,j}+\\nu_{i,j+1}+\\nu_{i,j-1}-4\\nu_{i,j}).\n\\end{align*}\nMore generally, if $\\mathcal{D}$ is a periodic $scalar$ function that is defined at all of the face center points and $\\vec{f} \\in \\vec{\\mathcal{E}}_{per}$, \nassuming point-wise multiplication, we may define\n\\begin{equation*}\n\\nabla_{h}\\cdot(\\mathcal{D}\\vec{f})_{i,j}\n=d_{x}(\\mathcal{D}f^{x})_{i,j}+d_{y}(\\mathcal{D}f^{y})_{i,j}.\n\\end{equation*}\nSpecifically, if $\\nu\\in\\mathcal{C}_{per}$, then \n$\\nabla_{h}\\cdot(\\mathcal{D}\\nabla_{h}):\\mathcal{C}_{per}\\to\\mathcal{C}_{per}$ \nis defined point-wise via\n\\begin{equation*}\n\\nabla_{h}\\cdot(\\mathcal{D}\\nabla_h\\nu)_{i,j}\n=d_{x}(\\mathcal{D}D_{x}\\nu)_{i,j}+d_{y}(\\mathcal{D}D_{y})_{i,j}.\n\\end{equation*}\n\nNow we are ready to introduce the following grid inner products:\n\\begin{align*}\n&\\langle\\nu,\\xi\\rangle_{\\Omega}:=h^{2}\\sum_{i,j=1}^{N}\\nu_{i,j}\\xi_{i,j},\\quad\n \\nu, \\xi \\in \\mathcal{C}_{per}, \\\\\n&[\\nu,\\xi]_{x}:=\\langle a_{x}(\\nu\\xi),1\\rangle_{\\Omega},\\quad \n \\nu, \\xi \\in \\mathcal{E}_{per}^{x},\\\\\n&[\\nu,\\xi]_{y}:=\\langle a_{y}(\\nu\\xi),1\\rangle_{\\Omega},\\quad \n \\nu, \\xi \\in \\mathcal{E}_{per}^{y}.\\\\\n&[\\vec{f}_{1},\\vec{f}_{2}]_{\\Omega}\n:=[f_{1}^{x},f_{2}^{x}]_{x}\n +[f_{1}^{y},f_{2}^{y}]_{y},\\quad \n\\vec{f}_{i}=(f_{i}^{x},f_{i}^{y}) \\in \\vec{\\mathcal{E}}_{per},\\quad i=1,2.\n\\end{align*}\n\nWe define the following norms for cell-centered functions. \nIf $\\nu\\in\\mathcal{C}_{per}$, then $\\|\\nu\\|_{2}^{2}:=\\langle\\nu,\\nu\\rangle_{\\Omega}$; $\\|\\nu\\|_{p}^{p}:=\\langle|\\nu|^{p},1\\rangle_{\\Omega}$, for $1\\leq p\\leq \\infty$, \nand $\\|\\nu\\|_{\\infty}:=\\max\\limits_{1\\leq i,j\\leq N}|\\nu_{i,j}|$. The norms of the gradient are defined as follows: for $\\nu\\in\\mathcal{C}_{per}$,\n\\begin{equation*}\n\\|\\nabla_{h}\\nu\\|_{2}^{2}\n:=[\\nabla_{h}\\nu,\\nabla_{h}\\nu]_{\\Omega}\n =[D_{x}\\nu,D_{x}\\nu]_{x}\n +[D_{y}\\nu,D_{y}\\nu]_{y},\n\\end{equation*}\nand, more generally, for $1\\leq p \\leq \\infty$,\n\\begin{equation*}\n\\|\\nabla_{h}\\nu\\|_{p}\n:=\\left(\n[|D_{x}\\nu|^{p},1]_{x}+\n[|D_{y}\\nu|^{p},1]_{y}\\right)^{\\frac{1}{p}},\n\\end{equation*}\nHigher order norms can be similarly introduced; for example,\n\\begin{equation*}\n\\|\\nu\\|_{H_{h}^{1}}^{2}:=\\|\\nu\\|_{2}^{2}+\\|\\nabla_{h}\\nu\\|_{2}^{2},\\quad \n\\|\\nu\\|_{H_{h}^{2}}^{2}:=\\|\\nu\\|_{H_{h}^{1}}^{2}+\\|\\Delta_{h}\\nu\\|_{2}^{2}.\n\\end{equation*}\n\nTo facilitate the convergence analysis, we need to introduce a discrete analogue of the space $H_{per}^{-1}(\\Omega)$, as outlined in \\cite{Wang2011Energy}. \nSuppose that $\\mathcal{D}$ is a positive, periodic scalar function defined at all of the face center points. \nFor any $\\phi\\in\\mathcal{C}_{per}$, there exists a unique $\\psi\\in\\mathring{\\mathcal{C}}_{per}$, that solves\n\\begin{equation}\n\\mathcal{L}_{\\mathcal{D}}(\\psi):=-\\nabla_{h}\\cdot\\left(\\mathcal{D}\\nabla_{h}\\psi\\right)=\\phi-\\bar{\\phi},\n\\end{equation}\nwhere, recall, $\\bar{\\phi}:=|\\Omega|^{-1}\\langle\\phi,1\\rangle_{\\Omega}$. We equip this space with a bilinear form: for any $\\phi_{1},\\phi_{2}\\in\\mathring{\\mathcal{C}}_{per}$, define\n\\begin{equation}\n\\langle\\phi_{1},\\phi_{2}\\rangle_{\\mathcal{L}_{\\mathcal{D}}^{-1}}\n:=[\\mathcal{D}\\nabla_{h}\\psi_{1},\\nabla_{h}\\psi_{2}]_{\\Omega},\n\\end{equation}\nwhere $\\psi_{i}\\in\\mathring{\\mathcal{C}}_{per}$ is the unique solution to \n\\begin{equation}\n\\mathcal{L}_{\\mathcal{D}}(\\psi_{i}):=-\\nabla_{h}\\cdot(\\mathcal{D}\\nabla_{h}\\psi_{i})=\\phi_{i},\\quad i=1,2.\n\\end{equation}\nThe following identity is easy to prove via summation-by-parts:\n\\begin{equation}\n\\langle\\phi_{1},\\phi_{2}\\rangle_{\\mathcal{L}_{\\mathcal{D}}^{-1}}\n=\\langle\\phi_{1},\\mathcal{L}_{\\mathcal{D}}^{-1}(\\phi_{2})\\rangle_{\\Omega}\n=\\langle\\mathcal{L}_{\\mathcal{D}}^{-1}(\\phi_{1}),\\phi_{2}\\rangle_{\\Omega},\n\\end{equation}\nand since $\\mathcal{L}_{\\mathcal{D}}$ is symmetric positive definite, $\\langle\\cdot,\\cdot\\rangle_{\\mathcal{L}_{\\mathcal{D}}^{-1}}$ is \nan inner product on $\\mathring{\\mathcal{C}}_{per}$. When $\\mathcal{D}\\equiv 1$, we drop the subscript and write $\\mathcal{L}_{1} = \\mathcal{L}$, \nand in this case we usually write $\\langle\\cdot,\\cdot\\rangle_{\\mathcal{L}_{\\mathcal{D}}^{-1}}=:\\langle\\cdot,\\cdot\\rangle_{-1,h}$. \nIn the general setting, the norm associated to this inner product is denoted as \n$\\|\\phi\\|_{\\mathcal{L}_{\\mathcal{D}}^{-1}}:=\\sqrt{\\langle\\phi,\\phi\\rangle_{\\mathcal{L}_{\\mathcal{D}}^{-1}}}$,\n for all $\\phi \\in \\mathring{\\mathcal{C}}_{per}$, but, if $\\mathcal{D}\\equiv 1$, we write $\\|\\cdot\\|_{\\mathcal{L}_{\\mathcal{D}}^{-1}}=:\\|\\cdot\\|_{-1,h}$.\n\nWith the preparations above, we turn to discuss the discrete energy and the fully discrete scheme. \nDefine the discrete energies $E,E_{c},E_{e}:\\mathcal{C}_{per}\\times\\mathcal{C}_{per}\\to\\mathbb{R}$ as\n\\begin{equation*}\n\\begin{aligned} \nE\\left(\\phi,\\rho\\right)= & \nh^{2}\\sum_{i,j=1}^{N} \\Big(\\frac{f\\left(\\phi_{i,j}\\right)}{\\varepsilon}\n+\\frac{\\varepsilon}{2}\\left|\\nabla_{h}\\phi_{i,j}\\right|^{2}\n+\\frac{\\eta^{2}}{2}\\left|\\Delta_{h}\\phi_{i,j}\\right|^{2}\n+\\frac{\\xi}{2}\\left|\\nabla_{h}\\rho_{i,j}\\right|^{2} \n\\\\\n & \\qquad \\quad \n+\\frac{\\varepsilon}{2}\\left(\\rho_{i,j}- {\\cal A} |\\nabla_{h}^\\delta \\phi|_{i,j} \\right)^{2}\n+\\beta H (\\rho_{i,j} ) \\Big), \n\\end{aligned} \n\\end{equation*}\nin which ${\\cal A} |\\nabla_{h}^\\delta \\phi|$ is defined as \n$$\n ( {\\cal A} |\\nabla_{h}^\\delta \\phi|_{i,j} )^2 = \\frac12 ( (D_x \\phi)_{i+\\frac12, j}^2 + (D_x \\phi)_{i-\\frac12, j}^2 + (D_y \\phi)_{i, j+\\frac12}^2 + (D_y \\phi)_{i, j-\\frac12}^2) + \\delta^2 . \n$$ \nThe relevant discrete convex and concave energy functionals $E_{c}, E_{e}:\\mathcal{C}_{per}\\times\\mathcal{C}_{per}\\to\\mathbb{R}$ are given by\n\\begin{align}\nE_{c}\\left(\\phi,\\rho\\right)\n=& h^{2}\\sum_{i,j=1}^{N}\\left(\\frac{1}{4\\varepsilon} (\\phi_{i,j}-\\frac{1}{2} )^{4}\n+\\frac{1}{64\\varepsilon}\n +\\frac{\\varepsilon}{2}|\\nabla_{h} \\phi_{i,j}|^{2}\n +\\frac{\\eta^{2}}{2}|\\Delta_{h}\\phi_{i,j}|^{2}\n +\\beta H\\left(\\rho_{i,j}\\right)\n +\\frac{\\xi}{2}|\\nabla_{h}\\rho_{i,j}|^{2}\\right. \\nonumber \\\\\n &\\left.+\\frac{\\alpha}{2}\\left(\\left(\\rho_{i,j}- {\\cal A} |\\nabla_{h}^\\delta \\phi |_{i,j} \\right)^{2}\n + (\\sqrt{2}-1 )\\rho_{i,j}^{2}\n +\\frac{1}{\\delta}|\\nabla_{h}\\phi_{i,j}|^{2}\\right)\\right), \\label{discrete-energy-convex} \\\\\nE_{e}\\left(\\phi,\\rho\\right)\n=& h^{2}\\sum_{i,j=1}^{N}\\left(\\frac{1}{8\\varepsilon} (\\phi_{i,j}-\\frac{1}{2} )^{2}\n +\\frac{\\alpha}{2}\\left( (\\sqrt{2}-1 )\\rho_{i,j}^{2}\n +\\frac{1}{\\delta}|\\nabla_h \\phi_{i,j} |^2 \\right)\\right) . \\label{discrete-energy-concave} \n\\end{align}\nIn particular, a Hessian matrix could be similar formulated as the one given by~\\eqref{Hessian-H-1}, for the following discrete function \n\\begin{equation} \n e_{c_2, h} (\\rho, v_1, v_2, v_3, v_4 ) \n = \\frac{\\alpha}{2} \\Big( (\\rho - \\sqrt{ \\frac12 ( v_1^2 + v_2^2 + v_3^2 + v_4^2) + \\delta^2} )^{2}\n + (\\sqrt{2}-1 ) \\rho^{2}\n +\\frac{1}{2 \\delta} ( v_1^2 + v_2^2 + v_3^2 + v_4^2) \\Big) . \n\\end{equation} \nA careful calculation reveals that, the corresponding $5 \\times 5$ Hessian matrix is diagonally dominated, therefore non-negative definite. This in turn leads to the convexity of the following discrete functional: \n\\begin{equation} \n E_{c_2, h} (\\rho, v_1, v_2, v_3, v_4 ) \n = \\frac{\\alpha}{2} h^{2} \\sum_{i,j=1}^{N}\n \\left( (\\rho_{i,j}- {\\cal A} |\\nabla_{h}^\\delta \\phi |_{i,j} )^{2}\n + (\\sqrt{2}-1 )\\rho_{i,j}^{2}\n +\\frac{1}{\\delta}|\\nabla_{h}\\phi_{i,j}|^{2}\\right) . \n\\end{equation} \nThe convexity analysiss for the other parts of $E_c$ and $E_e$ is more straightforward. \n\nWe follow the idea of convexity splitting and consider the following semi-implicit, \nfully discrete scheme: given $\\phi^{n}, \\rho^{n} \\in \\mathcal{C}_{per}$, find \n$\\phi^{n+1}, \\rho^{n+1},\\mu_{\\phi}^{n+1},\\mu_{\\rho}^{n+1} \\in \\mathcal{C}_{per}$, \nsuch that\n\\begin{subequations}\\label{cs_1st}\n\\begin{align}\n\\frac{\\phi^{n+1}-\\phi^{n}}{\\Delta t} &=\\mathcal{M}\\Delta_{h} \\mu_{\\phi}^{n+1},\\label{cs_1st_phi}\\\\\n\\frac{\\rho^{n+1}-\\rho^{n}}{\\Delta t} &= \\mathcal{M}\\Delta_{h}\\mu_{\\rho}^{n+1},\\label{cs_1st_rho}\\\\\n\\mu_{\\phi}^{n+1} &= \\delta_{\\phi} E_{c}\\left(\\phi^{n+1},\\rho^{n+1}\\right)\n-\\delta_{\\phi}E_{e}\\left(\\phi^{n},\\rho^{n}\\right),\\label{cs_1st_mu1}\\\\\n\\mu_{\\rho}^{n+1} &= \\delta_{\\rho} E_{c}\\left(\\phi^{n+1},\\rho^{n+1}\\right)\n-\\delta_{\\rho}E_{e}\\left(\\phi^{n},\\rho^{n}\\right),\\label{cs_1st_mu2}\n\\end{align}\n\\end{subequations}\nwhere\n\\begin{align}\n\\mu_{\\phi}^{n+1}=\n&\\frac{1}{\\varepsilon} (\\phi^{n+1}-\\frac{1}{2} )^{3}\n- (\\varepsilon+\\alpha+\\frac{\\alpha}{\\delta} )\\Delta_h \\phi^{n+1}\n+\\eta^{2} \\Delta_h^{2} \\phi^{n+1}\n+\\alpha \\nabla_h \\cdot \\left( {\\cal A} ( \\frac{\\rho^{n+1} }{ {\\cal A} |\\nabla_h^\\delta \\phi^{n+1}|} ) \\nabla_h \\phi^{n+1} \\right) \\nonumber\\\\\n&-\\frac{1}{4\\varepsilon} (\\phi^{n}-\\frac{1}{2} )\n+\\frac{\\alpha}{\\delta} \\Delta_h \\phi^{n}, \\label{mu_phi}\\\\\n\\mu_{\\rho}^{n+1}=\n&-\\xi \\Delta_h \\rho^{n+1}\n+ \\beta\\left(\\ln \\rho^{n+1} - \\ln \\left(1-\\rho^{n+1}\\right)\\right)\n+\\sqrt{2}\\alpha\\rho^{n+1}\n-\\alpha {\\cal A} \\left|\\nabla_h \\phi^{n+1}\\right|\n-\\alpha(\\sqrt{2}-1) \\rho^{n}. \\label{mu_rho}\n\\end{align}\nNotice that $\\nabla_h \\cdot \\left( {\\cal A} ( \\frac{\\rho }{ {\\cal A} |\\nabla_h \\phi|} ) \\nabla_h \\phi \\right)$ is evaluated as follows \n\\begin{align} \n & \n \\nabla_h \\cdot \\left( {\\cal A} ( \\frac{\\rho }{ {\\cal A} |\\nabla_h^\\delta \\phi|} ) \\nabla_h \\phi \\right) \n \\nonumber \n\\\\\n =& \\frac{1}{h} \\Big( \\frac12 ( \\frac{\\rho_{i,j} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i,j} |} \n + \\frac{\\rho_{i+1,j} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i+1,j} |} ) ( D_x \\phi )_{i+\\frac12, j} \n - \\frac12 ( \\frac{\\rho_{i,j} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i,j} |} \n + \\frac{\\rho_{i-1,j} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i-1,j} |} ) ( D_x \\phi )_{i-\\frac12, j} \\Big) \n \\nonumber \n\\\\\n & \n + \\frac{1}{h} \\Big( \\frac12 ( \\frac{\\rho_{i,j} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i,j} |} \n + \\frac{\\rho_{i,j+1} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i,j+1} |} ) ( D_y \\phi )_{i, j+\\frac12} \n - \\frac12 ( \\frac{\\rho_{i,j} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i,j} |} \n + \\frac{\\rho_{i,j-1} }{ {\\cal A} |\\nabla_h^\\delta \\phi_{i,j-1} |} ) ( D_y \\phi )_{i, j-\\frac12} \\Big) . \n\\end{align} \n\nIt is observed that the finite difference scheme is a system of nonlinear equations with respect to $\\phi^{n+1}$ and $\\rho^{n+1}$, so we that have to solve it iteratively. \nThe theoretical properties of this scheme are analyzed in the next few sections.\n\\subsection{The positivity-preserving property}\nOf course, a point-wise bound for the grid function $\\rho^{n+1}$, namely, $0<\\rho_{i,j}^{n+1}< 1$, is needed to make sure the numerical scheme is well-defined. \nThe main theoretical result is stated below, which assures that there exists a unique numerical solution for $\\eqref{cs_1st_phi}$ and $\\eqref{cs_1st_rho}$, \nso that the given bound is satisfied. \n\\begin{theorem} \\label{thm: positivity} \nGiven $\\phi^{n},\\rho^{n}\\in\\mathcal{C}_{per}$, with $\\|\\rho^{n}\\|_{\\infty}\\leq M$, for some $M\\geq 0$, and $ 0 < \\overline{\\rho^{n}} <1$, \nthere exists a unique solution $\\phi^{n+1},\\rho^{n+1}\\in\\mathcal{C}_{per}$ to $\\eqref{cs_1st}$, with $0 < \\rho^{n+1} <1$ at a point-wise level. \n\\end{theorem}\n\nBefore the proof of the positivity-preserving property, we recall the following lemma, cited from~\\cite{Chen2019Pos}. \n\n\\begin{lemma} \\label{lem: Poisson} \\cite{Chen2019Pos} \nSuppose that $\\phi\\in \\mathring{\\mathcal{C}}_{\\mathrm{per}}$ and $\\|\\phi\\|_{\\infty}\\leq C_1$, then we have the following estimate:\n\\begin{equation}\\label{LEM}\n\\begin{aligned}\n\\| ( - \\Delta_h )^{-1}\\phi\\|_{\\infty}\\leq C_{2} C_1 ,\n\\end{aligned}\n\\end{equation}\nwhere $C_2$ depends only on $\\Omega$.\n\\end{lemma}\n\nIn addition, a few more preliminary estimates are needed for the positivity-preserving analysis. The following discrete energy functional is introduced\n\\begin{equation} \n\\begin{aligned} \n\\mathcal{J}^{n}\\left(\\phi,\\rho\\right):=\n&\\frac{1}{2\\mathcal{M}\\Delta t}\\|\\phi-\\phi^{n}\\|_{-1,h}^{2}\n+\\frac{1}{2\\mathcal{M}\\Delta t}\\|\\rho-\\rho^{n}\\|_{-1,h}^{2} \\\\\n&+\\frac{1}{4\\varepsilon} \\| \\phi-\\frac{1}{2} \\|_4^4 \n+\\frac{\\varepsilon}{2}\\left\\|\\nabla_{h}\\phi\\right\\|_{2}^{2}\n+\\frac{\\eta^{2}}{2}\\left\\|\\Delta_{h}\\phi\\right\\|_{2}^{2}\n+\\frac{\\xi}{2}\\left\\|\\nabla_{h}\\rho\\right\\|_{2}^{2} \\\\\n&+\\frac{\\alpha}{2}\\left\\{\\left\\|\\rho- {\\cal A} \\left|\\nabla_{h}^\\delta \\phi\\right|\\right\\|_{2}^{2}\n+ (\\sqrt{2}-1 ) \\| \\rho \\|_{2}^{2}\n+\\frac{1}{\\delta}\\left\\|\\nabla_{h}\\phi\\right\\|_{2}^{2}\\right\\} \\\\\n&+\\beta\\left\\langle\\rho,\\ln\\rho\\right\\rangle_{\\Omega}\n+\\beta\\left\\langle 1-\\rho,\\ln\\left(1-\\rho\\right)\\right\\rangle_\\Omega \n + \\langle \\phi , f_\\phi^n \\rangle_\\Omega \n + \\langle \\rho, f_\\rho^n \\rangle_\\Omega , \\\\\nf_\\phi^n = & -\\frac{1}{4\\varepsilon} ( \\phi^{n} -\\frac{1}{2} ) \n-\\frac{\\alpha}{\\delta} \\Delta_{h}\\phi^{n} , \\quad \n f_\\rho^n = - (\\sqrt{2}-1 )\\alpha \\rho^n . \n\\end{aligned} \n\\label{J-defi-1} \n\\end{equation} \n\n\\begin{lemma} \\label{lem: J} \nSet $M_1^* = \\max | f_\\phi^n |$, $M_2^* = \\max | f_\\rho^n| \\le ( \\sqrt{2}-1 ) \\alpha M$. We notice that $M_1^*$ may be $h$, $\\varepsilon$, $\\delta$ and $\\phi^n$ dependent. The following inequality is available for the lower bound of $\\mathcal{J}^{n} (\\phi, \\rho)$, for $0 < \\rho <1$: \n\\begin{eqnarray} \\label{lem: est-J} \n \\mathcal{J}^{n} (\\phi, \\rho) \\ge \n \\frac{1}{8 \\varepsilon} \\| \\phi-\\frac{1}{2} \\|_4^4 - M_3^n , \\quad \n M_3^n = ( \\frac12 ( M_1^* )^2 + \\frac12 M_1^* + \\frac12 \\varepsilon + \\beta \\ln 2 + M_2^* ) | \\Omega | , \n \\label{est-J-0} \n\\end{eqnarray}\n\\end{lemma}\n\n\\begin{proof} \nIn the expansion of $\\mathcal{J}^{n} (\\phi, \\rho)$, it is observed that all the terms are non-negative, except for the last four term. Then we get \n\\begin{equation} \n\\mathcal{J}^{n} (\\phi,\\rho) \\ge \n \\frac{1}{4\\varepsilon} \\| \\phi-\\frac{1}{2} \\|_4^4 \n + \\langle \\phi , f_\\phi^n \\rangle_\\Omega \n +\\beta ( \\left\\langle\\rho,\\ln\\rho\\right\\rangle_{\\Omega}\n+ \\left\\langle 1-\\rho,\\ln\\left(1-\\rho\\right)\\right\\rangle_\\Omega ) \n + \\langle \\rho, f_\\rho^n \\rangle_\\Omega . \\label{est-J-1} \n\\end{equation} \nFor the third and fourth terms on the right hand side of~\\eqref{est-J-1}, the following point-wise lower bound is available: \n\\begin{equation} \n\\begin{aligned} \n &\n \\rho \\ln \\rho + ( 1-\\rho) \\ln (1-\\rho) \\ge 2 \\cdot \\frac12 \\ln \\frac12 \n = - \\ln 2 , \\quad \\mbox{for $0 < \\rho < 1$} , \n\\\\\n & \n \\mbox{which in turn leads to} \\quad \n \\left\\langle\\rho,\\ln\\rho\\right\\rangle_{\\Omega}\n+ \\left\\langle 1-\\rho,\\ln\\left(1-\\rho\\right)\\right\\rangle_\\Omega \n\\ge - | \\Omega | \\ln 2 . \n\\end{aligned} \n \\label{est-J-2} \n\\end{equation} \nFor the last term on the right hand side of~\\eqref{est-J-1}, the following lower bound is valid, based on the fact that $0 < \\rho < 1$: \n\\begin{equation} \n \\langle \\rho, f_\\rho^n \\rangle_\\Omega \\ge - \\| f_\\rho^n \\|_\\infty \\cdot \\| {\\bf 1} \\|_1 \n \\ge - M_2^* | \\Omega | . \\label{est-J-3} \n\\end{equation} \nFor the first two terms on the right hand side of~\\eqref{est-J-1}, we begin with the following estimate: \n\\begin{eqnarray} \n\\begin{aligned} \n \\langle \\phi , f_\\phi^n \\rangle_\\Omega \n =& \\langle \\phi - \\frac12 , f_\\phi^n \\rangle_\\Omega \n + \\frac12 \\langle {\\bf 1} , f_\\phi^n \\rangle_\\Omega \n \\ge - \\frac12 ( \\| \\phi - \\frac12 \\|_2^2 + \\| f_\\phi^n \\|_2^2 ) \n - \\frac12 M_1^* | \\Omega | \n\\\\\n \\ge& \n - \\frac12 \\| \\phi - \\frac12 \\|_2^2 - \\frac12 ( M_1^* )^2 | \\Omega | \n - \\frac12 M_1^* | \\Omega | . \n\\end{aligned} \n\\label{est-J-4-1} \n\\end{eqnarray} \nMeanwhile, the following lower bound is a direct application of Cauchy inequality: \n\\begin{equation} \n \\frac{1}{8\\varepsilon} \\| \\phi-\\frac{1}{2} \\|_4^4 \n - \\frac12 \\| \\phi - \\frac12 \\|_2^2 \\ge -\\frac12 \\varepsilon | \\Omega | , \n \\label{est-J-4-2} \n\\end{equation} \nand its combination with~\\eqref{est-J-4-1} yields \n\\begin{eqnarray} \n \\frac{1}{8\\varepsilon} \\| \\phi-\\frac{1}{2} \\|_4^4 +\n \\langle \\phi , f_\\phi^n \\rangle_\\Omega \n \\ge - ( \\frac12 ( M_1^* )^2 + \\frac12 M_1^* + \\frac12 \\varepsilon ) | \\Omega | . \n\\label{est-J-4-3} \n\\end{eqnarray} \n Finally, a substitution of~\\eqref{est-J-2}, \\eqref{est-J-3} and \\eqref{est-J-4-3} into \\eqref{est-J-1} results in~\\eqref{est-J-0}. This completes the proof of Lemma~\\ref{lem: J}. \n\\end{proof} \n\nNow we proceed into the proof of Theorem~\\ref{thm: positivity}. \n\n\\begin{proof}\nWe denote $M_0^n = \\mathcal{J}^{n} (\\phi^n, \\rho^n)$, a fixed constant with given $(\\phi^n, \\rho^n)$. The numerical solution of $\\eqref{cs_1st}$ is a minimizer of the discrete energy functional $\\mathcal{J}^{n}\\left(\\phi,\\rho\\right)$ (defined in~\\eqref{J-defi-1}), over the admissible set\n\\begin{equation} \n\\begin{aligned} \n & \nA_{h}:=\\left\\{\\phi,\\rho\\in\\mathcal{C}_{per}| -A^* < \\phi < A^*, \\, 0 < \\rho < 1,\\left\\langle\\phi-\\bar{\\phi}_{0}\\right\\rangle_{\\Omega}=0,\\left\\langle\\rho-\\bar{\\rho}_{0}\\right\\rangle_{\\Omega}=0\\right\\} \\subset \\mathbb{R}^{N^2} \\times \\mathbb{R}^{N^2} , \n\\\\\n & \n A^* := \\max \\Big( ( 8 \\varepsilon (M_0^n + M_3 ^n ) )^\\frac14 h^{-\\frac12} , \\| \\phi^n \\|_\\infty \\Big) + 1 . \n\\end{aligned} \n \\label{defi-A-delta} \n\\end{equation}\nWe can observe that $\\mathcal{J}^{n}$ is a strictly convex function over this set.\n\nTo facilitate the analysis below, we transform the minimization problem into an equivalent one. Consider the functional\n\\begin{align}\n\\mathcal{F}^{n}\\left(\\phi,\\rho\\right)\n:=&\\mathcal{J}^{n}\\left(\\phi+\\bar{\\phi}_{0},\\rho+\\bar{\\rho}_{0}\\right)\\nonumber\\\\\n=&\\frac{1}{2\\mathcal{M}\\Delta t}\\|\\phi+\\bar{\\phi}_{0}-\\phi^{n}\\|_{-1,h}^{2}\n+\\frac{1}{2\\mathcal{M}\\Delta t}\\|\\rho+\\bar{\\rho}_{0}-\\rho^{n}\\|_{-1,h}^{2}\\nonumber\\\\\n&+\\frac{1}{4\\varepsilon} \\| (\\phi+\\bar{\\phi}_{0}-\\frac{1}{2} )^{2} \\|_{2}^{2}\n+\\frac{\\varepsilon}{2}\\left\\|\\nabla_{h}\\phi\\right\\|_{2}^{2}\n+\\frac{\\eta^{2}}{2}\\left\\|\\Delta_{h}\\phi\\right\\|_{2}^{2}\n+\\frac{\\xi}{2}\\left\\|\\nabla_{h}\\rho\\right\\|_{2}^{2}\\nonumber\\\\\n&+\\frac{\\alpha}{2}\\left\\{\\left\\|\\rho+\\bar{\\rho}_{0}\n- {\\cal A} |\\nabla_{h}^\\delta \\phi| \\right\\|_{2}^{2}\n+ (\\sqrt{2}-1 )\\left\\|\\rho+\\bar{\\rho}_{0}\\right\\|_{2}^{2}\n+\\frac{1}{\\delta}\\left\\|\\nabla_{h}\\phi\\right\\|_{2}^{2}\\right\\}\\nonumber\\\\\n&+\\beta\\left\\langle\\rho+\\bar{\\rho}_{0},\\ln\\left(\\rho+\\bar{\\rho}_{0}\\right)\\right\\rangle_{\\Omega}\n+\\beta\\left\\langle 1-\\rho-\\bar{\\rho}_{0},\\ln\\left(1-\\rho-\\bar{\\rho}_{0}\\right)\\right\\rangle_{\\Omega}\\nonumber\\\\\n& + \\langle\\phi+\\bar{\\phi}_{0}, f_\\phi^n \\rangle_\\Omega \n + \\langle\\rho+\\bar{\\rho}_{0}, f_\\rho^n \\rangle_{\\Omega},\n\\end{align}\ndefined on the set\n\\begin{equation*}\n\\mathring{A}_{h}:=\n\\left\\{\\left(\\phi,\\rho\\right)\\in\\mathring{\\mathcal{C}}_{per}\\times\\mathring{\\mathcal{C}}_{per} | -A^* - \\bar{\\phi}_0 < \\phi < A^*- \\bar{\\phi}_0 , -\\bar{\\rho}_{0} < \\rho < 1-\\bar{\\rho}_{0}\\right\\}\n\\subset\\mathbb{R}^{N^{2}}\\times\\mathbb{R}^{N^{2}}.\n\\end{equation*}\nIf $(\\phi, \\rho) \\in\\mathring{A}_{h}$ minimizes $\\mathcal{F}^{n}$, then $(\\varphi, \\varrho) :=( \\phi+\\bar{\\phi}_{0}, \\rho + \\bar{\\rho}_0) \\in A_{h}$ minimizes $\\mathcal{J}^{n}$ and $vice~versa$. \nNext, we prove that there exists a minimizer of $\\mathcal{F}^{n}$ over the domain $\\mathring{A}_{h}$. Consider the following closed domain: for $\\delta_0 \\in\\left(0,\\frac{1}{2}\\right)$\n\\begin{equation*} \n\\begin{aligned} \n\\mathring{A}_{h,\\delta_0}:= & \\left\\{\\left(\\phi,\\rho\\right)\\in\\mathring{\\mathcal{C}}_{per}\\times\\mathring{\\mathcal{C}}_{per} | -A^* - \\bar{\\phi}_0 + \\delta_0 \\le \\phi \\le A^*- \\bar{\\phi}_0 - \\delta_0 , \\, \\delta_0 -\\bar{\\rho}_{0}\\leq \\rho \\leq 1-\\delta_0 -\\bar{\\rho}_{0}\\right\\} \n\\\\\n & \n \\subset\\mathbb{R}^{N^{2}}\\times\\mathbb{R}^{N^{2}}. \n\\end{aligned} \n\\end{equation*}\nSince $\\mathring{A}_{h,\\delta_0}$ is a bounded, compact, and convex set in the subspace $\\mathring{\\mathcal{C}}_{per}$, \nthere exists a (not necessarily unique) minimizer of $\\mathcal{F}^{n}$ over $\\mathring{A}_{h,\\delta_0}$. The key point of the positivity analysis is that, such a minimizer could not occur on the boundary of $\\mathring{A}_{h,\\delta_0}$, if $\\delta_0$ is sufficiently small. To be more explicit, by the boundary of $\\mathring{A}_{h,\\delta_0}$, \nwe mean the locus of points $\\left(\\phi,\\psi\\right)\\in\\mathring{A}_{h,\\delta_0}$ such that $\\psi+\\bar{\\rho}_{0} =\\delta_0$ or $1-\\delta_0$, or $\\phi+\\bar{\\phi}_{0} =-A^* + \\delta_0$ or $A^* -\\delta_0$, precisely.\n\nTo get a contradiction, suppose that the minimizer of $\\mathcal{F}^{n}$, call it $\\left(\\phi^{\\star},\\rho^{\\star}\\right)$, occurs at a boundary point of $\\mathring{A}_{h,\\delta_0}$. \nThere is at least one grid point $\\vec{\\alpha}_{0}=\\left(i_{0},j_{0}\\right)$ such that $\\rho_{\\vec{\\alpha}_{0}}^{\\star}+\\bar{\\rho}_{0} = \\delta_0$ or $1-\\delta_0$, or $\\phi_{\\vec{\\alpha}_{0}}^{\\star} +\\bar{\\phi}_{0} =-A^* + \\delta_0$ or $A^* -\\delta_0$. Let us assume that $\\rho_{\\vec{\\alpha}_{0}}^{\\star}+\\bar{\\rho}_{0}=\\delta_0$, \nand denote by $\\vec{\\alpha}_{1}=\\left(i_{1},j_{1}\\right)$ as the grid point at which $\\rho^{\\star}$ achieves its maximum. By the fact that $\\bar{\\rho^{\\star}}=0$, it is obvious that $\\rho_{\\vec{\\alpha}_{1}}^\\star \\ge 0$ and \n\\begin{equation*}\n1-\\delta_0 \\geq \\rho_{\\vec{\\alpha}_{1}}^{\\star}+\\bar{\\rho}_{0}\\geq \\bar{\\rho}_{0}.\n\\end{equation*}\nSince \n$\\mathcal{F}^{n}$ is smooth over $\\mathring{A}_{h,\\delta}$, for all \n$\\left(\\varphi,\\psi\\right)\\in\\mathring{\\mathcal{C}}_{per}$, the directional derivative is \n\\begin{align}\n&\\mathrm{d}_{s}\\mathcal{F}^{n}\\left(\\phi^{\\star}+s\\varphi,\\rho^{\\star}+s\\psi\\right)|_{s=0}\\nonumber\\\\\n=&\\frac{1}{\\mathcal{M}\\Delta t}\\left\\langle\\left(-\\Delta_{h}\\right)^{-1}\n\\left(\\phi^{\\star}+\\bar{\\phi}_{0}-\\phi^{n}\\right),\\varphi\\right\\rangle_{\\Omega}\n+\\frac{1}{\\mathcal{M}\\Delta t}\\left\\langle\\left(-\\Delta_{h}\\right)^{-1}\n\\left(\\rho^{\\star}+\\bar{\\rho}_{0}-\\rho^{n}\\right),\\psi\\right\\rangle_{\\Omega}\\nonumber\\\\\n&+\\frac{1}{2\\varepsilon} \\langle\\phi^{\\star}+\\bar{\\phi}_{0}-\\frac{1}{2},\\varphi \\rangle_{\\Omega}\n-\\varepsilon\\left\\langle\\Delta_{h}\\phi^{\\star},\\varphi\\right\\rangle_{\\Omega}\n+\\eta^{2}\\left\\langle\\Delta_{h}^{2}\\phi^{\\star},\\varphi\\right\\rangle_{\\Omega}\n-\\xi\\left\\langle\\Delta_{h}\\rho^{\\star},\\psi\\right\\rangle_{\\Omega}\\nonumber\\\\\n&+\\beta\\left\\langle\\ln\\left(\\rho^{\\star}+\\bar{\\rho}_{0}\\right)\n-\\ln\\left(1-\\rho^{\\star}-\\bar{\\rho}_{0}\\right),\\psi\\right\\rangle_{\\Omega}\n-\\alpha\\left\\langle \\nabla_{h}\\cdot \\Big( {\\cal A} ( \\frac{\\rho^{\\star}+\\bar{\\rho}_{0} }{ {\\cal A} |\\nabla_{h}^\\delta \\phi^{\\star}|} ) \\nabla_h \\phi^{\\star} \\Big) ,\\varphi\\right\\rangle_{\\Omega}\\nonumber\\\\\n&-\\alpha\\left\\langle {\\cal A} |\\nabla_{h}^\\delta \\phi^{\\star}|,\\psi\\right\\rangle_{\\Omega}\n+\\alpha (\\sqrt{2}-1 )\\left\\langle\\rho^{\\star}+\\bar{\\rho}_{0},\\psi\\right\\rangle_{\\Omega}\n-\\frac{\\alpha}{\\delta}\\left\\langle\\Delta_{h}\\phi^{\\star},\\varphi\\right\\rangle_{\\Omega} \n+ \\langle f_\\phi^n ,\\varphi \\rangle_{\\Omega} \n+ \\langle f_\\rho^n, \\psi \\rangle_{\\Omega} . \n\\end{align}\nHere, we take the direction $\\varphi,\\psi\\in\\mathring{C}_{per}$, such that\n\\begin{equation*}\n\\varphi = 0,\\quad\\psi=\\delta_{i,i_{0}}\\delta_{j,j_{0}}-\\delta_{i,i_{1}}\\delta_{j,j_{1}}.\n\\end{equation*}\nThen the derivative may be expressed as\n\\begin{align}\\label{estimate_direction}\n&\\frac{1}{h^{2}}\\mathrm{d}_{s}\\mathcal{F}^{n}\\left(\\phi^{\\star},\\rho^{\\star}+s\\psi\\right)|_{s=0}\\nonumber\\\\\n=&\\frac{1}{\\mathcal{M}\\Delta t}\\left(-\\Delta_{h}\\right)^{-1}\n\\left(\\rho^{\\star}+\\bar{\\rho}_{0}-\\rho^{n}\\right)_{\\vec{\\alpha}_{0}}\n-\\frac{1}{\\mathcal{M}\\Delta t}\\left(-\\Delta_{h}\\right)^{-1}\n\\left(\\rho^{\\star}+\\bar{\\rho}_{0}-\\rho^{n}\\right)_{\\vec{\\alpha}_{1}}\n-\\xi\\left(\\Delta_{h}\\rho_{\\vec{\\alpha}_{0}}^{\\star}-\\Delta_{h}\\rho_{\\vec{\\alpha}_{1}}^{\\star}\\right)\\nonumber\\\\\n&+\\beta\\ln\\left(\\rho_{\\vec{\\alpha}_{0}}^{\\star}+\\bar{\\rho}_{0}\\right)\n-\\beta\\ln\\left(1-\\rho_{\\vec{\\alpha}_{0}}^{\\star}-\\bar{\\rho}_{0}\\right)\n-\\beta\\ln\\left(\\rho_{\\vec{\\alpha}_{1}}^{\\star}+\\bar{\\rho}_{0}\\right)\n+\\beta\\ln\\left(1-\\rho_{\\vec{\\alpha}_{1}}^{\\star}-\\bar{\\rho}_{0}\\right)\\nonumber\\\\\n&-\\alpha\\left( {\\cal A} |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{0}}^{\\star}| \n- {\\cal A} |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{1}}^{\\star}|\\right)\n+\\alpha (\\sqrt{2}-1 )\\left(\\rho_{\\vec{\\alpha}_{0}}^{\\star}-\\rho_{\\vec{\\alpha}_{1}}^{\\star}\\right)\n + ( f_\\rho^n )_{\\vec{\\alpha}_{0}} - (f_\\rho^n )_{\\vec{\\alpha}_{1}} . \n\\end{align}\nFor simplicity, now let us write $\\varrho^{\\star}:=\\rho^{\\star}+\\bar{\\rho}_{0}$. Since \n$\\varrho_{\\vec{\\alpha}_{0}}^{\\star}=\\delta_0$ and $\\varrho_{\\vec{\\alpha}_{1}}^{\\star}\\geq\\bar{\\rho}_{0}$, we have\n\\begin{equation}\\label{estimate_log}\n\\ln\\left(\\varrho_{\\vec{\\alpha}_{0}}^{\\star}\\right)\n-\\ln\\left(1-\\varrho_{\\vec{\\alpha}_{0}}^{\\star}\\right)\n-\\ln\\left(\\varrho_{\\vec{\\alpha}_{1}}^{\\star}\\right)\n+\\ln\\left(1-\\varrho_{\\vec{\\alpha}_{1}}^{\\star}\\right) \n\\le \\ln \\frac{\\delta_0}{1-\\delta_0}\n-\\ln\\frac{\\bar{\\rho}_{0}}{1-\\bar{\\rho}_{0}}.\n\\end{equation}\nSince $\\varrho^{\\star}$ takes a minimum at the grid point $\\vec{\\alpha}_{0}$, with \n$\\varrho_{\\vec{\\alpha}_{0}}^{\\star}=\\delta\\leq\\varrho_{i,j}^{\\star}$, for any $\\left(i,j\\right)$, \nand a maximum at the grid point $\\vec{\\alpha}_{1}$, with \n$\\varrho_{\\vec{\\alpha}_{1}}^{\\star}\\geq\\varrho_{i,j}^{\\star}$, for any $\\left(i,j\\right)$,\n\\begin{equation}\\label{estimate_star}\n\\Delta_{h}\\rho_{\\vec{\\alpha}_{0}}^{\\star}\\geq 0,\\quad\n\\Delta_{h}\\rho_{\\vec{\\alpha}_{1}}^{\\star}\\leq 0,\\quad\n\\rho_{\\vec{\\alpha}_{0}}^{\\star}-\\rho_{\\vec{\\alpha}_{1}}^{\\star}\\leq 0.\n\\end{equation}\nFor the numerical solution $\\rho^{n}$ at the previous time step, the a priori assumption \n$\\left\\|\\rho^{n}\\right\\|_{\\infty}\\leq M$ indicates that\n\\begin{equation}\\label{estimate_n}\n-2M\\leq\\rho_{\\vec{\\alpha}_{0}}^{n}-\\rho_{\\vec{\\alpha}_{1}}^{n}\\leq 2M.\n\\end{equation}\nAccording to Lemma~\\ref{lem: Poisson}, we obtain\n\\begin{equation}\\label{estimate_lap}\n- 4M C_2 \\leq\\left(-\\Delta_{h}\\right)^{-1}\\left(\\varrho^{\\star}-\\rho^{n}\\right)_{\\vec{\\alpha}_{0}}\n-\\left(-\\Delta_{h}\\right)^{-1}\\left(\\varrho^{\\star}-\\rho^{n}\\right)_{\\vec{\\alpha}_{1}}\\leq 4M C_2.\n\\end{equation}\nDenote $C_{3}= \\max\\left\\{ {\\cal A} |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{0}}^{\\star}|, {\\cal A} |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{1}}^{\\star}|\\right\\}$. Based on the fact that $-A^* - \\bar{\\phi}_0 < \\phi^* < A^*- \\bar{\\phi}_0$ at a point-wise level, we conclude that \n\\begin{equation} \n {\\cal A} |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{0}}^{\\star}|,\\, \\, \n {\\cal A} |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{1}}^{\\star} | \n \\le \\frac{2 A^*}{h} + 1 , \\quad \n \\mbox{so that} \\, \\, \\, C_3 \\le \\frac{2 A^*}{h} + 1 . \n\\end{equation} \nThen we have \n\\begin{equation}\\label{estimate_phi-1}\n-\\alpha C_{3}\\leq\n-\\alpha {\\cal A} \\left(|\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{0}}^{\\star}| \n- |\\nabla_{h}^\\delta \\phi_{\\vec{\\alpha}_{1}}^{\\star}|\\right) \\le \\alpha C_{3}.\n\\end{equation}\nConsequently, a substitution of $\\eqref{estimate_log},\\eqref{estimate_star},\\eqref{estimate_n},\\eqref{estimate_lap},\\eqref{estimate_phi-1}$ into $\\eqref{estimate_direction}$ yields \nthe following bound on the directional derivative:\n\\begin{equation*}\n\\frac{1}{h^{2}}\\mathrm{d}_{s}\\mathcal{F}^{n}\\left(\\phi^{\\star},\\rho^{\\star}+s\\psi\\right)|_{s=0}\n\\leq \\beta \\ln \\frac{\\delta_0}{1-\\delta_0} - \\beta\\ln\\frac{\\bar{\\rho}_{0}}{1-\\bar{\\rho}_{0}}\n+ 4M C_2 (\\mathcal{M}\\Delta t)^{-1}+2\\alpha C_{3}+ 2 M_2 .\n\\end{equation*}\nWe denote $D_0 = 4M C_2 (\\mathcal{M}\\Delta t)^{-1}+2\\alpha C_{3}+ 2 M_2$. Notice that $D_0$ is a constant for fixed $\\Delta t$ and $h$, though it becomes singular as $\\Delta t\\to 0$ and $h\\to 0$. On the other hand, for any fixed $\\Delta t$ and $h$, we may choose $\\delta_0 \\in\\left(0,1\/2\\right)$ sufficiently small so that\n\\begin{equation}\\label{estimate_C}\n\\beta \\ln \\frac{\\delta_0}{1-\\delta_0} - \\beta\\ln\\frac{\\bar{\\rho}_{0}}{1-\\bar{\\rho}_{0}} \n+ D_0 < 0.\n\\end{equation}\nThis in turn leads to the following inequality, provided $\\delta_0$ satisfies $\\eqref{estimate_C}$,\n\\begin{equation}\n\\frac{1}{h^{2}}\\mathrm{d}_{s}\\mathcal{F}^{n}\\left(\\phi^{\\star},\\rho^{\\star}+s\\psi\\right)|_{s=0} < 0.\n\\end{equation}\nAs before, this contradicts the assumption that $\\mathcal{F}^{n}$ has a minimum at $\\left(\\phi^{\\star},\\rho^{\\star}\\right)$, since the directional derivative is negative in a direction pointing into the interior of $\\mathring{A}_{h,\\delta_0}$.\n\nUsing very similar arguments, we can also prove that the global minimum of $\\mathcal{F}^{n}$ over $\\mathring{A}_{h,\\delta_0}$ could not occur at a boundary point $\\left(\\phi^{\\star},\\rho^{\\star}\\right)$ such that $\\rho_{\\vec{\\alpha}_{0}}^{\\star}+\\bar{\\rho}_{0}=1-\\delta_0$, for some $\\vec{\\alpha}_{0}$, so that the grid function $\\rho^{\\star}$ has a global maximum at $\\vec{\\alpha}_{0}$. The details are left to interested readers. \n\nMoreover, if the global minimum of $\\mathcal{F}^{n}$ over $\\mathring{A}_{h,\\delta_0}$ could occurs at a boundary point $\\left(\\phi^{\\star},\\rho^{\\star}\\right)$ such that $\\varphi_{\\vec{\\alpha}_{0}}^* = \\phi_{\\vec{\\alpha}_{0}}^{\\star}+\\bar{\\phi}_{0}=A^*-\\delta_0$. In turn, we apply Lemma~\\ref{lem: J} and obtain \n\\begin{eqnarray} \n\\begin{aligned} \n \\mathcal{F}^{n} (\\phi^* , \\rho^*) = & \\mathcal{J}^{n} (\\varphi^*, \\varrho^*) \n \\ge \\frac{1}{8 \\varepsilon} \\| \\varphi^* -\\frac{1}{2} \\|_4^4 - M_3^n \n \\ge \\frac{1}{8 \\varepsilon} h^2 \\cdot ( \\varphi_{\\vec{\\alpha}_{0}}^* - \\frac12 )^4 \n - M_3^n \n\\\\\n \\ge & \n \\frac{1}{8 \\varepsilon} h^2 ( A^* - \\delta_0 - \\frac12 )^4 - M_3^n \n > \\frac{1}{8 \\varepsilon} h^2 ( A^* - 1 )^4 - M_3^n \n\\\\\n \\ge & \n \\frac{1}{8 \\varepsilon} h^2 \\cdot \n ( 8 \\varepsilon (M_0^n + M_3 ^n ) ) h^{-2} - M_3^n \n = M_0^n = \\mathcal{J}^{n} ( \\phi^n, \\rho^n) , \n\\end{aligned} \n\\label{positivity-2-1} \n\\end{eqnarray}\nin which the definition of $A^*$ (in~\\eqref{defi-A-delta}) has been recalled. This contradicts the assumption that $\\mathcal{F}^{n}$ has a minimum at $\\left(\\phi^{\\star},\\rho^{\\star}\\right)$. \n\nUsing similar arguments, a minimization point cannot occur at a boundary point $\\left(\\phi^{\\star},\\rho^{\\star}\\right)$ such that $\\varphi_{\\vec{\\alpha}_{0}}^* = \\phi_{\\vec{\\alpha}_{0}}^{\\star}+\\bar{\\phi}_{0}=-A^*+ \\delta_0$. The details are left to interested readers. \n\nA combination of above four facts have indicated that, the global minimum of $\\mathcal{F}^{n}$ over $\\mathring{A}_{h,\\delta_0}$ could only possibly occur at interior point $\\left(\\phi,\\rho\\right)\\in\\left(\\mathring{A}_{h,\\delta_0}\\right)^{\\circ} \\subset\\left(\\mathring{A}_{h}\\right)^{\\circ}$. We conclude that there must be a solution $\\left(\\phi,\\rho\\right)\\in A_{h}$ that minimizes $\\mathcal{J}^{n}$ over $A_{h}$, which is equivalent to the numerical solution of $\\eqref{cs_1st}$. The existence of the numerical solution is established. \n\nIn addition, since $\\mathcal{J}^{n}$ is strictly convex function over $A_{h}$, the uniqueness analysis for this numerical solution is straightforward. The proof of Theorem~\\ref{thm: positivity} is completed.\n\\end{proof}\n\\subsection{Unconditional energy stability}\n\\begin{theorem}[Energy stability] \\label{thm: energy stability} \nFor $n\\geq 1$, the numerical scheme $\\eqref{cs_1st}$ is unconditionally energy stable, i.e. \n\\begin{equation*}\nE\\left(\\phi^{n+1},\\rho^{n+1}\\right)\\leq E\\left(\\phi^{n},\\rho^{n}\\right).\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathcal{L}=-\\Delta_{h}$. Due to the mass conservation, $\\mathcal{L}^{-1}\\left(\\phi^{n+1}-\\phi^{n}\\right)$ and $\\mathcal{L}^{-1}\\left(\\rho^{n+1}-\\rho^{n}\\right)$ \nare well-defined. Taking a discrete inner product with $\\eqref{cs_1st_phi}$, $\\eqref{cs_1st_rho}$, $\\eqref{cs_1st_mu1}$, $\\eqref{cs_1st_mu2}$ by $\\mathcal{L}^{-1}\\left(\\phi^{n+1}-\\phi^{n}\\right)$, $\\mathcal{L}^{-1}\\left(\\rho^{n+1}-\\rho^{n}\\right)$, $\\phi^{n+1}-\\phi^{n}$ and $\\rho^{n+1}-\\rho^{n}$, respectively, yields the following estimate \n\\begin{align}\n0 =& \\frac{1}{\\mathcal{M}\\Delta t}\\left\\langle\\phi^{n+1}-\\phi^{n}, \n\\mathcal{L}^{-1}\\left(\\phi^{n+1}-\\phi^{n}\\right)\\right\\rangle_{\\Omega}\n+\\frac{1}{\\mathcal{M}\\Delta t}\\left\\langle\\rho^{n+1}-\\rho^{n}, \n\\mathcal{L}^{-1}\\left(\\rho^{n+1}-\\rho^{n}\\right)\\right\\rangle_{\\Omega}\\nonumber\\\\\n &+\\left\\langle\\delta_{\\phi} E_{c}\\left(\\phi^{n+1},\\rho^{n+1}\\right)\n -\\delta_{\\phi}E_{e}\\left(\\phi^{n},\\rho^{n}\\right),\\phi^{n+1}-\\phi^{n}\\right\\rangle_{\\Omega}\\nonumber\\\\\n &+\\left\\langle\\delta_{\\rho} E_{c}\\left(\\phi^{n+1},\\rho^{n+1}\\right)\n -\\delta_{\\rho}E_{e}\\left(\\phi^{n},\\rho^{n}\\right),\\rho^{n+1}-\\rho^{n}\\right\\rangle_{\\Omega}\\nonumber\\\\\n\\geq & E\\left(\\phi^{n+1},\\rho^{n+1}\\right)-E\\left(\\phi^{n},\\rho^{n}\\right).\\label{1st_energy_product}\n\\end{align}\nHence that\n\\begin{equation*}\nE\\left(\\phi^{n+1},\\rho^{n+1}\\right)\\leq E\\left(\\phi^{n},\\rho^{n}\\right).\n\\end{equation*}\nThis completes the proof.\n\\end{proof}\n\n\\subsection{Optimal rate convergence analysis}\nLet $\\Phi$ and $\\Psi$ be the exact solution for the binary fluid-surfactant system $\\eqref{bfs}$.\nWith initial data with sufficient regularity, we could assume that the exact solution \nhas regularity of class $\\mathcal{R}_{1}$ and $\\mathcal{R}_{2}$:\n\\begin{subequations}\\label{regular}\n\\begin{align}\n&\\Phi\\in \\mathcal{R}_{1}:=H^{2}\\left(0,T;C_{per}\\left(\\Omega\\right)\\right)\n\\cap L^{\\infty}\\left(0,T;C_{per}^{8}\\left(\\Omega\\right)\\right),\\\\\n&\\Psi\\in \\mathcal{R}_{2}:=H^{2}\\left(0,T;C_{per}\\left(\\Omega\\right)\\right)\n\\cap L^{\\infty}\\left(0,T;C_{per}^{6}\\left(\\Omega\\right)\\right).\n\\end{align}\n\\end{subequations}\nDefine $\\Phi_{N}\\left(\\cdot,t\\right):=\\mathcal{P}_{N}\\Phi\\left(\\cdot,t\\right)$, and $\\Psi_{N}\\left(\\cdot,t\\right):=\\mathcal{P}_{N}\\Psi\\left(\\cdot,t\\right)$, \nthe spatial Fourier projection of the exact solutions into $\\mathcal{B}^{K}$, the space of trigonometric polynomials of degree to and including $K$ (with $N=2K+1$). \nThe following projection approximation is standard: if $\\Phi \\in L^{\\infty}\\left(0,T;H_{per}^{l}\\left(\\Omega\\right)\\right)$ \nfor some $l\\in \\mathbb{N}$,\n\\begin{equation}\\label{err_for_Phi}\n\\left\\|\\Phi_{N}-\\Phi\\right\\|_{L^{\\infty}\\left(0,T;H^{k}\\left(\\Omega\\right)\\right)}\n\\leq Ch^{l-k}\\left\\|\\Phi\\right\\|_{L^{\\infty}\\left(0,T;H^{l}\\left(\\Omega\\right)\\right)},\\quad 0\\leq k\\leq l.\n\\end{equation}\nBy $\\Phi_{N}^{m},\\Phi^{m}$, \nwe denote $\\Phi_{N}\\left(\\cdot,t_{m}\\right)$ and $\\Phi\\left(\\cdot,t_{m}\\right)$, respectively, \nwith $t_{m}=m\\cdot\\Delta t$. Since $\\Phi_{N}\\in\\mathcal{B}^{m}$, \nthe mass conservative property is available at the discrete level:\n\\begin{equation*}\n\\overline{\\Phi_{N}^{m}}\n=\\frac{1}{|\\Omega|}\\int_{\\Omega}\\Phi_{N}\\left(\\cdot,t_{m}\\right)\\mathrm{d}\\bm{x}\n=\\frac{1}{|\\Omega|}\\int_{\\Omega}\\Phi_{N}\\left(\\cdot,t_{m+1}\\right)\\mathrm{d}\\bm{x}\n=\\overline{\\Phi_{N}^{m+1}}, \\quad m\\in\\mathbb{N}.\n\\end{equation*}\nWe have a similar result about $\\Psi$. On the other hand, the solution of $\\eqref{bfs}$ is also mass conservative at the discrete level:\n\\begin{equation}\n\\overline{\\phi^{m}}=\\overline{\\phi^{m+1}}, \\quad\n\\overline{\\rho^{m}}=\\overline{\\rho^{m+1}}, \\quad m\\in\\mathbb{N}.\n\\end{equation}\nAs indicated before, we use the mass conservative interpolation for the initial data: \n$\\phi^{0}=\\mathcal{P}_{h}\\Phi_{N}\\left(\\cdot,t=0\\right)$ and \n$\\rho^{0}=\\mathcal{P}_{h}\\Psi_{N}\\left(\\cdot,t=0\\right)$, that is \n\\begin{equation}\n\\phi^{0}_{i,j} = \\mathcal{P}_{h} ( \\Phi_{N} ( \\cdot, t=0) )_{i,j} \n:=\\Phi_{N}\\left(x_{i}, y_{j},t=0\\right),\\quad\n\\rho^{0}_{i,j} = \\mathcal{P}_{h} ( \\Psi_{N} ( \\cdot, t=0) )_{i,j} \n :=\\Psi_{N}\\left( x_{i}, y_{j},t=0\\right). \\label{interpolation-1} \n\\end{equation}\nThe error grid function is defined as \n\\begin{equation}\n\\tilde{\\phi}^{m}:=\\mathcal{P}_{h}\\Phi_{N}^{m}-\\phi^{m}, \\quad\n\\tilde{\\rho}^{m}:=\\mathcal{P}_{h}\\Psi_{N}^{m}-\\rho^{m}, \\quad m\\in\\mathbb{N} , \n\\label{error function-1} \n\\end{equation}\nin which a similar interpolation formula could be applied to ${\\cal P}_h$ as in~\\eqref{interpolation-1}. Therefore, it follows that $\\overline{\\tilde{\\phi}^{m}}=0$ and $\\overline{\\tilde{\\rho}^{m}}=0$, for any $m\\in\\mathbb{N}$, so that the discrete norm $\\left\\|\\cdot\\right\\|_{-1,h}$ is well defined for the numerical error grid function.\n\\begin{theorem} \\label{thm: convergence} \nGiven initial data $\\Phi(\\cdot,t=0)\\in C_{per}^{8}(\\Omega)$ and $\\Psi(\\cdot,t=0)\\in C_{per}^{6}(\\Omega)$, \nsuppose the exact solutin for binary fluid-surfactant system $\\eqref{bfs}$ is of regularity class $\\mathcal{R}=\\mathcal{R}_{1}\\times\\mathcal{R}_{2}$.\nThen, provided that $\\Delta t$ is sufficiently small, for all positive integers n, such that $t_n\\leq T$, we have\n\\begin{equation}\n\\|\\tilde{\\phi}^{n+1}\\|_{-1,h} + \\|\\tilde{\\rho}^{n+1}\\|_{-1,h} \n+\\left(\\mathcal{M}\\Delta t\\sum_{k=0}^{n+1}\n\\left(\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{k}\\|_{2}^{2}\n+\\varepsilon\\|\\nabla_{h}\\tilde{\\phi}^{k}\\|_{2}^{2}\n+\\xi\\|\\nabla_{h}\\tilde{\\rho}^{k}\\|_{2}^{2}\\right)\\right)^{\\frac{1}{2}}\n\\leq C\\left(\\Delta t+h^{2}\\right) , \n\\end{equation}\nwhere $C>0$ is independent of n, $\\Delta t$, and h.\n\\end{theorem}\n\n\\begin{proof}\nA carefully consistency analysis indicates the following truncation error estimate:\n\\begin{subequations}\\label{estimate_1st}\n\\begin{align}\n\\frac{\\Phi^{n+1}_{N}-\\Phi^{n}_{N}}{\\Delta t}\n=&\\mathcal{M}\\Delta_{h}\\left(\\eta^{2}\\Delta_{h}^{2}\\Phi^{n+1}_{N}\n-\\varepsilon\\Delta_{h}\\Phi^{n+1}_{N}\n+\\frac{1}{\\varepsilon} (\\Phi^{n+1}_{N}-\\frac{1}{2} )^{3}\n+\\alpha\\nabla_{h} \\cdot \\left( {\\cal A} ( \\frac{\\Psi^{n+1}_{N}}\n{{\\cal A} |\\nabla_{h}^\\delta \\Phi^{n+1}_{N}|} ) \\nabla_{h}\\Phi^{n+1}_{N} \\right)\\right.\\nonumber\\\\\n&\\left. -\\alpha\\left(1+\\frac{1}{\\delta}\\right)\\Delta_{h}\\Phi^{n+1}_{N}\n-\\frac{1}{4\\varepsilon} (\\Phi^{n}_{N}-\\frac{1}{2} )\n-\\frac{\\alpha}{\\delta}\\Delta_{h}\\Phi^{n}_{N}\\right)+\\tau_{\\phi},\n\\label{estimate_phi} \\\\\n\\frac{\\Psi^{n+1}_{N}-\\Psi^{n}_{N}}{\\Delta t}=&\\mathcal{M}\\Delta_{h}\n\\left(-\\xi\\Delta_{h}\\Psi_{N}^{n+1}\n+\\beta H^{\\prime}\\left(\\Psi^{n+1}_{N}\\right)\n+\\sqrt{2}\\alpha \\Psi^{n+1}_{N} - \\alpha {\\cal A} | \\nabla_h^\\delta \\Phi^{n+1}_{N} |\n-\\alpha\\left(\\sqrt{2}-1\\right)\\Psi^{n}_{N}\\right)+\\tau_{\\rho},\n\\label{estimate_rho}\n\\end{align}\n\\end{subequations}\nwith $\\left\\|\\tau^{n}\\right\\|_{-1,h}\\leq C\\left(\\Delta t+h^{2}\\right)$. Observe that we have dropped the operator $\\mathcal{P}_{h}$, which should appear in front of $\\Phi_{N}$, \nfor simplicity of presentation. \n\nSubtracting the numerical scheme $\\eqref{cs_1st}$ from $\\eqref{estimate_1st}$ gives\n\\begin{subequations}\\label{error_1st}\n\\begin{align}\n\\frac{\\tilde{\\phi}^{n+1}_{N}-\\tilde{\\phi}^{n}_{N}}{\\Delta t}\n=&\\mathcal{M}\\Delta_{h}\\left(\\eta^{2}\\Delta_{h}^{2}\\tilde{\\phi}^{n+1}\n-\\varepsilon\\Delta_{h}\\tilde{\\phi}^{n+1}\n+\\frac{1}{\\varepsilon} (\\Phi^{n+1}_{N}-\\frac{1}{2} )^{3}\n-\\frac{1}{\\varepsilon} (\\phi^{n+1}-\\frac{1}{2} )^{3}\\right.\\nonumber\\\\\n&\\left.+\\alpha \\nabla_{h} \\cdot \\left( {\\cal A} ( \\frac{\\Psi^{n+1}_{N}}\n{{\\cal A} |\\nabla_{h}^\\delta \\Phi^{n+1}_{N}|} ) \\nabla_{h}\\Phi^{n+1}_{N} \\right) \n-\\alpha \\nabla_{h} \\cdot \\left( {\\cal A} ( \\frac{\\rho^{n+1}}\n{{\\cal A} |\\nabla_{h}^\\delta \\phi^{n+1} |} ) \\nabla_{h} \\phi^{n+1} \\right) \\right.\\nonumber\\\\\n&\\left.-\\alpha (1+\\frac{1}{\\delta} )\\Delta_{h} \\tilde{\\phi}^{n+1} \n-\\frac{1}{4\\varepsilon}\\tilde{\\phi}^{n}\n-\\frac{\\alpha}{\\delta}\\Delta_{h}\\tilde{\\phi}^{n}\\right)+\\tau_{\\phi},\n\\label{error_phi}\\\\\n\\frac{\\tilde{\\rho}^{n+1}-\\tilde{\\rho}^{n}}{\\Delta t}=\n&\\mathcal{M}\\Delta_{h}\n\\left(-\\xi\\Delta_{h}\\tilde{\\rho}^{n+1}\n+\\beta H^{\\prime}\\left(\\Psi^{n+1}_{N}\\right)\n-\\beta H^{\\prime}\\left(\\rho^{n+1}\\right)\n+\\sqrt{2}\\alpha \\tilde{\\rho}^{n+1}\\right.\\nonumber\\\\\n&\\left.-\\alpha {\\cal A} | \\nabla_h^\\delta \\Phi^{n+1}_{N} | \n + \\alpha {\\cal A} | \\nabla_h^\\delta \\phi^{n+1}|\n-\\alpha (\\sqrt{2}-1 )\\tilde{\\rho}^{n}\\right)+\\tau_{\\rho}.\n\\label{error_rho}\n\\end{align}\n\\end{subequations}\nSince the numerical error function has zero-mean, we see that both $\\left(-\\Delta_{h}\\right)^{-1}\\tilde{\\phi}^{m}$ and $\\left(-\\Delta_{h}\\right)^{-1}\\tilde{\\rho}^{m}$ are well-defined, \nfor any $k\\geq 0$. Taking a discrete inner product with $\\eqref{estimate_phi}$ and $\\eqref{estimate_rho}$ by $2\\left(-\\mathcal{M}\\Delta_{h}\\right)^{-1}\\tilde{\\phi}^{n+1}$ \nand $2\\left(-\\mathcal{M}\\Delta_{h}\\right)^{-1}\\tilde{\\rho}^{n+1}$, respectively, yields\n\\begin{align*}\n&\\frac{1}{\\mathcal{M}\\Delta t}\\left(\n\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\phi}^{n+1}-\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}-\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\\right)\\nonumber\\\\\n&+2\\eta^{2}\\langle\\tilde{\\phi}^{n+1},\\Delta_{h}^{2}\\tilde{\\phi}^{n+1}\\rangle_{\\Omega}\n-2\\varepsilon\\langle\\tilde{\\phi}^{n+1},\\Delta_{h}\\tilde{\\phi}^{n+1}\\rangle_{\\Omega}\n-2\\xi\\langle\\tilde{\\rho}^{n+1},\\Delta_{h}\\tilde{\\rho}^{n+1}\\rangle_{\\Omega}\\nonumber\\\\\n&+\\frac{2}{\\varepsilon}\\left\\langle\\tilde{\\phi}^{n+1}, (\\Phi^{n+1}_{N}-\\frac{1}{2} )^{3}\n- (\\phi^{n+1}-\\frac{1}{2} )^{3}\\right\\rangle_{\\Omega}\n-2\\alpha (1+\\frac{1}{\\delta} ) \\langle\\tilde{\\phi}^{n+1}, \\Delta_{h}\\tilde{\\phi}^{n+1} \\rangle_{\\Omega} \\nonumber\\\\\n&+2\\alpha\\left\\langle\\tilde{\\phi}^{n+1}, \\nabla_{h} \\cdot \\left( \n {\\cal A} ( \\frac{\\Psi^{n+1}_{N}}\n{{\\cal A} |\\nabla_{h}^\\delta \\Phi^{n+1}_{N}|} ) \\nabla_{h}\\Phi^{n+1}_{N} \n- {\\cal A} ( \\frac{\\rho^{n+1}}\n{{\\cal A} |\\nabla_{h}^\\delta \\phi^{n+1} |} ) \\nabla_{h} \\phi^{n+1} \\right) \\right\\rangle_{\\Omega} \n\\\\\n& +2\\beta \\left\\langle\\tilde{\\rho}^{n+1},H^{\\prime}\\left(\\Psi^{n+1}_{N}\\right)\n-H^{\\prime}\\left(\\rho^{n+1}_{N}\\right)\\right\\rangle_{\\Omega} \n +2\\sqrt{2}\\alpha\\left\\langle\\tilde{\\rho}^{n+1}, \\tilde{\\rho}^{n+1} \\right\\rangle_{\\Omega}\n-2\\alpha \\left\\langle \\tilde{\\rho}^{n+1}, {\\cal A} | \\nabla_h^\\delta \\Phi^{n+1}_{N}| \n- {\\cal A} | \\nabla_h^\\delta \\phi^{n+1}|\\right\\rangle_{\\Omega}\\nonumber\\\\\n=&\\frac{1}{2\\varepsilon} \\langle\\tilde{\\phi}^{n+1},\\tilde{\\phi}^{n} \\rangle_{\\Omega}\n+\\frac{2\\alpha}{\\delta} \\langle\\tilde{\\phi}^{n+1},\\Delta_{h}\\tilde{\\phi}^{n} \\rangle_{\\Omega}\n+2\\alpha (\\sqrt{2}-1 )\\left\\langle\\tilde{\\rho}^{n+1},\\tilde{\\rho}^{n}\\right\\rangle_{\\Omega} \\\\\n& \n+\\frac{2}{\\mathcal{M}} ( \\langle\\tilde{\\phi}^{n+1},\\tau_{\\phi}^{n} \\rangle_{-1,h}\n+ \\langle \\tilde{\\rho}^{n+1},\\tau_{\\rho}^{n} \\rangle_{-1,h} ) . \n\\end{align*}\nThe estimate for the terms associated with the surface diffusion is straightforward:\n\\begin{align*}\n\\langle\\tilde{\\phi}^{n+1},\\Delta_{h}^{2}\\tilde{\\phi}^{n+1}\\rangle_{\\Omega}\n&=\\|\\Delta_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}, \\quad \n-\\langle\\tilde{\\phi}^{n+1},\\Delta_{h}\\tilde{\\phi}^{n+1}\\rangle_{\\Omega}\n =\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2},\\\\\n-\\langle\\tilde{\\rho}^{n+1},\\Delta_{h}\\tilde{\\rho}^{n+1}\\rangle_{\\Omega}\n&=\\|\\nabla_{h}\\tilde{\\rho}^{n+1}\\|_{2}^{2}.\n\\end{align*}\nFor the nonlinear inner product, we have the following result\n\\begin{equation}\n2\\beta \\left\\langle\\tilde{\\rho}^{n+1},H^{\\prime}\\left(\\Psi^{n+1}_{N}\\right)\n-H^{\\prime}\\left(\\rho^{n+1}_{N}\\right)\\right\\rangle_{\\Omega}\\geq 0,\n\\end{equation}\ndue to the fact that the logarithmic function is an increasing function. \nSimilarly, the convexity of the nonlinear functional $g_1 (\\phi) = h^2 \\sum_{i,j=1}^N ( \\phi_{i,j} - \\frac12 )^4$ and $g_2 (\\phi, \\rho) = h^2 \\sum_{i,j=1}^N (\\rho_{i,j} - {\\cal A} | \\nabla_h^\\delta \\phi_{i,j} | )^2 + (\\sqrt{2} -1) \\| \\rho \\|_2^2 + \\frac{1}{\\delta} \\| \\nabla_h \\phi \\|_2^2$ leads to the following inequalities: \n\\begin{align}\n&\\frac{2}{\\varepsilon} \\left\\langle\\tilde{\\phi}^{n+1}, (\\Phi^{n+1}_{N}-\\frac{1}{2} )^{3} - (\\phi^{n+1}-\\frac{1}{2} )^{3}\\right\\rangle_{\\Omega} \\ge 0 , \\nonumber \\\\\n& 2\\alpha\\left\\langle \\tilde{\\phi}^{n+1},\\nabla_{h} \\cdot \\left( \n {\\cal A} ( \\frac{\\Psi^{n+1}_{N}}\n{{\\cal A} |\\nabla_{h}^\\delta \\Phi^{n+1}_{N}|} ) \\nabla_{h}\\Phi^{n+1}_{N} \n- {\\cal A} ( \\frac{\\rho^{n+1}}\n{{\\cal A} |\\nabla_{h}^\\delta \\phi^{n+1} |} ) \\nabla_{h} \\phi^{n+1} \\right) \\right\\rangle_{\\Omega} \n-2\\alpha (1+\\frac{1}{\\delta} )\n \\langle \\tilde{\\phi}^{n+1}, \\Delta_{h}\\tilde{\\phi}^{n+1} \\rangle_{\\Omega} \n \\nonumber \\\\ \n&+2 \\sqrt{2}\\alpha\\left\\langle\\tilde{\\rho}^{n+1}, \\tilde{\\rho}^{n+1} \\right\\rangle_{\\Omega}\n-2\\alpha\\left\\langle\\tilde{\\rho}^{n+1},{\\cal A} | \\nabla_h^\\delta \\Phi^{n+1}_{N}| \n- {\\cal A} | \\nabla_h^\\delta \\phi^{n+1}|\\right\\rangle_{\\Omega} \\geq 0 . \n\\end{align}\nThen we arrive at the following estimate: \n\\begin{align*}\n&\\frac{1}{\\mathcal{M}\\Delta t}\\left(\n\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\phi}^{n+1}-\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}-\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\\right)\\nonumber\\\\\n&+2\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+2\\varepsilon\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+2\\xi\\|\\nabla_{h}\\tilde{\\rho}^{n+1}\\|_{2}^{2}\\nonumber\\\\\n\\leq &\\frac{1}{2\\varepsilon}\\left\\langle\\tilde{\\phi}^{n+1},\\tilde{\\phi}^{n}\\right\\rangle_{\\Omega}\n+\\frac{2\\alpha}{\\delta}\\left\\langle\\tilde{\\phi}^{n+1},\\Delta_{h}\\tilde{\\phi}^{n}\\right\\rangle_{\\Omega}\n+2\\alpha\\left(\\sqrt{2}-1\\right)\\left\\langle\\tilde{\\rho}^{n+1},\\tilde{\\rho}^{n}\\right\\rangle_{\\Omega}\\nonumber\n\\\\\n&+\\frac{2}{\\mathcal{M}}\\left\\langle\\tilde{\\phi}^{n+1},\\tau_{\\phi}^{n}\\right\\rangle_{-1,h}\n+\\frac{2}{\\mathcal{M}}\\left\\langle\\tilde{\\rho}^{n+1},\\tau_{\\rho}^{n}\\right\\rangle_{-1,h}.\n\\end{align*}\nMeanwhile, for the inner product associated with the concave part, the following inequalities could be derived: \n\\begin{subequations}\n\\begin{align}\n\\frac{1}{2\\varepsilon}\\left\\langle\\tilde{\\phi}^{n+1},\\tilde{\\phi}^{n}\\right\\rangle_{\\Omega}\n&\\leq\\frac{\\varepsilon}{2}\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\frac{1}{8\\varepsilon^{3}}\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2},\n\\\\\n2\\left(\\sqrt{2}-1\\right)\\alpha\\left\\langle\\tilde{\\rho}^{n+1},\\tilde{\\rho}^{n}\\right\\rangle_{\\Omega}\n&\\leq\\xi\\|\\nabla_{h}\\tilde{\\rho}^{n+1}\\|_{2}^{2}\n+\\frac{\\alpha^{2}\\left(\\sqrt{2}-1\\right)^{2}}{\\xi}\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2},\n\\\\\n\\frac{2}{\\mathcal{M}}\\left\\langle\\tilde{\\phi}^{n+1},\\tau_{\\phi}^{n}\\right\\rangle_{-1,h}\n&\\leq\\frac{1}{\\mathcal{M}}\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tau_{\\phi}^{n}\\|_{-1,h}^{2},\n\\\\\n\\frac{2}{\\mathcal{M}}\\left\\langle\\tilde{\\rho}^{n+1},\\tau_{\\rho}^{n}\\right\\rangle_{-1,h}\n&\\leq\\frac{1}{\\mathcal{M}}\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tau_{\\rho}^{n}\\|_{-1,h}^{2},\n\\\\\n-\\frac{2\\alpha}{\\delta}\\left\\langle\\tilde{\\phi}^{n+1},\\Delta_{h}\\tilde{\\phi}^{n}\\right\\rangle_{\\Omega}\n&=\\frac{2\\alpha}{\\delta}\\left\\langle\\nabla_{h}\\tilde{\\phi}^{n+1},\n\\nabla_{h}\\tilde{\\phi}^{n}\\right\\rangle_{\\Omega}\\nonumber\n\\\\\n&\\leq\\frac{\\varepsilon}{2}\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\frac{2\\alpha^{2}}{\\varepsilon\\delta^{2}}\\|\\nabla_{h}\\tilde{\\phi}^{n}\\|_{2}^{2}\\nonumber\n\\\\\n&\\leq\\frac{\\varepsilon}{2}\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\frac{\\alpha^{2}}{\\varepsilon\\delta^{2}}\\|\\tilde{\\phi}^{n}\\|_{-1}^{\\frac{2}{3}}\n\\|\\Delta_{h}\\tilde{\\phi}^{n}\\|_{2}^{\\frac{4}{3}}\\nonumber\n\\\\\n&\\leq\\frac{\\varepsilon}{2}\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\frac{8\\alpha^{6}}{\\eta^{4}\\varepsilon^{3}\\delta^{6}}\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{n}\\|_{2}^{2}.\n\\end{align}\n\\end{subequations}\nTherefore, we obtain \n\\begin{align}\n&\\frac{1}{\\mathcal{M}\\Delta t}\\left(\n\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\phi}^{n+1}-\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}-\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\\right)\\nonumber\n\\\\\n&+2\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+2\\varepsilon\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+2\\xi\\|\\nabla_{h}\\tilde{\\rho}^{n+1}\\|_{2}^{2}\\nonumber\n\\\\\n\\leq &\\frac{\\varepsilon}{2}\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\frac{1}{8\\varepsilon^{2}}\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\xi\\|\\nabla_{h}\\tilde{\\rho}^{n+1}\\|_{2}^{2}\n+\\frac{\\alpha^{2}\\left(\\sqrt{2}-1\\right)^{2}}{\\xi}\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tau_{\\phi}^{n}\\|_{-1,h}^{2}\\nonumber\n\\\\\n&+\\frac{1}{\\mathcal{M}}\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tau_{\\rho}^{n}\\|_{-1,h}^{2}\n+\\frac{\\varepsilon}{2}\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\frac{8\\alpha^{6}}{\\eta^{4}\\varepsilon^{3}\\delta^{6}}\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{n}\\|_{2}^{2} ,\n\\end{align}\nwhich in turn gives \n\\begin{align}\n&\\frac{1}{\\mathcal{M}\\Delta t}\\left(\n\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}-\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\n\\right) \\nonumber \n\\\\\n&+2\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\varepsilon\\|\\nabla_{h}\\tilde{\\phi}^{n+1}\\|_{2}^{2}\n+\\xi\\|\\nabla_{h}\\tilde{\\rho}^{n+1}\\|_{2}^{2}\\nonumber\n\\\\\n\\leq &\\left(\\frac{1}{8\\varepsilon^{2}}+\\frac{8\\alpha^{6}}{\\eta^{4}\\varepsilon^{3}\\delta^{6}}\\right)\n\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}\n+\\frac{\\alpha^{2}\\left(\\sqrt{2}-1\\right)^{2}}{\\xi}\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}\n+\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{n}\\|_{2}^{2}\\nonumber\n\\\\\n&+\\frac{1}{\\mathcal{M}}\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tau_{\\phi}^{n}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}^{2}\n+\\frac{1}{\\mathcal{M}}\\|\\tau_{\\rho}^{n}\\|_{-1,h}^{2}. \\label{convergence-f} \n\\end{align}\nFinally, an application of a discrete Gronwall inequality results in the desired convergence estimate:\n\\begin{equation}\n\\|\\tilde{\\phi}^{n+1}\\|_{-1,h}\n+\\|\\tilde{\\rho}^{n+1}\\|_{-1,h}\n+\\left(\\mathcal{M}\\Delta t\\sum_{k=0}^{n+1}\n\\left(\\eta^{2}\\|\\Delta_{h}\\tilde{\\phi}^{k}\\|_{2}^{2}\n+\\varepsilon\\|\\nabla_{h}\\tilde{\\phi}^{k}\\|_{2}^{2}\n+\\xi\\|\\nabla_{h}\\tilde{\\rho}^{k}\\|_{2}^{2}\\right)\\right)^{\\frac{1}{2}}\n\\leq C\\left(\\Delta t+h^{2}\\right) , \n\\end{equation}\nwhere $C>0$ is independent of $\\Delta t, h$ and $n$. This completes the proof of Theorem~\\ref{thm: convergence}. \n\\end{proof}\n\n\n\\begin{remark} \nIn the application of the discrete Gronwall inequality, we see that the growth constants for $\\|\\tilde{\\phi}^{n}\\|_{-1,h}^{2}$ and $\\|\\tilde{\\rho}^{n}\\|_{-1,h}^{2}$ terms, given by $\\frac{1}{8\\varepsilon^{2}}+\\frac{8\\alpha^{6}}{\\eta^{4}\\varepsilon^{3}\\delta^{6}}$ and $\\frac{\\alpha^{2}\\left(\\sqrt{2}-1\\right)^{2}}{\\xi}$, respectively, depend singularly on $\\varepsilon$, $\\eta$, $\\delta$ and $\\xi$. In turn, it would be reasonable to require that \n$$\n \\Big( \\frac{1}{8\\varepsilon^{2}}+\\frac{8\\alpha^{6}}{\\eta^{4}\\varepsilon^{3}\\delta^{6}} \\Big) {\\cal M} \\Delta t \\le 1 , \\quad \n \\frac{\\alpha^{2}\\left(\\sqrt{2}-1\\right)^{2}}{\\xi} {\\cal M} \\Delta t \\le 1 , \n$$\nso that a singular convergence constant is avoided. In other words, the time step size $\\Delta t$ should be bounded by a given constant, dependent on $\\varepsilon$, $\\eta$, $\\delta$ and $\\xi$, to present a singular convergence constant at a theoretical level; this requirement refers to the condition that ``provided that $\\Delta t$ is sufficiently small\" in the statement of Theorem~\\ref{thm: convergence}. Meanwhile, such a requirement is only associated with a theoretical analysis, and this requirement may not be necessary in the practical computations to preserve a numerical convergence. \n\\end{remark} \n\n\n\\begin{remark} \nAs the regularization parameter $\\delta \\to 0$, the positivity-preserving property and the energy stability estimates, as established in Theorems~\\ref{thm: positivity} and \\ref{thm: energy stability}, are still valid. In fact, these two theoretical properties are available even with $\\delta=0$. On the other hand, the optimal rate convergence estimate, as established in Theorem~\\ref{thm: convergence}, is only available for a fixed $\\delta >0$, due to the singularly-$\\delta$-dependent convergence constant appearing in~\\eqref{convergence-f}. In other words, the convergence constant in Theorem~\\ref{thm: convergence} depends singularly on $\\delta$, and such a convergence estimate would not be theoretically justified as $\\delta \\to 0$, although the numerical convergence has also been verified in various numerical experiments. \n\\end{remark} \n\n\n\\begin{remark} \nIn the proposed numerical scheme~\\eqref{cs_1st}, we take $M_1 = M_2 = {\\cal M}$, and $M (\\rho) \\equiv 1$, for simplicity of presentation. In case of a $\\rho$-dependent mobility function $M (\\rho)$, the positivity-preserving property and energy stability are still valid, as long as $M (\\rho) >0$ is available at a point-wise level. Meanwhile, the corresponding convergence analysis and error estimate are expected to face certain theoretical difficulties in the case of a non-constant mobility function, due to the highly nonlinear and singular nature of the chemical potential. The theoretical justification of this convergence analysis will be left to the future works, and some techniques of rough error estimate and refined error estimate, as reported in a recent work~\\cite{LiuC20a} to analyze the non-constant-mobility Poisson-Nernst-Planck system, may have to be applied in this future work. \n\\end{remark} \n\n\\section{Numerical experiments} \\label{sec: numerical results} \n\nIn this section, we preform a few two-dimensional numerical simulations using the proposed scheme $\\eqref{cs_1st}$. The mass conservation, energy decay, positivity of the numerical solution, as well as the numerical accuracy, will be demonstrated in these computations. To achieve this goal, we will present two numerical examples with different initial conditions. \n\n\\subsection{Accuracy test}\n\nHere, we take the domain $\\Omega=(0,8)^2$, and choose the parameters as follows \n\\begin{align*}\n&\\varepsilon=0.05,\\quad \\alpha=0.001,\\quad \\beta=0.02,\\quad \\delta=0.001, \\\\\n&\\eta=0.05,\\quad \\xi=0.05,\\quad M_{1}=0.01,\\quad M_{2}=0.01. \n\\end{align*}\nThe initial data are set as \n\\begin{equation}\\label{CS:test}\n\\left\\{\n\\begin{aligned}\n&\\phi_{0}(x,y)=0.5+0.2\\cos\\frac{4\\pi x}{8}\\cos \\frac{4\\pi y}{8},\\\\\n&\\rho_{0}(x,y)=0.5+0.2\\sin\\frac{4\\pi x}{8}\\sin \\frac{4\\pi y}{8}.\n\\end{aligned}\n\\right.\n\\end{equation}\nIt is obvious that the initial data are subject to periodic boundary condition. \nThis example is designed to study the numerical accuracy in time and space. \nIn order to test the first order convergence rate in time and second order convergence rate in space, we use a linear refinement pate, i.e. $\\Delta t = Ch^{2}$, $C = 0.01$. The global error is expected to be $O(\\Delta t)+O(h^{2})=O(h^{2})$ under the discrete $L^{2}$ norm. Since an exact solution is not available, we compute the Cauchy difference instead of directly calculating the numerical error, which is defined as $\\delta_{u} = u_{h_{f}}-\\mathcal{I}_{c}^{f}(u_{h_{c}})$, where $\\mathcal{I}_{c}^{f}$ is a bilinear interpolation operator. \nThis requires a relatively coarse solution, parametrized by $h_{c}$, \nand a relatively fine solution, parametrized by $h_{f}$, where $h_{c}=2h_{f}$, at the same final time. The discrete $L^{2}$ norms of Cauchy difference and the convergence rates are displayed in Table $\\ref{CS:error}$. These results confirm the expected convergence rate. \n\\begin{table}[h!]\n \\centering\n \\caption{The discrete $L^{2}$ error and convergence rate at $t = 0.1$ \n with initial data $\\eqref{CS:test}$ and the given parameters.}\\label{CS:error}\n \\vskip 0.2cm\n \\begin{tabular}{lcccc}\n \\toprule\n Grid sizes &Error$(\\phi)$ & Rate & Error$(\\rho)$ &Rate\\\\\n \\midrule\n $16\\times 16$ & 1.93E--01 & -- & 1.88E--01 & -- \\\\\n $32\\times 32$ & 5.07E--02 & 1.93 & 4.86E--02 & 1.95\\\\\n $64\\times 64$ & 1.28E--02 & 1.98 & 1.23E--02 & 1.99\\\\\n $128\\times 128$ & 3.21E--03 & 2.00 & 3.07E--03 & 2.00\\\\\n $256\\times 256$ & 8.04E--04 & 2.00 & 7.68E--04 & 2.00\\\\\n \\bottomrule\n \\end{tabular}\n \\end{table}\n\n\\subsection{Spinodal decomposition}\nIn this example, we study the phase separation phenomenon, so called spinodal decomposition. Usually we describe this process as a thermal quench, which is considered that an initially homogeneous mixture is thrust into a two-phase region. In this case, the spinodal decomposition occurs and leads the system from the homogeneous to two-phase state. \nWe take the domain as $\\Omega = (0,2\\pi)^2$. The initial data are given by \n\\begin{equation}\\label{example: spinodal}\n\\left\\{\n\\begin{aligned}\n&\\phi_{0}(x,y)=0.4+0.1\\mbox{rand}(x,y), \\\\\n&\\rho_{0}(x,y)=0.4+0.1\\mbox{rand}(x,y), \n\\end{aligned}\n\\right.\n\\end{equation}\nwhere rand$(x,y)$ is a random number in $[-1,1]$ and has zero mean. The parameters are chosen as follows \n\\begin{equation}\\label{eqn: parameters}\n\\varepsilon = 0.02,~\\alpha = 0.02,~\\beta = 0.02,~\\eta = 0.02,~\\delta = 0.01,~\\xi = 0.02,~\\mathcal{M} = 0.01. \n\\end{equation} \nFrom Figures $\\ref{fig: spinodal t=0}$ to $\\ref{fig: spinodal t=700}$, we display the snapshots of coarsening dynamics. Initially, the two fluids are well mixed, and they sooner start to decompose and accumulate. We observe that a relatively high value of the concentration variable $\\rho$ gathers at the interface between the two different fluids. A monotone decay evolution of the physical energy is illustrated in Figure $\\ref{fig: energy}$. \n\n\\begin{figure}[h]\n \\centering\n \\subfigure[$\\phi(t=0)$]{\n \\includegraphics[width=0.4\\textwidth]{phi0_0} \n}\n \\subfigure[$\\rho(t=0)$]{\n \\includegraphics[width=0.4\\textwidth]{rho0_0}\n}\n \\caption{Snapshots of the phase variables $\\phi$ and $\\rho$, taken at $t=0$ for Example $\\ref{example: spinodal}$.}\n \\label{fig: spinodal t=0}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\subfigure[$\\phi(t=0.5)$]{\n \\includegraphics[width=0.4\\textwidth]{phi0_5} \n}\n \\subfigure[$\\rho(t=0.5)$]{\n \\includegraphics[width=0.4\\textwidth]{rho0_5}\n}\n \\caption{Snapshots of the phase variables $\\phi$ and $\\rho$, taken at $t=0.5$ for Example $\\ref{example: spinodal}$.}\n \\label{fig: spinodal t=0.5}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\subfigure[$\\phi(t=2)$]{\n \\includegraphics[width=0.4\\textwidth]{phi2} \n}\n \\subfigure[$\\rho(t=2)$]{\n \\includegraphics[width=0.4\\textwidth]{rho2}\n}\n \\caption{Snapshots of the phase variables $\\phi$ and $\\rho$, taken at $t=2$ for Example $\\ref{example: spinodal}$.}\n \\label{fig: spinodal t=2}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\subfigure[$\\phi(t=10)$]{\n \\includegraphics[width=0.4\\textwidth]{phi10} \n}\n \\subfigure[$\\rho(t=10)$]{\n \\includegraphics[width=0.4\\textwidth]{rho10}\n}\n \\caption{Snapshots of the phase variables $\\phi$ and $\\rho$, taken at $t=10$ for Example $\\ref{example: spinodal}$.}\n \\label{fig: spinodal t=10}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\subfigure[$\\phi(t=40)$]{\n \\includegraphics[width=0.4\\textwidth]{phi40} \n}\n \\subfigure[$\\rho(t=40)$]{\n \\includegraphics[width=0.4\\textwidth]{rho40}\n}\n \\caption{Snapshots of the phase variables $\\phi$ and $\\rho$, taken at $t=40$ for Example $\\ref{example: spinodal}$.}\n \\label{fig: spinodal t=40}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\subfigure[$\\phi(t=700)$]{\n \\includegraphics[width=0.4\\textwidth]{phi700} \n}\n \\subfigure[$\\rho(t=700)$]{\n \\includegraphics[width=0.4\\textwidth]{rho700}\n}\n \\caption{Snapshots of the phase variables $\\phi$ and $\\rho$, taken at $t=700$ for Example $\\ref{example: spinodal}$.}\n \\label{fig: spinodal t=700}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=1\\textwidth]{energy.eps}\n \\caption{Time evolution of the free energy functional for Example $\\ref{example: spinodal}$, which shows a monotone decay.}\n \\label{fig: energy}\n\\end{figure}\n\n\\section{Conclusions} \\label{sec: conclusion}\n\nIn this paper, we propose and analyze a first order (in time) accurate, convex splitting scheme for the binary fluid-surfactant phase field model. \nThe multi-phase structure and the singularity associated with the 1-Laplacian part makes the whole system very challenging, at both the theoretical and numerical levels. \nTo overcome this subtle difficulty, we make an observation for a non-standard convex-concave decomposition of the free energy. \nIn addition, the singular nature of the logarithmic function around the limit values prevents the numerical solution approaching these limit values, \nso that the positivity property is preserved for the numerical scheme. \nAs a result, the convex structure of the implicit part guarantees the unique solvability and energy stability of the proposed numerical scheme. \nFurthermore, an optimal rate convergence analysis is carefully derived, which is the first such result in this area. \nSome numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.\n\n\n\\section{Acknowledgements}\n\nThe authors greatly appreciate many helpful discussions with Professor Hui Zhang, in particular for his insightful suggestions and comments. \nThis work is supported in part by the grants NSF DMS-2012669 (C.~Wang),\nNSFC-11871105, the Science Challenge Project TZ2018002 \nand the Fundamental Research Funds for the Central Universities (Z.~Zhang). \n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:Introduction}}\nWithin the realm of solid-state electronic structure theory, it is now well-appreciated that how Bloch orbitals change their phase as a function of crystal momentum (i.e. Berry phase\\cite{berry1984quantal}) is fundamentally tied to the geometry and topology of a given material and has direct consequences as far as experimental observables, such as the anomalous Hall effect\\cite{haldane_berry_2004,zhang_experimental_2005,xiao_berry_2010,gritsev_dynamical_2012}. \nFurthermore, in the world of electronic transport, it is well understood that when a current runs through a molecule or through a material, geometric magnetic forces (which follows from Berry phase) can emerge leading to current-induced spin-orbit torques\\cite{manchon_current-induced_2019,brataas_current-induced_2012} or runaway vibrational motion\\cite{Lu2010_blowing}.\nHowever, in the context of molecular systems far from a metal surface, the experimental consequences of Berry phase remain murkier. \nMore precisely, as far as molecular electronic structure is concerned, one considers Berry phase only around a conical intersection (CI) where the integral of the phase change around the CI is non-zero (and a multiple of $\\pi$) which is known as molecular Aharonov-Bohm physics\\cite{yarkony_determining_1998,yarkony_diabolical_1996,xie_nonadiabatic_2016,guo_accurate_2016,xie_constructive_2017,yuan_observation_2018,xie2019up,xie_quantum_2020}; such interference effects have now been measured for some realistic molecules\\cite{xie_nonadiabatic_2016,xie2019up}. \nAnd yet, for many systems, numerical investigations of non-adiabatic dynamics suggest that semiclassical simulations (e.g. Tully's surface hopping method\\cite{tully1990molecular}) can recover many observables after wavepackets approach a CI \\emph{even though these methods do not include geometric phase} \\cite{ryabinkin_geometric_2013,ryabinkin_geometric_2017,ryabinkin_analysis_2014}.\nMoreover, recent analysis based on exact factorization\\cite{abedi2010exact,agostini2015exact} has argued that the overall effect of a molecular Berry phase disappears under a gauge transformation\\cite{juanes-marcos_theoretical_2005,min_is_2014,agostini2018exact,curchod2017dynamics}, except in the limit of non-zero circular nuclear currents\\cite{requist_molecular_2016}.\nIn the end, for theoretical chemists who think about molecular motion (rather than condensed matter physicists who think about periodic electronic structure), the experimental importance or relevance of molecular Berry phase still remains unclear. \n\nNow one key assumption is usually made in the context of molecular non-adiabatic dynamics: almost always we assume that the electronic Hamiltonian is \\emph{real}-valued.\nHowever, when the electronic Hamiltonian is \\emph{complex}-valued, it is known that a non-zero Berry phase yields a Lorentz-like magnetic force arising from the imaginary part of the derivative coupling ($\\vec{d}$) vector\\cite{berry_chaotic_1993,takatsuka_exploring_2011,takatsuka_lorentz-like_2017}. \nThis so-called geometric magnetic force acts as the first order correction to the Born-Oppenheimer approximation and will affect nuclear motion whenever the electronic Hamiltonian is not real-valued\\cite{mead1979noncrossing,mead1979determination}, e.g. for molecular systems with complex spin-orbit coupling\\cite{matsika_effects_2001,matsika_effects_2001-1,matsika_spin-orbit_2002}. \nNevertheless, even though derivative couplings ($\\vec{d}$) are often large (e.g. whenever the energy gap between levels becomes small), most chemists have always assumed that they can ignore such a magnetic force even in the context of spin-dependent nuclear phenomena (e.g. spin-vibronic intersystem crossing)\\cite{penfold_spin-vibronic_2018, talotta2020internal}.\nTraditional surface hopping does not account for a Berry magnetic force directly or indirectly\\cite{tully1990molecular}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics{fig00.pdf}\n \\caption{Schematic diagram of a molecular system (e.g. bacteriochlorophyll) under illumination by laser fields at frequencies $\\omega$ (red) and $2\\omega$ with phase difference $+\\Delta\\phi$ (blue) and $-\\Delta\\phi$ (green). Any detected difference in the resulting molecular dynamics can be attributed to Berry magnetic force exclusively. }\n \\label{fig:schematic}\n\\end{figure}\nIn a future set of articles, we will suggest a set of \\emph{complex}-valued Hamiltonians and chemical reactions for illustrating how Berry magnetic force can lead to spin-dependent nuclear motion. \nFor the present letter, however, our goal is to demonstrate that Berry magnetic force should also be experimentally observable for a spin-less system with a \\emph{real}-valued electronic Hamiltonian, provided that we have a strong laser source.\nIn order to generate a necessarily complex-valued Hamiltonian from a real-valued Hamiltonian, we will assume that the spin-less molecular system is coupled to multiple laser fields that are periodic in time with frequencies $\\omega$ and $2\\omega$ (see Fig.~\\ref{fig:schematic}); such a system should be easily realizable with frequency doubling.\nIn such a case, using Floquet theory\\cite{shirley_solution_1965}, one can expand the electronic wavefunction in a Floquet state basis (the electronic states dressed by $e^{im\\omega t}$ for an integer $m$) and recast the explicitly time-dependent real-valued Hamiltonian ($\\hat{H}(t)$) into a time-independent complex-valued Floquet Hamiltonian ($\\hat{H}_F$).\nFurthermore, the complex-valued nature of the Floquet Hamiltonian can be tuned by the phase difference between the two frequency components $\\omega$ and $2\\omega$ and, as we will show, this phase difference gives experimentalists the capacity to measure the dynamics of $\\hat{H}_F$ and $\\hat{H}_F^*$ independently.\nBy comparing the dynamics of $\\hat{H}_F$ and $\\hat{H}_F^*$, one can definitively isolate the effect of Berry phase on a laser-driven molecular system. \n\nNow using a laser-driven periodic potential (together with Floquet theory) to engineer materials to achieve a given band structure with desirable Berry phase is hardly new in the context of designing band structure\\cite{flaschner2016experimental,oka2019floquet}; moreover, observing the\npresence of a non-trivial Berry phase is considered a fingerprint of a light-induced conical intersection (LICI) in photodissociation dynamics\\cite{halasz2011conical,halasz2012light,halasz2015direct,bouakline2018unambiguous,kim2012control}.\nThat being said, to our knowledge, the experimental ramifications of Berry force per se have not been explored in the literature in the context of laser-driven nuclear dynamics.\nFor the most part, heretofore semiclassical simulations of laser-driven dynamics have either centered around monochromatic periodic potentials (for which the Floquet Hamiltonian is real-valued and there is no Berry geometric force\\cite{kim2015ab,makhov2018floquet,maitra2002floquet,fiedlschuster_floquet_2016,fiedlschuster_surface_2017,zhou_nonadiabatic_2020,chen2020proper,restrepo_driven_2016}), or focused on electronic symmetry breaking of charge transport driven by an $\\omega+2\\omega$ field (where Berry force is not a key factor in these calculations)\\cite{franco2006laser,franco2008femtosecond,franco2008laser}.\nThe goal of this letter is to establish a clear set of guidelines outlining (1) how to construct a Hamiltonian (or search for a molecular system) for which Berry force effects should be strong, and (2) how to experimentally observe Berry force using a continuous wave laser source.\n\n\n\n\\section{Floquet theory and Berry curvature}\nLet us briefly review Floquet theory as applied to solving periodically driven nuclear-electronic system. Consider a real-valued electronic Hamiltonian that is periodic in time i.e. $\\hat{H}(t)=\\hat{H}(t+T_0)$ in a diabatic electronic basis $\\ket{j}$ for $j=1,\\cdots,N$. The corresponding time-independent Floquet Hamiltonian can be determined by taking the Fourier transform of $\\hat{H}_F=\\hat{H}-i\\hbar\\frac{\\partial}{\\partial t}$:\n\\begin{equation}\n\\bra{k n}\\hat{H}_{F}\\ket{j m}\n=\\frac{1}{T_{0}}\\int_{0}^{T_{0}}dt\\bra{k}\\hat{H}(t)\\ket{j}e^{-i(n-m)\\omega t} + \\delta_{jk}\\delta_{m n}n\\hbar\\omega\n\\label{eq:floquettransform}\n\\end{equation}\nHere $\\omega = 2\\pi\/T_{0}$ and $\\ket{j m}=\\ket{j}e^{im\\omega t}$ is an element of the diabatic Floquet state basis. \nWithout loss of generality, we consider a time-periodic Hamiltonian written in a diabatic electronic basis of the form $H(t)=H_0+2\\sum_{k=1}^\\infty{H}_k\\cos(k\\omega t+\\phi_k)$ where $H_k$ is a real-valued $N\\times N$ block matrix. \nThe corresponding Floquet Hamiltonian matrix takes the form of\n\\begin{equation}\\label{eq:H_F}\nH_F = \\left(\n\\begin{array}{c c c c c}\n\\ddots & & & & \\\\\n & H_0+\\hbar\\omega & H_1e^{i\\phi_1} & H_2e^{i\\phi_2} & \\\\\n & H_1^{\\dag}e^{-i\\phi_1} & H_0 & H_1e^{i\\phi_1} & \\\\\n & H_2^{\\dag}e^{-i\\phi_2} & H_1^{\\dag}e^{-i\\phi_1} & H_0-\\hbar\\omega & \\\\\n & & & & \\ddots \\\\\n\\end{array} \\right)\n\\end{equation}\nIn the Floquet diabatic representation, the electronic wavefunction can be written as $\\ket{\\Psi_F(t)}=\\sum_{j}\\sum_{m=-\\infty}^{\\infty}C_{j m}(t)\\ket{j m}$, and the Schr\\\"{o}dinger equation becomes $i\\hbar\\frac{\\partial}{\\partial t}C = {H}_FC$ with a complex-valued Floquet Hamiltonian ${H}_F$. \nNote that here the Floquet expansion dresses the electronic diabatic states and does not affect the nuclear degrees of freedom.\n\nNext, in order to account for nuclear motion, we let $\\hat{H}_F=\\hat{H}_F(\\vec{R})$ depend on a nuclear coordinate $\\vec{R}$.\nThe Floquet Hamiltonian can be diagonalized by solving $\\hat{H}_F(\\vec{R})|\\Phi^{J}(\\vec{R})\\rangle =V_F^{J}(\\vec{R})|\\Phi^{J}(\\vec{R})\\rangle$. \nHere $V_F^{J}(\\vec{R})$ denotes the Floquet quasi-energy surface and $|\\Phi^{J}(\\vec{R})\\rangle=\\sum_{j}\\sum_{m=-\\infty}^{\\infty}G^{J}_{j m}(\\vec{R})\\ket{j m}$ are the Floquet adiabatic states. \nThen, within the Floquet adiabatic representation, we can propagate non-adiabatic nuclear dynamics where the potential energy surfaces are $V_F^{J}(\\vec{R})$ and the derivative coupling between Floquet adiabatic states $J$ and $K$ is\n\\begin{equation}\n \\vec{d}_{JK}=\\sum_{j}\\sum_{m=-\\infty}^{\\infty}G^{J*}_{j m}\\frac{\\partial}{\\partial\\vec{R}}G^{K}_{j m} = \\frac{\\bra{G^{J}}\\nabla_{\\vec{R}} H_{F}\\ket{G^{K}}}{\\epsilon_{K} - \\epsilon_{J}}\n \\label{eq:derivcoup}\n\\end{equation}\n\nWith the complex-valued Floquet Hamiltonian, it is well known that the Berry curvature near the derivative coupling region yields an effective \"magnetic\" force on adiabatic surface $J$\\cite{sakurai2014modern, shankar2012principles, miao2019extension} $\\vec{F}_{J}^{mag} = \\hbar\\frac{\\vec{P}}{M}\\times\\vec{B}_{J}$\nwhere $\\vec{P}$ is the nuclear momentum and $M$ is the nuclear mass. \nHere $\\vec{B}_{J}$ is defined as the Berry curvature of Floquet adiabat $J$\n\\begin{equation}\n \\vec{B}_{J} = -i\\sum_{K\\neq J}\\vec{d}_{JK}\\times \\vec{d}_{KJ}.\n \\label{eq:berrycurvature}\n\\end{equation}\nNote that $\\vec{d}_{JK}=-\\vec{d}_{KJ}^*$ and the \"magnetic\" force arise from the imaginary part of the derivative coupling;\nexplicitly \n\\begin{equation}\n \\vec{F}_{J}^{mag} =\\hbar\\frac{\\vec{P}}{M}\\times\\vec{B}_{J}= 2\\hbar\\text{Im}\\sum_{K\\neq J}\\vec{d}_{JK}(\\frac{\\vec{P}}{M}\\cdot\\vec{d}_{KJ}).\n\\end{equation}\n\n\\section{Sufficient condition for observing Berry force}\\label{sec:condition}\nFor a spectroscopist who wants to observe Berry curvature directly through nuclear dynamics, we will now establish a practical framework.\nTo begin this analysis, note that the Berry curvature of the Floquet Hamiltonian remains unchanged under two transformations: \n(1) adding an arbitrary phase factor to a spatial electronic basis function: $\\ket{j}\\rightarrow\\ket{j}e^{i\\theta_j}$ for $\\theta_{j}\\in[0,2\\pi)$; \n(2) translating a Floquet diabatic basis function in time for arbitrary $\\eta$: $e^{im\\omega t}\\rightarrow e^{im\\omega (t+\\eta\/\\omega)}$ .\nIf these operations are applied, a block of the Floquet Hamiltonian in \\eqref{eq:H_F} transforms as follows \n\\begin{equation}\n H_k\\rightarrow\\tilde{H}_k = U^\\dagger H_k Ue^{ik\\eta}\n \\label{eq:shifttransform}\n\\end{equation}\nSuch a transformation (replacing $H_{k}$ with $\\tilde{H_{k}}$) has no effect on the Berry curvature in Eqs.~\\ref{eq:derivcoup} and \\ref{eq:berrycurvature}, which will now allow us to isolate a set of sufficient conditions for isolating a Berry force.\n\n\n\\subsection*{Case \\#1: Monochromatic Laser}\nThe simplest case to consider is the case of a monochromatic laser of the form $H(t)=H_0+H_1\\cos(\\omega t+\\phi_1)$\\cite{bajo2012mixed}.\nIn such a case, the Floquet Hamiltonian will be block tridiagonal (i.e. $H_1\\neq0$, but $H_{2}=H_{3}=H_{4}=\\dots=0$ in \\eqref{eq:H_F}). \nIf we now apply a transform of the form of \\eqref{eq:shifttransform}, letting $\\theta_j=0$ for all $j$ and $\\eta=-\\phi_1$, and noting that $\\cos(\\omega t)$ yields \\emph{real} Fourier transform components ($\\frac{1}{2}H_1$ and $\\frac{1}{2}H_1$), we find that a complex-valued $H_F$ can always be transformed to a real-valued $\\tilde{H}_F$. In other words, if a molecule and material is illuminated by a monochromatic laser, the Berry curvature is still strictly zero. \n\n\n\n\\subsection*{Case \\#2: 2 Frequencies, 2 Electronic States}\nNext, we focus on the case where the laser profile includes two periodic frequency components. \nSpecifically, we consider two electronic states ($N=2$) coupled through the laser excitations only (i.e. $H_0$ is diagonal). \nWe find that $H_{F}$ cannot be transformed to a real-valued $\\tilde{H}_F$ (except for a trivial case when $\\phi_{2}-2\\phi_{1}=0\\mod\\pi$), suggesting that one should be able to observe non-zero Berry force effect in this case.\nHowever, if a system contains only two electronic states, there is only one energy difference that can be made resonant with one incoming frequency (and we have already discussed why Berry curvature is zero in the case of a monochromatic laser.) Thus, observing non-zero Berry curvature in the presence of only two electronic states would require higher order non-resonant light-matter interactions, for example, multiphoton absorption. Such effects are usually very weak and very likely will not yield robust experimental signals. \n\n\n\n\\subsection*{Case \\#3: 2 Frequencies, 3 Electronic States}\nTo observe a significant Berry force effect, we find we require a minimal model of three electronic states ($\\ket{0}, \\ket{1}, \\ket{2}$) with three bright transitions ($\\ket{0}\\rightarrow\\ket{1}$, $\\ket{1}\\rightarrow\\ket{2}$ and $\\ket{0}\\rightarrow\\ket{2}$). \nIn this case, the complex-valued Floquet Hamiltonian cannot be made real-valued (except for the trivial case when $\\phi_{2}-2\\phi_{1}=0~\\text{mod}~\\pi$), such that a non-zero Berry curvature is expected.\nNote that all three electronic transitions must be bright: if, for instance, the $\\ket{1}\\rightarrow\\ket{2}$ transition is dark, (i.e. $(H_k)_{12}=0$ in \\eqref{eq:H_F}), the Berry curvature will again likely be small as in case \\#2 above. After all, with only two bright transitions and two frequencies, one can make a reasonable rotating wave approximation (RWA) on resonance and reduce the Floquet Hamiltonian to be effectively real-valued.\nThus, we are led to a Hamiltonian of the following mathematical form:\n\\begin{equation}\\label{eq:tdhamiltonian}\n H(t) = H_0 + H_1 \\cos\\omega t + H_2 \\cos(2\\omega t+ \\Delta\\phi).\n\\end{equation}\nwhere $\\Delta\\phi\\in[-\\pi,\\pi]$. \n\nFor the spectroscopist, control over the phase difference $\\Delta\\phi$ will allow one to isolate Berry curvature. \nWhen $\\Delta\\phi=0$ or $\\pm\\pi$, based on the above analysis, the Berry curvature is $0$. \nHowever, if $\\alpha$ is a constant not equal to $0$ or $\\pm\\pi$, choosing $\\Delta\\phi=\\alpha$ vs $\\Delta\\phi=-\\alpha$ is equivalent to choosing $\\hat{H}_{F}$ vs $\\hat{H}^{*}_{F}$.\nMost importantly, because two linear operators that are complex conjugate to each other, i.e. $\\hat{H}_{F}$ vs $\\hat{H}^{*}_{F}$, will have identical eigenvalues, any difference in the resulting nuclear dynamics must reflect differences in the phases of the eigenvectors, i.e. Berry force.\nThus, by comparing the dynamics of $\\hat{H}_F$ and $\\hat{H}_F^*$, the effect of Berry phase on a laser-driven molecular system can be isolated.\n\n\n\n\n\\section{Model and results}\n\nFor a numerical demonstration of a Berry force on nuclear dynamics using the Hamiltonian in \\eqref{eq:tdhamiltonian}, we consider the following model with three electronic diabatic states $|j=0,1,2\\rangle$ coupled to two nuclear degrees of freedom $\\vec{R}=(x,y)$. \n$H_0$ is taken to be diagonal and of the form (see \\eqref{eq:tdhamiltonian})\n\\begin{align}\n\\bra{0}H_0(\\vec{R})\\ket{0} &= A\\tanh(x),\\nonumber \\\\\n\\label{eq:T1st}\n\\bra{1}H_0(\\vec{R})\\ket{1} &= \\hbar\\omega-A\\tanh(x), \\\\\\nonumber\n\\bra{2}H_0(\\vec{R})\\ket{2} &= 0.2A + 2\\hbar\\omega \n\\end{align}\nwith $A = 0.02$ and $\\hbar\\omega = 0.5$. \nWe assume that the only coupling between the diabatic electronic states is caused by the coupling between the transition dipole moments and the external electric fields, which we take to be of the form \n\\begin{equation}\n\\bra{j}H_1(\\vec{R})\\ket{k} =\\bra{j}H_2(\\vec{R})\\ket{k} = De^{-(x^2+y^2)\/2\\sigma^2}.\n\\label{eq:diabaticcoupling}\n\\end{equation}\nfor all $j\\neq k$ with $\\sigma = 1.0$ and $D=0.01$.\n\nFor convergence only, the eigenenergies of the Hamiltonian $H_0(\\vec{R})$ have been chosen to mimic scattering potentials. Although all the theory above in Sec.~\\ref{sec:condition} is completely general, working with a scattering Hamiltonian (where the diabatic Floquet states $|j m\\rangle$ have constant asymptotic energies) will allow for a very simple visualization of the Floquet dynamics below. Moreover, for this Hamiltonian, invoking the RWA (where we only keep the three diabatic states $\\ket{j m} \\in \\{\\ket{02}$, $\\ket{11}, \\ket{20}\\}$) is a fairly good approximation. For a visualization of the relevant diabatic and adiabatic energies, see Fig. \\ref{fig: dia_ad_states}. Henceforward, the Floquet adiabatic energies will be labeled as $\\ket{\\epsilon_{0}^{F}}$, $\\ket{\\epsilon_{1}^{F}}$ and $\\ket{\\epsilon_{2}^{F}}$. \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics{fig0.pdf}\n \\caption{(a) The three diabatic Floquet states $\\ket{02}$ (electronic state $\\ket{0}$ dressed with $2$ photons), $\\ket{11}$ (electronic state $\\ket{1}$ dressed with $1$ photon) and $\\ket{20}$ (electronic state $\\ket{2}$ dressed with $0$ photons) approach each other near $x=0$. All other Floquet states have quasi-energies greater than 1.498 or smaller than 0.502 and are not dynamically relevant. (b) The energies of the three Floquet adiabatic states $\\ket{\\epsilon_{0}^{F}}$, $\\ket{\\epsilon_{1}^{F}}$ and $\\ket{\\epsilon_{2}^{F}}$, clearly show a complicated avoided crossing. All energies are evaluated at $y = 0$.}\n \\label{fig: dia_ad_states}\n\\end{figure}\n\n\nTo propagate the laser driven electron-nuclear system, we integrate the time-dependent Schr\\\"{o}dinger equation (TDSE) with the total Hamiltonian $\\hat{H}_\\text{tot}(\\vec{R},t)=-\\frac{\\hbar^2}{2M}\\frac{\\partial^2}{\\partial R^2}+\\hat{H}(\\vec{R},t)$ using a Tr\\\"{o}tter decomposition\\cite{kosloff1988time}. \nThe nuclear mass is chosen to be $M=1000~\\text{a.u.}$.\nWe assume the initial nuclear wavepacket is on diabat \\ket{2} and of the form \n$\\ket{\\Psi(x, y, 0)}={\\cal N}\\exp[-\\frac{(x-x_{0})^2}{2\\sigma_{x}^2} + ip^{x}_{0}(x-x_{0})-\\frac{(y-y_{0})^2}{2\\sigma_{y}^2} + ip^{y}_{0}(y-y_{0})] \\ket{2}$.\nHere, $x_{0}, y_{0}, p_{0}^{x}, p_{0}^{y}$ are the initial positions and momenta along the $x$ and $y$ directions respectively, and $\\sigma_{x}=\\sigma_{y}=1$ represent the width of the Gaussian along $x$ and $y$ direction.\nNote that solving the TDSE with the total Hamiltonian is equivalent to the exact propagation of $i\\hbar\\frac{\\partial}{\\partial t}C = {H}_FC$ with a time-independent Floquet Hamiltonian. \n\n\n\\subsection{Floquet-based Berry curvature}\n\\begin{figure}[tbhp]\n \\centering\n \\includegraphics{fig22.pdf}\n \\caption{(a) The maximal Berry curvature found in the $y<0$ half-plane for Floquet adiabatic state $\\ket{\\epsilon_{0}^{F}}$ as a function of $\\Delta\\phi$. Note that the Berry curvature becomes zero at $\\Delta\\phi=-\\pi,0,\\pi$ and reaches the largest magnitude around $\\Delta\\phi\\approx\\pm3\\pi\/8$. The map of the Berry curvature in the nuclear coordinate is plotted for phase differences (b) $\\Delta\\phi=-3\\pi\/8$ and (c) $\\Delta\\phi=+3\\pi\/8$. Note that for $\\Delta\\phi=\\pm3\\pi\/8$, the Berry curvatures are the same in magnitude, but opposite in sign, leading to opposite Berry magnetic forces. Note further that these Berry curvature plots have excluded the contribution from the trivial crossing at $x=0$ and $x=\\pm \\text{atanh}(0.2)$, which are zero in principle (when evaluated exactly).}\n \\label{fig: berrycurvature2d}\n\\end{figure}\nBefore running dynamics, we analyze the Berry curvature (in the $z$ direction) of the Floquet Hamiltonian as calculated by \\eqref{eq:berrycurvature}.\nFor this 2D model, the Berry curvature is antisymmetric with respect to the $x$ axis, see Fig.~\\ref{fig: berrycurvature2d} (b) and (c). In agreement with the theory presented above, the computed Berry curvatures for phase differences $+\\Delta\\phi$ and $-\\Delta\\phi$ have the same magnitude but opposite sign. \nFor a nuclear wavepacket moving in the $x-y$ plane, equal and opposite Berry curvatures are equivalent to equal and opposite effective magnetic fields, so that we expect the nuclear wavepackets will move in different directions for Hamiltonians specified by phase differences $\\pm\\Delta\\phi$.\n\nNote that the magnitude of the Berry curvature varies dramatically as a function of space. In order to best characterize how one phase difference $\\Delta\\phi$ determines the overall Berry curvature, in Fig.~\\ref{fig: berrycurvature2d}(a), we report the maximum (signed) Berry curvature sampled over the lower half-plane ($y<0$) as a function of $\\Delta\\phi$. We find the largest difference in the Berry magnetic force when comparing $\\Delta\\phi\\approx3\\pi\/8$ vs $\\Delta\\phi\\approx-3\\pi\/8$.\nAs a sidenote, the Berry curvature is indeed zero when $\\Delta\\phi=-\\pi,0,\\pi$---in agreement with the analytic theory discussed in Sec.~\\ref{sec:condition}. \n\n\n\n\\subsection{Distinct transmission and reflection probabilities as induced by opposite Berry forces}\nTo observe the consequence of a large Berry magnetic force, we compare the transmission and reflection probabilities of the wavepacket in the presence of the $\\omega$ and $2\\omega$ CW lasers with phase difference $\\Delta\\phi=-3\\pi\/8$ and $\\Delta\\phi=+3\\pi\/8$; see Fig~\\ref{fig: berrycurvature2d}(a). \nWe initialize an incoming wavepacket centered at $(x_{0}, y_{0}) = (-2, -6)$ with the initial momentum $(p_{0}^{x}, p_{0}^{y}) = (6.7, 9)$. \nWe choose the initial conditions so that both (i) the wavepacket will pass through the non-zero Berry curvature region (see Figs.~\\ref{fig: berrycurvature2d} (b) and (c)) and (ii) the wavepacket momentum will be slow enough so that the asymptotic wavepacket is not sensitive to the initial position of the wavepacket or the initial phase of the $\\omega$ and $2\\omega$ CW waves (see discussion in Ref~\\citenum{zeyu2020arobust}). \n\nFinally, let us analyze the nuclear dynamics. As shown in Fig.~\\ref{fig: wfnks}, after scattering, the asymptotic wavepackets for the two choices of $\\Delta\\phi$ are significantly different both in their spatial distributions (Figs.~\\ref{fig: wfnks}(a,b)) and in their momentum distributions (Figs.~\\ref{fig: wfnks}(c,d)). \nFirst, we find that, after scattering, the wavepacket on state 1 (red) moves in different directions as a function of $\\Delta \\phi$ ($x<0$, $p_x<0$ for $\\Delta\\phi=-3\\pi\/8$ and $x>0$, $p_x>0$ for $\\Delta\\phi=+3\\pi\/8$),which is demonstrable proof that the Berry magnetic force have a strong influence on turning and guiding nuclear dynamics. \nSecond, if we focus on the total transmission ($x>0$) and reflection ($x<0$) probabilities (after adding up the contributions from all three electronic states), we find that the bifurcation forward and backward in the x-direction is very different depending on $\\Delta \\phi$. In particular, for $\\Delta\\phi=-3\\pi\/8$, we find that $\\text{Prob}(x<0)=0.44$ and $\\text{Prob}(x>0)=0.56$; vice versa, for $\\Delta\\phi=+3\\pi\/8$, we find $\\text{Prob}(x<0)=0.939$, which implies almost complete reflection. \nAnd at the same time, the transition probabilities from state 0 to state 1 (red) and 2 (blue) are significantly different as well (see Table~\\ref{table1} and the color contours in Fig.~\\ref{fig: wfnks}). \nAltogether, these significant differences suggest that the Berry magnetic force effect should be able to promote or suppress a chemical reaction, which should indeed be easy to observe experimentally. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ lcc }\n & $\\Delta\\phi=-3\\pi\/8$ & $\\Delta\\phi=+3\\pi\/8$ \\\\\n \\hline\n $\\text{Prob}(0\\rightarrow0)$ & $0.440$ & $0.441$ \\\\ \n $\\text{Prob}(0\\rightarrow1)$ & $0.355$ & $0.061$ \\\\ \n $\\text{Prob}(0\\rightarrow2)$ & $0.205$ & $0.498$ \\\\ \n \\hline\n\\end{tabular}\n\\caption{\\label{table1} The transition probabilities for $\\Delta\\phi=\\pm3\\pi\/8$ as calculated in Fig.~\\ref{fig: wfnks}.}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics{fig54.pdf}\n\\caption{Visualization of the transmitting wavepackets in (a)(b) real space and (c)(d) momentum space, corresponding to different diabatic electronic states $\\ket{0}$ (Yellow), $\\ket{1}$ (Red), and $\\ket{2}$ (Blue). For all the panels, the dashed lines represent the trajectories for each states, the dotted black lines represent the $x = 0$ and $p_x=0$ line respectively.\nFor diabatic states $\\ket{0}$, the transmitting wavepackets for $\\Delta\\phi = -3\\pi\/8$ and $\\Delta\\phi = +3\\pi\/8$ are approximately the same. However, on diabatic state $1$ and $2$, the transmitting wavepackets have significant differences for $\\Delta\\phi = -3\\pi\/8$ and $\\Delta\\phi = +3\\pi\/8$ in spatial distribution and momentum distribution. \nMoreover, as qualitatively shown by the transparency of the contour colors, the asymptotic probabilities for opposite phase differences are very different. When we combine the probabilities on all states for $x<0$ (or equivalently $p_x$<0) vs $x>0$ ($p_{y}>0$), we find that for $\\Delta\\phi = -3\\pi\/8$, Prob($x<0$) = 0.440 and Prob($x>0$) = 0.560; for $\\Delta\\phi = +3\\pi\/8$, Prob($x<0$) = 0.939 and Prob($x>0$) = 0.061.\nThese differences can arise only due to the Berry magnetic force. \n}\n\\label{fig: wfnks}\n\\end{figure*}\n\n\n\\section{Discussion\\label{sec:Discussion}}\nWe have isolated one class of Hamiltonians whose dynamics clearly demonstrate large effects as caused by the presence of light-induced Berry forces. In order to realize such a Hamiltonian (or a similar Hamiltonian) within a realistic spectroscopic experiment, there are three major questions that we must now address. First, based on the theory in Sec.~\\ref{sec:condition}, the experiment must be carried out with molecules or materials that have three bright transitions, i.e. two absorption bands at frequencies $\\omega$ and $2\\omega$ and one interband transition. What molecules or materials should we choose to satisfy such a requirement?\nSecond, the intensity of the laser source must be strong enough such that the transition dipolar coupling of the system leads to transitions between diabatic dressed states; how much power must the laser produce in practice?\nThird, the experimental measurements must be sensitive to the nuclear dynamics in order to exhibit different signals induced by Berry magnetic force effects; what observables should be measured?\nWe will now address these practical questions in detail.\n\n\\subsection{Molecules and Materials}\nAs far as target molecules and materials, we can envision several possible candidates for the experiment proposed above:\n\\begin{enumerate}\n \\item Hybrid metal nanostructures: \n Cyltrimethylammonium bromide (CTAB) is widely used as a surfactant ligand in metallic nanoparticle (NP) synthesis and fortuitously has vibrational bands at $1500~\\text{cm}^{-1}$ and $3000~\\text{cm}^{-1}$\\cite{guivar2015preparation,ding2017plasmon}. Moreover for gold nanoparticles capped with CTAB (CTAB@AuNPs), we can expect very strong absorption for all transitions due to the coupling to plasmons. As such, illuminating a CTAB-coated hybrid nanostructure with IR lasers is one possibility for realizing the experiment above.\n \n \\item Photosynthetic complexes: \n As another example of bright molecules with a fortuitous energy spacing, many light-absorbing components within a photosynthetic bacteria, such as bacteriochlorophyll (BChl) and bacteriophytochromes, have separate absorption bands around $400~\\text{nm}$ (Soret band) and $800~\\text{nm}$ (the $Q_y$ band)\\cite{oren2011characterization, lenngren2018coordination}. In a heterogeneous environment, without symmetry, there is no reason to expect that the Soret band to $Q_{y}$ band should be forbidden.\n \n \\item Quantum dot(QD)--molecule complexes: \n Lastly, rather than relying on a fortuitous alignment of energies, another approach for generating bright transitions at frequencies $\\omega$ and $2\\omega$ is to match an adsorbate with a quantum dot of the optimal radius. After all, the electronic properties of a colloidal QD, especially the exciton energy and the transition dipole moment, can be tuned by changing the core size and capping ligands. As an example, oleate-capped colloidal PbSe QDs with a diameter of $6.5~\\text{nm}$ have an exciton energy around $5500-6000~\\text{cm}^{-1}$ while the oleate ligand itself has a vibrational mode that absorbs strongly at $2900~\\text{cm}^{-1}$ \\cite{abelson2020collective}. Thus, matching a ligand with a tunable QD is another attractive approach to generating three bright transitions: two at frequency $\\omega$ and one at frequency $2\\omega$.\n \n\\end{enumerate}\n\n\\subsection{Laser Source Intensity}\nWe now turn our attention to the intensity of the laser source as required to observe a Berry force effect in the proposed experiment.\nAs shown in Fig~\\ref{fig: wfnks}, when the initial diabatic state is 0, the dynamical difference of the nuclear wavepackets between the $\\Delta\\phi=\\pm3\\pi\/8$ cases is most significant on diabats 1 and 2.\nTherefore, in addition to having non-zero Berry curvature,\nanother key requirement for observing Berry force is that the light-induced diabatic coupling (characterized by $D$ in \\eqref{eq:diabaticcoupling}) must be strong enough to induce meaningful jumps between diabatic dressed states---for the model Hamiltonian above, we estimate that a lower bound $D>\\sqrt{p^x_0A\/2\\pi M}\\approx10^{-3}~\\text{a.u.} \\approx 0.02~\\text{eV}$ will yield a reasonable diabatic transition probability according to the Landau--Zener formula (and a smaller coupling may work as well). \nMore generally, given thermal motion at room temperature, a diabatic coupling on the order of $0.02~\\text{eV}$ should be quite sufficient for ensuring transitions between diabatic states.\nFor instance, within the Marcus model of electron transfer, the key Massey parameter that dictates the probability of a transition between diabats is $W\\approx 2\\pi H_{ab}^2\/\\hbar\\Omega\\sqrt{E_R k_B T}$ where $H_{ab}$ is the diabatic coupling, $\\Omega$ is the nuclear frequency of the reaction coordinate, and $E_R$ is the reorganization energy\\cite{massey1949collisions,tully2012perspective}. \nAt room temperature $k_B T=0.025~\\text{eV}$, a typical nuclear frequency might be $\\hbar\\Omega\\approx0.01~\\text{eV}$ and a typical reorganization energy is about $E_R\\approx1.0~\\text{eV}$. \nTherefore, one might estimate $H_{ab}>\\sqrt{0.01\\sqrt{0.025}\/2\\pi}\\approx0.016~\\text{eV}$ should imply $W>1$ and a large probability to switch between diabats.\n\nNow, for most photoexcitation experiments, the light-induced diabatic coupling is given by $D=\\mu E_L$, where ${\\mu}$ is the transition dipole moment, ${E}_L$ is the peak electric field strength, and the average laser power is $P_L=\\alpha_L\\frac{c\\epsilon_0}{2}|\\frac{D}{\\mu}|^2$ where $\\alpha_L$ is the focus area. \nGiven that the transition dipole moment of the above candidates ranges from $\\mu\\approx4.5~\\text{Debye}$ (Bchl molecules\\cite{knox2003dipole,oviedo2011transition}) to $\\mu\\approx1000~\\text{Debye}$ (colloidal QD\\cite{sabaeian2014investigation}) and assuming that the laser focus area is $0.003~\\text{mm}^2$ (with a beam waist radius of $30~\\mu m$), we can then estimate the necessary laser power that one would need to see a reasonable Berry force effect as somewhere between $1.0$ and $10^{4}$ Watts (see Table~\\ref{table2}).\nNote that these requirements are somewhat different from the conditions which have historically been applied to create LICIs in gas phase photodissociation experiments; for those experiments, one usually applies a laser pulse having a duration around $10-100~\\text{fs}$ and a peak intensity of the order of $10^{12}~\\text{W\/cm}^2$\\cite{corrales2014control,halasz2015direct,kim2012control}). By contrast, for the model proposed above with Floquet theory, the experiments will require a CW laser field (or perhaps a long plateau pulse on the order of ns) with much lower power. These requirements should be realizable given today's laser sources. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ clr } \n parameters & & values \\\\\n \\hline\n $\\mu$&[$\\text{Debye}$] & $4.5-1000$ \\\\ \n $E_L$&[$\\text{V\/m}$] & $1.1\\times10^8-5.1\\times10^5$ \\\\ \n $P_L$&[$\\text{W}$] & $5.1\\times10^{4}-1.0\\times10^{0}$ \\\\ \n \\hline\n\\end{tabular}\n\\caption{\\label{table2}An estimate of the electric field strength and laser power necessary for observing a reasonably strong Berry force effect given a diabatic coupling between light-dressed Floquet states to be $0.02~\\text{eV}$.\nA smaller diabatic coupling (i.e. a lower electric field strength and laser power) would likely lead to a detectable Berry magnetic force, but the effect might not be strong. }\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\subsection{Physical Observables}\nAs far as experimental measurements are concerned, we must emphasize that any experimental difference between $H_F$ and $H_F^*$ must reflect a Berry force effect. Thus, one can imagine several different experimental approaches for isolating Berry phase in practice.\n\\begin{enumerate}\n \\item Velocity map imaging (VMI): \n For a gas phase photodissociation reaction, VMI is one technique for quantifying the kinetic energy distribution of the nuclear fragments that are generated\\cite{corrales2014control}. Thus, if a given photodissociation channel can be strongly activated by exposure to two frequencies $\\omega$ and $2\\omega$ within the lifetime of a molecular beam experiment, VMI should be able to directly quantify fragment momenta as a function of $\\Delta\\phi$ and in so doing isolate a Berry curvature effect.\n \n \\item Fluorescence emission spectrum: \n Next, steady-state fluorescence spectra do reflect some degree of excited state dynamics. After all, when exposed to continuous illumination, a molecule or material can go through several transformations before emitting a photon. In this regard, if a molecule or material relaxes differently depending on the Berry curvature, one should expect to measure different emission spectra. In particular, very often excited state dynamics can be probed by measuring the anisotropy of the emission spectra, and this represents another experimental measurement for probing Berry phase effects.\n \n \\item Photo-induced current measurement: \n When a QD is placed in a nanojunction in the presence of a light field, it is well known that the shape of the current-voltage (I-V) curve is sensitive to the illumination (usually using a single-frequency laser that leads to photo-assisted electron tunneling processes\\cite{meyer2007photon,braakman2013photon}. Although the experiments are very difficult, in principle, one can imagine that the photon-induced current can pass through a QD-ligand nanojunction with three bright transitions in the presence of two light fields of frequencies $\\omega$ and $2\\omega$. Does the Berry magnetic force lead to a strong modified I-V curve?\n\\end{enumerate}\n\n\n\\section{Conclusions}\nIn conclusion, physical chemists today have the necessary laser power and non-linear optics such that they should be able to determine whether or not Berry force can have a meaningful effect on chemical dynamics. Here we have proposed the simplest set of experiments to make such a determination.\nThese experiments require a material with three bright transitions $\\ket{0} \\xrightarrow[]{\\omega} \\ket{1}$, $\\ket{1} \\xrightarrow[]{\\omega} \\ket{2}$, $\\ket{0} \\xrightarrow[]{2\\omega} \\ket{2}$ as well as two strong CW at frequencies $\\omega$ and $2\\omega$. In the presence of these two laser fields, the key control parameter is the difference in phase $\\Delta\\phi$. If identical experiments are carried out for $\\Delta\\phi$ vs $-\\Delta\\phi$, any detected difference can and must be attributed to Berry force alone. If chemical spectroscopists can indeed isolate such differences and connect to the Berry theory of geometric phase,\nsuch a connection will not only help merging chemical physics and quantum optics, it may also lead to a new understanding of quantum control and photo-chemical catalysis.\n\n\\section*{acknowlegment}\n{This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0019397. This research also used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231. We thank Abraham Nitzan, Jessica Anna, Qi Ou, and Shaojie Liu for very helpful discussions.}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nEinstein's equations\\index{Einstein equations} \nhave, for much of the history of general\nrelativity, been explored very fruitfully in terms of their\nconcise and elegant statements characterizing the geometry\nof four dimensional pseudo-riemannian geometries. Such\ngeometries depict possible physical spacetimes containing\nonly ``the gravitational field itself.'' The variety\nand properties of these ``empty'' spacetimes is truly\nastonishing. Quasi-local geometric entities such as trapped \nsurfaces and event horizons have become familiar. \nIt is now firmly established that large-scale\ntopological and geometrical features\nof spacetime are, indeed, subjects\nof physical inquiry. The nature and\ndistribution of ``matter'' at stellar scales and upward\nhas also brought particle physics and hydrodynamics to the\nfore.\n\nDuring these years, however, a steady development of the\n``space-plus-time'' or $3+1$ view of spacetime geometry\nhas also occured. Here one views general relativity\nas ``geometrodynamics''\n\\index{geometrodynamics} in the parlance of John\nWheeler~\\cite{Wheeler}. The emphasis, in the canonical \nor Hamiltonian\nexplication of geometrodynamics\\index{geometrodynamics}\n\\index{Hamiltonian dynamics} given by Arnowitt,\nDeser, and Misner (``ADM'')~\\cite{ADMDir} and by \nDirac~\\cite{Dir58,Dir59},\nis on the evolving intrinsic and extrinsic geometry \nof spacelike hypersurfaces which determine, by knowledge\nof the appropriate initial data and by classical \ncausality, the spacetime ``ahead'' (and ``behind''),\nif spacetime is globally hyperbolic, an assumption\nwe adopt throughout.\n\nUnderlying and preceding geometrodynamics\n\\index{geometrodynamics} and Hamiltonian\n\\index{Hamiltonian dynamics}\nmethods, however, was the basic realization that four \nof the ten Einstein vacuum equations\\index{Einstein equations!vacuum}\nare nonlinear\nconstraints\\index{constraint!nonlinear} \non the initial Cauchy data\\index{Cauchy data}, \nwhich play\nsuch a decisive role in defining the later canonical \nformalism~\\cite{Lic}. The Cauchy problem\\index{Cauchy problem}, \nconstraints plus\nevolution, was shown to be well-posed in the modern\nsense of nonlinear partial differential \nequations~\\cite{FB52,CB56,Lic44,CBY80}. \nThis train of progress was\nmarked by early work of Darmois~and Lichnerowicz~\\cite{Lic44},\nand brought to\ndefinitive development by one of us~\\cite{FB52,CB56}.\n\nAt this writing, with accurate three-dimensional simulations\nusing the full Einstein equations, \n\\index{Einstein equations} with and without the\npresence of stress-energy sources, becoming essential for\nrealistic studies of gravity waves, high energy astrophysics,\nand early cosmology, studies of Einstein's equations\nof evolution in $3+1$ form have blossomed~[11-28]\n(an incomplete sample --- see also~\\cite{AY99}).\nHyperbolic forms, especially first-order symmetrizable forms \npossessing only physically causal directions of propagation,\nhave undergone very significant development in the past \nfew years.\n\nSpecifically, in this chapter we describe in detail\nseveral explicitly causal hyperbolic formulations of\nEinstein's dynamical $3+1$ equations\n\\index{Einstein equations!hyperbolic} by following a\npath that can be viewed as lighted by the\n``slicing function,''\n$\\alpha$ --- the quantity that relates the lapse $N$\\index{lapse}\nto the determinant of the spatial metric $\\bar{g}$\n\\index{metric!spatial}\nthrough $N = \\bar{g}^{1\/2} \\alpha$. This representation of the\nlapse function was presented in~\\cite{CBR83}.\nThe slicing function\nallows us to demonstrate explicitly that no foliation of spacetime by\nspatial time-slices can be an obstacle to the causal\nhyperbolic forms of the dynamical Einstein equations. \n\\index{Einstein equations!hyperbolic} The slicing\nfunction plays an essential role (1) in a more precise form of \nthe canonical action principle\\index{action principle!canonical} \nand canonical dynamics for\ngravity, (2) leads to a recasting of the Bianchi identities\n\\label{Bianchi identities}\n$\\nabla_\\beta G^\\beta\\mathstrut_\\alpha \\equiv 0$ as a \nwell-posed system for the evolution of the gravitational\nconstraints\\index{constraint!vacuum} in vacuum and also (3) of $\\nabla_\\beta\nT^\\beta\\mathstrut_\\alpha \\equiv 0$ as a well-posed system for\nevolution of the energy and momentum components of the stress\ntensor in the presence of matter, (4) in an explicit display\nof four hyperbolic formulations of Einstein's equations\n\\index{Einstein equations!hyperbolic} with only\nphysical characteristics, and (5) even in providing guidance to a new\nelliptic ``conformal thin sandwich'' form of the initial value\nconstraints\\index{constraint!initial value!conformal thin sandwich}. \n\nWe recall that the proof of the existence of a causal evolution in local\nSobolev spaces of $\\bar{g}$ and its extrinsic curvature (second\nfundamental tensor) $K$ into an Einsteinian spacetime does not result\ndirectly from the equations giving the time derivatives of $\\bar{g}$\nand $K$ in terms of space derivatives of these quantities in a\nstraightforward $3+1$ decomposition of the Ricci tensor of the\nspacetime metric, which contains also the lapse and shift\ncharacterizing the time lines. These equations do not appear as a\nhyperbolic system for arbitrary lapse and shift, in spite of the fact\nthat their characteristics are only the light cone and the time \naxis~\\cite{Fri96}.\n\nWe now turn to notational matters, conventions, and to the $3+1$\ndecomposition of the Riemann and Ricci tensors. We assume here and\nthroughout the sequel that the spacetime $V=M\\times\\mathbb{R}$ is\nendowed with a metric $g$ of signature $\\left( -, +, +, + \\right)$ and\nthat the time slices are spacelike, that is, have signature\n$\\left( +, +, + \\right)$. These assumptions are not restrictive for\nglobally hyperbolic\\index{globally hyperbolic} (pseudo-riemannian)\nspacetimes. \n\nWe choose on $V$ a moving coframe\\index{moving coframe} such that the\ndual vector frame has a time axis orthogonal to the slices $M_t$ while\nthe space axes are tangent to them. Specifically, we set\n\\begin{eqnarray}\n\\theta^0 &=& d t \\; , \\nonumber\\\\\n\\theta^i &=& d x^i + \\beta^i dt \\; ,\n\\label{basis}\n\\end{eqnarray}\nwith $t\\in\\mathbb{R}$ and $x^i$, $i=1,2,3$ local coordinates on\n$M$. The Pfaff or convective derivatives $\\partial_\\alpha$ with\nrespect to $\\theta^\\alpha$ are\n\\begin{eqnarray}\n\\partial_0 &\\equiv& \\frac{\\partial}{\\partial t} - \\beta^i \\partial_i\n \\nonumber\\\\\n\\partial_i &\\equiv& \\frac{\\partial}{\\partial x^i}\n\\label{co-basis}\n\\end{eqnarray}\nIn this coframe, the metric $g$ reads\n\\begin{equation}\nds^2 = g_{\\alpha \\beta} \\theta^\\alpha \\theta^\\beta\n\\equiv -N^2 (\\theta^0)^2 + g_{i j} \\theta^i \\theta^j \\; .\n\\end{equation}\nThe $t$-dependent scalar $N$ and space vector $\\beta$ are\ncalled the lapse function and shift vector of the slicing.\nThese quantities were explicitly identified in~\\cite{CB56}\nand play prominent roles in all subsequent $3+1$ formulations.\nAny spacetime tensor decomposes into sets of time dependent\nspace tensors by projections on the tangent space or the normal to\n$M_t$. \n\nWe define for any $t$-dependent space tensor $T$ another such tensor\nof the same type, $\\bar{\\partial}_0 T$, by setting\n\\begin{equation}\n\\bar{\\partial}_0 \\equiv \\frac{\\partial}{\\partial t} - \\pounds_\\beta \\; ,\n\\end{equation}\nwhere $\\pounds_\\beta$ is the Lie derivative on $M_t$ with \nrespect to $\\beta$.\n\nNotice that in our foliation-adapted basis\\index{foliation-adapted \nbasis}~(\\ref{basis}) and~(\\ref{co-basis}), that if $g$ \ndenotes the spacetime metric\nand $\\bar{g}$ the space metric, then we have \n($g_{0 i} = g^{0 i} = 0$ in our frames):\n\\begin{equation}\ng_{i j} = \\bar{g}_{i j} \\; ; \\quad g^{i j} = \\bar{g}^{i j} \\; .\n\\end{equation}\n(Greek indices range $\\{0,1,2,3\\}$ while latin ones are\npurely spatial.) Hence, {\\em no overbars will be used to denote\ncomponents of the spatial metric}. On the other hand, for\nthe determinants, we have $(-\\det g) = N^2 (\\det \\bar{g})$.\nTherefore we shall use overbars on spatial metric determinants;\nfor example $\\bar{g}^{1\/2} \\equiv (\\det \\bar{g})^{1\/2}$.\n\nLikewise, to distinguish the purely spatial components of the\nspacetime Ricci tensor (say), we shall write $R_{i j}(g)$,\nwhile for the space Ricci tensor we shall write $R_{i j}(\\bar{g})$.\nIn general, of course, $R_{i j}(g) \\neq R_{i j}(\\bar{g})$. The\nLevi-Civita connection of $g$ is denoted by $\\nabla$ and that\nof $\\bar{g}$ by $\\bar\\nabla$.\n\nWith the convention\n\\begin{equation}\n\\nabla_\\alpha \\sigma_\\beta \\equiv \\partial_\\alpha \\sigma_\\beta\n - \\sigma_\\rho \\gamma^\\rho\\mathstrut_{\\beta \\alpha} \\; ,\n\\end{equation}\nand the definitions\n\\begin{eqnarray}\n\\gamma^\\alpha\\mathstrut_{\\beta \\gamma} &=& \n \\Gamma^\\alpha\\mathstrut_{\\beta \\gamma} + g^{\\alpha \\delta}\n C^\\varepsilon\\mathstrut_{\\delta (\\beta} g_{\\gamma) \\varepsilon}\n - \\frac{1}{2} C^\\alpha\\mathstrut_{\\beta \\gamma} \\; , \\\\\nd\\theta^\\alpha &=& -\\frac{1}{2} C^\\alpha\\mathstrut_{\\beta \\gamma}\n \\theta^\\beta \\wedge \\theta^\\gamma \\; ,\n\\end{eqnarray}\nwe have for the connection coefficients ($\\Gamma$ denotes an \nordinary Christoffel symbol)\n\\begin{equation}\n\\gamma^i\\mathstrut_{j k} = \\Gamma^i\\mathstrut_{j k}(g)\n = \\Gamma^i\\mathstrut_{j k}(\\bar{g}) \\\\\n\\end{equation}\n\\begin{equation}\n\\gamma^i\\mathstrut_{0 k} = - N K^i\\mathstrut_k \\;, \\quad\n \\gamma^i\\mathstrut_{j 0} = - N K^i\\mathstrut_k \n + \\partial_j \\beta^i \\;, \\quad\n \\gamma^0\\mathstrut_{i j} = - N^{-1} K_{i j}\n\\end{equation}\n\\begin{equation}\n\\gamma^i\\mathstrut_{0 0} = N \\partial^i N \\; , \\quad\n \\gamma^0\\mathstrut_{0 i} = \\gamma^0\\mathstrut_{i 0}\n = \\partial_i \\log N \\; , \\quad\n \\gamma^0\\mathstrut_{0 0} = \\partial_0 \\log N \\; .\n\\end{equation}\nObserve that if $\\alpha$ is a space scalar of weight $-1$,\nwe have\n\\begin{eqnarray}\n\\bar\\nabla_i \\alpha &=& \\partial_i \\alpha + \\alpha \n \\Gamma^k\\mathstrut_{k i}(g) \\,=\\, \\partial_i \\alpha\n + \\alpha \\partial_i \\log \\bar{g}^{1\/2} \\; , \\\\\n\\pounds_\\beta \\alpha &=& \\beta^i \\bar{\\nabla}_i \\alpha\n + \\alpha \\bar{\\nabla}_i \\beta^i \\; .\n\\end{eqnarray}\n\nThe Riemann tensor is fixed by\n\\begin{equation}\n\\left( \\nabla_\\alpha \\nabla_\\beta - \\nabla_\\beta \\nabla_\\alpha \\right)\n V^\\gamma = V^\\delta R^\\gamma\\mathstrut_{\\delta \\alpha \\beta}\n\\end{equation}\nwhile the Ricci tensor is $R_{\\delta \\beta} \\equiv\nR^\\gamma\\mathstrut_{\\delta \\gamma \\beta}$.\n\nThe $3+1$ decompositions of the Riemann and Ricci tensors are\n\\begin{eqnarray}\nR_{i j k l}(g) &=& R_{i j k l}(\\bar{g}) + 2 K_{i [k} K_{l] j} \\; ,\\\\\nR_{0 i j k}(g) &=& 2 N \\bar{\\nabla}_{[j} K_{k] i} \\; ,\\\\\nR_{0 i 0 j}(g) &=& N \\left( \\bar{\\partial}_0 K_{i j} + N K_{i k} K^k\\mathstrut_j\n + \\bar{\\nabla}_i \\partial_j N \\right) \\; .\n\\end{eqnarray}\nOne can then obtain for the Ricci tensor\n\\begin{eqnarray}\nR_{i j}(g) &=& R_{i j}(\\bar{g}) - N^{-1} \\bar{\\partial}_0 K_{i j} + K K_{i j}\n - 2 K_{i k} K^k\\mathstrut_j - N^{-1} \\bar{\\nabla}_i \\partial_j N\n \\; ,\\\\\nR_{0 j}(g) &=& N \\left( \\partial_j K - \\bar{\\nabla}_h K^h\\mathstrut_j\n \\right) \\; ,\\\\\nR_{0 0}(g) &=& N \\left( \\partial_0 K - N K_{i j} K^{i j} +\n \\triangle_{\\bar{g}} N \\right) \\; ,\n\\end{eqnarray}\nwhere $K\\equiv K^i\\mathstrut_i$ and \n$\\triangle_{\\bar{g}} \\equiv g^{i j} \\bar{\\nabla}_i \\bar{\\nabla}_j$.\nFinally, we note\n\\begin{equation}\nG^0\\mathstrut_0 = \\frac{1}{2} \\left( K_{i j} K^{i j} - K^2 -\n R(\\bar{g}) \\right) \\; .\n\\end{equation}\n\n\\section{Every Time Slicing Is ``Harmonic''}\n\\label{Sec:EverySlicing}\nThe standard statement of the harmonic time-slicing\n\\index{harmonic time-slicing}\ncondition is, that on a $t=\\mbox{const.}$ time slice,\n$\\bar{\\partial}_0 [(-g)^{1\/2} g^{00}]=0$. (This \nis equivalent, in a coordinate basis, \nto $\\partial_\\mu [(-g)^{1\/2} g^{\\mu t}]=0$.) Friedrich observed\nin~\\cite{Fri85} that the right hand side of these equations could be a \ngiven function of $(t , x^i)$\\footnote{Just as in\nelectrodynamics, $\\nabla^\\mu A_\\mu = 0 \\rightarrow \n\\nabla^\\mu {A_\\mu}^{\\prime} = \\ell(t ,x) \\neq 0$ is\nperfectly acceptable as a ``Lorentz gauge'' if $\\ell$ \nis known.}. (See also~\\cite{CBY95}.) Therefore, the \nstandard harmonic condition expressed in \n$3+1$ form, $\\bar{\\partial}_0 N \n+ N^2K = 0$, can be written as\na generalized ``harmonic'' condition\n\\index{generalized harmonic condition}\n\\begin{equation}\n\\bar{\\partial}_0 N + N^2 K = Nf \\; ,\n\\label{GeneralHarmonic}\n\\end{equation}\nwhere $f(t,x)$ is a known function. \nSpecifically, introduce $\\alpha(x,t)$ such that \n$\\bar{\\partial}_0 \\log \\alpha = f$, then~(\\ref{GeneralHarmonic})\nbecomes\n\\begin{equation}\n\\bar{\\partial}_0 N + N^2 K = N \\bar{\\partial}_0 \\log \\alpha \\; ,\n\\label{SpecialHarmonic}\n\\end{equation}\nfrom which the identity\n\\begin{equation}\n\\bar{\\partial}_0 \\log \\bar{g}^{1\/2} = - N K \\; ,\n\\end{equation}\nallows us to see that\n\\begin{equation}\nN = \\bar{g}^{1\/2} \\alpha \\; .\n\\label{slicingfunction}\n\\end{equation}\nWe shall call $\\alpha(x,t)$ the ``slicing function;''\nit is a freely given scalar density of weight $-1$.\n\nIt is clear that any $N>0$ on a given time slice $t=t_0$\ncan be written in the form $N_{t_0}= \\bar{g}^{1\/2}_{t_0}\n\\alpha (t_0, x)$ for some $\\alpha > 0$ provided that \n${g}_{ij} (t_0,x)$ is a proper riemannian metric.\n\\index{metric!riemannian} Introducing \n``harmonic'' time-slicing is thus a simple matter. It is not,\nhowever, known at present how to construct a specific long-time\nfoliation from general rules telling how to specify $\\alpha (t,x)$.\nHowever, many foliations can be constructed in a ``step-by step''\nfashion (numerical time steps) provided certain obvious\nconditions are met. For example, an elliptic condition on $N$\ncan determine $\\alpha(t)$ on a sequence of time slices\nif the condition does not couple to variables that disturb the\ncharacteristic directions of the hyperbolic equations.\n(The same is true for the shift vector $\\beta^i$.) \nAlternatively, we can try educated guesses for $\\alpha(t,x)$.\n\nAs it stands,~(\\ref{SpecialHarmonic}) is clearly a speed zero\n(with respect to $\\partial_0$) hyperbolic equation. However,\nthis equation and Einstein's equations\\index{Einstein equations} \nlead to a \nsecond order equation in space and time that propagates $N$\nalong the light cone. This result brings\ninto sharp relief the congruence of~(\\ref{SpecialHarmonic}) with\nthe propagation on the light cone of other variables.\n\nThe trace of $R_{i j}(g)$ gives an equation for $\\bar{\\partial}_0 K$~\\cite{CBY95}\n\\begin{equation}\n\\bar{\\partial}_0 K = -\\triangle_{\\bar{g}} N + \\left[ R(\\bar{g}) + K^2 - \n R^k\\mathstrut_k(g) \\right] N \\; ,\n\\label{bpzK}\n\\end{equation}\nwhere $\\triangle_{\\bar{g}}$ in (\\ref{bpzK}) denotes the Laplacian\n$g^{ij} \\bar{\\nabla}_i \\bar{\\nabla}_j$. Taking the time\nderivative of~(\\ref{SpecialHarmonic}) and eliminating\n$\\bar{\\partial}_0 K$ with~(\\ref{bpzK})\nshows that $N$ obeys the non-linear wave equation\n\\begin{equation}\n\\bar{\\Box\\mbox{\\small\\mathstrut}}_g N + R^k\\mathstrut_k(g) N - R(\\bar{g})N - N \\bar{\\partial}_0 \\log \\alpha \n\t+ (\\bar{\\partial}_0^2 \\log \\alpha)\nN^{-1}=0 \\;,\n\\label{NLWaveEQ}\n\\end{equation}\nwhere we wrote our wave operator or ``d'Alembertian'' as $\\bar{\\Box\\mbox{\\small\\mathstrut}}_g\n= -(N^{-1}\\bar{\\partial}_0)^2 + \\triangle_{\\bar{g}}$. The characteristic cone\nof $\\bar{\\Box\\mbox{\\small\\mathstrut}}_{g}$ is clearly the physical light cone $(c=1)$.\nThe equation~(\\ref{NLWaveEQ}) {\\em per se}\nwill not be used explicitly in the sequel.\n\nSubstitutions of the form $N_\\lambda = \\bar{g}^{\\lambda\/2}\n\\alpha_{\\lambda} \\; \n(\\lambda > 0)$ have also been considered~\\cite{FrR96}. \nHowever, after working out \nthe wave equation analogous to~(\\ref{NLWaveEQ})\nthat $N_\\lambda$ obeys, one finds\nthat the local proper propagation speed of \n$N_\\lambda$ is $\\sqrt{\\lambda}$. This behavior may or may not\nspoil the propagation of system variables other than\n$N$, but if $\\lambda \\neq 1$ and the system is hyperbolic, one will\nalways find that the characteristic directions of the system will\nnot all be physical ones. That is, in vacuum gravity, there will \nbe some variables that propagate neither on the light cone (speed = 1)\nnor along the axis parallel to $\\bar{\\partial}_0$ (speed $= 0$) which is\northogonal to $t =$ const. The variables not propagating in physical\ndirections are gauge variables and one will not have physical\ncriteria for their boundary values on characteristic surfaces.\nOn the other hand, with $\\lambda = 1$, one has fulfilled \na necessary condition that physical and gauge variables propagate \ntogether in the same directions.\n\nIn the following sections, whenever we consider hyperbolic systems,\nwe will focus on first-order symmetric (or symmetrizable) \nhyperbolic (``FOSH'')\\index{FOSH}\n\\index{first-order symmetric hyperbolic}\nequations possessing {\\em only} physical\ncharacteristic directions\\index{physical characteristic directions}. \nWe understand ``FOSH''\\index{FOSH}\n\\index{first-order symmetric hyperbolic} in\n{\\em this restricted physical sense only} in this paper, and likewise for\nother uses of the term ``hyperbolic.''\n\n\\section{Canonical Action and Equations of Motion}\n\nChoice of the slicing function \\index{slicing function}\n$\\alpha$ in~(\\ref{slicingfunction})\nis arbitrary ($\\alpha > 0$). That $\\alpha$ is freely chosen while $N$\nmust satisfy an equation of motion~(\\ref{SpecialHarmonic}) \nsuggests that it should be regarded as the\nundetermined\\footnote{The multipliers associated with gauge\nfreedom or, as here, with spacetime coordinate freedom, can be\nfreely chosen because they are not determined by physical conditions.\nHence, they are not true ``Lagrange multipliers.''}\nmultiplier in the canonical action principle \n\\index{action principle!canonical} of\nArnowitt, Deser, and Misner (``ADM'')~\\cite{ADMDir}.\nHere we follow~\\cite{AY98}.\n\nTo draw some lessons for the canonical formalism, let\nus first express the $3+1$ evolution equations in \ntheir standard geometrical form (see: with zero shift~\\cite{Lic44},\narbitrary lapse\\index{lapse} and shift~\\cite{CB56},\nspacetime perspective~\\cite{Yor79}):\n\\begin{equation}\n\\dot{g}_{i j} \\equiv - 2 N K_{i j} \\; ,\n\\label{gdot}\n\\end{equation}\n\\begin{equation}\n\\dot{K}_{i j} \\equiv N \\left( -R_{i j}(g) + {R}_{i j}(\\bar{g})\n\t+ N K_{i j} - K_{i k} K^k_j \n\t- N^{-1} \\bar{\\nabla}_i \\partial_j N \\right) \\; ,\n\\label{Kdot}\n\\end{equation}\nwhere $\\dot{(\\;)} \\equiv \\bar{\\partial}_0(\\; )$.\n\nA brief look at (\\ref{Kdot}) shows that forming the combination \n${\\cal R}_{i j} = R_{i j}(g) - g_{i j} R^k\\mathstrut_k(g)$\nleads to an equation of motion for the\nADM canonical momentum\n\\begin{equation}\n\\pi^{i j} = \\bar{g}^{1\/2} \\left( K g^{i j} - K^{i j} \\right)\n\\end{equation}\nthat contains no constraints.\\index{constraint} \n(In this section we choose units\nin which $16\\pi G = c=1$.) Indeed, using\n(\\ref{gdot}) and (\\ref{Kdot}), we obtain the\nidentity\n\\begin{eqnarray}\n\\dot{\\pi}^{i j} &\\equiv& N \\bar{g}^{1\/2} \n\t\\left( R(\\bar{g}) g^{i j} - R^{i j}(\\bar{g}) \\right)\n\t- N \\bar{g}^{-1\/2} \\left( 2 \\pi^{i k} \\pi^j\\mathstrut_k \n\t- \\pi \\pi^{i j} \\right) \\nonumber\\\\\n\t& & + \\bar{g}^{1\/2} \\left( \\bar{\\nabla}^i \\bar{\\nabla}^j N\n\t- g^{i j} \\bar{\\nabla}_k \\bar{\\nabla}^k N \\right)\n\t+ N \\bar{g}^{1\/2} \\left[ {\\cal R}^{i j} \\right] \\; .\n\\label{pidot}\n\\end{eqnarray}\n{}From the identity $\\dot{g}_{i j} = -2 N K_{i j}$, we have\n\\begin{equation}\n\\dot{g}_{i j} \\equiv N \\bar{g}^{-1\/2} \n\t\\left( 2 \\pi_{i j} - \\pi g_{i j} \\right) \\; .\n\\label{gdot2}\n\\end{equation}\n\nWe now come to a crucial observation.\nWere the canonical equation for $\\dot{\\pi}^{i j}$ to be \ndictated by vanishing of the spatial part of the\nEinstein tensor, $G^{i j}(g) = 0$, as it is in the conventional ADM \nanalysis~\\cite{ADMDir}, then the identity\n\\begin{equation}\nG_{i j}(g) + g_{i j} G^0\\mathstrut_0 (g) \\equiv R_{i j}(g)\n\t- g_{i j} R^k\\mathstrut_k(g) \\equiv {\\cal R}_{ij}\n\\label{calR}\n\\end{equation}\nshows that a Hamiltonian constraint\\index{constraint!Hamiltonian} \nterm\n$\\sim\\bar{g}^{1\/2} G^0\\mathstrut_0$\nremains in the $\\dot{\\pi}^{i j}$ equation~(\\ref{pidot}).\nThis would mean that the validity of the $\\dot{\\pi}^{i j}$\nequation would be restricted to the subspace on which the \nHamiltonian constraint\n\\index{constraint!Hamiltonian} is satisfied (i.e., vanishes).\n\nThough the ADM derivation of the $\\dot{\\pi}^{i j}$\nequation, found by varying $g_{i j}$ in their canonical\naction\\index{action} ($\\beta^i$ is the shift vector)\n\\begin{equation}\nS \\left[ g, \\pi ; N, \\beta \\right) \n\t= \\int d^4x \\left( \\pi^{i j} \\dot{g}_{i j}\n\t- N {\\cal H} \\right) \\; ,\n\\end{equation}\nwith $N(t,x) \\; , \\; \\beta^i(t,x)$ and $\\pi^{ij}$ held fixed,\nis of course perfectly correct, another point\nof view is possible. [We are ignoring boundary terms,\na subject not of interest here, and we note that the\nmomentum constraint term \\index{constraint!momentum}\n$-\\beta^i {\\cal H}_i$ \n(${\\cal H}_i = g^{1\/2} {\\cal C}_i$, ${\\cal C}_i = 2 N R^0\\mathstrut_i$) \nis contained in\n$\\pi^{i j} \\dot{g}_{i j}$ \n($\\dot{\\left( \\; \\right)} \\equiv \\bar{\\partial}_0 $)\nupon integration by parts.]\n(The slicing density\n\\index{slicing function} $\\alpha$ has also been used prominently\nin the action\\index{action} by Teitelboim~\\cite{Tei82},\nwho simply set $\\alpha = 1$ ($N=\\bar{g}^{1\/2}$), and by \nAshtekar~\\cite{Ash88,Ash87} for other purposes.)\n\nWe have explained that $\\alpha$ can be regarded as a free \nundetermined multiplier while $N$ is a\n{\\em dynamical variable} (a conclusion also reached by\nAshtekar for other reasons~\\cite{Ash88,Ash87}) \nthat determines\nthe proper time $N \\delta t$ between slices $t=t^\\prime$\nand $t=t^\\prime+\\delta t$. $N$ is determined from \n$\\alpha(t,x)$ and $\\bar{g}^{1\/2}$ found by solving the\ninitial value constraint equations\\index{constraint!initial value}.\n(See the treatment of the constraints in the final section\nof this article and in~\\cite{Yor79,Yor73,OMY74a,CBY80}.)\nMotivated by this viewpoint,\nwe alter the undetermined multiplier $N$ in the ADM\naction principle\\index{action principle!ADM} \nto $\\alpha$, where the Hamiltonian\ndensity $\\tilde{\\cal H}$ is \n(with ${\\cal H} \\equiv 2 \\bar{g}^{1\/2} G^0\\mathstrut_0(g)$\nbeing the ADM Hamiltonian density of weight $+1$)\n\\begin{equation}\n\\tilde{\\cal H} \\equiv \\bar{g}^{1\/2} {\\cal H} = \n\t\\pi_{i j} \\pi^{i j} - \\frac{1}{2} \\pi^2 - \\bar{g} \n\tR(\\bar{g}) \\; ,\n\\end{equation}\na scalar density of weight $+2$ and \na rational function of the metric. The action\\index{action} \nbecomes\n\\begin{equation}\nS\\left[\\bar{g}, \\pi; \\alpha, \\beta \\right) =\n\t\\int d^4 x \\left( \\pi^{i j} \\dot{g}_{i j}\n\t- \\alpha \\tilde{\\cal H} \\right) \\; .\n\\label{action}\n\\end{equation}\nThe modified action {\\em principle}\n\\index{action principle!modified} for the canonical\nequations that we propose in~(\\ref{action}) is to vary $\\pi^{i j}$ and\n$g_{i j}$, with $\\alpha(t,x)$ and $\\beta^i(t,x)$ as\nfixed undetermined multipliers. From\n\\begin{eqnarray}\n\\delta \\tilde{\\cal H} &=& \\left( 2 \\pi_{i j} - g_{i j} \\pi \\right)\n\t\\delta \\pi^{i j} \n\t+ \\left( 2 \\pi^{i k} \\pi^j\\mathstrut_k - \\pi \\pi^{i j} \n\t+ \\bar{g} R^{i j}(\\bar{g}) - \\bar{g} g^{i j} R(\\bar{g}) \\right) \n\t\\delta g_{i j} \t\\nonumber\\\\\n\t& & - \\bar{g} \\left( \\bar{\\nabla}^i \\bar{\\nabla}^j \\delta g_{i j}\n\t- g^{i j} \\bar{\\nabla}_k \\bar{\\nabla}^k \\delta g_{i j} \\right) \\; ,\n\\end{eqnarray}\nwe obtain the canonical equations\n\\begin{eqnarray}\n\\dot{g}_{i j} = \\alpha \\frac{\\delta \\tilde{\\cal H}}{\\delta \\pi^{i j}}\n\t&=& \\alpha \\left( 2 \\pi_{i j} - \\pi g_{i j}\\right)\n\t\\equiv -2 N K_{i j} \\; , \\\\\n\\label{Canonicalgdot}\n\\dot{\\pi}^{i j} = -\\alpha \\frac{\\delta \\tilde{\\cal H}}{\\delta g_{i j}}\n\t&=& - \\alpha \\bar{g} \\left( R^{i j}(\\bar{g}) - R(\\bar{g}) \n\tg^{i j} \\right )\n\t- \\alpha \\left( 2 \\pi^{i k} \\pi^j\\mathstrut_k - \\pi \\pi^{i j} \\right)\n\t\\nonumber\\\\\n\t& & + \\bar{g} \\left( \\bar{\\nabla}^i \\bar{\\nabla}^j \\alpha - \n\t\\bar{g}^{i j} \\bar{\\nabla}_k \\bar{\\nabla}^k \\alpha \\right) \\; .\n\\label{Canonicalpidot}\n\\end{eqnarray}\nEquation (\\ref{Canonicalpidot}) for $\\dot{\\pi}^{i j}$\nis the identity (\\ref{pidot}) with ${\\cal R}^{i j}=0$,\nwhich is equivalent to $R_{i j}(g)=0$. Thus, (\\ref{Canonicalpidot}) is a \n``strong'' equation unlike its ADM counterpart, which requires\nin addition the imposition of a constraint\\index{constraint}: \n${\\cal H}=0$.\n\nIn the present formulation, the canonical equations of \nmotion hold everywhere on phase space with any \nparameter time $t$, a necessary condition for the issue\nof ``constraint evolution''\n\\index{constraint evolution} even to be {\\em discussed} \nin the Hamiltonian\n\\index{Hamiltonian dynamics} framework. (See below in \nSect.~\\ref{ContractedBianchi}.)\n\nIf we define the ``smeared'' Hamiltonian as the integral\nof the Hamiltonian density,\n\\begin{equation}\n\\tilde{\\cal H}_\\alpha = \\int d^3 x^\\prime \\alpha(t,x^\\prime) \n\t\\tilde{\\cal H} \\;,\n\\end{equation}\nthe equation of motion for a general functional\n$F[\\bar{g},\\pi;t,x)$ anywhere on the phase space\nis \n\\begin{equation}\n\\dot{F}\\left[ \\bar{g}, \\pi; t, x \\right) = \n\t- \\left\\{\\tilde{\\cal H}_\\alpha, F\\right\\} \n\t+ \\tilde{\\partial}_0 F \\; ,\n\\label{FunctionalEOM}\n\\end{equation}\nwhere $\\dot{\\left( \\; \\right)}$ denotes our total time\nderivative and $\\tilde{\\partial}_0$ is a ``partial'' \nderivative of the form $\\partial_t - \\pounds_\\beta$\nacting only on explicit spacetime dependence. The \nPoisson bracket is\n\\begin{equation}\n\\left\\{F,G\\right\\} = \\int d^3 x \n\t\\left( \\frac{\\delta F}{\\delta g_{i j}(t,x)}\n\t\\frac{\\delta G}{\\delta \\pi^{i j}(t,x)}\n\t- \n\t\\frac{\\delta G}{\\delta g_{i j}(t,x)}\n\t\\frac{\\delta F}{\\delta \\pi^{i j}(t,x)} \\right) \\; ,\n\\label{PoissonBracket}\n\\end{equation}\nand one sees that \ntime evolution is generated by the Hamiltonian vector field\n\\begin{eqnarray}\n{\\cal X}_{\\tilde{\\cal H}_\\alpha} &=& \\int d^3 x \\left\\{ \n\t\\alpha ( 2 \\pi_{i j} - \\pi g_{i j} )\n\t\\frac{\\delta}{\\delta g_{i j}} \n\t- [ \\alpha \\bar{g} ( R^{i j}(\\bar{g}) \n\t- R(\\bar{g}) g^{i j}) \\right.\n\\nonumber\\\\\n\t& & + \\left. \\alpha ( 2 \\pi^{i k} \\pi^j\\mathstrut_k \n\t- \\pi \\pi^{i j} )\n\t- \\bar{g} ( \\bar{\\nabla}^i \\bar{\\nabla}^j \\alpha \n\t- g^{i j} \\bar{\\nabla}_k \\bar{\\nabla}^k \\alpha ) ] \n\t\\frac{\\delta}{\\delta \\pi^{i j}} \\right\\} \\; .\n\\label{HamVecField}\n\\end{eqnarray}\nBecause it does not contain any explicit constraint\\index{constraint}\ndependence,~(\\ref{HamVecField}) is a valid time evolution\noperator on the entire phase space.\nIt is clear that the $\\left( \\dot{\\bar{g}}, \\dot{\\pi} \\right)$\nequations come from~(\\ref{HamVecField}) applied to the\ncanonical variables. The harmonic time slicing\n\\index{harmonic time-slicing} \nequation~(\\ref{SpecialHarmonic}) results from application \nof~(\\ref{HamVecField})\nto $N$, and the wave equation for $N$ comes from a repeated \napplication of~(\\ref{HamVecField}) to~(\\ref{SpecialHarmonic}).\n\nEvolution equations for the ``constraints'' \n\\index{constraint!evolution equations} are computed to be\n\\begin{eqnarray}\n\\bar{\\partial}_0\\tilde{\\cal H} &=& - \\left\\{ \\tilde{\\cal H}_\\alpha, \n \\tilde{\\cal H} \\right\\}\n\t= \\alpha \\bar{g} g^{i j} \\partial_i {\\cal H}_j \n\t+ 2 \\bar{g} g^{i j} {\\cal H}_i \\bar{\\nabla}_j \\alpha \\; , \n\\label{newConstraint1}\\\\\n\\bar{\\partial}_0{\\cal H}_j &=& - \\left\\{ \\tilde{\\cal H}_\\alpha, {\\cal H}_j \\right\\}\n\t= \\alpha \\partial_j \\tilde{\\cal H} \n\t+ 2 \\tilde{\\cal H} \\partial_j \\alpha \\; ,\n\\label{newConstraint2}\n\\end{eqnarray}\nwhere $\\bar{\\nabla}_j \\alpha = \\partial_j \\alpha \n\t+ \\alpha \\bar{g}^{-1\/2} \\partial_i \\bar{g}^{1\/2}$.\nThese are well-posed evolution \nequations for the constraints,\n\\index{constraint!evolution equations} and they are equivalent\nto the twice-contracted Bianchi\nidentities\\index{Bianchi identities!twice-contracted}\nwhen ${\\cal R}_{i j}=0$ or $R_{i j}=0$\n(see below).\n\nThese results shed new light\non the Dirac ``algebra''\\index{Dirac algebra}\nof constraints~\\cite{Tei73}.\nIt is well known that the\nDirac algebra is not the spacetime diffeomorphism\nalgebra. This can be seen from the fact that\nwhile the action~(\\ref{action})\\index{action} is invariant under\ntransformations generated by ${\\cal H}_j$ and \n$\\tilde{\\cal H}$,~\\cite{Tei77}\nthe equations of motion that follow from this action\\index{action} are\n$R_{i j}(g)=0$ even when ${\\cal H}_j$ and $\\tilde{\\cal H}$\ndo not vanish. These equations of motion are preserved by\nspatial diffeomorphisms and time translations along\ntheir flow in phase space, whereas a general spacetime\ndiffeomorphism applied to $R_{i j}(g)=0$ would mix in the \nconstraints.\\index{constraint}\n\nA second important view of the Dirac algebra\nresults from the direct and beautiful dynamical\nmeaning of its once-smeared form. \nEquations~(\\ref{newConstraint1}) and~(\\ref{newConstraint2})\nexpress {\\em consistency} of the constraints\\index{constraint} as a \n{\\em well posed} initial-value problem. If the constraint\nfunctions vanish in some region on an intial time slice, \nthey continue to do\nso under evolution by the Hamiltonian vector field into the \ndomain of dependence \nof that initial region. This mechanism\nfollows from the dual role of $\\tilde{\\cal H}$ as a \nconstraint\\index{constraint} and as part of the generator of\ntime translations of functionals of the canonical\nvariables anywhere on the phase space.\n\nLet us take note that the Hamiltonian constraint\n\\index{constraint!Hamiltonian}\n{\\em per se} does not express the dynamics of \nthe theory; the equation of dynamics \nis~(\\ref{FunctionalEOM}). In its ``altered'' \nrole, the Hamiltonian constraint function simply\nvanishes as an initial value condition\n\\index{constraint!initial value}, from which \n$\\bar{g}^{1\/2}$ is determined as in the initial value\nproblem.~\\cite{Yor79} Then $N$ can be constructed\n{}from $\\alpha$. The Hamiltonian constraint,\n\\index{constraint!Hamiltonian} once\nsolved, remains so according to the results\nembodied in~(\\ref{newConstraint1}) and~(\\ref{newConstraint2}).\n\n\\section{Contracted Bianchi Identities}\n\\label{ContractedBianchi}\n\nThe results on canonical dynamics that follow on using $\\alpha$ \nas an undetermined multiplier are also reflected in the\nmanner in which the twice-contracted Bianchi identities,\n\\index{Bianchi identities!twice-contracted}\n\\begin{equation}\n\\nabla_\\beta G^\\beta\\mathstrut_\\alpha \\equiv 0 \\; ,\n\\label{BiID}\n\\end{equation}\ncan be written as a first-order symmetrizable hyperbolic\nsystem~\\cite{AY98}. \n(In the absence of hyperbolic form,~(\\ref{BiID}) is\npractically useless in providing physical equations of \nmotion for the constraints\\index{constraint!evolution equations} \nwhen they are {\\em not}\nsatisfied.) Likewise, this system extends to\nmatter (see below). (Frittelli obtained well-posedness\nfor~(\\ref{BiID}) by other methods~\\cite{Fri97}.)\n\nWe recall that the equations of motion of the canonical\nmomenta in vacuum are\n\\begin{equation}\n{\\cal R}_{ij} \\equiv R_{ij}(g) - g_{ij} R^k\\mathstrut_k(g) = 0 \\; ,\n\\label{CalRij}\n\\end{equation}\nwhile the weight zero Hamiltonian constraint\n\\index{constraint!Hamiltonian} is\n\\begin{equation}\nC = 2G^0\\mathstrut_0(g) = K_{ij}K^{ij} - K^2 - R(\\bar{g}) = 0 \\; ,\n\\label{C}\n\\end{equation}\nand the weight zero one-form momentum constraint\n\\index{constraint!momentum} is \n\\begin{equation}\nC_i = 2NR^0\\mathstrut_i (g) = 2\\bar{\\nabla}^j (K_{ij} - Kg_{ij})\n= 0 \\; .\n\\label{Ci}\n\\end{equation}\nRecall the identity~(\\ref{calR}):\n\\begin{equation}\nG_{ij}(g) + g_{ij}G^0\\mathstrut_0(g) \\equiv R_{ij}(g) - g_{ij}\nR^k\\mathstrut_k(g) \\equiv {\\cal R}_{ij} \\; .\n\\label{Gij}\n\\end{equation}\nCombining~(\\ref{CalRij}),~(\\ref{C}),~(\\ref{Ci}), and~(\\ref{calR}) \nwith~(\\ref{BiID}) gives \nthe twice-contracted Bianchi identities\n\\index{Bianchi identities!twice-contracted} as a FOSH\\index{FOSH}\n\\index{first-order symmetric hyperbolic} system\n\\begin{equation}\n\\dot{C} - N \\bar{\\nabla}^j C_j \\equiv 2\\left( C_j \\bar{\\nabla}^j \nN + NKC - NK^{ij}[{\\cal R}_{ij}] \\right) \\; ,\n\\label{ConWave1}\n\\end{equation}\n\\begin{equation}\n\\dot{C}_j - N \\bar{\\nabla}_j C \\equiv 2 \\left( C \\bar{\\nabla}_j N\n+ \\frac{1}{2} NKC_j - \\bar{\\nabla}^i(N[{\\cal R}_{ij}]) \\right) \\; .\n\\label{ConWave2}\n\\end{equation} \nSubstituting ${\\cal H}_i = \\bar{g}^{1\/2} C_i$,\n$\\tilde{\\cal H} = \\bar{g} C$, and setting the equations of\nmotion ${\\cal R}_{ij}=0$ in~(\\ref{ConWave1}) and~(\\ref{ConWave2}) \nyields the \nevolution equations of the unsmeared constraints\n\\index{constraint!evolution equations} as \nin~(\\ref{newConstraint1}) and~(\\ref{newConstraint2}).\n\nSimilar considerations show how to put the ``matter conservation''\nequations $\\nabla_\\beta T^{\\alpha \\beta} = 0$ into well-posed form.\nThis was also carried out by one of us (YCB) and \nNoutchegueme~\\cite{CBN86}\nbut the results obtained here are more immediately physical.\nUnlike~\\cite{CBN86}, we use the\nenergy density $\\varepsilon = -T^0\\mathstrut_0$ rather than $\\rho^{00}\n\t(\\rho^\\alpha\\mathstrut_\\beta \\equiv T^\\alpha\\mathstrut_\\beta \n\t- \\frac{1}{2} \n\t\\delta^\\alpha\\mathstrut_\\beta T^\\mu\\mathstrut_\\mu)$ to obtain \nthis result. It\nis clear that such a result is possible because\n\\begin{equation}\nH_{\\alpha \\beta} \\equiv {\\kappa}^{-1} G_{\\alpha \\beta}(g)-T_{\\alpha \\beta}\n\\end{equation}\nvanishes as Einstein's equation and in any case satisfies\n$\\nabla_\\beta H^\\beta\\mathstrut_\\alpha = 0$. \nWe can treat $H_{\\alpha \\beta}$\nas we did $G_{\\alpha \\beta}$ above. $({\\kappa} = 8\\pi G \\; ; \\; c=1.)$\nThe result is nevertheless of interest as it presents the\ncontinuity and relativistic Euler equations\\index{Euler equations}\nof matter in a well-posed form.\n\nStraightforwardly expanding $\\nabla_\\beta T^\\beta\\mathstrut_0 = 0$ and \n$\\nabla_\\beta T^\\beta\\mathstrut_i = 0$ gives the continuity and Euler\nequations \\index{Euler equations}\n(cf.~\\cite{Yor79}, p. 89), with $\\varepsilon \\equiv -T^0\\mathstrut_0$\nand the matter current one-form $j_i \\equiv NT^0\\mathstrut_i$. The\ncontinuity equation is\n\\begin{equation}\n\\bar{\\partial}_0 \\varepsilon + N \\bar{\\nabla}^i j_i = \nN(K_{ij}T^{ij} + K \\varepsilon - 2j_i a^i) \\; ,\n\\label{hatpar1}\n\\end{equation} \nwhere $a_i \\equiv \\bar{\\nabla}_i \\log N$ is the acceleration\nof observers at rest in a given time-slice. Likewise, we\nfind for Euler's equation\\index{Euler equations}\n\\begin{equation}\n\\bar{\\partial}_0 j_i + N \\bar{\\nabla}_j T^j\\mathstrut_i \n\t= N (K^j\\mathstrut_i - \n\tT^j\\mathstrut_i a_j - \\varepsilon a_i) \\; .\n\\label{hatpar2}\n\\end{equation}\nThe divergence term on the left side of~(\\ref{hatpar2}) spoils\nthe well-posed FOSH\\index{FOSH}\n\\index{first-order symmetric hyperbolic} \nform we seek. However, if we use the\nidentity $G_{ij}(g) + g_{ij} G^0\\mathstrut_0 (g) \\equiv R_{ij}\n(g) - g_{ij} R^k\\mathstrut_k(g)$ and the Einstein equations\n\\index{Einstein equations}\n${\\kappa}^{-1} G_{\\alpha \\beta} - T_{\\alpha \\beta} = 0$, or\n${\\kappa}^{-1} R_{\\alpha \\beta} = \\rho_{\\alpha \\beta}$,\nwe obtain $(\\varepsilon = -T^0\\mathstrut_0 \\; , \\; j_i = N T^0\\mathstrut_i)$\n\\begin{equation}\nT^j\\mathstrut_i - \\delta^j\\mathstrut_i \\varepsilon \n\t= (\\rho^j\\mathstrut_i - \\delta^j\\mathstrut_i\n\t\\rho^k\\mathstrut_k) \\equiv {\\cal S}^j\\mathstrut_i \\; .\n\\end{equation}\nThen~(\\ref{hatpar1}) and~(\\ref{hatpar2}) obtain well-posed\nform (if ${\\cal S}^j\\mathstrut_i$ is assumed known),\n\\begin{equation}\n\\bar{\\partial}_0 \\varepsilon + N \\bar{\\nabla}^i j_i = \n-2j_i \\bar{\\nabla}^i N + 2NK\\varepsilon + NK^{ij}\n{\\cal S}_{ij} \\; ,\n\\label{hatpar3}\n\\end{equation}\n\\begin{equation}\n\\bar{\\partial}_0 j_i + N \\bar{\\nabla}_i \\varepsilon = \n-2\\varepsilon \\bar{\\nabla}_i N + NKj_i - \\bar{\\nabla}^j\n(N {\\cal S}_{ij}) \\; .\n\\label{hatpar4}\n\\end{equation}\n\nBy combining~(\\ref{hatpar3}) plus~(\\ref{Ci}), and~(\\ref{hatpar4})\nplus~(\\ref{Gij}), we obtain expressions of gravity constraint\nevolution\\index{constraint!evolution equations} in the presence of matter,\n\\begin{equation}\n\\bar{\\partial}_0 C^T - N \\bar{\\nabla}^i C^T_i = \n2 \\left( C^T_j \\bar{\\nabla}^j N + NKC^T - NK^{ij} ({\\kappa}^{-1}\n{\\cal R}_{ij} - {\\cal S}_{ij}) \\right) \\; ,\n\\end{equation}\n\\begin{equation}\n\\bar{\\partial}_0 C^T_j - N \\bar{\\nabla}_j C^T = 2 \\left[ C^T\n\\bar{\\nabla}_j N + \\frac{1}{2} NKC^T_j - \\bar{\\nabla}^i\n\\left( N({\\kappa}^{-1} {\\cal R}_{ij} - {\\cal S}_{ij}) \\right)\n\\right]\n\\end{equation}\nwhere $C^T \\equiv C + 2 \\varepsilon$ and $C^T_j = C_j - 2j_j$.\nThis is just the form we would anticipate on the basis of\nHamiltonian dynamics\\index{Hamiltonian dynamics} \nand the form~(\\ref{Ci}) and~(\\ref{Gij})\nof the vacuum constraints\\index{constraint!vacuum}. \nThus, for gravity plus a matter field,\nwe obtain results analogous to~(\\ref{newConstraint1})\nand~(\\ref{newConstraint2}) for the total system. If there are no \nviolations of\nconstraints\\index{constraint violation}, \nthen $C^T = 0 \\; , \\; C^T_j = 0$, while the\ndynamical gravity equation is ${\\kappa}^{-1} {\\cal R}_{ij}\n- {\\cal S}_{ij} = 0$.\n\n\\section{Wave Equation for $K_{i j}$}\n\nEinstein's equations,\\index{Einstein equations} \nviewed mathematically as a system\nof second-order partial differential equations for the metric,\ndo not form a hyperbolic system without modification and\nare not manifestly well-posed, though, of course, physical\ninformation does propagate at the speed of light. A \nwell-posed hyperbolic system admits unique solutions \ndepending continuously on the initial data and seems to be \nrequired for robust, stable numerical integration and\nfor full treatment by the methods of modern analysis, for \nexample, exploitation of energy estimates. The\nwell-known traditional approach achieves hyperbolicity\nthrough special coordinate choices.\\footnote{The classic\nsecond-order fully harmonic form was given in~\\cite{FB52,CB56} and\ndiscussed, for example, in~\\cite{CBY80}. It will not be discussed\nin this article. A FOSH form based on these equations was given\nfirst by Fischer and Marsden in~\\cite{FiM72}.}\nThe formulation described here permits coordinate gauge\nfreedom. Because these exact nonlinear theories incorporate\nthe constraints, they are natural starting points for \ndeveloping gauge-invariant perturbation theory.\n\nConsider a globally hyperbolic manifold \n$V = \\Sigma \\times {\\mathbb R}$ with the metric as given\nin the introduction. To achieve hyperbolicity for the\n$3+1$ equations, we proceed as follows.\n\nBy taking a time derivative of $R_{i j}(g)$ and subtracting\nappropriate spatial covariant derivatives of the momentum\nconstraints\\index{constraint!momentum}, one of us (YCB) and \nT.~Ruggeri~\\cite{CBR83} (see also~\\cite{CBY95}, \nwhere the shift is not set to zero) obtained\nan equation with a wave operator acting on the extrinsic\ncurvature.\\index{extrinsic curvature} In vacuum, one finds\n\\begin{equation}\n\t\\bar{\\partial}_0 R_{i j}(g) - \\bar{\\nabla}_i R_{0 j} - \\bar{\\nabla}_j R_{0 i}\n\t= N \\bar{\\Box\\mbox{\\small\\mathstrut}}_g K_{i j} + J_{i j} + S_{i j} = 0 \\; ,\n\\label{bpzRij}\n\\end{equation}\nwhere $\\bar{\\Box\\mbox{\\small\\mathstrut}}_g = - \\left( N^{-1} \\bar{\\partial}_0 \\right)^2\n\t+ \\bar{\\nabla}_k \\bar{\\nabla}^k$, $J_{i j}$ consists\nof terms at most first order in derivatives of $K_{i j}$, \nsecond order in derivatives of $g_{i j}$, and second order\nin derivatives of $N$, and\n\\begin{equation}\n\tS_{i j} = -N^{-1} \\bar{\\nabla}_i \\bar{\\nabla}_j (\\bar{\\partial}_0 N \n\t+ N^2 K)\\; .\n\\end{equation}\nThe term $S_{i j}$ is second order in derivatives of $K_{i j}$\nand would spoil hyperbolicity of the wave operator $\\bar{\\Box\\mbox{\\small\\mathstrut}}$\nacting on $K_{i j}$. Hyperbolicity is achieved by setting\n$N = \\bar{g}^{1\/2} \\alpha (t,x)$, or\n\\begin{equation}\n\t\\bar{\\partial}_0 N + N^2 K = \\bar{g}^{1\/2} \\bar{\\partial}_0 \\alpha(t,x) \\;,\n\\end{equation}\nas discussed in Sect.~\\ref{Sec:EverySlicing}.\nThe resulting equation combined with~(\\ref{gdot}) \nforms a quasi-diagonal hyperbolic\nsystem for the metric $g_{i j}$ \nwith principal operator $\\bar{\\partial}_0 \\bar{\\Box\\mbox{\\small\\mathstrut}}$. This system\ncan also be put in first order symmetric hyperbolic \nform~\\cite{CBY95,AACbY95}, by the introduction of sufficient auxiliary\nvariables and by use of the equation for $R_{0 0}$ (thus\nincorporating the Hamiltonian constraint\n\\index{constraint!Hamiltonian}). The Cauchy\ndata\\index{Cauchy data} for the system \n(in vacuum)~\\cite{CBR83,CBY95}\nare\n(1) $(\\bar{g}, K)$ such that the constraints\n\t$R_{0 i} = 0$, \n$G^0\\mathstrut_0 = 0$ hold on the initial slice;\n\\index{constraint!momentum}\n\\index{constraint!Hamiltonian}\n\\index{constraint!initial value}\n(2) $\\bar{\\partial}_0 K_{i j}$ such that $R_{i j}=0$ on the intial slice;\nand\n(3) $N>0$ arbitrary on the initial slice.\nNote that the shift $\\beta^k(x,t)$ is arbitrary. Using\nthe Bianchi identities\\index{Bianchi identities}, \none can prove~\\cite{CBR83,CBY95} that this system\nis fully equivalent to the Einstein equations.\n\\index{Einstein equations!equivalence}\nThe point is that quasi-diagonal Leray~\\cite{leray}\nhyperbolic systems have well posed Cauchy problems\\index{Cauchy problem}\nand therefore unique solutions for given initial data.\nBecause every solution of the Einstein equations\nalso satisfies the $\\bar{\\Box\\mbox{\\small\\mathstrut}} K_{i j}$ equation\nin particular and provides initial data for it,\nuniqueness implies, conversely, that if the initial\ndata for the $\\bar{\\Box\\mbox{\\small\\mathstrut}} K_{i j}$ equation are Einsteinian,\nall solutions of Einstein's equations, and only these,\nare captured. The restriction on the initial value\nof $\\bar{\\partial}_0 K_{i j}$ prevents the higher derivative from\nintroducing spurious unphysical solutions.\n\nAll variables propagate either with characteristic\nspeed zero or the speed of light. The only variables\nwhich propagate at the speed of light have the dimensions\nof curvature, and one sees that this is a theory of\npropagating curvature.\\index{propagating curvature} However, a FOSH\n\\index{first-order symmetric hyperbolic} system that \npropagates curvature is more transparent in the ``Einstein-Bianchi''\n\\index{Einstein equations!hyperbolic!Einstein-Bianchi}\nform (next section).\n\nIn the above formulation, the shift and $\\alpha$ are arbitrary.\nThis and our other systems (except the Einstein-Christoffel\n\\index{Einstein equations!hyperbolic!Einstein-Christoffel}\nsystem in Sect.~\\ref{EinsteinChristoffel}) are\nmanifestly spatially covariant and all time slicings\n(using $\\alpha$) are allowed. Spacetime covariance is\ntherefore present, but not completely manifest.\n\nBy taking another time derivative and adding an appropriate\nderivative of $R_{0 0}$, one finds (in vacuum)~\\cite{AY98}\n\\begin{equation}\n\t\\bar{\\partial}_0 \\bar{\\partial}_0 R_{i j} - \\bar{\\partial}_0 \\bar{\\nabla}_i R_{0 j}\n\t+ \\bar{\\partial}_0 \\bar{\\nabla}_j R_{0 i} + \\bar{\\nabla}_i\n\t\\bar{\\nabla}_j R_{0 0} = \\bar{\\partial}_0 (N \\bar{\\Box\\mbox{\\small\\mathstrut}} K_{i j})\n\t+ {\\cal J}_{i j} = 0 \\;,\n\\end{equation}\nwhere ${\\cal J}_{i j}$ consists of terms at most third\norder in derivatives of $g_{i j}$ and second order in\nderivatives of $K_{i j}$. Together with $\\bar{\\partial}_0 g_{i j}$,\nthese form a system for $(\\bar{g}, {K})$\nwhich is hyperbolic non-strict in the sense of Leray-Ohya.~\\cite{LeO67}\nHere, the lapse\\index{lapse} itself, as well as the shift, is arbitrary\n$(N>0)$. The Cauchy data\\index{Cauchy data} of the previous \nform (in vacuum)\nmust be supplemented by $\\bar{\\partial}_0 \\bar{\\partial}_0 K_{i j}$ such that\n$\\bar{\\partial}_0 R_{i j} = 0$ on the initial slice. This guarantees\nthat the system is fully equivalent to Einstein's theory\n\\index{Einstein equations!equivalence}\n(except that its solutions are not in Sobolev spaces~\\cite{AY98}).\nThis system does not have a first order symmetric hyperbolic\nformulation, but has been used very effectively in perturbation\ntheory~\\cite{AAL98} and in other applications~\\cite{YCB97a,YCB97b}.\n\n\\section{Einstein-Bianchi Hyperbolic System}\n\nTo obtain a first order symmetric hyperbolic system, one can \nuse the Riemann tensor of the spacetime metric. It satisfies the\nBianchi identities for the spacetime geometry\n\\begin{equation}\n\\nabla_\\alpha R_{\\beta \\gamma \\lambda \\mu}\n + \\nabla_\\beta R_{\\gamma \\alpha \\lambda \\mu}\n + \\nabla_\\gamma R_{\\alpha \\beta \\lambda \\mu} \\equiv 0 \\; .\n\\end{equation}\nThese identities imply by contraction and use of the symmetries of the\nRiemann tensor\n\\begin{equation}\n\\nabla_\\alpha R^\\alpha\\mathstrut_{\\mu \\beta \\gamma} +\n \\nabla_\\gamma R_{\\beta \\mu} + \\nabla_\\beta R_{\\gamma \\mu}\n \\equiv 0 \\; .\n\\end{equation}\nIf the Ricci tensor $R_{\\alpha \\beta}$ satisfies the Einstein\nequations ($\\kappa = c = 1$)\n\\begin{equation}\nR_{\\alpha \\beta} = \\rho_{\\alpha \\beta} \\; ,\n\\end{equation}\nthen the previous identities imply the equations\n\\begin{equation}\n\\nabla_\\alpha R^\\alpha\\mathstrut_{\\mu \\beta \\gamma} =\n \\nabla_\\beta \\rho_{\\gamma \\mu} -\n \\nabla_\\gamma \\rho_{\\beta \\mu} \\; .\n\\end{equation}\n\nThe first equations with $(\\alpha \\beta \\gamma) = (i j k)$\nand the last one with $\\mu=0$ do not contain derivatives\nof the Riemann tensor transverse to $M_t$. They are\nconsidered as ``constraints'' and will be identically satisfied\n(initially) in our method. They remain satisfied in an exact\nintegration. All detail and rigor concerning this elegant\nsystem is given in~\\cite{CBYA98,ACBY97},\nto which the reader is referred.\nIt has 66 equations, just as do the Einstein-Ricci first\norder curvature equations.\n\nThe system we are now developing~\\cite{CBY97} is similar to an analogous\nsystem obtained by H.~Friedrich~\\cite{Fri96} that is based on the\nWeyl tensor. The Weyl tensor system is causal but with\nadditional unphysical characteristics. \n\nWe wish first to show that the remaining equations are, for $n=3$\nin the vacuum case, when $g$ is given, a symmetric first order\nhyperbolic system for the double two-form \n$R_{\\alpha \\beta \\lambda \\mu}$. For this purpose, following\nBel~\\cite{Bel58,Bel61} we introduce two pairs of ``electric''\nand ``magnetic'' space tensors associated with a spacetime\ndouble two-form $A$,\n\\begin{eqnarray}\nN^2 E_{i j}(g) &\\equiv& A_{0 i 0 j} \\\\\nD_{i j}(g) &\\equiv& \\frac{1}{4} \\epsilon_{i h k} \\epsilon_{j l m}\n A^{h k l m} \\\\\nN H_{i j}(g) &\\equiv& \\frac{1}{2} \\epsilon_{i h k} \n A^{h k}\\mathstrut_{0 j} \\\\\nN B_{j i}(g) &\\equiv& \\frac{1}{2} A_{0 j}\\mathstrut^{h k} \\epsilon_{i h k}\n\\end{eqnarray}\nwhere $\\epsilon_{i j k}$ is the volume form of $\\bar{g}$.\nIt results from the symmetry of the Riemann tensor $R$ with\nrespect to its first and second pairs of indices ($R$ is a\n``symmetric double two-form'') that if $A \\equiv R$, then $E$\nand $D$ are symmetric while $H_{i j} = B_{j i}$. A useful\nidentity for a symmetric double two-form like $R$, with a tilde\nrepresenting the spacetime double dual, is (``Lanczos identity'')\n\\begin{equation}\n\\tilde{R}_{\\alpha \\beta \\lambda \\mu} + R_{\\alpha \\beta \\lambda \\mu}\n = C_{\\alpha \\lambda} \\, g_{\\beta \\mu} -\n C_{\\alpha \\mu} \\, g_{\\beta \\lambda} +\n C_{\\beta \\mu} \\, g_{\\alpha \\lambda} -\n C_{\\beta \\lambda} \\, g_{\\alpha \\mu} \\; ,\n\\end{equation}\nwhere $C_{\\alpha \\beta} = R_{\\alpha \\beta} - (1\/4) g_{\\alpha \\beta}\nR$.\nIt follows that when $R_{\\alpha \\beta} = \\lambda \\, g_{\\alpha \\beta}$,\nthen $E = -D$ and $H = B$. In order to avoid introducing unphysical\ncharacteristics, and to be able to extend the treatment to the\nnon-vacuum case, {\\em we do not use these properties in the evolution\nequations}, but write them as a first order system for an arbitrary\ndouble two-form $A$, as follows:\n\\begin{eqnarray}\n& \\nabla_0 A_{h k 0 j} + \\nabla_k A_{0 h 0 j}\n - \\nabla_h A_{0 k 0 j} = 0 \\; , & \\\\\n& \\nabla_0 A^0\\mathstrut_{i 0 j} + \\nabla_h A^h_{i 0 j}\n = \\nabla_0 \\, \\rho_{j i} - \\nabla_j \\, \\rho_{0 i} \\; ,&\n\\end{eqnarray}\nand analogous equations with the pair $(0\\, j)$ replaced by $(l\\,m)$.\nOne obtains a first order system for the unknowns $E$, $H$, $D$,\nand $B$ by using the relations inverse to the definitions\nabove. The principal parts of these equations, all with one\ndefinite index fixed on $E$, $H$, $D$, and $B$, are identical\nto the corresponding Maxwell equations. The characteristic matrix\nof this ``Maxwell'' part of the system has determinant\n$-N^6 (\\xi_0 \\xi^0) \\, (\\xi_\\alpha \\xi^\\alpha)^2$.\nThe system obtained has a principal matrix consisting of 6\nidentical 6 by 6 blocks around the diagonal, which are symmetrizable\nand hyperbolic. Hence, the system is symmetric hyperbolic, when\n$g$ is a given metric such that $\\bar{g}$ is properly\nriemannian and $N>0$.\n\nTo relate the Riemann tensor to the metric $\\bar{g}$ we use the\ndefinition \n\\begin{equation}\n\\bar{\\partial}_0 g_{i j} = -2 N K_{i j}\n\\end{equation}\nand we use the $3+1$ identities given in the Introduction.\n{\\em Note that in this section all $\\Gamma$'s are spatial.}\n\nWe next choose $N=\\bar{g}^{1\/2} \\alpha(t,x)$. We generalize\nsomewhat the ideas used\nby Friedrich (see~\\cite{Fri96}) for the Weyl tensor to write a\nsymmetric hyperbolic system for $K$ and $\\Gamma$, namely\nwe obtain equations relating $\\Gamma$ and $K$, for a given\ndouble two-form $A$, and by considering the definition of $K$ and the \n$3+1$ decomposition of the Riemann tensor,\nreplacing in these identities the Riemann\ntensor by $A$. To deduce from this\nsystem a symmetric hyperbolic first order system, with the\nalgebraic form of the harmonic gauge $N=\\bar{g}^{1\/2} \\alpha$,\none uses the fact that in this gauge one has\n\\begin{equation}\n\\Gamma^h\\mathstrut_{i h} = \\partial_i \\log N -\n \\partial_i \\log \\alpha \\; .\n\\end{equation}\nWe obtain\n\\begin{eqnarray}\n\\bar{\\partial}_0 \\Gamma^h\\mathstrut_{i j} + N \\bar{\\nabla}^h K_{i j}\n &=& N K_{i j} g^{h k} \\left( \\Gamma^m\\mathstrut_{m k}\n + \\partial_k \\log \\alpha \\right) \\nonumber\\\\\n& & - 2 N K^h\\mathstrut_{(i} \\left( \\Gamma^m\\mathstrut_{j) m}\n + \\partial_{j)} \\log\\alpha \\right) \\\\\n& & - N \\left( \\epsilon^k\\mathstrut_{( j}\\mathstrut^h B_{i) k}\n + H_{k (i} \\epsilon^k\\mathstrut_{j)}\\mathstrut^h \\right) \\; ,\n\\nonumber\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\bar{\\partial}_0 K_{i j} + N \\partial_h \\Gamma^k \\mathstrut_{i j}\n &=& N \\left[ \\Gamma^m\\mathstrut_{i h} \n \\Gamma^h\\mathstrut_{j m} - \\left(\n \\Gamma^h\\mathstrut_{i h} + \\partial_i \\log \\alpha \\right)\n \\left( \\Gamma^k\\mathstrut_{j k} + \\partial_j \\log\\alpha\n \\right) \\right] \\nonumber\\\\\n& & - N \\left( \\partial_i \\partial_j \\log\\alpha -\n \\Gamma^k\\mathstrut_{i j} \\partial_k \\log \\alpha \\right) \\\\\n& & - N \\left[ D_{(i j)} + E_{(i j)} - D^k\\mathstrut_k g_{i j}\n - K K_{i j} \\right] \\; . \\nonumber\n\\end{eqnarray}\nThe system obtained for $K$ and $\\Gamma$ has a \ncharacteristic matrix composed of 6 blocks around the diagonal,\neach block a 4 by 4 matrix that is symmetrizable hyperbolic,\nwith characteristic polynomial $-N^4 (\\xi_0 \\xi^0) \\, \n (\\xi_\\alpha \\xi^\\alpha)$.\n\nThe whole system for $A$, $K$, $\\Gamma$, $\\bar{g}$ is\nsymmetrizable hyperbolic, with characteristics the light cone and the\nnormal to $M_t$. It is somewhat involved to prove that a solution of\nthe constructed system satisfies the Einstein equations if the initial\ndata satisfy the constraints, but we can argue as follows. We consider\nthe vacuum case with initial data satisfying the Einstein\nconstraints. These initial data determine the initial values of\n$\\Gamma$, and also, if $\\beta$ and $N$ are known at $t=0$, the\ninitial values of $A_{i j h m}$, $A_{j h i 0}$, $A_{i 0 j h}$ by\nusing the decomposition formulas. (We set A equal to the Riemann\ntensor on the initial surface.) We use the Lanczos formula to\ndetermine $A_{i0j0}$ initially. We know that our symmetrizable\nhyperbolic system has one and only one solution. Because a solution of\nEinstein's equations with $N=\\bar{g}^{1\/2} \\alpha$, proved to exist in\nthe section on $\\bar{\\Box\\mbox{\\small\\mathstrut}} K_{i j}$, satisfies together with its Riemann\ntensor the present system and takes the same initial values, that\nsolution coincides with the solution of the present system in their\ncommon domain of existence.\n\n\\section{Einstein-Christoffel System}\n\\label{EinsteinChristoffel}\n\nThe first-order form of the wave equation for $K_{i j}$, the\nEinstein-Ricci\\index{Einstein equations!hyperbolic!Einstein-Ricci} \nsystem~\\cite{CBY95,AACbY95}, to which\nwe have alluded, has 66 equations, the correct number for a\ncurvature system as does the Einstein-Bianchi system\nof the previous section. It is symmetric hyperbolic\n\\index{symmetric hyperbolic} and therefore\nwell-posed. But it is natural to ask whether there is a simpler\nfirst-order system of fewer variables that is perhaps closer\nin form to~(\\ref{gdot}) and~(\\ref{Kdot}). Frittelli and \nReula~\\cite{FrR94,FrR96} proposed\nsuch a system, but their system has unphysical characteristics\nand is not written fully in terms of geometric variables. Here we deduce\na different system having only physical characteristics and\nexpressed in geometric variables. We understand (private\ncommunication to JWY) that James Bardeen\nhas likewise obtained a similar system improving that found\nin~\\cite{BM92,BMSS95}. \nWhile our derivation~\\cite{AY99} can proceed systematically by direct\nconstruction of an energy norm and the characteristic speeds,\na more heuristic derivation is indicated here from the structure\nof the wave equation for $K_{i j}$. Not that in this section\nall $\\Gamma$'s are spatial.\n\nIn the dynamical spacetime Ricci tensor $R_{i j}(g)$,~(\\ref{Kdot}),\none has a dynamical equation for the extrinsic curvature $K_{i j}$\n\\index{extrinsic curvature}\nin terms of spatial derivatives of the spatial Christoffel\nsymbols. When the lapse\\index{lapse} $N$ is replaced by the \nslicing density\\index{slicing function}\nthrough $\\alpha \\bar{g}^{1\/2}$, the differentiated Christoffel\nterms become\n\\begin{equation}\n\t\\partial_k \\Gamma^k\\mathstrut_{i j}\n\t- \\partial_j \\Gamma^k\\mathstrut_{i k}\n\t- \\partial_i \\Gamma^k\\mathstrut_{j k} \\; .\n\\end{equation}\nThis may be read as the divergence of a linear combination of\nChristoffel symbols, which puts the $R_{i j}(\\bar{g})$ equation \nin a form reminiscent of the structure of one of the first-order\nequations for a free wave, namely\n\\begin{equation}\n\\partial_0 u + \\partial^k v_k = 0\\;.\n\\label{udot}\n\\end{equation}\nWe would have a symmetric hyperbolic system if there were \nan analog of the other equation for a wave,\n\\begin{equation}\n\\partial_0 v_k + \\partial_k u = 0 \\; .\n\\label{vdot}\n\\end{equation}\nSome manipulation quickly leads to the conclusion that $K_{i j}$\nand the Christoffel combination above are not paired in a \nsymmetric hyperbolic system like $u$ and $v_k$.\n\nRecall however, that the free wave equation $(\\partial_0)^2 u\n\t- \\partial^k \\partial_k u = 0$ is obtained by taking a \ntime derivative of~(\\ref{udot}) and subtracting the\ndivergence of~(\\ref{vdot}). In obtaining the wave equation\nfor $K_{i j}$~(\\ref{bpzRij}) we have taken a time derivative of $R_{i j}$\nand subtracted a (suitably symmetrized) divergence of the\nmomentum constraint\\index{constraint!momentum} $R_{0 i}$. \nThis motivates the speculation\nthat in gravity the ``other'' equation should be related to the\nmomentum constraint\\index{constraint!momentum} and its sole spatial derivative should be\n$\\partial_k K_{i j}$.\n\nFollowing this idea about the second equation leads one to \nconsider an equation of the form\n\\begin{equation}\ng_{k i} R_{j 0} + g_{k j} R_{i 0} = -\\bar{\\partial}_0 f_{k i j}\n\t- \\partial_k (N K_{i j}) + l.o._{k i j}\n\\label{mom1}\n\\end{equation}\nwhere $l.o._{k i j}$ are lower order terms involving no\nderivatives of $f_{k i j}$ or $K_{i j}$. One must choose\n$f_{k i j}$ from a linear combination of spatial derivatives\nof the metric. Introduce\n\\begin{equation}\n{\\cal G}_{k i j} = \\partial_k g_{i j}\n\\label{calGkij}\n\\end{equation}\nand use the identity\n\\begin{equation}\n\\bar{\\partial}_0(\\partial_k g_{i j}) = -\\partial_k (2 N K_{i j})\n\\end{equation}\nto find that\n\\begin{equation}\nf_{k i j} = \\frac{1}{2} {\\cal G}_{k i j} -\n\tg_{k (i} g^{r s} \\left( {\\cal G}_{|r s| j)}\n\t- {\\cal G}_{j) r s} \\right)\n\\end{equation}\nproduces the correct coordinate derivatives occuring in the\nmomentum constraints\\index{constraint!momentum}. \nThe lower order terms are those terms\nnecessary to complete~(\\ref{mom1}) into an identity\nand take the form\n\\begin{eqnarray}\n\tl.o._{k i j} &=& 2 N K_{k (i} g^{r s} \\left( {\\cal G}_{|r s| j)}\n\t- {\\cal G}_{j) r s} \\right) \\nonumber\\\\\n& &\n\t+ 2 g_{k (i} \\Bigl[ K_{j) m} \\partial^m N - K \\partial_{j)}\n\tN \\nonumber\\\\\n& & + N K_{j) m} g^{r s} \\Gamma^m\\mathstrut_{r s}({\\cal G})\n\t+ \\frac{1}{2} N \\left( {\\cal G}_{j) r s} - 2 {\\cal G}_{| r s | j)}\n\t\\right) K^{r s} \\Bigr] \\; ,\n\\label{lokij}\n\\end{eqnarray}\nwhere the spatial Christoffel symbols are constructed from \n${\\cal G}_{i j k}$,\n\\begin{equation}\n\\Gamma_{k i j}({\\cal G}) \\equiv (1\/2) \\left( {\\cal G}_{j k i}\n\t+ {\\cal G}_{i k j} - {\\cal G}_{k i j} \\right) \\; .\n\\label{bGam_def}\n\\end{equation}\n(It is clear\nthat only one of the three-index symbols ${\\cal G}_{k i j}$,\n$\\Gamma_{k i j}$, and $f_{k i j}$ is needed,\nsay $f_{k i j}$. The necessary algebra will not be reproduced\nhere.)\n\nOne then easily verifies that by expressing~(\\ref{bGam_def}) in\nterms of derivatives of the metric (assuming a metric compatible\n\\index{metric compatibility}\nconnection), we can manipulate it to take the form of a \ndivergence of $f_{k i j}$ plus lower order terms. The\ndynamical Ricci equation becomes\n\\begin{equation}\nR_{i j} = -N^{-1} \\bar{\\partial}_0 K_{i j} - \\partial^k f_{k i j}\n\t+ l.o._{i j} \\; ,\n\\label{Rij3}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nl.o._{i j} &=& K K_{i j} - 2 K_{i k} K^k\\mathstrut_j - \\alpha^{-1}\n\t\\left( \\partial_i \\partial_j - \\Gamma^k\\mathstrut_{i j}\n\t({\\cal G}) \\partial_k \\right) \\alpha \\nonumber\\\\\n\t& & - \\left( \\Gamma^k\\mathstrut_{k i} ({\\cal G})\n\t+ \\alpha^{-1} \\partial_i \\alpha \\right)\n\t\\left( \\Gamma^m\\mathstrut_{m j}({\\cal G})\n\t+ \\alpha^{-1} \\partial_j \\alpha \\right) \\\\\n\t& & + 2 \\Gamma^k\\mathstrut_{m k}({\\cal G})\n\t\\Gamma^m\\mathstrut_{i j}({\\cal G})\n\t- \\Gamma^k\\mathstrut_{m j}({\\cal G})\n\t\\Gamma^m_{i k}({\\cal G}) \\nonumber\\\\\n\t& & + g^{k r} g^{s m} \\left[ {\\cal G}_{k r s} f_{m i j}\n\t+ {\\cal G}_{k m (i} {\\cal G}_{j) r s}\n\t- {\\cal G}_{k r s} {\\cal G}_{(i j) m} \\right]\\; .\\nonumber\n\\end{eqnarray}\nTogether with~(\\ref{gdot}),~(\\ref{mom1}) \nand~(\\ref{Rij3}) constitute a symmetric hyperbolic system for the\nevolution of $g_{i j}$, $K_{i j}$ and $f_{k i j}$. \n\nNote\nthat once $f_{k i j}$ or, equivalently, ${\\cal G}_{k i j}$\nare introduced as variables, the relation~(\\ref{bGam_def}) becomes\nan initial condition and does not {\\em a priori} hold\nfor all time. Equation~(\\ref{mom1}) can be\nrelated to metric compatibility \\index{metric compatibility}\nby putting it in the form\n\\begin{eqnarray}\n4 g_{k (i} R_{j) 0} &=& \\bar{\\partial}_0 {\\cal G}_{k i j} + \\partial_k\n\t\\left( 2 N K_{i j} \\right) \\\\\n& & - 4 g_{k (i} N \\bar{\\nabla}^m \\left( K_{j) m} - g_{j) m} K \\right)\n\t\\; . \\nonumber\n\\label{R0k_gg}\n\\end{eqnarray}\nHere, one sees that if the momentum constraint\n\\index{constraint!momentum} is satisfied\nfor all time, then\n\\begin{equation}\n{\\cal G}_{k i j} = 2 \\Gamma_{(i j) k} =\n\t\\partial_k g_{i j}\n\\end{equation}\nand the connection is metric compatible.\\index{metric compatibility} \nIf the momentum\nconstraint\\index{constraint!momentum} \nis violated, metric compatibility\n\\index{metric compatibility} is sacrificed.\nThis shows the price paid to achieve a symmetric\nhyperbolic system that is close to the canonical equations:\nthe momentum constraints\\index{constraint!momentum} \nbecome dynamical, and metric\ncompatibility is lost if the latter are violated. \n\\index{metric compatibility}\n\n\\section{Conformal ``Thin Sandwich'' Data for the Initial Value\nProblem}\n\\label{ConformalTS}\n\nThe standard approach to the initial value problem\n\\index{initial value problem} is the ``conformal\nmethod,''\\index{initial value problem!conformal method} \nthe fundamental rudiments of which were introduced by\nLichnerowicz~\\cite{Lic44}. The essentially complete form was developed by\ntwo of us (YCB and JWY), see~\\cite{Yor73,CBY80}. \nBasic theorems were obtained\nby us and by O'Murchadha~\\cite{OMY73,OMY74a,OMY74b}, \nand Isenberg and Moncrief~\\cite{IsM94}.\nThe older approach concentrates (in the vacuum\ncase to which we restrict ourselves) on the construction of the\nspatial metric\\index{metric!spatial} \n$g_{ij} = \\psi^4 \\gamma_{ij}$ and the traceless\npart $A^{ij}= \\psi^{-10} \\lambda^{ij}$\nof the extrinsic curvature\\index{extrinsic curvature}, where $K_{ij} = A_{ij}\n+ \\frac{1}{3} g_{ij}K$. Here, $\\gamma_{ij}$ is a proper\nriemannian metric\\index{metric!riemannian} \ngiven freely, $\\lambda^{ij}$ is constructed\nby a tensor decomposition method~\\cite{Yor73,Yor74}, \nand $K$ is given freely\n(not conformally transformed). Note that one may as well assume \n$\\det(\\gamma_{ij}) = 1$ because only the conformal equivalence \nclass of the metric\\index{metric!conformal equivalence class} \nmatters: the entire method is ``conformally\ncovariant.''\\index{conformal covariance}\\\\\n\nN.B.: In Sect.~\\ref{ConformalTS}, only spatial metrics\n\\index{metric!spatial} will be used. Therefore,\n{\\em all overbars are dropped in this final section}.\\\\\n\nHere, we discuss a new interpretation of the four Einstein vacuum\n\\index{Einstein equations!vacuum}\ninitial-value constraints\\index{constraint!initial value}. \n(The presence of matter would add\nnothing new to the analysis.) Partly in the spirit of a ``thin\nsandwich'' viewpoint, this approach is based on prescribing the\n{\\em conformal} metric\\index{metric!conformal}~[1] \non each of two nearby spacelike \nhypersurfaces (``time slices'' $t=t^{\\prime}$ and $t = t^{\\prime}\n+ \\delta t$) that make a ``thin sandwich'' (TS). {\\em Essential use\nis made of the understanding of the slicing function in general\nrelativity}. The new formulation could prove useful both \nconceptually and in practice, as a way to construct initial data\nin which one has a hold on the input data different from that in the\ncurrently accepted approach. The new approach allows us to \n{\\em derive} from its dynamical and metrical foundations the \nimportant scaling law ${A}^{ij} = \\psi^{-10} \\lambda^{ij}$ for the\ntraceless part of the extrinsic curvature.\n\\index{extrinsic curvature} This rule is simply\npostulated in the one-hypersurface approach.\n\nThe constraint equations\\index{constraint!vacuum} \non $\\Sigma$ are, in vacuum,\n\\begin{equation}\n\\nabla_j (K^{ij} - K g^{ij}) = 0 \\; ,\n\\end{equation}\n\\begin{equation}\nR(g) - K_{ij}K^{ij} + K^2 = 0\n\\end{equation}\nwhere $R(g)$ is the spatial scalar curvature of $g_{ij},\\; \\nabla_j$\nis the Levi-Civita connection of $g_{ij}$; and $K$ is the trace of\n$K_{ij}$, also called the ``mean curvature'' of the slice. \n\nThe time derivative of the spatial metric\n\\index{metric!spatial} $g_{ij}$ is related to\n$K_{ij}, \\; N$, and the shift vector $\\beta^i$ by\n\\begin{equation}\n\\partial_t g_{ij} \\equiv -2NK_{ij} + (\\nabla_i \\beta_j + \\nabla_j\n\\beta_i) \\; ,\n\\label{gdotKNbeta}\n\\end{equation}\nwhere $\\beta_j = g_{j i} \\beta^i$. The fixed spatial coordinates\n$x$ of a point on the ``second'' hypersurface, as evaluated\non the ``first'' hypersurface, are displaced by \n$\\beta^i(x) \\delta t$ with respect to those on\nthe first hypersurface, with an orthogonal link from the first \nto the second surface as a fiducial reference:\n$\\beta_i = \\frac{\\partial}{\\partial t}\n* \\frac{\\partial}{\\partial x^i}$,\nwhere $*$ is the physical spacetime inner product of the\nindicated natural basis four-vectors. The essentially arbitrary\ndirection of $\\frac{\\partial}{\\partial t}$\nis why $N(x)$ and $\\beta^i(x)$ appear in the TS formulation.\nIn contrast, the tensor $K_{i j}$ is always determined\nby the behavior of the unit normal on one slice and therefore\ndoes not possess the kinematical freedom, i.e., the gauge variance, \nof $\\frac{\\partial}{\\partial t}$. Therefore, $N$\nand $\\beta^i$ do not appear in the one-hypersurface IVP for\n$(\\Sigma, {g}, {K})$.\n\nTurning now to the conformal metrics\\index{metric!conformal} \nin the IVP, we recall that\ntwo metrics $g_{i j}$ and $\\gamma_{i j}$ are conformally \nequivalent\\index{conformal equivalence} \nif and only if there is a scalar $\\psi > 0$ such\nthat $g_{i j} = \\psi^4 \\gamma_{i j}$. The conformally\ninvariant representative of the entire conformal equivalence\nclass\\index{metric!conformal equivalence class}, \nin three dimensions, is the weight $(-2\/3)$ \nunit-determinant ``conformal metric''\\index{metric!conformal} \n$\\hat{g}_{i j} = g^{-1\/3} g_{i j} = \\gamma^{-1\/3} \\gamma_{i j}$\nwith $g = \\mbox{det}(g_{i j})$ and \n$\\gamma = \\mbox{det}(\\gamma_{i j})$. Note particularly\nthat for any small perturbation, $g^{i j} \\delta \\hat{g}{i j} = 0$.\nWe will use the important relation\n\\begin{equation}\ng^{i j} \\partial_t \\hat{g}_{i j} \n = \\gamma^{i j} \\partial_t \\hat{g}_{i j}\n = \\hat{g}^{i j} \\partial_t \\hat{g}_{i j}\n = 0 \\; .\n\\label{ImportantRelation}\n\\end{equation}\n\nIn the following, rather than use the mathematical apparatus \nassociated with conformally weighted objects such as \n$\\hat{g}_{i j}$, we find it simpler to use ordinary scalars\nand tensors to the same effect. Thus, let the role of \n$\\hat{g}_{i j}$ on the first surface be played by a given metric\n$\\gamma_{i j}$ such that the physical metric that satisfies\nthe constraints is $g_{i j} = \\psi^4 \\gamma_{i j}$ for\nsome scalar $\\psi > 0$. (This corresponds to ``dressing''\nthe initial unimodular conformal metric\\index{metric!conformal} \n$\\hat{g}_{i j}$ with \nthe correct determinant factor $g^{1\/3} = \\psi^4 \\gamma^{1\/3}$.\nThis process does not alter the conformal equivalence class of\nthe metric.)\\index{metric!conformal equivalence class} \nThe role of the conformal metric\\index{metric!conformal} on the second\nsurface is played by the metric \n$\\gamma^\\prime_{i j} = \\gamma_{i j} + u_{i j} \\delta t$,\nwhere, in keeping with (\\ref{ImportantRelation}), the \nvelocity tensor $u_{i j} = \\partial_t \\gamma_{i j}$\nis chosen such that\n\\begin{equation}\n\\gamma^{i j} u_{i j} \n = \\gamma^{i j} \\partial_t \\gamma_{i j} = 0 \\; .\n\\end{equation}\nThen, to first order in $\\delta t$, $\\gamma^\\prime_{i j}$ and \n$\\gamma_{i j}$ have equal determinants, as desired;\nbut $\\gamma_{i j}$ and $\\gamma^\\prime_{i j}$ are not\nin the same conformal equivalence class in general.\n\\index{metric!conformal equivalence class}\n\nWe now examine the relation between the covariant derivative\noperators $D_i$ of $\\gamma_{i j}$ and $\\nabla_i$ of\n$g_{i j}$. The relation is determined by\n\\begin{equation}\n\\Gamma^i\\mathstrut_{j k} (g) = \\Gamma^i\\mathstrut_{j k}(\\gamma)\n + 2 \\psi^{-1} \\left( 2 \\delta^i\\mathstrut_{( j} \n \\partial_{k )} \\psi - \\gamma^{i l} \\gamma_{j k}\n \\partial_l \\psi \\right) \\; ,\n\\label{CDrelation}\n\\end{equation}\n{}from which follows the scalar curvature relation first\nused in an initial-value problem by Lichnerowicz~\\cite{Lic},\n\\begin{equation}\nR(g) = \\psi^{-4} R(\\gamma) - 8 \\psi^{-5} \n \\triangle_\\gamma\\psi \\;,\n\\end{equation}\nwhere $\\triangle_\\gamma \\equiv \\gamma^{k l} D_k D_l \\psi$\nis the scalar Laplacian associated with \n$\\gamma_{i j}$.\n\nNext, we solve~(\\ref{gdotKNbeta}) for its traceless part\n\\begin{equation}\n\\partial_t g_{i j} - \\frac{1}{3} g_{i j} g^{k l}\n\t\\partial_t g_{k l} \\equiv V_{i j}\n\t= -2 N A_{i j} + (L_g \\beta)_{i j}\n\\label{TracelessPart}\n\\end{equation}\nwith $A_{i j} \\equiv K_{i j} - (1\/3) K g_{i j}$\nand\n\\begin{equation}\n(L_g \\beta)_{i j} \\equiv \\nabla_i \\beta_j + \\nabla_j \\beta_i\n\t- (2\/3) g_{i j} \\nabla^k \\beta_k \\;.\n\\label{Lgbeta}\n\\end{equation}\n\nExpression (\\ref{Lgbeta}) vanishes, for non-vanishing $\\beta^i$,\nif and only if $g_{i j}$ admits a conformal Killing vector\n$\\beta^i = k^i$. Clearly, $k^i$ would also be a conformal Killing \nvector of $\\gamma_{i j}$, or of any metric conformally\nequivalent\\index{conformal equivalence} \nto $g_{i j}$, with no scaling of $k^i$. This teaches\nus that $\\beta^i$ does not scale. That $\\beta^i$ does not scale\nalso follows because, as generator of a spatial diffeomorphism,\nit is not a dynamical variable. We take the latter ``rule'' \nas a matter of principle.\n\nIt is clear in (\\ref{TracelessPart}) that the left hand side\n$u_{i j}$ satisfies $u_{i j} = \\psi^4 V_{i j}$ because the\nterms in $\\dot\\psi$ cancel out. Furthermore, a straightforward\ncalculation shows that\n\\begin{equation}\n(L_g \\beta)_{i j} = \\psi^4 \\left[ L_\\gamma (\\psi^{-4} \\beta)\\right]_{i j}\n\\, ; \\quad\n(L_g \\beta)^{i j} = \\psi^4 (L_\\gamma \\beta)^{i j}\\; ,\n\\end{equation}\nwhere $\\psi^{-4} \\beta_j = \\gamma_{i j} \\beta^i$.\nNext, we note that the lapse function $N$\\index{lapse}\nhas essential non-trivial\nconformal behavior. This is a new element in the IVP analysis.\nThe slicing function \n$\\alpha(t,x)>0$ can replace the lapse function $N$,\\index{lapse}\n\\begin{equation}\nN = g^{1\/2} \\alpha \\; .\n\\end{equation}\n(The treatment here extends the one in~\\cite{Yor99} in \na simple but interesting way.)\nWe have concluded that $\\alpha$ is not a dynamical variable\nand therefore does not scale. Furthermore, without loss of generality,\nwe can set $\\mbox{det} (\\gamma_{i j}) = 1$ and thus\n$g^{1\/2} = \\psi^6$. Then\n\\begin{equation}\nN = \\psi^6 \\alpha \\; .\n\\end{equation}\nFinally, we fix $K$ and require that it does not scale, \nas in the standard treatment of\nthe IVP~\\cite{Yor72}. \nThis step is absolutely essential for geometric\nconsistency, as we shall see.\n\nNext we solve (\\ref{CDrelation}) for $A^{i j}$, using the scaling\nrules established above, and find\n\\begin{equation}\nA^{i j} = \\psi^{-10} \\left\\{ \\frac{1}{2\\alpha} \\left[\n\t(L_\\gamma \\beta)^{i j} - u^{i j}\\right] \\right\\}\\;.\n\\end{equation}\nThe momentum constraint\\index{constraint!momentum} becomes\n\\begin{equation}\nD_j \\left[ \\frac{1}{2\\alpha} (L_\\gamma \\beta)^{i j}\\right]\n\t= D_j \\left[ \\frac{1}{2 \\alpha} u^{i j}\\right]\n\t+ \\frac{2}{3} \\psi^6 \\gamma^{i j} \\partial_j K \\; ,\n\\label{MomConD}\n\\end{equation}\nwhile the Hamiltonian constraint becomes~\\cite{Yor73}\n\\index{constraint!Hamiltonian}\n\\begin{equation}\n8 \\triangle_\\gamma \\psi - R(\\gamma) \\psi + (\\gamma_{i k} \\gamma_{j l})\n\tA^{i j} A^{k l} \\psi^{-7} - (2\/3) K^2 \\psi^5 = 0\\;.\n\\label{HamConA}\n\\end{equation}\nThe unknowns $(\\psi,\\beta)$ obey equations of the same form as \ndo the conformal scalar potential $\\phi$ and the vector\npotential $W^i$ in\nthe standard analysis~\\cite{Yor73,CBY80}, but {\\em no tensor splittings} are\nrequired. Further, (\\ref{MomConD}) and (\\ref{HamConA}) are\ncoupled in only one direction when $K = \\mbox{const}$.\n\nNow we note two interesting consequences of this approach. First\nwe see that from $N = \\psi^6 \\alpha$, we have identically\n\\begin{equation}\nN = g^{1\/2} \\alpha\n\\end{equation}\nas a consequence of the method. Therefore, time slices $t$ \nand $t+\\delta t$ have a relation that is manifestly ``harmonic:''\n\\begin{equation}\n\\bar{\\partial}_0 N + N^2 K = N \\bar{\\partial}_0 \\log \\alpha\\;,\n\\end{equation}\na result that is fully consistent with our previous discussions\nand requiring that $K$ be a fixed, non-scaling, variable.\n\nFinally, we can establish the final relationships between the\nfull riemannian metrics\\index{metric!riemannian} $g_{i j}(t)$ and \n$g^\\prime_{i j} = g_{i j}(t+\\delta t)$ on the two manifestly\nharmonically related slices $t$ and $t+\\delta t$. As \nin~(\\ref{gdotKNbeta}),\n\\begin{equation}\n\\partial_t g_{i j} = \\partial_t (\\psi^4 \\gamma_{i j})\n\t= g_{i k} g_{j l} \\left[ -2 N (A^{k l} + \\frac{1}{3}\n\tg^{k l} K) + (\\nabla^k \\beta^l + \\nabla^l \\beta^k)\\right] \\;.\n\\label{ggprimeRelation}\n\\end{equation}\nWorking out~(\\ref{ggprimeRelation}) gives\n\\begin{eqnarray}\n\\partial_t g_{i j} &=& \\psi^4 \\left[ u_{i j} + \\gamma_{i j}\n\t\\partial_t (\\psi \\log \\psi) \\right] \\nonumber\\\\\n\t&=& V_{i j} + g_{i j} \\partial_t (\\psi \\log \\psi) \\; ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\partial_t (\\psi \\log \\psi) &=& \\frac{2}{3} \\left( D_k \\beta^k\n\t+ 6 \\beta^k \\partial_k \\log \\psi - \\alpha K \\psi^6 \\right)\n\\nonumber\\\\\n\t&=& \\partial_t (g\/\\gamma)^{1\/2} = \\partial_t (g)^{1\/3}\n\t= \\frac{2}{3} \\left( \\nabla_k \\beta^k - N K \\right) \\; .\n\\end{eqnarray}\nHence, $\\partial_t \\psi$ and $\\partial_t g_{i j}$ are fully\ndetermined and we note that the no-scaling rules for $\\beta^k$ \nand $K$ were essential.\n\n\\section*{Acknowledgments}\nThis paper is based on a lecture given by James W. York Jr.\nat the $2^\\mathrm{nd}$ Samos meeting, 1 September, 1998.\n\nThe authors thank the supporters of the $2^{\\mathrm{nd}}$ Samos\nMeeting and its hosts, especially\nSpiros Cotsakis, for their warm hospitality and the beautiful\nsetting of the meeting. They thank\nMark Hannam and Sarah and Mark Rupright for help in preparing\nthe manuscript. AA and JWY acknowledge support from National\nScience Foundation grant PHY-9413207.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} %\n\n\nMany sequential decision making problems, such as automated health-care, robotics, and online recommendations are high-stakes in terms of health, safety, or finance \\citep{liao2020off,brown2020safe,theocharous2020reinforcement}.\nFor such problems, collecting new data to evaluate the performance of a new decision rule, called an evaluation policy $\\pi_e$, may be expensive or even dangerous if $\\pi_e$ results in undesired outcomes. \nTherefore, one of the most important challenges in such problems is the estimation of the performance $J(\\pi_e)$ of the policy $\\pi_e$ \\textit{before its deployment}.\n\nMany off-policy evaluation (OPE) methods enable estimation of $J(\\pi_e)$ with historical data collected using an existing decision rule, called a behavior policy $\\pi_b$. \nOne popular OPE technique is trajectory-based importance sampling (IS) \\citep{precup2000eligibility}.\nWhile this method is both non-parametric and provides unbiased estimates of $J(\\pi_e)$, it suffers from the \\textit{curse of horizon} and can have variance exponential in the horizon length \\citep{jiang2016doubly,guo2017using}.\nTo mitigate this problem, recent methods use stationary distribution importance sampling (SIS) to adjust the \\textit{stationary distribution} of the Markov chain induced by the policies, instead of the individual trajectories \\citep{liu2018breaking,gelada2019off,nachum2020reinforcement}.\nThis requires (parametric) estimation of the ratio between the stationary distribution induced by $\\pi_e$ and $\\pi_b$.\nUnfortunately, estimating this ratio accurately can require \\textit{unverifiably} strong assumptions on the parameters \\citep{jiang2020minimax}, and often requires solving non-trivial min-max saddle point optimization problems \\citep{yang2020off}.\nConsequently, if the parameterization is not rich enough, then it may not be possible to represent the distribution ratios accurately, and when using rich function approximators (such as neural networks) then the optimization procedure may get stuck in sub-optimal saddle points.\nIn practice, these challenges can introduce error when estimating the distribution ratio, potentially leading to arbitrarily biased estimates of $J(\\pi_e)$, even when an infinite amount of data is available. \n\nIn this work, we present a new perspective on the bias-variance trade-off for OPE that bridges the unbiasedness of IS and the often lower variance of SIS. \nParticularly, we show that\n\\begin{itemize}\n \\item There exists a \\textit{spectrum} of OPE estimators whose end-points are IS and SIS, respectively.\n \\item Estimators in this spectrum can have lower mean-squared error than both IS and SIS.\n \\item This spectrum can also be established for doubly-robust and weighted version of IS and SIS.\n\\end{itemize} \n\nIn Sections \\ref{section:generic} and \\ref{section:accumulate} we show how trajectory-based and distribution-based methods can be combined.\nThe core idea establishing the existence of this spectrum relies upon first splitting individual trajectories into two parts and then computing the probability of the first part using SIS and IS for the latter.\nIn Section \\ref{section:wdr}, we introduce weighted and doubly-robust extensions of the spectrum.\nFinally, in Section \\ref{section:exp}, we present empirical case studies to highlight the effectiveness of these new estimators.\n\n\n\\section{Background} %\n\n\\textbf{Notation:} \nA Markov decision process (MDP) is a tuple $(\\mathcal S, \\mathcal A, r, T, \\gamma, d_1)$, where $\\mathcal S$ is the state set, $\\mathcal A$ is the action set, $r$ is the reward function, $T$ is the transition function, $\\gamma$ is the discounting factor, and $d_1$ is the initial state distribution. \nAlthough our results extend to the continuous setting, for simplicity of notation we assume that $\\mathcal S$ and $\\mathcal A$ are finite.\nA policy $\\pi$ is a distribution over $\\mathcal A$, conditioned on the state.\nStarting from initial state $S_1 \\sim d_1$, policy $\\pi$ interacts with the environment iteratively by sampling action $A_t$ at every time step $t$ from $\\pi(\\cdot | S_t)$.\nThe environment then produces reward $R_t$ with the expected value $r(S_t, A_t)$, and transitions to the next state $S_{t+1}$ according to $T(\\cdot | S_t, A_t)$.\nLet $\\bm{\\tau} \\coloneqq (S_1, A_1, R_1, S_2, ..., S_L, A_L, R_L)$ be the sequence of random variables corresponding to a trajectory sampled from $\\pi$, where $L$ is the horizon length.\nLet $p_{\\pi}$ denote the distribution of $\\bm{\\tau}$ under $\\pi$. \n\n\\textbf{Problem Statement:} \nThe performance of any policy $\\pi$ is given by its value defined by the expected discounted sum of rewards $J(\\pi) \\coloneqq \\mathbf{E}_{\\bm{\\tau} \\sim p_\\pi}[\\sum_{t=1}^L \\gamma^{t-1} R_t]$.\nThe infinite horizon setting can be obtained by letting $L \\rightarrow \\infty$. \nIn general, for any random variable, we use the superscript of $i$ to denote the trajectory associated with it. \nThe goal of the off-policy policy evaluation (OPE) problem is to estimate the performance $J(\\pi_e)$ of an evaluation policy $\\pi_e$ using only a batch of historical trajectories $D \\coloneqq \\{\\tau^{i}\\}_{i=1}^m$ collected from a different behavior policy $\\pi_b$.\nThis problem is challenging because $J(\\pi_e)$ must be estimated using only observational, off-policy data from the deployment of a different behavior policy $\\pi_b$.\nAdditionally, this problem might not be feasible if the data collected using $\\pi_b$ is not informative about the outcomes possible under $\\pi_e$.\nTherefore, to make the problem tractable, we make the following standard support assumption, which implies that any outcome possible under $\\pi_e$ also has non-zero probability of occurring under $\\pi_b$.\n\n\\begin{ass}\n For all $s \\in \\mathcal S$ and $ a \\in \\mathcal A$, the ratio $\\frac{\\pi_e(a|s)}{\\pi_b(a|s)} < \\infty$. \\thlabel{ass:support}\n\\end{ass}\n\n\\textbf{Trajectory-Based Importance Sampling:}\nOne of the earliest methods for estimating $J(\\pi_e)$ is trajectory-based importance sampling.\nThis method corrects the difference in distribution of $\\pi_b$ and $\\pi_e$ by re-weighting the trajectories from $\\pi_b$ in $D$ by the probability ratio of the trajectory under $\\pi_e$ and $\\pi_b$, i.e. %\n$\\frac{p_{\\pi_e}(\\tau)}{p_{\\pi_b}(\\tau)} = \\prod_{t=1}^L \\frac{\\pi_e(A_t | S_t)}{\\pi_b(A_t | S_t)}$.\nLet the single-step action likelihood ratio be denoted $\\bm{\\rho}_t \\coloneqq \\frac{\\pi_e(A_t|S_t)}{\\pi_b(A_t|S_t)}$ and the likelihood ratio from steps $j$ to $k$ be denoted $\\bm{\\rho}_{j:k} \\coloneqq \\prod_{t=j}^k \\bm{\\rho}_t$.\nThe full-trajectory importance sampling (IS) estimator and the per-decision importance sampling (PDIS) estimator \\citep{precup2000eligibility} can then be defined as:\n\\begin{align}\n \\text{IS}(D) \\coloneqq \\frac{1}{m} \\sum_{i=1}^m \\rho_{1:L}^i \\sum_{t=1}^L \\gamma^{t-1} R_t^i , && \\text{PDIS}(D) \\coloneqq \\frac{1}{m} \\sum_{i=1}^m \\sum_{t=1}^L \\gamma^{t-1} \\rho_{1:t}^i R_t^i .\n\\end{align}\nIt was shown by \\citet{precup2000eligibility} that under \\thref{ass:support}, $\\text{IS}(D)$ and $\\text{PDIS}(D)$ are unbiased estimators of $J(\\pi_e)$. That is, $J(\\pi_e) = \\mathbf{E}_{\\bm{\\tau} \\sim p_{\\pi_b}}[\\text{IS}(\\bm{\\tau})] = \\mathbf{E}_{\\bm{\\tau} \\sim \\pi_b}[\\text{PDIS}(\\bm{\\tau})]$.\nUnfortunately, however, both IS and PDIS directly depend on the product of importance ratios and thus can often suffer from exponentially high-variance in the horizon length $L$, known as the ``curse of horizon'' \\citep{jiang2016doubly, guo2017using, liu2018breaking}. \n\n\\textbf{Distribution-Based Importance Sampling:}\nTo eliminate the dependency on trajectory length, recent works apply importance sampling over the state-action space rather than the trajectory space.\nFor any policy $\\pi$, let $d_t^\\pi$ denote the induced state-action distribution at time step $t$, i.e. $d_t^\\pi(s,a) = p_{\\pi}(S_t =s , A_t = a)$.\nLet the average state-action distribution be\n$d^\\pi(s,a) \\coloneqq (\\sum_{t=1}^L \\gamma^{t-1} d_t^\\pi(s, a)) \/ (\\sum_{t=1}^L \\gamma^{t-1}$).\nThis gives the likelihood of encountering $(s,a)$ when following policy $\\pi$ and averaging over time with $\\gamma$-discounting.\nLet $(S, A) \\sim d^{\\pi}$ and $(S, A) \\sim d_t^\\pi$ denote that $(S,A)$ are sampled from $d^{\\pi}$ and $d_t^\\pi$ respectively.\nThe performance of $\\pi_e$ can be expressed as,\n \\begin{align}\n J(\\pi_e) &= \\mathbf{E}_{\\bm{\\tau} \\sim p_{\\pi_e}}\\left[\\sum_{t=1}^L \\gamma^{t-1} R_t\\right] = \\sum_{s,a} \\sum_{t=1}^L \\gamma^{t-1}\\ d_t^{\\pi_e}(s, a) r(s,a)\n = \\left(\\sum_{t=1}^L \\gamma^{t-1}\\right) \\sum_{s,a} d^{\\pi_e}(s,a) r(s,a) \\\\\n &\\overset{(a)}{=} \\left(\\sum_{t=1}^L \\gamma^{t-1}\\right) \\sum_{s,a} d^{\\pi_b}(s,a) \\frac{d^{\\pi_e}(s,a)}{d^{\\pi_b}(s,a)} r(s,a)\n = \\sum_{s,a} \\sum_{t=1}^L \\gamma^{t-1} d_t^{\\pi_b}(s,a) \\frac{d^{\\pi_e}(s,a)}{d^{\\pi_b}(s,a)} r(s,a),\n \\\\\n &= \\mathbf{E}_{\\bm \\tau \\sim p_{\\pi_b}}\\left[\\sum_{t=1}^L \\gamma^{t-1} \\frac{d^{\\pi_e}(S_t,A_t)}{d^{\\pi_b}(S_t,A_t)} R_t\\right],\n \\end{align} \\label{eq: SIS_target}\nwhere (a) is possible due to \\thref{ass:support}. \nUsing this observation, recent works have considered the following stationary-distribution importance sampling estimator \\citep{liu2018breaking,yang2020off,jiang2020minimax},\n\\begin{align}\n \\text{SIS}(D) &\\coloneqq \\frac{1}{m} \\sum_{i=1}^m \\sum_{t=1}^L \\gamma^{t-1} w(S_t^i, A_t^i) R_t^i, \\label{eq:SIS}\n\\end{align}\nwhere $w(s,a) \\coloneqq \\frac{d^{\\pi_e}(s,a)}{d^{\\pi_b}(s,a)}$ is the distribution correction ratio.\nNotice that SIS$(\\tau)$ marginalizes over the product of importance ratios \n$\\rho_{1:t}$, and thus can help in mitigating variance's dependence on horizon length for PDIS and IS estimators. \nWhen an unbiased estimate of $w$ is available, then SIS($\\tau$) is also an unbiased estimator, i.e., $\\mathbf{E}_{\\bm{\\tau}\\sim\\pi_b}[\\text{SIS}(\\tau)] = J(\\pi_e)$.\nUnfortunately, such an estimate of $w$ is often not available.\nFor large-scale problems, parametric estimation $w$ is required in practice and we replace the true density ratios $w$ with an estimate $\\hat{w}$.\nHowever, estimating $w$ accurately may require both a non-verifiable strong assumption on the parametric function class, and global solution to a non-trivial min-max optimization problem \\citep{jiang2020minimax,yang2020off}.\nWhen these conditions are not met, SIS estimates can be arbitrarily biased, even when an infinite amount of data is available. \n\n\\section{Combining Trajectory-Based and Density-Based Importance Sampling} \\label{section:generic}\n\nTrajectory-based and distribution-based importance sampling methods are typically presented as alternative methods of applying importance sampling for off-policy evaluation.\nHowever, in this section we show that the choice of estimator is not binary, and these two styles of computing importance weights can actually be combined into a single importance sampling estimate.\nFurthermore, using this combination, in the next section, we will derive a spectrum of estimators that allows interpolation between the trajectory-based PDIS and distribution-based SIS, which will often allow us trade-off between the strengths and weaknesses of these methods.\n\nIntuitively, trajectory-based and distribution-based importance sampling provide two different ways of correcting the distribution mismatch under the evaluation and behavior policies.\nTrajectory-based importance sampling corrects the distribution mismatch by examining how likely policies are to take the same sequence of actions and thus applies the action likelihood ratio as the correction term.\nDistribution-based importance sampling corrects the mismatch by how likely policies are to visit the same state and action pairs---while remaining agnostic to \\textit{how} they arrived---and applies the distribution ratio as the importance weight.\nHowever, using distribution ratio and action likelihood ratio correction terms are not mutually exclusive, and one can draw on both types of correction terms to derive combined estimators. \n\n\\begin{SCfigure}[][t]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{Images\/actual_two_rooms.jpeg} \n \\caption{Illustration of two room domain. The domain consists of two rooms, the left room and the right room separated by a connecting door.\n $\\pi_b$ and $\\pi_e$ are two different policies that move from the left room to the right room.\n Note that, although $\\pi_b$ and $\\pi_e$ have two different behaviors in the left room and right room, both pass through the connecting door.}\n\\end{SCfigure}\n\\label{fig:2rooms}\n\n\nTo build intuition for why likelihood ratios and distribution ratios can naturally be combined, we consider the two rooms domain shown in Figure \\ref{fig:2rooms}.\nIn this example, there are two policies $\\pi_b, \\pi_e$ which have different strategies for navigating from the first room to the second room.\nNote that while the behavior of the two policies are very different in the left room, both policies must pass through the connecting door to get to the right room at some point in time.\nConditioning on having passed through the connecting door at a point in time, all parts of the trajectory that occur in the right room are independent from what has occurred in the left room by the Markov property.\nThus, when considering a reward $R_t$ that occurs in the right room, it is natural to consider the probability of reaching the door and then the probability of the action sequence policy in the right room under each policy.\n\nNow, we formalize this intuition and show how trajectory-based and density-based importance sampling can be combined in the same estimator.\nGiven a trajectory $\\bm{\\tau}$, we can consider $(S_z, A_z)$, the state and action at time $z$ in the trajectory.\nBy conditioning on $(S_z, A_z)$, trajectory $\\bm{\\tau}$ can be separated into two conditionally independent partial trajectories $\\bm{\\tau}_{0:z}$ and $\\bm{\\tau}_{z+1, L}$ by the Markov property.\nSince the segments of $\\bm{\\tau}$ before and after time $z$ are conditionally independent, then $\\rho_{1:z}$, the likelihood ratio for the trajectory before time $z$, is conditionally independent from $\\rho_{z+1:L}$ and from $R_t$ for all $t \\geq z$.\nFormally, let $(S_z, A_z) \\sim d_z^{\\pi_b}$, then,\n\\begin{align}\n J(\\pi_e) &= \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}}[\\text{PDIS}(\\bm{\\tau})] \n = \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}} \\left[\\sum_{t=1}^L \\gamma^{t-1} \\rho_{1:t} R_t\\right] \\\\\n &= \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}} \\left[\\sum_{t=1}^{z} \\gamma^{t-1} \\rho_{1:t} R_t\\right] + \\mathbf{E}_{\\substack{\\scriptscriptstyle(S_{z}, A_{z}) \\\\ \\scriptscriptstyle \\sim d_z^{\\pi_b}}}\\left[\\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}}\\left[ \\sum_{t=z+1}^L \\gamma^{t-1} \\rho_{1:z} \\rho_{z+1:t} R_t \\middle| S_{z}, A_{z} \\right]\\right] \\\\\n &= \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}} \\left[\\sum_{t=1}^{z} \\gamma^{t-1} \\rho_{1:t} R_t\\right] + \\mathbf{E}_{\\substack{\\scriptscriptstyle (S_{z}, A_{z}) \\\\ \\scriptscriptstyle \\sim d_z^{\\pi_b}}}\\left[ \\sum_{t=z+1}^L \\gamma^{t-1} \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}}\\left[\\rho_{1:z} \\middle| S_{z}, A_{z}\\right] \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim \\pi_b} \\left[\\rho_{z+1:t} R_t \\middle| S_{z}, A_{z} \\right]\\right] \\\\\n &\\overset{(a)}{=} \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}} \\left[\\sum_{t=1}^{z} \\gamma^{t-1} \\rho_{1:t} R_t\\right] + \\mathbf{E}_{\\substack{\\scriptscriptstyle (S_{z}, A_{z}) \\\\ \\scriptscriptstyle \\sim d_z^{\\pi_b}}}\\left[ \\sum_{t=z+1}^L \\gamma^{t-1} \\frac{d_z^{\\pi_e}(S_{z}, A_{z})}{d_z^{\\pi_b}(S_{z}, A_{z})} \\mathbf{E}_{\\scriptscriptstyle\\bm{\\tau} \\sim p_{\\pi_b}} \\Big[\\rho_{z+1:t} R_t \\Big| S_{z}, A_{z} \\Big]\\right] \\\\\n &= \\mathbf{E}_{\\scriptscriptstyle \\bm{\\tau} \\sim p_{\\pi_b}} \\left[\\sum_{t=1}^{z} \\gamma^{t-1} \\rho_{1:t} R_t + \\sum_{t=z+1}^L \\gamma^{t-1} \\frac{d_z^{\\pi_e}(S_z, A_z)}{d_z^{\\pi_b}(S_z, A_z)} \\rho_{z+1:t} R_t \\right], \\label{interp}\n\\end{align}\n\nwhere (a) follows from the following \\thref{lemma:1}, which states that the expected value of product likelihood ratios $\\rho_{1:z}$ conditioned on $(S_z, A_z)$ is equal to the time-dependent state-action distribution ratio for $(S_z, A_z)$. We provide a detailed proof of \\thref{lemma:1} in Appendix \\ref{append:lemma1}.\n\n\\begin{prop}[\\citep{liu2018breaking}] \n Under \\thref{ass:support}, $\\mathbf{E}_{\\bm{\\tau} \\sim p_{\\pi_b}} [\\rho_{1:t} | S_{t} = s, A_{t} = a] = \\frac{d_t^{\\pi_e}(s,a)}{d_t^{ \\pi_b}(s, a)}$.\n \\thlabel{lemma:1}\n\\end{prop}\n\n\nObserve that Eq \\eqref{interp} is indexed by time $z$.\nIntuitively, $z$ can be thought of as the time to switch from using distribution ratios to action likelihood ratios in the importance weight.\nSpecifically, the distribution ratios are used to estimate the probability of being in state $S_z$ and taking action $A_z$ at time $z$ and action likelihood ratios are used to correct for the probability of actions taken after time $z$.\nFurther observe that $z$ does not have to be a fixed constant---$z(t)$ can be a function of $t$ so that each reward in the trajectory $R_t$ can utilize a different switching time.\nIn the next section, we show that by using a function $z(t)$ that allows the switching time to be time-dependent, we are able to \nfurther marginalize over time and\ncreate an estimator that interpolates between \\textit{average} state-action distribution ratios $w(s,a) = \\frac{d^{\\pi_e}(s,a)}{d^{\\pi_b}(s,a)}$, rather than time-dependent distribution ratios $\\frac{d_t^{\\pi_e}(s,a)}{d_t^{\\pi_b}(s,a)}$.\n\n\n\\section{Bias-Variance Trade-off using \\texorpdfstring{$\\bm{n}$}{n}-step Interpolation Between PDIS and SIS} \\label{section:accumulate} %\n\n\\begin{figure*}[t!]\n \\subfloat[PDIS]{%\n \\includegraphics[trim=20 40 50 30,clip, width=0.3\\textwidth]{Images\/PDIS.jpeg}}\n\\hspace{\\fill}\n \\subfloat[$\\text{SOPE}_n$]{%\n \\includegraphics[trim=10 40 10 30,clip, width=0.3\\textwidth]{Images\/SOPE.jpeg}}\n\\hspace{\\fill}\n \\subfloat[SIS]{%\n \\includegraphics[trim=30 40 30 30,clip, width=0.3\\textwidth]{Images\/SIS.jpeg}}\\\\\n \\caption{Illustrations of the PDIS, $\\text{SOPE}_n$ and SIS estimators. The dotted blue line represents an example trajectory drawn from $\\pi_e$, and the solid red line represents an example trajectory from $\\pi_b$. All three importance sampling methods work by re-weighting each reward $R_t$ in the trajectory from $\\pi_b$. (a) Trajectory-based PDIS works by re-weighting each reward by $\\frac{p_{\\pi_e}(\\tau_{1:t})}{p_{\\pi_b}(\\tau_{1:t})}$, the probability ratio of the sub-trajectory leading up to $R_t$ under the $\\pi_b$ and $\\pi_e$, respectively. This factors into $\\rho_{1:t}$, the product of $t$ action likelihood ratios. (c) Distribution-based SIS considers the probability of encountering $(S_t, A_t)$ under $\\pi_e$ and $\\pi_b$, and re-weights $R_t$ by $\\frac{d^{\\pi_e}(S_t, A_t)}{d^{\\pi_b}(S_t, A_t)}$,\n (b) $\\text{SOPE}_n$ combines trajectory and distribution importance sampling weights by considering the probability of each policy visiting $(S_{t-n}, A_{t-n})$, the state-action pair $n$ steps in the past, and additionally the probability of the sub-trajectory $\\tau_{t-n+1:t}$ from $n$ steps in the past to $t$. Thus, $\\text{SOPE}_n$ re-weights $R_t$ by $\\frac{d^{\\pi_e}(S_{t-n}, A_{t-n})}{d^{\\pi_b}(S_{t-n}, A_{t-n})} \\rho_{t-n+1:t}$.\n }\\label{fig:reweighting}\n\\end{figure*} \n\n\n\nWe now build upon the ideas from Section \\ref{section:generic} to derive a spectrum of off-policy estimators that allows for interpolation between the trajectory-based PDIS and distribution-based SIS estimators.\nThis spectrum contains PDIS and SIS at the endpoints and allows for smooth interpolation between them to obtain new estimators that can often trade-off the strengths and weaknesses of PDIS and SIS.\nAn illustration of the key idea can be found in Figure \\ref{fig:reweighting}.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.95\\textwidth]{Images\/interp2.png}\n \\caption{\n On the left side of the figure, we show an illustration the $\\text{SOPE}_n$ spectrum of estimators.\n For the purpose of this illustration, consider that only at the last time step there is a non-zero reward $R_L$.\n The $\\text{SOPE}_n$ spectrum allows for control of how much an estimate depends on distribution ratios vs action likelihood ratios.\n Notice that $\\text{SOPE}_0$ results in SIS, $\\text{SOPE}_L$ results in PDIS estimator, and other values of $n$ result in new interpolated estimators.\n As an analogy, consider the backup-diagram \\citep{SuttonBarto2} for the n-step q-estimate as illustrated on the right-hand side of the solid vertical line. \n Notice that in the $n$-step q-estimate, returns are backed up from possible \\textit{future outcomes}, whereas in the $n$-step interpolation estimators the probabilities are `backed-up' from the possible \\textit{histories}. \n (In the diagram, bias-variance characterization of PDIS and SIS is based on typical practical observations \\citep{voloshin2019empirical,fu2021benchmarks}, however it is worth noting that SIS is not biased when oracle density ratios are available, and there are also edge cases, particularly for short horizon problems, where SIS can have higher variance than PDIS \\citep{liu2020understanding,metelli2020importance}).}\n \\label{fig:nstep}\n\n\\end{figure}\n\n\n\n\n\n\n\nOne simple way to perform this trade-off is to control the number of terms in the product in the action likelihood ratio for each reward $R_t$.\nSpecifically, for any reward $R_t$, we propose including only the $n$ most recent action likelihood ratios $\\bm{\\rho}_{t-n+1:t}$ in the importance weight, rather than $\\bm{\\rho}_{1:t}$.\nThus, the overall importance weight becomes the re-weighted probability of visiting $(S_{t-n}, A_{t-n})$, followed by the re-weighted probability of taking the last $n$ actions leading up to reward $R_t$.\nThis reduces the exponential impact that horizon length $L$ has on the variance of PDIS, and provides control over this reduction via the parameter $n$. %\nTo get an estimator to perform this trade-off, we start with the derivation in \\eqref{interp} with $z(t) = t - n$, then accumulate the time-dependent state-action distributions $d_t$ over time. \nThe final expression for the finite horizon setting requires some additional constructs and is thus presented along with its derivations and additional discussion in Appendix \\ref{append:NStep}. \nIn the following we present the result for the infinite horizon setting,\n\\begin{align}\n J(\\pi_e) = \\mathbf{E}_{\\bm{\\tau} \\sim p_{\\pi_b}} \\left[ \\sum_{t=1}^n \\gamma^{t-1} \\rho_{1:t} R_t + \\sum_{t=n+1}^\\infty \\gamma^{t-1} \\frac{d^{\\pi_e}(S_{t-n}, A_{t-n})}{d^{\\pi_b}(S_{t-n}, A_{t-n})} \\rho_{t-n+1:t} R_t\\right]. \\label{eq:nstep_final}\n\\end{align}\n\nUsing the sample estimate of \\eqref{eq:nstep_final}, we obtain the Spectrum of Off-Policy Estimators ($\\text{SOPE}_n$),\n\\begin{equation} \\label{eq:nstep_est}\n\\text{SOPE}_n(D) = \\frac{1}{m} \\sum_{i=1}^{m} \\left(\\sum_{t=1}^{n} \\gamma^{t-1} \\rho_{1:t}^i R_t^i + \\sum_{t=n+1}^\\infty \\gamma^{t-1} \\hat w(S_{t-n}^i, A_{t-n}^i) \\rho_{t-n+1:t}^i R_t^i\\right).\n\\end{equation}\n\n\n\\begin{rem}\nNote that since we generally do not have access to the true density ratios, in practice we substitute $w$ with the estimated density ratios $\\hat w$ similarly as in SIS.\nSince $\\text{SOPE}_n$ is agnostic to how $\\hat w$ is estimated, it can readily leverage existing and new methods for estimating $\\hat w$.\n\\end{rem}\n\n \nObserve that $\\text{SOPE}_n$ doesn't just give a single estimator, but a spectrum of off-policy estimators indexed by $n$.\nAn illustration of this spectrum can be seen in Figure \\ref{fig:nstep}. \nAs $n$ decreases, the number of terms in the action likelihood ratio decreases, and $\\text{SOPE}_n$ depends more on the distribution correction ratio and is more like SIS.\nLikewise as $n$ increases, the number of terms in the action likelihood ratio increases, and $\\text{SOPE}_n$ is closer to PDIS.\nFurther note that that for the endpoint values of this spectrum, $n=0$ and $n=L$, $\\text{SOPE}_n$ gives the SIS and PDIS estimators exactly (for PDIS, horizon length needs to be $L$ instead of $\\infty$ for the estimator to be well defined),\n\\begin{align*}\n \\text{SOPE}_0(D) &= \\frac{1}{m} \\sum_{i=1}^{m} \\sum_{t=1}^{L} \\gamma^{t-1} w(S_t^i, A_t^i) R_t^i = \\text{SIS}(D), \\\\\n \\text{SOPE}_L(D) &= \\frac{1}{m} \\sum_{i=1}^{m} \\sum_{t=1}^L \\gamma^{t-1} \\rho_{1:t}^i R_t^i = \\text{PDIS}(D).\n\\end{align*}\n\n\n\n\n\n\n\\section{Doubly-Robust and Weighted IS Extensions to $\\text{SOPE}_n$} \\label{section:wdr}\n\n\n\n\n\n\nAn additional advantage of $\\text{SOPE}_n$ is that it can be readily extended to obtain a spectrum for other estimators.\nFor instance, to mitigate variance further a popular technique is to leverage domain knowledge from (imperfect) models using doubly-robust estimators \\citep{jiang2016doubly,jiang2020minimax}. \nIn the following we can create a doubly robust version of the $\\text{SOPE}_n$ estimator. \n\nBefore moving further, we introduce some additional notation. Let,\n\\begin{align}\n w(t,n) \\coloneqq \\begin{cases}\n \\frac{d^{\\pi_e}(S_{t-n},A_{t-n})}{d^{\\pi_b}(S_{t-n},A_{t-n})}\\left(\\prod_{j=0}^{n-1} \\frac{\\pi_e(A_{t-j}|S_{t-j})}{\\pi_b(A_{t-j}|S_{t-j})}\\right) & \\text{if } t > n \\\\\n \\prod_{j=1}^{t} \\frac{\\pi_e(A_{j}|S_{j})}{\\pi_b(A_{j}|S_{j})} & 1 \\leq t \\leq n\n \\\\\n 1 & \\text{otherwise}.\n \\end{cases}\n\\end{align}\n\nLet $q$ be an estimate for the q-value function for $\\pi_e$, computed using the (imperfect) model.\nFor brevity, we make the random variable $\\bm{\\tau} \\sim p_{\\pi_b}$ implicit for the expectations in this section. \nFor a given value of $n$, performance \\eqref{eq:nstep_final} of $\\pi_e$ can then\nbe expressed as,\n\\begin{align}\n J(\\pi_e) = \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} R_t \\right] .\n\\end{align}\nWe now use this form to create a spectrum of doubly-robust estimators,\n{\n\\footnotesize\n\\begin{align}\n J(\\pi_e) &= \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} R_t \\right] + \\underbrace{\\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} q(S_t,A_t) \\right] - \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} q(S_t,A_t) \\right] }_{=0}\n \\\\\n &\\overset{(a)}{=} \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} R_t \\right] + \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t-1,n)\\gamma^{t-1} q(S_t,A_t^{\\pi_e}) \\right] - \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} q(S_t,A_t)\\right]\n \\\\\n &= \\mathbf{E}\\left[ w(0,n)\\gamma^0 q(S_1,A_1^{\\pi_e}) \\right] + \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} \\Big( R_t + \\gamma q(S_{t+1},A_{t+1}^{\\pi_e}) - q(S_t,A_t) \\Big) \\right]\n \\\\\n &= \\mathbf{E}\\Big[ q(S_1,A_1^{\\pi_e}) \\Big] + \\mathbf{E}\\left[ \\sum_{t=1}^{\\infty} w(t,n)\\gamma^{t-1} \\Big( R_t + \\gamma q(S_{t+1},A_{t+1}^{\\pi_e}) - q(S_t,A_t) \\Big)\\right], \\label{eqn:DRn} \n\\end{align}\n}\nwhere in (a) we used the notation $A_t^{\\pi_e}$ to indicate the $A_t \\sim \\pi_e(\\cdot | S_t)$.\nUsing $A_t^{\\pi_e}$ eliminates the need for correcting $A_t$ sampled under $\\pi_b$.\nWe define $\\text{DR-SOPE}_n(D)$ to be the sample estimate of \\eqref{eqn:DRn}, i.e., a doubly-robust form for the $\\text{SOPE}_n(D)$ estimator. \nIt can now be observed that existing doubly-robust estimators are end-points of $\\text{DR-SOPE}_n(D)$ (for trajectory-wise settings, horizon length needs to be $L$ instead of $\\infty$ for the estimator to be well defined),\n\n\\begin{align}\n &\\text{DR-SOPE}_L(D) = \\text{Trajectory-wise DR \\citep{jiang2016doubly, thomas2016data}},\n \\\\\n &\\text{DR-SOPE}_0(D) = \\text{State-action distribution DR \\citep{jiang2020minimax,kallus2020double}}.\n\\end{align}\n\nA variation of PDIS that can often also help in mitigating the variance of PDIS method is the Consistent Weighted Per-Decision Importance Sampling estimator (CWPDIS) \\citep{thomas2015safe}.\nCWDPIS renormalizes the importance ratio at each time with the sum of importance weights, which causes CWPDIS to be biased (but consistent) and often have lower variance than PDIS.\n\\begin{align*}\n \\text{CWPDIS}(D) := \\sum_{t=1}^L \\gamma^{t-1} \\frac{\\sum_{i=1}^m \\rho_{1:t}^{i} R_t^i}{\\sum_{i=1}^m \\rho_{1:t}^{i}}.\n\\end{align*}\nSimilar $\\text{DR-SOPE}_n$, we can create a weighted version of $\\text{SOPE}_n$ estimator that interpolates between a weighted-version of SIS and CWPDIS,\n\\begin{equation}\n \\text{W-SOPE}_n(D) \\coloneqq \\sum_{t=1}^{n} \\left(\\gamma^{t-1} \\sum_{i=1}^m \\frac{ \\rho_{1:t}^i}{\\sum_{i=1}^m \\rho_{1:t}^i} R_t^i\\right) + \\sum_{t=n+1}^\\infty \\left(\\gamma^{t-1} \\sum_{i=1}^m \\frac{ w(S_{t-n}^i, A_{t-n}^i) \\rho_{t-n+1:t}^i}{\\sum_{i=1}^m w(S_{t-n}^i, A_{t-n}^i) \\rho_{t-n+1:t}^i} R_t^i\\right). \\label{wnstep}\n\\end{equation}\n\nSince, unlike PDIS, CWPDIS is a biased (but consistent) estimator, $\\text{W-SOPE}_n$ interpolates between two biased estimators as endpoints.\nNonetheless, we show experimentally in Section \\ref{section:exp} that in practice $\\text{W-SOPE}_n$ estimators for intermediate values of $n$ can still outperform weighted-SIS and CWPDIS.\n\n\\section{Experimental Results} \\label{section:exp}\n\n\n\n\n\\begin{figure}[t]\n\\subfloat[$\\text{SOPE}_n$ on Graph Domain]{%\n \\includegraphics[clip,width=\\textwidth]{Images\/graph_unweighted.png}%\n}\n\n\\subfloat[$\\text{SOPE}_n$ on Toy Mountain Car Domain]{%\n \\includegraphics[clip,width=\\textwidth]{Images\/toymc_unweighted.png}%\n}\n \\caption{Experimental results from evaluating the $\\text{SOPE}_n$ estimator on the Graph and Toy Mountain Car domains.\n The $x$-axis for each plot indicates the value of $n$ in the $\\text{SOPE}_n$ estimate.\n The shaded regions denote 95\\% confidence regions on the mean of MSE. \n Recall that $\\text{SOPE}_0$ gives SIS and $\\text{SOPE}_L$ gives PDIS.\n The evaluation and behavior policies are $\\pi_e(a = 0) = 0.9$ and $\\pi_b(a = 0) = 0.5$ for the experiments on the Graph Domain and and $\\pi_e(a = 0) = 0.5$ and $\\pi_b(a = 0) = 0.6$ for the Toy Mountain Car domain.\n In both these domains, we can see that there exist interpolating estimators in the $\\text{SOPE}_n$ spectrum that outperform SIS and PDIS, and that the $\\text{SOPE}_n$ spectrum empirically performs a bias-variance trade-off.\n }\n\\label{fig:unweighted}\n\\end{figure}\n\n\\begin{figure}[t]\n\\subfloat[$\\text{W-SOPE}_n$ on Graph Domain]{%\n \\includegraphics[clip,width=\\textwidth]{Images\/graph_vary_traj.png}%\n}\n\n\\subfloat[$\\text{W-SOPE}_n$ on Toy Mountain Car Domain]{%\n \\includegraphics[clip,width=\\textwidth]{Images\/toymc_vary_traj.png}%\n}\n \\caption{\n Experimental results from evaluating the $\\text{W-SOPE}_n$ estimator on the Graph and Toy Mountain Car domains.\n The $x$-axis for each plot indicates the value of $n$ in the $\\text{SOPE}_n$ estimate.\n The shaded regions denote 95\\% confidence regions on the mean of MSE. \n Recall that $\\text{W-SOPE}_0$ gives weighted-SIS and $\\text{W-SOPE}_L$ gives CWPDIS.\n The evaluation and behavior policies are $\\pi_e(a = 0) = 0.9$ and $\\pi_b(a = 0) = 0.7$ for the experiments on the Graph Domain and and $\\pi_e(a = 0) = 0.5$ and $\\pi_b(a = 0) = 0.9$ for the Toy Mountain Car domain.\n In both these domains, we can see that although we do not get as clean of a bias-variance trade-off as when we use $\\text{SOPE}_n$, there still exist interpolating estimators in the $\\text{W-SOPE}_n$ spectrum that outperform SIS and PDIS.\n}\n\\vspace{-5pt}\n\\label{fig:weighted}\n\\end{figure}\n\n\n\nIn this section, we present experimental results showing that interpolated estimators within the $\\text{SOPE}_n$ and $\\text{W-SOPE}_n$ spectrums can outperform the SIS\/weighted-SIS and PDIS\/CWPDIS endpoints.\nIn each experiment, we evaluate $\\text{SOPE}_n$ and $\\text{W-SOPE}_n$ for different values of $n$ ranging from $0$ to $L$.\nThis allows us to compare the different estimators we get for each $n$ and see trends of how the performance changes as $n$ varies.\nAdditionally, we plot estimates of the bias and the variance for the different values of $n$ to further investigate the properties of estimators in this spectrum.\n\nFor our experiments, we utilize the environments and implementations of baseline estimators in the Caltech OPE Benchmarking Suite (COBS) \\citep{voloshin2019empirical}.\nIn this section, we present results on the Graph and Toy Mountain Car environments.\nTo obtain an estimate of the density ratios $\\hat w$, we use COBS's implementation of infinite horizon methods from \\citep{liu2018breaking}.\nFull experimental details and additional experimental results can be found in Appendix \\ref{append:experiments}. \nAdditional experiments include an investigation on the impact on the degree of $\\pi_e$ and $\\pi_b$ mismatch on $\\text{SOPE}_n$ and $\\text{W-SOPE}_n$, as well as additional experiments on the Mountain Car domain.\n\nThe experimental results for the $\\text{SOPE}_n$ and $\\text{W-SOPE}_n$ estimators can be seen in Figures \\ref{fig:unweighted} and \\ref{fig:weighted} respectively.\nWe observe that for both $\\text{SOPE}_n$ and $\\text{W-SOPE}_n$, the plots of mean-squared error (MSE) have a U-shape indicating that there exist interpolated estimators within the spectrum with lower MSE than the endpoints.\nAdditionally, from the bias and variance plots, we can see that $\\text{SOPE}_n$ performs a bias-variance trade-off in these experiments.\nWe observe that as $n$ increases and the estimators become closer to PDIS, the bias decreases but the variance increases.\nLikewise, as $n$ decreases and the estimators become closer to SIS, the variance decreases but the bias increases.\nThis bias-variance trade-off trend is very notable for the unweighted $\\text{SOPE}_n$ which trades-off between biased SIS and unbiased PDIS endpoints.\nHowever, we still can see this trend even with the $\\text{W-SOPE}_n$ estimator, although the trade-off is not as clean because $\\text{W-SOPE}$ interpolates between biased SIS and the also biased (but consistent) CWPDIS.\n\nFinally, note that our plots also show the results for different batch sizes of historical data.\nIn our plots, as batch size increases, for some domains the PDIS\/CWPDIS endpoints eventually outperform the SIS\/weighted-SIS endpoints.\nHowever, even in this case, there still exist interpolated estimators that outperform both endpoints.\n\n\n\n\\section{Related Work}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nOff-policy evaluation (also related to counterfactual inference in the causality literature \\citep{pearl2009causality}) is one the most crucial aspects of RL, and importance sampling \\citep{metropolis1949monte, horvitz1952generalization} plays a central role in it.\n\\cite{precup2000eligibility} first introduced IS, PDIS, and WIS estimates for OPE.\nSince then there has been a flurry of research in this direction: using partial-models to develop doubly robust estimators \\citep{jiang2016doubly, thomas2016data}, using multi-importance sampling \\citep{papini2019optimistic,metelli2020importance}, estimating the behavior policy \\citep{hanna2019importance}, clipping importance ratios \\citep{bottou2013counterfactual, thomas2015higha, munos2016safe, schulman2017proximal}, dropping importance ratios \\citep{guo2017using}, importance sampling the entire return distribution \\citep{chandak2021universal}, importance resampling of trajectories \\citep{schlegel2019importance}, emphatic weighting of TD methods \\citep{mahmood2015emphatic,hallak2016generalized, patterson2021investigating}, and estimating state-action distributions \\citep{hallak2017consistent, liu2018breaking, gelada2019off, xie2019towards, nachum2020reinforcement, yang2020off, zhang2020gendice, jiang2020minimax, uehara2020minimax}.\n\n\n\n\nPerhaps the most relevant to our work are the recent works by \\cite{liu2020understanding} and \\cite{rowland2020conditional} that use the conditional IS (CIS) framework to show how IS, PDIS, and SIS are special instances of CIS.\nSimilarly, our proposed method for combining trajectory and density-based importance sampling also falls under the CIS framework.\n\\cite{liu2020understanding} also showed that in the finite horizon setting, none of IS, PDIS, or SIS has variance \\textit{always} lesser than the other. \nSimilarly, \\cite{rowland2020conditional} used sufficient conditional functions to create new off-policy estimators and showed that return conditioned estimates (RCIS) can provide optimal variance reduction.\nHowever, using RCIS requires a challenging task of estimating \\textit{density ratios for returns} (not state-action pair)\nand \\cite{liu2020understanding} established a negative result that estimating these ratios using linear regression may result in the IS estimate itself.\n\n Our analysis complements these recent works by showing that there exists interpolated estimators that can provide lower variance estimates than any of IS, PDIS, or SIS.\nOur proposed estimator $\\text{SOPE}_n$ provides a natural interpolation technique to trade-off between the strengths and weaknesses of these trajectory and density based methods.\nAdditionally, while it is known that $q^\\pi(s,a)$ and \n$d^\\pi(s,a)$ have a primal-dual connection \\citep{wang2007dual}, our time-based interpolation technique also sheds new light on connections between their n-step generalizations.\n\n\\section{Conclusions} \\label{conclusions}\n\n\nWe present a new perspective in off-policy evaluation connecting two popular estimators, PDIS and SIS, and show that PDIS and SIS lie as endpoints on the Spectrum of Off-Policy Estimators $\\text{SOPE}_n$ which interpolates between them.\nAdditionally, we also derive a weighted and doubly robust version of this spectrum of estimators.\nWith our experimental results, we illustrate that estimators that lie on the interior of the $\\text{SOPE}_n$ and $\\text{W-SOPE}_n$ spectrums can be used outperform their endpoints SIS\/weighted-SIS and PDIS\/CWPDIS.\n\nWhile we are able to show there exist $\\text{SOPE}_n$ estimators that are able to outperform PDIS and SIS, it remains as future work to devise strategies to automatically select $n$ to trade-off bias and variance.\nFuture directions may include developing methods to select $n$ or combine all estimators for all $n$ using $\\lambda$-trace methods \\citep{SuttonBarto2} to best trade-off bias and variance. \n\n\n\nFinally, like all off-policy evaluation methods, our approach carries risks if used inappropriately. When using OPE for sensitive or safety-critical applications such as medical domains, caution should be taken to carefully consider the variance and bias of the estimator that is used. In these cases, high-confidence OPE methods \\citep{thomas2015higha} may be more appropriate.\n\n\n\\section{Acknowledgement}\n\nWe thank members of the Personal Autonomous Robotics Lab (PeARL) at the University of Texas at Austin for discussion and feedback on early stages of this work.\nWe especially thank Jordan Schneider, Harshit Sikchi, and Prasoon Goyal for reading and giving suggestions on early drafts.\nWe additionally thank Ziyang Tang for suggesting an additional marginalization step in the main proof that helped us unify the results for finite and infinite horizon setting.\nThe work also benefited from feedback by Nan Jiang during initial stages of this work.\nWe would also like to thank the anonymous reviewers for their suggestions which helped improve the paper.\n\nThis work has taken place in part in the Personal Autonomous Robotics Lab (PeARL) at The University of Texas at Austin. PeARL research is supported in part by the NSF (IIS-1724157, IIS-1638107, IIS-1749204, IIS-1925082), ONR (N00014-18-2243), AFOSR (FA9550-20-1-0077), and ARO (78372-CS). This research was also sponsored by the Army Research Office under Cooperative Agreement Number W911NF-19-2-0333, a gift from Adobe, NSF award \\#2018372, and the DEVCOM Army Research Laboratory under Cooperative Agreement W911NF-17-2-0196 (ARL IoBT CRA). The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.\n\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzpktf b/data_all_eng_slimpj/shuffled/split2/finalzzpktf new file mode 100644 index 0000000000000000000000000000000000000000..ffcf53b25c06663d0b49c19389f9b4219907fc06 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzpktf @@ -0,0 +1,5 @@ +{"text":"\n\\chapter*{Preface to Internet Supplement}\nThis document is an \ninternet supplement to my book ``Partially Observed Markov Decision Processes -- From Filtering to Controlled Sensing'' published\nby Cambridge University Press in 2016.\\footnote{Online ISBN:9781316471104\nand \nHardback ISBN:9781107134607}\n\nThis internet supplement contains exercises, examples and case studies.\nThe material appears in this internet supplement (instead of the book) so that it can be updated.\n This document will evolve over time and further discussion and examples will be added.\n\n\n\n\nThe website \\url{http:\/\/www.pomdp.org} contains downloadable software for solving POMDPs and several examples of POMDPs.\nI have found that by interfacing the POMDP solver with Matlab, one can solve several interesting types of POMDPs such as those \nwith nonlinear costs (in terms of the information state) and bandit problems. \n\nI have given considerable thought into designing the exercises and case studies in this internet supplement.\nThey are mainly mini-research type exercises rather than simplistic drill type exercises. Some of the problems\nare extensions of the material in the book. As can be seen from the content list, this document also contains some short (and in some cases, fairly incomplete) case studies which will be made more detailed over time.\nThese case studies were put in this internet supplement in order to keep the size of the book manageable.\nAs time progresses, I hope to incorporate additional case studies and other pedagogical notes to this document to assist in understanding some of the material in the book. Time permitting, future plans include adding a detailed discussion on structural results for POMDP games; structural results for quasi-variational\ninequalities, etc.\n\n\nTo avoid confusion in numbering, the equations in this internet supplement are numbered consecutively starting from (1) and not chapter wise.\nIn comparison, the equations in the book are numbered chapterwise.\n\n\n\n\n This internet supplement document is work in progress and will be updated periodically.\nI welcome constructive comments from readers of the book and this internet supplement.\nPlease email me at {\\tt vikramk@ece.ubc.ca} \\\\\n\\\\ \\\\\n\n\\hfill\n\\begin{minipage}{6cm}\nVikram Krishnamurthy,\\\\\n 2016\n \\end{minipage}\n\n\\newpage\n\n\\setcounter{chapter}{1}\n\n\n\\chapter{Stochastic State Space Models} \n\n\n\\begin{compactenum}\n\n\n\\item Theorem \\ref{thm:pf} dealt with the stationary distribution and eigenvalues of a stochastic matrix (transition probability matrix of a Markov chain).\nParts of Theorem \\ref{thm:pf} can be shown via elementary linear algebra.\n\n{\\bf Statement 2}:\n Define spectral radius $\\bar{\\lambda}(P) = \\max_i|\\lambda_i|$ \\\\\n{\\em Lemma} : $\\bar{\\lambda}(P) \\leq \\|P\\|_\\infty$ where $\\|P\\|_\\infty =\\max_i \\sum_j P_{ij} $\\\\\n{\\em Proof}: For all eigenvalues $\\lambda$, $|\\lambda | \\|x\\| = \\|\\lambda x\\| = \\|P x \\| \\leq \\|P \\| \\|x \\| \\implies\n |\\lambda| < \\|P\\|$.\n\nFor a stochastic matrix, $\\|P\\|_\\infty =1 $ and $P$ has an eigenvalue at 1. So $\\bar{\\lambda}=1$.\n\n\n{\\bf Statement 3}: For non-negative matrix $P$, $P^\\prime \\pi = \\pi $ implies $P^\\prime|\\pi| = |\\pi|$ where $|\\pi|$ denotes the vector with element-wise absolute values. \\\\\nProof: $|\\pi| = |P^\\prime \\pi| \\leq |P^\\prime| |\\pi| = P^\\prime |\\pi|$\nSo $P^\\prime|\\pi| - |\\pi| \\geq 0$.\\\\\nBut $P^\\prime|\\pi| - |\\pi| > 0$ is impossible, since it implies $1^\\prime P^\\prime |\\pi| > 1^\\prime |\\pi|$,\ni.e., $1^\\prime |\\pi| > 1^\\prime |\\pi|$.\n\n\n\\item Farkas' lemma \\index{Farkas' lemma} is a widely used result in linear algebra. It states: Let $M$ be an $m \\times n$ matrix and $b$ an $m$-dimensional vector. Then only one of the following statements is true:\n\\begin{enumerate}\n\\item There exists a vector $x\\in {\\rm I\\hspace{-.07cm}R}^n$ such that $M x = b$ and $ x \\geq 0$. \n\\item There exists a vector $y\\in {\\rm I\\hspace{-.07cm}R}^m$ such that $M^\\prime y \\geq 0$ and $b^\\prime y < 0$.\n\\end{enumerate}\nHere $x\\geq 0$ means that all components of the vector $x$ are non-negative.\n\nUse Farkas lemma to prove that every transition matrix $P$ has a stationary distribution. That is, for any $X \\timesX$ stochastic matrix $P$, there\nexists a probability vector $\\pi$ such that $P^\\prime \\pi = \\pi$. (Recall a probability vector $\\pi$ satisfies\n$\\pi(i) \\geq 0$, $\\sum_i \\pi(i) = 1$).\n\nHint: Write alternative (a) of Farkas lemma as\n$$ \\begin{bmatrix} (P - I)^\\prime \\\\ \\mathbf{1}^\\prime \\end{bmatrix} \\pi = \\begin{bmatrix} 0_{X} \\\\ 1 \\end{bmatrix} , \\quad \\pi > 0$$\nShow that this has a solution by demonstrating that alternative (b) does not have a solution.\n\n\n\n\n\n\\item\nUsing the maneuvering target model of Chapter \\ref{chp:manuever}, \\index{maneuvering target}\nsimulate the dynamics and measurement process of a target with the following specifications:\\\\\n{\\small\n\\begin{tabular}{|l|ll|} \\hline \\hline\nSampling interval & $\\Delta = 7$ s & \\\\ \\hline\nNumber of measurements & $N = 50$ &\\\\ \\hline\nInitial target position & $(-500 , -500)'$ m &\\\\ \\hline\nInitial target velocity & $(0.0, 5.0)'$ m\/s &\\\\ \\hline\nTransition probability matrix & $P_{ij} = \n\\left\\{\\begin{array}{cc} 0.9 \\mbox{ if } i=j\\\\\n 0.05 \\mbox{ otherwise }\\end{array}\\right.$ &\\\\\n\\hline \nManeuver commands (three)& $f \\mc = \\left( \\begin{array}{llll}\n 0 & 0 & 0 & 0\n \\end{array} \\right)'$ &(straight)\n \\\\\n& $f \\mc = \\left( \\begin{array}{llll}\n -1.225 & -0.35 & 1.225 & 0.35\n \\end{array} \\right)'$ &(left turn)\n \\\\\n& $f \\mc = \\left( \\begin{array}{llll}\n 1.225 & 0.35 & -1.225 & -0.35\n \\end{array} \\right)'$ &(right turn) \\\\ \\hline\nObservation matrix & $H=I_{4\\times 4}$ &\\\\ \\hline\nProcess noise & $Q=(0.1)^2 I_{4\\times 4}$ &\\\\ \\hline\nMeasurement noise & $R= \\mbox{diag}( 20.0^2, 1.0^2, 20.0^2, 1.0^2)$ &\\\\ \\hline\nMeasurement volume $V$ & $[-1000, 1000]$ m in $x$ and $y$ position &\\\\ \n & $[-10.0, 10.0]$ m\/s in $x$ and $y$ velocity &\n\\\\ \\hline \\hline\n\\end{tabular} }\n\n\\item Simulate the optimal predictor via the composition method. The composition method is discussed in \\S \\ref{sec:jmlspred}.\n\n\n\\item As should be apparent from an elementary linear systems course, the algebraic Lyapunov equation (\\ref{eq:algebraic_lyapunov})\nis intimately linked with the stability of a linear discrete time system. Prove that $A$ has all its eigenvalues strictly inside the unit circle\niff for every positive definite matrix $Q$, there exists a positive definite matrix $\\Sigma_\\infty$ such that (\\ref{eq:algebraic_lyapunov}) holds.\n\n\\item Theorem \\ref{thm:dobproperty} states that \n$|\\lambda_2| \\leq \\dob(P) $. That is, the Dobrushin coefficient upper bounds the second largest eigenvalue modulus of a stochastic matrix $P$. \nShow that \n$$\n\\log |\\lambda_2| = \\lim_{k\\rightarrow \\infty} \\frac{1}{k} \\log \\dob(P^k) $$\n\n\n\\item \nOften for sparse transition matrices, $\\dob(P)$ is typically equal to 1 and therefore not useful since it provides a trivial upper bound for $|\\lambda_2|$.\nFor example, consider a random walk characterized by the tridiagonal transition matrix\n$$ P = \\begin{bmatrix} \nr_0 & p_0 & 0 & 0 & \\cdots & 0 \\\\\nq_1 & r_1 & p_1 & 0 & \\cdots & 0 \\\\\n0 & q_2 & r_2 & p_2 & \\cdots & 0 \\\\\n\\vdots & & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n0 & \\cdots & 0 & q_{X-1} & r_{X-1} & p_{X-1} \\\\\n0 & \\cdots & 0 & 0 & q_{X} & r_{X} \\end{bmatrix}\n$$\nThen using Property 3 of $\\dob(\\cdot)$ above, clearly\n$\\sum_l \\min \\{ P_{il}, P_{jl} \\} = 0$, implying that\n $\\dob(P) = 1$. So for this example, the Dobrushin coefficient does not say anything about the initial condition being forgotten geometrically fast.\n \n For such cases, it is often useful to consider the Dobrushin coefficient of powers of $P$.\nIn the above example, clearly every state communicates with every other state in at least $X$ time points. So\n $P^X$ has strictly positive elements. Therefore $\\dob(P^X)$ is strictly smaller than 1 and is a useful bound. Geometric ergodicity follows\n by consider blocks of length $X$, i.e., \n $$ \\dvar{ { P^X}^\\prime \\pi} {{P^X}^\\prime \\bar{\\pi} } \\leq \\dob(P^X) \\dvar{\\pi}{\\bar{\\pi}} $$\n\n\\item Show that the inhomogeneous Markov chain with transition matrix\n$$ P(2n-1) = \\begin{bmatrix} 0.5 & 0.5 \\\\ 1 & 0 \\end{bmatrix} ,\\quad P(2n) = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix}$$\nis weakly ergodic.\n\n\n\\item {\\bf Wasserstein distance.} \n\\index{Wasserstein distance}\nAs mentioned in \\S \\ref{sec:modelnotes} of the book, the Dobrushin coefficient is a special case of a more general coefficient of ergodicity.\nThis general definition is in terms of the \nWasserstein metric which we now define: Let $d$ be a metric on the state space $\\mathcal{X} = \\{e_1,e_2,\\ldots,\\}$ where the state space is possibly denumerable.\nConsider the bivariate random vector $(x,y) \\in \\mathcal{X}\\times \\mathcal{X}$ with marginals $\\pi_x$ and $\\pi_y$, respectively.\n\nDefine the Wasserstein distance as\n$$ d(\\pi_x,\\pi_y) = \\inf \\E\\{ d(x,y) \\} $$\nwhere the infimum is over the joint distribution of $(x,y)$.\n\n\\begin{compactenum}\n\\item Show that the variational distance is a special case of the Wasserstein distance obtained by choosing $d(x,y)$ as the discrete metric\n$$ d(x,y) = \\begin{cases} 1 & x \\neq y \\\\ 0 & x = y .\\end{cases} $$\n\\item Define the coefficient of ergodicity associated with the Wasserstein distance as \n$$ \\dob(P) = \\sup_{i \\neq j} \\frac{d( P^\\prime e_i, P^\\prime e_j)}{d(e_i,e_j)} $$\nShow that the Dobrushin coefficient is a special case of the above coefficient of ergodicity corresponding to the discrete metric.\n\n\\item Show that the above coefficient of ergodicity satisfies properties 2, 4 and 5 of Theorem \\ref{thm:dobproperty}.\n\\end{compactenum}\n\n\\item {\\bf Ultrametric transition matrices.} \\index{ultrametric matrix}\nIt is trivial to verify that $P^n$ is a stochastic matrix for any integer $n \\geq 0$.\nUnder what conditions is $P^{1\/n}$ a stochastic matrix?\nA symmetric ultrametric stochastic matrix $P$ defined in \\S \\ref{sec:blackwelldom} of the book satisfies this property.\n\n\n\\begin{comment}\n\\item \nThe MATLAB code to generate a discrete valued\n random variable with pmf {\\tt prob} is\n {\\small \n\\begin{lstlisting}\nfunction rand_number = simulatestate(prob);\nstatedim = numel(prob); \nmm = max(prob);\nc = mm * statedim;\naccept = false;\nwhile accept == false\n y = floor(statedim * rand) + 1; \n if rand <= prob(y) \/ mm\n rand_number = y;\n accept = true;\n end\nend\n\\end{lstlisting}} \\end{comment}\n\\end{compactenum}\n\n\\chapter{Optimal Filtering} \n\n\\section{Problems}\n\\begin{compactenum}\n\\item Standard drill exercises include: \n\\begin{compactenum}\n\\item Compare via simulations the recursive least squares with the Kalman filter\n\\item Compare via simulations the recursive least square and the least mean squares\n (LMS) algorithm with a HMM filter when tracking a slow Markov chain. Note that Chapter \\ref{sec:hmmsa} of the book gives performance bounds on how well a LMS algorithm can track a slow Markov chain.\n \\item\nAnother standard exercise is to try out variations of the particle filter with different importance distributions and resampling strategies on different models.\nCompare via simulations the cubature filter, unscented Kalman filter and a particle filter for a bearings only target tracking model.\n\\item\nA classical result involving the Kalman filter is the so called innovations state space model representation and the associated spectral factorization\nproblem for the Riccati equation, see \\cite{AM79}.\n\\item {\\bf Posterior Cramer Rao bound.} The posterior Cramer Rao bound \\cite{TMN98} for filtering can be used to compute a lower bound to the mean square error. \\index{posterior Cramer Rao bound}\nThis requires twice differentiability of the logarithm of the joint density. For HMMs, one possibility is to consider the Weiss-Weinstein\nbounds \\index{Weiss-Weinstein bounds}, see \\cite{RO05}. Chapter~\\ref{chp:filterstructure}\n of the book gives more useful sample path bounds on the HMM filter using\nstochastic dominance.\n\\end{compactenum}\n\n\\item {\\bf Bayes' rule interpretation of Lasso.}\\cite{PC08} \\index{Lasso} Suppose that \nthe state $x \\in {\\rm I\\hspace{-.07cm}R}^X$ is a random variable with prior pdf\n$$ p(x) = \\prod_{j=1}^X \\frac{\\lambda}{2} \\, \\exp\\left( -\\lambda x(j) \\right) .$$\nSuppose $x$ is observed via the observation equation\n$$ y = F x + v, \\qquad v \\sim \\normal(0,\\sigma^2 I) $$\nwhere $A$ is a known $n \\times X$ matrix.\nThe variance $\\sigma^2$ is not known and has a prior pdf $p(\\sigma^2)$. Then show that the posterior of $(x,\\sigma^2)$ given \nthe observation $y$ is of the form\n$$ \np(x,\\sigma^2| y) \\propto\np(\\sigma^2) \\, (\\sigma^2)^{-\\frac{n+1}{2}} \\exp\\big( - \\frac{1}{2\\sigma^2}\\, \\text{Lasso}(x,y,\\mu)\\,\\big) $$\nwhere $\\mu = 2 \\sigma^2 \\lambda$ and \n$$ \\text{Lasso}(x,y,\\mu) = \\|y-F x\\|^2 + \\mu \\|x\\|_1 .$$\nTherefore for fixed $\\sigma^2$, computing the mode $\\hat{x}$ of the posterior is equivalent to computing the minimizer $\\hat{x}$ of $\\text{Lasso}(x,y,\\mu) $.\n\nThe resulting Lasso (least absolute shrinkage and selection operator) estimator $\\hat{x}$ was proposed in \\cite{Tib96} which is one of the most influential papers \n in statistics since the 1990s. Since $\\text{Lasso}(x,y,\\mu) $ is convex in $x$ it can be computed efficiently via convex optimization algorithms.\n\n\\item Show that if $X \\leq Y$ (with probability 1), then $\\E\\{X|Z\\} \\leq \\E\\{Y|Z\\}$ for any information $Z$.\n\\item Show that for a linear Gaussian system (\\ref{eq:filter_linears}), (\\ref{eq:filter_linearobs}),\n$$ p(y_k|y_{1:k-1}) = \\normal(y_k - y_{k|k-1}, H_k \\Sigma_{k|k-1} H^\\prime_k + R_k ) $$\nwhere $y_{k|k-1}$ and $\\Sigma_{k|k-1}$ are defined in (\\ref{eq:statepred}), (\\ref{eq:covpred}), respectively.\n\n\n\\item Simulate in Matlab the HMM filter, and fixed lag smoother. Study empirically how the error probability of the estimates\ndecreases with lag. (The filter is a fixed lag smoother with lag of zero). Please also refer to \\cite{GSV05} for a very nice analysis\nof error probabilities.\n\n\\item Consider a HMM where the Markov chain evolves slowly with transition matrix $P = I + \\epsilon Q$ where $\\epsilon $ is a small\npositive constant and \n$Q$ is a generator matrix. That is $Q_{ii} < 0$, $Q_{ij} > 0$ and each row of $Q$ sums to zero.\nCompare the performance of the HMM filter with the recursive least squares algorithm (with an appropriate\nforgetting factor chosen) for estimating the underlying state.\n\n\n\\item Consider the following Markov modulated auto-regressive time series model: \n$$ z_{k+1} = F(r_{k+1})\\,z_k + \\Gamma(r_{k+1})\\,w_{k+1} + f(r_{k+1})\\,u_{k+1} $$\nwhere $w_{k} \\sim \\normal(0,1)$, $u_k$ is a known exogenous input.\nAssume the sequence $\\{z_k\\}$ is observed. \n Derive an optimal filter for the underlying Markov chain $r_k$. \n (In comparison to a jump Markov linear system, $z_k$ is observed without noise in this problem. The optimal filter is very\n similar to the HMM filter).\n \n \n \\item Consider a Markov chain $x_k$ corrupted by iid zero mean Gaussian noise and a sinusoid:\n $$y_k = x_k + \\sin(k\/100) + v_k$$\nObtain a filtering algorithm for extracting $x_k$ given the observations.\n \n \\item {\\bf Image Based Tracking.} The idea is to estimate the coordinates $z_k$ of the target by measuring its orientation $r_k$ in noise.\n For example an imager can determine which direction an aircraft's nose is pointing thereby giving useful information about which\n direction it can move.\n Assume that the target's orientation evolves according to a finite state Markov chain. (In other words, the imager quantizes the target\n orientation to one of a finite number of possibilities.) Then the model for the filtering problem is\n \\[\n \\begin{split}\n z_{k+1} &= F(r_{k+1})\\,z_k + \\Gamma(r_{k+1})\\,w_{k+1} \\\\\ny_k &\\sim p(y | r_k) \n \\end{split} \\]\n Derive the filtering expression for $\\E\\{z_k | y_1,\\ldots,y_k\\}$. The papers \\cite{SSD93,KE97a,EE99} consider image based filtering. \\index{image-based tracking}\n \n \\item Consider a jump Markov linear system. Via computer simulations, compare the IMM algorithm, Unscented Kalman filter and particle filter.\n \n \n\n \n \\item {\\bf Radar pulse train de-interleaving.} \\index{de-interleaving} \n \\index{jump Markov linear system! pulse de-interleaving} \n In radar signal processing, radar pulses are received from multiple periodic sources.\n It is of interest to estimate the periods of these sources. For example, suppose:\n \\begin{compactitem} \n\\item source 1 pulses are received at times\n$$ 2, 7, 12, 17, 22 , 27, 32, 37, 42,\\ldots , \\quad \\text{ (period = 5, phase =2)} $$\n\\item source 2 pulses are received at times\n$$ 4, 15, 26, 37, 48, 59, 70, 81,\\ldots , \\quad \\text{ (period = 11, phase = 4) }.$$\\end{compactitem} The interleaved signal consists of pulses at times\n$$2,4,7,12,15,22,26,27, 32, 37, 42,\\ldots. $$ So the above interleaved signal contains time of arrival information. \nNote that at time 37, pulses are received from both sources; but it is assumed that there is no amplitude information - so the received signal\nis simply a time of arrival event at time 37.\nAt the receiver, interleaved signal (time of arrivals) is corrupted by jitter noise (modeled as iid noise).\n So the noisy received signal are, for example, $$2.4, \\; 4.1, \\; 6.7, \\; 11.4, \\; 15.5, \\; 21.9, \\; 26.2, \\; 27.5, \\; 30.9, \\; 38.2,\\; 43.6, \\ldots.$$\nGiven this noisy interleaved signal, the de-interleaving problem aims to determine which pulses came from which source.\nThis can be done by estimating the periods (namely, 5 and 11) and phases (namely, 2 and 4) of the 2 sources.\n\n\n\n\n\nThe de-interleaving problem can be formulated as a jump Markov linear system.\nDefine the state \n $x_k' = (T', \\tau_k')$,\nconsists of the periods $T' = (T^{(1)}, \\ldots, T^{(N)})$ of the $N$ sources and \n$\\tau_k'= (\\tau_k^{(1)}, \\ldots, \\tau_k^{(N)})$, where $\\tau_k^{(i)}$ denotes the last\ntime source $i$ was active up to and including the arrival of the $k$th pulse.\nLet $\\tau_1 =( \\phi^{(1)}, \\ldots, \\phi^{(N)})$, be the phases of periodic pulse-train sources.\nThen\n\\begin{equation}\n \\tau_{k+1}^i = \\left\\{\\begin{array}{cc} \\tau_k^i + T^i & \\mbox{if $(k+1)$th pulse is due to source $i$} \\\\\n \\tau_k^i & \\mbox{otherwise} \\end{array} \\right.; \\;\\;\\tau_1^i = \\phi^{(i)}.\n \\label{eq:basic}\\end{equation}\n \nLet $e_i, i=1,\\ldots, N$, be the unit $N$-dimensional vectors with $1$ in the $i$th\nposition. Let $r_k \\in \\{1,\\ldots,N\\}$ denote the active source at time $k$. Then one can express the time of arrivals as\nthe jump Markov linear system\n\\begin{align*} x_{k+1} & = A(r_{k+1}) x_k + w_{k} \\\\\ny_k &= C(r_k) x_k + v_k \\end{align*}\nwhere \n$$\nA(r_{k+1}) = \\begin{bmatrix}\nI_N & 0_{N\\times N} \\\\\n\\mbox{diag}(e_{r_{k+1}}) & I_N\n\\end{bmatrix},\\;\nC(r_k) = \n\\begin{bmatrix}\n 0_{1\\times N} & e_{r_k}'\n\\end{bmatrix}\n$$\nNote that $r_k $ is a periodic process and so has transition probabilities $$P_{i,i+1} = 1, \\; \\text{ for $i< N$, and }\\, P_{N,1} = 1. $$\n$v_k$ denotes the measurement (jitter) noise; while $w_k$ can be used to model time varying periods.\\\\\n{\\em Remark}: Obviously, there are identifiability issues; for example, if $\\phi^{(1)} = \\phi^{(2)}$ and $T^{(1}$ is a multiple of $T^{(2)}$ then\nit is impossible to detect source 1.\n \n \\item {\\bf Narrowband Interference and JMLS.} \\index{jump Markov linear system! narrowband interference} Narrowband interference corrupting a Markov chain can be modeled as a jump Markov linear\n system. Narrowband interference can be modeled as an auto-regressive (AR) process with poles close to the unit circle: for example\n$$i_k = a\\, i_{k-1} + w_k $$\nwhere $a = 1- \\epsilon$ and $\\epsilon $ is a small positive number.\nConsider the observation model\n$$ y_k = x_k + i_k + v_k $$\nwhere $x_k$ is a finite state Markov chain, $i_k$ is narrowband interference and $v_k$ is observation noise. Show that the above\nmodel can be represented as a jump Markov linear system.\n \n \\item {\\bf Bayesian estimation of Stochastic context free grammar.} \\index{stochastic context free grammars}\n First some perspective: HMMs with a finite observation space are also called regular grammars.\n They are a subset of a more general class of models called stochastic context free grammars as depicted by Chomsky's \n hierarchy in Figure \\ref{fig:chomsky}.\n \\begin{figure}[h] \\centering\n\\includegraphics[scale=0.8]{chomsky.pdf}\n\\caption{The Chomsky hierarchy of languages} \\label{fig:chomsky}\n\\end{figure} \n\n Stochastic context free grammars (SCFGs) provide a powerful modeling tool for strings of alphabets and are used widely in\n natural language processing \\cite{MS99a}. For example, consider the randomly generated string $a^n c^m b^n$ where $m,n$ are non-negative integer valued random variables. Here $a^n$ means the alphabet $a$ repeated $n$ times.\n The string $a^n c^m b^n$\n could model the trajectory of a target that moves $n$ steps north and then an arbitrary number of steps east or west and then\n $n$ steps south, implying that the target performs a U-turn.\nA basic course in computer science would show (using a pumping lemma) \nthat such strings cannot be generated exclusively using a Markov chain (since the memory $n$ is variable).\n\n If the string $a^n c^m b^n$ was observed in noise, then Bayesian estimation\n(stochastic parsing) algorithms can be used to estimate the underlying string. Such meta-level tracking algorithms have polynomial computational\ncost (in the data length) and\nare useful for estimating {\\em trajectories} of targets (given noisy position and velocity measurements). They allow a human radar operator\nto interpret tracks and can be viewed as middleware in the human-sensor interface.\nSuch stochastic context free grammars generalize HMMs and facilitate modeling complex spatial trajectories of targets.\n\nPlease refer to \\cite{MS99a} for Bayesian signal processing algorithms and EM algorithms for stochastic context free grammars.\n\\cite{FK13,FK14} gives examples of meta-level target tracking using stochastic context free grammars.\n \n \n \\item {\\bf Kalman vs HMM filter}. \n A Kalman filter is the optimal state estimator for the linear Gaussian state space model\n \\begin{align*} x_{k+1} &= F x_k + w_k, \\\\\ny_k & = H^\\prime x_k + v_k . \\end{align*}\nwhere $w$ and $v$ are mutually independent iid Gaussian processes.\n\n\nRecall from (\\ref{eq:intro_hmms}), (\\ref{eq:intro_hmmobs}) that for a Markov chain with state space \n$\\mathcal{X} = \\{e_1,\\ldots, e_{X}\\}$ of unit vectors, an HMM can be expressed as\n\\begin{align*} x_{k+1} &= P^\\prime x_k + w_k, \\\\\ny_k & = H^\\prime x_k + v_k . \\end{align*}\nA key difference is that in (\\ref{eq:intro_hmms}), $w$ is no longer i.i.d; instead it is \n a martingale difference process:\n $\\E\\{w_k|x_0,x_1,\\ldots,x_{k}\\} = 0$.\n \nFrom \\S \\ref{sec:linearminvar} of the book, it follows that the Kalman filter is the minimum variance linear estimator for the above HMM.\nOf course the optimal {\\em linear} estimator (Kalman filter) can perform substantially worse than the optimal estimator (HMM filter).\nCompare the performance of the HMM filter and Kalman filter numerically for the above example.\n\n \n \\item {\\bf Interpolation of a HMM.} \\index{interpolation of HMM} Consider a Markov chain $x_k$ with transition matrix $P$ where the discrete time clock ticks at intervals of 10 seconds.\n Assume noisy measurements are obtained of at each time $k$. Devise a smoothing algorithm to estimate the state of the Markov chain at 5 second intervals.\n (Note: Obviously on the 5 second time scale, the transition matrix is $P^{1\/2}$. For this to be a valid stochastic matrix\n it is sufficient that $P$ is a symmetric ultrametric matrix or more generally $P^{-1}$ is an M-matrix \\cite{HL11}; see also \\S \\ref{sec:blackwelldom}\n of the book.)\n \n \n \n \\end{compactenum}\n \\section{Case Study. Sensitivity of HMM filter to transition matrix} Almost an identical proof to that of geometric ergodicity proof of the HMM filter in \\S \\ref{sec:hmmforget} can be used to obtain expressions for the sensitivity of the HMM filter to the HMM parameters.\n \\index{HMM filter! sensitivity bound to transition matrix}\n \n{\\bf Aim}: We are interested in\na recursion for $\\|\\pi_k - \\underline{\\pi}_k\\|_1$ when $\\pi_k$ is updated with HMM filter using \ntransition matrix $P$ and $\\underline{\\pi}_k$ is updated with HMM filter using transition matrix $\\underline{\\tp}$. That is,\nwe want an expression\nfor \n\\begin{equation} \\|T(\\pi,y;P) - T(\\underline{\\pi},y; \\underline{\\tp}) \\|_1 \\text{ in terms of } \\|\\pi - \\underline{\\pi}\\|_1. \\label{eq:hmmsensc} \\end{equation}\nSuch a bound if useful when the HMM filter is implemented with an incorrect transition matrix $\\underline{\\tp}$ instead of actual transition matrix $P$.\nThe idea is that when $P $ is close to $\\underline{\\tp} $ then $T(\\pi,y;P) $ is close to $T(\\pi,y;P) $.\n\nA special case of (\\ref{eq:hmmsensc}) is to obtain an expression for \n\\begin{equation} \\|T(\\pi,y;P) - T(\\pi,y; \\underline{\\tp}) \\|_1 \\label{eq:hmmsensc2}\\end{equation} that is when both HMM filters have the same initial belief $\\pi$ but are updated\nwith different transition matrices, namely $P$ and $\\underline{\\tp}$. \n\n The theorem below obtains expressions for both (\\ref{eq:hmmsensc}) and (\\ref{eq:hmmsensc2}).\n \n \\begin{theorem-non} Consider a HMM with transition matrix $P$ and state levels $g$.\n Let $\\epsilon > 0$ denote the user defined parameter.\n Suppose $\\|\\underline{\\tp}-P\\|_1 \\leq \\epsilon$, where $\\|\\cdot\\|_1$ denotes the induced 1-norm for matrices.\\footnote{\nThe three statements $\\| P^\\prime \\pi - \\underline{\\tp}^\\prime \\pi \\|_1 \\leq \\epsilon$, $\\|\\underline{\\tp}-P\\|_1 \\leq \\epsilon$ and $\\sum_{i=1}^X \\| (P^\\prime - \\underline{\\tp}^\\prime)_{:,i} \\|_1 \\pi(i) \\leq \\epsilon$\nare all equivalent since $\\|\\pi\\|_1 = 1$.}\nThen\n\\begin{compactenum}\n\\item The expected absolute deviation between one step of filtering using $P$ versus $\\underline{\\tp}$ is upper bounded as:\n\\begin{multline} \\E_{y} \\left| g^\\prime \\left( T(\\pi,y;P) - T(\\pi,y;\\underline{\\tp}) \\right) \\right| \\leq \n\\epsilon \\sum_y \\max_{i,j} g^\\prime (I - T(\\pi,y;\\underline{\\tp}) \\mathbf{1}^\\prime) B_y (e_i - e_j)\n\\label{eq:ebound}\n\\end{multline}\n\n\\item The sample paths of the filtered posteriors and conditional means have the following explicit bounds at each time $k$:\n\\begin{multline} \\label{eq:samplepath1}\n\\| \\pi_k - \\underline{\\pi}_k\\|_1 \\leq \\frac{\\epsilon}{ \\max\\{ F(\\underline{\\pi}_{k-1},y_k) - \\epsilon, \\, \\mu(y_k) \\}} + \\frac{ \\dob(\\underline{\\tp}) \\, \\|\\pi_{k-1} - \\underline{\\pi}_{k-1} \\|_1}{F(\\underline{\\pi}_{k-1},y_k)}\n\n\\end{multline}\nHere $\\dob(\\underline{\\tp})$ denotes the Dobrushin coefficient of the transition matrix $\\underline{\\tp}$ and $\\underline{\\pi}_k$ is the posterior\ncomputed using the HMM filter with $\\underline{\\tp}$, and\n\\begin{equation} \nF(\\underline{\\pi},y) = \\frac{\\mathbf{1}^\\prime B_{y} \\underline{\\tp}^\\prime \\underline{\\pi}}{\\max_i B_{i,y}}, \\quad\n\\mu(y) = \\frac{ \\min_i B_{iy} } { \\max_i B_{iy} } .\n\\end{equation}\n\\end{compactenum}\n\\end{theorem-non} \n\n\n The above theorem gives explicit upper bounds between the filtered distributions using \n transition matrices $\\underline{\\tp}$ and $\\bar{\\tp}$. The $\\E_y$ in (\\ref{eq:ebound}) is with respect to the \n measure $\\sigma(\\pi,y;P) = \\mathbf{1}^\\prime B_y P^\\prime \\pi$ which corresponds to $\\mathbb{P}(y_k=y| \\pi_{k-1} = \\pi)$. \n \n \n\\begin{proof}\nThe triangle inequality for norms yields\n\\begin{align}\n& \\dvar{\\pi_{k+1} } { \\underline{\\pi}_{k+1} } = \\dvar{ T(\\pi_k,y_{k+1};P) }{ T(\\underline{\\pi}_k,y_{k+1};\\underline{\\tp}) } \\nonumber \\\\\n& \\leq \n\\dvar{ T(\\pi_k,y_{k+1};P) } { T(\\pi_k,y_{k+1};\\underline{\\tp}) } \\nonumber \\\\ & \\hspace{1cm} + \\dvar{ T(\\pi_k,y_{k+1};\\underline{\\tp}) } { T( \\underline{\\pi}_k,y_{k+1}; \\underline{\\tp})}. \n\\label{eq:triangle}\n\\end{align}\n\n{\\bf Part 1}: Consider the first normed term in the right hand side of (\\ref{eq:triangle}).\nApplying (\\ref{eq:rhshmmgeom}) with \n $\\pi = P^\\prime \\pi_k$ and $\\pi^0 = \\underline{\\tp}^\\prime \\pi_k$ yields\n\\begin{multline*}\n g^\\prime (T(\\pi_k,y;P) - T(\\pi_k,y;\\underline{\\tp})) = \\frac{1}{\\sigma(\\pi,y;P)}\n g^\\prime \\left[ I - T(\\pi,y,\\underline{\\tp}) \\mathbf{1}^\\prime\\right] B_y (P - \\underline{\\tp})^\\prime \\pi \n\\end{multline*}\nwhere $\\sigma(\\pi,y;P) = \\mathbf{1}^\\prime B_y P^\\prime \\pi$.\n Then Lemma \\ref{lem:cs}(i) yields \n\\begin{multline*} g^\\prime (T(\\pi_k,y;P) - T(\\pi_k,y;\\underline{\\tp}))\\\\ \\leq \n\\max_{i,j} \\frac{1}{\\sigma(\\pi,y;P)} g^\\prime \\left[ I - T(\\pi,y,\\underline{\\tp}) \\mathbf{1}^\\prime\\right] B_y (e_i - e_j) \\dvar{P^\\prime \\pi}{ \\underline{\\tp}^\\prime \\pi} \\end{multline*}\nSince $\\dvar{P^\\prime \\pi}{ \\underline{\\tp}^\\prime \\pi} \\leq \\epsilon$, taking expectations with respect to the measure $\\sigma(\\pi,y;P)$, completes the proof\nof the first assertion.\n \n{\\bf Part 2}:\nApplying Theorem \\ref{thm:bayesbackvar}(i) with the notation $\\pi = P^\\prime \\pi_k$ and $\\pi^0 = \\underline{\\tp}^\\prime \\pi_k$ yields\n\\begin{align} &\\dvar{T(\\pi_k,y;P) } { T(\\pi_k,y;\\underline{\\tp}) } \\leq \n \\frac{ \\max_i B_{i,y} \\dvar{P^\\prime \\pi_k}{\\underline{\\tp}^\\prime \\pi_k} } { \\mathbf{1}^\\prime B_{y} \\underline{\\tp}^\\prime \\pi_k } \\nonumber\\\\\n& \\leq \\frac{\\epsilon}{2}\\,\\frac{ \\max_i B_{i,y} } { \\mathbf{1}^\\prime B_{y} \\underline{\\tp}^\\prime \\pi_k } \n\\leq \\frac{ \\max_i B_{i,y} \\, \\epsilon\/2} { \\max\\{ \\mathbf{1}^\\prime B_{y} \\underline{\\tp}^\\prime \\underline{\\pi}_k - \\epsilon \\max_i B_{iy}, \\min_i B_{iy} \\}}.\n\\label{eq:term1}\n\\end{align}\nThe second last inequality follows from the construction of $\\underline{\\tp}$ satisfying (\\ref{eq:con2}) (recall the variational norm is half the $l_1$ norm). The last inequality follows from \nTheorem \\ref{thm:bayesbackvar}(ii).\n\nConsider the second normed term in the right hand side of (\\ref{eq:triangle}).\nApplying Theorem \\ref{thm:bayesbackvar}(i) with notation $\\pi = \\underline{\\tp}^\\prime \\pi_k$ and $\\pi^0 = \\underline{\\tp}^\\prime \\underline{\\pi}_k$ yields\n\\begin{multline} \\dvar{ T(\\pi_k,y;\\underline{\\tp}) } { T( \\underline{\\pi}_k,y; \\underline{\\tp}) } \n \\leq \n \\frac{ \\max_i B_{i,y} \\dvar{\\underline{\\tp}^\\prime \\pi_k}{\\underline{\\tp}^\\prime \\underline{\\pi}_k} } { \\mathbf{1}^\\prime B_{y} \\underline{\\tp}^\\prime \\underline{\\pi}_k } \\\\ \\leq\n\\frac{ \\max_i B_{i,y} \\, \\dob(\\underline{\\tp}) \\, \\dvar{ \\pi_k}{\\underline{\\pi}_k} } { \\mathbf{1}^\\prime B_{y} \\underline{\\tp}^\\prime \\underline{\\pi}_k } \\label{eq:term2}\n \\end{multline}\n where the last inequality follows from the submultiplicative property of the Dobrushin coefficient.\nSubstituting (\\ref{eq:term1}) and (\\ref{eq:term2}) into the right hand side of the triangle inequality (\\ref{eq:triangle}) proves the result.\n\\end{proof}\n \n \n \n \\section{Case Study. Reference Probability Method for Filtering}\nWe describe here the so called {\\em reference probability method} for deriving the un-normalized filtering recursion (\\ref{eq:unnormalized}).\nThe main idea is to start with the joint probability mass function of all observations and states until time $k$, namely, $p(x_{0:k}, y_{1:k})$.\nSince this joint density contains all the information we need, it is not surprising that by suitable marginalization and integration, the filtering\nrecursion and hence the conditional\nmean estimate can be computed.\n\nGiven the relatively straightforward derivations of the filtering recursions given in Chapter \\ref{sec:filtermain} of the book, the reader might wonder why we present\nyet another derivation. The reason is that in more complicated filtering problems, the reference probability method gives a systematic way of deriving\nfiltering expressions.\nIt is used extensively in \\cite{EAM95} to derive filters in both discrete and continuous time.\nIn continuous time,\nthe reference probability measure is extremely useful -- it yields the so called Duncan-Mortenson-Zakai equations for nonlinear filtering.\n\n\\subsubsection{The Engineering Version}\nSuppose the state and observation processes $\\{x_k\\}$ and $\\{y_k\\}$ are in a probability space with probability measure $\\mathbb{P}$.\nSince the state and observation noise processes are iid, under $\\mathbb{P}$, we have the following factorization:\n\\begin{align} & p(x_{0:k}, y_{1:k}) = \\prod_{n=1}^k p(y_n|x_n)\\, p(x_n| x_{n-1})\\, \\pi_0(x_0) \n \\label{eq:factor}\\\\ \n &\\propto \\prod_{n=1}^k p_v\\left(D_n^{-1}(x_n)\\left[y_n - H_n(x_k)\\right] \\right)\n p_w\\left(\\Gamma_{n-1}^{-1}(x_{n-1})\\,\\left[x_{n}- F_{n-1}(x_{n-1})\\right]\\right) \\, \\pi_0(x_0) \\nonumber\n\\end{align}\nStarting with $p(x_{0:k}, y_{1:k})$, the \nconditional expectation of any function $\\phi(x_k)$ is\n\\begin{equation}\n\\E\\{ \\phi(x_k) | y_{1:k}\\} = \\frac{ \\int \\phi(x_k) p(x_{0:k}, y_{1:k}) dx_{0:k}}{\\int p(x_{0:k}, y_{1:k}) dx_{0:k}} \n= \\frac{ \\int_\\mathcal{X} \\phi(x_k) \\left[ \\int p(x_{0:k}, y_{1:k}) dx_{0:k-1} \\right] dx_k}{\\int p(x_{0:k}, y_{1:k}) dx_{0:k}} \n\\label{eq:erp}\n\\end{equation}\nThe main idea then is to define the term within the square brackets in the numerator as \n the un-normalized density \n$ {q}_k(x_k) = \\int p(x_{0:k}, y_{1:k}) dx_{0:k-1}$.\n(Of course then ${q}_k(x_k) = p(x_k, y_{1:k})$).\nWe now derive the recursion (\\ref{eq:unnormalized}) for the un-normalized density ${q}_k$:\n\\begin{align*}\n\\int_\\mathcal{X} \\phi(x_k) {q}_k(x_k) dx_k&= \\int_\\mathcal{X} \\phi(x_k) \\int p(x_{0:k}, y_{1:k}) dx_{0:k-1} dx_k \\\\\n &\\hspace{-1.5cm}= \\int_\\mathcal{X} \\int_\\mathcal{X} \\phi(x_k) p(y_k| x_k) p(x_k| x_{k-1}) \\left[\\int p(x_{0:k-1},y_{1:k-1}) dx_{0:k-2} \\right]\\, dx_{k-1} dx_k\n \\\\\n &\\hspace{-1.5cm}= \\int_\\mathcal{X} \\int_\\mathcal{X} \\phi(x_k) p(y_k| x_k) p(x_k| x_{k-1}) {q}_{k-1}(x_{k-1}) dx_{k-1} dx_k\n \\end{align*}\nwhere the second equality follows from (\\ref{eq:factor}). \nSince the above holds for any test function $\\phi$, it follows that the integrands within the outside integral are equal, thereby yielding the\nun-normalized filtering\nrecursion (\\ref{eq:unnormalized}). \n\n\n\n\\subsubsection{Interpretation as Change of Measure}\nWe now interpret the above derivation as the engineering version of the reference probability method.\\footnote{In continuous time, the change of measure of a random process involves Girsanov's theorem, see \\cite{EAM95}. Indeed the Zakai form of the continuous time filters in the appendix of the book can be derived in a fairly\nstraightforward manner using Girsanov's theorem.}\nDefine a new probability measure $\\bar{\\mathbb{P}}$ as having associated density\n$$q} % reference prob method density under \\bar{P(x_{0:k}, y_{1:k}) = \\prod_{n=1}^k p_v(y_n) \\,\n p_w(x_n) \\, \\pi_0(x_0) . $$\n The above equation is tantamount to saying that under this new measure $ \\bar{\\mathbb{P}}$, the processes $\\{x_k\\}$ and $\\{y_k\\}$ are iid sequences\nwith density functions $p_w$ and $p_v$, respectively. $\\bar{\\mathbb{P}}$ will be called the {\\em reference probability measure} - under this measure, due to the iid nature of $\\{x_k\\}$ and $\\{y_k\\}$, the filtering\nrecursion can be derived conveniently, as we now describe.\n\nLet\n $\\bar{\\E}$ denote expectation associated with measure $\\bar{P}$, so that for any function $\\phi(x_k)$, the conditional expectation is\n $$ \\bar{\\E}\\{ \\phi(x_k) | y_{1:k}\\} = \\int \\phi(x_k) q} % reference prob method density under \\bar{P(x_{0:k}, y_{1:k}) dx_{0:k} $$\n Obviously, to obtain the expectation $\\E\\{\\phi(x_k)| y_{1:k}\\}$ under the probability\nmeasure $\\mathbb{P}$, it follows from (\\ref{eq:erp}) that\n \\begin{align}\n \\E\\{ \\phi(x_k) | y_{1:k}\\} &= \\frac{\\int \\phi(x_k) \\Lambda_k q} % reference prob method density under \\bar{P(x_{0:k}, y_{1:k}) dx_{0:k}}\n {\\int \\Lambda_k q} % reference prob method density under \\bar{P(x_{0:k}, y_{1:k}) dx_{0:k}} ,\n \\quad \\text{ where } \\Lambda_k = \\frac{p(x_{0:k}, y_{1:k}) }{ q} % reference prob method density under \\bar{P(x_{0:k}, y_{1:k}) } \\\\\n &= \\frac{\\bar{\\E}\\{ \\Lambda_k\\phi(x_k) | y_{1:k} \\} }{ \\bar{\\E}\\{ \\Lambda_k| y_{1:k}\\}}\n \\nonumber\n \\end{align}\n \n\n\nThe derivation then proceeds as follows. \\begin{align*}\n& \\int_\\mathcal{X} {q}_k(x) \\phi(x) dx =\n\\bar{\\E}\\{\\Lambda_k \\phi(x_k) | \\mathcal{Y}_k\\} \\qquad \\text{ (definition of ${q}_k$) } \\\\ \n&= \\int \\frac{p(x_{0:k},y_{1:k})}{q} % reference prob method density under \\bar{P(x_{0:k},y_{1:k})} \\phi(x_k) q} % reference prob method density under \\bar{P(x_{0:k},y_{1:k}) dx_{0:k} \\\\\n&= \\int \\frac{p(x_{0:k-1},y_{1:k-1})}{q} % reference prob method density under \\bar{P(x_{0:k-1},y_{1:k-1})} \n\\frac{p(y_k | x_k) p(x_k| x_{k-1})}{\\cancel{p_v(y_k) } \\cancel{ p_w(x_k)} }\n\\phi(x_k) \\cancel{p_v(y_k)} \\cancel{p_w(x_k) } q} % reference prob method density under \\bar{P(x_{0:k-1},y_{1:k-1}) dx_{0:k} \\\\\n&= \\int \\frac{p(x_{0:k-1},y_{1:k-1})}{q} % reference prob method density under \\bar{P(x_{0:k-1},y_{1:k-1})} \n\\left[\\int_\\mathcal{X} {p(y_k | x_k) p(x_k| x_{k-1})} \\phi(x_k) dx_k\\right] q} % reference prob method density under \\bar{P(x_{0:k-1},y_{1:k-1}) dx_{0:k-1} \n\\\\\n&= \\bar{\\E}\\{ \\Lambda_{k-1} \\left[\\int_\\mathcal{X} {p(y_k | x_k) p(x_k| x_{k-1})} \\phi(x_k) dx_k\\right] | y_{1:k-1} \\} \\\\\n&= \\int {q}_{k-1}(x_{k-1}) \\left[\\int_\\mathcal{X} {p(y_k | x_k) p(x_k| x_{k-1})} \\phi(x_k) dx_k\\right] dx_{k-1}\n\\end{align*}\nwhere the last equality follows from the definition of ${q}$ in the first equality.\n\nSince this holds for any test function $\\phi(x)$, we have that the material inside the integral in the left and right hand side are equal.\nSo \n$$ \\pi_k(x_k ) = p(y_k| x_k) \\int_\\mathcal{X} {q}_{k-1}(x_{k-1}) p(x_k| x_{k-1}) dx_{k-1}. $$\n\n\n \n \n\n\\chapter{Algorithms for Maximum Likelihood Parameter Estimation} \n\n\\begin{compactenum}\n\n\\item A standard drill exercise involves deriving the Cram\\'er-Rao bound in terms of the Fisher information matrix; see wikipedia or any book\nin statistical signal processing for an elementary\ndescription.\n\n\\item {\\bf Minorization Maximization Algorithm (MM Algorithm).} The EM algorithm is a special case of the MM algorithm\\footnote{MM can also be used equivalently to denote majorization minimization}; see \\cite{HL04} for a nice tutorial on MM \n\\index{Minorization Maximization algorithm} \nalgorithms. MM algorithms constitute a general purpose method for optimization and are not restricted just to maximum likelihood estimation.\n\nThe main idea behind the MM algorithm is as follows: Suppose we wish to compute the maximizer $\\th^*$ of a function $\\phi(\\theta)$.\nThe idea is to construct a minorizing function $g(\\theta, \\theta^{(m)})$ such that\n\\begin{equation} \\begin{split}\ng(\\theta, \\theta^{(m)}) & \\leq \\phi(\\theta) \\quad \\text{ for all } \\theta \\\\\n g(\\theta^{(m)} , \\theta^{(m)} ) &= \\phi(\\theta^{(m)}) . \\label{eq:minorf}\n\\end{split} \\end{equation}\nThat is, the minorizing function $g(\\theta, \\theta^{(m)}) $ lies above $ \\phi(\\theta)$ and is a tangent to it at the point $\\theta^{(m)}$.\nHere $$\\theta^{(m)} = \\operatornamewithlimits{argmax}_\\th g(\\th^{(m-1)}, \\th) $$\n denotes the estimate of the maximizer at iteration $m$ of MM algorithm.\n\nThe property (\\ref{eq:minorf})\nimplies that successive iterations of the MM algorithm yield $$\\phi(\\theta^{(m+1)}) \\geq \\phi(\\theta^{(m)}). $$\nIn words, successive iterations of the MM algorithm yield increasing values of the objective function which is a very useful property for a general\npurpose numerical optimization algorithm.\nThis is shown straightforwardly as follows:\n\\begin{align*} \\phi(\\th^{(m+1}) &= \\phi(\\th^{(m+1}) - g(\\theta^{(m+1)}, \\theta^{(m)}) + g(\\theta^{(m+1)}, \\theta^{(m)}) \\\\\n& \\stackrel{a}{\\geq} \\phi(\\th^{(m+1}) - g(\\theta^{(m+1)}, \\theta^{(m)}) + g(\\theta^{(m)}, \\theta^{(m)}) \\\\\n& \\stackrel{b}{\\geq} \\phi(\\th^{(m}) - \\cancel{g(\\theta^{(m)}, \\theta^{(m)}) } + \\cancel{g(\\theta^{(m)}, \\theta^{(m)})}\n \\end{align*}\nInequality (a) follows since $g(\\theta^{(m+1)}, \\theta^{(m)}) \\geq g(\\theta^{(m)}, \\theta^{(m)})$ by definition since $\\theta^{(m+1)} =\n\\operatornamewithlimits{argmax}_\\th g(\\theta, \\theta^{(m)}) $. Inequality (b) follows from (\\ref{eq:minorf}).\n\nThe EM algorithm is a special case of the MM algorithm where \\index{EM algorithm} \\index{Minorization Maximization algorithm! EM algorithm}\n$$ g(\\th,\\theta^{(m)} = Q(\\th,\\theta^{(m)}) - Q(\\theta^{(m)},\\theta^{(m)}), \\quad\n\\phi(\\th) = \\logl(\\th) - \\logl(\\th^{(m)} ) $$\nHere $\\logl(\\theta) = \\log p(y_{1:N} | \\theta)$ is the log likelihood which we want to maximize to compute the MLE\nand $Q(\\th,\\theta^{(m)}) $ is the auxiliary log likelihood defined\nin (\\ref{eq:auxl0}) which is maximized in the M step of the EM algorithm.\n\nIndeed the minorization property (\\ref{eq:minorf}) was established for the EM algorithm in Lemma \\vref{lem:em} of the book by using Jensen's inequality.\n\n\\item {\\bf EM algorithm in more elegant (abstract) notation.}\nLet $\\{P_\\theta\\,,\\,\\theta\\in \\Theta\\}$ be a family of probability\nmeasures on a measurable space $(\\Omega, {\\cal F})$ all absolutely\ncontinuous with respect to a fixed probability measure $P_0$,\nand let ${\\cal Y} \\subset {\\cal F}$. The likelihood function for\ncomputing an estimate of the parameter $\\theta$ based on the\ninformation available in ${\\cal Y}$ is\n\\begin{displaymath} \n L(\\theta) = \\E_0[ \\frac{dP_\\theta}{dP_0}\\mid {\\cal Y}]\\ , \n\\end{displaymath}\nand the MLE estimate is defined by\n\\begin{displaymath} \n \\widehat{\\theta}\\in \\operatornamewithlimits{argmax}_{\\theta\\in \\Theta} L(\\theta)\\ . \n\\end{displaymath}\nIn general, the MLE is difficult to compute directly,\nand the EM algorithm provides an iterative approximation method~:\n\\begin{description}\n \\item{{\\it Step 1.}} Set $p=0$ and choose $\\widehat{\\theta}_0$.\n\n \\item{{\\it Step 2.}} (E--step) Set $\\theta' = \\widehat{\\theta}_p$\nand compute $Q(\\cdot,\\theta')$, where\n\\begin{displaymath} \n Q(\\theta,\\theta') \n = \\E_{\\theta'}[ \\log\\frac{dP_\\theta}{dP_{\\theta'}}\\mid {\\cal Y}]\\ . \n\\end{displaymath}\n\n \\item{{\\it Step 3.}} (M--step) Find\n\\begin{displaymath} \n \\widehat{\\theta}_{p+1}\\in \\operatornamewithlimits{argmax}_{\\theta \\in \\Theta} Q(\\theta,\\theta')\\ . \n\\end{displaymath}\n\n \\item{{\\it Step 4.}} Replace $p$ by $p+1$ and repeat beginning\nwith Step~2, until a stopping criterion is satisfied.\n\\end{description}\nThe sequence generated $\\{\\widehat{\\theta}_p\\,,\\,p\\geq 0\\}$\ngives non--decreasing values of the likelihood function~:\nindeed, it follows from Jensen's inequality that\n\\begin{displaymath} \n \\log L(\\widehat{\\theta}_{p+1}) - \\log L(\\widehat{\\theta}_p) \n \\geq Q(\\widehat{\\theta}_{p+1},\\widehat{\\theta}_p) \n \\geq Q(\\widehat{\\theta}_p,\\widehat{\\theta}_p) = 0\\ , \n\\end{displaymath}\nwith equality if and only if $\\widehat{\\theta}_{p+1} = \\widehat{\\theta}_p$.\n\n\\item {\\bf Forward-only EM algorithm for Linear Gaussian Model.} In \\S 4.4 of the book, we described a forward-only EM algorithm for ML parameter estimation of the a HMM.\nForward-only EM algorithms can also be constructed for maximum likelihood estimation of the parameters of a linear Gaussian state space model\n\\cite{EK99}. These involve computing filters for functionals of the state and use Kalman filter estimates.\n\n\\item {\\bf Sinusoid in HMM.} Consider a sinusoid with amplitude $A$ and phase $\\phi$. It is observed as\n$$y_k = x_k + A\\, \\sin(k\/100 + \\phi) + v_k$$\nwhere $v_k$ is an iid Gaussian noise process.\nUse the EM algorithm to estimate $A, \\phi$ and the parameters of the Markov chain and noise variance.\n\n\\item In the forward-only EM algorithm of \\S 4.4, the filters for the number of jumps involves $O(X^4)$ computations at each time while filters for the duration time involve\n$O(X^3)$ at each time. Is it possible to reduce the computational cost by approximating some of these estimates? \n\n\\item Using computer simulations, compare the methods of moments estimator for a HMM in \\S 4.5 with the maximum likelihood estimator in terms of efficiency. That is generate several $N$ point trajectories of an HMM with a fixed set of parameters, then compute the variance\nof the estimates. (Of course, instances where the MLE\nthe algorithm converges to local maxima should be eliminated from the computation).\n\n\n\\item Non-asymptotic statistical inference using concentration of measure if very popular today.\nAssuming the likelihood is a Lipschitz function of the observations, and the observations are Markovian, show that\nthe likelihood function concentrates to the Kullback Leibler function.\n\n\\item {\\bf EM Algorithm for State Estimation.} The EM algorithm was used in Chapter \\ref{chp:mle} as a numerical algorithm\nfor maximum likelihood {\\em parameter} estimation. It turns out that the EM algorithm can be used for {\\em state} estimation, particularly for\na jump Markov linear system (JMLS). Recall from \\S \\ref{sec:particle} that a JMLS has model\n\\begin{align*}\nz_{k+1} &= F(\\mc_{k+1})\\,z_k + \\Gamma(\\mc_{k+1})\\,w_{k+1} + f (\\mc_{k+1})\\,u_{k+1} \\\\\ny_k &= H(\\mc_k)\\,z_k + D(\\mc_k)\\,v_k + g(\\mc_k)u_k. \n\\end{align*}\nAs described in \\S \\ref{sec:particle}, \n the optimal filter for a JMLS is computationally intractable.\n In comparison for a JMLS, the EM algorithm can be used to estimate the MAP (maximum aposteriori state estimate).\nsystem (assuming the parameters of the JMLS are known). Show how \none can compute this MAP state estimate $\\max_{z_{1:k},r_{1:k}} P(y_{1:k} | z_{1:k},r_{1:k})$ using the EM algorithm.\nIn \\cite{LK99} is shown that the resulting EM algorithm involves the cross coupling of a Kalman and HMM smoother.\nA data augmentation algorithm in similar spirit appears in \\cite{DLK00}. \\index{jump Markov linear system! EM algorithm for state estimation}\n\n\n\\item {\\bf Quadratic Convergence of Newton Algorithm.}\n\n\\index{Newton algorithm! quadratic convergence}\nWe start with some definitions:\nGiven a sequence $\\{\\th^{(n)}\\}$ generated by an optimization algorithm, the {\\em order} of convergence is $p$ if \n\\begin{equation} \\beta = \\limsup_{n \\rightarrow \\infty} \\frac{\\|\\th^{(n+1) } - \\th^*\\|} {\\|\\th^{(n) } - \\th^*\\|^p} \\text{ exists } \n\\label{eq:convergenceorder}\n\\end{equation}\nAlso if $p=1$ and $\\beta < 1$, the sequence is said to converge linearly to $\\th^*$ with {\\em convergence ratio (rate)} $\\beta$.\nMoreover, the case $p=1$ and $\\beta = 0$ is referred to as superlinear convergence.\n\n\n\\begin{compactenum}\n\\item\nRecall that the Newton Raphson algorithm computes the MLE iteratively as\n$$ \\th^{(n+1)} = \\th^{(n)} + \\bigl({\\nabla^2} \\logl (\\th^{(n)}) \\big)^{-1} \\nabla \\logl(\\th^{(n)}) $$\n The Newton Raphson algorithm has quadratic order of convergence in the following sense.\nSuppose the log likelihood $\\logl(\\th)$ is twice continuous differentiable and that at a local maximum $\\th^*$, the Hessian\n$\\nabla_\\th^2\\logl$ is positive definite. Then if started sufficient close to $\\th^*$, Newton Raphson converges to $\\th^*$ at a quadratic rate.\nthat the model estimates satisfy $\\th^{(n)}$ satisfy\n$$\\| \\th^{(n+1)} - \\th^* \\| \\leq \\beta \\| \\th^{(n)} - \\th^*\\|^2 $$\nfor some constant $\\beta$.\n\nThis is shown straightforwardly (see any optimization textbook) as follows:\n\\begin{equation}\n\\begin{split} \\|\\th^{(n+1)} - \\th^* \\| & = \\| \\th^{(n)} - \\th^* + \\bigl({\\nabla^2} \\logl (\\th^{(n)}) \\big)^{-1} \\nabla \\logl(\\th^{(n)}) \\|\n\\\\ \n&= \\| \\bigl({\\nabla^2} \\logl (\\th^{(n)}) \\big)^{-1} \\biggl( \\nabla \\logl(\\th^{(n)}) - \\nabla \\logl(\\th^*) - \\nabla^2 \\logl (\\th^{(n)}) \\big(\\th^{(n) }- \\th^* \\big)\n\\biggr)\n\\end{split}\n\\end{equation}\nFor $\\|\\th^{(n)} - \\th^*\\| < \\rho$, it is clear from a Taylor series expansion that\n$$ \\|\\nabla \\logl(\\th^*) - \\nabla \\logl(\\th^{(n)}) - \\nabla^2 \\logl (\\th^{(n)}) \\big(\\th^* - \\th^{(n)} \\big) \\| \\leq \\beta_1\\| \\th^{(n)} - \\th^*\\|^2$$\nfor some positive constant $\\beta_1$.\nAlso, $\\| \\bigl({\\nabla^2} \\logl (\\th^{(n)}) \\big)^{-1} \\| \\leq \\beta_2$.\n\n\\item The convergence order and rate\nof the EM algorithm has been studied in great detail since the early 1980s; there are numerous papers in the area; see \\cite{XJ96} and the\nreferences therein.\nThe EM algorithm has linear convergence order, i.e., $p=1$ in (\\ref{eq:convergenceorder}). \nPlease see\n\\cite{MXJ00} and the references therein for examples where EM exhibits superlinear convergence.\n\n\\end{compactenum}\n\\end{compactenum}\n\n\\chapter{Multi-agent Sensing: Social Learning and Data Incest} \n\n\\section{Problems}\n\\begin{enumerate}\n\n\\item A substantial amount of insight can be gleaned by actually simulating the setup (in Matlab) of the social learning\nfilter for both the random variable and Markov chain case. Also simulate the risk-averse social learning filter discussed in \\S \\ref{sec:classicalsocial}\nof the book.\n\n\n\\item {\\bf CVaR Social Learning Filter.} Consider the risk averse social learning \\index{CVaR social learning filter}\ndiscussed in \\S \\ref{sec:classicalsocial}.\nSuppose\nagents choose their actions $a_{k}$ to minimize the CVaR risk averse measure \n$$\na_{k} = {\\underset{a \\in \\mathcal{A}}{\\text{argmin}}} \\{ {\\underset{z \\in \\mathbb{R}}{\\text{min}}} ~ \\{ z + \\frac{1}{\\alpha} \\mathbb{E}_{y_{k}}[{\\max} \\{ (c(x_{k},a)-z),0 \\rbrace] \\} \\} \n$$\nHere $\\alpha \\in (0,1]$ reflects the degree of risk-aversion for the agent (the smaller $\\alpha$ is, the more risk-averse the agent is). \nShow that the structural result Theorem \\ref{thm:monotone} continues to hold for the CVaR social learning filter.\nAlso show that for sufficiently risk-averse agents (namely, $\\alpha$ close to zero), social learning ceases and agents always herd.\n\nGeneralize the above result to any coherent risk measure.\n\n\\item The necessary and sufficient condition given in Theorem \\ref{thm:sufficient} for exact data incest removal requires\n that \n$$ A_n(j,n)=0 \\implies w_n(j)= 0, \\text{ where } \\quad w_n = T_{n-1}^{-1} t_n,\n$$\nand $T_n = \\text{sgn}((\\mathbf{I}_n-A_n)^{-1}) = \\begin{bmatrix} T_{n-1} & t_n \\\\ 0_{1 \\times n-1} & 1 \\end{bmatrix} $ is the transitive closure matrix.\nThus the condition depends purely on the adjacency matrix. Discuss what types of matrices satisfy the above condition.\n\n\\item Theorem \\ref{thm:sufficient} also applies to data incest where the prior and likelihood are Gaussian. The posterior is then\nevaluated by a Kalman filter. Compare the performance of exact data incest removal with the covariance intersection algorithm\nin \\cite{CAM02} which assumes no knowledge of the correlation structure (and hence of the network).\n\n\\item Consensus algorithms \\cite{SM04} have been extremely popular during the last decade and there are numerous papers in the area.\n They are non-Bayesian and seek to compute,\nfor example, the average over measurements observed at a number of nodes in a graph. It is worthwhile comparing the performance\nof the optimal Bayesian incest removal algorithms with consensus algorithms.\n\n\n\\item The data incest removal algorithm in \\S \\ref{sec:incest} of the book arises assumes that agents do not send additional information\napart from their incest free estimates. Suppose agents are allowed to send a fixed number of labels of previous agents from whom\nthey have received information. What is the minimum about of additional labels the agents need to send in order to completely remove data incest.\n\n\\item Quantify the bias introduced by data incest as a function of the adjacency matrix.\n\n\\item Prospect theory \\index{prospect theory} (pioneered by the psychologist Kahneman \\cite{KT79} who won the 2003 Nobel prize in economics) is a behavioral economic theory\nthat seek to model how humans make decisions amongst probabilistic alternatives. (It is an alternative to expected utility theory considered\nin the social learning models of this chapter.) The main features are:\n\\begin{compactenum}\n\\item Preference is an S-shaped curve with reference point $x=0$\n\\item The investor maximizes the expected value $V(x)$ where $V$ is a preference and $x$ is the change in wealth.\n\n\\item Decision maker employ decision weight $w(p)$ rather than objective probability $p$, where the weight function $w(F)$ has a reverse\nS shape where $F$ is the cumulative probability.\n\\end{compactenum}\nConstruct a social learning filter where the utility function satisfies the above assumptions. Under what conditions do information cascades\noccur?\n\n\\item {\\bf Rational Inattention.} Another powerful way for modeling the behavior of (human) decision makers is in terms of rational inattention. \nSee the seminal work of \\cite{Sim03} where essentially the ability of the human to absorb information is modeled via the \ninformation theoretic capacity of a communication channel.\\index{rational inattention}\n\n\\item There are several real life experiments that seek to understand how humans interact in decision making. See for example \\cite{BOL10}\nand \\cite{KH15}. In \\cite{BOL10}, four models are considered. How can these models be linked to social learning?\n\\end{enumerate}\n\n\n\n\n\\section{Social Learning with limited memory} \\index{social learning! limited memory}\nHere we briefly describe a variation of the vanilla social learning protocol. In order to mitigate herding, assume that agents randomly\nsample only a fixed number of previous actions. The aim below is to describe the resulting setup; see \\cite{SS97} for a detailed discussion.\n\n\nLet the variable $\\theta \\in \\{1,2\\}$ denote the states. Let $a \\in \\{1,2\\} $ denote the action alphabet and $y \\in \\{1,2\\} $ denote the observation alphabet. In this model of social learning with limited memory, it is assumed that each agent (at time $t \\geq N+1$) observes \\textit{only} $N$ randomly selected actions from the history $h_{t}= \\{a_{1},a_{2},\\hdots,a_{t-1} \\}$. In the periods $t \\leq N$, each agent acts according to his private belief. This phase is termed as the seed phase in the model. \\\\\n\nLet $z_{t}^{(1)}$ denote the number of times action $1$ is chosen until time $t$, i.e, $$z_{t}^{(1)} = \\sum_{j=1}^{t} I(a_{j}=1).$$ Let $\\hat{z}_{t}^{(1)}$ denote the number of times action $1$ is chosen in a sample of $N$ randomly observed actions in the past, i.e, $\\hat{z}_{t}^{(1)} = \\sum_{j=1}^{N} I(a_{j}=1)$. \\\\\n\nThe social learning protocol with limited memory is as follows:\n\\begin{itemize}\n\\item[1.)] \\textit{Private belief update}: Agent $t$ makes two observations at each instant $t(>N)$. These observations correspond to a noisy private signal $y_{t}$ and a sample of $N$ past actions from the history $h_{t}$ sampled uniformly randomly. Let $B_{y_{t}}$ and $D_{z_{t}=k}$ denote the probability of observing $y_{t}$ and $(\\hat{z}_{t}^{(1)}=k)$ respectively. The private belief is updated as follows.\n\nFor each draw from the past, the probability of observing action 1 is $z_{t}^{(1)}\/ (t-1)$. So the\nprobability that at time $t$, action $1$ occurs $k$ times in a random sample of $N$ observed actions is\n\\begin{equation*}\n\\mathbb{P}(\\hat{z}_{t}^{(1)} = k|z_{t}^{(1)}) = \\frac{N!}{(N-k)!k!} \\left( \\frac{z_{t}^{(1)}}{t}\\right) ^{k} \\left(1-\\frac{z_{t}^{(1)}}{t}\\right)^{N-k}\n\\end{equation*}\nTherefore,\nthe number of times action `$1$' is chosen in the sample, $\\hat{z}_{t}^{(1)}$, has a distribution that depends on $\\theta$ according to:\n\\begin{equation*}\n\\mathbb{P}(\\hat{z}_{t}^{(1)} = k|\\theta) = \\sum_{z_{t}^{(1)}=1}^{t} \\mathbb{P}(\\hat{z}_{t}^{(1)} = k|z_{t}^{(1)}) \\mathbb{P}(z_{t}^{(1)}|\\theta)\n\\end{equation*}\n\nAfter obtaining a private noisy signal $y_{t}$, and having observed ($\\hat{z}_{t}^{(1)}=k$), the belief $\\pi_t = [\\pi_t(1), \\pi_t(2)]^\\prime$ where\n$\\pi_t(i) = \\mathbb{P}(\\th=i| \\hat{z}_{t}^{(1)},y_t)$\nis updated by agent $t$ as:\n\\begin{equation*}\n{\\hspace{1.5cm}}\\pi_{t} = \\frac{B_{y_{t}}D_{z_{t}=k}}{\\textbf{1}'B_{y_{t}}D_{z_{t}=k}}.\n\\end{equation*}\nHere $B$ and $D$ are the observation likelihoods of $y_t$ and $\\hat{z}_{t}^{(1)}$ given the state:\n\\[ ~~B_{y_{t}}=\\text{diag}(\\mathbb{P}(y_{t}|\\theta=i),i\\in \\{1,2\\}), \\quad \nD_{z_{t}=k} = \\begin{bmatrix}\n\\mathbb{P}(\\hat{z}_{t}^{(1)}=k|\\theta=1) \\\\\n\\mathbb{P}(\\hat{z}_{t}^{(1)}=k|\\theta=2)\n\\end{bmatrix}\n\\]\n\n\\item[2.)] \\textit{Agent's decision}: With the private belief $\\pi_{t}$, the agent $t$ makes a decision as:\n\\begin{equation*}\na_{t} = {\\underset{a\\in \\{1,2\\}}{\\text{argmin}}}~ c_{a}^{T} \\pi_{t}\n\\end{equation*}\nwhere $c_{a}$ denotes the cost vector.\n\n\\item[3.)] \\textit{Action distribution}: The distribution of actions $\\mathbb{P}(z_{t}^{(1)}|\\theta)$ in the two states $\\theta = 1,2$ is assumed to be common knowledge at time $t$. It is updated after the decision of agent $t$ as follows. \\\\\n\nThe probability of $(a_{t}=1)$ in period $t$ depends on the actual number of `$1$' actions $z_{t}^{(1)}$ and on the state according to:\n\\begin{equation*}\n{\\hspace{-0.75cm}}\\mathbb{P}(a_{t}=1|z_{t}^{(1)}=n,\\theta) = \\sum_{k=0}^{N} \\sum_{i=1}^{2} \\mathbb{P}(a_{t}=1|y=i,\\hat{z}_{t}^{(1)}=k,z_{t}^{(1)}=n,\\theta)~ \\mathbb{P}(y=i|\\theta)~\\mathbb{P}(\\hat{z}_{t}^{(1)} = k|z_{t}^{(1)}=n)\n\\end{equation*}\n\nwhere,\n\\[ \\mathbb{P}(a_{t}=1|y=i,\\hat{z}_{t}^{(1)}=k,z_{t}^{(1)}=n,\\theta) = \\left\\{ \\begin{array}{ll}\n 1 & \\mbox{if $c_{1}^{T}B_{y=i}D_{z_{t}=k} < c_{2}^{T}B_{y=i}D_{z_{t}=k}$};\\\\\n 0 & \\mbox{otherwise}.\\end{array} \\right. \\] \n\n\nAfter agent $t$ takes an action, the distribution is updated as:\n\\begin{equation}{\\label{eq:dis}}\n{\\hspace{-0.75cm}}\\mathbb{P}(z_{t+1}^{(1)}=n|\\theta) = \\mathbb{P}(z_{t}^{(1)}=n|\\theta)(1-\\mathbb{P}(a_{t}=1|z_{t}^{(1)}=n,\\theta)) + \\mathbb{P}(z_{t}^{(1)}=n-1|\\theta)\\mathbb{P}(a_{t}=1|z_{t}^{(1)}=n,\\theta)\n\\end{equation}\n\\end{itemize}\n\nAccording to equation~\\eqref{eq:dis}, the sufficient statistic $\\mathbb{P}(z_{t}^{(1)}|\\theta)$ is growing with time $t$. It is noted that this has $(t-2)$ numbers at time $t$ and hence grows with time. $\\mathbb{P}(z_{t+1}^{(1)}=n|\\theta)$ in equation~\\eqref{eq:dis} is used to compute $D_{z_{t+1}}$.\n\n\nWith the above model, consider the following questions:\n\\begin{compactenum}\n\\item Show that there is asymptotic herding when $N =1$.\n\\item Show that for $N=2A$, reduction in the historical information will improve social learning. Also, comment on whether there is herding when $N=2$.\n\\item Show that as $N$ increases, the convergence to the true state is slower.\nHint: Even though more observations are chosen, greater weight on the history precludes the use of private information.\n\\end{compactenum}\n\n\\chapter{Fully Observed Markov Decision Processes} \n\n\n\\section{Problems}\n\\begin{compactenum}\n\n\\item The following nice example from \\cite{KV86} gives a useful motivation for feedback control in stochastic systems. It shows that for\nstochastic systems, using feedback control can result in behavior that cannot be obtained by an open loop system.\n\\index{why feedback control?}\n\\begin{compactenum}\n\\item First, recall from undergraduate control courses that for a deterministic linear time invariant system with forward transfer function $G(z^{-1})$ and negative feedback $H(z^{-1})$, the equivalent transfer function\nis $\\frac{G(z^{-1})}{1+G(z^{-1}) H(z^{-1}) }$. So an open loop system with this equivalent transfer function is identical to a feedback system.\n\n\\item More generally, consider the deterministic system $$x_{k+1} = \\phi(x_k, u_k), y_k = \\psi(x_k,u_k) $$ Suppose the actions are given\nby a policy of the form $$u_k = \\mu(x_{0:k},y_{1:k})$$\nThen clearly, the open loop system, $$x_{k+1} = \\phi(x_k, \\mu(x_{0:k},y_{1:k})), y_k = \\psi(x_k,\\mu(x_{0:k},y_{1:k})) $$ generates the same\nstate and observation sequences. \n\nSo for a deterministic system (with fully specified model), open and closed loop behavior are identical.\n\n\\item Now consider a fully observed stochastic system with feedback:\n\\begin{equation}\n\\begin{split} x_{k+1} &= x_k + u_k + w_k , \\\\\n u_k &= - x_k \\end{split} \\end{equation}\n where $w_k$ is iid with zero mean and variance $\\sigma^2$ (as usual we assume $x_0$ is independent of $\\{w_k\\}$.)\n Then $x_{k+1} = w_k$ and so $u_k = -w_{k-1}$ for $k=1,2,\\dots$.\n Therefore $\\E\\{x_k\\} = 0$ and $\\operatorname{Var}\\{x_k^2\\} = \\sigma^2$.\n \n\\item Finally, consider an open loop stochastic system where $u_k$ is a deterministic sequence:\n$$\n x_{k+1}= x_k + u_k + w_k $$\n Then $\\E\\{x_k\\} = \\E\\{x_0\\} + \\sum_{n=0}^{k-1} u_k $ and $\\operatorname{Var}\\{x_k^2\\} = \\E\\{x_0^2\\} + k \\sigma^2$.\n Clearly, it is impossible to construct a deterministic input sequence that yields a zero mean state with variance $\\sigma^2$.\n \\end{compactenum}\n\n\\item {\\bf Trading of call options.} An investor buys a call option at a price $p$. He has $N$ days to exercise this option. If the investor exercises the option when the stock price is $x$, he\ngets $x-p$ dollars. The investor can also decide not the exercise the option at all.\n\nAssume the stock price evolves as $x_k = x_0 + \\sum_{n=1}^k w_n $ where $\\{w_n \\}$ is in iid process.\nLet $\\tau$ denote the day the investor decides to exercise the option. Determine the optimal investment strategy to maximize\n$$\\E\\{ (x_\\tau - p) I(\\tau \\leq T) \\}. $$\nThis is an example of a fully observed stopping time problem. Chapter \\ref{ch:pomdpstop} considers more general stopping time POMDPs.\n\nNote: Define $s_k \\in \\{0,1\\}$ where $s_k = 0$ means that the option has not been exercised until time $k$.\n$s_k = 1$ means that the option has been exercised before time $k$. Define the state $z_k = (x_k,s_k)$.\n\nDenote the action $u_k = 1$ to exercise option and $u_k = 0$ means do not exercise option.\nThen the dynamics are\n$$ s_{k+1} = \\max\\{ s_k,u_k\\} , \\quad x_{k+1} = x_k + w_k $$\nThe reward at each time $k$ is $r(z_k,u_k,k) = (1-s_k) u_k (x_k-p)$ and the problem can be formulated as\n$$\\max_\\mu \\E\\{ \\sum_{k=1}^N r(z_k,u_k,k)\\} $$\n\n\\item\nDiscounted cost problems can also be motivated as stopping time problems (with a random termination time).\nSuppose at each time $k$, the MDP can terminate with probability $1 - \\rho$ or continue with probability $\\rho$.\nLet $\\tau$ denote the random variable for the termination time. Consider the undiscounted cost MDP\n\\begin{align*}\n \\E_{{\\boldsymbol{\\mu}}} \\left\\{ \\sum_{k=0}^{\\tau} c(x_k,u_k) \\mid x_0 = i\\right\\} &= \n\\E_{{\\boldsymbol{\\mu}}} \\left\\{ \\sum_{k=0}^{\\infty} I(k \\leq \\tau) \\, c(x_k,u_k) \\mid x_0 = i\\right\\} \\\\\n&= \\E_{{\\boldsymbol{\\mu}}} \\left\\{ \\sum_{k=0}^{\\infty} \\rho^k c(x_k,u_k) \\mid x_0 = i\\right\\}.\n\\end{align*}\nThe last equality follows since $\\mathbb{P}(k \\leq \\tau) = \\rho^k$. \n\n\n\n\\begin{comment}\n\n\\item Deterministic and stochastic LQ control\n\n\\item Modified Policy iteration\n\n\\item Afriat's test for a single jump change in utility function.\n\n\\end{comment}\n\n\\item We discussed risk averse utilities and dynamic risk measures briefly in \\S \\ref{sec:riskaverse}. Also \\S \\ref{subsec:afriat} discussed revealed\npreferences for constructing a utility function from a dataset. \nGiven a utility function $U(x)$,\na widely used measure for the degree of risk aversion is the Arrow-Pratt risk aversion coefficient which is defined as \n$$ a(x) =- \\frac{d^2 U\/dx^2}{ dU\/dx} .$$\nThis is often termed as an absolute risk aversion measure, while $x\\, a(x)$ is termed a relative risk aversion measure. \nCan this risk averse coefficient be used for mean semi-deviation risk, conditional value at risk( CVaR) and exponential risk?\n\n\n\\item A classical result involving utility functions is the following \\cite[pp.42]{HS84}:\nA rational decision maker who compares random variables only according to their means and variances must have preferences consistent\nwith a quadratic utility function. Prove this result.\n\n\\end{compactenum}\n\n\n\n\n\\section{Case study. Non-cooperative Discounted Cost Markov games} \\index{Markov game} \\label{sec:markovgame}\n\\S \\ref{sec:mdpdiscount} of the book dealt with infinite horizon discounted MDPs.\nBelow we introduce briefly some elementary ideas in non-cooperative infinite horizon discounted Markov games. There are several excellent \nbooks in the area\n\\cite{KV12,BO91}. \n\nMarkov games can be viewed as a multi-agent decentralized extension of MDPs. They arise in a variety of applications including dynamic spectrum allocation, financial models and smart grids. Our aim here is to consider some simple cases where the Nash\nequilibrium can be obtained by solving a linear programming problem.\\footnote{The reader should be cautious with decentralized stochastic\ncontrol. The famous Witsenhausen's counterexample formulated in the 1960s shows that even a deceptively simple toy problem in decentralized\nstochastic control can be very difficult to solve, see \\url{https:\/\/en.wikipedia.org\/wiki\/Witsenhausen\\%27s_counterexample} \\index{Witsenhausen's counterexample}}\n\nConsider the following infinite horizon discounted cost two-payer Markovian game.\nThere are two decision makers (players) indexed by $l=1,2$.\n\\begin{compactitem}\n\\item Let $\\act1_k \\in \\,\\mathcal{U} $ and $\\act2_k\\in \\,\\mathcal{U} $ denote the action of player 1 and player 2, respectively, at time $k$. For convenience we assume\nthe same action space for both players.\n\\item The cost incurred\nby player $l \\in \\{1,2\\}$ for state $x$, actions $\\act1,\\act2$ is $c_l(x,\\act1,\\act2)$.\n\\item The transition probabilities of the Markov process $x$ depends on the actions of both players:\n$$ P_{ij}(\\act1,\\act2) = \\mathbb{P}(x_{k+1} = j | x_k = i, \\act1_k = \\act1, \\act2_k = \\act2) $$\n\\item \nDefine the policies for the stationary (randomized) Markovian policies for two players as $\\pol1$, $\\pol2$, respectively.\nSo $\\act1_k $ is chosen from probability distribution $\\pol1(x_k)$ and $\\act2_k $ is chosen from probability distribution $\\pol2(x_k)$.\nFor convenience denote the class of stationary Markovian policies as ${\\boldsymbol{\\policy}}_S$.\n\\item The cumulative cost incurred by each player $l \\in \\{1,2\\}$ is \n\\begin{equation} \\Jc{l}_{\\pol1,\\pol2}(x) = \\E\\big\\{ \\sum_{k=0}^\\infty \\rho^k c_l(x_k,\\act1_k,\\act2_k) \\vert x_0 = x\\big\\} \n\\label{eq:cumcostgame} \\end{equation}\nwhere as usual $\\rho \\in (0,1)$ is the discount factor.\n\\end{compactitem}\n\nThe non-cooperative assumption in game theory is that the players are interested in minimizing their individual cumulative costs only; they do not\ncollude.\n\n\\subsection{Nash equilibrium of general sum Markov game} \\index{Markov game! Nash equilibrium}\n \\index{game theory! Markov game}\nAssume that each player has complete knowledge of the other player's cost function. Then the policies\n${\\pol1}^*, {\\pol2}^*$ of the non-cooperative infinite horizon Markov game constitute a Nash equilibrium if \n\\begin{equation} \\label{eq:nasheq}\n\\begin{split}\n \\Jc{1}_{{\\pol1}^*,{\\pol2}^*}(x) &\\leq \\Jc{1}_{\\pol1,{\\pol2}^*}(x), \\quad \\text{ for all } \\pol1 \\in {\\boldsymbol{\\policy}}_S \\\\\n \\Jc{1}_{{\\pol1}^*,{\\pol2}^*}(x) &\\leq \\Jc{1}_{ {\\pol1}^*,{\\pol2}}(x), \\quad \\text{ for all } \\pol2 \\in {\\boldsymbol{\\policy}}_S.\n \\end{split} \\end{equation}\nThis means that unilateral deviations from ${\\pol1}^*, {\\pol2}^*$ result in either player being worse off (incurring a larger cost).\nSince in a non-cooperative game collusion is not allowed, there is no rational reason for players to deviate from the Nash equilibrium\n(\\ref{eq:nasheq}).\n\n\nIn game theory, two important issues are: \n\\begin{compactenum}\n\\item {\\em Does a Nash equilibrium exist? } \nFor the above discounted cost game with finite action and state space, the answer is \"yes\".\n\\begin{theorem}\nA discounted Markov game has at least one Nash equilibrium within the class of Markovian stationary (randomized) policies.\n\\end{theorem}\nThe proof is in \\cite{FV12} and involves Kakutani's fixed point theorem.\\footnote{Existence proofs for equilibria involve using either Kakutani's fixed point theorem (which generalizes Brouwer's fixed point theorem to set valued correspondences) or Tarski's fixed point theorem (which applies to supermodular games). Please see \\cite{MWG95} for a nice intuitive visual illustration\nof these fixed point theorems.}\n\n\\item {\\em How can the Nash equilibria be computed?} Define the randomized policy of player 1 (corresponding to $\\pol1$) and player 2 \n(corresponding to $\\pol2$) as \n$$p(i,\\act1) = \\mathbb{P}(\\act1_k = \\act1 | x_k = i), \\quad q(i,\\act2) = \\mathbb{P}(\\act2_k = \\act2 | x_k = i) $$\nThen for an infinite horizon discounted cost Markov game, the Nash equilibria $(p^*,q^*)$ are global optima of the following non-convex optimization problem:\n\\begin{equation} \\begin{split} & \\text{ Compute } \\max \\sum_{l=1}^2 \\sum_{i=1}^X \\alpha_i \\bigg( \\underline{V}^{(l)}(i)\n- \\sum_{\\act1,\\act2} c_l(i,\\act1,\\act2) p(i,\\act1) q(i,\\act2) \\\\ & - \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act1,\\act2} P_{ij}(\\act1,\\act2) p(i,\\act1) q(i,\\act2) \\underline{V}^{(l)}(j) \\bigg) \\\\ &\\text{ with respect to $(\\underline{V}^{(1)},\\underline{V}^{(2)},p, q)$} \\\\\n \\text{ subject to } &\\underline{V}^{(1)}(i) \\leq \\sum_{\\act2} c(i,\\act1,\\act2) q(i,\\act2) + \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act2} P_{ij}(\\act1,\\act2) q(i,\\act2) \\underline{V}^{(1)}(j) ,\\\\\n&\\underline{V}^{(2)}(i) \\leq \\sum_{\\act2} c(i,\\act1,\\act2) p(i,\\act1) + \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act1} P_{ij}(\\act1,\\act2) p(i,\\act1) \\underline{V}^{(2)}(j) ,\\\\\n& q(i,\\act2) \\geq 0, \\quad \\sum_{\\act2} q(i,\\act2) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act2 = 1,\\ldots,U\n\\\\\n& p(i,\\act1) \\geq 0, \\quad \\sum_{\\act1} p(i,\\act1) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act1 = 1,\\ldots,U.\n\\end{split} \\label{eq:genmarkovgame} \\end{equation}\n In general, solving the non-convex optimization problem (\\ref{eq:genmarkovgame}) is difficult; there\n can be multiple global optima (each corresponding to a Nash equilibrium) and multiple local optima. In fact there is a fascinating property that if all\n the parameters (transition probabilities, costs) are rational numbers, the Nash equilibrium policy can involve irrational numbers. This points to the \n fact that in general one can only approximately compute the Nash equilibrium.\n\n\\index{Markov game! general sum}\n\\end{compactenum}\n\n{\\bf Proof}.\nFirst write (\\ref{eq:genmarkovgame}) in more abstract but intuitive notation in terms of the randomized policies $p,q$ as\n\\begin{equation} \\begin{split} & \n \\max \\sum_{l=1}^2 \\alpha^\\prime \\bigg( \\underline{V}^{(l)} - c_l(p,q) - \\rho P(p,q) \\underline{V}^{(l)} \\biggr) \\\\\n\\text{ subject to } \\; & \\underline{V}^{(1)} \\leq c_1(\\act1,q) + \\rho P(\\act1,q) \\underline{V}^{(1)} , \\quad \\act1 = 1,\\ldots,U \\\\\n & \\underline{V}^{(2)} \\leq c_2(p,\\act2) + \\rho P(p,\\act2) \\underline{V}^{(2)} , \\quad \\act2 = 1,\\ldots,U \\\\\n& p , q \\text{ valid pmfs } \n \\end{split} \\label{eq:genmarkovgame2} \\end{equation}\n It is clear from the constraints that the objective function is always $\\leq 0$. In fact the maximum is attained when the objective function\n is zero, in which case the constraints hold with equality. When the constraints hold at equality, they satisfy\n $$ V_*^{(l)} = \\big(I - \\rho P(p^*,q^*) \\big)^{-1} c_l(p^*,q^*) , \\quad l = 1,2. $$\n This serves as definition of $V_*^{(l)}$ and \nis equivalent to saying\\footnote{This holds since from (\\ref{eq:cumcostgame}), $ \\Jc{l}_{p^*,q^*}(x) = c_l(p^*,q^*) + \n \\rho P c_l(p^*,q^*) + \\rho^2 P^2 c_l(p^*,q^*) + \\cdots + $. Indeed a similar expression holds for discounted cost MDPs.}\n that $V_*^{(l)}$ is the infinite horizon cost attained by the policies $(p^*,q^*)$. That is,\n\\begin{equation} V_*^{(l)} = c_l(p^*,q^*) + \\rho P(p^*,q^*) V_*^{(l)} \\implies \n \\Jc{l}_{p^*,q^*}(x) = V_*^{(l)}. \\label{eq:neeqa} \\end{equation}\n\n \nAlso setting $\\underline{V}^{(l)} =V_*^{(l)}$, the constraints in (\\ref{eq:genmarkovgame2}) satisfy\n$$ V_*^{(1)} \\leq c_1(p,q^*) + \\rho P(p,q^*)V_*^{(1)} , \\quad V_*^{(2)} \\leq c_2(p^*,q) + \\rho P(p^*,q) V_*^{(2)} $$\nimplying that\n\\begin{equation} \\Jc{1}_{p,q^*}(x) \\geq V_*^{(1)}, \\quad \\Jc{2}_{p^*,q}(x) \\geq V_*^{(2)}. \\label{eq:neeqb} \\end{equation}\n(\\ref{eq:neeqa}) and (\\ref{eq:neeqb}) imply that $(p^*,q^*)$ constitute a Nash equilibrium. \\qed\n \n{\\em Remark}.\nThe reader should compare the above proof with the linear programming formulation for a discounted cost MDP. In that derivation we started with\na similar constraint\n\\begin{equation} \\underline{V} \\leq c_1(\\act1) + \\rho P(\\act1) \\underline{V} . \\label{eq:edmdpc} \\end{equation}\n This implies that $\\underline{V} < V$ where $V$ denotes the unique value function of Bellman's equation. Therefore the objective was to find $\\max \\alpha^\\prime \\underline{V}$\nsubject to (\\ref{eq:edmdpc}). \nSo in MDP case we obtain a linear program.\nIn the dynamic game case, in general, there is no value function to clamp (upper bound) $\\underline{V}$.\n\n\n\n\n\n\\subsection{Zero-sum discounted Markov game} \\index{Markov game! zero sum}\nWith the above brief introduction,\nthe main aim below is to give special cases of {\\em zero-sum} Markov games where the Nash equilibrium can be \ncomputed via linear programming. (Recall \\S \\ref{sec:vipilp} of the book shows how a discounted cost MDP can be solved via linear programming.)\n\nA discounted Markovian game is said to be zero sum\\footnote{A constant sum game \n $c_1(x,\\act1,\\act2) + c_2(x,\\act1,\\act2) = K$ for constant $K$ is equivalent to a zero sum game.\n Define $\\bar{c}_l(x,\\act1,\\act2) = c_l(x,\\act1,\\act2)+K\/2 $, $l=1,2$, resulting in a zero sum game in terms of $\\bar{c}_l$.}\n if \n$$ c_1(x,\\act1,\\act2) + c_2(x,\\act1,\\act2) = 0.$$ \nThat is, \n$$c(x,\\act1,\\act2) \\stackrel{\\text{defn}}{=} c_1(x,\\act1,\\act2) = - c_2(x,\\act1,\\act2) .$$\nFor a zero sum game, the Nash equilibrium (\\ref{eq:nasheq}) becomes a saddle point:\n$$J_{{\\pol1}^*,{\\pol2}}(x) \\leq J_{{\\pol1}^*,{\\pol2}^*}(x) \\leq J_{{\\pol1},{\\pol2}^*}(x),\n$$\nthat is, it is a minimum in the $\\pol1$ direction and a maximum in the $\\pol2$ direction.\n\n\nA well known result from the 1950s due to Shapley is: \\index{Markov game! Shapley's theorem}\n\\begin{theorem}[Shapley] A zero sum infinite horizon discounted cost Markov game has a unique value function, even though\nthere could be multiple Nash equilibria (saddle points). Thus all the Nash equilibria are equivalent.\n\\end{theorem}\nThe value function of the zero-sum game is \n$$ J_{{\\pol1}^*,{\\pol2}^*}(i) = V(i)$$ where\n$V$ satisfies an equation that resembles dynamic programming:\n\\begin{equation} V(i) = \\val \\big[ (1-\\rho) c(i,\\act1,\\act2) + \\rho \\sum_{j} P_{ij} (\\act1,\\act2) V(j) \\big]_{\\act1,\\act2} \\label{eq:dpzg} \\end{equation}\nHere $\\val[M]_{\\act1,\\act2}$ denotes the value of the matrix\\footnote{A zero sum matrix game is of the form: Given a $m\\times n$ \nmatrix $M$, determine the Nash equilibrium\n$$ (x^*, y^*) = \\operatornamewithlimits{argmax}_x \\operatornamewithlimits{argmin}_y y^\\prime M x, \\quad \\text{ where } x, y \\text{ are probability vectors }$$\nThe value of this matrix game is $\\val[M] = {y^*}^\\prime M x^*$ and is computed as the solution of a linear programming (LP) problem as follows:\nClearly $ \\max_ x \\min_y y^\\prime M x = \\max_x \\min_{i} e_i^\\prime M x $ where $e_i$, $i=1,2,\\ldots,m$ denotes the unit $m$-dimensional vector with 1 in the $i$-th position. This follows since a linear function is minimized at its extreme points. So the minimization over continuum has been reduced\nto one over a finite set. Denoting $z = \\min_{i} e_i^\\prime M x $, the value of the game is the solution of the following LP:\n\\begin{equation} \\val[M] = \\begin{cases}\n \\text{ Compute } \\max z \\\\\n z < e_i^\\prime M x , \\quad i =1,2,\\ldots, m, \\\\\n \\mathbf{1}^\\prime x = 1 , \\quad x_j \\geq 0, j=1,2\\ldots,n \\end{cases}\n\\end{equation}\n}\n game with elements $M(\\act1,\\act2)$.\nEven though for a specific vector $V$, the $\\val[\\cdot]$ in the right hand side of (\\ref{eq:dpzg}) can be evaluated by solving an LP, it is not useful for the Markov zero sum\ngame, since we have a functional equation in the variable $V$. So solving a zero sum Markov game is difficult in general.\n\n\\subsubsection*{Nash Equilibrium as a Non-convex Bilinear Program} \\index{Markov game! Nash equilibrium as bilinear program}\nTo give more insight, as we did in the discounted cost MDP case, let us formulate computing the Nash equilibrium (saddle point) of the zero sum Markov game as an optimization problem. In the MDP case we obtained a LP; for the Markov game (as shown below) we obtain a non-convex\nbilinear optimization problem.\n\nDefine the randomized policy of player 1 (minimizer) and player 2 (maximizer) as \n$$p(i,\\act1) = \\mathbb{P}(\\act1_k = \\act1 | x_k = i), \\quad q(i,\\act2) = \\mathbb{P}(\\act2_k = \\act2 | x_k = i) $$\nIn complete analogy to the discounted MDP case in (\\ref{eq:discountprimal}), player 2 optimal strategy $q^*$ is the solution of the bilinear program\n\\begin{equation} \\begin{split} & \\max \\sum_i \\alpha_i \\underline{V}(i) \\;\\text{ with respect to $(\\underline{V}, q)$} \\\\\n& \\text{ subject to } \\underline{V}(i) \\leq \\sum_{\\act2} c(i,\\act1,\\act2) q(i,\\act2) + \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act2} P_{ij}(\\act1,\\act2) q(i,\\act2) \\underline{V}(j) ,\\\\\n& q(i,\\act2) \\geq 0, \\quad \\sum_{\\act2} q(i,\\act2) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act2 = 1,2,\\ldots,U.\n\\end{split} \\label{eq:gdiscountprimal} \\end{equation}\nBy symmetry, player 1 optimal strategy $p^*$ is the solution of the bilinear program\n\\begin{equation} \\begin{split} & \\min \\sum_i \\alpha_i \\underline{V}(i) \\;\\text{ with respect to $(\\underline{V}, p)$} \\\\\n& \\text{ subject to } \\underline{V}(i) \\geq \\sum_{\\act2} c(i,\\act1,\\act2) p(i,\\act1) + \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act1} P_{ij}(\\act1,\\act2) p(i,\\act1) \\underline{V}(j) ,\\\\\n& p(i,\\act1) \\geq 0, \\quad \\sum_{\\act1} p(i,\\act1) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act1 = 1,2,\\ldots,U.\n\\end{split} \\label{eq:gdiscountprimal2} \\end{equation}\nThe key difference between the above discounted Markov game problem and the discounted MDP (\\ref{eq:discountprimal}) is that the above equations\nare no longer LPs. Indeed the constraints are {\\em bilinear} in $ (\\underline{V},q)$ and $(\\underline{V},p)$. So the constraint set for a zero-sum Markov game is non-convex. Despite (\\ref{eq:gdiscountprimal}) and (\\ref{eq:gdiscountprimal2}) being nonconvex, in light of Shapley's theorem all local minima are global minima.\n\nFinally (\\ref{eq:gdiscountprimal}) and (\\ref{eq:gdiscountprimal2}) can be combined into a single optimization problem. To summarize, the (randomized) Nash equilibrium $p^*,q^*$ of \na zero-sum Markov game is the solution of the following bilinear (noconvex) optimization problem:\n\\begin{equation} \\begin{split} & \\max \\sum_i \\alpha_i \\big( \\underline{V}^{(1)}(i) -\\underline{V}^{(2)}(i) \\big) \\;\\text{ with respect to $(\\underline{V}^{(1)},\\underline{V}^{(2)},p, q)$} \\\\\n \\text{ subject to } &\\underline{V}^{(1)}(i) \\leq \\sum_{\\act2} c(i,\\act1,\\act2) q(i,\\act2) + \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act2} P_{ij}(\\act1,\\act2) q(i,\\act2) \\underline{V}^{(1)}(j) ,\\\\\n&\\underline{V}^{(2)}(i) \\geq \\sum_{\\act2} c(i,\\act1,\\act2) p(i,\\act1) + \n\\rho \\sum_{j \\in \\mathcal{X}} \\sum_{\\act1} P_{ij}(\\act1,\\act2) p(i,\\act1) \\underline{V}^{(2)}(j) ,\\\\\n& q(i,\\act2) \\geq 0, \\quad \\sum_{\\act2} q(i,\\act2) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act2 = 1,2,\\ldots,U\n\\\\\n& p(i,\\act1) \\geq 0, \\quad \\sum_{\\act1} p(i,\\act1) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act1 = 1,2,\\ldots,U.\n\\end{split} \\label{eq:cgdiscountprimal2} \\end{equation}\n\n\n\\subsubsection*{Special cases where computing Nash Equilibrium is an LP} \\index{Markov game! linear programming}\nWe now give two special examples of zero-sum Markov games that can be solved as a linear programming problem (LP); single controller\ngames and switched controller games. In both cases the bilinear terms in (\\ref{eq:cgdiscountprimal2}) vanish and the computing the Nash\nequilibrium reduces to solving linear programs.\n\n\\subsection{Example 1. Single Controller zero-sum Markov Game} \\index{Markov game! single controller}\nIn a single controller Markov game, the transition probabilities are controlled by one player only; we assume that this is player 1. So \n$$ P_{ij}(\\act1,\\act2) = P_{ij}(\\act1) = \\mathbb{P}(x_{k+1} = j | x_k = i, \\act1_k = \\act1) $$\n Due to this assumption, the bilinear constraint in (\\ref{eq:gdiscountprimal}) becomes {\\em linear}, namely \n$$ \\underline{V}(i) \\leq \\sum_{\\act2} c(i,\\act1,\\act2) q(i,\\act2) + \n\\rho \\sum_{j \\in \\mathcal{X}} P_{ij}(\\act1) \\underline{V}(j) $$\nsince $\\sum_{\\act2} q(i,\\act2) = 1$. Therefore (\\ref{eq:gdiscountprimal}) is now an LP which can be solved for $q^*$, namely:\n \\begin{equation} \\begin{split} & \\max_{\\underline{V}} \\sum_i \\alpha_i \\underline{V}(i) \\;\\text{ with respect to $(\\underline{V}, q)$} \\\\\n& \\text{ subject to } \\underline{V}(i) \\leq \\sum_{\\act2} c(i,\\act1,\\act2) q(i,\\act2) + \n\\rho \\sum_{j \\in \\mathcal{X}} P_{ij}(\\act1) \\underline{V}(j) , \\\\\n& q(i,\\act2) \\geq 0, \\quad \\sum_{\\act2} q(i,\\act2) = 1, \\quad i = 1,2,\\ldots, X ,\\; \\act2 = 1,2,\\ldots,U.\n\\end{split} \\label{eq:zdiscountprimal} \\end{equation}\nSolving the above LP yields the Nash equilibrium policy $\\pol2$ for player 2.\n\nThe dual problem to (\\ref{eq:zdiscountprimal}) is the linear program\n\\begin{equation} \\begin{split}\n \\text{ Minimize } & \\sum_{i\\in \\mathcal{X}} \n z(i) \n\n \\;\\text{ with respect to $(z, p)$} \\nonumber \\\\\n\\text{ subject to } & p({i,\\act1}) \\geq 0, \\quad i \\in \\mathcal{X}, u \\in \\,\\mathcal{U} \\\\\n& \\sum_{\\act1} p({j,\\act1} )= \\rho\\, \\sum_{i} \\sum_{\\act1} P_{ij}(\\act1)\\, p({i,\\act1}) + \\alpha_j ,\\; j \\in \\mathcal{X}. \\\\\n& z(i) \\geq \\sum_{\\act1} p(i,\\act1)\\, c(i,\\act1,\\act2)\n\\end{split}\n\\label{eq:gdiscountdual}\n\\end{equation}\n The above dual gives the randomized Nash equilibrium policy $p^*$ for player 1.\n \n \\subsection{Example 2. Switching Controller Markov Game} \\index{Markov game! switched controller}\n This is a special case of a zero sum Markov game where the state space $\\mathcal{X}$ is partitioned into disjoint sets $\\sr1,\\sr2$ such\n that $\\sr1 \\cup \\sr2 = \\mathcal{X}$ and \n $$ P_{ij}(\\act1,\\act2) = \\begin{cases} P_{ij}(\\act1), & i \\in \\sr1 \\\\\n \t\t\t\t\t\t\t P_{ij}(\\act2), & i \\in \\sr2 \\end{cases} $$\nSo for states in $\\sr1$, controller 1 controls that transition matrix, while for states in $\\sr2$, controller 2 controls the transition matrix.\n\n\n\nObviously for $ i \\in\\sr2$, (\\ref{eq:gdiscountprimal}) becomes an linear program while for $i \\in \\sr1$, (\\ref{eq:gdiscountprimal2}) becomes a\nlinear program.\nAs discussed in \\cite{FV12}, the Nash equilibrium can be computed by solving a finite sequence of linear programming problems.\n\n\n \n\\chapter{Partially Observed Markov Decision Processes (POMDPs)} \n\nSeveral well studied instances of POMDPs and their parameter files can be found at \\url{http:\/\/www.pomdp.org\/examples\/}\n\\\\ \\\\\n\n\\begin{compactenum}\n\\item Much insight can be gained by simulating the dynamic programming recursion for a 3-state POMDP. The belief\nstate needs to be quantized to a finite grid. We also strongly recommend using the exact POMDP solver in \\cite{POMDP} to gain\ninsight into the piecewise linear concave nature of the value function.\n\n\\item Implement Lovejoy's suboptimal algorithm and compare its performance with the optimal policy.\n\n\\item {\\bf Tiger problem}: \\index{POMDP tiger problem} This is a colorful name given to the following POMDP problem.\n\nA tiger resides behind one of two doors, a left door $(l)$ and a right door $(r)$.\nThe state $x \\in \\{l,r\\}$ denotes the position of a tiger. The action $u \\in \\{l,r,h\\}$ denotes a human either opening the left door $(l)$,\nopening the right door $(r)$, or simply hearing $(h)$ the growls of the tiger. If the human opens a door, he gets a perfect measurement\nof the position of the tiger (if the tiger is not behind the door he opens, then it must be behind the other door). If the human chooses action\n$h$ then he hears the growls of the tiger which gives noisy information about the tiger's position. Denote the probabilities\n$B_{ll}(h) = p$, $B_{rr}(h) = q$.\n\nEvery time the human chooses the action to open a door, the problem resets and the tiger is put with equal probability behind one of the doors.\n(So the transition probabilities for the actions $l$ and $r$ are $0.5$). \n\nThe cost of opening the door behind where the tiger is hiding is $\\alpha$, possibly reflecting injury from the tiger. The cost\nof opening the other door is $-\\beta$ indicating a reward. Finally the cost of hearing and not opening a door is $\\gamma$.\n\nThe aim is to minimize the cost (maximize the reward) over a finite or infinite horizon.\nTo summarize, the POMDP parameters of the tiger problem are:\n\\begin{align*} \\mathcal{X} &= \\{l,r\\}, \\mathcal{Y}= \\{l,r\\}, \\,\\mathcal{U} = \\{l, r, h\\}, \\\\\nB(l) &= B(r) = I_{2 \\times 2}, B(h) = \\begin{bmatrix} p & 1-p \\\\ 1-q & q \\end{bmatrix} \\\\\nP(l) &= P(r) = \\begin{bmatrix} 0.5 & 0.5 \\\\ 0.5 & 0.5 \\end{bmatrix} ,\nP(h) = I_{2 \\times 2}, \\\\\nc_l &= (\\alpha, -\\beta)^\\prime, c_r = (-\\beta, \\alpha)^\\prime, c_h = (\\gamma, \\gamma)^\\prime\n\\end{align*}\n\n\n\n\n\n\\item {\\bf Open Loop Feedback Control.} As described in \\S \\ref{sec:olfc}, open loop feedback control is a useful suboptimal scheme for solving\nPOMDPs. Is it possible to exploit knowledge that the value function of a POMDP is piecewise linear and concave in\nthe design of an open loop feedback controller? \n\n\n\\item Finitely transient policies were discussed in \\S 7.6. For a 2-state, 2-action, 2-observation POMDP, give an example of POMDP parameters that yield a finitely transient policy with $n^* = 2$.\n\n\n\\item {\\bf Uniform sampling from Belief space.} Recall that the belief space $\\Pi(\\statedim)$ is the unit $X-1$ dimensional simplex.\nShow that a convenient way of sampling uniformly from $\\Pi(\\statedim)$ is to use \nthe Dirichlet distribution $$\\pi_0(i) = \\frac{x_i}{\\sum_{j=1}^X x_j}, \\quad \\text{ where } x_i \\sim \\text{ unit exponential distribution. } $$ \\index{Dirichlet distribution}\n \n\n\\item {\\bf Adaptive Control of a fully observed MDP formulated as a POMDP problem.}\n\\label{prob:adaptive control} \\index{adaptive control of MDP as a POMDP}\nConsider a fully observed MDP with transition matrix $P(u)$ and cost $c(i,u)$, where $u\\in \\{1,2,\\ldots,U\\}$ denotes the action. Suppose the true transition matrices \n$P(u)$ are not known.\nHowever, it is known apriori that they belong to a known finite set of matrices $P(u,\\th)$ where $\\th \\in \\{1,2, ,\\ldots, L\\}$. As data accumulates, the controller\nmust simultaneously control the Markov chain and also estimate the transition matrices.\n\n\nThe above problem can be formulated straightforwardly as a POMDP.\nLet $\\th_k$ denote the parameter process. Since the parameter $\\th_k = \\th$ does not evolve with time, it has identity transition matrix.\nNote that $\\th$ is not known; it is partially observed since we only see the sample path realization of the Markov chain $x$ with transition matrix\n$P(u,\\th)$.\\\\\n{\\bf Aim}: Compute the optimal policy\n$$ \\mu^* = \\operatornamewithlimits{argmin}_\\mu J_\\mu(\\pi_0) = \\E\\{ \\sum_{k=0}^{N-1} c\\big(x_k, u_k\n \\big) | \\pi_0 \\} $$\nwhere $\\pi_0$ is the prior pmf of $\\th$. The key point here is that as in a POMDP (and unlike an MDP),\nthe action $u_k$ will now depend on the history of past actions and the trajectory of the Markov chain as we will now describe.\n\n{\\bf Formulation}:\nDefine the augmented state $(x_k, \\th_k)$. Since $\\th_k =\\th$ does not evolve, clearly the augmented state has transition probabilities\n$$ \\mathbb{P}(x_{k+1} = j, \\th_{k+1} = m | x_{k} = i, \\th_k = l, u_k = u) = P_{ij}(u,l)\\, \\delta(l-m), \\quad \nm=1,\\ldots, L. $$\nAt time $k$, denote the history as\n$\\mathcal{H}_k = \\{x_0,\\ldots,x_k, u_1,\\ldots,u_{k-1}\\}$. Then define the belief state which is the posterior pmf of the model\nparameter estimate:\n$$ \\pi_k(l) = \\mathbb{P}( \\th_k = l| \\mathcal{H}_k) , \\qquad l = 1,2.\\ldots, L.$$\n\\begin{compactenum}\n\\item Show that the posterior is updated via Bayes' formula as\n\\begin{equation}\n\\begin{split}\n \\pi_{k+1}(l) &= T(\\pi_k, x_k,x_{k+1},u_k)(l) \\stackrel{\\text{defn}}{=} \\frac{P_{x_k,x_{k+1}}(u_k, l) \\, \\pi_k( l)}\n{\\sigma(\\pi_k,x_k,x_{k+1})}, \\; l = 1,2.\\ldots, L \\\\ \\text{ where } \\; &\n\\sigma(\\pi_k,x_k,x_{k+1},u_k) = \\sum_m P_{x_k,x_{k+1}}(u_k, m)\\, \\pi_k( m). \\end{split}\n\\end{equation}\nNote that $\\pi_k$ lives in the $L-1$ dimensional unit simplex.\n\nDefine the belief state as $(x_k, \\pi_k)$. The actions are then chosen as\n$$u_{k} = \\mu_k(x_k,\\pi_k) $$\n\nThen the optimal policy $\\mu_k^*(i,\\pi)$ satisfies Bellman's equation\n\\begin{equation} \\begin{split}\nJ_k(i,\\pi) &= \\min_u Q_k( i, u, \\pi) , \\quad \\mu^*_k(i,\\pi) = \\operatornamewithlimits{argmin}_u Q_k( i, u, \\pi) \\\\\nQ_k(i,u,\\pi) &= \nc(i,u) + \\sum_{j} J_{k+1}\\big(j, T(\\pi,i,j,u) \\big) \\, \n\\sigma(\\pi,i,j,u)\n \\end{split}\n\\end{equation} \ninitialized with the terminal cost $J_N(i,\\pi) = c_N(i)$.\n\n\\item Show that the value function $J_k$ is piecewise linear and concave in $\\pi$. Also show how the exact POMDP solution algorithms \nin Chapter \\ref{ch:pomdpbasic}\ncan be used to compute\nthe optimal policy. \n\\end{compactenum}\nThe above problem is related to the concept of {\\em dual control} which dates back to the 1960s \\cite{Fel65}; see also\n\\cite{Lov93} for the use of Lovejoy's suboptimal algorithm to this problem. \\index{dual control}\nDual control relates to the tradeoff between estimation and control: if the controller is uncertain about the model\n parameter, it needs to control \nthe system more aggressively in order to probe the system to estimate it; if the controller is more certain about the model parameter, \nit can deploy a less aggressive control.\nIn other words, initially the controller explores and as the controller becomes more certain it exploits. Multi-armed bandit problems optimize the\ntradeoff between exploration and exploitation. \\index{dual control}\n\n\n\\item{\\bf Optimal Search and Dynamic (Active) hypothesis testing.} \\index{dynamic hypothesis testing} \\index{active hypothesis testing} In \\S 7.7.4 of the book, we considered the classical optimal search problem where the objective was to search for a non-moving target amongst a finite number of cells.\nA crucial assumption was that there are no false alarms; if an object is not present in a cell and the cell is searched, the observation recorded\nis $\\bar{F}$ (not found). \n\nA generalization of this problem is studied in \\cite{Cas95}. Assume there are $\\,\\mathcal{U} = \\{1,2,\\ldots,U\\}$ cells. When cell $u$ is searched\n\\begin{compactitem}\n\\item If the target is in cell $u$ then an observation $y$ is generated with pdf or pmf $\\phi(y)$ if \nthe target is in cell $u$\n\\item If the target is {\\bf not} in cell $u$, then an observation $y$ is generated with pdf or pmf $\\bar{\\phi}(y)$. \n(Recall in classical search $\\bar{\\phi}(y)$ is dirac measure on the observation symbol $\\bar{F}$.)\n\\end{compactitem}\nThe aim is to determine the optimal search policy ${\\boldsymbol{\\mu}}$ over a time\nhorizon $N$\nto maximize\n$$ J_{\\boldsymbol{\\mu}} = \\E_{{\\boldsymbol{\\mu}}} \\max_{u \\in \\{1,\\ldots, U\\}} \\pi_N (u) \\} $$\nat the final time $N$.\n\nAssume the pdf or pmf $\\bar{\\phi}(y)$ is symmetric in $y$, that is $\\bar{\\phi}(y) = \\bar{\\phi}(b-y)$ for some real constant $b$. Then \\cite[Proposition 3] {Cas95} shows the nice\nresult that the optimal policy is to search either of the two most likely locations given the belief $\\pi_k$.\n\nThe above problem can be viewed as an active hypothesis testing problem, which is an instance of a controlled\nsensing problem.\nThe decision maker seeks to adaptively select the most informative sensing action for making a decision in a hypothesis testing problem. Active hypothesis testing goes all the way\nback to the 1959 paper by \nChernoff \\cite{Che59}.\nFor a more general and recent take of active hypothesis testing please see~\\cite{NJ13}. \n\\end{compactenum}\n\n\n\\chapter{POMDPs in Controlled Sensing and Sensor Scheduling} \n\n\\begin{compactenum}\n\n\\item {\\bf Optimal Observer Trajectory for Estimating a Markovian Target.} This problem is identical to the search problem described in\n\\S \\ref{chp:optimal search}. A target moves in space according to a Markov chain. (For convenience assume $X$-cells in two dimensional\nspace.\nA moving observer (sensor) measures the target's state (position) in noise. Assume that the noise depends on the relative distance between the target\nand the observer. \\index{optimal observer trajectory}\nHow should the observer move amongst the $X$-cells in order to locate where the target is? \nOne metric that has been used in the literature \\cite{LI99} is the stochastic observability (which is related to the mutual information) of the target; see also \\S \\ref{sec:radarkf}. The aim of the observer is to move so as to maximize\nthe stochastic observability of the target. As described in \\S \\ref{chp:optimal search}, the problem is equivalent to a POMDP.\n\nA more fancy version of the setup involves multiple observers (sensors) that move within the state space and collaboratively seek to locate the \ntarget. Assume that the observers exchange information about their observations and actions. The problem can again be formulated as a POMDP with a larger action and observation space.\n\nSuppose the exchange of information between the observers occurs over a noisy communication channel where the error probabilities\nevolve according to a Markov chain as in \\S \\ref{sec:minh}. Formulate the problem as a POMDP.\n\n\n\\item {\\bf Risk averse sensor scheduling.} As described in \\S \\ref{sec:nonlinearpomdpmotivation}, in controlled sensing applications, one is interested in incorporating\nthe uncertainty in the state estimate into the instantaneous cost. This cannot be modeled using a linear cost since the uncertainty is minimized\nat each vertex of the simplex $\\Pi(\\statedim)$. In \\S \\ref{sec:nonlinearpomdpmotivation}, quadratic functions of the belief were used to model the conditional\nvariance. A more principled alternative is to use dynamic coherent risk measures; recall three examples of such risk measures were discussed\nin \\S \\ref{sec:riskaverse}. \n\nDiscuss how open loop feedback control can be used for a POMDP with dynamic coherent risk measure.\n\n \\item {\\bf Sensor Usage Constraints.} The aim here is to how the POMDP formulation of a controlled sensing problem can be modified \n straightforwardly to incorporate sensing \n constraints \n on the total usage of particular sensors.\nSuch constraints are often used in \nsensor resource management.\n\n\\begin{compactenum}\n\\item Consider a $N$ horizon problem \\index{sensor management! usage constraints}\nwhere sensor 1 can be used at most $L$ times\nwhere $L \\leq N$. For notational simplicity, assume that there are two sensors, so $\\,\\mathcal{U} = \\{1,2\\}$.\nAssume that there are no constraints on the usage of the other sensors.\n\nFor notational convenience we consider rewards denoted as $R(\\pi,u) = \\sum_{i=1}^X R(i,u) \\pi(i) $ instead of costs $C(\\pi,u)$ expressed in terms of the belief state\n$\\pi$.\nShow that Bellman's equation is given by\n\\begin{multline*}\n V_{n+1}(\\pi,l) = \\max\\{ R(\\pi,1) + \\sum_y V_{n}(T(\\pi,y,1), l-1) \\sigma(\\pi,y,1), \\\\\nR(\\pi,2) + \\sum_y V_{n}(T(\\pi,y,2), l) \\sigma(\\pi,y,2)\\}\n\\end{multline*}\nwith boundary condition $V_n(\\pi,0) = 0$, $n=0,1,\\ldots, N$.\n\n\\item If the constraint is that sensor 1 needs to be used exactly $L$ times, then show that the following additional boundary condition needs to be included:\n$$V_n(\\pi,n) = R(\\pi,1) + \\sum_y V_{n-1}(T(\\pi,y,1), n-1) \\sigma(\\pi,y,1) , \\; \\text{ for } n=1,\\ldots,L .$$\n\n\\item In terms of the POMDP solver software, the constraint for using sensor~1 at most $L$ times is easily incorporated by augmenting the state space. Define the controlled finite state process $r_k \\in \\{0,2,\\ldots,L\\}$ with $(L+1 )\\times (L+1)$ transition matrices\n$$ Q(1) = \\begin{bmatrix} 0 & 1 & 0 & \\cdots & 0\\\\\n 0 & 0 & 1 & \\cdots & 0 \\\\\n \\vdots & \\vdots& \\vdots & \\ddots & 1 \\\\\n 0 & 0 & 0 & \\cdots & 1 \\end{bmatrix}, \\quad Q(2) = I .$$\nThen define the POMDP with:\n\\begin{compactitem}\n\\item transition matrices $P(1) \\otimes Q(1)$ and $P(2) \\otimes Q(2)$,\n\\item observation probabilities\n$p(y | x, r,u) = p(y| x, u)$, \n\\item rewards $R(x,r,u) = R(x,u)$ for $r> 0$ and $R(x,r=0,u) = 0$.\n\\end{compactitem}\n\n\n\nIn the problems for Chapter \\ref{ch:pomdpstop}, we consider a simpler version of the above problem for optimal measurement selection of a HMM.\nIn that simpler case, one can develop structural results for the optimal policy.\n\\end{compactenum}\n\n\\begin{comment}\nLet $S_1 = \\{e_1,\\ldots,e_{N_1+1}\\}$ denote the set of $N_1+1$\ndimensional unit vectors, where $e_i$ has \n 1 in the $i$-th position. \nWe will use process $n_k$ to denote the number of times sensor 1 is used.\nLet $n_k = e_i$ \nif sensor 1 has been used $i-1$ times up to time $t$.\nThen the process $n_k \\in S_1$ can be modelled as follows:\nIf sensor 1 is used (i.e., $u_k =1$) and $n_k = e_i$,\nthen $n_{k+1}$ jumps to state $e_{i+1}$.\nIf any other sensor is used then $n_{k+1}=n_k = e_i$.\nThus $n_k$ is a deterministic\nMarkov chain with dynamics given by\n\\begin{equation}\nn_k = Q(u_k) n_{k-1}, \\qquad n_0 = e_1\n\\end{equation}\nwhere the transition probability matrix $Q(\\cdot)$ is defined as\n $$Q(u_k = 1) = \n\\begin{bmatrix} 0 & 1 & 0 & \\cdots & 0\\\\\n 0 & 0 & 1 & \\cdots & 0 \\\\\n \\vdots & \\vdots& \\vdots & \\ddots & 1 \\\\\n 0 & 0 & 0 & \\cdots & 1 \\end{bmatrix}\n\\quad \\text{and } Q(u_k) = I_{(N_1+1)\\times (N_1+1)} \\text { if $u_k \\neq 1$}$$\n\n\nThe action space $U_{k,n_k}$ is defined as follows:\n$$\nU_{k,n_k} =\n\\begin{cases} \n \\{2,\\ldots,U\\} &\\text{ if } n_k = e_{N_1+1}\\\\\n \\{1,\\ldots,U\\} & \\text{ if } n_k \\neq e_{N_1+1} \\end{cases}\n$$\n\nThe one-step ahead scheduling policy is given by\n$\nV_N(\\pi_N,n_N) = \\epsilon _{N}(\\pi ^{z_{N},r_{N}})\n$\nand for $t = N-1, N-2, \\ldots, 0$\n\\begin{equation} \\label{eq:mgment}\nV_{k}(\\pi ^{z_{k},r_{k}},n) = \\min_{u\\in U_{k+1,n_{k+1}}} \\left[\\; \\epsilon\n_{k}(u,\\pi _{k}^{z_{k},r_{k}}) + \\E_{y_{k+1}}\\left\\{ V_{k+1}\\left( T(\\pi\n_{k}^{z_{k},r_{k}},y_{k+1},u), Q^{\\prime} n\\right) \\right\\} \\right]\n\\end{equation}\nThe above dynamic programming recursion can be recast into a form\nsimilar to that for a Jump Markov linear\nsystem by the following coordinate change:\nConsider the augmented Markov chain $(r_k,n_k)$. This has transition\nprobability matrix \n$ P \\otimes Q$ where $P$ denotes\nthe transition probability matrix of $r_k$ and\n$\\otimes$ denotes tensor (Kronecker product).\nBecause $n_k$ is a fully observed Markov chain, the information state of \n$(x_k,n_k)$ is $\\pi_k^{r_k} \\otimes n_k$ .\n This augmented information state \n is identical to that of a jump Markov linear\nsystem (with larger state space) and can be computed via the \n IMM algorithm in \\S \\ref{sec:IMM} or particle filtering algorithm. Thus the one-step ahead\nalgorithm described above can be used for a practical suboptimal\nsolution. \\end{comment}\n\n\n\n\\item As described in \\S \\ref{sec:nonlinearpomdpmotivation}, in controlled sensing it makes sense to choose a cost that is nonlinear in the belief state $\\pi$ in order to penalize uncertainty in the state estimate.\nOne choice of a nonlinear cost that has zero cost at the vertices of the belief space is \n$$C(\\pi,u) = \\min_{ i \\in \\{1,\\ldots,X\\}} \\pi(i) . $$\nThis cost $C(\\pi,u)$ is piecewise linear and concave in $\\pi \\in \\Pi(\\statedim)$ where $\\Pi(\\statedim)$ denotes the belief space.\n\nSince $C(\\pi,u)$ is positively homogeneous, show that the value function is piecewise linear and concave for any finite horizon $N$.\nHence the optimal POMDP solvers of Chapter \\ref{ch:pomdpbasic} can be used to solve this nonlinear cost POMDP exactly and therefore compute the optimal\n policy.\n\n\\end{compactenum}\n\n\\chapter{Structural Results for Markov Decision Processes} \n\n\\begin{compactenum}\n\n\n\\item {\\bf Supermodularity, Single Crossing Condition \\& Interval Dominance Order.} \\index{supermodular} \\index{single crossing} \\index{interval dominance order}\nA key step in establishing structural results for MDPs is to give sufficient conditions for $u^*(x) = \\operatornamewithlimits{argmax}_u \\phi(x,u)$ to be increasing in $x$.\nIn \\S \\ref{chp:fullsupermod} of Chapter \\ref{chp:monotonemdp} we gave two conditions, namely supermodularity and the single crossing condition (which\nis a more general condition than supermodularity).\nMore recently, the interval dominance order has been introduced in \\cite{QS09} as an even more general condition.\nAll three conditions boil down to the following statement:\n\\begin{equation} \\phi(x+1,u+1) - \\phi(x+1,u) \\geq \\rho(u) \\, \\big( \\phi(x,u+1) - \\phi(x,u) \\big) \\label{eq:intdominance} \\end{equation}\nwhere $\\rho(u)$ is a strictly positive function of $u$. In particular,\n\\begin{compactitem}\n\\item Choosing $\\rho(u)=1$ in (\\ref{eq:intdominance}) yields the supermodularity condition.\n\\item If there exists a fixed positive constant $\\rho(u)$ such that (\\ref{eq:intdominance}) holds, then the single crossing condition holds.\n\\item If there exists a positive function $\\rho(u)$ that is increasing\\footnote{Recall that in the book we use increasing in the weak sense to mean non-decreasing} in $u$, then (\\ref{eq:intdominance}) yields the interval dominance order condition (actually this is a sufficient condition for interval dominance, see \\cite{QS09} for details).\n\\end{compactitem}\nNote that single crossing and interval dominance are ordinal properties in the sense that they are preserved by monotone transformations.\n\nThe sum of supermodular functions is supermodular. Unfortunately, in general, the um of single crossing functions is not single crossing; however, see\n\\cite{QS12} for some results.\nDiscuss if the interval dominance order holds for sums of functions. Can it be used to develop structural results for an MDP?\n\n\\item Clearly, in general, the sum of single crossing functions is not single crossing. Even a constant plus a single crossing function\nis not necessarily single crossing. Sketch the curve of a single crossing function which wiggles close to zero. Then adding a positive constant\nimplies that the curve will cross zero more than once.\nAlso the sum of a supermodular plus single crossing is not single crossing. In terms of $\\phi(x) = f(x,u+1) - f(x,u)$, supermodular implies\n$\\phi(x) $ is increasing in $x$. Clearly the sum of an increasing function and a single crossing is not single crossing in general.\n\n\n\n\\item {\\bf Invariance of optimal policy to costs.} Recall that Theorem \\ref{thm:mdpmonotone} require that the MDP costs satisfy assumptions\n(A1) and (A3) for the optimal policy to be monotone.\nShow that for a discounted cost infinite horizon MDP, assumption (A1) and (A3) can be relaxed as follows:\n\nThere exists a single vector $\\phi \\in {\\rm I\\hspace{-.07cm}R}^X$ such that for every action $u \\in \\,\\mathcal{U}$, \n\\begin{compactenum}\n\\item[(A1')] $(I - \\rho P(u)) \\phi $ is a vector with increasing elements.\n(Recall $\\rho$ is the discount factor.)\n\\item[(A3')] $ (P(u+1) - P(u)) \\phi$ is a vector with decreasing elements.\n\\end{compactenum}\nIn other words the structure of the transition matrix is enough to ensure a monotone policy and no assumptions are required on the cost\n(of course the costs are assumed to be bounded)\n\n{\\em Hint}: Define the new value function\n$\\bar{V}(i) = V(i) - \\phi(i)$ . Clearly the optimal policy remains unchanged and $\\bar{V}$ satisfies Bellman's equation\n$$ \\bar{V}(i) = \\min_u \\{ c(i,u) - \\phi(i) + \\rho \\sum_j \\phi(j) P_{ij}(u) + \\rho \\sum_j \\bar{V}(j) P_{ij}(u)\\} $$\nwhere $\\rho \\in (0,1)$ denotes the discount factor.\n\n\\item {\\bf Myopic lower bound to optimal policy.}\nRecall that supermodularity of the transition matrix (A4) was a key requirement for the optimal policy to be monotone.\nIn particular, Theorem \\ref{thm:mdpmonotone} shows that $Q(i,u)$ is submodular, i.e., \n$Q(i,u+1) - Q(i,u)$ is decreasing in $i$.\nSometimes supermodularity of the transition matrix is too much to ask for. \nConsider instead of (A4) the relaxed condition\n\\begin{compactitem}\n\\item[(A4')]$P_i(u+1) \\geq_s P_i(u)$ for each row $i$.\n\\end{compactitem}\nShow that (A4') together with (A1), (A2) implies that \n$$ \\sum_j P_{ij}(u+1) V(j) \\leq \\sum_j P_{ij}(u) V(j) $$\nDefine the myopic policy $\\policyl(i) = \\operatornamewithlimits{argmin}_u c(i,u)$.\nShow that \nunder (A1), (A2), (A4'),\n$\\mu^*(i) \\geq \\underline{\\mu}(i)$. In other words, the myopic policy $\\policyl$ forms a lower bound to the optimal policy $\\mu^*$.\n\n\\item {\\bf Monotone policy iteration algorithm.} Suppose an MDP has a monotone policy.\n If the MDP parameters are known,\nthen the policy iteration algorithm of \\S \\ref{sec:vipilp} can be used. If the policy $\\mu_{n-1}$ at iteration $n-1$ is monotone then show that under the assumptions of (A1), (A2) of Theorem \\ref{thm:mdpmonotone}, the policy evaluation step yields $J_{\\mu_{n-1}}$ as a decreasing vector. Also show that under (A1)-(A4), (a similar proof to Theorem \\ref{thm:mdpmonotone}) implies that the policy improvement\n step yields $\\mu_n$ that is monotone. So the policy iteration algorithm will automatically be confined to monotone policies if initialized\n by a monotone policy.\n\n\n\n\\item {\\bf Stochastic knapsack problem.} Consider the following version of the stochastic knapsack problem;\\footnote{The classical NP hard knapsack problem deals with\n$U$ items with costs $c(1),c(2),\\ldots,c(U)$ and lifetimes $t_1,t_2,\\ldots t_U$. The aim is to compute \nthe minimum cost subset of these items whose total lifetime is at most~$T$.}\nsee \\cite{Ros83} and also \\cite{CR14a}. \\index{stochastic knapsack problem}\nA machine must operate for $T$ time points. Suppose that one specific component of the machine fails intermittently.\nThis component is replaced when it fails. There are $U$-possible brands one can choose to replace this component when it fails.\nBrand $u\\in \\{1,2\\ldots,U\\}$ costs $c_u$ and has an operating lifetime that is exponentially distributed with rate $\\lambda_u$. \nThe aim is to minimize the expected total cost incurred by replacing the failed component so that the machine operates for $T$ time points.\n\nSuppose a component has just failed. Let $t$ denote the remaining time left to operate the machine.\nThe optimal policy for deciding which of the $U$ possible brands to choose the replacement satisfies Bellman's equation\n\\begin{align*} Q(t,u) &= c(u) + \\int_0^t V(t-\\tau) \\, \\lambda_u e^{-\\lambda_u \\tau} d\\tau , \\quad Q(0,u) = 0, \\\\\nV(t) &= \\min_{u \\in \\{1,2,\\ldots,U\\}} Q(t,u) , \\quad \\policy^*(t) = \\operatornamewithlimits{argmin}_{u \\in \\{1,2,\\ldots,U\\}} Q(t,u)\n\\end{align*}\nShow that if $\\lambda_u c(u)$ is decreasing with $u$, then $Q(t,u)$ is submodular.\nIn particular, show that \n$$\\frac{d}{dt} Q(t,u) = \\lambda_u c(u) $$\nTherefore, the optimal policy $\\policy^*(t)$ has the following structure:\nUse brand 1 when the time remaining is small, then switch to brand 2 when the time increases, then brand 3, etc.\n\nGeneralize the above result to the case when time $k$ is discrete and the brand $u$ has life time pmf $p(k,u)$, $k=0,1\\ldots$. \nThen Bellman's equation reads\n\\begin{align*} Q(n,u) &= c(u) + \\sum_{k=0}^n V(n-k) \\, p(k,u) \\\\\n V(n) &= \\min_{u \\in \\{1,2,\\ldots,U\\}} Q(n,u), \\quad \\policy^*(n) = \\operatornamewithlimits{argmin}_{u \\in \\{1,2,\\ldots,U\\}} Q(n,u)\n \\end{align*}\n What are sufficient conditions in terms of submodularity of the lifetime pmf $p(k,u)$ for the optimal policy to be monotone?\n \n \n \\item {\\bf Monotonicity of optimal policy with respect to horizon.} \\index{effect of planning horizon} Show that the following result holds for a \n finite horizon MDP.\n If $Q_n(i,u)$ is supermodular in $(i,u,n)$ then $V_n(i) = \\max_u Q_n(i,u)$ is supermodular in $i,u$.\n Note that checking supermodularity with respect to $(i,u,n)$ is pairwise: so it suffices to check\n supermodularity with respect to $(i,u)$, $(i,n)$ and $(u,n)$.\n \nWith the above result, consider a finite horizon MDP satisfies the assumptions (A1)-(A4) of \\S \\ref{sec:monotonecond}.\nUnder what further conditions is \n $\\mu_n^*(i)$ is increasing in $n$ for fixed $i$? What does this mean intuitively?\n \n \n \\item {\\bf Monotone Discounted Cost Markov Games.} \\index{structural result! Markov game} \\index{Markov game! structural result} \n \\index{Markov game! Nash equilibrium! structural result} \n In \\S \\vref{sec:markovgame} of this internet supplement we briefly described the formulation of infinite horizon discounted cost Markov games.\n Below we comment briefly on structural results for the Nash equilibrium of such games.\n \n Consider the infinite horizon discounted cumulative cost of (\\ref{eq:cumcostgame}).\n The structural results developed in this chapter for MDPs extend straightforwardly to \n infinite horizon discounted cost Markov games.\n The assumptions (A1) to (A4) of \\S \\ref{sec:monotonecond} of the book need to be extended as follows:\n \n \\begin{description} \n\\item[(A1)] Costs $c(x,u,u^-)$ are decreasing in $x$ and $u^-$. Here $u^-$ denotes the actions of others players.\n\\item[(A2)] $P_i(u,u^-) _{s}{\\le} P_{i+1}(u,u^-)$ for each $i$ and fixed $u,u^-$. \nHere $P_i(u,u^-)$ denotes the $i$-th row of the transition matrix for action $u,u^-$.\n\\item[(A3)] $c(x,u,u^-)$ is submodular in $(x,u)$ and $(u,u^-)$\n\\item[(A4)] $P_{ij}(u,u^-)$ is tail-sum supermodular in $(i,u,u^-)$. That is,\n$$\\sum_{j\\geq l} \\big(P_{i j}(u+1,u^-) - P_{i j}(u,u^-) \\big) \\text{ is increasing in } i .$$\n\\end{description}\n\\begin{theorem}\nUnder conditions (A1)-(A4), there exists a pure Nash equilibrium $({\\pol1}^*, {\\pol2}^*)$ such that the pure policies\n${\\pol1}^*$ and ${\\pol2}^*$ are increasing in state $i$. \n\\end{theorem}\nContrast this with the case of a general Markov game (\\S \\ref{sec:markovgame} of this internet supplement) where one can only\nguarantee the existence of a randomized Nash equilibrium\nin general.\n \n The proof of the above theorem is as follows. First for any increasing fixed policy $\\pol2$ for player 2, one can show via an identical proof to Theorem\n \\ref{thm:mdpmonotone}, the optimal policy ${\\pol1}^*(x,\\pol2(x))$ is increasing in $x$. Similarly, for any increasing fixed policy $\\pol1$ for player 1, \n ${\\pol2}^*(x,\\pol1(x))$ is increasing in $x$. These are obtained as the solution of Bellman's equation. In game theory, these are called best response strategies.\n Therefore the vector function $[{\\pol1}^*(x), {\\pol2}^*(x)]$ is increasing in $x$.\nIt then follows from Tarski's fixed point theorem\\footnote{Let $X$ denote a compact lattice and $f: X \\rightarrow X$ denote an increasing function.\nThen there exists a fixed point $x^* \\in X$ such that $f(x^*) = x^*$} that such a function has a fixed point. Clearly this fixed point is a Nash equilibrium since any unilateral deviation\nmakes one of the players worse off.\n\nActually for submodular games a lot more holds. The smallest and largest Nash equilibria are pure (non-randomized) and satisfy the monotone property\nof the above theorem. These can be obtained\nvia a best response algorithm the simply iterates the best responses ${\\pol1}^*(x,\\pol2(x))$ and ${\\pol2}^*(x,\\pol1(x))$ until convergence.\nThere are numerous papers and books in the area.\n \n\\end{compactenum}\n\n\\chapter{Structural Results for Optimal Filters} \n\n\\begin{compactenum}\n\n\\item In the structural results presented in the book, we have only considered first order stochastic dominance and monotone likelihood ratio dominance (MLR) since they are sufficient\nfor our purposes. Naturally there are many other concepts of stochastic dominance \\cite{MS02}. Show that \n$$ \\text{ MLR } \\implies \\text{Hazard rate order} \\implies \\text{first order} \\implies \\text{second order} $$\nEven though second order stochastic dominance is useful for concave decreasing functions (such as the value function of a POMDP),\njust like first order dominance, it cannot cope with conditioning (Bayes' rule).\n\n\\item Consider a reversible Markov chain with transition matrix $P$, initial distribution $\\pi_0$ and stationary distribution $\\pi_\\infty$.\nSuppose $\\pi_0 \\leq_r \\pi_\\infty$. Show that if $P$ has rows that are first order increasing then $\\pi_n \\leq_r \\pi_\\infty$. \n\n\n\n\\item {\\bf TPn matrix.} A key assumption \\ref{A3} in the structural results is that the transition matrix $P$ is TP2. More generally, suppose $n =2,3,\\ldots$.\nThen a $X\\times X$ \nmatrix $P$ is said to be totally positive of order $n$ (denoted as TPn) if for each $k \\leq n$, all the $k \\times k$ minors of $P$ are non-negative.\n\n\n\\item {\\bf TP2 matrix properties.\\footnote{Note that a TP2 matrix does not need to be a square matrix; we consider $P$ to be square here since it is a transition probability matrix.}}\n\\S \\ref{sec:assumpdiscussion} gave some useful properties of TP2 matrices.\n\nSuppose the $X \\times X$ stochastic matrix $P$ is TP2. \n\\begin{compactenum}\n\\item Show that this implies that the elements satisfy\n\\begin{align*} & P_{11} \\geq P_{21} \\geq \\cdots \\geq P_{X1} \\\\\n & P_{1X} \\geq P_{2X} \\geq \\cdots \\geq P_{XX} \\end{align*}\n \\item\n Suppose $P$ has no null columns. Show that if $P_{ij} = 0$, then either $P_{kl} = 0$ for $k \\leq i$ and $l \\geq j$, or\n $P_{kl} = 0$ for $k \\geq i$ and $l \\leq j$.\n \n \\item Show that \n $$ e_1^\\prime ({P^{n}})^\\prime e_1 \\downarrow n, \\qquad e_X^\\prime ({P^{n}})^\\prime e_1 \\uparrow n.\n $$\nAlso show that for each $n$,\n$$ e_1^\\prime ({P^{n}})^\\prime e_i \\downarrow i, \\qquad e_X^\\prime ({P^{n}})^\\prime e_i \\uparrow i\n$$\n\n\n \\end{compactenum}\n Please see \\cite{KK77} for several other interesting properties of TP2 matrices.\n\n\\item MLR dominance is intimately linked with the TP2 property. Show that \n$$\\pi_1 \\leq_r \\pi_2 \\iff \n\\begin{bmatrix} \\pi_1^\\prime \\\\ \\pi_2^\\prime \\end{bmatrix} \\text{ is TP2 .} $$\n\n\\item{\\bf Properties of MLR dominance.} Suppose $X$ and $Y$ are random variables and recall that $\\geq_r$ denotes MLR dominance.\\footnote{Stochastic dominance is a property of the distribution of a random variable and has nothing to do with the random variable itself. Therefore in the book, we defined stochastic dominance in terms of the pdf or pmf. Here to simplify notation we use the random variable instead of its distribution.}\n\\begin{compactenum}\n\\item Show that $X\\geq_r Y$ is equivalent to \n$$ \\{ X | X \\in A\\} \\geq_s \\{Y | Y \\in A\\} $$\nfor all events $A$ with $P(X \\in A) > 0$ and $P(Y\\in A) > 0$ where $\\geq_s$ denotes first order dominance. This property is due to \\cite{Whi82}.\n\\item Show that $X\\geq_r Y$ implies that $g(X) \\geq_r g(Y)$ for any increasing function $ g$.\n\\item Show that $X \\geq_r Y$ implies that $\\max\\{X,c\\} \\geq_r \\max\\{Y,c\\}$ for any positive constant $c$.\n\n\\item Under what conditions does $X \\geq_r Y$ imply that $-X \\leq_r -Y$?\n\\end{compactenum}\nDo the above two properties hold for first order dominance?\n\n\n\\item {\\bf MLR monotone optimal predictor.} Consider the HMM predictor given by the Chapman Kolmogorov equation\n$ \\pi_k = P^\\prime \\pi_{k-1}$.\nShow that if $P$ is a TP2 matrix and $\\pi_0 \\leq_r \\pi_1$, then \n$\\pi_0 \\leq_r \\pi_1 \\leq_r \\pi_2 \\leq_r \\ldots$.\n\n\\item {\\bf MLR constrained importance sampling.}\nOne of the main results of this chapter was to construct reduced complexity HMM filters that provably form lower and upper bounds to the optimal HMM filter\nin the MLR sense. \nIn this regard, consider the following problem. Suppose it is known that \n$ \\underline{\\tp}^\\prime \\pi \\leq_r P^\\prime \\pi$. Then given the reduced complexity computation of $\\underline{\\tp}^\\prime \\pi$,\nhow can this be exploited to compute $P^\\prime \\pi$?\n\nIt is helpful to think of the following toy example:\nSuppose it is known that $x^\\prime p \\leq 1$ for a positive vector $x$ and probability vector $p$. How can this constraint be exploited to actually compute the inner \nproduct \n$x^\\prime p$? Obviously from a deterministic point of view there is little one can do to exploit this constraint. \\index{constrained importance sampling}\nBut one can use constrained important sampling: one simple estimator is as follows:\n$$ \\frac{1}{N} \\sum_{i=1}^N x_i I(x_i \\leq 1)$$ \nwhere index $i$ is simulated iid from probability vector $p$.\nIn \\cite{KR14} a more sophisticated constrained importance sampling approach is used to estimate $P^\\prime \\pi$ by exploiting the constraint\n$ \\underline{\\tp}^\\prime \\pi \\leq_r P^\\prime \\pi$.\n\n\n\n\\item {\\bf Posterior Cramer Rao bound.} The posterior Cramer Rao bound \\cite{TMN98} for filtering can be used to compute a lower bound to the mean square error. \\index{posterior Cramer Rao bound}\nThis requires twice differentiability of the logarithm of the joint density. For HMMs, one possibility is to consider the Weiss-Weinstein\nbounds \\index{Weiss-Weinstein bounds}, see \\cite{RO05}. Alternatively, the analysis of \\cite{GSV05} can be used. Compare these \nwith the sample path bounds for the HMM filter obtained in this chapter.\n\n\n \\item \\index{shifted likelihood ratio order} The shifted likelihood ratio order is a stronger order than the MLR order. Indeed, $p> q$ in the shifted likelihood ratio order sense\nif $p_i\/q_{i+j}$ is increasing in $i$ for any $j$. (If $j=0$ it coincides with the standard MLR order.) What additional assumptions are required\nto preserve the shifted likelihood ratio order under Bayes' rule? Show that \nthe shifted likelihood ratio order is closed under convolution. How can this property be exploited to bound an optimal filter?\n\n\n\\item In deriving sample path bounds for the optimal filter, we did not exploit the fact that $T(\\pi,y)$ increases with $y$.\nHow can this fact be used in bounding the sample path of an optimal filter?\n\n\n\n\\item {\\bf Neyman-Pearson Detector} \\index{Neyman-Pearson detector}\nHere we briefly review elementary Neyman-Pearson detection theory and show the classical result that MLR dominance results in a threshold optimal detector.\n\n\nGiven the observation $x$ of a random variable, we wish to decide if $x$ is from pdf $f$ or $g$. To do this, we construct a decision policy $\\phi(x)$. The detector\ndecides\n\\begin{equation} \\begin{split} f \\quad \\text{ if } \\phi(x) & = 0 \\\\\n g \\quad \\text{ if } \\phi(x) & = 1 \\end{split} \\label{eq:detectornp} \\end{equation}\nThe performance of the decision policy $\\phi$ in (\\ref{eq:detectornp}) is determined in terms of two metrics:\n\\begin{compactenum}\n\\item $\\mathcal{P} = \\mathbb{P}( \\text{ reject } f | f \\text{ is true }) $\n\\item $\\mathcal{Q} = \\mathbb{P}( \\text{ reject } f | f \\text{ is false }) $\n\\end{compactenum}\nClearly for the decision policy $\\phi(\\cdot)$ in (\\ref{eq:detectornp}),\n$$ \\mathcal{P} = \\int_{\\rm I\\hspace{-.07cm}R} f(x) \\phi(x) dx, \\quad \\mathcal{Q} = \\int_{\\rm I\\hspace{-.07cm}R} g(x) \\phi(x) dx. $$\nThe well known Neyman-Pearson detector seeks to determine the optimal decision policy $\\phi^*$ that maximizes $\\mathcal{Q}$ subject to\nthe constraint $\\mathcal{P} \\leq \\alpha$ for some user specified $\\alpha \\in (0,1]$.\nThe main result is\n\\begin{theorem-non}[Neyman-Pearson lemma]\nAmongst all decision rules $\\phi$ such that $\\mathcal{P} \\leq \\alpha$, the decision rule $\\phi^*$ which maximizes $\\mathcal{Q}$ is given by\n$$ \\phi^*(x) = \\begin{cases} 0 & \\frac{f(x)}{g(x)} \\geq c \\\\\n\t\t1 & \\frac{f(x)}{g(x)} < c \\end{cases} $$\nwhere $c$ is chosen so that $\\mathcal{P} = \\alpha$.\n\\end{theorem-non}\n\\begin{proof}\nClearly for any $x \\in {\\rm I\\hspace{-.07cm}R}$,\n$$ \\big(\\phi^*(x) - \\phi(x) \\big) \\big( c g(x) - f(x) \\big) \\geq 0. $$\nPlease verify the above inequality by showing that if $\\phi^*(x) =1$ then both the terms in the above product are nonnegative; while if $\\phi^*(x) = 0$, then\nboth the terms are nonpositive.\nTherefore,\n$$\nc \\bigg( \\int \\phi^*(x) g(x) dx - \\int \\phi(x) g(x) dx\\bigg) \\geq \\int \\phi^*(x) f(x) dx - \\int \\phi(x) f(x) dx \n$$\nThe right hand side is non-negative since by construction $ \\int \\phi^*(x) f(x) dx = \\alpha$ , while $ \\int \\phi(x) f(x) dx \\leq \\alpha$.\n\\end{proof}\n\n{\\bf Threshold structure of optimal detector.} \\index{Neyman-Pearson detector! optimal threshold structure}\nLet us now give conditions so that the optimal Neyman-Pearson decision policy is a threshold policy:\nSuppose now that $f$ MLR dominates $g$, that is $f(x)\/g(x) \\uparrow x$. \nThen clearly\n\\begin{equation} \\phi^*(x) = \\begin{cases} 0 & x \\geq x^*\\\\\n\t\t\t\t\t1 & x < x^* \\end{cases} \\label{eq:npt}\\end{equation}\n\t\t\t\t\twhere threshold $x^*$ satisfies\n\t\t\t\t\t$$ \\int_{-\\infty}^{x^*} f(x) dx = \\alpha $$\nThus if $f \\geq_r g$, then the optimal detector (in the Neyman-Pearson sense) is the threshold detector (\\ref{eq:npt}).\n\n\n\\end{compactenum}\n\n\n\\chapter{Monotonicity of Value Function for POMDPs} \n\n\\begin{compactenum}\n\n\n\\item Theorem \\ref{thm:pomdpmonotoneval} is the main result of the chapter and it gives conditions under which the value function\nof a POMDP is MLR decreasing. Condition \\ref{A1} was the main assumption on the possibly non-linear cost.\nGive sufficient conditions for a quadratic cost $1-\\pi^\\prime \\pi + c_u^\\prime \\pi$ to satisfy \\ref{A1}. Under what conditions does the entropy\n$-\\sum_i \\pi(i) \\log \\pi(i) + c_u^\\prime \\pi$ satisfy \\ref{A1}.\n\n\\item The shifted likelihood ratio order is a stronger order than the MLR order. Indeed, $p> q$ in the shifted likelihood ratio order sense\nif $p_i\/q_{i+j}$ is increasing in $i$ for any $j$. If $j=0$ it coincides with the standard MLR order. \n(Recall also the problem in the previous chapter which says that the shifted likelihood ratio order is closed under convolution.)\nBy using the shifted likelihood ratio order, what further results on the value function\n$V(\\pi)$ can one get by using Theorem \\ref{thm:pomdpmonotoneval}. \n\n\\item Theorem \\ref{thm:pomdp2state} gives sufficient conditions for a 2-state POMDP to have a threshold policy. We have assumed that \nthe observation probabilities are not action dependent. How should the assumptions and proof be modified to allow for action\ndependent observation probabilities?\n\n\\item How can Theorem \\ref{thm:pomdp2state} be modified if dynamic risk measures of \\S \\ref{sec:riskaverse} are considered?\n(see also \\S \\ref{sec:risk}).\n\n\n\\item {\\bf Finite dimensional characterization of Gittins index for POMDP bandit} \\cite{KW09}: \\S \\ref{sec:POMDPbandit} dealt with\nPOMDP multi-armed bandit problem.\nConsider a POMDP bandit where the Gittins index (\\ref{eq:gittindef}) is characterized as the solution of Bellman's equation\n(\\ref{eq:bellmanbandit}).\n Since the value function of a POMDP is piecewise linear and concave (and therefore a finite dimensional characterization), it follows that \na value iteration algorithm for (\\ref{eq:bellmanbandit}) that characterizes the Gittins index also has a finite dimensional characterization. Obtain an expression for \nthis finite dimensional characterization for the Gittins index (\\ref{eq:gittindef}) for a horizon $N$ value iteration algorithm.\n\n\\item \\S \\ref{sec:POMDPbandit} of the book deals with structural results for POMDP bandits. Consider the problem where several searchers are looking\nfor a stationary target. Only one searcher can operate at a given time and the searchers cannot receive state estimate information from other searchers or a base-station. The base station simply sends a 0 or 1 signal to each searcher telling them when to operate and when to shut down.\nWhen it operates, the searcher obtains moves according to a Markov chain and obtains noisy information about the target.\nShow how the problem can be formulated as a POMDP multi-armed bandit.\n\nShow how a radar seeking to hide its emissions (low probability of intercept radar) can be formulated approximately as a POMDP bandit.\n\n\n\\item How does the structural result for the Gittins index for a POMDP bandit specialize to that of a full observed Markov decision\nprocess bandit problem?\n\n\n\n\\item Consider Problem \\vref{prob:adaptive control} of Chapter \\ref{ch:pomdpbasic} where optimal adaptive control of a fully observed MDP was \nformulated as a POMDP. Give conditions that ensure that the value function $J_k(i,\\pi)$ is MLR decreasing in $\\pi$ and also monotone in $i$. What are the implications of this monotonicity in terms of dual control (i.e., exploration vs exploitation)?\n\n\n\\item {\\bf Optimality of Threshold Policy for 2-state POMDP} \\index{POMDP! optimality of threshold policy}\nRecall that Theorem \\ref{thm:pomdp2state} in the book gave sufficient conditions for the optimal policy of a 2-state POMDP to be a threshold.\nConsider the proof of Theorem \\ref{thm:pomdp2state} in Appendix 11.A of the book. The last step involved going from (\\ref{eq:subex}) \nto a simpler expression via tedious but elementary steps. Here we specify what these steps are.\n\nStart with (\\ref{eq:subex}) in the book:\n\\begin{align}\n\\begin{aligned}\\label{eq:i3}\nI_3 &= \\left[\\sigma(\\bar{\\pi},y,2) + \\sigma(\\bar{\\pi},y,1)\\cfrac{T(\\bar{\\pi},y,1) - T(\\pi,y,2)}{T(\\pi,y,2) - T(\\bar{\\pi},y,2)} + \\sigma(\\pi,y,1)\\cfrac{T(\\pi,y,2) - T(\\pi,y,1)}{T(\\pi,y,2) - T(\\bar{\\pi},y,2)}\\right]\\\\\n&=\\cfrac{I_{31} + I_{32} + I_{33}}{\\sigma(\\pi,y,2)\\left(T(\\pi, y, 2) - T(\\bar{\\pi}, y, 2)\\right)}\\\\\nI_{31} &= \\sigma(\\pi,y,2)\\sigma(\\bar{\\pi},y,1)\\left(T(\\bar{\\pi},y,1) - T(\\pi,y,2)\\right)\\\\ \nI_{32} &= \\sigma(\\pi,y,2)\\sigma(\\bar{\\pi},y,2)\\left(T(\\pi,y,2) - T(\\bar{\\pi},y,2)\\right)\\\\ \nI_{33} &= \\sigma(\\pi,y,2)\\sigma(\\pi,y,1)\\left(T(\\pi,y,2) - T(\\pi,y,1)\\right) \n\\end{aligned}\n\\end{align}\n\nThe second element of HMM predictors $P(a)^\\prime\\pi$ and $(P(a)^\\prime\\bar{\\pi})$ are denoted by $b_{a2}$, $b_{a1}, a = 1, 2$ respectively. Here $b_{a2}$ is defined as follows\n\\begin{align}\\label{eq:sim_fil}\nb_{a2} = (1-\\pi(2))P_{12}(a) + \\pi(2)P_{22}(a). \n\\end{align}\n\nConsider the following simplification of the term $I_{31}$ by using $b_{a2}$ and $b_{a1}$.\n\\begin{align}\\label{eq:i31}\n\\begin{aligned}\nI_{31} = &(B_{1y}(1-b_{22}) + B_{2y}b_{22})B_{2y}b_{11} - (B_{1y}(1-b_{11}) + B_{2y}b_{11})B_{2y}b_{22}\\\\\n = &B_{1y}B_{2y}(b_{11} - b_{22}) \n\\end{aligned}\n\\end{align}\nSimilarly, $I_{32}$ and $I_{33}$ are simplified as follows\n\\begin{align}\\label{eq:i3233}\n\\begin{aligned}\nI_{32} = B_{1y}B_{2y}(b_{22} - b_{21}), I_{33} = B_{1y}B_{2y}(b_{22} - b_{12}) \n\\end{aligned}\n\\end{align}\nSubstituting \\eqref{eq:i31}, \\eqref{eq:i3233} in \\eqref{eq:i3} yields the following\n\\begin{align}\\label{eq:i3f}\n\\begin{aligned}\nI_3 &= B_{1y}B_{2y}\\cfrac{b_{11} + b_{22} - b_{21} - b_{12}}{\\sigma(\\pi,y,2)\\left(T(\\pi, y, 2) - T(\\bar{\\pi}, y, 2)\\right)}\n\\end{aligned}\n\\end{align}\nSubstituting \\eqref{eq:sim_fil} for $b_{ij}$ and some trivial algebraic manipulations yield the following\n\\begin{align}\\label{eq:i3f}\n\\begin{aligned}\nI_3 &= B_{1y}B_{2y}(\\pi(2) - \\bar{\\pi}(2))\\cfrac{P_{22}(2) - P_{12}(2) - (P_{22}(1) - P_{12}(1))}{\\sigma(\\pi,y,2)\\left(T(\\pi, y, 2) - T(\\bar{\\pi}, y, 2)\\right)}.\n\\end{aligned}\n\\end{align}\n\n\\item Consider the following special case of a POMDP. Suppose the prior belief $\\pi_0 \\in \\Pi(\\statedim) $ is known. From time 1 onwards,\nthe state is fully observed. How can the structural results in this chapter be used to characterize the optimal policy?\n\\end{compactenum}\n\n\n\\chapter{Structural Results for Stopping Time POMDPs} \n\n\\section{Problems}\nMost results in stopping time POMDPs in the literature use the fact that the stopping set is convex\n(namely, Theorem \\ref{thm:pomdpconvex}). \nRecall that the only requirements of \nTheorem \\ref{thm:pomdpconvex} \nare that the value function is convex and the stopping cost is linear.\nAnother important result for finite horizon POMDP stopping time problems is the nested \nstopping set property $\\mathcal{S}_0 \\subseteq \\mathcal{S}_1 \\subseteq \\mathcal{S}_2 \\ldots$. The following exercises discuss both these aspects.\n\n\\begin{compactenum}\n\\item {\\bf Nested stopping set structure.} \\index{stopping time POMDP! nested stopping sets}\nConsider the stopping time POMDP dynamic programming equation\n$$ V(\\pi) = \\min\\{c_1^\\prime \\pi, c_2^\\prime \\pi + \\sum_y V( T(\\pi,y,u) ) \\sigma(\\pi,y,u) \\}.$$\nDefine the stopping set as\n$$ \\mathcal{S} = \\{ \\pi: c_1^\\prime \\pi \\leq c_2^\\prime \\pi + \\sum_y V( T(\\pi,y,u) ) \\sigma(\\pi,y,u) \\} = \\{\\pi: \\mu^*(\\pi) = 1 \\text{ (stop) } \\}$$\nRecall the value iteration algorithm is\n$$ V_{n+1}(\\pi) = \\min\\{c_1^\\prime \\pi, c_2^\\prime \\pi + \\sum_y V_n( T(\\pi,y,u) ) \\sigma(\\pi,y,u) \\}, \\quad V_0(\\pi) = 0. $$\nDefine the stopping sets $\\mathcal{S}_n = \\{\\pi: c_1^\\prime \\pi \\leq c_2^\\prime \\pi + \\sum_y V_n( T(\\pi,y,u) ) \\sigma(\\pi,y,u) \\}$.\n\nShow that the stopping sets satisfy $\\mathcal{S}_0 \\subseteq \\mathcal{S}_1 \\subseteq \\mathcal{S}_2 \\ldots$ implying that $$\\mathcal{S} = \\cup_{n} \\mathcal{S}_n $$\n\n\n\\item {\\bf Explicit characterization of stopping set.} Theorem \\ref{thm:pomdpconvex} showed that for a stopping time POMDP, the stopping set $\\mathcal{S}$ is convex.\nBy imposing further conditions, the set $\\mathcal{S}$ can be determined explicitly.\nConsider the following set of belief states\n\\begin{equation} \\mathcal{S}^o= \\{ \\pi: c_1^\\prime \\pi \\leq c_2^\\prime \\pi + c_1^\\prime P^\\prime \\pi \\} \\label{eq:stopsetoo} \\end{equation}\nSuppose the transition matrix $P$ and observation probabilities $B$ of the stopping time POMDP satisfy the following property:\n\\begin{equation} \\pi \\in \\mathcal{S}^o \\implies T(\\pi,y) \\in \\mathcal{S}^o, \\quad \\forall y \\in \\mathcal{Y} . \\label{eq:onesteppp}\n\\end{equation}\n\n\\begin{compactenum}\n\\item Prove that $\\mathcal{S}^o = \\mathcal{S}$. Therefore, the hyperplane $c_1^\\prime \\pi = c_2^\\prime \\pi + c_1^\\prime P^\\prime \\pi $ determines the stopping set $\\mathcal{S}$.\n\n\nThe proof proceeds in two steps: First prove by induction on the value iteration algorithm that for $\\pi \\in \\mathcal{S}^o$, $V_n(\\pi) = c_1^\\prime \\pi$, for $n=1,2\\ldots$. \\index{stopping time POMDP! characterization of stopping set}\n\nSecond, consider a belief $\\pi $ such that the optimal policy goes one step and then stops. This implies that the value function is $V(\\pi) = c_2^\\prime \\pi + c_1^\\prime P^\\prime \\pi $. Therefore clearly $c_2^\\prime \\pi + c_1^\\prime P^\\prime \\pi \n< c_1^\\prime \\pi$. This implies that $\\pi \\notin \\mathcal{S}^o$. So for any belief $\\pi $ such that $\\policy^*(\\pi) $ goes one step and stops, then $\\pi \\notin \\mathcal{S}^o$. Therefore, for any belief $\\pi$ such that $\\policy^*(\\pi) $ goes more than one step and stops, then $\\pi \\notin \\mathcal{S}^o$.\n\nThe two steps imply that $\\mathcal{S}^o = \\mathcal{S}$.Therefore that the stopping set is explicitly given by the polytope in (\\ref{eq:stopsetoo}).\n\n\\item Give sufficient conditions on $P$ and $B$ so that condition (\\ref{eq:onesteppp}) holds for a stopping time POMDP.\n\\end{compactenum}\n\n\\item \n\nShow that an identical, proof to Theorem \\ref{thm:pomdpconvex}\nimplies that the stopping sets $\\mathcal{S}_n$, $n=1,2,\\ldots$ are convex for a finite horizon problem.\n\n\n\\item {\\bf Choosing a single sample from a HMM.} \\index{multiple stopping problem}\nSuppose a Markov chain $x_k$ is observed in noise sequentially over time as $y_k \\sim B_{x_k,y}$, $k=1,2\\ldots,N$.\n Over a horizon of length $N$, I need to choose a single observation $y_k$ to maximize\n$\\E\\{y_k\\}$, $k \\in 1,\\ldots,N$. If at time $k$ I decide to choose observation $y_k$, then I get reward \n$\\E\\{y_k\\}$ and the problem stops.\nIf I decide not to choose observation $y_k$, then I can use it to update my estimate of the state and proceed to the next time\ninstant. However, I am not allowed to choose $y_k$ at a later time.\n\n\\begin{compactenum}\n\\item Which single observation should I choose?\n\n Show that Bellman's equation becomes\n$$ V_{n+1}(\\pi) = \\max_{u\\in \\{1,2\\} } \\{ r^\\prime \\pi , \\sum_y V_n( T(\\pi, y)) \\sigma(\\pi,y) \\} $$\nwhere the elements of $r$ are $r(i) = \\sum_{y} y B_{iy}$, $i=1,\\ldots,X$.\nHere $u=1$ denotes choose an observation, while $u=2$ denotes do not choose an observation.\n\n\n\\item Show using an identical proof to Theorem \\ref{thm:pomdpconvex} that the region of the belief space $\\mathcal{S}_n = \\{\\pi: \\mu^*(\\mu) = 1\\}$ is convex.\nMoreover if (\\ref{A2},\\ref{A3}) hold, show that $e_1$ belongs to $\\mathcal{S}_n$. \nAlso show that $\\mathcal{S}_0 \\subseteq \\mathcal{S}_1 \\subseteq \\mathcal{S}_2 \\ldots$.\n\n\\item {\\bf Optimal Channel sensing.}\nAnother interpretation of the above problem is as follows: The quality $x_k$ of a communication channel is observed in noise. I need to\ntransmit a packet using this channel. If the channel is in state $x$, I incur a cost $c(x)$ for transmission. Given $N$ slots, when should\nI transmit? \\index{optimal channel sensing}\n\\end{compactenum}\n\n\\item {\\bf Optimal measurement selection for a Hidden Markov Model (Multiple stopping problem)}. \\index{multiple stopping problem}\n The following problem generalizes the previous problem as follows. \\index{multiple stopping problem}\n I need to choose the best $L$ observations of a Hidden Markov model in a horizon of length $N$\nwhere $L \\leq N$? If I select observation $k$ then I get a reward $\\E\\{y_k\\}$, if I reject the observation then I get no reward.\nIn either case, I use the observation $y_k$ to update my belief state. (This problem is also called the multiple stopping problem in \n\\cite{Nak95}.)\nShow that Bellman's dynamic programming recursion reads:\n\\begin{multline*} V_{n+1}(\\pi,l) = \\max\\{ r^\\prime \\pi + \\sum_y V_{n}(T(\\pi,y),l-1) \\sigma(\\pi,y) , \\\\ \\sum_y V_{n}(T(\\pi,y),l) \\sigma(\\pi,y) \\} , \\quad n=1,\\ldots,N \\end{multline*}\nwith initial condition $V_n(\\pi,0) = 0$, $n=0,1,\\ldots$ and boundary conditions\n$$V_{n}(\\pi,n) = r^\\prime \\pi + \\sum_y V_{n-1} ( T(\\pi,y), n-1) \\sigma(\\pi,Y) , \\quad n=1,\\ldots,L. $$\nThe boundary condition says that if I have only $n$ time points left to make $n$ observations, then I need to make an observation\nat each of these $n$ time points. Obtain a structural result for the optimal measurement selection policy. (Notice that the actions\ndo no affect the evolution of the belief state $\\pi$, they only affect $l$, so the problem is simpler than a full blown POMDP.)\n\n\n\\item {\\bf Separable POMDPs.} \\index{separable POMDPs} \nRecall that the action space is denoted as $\\,\\mathcal{U} = \\{1,2,\\ldots,U\\}$.\nIn analogy to \\cite[Chapter 7.4]{HS84}, define a POMDP to be separable if:\nthe exists a subset $\\bar{\\,\\mathcal{U}}= \\{1,2\\ldots,\\bar{U}\\}$ of the action space $\\,\\mathcal{U}$ such that for $u \\in \\bar{\\,\\mathcal{U}}$\n\\begin{compactenum}\n\\item The cost is additively separable: $c(x,u) = \\phi(u) + g(x)$ for some scalars $\\phi(u)$ and $g(x)$.\n\\item The transition matrix $P_{ij}(u)$ depends only on $j$. That is the process evolves independently of the previous state.\n\\end{compactenum}\nAssuming that the actions $u \\in \\bar{\\,\\mathcal{U}}$ are ordered so that $\\phi(1) < \\phi(2) < \\ldots < \\phi(\\bar{U})$, clearly it is never\noptimal to pick actions $2,\\ldots,\\bar{U}$. So solving the POMDP involves choosing between actions $\n\\{1, \\bar{U}+1,\\ldots, U\\}$.\nSo from Theorem \\ref{thm:pomdpconvex}, the set of beliefs where the optimal policy \n$\\mu^*(\\pi) = 1$ is convex.\n\nSolving for the optimal policy for which the actions $\\{\\bar{U}+1,\\ldots, U\\}$ arise is still as complex as a solving a standard POMDP. However, the \nbounds proposed in Chapter \\ref{chp:myopicul} can be used. \n\nConsider the special case of the above model where $\\bar{\\,\\mathcal{U}} = \\,\\mathcal{U} $ and instead of (a), $c(x,u)$ are arbitrary costs. Then\nshow that the optimal policy is a linear threshold policy.\n\\end{compactenum}\n\n\\section{Case Study: Bayesian Nash equilibrium of one-shot global game for coordinated sensing} \\index{coordinated sensing}\n\\index{Bayesian global game}\n\\index{Bayesian Nash equilibrium (BNE)}\n \\index{game theory! global game}\nThis section gives a short description of Bayesian global games.\nThe ideas involve MLR dominance of posterior distributions and supermodularity and serves as a useful illustration of the \nstructural results developed in the chapter.\n\nWe start with some perspective:\n Recall that in the classical Bayesian social learning, agents act sequentially in time. The global games model that has been studied\nin economics during the last two decades, considers multiple agents that act simultaneously by predicting the behavior\nof other agents. \nThe theory of global games was first introduced in \\cite{CD93} as a tool for refining equilibria in economic game theory;\nsee \\cite{MS00} for an excellent exposition.\nGlobal games\nrepresent a useful method for decentralized coordination amongst agents; they \nhave been used to model speculative currency attacks and regime change in social systems, see\n\\cite{MS00,KLM07,AHP07}. Applications in sensor networks and cognitive radio appear in \\cite{Kri08,Kri09}.\n\n\\subsection{Global Game Model}\nConsider a continuum of agents in which each agent $i$ obtains noisy measurements $Y^{(i)}$\nof an underlying state of nature $X$. Here \n$$ Y^{(i)} = X + W^{(i)} , \\quad X \\sim \\pi,\\;\nW^{(i)} \\sim p_{W}(\\cdot) $$\n\nAssume all agents have the same noise distribution $p_W$.\n Based on its observation $y^{(i})$, each agent \ntakes an action $u^i \\in \\{1, 2\\}$ to optimize its expected reward \\begin{equation} \\label{eq:utilityi}\n R(X,\\alpha,u=2) = X + \n f(\\alpha), \\quad\n R(X,u=1) = 0 \n \\end{equation}\nHere $\\alpha\\in [0,1]$ denotes the fraction of agents that choose action 2 and $f(\\alpha)$ is a user specified function. We will call\n$f$ the congestion function for reasons explained below.\n\nAs an illustrative\nexample, suppose $x$ (state of nature) denotes the quality of a social group and $y^{(i})$ denotes the measurement of this quality by agent $i$. The action $u^i = 1$ means that agent $i$ decides not to join the social group, while $u^i = 2$ means that agent $i$ joins the group.\nThe utility function $R(u^i=2,\\alpha)$ for joining the social group depends on $\\alpha$, where $\\alpha$ is the fraction of people who decide to join the group. \nIf $\\alpha \\approx 1$, i.e., too many people join the group, then the utility to each agent is small since the group is too congested and agents do not receive sufficient individual service.\nOn the other hand, if $\\alpha \\approx 0$, i.e., too few people join the group, then the utility is also small since there is not enough social interaction.\nIn this case the congestion function $f(\\alpha)$ would be chosen as a quasi-concave function of $\\alpha$ (that increases with $\\alpha$ up to a certain value of $\\alpha$ and then decreases with $\\alpha$).\n\nSince each agent is rational, it uses its observation $y^{(i)}$ to predict $\\alpha$, i.e., the fraction of other agents \nthat choose action 2. The main question is: {\\em What is the optimal strategy for each agent $i$ to maximize its expected reward?}\n\n\\subsection{Bayesian Nash Equilibrium}\nLet us now formulate this problem:\nEach agent\nchooses its action $u \\in \\{1 , 2 \\}$ based on a (possibly randomized) strategy $\\mu^{(i)}$ that maps the current\nobservation $Y^{(i)}$ to the action $u$. In a global game we are interested\nin {\\em symmetric strategies}, i.e., where all choose the same\nstrategy denoted as $\\mu$. \nThat is, each agent $i$ deploys the strategy $$ \\mu:Y^{(i)} \\rightarrow\\{1 ,2 \\} . $$\n(Of course, the action $\\mu(Y^{(i)})$ picked by individual agents $i$ \ndepend on their random observation $Y^{(i)}$. So the actions picked\nare not necessarily identical even though the strategies are identical).\n\n\nLet $\\alpha(x)$ denote the fraction of agents that\nselect action $u=2$ (go) given the quality of music $X = x$.\nSince we are considering an infinite number of agents that behave independently, \n$\\alpha(x)$ is also (with probability 1) the \nconditional probability that an agent receives signal $Y^{(i)}$\nand decides to pick $u=2$, given\n$X$. So\n\\begin{equation} \\label{eq:alfagdef}\n\\alpha(x) = P(\\mu(Y) =2 | X=x) .\n\\end{equation}\n\nWe can now define the Bayesian Nash equilibrium (BNE) of the global game.\nFor each agent $i$ \ngiven its observation $Y^{(i)}$, the goal is to choose a strategy to optimize its local reward. That is, \nagent $i$ seeks to compute strategy $\\mu^{(i),*}$ such that\n\\begin{equation} \\label{eq:localpolicy}\n\\mu^{(i),*}(Y^{(i)})\\in \\{1 \\text{ (stay) } ,2 \\text{ (go) }\\} \\text{ maximizes }\n\\E [R(X,\\alpha(X),\\mu^{(i)}(Y^{(i)}) )|Y^{(i)} ].\n \\end{equation}\nHere $ R(X,\\alpha(X),u )$ is defined as in (\\ref{eq:utilityi}) with $\\alpha(X)$\ndefined in (\\ref{eq:alfagdef}). \n\nIf such\na strategy $\\mu^{(i),*}$ in (\\ref{eq:localpolicy})\n exists and is the same for all agents $i$,\n then they constitute a {\\em symmetric} BNE for the global game.\nWe will use the notation $\\mu^*(Y)$ to denote this symmetric BNE.\n\n\n\\noindent {\\em Remark}: Since we are dealing with an incomplete information game, players\nuse randomized strategies. If a BNE exists, then a pure (non-randomized) version exists straightforwardly\n(see Proposition 8E.1, pp.225 in \\cite{MWG95}). Indeed, with $y^{(i)}$ denoting realization\nof random variable $Y^{(i)}$, \n$$ \\E[R\\big(X,\\alpha(X),\\mu(Y^{(i)}) \\big)| Y^{(i)} = y^{(i)}] = \\sum_{u=1}^2 \\E[ R(X,\\alpha(X),u)|Y^{(i)}= y^{(i)}] P(u|Y^{(i)}= y^{(i)}) .$$\nSince a linear combination is maximized at its extreme values,\n the optimal (BNE) strategy is to choose $P(u^*|Y^{(i)} = y^{(i)}) = 1$ where \n\\begin{equation}\\label{eq:bang} u^* = \\mu^*(y^{(i)}) = \\operatornamewithlimits{argmax}_{u \\in \\{1,2\\}} \\E[R(X,\\alpha(X),u)|Y^{(i)}= y^{(i)}] .\\end{equation}\n\nFor notational convenience denote\n$$ R(y,u) = \\E[R(X,\\alpha(X),u)|Y^{(i)}= y^{(i)}] $$\n\n\\subsection{Main Result. Monotone BNE} \\index{monotone Bayesian Nash equilibrium}\nWith the above description, we will now give sufficient conditions for the BNE $\\mu^*(y)$ to be monotone increasing in $y$ (denoted\n$\\mu^*(y) \\uparrow y$).\nThis implies that the BNE is a threshold policy of the form:\n$$ \\mu^*(y) = \\begin{cases} 1 & y \\leq y^* \\\\\n\t\t\t\t\t2 & y > y^* \\end{cases} $$\nBefore proving this monotone structure, first note that $\\mu^*(y) \\uparrow y$ implies that $\\alpha(x)$ in (\\ref{eq:alfagdef}) becomes\n$$ \\alpha(x) = P( y > y^* | X = x) = P(x + w > y^*) = P( w > y^* - x) = 1- F_W(y^*-x)$$\n\nClearly from (\\ref{eq:bang}), a sufficient condition for $\\mu^*(y) \\uparrow y$ is that\n$$R(y,u ) = \\int R(x, \\alpha(x) ,u ) \\,p( x | y) dx $$\nis supermodular in $(y,u)$ that is \n$$ R(y,u+1) - R(y,u) \\uparrow y.$$\nSince $R(X,u=0)$ it follows that $R(y,1) = 0$. So it suffices that $R(y,2) \\uparrow y $.\n\n\\begin{compactenum}\n\\item What are sufficient conditions on the noise pdf $p_W(\\cdot)$, and congestion function $f(\\cdot)$ in\n(\\ref{eq:utilityi}) so that $R(y,2) \\uparrow y$ and so BNE $\\mu^*(y) \\uparrow y$?\n\nClearly sufficient conditions for $R(y,2) \\uparrow y$ are:\n\\begin{compactenum}\n\\item $p(x|y)$ is MLR increasing in $y$,\n\\item ${\\rm I\\hspace{-.07cm}R}(x,\\alpha(x),2)$ is increasing in $x$.\n\\end{compactenum}\n\nBut we know that $p(x|y)$ is MLR increasing in $y$ if the noise distribution is such that $p_W(y - x)$ is TP2 in $x,y$\n\nAlso $R(x,\\alpha(x),2)$ is increasing in $x$ if its derivative wrt $x$ is positive. That is,\n$$ \\frac{d}{dx} R(x,\\alpha(x),2) = 1 + \\frac{df}{d\\alpha} \\frac{d \\alpha} {dx} = 1 + \\frac{df}{d\\alpha} p_W(y^*-x) > 0 $$\n\nTo summarize: The BNE $\\mu^*(y) \\uparrow y$ if the following two conditions hold:\n\\begin{compactenum}\n\\item $p(y|x) = p_W(y-x)$ is TP2 in $(x,y)$\n\\item $$ \\frac{df}{d\\alpha} > -\\frac{1}{p_W(y^*-x) } $$\n\\end{compactenum}\nNote that a sufficient condition for the second condition is that \n$$ \\frac{df}{d\\alpha} > -\\frac{1}{\\max_w p_W(w) } $$\n\n\\item Suppose $W$ is uniformly distributed in $[-1.1]$. Then using the above conditions show that a sufficient condition on the congestion function $f(\\alpha)$ \nfor the BNE to be monotone is that $df\/d\\alpha > - 2$.\n\n\n\\item Suppose $W$ is zero mean Gaussian noise with variance $\\sigma^2$. Then using the above conditions show that a sufficient condition on the congestion function $f(\\alpha)$ \nfor the BNE to be monotone is that $df\/d\\alpha > - \\sqrt{2 \\pi} \\sigma$.\n\n \n\\subsection{One-shot HMM Global Game} \\index{HMM global game}\nSuppose that $X_0 \\sim \\pi_0$, and given $X_0$, $X_1$ is obtained by simulating from transition matrix $P$. The observation for agent\n$i$ is obtained as the HMM observation\n$$ Y^{(i)} = X_1 + W^{(i)}, \\quad W^{(i)} \\sim p_W(\\cdot). $$ \nIn analogy to the above\nderivation, characterize the BNE of the resulting one-shot HMM global game. (This will require assuming that $P$ is TP2.)\n\n\\end{compactenum}\n\n\n\n\n\\chapter{Stopping Time POMDPs for Quickest Change Detection} \n\n\\begin{compactenum}\n\n\\item For classical detection theory, a ``classic\" book is the multi-volume \\cite{Van68}.\n\\item As mentioned in the book, \nthere are two approaches to quickest change detection: Bayesian and minimax.\nChapter \\ref{chp:stopapply} of the book deals with Bayesian quickest detection which assumes that the change point distribution is known (e.g. phase distribution). The focus of Chapter \\ref{chp:stopapply} was to determine the structure of the optimal policy of the \nBayesian detector by showing that the problem is a special case of a stopping time POMDP.\n\\cite{VB13} uses nonlinear renewal theory to analyze the performance of the optimal Bayesian detector.\n\nThe minimax formulation for quickest detection assumes that the change point is either deterministic or has an unknown distribution.\nFor an excellent starting point on performance analysis of change detectors with minimax formulations please see \\cite{TM10} and \\cite{PH08}.\nThe papers \\cite{Lor71,Mou86} gives a lucid description of the analysis of change detection in this framework.\n\n\\item {\\bf Shiryaev Detection Statistic.} \nIn the classical Bayesian formulation of quickest detection described in \\S \\ref{sec:convexstop},\na two state Markov chain is considered to model geometric distributed change times. Recall (\\ref{eq:tpqdp}), namely,\n\\begin{equation} P = \\begin{bmatrix} 1 & 0 \\\\ 1- P_{22} & P_{22} \\end{bmatrix} , \\; \\pi_0 = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} , \\quad\n\\tau^0 = \\inf\\{ k: x_k = 1\\}. \\end{equation}\nwhere $1-P_{22}$ is the parameter of the geometric prior.\n\nIn classical detection theory, the belief state $\\pi_k$ is written in terms of the {\\em Shiryaev detection\nstatistic} $r_k$ which is defined as follows: \\index{Shiryaev detection statistic}\n\\begin{equation}\n r_k \\stackrel{\\text{defn}}{=} \\frac{1}{1-P_{22}}\\times \\frac{\\pi_k(2) }{1 - \\pi_k(2)} \\end{equation}\nClearly $r_k$ is an increasing function of $\\pi_k(2)$ and so all the monotonicity results in the chapter continue to hold. \nIn particular Corollary \\ref{cor:qdclassical} in the book holds for $r_k$ implying a threshold policy in terms of $r_k$.\n\nIn terms of the Shiryaev \nstatistic $r_k$, it is straightforward to write\nthe belief state update (HMM filter for 2 state Markov chain) as a function of the likelihood ratio as follows:\n\\begin{empheq}[box=\\fbox]{equation}\nr_k = \\frac{1}{1-p} \\left( r_{k-1} + 1 \\right) L(y_k) \\label{eq:shirstat}\n\\end{empheq}\nwhere\n$$ p = 1 - P_{22}, \\quad L(y_k ) = \\frac{B_{2y_k}}{ B_{1y_k}} \\text{ (likelihood ratio) }$$\nIn (\\ref{eq:shirstat}) by choosing $p \\rightarrow 0$, the Shiryaev detection statistic converges to the so called {\\em Shiryaev-Roberts detection statistic}.\nNote that as $p \\rightarrow 0$, the Markov chain becomes a slow Markov chain. We have analyzed in detail how to track the state of such a slow\nMarkov chain via a stochastic approximation algorithm in Chapter \\ref{chp:markovtracksa} of the book. \\index{Shiryaev-Roberts detection statistic}\n\nThe Shiryaev-Roberts detector for change detection reads:\n\\begin{compactenum}\n\\item Update the Shiryaev-Roberts statistic \n$$ r_k = \\left( r_{k-1} + 1 \\right) L(y_k) $$\n\\item If $r_k \\geq r^*$ then stop and declare a change.\nHere $r^*$ is a suitably chosen detection threshold.\n\n\n\\end{compactenum}\nPlease see \\cite{PT12} for a nice survey description of minimax change detection and also the sense in which the above Shiryaev-Roberts detector\nis optimal.\n\n\n\\item {\\bf Classical Bayesian sequential detection.} \n\\index{classical sequential detection}\n This problem shows that classical Bayesian sequential detection is a trivial case of the results\ndeveloped in Chapter \\ref{ch:pomdpstop}.\n\n\n\nConsider a random variable $x \\in \\{1,2\\}$.\nSo the transition matrix is $P = I$. Given noisy observations $y_k \\simB_{x y}$, the aim is to decide if the underlying state is either 1 or 2. Taking stop action 1 declares that the state is 1 and stops. Taking stop action 2 declares that the state is 2 and stops. \nTaking action 3 at time $k$ simply takes another measurement $y_{k+1}$.\nThe misclassification costs are: $$c(x=2,u=1) = c(x=1,u=2) = L.$$ \nThe cost of taking an additional measurement is $c(x,u=3) = C$. What is the optimal policy $\\mu^*(\\pi)$?\n\nSince $P = I$, show that the dynamic programming equation reads\n\\begin{align*}\n V(\\pi) &= \\min\\{ \\pi_2 L, \\;\\pi_1 L, \\; C + \\sum_y V(T(\\pi,y)) \\sigma(\\pi,y) \\} \\\\\n T(\\pi,y) &= \\frac{B_y \\pi}{\\mathbf{1}^\\prime B_y \\pi} , \\quad \\sigma(\\pi,y) = \\mathbf{1}^\\prime B_y \\pi ,\n \\end{align*}\nwhere $\\pi = [\\pi(1), \\pi(2)]^\\prime$ is the belief state. Note that $y \\in \\mathcal{Y}$ where $\\mathcal{Y}$ can be finite or\n continuum (in which case $\\sum$ denotes integration over $\\mathcal{Y}$).\n\nFrom Theorem \\ref{thm:pomdpconvex} we immediately know that the stopping sets $$\\mathcal{R}_1 = \\{\\pi: \\mu^*(\\pi) = 1\\}, \\text{ and }\n\\mathcal{R}_2 = \\{\\pi: \\mu^*(\\pi) = 2\\}$$ are convex sets. Since the belief state is two dimensional, in terms of the second component $\\pi(2)$, $\\mathcal{R}_1 $ and \n$\\mathcal{R}_2$ are intervals in the unit interval $[0,1]$. Clearly $\\pi(2) = 0 \\in \\mathcal{R}_1$ and $\\pi(2)=1$ in $\\mathcal{R}_2$.\nTherefore $\\mathcal{R}_1 = [0,\\pi_1^*]$ and $\\mathcal{R}_2 = [\\pi_2^*,1]$ for some $\\pi_1^* \\leq \\pi_2^*$. So the continue region is\n$[\\pi_1^*,\\pi_2^*]$. \n\nOf course, Theorem \\ref{thm:pomdpconvex} is much more general since it does not require $X=2$ states and $x_k$ can evolve\naccording to a Markov chain with transition matrix $P$ (whereas in the simplistic setting above, $x$ is a random variable).\n\n\\item {\\bf Stochastic Ordering of Passage Times for Phase-Distribution.} \\index{passage time}\nIn quickest detection, we formulated the \nchange point $\\tau^0$ to have a \n{\\em phase type (PH) distribution}. \nA systematic investigation of the statistical properties of PH-distributions can be found in \\cite{Neu89}.\nThe family of all PH-distributions forms a dense subset for the set of all distributions\n\t\\cite{Neu89} i.e., for any given distribution function $F$ such that $F(0) = 0$, one can find a sequence of PH-distributions\t\n$\\{F_n , n\t\\geq\t1\\}$\t to\t\tapproximate\t$F$\tuniformly over $[0, \\infty)$.\nThus PH-distributions can be used to approximate change points with an arbitrary distribution. This is done by\n\tconstructing a multi-state Markov chain as follows:\nAssume state `1' (corresponding to belief $e_1$) is an absorbing state\nand denotes the state after the jump change. The states $2,\\ldots,X$ (corresponding to beliefs $e_2,\\ldots,e_X$) can be viewed as a single composite state that $x$ resides in before the jump. \nTo avoid trivialities, assume that the change occurs after at least one measurement. So the initial distribution $\\pi_0$ satisfies $\\pi_0(1) = 0$.\nThe \ntransition probability matrix is of the form\n\\begin{equation} \\label{eq:phmatrix}\nP = \\begin{bmatrix} 1 & 0 \\\\ \\underline{P}_{(X-1)\\times 1} & \\bar{P}_{(X-1)\\times (X-1)} \\end{bmatrix}.\n\\end{equation}\nThe {\\em first passage time} $\\tau^0$ to state 1 denotes the time at which $x_k$ enters the absorbing state 1:\n \\begin{equation} \\tau^0 = \\min\\{k: x_k = 1\\}. \\label{eq:tau} \\end{equation}\n As described in \\S \\ref{sec:qdphform} of the book,\nthe distribution of $\\tau^0$ is determined by choosing the transition probabilities $\\underline{P}, \\bar{P}$ in \n (\\ref{eq:phmatrix}). \n The \ndistribution of the absorption time to state 1 is denoted by $$\\nu_k = \\mathbb{P}(\\tau^0 = k)$$ and given by\n\\begin{equation} \\label{eq:nu}\n \\nu_0 = \\pi_0(1), \\quad \\nu_k = \\bar{\\pi}_0^\\prime \\bar{P}^{k-1} \\underline{P}, \\quad k\\geq 1, \\end{equation}\n where $\\bar{\\pi}_0 = [\\pi_0(2),\\ldots,\\pi_0(X)]^\\prime$.\n\n\n{\\bf Definition. Increasing Hazard Rate}: A pmf $p$ is said to be increasing hazard rate (IHR) if \\index{increasing hazard rate}\n$$ \\frac{\\bar{F}_{i+1}}{{\\bar{F}_{i}}} \\downarrow i, \\quad \\text{ where } \\bar{F}_i = \\sum_{j=i}^\\infty p_j$$\n\n\n{\\bf Aim}. Show that if the transition matrix $P$ in (\\ref{eq:phmatrix}) is TP2 and initial condition $\\pi_0 = e_X$, then the passage time distribution $\\nu_k$ \nin (\\ref{eq:nu})\nsatisfies the increasing hazard rate (IHR)\nproperty; see \\cite{Sha88} for a detailed proof.\n\n\\item {\\bf Order book high frequency trading and social learning.} \\index{order book high frequency trading}\nAgent based models for high frequency trading with an order book have been studied a lot recently \\cite{AS08}. Agents trade (buy or sell) stocks by exploiting information about the decisions of previous agents (social learning) via an order book in addition to a private (noisy) signal they receive on the value of the stock. We are interested in the following: (1) Modeling the dynamics of these risk averse agents, (2) Sequential detection of a market shock based on the behavior of these agents.\n\nThe agents perform social learning according to the protocol in \\S \\ref{sec:herdaa} of the book. A market maker needs to decide based on the actions of the agents if there\nis a sudden change (shock) in the underlying value of an asset. Assume that the shock occurs with a phase distributed change time.\nThe individual agents perform social learning with a CVaR social learning filter as in \\S \\ref{sec:classicalsocial} of the book. The market maker aims\nto determine the shock as soon as possible.\n\nFormulate this decision problem as a quickest detection problem. Simulate the value function and optimal policy. Compare it with the \nmarket maker's optimal policy\nobtained when the agents perform risk\nneutral social learning. See \\cite{KB16} for details.\n\\end{compactenum}\n\n\\chapter{Myopic Policy Bounds for POMDPs and Sensitivity} \n\n\\begin{compactenum}\n\n\\item To obtain upper and lower bounds to the optimal policy, the key idea was to change the cost vector but still preserve the optimal policy. \\cite{KP15} gives a complete description of this idea.\nWhat if a nonlinear cost was subtracted from the costs thereby still keeping the optimal policy the same. Does that allow for larger regions of the belief space\nwhere the upper and lower bounds coincide?\nIs it possible to construct different transition matrices that yield the same optimal policy? \n\n\n\n\\item{\\bf First order dominance of Markov chain sample paths.} In \\S \\ref{sec:tpcp} of the book we defined the importance concept of copositive dominance\nto say that if two transition matrices $P_1$ and $P_2$ satisfy $P_1 \\preceq P_2$ (see Definition \\ref{def:lR}), then the one step ahead predicted belief satisfies the MLR dominance property\n$$ P_1^\\prime \\pi \\leq_r P_2 ^\\prime \\pi. $$ \nIf we only want first order stochastic dominance, then the following condition suffices: \nLet $U$ denote the $X \\times X $ dimensional triangular matrix with elements \n$U_{ij} = 0, i > j$ and $U_{ij} = 1, i \\leq j$.\n\\begin{compactenum}\n\\item \nShow the following result:\n$$ P_1 U \\geq P_2 U \\implies P_1^\\prime \\pi_1 \\geq_s P_2^\\prime \\pi_2\\; \\text{ if } \\pi_1 \\geq_s \\pi_2. $$\n\\item\n Consider the following special case of a POMDP. Suppose the prior belief $\\pi_0 \\in \\Pi(\\statedim) $ is known. From time 1 onwards,\nthe state is fully observed. How can the structural results in this chapter be used to characterize the optimal policy?\n\\end{compactenum}\n\n\\item In \\cite{Lov87} it is assumed that one can construct a POMDP with observation matrices $B(1),B(2)$ such that \n(i) $T(\\pi,y,2) \\geq_r T(\\pi,y,1)$ for each $y$ and (ii) $\\sigma(\\pi,2) \\geq_s \\sigma(\\pi,1) $. Prove that it is impossible\nto construct an example that satisfies (i) and (ii) apart from the trivial case where $B(1) = B(2)$. Therefore Theorem \\ref{th:theorem1udit} does not apply\nwhen the transition probabilities are the same and only the observation probabilities are action dependent. For such cases, Blackwell dominance is used.\n\n\n\\item {\\bf Extensions of Blackwell dominance idea in POMDPs to more general cases.}\nBlackwell dominance \\index{Blackwell dominance} was used in \\S \\ref{sec:blackwelldom} of the book to construct myopic policies that bound the optimal policy of a POMDP.\nBut Blackwell dominance\nis quite finicky. In Theorem \\ref{thm:compare2} we assumed that the POMDP has dependency structure $x \\rightarrow y^{(2)} \\rightarrow y^{(1)}$. That is, the observation distributions are\n$B(2) = p(y^{(2)}|x)$ and $B(1) = p(y^{(1)}|y^{(2)})$.\n\n\\begin{compactenum}\n\\item \nRecall the proof of Theorem \\ref{thm:compare2} which is written element wise below for maximum clarity:\n\\[\n\\begin{split}\n T_j(\\pi,y^{(1)},1) = \\frac{\\sum_{y^{(2)}} \\sum_i \\pi(i) P_{ij} p(y^{(2)}|j,2) p(y^{(1)} | y^{(2)})}\n{\\sum_m \\sum_{y^{(2)}} \\sum_i \\pi(i) P_{im} p(y^{(2)}|m,2)p(y^{(1)} | y^{(2)}) } \\\\\n= \n \\frac{\\sum_{y^{(2)}} \\dfrac{ \\sum_i \\pi(i) P_{ij} p(y^{(2)}|j,2) }\n{ \\cancel{ \\sum_i \\sum_m \\pi(i) P_{im} p(y^{(2)}|m,2) }} \\cancel{ \\sum_i \\sum_m \\pi(i) P_{im} p(y^{(2)}|m,u)} \\,\n p(y^{(1)} | y^{(2)})}\n{\\sum_m \\sum_{y^{(2)}} \\sum_i \\pi(i) P_{im} p(y^{(2)}|m,2) p(y^{(1)} | y^{(2)}) } \\\\ =\n \\frac{\\sum_{y^{(2)}} T_j(\\pi,y^{(2)},2) \\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)})}{ \\sum_{y^{(2)}} \\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)})}\n\\end{split}\n\\]\n\nThen clearly $ \\frac{ \\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)})}{ \\sum_{y^{(2)}} \\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)})}$\nis a probability measure w.r.t $y^{(2)}$.\n\n\\item Consider now the more general POMDP where $p(y^{(1)}| y^{(2)},x)$ depends on the state $x$. (In Theorem \\ref{thm:compare2}\nthis was functionally independent of $x$.)\nThen \n$$ T_j(\\pi,y^{(1)},1) = \\frac{\\sum_{y^{(2)}} T_j(\\pi,y^{(2)},2) \\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)},j)}{ \\sum_{y^{(2)}} \\sum_m\\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)},m)} $$\nNow \n$$ \\frac{ \\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)},j)}{ \\sum_{y^{(2)}} \\sum_m\\sigma(\\pi,y^{(2)},2) p(y^{(1)}| y^{(2)},m)} $$ is no longer a probability measure w.r.t. $y^{(2)}$.\nThe proof of Theorem \\ref{thm:compare2} no longer holds.\n\n\\item Next consider the case where the observation distribution is $p(y^{(2)}_k | x_k, x_{k-1})$ and $p(y^{(1)} | y^{(2)})$. Then the proof\nof Theorem \\ref{thm:compare2} continues to hold.\n\n\\end{compactenum}\n\n\n\\item {\\bf Blackwell dominance implies higher channel capacity.} Show that if $B(1)$ Blackwell dominates $ B(2) $, i.e., $B(2) = B(1) Q$ for some stochastic matrix $Q$, then the capacity of a channel\nwith likelihood probabilities given by $B(1)$ is higher than that with likelihood probabilities $B(2)$. \n\n\\item \nRecall that the structural result involving Blackwell dominance deals with action dependent observation probabilities but assumes identical transition matrices for the various actions.\nShow that copositive dominance and Blackwell dominance can be combined to deal with a POMDP with action dependent transition and\nobservation probabilities of the form:\n\\\\ Action $u = 1$: $P^2, B$.\n\\\\ Action $u=2$: $P, B^2$.\n\\\\\nGive numerical examples of POMDPs with the above structure.\n\\end{compactenum}\n\n\n\\newpage\n\\chapter{Part IV. Stochastic Approximation and Reinforcement Learning}\n\nHere we present three case studies of stochastic approximation algorithms.\nThe first case study deals with online HMM parameter estimation and extends the method described in Chapter \\ref{chp:markovtracksa}.\nThe second case study deals with reinforcement learning of equilibria in repeated games. The third case study deals with discrete stochastic optimization\n(recall \\S \\ref{sec:dopt} gave two algorithms)\nand provides a simple example of such an algorithm. \n\n\n\n\\section{Case Study. Online HMM parameter estimation} \\index{online HMM estimation} \nRecall from Chapter \\ref{chp:markovtracksa} that estimating the parameters of a HMM in real time is motivated by adaptive control of a POMDP.\nThe parameter estimation algorithm can be used to estimate the parameters of the POMDP\nfor a fixed policy; then the policy can be updated using dynamic programming (or approximation) based on the parameters and so on.\n\n\nThis case study outlines several algorithms for recursive estimation of HMM parameters. The reader should implement these \nalgorithms in Matlab to get a good feel for how they work.\n\nConsider the loss function for $N$ data points of a HMM or Gaussian state space model:\n\\begin{equation} J_N(\\th) = \\E\\{ \\sum_{k=1}^N c_\\th(x_k,y_k,\\pi_k^\\theta) \\} \\label{eq:lossfunction} \\end{equation}\nwhere $x_k$ denotes the state, $y_k$ denotes the observation, $\\pi_k^\\th$ denotes the belief state, and $\\th$ denotes the model variable.\n\n\nThe aim is to determine the model $\\th$ that maximizes this loss function.\n\nAn offline gradient algorithm operates iteratively to minimize this loss as follows:\n\\begin{equation} \\th^{(l+1)} = \\th^{(l)} - \\epsilon \\nabla_\\th J_N(\\th)\\big\\vert_{\\th = \\th^{(l)}} \\label{eq:offlinegrad} \\end{equation}\nThe notation $ \\vert_{\\th = \\th^{(l)}}$ above means that the derivatives are evaluated at $\\th = \\th^{(l)}$.\n\nAn offline Newton type algorithm operates iteratively as follows:\n\\begin{equation} \\th^{(l+1)} = \\th^{(l)} - \\big[ \\nabla_\\th^2 J_N(\\th) \\big]^{-1} \\nabla_\\th J_N(\\th)\\big\\vert_{\\th = \\th^{(l)}} \n\\label{eq:offlinenewton}\n\\end{equation}\n\n\\subsection{Recursive Gradient and Gauss-Newton Algorithms}\nA \nrecursive online version of the above gradient algorithm (\\ref{eq:offlinegrad}) is \n\\begin{empheq}[box=\\fbox]{equation} \\begin{split}\n & \\th_{k} = \\th_{k-1} - \\epsilon \\, \\nabla_\\th c_\\th(x_k,y_k,\\pi^\\th_k)\\big\\vert_{\\th = \\th_{k-1}} \\, \n \\\\ & \\pi_k^{\\th_{k-1}} = T( \\pi_{k-1}^{\\th_{k-1}}, y_k; \\th_{k-1}) \n \\end{split} \\label{eq:onlinegrad}\n\\end{empheq}\nwhere $ T( \\pi_{k-1}^{\\th_{k-1}}, y_k; \\th_{k-1})$ is the optimal filtering recursion at time $k$ using prior $\\pi_{k-1}^{\\th_{k-1}}$,\nmodel $\\th_{k-1}$ and observation $y_k$. The notation $ \\vert_{\\th = \\th_{k-1}}$ above means that the derivatives are evaluated at $\\th = \\th_{k-1}$\nFinally, $\\epsilon$ is a small positive step size.\n\nThe recursive Gauss Newton algorithm is an online implementation of (\\ref{eq:offlinenewton}) and reads\n\\begin{empheq}[box=\\fbox]{equation} \\begin{split}\n & \\th_{k} = \\th_{k-1} - \\mathcal{I}_k^{-1} \\, \\nabla_\\th c_\\th(x_k,y_k,\\pi_k^{\\th})\\big\\vert_{\\th = \\th_{k-1}} \\\\\n& \\mathcal{I}_k = \\mathcal{I}_{k-1} + \\epsilon \\nabla^2 c_\\th(x_k,y_k,\\pi_k^{\\th})\\big\\vert_{\\th = \\th_{k-1}} \\\\\n & \\pi_k^{\\th_{k-1}} = T( \\pi_{k-1}^{\\th_{k-1}}, y_k; \\th_{k-1}) \n \\end{split} \\label{eq:recgaussnewton}\n\\end{empheq}\nNote that the above recursive Gauss Newton is a stochastic approximation algorithm with a matrix step size $\\mathcal{I}_k$.\n\n\\subsection{Justification of (\\ref{eq:onlinegrad})}\nBefore proceeding with examples, we give a \n heuristic derivation of (\\ref{eq:onlinegrad}). Write (\\ref{eq:offlinegrad}) as \n$$ \\th_{k}^{(k)} = \\th_k^{(k-1)} - \\epsilon \\nabla_\\th J_N(\\th)\\big\\vert_{\\th = \\th_k^{(k-1)}} $$\nHere the subscript $k$ denotes the estimate based on observations $y_{1:k}$. The superscript $(k)$ denotes the iteration of the offline optimization\nalgorithm.\n\nSuppose that at each iteration $k$ we collect one more observation. Then the above algorithm becomes\n\\begin{equation} \\th_{k}^{(k)} = \\th_{k-1}^{(k-1)} - \\epsilon \\nabla_\\th J_k(\\th)\\big\\vert_{\\th = \\th_{k-1}^{(k-1)}} \\label{eq:onoff}\\end{equation}\nIntroduce the convenient notation\n$$ \\th_k = \\th_{k}^{(k)} . $$\nNext we use the following two crucial approximations:\n\\begin{compactitem}\n\\item First, make the inductive assumption that $\\th_{k-1}$ minimized $J_{k-1}(\\th)$ so that \n$$ \\nabla_\\th J_{k-1}(\\th)\\big\\vert_{\\th = \\th_{k-1}} = 0 $$ Then from (\\ref{eq:lossfunction}) it follows that\n\\begin{equation} \\nabla_\\th J_{k}(\\th)\\big\\vert_{\\th = \\th_{k-1}} = \\nabla_\\th \\E\\{ c_\\th(x_k,y_k,\\pi^\\th_k) \\}\\big\\vert_{\\th = \\th_{k-1}} \n\\label{eq:approx1}\n\\end{equation}\n\\item Note that evaluating the right hand side of (\\ref{eq:approx1}) requires running a filter and its derivates wrt $\\th$ from time 0 to $k$ for fixed model $\\th_{k-1}$. We want a recursive\napproximation for this. It is here that the\nsecond approximation is used. We revlaute the filtering recursion using\nthe a sequence of available model estimates $\\th_t$, $t=1,\\ldots, k$ at each time $t$. In other words, we make the approximation\n\\begin{equation} \\pi_k^{\\th_{k-1}} = T( \\pi_{k-1}^{\\th_{k-1}}, y_k; \\th_k) , k = 1,2,,\\ldots, \\label{eq:approx2} \\end{equation}\n\n\\end{compactitem}\nTo summarize, introducing approximations (\\ref{eq:approx1}) and (\\ref{eq:approx2}) in (\\ref{eq:onoff}) yields the online gradient algorithm (\\ref{eq:onlinegrad}). The derivation of the Gauss-Newton algorithm is similar.\n\n\\subsection{Examples of online HMM estimation algorithm}\nWith the algorithms (\\ref{eq:onlinegrad}) and (\\ref{eq:recgaussnewton}) we can obtain several types of online HMM parameter estimators by choosing different loss functions $J$ in (\\ref{eq:lossfunction}). Below we outline two popular choices.\n\n\\subsubsection*{1. Recursive EM algorithm\\footnote{This name is a misnomer. More accurately the algorithm below is a stochastic approximation algorithm that seeks to approximate the EM algorithm}} \\index{online HMM estimation! recursive EM} \nRecall from the EM algorithm, that the auxiliary likelihood for fixed parameter $\\bar{\\th}$ is \n$$Q_n(\\th,\\bar{\\th}) = \\E\\{ \\log (p(x_{0:n},y_{1:n}| \\th) | y_{1:n}, \\bar{\\th})\\} = \\E\\{ \\sum_{k=1}^n \\log p_\\th(x_k,y_k| x_{k-1}) | y_{1:n}, \\bar{\\th} \\} $$\nWith $\\th^o$ denoting the true model, $\\th$ denoting the model variable, and $\\bar{\\model}$ denoting a fixed model value, define\n$$J_n(\\th,\\bar{\\model}) = \\E_{y_{1:n}}\\{ Q_n(\\th, \\bar{\\th}) | \\th^o\\}. $$\nTo be more specific, \nfor a HMM, from (\\ref{eq:emqfn}),\nin the notation of (\\ref{eq:lossfunction}),\n\\begin{equation} c_\\th (y_k,\\bar{\\model}) = \n\\sum_{i=1}^X \\pi_{k|n}^{\\bar{\\th}}(i) \\; \\log B^\\theta_{iy_k}\n+ \\sum_{i=1}^X \\sum_{j=1}^X \\pi^{\\bar{\\model}}_{k|n} (i,j )\n\\log P^\\theta_{ij} . \\label{eq:receminst}\n\\end{equation}\nwhere $P^\\theta$ denotes the transition matrix and $B^\\theta$ is the observation matrix\nand $\\bar{\\model}$ is a fixed model for which the smoothed posterior $\\pi^{\\bar{\\model}}_{k|n}$ is computed.\n\nNote that $c_\\th$ is a reward and not a loss; our aim is to maximize $J_n$.\nThe idea then is to implement a Gauss-Newton stochastic gradient algorithm for maximizing $J_n(\\th,\\bar{\\model})$ for fixed model $\\bar{\\model}$,\nthen update $\\bar{\\model}$ and so on. This yields the following {\\em recursive EM algorithm}:\n\n\n\\begin{compactenum}\n\\item For $ k = n \\Delta +1, \\ldots (n+1) \\Delta$ run\n\\begin{equation} \\label{eq:recem} \n\\begin{split} \\th_{k} &= \\th_{k-1} + \\mathcal{I}_k^{-1} \n\\sum_i \\nabla_\\model c_\\th (y_k, \\bar{\\model}_{n-1}) \\pi_k^{\\bar{\\model}_{n}}(i) \\\\\n \\mathcal{I}_k & = \\mathcal{I}_{k-1} + \\epsilon \\sum_i \\nabla^2 c_\\th(i,y_k,\\pi_k^{\\th})\\big\\vert_{\\th = \\th_{k-1}} \\, \\pi_k^{\\bar{\\model}_{n}}(i) \\\\\n \n \\pi^{\\bar{\\model}_{n-1}}_{k} & = T(\\pi^{\\bar{\\model}_{n-1}}_{k-1}, y_k; \\th_{k-1}) \\quad \\text{ (HMM filter update) } \n\\end{split}\n\\end{equation}\nHere $\\pi_{k|n}^{\\bar{\\th}}$ and $\\pi^{\\bar{\\model}}_{k|n} (i,j )$ in (\\ref{eq:receminst}) are replaced by filtered estimates\n$\\pi_{k}^{\\bar{\\th}}$ and $\\pi^{\\bar{\\model}}_{k-1} (i) P^{\\bar{\\model}}_{ij} B^{\\bar{\\model}}_{j }y_{k}$.\n\\item Then update \n$ \\bar{\\model}_{n+1} = \\theta_{(n+1)\\Delta} $, set $n$ to $n+1$ and go to step 1.\n\\end{compactenum}\n\n \n\n\nTo ensure that the transition matrix estimates are a valid stochastic matrix, one can parametrize it in terms of spherical coordinates,\nsee~(\\ref{eq:alfadef}).\n\nAs an illustrative example, suppose we wish to estimate the $X$-dimension vector of state levels $g = (g(1),g(2),\\ldots,g(X))^\\prime $\n of a HMM in zero mean Gaussian noise \nwith known variance $\\sigma^2$. Assume the transition matrix $P$ is known.\nThen $\\th = g$ and \n$$ c_\\th (y_k,\\bar{\\model}) \n= - \\frac{1}{2 \\sigma^2} \\sum_{i} \\pi_k^{\\bar{\\model}}(i) \\big( y_k - g(i) \\big)^2 + \\text{ constant }$$\n\n\n\n \\subsubsection*{2. Recursive Prediction Error (RPE)} \\index{online HMM estimation! recursive prediction error} \n Suppose $g$ is the vector of state levels of the underlying Markov chain and $P$ the transition matrix. Then the model to estimate\n is $\\th = (g,P)$. \n Offline prediction error methods seek to find the model $\\th$ that minimizes the loss function\n $$ J_N(\\th) = \\E\\{ \\sum_{k=1}^N (y_k - g^\\prime \\pi_{k|k-1}) ^2\\} $$ \n So squared prediction error at each time $k$ is \n\\begin{equation} c_\\th(x_k,\\theta_k,\\pi^\\theta_k) = \\big(y_k - g^\\prime P^\\prime \\pi^\\th_{k-1}\\big)^2 \\label{eq:rpecost} \\end{equation}\nNote that unlike (\\ref{eq:lossfunction}) there is no conditional expectation in the loss function.\n Note the key difference compared to the recursive EM. In the recursive \nEM $c_\\th (x_k,y_k) $ is functionally independent of $\\pi^\\th$ and hence the recursive EM does not involve derivatives (sensitivity) of the HMM filter. In comparison, the RPE cost (\\ref{eq:rpecost}) involves derivatives of $\\pi^\\th_{k-1}$ with respect to $\\th$.\n Then the derivatives with respect to $\\th$ can be evaluated as in \\S \\ref{sec:rmle}.\n \n \n \\subsubsection*{3. Recursive Maximum likelihood} This was discussed in \\S \\ref{sec:rmle}. The cost function is\n$$c_\\th(x_k,\\theta_k,\\pi^\\theta_k) =\\log \\left[ \\mathbf{1}^\\prime B_{y_k}(\\th) \\pi_{k|k-1}^\\th \\right]\n$$\n \\index{online HMM estimation! recursive maximum likelihood} \n\n\nRecursive versions of the method of moment estimation algorithm for the HMM parameters is presented in \\cite{MKW15}.\n\n\\section{Case Study. Reinforcement Learning of Correlated Equilibria} \\index{reinforcement learning of correlated equilibria}\nThis case study illustrates the use of stochastic approximation algorithms for learning the correlated equilibrium in a repeated\ngame. Recall in Chapter \\ref{chp:markovtracksa} we used the ordinary differential equation analysis of a stochastic approximation algorithm to \ncharacterize where it converges to. For a game, we will show that the stochastic approximation algorithm converges to\na differential inclusion (rather than a differential equation). Differential inclusions are generalization of ordinary differential equations (ODEs) and arise naturally in game-theoretic learning, since the strategies according to which others play are unknown. Then by a straightforward\nLyapunov function type proof, we show that the differential inclusion converges to the set of correlated equilibria of the game, implying that the stochastic approximation algorithm also converges to the set of correlated equilibria.\n\n\\subsection{Finite Game Model}\nConsider a finite action static game\\footnote{For notational convenience we assume two players with identical action spaces.} comprising two players $l=1,2$ with\ncosts\n$c_l(\\act1,\\act2)$ where $\\act1, \\act2 \\in \\{1,\\ldots,U\\}$. Let $p$ and $q$ denote the randomized policies (strategies) of the two players:\n$p(i) = \\mathbb{P}( \\act1 = i) $ and $q(i) = \\mathbb{P}(\\act2=i)$. So $p,q$ are $U$ dimensional probability vectors that live in the $U-1$ dimensional unit simplex $\\Pi$. Then the policies\n$(p^*,q^*)$ constitute a Nash equilibrium if the following inequalities hold:\n\\begin{empheq}[box=\\fbox]{equation} \n\\begin{split}\n \\sum_{\\act1,\\act2} c_1(\\act1,\\act2)\\, p^*(\\act1) \\, q^*(\\act2) \\leq \\sum_{\\act2} c_1(u,\\act2) \\, q^*(\\act2) , \\quad u =1,\\ldots,U \\\\\n \\sum_{\\act1,\\act2} c_2(\\act1,\\act2) \\, p^*(\\act1) \\,q^*(\\act2) \\leq \\sum_{\\act1} c_2(\\act1,u) \\, p^*(\\act1) , \\quad u =1,\\ldots,U.\n \\end{split} \\label{eq:staticnashone}\n \\end{empheq}\nEquivalently, $(p^*,q^*)$ constitute a Nash equilibrium if for all policies $p,q \\in \\Pi$,\n\\begin{equation} \n\\begin{split}\n \\sum_{\\act1,\\act2} c_1(\\act1,\\act2) \\, p^*(\\act1) \\, q^*(\\act2) \\leq \\sum_{\\act1,\\act2} c_1(\\act1,\\act2)\\, p(\\act1) \\, q^*(\\act2) \\\\\n \\sum_{\\act1,\\act2} c_2(\\act1,\\act2) \\, p^*(\\act1)\\, q^*(\\act2) \\leq \\sum_{\\act1,\\act2} c_2(\\act1,\\act2) \\, p^*(\\act1) \\, q(\\act2) \n \\end{split} \\label{eq:staticnash}\n \\end{equation}\nThe first inequality in (\\ref{eq:staticnash}) says that if player 1 cheats and deploys policy $p$ instead of $p^*$, then it is worse off\nand incurs an higher cost. The second inequality says that same thing for player 2. So in a non-cooperative game, since collusion is not allowed, there is no rational reason for any of the players to unilaterally deviate from the Nash equilibrium\n$p^*,q^*$.\n\nBy a standard application of Kakutani's fixed point theorem, it can be shown that for a finite action game, at least\none Nash equilibrium always exists. However, computing it can be difficult since the above constraints are bilinear and therefore nonconvex.\n\n\n\n\\subsection{Correlated Equilibrium} The Nash equilibrium assumes that the player's act independently. The correlated equilibrium is a generalization of the Nash equilibrium. The two players now choose their action from the joint probability distribution\n$\\pi(\\act1,\\act2) $ where $$\\pi(i,j) = \\mathbb{P}(\\act1 = i,\\act2=j).$$ \\index{correlated equilibrium}\nHence the actions of the players are correlated. Then the policy $\\pi^*$ is said to be a correlated equilibrium if\n\\begin{empheq}[box=\\fbox]{equation} \n\\begin{split}\n \\sum_{\\act2} c_1(\\act1,\\act2)\\, \\pi^*(\\act1,\\act2) \\leq \\sum_{\\act2} c_1(u,\\act2)\\, \\pi^*(\\act1,\\act2) \\\\\n \\sum_{\\act1} c_2(\\act1,\\act2) \\pi^*(\\act1,\\act2) \\leq \\sum_{\\act1} c_2(\\act1,u) \\,\\pi^*(\\act1,\\act2) \n \\end{split} \\label{eq:staticcoeq}\n \\end{empheq}\n Define the set of correlated equilibria as\n \\begin{equation} \\label{eq:coreqset}\n \\mathcal{C} = \\bigg\\{\\pi: \\text{(\\ref{eq:staticcoeq}) holds and $\\pi(\\act1,\\act2)\\geq 0, \\sum_{\\act1,\\act2}\n \\pi(\\act1,\\act2)=1$ } \\bigg\\} \\end{equation}\n{\\em Remark}: In the special case where the players act independently, the correlated equilibrium specializes to a Nash equilibrium.\nIndependence implies the joint distribution $\\pi^*(\\act1,\\act2)$ becomes the product of marginals:\nso $\\pi^*(\\act1,\\act2)= p^*(\\act1) q^*(\\act2) $. Then clearly (\\ref{eq:staticcoeq}) reduces to the definition (\\ref{eq:staticnashone})\nof a Nash equilibrium. Note that the set of correlated equilibria specified by (\\ref{eq:coreqset}) is a convex polytope in $\\pi$.\n\n\n\\subsubsection{Why Correlated Equilibria?} \\index{game theory! correlated equilibrium}\n\nJohn F. Nash proved in his famous paper~\\cite{Nas51} that every game with a finite set of players and actions has at least one mixed strategy Nash equilibrium.\nHowever, as asserted by Robert J. Aumann\n\\footnote{Robert J. Aumann was awarded the Nobel Memorial Prize in Economics in 2005 for his work on conflict and cooperation through game-theoretic analysis. He is the first to conduct a full-fledged formal analysis of the so-called infinitely repeated games.}\nin the following extract from~\\cite{Aum87}, ``Nash equilibrium does make sense if one starts by assuming that, for some specified reason, each player knows which strategies the other players are using.'' Evidently, this assumption is rather restrictive and, more importantly, is rarely true in any strategic interactive situation. He adds:\n\\begin{quote} {\\em\n``Far from being inconsistent with the Bayesian view of the world, the notion of equilibrium is an unavoidable consequence of that view. It turns out, though, that the appropriate equilibrium notion is not the ordinary mixed strategy equilibrium of Nash (1951), but the more general notion of correlated equilibrium.''} -- Robert J. Aumann\n\\end{quote}\nThis, indeed, is the very reason why correlated equilibrium~\\cite{Aum87} best suits and is central to the analysis of strategic decision-making. \n\n\nThere is much to be said about correlated equilibrium; see Aumann~\\cite{Aum87} for rationality arguments. Some advantages that make it ever more appealing include:\n\\begin{enumerate}\n \\item \\emph{Realistic:} Correlated equilibrium is realistic in multi-agent learning. Indeed, Hart and Mas-Colell observe in~\\cite{HM01b} that for most simple adaptive procedures, ``\\ldots there is a natural coordination device: the common history, observed by all players. It is thus reasonable to expect that, at the end, independence among players will not obtain;''\n\n \\item \\emph{Structural Simplicity:} The correlated equilibria set constitutes a compact convex polyhedron, whereas the Nash equilibria are isolated points at the extrema of this set~\\cite{NCH04}. Indeed from (\\ref{eq:coreqset}), the set of correlated equilibria is a convex polytope in $\\pi$.\n\n \\item \\emph{Computational Simplicity:} Computing correlated equilibrium only requires solving a linear feasibility problem (linear program with null objective function) that can be done in polynomial time, whereas computing Nash equilibrium requires finding fixed points;\n\n \\item \\emph{Payoff Gains:} The coordination among agents in the correlated equilibrium can lead to potentially higher payoffs than if agents take their actions independently (as required by Nash equilibrium)~\\cite{Aum87};\n\n \\item \\emph{Learning:} There is no natural process that is known to converge to a Nash equilibrium in a general non-cooperative game that is not essentially equivalent to exhaustive search. There are, however, natural processes that do converge to correlated equilibria (the so-called law of conservation of coordination~\\cite{HM03}), e.g., \\index{regret-matching procedure} regret-matching~\\cite{HM00}.\n\\end{enumerate}\n\n\n\nExistence of a centralized coordinating device neglects the distributed essence of social networks. Limited information at each agent about the strategies of others further complicates the process of computing correlated equilibria. In fact, even if agents could compute correlated equilibria, they would need a mechanism that facilitates coordinating on the same equilibrium state in the presence of multiple equilibria---each describing, for instance, a stable coordinated behavior of manufacturers on targeting influential nodes in the competitive diffusion process~\\cite{TAM12}. This highlights the significance of adaptive learning algorithms \\index{game theory! adaptive learning} that, through repeated interactive play and simple strategy adjustments by agents, ensure reaching correlated equilibrium. The most well-known of such algorithms, fictitious play, was first introduced in 1951~\\cite{Rob51}, and is extensively treated in~\\cite{FL98}. It, however, requires monitoring the behavior of all other agents that contradicts the information exchange structure in social networks. The focus below is on the more recent regret-matching learning algorithms~\\cite{BHS06,Cah04,HM00,HM01b}. \n\nFigure~\\ref{fig:strategy-sets} illustrates how the various notions of equilibrium are related in terms of the relative size and inclusion in other equilibria sets. As discussed earlier in this subsection, dominant strategies and pure strategy Nash equilibria do not always exist---the game of ``Matching Pennies'' being a simple\nexample. Every finite game, however, has at least one mixed strategy Nash equilibrium. Therefore, the ``nonexistence critique'' does not apply to any notion that generalizes the mixed strategy Nash equilibrium in Figure~\\ref{fig:strategy-sets}. A Hannan consistent strategy (also known as ``universally consistent'' strategies~\\cite{FL95}) is one that ensures, no matter what other players do, the player's average payoff is asymptotically no worse than if she were to play any \\emph{constant} strategy for in all previous periods. Hannan consistent strategies guarantee no asymptotic external regrets and lead to the so-called ``coarse correlated equilibrium''~\\cite{MV78} notion that generalizes the Aumann's correlated equilibrium.\n\n\n\n\\begin{figure}[h]\n\\begin{tikzpicture}[decoration=zigzag]\n \n \\filldraw[fill=gray!10, draw=gray!80] (0,0) ellipse (8.0cm and 4.0cm);\n \\path[postaction={decorate,\n decoration={\n raise=-1em,\n text along path,\n text={|\\tt|Hannan Consistent},\n text align=center,\n },\n }] (0,0) ++(110:8.0cm and 4.0cm) arc(110:70:8.0cm and 4.0cm);\n \\filldraw[fill=gray!10, draw=gray!80] (0,0) ellipse (6.4cm and 3.2cm);\n \\path[postaction={decorate,\n decoration={\n raise=-1em,\n text along path,\n text={|\\tt|Correlated Equilibria},\n text align=center,\n },\n }] (0,0) ++(110:6.4cm and 3.2cm) arc(110:70:6.4cm and 3.2cm);\n \\filldraw[fill=gray!20, draw=gray!80] (0,0) ellipse (4.8cm and 2.4cm);\n \\path[postaction={decorate,\n decoration={\n raise=-1em,\n text along path,\n text={|\\tt|Randomized Nash},\n text align=center,\n },\n }] (0,0) ++(110:4.8cm and 2.4cm) arc(110:70:4.8cm and 2.4cm);\n \\filldraw[fill=gray!30, draw=gray!80] (0,0) ellipse (3.2cm and 1.6cm);\n \\path[postaction={decorate,\n decoration={\n raise=-1em,\n text along path,\n text={|\\tt|Pure Nash},\n text align=center,\n },\n }] (0,0) ++(110:3.2cm and 1.6cm) arc(110:70:3.2cm and 1.6cm);\n \\filldraw[fill=gray!40, draw=gray!80] (0,0) ellipse (1.6cm and 0.8cm) node\n {\\texttt{Dominant Strategies}};\n\\end{tikzpicture}\n\\caption{Equilibrium notions in non-cooperative games. Enlarging the equilibria set weakens the behavioral sophistication on the player's part to distributively reach equilibrium through repeated plays of the game.}\n\\label{fig:strategy-sets}\n\\end{figure}\n\n\\subsection{Reinforcement Learning Algorithm}\nTo describe the learning algorithm and the concept of regret, it is convenient to deal with rewards rather than costs.\nEach agent $l$ has utility reward $r_l(u^{(l)},u^{-l})$ where $u^{(l)}$ denotes the action of agent $l$ and $u^{-l}$ denotes the action of the other agents.\nThe action space for each agent $l$ is $\\{1,2,\\ldots, U\\}$.\nDefine the inertia parameter\n\\begin{equation} \\mu \\geq U \\big( \\max r_l(u, u^{-l}) - \\min r_l(u, u^{-l}) \\big) \\label{eq:inertia}\\end{equation}\n\nEach agent then runs the regret matching Algorithm \\ref{alg:rm}. Algorithm \\ref{alg:rm} assumes that once a decision is made by an agent, it is observable by all other agents. However, agent $l$ does not know the utility function of other agents. Therefore, a learning algorithms\nsuch as Algorithm \\ref{alg:rm} is required to learn the correlated equilibria.\n\nThe assumption that the actions of each agent are known to all other agents can be relaxed; see \\cite{HM01b} for \"blind\" algorithms that do not require this. \\index{regret matching algorithm} \\index{reinforcement learning of correlated equilibria! regret matching algorithm}\n\n\\begin{algorithm}\nEach agent $l$ with utility reward $r_l(u^{(l)},u^{-l})$ independently executes the following:\n\\begin{compactenum}\n \\item \\bf Initialization: \\rm\n Choose action $u_0^{(l)}\\in \\{1,\\ldots,U\\}$ arbitrarily. \n Set $R^l_1=0.$\n \\item\n Repeat for $n=1,2,\\ldots$, the following steps:\\\\\n { \\bf Choose Action:} $u^{(l)}_{n} \\in \\{1,\\ldots,U\\}$ with probability\n \\begin{align}\n \\label{rmtrans} \\mathbb{P}(u^{(l)}_{n}=j| u_{n-1}^{(l)}=i,R^l_n)\n &= \\begin{cases}\n \\frac{ |R_n^l(i,j)|^+} {\\mu } & j\\neq i, \\\\\n 1-\\sum_{m\\neq i}\\frac{ |R_n^l(i,m)|^+}{\\mu} & j=i\n\\end{cases}\n \\end{align}\nwhere inertia parameter $\\mu$ is defined in (\\ref{eq:inertia}) and $|x|^+ \\stackrel{\\text{defn}}{=} \\max\\{x,0\\}.$\n\n{\\bf Regret Update}: \n Update the $U \\times U$ regret matrix $R^l_{n+1}$ as\n \\begin{equation}\n \\label{sa-d}\n \\begin{array}{ll}&\\!\\!\\!\\displaystyle\n R^l_{n+1}(i,j) =R^l_{n}(i,j)+\\epsilon\\left(I\\{u^{(l)}_n=i\\}\\big(r_l(j,u^{-l}_n)-r_l(i,u^{-l}_n)\\big) -R^l_{n}(u,j)\\right).\n \n \\end{array}\n \\end{equation}\n Here $\\varepsilon \\ll1$ denotes a constant positive step size.\n\\end{compactenum}\n\\caption{Regret Matching Algorithm for Learning Correlated Equilibrium} \\label{alg:rm}\n\\end{algorithm}\n\n\n\\subsubsection{Discussion and Intuition of Algorithm \\ref{alg:rm}}\n{\\em 1. Adaptive Behavior:} In~(\\ref{sa-d}), $\\epsilon$ serves as a forgetting factor to foster adaptivity to the evolution of the non-cooperative game parameters. That is, as agents repeatedly take actions, the effect of the old underlying parameters on their current decisions vanishes.\n\n\\noindent {\\em 2. Inertia:} The choice of $\\mu$ guarantees that there is always a positive probability of playing the same action as the last period. Therefore, $\\mu$ can be viewed as an ``inertia'' parameter: A higher $\\mu$\nyields switching with lower probabilities. It plays a significant role in breaking away from bad cycles. It is worth emphasizing that the speed of convergence to the correlated equilibria set is closely related to this inertia parameter.\n\n\\noindent {\\em 3. Better-reply vs. Best-reply:} In light of the above discussion, the most distinctive feature of the regret-matching procedure, that differentiates it from other works such as~\\cite{FL99a}, is that it implements a better-reply rather than a best-reply strategy\\footnote{This has the additional effect of making the behavior\ncontinuous, without need for approximations~\\cite{HM00}.}. This inertia assigns positive probabilities to any actions that are just better. Indeed, the behavior of a regret-matching decision maker is very far from that of a rational decision maker that makes optimal decisions given his (more or less well-formed) beliefs about the environment. Instead, it resembles the model of a reflex-oriented individual that reinforces decisions with ``pleasurable'' consequences~\\cite{HM01b}.\n\nWe also point out the generality of Algorithm \\ref{alg:rm}, by noting that it can be easily transformed into the well-known\n{\\em fictitious play} algorithm by choosing $u^{(l)}_{n+1}=\\arg\\max_k R^l_{n+1}(i,j)$ deterministically, where $u_n^{(l)}=i$,\nand the extremely simple {\\em best response} algorithm by further specifying $\\epsilon=1$.\n\n\n\n\n\\noindent {\\em 4. Computational Cost:} The computational burden (in terms of calculations per iteration) of the regret-matching algorithm does not grow with the number of agents and is hence scalable. At each iteration, each agent needs to execute two multiplications, two additions, one comparison and two table lookups (assuming random numbers are stored in a table) to calculate the next decision. Therefore, it is suitable for implementation in sensors with limited local computational capability.\n\n\\noindent {\\em 5. Global performance metric}\nFinally, we introduce a metric for the global behavior of the system. \nThe global behavior $z_n$ at time $k$ is defined as the empirical frequency of joint play of all agents up to period $k$. Formally,\n\\begin{equation}\n\\label{eq:global-behavior}\nz_n = \\sum_{\\tau\\leq k} (1-\\epsilon)^{k-\\tau} e_{\\mathbf{u}_\\tau}\n\\end{equation}\nwhere $e_{\\mathbf{u}_\\tau}$ denotes the unit vector with the element corresponding to the joint play $\\mathbf{u}_\\tau$ being equal to one. Given $z_n$, the average payoff accrued by each agent can be straightforwardly evaluated, hence the name global behavior. It is more convenient to define $z_n$ via the stochastic approximation recursion\n\\begin{equation}\n\\label{eq:global-behavior-SA}\nz_n = z_{k-1} + \\epsilon \\left[ e_{\\mathbf{u}_n} - z_{k-1}\\right].\n\\end{equation}\n\nThe global behavior $z_n$ is a system ``diagnostic'' and is only used for the analysis of the emergent collective behavior of agents. That is, it does not need to be computed by individual agents.\nIn real-life application such as smart sensor networks, however, a network controller can monitor $z_n$ and use it to adjust agents' payoff functions to achieve the desired global behavior.\n\n\n\\subsection{Ordinary Differential Inclusion Analysis of Algorithm \\ref{alg:rm}} \\index{ordinary differential inclusion analysis}\nRecall from Chapter \\ref{chp:markovtracksa} that the dynamics of a stochastic approximation algorithm can be characterized by an\nordinary differential equation obtained by averaging the equations in the algorithm. In particular,\nusing Theorem \\ref{thm:odeweak} of Chapter \\ref{chp:markovtracksa}, the estimates generated by the stochastic approximation\nalgorithm converge weakly to the averaged system corresponding to (\\ref{sa-d}) and\n(\\ref{eq:global-behavior-SA}), namely,\n\\begin{equation}\n\\begin{split}\n\\frac{d R(i,j)}{dt} & = \\E_{\\pi} \\left\\{ I(u_t= i ) \\big( r_l(j,u^{-l}) - r_l(i,u^{-l}) \\big) - R(i,j) \\right\\} \\\\\n&= \\sum_{u^{-l}} \\biggl[ \\pi(i | u^{-l}) \\,\\bigg( r_l(j,u^{-l}) - r_l(i,u^{-l}) \\bigg) \\biggr] \\pi(u^{-l}) - R(i,j) \\\\\n\\frac{dz}{dt} &= \\pi(i|u^{-l})\\, \\pi(u^{-l}) - z \n\\end{split} \\label{eq:avgame1}\n\\end{equation}\nwhere $\\pi(u^{(l)}, u^{-l}) = \\pi(u^{-l} | u^{(l)}) \\pi( u^{(l)} ) $ is the stationary distribution of the Markov process $(u^{(l)}, u^{-l})$.\n\nNext note that the transition probabilities in (\\ref{rmtrans}) of $u_n^{(l)}$ given $R_n$ are conditionally independent of $u_n^{-l}$. So given\n$R_n$, $\\pi(i | u^{-l}) = \\pi(i)$.\nSo given the transition probabilities in (\\ref{rmtrans}), clearly the stationary distribution $\\pi(u^{(l)})$ satisfies the linear algebraic equation\n$$\n\\pi(i) = \\pi(i) \\biggl[ 1 - \\sum_{j\\neq i} \\frac{ | R(j,i)|^+ }{ \\mu} \\biggr] + \\sum_{j\\neq i} \\pi(j) \\frac{|R(i,j)|^+}{\\mu} .\n$$\nwhich after cancelling out $\\pi(i)$ on both sides yields\n\\begin{equation}\n\\sum_{i\\neq j} \\pi(i) |R(i,j)|^+ = \\sum_{i\\neq j} \\pi(j) |R(j,i)|^+ \\label{eq:avgame2}\n\\end{equation}\nTherefore the stationary distribution $\\pi$ is functionally independent of the inertia parameter~$\\mu$.\n\nFinally note that as far as player $l$ is concerned, the strategy $\\pi(u^{-l})$ is not known. All is known is that $\\pi(u^{-l})$ is a valid pmf.\nSo we can write\nthe averaged dynamics of the regret matching Algorithm \\ref{alg:rm} as\n \\begin{equation} \\displaystyle\n \\begin{split}\n& \\begin{rcases} \n \\dfrac{d R(i,j)}{dt} \\in \\displaystyle \\sum_{u^{-l}} \\biggl[ \\pi(i ) \\,\\bigg( r_l(j,u^{-l}) - r_l(i,u^{-l}) \\bigg) \\biggr] \\pi(u^{-l}) - R(i,j) \n\\\\\n \\dfrac{dz}{dt} \\in \\pi(i)\\, \\pi(u^{-l}) - z \\end{rcases} \\; \\pi(u^{-l}) \\in \\text{ valid pmf} \\\\\n&\\sum_{i\\neq j} \\pi(i) |R(i,j)|^+ = \\sum_{i\\neq j} \\pi(j) |R(j,i)|^+ \\end{split} \\label{eq:odi}\n\\end{equation}\nThe above averaged dynamics \n constitute an algebraically constrained ordinary differential inclusion.\\footnote{Differential inclusions \\index{differential inclusion} are a generalization of the concept of ordinary differential equations. A generic differential inclusion is of the form $dx\/dt \\in \\mathcal{F}(x,t)$, where $\\mathcal{F}(x,t)$ specifies a family of trajectories rather than a single trajectory as in the ordinary differential equations $dx\/dt = F(x,t)$.}\nWe refer the reader to \\cite{BHS05,BHS06} for an excellent exposition of the use of differential inclusions for analyzing game theoretical type learning algorithms.\n\n{\\em Remark}:\nThe asymptotics of a stochastic approximation algorithm is typically captured by an ordinary differential equation (ODE). Here, although agents observe $u^{-l}$, they are oblivious to the strategies $\\pi(u^{-l})$ from which $u^{-l}$ has been drawn. Different strategies\n$\\pi(u^{-l})$ result in \ndifferent trajectories of $R_n$. Therefore, $R_t$ and $z_t$ are specified by a differential inclusions rather than ODEs \\index{ordinary differential equation}.\n\n\n\\subsection{Convergence of Algorithm \\ref{alg:rm} to the set of correlated equilibria}\nThe previous subsection says that the regret matching Algorithm \\ref{alg:rm} behaves asymptotically as an \nalgebraically constrained differential inclusion (\\ref{eq:odi}). So we only need to analyze the behavior of this differential inclusion to characterize the\nbehavior of the regret matching algorithm. \n\n\\begin{theorem}\nSuppose every agent follows the ``regret-matching''Algorithm~\\ref{alg:rm}. Then as $t\\to\\infty$:\n(i) $R(t) $ converges to the negative orthant in the sense that\n\\begin{equation}\n\\dis\\big[ R(t),\\mathbb{R}^-\\big] = \\inf_{\\boldsymbol{r} \\in {\\rm I\\hspace{-.07cm}R}^-}\\big\\| R(t) - \\boldsymbol{r}\\big\\| \\Rightarrow 0;\n\\end{equation}\n\n(ii) $z(t) $ converges to the correlated equilibria set $\\mathcal{C}$ in the sense that\n\\begin{equation}\n\\label{eq:convergence-1}\n\\dis [ z(t),\\mathcal{C} ] = \\inf_{\\boldsymbol{z} \\in \\mathcal{C}}\\left\\| z(t) - \\boldsymbol{z}\\right\\| \\Rightarrow 0.\n\\end{equation}\n\\end{theorem}\nThe proof below shows the simplicity and elegance of the ordinary differential equation (inclusion) approach for analyzing stochastic approximation algorithm. Just a few elementary lines\nbased on the Lyapunov function yields the proof.\n\n\\begin{proof}\n\nDefine the Lyapunov function \\index{Lyapunov function}\n\\begin{equation}\nV\\big(R\\big) = \\frac{1}{2}\\big( \\textmd{dist}\\big[R,{\\rm I\\hspace{-.07cm}R}^-\\big] \\big)^2= \\frac{1}{2}\\sum_{i,j} \\big(\\big| R(i,j)\\big|^+\\big)^2.\n\\end{equation}\nEvaluating the time-derivative and substituting for $dR(i,j) \/ dt$ from (\\ref{eq:odi}) we obtain\n\\begin{align}\n{d\\over dt}V\\big(R\\big) &= \\sum_{i,j} \\big| R(i,j)\\big|^+ \\cdot {d\\over dt} R(i,j) \\nonumber\\\\\n& = \\sum_{i,j} \\big| r(i,j)\\big|^+ \\Big[ ( r_l\\big(j,u^{-l}\\big) - r_l\\big(i,u^{-l}\\big)) \\pi(i) - R(i,j) \\Big]\\nonumber\\\\\n& = \\underbrace{\\sum_{i,j} \\big| R(i,j)\\big|^+ \\big( r_l \\big(j,u^{-l}\\big) - r_l\\big(i,u^{-l}\\big)\\big) \\pi(i)}_{= 0\\;\\text{from (\\ref{eq:avgame2}) }} - \\sum_{i,j} \\big| R(i,j)\\big|^+ R(i,j)\\nonumber\\\\\n& = -2 V\\big(R\\big).\n\\end{align}\nIn the last equality we used\n\\begin{equation}\n\\sum_{i,j} \\big| R(i,j)\\big|^+ R(i,j) = \\sum_{i,j} \\big( \\big| R(i,j)\\big|^+ \\big)^2 = 2 V\\big(R\\big).\n\\end{equation}\nThis completes the proof of the first assertion, namely that Algorithm \\ref{alg:rm} eventually generates regrets that are non-positive.\n\nTo prove the second assertion, from Algorithm \\ref{alg:rm}, the elements of the regret matrix are\n\\begin{align}\nR_k(i,j) &= \\epsilon \\sum_{\\tau \\leq k} (1-\\epsilon)^{k-\\tau} \\bigg[ r_l \\big(j,u_\\tau^{-l}\\big) - r_l \\big(u^{(l)}_\\tau, u^{-l}_\\tau)\\bigg]\nI(u_\\tau^{(l)} = i) \\nonumber\\\\\n& = \\sum_{u^{-l}} z (i,u^{-l}) \\big[ r_l(j,u^{-l}) - r_l(i, u^{-l})\\big] \n\\end{align}\nwhere $z(i,u^{-l})$ denotes the empirical distribution of agent $l$ choosing action $i$ and the rest playing $u^{-l}$. \nOn any convergent subsequence $\\lbrace z_{\\underline{k}}\\rbrace_{\\underline{k}\\geq 0}\\rightarrow {\\pi}$, then\n\\begin{equation}\n\\label{eq:thrm-5-3}\n\\lim_{k\\to\\infty} R_k(i,j) = \\sum_{u^{-l}} \\pi(i, u^{-l}) \\big[ r_l (j,u^{-l}) - r_l(i,u^{-l})\\big]\n\\end{equation}\nwhere $\\pi(i,u^{-l})$ denotes the probability of agent $l$ choosing action $i$ and the rest playing $u^{-l}$.\nThe first assertion of the theorem proved that the regrets converge to non-positive values (negative orthant). Therefore (\\ref {eq:thrm-5-3})\nyields that \n$$ \\sum_{u^{-l}} \\pi(i, u^{-l}) \\big[ r_l (j,u^{-l}) - r_l(i,u^{-l})\\big] \\leq 0 $$\nimplying that $\\pi$ is a correlated equilibrium.\n\\end{proof}\n\n\\subsection{Extension to switched Markov games} \\index{reinforcement learning of correlated equilibria! switched Markov game}\nConsider the case now where rewards $r_l(u^{(l)},u^{-l})$ evolve according to an unknown Markov chain $\\th_n$. Such a time varying\ngame can result from utilities in a social network evolving with time or the number of players changing with time. \nThe reward for agent $l$ is now $r_l(u^{(l)},u^{-l}, \\th_n)$. The aim is to track the set of correlated equilibria $\\mathcal{C}(\\th_n)$; that is\nuse the regret matching algorithm \\ref{alg:rm} so that agents eventually deploy strategies from $\\mathcal{C}(\\th_n)$.\nIf $\\th_n$ evolves with transition matrix $I + \\epsilon^2 Q$ (where $Q$ is a generator), then it is on a slower time scale than the dynamics of the regret\nmatching Algorithm \\ref{alg:rm}. Then a more general proof in the spirit of Theorem \\ref{ode-lim} yields that the regret matching algorithm can track the time\nvarying correlated equilibrium set $\\mathcal{C}(\\th_n)$. Moreover, in analogy to \\S \\ref {sec:fastmc}, if the transition matrix for $\\th_n$ is $I + \\epsilon Q$, then the asymptotic dynamics\nare given by a switched Markov differential inclusion, see \\cite{KMY08,NKY13}.\n\n\n\\section{Stochastic Search-Ruler Algorithm} \\index{discrete stochastic optimization! stochastic ruler}\n\n\nWe discuss two simple variants of Algorithm \\ref{alg:RS} that\nrequire less restrictive conditions for convergence than condition (O). Assume $c_n(\\th)$ are uniformly bounded for $\\th \\in \\Theta$.\nNeither of the algorithms given below are particularly novel; but they are useful from a pedagogical point of view.\n\n It is convenient to normalize the objective (\\ref{eq:discobj}) as follows:\n Let $\\alpha \\leq c_n(\\th) \\leq \\beta$ where $\\alpha$ denotes a finite lower\nbound and $\\beta>0$ denotes a finite upper bound. Define the\nnormalized costs $m_n(\\th)$ as\n\\begin{equation} \\label{eq:mn}\nm_n(\\th) = \\frac{c_n(\\th) -\\alpha}{\\beta-\\alpha} , \\quad\n\\text{ where } 0 \\leq m_n(\\th) \\leq 1. \n\\end{equation}\nThen the stochastic optimization problem (\\ref{eq:discobj}) is equivalent to \n\\begin{equation}\n\\th^* = \n\\arg \\min_{\\th \\in \\Theta}m(\\th) \\text{ where } m(\\th) = \\E\\{m_n(\\th)\\} \n\\label{eq:scale}\n\\end{equation}\nsince scaling the cost function does not affect the minimizing solution.\nRecall $\\Theta = \\{1,2,\\ldots,S\\}$.\n\n\nDefine the loss function\n\\begin{equation} Y_n(\\th,u_n) = I\\left(m_n(\\th) - u_n\\right)\n\\text{ where } I(x) = \\begin{cases} 1 & \\text{ if } x> 0 \\\\\n 0 & \\text{ otherwise } \\end{cases} \\label{eq:yn}\\end{equation}\nHere\n$u_n $ is a independent uniform random number in $[0,1]$. The uniform random number $u_n$ is a stochastic ruler against which the candidate\n$m_n(\\th)$ is measured.\nThe result was originally used\nin devising stochastic ruler optimization algorithms \\cite{AA01} -- although here we \npropose a more efficient algorithm than the stochastic ruler.\nApplying Algorithm \\ref{alg:RS} to the cost function $ \\E\\{Y_n(\\th,u_n)\\}$ defined \nin (\\ref{eq:yn}) yields the following stochastic search-ruler algorithm:\n\n\\begin{algorithm}\n\\caption{Stochastic Search-Ruler}\n\\label{alg:SR}\nIdentical to Algorithm \\ref{alg:RS} with $c_n(\\th_n)$ and\n$c_n(\\tilde{\\theta}_n)$ replaced by $Y_n(\\th_n,u_n)$ and\n$Y_n(\\tilde{\\th}_n,\\tilde{u}_n)$. Here $u_n$ and $\\tilde{u}_n$ are \nindependent uniform random numbers in $[0,1]$.\n\\end{algorithm}\n\nAnalogous to Theorem \\ref{thm:RS} we have the following result:\n\n\\begin{theorem} \\label{thm:SR} Consider the discrete stochastic optimization problem (\\ref{eq:discobj}).\nThen the Markov chain $\\{\\th_n\\}$ generated by Algorithm \\ref{alg:SR} has \nthe following property for its stationary distribution $\\pi_\\infty$:\n\\begin{equation} \\frac{\\pi_\\infty(\\th^*)}{\\pi_\\infty(\\th)} =\n \\frac{m(\\th)}{m(\\th^*)}\n\\frac{ (1 - m(\\th^*))}{ (1-m(\\th)) } > 1 . \\label{eq:ratio}\\end{equation}\n\\end{theorem}\n\n The theorem says that Algorithm \\ref{alg:SR} is attracted to set the global minimizers $\\mathcal{G}$. It spends more time in $\\mathcal{G}$\nthan any other candidates. The restrictive condition (O) is not required for Algorithm \\ref{alg:SR} to be attracted to $\\mathcal{G}$.\n Theorem \\ref{thm:SR}\ngives an explicit representation of the discriminative power of the algorithm\nbetween the optimizer $\\th^*$ and any other candidate $\\th$ in terms of the normalized\nexpected costs $m(\\th)$ and $m(\\th^*)$.\n Algorithm \\ref{alg:SR} is more efficient than the stochastic ruler algorithm of \\cite{And99} when the candidate samples are chosen with equal probability.\nThe stochastic ruler algorithm of \\cite{And99} has asymptotic efficiency\n$\\pi(\\th^*)\/\\pi(\\th) = (1 - m(\\th^*))\/ (1-m(\\th)) $.\nSo Algorithm \\ref{alg:SR} has the additional improvement in efficiency due to the \nadditional multiplicative term $m(\\th)\/m(\\th^*)$ in (\\ref{eq:ratio}).\n\n\n\\noindent {\\em Variance reduction using common random numbers}:\nA more efficient implementation of Algorithm \\ref{alg:SR} can be obtained\nby using variance reduction based on common random numbers (discussed in Appendix \\ref{sec:crn} of the book) as follows:\nSince $u_n$ is uniformly distributed in $[0,1]$, so is $1-u_n$.\nSimilar to Theorem \\ref{thm:SR}\nit can be shown that \nthe optimizer\n $\\th^*$ is the minimizing solution of\nthe following stochastic optimization problem\n$\n\\th^* = \\arg \\min_\\th \\E\\{Z_n(\\th,u_n)\\} $\nwhere \n\\begin{equation}\nZ_n(\\th,u_n) = \\frac{1}{2} \\left[ Y_n(\\th,u_n) + Y_n(\\th,1-u_n)\\right]\n\\label{eq:zn} \\end{equation}\nwhere the normalized sample cost $m_n(\\th)$ is defined in (\\ref{eq:scale}).\nApplying Algorithm \\ref{alg:SR} with $ Z_n(\\th_n,u_n)$ and $Z_n(\\tilde{\\th}_n,u_n)$ replacing $ Y_n(\\th_n,u_n)$ and $Y_n(\\tilde{\\th}_n,u_n)$,\nrespectively,\nyields the variance reduced search-ruler algorithm.\n\nIn particular, since the indicator function $I(\\cdot)$ in (\\ref{eq:yn}) is a monotone function of its argument, it follows that \n$\\operatorname{Var}\\{Z_n(\\th,u_n)\\} \\leq \\operatorname{Var}\\{Y_n(\\th,u_n)\\} $.\n As a result one would expect that the stochastic\noptimization algorithm using $Z_n$ would converge faster.\n\n\\begin{proof}\nWe first show that \n $\\th^*$ defined in (\\ref{eq:scale}) is the minimizing solution of\nthe stochastic optimization problem\n$\n\\th^* = \\arg \\min_\\th \\E\\{Y_n(\\th,u_n)\\}$.\n Using the smoothing property of conditional expectations (\\ref{eq:lieeng})\n yields \\index{smoothing property of conditional expectation}\n\\begin{align*}\n\\E\\{ I\\left(m_n(\\th) - u_n\\right) \\} &= \n\\E\\{\\E\\{I\\left(m_n(\\th) - u_n\\right)| m_n(\\th) \\} \\} \\\\\n&\\hspace{-1.3cm}= \\E\\{ \\mathbb{P}(u_n < m_n(\\th))\\} = \\E\\{m_n(\\th)\\} = m(\\th)\\end{align*}\n The second equality follows since expectation of an indicator function is probability,\n the third equality holds because $u_n$ is a uniform random number in [0,1] so\n that $\\mathbb{P}(u_n u_n)\n\\end{align*}\nFinally, for this transition matrix, it is easily verified that \n\\begin{equation} \\pi_\\infty(\\th) = \\kappa (1 - m(\\th)) \\prod_{j\\neq \\th} m(j) \\end{equation}\nis the invariant distribution\n where $\\kappa$ denotes a normalization constant.\n Hence \n$$\\frac{\\pi_\\infty(\\th^*)}{ \\pi_\\infty(\\th)} = \\frac{m(\\th)}{m(\\th^*)}\\frac{ (1 - m(\\th^*))}{ (1-m(\\th)) }\n= \\frac{ 1\/m(\\th^*)-1}{1\/m(\\th) - 1} > 1$$\nsince $m(\\th^*)$ is the global minimum\nand therefore $ m(\\th^*) < m(\\th)$ for $\\th \\in \\Theta - \\mathcal{G}$.\n\\end{proof}\n\n\n\\begin{comment}\n\\section{Problems}\n\\begin{compactenum}\n\\item Classical RLS vs LMS\n\\item Convergence of UCB algorithm and comparison with Thompson's sampling.\n\n\n\\item Polyak averaging for stochastic approximations\n\n\\item A detection aided test to adapt step size (kim \\& felisa)\n\n\\item Consensus diffusion algorithms for discrete stochastic optimization\n\n\\item Design of optimal observer trajectory\n\n\n\\end{compactenum} \\end{comment}\n\n\\newpage\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFeature selection has been widely investigated and used by the machine learning and data mining community. In this context, a feature, also called attribute or variable, represents a property of a process or system than has been measured, or constructed from the original input variables. The goal of feature selection is to select the smallest feature subset given a certain generalization error, or alternatively finding the best feature subset with $k$ features, that yields the minimum generalization error. Additional objectives of feature selection are: (i) improve the generalization performance with respect to the model built using the whole set of features, (ii) provide a more robust generalization and a faster response with unseen data, and (iii) achieve a better and simpler understanding of the process that generates the data \\cite{Kohavi97,Guyon03}. We will assume that the feature selection method is used either as a preprocessing step or in conjunction with a learning machine for classification or regression purposes.\nFeature selection methods are usually classified in three main groups: wrapper, embedded and filter methods \\cite{Guyon03}. Wrappers \\cite{Kohavi97} use the induction learning algorithm as part of the function evaluating feature subsets. The performance is usually measured in terms of the classification rate obtained on a testing set, i.e., the classifier is used as a black box for assessing feature subsets. Although these techniques may achieve a good generalization, the computational cost of training the classifier a combinatorial number of times becomes prohibitive for high dimensional datasets. In addition, many classifiers are prone to over-learning and show sensitiveness to initialization. Embedded methods \\cite{Lal06}, incorporate knowledge about the specific structure of the class of functions used by a certain learning machine, e.g. bounds on the leave-one-out error of SVMs \\cite{Weston00}. Although usually less computationally expensive than wrappers, embedded methods still are much slower than filter approaches, and the features selected are dependent on the learning machine. Filter methods \\cite{Duch03} assume complete independence between the learning machine and the data, and therefore use a metric independent of the induction learning algorithm to assess feature subsets. Filter methods are relatively robust against overfitting, but may fail to select the best feature subset for classification or regression.\nIn the literature, several criteria have been proposed to evaluate single features or feature subsets, among them: inconsistency rate \\cite{Huang2003Inconsistency}, inference correlation \\cite{Mo2011Inference}, classification error \\cite{Estevez1998Clas}, fractal dimension \\cite{Mo2012Fractal}, distance measure \\cite{Sebban2002Dist,Bins2001Dist}, etc. Mutual information (MI) is a measure of statistical independence, that has two main properties. First, it can measure any kind of relation between random variables, including nonlinear relationships \\cite{Cover06}. Second, MI is invariant under transformations in the feature space that are invertible and differentiable, e.g. translations, rotations and any transformation preserving the order of the original elements of the feature vectors \\cite{Kullback97,Kullback51}. Many advances in the field have been reported in the last 20 years since the pioneer work of Battiti \\cite{Battiti94}. Battiti defined the problem of feature selection as the process of selecting the $k$ most relevant variables from an original feature set of $m$ variables, $k0$. In this case, $x_{2}$ and $x_{3}$ interact positively to predict $C$, and this yields a positive value of the multi-information among these variables. The multi-information among the variables $x_{1}$, $x_{4}$ and $C$ is given by: $I(x_{1};x_{4};C)=I(\\{x_{1},x_{4}\\};C)-I(x_{1};C)-I(x_{4};C)$. The relevance of individual features $x_{1}$ and $x_{4}$ is the same, i.e., $I(x_{1};C)=I(x_{4};C)>0$. In this case the joint information provided by $x_{1}$ and $x_{4}$ with respect to $C$ is the same as that of each variable acting separately, i.e., $I(\\{x_{1},x_{4}\\};C)=I(x_{1};C)=I(x_{4};C)$. This yields a negative value of the multi-information among these variables. We can deduce that the interaction between $x_{1}$ and $x_{4}$ does not provide any new information about $C$. Let us consider now the multi-information among $x_{1}$, $x_{2}$ and $C$, which is zero: $I(x_{1};x_{2};C)=I(\\{x_{1},x_{2}\\};C)-I(x_{1};C)-I(x_{2};C)=0$. Since feature $x_{2}$ only provides information about $C$ when interacting with $x_{3}$, then $I(\\{x_{1},x_{2}\\};C)=I(x_{1};C)$. In this case, features $x_{1}$ and $x_{2}$ do not interact in the knowledge of $C$.\n\nFrom the viewpoint of feature selection, the value of the multi-information (positive, negative or zero) gives rich information about the kind of interaction there is among the variables. Let us consider the case where we have a set of already selected features $S$ and a candidate feature $f_{i}$, and we measure the multi-information of these variables with the class variable $C$, $I(f_{i};S;C)=I(S;C|f_{i})-I(S;C)$. When the multi-information is positive, it means that feature $f_i$ and $S$ are complementary. On the other hand, when the multi-information is negative, it means that by adding $f_{i}$ we are diminishing the dependence between $S$ and $C$, because $f_{i}$ and $S$ are redundant. Finally, when the multi-information is zero, it means that $f_{i}$ is irrelevant with respect to the dependency between $S$ and $C$.\n\nThe mutual information between a set of $m$ features and the class variable $C$ can be expressed compactly in terms of multi-information as follows:\n\\begin{equation}\nI(\\left\\{ x_{1},x_{2},...,x_{m}\\right\\} ;C)=\\sum_{k=1}^{m}\\sum_{\\begin{array}{c}\\forall S\\subseteq\\{x_{1},...,x_{m}\\}\\\\ |S|=k\\end{array}}I([S\\cup C]),\\label{eq:Multiple}\n\\end{equation}\nwhere $I([S\\cup C])=I(s_{1};s_{2};\\cdots;s_{k};C).$ Note that the sum on the right side of eq. (\\ref{eq:Multiple}), is taken over all subsets $S$ of size $k$ drawn from the set $\\{x_{1},...,x_{m}\\}$.\n\n\n\\section{Relevance, Redundancy and Complementarity}\n\nThe filter approach to feature selection is based on the idea of relevance, which we will explore in more detail in this section. Basically the problem is to find the feature subset of minimum cardinality that preserves the information contained in the whole set of features with respect to $C$. This problem is usually solved by finding the relevant features and discarding redundant and irrelevant features. In this section, we review the different definitions of relevance, redundancy and complementarity found in the literature.\n\n\\subsection{Relevance}\n\nIntuitively, a given feature is relevant when either individually or together with other variables, it provides information about $C$. In the literature there are many definitions of relevance, including different levels of relevance \\cite{Bell00,Battiti94,Guyon03,Kohavi97,Yu04,Peng05,Almuallim92,Almuallim91,Brown2012,Davies94}. Kohavi and John \\cite{Kohavi97} used a probabilistic framework to define three levels of relevance: strongly relevant, weakly relevant, and irrelevant features, as shown in Table \\ref{tab:relevance}. Strongly relevant features provide unique information about $C$, i.e., they cannot be replaced by other features. Weakly relevant features provide information about $C$, but they can be replaced by other features without losing information about $C$. Irrelevant features do not provide information about $C$, and they can be discarded without losing information. A drawback of the probabilistic approach is the need of testing the conditional independence for all possible feature subsets, and estimating the probability density functions (pdfs) \\cite{Raudy1991}.\n\nAn alternative definition of relevance is given under the framework of mutual information \\cite{Somol04,Bell00,Kojadinovic05,Koller96,Yu04,Kwak02b,Fleuret04,Tishby99}. An advantage of this approach is that there are several good methods for estimating MI. The last column of Table \\ref{tab:relevance} shows how the three levels of individual relevance are defined in terms of MI.\n\\begin{table}[H]\n\\caption{\\label{tab:relevance}Levels of relevance for candidate feature $f_{i}$, according to probabilistic framework \\cite{Kohavi97} and mutual information framework \\cite{Meyer08}}\n\\centering{}\\begin{tabular}{>{\\centering}m{2cm}|c|>{\\centering}m{3.7cm}|>{\\centering}m{2.8cm}}\n\\hline Relevance Level & Condition & Probabilistic Approach & Mutual Information Approach\\tabularnewline \\hline \\hline Strongly\\\\\n Relevant & $\\nexists$ & $p(C|f_{i},\\neg f_{i})\\neq p(C|\\neg f_{i})$ & $I(f_{i};C|\\neg\n f_{i})>0$\\tabularnewline\n\\hline Weakly\\\\\n Relevant & $\\exists\\, S\\subset\\neg f_{i}$ & $p(C|f_{i},\\neg f_{i})=p(C|\\neg f_{i})$\\\\\n {$\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\wedge\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,$}\n$p(C|f_{i},S)\\neq p(C|S)$ & $I(f_{i};C|\\neg f_{i})=0$\\\\\n {$\\,\\,\\,\\,\\,\\,\\,\\,\\wedge\\,\\,\\,\\,\\,\\,\\,\\,$\n$I(f_{i};C|S)>0$}\\tabularnewline \\hline Irrelevant & $\\forall\\, S\\subseteq\\neg f_{i}$ &\n$p(C|f_{i},S)=p(C|S)$ & $I(f_{i};C|S)=0$\\tabularnewline \\end{tabular}\n\\end{table}\n\nThe definitions shown in Table \\ref{tab:relevance} give rise to several drawbacks, which are summarized as follows:\n\\begin{enumerate}\n\\item To classify a given feature $f_{i}$, as irrelevant, it is necessary to assess all possible subsets $S$ of $\\neg f_{i}$. Therefore this procedure is subject to the curse of dimensionality \\cite{Bellman61,Trunk1979}.\n\\item The definition of strongly relevant features is too restrictive. If two features provides information about the class but are redundant, then both features will be discarded by this criterion. For example, let $\\{x_{1},x_{2},x_{3}\\}$ be a set of 3 variables, where $x_{1}=x_{2}$, and $x_{3}$ is noise, and the output class is defined as $C=x_{1}$. Following the strong relevance criterion we have $I(x_{1};C|\\{x_{2},x_{3}\\})=$$I(x_{2};C|\\{x_{1},x_{3}\\})=$$I(x_{3};C|\\{x_{1},x_{2}\\})=0$.\n\\item The definition of weak relevance is not enough for deciding whether to discard a feature from the optimal feature set. It is necessary to discriminate between redundant and non-redundant features.\n\\end{enumerate}\n\n\\subsection{Redundancy}\n\nYu and Liu \\cite{Yu04} proposed a finer classification of features into weakly relevant but redundant and weakly relevant but non-redundant. Moreover, the authors defined the set of optimal features as the one composed by strongly relevant features and weakly relevant but non-redundant features. The concept of redundancy is associated with the level of dependency among two or more features. In principle we can measure the dependency of a given feature $f_{i}$ with respect to a feature subset $S\\subseteq\\text{\\ensuremath{\\neg}}f_{i}$, by simply using the MI, $I(f_{i};S)$.\nThis information theoretic measure of redundancy satisfies the following properties: it is symmetric, non-linear, non-negative, and does not diminish when adding new features \\cite{Meyer08}.\nHowever, using this measure it is not possible to determine concretely with which features of $S$ is $f_{i}$ redundant. This calls for more elaborated criteria of redundancy, such as the Markov blanket \\cite{Koller96,Yu04}, and total correlation \\cite{Watanabe1960}. The Markov blanket is a strong condition for conditional independence, and is defined as follows.\n\\begin{definition}[Markov blanket]\nGiven a feature $f_{i}$, the subset $M\\subseteq\\neg f_{i}$ is a Markov blanket\nof $f_{i}$ iff \\cite{Koller96,Yu04}:\n\\begin{equation}\np(\\{F\\backslash\\{f_{i}\\,,M\\},C\\}\\,\\text{|\\,}\\{f_{i}\\,,M\\}) = p(\\{F\\backslash\\{f_{i}\\,,M\\},C\\}\\,\\text{|\\,}M).\\label{eq:MB_prob}\n\\end{equation}\n\\end{definition}\n\nThis condition requires that M subsumes all the information that $f_{i}$ has about $C$, but also about all other features $\\{F\\backslash\\{f_{i}\\,,M\\}\\}$. It can be proved that strongly relevant features do not have a Markov blanket \\cite{Yu04}.\n\nThe Markov blanket condition given by Eq. (\\ref{eq:MB_prob}) can be rewritten in the context of information theory as follows \\cite{Meyer08}:\n\\begin{equation}\nI(f_{i};\\{C,\\neg {f_{i},M}\\}\\text{|\\,}M)=0.\\label{MBMI}\n\\end{equation}\n\nAn alternative measure of redundancy is the total correlation or multivariate correlation \\cite{Watanabe1960}. Given a set of features $F=\\{f_{1},...,f_{m}\\}$, the total correlation is defined as follows:\n\\begin{equation}\nC(f_{1};...;f_{m})=\\sum_{i=1}^{m}H(f_{i})-H(f_{1},...,f_{m}).\\label{eq:CORRT1}\n\\end{equation}\n\nTotal correlation measures the common information (redundancy) among all the variables in $F$. If we want to measure the redundancy between a given variable $f_{i}$ and any feature subset $S\\subseteq\\neg f_{i}$, then we can use the total correlation as:\n\\begin{equation}\nC(f_{i};S)=H(f_{i})+H(S)-H(f_{i},S),\n\\end{equation}\nhowever this corresponds to the classic definition of MI, i.e., $C(f_{i};S)=I(f_{i};S)$.\n\n\n\\subsection{Complementarity}\n\nThe concept of complementarity has been re-discovered several times \\cite{Meyer08,Bonev08,Brown2012,Vidal2003,Cheng2011}. Recently, it has become more relevant because of the development of more efficient techniques to estimate MI in high-dimensional spaces \\cite{Kraskov2004,Hero99}. Complementarity, also known as synergy, measures the degree of interaction between an individual feature $f_{i}$ and feature subset $S$ given $C$, through the following expression $(I(f_{i};S|C))$. To illustrate the concept of complementarity, we will start expanding the multi-information among $f_{i}$, $C$ and $S$. Decomposing the multi-information in its three possible expressions we have:\n\\begin{equation}\nI(f_{i};S;C)=\n\\begin{cases}\nI(f_{i};S|C)-I(f_{i};S)\\\\\nI(f_{i};C|S)-I(f_{i};C)\\\\\nI(S;C|f_{i})-I(S;C).\n\\end{cases}\\label{eq:INTERATC1}\n\\end{equation}\n\n\nAccording to eq. (\\ref{eq:INTERATC1}), the first row shows that the multi-information can be expressed as the difference between complementarity $(I(f_{i};S|C))$ and redundancy $(I(f_{i};S))$. A positive value of the multi-information entails a dominance of complementarity over redundancy. Analyzing the second row of eq. (\\ref{eq:INTERATC1}), we observe that this expression becomes positive when the information that $f_{i}$ has about $C$ is greater when it interacts with subset $S$ with respect to the case when it does not. This effect is called complementarity. The third row of eq. (\\ref{eq:INTERATC1}), gives us another viewpoint of the complementarity effect. The multi-information is positive when the information that $S$ has about $C$ is greater when it interacts with feature $f_{i}$ compared to the case when it does not interact. Assuming that the complementarity effect is dominant over redundancy, Fig. \\ref{fig:ResRel} illustrates a Venn diagram with the relationships among complementarity, redundancy and relevancy.\n\\begin{center}\n\\begin{figure}[h]\n\\centering{}\\includegraphics[scale=0.4]{fig2.eps}\n\\caption{\\label{fig:ResRel} Venn diagram showing the relationships among complementarity, redundancy and relevancy, assuming that the multi-information among $f_{i}$, $S$ and $C$ is positive.}\n\\end{figure}\n\\par\\end{center}\n\n\n\\section{Optimal Feature Subset}\n\nIn this section we review the different definitions of the optimal feature subset, $S_{opt}$, given in the literature, as well as the search strategies used for obtaining this optimal set. According to \\cite{Tsamardinos03c}, in practice the feature selection problem must include a classifier or an ensemble of classifiers, and a performance metric. The optimal feature subset is defined as the one that maximizes the performance metric having minimum cardinality. However, filter methods are independent of both the learning machine and the performance metric. Any filter method corresponds to a definition of relevance that employs only the data distribution \\cite{Tsamardinos03c}. Yu and Liu \\cite{Yu04} defined the optimal feature set as composed of all strongly relevant features and the weakly relevant but not redundant features. In this section we review the definitions of the optimal feature subset from the viewpoint of filter methods, in particular MI feature selection methods. The key notion is conditional independence, which allows defining the sufficient feature subset as follows \\cite{Bell00,GuyonFE}:\n\\begin{definition}\n$S\\subseteq F$ is a sufficient feature subset iff\n\\begin{equation}\np(C|F)=p(C|S).\\label{eq:SFS}\n\\end{equation}\n\\end{definition}\n\nThis definition implies that $C$ and $\\neg S$ are conditionally independent, i.e., $\\neg S$ provides no additional information about $C$ in the context of $S$. However, we still need a search strategy to select the feature subset $S$, and an exhaustive search using this criterion is impractical due to the curse of dimensionality.\n\nIn probability the measure of sufficient feature subset can be expressed as the expected value over $p(F)$ of the Kullback-Leibler divergence between $p(C|F)$ and $p(C|S)$ \\cite{Koller96}. According to Guyon \\textit{et al.} \\cite{GuyonFE}, this can be expressed in terms of MI as follows:\n\\begin{equation}\nDMI(S)=I(F;C)-I(S;C).\\label{eq:SFS}\n\\end{equation}\n\nGuyon \\textit{et al.} \\cite{GuyonFE} proposed solving the following optimization problem:\n\\begin{equation}\n\\min_{S\\subseteq F}|S|+\\lambda\\cdot DMI(S),\\label{eq:OSF}\n\\end{equation}\nwhere $\\lambda>0$ represents the Lagrange multiplier. If $S$ is a sufficient feature subset, then $DMI(S)=0$, and eq. (\\ref{eq:OSF}) is reduced to $\\min_{S\\subseteq F}|S|$. Since $I(F;C)$ is constant, eq. (\\ref{eq:OSF}) is equivalent to:\n\\begin{equation}\n\\min_{S\\subseteq F}|S|-\\lambda\\cdot I(S;C).\\label{eq:OFS_MI}\n\\end{equation}\n\nThe feature selection problem corresponds to finding the smallest feature subset that maximizes $I(S;C)$. Since the term $\\min_{S\\subseteq F}|S|$ is discrete, the optimization of (\\ref{eq:OFS_MI}) is difficult. Tishby \\textit{et al.} \\cite{Tishby99} proposed replacing the term $\\min_{S\\subseteq F}|S|$ with $I(F;S)$.\n\nAn alternative approach to optimal feature subset selection is using the concept of the Markov blanket (MB). Remember that the Markov blanket, $M$, of a target variable $C$, is the smallest subset of $F$ such that $C$ is independent of the rest of the variables $F\\backslash M$. Koller and Sahami \\cite{Koller96} proposed using MBs as the basis for feature elimination. They proved that features eliminated sequentially based on this criterion remain unnecessary. However, the time needed for inducing an MB grows exponentially with the size of this set, when considering full dependencies. Therefore most MB algorithms implement approximations based on heuristics, e.g. finding the set of $k$ features that are strongly correlated with a given feature \\cite{Koller96}. Fast MB discovery algorithms have been developed for the case of distributions that are faithful to a Bayesian Network \\cite{Tsamardinos03c,Tsamardinos03b}. However, these algorithms require that the optimal feature subset does not contain multivariate associations among variables, which are individually irrelevant but become relevant in the context of others \\cite{BrownyTsamardinos}. In practice, this means for example that current MB discovery algorithms cannot solve Example 1 due to the XOR function.\n\nAn important caveat is that both feature selection approaches, sufficient feature subset and MBs, are based on estimating the probability distribution of $C$ given the data. Estimating posterior probabilities is a harder problem than classification, e.g. in using a $0\\backslash 1$-loss function only the most probable classification is needed. Therefore, this effect may render some features contained in sufficient feature subset or in the MB of $C$ unnecessary \\cite{Tsamardinos03c,Torkkola,GuyonFE}.\n\n\\subsection{Relation between MI and Bayes error classification}\n\nThere are some interesting results relating the MI between a random discrete variable $f$ and a random discrete target variable $C$, with the minimum error obtained by maximum a posteriori classifier (Bayer classification error) \\cite{Hellman70,Cover06,Feder94}. The Bayes error is bounded above and below according to the following expression:\n\\begin{equation}\n1-\\frac{I(f;C)+\\log(2)}{\\log(|C|)}\\leq e_{bayes}(f)\\leqslant\\frac{1}{2}\\left(H(C)-I(f;C)\\right).\\label{eq:IM_lim_inf}\n\\end{equation}\nInterestingly, Eq. (\\ref{eq:IM_lim_inf}) shows that both limits are minimized when the MI, $I(f;C)$, is maximized.\n\n\n\\subsection{Search strategies}\n\\label{search}\nAccording to Guyon \\textit{et al.} \\cite{GuyonFE}, a feature selection method has three components: 1) Evaluation criterion definition, e.g. relevance for filter methods, 2) evaluation criterion estimation, e.g. sufficient feature selection or MB for filter methods, and 3) search strategies for feature subset generation. In this section, we briefly review the main search strategies used by MI feature selection methods. Given a feature set $F$ of cardinality $m$, there are $2^{m}$ possible subsets, therefore an exhaustive search is impractical for high-dimensional datasets. \n\nThere are two basic search strategies: optimal methods and sub-optimal methods \\cite{Webb02}. Optimal search strategies include exhaustive search and accelerated methods based on the monotonic property of a feature selection criterion, such as branch and bound. But optimal methods are impractical for high-dimensional datasets, therefore sub-optimal strategies must be used.\n\nMost popular search methods are sequential forward selection (SFS) \\cite{Whitney1971} and sequential backward elimination (SBE) \\cite{Marill193}. Sequential forward selection is a bottom-up search, which starts with an empty set, and adds new features one at a time. Formally, it adds the candidate feature $f_i$ that maximizes $I(S;C)$ to the subset of selected features $S$, i.e.,\n\\begin{equation}\nS=S\\cup\\{\\underset{f_{i}\\in F\\backslash S}{\\arg\\,\\max}(I(\\{S,f_{i}\\};C))\\}.\n\\end{equation}\n\nSequential backward elimination is a top-down approach, which starts with the whole set of features, and deletes one feature at a time. Formally, it starts with $S=F$, and proceeds deleting the less informative features one at a time, i.e,\n\\begin{equation}\nS=S\\backslash\\{\\underset{f_{i}\\in S}{\\arg\\,\\min}(I(\\{S\\backslash f_{i}\\};C)\\}.\n\\end{equation}\nUsually backward elimination is computationally more expensive than forward selection, e.g. when searching for a small subset of features. However, backward elimination can usually find better feature subsets, because most forward selection methods do not take into account the relevance of variables in the context of features not yet included in the subset of selected features \\cite{Guyon03}. Both kinds of searching methods suffer from the nested effect, meaning that in forward selection a variable cannot be deleted from the feature set once it has been added, and in backward selection a variable cannot be reincorporated once it has been deleted. Instead of adding a single feature at a time, some generalized forward selection variants add several features, to take into account the statistical relationship between variables \\cite{Webb02}. Likewise, the generalized backward elimination deletes several variables at a time. An enhancement may be obtained by combining forward and backward selection, avoiding the nested effect. The strategy ``plus-l-take-away-r'' \\cite{Stearns1976} adds to $S$ $l$ features and then removes the worst $r$ features if $l>r$, or deletes $r$ features and then adds $l$ features if $r0$. The authors considered a Naive Bayes classifier, which assumes independence between variables.\n\nEq. (\\ref{eq:CMIM}) allows deriving the Conditional Mutual Information Maximization (CMIM) criterion \\cite{Fleuret04}, when we consider only the first term on the right hand side of this equation and replace the mean operator with a minimum operator. CMIM discards the second term on the right hand side of eq.(\\ref{eq:CMIM}) completely, taking into account only one-to-one relationships among variables and neglecting the multi-information among $f_{i}, \\neg s_{j}$ and $C$ in the context of $s_j$ $\\forall j$. On the other hand, CMIM-2 \\cite{Vergara2010} criterion corresponds exactly to the first term on the right hand side of eq. (\\ref{eq:CMIM}). These methods are able to detect pairs of relevant variables that act complementarily in predicting the class. In general CMIM-2 outperformed CMIM in experiments using artificial and benchmark datasets \\cite{Vergara2010}.\n\nSo far we have reviewed feature selection approaches that avoid estimating MI in high-dimensional spaces.\nBonev \\textit{et al.} \\cite{Bonev08} proposed an extension of the MD criterion, called Max-min-Dependence (MmD), which is defined as follows:\n\\begin{equation}\nJ_{MmD}(f_{i})=I(\\{f_{i},S\\};C)-I(\\neg \\{f_{i},S\\};C).\n\\end{equation}\nThe procedure starts with the empty set $S=\\emptyset$ and sequentially generates $S^{t+1}$ as:\n\\begin{equation}\nS^{t+1}=S^{t}\\cup\\underset{f_{i}\\in F\\backslash S}{\\max}\\left(J_{MmD}(f_{i})\\right).\n\\end{equation}\nThe MmD criterion is heuristic, and is not derived from a principled approach. However, Bonev \\textit{et al.} \\cite{Bonev08} were one of the first in selecting variables estimating MI in high-dimensional spaces \\cite{Hero99}, which allows using set of variables instead of individual variables. Chow and Huang \\cite{Chow05} proposed combining a pruned Parzen window estimator with quadratic mutual information \\cite{Principe}, using Renyi entropies, to estimate directly the MI between the feature subset $S^{t}$ and the classes $C$, $I(S^{t};C)$, in an effective and efficient way.\n\n\\section{Open Problems}\nIn this section we present some open problems and challenges in the field of feature selection, in particular from the point of view of information theoretic methods. Here can be found a non-exhaustive list of open problems or challenges.\n\n\\begin{enumerate}\n\\item \\textbf{Further developing a unifying framework for information theoretic feature selection.}\nAs we reviewed in section \\ref{UF}, a unifying framework able to explain the advantages and limitations of successful heuristics has been proposed. This theoretical framework should be further developed in order to derive new efficient feature selection algorithms that include in their functional terms information related to the three types of features: relevant, redundant and complementary. Also a stronger connection between this framework and the Markov blanket is needed. Developing hybrid methods that combine maximal dependency with minimal conditional mutual information is another possibility.\n\n\\item \\textbf{Further improving the efficacy and efficiency of information theoretic feature selection methods in high-dimensional spaces.}\nThe computational time depends on the search strategy and the evaluation criterion \\cite{GuyonFE}. As we enter the era of Big Data, there is an urgent need for developing very fast feature selection methods able to work with millions of features and billions of samples. An important challenge is developing more efficient methods for estimating MI in high-dimensional spaces. Automatically determining the optimal size of the feature subset is also of interest, many feature selection methods do not have a stop criterion. Developing new search strategies that go beyond greedy optimization is another interesting possibility.\n\n\\item \\textbf{Further investigating the relationship between mutual information and Bayes error classification.}\nSo far lower and upper bounds for error classification have been found for the case of one random variable and the target class. Extending these results to the case of mutual information between feature subsets and the target class is an interesting open problem.\n\n\\item \\textbf{Further investigating the effect of a finite sample over the statistical criteria employed and in MI estimation.}\nGuyon \\textit{et al.} \\cite{GuyonFE} argued that feature subsets that are not sufficient may render better performance than sufficient feature subsets. For example, in the bio-informatics domain, it is common to have very large input dimensionality and small sample size \\cite{Saeys07}.\n\n\\item \\textbf{Further developing a framework for studying the relation between feature selection and causal discovery.}\nGuyon \\textit{et al.} \\cite{GuyonCFS} investigated causal feature selection. The authors argued that the knowledge of causal relationships can benefit feature selection and viceversa. A challenge is to develop efficient Markov blanket induction algorithms for non-faithful distributions.\n\n\\item \\textbf{Developing new criteria of statistical dependence beyond correlation and MI.}\nSeth and Principe \\cite{Seth2010} revised the postulates of measuring dependence according to Renyi, in the context of feature selection. An important topic is normalization, because a measure of dependence defined on different kinds of random variables should be comparable. There is no standard theory about MI normalization \\cite{Estevez09,Duch2006}. Another problem is that estimators of measures of dependence should be good enough, even when using a few realizations, in the sense of following the desired properties of these measures. Seth and Principe \\cite{Seth2010} argued that this property is not satisfied by MI estimators, because they do not reach the maximum value under strict dependence, and are not invariant to one-to-one transformations.\n\\end{enumerate}\n\n\n\\section{Conclusions}\nWe have presented a review of the state-of-the-art in information theoretic feature selection methods. We showed that modern feature selection methods must go beyond the concepts of relevance and redundance to include complementarity (synergy). In particular, new feature selection methods that assess features in context are necessary. Recently, a unifying framework has been proposed, which is able to retrofit successful heuristic criteria. In this work, we have further developed this framework, presenting some new results and derivations. The unifying theoretical framework allows us to indicate the approximations made by each method, and therefore their limitations. A number of open problems in the field are suggested as challenges for the avid reader.\n\n\\section{Acknowledgement}\nThis work was funded by CONICYT-CHILE under grant FONDECYT 1110701.\n\n\\bibliographystyle{spbasic}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nEnd-to-end (E2E) systems have become dominant in automatic speech recognition (ASR) because of their simplicity and better performance. \nIn addition to new advancements in model architectures \\cite{Han2020-CIC,Gulati2020-CCA,Weninger2021-DEA},\none of the major efforts is to make E2E ASR systems support online streaming applications with strict latency requirements. \nThe Recurrent Neural Network Transducer (RNN-T) has been a favorite E2E model architecture for online streaming because of its time-synchronous processing of input audio and superior performance over the CTC model \\cite{Graves2012-STW,Battenberg2017-ENT,Prabhavalkar2017-ACO}.\n\nThere have been significant improvements to RNN-T since it was proposed in \\cite{Graves2012-STW}, such as replacing the LSTM\/BLSTM encoder with Transformer \\cite{Yeh2019-TTE,Zhang2020-TTA}, Conformer \\cite{Gulati2020-CCA}, and ContextNet \\cite{Han2020-CIC}.\nA major difference between online streaming and offline batch-mode E2E model is that the former is subject to strict and often application-dependent latency constraints. \nTo reduce the deterministic latency incurred during inference, an online ASR system is only allowed to access limited future context.\nSince many popular E2E ASR systems are based on bidirectional long-range context modeling (BLSTM, Transformer, etc.) in the encoder, this is the primary reason that online E2E ASR systems generally underperform their offline counterparts. \nThe degradation in accuracy is largely determined by the accessed amount of future context.\nThere has been extensive research in effectively utilizing future context with limited latency for improving online E2E model performance \\cite{Yeh2019-TTE,Zhang2020-TTA,Li2020-OTC,Li2021-ABA,ChunkedMicrosoft,EMFormer}.\n\nFrom a deployment efficiency point of view, it is beneficial to have a single model able to serve multiple different applications: from offline batch-mode to online streaming under different latency requirements.\nUnfortunately, a model trained for the offline use case generally does not perform well in the online use case and vice versa. \nTherefore, there is a direction of research towards making a single model suitable for multiple use cases with different latency requirements \\cite{Tripathi2020-TTO,Gao2020-UAU,Audhkhasi2021-MMA,Yu2021-DMA,Kim2021-MMT}.\n\n\\textbf{Contributions of our paper:} We extend the dual-mode ASR work in \\cite{Yu2021-DMA} in several aspects that were not covered there or in similar works \\cite{Audhkhasi2021-MMA,Kim2021-MMT}:\n1. Comprehensive evaluation of different online streaming approaches (i.e., autoregressive and chunked attention) based on Conformer Transducer in dual-mode training on two very different data sets. \n2. Evaluation of different distillation approaches for offline-to-online distillation in dual-mode training and the importance of modeling the output shift between offline and online modes.\n3. Propose a dual-mode model trained with shared convolution (i.e., causal convolution) and normalization layers across modes.\n\n\n\\section{Methodology}\n\n\n\n\n\n\\subsection{Dual-mode Conformer Transducer}\n\nIn our paper, we use end-to-end ASR systems based on the Conformer Transducer (Conf-T) architecture, which combines the concept of the recurrent neural network transducer (RNN-T) \\cite{Graves2012-STW} with the Conformer encoder \\cite{Zhang2020-TTA,Gulati2020-CCA}.\nEach Conformer encoder block consists of feedforward, multi-head self-attention (MHSA) \\cite{Vaswani2017-AIA}, and convolution layers.\nFollowing the dual-mode approach \\cite{Yu2021-DMA}, a single Conformer Transducer model can operate in both online and offline mode.\nIn online mode, the outputs of convolutions and attention layers are calculated by masking the weights corresponding to future frames.\nMoreover, online and offline mode use different sets of (batch \/ layer) normalization parameters (running average statistics and scales \/ offsets). \nFor the convolutions, the alternative approach proposed in our paper is to simply use causal (left) padding everywhere.\n\nOffline and online Transducer outputs are calculated as $z_{\\text{on}} = M_{\\text{on}}^{\\theta^\\text{on}}(x)$, $z_{\\text{off}} = M_{\\text{off}}^{\\theta^\\text{off}}(x) \\in [0,1]^{T \\times U \\times K}$, where $M$ is the model, $\\theta^{\\text{on}} = [ \\theta; \\nu^\\text{on} ]$, $\\theta^{\\text{off}} = [ \\theta; \\nu^\\text{off} ]$ are the online and offline model weights, $\\nu^\\text{on}$, $\\nu^\\text{off}$ are the corresponding normalization parameters, and $T$, $U$, $K$ denote \\# frames, \\# tokens, and vocabulary size.\nIn dual-mode training \\cite{Yu2021-DMA}, the offline and online mode are trained jointly while knowledge transfer is done via in-place distillation from the offline to the online mode. \nMore precisely, the following loss is minimized:\n\\begin{equation}\n \n \\mathcal{L} = \\alpha \\mathcal{L}_\\text{trd}(y^*,z_\\text{on}) + \\beta \\mathcal{L}_\\text{trd}(y^*,z_\\text{off}) + \\gamma \\mathcal{L}_\\text{dist}(z_\\text{off},z_\\text{on}) ,\n \\label{eq:loss}\n\\end{equation}\nwhere $\\mathcal{L}_\\text{trd}$ is the transducer loss \\cite{Graves2012-STW}, $\\mathcal{L}_\\text{dist}$ is a distillation loss (cf.\\ Section \\ref{sec:dist}), \n$y^*$ are the training labels, and $\\alpha, \\beta, \\gamma \\geq 0$ are hyperparameters.\n\n\\subsection{Chunked Attention}\n\n\\label{sec:chunked_attn}\n\nTo adapt Transformer-like architectures for the streaming use case, the key part to consider is the MHSA block.\nThe MHSA can be made strictly online by using autoregressive attention \\cite{Vaswani2017-AIA}, i.e., every frame in the encoder can only attend to previous frames.\nThis constraint is efficiently implemented by adding $-\\infty$ (or a large negative value) to the attention logits at the `invalid' positions.\nThere are several ways to modify this under specified latency requirements, such as truncated lookahead~\\cite{Zhang2020-TTA} or contextual lookahead~\\cite{EMFormer}. \nWhile the first approach builds lookahead that accumulates through layers into a larger overall lookahead of the encoder, the latter increases the computational cost at inference by overlapping the input audio chunks. \nIn this work, we focus on the chunked attention approach~\\cite{ChunkedMicrosoft}, where we divide the input audio into non-overlapping chunks.\nFor each encoder input chunk, the MHSA query at each position uses as memory (key and values) all the other positions that belong to the same chunk or previous chunks.\nFigure~\\ref{fig:dual_mode_conformer_chunked_attn} shows the dual-mode Conformer encoder with chunked attention mask.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{Conf.pdf}\n \\caption{Schema of the dual-mode Conformer encoder with chunked\/global attention mask (added to logits), causal\/non-causal convolutions and dual-mode normalization with parameters $\\nu^{(\\cdot)}_{(\\cdot)}$. Dashed lines indicate optional components.\n \n }\n \\label{fig:dual_mode_conformer_chunked_attn}\n\\end{figure}\n\n\\subsection{Distillation}\n\n\\label{sec:dist}\n\nIn addition to architectural advancements and various approaches for effectively leveraging limited future context, knowledge distillation from offline to online model is another way of improving online model performance \\cite{Kurata2020-KDF}. \nWe compare the approaches proposed in \\cite{Panchapagesan2021-EKD, Yang2021-KDF} in the context of in-place distillation in dual-mode training.\n\nKnowledge distillation by using the Kullback-Leibler divergence (KLD) loss \\cite{Yu2013-KDR,Weninger2019-LAS} directly is inefficient for Transducers due to the large size of the output lattice.\nThe efficient KLD \\cite{Panchapagesan2021-EKD} and 1-best distillation \\cite{Yang2021-KDF} losses address this issue by restricting the calculation to a reduced output lattice.\nMoreover, in order to make these approaches work for offline-to-online distillation, we consider the potential emission delay between online and offline model via a tunable shift parameter $\\tau$, similar to \\cite{Yu2021-DMA,Yang2021-KDF}.\n\nThe efficient KLD loss \\cite{Panchapagesan2021-EKD} collapses the probability distribution of the tokens as follows:\n\\begin{equation}\n \n \n \n \n \n \\mathcal{L}_\\text{dist}^\\text{eff}\n = \\sum_{t,u} \\sum_{l \\in \\{y, \\varnothing, r\\}} p_\\text{off}(l | t,u) \\log \\frac{p_\\text{off}(l | t,u)}{p_\\text{on}(l | t-\\tau,u)} ,\n\\end{equation}\nwhere $y$, $\\varnothing$ and $r$ denote the correct, the blank, and all other labels, $p_{(\\cdot)}$ denotes the probability obtained from the Transducer output $z_{(\\cdot)}$, and $t$ and $u$ denote time frame and token indices.\n\nConversely, the 1-best distillation loss \\cite{Yang2021-KDF} takes into account the full probability distribution, but only along the 1-best path in the teacher lattice.\nWe extend this approach to in-place distillation by regenerating the 1-best path of the offline model on-the-fly in each training step. \nFor consistency, we also use KLD, not cross-entropy as in \\cite{Yang2021-KDF}:\n\\begin{equation}\n \n \n \n \\mathcal{L}_\\text{dist}^\\text{1-best}\n = \\sum_{(t,u) \\in \\text{1-best}} \\sum_{k=1}^K p_\\text{off}(k | t, u) \\log \\frac{p_\\text{off}(k | t, u)}{p_\\text{on}(k | t-\\tau, u)} ,\n\\end{equation}\nwhere $k$ is the index of a symbol in the vocabulary.\n\n\n\\section{Experiments and Results}\n\n\\subsection{Librispeech Data}\n\n\\subsubsection{Training recipe}\nWe first perform a comparative evaluation using the 100 hour training subset of the Librispeech \\cite{Panayotov2015-LAA} corpus.\nSpeed perturbation \\cite{Ko2015-AAF} with factors 0.9, 1.0 and 1.1 and SpecAugment \\cite{Park2019-SAA} are applied to improve generalization.\nThe topology of the Conformer Transducer and the training recipe are similar to \\cite{Higuchi2021-ACS}. \nThe encoder consists of a feature frontend that extracts 80-dimensional log-Mel features, two convolutional layers that perform downsampling on the time axis by a factor of 4, and 18 Conformer blocks with hidden dimension 256 and feed-forward dimension 1024. \nThe prediction network has a single LSTM layer with 256 hidden units, and the joint network has 256 units.\nThe vocabulary contains 30 characters.\nModels are trained for 300 epochs.\nThe training hyperparameters (especially learning rate schedule) were tuned for the offline model using a limited grid search on the clean development set of Librispeech, then applied to all other models (online and dual-mode) without further tuning.\nFor dual-mode training, the online and offline losses are weighted equally ($\\alpha = \\beta = 0.5$) and the distillation weight is set to $\\gamma=0$ (no distillation) or $\\gamma=0.01$.\nWe measure the word error rate (WER) on the `clean' and `other' test set of Librispeech.\nDecoding is done by beam search with beam size 8, without using an external language model.\n\n\\subsubsection{Online and offline baselines}\n\n\\begin{table}[t]\n \\caption{Single-mode baselines on Librispeech 100h (LA: lookahead).}\n \\label{tab:results_ls100_single_mode}\n \\centering\n \\begin{tabular}{l|l|cc}\n \\bf Mode & \\bf Attention & \\multicolumn{2}{c}{\\bf Test WER [\\%]} \\\\\n & & \\bf cln & \\bf other \\\\\n \\hline\n Online & Autoregressive & 9.6 & 26.9 \\\\\n Online & Autoreg.\\ LA & 8.4 & 24.8 \\\\\n Online & Chunked & 7.9 & 23.4 \\\\\n \\hline\n Offline (causal conv) & Full context & 6.3 & 18.4 \\\\\n \\quad (non-causal conv) & Full context & 6.3 & 18.3 \\\\\n Offline \\cite{Higuchi2021-ACS} & Full context & 6.8 & 18.9 \\\\\n \\end{tabular}\n\\end{table}\n\nThe results of our single-mode baselines are shown in \\tablename~\\ref{tab:results_ls100_single_mode}.\nOur offline Conformer Transducer system outperforms the reference result obtained by ESPnet \\cite{Higuchi2021-ACS}.\nWe also investigated the usage of causal (left padded) 1-D depthwise convolutions in the Conformer blocks in the offline model. \nThe WER was similar to the standard non-causal (centered) convolutions. \nHence, we chose to apply causal convolutions for offline mode as well, thereby simplifying the implementation compared to the original dual-mode Conf-T \\cite{Yu2021-DMA}.\n\nFor the online systems, we compare autoregressive attention, autoregressive attention with 12 frames ($\\approx$ 0.5 seconds) lookahead in the 9th encoder layer\\footnote{We did not observe significant performance differences when putting the lookahead in another encoder layer or distributing it across multiple encoder layers.}, and chunked attention (see Section \\ref{sec:chunked_attn}) with a chunk size of 25 frames ($\\approx$ 1 second).\nUsing autoregressive attention leads to a drastic WER increase compared to the offline model (52\\,\\% relative).\nHowever, the relative WER increase is still much smaller than the one reported in \\cite{Yu2021-DMA}, suggesting that our online baseline is competitive.\nAs expected, the lookahead reduces the gap between online and offline WER significantly.\nFurthermore, despite having the same average lookahead of about 0.5 seconds, the chunked attention performs better than the autoregressive attention with lookahead (6\\,\\% WER reduction (WERR)).\n\n\\begin{table}[t]\n \\caption{Librispeech 100h task: WER obtained by dual-mode systems in online and offline inference with and without efficient KLD distillation (loss weight $\\gamma$, shift $\\tau$).}\n \\label{tab:results_ls100_dual_mode}\n \\centering\n \\begin{tabular}{l|c|c|cc|cc}\n \\bf Online att. & \\bf $\\gamma$ & $\\tau$ & \\multicolumn{4}{c}{\\bf Test WER [\\%]} \\\\\n & & & \\multicolumn{2}{c|}{\\bf Online} & \\multicolumn{2}{c}{\\bf Offline} \\\\\n & & & \\bf cln & \\bf other & \\bf cln & \\bf other \\\\\n \\hline\n Autoreg. & 0.0 & -- & 9.0 & 25.2 & 7.2 & 21.8 \\\\\n Autoreg. & 0.01 & 0 & 9.0 & 25.8 & 7.2 & 21.2 \\\\\n Autoreg. & 0.01 & -6 & 8.4 & 24.2 & 7.0 & 20.6 \\\\\n \\hline\n Autoreg.\\ LA & 0.0 & -- & 7.7 & 23.1 & 6.8 & 19.9 \\\\\n Autoreg.\\ LA & 0.01 & 0 & 7.7 & 22.6 & 7.0 & 19.8 \\\\\n Autoreg.\\ LA & 0.01 & -6 & 7.5 & 22.2 & 6.7 & 19.8 \\\\\n \\hline\n Chunked & 0.0 & -- & 7.4 & 22.0 & 6.4 & 19.2 \\\\\n Chunked & 0.01 & 0 & 7.7 & 22.4 & 6.4 & 19.2 \\\\\n Chunked & 0.01 & -6 & \\bf 7.1 & 21.5 & \\bf 6.1 & 18.9 \\\\\n \\end{tabular}\n\\end{table}\n\n\n\\subsubsection{Dual-mode systems}\n\n\\tablename~\\ref{tab:results_ls100_dual_mode} shows the results obtained by dual-mode training for various types of online attention.\nCompared to the online baselines in \\tablename~\\ref{tab:results_ls100_single_mode}, the WER in online mode is improved by dual-mode training in all cases (6\\,\\%, 8\\,\\% and 6\\,\\% WERR on test\\_clean for autoregressive, autoregressive with lookahead and chunked attention, respectively).\nFurthermore, we observe that there is a consistent gain in online performance from using distillation with shift $\\tau=-6$, but no gain from distillation without shift.\nStill, the gain from distillation is diminished when lookahead is used, likely because this brings the online performance closer to the offline model and reduces the benefit of knowledge distillation.\nThe dual-mode system using autoregressive attention in the online mode improves on the WER of the corresponding single-mode online system by 12\\,\\% relative.\nConversely, the offline performance is degraded by 11\\,\\% relative.\nThe trend is similar for the autoregressive attention with lookahead, despite overall better performance.\nIn contrast, using chunked attention in the online mode avoids the degradation in offline mode, and the corresponding\ndual-mode system performs better than the single-mode baselines in both offline and online mode, achieving 10\\,\\% and 3\\,\\% relative WERR, respectively.\nThis is likely because the lookahead for a given frame is not constant in chunked attention,\nwhich makes the online prediction task more similar to the offline one and thus facilitates joint training.\n\nWe also investigate the impact of the shift $\\tau$ between offline teacher and online student in the in-place distillation with both efficient distillation and 1-best distillation.\nAs can be seen from \\figurename~\\ref{fig:ls100_shift}, gains from distillation can be achieved only with fairly large shifts (e.g., $\\tau=-6$ corresponds to a $\\approx$ 240\\,ms emission delay), which is consistent with the findings in \\cite{Yang2021-KDF}, while results are unstable for small shifts (in fact, the experiments with $\\tau=-2$ diverged).\nThe best result with efficient distillation is achieved at $\\tau=-6$ (8.4\\,\\% WER), whereas the 1-best distillation performs best with $\\tau=-8$.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{LS100_shift.pdf}\n \\caption{Dual-mode WER on Librispeech 100h task with efficient and 1-best distillation, varying the shift parameter $\\tau$.}\n \\label{fig:ls100_shift}\n\\end{figure}\n\n\\subsection{Medical Data}\n\nAdditionally, we conduct experiments on an internal data set which consists of conversational speech data in the medical domain (doctor-patient conversations).\nThe experiments are based on a training set of 1\\,k hours manually end-pointed and transcribed speech covering various medical specialties.\nWe measure WER on a speaker-independent test set consisting of 263\\,k words.\n\nThe model topology and training recipe are similar to the one used for Librispeech.\nIn the encoder, we use 16 Conformer blocks with hidden dimension 512 and feed-forward dimension 1024 after the frontend.\nThe prediction network consists of a single Transformer layer with the same dimensions, and the joint network has 512 units.\nThe vocabulary contains 2\\,k word-pieces.\n\n\\begin{table}[t]\n \\caption{Single-mode baselines on medical conversation data}\n \\label{tab:results_dax_single_mode}\n \\centering\n \\begin{tabular}{l|l|c}\n \\bf Mode & \\bf Attention & \\bf WER [\\%] \\\\\n \\hline\n \n \n \n \n \n Online & Autoreg.\\ LA & 14.7 \\\\\n Online & Chunked & 14.4 \\\\\n \\hline\n Offline (causal conv) & Full context & 13.1 \\\\\n \\quad (non-causal conv) & Full context & 13.2 \\\\\n \\end{tabular} \n\\end{table}\n\n\\tablename~\\ref{tab:results_dax_single_mode} shows the single-mode baselines. \nFor the online systems, we compare autoregressive attention with lookahead (12 frames) and chunked attention (24 frames).\nUnlike on Librispeech, a pure autoregressive model (without any lookahead) did not yield satisfactory performance.\nThe chunked attention improves the WER of the online system by 2\\,\\% relative compared to the autoregressive attention with 12 frames lookahead.\nStill, there remains a gap of about 9\\,\\% relative WER difference between the online and the offline system.\n\n\\begin{table}[th]\n \\caption{Medical conversation data: WER obtained by dual-mode systems in online and offline inference with and without efficient distillation (loss weight $\\gamma$, shift $\\tau$).}\n \\label{tab:results_dax_dual_mode}\n \\centering\n \\begin{tabular}{l|c|c|cc}\n \\bf Online att. & \\bf $\\gamma$ & $\\tau$ & \\multicolumn{2}{c}{\\bf WER [\\%]} \\\\\n & & & \\bf Online & \\bf Offline \\\\\n \\hline\n \n \n \n \n Autoreg.\\ LA & 0.0 & -- & 14.2 & 13.3 \\\\\n Autoreg.\\ LA & 0.01 & 0 & 14.2 & 13.3 \\\\\n Autoreg.\\ LA & 0.01 & -6 & 14.1 & 13.2 \\\\\n \\hline\n Chunked & 0.0 & -- & \\bf 13.7 & \\bf 12.9 \\\\\n Chunked & 0.01 & 0 & \\bf 13.7 & \\bf 12.9 \\\\\n Chunked & 0.01 & -6 & 13.8 & 13.0 \\\\\n \n \n \n \n \n \n \n \\end{tabular}\n\\end{table}\n\n\\tablename~\\ref{tab:results_dax_dual_mode} shows the results obtained by dual-mode training.\nWe use the efficient KLD loss in case of $\\gamma > 0$.\nAs in the Librispeech scenario, using chunked attention in the online mode helps improving both online and offline performance. The dual-mode system with chunked attention obtains 3.7\\,\\% \/ 3.1\\,\\% relative WERR compared to the one using autoregressive attention with lookahead, and 5.4\\,\\% \/ 1.6\\,\\% with respect to the corresponding single-mode online \/ offline system.\nHowever, unlike on Librispeech, we do not observe any gain from distillation, even with $\\tau=-6$.\nOne possible reason is that the performance difference between the online and offline models on the medical data set is small (about 9\\,\\% relative, see \\tablename~\\ref{tab:results_ls100_single_mode}) compared to that on the Librispeech data set (23\\,\\% relative, see \\tablename~\\ref{tab:results_ls100_single_mode}).\n\n\n\\subsection{Emission Timing}\n\n\\figurename~\\ref{fig:dax_latency} shows the time delay of transcriptions produced by our dual-mode models, with respect to a single-mode reference.\nWe compute this delay as the average time difference of all the matching couples of correct words. For each word we consider the time of the emitting frame of its last word-piece in the RNN-T output alignment. If the encoder is chunked, times are rounded up to the end of their corresponding encoder chunks.\nThe absolute emission delay is similar between autoregressive attention with lookahead and chunked attention models.\n\nAs can be seen in \\figurename~\\ref{fig:dax_latency}, dual-mode models typically emit faster than the single-mode ones.\nWhen the dual-mode model is trained without distillation, we observe a slightly lower delay in the chunked configuration compared with the autoregressive one, while a significant improvement in terms of emission delay ($\\approx$ 110\\,ms) is evident in both cases when the distillation is enabled.\nHowever, the latency gain from distillation vanishes when the teacher targets are shifted with a negative $\\tau$ value, because this configuration encourages later emission.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=.9\\linewidth]{Latency.pdf}\n \\caption{Emission delay of dual-mode systems vs.\\ single-mode online reference measured on medical data (lower means earlier).}\n \\label{fig:dax_latency}\n\\end{figure}\n\n\n\n\\subsection{Effect of Dual-mode Normalization and Joint Training}\n\nIn \\tablename~\\ref{tab:results_ls100h_dax_dual_single_norm}, we assess the importance of the dual-mode normalization layers in the Conformer blocks proposed by \\cite{Yu2021-DMA} vs.\\ simply sharing the normalization layers between online and offline mode.\nOn Librispeech (using chunked attention and efficient distillation with $\\tau=-6$), the online performance is very similar between dual and single-mode normalization, while there is a small degradation in offline WER.\nThe medical conversation task (using chunked attention but no distillation) shows a similar picture.\nSince we use causal convolutions for both online and offline mode as in the previous experiments, using a single set of normalization layers means that convolutional and feedforward components are identical to the single-mode Conformer, and the MHSA layers vary only the attention mask.\nThus, single-mode normalization considerably simplifies the implementation while yielding similar performance.\n\n\n\\begin{table}[t]\n \\caption{WER obtained by dual-mode systems in online and offline inference, using dual normalization layers (one for online and one for offline) or single normalization layers (shared between online and offline mode).}\n \\label{tab:results_ls100h_dax_dual_single_norm}\n \n \\centering\n \\begin{tabular}{l|c|c}\n \\bf Norm.\\ layers & \\multicolumn{2}{c}{\\bf WER [\\%]} \\\\\n & \\bf Online & \\bf Offline \\\\\n \\hline\n \\multicolumn{3}{c}{\\em Librispeech 100h (clean \/ other)} \\\\\n \\hline\n dual & 7.1 \/ 21.5 & 6.1 \/ 18.9 \\\\\n single & 7.1 \/ 21.3 & 6.3 \/ 19.0 \\\\\n \n \n \n \n \n \\hline\n \\multicolumn{3}{c}{\\em Medical conversation task} \\\\\n \\hline\n dual & 13.7 & 12.9 \\\\\n single & 13.7 & 13.0 \\\\\n \\end{tabular}\n\\end{table}\n\nMotivated by these results, we also investigated a further simplification of the dual-mode training where the attention mask for all MHSA layers is randomly chosen as the global (offline) or chunked (online) one for each line in the current mini-batch, instead of training both modes on the entire batch (joint training, cf.\\ Eq.\\ \\eqref{eq:loss}).\nThis is similar in spirit to the sampling techniques in \\cite{Yu2021-DMA,Audhkhasi2021-MMA,Kim2021-MMT}.\nThe advantage is that only one model (online or offline) is computed for each utterance, thus saving approximately 50\\,\\% of computation and memory requirement.\nWe found such dual-mode training to yield a single model for both online and offline mode that performed similar to the dedicated single-mode models (14.2\\,\\% \/ 13.3\\,\\% WER on the medical task).\nHowever, unlike joint training, it did not result in a sizable WER gain compared to the single-mode baselines.\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nIn this paper, we presented an in-depth study on the performance of dual-mode training for online Conformer Transducer architectures. \nWe could obtain significant WER improvements in online mode on both the Librispeech and a medical conversational speech task, even without in-place distillation, and match the performance of dedicated offline models.\nBest results in online mode were obtained using chunked attention.\nOur results also shed light on the importance of modeling emission delay when doing offline-to-online knowledge distillation: we found that distillation without shift is helpful for reducing latency, while distillation with shift can reduce the WER at the expense of emission delay.\nThe latter could potentially be mitigated by techniques such as FastEmit \\cite{Yu2021-FLS}.\nIn general, the gain from distillation depends on the online configuration (especially the lookahead) and the data set.\nFurthermore, we explored several modifications to the original training approach, and found a simplified version, where only the attention mask is exchanged between online and offline modes,\nto perform equally well as the original proposal \\cite{Yu2021-DMA}.\nIn future work, we will apply our findings to multi-mode ASR \\cite{Kim2021-MMT} for improving robustness of the online model in multiple latency requirements.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{#1}}\n\n\\def\\mbox{sh}{\\mbox{sh}}\n\\def\\mbox{ch}{\\mbox{ch}}\n\\def\\mbox{e}{\\mbox{e}}\n\\def\\,\\mbox{i}\\,{\\,\\mbox{i}\\,}\n\\def\\,\\mbox{\\scriptsize i}\\,{\\,\\mbox{\\scriptsize i}\\,}\n\\def\\lambda{\\lambda}\n\\def\\hbox{th}{\\hbox{th}}\n\\def\\vartheta_1{\\vartheta_1}\n\\def\\vartheta_3{\\vartheta_3}\n\\def\\vartheta_4{\\vartheta_4}\n\\def\\case#1#2{{\\textstyle{#1\\over #2}}}\n\\def\\W#1#2#3#4#5{W #1 \\! \\left(\\hspace{-1mm}\n \\begin{array}{cc}#5 & #4 \\\\ #2 & #3 \\end{array}\n \\hspace{-1mm}\\right)}\n\\def\\case#1#2{{\\textstyle{#1\\over #2}}}\n\n\\begin{document}\n\n\\title{Lattice Ising model in a field: E$_8$ scattering theory}\n\n\\author{\nV. V. Bazhanov\\thanks{\nIAS, Australian National University,\nTheoretical Physics and Mathematics,\nGPO Box 4, Canberra, ACT 2601, Australia, e-mail {\\vartheta_2 vvb105@phys.anu.edu.au}}\n\\thanks{\nOn leave of absence from the Institute\nfor High Energy Physics,\nProtvino, Moscow Region, 142284, Russia.} ,\nB. Nienhuis\\thanks{\nInstituut voor Theoretische Fysica,\nUniversiteit van Amsterdam,\nValckenierstraat~65,\n1018~XE Amsterdam,\nThe Netherlands, e-mail {\\vartheta_2 nienhuis@phys.uva.nl}}\n\\hspace{0.2mm} and\nS. O. Warnaar$^{\\mbox{\\footnotesize\\ddag}}$\\thanks{\nPresent address: Mathematics Department,\nUniversity of Melbourne, Parkville, Victoria 3052,\nAustralia, e-mail {\\vartheta_2 warnaar@mundoe.maths.mu.oz.au}}}\n\n\\date{\\ }\n\\maketitle\n\n\\begin{abstract}\nZamolodchikov found an integrable field theory\nrelated to the Lie algebra E$_8$, which describes the\nscaling limit of the Ising model in a magnetic field.\nHe conjectured that there also exist solvable lattice\nmodels based on E$_8$ in the universality class of the Ising model in a\nfield.\nThe dilute A$_3$ model is a solvable lattice model with\na critical point in the Ising universality class.\nThe parameter by which the model can be taken away\nfrom the critical point acts like a magnetic field by breaking\nthe $\\:\\mbox{\\sf Z} \\hspace{-0.82em} \\mbox{\\sf Z}\\,_2$ symmetry between the states.\nThe expected direct relation of the model with E$_8$\nhas not been found hitherto.\nIn this letter we study the thermodynamics of the dilute\nA$_3$ model and show that in the scaling limit it exhibits\nan appropriate E$_8$ structure, which naturally leads to the E$_8$\nscattering theory for massive excitations over the ground state.\n\\end{abstract}\n\n\\newlength{\\mathin}\n\\setlength{\\mathin}{\\mathindent}\n\\setcounter{page}{1}\n\\nsection{Introduction}\nSince the work \\cite{Zamolodchikov-A87b}\nby A.B. Zamolodchikov it is known that certain\nperturbations of conformal field theories (CFT's) lead to completely\nintegrable models of massive quantum field theory (QFT).\nThe\nexistence of non-trivial higher integrals\nof motion and other dynamical symmetries\n\\cite{Zamolodchikov-A88a,A-B-LC,Reshetikhin90d,LeClair,Eguchi}\nin such a QFT allows to compute the spectrum of the particles\nand their $S$-matrix explicitly.\nAt the same time, these QFT models can be obtained as the\nscaling limit of appropriate non-critical solvable lattice models\nin statistical mechanics\n(see \\cite{Baxter-book} for an introduction\nand references on solvable lattice models).\nIn the latter approach the spectrum and the $S$-matrices\ncan be calculated from the Bethe Ansatz equations for the\ncorresponding lattice model\n\\cite{Reshetikhin87a,Bazhanov90d,Reshetikhin91c}.\nThe natural problem arising in this connection is to find lattice\ncounterparts for all known integrable perturbed CFT's and\nvice versa. A description of known results of such\ncorrespondence lies outside the scope of this letter and we refer the\ninterested reader to [1-10]\nand references therein.\nHere we consider one\nparticularly important example of this correspondence associated with\nthe Ising model at its critical temperature in a magnetic field,\nhereafter referred to as the magnetic Ising model.\n\nA.B. Zamolodchikov has shown \\cite{Zamolodchikov-A89a}\nthat the $c=1\/2$ CFT (corresponding to the critical\nIsing model) perturbed with the spin operator $\\phi_{1,2}=\\phi_{2,2}$\nof dimension $(1\/16,1\/16)$ describes an exactly integrable QFT\ncontaining eight massive particles with a reflectionless factorised\n$S$-matrix. Up to normalisation the masses of these particles coincide\nwith the components $S_i$ of the Perron-Frobenius vector of the Cartan\nmatrix of the Lie algebra E$_8$:\n\\begin{equation}\n\\frac{m_i}{m_j} = \\frac{S_i}{S_j}.\n\\label{massratio}\n\\end{equation}\nThe element of the\n$S$-matrix describing the scattering of the lightest particles, with\nmass $m_1$,\nreads \\cite{Zamolodchikov-A89a}\n\\begin{equation}\nS_{1,1}(\\beta)=\\frac\n{\\hbox{th}\\left(\\frac{\\beta}{2} + \\,\\mbox{i}\\, \\frac{\\pi}{6} \\right)\n \\hbox{th}\\left(\\frac{\\beta}{2} + \\,\\mbox{i}\\, \\frac{\\pi}{5} \\right)\n \\hbox{th}\\left(\\frac{\\beta}{2} + \\,\\mbox{i}\\, \\frac{\\pi}{30} \\right)}\n{\\hbox{th}\\left(\\frac{\\beta}{2} - \\,\\mbox{i}\\, \\frac{\\pi}{6} \\right)\n \\hbox{th}\\left(\\frac{\\beta}{2} - \\,\\mbox{i}\\, \\frac{\\pi}{5} \\right)\n \\hbox{th}\\left(\\frac{\\beta}{2} - \\,\\mbox{i}\\, \\frac{\\pi}{30} \\right)},\n\\end{equation}\nwith $\\beta$ the rapidity.\nThe other elements are uniquely determined by the\nbootstrap program \\cite{Zamolodchikov-A89a}.\n\nThe aim of this letter is to show\nthat the above QFT describes the scaling limit of\nthe dilute A$_3$ model of\nWarnaar, Nienhuis and Seaton \\cite{WNS,WPSN}\nin the appropriate regime.\nIt should be noted that there were some earlier, rather strong\nindications supporting the above correspondence.\nAll these parts remarkably fit together with our results,\ncompleting a sequence of arguments which can be summarised as follows:\n\\begin{description}\n\\item[{\\rm (i)}]\nThe dilute A$_3$ model is an interaction-round-a-face model on\nthe square lattice with spins taking three values (detailed\ndefinitions are given in equations\n(\\ref{Incidence})-(\\ref{regimes})). Admissible values of the\nadjacent spins are\ndetermined by the incidence matrix (\\ref{Incidence}),\nwhich has largest eigenvalue\nequal to $1+\\sqrt{2}$.\n\\item[{\\rm (ii)}]\nThe model has two\nphysically distinct regimes of relevance to our discussion,\nhere denoted as {\\it i)} and {\\it ii)},\ndepending on the region of the spectral parameter or, equivalently, of\na sign of the Hamiltonian of the associated one-dimensional chain.\n(These are the regimes $2^+$ and $3^+$ of ref~\\cite{WPSN}, respectively).\nThe central charges and the conformal dimensions of the leading perturbation\ncomputed from exact expressions for the free energy and the local\nstate probabilities\nof the dilute A$_3$ model\nfor these two regimes read \\cite{WBN,WPSN}\n\\begin{equation}\ni) \\quad c=1\/2, \\quad \\Delta=1\/16; \\qquad ii) \\quad c=6\/5, \\quad\n\\Delta=15\/16.\n\\label{canddelta}\n\\end{equation}\n\\item[{\\rm (iii)}]\nIn ref~\\cite{Bazhanov90d,Bazhanov90b} Bazhanov and Reshetikhin\nproposed thermodynamic Bethe Ansatz equations (TBAE) related\nto the A-D-E Lie algebras, corresponding to\nnon-critical models in statistical mechanics.\nUsing standard\nthermodynamics calculations and the high level Bethe Ansatz (see\n\\cite{Reshetikhin87a} and references therein)\nthey computed: the central charges of the corresponding scaling field\ntheories, dimensions of the leading\nperturbations, the spectra and scattering amplitudes of the\nmassive excitations, expressing them through fused Boltzmann weights.\nIn particular, in the case relevant to our discussion\n($\\cal G$=E$_8$, $g=30$, $p=\\ell=1$,\nin the notation of \\cite{Bazhanov90d}) the\nexponents they found \\footnote{Note that the equations (5$\\cdot$1)\nand (5$\\cdot$4) in \\cite{Bazhanov90d}\nhave been misprinted. Correcting (5$\\cdot$1)\nto $c=c^{\\cal G}(l)+c^{\\cal G}(r-l-g)-c^{\\cal G}(r-g)+\n\\mbox{rank}\\:{\\cal G}$ yields the following result for\nthe central charge in (5$\\cdot$4):\n$c=2\\:\\mbox{rank}\\:{\\cal G}\/(g+2)$. Also the phrases\n``minimal unitary'', just before, and\n``by the operator $\\phi_{(1,3)}$'' just after (5$\\cdot$4)\nshould be deleted.}\nprecisely match (\\ref{canddelta})\nin both regimes.\nFurthermore, the TBAE allowing the calculation of\nthe largest eigenvalue of the incidence matrix of the underlying\nlattice model, gave in this case precisely the value $1+\\sqrt{2}$\n\\cite{Bazhanov-remark-in-Kuniba's-paper}.\n\\item[{\\rm (iv)}]\nFinally, the\nspectrum and $S$-matrix of the scaling field theory in regime\n$\\it i)\\\/$ found in \\cite{Bazhanov90d} from the\nhigh level Bethe Ansatz for E$_8$\ncoincide with those of Zamolodchikov's magnetic Ising model.\n\\end{description}\nAll the above arguments strongly suggest that the TBAE\nbased on the Lie algebra E$_8$ as\nproposed in \\cite{Bazhanov90d,Bazhanov90b},\nare those of the the dilute A$_3$ model.\n\nIn this paper we present the Bethe Ansatz equations (BAE)\nfor the non-critical, dilute A$_L$ model.\nAs these equations, at criticality, are very similar to those of the\nIzergin-Korepin model \\cite{IK,Vichirko}, it is not\nsurprising that, when specialised to $L=3$,\nthey do not display any explicit structure related\nto the root system of E$_8$.\nIt turns out however that this structure reveals itself\nin a quite complicated string structure of the solutions\nto the BAE.\nMotivated by an extensive numerical\ninvestigation of the BAE\nwe formulate an exact conjecture\nfor the thermodynamically significant strings.\nThis leads to TBAE, which, rewritten in a\nnew string basis precisely yield the E$_8$ based TBAE of\nref~\\cite{Bazhanov90d} discussed under (iii).\nAs a result of (iv) this\nfinalises the\ncorrespondence between the dilute A$_3$ model and the\nmagnetic Ising model.\n\n\\nsection{The dilute A models}\nThe dilute A$_L$ model, belonging to the more\ngeneral class of dilute A-D-E models,\nis an exactly solvable, restricted solid-on-solid\nmodel defined on the square lattice.\nEach site of the lattice can take one of $L$ possible\n(height) values, subject to the restriction that\nneighbouring sites of the lattice either have the\nsame height, or differ by $\\pm 1$.\nThis adjacency condition can be\nconveniently expressed by a so-called\nincidence matrix $M$:\n\\begin{equation}\nM_{a,b} = \\delta_{a,b-1} + \\delta_{a,b} + \\delta_{a,b+1}\n\\qquad a,b\\in \\{1,\\ldots,L\\},\n\\label{Incidence}\n\\end{equation}\nwhere we note that $M$ relates to the\nCartan matrix $C^{\\mbox{\\scriptsize A}_L}$ of\nthe Lie algebra A$_L$\nby $M=3 I - C^{\\mbox{\\scriptsize A}_L}$,\nwith $I$ the identity matrix.\nThe eigenvalues of the incidence matrix are\nfound to be\n\\begin{equation}\n\\Lambda_j = 1 + 2\\cos\n\\left(\\frac{\\pi j}{L+1} \\right) \\qquad j=1,\\ldots,L.\n\\end{equation}\nFor the case of interest here, $L=3$, we thus find the\nlargest eigenvalue to be $1+\\sqrt{2}$, in accordance with\nthe prediction for the E$_8$ TBAE as mentioned in (iii)\nof the introduction.\n\nUsing standard definitions of $\\vartheta_{i}(u,q)$-functions,\nsuppressing the dependence on the nome $q=\\mbox{e}^{-\\tau}$, $\\tau>0$,\nthe Boltzmann weights of the allowed height configurations of\nan elementary face of the lattice are\n\\setlength{\\mathindent}{0 cm}\n\\begin{eqnarray}\n\\lefteqn{\\W{}{a}{a}{a}{a}=\n\\frac{\\vartheta_1(6\\lambda-u)\\vartheta_1(3\\lambda+u)}{\\vartheta_1(6\\lambda)\\vartheta_1(3\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\hphantom{\\W{}{a}{a}{a}{a}}\n-\\left(\\frac{S(a+1)}{S(a)}\\frac{\\vartheta_4(2a\\lambda-5\\lambda)}{\\vartheta_4(2a\\lambda+\\lambda)}\n +\\frac{S(a-1)}{S(a)}\\frac{\\vartheta_4(2a\\lambda+5\\lambda)}{\\vartheta_4(2a\\lambda-\\lambda)}\\right)\n\\frac{\\vartheta_1(u)\\vartheta_1(3\\lambda-u)}{\\vartheta_1(6\\lambda)\\vartheta_1(3\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\W{}{a}{a}{a}{a\\pm 1}=\\W{}{a}{a\\pm 1}{a}{a}=\n\\frac{\\vartheta_1(3\\lambda-u)\\vartheta_4(\\pm 2a\\lambda+\\lambda-u)}{\\vartheta_1(3\\lambda)\\vartheta_4(\\pm 2a\\lambda+\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\W{}{a\\pm 1}{a}{a}{a}=\\W{}{a}{a}{a\\pm 1}{a}=\n\\left(\\frac{S(a\\pm 1)}{S(a)}\\right)^{1\/2}\n\\frac{\\vartheta_1(u)\\vartheta_4(\\pm 2a\\lambda-2\\lambda+u)}{\\vartheta_1(3\\lambda)\\vartheta_4(\\pm 2a\\lambda+\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\W{}{a}{a\\pm 1}{a\\pm 1}{a}=\\W{}{a}{a}{a\\pm 1}{a\\pm 1}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{ \\hphantom{\\W{}{a}{a\\pm 1}{a\\pm 1}{a}}\n=\\left(\\frac{\\vartheta_4(\\pm 2a\\lambda+3\\lambda)\\vartheta_4(\\pm 2a\\lambda-\\lambda)}\n {\\vartheta_4^2(\\pm 2a\\lambda+\\lambda)}\\right)^{1\/2}\n\\frac{\\vartheta_1(u)\\vartheta_1(3\\lambda-u)}{\\vartheta_1(2\\lambda)\\vartheta_1(3\\lambda)} }\n\\nonumber \\\\ & & \\label{Bweights} \\\\\n\\lefteqn{\\W{}{a}{a\\mp 1}{a}{a\\pm 1}=\n\\frac{\\vartheta_1(2\\lambda-u)\\vartheta_1(3\\lambda-u)}{\\vartheta_1(2\\lambda)\\vartheta_1(3\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\W{}{a\\pm 1}{a}{a\\mp 1}{a}=\n-\\left(\\frac{S(a-1)S(a+1)}{S^2(a)}\\right)^{1\/2}\n\\frac{\\vartheta_1(u)\\vartheta_1(\\lambda-u)}{\\vartheta_1(2\\lambda)\\vartheta_1(3\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\W{}{a\\pm 1}{a}{a\\pm 1}{a}=\n\\frac{\\vartheta_1(3\\lambda-u)\\vartheta_1(\\pm 4a\\lambda+2\\lambda+u)}{\\vartheta_1(3\\lambda)\\vartheta_1(\\pm 4a\\lambda+2\\lambda)}\n+\\frac{S(a\\pm 1)}{S(a)}\n\\frac{\\vartheta_1(u)\\vartheta_1(\\pm 4a\\lambda-\\lambda+u)}{\\vartheta_1(3\\lambda) \\vartheta_1(\\pm 4a\\lambda+2\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\hphantom{\\W{}{a\\pm 1}{a}{a\\pm 1}{a}}=\n\\frac{\\vartheta_1(3\\lambda+u)\\vartheta_1(\\pm 4a\\lambda-4\\lambda+u)}\n{\\vartheta_1(3\\lambda)\\vartheta_1(\\pm 4a\\lambda-4\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{\\hphantom{\\W{}{a\\pm 1}{a}{a\\pm 1}{a}}+\n\\left(\\frac{S(a\\mp 1)}{S(a)}\\frac{\\vartheta_1(4\\lambda)}{\\vartheta_1(2\\lambda)}\n-\\frac{\\vartheta_4(\\pm 2a\\lambda-5\\lambda)}{\\vartheta_4(\\pm 2a\\lambda+\\lambda)} \\right)\n\\frac{\\vartheta_1(u)\\vartheta_1(\\pm 4a\\lambda-\\lambda+u)}{\\vartheta_1(3\\lambda) \\vartheta_1(\\pm 4a\\lambda-4\\lambda)}}\n\\nonumber \\\\ & & \\nonumber \\\\\n\\lefteqn{S(a)=(-)^{\\displaystyle a} \\;\n\\frac{\\vartheta_1(4a\\lambda)}{\\vartheta_4(2a\\lambda)} \\, .}\n\\nonumber\n\\end{eqnarray}\nThe variable $\\lambda$ and the range of the spectral parameter $u$\nin the above weights are\ngiven by\\footnote{In \\cite{WPSN} two more regimes were\ndefined, which are omitted being of no relevance here.}\n\\setlength{\\mathindent}{\\mathin}\n\\begin{equation}\n\\lambda = \\frac{\\pi}{4} \\, \\frac{L+2}{L+1}\n\\qquad\n\\left\\{\n\\begin{array}{lll}\n00.\n\\label{Energy}\n\\end{equation}\nwhere $\\epsilon=-1$ for regime {\\it i)} and\n$\\epsilon=1$ for regime {\\it ii)} in (\\ref{regimes}).\n\nThe densities $\\rho_t$ are normalised such\nthat\n\\begin{equation}\n\\int_{-\\tau r\/\\pi}^{\\tau r\/\\pi}\n\\rho_t(\\alpha)\\: d\\alpha = N^{(t)}\/N.\n\\end{equation}\nTherefore, from equation (\\ref{Nsum}), we have\n\\begin{equation}\n\\sum_{t=0}^8\n\\int_{-\\tau r\/\\pi}^{\\tau r\/\\pi}\nn^{(t)} \\rho_t(\\alpha) \\: d\\alpha = 1.\n\\end{equation}\nThis relation together with equation (\\ref{TBAE}) for $t=0$\nimplies\n\\begin{equation}\n\\tilde{\\rho}_0(\\alpha) =0.\n\\label{zero}\n\\end{equation}\nHence we conclude that the strings of type 0 have no holes in any\nstate, and we eliminate $\\rho_0(\\alpha)$ from (\\ref{TBAE}).\nAfter a tedious calculation we find that the resulting integral\nequations can naturally be described in\nterms of the E$_8$ root system as follows.\n\nLet $C^{\\mbox{\\scriptsize E}_8}_{t,s}$ $t,s=1,\\ldots,8$\nbe the elements of the Cartan matrix for\nE$_8$, where we use the following enumeration of the nodes of the\ncorresponding Dynkin diagram:\n\\[\n\\setlength{\\unitlength}{0.008in}%\n\\begingroup\\makeatletter\n\\def\\x#1#2#3#4#5#6#7\\relax{\\def\\x{#1#2#3#4#5#6}}%\n\\expandafter\\x\\fmtname xxxxxx\\relax \\def\\y{splain}%\n\\ifx\\x\\y \n\\gdef\\SetFigFont#1#2#3{%\n \\ifnum #1<17\\tiny\\else \\ifnum #1<20\\small\\else\n \\ifnum #1<24\\normalsize\\else \\ifnum #1<29\\large\\else\n \\ifnum #1<34\\Large\\else \\ifnum #1<41\\LARGE\\else\n \\huge\\fi\\fi\\fi\\fi\\fi\\fi\n \\csname #3\\endcsname}%\n\\else\n\\gdef\\SetFigFont#1#2#3{\\begingroup\n \\count@#1\\relax \\ifnum 25<\\count@\\count@25\\fi\n \\def\\x{\\endgroup\\@setsize\\SetFigFont{#2pt}}%\n \\expandafter\\x\n \\csname \\romannumeral\\the\\count@ pt\\expandafter\\endcsname\n \\csname @\\romannumeral\\the\\count@ pt\\endcsname\n \\csname #3\\endcsname}%\n\\fi\n\\endgroup\n\\begin{picture}(250,76)(75,710)\n\\thicklines\n\\put( 80,740){\\circle*{10}}\n\\put(120,740){\\circle*{10}}\n\\put(160,740){\\circle*{10}}\n\\put(200,740){\\circle*{10}}\n\\put(240,740){\\circle*{10}}\n\\put(280,740){\\circle*{10}}\n\\put(320,740){\\circle*{10}}\n\\put(240,780){\\circle*{10}}\n\\put(240,740){\\line( 0, 1){ 40}}\n\\put( 80,740){\\line( 1, 0){240}}\n\\put( 80,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}1}}}\n\\put(120,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}2}}}\n\\put(160,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}3}}}\n\\put(200,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}4}}}\n\\put(240,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}5}}}\n\\put(280,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}6}}}\n\\put(320,710){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}7}}}\n\\put(255,775){\\makebox(0,0)[b]{\\smash{\\SetFigFont{14}{16.8}{bf}8}}}\n\\end{picture}\n\\]\nFurthermore, define the functions\n$K^{\\mbox{\\scriptsize E}_8}_{t,s}$,\n$A^{\\mbox{\\scriptsize E}_8}_{t,s}$,\n$a^{\\mbox{\\scriptsize E}_8}_{t,s}$\nand $s$ by their FT\n\\begin{eqnarray}\n\\hat{K}^{\\mbox{\\scriptsize E}_8}_{t,s}(x)&=&\n\\delta_{t,s} + \\hat{s}(x)\n\\left(C^{\\mbox{\\scriptsize E}_8}_{t,s} - 2\\delta_{t,s}\\right)\n\\nonumber \\\\\n\\hat{A}^{\\mbox{\\scriptsize E}_8}_{t,s}(x)&=&\n\\left[\\hat{K}^{\\mbox{\\scriptsize E}_8}(x)\\right]^{-1}_{t,s}\n\\nonumber \\\\\n\\hat{a}^{\\mbox{\\scriptsize E}_8}_{t,s}(x)&=&\n\\hat{s}(x)\n\\hat{A}^{\\mbox{\\scriptsize E}_8}_{t,s}(x)\\\\\n\\hat{s}(x) &=& \\frac{1}{2\\cosh x} \\nonumber\n\\end{eqnarray}\nWith these definitions, and after eliminating $\\rho_0$,\n the integral equations (\\ref{TBAE}) and\nthe energy expression (\\ref{Energy}) take the form\n\\begin{eqnarray}\na_{1,t}^{\\mbox{\\scriptsize E}_8}(\\alpha)\n&=& \\tilde{\\rho}_t(\\alpha)\n+ \\sum_{s=1}^8 A^{\\mbox{\\scriptsize E}_8}_{t,s} \\ast \\rho_s\n\\, (\\alpha) \\qquad t=1,\\ldots,8, \\nonumber \\\\\n\\frac{E}{N} &=&-\\epsilon\\sum_{t=1}^8\n\\int_{-\\tau r\/\\pi}^{\\tau r\/\\pi}\na_{1,t}^{\\mbox{\\scriptsize E}_8}(\\alpha) \\, \\rho_t(\\alpha)\n\\: d\\alpha +\\mbox{const}.\n\\label{BAE8}\n\\end{eqnarray}\n\nWe can now use (\\ref{BAE8}) to study the scaling limit of the\nmodel. In fact, all relevant calculations have already been\ncarried out in ref~\\cite{Bazhanov90d} and we only need\nto refer to the appropriate results therein.\nTo make the correspondence with ref~\\cite{Bazhanov90d}\nsomewhat more transparent, let us give the\nexpression for the equilibrium free energy $F(T)$ of\nthe one-dimensional spin chain\nat finite temperature $T$,\nas it follows from (\\ref{BAE8}) via standard TBA calculations\n\\cite{Yang-Yang},\n\\begin{equation}\n\\frac{F(T)}{N} = -\\sum_{t=1}^8\n\\int_{-\\tau r\/\\pi}^{\\tau r\/\\pi}\na_{1,t}^{\\mbox{\\scriptsize E}_8}(\\alpha) \\,\nT \\log \\left(1+\\mbox{e}^{-\\beta \\epsilon_t(\\alpha)}\\right)\n\\, d\\alpha +\\mbox{const},\n\\label{FE}\n\\end{equation}\nwhere $\\beta=1\/T$ is the inverse temperature.\nThe functions $\\epsilon_t=T\\log(\\tilde{\\rho}_t\/\\rho_t)$ are\nthe solutions of the integral equation\n\\begin{equation}\n\\epsilon \\delta_{1,t} s(\\alpha) =\nT \\log \\left(1+\\mbox{e}^{-\\beta \\epsilon_t(\\alpha)}\\right)\n- \\sum_{s=1}^8 K_{t,s}^{\\mbox{\\scriptsize E}_8}\n\\ast\nT \\log \\left(1+\\mbox{e}^{\\beta \\epsilon_s(\\alpha)}\\right)\n(\\alpha).\n\\label{NLIE}\n\\end{equation}\n\nThe above two equations are equivalent to\n(3$\\cdot$20) and (3$\\cdot$21) of ref~\\cite{Bazhanov90d},\nrespectively, with their $\\cal G$=E$_8$, $r=32$, $g=30$, $p=\\ell = 1$,\ntheir nome $q$ replaced by $q^{1\/2}$ and\nwith their $\\epsilon_j^a$ negated.\nThis last difference reflects the fact that our TBA equations are dual\nto those of ref~\\cite{Bazhanov90d} in the sence that the densities of\nstrings and holes are interchanged. From (\\ref{FE}) and (\\ref{NLIE})\nit follows that for $T=0$\n\\begin{equation}\n\\begin{array}{lll}\ni) & \\epsilon =-1 \\quad & \\epsilon_t(\\alpha) =\na_{1,t}^{\\mbox{\\scriptsize E}_8} \\\\\n& & \\\\\nii) & \\epsilon =+1 & \\epsilon_t(\\alpha) =\n-\\delta_{t,1} s(\\alpha).\n\\end{array}\n\\end{equation}\nThe functions $|\\epsilon_t(\\alpha)|$ are the energies\nof the excitations over the ground state.\n\nFor $\\epsilon=-1$ the ground state is formed by type 0\nstrings. As was remarked after equation (\\ref{zero}),\nthese strings have no holes for any state. Therefore the\nDirac sea is ``frozen'', and the excitations correspond to\nthe remaining eight string types. The phenomenon of ``freezing''\nof the Dirac sea which can be interpreted as the confinement of\n``holes'' has been first observed in\nthe TBAE calculations of ref~\\cite{Bazhanov}\nfor the RSOS models of\nAndrews, Baxter and Forrester~\\cite{A-B-F}.\n\nFor $\\epsilon=1$ the Dirac sea is formed by the type 1\nstrings, and the only excitations correspond to holes\nin the Dirac sea. These excitations are of the kink type.\n\nNow we consider the scaling limit. We introduce a dimensional\nspacing parameter $d$ for our chain and\ntake the limit $N\\to\\infty$, $d\\to 0$,\nkeeping the (dimensional) length of the chain $L=N d $ to\nbe macroscopically bigger than the correlation length:\n$L>>R_c= q^{-\\xi}d$, where $\\xi$ is the index of the correlation\nlength.\nIn the scaling limit we thus have $d\\sim q^{\\xi}$,\n$N>>q^{-\\xi}$, $q\\to 0$, and we\nobtain the massive relativistic spectrum\nof excitations.\nTo find this, one has to compute the energy dispersion law for the\nphysical excitations in the $q\\to 0$ limit keeping the\nrapidities $\\alpha$ of the order of $\\alpha_0=\\tau r\/\\pi$, where\nthe functions $|\\epsilon_t(\\alpha)|$ have their minima.\nTaking into account the correspondence in notation\ndiscussed after (\\ref{NLIE}), one gets from\n(4$\\cdot$1) and (4$\\cdot$2) of ref~\\cite{Bazhanov90d}\n\\begin{eqnarray}\ni) & & \\epsilon_t\n\\left(\\case{30}{\\pi} \\beta + \\alpha_0 \\right) = m_t \\cosh \\beta +\no(q^{\\xi}\n) \\nonumber \\\\\n& & m_t = \\mbox{const } S_t q^{\\xi}, \\quad \\xi = \\case{8}{15}\n\\nonumber \\\\\nii) & & \\left|\\epsilon_1\n\\left(\\case{2}{\\pi} \\beta + \\alpha_0 \\right)\n\\right| = m \\cosh \\beta +\no(q^{\\xi}) \\label{Mass} \\\\\n& & m = \\mbox{const } q^{\\xi}, \\quad \\xi = 8,\n\\nonumber\n\\end{eqnarray}\nwhere $\\beta$ here denotes the rapidity variable and\n$S_t$ was defined just before equation (\\ref{massratio}).\n\nUsing the scaling relation\n$\\xi = (2-2\\Delta)^{-1}$ it is\nseen that the values of $\\xi$ in (\\ref{Mass})\nlead exactly to the dimensions of the leading perturbations\nas given in (\\ref{canddelta}).\n\nThe values of the central charges of the corresponding\n(ultraviolet) conformal field theories listed in\n(\\ref{canddelta}) have also been previously calculated.\nFor regime $i)$ in \\cite{Bazhanov90d,K-M,AlZamolodchikov}\nand for regime $ii)$ in \\cite{Bazhanov90d}.\n\nFinally, the\n$S$-matrix for regime {\\it i)}, where all the string excitations\nfor $t=1\\ldots 8$ correspond to distinct particles, can be found\nstraightforwardly from equation (\\ref{BAE8}).\nThe result is \\cite{Bazhanov90d}\n\\begin{equation}\nS_{t,s}(\\beta) = \\exp\\left\\{ \\,\\mbox{i}\\,\n\\int_0^{\\infty} A_{t,s}^{\\mbox{\\scriptsize E}_8}(x) \\,\n\\frac{\\sin (30\\beta x\/{\\pi})}{x} \\: dx \\right\\},\n\\end{equation}\nwhich coincides with Zamolodichkov's E$_8$ $S$-matrix\n\\cite{Zamolodchikov-A89a}.\n\nFor regime {\\it ii)} the kink-kink $S$-matrix is of\nthe RSOS type related to the E$_7$ Lie algebra\n\\cite{Bazhanov90d}, and will be discussed elsewhere\n\\cite{BNW}.\n\n\n\\nsection{Summary and Conclusion}\nIn this paper we have established the final link between\nZamolodchikov's E$_8$ $S$-matrix of the critical Ising model\nin a field \\cite{Zamolodchikov-A89a}\nand its underlying lattice model.\nBy making a conjecture for the possible string\nsolutions of its Bethe Ansatz equations,\nwe have derived a system of thermodynamic BAE for the\ndilute A$_3$ lattice model of Warnaar et al. \\cite{WNS}.\nAfter a suitable transformation\nwe have recast these TBAE in terms of the root system\nof the Lie algebra E$_8$.\nThese E$_8$ TBAE are found to be precisely those conjectured\nearlier by Bazhanov and Reshetikhin \\cite{Bazhanov90d}, and\nusing their results, the correspondence\nbetween the dilute A$_3$ model and the E$_8$ $S$-matrix\nis made.\n\nTo conclude\nwe mention that two more remarkable integrable $\\phi_{1,2}$ perturbations\nof CFT's are known, notably those related to $S$-matrices with\nhidden E$_7$ (c=7\/10) and E$_6$ (c=6\/7) structure \\cite{F-Z}.\nLike the E$_8$ case, the underlying lattice models\nof these integrable QFT's correspond to models in the\ndilute A hierarchy.\nThe working for these two extra cases, corresponding\nto dilute A$_4$ and A$_6$, respectively, as well as some\nadditional results for the dilute A$_3$ model will be the\nsubject of a future publication \\cite{BNW}\n\n\\section*{Acknowledgements}\nWe wish to thank\nM.~T.~Batchelor, R.~J.~Baxter, U.~Grimm, P.~A.~Pearce,\nand N.~Yu.~Reshetikhin for interesting discussions.\nWe thank U.~Grimm, P.~A.~Pearce and Y.~K.~Zhou\nfor sending us their work prior to publication.\nOne the authors (VVB) thanks\nthe University of Amsterdam for hospitality during his\nvisit in the summer of 1992, when this work\nhas been initiated.\nThis work has been supported by the Stichting voor Fundamenteel\nOnderzoek der Materie (FOM).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nMore than two years ago N.Seiberg and E.Witten proposed a way\n\\cite{SW1,SW2} leading to considerable progress in\nunderstanding of the low-energy effective actions.\nAmong other things, it provided a non-trivial confirmation\nof the old {\\it belief} that the low-energy effective actions\n(the end-points of the renormalization group flows) fit into universality\nclasses depending on the vacuum structure of the theory.\nThe trivial (and generic) case is that of isolated vacua,\ni.e. when there are no marginal degrees of freedom and effective\ntheories on the moduli space of the vacua. If degenerated vacua form\nfinite-\\-dimensional varieties, the effective actions can be essentially\ndescribed in terms of $1d$ systems\n\\footnote{By $1d$ $(0+1)$ systems here we mean the conventional\nclassical\/quantum mechanical systems with {\\it finite}-dimensional phase\nspace (\\# of degrees of freedom is equal to the rank of the gauge group).}\n(representatives of the\nsame universality class), though the original\ntheory lives in a many-dimensional space-time. Moreover, these effective\ntheories can be integrable\n\\footnote{When degenerated vacua form\ninfinite-dimensional structures one should expect generalized integrability\n(with spectral curve substituted by spectral hypersurface).}.\nOriginally,\nthis phenomenon was studied in the context of $2d$ topological theories,\nwhile the Seiberg-Witten (SW) construction can be considered as\nan extension of this approach from two to four space-time\n(``world sheet'') dimensions.\nThe natural context where one encounters continuous degeneration of vacua in\nfour dimensions is {\\it supersymmetry}; the SW construction {\\it per se}\nconcerns the $4d$ ${\\cal N}=2$ SUSY Yang-Mills models. Integrable structure\nbehind the SW theory has been found in \\cite{GKMMM} and later examined in\ndetail in \\cite{SWI1}-\\cite{N}. Its intimate relation to the previously\nknown topological theories has been revealed recently in \\cite{MMM}\nwhere it was proven that the $4d$ prepotentials do satisfy the\nWDVV equations \\cite{W,DVV}.\nThe purpose of the present paper is to give more evidence\n(and more examples) confirming this relation.\n\nNaively, the SW low-energy effective action is not\ntopological: it describes propagating particles. This is,\nhowever, a necessary consequence of the vacuum degeneracy and\nunbroken supersymmetry. All the correlators in the low-energy\ntheory can be expressed through the correlators of ``holomorphic''\n(or ``pure topological'') subsector. Technically, propagators\nare contained in the $\\langle\\phi\\bar\\phi\\rangle$ correlators,\n(derivatives of) which are related by the ${\\cal N}=2$ supersymmetry\nto the $\\langle\\phi\\phi\\rangle$ ones. This is summarized in the statement\n(sometimes called special geometry) that the low-energy effective action is\n{\\it completely} expressed in terms of a {\\it holomorphic} prepotential\n$F(a_I)$, $\\frac{\\partial}{\\partial\\bar a_I} F = 0$. Even if one accepts the\nterm ``topological'' for this kind of models, what we consider is a $4d$\ntopological theory, {\\it a priori} very different from the familiar ($2d$)\nexamples. However, if one believes in the above-mentioned universality, it is\nclear that only the structure of the moduli space of vacua (but not the\nspace-time dimension) is essential. Since the vacuum variety in the SW\ntheory is finite-dimensional, one should expect that this theory is not too\nfar from the conventional topological models, devised to describe\nexactly this kind of vacua; as was shown in \\cite{MMM} and confirmed below in\nthe present paper this is indeed the case. The way to see this is to recall\nthat the prepotential of conventional topological theory satisfies the\nWDVV equations \\cite{W,DVV} and so does the prepotential of the SW theory\n\\cite{MMM} (similar claims were made in \\cite{BM,dBdW}).\n\nMathematically, the WDVV equations reflect the specific properties of\nthe moduli space of Riemann spheres with punctures \\cite{Ma,KoMa}, in the $2d$\ncontext these spheres being interpreted as the world sheets.\nNaturally, in the $4d$ context they could be substituted by\nthe four-dimensional world hypersurfaces, and naively instead of integrable\nsystems, associated with the spectral curves one would deal with\nthose, associated with spectral hypersurfaces -- with\nno reason for the standard WDVV eqs. to emerge. This is of course\ntrue for ``generic topological $4d$ models'', but this is not\nthe only relevant problem that can be addressed in four dimensions.\nIf one instead emphasizes the issue of vacuum structures and\nuniversality classes, then the {\\it standard} WDVV eqs should\nbe expected in the ({\\it not} the most degenerate from the $4d$ point\nof view) situation with finite-dimensional (in the\nspace of all fields) variety of vacua -- but the conventional\nderivation of these equations is not (at least, directly) applicable.\nMoreover, as argued in \\cite{MMM}, the SW example emphasizes\nthat the natural form (and thus the structure behind) the\nWDVV eqs is covariant under the linear transformations of\ndistinguished (flat) coordinates on the moduli space of vacua --\nwhat is true, but not {\\it transparent} in the conventional\npresentation. For us, all this implies that the true\norigin of the WDVV eqs is still obscure.\n\nOur goal in this paper is rather modest: given (relatively)\nexplicit expressions for particular SW prepotentials\nwe check that they indeed satisfy the WDVV eqs.\n\nIn sect.\\ref{eqs} we briefly remind what are the\nWDVV eqs and how they are related in \\cite{MMM} to\nthe algebra of meromorphic 1-differentials on spectral curves.\n\nThen in sect.\\ref{pert} we discuss the perturbative prepotentials\nin $4d$ and $5d$ ${\\cal N}=2$ SUSY Yang-Mills theories which\nare expected to satisfy the WDVV eqs themselves. We find that this is indeed\ntrue, but only when matter hypermultiplets\nare absent or belong to the first fundamental representation\nof the gauge group (the representation of the lowest dimension). Remarkably,\nthis is in agreement with the\nfact that no Seiberg-Witten curves are known beyond such cases,\nand there are no string model to produce them in the field theory limit.\n\nIn sect.\\ref{npsw} the WDVV eqs are derived at the non-perturbative\nlevel from the representation\nof the prepotential in terms of spectral Riemann surfaces.\n\nThe general reasoning is illustrated in sect.\\ref{Examples} by examples\nwhich include\n${\\cal N}=2$ SUSY YM models with classical simple gauge groups and the\nmatter hypermultiplets in the first fundamental representation.\nAlso the $5d$ $SU(N_c)$ pure gauge model is analyzed, and Seiberg-Witten\ntheory is shown to produce a prepotential, corrected as compared\nto the naive field-theory expectations. Exactly these corrections are\nnecessary to make the WDVV equations true.\n\nFinally, in sect.\\ref{Calogero} the $SU(N_c)$ model with the matter\nhypermultiplet in adjoint representation is analyzed in\ndetail. We demonstrate by explicit calculation that the WDVV equations\nare {\\it not} satisfied in this case. Technically the reason is that\nthe spectral curve is essentially non-hyperelliptic and the\nalgebra of 1-differentials (though existing and being closed) is\nnon-associative.\nThe deep reason for the breakdown of WDVV in its standard from\nis the appearance of new modulus: the parameter $\\tau$\nof the complex structure on {\\it bare} elliptic curve, which is interpreted\nas the ultraviolet (UV) coupling constant of the UV-finite $4d$ theory.\nFurther understanding of the relevant deformation of the WDVV\nequations (presumably related to the analogue of WDVV eqs for\nthe generating function of the elliptic Gromov-Witten classes\n\\cite{Getzler}) is important for further investigation of\nstring models (since $\\tau$ is the remnant of heterotic dilaton\nfrom that point of view).\n\n\\section{The full set of the WDVV equations \\label{eqs}}\n\\setcounter{equation}{0}\nThe WDVV equations are non-linear relations for the third derivatives\nof the prepotential\n\\ba\nF_{IJK} \\equiv \\frac{\\partial^3 F}{\\partial a_I\n\\partial a_J \\partial a_K}.\n\\ea\nwritten in the most convenient way in terms of matrices\n$F_I$, $(F_I)_{MN} \\equiv F_{IMN}$.\nThe {\\it moduli} $a_I$ are defined up to linear transformations\n(i.e. define the {\\it flat structure} on the {\\it moduli space})\nwhich leave the whole set (\\ref{WDVV}) invariant.\n\nAny linear combination\nof these matrices can be used to form a ``metric''\n\\ba\\label{metric}\nQ = \\sum_K q_KF_K.\n\\ea\nGenerically it is non-degenerate square matrix and can be\nused to raise the indices:\n\\ba\\label{3}\nC^{(Q)}_I \\equiv Q^{-1}F_I, \\ \\ \\ {\\rm or}\\ \\\n{C^{(Q)}}^M_{IN} = (Q^{-1})^{ML}F_{ILN}.\n\\ea\nThe WDVV equations state that for any given $Q$ all the\nmatrices $C^{(Q)}_I$ commute with each other:\n\\ba\nC^{(Q)}_IC^{(Q)}_J = C^{(Q)}_JC^{(Q)}_I \\ \\ \\ \\ \\ \\forall I,J\n\\label{WDVV}\n\\ea\nIn particular, if $Q = F_K$, then $C^{(K)}_I = F_K^{-1}F_I$ and \\cite{MMM}\n\\ba\nF_IF_K^{-1}F_J = F_JF_K^{-1}F_I, \\ \\ \\ \\forall I,J,K.\n\\label{WDVVikj}\n\\ea\nIf (\\ref{WDVVikj}) holds for some index $K$, say $K=0$,\nit is automatically true for any other $K$\n(with the only restriction for $F_K$ to be non-degenerate).\nIndeed,\n\\footnote{\nThis simple proof was suggested to us by A.Rosly.\n}\nsince $F_I = F_0 C_I^{(0)}$,\n\\ba\nF_IF_K^{-1}F_J =\nF_0 \\left(C_I^{(0)}(C_K^{(0)})^{-1} C_J^{(0)}\\right)\n\\ea\nis obviously symmetric w.r.t. the permutation $I\\leftrightarrow J$.\nOf course, it also holds for any other non-degenerate $Q$ in\n(\\ref{metric}).\n\nEqs.(\\ref{WDVV}) are just the associativity condition of the formal algebra\n\\ba\n\\varphi_I\\circ \\varphi_J = {C^{(Q)}}_{IJ}^K\n\\varphi_K, \\nonumber \\\\\n(\\ref{WDVV}) \\Leftrightarrow\n(\\varphi_I \\circ \\varphi_M )\\circ \\varphi_J =\n\\varphi_I \\circ (\\varphi_M \\circ \\varphi_J).\n\\label{alge}\n\\ea\nThe way to prove the WDVV eqs, used in \\cite{MMM} and\nin the present paper, is to obtain (\\ref{alge}) from\na family of algebras formed by a certain set of (meromorphic) $(1,0)$-forms on\nthe curves, associated with particular\n$4d$ Yang-Mills models:\n\\ba\ndW_I(\\zeta) dW_J(\\zeta) = {C^{(Q)}}_{IJ}^K dW_K(\\zeta)\ndQ(\\zeta) \\ {\\rm mod}\\ \\left(d\\omega(\\zeta),d\\lambda(\\zeta)\\right).\n\\label{algdiff} \\label{2.8}\n\\ea\nIf this algebra exists and is associative we get the WDVV equations in context\nof the SW approach.\nThe structure constants $C_{IJ}^K$ depend on the choice of the linear\ncombination $dQ(\\zeta) = \\sum_K q_KdW_K(\\zeta)$.\nIn all relevant cases\n$dW_I(\\zeta) \\sim P_I(\\zeta)$, where for appropriately chosen\ncoordinates $\\{\\zeta\\}$, $P_I(\\zeta)$ are polynomials\n(perhaps, of several variables), thus $dQ(\\zeta) \\sim\n{\\cal Q}'(\\zeta) = \\sum_K q_KP_K(\\zeta)$. Also\n$d\\omega(\\zeta) \\sim {\\cal P}'(\\zeta)$,\n$d\\lambda(\\zeta) \\sim \\dot{\\cal P}(\\zeta)$\nwith some polynomials\n${\\cal P}'(\\zeta)$, $\\dot{\\cal P}(\\zeta)$ and (\\ref{WDVV})\nis equivalent to a certain associative algebra of polynomials\n\\ba\nP_I P_J = {C^{(Q)}}_{IJ}^K P_K{\\cal Q}'\\ {\\rm mod}\\\n({\\cal P}',\\dot{\\cal P}).\n\\label{algpol}\n\\ea\nIn order to obtain the WDVV equations one still needs to express\nthe structure constants matrices $C_I^{(Q)}$ through $F_I$\n(which, in contrast to $C_I$, are independent of the ``metric''\n$Q$). In all relevant cases the third derivatives of the\nprepotential are expressed in terms of the same 1-differentials\nas in (\\ref{algdiff}):\n\\ba\nF_{IJK} = \\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{dW_IdW_JdW_K}{d\\lambda d\\omega},\n\\label{2.10}\n\\ea\nwhere $\\delta\\lambda \\wedge \\delta\\omega$ is the symplectic form\non the phase space of $1d$ integrable system, associated with given\n$4d$ model.\n\nFrom (\\ref{metric}) and (\\ref{2.10}),\n\\ba\nQ_{LK} \\equiv \\sum_M q_M F_{LKM} =\n\\stackreb{d\\omega = 0}{\\hbox{res}} \\frac{dW_LdW_KdQ}{d\\lambda d\\omega}\n\\ea\nand according to (\\ref{2.8})\n\\ba\nF_{IJK} = {C^{(Q)}}_{IJ}^LQ_{LK}.\n\\ea\nGiven (\\ref{2.8}) and (\\ref{2.10}) this provides the proof of the WDVV eqs.\n\nWe derive eq.(\\ref{2.10}) in detail in sect.\\ref{npsw} and consider\nparticular examples of the SW theory and associated algebras\n(\\ref{2.8}) in sect.\\ref{Examples}. Let us now make\nseveral comments.\n\nConventional derivations of the WDVV equations rely upon the\nassumption that at least one of the ``metrics'' $Q$\nin (\\ref{WDVV}) is flat. Under this assumption one can relate\nthe associativity equation (\\ref{WDVV}) to the flatness of\nthe connection\n\\ba\n\\partial_I \\delta^A_B + zC^A_{IB}\n\\ea\nwith {\\it arbitrary} spectral parameter $z$, and then to the theory\nof deformations of Hodge structures and the standard theory\nof quantum cohomologies.\nThere is no {\\it a priori} reason for any\n\\ba\nQ_{IJ} = \\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{dW_IdW_JdQ}{d\\lambda d\\omega},\n\\ea\nto be flat and in this context such kind of derivation -- even if\nexists -- would look somewhat artificial.\n\nIn our presentation the WDVV equations depend on a\n{\\it triple} of differentials on the spectral curve,\n$d\\omega$, $d\\lambda$, $dQ$. The first two, $d\\omega$ and $d\\lambda$,\ndescribe the symplectic structure $\\delta\\omega \\wedge \\delta\\lambda$\nof the relevant integrable system,\nthis is the pair, discussed in \\cite{DKN,KriW,KriPho,M} and in another\ncontext in \\cite{KM}. The choice of the third differential $dQ$\nis more or less arbitrary.\\footnote{\nIn the conventional Landau-Ginzburg topological models the spectral\ncurve is Riemann sphere. In this case $dQ$ can be considered as\n``dressed'' $d\\lambda$: $dQ(\\lambda) = q(\\lambda)d\\lambda$\nwith some {\\it polynomial} $q(\\lambda)$ and there is a distinguished choice,\n$q(\\lambda)=1$, i.e. $dQ = d\\lambda$ when the associated metric is\nflat \\cite{Losev,KriW}.\n}\nSometime (for example, for hyperelliptic\nspectral surfaces) the algebra (\\ref{algdiff})\ndepends only on the pair $dQ$, $d\\omega$ (or ${\\cal Q}'$,\n${\\cal P}'$ in the formulation (\\ref{algpol})).\nWe shall demonstrate below that just in this case the algebra (\\ref{2.8}) is\nassociative and the WDVV equations are fulfilled.\nThe pair of differentials in this case is essentially the same\nas arising in many related contexts: $(p,q)$\nin minimal conformal models, a\npair of polynomials in associated matrix models \\cite{Dou,FKN,KM}, a pair of\noperators in the Kac-Schwarz problem \\cite{Sch,KM}\nand in a bispectral problem in the\ntheory of KP\/Toda hierarchies \\cite{OrHar} etc. The simplest example of this\nstructure\nis the family of associative rings in the number theory, with multiplication\n$ab = qc + pd$ with co-prime $q$ and $p>q$ where $a,b,c$ belong to the field\nof residues modulo $p$, while $d$ is defined modulo $q$. All the algebras\nwith multiplication rule $a\\cdot_q b = c$ are isomorphic, but if considered\nas a {\\it family} over the ``moduli space'' of $q$'s, they form some new\nnon-trivial structure.\nIn this oversimplified example the ``moduli space'' is discrete,\nbut it becomes continuous as soon as one switches to the families of the\npolynomial rings (\\ref{algpol}). One can assume that it is this structure -- a\n{\\it family} of associative algebras (not just a {\\it single} associative\nalgebra) -- that stands behind the WDVV equations, and then it should be\nsearched for in the theory of quantum cohomologies, and in topological\ntheories. (Let us emphasize \\cite{MMM} that even if some particular metric\n$Q$ is flat, this is normally not so for all other $Q$'s.)\n\n\\section{The SW prepotentials. Perturbative examples \\label{pert}}\n\\setcounter{equation}{0}\n\\subsection{General formulas}\nIf the exact prepotential of the SW theory satisfies the WDVV\nequations, the same should be true for its perturbative part\n{\\it per se}, since in $4d$ SUSY YM theory the perturbative contribution\nto effective action is one-loop, pure logarithmic, and well\ndefined as a {\\it leading} term in $1\/a$ expansion.\nIn \\cite{MMM} we discussed this phenomenon for the pure gauge ${\\cal N}=2$\nSUSY model and in this section we are going to present more general examples.\n\nPerturbatively, the prepotential\nof the pure gauge ${\\cal N}=2$ SUSY YM model with the gauge group\n$G=SU(N_c)$ is given by\n\\ba\nF_0 = \\frac{1}{2}\\sum_{1\\leq rj>k}a_ia_j\na_k$ so that the complete perturbative prepotential satisfying the WDVV eqs\nis\n\\ba\\label{FARTC}\nF={1\\over 4}\\sum_{i,j}\\left({1\n\\over 3}a_{ij}^3-{1\\over 2}Li_3\\left(e^{-2a_{ij}}\\right)\\right)-\n{N_c\\over 4}\\sum_{i>j>k}a_ia_ja_k\n\\ea\nwhere $a_{ij}\\equiv a_i-a_j$.\nThe cubic terms in (\\ref{FARTC}) could be compared with the $U^3$-terms in\nthe perturbative part of\nthe prepotential of heterotic string, coming from the\nrequirement of modular invariance \\cite{HM}. We show in sect.5.2 how these\nterms can be obtained in the\nperturbative limit of the SW non-perturbative description of $5d$ theory.\n\n\\section{Non-perturbative proof of WDVV equations \\label{npsw}}\n\\setcounter{equation}{0}\n\\subsection{SW theory and the general theory of Riemann surfaces}\nNon-perturbatively, the SW construction implies that\nfor any $4d$ ${\\cal N}=2$ SUSY Yang-Mills model one associates\na (finite-dimensional) integrable system, its phase space is a family of\nabelian varieties (Hamiltonian tori) over a moduli space ${\\cal M}$,\nparametrized by the angle and action variables respectively.\nThe phase space itself is directly \"observable\" when the $4d$\nmodel is compactified to $3d$, i.e. defined in the space-time\n$R^3\\times S^1$ rather than $R^4$: then the angle variables\nof the integrable system are identified with the Wilson loops along\nthe compactified dimension in the $4d$ model \\cite{SW3}.\nIn the non-compactified case the only four-dimensional observables\nare the periods of the generating 1-form $dS=\"pdq\"$\nalong the non-contractable contours on the Hamiltonian tori. The period\nmatrix for the given system consists of the second derivatives of the\nprepotential while the $B$-periods of $dS$ are its first derivatives.\n$A$-periods of $dS$ define the {\\it flat} structure on moduli space.\n\nIn the context of SW theory one actually deals with specific\nintegrable systems associated with Riemann surfaces\n(complex curves). This can be understood from the point of view of\nstring theory, which relates the emergency of Yang-Mills theories\nfrom string compactifications to emergency of Hitchin-like integrable\nsystems from more general systems \\cite{DoMa} associated with the Calabi-Yau\nfamilies. In practice it means that the families of abelian varieties,\nrelevant for the SW theory are in fact the Jacobian tori of complex curves.\n\nIn the context of SW theory one starts with a {\\it bare} spectral curve,\nwith a holomorphic 1-form $d\\omega $, which is elliptic curve (torus)\n\\ba\\label{torus}\nE(\\tau):\\ \\ \\ \\ y^2 = \\prod_{a=1}^3 (x - e_a(\\tau)), \\ \\ \\\n\\sum_{a=1}^3 e_a(\\tau) = 0,\\ \\ \\ \\ \\ d\\omega = \\frac{dx}{y},\n\\ea\nwhen the YM theory is UV finite, or its degeneration $\\tau\n\\rightarrow i\\infty$ -- the double-punctured sphere (``annulus''):\n\\ba\nx\\rightarrow w\\pm\\frac{1}{w},\\ \\\ny \\rightarrow w\\mp\\frac{1}{w},\\ \\ \\ \\ \\ d\\omega = \\frac{dw}{w}\n\\ea\nFrom the point of view of integrable system, the Lax operator\n${\\cal L}(x,y)$ is defined as a function (1-differential) on the\n{\\it bare} spectral curve \\cite{KriCal}, while the {\\it full} spectral curve\n${\\cal C}$ is given by the Lax-eigenvalue equation: $\\det({\\cal L}(x,y) -\n\\lambda ) = 0$. As a result, ${\\cal C}$ arises as a ramified covering over the\n{\\it bare} spectral curve:\n\\ba\n{\\cal C}:\\ \\ \\ {\\cal P}(\\lambda; x,y) = 0\n\\ea\nIn the case of the gauge group $G=SU(N_c)$, the function ${\\cal P}$ is a\npolynomial of degree $N_c$ in $\\lambda$.\n\nThe function ${\\cal P}$ depends also on parameters (moduli)\n$s_I$, parametrizing the moduli space ${\\cal M}$. From the\npoint of view of integrable system, the Hamiltonians\n(integrals of motion) are some specific co-ordinates on\nthe moduli space. From\nthe four-dimensional point of view, the co-ordinates $s_I$ include $s_i$ --\n(the Schur polynomials of) the adjoint-scalar expectation values $h_k =\n\\frac{1}{k}\\langle{\\rm Tr} \\phi^k\\rangle$ of the vector ${\\cal N}=2$\nsupermultiplet, as well as $s_\\iota = m_\\iota$ -- the masses of the\nhypermultiplets.\n\nThe generating 1-form $dS \\cong \\lambda d\\omega$ is meromorphic on\n${\\cal C}$ (hereafter the equality modulo total derivatives\nis denoted by ``$\\cong$''). The prepotential is defined in terms of the\ncohomological class of $dS$:\n\\ba\na_I = \\oint_{A_I} dS, \\ \\ \\ \\ \\ \\\n\\frac{\\partial F}{\\partial a_I} = \\int_{B_I} dS \\nonumber \\\\\nA_I \\circ B_J = \\delta_{IJ}.\n\\label{defprep}\n\\ea\nThe cycles $A_I$ include the $A_i$'s wrapping around the handles\nof ${\\cal C}$ and $A_\\iota$'s, going around the singularities\nof $dS$ (for simplicity we assume that they are\nsimple poles, generalization is obvious).\nThe conjugate contours $B_I$ include the cycles $B_i$ and the\n{\\it non-closed} contours $B_\\iota$, ending at the singularities\nof $dS$ (see sect.5 of \\cite{IM2} for more details).\nThe integrals $\\int_{B_\\iota} dS$ are actually divergent, but\nthe coefficient of divergent part is equal to residue of $dS$\nat particular singularity, i.e. to $a_\\iota$. Thus the divergent\ncontribution to the prepotential is quadratic in $a_\\iota$, while\nthe prepotential is normally defined {\\it modulo} quadratic combination\nof its arguments (since only its {\\it third} derivatives appear in the\nWDVV equations). In\nparticular models $\\oint_{A,B} dS$ for some conjugate pairs of contours are\nidentically zero on entire ${\\cal M}$: such pairs are not included into our\nset of indices $\\{I\\}$.\n\nIn the context of SW theory with the data $\\{ {\\cal M}, {\\cal C}, dS\\}$\nthe number of independent $a_I$ is exactly the same as ${\\rm dim}_C{\\cal M}$.\nThis allows us to use the functions $a_I(s)$ as co-\\-ordinates on\nmoduli space ${\\cal M}$.\n\nThe self-consistency of the definition (\\ref{defprep}) of $F$,\ni.e. the fact that the matrix\n$\\frac{\\partial^2 F}{\\partial a_I\\partial a_J}$ is symmetric is guaranteed\nby the following reasoning.\nLet us take the derivative of $dS$ over moduli:\n\\ba\n\\frac{\\partial dS}{\\partial s_I} \\cong \\frac{\\partial\\lambda}\n{\\partial s_I} d\\omega = -\\frac{\\partial {\\cal P}}{\\partial s_I}\n\\frac{d\\omega}{{\\cal P}'} \\equiv dv_I.\n\\label{defdv}\n\\ea\nwhere prime denotes the $\\lambda$-derivative (for constant $x, d\\omega$ and\n$s_I$).\nNote that we did not include $\\tau$ into the set of moduli\n$\\{s_I\\}$, therefore we can take $s_I$-derivatives keeping $(x,y)$\n(and thus $d\\omega$) fixed: the variation of coordinate on bare curve would\nnot change the cohomology classes.\nThe variation of ${\\cal P}(\\lambda; x,y) = 0$\n\\ba\n{\\cal P}'d\\lambda + \\frac{\\partial{\\cal P}}{\\partial \\omega}\nd\\omega = 0,\n\\ea\nimplies that the zeroes of $d\\omega$ and ${\\cal P}'$ on ${\\cal C}$\ncoincide (since $d\\lambda$ and $d\\omega$ are supposed to have non-coinciding\nzeroes), and, therefore, $\\frac{d\\omega}{{\\cal P}'}$ is a {\\it holomorphic}\n1-differential on ${\\cal C}$. If $\\partial{\\cal P}\/\\partial s_I$ is\nnon-singular, $dv_I$ is also holomorphic\\footnote{Since curves with punctures\nand the corresponding meromorphic differentials can be obtained by\ndegeneration of smooth curves of higher genera we do not make any distinction\nbetween punctured and smooth curves below. We remind that the holomorphic\n1-differentials can have at most simple poles at the punctures while\nquadratic differentials can have certain double poles etc.}\n -- this is the case of $dv_i$, while\n$dv_\\iota$ have poles encircled by contours $A_\\iota$. The second derivative\n\\ba\n\\frac{\\partial^2 F}{\\partial a_I\\partial a_J} = F_{IJ}\n\\ea\nis essentially the period matrix of the punctured Riemann surface\n${\\cal C}$\n\\ba\n\\sum_J F_{IJ}\\oint_{A_J} dv_L=\\int_{B_I} dv_L +\\int_{{\\partial B_I\\over \\partial s_L}}\ndS\n\\ea\nThe second term at the r.h.s. can be non-vanishing if the contour $B_I$ is\nnot closed. The end-points of such contours (punctures) are included into the\nset of moduli (this actually implies that a local coordinate is specified in\nthe vicinity of the puncture). The derivative ${\\partial B_I\\over \\partial s_L}$ is\nactually a point, thus, $\\int_{{\\partial B_I\\over \\partial s_L}} dS={dS\\over\nd\\lambda}\\left(\\lambda= {\\partial B_I\\over \\partial s_L}\\right)$. This term also cancels\nthe possible divergency in $\\int_{B_I} dv_L$, making $F_{IJ}$ finite.\n\nAs any period matrix $F_{IJ}$ is symmetric:\n\\ba\n\\sum_{IJ} (F_{IJ} - F_{JI}) \\oint_{A_I}dv_K\\oint_{A_J}dv_L =\\\\=\n\\sum_I\\left(\\oint_{A_I}dv_K\\int_{B_I}dv_L -\n\\int_{B_I}dv_K\\oint_{A_I}dv_L\\right)\n+ \\sum_I\\left(\\oint_{A_I}dv_K \\int_{{\\partial B_I\\over \\partial s_L}}\n- \\int_{{\\partial B_I\\over \\partial s_K}} \\oint_{A_I}dv_L\\right)\n\\ea\nThe first sum at the r.h.s. is equal to\n\\ba\n\\sum_I\\left(\\oint_{A_I}dv_K\\int_{B_I}dv_L -\n\\int_{B_I}dv_K\\oint_{A_I}dv_L\\right)=\n{\\hbox{res}} \\left( v_K dv_L \\right) = 0\n\\ea\nsince all the singularities of $dv_K$ are already taken into account\nby inclusion of $A_\\iota$ into our set of $A$-cycles. The second sum is also\nzero because the only non-vanishing entries are those with $K=L$.\n\nIn this paper we also need the moduli derivatives of the prepotential.\nIt is easy to get:\n\\ba\n\\sum_{IJ} \\frac{\\partial F_{IJ}}{\\partial s_M}\n\\oint_{A_I}dv_K\\oint_{A_J}dv_L =\n\\sum_I\\left(\\oint_{A_I}dv_K\\int_{B_I}\\frac{\\partial dv_L}{\\partial s_M} -\n\\int_{B_I}dv_K\\oint_{A_I}\\frac{\\partial dv_L}{\\partial s_M}\\right) +\\Sigma\n_{KLM}=\n\\nonumber \\\\ =\n{\\hbox{res}} \\left(dv_K \\frac{\\partial v_L}{\\partial s_M}\\right)+\\Sigma_{KLM}\n\\label{prom1}\n\\ea\nwhere\n\\ba\\label{sigma}\n\\Sigma_{KLM}=\\sum_I\\left(\\oint_{A_I}dv_K {\\partial\\over\\partial s_M}\\int_{{\\partial B_I\\over \\partial\ns_L}}dS+ \\oint_{A_I}dv_K \\int_{{\\partial B_I\\over \\partial s_M}}dv_L-\n\\oint_{A_I}{\\partial dv_L\\over\\partial s_M} \\int_{{\\partial B_I\\over \\partial s_K}}dS\n\\right)\n\\ea\nis the contribution of the moduli dependence of punctures.\n\nThe residue in (\\ref{prom1}) does not need to\nvanish, since the derivatives w.r.t. the\nmoduli produces new singularities. Indeed, from (\\ref{defdv}) one obtains\n\\ba\n-\\frac{\\partial dv_L}{\\partial s_M} =\n\\frac{\\partial^2{\\cal P}}{\\partial s_L\\partial s_M}\n\\frac{d\\omega}{{\\cal P}'} +\n\\left(\\frac{\\partial{\\cal P}}{\\partial s_L}\\right)'\n\\left(-\\frac{\\partial{\\cal P}}{\\partial s_M}\\right)\n\\frac{d\\omega}{{\\cal P}'} -\n\\frac{\\partial{\\cal P}}{\\partial s_L}\n\\frac{\\partial{\\cal P}'}{\\partial s_M}\n\\frac{d\\omega}{({\\cal P}')^2} + \\frac{\\partial{\\cal P}}{\\partial s_L}\n\\frac{\\partial{\\cal P}}{\\partial s_M}\n\\frac{{\\cal P}''d\\omega}{({\\cal P}')^3}=\n\\nonumber \\\\\n= \\left[\n\\left(\\frac{ {\\partial{\\cal P}\/\\partial s_L}\n {\\partial{\\cal P}\/\\partial s_M} }{ {\\cal P}'} \\right)' +\n\\frac{\\partial^2{\\cal P}}{\\partial s_L\\partial s_M} \\right]\n\\frac{d\\omega}{{\\cal P}'},\n\\ea\nand this expression has new singularities (second order poles)\nat the zeroes of ${\\cal P}'$ (i.e. of $d\\omega$). Note,\nthat the contributions from the singularities of\n$\\partial{\\cal P}\/\\partial s_L$, if any, are already taken into\naccount in the l.h.s. of (\\ref{prom1}).\nPicking up the coefficient at the leading singularity, we obtain:\n\\ba\\label{resch}\n{\\hbox{res}} \\left(dv_K \\frac{\\partial v_L}{\\partial s_M}\\right) +\\Sigma_{KLM}=\n-\\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{\\partial{\\cal P}}{\\partial s_K}\n\\frac{\\partial{\\cal P}}{\\partial s_L}\n\\frac{\\partial{\\cal P}}{\\partial s_M}\n\\frac{d\\omega^2}{({\\cal P}')^3 d\\lambda} =\n\\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{dv_Kdv_Ldv_M}{d\\omega d\\lambda}\n\\ea\nThe integrals at the l.h.s. of (\\ref{prom1}) serve to convert\nthe differentials $dv_I$ into {\\it canonical} ones $dW_I$,\nsatisfying $\\oint_{A_I} dW_J = \\delta_{IJ}$. The same matrix\n$\\oint_{A_I} dv_J$ relates the derivative w.r.t. the moduli $s_I$\nand the periods $a_I$. Putting all together we obtain\n(see also \\cite{KriW}):\n\\ba\n\\frac{\\partial F_{IJ}}{\\partial s^K} =\n\\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{dW_IdW_Jdv_K}{d\\omega d\\lambda}; \\nonumber \\\\\n\\frac{\\partial^3 F}{\\partial a^I\\partial a^J\\partial a^K} =\n\\frac{\\partial F_{IJ}}{\\partial a^K} =\n\\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{dW_IdW_JdW_K}{d\\omega d\\lambda}\n\\label{resfor}\n\\ea\n\n\\subsection{Algebra of 1-differentials \\label{algebdiff}}\n\nWe consider particular examples of the algebra (\\ref{2.8}) in\nsect.\\ref{Examples}. Here we sketch some common features\nof these examples that seem to be relevant for existence of such\nalgebras and show that they always exist in the hyperelliptic case.\n\n{\\bf 1)} The linear space of differentials $dW_I$ in (\\ref{2.8})\n(or the space of $dv_I$ in (\\ref{defdv})) differs a little bit from the set\nof {\\it all} holomorphic 1-differentials\\footnote{For the gauge\ngroups $B_n$, $C_n$, $D_n$ one should take {\\it all} 1-differentials which\nare {\\it odd} under the involution $\\lambda\\to -\\lambda$.}\non punctured\n\\footnote{We mean here by\n\"punctured\" that the \"holomorphic differentials\" can have at most {\\it\nsimple} poles at punctures. } spectral curve ${\\cal C}$ for the following\nreasons. In elliptic (UV-finite) cases, when the bare spectral curve is\n$E(\\tau)$, $d\\omega$ is holomorphic on ${\\cal C}$, but it is not included\ninto the set $\\{dW_I\\}$, because the algebra of $dW_I$ is defined {\\it\nmodulo} $d\\omega$. Up to such possible corrections, the number of linearly\nindependent $dW_I$'s is $g + n$, where $g$ is the genus of ${\\cal C}$ and\n$n+1$ -- the number of punctures.\n\n{\\bf 2)}\nThe products $dW_IdW_J$ are holomorphic quadratic differentials\non the punctured curve ${\\cal C}$. There are $\\frac{(g+n)(g+n-1)}{2}$\n(or $\\frac{(g+n-1)(g+n-2)}{2}$) such products, but at most\n$3g-3+2n$ of them are linearly independent, since this is the\nquantity of independent quadratic differentials.\n\n{\\bf 3)} We want to obtain an associative algebra of 1-differentials\n$dW_I$,\n\\ba\ndW_I\\circ dW_J = \\sum_K C_{IJ}^K dW_K\n\\ea\nwhich is an ordinary algebra of multiplications\n{\\it modulo} $d\\omega$ and $d\\lambda$.\nIn addition to these two 1-differentials one should also\nspecify $dQ \\equiv \\sum_I q_IdW_I$.\nThen, any quadratic differential can be represented as their ``linear\ncombination'', namely:\n\\ba\ndW_I dW_J = \\left(\\sum_K C_{IJ}^K dW_K\\right) dQ\n+ d\\widetilde W_{IJ}^\\omega d\\omega +\nd\\widetilde W_{IJ}^\\lambda d\\lambda\n\\label{algmult}\n\\ea\nand existence of such a representation follows already from\nthe estimation of the number of independent adjustment parameters.\n\nFor example, if there are no punctures, all $dW_I$ are holomorphic (have no\npoles) and so does $d\\omega$ (which is not included into the set $\\{dW_I\\}$),\nwhile $d\\lambda$ has the second order pole in a single point (this is\nliterally the case for the Calogero model, see sect.6), then $d\\widetilde\nW_{IJ}^Q \\equiv \\sum_K C_{IJ}^K dW_K$ has $g-1$ adjustment parameters ($g-1$\nstructure constants $C_{IJ}^K$ with $K = 1,2,\\ldots,g-1$), while $d\\widetilde\nW_{IJ}^\\omega$ and $d\\widetilde W_{IJ}^\\lambda$ are holomorphic\ndifferentials, the latter one has the second order zero at the puncture,\ni.e. they form $g$- and $g-2$-parametric families respectively. Thus,\ntotally we get $(g-1) +g+ (g-2)$ adjustment parameters, exactly exhausting\nthe $3g-3$-dimensional set of independent holomorphic quadratic $dW_IdW_J$.\nAccurate analysis shows that the matching is preserved at punctured surface\nas well.\n\n\\subsection{Associativity and the WDVV equations}\nHowever, the existence of the closed algebra does not immediately imply its\nassociativity. Indeed, to check associativity one needs to consider the\nexpansion of products of the 3-differentials:\n\\ba\\label{47}\ndW_IdW_JdW_K=dW_{IJK}dQ^2+d^2\\widetilde{\\widetilde W^{\\omega}\n\\!\\!\\!}_{_{IJK}}d\\omega+\nd^2\\widetilde{\\widetilde W^{\\lambda}\\!\\!\\!}_{_{IJK}}d\\lambda\n\\ea\nwhere $dW_{IJK}dQ$ is a set of holomorphic 1-differentials not\ncontaining $d\\omega$. Now repeating the above counting, we get that there\nexist totally $(g-1)+(2g-1)+(3g-5)$, since $d^2\\widetilde{\\widetilde\nW^{\\omega}\\!\\!\\!\n}_{_{IJK}}$ are holomorphic quadratic differentials that do not contain\n$d\\lambda$, and $d^2\\widetilde{\\widetilde W^{\\lambda}\\!\\!\\!\n}_{_{IJK}}$ are holomorphic\nquadratic differentials with second order zeroes. Thus, for the $5g-5$\nindependent holomorphic 3-differentials we have too many, namely, $6g-7$\nparameters. Therefore, among the differentials $dW_{IJK}dQ$, $d^2\\widetilde\n{\\widetilde W^{\\lambda}\\!\\!\\!}_{_{IJK}}$ and $d^2\\widetilde\n{\\widetilde W^{\\lambda}\\!\\!\\!}_{_{IJK}}$ there are some relations that\ngenerally spoil associativity. We present in sect.6 the explicit example\nwhere associativity is indeed violated.\n\nThe situation simplifies different in the hyperelliptic case when there is\nthe additional ${\\bf Z}_2$-transformation $\\sigma: y\\to -y$. In this case,\nthe holomorphic differentials, ${P_k(\\lambda)d\\lambda\\over Y}$,\n$k=0,\\ldots g-1$ ($P_k$ denotes a polynomial of degree $k$) are {\\it odd}\nunder $\\sigma$ and their quadratic combinations, ${P_k(\\lambda)\n(d\\lambda)^2\\over Y^2}$,\n$k=0,\\ldots,2g-2$ are {\\it even}.\nThe remaining $g-2$ quadratic holomorphic differentials, ${P_k(\\lambda)\n(d\\lambda)^2\\over\nY}$, $k=0,\\ldots,g-3$ are $\\sigma$-odd and do not appear in the l.h.s. of\n(\\ref{algmult}). On the other hand, in the expansion (\\ref{algmult}),\none can throw away the term with $d\\lambda$ at the r.h.s., since $d\\lambda$\nis $\\sigma$-even and, therefore, the coefficient in front of it\nis not a linear combination of the holomorphic differentials $dW_I$. This\nrestores the correct counting of $2g-1$ adjusting parameters for $2g-1$\ndifferentials. It turns out that this restricted algebra spanned by the two\ndifferentials $dQ$ and $d\\omega$ is both closed and {\\it associative}.\nIndeed, if the term with $d\\lambda$ is omitted at the r.h.s. of (\\ref{47}),\nthere remain $3g-2$ adjusting parameters -- exactly as many as\nthere are holomorphic $\\sigma$-odd 3-differentials ${P_k(\\lambda)d\\lambda\\over\ny^3}$,\n$k=0,\\ldots,3g-3$ that can arise at the l.h.s. in the triple products of\n$dW_I$. In other words the algebra in hyperelliptic case is obviously\nassociative since it is isomorphic to the ring of polynomials\n$\\{ P_k(\\lambda) \\}$ -- multiplication of all polynomials (of one variable)\nmodulo some ideal.\n\nThe same counting can be easily repeated for the general surface with $n$\npunctures\\footnote{The simplest way to do this is to obtain all the\npunctures degenerating handles of the Riemann surface of higher genus.} and\nleads to the conclusion that the closed associative algebra does always exist\nfor the hyperelliptic case.\n\n\nAccording to sect.\\ref{eqs} the WDVV eqs are immediately\nimplied by the residue formula (\\ref{resfor}) and associativity of the\nalgebra (\\ref{algmult}) because the terms with $d\\omega$ at the r.h.s. of\n(\\ref{algmult}) do not contribute into (\\ref{resfor}).\n\n\\section{Examples \\label{Examples}}\n\\setcounter{equation}{0}\n\\subsection{Hyperelliptic curves in the Seiberg-Witten theory}\nThis section contains a series of examples of the algebras of 1-differentials\nand WDVV equations, arising in particular ${\\cal N}=2$ supersymmetric models\nin four and five dimensions.\nAccording to \\cite{GKMMM} and \\cite{SWI1}, models without matter\n(pure gauge theories) can be described in terms of integrable systems\nof the periodic Toda-chain family. The corresponding spectral curves\nare hyperelliptic, at least, for the\nclassical simple groups $G$. The function ${\\cal P}(\\lambda, w)$ is nothing\nbut\nthe characteristic polynomial of the dual affine algebra ${\\hat G}^{\\vee}$\nand looks like (parameter $s$ was introduced in (\\ref{adjot}))\n\\ba\n{\\cal P}(\\lambda, w) = 2P(\\lambda) - w - \\frac{Q_0(\\lambda)}{w},\n\\ \\ \\ \\ d\\omega={dw\\over w},\\ \\ \\ \\ Q_0(\\lambda)=\\lambda^{2s}\n\\label{curven}\n\\ea\nwhile\n\\ba\ndS = \\lambda \\frac{dw}{w}\n\\label{dSn}\n\\ea\nHere $P(\\lambda)$ is characteristic polynomial of the\nalgebra $G$ itself, i.e.\n\\ba\nP(\\lambda) = \\det(G - \\lambda I) =\n\\prod_i (\\lambda - \\lambda_i)\n\\ea\nwhere determinant is taken in the first fundamental representation\nand $\\lambda_i$'s are the eigenvalues of the algebraic element $G$.\nFor classical simple algebras we have:\n\\ba\nA_n:\\ \\ \\ P(\\lambda) = \\prod_{i=1}^{N_c}(\\lambda - \\lambda_i), \\ \\ \\\nN_c = n+1,\\ \\ \\ \\sum_{i=1}^{N_c} \\lambda_i = 0; \\ \\ \\\ns=0; \\nonumber \\\\\nB_n:\\ \\ \\ P(\\lambda) = \\lambda\\prod_{i=1}^n(\\lambda^2 - \\lambda_i^2);\n\\ \\ \\ s=2 \\nonumber \\\\\nC_n:\\ \\ \\ P(\\lambda) = \\prod_{i=1}^n(\\lambda^2 - \\lambda_i^2);\n\\ \\ \\ s=-2 \\nonumber \\\\\nD_n:\\ \\ \\ P(\\lambda) = \\prod_{i=1}^n(\\lambda^2 - \\lambda_i^2);\n\\ \\ \\ s=2\n\\label{charpo}\n\\ea\nFor exceptional groups, the curves arising as the characteristic polynomial\nof the dual affine algebras do not acquire the hyperelliptic form -- and\nthey are left beyond the scope of this paper (though they can still have\nenough symmetries to suit into the general theory of s.4).\n\nThe list (\\ref{charpo}) implies that one needs to analyze\nseparately the cases of $A_n$ and the other three series.\n\nAbove formulas are well adjusted for generalizations.\nIn particular, in order to include massive hypermultiplets in the\nfirst fundamental representation one can just change\n$Q_0(\\lambda)$ for $Q(\\lambda) = Q_0(\\lambda)\\prod_{\\iota = 1}^{N_f}\n(\\lambda - m_\\iota)$ if $G=A_n$ \\cite{HO} and $Q(\\lambda) = Q_0(\\lambda)\n\\prod_{\\iota = 1}^{N_f}(\\lambda^2 - m^2_\\iota)$ if $G=B_n,C_n,D_n$ \\cite{AS}\n(sometimes it can also make sense\nto shift $P(\\lambda)$ by an $m$-dependent polynomial $R(\\lambda)$, which does\nnot depend on $\\lambda_i$).\n\nThese cases of the matter hypermultiplets included can be described by the\nperiodic $XXX$-chain family \\cite{SWI2}.\n\nWorking with\nthese models it is sometimes convenient to\nchange the $w$-variable, $w \\ \\rightarrow \\ \\sqrt{Q(\\lambda)}w$\nso that the curve ${\\cal P}(\\lambda, w) = 0$ turns into\n\\ba\nw + \\frac{1}{w} = 2\\frac{P(\\lambda) + R(\\lambda)}{\\sqrt{Q(\\lambda)}}\n\\label{curven2}\n\\ea\n\nAnother possible generalization is the extension\nto $5d$ models \\cite{N}. Then it is enough to change only the\nformula (\\ref{dSn}),\n\\ba\ndS^{(5)} = \\log\\lambda\\ \\frac{dw}{w}\n\\label{dS5}\n\\ea\nand put $Q_0=\\lambda^{N_c\/2}$ while eq.(\\ref{curven}) remains intact.\nFrom the point of view of integrable systems this corresponds to changing\nthe Toda chain for the relativistic Toda chain.\n\nExamples that do not suit into this scheme are related to the\nUV-finite models, of which the simplest one is the\n$SU(N_c)$ gauge theory in $4d$ with massive hypermultiplet in the\nadjoint representation. The corresponding integrable system\nis the elliptic Calogero model, and our analysis in s.\\ref{Calogero}\nreveals that it does not satisfy WDVV equations: the closed algebra\nof holomorphic differentials still exists, but is not\nassociative -- in accordance with the general reasoning in s.4.\n\nTo conclude these short remarks, let us note that, as we shall see below,\nassociativity of algebras of the holomorphic 1-differentials on\nhyperelliptic curve reduces afterwards to associativity of the polynomial\nalgebras. Therefore, we consider here the reference pattern of such an\nalgebra. Namely, we look at the algebra of $N$ one-variable polynomials\n$p_{i}(\\lambda)$ of degree $N-1$ whose product defined by modulo\nof a polynomial $P'$ of degree $N$. More concretely, the algebra\nis defined as\n\\ba\\label{polass}\np_i(\\lambda)p_j(\\lambda)=C^k_{ij}p_k(\\lambda)W(\\lambda)+\\xi_{ij}(\\lambda)\nP'(\\lambda)\n\\ea\nwhere $W(\\lambda)$ is some fixed linear combination of $p_i(\\lambda)$:\n$W(\\lambda)\\equiv \\sum q_lp_l(\\lambda)$\nco-prime with $P'(\\lambda)$. Then, the l.h.s. of this relation is a\npolynomial of degree $2(N-1)$ with $2N-1$ coefficients which are to be\nadjusted by the free parameters in the r.h.s. Since $\\xi_{ij}$ is of degree\n$N-2$, there are exactly $N+(N-1)=2N-1$ free parameters. The same counting is\ncorrect for the triple products of the polynomials and the algebra\nis associative, since the condition $P'(\\lambda)=0$ leads to the\nfactor-algebra over the ideal $P'(\\lambda)$. The same argument does not work\nfor the algebra of holomorphic 1-differentials, since their product belongs\nto a different space (of quadratic differentials) -- see \\cite{MMM2}.\n\n\\subsection{Perturbative limit of the pure gauge theories}\nNow we discuss the simplest case of curves corresponding to\nthe perturbative formulas of sect.3.\nPerturbative limit of the curve (\\ref{curven2}) is\n\\ba\\label{pertcurv}\nw = 2\\frac{P(\\lambda)}{\\sqrt{Q(\\lambda)}}\n\\ea\nand the corresponding\n\\ba\\label{pertdS}\ndS^{(4)} = \\lambda d\\log\\left(\\frac{P(\\lambda)}{\\sqrt{Q(\\lambda)}}\\right);\n\\nonumber \\\\\ndS^{(5)} = \\log\\lambda\\ d\\log\\left(\\frac{P(\\lambda)}{\\sqrt{Q(\\lambda)}}\\right)\n\\ea\nWe consider first the simplest case of the pure gauge $SU(N_c)$ Yang-Mills\ntheory, i.e. $Q(\\lambda)=Q_0(\\lambda)=1$.\nThe both cases of $4d$ and $5d$ theories\ncan be treated simultaneously. Indeed, they are described by the same Riemann\nsurface (\\ref{pertcurv}) which is nothing but\nthe sphere with punctures\\footnote{These punctures emerge as a\ndegeneration of the handles of the hyperelliptic surface so that the\n$a$-cycles encircle the punctures.}, the only difference being the\nrestrictions imposed onto the set of moduli $\\{\\lambda_i\\}$'s --\nin the $4d$ case\nthey are constrained to satisfy the condition $\\sum_i^{N_c} \\lambda_i=0$,\nwhile in\nthe $5d$ one -- $\\prod_i^{N_c} \\lambda_i=1$.\nNow the set of the $N_c-1$ independent\ncanonical holomorphic differentials is\\footnote{It can be instructive to\ndescribe in more details why (\\ref{difff}) arises not only in $4d$, but also\nin $5d$ models. Strictly speaking, in the $5d$ case, one should consider\ndifferentials on annulus, not on sphere. Therefore, instead of\n${d\\lambda\\over\\lambda-\\lambda_i}$, one rather needs to take\n$\\displaystyle{\\sum_{m=-\\infty}^{+\\infty}{da\\over\na-a_i+2n}\\sim\\coth{a-a_i\\over 2}{da\\over\n2}={\\lambda+\\lambda_i\\over\\lambda-\\lambda_i}{d\\lambda\\over 2\\lambda}}$, where\n$\\lambda\\equiv e^a$. Considering now, instead of\n$\\displaystyle{d\\omega_i={d\\lambda\\over\n\\lambda-\\lambda_i}-{d\\lambda\\over\\lambda-\\lambda_{N_c}}={\\lambda_{iN_c}\nd\\lambda\\over (\\lambda-\\lambda_i)(\\lambda-\\lambda_{N_c}}}$, differentials\n$\\displaystyle{d\\omega_i={\\lambda+\\lambda_i\\over\\lambda-\\lambda_i}\n{d\\lambda\\over 2\\lambda} -{\\lambda+\\lambda_{N_c}\\over\\lambda-\\lambda_{N_c}}\n{d\\lambda\\over\n2\\lambda}= {\\lambda_{iN_c}d\\lambda\\over\n(\\lambda-\\lambda_i)(\\lambda-\\lambda_{N_c})}}$, we obtain exactly formula\n(\\ref{difff}).}\n\n\\ba\\label{difff}\nd\\omega_i=\\left({1\\over\\lambda-\\lambda\n_i}-{1\\over\\lambda -\\lambda_{N_c}}\\right)d\\lambda=\n{\\lambda_{iN_c}d\\lambda\\over\n(\\lambda -\\lambda_i)(\\lambda -\\lambda_{N_c})},\\ \\ \\ i=1,...,N_c,\n\\ \\ \\ \\lambda_{ij}\\equiv \\lambda_i-\\lambda_j\n\\ea\nand one can easily check that $\\lambda_i=a_i$ ($a$-periods) in the $4d$ case\nand $\\lambda_i=e^{a_i}$ in the $5d$ case, i.e. $\\sum_i a_i=0$.\n\nThe next step is to expand the product of two such differentials\n\\ba\nd\\omega_id\\omega_j=\\left(C_{ij}^kd\\omega_k\\right)\\left(q_ld\\omega_l\\right)+\n\\left(D^k_{ij}d\\omega_k\\right)d\\log P(\\lambda),\\ \\ \\ d\\log P(\\lambda)=\n\\sum_l^{N_c}{d\\lambda\\over\\lambda -\\lambda_l}\n\\ea\nConverting this expression to the normal form with\ncommon denominator $P(\\lambda)$,\none reproduces (\\ref{polass}) for polynomial algebra.\nTherefore, the structure constants $C_{ij}^k$\nform the closed associative algebra and, in\naccordance with our general formulas, can be expressed through the\nprepotential by residue formulas\n\\ba\nF_{ijk}=\\stackreb{d\\log P=0}{\\hbox{res}} {d\\omega_id\\omega_jd\\omega_k\\over d\\log\nP(\\lambda)d\\lambda} \\ \\ \\ \\hbox{for the $4d$ case},\\\\\nF_{ijk}=\\stackreb{d\\log P=0}{\\hbox{res}} {d\\omega_id\\omega_jd\\omega_k\\over d\\log\nP(\\lambda){d\\lambda\\over\\lambda}} \\ \\ \\ \\hbox{for the $5d$ case}\n\\ea\nThese residues can be easily calculated. The only technical trick is to\ncalculate the residues not at zeroes of $d\\log P$ but at poles of\n$d\\omega$'s. This can be done immediately since there are no contributions\nfrom the infinity $\\lambda=\\infty$.\nThe final results have the following form\n\\ba\n\\hbox{\\underline{for the $4d$ case}}:\\\\\nF_{iii}=\\sum_{k\\ne i}{1\\over a_{ik}}+{6\\over a_{iN_c}}+\n\\sum_{k\\ne N_c}{1\\over a_{kN_c}},\n\\\\\nF_{iij}= {3\\over a_{iN_c}}+{2\\over a_{jN_c}}+\\sum_{k\\ne i,j,N_c}\n{1\\over a_{kN_c}}-{a_{jN_c}\\over a_{iN_c}a_{ij}},\n\\ \\ \\ i\\ne j,\\\\\nF_{ijk}=2\\sum_{l\\ne N_c}{1\\over a_{lN_c}}-\\sum_{l\\ne i,j,k,N_c}\n{1\\over a_{lN_c}},\n\\ \\ \\ i\\ne j\\ne k;\n\\ea\n\n\\ba\n\\hbox{\\underline{for the $5d$ case}}:\\\\\n2F_{iii}=\\sum_{k\\ne i}\\coth a_{ik}+\n6\\coth a_{iN_c}+\\sum_{k\\ne N_c}\\coth\n a_{kN_c},\\\\\n2F_{iij}= -\\coth a_{ij}+\n4\\coth a_{iN_c}+2\\coth a_{jN_c}+\\sum_{k\\ne\ni,j,N_c}\\coth a_{kN_c}+N_c,\\ \\ \\ i\\ne j,\\\\\n2F_{ijk}= 2\\sum_{l\\ne N_c}\\coth a_{lN_c}-\\sum_{l\\ne i,j,k,N_c}\\coth\n a_{lN_c}+N_c,\\ \\ \\ i\\ne j\\ne k\n\\ea\nNow it is immediate\nto check that the prepotential in the $4d$ case really is given by the\nformula (\\ref{FA}), while in the $5d$ case -- by (\\ref{FARTC}).\nThese results are in perfect agreement with the results of s.3.\n\nNow let us turn to the case of massive hypermultiplets included and, for the\nsake of simplicity, restrict ourselves to the $4d$ case. Then, there\narise additional differentials corresponding to the derivatives of $dS$\n(\\ref{pertdS}) with respect to masses. They are of the form\n\\ba\\label{massdiff}\ndW_{\\iota}=-{1\\over 2}{d\\lambda\\over \\lambda -\\lambda_{\\iota}}\n\\ea\nAgain, full set of the differentials gives rise to\nthe closed associative algebra.\nAs earlier, this can be proved by reducing the system of differentials to the\npolynomial one (this time the common denominator should be\n$P(\\lambda)Q(\\lambda)$). Then, applying the residue formula, one reproduces\nformula (\\ref{26}) for the perturbative prepotential with $r=1$. In order to\nobtain (\\ref{26}) with arbitrary coefficient $r$, one needs to consider\ninstead of $\\sqrt{Q(\\lambda)}$ in (\\ref{pertcurv})-(\\ref{pertdS})\n$Q(\\lambda)^{r\/2}$. This only changes normalization of the differentials\n(\\ref{massdiff}): $dW_{\\iota}=-{r\\over 2}{d\\lambda\\over \\lambda -\n\\lambda_{\\iota}}$. Now it is straightforward\nto use the residue formula\\footnote{Let us note that, despite\nthe differentials\n$dW_{\\iota}$ have the pole at infinity, this does not contribute\ninto the residue formula because of the quadratic pole of $d\\lambda$ in the\ndenominator.}\n\n\\ba\\label{residueQ}\nF_{IJK}=\\stackreb{d\\log {P\\over\\sqrt{Q}}=0}{\\hbox{res}}\n{dW_IdW_JdW_K\\over d\\log {P(\\lambda)\\over\\sqrt{Q(\\lambda)}}d\\lambda},\n\\ \\ \\ \\ \\\n\\left\\{I,J,K,\\dots\\right\\}=\\left\\{i,j,k,\\ldots|\\iota,\\iota',\n\\iota'',\\ldots\\right\\}\n\\ea\nand obtain\n\\ba\\label{perth}\nF_{iii}=\\sum_{k\\ne i}{1\\over \\lambda_{ik}}+{6\\over\n\\lambda_{iN_c}}+ \\sum_{k\\ne N_c}{1\\over\n\\lambda_{kN_c}}-{r\\over 2}\\sum_{\\iota}{\\lambda_{iN_c}\\over\n\\lambda_{i{\\iota}}\\lambda_{{\\iota}N_c}},\n\\\\\nF_{iij}= {3\\over \\lambda_{iN_c}}+{2\\over \\lambda_{jN_c}}+\\sum_{k\\ne i,j,N_c}\n{1\\over \\lambda_{kN_c}}-\n{\\lambda_{jN_c}\\over \\lambda_{iN_c}\\lambda_{ij}}-{r\\over 2}\\sum_{\\iota}{1\\over\n\\lambda_{{\\iota}N_c}},\n\\ \\ \\ i\\ne j,\\\\\nF_{ijk}=2\\sum_{l\\ne N_c}{1\\over \\lambda_{lN_c}}-\\sum_{l\\ne i,j,k,N_c}\n{1\\over \\lambda_{lN_c}}-{r\\over 2}\\sum_{\\iota}{1\\over\\lambda_{{\\iota}N_c}},\n\\ \\ \\ i\\ne j\\ne k;\\\\\nF_{ii{\\iota}}= {r\\over 2}\\left({1\\over\\lambda_{i{\\iota}}}-{1\\over\\lambda_{{\\iota}N_c}}\n\\right);\\\\\nF_{ij{\\iota}}=-{r\\over 2}{1\\over\\lambda_{{\\iota}N_c}},\\ \\ \\ i\\ne j;\\\\\nF_{i{\\iota}{\\iota}}=-{r\\over 2}{\\lambda_{iN_c}\\over\\lambda_{i{\\iota}}\n\\lambda_{{\\iota}N_c}};\\\\\nF_{{\\iota}{\\iota}{\\iota}}=\n{r\\over 2}\\sum_i{1\\over\\lambda_{{\\iota}i}}+{r^2\\over 4}\\sum_{{\\iota}'}\n{1\\over\\lambda_{{\\iota}{\\iota}'}};\\\\\nF_{{\\iota}{\\iota}{\\iota}'}=\n-{r^2\\over 4}{1\\over \\lambda_{{\\iota}{\\iota}'}},\\ \\ \\ {\\iota}\\ne {\\iota}';\\\\\nF_{i{\\iota}{\\iota}'}=F_{{\\iota}{\\iota}'{\\iota}''}=0\n\\ea\nThese formulas immediately lead to the prepotential (\\ref{26}) upon the\nidentification $\\lambda_i=a_i$ and $\\lambda_{\\iota}=m_{\\iota}$.\n\nAt last, let us consider the case of other classical groups. We also take into\naccount the massive hypermultiplets. Let us also note that, in this case, the\ncurve is invariant w.r.t. the involution $\\rho: \\lambda\\to\n-\\lambda$. Among all the holomorphic differentials there is a subset of the\n$\\rho$-odd ones that can be obtained by differentiating the $\\rho$-odd\ngenerating differential $dS\\cong \\lambda\\left(\\sum_i {2\\lambda d\\lambda\\over\n\\lambda^2-a_i^2}-r\\sum_{\\iota} {\\lambda d\\lambda\\over \\lambda^2-m_{\\iota}^2}-\ns{d\\lambda\\over\\lambda}\\right)$ w.r.t. the moduli:\n\\ba\ndW_i={2a_i d\\lambda\\over \\lambda^2-a_i^2},\\ \\ \\ \\\ndW_{\\iota}=-{r m_{\\iota} d\\lambda\\over \\lambda^2-m_{\\iota}^2}\n\\ea\nLook at the algebra of these differentials:\n\\ba\\label{ch}\n{d\\lambda\\over \\lambda^2-\\lambda_I^2} {d\\lambda\\over \\lambda^2-\\lambda_J^2}\n\\sim C_{IJ}^K {d\\lambda\\over \\lambda^2-\\lambda_K^2} dW+\n\\xi_{IJ}(\\lambda)\n\\left(\\sum_i {2\\lambda d\\lambda\\over \\lambda^2-a_i^2}-r\\sum_{\\iota}\n{\\lambda d\\lambda\\over \\lambda^2-m_{\\iota}^2}-s{d\\lambda\\over\\lambda}\\right)\n\\ea\nTo understand that this algebra is closed and associative, it is sufficient\nto note that the l.h.s. of (\\ref{ch}) is a linear combination of the\ndifferentials ${(d\\lambda)^2\\over \\lambda^2-\\lambda_I^2}$ and ${(d\\lambda)^2\n\\over (\\lambda^2-\\lambda_I^2)^2}$ -- totally $2N\\equiv 2n+2N_f$ independent\nquadratic $\\rho$-even differentials. However, one should also take into\naccount the condition which cancels the pole at infinity of\n${(d\\lambda)^2\\over \\lambda^2-\\lambda_I^2}$. Thus, the number of\n{\\it holomorphic} $\\rho$-even quadratic differentials is equal to $2N-1$.\nThis quantity perfectly suits the $2N-1$ adjusting parameters in the r.h.s.\nof (\\ref{ch}) -- $N$ $\\rho$-odd differentials in the first term and $N-1$\n$\\rho$-even differentials in the second term. Indeed, even differentials are\n${\\lambda d\\lambda\\over \\lambda^2-\\lambda_I^2}$, i.e. have the pole at\ninfinity. The same does the\n${dw\\over w}=\\left(\\sum_i {2\\lambda d\\lambda\\over \\lambda^2-a_i^2}-\\sum_{\\iota}\n{r\\lambda d\\lambda\\over \\lambda^2-m_{\\iota}^2}-s{d\\lambda\\over\\lambda}\\right)$.\nHowever, since all these quantities can be rewritten as depending on\n$\\lambda^2$, the cancellation of the pole of the differentials leads to\ntheir large-$\\lambda$ behaviour ${d\\lambda\\over\\lambda^3}$ that\nsimultaneously cancels the pole of ${dw\\over w}$. Thus, there is exactly one\ncondition imposed onto $N$ $rho$-even differentials.\n\nThe associativity is checked in complete analogy with the above\nconsideration -- there are $3N-3$ holomorphic $\\rho$-odd cubic differentials\nand exactly $N+(2N-3)=3N-3$ adjusting parameters.\nThus, the algebra (\\ref{ch}) is really closed associative\\footnote{In sect.5.5\nwe show how this algebra reduces to the\npolynomial one.}. The application of the residue formula (\\ref{residueQ})\ngives now (\\ref{adjot}).\n\nNow we turn to the less trivial examples corresponding to full hyperelliptic\ncurves.\n\n\\subsection{Pure gauge model (Toda chain system)}\nThis example was analyzed in detail in ref.\\cite{MMM}.\n\nThe spectral curve has the form\n\\ba\\label{specTC}\n{\\cal P}(\\lambda,w)=2P(\\lambda)-w-{1\\over w},\\ \\ \\\nP(\\lambda ) \\equiv \\sum^{N_c}_{k=0} s_k\\lambda ^k\n\\ea\nand can be represented in the standard hyperelliptic form after the change of\nvariables $Y = \\frac{1}{2}\\left(w - \\frac{1}{w}\\right)$\n\\ba\nY^2 = P^2(\\lambda ) - 1\n\\ea\nand is of genus $g = N_c-1$\n\\footnote{One can take for the $g$ moduli\nthe set $\\{h_k\\}$ or instead the set of periods $\\{a_i\\}$.\nThis particular family is associated with the Toda-chain\nhierarchy, $N_c$ being the length of the chain.}.\nNote that $s_{N_c}=1$ and that the condition of the unit determinant of the\nmatrix in the group $SU(N_c)$ implies here that $s_{N_c-1}=0$ (this is\nequivalent to the condition $\\sum a_i=0$).\nThe generating differential $dS$ has the form\n\\ba\\label{dSTC}\ndS=\\lambda{dw\\over w}=\\lambda{dP\\over Y}\n\\ea\n\nNow let us discuss associativity of the algebra of holomorphic differentials\n\\ba\nd\\omega_i(\\lambda )d\\omega_j(\\lambda ) =\nC_{ij}^k d\\omega_k(\\lambda )\ndW(\\lambda ) \\ {\\hbox{mod}}\\ {dw\\over w}\n\\label{.}\n\\ea\nWe require that $dW(\\lambda)$ and $dP(\\lambda )$ are co-prime.\n\nIf the algebra (\\ref{.}) exists, the structure constants\n$C_{ij}^k$ satisfy the associativity condition.\nBut we still need to show that\nit indeed exists, i.e. that once $dW$ is given, one can find\n($\\lambda $-independent) $C_{ij}^k$. This is a simple exercise:\nall $d\\omega_i$ are linear combinations of\n\\ba\\label{match}\ndv_k(\\lambda ) = \\frac{\\lambda ^{k}d\\lambda }{Y}\n\\cong {\\partial dS\\over\\partial s_k}, \\ \\ \\ k=0,\\ldots,g-1\n\\ea\nwhere\n\\ba\ndv_k(\\lambda ) = \\sigma_{ki}d\\omega_i(\\lambda ), \\ \\ \\\nd\\omega_i = (\\sigma^{-1})_{ik}dv_k, \\ \\ \\\n\\sigma_{ki} = \\oint_{A_i}dv_k,\n\\label{sigmadef}\n\\ea\nalso $dW(\\lambda ) = q_kdv_k(\\lambda )$.\nThus, (\\ref{.}), after multiplying all 1-differentials by\n$Y$, reduces to the algebra of polynomials\n$\\left(\\sigma^{-1}_{ii'}\\lambda ^{i'-1}\\right)$ that is as usual closed and\nassociative. This means that $C_{ij}^k$ satisfy the\nassociativity condition\n\\ba\nC_iC_j = C_j C_i\n\\ea\nNow the residue formula has the form \\cite{MMM}:\n\\ba\nF_{ijk} = \\frac{\\partial^3F}{\\partial a_i\\partial a_j\n\\partial a_k} = \\frac{\\partial F_{ij}}{\\partial a_k} = \\nonumber \\\\\n= \\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_id\\omega_j\nd\\omega_k}{d\\lambda\\left(\\frac{dw}{w}\\right)} =\n\\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_id\\omega_j\nd\\omega_k}{d\\lambda\\frac{dP}{Y}} =\n\\sum_{\\alpha} \\frac{\\hat\\omega_i(\\lambda_\\alpha)\\hat\\omega_j\n(\\lambda_\\alpha)\\hat\\omega_k(\\lambda_\\alpha)}{P'(\\lambda_\\alpha)\n\/\\hat Y(\\lambda_\\alpha)}\n\\label{v}\n\\ea\nThe sum at the r.h.s. goes over all the $2g+2$ ramification points\n$\\lambda_\\alpha$ of the hyperelliptic curve (i.e. over the zeroes\nof $Y^2 = P^2(\\lambda )-1 = \\prod_{\\alpha=1}^N(\\lambda - \\lambda_\\alpha)$);\n\\ $d\\omega_i(\\lambda) = (\\hat\\omega_i(\\lambda_\\alpha) +\nO(\\lambda-\\lambda_\\alpha))d\\lambda$,\\ $\\ \\ \\ \\hat Y^2(\\lambda_\\alpha) =\n\\prod_{\\beta\\neq\\alpha}(\\lambda_\\alpha - \\lambda_\\beta)$.\n\nNow following the line of sect.2, we can\ndefine the metric:\n\\ba\n\\eta_{kl}(dW) =\n\\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_kd\\omega_l\ndW}{d\\lambda\\left(\\frac{dw}{w}\\right)} =\n\\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_kd\\omega_l\ndW}{d\\lambda\\frac{dP}{Y}} = \\\\ =\n\\sum_{\\alpha} \\frac{\\hat\\omega_k(\\lambda_\\alpha)\\hat\\omega_l\n(\\lambda_\\alpha)\\hat W(\\lambda_\\alpha)}{P'(\\lambda_\\alpha)\n\/\\hat Y(\\lambda_\\alpha)}\n\\label{vv}\n\\ea\nIn particular, for $dW = d\\omega_k$ (which is evidently co-prime with\n${dP\\over Y}$), $\\eta_{ij}(d\\omega_k) = F_{ijk}$ -- this choice immediately\ngive rise to the WDVV in the form (5) of paper \\cite{MMM}.\n\nGiven (\\ref{.}), (\\ref{v}) and (\\ref{vv}), one can check that\n\\ba\nF_{ijk} = \\eta_{kl}(dW)C_{ij}^k(dW).\n\\label{vvv}\n\\ea\nNote that $F_{ijk} = {\\partial^3F\\over\\partial a_i\\partial a_j\n\\partial a_k}$ at the l.h.s. of (\\ref{vvv}) is independent\nof $dW$! The r.h.s. of (\\ref{vvv}) is equal to:\n\\ba\n\\eta_{kl}(dW)C_{ij}^l(dW) =\n\\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_kd\\omega_l\ndW}{d\\lambda\\left(\\frac{dw}{w}\\right)} C_{ij}^l(dW)\n\\stackrel{(\\ref{.})}{=} \\\\ =\n\\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_k}\n{d\\lambda\\left(\\frac{dw}{w}\\right)}\n\\left(d\\omega_id\\omega_j - p_{ij}\\frac{dPd\\lambda}{Y^2}\\right) =\nF_{ijk} - \\stackreb{d\\lambda =0}{{\\hbox{res}}} \\frac{d\\omega_k}\n{d\\lambda\\left(\\frac{dP}{Y}\\right)}p_{ij}(\\lambda)\\frac{dPd\\lambda}{Y^2}\n= \\\\ = F_{ijk} - \\stackreb{d\\lambda =0}{{\\hbox{res}}}\n\\frac{p_{ij}(\\lambda )d\\omega_k(\\lambda)} {Y}\n\\ea\nIt remains to prove\nthat the last item is indeed vanishing for any $i,j,k$.\nThis follows from the\nfact that $\\frac{p_{ij}(\\lambda )d\\omega_k(\\lambda )}{Y}$\nis singular only at zeroes of $Y$, it is not singular at\n$\\lambda =\\infty$ because\n$p_{ij}(\\lambda)$ is a polynomial of low enough degree\n$g-2 < g+1$. Thus the sum of its residues at ramification points\nis thus the sum over {\\it all} the residues and therefore vanishes.\n\nThis completes the proof of associativity condition for any $dW$.\n\n\\subsection{$5d$ pure gauge theory (Relativistic Toda chain)}\nThe above proof can be almost literally transferred to the case of the\nrelativistic Toda chain system corresponding to the $5d$ $N=2$ SUSY pure gauge\nmodel with one compactified dimension \\cite{N}. The main reason is\nthat in the case of relativistic Toda chain the spectral curve is a minor\nmodification of (\\ref{specTC}) having the form \\cite[eq.(2.23)]{N}\n\\ba\\label{specRTC}\nw + {1\\over w} = \\left(\\zeta\\lambda\\right)^{-N_c\/2}P(\\lambda ),\n\\ea\nwhich can be again rewritten as a {\\em hyperelliptic} curve in terms of the\nnew variable $Y\\equiv \\left(\\zeta\\lambda\\right)^{N_c\/2}\\left(w-\n{1\\over w}\\right)$\n\\ba\nY^2 = P^2(\\lambda ) - 4\\zeta^{2N_c}\\lambda^{N_c}\n\\ea\nwhere $\\lambda \\equiv e^{2\\xi}$, $\\xi $ is the \"true\" spectral\nparameter of the relativistic Toda chain and $\\zeta$ is its coupling constant.\n\nThe difference with the $4d$ case is at the following two points.\nThe first is that this time $s_0\\sim \\prod e^{a_i}=1$\n\\footnote{This formula looks quite natural since the relativistic Toda chain\nis a sort of \"group generalization\" of the ordinary Toda chain.} but instead\n$s_{N_c-1}$ no longer vanishes and becomes a modulus.\n\nThe second new point is that the generating differential instead of\n(\\ref{dSTC}) is now\n\\ba\\label{dSRTC}\ndS^{(5)} = \\xi{dw\\over w} \\sim \\log\\lambda {dw\\over w}\n\\ea\nso that its periods are completely different from those of $dS^{(4)}$.\nHowever, the set of holomorphic differentials\n\\ba\ndv_k = {\\partial dS^{(5)}\n\\over\\partial s_k} \\cong {\\lambda^{k-1}d\\lambda\\over Y},\n\\ \\ \\ k=1,...,g\n\\ea\nliterally coincides with the set (\\ref{match}), because $k$ now runs through\n$\\ 1,\\ldots,g\\ $ instead of $\\ 0,\\ldots,g-1\\ $ but there is the extra factor of\n$\\lambda$ in the denominator. Thus, the algebra of differentials remains just\nthe same closed associative algebra, but the residue formula is a little\nmodified.\n\nIndeed, when passing from the first line of (\\ref{resfor}) to the second\nline, there will arise the extra factor $\\lambda$ due to the shift of the\nindex of differentials $dv_k$ by 1 in the $5d$ case as compared with the $4d$\none. Thus, finally the residue formula acquires the following form\n\\ba\nF_{ijk} =\n\\stackreb{d\\lambda=0}{{\\hbox{res}}} \\frac{d\\omega_id\\omega_j\nd\\omega_k}{\\left(\\frac{d\\lambda}{\\lambda}\\right)\\left(\\frac{dw}{w}\\right)} =\n\\stackreb{d\\lambda=0}{{\\hbox{res}}} \\frac{d\\omega_id\\omega_j\nd\\omega_k}{\\left(\\frac{d\\lambda}{\\lambda}\\right)\\left(\\frac{dP}{Y}\\right)} =\n\\sum_{\\alpha} \\lambda_{\\alpha}\\frac{\\hat\\omega_i(\\lambda_\\alpha)\\hat\\omega_j\n(\\lambda_\\alpha)\\hat\\omega_k(\\lambda_\\alpha)}{P'(\\lambda_\\alpha)\n\/\\hat Y(\\lambda_\\alpha)}\n\\ea\n\n\\subsection{Pure gauge theory with $B_n$, $C_n$ and $D_n$ gauge groups}\nNow we consider shortly the case of the pure gauge theory with the groups\n$G\\ne A_n$. This example is of importance, since it demonstrates that, in the\ncases when naively there is no closed associative algebra of holomorphic\n1-differentials, it really exists due to some additional symmetries of the\ncurve.\n\nThe curve in this case has the form (\\ref{curven}) with $Q_0(\\lambda)\n=\\lambda^{2s}$ as in\n(\\ref{charpo}), while the generating differential -- (\\ref{dSn}). After\nchange of variables\n\\ba\n2Y(\\lambda)=w-{Q_0(\\lambda)\\over w}\n\\ea\nthe curve acquires the standard hyperelliptic form\n\\ba\\label{hra}\nY^2=P^2(\\lambda)-Q_0(\\lambda)\n\\ea\nNaively, we can repeat the procedure applied in the previous examples. The\nnew problem, however, is that the genus of this curve now is $2n-1$ (see\n(\\ref{charpo})), while there are only $n$ moduli, and, therefore, $n$\nholomorphic differentials which can be obtained from the generating\ndifferential (\\ref{dSn}). Therefore, one might expect that the algebra formed\nby these differentials is too small to be closed (and associative).\n\nHowever, it turns out that, due to the additional $Z_2$-symmetry, this is not\nthe case. Indeed, let us note that the curve (\\ref{hra}) admits the\ninvolution $\\rho:\\lambda\\to -\\lambda$ (in addition to that $Y\\to -Y$).\nTherefore, in the complete set of holomorphic $\\sigma$-odd\ndifferentials on the\nhyperelliptic curve (\\ref{hra}) (see sect.3.4)\n$dv_i={\\lambda^i d\\lambda\\over Y}$,\n$i=0,\\ldots,2n-2$, there are $n-1$ $\\rho$-even and $n-1$ $\\rho$-odd\ndifferentials. Keeping in mind the rational examples of sect.5.2, we should\nlook at the subalgebra formed by the $\\rho$-odd differentials that only can be\nobtained from the generating differential $dS$, i.e. at $dv_i$ with $i=2k$ --\nor at the canonical $\\rho$-odd differentials $d\\omega^o_{i}$\n\\ba\\label{pred}\nd\\omega^o_id\\omega^o_j=C^k_{ij}d\\omega^o_kdW\\ \\ \\hbox{mod}\\;{dw\\over w}\n\\ea\nSince ${dw\\over w}$ is $\\rho$-even, it can be multiplied only by $\\rho$-even\nholomorphic differentials. On the other hand, there are $2n-2$ $\\rho$-even\nquadratic holomorphic differentials. This perfectly matches the number of the\nfree adjusting parameters at the r.h.s. of {\\ref{pred}), i.e. algebra\n(\\ref{pred}) is really closed (and associative, as one can obtain by the\nanalogous counting).\n\nThe fact of closeness and associativity of this\nalgebra can be demonstrated in\nanother and easier way -- like it has been done in our previous examples.\nIndeed, let us multiply all the differentials by $\\lambda Y$ that is the\ncommon denominator (since ${dw\\over w}={dP\\over Y}+\ns\\left(1-{P\\over Y}\\right){d\\lambda\\over \\lambda}$). Then, the algebra is\nreduced (as usual in the hyperelliptic case) to the polynomial algebra. More\nprecisely, this is the closed associative algebra of $n-1$ odd polynomials\n$p_i^o$ of degree $2n-3$ factorized over the ideal which is an even polynomial\n$\\lambda P'(\\lambda)$ of the degree $2n$:\n\\ba\np_i^o(\\lambda)p_j^o(\\lambda)=C^k_{ij}p_k^o(\\lambda)W(\\lambda)\n+\\xi_{ij}^e(\\lambda)\\lambda P'(\\lambda)\n\\ea\nwhere $\\xi_{ij}^e$ is some even polynomial. The degree of this expression is\n$4n-6$, at the l.h.s. of it there are $2n-3$ independent coefficients of the\neven polynomial of degree $4n-6$ with no constant term, while at the r.h.s.\nthere are $n-1$ adjusting parameters $C_{ij}^k$ and $n-2$ adjusting\ncoefficients of the polynomial $\\xi^e_{ij}(\\lambda)$, totally $2n-3$ free\nparameters. This counting proves that the described algebra is closed (and\nassociative as polynomial algebra).\n\n\\subsection{Hypermultiplets in the fundamental representation\n(Spin chains)}\nTo conclude the section, we consider the example of hyperelliptic curve\nassociated with the $SU(N_c)$ Yang-Mills theory with mass\nhypermultiplets.\n\nThe set of holomorphic 1-differentials $\\{d\\omega_i\\}$ relevant\nin the Toda chain case (s.5.3 and \\cite{MMM}) should\nbe now enlarged to include meromorphic ones, with the simple poles at\nthe points $m_{\\iota}^\\pm$. The spectral curve is given by:\n\\ba\\label{qcd}\n{\\cal P}(\\lambda,w) = 2P(\\lambda) -\nw - \\frac{Q(\\lambda)}{w}\n\\ea\nwhere\n\\ba\nP(\\lambda) = \\sum_{k=0}^{N_c} s_k\\lambda^k + R(\\lambda) \\ \\ \\ \\ \\ \\\nQ(\\lambda) = \\prod_{s=1}^{N_f} (\\lambda - m_{\\iota})\n\\ea\n$s_k$ are the Schur polynomials of the Hamiltonians $h_k$ and do not\ndepend on $m_{\\iota}$, while\n$R(\\lambda)$ is instead an $h_k$-independent polynomial in $\\lambda$\nof degree $N_c-1$, which is non-vanishing only for $N_f>N_c$\n\\cite{HO} (see also \\cite{SWI2}).\nIt is convenient to introduce another function $Y$\n\\ba\n2Y = w - \\frac{Q(\\lambda)}{w}, \\ \\ \\ \\ \\\nY^2 = P^2(\\lambda) - Q(\\lambda) = \\prod(\\lambda - \\lambda_\\alpha)\n\\ea\nThe generating differential\n\\ba\\label{dS1}\ndS = \\lambda\\frac{dw}{w} = \\lambda\\left(\n\\frac{dP}{Y} - \\frac{PdQ}{2QY} + \\frac{dQ}{2Q}\\right)\n\\ea\nsatisfies the defining property\n\\ba\n\\frac{\\partial dS}{\\partial a_k} = d\\omega_k\\ \\\n\\left(\\{ dW_a\\} = \\frac{Pol}{QY}d\\lambda\\right)\n\\ea\nand the algebra of multiplication acquires the form:\n\\ba\ndW_a dW_b = C_{ab}^c dW_cdW \\ {\\hbox{mod}}\\ \\frac{dw}{w}\n\\ea\n$a = 1,\\ldots,g+ N_f = \\{i,s\\}$, $g = N_c-1$. The general formulas in the\nparticular case (\\ref{qcd}) are\n\\ba\n{\\partial dS\\over \\partial a_I} \\cong\n{{\\partial P\\over\\partial a_I}- {1\\over 2w}{\\partial Q\\over\\partial a_I}\n\\over {P'-{Q'\\over 2w}}}{dw\\over w}\n\\equiv {\\cal T}_I{dw\\over {\\cal P}'w}\n\\ea\nand\n\\ba\n{dw\\over w} = {dP\\over Y} - {dQ\\over 2Y(P+Y)}\n\\nonumber \\\\\n\\left.Y{dw\\over w}\\right|_{Y=0} = dP - {dQ\\over 2P}\n\\ea\nThe calculations (along the lines of \\cite{MMM}) lead to the same conclusion:\n\\ba\na_k = \\oint_{A_k} dS, \\ \\ \\ \\\n\\oint \\frac{\\partial\\ dS}{\\partial s_l} = \\oint\n\\frac{\\lambda^ld\\lambda}{Y}\n\\ea\n\\ba\n\\frac{\\partial F_{ij}}{\\partial\n a_k} =\n\\frac{\\partial F_{ij}}{\\partial \\lambda_\\alpha}\n\\frac{\\partial\\lambda_\\alpha}{\\partial s_l}\n\\frac{\\partial s_l}{\\partial a_k}\n\\ea\nThe first derivative is (as usual for any hyperelliptic curve)\nequal to\n$\\hat\\omega_i\\hat\\omega_j(\\lambda_\\alpha)$, the\nlast one -- to $\\sigma^{-1}_{kl}$. In order to find the\nmiddle one, consider\n\\ba\n\\left.\\frac{\\partial}{\\partial s_l}\n\\left(P^2 - Q = \\prod_\\beta (\\lambda - \\lambda_\\beta)\\right)\n\\right|_{\\lambda = \\lambda_\\alpha}\n\\ea\nIt gives\n\\ba\n\\left. 2P\\frac{\\partial P}{\\partial s_l} =\n(2PP' - Q')\\frac{\\partial \\lambda_\\alpha}{\\partial s_l}\n\\right|_{\\lambda = \\lambda_\\alpha}\n\\ea\ni.e.\n\\ba\n\\frac{\\partial\\lambda_\\alpha}{\\partial s_l} = \\left.\n\\frac{\\lambda^ld\\lambda}{dP - \\frac{1}{2}\\frac{dQ}{P}}\n\\right|_{\\lambda = \\lambda_\\alpha}\n\\ea\nSince at $\\lambda = \\lambda_\\alpha$ $Y=0$, we obtain:\n\\ba\n\\frac{\\partial\\lambda_\\alpha}{\\partial s_l} =\n \\left.\\frac{\\lambda_\\alpha^ld\\lambda}{Y\\frac{dw}{w}}\n\\right|_{\\lambda = \\lambda_\\alpha} =\n\\left.\\sigma_{lk}\\frac{d\\omega_k}{\\frac{dw}{w}}\n\\right|_{\\lambda = \\lambda_\\alpha}\n\\ea\nPutting this all together we obtain:\n\\ba\n\\frac{\\partial F_{ij}}{\\partial a_k} =\n-\\stackreb{d\\lambda = 0}{\\hbox{res}}\n\\frac{d\\omega_id\\omega_jd\\omega_k}{d\\lambda\\frac{dw}{w}} =\n\\stackreb{d\\omega = 0}{\\hbox{res}}\n\\frac{d\\omega_id\\omega_jd\\omega_k}{d\\lambda\\frac{dw}{w}}\n\\ea\nSince differentials in the numerator have no poles, the\nintegration contour that goes around the zeroes of $d\\lambda$,\ncan be also considered as encircling the zeroes of\n$d\\omega = \\frac{dw}{w}$ -- and we reproduce the general\nresult (\\ref{resfor}).\n\nThe similar identity can be proved for the expressions containing\none derivative w.r.t. the remaining moduli $m_{\\iota} \\sim \\oint dS$,\nas we can still work with the period matrix. More concretely, since\n\\ba\\label{dW1}\ndW_{\\iota} \\cong \\frac{\\partial dS}{\\partial m_{\\iota}} \\cong -\n\\frac{d\\lambda}{w}\\frac{\\partial w}{\\partial m_{\\iota}} = -\n\\frac{d\\lambda}{Y}\\left(\\frac{Q}{2(P+Y)(\\lambda - m_{\\iota})} +\n\\frac{\\partial R}{\\partial m_{\\iota}}\\right)\n\\ea\nthe derivative of ramification point is\n\\ba\n\\frac{\\partial \\lambda_\\alpha}{\\partial m_{\\iota}} =\n-\\left. \\frac{Q}{2P}\\frac{d\\lambda}{Y\\frac{dw}{w}}\n\\left(\\frac{1}{\\lambda - m_{\\iota}} +\n\\frac{2P\\frac{\\partial R}{\\partial m_{\\iota}}}{Q}\\right)\\right|_{\\lambda =\n\\lambda_\\alpha} =-\n\\left. \\frac{d\\lambda}{Y\\frac{dw}{w}}\n\\left(\\frac{Q}{2(P+Y)(\\lambda - m_{\\iota})} +\n\\frac{\\partial R}{\\partial m_{\\iota}}\\right)\\right|_{\\lambda =\n\\lambda_\\alpha}\n=\\left.\\frac{dW_{\\iota}}{\\frac{dw}{w}}\n\\right|_{\\lambda = \\lambda_\\alpha}\n\\ea\nThus\n\\ba\n\\frac{\\partial F_{ij}}{\\partial m_{\\iota}} =\n\\frac{\\partial F_{ij}}{\\partial \\lambda_\\alpha}\n\\frac{\\partial\\lambda_\\alpha}{\\partial m_{\\iota}} =\n-\\stackreb{d\\lambda = 0}{\\hbox{res}}\n\\frac{d\\omega_i d\\omega_j dW_{\\iota}}{d\\lambda \\frac{dw}{w}}\n= \\stackreb{\\frac{dw}{w} = 0}{\\hbox{res}}\n\\frac{d\\omega_i d\\omega_j dW_{\\iota}}{d\\lambda \\frac{dw}{w}}\n\\ea\nagain in accordance with (\\ref{resfor}).\nNote, that according to our agreements the punctures $m_\\iota$\nare included into the set of zeroes of $d\\lambda$ (along with\nramification points $\\lambda_\\alpha$).\n\nDouble and triple-derivatives of ${\\cal F}$ with respect to the\nmasses $m_\\iota$ can not be obtained from the ordinary\nperiod matrices of the hyperelliptic curves. They can be only\nevaluated by the general method of s.4. This calculation,\nwhich is the subject of the next subsection s.\\ref{rfmd},\nhelps also to illustrate the possible subtleties of the general proof.\n\nLet us note that there are two natural choices of the generating differential\n$dS$ for the case of theory with fundamental matter. One choice is given by\nformula (\\ref{dS1}) used throughout this subsection. Another one is\ngiven by $dS=\\lambda{d\\hat w\\over\\hat w}=\\lambda d\\log{P-Y\\over P+Y}$,\nwhere $\\hat w$ is defined in another parameterization of the curve $\\hat w\n+{1\\over\\hat w}={P\\over\\sqrt{Q}}$. The latter choice is\nusually accepted in the literature \\cite{SW2,SWI2,HO,AS}, and has been used in\nthe course of our perturbative consideration of sect.5.2. Note that the\ndifferentials (\\ref{dW1})\n\\ba\ndW_{\\iota}^{(1)} \\cong\n-\\frac{d\\lambda}{Y}\\left(\\frac{P-Y}{2(\\lambda - m_{\\iota})} +\n\\frac{\\partial R}{\\partial m_{\\iota}}\\right)=\ndW^{(2)}_{\\iota}+{1\\over 2}{d\\lambda\\over\\lambda -m_{\\iota}}\n\\ea\nhave poles at $m_{\\iota}$ on one of the sheets, while the differentials\n$dW^{(2)}_{\\iota}$ associated with the other $dS$\npossess the poles on both sheets.\nHowever, these two choices give rise to the identical results.\n\n\n\\subsection{Residue formula for mass differentials \\label{rfmd}}\n\nTo make our consideration of concrete examples exhaustive, in this\nsubsection we discuss the subtle points arising in the course of the\nderivation of the residue formula when two or all the three differentials\n$dv_I$ are associated\nwith punctures (masses). This is the case, when the $\\Sigma$-term (\\ref{sigma})\ncontributes to the result. In what follows we demonstrate\nhow this specific case of the residue formula can be treated in the example\nof the theory with matter hypermultiplet in the fundamental representation,\nand we start with consideration of the perturbative limit. To match the\nnotations of sect.5.2, we use the\ngenerating differential $dS=\\lambda{d\\hat w\\over\\hat w}=-{1\\over 2}\\lambda\nd\\log{P-Y\\over P+Y}$ and, accordingly, the holomorphic differentials\n$dW_{\\iota}^{(2)} \\cong -{P\\over 2Y}\\frac{d\\lambda}{(\\lambda - m_{\\iota})} -\n\\frac{\\partial R}{\\partial m_{\\iota}}{d\\lambda\\over Y}$. Then,\nin the perturbative limit\nthis $dW_{\\iota}$ acquires the form (\\ref{massdiff}),\n$s_{\\iota}=\\lambda_{\\iota}$.\n\nFirst, we derive formula (\\ref{resch})\nin the perturbative limit with two mass\ndifferentials. The symmetry of the l.h.s. of (\\ref{resch})\nw.r.t. the permutations of\nindices is not obvious, since there are no symmetric terms in the l.h.s.\nof (\\ref{resch}) and one needs\nto consider the entire sum. Therefore, in illustrative purposes, we\ndiscuss here all possible arranging of indices in (\\ref{resch}) --\n$\\{KLM\\}=\\{\\iota,\\iota,\\iota'\\}$, $\\{KLM\\}=\\{\\iota,\\iota',\\iota\\}$ and\n$\\{KLM\\}=\\{\\iota',\\iota,\\iota\\}$ checking that they lead to the same result.\nNote that the term $\\oint_{A_I}{\\partial dv_L\\over\\partial s_M}\n\\int_{{\\partial B_I\\over \\partial s_K}}dS$ is zero in all the cases considered below.\nOther terms are equal to\n\\ba\\label{sigma1}\n\\Sigma_{KLM}^{(1)}\\equiv\n\\stackreb{\\frac{\\partial v_L}{\\partial \\lambda_M}=0}{\\hbox{res}}\n\\left(dv_K \\frac{\\partial v_L}{\\partial \\lambda_M}\\right) =\n\\left\\{\\begin{array}{cl}\n0 & \\hbox{when}\\ \\ \\ K=L =\\iota,\\ M=\\iota'\n\\\\\n0 & \\hbox{when}\\ \\ \\ K=M =\\iota,\\ L=\\iota'\n\\\\\n-{1\\over 4}{1\\over\\lambda_{\\iota\\iota'}} & \\hbox{when}\\ \\ \\ L=M =\\iota,\\\nK=\\iota'\n\\end{array} \\right.\n\\ea\n\n\\ba\n\\Sigma_{KLM}^{(2)}\\equiv\\sum_I\\oint_{A_I}dv_K {\\partial\\over\\partial s_M}\\int_{{\\partial\nB_I\\over \\partial \\lambda_L}}dS=\n\\left\\{\\begin{array}{cl}\n-{1\\over 4}{1\\over \\lambda_{\\iota\\iota'}}& \\hbox{when}\\ \\ \\\nK=L=\\iota,\\ M=\\iota'\n\\\\\n0 & \\hbox{when}\\ \\ \\ K=M=\\iota,\\ L=\\iota'\n\\\\\n0 & \\hbox{when}\\ \\ \\ L=M=\\iota,\\ K=\\iota'\n\\end{array}\n\\right.\n\\ea\n\n\\ba\n\\Sigma_{KLM}^{(3)}\\equiv \\sum_I\n\\oint_{A_I}dv_K \\int_{{\\partial B_I\\over \\partial \\lambda_M}}dv_L=\n\\left\\{\\begin{array}{cl}\n0 & \\hbox{when}\\ \\ \\ K=L =\\iota,\\ M=\\iota'\n\\\\\n-{1\\over 4}{1\\over \\lambda_{\\iota\\iota'}}& \\hbox{when}\\ \\ \\\nK=M =\\iota,\\ L=\\iota'\n\\\\\n0 & \\hbox{when}\\ \\ \\ L=M =\\iota,\\ K=\\iota'\n\\end{array}\n\\right.\n\\ea\nsince $\\oint_{A_I}dv_{\\iota}=-{1\\over 2}\\delta_{I\\iota}$,\n$\\int_{{\\partial B_{I}\\over \\partial \\lambda_{\\iota}}}dv_{\\iota'}=\n{1\\over 2}\\delta_{I\\iota}{1\\over \\lambda_{\\iota\\iota'}}$ and\n$\\int_{{\\partial B_I\\over \\partial \\lambda_{\\iota}}}dS=\n\\left.\\delta_{I\\iota}\\log\\left({P(\\lambda)\\over\n\\sqrt{Q(\\lambda)}}\\right)\\right|_{\\lambda=\n\\lambda_{\\iota}}$.\n\nThe sum of $\\Sigma^{(1)}+\\Sigma^{(2)}+\\Sigma^{(3)}$\nin each of the three cases $K=L$, $K=M$ and $L=M$\nis the same how it is\nrequired by the symmetricity w.r.t. the permutations of indices. This sum\nis actually equal to $F_{\\iota\\iota\\iota'}$ in (\\ref{perth}).\n\nAnalogously, one can check the residue formula for three coinciding mass\ndifferentials. Then, the result looks like\n\\ba\n\\Sigma_{\\iota\\iota\\iota}^{(1)}=-{1\\over 4}\\hbox{res} {1\\over (\\lambda-\n\\lambda_{\\iota})^2}=0\n\\\\\n\\Sigma_{\\iota\\iota\\iota}^{(2)}={1\\over 2}\\sum_i{1\\over\\lambda_{i\\iota }}\n+{1\\over 4}\\sum_{\\iota'\\ne\\iota}{1\\over\\lambda_{\\iota\\iota'}}\n+\\left.{1\\over 4}{1\\over\\lambda-\\lambda_{\\iota}}\n\\right|_{\\lambda\\to\\lambda_{\\iota}}\n\\\\\n\\Sigma_{\\iota\\iota\\iota}^{(3)}=-\\left.\n{1\\over 4}{1\\over\\lambda-\\lambda_{\\iota}}\n\\right|_{\\lambda\\to\\lambda_{\\iota}}\n\\ea\nand the sum of these terms is equal to $F_{\\iota\\iota\\iota}$ in\n(\\ref{perth}). Let us note that the only role of $\\Sigma^{(3)}$ is to cancel\nthe singularity arising in $\\Sigma^{(2)}$.\n\nNow we can turn to less trivial case and discuss the residue formula in the\nhyperelliptic case of sect.5.6. Again, we start with consideration of\n$F_{\\iota'\\iota\\iota}$. However, this time we restrict ourselves only to\nthe case $L=M$ -- the most convenient choice of order of the indices, since\nit is the case when the only first term in the l.h.s. of (\\ref{resch})\ncontributes. In the hyperelliptic case the residue acquires two different\ncontributions. The first one comes from the pole at $\\lambda=m_{\\iota}$\nand can be\nobtained with the help of expansion\n\\ba\\label{vsm}\n{\\partial v_{\\iota}\\over\\partial m_{\\iota}}\\stackreb{\\lambda\n\\to m_{\\iota}}{\\longrightarrow}{1\\over 2}{1\\over\\lambda\n-m_{\\iota}}+{\\cal O}\n(\\lambda- m_{\\iota})\n\\ea\nTherefore, the contribution of this pole is equal to that in\nthe perturbative case (\\ref{sigma1}):\n${1\\over 4}{1\\over \\lambda_{\\iota'\\iota}}$. However, now there emerge also\nsome contributions from the ramification points. Indeed,\n\\ba\n{\\partial dv_{\\iota}\\over\\partial m_{\\iota}} \\stackreb{\\lambda\\to \\lambda_{\\alpha}}\n{\\longrightarrow} {1\\over 2}\\left({P(\\lambda_{\\alpha})\\over\n2(\\lambda_{\\alpha}-m_{\\iota})}+{\\partial R(\\lambda_{\\alpha})\n\\over\\partial m_{\\iota}}\\right)\n{\\partial Y^2\\over\\partial m_{\\iota}}{d\\lambda\\over Y^3}\n={P(\\lambda_{\\alpha})\\over\n\\hat Y^3(\\lambda_{\\alpha})}\n\\left({P(\\lambda_{\\alpha})\\over\n2(\\lambda_{\\alpha}-m_{\\iota})}+{\\partial R(\\lambda_{\\alpha})\n\\over\\partial m_{\\iota}}\\right)\n{d\\lambda\\over (\\lambda-\\lambda_{\\alpha})^{3\/2}}\n\\ea\nsince ${\\partial Y^2\\over \\partial m_{\\iota}}=2P{\\partial R\\over\\partial m_{\\iota}}+\n{Q\\over (\\lambda-m_{\\iota})}$. Therefore,\n\\ba\n{\\partial v_{\\iota}\\over\\partial m_{\\iota}} \\stackreb{\\lambda\\to \\lambda_{\\alpha}}\n{\\longrightarrow}\n-2{P(\\lambda_{\\alpha})\\over \\hat Y^4(\\lambda_{\\alpha})}\n\\left({P(\\lambda_{\\alpha})\\over\n2(\\lambda_{\\alpha}-m_{\\iota})}+{\\partial R(\\lambda_{\\alpha})\n\\over\\partial m_{\\iota}}\\right)^2\n{1\\over \\sqrt{\\lambda-\\lambda_{\\alpha}}}\n\\ea\nand the full answer becomes\n\\ba\nF_{\\iota\\iota\\iota'}=\\Sigma_{\\iota'\\iota\\iota}^{(1)}=2\n\\sum_{\\alpha}\n{P(\\lambda_{\\alpha})\\over\\hat Y^4(\\lambda_{\\alpha})}\n\\left({P(\\lambda_{\\alpha})\\over\n2(\\lambda_{\\alpha}-m_{\\iota})}+{\\partial R(\\lambda_{\\alpha})\n\\over\\partial m_{\\iota}}\\right)^2\n\\left({P(\\lambda_{\\alpha})\\over\n2(\\lambda_{\\alpha}-m_{\\iota'})}+{\\partial R(\\lambda_{\\alpha})\n\\over\\partial m_{\\iota'}}\\right)\n-{1\\over 4}{1\\over \\lambda_{\\iota\\iota'}}\\label{fghb}\n\\ea\nThis result should be compared with the residue formula\n\\ba\nF_{\\iota\\iota\\iota'}=\\stackreb{{dw\\over w}=0}{\\hbox{res}} {(dv_{\\iota})^2\ndv_{\\iota'}\\over d\\lambda {dw\\over w}}=\\stackreb{{dw\\over w}=0}{\\hbox{res}}\n{1\\over Y^2P}\n\\left({P\\over\n2(\\lambda-m_{\\iota})}+{\\partial R\\over\\partial m_{\\iota}}\\right)^2\n\\left({P\\over\n2(\\lambda-m_{\\iota'})}+{\\partial R\\over\\partial m_{\\iota'}}\\right)\n{d\\lambda\\over\n\\left(\\log(P\/\\sqrt{Q})\\right)'}\n\\ea\nAs we did it earlier, instead of quite involved calculation of this residue\nat zeroes of ${dw\\over w}$, we calculate it in all other poles (i.e.\nzeroes of $d\\lambda$). Since $\\left(\\log(P\/\\sqrt{Q})\\right)'$ contains\n$\\left[(\\lambda-m_{\\iota})(\\lambda -m_{\\iota'})\\right]^{-1}$,\nonly ramification points and\n$\\lambda=m_{\\iota}$ contributes to the result. Using formula\n\\ba\\label{aux1}\n\\left.\\left(\\log(P\/\\sqrt{Q})\\right)'\\right|_{\\lambda=\\lambda_{\\alpha}}\n={\\hat Y^2(\\lambda_{\\alpha})\\over\n2P^2(\\lambda_{\\alpha})}\n\\ea\none can easily check that this really gives (\\ref{fghb}).\n\nAnalogously one can\nstudy in the hyperelliptic case the residue formula for the\nthree coinciding differentials $F_{\\iota\\iota\\iota}$. This time the\ncalculation is a little longer. Using (\\ref{vsm}), one gets for\n$\\Sigma^{(1)}_{\\iota\\iota\\iota}$\n\\ba\\label{sigma1mmm}\n\\Sigma^{(1)}_{\\iota\\iota\\iota}=\n2\\sum_{\\alpha}\n{P(\\lambda_{\\alpha})\\over\\hat Y^4(\\lambda_{\\alpha})}\n\\left({P(\\lambda_{\\alpha})\\over\n2(\\lambda_{\\alpha}-m_{\\iota})}+{\\partial R(\\lambda_{\\alpha})\n\\over\\partial m_{\\iota}}\\right)^3\n-{1\\over 2P(m_{\\iota})}{\\partial R(m_{\\iota})\\over\n\\partial m_{\\iota}}\n-\\left.{1\\over 4}{\\partial(P\/Y)\\over \\partial\\lambda}\\right|_{\\lambda=m_{\\iota}}\n\\ea\ni.e. the sum over ramification points looks similar\nto the case of two coinciding differentials. Other $\\Sigma$ terms are\neven simpler:\n\\ba\\label{sigma2mmm}\n\\Sigma^{(2)}_{\\iota\\iota\\iota}=\n-\\left.{1\\over 2}{\\partial P\/\\partial\\lambda\\over P}\\right|_{\\lambda=m_{\\iota}}\n-{1\\over 2P(m_{\\iota})}{\\partial R(m_{\\iota})\\over\n\\partial m_{\\iota}}\n+{1\\over 4}\\sum_{\\iota'\\ne\\iota}{1\\over\\lambda_{\\iota\\iota'}}\n+\\left.{1\\over 4}{1\\over\\lambda-\\lambda_{\\iota}}\n\\right|_{\\lambda\\to\\lambda_{\\iota}}\n\\ea\n\\ba\\label{sigma3mmm}\n\\Sigma^{(3)}_{\\iota\\iota\\iota}=-{1\\over 2P(m_{\\iota})}{\\partial R(m_{\\iota})\\over\n\\partial m_{\\iota}} -\\left.\n{1\\over 4}{1\\over\\lambda-\\lambda_{\\iota}}\n\\right|_{\\lambda\\to\\lambda_{\\iota}}\n\\ea\nThe sum of all the three terms should be compared with the residue formula\n\\ba\\label{resmmm}\nF_{\\iota\\iota\\iota}=\\stackreb{{dw\\over w}=0}{\\hbox{res}} {(dv_{\\iota})^3\n\\over d\\lambda {dw\\over w}}=-\\stackreb{{dw\\over w}=0}{\\hbox{res}}\n{1\\over Y^2P}\n\\left({P\\over\n2(\\lambda-m_{\\iota})}+{\\partial R\\over\\partial m_{\\iota}}\\right)^3\n{d\\lambda\\over\n\\left(\\log(P\/\\sqrt{Q})\\right)'}\n\\ea\nAgain taking residues at zeroes of $d\\lambda$ and using formulas\n(\\ref{aux1}) along with\n\\ba\n\\left.(\\lambda-m_{\\iota})\\left(\\log{P\\over\\sqrt{Q}}\\right)'\n\\right|_{\\lambda=m_{\\iota}}=-{1\\over 2}\n\\\\\n\\left[\\left.(\\lambda-m_{\\iota})\\left(\\log{P\\over\\sqrt{Q}}\\right)'\\right]'\n\\right|_{\\lambda=m_{\\iota}}=\\left.{\\partial P\/\\partial\\lambda\\over P}\n\\right|_{\\lambda=m_{\\iota}}-{1\\over 2}\n\\sum_{\\iota'\\ne\\iota}{1\\over\\lambda_{\\iota\\iota'}}\n\\ea\none finally proves that (\\ref{resmmm}) is really equal to the sum of\n(\\ref{sigma1mmm}), (\\ref{sigma2mmm}) and (\\ref{sigma3mmm}).\nContributions from the\nramification points into (\\ref{resmmm}) entirely originate\nfrom analogous term in $\\Sigma^{(1)}_{\\iota\\iota\\iota}$, and\nall the terms in $\\Sigma$'s proportional to $\\partial R\/\\partial m_{\\iota}$\ncome from the first order pole at $\\lambda=m_{\\iota}$. The remaining\nterms in $\\Sigma^{(1)}$ and $\\Sigma^{(2)}$ are equal to the residue of\nthe second order pole at $\\lambda=m_{\\iota}$.\n\n\\section{Hypermultiplet in the adjoint representation (Calogero\nsystem) \\label{Calogero}}\n\\setcounter{equation}{0}\n\\subsection{General formulas \\label{genCa} }\nTo conclude our examples, in this section we discuss in detail the case of\nCalogero system when\nthe closed algebra exists but is {\\it not} associative, how it was\ndiscussed in sect.3-4. The physics os this example is the\n$4d$ ${\\cal N}=2$ SUSY YM model with one matter hypermultiplet in the\nadjoint representation of the gauge group $SU(N_c)$ that\nis described in the SW\nframework by elliptic Calogero model \\cite{SWI3,IM1,IM2}. The bare\ncurve for the UV finite model is elliptic curve (\\ref{torus}) and its modulus\n$\\tau $ can be identified with the bare (UV) coupling constant of the YM\ntheory. The mass of the $4d$ hypermultiplet is proportional to the\ncoupling constant $g$ in Calogero-Moser model. The spectral curve of\nthe elliptic Calogero model\n\\ba\n\\det ({\\cal L}(x,y) - \\lambda ) = 0\n\\ea\nis given by\n\\ba\\label{calocurve}\ny^2 = (x-e_1(\\tau))(x - e_2(\\tau))(x-e_3(\\tau)) =\nx^3 - \\frac{1}{4}g_2(\\tau)x - \\frac{1}{4}g_3(\\tau) \\nonumber \\\\\n{\\cal P}(\\lambda ,x) = \\sum _{i=0}^{N_c}g^is_i{\\cal T}_i(\\lambda, x) =0\n\\nonumber \\\\\n{\\cal T}_0 = 1, \\ \\ \\ \\ {\\cal T}_1 = \\lambda,\\ \\ \\ \\ \\\n{\\cal T}_2 = \\lambda^2 - x,\n\\ \\ \\ \\ \\ {\\cal T}_3 = \\lambda ^3 - 3\\lambda x + 2y, \\ \\ \\ \\ \\\n{\\cal T}_4 = \\lambda^4 - 6\\lambda^2 x + 8\\lambda y - 3(x^2 + \\mu_4),\\nonumber \\\\\n{\\cal T}_5 = \\lambda^5 - 10\\lambda^3x + 20\\lambda^2y -\n15\\lambda(x^2 + \\mu_4) + 4xy, \\ \\ \\ \\ \\\n{\\cal T}_6 = \\lambda^6 - 15\\lambda^4x + 40\\lambda^3y\n- 45\\lambda^2(x^2 + \\mu_4) + 24\\lambda xy - 5,\n\\nonumber \\\\\n\\ldots\n\\label{explT}\n\\ea\n(we remind that $e_1+e_2+e_3=0$).\nUsing gradation $\\deg\\ \\lambda\\ = 1$, $\\deg\\ x\\ = 2$,\n$\\deg\\ y\\ =3$, $\\ldots$,\n${\\cal T}_i$ becomes a homogeneous polynomial of degree $i$.\nActually this is a ``semi''-gradation from the point of view of\n$x$ and $y$, because some coefficients also have non-vanishing\n(but always positive) degree: $\\deg\\ e_i(\\tau) = 2$,\n$\\deg\\ \\mu_4 = 4$ etc.\nAs functions of $\\lambda$, the ${\\cal T}_i$ behave similar to the\nSchur polynomials:\n\\ba\\label{T'}\n{\\cal T}_i' \\equiv \\frac{\\partial{\\cal T}_i}{\\partial \\lambda} =\ni{\\cal T}_{i-1}.\n\\ea\nAs a corollary, ${\\cal P}'$ is again a linear combination of\n${\\cal T}_i$:\n\\ba\n{\\cal P}' \\equiv \\frac{\\partial{\\cal P}}{\\partial\\lambda} =\n\\sum_i ig^is_i{\\cal T}_{i-1}.\n\\label{calP'}\n\\ea\nFor our purposes we can absorb $g_i$ in (\\ref{calocurve}) into\nthe moduli $s_i$.\n\nThe full spectral curve (\\ref{calocurve}) is a $N_c$-fold\ncovering of bare elliptic curve, which has genus $N_c$\n(not $\\sim N_c^2\/2$ as one could naively expect), due to specific\nproperties of the functions ${\\cal T}_i$ which may be demonstrated in\nthe Hitchin-like interpretation\nof Calogero model, when the matrix-valued moment map is restricted\nto have a pole with $N_c-1$ coinciding eigenvalues at it.\nIn the limit $x \\sim \\frac{1}{\\xi^2} \\rightarrow \\infty$,\nthe coordinate $\\lambda \\sim \\frac{1}{\\xi}$ on the $N_c-1$ sheets\nand $\\lambda \\sim -\\frac{N_c-1}{\\xi}$ on the last sheet. We shall\nrefer to these to kinds of sheets as ``$+$'' and ``$-$'' respectively.\nOn the $N_c-1$ ``$+$'' sheets all the functions ${\\cal T}_i$\nbehave as\n\\ba\n{\\cal T}_i \\sim \\lambda\n\\label{asympt}\n\\ea\nin the limit $\\lambda \\longrightarrow \\infty$, while on the ``$-$'' sheet\n${\\cal T}_i \\sim \\lambda^i$. Since ${\\cal P}$ and ${\\cal P'}$ are both\nlinear combinations of ${\\cal T}_i$, they have the same asymptotics\n$\\sim\\lambda^1$ on the ``$+$'' sheet.\n\nFor the generating form $dS=\\lambda d\\omega = \\lambda\n\\frac{dx}{y}$, we have\n\\ba\ndv_k = \\frac{\\partial dS}{\\partial s_k} =\n-\\frac{{\\cal T}_k d\\omega}{{\\cal P}'}, \\ \\ \\\nk = 0,1,2,\\ldots,N_c-2.\n\\label{calodiffs}\n\\ea\nThe complete basis of $N_c$ holomorphic 1-differentials on ${\\cal C}$\nconsists of $dv_k$ with $k=0,1,2,\\ldots,N_c-2,N_c-1$.\nThe holomorphic 1-differential\n$dv_{N_c-1}$ is (up to addition of other $dv_k$ with $k\\leq N_c-2$)\nequal to $d\\omega = \\frac{dx}{y}$. The only non-vanishing\nintegrals of $d\\omega$ are along the cycles on the {\\it bare}\nspectral curve $E(\\tau)$. The periods $a_i$ relevant for the\nSW theory are instead associated with the other cycles,\nwhich turn to be\ntrivial after projection on $E(\\tau)$ (see sect.8 of \\cite{IM2} for discussion\nof the $N_c=2$ example).\n\nIn other words, the polynomial ${\\cal T}_{N_c-1}$ is linearly equivalent\nto a combination of ${\\cal T}_i$ with $i R$, then we are 100\\% confident that source point $A$ is not within $R$ of query point $B$. As a result, there is no need to compute the exact distance between point $A$ and $B$. Otherwise, the exact distance computation will still be carried out for direct comparison. \nWhile many previous researches~\\cite{elkan2003using, ding2015yinyang, lin2012k, MakingKMeansFaster, KNNAdaptiveBound} gain success in directly porting the above point-based TI to optimize distance-related algorithms, they usually suffer from memory overhead and computations irregularity, which result in inferior performance. \n\\begin{figure*}[htb] \\small\n \\centering\n \\makebox{\\includegraphics[scale=0.8]{images\/TI-optimization.pdf}}\n \\caption{TI Optimization.}\n \\label{fig: TI Optimization.}\n\\end{figure*}\n\\vspace{-1em}\n\\subsection{Generalized Triangle Inequality (GTI)} \nAccD uses a novel Generalized TI (GTI) to remove redundant distance computation. It generalizes the traditional point-based TI while significantly reducing the overhead of bound computations. The traditional point-based TI focuses on tighter bound (more closer to the exact distance) to remove more distance computations, but it induces the extra bound computations, which could become the new performance bottleneck even after many distance calculations being removed. In contrast, GTI strikes a good balance between distance computation elimination and bound computation overhead. In particular, AccD highlights GTI from three perspectives: \\textit{Two-landmark bound computation}, \\textit{Trace-based bound computation}, and \\textit{Group-level bound computation}.\n\n\\paragraph{Two-landmark Bound Computation} \nTwo-landmark scheme aims at reducing the bound computation through effective distance reuse. In this case, the distance bound between two points can be measured through two landmarks as the reference points. As illustrated in Figure \\ref{fig: TI Optimization.}\\subfig{b}, the distance bound between point $A$ and $B$ can be computed based on $d(A, A_{ref})$, $d(B, B_{ref})$ and $d(A_{ref}, B_{ref})$ through Equation \\ref{equ: Two-landmark Bound Computation.}, where $A_{ref}$ and $B_{ref}$ are the landmark points for point $A$ and $B$, correspondingly. \n\\begin{equation} \\small\n\\label{equ: Two-landmark Bound Computation.}\n\\begin{aligned}\n lb(A, B) \\geq d(A_{ref}, B_{ref}) - d(A, A_{ref}) - d(B, B_{ref}) \\\\\n ub(A, B) \\leq d(A_{ref}, B_{ref}) + d(A, A_{ref}) + d(B, B_{ref})\n\\end{aligned}\n\\end{equation}\n\nOne representative application scenario of Two-landmark bound computation is KNN-join, where two disjoint sets of landmarks are selected for the query and target point set. In this case, much fewer bound computations are required compared with the one-landmark case (shown in Figure~\\ref{fig: TI Optimization.}\\subfig{a}). This can also be validated through a simple calculation. Assuming in KNN-join, we have $m$ query points, $n$ target points, $z_{qry}$ query landmarks, and $z_{trg}$ target landmarks. Also, we have $z_{qry}< d(c, d) + d_{max}(d, d')$, it is impossible that the points inside the group $A'$ and $B'$ can become the closest point of $c$ in the current iteration. Therefore, the distance computation between the point $c$ and all points inside these groups can be safely avoided.\n\nIn addition to distance saving, group-level bound computation offers another two benefits to facilitate the underlying hardware acceleration. First, the computation regularity on the remaining distance computation becomes higher compared with the point-level bound computation. Since points inside each group will share the commonality in computation, which facilitates the parallelization for acceleration. For example, point-level bound computation usually results in a large divergence of distance computation among different points, as shown in Figure~\\ref{fig: group-level bound computation -- computation regularity.}\\subfig{a}, which is a killer of parallelization and pipelining. However, in group-level bound computation, points inside the same source group will always maintain the same groups of target points for distance computation, as shown in Figure~\\ref{fig: group-level bound computation -- computation regularity.}\\subfig{b}. \n\\begin{figure} [ht] \\small\n \\centering\n \\makebox{\\includegraphics[width=0.75\\columnwidth]{images\/Data-Grouping.pdf}}\n \\caption{Bound Computation at (a) Point-level, (b) Group-level.}\n \\label{fig: group-level bound computation -- computation regularity.}\n\\end{figure}\n\\vspace*{-0.8em}\nSecond, group-level bound computation brings the benefit of reducing memory overhead. Assuming we have $m$ source points, $n$ target points, $z_{src}$ source groups, and $z_{trg}$ target groups. The memory overhead of maintaining distance bounds is $O(m\\times n)$ in the point-level bound computation case. However, in the group-level bound computation case, we only have to maintain distance bounds among groups, and the memory overhead is $O(z_{src}\\times z_{trg})$, where $z_{src} << m$ and $z_{trg}<< n$. Therefore, in terms of memory efficiency, group-level bound computation can outperform the point-level bound computation to a great extent.\n\n\\iffalse\n\\subsection{Input-adaptive Strategy Selection}\nTo efficiently handle the diverse range of incoming distance-related algorithms and their data inputs, AccD offers two types of grouping strategies: \\textbf{source grouping} and \\textbf{target grouping}, to distinguish points based on their required operations. To maximize the benefits from grouping, AccD provides three types of filtering options: \\textbf{point-to-point}, \\textbf{group-to-point}, and \\textbf{group-to-group}, which can reduce the distance computation at the different levels of granularity and maintain different levels of computation regularity. Each level of filtering have different computation elimination power and overhead to obtain, therefore it is essential to obtain substantial speedup by choosing the proper filtering strategies. To this end, AccD utilizes a decision-tree based approach to determine the grouping and the filtering strategy at the design time. \n\n\\begin{figure} [h] \\small\n \\centering\n \\makebox{\\includegraphics[width=0.75\\columnwidth]{images\/AccD-grouping-strategy.pdf}}\n \\caption{AccD Grouping Strategy.}\n \\label{fig: AccD Grouping Strategy.}\n\\end{figure}\nTo show the power of this comprehensive strategy selection of AccD, we demonstrate a small example. As shown in the Figure~\\ref{fig: AccD Grouping Strategy.}, AccD would take different grouping \\& filtering combination based on the proprieties of the source and target points. There are four conditions: 1) If the dataset comes with a small target set and a small source set, AccD will not conduct and grouping and only apply point-to-point filtering, since it is inexpensive to compute point-to-point distance; 2) If the dataset comes with a small target set and a large source set, AccD will group the points and conduct group-point filtering, since it is cheaper to execute group-based filtering; 3) If the dataset comes with a large target set and a small source set, AccD will group the target points and conduct group-point filtering, since it is also cheaper to execute group-based filtering; 4) If the dataset comes with a large target and a large source, AccD will apply group-group filtering to reduce the bound overhead and maintain computation regularity, since it can balance the computation reduction performance and bound overhead.\n\\vspace{-0.2em}\n\\figureautorefname{}\n\\fi\n\\section{Hardware Acceleration} \\label{sect: Architecture Design}\nAccD design is built on the CPU-FPGA architecture, which highlights its significant performance and energy efficiency, and has been widely adopted as the modern data center solution for high-performance computing and acceleration. The host-side application of AccD design is responsible for data grouping and distance computation filtering, which consists of complex operations and execution dependency, but lacks pipeline and parallelism. On the other hand, the FPGA-side of AccD design is built for accelerating the distance computations, which are composed of simple and vectorizable operations.\n\nWhile FPGA accelerator features with high computation capability, the memory bandwidth bottleneck constraints the overall design performance. Therefore, optimizing data placement and memory architecture is the key to improving memory performance. In addition, the OpenCL-based programming model adds a layer of architectural complexity of the kernel design and management, which is also critical to the design performance. AccD framework distinguishes itself by using a novel memory and kernel optimization strategy that is tailored for TI-optimized distance-related algorithms to benefit CPU-FPGA designs.\n\n\\subsection{Memory Optimization}\nAfter applying the GTI optimization to remove the redundant distance computation, each source point group will have different target groups as candidates for distance computation, as shown in Figure~\\ref{table: inter-group memory.}\\subfig{a}, where \\textbf{Source-grp} is ID of the source group, and \\textbf{Target-grp} is ID of the target group. However, this would raise two concerns about performance degradation. \n\\begin{table}[ht] \\small\n\\begin{minipage}[b]{0.5\\columnwidth}\n\\centering\n\\begin{tabular}{|| c | c ||}\n\\hline\n\\textbf{Source-grp}\n& \\makecell{\\ \\textbf{Target-grp}\\\\} \n\\\\ \n\\hline\n\\hline $s_1$ &\t$t_1$, $t_4$, $t_6$\n\\\\ \n\\hline $s_2$ &\t$t_8$, $t_{10}$, $t_{12}$\n\\\\\n\\hline ... &\t...\n\\\\\n\\hline $s_5$ & $t_2$, $t_4$, $t_6$\n\\\\\n\\hline $s_6$ & $t_8$, $t_{10}$, $t_{12}$\n\\\\\n\\hline\n\\end{tabular}\n\\caption*{(a)}\n\\end{minipage}\n\\hspace*{-\\textwidth} \\hfill\n\\begin{minipage}[b]{0.5\\columnwidth}\n\\centering\n\\label{table: Optimized Memory}\n\\begin{tabular}{|| c | c ||}\n\\hline\n\\textbf{Source-grp}\n& \\makecell{\\ \\textbf{Target-grp}\\\\} \n\\\\ \n\\hline\n\\hline $s_1$ &\t$t_2$, $t_4$, $t_6$\n\\\\ \n\\hline $s_5$ &\t$t_2$, $t_4$, $t_6$\n\\\\\n\\hline $s_2$ &\t$t_8$, $t_{10}$, $t_{12}$\n\\\\\n\\hline $s_6$ & $t_8$, $t_{10}$, $t_{12}$\n\\\\\n\\hline ... &\t...\n\\\\\n\\hline\n\\end{tabular}\n\\caption*{(b)}\n\\end{minipage}\n\\captionof{figure}{(a) Non-optimized inter-group memory access; (b) Optimized inter-group memory access.}\n\\label{table: inter-group memory.}\n\\end{table}\n\\vspace{-0.8em}\n\nThe first issue is inter-group memory irregularity and low data reuse. For example, the target group information ($t_1$, $t_4$, $t_6$) required by source group $s_1$ can not be reused by $s_2$. Since $s_2$ requires quite different target groups ($t_8$, $t_{10}$, and $t_{12}$) for distance computation, thus, additional costly memory access has to be carried out. To tackle this problem, AccD places the source groups to the continuous memory space to maximize the memory access efficiency, only if these source groups have the same set of target groups as candidates for distance computation. An example has been shown in Figure~\\ref{table: inter-group memory.}\\subfig{b}, where the source group $s_2$ and $s_6$ are placed side by side in the memory, since they have the same list of target groups ($t_8$, $t_{10}$, and $t_{12}$), which can take advantage of the memory temporal locality without issuing another memory access.\n\nThe second issue is intra-group memory irregularity. For example, points from \\textit{group 1}, \\textit{2}, and \\textit{3} have taken up the memory space at intervals, as shown in Figure~\\ref{fig: memory optimization.}\\subfig{a}. However, a group of points are usually accessed simultaneously due to GTI optimization. This would cause frequent inefficient memory access for fetching individual point distributed at the discontinuous memory address. To solve this issue, AccD offers a second memory optimization to re-organize the target\/source points inside the same target\/source group into continuous memory space within the same memory bank, as illustrated in Figure~\\ref{fig: memory optimization.}\\subfig{b}. This strategy can largely benefit memory coalescing and external memory bandwidth while minimizing the access contention, since points inside the same bank can be accessed efficiently and points inside different banks can be accessed in parallel. \n\\begin{figure}[ht]\n\\begin{minipage}[c]{0.3\\columnwidth}\n\\centering\n\\begin{tabular}{|| c | c ||}\n\\hline\n\\textbf{Group}\n& \\makecell{\\ \\textbf{Points}\\\\} \n\\\\ \n\\hline\n\\hline $Grp_1$ & $3$, $8$, $9$\n\\\\ \n\\hline $Grp_2$ &\t$5$, $6$, $7$ \n\\\\\n\\hline $Grp_3$ &\t$1$, $2$, $4$\n\\\\\n\\hline ... &\t...\n\\\\\n\\hline\n\\end{tabular}\n\\caption*{(a)}\n\\label{table: Group Point Mapping}\n\\end{minipage}\n\\qquad\n\\begin{minipage}[c]{0.3\\columnwidth}\n \\centering\n \\includegraphics[width=0.7\\columnwidth]{images\/AccD-memory-opt1.pdf}\n \\caption*{(b)}\n\\end{minipage} \n\\hfill \n\\begin{minipage}[c]{0.3\\columnwidth}\n \\centering\n \\includegraphics[width=0.92\\columnwidth]{images\/AccD-memory-opt2.pdf}\n \\caption*{(c)}\n\\end{minipage}\n\\captionof{figure}{(a) Group-point mapping; (b) Non-aligned intra-group memory; (c) Aligned intra-group memory.}\n\\label{fig: memory optimization.}\n\\end{figure}\n\\vspace*{-1em}\n\\subsection{Distance Computation Kernel}\nDistance computation takes the major time complexity in distance-related algorithms. In AccD, after TI filtering on CPU, the remaining distance computations are accelerated on FPGA. Points involved in the remaining distance computations are organized into two sets: source set and target set, which can be organized as two matrices, $Mat_{A}$ ($m\\times d$) and $Mat_{B}$ ($n\\times d$), respectively, where each row of these matrices represents a point with $d$ dimension. The distance computation between $Mat_A$ and $Mat_B$ can be decomposed into three parts, as shown in Equation~\\ref{equ: distance computation},\n\\begin{equation} \\small\n \\label{equ: distance computation}\n {(Mat_{A} - Mat_{B})}^2 = Mat_{A}^2 - 2 * Mat_{A} \\cdot Mat_{B} + Mat_{B}^2\n\\end{equation}\nwhere $Mat_{A}^2$ or $Mat_{B}^2$ only takes the complexity of $O(m\\times d)$ and $O(n\\times d)$, while $Mat_A\\times Mat_ B$ takes $O(m\\times n\\times d^2)$, which dominates the overall computation complexity. AccD spots an efficient way of accelerating $Mat_A\\cdot Mat_B$ through highly-efficient matrix-matrix multiplication, which can benefit the hardware implementation on FPGA.\n\\begin{figure} [ht!] \\small\n \\centering\n \\includegraphics[width=0.8\\columnwidth]{images\/mm-opt.pdf}\n \\caption{AccD Matrix-based Distance Computation.}\n \\label{fig: AccD Distance Computation.}\n\\end{figure}\n\nThe overall computation process can be described as Figure~\\ref{fig: AccD Distance Computation.}, the source ($Mat_A$) and target set ($Mat_B$) Row-wise Square Sum (RSS) is pre-computed through in a fully-parallel manner. And the vector multiplication between each source and target point is mapped to an OpenCL kernel thread for a fine-grained parallelization. Moreover, a block of threads, as highlighted in the \"red\" square box of Figure~\\ref{fig: AccD Distance Computation.}, is the kernel thread workgroup, which can share a part of the source and target points to increase the on-chip data locality. Based on this kernel organization, AccD hardware architectural design offers several tunable hyperparameters for performance and resource trade-off: the size of kernel block, the number of parallel pipeline in each kernel block, etc. To efficiently find the \"optimal\" parameters that can maximize overall performance while respecting the constraints, we harness the AccD explorer for efficient design space search, which is detailed in Section~\\ref{sect: Design Space Exploration}.\n\n\\section{AccD Compiler} \\label{sect: AccD Compiler}\nIn this section, we detail AccD compiler in two aspects: design parameters and constraints, and design space exploration.\n\\vspace{-0.2em}\n\\subsection{Design Parameters and Constraints}\nAccD uses a parameterized design strategy for better design flexibility and efficiency. It takes the design parameters and constraints from algorithm and hardware to explore and locate the \"optimal\" design point tailored for the specific application scenario. At the algorithm level, the number of groups affects distance computation filtering performance. At the hardware level, there are three parameters: 1) Size of computation block, which decides the size of data shared by a group of computing elements; 2) SIMD factor, which decides the number of computing elements inside each computation block; 3) Unroll factor, which tells the degree of parallelization in each single distance computation. In addition, there are several hardware constraints, such as the on-chip memory size, the number of logic units, and the number of registers. All of these parameters and constraints are included in our analytical model for design exploration.\n\n\\vspace{0.4em}\n\\subsection{Design Space Exploration} \n\\label{sect: Design Space Exploration}\nFinding the best combination of design configurations (a set of hyper-parameters) under the given constraints requires non-trivial efforts in the design space search. Therefore, we incorporate an AccD explorer in our compiler framework for efficient design space exploration (Figure~\\ref{fig: AccD Explorer}). AccD explorer takes a set of raw configurations (hyper-parameters) as the initial input, and generates the optimal configuration as the output through several iterations of the design configuration optimization process. In particular, AccD explorer consists of three major phases: \\textit{Configuration Generation and Selection}, \\textit{Performance and Resource Modeling}, \\textit{Constraints Validation}. \n\\begin{figure} [h] \\small\n \\centering\n \\includegraphics[height=10em]{images\/AccD-Compiler-workflow.pdf}\n \\caption{AccD Explorer.}\n \\label{fig: AccD Explorer}\n\\end{figure}\n\\vspace{-1.3em}\n\\paragraph{Configuration Generation and Selection}\nThe functionality of this phase depends on its input. There are two kinds of inputs: If the input is from the initial configurations, this phase will directly feed these configurations to the modeling phase for performance and resource evaluation; If the input is the result from the constraints validation in the last iteration, this phase will leverage the genetic algorithm to crossover the \"premium\" configurations kept from the last iteration, and generate a new set of configurations for the modeling phase. \n\\paragraph{Performance Modeling}\nPerformance modeling measures the design latency and bandwidth requirement based on the input design configurations. We formulate the design latency by using Equation~\\ref{equ: Latency model},\n\\begin{equation} \\small\n \\label{equ: Latency model}\n \\begin{multlined}\n Latency = Latency_{filt} + Latency_{comp}\n \\end{multlined}\n\\end{equation}\nwhere $Latency_{filt}$ and $Latency_{comp}$ are the time of the GTI filtering process and remaining distance computations, respectively. And they can be calculated as Equation~\\ref{equ: Runtime Filtering.},\n\\begin{equation} \\small\n\\label{equ: Runtime Filtering.}\n\\begin{aligned}\n Latency_{filt} &=\\frac{n_{trg\\_grp}\\times n_{src\\_grp} \\times src_{size}\\times trg_{size}\\times d}{n_{iteration}} \n \\\\\n Latency_{comp} &=\\frac{src_{size}\\times trg_{size}\\times ratio_{save}\\times d}{blk^2\\times frequency\\times unroll \\times simd}\n\\end{aligned}\n\\end{equation}\nwhere $n_{src\\_grp}$ and $n_{trg\\_grp}$ are the number of groups for source and target points, respectively; $src_{size}$ and $trg_{size}$ are the number of points inside source and target set, respectively; $d$ is the data dimensionality; $n_{iteration}$ is the number of grouping iteration; $blk$ is the size of computation kernel block; $frequency$ is the FPGA design clock frequency; $unroll$ is the distance computation unroll factor; $simd$ is the number of parallel worker threads inside each computation block; $ratio_{save}$ is the distance saving ratio through GTI filtering (Equation~\\ref{equ: Saving Ratio}),\n\\begin{equation} \\small\n \\label{equ: Saving Ratio}\n ratio_{save} = \\frac{n_{iteration}}{\\alpha} \\times \\sqrt{\\frac{src_{size}\\times trg_{size}}{n_{src\\_grp} \\times n_{trg\\_grp}}}\n\\end{equation}\nwhere the $\\alpha$ the density of points distribution. This formula also tells that increasing of number of iterations and number of points inside each group would improve the performance of GTI filtering performance. Also, the increase of points distribution density $\\alpha$, (\\textit{i.e.} points are closer to each other) will decrease the GTI filtering performance.\n\nTo get the required bandwidth $BW$ of the current design, we leverage Equation~\\ref{equ: required bandwith}, \n\\begin{equation} \\small\n\\label{equ: required bandwith}\n BW = \\frac{(src_{size} + trg_{size})\\times d \\times size_{data\\_type}}{Latency}\n\\end{equation}\nwhere the $size_{data\\_type}$ can be either 32-bit for \\textit{int} and \\textit{float} or 64-bit for \\textit{double}. \n\n\\paragraph{Resource Modeling}\nDirectly measuring the hardware resource usage of the accelerator design from high-level algorithm description is challenging because of the hidden transformation and optimization in hardware design suite. However, AccD uses a micro-benchmark based methodology to measure the hardware resource usage by analytical modeling. The major hardware resource consumption of AccD comes from the distance computation kernel, which depends on several design factors, including the kernel block size, the number of SIMD workers, etc.\n\nIn AccD resource analytical model, the design factors are classified into two categories: \\textit{dataset-dependent} and \\textit{dataset-independent} factors. The main idea behind the AccD resource modeling is to get the exact hardware resource consumption statistics through micro-benchmark on the hardware designs with different dataset-independent factors. For example, we can benchmark a single distance computation kernel with different sizes of computation block to get its resource statistics. Since this factor is dataset-independent, which can be decided before knowing the dataset details. However, to estimate the resource consumption for datasets with different sizes and dimensionalities, AccD leverages the formula-based approach to estimate the overall hardware resource consumption (Equation~\\ref{equ: Resource estimation}), which combines online information (\\textit{e.g.}, kernel organization, and dataset properties) and offline information (\\textit{e.g.}, miro-benchmark statistics).\n\\begin{equation} \\small\n \\label{equ: Resource estimation}\n \\begin{aligned}\n Resource_{est} = Resource_{single}\\times \\mathbf{ceil}(\\frac{src_{size}}{blk})\\times \\mathbf{ceil}(\\frac{trg_{size}}{blk}) \\\\\n \\end{aligned}\n\\end{equation}\nwhere the types of $Resource$ can be on-chip memory, computing units, and logic operation units; $Resource_{est}$ is the estimated overall usage of a certain type resource for the overall design; $Resource_{single}$ is the usage of a certain type of resource for only one distance computation kernel block.\n\n\\paragraph{Constraints Validation}\nConstraints validation is the third phase of AccD explorer, which checks whether the design resources consumption of a given configuration is within the budget of the given hardware platform. The input of this phase are the resource estimation results from resource modeling step. The design constraint inequalities are listed in Equation~\\ref{eqn: Resource Constraints}, which includes $Mem$ (the size of on-chip memory), $BW$ (the bandwidth of data communication between external memory and on-chip memory), $Computing\\_Unit$ (the number of computing units) and $Logic\\_Unit$ (the number of logic units):\n\\begin{equation} \\small\n\\label{eqn: Resource Constraints}\n\\begin{aligned}\n BW &\\leq BW_{max}, \n \\\\\n Mem &\\leq Mem_{max}, \n \\\\\n Computing\\_Unit &\\leq Computing\\_Unit_{max},\n \\\\\n Logic\\_Unit &\\leq Logic\\_Unit_{max}\n\\end{aligned}\n\\end{equation}\n\nConstraints validation phase will also discard the configurations that cannot match the design performance and constraints, and only keep the \"well-performed\" configurations for further optimization in the next iteration. The constraints validation phase will also record the modeling information of the best configuration statistics in the last iteration, which will be used to terminate the optimization process if the modeling results difference between the configurations in two consecutive iterations is lower than a predefined threshold. This strategy can also help to avoid unnecessary time cost. After termination of the AccD explorer, the \"best\" configuration with maximum design performance under the given constraints will be output as the \"optimal\" solution for the AccD design. \n\n\n\\section{Conclusion} \n\\vspace{-0.4\\baselineskip}\n\\label{sect: Conclusion}\nIn this paper, we present our AccD compiler framework to accelerate the distance-related algorithms on the CPU-FPGA platform. Specifically, AccD leverages a simple but expressive language construct (DDSL) to unify the distance-related algorithms, and an optimizing compiler to improve the design performance from algorithmic and hardware perspective systematically and automatically.\nRigorous experiments on three popular algorithms (K-means, KNN-join, and N-body simulation) demonstrate the AccD as a powerful and comprehensive framework for hardware acceleration of distance-related algorithms on the modern CPU-FPGA platforms.\n\\section{\\textbf{D}istance-related Algorithm \\textbf{D}omain-\\textbf{S}pecific \\textbf{L}anguage (DDSL)} \n\\label{sect: DDSL}\nDistance-related algorithms share commonalities across different application domains and scenarios, even though they look different in their high-level algorithmic description. Therefore, it is possible to generalize these distance-related algorithms. AccD framework defines a DDSL, which provides a high-level programming interface to describe distance-related algorithms in a unified manner. Unlike the API-based programming interface used in the TOP framework~\\cite{Topframework}, DDSL is built on C-like language and provides more flexibility in low-level control and performance tuning support, which is crucial for FPGA accelerator design.\n\nSpecifically, DDSL utilizes several constructs to describe the basic components (\\textbf{Definition}, \\textbf{Operation}, and \\textbf{Control}) of the distance-related algorithms, and also identify the potential parallelism and pipeline opportunities during the design time. We detail these constructs in the following part of this section. \n\n\\subsection{Data Construct}\nData construct is a basic \\textbf{Definition Construct}. It leverages \\code{DSet} primitive to indicate the name of the data variable, and the \\code{DType} primitive to notate the type characters of the defined variable. Data construct serves as the basis for AccD compiler to understand the algorithm description input, such as the data points that are used in the distance-related algorithms. An example of data constructs is shown in the code below, where we define the variable and dataset using DDSL data construct.\n\\vspace*{-0.3\\baselineskip}\n\\begin{lstlisting}\n\/* Define a single variable *\/\nDVar [setName] DType [Optional_Initial_Value];\n\/* Define the matrix of dataset *\/\nDSet [setName] DType [size] [dim];\n\\end{lstlisting}\n\\vspace*{-0.3\\baselineskip}\n\nIn most distance-related algorithms, the dataset can be defined as the source set and the target set. For example, in K-means, the source set is the set of data points, and the target set is the set of clusters. Currently, AccD supports several data types including \\textit{int} (32-bit), \\textit{float} (32-bit), \\textit{double} (64-bit) based on the users' requests, algorithm performance, and accuracy trade-offs.\n\n\\subsection{Distance Compute Construct}\nDistance computation is the core \\textbf{Operation Construct} for distance-related algorithms, which measures the exact distance between two different data points. This construct requires several fields, including data dimensionality, distance metrics, and weight matrix (if weighted distance is specified).\n\\begin{lstlisting}\nAccD_Comp_Dist(Input p1, Input p2, Output disMat, Output idMat, Dim dim, Met mtr, Weg mat)\n\\end{lstlisting}\n\\vspace{-1.5em}\n\\begin{table}[ht] \\small\n\\centering\n \\begin{tabular}{|| c | l ||} \n \\hline\n p1, p2 & Input data matrix. ($n_1 \\times d$, $n_2 \\times d$)\\\\ \n \\hline\n disMat & Output distance matrix. ($n_1 \\times n_2$)\\\\\n \\hline\n idMat & Output id matrix. ($n_1 \\times n_2$) \\\\\n \\hline\n dim & Dimensionality of input data point.\\\\\n \\hline\n mtr & Distance metric:\\code{(Weighted|Unweighted)}\\\\ \\hline\n mat & Weight matrix: Used for weighted distance ($1\\times d$)\\\\\n \\hline\n\\end{tabular}\n\\caption{Distance Compute Construct Parameters.}\n\\label{fig: Distance Compute Construct Parameters.}\n\\end{table}\n\n\\vspace*{-1.3\\baselineskip}\n\\subsection{Distance Selection Construct}\nDistance selection construct is an \\textbf{Operation Construct} for distance value selection and it returns the Top-K smallest or largest distances and their corresponding points ID number from the provided distance and ID list. This construct helps AccD compiler to understand the distances of users' interests.\n\\begin{lstlisting}\nAccD_Dist_Select(Input distMat, Input idMat, Output TopKMat, Range ran, Scope scp)\n\\end{lstlisting}\n\\begin{table}[h] \\small\n\\centering\n \\begin{tabular}{|| c | p{18em}||} \n \\hline\n TopKMat & Top-K id matrix ($n_1 \\times k$)\\\\ \n \\hline\n ran & Scalar value of \\code{K} (\\textit{e.g.}, K-means, KNN) or distance threshold (\\textit{e.g.}, N-body Simulation)\\\\\n \\hline\n scp & Top-K \\code{(smallest|largest)} values\\\\\n \\hline\n\\end{tabular}\n\\caption{Distance Selection Construct Parameters.}\n\\label{fig: Distance Selection Construct Parameters.}\n\\end{table}\n\\vspace*{-1.1\\baselineskip}\n\\subsection{Data Update Construct}\n\\vspace*{-0.2\\baselineskip}\nData update construct is an \\textbf{Operation Construct} for updating the data points based on the results from the prior constructs. For example, K-means updates the cluster centers by averaging the positions of the points inside. This construct requires the variable to be updated and additional information to finish this update, such as the point-to-cluster distances. The status of this data update will be returned after the completion of all its inside operations. The status variable is to tell whether the data update makes a difference or not.\n\\begin{lstlisting} \nAccD_Update(Update var, Input p1 ,..., Input pm, Status s)\n\\end{lstlisting}\n\\vspace*{-1.2\\baselineskip}\n\\begin{table}[ht] \\small\n\\centering\n \\begin{tabular}{|| c | l ||} \n \\hline\n upVar & Input data\/dataset to be updated \\\\\n \\hline\n {p1, ..., pm} & Additional information used in update\\\\ \n \\hline\n $S$ & Status of update operation.\\\\\n \\hline\n\\end{tabular}\n\\caption{Data Update Construct Parameters.}\n\\label{fig: Data Update Construct Parameters.}\n\\end{table}\n\n\\vspace*{-1.5\\baselineskip}\n\\subsection{Iteration Construct}\n\\vspace*{-0.5\\baselineskip}\nIteration construct is a top-level \\textbf{Control Construct}. It is used to describe the distance-related algorithms that require iteration, such as K-means. Iteration construct requires users to provide either the maximum number of iteration or other exit condition.\n\\begin{lstlisting}[mathescape=true]\nAccD_Iter(maxIterNum|exitCond){\n subConstruct $sc_1$;\n subConstruct $sc_2$;\n ...\n subConstruct $sc_n$;\n}\n\\end{lstlisting}\n\n\\vspace*{-0.5\\baselineskip}\n\\subsection{Example: K-means}\n\\vspace*{-0.2em}\nTo show the expressiveness of DDSL, we take K-means as an example. From the code shown below, with no more than 20 lines of code, DDSL can capture the key components of user-defined K-means algorithm, which is essential for AccD compiler to generate designs for CPU-FPGA platforms.\n\\vspace{0.2em}\n\\begin{lstlisting}\nDVar K int 10;\nDVar D int 20;\nDVar psize int 1400;\nDVar csize int 200;\nDSet pSet float psize D;\nDSet cSet float csize D;\nDSet distMat float psize csize;\nDSet idMat int psize csize;\nDSet pkMat int psize K;\nAccD_Iter(S){\n S = false;\n \/* Compute the inter-dataset distances *\/\n AccD_Comp_Dist(pSet, cSet, distMat, idMat, D, \"Unweighted L1\", 0);\n \/* Select the distances of interests *\/\n AccD_Dist_Select(distMat, idMat, K, \"smallest\", pkMat);\n \/* Update the cluster center *\/\n AccD_Update(cSet, pSet, pkMat, S)\n}\n\\end{lstlisting}\n\n\\section{Evaluation} \\label{sect: Evaluation}\nIn this section, we choose three representative benchmarks (K-means, KNN-join, and N-body Simulation) and evaluate their corresponding AccD designs on the CPU-FPGA platform.\n\n\\paragraph{K-means}\nK-means~\\cite{LloydKMeans,dataclustering50, efficientKmeans, coates2012learning, ray1999determination} clusters a set of points into several groups in an iterative manner. At each iteration, it first computes the distances between each point and all clusters, and then update the clusters based on the average position of their inside points. We choose it as our benchmark since it can show the benefits of AccD hierarchy (\\textbf{Trace-based + Group-level}) bound computation optimization on iterative algorithms with disjoint source and target set.\n\n\\paragraph{KNN-join}\nKNN-join Search~\\cite{altman1992introduction, KNNJoinsHybridApproach, KNNJoinsDataStreams} finds the Top-K nearest neighbor points for each point in the source set from the target set. It first computes the distances between each source point and all the target points. Then it ranks the K-smallest distances for each source point and gets its corresponding closest Top-K target points. KNN-join can help to demonstrate the effectiveness of AccD hybrid (\\textbf{Two-landmark + Group-level}) bound computation optimization on non-iterative algorithms. \n\n\\paragraph{N-body Simulation}\nN-body Simulation~\\cite{nylons2007fast, ida1992n} mimics the particle movement within a certain range of 3D space. At each time step, distances between each particle and its neighbors (within a radius $R$) are first computed, and then the acceleration and the new position of each particle will be updated based on these distances. While N-body simulation is also iterative, it has several differences compared with K-means algorithm: 1) N-body simulation has the same dataset (particles) for source and target set, whereas K-means operates on different source (point) and target (cluster) sets; 2) All points in the N-body simulation would change their positions according to the time variation, whereas in K-means only the target set (cluster) would change their positions during the center update; 3) N-body simulation has the same size of source and target set, whereas K-means target set (cluster) is much smaller than source set (point) in general. N-body simulation can help us to show the strength of AccD hybrid bound computation (\\textbf{Two-landmark + Trace-based + Group-level}) on iterative algorithms with the same source and target set.\n\n\n\\vspace{-0.3\\baselineskip}\n\\subsection{Experiment Setup}\n\\vspace{-0.2\\baselineskip}\n\\paragraph{Tools and Metrics} \nIn our evaluation, we use Intel Stratix 10 DE10-Pro~\\cite{DE10-Pro} as the FPGA accelerator and run the host side software program on Intel Xeon Silver 4110 processor~\\cite{intelXeon} (8-core 16-thread, 2.1GHz base clock frequency, 85W TDP). DE10-Pro FPGA has 378,000 Logic elements (LEs), 128,160 adaptive logic modules (ALM), 512,640 ALM registers, 648 DSPs, and 1,537 M20K memory blocks. We implement AccD design on DE10-Pro by using Intel Quartus Prime Software Suite~\\cite{IntelQuartus} with Intel OpenCL SDK included. To measure the system power consumption (Watt) accurately, we use the off-the-shelf Poniie PN2000 as the external power meter to get the runtime power of Xeon CPU and DE10 Pro FPGA.\n\\vspace{-1em}\n\\begin{table}[h] \\small\n\\centering\n\\caption{Implementation Description.}\n\\label{table: Implementation Description.}\n\\begin{tabular}{|| c | C{9em} | L{9em} ||}\n\\hline\n\\textbf{Name}\n& \\makecell{\\ \\textbf{Techniques}}\n& \\makecell{\\ \\textbf{Description}} \n\\\\ \n\\hline\n\\hline \\textbf{Baseline} & Standard Algorithm without any optimization, CPU. & Naive for-loop based implementation on CPU.\n\\\\ \n\\hline \\textbf{TOP} & Point-based Triangle-inequality Optimized Algorithms, CPU. & TOP \\cite{Topframework} optimized distance-related algorithm running on CPU.\n\\\\ \n\\hline \\textbf{CBLAS} & CBLAS library Accelerated Algorithms, CPU. & Standard distance-related algorithm with CBLAS~\\cite{openblas} acceleration.\n\\\\\n\\hline \\textbf{AccD} & Algorithmic-hardware co-design, CPU-FPGA platform. & GTI filtering and FPGA acceleration of distance computations.\n\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table*}[t] \\small\n\\centering\n \\begin{tabular}{|| l c c c || l c c || c c ||} \n\\hline\n \\multicolumn{4}{||c||}{\\textbf{K-means}} \n & \\multicolumn{3}{c||}{\\textbf{KNN-join}} \n & \\multicolumn{2}{c||}{\\textbf{N-body Simulation}}\\\\ \n\\hline\n \\textbf{Dataset} & \\textbf{Size} & \\textbf{Dimension} & \\textbf{\\#Cluster} \n &\n \\textbf{Dataset} & \\textbf{Dimension} & \\textbf{\\#Source} & \n \\textbf{Dataset} & \\textbf{\\#Particle} \\\\\n\\hline\nPoker Hand & 25,010 &\t11\t & 158 & \nHarddrive1\t& 64 & 68,411 &\nP-1 & 16,384 \n \\\\ \nSmartwatch Sens & 58,371\t& 12\t& 242 & \nKegg Net Directed\t & 24 & 53,413\t &\nP-2 & 32,768\n\\\\\nHealthy Older People & 75,128 & \t9\t & 274 & \n3D Spatial Network\t & 3 & 434,874 &\nP-3 & 59,049\n\\\\\nKDD Cup 2004 & 285,409 & \t74\t & 534 & \nKDD Cup 1998\t & 56 & 95,413\t &\nP-4 & 78,125 \n\\\\\nKegg Net Undirected & 65,554 &\t28\t& 256 & \nSkin NonSkin & 4 & 245,057 & \nP-5 & 177,147 \n\\\\\nIpums & 70,187\t& 60 &\t265 & \nProtein\t & 11 & 26,611 &\nP-6 & 262,144 \n \\\\\n \\hline\n \\end{tabular}\n \\caption{Datasets for Evaluation.}\n \\label{table: Evaluation Dataset}\n\\end{table*} \n\n\\begin{figure*}[h] \\small\n \\centering\n \\subfloat[]{\\includegraphics[width=0.33\\textwidth]{images\/speedup_1.pdf}}\n \\subfloat[]{\\includegraphics[width=0.33\\textwidth]{images\/speedup_2.pdf}}\n \\subfloat[]{\\includegraphics[width=0.33\\textwidth]{images\/speedup_3.pdf}}\n \\caption{Performance Comparison (TOP, CBLAS, AccD): (a) K-means (b) KNN-Join (c) N-body Simulation. Note: Speedup is normalized w.r.t Baseline.}\n \\label{fig: Speedup Performance Comparison.}\n\\end{figure*} \n\\begin{figure*}[ht!] \\small\n \\centering\n \\subfloat[]{\\includegraphics[width=0.33\\textwidth]{images\/power-efficiency-1.pdf}} \n \\subfloat[]{\\includegraphics[width=0.33\\textwidth]{images\/power-efficiency-2.pdf}} \n \\subfloat[]{\\includegraphics[width=0.33\\textwidth]{images\/power-efficiency-3.pdf}}\n \\caption{Energy Efficiency Comparison (TOP, CBLAS, AccD): (a) K-means. (b) KNN-Join. (c) N-body Simulation. Note: Energy Efficiency is normalized w.r.t Baseline.}\n \\label{fig: Energy Efficiency Comparison}\n\\end{figure*}\n\\vspace{-1em}\n\\paragraph{Implementations} \nThe CPU-based implementations consist of three types of programs: the naive for-loop sequential implementation without any optimization (selected as our \\textbf{Baseline} to normalize the speedup and energy-efficiency), the algorithm optimized by \\textbf{TOP}~\\cite{Topframework} framework and the algorithm optimized by \\textbf{CBLAS}~\\cite{openblas} computing library. Note that the TOP + CBLAS implementation is \\textbf{not} included in our evaluation, since after applying TOP point-based TI filtering, each point in the source set has a distinctive list of points from the target set for distance computation, whereas CBLAS requires uniformity in the distance computations. Therefore, it is challenging to combine TOP and CBLAS optimization.\n\n\\paragraph{Dataset} \nIn the evaluation, we use six datasets for each algorithm. The selected datasets can cover the wide spectrum of mainstream datasets, including datasets from UCI Machine Learning Repository~\\cite{UCI-dataset}, and datasets that have ever been used by previous papers~\\cite{Topframework, ding2015yinyang, chen2017sweet} in the related domains. Details of these datasets are listed in Table~\\ref{table: Evaluation Dataset}. Note that KNN-join algorithm will find the Top-1000 closest neighbors of each query point.\n\n\\subsection{Comparison with Software Implementation} \n\\vspace{-0.2\\baselineskip}\n\\label{sect: Comparison with Software Implementation}\n\\paragraph{Performance Comparison} \nAs shown in Figure~\\ref{fig: Speedup Performance Comparison.}, TOP, CBLAS, and AccD achieve average $9.12\\times$, $9.19\\times$ and $31.42\\times$ compared with Baseline across all algorithm and dataset settings, respectively. As we can see, AccD design can always maintain the highest speedup among these implementations. This largely dues to AccD GTI optimization in reducing distance computation and its efficient hardware acceleration of the distance computation on FPGA.\n\nWe also observe that TOP implementation shows its strength for large datasets. For example, on dataset 3D Spatial Network ($n=434,874$) in KNN-join, TOP implementation achieves $39.78\\times$ speedup. Since the fine-grained point-based TI optimization of TOP can reduce most (more than 90\\%) of the unnecessary distance computations, which benefits the overall performance to a great extent. Note that the intrinsic point distribution of the dataset would also affect the filtering performance of TOP, but in general, the larger dataset could lead TOP to spot and remove more redundant computations.\n\nWhat we also notice is that CBLAS implementation demonstrates its performance on datasets with relatively high dimensionality. For example, on dataset KDD Cup 2004 ($d=74$) in the K-means algorithm, CBLAS achieves $11.78\\times$ speedup over Baseline, which is higher than its performance on other K-means datasets. This is because, on high dimension dataset, CBLAS implementation can get more benefits of parallelized computing and more regularized memory access, whereas, in low dimension settings, the same optimization can only yield minor speedup.\n\nOur AccD design achieves a considerable speedup on datasets with large size and high dimensionality. For example, on dataset KDD Cup 2004 ($n=285,409, d=74$) and Ipums ($n=70,187, d=60$) in K-means, AccD achieves $51.61\\times$ and $66.61\\times$ speedup over Baseline, and also significantly higher than both TOP and CBLAS implementations. This conclusion can also be extended to KNN-join, such as $88.95\\times$ speedup on dataset KDD Cup 1998 ($n=95,413$, $d=56$). Since our AccD design can effectively reconcile the benefits from both the GTI optimization and the FPGA acceleration, where the former provides the opportunity to reduce the distance computation at the algorithm level, and the latter boosts the performance from hardware acceleration perspective. More importantly, our AccD design can balance the above two benefits to maximize the overall performance. \n\n\n\\paragraph{Energy Comparison} \nThe energy efficiency of AccD design is also significant. For example, on the K-means algorithm, AccD designs deliver an average $116.85\\times$ better energy efficiency compared with Baseline, which is significantly higher than TOP and CBLAS implementations. There are namely two reasons behind these results: 1) Much lower power consumption. AccD CPU-FPGA design only consumes $5w \\sim 17.12w$ across all algorithm and dataset settings, whereas Intel Xeon CPU consumes at least $20.9w$ and $42.49w$ on TOP and CBLAS implementations, respectively; 2) Considerable performance. AccD design achieves a much better speedup (more than $5\\times$ on average) compared with the TOP and CBLAS, which contributes to overall design energy-efficiency. \n\nAmong these implementations, CLBAS implementation has the lowest energy efficiency, since it relies on multi-core parallel processing capability of the CPU, which improves the performance at the cost of much higher power consumption (average $65.79w$). TOP only leverages the single-core processing capability of the CPU and achieves moderate performance with effective distance computation reduction, which results in less power consumption (average $25.59w$) and higher energy efficiency (average $9.12\\times$) compared with Baseline. Different from the TOP and CBLAS implementations, AccD design is built upon a low-power platform with considerable performance, which shows a far better energy-performance trade-off. \n\n\\vspace{-0.15\\baselineskip}\n\\subsection{Performance Benefits Analysis}\n\\vspace{-0.15\\baselineskip}\nTo analyze the performance benefits of AccD CPU-FPGA design in detail, we use K-means as the example algorithm for study. Specifically, we build four implementations for comparison: 1) TOP K-means on CPU; 2) TOP K-means on CPU-FPGA platform; 3) AccD K-means on CPU; 4) AccD K-means on CPU-FPGA platform. Note that TOP K-means is designed for sequential-based CPUs, and no publicly available TOP implementation on CPU-FPGA platforms. For a fair comparison, we implement TOP K-means on CPU-FPGA platform with memory optimizations (inter-group and intra-group memory optimization) and distance computation kernel optimization (Vector-Matrix multiplication). These optimizations improve the data reuse and memory access performance.\n\\vspace{-0.8\\baselineskip}\n\\begin{figure} [ht] \\small\n \\centering\n \\makebox{\\includegraphics[width=0.85\\columnwidth]{images\/KMeans-speedup-breakdown.pdf}}\n \\caption{AccD Performance Benefits Breakdown.}\n \\label{fig: AccD Performance Benefits Breakdown.}\n\\end{figure}\nWe compute the normalized speedup performance of each implementation w.r.t the naive for-loop based K-means implementation on CPU. \n\nAs shown in Figure~\\ref{fig: AccD Performance Benefits Breakdown.}, AccD K-means on CPU-FPGA platform can always deliver the best overall speedup performance among these implementations. We also observe that TOP K-means can achieve average $3.77\\times$ speedup on CPU, however, directly porting this optimization towards CPU-FPGA platform could even lead to inferior performance (average $2.63\\times$). Even though we manage to add several possible optimizations, applying such fine-grained TI optimization from TOP would still cause a large divergence of computation among points, leading to low data reuse and inefficient memory access. \n\nWe also notice that AccD design on CPU achieves lower speedup (average $2.69\\times$) compared with the TOP (average $3.77\\times$), since its coarse-grained GTI optimization spots a fewer number of unnecessary distance computations. However, when combining AccD design with CPU-FPGA platform, the benefits of AccD GTI optimization become prominent (average $37.37\\times$), since it can maintain computation regularity while reducing memory overhead to facilitate the hardware acceleration on FPGA. Whereas, applying optimization to maximize the algorithm-level benefits while ignoring hardware-level properties would result in poor performance, such as the TOP (CPU-FPGA) implementation. \nMoreover, comparison of AccD (CPU) and AccD (CPU-FPGA) can also demonstrate the effectiveness of using FPGA as the hardware accelerator to boost the performance of the algorithms, which can deliver additional $9.68\\times \\sim 15.71\\times$ speedup compared with the software-only solution.\n\n\n\n\n\n\n\n\\section{Introduction} \nDistance-related algorithm (\\textit{e.g.}, K-means~\\cite{LloydKMeans}, KNN~\\cite{altman1992introduction}, and N-body Simulation~\\cite{NBody-simulation}) plays a vital role in many domains, including machine learning, computational physics, etc. However, these algorithms often come with high computation complexity, leading to poor performance and limited applicability.\nTo improve their performance, FPGA-based acceleration gains lots of interests from both industry and research field, given its great performance and energy-efficiency. However, accelerating distance-related algorithms on FPGAs requires non-trivial efforts, including the hardware expertise, time and monetary cost. While existing works try to ease this process, they inevitably fall in short in one of the following aspects.\n\n\n\n\\textbf{Rely on problem-specific design and optimization while missing effective generalization.} There is no such unified abstraction to formalize the definition and optimization of distance algorithms systematically. Most of the previous hardware designs and optimizations~\\cite{KMeansMicroarray,lin2012k,kdtreeKMeanscolorimage, KNNfpgahls} are heavily coded for a specific algorithm (\\textit{e.g.}, K-means), which can not be shared with different distance-related algorithms. Moreover, these \"hard-coded\" strategies could also fail to catch up with the ever-changing upper-level algorithmic optimizations and the underlying hardware settings, which could result in a large cost of re-design and re-implementation during the design evolvement.\n\n \\textbf{Lack of algorithm-hardware co-design.} Previous algorithmic~\\cite{elkan2003using, ding2015yinyang} and hardware optimizations~\\cite{lin2012k, kdtreeKMeanscolorimage, KMeansMicroarray, multicoreKMeans, KNNfpgahls} are usually applied separately instead of being combined collaboratively. Existing algorithmic optimizations, most of which are based on \\textit{Triangle Inequality (TI)}~\\cite{elkan2003using, ding2015yinyang, Topframework, chen2017sweet}, are crafted for sequential-based CPU. Despite removing a large number of distance computations, they also incur high computation irregularity and memory overhead. Therefore, directly applying these algorithmic optimizations to massively parallel platforms without taking appropriate hardware-aware adaption could lead to inferior performance\n\n\\textbf{Count on FPGAs as the only source of acceleration.} \nPrevious works~\\cite{ParallelArchitecturesKNN, IPcoresKNN, ParameterizedKMeans, Lavenier00fpgaimplementation, KMeansMicroarray, KNNfpgahls} place the whole algorithm on the FPGA accelerator without considering the assists from the computing resource on the host CPU. As a result, their designs are usually limited by the on-chip memory and computing elements, and cannot fully exploit the power of the FPGA. Moreover, they miss the full performance benefits from the heterogeneous computing paradigm, such as using the CPU for complex logic and control operations while offloading the compute-intensive tasks to the FPGA.\n\n\\textbf{Lack of well-structured design workflow.} Previous works~\\cite{ParallelArchitecturesKNN, ParameterizedKMeans, kdtreeKMeanscolorimage, lin2012k, KNNfpgahls} follow the traditional way of hardware implementation and require intensive user involvement in hardware design, implementation, and extra manual tuning process, which usually takes long development-to-validation cycles. Also, the problem-specific strategy leads to a case-by-case design process, which cannot be widely applied to handle different problem settings.\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=1.6\\columnwidth]{images\/AccD-overview.pdf}\n \\caption{AccD Overview.}\n \\label{fig: AccD Workflow}\n\\end{figure*}\nTo this end, we present a compiler-based optimization framework, \\textit{AccD}, to automatically accelerate distance-related algorithms on the CPU-FPGA platform (shown in Figure~\\ref{fig: AccD Workflow}). \nFirst, AccD provides a \\textbf{Distance-related Domain-Specific Language} (\\textit{DDSL}) as a problem-independent abstraction to unify the description and optimization of various distance-related algorithms. With the assist of the \\textit{DDSL}, end-user can easily create highly-efficient CPU-FPGA designs by only focusing on high-level problem specification without touching the algorithmic optimization or hardware implementation. \n\nSecond, AccD offers a novel \\textbf{algorithmic-hardware co-optimization} scheme to reconcile the acceleration from both sides. At the algorithmic level, AccD incorporates a novel \\textit{Generalized Triangle Inequality (GTI)} optimization to eliminate unnecessary distance computations, while maintaining the computation regularity to a large extent. At the hardware level, AccD employs a \\textit{specialized data layout} to enforce memory coalescing and an \\textit{optimized distance computation kernel} to accelerate the distance computations on the FPGA. \n\nThird, AccD leverages both the host and accelerator side of the \\textbf{CPU-FPGA heterogeneous system for acceleration}. In particular, AccD distributes the algorithm-level optimization (\\textit{e.g.}, data grouping and distance computation filtering) to CPU, which consists of complex operations and execution dependency, but lacks pipeline and parallelism. On the other hand, AccD assigns hardware-level acceleration (\\textit{e.g.}, distance computations) to the FPGA, which is composed of simple and vectorizable operations. Such mapping successfully capitalizes the benefit of CPU for managing control-intensive tasks and the advantage of FPGA for accelerating computation-intensive workloads. \n\nLastly, AccD compiler integrates an intelligent \\textbf{Design Space Explorer} \\textit{(DSE)} to pinpoint the \"optimal\" design for different problem settings. In general, there is no existing \"one size fits all\" solution: the best configuration for algorithmic and hardware optimization would differ across different distance-related algorithms or different inputs of the same distance-related algorithm.\nTo produce a high-quality optimization configuration automatically and efficiently, DSE combines the design modeling (performance and resource) and Genetic Algorithm to facilitate the design space search.\n\nOverall, our contributions are:\n\\begin{itemize} \n \\item We propose the first optimization framework that can automatically optimize and generate high-performance and power-efficient designs of distance-related algorithms on CPU-FPGA heterogeneous computing platforms. \n \\item We develop a Domain-specific Language, DDSL, to unify different distance-related algorithms in an effective and succinct manner, laying the foundation for general optimizations across different problems.\n \n \\item We build an optimizing compiler for the DDSL, which automatically reconciles the benefits from both the algorithmic optimization on CPU and hardware acceleration on FPGA.\n \n \n \n \\item Intensive experiments on several popular algorithms across a wide spectrum of datasets show that AccD-generated CPU-FPGA designs could achieve $31.42\\times$ speedup and $99.63\\times$ better energy-efficiency on average compared with standard CPU-based implementations\n \n\\end{itemize}\n\n\\section{Overview of AccD Framework} \n\n\\section{Related Work} \\label{sect: Related Work}\n\\vspace*{-0.32em}\nPrevious research accelerates distance-related algorithms in two aspects: \\textit{Algorithmic Optimization} and \\textit{Hardware Acceleration}. More details are discussed in the following subsections.\n\n\\subsection{Algorithmic Optimization} \nFrom the algorithmic standpoint, previous research highlights two optimizations. The first one is KD-tree based optimization~\\cite{KD-TreeKMeans, efficientKmeans, KNNJoinsDataStreams, 5952342, Zhong:2013:GEI:2505515.2505749}, which relies on storing points in special data structures to enable nearest neighbor search without computing distances to all target points. These methods often deliver $3\\times \\sim 6\\times$ performance improvement~\\cite{KD-TreeKMeans, efficientKmeans, KNNJoinsDataStreams, 5952342, Zhong:2013:GEI:2505515.2505749} compared with the unoptimized versions in low dimensional space, while suffering from a serious performance degradation when handling large datasets with high dimension ($d \\geq 20$) due to their exponentially-increased memory and computation overhead.\n\nThe second one is TI based optimization~\\cite{elkan2003using, ding2015yinyang, Topframework, chen2017sweet}, which aims at replacing computation-expensive distance computations with cheaper bound computations, demonstrates its flexibility and scalability. It can not only reduce the computation complexity at different levels of granularity but is also more adaptive and robust to the datasets with a wide range of size and dimension. \nHowever, most existing works focus on one specific algorithm (\\textit{e.g.}, KNN~\\cite{chen2017sweet}, K-means~\\cite{elkan2003using, ding2015yinyang}, etc.), which lack extensibility and generality across different distance-related problems. \nAn exception is a recent work, TOP~\\cite{Topframework}, which builds a unified framework to optimize various distance-related problems with pure TI optimization on CPUs. Our work shares a similar high-level motivation with their work, but targets at a more challenging scenario: algorithmic and hardware co-optimization on CPU-FPGA platforms.\n\n\\subsection{Hardware Acceleration}\nFrom the hardware perspective, several FPGA accelerator designs have been proposed, but still suffer from some major limitations.\n\nFirst, previous FPGA designs are generally built for specific distance-related algorithm and hardware. For example, works from~\\cite{KMeansMicroarray, kdtreeKMeanscolorimage, lin2012k} target on KNN FPGA acceleration, while researches from~\\cite{KNNfpgahls, IPcoresKNN, ParallelArchitecturesKNN} focus on K-means. Moreover, previous designs~\\cite{lin2012k, KMeansMicroarray} usually assume that dataset can be fully fit into the FPGA on-chip memory, and they are only evaluated on a limited number of small datasets, for example, in~\\cite{lin2012k}, K-means acceleration is evaluated on a micro-array dataset with only 2,905 points. These designs often encounter portability issues when transferring to different settings. Besides, these \"hard-coded\" designs and optimizations create difficulties for a fair comparison among different designs, which hamper future studies in this direction. \n\nThe second problem with previous works is that they fail to incorporate algorithmic optimizations in the hardware design.\nFor example, works from~\\cite{KMeansMicroarray, ParallelArchitecturesKNN, kdtreeKMeanscolorimage, KNNfpgahls}, directly port the standard K-means and KNN algorithms to FPGA, and only apply hardware-level optimization. \nOne exception is a recent work~\\cite{KPynq}, which promotes to combine TI optimization and FPGA acceleration for K-means. It gives a considerable speedup compared to state-of-the-art methods, showcasing the great opportunity of applying algorithm-hardware co-optimization. Nevertheless, this idea is far from well-explored, possibly because it requires the domain knowledge and expertise from both the algorithm and hardware to combine both of them effectively. \n\nIn addition, previous works largely focus on the traditional hardware design flow, which requires a long implementation cycle and huge manual efforts. For example, works from~\\cite{ParameterizedKMeans, KMeansMicroarray, multicoreKMeans, kdtreeKMeanscolorimage, ICSICT2016, Lavenier00fpgaimplementation, adaptiveKNNPartialReconfiguration, adaptiveKNN} build the design based on VHDL\/Verilog design flow, which requires hardware expertise and over months of arduous development. In contrast, our AccD design flow brings significant advantages of programmability and flexibility due to its high-level OpenCL-based programming model, which minimizes the user involvement in the tedious hardware design process.\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\\section{I\/O Contention and Inherent Noise Errors}\\label{sec:data_collection_errors}\nWith the ability to estimate the amount of application and system modeling error, as well as detect outlier jobs,\nleftover error is caused by system contention or inherent noise. Both of these error classes are caused by aleatory\nuncertainty, since the model lacks deeper insight into jobs or the system, as opposed the OoD case where the model lacks\nsamples. While e.g., application error was predictable and explainable in terms of broad application behavior (e.g.,\nthis application is slow because it frequently writes to shared files), the impact of contention and noise on I\/O\nthroughput is caused by lower level, transient effects. Though it may be possible to observe and log such effects\nthrough microarchitectural hardware counters or network switch logs, such logging would require vast amounts of storage\nper job and would impact performance. Lack of practical logging ability makes the last two error categories typically\n\\textit{unobservable}. Furthermore, these two classes may only be separated in hindsight, and while I\/O noise levels\nmay be constant, the amount of I\/O contention on the system is unpredictable for a job that is about to run.\n\nThe question we ask in this section are: how can errors due to noise and contention be separated from errors due to poor\nmodeling or epistemic uncertainty? Is there a fundamental limit to how accurate I\/O models can become? Can system I\/O\nvariability be quantified? \n\n\\subsection{Establishing the bounds of I\/O modeling}\nTo separate the contention and noise from the first three classes of error, we develop a litmus test based on the test\nfrom Section~\\ref{sec:stationary_errors}. \nThere, by observing sets of duplicates, the performance of an ideal model was estimated for which $e_{app} = 0$. Comparing real models against this ideal model allowed us to calculate real model's $e_{app}$.\nA similar litmus test can be designed estimate the sum of contention and noise error, where only concurrent duplicates are\nobserved and both application behavior $j$ and global system behavior $\\zeta_g$ can be held static for each duplicate set:\n\\begin{tcolorbox}[left=0mm,right=0mm]\n\\vspace{-0.1cm}\n1. OoD jobs are removed and sets of duplicate jobs ran at the same time ($\\Delta t=0$) are collected;\n2. The mean I\/O throughput of each set is calculated; \n3. Duplicate error is calculated as before;\n4. The median error across all duplicates is reported. \n\\vspace{-0.1in}\n\\end{tcolorbox}\n\nIn the fifth column of Figure~\\ref{fig:teaser} we show the distribution of I\/O throughput differences $\\Delta \\phi$ and\ntiming differences $\\Delta t$ between all pairs of Cori duplicate jobs, weighted so that large duplicate sets are not\noverrepresented. The vertical strip on the left contains Cori duplicate jobs that were ran\nsimultaneously, largely because they were batched together. These jobs share $j$ and $\\zeta_g$, but may differ in\n$\\zeta_l$ and $\\omega$. Due to the denser sampling around 1 minute to 1 hour range, it is not immediately apparent\nhow the I\/O difference changes between duplicates ran at the same time and duplicates ran with a delay. By grouping\nduplicates from different $\\Delta t$ ranges and independently scaling them, a better understanding of duplicate I\/O\nthroughput distributions across timescales can be made, as shown in Figure~\\ref{fig:duplicates_over_time_kde}. For both\nsystems (Cori omitted due to lack of space), the distributions on the right contain jobs ran over large periods of time where\nglobal system impact $\\zeta_g$ might have changed, explaining the asymmetric shape of some of them. The left-most\ndistributions are similar, since variance only stems from contention $\\zeta_l$ and noise $\\omega$. While some\ndistributions (e.g., the $10^5$ to $10^6$ second) show complex multimodal behavior, all of the\ndistributions seem to contain the initial zero second ($0s$ to $1s$) distribution. \n\n\\begin{figure}[h]\n \\centering\n \\vspace{-0.5cm}\n \\includegraphics[width=0.8\\columnwidth]{figures\/kde_duplicates.pdf}\n \\caption{Distribution of errors for different periods between duplicate runs.}\n \\label{fig:duplicates_over_time_kde}\n \\vspace{-0.2cm}\n\\end{figure}\n\nBy fitting a normal distribution to the $\\Delta t=0$ distribution (0s to 1s) in Figure~\\ref{fig:duplicates_over_time_kde}, we can both\n(1) learn the lower limit on total modeling error and (2) learn the system's I\/O noise level, i.e., how much I\/O\nthroughput variance should jobs running on the system expect. However, upon closer inspection, the $\\Delta t=0$\ndistribution \\textit{does not} follow a normal distribution. This is surprising, since if noise is normally\ndistributed, independent over time, and it's effects are cumulative, the total impact is a sum of normal distributions,\nwhich should also be a normal distribution. The answer lies in how the concurrent ($\\Delta t=0$) duplicates are sampled.\nWhen observing duplicates, in general, duplicate sets have between 2 and hundreds of thousands of identical jobs in them.\nHowever, in duplicate sets with identical start times on Theta, 70\\% of the sets only have two identical jobs, and 96\\% have\n6 jobs or less, with similar results on Cori. The issue stems from how small (sub-30 samples) duplicate set\nerrors are calculated: when only a small number of jobs exist in the set, the mean I\/O throughput of the set is biased\nby the sampling, i.e., the estimated mean is closer to the samples than the real mean is. This causes the set I\/O\nthroughput variance to decrease and therefore duplicate error estimate will be reduced as well. Student's\n\\textit{t}-distribution describes this effect: when the true mean of a distribution is known, error calculations follow\na normal distribution. When the true mean is not known, the biased mean estimate makes the error follow the\n\\textit{t}-distribution. As the sample count reaches 30 and above, the \\textit{t}-distribution approaches the normal\ndistribution.\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n\nFinally, we seek to estimate the I\/O noise variance of a given system. Naively taking the variance of the\n\\textit{t}-distribution will produce a biased sample variance $\\sigma^2$, but it can be fixed by applying Bessel's\ncorrection as $\\frac{n}{n-1}\\sigma^2$. \nIn practical terms, a job\nrunning on Theta can expect an I\/O throughput within $\\pm5.71\\%$ of the predicted value \\textit{68\\% of the\ntime}, or within $\\pm 10.56\\%$ 95\\% of the time. For Cori, these values are $\\pm 7.21\\%$ and $\\pm 14.99\\%$, respectively.\nThis is a fundamental barrier not just to I\/O model improvement, but to predictable system usage in general. Although some insight into contention can be gained through low-level logging tools, noise cannot be overcome. I\/O practitioners\ncan use this litmus test to evaluate the noise levels of their systems, and ML practitioners should reconsider how they\nevaluate models, since some systems may be simply harder to model. \n\n\n\n\n\n\\endinput\n\n\n\n\\section{Discussion}~\\label{sec:discussion}\nDeveloping production-ready machine learning models that analyze HPC jobs and predict I\/O throughput is difficult: the\nspace of all application behaviors is large, HPC jobs are competing for resources, and the system changes over time. To efficiently improve these models, we present a taxonomy of HPC I\/O modeling errors that allows us to study the\ntypes of errors independently, quantify their impact, and identify the most promising avenues for model improvement. Our\ntaxonomy breaks errors into five categories: (1) application and (2) system modeling errors, (3) poor generalization, (4) \nresource contention, and (5) I\/O noise. We present litmus tests that quantify what percentage of existing error belongs\nto the classes and show that models improved by using the taxonomy are within several percentage points of an\nestimated best-case I\/O throughput modeling accuracy. We show that a large portion of I\/O throughput modeling error is irreducible and stems from I\/O variability. We provide tests that quantify the I\/O variability and establish an upper bound on how accurate models can become. Our test shows that jobs run on Theta and Cori can expect an I\/O throughput standard deviation of 5.7\\% and 7.2\\%, respectively.\n\n\\section{Applying the taxonomy} \\label{sec:framework}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=.99\\linewidth]{figures\/framework_updated.pdf}\n \\caption{Framework for applying the taxonomy (left), and the results from ALCF Theta and NERSC Cori systems.} \n \\label{fig:framework}\n \\vspace{-.2in}\n\\end{figure*}\n\nWe now illustrate how the proposed taxonomy can be used in practice. In Figure~\\ref{fig:framework} (left), we show the\nsteps a modeler can follow to evaluate the taxonomy on a new system. \nStep 1: The modeler splits the available data into training and test sets, and then trains and evaluates some baseline\nmachine learning model on the task of predicting I\/O throughput. This model does not have to be perfect, as the\ntaxonomy should reveal what are the main sources of error and approximately how much the quality of the model is at\nfault. \nStep 2.1: The modeler estimates application modeling errors by finding duplicate jobs and evaluating the mean predictor\nperformance on every set of duplicates. Assuming that the distribution of duplicate HPC jobs is representative of the\nwhole population of jobs, this step provides the modeler with a lower bound on the application modeling error. \nStep 2.2: By contrasting the baseline model error (Step 1) and the estimated application modeling error, the modeler can\nestimate the percentage of error that can be attributed to poor modeling. The modeler performs a hyperparameter or network\narchitecture search and arrives at a good model close to the bound. \nStep 3.1: The modeler estimates system modeling errors by exposing the job start time to a golden model.\nThis step requires that the modeler has developed a well-performing model in Step 2.2, that is, one that achieves\nclose to the estimated ideal performance. The test set error of the model serves as an estimate of the application\n+ system modeling lower bound. \nStep 3.2: The modeler explores adding sources of system data to improve the performance of the baseline\nmodel up to the estimated limit of application and system modeling. \nStep 4: The modeler identifies out-of-distribution samples using AutoDEUQ, calculates OoD error that stems\nfrom these samples, and removes them from the dataset. \nStep 5: The modeler estimates the error that can be attributed to contention and noise, as well as I\/O variance of the\nsystem. This estimate is made by observing the I\/O throughput differences between sets of concurrent duplicates, i.e.,\nduplicate jobs ran at around the same time. \n\nIn the middle and right portion of Figure~\\ref{fig:framework} we show the average baseline model error (inner pink circle\nsegment) of both ANL Theta and NERSC Cori systems, and how that error is broken down into different classes of error.\nWe do not focus on the cumulative error value of the two systems; instead, we focus on attributing the baseline model error\ninto the five classes of errors in the taxonomy (middle circle segments of the pie chart), and how much improved\napplication and system modeling can help reduce the cumulative error (outer segments of the pie chart). \nThe inner blue section of the two pie charts represents the estimated application modeling error,\nas arrived at in Step 2.1. The outer blue section represents how much of the error can be fixed through hyperparameter\nexploration, as explored in Step 2.2. \nThe inner green section represents the estimated system modeling error, derived in Step 3.1. Note that the total\npercentage of system modeling error is relatively small on both systems; i.e., I\/O contention, filesystem health,\nhardware faults, etc., do not have a dominant impact on I\/O throughput. The outer green circle segment represents the\npercentage of error that can be fixed by including system logs (LMT logs in our case), as described in Step 3.2. Only\nthe Cori pie chart has this segment, as Theta does not collect LMT logs. On Cori, the inclusion of LMT logs helps remove\nmost of the system modeling errors, pointing to the conclusion that including other logs (i.e., topology, networking)\nmay not help to significantly reduce errors. \nThe inner red segment represents the percentage of error that can be attributed to out-of-distribution samples of the\ntwo systems, as calculated in Step 4. \nFinally, the yellow circle segment represents the percentage of error that can be attributed to aleatory uncertainty.\nFor both Theta and Cori, this is a rather large amount, pointing to the fact that there exists a lot of innate noise in the\nbehavior of these systems, and setting a relatively high lower bound on ideal model performance. \n\nThe similarity between the modeling error estimates (Steps 2.1 and 3.1) and the actual updated model performance (Steps 2.2\nand 3.2) is surprising and serves as evidence for the quality of the error estimates. However, the estimates of the five error classes \\textit{do not} add up to 100\\%. The first three error estimates are just that - estimates,\nderived from a subset of data (duplicate HPC jobs) that do not necessarily follow the same distribution as the rest of\nthe dataset and may be biased. If we add the estimates, we see that on Theta 32.9\\% of the error is\nunexplained, and on Cori 13.5\\% of the error is unexplained.\nCori's lower unexplained error may be due to the fact that we have collected some 1.1M jobs compared to 100K on Theta. \n\n\\section{Introduction}\n\nAs scientific applications push to leverage ever more capable computational platforms, there is a critical need to\nidentify and address bottlenecks of all types.\nDue to the large data movements in these\napplications, the I\/O subsystem is often a major source of performance bottlenecks, and it is common for applications to\nattain only a small fraction of the peak I\/O rates~\\cite{luu:behavior}. These performance problems can severely limit the\nscalability of applications and are difficult to detect, diagnose, and fix. Data-driven machine learning-based models\nof I\/O throughput can help practitioners understand application bottlenecks (e.g.,~\\cite{isakov_sc20, moana,\n10.1007\/978-3-319-92040-5_10, 10.1145\/3337821.3337922, isakov_ross20, 10.1145\/3369583.3392678}), and have the potential to\nautomate I\/O tuning and other tasks. However, current machine learning-based I\/O models are not robust enough for production\nuse~\\cite{isakov_ross20}.\nA thorough investigation of \\emph{why} these models underperform when deployed on high\nperformance computing (HPC) systems will provide key insights and guidance on how to address their shortcomings. The\ngoal of our study is to help machine learning (ML)-driven I\/O modeling techniques make the transition from theory to\npractice. \n\nThere are several reasons why machine learning-based I\/O models underperform when deployed: poor modeling choices~\\cite{isakov_sc20,\n10.1145\/3369583.3392678}, concept drift in the data~\\cite{10.1145\/3337821.3337922}, and weak generalizatio\n~\\cite{isakov_ross20},\namong others. I\/O models are often opaque, and there is no established methodology for diagnosing the root\ncause of model errors. In this work, we present a taxonomy of ML-based I\/O modeling errors, as shown in\nFigure~\\ref{fig:teaser}. Through this taxonomy, we show that I\/O throughput prediction errors can be separated and\nquantified into five error classes: inadequate (1) application and (2) system models, (3) novel application or system\nbehaviors, (4) I\/O contention and (5) inherent noise. \nFor each class, we present data-driven litmus tests that estimate the portion of\nmodeling error caused by that class. The taxonomy enables independent study of each source of error \nand prescribes appropriate ML techniques to tackle the underlying sources of error. \n\n\n\nOur contributions in this work are as follows:\n\\begin{enumerate}[leftmargin=*]\n \\item We introduce a taxonomy of ML-based I\/O throughput modeling errors which consists of five causes.\n \n \n \n \n\n \\item We show that choice of machine learning model type, scaling model size, and hyperparameter tuning cannot reduce all potential errors.\n We present two litmus tests that quantify error due to poor application and\n system modeling.\n\n \\item We present a litmus test that estimates what portion of error is caused by\n \n \n rare jobs with previously unseen behavior, and apply uncertainty quantification methods to classify those jobs as\n out-of-distribution jobs.\n\n \\item We present a method for quantifying the impact of I\/O contention and noise on I\/O throughput, which (1)\n defines a fundamental limit in how accurate ML models can become,\n \n and (2) gives HPC system users and\n administrators a practical estimate of the I\/O throughput variance they should expect. We show that underlying\n system noise is the dominant source of errors, and not poor modeling or lack of application or system data. \n \n\n \\item We present a framework for how the taxonomy is practically applied to new systems and evaluate it on two\n leadership-class supercomputers: Argonne Leadership Computing Facility (ALCF) Theta and National Energy\n Research Scientific Computing Center (NERSC) Cori.\n\\end{enumerate}\n\n\n\n\\section{Global system modeling errors}\\label{sec:nonstationary_errors}\nThe second part of the approximation error in our taxonomy is the global system modeling error. This error \nrefers to I\/O climate and I\/O weather effects~\\cite{lockwood_pdsw17} that affect all jobs running on the system, and\ncorresponds to the second component in Equation~\\ref{eq:phi_breakdown}. While global and local system impacts have\ncomplex and overlapping effects, we suggest that factorizing the impact applied to all jobs versus the impact that is\ndependent on pairs of concurrent jobs is useful for modeling purposes. \nThe main difference between the two is that local system impacts cannot be predicted or modeled without knowledge of all\njobs running on the system, while global system impacts can. In other words, global system impacts can be expressed as a\nproperty of the system at a given time, effectively compressing a part of system behavior.\nWe now ask: How does I\/O contention impact job I\/O throughput prediction? What are the\nlimits of global system modeling? How can I\/O models approach this limit? \n\n\\subsection{Estimating limits of global system modeling}\nGlobal system impact $\\zeta_g(t)$ on job $j$ from Equation~\\ref{eq:phi_breakdown} can be formalized as some\nfunction $\\zeta_g(t) = g(J(t))$ where $J$ is the set of jobs running at time $t$. Since jobs have a start and end\ntime, given a dataset with a dense enough sampling of $J$, $g(J(t))$ can be calculated for every point in time. During periods of time where\nthe file system is suffering a service degradation, all jobs on the system will be impacted with varying severity. A\nmodel of the system does not need to understand how and why the degradation happened, it only needs to know the period\nwhen it lasted and how different types of jobs were impacted. This time-based model is useless for predicting\nfuture performance, and it's only utility is in evaluating how much of the degradation can be described as purely a\nfunction of time. A deployed model does not have data on the future and will still need to observe the system.\n\nTo evaluate the global system impact, a golden model that exhibits no global modeling error is developed, against which we\ncan compare other, `real' ML models. Since the global system impact of $\\zeta_g(t)$ only depends on time and does not need the set of all jobs $J$, only application behavior $j$ and the job start time\nfeature are exposed to the golden model. Here, a golden model is an XGBoost model fine tuned on a validation set and\nevaluated on a test set. Assuming that the golden model is exposed to enough jobs throughout the lifetime of the system,\nit will learn the impact of $\\zeta_g(t)$ even without having access to the underlying system features causing that\nimpact. This golden model is used in the following litmus test:\n\\begin{tcolorbox}[left=0mm,right=0mm]\n\\vspace{-0.1cm}\n1. The timing feature is\nadded to the Darshan-only (no Lustre or Cobalt) dataset; 2. A hyperparameter search is performed on a validation set and\napplication modeling error is removed; 3. the final model is evaluated on the test set, and it's error is reported.\n\\vspace{-0.1in}\n\\end{tcolorbox}\nIf the litmus test is correct, the golden model only suffers from the last three classes of errors: poor generalization, \nlocal system impact, and inherent noise. \nIn Figure~\\ref{fig:lustre_model_comparison} we evaluate a baseline model (blue) and a model enriched with the\njob start time (orange). The addition of this single feature has a large impact on error: on\nCori, the error drops $40\\%$, from 16.49\\% down to 10.02\\%, while on Theta the error drops by 30.8\\%. Note that to obtain high accuracy on the POSIX+start time, a much larger model is needed, i.e., one that can remember the I\/O weather\nthroughout the lifetime of the system. \n\n\\subsection{Improving modeling through I\/O visibility}\nWith an estimate of minimal error achievable assuming perfect application and global system modeling, we investigate\nwhether I\/O subsystem logs can help models approach this limit. Since Theta does not collect I\/O subsystem logs, we\nanalyze Cori, which collects both application and I\/O logs. Figure~\\ref{fig:lustre_model_comparison} shows the XGBoost\nperformance of three models: a baseline where $e_{app}=0$ (blue), the litmus test golden model where also $e_{system}=0$ (orange), and a\nLustre-enriched model (green). Cori's median absolute error is reduced by 40\\%, from\n16.49\\% down to 9.96\\%. The Lustre-enriched results are surprisingly close to the litmus test's predictions,\nand suggest that predictions cannot be improved through further I\/O insight since the litmus test's prediction is\nreached.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.95\\columnwidth]{figures\/lustre.pdf}\n \\caption{\n \n Error distribution of models trained on (1) POSIX, (2) POSIX + \n the start time feature, and (3) Darshan and Lustre}\n \\label{fig:lustre_model_comparison}\n \\vspace{-0.5cm}\n\\end{figure}\n\n\\endinput \n\n\n\\section{Generalization errors}\\label{sec:generalization_errors}\nThe remaining three classes of error are caused by lack of data and not poor modeling, as the top branch of the\ntaxonomy shows. While I\/O contention and inherent noise errors are examples of aleatory uncertainty and\nare caused by lack of insight into specific jobs, generalization errors stem from epistemic uncertainty, i.e., the lack\nof other logged jobs around a specific job of interest. \nTo motivate this section, in the third graph of\nFigure~\\ref{fig:teaser} we show error distribution of a model trained on data from January 2018 to July 2019. When \nevaluated on held-out data from the same period, the median absolute error is low (green line). Once the model is\ndeployed and evaluated on the data collected after the training period (July 2019 and after), median error spikes up (red\nline). \n\n\n\n\n\n\\subsection{Estimating generalization error}\n \n \n \n \n \n \n \n\n\nEstimating the amount of out-of-distribution error $e_{OoD}$ is important because any\nunaccounted OoD error will be classified as noise or contention. This will make systems that run a lot of novel jobs\nappear to be more noisy than they truly are. Because OoD and ID jobs will likely have a similar amount of I\/O and\ncontention noise, it is better to have false positives (ID jobs classified as OoD) than the other way, since false\nnegatives contribute to overestimating I\/O noise.\nTo estimate the impact of out-of-distribution jobs on error $e_{OoD}$, we aim to quantify the how much of error is\nepistemic, and how much is aleatory in nature, as shown in Figure~\\ref{fig:teaser} (upper right).\nThe leading paradigm for uncertainty quantification works by training an ensemble of models and evaluating all of the\nmodels on the test set. If the models make the same error, the sample has high aleatory uncertainty, but if the models\ndisagree, the sample has high epistemic uncertainty~\\cite{10.5555\/3295222.3295387}. The intuition is that predictions on\nout-of-distribution samples will vary significantly on the basis of the model architecture, whereas predictions on noisy\nsamples will exhibit the same amount of error. Since this method relies on ensemble model diversity, several works have explored\nincreasing diversity through different model hyperparameters~\\cite{Wenzel2020HyperparameterEF}, different\narchitectures~\\cite{Zaidi2020NeuralES}, or both~\\cite{autodeuq}. We choose to use AutoDEUQ~\\cite{autodeuq}, a method\nthat evolves an ensemble of neural network models and jointly optimizes both the architecture and hyperparameters of the\nmodels. AutoDEUQ's Neural Architecture Search (NAS) is compatible with the NAS search from\nsection~\\ref{sec:stationary_errors}, reducing the computational load of applying the\ntaxonomy. Figure~\\ref{fig:autodeuq} shows the distribution of epistemic (EU) and aleatory uncertainties (AU) of Theta\nand Cori test sets. For both systems, aleatoric uncertainty is significantly higher than epistemic uncertainty.\nFurthermore, \\textit{all} jobs seem to have AU larger than some about 0.05, hinting at the inherent noise present in the\nsystem. The inverse cumulative distributions on the top and right show the what percentage of total error is caused by\nAU \/ EU \\textit{below} that value. For example, for both systems 50\\% of all error is caused by jobs with EU below 0.04,\nwhile in case of AU, 50\\% of error is below AU=0.25. The low total EU is expected since the test set was drawn from the\nsame distribution as the training set, and increases on the 2020 set (omitted due to space concerns). \n\n\\begin{figure}[h]\n \\vspace{-0.2cm}\n \\centering \n \\includegraphics[width=0.95\\columnwidth]{figures\/uncertainty}\n \\caption{Distribution of job aleatory and epistemic uncertainties for the two systems, with marginal distributions\n and inverse cumulative error shown on the margins.} \n \\label{fig:autodeuq}\n \\vspace{-0.2cm}\n\\end{figure}\n\nEpistemic uncertainty does not directly translate into the out-of-distribution error $e_{OoD}$ from\nEquation~\\ref{eq:error_breakdown}. When a sample is truly OoD, it may not be possible to separate aleatory and epistemic\nuncertainty, since a good estimate of AU requires dense sampling around the job of interest. Therefore, we choose to\nattribute all errors of a sample marked as out-of-distribution to $e_{OoD}$. This error attribution requires classifying\nevery test set sample as either in- or out-of-distribution, but since EU estimates are continuous values, an EU\nthreshold which will separate OoD and ID samples is required. Although this threshold is specific to the dataset and\nmay require tuning, the quick drop or `shoulder' in inverse cumulative error around EU=0.1 in Figure~\\ref{fig:autodeuq}\nmakes the choice of an $e_{OoD}$ threshold robust. \nA litmus test that estimates the error due to out-of-order samples has the following steps:\n\\begin{tcolorbox}[left=0mm,right=0mm]\n\\vspace{-0.1cm}\n1. Run NAS and collect the best performing models; 2. Estimate the aleatory and epistemic uncertainty using AutoDEUQ; 3. Find a stable EU threshold and classify the samples as ID and OoD; 4. Calculate $e_{OoD}$ as the sum of OoD sample errors. \n\\vspace{-0.1in}\n\\end{tcolorbox}\nOn Theta, for an EU threshold of 0.24, .7\\% of the samples are classified as OoD, but constitute 2.4\\%\nof the errors, while on Cori 2.1\\% of error gets removed for the same EU threshold. In other words, the selected jobs have $3\\times$ larger average error than random samples. By manually exploring the types of jobs that do get removed, we confirm that these are typically rare or novel applications. \n\n\\endinput\n\n\n\\section{Datasets and experimental setup}\\label{sec:preliminaries}\nThis work is evaluated on two datasets, one collected from ALCF Theta\nsupercomputer in the period from 2017 to 2020, and one collected from NERSC Cori\nsupercomputer in the period from 2018 to 2019. Theta collects Darshan~\\cite{darshan} and Cobalt logs and consists\nof about 100K jobs with an I\/O volume larger than 1GiB, while Cori collects Darshan and Lustre Monitoring\nTools (LMT) logs, and consists of 1.1M jobs larger than 1GiB. \n\nDarshan is an HPC I\/O characterization tool that collects HPC job I\/O access patterns on both the POSIX and MPI-IO levels, and serves as our main insight\ninto application behavior. It collects POSIX aggregate job-level data, e.g., total number of bytes transferred, accesses made,\nread \/ write ratios, unique or shared files opened, distribution of accesses per access size, etc. \nMPI-IO is a library built on top of POSIX that offers\nhigher-level primitives for performing I\/O operations and may offer the model more insight into application behavior.\nDarshan collects MPI-IO information for jobs that use it, and all requests through MPI-IO are also visible on the POSIX level.\nThe Cobalt scheduler logs number of nodes and cores assigned to a job, job start and end times, job placement, etc.\nCobalt logs may be useful to the model since the number of cores a job allotted is not visible to Darshan. Darshan\nonly collects the number of job processes, which is commonly equal to or greater than the number of cores allocated to a job.\nLMT collects I\/O\nsubsystem information such as storage server load and file system utilization, and serves as our main insight into the\nI\/O subsystem state over time. \nLMT records the state of object storage servers (OSS) and\ntargets (OST), and metadata servers (MDS) and targets (MDT) of the Lustre scratch filesystem every 5 seconds. Some of\nthe features LMT collects are CPU and memory utilization of the OSS's and MDS's, number of bytes\ntransferred to and from the OSTs, the fullness of the system, or the number of metadata operations (e.g., \\texttt{open},\n\\texttt{close}, \\texttt{mkdir}, etc.) performed by the metadata targets, etc. \nLMT collects per-OSS\/OST\/MDS\/MDT logs, but since a job may be served by an arbitrary number of these I\/O nodes, only the\nminimum, maximum, mean and standard deviation are exposed to the ML model.\nOverall, models have access to 48 Darshan POSIX, 48 Darshan MPI-IO, 37 LMT, and 5 Cobalt features. \n\n\nThe ML models in this work are trained using supervised learning on the task of predicting the I\/O throughput of\nindividual HPC jobs. The error models are optimizing is:\n\\begin{equation}\n e(y, \\hat y) = \\frac{1}{n} \\sum^n_{i=1}\\left|log_{10} \\left(\\frac{y_i}{\\hat y_i}\\right)\\right|\n\\end{equation}\nwhere $y_i$ and $\\hat y_i$ are the $i$-th job's measured and predicted I\/O throughputs. \nBecause $log(x) = -log(1\/x)$, if a model overestimates or underestimates the I\/O throughput by the same\nrelative amount, the error remains the same. We use percentages to write errors, where, e.g., a -25\\% error specifies \nthat the model underestimated real I\/O throughput by 25\\%; however, some figures show the absolute error when model bias\nis not important. While models try to lower mean error, they report median values since some of the distributions have heavy tails that make mean estimates unreliable.\n\n\\section{Related work}\\label{sec:related}\nIn recent years, automating HPC I\/O system analysis through ML has received significant attention, with two prominent\ndirections: \n(1) workload clustering to better understand groups of HPC jobs and automate handling of whole groups, and \n(2) I\/O subsystem modeling and make predictions of HPC job I\/O time, I\/O throughput, optimal scheduling, etc. \nClustering HPC job logs has been explored in~\\cite{isakov_sc20, gauge, taxonomist} with the\ngoal of better understanding workload distribution, scaling I\/O expert effort more efficiently, and revealing hidden\ntrends and I\/O workload patterns. ML-based modeling has been used for predicting I\/O\ntime~\\cite{10.1007\/978-3-319-92040-5_10}, I\/O throughput~\\cite{10.1145\/3369583.3392678, isakov_sc20}, optimal filesystem\nconfiguration~\\cite{8752835, capes}, as well as for building black boxes of I\/O subsystems in order to apply ML model\ninterpretation techniques~\\cite{isakov_sc20}. While there have been some attempts at creating analytical models of I\/O\nsubsystems~\\cite{10.1109\/CLUSTER.2015.29}, most attempts are data-driven, and rely on HPC system logs to create models\nof I\/O \\cite{luu:behavior, 10.1145\/3369583.3392678, 10.1007\/978-3-319-92040-5_10, isakov_sc20, Wang2018IOMinerLA}. \nAlthough the challenges of developing accurate machine learning models are well known, the nature of the domain requires\nspecial consideration: I\/O subsystems have to service multiple competing jobs, their configuration evolves over time,\nthey have periods of increased variability, they experience occasional hardware faults, etc.~\\cite{1526010,\n10.1109\/SC.2018.00077, costa1}. Diagnosing this I\/O variability, where the performance of a job depends on external factors to\nthe job itself has been extensively studied~\\cite{moana, 1526010, 10.1145\/3322789.3328743,\n10.1007\/978-3-319-92040-5_10, costa1}. Finally, the deployment of I\/O models has been shown to require special consideration as\nthese models often significantly underperform on new applications~\\cite{10.1145\/3337821.3337922, isakov_ross20}. \nWhile different sources of model error have been studied individually, no prior work characterizes the relative impact\nof different sources of error on model accuracy.\n\n\n\\section{Application modeling errors}\\label{sec:stationary_errors}\nWhen an ML practitioner is tasked with a regression problem (predicting continuous values), the first model they\nevaluate will likely under-perform on the task, e.g., due to inadequate data preprocessing, architecture, or\nhyperparameters. Therefore, the model will suffer from \\textit{approximation errors}, which can be removed by tuning the\nmodel hyperparameters. Because inappropriate models may be unable to fit even easy tasks, this class of errors should\nbe solved before seeking, e.g., additional samples or sample features.\n\nFigure~\\ref{fig:teaser} shows that the approximation consists of the application and\nsystem approximation. This section asks whether I\/O models build faithful representations of application\nbehavior.\n The questions we\nask are as follows: What are the limits of I\/O application modeling? In practice, do I\/O models faithfully learn application\nbehavior? Can I\/O application modeling benefit from extra hyperparameter fine-tuning or new application features? \n\n\\subsection{Estimating limits of application modeling}\nWe develop an application modeling error litmus test whose objective is to separate the application modeling error $e_{app}$ from\nthe four other error classes in Equation~\\ref{eq:error_breakdown}. To do so, we seek a `golden model' that models\napplication behavior as accurately as possible given the inputs. When practical ML models are compared with this golden model, an\nestimate of the application modeling error can be made. \n\nTo build this `golden model', we rely on a property of synthetic datasets where the data-generating process can be freely\nand repeatedly sampled. When analyzing HPC logs, it is common to see records of the same application ran multiple times\non the same data, or data of the same format. For example, system benchmarks such as IOR~\\cite{ior} may be run\nperiodically to evaluate filesystem health and overall performance. We call these sets of repeated jobs\n\\textit{`duplicate jobs'}. Jobs are duplicates if they belong to the same application and all their\n\\textit{observable} application features are identical.\nNote that an application can have many different sets of duplicate jobs, likely ran with different input parameters.\nBecause the duplicate features fed to an ML model are identical, the model cannot distinguish between duplicates from\nthe same set, and will achieve best possible accuracy on the training set if it learns to predict the mean I\/O\nthroughput value of a set.\nA model that does not learn to predict a set's mean value suffers from application-modeling error.\n\nSets of duplicate jobs can be used to build a litmus test that evaluates the median absolute error of a model which has $e_{app}=0$. Any practical model with a median error $e^p$ \ncan then learn its application modeling error $e^p_{app}$ by comparing it with the median error\nof the `golden model' $e^g$ as $e^p_{app} = e^p - e^g$. The litmus test is administered as: \n\\begin{tcolorbox}[left=0mm,right=0mm]\n\\vspace{-0.1cm}\n1. Sets of duplicate jobs in the dataset are found; 2. The mean I\/O throughput of each set is calculated; \n3. This mean is subtracted from each duplicate's I\/O throughput in the set to calculate duplicate error, and Bessel's correction is applied~\\cite{bishop};\n4. The median error across all duplicates is reported. \n\\vspace{-0.1in}\n\\end{tcolorbox}\nThis average difference represents\nthe smallest average error \nthat a model $m(j)$ can achieve on \\textit{duplicate jobs}. Assuming that duplicate jobs are drawn\nfrom the same distribution of applications as the whole dataset, the duplicate median absolute error represents the lower bound on\nmedian absolute error a model can achieve on the whole dataset. Note that different applications may have different distributions of\nduplicate I\/O throughputs, as shown in the fourth column of Figure~\\ref{fig:teaser}. \nFor this litmus test to be accurate, a large sample of applications representative of the HPC system workload \nmust be acquired. When applied to Theta, 19010 duplicates (23.5\\% of the dataset) over 3509 sets show a median \nabsolute error of 10.01\\%. Cori has 504920 duplicates (54\\%) in 77390 sets show a median absolute error of 14.15\\%.\nIf the litmus test is correct, a model that has the same (but not lower!) median absolute error on the general dataset can be found.\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\\subsection{Minimizing application modeling error}\nThe next question is whether ML models can practically reach the estimate of the lower bound of the error. Several I\/O modeling works\nhave explored different types of ML models: linear regression~\\cite{isakov_sc20}, decision\ntrees~\\cite{10.1007\/978-3-319-58667-0_19}, gradient boosting machines~\\cite{isakov_sc20, 10.1007\/978-3-319-58667-0_19, xie1},\nGaussian processes~\\cite{10.1007\/978-3-319-92040-5_10}, neural networks~\\cite{10.1145\/3337821.3337922}, etc. Here, we explore two types of models: XGBoost~\\cite{xgboost}, an implementation\nof gradient boosting machines, and feedforward neural networks. These model types are chosen for their accuracy,\nscalability, and previous success in I\/O modeling. \n\nNeither type of model achieves ideal performance `out of the box'. XGBoost model performance can be improved through\nhyperparameter tuning, e.g., by exploring different (1) numbers of decision trees, (2) their depth, (3) the features each\ntree is exposed to, and (4) part of the dataset each tree is exposed to. Neural networks are more complex, since they\nrequire tuning hyperparameters (learning rate, weight decay, dropout, etc.), while also exploring different\narchitectures (number, size, type of layers, and their connectivity). In the case of XGBoost, we exhaustively explore four\nhyperparameters listed above, for a total of 8046 XGBoost models. \nIn the case of neural\nnetworks, exhaustive exploration is not feasible due to state space explosion, so we use\nAgEBO~\\cite{DBLP:journals\/corr\/abs-2010-16358}, a Network Architecture Search (NAS) method that trains populations of\nneural networks and updates each subsequent generation's hyperparameters and architectures through Bayesian logic.\n\nThe leftmost column of Figure~\\ref{fig:teaser} shows a heatmap of an exhaustive search over two parameters on the Theta \ndataset, with the other two parameters (\\% of columns and rows revealed to the trees) selected from the\nbest possible result found. \nThe best performing model has an error rate of 10.51\\% - very close to the predicted bound.\nThe search on Cori data arrives at a similar configuration (omitted due to space).\n\nIn the case of neural networks, Figure~\\ref{fig:dnn_nas} shows a scatter plot of test set errors of 10 generations\nof neural networks on the Cori system, with 30 networks per generation. The networks are evolved using a separate\nvalidation test to prevent leakage of the test set into the model parameters. Networks approach the estimated error\nlimit, and the best result achieves a median absolute error of 14.3\\%. The fact that (1) after extensive tuning both\ntypes of models asymptotically approach the estimated limit in model accuracy, and (2) NAS does little to improve\nmodels, since models improve predictions only 6 times (gold stars). This suggests that both types of ML models\nare impeded by the same barrier and that the architecture and the tuning of models are not the fundamental\nissue in achieving better accuracy, i.e., that the source of error lies elsewhere.\n\n\\begin{figure}[h]\n \\vspace{-0.3cm}\n \\centering\n \\includegraphics[width=\\columnwidth]{figures\/nas}\n \\caption{Results of the Neural Architecture Search (NAS), with the estimated error lower bound highlighted in red.}\n \\label{fig:dnn_nas}\n \\vspace{-0.3cm}\n\\end{figure}\n\n\n\\subsection{Increasing visibility into applications}\nWhile hyperparameter and architecture searches approach but do not surpass the litmus test's estimated lower bound on\nerror, this is not conclusive evidence that all application modeling error has been removed and that error stems\nfrom other sources. Possibly, there exist missing application features that might further reduce errors. We explore\ntwo such sets of features: MPI-IO logs and Cobalt scheduler logs. \n\nFigure~\\ref{fig:stationary_models} shows the absolute error distribution of the tuned models on three Theta datasets:\nPOSIX, POSIX + MPI-IO, and POSIX + Cobalt (Cori excluded because of the lack of Cobalt logs). None of the dataset enrichments\nhelp reduce error, corroborating the conclusion that application modeling is not a source of error for these models, and further insight into applications will not help. Adding Cobalt logs does reduce the error on\nthe training set, and our experiments show that the job start and end time features are the cause.\nOnce timing features are present in the dataset, no two jobs are duplicates due to small timing variations. Although\npreviously the ML model was not able to overfit the dataset due to the existence of duplicates, this is no longer the\ncase, and the ML model can differentiate and memorize each individual sample. In~\\cite{isakov_sc20} authors remove\ntiming features for a similar reason: ML models can learn Darshan's implementation of I\/O throughput calculation and\nachieve good predictions without having to observe job behavior.\n\n\\begin{figure}[h]\n \\centering\n \n \\includegraphics[width=0.85\\columnwidth]{figures\/models.pdf}\n \\vspace{-0.0in}\n \\caption{Error distributions for models trained on POSIX, POSIX + MPI-IO, and POSIX + Cobalt feature\n sets.} \n \\label{fig:stationary_models}\n \\vspace{-0.2in}\n\\end{figure}\n\n\n\\section{Modeling HPC applications and systems}\\label{sec:system}\nThe behavior of an HPC system is governed by both complex rules and inherent noise. By formalizing the system as a\nmathematical function (or, more generally, a stochastic process) with its inputs and outputs, the process may be decomposed into\nsmaller components more amenable to analysis. The I\/O throughput of a system running specific sets of applications may\nbe treated as a data-generating process from which I\/O throughput measurements are drawn. While building a perfect model\nof an HPC system may not be possible, it is useful to understand the inputs to the `true' process and the process's\nfunctional properties. The theoretical model of the process must include all causes that might affect a real HPC system,\nsuch as: how well a job uses the system, hardware and software configurations over the life of the system, resource\ncontention between concurrent jobs, inherent application-specific and system noise, as well as application-specific\nnoise sensitivity. Although many of these aspects are not observable in practice, the true\ndata-generating process takes these aspects into account.\n\nWe adapt the global modeling formulation from Madireddy et al.~\\cite{10.1007\/978-3-319-92040-5_10}, and formulate I\/O\nthroughput $\\phi(j)$ of a job $j$ as:\n\\begin{equation} \\label{eq:global_model}\n \\phi(j) = f(j, \\zeta, \\omega)\n\\end{equation}\nHere, $\\zeta$ represents system state (e.g., filesystem health, system configuration, node availability, etc.) and\nsystem behavior (e.g., the behavior of other applications co-located with the modeled application during its run,\ncontention from resource sharing, etc.) \\textit{at a given time}. $\\omega$ represents randomness acting on the\nsystem.\nThe system $\\zeta$ can be further decomposed as:\n\\begin{equation}\n \\zeta = \\zeta_g(t) + \\zeta_l(t, j)\n\\end{equation}\nThe component $\\zeta_g(t)$ represents the \\textit{global system impact} on all jobs running on the system (e.g., a\nservice degradation that equally impacts all jobs) and is only a function of time $t$. The component $\\zeta_l(t, j)$\nrepresents the \\textit{local system impact} on the I\/O throughput of job $j$ caused by resource contention and interactions with other jobs\nrunning on the system. Contrary to the $\\zeta_g(t)$ component, $\\zeta_l(t, j)$ is job-specific and depends on the behavior of the\ncurrent set of applications running on the system, the sensitivity of $j$ to resource contention and noise, etc. Without\nloss of generality, the I\/O throughput of a job from Equation~\\ref{eq:global_model} can be represented as:\n\\begin{equation} \\label{eq:phi_breakdown}\n\\begin{split}\n \n \\phi(j) &= f(j,\\; \\zeta_g(t),\\; \\zeta_l(t, j),\\; \\omega) \\\\\n &= f_a(j) + f_g(j, \\zeta_g(t)) + f_l(j, \\zeta_l(t, j)) + f_n(j, \\zeta, \\omega)\n\\end{split}\n\\raisetag{28pt}\n\\end{equation}\n\nHere, $f_a(j)$ represents job throughput on an idealized system where the job is alone on the system, the system does not change over time, and there is no resource contention. $f_g(j, \\zeta_g(t))$ represents how the evolving\nconfiguration of the system (hardware provisioning, software updates, etc.) affects a job's I\/O throughput. The $f_l(j,\n\\zeta_l(t, j))$ component represents the impact of resource contention and the sensitivity of $j$ to I\/O noise. \nFinally, $f_n(j, \\zeta, \\omega)$ represents the impact of inherent system noise (e.g., packets dropped) on the job. \n\n\\vspace{-0.17cm}\n\\subsection{Modeling assumptions}\nThe task of modeling a system's I\/O throughput is to predict the behavior of the system when executing a job from some\napplication on some data. Modeling I\/O throughput requires modeling both the HPC system and the jobs running on it.\nMachine learning models used in this work attempt to learn the true function $\\phi$ by mapping observable features of the\njob $j$ and the system $\\zeta$ to measured I\/O throughputs $\\phi(j)$. A model $m(j_o, \\zeta_o)$ is tasked with predicting throughput $\\phi(j)$, where $j_o$ and $\\zeta_o$ are the observable job and system features.\n\nWhen designing ML models, the choice of model architecture and model inputs has implicit assumptions about the process\nthat generates the data. When incorrect assumptions are made about the domain, the model will suffer errors that cannot\nbe fixed within that modeling framework. We investigate four common assumptions about the HPC domain.\n\n\\textbf{All data is in-distribution:}\na common assumption that practitioners make is that all model errors are the product of insufficiently trained models,\ninadequate model architectures, or missing discriminative \\textit{features}. However, some jobs in the\ndataset may be \\textit{Out of Distribution (OoD)}, that is, they may be collected at a different time or environment, or\nthrough a different process. The model may underperform on OoD jobs due to the lack of similar jobs in the training\nset and not due to lack of insight (features) into the job. The cause of the problem is \\textit{epistemic uncertainty (EU)}\n- the model suffers from lack of knowledge or \\textit{reducible uncertainty}, since a broader\ntraining set would make the OoD jobs in-distribution (ID). In the HPC domain, epistemic uncertainty is present in\ncases of rarely run or novel jobs or uncommon system states. Without considering the possibility that a portion of the\nerror is a product of epistemic uncertainty, practitioners will put effort into tuning models instead of collecting more\nunderrepresented jobs. Referring to Equation~\\ref{eq:global_model}, this assumption may be expressed as: deployment\ntime $j_d$ and $\\zeta_d$ are drawn from a different distribution from training time $j_t$ and $\\zeta_t$.\n\n\\textbf{Noise is absent:} \nall systems have some inherent noise that cannot be modeled and will impact predictions.\n\\textit{Aleatory uncertainty} (AU) refers to inherent uncertainty, either due to noise or missing features about jobs in the\ndataset.\nModeling errors due to aleatory uncertainty are\ndifferent from epistemic uncertainty because collecting more jobs may not reduce AU. HPC I\/O domain experts note that\ncertain systems do have significantly higher or lower I\/O noise~\\cite{wan1, xie2}, but quantifying noise in modern leadership computing systems is an under-explored subject. Understanding and characterizing system's inherent I\/O noise is key to account for the aleatory uncertainty\nin the ML-based I\/O models and better quantify the effect of this uncertainty on the I\/O throughput predictions. I\/O\nmodeling works rarely attempt to quantify ML model uncertainty~\\cite{madireddy_vae} even though an estimate of AU\nsignificantly helps in model selection. The assumption that noise is not present in the dataset can be expressed as\nfollows: The practitioner assumes that the process has the form of $\\phi(j) = f(j, \\zeta)$ instead of $\\phi(j) = f(j,\n\\zeta, \\omega)$. \n\n\n \n\\textbf{Sampling is independent:}\nrunning a job on a system can be viewed as sampling the combination of system state and application\nbehavior and measuring I\/O throughput.\nMost I\/O modeling works implicitly assume that multiple samples taken at the same time are independent of each other. The system is modeled as equally affecting\nall jobs running on it, that is, the placement of different jobs on nodes, the interactions between neighboring jobs,\nnetwork contention, etc. \\textit{do not affect the job}. This assumption can then be expressed as: the process has the\nform of $\\phi(j) = f(j, \\zeta_g(t), \\omega)$, not $\\phi(j) = f(j, \\zeta, \\omega)$. \n\n\\textbf{Process is stationary:}\na common assumption is that the data-generating process is stationary, and that the same job ran at\ndifferent times achieves the same I\/O throughput. As hardware fails, as new nodes are provisioned, and\nshared libraries get updates, the system evolves over time. This assumption is therefore incorrect, and ignoring it,\ne.g., by not exposing \\textit{when} a job is ran to the ML model may cause hard-to-diagnose errors. In other words, the\nassumption is that the process has the form of $\\phi(j) = f(j, \\omega)$ and $f_g(j, \\zeta_g(t)) = 0$.\n\n\n\\section{Classifying I\/O throughput prediction errors}\\label{sec:taxonomy}\nNo matter the problem to which machine learning is applied, a systematic characterization of\nthe sources of errors is crucial to improve model accuracy.\nWhile there is no substitute for `looking at the data' to understand the root\ncause of the problem, this approach does not scale for large datasets. We seek a systematic way to understand the barriers\nto greater accuracy and improve ML models applied to system data. \n\nThe key questions we ask in this work are: What are the impediments to the successful application of learning algorithms in\nunderstanding I\/O? Should ML practitioners focus on acquiring more data on HPC applications or the HPC system?\nHow much of the error stems from poor ML model architectures? How much of the error can be attributed to the dynamic\nnature of the system and the interactions between concurrent jobs? What fraction of jobs exhibit a truly novel I\/O\nbehavior compared to the jobs observed thus far? At what point are the applications \\textit{too novel}, so much so that\nusers should no longer trust the predictions of the I\/O model? We now describe five classes of errors and\ndive deeper into error attribution in Sections~\\ref{sec:stationary_errors}, \\ref{sec:nonstationary_errors},\n\\ref{sec:data_collection_errors} and \\ref{sec:generalization_errors}.\n\nThe lack of application and system observability, the inherent noise $\\omega$, and \nthe OoD jobs prevent ML models from fully capturing system behavior, causeing errors. We define the I\/O\nthroughput prediction error of a model $m$ in a job $j$ as: \n\\begin{equation} \n e(j) = \\phi(j, \\zeta, \\omega) - m(j_o, \\zeta_o)\n\\end{equation}\nFollowing the $\\phi(j)$ terms from Eq.~\\ref{eq:phi_breakdown} and including the out-of-distribution\nerror, the error can be broken down as follows: \n\\begin{equation}\\label{eq:error_breakdown}\n e(j) = e_{app} + e_{system} + e_{OoD} + e_{contention} + e_{noise}\n\\end{equation}\nHere, the application modeling error $e_{app}$ is caused by a poor model fit of application behavior ($f_a(j)$ component),\nthe global system error $e_{system}$ is caused by poor predictions of system dynamics ($f_g(j, \\zeta_g(t))$ component),\nthe out-of-distribution error $e_{OoD}$ is caused by weak model generalization on novel applications or system states,\nthe contention error $e_{contention}$ is caused by poor predictions of job interactions ($f_l(j, \\zeta_l(t, j))$ component),\nand the noise error $e_{noise}$ is caused by the inability of any model to predict inherent noise ($f_4(j, \\zeta, \\omega)$ component). \nThis leaves us with five classes of errors shown at the bottom of Figure~\\ref{fig:teaser}. While attributing such an error\non a per-job basis is difficult, we will show that estimating each component across a whole dataset is possible.\n\n\\input{figures\/taxonomy}\n\n\\vspace{-0.15cm}\n\\subsection{I\/O Model Error Taxonomy and Litmus Tests}\nErrors in Equation~\\ref{eq:error_breakdown} must be estimated in a specific order shown in Figure~\\ref{fig:teaser} due\nto the specifics of individual litmus tests. For example, before the effect of aleatory and epistemic\nuncertainty can be separated, a good model must be found~\\cite{autodeuq}. Similarly, before global and local system\nmodeling errors can be separated, OoD jobs must be identified. \n\n\n\\textbf{Application modeling errors: }\nML models can have varying expressivity and may not always have the correct structure or enough parameters to fit the\navailable data. Models whose structure or training prevents them from learning the shape of the data-generating process\nare said to suffer from \\textit{approximation errors}, which are further divided into \\textit{application} and \\textit{system\nmodeling errors}. \nApplication modeling errors are caused by poor predictions of application behavior which can be fixed through better I\/O models or hyperparameter searches. \nThe first column of Figure~\\ref{fig:teaser} illustrates the impact of application modeling errors\nwith an example hyperparameter search over two XGBoost parameters on the Theta dataset. \nThe search finds that 32 trees with depth of 21 perform best, while the XGBoost defaults use 100 trees of depth 6.\nApproximation errors cannot be classified\nas epistemic or aleatory in nature, because no new features or jobs are necessary to remove this error. To\nestimate AU and EU in the dataset, methods such as AutoDEUQ~\\cite{autodeuq} require first that an appropriate model\narchitecture is found and trained, so approximation errors are the first branch of the taxonomy. \n\n\\textbf{System modeling errors:}\nsystem behavior changes over time due to transient or long-term changes such as filesystem metadata\nissues, failing components, new provisions, etc.~\\cite{lockwood_pdsw17}. A model that is only aware of application\nbehavior, but not of system state implicitly assumes that the process is stationary. It will be forced to learn the\n\\textit{average} system response to I\/O patterns, and will suffer greater prediction errors during periods when system\nbehavior is perturbed. Errors that occur due to poor modeling of the global system component $\\zeta_g(t)$ are called\n\\textit{system modeling errors}. To illustrate this class of errors, two models are trained to predict I\/O throughput in the second column of Figure~\\ref{fig:teaser}, and each model's weekly average error is plotted against time.\nThe blue model is exposed only to application behavior, while the orange model also knows the \\textit{job start time}.\nDuring service degradations, the blue model has long periods of biased errors while the orange model does not, since it knows when degradations happen.\n\n\\textbf{Generalization errors: }\nML models generally perform well on data drawn from the same distribution from which their training set\nwas collected. When exposed to samples highly dissimilar from their training set, the same models tend to make\nmispredictions. These samples are called `out-of-distribution' (OoD) because they come from new, shifted,\ndistributions, or the training set does not have full coverage of the sample space.\nAs an example, the third column of Figure~\\ref{fig:teaser} shows model error before (green) and after (red) deployment, with the error\nsignificantly rising when the model is evaluated on data collected outside the training time span.\n\n\\textbf{Contention errors:}\na diverse and variable number of applications compete for compute, networking, and I\/O\nbandwidth on HPC systems and interact with each other through these shared\nresources~\\cite{10.1145\/3322789.3328743, 7877142}. Although the global system state will impact all jobs equally, the impact of resource sharing is specific to pairs\nof jobs that are interacting and is harder to observe and model. Prediction errors that occur due to lack of\nvisibility into job interactions are called \\textit{contention errors} and are shown in the fourth column of\nFigure~\\ref{fig:teaser}. Here, the I\/O throughputs of a number of identical runs (the same code and data) of different\napplications illustrate that some applications are more sensitive to contention than others, even when accounting for\nglobal system state.\n\n\\textbf{Inherent noise errors:}\nwhile hard to measure, resource sharing errors can potentially be removed through greater insight into the system and\nworkloads. What fundamentally cannot be removed are \\textit{inherent noise errors}: errors due to random behavior by the\nsystem (e.g., dropped packets, randomness introduced through scheduling, etc.). Inherent noise is problematic both\nbecause ML models are bound to make errors on samples affected by noise and because noisy samples may impede model\ntraining. The fifth column of Figure~\\ref{fig:teaser} shows the I\/O throughput and start time differences between pairs\nof identical jobs. The leftmost column contains jobs that ran at exactly the same time, which often experience 5\\%\nor more difference in I\/O throughput.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Appendix}\n\\label{sec:appendix}\n\nIn this section, we gather a few technical results that are used in the main results of the paper. \n\n\n\\begin{definition}[Type $K$ function]~\\cite{Smith:08}\nLet $g:S\\mapsto\\mathbb{R}^n$ be a function on $S\\subset\\mathbb{R}^n$. $g$ is said to be of type $K$ in $S$ if, for each $i \\in \\until{n}$, $g_i(z_1)\\leq g_i(z_2)$ holds true for any two points $z_1$ and $z_2$ in $S$ satisfying $z_1 \\leq z_2$ and $z_{1,i}=z_{2,i}$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lem:monotonicity-same-size}\nLet $\\map{g}{S}{\\mathbb{R}^N}$ and $\\map{h}{S}{\\mathbb{R}^N}$ be both of type $K$ over $S \\subset \\mathbb{R}^N$. Let $z_1(t)$ and $z_2(t)$\nbe the solutions to $\\dot{z}=g(z)$ and $\\dot{z}=h(z)$, respectively, starting from initial conditions $z_1(0)$ and $z_2(0)$ respectively. \nLet $S$ be positively invariant under $\\dot{z}=g(z)$ and $\\dot{z}=h(z)$.\nIf $g(z) \\leq h(z)$ for all $z \\in S$, and $z_1(0) \\leq z_2(0)$, then $z_1(t) \\leq z_2(t)$ for all $t \\geq 0$. \n\\end{lemma}\n\\begin{proof}\nBy contradiction, let $\\tilde{t} \\geq 0$ be the smallest time at which, there exists, say $k \\in \\until{N}$, such that $z_1(\\tilde{t}) \\leq z_2(\\tilde{t})$, $z_{1,k}(\\tilde{t}) = z_{2,k}(\\tilde{t})$, and \n\\begin{equation}\n\\label{eq:class-K-contradiction}\ng_k(z_1(\\tilde{t})) > h_k(z_2(\\tilde{t})).\n\\end{equation}\nSince $g(z)$ is of class $K$, $z_1(\\tilde{t}) \\leq z_2(\\tilde{t})$ and $z_{1,k}(\\tilde{t}) = z_{2,k}(\\tilde{t})$ imply that $g(z_1(\\tilde{t})) \\leq g(z_2(\\tilde{t}))$. This, combined with the assumption that $g(z) \\leq h(z)$ for all $z \\in S$ implies that $g(z_1(\\tilde{t})) \\leq h(z_2(\\tilde{t}))$, which contradicts \\eqref{eq:class-K-contradiction}.\n\\qed\n\\end{proof}\n\nLemma~\\ref{lem:monotonicity-same-size} is relevant because the basic dynamical system in our case is of type $K$. \n\n\\begin{lemma}\n\\label{lem:car-following-type-K}\nFor any $L > 0$, $m>0$, and $N \\in \\mathbb{N}$, the right hand side of \\eqref{eq:x-dynamics-in-Rn} is of type $K$ in $\\mathbb{R}_+^N$.\n\\end{lemma}\n\\begin{proof}\nConsider $\\tilde{x}, \\hat{x} \\in \\mathbb{R}_+^N$ such that $\\tilde{x} \\leq \\hat{x}$. \nIf $\\tilde{x}_i = \\hat{x}_i$ for some $i \\in \\until{N}$, then, according to \\eqref{eq:inter-vehicle-distance-Rn}, $y_i(\\tilde{x})-y_i(\\hat{x})= \\left(\\tilde{x}_{i+1} - \\hat{x}_{i+1} \\right) -\\left(\\tilde{x}_i-\\hat{x}_i \\right) = \\tilde{x}_{i+1} - \\hat{x}_{i+1}$ if $i \\in \\until{N-1}$, and is equal to $\\left(\\tilde{x}_{1} - \\hat{x}_{1} \\right) - \\left(\\tilde{x}_N-\\hat{x}_N \\right)= \\tilde{x}_{1} - \\hat{x}_{1}$ if $i=N$. In either case, $y_i(\\tilde{x}) \\leq y_i(\\hat{x})$, which also implies $y_i^m(\\tilde{x}) \\leq y_i^m(\\hat{x})$ for all $m > 0$.\n\\qed\n\\end{proof}\n\n\n\n\n\nIn order to state the next lemma, we need a couple of additional definitions. \n\n\\begin{definition}[Monotone Aligned and Monotone Opposite Functions]\nTwo strictly monotone functions $\\map{h}{\\mathbb{R}}{\\mathbb{R}}$ and $\\map{g}{\\mathbb{R}}{\\mathbb{R}}$ are said to be \\emph{monotone-aligned} if they are both either strictly increasing, or strictly decreasing. Similarly, the two functions are called \\emph{monotone opposite} if one of them is strictly increasing, and the other is strictly decreasing. \n\\end{definition}\n\n\n\\begin{lemma}\n\\label{lem:appendix-general-summation}\nLet $\\map{h}{\\mathbb{R}_+}{\\mathbb{R}}$ and $\\map{g}{\\mathbb{R}_+}{\\mathbb{R}}$ be strictly monotone functions. Then, for every $y \\in \\mathcal S_N^L$, $n \\in \\mathbb{N}$, $L>0$,\n\\begin{equation}\n\\label{eq:summation-general}\n\\sum_{i=1}^N h(y_i) \\left(g(y_{i+1}) - g(y_i) \\right)\n\\end{equation}\nis non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned. Moreover, \\eqref{eq:summation-general} is equal to zero if and only if $y=\\frac{L}{N} \\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\nFor $i \\in \\until{N}$, let $I_i$ be the interval with end points $g(y_i)$ and $g(y_{i+1})$. \nFor $i \\in \\until{N}$, let $f_i(z):=\\sgn{g(y_{i+1}) - g(y_i)} h(y_i) \\mathbf{1}_{I_i}(z)$. \nLet $g_{\\text{min}}:=\\min_{i \\in \\until{N}} g(y_i)$, and $g_{\\text{max}}:=\\max_{i \\in \\until{N}} g(y_i)$. With $f(z):=\\sum_{i=1}^N f_i(z)$,\n \\eqref{eq:summation-general} can then be written as:\n\\begin{equation}\n\\label{eq:line-integral}\n\\sum_{i=1}^N h(y_i) \\left(g(y_{i+1}) - g(y_i) \\right) = \\int_{g_{\\text{min}}}^{g_{\\text{max}}} f(z) \\, dz.\n\\end{equation}\nWe now show that, for every $z \\in [g_{\\text{min}}, g_{\\text{max}}] \\setminus \\{g(y_i): i \\in \\until{N}\\}$, $f(z)$ is non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned. \nThis, together with \\eqref{eq:line-integral}, will then prove the lemma.\n\nIt is easy to see that every $z \\in [g_{\\text{min}}, g_{\\text{max}}] \\setminus \\{g(y_i): i \\in \\until{N}\\}$ belongs to an even number of intervals in $\\{I_i: \\, i \\in \\until{N}\\}$, say $I_{\\ell_1}, I_{\\ell_2}, \\ldots$, with $\\ell_1 < \\ell_2 < \\ldots$ (see Figure \\ref{fig:intervals} for an illustration).\nWe now show that $f_{\\ell_1}(z) + f_{\\ell_2}(z)$ is non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned. The same argument holds true for $f_{\\ell_3}(z)+f_{\\ell_4}(z), \\ldots$. \nAssume that $g(y_{\\ell_1}) \\leq g(y_{\\ell_2})$; the other case leads to the same conclusion. By definition of $f_i$'s, $f_{\\ell_1}(z)=h(y_{\\ell_1})$ and $f_{\\ell_2}(z)=-h(y_{\\ell_2})$. $g(y_{\\ell_1}) \\leq g(y_{\\ell_2})$ implies that \n$f_{\\ell_1}(z)+f_{\\ell_2}(z)=h(y_{\\ell_1})-h(y_{\\ell_2})$ is non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned, with the equality holding true if and only if $y_{\\ell_1}=y_{\\ell_2}$. \n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=15.5cm]{intervals} \n \\caption{A schematic view of (a) $f_i(z), i=\\{1,2,3,4\\}$ and (b) $f(z)=\\sum_{i=1}^4f_i(z)$ for a $y\\in\\mathcal S_4^L$ ($L=1$) with $y_{\\text{min}} = y_2 < y_4 < y_3 < y_1 = y_{\\mathrm{max}}$ for a $m<1$. }\n \\label{fig:intervals}\n\\end{figure}\n\\qed\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lemma:t-n}\nFor $n \\in \\mathbb{N} \\setminus \\{1\\} $, let $\\psi_n$ be the n-fold convolution of $\\psi \\in \\Psi$. Then, \n$$\\int_{0}^t z \\, \\psi(z)\\psi_{n-1}(t-z)\\mathrm{d} z = \\frac{t}{n}\\psi_n(t) \\qquad \\forall \\, t \\geq 0$$\n\\end{lemma}\n\\begin{proof}\nLet $J_1,\\cdots,J_n$ be $n$ random variables, all with distribution $\\psi$. Therefore, the probability distribution function of the random variable $V:=\\sum_{i=1}^n J_i$ is $\\psi_n$. \nUsing linearity of the expectation, we get that \n\\begin{equation*}\nt=E\\left[\\sum_{i=1}^n J_i|V=t\\right]=\\sum_{i=1}^n E\\left[J_i|V=t\\right]=n \\, E\\left[J_1|V=t \\right]\n\\end{equation*}\ni.e., \n\\begin{equation}\n\\label{eq:D1-cond-sum}\nE\\left[J_1|V=t \\right] = \\frac{t}{n}\n\\end{equation}\nLet $f_{J_1|V}(j_1|t)$ denote the probability distribution function of $J_1|V$. By definition:\n\\begin{equation}\\label{eq:f-d1-z}\nf_{J_1|V}(j_1|t) = \\frac{f_{J_1,V}(j_1,t)}{\\psi_n(t)}=\\frac{\\psi(j_1)\\psi_{n-1}(t-j_1)}{\\psi_n(t)}\n\\end{equation}\nTherefore, using \\eqref{eq:D1-cond-sum} and \\eqref{eq:f-d1-z}, we get that\n\\begin{equation*}\nE[J_1|V=t] = \\int_0^t zf_{J_1|V}(z|t)\\, \\mathrm{d} z = \\int_0^t z\\frac{\\psi(z)\\psi_{n-1}(t-z)}{\\psi_n(t)} \\, \\mathrm{d} z = \\frac{t}{n}\n\\end{equation*}\nSimple rearrangement gives the lemma. \n\\qed\n\\end{proof}\n\n\nThe following is an adaptation of \\cite[Lemma 2.3.4]{Ross:96}. \n\\begin{lemma}\\label{lemma:busy-period-type1-prob}\nLet $a_1, \\cdots, a_{n-1}$ denote the ordered values from a set of $n-1$ independent uniform $(0,t)$ random variables. Let $\\tilde d_0=z \\geq 0$ be a constant and $\\tilde d_1, \\tilde d_2, \\cdots \\tilde d_{n-1}$ be i.i.d. non-negative random variables that are also independent of $\\{a_1, \\cdots, a_{n-1}\\}$, then \n\\begin{align*}\n\\Pr(\\tilde d_{k}+ \\cdots +\\tilde d_{n-1}\\leq a_{n-k},k=1,\\cdots,n-1 |\\tilde d_0+ \\cdots +\\tilde d_{n-1}=t, \\tilde{d}_0=z)\n = \\begin{cases} z\/t & z 0$, $m=1$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, the mean value of the busy period duration is equal to $\\bar \\psi \/ (L - \\lambda \\bar \\psi)$.\n \\end{lemma}\n \\begin{proof}\nA busy period, say of duration $B$, is initiated by the arrival of a vehicle, say $j$, when the system is idle. \nLet the number of vehicles that arrive during the busy period be $N_{bn}$. Note that $N_{bn}$ does not include the vehicle initiating the busy period. Therefore, the workload brought into the system during the busy period is equal to $w_B=\\sum_{i=j}^{j+N_{bn}} d_i$.\nThe expected value of $N_{bn}$ can be obtained by conditioning on the duration of the busy period: \n\\begin{equation}\\label{eq:nb-expectation}\nE[N_{bn}]=E\\left[E[N_{bn}|B]\\right]=E[\\lambda B]=\\lambda E[B]\n\\end{equation}\nwhere the second equality follows from the fact that the arrival process is a Poisson process. \nSince the event $\\{N_{bn}+1=n\\}$ is independent of $\\{d_{j+i},i > n\\}$, $N_{bn}+1$ is a stopping time for the sequence $\\{d_{j+i},i\\geq 1\\}$. Therefore, using Wald's equation, e.g., see \\cite[Theorem 3.3.2]{Ross:96}, and \\eqref{eq:nb-expectation}, the expected value of the workload $w_B$ added to the system during the busy period $B$ is given by: \n\\begin{equation}\\label{eq:wb-expectation}\nE[w_B]=(E[N_{bn}]+1) \\, \\bar \\psi=(\\lambda E[B]+1) \\, \\bar \\psi.\n\\end{equation}\n\n\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=5.5cm]{busy-period-new-notation} \n \\caption{(a) Queue length process and (b) workload process during a busy period. }\n \\label{fig:busy-period}\n\\end{figure}\n\nSince the workload decreases at a constant rate $L$ during a busy period, we have $B=w_B\/L$ (see Figure \\ref{fig:busy-period} for an illustration). Therefore, $E[B]=E[w_B]\/L$, which when combined with \\eqref{eq:wb-expectation}, establishes the lemma. \n\\qed\n\\end{proof}\n\n\n\n\n\\begin{remark}\n\\label{remark:waiting-time}\nSince the mean busy period duration is an upper bound on the mean waiting time, Lemma \\ref{lemma:busy-period-mean} also gives an upper bound on the mean waiting time. One can then use Little's law~\\cite{Kleinrock:75}\\footnote{Little's law has previously been used in the context of processor sharing queues, e.g., in \\cite{Altman.ea:06}.} to show that the mean queue length is upper bounded by $\\lambda\\bar \\psi \/ (L - \\lambda \\bar \\psi)$.\n\\end{remark}\n\nLet $\\mathcal{I}(t):=\\int_0^t\\delta_{\\{w(s)=0\\}}\\mathrm{d} s$ be the cumulative \\emph{idle time} up to time $t$. The following result characterizes the long run proportion of the idle time in the linear case.\n\\begin{proposition}\n\\label{prop:long-run-idle-time}\nFor any $\\lambda 0$, $\\varphi \\in \\Phi, \\psi \\in \\Psi$, the long-run proportion of time in which HTQ is idle is given by the following:\n$$\\lim_{t\\to\\infty} \\frac{\\mathcal{I}(t)}{t}=1-\\frac{\\lambda \\bar \\psi}{L}>0 \\quad a.s.$$\n\\end{proposition}\n\n\\begin{proof}\nHTQ alternates between busy and idle periods. Let $Z=I+B$ be the duration of a cycle that contains an idle period of length $I$ followed by a busy period of length $B$. Idle period, $I$, has the same distribution as inter-arrival times i.e. an exponential random variable with mean $1\/\\lambda$, and the mean value of $B$ is given in Lemma \\ref{lemma:busy-period-mean}. Note that duration of cycles, $Z$, are i.i.d. random variables. Thus, the busy-idle profile of the system is an alternating renewal process where renewals correspond to the moments at which the system gets idle. Suppose the system earns reward at a rate of one per unit of time when it is idle (and thus the reward for a cycle equals the idle time of that cycle i.e. $I$). Then, the total reward earned up to time $t$ is equal to the total idle time in $[0,t]$ (or $\\mathcal{I}(t)$), and by the result for renewal reward process (see \\cite{Ross:96}, Theorem 3.6.1), with probability one,\n$\\lim_{t\\to\\infty} \\mathcal{I}(t)\/t=E[I]\/(E[B]+E[I])$. \n\\qed\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Busy Period Distribution}\n\\label{sec:busy-period}\nIn this section, we compute the cumulative distribution function for the number of new arrivals during a busy period for a HTQ with constant service rate, say $p>0$. This could, e.g., correspond to \\eqref{eq:inter-vehicle-distance-dynamics} for $m=1$. However, our analysis in this section, is not restricted to this specific model, but applies to any HTQ with constant service rate $p$. \nThis cumulative distribution for the number of new arrivals during a busy period, while of independent interest, will be used to derive lower bounds on the throughput in the super-linear case in Section~\\ref{subsec:superlinear}. Our analysis is inspired by that of M\/G\/1 queue, e.g., see \\cite{Ross:96}, where our consideration for non-zero initial condition appears to be novel. \n\nLet us consider an arbitrary busy period spanning time interval $(0,t)$, without loss of generality. For non-zero initial condition, one has to distinguish between the first and subsequent busy periods. Let the workload at the beginning of the arbitrary busy period, denoted as $d_0$, be sampled from $\\theta$. The relationship between $\\theta$ and $\\psi$ is as follows. \nIf the system starts with a non-zero initial initial condition with initial workload $w_0>0$, then the value of the $d_0$ for the first busy period will be deterministic and equals $w_0$, and hence $\\theta=\\delta_{w_0}$. However, for subsequent busy periods, or if the initial condition is zero, $d_0$ is sampled from $\\theta=\\psi$.\nThe workload brought to the system by arriving vehicles, $\\{d_i\\}_{i=1}^\\infty$, equals to the distance that vehicles wish to travel and are sampled identically and independently from the distribution $\\psi$. When the system is busy, the workload decreases at a given constant rate $p > 0$. The busy period ends when the workload becomes zero. \n\\begin{remark}\nWe emphasize that $d_0$ denotes the workload at the beginning of a busy period (see Figure \\ref{fig:normalized-distance} for further illustration), and hence is not equal to zero when the queue starts from a zero initial condition. \n\\end{remark}\n\n \n\nIn order to align our calculations with the standard M\/G\/1 framework, where service rate is assumed to be unity, we consider normalized workloads, $\\tilde{d}_i:=d_i\/p$ for all $i \\in \\{0,1,\\cdots\\}$ (see Figure \\ref{fig:normalized-distance} for an illustration). Correspondingly, let the distributions for the normalized distances be denoted as $\\tilde{\\theta}$ and $\\tilde{\\psi}$. Let the arrival time of the $k$-th new vehicle during $(0,t)$ be denoted as $T_k$, and let $N_{bn}$ denote the number of arrivals in $(0,t)$, i.e., the total number of arrivals over the entire duration of the busy period, including the vehicle which initiates the busy period, is $N_{bn} + 1$. \n\n A busy period ends at time $t$, and $N_{bn}=n-1$ if and only if, \n\\begin{enumerate}[(i)]\n\\item $T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, \\qquad k=1,\\cdots,n-1$ \n\\item $\\tilde d_0+\\cdots+ \\tilde d_{n-1}=t$\n\\item There are exactly $n-1$ arrivals in $(0,t)$\n\\end{enumerate} \n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{normalized-distance} \n\\caption{Evolution of workload during first two busy periods for an HTQ with constant service rate $p$, and starting from a non-zero initial condition. In the first busy period, $d_0$ is equal to the workload $w_0$ associated with the non-zero initial condition. In the second busy period, $d_0$ is equal to the workload brought by the first vehicle that initiates that busy period.}\n\\label{fig:normalized-distance}\n\\end{center}\n\\end{figure}\n\n\n\nBy treating densities as if they are probabilities, we get:\n\\begin{align}\n\\label{eq:prop-busy-des}\n\\Pr & (B =t \\text{ and }N_{bn} = n -1)\\nonumber \\\\\n=& \\Pr(\\tilde d_0+\\cdots+ \\tilde d_{n-1}=t, n-1 \\text{ arrivals in $(0,t)$}, T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, k=1,\\cdots,n-1) \\nonumber \\\\\n=& \\int_0^t\\Pr(T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, k=1,\\cdots,n-1| n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_0+\\cdots+ \\tilde d_{n-1}=t, \\tilde d_0 =z) \\nonumber \\\\\n& \\times \\Pr(n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_1+\\cdots+ \\tilde d_{n-1}=t-z)\\, \\tilde \\theta(z) \\, \\mathrm{d} z \n\\end{align}\nwhere we recall that $B$ is the random variable corresponding to the busy period duration. By the independence of normalized distances and the arrival process, the second probability term in the integrand in \\eqref{eq:prop-busy-des} can be expressed as\n\\begin{align}\n\\label{eq:prop-busy-part2}\n\\Pr(n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_1+\\cdots+ \\tilde d_{n-1}=t-z)=e^{-\\lambda t}\\frac{(\\lambda t)^{n-1}}{(n-1)!} \\tilde \\psi_{n-1}(t- z) \n\\end{align}\nwhere $\\tilde\\psi_n$ is the $n$-fold convolution of $\\tilde\\psi$ with itself. \n\nIn the first probability term in \\eqref{eq:prop-busy-des}, it is given that the system receives $n-1$ arrivals in $(0,t)$ and since the arrival process is a Poisson process, the ordered arrival times, $\\{T_1,T_2,\\cdots,T_{n-1}\\}$, are distributed as the ordered values of a set of $n-1$ independent uniform $(0,t)$ random variables $\\{a_1,a_2,\\cdots,a_{n-1}\\}$ (see Theorem 2.3.1 in \\cite{Ross:96}). Thus,\n\\begin{align}\\label{eq:busy-period-given-n-prob}\n\\Pr & (T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, k=1,\\cdots,n-1| n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_0+\\cdots+ \\tilde d_{n-1}=t,\\tilde d_0 =z) \\nonumber \\\\\n& = \\Pr(a_k\\leq \\tilde d_0+\\cdots+\\tilde d_{k-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z)\n\\end{align}\n\n\n\n\n\n\n\n\nBy noting that $t-U$ will also be a uniform $(0,t)$ random variable whenever $U$ is, it follows that $a_1,\\cdots,a_{n-1}$ has the same joint distribution as $t-a_{n-1},\\cdots,t-a_1$. Thus, replacing $a_k$ with $a_{n-k}$ for $k\\in\\{1,\\cdots,n-1\\}$ in \\eqref{eq:busy-period-given-n-prob}, we get\n\\begin{align}\\label{eq:prob-manipulation}\n \\Pr&(a_k \\leq \\tilde d_0+\\cdots+\\tilde d_{k-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) \\nonumber \\\\\n & =\\Pr(t-a_{n-k} \\leq \\tilde d_0+\\cdots+\\tilde d_{k-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_{0} =z) \\nonumber \\\\\n & = \\Pr(t-a_{n-k} \\leq t-(\\tilde d_{k}+\\cdots+\\tilde d_{n-1}), k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) \\nonumber \\\\\n & = \\Pr(a_{n-k} \\geq \\tilde d_{k}+\\cdots+\\tilde d_{n-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) \\nonumber \\\\\n & = \\Pr(a_{n-k} \\geq \\tilde d_{k}+\\cdots+\\tilde d_{n-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) = \\begin{cases} z\/t & z1$ case}\nIn this section, we assume that system starts from an empty initial condition and then we derive a probabilistic lower bound on the throughput of the system. For our analysis we focus on busy period and the number of vehicles served in a busy period. \n\nWe first assume an upper bound on the queue length, i.e. $N(t)\\leq M \\; \\forall t\\geq0$, which by Lemma \\ref{lem:service-rate-bounds} gives a lower bound on the service rate of the system i.e. $s(y(t))\\geq L^mM^{1-m}\\; \\forall t\\geq 0$. Then, we use this lower bound on the service rate to compare the system with a system with the same temporal-spatial arrival process but a slower decrease rate of workload . We use this comparison as a tool to find a lower bound on the probability of achieving the assumed upper bound on the queue length. This result leads to a probabilistic lower bound on the throughput of the system which is stated in Theorem \\ref{}. Our analysis heavily depends on the probability distribution of busy periods which is described in the next section.\n\n\n \n \\subsubsection{From busy period to throughput}\n\n\n \n \\begin{proposition}\\label{prop:nbp-lower-bound}\n For any $m>1$, $w_0=0$, $L>0$, $\\lambda > 0$ $\\varphi \\in \\Psi$ and $M\\in\\{2,3,\\cdots\\}$, given $N(t)0$, then \n $$\\Pr(N_{bp}\\leq M)\\geq C^{(2)}_{L^m M^{1-m}}(M,\\lambda)\\qquad \\forall n\\in\\mathbb{N}$$\n \\end{proposition}\n \\begin{proof}\n \\mmmargin{service is not greater than this lower bound for all t because of idle time. revise it}\n For $m>1$, by Lemma \\ref{lem:service-rate-bounds}, and the upper bound given on the queue length,\n \\begin{equation}\n s(t)\\geq L^mM^{1-m} = p \\quad \\forall t>0\n \\end{equation} \n \n We consider another system with the same realization of temporal-spatial arrival process but fixed service rate of $p$ and name it as the \\emph{slower} system. Let $B_1$ and $B_2$ be random variables that denote the length of a busy period in actual and slower system, respectively. Since the realization of arrival processes are the same for both systems, and the decrease rate of workload in slower system is smaller it is clear that,\n \\begin{equation}\\label{eq:workload-actual-slower}\n w_1(t)\\leq w_2(t) \\quad \\forall t>0\n \\end{equation}\nwhere $w_1(t)$ and $w_2(t)$ denote the the workload processes in actual and slower system, respectively. \n\nNow, let $t_s$ be the time at which a (random) busy period in the slower system ends and $t_1$ be the time at which the next busy period in that system ends (see Figure \\ref{fig:b1-b2} for an illustration). Obviously, $w_1(t_s)=w_1(t_f)=0$. Also, by \\eqref{eq:workload-actual-slower}, $w_2(t_s)=w_2(t_f)=0$. Let $t_1\\in(t_s,t_f)$ be the time of the first arrival after $t=t_s$. Note that this time is the same in both systems. Therefore, in the slower system, the busy period that lies in $[t_s,t_f]$ has the length of $t_f-t_1$. Let $t_2\\in[t_1,t_f]$ be the time at which the actual system gets idle for the first time after $t=t_1$. We want to show that $t_2\\leq t_f$. By contradiction, assume $t_2> t_f$. Thus, $w_1(t_f)>0$, but $w_2(t_f)=0$. This contradicts \\eqref{eq:workload-actual-slower} and proves that $t_2\\leq t_f$. This implies that the longest busy period in actual system contained in $[t_s,t_f]$ is no longer than the busy period in actual system. This argument holds for any busy period in slower system. Thus, with probability one, \n \\begin{equation}\\label{eq:relation-b1-b2}\n B_1\\leq B_2\n \\end{equation} \n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=9cm]{b1-b2} \n\\caption{Workload process for two systems with the same realization temporal-spatial arrival process, a system with slower service rate (solid line) and a system with a faster service rate (dashed line).}\n\\label{fig:b1-b2}\n\\end{center}\n\\end{figure}\n\nNow, let $N_{bp}$ and $\\tilde N_{bp}$ be the number of customers served in a busy period in the actual system and slower system, respectively. Note that $N_{bp}$ and $\\tilde{N}_{bp}$ are Poisson random variables with mean $\\lambda B_1$ and $\\lambda B_2$, respectively and by \\eqref{eq:relation-b1-b2}, $\\lambda B_1\\leq \\lambda B_2$. Poisson random variable is stochastically increasing in its mean (see e.g. \\cite{Ross:96}), therefore\n\\begin{equation}\n\\tilde{N}_{bp}\\geq_{\\text{st}}{N}_{bp}\n\\end{equation} \nwhich establishes this proposition.\n\n\n\n \\end{proof}\n \n\\mmmargin{have we defined this notation ($n_0$) earlier?}\n \\begin{proposition}\\label{prop:nbn-lower-bound}\n For any $m>1$, $w_0>0$, \\mmcomment{$n_0\\in\\mathbb{N}$}, $L>0$, $\\lambda > 0$, $\\varphi \\in \\Psi$ and $M\\in\\{2,3,\\cdots\\}$, given $N(t)0$, then \n $$\\Pr(N_{bn}\\leq M)\\geq C^{(1)}_{L^m (M+n_0)^{1-m}}(M,\\lambda,w_0)\\qquad \\forall n\\in\\mathbb{N}$$\n \\end{proposition}\n \\begin{proof}\n The proof is very similar to the proof of Proposition \\ref{prop:nbp-lower-bound}; however, in this case the upper bound on the queue length is $M+n_0$, and the lower bound on the service rate is $L^m(M+n_0)^{1-m}$. \n \\end{proof}\n\n\n\n \n\n\n\n \n \\begin{lemma} \\label{lemma:st-order-queulength-nbp}\n For any $m>1$, $L>0$, $\\lambda>0$, $\\varphi \\in \\Psi$, $w_0=0$\n $$N_{bp}\\geq_{\\text{st}}N(t)\\quad \\forall t\\geq 0$$\n \\end{lemma}\n \\begin{proof}\n Conditioned on the length of busy period, $B$, \n \\begin{equation}\n N_{bp}\\leq M | B = b \\implies N(t)\\leq M | B = b \\quad \\forall t\\in[0,b]\n \\end{equation}\n where here $t=0$ denotes the time at which busy period starts. Then,\n \\begin{equation}\n \\Pr(N_{bp}\\leq M | B = b) \\leq \\Pr(N(t)\\leq M | B = b) \\quad \\forall t\\in[0,b]\n \\end{equation}\n and by taking the expectation of the above with respect to length o busy period, we get \n \\begin{equation}\n \\Pr(N_{bp}\\leq M) \\leq \\Pr(N(t)\\leq M) \\quad \\forall t>0\n \\end{equation}\n which establishes this lemma. \n \\end{proof} \n\n\n Now, we are ready to state the main result in the following theorem which gives a probabilistic lower bound on the throughput of the system for $m>1$.\n \\begin{theorem}\n Given $\\delta\\in(0,1)$, $L>0$, $m>1$, $\\varphi\\in\\Psi$, $w_0=0$, then\n $$\\lambda_{\\max}(m, L, \\varphi,\\delta)\\geq \\max_{M\\geq 2}\\;\\max\\{\\lambda>0 \\;| C^{(2)}_{L^m M^{1-m}}(M,\\lambda)\\geq1-\\delta\\}$$\n \\end{theorem}\n \\begin{proof}\nFor a given upper bound on the queue length, $M\\in\\{2,3\\cdots\\}$, \n \\begin{equation}\\label{eq:prob-bound-queue}\n \\Pr(N\\leq M) \\geq \\Pr(N_{bp}\\leq M) \\geq C^{(2)}_{L^m M^{1-m}}(M,\\lambda)\n \\end{equation}\n where the first inequality follows by Lemma \\ref{lemma:st-order-queulength-nbp} and the second inequality follows by Proposition \\ref{prop:nbp-lower-bound}. Therefore, based on Definition \\ref{def:throughput}, $\\tilde \\lambda = \\max\\{\\lambda>0 \\;| C_{L^m M^{1-m}}(M,\\lambda)=1-\\delta\\}$ gives a lower bound on $\\lambda_{\\max}$. In Definition \\ref{def:throughput}, however, only boundedness of queue length matters. Thus, one can maximize $\\tilde \\lambda$ over $M\\geq 2$ to get tighter lower bound. \n \n \n\n\n\n\n\n \\end{proof}\n \n \n \\begin{theorem}\n Given $\\delta_1,\\delta_2\\in(0,1)$, $L>0$, $m>1$, $\\varphi\\in\\Psi$, $w_0>0$, $n_0\\in\\mathbb{N}$ then\n $$\\lambda_{\\max}(m, L, \\varphi,\\delta)\\geq \\max_{M_1,M_2\\geq 2}\\;\\max\\{\\lambda>0 \\;| C^{(1)}_{L^m (M+n_0)^{1-m}}(M,\\lambda,w_0)C^{(2)}_{L^m (M_2)^{1-m}}(M,\\lambda)\\geq(1-\\delta_1)(1-\\delta_2)\\geq (1-\\delta)\\}$$\n \\end{theorem}\n \\begin{proof}\n\\mmcomment{work in progress ...} \n \n \\end{proof}\n \n In order to further clarify the analysis in this section, the following example shows the application of these results in order to probabilistically characterize the throughput of the system for a specific spatial distribution. \n \n \\begin{example}[Deterministic traveled distances] Assume that vehicles, upon arrival, wish to travel a deterministic distance of length $L>0$ i.e. $\\varphi(x)=\\delta_{\\{x=L\\}}$ and their motion is determined by a car following model with $m>1$. Also, further assume that the queue length does not exceed $M\\in\\{2,3,\\cdots\\}$. In this case, the lower bound on the service rate is $p=L^mM^{1-m}$. Application of \\eqref{eq:busy-period-and-nbp-joint} gives,\n \\begin{equation}\\label{eq:busy-period-joint-deterministic}\n D_p(t,N_{bp}=n) = \\int_0^te^{-\\lambda y}\\frac{(\\lambda y)^{n-1}}{n!}\\delta_{\\{y=nL\/p\\}}\\mathrm{d} y=\\begin{cases}0 & t\\leq nL\/p\\\\ e^{-\\lambda nL\/p}\\frac{(\\lambda nL\/p)^{n-1}}{n!}& t> nL\/p\\end{cases}\n \\end{equation}\nand by Lemma \\ref{lemma:st-order-queulength-nbp} and Proposition \\ref{prop:nbp-lower-bound},\n\\begin{equation}\n \\Pr(N\\leq M)\\geq \\Pr(N_{bp}\\leq M)\\geq C_{L^m M^{1-m}}(M,\\lambda) = \\sum_{i=1}^M e^{-\\lambda nL\/p}\\frac{(\\lambda nL\/p)^{n-1}}{n!} \\geq1-\\delta.\n\\end{equation}\nThe above inequality can be used in order to find the maximum throughput for which queue length remains bounded by $M$ with probability of at least $1-\\delta$.\n \\end{example}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper, we formulated and analyzed a novel horizontal traffic queue. A key characteristic of this queue is the state dependence of its service rate. We establish useful properties of the service rate dynamics. We also extend calculations for M\/G\/1 busy period distributions to our setting, even for non-empty initial condition. These results allow us to provide tight results for throughput in the linear case, and probabilistic bounds on queue length over finite time horizon in the super-linear case. We also study throughput under a batch release control policy, where the additional waiting induced by the control policy is interpreted as a perturbation to the arrival process. We provide lower bound on the throughput for a maximum permissible perturbation. In particular, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded as $m \\to 0^+$. Simulation results suggest a sharp phase transition in the throughput as the car-following behavior transitions from super-linear to sub-linear regime. \n\nIn future, we plan to sharpen our analysis to theoretically demonstrate the phase transition behavior. \nThis could include, for example, considering other release control policies with better perturbation properties. The increasing nature of the throughput in the super-linear regime for large values of $L$, as illustrated in Figure~\\ref{fig:throughput-simulations}, is possibly because the car-following model considered in this paper does not impose any explicit upper bounds on the speed of the vehicles. We plan to extend our analysis to such practical constraints, as well as to higher order, e.g., second order, car-following models, and models emerging from consideration of inter-vehicle distance beyond the vehicle immediately in front. The connections with processor sharing queue, as highlighted in this paper, suggest the possibility of utilizing the construct of measure-valued state descriptors~\\cite{Grishechkin:94,Gromoll.Puha.ea:02} to derive fluid and diffusion limits of the proposed horizontal traffic queues. In particular, one could interpret the measure-valued state descriptor to play the role of traffic density in the context of traffic flow theory. Along this direction, we plan to investigate connections between the fluid limit of horizontal traffic queues, and PDE models for traffic flow. \n\\section{Introduction}\n\\label{sec:introduction}\nWe consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart the queue after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the speed of a vehicle is proportional to a power $m>0$ of the distance to the vehicle in front. For a given initial condition, we define the throughput of such a queue as the largest arrival rate under which the queue length remains bounded. We provide rigorous analysis for the service rate, busy period distribution, and throughput of the proposed HTQ. \n\nOur motivation for studying HTQ comes from advancements in connected and autonomous vehicle technologies that allow to program individual vehicles with rules that can optimize system level performance. Within this application context, one can interpret the results of this paper as rigorously characterizing the impact of a parametric class of car-following behavior on system throughput. \n\n\nIn the linear case ($m=1$), i.e., when the speed of every vehicle is proportional to the distance to the vehicle directly in front, the periodicity of the road segment implies that the sum of the speeds of the vehicles is proportional to the total length of the road segment, i.e., it is constant. This feature allows us to exploit the equivalence between workload and queue length to show that, independent of the initial condition and almost surely, the throughput is the inverse of the time required by a solitary vehicle to travel average distance. \n\nIn the non-linear case ($m \\neq 1$), the cumulative service rate of HTQ queue is constant if and only if all the inter-vehicle distances are equal. For all other inter-vehicle configurations, we show that the service rate is strictly decreasing (resp., strictly increasing) in the super-linear, i.e., $m>1$ (resp., sub-linear, i.e., $m<1$) case. The service rate exhibits an another contrasting behavior in the sub- and super-linear regimes. In the super-linear case, the service rate is maximum (resp., minimum) when all the vehicles are co-located (resp., when the inter-vehicle distances are equal), and vice-versa for the sub-linear case. Using a combination of these properties, we prove that, when the length of the road segment is at most one, the throughput in the super-linear (resp., sub-linear) case is upper (resp., lower) bounded by the throughput for the linear case.\n\nWe prove the remaining bounds on the throughput for the non-linear case as follows. The standard calculations for joint distributions of duration and number of arrivals during a busy period for M\/G\/1 queue are extended to the HTQ setting, including for non-empty initial conditions. \nThese joint distributions are used to derive probabilistic upper bounds on queue length over finite time horizons for HTQ for the $m>1$ case. Such bounds are optimized to get lower bounds on throughput defined over finite time horizons. Simulation results show good comparison between such lower bounds and numerical estimates.\n \nWe also analyze throughput in the sub-linear and super-linear cases under perturbation to the arrival process, which is attributed to the additional expected waiting time induced by a release control policy that adds appropriate delay to the arrival times to ensure a desired minimum inter-vehicle distance $\\triangle>0$ at the time of a vehicle joining the HTQ. Since the minimum inter-vehicle distance is non-decreasing in between arrivals and jumps, this implies an upper bound on the queue length which is inversely proportional to $\\triangle$. We derive a lower bound on throughput for a given \ncombination of maximum allowable perturbation. In particular, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded, as $m \\to 0^+$. \n\n\n \nQueueing models have been used to model and analyze traffic systems. The focus here has been primarily on vertical queues, under which vehicles travel at maximum speed until they hit a congestion spot where all vehicles queue on top of each other.\nThe queue length and waiting time of a minor traffic stream at an unsignalized intersection where major traffic stream has high priority is studied in \\cite{Tanner:62} and \\cite{Heidemann:91}. \nIn \\cite{Heidemann:94}, a vertical single server queue is utilized to model the queue length distribution at signalized intersections.\nIn \\cite{Jain.Smith:97}, a state-dependent queuing system is used to model vehicular traffic flow where the service rate depends on the number of vehicles on each road link. \n\n\n On the other hand, the \\emph{horizontal traffic queue} terminology has been primarily used to study macroscopic traffic flow, e.g., see \\cite{Helbing:03}. While such models capture the macroscopic relationship between traffic flow and density, a rigorous description and analysis of an underlying queue model is lacking. Indeed, to the best of our knowledge, there is no prior work on the analysis of a traffic queue model that explicitly incorporates car-following behavior. \n\nThe proposed HTQ has an interesting connection with processor sharing (PS) queues, and this connection does not seem to have been documented before.\nA characteristic feature of PS queues is that all the outstanding jobs receive service simultaneously, while keeping the total service rate of the server constant. The simplest model is where the service rate for an individual job is equal to $1\/N$, where $N$ is the number of outstanding jobs. In our proposed system, one can interpret the road segment as a server simultaneously providing service to all the vehicles, with the service rate of an individual vehicle equal to its speed. This natural analogy between HTQ and PS queues, to the best of our knowledge, was reported for the first time in our recent work~\\cite{Motie.Savla.CDC15}.\nThe $1\/N$ rule applied to our setting implies that all the vehicles travel with the same speed. Clearly, such a rule, or even the general discriminatory PS disciplines, e.g., see \\cite{Kleinrock:67}, are not applicable to the car following models considered in this paper. Indeed, the proposed HTQ is best described as a state-dependent PS queue. \n\n\nIn the PS queue literature, the focus has been on the sojourn time and queue length distribution. For example, see \\cite{Ott:84} and \\cite{Yashkov:83} for M\/G\/1-PS queue and \n\\cite{Grishechkin:94} for G\/G\/1-PS queue. Fluid limit analysis for PS queue is provided in \\cite{Chen.ea:97} and \\cite{Gromoll.Puha.ea:02}. \nHowever, relatively less attention has been paid to the throughput analysis of state-dependent PS queues.\nIn \\cite{Moyal:08,Kherani.Kumar:02,Chen.Jordan:07}, throughput analysis for state-dependent PS queues is provided, where throughput is defined as the quantity of work achieved by the server per unit of time.\nStability analysis for a single server queue with workload-dependent service and arrival rate is provided in \\cite{Bambos.Walrand:89} and \\cite{Bekker:05}. However, the dependence of service rate on the system state in the HTQ proposed in the current paper is complex, and hence none of these results are readily applicable. \n\nIn summary, there are several novel contributions of the paper. First, we propose a novel horizontal traffic queue and place it in the context of processor-sharing queues and state-dependent queues. We establish monotonicity properties of service rates in between jumps (i.e., arrivals and departures), and derive bounds on change in service rates at jumps. \nSecond, we adapt busy period calculations for M\/G\/1 queue to our current setup, including for non-empty initial conditions. These results allow us to provide tight results for throughput in the linear case, and probabilistic bounds on queue length over finite time horizon in the super-linear case. We also study throughput under a batch release control policy, whose effect is interpreted as a perturbation to the arrival process. We provide lower bound on the throughput for a maximum permissible perturbation for sub- and super-linear cases. In particular, we show that, for sufficiently large perturbation, this lower bound grows unbounded as $m \\to 0^+$. It is interesting to compare our analytical results with simulation results, which suggest a sharp transition in the throughput from being unbounded in the sub-linear regime to being bounded in the super-linear regime. While our analytical results do not exhibit such a phase transition yet, their novelty is in providing rigorous estimates of any kind on the throughput of horizontal traffic queues under nonlinear car following models.\n\nThe rest of the paper is organized as follows. We conclude this section with key notations to be used throughout the paper. The setting for the proposed horizontal traffic queue and formal definition of throughput are provided in Section~\\ref{sec:problem-formulation}. Section~\\ref{sec:service-rate-monotonicity} contains useful properties on the dynamics in service rate in between and during jumps. Key busy period properties for the M\/G\/1 queue are extended to the HTQ case in Section~\\ref{sec:busy-period}. Throughput analysis is reported in Section~\\ref{sec:throughput-analysis}. Simulations are presented in \\ref{sec:simulations}. Concluding remarks and directions for future work are presented in Section~\\ref{sec:conclusions}. A few technical intermediate results are collected in the appendix. \n\n\\subsection*{Notations}\nLet $\\mathbb{R}$, $\\mathbb{R}_+$, and $\\mathbb{R}_{++}$ denote the set of real, non-negative real, and positive real numbers, respectively. \nLet $\\mathbb{N}$ be the set of natural numbers. \nIf $x_1$ and $x_2$ are of the same size, then $x_1 \\geq x_2$ implies element-wise inequality between $x_1$ and $x_2$. If $x_1$ and $x_2$ are of different sizes, then $x_1 \\geq x_2$ implies inequality only between elements which are common to $x_1$ and $x_2$ -- such a common set of elements will be specified explicitly. \nFor a set $\\mathcal{J}$, let $\\text{int}(\\mathcal{J})$ and $|\\mathcal{J}|$ denote the interior and cardinality of $\\mathcal{J}$, respectively.\nGiven $a \\in\\mathbb{R}$, and $b > 0$, we let $\\mod(a,b):=a-\\lfloor \\frac{a}{b}\\rfloor b$. Let $\\mathcal S_N^L$ be the $N-1$-simplex over $L$, i.e., $\\mathcal S_N^L=\\setdef{x \\in \\mathbb{R}_+^N}{\\sum_{i=1}^N x_i = L}$. When $L=1$, we shall use the shorthand notation $\\mathcal S_N$.\nWhen referring to the set $\\until{N}$, for brevity, we let the indices $i=-1$ and $i=N+1$ correspond to $i=N$ and $i=1$ respectively.\n Also, for $p, q \\in \\mathcal S_N$, we let\n$D(p || q)$ denote the K-L divergence of $q$ from $p$, i.e., $D(p || q):= \\sum_{i=1}^N p_i \\log\\left(p_i\/q_i\\right)$.\nWe also define a permutation matrix, $P^- \\in \\{0,1\\}^{N \\times N}$, as follows:\n\\begin{align*}\nP^-:=\\begin{bmatrix}\n\\mathbf{0}_{N-1}^T & 1 \\\\\nI_{N-1} & \\mathbf{0}_{N-1}\n\\end{bmatrix}\n\\end{align*}\nwhere $\\mathbf{0}_N$ and $\\mathbf{1}_N$ stand for vectors of size $N$, all of whose entries are zero and one, respectively. We shall drop $N$ from $\\mathbf{0}_N$ and $\\mathbf{1}_N$ whenever it is clear from the context.\n\n\n\n\n\\subsection{Linear Case: $m=1$}\n\\label{sec:linear}\nIn this section, we provide an exact characterization of throughput for the linear case, i.e., when $m=1$. Recall that, for $m=1$, the service rate $s(y)=\\sum_{i=1}^N y_i \\equiv L$ is constant.\n\n\n\n\\begin{proposition}\\label{prop:unstable}\nFor any $L > 0$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$ and :\n$$\n\\lambda_{\\text{max}}(L,m=1,\\varphi,\\psi,x_0,\\delta=0) \\leq L\/\\bar{\\psi} \\, .\n$$ \n\\end{proposition}\n\\begin{proof}\nBy contradiction, assume $\\lambda_{\\text{max}}>L\/\\bar{\\psi}$. Let $r(t):=\\sum_{i=1}^{A(t)} d_i$ be the workload added to the system by the $A(t)$ vehicles that arrive over $[0,t]$. Therefore, \n\\begin{equation}\\label{eq:workload-linear}\nw(t)=w_0+r(t)-L(t-\\mathcal{I}(t))\n\\end{equation}\nwhere $w_0$ is the initial workload. The process $\\{r(t), \\, t\\geq 0\\}$ is a renewal reward process, where the renewals correspond to arrivals of vehicles and the rewards correspond to the distances $\\{d_i\\}_{i=1}^{\\infty}$ that vehicles wish to travel in the system upon arrival before their departures. \nInter-arrival times are exponential random variables with mean $1\/\\lambda$, and the reward associated with each renewal is independently and identically sampled from $\\psi$, whose mean is $\\bar{\\psi}$. \nTherefore, e.g., \\cite[Theorem 3.6.1]{Ross:96} implies that, with probability one, \n\\begin{equation}\n\\label{eq:slln-renewal}\n\\lim_{t\\to\\infty}\\frac{r(t)}{t}=\\lambda \\bar{\\psi}\n\\end{equation}\nThus, for all $\\varepsilon \\in \\left(0,\\lambda \\bar{\\psi} -L\\right)$, there exists a $t_0\\geq0$ such that, with probability one,\n\\begin{equation}\\label{eq:s-t-lowerbound}\n\\frac{r(t)}{t}\\geq\\lambda \\bar \\psi-\\varepsilon\/2> L + \\varepsilon\/2 \\qquad \\forall \\, t \\geq t_0.\n\\end{equation}\nSince $w_0$ and $\\mathcal{I}(t)$ are both non-negative, \\eqref{eq:workload-linear} implies that $w(t)\\geq r(t) - Lt$ for all $t \\geq 0$. \nThis combined with \\eqref{eq:s-t-lowerbound} implies that, with probability one, $w(t) \\geq \\varepsilon t \/2$ for all $t \\geq t_0$, and hence \n$\\lim_{t\\to\\infty}w(t)= + \\infty$. This combined with \\eqref{eq:workload-upperbound} implies that, with probability one, $\\lim_{t\\to\\infty}N(t)=+\\infty$. \n\\qed\n\\end{proof}\n\n\\begin{theorem}\\label{thm:stable}\nFor any $L > 0$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$:\n$$\n\\lambda_{\\text{max}}(L,m=1,\\varphi,\\psi,x_0,\\delta=1) = L\/\\bar{\\psi} \\, .\n$$ \n\\end{theorem}\n\\begin{proof}\nAssume that for some $\\lambda < L\/\\bar{\\psi}$, there exists some initial condition $(x_0,n_0)$ such that the queue length grows unbounded with some positive probability. \nSince the workload brought by every vehicle is i.i.d., and the inter-arrival times are exponential, without loss of generality, we can assume that the queue length never becomes zero. That is, the idle time satisfies $\\mathcal{I}(t) \\equiv 0$. Moreover, \\eqref{eq:slln-renewal} implies that, for every $\\varepsilon \\in \\left(0, L- \\lambda \\bar{\\psi} \\right)$, there exists $t_0 \\geq 0$ such that, with probability one,\n\\begin{equation}\n\\label{eq:s-t-upperbound}\n\\frac{r(t)}{t} \\leq \\lambda \\bar{\\psi} + \\varepsilon\/2 < L - \\varepsilon\/2 \\qquad \\forall t \\geq t_0\n\\end{equation} \n\nCombining \\eqref{eq:workload-linear} with \\eqref{eq:s-t-upperbound}, and substituting $\\mathcal{I}(t) \\equiv 0$, we get $w(t) < w_0 - \\varepsilon t \/2$, which implies that workload, and hence queue length, goes to zero in finite time after $t_0$, leading to a contradiction. \nCombining this with the upper bound proven in Proposition~\\ref{prop:unstable} gives the result. \n\\qed\n\\end{proof}\n\n\\begin{remark}\nTheorem~\\ref{thm:stable} implies that the throughput in the linear case is equal to the inverse of the \ntime required to travel average total distance by a solitary vehicle in the system. In the linear case, the throughput can be characterized with probability one, independent of the initial condition of the queue.\n\\end{remark}\n\n\n\n\n\n\\subsection{Monotonicity of Throughput in $m$ and $x_0$}\n\\label{sec:non-linear}\n\n\nIn this section, we show the following monotonicity property of $\\lambda_{\\text{max}}$ with respect to $m$ for small values of $L$: for given $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, $L \\in (0,1)$, $\\varphi \\in \\Phi$, and $\\psi \\in \\Psi$, throughput is a monotonically decreasing function of $m$. \nFor this section, we rewrite \\eqref{eq:dynamics-moving-coordinates} in $\\mathbb{R}_+^N$, i.e., without projecting onto $[0,L]^N$. Specifically, let the vehicle coordinates be given by the solution of \n\\begin{equation}\n\\label{eq:x-dynamics-in-Rn}\n\\dot{x}_i = y_i^m, \\qquad x_i(0)=x_{0,i}, \\qquad i \\in \\until{N} \n\\end{equation} \nLet $X(t;x_0,m)$ denote the solution to \\eqref{eq:x-dynamics-in-Rn} at $t$ starting from $x_0$ at $t=0$. We will compare $X(t;x_0,m)$ under different values of $m$ and initial conditions $x_0$, over an interval of the kind $[0,\\tau)$, in between arrivals and departures. \nWe recall the notation that, if $x_{0}^1$ and $x_{0}^2$ are vectors of different sizes, then $x_{0}^1 \\leq x_{0}^2$ implies element-wise inequality only for components which are common to $x_{0}^1$ and $x_{0}^2$. In Lemma~\\ref{lemma:two-size-compare} and Proposition~\\ref{prop:queue-length-2m}, this common set of components corresponds to the set of vehicles common between $x_{0}^1$ and $x_{0}^2$.\n \n\n\\begin{lemma}\n\\label{lemma:two-size-compare}\nFor any $L \\in (0,1]$, $x^1_{0} \\in \\mathbb{R}_+^{n_1}$, $x^2_{0} \\in \\mathbb{R}_+^{n_2}$, $n_1, n_2 \\in \\mathbb{N}$,\n$$\nx^1_{0} \\leq x^2_{0}, \\, \\, n_2 \\leq n_1, \\, \\, 0 < m_2 \\leq m_1 \\implies X(t;x^1_{0},m_1) \\leq X(t;x^2_{0},m_2) \\qquad \\forall \\, t \\in [0,\\tau)\n$$ \n\\end{lemma}\n \\begin{proof}\nThe proof is straightforward when $n_1=n_2$. This is because, in this case, since $y_i \\leq L\\leq1$, $m_2 \\leq m_1$ implies $y_i^{m_2} \\geq y_i^{m_1}$ for all $i \\in \\until{n_1}$. Using this with Lemmas~\\ref{lem:monotonicity-same-size} and \\ref{lem:car-following-type-K} gives the result. \n \nIn order to prove the result for $n_2 < n_1$, we show that $X(t; x^1_{0}, m_1) \\leq X(t; x^2_{0}, m_1) \\leq X(t; x^2_{0}, m_2)$. Note that the second inequality follows from the previous case. Therefore, it remains to prove the first inequality. Let $(i_1, \\ldots, i_{n_2})$ be the set of indices of $n_2$ vehicles such that $0 \\leq x^2_{0,i_1} \\leq \\ldots \\leq x^2_{0,i_{n_2}} \\leq L$. Similarly, let $(i_1, i_1+1, \\ldots, i_2, i_2+1, \\ldots)$ be the indices of $n_1$ vehicles in the order of increasing coordinates in $x_0^1$. Our assumption on the initial condition implies that $x^1_{0,i_k} \\leq x^2_{0,i_k}$ for all $k \\in \\until{n_2}$. For brevity, let $x^1(t) \\equiv X(t;x_0^1,m_1)$, and $x^2(t) \\equiv X(t;x_0^2,m_1)$. It is easy to check that, for all $t \\in [0,\\tau)$, and all $k \\in \\until{n_2}$, \n\\begin{equation}\n\\label{eq:xdot-ub}\n\\dot{x}^1_{i_k} = \\left(x^1_{i_{k}+1} - x^1_{i_k}\\right)^{m_1} \\leq \\left(x^1_{i_{k+1}} - x^1_{i_k}\\right)^{m_1}\n\\end{equation}\nLet $t \\in [0,\\tau)$ be the first time instant when $x^1_{i_k}(t)=x^2_{i_k}(t)$ for some $k \\in \\until{n_2}$. Then, recalling $x^1_{i_{k+1}}(t) \\leq x^2_{i_{k+1}}(t)$, \\eqref{eq:xdot-ub} implies that $\\dot{x}_{i_k}^1(t) \\leq \\left(x^2_{i_{k+1}} - x^2_{i_k} \\right)^{m_1} = \\dot{x}^2_{i_k}(t)$. The result then follows from Lemma~\\ref{lem:monotonicity-same-size}. \n\\qed\n \\end{proof}\n \n \n\nLemma~\\ref{lemma:two-size-compare} is used to establish monotonicity of throughput as follows.\n\\begin{proposition}\n\\label{prop:queue-length-2m}\nFor any $L \\in (0,1]$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $\\delta \\in (0,1)$, $x^1_{0} \\in [0,L]^{n_1}$, $x^2_{0} \\in [0,L]^{n_2}$, $n_1, n_2 \\in \\mathbb{N}$:\n$$\nx^1_{0} \\leq x^2_{0}, \\, \\, n_2 \\leq n_1, \\, \\, 0 < m_2 \\leq m_1 \\implies \\lambda_{\\text{max}}(L,m_1,\\varphi,\\psi,x^1_0,\\delta) \\leq \\lambda_{\\text{max}}(L,m_2,\\varphi,\\psi,x^2_0,\\delta)\n$$\n\\end{proposition}\n\\begin{proof}\nFor brevity in notation, we refer to the queue corresponding to $m_1$, and initial condition $x_0^1$ as HTQ-S. We refer to the other queue as HTQ-F. Let $\\lambda$, $\\varphi$ and $\\psi$ common to HTQ-S and HTQ-F be given. Let $x^1(t) \\equiv X(t; x^1_0, m_1)$ and $x^2(t) \\equiv X(t; x^2_0, m_2)$, and let $N_s(t)$ and $N_f(t)$ be the queue lengths in the two queues at time $t$. It suffices to show that $N_s(t) \\geq N_f(t)$ for a given realization of arrival times, arrival locations, and travel distances. In particular, this also implies that the departure locations are also the same for every vehicle, including the vehicles present at $t=0$, in both the queues. \n\nIndeed, it is sufficient to show that $x^1(\\tau) \\leq x^2(\\tau)$ and $N_s(\\tau)\\geq N_f(\\tau)$ where $\\tau$ is the time of first arrival or departure from either HTQ-S or HTQ-F. Accordingly, we consider two cases, corresponding to whether $\\tau$ corresponds to arrival or departure. \n\nSince $x^1(t) \\leq x^2(t)$ for all $t \\in [0,\\tau)$ from Lemma~\\ref{lemma:two-size-compare}, and the departure locations of all the vehicles in HTQ-S and HTQ-F are identical, the first departure from HTQ-S can not happen before the first departure in HTQ-F. Therefore, $N_s(\\tau) \\geq N_f(\\tau)$. Since $x^1(\\tau^-) \\leq x^2(\\tau^-)$, and $x^2(\\tau)$ is a subset of $x^2(\\tau^-)$, we also have $x^1(\\tau) \\leq x^2(\\tau)$. \n\nWhen $\\tau$ corresponds to the time of the first arrival, since the arrivals happen at the same location in HTQ-S and HTQ-F, and since $x^1(\\tau^-) \\leq x^2(\\tau^-)$, rearrangement of the indices of the vehicles to include the new arrival at $t=\\tau$ implies that $x^1(\\tau) \\leq x^2(\\tau)$. Moreover, since $N_s(\\tau^-) \\geq N_f(\\tau^-)$, and the arrivals happen simultaneously in both HTQ-S and HTQ-F, we have $N_s(\\tau) \\leq N_f(\\tau)$. \n\\qed\n\\end{proof}\n\n\\begin{remark}\nProposition \\ref{prop:queue-length-2m} establishes monotonicity of throughput only for $L\\in(0,1]$. This is consistent with our simulation studies, e.g., as reported in Figure \\ref{fig:throughput-simulations}, according to which, the throughput is non-monotonic for large $L$. \n\\end{remark}\nFor the analysis of the linear car following model, we exploited the fact that the total service rate of the system is constant. However, for the nonlinear model, i.e., $m \\neq 1$, the total service rate depends on the number and relative locations of vehicles. The state dependent service rate of nonlinear models makes the throughput analysis much more complex. In the next section, we find probabilistic bound on the throughput in the super-linear case.\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Horizontal Traffic Queue (HTQ) Setup}\n\\label{sec:problem-formulation}\nConsider a periodic road segment of length $L$; without loss of generality, we assume it be a circle. Starting from an arbitrary point on the circle, we assign coordinates in $[0,L]$ to the circle in the clock-wise direction (See Figure \\ref{fig:htq}). Vehicles arrive on the circle according to a spatio-temporal process: the arrival process $\\{A(t),t\\geq0\\}$, is assumed to be a Poisson process with rate $\\lambda>0$, and the arrival locations are sampled independently and identically from a spatial distribution $\\varphi$ and mean value $\\bar \\varphi$. Without loss of generality, let the support of $\\varphi$ be $\\text{supp}(\\varphi)=[0,\\ell]$ for some $\\ell\\in[0,L]$. \nUpon arriving, vehicle $i$ travels distance $d_i$ in a counter-clockwise direction, after which it departs the system. \nThe travel distances $\\{d_i\\}_{i=1}^\\infty$ are sampled independently and identically from a spatial distribution $\\psi$ with support $[0,R]$ and mean value $\\bar \\psi$.\nLet the set of $\\varphi$ and $\\psi$ satisfying the above conditions be denoted by $\\Phi$ and $\\Psi$ respectively.\nThe stochastic processes for arrival times, arrival locations, and travel distances are all assumed to be independent of each other.\n\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=5cm]{htq-colorful} \n \\caption{Illustration of the proposed HTQ with three vehicles.}\n \\label{fig:htq}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Dynamics of vehicle coordinates between jumps}\nLet the time epochs corresponding to arrival and departure of vehicles be denoted as $\\{\\tau_1, \\tau_2, \\ldots\\}$. We shall refer to these events succinctly as \\emph{jumps}. We now formally state the dynamics under this car-following model. \nWe describe the dynamics over an arbitrary time interval of the kind $[\\tau_j,\\tau_{j+1})$. Let $N \\in \\mathbb{N}$ be the fixed number of vehicles in the system during this time interval. \nDefine the inter-vehicle distances associated with vehicle coordinates $x \\in [0,L]^N$ as follows:\n\\begin{equation}\n\\label{eq:inter-vehicle-distance-Rn}\ny_i(x) = \\mod \\left(x_{i+1} - x_i, L\\right), \\quad i \\in \\until{N} \n\\end{equation}\nwhere we implicitly let $x_{N+1} \\equiv x_1$ (See Figure \\ref{fig:htq} for an illustration). Note that the normalized inter-vehicle distances $y\/L$ are probability vectors. \nWhen inside the queue, the speed of every vehicle is proportional to a power $m >0$ of the distance to the vehicle directly in front of it. \nWe assume that this power $m > 0$ is the same for every vehicle at all times. Then, starting with $x(\\tau_{j}) \\in [0,L]^N$, the vehicle coordinates over $[\\tau_{j},\\tau_{j+1})$ are given by:\n\\begin{equation}\n\\label{eq:dynamics-moving-coordinates}\nx_i(t) = \\mod \\left( x_i(\\tau_{j}) + \\int_{\\tau_{j}}^t y_i^m(x(z)) \\, dz, L \\right), \\qquad \\forall \\, i \\in \\until{N}, \\quad \\forall \\, t \\in [\\tau_{j},\\tau_{j+1}) \\, ,\n\\end{equation}\n\n\\begin{remark}\nIt is easy to see that the clock-wise ordering of the vehicles is invariant under \\eqref{eq:inter-vehicle-distance-Rn}-\\eqref{eq:dynamics-moving-coordinates}. \n\\end{remark}\n\nThe dynamics in inter-vehicle distances is given by:\n\\begin{equation}\n\\label{eq:inter-vehicle-distance-dynamics}\n\\dot{y}_i = y^m_{i+1} - y_i^m, \\qquad i \\in \\until{N} \n\\end{equation}\nwhere we implicitly let $y_{N+1} \\equiv y_1$.\n\n\\subsection{Change in vehicle coordinates during jumps}\nLet $x(\\tau^-_{j}) =\\left(x_1(\\tau^-_{j}), \\ldots, x_N(\\tau^-_{j})\\right) \\in [0,L]^N$ be the vehicle coordinates just before the jump at $\\tau_{j}$. If the jump corresponds to the departure of vehicle $k \\in \\until{N}$, then the coordinates of the vehicles $x(\\tau_{j}) =\\left(x_1(\\tau_{j}), \\ldots, x_{N-1}(\\tau_{j})\\right) \\in [0,L]^{N-1}$ after re-ordering due to the jump, for $i \\in \\until{N-1}$, are given by:\n \\begin{equation*}\n\t\tx_i(\\tau_{j})=\\left\\{\\begin{array}{ll}\n\t\t\\displaystyle x_i(\\tau^-_{j}) \t& i\\in \\until{k-1} \\\\[15pt]\n\t\t\\displaystyle x_{i+1}(\\tau^-_{j})\t& i\\in \\{k+1, \\ldots, N-1\\}\\,.\\end{array}\\right.\n\t\\end{equation*}\n\n Analogously, if the jump corresponds to arrival of a vehicle at location $z \\in [0,\\ell]$ in between the locations of the $k$-th and $k+1$-th vehicles at time $\\tau_{j}^-$, then the coordinates of the vehicles $x(\\tau_{j}) =\\left(x_1(\\tau_{j}), \\ldots, x_{N+1}(\\tau_{j})\\right) \\in [0,L]^{N+1}$ after re-ordering due to the jump, for $i \\in \\until{N+1}$, are given by:\n \\begin{equation*}\n \\begin{split}\n x_{k+1}(\\tau_{j}) & = z \\\\\n\t\tx_i(\\tau_{j}) & =\\left\\{\\begin{array}{ll}\n\t\t\\displaystyle x_i(\\tau^-_{j}) \t& i\\in \\until{k} \\\\[15pt]\n\t\t\\displaystyle x_{i-1}(\\tau^-_{j})\t& i\\in \\{k+2, \\ldots, N+1\\}\\,.\\end{array}\\right.\n\t\t\\end{split}\n\t\\end{equation*}\n\n\\subsection{Problem statement}\nLet $x_0 \\in [0,L]^{n_0}$ be the initial coordinates of $n_0$ vehicles present at $t=0$. An HTQ is described by the tuple $\\left(L, m, \\lambda, \\varphi, \\psi, x_0\\right)$. \nLet \n$N(t;L, m, \\lambda, \\varphi, \\psi, x_0)$ be the corresponding queue length, i.e., the number of vehicles at time $t$ for an HTQ $\\left(L, m, \\lambda, \\varphi, \\psi, x_0\\right)$. For brevity in notation, at times, we shall not show the dependence of $N$ on parameters which are clear from the context.\n\n\n\nIn this paper, our objective is to provide rigorous characterizations of the dynamics of the proposed HTQ. A key quantity that we study is throughput, defined below. \n \n\\begin{definition}[Throughput of HTQ]\\label{def:throughput}\nGiven $L > 0$, $m > 0$, $\\varphi \\in \\Phi , \\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$ and $\\delta \\in [0,1)$, \nthe throughput of HTQ is defined as:\n\\begin{equation}\n\\label{eq:throughput-def}\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta):= \\sup \\left\\{\\lambda \\geq 0: \\, \\Pr \\left( N(t;L, m, \\lambda, \\varphi, \\psi, x_0) < + \\infty, \\quad \\forall t \\geq 0 \\right) \\geq 1 - \\delta \\right\\}.\n\\end{equation}\n\\end{definition}\n\n\nFigure~\\ref{fig:throughput-simulations} shows the complex dependency of throughput on key queue parameters such as $m$ and $L$. In particular, it shows that for every $L$, $\\varphi$, $\\psi$, $x_0$ and $\\varphi$, the throughput exhibits a phase transition from being unbounded for $m \\in (0,1)$ to being bounded for $m>1$. Moreover, Figure~\\ref{fig:throughput-simulations} also suggests that, for sufficiently small $L$, throughput is monotonically non-increasing in $m$, and that it is monotonically non-decreasing in $m>1$, for sufficiently large $L$. Also, it can be observed that initial condition can also affect the throughput. We now develop analytical results that match the throughput profile in Figure~\\ref{fig:throughput-simulations} as closely as possible. To that purpose, we will make extensive use of novel properties of \\emph{service rate} and \\emph{busy period} of the proposed HTQ, which could be of independent interest. \n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-4L-n0}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-5L-n100}}\\\\\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-4L-n0-unif}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-5L-n100-unif}}\n\\caption{Throughput for various combinations of $m$, $L$, and $n_0$. The parameters used in individual cases are: (a) $\\varphi=\\delta_{0}$, $\\psi = \\delta_L$, and $n_0=0$ (b) $\\varphi=\\delta_{0}$, $\\psi = \\delta_L$, and $n_0=100$ (c) $\\varphi=U_{[0,L]}$, $\\psi = U_{[0,L]}$, and $n_0=0$ (d) $\\varphi=U_{[0,L]}$, $\\psi = U_{[0,L]}$, and $n_0=100$. In all the cases, the locations of initial $n_0$ vehicles were chosen at equal spacing in $[0,L]$.\n \\label{fig:throughput-simulations}}\n\\end{figure}\n\n\n\n\n\n\n\\section{Service Rate Properties of the Horizontal Traffic Queue}\n\\label{sec:service-rate-monotonicity}\nFor every $y \\in \\mathcal S_N^L$, $N \\in \\mathbb{N}$, $L>0$, we let \n$y_{\\text{min}}:=\\min_{i \\in \\until{N}} y_i$, and $y_{\\mathrm{max}}:=\\max_{i \\in \\until{N}} y_i$ denote the minimum and maximum inter-vehicle distances respectively. It is easy to establish the following monotonicity properties of $y_{\\text{min}}$ and $y_{\\mathrm{max}}$.\n\n\\begin{lemma}[Inter-vehicle Distance Monotonicity Between Jumps]\n\\label{lem:vehicle-distance-monotonicity}\nFor any $y \\in \\mathcal S^L_N$, $N \\in \\mathbb{N}$, $L>0$, under the dynamics in \\eqref{eq:inter-vehicle-distance-dynamics}, for all $m>0$\n$$\n\\frac{d}{dt} y_{\\text{min}} \\geq 0 \\qquad \\& \\qquad \\qquad \\frac{d}{dt} y_{\\mathrm{max}} \\leq 0.\n$$\n\\end{lemma}\n\\begin{proof}\nLet $y_{\\text{min}}(t) = y_j(t)$, i.e., the $j$-th vehicle has the minimum inter-vehicle distance at time $t\\geq 0$.\nTherefore, \\eqref{eq:inter-vehicle-distance-dynamics} implies that $\\dot{y}_{\\min}(t) = \\dot{y}_j(t)=y_{j+1}^m(t) - y_j^m(t)\\geq 0$. One can similarly show that $y_{\\mathrm{max}}$ is non-increasing.\n\\qed\n\\end{proof}\n\nDue to the complex state-dependence of the departure process, the queue length process is difficult to analyze. We propose to study a related scalar quantity, called \\emph{workload} formally defined as follows, where we recall the notations introduced in \nSection~\\ref{sec:problem-formulation}.\n\n\\begin{definition}[Workload]\nThe workload associated with the HTQ at any instant is the sum of the distances remaining to be travelled by all the vehicles present at that instant. That is, if the current coordinates and departure coordinates of all vehicles are $x \\in [0,L]^N$ and $q \\in \\mathbb{R}_+^N$ respectively, with $q \\geq x$, then the workload is given by:\n$$\nw(x,q):= \\sum_{i=1}^N (q_i-x_i).\n$$ \n\\end{definition}\n\n\nSince the maximum distance to be travelled by any vehicle from the time of arrival to the time of departure is upper bounded by $R$, we have the following simple relationship between workload and queue length at any time instant:\n\\begin{equation}\\label{eq:workload-upperbound}\nw(t) \\leq N(t)\\, R \\, , \\qquad \\forall \\, t \\geq 0 \\, .\n\\end{equation}\nAn implication of \\eqref{eq:workload-upperbound} is that unbounded workload implies unbounded queue length in our setting. We shall use this relationship to establish an upper bound on the throughput.\nHowever, a finite workload does not necessarily imply finite queue length. In order to see this, consider the state of the queue with $N$ vehicles, all of whom have distance $1\/N$ remaining to be travelled. Therefore, the workload at this instant is $1\/N \\times N= 1$, which is independent of $N$. \n\nWhen the workload is positive, its rate of decrease is equal to \\emph{service rate} in between jumps, defined next.\n\n\\begin{definition}[Service Rate]\nWhen the HTQ is not idle, its instantaneous service rate is equal to the sum of the speeds of the vehicles present in the system at that time instant, i.e., $s(x)=\\sum_{i=1}^N y_i^m(x)$.\n\\end{definition}\n\n\nSince the service rate depends only on the inter-vehicle distances, we shall alternately denote it as $s(y)$. For $m=1$, $s(y)=\\sum_{i=1}^N y_i \\equiv L$, i.e., the service rate is independent of the state of the system, and is constant in between and during jumps. This property does not hold true in the nonlinear ($m \\neq 1$) case. Nevertheless, one can prove interesting properties for the service rate dynamics. We start by deriving bounds on service rate in between jumps. \n\n\\begin{lemma}[Bounds on Service Rates]\n\\label{lem:service-rate-bounds}\nFor any $y \\in \\mathcal S_N^L$, $N \\in \\mathbb{N}$, $L>0$, under the dynamics in \\eqref{eq:inter-vehicle-distance-dynamics},\n\\begin{enumerate}\n\\item $L^m N^{1-m} \\leq s(y) \\leq L^m$ if $m > 1$;\n\\item $L^m \\leq s(y) \\leq L^m N^{1-m}$ if $m \\in (0,1)$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nNormalizing the inter-vehicular distances by $L$, the service rate can be rewritten as \n\\begin{equation}\n\\label{eq:service-rate-normalized}\ns(y)=L^m \\sum_{i=1}^N \\left(\\frac{y_i}{L}\\right)^m. \n\\end{equation}\nTherefore, for $m > 1$, $s(y) \\leq L^m \\sum_{i=1}^N \\frac{y_i}{L}=L^m$. One can similarly show that, for $m \\in (0,1)$, $s(y) \\geq L^m$. In order to prove the remaining bounds, we note that $\\sum_{i=1}^N z_i^m$ is strictly convex in $z=[z_1, \\ldots, z_N]$ for $m > 1$, and that the minimum of $\\sum_{i=1}^N z_i^m$ over $z \\in \\mathcal S_N$ occurs at $z= \\mathbf{1}\/N$, and is equal to $N^{1-m}$. Similarly, for $m \\in (0,1)$, $\\sum_{i=1}^N z_i^m$ is strictly concave in $z$, and its maximum over $z \\in \\mathcal S_N$ occurs at $z= \\mathbf{1}\/N$, and is equal to $N^{1-m}$. Combining these facts with \\eqref{eq:service-rate-normalized}, and noting that $y\/L \\in \\mathcal S_N$, gives the lemma.\n\\qed\n\\end{proof}\n\n\\begin{lemma}[Service Rate Monotonicity Between Jumps]\n\\label{lem:service-rate-dynamics-between-jumps}\nFor any $y \\in \\mathcal S^L_N$, $N \\in \\mathbb{N}$, $L>0$, under the dynamics in \\eqref{eq:inter-vehicle-distance-dynamics}, \n$$\n\\frac{d}{dt} s(y) \\leq 0 \\quad \\text{ if } m > 1 \\qquad \\& \\qquad \\frac{d}{dt} s(y) \\geq 0\\quad \\text{ if } m \\in (0,1) \\, ,\n$$\nwhere the equality holds true if and only if $y=\\frac{L}{N}\\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\nThe time derivative of service rate is given by:\n\\begin{align}\n\\frac{d}{dt} s(y) & = \\frac{d}{dt} \\sum_{i=1}^N y_i^m = m \\sum_{i=1}^N y_i^{m-1} \\dot{y}_i \\nonumber \\\\ \n\\label{eq:service-rate-derivative}\n& = m \\sum_{i=1}^N y_i^{m-1} \\left( y_{i+1}^m - y_i^m\\right) \n\\end{align}\nwhere the second equality follows by \\eqref{eq:inter-vehicle-distance-dynamics}. The result then follows by application of Lemma~\\ref{lem:appendix-general-summation}, and by noting that $g(z)=z^m$ is a strictly increasing function for all $m>0$, and $h(z)=z^{m-1}$ is strictly decreasing if $m \\in (0,1)$, and strictly increasing if $m>1$.\n\\qed\n\\end{proof}\n\n\nThe following lemma quantifies the change in service rate due to departure of a vehicle. \n\n\\begin{lemma}[Change in Service Rate at Departures]\n\\label{lem:service-rate-jumps}\nConsider the departure of a vehicle that changes inter-vehicle distances from $y \\in \\mathcal S_N^L$ to $y^- \\in \\mathcal S_{N-1}^L$, for some $N \\in \\mathbb{N} \\setminus \\{1\\}$, $L > 0$. If $y_1 \\geq 0$ and $y_2 \\geq 0$ denote the inter-vehicle distances behind and in front of the departing vehicle respectively, at the moment of departure, then the change in service rate due to the departure satisfies the following bounds:\n\\begin{enumerate}\n\\item if $m > 1$, then $0 \\leq s(y^-) - s(y) \\leq (y_1+y_2)^m \\left(1-2^{1-m} \\right)$;\n\\item if $m \\in (0,1)$, then $0 \\leq s(y) - s(y^-) \\leq \\min\\{y_1^m,y_2^m\\}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nIf $m>1$, then $\\left(\\frac{y_1}{y_1+y_2}\\right)^m + \\left(\\frac{y_2}{y_1+y_2}\\right)^m \\leq \\frac{y_1}{y_1+y_2} + \\frac{y_2}{y_1+y_2} = 1$, i.e., $s(y^-)-s(y) = (y_1 + y_2)^m - y_1^m - y_2^m \\geq 0$. One can similarly show that $s(y)-s(y^-) \\geq 0$ if $m \\in (0,1)$.\n\nIn order to show the upper bound on $s(y^-)-s(y)$ for $m>1$, we note that the minimum value of $z^m + (1-z)^m$ over $z \\in [0,1]$ for $m > 1$ is $2^{1-m}$, and it occurs at $z=1\/2$. Therefore, \n\\begin{align*}\ns(y^-)-s(y) = (y_1 + y_2)^m - y_1^m - y_2^m & = (y_1 + y_2)^m \\left(1 - \\left(\\frac{y_1}{y_1+y_2}\\right)^m - \\left(\\frac{y_2}{y_1+y_2}\\right)^m \\right) \\\\\n& \\leq (y_1+y_2)^m \\left(1-2^{1-m} \\right)\n\\end{align*}\n\nThe upper bound on $s(y)-s(y^-)$ for $m \\in (0,1)$ can be proven as follows. Since $y_1^m \\leq (y_1 + y_2)^m$, $s(y) - s(y^-) = y_1^m + y_2 ^m - (y_1 + y_2)^m \\leq y_2^m$. \nSimilarly, $s(y)-s(y^-) \\leq y_1^m$. Combining, we get $s(y)-s(y^-) \\leq \\min \\{y_1^m, y_2^m\\}$. Note that, in proving this, we nowhere used the fact that $m \\in (0,1)$. However, this bound is useful only for $m \\in (0,1)$. \n\\qed\n\\end{proof}\n\n\\begin{remark}[Change in Service Rate at Arrivals]\n\\label{rem:service-rate-jump-arrival}\nThe bounds derived in Lemma~\\ref{lem:service-rate-jumps} can be trivially used to prove the following bounds for change in service rate at arrivals:\n\\begin{enumerate}\n\\item if $m > 1$, then $0 \\leq s(y) - s(y^+) \\leq (y_1+y_2)^m \\left(1-2^{1-m} \\right)$;\n\\item if $m \\in (0,1)$, then $0 \\leq s(y^+) - s(y) \\leq \\min\\{y_1^m,y_2^m\\}$,\n\\end{enumerate}\nwhere $y_1$ and $y_2$ are the inter-vehicle distances behind and in front of the arriving vehicle respectively, at the moment of arrival.\n\\end{remark}\n\n\nThe following lemma will facilitate generalization of Lemma~\\ref{lem:service-rate-dynamics-between-jumps}. In preparation for the lemma, let $f(y,m):=m \\sum_{i=1}^N y_i^{m-1} \\left( y_{i+1}^m - y_i^m\\right)$ be the time derivative of service rate, as given in \\eqref{eq:service-rate-derivative}.\n\n\\begin{lemma}\n\\label{lem:service-rate-dot-lower-bound}\nFor all $y \\in \\text{int}(\\mathcal S_N^L)$, $N \\in \\mathbb{N} \\setminus \\{1\\}$, $L > 0$:\n\\begin{equation}\n\\label{eq:service-rate-der-m}\n\\frac{\\partial}{\\partial m} f(y,m)|_{m=1} = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right) \\leq 0\n\\end{equation}\nAdditionally, if $L < e^{-2}$, then\n\\begin{equation}\n\\label{eq:service-rate-twice-der-m}\n\\frac{\\partial^2}{\\partial m^2} f(y,m)|_{m=1} \\geq 0\n\\end{equation}\nMoreover, equality holds true in \\eqref{eq:service-rate-der-m} and \\eqref{eq:service-rate-twice-der-m} if and only if $y = \\frac{L}{N}\\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\nTaking the partial derivative of $f(y,m)$ with respect to $m$, we get that \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) & = \\frac{f(y,m)}{m} + m \\sum_{i=1}^N \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right)\n\\end{align*}\nIn particular, for $m=1$: \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) |_{m=1} & = f(y,1) + \\sum_{i=1}^N \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\\\\n& = L \\sum_{i=1}^N \\frac{y_i}{L} \\log \\left(\\frac{y_{i-1}\/L}{y_i\/L} \\right) \\\\\n& = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right)\n\\end{align*}\nwhere, for the second equality, we used the trivial fact that $f(y,1)=0$. \nTaking second partial derivative of $f(y,m)$ w.r.t. $m$ gives:\n\\begin{align*}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) = & \\sum_{i=1}^N y_i^{m-1}\\log y_i \\left(y_{i+1}^m - y_i^m \\right) + \\sum_{i=1}^N y_i^{m-1} \\left(y_{i+1}^m \\log y_{i+1} - y_i^m \\log y_i \\right)\\\\\n& + \\sum_{i=1}^N \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right) \\\\\n& + m \\sum_{i=1}^N \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right)^2 - 4 y_i^{2m-1} \\log^2 y_i \\right)\n\\end{align*}\nIn particular, for $m=1$:\n\\begin{align}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) |_{m=1} = & \\sum_{i=1}^N \\left(y_{i+1} - y_i \\right) \\log y_i + \\sum_{i=1}^N \\left(y_{i+1} \\log y_{i+1} - y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^N \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^N \\left( y_{i+1} (\\log y_i + \\log y_{i+1})^2 - 4 y_i \\log^2 y_i\\right) \\nonumber \\\\\n\\label{eq:second-derivative-m-final-eq}\n= & \\sum_{i=1}^N \\log^2 y_i \\left(y_{i+1} - y_i \\right) + 2 \\sum_{i=1}^N \\log y_i \\left(y_{i+1} \\log y_{i+1} + y_{i+1} - y_i \\log y_i - y_i \\right) \\\\\n\\nonumber\n\\geq & \\, 0\n\\end{align}\nIt is easy to check that, $\\log z$, $\\log^2 z$ and $z + z \\log z$ are strictly increasing, strictly decreasing and strictly decreasing functions, respectively, for $z \\in (0,e^{-2})$. Therefore, Lemma~\\ref{lem:appendix-general-summation} implies that each of the two terms in \\eqref{eq:second-derivative-m-final-eq} is non-negative, and hence the lemma.\n\\qed\n\\end{proof}\n\n\\iffalse\n\n\\begin{lemma}\n\\label{lem:service-rate-dot-lower-bound}\nFor all $y \\in \\mathcal S_n^L$, $n \\in \\mathbb{N} \\setminus \\{1\\}$, $L > 0$:\n\\begin{equation}\n\\label{eq:service-rate-der-m}\n\\frac{\\partial}{\\partial m} f(y,m)|_{m=1} = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right) \\leq 0\n\\end{equation}\nAdditionally, if $L < e^{-2}$, then\n\\begin{equation}\n\\label{eq:service-rate-twice-der-m}\n\\frac{\\partial^2}{\\partial m^2} f(y,m)|_{m=1} \\geq 0\n\\end{equation}\nMoreover, equality holds true in \\eqref{eq:service-rate-der-m} and \\eqref{eq:service-rate-twice-der-m} if and only if $y = \\frac{L}{n}\\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\n\\ksmargin{need to exclude from the proof all $y$ such that $y_i=0$ for some $i \\in \\until{n}$}\nTaking the partial derivative of $f(y,m)$ with respect to $m$, we get that \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) & = \\frac{f(y,m)}{m} + m \\sum_{i=1}^n \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right)\n\\end{align*}\nIn particular, for $m=1$: \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) |_{m=1} & = f(y,1) + \\sum_{i=1}^n \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\\\\n& = L \\sum_{i=1}^n \\frac{y_i}{L} \\log \\left(\\frac{y_{i-1}\/L}{y_i\/L} \\right) \\\\\n& = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right)\n\\end{align*}\nwhere, for the second equality, we used the trivial fact that $f(y,1)=0$. \nTaking second partial derivative of $f(y,m)$ w.r.t. $m$ gives:\n\\begin{align*}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) = & \\sum_{i=1}^n y_i^{m-1}\\log y_i \\left(y_{i+1}^m - y_i^m \\right) + \\sum_{i=1}^n y_i^{m-1} \\left(y_{i+1}^m \\log y_{i+1} - y_i^m \\log y_i \\right)\\\\\n& + \\sum_{i=1}^n \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right) \\\\\n& + m \\sum_{i=1}^n \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right)^2 - 4 y_i^{2m-1} \\log^2 y_i \\right)\n\\end{align*}\nIn particular, for $m=1$:\n\\begin{align}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) |_{m=1} = & \\sum_{i=1}^n \\left(y_{i+1} - y_i \\right) \\log y_i + \\sum_{i=1}^n \\left(y_{i+1} \\log y_{i+1} - y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^n \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^n \\left( y_{i+1} (\\log y_i + \\log y_{i+1})^2 - 4 y_i \\log^2 y_i\\right) \\nonumber \\\\\n\\label{eq:second-derivative-m-final-eq}\n= & \\sum_{i=1}^n \\log^2 y_i \\left(y_{i+1} - y_i \\right) + 2 \\sum_{i=1}^n \\log y_i \\left(y_{i+1} \\log y_{i+1} + y_{i+1} - y_i \\log y_i - y_i \\right) \\\\\n\\nonumber\n\\geq & \\, 0\n\\end{align}\nIt is easy to check that, $\\log z$, $\\log^2 z$, and $z + z \\log z$ are strictly increasing, strictly decreasing and strictly decreasing functions, respectively, for $z \\in (0,e^{-2})$. Therefore, Lemma~\\ref{lem:appendix-general-summation} implies that each of the two terms in \\eqref{eq:second-derivative-m-final-eq} is non-negative, and hence the lemma.\n\\end{proof}\n\n\\fi\n\nLemma~\\ref{lem:service-rate-dot-lower-bound} implies that, for sufficiently small $L$, $f(y,m)$ is locally convex in $m$. One can use this property along with an exact expression for $\\frac{\\partial}{\\partial m}f(y,m)$ in Lemma~\\ref{lem:service-rate-dot-lower-bound} at $m=1$, and the fact that $f(y,1)=0$ for all $y$, to develop a linear approximation in $m$ of $f(y,m)$ around $m=1$. The following lemma derives this approximation, as also suggested by Figure~\\ref{fig:service-rate-dot-vs-m}.\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=5cm]{service-rate-dot-vs-m} \n \\caption{$f(y,m)$ vs. $m$ for a typical $y \\in \\mathcal S_{10}$.}\n \\label{fig:service-rate-dot-vs-m}\n\\end{figure}\n\n\\begin{lemma}\n\\label{lem:service-rate-dot-deltay-bound}\nFor a given $y \\in \\text{int}(\\mathcal S_N^L)$, $n \\in \\mathbb{N}$, $L \\in (0,e^{-2})$, there exists $\\underbar{m}(y) \\in [0,1)$ such that \n$$\n\\frac{d}{dt} s(y) \\geq 2 \\frac{(1-m)}{L} \\left(y_{\\mathrm{max}} - y_{\\text{min}} \\right)^2 \\, , \\qquad \\forall \\, m \\in [\\underbar{m}(y),1]\n$$\n\\end{lemma}\n\\begin{proof}\nFor a given $y \\in \\text{int}(\\mathcal S_N^L)$, the local convexity of $f(y,m):=\\frac{d}{dt} s(y)$ in $m$, and the expression of $\\frac{\\partial}{\\partial m}f(y,m)$ at $m=1$ in Lemma~\\ref{lem:service-rate-dot-lower-bound} implies that $\\frac{d}{dt} s(y) \\geq (1-m) L D\\left(\\frac{y}{L} || P^-\\frac{y}{L} \\right)$ for sufficiently small $m<1$. Pinsker's inequality implies $D\\left(\\frac{y}{L} || P^-\\frac{y}{L} \\right) \\geq \\frac{\\|y-P^-y\\|_1^2}{2 L^2}$. This, combined with the fact that $\\|y-P^-y\\|_1 \\geq 2(y_{\\mathrm{max}}-y_{\\text{min}})$ for all $y \\in \\text{int}(\\mathcal S_N^L)$, gives the lemma.\n\\qed\n\\end{proof}\n\n\n\\section{Simulations}\n\\label{sec:simulations}\nIn this section, we present simulation results on throughput analysis, and compare with our theoretical results from previous sections. \n \n \n\n\n\n \n\n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=5cm]{lambda-m-lowerUpper-w-phase-transition-separated-v2}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=5cm]{lambda-m-lowerUpper-uniform-w-phase-transition-separated-v2}}\n\\caption{Comparison between theoretical estimates of throughput from Theorem~\\ref{thm:superlinear-bound-empty}, and range of numerical estimates from simulations, for zero initial condition. The parameters used for this case are: $L = 1$, $\\delta = 0.1$, and (a) $\\varphi = \\delta_0$, $\\psi = \\delta_L$, (b) $\\varphi = U_{[0,L]}$, $\\psi=U_{[0,L]}$.\n \\label{fig:lambda-m-empty}}\n\\end{figure}\n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{lambda-m-busy-period-L100-v2} \n\\caption{Comparison between theoretical estimates of throughput from Theorem \\ref{thm:superlinear-bound-empty}, and range of numerical estimates from simulations, for zero initial condition. The parameters used for this case are: $L = 100$, $\\delta = 0.1$, $T=10$, and $\\varphi = \\delta_0$, $\\psi = \\delta_L$.}\n\\label{fig:superlinear-large-L}\n\\end{center}\n\\end{figure} \n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{phase-transition-non-empty-separated-v2} \n\\caption{Comparison between theoretical estimates of throughput from Theorem~\\ref{thm:superlinear-bound-non-empty}, and range of numerical estimates from simulations. The parameters used for this case are: $L = 1$, $\\delta = 0.1$, $\\varphi = \\delta_0$, $\\psi = \\delta_L$, $w_0=1$ and $n_0=4$, $x_1(0) =0.6, x_2(0) =0.7, x_3(0) =0.8, x_4(0) =0.9$.}\n\\label{fig:phase-transition-non-empty}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\nFigures~\\ref{fig:lambda-m-empty}, \\ref{fig:superlinear-large-L} and \\ref{fig:phase-transition-non-empty} show comparison between the lower bound on throughput over finite time horizons, as given by Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, and the corresponding numerical estimates from simulations. \n\n Figures~\\ref{fig:lambda-m-empty} and \\ref{fig:superlinear-large-L} are for zero initial condition, and Figure~\\ref{fig:phase-transition-non-empty} is for non-zero initial condition.\n\nFigures~\\ref{fig:sublinear-lowerbound} and \\ref{fig:superlinear-batch} show comparison between the lower bound on throughput as given by the bacth release control policy, as per Theorems~\\ref{thm:main-sub-linear} and \\ref{thm:main-batch-super-linear}, respectively, under a couple of representative values of maximum permissible perturbation $\\eta$. In particular, Figure~\\ref{fig:sublinear-lowerbound} demonstrates that the lower bound achieved from Theorem \\ref{thm:main-sub-linear} increases drastically as $m \\to 0^+$. Both the figures also confirm that the throughput indeed increases with increasing maximum permissible perturbation $\\eta$.\n\nIt is instructive to compare Figures~\\ref{fig:lambda-m-empty}(b) and \\ref{fig:superlinear-batch}(a), both of which depict throughput estimates for the sub-linear case, however obtained from different methods, namely busy period distribution and batch release control policy. Accordingly, one should bear in mind that the two bounds have different qualifiers attached to them: the bound in Figure~\\ref{fig:lambda-m-empty}(b) is valid probabilistically only over a finite time horizon, whereas the bound in Figure~\\ref{fig:superlinear-batch}(a) is valid with probability one, although under a perturbation to the arrival process. \n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{sublinear-lowerbound} \n\\caption{Theoretical estimates of throughput from Theorems~\\ref{thm:main-sub-linear} for different values of $\\eta$. The parameters used for this case are: $L = 1$, $\\varphi = U_{[0,L]}$, $\\psi = U_{[0,L]}$, and $w_0=0$ . Note that the vertical axis is in logarithmic scale.}\n\\label{fig:sublinear-lowerbound}\n\\end{center}\n\\end{figure}\n\n\n\n \n\n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=5cm]{sublinear-small-L}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=5cm]{superlinear-batch-new-v2}}\n\\caption{Theoretical estimates of throughput from Theorem \\ref{thm:main-batch-super-linear}, and numerical estimates from simulations for different The parameters used for this case are: $\\varphi=U_{[0,L]}$, $\\psi=U_{[0,L]}$, and (a) $L = 1$, (b) L = 100. \\label{fig:superlinear-batch}}\n\\end{figure}\n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{upper-bound-queue-length-linear-large-lambdas} \n\\caption{Comparison between the empirical expectation of the queue length and the upper bound suggested by Remark~\\ref{remark:waiting-time}. We let the simulations run up to time $t=80,000$. The parameters used for this case are: $L=1$, $m=1$, $\\varphi=\\delta_0$, $\\psi=\\delta_L$. For these values, we have $\\lambda_{\\text{max}}=1$.}\n\\label{fig:queue-length-linear}\n\\end{center}\n\\end{figure} \n\nFinally, Figure~\\ref{fig:queue-length-linear} shows a good agreement between queue length bound suggested by Remark~\\ref{remark:waiting-time}, and the corresponding numerical estimates in the linear case. \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{Throughput Bounds under Batch Release Control Policy}\nIn this section, we consider a \\emph{time-perturbed} version of the arrival process. \nFor a given realization of arrival times, $\\{t_1,t_2,\\cdots\\}$, consider a perturbation map $t_i^{\\prime} \\equiv t_i^{\\prime}(t_1,\\ldots,t_i)$ satisfying $t_i^{\\prime}\\geq t_i$ for all $i$, which prescribes the perturbed arrival times. The magnitude of perturbation is defined as $\\eta := E\\left(t_i^{\\prime}-t_i\\right)$, where the expectation is with respect to the Poisson process with rate $\\lambda$ that generates the arrival times.\n\nWe prove boundedness of the queue length under a specific perturbation map. This perturbation map is best understood in terms of a control policy that governs the release of arrived vehicles into HTQ. In order to clarify the implementation of the control policy, we decompose the proposed HTQ into two queues in series: denoted as HTQ1 and HTQ2, both of which have the same geometric characteristics as HTQ, i.e., a circular road segment of length $L$ (see Figure \\ref{fig:htq1-htq2} for illustrations). The original arrival process for HTQ, i.e. spatio-temporal Poisson process with rate $\\lambda$ and spatial distribution $\\varphi$ is now the arrival process for HTQ1. Vehicles remain stationary at their arrival locations in HTQ1, until released by the control policy into HTQ2. Upon released into HTQ2, vehicles travel according to \\eqref{eq:dynamics-moving-coordinates} until they depart after traveling a distance that is sampled from $\\psi$, as in the case of HTQ. The time of release of the vehicles into HTQ2 correspond to their perturbed arrival times $t_1^{\\prime}, t_2^{\\prime}, \\ldots$. The average waiting time in HTQ1 under the given release control policy is then the magnitude of perturbation in the arrival times. \n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=14cm]{htq1-htq2-decomposed} \n\\caption{Decomposition of HTQ into HTQ1 and HTQ2 in series.}\n\\label{fig:htq1-htq2}\n\\end{center}\n\\end{figure}\n\nWe consider the following class of release control policy, for which we recall from the problem setup in Section~\\ref{sec:problem-formulation} that $\\text{supp}(\\varphi)=[0,\\ell]$ for some $\\ell \\in [0,L]$. \n\n\n\\begin{definition}[Batch Release Control Policy $\\pi^b_\\triangle$]\n\\label{def:batch-release-control-policy}\nDivide $[0,\\ell]$ into sub-intervals, each of length $\\triangle$, enumerated as $1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil$. Let $T_1$ be the first time instant when HTQ2 is empty. At time $T_1$, release one vehicle each, if present, from all odd-numbered sub-intervals in $\\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$ simultaneously into HTQ2. Let $T_2$ be the next time instant when HTQ2 is empty. At time $T_2$, release one vehicle each, if present, from all even-numbered sub-intervals in $\\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$ simultaneously into HTQ2. Repeat this process of alternating between releasing one vehicle each from odd and even-numbered sub-intervals every time that HTQ2 is empty.\n\\end{definition}\n\n\n\n\\begin{remark}\n\\begin{enumerate}\n\\item Under $\\pi^b_{\\triangle}$, when vehicles are released into HTQ2, the inter-vehicle distances in the front and rear of each vehicle being released is at least equal to $\\triangle$. \n\\item The order in which vehicles are released into HTQ2 from HTQ1 under $\\pi^b_{\\triangle}$ may not be the same as the order of arrivals into HTQ1.\n\\end{enumerate}\n\\end{remark}\n\nIn the next two sub-sections, we analyze the performance of the batch release control policy for sub-linear and super-linear cases. \n\\subsubsection{The Sub-linear Case}\nIn this section, we derive a lower bound on throughput when $m\\in(0,1)$. We first derive a trivial lower bound in Proposition \\ref{prop:throughput-bound-sublinear} implied by Lemma \\ref{lem:service-rate-jumps} and Remark \\ref{rem:service-rate-jump-arrival}. Next, we improve this lower bound in Theorem \\ref{thm:main-sub-linear} under a under a batch release control policy, $\\pi_{\\triangle}^b$. \n\\begin{proposition}\n\\label{prop:throughput-bound-sublinear}\nFor any $L > 0$, $m\\in(0,1)$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$:\n$$\n \\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0) \\geq L^m\/\\bar{\\psi} \n$$\n\\end{proposition}\n\\begin{proof}\nRemark \\ref{rem:service-rate-jump-arrival} implies that, for $m \\in (0,1)$, the service rate does not decrease due to arrivals. Therefore, a simple lower bound on the service rate for any state is the service rate when there is only one vehicle in the system, i.e., $L^m$. Therefore, \nthe workload process is upper bounded as\n$w(t) = w_0+r(t)-\\int_0^ts(z)\\mathrm{d} z \\leq w_0+r(t) -L^m(t-\\mathcal{I}(t)), \\quad \\forall t\\geq 0$, \nwhere $r(t)$ and $\\mathcal{I}(t)$ denote the renewal reward and the idle time processes, respectively, as introduced in the proof of Proposition \\ref{prop:unstable}. Similar to the proof of Proposition \\ref{prop:unstable}, it can be shown that, if $\\lambda < L^m\/\\bar{\\psi}$, then the workload, and hence the queue length, goes to zero in finite time with probability one. \n\\qed\n\\end{proof}\n\nNext, we establish better throughput guarantees than Proposition~\\ref{prop:throughput-bound-sublinear}, under a batch release control policy, $\\pi_{\\triangle}^b$.\n\nThe next result characterizes the time interval between release of successive batches into HTQ2 under $\\pi_\\triangle^b$.\n\n\\begin{lemma}\n\\label{eq:successive-release-time-difference}\nFor given $\\lambda>0$, $\\triangle > 0$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m\\in(0,1)$, $x_0\\in[0,L]^{n_0}$, $L>0$, $n_0\\in \\mathbb{N}$, let $T_1$, $T_2$, $\\ldots$ denote the random variables corresponding to time of successive batch releases into HTQ2 under $\\pi_{\\triangle}^b$. Then, $T_1 \\leq \\frac{n_0R}{L^m}$, $T_{i+1}-T_i \\leq R\/\\triangle^m$ for all $i \\geq 1$, and $y_{\\text{min}}(t) \\geq \\triangle$ for all $t \\geq T_1$.\n\\end{lemma}\n\\begin{proof}\nSince the maximum distance to be traveled by every vehicle is upper bounded by $R$, the initial workload satisfies $w_0\\leq n_0R$. Since the minimum service rate for $m \\in (0,1)$ is $L^m$ (see proof of Proposition~\\ref{prop:throughput-bound-sublinear}), with no new arrivals, it takes at most $w_0\/L^m=n_0 R\/L^m$ amount of time for the system to become empty. This establishes the bound on $T_1$. \n\nLemma~\\ref{lem:vehicle-distance-monotonicity} implies that, under $\\pi^b_{\\triangle}$, the minimum inter-vehicle distance in HTQ2 is at least $\\triangle$ after $T_1$. \nThis implies that $y_{\\text{min}}(t) \\geq \\triangle$ for all $t \\geq T_1$, and hence the minimum speed of every vehicle in HTQ2 is at least $\\triangle^m$ after $T_1$. Since the maximum distance to be traveled by every vehicle is $R$, this implies that the time between release of a vehicle into HTQ2 and its departure is upper bounded by $R\/\\triangle^m$, which in turn is also an upper bound on the time required by all the vehicles released in one batch to depart from the system. \n\\qed\n\\end{proof}\nLet $N_1(t)$ and $N_2(t)$ denote the queue lengths in HTQ1 and HTQ2, respectively, at time $t$. Lemma~\\ref{eq:successive-release-time-difference} implies that, for every $\\triangle>0$, $N_2(t)$ is upper bounded for all $t \\geq T_1$. The next result identifies conditions under which $N_1(t)$ is upper bounded. \n\n\n\n\nFor $F>0$, let $\\Phi_F:=\\setdef{\\varphi \\in \\Phi}{\\sup_{x \\in [0,\\ell]} \\varphi(x) \\leq F}$. For subsequent analysis, we now derive an upper bound on the \\emph{load factor}, i.e., the ratio of the arrival and departure rates, associated with a typical sub-queue of HTQ1 among $\\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$. It is easy to see that, for every $\\varphi \\in \\Phi_F$, $F>0$, the arrival process into every sub-queue is Poisson with arrival rate upper bounded by $\\lambda F \\triangle$. Lemma~\\ref{eq:successive-release-time-difference} implies that the departure rate is at least $\\triangle^m\/2R$. Therefore, the load factor for every sub-queue is upper bounded as \n\\begin{equation}\n\\label{eq:load-factor-upper-bound}\n\\rho \\leq \\frac{2 R \\lambda F \\triangle}{\\triangle^m}=2 R \\lambda F \\triangle^{1-m}\n\\end{equation}\nIn particular, if \n\\begin{equation}\\label{eq:triangle-star}\n\\triangle < \\triangle^*(\\lambda):=\\left(2 R \\lambda F \\right)^{-\\frac{1}{1-m}},\n\\end{equation} \nthen $\\rho<1$. It should be noted that for $n_0<+\\infty$, by Lemma \\ref{eq:successive-release-time-difference}, $T_1<+\\infty$. The service rate is zero during $[0,T_1]$; however, since $T_1$ is finite, this does not affect the computation of load factor. \n\n\n\\begin{proposition}\n\\label{prop:pi1-policy}\nFor any $\\lambda>0$, $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m\\in(0,1)$, $x_0\\in[0,L]^{n_0}$, $L>0$, $n_0\\in \\mathbb{N}$, for sufficiently small $\\triangle$, $N_1(t)$ is bounded for all $t \\geq 0$ under $\\pi_{\\triangle}^b$, almost surely.\n\\end{proposition}\n\\begin{proof}\nBy contradiction, assume that $N_1(t)$ grows unbounded. This implies that there exists at least one sub-queue, say $ j \\in \\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$, such that its queue length, say $N_{1,j}(t)$, grows unbounded. In particular, this implies that there exists $t_0 \\geq T_1$ such that $N_{1,j}(t) \\geq 2$ for all $t \\geq t_0$. Therefore, for all $t \\geq t_0$, the ratio of arrival rate to departure rate for the $j$-th sub-queue is given by \\eqref{eq:load-factor-upper-bound}, which is a decreasing function of $\\triangle$, and hence becomes strictly less than one for sufficiently small $\\triangle$. A simple application of the law of large numbers then implies that, almost surely, $N_{1,j}(t)=0$ for some finite time, leading to a contradiction.\n\\qed \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nThe following result gives an estimate of the mean waiting time in a typical sub-queue in HTQ1 under the $\\pi_{\\triangle}^b$ policy.\n\n\\begin{proposition}\n\\label{eq:waiting-time}\nFor $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$, $m \\in (0,1)$, there exists a sufficiently small $\\triangle$ such that the average waiting time in HTQ1 under $\\pi_{\\triangle}^b$ is upper bounded as: \n\\begin{equation}\n\\label{eq:W-upper-bound}\nW \\leq R (2 R \\lambda F )^{\\frac{m}{1-m}} \\left(\\frac{2}{m^{\\frac{m}{1-m}}} + \\frac{m}{m^{\\frac{m}{1-m}}-m^{\\frac{1}{1-m}}} \\right).\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIt is easy to see that the desired waiting time corresponds to the system time of an M\/D\/1 queue with load factor given by \\eqref{eq:load-factor-upper-bound} along with the arrival and departure rates leading to \\eqref{eq:load-factor-upper-bound}. Note that, by Lemma \\ref{eq:successive-release-time-difference}, for finite $n_0$, the value of $T_1$ is finite and does not affect the average waiting time. Therefore, using standard expressions for M\/D\/1 queue~\\cite{Kleinrock:75}, we get that the waiting time in HTQ1 is upper bounded as follows for $\\rho <1$:\n\\begin{align}\nW \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m} \\frac{\\rho}{1-\\rho} & \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m} \\frac{1}{1-\\rho} \\nonumber \\\\ \n\\label{eq:waiting-time-upper-bound}\n& \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m -2 R \\lambda F \\triangle} \n\\end{align} \nIt is easy to check that the minimum of the second term in \\eqref{eq:waiting-time-upper-bound} over $\\big(0,\\triangle^*(\\lambda)\\big)$ occurs at $\\triangle = \\left(\\frac{m}{2 R \\lambda F} \\right)^{\\frac{1}{1-m}}$. Substitution in the right hand side of the first inequality in \\eqref{eq:waiting-time-upper-bound} gives the result.\n\\qed\n\\end{proof}\n\n\\begin{remark}\n\\label{rem:W-limit}\n\\eqref{eq:W-upper-bound} implies that, for every $R>0$, $F>0$, $\\lambda>0$, we have $W \\to 2R$ as $m \\to 0^+$. \n\\end{remark}\n\nWe extend the notation introduced in \\eqref{eq:throughput-def} to $\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta,\\eta)$ to also show the dependence on maximum allowable perturbation $\\eta$. This is not to be confused with the notation for $\\lambda_{\\text{max}}$ used in Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, where we used the notion of throughput over finite time horizons. We choose to use the same notations to maintain brevity. \n\nIn order to state the next result, for given $R>0$, $F>0$, $m \\in (0,1)$ and $\\eta \\geq 0$, let $\\tilde{W}(m,F,R,\\eta)$ \nbe the value of $\\lambda$ for which the right hand side of \\eqref{eq:W-upper-bound} is equal to $\\eta$, if such a $\\lambda$ exists and is at least $L^m\/\\bar{\\psi}$, and let it be equal to $L^m\/\\bar{\\psi}$, otherwise. The lower bound of $L^m\/\\bar{\\psi}$ in the definition of $\\tilde{W}$ is inspired by Proposition~\\ref{prop:throughput-bound-sublinear}. The next result formally states $\\tilde{W}$ as a lower bound on $\\lambda_{\\text{max}}$. \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\\begin{theorem}\\label{thm:main-sub-linear}\nFor any $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m \\in (0,1)$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, $L>0$, and maximum permissible perturbation $\\eta \\geq 0$, \n$$\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0,\\eta) \\geq \\tilde{W}(m,F,R,\\eta)\n$$\nIn particular, if $\\eta > 2R$, then $\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0,\\eta) \\to + \\infty$ as $m \\to 0^+$.\n\\end{theorem}\n\\begin{proof}\nConsider any $\\lambda \\leq\\tilde{W}(m,F,R,\\eta)$, and $\\triangle \\leq \\left(\\frac{m}{2 R \\lambda F} \\right)^{\\frac{1}{1-m}}$. Under policy $\\pi_\\triangle^b$, Lemma \\ref{eq:successive-release-time-difference} and Proposition \\ref{prop:pi1-policy} imply that, for finite $n_0$, $N_2(t)$ and $N_1(t)$ remain bounded for all times, with probability one. Also, for $\\lambda = \\tilde{W}(m,F,R,\\eta)$, by Proposition \\ref{eq:waiting-time} and the definition of $\\tilde{W}(m,F,R,\\eta)$, the introduced perturbation remains upper bounded by $\\eta$. Since the right hand side of \\eqref{eq:W-upper-bound} is monotonically increasing in $\\lambda$, perturbations remain bounded by $\\eta$ for all $\\lambda \\leq \\tilde{W}(m,F,R,\\eta)$. \nIn particular, by Remark \\ref{rem:W-limit}, we have $W\\to 2R$ as $m\\to 0^+$. In other words, as $m\\to 0^+$, the magnitude of the introduced perturbation becomes independent of $\\lambda$. \nTherefore, when $\\eta >2R$, and $m\\to 0^+$ throughput can grow unbounded while perturbation and queue length remains bounded.\n\\qed\n\\end{proof}\n\n\n\n\n\\begin{remark}\nWe emphasize that the only feature required in a batch release control policy is that, at the moment of release, the front and rear distances for the vehicles being released should be greater than $\\triangle$. The requirement of the policy in Definition~\\ref{def:batch-release-control-policy} for the road to be empty at the moment of release makes the control policy conservative, and hence affects the maximum permissible perturbation. In fact, for special spatial distributions, e.g., when $\\varphi$ is a Dirac delta function and the support of $\\psi$ is $[0,L-\\triangle])$, one can relax the conservatism to guarantee unbounded throughput for arbitrarily small permissible perturbation. \n\\end{remark}\n\n\\subsubsection{The Super-linear Case}\n\nIn this section, we study the throughput for the super-linear case under perturbed arrival process with a maximum permissible perturbation of $\\eta$. For this purpose, we consider the batch release control policy $\\pi_{\\triangle}^b$, defined in Definition \\ref{def:batch-release-control-policy}, for our analysis. Time intervals between release of successive batches, under $\\pi_{\\triangle}^b$, are characterized the same as Lemma \\ref{eq:successive-release-time-difference}. However, in the super linear case, by Lemma \\ref{lem:service-rate-bounds}, the initial minimum service rate is $L^mn_0^{1-m}$. Therefore, the time of first release is bounded as $T_1 \\triangle^*(\\lambda).\n\\end{equation}\nIt should be noted that since the batch release control policy iteratively releases from odd and even sub-queues, we need at least two sub-queues to be able to implement this policy. As a result, $\\triangle$ cannot be arbitrary large and $\\triangle<\\ell\/2$. This constraint gives the following bound on the admissible throughput under this policy\n\\begin{equation}\\label{eq:lambda-star-superlinear}\n\\lambda<\\lambda^*:=(\\ell\/2)^{m-1}\/2RF\n\\end{equation}\n The following result shows that for the above range of throughput, the queue length in HTQ1, $N_1(t)$, remains bounded at all times. \n \\begin{proposition}\n\\label{prop:pi1-policy-super-linear}\nFor any $\\lambda<\\lambda^*$, $\\triangle\\in\\big(\\triangle^*(\\lambda),\\ell\/2\\big]$, $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m>1$, $x_0\\in[0,L]^{n_0}$, $L>0$, $n_0\\in \\mathbb{N}$, $N_1(t)$ is bounded for all $t \\geq 0$ under $\\pi_{\\triangle}^b$, almost surely.\n\\end{proposition}\n\n\\begin{proof}\nThe proof is similar to proof of Proposition \\ref{prop:pi1-policy}. In particular, by \\eqref{eq:triangle-star-superlinear} and \\eqref{eq:lambda-star-superlinear}, one can show that load factor \\eqref{eq:load-factor-upper-bound} remains strictly smaller than one. This implies that no sub-queue in HTQ1 can grow unbounded, and $N_1(t)$ remains bounded for all times, with probability one. \n\\qed\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{eq:waiting-time-super-linear}\nFor any $\\lambda<\\lambda^*$, $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$, $m> 1$, the average waiting time in HTQ1 under $\\pi_{\\triangle}^b$ for $\\triangle = \\ell\/2$ is upper bounded as: \n\\begin{equation}\n\\label{eq:waiting-time-super-linear-eq}\nW \\leq \\frac{2R}{(\\ell\/2)^m}+\\frac{R}{(\\ell\/2)^m}\\frac{2R\\lambda F(\\ell\/2)^{1-m}}{1-2R\\lambda F(\\ell\/2)^{1-m}}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe proof is very similar to the proof of Proposition \\ref{eq:waiting-time}. Thus, we get the following bounds:\n\\begin{align*}\nW \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m} \\frac{\\rho}{1-\\rho} \\leq \\frac{2R}{\\triangle^m}+\\frac{R}{\\triangle^m}\\frac{2R\\lambda F\\triangle^{1-m}}{1-2R\\lambda F\\triangle^{1-m}}\n\\end{align*} \nThe right hand side of the above inequality is a decreasing function of $\\triangle$; therefore, $\\triangle = \\ell\/2$ minimizes it, and gives \\eqref{eq:waiting-time-super-linear-eq}. \n\\qed\n\\end{proof}\n\nLet $\\hat{W}(m,F,R,\\eta)$ \nbe the value of $\\lambda$ for which the right hand side of \\eqref{eq:waiting-time-super-linear-eq} is equal to $\\eta$, if such a $\\lambda\\leq \\lambda^*$ exists, and let it be equal to $\\lambda^*$ otherwise. Note that since the right hand side of \\eqref{eq:waiting-time-super-linear-eq} is monotonically increasing in $\\lambda$, for all $\\lambda\\leq \\hat{W}(m,F,R,\\eta)$ the introduced perturbation remains upper bounded by $\\eta$. \n\\begin{theorem}\\label{thm:main-batch-super-linear}\nFor any $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m >1$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, $L>0$, and maximum permissible perturbation $\\eta \\geq 0$, \n$$\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0,\\eta) \\geq \\hat{W}(m,F,R,\\eta)\n.$$\n\\end{theorem}\n\\begin{proof}\nFor any $\\lambda <\\hat{W}(m,F,R,\\eta)$, under $\\pi_\\triangle^b$, Lemma \\ref{eq:successive-release-time-difference} and Proposition \\ref{prop:pi1-policy-super-linear} imply that, for finite $n_0$, $N_2(t)$ and $N_1(t)$ remain bounded for all times, with probability one. Also, by Proposition \\ref{eq:waiting-time-super-linear} and the definition of $\\hat{W}(m,F,R,\\eta)$, the introduced perturbation remains upper bounded by $\\eta$. \n\\qed\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\\subsection{Throughput Bounds for the Super-linear Case from Busy Period Calculations}\\label{subsec:superlinear}\nIn this section, we derive lower bound on the throughput for the super-linear case. The next result computes a bound on the probability that the queue length of the HTQ satisfies a given upper bound over a given time interval, using the probability distribution functions from \\eqref{eq:G-def}.\nIn Propositions~\\ref{prop:nbp-lower-bound} and \\ref{prop:nbn-lower-bound}, for the sake of clarity, we add explicit dependence on $\\lambda$ to this probability distribution function. \n\n\n\n\\begin{proposition}\\label{prop:nbp-lower-bound}\nFor any $m>1$, $M\\in \\mathbb{N}$, $L>0$, $\\lambda >0$, $\\varphi\\in \\Phi$, $\\psi \\in \\Psi$, and zero initial condition $x_0=0$, the probability that the queue length is upper bounded by $M$ over a given time interval $[0,T]$ satisfies the following bound: \n\\begin{equation}\n\\label{eq:nbp-lower-bound}\n \\Pr \\big(N(t) \\leq M \\quad \\forall t \\in [0,T] \\big)\\geq \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{r,L^m M^{1-m}}(t,n,\\psi,\\lambda) \\, \\mathrm{d} t\n \\end{equation}\n \\end{proposition}\n \\begin{proof}\nLet us denote the current queueing system as HTQ-f. We shall compare queue lengths between HTQ-f and a slower queueing system HTQ-s, which starts from the same (zero) initial condition, and experiences the same realizations of arrival times, locations and travel distances. Let every incoming vehicle into HTQ-s and HTQ-f be tagged with a unique identifier. At time $t$, let $\\mathcal J(t)$ be the set of identifiers of vehicles present both in HTQ-s and HTQ-f, $\\mathcal J_{s\/f}(t)$ be the set of identifiers of vehicles present only in HTQ-s, and $\\mathcal J_{f\/s}(t)$ be the set of identifiers of vehicles present only in HTQ-f.\nLet $v^f_i$ denote the speed of the vehicle in HTQ-f with identifier $i \\in \\mathcal J(t) \\cup \\mathcal J_{f\/s}(t)$, as determined by the car-following behavior underlying \\eqref{eq:dynamics-moving-coordinates}. The vehicle speeds in HTQ-s are not governed by the car following behavior, but are rather related to the speeds of vehicles in HTQ-f as: \n \\begin{equation}\n \\label{eq:htq-s-speed}\n\t\tv^s_i(t) =\\left\\{\\begin{array}{ll}\n\t\t\\displaystyle v^f_i(t) \\frac{p}{v^f(t)} \\frac{|\\mathcal J(t)|}{|\\mathcal J(t)| + |\\mathcal J_{s\/f}(t)|} \t&i \\in \\mathcal J(t)\\\\[15pt]\n\t\t\\displaystyle \\frac{p}{|\\mathcal J(t)| + |\\mathcal J_{s\/f}(t)|}& i \\in \\mathcal J_{s\/f}(t)\\,\\end{array}\\right.\n\t\\end{equation}\nwhere $v^f(t):=\\sum_{i \\in \\mathcal J(t)}v_i^f(t)$ is the sum of speeds of vehicles in HTQ-f that are also present in HTQ-s at time $t$, and $p$ is a parameter to be specified. Indeed, note that $\\sum_{i \\in \\mathcal J(t) \\cup \\mathcal J_{s\/f}(t)} v_i^s(t) \\equiv p$, i.e., $p$ is the (constant) service rate of HTQ-s. \n\nConsider a realization where the number of arrivals into HTQ-s with $p=L^m M^{1-m}$ during any busy period overlapping with $[0,T]$ does not exceed $M$. We refer to such a realization as \\emph{event} in the rest of the proof. Since the maximum queue length during a busy period is trivially upper bounded by the number of arrivals during that busy period, conditioned on the event, we have \n\\begin{equation}\n\\label{eq:slow-queue-queue-length-bound}\nN_s(t) \\leq M, \\qquad t \\in [0,T]\n\\end{equation}\n\nConsider the union of departure epochs from HTQ-s and HTQ-f in $[0,T]$: $0=\\tau_0 \\leq \\tau_1 \\leq \\ldots$. If $\\mathcal J_{f\/s}(\\tau_k)=\\emptyset$ for some $k \\geq 0$, then $\\mathcal J_{f\/s}(t)=\\emptyset$ for all $t \\in (\\tau_k,\\tau_{k+1})$. Hence, the service rate for HTQ-f over the interval $(\\tau_k,\\tau_{k+1})$ is $v^f(t)$, which, conditioned on the event, is lower bounded by $L^m M^{1-m}=p$ by Lemma \\ref{lem:service-rate-bounds}.\nTherefore, $p\/v^f(t) \\leq 1$ over $(\\tau_k,\\tau_{k+1})$, and hence \\eqref{eq:htq-s-speed} implies that all the vehicles with identifiers in $\\mathcal J_f$ will travel slower in HTQ-s in comparison to HTQ-f. In particular, this implies that $\\mathcal J_{f\/s}(\\tau_{k+1})=\\emptyset$. This, combined with the fact that $\\mathcal J_{f\/s}(\\tau_0)=\\emptyset$ (both the queues start from the same initial condition), we get that, conditioned on the event, $\\mathcal J_{s\/f}(t) \\equiv \\emptyset$, and hence $N(t) \\leq N_s(t)$ over $[0,T]$. Combining this with \\eqref{eq:slow-queue-queue-length-bound} gives that, conditioned on the event, $N(t) \\leq M$ over $[0,T]$. \n\n\nWe now compute the probability of the occurrence of the event using busy period calculations from Section~\\ref{sec:busy-period}. The event can be categorized by the maximum number of busy periods, say $r \\in \\mathbb{N}$, that overlap with $[0,T]$, i.e., the $r$-th busy period ends after time $T$ (and each of these busy periods has at most $M$ arrivals). Since these busy periods are interlaced with idle periods, the probability of the $r$-th busy period ending after time $T$ is lower bounded by the probability that the sum of the durations of $r$ busy periods is at least $T$. \\eqref{eq:G-def} implies that the latter quantity is equal to $\\sum_{n=1}^M \\int_T^{\\infty} G_{r,L^m M^{1-m}}(t,n,\\psi,\\lambda) \\, \\mathrm{d} t$. The proposition then follows by noting that this is true for any $r \\in \\mathbb{N}$.\n\\qed\n \\end{proof}\n \n \\begin{remark}\nIn the proof of Proposition~\\ref{prop:nbp-lower-bound}, when deriving probabilistic upper bound on the queue length over a given time horizon $[0,T]$, we neglected the idle periods in $[0,T]$. This introduces conservatism in the bound on the right hand side of \\eqref{eq:nbp-lower-bound}. Since the idle period durations are distributed independently and identically according to an exponential random variable (since the arrival process is Poisson), one could incorporate them into \\eqref{eq:nbp-lower-bound} by taking convolution of $G$ with idle period distributions. Our choice for not doing so here is to ensure conciseness in the presentation of bounds in \\eqref{eq:nbp-lower-bound}. The resulting conservatism is also present in Proposition~\\ref{prop:nbn-lower-bound}, and carries over to Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, as well as to the corresponding simulations reported in Figures~\\ref{fig:lambda-m-empty}, \\ref{fig:superlinear-large-L} and \\ref{fig:phase-transition-non-empty}. \n\\end{remark}\n \n The next result generalizes Proposition~\\ref{prop:nbp-lower-bound} for non-zero initial condition. Note that the non-zero initial condition only affects the first busy period; all subsequent busy periods will necessarily start from with zero initial condition.\n\n \\begin{proposition}\\label{prop:nbn-lower-bound}\n For any $m>1$, $M\\in\\mathbb{N}$, $L>0$, $\\lambda >0$, $\\varphi\\in \\Phi$, $\\psi \\in \\Psi$, initial condition $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, with associated workload $w_0>0$, the probability that the queue length is upper bounded by $M+n_0$ over a given time interval $[0,T]$ satisfies the following: \n\n $$\\Pr \\big(N(t) \\leq M + n_0 \\quad \\forall t \\in [0,T] \\big)\\geq \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{L^m (M+n_0)^{1-m}}(\\delta_{w_0}) * G_{r-1,L^m M^{1-m}}(\\psi) (t,n,\\lambda) \\, \\mathrm{d} t$$\n \\end{proposition}\n \\begin{proof}\nThe proof is similar to the proof of Proposition \\ref{prop:nbp-lower-bound}; however, since we consider $M$ number of new arrivals in each of the busy periods, the \\emph{event} of interest is when the queue length in HTQ-s does not exceed $M+n_0$ and $M$ in the first and subsequent busy periods, respectively, while operating with constant service rates $L^m (M+n_0)^{1-m}$ and $L^m M^{1-m}$, respectively. \n\\qed\n \\end{proof}\n\n\n\n \n\n\n\n \n\n\n\n \n\n We shall use Propositions~\\ref{prop:nbp-lower-bound} and \\ref{prop:nbn-lower-bound} to establish probabilistic lower bound for a finite time horizon version of the throughput defined in Definition \\ref{def:throughput}: for $T>0$, let \n \\begin{equation*}\n\\label{eq:throughput-def-finite-horizon}\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta,T):= \\sup \\left\\{\\lambda \\geq 0: \\, \\Pr \\left( N(t;L, m, \\lambda, \\varphi, \\psi, x_0) < + \\infty, \\quad \\forall t \\in [0,T] \\right) \\geq 1 - \\delta \\right\\}.\n\\end{equation*}\n \n \n\n\\begin{theorem}\\label{thm:superlinear-bound-empty}\n For $L>0$, $m>1$, $\\varphi\\in\\Phi$, $\\psi\\in\\Psi$, $\\delta\\in(0,1)$, $T>0$, zero initial condition $x_0=0$, \n \\begin{equation}\n \\label{eq:sublinear-throughput-zero-initial-condition}\n \\lambda_{\\max}(L, m, \\varphi, \\psi, x_0,\\delta,T)\\geq \\sup_{M \\in \\mathbb{N}}\\;\\sup \\Big\\{\\lambda \\geq 0 \\; \\Big | \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{r,L^m M^{1-m}}(t,n,\\psi,\\lambda) \\, \\mathrm{d} t \\geq1-\\delta \\Big\\} \n \\end{equation}\n \\end{theorem}\n \\begin{proof}\nFollows from Proposition \\ref{prop:nbp-lower-bound}.\n\\qed\n \\end{proof} \n \n \n \n \\begin{theorem}\\label{thm:superlinear-bound-non-empty}\n For $L>0$, $m>1$, $\\varphi\\in\\Phi$, $\\psi\\in\\Psi$, $\\delta\\in(0,1)$, $T>0$, initial condition $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, with associated workload $w_0>0$, \n \\begin{multline*}\n \\lambda_{\\max}(L, m, \\varphi, \\psi, x_0,\\delta,T) \\\\ \\geq \\sup_{M \\in \\mathbb{N}}\\;\\sup \\Big \\{\\lambda>0 \\; \\Big | \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{L^m (M+n_0)^{1-m}}(\\delta_{w_0}) * G_{r-1,L^m M^{1-m}}(\\psi) (t,n,\\lambda)\\geq 1-\\delta \\Big\\} \n \\end{multline*}\n \\end{theorem}\n \\begin{proof}\n Follows from Proposition \\ref{prop:nbn-lower-bound}.\n \\qed\n \\end{proof}\n \n \\begin{remark}\n In Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, we implicitly assume the rather standard convention that supremum over an empty set is zero.\n \\end{remark}\n\n \n\n \n\n\n\n\n\\section{Throughput Analysis}\n\\label{sec:throughput-analysis}\n\n\\input{linear}\n\n\\input{nonlinear}\n\n\\input{super-linear}\n\n\n\\input{sublinear-new}\n\n\\section{Appendix}\n\\label{sec:appendix}\n\nIn this section, we gather a few technical results that are used in the main results of the paper. \n\n\n\\begin{definition}[Type $K$ function]~\\cite{Smith:08}\nLet $g:S\\mapsto\\mathbb{R}^n$ be a function on $S\\subset\\mathbb{R}^n$. $g$ is said to be of type $K$ in $S$ if, for each $i \\in \\until{n}$, $g_i(z_1)\\leq g_i(z_2)$ holds true for any two points $z_1$ and $z_2$ in $S$ satisfying $z_1 \\leq z_2$ and $z_{1,i}=z_{2,i}$.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lem:monotonicity-same-size}\nLet $\\map{g}{S}{\\mathbb{R}^N}$ and $\\map{h}{S}{\\mathbb{R}^N}$ be both of type $K$ over $S \\subset \\mathbb{R}^N$. Let $z_1(t)$ and $z_2(t)$\nbe the solutions to $\\dot{z}=g(z)$ and $\\dot{z}=h(z)$, respectively, starting from initial conditions $z_1(0)$ and $z_2(0)$ respectively. \nLet $S$ be positively invariant under $\\dot{z}=g(z)$ and $\\dot{z}=h(z)$.\nIf $g(z) \\leq h(z)$ for all $z \\in S$, and $z_1(0) \\leq z_2(0)$, then $z_1(t) \\leq z_2(t)$ for all $t \\geq 0$. \n\\end{lemma}\n\\begin{proof}\nBy contradiction, let $\\tilde{t} \\geq 0$ be the smallest time at which, there exists, say $k \\in \\until{N}$, such that $z_1(\\tilde{t}) \\leq z_2(\\tilde{t})$, $z_{1,k}(\\tilde{t}) = z_{2,k}(\\tilde{t})$, and \n\\begin{equation}\n\\label{eq:class-K-contradiction}\ng_k(z_1(\\tilde{t})) > h_k(z_2(\\tilde{t})).\n\\end{equation}\nSince $g(z)$ is of class $K$, $z_1(\\tilde{t}) \\leq z_2(\\tilde{t})$ and $z_{1,k}(\\tilde{t}) = z_{2,k}(\\tilde{t})$ imply that $g(z_1(\\tilde{t})) \\leq g(z_2(\\tilde{t}))$. This, combined with the assumption that $g(z) \\leq h(z)$ for all $z \\in S$ implies that $g(z_1(\\tilde{t})) \\leq h(z_2(\\tilde{t}))$, which contradicts \\eqref{eq:class-K-contradiction}.\n\\qed\n\\end{proof}\n\nLemma~\\ref{lem:monotonicity-same-size} is relevant because the basic dynamical system in our case is of type $K$. \n\n\\begin{lemma}\n\\label{lem:car-following-type-K}\nFor any $L > 0$, $m>0$, and $N \\in \\mathbb{N}$, the right hand side of \\eqref{eq:x-dynamics-in-Rn} is of type $K$ in $\\mathbb{R}_+^N$.\n\\end{lemma}\n\\begin{proof}\nConsider $\\tilde{x}, \\hat{x} \\in \\mathbb{R}_+^N$ such that $\\tilde{x} \\leq \\hat{x}$. \nIf $\\tilde{x}_i = \\hat{x}_i$ for some $i \\in \\until{N}$, then, according to \\eqref{eq:inter-vehicle-distance-Rn}, $y_i(\\tilde{x})-y_i(\\hat{x})= \\left(\\tilde{x}_{i+1} - \\hat{x}_{i+1} \\right) -\\left(\\tilde{x}_i-\\hat{x}_i \\right) = \\tilde{x}_{i+1} - \\hat{x}_{i+1}$ if $i \\in \\until{N-1}$, and is equal to $\\left(\\tilde{x}_{1} - \\hat{x}_{1} \\right) - \\left(\\tilde{x}_N-\\hat{x}_N \\right)= \\tilde{x}_{1} - \\hat{x}_{1}$ if $i=N$. In either case, $y_i(\\tilde{x}) \\leq y_i(\\hat{x})$, which also implies $y_i^m(\\tilde{x}) \\leq y_i^m(\\hat{x})$ for all $m > 0$.\n\\qed\n\\end{proof}\n\n\n\n\n\nIn order to state the next lemma, we need a couple of additional definitions. \n\n\\begin{definition}[Monotone Aligned and Monotone Opposite Functions]\nTwo strictly monotone functions $\\map{h}{\\mathbb{R}}{\\mathbb{R}}$ and $\\map{g}{\\mathbb{R}}{\\mathbb{R}}$ are said to be \\emph{monotone-aligned} if they are both either strictly increasing, or strictly decreasing. Similarly, the two functions are called \\emph{monotone opposite} if one of them is strictly increasing, and the other is strictly decreasing. \n\\end{definition}\n\n\n\\begin{lemma}\n\\label{lem:appendix-general-summation}\nLet $\\map{h}{\\mathbb{R}_+}{\\mathbb{R}}$ and $\\map{g}{\\mathbb{R}_+}{\\mathbb{R}}$ be strictly monotone functions. Then, for every $y \\in \\mathcal S_N^L$, $n \\in \\mathbb{N}$, $L>0$,\n\\begin{equation}\n\\label{eq:summation-general}\n\\sum_{i=1}^N h(y_i) \\left(g(y_{i+1}) - g(y_i) \\right)\n\\end{equation}\nis non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned. Moreover, \\eqref{eq:summation-general} is equal to zero if and only if $y=\\frac{L}{N} \\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\nFor $i \\in \\until{N}$, let $I_i$ be the interval with end points $g(y_i)$ and $g(y_{i+1})$. \nFor $i \\in \\until{N}$, let $f_i(z):=\\sgn{g(y_{i+1}) - g(y_i)} h(y_i) \\mathbf{1}_{I_i}(z)$. \nLet $g_{\\text{min}}:=\\min_{i \\in \\until{N}} g(y_i)$, and $g_{\\text{max}}:=\\max_{i \\in \\until{N}} g(y_i)$. With $f(z):=\\sum_{i=1}^N f_i(z)$,\n \\eqref{eq:summation-general} can then be written as:\n\\begin{equation}\n\\label{eq:line-integral}\n\\sum_{i=1}^N h(y_i) \\left(g(y_{i+1}) - g(y_i) \\right) = \\int_{g_{\\text{min}}}^{g_{\\text{max}}} f(z) \\, dz.\n\\end{equation}\nWe now show that, for every $z \\in [g_{\\text{min}}, g_{\\text{max}}] \\setminus \\{g(y_i): i \\in \\until{N}\\}$, $f(z)$ is non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned. \nThis, together with \\eqref{eq:line-integral}, will then prove the lemma.\n\nIt is easy to see that every $z \\in [g_{\\text{min}}, g_{\\text{max}}] \\setminus \\{g(y_i): i \\in \\until{N}\\}$ belongs to an even number of intervals in $\\{I_i: \\, i \\in \\until{N}\\}$, say $I_{\\ell_1}, I_{\\ell_2}, \\ldots$, with $\\ell_1 < \\ell_2 < \\ldots$ (see Figure \\ref{fig:intervals} for an illustration).\nWe now show that $f_{\\ell_1}(z) + f_{\\ell_2}(z)$ is non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned. The same argument holds true for $f_{\\ell_3}(z)+f_{\\ell_4}(z), \\ldots$. \nAssume that $g(y_{\\ell_1}) \\leq g(y_{\\ell_2})$; the other case leads to the same conclusion. By definition of $f_i$'s, $f_{\\ell_1}(z)=h(y_{\\ell_1})$ and $f_{\\ell_2}(z)=-h(y_{\\ell_2})$. $g(y_{\\ell_1}) \\leq g(y_{\\ell_2})$ implies that \n$f_{\\ell_1}(z)+f_{\\ell_2}(z)=h(y_{\\ell_1})-h(y_{\\ell_2})$ is non-negative if $h$ and $g$ are monotone-opposite, and is non-positive if $h$ and $g$ are monotone-aligned, with the equality holding true if and only if $y_{\\ell_1}=y_{\\ell_2}$. \n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=15.5cm]{intervals} \n \\caption{A schematic view of (a) $f_i(z), i=\\{1,2,3,4\\}$ and (b) $f(z)=\\sum_{i=1}^4f_i(z)$ for a $y\\in\\mathcal S_4^L$ ($L=1$) with $y_{\\text{min}} = y_2 < y_4 < y_3 < y_1 = y_{\\mathrm{max}}$ for a $m<1$. }\n \\label{fig:intervals}\n\\end{figure}\n\\qed\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lemma:t-n}\nFor $n \\in \\mathbb{N} \\setminus \\{1\\} $, let $\\psi_n$ be the n-fold convolution of $\\psi \\in \\Psi$. Then, \n$$\\int_{0}^t z \\, \\psi(z)\\psi_{n-1}(t-z)\\mathrm{d} z = \\frac{t}{n}\\psi_n(t) \\qquad \\forall \\, t \\geq 0$$\n\\end{lemma}\n\\begin{proof}\nLet $J_1,\\cdots,J_n$ be $n$ random variables, all with distribution $\\psi$. Therefore, the probability distribution function of the random variable $V:=\\sum_{i=1}^n J_i$ is $\\psi_n$. \nUsing linearity of the expectation, we get that \n\\begin{equation*}\nt=E\\left[\\sum_{i=1}^n J_i|V=t\\right]=\\sum_{i=1}^n E\\left[J_i|V=t\\right]=n \\, E\\left[J_1|V=t \\right]\n\\end{equation*}\ni.e., \n\\begin{equation}\n\\label{eq:D1-cond-sum}\nE\\left[J_1|V=t \\right] = \\frac{t}{n}\n\\end{equation}\nLet $f_{J_1|V}(j_1|t)$ denote the probability distribution function of $J_1|V$. By definition:\n\\begin{equation}\\label{eq:f-d1-z}\nf_{J_1|V}(j_1|t) = \\frac{f_{J_1,V}(j_1,t)}{\\psi_n(t)}=\\frac{\\psi(j_1)\\psi_{n-1}(t-j_1)}{\\psi_n(t)}\n\\end{equation}\nTherefore, using \\eqref{eq:D1-cond-sum} and \\eqref{eq:f-d1-z}, we get that\n\\begin{equation*}\nE[J_1|V=t] = \\int_0^t zf_{J_1|V}(z|t)\\, \\mathrm{d} z = \\int_0^t z\\frac{\\psi(z)\\psi_{n-1}(t-z)}{\\psi_n(t)} \\, \\mathrm{d} z = \\frac{t}{n}\n\\end{equation*}\nSimple rearrangement gives the lemma. \n\\qed\n\\end{proof}\n\n\nThe following is an adaptation of \\cite[Lemma 2.3.4]{Ross:96}. \n\\begin{lemma}\\label{lemma:busy-period-type1-prob}\nLet $a_1, \\cdots, a_{n-1}$ denote the ordered values from a set of $n-1$ independent uniform $(0,t)$ random variables. Let $\\tilde d_0=z \\geq 0$ be a constant and $\\tilde d_1, \\tilde d_2, \\cdots \\tilde d_{n-1}$ be i.i.d. non-negative random variables that are also independent of $\\{a_1, \\cdots, a_{n-1}\\}$, then \n\\begin{align*}\n\\Pr(\\tilde d_{k}+ \\cdots +\\tilde d_{n-1}\\leq a_{n-k},k=1,\\cdots,n-1 |\\tilde d_0+ \\cdots +\\tilde d_{n-1}=t, \\tilde{d}_0=z)\n = \\begin{cases} z\/t & z 0$, $m=1$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, the mean value of the busy period duration is equal to $\\bar \\psi \/ (L - \\lambda \\bar \\psi)$.\n \\end{lemma}\n \\begin{proof}\nA busy period, say of duration $B$, is initiated by the arrival of a vehicle, say $j$, when the system is idle. \nLet the number of vehicles that arrive during the busy period be $N_{bn}$. Note that $N_{bn}$ does not include the vehicle initiating the busy period. Therefore, the workload brought into the system during the busy period is equal to $w_B=\\sum_{i=j}^{j+N_{bn}} d_i$.\nThe expected value of $N_{bn}$ can be obtained by conditioning on the duration of the busy period: \n\\begin{equation}\\label{eq:nb-expectation}\nE[N_{bn}]=E\\left[E[N_{bn}|B]\\right]=E[\\lambda B]=\\lambda E[B]\n\\end{equation}\nwhere the second equality follows from the fact that the arrival process is a Poisson process. \nSince the event $\\{N_{bn}+1=n\\}$ is independent of $\\{d_{j+i},i > n\\}$, $N_{bn}+1$ is a stopping time for the sequence $\\{d_{j+i},i\\geq 1\\}$. Therefore, using Wald's equation, e.g., see \\cite[Theorem 3.3.2]{Ross:96}, and \\eqref{eq:nb-expectation}, the expected value of the workload $w_B$ added to the system during the busy period $B$ is given by: \n\\begin{equation}\\label{eq:wb-expectation}\nE[w_B]=(E[N_{bn}]+1) \\, \\bar \\psi=(\\lambda E[B]+1) \\, \\bar \\psi.\n\\end{equation}\n\n\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=5.5cm]{busy-period-new-notation} \n \\caption{(a) Queue length process and (b) workload process during a busy period. }\n \\label{fig:busy-period}\n\\end{figure}\n\nSince the workload decreases at a constant rate $L$ during a busy period, we have $B=w_B\/L$ (see Figure \\ref{fig:busy-period} for an illustration). Therefore, $E[B]=E[w_B]\/L$, which when combined with \\eqref{eq:wb-expectation}, establishes the lemma. \n\\qed\n\\end{proof}\n\n\n\n\n\\begin{remark}\n\\label{remark:waiting-time}\nSince the mean busy period duration is an upper bound on the mean waiting time, Lemma \\ref{lemma:busy-period-mean} also gives an upper bound on the mean waiting time. One can then use Little's law~\\cite{Kleinrock:75}\\footnote{Little's law has previously been used in the context of processor sharing queues, e.g., in \\cite{Altman.ea:06}.} to show that the mean queue length is upper bounded by $\\lambda\\bar \\psi \/ (L - \\lambda \\bar \\psi)$.\n\\end{remark}\n\nLet $\\mathcal{I}(t):=\\int_0^t\\delta_{\\{w(s)=0\\}}\\mathrm{d} s$ be the cumulative \\emph{idle time} up to time $t$. The following result characterizes the long run proportion of the idle time in the linear case.\n\\begin{proposition}\n\\label{prop:long-run-idle-time}\nFor any $\\lambda 0$, $\\varphi \\in \\Phi, \\psi \\in \\Psi$, the long-run proportion of time in which HTQ is idle is given by the following:\n$$\\lim_{t\\to\\infty} \\frac{\\mathcal{I}(t)}{t}=1-\\frac{\\lambda \\bar \\psi}{L}>0 \\quad a.s.$$\n\\end{proposition}\n\n\\begin{proof}\nHTQ alternates between busy and idle periods. Let $Z=I+B$ be the duration of a cycle that contains an idle period of length $I$ followed by a busy period of length $B$. Idle period, $I$, has the same distribution as inter-arrival times i.e. an exponential random variable with mean $1\/\\lambda$, and the mean value of $B$ is given in Lemma \\ref{lemma:busy-period-mean}. Note that duration of cycles, $Z$, are i.i.d. random variables. Thus, the busy-idle profile of the system is an alternating renewal process where renewals correspond to the moments at which the system gets idle. Suppose the system earns reward at a rate of one per unit of time when it is idle (and thus the reward for a cycle equals the idle time of that cycle i.e. $I$). Then, the total reward earned up to time $t$ is equal to the total idle time in $[0,t]$ (or $\\mathcal{I}(t)$), and by the result for renewal reward process (see \\cite{Ross:96}, Theorem 3.6.1), with probability one,\n$\\lim_{t\\to\\infty} \\mathcal{I}(t)\/t=E[I]\/(E[B]+E[I])$. \n\\qed\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Busy Period Distribution}\n\\label{sec:busy-period}\nIn this section, we compute the cumulative distribution function for the number of new arrivals during a busy period for a HTQ with constant service rate, say $p>0$. This could, e.g., correspond to \\eqref{eq:inter-vehicle-distance-dynamics} for $m=1$. However, our analysis in this section, is not restricted to this specific model, but applies to any HTQ with constant service rate $p$. \nThis cumulative distribution for the number of new arrivals during a busy period, while of independent interest, will be used to derive lower bounds on the throughput in the super-linear case in Section~\\ref{subsec:superlinear}. Our analysis is inspired by that of M\/G\/1 queue, e.g., see \\cite{Ross:96}, where our consideration for non-zero initial condition appears to be novel. \n\nLet us consider an arbitrary busy period spanning time interval $(0,t)$, without loss of generality. For non-zero initial condition, one has to distinguish between the first and subsequent busy periods. Let the workload at the beginning of the arbitrary busy period, denoted as $d_0$, be sampled from $\\theta$. The relationship between $\\theta$ and $\\psi$ is as follows. \nIf the system starts with a non-zero initial initial condition with initial workload $w_0>0$, then the value of the $d_0$ for the first busy period will be deterministic and equals $w_0$, and hence $\\theta=\\delta_{w_0}$. However, for subsequent busy periods, or if the initial condition is zero, $d_0$ is sampled from $\\theta=\\psi$.\nThe workload brought to the system by arriving vehicles, $\\{d_i\\}_{i=1}^\\infty$, equals to the distance that vehicles wish to travel and are sampled identically and independently from the distribution $\\psi$. When the system is busy, the workload decreases at a given constant rate $p > 0$. The busy period ends when the workload becomes zero. \n\\begin{remark}\nWe emphasize that $d_0$ denotes the workload at the beginning of a busy period (see Figure \\ref{fig:normalized-distance} for further illustration), and hence is not equal to zero when the queue starts from a zero initial condition. \n\\end{remark}\n\n \n\nIn order to align our calculations with the standard M\/G\/1 framework, where service rate is assumed to be unity, we consider normalized workloads, $\\tilde{d}_i:=d_i\/p$ for all $i \\in \\{0,1,\\cdots\\}$ (see Figure \\ref{fig:normalized-distance} for an illustration). Correspondingly, let the distributions for the normalized distances be denoted as $\\tilde{\\theta}$ and $\\tilde{\\psi}$. Let the arrival time of the $k$-th new vehicle during $(0,t)$ be denoted as $T_k$, and let $N_{bn}$ denote the number of arrivals in $(0,t)$, i.e., the total number of arrivals over the entire duration of the busy period, including the vehicle which initiates the busy period, is $N_{bn} + 1$. \n\n A busy period ends at time $t$, and $N_{bn}=n-1$ if and only if, \n\\begin{enumerate}[(i)]\n\\item $T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, \\qquad k=1,\\cdots,n-1$ \n\\item $\\tilde d_0+\\cdots+ \\tilde d_{n-1}=t$\n\\item There are exactly $n-1$ arrivals in $(0,t)$\n\\end{enumerate} \n\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{normalized-distance} \n\\caption{Evolution of workload during first two busy periods for an HTQ with constant service rate $p$, and starting from a non-zero initial condition. In the first busy period, $d_0$ is equal to the workload $w_0$ associated with the non-zero initial condition. In the second busy period, $d_0$ is equal to the workload brought by the first vehicle that initiates that busy period.}\n\\label{fig:normalized-distance}\n\\end{center}\n\\end{figure}\n\n\n\nBy treating densities as if they are probabilities, we get:\n\\begin{align}\n\\label{eq:prop-busy-des}\n\\Pr & (B =t \\text{ and }N_{bn} = n -1)\\nonumber \\\\\n=& \\Pr(\\tilde d_0+\\cdots+ \\tilde d_{n-1}=t, n-1 \\text{ arrivals in $(0,t)$}, T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, k=1,\\cdots,n-1) \\nonumber \\\\\n=& \\int_0^t\\Pr(T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, k=1,\\cdots,n-1| n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_0+\\cdots+ \\tilde d_{n-1}=t, \\tilde d_0 =z) \\nonumber \\\\\n& \\times \\Pr(n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_1+\\cdots+ \\tilde d_{n-1}=t-z)\\, \\tilde \\theta(z) \\, \\mathrm{d} z \n\\end{align}\nwhere we recall that $B$ is the random variable corresponding to the busy period duration. By the independence of normalized distances and the arrival process, the second probability term in the integrand in \\eqref{eq:prop-busy-des} can be expressed as\n\\begin{align}\n\\label{eq:prop-busy-part2}\n\\Pr(n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_1+\\cdots+ \\tilde d_{n-1}=t-z)=e^{-\\lambda t}\\frac{(\\lambda t)^{n-1}}{(n-1)!} \\tilde \\psi_{n-1}(t- z) \n\\end{align}\nwhere $\\tilde\\psi_n$ is the $n$-fold convolution of $\\tilde\\psi$ with itself. \n\nIn the first probability term in \\eqref{eq:prop-busy-des}, it is given that the system receives $n-1$ arrivals in $(0,t)$ and since the arrival process is a Poisson process, the ordered arrival times, $\\{T_1,T_2,\\cdots,T_{n-1}\\}$, are distributed as the ordered values of a set of $n-1$ independent uniform $(0,t)$ random variables $\\{a_1,a_2,\\cdots,a_{n-1}\\}$ (see Theorem 2.3.1 in \\cite{Ross:96}). Thus,\n\\begin{align}\\label{eq:busy-period-given-n-prob}\n\\Pr & (T_k\\leq \\tilde d_0+\\cdots+ \\tilde d_{k-1}, k=1,\\cdots,n-1| n-1 \\text{ arrivals in $(0,t)$}, \\tilde d_0+\\cdots+ \\tilde d_{n-1}=t,\\tilde d_0 =z) \\nonumber \\\\\n& = \\Pr(a_k\\leq \\tilde d_0+\\cdots+\\tilde d_{k-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z)\n\\end{align}\n\n\n\n\n\n\n\n\nBy noting that $t-U$ will also be a uniform $(0,t)$ random variable whenever $U$ is, it follows that $a_1,\\cdots,a_{n-1}$ has the same joint distribution as $t-a_{n-1},\\cdots,t-a_1$. Thus, replacing $a_k$ with $a_{n-k}$ for $k\\in\\{1,\\cdots,n-1\\}$ in \\eqref{eq:busy-period-given-n-prob}, we get\n\\begin{align}\\label{eq:prob-manipulation}\n \\Pr&(a_k \\leq \\tilde d_0+\\cdots+\\tilde d_{k-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) \\nonumber \\\\\n & =\\Pr(t-a_{n-k} \\leq \\tilde d_0+\\cdots+\\tilde d_{k-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_{0} =z) \\nonumber \\\\\n & = \\Pr(t-a_{n-k} \\leq t-(\\tilde d_{k}+\\cdots+\\tilde d_{n-1}), k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) \\nonumber \\\\\n & = \\Pr(a_{n-k} \\geq \\tilde d_{k}+\\cdots+\\tilde d_{n-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) \\nonumber \\\\\n & = \\Pr(a_{n-k} \\geq \\tilde d_{k}+\\cdots+\\tilde d_{n-1}, k=1,\\cdots,n-1|\\tilde d_0+\\cdots+\\tilde d_{n-1} =t,\\tilde d_0 =z) = \\begin{cases} z\/t & z1$ case}\nIn this section, we assume that system starts from an empty initial condition and then we derive a probabilistic lower bound on the throughput of the system. For our analysis we focus on busy period and the number of vehicles served in a busy period. \n\nWe first assume an upper bound on the queue length, i.e. $N(t)\\leq M \\; \\forall t\\geq0$, which by Lemma \\ref{lem:service-rate-bounds} gives a lower bound on the service rate of the system i.e. $s(y(t))\\geq L^mM^{1-m}\\; \\forall t\\geq 0$. Then, we use this lower bound on the service rate to compare the system with a system with the same temporal-spatial arrival process but a slower decrease rate of workload . We use this comparison as a tool to find a lower bound on the probability of achieving the assumed upper bound on the queue length. This result leads to a probabilistic lower bound on the throughput of the system which is stated in Theorem \\ref{}. Our analysis heavily depends on the probability distribution of busy periods which is described in the next section.\n\n\n \n \\subsubsection{From busy period to throughput}\n\n\n \n \\begin{proposition}\\label{prop:nbp-lower-bound}\n For any $m>1$, $w_0=0$, $L>0$, $\\lambda > 0$ $\\varphi \\in \\Psi$ and $M\\in\\{2,3,\\cdots\\}$, given $N(t)0$, then \n $$\\Pr(N_{bp}\\leq M)\\geq C^{(2)}_{L^m M^{1-m}}(M,\\lambda)\\qquad \\forall n\\in\\mathbb{N}$$\n \\end{proposition}\n \\begin{proof}\n \\mmmargin{service is not greater than this lower bound for all t because of idle time. revise it}\n For $m>1$, by Lemma \\ref{lem:service-rate-bounds}, and the upper bound given on the queue length,\n \\begin{equation}\n s(t)\\geq L^mM^{1-m} = p \\quad \\forall t>0\n \\end{equation} \n \n We consider another system with the same realization of temporal-spatial arrival process but fixed service rate of $p$ and name it as the \\emph{slower} system. Let $B_1$ and $B_2$ be random variables that denote the length of a busy period in actual and slower system, respectively. Since the realization of arrival processes are the same for both systems, and the decrease rate of workload in slower system is smaller it is clear that,\n \\begin{equation}\\label{eq:workload-actual-slower}\n w_1(t)\\leq w_2(t) \\quad \\forall t>0\n \\end{equation}\nwhere $w_1(t)$ and $w_2(t)$ denote the the workload processes in actual and slower system, respectively. \n\nNow, let $t_s$ be the time at which a (random) busy period in the slower system ends and $t_1$ be the time at which the next busy period in that system ends (see Figure \\ref{fig:b1-b2} for an illustration). Obviously, $w_1(t_s)=w_1(t_f)=0$. Also, by \\eqref{eq:workload-actual-slower}, $w_2(t_s)=w_2(t_f)=0$. Let $t_1\\in(t_s,t_f)$ be the time of the first arrival after $t=t_s$. Note that this time is the same in both systems. Therefore, in the slower system, the busy period that lies in $[t_s,t_f]$ has the length of $t_f-t_1$. Let $t_2\\in[t_1,t_f]$ be the time at which the actual system gets idle for the first time after $t=t_1$. We want to show that $t_2\\leq t_f$. By contradiction, assume $t_2> t_f$. Thus, $w_1(t_f)>0$, but $w_2(t_f)=0$. This contradicts \\eqref{eq:workload-actual-slower} and proves that $t_2\\leq t_f$. This implies that the longest busy period in actual system contained in $[t_s,t_f]$ is no longer than the busy period in actual system. This argument holds for any busy period in slower system. Thus, with probability one, \n \\begin{equation}\\label{eq:relation-b1-b2}\n B_1\\leq B_2\n \\end{equation} \n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=9cm]{b1-b2} \n\\caption{Workload process for two systems with the same realization temporal-spatial arrival process, a system with slower service rate (solid line) and a system with a faster service rate (dashed line).}\n\\label{fig:b1-b2}\n\\end{center}\n\\end{figure}\n\nNow, let $N_{bp}$ and $\\tilde N_{bp}$ be the number of customers served in a busy period in the actual system and slower system, respectively. Note that $N_{bp}$ and $\\tilde{N}_{bp}$ are Poisson random variables with mean $\\lambda B_1$ and $\\lambda B_2$, respectively and by \\eqref{eq:relation-b1-b2}, $\\lambda B_1\\leq \\lambda B_2$. Poisson random variable is stochastically increasing in its mean (see e.g. \\cite{Ross:96}), therefore\n\\begin{equation}\n\\tilde{N}_{bp}\\geq_{\\text{st}}{N}_{bp}\n\\end{equation} \nwhich establishes this proposition.\n\n\n\n \\end{proof}\n \n\\mmmargin{have we defined this notation ($n_0$) earlier?}\n \\begin{proposition}\\label{prop:nbn-lower-bound}\n For any $m>1$, $w_0>0$, \\mmcomment{$n_0\\in\\mathbb{N}$}, $L>0$, $\\lambda > 0$, $\\varphi \\in \\Psi$ and $M\\in\\{2,3,\\cdots\\}$, given $N(t)0$, then \n $$\\Pr(N_{bn}\\leq M)\\geq C^{(1)}_{L^m (M+n_0)^{1-m}}(M,\\lambda,w_0)\\qquad \\forall n\\in\\mathbb{N}$$\n \\end{proposition}\n \\begin{proof}\n The proof is very similar to the proof of Proposition \\ref{prop:nbp-lower-bound}; however, in this case the upper bound on the queue length is $M+n_0$, and the lower bound on the service rate is $L^m(M+n_0)^{1-m}$. \n \\end{proof}\n\n\n\n \n\n\n\n \n \\begin{lemma} \\label{lemma:st-order-queulength-nbp}\n For any $m>1$, $L>0$, $\\lambda>0$, $\\varphi \\in \\Psi$, $w_0=0$\n $$N_{bp}\\geq_{\\text{st}}N(t)\\quad \\forall t\\geq 0$$\n \\end{lemma}\n \\begin{proof}\n Conditioned on the length of busy period, $B$, \n \\begin{equation}\n N_{bp}\\leq M | B = b \\implies N(t)\\leq M | B = b \\quad \\forall t\\in[0,b]\n \\end{equation}\n where here $t=0$ denotes the time at which busy period starts. Then,\n \\begin{equation}\n \\Pr(N_{bp}\\leq M | B = b) \\leq \\Pr(N(t)\\leq M | B = b) \\quad \\forall t\\in[0,b]\n \\end{equation}\n and by taking the expectation of the above with respect to length o busy period, we get \n \\begin{equation}\n \\Pr(N_{bp}\\leq M) \\leq \\Pr(N(t)\\leq M) \\quad \\forall t>0\n \\end{equation}\n which establishes this lemma. \n \\end{proof} \n\n\n Now, we are ready to state the main result in the following theorem which gives a probabilistic lower bound on the throughput of the system for $m>1$.\n \\begin{theorem}\n Given $\\delta\\in(0,1)$, $L>0$, $m>1$, $\\varphi\\in\\Psi$, $w_0=0$, then\n $$\\lambda_{\\max}(m, L, \\varphi,\\delta)\\geq \\max_{M\\geq 2}\\;\\max\\{\\lambda>0 \\;| C^{(2)}_{L^m M^{1-m}}(M,\\lambda)\\geq1-\\delta\\}$$\n \\end{theorem}\n \\begin{proof}\nFor a given upper bound on the queue length, $M\\in\\{2,3\\cdots\\}$, \n \\begin{equation}\\label{eq:prob-bound-queue}\n \\Pr(N\\leq M) \\geq \\Pr(N_{bp}\\leq M) \\geq C^{(2)}_{L^m M^{1-m}}(M,\\lambda)\n \\end{equation}\n where the first inequality follows by Lemma \\ref{lemma:st-order-queulength-nbp} and the second inequality follows by Proposition \\ref{prop:nbp-lower-bound}. Therefore, based on Definition \\ref{def:throughput}, $\\tilde \\lambda = \\max\\{\\lambda>0 \\;| C_{L^m M^{1-m}}(M,\\lambda)=1-\\delta\\}$ gives a lower bound on $\\lambda_{\\max}$. In Definition \\ref{def:throughput}, however, only boundedness of queue length matters. Thus, one can maximize $\\tilde \\lambda$ over $M\\geq 2$ to get tighter lower bound. \n \n \n\n\n\n\n\n \\end{proof}\n \n \n \\begin{theorem}\n Given $\\delta_1,\\delta_2\\in(0,1)$, $L>0$, $m>1$, $\\varphi\\in\\Psi$, $w_0>0$, $n_0\\in\\mathbb{N}$ then\n $$\\lambda_{\\max}(m, L, \\varphi,\\delta)\\geq \\max_{M_1,M_2\\geq 2}\\;\\max\\{\\lambda>0 \\;| C^{(1)}_{L^m (M+n_0)^{1-m}}(M,\\lambda,w_0)C^{(2)}_{L^m (M_2)^{1-m}}(M,\\lambda)\\geq(1-\\delta_1)(1-\\delta_2)\\geq (1-\\delta)\\}$$\n \\end{theorem}\n \\begin{proof}\n\\mmcomment{work in progress ...} \n \n \\end{proof}\n \n In order to further clarify the analysis in this section, the following example shows the application of these results in order to probabilistically characterize the throughput of the system for a specific spatial distribution. \n \n \\begin{example}[Deterministic traveled distances] Assume that vehicles, upon arrival, wish to travel a deterministic distance of length $L>0$ i.e. $\\varphi(x)=\\delta_{\\{x=L\\}}$ and their motion is determined by a car following model with $m>1$. Also, further assume that the queue length does not exceed $M\\in\\{2,3,\\cdots\\}$. In this case, the lower bound on the service rate is $p=L^mM^{1-m}$. Application of \\eqref{eq:busy-period-and-nbp-joint} gives,\n \\begin{equation}\\label{eq:busy-period-joint-deterministic}\n D_p(t,N_{bp}=n) = \\int_0^te^{-\\lambda y}\\frac{(\\lambda y)^{n-1}}{n!}\\delta_{\\{y=nL\/p\\}}\\mathrm{d} y=\\begin{cases}0 & t\\leq nL\/p\\\\ e^{-\\lambda nL\/p}\\frac{(\\lambda nL\/p)^{n-1}}{n!}& t> nL\/p\\end{cases}\n \\end{equation}\nand by Lemma \\ref{lemma:st-order-queulength-nbp} and Proposition \\ref{prop:nbp-lower-bound},\n\\begin{equation}\n \\Pr(N\\leq M)\\geq \\Pr(N_{bp}\\leq M)\\geq C_{L^m M^{1-m}}(M,\\lambda) = \\sum_{i=1}^M e^{-\\lambda nL\/p}\\frac{(\\lambda nL\/p)^{n-1}}{n!} \\geq1-\\delta.\n\\end{equation}\nThe above inequality can be used in order to find the maximum throughput for which queue length remains bounded by $M$ with probability of at least $1-\\delta$.\n \\end{example}\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this paper, we formulated and analyzed a novel horizontal traffic queue. A key characteristic of this queue is the state dependence of its service rate. We establish useful properties of the service rate dynamics. We also extend calculations for M\/G\/1 busy period distributions to our setting, even for non-empty initial condition. These results allow us to provide tight results for throughput in the linear case, and probabilistic bounds on queue length over finite time horizon in the super-linear case. We also study throughput under a batch release control policy, where the additional waiting induced by the control policy is interpreted as a perturbation to the arrival process. We provide lower bound on the throughput for a maximum permissible perturbation. In particular, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded as $m \\to 0^+$. Simulation results suggest a sharp phase transition in the throughput as the car-following behavior transitions from super-linear to sub-linear regime. \n\nIn future, we plan to sharpen our analysis to theoretically demonstrate the phase transition behavior. \nThis could include, for example, considering other release control policies with better perturbation properties. The increasing nature of the throughput in the super-linear regime for large values of $L$, as illustrated in Figure~\\ref{fig:throughput-simulations}, is possibly because the car-following model considered in this paper does not impose any explicit upper bounds on the speed of the vehicles. We plan to extend our analysis to such practical constraints, as well as to higher order, e.g., second order, car-following models, and models emerging from consideration of inter-vehicle distance beyond the vehicle immediately in front. The connections with processor sharing queue, as highlighted in this paper, suggest the possibility of utilizing the construct of measure-valued state descriptors~\\cite{Grishechkin:94,Gromoll.Puha.ea:02} to derive fluid and diffusion limits of the proposed horizontal traffic queues. In particular, one could interpret the measure-valued state descriptor to play the role of traffic density in the context of traffic flow theory. Along this direction, we plan to investigate connections between the fluid limit of horizontal traffic queues, and PDE models for traffic flow. \n\\section{Introduction}\n\\label{sec:introduction}\nWe consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart the queue after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the speed of a vehicle is proportional to a power $m>0$ of the distance to the vehicle in front. For a given initial condition, we define the throughput of such a queue as the largest arrival rate under which the queue length remains bounded. We provide rigorous analysis for the service rate, busy period distribution, and throughput of the proposed HTQ. \n\nOur motivation for studying HTQ comes from advancements in connected and autonomous vehicle technologies that allow to program individual vehicles with rules that can optimize system level performance. Within this application context, one can interpret the results of this paper as rigorously characterizing the impact of a parametric class of car-following behavior on system throughput. \n\n\nIn the linear case ($m=1$), i.e., when the speed of every vehicle is proportional to the distance to the vehicle directly in front, the periodicity of the road segment implies that the sum of the speeds of the vehicles is proportional to the total length of the road segment, i.e., it is constant. This feature allows us to exploit the equivalence between workload and queue length to show that, independent of the initial condition and almost surely, the throughput is the inverse of the time required by a solitary vehicle to travel average distance. \n\nIn the non-linear case ($m \\neq 1$), the cumulative service rate of HTQ queue is constant if and only if all the inter-vehicle distances are equal. For all other inter-vehicle configurations, we show that the service rate is strictly decreasing (resp., strictly increasing) in the super-linear, i.e., $m>1$ (resp., sub-linear, i.e., $m<1$) case. The service rate exhibits an another contrasting behavior in the sub- and super-linear regimes. In the super-linear case, the service rate is maximum (resp., minimum) when all the vehicles are co-located (resp., when the inter-vehicle distances are equal), and vice-versa for the sub-linear case. Using a combination of these properties, we prove that, when the length of the road segment is at most one, the throughput in the super-linear (resp., sub-linear) case is upper (resp., lower) bounded by the throughput for the linear case.\n\nWe prove the remaining bounds on the throughput for the non-linear case as follows. The standard calculations for joint distributions of duration and number of arrivals during a busy period for M\/G\/1 queue are extended to the HTQ setting, including for non-empty initial conditions. \nThese joint distributions are used to derive probabilistic upper bounds on queue length over finite time horizons for HTQ for the $m>1$ case. Such bounds are optimized to get lower bounds on throughput defined over finite time horizons. Simulation results show good comparison between such lower bounds and numerical estimates.\n \nWe also analyze throughput in the sub-linear and super-linear cases under perturbation to the arrival process, which is attributed to the additional expected waiting time induced by a release control policy that adds appropriate delay to the arrival times to ensure a desired minimum inter-vehicle distance $\\triangle>0$ at the time of a vehicle joining the HTQ. Since the minimum inter-vehicle distance is non-decreasing in between arrivals and jumps, this implies an upper bound on the queue length which is inversely proportional to $\\triangle$. We derive a lower bound on throughput for a given \ncombination of maximum allowable perturbation. In particular, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded, as $m \\to 0^+$. \n\n\n \nQueueing models have been used to model and analyze traffic systems. The focus here has been primarily on vertical queues, under which vehicles travel at maximum speed until they hit a congestion spot where all vehicles queue on top of each other.\nThe queue length and waiting time of a minor traffic stream at an unsignalized intersection where major traffic stream has high priority is studied in \\cite{Tanner:62} and \\cite{Heidemann:91}. \nIn \\cite{Heidemann:94}, a vertical single server queue is utilized to model the queue length distribution at signalized intersections.\nIn \\cite{Jain.Smith:97}, a state-dependent queuing system is used to model vehicular traffic flow where the service rate depends on the number of vehicles on each road link. \n\n\n On the other hand, the \\emph{horizontal traffic queue} terminology has been primarily used to study macroscopic traffic flow, e.g., see \\cite{Helbing:03}. While such models capture the macroscopic relationship between traffic flow and density, a rigorous description and analysis of an underlying queue model is lacking. Indeed, to the best of our knowledge, there is no prior work on the analysis of a traffic queue model that explicitly incorporates car-following behavior. \n\nThe proposed HTQ has an interesting connection with processor sharing (PS) queues, and this connection does not seem to have been documented before.\nA characteristic feature of PS queues is that all the outstanding jobs receive service simultaneously, while keeping the total service rate of the server constant. The simplest model is where the service rate for an individual job is equal to $1\/N$, where $N$ is the number of outstanding jobs. In our proposed system, one can interpret the road segment as a server simultaneously providing service to all the vehicles, with the service rate of an individual vehicle equal to its speed. This natural analogy between HTQ and PS queues, to the best of our knowledge, was reported for the first time in our recent work~\\cite{Motie.Savla.CDC15}.\nThe $1\/N$ rule applied to our setting implies that all the vehicles travel with the same speed. Clearly, such a rule, or even the general discriminatory PS disciplines, e.g., see \\cite{Kleinrock:67}, are not applicable to the car following models considered in this paper. Indeed, the proposed HTQ is best described as a state-dependent PS queue. \n\n\nIn the PS queue literature, the focus has been on the sojourn time and queue length distribution. For example, see \\cite{Ott:84} and \\cite{Yashkov:83} for M\/G\/1-PS queue and \n\\cite{Grishechkin:94} for G\/G\/1-PS queue. Fluid limit analysis for PS queue is provided in \\cite{Chen.ea:97} and \\cite{Gromoll.Puha.ea:02}. \nHowever, relatively less attention has been paid to the throughput analysis of state-dependent PS queues.\nIn \\cite{Moyal:08,Kherani.Kumar:02,Chen.Jordan:07}, throughput analysis for state-dependent PS queues is provided, where throughput is defined as the quantity of work achieved by the server per unit of time.\nStability analysis for a single server queue with workload-dependent service and arrival rate is provided in \\cite{Bambos.Walrand:89} and \\cite{Bekker:05}. However, the dependence of service rate on the system state in the HTQ proposed in the current paper is complex, and hence none of these results are readily applicable. \n\nIn summary, there are several novel contributions of the paper. First, we propose a novel horizontal traffic queue and place it in the context of processor-sharing queues and state-dependent queues. We establish monotonicity properties of service rates in between jumps (i.e., arrivals and departures), and derive bounds on change in service rates at jumps. \nSecond, we adapt busy period calculations for M\/G\/1 queue to our current setup, including for non-empty initial conditions. These results allow us to provide tight results for throughput in the linear case, and probabilistic bounds on queue length over finite time horizon in the super-linear case. We also study throughput under a batch release control policy, whose effect is interpreted as a perturbation to the arrival process. We provide lower bound on the throughput for a maximum permissible perturbation for sub- and super-linear cases. In particular, we show that, for sufficiently large perturbation, this lower bound grows unbounded as $m \\to 0^+$. It is interesting to compare our analytical results with simulation results, which suggest a sharp transition in the throughput from being unbounded in the sub-linear regime to being bounded in the super-linear regime. While our analytical results do not exhibit such a phase transition yet, their novelty is in providing rigorous estimates of any kind on the throughput of horizontal traffic queues under nonlinear car following models.\n\nThe rest of the paper is organized as follows. We conclude this section with key notations to be used throughout the paper. The setting for the proposed horizontal traffic queue and formal definition of throughput are provided in Section~\\ref{sec:problem-formulation}. Section~\\ref{sec:service-rate-monotonicity} contains useful properties on the dynamics in service rate in between and during jumps. Key busy period properties for the M\/G\/1 queue are extended to the HTQ case in Section~\\ref{sec:busy-period}. Throughput analysis is reported in Section~\\ref{sec:throughput-analysis}. Simulations are presented in \\ref{sec:simulations}. Concluding remarks and directions for future work are presented in Section~\\ref{sec:conclusions}. A few technical intermediate results are collected in the appendix. \n\n\\subsection*{Notations}\nLet $\\mathbb{R}$, $\\mathbb{R}_+$, and $\\mathbb{R}_{++}$ denote the set of real, non-negative real, and positive real numbers, respectively. \nLet $\\mathbb{N}$ be the set of natural numbers. \nIf $x_1$ and $x_2$ are of the same size, then $x_1 \\geq x_2$ implies element-wise inequality between $x_1$ and $x_2$. If $x_1$ and $x_2$ are of different sizes, then $x_1 \\geq x_2$ implies inequality only between elements which are common to $x_1$ and $x_2$ -- such a common set of elements will be specified explicitly. \nFor a set $\\mathcal{J}$, let $\\text{int}(\\mathcal{J})$ and $|\\mathcal{J}|$ denote the interior and cardinality of $\\mathcal{J}$, respectively.\nGiven $a \\in\\mathbb{R}$, and $b > 0$, we let $\\mod(a,b):=a-\\lfloor \\frac{a}{b}\\rfloor b$. Let $\\mathcal S_N^L$ be the $N-1$-simplex over $L$, i.e., $\\mathcal S_N^L=\\setdef{x \\in \\mathbb{R}_+^N}{\\sum_{i=1}^N x_i = L}$. When $L=1$, we shall use the shorthand notation $\\mathcal S_N$.\nWhen referring to the set $\\until{N}$, for brevity, we let the indices $i=-1$ and $i=N+1$ correspond to $i=N$ and $i=1$ respectively.\n Also, for $p, q \\in \\mathcal S_N$, we let\n$D(p || q)$ denote the K-L divergence of $q$ from $p$, i.e., $D(p || q):= \\sum_{i=1}^N p_i \\log\\left(p_i\/q_i\\right)$.\nWe also define a permutation matrix, $P^- \\in \\{0,1\\}^{N \\times N}$, as follows:\n\\begin{align*}\nP^-:=\\begin{bmatrix}\n\\mathbf{0}_{N-1}^T & 1 \\\\\nI_{N-1} & \\mathbf{0}_{N-1}\n\\end{bmatrix}\n\\end{align*}\nwhere $\\mathbf{0}_N$ and $\\mathbf{1}_N$ stand for vectors of size $N$, all of whose entries are zero and one, respectively. We shall drop $N$ from $\\mathbf{0}_N$ and $\\mathbf{1}_N$ whenever it is clear from the context.\n\n\n\n\n\\subsection{Linear Case: $m=1$}\n\\label{sec:linear}\nIn this section, we provide an exact characterization of throughput for the linear case, i.e., when $m=1$. Recall that, for $m=1$, the service rate $s(y)=\\sum_{i=1}^N y_i \\equiv L$ is constant.\n\n\n\n\\begin{proposition}\\label{prop:unstable}\nFor any $L > 0$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$ and :\n$$\n\\lambda_{\\text{max}}(L,m=1,\\varphi,\\psi,x_0,\\delta=0) \\leq L\/\\bar{\\psi} \\, .\n$$ \n\\end{proposition}\n\\begin{proof}\nBy contradiction, assume $\\lambda_{\\text{max}}>L\/\\bar{\\psi}$. Let $r(t):=\\sum_{i=1}^{A(t)} d_i$ be the workload added to the system by the $A(t)$ vehicles that arrive over $[0,t]$. Therefore, \n\\begin{equation}\\label{eq:workload-linear}\nw(t)=w_0+r(t)-L(t-\\mathcal{I}(t))\n\\end{equation}\nwhere $w_0$ is the initial workload. The process $\\{r(t), \\, t\\geq 0\\}$ is a renewal reward process, where the renewals correspond to arrivals of vehicles and the rewards correspond to the distances $\\{d_i\\}_{i=1}^{\\infty}$ that vehicles wish to travel in the system upon arrival before their departures. \nInter-arrival times are exponential random variables with mean $1\/\\lambda$, and the reward associated with each renewal is independently and identically sampled from $\\psi$, whose mean is $\\bar{\\psi}$. \nTherefore, e.g., \\cite[Theorem 3.6.1]{Ross:96} implies that, with probability one, \n\\begin{equation}\n\\label{eq:slln-renewal}\n\\lim_{t\\to\\infty}\\frac{r(t)}{t}=\\lambda \\bar{\\psi}\n\\end{equation}\nThus, for all $\\varepsilon \\in \\left(0,\\lambda \\bar{\\psi} -L\\right)$, there exists a $t_0\\geq0$ such that, with probability one,\n\\begin{equation}\\label{eq:s-t-lowerbound}\n\\frac{r(t)}{t}\\geq\\lambda \\bar \\psi-\\varepsilon\/2> L + \\varepsilon\/2 \\qquad \\forall \\, t \\geq t_0.\n\\end{equation}\nSince $w_0$ and $\\mathcal{I}(t)$ are both non-negative, \\eqref{eq:workload-linear} implies that $w(t)\\geq r(t) - Lt$ for all $t \\geq 0$. \nThis combined with \\eqref{eq:s-t-lowerbound} implies that, with probability one, $w(t) \\geq \\varepsilon t \/2$ for all $t \\geq t_0$, and hence \n$\\lim_{t\\to\\infty}w(t)= + \\infty$. This combined with \\eqref{eq:workload-upperbound} implies that, with probability one, $\\lim_{t\\to\\infty}N(t)=+\\infty$. \n\\qed\n\\end{proof}\n\n\\begin{theorem}\\label{thm:stable}\nFor any $L > 0$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$:\n$$\n\\lambda_{\\text{max}}(L,m=1,\\varphi,\\psi,x_0,\\delta=1) = L\/\\bar{\\psi} \\, .\n$$ \n\\end{theorem}\n\\begin{proof}\nAssume that for some $\\lambda < L\/\\bar{\\psi}$, there exists some initial condition $(x_0,n_0)$ such that the queue length grows unbounded with some positive probability. \nSince the workload brought by every vehicle is i.i.d., and the inter-arrival times are exponential, without loss of generality, we can assume that the queue length never becomes zero. That is, the idle time satisfies $\\mathcal{I}(t) \\equiv 0$. Moreover, \\eqref{eq:slln-renewal} implies that, for every $\\varepsilon \\in \\left(0, L- \\lambda \\bar{\\psi} \\right)$, there exists $t_0 \\geq 0$ such that, with probability one,\n\\begin{equation}\n\\label{eq:s-t-upperbound}\n\\frac{r(t)}{t} \\leq \\lambda \\bar{\\psi} + \\varepsilon\/2 < L - \\varepsilon\/2 \\qquad \\forall t \\geq t_0\n\\end{equation} \n\nCombining \\eqref{eq:workload-linear} with \\eqref{eq:s-t-upperbound}, and substituting $\\mathcal{I}(t) \\equiv 0$, we get $w(t) < w_0 - \\varepsilon t \/2$, which implies that workload, and hence queue length, goes to zero in finite time after $t_0$, leading to a contradiction. \nCombining this with the upper bound proven in Proposition~\\ref{prop:unstable} gives the result. \n\\qed\n\\end{proof}\n\n\\begin{remark}\nTheorem~\\ref{thm:stable} implies that the throughput in the linear case is equal to the inverse of the \ntime required to travel average total distance by a solitary vehicle in the system. In the linear case, the throughput can be characterized with probability one, independent of the initial condition of the queue.\n\\end{remark}\n\n\n\n\n\n\\subsection{Monotonicity of Throughput in $m$ and $x_0$}\n\\label{sec:non-linear}\n\n\nIn this section, we show the following monotonicity property of $\\lambda_{\\text{max}}$ with respect to $m$ for small values of $L$: for given $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, $L \\in (0,1)$, $\\varphi \\in \\Phi$, and $\\psi \\in \\Psi$, throughput is a monotonically decreasing function of $m$. \nFor this section, we rewrite \\eqref{eq:dynamics-moving-coordinates} in $\\mathbb{R}_+^N$, i.e., without projecting onto $[0,L]^N$. Specifically, let the vehicle coordinates be given by the solution of \n\\begin{equation}\n\\label{eq:x-dynamics-in-Rn}\n\\dot{x}_i = y_i^m, \\qquad x_i(0)=x_{0,i}, \\qquad i \\in \\until{N} \n\\end{equation} \nLet $X(t;x_0,m)$ denote the solution to \\eqref{eq:x-dynamics-in-Rn} at $t$ starting from $x_0$ at $t=0$. We will compare $X(t;x_0,m)$ under different values of $m$ and initial conditions $x_0$, over an interval of the kind $[0,\\tau)$, in between arrivals and departures. \nWe recall the notation that, if $x_{0}^1$ and $x_{0}^2$ are vectors of different sizes, then $x_{0}^1 \\leq x_{0}^2$ implies element-wise inequality only for components which are common to $x_{0}^1$ and $x_{0}^2$. In Lemma~\\ref{lemma:two-size-compare} and Proposition~\\ref{prop:queue-length-2m}, this common set of components corresponds to the set of vehicles common between $x_{0}^1$ and $x_{0}^2$.\n \n\n\\begin{lemma}\n\\label{lemma:two-size-compare}\nFor any $L \\in (0,1]$, $x^1_{0} \\in \\mathbb{R}_+^{n_1}$, $x^2_{0} \\in \\mathbb{R}_+^{n_2}$, $n_1, n_2 \\in \\mathbb{N}$,\n$$\nx^1_{0} \\leq x^2_{0}, \\, \\, n_2 \\leq n_1, \\, \\, 0 < m_2 \\leq m_1 \\implies X(t;x^1_{0},m_1) \\leq X(t;x^2_{0},m_2) \\qquad \\forall \\, t \\in [0,\\tau)\n$$ \n\\end{lemma}\n \\begin{proof}\nThe proof is straightforward when $n_1=n_2$. This is because, in this case, since $y_i \\leq L\\leq1$, $m_2 \\leq m_1$ implies $y_i^{m_2} \\geq y_i^{m_1}$ for all $i \\in \\until{n_1}$. Using this with Lemmas~\\ref{lem:monotonicity-same-size} and \\ref{lem:car-following-type-K} gives the result. \n \nIn order to prove the result for $n_2 < n_1$, we show that $X(t; x^1_{0}, m_1) \\leq X(t; x^2_{0}, m_1) \\leq X(t; x^2_{0}, m_2)$. Note that the second inequality follows from the previous case. Therefore, it remains to prove the first inequality. Let $(i_1, \\ldots, i_{n_2})$ be the set of indices of $n_2$ vehicles such that $0 \\leq x^2_{0,i_1} \\leq \\ldots \\leq x^2_{0,i_{n_2}} \\leq L$. Similarly, let $(i_1, i_1+1, \\ldots, i_2, i_2+1, \\ldots)$ be the indices of $n_1$ vehicles in the order of increasing coordinates in $x_0^1$. Our assumption on the initial condition implies that $x^1_{0,i_k} \\leq x^2_{0,i_k}$ for all $k \\in \\until{n_2}$. For brevity, let $x^1(t) \\equiv X(t;x_0^1,m_1)$, and $x^2(t) \\equiv X(t;x_0^2,m_1)$. It is easy to check that, for all $t \\in [0,\\tau)$, and all $k \\in \\until{n_2}$, \n\\begin{equation}\n\\label{eq:xdot-ub}\n\\dot{x}^1_{i_k} = \\left(x^1_{i_{k}+1} - x^1_{i_k}\\right)^{m_1} \\leq \\left(x^1_{i_{k+1}} - x^1_{i_k}\\right)^{m_1}\n\\end{equation}\nLet $t \\in [0,\\tau)$ be the first time instant when $x^1_{i_k}(t)=x^2_{i_k}(t)$ for some $k \\in \\until{n_2}$. Then, recalling $x^1_{i_{k+1}}(t) \\leq x^2_{i_{k+1}}(t)$, \\eqref{eq:xdot-ub} implies that $\\dot{x}_{i_k}^1(t) \\leq \\left(x^2_{i_{k+1}} - x^2_{i_k} \\right)^{m_1} = \\dot{x}^2_{i_k}(t)$. The result then follows from Lemma~\\ref{lem:monotonicity-same-size}. \n\\qed\n \\end{proof}\n \n \n\nLemma~\\ref{lemma:two-size-compare} is used to establish monotonicity of throughput as follows.\n\\begin{proposition}\n\\label{prop:queue-length-2m}\nFor any $L \\in (0,1]$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $\\delta \\in (0,1)$, $x^1_{0} \\in [0,L]^{n_1}$, $x^2_{0} \\in [0,L]^{n_2}$, $n_1, n_2 \\in \\mathbb{N}$:\n$$\nx^1_{0} \\leq x^2_{0}, \\, \\, n_2 \\leq n_1, \\, \\, 0 < m_2 \\leq m_1 \\implies \\lambda_{\\text{max}}(L,m_1,\\varphi,\\psi,x^1_0,\\delta) \\leq \\lambda_{\\text{max}}(L,m_2,\\varphi,\\psi,x^2_0,\\delta)\n$$\n\\end{proposition}\n\\begin{proof}\nFor brevity in notation, we refer to the queue corresponding to $m_1$, and initial condition $x_0^1$ as HTQ-S. We refer to the other queue as HTQ-F. Let $\\lambda$, $\\varphi$ and $\\psi$ common to HTQ-S and HTQ-F be given. Let $x^1(t) \\equiv X(t; x^1_0, m_1)$ and $x^2(t) \\equiv X(t; x^2_0, m_2)$, and let $N_s(t)$ and $N_f(t)$ be the queue lengths in the two queues at time $t$. It suffices to show that $N_s(t) \\geq N_f(t)$ for a given realization of arrival times, arrival locations, and travel distances. In particular, this also implies that the departure locations are also the same for every vehicle, including the vehicles present at $t=0$, in both the queues. \n\nIndeed, it is sufficient to show that $x^1(\\tau) \\leq x^2(\\tau)$ and $N_s(\\tau)\\geq N_f(\\tau)$ where $\\tau$ is the time of first arrival or departure from either HTQ-S or HTQ-F. Accordingly, we consider two cases, corresponding to whether $\\tau$ corresponds to arrival or departure. \n\nSince $x^1(t) \\leq x^2(t)$ for all $t \\in [0,\\tau)$ from Lemma~\\ref{lemma:two-size-compare}, and the departure locations of all the vehicles in HTQ-S and HTQ-F are identical, the first departure from HTQ-S can not happen before the first departure in HTQ-F. Therefore, $N_s(\\tau) \\geq N_f(\\tau)$. Since $x^1(\\tau^-) \\leq x^2(\\tau^-)$, and $x^2(\\tau)$ is a subset of $x^2(\\tau^-)$, we also have $x^1(\\tau) \\leq x^2(\\tau)$. \n\nWhen $\\tau$ corresponds to the time of the first arrival, since the arrivals happen at the same location in HTQ-S and HTQ-F, and since $x^1(\\tau^-) \\leq x^2(\\tau^-)$, rearrangement of the indices of the vehicles to include the new arrival at $t=\\tau$ implies that $x^1(\\tau) \\leq x^2(\\tau)$. Moreover, since $N_s(\\tau^-) \\geq N_f(\\tau^-)$, and the arrivals happen simultaneously in both HTQ-S and HTQ-F, we have $N_s(\\tau) \\leq N_f(\\tau)$. \n\\qed\n\\end{proof}\n\n\\begin{remark}\nProposition \\ref{prop:queue-length-2m} establishes monotonicity of throughput only for $L\\in(0,1]$. This is consistent with our simulation studies, e.g., as reported in Figure \\ref{fig:throughput-simulations}, according to which, the throughput is non-monotonic for large $L$. \n\\end{remark}\nFor the analysis of the linear car following model, we exploited the fact that the total service rate of the system is constant. However, for the nonlinear model, i.e., $m \\neq 1$, the total service rate depends on the number and relative locations of vehicles. The state dependent service rate of nonlinear models makes the throughput analysis much more complex. In the next section, we find probabilistic bound on the throughput in the super-linear case.\n\n\n\n\n\n\n\n\n\n\n\n\\section{The Horizontal Traffic Queue (HTQ) Setup}\n\\label{sec:problem-formulation}\nConsider a periodic road segment of length $L$; without loss of generality, we assume it be a circle. Starting from an arbitrary point on the circle, we assign coordinates in $[0,L]$ to the circle in the clock-wise direction (See Figure \\ref{fig:htq}). Vehicles arrive on the circle according to a spatio-temporal process: the arrival process $\\{A(t),t\\geq0\\}$, is assumed to be a Poisson process with rate $\\lambda>0$, and the arrival locations are sampled independently and identically from a spatial distribution $\\varphi$ and mean value $\\bar \\varphi$. Without loss of generality, let the support of $\\varphi$ be $\\text{supp}(\\varphi)=[0,\\ell]$ for some $\\ell\\in[0,L]$. \nUpon arriving, vehicle $i$ travels distance $d_i$ in a counter-clockwise direction, after which it departs the system. \nThe travel distances $\\{d_i\\}_{i=1}^\\infty$ are sampled independently and identically from a spatial distribution $\\psi$ with support $[0,R]$ and mean value $\\bar \\psi$.\nLet the set of $\\varphi$ and $\\psi$ satisfying the above conditions be denoted by $\\Phi$ and $\\Psi$ respectively.\nThe stochastic processes for arrival times, arrival locations, and travel distances are all assumed to be independent of each other.\n\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=5cm]{htq-colorful} \n \\caption{Illustration of the proposed HTQ with three vehicles.}\n \\label{fig:htq}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Dynamics of vehicle coordinates between jumps}\nLet the time epochs corresponding to arrival and departure of vehicles be denoted as $\\{\\tau_1, \\tau_2, \\ldots\\}$. We shall refer to these events succinctly as \\emph{jumps}. We now formally state the dynamics under this car-following model. \nWe describe the dynamics over an arbitrary time interval of the kind $[\\tau_j,\\tau_{j+1})$. Let $N \\in \\mathbb{N}$ be the fixed number of vehicles in the system during this time interval. \nDefine the inter-vehicle distances associated with vehicle coordinates $x \\in [0,L]^N$ as follows:\n\\begin{equation}\n\\label{eq:inter-vehicle-distance-Rn}\ny_i(x) = \\mod \\left(x_{i+1} - x_i, L\\right), \\quad i \\in \\until{N} \n\\end{equation}\nwhere we implicitly let $x_{N+1} \\equiv x_1$ (See Figure \\ref{fig:htq} for an illustration). Note that the normalized inter-vehicle distances $y\/L$ are probability vectors. \nWhen inside the queue, the speed of every vehicle is proportional to a power $m >0$ of the distance to the vehicle directly in front of it. \nWe assume that this power $m > 0$ is the same for every vehicle at all times. Then, starting with $x(\\tau_{j}) \\in [0,L]^N$, the vehicle coordinates over $[\\tau_{j},\\tau_{j+1})$ are given by:\n\\begin{equation}\n\\label{eq:dynamics-moving-coordinates}\nx_i(t) = \\mod \\left( x_i(\\tau_{j}) + \\int_{\\tau_{j}}^t y_i^m(x(z)) \\, dz, L \\right), \\qquad \\forall \\, i \\in \\until{N}, \\quad \\forall \\, t \\in [\\tau_{j},\\tau_{j+1}) \\, ,\n\\end{equation}\n\n\\begin{remark}\nIt is easy to see that the clock-wise ordering of the vehicles is invariant under \\eqref{eq:inter-vehicle-distance-Rn}-\\eqref{eq:dynamics-moving-coordinates}. \n\\end{remark}\n\nThe dynamics in inter-vehicle distances is given by:\n\\begin{equation}\n\\label{eq:inter-vehicle-distance-dynamics}\n\\dot{y}_i = y^m_{i+1} - y_i^m, \\qquad i \\in \\until{N} \n\\end{equation}\nwhere we implicitly let $y_{N+1} \\equiv y_1$.\n\n\\subsection{Change in vehicle coordinates during jumps}\nLet $x(\\tau^-_{j}) =\\left(x_1(\\tau^-_{j}), \\ldots, x_N(\\tau^-_{j})\\right) \\in [0,L]^N$ be the vehicle coordinates just before the jump at $\\tau_{j}$. If the jump corresponds to the departure of vehicle $k \\in \\until{N}$, then the coordinates of the vehicles $x(\\tau_{j}) =\\left(x_1(\\tau_{j}), \\ldots, x_{N-1}(\\tau_{j})\\right) \\in [0,L]^{N-1}$ after re-ordering due to the jump, for $i \\in \\until{N-1}$, are given by:\n \\begin{equation*}\n\t\tx_i(\\tau_{j})=\\left\\{\\begin{array}{ll}\n\t\t\\displaystyle x_i(\\tau^-_{j}) \t& i\\in \\until{k-1} \\\\[15pt]\n\t\t\\displaystyle x_{i+1}(\\tau^-_{j})\t& i\\in \\{k+1, \\ldots, N-1\\}\\,.\\end{array}\\right.\n\t\\end{equation*}\n\n Analogously, if the jump corresponds to arrival of a vehicle at location $z \\in [0,\\ell]$ in between the locations of the $k$-th and $k+1$-th vehicles at time $\\tau_{j}^-$, then the coordinates of the vehicles $x(\\tau_{j}) =\\left(x_1(\\tau_{j}), \\ldots, x_{N+1}(\\tau_{j})\\right) \\in [0,L]^{N+1}$ after re-ordering due to the jump, for $i \\in \\until{N+1}$, are given by:\n \\begin{equation*}\n \\begin{split}\n x_{k+1}(\\tau_{j}) & = z \\\\\n\t\tx_i(\\tau_{j}) & =\\left\\{\\begin{array}{ll}\n\t\t\\displaystyle x_i(\\tau^-_{j}) \t& i\\in \\until{k} \\\\[15pt]\n\t\t\\displaystyle x_{i-1}(\\tau^-_{j})\t& i\\in \\{k+2, \\ldots, N+1\\}\\,.\\end{array}\\right.\n\t\t\\end{split}\n\t\\end{equation*}\n\n\\subsection{Problem statement}\nLet $x_0 \\in [0,L]^{n_0}$ be the initial coordinates of $n_0$ vehicles present at $t=0$. An HTQ is described by the tuple $\\left(L, m, \\lambda, \\varphi, \\psi, x_0\\right)$. \nLet \n$N(t;L, m, \\lambda, \\varphi, \\psi, x_0)$ be the corresponding queue length, i.e., the number of vehicles at time $t$ for an HTQ $\\left(L, m, \\lambda, \\varphi, \\psi, x_0\\right)$. For brevity in notation, at times, we shall not show the dependence of $N$ on parameters which are clear from the context.\n\n\n\nIn this paper, our objective is to provide rigorous characterizations of the dynamics of the proposed HTQ. A key quantity that we study is throughput, defined below. \n \n\\begin{definition}[Throughput of HTQ]\\label{def:throughput}\nGiven $L > 0$, $m > 0$, $\\varphi \\in \\Phi , \\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$ and $\\delta \\in [0,1)$, \nthe throughput of HTQ is defined as:\n\\begin{equation}\n\\label{eq:throughput-def}\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta):= \\sup \\left\\{\\lambda \\geq 0: \\, \\Pr \\left( N(t;L, m, \\lambda, \\varphi, \\psi, x_0) < + \\infty, \\quad \\forall t \\geq 0 \\right) \\geq 1 - \\delta \\right\\}.\n\\end{equation}\n\\end{definition}\n\n\nFigure~\\ref{fig:throughput-simulations} shows the complex dependency of throughput on key queue parameters such as $m$ and $L$. In particular, it shows that for every $L$, $\\varphi$, $\\psi$, $x_0$ and $\\varphi$, the throughput exhibits a phase transition from being unbounded for $m \\in (0,1)$ to being bounded for $m>1$. Moreover, Figure~\\ref{fig:throughput-simulations} also suggests that, for sufficiently small $L$, throughput is monotonically non-increasing in $m$, and that it is monotonically non-decreasing in $m>1$, for sufficiently large $L$. Also, it can be observed that initial condition can also affect the throughput. We now develop analytical results that match the throughput profile in Figure~\\ref{fig:throughput-simulations} as closely as possible. To that purpose, we will make extensive use of novel properties of \\emph{service rate} and \\emph{busy period} of the proposed HTQ, which could be of independent interest. \n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-4L-n0}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-5L-n100}}\\\\\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-4L-n0-unif}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=6cm]{phase-transition-5L-n100-unif}}\n\\caption{Throughput for various combinations of $m$, $L$, and $n_0$. The parameters used in individual cases are: (a) $\\varphi=\\delta_{0}$, $\\psi = \\delta_L$, and $n_0=0$ (b) $\\varphi=\\delta_{0}$, $\\psi = \\delta_L$, and $n_0=100$ (c) $\\varphi=U_{[0,L]}$, $\\psi = U_{[0,L]}$, and $n_0=0$ (d) $\\varphi=U_{[0,L]}$, $\\psi = U_{[0,L]}$, and $n_0=100$. In all the cases, the locations of initial $n_0$ vehicles were chosen at equal spacing in $[0,L]$.\n \\label{fig:throughput-simulations}}\n\\end{figure}\n\n\n\n\n\n\n\\section{Service Rate Properties of the Horizontal Traffic Queue}\n\\label{sec:service-rate-monotonicity}\nFor every $y \\in \\mathcal S_N^L$, $N \\in \\mathbb{N}$, $L>0$, we let \n$y_{\\text{min}}:=\\min_{i \\in \\until{N}} y_i$, and $y_{\\mathrm{max}}:=\\max_{i \\in \\until{N}} y_i$ denote the minimum and maximum inter-vehicle distances respectively. It is easy to establish the following monotonicity properties of $y_{\\text{min}}$ and $y_{\\mathrm{max}}$.\n\n\\begin{lemma}[Inter-vehicle Distance Monotonicity Between Jumps]\n\\label{lem:vehicle-distance-monotonicity}\nFor any $y \\in \\mathcal S^L_N$, $N \\in \\mathbb{N}$, $L>0$, under the dynamics in \\eqref{eq:inter-vehicle-distance-dynamics}, for all $m>0$\n$$\n\\frac{d}{dt} y_{\\text{min}} \\geq 0 \\qquad \\& \\qquad \\qquad \\frac{d}{dt} y_{\\mathrm{max}} \\leq 0.\n$$\n\\end{lemma}\n\\begin{proof}\nLet $y_{\\text{min}}(t) = y_j(t)$, i.e., the $j$-th vehicle has the minimum inter-vehicle distance at time $t\\geq 0$.\nTherefore, \\eqref{eq:inter-vehicle-distance-dynamics} implies that $\\dot{y}_{\\min}(t) = \\dot{y}_j(t)=y_{j+1}^m(t) - y_j^m(t)\\geq 0$. One can similarly show that $y_{\\mathrm{max}}$ is non-increasing.\n\\qed\n\\end{proof}\n\nDue to the complex state-dependence of the departure process, the queue length process is difficult to analyze. We propose to study a related scalar quantity, called \\emph{workload} formally defined as follows, where we recall the notations introduced in \nSection~\\ref{sec:problem-formulation}.\n\n\\begin{definition}[Workload]\nThe workload associated with the HTQ at any instant is the sum of the distances remaining to be travelled by all the vehicles present at that instant. That is, if the current coordinates and departure coordinates of all vehicles are $x \\in [0,L]^N$ and $q \\in \\mathbb{R}_+^N$ respectively, with $q \\geq x$, then the workload is given by:\n$$\nw(x,q):= \\sum_{i=1}^N (q_i-x_i).\n$$ \n\\end{definition}\n\n\nSince the maximum distance to be travelled by any vehicle from the time of arrival to the time of departure is upper bounded by $R$, we have the following simple relationship between workload and queue length at any time instant:\n\\begin{equation}\\label{eq:workload-upperbound}\nw(t) \\leq N(t)\\, R \\, , \\qquad \\forall \\, t \\geq 0 \\, .\n\\end{equation}\nAn implication of \\eqref{eq:workload-upperbound} is that unbounded workload implies unbounded queue length in our setting. We shall use this relationship to establish an upper bound on the throughput.\nHowever, a finite workload does not necessarily imply finite queue length. In order to see this, consider the state of the queue with $N$ vehicles, all of whom have distance $1\/N$ remaining to be travelled. Therefore, the workload at this instant is $1\/N \\times N= 1$, which is independent of $N$. \n\nWhen the workload is positive, its rate of decrease is equal to \\emph{service rate} in between jumps, defined next.\n\n\\begin{definition}[Service Rate]\nWhen the HTQ is not idle, its instantaneous service rate is equal to the sum of the speeds of the vehicles present in the system at that time instant, i.e., $s(x)=\\sum_{i=1}^N y_i^m(x)$.\n\\end{definition}\n\n\nSince the service rate depends only on the inter-vehicle distances, we shall alternately denote it as $s(y)$. For $m=1$, $s(y)=\\sum_{i=1}^N y_i \\equiv L$, i.e., the service rate is independent of the state of the system, and is constant in between and during jumps. This property does not hold true in the nonlinear ($m \\neq 1$) case. Nevertheless, one can prove interesting properties for the service rate dynamics. We start by deriving bounds on service rate in between jumps. \n\n\\begin{lemma}[Bounds on Service Rates]\n\\label{lem:service-rate-bounds}\nFor any $y \\in \\mathcal S_N^L$, $N \\in \\mathbb{N}$, $L>0$, under the dynamics in \\eqref{eq:inter-vehicle-distance-dynamics},\n\\begin{enumerate}\n\\item $L^m N^{1-m} \\leq s(y) \\leq L^m$ if $m > 1$;\n\\item $L^m \\leq s(y) \\leq L^m N^{1-m}$ if $m \\in (0,1)$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nNormalizing the inter-vehicular distances by $L$, the service rate can be rewritten as \n\\begin{equation}\n\\label{eq:service-rate-normalized}\ns(y)=L^m \\sum_{i=1}^N \\left(\\frac{y_i}{L}\\right)^m. \n\\end{equation}\nTherefore, for $m > 1$, $s(y) \\leq L^m \\sum_{i=1}^N \\frac{y_i}{L}=L^m$. One can similarly show that, for $m \\in (0,1)$, $s(y) \\geq L^m$. In order to prove the remaining bounds, we note that $\\sum_{i=1}^N z_i^m$ is strictly convex in $z=[z_1, \\ldots, z_N]$ for $m > 1$, and that the minimum of $\\sum_{i=1}^N z_i^m$ over $z \\in \\mathcal S_N$ occurs at $z= \\mathbf{1}\/N$, and is equal to $N^{1-m}$. Similarly, for $m \\in (0,1)$, $\\sum_{i=1}^N z_i^m$ is strictly concave in $z$, and its maximum over $z \\in \\mathcal S_N$ occurs at $z= \\mathbf{1}\/N$, and is equal to $N^{1-m}$. Combining these facts with \\eqref{eq:service-rate-normalized}, and noting that $y\/L \\in \\mathcal S_N$, gives the lemma.\n\\qed\n\\end{proof}\n\n\\begin{lemma}[Service Rate Monotonicity Between Jumps]\n\\label{lem:service-rate-dynamics-between-jumps}\nFor any $y \\in \\mathcal S^L_N$, $N \\in \\mathbb{N}$, $L>0$, under the dynamics in \\eqref{eq:inter-vehicle-distance-dynamics}, \n$$\n\\frac{d}{dt} s(y) \\leq 0 \\quad \\text{ if } m > 1 \\qquad \\& \\qquad \\frac{d}{dt} s(y) \\geq 0\\quad \\text{ if } m \\in (0,1) \\, ,\n$$\nwhere the equality holds true if and only if $y=\\frac{L}{N}\\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\nThe time derivative of service rate is given by:\n\\begin{align}\n\\frac{d}{dt} s(y) & = \\frac{d}{dt} \\sum_{i=1}^N y_i^m = m \\sum_{i=1}^N y_i^{m-1} \\dot{y}_i \\nonumber \\\\ \n\\label{eq:service-rate-derivative}\n& = m \\sum_{i=1}^N y_i^{m-1} \\left( y_{i+1}^m - y_i^m\\right) \n\\end{align}\nwhere the second equality follows by \\eqref{eq:inter-vehicle-distance-dynamics}. The result then follows by application of Lemma~\\ref{lem:appendix-general-summation}, and by noting that $g(z)=z^m$ is a strictly increasing function for all $m>0$, and $h(z)=z^{m-1}$ is strictly decreasing if $m \\in (0,1)$, and strictly increasing if $m>1$.\n\\qed\n\\end{proof}\n\n\nThe following lemma quantifies the change in service rate due to departure of a vehicle. \n\n\\begin{lemma}[Change in Service Rate at Departures]\n\\label{lem:service-rate-jumps}\nConsider the departure of a vehicle that changes inter-vehicle distances from $y \\in \\mathcal S_N^L$ to $y^- \\in \\mathcal S_{N-1}^L$, for some $N \\in \\mathbb{N} \\setminus \\{1\\}$, $L > 0$. If $y_1 \\geq 0$ and $y_2 \\geq 0$ denote the inter-vehicle distances behind and in front of the departing vehicle respectively, at the moment of departure, then the change in service rate due to the departure satisfies the following bounds:\n\\begin{enumerate}\n\\item if $m > 1$, then $0 \\leq s(y^-) - s(y) \\leq (y_1+y_2)^m \\left(1-2^{1-m} \\right)$;\n\\item if $m \\in (0,1)$, then $0 \\leq s(y) - s(y^-) \\leq \\min\\{y_1^m,y_2^m\\}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nIf $m>1$, then $\\left(\\frac{y_1}{y_1+y_2}\\right)^m + \\left(\\frac{y_2}{y_1+y_2}\\right)^m \\leq \\frac{y_1}{y_1+y_2} + \\frac{y_2}{y_1+y_2} = 1$, i.e., $s(y^-)-s(y) = (y_1 + y_2)^m - y_1^m - y_2^m \\geq 0$. One can similarly show that $s(y)-s(y^-) \\geq 0$ if $m \\in (0,1)$.\n\nIn order to show the upper bound on $s(y^-)-s(y)$ for $m>1$, we note that the minimum value of $z^m + (1-z)^m$ over $z \\in [0,1]$ for $m > 1$ is $2^{1-m}$, and it occurs at $z=1\/2$. Therefore, \n\\begin{align*}\ns(y^-)-s(y) = (y_1 + y_2)^m - y_1^m - y_2^m & = (y_1 + y_2)^m \\left(1 - \\left(\\frac{y_1}{y_1+y_2}\\right)^m - \\left(\\frac{y_2}{y_1+y_2}\\right)^m \\right) \\\\\n& \\leq (y_1+y_2)^m \\left(1-2^{1-m} \\right)\n\\end{align*}\n\nThe upper bound on $s(y)-s(y^-)$ for $m \\in (0,1)$ can be proven as follows. Since $y_1^m \\leq (y_1 + y_2)^m$, $s(y) - s(y^-) = y_1^m + y_2 ^m - (y_1 + y_2)^m \\leq y_2^m$. \nSimilarly, $s(y)-s(y^-) \\leq y_1^m$. Combining, we get $s(y)-s(y^-) \\leq \\min \\{y_1^m, y_2^m\\}$. Note that, in proving this, we nowhere used the fact that $m \\in (0,1)$. However, this bound is useful only for $m \\in (0,1)$. \n\\qed\n\\end{proof}\n\n\\begin{remark}[Change in Service Rate at Arrivals]\n\\label{rem:service-rate-jump-arrival}\nThe bounds derived in Lemma~\\ref{lem:service-rate-jumps} can be trivially used to prove the following bounds for change in service rate at arrivals:\n\\begin{enumerate}\n\\item if $m > 1$, then $0 \\leq s(y) - s(y^+) \\leq (y_1+y_2)^m \\left(1-2^{1-m} \\right)$;\n\\item if $m \\in (0,1)$, then $0 \\leq s(y^+) - s(y) \\leq \\min\\{y_1^m,y_2^m\\}$,\n\\end{enumerate}\nwhere $y_1$ and $y_2$ are the inter-vehicle distances behind and in front of the arriving vehicle respectively, at the moment of arrival.\n\\end{remark}\n\n\nThe following lemma will facilitate generalization of Lemma~\\ref{lem:service-rate-dynamics-between-jumps}. In preparation for the lemma, let $f(y,m):=m \\sum_{i=1}^N y_i^{m-1} \\left( y_{i+1}^m - y_i^m\\right)$ be the time derivative of service rate, as given in \\eqref{eq:service-rate-derivative}.\n\n\\begin{lemma}\n\\label{lem:service-rate-dot-lower-bound}\nFor all $y \\in \\text{int}(\\mathcal S_N^L)$, $N \\in \\mathbb{N} \\setminus \\{1\\}$, $L > 0$:\n\\begin{equation}\n\\label{eq:service-rate-der-m}\n\\frac{\\partial}{\\partial m} f(y,m)|_{m=1} = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right) \\leq 0\n\\end{equation}\nAdditionally, if $L < e^{-2}$, then\n\\begin{equation}\n\\label{eq:service-rate-twice-der-m}\n\\frac{\\partial^2}{\\partial m^2} f(y,m)|_{m=1} \\geq 0\n\\end{equation}\nMoreover, equality holds true in \\eqref{eq:service-rate-der-m} and \\eqref{eq:service-rate-twice-der-m} if and only if $y = \\frac{L}{N}\\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\nTaking the partial derivative of $f(y,m)$ with respect to $m$, we get that \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) & = \\frac{f(y,m)}{m} + m \\sum_{i=1}^N \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right)\n\\end{align*}\nIn particular, for $m=1$: \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) |_{m=1} & = f(y,1) + \\sum_{i=1}^N \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\\\\n& = L \\sum_{i=1}^N \\frac{y_i}{L} \\log \\left(\\frac{y_{i-1}\/L}{y_i\/L} \\right) \\\\\n& = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right)\n\\end{align*}\nwhere, for the second equality, we used the trivial fact that $f(y,1)=0$. \nTaking second partial derivative of $f(y,m)$ w.r.t. $m$ gives:\n\\begin{align*}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) = & \\sum_{i=1}^N y_i^{m-1}\\log y_i \\left(y_{i+1}^m - y_i^m \\right) + \\sum_{i=1}^N y_i^{m-1} \\left(y_{i+1}^m \\log y_{i+1} - y_i^m \\log y_i \\right)\\\\\n& + \\sum_{i=1}^N \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right) \\\\\n& + m \\sum_{i=1}^N \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right)^2 - 4 y_i^{2m-1} \\log^2 y_i \\right)\n\\end{align*}\nIn particular, for $m=1$:\n\\begin{align}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) |_{m=1} = & \\sum_{i=1}^N \\left(y_{i+1} - y_i \\right) \\log y_i + \\sum_{i=1}^N \\left(y_{i+1} \\log y_{i+1} - y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^N \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^N \\left( y_{i+1} (\\log y_i + \\log y_{i+1})^2 - 4 y_i \\log^2 y_i\\right) \\nonumber \\\\\n\\label{eq:second-derivative-m-final-eq}\n= & \\sum_{i=1}^N \\log^2 y_i \\left(y_{i+1} - y_i \\right) + 2 \\sum_{i=1}^N \\log y_i \\left(y_{i+1} \\log y_{i+1} + y_{i+1} - y_i \\log y_i - y_i \\right) \\\\\n\\nonumber\n\\geq & \\, 0\n\\end{align}\nIt is easy to check that, $\\log z$, $\\log^2 z$ and $z + z \\log z$ are strictly increasing, strictly decreasing and strictly decreasing functions, respectively, for $z \\in (0,e^{-2})$. Therefore, Lemma~\\ref{lem:appendix-general-summation} implies that each of the two terms in \\eqref{eq:second-derivative-m-final-eq} is non-negative, and hence the lemma.\n\\qed\n\\end{proof}\n\n\\iffalse\n\n\\begin{lemma}\n\\label{lem:service-rate-dot-lower-bound}\nFor all $y \\in \\mathcal S_n^L$, $n \\in \\mathbb{N} \\setminus \\{1\\}$, $L > 0$:\n\\begin{equation}\n\\label{eq:service-rate-der-m}\n\\frac{\\partial}{\\partial m} f(y,m)|_{m=1} = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right) \\leq 0\n\\end{equation}\nAdditionally, if $L < e^{-2}$, then\n\\begin{equation}\n\\label{eq:service-rate-twice-der-m}\n\\frac{\\partial^2}{\\partial m^2} f(y,m)|_{m=1} \\geq 0\n\\end{equation}\nMoreover, equality holds true in \\eqref{eq:service-rate-der-m} and \\eqref{eq:service-rate-twice-der-m} if and only if $y = \\frac{L}{n}\\mathbf{1}$.\n\\end{lemma}\n\\begin{proof}\n\\ksmargin{need to exclude from the proof all $y$ such that $y_i=0$ for some $i \\in \\until{n}$}\nTaking the partial derivative of $f(y,m)$ with respect to $m$, we get that \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) & = \\frac{f(y,m)}{m} + m \\sum_{i=1}^n \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right)\n\\end{align*}\nIn particular, for $m=1$: \n\\begin{align*}\n\\frac{\\partial}{\\partial m} f(y,m) |_{m=1} & = f(y,1) + \\sum_{i=1}^n \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\\\\n& = L \\sum_{i=1}^n \\frac{y_i}{L} \\log \\left(\\frac{y_{i-1}\/L}{y_i\/L} \\right) \\\\\n& = - L D\\left(\\frac{y}{L} || P^- \\frac{y}{L}\\right)\n\\end{align*}\nwhere, for the second equality, we used the trivial fact that $f(y,1)=0$. \nTaking second partial derivative of $f(y,m)$ w.r.t. $m$ gives:\n\\begin{align*}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) = & \\sum_{i=1}^n y_i^{m-1}\\log y_i \\left(y_{i+1}^m - y_i^m \\right) + \\sum_{i=1}^n y_i^{m-1} \\left(y_{i+1}^m \\log y_{i+1} - y_i^m \\log y_i \\right)\\\\\n& + \\sum_{i=1}^n \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i^{2m-1} \\log y_{i} \\right) \\\\\n& + m \\sum_{i=1}^n \\left( y_i^{m-1} y_{i+1}^m \\left(\\log y_i + \\log y_{i+1} \\right)^2 - 4 y_i^{2m-1} \\log^2 y_i \\right)\n\\end{align*}\nIn particular, for $m=1$:\n\\begin{align}\n\\frac{\\partial^2}{\\partial m^2} f(y,m) |_{m=1} = & \\sum_{i=1}^n \\left(y_{i+1} - y_i \\right) \\log y_i + \\sum_{i=1}^n \\left(y_{i+1} \\log y_{i+1} - y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^n \\left(y_{i+1} \\left(\\log y_i + \\log y_{i+1} \\right) - 2 y_i \\log y_i \\right) \\nonumber \\\\\n& + \\sum_{i=1}^n \\left( y_{i+1} (\\log y_i + \\log y_{i+1})^2 - 4 y_i \\log^2 y_i\\right) \\nonumber \\\\\n\\label{eq:second-derivative-m-final-eq}\n= & \\sum_{i=1}^n \\log^2 y_i \\left(y_{i+1} - y_i \\right) + 2 \\sum_{i=1}^n \\log y_i \\left(y_{i+1} \\log y_{i+1} + y_{i+1} - y_i \\log y_i - y_i \\right) \\\\\n\\nonumber\n\\geq & \\, 0\n\\end{align}\nIt is easy to check that, $\\log z$, $\\log^2 z$, and $z + z \\log z$ are strictly increasing, strictly decreasing and strictly decreasing functions, respectively, for $z \\in (0,e^{-2})$. Therefore, Lemma~\\ref{lem:appendix-general-summation} implies that each of the two terms in \\eqref{eq:second-derivative-m-final-eq} is non-negative, and hence the lemma.\n\\end{proof}\n\n\\fi\n\nLemma~\\ref{lem:service-rate-dot-lower-bound} implies that, for sufficiently small $L$, $f(y,m)$ is locally convex in $m$. One can use this property along with an exact expression for $\\frac{\\partial}{\\partial m}f(y,m)$ in Lemma~\\ref{lem:service-rate-dot-lower-bound} at $m=1$, and the fact that $f(y,1)=0$ for all $y$, to develop a linear approximation in $m$ of $f(y,m)$ around $m=1$. The following lemma derives this approximation, as also suggested by Figure~\\ref{fig:service-rate-dot-vs-m}.\n\n\\begin{figure}[htb!]\n \\centering\n \\includegraphics[width=5cm]{service-rate-dot-vs-m} \n \\caption{$f(y,m)$ vs. $m$ for a typical $y \\in \\mathcal S_{10}$.}\n \\label{fig:service-rate-dot-vs-m}\n\\end{figure}\n\n\\begin{lemma}\n\\label{lem:service-rate-dot-deltay-bound}\nFor a given $y \\in \\text{int}(\\mathcal S_N^L)$, $n \\in \\mathbb{N}$, $L \\in (0,e^{-2})$, there exists $\\underbar{m}(y) \\in [0,1)$ such that \n$$\n\\frac{d}{dt} s(y) \\geq 2 \\frac{(1-m)}{L} \\left(y_{\\mathrm{max}} - y_{\\text{min}} \\right)^2 \\, , \\qquad \\forall \\, m \\in [\\underbar{m}(y),1]\n$$\n\\end{lemma}\n\\begin{proof}\nFor a given $y \\in \\text{int}(\\mathcal S_N^L)$, the local convexity of $f(y,m):=\\frac{d}{dt} s(y)$ in $m$, and the expression of $\\frac{\\partial}{\\partial m}f(y,m)$ at $m=1$ in Lemma~\\ref{lem:service-rate-dot-lower-bound} implies that $\\frac{d}{dt} s(y) \\geq (1-m) L D\\left(\\frac{y}{L} || P^-\\frac{y}{L} \\right)$ for sufficiently small $m<1$. Pinsker's inequality implies $D\\left(\\frac{y}{L} || P^-\\frac{y}{L} \\right) \\geq \\frac{\\|y-P^-y\\|_1^2}{2 L^2}$. This, combined with the fact that $\\|y-P^-y\\|_1 \\geq 2(y_{\\mathrm{max}}-y_{\\text{min}})$ for all $y \\in \\text{int}(\\mathcal S_N^L)$, gives the lemma.\n\\qed\n\\end{proof}\n\n\n\\section{Simulations}\n\\label{sec:simulations}\nIn this section, we present simulation results on throughput analysis, and compare with our theoretical results from previous sections. \n \n \n\n\n\n \n\n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=5cm]{lambda-m-lowerUpper-w-phase-transition-separated-v2}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=5cm]{lambda-m-lowerUpper-uniform-w-phase-transition-separated-v2}}\n\\caption{Comparison between theoretical estimates of throughput from Theorem~\\ref{thm:superlinear-bound-empty}, and range of numerical estimates from simulations, for zero initial condition. The parameters used for this case are: $L = 1$, $\\delta = 0.1$, and (a) $\\varphi = \\delta_0$, $\\psi = \\delta_L$, (b) $\\varphi = U_{[0,L]}$, $\\psi=U_{[0,L]}$.\n \\label{fig:lambda-m-empty}}\n\\end{figure}\n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{lambda-m-busy-period-L100-v2} \n\\caption{Comparison between theoretical estimates of throughput from Theorem \\ref{thm:superlinear-bound-empty}, and range of numerical estimates from simulations, for zero initial condition. The parameters used for this case are: $L = 100$, $\\delta = 0.1$, $T=10$, and $\\varphi = \\delta_0$, $\\psi = \\delta_L$.}\n\\label{fig:superlinear-large-L}\n\\end{center}\n\\end{figure} \n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{phase-transition-non-empty-separated-v2} \n\\caption{Comparison between theoretical estimates of throughput from Theorem~\\ref{thm:superlinear-bound-non-empty}, and range of numerical estimates from simulations. The parameters used for this case are: $L = 1$, $\\delta = 0.1$, $\\varphi = \\delta_0$, $\\psi = \\delta_L$, $w_0=1$ and $n_0=4$, $x_1(0) =0.6, x_2(0) =0.7, x_3(0) =0.8, x_4(0) =0.9$.}\n\\label{fig:phase-transition-non-empty}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\nFigures~\\ref{fig:lambda-m-empty}, \\ref{fig:superlinear-large-L} and \\ref{fig:phase-transition-non-empty} show comparison between the lower bound on throughput over finite time horizons, as given by Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, and the corresponding numerical estimates from simulations. \n\n Figures~\\ref{fig:lambda-m-empty} and \\ref{fig:superlinear-large-L} are for zero initial condition, and Figure~\\ref{fig:phase-transition-non-empty} is for non-zero initial condition.\n\nFigures~\\ref{fig:sublinear-lowerbound} and \\ref{fig:superlinear-batch} show comparison between the lower bound on throughput as given by the bacth release control policy, as per Theorems~\\ref{thm:main-sub-linear} and \\ref{thm:main-batch-super-linear}, respectively, under a couple of representative values of maximum permissible perturbation $\\eta$. In particular, Figure~\\ref{fig:sublinear-lowerbound} demonstrates that the lower bound achieved from Theorem \\ref{thm:main-sub-linear} increases drastically as $m \\to 0^+$. Both the figures also confirm that the throughput indeed increases with increasing maximum permissible perturbation $\\eta$.\n\nIt is instructive to compare Figures~\\ref{fig:lambda-m-empty}(b) and \\ref{fig:superlinear-batch}(a), both of which depict throughput estimates for the sub-linear case, however obtained from different methods, namely busy period distribution and batch release control policy. Accordingly, one should bear in mind that the two bounds have different qualifiers attached to them: the bound in Figure~\\ref{fig:lambda-m-empty}(b) is valid probabilistically only over a finite time horizon, whereas the bound in Figure~\\ref{fig:superlinear-batch}(a) is valid with probability one, although under a perturbation to the arrival process. \n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{sublinear-lowerbound} \n\\caption{Theoretical estimates of throughput from Theorems~\\ref{thm:main-sub-linear} for different values of $\\eta$. The parameters used for this case are: $L = 1$, $\\varphi = U_{[0,L]}$, $\\psi = U_{[0,L]}$, and $w_0=0$ . Note that the vertical axis is in logarithmic scale.}\n\\label{fig:sublinear-lowerbound}\n\\end{center}\n\\end{figure}\n\n\n\n \n\n\n\n\\begin{figure}\n\\centering\n\\subfigure[]{\\includegraphics[width=5cm]{sublinear-small-L}}\n\\centering\n\\hspace{0.3in}\n\\subfigure[]{\\includegraphics[width=5cm]{superlinear-batch-new-v2}}\n\\caption{Theoretical estimates of throughput from Theorem \\ref{thm:main-batch-super-linear}, and numerical estimates from simulations for different The parameters used for this case are: $\\varphi=U_{[0,L]}$, $\\psi=U_{[0,L]}$, and (a) $L = 1$, (b) L = 100. \\label{fig:superlinear-batch}}\n\\end{figure}\n\n \\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=5cm]{upper-bound-queue-length-linear-large-lambdas} \n\\caption{Comparison between the empirical expectation of the queue length and the upper bound suggested by Remark~\\ref{remark:waiting-time}. We let the simulations run up to time $t=80,000$. The parameters used for this case are: $L=1$, $m=1$, $\\varphi=\\delta_0$, $\\psi=\\delta_L$. For these values, we have $\\lambda_{\\text{max}}=1$.}\n\\label{fig:queue-length-linear}\n\\end{center}\n\\end{figure} \n\nFinally, Figure~\\ref{fig:queue-length-linear} shows a good agreement between queue length bound suggested by Remark~\\ref{remark:waiting-time}, and the corresponding numerical estimates in the linear case. \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{Throughput Bounds under Batch Release Control Policy}\nIn this section, we consider a \\emph{time-perturbed} version of the arrival process. \nFor a given realization of arrival times, $\\{t_1,t_2,\\cdots\\}$, consider a perturbation map $t_i^{\\prime} \\equiv t_i^{\\prime}(t_1,\\ldots,t_i)$ satisfying $t_i^{\\prime}\\geq t_i$ for all $i$, which prescribes the perturbed arrival times. The magnitude of perturbation is defined as $\\eta := E\\left(t_i^{\\prime}-t_i\\right)$, where the expectation is with respect to the Poisson process with rate $\\lambda$ that generates the arrival times.\n\nWe prove boundedness of the queue length under a specific perturbation map. This perturbation map is best understood in terms of a control policy that governs the release of arrived vehicles into HTQ. In order to clarify the implementation of the control policy, we decompose the proposed HTQ into two queues in series: denoted as HTQ1 and HTQ2, both of which have the same geometric characteristics as HTQ, i.e., a circular road segment of length $L$ (see Figure \\ref{fig:htq1-htq2} for illustrations). The original arrival process for HTQ, i.e. spatio-temporal Poisson process with rate $\\lambda$ and spatial distribution $\\varphi$ is now the arrival process for HTQ1. Vehicles remain stationary at their arrival locations in HTQ1, until released by the control policy into HTQ2. Upon released into HTQ2, vehicles travel according to \\eqref{eq:dynamics-moving-coordinates} until they depart after traveling a distance that is sampled from $\\psi$, as in the case of HTQ. The time of release of the vehicles into HTQ2 correspond to their perturbed arrival times $t_1^{\\prime}, t_2^{\\prime}, \\ldots$. The average waiting time in HTQ1 under the given release control policy is then the magnitude of perturbation in the arrival times. \n\n\n\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[width=14cm]{htq1-htq2-decomposed} \n\\caption{Decomposition of HTQ into HTQ1 and HTQ2 in series.}\n\\label{fig:htq1-htq2}\n\\end{center}\n\\end{figure}\n\nWe consider the following class of release control policy, for which we recall from the problem setup in Section~\\ref{sec:problem-formulation} that $\\text{supp}(\\varphi)=[0,\\ell]$ for some $\\ell \\in [0,L]$. \n\n\n\\begin{definition}[Batch Release Control Policy $\\pi^b_\\triangle$]\n\\label{def:batch-release-control-policy}\nDivide $[0,\\ell]$ into sub-intervals, each of length $\\triangle$, enumerated as $1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil$. Let $T_1$ be the first time instant when HTQ2 is empty. At time $T_1$, release one vehicle each, if present, from all odd-numbered sub-intervals in $\\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$ simultaneously into HTQ2. Let $T_2$ be the next time instant when HTQ2 is empty. At time $T_2$, release one vehicle each, if present, from all even-numbered sub-intervals in $\\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$ simultaneously into HTQ2. Repeat this process of alternating between releasing one vehicle each from odd and even-numbered sub-intervals every time that HTQ2 is empty.\n\\end{definition}\n\n\n\n\\begin{remark}\n\\begin{enumerate}\n\\item Under $\\pi^b_{\\triangle}$, when vehicles are released into HTQ2, the inter-vehicle distances in the front and rear of each vehicle being released is at least equal to $\\triangle$. \n\\item The order in which vehicles are released into HTQ2 from HTQ1 under $\\pi^b_{\\triangle}$ may not be the same as the order of arrivals into HTQ1.\n\\end{enumerate}\n\\end{remark}\n\nIn the next two sub-sections, we analyze the performance of the batch release control policy for sub-linear and super-linear cases. \n\\subsubsection{The Sub-linear Case}\nIn this section, we derive a lower bound on throughput when $m\\in(0,1)$. We first derive a trivial lower bound in Proposition \\ref{prop:throughput-bound-sublinear} implied by Lemma \\ref{lem:service-rate-jumps} and Remark \\ref{rem:service-rate-jump-arrival}. Next, we improve this lower bound in Theorem \\ref{thm:main-sub-linear} under a under a batch release control policy, $\\pi_{\\triangle}^b$. \n\\begin{proposition}\n\\label{prop:throughput-bound-sublinear}\nFor any $L > 0$, $m\\in(0,1)$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$:\n$$\n \\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0) \\geq L^m\/\\bar{\\psi} \n$$\n\\end{proposition}\n\\begin{proof}\nRemark \\ref{rem:service-rate-jump-arrival} implies that, for $m \\in (0,1)$, the service rate does not decrease due to arrivals. Therefore, a simple lower bound on the service rate for any state is the service rate when there is only one vehicle in the system, i.e., $L^m$. Therefore, \nthe workload process is upper bounded as\n$w(t) = w_0+r(t)-\\int_0^ts(z)\\mathrm{d} z \\leq w_0+r(t) -L^m(t-\\mathcal{I}(t)), \\quad \\forall t\\geq 0$, \nwhere $r(t)$ and $\\mathcal{I}(t)$ denote the renewal reward and the idle time processes, respectively, as introduced in the proof of Proposition \\ref{prop:unstable}. Similar to the proof of Proposition \\ref{prop:unstable}, it can be shown that, if $\\lambda < L^m\/\\bar{\\psi}$, then the workload, and hence the queue length, goes to zero in finite time with probability one. \n\\qed\n\\end{proof}\n\nNext, we establish better throughput guarantees than Proposition~\\ref{prop:throughput-bound-sublinear}, under a batch release control policy, $\\pi_{\\triangle}^b$.\n\nThe next result characterizes the time interval between release of successive batches into HTQ2 under $\\pi_\\triangle^b$.\n\n\\begin{lemma}\n\\label{eq:successive-release-time-difference}\nFor given $\\lambda>0$, $\\triangle > 0$, $\\varphi \\in \\Phi$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m\\in(0,1)$, $x_0\\in[0,L]^{n_0}$, $L>0$, $n_0\\in \\mathbb{N}$, let $T_1$, $T_2$, $\\ldots$ denote the random variables corresponding to time of successive batch releases into HTQ2 under $\\pi_{\\triangle}^b$. Then, $T_1 \\leq \\frac{n_0R}{L^m}$, $T_{i+1}-T_i \\leq R\/\\triangle^m$ for all $i \\geq 1$, and $y_{\\text{min}}(t) \\geq \\triangle$ for all $t \\geq T_1$.\n\\end{lemma}\n\\begin{proof}\nSince the maximum distance to be traveled by every vehicle is upper bounded by $R$, the initial workload satisfies $w_0\\leq n_0R$. Since the minimum service rate for $m \\in (0,1)$ is $L^m$ (see proof of Proposition~\\ref{prop:throughput-bound-sublinear}), with no new arrivals, it takes at most $w_0\/L^m=n_0 R\/L^m$ amount of time for the system to become empty. This establishes the bound on $T_1$. \n\nLemma~\\ref{lem:vehicle-distance-monotonicity} implies that, under $\\pi^b_{\\triangle}$, the minimum inter-vehicle distance in HTQ2 is at least $\\triangle$ after $T_1$. \nThis implies that $y_{\\text{min}}(t) \\geq \\triangle$ for all $t \\geq T_1$, and hence the minimum speed of every vehicle in HTQ2 is at least $\\triangle^m$ after $T_1$. Since the maximum distance to be traveled by every vehicle is $R$, this implies that the time between release of a vehicle into HTQ2 and its departure is upper bounded by $R\/\\triangle^m$, which in turn is also an upper bound on the time required by all the vehicles released in one batch to depart from the system. \n\\qed\n\\end{proof}\nLet $N_1(t)$ and $N_2(t)$ denote the queue lengths in HTQ1 and HTQ2, respectively, at time $t$. Lemma~\\ref{eq:successive-release-time-difference} implies that, for every $\\triangle>0$, $N_2(t)$ is upper bounded for all $t \\geq T_1$. The next result identifies conditions under which $N_1(t)$ is upper bounded. \n\n\n\n\nFor $F>0$, let $\\Phi_F:=\\setdef{\\varphi \\in \\Phi}{\\sup_{x \\in [0,\\ell]} \\varphi(x) \\leq F}$. For subsequent analysis, we now derive an upper bound on the \\emph{load factor}, i.e., the ratio of the arrival and departure rates, associated with a typical sub-queue of HTQ1 among $\\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$. It is easy to see that, for every $\\varphi \\in \\Phi_F$, $F>0$, the arrival process into every sub-queue is Poisson with arrival rate upper bounded by $\\lambda F \\triangle$. Lemma~\\ref{eq:successive-release-time-difference} implies that the departure rate is at least $\\triangle^m\/2R$. Therefore, the load factor for every sub-queue is upper bounded as \n\\begin{equation}\n\\label{eq:load-factor-upper-bound}\n\\rho \\leq \\frac{2 R \\lambda F \\triangle}{\\triangle^m}=2 R \\lambda F \\triangle^{1-m}\n\\end{equation}\nIn particular, if \n\\begin{equation}\\label{eq:triangle-star}\n\\triangle < \\triangle^*(\\lambda):=\\left(2 R \\lambda F \\right)^{-\\frac{1}{1-m}},\n\\end{equation} \nthen $\\rho<1$. It should be noted that for $n_0<+\\infty$, by Lemma \\ref{eq:successive-release-time-difference}, $T_1<+\\infty$. The service rate is zero during $[0,T_1]$; however, since $T_1$ is finite, this does not affect the computation of load factor. \n\n\n\\begin{proposition}\n\\label{prop:pi1-policy}\nFor any $\\lambda>0$, $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m\\in(0,1)$, $x_0\\in[0,L]^{n_0}$, $L>0$, $n_0\\in \\mathbb{N}$, for sufficiently small $\\triangle$, $N_1(t)$ is bounded for all $t \\geq 0$ under $\\pi_{\\triangle}^b$, almost surely.\n\\end{proposition}\n\\begin{proof}\nBy contradiction, assume that $N_1(t)$ grows unbounded. This implies that there exists at least one sub-queue, say $ j \\in \\{1, 2, \\ldots, \\lceil \\frac{\\ell}{\\triangle} \\rceil\\}$, such that its queue length, say $N_{1,j}(t)$, grows unbounded. In particular, this implies that there exists $t_0 \\geq T_1$ such that $N_{1,j}(t) \\geq 2$ for all $t \\geq t_0$. Therefore, for all $t \\geq t_0$, the ratio of arrival rate to departure rate for the $j$-th sub-queue is given by \\eqref{eq:load-factor-upper-bound}, which is a decreasing function of $\\triangle$, and hence becomes strictly less than one for sufficiently small $\\triangle$. A simple application of the law of large numbers then implies that, almost surely, $N_{1,j}(t)=0$ for some finite time, leading to a contradiction.\n\\qed \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nThe following result gives an estimate of the mean waiting time in a typical sub-queue in HTQ1 under the $\\pi_{\\triangle}^b$ policy.\n\n\\begin{proposition}\n\\label{eq:waiting-time}\nFor $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$, $m \\in (0,1)$, there exists a sufficiently small $\\triangle$ such that the average waiting time in HTQ1 under $\\pi_{\\triangle}^b$ is upper bounded as: \n\\begin{equation}\n\\label{eq:W-upper-bound}\nW \\leq R (2 R \\lambda F )^{\\frac{m}{1-m}} \\left(\\frac{2}{m^{\\frac{m}{1-m}}} + \\frac{m}{m^{\\frac{m}{1-m}}-m^{\\frac{1}{1-m}}} \\right).\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nIt is easy to see that the desired waiting time corresponds to the system time of an M\/D\/1 queue with load factor given by \\eqref{eq:load-factor-upper-bound} along with the arrival and departure rates leading to \\eqref{eq:load-factor-upper-bound}. Note that, by Lemma \\ref{eq:successive-release-time-difference}, for finite $n_0$, the value of $T_1$ is finite and does not affect the average waiting time. Therefore, using standard expressions for M\/D\/1 queue~\\cite{Kleinrock:75}, we get that the waiting time in HTQ1 is upper bounded as follows for $\\rho <1$:\n\\begin{align}\nW \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m} \\frac{\\rho}{1-\\rho} & \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m} \\frac{1}{1-\\rho} \\nonumber \\\\ \n\\label{eq:waiting-time-upper-bound}\n& \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m -2 R \\lambda F \\triangle} \n\\end{align} \nIt is easy to check that the minimum of the second term in \\eqref{eq:waiting-time-upper-bound} over $\\big(0,\\triangle^*(\\lambda)\\big)$ occurs at $\\triangle = \\left(\\frac{m}{2 R \\lambda F} \\right)^{\\frac{1}{1-m}}$. Substitution in the right hand side of the first inequality in \\eqref{eq:waiting-time-upper-bound} gives the result.\n\\qed\n\\end{proof}\n\n\\begin{remark}\n\\label{rem:W-limit}\n\\eqref{eq:W-upper-bound} implies that, for every $R>0$, $F>0$, $\\lambda>0$, we have $W \\to 2R$ as $m \\to 0^+$. \n\\end{remark}\n\nWe extend the notation introduced in \\eqref{eq:throughput-def} to $\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta,\\eta)$ to also show the dependence on maximum allowable perturbation $\\eta$. This is not to be confused with the notation for $\\lambda_{\\text{max}}$ used in Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, where we used the notion of throughput over finite time horizons. We choose to use the same notations to maintain brevity. \n\nIn order to state the next result, for given $R>0$, $F>0$, $m \\in (0,1)$ and $\\eta \\geq 0$, let $\\tilde{W}(m,F,R,\\eta)$ \nbe the value of $\\lambda$ for which the right hand side of \\eqref{eq:W-upper-bound} is equal to $\\eta$, if such a $\\lambda$ exists and is at least $L^m\/\\bar{\\psi}$, and let it be equal to $L^m\/\\bar{\\psi}$, otherwise. The lower bound of $L^m\/\\bar{\\psi}$ in the definition of $\\tilde{W}$ is inspired by Proposition~\\ref{prop:throughput-bound-sublinear}. The next result formally states $\\tilde{W}$ as a lower bound on $\\lambda_{\\text{max}}$. \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\\begin{theorem}\\label{thm:main-sub-linear}\nFor any $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m \\in (0,1)$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, $L>0$, and maximum permissible perturbation $\\eta \\geq 0$, \n$$\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0,\\eta) \\geq \\tilde{W}(m,F,R,\\eta)\n$$\nIn particular, if $\\eta > 2R$, then $\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0,\\eta) \\to + \\infty$ as $m \\to 0^+$.\n\\end{theorem}\n\\begin{proof}\nConsider any $\\lambda \\leq\\tilde{W}(m,F,R,\\eta)$, and $\\triangle \\leq \\left(\\frac{m}{2 R \\lambda F} \\right)^{\\frac{1}{1-m}}$. Under policy $\\pi_\\triangle^b$, Lemma \\ref{eq:successive-release-time-difference} and Proposition \\ref{prop:pi1-policy} imply that, for finite $n_0$, $N_2(t)$ and $N_1(t)$ remain bounded for all times, with probability one. Also, for $\\lambda = \\tilde{W}(m,F,R,\\eta)$, by Proposition \\ref{eq:waiting-time} and the definition of $\\tilde{W}(m,F,R,\\eta)$, the introduced perturbation remains upper bounded by $\\eta$. Since the right hand side of \\eqref{eq:W-upper-bound} is monotonically increasing in $\\lambda$, perturbations remain bounded by $\\eta$ for all $\\lambda \\leq \\tilde{W}(m,F,R,\\eta)$. \nIn particular, by Remark \\ref{rem:W-limit}, we have $W\\to 2R$ as $m\\to 0^+$. In other words, as $m\\to 0^+$, the magnitude of the introduced perturbation becomes independent of $\\lambda$. \nTherefore, when $\\eta >2R$, and $m\\to 0^+$ throughput can grow unbounded while perturbation and queue length remains bounded.\n\\qed\n\\end{proof}\n\n\n\n\n\\begin{remark}\nWe emphasize that the only feature required in a batch release control policy is that, at the moment of release, the front and rear distances for the vehicles being released should be greater than $\\triangle$. The requirement of the policy in Definition~\\ref{def:batch-release-control-policy} for the road to be empty at the moment of release makes the control policy conservative, and hence affects the maximum permissible perturbation. In fact, for special spatial distributions, e.g., when $\\varphi$ is a Dirac delta function and the support of $\\psi$ is $[0,L-\\triangle])$, one can relax the conservatism to guarantee unbounded throughput for arbitrarily small permissible perturbation. \n\\end{remark}\n\n\\subsubsection{The Super-linear Case}\n\nIn this section, we study the throughput for the super-linear case under perturbed arrival process with a maximum permissible perturbation of $\\eta$. For this purpose, we consider the batch release control policy $\\pi_{\\triangle}^b$, defined in Definition \\ref{def:batch-release-control-policy}, for our analysis. Time intervals between release of successive batches, under $\\pi_{\\triangle}^b$, are characterized the same as Lemma \\ref{eq:successive-release-time-difference}. However, in the super linear case, by Lemma \\ref{lem:service-rate-bounds}, the initial minimum service rate is $L^mn_0^{1-m}$. Therefore, the time of first release is bounded as $T_1 \\triangle^*(\\lambda).\n\\end{equation}\nIt should be noted that since the batch release control policy iteratively releases from odd and even sub-queues, we need at least two sub-queues to be able to implement this policy. As a result, $\\triangle$ cannot be arbitrary large and $\\triangle<\\ell\/2$. This constraint gives the following bound on the admissible throughput under this policy\n\\begin{equation}\\label{eq:lambda-star-superlinear}\n\\lambda<\\lambda^*:=(\\ell\/2)^{m-1}\/2RF\n\\end{equation}\n The following result shows that for the above range of throughput, the queue length in HTQ1, $N_1(t)$, remains bounded at all times. \n \\begin{proposition}\n\\label{prop:pi1-policy-super-linear}\nFor any $\\lambda<\\lambda^*$, $\\triangle\\in\\big(\\triangle^*(\\lambda),\\ell\/2\\big]$, $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m>1$, $x_0\\in[0,L]^{n_0}$, $L>0$, $n_0\\in \\mathbb{N}$, $N_1(t)$ is bounded for all $t \\geq 0$ under $\\pi_{\\triangle}^b$, almost surely.\n\\end{proposition}\n\n\\begin{proof}\nThe proof is similar to proof of Proposition \\ref{prop:pi1-policy}. In particular, by \\eqref{eq:triangle-star-superlinear} and \\eqref{eq:lambda-star-superlinear}, one can show that load factor \\eqref{eq:load-factor-upper-bound} remains strictly smaller than one. This implies that no sub-queue in HTQ1 can grow unbounded, and $N_1(t)$ remains bounded for all times, with probability one. \n\\qed\n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{eq:waiting-time-super-linear}\nFor any $\\lambda<\\lambda^*$, $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$, $m> 1$, the average waiting time in HTQ1 under $\\pi_{\\triangle}^b$ for $\\triangle = \\ell\/2$ is upper bounded as: \n\\begin{equation}\n\\label{eq:waiting-time-super-linear-eq}\nW \\leq \\frac{2R}{(\\ell\/2)^m}+\\frac{R}{(\\ell\/2)^m}\\frac{2R\\lambda F(\\ell\/2)^{1-m}}{1-2R\\lambda F(\\ell\/2)^{1-m}}\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nThe proof is very similar to the proof of Proposition \\ref{eq:waiting-time}. Thus, we get the following bounds:\n\\begin{align*}\nW \\leq \\frac{2R}{\\triangle^m} + \\frac{R}{\\triangle^m} \\frac{\\rho}{1-\\rho} \\leq \\frac{2R}{\\triangle^m}+\\frac{R}{\\triangle^m}\\frac{2R\\lambda F\\triangle^{1-m}}{1-2R\\lambda F\\triangle^{1-m}}\n\\end{align*} \nThe right hand side of the above inequality is a decreasing function of $\\triangle$; therefore, $\\triangle = \\ell\/2$ minimizes it, and gives \\eqref{eq:waiting-time-super-linear-eq}. \n\\qed\n\\end{proof}\n\nLet $\\hat{W}(m,F,R,\\eta)$ \nbe the value of $\\lambda$ for which the right hand side of \\eqref{eq:waiting-time-super-linear-eq} is equal to $\\eta$, if such a $\\lambda\\leq \\lambda^*$ exists, and let it be equal to $\\lambda^*$ otherwise. Note that since the right hand side of \\eqref{eq:waiting-time-super-linear-eq} is monotonically increasing in $\\lambda$, for all $\\lambda\\leq \\hat{W}(m,F,R,\\eta)$ the introduced perturbation remains upper bounded by $\\eta$. \n\\begin{theorem}\\label{thm:main-batch-super-linear}\nFor any $\\varphi \\in \\Phi_F$, $F>0$, $\\psi \\in \\Psi$ with $\\text{supp}(\\psi)=[0,R]$, $R>0$, $m >1$, $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, $L>0$, and maximum permissible perturbation $\\eta \\geq 0$, \n$$\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta=0,\\eta) \\geq \\hat{W}(m,F,R,\\eta)\n.$$\n\\end{theorem}\n\\begin{proof}\nFor any $\\lambda <\\hat{W}(m,F,R,\\eta)$, under $\\pi_\\triangle^b$, Lemma \\ref{eq:successive-release-time-difference} and Proposition \\ref{prop:pi1-policy-super-linear} imply that, for finite $n_0$, $N_2(t)$ and $N_1(t)$ remain bounded for all times, with probability one. Also, by Proposition \\ref{eq:waiting-time-super-linear} and the definition of $\\hat{W}(m,F,R,\\eta)$, the introduced perturbation remains upper bounded by $\\eta$. \n\\qed\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\\subsection{Throughput Bounds for the Super-linear Case from Busy Period Calculations}\\label{subsec:superlinear}\nIn this section, we derive lower bound on the throughput for the super-linear case. The next result computes a bound on the probability that the queue length of the HTQ satisfies a given upper bound over a given time interval, using the probability distribution functions from \\eqref{eq:G-def}.\nIn Propositions~\\ref{prop:nbp-lower-bound} and \\ref{prop:nbn-lower-bound}, for the sake of clarity, we add explicit dependence on $\\lambda$ to this probability distribution function. \n\n\n\n\\begin{proposition}\\label{prop:nbp-lower-bound}\nFor any $m>1$, $M\\in \\mathbb{N}$, $L>0$, $\\lambda >0$, $\\varphi\\in \\Phi$, $\\psi \\in \\Psi$, and zero initial condition $x_0=0$, the probability that the queue length is upper bounded by $M$ over a given time interval $[0,T]$ satisfies the following bound: \n\\begin{equation}\n\\label{eq:nbp-lower-bound}\n \\Pr \\big(N(t) \\leq M \\quad \\forall t \\in [0,T] \\big)\\geq \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{r,L^m M^{1-m}}(t,n,\\psi,\\lambda) \\, \\mathrm{d} t\n \\end{equation}\n \\end{proposition}\n \\begin{proof}\nLet us denote the current queueing system as HTQ-f. We shall compare queue lengths between HTQ-f and a slower queueing system HTQ-s, which starts from the same (zero) initial condition, and experiences the same realizations of arrival times, locations and travel distances. Let every incoming vehicle into HTQ-s and HTQ-f be tagged with a unique identifier. At time $t$, let $\\mathcal J(t)$ be the set of identifiers of vehicles present both in HTQ-s and HTQ-f, $\\mathcal J_{s\/f}(t)$ be the set of identifiers of vehicles present only in HTQ-s, and $\\mathcal J_{f\/s}(t)$ be the set of identifiers of vehicles present only in HTQ-f.\nLet $v^f_i$ denote the speed of the vehicle in HTQ-f with identifier $i \\in \\mathcal J(t) \\cup \\mathcal J_{f\/s}(t)$, as determined by the car-following behavior underlying \\eqref{eq:dynamics-moving-coordinates}. The vehicle speeds in HTQ-s are not governed by the car following behavior, but are rather related to the speeds of vehicles in HTQ-f as: \n \\begin{equation}\n \\label{eq:htq-s-speed}\n\t\tv^s_i(t) =\\left\\{\\begin{array}{ll}\n\t\t\\displaystyle v^f_i(t) \\frac{p}{v^f(t)} \\frac{|\\mathcal J(t)|}{|\\mathcal J(t)| + |\\mathcal J_{s\/f}(t)|} \t&i \\in \\mathcal J(t)\\\\[15pt]\n\t\t\\displaystyle \\frac{p}{|\\mathcal J(t)| + |\\mathcal J_{s\/f}(t)|}& i \\in \\mathcal J_{s\/f}(t)\\,\\end{array}\\right.\n\t\\end{equation}\nwhere $v^f(t):=\\sum_{i \\in \\mathcal J(t)}v_i^f(t)$ is the sum of speeds of vehicles in HTQ-f that are also present in HTQ-s at time $t$, and $p$ is a parameter to be specified. Indeed, note that $\\sum_{i \\in \\mathcal J(t) \\cup \\mathcal J_{s\/f}(t)} v_i^s(t) \\equiv p$, i.e., $p$ is the (constant) service rate of HTQ-s. \n\nConsider a realization where the number of arrivals into HTQ-s with $p=L^m M^{1-m}$ during any busy period overlapping with $[0,T]$ does not exceed $M$. We refer to such a realization as \\emph{event} in the rest of the proof. Since the maximum queue length during a busy period is trivially upper bounded by the number of arrivals during that busy period, conditioned on the event, we have \n\\begin{equation}\n\\label{eq:slow-queue-queue-length-bound}\nN_s(t) \\leq M, \\qquad t \\in [0,T]\n\\end{equation}\n\nConsider the union of departure epochs from HTQ-s and HTQ-f in $[0,T]$: $0=\\tau_0 \\leq \\tau_1 \\leq \\ldots$. If $\\mathcal J_{f\/s}(\\tau_k)=\\emptyset$ for some $k \\geq 0$, then $\\mathcal J_{f\/s}(t)=\\emptyset$ for all $t \\in (\\tau_k,\\tau_{k+1})$. Hence, the service rate for HTQ-f over the interval $(\\tau_k,\\tau_{k+1})$ is $v^f(t)$, which, conditioned on the event, is lower bounded by $L^m M^{1-m}=p$ by Lemma \\ref{lem:service-rate-bounds}.\nTherefore, $p\/v^f(t) \\leq 1$ over $(\\tau_k,\\tau_{k+1})$, and hence \\eqref{eq:htq-s-speed} implies that all the vehicles with identifiers in $\\mathcal J_f$ will travel slower in HTQ-s in comparison to HTQ-f. In particular, this implies that $\\mathcal J_{f\/s}(\\tau_{k+1})=\\emptyset$. This, combined with the fact that $\\mathcal J_{f\/s}(\\tau_0)=\\emptyset$ (both the queues start from the same initial condition), we get that, conditioned on the event, $\\mathcal J_{s\/f}(t) \\equiv \\emptyset$, and hence $N(t) \\leq N_s(t)$ over $[0,T]$. Combining this with \\eqref{eq:slow-queue-queue-length-bound} gives that, conditioned on the event, $N(t) \\leq M$ over $[0,T]$. \n\n\nWe now compute the probability of the occurrence of the event using busy period calculations from Section~\\ref{sec:busy-period}. The event can be categorized by the maximum number of busy periods, say $r \\in \\mathbb{N}$, that overlap with $[0,T]$, i.e., the $r$-th busy period ends after time $T$ (and each of these busy periods has at most $M$ arrivals). Since these busy periods are interlaced with idle periods, the probability of the $r$-th busy period ending after time $T$ is lower bounded by the probability that the sum of the durations of $r$ busy periods is at least $T$. \\eqref{eq:G-def} implies that the latter quantity is equal to $\\sum_{n=1}^M \\int_T^{\\infty} G_{r,L^m M^{1-m}}(t,n,\\psi,\\lambda) \\, \\mathrm{d} t$. The proposition then follows by noting that this is true for any $r \\in \\mathbb{N}$.\n\\qed\n \\end{proof}\n \n \\begin{remark}\nIn the proof of Proposition~\\ref{prop:nbp-lower-bound}, when deriving probabilistic upper bound on the queue length over a given time horizon $[0,T]$, we neglected the idle periods in $[0,T]$. This introduces conservatism in the bound on the right hand side of \\eqref{eq:nbp-lower-bound}. Since the idle period durations are distributed independently and identically according to an exponential random variable (since the arrival process is Poisson), one could incorporate them into \\eqref{eq:nbp-lower-bound} by taking convolution of $G$ with idle period distributions. Our choice for not doing so here is to ensure conciseness in the presentation of bounds in \\eqref{eq:nbp-lower-bound}. The resulting conservatism is also present in Proposition~\\ref{prop:nbn-lower-bound}, and carries over to Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, as well as to the corresponding simulations reported in Figures~\\ref{fig:lambda-m-empty}, \\ref{fig:superlinear-large-L} and \\ref{fig:phase-transition-non-empty}. \n\\end{remark}\n \n The next result generalizes Proposition~\\ref{prop:nbp-lower-bound} for non-zero initial condition. Note that the non-zero initial condition only affects the first busy period; all subsequent busy periods will necessarily start from with zero initial condition.\n\n \\begin{proposition}\\label{prop:nbn-lower-bound}\n For any $m>1$, $M\\in\\mathbb{N}$, $L>0$, $\\lambda >0$, $\\varphi\\in \\Phi$, $\\psi \\in \\Psi$, initial condition $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, with associated workload $w_0>0$, the probability that the queue length is upper bounded by $M+n_0$ over a given time interval $[0,T]$ satisfies the following: \n\n $$\\Pr \\big(N(t) \\leq M + n_0 \\quad \\forall t \\in [0,T] \\big)\\geq \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{L^m (M+n_0)^{1-m}}(\\delta_{w_0}) * G_{r-1,L^m M^{1-m}}(\\psi) (t,n,\\lambda) \\, \\mathrm{d} t$$\n \\end{proposition}\n \\begin{proof}\nThe proof is similar to the proof of Proposition \\ref{prop:nbp-lower-bound}; however, since we consider $M$ number of new arrivals in each of the busy periods, the \\emph{event} of interest is when the queue length in HTQ-s does not exceed $M+n_0$ and $M$ in the first and subsequent busy periods, respectively, while operating with constant service rates $L^m (M+n_0)^{1-m}$ and $L^m M^{1-m}$, respectively. \n\\qed\n \\end{proof}\n\n\n\n \n\n\n\n \n\n\n\n \n\n We shall use Propositions~\\ref{prop:nbp-lower-bound} and \\ref{prop:nbn-lower-bound} to establish probabilistic lower bound for a finite time horizon version of the throughput defined in Definition \\ref{def:throughput}: for $T>0$, let \n \\begin{equation*}\n\\label{eq:throughput-def-finite-horizon}\n\\lambda_{\\text{max}}(L,m,\\varphi,\\psi,x_0,\\delta,T):= \\sup \\left\\{\\lambda \\geq 0: \\, \\Pr \\left( N(t;L, m, \\lambda, \\varphi, \\psi, x_0) < + \\infty, \\quad \\forall t \\in [0,T] \\right) \\geq 1 - \\delta \\right\\}.\n\\end{equation*}\n \n \n\n\\begin{theorem}\\label{thm:superlinear-bound-empty}\n For $L>0$, $m>1$, $\\varphi\\in\\Phi$, $\\psi\\in\\Psi$, $\\delta\\in(0,1)$, $T>0$, zero initial condition $x_0=0$, \n \\begin{equation}\n \\label{eq:sublinear-throughput-zero-initial-condition}\n \\lambda_{\\max}(L, m, \\varphi, \\psi, x_0,\\delta,T)\\geq \\sup_{M \\in \\mathbb{N}}\\;\\sup \\Big\\{\\lambda \\geq 0 \\; \\Big | \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{r,L^m M^{1-m}}(t,n,\\psi,\\lambda) \\, \\mathrm{d} t \\geq1-\\delta \\Big\\} \n \\end{equation}\n \\end{theorem}\n \\begin{proof}\nFollows from Proposition \\ref{prop:nbp-lower-bound}.\n\\qed\n \\end{proof} \n \n \n \n \\begin{theorem}\\label{thm:superlinear-bound-non-empty}\n For $L>0$, $m>1$, $\\varphi\\in\\Phi$, $\\psi\\in\\Psi$, $\\delta\\in(0,1)$, $T>0$, initial condition $x_0 \\in [0,L]^{n_0}$, $n_0 \\in \\mathbb{N}$, with associated workload $w_0>0$, \n \\begin{multline*}\n \\lambda_{\\max}(L, m, \\varphi, \\psi, x_0,\\delta,T) \\\\ \\geq \\sup_{M \\in \\mathbb{N}}\\;\\sup \\Big \\{\\lambda>0 \\; \\Big | \\sup_{r \\in \\mathbb{N}} \\, \\sum_{n=1}^M \\int_T^{\\infty} G_{L^m (M+n_0)^{1-m}}(\\delta_{w_0}) * G_{r-1,L^m M^{1-m}}(\\psi) (t,n,\\lambda)\\geq 1-\\delta \\Big\\} \n \\end{multline*}\n \\end{theorem}\n \\begin{proof}\n Follows from Proposition \\ref{prop:nbn-lower-bound}.\n \\qed\n \\end{proof}\n \n \\begin{remark}\n In Theorems~\\ref{thm:superlinear-bound-empty} and \\ref{thm:superlinear-bound-non-empty}, we implicitly assume the rather standard convention that supremum over an empty set is zero.\n \\end{remark}\n\n \n\n \n\n\n\n\n\\section{Throughput Analysis}\n\\label{sec:throughput-analysis}\n\n\\input{linear}\n\n\\input{nonlinear}\n\n\\input{super-linear}\n\n\n\\input{sublinear-new}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction: the double side of leptogenesis}\n\nA successful model of baryogenesis cannot be\nrealised within the Standard Model (SM) and, therefore,\nthe observed matter-antimatter asymmetry of the Universe\ncan be regarded as an evidence for new physics beyond the SM.\n\nThe discovery of neutrino masses and mixing in neutrino oscillation\nexperiments in 1998 \\cite{superk} has for the first time shown directly,\nin particle physics experiments, that the SM is indeed incomplete, since it \nstrictly predicts, in the limit of infinite cutoff, \nthat neutrinos are massless and, therefore, cannot oscillate.\n\nThis discovery has greatly raised the interest for\nleptogenesis \\cite{Fukugita:1986hr,reviews}, a model of baryogenesis that is a cosmological\nconsequence of the most popular way to extend the SM in order to\nexplain why neutrinos are massive but at the same time much lighter than all the\nother fermions: the seesaw mechanism~\\cite{seesaw}.\nAs a matter of fact, leptogenesis realises a highly non-trivial link\nbetween two completely independent experimental observations:\nthe absence of primordial antimatter in the observable Universe and the\nobservation that neutrinos mix and (therefore) have masses.\nIn fact, leptogenesis has a naturally built-in double-sided nature.\nOn one side, it describes a very early stage in the history of the Universe\ncharacterised by temperatures $T_{\\rm lep} \\gtrsim 100\\,{\\rm GeV}$,\nmuch higher than those probed by Big Bang Nucleosynthesis, $T_{\\rm BBN} \\sim (0.1$--$1)\\,{\\rm MeV}$;\non the other side, it complements low-energy neutrino experiments providing\na completely independent phenomenological tool\nto test models of new physics embedding the seesaw mechanism.\n\nIn this article we review the main features and results of leptogenesis.\nLet us give a brief outline.\nIn Section 2 we present the status of low-energy neutrino experiments measuring\nneutrino masses and mixing parameters, and we introduce the seesaw mechanism, which provides an elegant framework to explain them.\nIn Section 3 we discuss the vanilla leptogenesis scenario.\nIn Section 4 we show the importance of accounting for flavour effects for a correct calculation of the final asymmetry. In Section 5 we discuss the density matrix formalism which properly takes into account decoherence effects, which are\ncrucial to describe the transition from a one-flavoured regime to a fully flavoured regime.\nIn Section 6 we relax the assumption of hierarchical right-handed neutrino mass spectrum, and discuss how the asymmetry can be calculated in the degenerate limit.\nIn Section 7 we discuss different ways to improve the kinetic description\nbeyond the density matrix formalism.\nIn Section 8 we discuss other effects (thermal corrections, spectator processes, scatterings)\nthat have been considered and that can give in some cases important corrections.\nIn Section 9 we show how leptogenesis represents an important guidance to test models of new physics.\nFinally, in Section 10 we conclude outlining the prospects to test leptogenesis in future years.\n\n\\section{Neutrino masses and mixing}\n\nNeutrino oscillation experiments have established two fundamental\nproperties of neutrinos. The first one is that the neutrinos mix.\nThis means that the neutrino weak eigenfields\n$\\nu_{\\alpha}$ ($\\alpha=e,\\mu,\\tau$)\ndo not coincide with the neutrino mass eigenfields $\\nu_i$ ($i=1,2,3$) but\nare obtained applying to them a unitary transformation described\nby the $(3\\times 3)$ leptonic mixing matrix $U$,\n\\begin{equation}\n\\nu_{\\alpha} = \\sum_{i} \\, U_{\\alpha i}\\,\\nu_i \\, .\n\\end{equation}\nThe leptonic mixing matrix is usually parameterised in terms of 6 physical parameters,\nthree mixing angles, $\\theta_{12}$, $\\theta_{13}$ and $\\theta_{23}$, and three phases,\ntwo Majorana phases, $\\rho$ and $\\sigma$ and one Dirac phase $\\delta$,\n\\begin{equation}\\label{Umatrix} \\fl\nU=\\left( \\begin{array}{ccc}\nc_{12}\\,c_{13} & s_{12}\\,c_{13} & s_{13}\\,e^{-{\\rm i}\\,\\delta} \\\\\n-s_{12}\\,c_{23}-c_{12}\\,s_{23}\\,s_{13}\\,e^{{\\rm i}\\,\\delta} &\nc_{12}\\,c_{23}-s_{12}\\,s_{23}\\,s_{13}\\,e^{{\\rm i}\\,\\delta} & s_{23}\\,c_{13} \\\\\ns_{12}\\,s_{23}-c_{12}\\,c_{23}\\,s_{13}\\,e^{{\\rm i}\\,\\delta}\n& -c_{12}\\,s_{23}-s_{12}\\,c_{23}\\,s_{13}\\,e^{{\\rm i}\\,\\delta} &\nc_{23}\\,c_{13}\n\\end{array}\\right)\n\\, {\\rm diag}\\left(e^{i\\,\\rho}, 1, e^{i\\,\\sigma}\n\\right)\\, ,\n\\end{equation}\nwhere $s_{ij} \\equiv \\sin\\theta_{ij}$ and $c_{ij}\\equiv\\cos\\theta_{ij}$.\nA global analysis \\cite{Fogli:2011qn} of all existing neutrino data, prior to the results\nof a non-vanishing $\\theta_{13}$ from short baseline reactors, gives\n$\\theta_{12}=34^{\\circ}\\pm 1^{\\circ}$ for the solar mixing angle,\n$\\theta_{23}= 40.4{^{\\circ}}^{+4.6^{\\circ}}_{-1.8^{\\circ}}$\nfor the atmospheric mixing angle, and $\\theta_{13}=9.0^{\\circ}\\pm 1.3^{\\circ}$\nfor the reactor mixing angle, where the latter is mainly dominated by\nthe evidence for a non-vanishing $\\theta_{13}$ found by the T2K experiment \\cite{T2K}\nthat confirmed previous hints \\cite{hints}.\nRecently, the Daya Bay and Reno short-baseline reactor neutrino experiments\nrespectively found $\\theta_{13}= 8.8^{\\circ}\\pm 0.8^{\\circ}\\pm 0.3^{\\circ}$ \\cite{dayabay},\nand $\\theta_{13}=9.8^{\\circ}\\pm 0.6{^{\\circ}}\\pm 0.85^{\\circ}$ \\cite{reno}, confirming, at more than $5\\sigma$,\nthe previous results.\n\n\nThe second important property established by neutrino oscillation experiments\nis that neutrinos are massive. More specifically, defining the three neutrino masses\nin a way that $m_1 \\leq m_2 \\leq m_3$, neutrino oscillation experiments\nmeasure two mass-squared differences that we can indicate with\n$\\Delta m^2_{\\rm atm}$ and $\\Delta m^2_{\\rm sol}$ since historically the first\none has been first measured in atmospheric neutrino experiments and the second one in\nsolar neutrino experiments.\n Two options are currently allowed by previous\nexperiments. A first option is `normal ordering' (NO) and in this case\n\\begin{equation}\nm_3^2 - m^2_2 = \\Delta m^2_{\\rm atm} \\hspace{5mm} \\mbox{\\rm and}\n\\hspace{5mm}\nm_2^2 - m^2_1 = \\Delta m^2_{\\rm sol} \\, ,\n\\end{equation}\nwhile a second option is represented by `inverted ordering' (IO) and in this case\n\\begin{equation}\nm_3^2 - m^2_2 = \\Delta m^2_{\\rm sol} \\hspace{5mm} \\mbox{\\rm and} \\hspace{5mm}\nm_2^2 - m^2_1 = \\Delta m^2_{\\rm atm} \\, .\n\\end{equation}\nIt is convenient to introduce the atmospheric neutrino mass scale\n$m_{\\rm atm} \\equiv \\sqrt{\\Delta m^2_{\\rm atm}+\\Delta m^2_{\\rm sol}}\n= (0.049\\pm 0.001)\\,{\\rm eV}$ and the solar neutrino mass scale\n$m_{\\rm sol} \\equiv \\sqrt{\\Delta m^2_{\\rm sol}}=(0.0087 \\pm 0.0001)\\,{\\rm eV}$\n\\cite{Fogli:2011qn}.\n\nThe measurements of $m_{\\rm atm}$ and $m_{\\rm sol}$\nare not sufficient to fix all three neutrino masses.\nIf we express them in terms of the lightest neutrino\nmass $m_1$ we can see from Fig.~1 that while\n $m_2 \\geq m_{\\rm sol}$ and $m_3 \\geq m_{\\rm atm}$, the lightest\nneutrino mass can be arbitrarily small implying that the lightest neutrino\ncould be even massless.\n\nThe lower limits for $m_2$ and $m_3$\nare saturated when $m_1 \\ll m_{\\rm sol}$. In this case one has hierarchical\nneutrino models, either normal and in this case $m_2 \\simeq m_{\\rm sol}$\nand $m_3\\simeq m_{\\rm atm}$, or inverted, and in this case\n$m_2\\simeq \\sqrt {m^2_{\\rm atm}-m^2_{\\rm sol}} \\simeq m_3\\simeq m_{\\rm atm}$.\nOn the other hand, for $m_1\\gg m_{\\rm atm}$ one obtains the limit of quasi-degenerate neutrinos\nwhen all three masses can be arbitrarily close to each other.\n\nHowever, the lightest neutrino mass is upper bounded by absolute neutrino mass scale\nexperiments. Tritium beta decay experiments \\cite{Kraus:2004zw}\nplace an upper bound on the effective electron neutrino mass\n$m_{\\nu_e} \\lesssim 2\\,{\\rm eV}$ ($95 \\%$ C.L.) that translates\ninto the same upper bound on $m_1$. This is derived from model-independent\nkinematic considerations that apply independently of whether neutrinos have a Dirac or\nMajorana nature.\n\nNeutrinoless double beta decay ($0\\nu\\beta\\b$) experiments\nplace a more stringent upper bound on the effective $0\\nu\\beta\\b$ Majorana neutrino mass,\n$m_{ee}\\lesssim (0.34$--$0.78)\\,{\\rm eV}$\\,($95\\%$ C.L.) as obtained by the CUORICINO\nexperiment~\\cite{bb0n}\n\\footnote[1]{At $90\\%$~C.L. the CUORICINO result is $m_{ee}\\lesssim (0.27$--$0.57)\\,{\\rm eV}$. For comparison,\nthe bound from the Heidelberg-Moscow experiment is $m_{ee}\\lesssim (0.21$--$0.53)\\,{\\rm eV}$ ($90\\%$~C.L.) \n\\cite{HDM} while the EXO-200 experiment has recently found the upper \nbound $m_{ee}\\lesssim (0.14$--$0.38)\\,{\\rm eV}$ ($90\\%$~C.L.) \\cite{EXO}.}.\nIt translates into the following upper bound on $m_1$~\\cite{rodejohann}:\n\\begin{equation} \\fl\nm_1 \\leq m_{ee}\/(\\cos 2\\theta_{12}\\,\\cos^2 \\theta_{13}-\\sin^2\\theta_{13}) \n\\lesssim 3.45 \\, m_{ee} \\lesssim (1.2 \\textrm{--}2.7)\\,{\\rm eV} \n\\hspace{3mm} (95\\%\\, {\\rm C.L.}) \\, .\n\\end{equation}\nHere, the wide range is due to\ntheoretical uncertainties in the calculation\nof the involved nuclear matrix elements. However, this upper bound applies\nonly if neutrinos are of Majorana nature, which is the relevant case for us, since\nthe seesaw mechanism predicts Majorana neutrinos.\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{150mm}\n\\begin{center}\n\\centerline{\\psfig{file=numasses.eps,height=8cm,width=11cm}}\n\\caption{Neutrino masses $m_i$ versus the lightest neutrino mass $m_1$.\nThe three upper bounds (at $95\\%$ C.L.) discussed in the body text\nfrom absolute neutrino mass scales phenomenologies are also indicated.}\n\\end{center}\n\\end{minipage}\n\\end{center}\n\\end{figure}\n\nFrom cosmological observations,\nwithin the $\\Lambda$CDM model, one obtains a very stringent upper bound on the\nsum of the neutrino masses \\cite{WMAP7}\n\\begin{equation}\\label{WMAP7}\n\\sum_i\\, m_i \\lesssim 0.58\\,{\\rm eV} \\hspace{10mm} (95\\% \\, {\\rm C.L.})\\, .\n\\end{equation}\nThis translates into the\nmost stringent upper bound that we currently have on the\nlightest neutrino mass $m_1 \\lesssim 0.19\\,{\\rm eV}$ ($95\\%$ C.L.), an upper bound that\nalmost excludes quasi-degenerate neutrino models\n\\footnote[2]{For a more general discussion of neutrino mass bounds, \nin particular on theoretical assumptions and uncertainties, see for example \n\\cite{rodejohann,giunti,pastor}.}.\n\nA minimal extension of the SM, able to explain not only why neutrinos\nare massive but also why they are much lighter than all the other massive fermions,\nis represented by the seesaw mechanism \\cite{seesaw}.\nIn the minimal type I version, one adds\nright-handed neutrinos $N_{Ri}$ to the SM lagrangian with Yukawa couplings $h$\nand a Majorana mass term that violates lepton number\n\\begin{equation}\\label{lagrangian}\n\\mathcal{L} = \\mathcal{L}_{\\rm SM}\n+{\\rm i} \\, \\overline{N_{iR}}\\gamma_{\\mu}\\partial^{\\mu} N_{iR} -\n\\overline{\\ell_{\\alpha L}}\\,h_{\\alpha i}\\, N_{iR}\\, \\tilde{\\Phi} -\n {1\\over 2}\\,M_i \\, \\overline{N_{iR}^c} \\, N_{iR} +h.c. \\nonumber\n\\end{equation}\nFor definiteness we will consider the case of three RH neutrinos ($i=1,2,3$).\nThis is the most attractive case corresponding to have one RH neutrino for each generation,\nas for example nicely predicted by $SO(10)$ grand-unified models.\nNotice, however, that all current\ndata from low-energy neutrino experiments are also consistent\nwith a more minimal model with only two RH neutrinos that will be discussed in Section 9.\n\nAfter spontaneous symmetry breaking, a Dirac mass term $m_D=v\\,h$\nis generated by the Higgs vev $v$. In the seesaw limit, $M\\gg m_D$, the spectrum\nof neutrino masses splits into a light set given by the eigenvalues $m_1 \\eta_{B}^{\\rm CMB}$,\none obtains the allowed region in the plane $(m_1,M_1)$ shown in the left panel of Fig.~2.\n\\begin{figure}\n\\begin{center}\n\\psfig{file=massbounds.eps,width=0.4\\textwidth} \\hspace{5mm}\n\\psfig{file=BoundFlavoredK1Relaxw32.eps,width=0.4\\textwidth}\n\\end{center}\n\\caption{Left: Neutrino mass bounds in the vanilla scenario. Right:\n Relaxation of the lower bound on $M_1$ thanks\n to additional unbounded flavoured $C\\!P$ violating terms~\\cite{bounds}.}\n\\end{figure}\nOne can notice the existence of an upper bound on the light\nneutrino masses $m_1\\lesssim 0.12\\,{\\rm eV}$ \\cite{Buchmuller:2002jk,window},\nincompatible with quasi-degenerate\nneutrino mass models, and a lower bound on\n$M_1\\gtrsim 3\\times 10^9\\,{\\rm GeV}$ \\cite{Davidson:2002qv,Buchmuller:2002jk,window}\nimplying a lower bound on the\nreheat temperature $T_{\\rm reh}\\gtrsim 10^9\\,{\\rm GeV}$ \\cite{pedestrians}.\n\nAn important feature of vanilla leptogenesis is that the final asymmetry does not\ndirectly depend on the parameters of leptonic mixing matrix $U$.\nThat implies that one cannot establish a direct model-independent connection.\nIn particular the discovery\nof $C\\!P$ violation in neutrino mixing would not be a smoking gun for leptogenesis\nand, vice versa, a non-discovery would not rule out leptogenesis.\nHowever, within more restricted scenarios, for example imposing some conditions on\nthe neutrino Dirac mass matrix, links can emerge. In Section 9 we will discuss\nin detail the interesting case of $SO(10)$-inspired models.\n\nIn the past years different directions beyond the\nvanilla leptogenesis scenario were explored,\nusually aiming at evading the above-mentioned bounds.\nFor example, it was noticed that the Davidson-Ibarra bound eq.~(\\ref{CPbound})\nis strictly speaking evaded by an extra-term contribution $\\Delta \\varepsilon_1$ to the total\n$C\\!P$ asymmetry \\cite{hambyestrumia,bounds}. However, this extra-term vanishes\nif $M_3 \\simeq M_2$ and is suppressed as $\\Delta \\varepsilon_1 \\propto (M_1\/M_2)^2$. It can\nsignificantly relax the bounds on neutrino masses only when $|\\Omega_{ij}|^2 \\gtrsim (M_2\/M_1)^2$,\nimplying cancellations with a certain degree of fine tuning in the see-saw formula \nfor the neutrino masses. For example,\nin usual models with $|\\Omega_{ij}|^2 \\lesssim 10$, this extra-contribution can be safely neglected\nfor $M_2\\gtrsim 10\\,M_1$.\n\nThe development that proved to have the most important\nimpact on the final asymmetry, compared to a calculation within the vanilla scenario,\nis certainly the inclusion of flavour effects. For this reason we discuss\nthem in detail in the next Section.\n\n\\section{The importance of flavour}\n\nThe inclusion of flavour effects provides the most significant modification in\nthe calculation of the final asymmetry compared to the vanilla scenario. \nTwo kinds of flavour\neffects are neglected in the vanilla scenario:\nheavy neutrino flavour effects, how heavier RH neutrinos\ncan contribute to the final asymmetry, \nand lepton flavour effects,\nhow the flavour composition of the leptons quantum states produced in the RH neutrino decays\naffects the calculation of the final asymmetry.\nWe first discuss the two effects separately, and then we show\nhow their interplay can have very interesting consequences.\n\n\\subsection{Heavy neutrino flavour effects}\n\nIn the vanilla scenario the contribution to the final asymmetry from the\nheavier RH neutrinos is negligible because either the $C\\!P$ asymmetries are\nsuppressed in the hierarchical limit compared to\n$\\varepsilon_1^{\\rm max}$ (cf. Eq.~(\\ref{CPbound})) and\/or because,\neven assuming that a sizeable asymmetry (compared to the observed one) \nis produced at $T \\sim M_{2,3}$,\nit is later on washed out by the lightest RH neutrino inverse processes.\n\nHowever, as we anticipated, there is a particular case when,\neven neglecting the lepton flavour composition and assuming\na hierarchical heavy neutrino mass spectrum, the contribution\nto the final asymmetry from next-to-lightest RH neutrino decays can be\ndominant and explain the observed asymmetry \\cite{DiBari:2005st}. This case corresponds to a particular\nchoice of the orthogonal matrix such that $N_1$ is so weakly coupled\n(corresponding to $K_1 \\ll 1$) that its washout can be neglected.\nFor the same choice of the parameters, the $N_2$ total $C\\!P$ asymmetry\n$\\varepsilon_2$ is unsuppressed if $M_3\\lesssim 10^{15}\\,{\\rm GeV}$.\nIn this case a $N_2$-dominated scenario is realised.\nNotice that in this case the existence of a third (heavier) RH neutrino\nspecies is crucial in order to have a sizeable $\\varepsilon_2$.\n\nThe contribution from the two heavier RH neutrino species is also important\nin the quasi-degenerate limit when $\\delta_{i}\\equiv (M_i-M_1)\/M_1 \\ll 1,~i=2,3$.\nIn this case the $C\\!P$ asymmetries $\\varepsilon_{2,3}$ are not suppressed, and the\nwashout from the lighter RH neutrino species is moderate, with no exponential\nprefactor \\cite{Pilaftsis:1997jf,Blanchet:2006dq}.\n\n\\subsection{Lepton flavour effects}\n\nThe importance and generality of flavour effects in leptogenesis was fully highlighted in \\cite{Abada:2006fw,Nardi:2006fx}.\nTheir role was first discussed in \\cite{bcst} and \nincluded in specific scenarios in \\cite{endoh,Pilaftsis:2005rv,vives}.\n\nFor the time being, let us continue to assume that the final asymmetry is\ndominantly produced from the decays of the lightest RH neutrinos $N_1$,\nneglecting the contribution from the decays of the heavier RH neutrinos $N_2$ and $N_3$.\nIf $M_1\\gg 10^{12}\\,{\\rm GeV}$, the flavour composition\nof the quantum states of the leptons produced from $N_1$ decays\nhas no influence on the final asymmetry and a one-flavour\nregime holds \\cite{bcst,Abada:2006fw,Nardi:2006fx}.\nThis is because the lepton quantum states evolve coherently between the production\nfrom a $N_1$-decay\nand a subsequent inverse decay with a Higgs boson.\nIn this way the lepton flavour composition does not play any role.\n\nHowever, if $10^{12}\\,{\\rm GeV}\\gtrsim M_1 \\gtrsim 10^{9}\\,{\\rm GeV}$,\nduring the relevant period of generation of the asymmetry, the produced lepton\nquantum states will, on average, have an interaction with RH tauons before\nundergoing the subsequent inverse decay. In this way the tauon component of the lepton quantum\nstates is measured by the thermal bath and the coherent evolution breaks down\n\\cite{bcst,Abada:2006fw,Nardi:2006fx}.\nTherefore, at the subsequent inverse decays,\nthe lepton quantum states are an incoherent mixture of a tauon component and\nof a (still coherent) superposition of an electron and a muon component that\nwe can indicate with $\\tau^{\\bot}$.\n\nThe fraction of asymmetry stored in each flavour component is not proportional in general\nto the branching ratio of that component. This implies that the two\nflavour asymmetries, the tauon and the $\\tau^{\\bot}$ components,\nevolve differently and have to be calculated separately.\nIn this way the resulting final asymmetry can considerably differ\nfrom the result in the one-flavour regime. This can be indeed approximated by \nthe expression\n\\begin{equation}\\label{twofully}\nN^{\\rm f}_{B-L} \\simeq 2\\,\\varepsilon_1\\,\\kappa(K_1) +\n{\\Delta p_{1\\tau}\\over 2}\\,\\left[\\kappa(K_{1{\\tau}^{\\bot}_1})-\\kappa(K_{1\\tau})\\right] \\, ,\n\\end{equation}\nwhere $K_{1\\alpha}\\equiv p^0_{1\\alpha}\\,K_i$, the $p^0_{1\\alpha}$'s ($\\alpha=\\tau, \\tau^{\\bot}$) are the \ntree level probabilities that the leptons ${\\ell}_1$ and the anti-leptons $\\bar{\\ell}'_1$ \nproduced in the decays of the $N_1$'s \nare in the flavour $\\alpha$, while $\\Delta p_{1\\tau}$ is the difference \nbetween the probability to find ${\\ell}_1$ in the flavour $\\tau$ and that\none to find $\\bar{\\ell}'_1$ in the flavour $\\bar{\\tau}$. If we \ncompare this expression with the one-flavour regime result eq.~(\\ref{unflavoured})\none can see that if $\\Delta p_{1\\tau}=0$ (or if $K_{1\\tau}=K_{1\\tau^{\\bot}}$), \nthen the final symmetry is enhanced just by a factor $2$. However, in general\nleptons and anti-leptons have a different flavour composition \\cite{bcst,Nardi:2006fx}\nand in this case $\\Delta p_{1\\tau}\\neq 0$ and the final asymmetry can be much higher than in the one-flavour regime.\nThe most extreme case is when the total $B-L$ number is conserved (i.e. $\\varepsilon_1=0$)\nand still the second term in eq.~(\\ref{twofully}) \ncan be non-vanishing \\cite{Nardi:2006fx} and even explain the observed asymmetry \\cite{Blanchet:2006be}. \n\nIf $M_1\\lesssim 10^{9}\\,{\\rm GeV}$, even the coherence of the $\\tau^{\\bot}$\ncomponent is broken by the muon interactions between decays and inverse decays\nand a full three flavour regime applies. In the intermediate regimes\na density matrix formalism is necessary to properly describe decoherence\n\\cite{Abada:2006fw,Blanchet:2006ch,DeSimone:2006dd,Beneke:2010dz}.\n\nWe can briefly say that \nlepton flavour effects induce three major consequences that are all encoded \nin the expression (\\ref{twofully}) valid in the two fully flavoured regime. \ni) The washout can be considerably lower than in the unflavoured regime \\cite{Abada:2006fw,Nardi:2006fx}\nsince one can have that the asymmetry is dominantly produced in a flavour $\\alpha$ with $K_{1\\alpha}\\ll K_1$\nii) The leptonic mixing matrix enters directly in the calculation of the final asymmetry, more specifically\nand in particular the low-energy phases\ncontribute as a second source of $C\\!P$ violation in the flavoured $C\\!P$ asymmetries\n\\cite{Nardi:2006fx,Abada:2006ea,Blanchet:2006be,Pascoli:2006ci}. As an interesting phenomenological \nconsequence, the same source of $C\\!P$\nviolation that could take place in neutrino oscillations could be sufficient\nto explain the observed asymmetry \\cite{Blanchet:2006be,Pascoli:2006ci},\nthough under quite stringent conditions on the RH neutrino mass spectrum \nand with an exact determination requiring a density matrix calculation \\cite{Anisimov:2007mw}.\nNotice that this is a particular case realising the above-mentioned scenario with $\\varepsilon_1=0$ \\cite{Nardi:2006fx}. \nThis problem becomes particularly interesting\nin the light of the recent discovery of a non-vanishing $\\theta_{13}$ angle \\cite{hints,T2K,dayabay,reno},\na necessary condition to have $C\\!P$ violation in neutrino oscillations.\n In particular a calculation of the lower bound on $\\theta_{13}$ that necessarily requires\nthe use of density matrix equations.\niii) The flavoured $C\\!P$ asymmetries, given by\n\\begin{equation}\\label{flavouredCP} \\fl\n\\varepsilon_{i\\alpha}=\n{3\\over 16\\pi (h^{\\dagger}h)_{ii}}\\sum_{j\\neq i} \\left\\{ {\\rm Im}\n\\left[h_{\\alpha i}^{\\star}h_{\\alpha j}(h^{\\dagger}h)_{ij}\\right] {\\xi(x_j\/x_i)\\over\n\\sqrt{x_j\/x_i}}\\right. \\nonumber\n\\left.+{2\\over 3(x_j\/x_i-1)}{\\rm Im}\n\\left[h_{\\alpha i}^{\\star}h_{\\alpha j}(h^{\\dagger}h)_{ji}\\right]\\right\\} \\, ,\n\\end{equation}\ncontain a second term that conserves the total lepton number (it cancels exactly\nwhen summing over flavour in the total $C\\!P$ asymmetry and it, therefore, contribute to\n$\\Delta p_{1\\tau}$ in eq.~(\\ref{twofully}) but not to $\\varepsilon_1$),\nand therefore the upper bound in Eq.~(\\ref{CPbound})\ndoes not strictly apply to the flavoured $C\\!P$ asymmetries.\nAs a consequence, allowing for a mild cancelation in the neutrino mass matrix,\ncorresponding to $|\\Omega_{ij}|\\sim 1$, and also\nfor a mild RH neutrino mass hierarchy ($M_2\/M_1 \\sim 10$), the lower bound on $T_{\\rm reh}$\ncan be relaxed by about one order of magnitude, down to $10^8\\,{\\rm GeV}$\n\\cite{bounds}, as shown in the right panel of Fig.~2.\nHowever, for many models such as sequential dominated models \\cite{King:2003jb},\nthese cancellations do not occur, and lepton flavour effects cannot relax the\nlower bound on $T_{\\rm reh}$ \\cite{Blanchet:2006be}. One known exception is given\nby the inverse seesaw model~\\cite{Mohapatra:1986bd} which naturally explains large\ncancellations in the neutrino\nmass matrix and leads to large $C\\!P$ asymmetries thanks to an underlying\nlepton number symmetry. This leads to the relaxation of the lower bound on $T_{\\rm reh}$ \nby up to three orders of magnitude~\\cite{Antusch:2009gn,Racker:2012vw}\n(see more detailed discussion in the next Section).\n\nWhether or not the upper bound $m_i\\lesssim 0.1$~eV on neutrino masses found in the vanilla scenario\nstill holds in a flavoured $N_1$-dominated scenario, is relaxed, or even completely evaporates, is a controversial topic. \nIt still surely holds in the one-flavour regime for $M_1 \\gtrsim 10^{12}\\,{\\rm GeV}$ but it certainly does not hold in the two-fully flavoured regime\n\\cite{Abada:2006fw,JosseMichaux:2007zj,bounds}. \nHowever, it was found that the two-fully \nflavoured regime is not respected at large values $m_1\\gtrsim 0.1 \\,{\\rm eV}$\n\\cite{Blanchet:2006ch}.\nIn \\cite{DeSimone:2006dd} it was found \nthat it holds up to $m_1 \\sim 2\\,{\\rm eV}$, \nimplying in any case an upper bound much above current\nexperimental bounds and, therefore, uninteresting. \nIn \\cite{bounds}, including the information from low energy neutrino data and accounting for the Higgs asymmetry, it was found again that it \nholds only up to $m_1\\gtrsim 0.1 \\,{\\rm eV}$ and in this case a density matrix approach would be required for a conclusion.\n\n\\subsection{The interplay between lepton and heavy neutrino flavour effects}\n\nAs we have seen, when lepton flavour effects are neglected, the possibility\nthat the next-to-lightest RH neutrino decays contribute to the final\nasymmetry relies on a special case realising the $N_2$-dominated scenario\n\\cite{DiBari:2005st}. On the other hand,\nwhen lepton flavour effects are taken into account,\nthe contribution from heavier RH neutrinos\ncannot be neglected in a much more general situation. Even the contribution\nfrom the heaviest RH neutrinos can be sizeable (i.e. explain the observed asymmetry)\nand has to be taken into account in general.\n\nAs a result, the calculation of the final asymmetry becomes much more involved.\nAssuming hierarchical mass patterns and that the RH neutrino processes\noccur in one of the three different fully flavoured regimes, one has to consider\nten different mass patterns, shown in Fig.~2, that require specific multi-stage\nsets of classical Boltzmann equations for the calculation of the final asymmetry.\n\\begin{figure}\n\\begin{minipage}{140mm}\n\\begin{center}\n\\psfig{file=f4p1.eps,height=3cm,width=3cm} \\hspace*{2mm}\n\\psfig{file=f5p1.eps,height=3cm,width=3cm} \\hspace*{2mm}\n\\psfig{file=f5p2.eps,height=3cm,width=15mm} \\hspace*{2mm}\n\\psfig{file=f5p3.eps,height=3cm,width=15mm}\n\\end{center}\n\\begin{center}\n\\psfig{file=f6p1.eps,height=3cm,width=3cm}\n\\psfig{file=f6p2.eps,height=3cm,width=15mm}\n\\psfig{file=f6p3.eps,height=3cm,width=15mm}\n\\psfig{file=f6p4.eps,height=3cm,width=3cm}\n\\psfig{file=f6p5.eps,height=3cm,width=15mm}\n\\psfig{file=f6p6.eps,height=3cm,width=15mm}\n\\end{center}\n\\caption{The ten RH neutrino mass patterns corresponding to\nleptogenesis scenarios with\ndifferent sets of classical Boltzmann equations for the\ncalculation of the final asymmetry \\cite{problem}.}\n\\end{minipage}\n\\end{figure}\n\n\\subsubsection{The (flavoured) $N_2$-dominated scenario}\n\nAmong these 10 RH neutrino mass patterns,\nfor those three where the $N_1$ washout occurs in\nthe three-fully-flavoured regime, for $M_1 \\ll 10^{9}\\,{\\rm GeV}$,\nthe final asymmetry has necessarily to be produced by the\n$N_2$ either in the one or in the two-fully-flavoured\nregime (for $M_2 \\gg 10^9\\,{\\rm GeV}$). They have some particularly attractive\nfeatures and realise a `flavoured $N_2$-dominated scenario' \\cite{vives}\n(it corresponds to the fifth, sixth and seventh mass pattern in Fig.~3).\n\nWhile in the unflavoured approximation the lightest RH neutrino washout\nyields a global exponential washout factor, when lepton flavour effects are\ntaken into account, the asymmetry produced by the\nheavier RH neutrinos, at the $N_1$ washout, gets distributed\ninto an incoherent mixture of charged lepton flavour eigenstates \\cite{vives}.\nIt turns out that the $N_1$ washout in one of the three flavours is negligible, corresponding to have\nat least one $K_{1\\alpha}\\lesssim 1$,\nin quite a wide region of the parameter space \\cite{bounds}.\nIn this way, accounting for flavour effects, the region of applicability of the\n$N_2$-dominated scenario enlarges considerably, since it is not\nnecessary that $N_1$ fully decouples but it is sufficient that it decouples\njust in a specific lepton flavour \\cite{vives}. \nThe unflavoured $N_2$-dominated scenario is recovered\nin the limit where all three $K_{1\\alpha}\\lesssim 1$ \nand either $M_2\\gtrsim 10^{12}\\,{\\rm GeV}$ or for \n$10^{12}\\,{\\rm GeV} \\gtrsim M_2\\gtrsim 10^{9}\\,{\\rm GeV}$ and $K_2 \\lesssim 1$. \n\nRecently, it has also been realised that,\naccounting for the Higgs and for the quark asymmetries, the dynamics of the flavour asymmetries\ncouple and the lightest RH neutrino washout in a particular flavour can be circumvented even when $N_1$ is strongly\ncoupled in that flavour \\cite{Antusch:2010ms}.\nAnother interesting effect arising in the $N_2$-dominated scenario is {\\em phantom leptogenesis}.\nThis is a pure quantum-mechanical effect that for example\nallows parts of the electron and of the muon asymmetries, the phantom terms,\nto undergo a weaker washout at the production than the total asymmetry.\nIt has been recently shown that phantom terms\nassociated to a RH neutrino species $N_i$ with $M_i \\gg 10^{9}\\,{\\rm GeV}$\nare present not just in the $N_2$-dominated scenario \\cite{Blanchet:2011xq}.\nHowever, it should be noticed that phantom terms produced by the\nlightest RH neutrinos\ncancel with each other and thus do not contribute to the final asymmetry, though\nthey can induce flavoured asymmetries much larger than the total asymmetry,\nsomething potentially relevant in active-sterile neutrino oscillations~\\cite{activesterile}.\n\n\\subsubsection{Heavy neutrino flavour projection}\n\nEven assuming a strong RH neutrino mass hierarchy, a coupled $N_1$ in all lepton flavours\n($K_{1\\alpha} \\gg 1$ for any $\\alpha$) and\n$M_1\\gtrsim 10^{12}\\,{\\rm GeV}$ (it corresponds to the first panel in Fig.~2), the asymmetry produced by the\nheavier RH neutrino decays at $T\\sim M_i$, in particular by the $N_2$'s decays,\ncan be large enough to explain the observed asymmetry by avoiding most of the washout from \nthe lightest RH neutrino\nprocesses. This is because, in general, there is an `orthogonal' component\nthat escapes the $N_1$ washout \\cite{bcst,Engelhard:2006yg} while the remaining\n`parallel' component undergoes the usual exponential washout.\nFor a mild mass hierarchy, $\\delta_3 \\lesssim 10$,\neven the asymmetry produced by the $N_3$'s decays can be large enough to\nexplain the observed asymmetry and escape\n the $N_1$ and $N_2$ washout. Heavy neutrino flavour projection is also occurring\n when $10^{12}\\,{\\rm GeV}\\gtrsim M_1 \\gtrsim 10^9 \\,{\\rm GeV}$, in this case \n in the $e$--$\\mu$ plane.\n\nWhen the effect of heavy neutrino flavour projection is taken into account jointly\n with an additional contribution to the flavoured $C\\!P$ asymmetries $\\varepsilon_{2\\alpha}$\n that is not suppressed when the heaviest RH neutrino mass $M_3\\gtrsim 10^{15}\\,{\\rm GeV}$,\nthis can lead to the possibility of a dominant contribution from the next-to-lightest RH neutrinos\neven in an effective two RH neutrino model that can be regarded as \nas a limit case in a (more appealing) three RH neutrino case with \n$M_3\\gtrsim 10^{15}\\,{\\rm GeV}$ \\cite{Antusch:2010ms}.\n\n\\subsubsection{The problem of the initial conditions in flavoured leptogenesis}\n\nAs we have seen, in the vanilla scenario the unflavoured assumption reduces the problem\nof the dependence on the initial conditions to simply imposing\nthe strong washout condition\n$K_1 \\gg 1$. In other words, there is a full equivalence between\nstrong washout and independence of the initial conditions.\n\nWhen (lepton and heavy neutrino) flavour effects are considered, the situation is much more\ninvolved and for example imposing strong washout conditions on all flavoured\nasymmetries ($K_{i\\alpha}$) is not enough to guarantee independence of the initial conditions.\nPerhaps the most striking consequence is that in a traditional\n$N_1$-dominated scenario there is no condition that can guarantee\n independence of the initial conditions. The only possibility to have independence\n of the initial conditions is represented by a tauon $N_2$-dominated scenario~\\cite{problem},\n i.e. a scenario where the asymmetry is dominantly produced\n from the next-to-lightest RH neutrinos, and therefore $M_2\\gg 10^{9}\\,{\\rm GeV}$,\n in the tauon flavour. The condition $M_1 \\ll 10^{9}\\,{\\rm GeV}$\n is also important to have projection on the orthonormal three lepton flavour\n basis before the lightest RH neutrino washout \\cite{Engelhard:2006yg}.\n\n \\section{Density matrix formalism}\n\nAs explained in the last Section, leptogenesis is sensitive to the temperature\nrange at which the asymmetry is produced. This determines whether the lepton quantum states produced in RH neutrino\ndecays either remain coherent or undergo decoherence and get projected in flavour space before scattering in inverse\nprocesses. Moreover, since the lepton states produced in heavy neutrino decays differ in general\nfrom the lepton flavour eigenstates, lepton flavour\noscillations can also in principle arise in a similar way as neutrino oscillations happen in\nvacuum or in a medium. In order to treat the problem of flavour oscillations and\npartial loss of decoherence in a consistent way, one has to extend the classical\nBoltzmann framework to account for these intrinsically quantum effects.\nThe formalism of the density matrix is appropriate for this purpose~\\cite{Sigl:1992fn}. The\ndensity matrix for leptons with momentum ${\\mathbf p}$ is defined as\n\\begin{equation}\n\\rho_{\\ell}({\\mathbf p})=\\left(\\begin{array}{cc}\\langle a_{\\alpha}^{\\dagger}({\\mathbf p})\\,a_{\\alpha}({\\mathbf p})\\rangle &\n\\langle a_{\\beta}^{\\dagger}({\\mathbf p})\\, a_{\\alpha}({\\mathbf p})\\rangle \\\\\n\\langle a_{\\alpha}^{\\dagger}({\\mathbf p})\\,a_{\\beta}({\\mathbf p})\\rangle &\n\\langle a_{\\beta}^{\\dagger}({\\mathbf p})\\, a_{\\beta}({\\mathbf p})\\rangle \\end{array}\\right) \\, ,\n\\end{equation}\nwith $a\\,(a^{\\dagger})$ denoting\nthe annihilation (creation) operator, i.e. it is the expectation\nvalue (to be understood in statistical terms) of the generalised number operator.\nFor anti-leptons one has analogously a density matrix $\\rho_{\\bar{\\ell}}({\\mathbf p})$.\nThe diagonal elements of the density matrix contains\nnothing else than the occupation numbers of the two flavoured leptons, and the off-diagonal\nelements encode flavour correlations.\n\nLet us consider an $N_1$-dominated scenario for simplicity. Moreover, let us consider\na momentum integrated description, introducing the matrix of lepton number densities\n$N_{\\ell}\/R^3=g_{\\ell}\\int\\,{d^3p\\over (2\\pi)^3} \\, \\rho_{\\ell}({\\mathbf p})$, where $R$ is the \nscale factor. \nWithin the density matrix formalism,\nthe asymmetry can be calculated from the following\ndensity matrix equation in the flavour space $\\tau$--$\\tau^{\\bot}$\n~\\cite{Abada:2006fw,DeSimone:2006dd,Blanchet:2008hg,\nBeneke:2010dz,Blanchet:2011xq}\n\\begin{eqnarray} \\label{densmateq}\n{dN^{B-L}_{\\alpha\\beta} \\over dz} &=&\n\\varepsilon^{(1)}_{\\alpha\\beta}\\,D_1\\,(N_{N_1}-N_{N_1}^{\\rm eq})-{1\\over 2}\\,W_1\\,\\left\\{{\\mathcal P}^{0(1)} , N^{B-L}\\right\\}_{\\alpha\\beta}\\nonumber \\\\ \\nonumber\n&+& {\\rm i}\\,{{\\rm Re}(\\Lambda_{\\tau})\\over H\\, z}\\left[\\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & 0\n\\end{array}\\right),N^{\\ell +\\bar{\\ell}}\\right]_{\\alpha\\beta} \\\\\n& - & {{\\rm Im}(\\Lambda_{\\tau})\\over H\\, z}\\left[\\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & 0\n\\end{array}\\right),\\left[\\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & 0\n\\end{array}\\right),N^{B-L} \\right]\\right]_{\\alpha\\beta} \\, ,\n\\end{eqnarray}\nwhere we defined $N^{\\ell +\\bar{\\ell}}_{\\alpha\\beta} \\equiv N^{{\\ell}}_{\\alpha\\beta} +N^{{\\bar{\\ell}}}_{\\alpha\\beta}$ and\nwhere $\\mathcal{P}^{0(1)}$ is a matrix projecting the\nlepton quantum states along the flavour `$1$' of a lepton ${\\ell}_1$\nproduced from the decay of a RH neutrino $N_1$ . The $C\\!P$ asymmetry\nmatrix is a straightforward generalization of Eq.~(\\ref{flavouredCP})~\\cite{Abada:2006ea,Beneke:2010dz,Blanchet:2011xq}.\nThe real and imaginary parts of the tau-lepton self-energy are respectively given by~\n\\cite{Weldon:1982bn,Cline:1993bd}\n\\begin{equation}\n{\\rm Re}(\\Lambda_{\\tau})= {f_{\\tau}^2\\over 64} T \\, \\hspace{10mm} \\mbox{\\rm and}\n\\hspace{10mm} {\\rm Im}(\\Lambda_{\\tau})= 8\\times 10^{-3}f_{\\tau}^2 T \\, ,\n\\end{equation}\nwhere $f_{\\tau}$ is the tauon Yukawa coupling.\nThe commutator structure in the third term on the RHS of Eq.~(\\ref{densmateq})\naccounts for oscillations in flavour space driven by the real part of\nthe self energy, and the double commutator accounts for damping of the off-diagonal terms\ndriven by the imaginary part\nof the self energy.\n\nIn order to close the system of equations, we also need an\nequation for the matrix $N_{\\ell +\\bar{\\ell}}$, which is given by\n\\begin{equation}\n{dN^{\\ell +\\bar{\\ell}}_{\\alpha\\beta} \\over dz} =\n-{{\\rm Re}(\\Lambda_{\\tau})\\over H\\, z}(\\sigma_2)_{\\alpha\\beta} N^{B-L}_{\\alpha\\beta} -S_g \\,\n(N^{\\ell +\\bar{\\ell}}_{\\alpha\\beta}-2\\,N_{\\ell}^{\\rm eq }\\delta_{\\alpha\\beta}) \\, ,\n\\end{equation}\nwhere $S_g\\equiv \\Gamma_g\/(Hz)$ accounts for gauge interactions. As shown in \\cite{Beneke:2010dz}, this term\nhas the effect of damping the flavour oscillations. This can be understood by noticing that gauge\ninteractions force $N^{\\ell +\\bar{\\ell}}_{\\alpha\\beta}=2\\,N_{\\ell_1}^{\\rm eq }\\delta_{\\alpha\\beta}$, which in turn renders the oscillatory\nterm Eq.~(\\ref{densmateq}) negligible.\n\nEq.~(\\ref{densmateq}) should be solved in any of the intermediate regimes where lepton states\nare partially coherent. Actually the range of relevance of the density matrix equation might be more\nimportant than previously believed. Indeed, it was found in~\\cite{Beneke:2010dz} that in order to\nrecover the unflavoured regime, one should have masses well above $10^{13}$~GeV.\n\nAs already discussed, the contribution from heavier RH neutrinos cannot be neglected in\ngeneral. Therefore, the density matrix equation (\\ref{densmateq}) should be extended to\naccount for such effects; this was done in~\\cite{Blanchet:2011xq}, where a general equation\nwas presented, valid for any RH neutrino decays and any temperature range. Flavour projection effects\nas well as phantom terms are readily taken into account in this framework.\n\n\n\\section{Limit of quasi-degenerate heavy neutrinos}\n\nIf $\\delta_2 \\ll 1$, the $C\\!P$ asymmetries $\\varepsilon_{1,2}$\nget resonantly enhanced as $\\varepsilon_{1,2}\\propto 1\/\\delta_2$ \\cite{flanz,Covi:1996wh,buchplumi1}.\nIf, more stringently, $\\delta_2\\lesssim 10^{-2}$, then\n$\\eta_B \\propto 1\/\\delta_2$ and the degenerate limit is obtained \\cite{Blanchet:2006dq}.\nIn this limit the lower bounds on $M_1$ and on $T_{\\rm reh}$\nget relaxed proportionally to $\\delta_2$ and at the resonance they completely disappear\n\\cite{Pilaftsis:1997jf,Pilaftsis:2003gt}. The upper bound on $m_1$ also \ndisappears in this extreme case. In a more realistic case where the degeneracy of the RH neutrino masses\nis comparable to the degeneracy of the light neutrino masses, as typically occurring in \nmodels with flavour symmetries, the upper bound $m_1 \\lesssim 0.1\\,{\\rm eV}$ obtained in the hierarchical case\nimposing the validity of Boltzmann equations, gets relaxed to $m_1 \\lesssim 0.4\\,{\\rm eV}$\n\\cite{hambyestrumia,bounds} in the case $M_3\\gg M_2$, while it is basically unchanged\nif $M_3=M_2$ \\cite{bounds}. The difference is due to the fact that the relaxation is mainly \nto be ascribed to the extra-term in the $C\\!P$ asymmetry mentioned at the end of Section 3. This \nextra-term, subdominant in the hierarchical case, can become dominant (if $M_3\\gg M_2$) in the quasi-degenerate case\n($\\delta_2 \\equiv (M_2-M_1)\/M_1 \\ll 1$)\nand it grows with the absolute neutrino mass scale instead of being suppressed as the usual term \\cite{hambyestrumia}. \nOn the other hand this extra-terms vanishes exactly when $M_2=M_3$, \na more reasonable assumption for $\\delta_2 \\ll 1$. \n\nIn the full three-flavour regime, the contributions from all quasi-degenerate\nRH neutrinos should be taken into account and in this case the final asymmetry can be calculated as\n\\begin{equation}\nN^{\\rm f}_{B-L}=\\sum_{i,\\alpha} \\varepsilon_{i\\alpha}\\, \\kappa\\left(\\sum_j K_{j\\alpha}\\right) \\, .\n\\end{equation}\nNotice that, for each lepton flavour $\\alpha$, the washout in the degenerate limit\nis described by the sum of the\nflavoured decay parameters for each RH neutrino species.\n\nThe simplest way to obtain a quasi-degenerate RH neutrino mass spectrum is to postulate the\nexistence of a slightly broken lepton number symmetry in the lepton sector~\\cite{Branco:1988ex,Shaposhnikov:2006nn,\nKersten:2007vk}. Assuming the existence of\nonly two RH neutrinos at first for simplicity, it is possible to write down Yukawa couplings and\na Majorana mass term in a way that conserves lepton number. In the ``flavour''\nbasis, which we denote by a prime, one RH neutrino can be assigned lepton number +1,\nand the other one -1, so that the seesaw mass matrix from Eq.~(\\ref{lagrangian}) takes the form\n\\begin{equation}\\label{inverse}\nM_{\\nu}=\\left(\\begin{array}{ccc} 0 & h'_{\\alpha 1}v^2 & 0\\\\\nh^{'T}_{\\alpha 1}v^2 & 0 & M \\\\\n0 & M & 0 \\end{array}\\right) \\, ,\n\\end{equation}\nwhich conserves lepton number. Rotating to the RH neutrino mass basis, one finds that\n\\begin{equation}\nM_{\\nu}=\\left(\\begin{array}{ccc} 0 & h'_{\\alpha 1}v^2 & {\\rm i} h'_{\\alpha 1} v^2\\\\\nh^{'T}_{\\alpha 1}v^2 & M & 0 \\\\\n{\\rm i} h^{'T}_{\\alpha 1} v^2 & 0 & M \\end{array}\\right) \\, .\n\\end{equation}\nIt can be seen that the two RH neutrinos are exactly degenerate in this limit, and one\ncan easily show that the neutrino mass matrix $m_{\\nu}$ in Eq.~(\\ref{seesaw}) vanishes\nidentically. In other words, having non-zero neutrino masses requires a small\nbreaking of the lepton number symmetry, which automatically splits the two RH neutrinos\ninto a quasi-Dirac fermion pair.\n\nThere are different ways to implement the breaking of the\nlepton number symmetry, thus generating non-zero neutrino masses. For instance,\nin the above two-RH neutrino model, we can write:\n\\begin{equation}\\label{breaking}\nM_{\\nu}=\\left(\\begin{array}{ccc} 0 & h'_{\\alpha 1}v^2 & \\epsilon_{\\alpha}v^2\\\\\nh^{'T}_{\\alpha 1}v^2 & \\mu_1 & M \\\\\n\\epsilon^{T}_{\\alpha}v^2 & M & \\mu_2 \\end{array}\\right) \\, ,\n\\end{equation}\nwhich implies that the full light neutrino mass matrix is given by~\\cite{Gavela:2009cd,Blanchet:2009kk}\n\\begin{equation}\\label{nuinverse}\nm_{\\nu}\\simeq v^2\\left(\\epsilon {1\\over M} h^{'T} + h' {1\\over M} \\epsilon^T\\right) - v^2\n\\left(h'{1\\over M}\\mu_2 {1\\over M} h^{'T} \\right) \\, ,\n\\end{equation}\nproportional to the breaking parameters $\\epsilon$ and $\\mu_2$, as expected.\nIf the lepton number symmetry is broken as in the first term in Eq.~(\\ref{nuinverse}), it\nis referred to as `linear'~\\cite{Barr:2003nn}; if the second term is at work, it is referred to\nas inverse\/double\nseesaw mechanism~\\cite{Mohapatra:1986bd}. In the following, we\nwill refer to these models for simplicity as\n`inverse seesaw models'.\nNo matter how the lepton number symmetry is broken, the bottom line is that these\nmodels fall in the category of \\emph{low-scale seesaw\nmodels}, where the size of the Yukawa couplings is not necessarily suppressed if the RH neutrino\nmass scale is lowered to the electroweak scale. This can lead to interesting non-unitarity effects in neutrino\noscillation experiments~\\cite{FernandezMartinez:2007ms,Antusch:2006vwa}, as well\nas observable lepton flavour violating rates in experiments looking for $\\mu \\to e\\gamma$,\n$\\tau \\to \\mu (e)\\gamma$, $\\mu \\to eee$ or $\\mu \\to e$ conversion in nuclei~\\cite{Pilaftsis:2005rv}.\nNote that, within the orthogonal parameterisation Eq.~(\\ref{casas}), the inverse seesaw model\nwith two RH neutrinos is obtained in the limit $|\\Omega| \\to \\infty$~\\cite{Asaka:2008bj}.\n\nThe simple model in Eq.~(\\ref{inverse}) can be trivially\nextended to have a third massive RH neutrino of mass still conserving lepton number.\nIt would have zero lepton\nnumber and be decoupled from leptons. Without further assumptions, the other RH neutrino mass\nscale is independent of $M$, and therefore it can be much lower or much higher.\n\nLeptogenesis in the context of these low-scale seesaw models was intensely studied in recent\nyears. It was found in~\\cite{Pilaftsis:2005rv} that it is possible to have at the same time\nsuccessful leptogenesis and low energy observable effects (beyond standard neutrino oscillation phenomenology)\nsuch as, for example, charged lepton flavour violation processes. \nThis is possible in a model with three RH neutrinos and with the help\nof very large flavour effects (hence the name `resonant $\\tau$-leptogenesis').\n\nMore recently, this possibility was examined in the context of a two-RH\nneutrino model, and it was found that, in the limit $M_2-M_1 \\gg \\Gamma_{1,2}$, leptogenesis does\nnot allow large enough Yukawa couplings to have non-trivial consequences at low energies~\\cite{Asaka:2008bj}.\nThis conclusion was re-examined in \\cite{Blanchet:2009kk}\nrelaxing the requirement on the mass splitting, and allowing for more extreme quasi-degeneracies. In the\nlimit $M_2-M_1 \\ll \\Gamma_{1,2}$, it was argued that the decay parameter $K$ should be replaced by\nan effective decay parameter $K^{\\rm eff}_{\\alpha} \\propto K_{\\alpha} (M_2-M_1)^2\/\\Gamma_1^2$\nwhich depends explicitly on the small breaking of the lepton number\nsymmetry. As a matter of fact,\nit is expected that the washout of lepton number vanishes in the limit\nof lepton number conservation, and in \\cite{Blanchet:2009kk} it was rigorously derived from the negative interference\nbetween the two RH neutrinos exchanged in the $\\Delta L=2$ process $\\ell \\Phi \\to \\bar{\\ell}\\Phi^{\\dagger}$.\n\nA more controversial issue is the behavior of the $C\\!P$ asymmetry parameter in the limit\n$M_2-M_1 \\ll \\Gamma_{1,2}$, which is directly related to the form of the regulator for the RH neutrino\npropagator in the self-energy diagram. The reason is that the location of the pole for the RH neutrino determines\nthe maximum enhancement of the $C\\!P$ asymmetry. Following \\cite{Anisimov:2005hr}, Ref.~\\cite{Blanchet:2009kk}\nuses\n\\begin{eqnarray}\\label{CPdeg}\n\\varepsilon_{i\\alpha} &\\simeq& {1\\over 8\\pi (h^{\\dagger}h)_{ii}}\\sum_{j\\neq i}\n\\left\\{ {\\rm Im}\\left[\nh_{\\alpha i}^{\\star}h_{\\alpha j}(h^{\\dagger}h)_{ij}\\right] +\n{\\rm Im}\\left[h_{\\alpha i}^{\\star}h_{\\alpha j}(h^{\\dagger}h)_{ji}\\right]\\right\\} \\nonumber\\\\\n&&\\times{M_j^2-M_i^2 \\over (M_j^2-M_i^2)^2\n+(M_i\\Gamma_i -M_j\\Gamma_j)^2}\n\\end{eqnarray}\nwith the regulator given by the difference $M_i\\Gamma_i -M_j\\Gamma_j$,\nwhereas \\cite{Pilaftsis:2005rv} finds the regulator $M_i\\Gamma_i$. Within \nthe inverse seesaw model considered (with two RH neutrinos)\nthe decay rates $\\Gamma_1$ and $\\Gamma_2$ are predicted to be equal\nin the lepton number conserving limit. Therefore, a regulator $M_1\\Gamma_1 -M_2\\Gamma_2$ allows for a much\nlarger enhancement of the $C\\!P$ asymmetry than $M_1\\Gamma_1$, implying that\nleptogenesis is compatible with observable lepton flavour violation rates~\\cite{Blanchet:2009kk}.\nThe precise value of the $C\\!P$ asymmetry in the regime $M_2-M_1 \\ll \\Gamma_{1,2}$, which is especially relevant in inverse\nseesaw models, is currently still an open issue, whose resolution presumably lies beyond the classical Boltzmann approach\n(see next Section).\n\nNote that the Weinberg-Nanopoulos~\\cite{Nanopoulos:1979gx} requirement, that at least\ntwo couplings should violate lepton (or baryon) number to have a generation of asymmetry,\nis satisfied with both regulators in the inverse seesaw model.\nIndeed, when four out of the five lepton-number-violating couplings in Eq.~(\\ref{breaking})\nare turned off, the $C\\!P$ asymmetry vanishes with both regulators,\nas it should~\\cite{Blanchet:2009kk}. However, the limit of all couplings taken\nsimultaneously to zero is not well-behaved for the regulator in Eq.~(\\ref{CPdeg}),\nas noted in \\cite{Deppisch:2010fr}.\n\nA different scenario was considered in \\cite{Antusch:2009gn} but still within the inverse seesaw framework. There,\na third almost decoupled RH neutrino was added, with a mass $M_1\\ll M$. \nLepton number violation was included in the coupling of the lightest\nRH neutrino, such that it can decay producing a lepton asymmetry. Therefore, in this case, the quasi-degenerate\npair of RH neutrinos is not responsible for the generation of asymmetry. Nonetheless, their large couplings to leptons\n(leading for instance to non-unitarity effects in neutrino oscillations) imply that the flavoured $C\\!P$ asymmetry, more precisely\nthe lepton number conserving part in Eq.~(\\ref{flavouredCP}), can be large even for TeV-scale RH neutrino masses.\nHowever, this scenario of `non-unitarity driven leptogenesis' has intrinsically large lepton\nflavour violating interactions that lead to flavour equilibration~\\cite{AristizabalSierra:2009mq}. Using the\ncross-sections for flavour violating interactions obtained in~\\cite{Pilaftsis:2005rv}, one finds that\nthe asymmetry cannot be generated in the right amount if $M_1\\lesssim 10^8$~GeV~\\cite{Antusch:2009gn}. \nHowever, these cross-sections were found recently to be significantly more suppressed than in~\\cite{Pilaftsis:2005rv}, \nleading to an interesting\nrelaxation of the bounds down to $M_1\\gtrsim 10^6$~GeV~\\cite{Racker:2012vw}.\n\nIt is worth mentioning that leptogenesis was investigated in the context of the inverse seesaw model also when\nlepton number is exactly conserved \\cite{GonzalezGarcia:2009qd}.\nIn this case, the observed baryon asymmetry can be generated\nif leptogenesis occurs during the electroweak phase transition, when the sphaleron rate progressively\ngoes out of equilibrium. The lepton flavour asymmetries, generated exclusively thanks to flavour effects (again the second\nterm in Eq.~(\\ref{flavouredCP})),\nare then converted into a baryon asymmetry before total washout.\n\nAnother example where leptogenesis occurs in the resonant regime\nis radiative leptogenesis~\\cite{GonzalezFelipe:2003fi,Branco:2005ye}. There, RH neutrinos are assumed exactly\ndegenerate at some high scale (for instance the GUT scale), and\nsmall RH neutrino mass splittings are generated by the running of the seesaw parameters.\n\nIn these low-scale seesaw models with TeV-scale RH neutrinos,\nit is natural to wonder whether some interesting signatures could be observed at collider experiments such as the LHC.\nUnfortunately, it seems that within the simplest type-I seesaw, the prospects are\nrather dim~\\cite{Kersten:2007vk,Ibarra:2011xn}.\nThe main problem is that the mixing of RH neutrinos with light neutrinos has strong upper\nlimits from rare (lepton-flavour-violating) decays~\\cite{FernandezMartinez:2007ms}, which prevents an\nimportant production of RH neutrinos at the LHC.\nHowever, in extended models such as Type II seesaw, Type III seesaw, left-right symmetric\nmodels, or simply an extra $U(1)_{B-L}$, \nthe prospects are much more encouraging (see \\cite{hambye} in this Issue).\n\n\n\\section{Improved kinetic description}\n\nWe have already discussed the density matrix formalism, which goes beyond the traditional\nkinetic treatment with Boltzmann (rate) equations. This Section is devoted to other\nkinetic effects that are important both for an estimation of the\ntheoretical uncertainties in the calculation and for a better conceptual understanding of\nthe minimal leptogenesis framework.\n\n\\subsection{Momentum dependence}\n\nWithin the vanilla scenario, the final asymmetry is computed by solving classical Boltzmann equations\nfor the RH neutrino and lepton \\emph{number densities}, so-called rate equations. These\nare obtained from the Boltzmann equations for the \\emph{distribution\nfunction} integrating over momenta with some approximations (see below).\nOne can then wonder what is the theoretical error introduced by this integrated description.\n\nGiven a particle species $X$, the number density is obtained by integrating\nthe distribution function over momentum,\n\\begin{equation}\nn_X ={g_X\\over(2\\pi)^3}\\, \\int \\, d^3 \\, p_X \\, f_X \\, ,\n\\end{equation}\nwhere $g_X$ is the number of degrees of freedom of particle $X$.\nFor leptogenesis with decays and inverse decays,\nthe system of Boltzmann equations (one for the RH neutrino and one for lepton number)\nin the expanding Friedmann-Robertson-Walker Universe\nis given by\n\\begin{eqnarray}\n{\\partial f_N\\over \\partial t} - |{\\bf p}_N| H {\\partial f_N\\over \\partial |{\\bf p}_N|}&=& \\mathcal{C}_D[f_N]\\\\\n{\\partial f_{\\ell-\\bar{\\ell}}\\over \\partial t} - |{\\bf p}_{\\ell}| H\n{\\partial f_{\\ell-\\bar{\\ell}}\\over \\partial |{\\bf p}_{\\ell}|}&=& \\mathcal{C}_D[f_{\\ell-\\bar{\\ell}}] \\, ,\n\\label{momentumke}\n\\end{eqnarray}\nwhere the collision integrals on the right-hand side are defined as\n\\begin{eqnarray} \\fl\n\\mathcal{C}[f_A, A \\leftrightarrow B\\, C] &=& {1\\over 2 E_A}\\int {d^3 p_B \\over 2E_B (2\\pi)^3}\n{d^3 p_C \\over 2E_C (2\\pi)^3} (2\\pi)^4 \\delta^4(p_A - p_B - p_C) \\nonumber\\\\\n&&\\times \\left[ f_B f_C(1-f_A)|\\mathcal{M}( B\\,C\\to A)|^2 - \\right.\\\\\n&&\\left. f_A(1-f_B)(1-f_C)|\\mathcal{M}(A\\to B\\,C)|^2\\right]\\nonumber \\,.\n\\end{eqnarray}\nNote that the double-counting problem is\nsolved here in the same way as for the integrated Boltzmann equations, by consistently including\nthe resonant part of the $\\Delta L=2$ scatterings. \nIn order to recover the usual Boltzmann equations~(\\ref{dlg1})--(\\ref{unflke}),\none has to introduce three approximations:\n(i) kinetic equilibrium for the RH neutrinos, which can be expressed as $f_N\/f_N^{\\rm eq} = n_N\/n_N^{\\rm eq}$;\n(ii) Maxwell-Boltmann distributions for RH neutrinos, leptons and Higgs fields;\n(iii) neglect Pauli blocking and Bose enhancement factors.\n\nRH neutrinos are only coupled to the thermal bath via their Yukawa couplings. It is therefore\nclear that in the weak washout regime, $K_1\\ll 1$, the assumption of kinetic equilibrium is not a very good one,\nand, indeed, it was found that the lepton asymmetry computed with the above equations can differ by\nup to 50\\% in the weak\nwashout regime compared to the usual treatment with integrated equations~\\cite{Basboll:2006yx,HahnWoernle:2009qn,\nGarayoa:2009my}.\nHowever, as expected, in the strong washout regime, the above-mentioned approximations\nare very good and the integrated rate equations can be used safely.\n\n\\subsection{Non-equilibrium formalism}\n\nLeptogenesis is an intrinsic non-equilibrium problem. One of the Sakharov's conditions is indeed that\na departure from thermal equilibrium is necessary to produce the baryon asymmetry. Therefore,\nit does not come as a surprise that recently a huge effort~\\cite{Buchmuller:2000nd,desimone,qke,qke2,garny,beneke,\nBeneke:2010dz,Garny:2011hg,Garbrecht:2011aw}\nwas made to understand leptogenesis\nwithin non-equilibrium quantum field theory, also known as the closed-time-path (CTP) or\nKeldysh-Schwinger formalism. This more rigorous, though far more complex, approach has the advantage\nof taking into account quantum effects that are completely missed by the usual approach,\nlike memory effects and off-shell effects. Moreover, it allows the straightforward inclusion\nof flavour oscillations and decoherence~\\cite{Beneke:2010dz}, it has the advantage of being able to consistently\naccount for finite density corrections, and it has no double counting problem.\n\nConcretely, in the non-equilibrium framework,\none needs to find the equations of motion for the two-point correlation functions\n(i.e. the propagators or Green's functions) of RH neutrinos\nand leptons from the general Schwinger-Dyson equation on the CTP,\n\\begin{eqnarray}\nS_{\\ell}^{-1}(x,y)&=& S^{-1}_{\\ell 0}(x,y) -\\Sigma_{\\ell}(x,y)\\, ,\\\\\nS^{-1}(x,y)&=& S^{-1}_{0}(x,y) -\\Sigma_{N}(x,y)\\, ,\n\\end{eqnarray}\nwhich are obtained from the variational principle on the effective action, a functional of the\nfull propagators $\\Delta_{\\phi}$, $S_{\\ell}$ and $S$, for the Higgs, lepton and RH neutrino, respectively.\nIn the above equations, the subscript $0$ denotes the free propagators, and $\\Sigma_{\\ell}$ and\n$\\Sigma_N$ are the self-energies for the leptons and RH neutrinos, respectively.\nNote that the self-energies are themselves functions of the propagators. For instance, the one-loop\nlepton self-energy depends on the RH neutrino and Higgs propagators:\n\\begin{equation}\n\\Sigma_{\\ell}^{\\alpha \\beta}(x,y)=-h_{\\alpha i}h_{j \\beta}^{\\dagger} P_R S^{ij}(x,y)P_L \\Delta_{\\phi}(y,x) \\, .\n\\end{equation}\nIt usually proves convenient\nto decompose any two-point function $D(x,y)$ into a \\emph{spectral}, $D_{\\rho}$, and a \\emph{statistical}\ncomponent, $D_F$:\n\\begin{equation}\nD(x,y)=D_F(x,y)-{i\\over 2} {\\rm sgn}(x^0-y^0)D_{\\rho}(x,y) \\, .\n\\end{equation}\nConvoluting the Schwinger-Dyson equations\nwith the full propagator, we finally arrive at a system of two coupled integro-differential\nequations, the so-called Kadanoff-Baym equations:\n\\begin{eqnarray}\ni\\slashed{\\partial}_x {S_{\\ell}}_F^{\\alpha\\beta}(x,y)&=&\\int_0^{x^0} d^4 z {\\Sigma_{\\ell}}_{\\rho}^{\\alpha\\gamma}(x,z)\n{S_{\\ell}}_F^{\\gamma\\beta}(z,y) \\nonumber\\\\\n&& -\\int_0^{y^0} d^4 z {\\Sigma_{\\ell}}_{F}^{\\alpha\\gamma}(x,z)\n{S_{\\ell}}_{\\rho}^{\\gamma\\beta}(z,y)\\, , \\\\\ni\\slashed{\\partial}_x {S_{\\ell}}_{\\rho}^{\\alpha\\beta}(x,y)&=&\\int_{y^0}^{x^0} d^4 z {\\Sigma_{\\ell}}_{\\rho}^{\\alpha\\gamma}(x,z)\n{S_{\\ell}}_{\\rho}^{\\gamma\\beta}(z,y) \\, .\n\\end{eqnarray}\nThe corresponding equations for the RH neutrino two-point function are obtained by changing $\\slashed{\\partial}_x \\to\n\\slashed{\\partial}_x -M$, and $S^{\\alpha \\beta} \\to\nS^{ij}$, where $\\alpha, \\beta= e,\\mu,\\tau$ are lepton flavours, and $i,j=1,2,3$ are RH neutrino flavours.\nIt can be noticed that the Kadanoff-Baym equations contain an integration over the entire history of the system, a `memory'\nintegral, which encodes all previous interactions with momentum and spin correlations. An attempt\nof studying memory effects in the context of leptogenesis~\\cite{desimone}\nhas found that large effects could arise in the resonant limit in the weak washout regime.\nIn order to recover a Markovian description of\nthe system, characterized by uncorrelated initial states at every timestep, one has to perform a gradient expansion\n(for an alternative approach, see~\\cite{qke2}),\nrelying on the fact that the microscopic timescale $t_{\\rm mic}\\sim 1\/M_i$ is much smaller than the macroscopic timescales\n$t_{\\rm mac}\\sim 1\/\\Gamma_i,1\/H$. This is also known as the molecular chaos approximation.\n\nIn leptogenesis, one needs to compute the evolution of the lepton number density. The latter\nis given by the average expectation value of the zeroth component of the lepton number current,\ngiven by\n\\begin{equation}\nj^{\\mu}_{L\\alpha\\beta}(x)=-{\\rm tr}\\left[\\gamma^{\\mu}S_{\\ell \\alpha\\beta}(x,x)\\right] \\, .\n\\end{equation}\nOne then obtains for the lepton number density\n\\begin{eqnarray}\nn_{L\\alpha\\beta}(t)&=& i\\int {d^3 p\\over (2\\pi)^3} \\int_0^t dt' \\int_0^{t'} dt'' {\\rm tr}\n\\left[{{\\Sigma_{\\ell}}_{\\rho}}_p^{\\alpha\\beta} (t',t'') {{S_{\\ell}}_F}_p(t'',t') \\right.\\nonumber \\\\\n&& \\left. -{{\\Sigma_{\\ell}}_F}_p^{\\alpha \\beta} (t',t'') {{S_{\\ell}}_{\\rho}}_p(t'',t')\\right] \\, ,\n\\end{eqnarray}\nafter switching to momentum space. From this master equation, one then needs to\ninput the equilibrium expression for the lepton and\nHiggs propagators, as well as the non-equilibrium Majorana neutrino propagator.\nIn order to obtain a Boltzmann-like equation, two further simplifications are required.\nFirst, the quasi-particle ansatz, also known as the on-shell approximation, which states:\n\\begin{equation}\nD_{\\rho}(X,p)=2\\pi\\, {\\rm sgn}(p^0)\\delta(p^2-m^2) \\, ,\n\\end{equation}\nwhere $X\\equiv (x+y)\/2$ is the central coordinate. Then, one can express\nthe statistical propagator for the occupation number using the so-called Kadanoff-Baym ansatz,\n\\begin{equation}\nD_F(X,p)=\\left[f(X,p)+{1\\over 2}\\right]D_{\\rho}(X,p) \\, ,\n\\end{equation}\nwhich is chosen such that, in equilibrium, the correct \\emph{fluctuation dissipation relation} between $D_F^{\\rm eq}(p)$ and $D_{\\rho}^{\\rm eq}(p)$ \nis automatically obtained (see \\cite{qke} for more details).\nWith these approximations, one arrives at a Boltzmann equation which includes finite density\neffects. This equation would be similar to the kinetic equation~(\\ref{momentumke}), with however\nthe important new property:\n\\begin{equation}\\label{property}\n|\\mathcal{M}( N\\to \\ell \\Phi)|^2 = |\\mathcal{M}( \\ell \\Phi \\to N)|^2 =\n|\\mathcal{M}_0( N\\leftrightarrow \\ell \\Phi)|^2 (1+\\varepsilon(p,T)) \\,,\n\\end{equation}\nwhere subscript `0' denotes tree level, and~\\cite{garny}\n\\begin{equation}\n\\varepsilon (p, T)=\\varepsilon \\times \\left(1+\\int {d \\Omega\\over 4\\pi}[f_{\\phi}^{\\rm eq}(E_1)\n-f_{\\ell}^{\\rm eq}(E_2)]\\right) \\,\n\\end{equation}\nwhere $\\varepsilon$ is the $C\\!P$ asymmetry defined in Eq.~(\\ref{CPas}), and\n$E_{1,2}={1\\over 2} \\left[(M^2+ p^2)^{1\\over 2} \\pm p \\cos\\theta\\right]$.\nNote that the above property in Eq.~(\\ref{property}) explicitly avoids\nthe double counting problem, which plagues the momentum-dependent description of the\nlast Subsection (as well as the usual integrated one).\nAs it can be seen, the\n$C\\!P$ asymmetry parameter now includes finite density effects via a dependence on the Higgs and lepton\ndistribution functions. This result\nagrees with that one found using thermal field theory in the real time formalism, when the right convention\nis used \\cite{garny}, as well as in the imaginary time formalism as recently obtained~\\cite{Kiessig:2011fw}.\nIt, however, disagrees with an earlier work based on thermal field theory in the real time formalism, where\na term quadratic in the distribution functions was found~\\cite{giudice}.\n\n\n\nOne should then wonder whether the non-equilibrium formalism described above leads to substantial\ndifferences in the leptogenesis predictions for the baryon asymmetry. It turns out\nthat the modifications are very\nimportant (up to an order of magnitude) in the regime where $M_i\/T\\ll 1$, but they tend to vanish\nin the non-relativistic regime $M_i\/T \\gg 1$. In other words, major changes are expected mainly in the\nweak washout regime for a vanishing initial RH neutrino abundance, because in this case the asymmetry is produced\nin both regimes: the initial asymmetry is produced with the wrong sign at early times and is compensated\nby the right-sign asymmetry at later times. This was confirmed in~\\cite{beneke}, where it was found\nthat the sign of the final asymmetry could even get changed by finite density effects.\nIn all other cases, corrections are quantitatively small, in particular in the strong washout regime,\n$K_i\\gg 1$,\nwhere the asymmetry is produced exclusively in the non-relativistic regime~\\cite{garny,beneke}.\n\n\n\\subsubsection{Resonant limit}\n\nIt is interesting to study whether the above quantum kinetic formalism can provide some insight\nonto the $C\\!P$ asymmetry produced\nin the extreme quasi-degenerate limit, $|M_1 -M_2|\\ll M_{1,2}$. This formalism includes off-shell\neffects, as well as the proper inclusion of coherent\ntransitions $N_i \\to N_j$, both of which should be important when $|M_1 -M_2|\\sim \\Gamma_{1,2}$. This\nproblem attracted some attention recently~\\cite{Garny:2011hg,Garbrecht:2011aw}.\n\nIn \\cite{Garny:2011hg} it was found that the enhancement of the $C\\!P$ asymmetry\nderived within the Kadanoff-Baym (KB) formalism, \n\\begin{equation}\nR^{\\rm KB}={M_1 M_2 (M_2^2-M_1^2)\\over (M_2^2-M_1^2)^2+(M_1\\Gamma_1+M_2\\Gamma_2)^2} \\, ,\n\\end{equation}\ndiffers from the Boltzmann result, \n\\begin{equation}\nR^{\\rm BE}={M_1 M_2 (M_2^2-M_1^2)\\over (M_2^2-M_1^2)^2+(M_1\\Gamma_1-M_2\\Gamma_2)^2} \\, ,\n\\end{equation}\nspecifically because of a contribution from coherent RH neutrino oscillations. \nThis would prevent any additional enhancement when $\\Gamma_1 \\sim \\Gamma_2$, as predicted by\nthe inverse seesaw model. In such a case, the new region in the parameter space of successful\nleptogenesis explored in~\\cite{Blanchet:2009kk} is not available.\n\n\n\\subsubsection{Adding flavour}\n\nWe saw in Section 5 how the first kind of `quantum' effects were included,\nnamely flavour oscillations and decoherence, by switching from classical Boltzmann equations\nfor lepton number densities to evolution equations for the density matrix. For the lightest\nRH neutrino and two relevant lepton flavours, we presented Eq.~(\\ref{densmateq}). The CTP\nformalism can also account for a flavour matrix structure, through the\nlepton propagators and self-energies. Reassuringly, the structure of\nEq.~(\\ref{densmateq}) was also found within this formalism~\\cite{Beneke:2010dz}.\n\n\\section{Other corrections}\n\n\\subsection{Thermal effects}\n\nLeptogenesis occurs in the very hot thermal bath of the early Universe. Leptons and Higgs fields\nhave very fast interactions with the thermal bath due to their gauge couplings. This gives them an\neffective mass which is proportional to temperature~\\cite{Weldon:1982bn}. This effective mass allows processes\nwhich were otherwise kinematically forbidden to occur, such as $C\\!P$-violating Higgs decays to\nRH neutrinos, when $T\\gg M$. Thermal effects have therefore a direct impact on leptogenesis, and the\nfirst study to try and quantify them \\cite{giudice} employed a real time formalism and\nhard thermal loop resummation. It was\nfound that the $C\\!P$ violating parameter has a strong temperature dependence, and\nthat in the weak washout regime, there\ncould be important differences with the zero-temperature treatment. On the other hand, in the strong\nwashout regime, $K\\gg 1$, the usual results from a vacuum calculation are recovered with\na good accuracy. Recently, a new study came out using the imaginary time formalism~\\cite{Kiessig:2011fw},\nand the $C\\!P$\nasymmetry parameter was found to differ from~\\cite{giudice}, and to agree with more recent attempts\nwith non-equilibrium quantum field theory (see previous Section).\n\n\\subsection{Spectator processes}\n\nChemical equilibrium holds among Standard Model particles in the early Universe above the electroweak phase\ntransition thanks to gauge interactions. On the other hand, Yukawa interactions only force some new conditions\n(between left- and right-handed fermions)\nwhen the temperature is high enough, depending on the size of the Yukawa coupling. If one imposes\nadditionally hypercharge\nneutrality and the effect of electroweak and strong sphaleron equilibrium, one arrives at a set\nof relations among the chemical potentials (or among the asymmetries) of leptons, Higgs and\nbaryons~\\cite{Nardi:2005hs}.\nThese processes, although not directly involved in the leptogenesis process, hence the name `spectator\nprocesses'~\\cite{Buchmuller:2001sr},\nhave an indirect effect\non the final asymmetry through a modified washout. The first effect is the inclusion of the Higgs asymmetry\nas a new contribution to the washout. The second effect is that the asymmetry originally produced in leptons of a\nparticular flavour gets redistributed into the other flavours following precise relations~\\cite{bcst}.\nThis means\nthat the set of Boltzmann equations to solve for the different lepton flavours are now coupled to each other\nvia a flavour coupling matrix.\n\nWithin the $N_1$-dominated scenario,\nthe overall effect of spectator processes is usually subdominant compared to flavour effects, but it can change\nthe final result by as much as 40\\% depending on the temperature at which the asymmetry\nis produced~\\cite{Nardi:2005hs,JosseMichaux:2007zj}.\nSupersymmetry has new degrees of freedom and new constraints can be derived. A full study was performed recently\nin~\\cite{Fong:2010qh} and the overall effect was found to be again of order one.\nWithin the $N_2$-dominated scenario, however, potentially much bigger effects are possible \\cite{Antusch:2010ms}.\n\n\\subsection{Scattering processes}\n\nIt can be shown that leptogenesis is well-described in the strong washout regime, $K\\gg 1$, by just\ndecays and inverse decays. The reason is that in this regime\nthe asymmetry is produced at relatively late times, at $T\\ll M$, with no dependence on the dynamics\nhappening at $T\\gtrsim M$ when scattering\nprocesses are important. However, they should be included\nin the weak washout regime for initial vanishing RH neutrino abundance. For instance, $\\Delta L=1$ Higgs-mediated\nscatterings involving top quarks, such as $\\ell N \\leftrightarrow Q_3 \\bar{t}$, contribute to the washout\nand the $C\\!P$ asymmetry~\\cite{Abada:2006ea}, and their inclusion is crucial to have a correct estimation\nof the final asymmetry in the weak washout regime~\\cite{Nardi:2007jp}. Scatterings involving gauge\nbosons, such as $\\ell N\\leftrightarrow \\Phi^{\\dagger}A$, have also been included, especially their contribution\nto the $C\\!P$ asymmetric source term~\\cite{Fong:2010bh}. It was found that the factorization of the $C\\!P$ asymmetry\nfrom decays and scatterings involving top quarks does not happen with scatterings involving gauge bosons.\nIn particular, there is a new source of lepton-number-conserving $C\\!P$ asymmetry.\n\n\n\\section{Testing new physics with leptogenesis}\n\nThe seesaw mechanism with three RH neutrinos extends the Standard Model by introducing eighteen new parameters.\nOn the other hand, low-energy\nneutrino experiments can only potentially test the nine parameters in the low-energy\nneutrino mass matrix $m_{\\nu}$. Nine high-energy parameters, those characterising the properties\nof the three RH neutrinos, the three masses and the six parameters\nencoded in the seesaw orthogonal matrix of Eq.~(\\ref{casas}),\nbasically fixing, together with the light neutrino masses, the three lifetimes and the three total $C\\!P$ asymmetries,\nare not tested by low-energy neutrino experiments.\nQuite interestingly, the requirement of successful leptogenesis,\n\\begin{equation}\n\\eta_B(m_{\\nu},\\Omega,M_i)=\\eta_{B}^{\\rm CMB} \\, ,\n\\end{equation}\nprovides an additional constraint on a combination\nof both low-energy neutrino parameters and high-energy neutrino parameters.\nHowever, just one additional constraint would not seem sufficient to over-constrain the parameter\nspace leading to testable predictions.\nIn spite of this observation, as we have seen, in the vanilla leptogenesis scenario\none can derive an upper bound on the neutrino masses. The reason is that, within this scenario,\nthe dependence of $\\eta_B$ on the six parameters related to the properties of the two heavier RH neutrinos\ncancels out. In this way the asymmetry depends on a reduced\nsubset of high-energy parameters (just three instead of nine).\nAt the same time, the final asymmetry gets strongly suppressed when\nthe absolute neutrino mass scale it is larger than the atmospheric neutrino mass scale.\nFor all these reasons, by maximising the final asymmetry over the high energy parameters and\nby imposing successful leptogenesis, an upper bound on the neutrino masses is found.\n\nWhen flavour effects are considered, the vanilla leptogenesis\nscenario holds only under very special conditions, as we have seen.\nIn general, the final asymmetry depends also on the parameters in the leptonic mixing matrix.\nTherefore, accounting for flavour effects,\none could naively hope to derive definite predictions on the leptonic mixing matrix as well,\nin addition to the upper bound on the absolute neutrino mass scale.\nHowever, the situation is quite different when flavour effects are taken into account.\nThis is because the final asymmetry depends, in general, also on the six parameters\nrelated describing the two heavier RH neutrino properties and\nthat were cancelling out in the calculation of the final asymmetry in the vanilla\nscenario, and this goes at the expense of predictability.\n\nFor this reason, in a general scenario with three RH neutrinos and flavour effects included, it is not possible\nto derive any prediction on low-energy neutrino parameters.\nAs we discussed, even whether the upper bound $m_i\\lesssim 0.1$~eV on neutrino masses still holds\nor not is an open issue and a precise value seems to depend on a precise account of \nmany different subtle effects and in particular it necessarily requires a density matrix formalism. \n\nIn order to gain predictive power, two possibilities have been explored in the past years.\n\nA first possibility is to consider non-minimal scenarios giving rise to additional phenomenological constraints.\nFor example, as discussed in Section 6, the inverse seesaw model, which technically can be still regarded as part\nof the minimal type I seesaw model, allows for non-trivial phenomenologies at low energy beyond standard neutrino\noscillations, such as non-unitarity effects and observable lepton flavour violation. An experimental\nobservation of any of these signatures would be of great value to reduce the freedom in\nthe choice of seesaw parameters.\nIn recent years, during the Large Hadron Collider era, it has been also intensively\nexplored the possibility that, within a non-minimal version of the seesaw mechanism,\none can have successful low scale leptogenesis together with collider signatures.\nIt has also been noticed that\nin the supersymmetric version of the seesaw, the branching ratios of lepton-flavour-violating processes\nor electric dipole moments are typically enhanced, and hence the existing experimental bounds\nfurther constrain the seesaw parameter space \\cite{Pascoli:2003uh,Dutta:2003my}.\n\nA second possibility is to search for a reasonable\nscenario where the final asymmetry depends only on a reduced set of independent parameters\nover-constrained by the successful leptogenesis condition, as with the vanilla scenario. \nFrom this point of view, the account of\nflavour effects has opened very interesting new opportunities or even re-opened old attempts\nthat fail within a strict unflavoured scenario. Let us briefly discuss some\nof the main ideas that have been proposed within this second possibility, the first one being\ncovered elsewhere in this Issue~\\cite{hambye}.\n\n\\subsection{Two-RH neutrino model}\n\nA phenomenological possibility that attracted great attention is the two-RH\nneutrino model \\cite{Frampton:2002qc},\nwhere the third RH neutrino is either absent or\neffectively decoupled in the seesaw formula.\nThis necessarily happens when $M_3\\gg 10^{14}\\,{\\rm GeV}$, implying that the\nlightest left-handed neutrino mass $m_1$ has to vanish. It can be shown that the number of parameters\ndecreases from 18 to 11 in this case.\nIn particular the orthogonal matrix is parameterised in terms of just one complex angle.\n\nIn leptogenesis the two-RH neutrino model has been traditionally considered as a sort of benchmark case\nfor the $N_1$-dominated scenario, where the final asymmetry is dominated by the contribution\nfrom the lightest RH neutrinos \\cite{Abada:2006ea,Blanchet:2006dq,petcovmolinaro}. However, recently, as we anticipated already, it has been\nshown that there are some regions in the one complex angle parameter space\nthat are $N_2$-dominated~\\cite{Antusch:2011nz} and that\ncorrespond to so-called light sequential dominated models \\cite{King:2003jb}.\n\nIt should be said that even though the number of parameters is highly reduced,\nin a general two-RH neutrino model it is still not possible to make predictions on the low-energy neutrino\nparameters. To this extent, one should further reduce the parameter space, for example assuming texture zeros\nin the neutrino Dirac mass matrix.\n\n\\subsection{$SO(10)$-inspired models}\n\n In order to gain predictive power, one can\nimpose conditions within some model of new physics embedding the seesaw mechanism.\nAn interesting example is represented by the `$SO(10)$-inspired leptogenesis scenario' \n\\cite{buchplumi,Branco:2002kt,Akhmedov:2003dg},\nwhere $SO(10)$-inspired conditions are imposed on the neutrino Dirac mass matrix $m_D$.\nIn the basis where the charged leptons mass matrix and the Majorana mass matrix are diagonal,\nin the bi-unitary parametrisation, one has $m_D = V_L^{\\dagger}\\,D_{m_D}\\,U_R$,\nwhere $D_{m_D}\\equiv {\\rm diag}({\\lambda_1,\\lambda_2,\\lambda_3})$ is the diagonalised neutrino Dirac mass matrix\nand the mixing angles in $V_L$ are of the order\nof the mixing angles in the CKM matrix $V_{CKM}$.\nThe $U_R$ and three $M_i$ can then be calculated from $V_L$, $U$ and $m_i$,\nsince the seesaw formula Eq.~(\\ref{seesaw}) directly leads to the Takagi factorisation of\n$M^{-1} \\equiv D^{-1}_{m_D}\\,V_L\\,U\\,D_m\\,U^T\\,V_L^T\\,D^{-1}_{m_D}$,\nor explicitly $M^{-1} = U_R\\,D_M^{-1}\\,U_R^T$.\n\nIn this way the RH neutrino masses and the matrix $U_R$ are expressed in terms of the\nlow-energy neutrino parameters, of the eigenvalues $\\lambda_i$ and of the parameters in $V_L$.\nTypically one obtains a very hierarchical spectrum $M_1 \\sim 10^{5}\\,{\\rm GeV}$ and $M_{2}\\sim 10^{11}\\,{\\rm GeV}$,\nthe asymmetry produced from the lightest RH neutrino decays is by far unable to explain the\nobserved asymmetry \\cite{Branco:2002kt}. \n\nHowever, when the $N_2$ produced asymmetry is taken into account,\nsuccessful ($N_2$-dominated) leptogenesis can be attained \\cite{SO10lep1}. \nIn this case, imposing the leptogenesis bound\nand considering that the final asymmetry does not depend on $\\lambda_1$ and on $\\lambda_3$, one obtains\nconstraints on all low-energy neutrino parameters, which have some dependence on the\nparameter $\\lambda_2$ typically parameterised in terms of $\\alpha_2\\equiv \\lambda_2\/m_c$, where $m_c$\nis the charm quark mass. Some examples of the constraints on the low-energy neutrino parameters\nare shown in Fig.~3. They have been obtained scanning over the $2\\sigma$ ranges of the allowed values of the\nlow-energy parameters and over the parameters\nin $V_L$ assumed to be $I< V_L < V_{CKM}$ and\nfor three values of $\\alpha_2=5,4, 1$.\nIt is particularly interesting that when the independence of the initial conditions\nis imposed, negative values of $J_{CP}$ seem to be favoured \\cite{prep} establishing\na connection between the sign of $J_{CP}$ and of the matter-antimatter asymmetry.\n\n\nA supersymmetric version of this scenario including the renormalization group evolution\nof all the relevant couplings was also studied in~\\cite{Blanchet:2010td}, and\nincluding a type II contribution to the seesaw mechanism\nfrom a triplet Higgs in left-right symmetric models in~\\cite{Abada:2008gs}.\n\\begin{figure}\n\\begin{center}\n \\mbox{\\epsfig{figure=m1th13NOGLOBAL.eps,width=55mm,height=55mm}}\n \\vspace*{-20pt}\n \\mbox{\\epsfig{figure=m1th23NOGLOBAL.eps,width=55mm,height=55mm}} \\\\\n \\vspace*{15pt}\n \\mbox{\\epsfig{figure=th13th23NOGLOBAL.eps,width=55mm,height=55mm}}\n \\vspace*{-5pt}\n \\mbox{\\epsfig{figure=th13deltaNOGLOBAL.eps,width=55mm,height=55mm}}\n\\caption{Constraints on some of the low-energy neutrino parameters in the $SO(10)$-inspired\n scenario for normal ordering and $I< V_L < V_{CKM}$~\\cite{DiBari:2010ux}. The yellow, green and red points\n correspond respectively to $\\alpha_2=5,4,1$.}\n\\end{center}\n\\end{figure}\n\nTogether with strongly hierarchical RH neutrino mass patterns,\nthere exist also `level crossing' regions where at least two RH neutrino masses \nare arbitrarily close to each other, in a way that the lightest RH neutrino mass is uplifted and \n$C\\!P$ asymmetries are resonantly enhanced to some lever. Also in this case, imposing the successful\nleptogenesis condition, one then obtains\nconditions, though quite fine tuned ones, \non the low energy neutrino parameters \\cite{Akhmedov:2003dg}. \n\n\\subsection{Discrete flavour symmetries}\n\n An account of heavy neutrino flavour effects is also important when leptogenesis is embedded within\ntheories that try to explain tribimaximal mixing for the leptonic\nmixing matrix via flavour symmetries. It has been shown in particular that,\nif the symmetry is unbroken, then the $C\\!P$ asymmetries of the RH neutrinos would exactly\nvanish. On the other hand, when the symmetry is broken, for the naturally expected\nvalues of the symmetry breaking parameters, then the observed\nmatter-antimatter asymmetry can be successfully reproduced \n\\cite{Jenkins:2008rb,Bertuzzo:2009im,Hagedorn:2009jy,aristizabal}.\nIt is interesting that in a minimal picture based on an $A4$ symmetry, one has a RH neutrino mass spectrum with\n$10^{15}\\,{\\rm GeV} \\gtrsim M_3 \\gtrsim M_2 \\gtrsim M_1 \\gg 10^{12}\\,{\\rm GeV}$. One has therefore\nthat all the asymmetry is produced in the unflavoured regime and that the mass spectrum\nis only mildly hierarchical (it has actually the same kind of hierarchy as light neutrinos).\nAt the same time, the small symmetry breaking imposes\na quasi-orthogonality of the three lepton quantum states produced in the RH neutrino\ndecays. Under these conditions the washout of the asymmetry produced by one RH neutrino species\nfrom the inverse decays of a lighter RH neutrino species is essentially negligible. The final\nasymmetry then receives a non-negligible contribution from the decays of all three RH neutrinos species.\n\n\\subsection{Supersymmetric models}\n\nWithin a supersymmetric vanilla framework the final asymmetry\nis only slightly modified compared to the non-supersymmetric calculation \\cite{proceedings}.\nHowever, supersymmetry introduces a conceptually important issue: the stringent\nlower bound on the reheat temperature, $T_{\\rm reh}\\gtrsim 10^{9}\\,{\\rm GeV}$,\nis typically marginally compatible with an upper bound\nfrom the avoidance of the gravitino problem $T_{\\rm reh}\\lesssim 10^{6-10}\\,{\\rm GeV}$, with the\nexact value depending on the parameters of the model \\cite{Khlopov:1984pf,Ellis:1984eq,Kawasaki:2008qe}.\nIt is quite remarkable\nthat the solution of such an issue inspired an intense research activity on supersymmetric\nmodels able to reconcile minimal leptogenesis and the gravitino problem. Of course, on the\nleptogenesis side, some of the discussed extensions beyond the vanilla scenario that relax the RH neutrino\nmass bounds also relax the $T_{\\rm reh}$ lower bound. However, notice that in the $N_2$ dominated\nscenario, while the lower bound on $M_1$ simply disappears, there is still a lower bound\non $T_{\\rm reh}$ that is even more stringent, $T_{\\rm reh}\\gtrsim 6\\times 10^{9}\\,{\\rm GeV}$ \\cite{DiBari:2005st}.\n\nAs we mentioned already, with flavour effects one has the possibility to relax the lower bound\non $T_{\\rm reh}$ if a mild hierarchy in the RH neutrino masses\nis allowed together with a mild cancellation in the seesaw formula \\cite{bounds}.\nHowever for most models, such as sequential dominated models \\cite{King:2003jb},\nthis solution does not work. A major modification introduced by supersymmetry\nis that the critical value of the mass of the decaying RH neutrinos\nsetting the transition from an unflavoured regime to a two-flavour regime\nand from a two-flavour regime to a three flavour regime is enhanced by a factor\n$\\tan^2\\beta$~\\cite{Abada:2006fw,Antusch:2006cw}.\nThis has a practical relevance in the calculation of the asymmetry within supersymmetric models\nand it is quite interesting that leptogenesis becomes sensitive to such a relevant\nsupersymmetric parameter. Recently, a refined analysis,\nmainly discussing how the asymmetry is distributed among all particle species,\nhas shown different subtle effects in the calculation of the final asymmetry\nwithin supersymmetric models finding corrections below ${\\cal O}(1)$ \\cite{Fong:2010qh}.\n\n\\section{Future prospects for testing leptogenesis}\n\nIn 2011 two important experimental results have been announced that,\nif confirmed, can be interpreted as positive for future tests of leptogenesis.\n\nThe first result is the discovery of a non-vanishing\n$\\theta_{13}\\simeq 9^{\\circ}$ (cf. Section 2) confirmed now\nby various experiments (both long-baseline and reactor).\nAn important consequence of the measurement of such a `large'\n$\\theta_{13}$ is the encouraging prospects for the discovery\nof the neutrino mass ordering (either normal or inverted)\nin current or near-future neutrino oscillation experiments such as T2K and NO$\\nu$A. \nFor instance, if the ordering is found\nto be inverted in these experiments, we should expect a signal in $0\\nu\\beta\\beta$ experiments\nin the next decade. If it is not found, the Majorana nature of neutrinos, and therefore\nthe seesaw mechanism, would be ruled out.\n\nMoreover, such a large value of $\\theta_{13}$ opens the possibility of a measurement of\nthe neutrino oscillation $C\\!P$-violating invariant\n$J_{CP}\\propto \\sin\\theta_{13}\\,\\sin\\delta$ during next years.\nThis would have some direct model-independent consequences.\nIf a non-vanishing and close-to-maximal value is found ($|\\sin\\delta| \\sim 1$),\neven the small contribution to the final asymmetry uniquely stemming\nfrom a Dirac phase could be sufficient to reproduce the observed final\nasymmetry.\nMore generally, the presence of $C\\!P$ violation at low energies would\ncertainly support the presence of $C\\!P$ violation at high energies as well, since\ngiven a generic theoretical model predicting the neutrino Dirac mass matrix $m_D$,\nthese are in general both present.\n\n\nA more practical relevance of such a measurement of $J_{CP}$, even if in the end it\nindicates a vanishing value within the experimental error, is that it\nwill provide an additional constraint on specific models embedding the seesaw,\nsatisfying successful leptogenesis and able to make predictions on the low energy neutrino parameters.\nIn this way, the expected improvements in low-energy neutrino experiments, made easier by\na non-vanishing $\\theta_{13}$, will test the models more and more stringently.\n\nThe second important experimental result of 2011 is the hint for\nthe existence of the Higgs boson reported by the ATLAS and CMS collaborations.\nThere are at least two reasons why this hint can be also interpreted as positive for leptogenesis.\nFirst, because the whole leptogenesis mechanism relies on\nthe Yukawa coupling between Higgs, lepton and RH neutrino. Second,\nbecause the measurement\nof the Higgs boson mass could in future open opportunities for additional\nphenomenological information to be imposed on leptogenesis scenarios,\nfor example relying on the requirement of Standard Model electroweak \nvacuum stability \\cite{giudiceetal}.\nIn this respect, it is interesting to note that the current value of 125 GeV is compatible \nwith reheating temperatures as high as $10^{15}\\,{\\rm GeV}$\nas needed by thermal leptogenesis, and it is also compatible with the requirement that the\nYukawa couplings of the RH neutrinos do not destabilise the Higgs potential \nwhen RH neutrino masses $M_i$, assumed to be quasi-degenerate, are smaller than $10^{14}$~GeV.\n\nWhat are other possible future experimental developments\nthat could further support the idea of leptogenesis?\nAn improved information from absolute neutrino mass scale experiments,\nboth on the sum of the neutrino masses from cosmology and on $m_{ee}$\nfrom $0\\nu\\beta\\b$ experiments could be crucial. For instance,\nif cosmology provides a measurement of the neutrino mass\n$0.01\\,{\\rm eV} \\lesssim m_1 \\lesssim 0.2\\,{\\rm eV}$\nin the next few years, as it is reasonable to expect, then a positive signal\nin $0\\nu\\beta\\b$ experiments must be found, otherwise Majorana neutrinos and the seesaw\nmechanism will be disfavored.\nOn the other hand, in case of a positive signal in $0\\nu\\beta\\b$ experiments, we\nwill be able to say that minimal leptogenesis with hierarchical heavy neutrino\nmasses works in a optimal neutrino mass window\n$10^{-3}\\,{\\rm eV}\\lesssim m_1 \\lesssim 0.1\\,{\\rm eV}$ \\cite{window,bounds,problem}, where\nindependence of the initial conditions is more easily obtained and where we know that successful\nleptogenesis can be safely obtained from existing calculations using Boltzmann equations.\nMoreover, a determination of the allowed region in the plane $m_{ee}$--$\\sum_i\\,m_i$\ncould provide an additional test of specific leptogenesis scenarios, such as, for example, \n$SO(10)$-inspired scenarios.\n\nIn conclusion, we are living in an exciting time where new experimental information\nis coming, which is providing and will continue to provide crucial tests for new\nphysics models, in particular leptogenesis, in the near future.\n\n\\section*{Acknowledgements}\n\nWe wish to thank A.~Abada, W.~Buchm\\\"{u}ller, S.~Davidson, M.~Drewes, T.~Hambye, \nD.A.~Jones, S.F.~King, L.~Marzola, A.~Pilaftsis for useful comments and discussions.\nWe also wish to thank A.~Hohenegger for enlightening discussions about the closed-time-path formalism.\nPDB also wishes the Laboratoire de Physique Th\\'{e}orique, Universit\\'{e} de Paris-Sud 11 (Orsay) \nfor the warm hospitality during the completion of this work.\nSB acknowledges support from the Swiss National Science Foundation, under the Ambizione grant\nPZ00P2\\_136947. PDB acknowledges financial support from the NExT\/SEPnet Institute, \nfrom the STFC Rolling Grant ST\/G000557\/1 and from the EU FP7 ITN INVISIBLES \n(Marie Curie Actions, PITN- GA-2011- 289442).\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\\subsubsection{Computation of $\\PLQ{\\phi_{il}}$}\n\n\n\\xhdr{Variational M-Step}\nIn the E-step, we introduced the variational distribution $Q(F)$ parameterized\nby $\\phi$ and approximated the posterior distribution $P(F| A, \\mu, \\Theta)$ by\nmaximizing $\\mathcal{L}_{Q}(\\mu, \\Theta)$ over $\\phi$.\nIn the M-step, we now fix $Q(F)$,\n\\emph{i.e.}\\xspace, fix the variational parameters $\\phi$, and update the model parameters\n$\\mu$ and $\\Theta$ to maximize $\\mathcal{L}_{Q}$.\n\n\nFirst, in order to maximize $\\mathcal{L}_{Q}(\\mu, \\Theta)$ with respect to $\\mu$, we need to maximize $\\mathcal{L}_{\\mu_{l}} = \\sum_{i} \\EXATTR{il}{\\log P(F_{il} |\n\\mu_{l})}$ for each $\\mu_{l}$.\nBy definitions in \\EQ{eq:jointlik} and (\\ref{eq:qdef}), we obtain\n\\[\n \\mathcal{L}_{\\mu_{l}} = \\sum_{i} \\left( \\phi_{il} \\mu_{il} + (1-\\phi_{il})(1-\\mu_{il}) \\right) \\,.\n\\]\nThen $\\mathcal{L}_{\\mu_{l}}$ is maximized when\n\\[\n \\frac{\\partial \\mathcal{L}_{\\mu_{l}}}{\\partial \\mu_{l}} = \\sum_{i} \\phi_{il} - N = 0\n\\]\nwhere $\\mu_{l} = \\frac{1}{N} \\sum_{i} \\phi_{il}$.\n\nSecond, to maximize $\\mathcal{L}_{Q}(\\mu, \\Theta)$ with respect to $\\Theta_{l}$, we\nmaximize $\\mathcal{L}_{\\Theta} = \\EXATTR{}{\\log P(A, F | \\mu, \\Theta) - \\log Q(F)}$.\nWe first obtain the gradient\n\\begin{align}\n \\nabla_{\\Theta_{l}} \\mathcal{L}_{\\Theta} = \\sum_{i, j} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(A_{ij} |F_{i}, F_{j}, \\Theta)}\n \\label{eq:mstep-theta}\n\\end{align}\nand then use a gradient-based method to optimize $\\mathcal{L}_{Q}(\\mu, \\Theta)$ with regard to $\\Theta_{l}$.\nAlgorithm~\\ref{alg:mstep} gives details for optimizing $\\mathcal{L}_{Q}(\\mu, \\Theta)$ over $\\mu$ and $\\Theta$.\n\n\n\\xhdr{Speeding up \\textsc{MagFit}\\xspace}\nSo far we described how to apply the variational EM algorithm to MAG model parameter estimation. However, both E-step and M-step are infeasible when the number of nodes $N$ is large. In particular, in the E-step, for each update of $\\phi_{il}$, we have to\ncompute the expected log-likelihood value of every entry in the $i$-th row and\ncolumn of the adjacency matrix $A$.\nIt takes $O(LN)$ time to do this, so overall $O(L^2 N^2)$ time is needed to update all $\\phi_{il}$. Similarly, in\nthe M-step, we need to sum up the gradient of $\\Theta_{l}$ over every pair of\nnodes (as in \\EQ{eq:mstep-theta}). Therefore, the M-step requires $O(L\nN^2)$ time and so it takes $O(L^2 N^2)$ to run a single iteration of EM. Quadratic dependency in the number of attributes $L$ and the number of nodes $N$ is infeasible for the size of the networks that we aim to work with here.\n\nTo tackle this, we make the following observation. Note that both\n\\EQ{eq:mstep-theta} and computation of $\\PLQ{\\phi_{il}}$ involve the sum of\n\\rev{expected values of the log-likelihood or the gradient.}\nIf we can quickly approximate this sum of the expectations,\nwe can dramatically reduce the computation time. As real-world networks are\nsparse in a sense that most of the edges do not exist in the network, we can\nbreak the summation into two parts --- a fixed part that ``pretends'' that the\nnetwork has no edges and the adjustment part \\rev{that} takes into account the edges\nthat actually exist in the network.\n\n\nFor example, in the M-step we can separate \\EQ{eq:mstep-theta} into two parts,\nthe first term\\hide{left part} that considers an empty graph and the second\nterm\\hide{the adjustment part} that accounts for the edges that actually\noccurred in the network:\n\\begin{small}\n\\begin{align}\n & \\nabla_{\\Theta_{l}} \\mathcal{L}_{\\Theta} =\n \\sum_{i, j} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(0 |F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n & \\quad + \\sum_{A_{ij} = 1} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(1 |F_{i}, F_{j}, \\Theta) - \\log P(0 |F_{i}, F_{j}, \\Theta)} \\,.\n \\label{eq:fastmstep1}\n\\end{align}\n\\end{small}\nNow we approximate the first term that computes the gradient pretending that\nthe graph $A$ has no edges:\n\\begin{align}\n & \\sum_{i, j} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n & = \\nabla_{\\Theta_{l}} \\mathbb{E}_{Q_{i, j}}[\\sum_{i, j} \\log P(0 | F_{i}, F_{j}, \\Theta)] \\nonumber \\\\\n & \\approx \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{N(N-1) \\mathbb{E}_{F} [\\log P(0 | F, \\Theta)]} \\nonumber \\\\\n & = \\nabla_{\\Theta_{l}} N(N-1) \\mathbb{E}_{F} [\\log P(0 | F, \\Theta)] \\,.\n \\label{eq:fastmstep2}\n\\end{align}\nSince each $F_{il}$ follows the Bernoulli distribution with parameter\n$\\mu_{l}$, \\EQ{eq:fastmstep2} can be\n\\rev{computed in $O(L)$ time.}\nAs the second term in \\EQ{eq:fastmstep1} requires only $O(LE)$ time, the computation\ntime of the M-step is reduced from $O(LN^2)$ to\n\\rev{$O(LE)$.}\nSimilarly we reduce the computation time of the E-step from $O(L^2 N^2)$ to\n$O(L^2 E)$\n(see Appendix for details).\nThus overall we reduce the computation time of \\textsc{MagFit}\\xspace from $O(L^2 N^2)$\n\\rev{to $O(L^2 E)$.} \n\n\\subsection{Synthetic Network}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\xhdr{Convergence of \\textsc{MagFit}\\xspace}\nFirst, we briefly evaluate the convergence of the \\textsc{MagFit}\\xspace algorithm. For this\nexperiment, we use synthetic MAG networks with $N = 1024$ and $L = 4$.\nFigure~\\ref{fig:convll} illustrates that the objective function $\\mathcal{L}_{Q}$, \\emph{i.e.}\\xspace, the\nlower bound of the log-likelihood, nicely converges with the number of EM\niterations. While the log-likelihood converges, the model parameters $\\mu$ and\n$\\Theta$ also nicely converge. Figure~\\ref{fig:convmu} shows convergence of\n$\\mu_1, \\dots, \\mu_4$, while Fig. \\ref{fig:convtheta} shows the convergence of\nentries $\\Theta_l[0,0]$ for $l=1,\\dots,4$. Generally, in 100 iterations of EM,\nwe obtain stable parameter estimates.\n\nWe also compare the runtime of the fast \\textsc{MagFit}\\xspace to the naive version where we\ndo not use speedups for the \n\\rev{algorithm.}\nFigure~\\ref{fig:scale} shows the\nruntime as a function of the number of nodes in the network.\nThe runtime of the naive algorithm scales quadratically $O(N^2)$, while\nthe fast version runs in near-linear time.\n\\rev{For example, on 4,000 node network, the fast algorithm runs about 100 times faster than the naive one.}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Convergence of $\\mathcal{L}_{Q}(\\mu, \\Theta)$\\label{fig:convll}]{\\includegraphics[width=0.235\\textwidth]{FIG\/PowL4-Conv-LL.eps}}\n\\subfigure[Convergence of $\\mu_{l}$'s \\label{fig:convmu}]{\\includegraphics[width=0.235\\textwidth]{FIG\/PowL4-Conv-Mu.eps}}\n\\subfigure[Convergence of $\\Theta_{l}{[0, 0]}$'s\\label{fig:convtheta}]{\\includegraphics[width=0.235\\textwidth]{FIG\/PowL4-Conv-Mtx.eps}}\n\\subfigure[\\rev{Run time}\\label{fig:scale}]{\\includegraphics[width=0.235\\textwidth]{FIG\/Scale2.eps}}\n\\vspace{-3mm}\n\\caption{{Parameter convergence and scalability.}}\n\\vspace{-5mm}\n\\end{figure}\n\nBased on these experiments, we conclude that the variational EM gives robust\nparameter estimates. We note that the \\textsc{MagFit}\\xspace optimization problem is\nnon-convex, however, in practice we observe fast convergence and good fits.\nDepending on the initialization \\textsc{MagFit}\\xspace may converge to different solutions but\nin practice solutions tend to have comparable log-likelihoods and consistently\ngood fits. Also, the method nicely scales to networks with up to hundred\nthousand nodes.\n\n\\xhdr{Experiments on real data}\nWe proceed with experiments on real datasets. We use the LinkedIn social\nnetwork~\\cite{jure08microevol} at the time in its evolution when it had $N =$\n4,096 nodes and $E =$ 10,052 edges.\nWe also use the Yahoo!-Answers question answering social network, again from\nthe time when the network had $N =$ 4,096, $E =$ 5,678~\\cite{jure08microevol}.\nFor our experiments we choose $L = 11$, which is roughly $\\log N$ as it has\nbeen shown that this is the optimal choice for $L$~\\cite{mh10mag}.\n\nNow we proceed as follows. Given a real network $A$, we apply \\textsc{MagFit}\\xspace to\nestimate MAG model parameters $\\hat{\\Theta}$ and $\\hat{\\mu}$. Then, given these\nparameters, we generate a synthetic network $\\hat{A}$ and compare how well\nsynthetic $\\hat{A}$ mimics the real network $A$.\n\n\\xhdr{Evaluation} To measure the level of agreement between synthetic $\\hat{A}$\nand the real $A$, we use several different metrics. First, we evaluate how well\n$\\hat{A}$ captures the structural properties, like degree distribution and\nclustering coefficient, of the real network $A$. We consider the following\nnetwork properties:\n\n\\vspace{-3mm}\n\\begin{itemize}\n \\itemsep-3pt \\topsep-5pt \\partopsep-5pt\n \\item {\\em In\/Out-degree distribution (InD\/OutD)} is a histogram of the\n number of in-coming and out-going links of a node.\n \n \\item {\\em Singular values (SVal)} indicate the singular values of the\n adjacency matrix versus their rank.\n \n \\item {\\em Singular vector (SVec)} represents the distribution of\n components in the left singular vector associated with the largest\n singular value.\n \n \\item {\\em Clustering coefficient (CCF)} represents the degree versus the\n average (local) clustering coefficient of nodes of a given\n degree~\\cite{watts98smallworld}.\n \\item {\\em Triad participation (TP)} indicates the number of triangles\n that a node is adjacent to. It measures the transitivity in networks.\n\\end{itemize}\n\\vspace{-3mm}\n\nSince distributions of the above quantities are generally heavy-tailed, we plot\nthem in terms of complementary cumulative distribution functions ($P(X>x)$ as a\nfunction of $x$). Also, to indicate the scale, we do not normalize the\ndistributions to sum to 1.\n\n\n\\newcommand{\\textit{KS}}{\\textit{KS}}\n\\newcommand{\\textit{L2}}{\\textit{L2}}\n\nSecond, to quantify the discrepancy of network properties between real and\nsynthetic networks, we use a variant of Kolmogorov-Sminorv (KS) statistic and\nthe $L2$ distance between different distributions.\nThe original KS statistics is not appropriate here since if the distribution\nfollows a power-law then the original KS statistics is usually dominated by the\nhead of the distribution. We thus consider the following variant of the KS\nstatistic: $\\textit{KS}(D_{1}, D_{2}) = \\max_{x} | \\log D_{1}(x) - \\log\nD_{2}(x)|$~\\cite{mh11kronem}, where $D_{1}$ and $D_{2}$ are two complementary\ncumulative distribution functions.\nSimilarly, we also define a variant of the $L2$ distance on the log-log scale,\n$\\textit{L2}(D_{1}, D_{2}) = \\sqrt{\\frac{1}{\\log b - \\log a} \\left( \\int_{a}^{b}\n\\left(\\log D_{1}(x) - \\log D_{2}(x)\\right)^{2} \\, d (\\log x)\\right)} $ where\n$[a, b]$ is the support of distributions ${D_{1}}$ and ${D_{2}}$. Therefore, we\nevaluate the performance with regard to the recovery of the network properties\nin terms of the $\\textit{KS}$ and $\\textit{L2}$ statistics.\n\n\\newcommand{\\textit{TPI}\\xspace}{\\textit{TPI}\\xspace}\n\\newcommand{\\textit{LL}\\xspace}{\\textit{LL}\\xspace}\n\nLast, since MAG generates a probabilistic adjacency matrix $P$, we also\nevaluate how well $P$ represents a given network $A$. We use the following two\nmetrics:\n\n\\vspace{-3mm}\n\\begin{itemize}\n \\itemsep-3pt \\topsep-5pt \\partopsep-5pt\n \\item {\\em Log-likelihood (\\textit{LL})} measures the possibility that\n the probabilistic adjacency matrix $P$ generates network $A$: $LL=\n \\sum_{i j} \\log (P_{ij}^{A_{ij}} (1-P_{ij})^{1-A_{ij}})$.\n \\item {\\em True Positive Rate Improvement (\\textit{TPI}\\xspace)} represents the\n improvement of the true positive rate over a random graph: $TPI =\n \\sum_{A_{ij}=1}P_{ij} \/ \\frac{E^2}{N^2}$. \\textit{TPI}\\xspace\\xspace indicates how\n much more probability mass is put on the edges compared to a random\n graph (where each edge occurs with probability $E\/N^2$).\n\\end{itemize}\n\\vspace{-3mm}\n\n\n\\begin{figure}[t]\n \\centering\n \\subfigure[In-degree]{\\includegraphics[width=0.235\\textwidth]{FIG\/LinkedIn4K-7-InDeg.eps}}\n \n \\subfigure[Out-degree]{\\includegraphics[width=0.235\\textwidth]{FIG\/LinkedIn4K-7-OutDeg.eps}}\n \n \\subfigure[Singular value]{\\includegraphics[width=0.235\\textwidth]{FIG\/LinkedIn4K-7-Sval.eps}}\n \n \\subfigure[Singular vector]{\\includegraphics[width=0.235\\textwidth]{FIG\/LinkedIn4K-7-Svec.eps}}\n \n \\subfigure[Clustering coefficient]{\\includegraphics[width=0.235\\textwidth]{FIG\/LinkedIn4K-7-Ccf.eps}}\n \n \\subfigure[Triad participation]{\\includegraphics[width=0.235\\textwidth]{FIG\/LinkedIn4K-7-Triad.eps}}\n \n \\caption{The recovered network properties by the MAG model\\xspace~and the Kronecker graphs\n model on the LinkedIn network.\n For every network property, MAG model\\xspace~outperforms the Kronecker graphs model.}\n \\label{fig:realplot}\n \\vspace{-3mm}\n\\end{figure}\n\n\\begin{table}[t]\n\\caption{\\textit{KS}~and \\textit{L2}~of MAG and the Kronecker graphs model on the LinkedIn\nnetwork. MAG exhibits 50-70\\% better performance than Kronecker graphs model.}\n\\label{tbl:linkedinks} \\centering \\small\n\\begin{tabular}{c||c|c|c|c|c|c||c}\n\\multicolumn{8}{l}{\\small{}}\\\\\n {\\bf \\textit{KS}} & InD & OutD & SVal & SVec & TP & CCF & Avg \\\\ \\hline \\hline\n MAG & \\small 3.70 & 3.80 & 0.84 & 2.43 & 3.87 & 3.16 & 2.97 \\\\ \\hline\n Kron & 4.00 & 4.32 & 1.15 & 7.22 & 8.08 & 6.90 & 5.28 \\\\\n \n \\multicolumn{8}{l}{\\bf \\textit{L2}} \\\\ \\hline \\hline\n MAG & 1.01 & 1.15 & 0.46 & 0.62 & 1.68 & 1.11 & 1.00 \\\\ \\hline\n Kron & 1.54 & 1.57 & 0.65 & 6.14 & 6.00 & 4.33 & 3.37\n\\end{tabular}\n\\vspace{-3mm}\n\\end{table}\n\n\\xhdr{Recovery of the network structure}\nWe begin our investigations of real networks by comparing the performance of\nthe MAG model to that of the Kronecker graphs model~\\rev{\\cite{jure10kronecker}, which}\noffers a state of the art baseline for\nmodeling the structure of large networks.\nWe use evaluation methods described in the previous section where \\hide{a given\nreal-world network $A$, }we fit both models to a given real-world network $A$\nand generate synthetic $\\hat{A}_{MAG}$ and $\\hat{A}_{Kron}$. Then we compute\nthe structural properties of all three networks and plot them in\nFigure~\\ref{fig:realplot}. Moreover, for each of the properties we also compute\n\\textit{KS}~and \\textit{L2}~statistics and show them in Table~\\ref{tbl:linkedinks}.\n\nFigure~\\ref{fig:realplot} plots the six network properties described above for\nthe LinkedIn network and the synthetic networks generated by fitting MAG and\nKronecker models to the LinkedIn network.\nWe observe that MAG can successfully produce synthetic networks that match the\nproperties of the real network.\nIn particular, both MAG and Kronecker graphs models capture the degree\ndistribution of the LinkedIn network well. However, MAG model\\xspace~performs much better\nin matching spectral properties of graph adjacency matrix as well as the local\nclustering of the edges in the network.\n\nTable~\\ref{tbl:linkedinks} shows the \\textit{KS}~and \\textit{L2}~statistics for each of the\nsix structural properties plotted in Figure~\\ref{fig:realplot}. Results confirm\nour previous visual inspection. The MAG model\\xspace~is able to fit the network structure\nmuch better than the Kronecker graphs model. In terms of the average\n\\textit{KS}~statistics, we observe 43\\% improvement, while observe even greater\nimprovement of 70\\% in the \\textit{L2}~metric. For degree distributions and the\nsingular values, MAG outperforms Kronecker for about 25\\% while the improvement\non singular vector, triad participation and clustering coefficient is 60 $\\sim$\n75\\%.\n\nWe make similar observations on the Yahoo!-Answers network but omit the results\nfor brevity. We include them\nin Appendix.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}{ccc}\n \\includegraphics[width=0.15\\textwidth]{FIG\/Homophily.eps} &\n \\includegraphics[width=0.13\\textwidth]{FIG\/Heterophily.eps} &\n \\includegraphics[width=0.13\\textwidth]{FIG\/CorePeri.eps} \\\\\n \\includegraphics[width=0.1\\textwidth]{FIG\/HomophilyT.eps} &\n \\includegraphics[width=0.1\\textwidth]{FIG\/HeterophilyT.eps} &\n \\includegraphics[width=0.1\\textwidth]{FIG\/CorePeriT.eps} \\\\\n \\small (a) Homophily & \\small (b) Heterophily & \\small (c) Core-Periphery \\\\\n\\end{tabular}\n \\caption{Structures in which a node attribute can affect link affinity.\n The widths of arrows\n correspond to the affinities towards link formation.}\n \\label{fig:structure}\n\\vspace{-3mm}\n\\end{figure}\n\nWe interpret the improvement of the MAG over Kronecker graphs model in the\nfollowing way. Intuitively, we can think of Kronecker graphs model as a version\nof the MAG model where all affinity matrices $\\Theta_l$ are the same and all\n$\\mu_l=0.5$.\nHowever, real-world networks may include various types of structures and thus\ndifferent attributes may interact in different ways. For example,\nFigure~\\ref{fig:structure} shows three possible linking affinities of a binary\nattribute. Figure~\\ref{fig:structure}(a) shows a homophily (love of the same)\nattribute affinity and the corresponding affinity matrix $\\Theta$. Notice large\nvalues on the diagonal entries of $\\Theta$, which means that link probability\nis high when nodes share the same attribute value.\n\\rev{The top of each figure}\ndemonstrates that there will be many links between nodes that have the value of\nthe attribute set to ``0'' and many links between nodes that have the value\n``1'', but there will be few links between nodes where one has value ``0'' and\nthe other ``1''.\nSimilarly, Figure~\\ref{fig:structure}(b) shows a heterophily (love of the\ndifferent) affinity, where nodes that do not share the value of the attribute\nare more likely to link, which gives rise to near-bipartite networks. Last,\nFigure~\\ref{fig:structure}(c) shows a core-periphery affinity, where links are\nmost likely to form between ``0'' nodes (\\emph{i.e.}\\xspace, members of the core) and least\nlikely to form between ``1'' nodes (\\emph{i.e.}\\xspace, members of the periphery). Notice that\nlinks between the core and the periphery are more likely than the links between\nthe nodes of the periphery.\n\nTurning our attention back to MAG and Kronecker models, we note that\nreal-world networks globally exhibit nested core-periphery\nstructure~\\cite{jure10kronecker} (Figure \\ref{fig:structure}(c)). While there\nexists the core (densely connected) and the periphery (sparsely connected) part\nof the network, there is another level of core-periphery structure inside the\ncore itself. On the other hand, if viewing the network more finely, we may also\nobserve the homophily which produces local community structure. MAG can model\nboth global core-periphery structure and local homophily communities, while the\nKronecker graphs model cannot express the different affinity types because it\nuses only one initiator matrix.\n\n\n\n\nFor example, the LinkedIn network consists of 4 core-periphery affinities, 6\nhomophily affinities, and 1 heterophily affinity matrix. Core-periphery\naffinity models active users who are more likely to connect to others.\nHomophily affinities model people who are more likely to connect to others in\nthe same job area. Interestingly, there is a heterophily affinity which results\nin bipartite relationship. We believe that the relationships between job\nseekers and recruiters or between employers and employees leads to this\nstructure.\n\n\n\\begin{table}[t]\n \\caption{{\\textit{LL} and \\textit{TPI}\\xspace values for LinkedIn (\\textit{LI}) and\n Yahoo!-Answers (\\textit{YA}) networks}} \\label{tbl:linkacc}\n\\centering \\small\n \\vspace{-2mm}\n\\begin{tabular}{c||c|c||c|c}\n\\multicolumn{5}{l}{}\\\\\n & \\textit{LL}(\\textit{LI}) & \\textit{TPI}\\xspace(\\textit{LI}) & \\textit{LL}(\\textit{YA}) & \\textit{TPI}\\xspace(\\textit{YA}) \\\\ \\hline \\hline\n MAG & -47663 & 232.8 & -33795 & 192.2 \\\\ \\hline\n Kron & -87520 & 10.0 & -48204 & 5.4\n\\end{tabular}\n \\vspace{-3mm}\n\\end{table}\n\n\\xhdr{TPI and LL}\nWe also compare the \\textit{LL} and \\textit{TPI}\\xspace~values of MAG and Kronecker models on\nboth LinkedIn and Yahoo!-Answers networks. Table~\\ref{tbl:linkacc} shows that\nMAG outperforms Kronecker graphs by surprisingly large margin.\n\\rev{In \\textit{LL} metric, the MAG model shows $50\\sim\\,60$ \\% improvement over the Kronecker model.}\nFurthermore, in \\textit{TPI}\\xspace~metric, the MAG model shows $23\\sim\\,35$ times better\naccuracy than the Kronecker model. From these results, we conclude that the\nMAG model\\xspace~achieves a superior probabilistic representation of a given network.\n\n\n\\xhdr{Case Study: \\textit{AddHealth} network}\nSo far we considered node attributes as {\\em latent} and we inferred the\naffinity matrices $\\Theta$ as well as the attributes themselves. Now, we\nconsider the setting where the node attributes are already given and we only\nneed to infer affinities $\\Theta$. Our goal here is to study how real\nattributes explain the underlying network structure.\n\nWe use the largest high-school friendship network ($N =$ 457, $E =$ 2,259) from\nthe National Longitudinal Study of Adolescent Health (\\textit{AddHealth})\ndataset. The dataset includes more than 70 school-related attributes for each\nstudent. Since some attributes do not take binary values, we binarize them\nby taking value 1 if the value of the attribute is less than the\nmedian value. Now we aim to investigate which attributes affect the friendship\nformation and how.\n\n\nWe set $L=7$ and consider the following methods for selecting a subset of 7\nattributes:\n\n\\vspace{-3mm}\n\\begin{itemize}\n \\itemsep-3pt \\topsep-5pt \\partopsep-5pt\n \\item {\\em R7}: Randomly choose 7 real attributes and fit the model (\\emph{i.e.}\\xspace,\n only fit $\\Theta$ as attributes are given).\n \\item {\\em L7}: Regard all 7 attributes as latent (\\emph{i.e.}\\xspace, not given) and\n estimate $\\mu_l$ and $\\Theta_l$ for $l=1,\\dots,7$.\n \\item {\\em F7}: Forward selection. Select attributes one by one. At each\n step select an additional attribute that maximizes the overall\n log-likelihood (\\emph{i.e.}\\xspace, select a real attribute and estimate its\n $\\Theta_l$).\n \\item {\\em F5+L2}: Select 5 real attributes using forward selection.\n Then, we infer 2 more latent attributes.\n\\end{itemize}\n\\vspace{-3mm}\n\n\nTo make the \\textsc{MagFit}\\xspace work with fixed real attributes (\\emph{i.e.}\\xspace, only infer $\\Theta$)\nwe fix $\\phi_{il}$ to the values of real attributes. In the E-step we\nthen optimize only over the latent set of $\\phi_{il}$ and the M-step remains as\nis.\n\n\n\\xhdr{{\\em AddHealth} network structure}\nWe begin by evaluating the recovery of the network structure.\nFigure~\\ref{fig:addhealth} shows the recovery of six network properties for\neach attribute selection method. We note that each method manages to recover\ndegree distributions as well as spectral properties (singular values and\nsingular vectors) but the performance is different for clustering coefficient\nand triad participation.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.235\\textwidth]{FIG\/School2MedComp-InDeg.eps}\n\\includegraphics[width=0.235\\textwidth]{FIG\/School2MedComp-OutDeg.eps}\n\\includegraphics[width=0.235\\textwidth]{FIG\/School2MedComp-Sval.eps}\n\\includegraphics[width=0.235\\textwidth]{FIG\/School2MedComp-Svec.eps}\n\\includegraphics[width=0.235\\textwidth]{FIG\/School2MedComp-Ccf.eps}\n\\includegraphics[width=0.235\\textwidth]{FIG\/School2MedComp-Triad.eps}\n\\vspace{-8mm}\n\\caption{{Properties of the AddHealth network.}}\n\\vspace{-5mm}\n\\label{fig:addhealth}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering \\small\n \\caption{Performance of different selection methods.}\n \\label{tbl:addhealthks}\n\\begin{tabular}{l|c|c|c|c|c|c||c}\n\\multicolumn{8}{l}{\\small{}}\\\\\n {\\bf \\textit{KS}} & InD & OutD & SVal & SVec & TP & CCF & Avg \\\\ \\hline \\hline\n R7 &1.00 & 0.58 & 0.48 & 2.92 & 4.52 & 4.45 & 2.32 \\\\ \\hline\n F7 &2.32 & 2.80 & 0.30 & 2.68 & 2.60 & 1.58 & 2.05 \\\\ \\hline\n F5+L2&3.45 & 4.00 & 0.26 & 0.95 & 1.30 & 3.45 & 2.24 \\\\ \\hline\n L7 &1.58 & 1.58 & 0.18 & 2.00 & 2.67 & 2.66 & 1.78 \\\\\n \n \\multicolumn{8}{l}{\\bf \\textit{L2}} \\\\ \\hline \\hline\n R7 &0.25 & 0.16 & 0.25 & 0.96 & 3.18 & 1.74 & 1.09 \\\\ \\hline\n F7 &0.71 & 0.67 & 0.18 & 0.98 & 1.26 & 0.78 & 0.76 \\\\ \\hline\n F5+L2&0.80 & 0.87 & 0.13 & 0.34 & 0.76 & 1.30 & 0.70 \\\\ \\hline\n L7 &0.29 & 0.27 & 0.10 & 0.64 & 0.75 & 1.22 & 0.54\n \n\\end{tabular}\n \\vspace{-5mm}\n\\end{table}\n\nTable~\\ref{tbl:addhealthks} shows the discrepancies in the 6 network properties\n(\\textit{KS}~and \\textit{L2}~statistics) for each attribute selection method. As expected,\nselecting 7 real attributes at random (R7) performs the worst. Naturally, L7\nperforms the best (23\\% improvement over R7 in \\textit{KS}~and 50\\% in \\textit{L2}) as it has\nthe most degrees of freedom\\hide{ ($(1+4)\\cdot7=35$ parameters)}. It is\nfollowed by F5+L2 (the combination of 5 real and 2 latent attributes) and F7\n(forward selection).\n\nAs a point of comparison we also experimented with a simple logistic regression\nclassifier where given the attributes of a pair of nodes we aim to predict an\noccurrence of an edge. Basically, given network $A$ on $N$ nodes, we have $N^2$\n(one for each pair of nodes) training examples: $E$ are positive (edges) and $N^2-E$ are negative (non-edges).\nHowever, the model performs poorly as it gives 50\\% worse \\textit{KS}~statistics than\nMAG. The average \\textit{KS}~of logistic regression under R7 is 3.24 (vs. 2.32 of MAG) and the same statistic under F7 is 3.00 (vs. 2.05 of MAG). Similarly, logistic regression gives 40\\% worse \\textit{L2}~under R7\nand 50\\% worse \\textit{L2}~under F7.\nThese results demonstrate that using the same\nattributes MAG heavily outperforms logistic regression.\n\\rev{We understand that this performance difference arises because the connectivity between a pair of nodes depends on some factors other than the linear combination of their attribute values.}\n\n\\begin{table}[t]\n \\caption{{\\textit{LL} and \\textit{TPI}\\xspace~for the {\\em AddHealth} network.}}\n\\label{tbl:addhealthacc}\n\\centering \\small\n\\begin{tabular}{c|c|c|c|c}\n\\multicolumn{5}{l}{\\small{}}\\\\\n & R7 & F7 & F5+L2 & \\hide{Kron & }L7 \\\\ \\hline \\hline\n \\textit{LL} & -13651 & -12161 & -12047 & \\hide{-11629 &} -9154 \\\\ \\hline\n \\textit{TPI}\\xspace & 1.0 & 1.1 & 1.9 & \\hide{3.0 &} 10.0\n\\end{tabular}\n\\vspace{-5mm}\n\\end{table}\n\nLast, we also examine the \\textit{LL} and \\textit{TPI}\\xspace values and compare them to the\nrandom attribute selection R7 as a baseline. Table~\\ref{tbl:addhealthacc} gives\nthe results. Somewhat contrary to our previous observations, we note that F7\nonly slightly outperforms R7, while F5+L2 gives a factor 2 better \\textit{TPI}\\xspace than R7.\nAgain, L7 gives a factor 10 improvement in \\textit{TPI}\\xspace and overall best performance.\n\n\\begin{table}[t]\n \\caption{{Affinity matrices of 5 AddHealth attributes.}}\n \\label{tbl:attrs}\n\\centering \\small \\vspace{-2mm}\n\\begin{tabular}{ l | l }\n\\multicolumn{2}{l}{\\small{}}\\\\\n \\hide{$\\mu$ &} Affinity matrix & Attribute description \\\\ \\hline \\hline\n \\hide{0.624 &} [0.572\\,0.146;\\,0.146\\,0.999] & School year (0 if $\\geq$ 2) \\\\ \\hline\n \\hide{0.513 &} [0.845\\,0.332;\\,0.332\\,0.816] & Highest level math (0 if $\\geq$ 6) \\\\ \\hline\n \\hide{0.502 &} [0.788\\,0.377;\\,0.377\\,0.784] & Cumulative GPA (0 if $\\geq$ 2.65) \\\\ \\hline\n \\hide{0.078 &} [0.999\\,0.246;\\,0.246\\,0.352] & AP\/IB English (0 if taken) \\\\ \\hline\n \\hide{0.672 &} [0.794\\,0.407;\\,0.407\\,0.717] & Foreign language (0 if taken)\n \n \n\\end{tabular}\n\\vspace{-5mm}\n\\end{table}\n\n\\xhdr{Attribute affinities}\nLast, we investigate the structure of attribute affinity matrices to illustrate\nhow MAG model can be used to understand the way real attributes interact in\nshaping the network structure. We use forward selection (F7) to select 7 real\nattributes and estimate their affinity matrices. Table~\\ref{tbl:attrs} reports\nfirst 5 attributes selected by the forward selection.\n\nFirst notice that {\\em AddHealth} network is undirected graph and that the\nestimated affinity matrices are all symmetric. This means that without a priori\nbiasing the fitting towards undirected graphs, the recovered parameters obey\nthis structure.\nSecond, we also observe that every attribute forms a homophily structure in a\nsense that each student is more likely to be friends with other students of the\nsame characteristic. For example, people are more likely to make friends of the\nsame school year. Interestingly, students who are freshmen or sophomore are\nmore likely (0.99) to form links among themselves than juniors and seniors (0.57).\nAlso notice that the level of advanced courses that each student takes as well\nas the GPA affect the formation of friendship ties. Since it is difficult for\nstudents to interact if they do not take the same courses, the chance of the\nfriendships may be low. We note that, for example, students that\ntake advanced placement (AP) English courses are very likely to form links.\nHowever, links between students who did not take AP English are nearly as\nlikely as links between AP and non-AP students. Last, we also observe\nrelatively small effect of the number of foreign language courses taken on the\nfriendship formation.\n\n\n\\section{Variational EM Algorithm}\n\n\nIn Section~\\ref{sec:problem}, we proposed a version of MAG model\\xspace~by introducing a generative Bernoulli model for node attributes and formulated the problem to solve.\nIn the following Section~\\ref{sec:algorithm}, we gave a sketch of \\textsc{MagFit}\\xspace~that used the variational EM algorithm to solve the problem.\nHere we provide how to compute the gradients of the model parameters ($\\phi$, $\\mu$, and $\\Theta$) for the of E-step and M-step that we omitted in Section~\\ref{sec:algorithm}.\nWe also give the details of the fast \\textsc{MagFit}\\xspace~in the following.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Variational E-Step}\n\nIn the E-step, the MAG model\\xspace~parameters $\\mu$ and $\\Theta$ are given and we aim to find the optimal variational parameter $\\phi$\nthat maximizes $\\mathcal{L}_{Q}(\\mu, \\Theta)$ as well as minimizes the mutual information factor $\\mbox{MI}(F)$.\nWe randomly select a batch of entries in $\\phi$ and update the selected entries by their gradient values of the objective function $\\mathcal{L}_{Q}(\\mu, \\Theta)$.\nWe repeat this updating procedure until $\\phi$ converges.\n\nIn order to obtain $\\nabla_{\\phi} \\left( \\mathcal{L}_{Q}(\\mu, \\Theta) - \\lambda \\mbox{MI}(F) \\right)$,\nwe compute $\\PLQ{\\phi_{il}}$ and $\\frac{\\partial \\mbox{MI}}{\\partial \\phi_{il}}$ in turn as follows.\n\n\\xhdr{Computation of $\\PLQ{\\phi_{il}}$}\nTo calculate the partial derivative $\\PLQ{\\phi_{il}}$, we begin by restating $\\mathcal{L}_{Q}(\\mu, \\Theta)$ as a function of one specific parameter $\\phi_{il}$ and differentiate this function over $\\phi_{il}$.\nFor convenience, we denote $F_{-il} = \\{F_{jk}: j \\neq i, k \\neq l\\}$\nand\n$\nQ_{-il} = \\prod_{j \\neq i, k \\neq l} Q_{jk}\n$.\nNote that $\\sum_{F_{il}} Q_{il}(F_{il}) = 1$ and $\\sum_{F_{-il}} Q_{-il}(F_{-il})$ because both are the sums of probabilities of all possible events.\nTherefore, we can separate $\\mathcal{L}_{Q}(\\mu, \\Theta)$ in \\EQ{eq:lqdef} into the terms of $Q_{il}(F_{il})$ and $Q_{-il}(F_{-il})$:\n\\begin{align}\n& \\mathcal{L}_{Q}(\\mu, \\Theta) \\nonumber \\\\\n& = \\EXATTR{}{\\log P(A, F | \\mu, \\Theta) - \\log Q(F)} \\nonumber \\\\\n& = \\sum_{F} Q\\left(F\\right) \\left( \\log P\\left(A, F | \\mu, \\Theta\\right) - \\log Q\\left(F\\right) \\right) \\nonumber \\\\\n& = \\sum_{F_{-il}} \\sum_{F_{il}} Q_{-il}(F_{-il}) Q_{il}(F_{il}) \\nonumber \\\\\n& \\quad \\times \\left( \\log P\\left(A, F | \\mu, \\Theta\\right) - \\log Q_{il}\\left(F_{il}\\right) - \\log Q_{-il}\\left(F_{-il}\\right) \\right) \\nonumber \\\\\n& = \\sum_{F_{il}} Q_{il}(F_{il}) \\left( \\sum_{F_{-il}} Q_{-il}\\left(F_{-il}\\right) \\log P\\left(A, F | \\mu, \\Theta\\right) \\right) \\nonumber \\\\\n& \\quad \\quad\n- \\sum_{F_{il}} Q_{il}(F_{il}) \\log Q_{il}(F_{il}) \\nonumber \\\\\n& \\quad \\quad\n- \\sum_{F_{-il}} Q_{-il}(F_{-il}) \\log Q_{-il}(F_{-il}) \\nonumber \\\\\n& = \\sum_{F_{il}} Q_{il}(F_{il}) \\EXATTR{-il}{\\log P\\left(A, F | \\mu, \\Theta\\right)} \\nonumber \\\\\n& \\quad \\quad + \\mathcal{H} (Q_{il}) + \\mathcal{H} (Q_{-il})\n\\label{eq:lqsep}\n\\end{align}\nwhere $\\mathcal{H}(P)$ represents the entropy of distribution $P$.\n\nSince we compute the gradient of $\\phi_{il}$, we regard the other variational parameter $\\phi_{-il}$ as a constant so $\\mathcal{H} (Q_{-il})$ is also a constant.\nMoreover, as $\\EXATTR{-il}{\\log P\\left(A, F | \\mu, \\Theta\\right)}$ integrates out all the terms\nwith regard to $\\phi_{-il}$, \nit is a function of $F_{il}$.\nThus, for convenience, we denote $\\EXATTR{-il}{\\log P\\left(A, F | \\mu, \\Theta\\right)}$ as $\\log \\PFUNC{F_{il}}$.\nThen, since $F_{il}$ follows a Bernoulli distribution with parameter $\\phi_{il}$, by \\EQ{eq:lqsep}\n\\begin{align}\n\\mathcal{L}_{Q}(\\mu, \\theta) & = (1 - \\phi_{il}) \\left( \\log \\PFUNC{1} - \\log (1-\\phi_{il}) \\right) \\nonumber \\\\\n& \\quad + \\phi_{il} \\left( \\log \\PFUNC{0} - \\log \\phi_{il} \\right) + const \\, .\n\\label{eq:lqonephi}\n\\end{align}\n\nNote that both $\\PFUNC{0}$ and $\\PFUNC{1}$ are constant. Therefore,\n\\begin{equation}\n\\PLQ{\\phi_{il}} = \\log \\frac{\\PFUNC{0}}{\\phi_{il}} - \\log \\frac{\\PFUNC{1}}{1-\\phi_{il}} \\, .\n\\label{eq:eqonephi}\n\\end{equation}\n\\hide{\nFrom \\EQ{eq:eqonephi}, the optimal $\\phi_{il}$ is as follows:\n\\begin{equation}\n\\phi_{il} = \\frac{\\PFUNC{0}}{\\PFUNC{0} + \\PFUNC{1}} \\, .\n\\label{eq:updatephi}\n\\end{equation}\n}\n\n\nTo complete the computation of $\\PLQ{\\phi_{il}}$,\nnow we focus on the value of $\\PFUNC{F_{il}}$ for $F_{il} = 0, 1$.\nBy \\EQ{eq:jointlik}~and the linearity of expectation,\n$\\log \\PFUNC{F_{il}}$ is separable into small tractable terms as follows:\n\\begin{align}\n\\log \\PFUNC{F_{il}}\n& = \\EXATTR{-il}{\\log P(A, F | \\mu, \\Theta)} \\nonumber \\\\\n& = \\sum_{u, v} \\EXATTR{-il}{\\log P(A_{uv} | F_{u}, F_{v}, \\Theta)} \\nonumber \\\\\n& \\quad + \\sum_{u, k} \\EXATTR{-il}{\\log P(F_{uk} | \\mu_{k})}\n\\label{eq:ptildesep}\n\\end{align}\nwhere $F_{i} = \\{F_{il}: l = 1, 2, \\cdots, L \\}$.\nHowever, if $u, v \\neq i$, then $\\EXATTR{-il}{\\log P(A_{uv} | F_{u}, F_{v}, \\Theta)}$ is a constant,\nbecause the average over $Q_{-il}(F_{-il})$ integrates out all the variables $F_{u}$ and $F_{v}$.\nSimilarly, if $u \\neq i$ and $k \\neq l$, then $\\EXATTR{-il}{\\log P(F_{uk} | \\mu_{k})}$ is a constant.\nSince most of terms in \\EQ{eq:ptildesep}~are irrelevant to $\\phi_{il}$, $\\log \\PFUNC{F_{il}}$~is simplified as\n\\begin{align}\n\\log \\PFUNC{F_{il}}\n& = \\left(\\sum_{j} \\EXATTR{-il}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)}\\right) \\nonumber \\\\\n& \\quad + \\left(\\sum_{j} \\EXATTR{-il}{\\log P(A_{ji} | F_{j}, F_{i}, \\Theta)}\\right) \\nonumber \\\\\n& \\quad + \\log P(F_{il} | \\mu_{l}) + C \n\\label{eq:ptildesim}\n\\end{align}\nfor some constant $C$.\n\nBy definition of $P(F_{il} | \\mu_{l})$ in \\EQ{eq:jointlik}, the last term in \\EQ{eq:ptildesim} is \n\\begin{equation}\n\\log P(F_{il} | \\mu_{l}) = F_{il} \\log \\mu_{l} + (1-F_{il}) \\log (1 - \\mu_{il})\n\\, .\n\\label{eq:logmu}\n\\end{equation}\n\nWith regard to the first two terms in \\EQ{eq:ptildesim},\n\\[\n\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)\n= \\log P(A_{ji} | F_{i}, F_{j}, \\Theta^{T}) \\, .\n\\]\nHence, the methods to compute the two terms are equivalent.\nThus, we now focus on the computation of $\\EXATTR{-il}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)}$.\n\nFirst, in case of $A_{ij} = 1$, by definition of $P(A_{ij} | F_{i}, F_{j})$ in \\EQ{eq:jointlik},\n\\begin{align}\n& \\EXATTR{-il}{\\log P(A_{ij} = 1 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& = \\EXATTR{-il}{\\sum_{k} \\log \\Theta_{k} [F_{ik}, F_{jk}]} \\nonumber \\\\\n& = \\EXATTR{jl}{\\log \\Theta_{l}[F_{il}, F_{jl}]} + \\sum_{k \\neq l} \\EXATTR{ik, jk}{\\log \\Theta_{k} [F_{ik}, F_{jk}]} \\nonumber \\\\\n& = \\EXATTR{jl}{\\log \\Theta_{l}[F_{il}, F_{jl}]} + C'\n\\label{eq:logedgeprob1}\n\\end{align}\nfor some constant $C'$ where $Q_{ik, jk}(F_{ik}, F_{jk}) = Q_{ik}(F_{ik}) Q_{jk}(F_{jk})$,\nbecause $\\EXATTR{ik, jk}{\\log \\Theta_{k}[F_{ik}, F_{jk}]}$ is constant for each $k$.\n\nSecond, in case of $A_{ij} = 0$,\n\\begin{align}\nP(A_{ij} = 0 | F_{i}, F_{j}, \\Theta)\n& = 1 - \\prod_{k} \\Theta_{k} [F_{ik}, F_{jk}]\n\\, .\n\\label{eq:logedgeprob0}\n\\end{align}\nSince $\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)$ is not separable in terms of $\\Theta_{k}$, it takes $O(2^{2L})$ time to compute $\\EXATTR{-il}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)}$ exactly.\nWe can reduce this computation time to $O(L)$ by applying Taylor's expansion of $\\log (1 - x) \\approx -x - \\frac{1}{2}x^{2}$ for small $x$:\n\\begin{align}\n& \\EXATTR{-il}{\\log P(A_{ij} = 0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& \\approx \\EXATTR{-il}{-\\prod_{k} \\Theta_{k}[F_{ik}, F_{jk}] - \\frac{1}{2} \\prod_{k} \\Theta^{2}_{k}[F_{ik}, F_{jk}]} \\nonumber \\\\\n& = -\\EXATTR{jl}{\\Theta_{l}[F_{il}, F_{jl}]} \\prod_{k \\neq l} \\EXATTR{ik, jk}{\\Theta_{k}[F_{ik}, F_{jk}]} \\nonumber \\\\\n& \\quad \\quad - \\frac{1}{2} \\EXATTR{jl}{\\Theta^{2}_{l}[F_{il}, F_{jl}]} \\prod_{k \\neq l} \\EXATTR{ik, jk}{\\Theta^{2}_{k}[F_{ik}, F_{jk}]}\n\\label{eq:logedgeprob0approx}\n\\end{align}\nwhere each term can be computed by\n\\begin{align*}\n\\EXATTR{il}{Y_{l}[F_{il}, F_{jl}]} & = \\phi_{jl} Y_{l}[F_{il}, 0] + (1-\\phi_{jl}) Y_{l}[F_{il}, 1] \\\\\n\\EXATTR{ik, jk}{Y_{k}[F_{ik}, F_{jk}]} & = [\\phi_{ik}~~\\phi_{jk}] \\cdot Y_{k} \\cdot [1-\\phi_{ik} ~~ 1-\\phi_{jk}]^{T}\n\\end{align*}\nfor any matrix $Y_{l}, Y_{k} \\in \\mathbb{R}^{2 \\times 2}$.\n\nIn brief, for fixed $i$ and $l$, we first compute $\\EXATTR{-il}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)}$ for each node $j$ depending on whether or not $i \\rightarrow j$ is an edge.\nBy adding $\\log P(F_{il} | \\mu_{l})$, we then acheive the value of $\\log \\PFUNC{F_{il}}$ for each $F_{il}$.\nOnce we have $\\log \\PFUNC{F_{il}}$, we can finally compute $\\PLQ{il}$.\n\n\n\n\\xhdr{Scalable computation}\nHowever, as we analyzed in Section~\\ref{sec:algorithm}, the above E-step algorithm requires $O(LN)$ time for each computation of $\\PLQ{\\phi_{il}}$ so that the total computation time is $O(L^{2}N^{2})$, \nwhich is infeasible when the number of nodes $N$ is large.\n\nHere we propose the scalable algorithm of computing $\\PLQ{\\phi_{il}}$\nby further approximation.\nAs described in Section~\\ref{sec:algorithm}, we quickly approximate the value of $\\PLQ{\\phi_{il}}$ as if the network would be empty, \nand adjust it by the part where edges actually exist.\nTo approximate $\\PLQ{\\phi_{il}}$ in empty network case, \nwe reformulate the first term in \\EQ{eq:ptildesim}:\n\\begin{align}\n& \\sum_{j} \\EXATTR{-il}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)} = \\sum_{j} \\EXATTR{-il}{\\log P(0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& \\quad + \\sum_{A_{ij} = 1} \\EXATTR{-il}{\\log P(1 | F_{i}, F_{j}, \\Theta) - \\log P(0 | F_{i}, F_{j}, \\Theta)}\n\\label{eq:fastestep1_x}\n\\end{align}\nHowever, since the sum of \\textit{i.i.d.} random variables can be approximated in terms of the expectaion of the random variable,\nthe first term in \\EQ{eq:fastestep1_x} can be approximated as follows:\n\\begin{align}\n& \\sum_{j} \\EXATTR{-il}{\\log P(0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& = \\EXATTR{-il}{\\sum_{j} \\log P(0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& \\approx \\EXATTR{-il}{(N-1) \\mathbb{E}_{F_{j}}[ \\log P(0 | F_{i}, F_{j}, \\Theta)]} \\nonumber \\\\\n& = (N-1) \\mathbb{E}_{F_{j}} [\\log P(0 | F_{i}, F_{j}, \\Theta)]\n\\label{eq:fastestep2_x}\n\\end{align}\nAs $F_{jl}$ marginally follows a Bernoulli distribution with $\\mu_{l}$, we can compute \\EQ{eq:fastestep2_x} by using \\EQ{eq:logedgeprob0approx} in $O(L)$ time.\nSince the second term of \\EQ{eq:fastestep1_x} takes $O(L N_{i})$ time where $N_{i}$ represents the number of neighbors of node $i$,\n\\EQ{eq:fastestep1_x} takes only $O(L N_{i})$ time in total.\nAs in the E-step we do this operation by iterating for all $i$'s and $l$'s,\nthe total computation time of the E-step eventually becomes $O(L^2 E)$,\nwhich is feasible in many large-scale networks.\n\n\n\\xhdr{Computation of $\\PMI{\\phi_{il}}$}\nNow we turn our attention to the derivative of the mutual information term.\nSince $\\mbox{MI}(F) = \\sum_{l \\neq l'} \\mbox{MI}_{ll'}$,\nwe can separately compute the derivative of each term $\\frac{\\partial \\mbox{MI}_{ll'}}{\\partial \\phi_{il}}$.\nBy definition in \\EQ{eq:midef} and Chain Rule,\n\\begin{align}\n& \\frac{\\partial \\mbox{MI}_{ll'}}{\\partial \\phi_{il}} = \n\\sum_{x, y \\in \\{0, 1\\}} \\frac{\\partial p_{ll'}(x,y)}{\\partial \\phi_{il}} \\log \\frac{p_{ll'}(x, y) }{p_{l}(x)p_{l'}(y)} \\nonumber \\\\\n& \\quad + \\frac{\\partial p_{ll'}(x, y)}{\\partial \\phi_{il}} \n+ \\frac{p_{ll'}(x, y)}{p_{l}(x)} \\frac{\\partial p_{l}(x)}{\\partial \\phi_{il}} \n+ \\frac{p_{ll'}(x, y)}{p_{l'}(y)} \\frac{\\partial p_{l'}(y)}{\\partial \\phi_{il}} \\,.\n\\label{eq:partialmi}\n\\end{align}\nThe values of $p_{ll'}(x, y)$, $p_{l}(x)$, and $p_{l'}(y)$ are defined in \\EQ{eq:midef}.\nTherefore, in order to compute $\\frac{\\partial \\mbox{MI}_{ll'}}{\\partial \\phi_{il}}$,\nwe need the values of $\\frac{\\partial p_{ll'}(x, y)}{\\partial \\phi_{il}}$, $\\frac{\\partial p_{l}(x)}{\\partial \\phi_{il}}$, and $\\frac{\\partial p_{l'}(y)}{\\partial \\phi_{il}}$.\nBy definition in \\EQ{eq:midef},\n\\begin{align*}\n& \\frac{\\partial p_{ll'}(x, y)}{\\partial \\phi_{il}} = Q_{il'}(y) \\frac{\\partial Q_{il}}{\\partial \\phi_{il}} \\\\\n& \\frac{\\partial p_{l}(x)}{\\partial \\phi_{il}} = \\frac{\\partial Q_{il}}{\\partial \\phi_{il}} \\\\\n& \\frac{\\partial p_{l'}(y)}{\\partial \\phi_{il}} = 0\n\\end{align*}\nwhere $\\frac{\\partial Q_{il}}{\\partial \\phi_{il}} |_{F_{il}=0} = 1$\nand $\\frac{\\partial Q_{il}}{\\partial \\phi_{il}} |_{F_{il}=1} = -1$.\n\nSince all terms in $\\frac{\\partial \\mbox{MI}_{ll'}}{\\partial \\phi_{il}}$ are tractable, we can eventually compute $\\PMI{\\phi_{il}}$.\n\n\n\n\n\n\n\\subsection{Variational M-Step}\n\nIn the E-Step, \nwith given model parameters $\\mu$ and $\\Theta$, we updated the variational parameter $\\phi$\nto maximize $\\mathcal{L}_{Q}(\\mu, \\Theta)$ as well as to minimize the mutual information between every pair of attributes.\nIn the M-step,\nwe basically fix the approximate posterior distribution $Q(F)$, \\emph{i.e.}\\xspace~fix the variational parameter $\\phi$,\nand update the model parameters $\\mu$ and $\\Theta$ to maximize $\\mathcal{L}_{Q}(\\mu, \\Theta)$.\n\n\nTo reformulate $\\mathcal{L}_{Q}(\\mu, \\Theta)$\nby \\EQ{eq:jointlik},\n\\begin{align}\n& \\mathcal{L}_{Q}(\\mu, \\Theta) \\nonumber \\\\\n& = \\EXATTR{}{\\log P(A, F | \\mu, \\Theta) - \\log Q(F)} \\nonumber \\\\\n& = \\EXATTR{}{\\sum_{i, j} P(A_{ij} | F_{i}, F_{j}, \\Theta)\n+ \\sum_{i, l} P(F_{il} | \\mu_{l}) } + \\mathcal{H}(Q) \\nonumber \\\\\n& = \\sum_{i, j} \\EXATTR{i, j}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& \\quad + \\sum_{l} \\left( \\sum_{i} \\EXATTR{il}{\\log P(F_{il} | \\mu_{l})} \\right)\n+ \\mathcal{H}(Q)\n\\label{eq:mstep}\n\\end{align}\nwhere $Q_{i, j}(F_{\\{i\\cdot\\}}, F_{\\{j\\cdot\\}})$ represents $\\prod_{l} Q_{il}(F_{il}) Q_{jl}(F_{jl})$.\n\nAfter all, $\\mathcal{L}_{Q}(\\mu, \\Theta)$ in \\EQ{eq:mstep} is divided into the following terms:\na function of $\\Theta$, a function of $\\mu_{l}$, and a constant.\nThus, we can exclusively update $\\mu$ and $\\Theta$. Since we already showed how to update $\\mu$ in Section~\\ref{sec:algorithm}, here we focus on the maximization of $\\mathcal{L}_{\\Theta} = \\EXATTR{}{\\log P(A, F | \\mu, \\Theta) - \\log Q(F)}$ using the gradient method.\n\n\n\n\\xhdr{Computation of $\\nabla_{\\Theta_{l}} \\mathcal{L}_{\\Theta}$}\nTo use the gradient method, we need to compute the gradient of $\\mathcal{L}_{\\Theta}$:\n\\begin{align}\n\\nabla_{\\Theta_{l}} \\mathcal{L}_{\\Theta} = \\sum_{i, j} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(A_{ij} |F_{i}, F_{j}, \\Theta)} \\, .\n\\label{eq:mstep-theta_x}\n\\end{align}\nWe separately calculate the gradient of each term in $\\mathcal{L}_{\\Theta}$ as follows:\nFor every $z_{1}, z_{2} \\in \\{0, 1\\}$,\nif $A_{ij} = 1$,\n\\begin{align}\n& \\frac{\\partial \\EXATTR{i, j}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)}}{\\partial \\Theta_{l}[z_{1}, z_{2}]} \\bigg|_{A_{ij} = 1} \\nonumber \\\\\n& = \\frac{\\partial}{\\partial \\Theta_{l}[z_{1}, z_{2}]}\n\\EXATTR{i, j}{\\sum_{k} \\log \\Theta_{k}[F_{ik}, F_{jk}]} \\nonumber \\\\\n& = \\frac{\\partial}{\\partial \\Theta_{l}[z_{1}, z_{2}]} \\EXATTR{i, j}{\\log \\Theta_{l}[F_{il}, F_{jl}]} \\nonumber \\\\\n& = \\frac{Q_{il}(z_{1}) Q_{jl}(z_{2})}{\\Theta_{l}[z_{1}, z_{2}]} \\, .\n\\label{eq:mstep-theta1}\n\\end{align}\nOn the contrary, if $A_{ij} = 0$,\nwe use Taylor's expansion as used in \\EQ{eq:logedgeprob0approx}:\n\\begin{align}\n& \\frac{\\partial \\EXATTR{i, j}{\\log P(A_{ij} | F_{i}, F_{j}, \\Theta)}}{\\partial \\Theta_{l}[z_{1}, z_{2}]} \\bigg|_{A_{ij} = 0} \\nonumber \\\\\n& \\approx \\frac{\\partial}{\\partial \\Theta_{l}} \\EXATTR{i, j}{- \\prod_{k} \\Theta_{k}[F_{ik}, F_{jk}] - \\frac{1}{2} \\prod_{k} \\Theta^{2}_{k}[F_{ik}, F_{jk}]} \\nonumber \\\\\n& = -Q_{il}(z_{1})Q_{jl}(z_{2}) \\prod_{k \\neq l} \\EXATTR{ik, jk}{\\Theta_{k}[F_{ik}, F_{jk}]} \\nonumber \\\\\n& \\quad \\quad - Q_{il}(z_{1})Q_{jl}(z_{2})\\Theta_{k}[z_{1}, z_{2}] \\prod_{k \\neq l} \\EXATTR{ik, jk}{\\Theta^{2}_{k}[F_{ik}, F_{jk}]}\n\\label{eq:mstep-theta2}\n\\end{align}\nwhere $Q_{il, jl}(F_{il}, F_{jl}) = Q_{il}(F_{il}) Q_{jl}(F_{jl})$.\n\nSince\n\\[\n\\EXATTR{ik, jk}{f(\\Theta)} = \\sum_{z_{1}, z_{2}} Q_{ik}(z_{1})Q_{jk}(z_{2})f\\left(\\Theta[z_{1}, z_{2}]\\right)\n\\]\nfor any function $f$\nand we know each function values of $Q_{il}(F_{il})$ in terms of $\\phi_{il}$,\nwe are able to achieve the gradient $\\nabla_{\\Theta_{l}}\\mathcal{L}_{\\Theta}$ by \\EQ{eq:mstep-theta_x}~$\\sim$~(\\ref{eq:mstep-theta2}).\n\n\\xhdr{Scalable computation}\nThe M-step requires to sum $O(N^{2})$ terms in \\EQ{eq:mstep-theta_x} where each term takes $O(L)$ time to compute.\nSimilarly to the E-step, here we propose the scalable algorithm by separating \\EQ{eq:mstep-theta_x} into two parts, the fixed part for an empty graph and the adjustment part for the actual edges:\n\\begin{align}\n& \\nabla_{\\Theta_{l}} \\mathcal{L}_{\\Theta} =\n\\sum_{i, j} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(0 |F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& \\quad + \\sum_{A_{ij} = 1} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(1 |F_{i}, F_{j}, \\Theta) - \\log P(0 |F_{i}, F_{j}, \\Theta)} \\, .\n\\label{eq:fastmstep1_x}\n\\end{align}\nWe are able to approximate the first term in \\EQ{eq:fastmstep1_x}, the value for the empty graph part, as follows:\n\\begin{align}\n& \\sum_{i, j} \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\log P(0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& = \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{\\sum_{i, j} \\log P(0 | F_{i}, F_{j}, \\Theta)} \\nonumber \\\\\n& \\approx \\nabla_{\\Theta_{l}} \\EXATTR{i, j}{N(N-1) \\mathbb{E}_{F} [\\log P(0 | F, \\Theta)]} \\nonumber \\\\\n& = \\nabla_{\\Theta_{l}} N(N-1) \\mathbb{E}_{F} [\\log P(0 | F, \\Theta)] \\,.\n\\label{eq:fastmstep2_x}\n\\end{align}\nSince each $F_{il}$ marginally follows the Bernoulli distribution with $\\mu_{l}$, \\EQ{eq:fastmstep2_x} is computed by \\EQ{eq:mstep-theta2} in $O(L)$ time.\nAs the second term in \\EQ{eq:fastmstep1_x} requires only $O(LE)$ time,\nthe computation time of the M-step is finally reduced to $O(LE)$ time.\n\n\n\n\\section{Experiments}\n\n\\subsection{Yahoo!-Ansers Network}\n\nHere we add some experimental results that we omitted in Section~\\ref{sec:experiments}.\nFirst, Figure~\\ref{fig:answerplot} compares the six network properties of Yahoo!-Answers network and the synthetic networks generated by MAG model\\xspace~and Kronecker graphs model fitted to the real network.\nThe MAG model\\xspace~in general shows better performance than the Kronecker graphs model. Particularly, the MAG model\\xspace~greatly outperforms the Kronecker graphs model in local-clustering properties (clustering coefficient and triad participation).\n\nSecond, to quantify the recovery of the network properties, we show the \\textit{KS}~and \\textit{L2}~statistics for the synthetic networks generated by MAG model\\xspace~and Kronecker graphs model in Table~\\ref{tbl:answerks}.\nThrough Table~\\ref{tbl:answerks}, we can confirm the visual inspection in Figure~\\ref{fig:answerplot}.\nThe MAG model\\xspace~shows better statistics than the Kronecker graphs model in overall\nand there is huge improvement in the local-clustering properties.\n\n\n\\begin{figure}[t]\n \\centering\n \\subfigure[In-degree]{\\includegraphics[width=0.235\\textwidth]{FIG\/Answer4K-7-InDeg.eps}}\n \\subfigure[Out-degree]{\\includegraphics[width=0.235\\textwidth]{FIG\/Answer4K-7-OutDeg.eps}}\n \\subfigure[Singular value]{\\includegraphics[width=0.235\\textwidth]{FIG\/Answer4K-7-Sval.eps}}\n \\subfigure[Singular vector]{\\includegraphics[width=0.235\\textwidth]{FIG\/Answer4K-7-Svec.eps}}\n \\subfigure[Clustering coefficient]{\\includegraphics[width=0.235\\textwidth]{FIG\/Answer4K-7-Ccf.eps}}\n \\subfigure[Triad participation]{\\includegraphics[width=0.235\\textwidth]{FIG\/Answer4K-7-Triad.eps}}\n \\caption{The recovered network properties by the MAG model\\xspace~and the Kronecker graphs\n model on the Yahoo!-Answers network.\n For every network property, MAG model\\xspace~outperforms the Kronecker graphs model.}\n \\label{fig:answerplot}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\textit{KS}~and \\textit{L2}~for MAG and Kronecker model fitted to Yahoo!-Answers network}\n\\label{tbl:answerks}\n\\centering \\small\n\\begin{tabular}{c||c|c|c|c|c|c||c}\n\\multicolumn{8}{l}{\\small{}}\\\\\n {\\bf \\textit{KS}} & InD & OutD & SVal & SVec & TP & CCF & Avg \\\\ \\hline \\hline\nMAG & 3.00 & 2.80 & 14.93 & 13.72 & 4.84 & 4.80 & 7.35 \\\\ \\hline\nKron & 2.00 & 5.78 & 13.56 & 15.47 & 7.98 & 7.05 & 8.64 \\\\ \\hline\n \\multicolumn{8}{l}{\\bf \\textit{L2}} \\\\ \\hline \\hline\nMAG & 0.96 & 0.74 & 0.70 & 6.81 & 2.76 & 2.39 & 2.39 \\\\ \\hline\nKron & 0.81 & 2.24 & 0.69 & 7.41 & 6.14 & 4.73 & 3.67 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{AddHealth Network}\n\nWe briefly mentioned the logistic regression method in AddHealth network experiment.\nHere we provide the details of the logistic regression and full experimental results of it.\n\nFor the variables of the logistic regression, we use a set of real attributes in the AddHealth network dataset.\nFor such set of attributes, we used F7 (forward selection) and R7 (random selection) defined in Section~\\ref{sec:experiments}.\nOnce the set of attributes is fixed, we come up with a linear model:\n\\begin{align*}\nP(i \\rightarrow j) = \n\\frac{\\exp(c + \\sum_{l} \\alpha_{l} F_{il} + \\sum_{l} \\beta_{l} F_{jl}) }\n{1 + \\exp(c + \\sum_{l} \\alpha_{l} F_{il} + \\sum_{l} \\beta_{l} F_{jl}) } \\,.\n\\end{align*}\n\nTable~\\ref{tbl:addhealthlogit} shows the \\textit{KS}~and \\textit{L2}~statistics for logistic regression methods under R7 and F7 attribute sets.\nIt seems that the logistic regression succeeds in the recovery of degree distributions.\nHowever, it fails to recover the local-clustering properties (clustering coefficient and triad participation) for both sets.\n\n\\begin{table}\n\\caption{\\textit{KS}~and \\textit{L2}~for logistic regression methods fitted to AddHealth network}\n\\label{tbl:addhealthlogit}\n\\centering \\small\n\\begin{tabular}{c||c|c|c|c|c|c||c}\n\\multicolumn{8}{l}{\\small{}}\\\\\n {\\bf \\textit{KS}} & InD & OutD & SVal & SVec & TP & CCF & Avg \\\\ \\hline \\hline\nR7 & 2.00 & 2.58 & 0.58 & 3.03 & 5.39 & 5.91 & 3.24 \\\\ \\hline\nF7 & 1.59 & 1.59 & 0.52 & 3.03 & 5.43 & 5.91 & 3.00 \\\\ \\hline\n \\multicolumn{8}{l}{\\bf \\textit{L2}} \\\\ \\hline \\hline\nR7 & 0.54 & 0.58 & 0.29 & 1.09 & 3.43 & 2.42 & 1.39 \\\\ \\hline\nF7 & 0.42 & 0.24 & 0.27 & 1.12 & 3.55 & 2.09 & 1.28 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\hide{\n\\newpage\n\\section{General Questions}\n\nMAG model\\xspace~provides a class of social and information network model defined by\nmultiplicative node attributes and their similarity matrices. We come up with\ninteresting questions on top of this MAG model\\xspace~as follows:\n\n\\begin{itemize}\n\\item{\\em{Parameter estimation}:} Given a network, how can we estimate the\n node attributes and their similarity matrices in MAG model\\xspace?\n\\item{\\em{MAG-PCA}:} Given a network, how can we estimate the node\n attributes such that the attributes for each node can be characterized\n by a few components? Also, how can we estimate their similarity\n matrices under this setting?\n\\item{\\em{Node attribute inference}:} Given a network and a part of node\n attributes, how can we infer the missing part of the attributes as well\n as the similarity matrices in MAG model\\xspace?\n\\end{itemize}\n\n\n\n\\xhdr{Parameter estimation}\n\nTo simplify the model, here we reduce the categorical attributes to only binary\ncases. Basically, we want to estimate these binary attributes for each node and\nthe underlying similarity matrices, given a network. However, this is an\nNP-hard combinatorial problem when we find the binary attributes for a large\nnumber of nodes such that the estimated attributes maximize the likelihood\ncombined with the similarity matrices. To solve this difficult problem, we can\nformulate the problem such that each attribute allows the value between 0 and 1\nrather than 0 or 1 itself. This relaxation is very natural in a sense that each\nvalue between 0 and 1 represents the probability and its actual binary\nattribute can be regarded as a Bernoulli random variable.\n\nHere we are interested in two types of problems.\n\\begin{itemize}\n\\item If we believe that each attribute is defined in the whole population,\n independently of the individual nodes, then the parameter $\\mu_{l}$ of\n the Bernoulli random variable for each attribute $a_{l}$ is shared by\n all nodes in the network.\n\\item On the other hand, if we allow the differences between the nodes,\n then each attribute $a_{l}$ for each node $i$ has a different Bernoulli\n parameter, $\\mu_{il}$.\n\\end{itemize}\n\nIn both problem settings, we eventually find how each attribute interacts\nbetween nodes (\\emph{i.e.}\\xspace~$\\Theta_{l}$) and the generative model for the attributes of\na node (\\emph{i.e.}\\xspace~$\\mu_{l}$ or $\\mu_{il}$).\n\nThis is important in the following senses:\n\\begin{itemize}\n\\item Fits the generative model to a given network : Possible to generate a\n synthetic network with the same characteristics\n\\item Compares two networks under the MAG model\\xspace : Possible to quantify the\n informative discrepency between the two networks\n\\item Characterizes each node of the network via the posterior attribute\n distribution\n\\end{itemize}\n\n\n\n\n\n\\xhdr{MAG-PCA}\n\nWe are interested in the following types of problem.\n\\begin{itemize}\n\\item We assume that there are only a few possible binary attribute vectors\n (combinations) in the MAG model\\xspace and the attribute vector for each node\n belongs to those candidates with some assignment probabilities. We want\n to estimate the candidate vectors as well as the assignment probability\n for each node.\n\\item We relax the previous problem so that the the attribute vector allows\n the values between 0 and 1, \\emph{i.e.}\\xspace~each component in the attribute vector\n represents the probability. This problem may be more robust and\n tractable than the previous problem using binary attributes.\n\\end{itemize}\n\nThis is important in the following senses:\n\\begin{itemize}\n\\item Relates to PCA\n\\item Nicely explains the continuous-valued attributes which are not\n necessarily multiplicative each other in terms of similarities\n\\end{itemize}\n\n\n\n\n\\xhdr{Node attribute inference}\n\nWe are interested in the following problems.\n\\begin{itemize}\n\\item Given a network and the attributes for a part of nodes, how can we\n infer the similarity matrices and the attributes for the other nodes\n that we cannot observe?\n\\item Given a network and the attributes for every node with some missing\n entries, how can we infer the similarity matrices as well as the\n missing attributes?\n\\end{itemize}\n\nNote that the attributes are not limited to binary cases and, moreover, they\nare not necessarily categorical. We will present the methods to bring general\nattributes into our MAG model\\xspace~framework.\n\nThis is important in the following senses:\n\\begin{itemize}\n\\item Plugs a network structure into data inference problem. What will be\n the information gain compared to not using the network structure?\n\\end{itemize}\n\nThis problem will be particularly interesting when the network data is noisy or\npartially observed as well as when the attribute observations are noisy.\n\n\n\\subsection{Related Work}\n\\xhdr{Random Dot Product Model}\n\n\\xhdr{Latent Dirchelt Allocation (LDA) based model}\n\n\\xhdr{Kronecker graphs model} Kronecker graphs model~\\cite{TOCITE} relates to\nour work in a sense that this model is a special case of MAG model\\xspace~\\cite{TOCITE}.\nWhile estimating a single small initiator matrix and a node mapping in the\nKronecker graphs model, we estimate the parameters involved in the generative\nmodel for node attributes as well as the multiple similarity matrices\nassociated with the node attributes in the MAG model\\xspace.\n\n\\xhdr{Mixed Membership Stochastic Blockmodel (MMSB)}\n\\begin{itemize}\n\\item{MMSB} : Simple attribute, arbtrary interactions between memberships\n\\item{MAG} : Allows a rich set of attributes, hierarchical interactions\n with regard to each attribute\n\\end{itemize}\n\n}\n\n\\section{Introduction}\n\\label{sec:intro}\n\\input{010intro}\n\n\\section{Multiplicative Attribute Graphs}\n\\label{sec:problem}\n\\input{020problem}\n\n\\section{MAG Parameter Estimation}\n\\label{sec:algorithm}\n\\input{030algorithm}\n\n\\section{Experiments}\n\\label{sec:experiments}\n\\input{040experiments}\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\input{050conclusion}\n\n\\subsubsection*{Acknowledgements}\nResearch was in-part supported by NSF\nCNS-1010921, \nNSF IIS-1016909, \nAFRL FA8650-10-C-7058,\nLLNL DE-AC52-07NA27344,\nAlbert Yu \\& Mary Bechmann Foundation, IBM,\nLightspeed, Yahoo and the Microsoft Faculty Fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzppwr b/data_all_eng_slimpj/shuffled/split2/finalzzppwr new file mode 100644 index 0000000000000000000000000000000000000000..8cc00f6bcd45c31b8861f5215862ff11a1d1e02b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzppwr @@ -0,0 +1,5 @@ +{"text":"\n\\section{Introduction}\n\\label{sec:Introduction}\nSleepiness is a state identified by reduced alertness that varies according to a circadian rhythm, i.\\,e., with the time of the day~\\cite{Dement.1982,Killgore.2010}. Its detection is important for safety applications, as it has been shown, for example, that sleepiness impacts driving performance, even more so than fatigue~\\cite{Philip.2005}. \n\\cite{lyznicki1998sleepiness}\nMost systems that aim to detect a sleepy driver rely on signals derived from interaction with the vehicle, such as abnormal steering behaviour, failures in lane keeping or irregular use of the pedals~\\cite{Mashko.2015}. Research dealing with automatic sleepiness recognition has investigated methods to derive the state based on different bio-signals. Performing visual analysis of a subject's face, e.\\,g., measuring blinking, can serve in assessing sleepiness but may be negatively affected by changing environmental parameters, such as illumination~\\cite{F.Friedrichs.2010}. While electroencephalography (EEG) has also been shown to be a robust approach to the problem~\\cite{Balandong.2018}, it is far more intrusive and can only be achieved with special equipment and professional setup. In contrast to this, performing acoustic analysis of speech is non-obtrusive and does not need as much sensor application and calibration effort~\\cite{Krajewski09-ASD}. \n\nIn order to define a suitable target for the automatic analysis of sleepiness, the state can be described by the Karolinska Sleepiness Scale (KSS) in terms of ratings ranging from 1 to 9, sometimes additionally extended by an extra category 10 for extreme sleepiness. A binary classification of sleepy speech has been previously performed as part of the INTERSPEECH 2011 Speaker State Challenge~\\cite{Schuller11-TI2}. The subchallenge of INTERSPEECH 2019's \\mbox{\\textsc{ComParE }} challenge deals with the detection of continuous sleepiness, and the sleepiness of a speaker is assessed as a regression problem~\\cite{Schuller19-TI2}. In the challenge baseline, the problem was approached in a number of ways, including a traditional acoustic feature extraction pipeline, Bag-of-Audio-Words, and a deep recurrent autoencoder framework. Here, the unsupervised sequence to sequence model \\mbox{\\textsc{auDeep }} achieved the strongest results. \n\nDeep learning models, such as \\mbox{\\textsc{auDeep }}, that process raw or low level inputs yield state-of-the-art results for a wide range of machine learning problems~\\cite{young2018recent, amodei2016deep, he2016deep}. Many of these approaches consider inputs as a whole, treating every part with the same importance. Often, however, some pieces of the input contain more information pertinent to solving the task at hand than others. A popular approach that takes this notion into consideration can be found with attention mechanisms, such as the one introduced by Bahdanau~et\\,al.~\\cite{bahdanau} for machine translation. Compared to regular sequence to sequence autoencoders where all information is compressed into the last hidden state of the encoder, the dynamic context vector in the attention model retains information about all hidden states of the encoder and their alignment to the current decoding step. Since their introduction, attention mechanisms have also been adapted to speech recognition~\\cite{chorowski2015attention, bahdanau2016end}, visual image captioning~\\cite{xu2015show} or question answering~\\cite{anderson2018bottom}, and speech emotion classification~\\cite{huang2017deep}. \n\nMotivated both by the effectiveness of recurrent autoencoders for acoustic analysis of sleepiness from speech as well as by the improvements to sequence to sequence models achieved with attention mechanisms, we evaluate the impacts of combining the two approaches for the detection of continuous sleepiness on the respective 2019 INTERSPEECH \\mbox{\\textsc{ComParE }} sub-challenge.\n\n\n\\begin{figure*}[ht!]\n \\centering\n \\includegraphics[width=.85\\linewidth]{figs\/sys-overview-s2s.pdf}\n \\caption{An overview of our framework for unsupervised representation learning with sequence to sequence recurrent autoencoders. Except for the SVR training, the approach is fully unsupervised. A detailed description of the procedure is given in~\\Cref{sec:approach}.}\n \\label{fig:sys-overview-s2s}\n\\end{figure*}\n\\section{Dataset}\\label{sec:dataset}\nWe use a subset of the SLEEP Corpus that was employed in the 2019 edition of the INTERSPEECH Computational Paralinguistics Challenge (\\mbox{\\textsc{ComParE }})~\\cite{Schuller19-TI2}. The corpus contains speech recordings of 915 individuals (364 females, 551 males) at varying levels of sleepiness. The audio files have a sample rate of 16\\,kHz with a 16\\,bit quantisation. Each file is annotated with a KSS-score, ranging from 1 to 9, with 9 denoting extreme sleepiness. The labels were derived by averaging self-report with the scores of two external observers. The dataset is split into three partitions with 5\\,564, 5\\,328, and 5\\,570 samples. \n\n\n\\section{Approach}\\label{sec:approach}\nA high-level overview of our proposed approach is depicted in~\\Cref{fig:sys-overview-s2s}. First, Mel-spectrograms are extracted from audio signals. Recurrent autoencoders (AEs), both with and without attention mechanism, are then trained on the Mel-spectrograms. Afterwards, the learnt representations are obtained from the AEs. We then fuse the representations and classify them.\n\n\n\n\n\\subsection{Spectrogram Extraction}\\label{ssec:spectrogram-extraction}\nFirst, the Mel-spectrograms of audio recordings are extracted using periodic Hamming windows with width $w$ and overlap $0.5w$. From these, a given number of log-scaled Mel-frequency\nbands are then computed. Finally, we normalise the Mel-spectra to have values in [-1; 1], since the outputs of the recurrent sequence to sequence autoencoders are constrained to this interval.\n\n\\subsection{Autoencoder Architecture without Attention}\\label{ssec:ae-wo-att}\nFor this architecture, we utilise \\textsc{auDeep}\\footnote{\\url{https:\/\/github.com\/auDeep\/auDeep}}~\\cite{Amiriparian17-STS,Freitag18-AUL}, our recurrent sequence to sequence autoencoder. For the representation learning with our framework, we can adjust a range of autoencoder parameters, including the direction (e.\\,g., uni- or bidirectional) of the encoder and decoder RNNs, types of RNN cells, e.\\,g., gated recurrent units (GRUs), or long short-term memory (LSTM) cells, and the number of hidden layers and units. To use LSTM-RNNs in the decoder, the LSTM cell is modified to work with a context vector similar to the GRUs in the encoder-decoder model proposed by Cho~et\\,al.~\\cite{seq2seq_cho}. The weight matrices $C_i, C_f, C_o$, and $C_z$ are added to the input $z_t$, input gate $i_t$, forget gate $f_t$, and output gate $o_t$ to enable an LSTM \ncell \nto work with the context vector:\n\\begin{equation}\n\\begin{aligned}\n z_t &= \\tanh \\left(W_zx_t+R_z y_{t-1} + C_z c + b_z \\right) \\\\\ni_t &= \\sigma \\left(W_ix_t + R_iy_{t-1} + C_i c + p_i \\odot c_{t-1} + b_i\\right)\\\\\nf_t &= \\sigma \\left(W_fx_t + R_fy_{t-1} + C_f c + p_f \\odot c_{t-1} + b_f\\right)\\\\\no_t &= \\sigma \\left(W_ox_t + R_oy_{t-1} + C_o c + p_o \\odot c_t + b_o\\right).\n\\end{aligned}\n\\end{equation}\n\nFor each input sequence, the initial hidden state vector of the encoder is zero-padded. The last concatenated hidden state vector of the encoder $h_T^{e} = \\left[\\overrightarrow{h}_{T} \\overleftarrow{h}_{T}\\right]^T$ is then passed through a fully connected layer with $\\tanh$ activation which has the same number of units as the decoder RNN. The output of this layer represents the context vector and is used as the first hidden state vector of the decoder $h^{d}_0$. During the feature extraction, the context vector also represents the feature vector.\nThe outputs of the decoder are passed through a fully connected projection layer with $\\tanh$ activation at each time step in order to map the decoder output dimensionality to the target dimensionality. The weights of this output projection are shared across time steps.\nFor the network training the teacher forcing algorithm~\\cite{lamb2016professor} is applied. Following this method, instead of feeding the decoder with the predicted output at time step $t-1$ ($\\hat{y}_{t-1}$), the expected decoder output at time step $t-1$ ($y_{t-1}$) is fed as an input to the decoder.\nThis means that the decoder input is the same original spectrogram, only shifted by one step in time. Instead of the first step, the zero vector is inserted and the frequency vector at the last step is removed. \nDuring the training, the autoencoder learns to reconstruct the reversed input spectrogram~\\cite{Amiriparian17-STS, seq2seq_sutskever}. Mean squared error (MSE) is used as the loss function to compare the reversed source spectrogram with the concatenated spectrogram obtained from the projection layer. \n\n\n\\begin{figure*}[tph!]\n\\centering\n\\includegraphics[width=.90\\linewidth]{figs\/Figure2_cropped.pdf}\n\\caption{Schematic structure of our attention-based autoencoder with stacked encoder and decoder RNNs. Feature vectors 1 and 2 are extracted from the activations of the fully connected layer of the encoder RNN and the last hidden state of the decoder RNN, respectively. A detailed account of the proposed architecture is provided in~\\Cref{ssec:ae-w-att}.} \\label{fig:att-model}\n\\end{figure*}\n\n\\subsection{Autoencoder Architecture With Attention}\\label{ssec:ae-w-att}\n\nIn the second model, we add an attention mechanism to the autoencoder architecture. Here, encoder and decoder have almost the same structure as the baseline autoencoder. The problem of the sequence to sequence model is that the encoder must map all essential information of the input sequence to a fixed-length vector. This may not be enough to represent a long input sequence. To circumvent this, Bahdanau et\\,al.~\\cite{bahdanau} have introduced an attention mechanism that includes dynamic computation of the context vector. At each time step of the decoder, the attention mechanism enables to choose the hidden state vectors of the encoder that contain the most significant information to generate the context vector $c_t$ which is used to generate the output $y_t$. This computation is based on all hidden state vectors of the encoder and the last hidden state vectors of the decoder. The context vector $c_i$ is the linear combination of the hidden state vectors of the encoder $h^{e}_j$:\n\\begin{gather}\n\tc_i = \\sum_{j=1}^{T_x} \\alpha_{ij} h_j^{e}. \n\\end{gather}\nHere, the weights $\\alpha_{ij}$ are numbers between 0 and 1 and define which hidden states $h^{e}_j$ have the biggest influence on $y_i$. $\\alpha_{ij}$ is calculated with the softmax-normalised inner activation of the alignment model $e_{ij}$:\n\\begin{gather}\n\t\\alpha_{ij}=\\frac{\\exp({e_{ij}})}{\\sum_{k=1}^{T_x} \\exp({e_{ik}})}. \n\\end{gather}\nThe alignment model $e_{ij} = a(h^e_{i-1}, h^d_j)$ is a small feedforward neural network trained together with the sequence to sequence model using backpropagation and is defined as\n\\begin{gather}\n v^\\top_{a_i}\\tanh \\left( W_a h^{d}_{i-1} + U_a h^e_j \\right),\n\\end{gather}\nwhere $v_a\\in\\mathbb{R}^l$ is a weight vector, and $W_a\\in\\mathbb{R}^{l\\times n}$ and $U_a\\in\\mathbb{R}^{l\\times 2n}$ are weight matrices, and $l$ is the number of units in the alignment model.\n\n\n\nThe difference to the autoencoder described in \\Cref{ssec:ae-wo-att} is that the context vector is calculated dynamically using the alignment model and is not solely used as a feature vector. If we compute the feature vector from the last hidden state vector of the encoder, the attention mechanism will have less influence on the feature extraction. The feature vector, however, should contain information of all context vectors and, therefore, is extracted after their creation. For this reason, an additional set of feature vectors is extracted from the hidden state vector of the last decoder layer at the last time-step. The attention mechanism in the sequence to sequence model is visualised in~\\Cref{fig:att-model}.\n\n\\section{Experimental Settings}\\label{sec:experimental-settings}\nFor all experiments, we train autoencoder models on the sleepiness spectrograms and then extract features for the three partitions. These feature vectors are used to train a linear support vector regressor (SVR) for which we optimise the complexity parameter on a logarithmic scale of $10^{-5}$ to 1. The complexity parameter is chosen based on the Spearman's correlation coefficient ($\\rho$) achieved on the development partition.\n\nOur proposed autoencoder approaches contain a large amount of adjustable hyperparameters (cf.~\\Cref{ssec:ae-w-att,ssec:ae-wo-att}), which prohibits an exhaustive exploration of the parameter space. For this reason, we choose suitable values for the hyperparameters in multiple stages, using the results of our initial experiments to bootstrap the process. In preliminary experiments, we test different configurations for the spectrograms that are used as input for the autoencoders. We arrived at using Mel-spectrograms with 160 and \\{128, 256\\} Mel-bands extracted from the audio samples for our autoencoders with and without attention, respectively. \nFor the fast Fourier transform (FFT), we apply Hamming windows of 40\\,ms width and 20\\,ms overlap. We further experimented with the architecture of our recurrent autoencoder models. Here, we tested one and two layer variants for both encoder and decoder using either GRU or LSTM cells with 128, 256, or 512 hidden units. Moreover, bidirectional and unidirectional encoders were compared. For the attention autoencoder, models with two-layer bidirectional encoders and two-layer unidirectional decoders worked best with 512 hidden units. For the results presented herein, we therefore settled on this architecture with the choice of either GRU or LSTM layers (cf.~\\Cref{tab:attention-results}). Furthermore, we introduce an additional RNN layer in the decoder that serves to produce hidden states for the attention mechanism. The architecture of those models is visualised in~\\Cref{fig:att-model}. We evaluate features extracted from both the last hidden state of the encoder and decoder. All attention models are trained using a batch size of 256 with the Adam optimiser~\\cite{adam} and the learning rate set to $10^{-4}$ for a maximum of 40 epochs. The model checkpoints at 20, 25, 30, 35, and 40 epochs further serve as feature extractors.\nFor the \\mbox{\\textsc{auDeep }} experiments (cf.~\\Cref{ssec:ae-wo-att}), we found the best configuration with 2 hidden layers each with 256 hidden units. We then optimise the direction of the encoder and decoder and adjust the RNN cell type. Additionally, we filter some of the background noise in the recordings by clipping amplitudes below \\{-40, -50, -60, -70\\}\\,dB thresholds, and fuse them together resulting in five different feature vectors for each data partition. In~\\cite{Amiriparian18-DUR,Amiriparian18-AFO}, we have shown the effectiveness of our amplitude clipping approach for various audio processing tasks.\n\n\\section{Results and Discussion}\n\\label{sec:results}\nAll results obtained with our autoencoders and their early fusion are shown in~\\Cref{tab:attention-results,tab:audeep-results,tab:fusion-results}. In the attention model (cf.~\\Cref{tab:attention-results}), features from the fully connected layer of the encoder RNN (fc$_{enc}$) generalise better when GRUs are applied ($\\rho_{devel} = .250$, $\\rho_{test} = .314$), and the features from the last hidden state of the decoder RNN (state$_{dec}$) perform better with LSTM cells ($\\rho_{devel} = .324$, $\\rho_{test} = .311$). \nFrom both autoencoder approaches, the recurrent model without attention shows the best performance on the test partition ($\\rho_{test} = .359$), whilst the attention model achieves the highest results on the development partition ($\\rho_{devel} = .324$). The result \nimplies \npossible overfitting of the attention model on the development data. This issue is not strongly present in our model without attention, and we hypothesise that this is mainly because of the filtering of some of the background noise found in the audio data by clipping amplitudes below a certain threshold. In~\\Cref{tab:audeep-results}, we provide the highest achieved results with various thresholds and hyperparameter combinations. Furthermore, we fuse the best performing attention feature set on the development set ($\\rho_{devel} = .324$, $\\rho_{test} = .311$) with all non-attention (\\textsc{auDeep}) features to analyse the complementarity of the learnt representations. The results in~\\Cref{tab:fusion-results} demonstrate an improvement of all results after early fusion. The highest improvement on the test partition after fusion is achieved when the best attention feature set is combined with the fourth \\mbox{\\textsc{auDeep }} feature with GRUs, \nand the \nunidirectional encoder and bidirectional encoder trained on Mel-spectrograms with 256 Mel-band (FFT window width of 80\\,ms and overlap of 40\\,ms) and -60\\,dB amplitude clipping (cf.~\\Cref{tab:fusion-results}). It is worth mentioning that the dimensionality of the attention features are either $1\/2$ or $1\/8$ of the \\mbox{\\textsc{auDeep }} features, leading to a faster classifier training. Moreover, the training process with attention autoencoders can be performed faster, as the encoder RNN is relieved from encoding all information in the whole input sequence of the Mel-spectrograms into a fixed-length vector~\\cite{Amiriparian19-DRL}.\nWe further compare our best performing approaches with \nthe \nbest challenge baselines~\\cite{Schuller19-TI2}, the winner of the challenge who combined Fisher vectors with baseline features~\\cite{Gosztolya.2019}, and the runner-up who utilised a fusion of convolutional neural networks (CNNs) and RNNs~\\cite{yeh2019using} (cf.~\\Cref{tab:compare_results}). \n\n\n\\begin{table}[tp]\n \\centering\n \\caption{Performance comparison of the features obtained from the fully connected layer of the encoder (fc$_{enc}$) and the last hidden state of the decoder (state$_{dec}$) in our attention autoencoder. id: feature identifier, Dim.: feature dimensionality.}\n \\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{rrlrrrrr}\n \\toprule\n \\multicolumn{4}{c}{Parameters} & \\multicolumn{2}{c}{fc$_{enc}$} & \\multicolumn{2}{c}{state$_{dec}$} \\\\\n \\midrule\n id & Epoch & Cell & Dim. & $\\rho_{devel} $ & $\\rho_{test}$ & $\\rho_{devel} $ & $\\rho_{test}$ \\\\\n \\midrule\n 1 & 20 & GRU & 512 & .262 & .308 & .276 & .294 \\\\\n 2 & 25 & GRU & 512 & .258 & .308 & .278 & .298 \\\\\n 3 & 30 & GRU & 512 & .250 & .314 & .267 & .292 \\\\\n 4 & 35 & GRU & 512 & .260 & .312 & .266 & .298 \\\\\n 5 & 40 & GRU & 512 & .253 & .307 & .265 & .288 \\\\\n \\midrule\n 6 & 20 & LSTM & 512 & .293 & .294 & .318 & .303 \\\\\n 7 & 25 & LSTM & 512 & .303 & .276 & \\textbf{.324} & \\textbf{.311} \\\\\n 8 & 30 & LSTM & 512 & .298 & .263 & .322 & .305 \\\\\n 9 & 35 & LSTM & 512 & .303 & .285 & .289 & .304 \\\\\n 10 & 40 & LSTM & 512 & .307 & .302 & .288 & .299 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:attention-results}\n\\end{table}\n\n\\begin{table}[tp]\n \\centering\n \\caption{Results obtained from our autoencoder without attention. id: feature identifier, w: width of the Hamming window, Mel: number of Mel-bands, Clip: clipped amplitudes below a certain threshold to filter some noise from audio, Dim.: feature dimensionality, Dir.: direction of the encoder-decoder.}\n \n \n \\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{rrrrrrrrr}\n \\toprule\n \\multicolumn{8}{c}{Autoencoders without attention} \\\\\n \\midrule\n id & w & Mel & Clip & Cell & Dim. & Dir. & {$\\rho_{devel}$} & {$\\rho_{test}$} \\\\\n \\midrule\n 1 & 0.08 & 256 & fused & GRU & 4\\,096 & uni-bi & .286 & .338 \\\\\n 2 & 0.08 & 256 & -70\\,dB & GRU & 1\\,024 & bi-uni & \\textbf{.283} & \\textbf{.359}\\\\\n 3 & 0.06 & 256 & -70\\,dB & GRU & 1\\,024 & bi-uni & .281 & .357 \\\\\n 4 & 0.08 & 256 & -60\\,dB & GRU & 1\\,024 & uni-bi & .278 & .331 \\\\\n 5 & 0.04 & 256 & -70\\,dB & GRU & 1\\,024 & uni-bi & .278 & .340 \\\\\n 6 & 0.06 & 256 & fused & GRU & 4\\,096 & bi-uni & .277 & .346 \\\\\n 7 & 0.06 & 256 & -70\\,dB & GRU & 1\\,024 & uni-bi & .277 & .348 \\\\\n 8 & 0.06 & 128 & -70\\,dB & LSTM & 1\\,024 & uni-bi & .277 & .317 \\\\\n 9 & 0.08 & 128 & -60\\,dB & GRU & 1\\,024 & bi-uni & .275 & .324 \\\\\n 10 & 0.04 & 256 & -60\\,dB & GRU & 1\\,024 & uni-bi & .275 & .336 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:audeep-results}\n\\end{table}{}\n\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:conclusions}\nIn~\\Cref{ssec:ae-wo-att}, we introduced a novel attention mechanism for recurrent sequence to sequence autoencoders to process audio signals in an unsupervised manner\\footnote{Our audio-based attention framework (\\textsc{auttention}), and all codes to reproduce the attention results are provided here: \\mbox{\\url{https:\/\/github.com\/auttention\/SleepyAttention}}}. We have demonstrated the suitability of two fully unsupervised representation learning techniques for continuous sleepiness recognition, and their superior performance to expert-designed, hand-crafted features (cf.~\\Cref{tab:compare_results}). Furthermore, we conducted a feature fusion strategy and demonstrated the complementarity of the learnt representations from both autoencoder architectures (cf.~\\Cref{sec:results}). In future work, we will be evaluating our systems over a wide range of audio recognition tasks. We also want to utilise dimensionality reduction techniques~\\cite{Amiriparian17-FSI} to cope with the high-dimensionality of our autoencoder features. Finally, we want to combine our methods with CNN-based representation learning systems, such as deep convolutional generative adversarial networks~\\cite{radford2015unsupervised}.\n\n\n\\begin{table}[tp]\n \\centering\n \\caption{Results of our early fusion experiments with the best attention result (id$_{att}$ = 7) and all results provided in~\\Cref{tab:audeep-results}. id$_{att}$ and id$_{audeep}$: identifiers for the attention feature and \\mbox{\\textsc{auDeep }} features which are fused. $C_{SVR}$: Complexity of the SVR which is optimised on the development partition after fusion.}\n \\begin{tabular}{rr|rrr}\n \\toprule\n \\multicolumn{5}{c}{Early fusion} \\\\\n \\midrule\n id$_{att}$ &\n id$_{audeep}$ &\n $C_{SVR}$ &\n {$\\rho_{devel}$} &\n {$\\rho_{test}$} \\\\\n \\midrule\n 7 & 1 & $10^{-3}$ & .315 & .359 \\\\\n 7 & 2 & $10^{-2}$ & .336 & .360 \\\\\n 7 & 3 & $10^{-2}$ & .334 & .365 \\\\\n 7 & 4 & $10^{-1}$ & \\textbf{.320} & \\textbf{.367} \\\\\n 7 & 5 & $10^{-2}$ & .333 & .349 \\\\\n 7 & 6 & $10^{-3}$ & .319 & .363 \\\\\n 7 & 7 & $10^{-2}$ & .326 & .361 \\\\\n 7 & 8 & $10^{-2}$ & .333 & .341 \\\\\n 7 & 9 & $10^{-2}$ & .340 & .351 \\\\\n 7 & 10 & $10^{-2}$ & .339 & .357 \\\\\n \\bottomrule\n \\end{tabular}\n \\label{tab:fusion-results}\n\\end{table}{}\n\n\n\\begin{table}[tp]\n \\centering\n \\caption{Comparison of our best performing models with best performing challenge baselines and the challenge winner. $Dim.$: feature dimensionality of each system.}\n \\resizebox{\\columnwidth}{!}{\n \\begin{tabular}{lrrr}\n \\toprule\n System & Dim. & $\\rho_{dev}$ & $\\rho_{test}$ \\\\\n \\midrule\n \\multicolumn{4}{l}{\\textbf{Challenge Winner}~\\cite{Gosztolya.2019}} \\\\\n \\mbox{\\textsc{ComParE }} + BoAW + Fisher vectors & -- & -- & \\textbf{.383} \\\\\n \\multicolumn{4}{l}{\\textbf{Runner-up}~\\cite{yeh2019using}} \\\\\n CNNs and BLSTMs with attention & -- & .373 & .369 \\\\ \n \n \\midrule\n \\multicolumn{4}{l}{\\textbf{Best Challenge Baselines}~\\cite{Schuller19-TI2}} \\\\\n \\mbox{\\textsc{ComParE }} & 6\\,373 & .251 & .314 \\\\\n Bag-of-Audio-Words & 500 & .250 & .304 \\\\\n Autoencoders & 1\\,024 & .243 & .325 \\\\\n Late fusion of best & -- & -- & .343 \\\\\n \\midrule\n \\multicolumn{4}{l}{\\textbf{Best of Our Proposed Approaches}} \\\\\n With attention & 512 & .324 & .311 \\\\\n Without attention & 4\\,096 & .286 & .338 \\\\\n Early fusion & 4\\,608 & \\textbf{.320} & \\textbf{.367} \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:compare_results}\n\\end{table}{}\n\\section{Acknowledgements}\nThis research was partially supported by BMW AG and Deutsche Forschungsgemeinschaft (DFG) under grant agreement No. 421613952 (ParaStiChaD).\n\\clearpage\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{CONCLUSIONS}\n\nFirst results from simulations using the non-perturbatively\\ improved Wilson action\nand currents in the quenched approximation show that lattice artefacts\nare drastically reduced. Unlike the case of unimproved actions, the\nresults for the vector meson mass are practically independent of~$a$\nat $\\beta=6.0$ and~6.2, so that residual~$a^2$ effects are around\n$2\\%$ at $a\\simeq0.1\\,{\\rm fm}$. \nA real test of the scaling behaviour is still lacking and will only\nbecome possible when more and different values of the lattice spacing\nare studied.\n\nThe analysis of data for the pseudoscalar decay constant has shown\nthat a non-perturbative determination of $b_{\\rm A}$ is required for quark\nmasses above $m_{\\rm strange}$. The improvement coefficient $c_{\\rm A}$ has\na large influence on the scaling behaviour, which motivates further\ninvestigation. On the whole, it appears that non-perturbative\\ improvement leads to\nbetter agreement between the continuum result for $f_K$ and the\nexperimental value.\n\nThe systematic nature of non-perturbative\\ improvement makes it easily applicable\nto other situations: results for $c_{\\rm sw}$ computed for two flavours of\ndynamical quarks have been reported\\,\\cite{Karl_Rainer_lat97}.\nFurthermore, the formalism has been applied to quenched QCD with an\nimproved gauge action\\,\\cite{KlasEd_lat97}.\n\nI thank P. Rowland for help with the UKQCD data and also M. Guagnelli,\nT. Mendes and G. Schierholz for communicating their data prior to\npublication. I wish to thank M. L\\\"uscher and R. Sommer for many\nhelpful discussions.\n\n\\section{DECAY CONSTANTS}\n\nFrom the definitions of the renormalised and improved currents,\neq.\\,(\\ref{EQ:currents}), one sees that in order to obtain O($a$)\nimproved matrix elements of axial and vector currents one requires\nknowledge of the improvement coefficients $c_{\\rm A},b_{\\rm A},c_{\\rm V},b_{\\rm V}$ and the\nrenormalisation factors $Z_{\\rm A}$ and $Z_{\\rm V}$. In this section we will focus\non the pseudoscalar decay constant $(f_{\\rm PS})_{\\rm R}$, which we write as\n\\begin{equation}\n (f_{\\rm PS})_{\\rm R} = Z_{\\rm A}(1+b_{\\rm A} am_{\\rm q})\\big\\{f_{\\rm PS}^{(0)}+c_{\\rm A} g_{\\rm PS}^{(0)}\\big\\}, \n\\end{equation}\nwhere the unrenormalised matrix elements are parametrised as\n\\begin{eqnarray}\n \\mpsf_{\\rm PS}^{(0)} & = & \\langle0|A_4(0)|{\\rm PS}\\rangle \\\\\n g_{\\rm PS}^{(0)} & = & \\langle0|P(0)|{\\rm PS}\\rangle.\n\\end{eqnarray}\nA major advantage of non-perturbative\\ improvement is that the renormalisation\nfactor $Z_{\\rm A}$ has been determined with an accuracy of around\n1\\%. At $\\beta=6.0$ and 6.2 one finds\\,\\cite{paperIV}\n\\begin{equation}\n Z_{\\rm A}=\\left\\{\\begin{array}{lr}\n\t0.791(9), & \\quad\\beta=6.0 \\\\\n\t0.807(8), & \\quad\\beta=6.2 \n\t \\end{array}\n \\right.\n\\end{equation}\nNote that the error on $Z_{\\rm A}$ must be combined with the statistical\nerror on $f_{\\rm PS}^{(0)}+c_{\\rm A} g_{\\rm PS}^{(0)}$.\n\nThe improvement coefficient $c_{\\rm A}$ has also been determined\nnon-perturbatively\\,\\cite{paperIII}:\n\\begin{equation}\n c_{\\rm A}=\\left\\{\\begin{array}{lr}\n -0.083, & \\quad\\beta=6.0 \\\\\n -0.037, & \\quad\\beta=6.2\n \\end{array}\n \\right.\n\\end{equation}\nThe effect of $c_{\\rm A}$ on the decay constant can be studied by comparing\n$f_{\\rm PS}^{(0)}$ with $f_{\\rm PS}^{(0)}+c_{\\rm A}g_{\\rm PS}^{(0)}$. Here the contribution from $c_{\\rm A}g_{\\rm PS}^{(0)}$\nleads to a decrease in the pseudoscalar decay constant of $\\sim4\\%$ at\n$\\beta=6.2$ and even $\\sim15\\%$ at $\\beta=6.0$. This effect is\nparticularly pronounced for $\\beta<6.2$.\n\nThe improvement coefficient $b_{\\rm A}$ is relevant for the determination of\nthe kaon decay constant $f_K$. However, unlike $b_{\\rm V}$, the coefficient\n$b_{\\rm A}$ has so far not been determined non-perturbatively. In one-loop perturbation\ntheory one finds\\,\\cite{paperV}\n\\begin{equation}\n b_{\\rm A}=1+0.1522g_0^2+O(g_0^4).\n\\end{equation}\nIn order to study the influence of $b_{\\rm A}$, one can evaluate $(f_{\\rm PS})_{\\rm R}$\naround the strange quark mass for different choices of $b_{\\rm A}$. Here we\ncompare\n\\begin{itemize}\n\\item $b_{\\rm A}=0$\n\\item $b_{\\rm A} = 1 + 0.1522g_0^2$\n\\item $b_{\\rm A}=b_{\\rm V}$\n\\end{itemize}\nThe seemingly {\\it ad hoc\\\/} choice of $b_{\\rm A}=b_{\\rm V}$ (here we use the non-perturbative\\\ndetermination of $b_{\\rm V}$) is motivated by the observation that the\none-loop coefficients for $b_{\\rm A}$ and $b_{\\rm V}$ are approximately\nequal\\,\\cite{paperV}. By applying different choices of $b_{\\rm A}$ to the\nanalysis of the UKQCD data around $m_K$, one finds that $b_{\\rm A}$ leads to\nan increase in $(f_{\\rm PS})_{\\rm R}$ of at most $\\sim2\\%$ at $\\beta=6.2$ and\n$\\sim3\\%$ at $\\beta=6.0$, which is fairly small. Since the choice\n$b_{\\rm A}=b_{\\rm V}$ gives essentially the same mass behaviour compared\nto choosing $b_{\\rm A} = 1 + 0.1522g_0^2$, one concludes that perturbative\nestimates of $b_{\\rm A}$ are quite acceptable for quark masses up to and\naround $m_{\\rm strange}$.\n\nWe now analyse the scaling behaviour of $(f_{\\rm PS})_{\\rm R}$. In\nFig.\\,\\ref{Fig:fpsvsmps2} we plot $(f_{\\rm PS})_{\\rm R} r_0$ versus $(m_{\\rm PS} r_0)^2$,\nwhere $r_0$ is the hadronic radius defined in\\,\\cite{rainer_r0}. Data\nfor $r_0\/a$ were taken from\\,\\cite{how_r0}. If lattice effects are\nsmall the data in Fig.\\,\\ref{Fig:fpsvsmps2} should lie on a universal\ncurve. The results for $f_Kr_0$ computed at $\\beta=6.0$ and~6.2 show a\nslight dependence on the lattice spacing. In order to study residual\nlattice artefacts, we employ a similar procedure as in the case of the\nvector mass and extrapolate $(f_{\\rm PS})_{\\rm R} r_0$ to $(m_K\nr_0)^2$. Fig.\\,\\ref{Fig:fK} shows the resulting values of $f_Kr_0$ as\na function of $(a\/r_0)^2$. There is good agreement between the data\nfrom all three collaborations, and in principle their results could be\ncombined. Furthermore, it appears that a linear extrapolation in $a^2$\nyields a continuum result which is compatible with the experimentally\nobserved value (although there is {\\it a priori\\\/} no reason why the\nquenched approximation should reproduce the measured result). In\ncontrast, the authors of\\,\\cite{GF11_decay} have found that, for the\nunimproved action, the continuum value of $f_K$ is significantly lower\nthan the experimental result.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{fpi_r0.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{The pseudoscalar decay constant as a function of $m_{\\rm PS}^2$ in\nunits of $r_0$ for $\\beta=6.0$ (open symbols) and $\\beta=6.2$ (full\nsymbols). Asterisks denote the experimental values for $f_\\pi$ and\n$f_K$ using $r_0=0.5\\,{\\rm fm}$.} \n\\label{Fig:fpsvsmps2}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{fK_r0_vs_r0_notad.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{$f_Kr_0$ plotted versus $(a\/r_0)^2$. The asterisk denotes the\nexperimental result.}\n\\label{Fig:fK}\n\\end{figure}\n\nFrom the slope in $(a\/r_0)^2$ one estimates that residual lattice\nartefacts in $f_K r_0$ amount to ca. 10\\% at\n$a\\simeq0.1\\,{\\rm fm}$. Compared to the previously discussed case of the\nvector meson, this is a fairly large correction. Given the substantial\ncontribution of $c_{\\rm A}g_{\\rm PS}^{(0)}$ to $(f_{\\rm PS})_{\\rm R}$ at $\\beta=6.0$, one can ask\nwhether $c_{\\rm A}$ has a large effect on the scaling behaviour. If one\nformulates a similar improvement condition for $c_{\\rm A}$ as in\neq.\\,(\\ref{EQ:phenimp}), for instance\n\\begin{equation}\n f_K r_0 = {\\rm const.}\n\\end{equation}\none can study how well it is satisfied for different choices of\n$c_{\\rm A}$. Using the UKQCD data it turns out that the above condition\n``favours'' smaller values of $c_{\\rm A}$ at the lower end of the\n$\\beta$-range. However, at this stage one should not jump to\nconclusions before a more thorough investigation of improvement\nconditions has been performed.\n\n\\section{INTRODUCTION}\n\nOne of the most important systematic effects in lattice simulations of\nQCD is the finiteness of the lattice spacing~$a$. It is well known\nthat physical observables computed using the Wilson action are subject\nto corrections of order~$a$, which can be rather large. In order to\nobtain reliable results in the continuum limit it is desirable to\nreduce lattice artefacts, either through the Symanzik improvement\nprogramme\\,\\cite{SymanzikI,SymanzikII}, or by employing a\nrenormalisation group approach\\,\\cite{HasNied94,Has_lat97}. Recently\nthe ALPHA Collaboration has carried out the Symanzik improvement\nprogramme to leading order through a non-perturbative determination of\nthe O($a$) improved fermion action and isospin\ncurrents\\,\\cite{Alpha_lat96,paperIII,paperIV,marco_lat97}. This\napproach should lead to the complete removal of lattice artefacts of\norder~$a$ in spectral quantities and matrix elements of local\ncurrents, so that the remaining cutoff effects are of order~$a^2$. The\nnon-perturbatively\\ improved action has already been employed in a number of\nsimulations in the quenched\napproximation\\,\\cite{QCDSF_97,par_lat97,mendes_lat97}. Here, we\nassess the impact of non-perturbative\\ improvement on the calculation of the mass\nspectrum and decay constants in the light hadron sector, analyse the\nscaling behaviour and estimate the size of residual lattice artefacts.\n\nThe general expression of the O($a$) improved fermion action\nreads\\,\\cite{SW85} \n\\begin{eqnarray}\nS_F^I[U,\\overline{\\psi},\\psi] & = & S_F^W[U,\\overline{\\psi},\\psi] \\nonumber\\\\\n & & \\hspace{-1.5cm}+\\,c_{\\rm sw}{\\textstyle\\frac{ia}{4}}\\sum_{x,\\mu,\\nu}\n \\overline{\\psi}(x)\\sigma_{\\mu\\nu}F_{\\mu\\nu}(x)\\psi(x),\n\\label{EQ:currents}\n\\end{eqnarray}\nwhere $S_F^W$ is the (unimproved) Wilson fermion action and $c_{\\rm sw}$ is\nan improvement coefficient. The improved and renormalised axial and\nvector currents are defined as\\,\\cite{paperI}\n\\begin{eqnarray}\n(A_{\\rm R})_\\mu^a\\hspace{-0.2cm} & = &\\hspace{-0.2cm}\n Z_{\\rm A}(1+b_{\\rm A} am_{\\rm q})\\big\\{A_\\mu^a+c_{\\rm A} a\\partial_\\mu\nP^a\\big\\}, \\\\\n(V_{\\rm R})_\\mu^a\\hspace{-0.2cm} & = &\\hspace{-0.2cm}\n Z_{\\rm V}(1+b_{\\rm V} am_{\\rm q})\\big\\{V_\\mu^a+c_{\\rm V} a\\partial_\\nu\nT_{\\mu\\nu}^a\\big\\},\n\\end{eqnarray}\nwhere $Z_{\\rm A}, Z_{\\rm V}$ are the renormalisation factors of the respective\ncurrents, and $b_{\\rm A},c_{\\rm A},b_{\\rm V}$ and $c_{\\rm V}$ are further improvement\ncoefficients. The unimproved currents and densities are defined as\n\\begin{eqnarray}\nA_\\mu^a & = & \\overline{\\psi}\\gamma_\\mu\\gamma_5\\frac{\\tau^a}{2}\\psi,\\quad\n~V_\\mu^a = \\overline{\\psi}\\gamma_\\mu\\frac{\\tau^a}{2}\\psi,\\nonumber\\\\\nP^a & = & \\overline{\\psi}\\gamma_5\\frac{\\tau^a}{2}\\psi,\\qquad\nT_{\\mu\\nu}^a = i\\overline{\\psi}\\sigma_{\\mu\\nu}\\frac{\\tau^a}{2}\\psi,\n\\end{eqnarray}\nwhere $\\tau^a$ are the Pauli matrices acting in flavour space.\nThe normalisations $Z_{\\rm A},Z_{\\rm V}$ and improvement coefficients $c_{\\rm sw}, c_{\\rm A},\nc_{\\rm V}, b_{\\rm V}$ have been determined non-perturbatively\\ for bare couplings $g_0$ in the\nrange $0\\leq g_0\\leq1$\\,\\cite{paperIII,paperIV,marco_lat97}. Results\nand proposals for non-perturbative determinations of $b_{\\rm A}$ at\n$g_0\\simeq1$ have been\nreported\\,\\cite{MRSSTT97,giulia_lat97,ssharpe_lat97}. Furthermore, all\nof the above improvement coefficients have been calculated in\nperturbation theory to one-loop order\\,\\cite{paperII,paperV}.\n\nThe parameters of the simulations discussed here are listed in\nTable\\,\\ref{Tab:simpar}. In addition to the non-perturbative\\ value of $c_{\\rm sw}$, the\nQCDSF collaboration have also used unimproved Wilson fermions for a\ndirect comparison\\,\\cite{QCDSF_97}. UKQCD have also used the tadpole\nimproved value of $c_{\\rm sw}$ on the same set of configurations and at\n$\\beta=5.7$\\,\\cite{UKQCD_lat96,tadpole_comp}, where a non-perturbatively\\\ndetermined value of $c_{\\rm sw}$ is not available. Table\\,\\ref{Tab:simpar}\nshows that all collaborations have used the same two values of\n$\\beta$. This implies that one cannot test as yet whether non-perturbative\\\nimprovement indeed leads to a scaling behaviour of physical\nobservables which is consistent with O($a^2$) corrections. However, if\none {\\it assumes\\\/} this to be the leading scaling behaviour after\nimprovement, one can still estimate the size of residual lattice\nartefacts at a given value of~$a$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\caption{Simulations using non-perturbative\\\timprovement. The number of exceptional\nconfigurations discarded from the ensemble is shown in brackets.}\n\\begin{tabular}{lccr}\n\\hline\nCollab.\t& $\\beta$ & $L^3\\cdot T$ & Statistics \\\\\n\\hline\nQCDSF\\,\\cite{QCDSF_97}\n\t& 6.0\t & $16^3\\cdot48$ & $\\sim1000$~~~\t\\\\\n\t& \t & $24^3\\cdot48$ & $\\sim200$~~~\t\\\\\n\t& 6.2\t & $24^3\\cdot48$ & $\\sim300$~~~\t\\\\\n\\hline\nUKQCD\\,\\cite{par_lat97}\n\t& 6.0 & $16^3\\cdot48$ & 497(3)\t\\\\\n\t&\t & $32^3\\cdot64$ & 70(1)\t\\\\\n\t& 6.2\t & $24^3\\cdot48$ & 251~~~\t\\\\\n\\hline\nAPETOV\\,\\cite{mendes_lat97}\n\t& 6.0\t & $16^3\\cdot48$ & 50(1)\t\\\\\n\t& 6.2\t & $24^3\\cdot48$ & 50~~~~\t\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{Tab:simpar} \n\\vspace{-0.8cm}\n\\end{table}\n\n\n\n\n\n\n\\section{SPECTRAL QUANTITIES}\n\nIn Fig.\\,\\ref{Fig:edin} we show the Edinburgh plot. In addition to the\nnon-perturbatively\\ improved data, we also display the tadpole improved results by\nUKQCD at $\\beta=5.7$ and the data by GF11 using the unimproved action\nat $\\beta=5.7, 5.93, 6.17$\\,\\cite{GF11_spec}. One observes that\nimprovement yields consistently lower values for $m_N\/m_\\rho$ compared\nto the unimproved action. The most dramatic effect is observed at\n$\\beta=5.7$ when one compares the unimproved results (full triangles)\nto the tadpole improved ones (full circles). In fact, it seems that\nthe mass behaviour obtained using tadpole improvement at $\\beta=5.7$\nis indistinguishable from the non-perturbatively\\ improved action. However, as we\nshall see later, the residual lattice artefacts in the tadpole\nimproved data at $\\beta=5.7$ are still large. Thus, the Edinburgh plot\ndisguises rather than exposes lattice artefacts and should therefore\nnot be used to draw conclusions about the scaling behaviour.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{edin2.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{Edinburgh plot.}\n\\label{Fig:edin}\n\\end{figure}\n\nIt is a well-known fact that lattice simulations using unimproved\nactions fail to reproduce the experimentally observed behaviour of the\nvector-pseudoscalar mass splitting, i.e. $m_{\\rm V}^2-m_{\\rm PS}^2\\sim {\\rm\nconst}$, which holds up to the mass of the charm quark. This is\nusually ascribed to the influence of lattice artefacts in the\ncomputation of these splittings. Data for $m_{\\rm V}^2-m_{\\rm PS}^2$ by\nUKQCD\\,\\cite{par_lat97} and QCDSF\\,\\cite{QCDSF_97} using the\nnon-perturbatively\\ improved action show that the splittings are close to the\nexperimental values for the $(\\rho,\\pi)$ and $(K^*,K)$ systems (see\nFig.\\,\\ref{Fig:splitting} for a plot of the UKQCD data). Furthermore,\nby comparing the results at $\\beta=6.0$ and~6.2 one observes that the\ndependence on the lattice spacing is small. However, for quark masses\nabove $m_{\\rm strange}$ the two collaborations see a slight downward\ntrend in the data, so that one can expect that the $D^*-D$ splitting\nis still not reproduced correctly.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{mass_splitting.eps}{90mm}\n\\vspace{-4.3cm}\n\\caption{UKQCD data for the vector-pseudo-scalar mass splitting\nin units of $m_K^2$. Asterisks denote the experimentally observed\nvalues.} \n\\vspace{-0.3cm}\n\\label{Fig:splitting}\n\\end{figure}\n\nWe now discuss the chiral limit and the critical value of the hopping\nparamter, $\\kappa_{\\rm c}$. Usually $\\kappa_{\\rm c}$ is defined at the point where the\npseudoscalar mass vanishes, $m_{\\rm PS}=0$. In accordance with the quark\nmass behaviour of $m_{\\rm PS}^2$ implied by the PCAC relation, one can\ndetermine $\\kappa_{\\rm c}$ from a linear fit to\n\\begin{equation}\n (am_{\\rm PS})^2 = a^2B\\frac{1}{2}\\left(\\frac{1}{\\kappa}-\\frac{1}{\\kappa_{\\rm c}}\n \\right) = a^2B\\,m_{\\rm q}.\n\\label{EQ:mps2}\n\\end{equation}\nBoth QCDSF and UKQCD have reported that this linear ansatz in\n$1\/\\kappa$ results in poor fits with large correlated $\\chi^2\/{\\rm\ndof}$ and have therefore resorted to using model functions which also\ncontain quadratic terms in $1\/\\kappa$. However, in order to be\nconsistent with O($a$) improvement, the quark mass $m_{\\rm q}$ in\neq.\\,(\\ref{EQ:mps2}) should be replaced by\n\\begin{equation}\n\\widetilde{m}_{\\rm q}=m_{\\rm q}(1+b_{\\rm m} am_{\\rm q}),\n\\end{equation}\nwhere $b_{\\rm m}=-\\frac{1}{2}-0.0962g_0^2$ in one-loop perturbation\ntheory\\,\\cite{paperV}. This modification of the fitting ansatz in the\ndetermination of $\\kappa_{\\rm c}$ has so far not been used.\n\nAnother method defines $\\kappa_{\\rm c}$ at the point where the quark mass defined\nthrough the PCAC relation vanishes, i.e. $m_{\\rm PCAC}=0$. Here it is\nimportant to realise that \neven after non-perturbative\\ improvement\nchiral symmetry is only approximately restored, so that\n\\begin{eqnarray}\n &\\partial_\\mu &\\hspace{-0.3cm} \\big\\{A_\\mu(x)+c_{\\rm A} a\\partial_\\mu P(x)\\big\\}\n\t\\nonumber\\\\\n & &\\hspace{1.0cm} =\\hspace{0.1cm} 2m_{\\rm PCAC}P(x) + {\\rm O}(a^2).\n\\end{eqnarray}\nTherefore, the values of $\\kappa_{\\rm c}$ determined by requiring either\n$m_{\\rm PS}^2=0$ or $m_{\\rm PCAC}=0$ will differ by terms of order\n$a^2$. The compilation of results in Table\\,\\ref{Tab:kappac} shows\nthat in the range of $\\beta$ under study the difference in $\\kappa_{\\rm c}$ using\neither method is of the order of $10^{-4}$ and thus statistically\nsignificant.\n\n\\begin{table}[tb]\n\\begin{center}\n\\caption{Values for $\\kappa_{\\rm c}$ determined at $m_{\\rm PS}^2=0$ and at $m_{\\rm\nPCAC}=0$.} \n\\begin{tabular}{cr@{.}lr@{.}lr@{.}l}\n\\hline\n\\hline\n\t& \\multicolumn{6}{c}{$m_{\\rm PS}^2=0$} \\\\\n\\cline{2-7}\n$\\beta$ & \\multicolumn{2}{c}{UKQCD} & \\multicolumn{2}{c}{QCDSF}\n & \\multicolumn{2}{c}{APETOV} \\\\\n\\hline\n6.0 & 0&135335${}^{+20}_{-17}$ & 0&13531(1)\n\t& \\multicolumn{2}{c}{--} \\\\[0.1cm]\n6.2 & 0&135895${}^{+14}_{-55}$ & 0&13589(2) & 0&135861(19) \\\\[0.05cm]\n\\hline\n\\hline\n\t& \\multicolumn{6}{c}{$m_{\\rm PCAC}=0$} \\\\\n\\cline{2-7}\n$\\beta$ & \\multicolumn{3}{c}{ALPHA} & \\multicolumn{3}{c}{APETOV} \\\\[0.05cm]\n\\hline\n6.0\t& \\multicolumn{3}{c}{0.135196(14)}\n\t& \\multicolumn{3}{c}{--}\t\\\\[0.1cm]\n6.2\t& \\multicolumn{3}{c}{0.135795(13)}\n\t& \\multicolumn{3}{c}{0.135802(6)}\t\\\\[0.05cm]\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{Tab:kappac}\n\\vspace{-0.8cm}\n\\end{table}\n\nWe now analyse the scaling behaviour of the mass of the vector meson\nby comparing the approach to the continuum limit for unimproved and\nimproved actions. To this end, we note that the continuum limit should\nbe studied for constant physical volume, so that finite-volume effects\ndo not distort the scaling behaviour. Since chiral extrapolations are\npoorly understood, we will use the available lattice data for the\nvector and pseudoscalar masses $am_{\\rm V}, am_{\\rm PS}$ as well as the string\ntension $a\\sqrt{\\sigma}$ in order to interpolate the dimensionless ratio\n$m_{\\rm V}\/\\sqrt{\\sigma}$ to $m_{\\rm PS}\/\\sqrt{\\sigma}=1.125$. For $\\sqrt{\\sigma}=440\\,{\\rm MeV}$ this implies\n$m_{\\rm PS}=m_K=495\\,{\\rm MeV}$. If $m_{\\rm PS}^2$ and $m_{\\rm V}$ are linear functions of\nthe quark masses then one expects that $m_{\\rm V}\\simeq m_{K^*}=892\\,{\\rm MeV}$\nor $m_{\\rm V}\/\\sqrt{\\sigma}\\simeq2.027$. In Fig.\\,\\ref{Fig:mKstar} we have plotted\nthe interpolated data for $m_{\\rm V}\/\\sqrt{\\sigma}$ versus $a\\sqrt{\\sigma}$ for\nunimproved\\,\\cite{GF11_spec} and improved\nactions\\,\\cite{par_lat97,UKQCD_lat96,tadpole_comp}. For the unimproved\naction one observes that $m_{\\rm V}\\simeq m_{K^*}$ is satisfied, but only\nafter the extrapolation to the continuum limit. In contrast, the data\nobtained using the non-perturbatively\\ or tadpole improved actions at $\\beta=6.0,\n6.2$ show very little dependence on~$a$ and are rather close to\n$m_{\\rm V}\\simeq m_{K^*}$. This, however, can no longer be claimed for the\ntadpole improved data at $\\beta=5.7$, where instead one observes large\nresidual lattice artefacts. From the slope of the linear fit to\n$m_{\\rm V}\/\\sqrt{\\sigma}$ for the unimproved data, we infer the size of residual\nlattice artefacts at $a\\simeq0.1\\,{\\rm fm}$ to be $(12\\pm1)\\%$. For the\nnon-perturbatively\\ improved data we estimate the leading corrections of O($a^2$)\nto be only $\\simeq2\\%$.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{mK.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{Scaling behaviour of the vector mass at $m_{\\rm PS}=m_K$. The\ndashed line indicates the extrapolated result using the GF11 data.}\n\\label{Fig:mKstar}\n\\end{figure}\n\nWe can now turn the tables and ask how well various prescriptions to\nfix $c_{\\rm sw}$ satisfy the ``phenomenological'' improvement condition\n\\begin{equation}\n{m_{\\rm V}}\/{\\sqrt{\\sigma}}={\\rm const}\n\\label{EQ:phenimp}\n\\end{equation}\nBesides the data obtained using tadpole and non-perturbative\\ estimates for $c_{\\rm sw}$,\none can use further spectrum data obtained using\n$c_{\\rm sw}=0$\\,\\cite{GF11_spec} and $c_{\\rm sw}=1$\\,\\cite{strange}. In\nFig.\\,\\ref{Fig:mK_vs_csw} we plot $m_{\\rm V}\/\\sqrt{\\sigma}$ at $m_{\\rm PS}=m_K$ versus\n$c_{\\rm sw}$. From the plot one infers that the condition (\\ref{EQ:phenimp})\nis satisfied within statistical errors if\n\\begin{equation}\n c_{\\rm sw}\\,\\,\\raisebox{-.6ex}{$\\stackrel{\\textstyle{>}}{\\sim}$}\\,\\, c_{\\rm sw}^{\\rm tadpole}\\equiv\n u_0^{-3},\\qquad\\beta\\,\\,\\raisebox{-.6ex}{$\\stackrel{\\textstyle{>}}{\\sim}$}\\,\\,6.0,\n\\label{EQ:cswtad}\n\\end{equation}\nwhere $u_0$ denotes the average link variable. Thus, with the present\nstatistical accuracy, already the tadpole improved estimate of $c_{\\rm sw}$\nleads to a large reduction of O($a$) effects, provided the lattice\nspacing is not too large. Fig.\\,\\ref{Fig:mK_vs_csw} also shows that in\norder to satisfy (\\ref{EQ:phenimp}) at $\\beta=5.7$ one would have to\ngo to much larger values of $c_{\\rm sw}$ than those implied by the tadpole\nprescription. This, however, appears impossible due to the appearance\nof exceptional configurations\\,\\cite{paperIII}.\n\n\\begin{figure}[tb]\n\\vspace{-1.0cm}\n\\ewxy{mK_vs_csw.eps}{90mm}\n\\vspace{-4.5cm}\n\\caption{$m_{\\rm V}\/\\sqrt{\\sigma}$ as a function of $c_{\\rm sw}$. The sets of\npoints from right to left have been obtained using non-perturbative, tadpole,\ntree-level and no improvement. Data at $\\beta=5.7$ are shown only for\nunimproved and tadpole improved actions. The dashed line shows the\nextrapolated GF11 data.}\n\\label{Fig:mK_vs_csw}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe characterization of clock performance by means of average measures such as Allan variance, Hadamard variance, Theo, and modified forms of such variances, is widely applied within the time and frequency community as well as by most clock and oscillator fabricators. Such variances are measured by comparing the times $t_k$ on a device under test, with the times at regular intervals $k\\tau_0$ on a perfect reference, or at least on a better reference. Imperfections in performance of the clock under test are studied by analyzing noise in the time deviation sequence $x_k=t_k-k\\tau_0$, or the fractional frequency difference during the sampling interval $\\tau=s\\tau_0$:\n\\be\n\\Delta_{k,s}^{(1)}=(x_{k+s}-x_k)\/(s\\tau_0).\n\\ee\nThe frequency spectrum of fractional frequency differences can usually be adequately characterized by linear superposition of a small set of types of power law noise. The frequency spectrum of the fractional frequency differences of a particular noise type is given by a one-sided spectral density \n\\be\\label{onesided}\nS_y(f)=h_{\\alpha}f^{\\alpha},\\quad f>0.\n\\ee\n(The units of $S_y(f)$ are Hz$^{-1}$.) For the common power-law noise types, $\\alpha$ varies in integral steps from +2 down to -2 corresponding respectively to white phase modulation, flicker phase modulation, white frequency modulation, flicker frequency modulation, and random walk of frequency. \n\n\tSimulation of clock noise can be extremely useful in testing software algorithms that use various windowing functions and Fourier transform algorithms to extract spectral density and stability information from measured time deviations, and especially in predicting the probability for observing a particular value of some clock stability variance. This paper develops a simple simulation method for a time difference sequence that guarantees the spectral density will have some chosen average power law dependence. Expressions for the common variances and their modified forms are derived here that agree with expressions found in the literature, with some exceptions. This approach also leads to predictions of probabilities for observing a variance of a particular type at particular values of the sampling time. A broad class of probability functions naturally arises. These only rarely correspond to chi-squared distributions.\n\n\tThis paper is organized as follows. Sect. 2 introduces the basic simulation method, and Sect 3 applies the method to the overlapping Allan variance. Sect. 4 shows how diagonalization of the averaged squared second-difference operator, applied to the simulated time series, leads to expressions for the probability of observing a value of the variance for some chosen value of the sampling or averaging time. Expressions for the mean squared deviation of the mean of the variance itself are derived in Sect. 6. The approach is used to discuss the modified Allan variance in Sect. 7, and the non-overlapping form of the Allan variance is treated in Sect. 8. Appendix 1 discusses evaluation of a contour integral for the probability, first introduced in Sect. 4, in the general case.\n\n\\section{Discrete Time Series}\n\n\tWe imagine the noise amplitudes at Fourier frequencies $f_m$ are generated by a set of $N$ normally distributed random complex numbers $w_n$ having mean zero and variance $\\sigma$, that would by themselves generate a simulated spectrum for white phase noise. These random numbers are divided by a function of the frequency, $\\vert f_m \\vert^{\\lambda}$, producing a spectral density that has the desired frequency characteristics. For ordinary power law noise, the exponent $\\lambda$ is a multiple of $1\/2$, but it could be anything. The frequency noise is then transformed to the time domain, producing a time series with the statistical properties of the selected power law noise. The Allan variance, Modified Allan variance, Hadamard Variance, variances with dead time, and other quantities of interest can be calculated using either the frequency noise or the time series. \n\n\tIn the present paper we discuss applications to calculation of various versions of the Allan variance. Of considerable interest are results for the probability of observing a value of the Allan variance for particular values of the sampling time $\\tau$ and time series length $N$. The derivations in this paper are theoretical predictions. A natural frequency cutoff occurs at $f_h=1\/(2 \\tau_0)$, where $\\tau_0$ is the time between successive time deviations. This number is not necessarily related in an obvious way to some hardware bandwidth. The measurements are assumed to be made at the times $k\\tau_0$, and the time errors or residuals relative to the reference clock are denoted by $x_k$. The averaging or sampling time is denoted by $\\tau=s\\tau_0$, where $s$ is an integer. The total length of time of the entire measurement series is $T=N\\tau_0$. The possible frequencies that occur in the Fourier transform of the time residuals are\n\\be\nf_m=\\frac{m}{N\\tau_0}\\,.\\quad\\quad -\\frac{N}{2}+1 \\le m \\le \\frac{N}{2}\\,.\n\\ee \n\n\t{\\it Noise Sequences}\\\/. In order that a set of noise amplitudes in the frequency domain represent a real series in the time domain, the amplitudes must satisfy the reality condition\n\\be\nw_{-m}=(w_m)^{\\ast}.\n\\ee\n$N$ random numbers are placed in $N\/2$ real and $N\/2$ imaginary parts of the positive and negative frequency spectrum. Thus if $w_m=u_m+iv_m$ where $u_m$ and $v_m$ are independent uncorrelated random numbers, then $(w_m)^{\\ast}=u_m-iv_m$. Since the frequencies $\\pm 1\/2\\tau_0$ represent essentially the same contribution, $v_{N\/2}$ will not appear. We shall assume the variance of the noise amplitudes is such that\n\\be\\label{wmaverage}\n\\big<(w_m)^*w_n\\big>=\\big\\delta_{mn}=2\\sigma^2 \\delta_{mn};\\quad m\\ne 0,N\/2.\n\\ee \nAlso, $==0 {\\rm\\ for\\ }m\\ne N\/2.$\nThe index $m$ runs from $-N\/2+1$ to $N\/2$. In order to avoid division by zero, we shall always assume that the Fourier amplitude corresponding to zero frequency vanishes. This only means that the average of the time residuals in the time series will be zero, and has no effect on any variance that involves time differences.\n\n\tWe perform a discrete Fourier transform of the frequency noise and obtain the amplitude of the $k^{th}$ member of the time series for white PM:\n\\be\nx_k=\\frac{\\tau_0^2}{\\sqrt{N}}\\sum_{m=-N\/2+1}^{N\/2} e^{-\\frac{2\\pi i m k}{N}}w_m.\n\\ee\nThe factor $\\tau_0^2$ is inserted so that the time series will have the physical dimensions of time if $w_m$ has the dimensions of frequency. We then multiply each frequency component by $\\vert f_0\/f_m \\vert^{\\lambda}$. This will generate the desired power-law form of the spectral density. The time series will be represented by\n\\be\nX_k=\\frac{\\tau_0^2}{\\sqrt{N}}\\sum_{m=-N\/2+1}^{N\/2}\\frac{\\vert f_0 \\vert^{\\lambda}}{\\vert f_m \\vert^{\\lambda}} e^{-\\frac{2\\pi i m k}{N}}w_m.\n\\ee\nThe constant factor $\\vert f_0 \\vert^{\\lambda}$ has been inserted to maintain the physical units of the time series. The noise level is determined by $f_0$. For this to correspond to commonly used expressions for the one-sided spectral density, Eq. (\\ref{onesided}), we shall assume that\n\\be\n\\frac{\\tau_0^2\\vert f_0 \\vert^{\\lambda}}{\\sqrt{N}}=\\sqrt{\\frac{h_{\\alpha}}{16 \\pi^2 \\sigma^2 (N \\tau_0)}}\\,.\n\\ee\nWe shall show that if $2 \\lambda = 2-\\alpha$ the correct average spectral density is obtained. The simulated time series is \n\\be\\label{timeseries}\nX_k=\\sqrt{\\frac{h_{\\alpha}}{16 \\pi^2 \\sigma^2 (N \\tau_0)}}\n\\sum_m \\frac{e^{-\\frac{2\\pi i m k}{N}}}{\\vert f_m \\vert^{\\lambda}}w_m\\,.\n\\ee\nThe average (two-sided) spectral density of the time residuals is obtained from a single term in Eq. (\\ref{timeseries}):\n\\be\ns_x(f_m)=\\frac{h_{\\alpha}}{16 \\pi^2 \\sigma^2 (N \\tau_0) f_m^{2\\lambda}}\\bigg<\\frac{w_mw_m^*}{\\Delta f}\\bigg>=\\frac{h_{\\alpha}}{8\\pi^2 f_m^{2\\lambda}}\n\\ee\nwhere $\\Delta f = 1\/(N\\tau_0)$ is the spacing between successive allowed frequencies. The average (two-sided) spectral density of fractional frequency fluctuations is given by the well-known relation\n\\be\ns_y(f)=(2 \\pi f)^2 s_x(f)\\,,\n\\ee\nand the one-sided spectral density is\n\n\\be\nS_y(f)=\\cases{0, & $f < 0$;\\cr\n\t\t2 s_y(f)= h_{\\alpha}f^{\\alpha},& $f>0$\\,,\\cr}\n\\ee\nwhere $2 \\lambda = 2 -\\alpha$.\n\\section{Overlapping Allan Variance}\n\tConsider the second-difference operator defined by\n\\be\n\\Delta_{j,s}^{(2)}=\\frac{1}{\\sqrt{2 \\tau^2}}(X_{j+2s}-2X_{j+s}+X_j).\n\\ee\nThe fully overlapping Allan variance is formed by averaging the square of this quantity over all possible values of $j$ from 1 to $N-2s$. Thus\n\\be\\label{overlapvar}\n\\sigma_y^2(\\tau)=\\bigg<\\frac{1}{N-2s}\\sum_{j=1}^{N-2s}(\\Delta_{j,s}^{(2)})^2 \\bigg>.\n\\ee\nIn terms of the time series, Eq. (\\ref{timeseries}), the second difference can be reduced using elementary trigonometric identities:\n\\eject\n\\ba\\label{del2ave}\n\\Delta_{j,s}^{(2)}=\\sqrt{\\frac{h_{\\alpha}}{32 \\pi^2 \\tau^2 \\sigma^2 (N\\tau_0)}}\\hbox to 2.3in{}\\nonumber\\\\\n\\times \\sum_m\\frac{w_m}{\\vert f_m \\vert^{\\lambda}}\\bigg(e^{-\\frac{2 \\pi i m(j+2s)}{N}}-2e^{-\\frac{2 \\pi i m(j+s)}{N}}+e^{-\\frac{2 \\pi i m(j)}{N}}\\bigg)\\nonumber\\\\\n=-\\sqrt{\\frac{h_{\\alpha}}{2 \\pi^2 \\tau^2 \\sigma^2 (N \\tau_0)}}\\sum_m\\frac{w_m}{\\vert f_m \\vert^{\\lambda}}e^{-\\frac{2 \\pi i m(j+s)}{N}}\\bigg(\\sin\\frac{\\pi m s}{N}\\bigg)^2\\,.\n\\ea\nWe form the averaged square of $\\Delta_{j,s}^{(2)}$ by multiplying the real quantity times its complex conjugate, then averaging over all possible values of $j$.\n\\ba\n\\sigma_y^2(\\tau)=\\frac{h_{\\alpha}}{2 \\pi^2 \\tau^2 (N\\tau_0)(N-2s)}\n\\sum_{m,n,j}\\bigg<\\frac{w_m w_n^*}{\\sigma^2}\\bigg>\n\\frac{\\bigg(\\sin\\big(\\frac{\\pi m s}{N}\\big)\\sin\\big(\\frac{\\pi n s}{N}\\big)\\bigg)^2}{\\vert f_m f_n \\vert^{\\lambda}}\\nonumber\\\\\n\\hbox to -.5in{}\\times e^{-\\frac{2 \\pi i (m-n)(j+s)}{N}}\\,.\n\\ea\nThe average of the product of random variables only contributes $2 \\sigma^2$ when $m=n$ (see Eq. (\\ref{wmaverage}), except when $m=n=N\/2$ where $\\big<\\vert w_{N\/2} \\vert^2\\big>=\\sigma^2$.\nThe Allan variance reduces to\n\\be\n\\sigma_y^2(\\tau)=\\frac{h_{\\alpha}}{\\pi^2 \\tau^2(N\\tau_0)}\\bigg(\\sum_m \\frac{\\bigg(\\sin\\big(\\frac{\\pi m s}{N}\\big)\\bigg)^4}{\\vert f_m \\vert^{2\\lambda}} +\\frac{1}{2}\\frac{\\bigg(\\sin\\big(\\frac{\\pi s}{2}\\big)\\bigg)^4}{\\big(f_{N\/2}\\big)^{2\\lambda}}\\bigg)\\,,\n\\ee\nsince every term in the sum over $j$ contributes the same amount. The zero frequency term is excluded from the sum. For convenience we introduce the abbreviation\n\\be\nK=\\frac{2 h_{\\alpha}}{\\pi^2 \\tau^2 (N \\tau_0)}\\,. \n\\ee\nIf we sum over positive frequencies only, a factor of 2 comes in except for the most positive frequency and so\n\\be\n\\sigma_y^2(\\tau)=K\\bigg(\\sum_{m>0}^{N\/2-1}\\frac{\\bigg(\\sin\\big(\n\\frac{\\pi m s}{N}\\big)\\bigg)^4}\n{f_m^{2\\lambda}}+\\frac{1}{4}\\frac{\\bigg(\\sin\\frac{\\pi s}{2}\\bigg)^4}{(f_{N\/2})^{2\\lambda}} \\bigg).\n\\ee\nIf the frequencies are spaced densely enough to pass from the sum to an integral, then $\\Delta f= (N \\tau_0)^{-1}$ and\n\\be\n\\frac{1}{N\\tau_0}\\sum_mF(\\vert f_m \\vert)\\rightarrow\\int F(\\vert f \\vert)\\,df\n\\ee\nand we obtain the well-known result\\cite{barnes71}\n\\be\n\\sigma_y^2(\\tau)=2\\int_0^{f_h}\\frac{S_y(f)\\,df}{\\pi^2\\tau^2f^2}\\bigg(\\sin(\\pi f \\tau)\\bigg)^4\\,.\n\\ee\nSimilar arguments lead to known expressions for the non-overlapping version of the Allan variance as well as for the modified Allan variance. These will be discussed in later sections.\n\\section{Confidence Estimates}\n\tIn the present section we shall develop expressions for the probability of observing a particular value $A_o$ for the overlapping Allan variance in a single measurement, or in a single simulation run. $A_o$ is a random variable representing a possible value of the overlapping variance. We use a subscript ``o\" to denote the completely overlapping case. To save writing, we introduce the following abbreviations:\n\\ba\\label{FGdefs}\nF_m^j=\\frac{\\bigg(\\sin\\bigg(\\frac{\\pi m s}{N}\\bigg)\\bigg)^2}{\\vert f_m \\vert^{\\lambda}}\\cos\\bigg(\\frac{2\\pi m(j+s)}N{}\\bigg)\\nonumber\\\\\nG_m^j=\\frac{\\bigg(\\sin\\bigg(\\frac{\\pi m s}{N}\\bigg)\\bigg)^2}{\\vert f_m \\vert^{\\lambda}}\\sin\\bigg(\\frac{2\\pi m(j+s)}N{}\\bigg)\n\\ea\nThe dependence on $s$ is suppressed, but is to be understood. We write the second difference in terms of a sum over positive frequencies only, keeping in mind that the most positive and the most negative frequencies only contribute a single term since $\\sin(\\pi(j+s))=0$. The imaginary contributions cancel, and we obtain\n\\be\\label{diffFG}\n\\del2{j}=\\sqrt{K}\\sum_{m>0}\\big(F_m^j \\frac{u_m}{\\sigma}+G_m^j\\frac{v_m}{\\sigma}\\big)\\,.\n\\ee\nThere is no term in $v_{N\/2}$. It is easy to see that the overlapping Allan variance is given by\n\\be\\label{avarFG}\n\\sigma_y^2(\\tau)=\\frac{K}{N-2s}\\sum_j\\sum_{m>0}\\bigg((F_m^j)^2+(G_m^j)^2 \\bigg)\\,.\n\\ee\nTo compute the probability that a particular value $A_o$ is observed for the Allan variance, given all the possible values that the random variables $u_1,v_1,...u_{N\/2}$ can have, we form the integral\n\\be\n\\label{probdef}\nP(A_o)=\\int\\delta\\bigg(A_o-\\frac{1}{N-2s}\\sum_j \\big(\\Delta_{j,s}^{(2)}\\big)^2\\bigg)\\prod_{m>0}\\bigg(e^{-\\frac{u_m^2+v_m^2}{2\\sigma^2}}\\frac{du_m dv_m}{2\\pi \\sigma^2}\\bigg)\\,.\n\\ee\nThe delta function constrains the averaged second difference to the specific value $A_o$ while the random variables $u_1,v_1,...u_m,v_m,...u_{N\/2}$ range over their (normally distributed) values. There is no integral for $v_{N\/2}$. Inspecting this probability and the Eq. (\\ref{diffFG})for the second difference indicates that we can dispense with the factors of $\\sigma^{-1}$ and work with normally distributed random variables having variance unity. Henceforth we set $\\sigma=1$.\n\nThe exponent involving the random variables is a quadratic form that can be written in matrix form by introducing the $N-1$ dimension column vector $U$ (the zero frequency component is excluded)\n\\be\nU^T=[u_1\\,v_1\\,...u_m\\,v_m,...v_{N\/2-1},u_{N\/2}]\\,.\n\\ee\nThen\n\\be\n\\halb \\sum_{m>0}(u_m^2+v_m^2)=\\halb U^TU=\\halb U^T {\\bf 1} U,\n\\ee\nwhere ${\\bf 1}$ represents the unit matrix. The delta-function in Eq. (\\ref{probdef}) can be written in exponential form by introducing one of its well-known representations, an integral over all angular frequencies $\\omega$:\\cite{lighthill}\n\\be\nP(A_o)=\\int_{-\\infty}^{\\infty} \\frac{d \\omega}{2 \\pi}e^{i\\omega\\big(A_o-\\frac{1}{N-2s}\\sum_j \\big(\\Delta_{j,s}^{(2)}\\big)^2 \\big)}\\prod_{m>0}\\bigg(e^{-\\frac{u_m^2+v_m^2}{2\\sigma^2}}\\frac{du_m dv_m}{2\\pi \\sigma^2}\\bigg)\\,.\n\\ee\nThe contour of integration goes along the real axis in the complex $\\omega$ plane. \n\n\tThe squared second difference is a complicated quadratic form in the random variables $u_1,v_1,...u_m,v_m,...u_{N\/2}$. If this quadratic form could be diagonalized without materially changing the other quadratic terms in the exponent, then the integrals could be performed in spite of the imaginary factor $i$ in the exponent. To accomplish this we introduce a column vector $C^j$ that depends on $j,m,s,N$ and whose transpose is \n\\ba\n(C^j)^T=[F_1^j,G_1^j,...F_m^j,G_m^j,...G_{N\/2-1}^j,F_{N\/2}^j]\\,.\n\\ea\nDependence on $s$ is not written explicitly but is understood. The column vector has $N-1$ real components. It contains all the dependence of the second difference on frequency and on the particular power law noise. We use indices $\\{m,n\\}$ as matrix (frequency) indices. The (scalar) second difference operator can be written very compactly as a matrix product\n\\be\n\\del2{j}=\\sqrt{K} (C^j)^TU=\\sqrt{K}U^TC^j.\n\\ee\nThen \n\\be\n\\frac{1}{N-2s}\\sum_j\\bigg(\\Delta_{j,s}^{(2)} \\bigg)^2=U^T\\bigg(\\frac{K}{N-2s}\\sum_j C^j(C^j)^T\\bigg)U.\n\\ee\nThe matrix \n\\be\\label{hoverlapping}\nH_o=\\frac{K}{N-2s}\\sum_jC^j(C^j)^T\n\\ee\nis real and symmetric. $H_o$ is also Hermitian and therefore has real eigenvalues. A real symmetric matrix can be diagonalized by an orthogonal transformation,\\cite{Strang,Stoll} which we denote by $O$. Although we shall not need to determine this orthogonal transformation explicitly, it could be found by first finding the eigenvalues $\\epsilon$ and eigenvectors $\\psi_{\\epsilon}$ of $H_o$, by solving the equation \n\\be\nH_o\\psi_{\\epsilon}=\\epsilon \\psi_{\\epsilon}\\,.\n\\ee\nThe transformation $O$ is a matrix of dimension $(N-1)\\times (N-1)$ consisting of the components of the normalized eigenvectors placed in columns. Then\n\\be\nH_oO=OE\\,,\n\\ee\nwhere $E$ is a diagonal matrix with entries equal to the eigenvalues of the matrix $H_o$. Then since the transpose of an orthogonal matrix is the inverse of the matrix,\n\\be\nO^TH_oO=E\\,.\n\\ee\nThe matrix $H_o$ is thus diagonalized, at the cost of introducing a linear transformation of the random variables:\n\\be\n\\frac{K}{N-2s}\\sum_j\\big(\\del2{j}\\big)^2=U^TH_oU=U^TOO^TH_oOO^TU=(U^TO)E(O^TU)\\,.\n\\ee\nWe introduce $N-1$ new random variables by means of the transformation:\n\\be\nV=O^TU\\,.\n\\ee\nThen the term in the exponent representing the Gaussian distributions is\n\\be\n-\\halb U^T {\\bf 1 }U=-\\halb U^TO1O^TU=-\\halb V^T {\\bf 1}V=-\\halb \\sum_{n=1}^{N-1}V_n^2\\,.\n\\ee\nThe Gaussian distributions remain basically unchanged. \n\nFurther, the determinant of an orthogonal matrix is $\\pm 1$, because the transpose of the matrix is also the inverse:\n\\be\n\\det(O^{-1}O)=1=\\det(O^TO)=\\big(\\det(O)\\big)^2.\n\\ee\nTherefore, changes in the volume element are simple since the volume element for the new variables is\n\\ba\ndV_1 dV_2 ...dV_{N-1}=\\bigg|\\det \\bigg( \\frac{\\partial V_m}{\\partial U_n}\\bigg)\\bigg|dU_1dU_2...dU_{N-1}\\nonumber\\\\\n=\\vert \\det(O)\\vert dU_1 dU_2 ...dU_{N-1}\\nonumber\\\\ \n=dU_1 dU_2...dU_{N-1}.\n\\ea\nAfter completing the diagonalization, \n\\be\n\\frac{1}{N-2s}\\sum_j\\big(\\Delta_{j,s}^{(2)}\\big)^2=\\sum_i \\epsilon_i V_i^2\\,.\n\\ee\nThe probability is therefore\n\\be\nP(A_o)=\\int\\frac{d\\omega}{2\\pi}e^{i\\omega\\big(A_o-\\sum_k\\epsilon_k V_k^2\\big)}\\prod_i\\bigg( e^{-\\frac{V_i^2}{2}}\\frac{dV_i}{\\sqrt{2\\pi}}\\bigg)\\,.\n\\ee\nAn eigenvalue of zero will not contribute in any way to this probability since the random variable corresponding to a zero eigenvalue just integrates out. \n\nLet the eigenvalue $\\epsilon_i$ have multiplicity $\\mu_i$, by which is meant that the eigenvalue $\\epsilon_i$ is repeated $\\mu_i$ times. Integration over the random variables then gives a useful form for the probability:\n\\be\\label{probint}\nP(A_o)=\\int_{-\\infty}^{+\\infty} \\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A_o}}{\\prod_i(1+2i\\epsilon_i \\omega)^{\\mu_i\/2}}\\,.\n\\ee\n\nFinally the contour integral may be deformed and closed in the upper half complex plane where it encloses the singularities of the integrand. This is discussed in Appendix 1. Knowing the probability, one may integrate with respect to the variance to find the cumulative distribution, and then find the limits corresponding to a 50\\% probability of observing the variance. \n\n\t{\\it Properties of the eigenvalues\\\/}. First, it is easily checked that the probability is correctly normalized by integrating over all $A_o$ and using properties of the delta-function:\n\\ba\n\\int P(A_o)dA_o=\\int_{-\\infty}^{+\\infty}\n\\frac{d \\omega}{2\\pi} \n\\frac{\\int e^{i\\omega A_o}dA_o}\n\t{\\prod_i(1+2 i \\epsilon_i \\omega)^{\\mu_i\/2}}\n=\\int_{-\\infty}^{+\\infty}\n\\frac{\\delta(\\omega) d\\omega }{\\prod_i(1+2 i \\epsilon_i \\omega)^{\\mu_i\/2}}\\nonumber\\\\\n=\\int_{-\\infty}^{+\\infty}d \\omega \\delta(\\omega)=1\\,.\n\\ea\nSecond, the eigenvalues are all either positive or zero. The eigenvalue equation for the eigenvector labeled by $\\epsilon$ is:\n\\be\n\\frac{K}{N-2s}\\sum_j C^j(C^j)^T\\psi_{\\epsilon}=\\epsilon \\psi_{\\epsilon}\\,.\n\\ee\nMultiply on the left by $\\psi_{\\epsilon}^T$; assuming the vector has been normalized, we obtain\n\\be\n\\epsilon=\\frac{K}{N-2s}\\sum_j\\bigg((C^j)^T\\psi_{\\epsilon}\\bigg)^2\\ge 0\\,.\n\\ee\nThus every eigenvalue must be positive or zero.\n\nNext let us calculate the trace of $H_o$. Since the trace is not changed by an orthogonal transformation,\n\\ba\n{\\rm Trace}(O^TH_oO)={\\rm Trace}(H_oOO^T)={\\rm Trace}(H_oOO^{-1})\\nonumber\\\\\n={\\rm Trace}(H_o)=\\sum_i \\epsilon_i\\,.\n\\ea\nThe sum of the diagonal elements of $H_o$ equals the sum of the eigenvalues of $H_o$. If we then explicitly evaluate the sum of the diagonal elements of $H_o$ we find\n\\ba\n\\sum_i \\epsilon_i=\\frac{K}{N-2s}\\sum_j{\\rm Trace}\\big(C^j (C^j)^T)\\nonumber\\\\\n=\\frac{K}{N-2s}\\sum_j\\sum_{m>0}\\bigg((F_m^j\\big)^2+\\big(G_m^j)^2)\\bigg)\\nonumber\\\\\n=K\\sum_{m>0}\n\\frac{\\bigg(\\sin\\frac{\\pi m s}{N}\\bigg)^4}\n{\\vert f_m \\vert^{2-2\\alpha}}=\\sigma_y^2(\\tau)\\,.\n\\ea\nEvery term labeled by $j$ contributes the same amount. We obtain the useful result that {\\it the overlapping Allan variance is equal to the sum of the eigenvalues of the matrix $H_o$.} Similar results can be established for many of the other types of variances.\n\n {\\it Distribution of eigenvalues\\\/}. The eigenvalue equation $H_o\\psi_{\\mu}=\\epsilon \\psi_{\\mu}$ produces many zero eigenvalues, especially when $\\tau$ is large. The dimension of the matrix $H_o$ is therefore much larger than necessary. Numerical calculation indicates that for the completely overlapping Allan variance, the eigenvalue equation has a total of $N-1$ eigenvalues, but only $N-2s$ non-zero eigenvalues; the number of significant eigenvalues is in fact equal to the number of terms in the sum over $j$ in the equations:\n\\be\n\\frac{K}{N-2s}\\sum_n\\sum_j^{N-2s}(C_m)^j(C_n)^j\\psi_{n}=\\epsilon \\psi_m\\,.\n\\ee\nThe factorized form of $H_o$, that arises on squaring a difference operator, permits the reduction of the size of the matrix that is to be diagonalized. We introduce the quantities\n\\be\n\\phi_{\\mu}^j=\\sum_n(C_n)^j\\psi_{n\\mu}\\,.\n\\ee\nWe are using the Greek index $\\mu$ to label a non-zero eigenvalue and the index $\\nu$ to label a zero eigenvalue. The eigenvalue equation becomes\n\\be\n\\frac{K}{N-2s} \\sum_j(C_m)^j\\phi_{\\mu}^j=\\epsilon \\psi_{m\\mu}\\,.\n\\ee\nMultiply by $(C_m)^l$ and sum over the frequency index $m$. Then\n\\be\\label{eigenoverlapJ}\n\\frac{K}{N-2s}\\sum_{m,j}(C_m)^l(C_m)^j\\phi_{\\mu}^j=\\epsilon\\phi_{\\mu}^l\\,.\n\\ee\nThis is an eigenvalue equation with reduced dimension $N-2s$ rather than $N-1$, since the number of possible values of $j$ is $N-2s$. The eigenvalue equation can be written in terms of a reduced matrix $H_{red}$, given by\n\\be\n(H_{red})^{lj}=\\frac{K}{N-2s}\\sum_m(C_m)^l(C_m)^j\\,.\n\\ee\nThe indices $l,j$ run from 1 to $N-2s$. Eigenvalues generated by Eq. (\\ref{eigenoverlapJ}) are all non-zero. To prove this, multiply Eq. (\\ref{eigenoverlapJ}) by $\\phi_{\\mu}^l$ and sum over $l$. We obtain\n\\be\n\\frac{K}{N-2s}\\sum_m\\bigg(\\sum_l (C_m)^l\\phi_{\\mu}^l \\bigg)^2=\\epsilon\\sum_l\\big(\\phi_{\\mu}^l \\big)^2\\,.\n\\ee\nThe eigenvalue cannot be zero unless\n\\be\\label{zerocondition}\n\\sum_l (C_m)^l\\phi_{\\mu}^l =0\n\\ee\nfor every $m$. The number of such conditions however is larger than the number $N-2s$ of variables, so the only way this can be satisfied is if $\\phi_{\\mu}^l=0$, a trivial solution. Therefore to obtain normalizable eigenvectors from Eq. (\\ref{eigenoverlapJ}), the corresponding eigenvalues must all be positive.\n This is true even through some of these conditions may be trivially satisfied if the factor $\\sin(\\pi m s\/N)$ vanishes, which happens sometimes when\n\\be\\label{lossequation}\nms=M N\n\\ee\nwhere M is an integer. Every time a solution of Eq. (\\ref{lossequation}) occurs, two equations relating components of $\\phi_{\\mu}^l$ are lost. Suppose there were $n$ solutions to Eq. (\\ref{lossequation}); then the number of conditions lost would be $2n$. The number of variables is $N-2s$ and the number of conditions left in Eq. (\\ref{zerocondition}) would be $N-1-2n$. The excess of conditions over variables is thus\n\\be\nN-1-2n-(N-2s)=2(s-n)-1\\,.\n\\ee\nIn Appendix 2 we prove that under all circumstances $2(s-n)-1>0$.\n\n\tWe temporarily drop the subscript $o$ since the remainder of the results in this section are valid for any of the variances. If the eigenvalues are found and the appropriate matrix is diagonalized, we may compute the probability for observing a value of the overlapping variance, denoted by the random variable $A$, by\n\n\\ba\nP(A)=\\int_{-\\infty}^{\\infty}\\frac{d\\omega}{2\\pi}e^{i\\omega\\big(A-V^TEV\\big)}\\prod_i\\bigg(\\frac{e^{-V_i^2\/2}dV_i}{\\sqrt{2\\pi}}\\bigg)\\nonumber\\\\\n=\\int\\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A}}{\\prod(1+2i\\epsilon_i \\omega)^{\\mu_i\/2}}\\,.\n\\ea\n{\\it Case of a single eigenvalue\\\/}. If a single eigenvalue occurs once only, the general probability expression, Eq. (\\ref{probint}), has a single factor in the denominator:\n\\be\nP(A)=\\frac{1}{2\\pi}\\int_{-\\infty}^{\\infty}\\frac{d \\omega e^{i\\omega A}}{\\sqrt{1+2i\\omega\\epsilon}}\\,.\n\\ee\nThe integral is performed by closing the contour in the upper half complex $\\omega$ plane. There is a branch point on the imaginary axis at $\\omega=i\/(2\\epsilon)$, and a branch line from that point to infinity. Evaluation of the integral gives:\n\\be\\label{probmod}\nP(A)=\\frac{1}{\\sqrt{2\\pi\\sigma_y^2(\\tau)}}\\frac{e^{-A\/(2 \\sigma_y^2(\\tau))}}{\\sqrt{A}}\n\\ee\nThis is a chi-squared distribution with exactly one degree of freedom. The computation of the confidence interval for a given $s$ is simple. The cumulative probability obtained from Eq. (\\ref{probmod}) is\n\\be\n\\Phi(A)=\\int_0^{A}\\frac{1}{\\sqrt{2\\pi\\sigma_y^2(\\tau)}}\\frac{e^{-x\/(2 \\sigma_y^2(\\tau))}}{\\sqrt{x}}dx={\\rm erf}\\big(\\frac{A}{\\sqrt{2 \\sigma_y^2(\\tau)}} \\big)\\,.\n\\ee\nThe $\\pm 25$\\% limits on the probability of observing a value $A$ are then found to be $1.323 \\sigma_y^2$ and $0.1015 \\sigma_y^2$, respectively. An example of this is plotted in Figure 2. ($A$ is a variance, not a deviation.)\n\n{\\it Case of two distinct non-zero eigenvalues\\\/}. For the overlapping variance, when $s$ has its maximum value $N\/2-1$ there are two unequal eigenvalues. The probability integral can be performed by closing the contour in the upper half plane and gives the expression\n\\be\\label{twoeigenvalues}\nP(A)=\\frac{1}{2\\sqrt{\\epsilon_1 \\epsilon_2}}e^{-\\frac{A}{4}\\big(\\frac{1}{\\epsilon_1}+\\frac{1}{\\epsilon_2}\\big)}I_0\\bigg(\\frac{A}{4}\\big(\\frac{1}{\\epsilon_2}-\\frac{1}{\\epsilon_1} \\big) \\bigg)\\,.\n\\ee\nwhere $I_0$ is the modified Bessel function of order zero. The probability is correctly normalized. It differs from a chi-squared distribution in that the density does not have a singularity at $A=0$. This is illustrated in Figure 1. \n\nEvaluation of the contour integral when there are more than two distinct eigenvalues is discussed in Appendix 1. If an eigenvalue occurs an even number $2n$ times, the corresponding singularity becomes a pole of order $n$ and a chi-squared probability distribution may result; this has only been observed to occur for white PM.\n\n\n\\section{Variance of the Variance}\n\tOne of the goals of this investigation is to estimate the uncertainty in the variance when some value of the variance is measured. This can be rigorously defined if the probability $P(A_o)$ is available, but it may be difficult to evaluate the integral in Eq. (\\ref{probint}). In this case, one might be interested in a measure such as the rms value of the variance, measured with respect to the mean variance $\\sigma_y^2(\\tau)$. One method for obtaining this measure is obtained from the probability without introducing a Fourier integral representation for the delta function:\n\\ba\n\\int A_o^2 P(A_o)dA_o=\\int A_o^2\\delta(A_o-\\sum_i \\epsilon_i V_i^2)\n\\prod_m \\bigg(e^{-V_m^2\/2}\\frac{dV_m}{\\sqrt{2\\pi}} \\bigg)dA_o \\nonumber\\\\\n=\\int \\bigg(\\sum_i \\epsilon_i V_i^2 \\bigg)^2\\delta(A_o-\\sum_i \\epsilon_i V_i^2)\\prod_m \\bigg(e^{-V_m^2\/2}\\frac{dV_m}{\\sqrt{2\\pi}} \\bigg)dA_o \\nonumber\\\\\n=\\int \\bigg(\\sum_i \\epsilon_i V_i^2 \\bigg)^2\n\\prod_m \\bigg(e^{-V_m^2\/2}\\frac{dV_m}{\\sqrt{2\\pi}} \\bigg)\\,.\n\\ea\nExpanding the square in the integrand, there are $N$ fourth-order terms and $N(N-1)$ cross terms so we get\n\\ba\n\\int A_o^2 P(A_o)dA_o=\\sum_i \\epsilon_i^2 \\big< V^4 \\big> + 2 \\sum_{i \\ne j}\\epsilon_i\\epsilon_j \\big^2\\nonumber\\\\\n=3\\sum_i \\epsilon^2+2 \\sum_{i \\ne j} \\epsilon_i \\epsilon_j\\,.\n\\ea\nTo obtain the variance of the variance, we must subtract\n\\be\n\\big^2=\\sum_i \\epsilon_i^2+2 \\sum_{i \\ne j} \\epsilon_i \\epsilon_j\\,.\n\\ee\nThe result is\n\\be\\label{varofvar}\n\\big-\\big^2=2\\sum_i \\epsilon_i^2\\,.\n\\ee\nTherefore {\\it the rms deviation of the variance from the mean variance equals $\\sqrt{2}$ times the square root of the sum of the squared eigenvalues.} This result is not very useful if there are only a small number of eigenvalues, since the factor $\\sqrt{2}$ can make the rms deviation from the mean of the variance larger than the variance. This is because the probability distribution of the variance is not a normal distribution, and can lead to unreasonably large estimates of confidence intervals.\n\nIf the eigenvalues or the probabilities are not readily available, a similar confidence estimate can be obtained from an alternative form of Eq. (\\ref{varofvar}) by considering the trace of the square of $H_o$:\n\\be\\label{varofvar2}\n2\\sum_i\\epsilon_i^2=2\\sum_{i,j}\\big(\\sum_{m>0}(F_m^iF_m^j+G_m^iG_m^j) \\big)^2\\,.\n\\ee\nTo prove this, consider diagonalization of $H_o$ by the orthogonal transformation $O$, and compute\n\\ba\n2\\sum_i\\epsilon_i^2=2{\\rm Trace}(E^2)=2{\\rm Trace}(O^TH_oOO^TH_oO)=2{\\rm Trace}H_o^2\\nonumber\\\\\n=\\frac{2K^2}{(N-2s)^2}\\sum_{m,n}\\bigg(\\sum_i\\big((C_m)^i(C_m)^i\\big)\\sum_j\\big((C_n)^j(C_n)^j\\big) \\bigg)\\nonumber\\\\\n=\\frac{2K^2}{(N-2s)^2}\\sum_{i,j}\\bigg(\\sum_m\\big((C_m)^i(C_m)^j\\big) \\sum_n\\big((C_n)^j(C_n)^i \\big) \\bigg)\\nonumber\\\\\n=\\frac{2K^2}{(N-2s)^2}\\sum_{i,j}\\bigg(\\sum_m\\big(F_m^iF_m^j+G_m^iG_m^j \\big) \\bigg)^2\\,.\n\\ea\nIf there are not too many terms in the sums over $i$ and $j$ then the sum over the frequency index in Eq. (\\ref{varofvar2}) becomes\n\\be\n\\sum_m\\big(F_m^iF_m^j+G_m^iG_m^j \\big)=\\sum_{m>0}\\frac{\\big(\\sin\\frac{\\pi m s}{N}\\big)^4}{f_m^{2\\lambda}}\\cos\\frac{2\\pi m(i-j)}{N}\\,.\n\\ee\nThis leads to some useful approximation schemes, but these will not be discussed further in this paper.\n\nUsually the variance of the variance is larger than the range of possible values of the variance computed from the $\\pm 25$\\% limits obtained from calculated probabilities. An example is given in Figure 1, where the confidence intervals are plotted for the non-overlapping variance for $N=1024$. For $s$ ranging between 342 and 511 there is only one eigenvalue, so the rms deviation of the variance from the variance is $\\sqrt{2}$ times the variance. This gives an upper limit on the confidence interval that is 1.414 times the variance, larger than that obtained from the actual probability distribution. The medium heavy lines in Figure 1 show the true $\\pm 25$\\% probability limits obtained from calculations such as Eq. (\\ref{probfunction})\n\\begin{figure}\\label{varsim}\n\\centering\n\\includegraphics[width=4.5 truein]{figure1.eps}\n\\caption{Comparisons of average non-overlapping Allan variance (light line) with rms deviations from the mean of the variance (heavy lines) and true $\\pm 25$\\% limits (medium heavy lines) for $N=1024$ data items in the time series. Results for the Allan variance for two independent simulation runs are plotted for comparison.}\n\\end{figure}\n\\section{Modified Allan Variance}\n\tThe modified Allan variance is defined so that averages over time are performed before squaring and averaging. We use a subscript $m$ to distinguish this form from the overlapping form of the variance. The definition is \n\\be\n\\sigma_m^2(\\tau)=\\bigg<\\bigg(\\frac{1}{s}\\sum_{j=1}^s \\del2{j} \\bigg)^2 \\bigg>\\,.\n\\ee\nSumming the expression for the second-difference given in Eq. (\\ref{del2ave}),\n\\ba\n\\frac{1}{s}\\sum_j^s \\del2{j} =-\\sqrt{\\frac{h_{\\alpha}}{2 s^2 \\pi^2 \\tau^2 \\sigma^2(N\\tau_0)}}\\sum_m\\frac{w_m}{\\vert f_m\\vert^{\\lambda}}\\big(\\sin\\frac{\\pi m s}{N} \\big)^2\\nonumber\\\\\n\\times\\bigg(\\frac{\\sin\\frac{\\pi m s}{N}}{\\sin \\frac{\\pi m}{N}} \\bigg)e^{-\\frac{\\pi i m (3s+1)}{N}}\\,.\n\\ea\nSquaring this, we write the complex conjugate of one factor and obtain\nthe ensemble average\n\\be\\\n\\sigma_m^2(\\tau)=\\frac{h_{\\alpha}}{2 s^2 \\pi^2 \\tau^2 \\sigma^2(N\\tau_0)}\n\\sum_{m,n}\\frac{\\big}{\\vert f_m f_n\\vert^{\\lambda}}\\frac{\\bigg(\\sin \\frac{\\pi m s}{N}\\bigg)^6}{\\big(\\sin \\frac{\\pi m}{N}\\big)^2}\\,.\n\\ee\nUsing Eq. (\\ref{wmaverage})and writing the result in terms of a sum over positive frequencies only,\n\\be\\label{modsigma}\n\\sigma_m^2(\\tau)=2\\sum_{m>0}\\frac{S_y(f_m)}{\\pi^2 s^2 \\tau^2 f_m^2 (N \\tau_0)}\n\\frac{\\bigg(\\sin \\frac{\\pi m s}{N}\\bigg)^6}{\\bigg(\\sin\\frac{\\pi m}{N}\\bigg)^2}\\,.\n\\ee\nPassing to an integral when the frequency spacing is sufficiently small,\n\\be\n\\sigma_m^2(\\tau)=2\\int_0^{f_h}\\frac{df S_y(f)}{\\pi^2 s^2 \\tau^2 f^2} \n\\frac{\\big(\\sin( \\pi f \\tau)\\big)^6}{\\big(\\sin(\\pi f\\tau_0)\\big)^2}\\,.\n\\ee\nThis agrees with previously derived expressions\\cite{sullivanetal}. The appearance of the sine function in the denominator of this expression makes the explicit evaluation of the integral difficult. In Table 1 we give the leading contributions to the modified Allan variance for very large values of the averaging time $\\tau$. In general there are additional oscillatory contributions with small amplitudes that are customarily neglected. These results do not agree however with those published in\\cite{lesage84}; the leading terms do agree with those published in\\cite{howe99}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n\\vbox to .3in{}\nNoise Type&$S_y(f)$& Mod $\\sigma_y^2(\\tau)$&$\\lambda$&$\\alpha$\\\\ \n\\vbox to -.3in{}&&&& \\\\ \\hline\n\\vbox to .0in{}&&&&\\\\ \n White PM& $h_2 f^2$ &$\\mathstrut$ $ \\frac{3h_2}{\\pi^2 \\tau^3}\\big(1 +\\frac{5}{18 s^3}\\big)$&0&2\\\\ \n\\vbox to .03in{}&&&& \\\\ \\hline\n\\vbox to .03in{}&&&&\\\\\nFlicker PM& $h_1 f $& $\\frac{3h_1 \\ln(256\/27)}{8\\pi^2\\tau^2}$&$\\frac{1}{2}$&1 \\\\\n\\vbox to .03in{}&&&& \\\\ \\hline\n\\vbox to .03in{}&&&&\\\\\n White FM&$h_0$& $\\frac{h_0}{4\\tau}\\big(1+\\frac{1}{2 s^2}\\big)$&1& 0 \\\\ \n\\vbox to .02in{}&&&& \\\\ \\hline\n\\vbox to .02in{}&&&&\\\\\n Flicker FM&$ h_{-1}f^{-1}$&$ 2 h_{-1}\\ln\\big(\\frac{3^{27\/16}}{4}\\big)$&$\\frac{3}{2}$& -1 \\\\ \n\\vbox to .02in{}&&&& \\\\ \\hline\n\\vbox to .02in{}&&&&\\\\\nRandom Walk&$h_{-2}f^{-2}$&$h_{-2}\\pi^2 \\tau\\big(\\frac{11}{20}+\\frac{1}{12 s^2}+\\frac{1}{40s^4} \\big) $&2& -2 \\\\ \n\\vbox to .2in{}&&&& \\\\ \\hline\n\\end{tabular}\n\\caption{Asymptotic expressions for the Modified Allan Variance, in the limit of large sampling times $\\tau = s\\tau_0$.}\n\\end{center}\n\\end{table}\n\n{\\it Eigenvalue structure for modified Allan variance\\\/}.\n\tFor the modified case, instead of Eq. (\\ref{hoverlapping}) there are two summations; we use a subscript $m$ on $H$ to denote the modified form:\n\\be\nH_m=\\frac{K}{s^2}\\sum_j C^j\\sum_k(C^k)^T\\,.\n\\ee\nWe then seek solutions of the eigenvalue equation\n\\be\nH_m\\psi_{\\epsilon}=\\frac{K}{s^2}\\sum_j C^j\\sum_k( C^k)^T\\psi_{\\epsilon}=\\epsilon \\psi_{\\epsilon}\\,.\n\\ee\nMultiply by $\\sum_l(C^l)^T$ and sum over the frequency index. \n\\be\n\\frac{K}{s^2}\\sum_{l,j,m}\\big((C_m)^l)^T(C_m)^j\\big(\\sum_k(C^k)^T\\psi_{\\epsilon}\\big)=\\epsilon\\big(\\sum_l (C^l)^T \\psi_{\\epsilon}\\big)\\,.\n\\ee \nThe matrix equation has been reduced to a scalar equation for the quantity\n\\be\n\\phi=\\sum_k (C^k)^T\\psi_{\\epsilon}\\,.\n\\ee \nHere the matrix product of $(C^k)^T$ with $\\psi_{\\epsilon}$ entails a sum over all frequencies. Since we have a scalar eigenvalue equation for $\\epsilon$, there can be one and only one eigenvalue, which is easily seen to be the same as given by Eq. (\\ref{modsigma}) for each sampling time $\\tau $:\n\\be\n\\epsilon=\\frac{K}{s^2}\\sum_{l,k,m}\\big((C_m)^l\\big)^T (C_m)^k=\\sigma_m^2(\\tau)\\,.\n\\ee\nThe probability distribution will be of the form of Eq. (\\ref{probmod}), a chi-squared distribution with one degree of freedom.\n\n\\section{Non-overlapping Allan variance}\n\n\tIn the non-overlapping form of the Allan variance, the only change with respect to Eq. (\\ref{overlapvar}) is that the sum over $j$ goes from 1 in steps of $s$ up to $j_{max} \\le N-2s$. We denote the number of values of $j$ by $n_{max}$. The average non-overlapping variance is the same as that for the overlapping case, but the probability distributions are different. The matrix $H_{no}$ takes the form\n\\be\n(H_{no})_{mn}=\\frac{K}{n_{max}}\\sum_{j=1,1+2s..}^{j_{max}}(C_m)^j(C_n)^j\\,.\n\\ee\nThe subscript ``no\" labels a non-overlapping variance, and the indices $m$ and $n$ label frequencies. Here there are only $n_{max}$ terms in the sum since the values of $j$ skip by $s$. $n_{max}$ is given by:\n\\be\nn_{max}=\\lfloor \\frac{N-1}{s}\\rfloor -1\\,,\n\\ee\nwhere $\\lfloor Q \\rfloor $ denotes the largest integer less than or equal to $Q$. In order to diagonalize $H_{no}$ and compute probabilities, we look for eigenvalues by seeking solutions of:\n\\be\\label{eigenno}\n\\sum_n(H_{no})_{mn}\\psi_n=\\epsilon \\psi_m.\n\\ee\nWe define a reduced eigenvector by\n\\be\n\\phi_j=\\sum_n(C_n)^j\\psi_n.\n\\ee\nThe eigenvalue equation reduces to\n\\be\n\\frac{K}{n_{max}}\\sum_j(C_m)^j\\phi_j=\\epsilon \\psi_m.\n\\ee\nMultiply this equation by $(C_m)^l$ and sum over the frequency labels $m$. Then\n\\be\\label{eigennonoverlap}\n\\frac{K}{n_{max}}\\sum_j\\sum_m(\\big(C_m)^l(C_m)^j\\big) \\phi_j=\\epsilon \\phi_l\\,.\n\\ee\nLet\n\\be\nJ_{lj}=\\frac{K}{n_{max}}\\sum_m(C_m)^l (C_m)^j=\\frac{K}{n_{max}}(C^l)^T C^j;\n\\ee\nthe sum over frequencies is accomplished by the matrix multiplication.\nJust as for the overlapping case, if we calculate the trace of the matrix $J_{lj}$,\nwe find that since each term in the sum over $j$ contributes the same amount,\n\\be\n{\\rm Trace}(J_{lj})= \\frac{K}{n_{max}}\\sum_j C_j^TC_j=\\sigma_{no}^2(\\tau)\\,.\n\\ee\nBecause the trace remains unchanged under an orthogonal transformation, the non-overlapping Allan variance will be equal to the sum of the eigenvalues of Eq. (\\ref{eigennonoverlap}). The probability functions for a given $\\tau$ can differ from those for the overlapping or modified cases because the number of eigenvalues and their multiplicities may be different. The eigenvalues are still all greater than or equal to zero.\n\n\tIf the eigenvalues are found and the matrix $J_{lj}$ is diagonalized, we may compute the probability for observing a value of the overlapping variance, denoted by the random variable $A_{no}$, by\n\\ba\nP(A_{no})=\\int_{-\\infty}^{\\infty}\\frac{d\\omega}{2\\pi}e^{i\\omega\\big(A_{no}-U^TJU\\big)}\\prod_i\\frac{e^{-U_i^2\/2}}{\\sqrt{2\\pi}}\\nonumber\\\\\n=\\int\\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A_{no}}}{\\prod(1+2i\\epsilon_i \\omega)^{\\mu_i\/2}}\\,.\n\\ea\nFor example, when there are only two distinct non-zero eigenvalues, The matrix $J_{lk}$ will have elements\n\\ba\nJ_{1,1}=J_{1+s,1+s}=\\frac{1}{2} \\sigma_{no}^2;\\hbox to 1in{}\\nonumber\\\\\nJ_{1,1+s}=J_{1+s,1}=\\int_0^{f_h} \\frac{S_y(f) df }{(\\pi \\tau f)^2} \\big(\\sin(\\pi \\tau f)\\big)^4\\cos(2\\pi \\tau f)\\,.\n \\ea\nThe eigenvalues are then\n\\ba\n\\epsilon_1=\\frac{1}{2}\\sigma_{no}^2+ |J_{1,1+s}|\\nonumber\\\\\n\\epsilon_2=\\frac{1}{2}\\sigma_{no}^2- |J_{1,1+s}|\\,.\n\\ea\nThe probability is of the form of Eq. (\\ref{twoeigenvalues}).\n\\begin{figure}\n\\label{flickersim}\n\\centering\n\\includegraphics[width=5 truein]{figure2.eps}\n\\caption{Simulation of Allan variance with $N=64$, $s=19$. For this case there are two distinct eigenvalues in the matrix for the non-overlapping case. The variance for $s=19$ was extracted from each of 4000 independent runs and a histogram of the values obtained was constructed for comparison with the probability, Eq. (\\ref{twoeigenvalues}). Chi-squared distributions with 1, 2, and 3 degrees of freedom are plotted for comparison.} \n\\end{figure}\nFigure 1 shows an example of this for flicker FM noise, for $N=64$ items in the time series. The variance corresponding to $\\tau=19 \\tau_0$ was extracted from each of 4000 independent simulation runs and a histogram of the resulting values was plotted. Good agreement with the predicted probability distribution can be seen.\n\n\\section{Other variances}\n\n\tThe analysis methods developed in this paper can be extended to other variances. For example, the Theo variance\\cite{Howe03} is defined by an overlapping average,\n\\ba\\label{theo}\n\\sigma_{Th}^2(s,\\tau_0,N)=\\frac{1}{N-s}\\sum_{i=1}^{N-m}\\frac{4}{3\\tau^2}\\times \\hbox to 1.5in{}\\nonumber\\\\\n\\sum_{\\delta=0}^{s\/2-1}\\frac{1}{s\/2-\\delta}\\bigg((X_i-X_{i-\\delta+s\/2})- \n\t(X_{i+s}-X_{i+\\delta+s\/2})\\bigg)^2\\,.\n\\ea\nwhere $s$ is assumed to be even, and $\\tau=3s\\tau_0\/4$. For any of the power law noises, and after passing to an integral, this expression can be transformed with the aid of elementary trigonometric identities to\n\\be\n\\sigma_{Th}^2(\\tau,\\tau_0,N)=\\frac{8}{3}\\int_0^{f_h}\\frac{S_y(f)df}{(\\pi \\tau f)^2}\\sum_{\\kappa=1}^{2\\tau\/3\\tau_0}\\frac{1}{\\kappa}\\bigg(\\sin(\\pi f \\kappa \\tau_0)\\sin\\big(\\pi f (\\frac{4}{3}\\tau-\\kappa \\tau_0)\\big)\\bigg)^2\\,.\n\\ee\nThe integer $\\kappa$ in the denominator makes this variance more difficult to evaluate, but a factorized quadratic form can still be constructed and the eigenvalue structure and resulting probabilities can be analyzed.\n\nThe Hadamard variance is defined in terms of a third difference, and is widely used to characterize clock stability when clock drift is a significant issue\\cite{hutsell95}. A third difference operator may be defined as\n\\be\n\\Delta_{j,s}^{(3)}=\\frac{1}{\\sqrt{6\\tau^2}}\\big(X_{j+3s}-3X_{j+2s}+X_{j+s}-X_j \\big)\\,.\n\\ee\nThe completely overlapping Hadamard variance is the average of the square of this third difference:\n\\ba\n\\sigma_H^2(\\tau)=\\frac{1}{N-3s}\\sum_{j=1}^{N-3s}\\big(\\Delta_{j,s}^{(3)}\\big)^2\\nonumber\\\\\n\\rightarrow\\frac{8}{3}\\int_0^{f_h}\\frac{S_y(f)df}{(\\pi \\tau f)^2}\\big(\\sin(\\pi \\tau f)\\big)^6\\,.\n\\ea\n\n\tThese methods can be applied to cases in which there is dead time between measurements of average frequency during the sampling intervals. Suppose for example that the measurements consist of intervals of length $\\tau=s\\tau_0$ during which an average frequency is measured, separated by dead time intervals of length $D-\\tau$ during which no measurements are available. Let the index $j$ label the measurement intervals with $j=1,2,...N$. An appropriate variance can be defined in terms of the difference between the average frequency in the $j^{th}$ interval and that in the interval labeled by $j+r$: \n\\be\n\\Delta_{j,r,s}^{(2)}=\\frac{1}{\\sqrt{2}}\\big(\\overline y_{j+r,s}-\\overline y_{j,s}\\big)\\,,\n\\ee\nwhere $\\overline y_{j,s}$ is the average frequency in the interval $j$ of length $s \\tau_0$.\nThen an appropriate variance can be defined as\n\\be\n\\Psi(\\tau,D)=\\bigg<\\big(\\Delta_{j,r,s}^{(2)} \\big)^2 \\bigg>\\,.\n\\ee\nIf the measurements are sufficiently densely spaced that it is possible to pass to an integral, this can be shown to reduce to\n\\be\n\\Psi(\\tau,D)=\\frac{2}{D\\tau}\\int_0^{f_h}df \\frac{S_y(f)}{(\\pi f)^2}\\big(\\sin(\\pi f r D)\\big)^2 \\big( \\sin(\\pi f \\tau)\\big)^2\\,.\n\\ee\nWhen $D=\\tau$ and $r=1$ there is no real dead time and this variance reduces to the ordinary Allan variance. \n\n\\section{Summary and Conclusion}\n\n\tIn this paper a method of simulating time series for the common power-law noises has been developed and applied to several variances used to characterize clock stability. These include overlapping and non-overlapping forms of the Allan variance, and the Modified Allan variance. Diagonalization of quadratic forms for the average variances leads to expressions for the probabilities of observing particular values of the variance for a given sampling time $\\tau=s \\tau_0$. The probabilities are expressed in terms of integrals depending on the eigenvalues of matrices formed from squares of the second differences that are used to define the variance. Generally speaking, the number of eigenvalues is equal to the number of terms occurring in the sum used to define averages of the second-difference operator, and this number gets smaller as the sampling time $\\tau$ gets larger. The probability distribution $P(A)$ for some variance $A$ is useful in estimating the $\\pm25$\\% confidence interval about the average variance. The eigenvalues are usually distinct; only for white PM have eigenvalues been observed to occur with multiplicities other than unity. This must happen in order that chi-square probability distributions result. Methods for computing the probabilities have been presented in a few useful cases. Results presented in this paper are confined to various forms of the Allan variance.\n\n\tOther methods of simulating power law noise have been published; the present approach differs from that of\\cite{kasdinwalter} in that no causality condition is imposed. The application of the methods developed here will be applied to other variances in future papers. \n\n\\section{Appendix 1. Evaluation of contour integrals for probability\\\/}\n\n\tIn almost all cases except for white PM, the eigenvalues are all distinct. We show here how then the probability function, Eq. (\\ref{probint}), can be reduced to a sum of real integrals. For each of the eigenvalues, we introduce the quantity\n\\be\nr_k=1\/(2\\epsilon_k)\\,.\n\\ee \nThe integral becomes\n\\be\\label{contour2}\nP(A)=\\prod_k\\big(\\frac{\\sqrt{r_k}}{e^{\\pi i\/4}} \\big)\\int\\frac{d\\omega}{2\\pi}\\frac{e^{i\\omega A}}{\\prod_k(\\omega-ir_k)^{1\/2}}\\,.\n\\ee\t\nBy Jordan's lemma\\cite{smirnov}, the contour of integration can be deformed into the upper half plane. The addition of a circular arcs at radius $\\vert\\omega\\vert=\\infty$ contributes nothing. Each of the square root factors in Eq. (\\ref{contour2}) has a branch point at $\\omega=ir_k$ with a branch line extending to $+i\\infty$ along the imaginary axis. We define the complex argument of each such factor by\n\\be\n-\\frac{3\\pi}{2} < \\arg(\\omega-ir_k)<\\frac{\\pi}{2}\\,.\n\\ee\n\\begin{figure}\n\\label{contourplot}\n\\centering\n\\includegraphics[width=3. truein]{figure3.eps}\n\\caption{Contour deformed to run along branch line on the $y-$axis. A pair of segments is shown with an odd number of singularities below the segments. The branch line required by square root factors of the form $\\sqrt{\\omega-r_k}$ is defined by the angles--either $\\pi\/2$ or $-3\\pi\/2$--of the directed segments relative to the Re$(\\omega)$ axis. Contributions from pairs of segments that are above an even number of branch points cancel out.}\n\\end{figure}\n\n\nAll branch lines extend along the positive $y-$axis to infinity. The largest eigenvalues give singularities closest to the real axis. Fig. 3 illustrates the resulting contour. Around each branch point is a circular portion which contributes nothing because as the radius $\\delta$ of each circle approaches zero, the contribution to the integral approaches zero as $\\sqrt{\\delta}$. The integral then consists of straight segments where $\\omega = iy \\pm \\delta$, where $\\delta$ approaches zero. Two such straight segments are illustrated in Fig. 3, one on each side of the branch line along the $y-$axis. Suppose the interval of interest is placed so that out of a total of $M$ eigenvalues, $n$ of them are below the interval (three are shown in the figure). For the contribution to the integral on the left, $y$ is decreasing and we can account for this by reversing the limits and introducing a minus sign. The phase factor contributed by $n$ factors with branch points below the interval is\n\\be\n\\frac{1}{\\big(e^{-\\frac{3\\pi i}{4}}\\big)^n}=\\bigg(e^{\\frac{3\\pi i}{4}} \\bigg)^n\\,.\n\\ee\nThe contribution from branch points above the interval of interest is the same on both sides of the $y-$axis, and is\n\\be\n\\bigg(e^{\\frac{i\\pi}{4}}\\bigg)^{M-n}\\,.\n\\ee\nThe factor in front of the integral sign in Eq. (\\ref{contour2}) is the same on both sides of the $y-$axis and is\n\\be\n\\bigg(e^{-\\frac{i\\pi}{4}}\\bigg)^M\\,.\n\\ee\nThe total phase factor of this contribution, including a factor $i$ that comes from setting $\\omega=iy$ is thus\n\\be\n-i\\bigg(e^{-\\frac{i\\pi}{4}}\\bigg)^M\\bigg(e^{\\frac{i\\pi}{4}}\\bigg)^{M-n}\\bigg(e^{\\frac{3\\pi i}{4}} \\bigg)^n=-i\\bigg(e^{\\frac{\\pi i}{2}} \\bigg)^n\\,.\n\\ee\nFor the integral along the segment on the positive side of the $y-$axis, the only difference is that the phase of each of the $n$ contributions from branch points below the interval changes from $3\\pi\/4$ to $-\\pi\/4$. The phase factor for this part of the contour is thus\n\\be\n+i\\bigg(e^{-\\frac{i\\pi}{4}}\\bigg)^M\\bigg(e^{\\frac{i\\pi}{4}}\\bigg)^{M-n}\\bigg(e^{\\frac{-\\pi i}{4}} \\bigg)^n=+i\\bigg(e^{\\frac{-\\pi i}{2}} \\bigg)^n\\,.\n\\ee\nIf $n$ is even, the two contributions cancel. If $n=2m+1$ is odd, then the contributions add up with a factor $2(-1)^m$. The probability is thus always real and consists of contributions with alternating signs, with every other interval left out.\n\nIn summary, the contour integral contributions from portions of the imaginary axis in the complex $\\omega$ plane that have an even number of branch points below the interval will not contribute to the integral. For example, if there are four distinct eigenvalues the probability will reduce to \n\\ba\\label{probfunction}\nP(A)=\\frac{\\sqrt{r_1 r_2 r_3 r_4}}{\\pi}\\int_{r_1}^{r_2}\\frac{e^{-yA}dy}{\\sqrt{(y-r_1)(r_2-y)(r_3-y)(r_4-y)}}\\nonumber\\\\\n-\\frac{\\sqrt{r_1 r_2 r_3 r_4}}{\\pi}\\int_{r_3}^{r_4}\\frac{e^{-yA}dy}{\\sqrt{(y-r_1)(y-r_2)(y-r_3)(r_4-y)}}\\,.\n\\ea\nSuch results have been used to evaluate the probabilities for certain sampling intervals for flicker fm noise in Sect. 4.\n\n\\section{Appendix 2. Proof that Eq. (\\ref{eigenoverlapJ}) generates positive eigenvalues}\n\nIn this Appendix we show that under all circumstances Eq. (\\ref{lossequation}) gives rise to positive eigenvalues. \n\nObviously if $N$ is a prime number Eq. (\\ref{lossequation}) can never be satisfied ($n=0$). Consider the case $s=N-1$. Then $m=MN$ and there are no solutions since $m \\le N\/2$. In general both $m$ and $s$ may contain factors that divide $M$ or $N$. Suppose $s=ab$ where at least one of the factors $a,b$ is greater than 1, where $a$ divides $M$ and $b$ divides $N$. Then let\n\\be\nM=am_a,\\quad N=bn_b\\,.\n\\ee\nThen $m=m_a n_b$. $m=m_a n_b>N\/2=bn_b\/2$ cannot happen because $m \\le N\/2$.\nIf $m_a n_b=N\/2=b n_b$, then the number of solutions is $n=1$ and\n$2(s-n)-1=2(ab-1)-1 \\ge 2(2-1)-1>0$. If $m=m_a n_b < N\/2=b n_b\/2$, then there may be solutions $m=m_a n_b, m=2 m_a n_b, m=3m_a n_b,...$up to $b n_b\/2$. The number of such solutions is\n\\be\nn=\\lfloor \\frac{bn_b}{m_a n_b}\\rfloor=\\lfloor \\frac{b}{m_a}\\rfloor\n\\ee\nwhere $\\lfloor x \\rfloor$ means the largest integer less than or equal to $x$. The excess of conditions is then\n\\be\n2(s-n)-1=2(a b -\\lfloor \\frac{b}{m_a} \\rfloor) > 1\\,,\n\\ee\nwhich proves the assertion.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nIn traditional drug development, safety and efficacy of potential new drugs have been evaluated separately. A phase I trial first finds the maximum tolerated dose, the dose associated with some predetermined probability of observing a Dose-Limiting Toxicity (DLT). This dose is then carried forward to phase II, where efficacy is evaluated, with limited borrowing of information between the two evaluations of safety and efficacy. An alternative option is a seamless Phase I\/II trial where safety and efficacy are evaluated simultaneously, with the aim to find the Optimum Biological Dose (OBD). The main advantage of collecting information on both is the increase in chances of finding a dose that is both safe and efficacious, by allowing for more sharing of information. \\\\\n\nThere are multiple methods that have been proposed to design such a trial that range in complexity, for example the relatively simple model-assisted toxicity and efficacy interval design of Lin and Yin (STEIN, \\citep{Lin2017}), or more complex model-based designs that can jointly model the efficacy-dose and toxicity-dose relationship such as the utility contour of Thall and Cook \\citep{Thall2004}, or the approach based on toxicity and efficacy odds ratios by Lin et al. \\citep{Yin2006}, or the bivariate CRM for competing outcomes of Braun \\citep{Braun2002} amongst others. These methods differ in both their approaches to inference of the bivariate (or trinary) outcomes, and the decision criteria based on this inference. A standard trial setting assumes that as each cohort of patients enters the trial, responses from all previous cohorts are available to inform the decision of the next dose assignment. The next cohort is assigned the dose in order to collect more information at the current estimate of the OBD.\\\\\n\nHowever, it is not always the case that the complete outcomes of all previous cohorts are available before the next cohort is assigned. Particularly in Oncology, where there are multiple cycles of treatment, both efficacy and safety outcomes may have a delayed onset. There are a limited number of phase I designs that account for late onset toxicities, but an even more limited number of designs that account for late onset outcomes in both safety and efficacy. The main contributions to dose-finding trial designs incorporating late onset toxicities include the interval censored approach of Sinclair and Whitehead (ISCDP, \\citep{Sinclair2014}), the approach of including a cycle effect in a proportional odds mixed effects model by Dousseau et al. (POMM, \\citep{Doussau2013}) and the Time-to-Event version of the Continual Reassessment Method by Cheung and Chappell (TITE-CRM, \\citep{YingKuenCheung2000}). When including late-onset efficacy as well, there is a limited number of approaches, for example the method jointly modelling the time-to-event of efficacy and toxicity of Yuan and Yin ($A_T\/A_E$, \\citep{Yuan2009}), or the model assisted approach of Liu and Johnson \\citep{Liu2016}.\\\\\n\nThe setting of late onset toxicities and efficacies is challenging, hence the very few methods that exist for such a trial, yet extremely relevant when multiple cycles of treatment are given. Using one of the many available methods for binary outcomes will give a much longer trial duration, as all cycles of treatment must be completed before the next dose assignment.\\\\\n\nIn this paper we investigate a joint time to event CRM for the implementation of a phase I\/II trial design with delayed onset outcomes for both safety and efficacy, in comparison to the existing methods. In Section~\\ref{sec:methods} we introduce the methods and setting, presenting results of their application in Section \\ref{sec:results} before concluding with a discussion in Section~\\ref{sec:discussion}.\\\\\n\\section{Methods} \\label{sec:methods}\nWe consider the setting where there are $J$ dose levels with values $d_1, \\ldots , d_j, \\ldots , d_J$. Patients enter the trial in cohorts. Each cohort is assigned to a dose, and followed up for $\\tau$ cycles of treatment. A DLT may occur at any time during the follow up period, in which case the patient leaves the trial. The patient may also observe an efficacy response at any time during their follow up, which is censored if the patient observes a DLT.\\\\\n\nThe choice of dose to assign to the next cohort, and the final dose recommendation, are chosen based on the design. Here we outline the proposed Joint TITE-CRM design, the $AT\/AE$ design of Yuan \\& Yin \\citep{Yuan2009} and the model-assisted design of Liu \\& Johnson \\citep{Liu2016}. For comparability, in each case, the trial proceeds in the following way\\\\\n\\begin{enumerate}\n\\item The first cohort of patients is assigned to the lowest dose.\n\\item After one cycle of follow up, if no DLT is observed, escalate to next highest dose and continue after each cycle until a DLT is observed or the highest dose is reached.\n\\item The relevant models are fitted to the currently observed responses (which will be from one cycle of follow up for the last cohort, two cycles for the last but one cohort etc.) and posterior distributions are updated. \\label{item:pp}\n\\item A set of `admissible' doses are calculated based on the models fitted.\n\\item The best dose of the `admissible' set according to some criterion of the design is chosen to assign to the next cohort (subject to certain prespecified rules discussed in Section~\\ref{sec:rules}). \\label{item:crit}\n\\item After one cycle of follow up, return to step \\ref{item:pp}. The trial is stopped when one of the pre-specified stopping rules is triggered. \n\\end{enumerate}\n\nIt is worthwhile to note that the form of the prior and posterior in step \\ref{item:pp} and criterion in step \\ref{item:crit} are the only aspects unique to each design.\n\n\\subsection{Joint Model TITE-CRM} \\label{sec:j_titecrm}\nBased on the TITE-CRM \\citep{YingKuenCheung2000} and similar to the bCRM of Braun \\cite{Braun2002}, we modify the procedure to include both safety and efficacy, whilst keeping the structure of the TITE-CRM.\\\\\n\nWe use a two-parameter logistic model for both efficacy and toxicity outcomes:\n\\[\nF(d,\\mathbf{\\beta_E})= \\frac{exp(\\beta_{E,0} + \\beta_{E,1} d)}{1+exp(\\beta_{E,0} + \\beta_{E,1} d)},\n\\]\n\\[\nF(d,\\mathbf{\\beta_T})= \\frac{exp(\\beta_{T,0} + \\beta_{T,1} d)}{1+exp(\\beta_{T,0} + \\beta_{T,1} d)},\n\\]\n\nWe then must give weights to both efficacy and toxicity observations based on their follow ups, and a weighted dose response model is used for each:\n\\[ \n{G(d,w{(E)},\\beta_E) = w^{(E)} F(d,\\beta_E)},\n\\]\n\\[ \n{G(d,w{(T)},\\beta_T) = w^{(T)} F(d,\\beta_T)},\n\\]\nwhere {$0 \\leq w^{(E)}, w^{(T)} \\leq 1$} are functions of time-to-event of a patient response. We use $w^{(T)}_{i,n}=u_{i,n}\/\\tau$, where $u_{i,n}$ is the current number of cycles patient $i$ has been observed for, unless a DLT is observed in which case $w^{(T)}_{i,n}=1$. In a similar fashion, we use $w^{(E)}_{i,n}=u_{i,n}\/\\tau$, where $u_{i,n}$ is the current number of cycles patient $i$ has been observed for, unless a DLT is observed before an efficacy outcome can be observed in which case $w^{(E)}_{i,n}=(\\mbox{DLT time - entry time})\/\\tau$, and $w^{(E)}_{i,n}=1$ if an efficacy outcome is observed.\\\\\n\nWe consider all binary combinations of efficacy and toxicity outcomes (as opposed to trinary sometimes used in such applications), related by a Gumbel Model \\citep{Thall2004}\n\n\\begin{align*}\n\\pi_{a,b}= &(G(d,w{(E)},\\beta_E) )^a (1-G(d,w{(E)},\\beta_E) )^{1-a} (G(d,w{(T)},\\beta_T) )^b (1-G(d,w{(T)},\\beta_T) )^{1-b} + \\\\ &(-1)^{a+b} (G(d,w{(E)},\\beta_E) ) (1-G(d,w{(E)},\\beta_E) ) G(d,w{(T)},\\beta_T) (1-G(d,w{(T)},\\beta_T) ) \\left( \\frac{e^{\\psi}-1}{e^{\\psi}+1} \\right),\n\\end{align*}\n\n\nwhere $\\pi_{a,b}$ is the probability of observing the binary combination of efficacy outcome ($a=0$ for no efficacy observed, $a=1$ for efficacy observed) and toxicity outcome ($b=0$ for no toxicity observed, $b=1$ for toxicity observed). \\\\\n\nThe likelihood is then based on the categorical variable $\\pi_{a,b}$ which can be in one of four states,\n\n\n\\begin{equation} \\label{eq:LIK}\n\\mathcal{L}(\\beta) = \\prod_{i=1}^{n} \\prod_{a=0}^{1} \\prod_{b=0}^{1} \\{\\pi_{a,b}(d_{[i]},\\beta)\\}^{\\mathbb{I}(Y=(a,b))}.\n\\end{equation}\n\nPriors are elicited on $\\psi$, $\\beta_{E,0}$, $\\beta_{E,1}$, $\\beta_{T,0}$, $\\beta_{T,1}$, and the likelihood (\\ref{eq:LIK}) is used to update the joint posterior for all of the parameters using a Gibbs Sampler.The next dose is then chosen based on a utility function of $\\pi_E$ and $\\pi_T$. \\\\\n\nThe two main components of the design are the model (either dose-response, survival or assisting model) and the utility function. The Joint TITE-CRM and the Model-assisted method in our implementations use the same utility function, as a result of a sensitivity analysis conducted on this choice. \\\\\n\nWe vary the weights $w_1$ and $w_2$ in the linear utility function, and also consider the more complex utility contour method suggested by Thall and Cook \\citep{Thall2004}. \\\\\n\nIt was found that there was negligible difference in the outcomes for any of the alternative utilities. This led to use using the simple linear utility used by Liu \\& Johnson \\citep{Liu2016} in our implementation of the Joint TITE-CRM, with the same weights. \\\\\n\n\\begin{equation}\nU(\\pi_E, \\pi_T) = \\pi_E - \\omega_1 \\pi_T- \\omega_2 \\pi_T \\mathds{I}(\\pi_T > \\phi_T). \\label{eq:utility}\n\\end{equation}\n\nThis uses two weights, $\\omega_1$ and $\\omega_2$ and a toxicity threshold $\\phi_T$, above which we penalise doses that have a higher probability of DLT than this threshold. The next dose is chosen to maximise the utility out of a set of admissible doses.\\\\\n\nThis utility is simple to implement and to interpret. Using the same utility function for the two different designs also allows the results to be interpreted in terms of the inference used.\n\n\\subsection{Joint Model CRM} \\label{sec:j_crm}\nFor comparison, we also consider a non-time-to-event method. For this implementation, all $\\tau$ cycles must be observed from the previous cohort before the next cohort is assigned.\n\nThis is equivalent to setting $ w^{(E)}= w^{(T)}=1$ for each patient in the above Joint TITE-CRM, effectively removing the time-to-event element of the design.\n\n\n\\subsection{$A_T$\/$A_E$ Design} \\label{sec:ATAE}\nWe here give an overview of the $A_T$\/$A_E$ design, for further details we refer the reader to the original proposal by Yuan \\& Yin \\citep{Yuan2009}.\\\\\n\nThis design fits survival models to the time to event data for both efficacy and toxicity outcomes, assuming a Weibull distribution.\n\n\\[\nS_T(t|d) = \\exp\\{-\\lambda_T t^{\\alpha_T} \\exp (\\beta_T d) \\},\n\\]\n\n\\[\nS_E(t|d) = \\exp\\{-\\lambda_E t^{\\alpha_E} \\exp (\\beta_E d) \\},\n\\]\n\n\\[\nS^*_E(t|d) = 1-\\pi + \\pi S_E(t|d).\n\\]\n\nThe bivariate Time to Event data is then modelled as\n\n\\[\nS(t_T,t_E|d)= \\{S_T(t_T|d)^{-1\/\\phi} + S_E(t_E|d)^{-1\/\\phi} -1\\}^{-\\phi}.\n\\]\nIn order to compute the likelihood, define $y_T=\\min (t_T,c_T)$ and $\\Delta_T = \\mathbb{I}(t_T \\leq c_T)$ where $c_T$ is the censoring time, and define $y_E$ and $\\Delta_E$ similarly for efficacy. The likelihood is then\n\\[\nL(\\theta | data_i) = L_1^{\\Delta_T \\Delta_E} L_2^{\\Delta_T (1-\\Delta_E)} L_3^{(1-\\Delta_T) \\Delta_E} L_4^{(1-\\Delta_T) (1-\\Delta_E)},\n\\]\nwhere\n\\[\nL_1=\\pi \\frac{\\partial^2 S(y_T,y_E|d)}{\\partial y_T \\partial y_E},\n\\]\n\n\\[\nL_2=-(1-\\pi) \\frac{\\partial S_T(y_T|d)}{\\partial y_T} - \\pi \\frac{\\partial S(y_T,y_E|d)}{\\partial y_T},\n\\]\n\n\\[\nL_3= -\\pi \\frac{\\partial S(y_T,y_E|d)}{\\partial y_E},\n\\]\n\n\\[\nL_4=(1-\\pi) \\partial S_T(y_T|d) + \\pi S(y_T,y_E|d).\n\\]\n\n\n$\\lambda_E$, $\\alpha_E$, $\\beta_E$, $\\lambda_T$, $\\alpha_T$ and $\\beta_T$ are assigned independent gamma priors and $\\pi$ and $\\phi$ uniform priors. The joint posterior distribution is then updated using a Gibbs Sampler.\\\\\n\nThe criterion used for decision making is a ratio of the area under the curve (AUC) for the survival for toxicity and efficacy:\n\n\\[\n\\frac{A_T}{A_E}= \\frac{\\alpha_T^{-1}\\{\\lambda_T \\exp (\\beta_T d)\\}^{-1\/\\alpha_T} \\Gamma \\{ \\alpha_T^{-1},\\lambda_T \\exp (\\beta_T d) \\tau^{\\alpha_T} \\}}{(1-\\pi)\\tau + \\pi \\alpha_E^{-1}\\{\\lambda_E \\exp (\\beta_E d)\\}^{-1\/\\alpha_E} \\Gamma \\{ \\alpha_E^{-1},\\lambda_E \\exp (\\beta_E d) \\tau^{\\alpha_E} \\}},\n\\] \n\nwhere $\\Gamma (a,b)$ is the incomplete gamma function and $\\tau$ is the follow up time. The dose in the admissible dose set that maximises the $A_T$\/$A_E$ is chosen for the next cohort of patients.\n\\subsection{Model-Assisted}\nThe model-assisted method by Liu and Johnson \\citep{Liu2016} uses a Bayesian dynamic model as follows:\n\\begin{align*}\ny_{T,i}|d=d_j &\\sim \\mbox{Bern} (p_{T,j}) \\\\\np_{T,j} &= p_{T,j-1} + ( 1-p_{T,j-1} )\\beta_{T,j} , \\hspace{10pt} j=2,\\ldots J \\\\\np_{T,1} &= \\beta_{T,1} \\\\\n\\beta_{T,j} &\\sim \\mbox{Beta} (a_{T,j},b_{T,j}) \\hspace{10pt} j=1,\\ldots J.\n\\end{align*}\nThe subscript $T$ indicates toxicity, with the equivalent for efficacy labelled $E$. The specification of the $p_{T,j}$ ensures monotonicity, since there is no dose-response model.\\\\\n\nThe likelihood is given as \n\\[\nL(\\mathbf{y} | \\mathbf{\\beta_T},\\mathbf{\\beta_E}) = \\prod_{k=\\{E,T\\}} \\prod_{i=1}^{n} \\left\\{ 1- \\prod_{r=1}^{j[i]} (1-\\beta_{k,r}) \\right\\}^{y_{k,j}} \\prod_{r=1}^{j[i]} (1 - \\beta_{k,r})^{1-y_{k,j}},\n\\]\nwhich is used to update the posterior distribution. The same utility function as is used in the Joint TITE-CRM (Equation~\\ref{eq:utility}) is used to assign the next dose.\n\n\\subsection{Rules} \\label{sec:rules}\n\\subsubsection{Admissible Dose Set} \\label{sec:admis}\nThe admissible dose set is calculated as all doses satisfying the following criteria:\n\\[\n\\mathbb{P} (\\pi_T < \\pi_T^*) > q_T \n\\hspace{15pt} \\mbox{\\&} \\hspace{15pt} \n\\mathbb{P} (\\pi_E > \\pi_E^*) > q_E.\n\\]\nwhere $\\pi_T^*$ and $\\pi_E^*$ are the target probabilities for efficacy and toxicity in the entire follow-up $\\tau$.\\\\\n\nThese constraints ensure that for both designs, the escalation proceeds to a promising dose that is considered neither futile or unsafe.\n\\subsubsection{Stopping \\& Enforcement Rules} \\label{sec:stop_enf_rules}\nAs well as ensuring any assigned dose is admissible, we also apply a set of enforcement and stopping rules.\nFor any given dose $d_j$, $p_{1,d_j}$ is the $P(DLT)$ in the first cycle, $p_{\\tau,d_j}$ is the $P(DLT)$ in the full follow up of $\\tau$.\\\\\nEnforcement rules are to ensure the safety in the first cycle of treatment.\n\\textbf{\\textit{Enforcement Rules:}}\n\\begin{enumerate}\n\\item \\textbf{Hard Safety}: This rule ensures that if there is a very high probability that the toxicity of an experimented dose exceeds 0.3 in the first cycle, then that dose and all above are excluded from any further experimentation.(i.e. dose $d_j$ and all above are excluded when $P(p_{1,d_j}>0.3)>\\zeta$ for some threshold $\\zeta$). In this implementation we use a threshold for excessive toxicity of $\\zeta =0.95$, with a $Beta(1,1)$ prior for Binomial responses. This is equivalent to if there are at least 3 DLT responses out of 3 patients, at least 4 DLT responses out of 6 patients, or at least 5 DLT responses out of 9 patients, then all dose assignments must be strictly lower than that dose for the rest of the study. If the lowest dose is excluded then the trial stops with no dose recommendation made.\n\\item \\textbf{K-fold Skipping Doses}: The next dose assignment must be no more than a 2-fold-rise in the value of the highest experimented dose so far.\n\\end{enumerate}\n\nStopping rules are to define when we may stop the trial, either for futility\/safety issues or because we have a level of certainty about the estimated OBD.\n\n\\textbf{\\textit{Stopping Rules:}}\n\\begin{enumerate}\n\\item \\textbf{No Admissible Doses}: If no doses satisfy the two constraints in Section~\\ref{sec:admis}, then the trial is stopped, either for futility or safety.\n\\item \\textbf{Lowest Dose Deemed Unsafe}: If $P(p_{1,d_1}>30\\%)>0.80$ and at least one cohort of patients has been assigned to dose $d_1$, the trial is stopped. \\label{rule:low_unsafe}\n\\item \\textbf{Highest Dose Deemed Very Safe}: If $P(p_{1,d_J}\\leq 30\\%)>0.80$ and at least one cohort of patients has been assigned to dose $d_J$, the trial is stopped. \\label{rule:hi_safe}\n\\item \\textbf{Sufficient Information}: If a dose is recommended for the next cohort on which $C_{suff}$ cohorts have already been assigned in the escalation, excluding backfilled cohorts, the trial is stopped. \\label{rule:suff}\n\\item \\textbf{Precision}: If the safety and efficacy profile are both estimated precisely enough, the trial is stopped. This precision is defined as $CV(MTD)<30\\%$ and $CV(d_{[\\pi^*_E]})<30\\%$, with the coefficient of variation calculated as an adjusted median absolute deviation divided by the median and $d_{[\\pi^*_E]}$ is the dose associated with the target efficacy. Both of these must be satisfied before the stopping rule is enforced. This stopping rule is only used once at least $C_{suff}$ cohorts of patients have had at least one cycle of treatment in the escalation, on any dose. This rule is not applicable for the model-assisted method. \\label{rule:prec}\n\\item \\textbf{Hard Safety}: If the lowest dose is considered unsafe according to the hard safety enforcement rule, the trial is stopped.\n\\item \\textbf{Maximum Patients}: If the maximum number of patients ($n=n_{\\mbox{max}}$) have been recruited, the trial is stopped.\n\\end{enumerate}\n\nWe consider two settings; a realistic setting where all enforcement and stopping rules are applied, and a theoretical setting where stopping rules \\ref{rule:low_unsafe}, \\ref{rule:hi_safe}, \\ref{rule:suff}, \\ref{rule:prec} are not applied. This second setting is to investigate the behaviour of the designs without restrictions.\n\n\\section{Simulations} \\label{sec:sims}\nIn order to compare the joint TITE-CRM to both the $A_T\/A_E$ design and the model-assisted design, we conduct simulation studies in a range of scenarios.\n\n\\subsection{Setting} \\label{sec:setting}\nWe consider the setting of six doses to investigate: 1.5MBq , 2.5MBq , 3.5MBq , 4.5MBq , 6.0MBq , 7.0MBq, with a follow-up period of $\\tau=3$ cycles. Each cycle lasts 6 weeks, although for simplicity of time-to-event outcomes, we measure the time to event in units of cycles. The variables for the rules are $n_{max}=60$ and $C_{suff}=30$.\n\\\\\n\nWe use a combination of five safety scenarios and four efficacy scenarios, giving 20 scenarios in total. We then extend to consider different efficacy patterns in Section~\\ref{sec:eff_time_trends}. The five safety scenarios represent four scenarios where there is at least one safe dose and one where no doses are safe. We give the probability of DLT in cycle 1 and in the full follow up period of 3 cycles, with $\\pi_T^*=0.391$. It is assumed that the probability of DLT decreases by a factor of 1\/3 in subsequent cycles, conditional on survival through the previous cycles. In T.1, dose levels 4 and above are unsafe; in T.2, the highest dose is unsafe; in T.3, all doses are unsafe; in T.4, dose levels 2 and above are unsafe; and in T.5 all doses are well below target safety. The four efficacy scenarios are described by the probability of efficacy in the whole follow up period, with the lower bound on target efficacy as $\\pi_T^*=0.2$. In E.1, dose levels 2 and above are efficacious, with efficacy increasing with dose; in E.2, all dose levels are efficacious with a plateau reached at dose level 3; in E.3, no dose levels are efficacious; and in E.4, dose levels 4 and above are efficacious.\\\\\n\nIn the main implementations, it is assumed the probability of efficacy in the total follow up is three times the probability of efficacy in the first cycle. In Section~\\ref{sec:eff_time_trends}, we alter this assumption.\\\\\n\nScenarios are referred to as Ex.Ty where x is the efficacy scenario, y is the safety scenario. Table~\\ref{tab:scens} gives the individual safety and efficacy scenarios, while Table~\\ref{tab:scens_ut} gives the utility and $AT\/AE$ values for each of the 20 combination scenarios.\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{lllllll}\n \\hline\n\\hline\nSafety & 1.5MBq & 2.5MBq & 3.5MBq & 4.5MBq & 6.0MBq & 7.0MBq \\\\ \n \\hline\nT1 (cycle 1)& 0.100 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 \\\\ \nT1 (full follow up)& 0.140 & 0.270 & 0.391 & 0.503 & 0.606 & 0.701 \\\\ \n T2 (cycle 1)& 0.100 & 0.130 & 0.160 & 0.200 & 0.250 & 0.400 \\\\ \n T2 (full follow up) & 0.140 & 0.180 & 0.219 & 0.270 & 0.332 & 0.503 \\\\ \n T3 (cycle 1)& 0.400 & 0.450 & 0.500 & 0.550 & 0.600 & 0.650 \\\\ \n T3 (full follow up) & 0.503 & 0.556 & 0.606 & 0.655 & 0.701 & 0.746 \\\\ \n T4 (cycle 1)& 0.300 & 0.400 & 0.450 & 0.500 & 0.550 & 0.600 \\\\ \n T4 (full follow up)& 0.391 & 0.503 & 0.556 & 0.606 & 0.655 & 0.701 \\\\ \n T5 (cycle 1)&0.100 &0.120 &0.140 &0.160 &0.180 &0.200 \\\\ \n T5 (full follow up)&0.140 &0.166 &0.193 &0.219 &0.245& 0.270 \\\\ \n \\hline\n\\hline\nEfficacy & 1.5MBq & 2.5MBq & 3.5MBq & 4.5MBq & 6.0MBq & 7.0MBq \\\\ \n \\hline\nE1 & 0.200 & 0.300 & 0.400 & 0.500 & 0.600 & 0.700 \\\\ \n E2 & 0.300 & 0.400 & 0.500 & 0.500 & 0.500 & 0.500 \\\\ \n E3 & 0.100 & 0.120 & 0.140 & 0.160 & 0.180 & 0.200 \\\\ \n E4 & 0.100 & 0.150 & 0.200 & 0.300 & 0.500 & 0.700 \\\\ \n \\hline\n\\end{tabular}\n\\caption{Scenario definitions for efficacy and toxicity.}\n\\label{tab:scens}\n\\end{table}\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{rrrrrrr} \\hline\nScenario & 1.5MBq & 2.5MBq & 3.5MBq & 4.5MBq & 6.0MBq & 7.0MBq \\\\ \\hline \\hline\n E1.T1 Utility & \\textit{0.15} & \\textit{0.21} & \\textbf{0.27 }& -0.22 & -0.26 & -0.30 \\\\ \n E1.T1 $AT\/AE$ & \\textit{0.99} & \\textit{0.93} & \\textbf{0.86} & 0.79 & 0.71 & 0.61 \\\\ \n \\hline\nE1.T2 Utility & \\textit{0.15} & \\textit{0.24} & \\textit{0.33} & \\textit{0.41} & \\textbf{0.49} & -0.02 \\\\ \n E1.T2 $AT\/AE$ & \\textit{0.99} & \\textit{1.02} & \\textit{1.05} & \\textit{1.06} & \\textbf{1.07} & 0.92 \\\\ \n \\hline\nE1.T3 Utility & -0.52 & -0.49 & -0.46 & -0.43 & -0.40 & -0.36 \\\\ \n E1.T3 $AT\/AE$ & 0.65 & 0.63 & 0.61 & 0.59 & 0.57 & 0.54 \\\\ \n \\hline\nE1.T4 Utility & \\textit{0.07} & -0.42 & -0.39 & -0.36 & -0.33 & -0.30 \\\\ \n E1.T4 $AT\/AE$ & \\textit{0.76} & 0.69 & 0.67 & 0.66 & 0.64 & 0.61 \\\\ \n \\hline\nE1.T5 Utility & \\textit{0.15} & \\textit{0.25} & \\textit{0.34} & \\textit{0.43} & \\textit{0.52} & \\textit{0.61} \\\\ \n E1.T5 $AT\/AE$ & \\textit{0.99} & \\textit{1.03} & \\textit{1.07} & \\textit{1.12} & \\textit{1.17} & \\textit{1.23} \\\\ \n \\hline\nE2.T1 Utility & \\textit{0.25} & \\textit{0.31} & \\textbf{0.37} & -0.22 & -0.36 & -0.50 \\\\ \n E2.T1 $AT\/AE$ & \\textit{1.05} & \\textit{0.99} & \\textbf{0.92} & 0.79 & 0.66 & 0.53 \\\\ \n \\hline\nE2.T2 Utility & \\textit{0.25} & \\textit{0.34} &\\textbf{ 0.43} & 0.41 & 0.39 & -0.22 \\\\ \n E2.T2 $AT\/AE$ & \\textit{1.05} & \\textit{1.08} & \\textbf{1.12} & 1.06 & 0.99 & 0.79 \\\\ \n \\hline\nE2.T3 Utility & -0.42 & -0.39 & -0.36 & -0.43 & -0.50 & -0.56 \\\\ \n E2.T3 $AT\/AE$ & 0.69 & 0.67 & 0.66 & 0.59 & 0.53 & 0.46 \\\\ \n \\hline\nE2.T4 Utility & \\textbf{0.17} & -0.32 & -0.29 & -0.36 & -0.43 & -0.50 \\\\ \n E2.T4 $AT\/AE$ & \\textbf{0.81} & 0.74 & 0.72 & 0.66 & 0.59 & 0.53 \\\\ \n \\hline\nE2.T5 Utility & \\textit{0.25} & \\textit{0.35} & \\textit{0.44 }& \\textit{0.43} & \\textit{0.42} & \\textit{0.41} \\\\ \n E2.T5 $AT\/AE$ & \\textit{1.05} & \\textit{1.10} & \\textit{1.14} & \\textit{1.12} & \\textit{1.09} & \\textit{1.06} \\\\ \n \\hline\nE3.T1 Utility & 0.05 & 0.03 & 0.01 & -0.56 & -0.68 & -0.80 \\\\ \n E3.T1 $AT\/AE$ & 0.94 & 0.84 & 0.74 & 0.64 & 0.54 & 0.44 \\\\ \n \\hline\nE3.T2 Utility & 0.05 & 0.06 & 0.07 & 0.07 & 0.07 & -0.52 \\\\ \n E3.T2 $AT\/AE$ & 0.94 & 0.92 & 0.89 & 0.86 & 0.81 & 0.65 \\\\ \n \\hline\nE3.T3 Utility & -0.62 & -0.67 & -0.72 & -0.77 & -0.82 & -0.86 \\\\ \n E3.T3 $AT\/AE$ & 0.62 & 0.57 & 0.53 & 0.48 & 0.43 & 0.38 \\\\ \n \\hline\nE3.T4 Utility & -0.03 & -0.60 & -0.65 & -0.70 & -0.75 & -0.80 \\\\ \n E3.T4 $AT\/AE$ & 0.72 & 0.63 & 0.58 & 0.53 & 0.48 & 0.44 \\\\ \n \\hline\nE3.T5 Utility & 0.05 & 0.07 & 0.08 & 0.09 & 0.10 & \\textit{0.11} \\\\ \n E3.T5 $AT\/AE$ & 0.94 & 0.93 & 0.92 & 0.90 & 0.89 & \\textit{0.88} \\\\ \n \\hline\nE4.T1 Utility & 0.05 & 0.06 & \\textit{0.07} & -0.42 & -0.36 & -0.30 \\\\ \n E4.T1 $AT\/AE$ & 0.94 & 0.85 & \\textit{0.76} & 0.69 & 0.66 & 0.61 \\\\ \n \\hline\nE4.T2 Utility & 0.05 & 0.09 & \\textit{0.13} & \\textit{0.21} & \\textbf{0.39} & -0.02 \\\\ \n E4.T2 $AT\/AE$ & 0.94 & 0.93 & \\textit{0.92} & \\textit{0.93} & \\textbf{0.99} & 0.92 \\\\ \n \\hline\nE4.T3 Utility & -0.62 & -0.64 & -0.66 & -0.63 & -0.50 & -0.36 \\\\ \n E4.T3 $AT\/AE$ & 0.62 & 0.58 & 0.54 & 0.52 & 0.53 & 0.54 \\\\ \n \\hline\nE4.T4 Utility & -0.03 & -0.57 & -0.59 & -0.56 & -0.43 & -0.30 \\\\ \n E4.T4 $AT\/AE$ & 0.72 & 0.64 & 0.60 & 0.58 & 0.59 & 0.61 \\\\ \n \\hline\nE4.T5 Utility & 0.05 & 0.10 & \\textit{0.14} & \\textit{0.23} &\\textit{ 0.42} & \\textit{0.61} \\\\ \n E4.T5 $AT\/AE$ & 0.94 & 0.94 & \\textit{0.95} & \\textit{0.98} & \\textit{1.09} & \\textit{1.23} \\\\ \n \\hline\n\\end{tabular}\n\\caption{Utility for the twenty scenarios, with OBD highlighted in \\textbf{bold} and acceptable doses highlighted in \\textit{italics}. Note that the $AT\/AE$ criteria has no penalty for exceeding a safety threshold and hence may give higher values for unsafe doses.}\n \\label{tab:scens_ut}\n\\end{table}\n\nIn Table~\\ref{tab:scens_ut}, it can be seen that the utility criterion and $AT\/AE$ criterion do not always agree on the OBD. For example in scenarios E1.T1 and E2.T1, the 3.5MBq dose is the OBD according to the utility criterion, and the 1.5MBq dose is the OBD according to the $AT\/AE$ criterion. We have chosen to designate the 3.5MBq dose as the true OBD as this is more realistic, with a P(DLT) equal to target and a higher P(efficacy) than the lower dose. This does however highlight an issue concerning the $AT\/AE$ criteria, in that in maximizing the ratio between the two areas under the curves, it does not target any specific safety level. Hence when ratios are similar across doses, even when values are not, there may be difficulties in selection of the optimal dose.\n\n\\subsection{Data Generation} \\label{sec:data_gen}\nTo generate the event times for toxicities and efficacies, we use a model that is not based on the assumptions of any of the methods. We use a bivariate log-normal distribution for survival times, with parameters matched to the first cycle and full follow up probabilities of efficacy and toxicity, with a correlation parameter of -1\/2. This indicates that there is a negative association between the event times. This is a simple yet effective mechanism to generate data that allows for easy specification of probabilities at dose levels that do not themselves follow any specific parametric dose-response model.\\\\\n\\[\n\\begin{pmatrix}\nt_T \\\\\nt_E\n\\end{pmatrix} \\sim Lognormal_2 \\left( \\begin{pmatrix}\n\\mu_T\\\\\n\\mu_E\n\\end{pmatrix} , \\begin{pmatrix}\n\\sigma^2_T & -\\frac{\\sigma_T \\sigma_E}{2}\\\\\n-\\frac{\\sigma_T \\sigma_E}{2} & \\sigma^2_E\n\\end{pmatrix} \\right).\n\\]\n\n\n\\subsection{Prior distributions} \\label{sec:prior}\nSince all methods are Bayesian, we must specify prior distributions for the parameters of the models.\\\\\n\\subsubsection{Joint Model TITE-CRM \\& Joint Model CRM}\nHere we use priors\n\\[\n\\begin{pmatrix}\n\\beta_0 \\\\\n\\log (\\beta_1)\n\\end{pmatrix} \\sim N_2 \\left( \\begin{pmatrix}\nc_1\\\\\nc_2\n\\end{pmatrix} , \\begin{pmatrix}\nv_1 & 0 \\\\\n0 & v_2 \n\\end{pmatrix} \\right).\n\\]\nfor both toxicity and efficacy. Values of hyper-parameters used are:\n$c_{1,T}=\\log (1\/16) $, $c_{2,T}=\\log (1\/4)$, $v_{1,T}=1$, $v_{2,T}=2$ and $c_{1,E}= -3 $, $c_{2,E}=-0.2$, $v_{1,E}=1$, $v_{2,E}=1$. The values for toxicity are calibrated over a range of scenarios. The values for efficacy are chosen so that all doses have prior mean $(\\pi_E)>$ 0.3 and this increases with dose, with average effective prior sample size of one patient per dose level. We used a vague prior for $\\phi \\sim N(0,100)$.\n\n\\subsubsection{$A_T$\/$A_E$ Design}\nHere, we follow the priors implemented by Yuan \\& Yin \\citep{Yuan2009}, with some minor modification to fit our setting. $\\lambda_E$, $\\alpha_E$, $\\beta_E$, $\\lambda_T$, $\\alpha_T$ and $\\beta_T$ are assigned independent Gamma priors with shape and rate parameters of 0.1. These are constrained to be less than 3, 4, 1, 2, 3 and 1 respectively. The modification is of the upper bound of the $\\beta$ to account for the differing values of doses in our implementation to the original implementation. The uniform priors for $\\pi$ and $\\phi$ are $\\pi\\sim U(0.6,1)$ and $\\phi\\sim U(0,5)$.\n\\subsubsection{Model-Assisted}\nHere we use the priors suggested by Liu \\& Johnson \\cite{Liu2016} of $\\mathbf{a_E}=(0.2,0.3,0.4,0.45,0.5,0.6)$, $\\mathbf{b_E}=(0.95,0.9,0.8,0.75,0.7,0.65)$, $\\mathbf{a_T}=(0.05,0.1,0.2,0.25,0.3,0.35)$ and $\\mathbf{b_T}=(0.95,0.9,0.8,0.75,0.7,0.65)$.\n\n\\subsection{Results} \\label{sec:results}\nWe conducted 1000 simulations in each of the 20 scenarios. Figure~\\ref{fig:correct} shows the percentage of simulations that select the true OBD if there is one, or the trial is stopped correctly when no true OBD exists in the experimental dose set. The most noticeable feature is that the model-assisted method gives a lower proportion of the correct recommendations than the two model-based methods. In scenarios where all doses are too safe, the model-assisted method performs poorly, a reflection on the cautious escalation so often apparent in model-assisted designs. The overall performance of the two model based methods is similar, although with some differences in individual scenarios. For example the performance in E2.T4, where the lowest dose is the OBD, the Joint TITE-CRM method vastly outperforms the $AT\/AE$ design, which too often stops early for no admissible doses. \\\\\n\nInterestingly, the percentage of acceptable selections does not follow such a similar pattern. Figure~\\ref{fig:acceptable} displays these results. Here we define an `acceptable' selection as one that is both safe and efficacious, regardless if it the best in terms of utility criterion. The Joint TITE-CRM is the best performing overall. The Model-assisted method performs poorly for example in E3.T5, where the acceptable selections are either the highest dose or stopping because the highest dose is too safe.\\\\\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Correct_comparison_YYLJ_TTE_v4anoTITE.pdf}\n \\caption{Proportion of correct selections across scenarios} \\label{fig:correct}\n \\end{figure}\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Acceptable_comparison_YYLJ_TTE_v4anoTITE.pdf}\n \\caption{Proportion of acceptable selections across scenarios} \\label{fig:acceptable}\n \\end{figure}\n\n\nIn terms of sample size, the model assisted method has a higher sample sample size on average across scenarios, as shown in Figure~\\ref{fig:ssize}. The two model-based methods have very similar sample sizes.\\\\\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{SSize_comparison_YYLJall_TTEv4noTITE.pdf}\n \\caption{Mean sample size across scenarios} \\label{fig:ssize}\n \\end{figure}\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Duration_comparison_YYLJall_TTEv4noTITE.pdf}\n \\caption{Mean trial duration across scenarios} \\label{fig:dur}\n \\end{figure}\nHowever, although the patterns are similar between the two model-based methods in terms of total sample sizes, correct and acceptable selections, there is a difference in the number of patients treated at unsafe doses, with the $AT\/AE$ method assigning more patients to unsafe doses than both other methods in almost all scenarios. The model assisted method has much fewer patients assigned to unsafe doses, again due to the cautious escalation behaviour of model-assisted methods. \\\\\n\nAlthough the performance of the Joint CRM is similar to that of the Joint TITE-CRM, with slight improvement, this is at the cost of a much greater average trial length. Figure~\\ref{fig:dur} indicated this relationship, with an average trial duration of more than twice that of the Joint TITE-CRM. This is a sure indication of the need for trial designs that can use partial information on cohorts effectively.\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Unsafe_comparison_YYLJ_TTE_v4anoTITE.pdf}\n \\caption{Number of patients assigned to unsafe doses across scenarios} \\label{fig:unsafe}\n \\end{figure}\n\n\nThe overall performance of the three methods in the setting of realistic stopping and enforcement rules can be summarised as follows. The cautious escalation of the model-assisted method gives rise to a poorer performance in terms of selections of correct and acceptable doses, with larger sample sizes and fewer patients exposed to unsafe doses. The two model based designs perform better, with similar percentages of correct and acceptable recommendations, but with the $AT\/AE$ design assigning more patients to unsafe doses, showing a more aggressive escalation that is not rewarded with better selection performance. Additionally, the $AT\/AE$ design is substantially more complex and computationally intensive than both other methods, a cost that does not increase the performance.\\\\\n\nWhen the stopping rules are relaxed to investigate the operating characteristics of the designs more closely, note that since stopping rule \\ref{rule:hi_safe} is no longer enforced, there is no longer a correct selection in safety scenario 5 and so all methods give 0\\% correct selection. We see that in most scenarios where there is a true OBD (e.g. E1.T2, E2.T4, E4.T2) the Joint TITE-CRM gives a better performance of correct selections. When there are no admissible doses (e.g. E3.T1, E3.T2) the Joint TITE-CRM performs worse than the other two time-to-event methods. \n\nIt is noticeable that when considering acceptable selections, the Joint TITE-CRM performs the best out of the selection in most scenarios. In the absence of the early stopping rules, this provides evidence that the Joint TITE-CRM is benefiting from the model-based inference that the model-assisted method lacks. However, the $AT\/AE$ method also performs worse than the Joint TITE-CRM. This may be due to the deviations from assumptions on the survival model used in the inference and data generation.\n\nThe sample size is of course considerably larger when the stopping rules are relaxed, with many scenarios reaching nearly the maximum sample size on average. The Joint TITE-CRM has a larger sample size on average, which tallies up to the observation that this design had fewer correct stoppings for no admissible doses. This suggests that in general, this method is more willing to label a dose as admissible than the other methods.\\\\\n\nThe mean trial duration is also drastically increased for all methods, with the Joint CRM having an average duration of up to seven years, compared to the other methods giving less than three years.\\\\\n\nWith the relaxed stopping rules, the $AT\/AE$ method once again has a large number of patients assigned to unsafe doses, indicating a more aggressive escalation. The model-assisted method has much fewer patients assigned to unsafe doses, due to the more cautious escalation.\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Correct_comparison_YYLJ_TTE_v4anoTITEnoSTOP.pdf}\n \\caption{Proportion of correct selections across scenarios, lesser stopping} \\label{fig:correctnoSTOP}\n \\end{figure}\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Acceptable_comparison_YYLJ_TTE_v4anoTITEnoSTOP.pdf}\n \\caption{Proportion of acceptable selections across scenarios lesser stopping} \\label{fig:acceptablenoSTOP}\n \\end{figure}\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{SSize_comparison_YYLJall_TTEv4noTITEnoSTOP.pdf}\n \\caption{Mean sample size across scenarios lesser stopping} \\label{fig:ssizenoSTOP}\n \\end{figure}\n\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Duration_comparison_YYLJall_TTEv4noTITEnoSTOP.pdf}\n \\caption{Mean trial duration across scenarios lesser stopping} \\label{fig:durnoSTOP}\n \\end{figure}\n \n \n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Unsafe_comparison_YYLJ_TTE_v4anoTITEnoSTOP.pdf}\n \\caption{Number of patients assigned to unsafe doses across scenarios lesser stopping} \\label{fig:unsafenoSTOP}\n \\end{figure}\n\n\n\\subsection{Efficacy Time Trend} \\label{sec:eff_time_trends}\nIn the main implementations, the data generation of the efficacy times assumed that the probability of efficacy in the first cycle is one third of the probability of efficacy in the entire follow up of three cycles. However, it may be the case that this is actually higher or lower than one third.\\\\\n\nTo investigate the effect of efficacy occurring at various points in the follow up, we consider three efficacy patterns. In efficacy pattern 1, the probability of efficacy in cycle 1 is 1\/3 of the total probability; in efficacy pattern 2, it is 1\/6 and in efficacy pattern 3 in is 1\/2. This is to investigate the effect of varying time trends in efficacy.\\\\\n\nIt is possible that this change in efficacy time trend will affect the performance of the three methods that us the time-to-event outcomes, and so we investigate the difference that this can make in a selection of scenarios.\n\n \\begin{figure}\n \\includegraphics[width=1\\linewidth]{Eff_patternsALL.pdf}\n \\caption{Operating characteristics for the three efficacy time trends} \\label{fig:eff_trends}\n \\end{figure}\n\nFigure~\\ref{fig:eff_trends} shows the operating characteristics of the three time to event methods when the time trend is varied. There is no difference in the number of patients assigned to unsafe doses, and very little difference to the overall sample size. In terms of selections, the $AT\/AE$ methods of Yuan \\& Yin is most variable to the time trend, since this method is more heavily dependent on the survival curve. The Joint TITE-CRM and the model-assisted method of Liu \\& Johnson are more robust to efficacy time trends. \n\n\n\n\\section{Discussion} \\label{sec:discussion}\nIn this work, we have compared three designs for a dose-finding trial that uses both efficacy and toxicity outcomes in the form of time-to-event responses. The comparison has highlighted the impact of both the inference of dose-response relationship and the decision criteria. The challenge presented by a trial using both toxicity and efficacy outcomes is compounded by the late onset nature of the responses.\\\\\n\nThe decision criteria used in the $AT\/AE$ design, whilst intuitive for survival outcomes, does somewhat deviate from the objective of the dose-finding trial. Without penalty for unsafe doses, it gave an aggressive escalation path in many scenarios. This leads on from the point made regarding Table~\\ref{tab:scens_ut}, that without targeting any safety level, scenarios with similar true ratios across levels present very challenging for this design. Additionally, the increased computational intensity of this design is not rewarded by an increase in performance.\\\\\n\nThe model-assisted method and the Joint TITE-CRM implemented both used the same utility criterion, which allowed a comparison between the simple inference of Liu \\& Johnson \\cite{Liu2016} and the joint logistic model. Here the increase in complexity is rewarded with increased performance. The Joint TITE-CRM is recommended as an alternative that balances complexity and performance across a range of scenarios.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLight is a very peculiar form of energy that constantly travels from one point to another, which makes it difficult to store or `freeze' it in one place. This, however, can be effectively overcome by using optical resonators that utilize feedback mechanisms together with wave interference effects to recycle light along periodic trajectories. Depending on the resonator's geometry, these trajectories may intercept each other in opposite directions forming standing-wave patterns with a vanishing Poynting vector. Alternatively, they may form closed loops that support degenerate circulating traveling-wave modes (clockwise (CW) or counterclockwise (CCW)) with a non-vanishing Poynting vector along the loop direction. This ability to confine and trap light has enabled several scientific breakthroughs over the past few decades. Importantly, recent technological progress in micro- and nano-fabrication has enabled the realization of on-chip optical resonators with spatial dimensions comparable to the wavelength of the trapped light, or even smaller \\citep{Iwanaga-PR} with a wide range of applications including microlasers \\cite{Loncar2002APL,Wu2004APL,McCall1992APL,Sandoghdar1996PRA,Miao2016S,Zhang2020LSA,Peng2016PNAS} and sensing \\cite{Foreman2015AOP,Yalcin2006IEEE,Wiersig2016PRA,Chen2017N,Hodaei2017N,Zhong2019PRL}, just to mention few examples. Despite the large variety in their designs (microrings, microdisks, photonic crystals, Bragg structures, etc.), sizes, and material systems, optical resonators are typically classified into one of the aforementioned categories, i.e. standing- or traveling-wave devices \\cite{Vahala-OM,Vahala2003N,Saleh-FP}. This classification is generally accepted as complete. Thus, research in the field of optical resonators has focused on applications of these resonators and implementations of novel designs with unique features. In particular, standing-wave resonators can be engineered to support small mode volumes and high quality factors, which makes them perfect choice for engineering quantum light-matter interactions \\cite{John1990PRL,Chang2006PRL,Louyer2005PRA,Moreau2001APL,Vuckovic2003APL,Bennett2005APL,Strauf2007NPhot,Pelton2002PRL,Zhong2021PRR}. On the other hand, traveling-wave resonators are the preferred platform for (classical and quantum) nonlinear optics due to the ability to engineer the interaction between different wave components and the unidirectional propagation properties of these modes which facilitates the input and output coupling \\cite{Kues2019NPhot,Kippenberg2018S}. In addition, it was shown previously that the interaction between an atom and a standing-wave pattern of light depends on whether the latter is generated in a standing-wave resonator or as a result of interference between counter-propagating waves in a traveling-wave resonator \\cite{Shore1991JOSAB}.\n\nIn this work we show that this classification scheme for optical resonators (as standing- or traveling-wave resonators) is not complete. Instead, we reveal a new type of optical modes supported by certain resonator structures that represents a missing link between these two categories. Specifically, we propose a new resonator concept that supports an optical mode exhibiting hybrid standing- and traveling-wave patterns simultaneously.\n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=6.2in]{Fig-Schematic.png}\n\t\\caption{The concept of a hybrid-wave resonator. A resonator structure that supports two degenerate standing-wave modes such that (A) mode $M_1$ resides in domain $D_0 \\cup D_1$, and (B) mode $M_2$ resides in domain $D_0 \\cup D_2$. (C) It is possible that a proper linear superposition between $M_1$ and $M_2$ can result in new modes that preserve the standing-wave character in domains $D_{1,2}$ while at the same time form a traveling wave in domain $D_0$. In the figure, standing and traveling waves are schematically represented by interference fringes and uniform field distributions respectively.}\n\t\\label{Fig-Schematic}\n\\end{figure*}\n\nThis article is organized as follows. First, we introduce a general concept outlining the behavior of the proposed resonator without a reference to a particular structure. Afterwards, we discuss an implementation based on standard chip-scale photonics technology. To gain insight into the modal structure of the proposed resonator, we present a detailed analysis of its modal features using a scattering matrix approach. Next, we confirm our results by using full-wave finite element simulations. Finally, we investigate the effect of local perturbation due to a small scatterer on the eigenmodes of the proposed hybrid-wave resonator.\n\n\\section{General concept} \nWe start by considering a generic concept of optical resonators that supports modes overlapping only partially with the physical structure of the resonator. For simplicity, assume that such a resonator can be divided into three domains (extension to more domains is straightforward): $D_0$, $D_1$ and $D_2$ (the exact definition of the domain boundaries is not important). Furthermore, assume that it supports two degenerate standing-wave modes $M_{1,2}$ such that the field distribution of $M_1$ mainly resides in $D_0 \\cup D_1$, and similarly the field associated with mode $M_2$ resides in $D_0 \\cup D_2$ as shown schematically in Figure \\ref{Fig-Schematic}A and \\ref{Fig-Schematic}B respectively. The continuity conditions for the electromagnetic field across domain boundaries dictate that the field distributions associated with modes $M_{1,2}$ must be different across domain $D_0$. Let us now consider a general superposition of these two modes. It may be anticipated that, a new set of basis (owing to the degeneracy, there are infinitely many bases) can be constructed such that the standing-wave nature of modes $M_{1,2}$ in domains $D_{1,2}$ remain unaltered and yet the field distribution in $D_0$ forms a traveling-wave pattern due to a particular linear superposition of modes $M_{1,2}$ (Figure \\ref{Fig-Schematic}C), akin to the relation $\\cos(kz)+i \\sin(kz)=\\exp(ikz)$. Such a `mutant' resonator, if it exists, will support a `mutant' optical mode that, in some properly chosen basis, exhibits purely standing- and traveling-wave patterns at the same time---a feature that, to the best of our knowledge, has never been discussed before. We will refer to such a resonator as a hybrid-wave resonator. So far we have kept the discussion abstract. In what follows, we show that this abstract concept can be implemented in realistic optical resonator designs. In the main text, we focus on chip-scale devices but we note that it is straightforward to extend the discussion to implementations using free-space optics. \n\n\\section{Integrated photonics implementation} \n\nTo demonstrate that the concept discussed above can be realized by standard optical components, here we consider an implementation based on integrated photonics as shown in Figure \\ref{Fig-Resonator}A (possible realizations based on free-space optics are discussed in Appendix A). The structure consists of three open ring sections connected by two beam splitters (labeled as $\\text{BS}_{1,2}$). The outer rings here act as Sagnac loop reflectors \\cite{Li2019LPR}. We note that even though variants of this geometry have been considered before for building various optical devices for different applications such as sensing, lasing, and information processing \\cite{Simova2005JOSAB, Das2003OE, Parared2022APR,Shamy2022P,Moslehi1984IEEE}, the peculiar feature that we highlight in this work has escaped attention. \n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=5.4in]{Fig-Resonator.png}\n\t\\caption{An implementation of a hybrid-wave resonator in integrated photonic platforms. (A) A hybrid-wave resonator can be constructed by deforming a ring resonator to introduce two 50\/50 beam splitters, BS$_{1,2}$. The resonator can be divided into three domains: $D_0=D_0' \\cup D_0''$ is the union of the left and right side rings, and domains $D_1$ and $D_2$ represent the top and bottom middle sections, respectively. The field amplitudes at each location and their traveling directions are indicated on the figure and labeled as $a_i$ and $b_i$, $i=1,2,3,4$. (B) Resonant frequencies (horizontal axis) as a function of the beam splitter coupling coefficient $\\kappa$ (vertical axis). Note that each $\\kappa$ value represents an independent resonator structure. Horizontal dashed lines indicate values of $\\kappa$ for which the spectrum is doubly degenerate. The limit $\\kappa=0$ corresponds to conventional microring resonator with two degenerate traveling-wave modes in clockwise and counterclockwise direction while the limit $\\kappa=1$ corresponds to degenerate `knotted' modes. Hybrid-wave modes exist for $\\kappa=1\/\\sqrt{2}$ as discussed in detail in the main text.}\n\t\\label{Fig-Resonator}\n\\end{figure*}\n\nTo investigate the modal structure of this resonator, we will employ a scattering matrix analysis along the junctions indicated in Figure \\ref{Fig-Resonator}A. Away from the beam splitter junctions, the field amplitudes can be decomposed into two traveling waves in opposite directions as shown in Figure \\ref{Fig-Resonator}A. Within the context of scattering matrix formalism \\cite{Yariv2000EL,Van-OMR,Saleh-FP}, the relations between these amplitudes are given by:\n\\begin{equation}\\label{Eq-Matrix_equation}\n\t\\begin{aligned}\n\\relax [a_1, b_1]^T = S_c [a_4, b_2]^T, \\\\\n\t\t[a_4, b_2]^T = S_c [a_3, b_3]^T, \\\\\n\t\t[a_3, b_3]^T = S_c [a_2, b_4]^T, \\\\\n\t\t[a_2, b_4]^T = S_c [a_1, b_1]^T,\n\t\\end{aligned}\n\\end{equation} \nwhere $S_c=\\exp{(i \\phi\/4)} S_b$ and $S_b =\\begin{bmatrix}\n\t\\tau & i \\kappa\\\\\n\ti \\kappa & \\tau\n\\end{bmatrix}$ is the scattering matrix of each beam splitter. Here, $\\tau$ and $\\kappa$ are the field transmission and coupling coefficient of each beam splitter, and in the absence of any loss, they satisfy $\\tau^2+\\kappa^2=1$. For the special case of a 50\/50 beam splitter, which is relevant to our discussion later, $\\tau=\\kappa=1\/\\sqrt{2}$. The phase term $\\phi$ is defined as $\\phi=2\\pi n_\\text{eff}Lf\/c$, where $n_\\text{eff}$ is the effective guiding index, $L$ is total length across the perimeter of the resonator, $f$ is the frequency and $c$ is the speed of light in vacuum. The numerical factor $1\/4$ in the expression of $S_c$ arises because each wave component travels one quarter the length structure between any two consecutive junctions. In the absence of dispersion, $\\phi$ is a linear function of $f$ and thus can be used directly to determine the resonant frequencies (the effect of dispersion is considered later in the full-wave simulations). \n\n\n\\begin{table*}[!t]\n\t\\caption{Field components associated with the degenerate eigenmodes of the structure shown in Figure \\ref{Fig-Resonator} as expressed in the three different bases $B_{1,2,3}$. } \n\t\\centering\n\t\\begin{threeparttable}\n\t\\begin{tabular}{|p{0.5in}<{\\centering}|p{0.5in}<{\\centering}|p{0.5in}<{\\centering}|p{0.5in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|p{0.7in}<{\\centering}|}\n\t\t\\hline\n\t\\multicolumn{2}{|p{1.0in}<{\\centering}|}{Domains} & \\multicolumn{2}{c|}{$D_0'$} & \\multicolumn{2}{c|}{$D_1$} & \\multicolumn{2}{c|}{$D_0''$} & \\multicolumn{2}{c|}{$D_2$} \\\\ \\hline\n\t\t\n\t\\multicolumn{2}{|p{1.0in}<{\\centering}|}{Field components} & $a_1$ & $b_1$ & $a_2$ & $b_2$ & $a_3$ & $b_3$ & $a_4$ & $b_4$ \\\\ \\hline\n\t\t\n\t\\multirow{2}{0.2in}{$B_1$} &$M_1^{(1)}$ & $1$ & $1$ & $e^{i\\frac{\\phi_m+\\pi}{4}}$ & $ie^{i\\frac{3\\phi_m+\\pi}{4}}$ & $ie^{i\\frac{\\phi_m}{2}}$ & $ie^{i\\frac{\\phi_m}{2}}$ & $ie^{i\\frac{3\\phi_m+\\pi}{4}}$ & $e^{i\\frac{\\phi_m+\\pi}{4}}$ \\\\ \\cline{2-10}\n\t\t\n\t&$M_2^{(1)}$ & $1$ & $-1 $& $e^{i\\frac{\\phi_m-\\pi}{4}}$ & $e^{i\\frac{3\\phi_m+\\pi}{4}}$ & $-ie^{i\\frac{\\phi_m}{2}}$ & $ie^{i\\frac{\\phi_m}{2}}$ & $-e^{i\\frac{3\\phi_m+\\pi}{4}}$ & $-e^{i\\frac{\\phi_m-\\pi}{4}}$ \\\\ \\hline\n\t\t\n\t\\multirow{2}{0.2in}{$B_2$} &$M_1^{(2)}$ & $1$& $-i$ & $\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ & $i \\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & $e^{i\\frac{\\phi_m}{2}}$ & $i e^{i\\frac{\\phi_m}{2}}$ &\\cellcolor{zcolor} $0$ & \\cellcolor{zcolor}$0$ \\\\ \\hhline{|~|---------|}\n\t\t\n\t&$M_2^{(2)}$ & $1$ & $i$ &\\cellcolor{zcolor}$0$& \\cellcolor{zcolor} $0$ & $-e^{i\\frac{\\phi_m}{2}}$ & $i e^{i\\frac{\\phi_m}{2}}$ & $-\\sqrt{2}e^{i\\frac{3\\phi_m}{4}}$ & $i \\sqrt{2}e^{i\\frac{\\phi_m}{4}}$ \\\\ \\hline\n\t\t\n\t\\multirow{2}{0.2in}{$B_3$} &$M_1^{(3)}$ &\\cellcolor{tcolor} $2$ & \\cellcolor{tcolor} $0$ &$\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ & $i\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & \\cellcolor{tcolor} $0$ & \\cellcolor{tcolor} $2ie^{i\\frac{\\phi_m}{2}}$ & $-\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & $i\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ \\\\ \\hhline{|~|---------|}\n\t\t\n\t&$M_2^{(3)}$ & \\cellcolor{tcolor} $0$ & \\cellcolor{tcolor} $2$ & $i\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ & $-\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & \\cellcolor{tcolor} $2ie^{i\\frac{\\phi_m}{2}}$ &\\cellcolor{tcolor} $0$ & $i\\sqrt{2} e^{i\\frac{3\\phi_m}{4}}$ & $\\sqrt{2} e^{i\\frac{\\phi_m}{4}}$ \\\\ \\hline\n\t\\end{tabular}\n\t\\begin{tablenotes}\n \\item Note: In this table, pink cells indicate the regions with traveling waves while uncolored cells denote standing waves, and blue cells indicate the regions where the field vanishes. And $\\phi_m=(2m+1)\\pi$, where $m$ is an integer.\n \\end{tablenotes}\n\\end{threeparttable}\n\t\\label{Table1}\n\\end{table*}\n\nBy successive substitution of the right side of each line in Eq. (\\ref{Eq-Matrix_equation}) from the expression in the next line, we find:\n\\begin{equation} \\label{Eq-Consistency}\n\t[a_1,b_1]^T= S_c^4 [a_1,b_1]^T= e^{i \\phi} S_b^4 [a_1,b_1]^T.\n\\end{equation} \nThe resonant modes can then be obtained by imposing a consistency condition requiring the operator $ \\exp (i \\phi) S_b^4$ to have an eigenvalue equal to unity \\cite{Hecht-Optics}. In order to find the values of $\\phi$ that satisfy the consistency condition and hence obtain the eigenfrequencies, we first note that the eigenvalues of $S_b$ are given by $\\lambda_{1,2}=\\exp(\\pm i \\theta)$ and the corresponding eigenvectors are $\\vec{v}_{1,2}=[1,\\pm1]^T$. Here, $\\theta=\\arcsin(\\kappa) \\in [0,\\pi\/2]$ since $\\tau$ and $\\kappa$ take only positive values. The consistency condition then reduces to $\\exp(i\\phi) \\exp(\\pm i4\\theta)=1$, which has two solutions given by $\\phi_m^\\pm=\\pm 4\\theta+2m\\pi$, where $m$ is an integer. In order to better understand this result, we recall that in the absence of beam splitters ($\\kappa=0$) the eigenfrequencies are doubly degenerate (for each resonant frequency there are two modes, one propagating in the CW and the other in the CCW direction) and are given by $\\phi_m=2m\\pi$. Introducing the beam splitters results in coupling between CW and CCW modes and thus lifts the degeneracy. As a result, each degenerate pair described by $\\phi_m$ splits into two modes: blue-shifted $\\phi_m^+$ and red-shifted $\\phi_m^-$. Interestingly, for identical 50\/50 beam splitters, i.e. when $\\kappa=1\/\\sqrt{2}$ corresponding to $\\theta=\\pi\/4$, the eigenmodes associated with $S_b^4$ become degenerate (this is not the case for the eigenvectors of $S_b$) and hence the eigenmodes of the resonators form degenerate pairs satisfying the resonant conditions $\\phi_m^+=\\phi_{m+1}^-=(2m+1)\\pi \\equiv\\phi_m$, as shown in Figure \\ref{Fig-Resonator}B. Before we proceed, we emphasize that the above-predicted degeneracy is not a result of a particular geometric symmetry. For instance, the length of any of the curved sections in the four domains ( $D_0'$, $D_0''$, $D_1$ and $D_2$) can be increased by a multiple of the operation wavelength without affecting the degeneracy despite the fact that it will break part of the geometric symmetries of the structure. \n\nWe now investigate the eigenmode structure associated with these newly-formed degenerate modes. In principle, these eigenmodes can be expressed in any basis of the eigenvectors of $S_b^4$. Choosing a particular basis fixes the vector $[a_1,b_1]^T$ which can be then used to obtain all other field components through Eq. (\\ref{Eq-Matrix_equation}). Table \\ref{Table1} lists the field values for three different bases given by: (1) $B_1=\\{\\vec{v}_{1,2}\\}$, $\\vec{v}_{1,2}=[1,\\pm 1]^T$; (2) $B_2=\\{\\vec{v}_{3,4}\\}$, $\\vec{v}_{3,4}=[1,\\mp i]^T$; and (3) $B_3=\\{\\vec{v}_{5,6}\\}$, $\\vec{v}_{5}=[2,0]^T$, $\\vec{v}_{6}=[0,2]^T$. These bases are related via the linear transformations: $\\vec{v}_{3,4}=\\left[(1\\mp i)\\vec{v}_1+(1\\pm i)\\vec{v}_2 \\right]\/2$ and $\\vec{v}_{5,6}=\\vec{v}_1\\pm\\vec{v}_2$. Expressed differently, $\\vec{v}_{5,6}$ can be also written as $\\vec{v}_{5}=\\vec{v}_3 + \\vec{v}_4$ and $\\vec{v}_{6}=i(\\vec{v}_3 - \\vec{v}_4)$.\n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=6.5in]{Fig-Eigenmode.png}\n\t\\caption{Eigenmodes of the hybrid-wave resonator presented in Figure \\ref{Fig-Resonator}. (A) and (B) are plots of electric field component perpendicular to the resonator's plane ($|E_z|$) associated with the two degenerate standing-wave modes $M_{1,2}^{(2)}$ which resides in domain $D_0 \\cup D_1$ and $D_0 \\cup D_2$, respectively, where $D_0=D_0' \\cup D_0''$. These corresponds to basis $B_2$. On the other hand, (C) and (D) depict the field distribution corresponding to $M_{1}^{(3)}=M_1^{(2)} + M_2^{(2)}$ and $M_{2}^{(3)}=M_1^{(2)} - M_2^{(2)}$, which feature a hybrid standing- and traveling-wave character. The white arrows in (C) and (D) indicate the traveling direction of traveling wave. The Poynting vectors (black arrows in inset of (C)) circulate around the loop in domain $D_0$ and vanish in domain $D_{1,2}$. The details of the geometry and material parameters used in these simulations are presented in Appendix C.}\n\t\\label{Fig-Eigenmode}\n\\end{figure*}\n\nTable \\ref{Table1} lists the field components associated with the degenerate eigenmodes of the structure shown in Figure \\ref{Fig-Resonator} as expressed in the three different bases $B_{1,2,3}$. The modes are expressed by the vector $M_i^{(j)} \\equiv [a_1, b_1, a_2, b_2, a_3,$ $b_3, a_4, b_4]$ in each basis, where $i$ represents the mode number and $j$ denote the basis number. Note that there is a pure standing wave whenever the field components belonging to any domain have the same amplitude. On the other hand, if one of the field components vanishes, the wave is traveling. Evidently, in basis $B_3$ the eigenmodes exhibit a hybrid-wave character with both standing and traveling waves coexisting as part of the same mode. Obviously, modes $M_{1,2}^{(1)}$ represent a standing wave that extends all over the structure. On the other hand, modes $M_{1,2}^{(2)}$ represent a standing wave that covers only part of the structure. Thus modes $M_{1,2}^{(3)}$ represent potential candidates for satisfying the conditions necessary for generating hybrid-wave modes. Indeed, this is confirmed by the field distribution of modes $M_{1,2}^{(3)}$ which exhibits the dual character of traveling and standing waves covering different domains of the resonators at the same time.\n\nAn important observation here is that for the perfectly closed resonator with no loss, the hybrid-wave modes occur only when the beam splitter is $50\/50$. A slight deviation from this condition removes the degeneracy and destroys the hybrid nature of the modes. This may seem to pose a challenge for experimentally observing these modes. However, realistic resonators are not perfectly closed but rather have losses due to optical absorption, radiation to free space, or due to the coupling to input and output channels. In turn, this will introduce an upper limit on the resonator's quality factor and result in a finite bandwidth of operation, which relaxes the above constraint as discussed in detail in Appendix B and C.\n\nIn order to verify the above predictions, we perform a finite element method (FEM) full-wave simulation (using COMSOL software package) of a realistic implementation for the structure shown in Figure \\ref{Fig-Eigenmode}. The details of the geometry and material parameters used in our simulations are presented in Appendix C. The eigenmodes generated by COMSOL package are in the basis $B_1$ and are shown in Appendix C. Figure \\ref{Fig-Eigenmode}A and \\ref{Fig-Eigenmode}B show the electric field distributions associated with the modes as represented in basis $B_2$ which are generated via linear superposition of the degenerate modes $M_1^{(1)}$ and $M_2^{(1)}$. On the other hand, the field distributions in basis $B_3$ are depicted in Figure \\ref{Fig-Eigenmode}C and \\ref{Fig-Eigenmode}D. The nature of the waves can be deduced from the field distribution. Standing waves are visible through their interference pattern while traveling waves are characterized by uniform fields without interference. These plots are in agreement with the field distributions expected from Table \\ref{Table1} and indeed confirm the results obtained using the scattering matrix analysis above. The field distribution of modes $M_{1,2}^{(3)}$ deserves more attention. At the center of the middle sections (domains $D_{1,2}$), the field features a standing wave while at the center of domains $D_0'$ and $D_0''$, they feature traveling waves. At the beam splitter junction, however, the field is neither a pure standing nor traveling wave. These regions represent transition domains where the wave gradually changes its character. From the Poynting vector point of view, this remarkable mode structure is enabled by the beam splitter junctions acting as interferometric mirrors for domains $D_{1,2}$ while at the same time recirculating the power incident on them from domains $D_0$ in a closed loop. Importantly, the standing-traveling wave nature observed in Figure \\ref{Fig-Eigenmode}C and \\ref{Fig-Eigenmode}D are characteristic of the eigenmodes of a single resonator structure and not associated with a particular steady state solution under certain engineered excitation\\cite{Schmid2011PRA}. Equally important is the fact that the traveling waves in the presented structure are part of the quasi-bound state within the resonator's boundaries and not part of the leaked radiation waves outside the resonator as in the case of a finite Fabry--Perot or photonic crystal geometry for example. From a practical point of view, mapping these field distributions experimentally can be done only by using near-field probes which is possible but not an easy task. In Appendix E, we discuss a more practical scheme for accessing these modes by using input\/output waveguides ports evanescently coupled to the various sections of the resonator.\t\n\n\\section{Local perturbations and sensing applications} \nIn this section, we investigate the effect of local perturbation due to a small scatterer on the eigenmodes of the proposed hybrid-wave resonator---a problem relevant to sensing applications \\cite{Foreman2015AOP}. Given that the modes of the resonator shown in Figure \\ref{Fig-Resonator}A can be written in various bases, only one of which demonstrates the hybrid-wave character, one may wonder if this feature will have any consequences under more general conditions where the particular mode is not selectively excited. This section demonstrates that this is indeed the case. To illustrate this, we consider the situation where a scatterer (nanoparticle or a fiber tip for instance) is located within the evanescent field of the hybrid-wave resonator. In particular, we investigate the two scenarios where the scatterer is located either in the traveling- or standing-wave domains, as shown in Figure \\ref{Fig-Particle-frequency}A.\n\n\\begin{figure*}[!t]\n\t\\includegraphics[width=5.5in]{Fig-Particle-frequency.png}\n\t\\caption{Effect of local perturbation. (A) A schematic of the resonator structure with nanoparticles scatterers added in the traveling wave (case 1) or standing waves (case 2) regions. In case 1, the presence of the scatterer generates two new optical modes that exhibit either a node (B) or an antinode (C) at the location of the particle with corresponding frequency shifts as shown in (D). In this case, the frequency shifts are independent of the particle location as long as it resides in the traveling wave domain. In case 2, the scatter leaves one of the modes intact with zero frequency shift (E) while at the same time introduces a perturbation to the second mode (F) with a frequency shift that varies with the location of the particle as expected (G). In both cases, $\\delta=2\\pi$ corresponds to a distance of 0.5 $\\mu$m along the perimeter. A rigorous derivation of these results as well as their intuitive explanations are discussed in the main text. In the figure, $\\Delta f_j=f_{j,m}-f_m$ with $j=1,2$. The scatterers in panels (B), (C), (E) and (F) are indicated by small white circles. }\n\t\\label{Fig-Particle-frequency}\n\\end{figure*}\n\nBefore we proceed, it is useful to review the situation for purely traveling-wave resonators (such as microring and microdisk arrangements) and purely standing-wave resonators (such as Bragg and photonic crystal arrangements). In the former, the scatterer breaks the rotational symmetry of the geometry and introduces coupling between the clockwise and counterclockwise modes, leading to a splitting of the eigenfrequency \\cite{Zhu2017PR,Zhu2010NP,Ozdemir2014PNAS}. Importantly, this behavior is independent of the location of the scatterer. In the latter case, however, the situation is quite different. An optical mode that has an electric field node at the scatterer location along the resonator direction will not be affected by its presence. On the other hand, a mode that exhibits an antinode at the location of the scatterer will experience a shift in its eigenfrequency \\cite{Lalouat2007PRB,He2013NJP}. \n\nTo this end, we consider a small perturbation caused by a scatterer having a scattering matrix (Appendix F): \n\\begin{equation} \\label{Eq-Sp}\n\tS_p=e^{i\\phi_p}\\begin{bmatrix}\n\t\tt & i r\\\\\n\t\ti r & t\n\t\\end{bmatrix},\n\\end{equation}\nwhere $r$ and $t$ are reflection and transmission coefficients which are taken to be real numbers satisfying $r^2+t^2=1$ (i.e. no loss); and $\\phi_p=\\arcsin(r)$ is an overall additional phase.\n\n\\textbf{Scatterer located along the traveling-wave domain:}\nHere, we assume a scatterer located along the traveling-wave domain, say $D''_0$, at a fixed distance from the resonator waveguide edge, as shown case 1 in Figure \\ref{Fig-Particle-frequency}A. The angular position of the scatterer is defined by distance $l$ (Figure \\ref{Fig-Particle-frequency}A) and the corresponding phase shift $\\delta\\equiv 2\\pi n_\\text{eff} l f\/c$. By following the same approach used in deriving Eq. (\\ref{Eq-Matrix_equation}), we find that the resonant frequencies (see Appendix G):\n\\begin{equation}\\label{Eq-Particle-traveling}\n\t\\begin{cases}\n\t\t\\phi_{1,m}=\\phi_m, \\\\\n\t\t\\phi_{2,m}=\\phi_m-2\\phi_p.\\\\\n\t\\end{cases}\n\\end{equation}\nwhere $\\phi_{j,m}$ with $j=1,2$ indicates the new eigenfrequencies (due to the perturbation introduced by the scatterer) branched from the unperturbed eigenfrequency $\\phi_m$. In other words, the perturbation shifts the frequency of only one mode while leaving that of the other unchanged. This can be explained by the fact that the new modes, arising because of the perturbation, both exhibit a standing-wave pattern, with the node of one mode and the antinode of the other located at the position of the scatterer. Moreover, Eq. (\\ref{Eq-Particle-traveling}) does not depend on the location of the scatterer along the perimeter of the resonator, as long as it lies in the traveling-wave domain. This behavior is exactly identical to the case of a scatterer introduced in the vicinity of traveling-wave resonator such as a microring or microdisk geometry \\cite{He2013NJP,Zhu2010OE}. \n\nThese predictions are confirmed by performing full-wave simulations, where the scattering was introduced via a nanoparticle. Figure \\ref{Fig-Particle-frequency}B and \\ref{Fig-Particle-frequency}C depict the field distribution of the perturbed modes around the particle. Note that, as expected, the particle modifies the field distribution and creates a standing-wave pattern. Moreover, the node of the first mode and the antinode of the second mode coincide with the particle location along the perimeter of the resonator, which is consistent with our theoretical predictions. As a result, the eigenfrequency of the first mode remains unchanged while that of the second mode experiences a constant shift that does not depend on the particle location, as shown in Figure \\ref{Fig-Particle-frequency}D. As a side note, we remark that the blue dots representing the simulation data in Figure \\ref{Fig-Particle-frequency}D do not exactly coincide with the zero axis as predicted by our scattering matrix analysis but rather exhibit a small shift. This can be explained by recalling that, in our analysis, we treat the particle as a Rayleigh scatterer, whereas in reality higher-order multipole terms must be considered in order to obtain more accurate results. To confirm this, we have performed additional numerical simulations for different particle sizes and indeed observed that this frequency shift decreases as the particle size is reduced (in fact we could not resolve the frequency shift for particles with radii less than 30 nm).\n\n\\textbf{Scatterer located along the standing-wave domain:} Next, we consider the case when the scatterer is located in the standing-wave region, for instance in domain $D_2$, as shown by case 2 in Figure \\ref{Fig-Particle-frequency}A. The resonant frequencies are given by (see Appendix G):\n\\begin{equation}\\label{Eq-Particle-standing}\n\t\\begin{cases}\n\t\t\\phi_{1,m}=\\phi_m, \\\\\n\t\t\\phi_{2,m}=\\phi_m-2\\phi_p [1+(-1)^{m+1}\\cos2\\delta].\\\\\n\t\\end{cases}\n\\end{equation}\nThe mode corresponding to the eigenfrequency $\\phi_{1,m}$ (Figure \\ref{Fig-Particle-frequency}E) is associated with $M_1^{(2)}$ in Figure \\ref{Fig-Eigenmode}A, in which the electric field is zero at domain $D_2$, leading to an unperturbed resonant frequency after adding the particle at domain $D_2$. On the other hand, the eigenfrequency $\\phi_{2,m}$ corresponds to a perturbation of mode $M_2^{(2)}$ in Figure \\ref{Fig-Eigenmode}A. Since mode $M_2^{(2)}$ is a pure standing wave at domain $D_2$, the eigenfrequency $\\phi_{2,m}$ varies with the angular position of the scatterer (Figure \\ref{Fig-Particle-frequency}F and \\ref{Fig-Particle-frequency}G). In Eq. (\\ref{Eq-Particle-standing}), when $m$ is an odd number, it is an antinode at the middle of $D_2$ domain ($\\delta=0$), $\\phi_{2,m}$ experiences the maximum frequency shift $-4\\phi_p$ from $\\phi_m$; when $m$ is an even number, it is a node at the middle of $D_2$ domain, the scatterer will not alter the field much and the resonant frequency will stay the same, i.e., $\\phi_{2,m}=\\phi_m$ at $\\delta=0$. In both scenarios, $\\phi_{2,m}$ will oscillate between $\\phi_m$ an $\\phi_m-4\\phi_p$ as a function of $\\delta$. In our simulation, the fact that $m$ is an odd number can be determined from Figure \\ref{Fig-Eigenmode}, and it is also verified by the electric field around the particle are shown in Figure \\ref{Fig-Particle-frequency}F. The two eigenfrequencies varying with $\\delta$ are shown in Figure \\ref{Fig-Particle-frequency}G, consistent with Eq. (\\ref{Eq-Particle-standing}). \n\nFrom the above analysis, it is clear that a resonator exhibiting hybrid-wave modes will respond very differently to perturbations affecting the standing- or traveling-wave zones. In terms of applications, this can be useful in a number of ways. For instance, the larger splitting in the location of the field maxima in the standing-wave zone can be utilized for selective sensing by functionalizing \\cite{Vollmer2021NatRev} this exact location with receptors that can bind only to a particular molecule while at the same time use the traveling-wave zone for excitation and collection. On the other hand, one can instead use both zones for detecting the presence of more than one molecule (but only one at a time). This can be achieved by attaching different receptors to each zone and inferring the presence of a particular substance by measuring the degree of splitting. We plan to investigate these possibility in future works.\n\n\\section{Conclusion}\nIn conclusion, we have proposed a new concept for optical resonators that exhibit simultaneously co-existing standing and traveling waves as part of the field distribution of the same optical mode but occupying different locations along the resonator geometry. In addition, we have presented a specific example of a structure that implements this concept and verified its standing and traveling wave nature by using scattering matrix analysis and FEM full-wave simulations. We have investigated the robustness of the hybrid-wave feature and shown that the openness of the system allows for a larger bandwidth of operation and thus facilitates experimental observation. In addition, we have described a practical experiential scheme for probing the hybrid-wave nature by using several waveguide channels attached to various sections of the resonator geometry. Furthermore, we have discussed the implication of the hybrid-wave nature for sensing applications by investigating how the eigenmodes of such hybrid-wave resonator interact with a small scatterer located at different sections of the structure, demonstrating that the system's response can be very different depending on the location of the scatterer along the standing- or traveling-wave sections. Another arena where hybrid-wave modes may prove useful is optical manipulation and trapping of particles. For instance, it is expected that a nanoparticle located in the traveling-wave zone will experience radiation pressure and lateral force acting in the direction towards the resonator, while a similar particle located in the standing-wave zone will in addition be subject to a trapping force along the perimeter of the resonator. Furthermore, actively tuning the beam-splitting values may allow for controlling the behavior of the resonator in real time and thus controlling its interaction with nanoparticles.\t\n\nIn addition, the existence of the hybrid-wave modes identified above, which to the best of our knowledge has not been known before, raises several fundamental questions in photonics, nonlinear and quantum optics applications. For instance, it is not a priori clear how such a resonator will behave under nonlinear conditions. Does it exhibit different nonlinear bistability responses than those observed in conventional ring resonators \\cite{Braginsky1989PLA}? Does it provide any new features in terms of frequency comb generation? Can soliton crystals \\cite{Cole2017NPhot} form in the presence of hybrid-wave modes? Along similar lines, it is not clear to what extent the presence of the hybrid-wave modes will impact the dynamics and instability features of laser devices made of such resonators. In the quantum domain, it would be interesting to explore how quantum emitters located inside or in the vicinity of such resonators will behave. How would spontaneous emission and superradiance scale in different sections of the resonators? It has been shown previously that the interaction between atoms and electromagnetic waves featuring a standing field pattern depends on the type of resonator \\cite{Shore1991JOSAB} (standing or traveling wave resonator). What makes these exploratory questions particularly interesting is that the proposed resonator exhibit transition regions (the beam splitter regions in Figure \\ref{Fig-Resonator}A that interpolates between the traveling- and standing-wave domains). Light-matter interaction in this region is expected to differ from its typical behavior in standard traveling- and standing-wave resonators, which may lead to interesting new effects. At the engineering level, our work also raises interesting questions. For instance, is there a fundamental size limit on building hybrid-wave resonators? Can one implement a small volume hybrid-wave mode? What would be the modes of these structures when implemented in material platforms that support plasmonic resonances? We plan to investigate these open questions as well as implementations in other platforms such as acoustics \\cite{Ingard1953JASA} and microwave \\cite{Pozar1990Microwave} in future works.\n\n\\begin{acknowledgements}\nThis project is supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative\t(MURI) Award on Programmable systems with non-Hermitian quantum dynamics\t(Award No. FA9550-21-1-0202). R.E. also acknowledges support from Army Research Office (ARO) (Grant No. W911NF-17-1-0481), National Science Foundation (NSF) (Grant No. ECCS 1807552), Henes Center for Quantum Phenomena, and the Alexander-von-Humboldt Foundation. S.K.O. also acknowledges support from ARO (Grant No. W911NF-16-1-0013, W911NF-17-1-0481), NSF (Grant No. ECCS 1454531, DMR-1420620, ECCS 1807552, ECCS 1757025, ECCS 1807485).\n\\end{acknowledgements}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzqgru b/data_all_eng_slimpj/shuffled/split2/finalzzqgru new file mode 100644 index 0000000000000000000000000000000000000000..cb7789135f40afaf493e402557f78763c06f1c31 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzqgru @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\input{text\/sec_intro}\n\n\\section{Observations}\n\\label{sec:obs}\n\\input{text\/sec_obs}\n\n\\section{Construction of the Wideband\\\\Data Set}\n\\label{sec:wb}\n\\input{text\/sec_wb}\n\\vspace{1in}\n\\section{Results \\& Discussion}\n\\label{sec:results}\n\\input{text\/sec_results}\n\n\\section{Summary \\& Conclusions}\n\\label{sec:conclusion}\n\\input{text\/sec_conclusion}\n\n\n\\input{text\/sec_acknowledgement}\n\n\\facilities{Arecibo, GBT}\n\n\\software{\\texttt{ENTERPRISE} \\citep{enterprise}, \\texttt{libstempo} \\citep{libstempo}, \\texttt{matplotlib} \\citep{matplotlib}, \\texttt{nanopipe} \\citep{nanopipe}, \\texttt{PSRCHIVE} \\citep{vS11}, \\texttt{PTMCMC} \\citep{ptmcmc}, \\texttt{PulsePortraiture} \\citep{pulseportraiture}, \\texttt{PyPulse} \\citep{pypulse}, \\texttt{Tempo} \\citep{tempo}, \\texttt{Tempo2} \\citep{tempo2}, \\texttt{tempo\\_utils}\\footnote{\\href{https:\/\/github.com\/demorest\/tempo\\_utils}{https:\/\/github.com\/demorest\/tempo\\_utils}}}\n\\\\\\\\\n\\input{text\/sec_contribution}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Overview}\n\\label{subsec:wideband_overview}\n\nThe measurement of TOAs from pulsar data with a large instantaneous bandwidth was first developed in \\citet{Liu14} and \\citet{PDR14}, and further explored in \\citet{PennucciPhDT} and \\citet{Pennucci19}.\nWe refer the reader to those works for details and here briefly summarize the important points.\n\nA single narrowband TOA corresponds to the time of arrival of a pulse profile observed in a single frequency channel\\footnote{Another similar protocol used in the pulsar timing community is to produce band-averaged TOAs, in which the detected profiles are summed over the observing bandwidth, creating a single profile from which to extract the TOA.} (sometimes referred to as a ``subband''); in contrast, a single wideband measurement is composed of both the time of arrival of a pulse at some reference frequency and an estimate of the dispersion measure at the time of observation.\nThe difference can be conceptualized thusly: narrowband TOAs from a single subintegration are like the individual, scattered measurements around a linear relationship, whereas the fitted intercept and slope to this relationship are like the wideband TOA and DM, respectively.\nThe log-likelihood function for the wideband measurements is reproduced in Section~\\ref{subsec:wb_toa_lnlike}.\n\nThe second important difference in the new wideband data set is not fundamental to the measurement of the TOA.\nHeretofore we have used a single, frequency-independent template profile for each receiver band to generate narrowband TOAs and have used FD parameters \\citep{Arzoumanian2015b} to account for constant phase offsets originating from the mismatch between the template and the evolving shape of the profiles.\nFor the measurement of wideband TOAs, we explicitly account for pulse profile evolution by using a high-fidelity, noise-free, frequency-dependent model for each receiver band.\nSee Section~\\ref{subsec:prof_evol} for a brief description of how these models are created.\n\nAlthough the narrowband and wideband data sets were developed in parallel, the established techniques in preparing the former allowed us to use some information from its final products to facilitate the production of the latter.\nIn particular, some of the curating performed, including flagging bad epochs, as well as the initial timing, was borrowed from the narrowband analysis.\nIn this way, the wideband data set is not completely independent, as is detailed in Sections~\\ref{subsec:wb_cleaning}~\\&~\\ref{subsec:timing}.\n\nIt is important to underscore that the wideband data set for each pulsar is composed of TOAs that are paired with estimates of the instantaneous DM.\nWhat makes the analysis of the wideband data set truly unique is that these DM estimates inform the portion of the timing model that accounts for DM variability (for our analyses, this is ``DMX''; see Section~\\ref{subsec:timing}).\nIn Section~\\ref{subsec:timing}, we describe our approach, with greater detail in Appendix~\\ref{sec:wb_like}; the results are examined in Section~\\ref{sec:results}.\n\nPublicly available code\\footnote{\\url{https:\/\/github.com\/pennucci\/PulsePortraiture}} is used for both the generation of frequency-dependent templates and the measurement of the wideband TOAs \\citep{pulseportraiture}.\n\\\\\n\\subsection{Wideband TOA Log-Likelihood Function}\n\\label{subsec:wb_toa_lnlike}\n\nAll of our narrowband and wideband TOAs are measured using what is now referred to as the ``Fourier phase-gradient shift algorithm'' \\citep[][historically known as ``FFTFIT'']{Taylor92}, which makes use of the Fourier shift theorem to achieve a phase offset precision much better than a single rotational phase bin, and which is computationally efficient by virtue of avoiding the time-domain cross-correlation calculation between the data and template pulse profiles.\nWe use a similar notation as Appendix~B of \\nineyr, but see also \\citet{DemorestPhDT} and \\citet{PDR14} for details of what follows.\nThe time-domain model has the assumed form\n\\begin{equation}\n\\label{eqn:toa_signal}\n D(\\nu, \\varphi) = B(\\nu) + a(\\nu)\\,T(\\nu, \\varphi-\\phi(\\nu)) + N(\\nu),\n\\end{equation}\nThat is, for each subintegration in an observation, we assume that the data profiles $D$ as a function of rotational phase $\\varphi$ and frequency $\\nu$ can be described by a template $T$ that is shifted in phase by $\\phi$ and scaled in amplitude by $a$, with added Gaussian-distributed phase-independent noise $N$; the term $B$ represents the bandpass shape.\nAfter discretizing these quantities, taking the discrete Fourier transform (DFT), making use of the Fourier shift theorem, and rearranging terms, we can reformulate Equation~\\ref{eqn:toa_signal} into our TOA log-likelihood,\n\\begin{equation}\n\\label{eqn:toa_lnlike}\n \n \\chi^{2} = \\sum_{n,k} \\frac{\\lvert d_{nk} - a_{n} t_{nk}e^{-2\\pi ik\\phi_n}\\rvert^{2}}{\\sigma^2_n}.\n\\end{equation}\nIn Equation~\\ref{eqn:toa_lnlike}, the integer index $k$ is the Fourier frequency (conjugate to rotational phase or time), $t_{nk}$ is the DFT of the template profile for the frequency channel indexed by $n$ (with frequency center $\\nu_n$), $a_{n}$ is the scaling amplitude parameter for the template, $\\phi_n$ is the phase offset for the template, and $d_{nk}$ is the DFT of the data profile for frequency channel $n$, which has the corresponding Fourier-domain noise level $\\sigma^2_n$\\,\\footnote{$\\sigma^2_n$ is the noise for either the real or imaginary part, and is larger than its (real) time-domain counterpart by a factor of $n_{\\textrm{bin}}\/2$.}.\n\nFor conventional TOAs, the optimization of this function takes place on an individual channel basis, in which case there is no index $n$ in Equation~\\ref{eqn:toa_lnlike} over which a summation occurs.\nMoreover, for our narrowband TOAs, $t_{nk}$ is not a function of $n$; that is, profile evolution is not accounted for by changing the shape of the template across a single receiver's frequency band.\nInstead, in \\twelveyr, a single template profile is used for each receiver band and constant phase offsets arising from the mismatch between the template shape and the evolving pulse shape are accounted for via FD parameters in the narrowband timing models.\n\nThe crucial difference in wideband TOAs is that the phase offsets $\\phi_n$ in Equation~\\ref{eqn:toa_lnlike} are constrained to follow the cold-plasma dispersion law, proportional to $\\nu^{-2}$:\n\\begin{equation}\n\\label{eqn:wb_constraint}\n \\phi_n(\\nu_n) = \\phi_\\circ}%{\\phi^\\circ_{ref} + \\frac{K\\times\\textrm{DM}}{P_s}\\Big(\\nu_n^{-2}-\\nu_{\\phi_\\circ}^{-2}\\Big),\n\\end{equation}\nwhere $P_s$ is the instantaneous spin period of the pulsar, $K$ is the dispersion constant (a combination of fundamental physical constants approximately equal to $4.148808 \\times 10^3$~MHz$^2$~cm$^3$~pc$^{-1}$~s), $\\textrm{DM}$ is the dispersion measure, and $\\phi_\\circ}%{\\phi^\\circ_{ref}$ is the phase offset at reference frequency $\\nu_{\\phi_\\circ}$.\nEquation~\\ref{eqn:toa_lnlike} can be recast using the maximum-likelihood values of $a_{n}$ and rewritten as a function of only the two parameters $\\phi_\\circ}%{\\phi^\\circ_{ref}$ and DM (see \\citet{PDR14}), which can then be readily optimized numerically.\nWe calculate the parameter uncertainties using the Fisher matrix and choose $\\nu_{\\phi_\\circ}$ such that there is zero covariance between the DM and $\\phi_\\circ}%{\\phi^\\circ_{ref}$, the latter of which is directly related to the TOA.\n\nAdditional terms to the wideband TOA log-likelihood are currently being explored (Pennucci et al., \\textit{in prep.}), which include accounting for pulse broadening from multi-path propagation through the turbulent ISM (i.e., ``scattering'') in a similar fashion to \\citet{lkd+17}, as well incorporating a higher-order delay term besides $\\nu^{-2}$, the motivation for which are discrete ISM ``events'' \\citep{Lam2018}.\nThe low-frequency, high-cadence capabilities offered by CHIME\/Pulsar will make tracking the interstellar weather in this way an exciting endeavor, following in the footsteps of studies like \\citet{Ramachandran06} and \\citet{Driessen19} (long-term ISM tracking of B1937$+$21 and the Crab pulsar, respectively).\n\n\\subsection{Frequency-dependent Template Profiles}\n\\label{subsec:prof_evol}\n\nThe evolving template $t_{nk}$ in Equation~\\ref{eqn:toa_lnlike} can be freely chosen, and in this work we employ the modeling method from \\citet{Pennucci19}, which describes a generalized, frequency-dependent version of our usual protocol for making template profiles.\nIn contrast, to make the conventional noise-free templates used in \\twelveyr\\ for narrowband TOA measurement, all profiles for each combination of pulsar and receiver are averaged together to build a single, high S\/N mean profile, which is then smoothed.\nWe direct the reader to \\citet{Pennucci19} for details, but we summarize its novel procedure as follows.\n\nAn analogous averaging of the data for each combination of pulsar and receiver is performed, but frequency resolution is maintained to arrive at a high S\/N mean ``portrait'' (a collection of nominally aligned mean pulse profiles across a contiguous frequency band); only the PUPPI and GUPPI data were averaged for this purpose.\nA principal component analysis is performed on the average portrait, and the most significant, highest S\/N eigenvectors (and mean profile) are smoothed to become noise-free basis functions (``eigenprofiles'').\nThe mean-profile-subtracted profiles from the average portrait are projected onto each of the eigenprofiles, producing a set of coefficients for each.\nThese coefficients are simultaneously fit to a slowly varying spline function that is parameterized by frequency and encapsulates the evolution of the pulse profile shape.\n\nIn this manner, a template profile $T$ at any frequency $\\nu$ can be constructed by evaluating the $n_{\\textrm{\\scriptsize{eig}}}$ coefficient spline functions $B_i$ at $\\nu$, linearly combining the eigenprofiles $\\hat{e}_i$ using these coefficients, and adding the result to the mean profile $\\widetilde{p}$,\n\\begin{equation}\n\\label{eqn:Tprof}\n T(\\nu) = \\sum^{n_{\\textrm{\\scriptsize{eig}}}}_{i=1} B_{i}(\\nu) \\, \\hat{e}_i ~ + ~ \\widetilde{p}.\n\\end{equation}\n\n\nIn summary, a single model for generating high-fidelity, noise-free template profiles is composed of the smoothed mean profile, the smoothed basis eigenprofiles, and a function to describe the profile evolution curve in that basis.\n\nThese models were made for each combination of pulsar and receiver, and then used to measure wideband TOAs according to Equation~\\ref{eqn:toa_lnlike}.\nThe modeling procedure attempts to guess the true, unknown profile alignment by starting with the same Occam assumption used in the narrowband analysis: there is no profile evolution, neither in the shape nor alignment of the profiles.\nThis assumption is used to initially align and average the profile data by using the fixed, mean profile shape as a reference for the alignment.\nAfter iteratively aligning and averaging the profile data and then creating a model, it should not come as a surprise that the absolute, average DM measured in each receiver band will differ slightly.\nWe minimize this difference by measuring the weighted-mean DM offset relative to the DM measured in the lowest frequency band.\nThe DM offset was then applied as a rotation proportional to $\\nu^{-2}$ to the average portrait, the profile evolution model was recreated, and the TOAs were remeasured; this process was iterated a total of three times.\nThe reference DM choice was made relative to the lowest frequency band because, except in the cases of Arecibo pulsars observed only at L- and S-bands, this will be a frequency band with lower fractional bandwidth than L-band, but from which reasonably precise DM measurements are made.\nThis choice gave better modeling results than rotating the averaged low frequency data relative to the L-band alignment, which may be ambiguous due to profile evolution.\nFor the other sources, S-band generally does not give precise DM measurements, and so L-band is used as the reference.\nSee Section~\\ref{subsec:model_results} for more discussion on this topic.\n\nThe initial set of wideband TOAs used in the timing and noise analyses were measured with these DM-aligned models, and instrumental time offsets were applied to TOAs from ASP and GASP profiles, as detailed in Appendix~A of \\nineyr.\nMetadata in the TOA files take the form of ``flags'', which get appended to each TOA line in the files.\nA number of new TOA flags have been added to aid wideband timing analyses, and a few of the usual TOA flags have different meanings from their narrowband TOA counterparts; these are listed in the top portion of Table~\\ref{tab:toa_flags}.\n\nThe choice of DM alignment in wideband profile models is analogous to the ambiguity of absolute phase between TOAs measured with different template profiles in the narrowband analysis.\nThose constant phase offsets are modeled in the timing model with so-called ``JUMP'' parameters and are also present in the wideband analysis.\nOur fiducial DM alignment is an attempt at getting the simplest profile evolution models, but a new, analogous timing model parameter is necessary when using multi-band DM measurements as data for the timing model.\nTo this end, we implemented ``DMJUMP'' parameters for wideband timing in the extended likelihood introduced in Section~\\ref{subsec:timing}.\nAppendix~\\ref{sec:wb_like} contains details about how these parameters influence the timing model.\n\n\\input{tables\/tab_toa_flags}\n\n\\subsection{Cleaning \\& Curating the Wideband Data Set}\n\\label{subsec:wb_cleaning}\n\nThe narrowband data set was prepared in advance of the wideband data set, and as a part of its creation we kept track of bad observations that were corrupted by instrumentation or calibration issues, or were so affected by RFI that we excised them outright (250 of 11,178 observations). \nThese observations (which are included in the narrowband data set as commented TOAs with the flag \\texttt{-cut badepoch}) were simply not introduced into the wideband pipeline.\nThere are also a small number of observations (36) for which data were taken on a pulsar using a different receiver than usual, often for testing purposes (these are included in the narrowband data set as commented TOAs with the flag \\texttt{-cut orphaned}).\nThese data are generally not sufficient to create good profile evolution models, and would add very few degrees of freedom; we similarly excluded them from the wideband analysis at the start.\n\nThere was one other additional step in curating the profile data set used to make wideband TOAs.\nUpon finishing the modeling procedure described in Section~\\ref{subsec:prof_evol}, we calculated goodness-of-fit statistics for each profile in the data set based on its predicted pulse shape from the corresponding model.\nProfiles in a given subintegration were zero-weighted if their goodness-of-fit exceeded a threshold ($\\chi^2_{\\textrm{reduced}} > 1.25$), which was empirically determined after examining the distributions for each combination of pulsar and receiver.\n\nFor most combinations, the number of discarded profiles in this manner was of order a few percent.\nAfter zero-weighting these profiles, the data were re-averaged and the profile evolution models were recreated.\nThis step was necessary because, as with the ADC artifact mentioned in Section~\\ref{sec:obs}, unmitigated RFI can corrupt the modeling procedure.\nMore general RFI-flagging techniques based on template-matching using the wideband profile models are in development within NANOGrav and elsewhere (MeerTime collaboration, private communication).\nSuch techniques could potentially identify irregularities in the profiles, be it from RFI or other sources, earlier in the reduction pipeline.\n\nThe remainder of the cleaning of the wideband data set was performed on the measured TOAs; any TOAs ``cut'' from further analysis were given one of the flags listed in Table~\\ref{tab:toa_flags}, but are included as commented TOAs in the publicly available text files.\nMost of the cuts described in the table have counterparts in the preparation of the narrowband data set, and we refer the reader to \\twelveyr\\ for details beyond those offered in the table and those that follow.\n\nThe S\/N threshold used for the wideband TOAs was set at $25$, compared to the value of $8$ used for narrowband TOAs.\nThe main reason for this was empirical and related to the fact that the estimated S\/N for wideband TOAs is subject to significant bias in the low S\/N regime, favoring a higher threshold than is naively derived.\nWe justify this choice in Appendix~\\ref{sec:low_snr}.\n\nNote that in \\elevenyr\\ and \\twelveyr\\ a numerical TOA outlier analysis is performed \\citep{Vallisneri2017}.\nSome of the narrowband TOAs identified in this way are from profiles corrupted by RFI or instrumental problems that were not otherwise identified.\nOur goodness-of-fit filter of the profile data described earlier served a similar purpose, and no separate outlier TOA analysis was performed.\nWe found that after filtering the profiles in this way and thresholding the TOAs based on the S\/N cutoff of $25$, the initial timing results were remarkably clean; there were only a handful of additional TOAs that were culled based on a large timing residual ($>100~\\mu$s) or were otherwise identified by eye (see Table~\\ref{tab:toa_flags}).\n\n\\input{tables\/tab_ntoa_nchan}\n\nOverall, despite the procedural differences in preparing the two data sets, the quality control for the wideband data set resulted in $\\sim$~16\\% more profiles used for TOA measurement, as can be seen in Table~\\ref{tab:ntoa_nchan}.\nThis difference is largely due to the inclusion of low S\/N ratio profiles that are discarded in the narrowband data set (see Appendix~\\ref{sec:low_snr}); as such, it is unsurprising that these additional data in general do not carry a proportionally large impact on the timing results, as will be shown.\nHowever, see Section~\\ref{subsec:pulsar_results} for specific examples.\n\nAfter curation, the resulting wideband data set has 12,598 TOAs, corresponding to 480,474 profiles; this is compared to the 415,122 TOAs in the narrowband data set, a factor of $\\sim$~33 larger in TOA volume, which will only grow as the ASP and GASP TOAs become a fractionally smaller subset of the entire data set, and as new wideband facilities and receivers come into use. \nNote, however, that the overall wideband data set volume is only a factor of $33\/2 \\sim 16$ smaller, after including the DM measurements in the analysis.\n\n\\begin{table*}[th!]\n \\caption{Basic Pulsar Parameters and TOA Statistics\\label{tab:toa_summary}}\n\\centering\n\\input{tables\/tab_toa_summary}\n\\end{table*}\n\n\\input{text\/fig_wb_toa_dm_err}\n\nA summary of the TOA uncertainties are presented in two forms.\nFirst, the median uncertainties are listed, along with other basic pulsar parameters, in Table~\\ref{tab:toa_summary}.\nThere is an analogous table in \\twelveyr\\ for the narrowband TOAs; in both cases, the uncertainties have been scaled to estimate the median TOA uncertainty from a 1800~second observation of the pulsar with 100~MHz of bandwidth.\nOverall, the values are comparable to their narrowband counterparts, but differences may be attributable to any of: unmodeled profile evolution in the narrowband data set, the inclusion of very low S\/N profiles in the wideband data set, the additional fit parameter (DM) in the wideband measurement, or other subtle discrepancies.\nSecond, in Figure~\\ref{fig:wb_toa_dm_err} we graphically present the ``raw'' median TOA and DM uncertainties with central intervals covering the central 68\\% of the distribution, ranking pulsars by their median PUPPI or GUPPI L-band TOA uncertainty.\nWe use ``raw'' to mean that these are the formal, estimated uncertainties from the template-matching procedure, which do not include any other sources of uncertainty and are not scaled in any way.\nIt is obvious from this plot that, depending on the pulsar, the improvement in raw TOA precision after moving from ASP and GASP to PUPPI and GUPPI is a factor of 2--3 or more in many cases, but the DM precision improves by an order of magnitude or more in all receiver bands except 327 and 430~MHz.\nThis improvement is due to the increase in bandwidth covered by PUPPI and GUPPI (see Table~\\ref{tab:observing_systems}).\n\n\\subsection{Obtaining Timing Solutions}\n\\label{subsec:timing}\n\nWe used the 12.5-year data set results from \\twelveyr\\ as initial timing solutions instead of deriving completely new timing results from the extended baselines of the 11-year data set.\nThis was done in part to facilitate comparisons and in part to reduce the need for redundant analyses.\nSpecifically, any new spin, astrometric, or binary timing model parameters found to be significant in \\twelveyr\\ were retained, but FD parameters were removed, as were the parameters that describe the DM model, called DMX.\n\nDMX is a piecewise-constant characterization of DM variability that is part of the timing model.\nSimpler models of DM variability, such as low-order polynomials, do not describe the data well, but more advanced models, such as those that use a stochastic description of variability \\citep[e.g., as a Gaussian process,][]{Lentati13}, are currently being investigated.\nThe criteria for dividing up the TOAs into DMX epochs defined by Modified Julian Dates (MJDs) can be found in \\twelveyr.\nFor each DMX epoch, a DM is measured based on the $\\nu^{-2}$ dependence of the TOAs that fall within the epoch, and all of these DMX model parameters are measured simultaneously with the fit for the rest of the timing model.\n\nIf we were to ignore the wideband DM measurements, the wideband TOA data set would be significantly hampered in the following ways.\nThere are a large number of DMX epochs which contain data from a single receiver.\nIn the cases where such an epoch has a single wideband TOA (instead of the dozens of analogous narrowband TOAs), the corresponding single DMX parameter removes the single degree of freedom, artificially zeroing out the timing residual for this epoch.\nIf there are a few wideband TOAs from the same receiver band in such an epoch, they will have similar reference frequencies, and so the DMX parameter will be poorly constrained and perhaps biased.\nFinally, even for the majority of DMX epochs for which there are multi-frequency wideband TOAs from dual receiver observations, DMX only has access to the TOAs, their uncertainties, and reference frequencies.\nThat is, the information about the dispersive delays across the individual receiver bands (captured by the wideband DM measurements, or, equivalently, the multi-frequency TOAs in the narrowband data set) is lost, and DMX only sees the dispersive delay between the bands.\nAs can be seen in most pulsars' DM and DMX time series (see Appendix~\\ref{sec:resid}), the wideband TOAs and their inter-band dispersive delay carry more weight in the DMX model than do the intra-band delays characterized by their corresponding wideband DM measurements.\nHow much more so depends on the pulsar and receiver bands in question, but it is important to highlight that disregarding the DM data is not a viable option for analyzing this data set.\nIndeed, we attempted several such analyses, which yielded significantly worse results in many pulsars.\n\nTherefore, not only was it appropriate, but it was also necessary to expand the likelihood used to fit our timing models so that the wideband DM measurements inform the DM model.\nIn effect, in the new likelihood, the wideband DM measurements influence the timing model as prior information on the DMX values.\nEach of the TOAs falling within a DMX epoch have a corresponding DM measurement; the weighted average of these measurements is used as the mean of a Gaussian prior on the DMX value for that epoch, while the standard error of the weighted average is the prior distribution's standard deviation.\nThe details of this new likelihood and its implementation in the pulsar timing software packages \\texttt{Tempo} \\citep{tempo} and \\texttt{ENTERPRISE} \\citep{enterprise} can be found in Appendix~\\ref{sec:wb_like}.\n\n\\afterpage{\\clearpage}\n\nThe timing models from \\twelveyr\\ were first refit with \\texttt{Tempo} using the wideband TOAs only, omitting the DM measurements, to setup the DMX epochs and to get initial DMX values.\nIncluding the DM measurements at this point sometimes resulted in poor timing results because there is currently no way to fit the DMJUMP parameters simultaneously with the timing model within \\texttt{Tempo}.\nIt is at this stage that TOAs were excluded from further analysis if they did not meet the frequency ratio criterion described in Table~\\ref{tab:toa_flags} or if the entire epoch was removed based on a new analysis performed in \\twelveyr\\ (also mentioned in Table~\\ref{tab:toa_flags}).\n\nThe wideband TOAs, DMs, and timing models were then subject to a Bayesian analysis with \\texttt{ENTERPRISE} using the new wideband likelihood.\nThis analysis optimizes the probability of the observed data by characterizing the noise in the timing residuals, which has both white and red components, much in the same way as in \\twelveyr, \\elevenyr, and \\nineyr, with a few important differences:\n\n\\textit{No ECORR --}\nThere is one parameter in the standard white noise model that is not used in the wideband analyses.\nThis parameter, called ECORR, accounts for the (assumed 100\\%) correlation between multi-frequency TOAs taken at the same time and is used in the narrowband analyses of \\nineyr, \\elevenyr, and \\twelveyr\\ \\citep{Arzoumanian2014,Arzoumanian2015b}.\nSince wideband TOAs effectively consolidate the many narrowband TOAs into one, any physical effects contributing to this parameter (such as pulse jitter or ISM effects; see Section~\\ref{subsubsec:wn_compare}) would be absorbed by the standard EQUAD noise parameter, which is added in quadrature to the measured TOA uncertainty \\citep{Edwards06,Lentati14}.\nAlternatively, any effects contributing to ECORR in the narrowband analysis may be modeled by a larger and shallower red noise process in the wideband analysis.\nA comparison of the detected excess white noise in the two data sets is presented in Section~\\ref{subsubsec:wn_compare}.\n\n\\textit{DMEFAC \\& DMJUMP --}\nTwo additional parameters are needed in the new wideband likelihood.\nThe first, which we call ``DMEFAC'', is analogous to the standard TOA EFAC: it is a factor that scales the estimated wideband DM measurement uncertainty.\nIn a similar fashion to the other white noise parameters, a DMEFAC is assigned for each combination of receiver and backend in each pulsar's noise model.\nThe second was introduced in Section~\\ref{subsec:prof_evol}, which we call ``DMJUMP''.\nThis parameter is analogous to standard JUMP parameters, but instead of modeling an achromatic phase offset between TOAs measured in different receiver bands, DMJUMP is a DM offset between wideband DMs measured in different bands.\nThese parameters account for the differences in alignment between profile evolution models in disparate bands, and amount to making a choice for the absolute DM.\nIt is important to stress that this ambiguity in absolute DM, as well as the offsets in DMs measured in disparate bands, exist also in the narrowband analyses; in \\twelveyr, the choice of having fixed templates in each band, coupled with using FD parameters to account for constant TOA biases as a function of frequency, amount to addressing the analogous problems.\nWe assign one DMJUMP parameter per receiver in each pulsar's timing model, since the profile evolution models are independent of backend.\nIt may seem that we should use one less DMJUMP parameter than there are receivers in each pulsar's analysis, as is done for standard phase JUMP parameters.\nHowever, because the DMX model is separately informed by the TOAs, it is not an overdetermined problem.\nThis fact is borne out by examining the posterior chains; although we see that the DMJUMP parameters are often highly covariant, they are not completely degenerate.\nWe used a uniform prior distribution on DMJUMP parameters in the range $[-0.01,0.01]~\\textrm{cm}^{-3}~\\textrm{pc}$; virtually all of the values are $\\lvert\\textrm{DMJUMP}\\rvert < 0.004~\\textrm{cm}^{-3}~\\textrm{pc}$.\n\n\\textit{White noise priors --}\nIn the analyses of all of our other data sets, we have used large, uniform priors on EFAC between 0.1 and 10.0.\nEFAC was originally implemented to account for instances when the profile data poorly matched the template profile in the TOA fit, which would underestimate the TOA uncertainty.\nIn the present analysis, we expect EFAC to be near 1.0 because we are using evolving profile templates and have carefully excised RFI at a number of stages in the pipeline.\nWe have found that allowing extreme EFAC values can inadvertently over- or down-weight subsets of the data when it is not justified.\nOne reason for this is that there is a larger amount of covariance between EFAC and EQUAD parameters in the wideband analysis because the formal TOA uncertainties (of which there are far fewer) are more homoscedastic; EFAC and EQUAD parameters can only be differentiated if there is variance in the uncertainties.\nEquation~\\ref{eqn:N_matrix} describes how EFAC and EQUAD parameters are related and affect the TOA measurement uncertainty.\nTherefore, we used a Gaussian prior on all EFAC parameters with a mean of 1.0 and standard deviation of 0.25; for similar reasons, we applied the same prior to DMEFAC parameters.\nThis choice is further justified in Appendix~\\ref{sec:low_snr}, where we show that the estimated TOA and DM uncertainties based on calculating the Fisher matrix of Equation~\\ref{eqn:toa_lnlike} are accurate down to very low S\/N.\nIt should also be noted that these uncertainties, being based on the Fisher information matrix, are equal to the Cram\\'{e}r-Rao lower bound, which motivates the continued use of EFAC parameters.\nWe use the same prior on EQUAD parameters as is used for both EQUAD and ECORR in \\twelveyr, which is a uniform distribution on log$_{10}$(EQUAD~[s])~$\\in [-8.5, -5.0]$.\nDue to our use of non-uniform priors for EFAC and DMEFAC parameters, we refer to all point estimates from the noise modeling as maximum a posteriori (MAP) values, instead of maximum-likelihood values.\n\n\\textit{Red noise priors --}\nWe use the exact same red noise model and priors as in \\twelveyr, but because the determination of red noise significance differs slightly from \\elevenyr\\ and \\nineyr, and because it will be relevant in the discussion of results, we summarize it here.\nThe red noise is assumed to be a stationary Gaussian process, which we parameterize with a power-law power spectral density $P$ of the form\n\\begin{equation}\n \\label{eqn:red_pl}\n P(f_m) = A^2_{\\small \\textrm{red}} \\left(\\frac{f_m}{1~\\textrm{yr}^{-1}}\\right)^{\\gamma_{\\small \\textrm{red}}},\n\\end{equation}\nwhere $A_{\\small \\textrm{red}}$ is the amplitude of the red noise at a frequency of 1\\,yr$^{-1}$ in units of $\\mu$s~yr$^{1\/2}$, and $\\gamma_{\\small \\textrm{red}}$ is the spectral index.\nThe spectrum is evaluated at thirty linearly spaced frequencies $f_m$ indexed by $m$, incremented by 1\/$T_{\\textrm{span}}$, where $T_{\\textrm{span}}$ is the span of the pulsar's data set.\nThe prior on the red noise amplitude is uniform on log$_{10}$($A_{\\small \\textrm{red}}$~[yr$^{3\/2}$])~$\\in [-20, -12]$, whereas the prior on the red noise index has been constrained in both 12.5-year analyses to be uniform on $\\gamma \\in [-7, -1.2]$.\nA pulsar is deemed to have ``significant red noise'' in these analyses if the Savage-Dickey density ratio \\citep[a proxy for the Bayes factor,][]{dickey1971} estimated from the posterior distribution of log$_{10}$($A_{\\small \\textrm{red}}$) is greater than one hundred.\nVery low-index red noise is thought to primarily arise from imperfect modeling of various effects from the ISM \\citep{ShannonCordes2017} and will be covariant with the white noise parameters.\nIncluding shallow red noise instead of modeling it with only white noise parameters will not significantly change the timing model.\nThe analyses here and in \\twelveyr\\ are only indicative of the presence of red noise, which may or may not be wholly intrinsic to the pulsar; a comparison of the red noise models is presented in Section~\\ref{subsubsec:rn_compare}.\nAdvanced noise modeling of the 11- and 12.5-year data sets, in which we explore bespoke models for each pulsar specifically in the context of GW analyses, is underway and will be presented elsewhere (Simon et al. \\textit{in prep.}).\n\nUpon completion of the noise analysis, following the same protocol as in \\twelveyr, the MAP noise model is included as fixed parameters in the timing model, which is re-optimized using the generalized least squares implementation of \\texttt{Tempo}, now using the augmented, wideband likelihood.\nThe large majority of the reduced chi-squared (goodness-of-fit) values fall between 0.9 and 1.1, with a few larger values.\nSome of these are to be expected because the additional DM data may not be particularly informative, or they may not be modeled well by DMX (e.g., see Section~\\ref{subsec:DM_nu}).\nAs in \\twelveyr, we examined the significance of adding and removing various timing model parameters, but after finding no strong evidence favoring change, we kept the identical set of timing model parameters for ease of comparison.\nThe differences with respect to crossing the significance threshold for including or excluding parameters are marginal, and in several cases are a function of the difference in red noise model (see Section~\\ref{subsubsec:rn_compare}).\n\nThe timing models are summarized in Table~\\ref{tab:timing_model_summary}, which also lists the Bayes factor, $B$, indicating the significance of red noise.\nThere is an analogous table in \\twelveyr\\ containing the results from the analyses of the narrowband data set.\nAs mentioned in Table~\\ref{tab:toa_flags}, we removed ASP and GASP TOAs that were taken simultaneously with concurrent PUPPI or GUPPI observations from the final TOA data sets.\nThe final timing models with noise parameters, curated wideband TOAs, and related auxiliary files are the furnished products comprising this data release.\nWe present the timing residuals and DM time series for these data in Appendix~\\ref{sec:resid}, which includes visual comparisons with the counterpart averaged residuals and DMX models from \\twelveyr.\n\n\\begin{table*}[th!]\n\\centering\n\\caption{Summary of Timing Model Fits\\label{tab:timing_model_summary}}\n\\input{tables\/tab_timing_model_summary}\n\\end{table*}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Average Portraits \\& Flux Density Measurements}\n\\label{subsec:port_results}\n\nA by-product of the profile evolution modeling procedure is a calibrated high S\/N average portrait with a nominal profile alignment and full polarization information.\nThe polarization portraits contain a wealth of information and are of interest to model in their own right; their models could potentially be used to improve the TOA measurement in cases of significant polarization.\nFor sufficiently polarized, large bandwidth, high S\/N data, the rotation measure (RM) could be measured as part of the wideband TOA measurement.\nSuch a development would combine the techniques summarized in Section~\\ref{subsec:prof_evol} with those from \\citet{vS06}, \\cite{vS13}, and \\citet{Oslowski13}, and is an active field of research.\n\nWe also estimated the phase- and frequency-averaged flux density for each of our PUPPI and GUPPI TOA measurements; ASP and GASP data were excluded because the profile data from which TOA measurements were made had been rescaled from their original flux calibration (see \\nineyr\\ for details).\nThe two main assumptions that go into the estimate and its formal, statistical uncertainty are that the profile evolution model sufficiently describes the data (i.e., no model error) and that it has a correct baseline of zero flux density; all phases contribute to the measurement.\nThe frequency-averaged flux density and uncertainty are calculated from the weighted-mean of the phase-averaged flux densities.\nSince the scaling parameters $a_n$ enter the calculation in the same way as for the S\/N estimate, the flux density estimates may contain similar biases (see Appendix~\\ref{sec:low_snr}).\nThe relevant flags for these measurements are listed in Table~\\ref{tab:toa_flags}, including a reference frequency for the flux density estimate.\nNo additional sources of uncertainties are considered, and the interpretation of these measurements should be treated with caution.\n\n\\subsection{Profile Evolution Models}\n\\label{subsec:model_results}\n\nWe find that for the majority of our pulsars, the profile evolution model for a given receiver band requires a single eigenprofile (62 of 102 pulsar-receiver combinations), which can be thought of as the gradient of the mean profile.\nMost of the remainder required two (20 of 102) or zero (13 of 102; i.e., those data are consistent with a constant, non-evolving profile).\nThe few cases in which more than three basis eigenprofiles are used to describe profile evolution arise in two very high S\/N pulsars (3 of 102 have three, the remaining 4 cases have more). \nB1937$+$21 shows spectral leakage from the overlapping, finite-attenuation filters used to subband the data\\footnote{We note that a better choice of filter appears to drastically improve this situation \\citep{Bailes20}.}, which results in the increased number of eigenprofiles in three of its models, and the imperfect correction of the ADC artifact image described in Section~\\ref{sec:obs} has the same consequence for one model for J1713$+$0747.\nRemoving the perhaps spurious eigenprofiles for these pulsars does not appear to significantly change the timing results in Section~\\ref{sec:results}, so we leave them for completeness.\nFurthermore, these two pulsars are observed with both observatories at L-band, and we find that the first two eigenprofiles (which contribute the most to profile evolution) are qualitatively the same between the models from each receiver\n\nProfile broadening from scattering in the ISM or other drastic, intrinsic profile evolution may be responsible for second and third eigenprofiles in the cases where either of those are detected.\nHowever, ``incorrect'' profile alignment with respect to a constant rotation proportional to $\\nu^{-2}$ (corresponding to a small, constant DM offset, generally not larger than, but at most a few times $\\sim$10$^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$) may also be the culprit for additional eigenprofiles.\n\nIt is important to highlight that this subtle issue exists in the narrowband analysis as well; the implicit assumption there is perhaps the most parsimonious one, that the profile shape does not evolve with frequency and that the profiles are aligned in phase.\nThe choice of profile alignment sets the value of the absolute DMs measured and will not have an effect on the timing analyses, though a detailed study of this question is beyond the scope of this paper.\nMore interesting questions about disentangling profile evolution from ISM variations and possible magnetospheric effects are still open \\citep{Hassall12}.\nA possible future development in the context of the present work is to take a similarly parsimonious approach and simultaneously model profile evolution across all observed bands while minimizing the number of significant eigenprofiles as a function of dispersive rotation.\nFurthermore, the underlying physical description of the observed profile evolution also warrants its own investigation.\n\nOne might expect a correlation between the total number of eigenprofiles for each pulsar and the number of FD parameters in the timing models from \\twelveyr.\nWe see a rough correspondence between these two numbers, but its interpretation is dubious.\nFor example, the FD parameters for B1855$+$09 (a.k.a. J1857$+$0943) from \\twelveyr\\ account for an approximate 20~$\\mu$s delay across the profiles in its 430~MHz band, purportedly from unmodeled frequency evolution of the profile shape.\nCareful inspection reveals that its 430~MHz profiles show no evidence for profile evolution, neither in the number of significant eigenprofiles (zero), nor in the profile residuals after subtracting the model, nor by direct comparison of the profiles, whereas there is prominent profile evolution across the L-wide bandwidth.\nEven though the 430~MHz band is a factor of three lower in frequency than L-wide, the latter's narrowband TOAs will be more influential in DM estimation.\nThis can be understood by the much larger fractional bandwidth of the L-wide receiver (see Table~\\ref{tab:observing_systems}): although the dispersive delay across both receiver bands is comparable, the median raw wideband TOA uncertainty from L-wide is an order of magnitude more precise, and its median raw wideband DM uncertainty is $\\sim$5 times smaller (see Figure~\\ref{fig:wb_toa_dm_err}).\nThe spurious FD prediction may arise from the interplay between the relative weighting of the L-band and 430~MHz data in the DMX model, the covariance between FD parameters and DMX values, or perhaps something more interesting; most likely, the FD parameters are filling in for the role of DMJUMP, as mentioned in Section~\\ref{subsec:timing}.\nThe details are beyond the scope of this paper and are under investigation elsewhere.\n\n\\subsection{Frequency-dependent DMs}\n\\label{subsec:DM_nu}\n\nFor a handful of our highest DM pulsars, the DM time series from each frequency band appear significantly different from one another.\nThese trends are apparent in the panels second from the top in Appendix~\\ref{sec:resid} for pulsars J1600$-$3053, J1643$-$1224, J1747$-$4036, and J1903$+$0327 (Figures \\ref{fig:summary-J1600-3053}, \\ref{fig:summary-J1643-1224}, \\ref{fig:summary-J1747-4036}, and \\ref{fig:summary-J1903+0327}, with DMs $\\sim$~52.3, 62.3, 153.0, and 297.5~$\\textrm{cm}^{-3}~\\textrm{pc}$, respectively).\nIt is also readily apparent in these panels, and in many other pulsars' DM time series, that the DM measurements are only significant after the switchover from the older generation of backend instruments (ASP and GASP) to the newer ones (PUPPI and GUPPI) due to their ability to process a larger bandwidth in real time (see Table~\\ref{tab:observing_systems}).\n\nAll four of these pulsars have clear pulse broadening in the form of frequency-dependent tails on the trailing edges of their profile components.\nTo estimate the amount of scattering present in these pulsars, we decomposed their concatenated average portraits into a small number of fixed Gaussian components and an evolving one-sided exponential function \\citep{PDR14}.\nIn this way we estimated the scattering timescale $\\tau$ at 1400~MHz for each of these four pulsars to be $\\tau_{\\textrm{1400}} \\sim$~26, 52, 22, and 130~$\\mu$s, respectively.\n\nIf the scattering timescale is changing with time and is not accounted for in the TOA measurement, the wideband DM measurements will be biased similarly as a function of time.\nAs mentioned in Section~\\ref{subsec:wb_toa_lnlike}, a forthcoming publication will present extensions to the wideband TOA measurement that will be better able to segregate time-variable profile broadening from classical DM variations (Pennucci et al. \\textit{in prep.}).\nThe scattering timescale scales more steeply with frequency than does the dispersive delay (approximately, $\\tau \\propto \\nu^{-4}$), and therefore the wideband DMs measured at lower frequencies will incur a greater bias, since the centroids of scattered pulse components shift by a greater amount.\nHowever, one expects that these biases, even if they are different in magnitude, will be correlated in time.\nConditioned on that assumption, it is difficult to explain the DM time series of these pulsars arising solely from time-variable scattering.\nIn all four instances, there are periods of correlation \\textit{and} anti-correlation between the DM time series measured in each frequency band.\n\nThis sort of behavior is, however, predicted by the phenomenon of ``frequency-dependent DM'' \\citep{Cordes16}, and very similar behavior has been seen in at least one other (canonical) pulsar \\citep{Lam19,Donner19}, although earlier indications existed in B1937$+$21 \\citep{DemorestPhDT,Ramachandran06,Cordes90} and in sparse multi-frequency measurements of the highest DM pulsar \\citep{Pennucci15}.\nThe dispersion measure is defined as the path integral of the free-electron density sampled by a propagating electromagnetic wave.\nDue to the refractive nature of the ISM, the path will vary as a function of the frequency of the wave, and due to the density inhomogeneities in the ISM, the integrated density -- the DM -- will therefore also be a function of frequency.\nHowever, these differences are expected to be small, with root-mean-square (RMS) values typically $\\ll 10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$, and thus only high-precision observations (e.g., bright MSPs, or bright low-frequency sources) of high-DM pulsars over long periods of time are expected to convincingly show this phenomenon.\n\nTo substantiate the claim that the DM trends seen in these four pulsars may arise from this peculiar ISM effect, we can calculate the predicted RMS difference between DMs measured at a fiducial frequency $\\nu$ and a lower frequency $\\nu'$, $\\sigma_{\\overline{\\textrm{DM}}}(\\nu,\\nu')$, using Equations~12 and 15 of \\citet{Cordes16}.\nUsing our rough scattering timescales to estimate the scintillation bandwidths at $\\nu$, and using the appropriate frequencies for each pulsar, we find $\\sigma_{\\overline{\\textrm{DM}}}(\\nu,\\nu') \\approx 2,~4,~2,~\\textrm{and~} 3~\\times~10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$ for J1600$-$3053, J1643$-$1224, J1747$-$4036, and J1903$+$0327, respectively.\nThese values are all within a factor of $\\sim$~2--3 of the RMS differences measured in the observed DM time series: 0.6, 1.7, 2.8, and 5.9~$\\times~10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$, respectively, where we only considered the PUPPI and GUPPI data for these measurements.\nGiven that this quick assessment involves the assumptions that the density inhomogeneities in the ISM are Kolmogorov in nature, and that the scattering occurs in a single thin screen, we find this level of agreement suggestive.\nA more in depth analysis is beyond the scope of this work, but these results indicate that long-term timing of high-DM MSPs in the context of PTA experiments offer a unique opportunity to study this phenomenon, as well as time-variable scattering; the low-frequency, high-cadence observations of CHIME\/Pulsar are especially promising in these areas.\n\n\nIn the two largest DM pulsars (J1747$-$4036 and J1903$+$0327), there are obvious chromatic trends in the timing residuals from \\twelveyr\\ that are ameliorated in the wideband analysis.\nThe narrowband noise analyses compensate for this by having larger white noise parameters and slightly larger, shallower red noise, which helps to explain the timing improvements seen in the wideband data set.\nSimilarly, because the ISM effects appear as apparently chromatic DM measurements in the wideband data set, the DMEFAC parameters are larger than expected ($\\sim$1.5$-$2.0).\nThat is, the boilerplate DMX model may not be good representation of these data, even with DMEFAC and DMJUMP parameters, and more advanced DM and noise models are required.\n\nIn addition to these four pulsars, there are four more in our sample that have a DM $\\gtrsim$~50~$\\textrm{cm}^{-3}~\\textrm{pc}$: J0340$+$4130, B1937$+$21, J1946$+$3417, and B1953$+$29 (a.k.a. J1955$+$2908; Figures \\ref{fig:summary-J0340+4130}, \\ref{fig:summary-B1937+21}, \\ref{fig:summary-J1946+3417}, and \\ref{fig:summary-B1953+29}, with DMs $\\sim$~49.6, 71.1, 110.2, and 104.5~$\\textrm{cm}^{-3}~\\textrm{pc}$, respectively).\nNone of their DM time series show the clear chromatic trends seen in the other four, but they all have some amount of additional variance that inflates their DMEFAC parameters.\nUsing the measured scintillation parameters from \\citet{Levin16} for the three lower DM pulsars and repeating similar calculations as above, we find that the RMS differences predicted from \\citet{Cordes16} are much smaller ($\\sim$ an order of magnitude or more) than what is seen in the data.\nWe could not find a published value for J1946$+$3417, so we estimated its scattering timescale by modeling its profile with Gaussian components in the same fashion as the first four pulsars and find $\\tau_{\\textrm{1400}} \\sim 64~\\mu$s.\nThe predicted and observed RMS DM differences are again similar, $\\sim$~4 and 2~$\\times~10^{-3}~\\textrm{cm}^{-3}~\\textrm{pc}$, respectively, and so frequency dependent DM effects could also be playing a role here.\n\nA couple of other well-timed pulsars with intermediate DM values show the same kind of extra DM variance (e.g., J0613$-$0200; Figure~\\ref{fig:summary-J0613-0200}, DM $\\sim$~38.8~$\\textrm{cm}^{-3}~\\textrm{pc}$, and see also Section~\\ref{subsubsec:rn_compare}).\nNeither frequency-dependent dispersion nor time-variable scattering (in the form of profile broadening) appears to be playing a role here or in the three pulsars mentioned above.\nAnother subtle effect may be at play in some of these pulsars, which is a manifestation of short-timescale variations referred to as ``pulse jitter''.\nPulse jitter arises due to the fact that any finite collection of real single pulses will produce a mean profile with a slightly different shape and location between realizations, despite the long-term stability of the average profile \\citep{Helfand75}.\nFor broadband observations that are significantly influenced by pulse jitter, the wideband DM estimate will be biased (cf. Parthasarathy et al. \\textit{in prep.}).\nDepending on the frequency dependence of pulse jitter \\citep{Lam19b}, the bias may also be strongly frequency dependent.\nAlternatively, the ``finite scintle effect'', in which the relevant time-variable scattering effects occur in the strong diffractive regime \\citep{CordShan10,Cordes90}, can have the same effect as pulse jitter and similarly bias the DM measurements.\nAn investigation of the observed DM variance in some of our pulsars is beyond the scope of this paper, but it will play a role in considerations of future wideband data sets.\n\n\\subsection{Comparison of Collective Timing Results}\n\\label{subsec:timing_results}\n\nIn this section we give an overview of the wideband timing results by assessing the overall characteristics in comparison with the those found in \\twelveyr\\ and addressing a few pulsars individually.\nBesides those already mentioned, and those that will be included in Section~\\ref{subsec:pulsar_results}, specific interesting, astrophysical results from the 12.5-year data set can be found in \\twelveyr; in particular, these include new or improved astrometric and binary timing model parameters.\nThe timing residuals from both data sets are presented in Appendix~\\ref{sec:resid}.\n\n\\subsubsection{Timing Model Parameters}\n\\label{subsubsec:param_compare}\n\n\\input{text\/fig_param_compare}\n\nAs mentioned in Section~\\ref{subsec:timing}, the set of spin, astrometric, and binary timing model parameters used in our analyses is identical to that in \\twelveyr; the phrase ``timing model parameters'' used for the remainder of the text refers to this collection, excluding DMX parameters, which are compared separately.\nThe ensemble of differences in these parameters is shown in Figure~\\ref{fig:param_compare}.\nThe differences between parameter values are plotted, where each difference has been normalized by the parameter uncertainty from \\twelveyr\\ ($\\sigma_{\\textrm{NB}}$), and the ``error bar'' on each difference has a length equal to the ratio of parameter uncertainties, with the uncertainty from \\twelveyr\\ in the denominator (i.e., $\\sigma_{\\textrm{WB}}\/\\sigma_{\\textrm{NB}}$).\nSuch a convention allows us to discuss the relative differences we see in parameters and address their consistency without having to reference their absolute units.\nIn the discussions that follow, we suppress the subscript on $\\sigma_{\\textrm{NB}}$ and use the standalone symbol $\\sigma$ to refer to the units of these normalized differences.\nJUMP parameters are not included in Figure~\\ref{fig:param_compare}, as they are not meaningful, and parameters that reference an epoch were excluded if the epochs differed.\nGenerally, this was not the case; 526 of 549 total parameters (96\\%), not including JUMPs or DMX parameters, were directly compared.\n\nAt a glance, we see that the timing model parameters are in very good agreement, almost entirely $<2\\sigma$ different (99\\% of the parameters), and with very similar parameter uncertainties.\nThe cases in which red noise is detected, or is detected in only one analysis, can be harder to interpret (these parameters are semi-transparent in Figure~\\ref{fig:param_compare}); due to covariance with the MAP red noise model, especially if the red noise is shallow, the parameter uncertainties can differ by a large factor.\nOnly 1 of the 526 parameters (0.2\\%) is $>5\\sigma$ away, while in total 4 (0.8\\%) are $>3\\sigma$ away; for context, if these were independent experiments and we interpreted these differences as random samples from the unit normal distribution, we would ``expect'' $\\sim$~0 deviations $>5\\sigma$ and $\\sim$~1 deviation $>3\\sigma$.\nThe only parameter larger than $5\\sigma$ different is a known peculiarity in our data set (see Section~\\ref{subsubsec:J1640+2224}).\nTwo of the remaining three parameters with a difference larger than $3\\sigma$ belong to J2234$+$0611, but also have larger uncertainties by $\\sim$30\\% (see Section~\\ref{subsubsec:J2234+0611}).\nThe last differing parameter is the parallax measurement of the black widow pulsar J2234$+$0944 (see Section~\\ref{subsubsec:J2234+0944}).\nNevertheless, the parameters agree remarkably well across the board, even in cases where red noise is detected.\n\n\\subsubsection{DMX Parameters}\n\\label{subsubsec:dmx_compare}\n\n\\input{text\/fig_dmx_compare}\n\nWe compare the mean-subtracted DMX model parameters in Figure~\\ref{fig:dmx_compare}, which has the same presentation as Figure~\\ref{fig:param_compare}.\nOf the 4,685 differences, 13 (0.3\\%) are $>5\\sigma$ away ($\\sim$~0 ``expected''), a total of 104 (2.2\\%) are $>3\\sigma$ away ($\\sim$~6 ``expected''), and 94\\% agree to better than $2\\sigma$.\nB1937$+$21 is responsible for 35 of the differences $>3\\sigma$, which are due to the scatter in its DM measurements and the large influence they have on the DMX model (see Sections~\\ref{subsec:DM_nu}~\\&~\\ref{subsubsec:B1937+21}).\nAnother 37 of these belong to the combination of J1713$+$0747 (see below), J1903$+$0327 (Section~\\ref{subsec:DM_nu}), and J2234$+$0944 (Section~\\ref{subsubsec:J2234+0944}).\nThe remaining 32 differences are distributed among 11 pulsars for which we have no particular suspicions.\n\nBesides the influence of differing red noise models that was already mentioned, the FD parameters in the narrowband data set are covariant with all DMX parameters, which makes the interpretation of the uncertainty ratios in Figure~\\ref{fig:dmx_compare} difficult.\nNevertheless, 92\\% of the DMX uncertainties agree to within a factor of 1.5, 99\\% agree to within a factor of two, and the median DMX uncertainty for each pulsar is comparable between the data sets.\n\nThe number of DMX parameters often differs slightly between the data sets by one, two, or three parameters; there are 10, 6, and 4 such instances, respectively, with 26 pulsars having the same number of DMX parameters.\nThe discrepancies in the number of DMX epochs arise from the slight differences in curating the data sets (Section~\\ref{subsec:wb_cleaning}).\nThe exception to this is J1713$+$0747, where we opted to use a higher density of DMX bins during and after the second dip in its DM time series \\citep{Lam2018}, resulting in 37 additional DMX values; a similar binning exception is made for the same reason in \\twelveyr.\nHowever, this means the relevant DMX epochs in \\twelveyr\\ average over a greater span of time and will be biased in comparison to their counterparts here.\nThe DMX time series are plotted in the topmost panels of the figures in Appendix~\\ref{sec:resid}.\nIn most instances, the DMX parameters from \\twelveyr\\ are hidden by those from the present analysis, demonstrating their close agreement.\n\n\\subsubsection{Excess White Noise}\n\\label{subsubsec:wn_compare}\n\n\\input{text\/fig_wn_compare}\n\nA pulse TOA is a proxy for the moment a fixed point of longitude on the neutron star passes over the line of sight.\nThere are a number of sources of uncertainty that obfuscate and bias the determination of this moment in time, even if the formal arrival time of the pulse can be very precisely determined.\nThese additional uncertainties introduce either time-uncorrelated (white) scatter or time-correlated (red) trends into the timing residuals, and can originate from a wide variety of sources local to the observatory, Earth, the solar system, the pulsar, or the intervening ISM.\nFor thorough reviews of the sources of these uncertainties, we direct the reader to \\citet{Verbiest18} and \\citet{CordShan10}.\nHere, we compare the excess white noise seen in both data sets, followed by the red noise in Section~\\ref{subsubsec:rn_compare}.\n\nThe formal TOA uncertainties are scaled in both analyses by EFAC parameters, which do not have a straightforward interpretation with respect to physical, excess noise; nominally, EFAC parameters account for misestimation of the system noise level or template matching errors.\nComparing the EFAC parameters is not very enlightening, particularly because the narrowband analysis uses fixed, non-evolving templates and uses a much broader prior on EFAC.\nThe wideband analysis also uses DMEFAC parameters, which can absorb some excess noise that might be modeled by EFAC in the narrowband analysis.\nAs mentioned in Section~\\ref{subsec:timing}, there is a difference between the two analyses in how white noise is modeled in the timing residuals.\nEQUAD and ECORR parameters capture the additional variance in the narrowband analysis, but because ECORR accounts for fluctuations that are completely correlated for simultaneously obtained measurements (i.e., narrowband TOAs), it cannot be differentiated from EQUAD in the wideband analysis and is therefore left out of the noise model.\n\nTo effectively compare the white noise, we plot the MAP EQUAD parameters from our analyses against the quadrature sum of the corresponding maximum-likelihood EQUAD and ECORR parameters in Figure~\\ref{fig:wn_compare}.\nIn the figure, points appear more opaque in proportion to how constrained the posterior distribution of the parameter is.\nThere is a clear correspondence over two orders of magnitude in the white noise parameters, suggesting that both analyses see very similar white noise.\n\nAlthough the integrated pulse profile shapes of MSPs are secularly stable \\citep[][with some exceptions, e.g., \\citet{slk+16}]{Brook18}, they vary minutely (indeed) on short timescales due pulse jitter (see Section~\\ref{subsec:DM_nu}).\nPulse jitter contributes additional uncertainty to the TOA and is expected to manifest in ECORR parameters, though the measured ECORR values exceed the predicted level of jitter \\citep{Lam16}.\nJitter is thought to be weakly or modestly dependent on frequency \\citep{Shannon14,Lam19b}, and its effects can only be reduced by longer integration times or actively accounting for shape change \\citep{Oslowski11}.\nOn the other hand, the various ISM effects that can contribute to EQUAD have a stronger (and mostly pulsar-independent) frequency dependence.\nFrom this perspective, analyzing narrowband TOAs may help to discriminate between sources of excess white noise, although using evolving profile templates would be an improvement to the overall approach.\nIn this way, both forms of analysis may contribute to arriving at the best results for a given pulsar.\nFor example, if some of our MSPs have large white noise because of time-variable scattering, then the pulse broadening can be included as part of the wideband TOA measurement.\n\n\\subsubsection{RMS Timing Residual}\n\\label{subsubsec:rms_compare}\n\n\\input{text\/fig_rms_compare}\n\nA second metric for gauging the overall level of noise is the RMS timing residual (see Table~\\ref{tab:timing_model_summary}).\nIn Figure~\\ref{fig:rms_compare} we compare the RMS values between the two analyses, taking care to use the averaged residuals from the narrowband data set and the whitened set of residuals whenever red noise was detected in either of the analyses.\nAlmost no pulsars differ by more than a factor of 1.5, with a number of exceptions explained as part of Section~\\ref{subsec:pulsar_results}, and half of them agree to within a factor of 1.1.\nHowever, the RMS residual from either analysis can be very sensitive to the exact noise model, which is fixed in the final optimization of the timing model.\nThe noise analyses explore the logarithm of the EQUAD, ECORR, and red noise amplitude parameters, so small statistical deviations in the best-fit parameters arising from the Monte Carlo analysis can lead to rather different RMS values.\nThe RMS should be thought of as a random variable, whose variance is influenced by the posterior distributions of the noise parameters.\nThis is true even when the noise model parameters are constrained, and it underscores the need for advanced noise modeling techniques.\nNevertheless, it is encouraging that 31 of the 47 pulsars show some amount of improvement and that all but five pulsars have RMS residuals no more than $\\sim10\\%$ larger than their narrowband counterparts.\n\n\n\n\n\\subsubsection{Detection of Red Noise}\n\\label{subsubsec:rn_compare}\n\n\\input{text\/fig_rn_compare}\n\nA final, and perhaps most crucial, litmus test for the wideband analyses is the detection of red noise in individual pulsars.\nObviously, the presence of red noise in the wideband data set (or lack thereof), in relation to what is seen in the narrowband data set, guides our expectations of full-scale GW analyses, which heretofore have only been vetted on our narrowband data sets.\nWe introduced the red noise model in Section~\\ref{subsec:timing}; there are additional details in Appendix~\\ref{sec:wb_like}, \\nineyr, \\elevenyr, and \\twelveyr.\nHere we discuss our findings in contrast to those from the narrowband analysis.\n\nIn Figure~\\ref{fig:rn_compare} we show the significantly detected power-law red noise in our analyses compared to those from \\twelveyr.\nWe again find the level of agreement between the data sets reassuring.\nRecall from Section~\\ref{subsec:timing} that a pulsar is deemed to have ``significant red noise'' if the estimated Bayes factor is above one hundred (see Table~\\ref{tab:timing_model_summary} for Bayes factors).\nThirteen pulsars have detected red noise in both analyses, one pulsar has significant red noise detected in just the narrowband analysis (J0613$-$0200), and two black widow pulsars, which are not shown in the plot, are treated differently and not discussed further here (J0023$+$0923 and J2234$+$0944; see Section~\\ref{subsubsec:J2234+0944}).\nTen of these pulsars (plus J1713$+$0747) had detected red noise in \\elevenyr; J1744$-$1134, J1853$+$1303, and J2317$+$1439 are new detections in \\twelveyr, which are all have significant red noise here.\n\nIt is thought that unmitigated ISM effects can manifest as shallow-spectrum red noise \\citep[][]{ShannonCordes2017,CordShan10,FosterCordes90,Rickett90}, which we indicate in Figure~\\ref{fig:rn_compare} for $\\gamma_{\\textrm{\\scriptsize red}} > -3$.\nJ0613$-$0200's DM is in the top third of our sample ($\\sim$ 38.8~$\\textrm{cm}^{-3}~\\textrm{pc}$, respectively), and has fairly shallow red noise in its narrowband analyses.\nRed noise is only marginally favored in its wideband analysis, as indicated by the Bayes factors of $\\sim$~15 in Table~\\ref{tab:timing_model_summary}.\nWhen red noise is included in the wideband analysis (the dashed-dotted lines in Figure~\\ref{fig:rn_compare}), the MAP model has the same index, a slightly smaller amplitude, and similar white noise parameters, than in \\twelveyr.\nWithout red noise, the corresponding wideband white noise EQUAD parameters are statistically unchanged.\nThis suggests that the wideband analysis might be able to mitigate some of the ISM-induced red noise.\n\n\nIntrinsic spin noise in pulsars has been modeled in the literature as a random walk in phase, frequency, or frequency derivative, with corresponding power spectral indices of $-2$, $-4$, and $-6$, respectively, as well as arising from chaotic behavior \\citep[e.g.,][]{Harding90}.\nThe lighter gray region in Figure~\\ref{fig:rn_compare} represents the best fit index ($\\gamma_{\\textrm{\\scriptsize spin}} = -4.46 \\pm 0.16$) for timing noise seen across pulsars of all types from \\citet{Lam17a}, consistent with a mixture of random walks \\citep[e.g.,][]{DAlessandro95,CordesDowns85}.\nThe scatter in this best fit relation, however, is large enough to essentially cover the range of observed spectra.\nIt is therefore difficult to interpret the spread of red noise we have detected, particularly because we suspect that some of the pulsars with shallow red noise are dominated by contributions from the ISM, whereas others may have a mix of contributions.\nCoexisting with the red noise intrinsic to the pulsar and that from the ISM, there is a contribution from the background of stochastic, low-frequency GWs, which is thought to have a steep power-law index ($\\gamma_{\\textrm{\\scriptsize GWB}} = -13\/3$; \\citet{Jaffe2003,Phinney2001}), indicated by a dotted vertical line in the figure.\nFor scale, the dashed vertical line indicates the 95\\% upper limit on the amplitude of the GW background from analyzing the 11-year data set \\citep{Arzoumanian2018b}.\nA more recent search for the stochastic GW background in the 12.5-year narrowband data set is presented in \\citet{Arzoumanian20}.\n\n\n\n\\subsection{Additional Discussion of Individual Pulsars}\n\\label{subsec:pulsar_results}\n\nThe results from a number of pulsars, some of which have been previously mentioned, deserve additional comments, caveats, or emphasis, which we detail here.\nIn addition, for the simple purpose of highlighting one example of generally good, comprehensive agreement with the narrowband results, and one example of where perhaps wideband timing did not prove beneficial, we direct the reader to J0931$-$1902 and J1910$+$1256, respectively.\n\n\\subsubsection{J0931$-$1902}\n\\label{subsubsec:J0931-1902}\n\nJ0931$-$1902 has the distinction of having the largest fractional difference between the number of pulse profiles used in the wideband analysis and the number of TOAs in the narrowband analysis; its wideband data set makes use of 81\\% more profiles (see Table~\\ref{tab:ntoa_nchan}).\nAgain, this difference arises because of the S\/N threshold used in the narrowband analysis; because this pulsar is fairly weak and scintillates, a large number of its low S\/N profiles get individually discarded in the narrowband analysis, even though they combine to yield useful wideband TOAs.\nJ0931$-$1902 is the second worst pulsar in our data set in terms of raw L-band timing precision (see Figure~\\ref{fig:wb_toa_dm_err}), but is somewhere in the middle in terms of RMS ($\\sim$~440~ns in both data sets).\nThere is absolutely nothing else different about its results from the narrowband analysis -- except that its timing model parameters are all $\\sim$~15\\% more precise in the wideband analysis.\nAt least two other pulsars show this level of improvement that is most likely attributable to a similar explanation -- J0340$+$4130 and J0740$+$6620 -- although their differences in data volume are not extreme.\nThese improvements underscore the benefit of using the wideband TOA approach for salvaging all information contained in less bright or scintillating pulsars.\n\n\n\\subsubsection{J1640$+$2224}\n\\label{subsubsec:J1640+2224}\n\nThe difference in J1640$+$2224's ecliptic longitude is the lone culprit referred to earlier for being very different ($\\sim$~6$\\sigma$) from its counterpart in the narrowband analysis.\nHowever, this is a known anomaly to us, albeit of unknown origin; we have previously compared timing results from different timing software using the exact same data sets, and J1640$+$2224's ecliptic longitude was the single outlier to be significantly different (see also the comparison between \\texttt{Tempo} and \\texttt{PINT} \\citep{PINT} in \\twelveyr).\nThe published position from Very Long Baseline Interferometry \\citep{Vigeland18} is not precise enough to discern between the two measurements.\nHowever, it should be noted that the value from \\twelveyr\\ is better than 1$\\sigma$ consistent with the extrapolated value from \\elevenyr, whereas the value from the wideband analysis is $\\sim$~2$\\sigma$ consistent with the extrapolated value from \\nineyr.\n\\citet{Fonseca2016} followed up on \\nineyr\\ and suspected that J1640$+$2224 is a massive neutron star \\citep[see also][]{Deng20}; the improvements on the mass measurements will be presented elsewhere.\n\n\\subsubsection{J1643$-$1224}\n\\label{subsubsec:J1643-1224}\n\nWe have already discussed J1643$-$1224 at some length in Section~\\ref{subsec:DM_nu}.\nIt is worth emphasizing, though, that some of the complexity and chromatic dependence seen in the DM measurements and timing residuals of this pulsar almost certainly arise from the fact that it lies directly behind the HII region Sh 2-27 associated with $\\zeta$-Ophiuchi \\citep{Ocker20}.\nThis association may also be responsible for a protracted decrease in its flux density \\citep{Maitia03}.\nIn addition to the confounding factors of the ISM, at least one intrinsic profile shape change event is thought to have occurred in this pulsar around February 2015 \\citep{slk+16}.\nAlthough we see the corresponding discrete perturbation in J1643$-$1224's timing residuals at this time, the follow-up analysis by \\citet{Brook18} on our 11-year data set argues that ISM effects cannot be ruled out.\nThe $\\sim$~45\\% improvement in its RMS timing residual seen in Figure~\\ref{fig:rms_compare} is almost certainly a result of the mitigation of the chromatic structure in its residuals; see the discussion at the end of Section~\\ref{subsec:DM_nu}.\n\n\\subsubsection{J1747$-$4036}\n\\label{subsubsec:J1747-4036}\n\nSimilarly, we have already discussed J1747$-$4036 in Section~\\ref{subsec:DM_nu}.\nJ1747$-$4036 also stands out in Figure~\\ref{fig:rms_compare}, with the same level of improvement in RMS residual as J1643$-$2224 due to the mitigation of chromatic structure in the timing residuals.\n\n\\subsubsection{J1910$+$1256}\n\\label{subsubsec:J1910+1256}\n\nJ1910$+$1256 has a significantly worse RMS wideband timing residual in Figure~\\ref{fig:rms_compare}, by just over a factor of two.\nWe find a significant L-band EQUAD detected in the wideband analysis, whereas the posterior distributions for the L-band EQUAD and ECORR are consistent with upper-limits.\nInterestingly, the white noise parameters for S-band are significantly measured in both analysis and are of similar amplitude.\nJ1910$+$1256 is in the top ten pulsars by raw L-band timing precision (Figure~\\ref{fig:wb_toa_dm_err}), with the median L-band PUPPI TOA having a precision just above 100~ns; the MAP PUPPI L-band EQUAD is more than three times larger.\nThe source of this discrepancy has not been determined but despite the difference, the timing model parameters are no more than $\\sim$~10\\% worse than their narrowband counterparts.\n\n\\subsubsection{B1937$+$21}\n\\label{subsubsec:B1937+21}\n\nB1937$+$21 (a.k.a. J1939$+$2134) presents a special set of challenges for the wideband analysis, being the brightest pulsar in the data set with the smallest formal measurement uncertainties by a considerable margin (see Figure~\\ref{fig:wb_toa_dm_err}).\nAs mentioned in Section~\\ref{subsec:prof_evol}, its profile modeling is contaminated by spectral leakage because it is so bright, although we do not believe this meaningfully affects the timing results.\nAs mentioned in Section~\\ref{subsec:DM_nu}, it has a substantial amount of scatter in its wideband DM measurements once the long-term trend is removed; this results in the highest DMEFAC parameters in the data set as well as the worst goodness-of-fit value for its timing model, due to the additional contribution from the DM model.\nThe restrictive Gaussian prior (see Section~\\ref{subsec:timing}) inhibits the DMEFAC parameters from taking even larger values, which would encapsulate more of the variance in the DM time series.\nRelaxing the prior is not physically motivated, and so these results direct us to implement an additional DM model parameter in future analyses, one that is analogous to the standard EQUAD parameter.\nGiven that both pulse jitter and variable diffractive interference in the ISM (i.e., the ``finite scintle effect'') play a roll in the observations of this pulsar \\citep{Lam19b}, it is feasible that both effects serve to bias the wideband DM estimates, resulting in extra variance in the DM time series.\nDespite the additional variance in the wideband DM time series, the astrometric timing model parameters are in very good agreement ($<$~1$\\sigma$) with similar uncertainties.\n\n\\subsubsection{J1946$+$3417}\n\\label{subsubsec:J1946+3417}\n\nJ1946$+$3417 is one of the two new pulsars in this data set, which has already been discussed in Section~\\ref{subsec:DM_nu} due to it having the third largest DM in the data set ($\\sim$110.2~$\\textrm{cm}^{-3}~\\textrm{pc}$).\nIt has the distinction of showing the single largest difference in RMS in either direction, seen in Figure~\\ref{fig:rms_compare}; the wideband RMS timing residual is a factor of three smaller.\nBoth analyses examine the same amount of data and the wideband raw timing precision is $\\sim$~10\\% better (see Table~\\ref{tab:toa_summary} and the equivalent table in \\twelveyr).\nThe Bayes factor for red noise in the narrowband analysis is $\\sim$~59, whereas it is not at all favored in the wideband analysis.\nThe preferred red noise model is large and shallow, and as a result of it not being included, the narrowband white noise parameters are larger than their wideband counterparts.\nThe timing model parameters agree to $\\le 1\\sigma$, but with $\\sim$~10\\% larger uncertainties in the wideband analysis.\nIt should also be noted that J1946$+$3417 is an astrophysically interesting source, as it is one of the few eccentric binary MSPs in the field and also contains a massive neutron star \\citep[][all of which also make note of J2234$+$0611]{Barr17,Jiang15,Antoniadis14,Freire14}.\n\n\n\\subsubsection{J2043$+$1711}\n\\label{subsubsec:J2043+1711}\n\nJ2043$+$1711 has the highest sub-threshold Bayes factor in Table~\\ref{tab:timing_model_summary}, $B\\sim50$.\nIn repeated analyses, the statistic $B$ was noisy enough to sometimes cross our significance threshold.\nThe narrowband analysis also favors red noise, but with a lower Bayes factor~$\\sim$~26.\nThe difference between the analyses may arise from the amount of data examined.\nAs can be seen in Table~\\ref{tab:ntoa_nchan}, J2043$+$1711's wideband data set is $\\sim$~70\\% larger than its narrowband counterpart, the third largest difference, which is due to its scintillation characteristics combined with the S\/N ratio cut off in the narrowband analysis.\nJ2043$+$1711 has a fairly low DM ($\\sim$20.8~$\\textrm{cm}^{-3}~\\textrm{pc}$), a timing baseline of six years and, importantly, it has been included in our high-cadence observations at Arecibo since 2015, which has increased its data volume by $\\sim$~70\\% since the 11-year data set.\nThe narrow features in its profile enable this pulsar to be timed very precisely when it is detected (see Tables~\\ref{tab:toa_summary}~\\&~\\ref{tab:timing_model_summary}, and Figure~\\ref{fig:wb_toa_dm_err}), and so we expect the emerging red noise in this pulsar to be significantly detected in the near future.\n\n\\subsubsection{J2234$+$0611}\n\\label{subsubsec:J2234+0611}\n\nAt face value, J2234$+$0611 is the best timed pulsar in the data set: the narrowband and wideband timing RMS values are $\\sim$~60 and 35~ns, respectively (Figure~\\ref{fig:rms_compare}), with no preference for red noise in either analysis.\nThis is partially due to it only having a timing baseline 3.4~years in length, although it is in the top ten pulsars by raw L-band timing precision.\nBoth analyses detect excess white noise, although the wideband analysis measures a significantly larger EQUAD in the 430~MHz band; this results in an overweighting of the wideband L-band data, which may explain the significantly smaller RMS value.\nWe make special mention of this pulsar also because it stands out for its level of disagreement in its timing model parameters with \\twelveyr, as mentioned in Section~\\ref{subsubsec:param_compare}.\nAll of its wideband timing model parameters have larger uncertainties by $\\sim$~30$-$40\\%, but no other pulsar shows quite this level of disagreement.\nIts ecliptic latitude and parallax measurements are $\\sim$~3.5$\\sigma$ different from their narrowband analysis counterparts.\nThe fact that J2234$+$0611 has a relatively short timing baseline but has significantly measured secular binary parameters is a testament to its timing precision.\nIn fact, additional modeling of its binary orbit is necessary, which was carried out in \\citet{Stovall2019} with an additional 1.5~years of data, most of which were NANOGrav observations collected beyond the cutoff of the present data set.\nAlong with the Shapiro delay and annual orbital parallax, \\citet{Stovall2019} were able to determine the 3-D orbital geometry of the binary.\nWe are confident that the discrepancies seen here will be resolved with the implementation of the \\citet{Stovall2019} timing solution in future data sets.\n\n\\subsubsection{J2234$+$0944}\n\\label{subsubsec:J2234+0944}\n\nJ2234$+$0944 was previously mentioned in Sections~\\ref{subsubsec:param_compare}, \\ref{subsubsec:dmx_compare}, and \\ref{subsubsec:rn_compare}.\nThis pulsar is one of four black widow pulsars in the data set (along with J0023$+$0923, J0636$+$5128, and J2214$+$3000), and one of two (along with J2214$+$3000) that do not show orbital or secular variability according to \\citet{BakNielsen20}, who studied three of these systems over $\\sim~8$~year baselines (J0636$+$5128 was not part of their study).\nThis pulsar has very significantly detected ``red noise'' in the wideband analysis, but no indication of it in the narrowband analysis, according to the Bayes factors in Table~\\ref{tab:timing_model_summary} and its analog in \\twelveyr.\nHowever, this ``red noise'' is specious; the preferred model is extremely shallow and the power-law fit to the function in Equation~\\ref{eqn:red_pl} is dominated by the frequencies higher than 1\\,yr$^{-1}$, reflecting the short timing baseline ($T_\\textrm{span} \\sim 4.0$~yr).\nBased on this reasoning, we exclude the red noise component from J2234$+$0944's analysis; we similarly excluded the ``red noise'' seen in J0023$+$0923 for the same reason.\nThe origin of the excess noise seen in the wideband data set is not known; it could be a sign of variability \\citep{Torres17}, but the findings of \\citet{BakNielsen20} refute this.\nInterestingly, the black widow J2214$+$3000 had a very similar issue in \\elevenyr, when it had a similar timing baseline, which was resolved with the additional data in this data set and a reparameterization of its timing model.\nFinally, \\twelveyr\\ reports that the parallax measurement is no longer significant, in contrast to \\elevenyr; this loss of significance in \\twelveyr\\ is marginal and not the case in the wideband analysis.\nDespite being $\\sim$~3$\\sigma$ different from \\twelveyr, J2234$+$0944's parallax is significantly measured and is $< 1\\sigma$ consistent with the value from \\elevenyr.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\nInteracting particle systems have attracted a lot of attention\nbecause of their versatile modelling power (see for instance\n\\cite{Durrett99,Liggett97}). However, most available results deal\nwith their asymptotic behavior, and relatively few theorems\ndescribe their transient regime. In particular, central limit\ntheorems for random fields have been available for a long time\n\\cite{Malyshev75,Neaderhouser78,Bolthausen82,Newman80,\nTakahata83,CometsJanzura,ChenShao04}, diffusion approximations and\ninvariance principles have an even longer history\n(\\cite{EthierKurtz} and references therein), but those\nfunctional central limit theorems that describe the\ntransient behavior of an interacting particle system are usually much\nless general than their fixed-time counterparts.\nExisting results (see\n\\cite{HolleyStrook78,HolleyStrook79,KipnisVaradhan,SethuramanXu})\nrequire rather stringent hypotheses:\nspin flip dynamics on $\\mathbb{Z}$, reversibility,\nexponential ergodicity, stationarity\\ldots (see Holley and Strook's\ndiscussion in the introduction of \\cite{HolleyStrook79}).\nThe main objective of this article is to prove a functional central\nlimit theorem for interacting particle systems,\nunder very mild hypotheses, using some new\ntechniques of weakly dependent random fields.\n\nOur basic reference on interacting particle systems\nis the textbook by Liggett \\cite{Liggett}, and\nwe shall try to keep our notations as close to his as possible:\n$S$ denotes the (countable) set of sites, $W$ the (finite) set of\nstates, ${\\cal X}=W^S$ the set of configurations, and\n$\\{\\eta_t\\,,\\;t\\geq 0\\}$ an interacting particle system, i.e. a\nFeller process with values in ${\\cal X}$. If $R$ is a finite\nsubset of $S$, an empirical process is defined by counting how\nmany sites of $R$ are in each possible state at time $t$. This\nempirical process will be denoted by $N^R = \\{N^R_t\\,,\\;t\\geq\n0\\}$, and defined as follows.\n$$\nN^R_t = (N^R_t(w))_{w\\in W}\\;,\\quad N^R_t(w) = \\sum_{x\\in R}\n\\mathbb{I}_w(\\eta_t(x))\\;,\n$$\nwhere $\\mathbb{I}_w$ denotes the indicator function of state $w$. Thus\n$N^R_t$ is a $\\mathbb{N}^W$-valued stochastic process, which is not\nMarkovian in general. Our goal is to show that, under suitable\nhypotheses, a properly scaled version of $N^R$ converges to a\nGaussian process as $R$ increases to $S$. The hypotheses will be\nprecised in sections \\ref{basic} and \\ref{covar} and the main result\n(Theorem \\ref{cltips2}) will be stated and proved in section\n\\ref{clt}. Here is a loose description of our assumptions. Dealing\nwith a sum of random variables, two hypotheses can be made for a\ncentral limit theorem: weak dependence and identical distributions.\n\\begin{enumerate}\n\\item {\\it Weak dependence:} In order to give it a sense, one has\nto\n define a distance between sites, and therefore a graph\n structure. We shall first suppose that this (undirected) graph\n structure has bounded degree. We shall assume also finite range\n interactions: the configuration can\n simultaneously change only on a bounded set of sites, and its value\n at one site can influence transition rates only up to a fixed\n distance (Definition \\ref{defetenduefinie}).\n Then if $f$ and $g$ are two functions whose dependence on\n the coordinates decreases exponentially fast with the distance from\n two distant finite sets $R_1$ and $R_2$,\n we shall prove that the covariance between\n $f(\\eta_s)$ and $g(\\zeta_t)$ decays exponentially fast in the\n distance between $R_1$ and $R_2$ (Proposition\n \\ref{propinegcovfini}). The central limit theorem\n \\ref{cltips2} will actually be proved in a much narrower setting,\n that of group invariant dynamics on a transitive graph (Definition\n \\ref{definvariant}). However we believe that a covariance inequality\n for general finite range interacting particle systems is of\n independent interest. Of course the bound of Proposition\n \\ref{propinegcovfini} is not uniform in time, without further\n assumptions.\n \n \n \n\\item {\\it Identical distributions:} In order to ensure that the\nindicator processes $\\{\\mathbb{I}_w(\\eta_t(x))\\,,\\;t\\geq 0\\}$ are\nidentically distributed, we shall assume that the set of sites $S$\nis endowed with a transitive graph structure (see\n\\cite{GodsilRoyle} as a general reference), and that both the\ntransition rates and the initial distribution are invariant by the\nautomorphism group action. This generalizes the notion of\ntranslation invariance, usually considered in $\\mathbb{Z}^d$\n(\\cite{Liggett} p.~36), and can be applied to non-lattice graphs\nsuch as trees. Several recent articles have shown the interest of\nstudying random processes on graph structures more general than\n$\\mathbb{Z}^d$ lattices: see e.g.\n\\cite{Haggstrometal02,HaggstromPeres99,Haggstrometal00}, and\n for general references \\cite{Peres99,Woess00}.\n\\end{enumerate}\nAmong the potential applications of our result, we chose to focus on\nthe hitting time of a prescribed level by a linear combination of\nthe empirical process. In \\cite{ParoissinYcart04}, such hitting\ntimes were considered in the application context of reliability.\nIndeed the sites in $R$ can be viewed as components of a coherent\nsystem and their states as degradation levels. Then a linear\ncombination of the empirical process is interpreted as the global\ndegradation of the system, and by Theorem \\ref{cltips2}, it is\nasymptotically distributed as a diffusion process if the number of\ncomponents is large. An upper bound for the degradation level can be\nprescribed: the system is working as soon as the degradation is\nlower, and fails at the hitting time. More precisely, let $f~:\nw\\mapsto f(w)$ be a mapping from $W$ to $\\mathbb{R}$. The total degradation\nis the real-valued process $D^R=\\{D^R_t\\,,\\;t\\geq 0\\}$, defined by:\n$$\nD^R_t = \\sum_{w\\in W} f(w)N^R_t(w).\n$$\nIf $a$ is the prescribed level, the failure time of the system\nwill be defined as the random variable\n$$\nT^R_a = \\inf\\{t\\,\\geq 0\\,,\\; D^R_t\\geq a\\,\\}.\n$$\nUnder suitable hypotheses, we shall prove that $T^R_a$ converges\nweakly to a normal distribution, thus extending Theorem 1.1 of\n\\cite{ParoissinYcart04} to systems with dependent components.\nIn reliability (see \\cite{BarlowProschan} for a general reference),\ncomponents of a coherent system are usually considered as\nindependent. The reason seems to be mathematical convenience rather\nthan realistic modelling. Models with dependent components have been\nproposed in the setting of stochastic Petri nets\n\\cite{LiuChiou01,Volovoi04}. Observing that a Markovian Petri net can\nalso be interpreted as an interacting particle system, we believe that\nthe model studied here is versatile enough to be used in practical\napplications.\n\nThe paper is organized as follows. Some basic facts about\ninteracting particle systems are first recalled in section\n\\ref{basic}. They are essentially those of sections I.3 and I.4 of\n\\cite{Liggett}, summarized here for sake of completeness, and in\norder to fix notations. The covariance inequality for finite range\ninteractions and local functions will be given in section\n\\ref{covar}. Our main result, Theorem \\ref{cltips2}, will be stated\nin section \\ref{clt}. Some examples of transitive graphs are\nproposed in section \\ref{graphs}. The application to hitting times\nand their reliability interpretation is the object of section\n\\ref{hitting}. In the proof of Theorem \\ref{cltips2}, we need a\nspatial CLT for an interacting particle system at fixed time, i.e. a\nrandom field. We thought interesting to state it independently in\nsection \\ref{cltfields}: Proposition \\ref{pro2r} is in the same vein\nas the one proved by Bolthausen \\cite{Bolthausen82} on\n$\\mathbb{Z}^d$, but it uses a somewhat different technique. All\nproofs are postponed to section \\ref{proofs}.\n\\section{Main notations and assumptions}\n\\label{basic}\nIn order to fix notations, we briefly recall the basic\nconstruction of general interacting particle systems, described in\nsections I.3 and I.4 of Liggett's book \\cite{Liggett}.\n\nLet $S$ be a countable set of sites, $W$ a finite set of states,\nand ${\\cal X}=W^S$ the set of configurations, endowed with its\nproduct topology, that makes it a compact set. One defines a\nFeller process on ${\\cal X}$ by specifying the local transition\nrates: to a configuration $\\eta$ and a finite set of sites $T$ is\nassociated a nonnegative measure $c_T(\\eta,\\cdot)$ on $W^T$.\nLoosely speaking, we want the configuration to change on $T$ after\nan exponential time with parameter\n$$\nc_{T,\\eta} = \\sum_{\\zeta\\in W^T} c_T(\\eta,\\zeta).\n$$\nAfter that time, the configuration becomes equal to $\\zeta$ on\n$T$, with probability $c_T(\\eta,\\zeta)\/c_{T,\\eta} $. Let\n$\\eta^\\zeta$ denote the new configuration, which is equal to\n$\\zeta$ on $T$, and to $\\eta$ outside $T$. The infinitesimal\ngenerator should be:\n\\begin{equation}\n\\label{defgenerateur} \\Omega f(\\eta) = \\sum_{T\\subset S}\n\\sum_{\\zeta\\in W^T} c_T(\\eta,\\zeta)(f(\\eta^\\zeta)-f(\\eta)).\n\\end{equation}\nFor $\\Omega$ to generate a Feller semigroup acting on continuous\nfunctions from $X$ into $\\mathbb{R}$, some hypotheses have to be imposed\non the transition rates $c_T(\\eta,\\cdot)$.\n\nThe first condition is that the mapping $\\eta\\mapsto\nc_T(\\eta,\\cdot)$ should be continuous (and thus bounded, since\n${\\cal X}$ is compact). Let us denote by $c_T$ its supremum norm.\n$$\nc_T = \\sup_{\\eta\\in X} \\, c_{T,\\eta}.\n$$\nIt is the maximal rate of change of a configuration on $T$. One\nessential hypothesis is that the maximal rate of change of a\nconfiguration at one given site is bounded.\n\\begin{equation}\n\\label{hyp0} B = \\sup_{x\\in \\,S}\\ \\sum_{T\\ni\\, x} c_T <\\infty.\n\\end{equation}\nIf $f$ is a continuous function on ${\\cal X}$, one defines\n$\\Delta_f(x)$ as the degree of dependence of $f$ on $x$:\n$$\n\\Delta_f(x) = \\sup\\{\\,|f(\\eta)-f(\\zeta)|\\,,\\;\\eta,\\zeta\\in X\n\\mbox{ and } \\eta(y) = \\zeta(y)\\; \\forall\\, y\\neq x\\,\\}.\n$$\nSince $f$ is continuous, $\\Delta_f(x)$ tends to $0$ as $x$ tends to\ninfinity, and $f$ is said to be {\\it smooth} if $\\Delta_f$ is\nsummable:\n$$\n\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert f\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert = \\sum_{x\\,\\in\\, S} \\Delta_f(x) <\\infty .\n$$\nIt can be proved that if $f$ is smooth, then $\\Omega f$ defined by\n(\\ref{defgenerateur}) is indeed a continuous function on ${\\cal\nX}$ and moreover:\n$$\n\\|\\Omega f\\| \\leq B \\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert f\\vert\\hspace{-0.6mm}\\vert\\hspace{-0.6mm}\\vert .\n$$\nWe also need to control the dependence of the transition rates on\nthe configuration at other sites. If $y\\in S$ is a site, and\n$T\\subset S$ is a finite set of sites, one defines\n\n$$\nc_T(y) = \\sup \\{ \\, \\| c_T(\\eta_1,\\,\\cdot\\,) -\nc_T(\\eta_2,\\,\\cdot\\,)\\|_{tv}\\,,\\; \\eta_1(z)=\\eta_2(z)\\;\\forall\\,\nz\\neq y\\,\\} ,\n$$\nwhere $\\|\\,\\cdot\\,\\|_{tv}$ is the total variation norm:\n$$\n\\| c_T(\\eta_1,\\,\\cdot\\,) - c_T(\\eta_2,\\,\\cdot\\,)\\|_{tv} =\n\\frac{1}{2} \\sum_{\\zeta \\in W^T} | c_T(\\eta_1,\\zeta) -\nc_T(\\eta_2,\\zeta)|.\n$$\nIf $x$ and $y$ are two sites such that $x\\neq y$, the {\\it influence} of\n$y$ on $x$ is defined as:\n$$\n\\gamma(x,y) = \\sum_{T\\,\\ni\\, x} c_T(y).\n$$\nWe will set $\\gamma(x,x)=0$ for all $x$. The influences\n$\\gamma(x,y)$ are assumed to be summable:\n\\begin{equation}\n\\label{hypl1} M = \\sup_{x\\in\\, S}\\ \\sum_{y\\in\\, S} \\gamma(x,y) <\n\\infty.\n\\end{equation}\nUnder both hypotheses (\\ref{hyp0}) and (\\ref{hypl1}), it can be\nproved that the closure of $\\Omega$ generates a Feller semigroup\n$\\{ S_t\\,,\\,t\\geq 0\\}$ (Theorem 3.9 p.~27 of \\cite{Liggett}). A\ngeneric process with semigroup $\\{S_t\\,,\\,t\\geq 0\\}$ will be\ndenoted by $\\{ \\eta_t\\,,\\,t\\geq 0\\}$. Expectations relative to its\ndistribution, starting from $\\eta_0=\\eta$ will be denoted by\n$\\mathbb{E}_\\eta$. For each continuous function $f$, one has:\n$$\nS_t f(\\eta) = \\mathbb{E}_\\eta[ f(\\eta_t) ] =\n\\mathbb{E}[f(\\eta_t)\\,|\\,\\eta_0=\\eta].\n$$\nAssume now that $W$ is ordered, (say $W=\\{1,\\ldots,n\\}$). Let\n${\\cal M}$ denote the class of all continuous functions on $X$\nwhich are monotone in the sense that $f(\\eta)\\leq f(\\xi)$ whenever\n$\\eta\\leq \\xi$. As it was noticed by Liggett (1985) it is\nessential to take advantage of monotonicity in order to prove\nlimit theorems for particle systems. The following theorems\ndiscuss a number of ideas related to monotonicity.\n\\begin{theo}[Theorem 2.2 Liggett, (1985)]\\label{thm1}\nSuppose $\\eta_t$ is a Feller process on $X$ with semigroup $S(t)$.\nThe following statement are equivalent :\n\\begin{enumerate}\n\\item[(a)] $f\\in {\\cal M}$ implies $S(t)f\\in {\\cal M}$, for all\n$t\\geq 0$\n\\item[(b)] $\\mu_1\\leq \\mu_2$ implies $\\mu_1S(t)\\leq \\mu_2S(t)$ for\nall $t\\geq 0$.\n\\end{enumerate}\nRecall that $\\mu_1\\leq \\mu_2$ provided that $\\int fd\\mu_1\\leq \\int\nfd\\mu_2$ for any $f\\in {\\cal M}$.\n\\end{theo}\n\\begin{defi}\nA Feller process is said to be monotone (or attractive) if the\nequivalent conditions of Theorem \\ref{thm1} are satisfied.\n\\end{defi}\n\\begin{theo}[Theorem 2.14 Liggett, (1985)]\\label{thm2}\nSuppose that $S(t)$ and $\\Omega$ are respectively the semigroup\nand the generator of a {\\bf monotone} Feller process on $X$.\nAssume further that $\\Omega$ is a {\\bf{bounded}} operator. Then\nthe following two statements are equivalent:\n\\begin{enumerate}\n\\item[(a)] $\\Omega fg\\geq f\\Omega g+ g\\Omega f$, for all $f$,\n$g\\in {\\cal M}$\n\\item[(b)] $\\mu S(t)$ has positive correlations whenever $\\mu$\ndoes.\n\\end{enumerate}\nRecall that $\\mu$ has positive correlation if $\\int fgd\\mu\\geq\n\\left(\\int fd\\mu\\right)\\left(\\int gd\\mu\\right)$ for any $f,g\\in\n\\,{\\cal M}$.\n\\end{theo}\nThe following corollary gives conditions under which the positive\ncorrelation property continue to hold at later times if it holds\ninitially.\n\\begin{coro}\\label{corligg}[Corollary 2.21 Liggett, (1985)]\nSuppose that the assumptions of Theorem \\ref{thm2} are satisfied\nand that the equivalent conditions of Theorem \\ref{thm2} hold. Let\n$\\eta_t$ be the corresponding process, where the distribution of\n$\\eta_0$ has positive correlations. Then for $t_10$, there exists $c$ such that~:\n$$\n|\\{x\\in S\\,,\\; d(x,R)=n\\}|\\leq |\\{y\\in S\\,,\\; d(x,R)\\leq n\\}|\\leq\nce^{n\\varepsilon}.\n$$\nWhat follows is written in the general case, using\n(\\ref{borneboule}). It applies to the amenable case replacing\n$\\rho$ by $\\varepsilon$, for any $\\varepsilon>0$.\n\nWe are going to deal with smooth functions, depending weakly on\ncoordinates away from a fixed finite set $R$. Indeed, it is not\nsufficient to consider functions depending only on coordinates in\n$R$, because if $f$ is such a function, then for any $t>0$, $S_tf$\nmay depend on all coordinates.\n\\begin{defi}\n\\label{defmainlylocated} Let $f$ be a function from $S$ into\n$\\mathbb{R}$, and $R$ be a finite subset of $S$. The function $f$ is said\nto be {\\rm mainly located on} $R$ if there exists two constants\n$\\alpha$ and $\\beta>\\rho$ such that $\\alpha>0$, $\\beta>\\rho$ and\nfor all $x\\in \\mathbb{R}$:\n\\begin{equation}\n\\label{mainlylocated} \\Delta_f(x) \\leq \\alpha e^{-\\beta d(x,R)}.\n\\end{equation}\n\\end{defi}\nSince $\\beta>\\rho$, the sum $\\sum_x \\Delta_f(x)$ is finite. Therefore a\nfunction mainly located on a finite set is necessarily smooth.\n\nThe system we are considering will be supposed to have finite\nrange interactions in the following sense (cf. Definition 4.17,\np.~39 of \\cite{Liggett}).\n\\begin{defi}\n\\label{defetenduefinie} A particle system defined by the rates\n$c_T(\\eta,\\cdot)$ is said to have {\\it finite range interactions}\nif there exists $k>0$ such that if $d(x,y)>k$:\n\\begin{enumerate}\n\\item $c_T = 0$ for all $T$ containing both $x$ and $y$\\;, \\item\n$\\gamma(x,y)=0$.\n\\end{enumerate}\n\\end{defi}\nThe first condition imposes that two coordinates cannot\nsimultaneously change if their distance is larger than $k$. The\nsecond one says that the influence of a site on the transition\nrates of another site cannot be felt beyond distance $k$.\n\nUnder these conditions, we prove the following covariance\ninequality.\n\n\\begin{prop}\n\\label{propinegcovfini} Assume (\\ref{hyp0}) and (\\ref{hypl1}).\nAssume moreover that the process is of finite range. Let $R_1$ and\n$R_2$ be two finite subsets of $S$.\n Let $\\beta$\nbe a constant such that $\\beta>\\rho$. Let $f$ and $g$ be two\nfunctions mainly located on $R_1$ and $R_2$, in the sense that\nthere exist positive constants $\\kappa_f,\\kappa_g$ such that,\n$$\n\\Delta_f(x) \\leq \\kappa_f e^{-\\beta d(x,R_1)} \\quad\\mbox{and}\\quad\n\\Delta_g(x) \\leq \\kappa_g e^{-\\beta d(x,R_2)}.\n$$\nThen for all positive reals $s,t$,\n\\begin{equation}\n\\label{inegcovariancefini} \\sup_{\\eta\\in X}\n\\Big|\\cov_\\eta(f(\\eta_s),g(\\eta_t))\\Big| \\leq\nC\\kappa_f\\kappa_g(|R_1|\\wedge|R_2|)e^{D(t+s)}e^{-({\\beta-\\rho})d(R_1,R_2)}\\;,\n\\end{equation}\nwhere\n$$\nD = 2Me^{(\\beta+\\rho) k} \\quad\\mbox{and}\\quad C = \\frac{2\nBe^{\\beta k}}{D}\\left(1+\\frac{e^{\\rho\nk}}{1-e^{-\\beta+\\rho}}\\right).\n$$\n\\end{prop}\n\\paragraph{Remark.} Shashkin \\cite{Shashkin} obtains a similar\ninequality for random fields indexed by $\\mathbb{Z}^d$.\n\n\\\n\nWe now consider a {\\it transitive graph}, such that the group of\nautomorphism acts transitively on $S$ (see chapter 3 of\n\\cite{GodsilRoyle}). Namely we need that\n\\begin{itemize}\n \\item for any $x$ and $y$ in $S$ there exists $a$ in $Aut(S)$, such that\n $a(x)=y$.\n \\item for any $x$ and $y$ in $S$ and any radius $n$, there exists $a$ in $Aut(S)$, such that\n $a(B(x,n))=B(y,n)$.\n\\end{itemize}\nAny element $a$ of the automorphism group acts on configurations,\nfunctions and measures on ${\\cal X}$ as follows:\n\\begin{itemize}\n\\item {\\it configurations:} $a\\cdot \\eta(x) = \\eta(a^{-1}(x))$, \\item\n{\\it functions:} $a\\cdot f(\\eta) = f(a\\cdot \\eta)$, \\item {\\it\nmeasures:} $\\int f\\, d(a\\cdot \\mu) = \\int(a\\cdot f)\\,d\\mu$.\n\\end{itemize}\nA probability measure $\\mu$ on ${\\cal X}$ is invariant through the\ngroup action if $a\\cdot \\mu = \\mu$ for any automorphism $a$, and\nwe want this to hold for the probability distribution of $\\eta_t$\nat all times $t$. It will be the case if the transition rates are\nalso invariant through the group action. In order to avoid\nconfusions with invariance in the sense of the semigroup\n(Definition 1.7, p.~10 of \\cite{Liggett}), invariance through the\naction of the automorphism group of the graph will be\nsystematically referred to as ``group invariance'' in the sequel.\n\\begin{defi}\n\\label{definvariant} Let $G$ be the automorphism group of the\ngraph. The transition rates $c_T(\\eta,\\cdot)$ are said to be {\\rm\ngroup invariant} if for any $a\\in G$,\n$$\nc_{a(T)}(a\\cdot \\eta,a\\cdot \\zeta) = c_T(\\eta,\\zeta).\n$$\n\\end{defi}\nThis definition extends in an obvious way that of translation\ninvariance on $\\mathbb{Z}^d$-lattices (\\cite{Liggett}, p.~36).\n\\\\\n\\\\\n{\\bf{Remark.}}\n Observe\nthat for rates which are both finite range and group invariant,\nthe hypotheses (\\ref{hyp0}) and (\\ref{hypl1})\nare trivially satisfied. In that case, it is easy to check that\nthe semi-group $\\{S_t\\,,\\;t\\geq 0\\}$ commutes with the\nautomorphism group. Thus if $\\mu$ is a group invariant measure,\nthen so is $\\mu S_t$ for any $t$ (see \\cite{Liggett}, p.~38). In\nother terms, if the distribution of $\\eta_0$ is group invariant,\nthen that of $\\eta_t$ will remain group invariant at all times.\n\n \\section{Functional CLT}\n\\label{clt} Our functional central limit theorem requires that all\ncoordinates of the interacting particle system $\\{\\eta_t\\,,\\;t\\geq\n0\\}$ are identically distributed.\n\n\\vskip 2mm\\noindent\nLet $(B_n)_{n\\geq 1}$ be an increasing sequence of finite subsets of $S$\nsuch that\n\\begin{equation}\\label{slc}\nS=\\bigcup_{n=1}^{\\infty}B_n,\\qquad \\lim_{n\\rightarrow\n+\\infty}\\frac{|\\partial B_n|}{|B_n|}=0\\;,\n\\end{equation}\nrecall that $|\\;\\cdot\\,|$ denotes the cardinality and $\n\\partial B_n=\\{x\\in B_n\\;,\\,\\exists\\,y\\not\\in B_n,\\, d(x,y)=1\\}\n$.\n\\begin{theo}\n\\label{cltips2} Let $\\mu=\\delta_{\\eta}$ be a Dirac measure where\n$\\eta\\in {\\cal X}$ fulfills $\\eta(x)=\\eta(y)$ for any $x,y\\in S$.\nSuppose that the transition rates are group invariant. Suppose\nmoreover that the process is of finite range, monotone and\nfulfilling the requirements of Corollary \\ref{corligg}. Let\n$(B_n)_{n\\geq 1}$ be an increasing sequence of finite subsets of\n$S$ fulfilling (\\ref{slc}). Then the sequence of processes\n$$\n\\left\\{\\frac{N_t^{B_n}-\\mathbb{E}_{\\mu}N_t^{B_n}}{\\sqrt{|B_n|}}\\,,\\;t\\geq\n0\\right\\},\\qquad \\mbox{ for } n=1,2,\\ldots$$ converges in\n${D([0,T])}$ as $n$ tends to infinity, to a centered Gaussian,\nvector valued process\\\\ $(B(t,w))_{t\\ge\\,0,\\,w\\in\\, W}$ with\ncovariance function $\\Gamma$ defined, for $w,\\, w'\\in\\, W$, by\n$$\n\\Gamma_\\mu(s,t)(w,w')=\\sum_{x\\in\\, S}\\cov_{\\mu}\\left(\n\\mathbb{I}_{w}(\\eta_s(x)),\\mathbb{I}_{w'}(\\eta_t(x))\\right).\n$$\n\\end{theo}\n\\paragraph{Remark.} One may wonder wether such results can extend under\nmore general initial distributions. The point is that the covariance\ninequality do not extend simply by integration with respect to\ndeterministic configurations. We are thankful to Pr. Penrose for\nstressing our attention on this important restriction. Monotonicity\nallows to get ride of this restriction.\n\n\\section{Examples of graphs}\n\\label{graphs}\nBesides the classical lattice graphs in $\\mathbb{Z}^d$ and their groups of\ntranslations, which are considered by most authors (see\n\\cite{DurrettLevin94,Liggett,Liggett97}), our setting applies to a\nbroad range of graphs. We propose some simple examples of\nautomorphisms on trees, which\ngive rise to a large variety of non classical situations.\n\\vskip 2mm\nThe simplest example\n corresponds to regular trees defined as\nfollows. Consider the non-commutative free group $S$ with finite\ngenerator set $G$. Impose that each generator $g$ is its own\ninverse ($g^2=1$). Now consider $S$ as a graph, such that $x$ and\n$y$ are connected if and only if there exists $g \\in G$ such that\n$x=yg$. Note that $S$ is a regular tree of degree equal to the\ncardinality $r$ of $G$. The size of spheres is exponential:\n$\\left|\\{y\\,,\\, d(x,y)=n\\}\\right|=r^n$. Now consider the group\naction of $S$ on itself: $x\\cdot y=xy$: this action is transitive\non $S$ (take $a=yx$). \\vskip 2mm From this basic example it is\npossible to get a large class of graphs by adding relations\nbetween generators; for example take the tree of degree $4$,\ndenote by $a$, $b$, $c$, and $d$ the generators, and add the\nrelation $ab=c$. Then, the corresponding graph is a regular tree\nof degree $4$ were nodes are replaced by tetrahedrons. The spheres\ndo not grow at rate $4^n$: $\\left|\\{y\\,,\\,\nd(x,y)=n\\}\\right|=4\\cdot3^{n\/2}$ if $n$ is even and $\\left|\n\\{y\\,,\\, d(x,y)=n\\}\\right|=6\\cdot3^{(n-1)\/2}$ if $n$ is odd.\n\n\\begin{figure}[ht]\n\\vskip 7truecm \\vbox{ \\centerline{ \\psset{unit=1cm}\n\\pspicture(2.5,2.5)(5,-0.5) \\psline(1,7)(2,7.5)(3,7)(2,6.5)(1,7)\n\\psline(1,7)(9,7) \\psline(2,6.5)(2,5.5)\n\\psline(1,5)(2,5.5)(3,5)(2,4.5)(1,5) \\psline(2,7.5)(2,6.5)\n\\psline(3,7)(4,7)\n\\psline[showpoints=true](4,7)(5,7.5)(6,7)(5,6.5)(4,7)\n\\psline(6,7)(7,7)\n\\psline[showpoints=true](7,7)(8,7.5)(9,7)(8,6.5)(7,7)\n\\psline(2,4.5)(2,6) \\psline(1,5)(3,5) \\psline(5,7.5)(5,4.5)\n\\psline[showpoints=true](4,5)(5,5.5)(6,5)(5,4.5)(4,5)\n\\psline(4,5)(6,5) \\psline(8,7.5)(8,6.5) \\rput(1,7){$\\bullet$}\n\\rput(0.75,7.3){$db$} \\psline[linewidth=3pt](1,7)(0,7)\n\\psline[linewidth=3pt](1.09,5)(0,5)\n \\rput(2,7.5){$\\bullet$} \\rput(1.75,7.8){$da$}\n\\psline[linewidth=3pt](2,7.5)(2,8.5)\n \\rput(3,7){$\\bullet$} \\rput(3.2,7.3){$d$}\n\\rput(4.2,7.3){$1$} \\psline[linewidth=3pt](3,7)(4,7)\n\\psline[linewidth=3pt](6,7)(7,7)\n\\psline[linewidth=3pt](2,6.5)(2,5.5)\n\\psline[linewidth=3pt](5,6.5)(5,5.5)\n \\rput(5.2,7.8){$a$}\n\\psline[linewidth=3pt](5,7.5)(5,8.5) \\rput(6.2,7.3){$b$}\n\\rput(2,6.5){$\\bullet$} \\rput(4.75,6.3){$c$} \\rput(1.75,6.3){$dc$}\n\\rput(2,5.5){$\\bullet$} \\rput(2.35,5.8){$dcd$}\n\\rput(2,4.5){$\\bullet$} \\rput(2.41,4.3){$dcdc$}\n\\rput(2,4.5){$\\bullet$} \\rput(1,5){$\\bullet$} \\rput(1,5.3){$dcda$}\n\\rput(3,5){$\\bullet$} \\rput(3.1,5.3){$dcdb$}\n \\psline[linewidth=3pt](2,4.5)(2,3.5)\n \\rput(4.7,5.8){$cd$}\\rput(4.6,4.3){$cdc$}\n\\psline[linewidth=3pt](5,4.5)(5,3.5)\n \\rput(5.999,5.3){$cdb$}\n \\psline[linewidth=3pt](6,5)(7,5)\n\\rput(3.999,5.3){$cda$} \\psline[linewidth=3pt](4,5)(3.75,5)\n\\psline[linewidth=3pt](2.9,5)(3.25,5)\n \\rput(4,7){$\\bullet$} \\rput(7.1,7.3){$bd$}\n\\rput(8.4,7.8){$bda$} \\psline[linewidth=3pt](8,7.5)(8,8.5)\n \\rput(9.1,7.3){$bdb$}\n\\psline[linewidth=3pt](9,7)(10,7)\n \\rput(8.4,6.3){$bdc$}\n \\psline[linewidth=3pt](8,6.5)(8,5.5)\n\\endpspicture\n} } \\vskip -3truecm\n \\caption{Graph structure of the tree with tetrahedron\ncells. The graph consists in a regular tree of degree 4 (bold\nlines), where nodes have been replaced by tetrahedrons.\nAutomorphisms in this graph correspond to composition of\nautomorphisms exchanging couples of branches of the tree (action\nof generator $a$ for example) and displacements in the subjacent\nregular tree.}\n\\end{figure}\n\n\\section{CLT for hitting times}\\label{hitting}\nIn this section we consider the case where $W$ is ordered, the\nprocess is monotone and satisfies the assumptions in Theorem\n\\ref{cltips2}, the initial condition is fixed and $f$ is an\nincreasing function from $W$ to $\\mathbb{R}$. In the reliability\ninterpretation, $f(w)$ measures a level of degradation for a\ncomponent in state $w$. The total degradation of the system in state\n$\\eta$ will be measured by the sum $\\sum_{x \\in B_n} f(\\eta(x))$. So\nwe shall focus on the process $D^{(n)} = \\{D_t^{(n)}\\,,\\,t\\geq 0\\}$,\nwhere $D_t^{(n)}=D_t^{B_n}$ is the total degradation of the system\nat time $t$ on the set $R=B_n$:\n$$\nD_t^{(n)} = \\sum_{x\\, \\in\\, B_n} f(\\eta_t(x)).\n$$\nIt is natural to consider the instants at which $D_t^{(n)}$\nreaches a prescribed level of degradation. Let $k=(k(n))$ be a\nsequence of real numbers. Our main object is the {\\it failure\ntime} $T_n$, defined as:\n$$\nT_n = \\inf\\{t\\geq 0\\,,\\; D_t^{(n)}\\geq k(n)\\}.\n$$\nIn the particular case where\n$W=\\{\\mbox{working},\\,\\mbox{failed}\\}$ (binary components), and\n$f$ is the indicator of a failed component, then $D_t^{(n)}$\nsimply counts the number of failed components at time $t$, and our\nsystem is a so-called ``$k$-out-of-$n$'' system\n\\cite{BarlowProschan}.\n\nLet $w_0$ be a particular state (in the reliability $w_0$ could be\nthe ``perfect state'' of an undergrade component). Let $\\eta$ be the\nconstant configuration where all components are in the perfect state\n$w_0$, for all $ x\\in S$. Our process starts from that configuration\n$\\eta$, which is obviously group invariant. We shall denote by\n$m(t)$ (respectively, $v(t)$) the expectation (resp., the variance)\nof the degradation at time $t$ for one component.\n$$\nm(t) = \\mathbb{E}[f(\\eta_t(x))\\,|\\,\\eta_0=\\eta]\\;,\\qquad\nv(t)=\\lim_{n\\to\\infty}\\frac{\\Var D_t^{(n)}}{|B_n|}.\n$$\nThese expressions do not depend on $x\\in S$, due to group\ninvariance.\n\nThe average degradation $D_t^{(n)}\/|B_n|$ converges in probability\nto its expectation $m(t)$. We shall assume that $m(t)$ is strictly\nincreasing on the interval $[0,\\tau]$, with $0<\\tau\\leq+\\infty$ (the\ndegradation starting from the perfect state increases on average).\nMathematically, one can assume that the states are ranked in\nincreasing order, the perfect state being the lowest. This yields a\npartial order on configurations. If the rates are such that the\ninteracting particle system is monotone (see \\cite{Liggett}), then\nthe average degradation increases. In the reliability\ninterpretation, assuming monotonicity is quite natural: it amounts\nto saying that the rate at which a given component jumps to a more\ndegraded state is higher if its surroundings are more degraded.\n\nWe consider a ``mean degradation level'' $\\alpha$, such\nthat $m(0)<\\alpha0$ and $\\beta>\\rho$, then for all $y\\in S$,\n$$\n|(\\exp(t\\Gamma)u)(y)| \\leq \\alpha\\exp(2tMe^{(\\beta+\\rho)\nk})\\,e^{-\\beta d(y,R)}.\n$$\n\\end{lemm}\nThis lemma, together with Proposition \\ref{propsmooth}, justifies\nDefinition \\ref{defmainlylocated}. Indeed, if $f$ is mainly\nlocated on $R$, then by (\\ref{ineg1}) and Lemma\n\\ref{lemmetechniquegamma}, $S_tf$ is also mainly located on $R$,\nand the rate of exponential decay $\\beta$ is the same for both\nfunctions.\n\\\\\n{\\bf{Proof of Lemma \\ref{lemmetechniquegamma}.}} Recall that\n$$\n\\Gamma u(y) = \\sum_{x\\in S} u(x)\\gamma(x,y).\n$$\nObserve that if $\\gamma(x,y)>0$, then the distance from $x$ to $y$\nmust be at most $k$ and thus the distance from $x$ to $R$ is at\nleast $d(y,R)-k$. If $u(x)\\leq \\alpha e^{-\\beta d(x,R)}$ then:\n$$\n\\Gamma u(y) \\leq 2\\alpha e^{\\rho k}e^{-\\beta(d(y,R)-k)}M = 2\\alpha\ne^{(\\beta +\\rho)k} M e^{-\\beta d(y,R)}.\n$$\nHence by induction,\n$$\n\\Gamma^n u(y) \\leq \\alpha 2^ne^{(\\beta+\\rho) kn} M^{n} e^{-\\beta\nd(y,R)}.\n$$\nThe result follows immediately.\\,\\, $\\Box$\n\\vskip 3mm\nTogether with (\\ref{inegcovariancest}), Lemma\n\\ref{lemmetechniquegamma} will be the key ingredient in the proof\nof our covariance inequality.\n\\\\\n{\\bf{End of the proof of Proposition \\ref{propinegcovfini}.}}\nBeing mainly located on finite sets, the functions $f$ and $g$ are\nsmooth. By (\\ref{inegcovariancest}), the covariance of $f(\\eta_s)$\nand $g(\\eta_t)$ is bounded by $M(s,t)$ with:\n$$\nM(s,t)= \\sum_{y,z\\in\\, S} \\left(\\sum_{T\\ni\\, y,z} c_T\\right)\n\\int_0^s\n(\\exp(\\tau\\Gamma)\\Delta_f)(y)(\\exp(\\tau\\Gamma)\\Delta_{S_{t-s}g})(z)\\,d\\tau.\n$$\nLet us apply Lemma \\ref{lemmetechniquegamma} to $\\Delta_f$ and\n$\\Delta_{S_{t-s}g}$.\n\\begin{equation}\n\\label{maj1} (\\exp(\\tau\\Gamma)\\Delta_f)(y) \\leq \\kappa_f\\exp(\\tau\nMe^{(\\beta+\\rho) k})\n e^{-\\beta d(y,R_1)}\n= \\kappa_f e^{D\\tau}e^{-\\beta d(y,R_1)}.\n\\end{equation}\nThe last bound, together with (\\ref{ineg1}), gives\n$$\n\\Delta_{S_{t-s}g}(x)\\leq (\\exp((t-s)\\Gamma)\\Delta_g)(x) \\leq\n\\kappa_g e^{D(t-s)}e^{-\\beta d(x,R_2)}.\n$$\nTherefore~:\n\\begin{equation}\n\\label{maj2} (\\exp(\\tau\\Gamma)\\Delta_{S_{t-s}g})(z)\\leq \\kappa_g\ne^{D(\\tau+t-s)} e^{-\\beta d(z,R_2)}.\n\\end{equation}\nInserting the new bounds (\\ref{maj1}) and (\\ref{maj2}) into\n$M(s,t)$, we obtain\n$$\nM(s,t)\\leq \\sum_{y,z\\in\\, S} \\left(\\sum_{T\\ni\\, y,z} c_T\\right)\n\\kappa_f\\kappa_g e^{-\\beta(d(y,R_1)+d(z,R_2))} \\int_0^s\ne^{D(2\\tau+t-s)}\\,d\\tau.\n$$\nNow if $d(y,z)>k$ and $y,z\\in T$, then $c_T$ is null by Definition\n\\ref{defetenduefinie}. Remember moreover that by hypothesis\n(\\ref{hyp0}):\n$$\nB=\\sup_{u\\in S} \\sum_{T\\ni u} c_T <\\infty.\n$$\nTherefore~:\n\\begin{equation}\\label{avantderniere}\nM(s,t) \\leq \\kappa_f\\kappa_g\\frac{Be^{D(s+t)}}{2D}\\sum_{y\\in S}\n\\sum_{d(y,z)\\leq k} e^{-\\beta(d(y,R_1)+d(z,R_2))}.\n\\end{equation}\nIn order to evaluate the last quantity, we have to distinguish two\ncases. \\\\\n\\\\\n$\\bullet$ If $d(R_1,R_2)\\leq k$, then\n\\begin{eqnarray*}\n\\sum_{y\\in S} \\sum_{d(y,z)\\leq k}\ne^{-\\beta(d(y,R_1)+d(z,R_2))}&\\leq& 2e^{\\rho k}\\sum_{y\\in S}\ne^{-\\beta d(y,R_1)}\n\\\\\n&\\leq & 2e^{\\rho k}\\sum_{n\\in\\mathbb{N}}\\sum_{y\\in S} e^{-\\beta\nd(y,R_1)}\\mathbb{I}_{d(y,R_1)=n}\n\\\\\n&\\leq& 4|R_1|e^{\\rho k}\\sum_{n=0}^\\infty e^{(\\rho-\\beta )n}\n\\\\\n&\\leq & \\frac{4|R_1| e^{\\rho k}}{1-e^{-(\\beta-\\rho)}}\\\\\n& \\leq & |R_1|\\frac{4 e^{(\\rho+\\beta)\nk}}{1-e^{-(\\beta-\\rho)}}e^{-\\beta d(R_1,R_2)}\\\\\n& \\leq & |R_1|\\frac{4 e^{(\\rho+\\beta)\nk}}{1-e^{-(\\beta-\\rho)}}e^{-(\\beta - \\rho) d(R_1,R_2)}\n\\end{eqnarray*}\n$\\bullet$ If $d(R_1,R_2)>k$, then we have, noting that\n$d(y,R_1)+d(z,R_2)\\geq d(R_1,R_2)-d(y,z)$ and that $d(y,z)\\leq k$,\n\n\\begin{eqnarray*}\n{\\lefteqn{\\sum_{y\\in S} \\sum_{d(y,z)\\leq k}\ne^{-\\beta(d(y,R_1)+d(z,R_2))}}}\\\\\n&&\\leq \\sum_{d(y,R_1)\\leq d(R_1,R_2)-k} \\sum_{d(y,z)\\leq\nk}e^{-\\beta (d(R_1,R_2)-k)} +\\sum_{d(y,R_1)\\geq d(R_1,R_2)-k}\n\\sum_{d(y,z)\\leq k}e^{-\\beta d(y,R_1)}\n\\\\[2ex]\n&&\\leq 4 |R_1|\\,e^{\\rho (d(R_1,R_2)-k)}e^{\\rho k}e^{-\\beta\n(d(R_1,R_2)-k)} +4|R_1|e^{\\rho k}\\sum_{n\\geq d(R_1,R_2)-k}\ne^{(\\rho-\\beta )n}\n\\\\[2ex]\n&&\\leq 4|R_1|\\, e^{\\beta\nk}\\left(1+\\frac{1}{1-e^{-(\\beta-\\rho)}}\\right)e^{-(\\beta-\\rho)d(R_1,R_2)}.\n\\end{eqnarray*}\n\nBy inserting the latter bound into (\\ref{avantderniere}), one\nobtains,\n$$\nM(s,t)\\leq\nC\\kappa_f\\kappa_g|R_1|e^{D(t+s)}e^{-({\\beta-\\rho})d(R_1,R_2)}\\;,\n$$\nwith~:\n$$\nC = \\frac{2B}{D}e^{\\beta k}\\left(1+\\frac{e^{\\rho\nk}}{1-e^{-\\beta+\\rho}}\\right).\\qquad \\Box\n$$\nThe covariance inequality (\\ref{inegcovariancefini}) implies that\nthe covariance between two functions essentially located on two\ndistant sets decays exponentially with the distance of those two\nsets, whatever the instants at which it is evaluated. However the\nupper bound increases exponentially fast with $s$ and $t$. In the\ncase where the process $\\{\\eta_t\\,,\\;t\\geq 0\\}$ converges at\nexponential speed to its equilibrium, it is possible to give a\nbound that increases only in $t-s$, thus being uniform in $t$ for\nthe covariance at a given instant $t$.\n\n\\subsection{Proof of Theorem \\ref{cltips2}}\n\\subsubsection{Finite dimensional laws}\nLet ${\\cal G}=(S,E)$ be a transitive graph and $Aut({\\cal G})$ be\nthe automorphism group of ${\\cal G}$. Let $\\mu$ be a probability\nmeasure on ${\\cal X}$ invariant through the automorphism group\naction. Let $(\\eta_t)_{t\\geq 0}$ be an interacting particle system\nfulfilling the requirements of Theorem \\ref{cltips2}. Recall that\n$\\{S_t\\,,\\;t\\geq 0\\}$ denotes the semigroup and $\\mu S_t$ the\ndistribution of $\\eta_t$, if the distribution of $\\eta_0$ is $\\mu$.\n\\begin{prop}\\label{fidi0}\nLet $(B_n)_n$ be an increasing sequence of finite subsets of $S$\nfulfilling {\\rm (\\ref{slc})}. Let assumptions of Theorem\n\\ref{cltips2} hold.\n Then for any fixed positive real numbers $t_1\\leq\nt_2\\leq\\cdots\\leq t_k$, the random vector\n$$\\frac{1}{\\sqrt{|B_n|}}\\left(N_{t_1}^{B_n}-\\mathbb{E}_{\\mu} N_{t_1}^{B_n},\nN_{t_2}^{B_n}-\\mathbb{E}_{\\mu} N_{t_2}^{B_n}\n,\\ldots,N_{t_k}^{B_n}-\\mathbb{E}_{\\mu} N_{t_k}^{B_n}\\right)$$ converges\nin distribution, as $n$ tends to infinity, to a centered Gaussian\nvector with covariance matrix $(\\Gamma_{\\mu}(t_i,t_j))_{1\\leq\ni,j\\leq k}$.\n\\end{prop}\n{\\bf{Proof of Proposition \\ref{fidi0}.}} We will only study the\nconvergence in distribution of the vector\n$$\\frac{1}{\\sqrt{|B_n|}}\\left(N_{t_1}^{B_n}-\\mathbb{E}_{\\mu} N_{t_1}^{B_n},\nN_{t_2}^{B_n}-\\mathbb{E}_{\\mu} N_{t_2}^{B_n}\\right)\\;,$$ the general\ncase being similar.\n For $i=1,2$, we denote by $\\alpha_i=(\\alpha_i(w))_{w\\in W}$ two\n fixed vectors of $\\mathbb{R}^{|W|}$. We have, denoting by $\\cdot$ the usual\n scalar product,\n \\begin{eqnarray*}\n{\\lefteqn{\\frac{1}{\\sqrt{|B_n|}}\\sum_{i=1}^2 \\alpha_i \\cdot\n\\left(N_{t_i}^{B_n}-\\mathbb{E}_{\\mu}\nN_{t_i}^{B_n}\\right)}}\\\\\n&& = \\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in\nB_n}\\left(\\sum_{i=1}^2\\left(\\sum_{w\\in W}\n\\alpha_i(w)(\\mathbb{I}_w(\\eta_{t_i}(x))-\\mathbb{P}_{\\mu}(\\eta_{t_i}(x)=w))\\right)\\right)\\\\\n&&= \\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in B_n}Y_x,\n\\end{eqnarray*}\n where $(Y_x)_{x\\in S}$ is the random field defined by\n\\begin{equation}\\label{Ydef}\nY_x = \\sum_{i=1}^2\\left(\\sum_{w\\in W}\n\\alpha_i(w)(\\mathbb{I}_w(\\eta_{t_i}(x))-\\mathbb{P}_{\\mu}(\\eta_{t_i}(x)=w))\\right)=:F_1(\\eta_{t_1}(x))\n+ F_2(\\eta_{t_2}(x)).\n\\end{equation}\nThe purpose is then to prove a central limit theorem for the sum\n$\\sum_{x\\in B_n}Y_x$. For this, we shall study the nature of the\ndependence of $(Y_x)_{x\\in S}$.\n\nLet $R_1$ and $R_2$ be two finite and disjoints subsets of $S$.\nLet $k_1$ and $k_2$ be two\n real valued functions defined respectively on\n$\\mathbb{R}^{|R_1|}$ and $\\mathbb{R}^{|R_2|}$. Let\n $K_1$, $K_2$ be two real valued functions,\ndefined respectively on $W^{R_1}$ and $W^{R_2}$, by\n$$K_j(\\nu,\\eta)=k_j(F_1(\\nu(x))+ F_2(\\eta(x)),\\ x\\in R_j),\\ \\\n\\ j=1,2. $$\nLet ${\\cal L}$ be the class of real valued Lipschitz functions $f$ defined\non $\\mathbb{R}^n$, for some positive integer $n$, for which\n$$\n\\Lip f :=\\sup_{x\\neq y}\\frac{\\textstyle\n\\left|f(x)-f(y)\\right|}{\\textstyle\\sum_{i=1}^n|x_i-y_i|}<\\infty.\n$$\nWe assume that $k_1$ and $k_2$ belong to ${\\cal L}$.\n Recall that\n\\begin{eqnarray*}\n\\Cov_{\\eta}(k_1(Y_x,\\, x\\in R_1), k_2(Y_x,\\,x \\in\nR_2))&=&\\Cov_{\\eta}\\left(K_1(\\eta_{t_1},\\eta_{t_2}),\nK_2(\\eta_{t_1},\\eta_{t_2})\\right) \\\\\n\\end{eqnarray*}\nBut\n$$|K_1(\\eta_{t_1},\\eta_{t_2})-K_1(\\eta'_{t_1},\\eta_{t_2})|\\leq\n4\\Lip k_1\\sum_{w\\in W}|\\alpha_1(w)|\\sum_{x\\in\nR_1}|\\eta_{t_1}(x)-\\eta'_{t_1}(x)|\n$$\nDenote $ A_1(W)=4\\Lip k_1\\sum_{w\\in W}|\\alpha_1(w)|$. Then, the\nfunctions\n$$\n\\eta_{t_1}\\longrightarrow (\\Lip k_1)A_1(W) \\sum_{x\\in\nR_1}\\eta_{t_1}(x) \\pm\\,\\,K_1(\\eta_{t_1},\\eta_{t_2})\n$$\nare increasing. Hence, the functions\n$$\nG_1^{\\pm}:\\,(\\eta_{t_1},\\eta_{t_2})\\longrightarrow \\Lip k_1\n\\sum_{x\\in R_1}\\left(A_1(W)\\eta_{t_1}(x)+A_2(W)\\eta_{t_2}(x)\\right)\n\\pm\\,\\,K_1(\\eta_{t_1},\\eta_{t_2})\n$$\nare increasing coordinate by coordinate. This also holds for,\n$$\nG_2^{\\pm}:\\,(\\eta_{t_1},\\eta_{t_2})\\longrightarrow \\Lip k_2\n\\sum_{x\\in R_2}(A_1(W)\\eta_{t_1}(x)+A_2(W)\\eta_{t_2}(x))\n\\pm\\,\\,K_2(\\eta_{t_1},\\eta_{t_2}).\n$$\nUnder assumptions of Theorem \\ref{thm2} and of its Corollary\n\\ref{corligg}, the vector $(\\eta_{t_1},\\eta_{t_2})$ has positive\ncorrelation so that\n$$\n\\Cov_{\\eta}(G_1^{\\pm}(\\eta_{t_1},\\eta_{t_2}),\nG_2^{\\pm}(\\eta_{t_1},\\eta_{t_2}))\\geq 0.\n$$\nThis gives\n\\begin{eqnarray*}\n&&\\left|\\Cov_{\\eta}(k_1(Y_x,\\, x\\in R_1), k_2(Y_x,\\,x \\in\nR_2))\\right|\\\\\n&& \\leq \\Lip k_1\\Lip k_2 \\sum_{x\\in R_1}\\sum_{y\\in\nR_2}\\Cov_{\\eta}(A_1(W)\\eta_{t_1}(x)+A_2(W)\\eta_{t_2}(x),A_1(W)\\eta_{t_1}(y)+A_2(W)\\eta_{t_2}(y)).\n\\end{eqnarray*}\n From this bilinear formula, we now apply Proposition\n\\ref{propinegcovfini} and obtain the following covariance\ninequality: for finite subsets $R_1$ and $R_2$ of $S$, we have\nletting $\\delta=\\beta-\\rho$,\n$$\\left|\\Cov_{\\eta}\\left(K_1(\\eta_{t_1},\\eta_{t_2}),K_2(\\eta_{t_1},\\eta_{t_2})\\right)\\right|\\leq C_{\\delta} \\Lip k_1\\Lip\nk_2\\left(|R_1|\\wedge|R_2|\\right)\\exp\\left(-\\delta\nd(R_1,R_2)\\right),$$ where $C_{\\delta}$ is a positive constant\ndepending on $\\beta$ and not depending on $R_1$, $R_2$, $k_1$ and\n$k_2$.\n\\\\\nWe then deduce from Proposition \\ref{pro2r} that\n$\\frac{1}{\\sqrt{|B_n|}}\\sum_{x\\in B_n}Y_x$ converges in distribution\nto a centered normal law as soon as the quantity\n$\\Var_{\\mu}(\\sum_{x\\in B_n}Y_x)\/|B_n|$ converges as $n$ tends to\ninfinity to a finite number $\\sigma^2$. This variance converges if\nthe requirements of Proposition \\ref{slpro2} are satisfied. For\nthis, we first check the condition of invariance\n(\\ref{slstationarity}):\n$$\n\\cov_{\\mu}(Y_x,Y_y)=\\cov_{\\mu}(Y_{a(x)},Y_{a(y)}),\n$$\nfor any automorphism $a$ of ${\\cal G}$ and for $Y_x$ as defined by\n(\\ref{Ydef}). We recall that the initial distribution is a Dirac\ndistribution on the configuration $\\eta$. Then it has positive\ncorrelations. We have supposed that $\\eta(x)=\\eta(y)$ for all\n$x,y\\in S$, hence $a\\cdot \\mu=\\mu$ and the group invariance\nproperty of the transition rates proves that $\\mu=\\delta_{\\eta}$\nfulfills (\\ref{essai}) below and then (\\ref{slstationarity}) will\nhold. Condition (\\ref{essai}) is true thanks to the following\nestimations valid for any suitable real valued functions $f$ and\n$g$,\n\\begin{eqnarray}\\label{essai}\n{\\lefteqn{\\mathbb{E}_{\\mu}(f(\\eta_{t_1})g(\\eta_{t_2}))}}\n{\\nonumber}\\\\\n&& =\\int d\\mu(\\eta)S_{t_1}\\left(fS_{t_2-t_1}g\\right)(\\eta)\n{\\nonumber}\n\\\\\n&&= \\int d\\mu(\\eta)\\, a\\cdot\nS_{t_1}\\left(fS_{t_2-t_1}g\\right)(\\eta)\\ \\ \\ {\\mbox{since}}\\ \\ \\ \\\na\\cdot\\mu=\\mu {\\nonumber}\n\\\\\n&&=\\int d\\mu(\\eta) S_{t_1}\\left((a\\cdot f)S_{t_2-t_1}(a\\cdot\ng)\\right)(\\eta)\\\n\\ \\ {\\mbox{since}}\\ \\ \\ \\ a\\cdot (S_sf)=S_s(a\\cdot f) {\\nonumber}\\\\\n&& = \\mathbb{E}_{\\mu}((a\\cdot f)(\\eta_{t_1})(a\\cdot\ng)(\\eta_{t_2}))=\\mathbb{E}_{\\mu}(f(a\\cdot\\eta_{t_1})g(a\\cdot\\eta_{t_2})).\n\\end{eqnarray}\n Hence Proposition \\ref{slpro2} applies and gives\n\\begin{eqnarray*}\n{\\lefteqn{\\sigma^2=\\sum_{z\\in S}\\cov_{\\mu}(Y_0,Y_z)}} \\\\\n&& = \\sum_{i,j=1}^2\\sum_{w,w'\\in W}\\alpha_i(w)\\alpha_j(w')\\sum_{z\\in\nS}\\cov_{\\mu}\\left(\\mathbb{I}_w(\\eta_{t_i}(0)),\\mathbb{I}_{w'}(\\eta_{t_i}(z))\\right)\n\\\\\n&&= \\sum_{i,j=1}^2 \\alpha_i^t \\Gamma_{\\mu}(t_i,t_j) \\alpha_j,\n\\end{eqnarray*}\nwhere $\\Gamma_{\\mu}(t_i,t_j)$ is the covariance matrix as defined in\nTheorem \\ref{cltips2}; with this we complete the proof of\nProposition \\ref{fidi0}.\n\n\\subsubsection{Tightness}\\label{secwdep}\n\n\n\nFirst we establish covariance inequalities for the counting\nprocess. Denote $g_{s,t,w}(\\eta,y)=\\mathbb{I}_w(\\eta_{\nt}(y))-\\mathbb{I}_w(\\eta_{s}(y))$ and for any multi-index ${\\bf\ny}=(y_1,\\ldots,y_u)\\in S^u$, for any state vector ${\\bf\nw}=(w_1,\\ldots, w_u)\\in W^u$, $\\Pi_{{\\bf y},{\\bf w}}=\\prod_{\n\\ell=1}^ug_{s,t,w_\\ell}(\\eta,y_\\ell)$.\n Following (\\ref{inegcovariancefini}), for $\\beta>\\rho$, for any\n$r$-distant finite multi-indices ${\\bf y}\\in S^u$ and ${\\bf z}\\in\nS^v$ ,\n for any times $0\\le s\\le t\\leq T$ and\n for any state vectors ${\\bf w}\\in W^u$\nand $ {\\bf w'}\\in W^v$\n\\begin{equation}\\label{inegcovsimp}\n\\left|\\cov_\\eta\\left(\\Pi_{{\\bf y},{\\bf w}},\\Pi_{{\\bf z},{\\bf w'}}\n\\right)\\right|\\leq 4C (u\\wedge v)e^{2DT }e^{-(\\beta-\\rho) r}\\equiv\nc_0 (u\\wedge v)e^{-c r} ,\n\\end{equation}\nfor $c=\\beta-\\rho$ and $c_0=\\frac{\\textstyle\n4Be^{2DT}e^{-(\\beta-\\rho)r}(2-e^{-c})}{\\textstyle Me^{\\rho\nk}(1-e^{-c})}$.\n\\begin{lemm} \\label{cor:covx}There exist $\\delta_0>0$ and $K_\\Omega>0$ such that for\n$|s-t|<\\delta_0$:\n\\begin{equation}\\label{eqn:covx1}\n\\left|\\cov_\\eta\\left(\\Pi_{{\\bf x},{\\bf w}},\\Pi_{{\\bf y},{\\bf w'}}\n\\right)\\right|\\leq K_\\Omega |t-s|.\n\\end{equation}\n\\end{lemm}\n\n\\begin{proof} Denote $f(\\eta)=\\mathbb{I}_w(\\eta(x))$ then\n$g_{t+h,t,w}(\\eta,x)=S_hf(\\eta_{t})-f(\\eta_t)$; the properties of\nthe generator $\\Omega$ imply that\n$$\n\\lim_{h\\rightarrow 0} \\frac{S_{h}f(\\eta)-f(\\eta)}h= \\Omega f(\\eta)\n$$\nBut\n\\begin{eqnarray*}\n|\\Omega f(\\eta)|&\\leq& \\sum_{T\\subset S}\\sum_{\\zeta \\in\nW^T}c_T(\\eta,\\zeta)|f(\\eta^\\zeta)-f(\\eta)|\n\\\\ &\\leq&\\sum_{T\\subset S, x \\in T}c_T(\\eta)\\leq \\sum_{T\\subset S, x \\in T}c_T\\leq C_\\Omega\n\\end{eqnarray*}\nso that for $h>0$ tending to zero\n$$\n|g_{s,s+h,w}(\\eta,x)|\\leq C_\\Omega h+o(h)\n$$\nBecause $\\Omega$ is group invariant, the remainder term is uniform\nwith respect to index $x$, so that we find convenient $\\delta_0$\nand $K_\\Omega$ uniformly with respect to location.\n\\end{proof}\n\\noindent From inequality (\\ref{inegcovsimp}) and lemma\n\\ref{cor:covx}, we deduce the following moment inequality:\n\\begin{prop}\\label{pro:moment}\nChoose $l$ and $c$ such that $\\rho(2l-1)n_0$ :\n\\begin{equation*}\n\\mathbb{P} (w(\\delta,N^{B_n})\\geq \\varepsilon)\\leq \\eta\n\\end{equation*}\nDefine $n_0$ as the smallest integer such that\n$|B_{n_0}|>\\delta^{-1-\\rho\/c}$, then for $n>n_0$, $|t-s|<\\delta$,\n$l=2$ and $c>3\\rho$, Proposition \\ref{pro:moment} yields:\n\\begin{equation*}\n\\mathbb{E} (N_t^{B_n}-N_s^{B_n})^{4}\\leq C \\delta^{2(1-\\frac{\\rho}{c})}\n\\end{equation*}\nand we now follow the proof in Billingsley \\cite{Billingsley} to\nconclude.\n\\subsection{Proof of Theorem \\ref{clthitting}}\n\nThe proof is close to that of the analogous result in\n\\cite{ParoissinYcart04}. The convergence in distribution of\n$Z_n=(Z_n(t))_{t\\ge0}$, where $Z_n(t)=(D_t^{(n)}-|B_n|\\cdot\nm(t))\/\\sqrt{ |B_n|}$, does not directly imply the CLT for $T_n$. The\nSkorohod-Dudley-Wichura representation theorem is a much stronger\nresult (see Pollard \\cite{Pollard}, section IV.3). It implies that\nthere exist versions $Z^{*}_n$ of $Z_n$ and non-decreasing\nfunctions $\\phi_n$ such that for any fixed $s$ such that for $Z^*$,\nlimit in distribution of $Z_n$:\n\\begin{equation*}\n\\lim_{n \\rightarrow \\infty} \\sup_{0 \\leq t \\leq s} \\left|\nZ_n^{*}(t) - Z^*(\\phi_n(t)) \\right| = 0 \\quad a.s.\n\\end{equation*}\nand:\n\\begin{equation*}\n\\lim_{n \\rightarrow \\infty} \\sup_{0 \\leq t \\leq s} \\left|\n\\phi_n(t) - t \\right| = 0 \\quad a.s.\n\\end{equation*}\nSince $Z^*$ has continuous paths, it is uniformly continuous on\n$[0,s]$, and hence:\n\\begin{equation}\\label{unif}\n\\lim_{n \\rightarrow \\infty} \\sup_{0 \\leq t \\leq s} \\left| Z^*_n(t)\n- Z^*(t) \\right| = 0 \\quad a.s.\\;,\n\\end{equation}\nWe shall first use (\\ref{unif}) to prove that the distributions of\n$\\sqrt{ |B_n|}(T_n-t_\\alpha)$ are a tight sequence. Let $c$ be a\npositive constant. On the one hand, if\n$D^{(n)}_{t_\\alpha+c\/\\sqrt{ |B_n|}} \\geq k(n)$, then $T_n \\leq\nt_\\alpha+c\/\\sqrt{ |B_n|}$. Thus:\n\\begin{eqnarray*}\n{\\mathbb{P} [ \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq c]}\n &\\geq&{\\mathbb{P} [D^{(n)}_{t_\\alpha+c\/\\sqrt{ |B_n|}}\\geq k(n)]} \\\\[1ex]\n &=&{\\mathbb{P} [Z^*_n(t_\\alpha+c\/\\sqrt{ |B_n|})\\geq\n \\sqrt{ |B_n|}(\\alpha-m(t_\\alpha+c\/\\sqrt{ |B_n|}))+o(1)]} \\\\[1ex]\n &=&{\\mathbb{P} [Z^*_n(t_\\alpha+c\/\\sqrt{ |B_n|})\\geq\n -cm'(t_\\alpha)+o(1)]} \\\\[1ex]\n &=&{\\mathbb{P} [Z^*(t_\\alpha)\\geq\n -cm'(t_\\alpha)]+o(1)}\\;,\n\\end{eqnarray*}\nusing (\\ref{unif}) and the continuity of $Z^*$. Since\n$m'(t_\\alpha)>0$, we obtain that:\n\\begin{equation}\n\\label{liminf} \\lim_{c \\rightarrow \\infty} \\liminf_{n \\rightarrow\n\\infty} \\mathbb{P} [\n \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq c] = 1 .\n\\end{equation}\nOn the other hand, we have:\n$$\n\\mathbb{P} [ \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq -c]\n =\\mathbb{P} [\\exists t \\leq t_\\alpha-c\/\\sqrt{ |B_n|}\\;,\\, Z^*_n(t)\\geq\n \\sqrt{ |B_n|}(\\alpha-m(t))+o(1)] .\n$$\nBut since the function $m$ is increasing, for all $t\\leq\nt_\\alpha-c\/\\sqrt{ |B_n|}$ we have:\n\\begin{equation*}\n\\sqrt{ |B_n|}(\\alpha-m(t)) \\geq \\sqrt{\n|B_n|}(\\alpha-m(t_\\alpha-c\/\\sqrt{ |B_n|})) = cm'(t_\\alpha)+o(1) .\n\\end{equation*}\nHence:\n\\begin{eqnarray*}\n\\displaystyle{\\mathbb{P} [ \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq -c]}\n&\\leq&\\displaystyle{\\mathbb{P} [\\exists t \\leq t_\\alpha-c\/\\sqrt{\n|B_n|}\\,, Z^*_n(t)\\geq\n cm'(t_\\alpha)+o(1)]} \\\\[1ex]\n&\\leq&\\displaystyle{ \\mathbb{P} [\\exists t \\leq t_\\alpha\\,, Z^*_n(t)\\geq\n cm'(t_\\alpha)+o(1)]} \\\\[1ex]\n&=&\\displaystyle{ \\mathbb{P} [\\exists t \\leq t_\\alpha\\,, Z^*(t)\\geq\n cm'(t_\\alpha)+o(1)]+o(1)} .\n\\end{eqnarray*}\nThe process $Z$ being \\textsl{a.s.} bounded on any compact set\nand $m'(t)$ being positive on $[0,\\tau]$, we deduce that:\n\\begin{equation}\n\\label{limsup} \\lim_{c \\rightarrow \\infty} \\limsup_{n \\rightarrow\n\\infty} \\mathbb{P} [\n \\sqrt{ |B_n|}(T_n - t_\\alpha) \\leq -c] = 0 .\n\\end{equation}\nNow (\\ref{liminf}) and (\\ref{limsup}) mean that the sequence of\ndistributions of $(\\sqrt{ |B_n|}(T_n-t_\\alpha))$ is tight. Hence\nto conclude it is enough to check the limit. \\vskip 1mm Using\nagain (\\ref{unif}), together with the almost sure continuity of\n$Z$ yields:\n\\begin{eqnarray*}\nD^{(n)}_{t_\\alpha+c\/\\sqrt{ |B_n|}}\n & = & |B_n|m(t_\\alpha+u\/\\sqrt{ |B_n|})\n +\\sqrt{ |B_n|}Z^*(t_\\alpha+u\/\\sqrt{ |B_n|})+o(\\sqrt{ |B_n|}) \\quad a.s. \\\\[1ex]\n & = & |B_n|\\alpha +u\\sqrt{ |B_n|}m'(t_\\alpha)\n +\\sqrt{ |B_n|}Z^*(t_\\alpha)+o(\\sqrt{ |B_n|}) \\quad a.s.\n\\end{eqnarray*}\nTherefore:\n\\begin{eqnarray*}\n\\inf \\left\\{ u \\;; D^{(n)}_{t_\\alpha+u\/\\sqrt{ |B_n|}} \\geq k(n) \\right\\}\n & = & \\inf \\Big\\{ u \\;; u\\sqrt{ |B_n|}m'(t_\\alpha)\n +\\sqrt{ |B_n|}Z^*(t_\\alpha)+o(\\sqrt{ |B_n|}) \\geq 0\\Big\\}\\\\[1ex]\n & = & - \\frac{Z^*(t_\\alpha)}{m'(t_\\alpha)} +o(1) .\n\\end{eqnarray*}\nThe distribution of $-Z^*(t_\\alpha)\/m'(t_\\alpha)$ is normal with\nmean $0$ and variance $\\sigma^2_\\alpha$, hence the result.\n\\subsection{Proof of Proposition \\ref{pro2r}}\nLet ${\\cal F}_{2,3}$ be\nthe set of real valued functions $h$\n defined on $\\mathbb{R}$, three times differentiable, such\n that $h(0)=0$, $\\|h''\\|_{\\infty}<+\\infty$, and\n $\\|h^{(3)}\\|_{\\infty}<+\\infty$. For a function $h\\in {\\cal\n F}_{2,3}$, we will denote by $b_2$ and $b_3$ the supremum norm of\n its second and third derivatives.\nWe first need the following lemma.\n\\begin{lemm}\\label{lem1}\nLet $h$ be a fixed function of the set ${\\cal F}_{2,3}$. Let\n$R$ be a fixed and finite subset of $S$. Let $r$ be a fixed\npositive real. For any $x\\in R$, let $V_x=B(x,r)\\cap R$.\nLet $(Y_x)_{x\\in S}$ be a real valued random field. Suppose that,\nfor any $x\\in S$, $\\mathbb{E} Y_x=0$ and $\\mathbb{E} Y_x^2<+\\infty$. Let\n$Z(R)=\\sum_{x\\in R}Y_x$. Then\n\\begin{eqnarray}\\label{cov1}\n{\\lefteqn{\\left|\\mathbb{E}(h(Z(R))) -\\Var Z(R)\\int_0^1t\n\\mathbb{E}(h''\\left(tZ(R)\\right))dt\\right|}} \\nonumber\\\\ &&\\leq\n\\int_0^1\\hspace{-1.5mm}\\sum_{x\\in\nR}\\left|\\Cov\\left(Y_x,h'(tZ(V_x^c))\\right)\\right|dt + 2\\sum_{x\\in\nR}\\mathbb{E} |Y_x||Z(V_x)|\\left[b_2\\wedge b_3|Z(V_x)|\\right] \\nonumber\\\\\n&& + b_2 \\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right| + b_2 \\sum_{x\\in\nR}\\left|\\Cov(Y_x, Z(V^c_x))\\right|,\n\\end{eqnarray}\nwhere $V^c_x=R\\setminus V_x$.\n\\end{lemm}\n{\\bf{Remark.}} For an independent random field $(Y_x)_{x\\in S}$,\nfulfilling $\\sup_{x\\in S}\\mathbb{E} Y_x^4<+\\infty$, Lemma \\ref{lem1}\napplied with $V_x=\\{x\\}$, ensures $$ \\left|\\mathbb{E}(h(Z(R))) -\\Var\nZ(R)\\int_0^1t \\mathbb{E}(h''\\left(tZ(R)\\right))dt\\right|\\leq 2 \\sum_{x\\in\nR}\\mathbb{E} |Y_x|^2\\left(b_2\\wedge b_3|Y_x|\\right) +\nb_2\\sqrt{|R|}\\sup_{x\\in S}\\|Y_x^2\\|_2 . $$\n{\\bf{Proof of Lemma \\ref{lem1}.}} We have,\n\\begin{eqnarray*}\\label{d1}\n{\\lefteqn{h(Z(R)) = Z(R)\\int_0^1 h'(tZ(R))dt =\n\\int_0^1\\left(\\sum_{x\\in R}Y_xh'(tZ(R))\\right)dt}} {\\nonumber}\\\\\n&& = \\int_0^1\\left(\\sum_{x\\in R}Y_xh'(tZ(V_x^c))\\right)dt +\n\\int_0^1\\left(\\sum_{x\\in\nR}Y_x\\left(h'(tZ(R))-h'(tZ(V_x^c))-tZ(V_x)h''(tZ(R))\\right)\\right)dt{\\nonumber}\\\\\n&& + \\sum_{x\\in R}Y_xZ(V_x)\\int_0^1th''(tZ(R))dt - \\sum_{x\\in\nR}\\mathbb{E}\\left(Y_xZ(V_x)\\right)\\int_0^1th''(tZ(R))dt\\\\ && + \\sum_{x\\in\nR}\\mathbb{E}\\left(Y_xZ(V_x)\\right)\\int_0^1th''(tZ(R))dt\n-\\sum_{x\\in R}\\mathbb{E}\\left(Y_xZ(R)\\right)\\int_0^1th''(tZ(R))dt {\\nonumber}\\\\\n&& + \\sum_{x\\in\nR}\\mathbb{E}\\left(Y_xZ(R)\\right)\\int_0^1th''(tZ(R))dt{\\nonumber}.\n\\end{eqnarray*}\nWe take expectation in the last equality. The obtained formula,\ntogether with the following estimations, proves Lemma \\ref{lem1}.\n\\begin{eqnarray*}\n{\\lefteqn{\\left|h'(tZ(R))-h'(tZ(V_x^c))-tZ(V_x)h''(tZ(R))\\right|}}\n\\\\ && \\leq\n\\left|h'(tZ(R))-h'(tZ(V_x^c))-tZ(V_x)h''(tZ(V_x^c))\\right|+\n|Z(V_x)||h''(tZ(R))-h''(tZ(V_x^c))| \\\\ && \\leq 2\n|Z(V_x)|\\left(b_2\\wedge b_3|Z(V_x)|\\right).\\qquad\\Box\n\\end{eqnarray*}\n\n\\vskip 3mm\\noindent\nOur purpose now is to control the right hand side of the bound\n(\\ref{cov1}) for a random field $(Y_x)_{x\\in S}$ fulfilling the\ncovariance inequality (\\ref{slcov}) and the requirements of\nProposition \\ref{pro2r}.\n\\begin{coro}\\label{cor1}\nLet $h$ be a fixed function of the set ${\\cal F}_{2,3}$. Let\n$R$ be a finite subset of $S$. For any $x\\in R$ and for any\npositive real $r$, let $V_x=B(x,r)\\cap R$. Let $(Y_x)_{x\\in\nS}$ be a real valued random field, fulfilling the covariance\ninequality (\\ref{slcov}). Suppose that, for any $x\\in S$, $\\mathbb{E}\nY_x=0$ and $\\sup_{x\\in S}\\|Y_x\\|_{\\infty}< M$, for some positive\nreal $M$. Recall that $Z(R)=\\sum_{x\\in R}Y_x$. Then, for\nany $\\delta>0$, there exists a positive constant $C({\\delta},M)$\nindependent of $R$, such that\n\\begin{eqnarray*}\n{\\lefteqn{\\sup_{h\\in {\\cal F}_{2,3}}\\left|\\mathbb{E}(h(Z(R))) -\\Var\nZ(R)\\int_0^1t \\mathbb{E}(h''\\left(tZ(R)\\right))dt\\right|}} \\\\ && \\leq\nC({\\delta},M)\\left\\{ b_2 |R|e^{-\\delta r}+ b_3 |R| \\kappa_r\n+ b_2|R|^{1\/2}\\kappa_r\\left(\\sum_{k=[3r]}^{\\infty}\\kappa_k e^{-\\delta(k-2r)}\\right)^{1\/2}\\right.\\\\\n&& \\left. +\nb_2|R|^{1\/2}\\kappa_{3r}\\left(\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k\\right)^{1\/2}\\right\\},\n\\end{eqnarray*}\nrecall that $\\sup_{x\\in S}|B(x,n)|\\leq \\kappa_n$.\n\\end{coro}\n{\\bf{Proof of Corollary \\ref{cor1}}}\n\\\\\nWe have $$V_x^c=\\{y\\in S,\\ \\ d(x,y)\\geq r\\}\\cap R. $$ Hence\n$$d(\\{x\\}, V^c_x)\\geq r. $$ The last bound together with\n(\\ref{slcov}), proves that\n\\begin{eqnarray}\\label{t1}\n\\sum_{x\\in R}\\left|\\Cov\\left(Y_x,h'(tZ(V_x^c))\\right)\\right|&\\leq\n& C_{\\delta} b_2 \\sum_{x\\in R}(|V_x^c|\\wedge 1)e^{-\\delta d(\\{x\\},\nV^c_x)} \\nonumber\n\\\\ & \\leq & C_{\\delta} b_2 |R|e^{-\\delta r}.\n\\end{eqnarray}\nIn the same way, we prove that\n\\begin{eqnarray}\\label{t2}\nb_2 \\sum_{x\\in R}\\left|\\Cov(Y_x, Z(V^c_x))\\right| & \\leq &\nC_{\\delta} b_2 |R|e^{-\\delta r}.\n\\end{eqnarray}\nNow\n\\begin{eqnarray}\\label{t3}\n\\sum_{x\\in R}\\mathbb{E} |Y_x||Z(V_x)|\\left(b_2\\wedge b_3|Z(V_x)|\\right)\n&\\leq & b_3 M |R| \\sup_{x\\in S} \\mathbb{E}|Z(V_x)|^2 {\\nonumber}\\\\\n&\\leq &\n b_3 M |R|\\kappa_r \\sup_{y\\in S}\n\\sum_{z\\in S}|\\Cov(Y_y,Y_z)|\n\\end{eqnarray}\nThe last bound is obtained since $|V_x|\\leq \\kappa_r$ and\n$\\sup_{y\\in S} \\sum_{z\\in S}|\\Cov(Y_y,Y_z)|<\\infty$ (the proof of\nthe last inequality is done along the same lines as that of\nProposition \\ref{slpro2}) .\n\\\\\nIt remains to control $$\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|.$$ For this, we\nargue as Bolthausen \\cite{Bolthausen82}. We have\n\\begin{eqnarray*}\n\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|^2 &=&\n\\Var(\\sum_{x\\in R}Y_xZ(V_x)) \\\\ & =& \\sum_{x\\in R}\\sum_{y\\in\nR}\\Cov(Y_xZ(V_x), Y_yZ(V_y)).\n\\end{eqnarray*}\nHence, since $V_x\\subset B(x,r)$,\n\\begin{eqnarray}\\label{i1}\n\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|^2\\leq \\sum_{x\\in\nR}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|.\n\\end{eqnarray}\nWe have,\n\\begin{eqnarray}\\label{second}\n \\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\n \\leq \\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\geq 3r}+\n\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}.\n\\end{eqnarray}\nWe begin by controlling the first term. The covariance inequality\n(\\ref{slcov}) together with some elementary estimations, ensures\n\\begin{eqnarray*}\n\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\geq 3r} & \\leq\n& \\sum_{k=[3r]}^{\\infty}\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{k\\leq d(x,y)< k+1} \\\\ & \\leq & 2 M^2\nC_{\\delta}\\sum_{k=[3r]}^{\\infty}e^{-\\delta\nd(\\{x,x'\\},\\{y,y'\\})}\\mathbb{I}_{k\\leq d(x,y)< k+1}\\\\ & \\leq & 2 M^2\nC_{\\delta}\\sum_{k=[3r]}^{\\infty}e^{-\\delta(k-2r)}\\mathbb{I}_{ d(x,y)<\nk+1},\n\\end{eqnarray*}\nthe last bound is obtained since, for any $x'\\in B(x,r)$ and\n$y'\\in B(y,r)$, we have, $$ d(\\{x,x'\\},\\{y,y'\\})+2r \\geq\nd(\\{x,x'\\},\\{y,y'\\})+ d(x,x')+ d(y,y')\\geq d(x,y).$$ Hence,\n\\begin{eqnarray}\\label{i2}\n{\\lefteqn{\\sum_{x\\in R}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\geq\n3r}}}\\nonumber \\\\ &&\\leq 2 M^2\nC_{\\delta}\\kappa_r^2\\sum_{k=[3r]}^{\\infty}\\sum_{x\\in R}\\sum_{y\\in\nR}e^{-\\delta(k-2r)} \\mathbb{I}_{y\\in B(x,k+1)}\\nonumber\n\\\\\n&&\\leq 2 M^2\nC_{\\delta}|R|\\kappa_r^2\\sum_{k=[3r]}^{\\infty}\\kappa_{k+1}e^{-\\delta(k-2r)}.\n\\end{eqnarray}\nWe now control the second term in (\\ref{second}). Inequality\n(\\ref{slcov}) and the fact that\\\\ $d(\\{x\\},\\{x',y,y'\\})\\leq\nd(\\{x\\},\\{x'\\})$, ensure\n\\begin{eqnarray*}\n{\\lefteqn{\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}}}\\\\\n &&\\leq \\left|\\Cov(Y_x,\nY_{x'}Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}+\n\\left|\\Cov(Y_x,Y_{x'})\\right|\\left|\\Cov(Y_y,Y_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq\n3r}\\\\ && \\leq 2M^2C_{\\delta}e^{-\\delta\nd(\\{x\\},\\{x',y,y'\\})}\\mathbb{I}_{d(x,y)\\leq 3r}.\n\\end{eqnarray*}\nWe deduce, using the last bound, that\n\\begin{eqnarray}\\label{bff}\n{\\lefteqn{\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}}}\\nonumber\\\\\n &&\\leq \\sum_{k=1}^{[3r]+1}\\left|\\Cov(Y_xY_{x'},\nY_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq 3r}\\mathbb{I}_{k-1\\leq\nd(\\{x\\},\\{x',y,y'\\})< k}\\nonumber\\\\ && \\leq\n2M^2C_{\\delta}\\sum_{k=1}^{[3r]+1}e^{-\\delta (k-1)}\\mathbb{I}_{d(x,y)\\leq\n3r}\\mathbb{I}_{d(\\{x\\},\\{x',y,y'\\})< k}.\n\\end{eqnarray}\nWe have $$\\mathbb{I}_{d(\\{x\\},\\{x',y,y'\\})\\leq k}\\leq\n\\mathbb{I}_{d(\\{x\\},\\{x'\\})\\leq k}+ \\mathbb{I}_{d(\\{x\\},\\{y\\})\\leq\nk}+\\mathbb{I}_{d(\\{x\\},\\{y'\\})\\leq k}.$$ Hence, we check that,\n\\begin{eqnarray}\\label{bf}\n\\sum_{x\\in R}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\mathbb{I}_{d(x,y)\\leq 3r}\\mathbb{I}_{d(\\{x\\},\\{x',y,y'\\})\\leq k} \\leq\n3|R|\\kappa_{3r}^2\\kappa_k.\n\\end{eqnarray}\nWe obtain combining (\\ref{bff}) and (\\ref{bf}),\n\\begin{multline}\n \\sum_{x\\in R}\\sum_{x'\\in B(x,r)}\\sum_{y\\in R}\\sum_{y'\\in\nB(y,r)}\\left|\\Cov(Y_xY_{x'}, Y_yY_{y'})\\right|\\mathbb{I}_{d(x,y)\\leq\n3r}\\\\\n\\label{i3} \\leq\n6e^{\\delta}M^2C_{\\delta}|R|\\kappa_{3r}^2\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k.\n\\end{multline}\nWe collect the bounds (\\ref{i1}), (\\ref{i2}) and (\\ref{i3}), we\nobtain,\n\\begin{eqnarray}\\label{t4}\n{\\lefteqn{\\mathbb{E}\\left|\\sum_{x\\in\nR}\\left(Y_xZ(V_x)-\\mathbb{E}(Y_xZ(V_x))\\right)\\right|}}\\nonumber\\\\\n&&\\leq\nC(\\delta,M)|R|^{1\/2}\\left\\{\\kappa_r\\left(\\sum_{k=[3r]}^{\\infty}\\kappa_{k+1}e^{-\\delta(k-2r)}\\right)^{1\/2}\n+ \\kappa_{3r} \\left(\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k\\right)^{1\/2}\\right\\}.\n\\end{eqnarray}\nFinally, the bounds (\\ref{t1}), (\\ref{t2}), (\\ref{t3}),\n(\\ref{t4}), together with Lemma \\ref{lem1} prove Corollary\n\\ref{cor1}. \\ \\ \\ \\ $\\Box$\n\\vskip 3mm\\noindent\n {\\bf{End of the proof of Proposition \\ref{pro2r}.}}\nWe apply Corollary \\ref{cor1} to the real and imaginary parts of\nthe function $x\\rightarrow \\exp(iux\/{\\sqrt{|B_n|}})-1$. Those\nfunctions belong to the set ${\\cal F}_{2,3}$, with\n$b_2=\\frac{\\textstyle u^2}{\\textstyle |B_n|}$ and\n$b_3=\\frac{\\textstyle |u|^3}{\\textstyle |B_n|^{3\/2}}$.\n\\\\\nWe obtain, noting by $\\phi_n$ the characteristic function of the\nnormalized sum $Z(B_n)\/{\\sqrt{|B_n|}}$,\n\\begin{eqnarray*}\n{\\lefteqn{\\left|\\phi_n(u)-1+\\frac{\\Var Z(B_n)}{|B_n|}u^2\n\\int_0^1t\\phi_n(tu)dt\\right|}}\n\\\\ && \\leq C({\\delta},M,u)\\left\\{\ne^{-\\delta r}+ \\frac{\\kappa_r}{\\sqrt{|B_n|}} +\n\\frac{\\kappa_r}{\\sqrt{|B_n|}}\\left(\\sum_{k=[3r]}^{\\infty}\\kappa_k e^{-\\delta(k-2r)}\\right)^{1\/2}\\right.\\\\\n&& \\left.\n+\\frac{\\kappa_{3r}}{\\sqrt{|B_n|}}\\left(\\sum_{k=1}^{[3r]+1}e^{-\\delta\nk}\\kappa_k\\right)^{1\/2}\\right\\}.\n\\end{eqnarray*}\nLet $\\delta$ be a fixed positive real such that\n$\\delta>12\\rho$, recall that\n$$\\sup_{x\\in S}|B(x,r)|\\leq 2 e^{r\\rho}=:\\kappa_r.$$ Hence\n\\begin{eqnarray*}\n{\\lefteqn{\\left|\\phi_n(u)-1+\\frac{\\Var Z(B_n)}{|B_n|}u^2\n\\int_0^1t\\phi_n(tu)dt\\right|}}\n\\\\ && \\leq C(\\delta,M,u)\\left\\{\ne^{-\\delta r}+\n\\frac{e^{r\\rho}}{\\sqrt{|B_n|}}+\\frac{e^{(\\rho+\\delta)r}}{\\sqrt{|B_n|}}\\left(\\sum_{k=[3r]}^{\\infty}e^{-(\\delta-\\rho)k}\\right)^{1\/2}\n +\\frac{e^{3\\rho\nr}}{\\sqrt{|B_n|}}\\left(\\sum_{k=1}^{[3r]+1}e^{-(\\delta-\\rho)\nk}\\right)^{1\/2}\\right\\}\\\\ &&\\leq\nC(M,\\rho,\\delta,u)\\left(e^{-\\delta\nr}+\\frac{e^{3r\\rho}}{\\sqrt{|B_n|}}+\n\\frac{e^{-(\\delta-5\\rho)r\/2}}{\\sqrt{|B_n|}}\\right).\n\\end{eqnarray*}\nFor a suitable choice of the sequence $r$ (for example we can take\n$r=\\frac{2}{\\delta}\\ln|B_n|$), the right hand side of the last\nbound tends to $0$ an $n$ tends to infinity:\n\\begin{equation}\\label{limf}\n\\lim_{n\\rightarrow \\infty}\\left|\\phi_n(u)-1+\\frac{\\Var\nZ(B_n)}{|B_n|}u^2 \\int_0^1t\\phi_n(tu)dt\\right|=0.\n\\end{equation}\nWe now need the following lemma.\n\\begin{lemm}\\label{lem2} Let $\\sigma^2$ be a positive real. Let $(X_n)$\nbe a sequence of real valued random variables such that\n$\\sup_{n\\in \\mathbb{N}}\\mathbb{E} X_n^2<+\\infty$. Let $\\phi_n$ be the\ncharacteristic function of $X_n$. Suppose that for any $u\\in \\mathbb{R}$,\n\\begin{equation}\\label{cara}\n\\lim_{n\\rightarrow\n+\\infty}\\left|\\phi_n(u)-1+\\sigma^2\\int_0^ut\\phi_n(t)dt\\right|=0.\n\\end{equation}\nThen, for any $u\\in \\mathbb{R}$, $$\\lim_{n\\rightarrow\n+\\infty}\\phi_n(u)=\\exp(-\\frac{u^2\\sigma^2}{2}). $$\n\\end{lemm}\n{\\bf{Proof of Lemma \\ref{lem2}.}}\n Lemma \\ref{lem2} is a variant of Lemma 2 in\nBolthausen \\cite{Bolthausen82}. The Markov inequality and the\ncondition $\\sup_{n\\in \\mathbb{N}}\\mathbb{E} X_n^2<+\\infty$ imply that the\nsequence $(\\mu_n)_{n\\in \\mathbb{N}}$ of the laws of $(X_n)$ is tight.\nTheorem 25.10 in Billingsley \\cite{Billingsley} proves the\nexistence of a subsequence $\\mu_{n_k}$ and a probability measure\n$\\mu$ such that $\\mu_{n_k}$ converges weakly to $\\mu$ as $k$ tends\nto infinity. Let $\\phi$ be the characteristic function of $\\mu$.\nWe deduce from (\\ref{cara}) that, for any $u\\in \\mathbb{R}$, $$\n\\phi(u)-1+\\sigma^2\\int_0^ut\\phi(t)dt=0, $$ or equivalently, for\nany $u\\in \\mathbb{R}$, $$ \\phi'(u)+\\sigma^2u\\phi(u)=0. $$ We obtain,\nintegrating the last equation, that for any $u\\in \\mathbb{R}$, $$\n\\phi(u)= \\exp(-\\frac{\\sigma^2 u^2}{2}). $$ The proof of Lemma\n\\ref{lem2} is completed by using Theorem 25.10 in Billingsley\n\\cite{Billingsley} and its corollary. \\,\\,\\, $\\Box$ \\vskip 1mm\nProposition \\ref{pro2r} follows from (\\ref{limff}), (\\ref{limf})\nand Lemma \\ref{lem2}.\\ \\ \\ $\\Box$ \\vskip 3mm\n\\subsection{Proof of Proposition \\ref{slpro2}.}\nWe deduce from\n(\\ref{slcov}) that for any positive real $\\delta$ there\nexists a positive constant $C_{\\delta}$ such that for different\nsites $x$ and $y$ of $S$,\n\\begin{equation}\\label{slc1}\n\\left|\\Cov(Y_x,Y_y)\\right|\\leq C_{\\delta} e^{-\\delta d(x,y)}.\n\\end{equation}\nHence, the first conclusion of Proposition \\ref{slpro2} follows\nfrom the bound (\\ref{slc1}), together with the following\nelementary calculations, for $\\rho<\\delta$,\n\\begin{eqnarray}\\label{slcf}\n\\sum_{z\\in S}|\\Cov(Y_0,Y_{z})| &\\leq & C_{\\delta} \\sum_{z\\in\nS}\\exp(-\\delta d(0,z)){\\nonumber}\\\\\n& \\leq & C_{\\delta}\\sum_{z\\in S}\\sum_{r=0}^{\\infty}\\exp(-\\delta\nd(0,z))\\mathbb{I}_{r \\leq d(0,z)< r+1} {\\nonumber}\\\\ & \\leq &\nC_{\\delta}\\sum_{r= 0}^{\\infty}\\exp(-\\delta\nr)\\sum_{z\\in S}\\mathbb{I}_{d(0,z)< r+1}{\\nonumber}\\\\\n& \\leq & C_{\\delta}\\sum_{r=0}^{\\infty}\\exp(-\\delta\nr)|B(0,r+1)|{\\nonumber}\\\\ & \\leq & C(\\delta,\\rho)\\sum_{r=\n0}^{\\infty}\\exp(-(\\delta -\\rho)r),\n\\end{eqnarray}\nwhere $C(\\delta,\\rho)$ is a positive constant depending on\n$\\delta$ and $\\rho$.\n\\\\\nWe now prove the second part of Proposition \\ref{slpro2}. Thanks\nto (\\ref{slc}), we can find a sequence $u=(u_n)$ of positive real\nnumbers such that\n\\begin{equation}\\label{sllimd}\n\\lim_{n\\rightarrow +\\infty}u_n=+\\infty, \\, \\lim_{n\\rightarrow\n+\\infty} \\frac{|\\partial B_n|}{|B_n|}\\exp(\\rho u_n)=0.\n\\end{equation}\nLet $(\\partial _{u}B_n)_n$ be the sequence of subsets of $S$\ndefined by\n$$\\partial _{u}B_n=\\{s\\in B_n\\,:\\, d(s,\\partial B_n)< u_n\\}. $$\nThe bound (\\ref{borneboule}) gives\n$$ |\\partial _{u}B_n|\\leq 2 |\\partial B_n| e^{u_n\\rho},\n$$\nwhich together with the suitable choice of the sequence $(u_n)$\nensures\n\\begin{equation}\\label{sllimt3}\n\\lim_{n\\rightarrow +\\infty}\\frac{|\\partial _{u}B_n|}{|B_n|}=0,\n\\end{equation}\nwe shall use this fact below without further comments. Let\n$B^{u}_n=B_n\\setminus \\partial _{u}B_n$. We decompose the quantity\n$\\Var\\,S_n$ as in Newman \\cite{Newman80}:\n\\begin{eqnarray*}\\label{sldeco}\n\\frac{1}{|B_n|}\\Var\\, S_n & = & \\frac{1}{|B_n|}\\sum_{x\\in\nB_n}\\sum_{y\\in B_n}\\Cov\\left(Y_x,Y_y\\right) = T_{1,n} + T_{2,n}\n+T_{3,n},\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\nT_{1,n} & = & \\frac{1}{|B_n|}\\sum_{x\\in B^{u}_n}\\,\\,\\sum_{y\\in\nB_n\\setminus B(x,u_n)}\\Cov\\left(Y_x,Y_y\\right),\\\\\nT_{2,n} & = & \\frac{1}{|B_n|}\\sum_{x\\in B^{u}_n}\\,\\,\\sum_{y\\in\nB_n\\cap B(x,u_n)}\\Cov\\left(Y_x,Y_y\\right),\\\\ T_{3,n} & = &\n\\frac{1}{|B_n|}\\sum_{x\\in \\partial_u B_n}\\sum_{y\\in\nB_n}\\Cov\\left(Y_x,Y_y\\right).\n\\end{eqnarray*}\n{\\bf{Control of $T_{1,n}$.}} We have, since $|B^{u}_n| \\leq |B_n|$\nand applying (\\ref{slc1})\n\\begin{eqnarray}\\label{slt1bisbis}\n|T_{1,n}| &\\leq & \\sup_{x\\in S} \\sum_{y\\in S\\setminus B(x,u_n)\n}\\left|\\Cov(Y_x,Y_y)\\right| \\leq C_\\delta \\sup_{x\\in S}\n\\sum_{y\\in S\\setminus B(x,n)}\\exp(-\\delta d(x,y)).\n\\end{eqnarray}\nFor any fixed $x\\in S$, we argue as for (\\ref{slcf}) and we obtain\nfor $\\rho<\\delta$,\n\\begin{equation}\\label{slt1bis}\n\\sum_{y\\in S\\setminus B(x,n)}\\exp(-\\delta d(x,y)) \\leq\nC(\\delta)\\sum_{r= [u_n]}^{\\infty}\\exp(-(\\delta-\\rho) r) \\leq\nC(\\delta,\\rho)\\exp(-(\\delta-\\rho)u_n)\n\\end{equation}\n We obtain, collecting (\\ref{slt1bisbis}), (\\ref{slt1bis}) together with\nthe first limit in (\\ref{sllimd}) :\n\\begin{equation}\\label{sllimT1}\n\\lim_{n\\rightarrow +\\infty} T_{1,n}=0.\n\\end{equation}\n\\\\\n{\\bf{Control of $T_{3,n}$.}} We obtain using (\\ref{slc1}) :\n\\begin{eqnarray}\\label{slt3bis}\n|T_{3,n}| &\\leq & \\frac{|\\partial_u B_n|}{|B_n|} \\sup_{x\\in S}\n\\sum_{y\\in S}\\left|\\Cov(Y_x,Y_y)\\right|.\n\\end{eqnarray}\nThe last bound, together with the limit (\\ref{sllimt3})\ngives\n\\begin{equation}\\label{sllimT3}\n\\lim_{n\\rightarrow +\\infty}T_{3,n}=0.\n\\end{equation}\n \\\\ {\\bf{Control of $T_{2,n}$.}}\n We deduce using the following implication, if $x\\in\nB^{u}_n$ and $y$ is not belonging to $B_n$ then $d(x,y)\\geq u_n$,\nthat\n\\begin{eqnarray*}\nT_{2,n} & = & \\frac{1}{|B_n|}\\sum_{x\\in B^{u}_n}\\,\\,\\sum_{y\\in\nB(x,u_n)}\\Cov(Y_x,Y_y)\n\\end{eqnarray*}\nWe claim that,\n\\begin{equation}\\label{ccc}\n\\sum_{y\\in B(x,u_n)}\\Cov(Y_x,Y_y) = \\sum_{z\\in\nB(0,u_n)}\\Cov\\left(Y_0,Y_z\\right),\n\\end{equation}\nin fact, since the graph ${\\cal G}$ is transitive, there exits an\nautomorphism $a_x$, such that $a_x(x)=0$ ($0$ is a fixed vertex in\n$S$). Equality (\\ref{slstationarity}) gives\n$$\n\\sum_{y\\in B(x,u_n)}\\Cov(Y_x,Y_y)=\\sum_{y\\in\nB(x,u_n)}\\Cov(Y_0,Y_{a_x(y)}).\n$$\nNow, Lemma 1.3.2 in Godsil and Royle \\cite{GodsilRoyle} yields that $d(x,y)=\nd(a_x(x), a_x(y))=d(0,a_x(y))$. From this we deduce that $y\\in\nB(x,u_n)$ if and only if $a_x(y)\\in B(0,u_n)$. From above, we\nconclude that,\n$$\n\\sum_{y\\in B(x,u_n)}\\Cov(Y_x,Y_y)=\\sum_{a_x(y)\\in\nB(0,u_n)}\\Cov(Y_0,Y_{a_x(y)})=\\sum_{z\\in B(0,u_n)}\\Cov(Y_0,Y_{z}),\n$$\nwhich proves (\\ref{ccc}). Consequently,\n\\begin{eqnarray*}\nT_{2,n} & = & \\frac{|B^{u}_n|}{|B_n|}\\sum_{z\\in B(0,u_n)\n}\\Cov(Y_0,Y_z).\n\\end{eqnarray*}\nThe last equality together with the first limit in (\\ref{sllimd})\nand (\\ref{sllimt3}), ensures\n\\begin{equation}\\label{sllimT2}\n\\lim_{n\\rightarrow +\\infty} T_{2,n}= \\sum_{z\\in S}\\Cov(Y_0,Y_z).\n\\end{equation}\nThe second conclusion of Proposition \\ref{slpro2} is proved by\ncollecting the limits (\\ref{sllimT1}), (\\ref{sllimT3}) and\n(\\ref{sllimT2}). $\\Box$\n\\paragraph{Acknowledgements.} We wish to thank Professor Mathew Penrose for\nhis important remarks which helped us to derive the present version\nof this work. He mentioned an error in a previous draft for this\nwork, see the Remark following Theorem \\ref{cltips2}. We also thank\nDavid Coupier for his precious comments.\n\\bibliographystyle{plain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDepth First Search (DFS) is a very well-known method for visiting the vertices and edges of a directed or undirected graph. DFS differs from other ways of traversing the graph such as Breadth First Search (BFS) by the following DFS protocol: Whenever two or more vertices were discovered by the search method and have unexplored incident (out)edges, an (out)edge incident on the most recently discovered such vertex is explored first. This DFS traversal produces a rooted spanning tree (forest), called DFS tree (forest) along with assigning an index to every vertex $v$ i.e., the time vertex $v$ is discovered for the first time during DFS. We call it depth-first-index (DFI($v$)). Let $G=(V,E)$ be a graph on $n=|V|$ vertices and $m=|E|$ edges where $V =\\{v_1, v_2, \\cdots, v_n\\}$. It takes $O(m+n)$ time to perform a DFS traversal of $G$ and to generate its DFS tree (forest) with DFIs of all the vertices. The DFS rule confers a number of structural properties on the resulting graph traversal that cause DFS to have a large number of applications. These properties are captured in the DFS tree (forest), and can be used crucially to design efficient algorithms for many basic and fundamental algorithmic graph problems, namely, biconnectivity~\\cite{Tarjan72}, $2$-edge connectivity~\\cite{Tarjan74}, strongly connected components~\\cite{Tarjan72}, topological sorting~\\cite{Tarjan72}, dominators~\\cite{Tarjan741}, {\\it st}-numbering~\\cite{EvenT76} and planarity testing~\\cite{HopcroftT74} among many others.\n\nThere are two versions of DFS studied in the literature. In the lexicographically smallest DFS or lex-DFS problem, when DFS looks for an unvisited vertex to visit in an adjacency list, it picks the \"first\" unvisited vertex where the \"first\" is with respect to the appearance order in the adjacency list. The resulting DFS tree will be unique. In contrast to lex-DFS, an algorithm that outputs some DFS numbering of a given graph, treats an adjacency list as a set, ignoring the order of appearance of vertices in it, and outputs a vertex ordering $Q$ such that there exists some adjacency ordering $R$ such that $Q$ is the DFS numbering with respect to $R$. We say that such a DFS algorithm performs general-DFS. In this work, we focus only on lex-DFS, thus, given a source vertex, the DFS tree is always unique. Given the lex-DFS tree, the non-tree edges of a given directed graph can be classified into four categories as follows. An edge directed from a vertex to its ancestor in the tree is called a back edge. Similarly, an edge directed from a vertex to its descendant in the tree is called a forward edge. Further, an edge directed from right to left in the DFS tree is called a cross edge. The remaining edges directed from left to right in the tree are called anti-cross edges. In the undirected graphs, there are no cross edges. Note that, we can store the complete DFS tree explicitly using $O(n \\lg n)$ bits by storing pointers between nodes. In what follows, we formally define the problem which we call the {\\sf{DFS-Indexing}}\\ problem.\n\n\\begin{center}\n\\fbox{\\begin{minipage}{11cm}\n\\noindent{{\\sf{DFS-Indexing}}\\ problem}\\\\\n\\noindent {\\em Input}: A directed or undirected graph $G=(V,E)$ where $|V|=n$, $|E|=m$, and a source vertex $v_s$, preprocess $G$ and answer the following queries with respect to the DFS tree $T$ rooted at $v_s$:\n\\begin{enumerate}\n\\item Given any pair of vertices $v_i$ and $v_j$,\n\\begin{enumerate}\n\\item Who is visited first in the DFS traversal of $G$?\n\\item Is $v_i$ an ancestor of $v_j$ in $T$?\n\\end{enumerate}\n\\item Given $v_i$, \n\\begin{enumerate}\n\\item Return the parent of $v_i$ in $T$.\n\\item Return the number of children (if any) of $v_i$ in $T$.\n\\item Enumerate all the children (if any) of $v_i$ in $T$.\n\\item Return the DFI of $v_i$.\n\\end{enumerate}\n\\item Enumerate the order in which vertices of $G$ are visited in the DFS.\n\\item Given $1\\leq i \\leq n$, return the vertex with DFI $i$.\n\\end{enumerate}\n\\end{minipage}}\n\\end{center} \n\nWe study the {\\sf{DFS-Indexing}}\\ problem in two well-known models: the {\\it indexing} and {\\it encoding} models~\\cite{Navarro}. In the indexing model, we wish to build an index {\\it ind} after preprocessing the input graph $G$ such that queries can be answered using both {\\it ind} and $G$ whereas in the encoding model, we seek to build a data structure {\\it encod} after preprocessing the input graph $G$ such that queries have to be answered using {\\it encod} only. Typically the parameters of interest are (i) query time, (ii) space consumed (in bits) by {\\it ind} and {\\it encod} resp. and (iii) the preprocessing time and space. We address all these issues in our paper for the {\\sf{DFS-Indexing}}\\ problem, assuming our computational model is a Random-Access-Machine with constant time operations on $O(\\lg n)$-bit words. In both models, it is not hard to see that using $O(n \\lg n)$ bits, we can answer all the queries of the {\\sf{DFS-Indexing}}\\ problem in the optimal $O(1)$ time except the query of 3 which takes $O(n)$ time. \nOur main objective here is to beat this trivial $O(n \\lg n)$ bit space bound without compromising too much on the query time. \n\nThe motivation for studying this question mainly stems from the rise of the ``big data'' phenomenon and its implications. To illustrate, the rate at which we store data is increasing even faster than the speed and capacity of computing hardware. Thus, if we want to use the stored data efficiently, we need to represent it in sophisticated ways. Many applications dealing with huge data structures can benefit from keeping them in compressed form. Compression has many advantages: it can allow a representation to fit in main memory rather than swapping out to disk, and it improves cache performance since it allows more data to fit into the cache. However, such a data structure is only handy if it allows the application to perform fast queries to the data, and this is the direction we want to explore for the DFS tree. More specifically, we are interested in representing the DFS tree of a given graph compactly while supporting all the queries mentioned above efficiently.\n\n\\subsection{Representation of the Input Graph}\nWe assume that the input graphs $G=(V,E)$ are represented using the {\\it adjacency array} format, i.e., $G$ is given by an array of length $|V|$ where the $i$-th entry stores a pointer to an array that stores all the neighbors of the $i$-th vertex. For the directed graphs, we assume that the input representation has both in\/out adjacency array for all the vertices i.e., for directed graphs, every vertex $v$ has access to two arrays, one array is for all the in-neighbors of $v$ and the other array is for all the out-neighbors of $v$. This form of input graph representation has now become somewhat standard and was recently used in plenty of other works~\\cite{Banerjee2018,Cha_thesis,CTW,Chakraborty00S18,ChakrabortyRS17,ChakrabortyS19}. Throughout this paper, we call a graph sparse when $m=O(n)$, and dense otherwise (i.e., $m=\\omega(n)$).\n\n\\subsection{Our Main Results and Organization of the Paper}\nWe start by mentioning some preliminary results that will be used throughout the paper in Section~\\ref{prelim}. Section~\\ref{main_algo} contains the description of our main index for solving the {\\sf{DFS-Indexing}}\\ problem in the indexing model. Our main results here can be summarized as follows,\n\n\n\\begin{theorem}\\label{sparsecase}\nIn the indexing model, given any sparse (dense resp.) undirected or directed graph $G$, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing algorithm which outputs a data structure of size $O(n)$ ($O(n \\lg (m\/n))$ resp.) bits, using which the queries 1(a), 1(b), 2(d) and 4 can be reported in $O(\\lg n)$ time, 2(a) and 2(b) \nin $O(1)$ time, 2(c) in time proportional to the number of solutions, and finally 3 can be solved in $O(n)$ time resp. for the {\\sf{DFS-Indexing}}\\ problem.\n\\end{theorem} \n\nWe want to emphasize that obtaining better results for sparse graphs is not only interesting from theoretical perspective\nbut also from practical point of view as these graphs do appear very frequently in most of the realistic network scenario in real world applications, e.g., Road networks and the Internet.\n\nIn Section~\\ref{encoding}, we provide the detailed proof of our index in the encoding model. This contains a space lower bound for any index for the {\\sf{DFS-Indexing}}\\ problem, followed by an index whose size asymptotically matches the lower bound and has efficient query time. We summarize our main results~below.\n\n\\begin{theorem}\\label{encoding_lower_bound}\nIn the encoding model, the size of any data structure for the {\\sf{DFS-Indexing}}\\ problem must be $\\Omega(n \\lg n)$ bits. On the other hand, given any (un)directed graph, there exists an $O(m+n)$ time and $O(n \\lg n)$ bits preprocessing scheme that outputs an index of size $(1+\\epsilon)n \\lg n+2n+o(n)$ bits (for any constant $\\epsilon > 0$), using which the queries 1(a), 1(b), 2(a), 2(b), 2(d) can be reported in $O(1)$ time, 2(c) in time proportional to the number of solutions, 3 in $O(n\/\\epsilon)$ time, and finally 4 in $O(1\/\\epsilon)$ time resp. for the {\\sf{DFS-Indexing}}\\ problem in this setting.\n\\end{theorem}\n\n\nBuilding on all these aforementioned results, we also show a host of applications of our techniques in designing indices for other fundamental graph problems in Appendix~\\ref{appendix}. \nFinally, we conclude in Section~\\ref{conclude} with some open problems and possible future directions to explore further.\n\n{\\bf Remark.} At this point we want to emphasize that our results are more general, i.e., they can be extended to store any arbitrary labeled tree (arising from some underlying graph) along with the mechanism for fast querying. This method is very useful as many graph algorithms (like shortest path, minimum spanning tree, biconnectivity etc) induce a tree structure which is used subsequently during the execution of the algorithm. Hence, we can use our technique to store and query those trees compactly as well as efficiently. Thus, we also believe that our algorithm may find many other potential interesting applications. However, we chose to provide all the details in terms of DFS as DFS is very widely popular graph traversal technique and is used as the backbone for multiple fundamental algorithms, yet there is no explicit indexing scheme for storing DFS tree compactly. In Appendix~\\ref{app}, \nwe show how one can extend these techniques to design indexing schemes for a variety of other classical and fundamental graph problems.\n\n\\subsection{Related Works}\nThere already exists a large body of work concerning compactly representing various specific classes of graphs, for example planar, constant genus graphs etc~\\cite{Acan,BlandfordBK03,FerresSG0N17,MunroN16,MunroR01,Navarro,YamanakaN10}. All of these works are able to store an $n$-vertex unlabeled planar graph in $O(n)$ bits, and some of them even allow for $O(1)$-time neighbor queries. Generally what is meant by unlabeled is that the algorithm is free to choose an ordering on the vertices (integer labels from $1$ to $n$). Our setting here is slightly different as we work with graphs whose vertices are labeled, and matches closely with~\\cite{BarbayAHM07}. Also we want to support more complex queries whereas the previous works only focused on adjacency queries mostly. Even though DFS being such a widely known method, and having many applications, to the best of our knowledge, we are not aware of any previous work focusing on compactly representing the DFS tree with efficient query support.\n\n\n\\section{Preliminaries} \\label{prelim}\n{\\bf Rank-Select.} We make use of the following theorem:\n\\begin{theorem}\\cite{Clark96}\n \\label{staticrs}\nWe can store a bitstring $B$ of length $n$ with additional $o(n)$ bits such that rank and select operations (defined below) can be supported in $O(1)$ time. Such a structure can also be constructed from the given bitstring in $O(n)$ time and space.\n\\end{theorem}\n\nFor any $ a\\in \\{0,1\\}$, the rank and select operations are defined as follows :\n\\begin{itemize}\n \\item $rank_a(B,i)$ = the number of occurrences of $a$\n\n in $B[1,i]$, for $1\\leq i\\leq n$;\n \\item $select_a(B,i)$ = the position in $B$ of the $i$-th occurrence of $a$, for $1\\leq i\\leq n$.\n\\end{itemize}\n\nWhen the bitvector $B$ is sparse, the space overhead of $o(n)$ bits can be avoided by using the following theorem, which will also be used later in our paper. \n\\begin{theorem}\\cite{Navarro}\n\\label{sparse}\nWe can store a bitstring $B$ of length $n$ with $m$ $1$s using $m\\lg(n\/m)+O(m)$ bits such that $select_1(B,1)$ can be supported in $O(1)$ time, $select_0(B,1)$ in $O(\\lg m)$ time, and both the rank queries ($rank_1(B,i)$ and $rank_0(B,i)$) can be supported in $O(\\text{min}(\\lg m,\\lg n\/m))$ time. Such a structure can also be constructed from $B$ in $O(n)$ time and space.\n\\end{theorem}\n\n{\\bf Permutation.} We also use the following theorem:\n\\begin{theorem}\\cite{MunroRRR12}\n\\label{perm}\nA permutation $\\pi$ of length $n$ can be represented using $(1+\\epsilon)n \\lg n$ bits so that $\\pi(i)$ is answered in $O(1)$ time and $\\pi^{-1}$ in time $O(1\/\\epsilon)$ for any constant $\\epsilon > 0$. Such a representation can be constructed using $O(n)$ time and space.\n\\end{theorem}\n\n{\\bf Succinct Tree Representation.} We need following result from~\\cite{FarzanM14}.\n\\begin{theorem}~\\cite{FarzanM14}\n\\label{succ_tree}\nThere exists a data structure to succinctly encode an ordered tree with $n$ nodes using $2n+o(n)$ bits such that, given a node $v$, (a) child($v$,$i$): $i$-th child of $v$, (b) degree($v$): number of children of $v$, (c) depth($v$): depth of $v$, (d) $select_{pre}$($v$): position of $v$ in preorder, (e) $LA(v,i)$: ancestor of $v$ at level $i$ can be supported in $O(1)$ time among many others. Such a structure can also be constructed in $O(n)$ time and space.\n\\end{theorem}\n\n\n\n\n\n\\section{Algorithms in the Indexing Model}\\label{main_algo}\nIn this section, we provide the main algorithmic ideas needed for the solution of the {\\sf{DFS-Indexing}}\\ problem in the indexing model. We start by describing the preprocessing procedure which is followed by the query algorithms.\n\\subsection{Preprocessing Step}\nWe first describe our algorithms for undirected graphs, and later mention the modifications required for the case of directed graphs. The preprocessing step of the algorithm is divided into two parts. In the first part, we perform a DFS of the input graph $G$ along with storing some necessary data structures. In the second step, we perform a partition of the DFS tree of $G$ using the well-known ``tree covering technique'' of the succinct data structures world~\\cite{FarzanM11}, and also store some auxiliary data structures. Later, in the final step of our algorithm, we show how to use these data structures to answer the required queries. In what follows, we describe each step in detail.\n\n{\\bf Step 1: Creating Parent-Child Array using Unary Degree Sequence Array.} The main idea of this step is to perform a DFS traversal of $G$ and store in a {\\it compact way} the parent-child relationship of the DFS tree $T$. The way we achieve this is by using three bitvectors of length $O(m+n)$ bits.\nRecall that, our input graphs $G=(V,E)$ are represented using the standard adjacency array. Central to our preprocessing algorithm is an encoding of the degrees of the vertices in unary. As usual, let $V =\\{v_1, v_2, \\cdots, v_n\\}$ be the vertex set of $G$. The unary degree sequence encoding $D$ of the undirected graph $G$ has $n$ $1$s to represent the $n$ vertices and each $1$ is followed by a number of $0$s equal to its degree. Moreover, if $d$ is the degree of vertex $v_i$, then $d$ $0$s following the $i$-th $1$ in the $D$ array corresponds to $d$ neighbors of $v_i$ (or equivalently the edges from $v_i$ to the $d$ neighbors of $v_i$) in the same order as in the adjacency array of $v_i$. Clearly $D$ uses $n+2m$ bits and can be obtained from the neighbors of each vertex in $O(m+n)$ time. Now using {\\it rank\/select} queries of Theorem~\\ref{staticrs} in Section~\\ref{prelim}, the $j$-th outgoing edge of vertex $v_i$ can be identified with the position $p = select_1(D,i)+j$ of $D$ ($1 \\le j \\le degree(v_i)$ where $degree(v_i)$ denotes the degree of the vertex $v_i$). From a position $p$, we can obtain an endpoint of the corresponding edge by $i = rank_1(D, p)$, and the other endpoint is the $j$-th neighbor of $v_i$ where $j = p - select_1(D, i)$.\n\nWe also use two bitvectors $E, P$ of the same length where every bit is initialized to $0$, and the bits in $E, P$ are in one-to-one correspondence with bits in $D$. The bitvector $E$ will be used to mark the tree edges of the DFS tree $T$, and the bitvector $P$ to mark the unique parent of every vertex in $T$.\nThe marking is carried out while performing a DFS of $G$ in the preprocessing step. I.e., if $(v_i, v_j)$ is an edge in the DFS tree where $v_i$ is the parent of $v_j$, and suppose $k$ is the index of the edge $(v_i, v_j)$ in $D$, then the corresponding location in $E$ is marked as $1$ during DFS. At the same time, we scan the adjacency array of $v_j$ to find the position of $v_i$ (as $G$ is undirected, there will be two entries for each edge in the adjacency array), and suppose $t$ is the index of the edge $(v_j,v_i)$ in $D$, then the corresponding location in $P$ is marked as $1$ during DFS. Thus, assuming $G$ is a connected graph, once DFS finishes traversing $G$, the number of ones in $E$ is exactly the number of tree edges (which is $n-1$) and the number of ones in $P$ will be $n-1$ as root does not have any parent. \n\nThe parent of $v_i$ in $T$ is computed in $O(1)$ time\nas follows.\nLet $v_r$ be the root of $T$. Then if $i > r$\n(resp. $i \\pi(j)$). We enumerate the vertex ordering as traversed in the DFS order by invoking $\\pi^{-1}(1)$, then $\\pi^{-1}(2)$, and so on till $\\pi^{-1}(n)$. We answer 1(b) in affirmative by checking if $LA(v_j,depth(v_i))$ matches with $v_i$, otherwise no. To answer 2(a), we return $LA(v_i,depth(v_i)-1)$. We return the answer of 2(b) by using the query $degree(v_i)$. Finally, we enumerate the children of a node $v_i$ as requested in query 2(c) by using the query $child(v_i,1)$ till $child(v_i,degree(v_i))$. Hence we obtain the results mentioned in Theorem~\\ref{encoding_lower_bound}.\n\n\n\\section{Conclusion}\\label{conclude}\nIn this paper, we provided procedures for compactly storing the DFS tree for any graph with efficiently supporting various queries in the indexing and encoding models, and showed how to extend these techniques to design indexing schemes for other fundamental and basic graph problems. With some work, our algorithm can be extended for indexing BFS tree (and other graph search tree also) as well while supporting similar types of queries. Also, as mentioned previously, our results are more general, and can be used in other situations as well. \n\n\nThis work opens up many possible future directions to explore. Can we further improve the query time while keeping the space bound same in the indexing model? Can we prove a space lower bound in the indexing model? Can we design compact data structures for indexing problems like maximum flow?\nFinally, we conclude by remarking that using \n\\cite{Banerjee2018,ChakrabortyRS17}, we can improve the preprocessing space of our algorithms\nto $O(n)$ bits (from $O(n \\lg n)$ bits) with marginal\nincrement in the preprocessing time. \n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThis is a contribution to the 3rd International Symposium about Quantum Mechanics based on a ``Deeper Level Theory'', but the message I want to present is that there is no need for a deeper theory. In my view bare quantum mechanics provides a good explanation of all that we see around us. In fact, it seems that this series of the Symposia leads in this direction. The quote from the objectives of the EmerQuM (2011) reads:\n\\begin{quote}\nThe theme of ``emergent quantum mechanics'' is, we believe, an appropriate present-day topic, which can both serve as ... a means to point towards promising future directions in physics. We intend to bring together many of those physicists who are interested in or work on attempts to understanding quantum mechanics as emerging from a suitable classical (or, more generally: deeper level) physics...\n\\end{quote}\nThe objectives of EmQM13 (2013) were not very different:\n\\begin{quote}\nThe symposium invites the open exploration of an emergent quantum mechanics, a possible ``deeper level theory'' that interconnects three fields of knowledge: emergence, the quantum, and information. Could there appear a revised image of physical reality from recognizing new links between emergence, the quantum, and information? Could a novel synthesis pave the way towards a 21st century, ``super-classical'' physics? ...\n\\end{quote}\nHowever, the first objective of EmQM15 (2015) is changed:\n\\begin{quote}\nThe symposium invites the open exploration of the quantum state as a reality. The resurgence of interest in ontological quantum theory, including both deterministic and indeterministic approaches, challenges long held assumptions...\n\\end{quote}\n\nI believe the quantum state is a reality. It is the reality. The only fundamental physical ontology is the quantum wave function.\n\nPlease allow me a sociology of science speculation for why this opinion is not in the consensus today. At the birth of quantum mechanics there was at least a decade of quantum mechanics without the wave function and the Scr\\\"odinger equation of its evolution. During this time Bohr with his charismatic behaviour convinced the physics community that the reality behind quantum mechanics cannot be grasped by human beings. A half century later, physicists working on the foundations of quantum mechanics expressed growing dissatisfaction with Bohr's view, they followed Bell \\cite{BellBeab} in search of ``local beables''. The ingenious model of Spekkens \\cite{Spek} gave a hope that our familiar everyday reality will emerge from a ``deeper level theory''.\n\nHowever, the research triggered by these developments cooled down the expectations. It suggested that whatever the ``deeper theory'' is, it cannot be simpler than the standard quantum theory \\cite{Leifer} and it is strongly suggested that the quantum wave function is ``ontic'' \\cite{PBR,Hardy}, i.e., it is the ontology of quantum mechanics.\n\nMoreover, it was claimed that just a ``free will'' assumption is enough for a simple proof of an ontic nature of the quantum wave function \\cite{CR}. In my view, however, this argument is a metaphysical error. In science we {\\it assume} ``free will'' of an external agent who tests theories we construct, so nothing can be derived from the ``free will'' assumption.\n\n It is considered ``politically incorrect'' to claim that physicists, at large, understand Nature. This was the view a little more than a century ago, the view which was proved to be very wrong. Two revolutions in physics took place in the last century: relativity theory and quantum theory. Classical physics provided very plausible explanations of many observable phenomena, but it was shown to be wrong in more precise experiments.\n\n Today there are no paradoxes similar to those encountered by classical physics. However, the ontology of the quantum wave function seems very far from what we observe with our senses, and this is what led the majority to believe that it cannot be the true description of Nature. I, however, think that the classical understanding of Nature at the beginning of the last century is roughly correct also for quantum theory. An atom is not made of point particles moving on definite trajectories, but is a highly entangled quantum wave of electrons, protons, etc. This quantum wave is, however, not relevant for the interaction with other systems, it exerts forces on other systems very much like a point-like body. The wave structure provides an explanation of the spectrum, which classical mechanics does not, but for explaining pressure or other macroscopic features, classical approximations work fine.\n\nIn the following I will present the view that all that exists is the wave function of the Universe which evolves according to the Schr\\\"odinger equation and without collapses during quantum measurements. This is the Many-worlds interpretation (MWI) of quantum mechanics \\cite{Everett}.\n\n\n\n\\section{Many worlds in a single Universe}\n\n\n\n\n In my view \\cite{SEP} the MWI is the only interpretation which can be viewed as a possible final theory of physics. It is a deterministic theory without action at a distance as all physics we know today. It is (still) not widely accepted since it requires a radical change in our view of Nature: there are many worlds which exist in parallel at the same space and time as our own.\n\nAs a physical theory, the MWI is a theory about the Wave Function of the physical Universe. The equations are those of the standard quantum mechanical formalism: the Schr\\\"odinger equation or its relativistic generalizations. The theory postulates that the only ontology is this Wave Function which evolves in a unitary way, i.e., there is no collapse of the Wave Function. Since the collapse process is the only random element in physics and also the only source for action at a distance, randomness and action at a distance are eliminated from physics in the MWI.\n\nClearly, the most general and complete description of Nature is not the Wave Function of all particles in the Universe. Such a picture does not describe the creation and annihilation of particles and field theory is required. In such a framework a wave functional describing the amplitude for the local fields at each point in space replaces the Wave Function. However, I do not foresee that this entails significant conceptual changes or difficulties. Surprisingly, little effort has been made to look for a rigorous definition of such a functional. Appreciation of the success of the MWI in resolving the paradoxes of standard quantum mechanics may lead to further research in this direction.\n\nIn spite of the name, I view the MWI as a theory of a single Wave Function of a single physical Universe. This is the only ontology. Observables, which are frequently considered as basic building blocks of the standard formalism (and which are the\nbasis of algebraic approaches to quantum theory) are not part of the ontology. Thus, the uncertainty relations of quantum observables do not lead to randomness, the theory remains deterministic since the Wave Function evolves deterministically \\cite{qmdet}.\n\nThere are two parts to the MWI: physics and interpretation. Physics, the theory of the evolution of the Wave Function, is rigorously defined and tested up to the maximally possible precision. It is a ``good'' theory: it has no paradoxes, it is complete, it contradicts neither the spirit nor the letter of the special theory of relativity. As a physical theory it is clearly simpler than the spontaneous collapse theories \\cite{Pearle,GRW}, Bohmian mechanics \\cite{deBroglie,Bohm}, the consistent histories approach \\cite{Grif}, or underlying probabilistic theories \\cite{Spek}. Moreover, recent analyses of possibilities for alternative interpretations show that they cannot be more parsimonious than the theory of the quantum Wave Function, see a recent experimental study \\cite{Rigbauer}.\n\nThe difficult part of the MWI is the interpretation, the explanation of how the Wave Function of the Universe describes the world we observe in 3-space while the Wave Function ``lives'' in the configuration space of $3N$ dimensions ($N$ is the number of particles in the world).\nThe role of interpretation is much larger in the MWI than in other interpretations of quantum mechanics. In most other interpretations the connection between ontology, e.g. Bohmian trajectories, or values of observables, is simple, transparent and immediate, so frequently it is not discussed at all, while in the MWI, it is the main part to be analyzed. Interpretation belongs to the realm of human science with different standards and methodology. It has a much wider range of acceptable approaches. FAPP (for all practical purposes) \\cite{Bell} definitions are good enough for discussing our experiences.\n\n\nThe main difficulty of connecting our experience with the ontology in the MWI is that it corresponds not just to our current experience, but also to a multitude of other experiences in parallel worlds. Clearly, there are very many parallel worlds now, although the number of worlds is not rigorously defined. However, even in the gedanken case in the framework of the MWI in which the Wave Function now describes just one world in the Universe, the correspondence to our experience in this world is not obvious. I will start with the analysis of such a single world.\n\n\\section{A World}\n\nThe concept of a world in the MWI does not belong to the physics part. It is not like a particle: the form of the Wave Function includes information about the number and the type of particles in the physical Universe. Since the Wave Function is all there is, it must also have information about worlds, but the definition of what a world means is not written in the Wave Function. It belongs to human science. A world is what a layman understands by the word ``world''. A quantum physicist might have a confusing concept of a world, he can think of a multiverse \\cite{Deut}, about particles in a superposition, etc., so the concept might not be clear. A world in the MWI is defined as a world of classical physics:\n\n\\begin{quote}\nA world is a collection of objects in the Universe in definite states.\n\\end{quote}\n\n\n\nIn classical physics we can imagine that all particles have definite positions and velocities.\nHowever, the positions of every molecule of Earth's atmosphere are not really relevant for describing a world in the MWI. It is enough to specify definite states of all macroscopic objects. Moving to the quantum domain, we do not even have a description in terms of positions and velocities of particles. In the standard approach, the notion of observables plays an important role. Our experience is described through a set of values of the observables. An observer is aware only of measured observables and their measurements collapse the Wave Function to definite eigenstates which provide the correspondence.\n\n\nDavid Bohm, in support of his interpretation, noted that in every experiment we read the values of observables by observing positions in space of the pointers of the measuring devices. Definite temperature corresponds to a particular position of a thermometer, etc. So, the world is specified by well localized positions of all macroscopic objects. The world is a concept in 3-space. It is a crucial point for explaining the connection between the quantum wave and our experience. In a theory with collapses at each quantum measurement there is a single world in which the Wave Function is a product of wave functions of all macroscopic objects in 3-space. The classical picture of interaction according to which an object creates fields in 3-space which cause accelerations of other objects present in the location of the field is valid here. (Recently I found a local explanation of this form for an apparent counter example to this claim, the Aharonov-Bohm effect \\cite{VAB}.)\n\nNeuroscience does not have a clear explanation of human experiences. Considering experience as information flow allows to speculate that it might be defined in terms of spin wave function \\cite{Albert92}. This can lead to some modification of the picture, but apparently will not change it dramatically and currently there are no signs that our brain operates with spins. The picture I present here is that our experiences supervene upon the space distribution of matter, so the Wave Function which allows us to build a three-dimensional picture of particles in 3-space specifies human experiences.\nThe density of particles is gauge independent and also properly transforms between different Lorentz observers. Thus, the explanation of our experience is unaffected by the ``narratability failure'' problem \\cite{Albert13}, the Wave Function description might be different for different Lorentz observers, but the local description remains the same.\n\n\nIn fact, in the standard approach with collapses at every quantum measurement ensuring a single world, the problem of correspondence between the ontology and our experience is not considered problematic. (The collapse process, however, is a very serious problem for the physics part of the theory.) This connection between the collapsed Wave Function and our experience will be the basis of the connection with the experience in the framework of the MWI.\n\nThe exact Wave Function of a world is not rigorously defined. It is based on collapse, but there is no precise definition when exactly it happens. The collapse avoids superpositions of macroscopic objects in macroscopically different states, but the word ``macroscopic'' is a FAPP concept. Von Neumann proved that there is a very large flexibility where we put the cut between quantum and classical, i.e. when exactly the collapse takes place. In the collapsed wave we can see the classical world. The three-dimensional picture of places where the wave density is large is the picture of the world we see around us. Instead of an {\\it ad hoc} von Neumann collapse, we can consider the spontaneously collapsed Wave Function of the GRW-Pearle type theory. The quantum wave resides essentially in 3-dimensions. It is connected to our experience in a transparent way. The macroscopic parts of objects collapse to pure unentangled states due to strong interaction with the environment. States of electrons in atoms, and atom states in molecules, etc. remain entangled, but micro states of particles are not relevant to the description of a world.\n\nThe state of an electron in an atom is not relevant for the description of a world, but when we describe a quantum interference experiment with single particles, their state is important. One way to avoid description of micro objects is to note that the macroscopic description of macroscopic preparation and detection devices in the experiment includes the information about these particles. I think that it is more convenient to add a description of such micro particles to the picture of the world since it provides a clear characterization of the weak coupling of the particles with the environment. However, such micro particles require a special treatment, in particular, adding a backward evolving wave function specified by the final measurement \\cite{tisy,Vpast}.\n\n\nLet me summarize the main points. In a world, by definition, all macroscopic objects have definite macroscopic states. In a world, there cannot be a Schr\\\"odinger's cat in a closed box in a superposition of being alive and dead.\nI add to the mathematical formalism of the Universal Wave Function a postulate regarding the connection to our experience. The connection between experience and the world wave function which is an element of the Universal Wave Function is the same as in the collapse theories in which there is only one world wave function and single experience of every observer. The Wave Function of a world $|\\Psi_{\\rm world}\\rangle$ is a product state of well localized (in 3-space) quantum states of all ``macroscopic'' objects multiplied by possibly entangled states of microscopic systems irrelevant for the macroscopic description of the world.\n\n\n\\section{The Wave Function of the Universe}\n\nThe correspondence between the Wave Function of a single world and our experience may not be as transparent as the correspondence between our experience and a classical world of particles moving on trajectories, or a world of Bohmian trajectories, but I still find it satisfactory. I do not believe that this is the picture of the Universe because there is no satisfactory physics which leads to such Wave Function. The collapse process which is required here is foreign to all physics that I know. It has some kind of action at a distance and randomness and nobody has found elegant and simple equations for the collapse process. Physics forces me to reject the idea of collapse, and therefore, after every quantum measurement a superposition of wave functions of the single world wave function type is created:\n\\begin{equation}\\label{Univ}\n| \\Psi_{\\rm Universe}\\rangle=\\sum \\alpha_i | \\Psi_{{\\rm world}~i}\\rangle.\n\\end{equation}\n\nThe Wave Funciton of the Universe is a wave function in a configuration space $|\\Psi_{\\rm Universe} (x_1, x_2,...x_N)\\rangle$. Each one of the wave functions of the worlds is essentially a product of well localized wave functions of macroscopic objects in 3-space $|\\Psi_{{\\rm world}~i}\\rangle= \\prod_j |\\Phi_i(X_j)\\rangle\\ |\\psi_i\\rangle$, where $|\\psi_i\\rangle$ signifies the wave function of other degrees of freedom: relative coordinates of electrons in atoms, weakly coupled particles like neutrinos, etc. Locality and strength of interactions (with a buzz word ``decoherence'') ensure approximate uniqueness of the decomposition.\n\nDifferent worlds must have different classical descriptions and therefore they correspond to orthogonal wave functions. Energies are bounded, so even well localized wave functions of macroscopic objects must have nonvanishing tails at macroscopic distances. This makes the above statement of orthogonality not so obvious. Most probably, the world wave functions of different worlds can be constructed to be orthogonal, but even if not, since the concept of a world is a human FAPP concept, a tiny overlap can and should be neglected.\n\nThe Wave Function of the Universe $| \\Psi_{\\rm Universe}\\rangle$ can be decomposed into a superposition of the world wave functions $| \\Psi_{{\\rm world}~i}\\rangle$. The reason we do not experience superpositions is not because they do not exist, but because {\\it we} are not capable of experiencing several different states simultaneously. The phrase ``different states'' means different places and the locality of physical interactions prevents conscious nonlocal creatures. A nonlocal creature which behaves differently depending on the relative phase $\\phi$ between parts at different locations, $\\frac{1}{\\sqrt 2} (|A\\rangle + e^{i\\phi} |B\\rangle)$, cannot be useful because local interactions wash out the relative phase almost immediately. This is the argument for the ``preferred basis'' of the decomposition of the Wave Function, it has to be the local basis.\n\nAnother argument for why the Universe cannot support worlds with nonlocal creatures being in a superposition of two locations is that such a creature will need much more resources. If such a nonlocal creature would like to eat, it will need two dinners instead of one. Both parts of the creature $ |A\\rangle$ and $|B\\rangle$ have to eat, but a superposition of a dinner in two locations $\\frac{1}{\\sqrt 2} (|A_{\\rm food}\\rangle + |B_{\\rm food}\\rangle)$ cannot provide a dinner for the creature through local interactions. Such interactions cannot transform a product state of the creature and the food into the required state $\\frac{1}{\\sqrt 2} (|A\\rangle|A_{\\rm food}\\rangle + e^{i\\phi} |B\\rangle|B_{\\rm food}\\rangle)$.\n\nThe {\\it stability} of world wave functions $| \\Psi_{{\\rm world}~i}\\rangle$ is the key issue. Of course, at every quantum measurement the world wave function splits to several different (well localized) world wave functions. But this does not happen too fast and too often. If no observer has the time to be aware of a particular world, we have no reason to define such a world: remember, `world' is a human concept which is supposed to help explaining our experience.\nI do want to consider worlds at a particular time, but it is not enough that the picture drawn from a wave function looks like a world (all macroscopic objects are well localized) at this moment. If, immediately after, it does not look like a world, such wave function should not be considered as corresponding to a world.\n\nWe can ask what was the past and what will be the future of a world. Every quantum measurement splits the world, so a world in a particular time corresponds to a multitude of worlds in the future. Going backward in time we have a unique past. For many worlds at present there is the same past. So, if we consider worlds for periods of time we obtain ``overlapping'' worlds: they overlap in the past. (If we want to add some micro particles for describing the world, then splitting for them happens at the measurement {\\it after} the present time of the world we consider \\cite{tisy}.)\n\nAnother important concept which has to be clarified is ``I''. In which world am I? If I perform a quantum measurement what happens to me? If somebody else performs a quantum measurement will I be changed?\n\nBy definition, I have to be in a well defined state. All parts of my body, neurons in my brain, etc. have to be well localized. When I perform a quantum measurement with a few possible results and I observe an outcome, a few worlds are created with a different ``I'' in each world. All these ``I''s are descendants of the ``I'' before the measurement, so they have identical memories of what happened before the quantum experiment, but they have different knowledge about the result of the experiment. ``I''s which have identical memories, but different locations of bodies, are also considered different. Such situations can be created when I am moved to different places, while asleep, according to the results of a quantum experiment \\cite{schizo}).\n\nIf somebody else makes a quantum measurement in a faraway location such that the information about the result does not arrive at my location, then I do not split. The world splits, but I remain part of all newly created worlds. There is no meaning for the question what is the result of this measurement in {\\it my} world. (This is one of the examples in which our language, developed during the time of a firm belief in existence of a single world, has difficulty in describing the situation.) So, it cannot be that different ``I''s are present in one world, but it is possible that the same ``I'' is present in several worlds.\n\nLet me summarize. I build the MWI on the basis of standard quantum mechanics. I ask physicists who accept Wave Function collapse to provide (a rough) description of the Wave Function of the physical Universe. This specifies what might be a world wave function. Next, I decompose the Wave Function of the Universe as a superposition of such world wave functions. I postulate the correspondence with the experience as the one in standard quantum mechanics. Thus, the presence of a term corresponding to a world $i$ in the decomposition, $\\alpha_i\\neq 0$, ensures the presence of such an experience. It is a question for quantum physicists what should be the world wave function to ensure stability and ``classicality'' of this world.\n\n\n The wave function of a world $|\\Psi_{{\\rm world}~i}\\rangle$ specifies what we ``feel'' in world $i$. What is the role of the coefficient $\\alpha_i$? The absolute value of the coefficient specifies the illusion of probability as will be discussed in the next section. The phase is not relevant to us. We assume that worlds are different enough that we cannot make interference experiments between them. The worlds differ by macroscopic differences of states of macroscopic objects. The FAPP meaning of the word ``macroscopic'' can be defined exactly by this property: the states are macroscopically different if we cannot observe interference between them in a realistic experiment. The phase of $\\alpha_i$ can be relevant for a creature having unlimited technological power to perform interference experiments with macroscopic objects, i.e., interference between different worlds.\n\n\n \\section{Measure of existence of the world and the illusion of probability}\n\n The measure of existence of the world, $\\mu_i=|\\alpha_i|^2$ , provides the illusion of probability in our world. Before discussing the qualitative issue, why $ |\\alpha_i|^2$ and not something else, I will discuss the meaning of probability in the MWI.\n\n I write ``illusion'' of probability, because I consider a genuine concept of probability to require that there are several options and only one takes place. In case of quantum measurements all possible outcomes take place.\n\nThere is no randomness in the basic laws of physics. But determinism by itself does not prevent the concept of probability. We might be ignorant of some details which specify the outcome. In quantum mechanics there are no such details (which usually are named ``hidden variables''). We might have complete information about the system and the measuring device and it still will not help us to know what will be the outcome. More precisely, we know that all outcomes will take place and we say that each outcome will correspond to one of the newly created worlds.\n\nIf there is a collapse of the Wave Function to one of the world wave functions $|\\Psi_{{\\rm world}~i}\\rangle$, then we do have a legitimate concept of probability. An experience of an observer supervenes upon the world wave function. In the MWI, every term in the decomposition (\\ref{Univ}) corresponds to such experience. By construction, our experience in each particular world is the same as our experience in a physical universe with the collapse law in which only one world exists. So, we have a complete illusion of probability: there is no difference between our experience and the experience of an agent with genuine probability.\n\n\nSince the MWI is a deterministic theory, there cannot be a genuine chance there, but we can arrange a special situation in which there will be a genuine ignorance probability of an agent about the outcome of a quantum experiment. Before, I pointed out that in quantum mechanics we can know everything about the past without knowing what will be an outcome in the future.\n Now I will show that in the MWI we might know everything about the present, but still be ignorant about the outcome of already performed experiment.\n\n Such a situation is not easy to understand because we are not used to considering a plurality of worlds. What might help is to imagine first a gedanken situation which is not related to quantum physics. Suppose that creatures with super technology land on Earth. They can do what we would consider to be miracles. The creatures can create copies of Earth with everything on it and add it to the Solar system. They show their ability to people on Earth and they say that tomorrow morning there will be three identical Earth planets with all inhabitants. In the morning I wake up and I do not know if I am the original, or I am one of the two copies. Three Earth worlds exist and there are three ``I''s. Each one does not know which ``I'' he is. The symmetry tells us that the probability to be each particular ``I'' is one third. Each ``I'' will bet one third on the fact that he is the original and not a copy.\n\n In the case of a quantum measurement I can arrange a similar situation without super technology \\cite{schizo}. Before performing a quantum experiment which has three possible outcomes I take a sleeping pill and ask my friends to move me while I am asleep to one of three rooms according to the result of the experiment. My friends, instead of performing the experiment themselves, can get instructions using the iPhone application ``Universe Splitter'' or the Tel Aviv World Splitter \\cite{TUWS}. When ``I'' in a particular room am awake, but still have closed eyes, I do not know in which room am I. It is an unusual situation: I, more precisely, every copy of me, might know everything about the Universe, but still be ignorant about the outcome. I am ignorant about self-location in a particular world. This is my privileged property. Any external observer does not have this probability concept. The question: ``In which world am I present?'' has no meaning for him. There are different ``I''s present in corresponding different worlds, but the external observer, even if he is aware that the splitting has occurred, belongs to all these worlds. Only when he contacts me, will he split his ``I'' according to the outcome of the quantum experiment.\n\n A widespread approach to probability is the readiness to put an intelligent bet on a particular outcome. Since the probability is relevant for all my descendants as they have a legitimate concept of ignorance probability, I suggest to associate it also with me at the time before the experiment. It is rational for me to place bets since all my descendants would like me to do so. They will get the rewards (and the losses) of the bet. Thus, considering the probability as an amount that an intelligent agent is ready to bet on a particular outcome \\cite{deFin}, we have a concept of probability in the framework of the MWI. The rule is that an experimentalist performing a quantum experiment should bet in proportion to the {\\it measure of existence} of the world with this outcome \\cite{SBAna}.\n\n In classical cases either a world exists or it does not. In case of several planet Earths, there is no way that (at a particular time) one planet exists more than the other. We must associate the same measure of existence with all planets: just 1. The definition of the measure of existence of a quantum world, $\\mu_i=|\\alpha_i|^2$, allows different values. I postulate, that we should bet according to this measure. This analog of the Born Rule in the framework of the MWI is sometimes named the Born-Vaidman rule \\cite{Tap}.\n\n Why accept this postulate? In my view, a good answer is that the records of the outcomes of quantum experiments in our world show that there is no general rule which leads to better results for an experimentalist betting on the outcomes of quantum experiments. One can add that the postulate is natural if we accept the idea of measure of existence. Indeed, arranging (artificially) that all worlds have equal measures of existence leads to a natural property of equal probability for all worlds as in the case of multiple ``classical'' worlds (compare with my three Earths example). The measure of existence of a world also quantifies the power of a world to interfere with other worlds if a quantum super technology will make interference experiments with macroscopic objects which are interference experiments between different worlds \\cite{schizo}.\n\n\n Another argument for accepting the Born-Vaidman Rule is that all the alternatives I try to imagine lead to contradiction with special relativity. If we have a wave function of a particle distributed in space, the sum of the probabilities to find it in some location adds to 1. Thus, if we can change the probability in one place by some local operation, this will change the probability in another place. Changed probability is a signal, since a large ensemble of identical systems with identical actions allows transmission of a message. Local operations at a local part $O$ can change almost all local aspects of the wave function except for changing its weight, $\\int_O\\Psi^*\\Psi dv$. So, it seems to be that the probability cannot depend on any other function of the wave function but its local weight.\n\n\n\n \\section{Conclusions}\n\nThe MWI, first, tells us what the ontology is: it is the evolving Wave Function. Then it finds us in this ontology, as stable waves in human shapes, and explains (through some natural postulates) our experience.\n\n The theory of the evolution of the Wave Function is an exact mathematical theory of a single physical Universe.\n Avoiding discussion of values of variables makes it free of paradoxes, action at a distance, and randomness, the features which are foreign to all known areas of physics.\n\nThe second part of the MWI, the interpretation, is good only as a FAPP theory. It allows various modifications, e.g., the original Everett relative state formulations provides a somewhat different decomposition to world wave functions.\n\nThe MWI is far from being the most clear explanation of our experience and although it can probably be improved, some other interpretations certainly explain it much better. However, these interpretations pay a very high price in spoiling the physical part of the theory which becomes so unnatural that most physicists are not ready to accept them.\n\nMy impression is that the majority of physicists are not ready to accept the MWI because in this framework {\\it we} are not in the center of the theory. The sun does not encircle us. Physicists are still hoping that there will be some new theory (interpretation) which will be better than the MWI. I doubt it, but I think that it should be at least accepted that meanwhile the MWI is the best alternative we have today.\n\nThis work has been supported in part by the Israel Science Foundation Grant No. 1311\/14.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe Grothendieck-Riemann-Roch theorem is a fundamental statement in\nalgebraic geometry. It describes the behavior of the Chern character\nfrom algebraic $K$-theory to suitable cohomology theories (for\ninstance Chow groups), with respect to the push-forward operation by\nproper maps. It provides a vast generalization of the classical\nRiemann-Roch theorem on Riemann surfaces and the\nHirzebruch-Riemann-Roch theorem on compact complex manifolds. In their\ndevelopment of arithmetic intersection theory, Gillet and Soul\\'e were\nlead to extend the Grothendieck-Riemann-Roch theorem to the context of\narithmetic varieties. In this setting, vector bundles are equipped\nwith additional smooth hermitian metrics, for which an extension of\nalgebraic $K$-theory can be defined. There is a theory of\ncharacteristic classes for hermitian vector bundles, with values in\nthe so-called arithmetic Chow groups \\cite{GilletSoule:vbhm}. In\nanalogy to the classical algebraic geometric setting, it is natural to\nask about the behavior of the arithmetic Chern character with respect\nto proper push-forward. This is the question first addressed by\nGillet-Soul\\'e \\cite{GilletSoule:aRRt} and later by\nGillet-R\\\"ossler-Soul\\'e\n\\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees}. These\nworks had to restrict to push-forward by morphisms which are smooth\nover the complex points of the arithmetic varieties. This assumption\nwas necessary in defining both push-forward on arithmetic $K$-theory\nand the arithmetic Chow groups. While on arithmetic Chow groups\npush-forward resides on elementary operations (direct images of cycles\nand fiber integrals of differential forms), they used holomorphic\nanalytic torsion on the arithmetic $K$-theory level\n\\cite{BismutKohler}.\n\nThe aim of this article is to extend the work of Gillet-Soul\\'e and\nGillet-R\\\"ossler-Soul\\'e to arbitrary projective morphisms of regular\narithmetic varieties. Hence we face to the difficulty of\nnon-smoothness of morphisms at the level of complex points. To\naccomplish our program, we have to introduce generalized arithmetic\nChow groups and arithmetic $K$-theory groups which afford proper\npush-forward functorialities for possibly non-smooth projective\nmorphisms. Loosely speaking, this is achieved by replacing smooth\ndifferential forms in the theory of Gillet and Soul\\'e by currents\nwith possibly non-empty wave front sets. To motivate the introduction\nof these currents, we remark that they naturally appear as\npush-forwards of smooth differential forms by morphisms whose critical\nset is non-empty. In this concrete example, the wave front set of the\ncurrents is controlled by the normal directions of the morphism. The\ndefinition of our generalized arithmetic Chow groups is a variant of\nthe constructions of Burgos-Kramer-K\\\"uhn\n\\cite{BurgosKramerKuehn:cacg}, specially their covariant arithmetic\nChow groups. As an advantage with respect to \\emph{loc. cit.}, the\npresentation we give simplifies the definition of proper push-forward,\nwhich is the main operation we have to deal with in the present\narticle. At the level of Chow groups, this operation relies on\npush-forward of currents and keeps track of the wave front sets. For\narithmetic $K$-groups, we replace the analytic torsion forms of\nBismut-K\\\"ohler by a choice of a generalized analytic torsion theory\nas developed in our previous work\n\\cite{BurgosFreixasLitcanu:GenAnTor}. While a generalized analytic\ntorsion theory is not unique, we proved it is uniquely determined by\nthe choice of a real genus. We establish an arithmetic\nGrothendieck-Riemann-Roch theorem for arbitrary projective morphisms,\nwhere this real genus replaces the $R$-genus of Gillet and Soul\\'e. We\ntherefore obtain the most general possible formulation of the\ntheorem. In particular, the natural choice of the 0 genus,\ncorresponding to what we called the homogenous theory of analytic\ntorsion, provides an exact Grothendieck-Riemann-Roch type formula,\nwhich is the formal translation of the classical algebraic geometric\ntheorem to the setting of Arakelov geometry. The present work is thus\nthe abutment of the articles \\cite{BurgosLitcanu:SingularBC} (by the\nfirst and third named authors) and\n\\cite{BurgosFreixasLitcanu:HerStruc}--\\cite{BurgosFreixasLitcanu:GenAnTor}.\n\nLet us briefly review the contents of this article. In section\n\\ref{section:GenArChow} we develop our new generalization of\narithmetic Chow groups, and consider as particular instances the\narithmetic Chow groups with currents of fixed wave front set. We study\nthe main operations, such as pull-back, push-forward and products. In\nsection \\ref{section:ArKTheory} we carry a similar program to\narithmetic $K$-theory. We also consider an arithmetic version of our\nhermitian derived categories \\cite{BurgosFreixasLitcanu:HerStruc},\nthat is specially useful to deal with complexes of coherent sheaves\nwith hermitian structures. The essentials on arithmetic characteristic\nclasses are treated in section \\ref{section:ArChar}. With the help of\nour theory of generalized analytic torsion, section\n\\ref{section:DirectImage} builds push-forward maps on the level of\narithmetic derived categories and arithmetic $K$-theory. The last\nsection, namely section \\ref{section:ARR}, is devoted to the statement\nand proof of the arithmetic Grothendieck-Riemann-Roch theorem for\narbitrary projective morphisms of regular arithmetic varieties. As an\napplication, we compute the main characteristic numbers of the\nhomogenous theory, a question that was left open in\n\\cite{BurgosFreixasLitcanu:GenAnTor}.\n\n\\section{Generalized arithmetic Chow groups}\\label{section:GenArChow}\n\nLet $(A,\\Sigma ,F_{\\infty})$ be an arithmetic ring\n\\cite{GilletSoule:ait}: that is, $A$ is an excellent regular\nNoetherian integral domain,\ntogether with a finite non-empty set of embeddings $\\Sigma$ of $A$\ninto ${\\mathbb C}$ and a linear \nconjugate involution $F_{\\infty}$ of the product ${\\mathbb C}^{\\Sigma}$ which\ncommutes with \nthe diagonal embedding of $A$. \nLet $F$ be the field of fractions of $A$.\nAn \\textit{arithmetic variety} $\\mathcal{X}$ is a\nflat and quasi-projective scheme over $A$ such that\n$\\mathcal{X}_{F}=\\mathcal{X}\\times \\Spec F$\nis smooth. Then $X_{{\\mathbb C}}:=\\coprod_{\\sigma \\in \\Sigma\n}\\mathcal{X}_{\\sigma }({\\mathbb C}) $ is a\ncomplex algebraic manifold, which is endowed with an\nanti-holomorphic automorphism $F_{\\infty}$. One also associates to\n$\\mathcal{X}$ the real variety\n$X=(X_{{\\mathbb C}},F_{\\infty})$. Whenever we have arithmetic varieties\n$\\mathcal{X},\\ \\mathcal{Y}, \\dots$ we will denote by $X_{{\\mathbb C}},\\\nY_{{\\mathbb C}}, \\dots$ the associated complex manifolds and by $X,\\\nY, \\dots$ the associated real manifolds.\n\nTo every regular arithmetic variety Gillet\nand Soul\\'e have associated arithmetic Chow groups, denoted $\\cha\n^{\\ast}(\\mathcal{X})$, and developed an\narithmetic intersection theory \\cite{GilletSoule:ait}. \n\nThe arithmetic Chow groups defined by Gillet and Soul\\'e are only\ncovariant for morphism that are smooth on the generic fiber. Moreover\nthey are not suitable to study the kind of singular metrics that\nappear naturally when dealing with non proper modular varieties. \nIn order to have arithmetic Chow groups that are covariant with\nrespect to arbitrary proper morphism, or that are suitable to treat\ncertain kind of singular metrics,\nin \\cite{BurgosKramerKuehn:cacg}\ndifferent kinds\nof arithmetic Chow groups are constructed, depending on the choice of a\nGillet sheaf of \nalgebras $\\mathcal{G}$ and a $\\mathcal{G}$-complex $\\mathcal{C}$. We\ndenote by $\\cha^{\\ast}(\\mathcal{X},\\mathcal{C})$ the arithmetic Chow groups\ndefined in \\emph{op. cit.} Section 4.\n\nThe basic example of a Gillet algebra is the Deligne complex of\nsheaves of differential forms with logarithmic singularities \n$\\mathcal{D}_{\\log}$, defined\nin \\cite[Definition 5.67]{BurgosKramerKuehn:cacg}; we refer to \\emph{op. cit.}\nfor the precise definition and properties. Therefore, to any\n$\\mathcal{D}_{\\log}$-complex we can associate arithmetic Chow\ngroups. In particular, considering $\\mathcal{D}_{\\log}$ itself as a\n$\\mathcal{D}_{\\log}$-complex, we obtain\n$\\cha^{\\ast}(\\mathcal{X},\\mathcal{D}_{\\log})$, \nthe arithmetic Chow groups defined in \\cite[Section\n6.1]{BurgosKramerKuehn:cacg}. When $\\mathcal{X}_{F}$ is projective,\nthese groups agree, up to a normalization factor, with the groups\ndefined by Gillet and Soul\\'e. \n\nThe groups $\\cha^{\\ast}(\\mathcal{X},\\mathcal{C})$ introduced in \\cite[Section\n6.1]{BurgosKramerKuehn:cacg} have several technical issues:\nthey depend on the sheaf structure of $\\mathcal{C}$ and not only on\nthe complex of global sections $\\mathcal{C}(X)$; moreover, they are\nnot completely satisfactory if the cohomology determined by\n$\\mathcal{C}$ does not satisfy a weak purity property; finally the\ndefinition of direct images is intricate. To overcome these\ndifficulties we introduce here a variant of the cohomological\narithmetic Chow groups that only depends on the complex of global\nsections of a $\\mathcal{D}_{\\log}$-complex. \n\n\\begin{definition}\\label{def:3} Let $\\mathcal{X}$ be an arithmetic variety,\n $X_{{\\mathbb C}}=\\mathcal{X}_{\\Sigma }$ the associated complex manifold and\n $X=(X_{{\\mathbb C}},F_{\\infty})$ the associated real manifold.\n A \\emph{$\\mathcal{D}_{\\log}(X)$-complex} is a graded complex of\n real vector spaces $C^{\\ast}(\\ast)$ provided with a morphism of\n graded complexes\n \\begin{displaymath}\n \\cmap\\colon \\mathcal{D}_{\\log}^{\\ast}(X,\\ast)\\longrightarrow\n C^{\\ast}(\\ast). \n \\end{displaymath}\n Given two $\\mathcal{D}_{\\log}(X)$-complexes $C$ and $C'$, we say that $C'$ is a $C$-complex if there is a commutative diagram of morphisms of graded complexes\n \\begin{displaymath}\n \t\\xymatrix{\n\t\t\\mathcal{D}_{\\log}^{\\ast}(X,\\ast)\\ar[r]^{c}\\ar[rd]^{c'}\t&C^{\\ast}(\\ast)\\ar[d]^{\\varphi}\\\\\n\t\t\t&C^{\\prime\\ast}(\\ast).\n\t}\n \\end{displaymath}\n In this situation, we say that $\\varphi$ is a morphism of $\\mathcal{D}_{\\log}(X)$-complexes.\n\\end{definition}\n\nWe stress the fact that a $\\mathcal{D}_{\\log}$-complex is a complex of\nsheaves while a $\\mathcal{D}_{\\log}(X)$-complex is a complex of vector\nspaces. \nIf $\\mathcal{C}$ is a $\\mathcal{D}_{\\log}$-complex\n of real vector spaces, then the complex of global sections\n $\\mathcal{C}^{\\ast}(X,\\ast)$ is a $\\mathcal{D}_{\\log}(X)$-complex.\nWe are mainly interested in the\n $\\mathcal{D}_{\\log}(X)$-complexes of Example \\ref{exm:1} made out of\n differential forms and \n currents. We will follow the conventions of \\cite[Section\n5.4]{BurgosKramerKuehn:cacg} regarding differential forms and\ncurrents. In particular, both the current associated to a differential form\nand the current associated to a cycle have implicit a power of the trivial\nperiod $2\\pi i$.\n\n\n\\begin{example} \\label{exm:1}\n\\begin{enumerate}\n\\item \\label{item:4} The Deligne complex $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)$ of\ndifferential forms on $X$ with arbitrary\nsingularities at infinity. Namely, if $E^{\\ast}(X_{{\\mathbb C}})$ is the Dolbeault\ncomplex \n(\\cite[Definition 5.7]{BurgosKramerKuehn:cacg}) of differential forms on\n$X_{{\\mathbb C}}$ then \n\\begin{displaymath}\n \\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X,\\ast)=\n\\mathcal{D}^{\\ast}(E^{\\ast}(X_{{\\mathbb C}}),\\ast)^{\\sigma},\n\\end{displaymath}\nwhere $\\mathcal{D}^{\\ast}(\\underline{\\ },\\ast)$ denotes the Deligne\ncomplex\n(\\cite[Definition 5.10]{BurgosKramerKuehn:cacg})\nassociated to a Dolbeault complex and $\\sigma $\nis the involution $\\sigma (\\eta)=\\overline\n{F_{\\infty}^{\\ast}\\eta}$ as in \\cite[Notation\n5.65]{BurgosKramerKuehn:cacg}.\n\nNote that $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)$ is the complex of\nglobal sections of the $\\mathcal{D}_{\\log}$-complex\n$\\mathcal{D}^{\\ast}_{{\\text{\\rm l, ll, a}}}$ that appears in \\cite[Section\n3.6]{BurgosKramerKuehn:accavb} with empty log-log singular locus. In\nparticular, by \\cite[Theorem \n3.9]{BurgosKramerKuehn:accavb} it satisfies the weak purity\ncondition. \n\\item \\label{item:5} The Deligne complex $\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\ast)$\n of currents on $X$. Namely, if $D^{\\ast}(X_{{\\mathbb C}})$ is the Dolbeault\n complex of currents on $X_{{\\mathbb C}}$ then\n\\begin{displaymath}\n \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{\\ast}(X,\\ast)=\n\\mathcal{D}^{\\ast}(D^{\\ast}(X_{{\\mathbb C}}),\\ast)^{\\sigma}.\n\\end{displaymath}\nNote that here we are considering arbitrary\ncurrents on $X_{{\\mathbb C}}$ and not extendable currents as in \n\\cite[Definition 6.30]{BurgosKramerKuehn:cacg}. \n\\item \\label{item:6} Let $T^{\\ast}X_{{\\mathbb C}}$ be the cotangent bundle of\n $X_{{\\mathbb C}}$. Denote by \n $T_{0}^{\\ast}X_{{\\mathbb C}}=T^{\\ast}X_{{\\mathbb C}}\\setminus X_{{\\mathbb C}}$ \n the cotangent bundle with the zero section deleted and let $S\\subset\n T_{0}^{\\ast}X_{{\\mathbb C}}$ be a closed conical subset that is invariant under\n $F_{\\infty}$. Let $D^{\\ast}(X_{{\\mathbb C}},S)$ be the\n complex of currents whose wave front set is contained in $S$\n \\cite[Section 4]{BurgosLitcanu:SingularBC}. The Deligne complex\n of currents on $X$ having the wave front set\n included in the fixed set $S$ is given by\n\\begin{displaymath}\n \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{\\ast}(X,S,\\ast)=\n\\mathcal{D}^{\\ast}(D^{\\ast}(X_{{\\mathbb C}},S),\\ast)^{\\sigma}.\n\\end{displaymath}\n\\end{enumerate}\nThe maps of complexes $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)\\to\n\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\ast)$ given by $\\eta\\mapsto\n[\\eta]$ is injective and makes of $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\ast)$ a $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)$-complex. We will use this map to identify\n$\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}$ with a subcomplex of\n$\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}$. Since $\\mathcal{D}^{\\ast}_{{\\text{\\rm a}}}(X,\\ast)=\n\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,\\emptyset,\\ast)$ and\n$\\mathcal{D}^{\\ast}_{D,{\\text{\\rm a}}}(X,\\ast)= \n\\mathcal{D}^{\\ast}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T^{\\ast}_{0}X,\\ast)$, examples\n\\ref{item:4} and \\ref{item:5} are particular cases of \\ref{item:6}. \n\\end{example}\n\\begin{remark}\nWith these examples at hand, we can specialize the definition of\n$C$-complex (Definition \\ref{def:3}) to $C=\\mathcal{D}_{{\\text{\\rm a}}}(X)$. When\ndealing with hermitian structures on sheaves on non-necessarily proper\nvarieties, we will to work with\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes rather than\n$\\mathcal{D}_{\\log}(X)$-complexes. \n\\end{remark}\nWe will also follow the notation of \\cite[Section\n3]{BurgosKramerKuehn:cacg} regarding complexes. In particular we will\nwrite\n\\begin{displaymath}\n \\widetilde C^{2p-1}(p)=C^{2p-1}(p)\/\\Img \\dd_{C}, \\quad\n \\Zr C^{2p}(p)=\\Ker \\dd_{C}\\cap C^{2p}(p).\n\\end{displaymath}\n\n\\begin{definition}\\label{def:1}\n The \\emph{arithmetic Chow groups with $C$ coefficients} are defined\n as \n \\begin{equation}\\label{pfformula}\n \\cha^p(\\mathcal{X},C)=\\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})\\times\n \\widetilde C^{2p-1}(p)\/\\sim\n \\end{equation}\n where $\\sim$ is the equivalence relation generated by\n \\begin{equation}\\label{equivdef}\n (a(g),0)\\sim (0,c(g)).\n \\end{equation}\n\\end{definition}\nIf $\\varphi:C\\rightarrow C'$ is a morphism of\n$\\mathcal{D}_{\\log}(X)$-complexes (so that $C'$ is a $C$-complex),\nthen there is a natural surjective morphism\n\\begin{equation}\n \\cha^{p}(\\mathcal{X},C)\\times\\widetilde{C^{\\prime}}^{2p-1}(p)\n \\longrightarrow\\cha^{p}(\\mathcal{X},C'). \\label{eq:10} \n\\end{equation}\nIntroducing the equivalence relation generated by\n\\begin{displaymath}\n ((0,c),0)\\equiv ((0,0),\\varphi(c)),\n\\end{displaymath}\nwe see that \\eqref{eq:10} induces an isomorphism\n\\begin{equation}\\label{eq:11}\n \\cha^{p}(\\mathcal{X},C)\\times\\widetilde{C}^{\\prime 2p-1}(p)\/\n \\equiv\\;\\overset{\\sim}{\\longrightarrow}\\cha^{p}(\\mathcal{X},C').\n\\end{equation}\n\nWe next unwrap Definition \\ref{def:1} in order to get simpler\ndescriptions of the arithmetic Chow groups associated to the complexes of\nExample \\ref{exm:1}. We start by recalling the construction of\n$\\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})$. The group of codimension $p$\narithmetic cycles is given by\n\\begin{displaymath}\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})=\n\\left\\{ (Z,\\widetilde g)\\in \\Zr^{p}(\\mathcal{X})\\times \n \\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus \\mathcal{Z}^{p},p)\\left\\vert\n \\begin{aligned}[c]\n \\dd_{\\mathcal{D}}\\widetilde g&\\in \\mathcal{D}_{\\log}^{2p}(X,p)\\\\\n \\cl(Z)&=[(\\dd_{\\mathcal{D}}\\widetilde g,\\widetilde g)]\n \\end{aligned}\n \\right.\\right\\},\n\\end{displaymath}\nwhere $\\Zr^{p}(\\mathcal{X})$ is the group of codimension $p$\nalgebraic cycles of $\\mathcal{X}$, $\\mathcal{Z}^{p}$ is the ordered\nsystem of codimension at least $p$ closed subsets of $X$,\n\\begin{displaymath}\n\\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus\n\\mathcal{Z}^{p},p) =\\lim_{\\substack{\\longrightarrow\\\\W\\in \\mathcal{Z}^{p}}}\n\\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus W,p), \n\\end{displaymath}\nand $\\cl(Z)$ and\n$[(\\dd_{\\mathcal{D}}\\widetilde g,\\widetilde g)]$ denote the class in\nthe real Deligne-Beilinson cohomology group\n$H^{2p}_{\\mathcal{D},\\mathcal{Z}^{p}}(X,{\\mathbb R}(p))$ with supports on\n$\\mathcal{Z}^{p}$ of the cycle $Z$ and the pair\n$(\\dd_{\\mathcal{D}}\\widetilde g,\\widetilde g)$ respectively.\n\nFor each codimension $p-1$ irreducible variety $W$ and each rational\nfunction $f\\in K(W)$, there is a class $[f]\\in\nH^{2p-1}_{\\mathcal{D}}(X\\setminus |\\dv f|,{\\mathbb R}(p))$. Hence a\nclass $\\bmap([f])\\in\n\\widetilde{\\mathcal{D}}_{\\log}^{2p-1}(X\\setminus\n\\mathcal{Z}^{p},p)$ that is denoted $\\mathfrak{g}(f)$. Then\n$\\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ is the group generated by the elements of\nthe form $\\diva(f)=(\\dv(f),\\mathfrak{g}(f))$.\n\nThen\n\\begin{displaymath}\n \\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})=\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\left\/\n\\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\right. .\n\\end{displaymath}\n\nWe will use the following well stablished notation.\nIf $B$ is any subring of ${\\mathbb R}$ we will denote by\n$\\Zr^{p}_{B}(\\mathcal{X})=\\Zr^{p}(\\mathcal{X})\\otimes B$, by\n$\\za^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})$ the group with the same\ndefinition as $\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ with\n$\\Zr^{p}_{B}(\\mathcal{X})$ instead of $\\Zr^{p}(\\mathcal{X})$, and we\nwrite\n$\\rata^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})=\n\\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\otimes B$. Finaly we write\n\\begin{equation*}\n \\cha^p_{B}(\\mathcal{X},\\mathcal{D}_{\\log})=\n \\za^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})\\left\/\n\\rata^{p}_{B}(\\mathcal{X},\\mathcal{D}_{\\log})\\right.\n\\end{equation*}\nNote that\n$\\cha^p_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\log})=\\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})\\otimes\n{\\mathbb Q}$ but, in general, $\\cha^p_{{\\mathbb R}}(\\mathcal{X},\\mathcal{D}_{\\log})\\not\n= \\cha^p(\\mathcal{X},\\mathcal{D}_{\\log})\\otimes {\\mathbb R}$. We will use the\nsame notation for all variants of the arithmetic Chow groups.\n\nNow, in the definition of $\\cha^{p}(\\mathcal{X},C)$ we can first\nchange coefficients and then take rational equivalence. We define\n\\begin{displaymath}\n \\za^p(\\mathcal{X},C)=\\za^p(\\mathcal{X},\\mathcal{D}_{\\log})\\times\n \\widetilde C^{2p-1}(p)\/\\sim\n\\end{displaymath}\nwhere again $\\sim$ is the equivalence relation generated by\n$(a(g),0)\\sim (0,c(g))$.\n\nThere are maps\n\\begin{alignat*}{2}\n\\zeta_{C}&\\colon\n\\za^{p}(\\mathcal{X},C)\\longrightarrow \\Zr^{p}(\\mathcal{X}),\n& \\zeta_{C}((Z,\\widetilde g),\\widetilde c)&=Z,\n\\\\\n\\amap_{C}&\\colon \\widetilde{C}^{2p-1}(p)\\longrightarrow\\za^{p}\n(\\mathcal{X},C),\n& \\amap_{C}(\\widetilde c)&=((0,0),-\\widetilde c),\\\\\n\\omega_{C}&\\colon \\za^{p}(\\mathcal{X},C)\\longrightarrow \\Zr C^{2p}(p), &\n\\qquad \\omega_{C}((Z,\\widetilde g),\\widetilde c)&=\n\\cmap(\\dd_{\\mathcal{D}}\\widetilde g)+\\dd_{C}\\widetilde c.\n\\end{alignat*}\nWe also consider the map \n$$\\bmap_{C}\\colon H^{2p-1}(C^{\\ast}(p))\\to \\za^{p} (\\mathcal{X},C)$$\nobtained by composing $\\amap_{C}$ \nwith the inclusion $H^{2p-1}(C^{\\ast}(p))\\to \\widetilde C^{2p-1}(p)$,\nand the map $$\\rho _{C}\\colon \\CH^{p,p-1}(\\mathcal{X})\\to\nH^{2p-1}(C^{\\ast}(p))$$ obtained by composing the regulator map $\\rho\n\\colon \\CH^{p,p-1}(\\mathcal{X})\\to H^{2p-1}_{\\mathcal{D}}(X,{\\mathbb R}(p))$\nin \\cite[Notation 4.12]{BurgosKramerKuehn:cacg}\nwith the map $\\cmap\\colon H^{2p-1}_{\\mathcal{D}}(X,{\\mathbb R}(p))\\to\nH^{2p-1}(C^{\\ast}(p))$. We will also denote by $\\rho _{C}$ the\nanalogous map with target $\\widetilde C^{2p-1}(p)$.\n\nThere are induced maps\n\\begin{align*}\n\\zeta_{C}&\\colon\n\\cha^{p}(\\mathcal{X},C)\\longrightarrow\\CH^{p}(\\mathcal{X}),\n\\\\\n\\amap_{C}&\\colon \\widetilde{C}^{2p-1}(p)\\longrightarrow\\cha^{p}\n(\\mathcal{X},C),\\\\\n\\omega_{C}&\\colon \\cha^{p}(\\mathcal{X},C)\\longrightarrow\\Zr C^{2p}(p).\n\\end{align*}\n\n\n\\begin{lemma}\\label{lemm:1}\n \\begin{enumerate}\n \\item \\label{item:1} Let $\\rata ^{p}(\\mathcal{X},C)$ denote the\n image of $\\rata \n ^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ in the group \n $\\za^p(\\mathcal{X},C)$. Then\n \\begin{displaymath}\n \\cha^p(\\mathcal{X},C)=\n \\za^{p}(\\mathcal{X},C)\\left\/\n \\rata^{p}(\\mathcal{X},C)\\right. . \n \\end{displaymath}\n \\item \\label{item:2} There is an exact sequence\n \\begin{equation}\\label{eq:3}\n 0\\to \\widetilde C^{2p-1}(p)\\overset{\\amap_{C}}{\\longrightarrow\n }\\za^{p}(\\mathcal{X},C)\n \\overset{\\zeta_{C}}{\\longrightarrow }\n \\Zr^{p}(\\mathcal{X}) \\to 0.\n \\end{equation}\n \\item \\label{item:3} There are exact sequences\n\\begin{equation}\\label{eq:1}\n\\CH^{p,p-1}(\\mathcal{X})\\overset{\\rho_{C}}{\\longrightarrow }\n\\widetilde{C}^{2p-1}(p)\\overset{\\amap_{C}}{\\longrightarrow }\n\\cha^p(\\mathcal{X},C) \n\\overset{\\zeta_{C}}{\\longrightarrow } \\CH^p(\\mathcal{X})\\to 0,\n\\end{equation}\nand\n\\begin{multline}\\label{eq:2}\n \\CH^{p,p-1}(\\mathcal{X})\\overset{\\rho_{C}}{\\longrightarrow }\n H^{2p-1}(C^{\\ast}(p))\\overset{\\bmap_{C}}{\\longrightarrow }\\cha^p(\\mathcal{X},C)\n \\overset{\\zeta_{C}\\oplus \\omega _{C}}{\\longrightarrow }\n \\\\ \\CH^p(\\mathcal{X})\\oplus\\Zr C^{2p}(p)\n\\longrightarrow H^{2p}(C^{\\ast}(p)) \\to 0.\n\\end{multline}\n \\end{enumerate}\n \\end{lemma}\n \\begin{proof}\n \\ref{item:1} Follows easily from the definition.\n\n \\ref{item:2} By \\cite[Proposition 5.5]{Burgos:CDB} there is an\n exact sequence\n \\begin{displaymath}\n 0\\to \\widetilde\n {\\mathcal{D}}_{\\log}^{2p-1}(X,p)\\overset{\\amap}{\\longrightarrow \n }\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\n \\overset{\\zeta}{\\longrightarrow }\n \\Zr^{p}(\\mathcal{X}) \\to 0. \n \\end{displaymath}\n From it and the definition of\n $\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})$ we derive the exactness of\n \\eqref{eq:3}. \n\n \\ref{item:3} Follows from the exact sequences of \\cite[Theorem\n 7.3]{Burgos:CDB} and the definition of $\\cha^p(\\mathcal{X},C)$.\n \\end{proof}\n\nThe contravariant functoriality of\n$\\cha^{\\ast}(\\mathcal{X},\\mathcal{D}_{\\log})$ is easily translated to\nother coefficients.\nLet $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a morphism of regular arithmetic\nvarieties. \nLet $C$ be a $\\mathcal{D}_{\\log}(X)$-complex and $C'$ a\n$\\mathcal{D}_{\\log}(Y)$-complex, such that\nthere exists a map of complexes $f^{\\ast}:C'^{\\ast}(\\ast)\\to\nC^{\\ast}(\\ast)$ that makes the \nfollowing diagram commutative:\n\\begin{displaymath}\n \\xymatrix{\n \\mathcal{D}_{\\log}^{\\ast}(Y)\\ar[r]^{f^{\\ast}}\\ar[d] & \\mathcal{D}_{\\log}^{\\ast}(X) \\ar[d] \\\\\n C'^{\\ast}(Y)\\ar[r]_{f^{\\ast}} & C^{\\ast}(X).\n}\n\\end{displaymath}\nThen we define\n$$f^{\\ast}((\\mathcal{Z},g),c)=(f^{\\ast}(\\mathcal{Z},g),f^{\\ast}(c)).$$\nIt is easy to see that this map is well defined, because the pull-back map\n$f^{\\ast}:\\cha^{\\ast}(\\mathcal{Y})\\to \\cha^{\\ast}(\\mathcal{X})$ (for\nthe Chow groups corresponding to \n$\\mathcal{D}_{\\log}$) is compatible to the map $\\amap$. \n\nBefore stating concrete examples of this contravariant functoriality\nwe need some notation (\\cite[Theorem 8.2.4]{Hormander:MR1065993},\n see also \\cite[Section 4]{BurgosLitcanu:SingularBC}). Let\n $f_{{\\mathbb C}}\\colon X_{{\\mathbb C}}\\to Y_{{\\mathbb C}}$ denote \nthe induced map of complex manifolds. Let $N_{f}$ be the set of normal\ndirections of $f_{{\\mathbb C}}$, that is\n \\begin{displaymath}\n N_{f}=\\{(f(x),\\xi)\\in T_{0}^{\\ast}Y_{{\\mathbb C}}\\mid \\dd\n f_{{\\mathbb C}}^{t}\\xi=0\\}. \n \\end{displaymath}\n Let $S\\subset\n T^{\\ast}_{0}Y_{{\\mathbb C}}$ be a closed conical subset invariant under\n $F_{\\infty}$. When\n $N_{f}\\cap S=\\emptyset$, the function $f$ is said to be transverse\n to $S$. In this case we write\n \\begin{displaymath}\n f^{\\ast}S=\\{(x,\\dd f_{{\\mathbb C}}^{t}\\xi)\\mid (f(x),\\xi)\\in S\\}.\n \\end{displaymath}\n It is a closed conical subset of $T^{\\ast}_{0}X_{{\\mathbb C}}$ invariant under\n $F_{\\infty}$.\n\n\\begin{proposition} \\label{prop:3}\n Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a morphism of regular arithmetic\n varieties.\n \\begin{enumerate}\n \\item There is a pull-back morphism $f^{\\ast}\\colon\n \\cha^{p}(\\mathcal{Y},\\mathcal{D}_{{\\text{\\rm a}}}(Y)) \\to\n \\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n \\item Let $N_{f}$ be the set of normal directions of $f_{{\\mathbb C}}$\n and $S\\subset\n T^{\\ast}_{0}Y_{{\\mathbb C}}$ a closed conical subset invariant under\n $F_{\\infty}$. If $N_{f}\\cap\n S=\\emptyset$, then there is a pull-back morphism\n \\begin{displaymath} f^{\\ast}\\colon\n \\cha^{p}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S)) \\to\n \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}S)).\n \\end{displaymath}\n \\item If $f_{F}$ is smooth (hence $N_{f}=\\emptyset$) then there is a\n pull-back morphism \n $$f^{\\ast}\\colon\n \\cha^{p}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y)) \\to\n \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)).$$\n \\end{enumerate}\n\\end{proposition}\n\n\nSimilarly the multiplicative properties of\n$\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\log})$ can be transferred to\nother coefficients. \nLet $C$, $C'$ and $C''$ be\n$\\mathcal{D}_{\\log}(X)$-complexes such that there is a commutative\ndiagram of morphisms of complexes\n\\begin{displaymath}\n \\xymatrix{\n \\mathcal{D}_{\\log}(X)\\otimes \\mathcal{D}_{\\log}(X) \\ar[r]^-{\\bullet} \\ar[d] & \\mathcal{D}_{\\log} (X)\\ar[d] \\\\\n C\\otimes C' \\ar[r]_-{\\bullet} & C''.\n}\n\\end{displaymath}\nThen we define a product\n$$\\cha^p(\\mathcal{X},C)\\times \\cha^q(\\mathcal{X},C')\n\\to \\cha^{p+q}_{{\\mathbb Q}}(\\mathcal{X},C'')$$\nby\n\\begin{equation}\\label{product}\n((\\mathcal{Z},g),c)\\cdot((\\mathcal{Z}',g'),c')=\n((\\mathcal{Z},g)\\cdot(\\mathcal{Z}',g'),c\\bullet\\cmap(\\omega(g'))+\n\\cmap(\\omega(g))\\bullet c'+\\dd_{C}c\\bullet c').\n\\end{equation}\nAs a consequence we obtain the following result.\n\\begin{proposition} \\label{prop:2}\n Let $\\mathcal{X}$ be a regular arithmetic variety.\n \\begin{enumerate}\n \\item $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$ is an\n associative commutative graded ring.\n \\item $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$ is a\n module over\n $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n \\item Let $S,S'$ be closed conic subsets of $T^{\\ast}_{0}X_{{\\mathbb C}}$ that are\n invariant under $F_{\\infty}$. Then\n $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))$ is a\n module over $\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n Moreover, if $S\\cap (-S')=\\emptyset$, there is a graded bilinear map\n \\begin{multline*}\n \\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\\times\n \\cha^q(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S')) \\longrightarrow \\\\\n \\cha^{p+q}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S\\cup S'\\cup (S+S'))).\n \\end{multline*}\n \\item The product is compatible with the pull-back of Proposition\n \\ref{prop:3}. \n \\end{enumerate}\n\\end{proposition}\n\n\n\nWe now turn our attention towards direct images. The definition of\ndirect images for general $\\mathcal{D}_{\\log}$-complexes is quite\nintricate, involving the notion of covariant $f$-pseudo-morphisms (see\n\\cite[Definition 3.71]{BurgosKramerKuehn:cacg}). By contrast, we will\ngive a another description of the groups\n$\\cha^{\\ast}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$ for which the\ndefinition of push-forward is much simpler.\n\nBy \\cite[3.8.2]{Burgos:Gftp} we know that any\n$\\mathcal{D}_{\\log}$-Green form is locally integrable. Therefore there\nis a well defined map\n\\begin{displaymath}\n \\varphi\\colon \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\log})\\longrightarrow \n \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n\\end{displaymath}\ngiven by $(Z,\\widetilde g)\\to (Z,\\widetilde{[g]})$ for any\nrepresentative $g$ of $\\widetilde g$. The previous map can be extended\nto a map\n\\begin{displaymath}\n \\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}\\colon\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))\\longrightarrow \n \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n\\end{displaymath}\ngiven by $\\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}((Z,\\widetilde g),\\widetilde\nh)=(Z,\\widetilde{[g]}+\\widetilde h)$.\n\nThe following result is clear.\n\\begin{lemma}\\label{lemm:3} The\n map $\\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}$ is an isomorphism. \n\\end{lemma}\n This lemma gives us a more concrete description of the\n group $\\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$. In fact,\n we will identify it with the group $\\Zr^{p}(\\mathcal{X})\\oplus\n \\widetilde {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)$ when necessary. Some\n care has to be taken when\n doing this identification. For instance\n \\begin{equation}\n \\label{eq:5}\n \\omega\n _{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}(\\varphi_{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}^{-1}(Z,\\widetilde\n g))= \\dd_{\\mathcal{D}}g+\\delta _{Z}.\n \\end{equation}\n \nLet now $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\nregular arithmetic varieties of relative dimension $e$. Using the\nabove identification, we define\n\\begin{equation}\\label{eq:4}\n f_{\\ast}\\colon\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))\\longrightarrow \n \\za^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y))\n\\end{equation}\nby $f_{\\ast}(Z,\\widetilde g)=(f_{\\ast}Z, \\widetilde{f_{\\ast} g})$,\nwhere $g$ is any representative of $\\widetilde g$, and $f_{\\ast}(g)$ is\nthe usual direct image of currents given by\n$f_{\\ast}(g)(\\eta)=g(f^{\\ast}\\eta)$. \n\n\\begin{proposition} \\label{prop:4}\n The map $f_{\\ast}$ in \\eqref{eq:4} sends the group $\n \\rata^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$ to the group\n $\\rata^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y))$. Therefore\n it induces a map\n \\begin{displaymath}\n \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))\\longrightarrow \n \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y)).\n \\end{displaymath}\n\\end{proposition}\n\nIn order to transfer this push-forward to other coefficients we\nintroduce two extra properties for a $\\mathcal{D}_{\\log}(X)$-complex $C$.\n\\begin{description}\n\\item{(H1)} There is a commutative diagram of injective morphisms of\n complexes \n\\begin{displaymath}\n \\xymatrix{\n \\mathcal{D}_{\\log} ^{\\ast}(X,\\ast)\\ar[r]^{\\cmap}\\ar[dr]&\n C^{\\ast}(\\ast)\\ar[d]^{\\cmap'} \\\\\n & \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}} ^{\\ast}(X,\\ast).\n}\n\\end{displaymath}\nSince $\\cmap'$ is injective we will usually identify $C$ with its image by\n$\\cmap'$.\n\\item{(H2)} The map $\\cmap'$ induces isomorphisms\n \\begin{align*}\n H^{n}(C^{\\ast}(p))\\cong H^{n}(\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{\\ast}(X,p))\n \\end{align*}\nfor all $p\\ge 0$ and $n=2p-1,2p$.\n\\end{description}\n\nThe conditions (H1) and (H2) have two consequences. First\nif $\\eta \\in D_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)$ is a current such that\n$\\dd_{\\mathcal{D}}\\eta\\in C^{2p}(p)$, there exist\n$a\\in\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-2}(X,p)$ such that \n$$\\dd_{\\mathcal{D}}a+\\eta\\in C^{2p-1}(p).$$\nSecond, the induced map $\\widetilde C^{2p-1}(p)\\to \\widetilde\n{\\mathcal{D}}^{2p-1}(X,p)$ is injective. \n\nLet $C$ be a $\\mathcal{D}_{\\log}(X)$-complex satisfying (H1).\nConsider the diagram\n\\begin{displaymath}\n \\xymatrix{\n \\cha^p(\\mathcal{X},C) \\ar[ddr]_{i}\\ar[dr]_{j}\\ar^{\\omega _{C}}[drr] &&\\\\\n & A \\ar[r] \\ar[d] & \\Zr C^{2p}(p)\\ar[d] \\\\\n & \\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)) \\ar[r]_-{\\omega} & \\Zr\n \\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p}(p) \n}\n\\end{displaymath}\nwhere $A$ is defined by the cartesian square, $i$ is induced by (H1)\nand $j$ is induced by $i$ and $\\omega _{C}$.\n\\begin{lemma} \\label{lemm:2}\n If $C$ also satisfies (H2) then $j$ is an isomorphism.\n\\end{lemma}\n\\begin{proof}\nBy the injectivity of $\\widetilde C^{2p-1}(p)\\to \\widetilde\n{\\mathcal{D}}^{2p-1}(X,p)$ and Lemma \\ref{lemm:1}~\\ref{item:3}, the map $i$ is\ninjective. Hence the map $j$ is injective and we only need to prove\nthat $j$ is surjective.\n\nLet $x=((Z,\\widetilde g),\\eta)\\in A$. This means that $(Z,\\widetilde\ng)\\in\\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$, $\\eta\\in \\Zr\nC^{2p}(p)$ and $\\omega _{\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)}(Z,\\widetilde\ng)=\\dd_{\\mathcal{D}}g+\\delta _{Z}=\\eta$. Let $g'\\in \n\\mathcal{D}^{2p-1}_{\\log}(X\\setminus |Z|,p)$ be a Green form for\n$Z$. Then $g-[g']\\in \\mathcal{D}^{2p-1}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,p)$ satisfies \n$\\dd_{\\mathcal{D}}(g-[g'])\\in C^{2p}(p)$. By (H2) there is $a\\in\n\\mathcal{D}^{2p-2}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,p)$ such that\n$g'':=g-[g']+\\dd_{\\mathcal{D}}a\\in C^{2p-1}(p)$.\nConsider the element $x'=((Z,\\widetilde g'),\\widetilde g'')\\in\n\\cha^{p}(\\mathcal{X},C)$. To see that $j(x')=x$ we have to check that \n$\\omega _{C}(x')=\\eta$ and $i(x')=(Z,\\widetilde g)$. We compute\n\\begin{gather*}\n \\omega _{C}(x')=\\cmap(\\dd_{\\mathcal{D}}g')+\\dd_{C}g''=\n \\dd_{\\mathcal{D}}[g']+\\delta _{Z}+\\dd_{\\mathcal{D}} g-\\dd_{\\mathcal{D}}[g']+\n \\dd_{\\mathcal{D}}\\dd_{\\mathcal{D}}a=\\eta,\\\\\n i(x')=(Z,([g']+g-[g']+\\dd_{\\mathcal{D}}a)^{\\sim})=(Z,\\widetilde g),\n\\end{gather*}\nconcluding the proof of the lemma.\n\\end{proof}\nWe can rephrase the lemma as follows.\n\\begin{theorem}\\label{pf=pb} \nLet $C$ be a $\\mathcal{D}_{\\log}(X)$-complex that satisfies (H1) and\n(H2). Then the map $\\varphi$ can be extended to an injective map \n\\begin{displaymath}\n \\varphi_{C}\\colon \\za^{p}(\\mathcal{X},C)\\longrightarrow \n \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde D_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{displaymath}\nMoreover\n\\begin{equation}\n \\label{eq:6}\n \\Img(\\varphi_{C})=\\left\\{(Z,\\widetilde g)\\in\n \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde\n {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p) \\mid \n \\dd_{\\mathcal{D}}g+\\delta _{Z}\\in\n C^{2p}(p)\\right\\}. \n\\end{equation}\n\\end{theorem}\nIn view of this theorem, if $C$ satisfies (H1) and (H2), we will\nidentify $\\za^{p}(\\mathcal{X},C)$ with the right hand side of equation\n\\eqref{eq:6}. \n\n\n\\begin{proposition}\\label{prop:1} The\n $\\mathcal{D}_{\\log}(X)$-complexes $\\mathcal{D}_{{\\text{\\rm a}}}(X)$,\n $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ and $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)$ satisfy\n (H1) and (H2). Therefore we can identify\n\\begin{align*}\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))&\\cong\n \\left\\{(Z,\\widetilde g) \\in \\Zr^{p}\\oplus \\widetilde\n {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n \\mid\n \\dd_{\\mathcal{D}}g+\\delta _{Z}\\in\n \\mathcal{D}^{2p}_{{\\text{\\rm a}}}(X,p)\\right\\}\\\\\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))&\\cong\n \\left\\{(Z,\\widetilde g) \\in \\Zr^{p}\\oplus \\widetilde\n {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p)\n \\mid \n \\dd_{\\mathcal{D}}g+\\delta _{Z}\\in\n \\mathcal{D}^{2p}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S,p)\\right\\}\\\\\n \\za^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))&\\cong\n \\Zr^{p}(\\mathcal{X})\\oplus \\widetilde\n {\\mathcal{D}}_{\\text{{\\rm cur}},{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n The result for $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ follows from the Poincar\\'e\n $\\overline \\partial$-Lemma for currents\n \\cite[Pag. 382]{GriffithsHarris:pag}. The result for\n $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ follows from the Poincar\\'e\n $\\overline \\partial$-Lemma for currents with fixed wave front set\n \\cite[Theorem 4.5]{BurgosLitcanu:SingularBC}. The statement for\n $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)$ is Lemma \\ref{lemm:3}.\n\\end{proof}\n\n\\begin{remark}\n The previous proposition gives us a more concrete description of the\n groups $\\za^{p}(\\mathcal{X},C)$ for $C=\\mathcal{D}_{{\\text{\\rm a}}}(X)$,\n $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ and $\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X)$ and\n also a more concrete description of the groups\n $\\cha^{p}(\\mathcal{X},C)$. In particular it easily implies that the\n groups $\\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$, $p\\ge 0$, agree\n (up to a \n normalization factor) with the arithmetic Chow groups of Gillet-Soul\\'e\n $\\cha^{p}(\\mathcal{X})$ and that the groups\n $\\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X))$, $p\\ge 0$, agree\n (again up to a \n normalization factor) with the\n arithmetic Chow groups of Kawaguchi-Moriwaki $\\cha^{p}_{D}(\\mathcal{X})$, introduced in\n \\cite{KawaguchiMoriwaki:isfav}. \n\\end{remark}\n\nWe want to transfer the push-forward of Proposition \\ref{prop:4} to\ncomplexes satisfying (H1) and (H2). \nLet again $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\nregular arithmetic varieties of relative dimension $e$. Let \n$C$ be a $\\mathcal{D}_{\\log}(X)$-complex and $C'$ a\n$\\mathcal{D}_{\\log}(Y)$-complex, both satisfying (H1) and (H2). We assume \nthere exists a map of complexes $f_{\\ast}\\colon C^{\\ast}(\\ast)\\to\nC'^{\\ast-2e}(\\ast-e)$ that makes the \nfollowing diagram commutative: \n\\begin{equation}\\label{eq:18}\n \\xymatrix{\n C^{\\ast}(\\ast)\\ar[r]^-{f_{\\ast}} \\ar[d] & C'^{\\ast-2e}(\\ast-e) \\ar[d] \\\\\n \\mathcal{D}_{\\text{{\\rm cur}}}^{\\ast}(X,\\ast)\\ar[r]_-{f_{\\ast}} & \\mathcal{D}_{\\text{{\\rm cur}}}^{\\ast-2e}(Y,\\ast-e).\n} \n\\end{equation}\nThen, for $p\\ge 0$, we define maps\n\\begin{displaymath}\n f_{\\ast}\\colon \\za^{p}(\\mathcal{X},C)\\to \\za^{p-e}(\\mathcal{Y},C') \n\\end{displaymath}\nthat, using the identification of Theorem \\ref{pf=pb} are given by\n\\begin{equation}\\label{dir_im_gen}\nf_{\\ast}(Z,\\widetilde g)=(f_{\\ast}Z,\\widetilde {f_{\\ast}(g)}).\n\\end{equation}\nBy Proposition \\ref{prop:4}, these maps send $\n\\rata^{p}(\\mathcal{X},C)$ to \n$\\rata^{p-e}(\\mathcal{Y},C')$. Therefore they\ninduce morphisms\n\\begin{displaymath}\n f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},C)\\longrightarrow \n \\cha^{p-e}(\\mathcal{Y},C').\n\\end{displaymath}\n\nBefore stating concrete examples of this covariant functoriality\nwe need some notation. Let\n $f_{{\\mathbb C}}\\colon X_{{\\mathbb C}}\\to Y_{{\\mathbb C}}$ denote \nthe induced proper map of complex manifolds.\n If $S\\subset\n T^{\\ast}_{0}X_{{\\mathbb C}}$ is a closed conical subset invariant under\n $F_{\\infty}$, then we write\n \\begin{displaymath}\n f_{\\ast}S=\\{(f(x),\\xi)\\in T^{\\ast}_{0}Y_{{\\mathbb C}}\\mid (x,\\dd\n f^{t}\\xi)\\in S\\}\\cup N_{f}. \n \\end{displaymath}\n It is a closed conical subset of $T^{\\ast}_{0}Y_{{\\mathbb C}}$ invariant under\n $F_{\\infty}$.\n\n \\begin{proposition}\\label{prop:8}\n Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$, $g\\colon \\mathcal{Y}\\to\\mathcal{Z}$ be proper morphism of\nregular arithmetic varieties of relative dimension $e$ and $e'$, and $S\\subset\n T^{\\ast}_{0}X_{{\\mathbb C}}$ a closed conical subset invariant under\n $F_{\\infty}$. Then there are maps\n \\begin{displaymath}\n f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\n \\to \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}S)),\n \\end{displaymath}\n and similarly\n \\begin{displaymath}\n \tg_{\\ast}\\colon \\cha^{p'}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}S))\n \\to \\cha^{p'-e'}(\\mathcal{Z},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Z,g_{\\ast}f_{\\ast}S)).\n \\end{displaymath}\n Furthermore, the relation $(g\\circ f)_{\\ast}=g_{\\ast}\\circ f_{\\ast}$ is satisfied.\n \\end{proposition}\n\nAs a particular case of the above proposition we obtain the following\ncases.\n\\begin{corollary}\\label{cor:1}\n Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\nregular arithmetic varieties of relative dimension $e$.\n\\begin{enumerate}\n\\item There are maps\n \\begin{displaymath}\n f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n \\to \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,N_{f})).\n \\end{displaymath}\n\\item If $N_{f}=\\emptyset$, then there are maps \n \\begin{displaymath}\n f_{\\ast}\\colon \\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n \\to \\cha^{p-e}(\\mathcal{Y},\\mathcal{D}_{{\\text{\\rm a}}}(Y)).\n \\end{displaymath}\n\\end{enumerate}\n\\end{corollary}\n\nThe functoriality and the multiplicative structures satisfy the\nfollowing compatibility properties.\n\n\\begin{theorem}\\label{thm:1}\n Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a morphism of regular arithmetic\n varieties. Let $N_{f}$ be the set of normal directions of $f_{{\\mathbb C}}$ and\n $S,\\,S'\\subset\n T^{\\ast}_{0}Y_{{\\mathbb C}}$ closed conical subsets invariant under\n $F_{\\infty}$. Then\n \\begin{displaymath}\n f^{\\ast}(S\\cup\n S'\\cup \n (S+S'))=f^{\\ast}(S)\\cup f^{\\ast}(\n S')\\cup \n (f^{\\ast}(S)+f^{\\ast}(S'))).\n \\end{displaymath}\n If $N_{f}\\cap(S\\cup S'\\cup S+S')=\\emptyset$ and $S\\cap\n (-S')=\\emptyset$ then $f^{\\ast}(S)\\cap\n (-f^{\\ast}(S'))=\\emptyset$. In\n this case, if $\\alpha \\in\n \\cha^p(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))$ and $\\beta \\in\n \\cha^q(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S'))$ then\n \\begin{displaymath}\n f^{\\ast}(\\alpha \\cdot \\beta )=f^{\\ast}(\\alpha )\\cdot\n f^{\\ast}(\\beta )\\in\n \\cha^{p+q}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S\\cup\n S'\\cup \n (S+S')))). \n \\end{displaymath}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:2}\n Let $f\\colon \\mathcal{X}\\to \\mathcal{Y}$ be a proper morphism of\n regular arithmetic \n varieties of relative dimension $e$. Let $N_{f}$ be the set of\n normal directions of $f_{{\\mathbb C}}$,\n $S\\subset\n T^{\\ast}_{0}X_{{\\mathbb C}}$ and $S'\\subset\n T^{\\ast}_{0}Y_{{\\mathbb C}}$ closed conical subsets invariant under\n $F_{\\infty}$. Then\n \\begin{displaymath}\n f_{\\ast}(S\\cup f^{\\ast}(\n S')\\cup \n (S+f^{\\ast}(S')))\\subset f_{\\ast}(S)\\cup\n S'\\cup \n (f_{\\ast}(S)+S')).\n \\end{displaymath}\n If $f_{\\ast}(S)\\cap (-S')=\\emptyset$ then $N_{f}\\cap\n S'=\\emptyset$ and $S\\cap (-f^{\\ast}(S'))=\\emptyset$. In this case, if\n $\\alpha \\in \n \\cha^p(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))$ and $\\beta \\in\n \\cha^q(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S'))$ then\n \\begin{displaymath}\n f_{\\ast}(\\alpha \\cdot f^{\\ast}(\\beta) )=f_{\\ast}(\\alpha )\\cdot\n \\beta \\in\n \\cha^{p+q-e}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(S)\\cup\n S'\\cup \n (f_{\\ast}(S)+S'))). \n \\end{displaymath} \n\\end{theorem}\n\n\\section{Arithmetic $K$-theory and derived categories}\\label{section:ArKTheory}\n\\subsection{Arithmetic $K$-theory}\nAs for arithmetic Chow groups, the arithmetic $K$-groups of\nGillet-Soul\\'e can be generalized to include more general\ncoefficients at the archimedean places. Because the\ndefinition of arithmetic $K$-groups involves hermitian vector bundles\nwhose metrics have arbitrary singularities at infinity, we are\nactually forced to consider $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex\ncoefficients. The reader is referred to\n\\cite[Sec. 4.2]{BurgosKramerKuehn:accavb} for the construction of the\ngroups $\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})$, which provide\nthe base of our extension (see also Definition \\ref{def:2} below). An\narithmetic Chern character allows to compare these generalized\narithmetic $K$-groups and the generalized arithmetic Chow groups. The\narithmetic Chern character is automatically compatible with pull-back\nand products whenever defined.\n\n\\begin{definition}\\label{def:2}\nLet $\\mathcal{X}$ be an arithmetic variety and $C^{\\ast}(\\ast)$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex with structure morphism $\\cmap:\\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X,\\ast)\\rightarrow C^{\\ast}(\\ast)$. The arithmetic $K$ group of $\\mathcal{X}$ with $C$ coefficients is the abelian group $\\widehat{K}_{0}(\\mathcal{X},C)$ generated by pairs $(\\overline{\\mathcal{E}},\\eta)$, where $\\overline{\\mathcal{E}}$ is a smooth hermitian vector bundle on $\\mathcal{X}$ and $\\eta\\in\\bigoplus_{p\\geq 0}\\widetilde{C}^{2p-1}(p)$, modulo the relations\n\\begin{displaymath}\n\t(\\overline{\\mathcal{E}}_{1},\\eta_{1})+(\\overline{\\mathcal{E}}_{2},\\eta_{2})=(\\overline{\\mathcal{E}},\\cmap(\\widetilde{\\ch}(\\overline{\\varepsilon}))+\\eta_{1}+\\eta_{2}),\n\\end{displaymath}\nfor every exact sequence\n\\begin{displaymath}\n\t\\overline{\\varepsilon}\\colon\\quad 0\\longrightarrow\\overline{\\mathcal{E}}_{1}\\longrightarrow\\overline{\\mathcal{E}}\\longrightarrow\\overline{\\mathcal{E}}_{2}\\longrightarrow 0\n\\end{displaymath}\nwith Bott-Chern secondary class $\\widetilde{\\ch}(\\overline{\\varepsilon})$.\n\\end{definition}\nAn equivalent construction can be given in the same lines as for the generalized arithmetic Chow groups. For this, observe there is a natural morphism\n\\begin{equation}\\label{eq:7}\n\t\\begin{split}\n\t\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})&\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C)\\\\\n\t\t[\\overline{\\mathcal{E}},\\eta]&\\longmapsto [\\overline{\\mathcal{E}},\\cmap(\\eta)]\n\t\\end{split}\n\\end{equation}\ninduced by $\\cmap:\\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X,\\ast)\\rightarrow C^{\\ast}(\\ast)$, and also\n\\begin{equation}\\label{eq:8}\n\t\\begin{split}\n\t\t\\bigoplus_{p\\geq 0}\\widetilde{C}^{2p-1}(p)&\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C)\\\\\n\t\t\\eta&\\longmapsto [0,\\eta].\n\t\\end{split}\n\\end{equation}\nOne easily sees the maps \\eqref{eq:7}--\\eqref{eq:8} induce a natural isomorphism of groups\n\\begin{equation}\\label{eq:9}\n\t\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})\\times\\bigoplus_{p\\geq 0}\\widetilde{C}^{2p-1}(p)\/\\equiv\\;\n\t\\overset{\\cong}{\\longrightarrow}\\widehat{K}_{0}(\\mathcal{X},C),\n\\end{equation}\nwhere $\\equiv$ is the equivalence relation generated by\n\\begin{displaymath}\n\t((\\overline{\\mathcal{E}},\\eta),0)\\equiv ((\\overline{\\mathcal{E}},0),\\cmap(\\eta)).\n\\end{displaymath}\nGeneralized arithmetic $K$-groups for suitable complexes have\npull-backs and products. Let $f:\\mathcal{X}\\to\\mathcal{Y}$ be a morphism\nof arithmetic varieties, and suppose given $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and\n$\\mathcal{D}_{{\\text{\\rm a}}}(Y)$ complexes $C$ and $C'$ respectively, for which\nthere is a commutative diagram \n\\begin{equation}\\label{eq:12}\n \\xymatrix{\n \\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(Y)\\ar[d]\\ar[r]^{f^{\\ast}}\t\t\n &\\mathcal{D}_{{\\text{\\rm a}}}^{\\ast}(X)\\ar[d]\\\\\n C^{\\prime\\ast}\\ar[r]_{f^{\\ast}}\n &C^{\\ast}.\n }\n\\end{equation}\nBy description \\eqref{eq:9} and the contravariant functoriality of\n$\\widehat{K}_{0}(\\underline{\\ },\\mathcal{D}_{{\\text{\\rm a}}})$, we see there is\nan induced morphism of groups \n\\begin{displaymath}\n\tf^{\\ast}:\\widehat{K}_{0}(\\mathcal{Y},C^{\\prime})\n \\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C).\n\\end{displaymath}\nFor this kind of functoriality, an analog statement to Proposition\n\\ref{prop:3} holds, and we leave to the reader the task of stating\nit. As for products, let $C$, $C'$ and $C''$ be\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes with a product $C\\otimes\nC'\\overset{\\bullet}{\\rightarrow} C''$ and a commutative diagram\n\\begin{equation}\\label{eq:19}\n\t\\xymatrix{\n\t\t\\mathcal{D}_{{\\text{\\rm a}}}(X)\\otimes\\mathcal{D}_{{\\text{\\rm a}}}(X)\\ar[r]^-{\\bullet}\\ar[d]\t&\\mathcal{D}_{{\\text{\\rm a}}}(X)\\ar[d]\\\\\n\t\tC\\otimes C'\\ar[r]_-{\\bullet}\t&C''.\n\t}\n\\end{equation}\nThen there is an induced product at the level of $\\widehat{K}_{0}$\n\\begin{displaymath}\n\t\\widehat{K}_{0}(\\mathcal{X},C)\\times\\widehat{K}_{0}(\\mathcal{X},C')\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C'')\n\\end{displaymath}\ndescribed by the rule\n\\begin{displaymath}\n\t[\\overline{\\mathcal{E}},\\eta]\\cdot [\\overline{\\mathcal{E}}',\\eta']=\n\t\t[\\overline{\\mathcal{E}}\\otimes\\overline{\\mathcal{E}}',c(\\ch(\\overline{\\mathcal{E}}))\\bullet\\eta'+c'(\\ch(\\overline{\\mathcal{E}}'))\\bullet\\eta\n\t\t+\\dd_{C}\\eta\\bullet\\eta'].\n\\end{displaymath}\nWith respect to this laws, the groups $\\widehat{K}_{0}$ enjoy of the\nanalogue properties to Proposition \\ref{prop:2}. \n\nFinally, we discuss on the arithmetic Chern character. If\n$\\mathcal{X}$ is a regular arithmetic variety, we recall there is an\nisomorphism of rings \\cite[Thm. 4.5]{BurgosKramerKuehn:accavb}, namely \n\\begin{displaymath}\n \\chh:\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}})_{{\\mathbb Q}}\\longrightarrow\n \\bigoplus_{p\\geq 0}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}).\n\\end{displaymath}\nIf $C$ is a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, by the presentations\n\\eqref{eq:11} of $\\cha^{\\ast}(\\mathcal{X},C)$ and \\eqref{eq:9} of\n$\\widehat{K}_{0}$ it is clear that $\\chh$ extends to an isomorphism\nof groups \n\\begin{displaymath}\n\t\\chh:\\widehat{K}_{0}(\\mathcal{X},C)_{{\\mathbb Q}}\\longrightarrow\\bigoplus_{p\\geq 0}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C).\n\\end{displaymath}\nSuppose now that $C,C',C''$ are $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes with\na product $C\\otimes C'\\rightarrow C''$ as above. Then, it is easily\nseen that there is a commutative diagram of morphisms of groups\n\\begin{displaymath}\n \\xymatrix{\n \\widehat{K}_{0}(\\mathcal{X},C)_{{\\mathbb Q}}\\times\\widehat{K}_{0}(\\mathcal{X},C')_{{\\mathbb Q}}\n \\ar[r]^{\\hspace{1cm}\\cdot}\\ar[d]_{(\\chh,\\chh)}\n &\\widehat{K}_{0}(\\mathcal{X},C'')_{{\\mathbb Q}}\\ar[d]^{\\chh}\\\\ \n \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C)\\times\\bigoplus_{p}\n \\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C')\\ar[r]^{\\hspace{1.3cm\\cdot}}\n &\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C''). \n }\n\\end{displaymath}\nThis is in particular true for complexes of currents with controlled wave front set, a result that we next record.\n \\begin{proposition}\nLet $S, S'$ be closed conical subsets of $T_{0}^{\\ast}X_{{\\mathbb C}}$\ninvariant under the action of complex conjugation, with $S\\cap\n(-S')=\\emptyset$. Define $T=S\\cup S'\\cup (S+S')$. If $\\alpha \\in\n\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))_{{\\mathbb Q}}$ and\n$\\beta \\in\n\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S'))_{{\\mathbb Q}}$, then\n\\begin{displaymath}\n \\chh(\\alpha )\\cdot \\chh(\\beta )=\\chh(\\alpha \\cdot \\beta )\\in\n \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T)). \n\\end{displaymath}\n\\end{proposition}\nAnother feature of the Chern character is its compatibility with\npull-back functoriality. Let $f:\\mathcal{X}\\rightarrow\\mathcal{Y}$ be\na morphism of regular arithmetic varieties. Let $C,C'$ be\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$ complexes,\nrespectively, together with a morphism of complexes\n$f^{\\ast}:C'\\rightarrow C$ satifying the commutativity\n\\eqref{eq:12}. Then there is a commutative diagram\n\\begin{displaymath}\n \\xymatrix{\n \\widehat{K}_{0}(\\mathcal{Y},C')_{{\\mathbb Q}}\\ar[r]^{f^{\\ast}}\\ar[d]_{\\chh}\n &\\widehat{K}_{0}(\\mathcal{X},C)_{{\\mathbb Q}}\\ar[d]^{\\chh}\\\\ \n \\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{Y},C')\\ar[r]^{f^{\\ast}}\n &\\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{X},C).\n }\n\\end{displaymath}\nAgain, the proof is a simple consequence for the known compatibility\nin the case of the $\\mathcal{D}_{{\\text{\\rm a}}}$ arithmetic Chow groups. In\nparticular we have the following proposition.\n\\begin{proposition}\\label{prop:6}\n Let $f\\colon \\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism of regular\n arithmetic varieties, and $S\\subset T_{0}^{\\ast}Y_{{\\mathbb C}}$ a closed\n conical subset, invariant under complex conjugation and disjoint\n with the normal directions $N_{f}$ of $f_{{\\mathbb C}}$. Then there is a\n commutative diagram\n\\begin{displaymath}\n \\xymatrix{\n \\widehat{K}_{0}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))_{{\\mathbb Q}}\\ar[r]^{f^{\\ast}}\\ar[d]_{\\chh}\t\n &\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S))_{{\\mathbb Q}}\\ar[d]^{\\chh}\\\\\n \\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))\\ar[r]^{f^{\\ast}}\n &\\cha_{{\\mathbb Q}}^{\\ast}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S)). \n }\n\\end{displaymath}\n\\end{proposition}\n\\subsection{Arithmetic derived categories}\nFor the problem of defining direct images on arithmetic $K$-theory it\nis useful to deal with arbitrary complexes of coherent sheaves instead\nof locally free sheaves. We therefore introduce an arithmetic\ncounterpart of our theory of hermitian structures on derived\ncategories of coherent sheaves, developed in\n\\cite{BurgosFreixasLitcanu:HerStruc}, and the $\\hDb$ categories in\n\\cite{BurgosFreixasLitcanu:GenAnTor}. We then compare this\nconstruction to the arithmetic $K$ groups. \n\n\\begin{definition}\nLet $S$ be a scheme. We denote by $\\Db(S)$ the derived category of cohomological complexes of quasi-coherent sheaves with bounded coherent cohomology.\n\\end{definition}\nIf $\\mathcal{X}$ is a regular arithmetic variety, every object of\n$\\Db(\\mathcal{X})$ is quasi-isomorphic to a bounded cohomological\ncomplex of locally free sheaves. One checks there is a well defined\nmap \n\\begin{displaymath}\n\t\\Ob\\Db(\\mathcal{X})\\longrightarrow K_{0}(\\mathcal{X})\n\\end{displaymath}\nthat sends an object $\\mathcal{F}^{\\ast}$ to the class\n$\\sum_{i}(-1)^{i}[\\mathcal{E}^{i}]$, where $\\mathcal{E}^{\\ast}$ is\nquasi-isomorphic to $\\mathcal{F}^{\\ast}$, and that is compatible with\nderived tensor products on $\\Db(\\mathcal{X})$ and the ring structure\non $K_{0}(\\mathcal{X})$. Our aim is thus to extend this picture and\nincorporate hermitian structures. \n\nLet us consider the complex quasi-projective manifold $X_{{\\mathbb C}}$\nassociated to an arithmetic variety $\\mathcal{X}$. It comes equipped\nwith the conjugate-linear involution $F_{\\infty}$. Recall the data\n$X=(X_{{\\mathbb C}},F_{\\infty})$ uniquely determines a smooth quasi-projective\nscheme $X_{{\\mathbb R}}$ over ${\\mathbb R}$, whose base change to ${\\mathbb C}$ is isomorphic\nto $X_{{\\mathbb C}}$, and such that its natural automorphism given by complex\nconjugation gets identified to $F_{\\infty}$. The abelian category of\nquasi-coherent (resp. coherent) sheaves over $X_{{\\mathbb R}}$ is equivalent\nto the category of quasi-coherent (resp. coherent) sheaves over\n$X_{{\\mathbb C}}$, equivariant with respect to the action of\n$F_{\\infty}$. Namely, giving a quasi-coherent (resp. coherent) sheaf\non $X_{{\\mathbb R}}$ is equivalent to giving a quasi-coherent (resp. coherent)\nsheaf $\\mathcal{F}$ on $X_{{\\mathbb C}}$, together with a morphism of\nsheaves\n\\begin{displaymath}\n\t\\mathcal{F}\\longrightarrow F_{\\infty\\ast}\\mathcal{F}\n\\end{displaymath}\ncompatible with the conjugate-linear morphism\n\\begin{displaymath}\n\tF_{\\infty}^{\\sharp}:{\\mathcal O}_{X_{{\\mathbb C}}}\\longrightarrow F_{\\infty\\ast}{\\mathcal O}_{X_{{\\mathbb C}}}\n\\end{displaymath}\ninduced by the morphism of ${\\mathbb R}$-schemes $F_{\\infty} \\colon\nX_{{\\mathbb C}}\\rightarrow X_{{\\mathbb C}}$. A similar condition characterizes\nmorphisms of quasi-coherent (resp. coherent) sheaves. Therefore, we\ndenote the bounded derived category of coherent sheaves on $X_{{\\mathbb R}}$\njust by $\\Db(X)$.\n\nThe theory of hermitian structures on the bounded derived category of\ncoherent sheaves on a complex algebraic manifold developed in\n\\cite{BurgosFreixasLitcanu:HerStruc} can be adapted to the real\nsituation of $\\Db(X)$, by considering hermitian structures invariant\nunder the action of complex conjugation. All the results in\n\\emph{loc. cit.} carry over to the real case. We denote by $\\oDb(X)$\nthe category whose objects are objects of $\\Db(X)$ endowed with a\nhermitian structure (\\emph{loc. cit.}, Def. 3.10) invariant under\ncomplex conjugation, and whose morphisms are just morphisms in\n$\\Db(X)$. Thus, every object $\\overline{\\mathcal{F}}^{\\ast}$ in $\\oDb(X)$ is\nrepresented by a quasi-isomorphism\n$\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, where\n$\\overline{\\mathcal{E}}^{\\ast}$ is a bounded complex of hermitian locally\nfree sheaves on $X_{{\\mathbb C}}$, equivariant under $F_{\\infty}$. There is an\nobvious forgetful functor $\\mathfrak{F}:\\oDb(X)\\rightarrow\\Db(X)$,\nthat makes of $\\oDb(X)$ a principal fibered category over $\\Db(X)$,\nwith structural group $\\overline{\\KA}(X)$, the group of hermitian structures\nover the 0 object \\cite[Def. 2.34,\nThm. 3.13]{BurgosFreixasLitcanu:HerStruc}.\n\nBase change to ${\\mathbb R}$ induces a covariant functor\n$\\Db(\\mathcal{X})\\rightarrow\\Db(X)$.\n\n\\begin{definition}\nWe define the category $\\oDb(\\mathcal{X})$ as the fiber product category\n\\begin{displaymath}\n \\xymatrix{\n \\Db(\\mathcal{X})\\times_{\\Db(X)}\\oDb(X)\\ar[r]\\ar[d]\t&\\oDb(X)\\ar[d]^{\\mathfrak{F}}\\\\\n \\Db(\\mathcal{X})\\ar[r]\t&\\Db(X).\t\n }\n\\end{displaymath}\nWe still denote by $\\mathfrak{F}$ the forgetful functor $\\oDb(\\mathcal{X})\\rightarrow\\Db(\\mathcal{X})$.\n\\end{definition}\nBy construction, $\\mathfrak{F}$ makes of $\\oDb(\\mathcal{X})$ a principal fibered category over $\\Db(\\mathcal{X})$, with structure group $\\overline{\\KA}(X)$. In particular, $\\overline{\\KA}(X)$ acts on $\\oDb(\\mathcal{X})$.\n\nThe Bott-Chern secondary character $\\cht$ can be defined at the level of $\\overline{\\KA}$ groups \\cite[Sec. 4, Def. 4.6]{BurgosFreixasLitcanu:HerStruc}. In our situation, we actually have a morphism of groups\n\\begin{equation}\\label{eq:13}\n\t\\cht:\\overline{\\KA}(X)\\longrightarrow\\bigoplus_{p}\\widetilde{\\mathcal{D}}_{{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{equation}\nMore generally, if $C$ is a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, we may consider a secondary Chern character with values in $\\bigoplus_{p}\\widetilde{C}^{2p-1}(p)$, that we denote $\\cht_{C}$. In particular, $\\overline{\\KA}(X)$ acts on $\\bigoplus_{p}\\widetilde{C}^{2p-1}(p)$ through $\\cht_{C}$.\n\\begin{definition}\nThe arithmetic derived category $\\hDb(\\mathcal{X},C)$ is defined as the cartesian product\n\\begin{displaymath}\n\t\\oDb(\\mathcal{X})\\times_{\\overline{\\KA}(X),\\cht_{C}}\\bigoplus_{p}\\widetilde{C}^{2p-1}(p).\n\\end{displaymath}\n\\end{definition}\n\\begin{remark}\\label{rem:1}\nIf $\\mathcal{X}$ is regular, then every object of $\\hDb(\\mathcal{X},C)$ can be represented by \n\\begin{displaymath}\n\t(\\mathcal{F}^{\\ast},\\overline{\\mathcal{E}}_{{\\mathbb C}}\\dashrightarrow\\mathcal{F}^{\\ast}_{{\\mathbb C}},\\widetilde{\\eta})\n\\end{displaymath}\nwhere $\\mathcal{E}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$ is any quasi-isomorphism from a bounded complex of locally free sheaves over $\\mathcal{X}$. Indeed, $\\mathcal{X}$ is assumed to be regular, so that $\\mathcal{F}^{\\ast}$ is quasi-isomorphic to a bounded complex of locally free sheaves $\\mathcal{E}^{\\ast}$. One then endows the $\\mathcal{E}^{i}$ with smooth hermitian metrics invariant under complex conjugation, and takes into account that $\\overline{\\KA}(X)$ acts transitively on the hermitian structures on $\\mathcal{F}^{\\ast}_{{\\mathbb C}}$. We introduce the simplified notation $(\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F},\\widetilde{\\eta})$ for such representatives.\n\\end{remark}\nThe categories $\\hDb(\\mathcal{X},C)$ are the arithmetic analogues to those we introduced in \\cite[Sec. 4]{BurgosFreixasLitcanu:GenAnTor}, and have similar properties. We are only going to review some of them.\n\nFor the next proposition, we recall that any group can be considered\nas a category, with morphisms given by the group law. This in\nparticular the case of $\\widehat{K}_{0}(\\mathcal{X},C)$.\n\\begin{theorem}\nIf $\\mathcal{X}$ is regular, there is a natural functor\n\\begin{displaymath}\n \\hDb(\\mathcal{X},C)\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C),\n\\end{displaymath}\nthat, at the level of objects is given by\n\\begin{displaymath}\n (\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F},\\widetilde{\\eta})\\longmapsto\n \\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},\\widetilde{\\eta}\\right],\n\\end{displaymath}\nand at the level of morphism sends any $f\\in \\Hom_{\\hDb(\\mathcal{X},C)}(A,B)$ to $[B]-[A]$.\n\\end{theorem}\n\\begin{proof}\nIt is enough to see that the defining assignment does not depend on\nthe representatives. For this, take two objects\n$(\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta})$\nand\n$(\\overline{\\mathcal{E}}^{\\prime\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta}^{\\prime})$\ngiving raise to the same class. We have an equivalence of objects\n\\begin{displaymath}\n \\begin{split}\n (\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta})\\sim&\n (\\overline{\\mathcal{E}}^{\\prime\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta}^{\\prime})\\\\\n &\\sim\n (\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta}^{\\prime}+c(\\widetilde{\\ch}(\\Id\\colon\n \\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast}))).\n \\end{split}\n\\end{displaymath}\nConsequently\n\\begin{displaymath}\n\t\\widetilde{\\eta}=\\widetilde{\\eta}^{\\prime}+c(\\widetilde{\\ch}(\\Id\\colon\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast})).\n\\end{displaymath}\nHence, the image of $(\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast},\\widetilde{\\eta})$ is equivalently written\n\\begin{displaymath}\n\t\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},\\widetilde{\\eta}^{\\prime}+c(\\widetilde{\\ch}(\\Id\\colon\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast}))\\right].\n\\end{displaymath}\nWe thus have to show the equality in $\\widehat{K}_{0}(\\mathcal{X},C)$\n\\begin{displaymath}\n\t[0,c(\\widetilde{\\ch}(\\Id\\colon\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{\\prime\\ast}))]\\overset{?}{=}\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{\\prime\n i},0\\right]-\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},0\\right].\n\\end{displaymath}\nBy \\cite[Lemma 3.5]{BurgosFreixasLitcanu:HerStruc}, the quasi-isomorphism $\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{E}}^{\\prime\\ast}$ inducing the identity on $\\mathcal{F}^{\\ast}$ can be lifted to a diagram\n\\begin{displaymath}\n \\xymatrix{\n &\\overline{\\mathcal{E}}^{''\\ast}\\ar[ld]_{a}\\ar[rd]^{b}\t&\\\\\n \\overline{\\mathcal{E}}^{\\ast}\t&\t&\\overline{\\mathcal{E}}^{\\prime\\ast},\n }\n\\end{displaymath}\nwhere $a$ and $b$ are quasi-isomorphisms and $\\ocone(a)$ is meager \\cite[Def. 2.9]{BurgosFreixasLitcanu:HerStruc}. On the one hand, by the characterization \\cite[Thm. 2.13]{BurgosFreixasLitcanu:HerStruc} of meager complexes, one can show\n\\begin{displaymath}\n\t\\left[\\sum_{i}(-1)^{i}\\ocone(a)^{i},0\\right]=0.\n\\end{displaymath}\nOn the other hand, we have an exact sequence of complexes\n\\begin{displaymath}\n\t0\\rightarrow\\overline{\\mathcal{E}}^{''\\ast}\\rightarrow\\ocone(a)\\rightarrow\\overline{\\mathcal{E}}^{\\ast}[1]\\rightarrow 0,\n\\end{displaymath}\nwhose constituent rows are orthogonally split. This shows\n\\begin{equation}\\label{eq:14}\n\t\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{'' i},0\\right]-\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{i},0\\right]=\\left[\\sum_{i}(-1)^{i}\\ocone(a)^{i},0\\right]=0.\n\\end{equation}\nSimilarly we have\n\\begin{equation}\\label{eq:15}\n\t\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{'i},0\\right]-\\left[\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{'' i},0\\right]=\\left[\\sum_{i}(-1)^{i}\\ocone(b)^{i},0\\right].\n\\end{equation}\nBut the complex underlying $\\ocone(b)$ is acyclic, so that\n\\begin{equation}\\label{eq:16}\n\t\\left[\\sum_{i}(-1)^{i}\\ocone(b)^{i},0\\right]=[0,c(\\widetilde{\\ch}(\\ocone(b))].\n\\end{equation}\nFinally, by \\cite[Def. 3.14, Thm. 4.11]{BurgosFreixasLitcanu:HerStruc} we have\n\\begin{equation}\\label{eq:17}\n\t\\widetilde{\\ch}(\\Id:\\overline{\\mathcal{F}}^{\\ast}\\rightarrow\\overline{\\mathcal{F}}^{'\\ast})=\\widetilde{\\ch}(\\ocone(b)).\n\\end{equation}\nPutting \\eqref{eq:14}--\\eqref{eq:17} together allows to conclude.\n\\end{proof}\n\\begin{notation}\nLet $\\mathcal{X}$ be a regular arithmetic variety. Then we still denote the image of an object $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]\\in\\Ob\\hDb(\\mathcal{X},C)$ by the morphism of the proposition by $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$. We call this image the class of $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$ in arithmetic $K$-theory.\n\\end{notation}\n\\begin{remark}\nTwo tightly isomorphic objects in $\\hDb(\\mathcal{X},C)$ have the same class in arithmetic $K$-theory.\n\\end{remark}\nLet $f:\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism of regular arithmetic varieties and let $C$, $C'$ be $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$ complexes respectively, with a commutative diagram as in \\eqref{eq:12}. Then there is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{Y},C')\\ar[r]^{f^{\\ast}}\\ar[d]\t&\\hDb(\\mathcal{X},C)\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{Y},C')\\ar[r]_{f^{\\ast}}\t&\\widehat{K}_{0}(\\mathcal{X},C).\n\t}\n\\end{displaymath}\nThe following statement is a particular case.\n\\begin{proposition}\nLet $f:\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism of regular arithmetic varieties, and $S\\subset T_{0}^{\\ast}Y_{{\\mathbb C}}$ a closed conical subset invariant under complex conjugation, disjoint with the normal directions $N_{f}$ of $f_{{\\mathbb C}}$. Then there is a commutative diagram\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))\\ar[r]^-{f^{\\ast}}\\ar[d]\t&\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X, f^{\\ast}(S)))\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,S))\\ar[r]_-{f^{\\ast}}\t&\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,f^{\\ast}(S))).\n\t}\n\\end{displaymath}\n\\end{proposition}\nIf $C,C',C''$ are $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complexes with a product $C\\otimes C'\\rightarrow C''$ compatible with the product of $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ and $\\mathcal{X}$ is regular, then there is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{X},C)\\times\\hDb(\\mathcal{X},C')\\ar[r]^-{\\otimes} \\ar[d]\t&\\hDb(\\mathcal{X},C'')\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{X},C)\\times\\widehat{K}_{0}(\\mathcal{X},C')\\ar[r]\t&\\widehat{K}_{0}(\\mathcal{X},C''),\n\t}\n\\end{displaymath}\nwhere the derived tensor product $\\otimes$ is defined by\n\\begin{displaymath}\n\t[\\overline{\\mathcal{F}},\\eta]\\otimes [\\overline{\\mathcal{G}},\\nu]=\n [\\overline{\\mathcal{F}}\\otimes\\overline{\\mathcal{G}},\n c(\\ch(\\overline{\\mathcal{F}}))\\bullet\\nu+\n \\eta\\bullet c'(\\ch(\\overline{\\mathcal{G}}))+\\dd_{C}\\eta\\bullet\\nu]\n\\end{displaymath}\n\\begin{proposition}\nLet $S, S'$ be closed conical subsets of $T_{0}^{\\ast}X_{{\\mathbb C}}$ invariant under the action of complex conjugation, with $S\\cap (-S')=\\emptyset$. Define $T=S\\cup S'\\cup (S+S')$. If $\\mathcal{X}$ is regular, then there is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\\times\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S'))\\ar[r]^-{\\otimes} \\ar[d]\t\n\t\t\t&\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T))\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S))\\times\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S'))\\ar[r]\t\n\t\t&\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,T)).\n\t}\n\\end{displaymath}\nFurthermore, it is compatible with pull-back $f^{\\ast}$ whenever defined.\n\\end{proposition}\nFinally, the class functor and the arithmetic Chern character on arithmetic $K$ groups, allow to extend it to arithmetic derived categories.\n\\begin{notation}\\label{not:1}\nLet $\\mathcal{X}$ be a regular arithmetic variety and $C$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$ complex. We denote by\n\\begin{displaymath}\n\t\\chh:\\hDb(\\mathcal{X},C)\\longrightarrow\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},C)\n\\end{displaymath}\nthe arithmetic Chern character on $\\hDb(\\mathcal{X},C)$, obtained as\nthe composition of the class functor and $\\chh$ on\n$\\widehat{K}_{0}(\\mathcal{X},C)$.\n\\end{notation}\n\\section{Arithmetic characteristic classes}\\label{section:ArChar}\nLet $\\mathcal{X}$ be a regular arithmetic variety. From the previous\nsections, there exists a natural functor \n\\begin{displaymath}\n \\oDb(\\mathcal{X})\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n\\end{displaymath}\nthat factors through $\\hDb(\\mathcal{X},C)$, and there is a ring isomorphism\n\\begin{displaymath}\n \\chh:\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))_{{\\mathbb Q}}\\longrightarrow\n \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)).\n\\end{displaymath}\nWe therefore obtain a functor\n\\begin{displaymath}\n \\chh:\\oDb(\\mathcal{X})\\longrightarrow\n \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n\\end{displaymath}\nautomatically satisfying several compatibilities with the operations\nin $\\oDb(\\mathcal{X})$ and distinguished triangles. More generally, in\nthis section we construct arithmetic characteristic classes attached\nto real additive or multiplicative genera. The case of the arithmetic\nTodd class will be specially relevant, since it is involved in the\narithmetic Riemann-Roch theorem. Our construction relies on the one\ngiven by Gillet-Soul\\'e \\cite{GilletSoule:vbhm}.\n\nLet $B$ be a subring of ${\\mathbb R}$ and $\\varphi\\in B[[x]]$ a real power series, defining an\nadditive genus. For each hermitian vector bundle $\\overline {\\mathcal{E}}$, in\n\\cite{GilletSoule:vbhm}, there is attached a class\n$\\widehat{\\varphi}(\\overline{\\mathcal{E}})\\in \\cha_{B}\n^{\\ast}(\\mathcal{X})$. By the isomorphism \\cite[Theorem\n3.33]{BurgosKramerKuehn:accavb} we obtain a class \n$\\widehat{\\varphi}(\\overline{\\mathcal{E}})\\in \\cha_{B}\n^{\\ast}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$. \n\n\nFor every finite complex of smooth hermitian vector\nbundles $\\overline{\\mathcal{E}}^{\\ast}$, we put\n\\begin{displaymath}\n \\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})=\n \\sum_{i}(-1)^{i}\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{i})\\in\n \\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)). \n\\end{displaymath}\nLet us now consider an object $\\overline{\\mathcal{F}}^{\\ast}$ in\n$\\oDb(\\mathcal{X})$. We choose an auxiliary quasi-isomor\\-phism\n$\\psi:\\mathcal{E}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, where\n$\\mathcal{E}^{\\ast}$ is a bounded complex of locally free sheaves on\n$\\mathcal{X}$. This is possible since $\\mathcal{X}$ is regular by\nassumption. We also fix auxiliary smooth hermitian metrics on the\nindividual terms $\\mathcal{E}^{i}$. We thus obtain an isomorphism\n$\\overline{\\psi}\\colon\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{F}}^{\\ast}$\nthat in general is not tight. The lack of tightness is measured by a\nclass $[\\overline{\\psi}_{{\\mathbb C}}]\\in\\overline{\\KA}(X)$, that we simply denote\n$[\\overline{\\psi}]$ \\cite[Sec. 3]{BurgosFreixasLitcanu:HerStruc}. Recall\nthat Bott-Chern secondary classes can be defined at the level of\n$\\overline{\\KA}(X)$ (see \\emph{loc. cit.} Sec. 4, and especially the\ncharacterization given in Prop. 4.6). In particular we have a class\n\\begin{displaymath}\n \\widetilde{\\varphi}(\\overline{\\psi}):=\\widetilde{\\varphi}([\\overline{\\psi}])\n \\in\\bigoplus_{p}\\widetilde{\\mathcal{D}}_{{\\text{\\rm a}}}^{2p-1}(X,p).\n\\end{displaymath}\n\\begin{lemma}\nThe class\n\\begin{equation}\\label{eq:22}\n \\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})+\n \\amap(\\widetilde{\\varphi}(\\overline{\\psi}))\\in\n \\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\n\\end{equation}\ndepends only on $\\overline{\\mathcal{F}}^{\\ast}$.\n\\end{lemma}\n\\begin{proof}\nLet\n$\\psi^{\\prime}\\colon\\mathcal{E}^{\\prime\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$\nbe another finite locally free resolution, and choose arbitrary\nmetrics on the $\\mathcal{E}^{\\prime i}$. We can construct a\ncommutative diagram of complexes in $\\Db(\\mathcal{X})$\n\\begin{equation}\\label{eq:27}\n \\xymatrix{\n &\\mathcal{E}^{\\prime\\prime\\ast}\\ar[ld]_{\\alpha}\\ar[rd]^{\\beta}\n &\\\\\n \\mathcal{E}^{\\ast}\\ar@{-->}[d]_{\\psi}\t&\n &\\mathcal{E}^{\\prime\\ast}\\ar@{-->}[d]^{\\psi^{\\prime}}\\\\ \n \\mathcal{F}^{\\ast}\\ar[rr]^{\\Id}\t&\t&\\mathcal{F}^{\\ast}, \n }\n\\end{equation}\nwhere $\\mathcal{E}^{\\prime\\prime}$ is also a finite complex of locally\nfree sheaves, that we endow with smooth hermitian metrics, and\n$\\alpha$, $\\beta$ are quasi-isomorphisms. Because the exact sequences\n\\begin{align*}\n\t&0\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\ast}\\rightarrow\\ocone(\\alpha)\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}[1]\\rightarrow 0,\\\\\n\t&0\\rightarrow\\overline{\\mathcal{E}}^{\\ast}\\rightarrow\\ocone(\\beta)\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}[1]\\rightarrow 0\n\\end{align*}\nhave orhtogonally split constituent rows, we find\n\\begin{align}\n\t&\\widehat{\\varphi}(\\ocone(\\alpha))=\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}),\\\\ \\label{eq:23}\n\t& \\widehat{\\varphi}(\\ocone(\\beta))=\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\prime\\ast}).\n\\end{align}\nAlso, the complexes $\\cone(\\alpha)$ and $\\cone(\\beta)$ are acyclic, so that\n\\begin{align}\n\t&\\widehat{\\varphi}(\\ocone(\\alpha))=\\amap(\\widetilde{\\varphi}(\\ocone(\\alpha)))=\\amap(\\widetilde{\\varphi}(\\overline{\\alpha})),\\\\\n\t&\\widehat{\\varphi}(\\ocone(\\beta))=\\amap(\\widetilde{\\varphi}(\\ocone(\\beta)))=\\amap(\\widetilde{\\varphi}(\\overline{\\beta})), \\label{eq:26}\n\\end{align}\nwhere we took into account the very definition of the class of an isomorphism in $\\oDb(X)$. From the relations \\eqref{eq:23}--\\eqref{eq:26} we derive\n\\begin{equation}\\label{eq:28}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t=\\amap(\\widetilde{\\varphi}(\\overline{\\alpha})-\\widetilde{\\varphi}(\\overline{\\beta}))=\\amap(\\widetilde{\\varphi}(\\overline{\\alpha}\\circ\\overline{\\beta}^{-1})),\n\\end{equation}\nwhere we plugged $\\widetilde{\\varphi}(\\overline{\\alpha}\\circ\\overline{\\beta}^{-1})=\\widetilde{\\varphi}(\\overline{\\alpha})-\\widetilde{\\varphi}(\\overline{\\beta})$ \\cite[Prop. 4.13]{BurgosFreixasLitcanu:HerStruc}. But by diagram \\eqref{eq:27} we have $\\overline{\\alpha}\\circ\\overline{\\beta}^{-1}=\\overline{\\psi}^{-1}\\circ\\overline{\\psi}^{\\prime}$. This fact combined with \\eqref{eq:28} implies\n\\begin{displaymath}\n\t\\begin{split}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t=&\\amap(\\widetilde{\\varphi}(\\overline{\\alpha})-\\widetilde{\\varphi}(\\overline{\\beta}))\\\\\n\t&=\\amap(\\widetilde{\\varphi}(\\overline{\\psi}^{-1}\\circ\\overline{\\psi}^{\\prime}))\\\\\n\t&\\hspace{0.4cm}=\\amap(\\widetilde{\\varphi}(\\overline{\\psi}^{\\prime}))-\\amap(\\widetilde{\\varphi}(\\overline{\\psi})).\n\t\\end{split}\n\\end{displaymath}\nThis completes the proof of the lemma.\n\\end{proof}\n\\begin{definition}\nThe notations being as above, we define\n\\begin{displaymath}\n \\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast}):=\n \\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})+\\amap(\\widetilde{\\varphi}(\\overline{\\psi}))\n \\in\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)),\n\\end{displaymath}\n\\end{definition}\nThe additive arithmetic characteristic classes are indeed additive with respect to direct sum, and are compatible with pull-back by morphisms of arithmetic varieties. However, the most important property of additive arithmetic characteristic classes is the behavior with respect to distinguished triangles. The reader may review \\cite[Def. 3.29, Thm. 3.33, Def. 4.17, Thm. 4.18]{BurgosFreixasLitcanu:HerStruc} for definitions and main properties, especially for the class in $\\overline{\\KA}(X)$ and secondary class of a distinguished triangle.\n\\begin{theorem}\\label{thm:3}\nLet us consider a distinguished triangle in $\\oDb(\\mathcal{X})$:\n\\begin{displaymath}\n\t\\overline{\\tau}\\colon\\quad \\overline{\\mathcal{F}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{G}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{H}}^{\\ast}\n\t\\dashrightarrow\\overline{\\mathcal{F}}^{\\ast}_{0}[1].\n\\end{displaymath}\nThen we have\n\\begin{displaymath}\n\t\\widehat{\\varphi}(\\mathcal{F}^{\\ast})-\\widehat{\\varphi}(\\mathcal{G}^{\\ast})+\\widehat{\\varphi}(\\mathcal{H}^{\\ast})\n\t=\\amap(\\widetilde{\\varphi}(\\overline{\\tau}))\n\\end{displaymath}\nin $\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$. In particular, if $\\overline{\\tau}$ is tightly distinguished, we have\n\\begin{displaymath}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{G}}^{\\ast})=\\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast})+\\widehat{\\varphi}(\\overline{\\mathcal{H}}^{\\ast}).\n\\end{displaymath}\n\\end{theorem}\n\\begin{proof}\nIt is possible to find a diagram\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\overline{\\eta}\\colon &\\mathcal{E}^{\\ast}\\ar@{-->}[d]^{f}\\ar[r]^{\\alpha}\t\t&\\mathcal{E}^{\\prime\\ast}\\ar@{-->}[d]^{g}\\ar[r]\t&\\cone(\\alpha)\\ar@{-->}[d]^{h}\\ar[r]\t&\\mathcal{E}^{\\ast}[1]\\ar@{-->}[d]^{f[1]}\\\\\n\t\t\\overline{\\tau}\\colon\t&\\mathcal{F}^{\\ast}\\ar@{-->}[r]\t&\\mathcal{G}^{\\ast}\\ar@{-->}[r]\t&\\mathcal{H}^{\\ast}\\ar@{-->}[r]\t&\\mathcal{F}^{\\ast}[1],\n\t}\n\\end{displaymath}\nwhere $\\mathcal{E}^{\\ast}$, $\\mathcal{E}^{\\prime\\ast}$ are bounded complexes of locally free sheaves and the vertical arrows are isomorphisms in $\\Db(\\mathcal{X})$. We choose arbitrary smooth hermitian metrics on the $\\mathcal{E}^{i}$, $\\mathcal{E}^{\\prime j}$, and put the orthogonal sum metric on $\\cone(\\alpha)$. Then, by construction of the arithmetic characteristic classes, we have\n\\begin{displaymath}\n\t\\begin{split}\n\t\t\\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{G}}^{\\ast})\n\t\t+\\widehat{\\varphi}(\\overline{\\mathcal{H}}^{\\ast})=&\n\t\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t\t+\\widehat{\\varphi}(\\ocone(\\alpha))\\\\\n\t\t&+a(\\widetilde{\\varphi}(\\overline{f})-\\widetilde{\\varphi}(\\overline{g})+\\widetilde{\\varphi}(\\overline{h})).\n\t\\end{split}\n\\end{displaymath}\nBecause the exact sequence\n\\begin{displaymath}\n\t0\\rightarrow\\overline{\\mathcal{E}}^{\\prime\\ast}\\rightarrow\\ocone(\\alpha)\\rightarrow\\overline{\\mathcal{E}}^{\\ast}[1]\\rightarrow 0\n\\end{displaymath}\nhas orthogonally split constituent rows, we observe\n\\begin{displaymath}\n\t\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\ast})-\\widehat{\\varphi}(\\overline{\\mathcal{E}}^{\\prime\\ast})\n\t\t+\\widehat{\\varphi}(\\ocone(\\alpha))=0.\n\\end{displaymath}\nMoreover, by \\cite[Thm 3.33 (vii)]{BurgosFreixasLitcanu:HerStruc} the equality\n\\begin{displaymath}\n\t\\widetilde{\\varphi}(\\overline{f})-\\widetilde{\\varphi}(\\overline{g})+\\widetilde{\\varphi}(\\overline{h})\n\t=\\widetilde{\\varphi}(\\overline{\\tau})-\\widetilde{\\varphi}(\\overline{\\eta}),\n\\end{displaymath}\nholds, and $\\widetilde{\\varphi}(\\overline{\\eta})=0$ since $\\overline{\\eta}$ is tightly distinguished. The theorem now follows.\n\\end{proof}\nWe may also say a few words on multiplicative arithmetic\ncharacteristic classes. We follow the discussion in\n\\cite[Sec. 5]{BurgosFreixasLitcanu:HerStruc}. Assume from now on that ${\\mathbb Q}\\subset B$. Let $\\psi\\in B[[x]]$ be a formal power series with \n$\\psi^{0}=1$. We denote also by $\\psi$ the associated multiplicative\ngenus. \nThen\n\\begin{displaymath}\n\t\\varphi=\\log(\\psi)\\in B[[x]]\n\\end{displaymath}\ndefines an additive genus, to which we can associate an addtive\narithmetic characteristic genus $\\widehat{\\varphi}$. Then, given an\nobject $\\overline{\\mathcal{F}}^{\\ast}$ in $\\oDb(\\mathcal{X})$, we have a\nwell defined class \n\\begin{displaymath}\n \\widehat{\\psi}_{m}(\\overline{\\mathcal{F}}^{\\ast}):=\\exp(\\widehat{\\varphi}(\\overline{\\mathcal{F}}^{\\ast}))\n \\in\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)).\n\\end{displaymath}\nObserve this construction uses the ring structure of\n$\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$, hence\nthe regularity of $\\mathcal{X}$ and the product structure of\n$\\mathcal{D}_{{\\text{\\rm a}}}(X)$. In case $\\psi$ has rational coefficients,\n$\\widehat{\\psi}(\\overline{\\mathcal{F}}^{\\ast})$ takes values in \n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$. We\nalso recall that there is a multiplicative secondary class associated\nto $\\psi$, denoted $\\widetilde{\\psi}_{m}$, that can be expressed in\nterms of $\\widetilde{\\varphi}$:\n\\begin{displaymath}\n \\widetilde{\\psi}_{m}(\\theta)=\\frac{\\exp(\\varphi(\\theta))-1}{\\varphi(\\theta)}\\widetilde{\\varphi}(\\theta),\n\\end{displaymath}\nwhere $\\theta$ is any class in $\\overline{\\KA}(X)$. If $\\overline{\\tau}$ is a distinguished triangle in $\\oDb(\\mathcal{X})$, we will simply write $\\widetilde{\\psi}_{m}(\\overline{\\tau})$ instead of $\\widetilde{\\psi}_{m}([\\overline{\\tau}])$.\n\n\\begin{theorem}\\label{thm:4}\nLet $\\psi$ be a multiplicative genus, with degree 0 component\n$\\psi^{0}=1$. Then, for every distinguished triangle in\n$\\oDb(\\mathcal{X})$ \n\\begin{displaymath}\n\t\\overline{\\tau}\\colon\t\\overline{\\mathcal{F}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{G}}^{\\ast}\\dashrightarrow\\overline{\\mathcal{H}}^{\\ast}\n\t\\dashrightarrow\\overline{\\mathcal{F}}[1]\n\\end{displaymath}\nwe have the relation\n\\begin{displaymath}\n\t\\widehat{\\psi}(\\overline{\\mathcal{F}}^{\\ast})^{-1}\\widehat{\\psi}(\\overline{\\mathcal{G}})\\widehat{\\psi}(\\overline{\\mathcal{H}})^{-1}-1=\n\t\\amap(\\widetilde{\\psi}_{m}(\\overline{\\tau})).\n\\end{displaymath}\nIn particular, if $\\overline{\\tau}$ is tightly distinguished, the equality\n\\begin{displaymath}\n\t\\widehat{\\psi}(\\overline{\\mathcal{G}}^{\\ast})=\\widehat{\\psi}(\\overline{\\mathcal{F}}^{\\ast})\\widehat{\\psi}(\\overline{\\mathcal{H}}^{\\ast})\n\\end{displaymath}\nholds.\n\\end{theorem}\n\\begin{proof}\nIt is enough to exponentiate the relation provided by Theorem \\ref{thm:3} and observe that, due to the mutliplicative law in $\\bigoplus_{p}\\cha^{p}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$, one has\n\\begin{displaymath}\n\t\\exp(\\amap(\\widetilde{\\varphi}(\\overline{\\tau})))=1+\\amap\\left(\\frac{\\exp(\\varphi(\\overline{\\tau}))-1}{\\exp(\\varphi(\\overline{\\tau}))}\\widetilde{\\varphi}(\\overline{\\tau})\\right).\n\\end{displaymath}\n\\end{proof}\n\\begin{example}\n\\begin{enumerate}\n\\item The arithmetic Chern class, is the additive class\n attached to the additive genus $\\ch(x)=e^{x}$. It coincides\n with the character $\\chh$ of the previous section. \n\\item The arithmetic Todd class, is the multiplicative class attached to the genus\n \\begin{displaymath}\n \\Td(x)=\\frac{x}{1-e^{-x}}.\n \\end{displaymath}\n Observe that the formal series of $\\Td(x)$ has constant coefficient 1.\n\\end{enumerate}\n\\end{example}\n\\begin{remark}\nIf $C$ is a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, we can define arithmetic\ncharacteristic classes with values in\n$\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},C)$, just taking their image by the\nnatural morphism\n$\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))\\rightarrow\n\\bigoplus_{p}\\cha^{p}_{B}(\\mathcal{X},C)$. We will use the same\nnotations to refer to these classes.\n\\end{remark}\n\n\n\\section{Direct images and generalized analytic torsion}\\label{section:DirectImage}\nIn the preceding section we introduced the arithmetic $K$-groups and\nthe arithmetic derived categories. They satisfy elementary\nfunctoriality properties and are related by a class functor. A missing\nfunctoriality is the push-forward by \\emph{arbitrary} projectives\nmorphisms of arithmetic varieties. Similarly to the work of\nGillet-R\\\"ossler-Soul\\'e\n\\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees},\nwe will define direct images after choosing a generalized analytic\ntorsion theory in the sense of\n\\cite{BurgosFreixasLitcanu:GenAnTor}. Our theory is more general in\nthat we don't require our morphisms to be smooth over the generic\nfiber. At the archimedean places, we are thus forced to work with\ncomplexes of currents with controlled wave front sets. In this level\nof generality, the theory of arithmetic Chow groups, arithmetic\n$K$-theory and arithmetic derived categories has already been\ndiscussed. Let us recall that a generalized analytic torsion theory is\nnot unique, but according to \\cite[Thm. 7.7 and\nThm. 7.14]{BurgosFreixasLitcanu:GenAnTor} it is classified by a real\nadditive genus.\n\nOur theory of generalized analytic torsion classes involves the notion of relative metrized complex \\cite[Def. 2.5]{BurgosFreixasLitcanu:GenAnTor}. In the sequel we will need a variant on real smooth quasi-projective schemes $X=(X_{{\\mathbb C}},F_{\\infty})$. With respect to \\emph{loc. cit.}, this amounts to imposing an additional invariance under the action of $F_{\\infty}$.\n\n\\begin{definition}\nA real relative metrized complex is a triple $\\overline{\\xi}=(\\overline{f}, \\overline{\\mathcal{F}}^{\\ast},\\overline{f_{\\ast}\\mathcal{F}^{\\ast}})$, where\n\\begin{itemize}\n\t\\item $\\overline{f}:X\\rightarrow Y$ is a projective morphism of real smooth quasi-projective varieties, together with a hermitian structure on the tangent complex $T_{f}$, invariant under the action of complex conjugation;\n\t\\item $\\overline{\\mathcal{F}}^{\\ast}$ is an object in $\\oDb(X)$;\n\t\\item $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ is an object in $\\oDb(Y)$ lying over $f_{\\ast}\\mathcal{F}^{\\ast}$.\n\\end{itemize}\n\\end{definition}\nThe following lemma is checked by a careful proof reading of the construction of generalized analytic torsion classes in \\cite{BurgosFreixasLitcanu:GenAnTor}.\n\\begin{lemma}\nLet $T$ be a theory of generalized analytic torsion classes. Then, for every real relative metrized complex $\\overline{\\xi}=(\\overline{f}\\colon X\\to Y,\\overline{\\mathcal{F}}^{\\ast},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast})$, $T(\\overline{\\xi})$ is a class of real currents,\n\\begin{displaymath}\n\tT(\\overline{\\xi})\\in\\bigoplus_{p}\\widetilde{\\mathcal{D}}^{2p-1}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,N_{f},p),\n\\end{displaymath}\nwhere $N_{f}$ is the cone of normal directions to $f$.\n\\end{lemma}\nIt will be useful to have an adaptation of the category\n$\\overline{\\Sm}_{\\ast\/{\\mathbb C}}$ \\cite[Sec. 5]{BurgosFreixasLitcanu:HerStruc} to\nreal quasi-projective schemes, that we denote\n$\\overline{\\Sm}_{\\ast\/{\\mathbb R}}$. In this category, the objects are real smooth\nquasi-projective schemes, and the morphisms are projective morphisms\nwith a hermitian structure invariant under complex conjugation. The\ncomposition law is then described in \\emph{loc. cit.}, Def. 5.7, with\nthe help of the hermitian cone construction. We will follow the\nnotation introduced in\n\\cite[Def. 2.12]{BurgosFreixasLitcanu:GenAnTor}. \n\\begin{notation}\n Let $\\overline{f}\\colon X\\to Y$ be in $\\overline{\\Sm}_{\\ast\/{\\mathbb R}}$, of pure\n relative dimension $e$, $C$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, $C'$ a\n $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$-complex and $f_{\\ast}:C\\rightarrow C'$ a\n morphism of fitting into the commutative diagram\n \\eqref{eq:18}. Assume furthermore that $C$ is a\n $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-module with a product law $\\bullet$ as in\n \\eqref{eq:19}. Then we put\n\\begin{displaymath}\n \\begin{split}\n \\overline{f}_{\\flat}:C^{\\ast}(\\ast)&\\longrightarrow C^{\\prime\\ast-2e}(p-e)\\\\\n \\eta&\\longmapsto f_{\\ast}(\\eta\\bullet \\Td(T_{\\overline{f}})).\n \\end{split}\n\\end{displaymath}\nThis morphism induces a corresponding morphism on\n$\\widetilde{C}^{\\ast}(\\ast)$, for which we use the same notation. \n\\end{notation}\nFor an arithmetic ring $A$, we introduce $\\overline{\\Reg}_{\\ast\/A}$ the\ncategory of quasi-proj\\-ective regular arithmetic varieties\nover $A$, with projective morphisms endowed (at the archimedean\nplaces) with a hermitian structure invariant under complex\nconjugation. By construction, there is a natural base change functor \n\\begin{displaymath}\n\t\\overline{\\Reg}_{\\ast\/A}\\longrightarrow\\overline{\\Sm}_{\\ast\/{\\mathbb R}}.\n\\end{displaymath}\nGiven $\\overline{f}\\colon\\mathcal{X}\\to\\mathcal{Y}$ a morphism in $\\overline{\\Reg}_{\\ast\/A}$ and objects $\\overline{\\mathcal{F}}^{\\ast}$, $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ in $\\oDb(\\mathcal{X})$ and $\\oDb(\\mathcal{Y})$, respectively, we may consider the corresponding real relative metrized complex, that we will abusively write $\\overline{\\xi}=(\\overline{f},\\overline{\\mathcal{F}},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast})$, and its analytic torsion class $T(\\overline{\\xi})$. We may also write $\\overline{f}_{\\flat}$ instead of $\\overline{f}_{{\\mathbb C}\\,\\flat}$, etc. \n\nWe are now in position to construct the arithmetic counterpart of \\cite[Eq. (10.6)]{BurgosFreixasLitcanu:GenAnTor}, namely the direct image functor on arithmetic derived categories, as well as a similar push-forward on arithmetic $K$-theory.\n\\begin{definition}\nLet $\\overline{f}\\colon\\mathcal{X}\\to\\mathcal{Y}$ be a morphism in\n$\\overline{\\Reg}_{\\ast\/A}$, $C$ a $\\mathcal{D}_{{\\text{\\rm a}}}(X)$-complex, $C'$ a\n $\\mathcal{D}_{{\\text{\\rm a}}}(Y)$-complex, both satisfying the\nhypothesis (H1) and (H2), and $f_{\\ast}:C\\rightarrow C'$ a\n morphism fitting into a commutative diagram like\n \\eqref{eq:18}.\n \\begin{enumerate}\n \\item We define the functor\n\\begin{displaymath}\n\t\\overline{f}_{\\ast}\\colon\\hDb(\\mathcal{X},C)\\longrightarrow\\hDb(\\mathcal{Y},C'),\n\\end{displaymath}\nacting on objects by the assignment\n\\begin{displaymath}\n\t[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]\\longmapsto [\\overline{f_{\\ast}\\mathcal{F}}^{\\ast},\\overline{f}_{\\flat}(\\widetilde{\\eta})-c'(T(\\overline{f},\\overline{\\mathcal{F}}^{\\ast},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}))].\n\\end{displaymath}\nHere $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ carries an arbitrary choice of hermitian structure. The action on morphisms of $\\overline{f}_{\\ast}$ is just the usual action $f_{\\ast}$ on morphisms of $\\Db(\\mathcal{X})$.\n\n\\item We define a morphism of groups\n\\begin{displaymath}\n\t\\begin{split}\n\t\tf_{\\ast}:\\widehat{K}_{0}(\\mathcal{X},C)&\\longrightarrow\\widehat{K}_{0}(\\mathcal{X},C')\\\\\n\t\t[\\overline{\\mathcal{E}},\\widetilde{\\eta}]&\\longmapsto [\\sum_{i}(-1)^{i}\\overline{\\mathcal{E}}^{\\prime i},\\overline{f}_{\\flat}(\\widetilde{\\eta})-T(\\overline{f},\\overline{\\mathcal{E}},\\overline{f_{\\ast}\\mathcal{E}}^{\\ast})],\n\t\\end{split}\n\\end{displaymath}\nwhere we choose an arbitrary quasi-isomorphism\n$\\mathcal{E}^{\\prime\\ast}\\dashrightarrow f_{\\ast}\\mathcal{E}$ and\narbitrary smooth hermitian metrics on the $\\mathcal{E}^{\\prime\n i}$. Here $f_{\\ast}\\mathcal{E}$ denotes the derived direct image of\nthe single locally free sheaf $\\mathcal{E}$.\n\n \\end{enumerate}\n\\end{definition}\nNotice the previous definition makes sense by the anomaly formulas satisfied by analytic torsion theories \\cite[Prop. 7.4]{BurgosFreixasLitcanu:GenAnTor}. \nBoth push-forwards are compatible through the class map from arithmetic derived categories to arithmetic $K$-theory.\n\\begin{theorem}\\label{thm:6}\nThere is a commutative diagram of functors\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\hDb(\\mathcal{X},C)\\ar[r]^-{\\overline{f}_{\\ast}}\\ar[d]\t&\\hDb(\\mathcal{Y},C')\\ar[d]\\\\\n\t\t\\widehat{K}_{0}(\\mathcal{X},C)\\ar[r]\t_-{\\overline{f}_{\\ast}}\t&\\widehat{K}_{0}(\\mathcal{Y},C').\n\t}\n\\end{displaymath}\nThis is in particular true for $C=\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,S)$ and $C'=\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(S))$, where $S\\subset T_{0}^{\\ast}X$ is a closed conical subset invariant under the action of complex conjugation.\n\\end{theorem}\n\\begin{proof}\nLet $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$ be an object in\n$\\hDb(\\mathcal{X},C)$. By Remark \\ref{rem:1}, we can suppose the\nhermitian structure on $\\overline{\\mathcal{F}}^{\\ast}$ is given by a\nquasi-isomorphism\n$\\overline{\\mathcal{E}}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, where\n$\\overline{\\mathcal{E}}^{\\ast}$ is a finite complex of locally free sheaves,\neach one endowed with a smooth hermitian metric. For every $i$, we\nhave to endow $f_{\\ast}\\mathcal{E}^{i}$ with a hermitian structure\ngiven by a finite locally free resolution\n$\\overline{\\mathcal{E}}^{i\\ast}\\dashrightarrow f_{\\ast}\\mathcal{E}^{i}$ and\na choice of arbitrary smooth hermitian metric on every piece\n$\\mathcal{E}^{ij}$. From the data\n$\\overline{\\mathcal{E}}^{i\\ast}\\dashrightarrow f_{\\ast}\\mathcal{E}^{i}$,\nevery $i$, the procedure of\n\\cite[Def. 3.39]{BurgosFreixasLitcanu:HerStruc} produces a hermitian\nstructure on $f_{\\ast}\\mathcal{E}^{\\ast}$, via the hermitian cone\nconstruction. Observe the construction of \\emph{loc. cit.} can be done\nin $\\oDb(\\mathcal{Y})$. Combined with the fixed quasi-isomorphism\n$\\mathcal{E}^{\\ast}\\dashrightarrow\\mathcal{F}^{\\ast}$, we thus obtain\na hermitian structure on $f_{\\ast}\\mathcal{F}^{\\ast}$, that we denote\nby $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$. \n\nThe class of $[\\overline{\\mathcal{F}},\\widetilde{\\eta}]$ in $\\hDb(\\mathcal{X},C)$ is thus\n\\begin{displaymath}\n\t\\sum_{i}(-1)^{i}[\\overline{\\mathcal{E}}^{i},0] + [0,\\widetilde{\\eta}].\n\\end{displaymath}\nIts image under $\\overline{f}_{\\ast}$ is\n\\begin{equation}\\label{eq:20}\n\t\\sum_{i}(-1)^{i}\\left((\\sum_{j}(-1)^{j}[\\overline{\\mathcal{E}}^{ij},0])-[0,c'(T(\\overline{f},\\overline{\\mathcal{E}}^{i}, \\overline{f_{\\ast}\\mathcal{E}}^{i}))]\\right) + [0,\\overline{f}_{\\flat}(\\widetilde{\\eta})].\n\\end{equation}\nThe class of $\\overline{f}_{\\ast}[\\overline{\\mathcal{F}},\\widetilde{\\eta}]$ in $\\hDb(\\mathcal{Y},C')$ is\n\\begin{equation}\\label{eq:21}\n\t[\\overline{f_{\\ast}\\mathcal{F}}^{\\ast},\\overline{f}_{\\flat}(\\widetilde{\\eta})-c'(T(\\overline{f},\\overline{\\mathcal{F}},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}))].\n\\end{equation}\nBy the choice of the hermitian structures on $\\overline{\\mathcal{F}}^{\\ast}$ and $\\overline{f_{\\ast}\\mathcal{F}}^{\\ast}$ and by Theorem \\cite[Prop. 7.6]{BurgosFreixasLitcanu:GenAnTor}, we have\n\\begin{displaymath}\n\tT(\\overline{f},\\overline{\\mathcal{F}},\\overline{f_{\\ast}\\mathcal{F}}^{\\ast})=\\sum_{i}(-1)^{i} T(\\overline{f},\\overline{\\mathcal{E}}^{i},\\overline{f_{\\ast}\\mathcal{E}}^{i}).\n\\end{displaymath}\nTherefore the class in arithmetic $K$-theory of \\eqref{eq:21} equals \\eqref{eq:20}.\n\\end{proof}\nThere are several compatibilities between direct images, inverse images and derived tensor product. We now state them without proof, referring the reader to \\cite[Thm. 10.7]{BurgosFreixasLitcanu:GenAnTor} for the details.\n\\begin{proposition}\\label{prop:5}\n Let $\\overline{f}\\colon\\mathcal{X}\\to\\mathcal{Y}$ and $\\overline{g}\\colon\\mathcal{Y}\\to \\mathcal{Z}$ be morphisms in $\\overline{\\Reg}_{\\ast\/A}$. Let $S\\subset T^{\\ast}X_{0}$ and $T\\subset T^{\\ast}Y_{0}$ be closed conical subsets.\n\\begin{enumerate}\n\t\\item (Functoriality of push-forward) We have the relation\n \t\t\\begin{displaymath}\n \t\t\t(\\overline{g}\\circ \\overline{f})_{\\ast}=\\overline{g}_{\\ast}\\circ\\overline{f}_{\\ast},\n \t\t\\end{displaymath}\n\t\tas functors $\\hDb(\\mathcal{X},S)\\rightarrow\\hDb(\\mathcal{Z},g_{\\ast}f_{\\ast}S)$.\n \\item (Projection formula) Assume that $T\\cap N_{f}=\\emptyset$ and that $T+f_{\\ast}S$ does not cross the zero section of $T^{\\ast}Y$. Let $[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$ be in $\\hDb(\\mathcal{X},S)$ and $[\\overline{\\mathcal{F}}^{\\prime\\ast},\\widetilde{\\eta}^{\\prime}]$ in $\\hDb(\\mathcal{Y},T)$. Then\n\t\\begin{displaymath}\n \t\t\\overline{f}_{\\ast}([\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}] \\otimes\n \t\tf^{\\ast}[\\overline{\\mathcal{F}}^{\\prime\\ast},\\widetilde{\\eta}^{\\prime}])\n \t\t= \\overline{f}_{\\ast}[\\overline{\\mathcal{F}},\\widetilde{\\eta}]\\otimes [\\overline{\\mathcal{F}}^{\\prime},\\widetilde{\\eta}^{\\prime}]\n \\end{displaymath}\n in $\\hDb(\\mathcal{Y},W)$, where\n \\begin{displaymath}\n \tW=f_{\\ast}(S + f^{\\ast}T)\\cup f_{\\ast}S\\cup f_{\\ast}f^{\\ast}T.\n \\end{displaymath}\n\t\\item There are analogous relations on the level of arithmetic $K$-theory, compatible with the class functor from arithmetic derived categories.\n\\end{enumerate}\n\\end{proposition}\n\\begin{remark}\nStrictly speaking, the equalities provided by the previous statement should be canonical isomorphisms, but as usual we abuse the notations and pretend they are equalities.\n\\end{remark}\n\\section{Arithmetic Grothendieck-Riemann-Roch}\\label{section:ARR}\n\\subsection{Statement and reductions}\n\\paragraph{Hermitian tangent complexes.} Let $\\overline{f}\\colon\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism in $\\overline{\\Reg}_{\\ast\/A}$. We explain how to construct the associated hermitian tangent complex $T_{\\overline{f}}$. This is an object in $\\oDb(\\mathcal{X})$, well defined up to tight isomorphism. \n\nBecause $\\mathcal{X}$, $\\mathcal{Y}$ are regular schemes and $f$ is projective, it is automatically a l.c.i. morphism. The tangent complex of $f$ is an object in $\\Db(\\mathcal{X})$, well defined up to isomorphism. Consider a factorization\n\\begin{displaymath}\n\t\\xymatrix{\n\t\t\\mathcal{X}\\ar@{^{(}->}[r]^-{i}\\ar[rd]\t&\\mathcal{Z}\\ar[d]^{\\pi}\\\\\n\t\t&\\mathcal{Y},\n\t}\n\\end{displaymath}\nwith $i$ being a closed regular immersion and $\\pi$ a smooth morphism. For instance, one may choose $\\mathcal{Z}={\\mathbb P}^{n}_{\\mathcal{Y}}$, for some $n$. Let us denote by $\\mathcal{I}$ the ideal defining the closed immersion $i$. Then $\\mathcal{I}\/\\mathcal{I}^{2}$ is a locally free sheaf on $\\mathcal{X}$ and, as customary, we define the normal bundle $N_{\\mathcal{X}\/\\mathcal{Z}}=(\\mathcal{I}\/\\mathcal{I}^{2})^{\\vee}$. There is a morphism of coherent sheaves\n\\begin{displaymath}\n\t\\varphi\\colon i^{\\ast}T_{\\mathcal{Z}\/\\mathcal{Y}}\\longrightarrow N_{\\mathcal{X}\/\\mathcal{Z}},\n\\end{displaymath}\nnamely the dual of the differential map $d:\\mathcal{I}\/\\mathcal{I}^{2}\\to i^{\\ast}\\Omega_{\\mathcal{Z}\/\\mathcal{Y}}$. We consider $T_{\\mathcal{Z}\/\\mathcal{Y}}$ as a complex concentrated in degree 0, and $N_{\\mathcal{X}\/\\mathcal{Z}}$ as a complex concentrated in degree one. We then put\n\\begin{displaymath}\n\tT_{f}:=\\cone(\\varphi)[-1].\n\\end{displaymath}\nThe isomorphism class of $T_{f}$ in $\\Db(\\mathcal{X})$ is independent of the factorization. \n\nThe base change to ${\\mathbb C}$ of $T_{f}$ is naturally isomorphic to the tangent complex $T_{f_{{\\mathbb C}}}:TX_{{\\mathbb C}}\\rightarrow f_{{\\mathbb C}}^{\\ast}Y_{{\\mathbb C}}$, which is equipped with a hermitian structure by assumption. Therefore, the data provided by the constructed complex $T_{f}$ and the hermitian structure on $\\overline{f}$ determine an object $T_{\\overline{f}}$ in $\\oDb(\\mathcal{X})$, which is well defined up to tight isomorphism. By Theorem \\ref{thm:4}, the arithmetic Todd class of $T_{\\overline{f}}$ is unambiguously defined.\n\\begin{definition}\nThe arithmetic Todd class of $\\overline{f}$ is\n\\begin{displaymath}\n \\widehat{\\Td}(\\overline{f}):=\\widehat{\\Td}(T_{\\overline{f}})\\in\n \\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X)). \n\\end{displaymath}\n\\end{definition}\n\\begin{theorem}\nLet $\\overline{f}\\colon\\mathcal{X}\\rightarrow\\mathcal{Y}$,\n$\\overline{g}\\colon\\mathcal{Y}\\rightarrow\\mathcal{Z}$ be morphisms in\n$\\overline{\\Reg}_{\\ast\/A}$. Then we have an equality \n\\begin{displaymath}\n \\widehat{\\Td}(\\overline{g}\\circ\\overline{f})=f^{\\ast}\\widehat{\\Td}(\\overline{g})\\cdot\\widehat{\\Td}(\\overline{f})\n\\end{displaymath}\nin $\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{X},\\mathcal{D}_{{\\text{\\rm a}}}(X))$.\n\\end{theorem}\n\\begin{proof}\nBy construction of $T_{\\overline{f}}$ and definition of the composition rule of morphisms in $\\overline{\\Reg}_{\\ast\/A}$, there a is tightly distinguished triangle in $\\oDb(\\mathcal{X})$\n \\begin{displaymath}\n\tT_{\\overline{f}}\\dashrightarrow T_{\\overline{g}\\circ \\overline{f}}\\dashrightarrow f^{\\ast}T_{\\overline{g}}\\dashrightarrow T_{\\overline{f}}[1].\n\\end{displaymath}\nWe conclude by an application of Theorem \\ref{thm:4}.\n\\end{proof}\n\\paragraph{Statement.} The arithmetic Grothendieck-Riemann-Roch theorem describes the behavior of the arithmetic Chern character with respect to the push-forward functor. Recall that the definition of the push-forward functor depends on the choice of a theory of generalized analytic torsion classes. In its turn, such a theory corresponds to a real additive genus. \n\\begin{theorem}\\label{thm:5}\nLet $\\overline{f}\\colon\\mathcal{X}\\rightarrow\\mathcal{Y}$ be a morphism in $\\overline{\\Reg}_{\\ast\/A}$. Fix a closed conical subset $W$ of $T_{0}^{\\ast}X$ and a theory of generalized analytic torsion classes $T$, whose associated real additive genus is $S$. Then, the derived direct image functor $$\\overline{f}_{\\ast}\\colon\\hDb(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,W))\\longrightarrow\\hDb(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(W)))$$ attached to $T$ satisfies the equation\n\\begin{equation}\\label{eq:29}\n\t\\chh(\\overline{f}_{\\ast}\\alpha)=f_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{f}))-\\amap(f_{\\ast}(\\ch(\\mathcal{F}^{\\ast}_{{\\mathbb C}})\\Td(T_{f_{{\\mathbb C}}})S(T_{f_{{\\mathbb C}}})),\n\\end{equation}\nfor every object\n$\\alpha=[\\overline{\\mathcal{F}}^{\\ast},\\widetilde{\\eta}]$. The equality\ntakes place in the arithmetic Chow group\n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(W)))$.\n\\end{theorem}\nRecall that the genus $S$=0 corresponds to the homogeneous theory\n$T^{h}$, which is characterized by satisfying an additional\nhomogeneity property \\cite[Sec. 9]{BurgosLitcanu:SingularBC}. Roughly\nspeaking, this condition is exactly the one guaranteeing an\narithmetic Grothendieck-Riemann-Roch theorem without\ncorrection term.\n\\begin{corollary}\nThe direct image functor $\\overline{f}^{h}_{\\ast}$ attached to the\nhomogeneous generalized analytic torsion theory $T^{h}$ satisfies an\nexact Grothendieck-Riemann-Roch type formula:\n\\begin{displaymath}\n \\chh(\\overline{f}^{h}_{\\ast}\\alpha)=f_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{f})).\n\\end{displaymath}\n\\end{corollary}\nA Grothendieck-Riemann-Roch type theorem in Arakelov geometry was\nfirst proven by Gillet-Soul\\'e \\cite{GilletSoule:aRRt}, for the degree\n1 part of the Chern character (namely the determinan of the\ncohomology) and under the restriction on the morphism $f$ to be smooth\nover ${\\mathbb C}$. Also they can only deal with hermitian vector\nbundles. They used the holomorphic analytic torsion\n\\cite{BismutGilletSoule:at}, \\cite{BismutGilletSoule:atII},\n\\cite{BismutGilletSoule:atIII} and deep results of Bismut-Lebeau\n\\cite{BismutLebeau:CiQm} on the compatibility of analytic torsion with\nclosed immersions. The holomorphic analytic torsion was later\ngeneralized by Bismut-K\\\"ohler \\cite{BismutKohler}, to the holomorphic\nanalytic torsion forms, that transgress the whole\nGrothendieck-Riemann-Roch theorem for K\\\"ahler submersions, at the\nlevel of differential forms. The extension of the arithmetic\nGrothendieck-Riemann-Roch theorem to the full Chern character and\ngenerically smooth morphisms was finally proven by\nGillet-R\\\"ossler-Soul\\'e\n\\cite{GilletRoesslerSoule:_arith_rieman_roch_theor_in_higher_degrees}. They\napplied the analogue to the Bismut-Lebeau immersion theorem, for\nanalytic torsion forms, established in the monograph by Bismut\n\\cite{Bismut:Asterisque}.\n\nTheorem \\ref{thm:5} provides an extension of the previous results in\nseveral directions. First, we allow the morphism to be an arbitrary\nprojective morphism, non necessarily generically smooth. Second, we\ncan deal with metrized objects in the bounded derived category of\ncoherent sheaves. Third, we provide all the possible forms of such a\ntheorem, by introducing our theory of generalized analytic torsion\nclasses, thus explaining the topological correction term. \n\n\\paragraph{Reductions.} The proof of our version of the arithmetic\nGrothendieck-Riemann-Roch theorem follows the pattern of the classical\napproach in algebraic geometry. Namely, it proceeds by factorization\nof the morphism $\\overline{f}$ into a regular closed immersion and a trivial\nprojective bundle projection. The advantage of working with the\nformalism of hermitian structures on the derived category of coherent\nsheaves makes the whole procedure more transparent and direct, in\nparticular avoiding the appearance of several secondary classes. Also\nthe cocyle type relation expressing the behaviour of generalized\nanalytic torsion with respect to composition of morphisms in\n$\\overline{\\Sm}_{\\ast\/{\\mathbb R}}$ (see the axioms\n\\cite[Sec. 7]{BurgosFreixasLitcanu:GenAnTor}) is well suited to this\nfactorization argument. \n\nBy the classification of generalized analytic torsion theories, it is\nenough to prove Theorem \\ref{thm:5} for the homogenous theory\n$T^{h}$. From now on we fix this choice, and hence all derived direct\nimage functors will be with respect to this theory. \n\nLet $\\overline{f}$ be a morphism in $\\overline{\\Reg}_{\\ast\/A}$, and consider a factorization\n\\begin{displaymath}\n \\xymatrix{\n \\mathcal{X}\\ar@{^{(}->}[r]^{i}\\ar[rd]^{f}\n &\\mathcal{{\\mathbb P}}^{n}_{\\mathcal{Y}}\\ar[d]^{\\pi}\\\\\n &\\mathcal{Y}.\n }\n\\end{displaymath}\nThe tangent complex of $\\pi$ is canonically isomorphic to $p^{\\prime\\ast}T_{{\\mathbb P}^{n}_{A}\/A}$, where \n\\begin{displaymath}\n\t\\xymatrix{\n\t\t{\\mathbb P}^{n}_{\\mathcal{Y}}\\ar[r]^-{p^{\\prime}}\\ar[d]_{\\pi}\t&{\\mathbb P}^{n}_{A}\\ar[d]^{\\pi^{\\prime}}\\\\\n\t\t\\mathcal{Y}\\ar[r]_-{p}\t&\\Spec A.\n\t}\n\\end{displaymath}\nWe may thus endow $\\pi$ with the pull-back by $p^{\\prime}$ of the Fubini-Study metric on $T_{{\\mathbb P}^{n}_{A}\/A}$ \\cite[Sec. 5]{BurgosFreixasLitcanu:GenAnTor}. We write $\\overline{\\pi}$ for the resulting morphism in $\\overline{\\Reg}_{\\ast\/A}$. By \\cite[Lemma 5.3]{BurgosFreixasLitcanu:HerStruc}, there exists a unique hermitian structure on $i$ such that $\\overline{f}=\\overline{\\pi}\\circ\\overline{i}$. Then we recall that we have the equality of functors\n\\begin{displaymath}\n\tf_{\\ast}=\\overline{\\pi}_{\\ast}\\circ\\overline{i}_{\\ast}\n\\end{displaymath}\nand the equality of arithmetic Todd genera\n\\begin{displaymath}\n\t\\widehat{\\Td}(\\overline{f})=i^{\\ast}\\widehat{\\Td}(\\overline{\\pi})\\widehat{\\Td}(\\overline{i}).\n\\end{displaymath}\n\\begin{lemma}\nIt is enough to prove Theorem \\ref{thm:5} for $\\overline{i}$ and $\\overline{\\pi}$ individually.\n\\end{lemma}\n\\begin{proof}\nLet us assume the theorem known for $\\overline{i}$ and for $\\overline{\\pi}$. Because the theory is known for $\\overline{\\pi}$, we may apply it to the object $\\overline{i}_{\\ast}\\alpha$ in $\\oDb({\\mathbb P}^{n}_{\\mathcal{Y}})$, to obtain\n\\begin{equation}\\label{eq:30}\n\t\\chh(\\overline{f}_{\\ast}\\alpha)=\\chh(\\overline{\\pi}_{\\ast}(\\overline{i}_{\\ast}\\alpha))=\n\t\t\\pi_{\\ast}(\\chh(\\overline{i}_{\\ast}\\alpha)\\widehat{\\Td}(\\overline{\\pi})).\n\\end{equation}\nBecause the theorem is known for $\\overline{i}$, we also have\n\\begin{equation}\\label{eq:31}\n\t\\begin{split}\n\t\t\\chh(\\overline{i}_{\\ast}\\alpha)\\widehat{\\Td}(\\overline{\\pi})=&i_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{i}))\\widehat{\\Td}(\\overline{\\pi})\\\\\n\t\t&=i_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{i})i^{\\ast}\\widehat{\\Td}(\\overline{\\pi}))\\\\\n\t\t&\\hspace{0.3cm}=i_{\\ast}(\\chh(\\alpha)\\widehat{\\Td}(\\overline{f})).\n\t\\end{split}\n\\end{equation}\nObserve we used the projection formula in arithmetic Chow groups\n(Theorem \\ref{thm:2}), and that this equality holds in\n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(Y,f_{\\ast}(W)))$,\nbecause the definition of direct image of wave front sets and the fact\nthat $\\pi$ is smooth yield $f_{\\ast}(W)=\\pi_{\\ast}(i_{\\ast}(W))$. We\nconclude by putting \\eqref{eq:30}--\\eqref{eq:31} together and using\nthe fact $f_{\\ast}=\\pi_{\\ast} i_{\\ast}$ on arithmetic Chow groups\n(Proposition \\ref{prop:8}).\n\\end{proof}\n\\begin{lemma} \\label{lemm:4}\nTheorem \\ref{thm:5} holds for $\\overline{i}$.\n\\end{lemma}\n\\begin{proof}\nIn \\cite[Thm 10.28]{BurgosLitcanu:SingularBC}, the authors prove the\ntheorem for direct images by closed immersions, defined on arithmetic\n$K$ groups. They suppose as well that the hermitian structure on\n$T_{i}$ is given by a smooth hermitian metric on\n$N_{\\mathcal{X}\/{\\mathbb P}^{n}_{\\mathcal{Y}}}$. Finally, the result in\n\\emph{loc,. cit.} holds in\n$\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}(\\mathcal{Y},\\mathcal{D}_{\\text{{\\rm cur}}, {\\text{\\rm a}}}(Y))$. \n\nBy the anomaly formula provided by Theorem \\ref{thm:4} and the anomaly\nformula of generalized analytic torsion theories\n\\cite[Prop. 7.4]{BurgosFreixasLitcanu:GenAnTor} for change of\nhermitian structure on the tangent complex, one can extend \\cite[Thm\n10.28]{BurgosLitcanu:SingularBC} to arbitrary hermitian structures on\n$i$, in particular the one we fixed. Also, a careful proof reading of\nthe proof in \\emph{loc. cit.} shows that the theorem can be adapted to\nallow\n$\\alpha\\in\\widehat{K}_{0}(\\mathcal{X},\\mathcal{D}_{\\text{{\\rm cur}},{\\text{\\rm a}}}(X,W))$,\ntaking values in $\\bigoplus_{p}\\cha^{p}_{{\\mathbb Q}}({\\mathbb P}^{n}_{Y},\\mathcal{D}_{\\text{{\\rm cur}},\n {\\text{\\rm a}}}({\\mathbb P}^{n}_{Y},i_{\\ast}W))$. Finally, to get the result for\n$\\overline{i}_{\\ast}$ on the arithmetic derived category, we use the class\nfunctor to arithmetic $K$-theory, the commutativity Theorem\n\\ref{thm:6} and that the Chern character on arithmetic derived\ncategory factors, by construction, through arithmetic $K$ groups (see\nNotation \\ref{not:1}).\n\\end{proof}\n\n\\begin{lemma}\nTo prove Theorem \\ref{thm:5} for $\\overline{\\pi}$, it is enough to prove it\nwhen $\\mathcal{Y}=\\Spec A$ and $\\alpha=\\overline{{\\mathcal O}(k)}$, $-n\\leq k\\leq 0$,\nwhere the chosen hermitian structure on ${\\mathcal O}(k)$ is the Fubini-Study\nmetric. \n\\end{lemma}\n\\begin{proof}\nThe derived category $\\Db({\\mathbb P}^{n}_{\\mathcal{Y}})$ is generated by\ncoherent sheaves of the form $\\pi^{\\ast}\\mathcal{G}\\otimes\np^{\\prime\\ast}{\\mathcal O}(k)$\n\\cite[Cor. 4.11]{BurgosFreixasLitcanu:GenAnTor}. By the anomaly\nformulas \\cite[Prop. 7.4, Prop. 7.6]{BurgosFreixasLitcanu:GenAnTor},\nwe thus reduce to prove the theorem for $\\alpha$ of the form\n$\\pi^{\\ast}\\overline{\\mathcal{G}}\\otimes p^{\\prime\\ast}\\overline{{\\mathcal O}(k)}$, for any\nmetric on $\\mathcal{G}$. On the one hand, by the formulas in Prop\n\\ref{prop:5}, the multiplicativity and pull-back functoriality\n(Prop. \\ref{prop:6}) of the Chern character, we have \n\\begin{displaymath}\n \\begin{split}\n \\chh(\\overline{\\pi}_{\\ast}\\alpha)=&\\chh(\\overline{\\mathcal{G}}\\otimes\\overline{\\pi}_{\\ast}p^{\\prime\\ast}\\overline{{\\mathcal O}(k)})\\\\\n =&\\chh(\\overline{\\mathcal{G}})\\chh(\\overline{\\pi}_{\\ast}p^{\\prime\\ast}\\overline{{\\mathcal O}(k)})\\\\\n &\\hspace{0.3cm}=\\chh(\\overline{\\mathcal{G}})\\chh(p^{\\ast}\\overline{\\pi}^{\\prime}_{\\ast}\\overline{{\\mathcal O}(k)})\\\\\n &\\hspace{0.6cm}=\\chh(\\overline{\\mathcal{G}})p^{\\ast}\\chh(\\overline{\\pi}^{\\prime}_{\\ast}\\overline{{\\mathcal O}(k)}).\n \\end{split}\n\\end{displaymath}\nHere we recall $\\pi^{\\prime}:{\\mathbb P}^{n}_{A}\\rightarrow\\Spec A$ is the\nstructure morphism, that we endow with the Fubini-Study metric. On the\nother hand, we similarly prove\n\\begin{displaymath}\n \\begin{split}\n \\pi_{\\ast}(\\chh(\\pi^{\\ast}\\overline{\\mathcal{G}}\\otimes p^{\\prime\\ast}\\overline{{\\mathcal O}(k)})\\widehat{\\Td}(\\overline{\\pi}))\n =&\\chh(\\overline{\\mathcal{G}})\\pi_{\\ast}(p^{\\prime\\ast}\\chh(\\overline{{\\mathcal O}(k)})p^{\\prime\\ast}\\widehat{\\Td}(\\overline{\\pi}^{\\prime}))\\\\\n &=\\chh(\\overline{\\mathcal{G}})p^{\\ast}\\pi^{\\prime}_{\\ast}(\\chh(\\overline{{\\mathcal O}(k)})\\widehat{\\Td}(\\overline{\\pi^{\\prime}})).\n \\end{split}\n\\end{displaymath}\nWe thus reduce to prove the theorem for $\\overline{\\pi}^{\\prime}$ and\n$\\alpha=\\overline{{\\mathcal O}(k)}$, as was to be shown.\n\\end{proof}\n\\paragraph{The case of projective spaces.} To prove Theorem\n\\ref{thm:5} in full generality, it remains to treat the case of the\nprojection $\\overline{\\pi}\\colon{\\mathbb P}^{n}_{A}\\rightarrow\\Spec A$, where we\nendow $\\pi$ with the Fubini-Study metric. Since this projection is the\npull-back of the projection\n$\\overline{\\pi}\\colon{\\mathbb P}^{n}_{{\\mathbb Z}}\\rightarrow\\Spec {\\mathbb Z}$ it is enough to\ntreat the case when $A={\\mathbb Z}$. \n\nFurthermore, we showed that\nit is enough to consider $\\alpha$ of the form $\\overline{{\\mathcal O}(k)}$, $-n\\leq\nk\\leq 0$, with the Fubini-Study metric as well and any metric of the\ndirect image $\\pi_{\\ast}{\\mathcal O}(k)$.\nIn \\cite[Def. 5.7]{BurgosFreixasLitcanu:GenAnTor}, we introduced the\nmain characteristic numbers of $T^{h}$,\n\\begin{displaymath}\n t_{n,k}^{h}=T^{h}(\\overline{\\pi},\\overline{{\\mathcal O}(k)},\\overline{\\pi_{\\ast}{\\mathcal O}(k)}),\\\n -n\\le k \\le 0,\n\\end{displaymath}\nwhere $\\pi_{\\ast}{\\mathcal O}(k)$ was endowed with its $L^{2}$ metric. Since we\nwill only consider the homogeneous analytic torsion we will shorthand\n$t^{h}_{n,k}=t_{n,k}$. \n\n We will\ndenote by $\\overline 0$ the trivial vector bundle with trivial hermitian\nstructure and, for any $X$, we denote by $\\overline {\\mathcal O}_{X}$ the structural\nsheaf with the metric $\\|1\\|=1$.\nFor\n$-n\\le k < 0$, the complex $\\overline{\\pi_{\\ast}{\\mathcal O}(k)}$ can be represented\nby $\\overline 0$, while \nthe complex $\\overline{\\pi_{\\ast}{\\mathcal O}(0)}$ can be represented by the\nstructural sheaf ${\\mathcal O}_{{\\mathbb Z}}$ with the metric $\\|1\\|=1\/n!$. Since this\nmetric depends on $n$ it will be simpler to consider the complexes\n$\\overline{\\pi_{\\ast}{\\mathcal O}(k)}'=\\overline{\\pi_{\\ast}{\\mathcal O}(k)}$ for $-n\\le k < 0$ and \n$\\overline{\\pi_{\\ast}{\\mathcal O}(k)}'=\\overline {\\mathcal O}_{{\\mathbb Z}}$.\nThen we write \n\\begin{displaymath}\n t'_{n,k}=T^{h}(\\overline{\\pi},\\overline{{\\mathcal O}(k)},\\overline{\\pi_{\\ast}{\\mathcal O}(k)}'),\\\n -n\\le k \\le 0.\n\\end{displaymath}\nThese characteristic numbers satify\n\\begin{displaymath}\n t'_{n,k}=\n \\begin{cases}\n t_{n,k},&\\text{ if }-n\\le k<0,\\\\\n t_{n,0}-(1\/2)\\log(n!),& \\text{ if }k=0.\n \\end{cases}\n\\end{displaymath} \nClearly it is equivalent to work with the characteristic numbers\n$t_{n,k}$ or with $t'_{n,k}$.\n\nIn order to finish the the proof of Theorem \\ref{thm:5} it only\nremains to show the following particular cases.\n\n\\begin{theorem}\\label{thm:7}\nFor every $0\\leq k\\leq n$, we have the equality in $\\cha^{1}(\\Spec\n{\\mathbb Z})={\\mathbb R}$ \n\\begin{equation}\\label{eq:32}\n \\amap(t'_{n,-k})=\\chh(\\overline{\\pi_{\\ast}{\\mathcal O}(-k)}')-\\\n \\pi_{\\ast}(\\chh(\\overline{{\\mathcal O}(-k)})\\widehat{\\Td}(\\overline{\\pi})).\n\\end{equation}\n\\end{theorem}\nThe proof will proceed by induction. The next proposition treats the first case.\n\\begin{proposition}\\label{prop:7}\nEquation \\eqref{eq:32} holds for $k=0$.\n\\end{proposition}\n\\begin{proof}\nThe proof exploits the behavior of generalized analytic torsion with\nrespect to composition of morphisms. Let us consider the diagram\n\\begin{displaymath}\n \\xymatrix{\n {\\mathbb P}^n_{{\\mathbb Z}} \\ar[rd]_{\\Id} \\ar[r]^{\\Delta\\ \\ \\ } & {\\mathbb P}^n_{{\\mathbb Z}}\n \\underset{{\\mathbb Z}}{\\times}{\\mathbb P}^n_{{\\mathbb Z}} \\ar[r]^{p_1}\\ar[d]^{p_2} &\n {\\mathbb P}^n_{{\\mathbb Z}} \\ar[d]^{\\pi}\\\\ \n & {\\mathbb P}^n_{{\\mathbb Z}} \\ar[r]_{\\pi_{1}} & \\Spec {\\mathbb Z} .}\n\\end{displaymath}\nThe Fubini-Studi metric on the tangent space $T_{{\\mathbb P}^{n}_{{\\mathbb C}}}$\nis invariant under complex conjugation. It induces a \nmetric on the tangent space of the product of projective spaces. On\neach morphism on the above diagram we consider the relative hermitian\nstructure deduced by the hermitian metrics on each tangent space. With\nthis choice \n\\begin{equation}\n \\label{eq:24}\n \\overline \\Id=\\overline p_{2}\\circ \\overline {\\Delta },\n\\end{equation}\nwhere the composition of relative hermitian structures is defined in\n\\cite[Definition 5.7]{BurgosFreixasLitcanu:HerStruc}. On the relative\ntangent bundle $T_{p_{2}}$\nwe consider the metric induced by the metric on $T_{{\\mathbb P}^{n}\\times\n {\\mathbb P}^{n}}$.\nSince the\nshort exact sequence\n\\begin{displaymath}\n 0\\longrightarrow\n \\overline T_{p_{2}}\\longrightarrow\n \\overline T_{{\\mathbb P}^{n}\\times {\\mathbb P}^{n}}\\longrightarrow\n p_{2}^{\\ast}\\overline T_{{\\mathbb P}^{2}} \\longrightarrow 0\n\\end{displaymath}\nis orthogonaly split,\nwe deduce that the metric we consider on $\\overline p_{2}$ agrees with the\nmetric on the vector bundle\n$\\overline T_{p_{2}}$ and this, in turn agrees with the metric on\n$p_{1}^{\\ast }T_{{\\mathbb P}^{n}}$. \n\nLet $\\overline Q$ be the tautological quotient bundle on ${\\mathbb P}^{n}_{{\\mathbb Z}}$ with\nany hermitian metric invariant under complex conjugation. We denote by\n$K$ the Koszul resolution of the diagonal and $\\overline K$ the same\nresolution with the induced metrics. Namely $\\overline K$ is the complex\n\\begin{displaymath}\n 0\\to p_{2}^{\\ast}\\Lambda ^{n}\\overline Q^{\\vee}\\otimes p_{1}^{\\ast}\\overline\n {{\\mathcal O}(-n)}\n\\to \\dots\\to p_{2}^{\\ast}\\overline Q^{\\vee}\\otimes p_{1}^{\\ast}\\overline\n {{\\mathcal O}(-1)}\\to \\overline {\\mathcal O}_{{\\mathbb P}^{n}\\times{\\mathbb P}^{n}}\\to 0,\n\\end{displaymath}\nand there is an isomorphism in the derived category $K\\to \\Delta\n_{\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}$. Since the theories of homogeneous torsion classes\nfor closed immersions and for projective spaces are compatible\n\\cite[Definition 6.2]{BurgosFreixasLitcanu:GenAnTor} the equation\n\\begin{equation}\n \\label{eq:25}\n T(\\overline p_{2},\\overline K, \\overline {\\mathcal O}_{{\\mathbb P}^{n}})+ \\overline p_{2\\flat}(T(\\overline \\Delta\n ,\\overline {\\mathcal O}_{{\\mathbb P}^{n}},\\overline K))=0\n\\end{equation}\nholds. Then\n\\begin{align*}\n T(\\overline p_{2},\\overline K, \\overline {\\mathcal O}_{{\\mathbb P}^{n}})&=\n T(\\overline p_{2},\\overline {\\mathcal O}_{{\\mathbb P}^{n}\\times {\\mathbb P}^{n}},\\overline {\\mathcal O}_{{\\mathbb P}^{n}})+\n \\sum_{i=1}^{n}(-1)^{i}T(\\overline p_{2}, p_{2}^{\\ast}\\Lambda ^{i}\\overline\n Q^{\\vee}\\otimes p_{1}^{\\ast}\\overline\n {{\\mathcal O}(-i)},\\overline 0)\\\\\n &=\\pi_{1}^{\\ast}T(\\overline \\pi ,\\overline {\\mathcal O}_{ {\\mathbb P}^{n}},\\overline\n {\\mathcal O}_{{\\mathbb Z}})\\bullet \\ch(\\overline {\\mathcal O}_{{\\mathbb P}^{n}})+\\\\\n &\\qquad \\qquad \\sum_{i=1}^{n}(-1)^{i}\\pi_{1}^{\\ast}T(\\overline \\pi ,\\overline {{\\mathcal O}\n (-i)},\\overline 0)\\bullet \\ch(\\Lambda ^{i}\\overline Q^{\\vee}) \\\\\n &=\\sum_{i= 0}^{n}(-1)^{i}t_{n,-i}\\bullet \\ch(\\Lambda ^{i}\\overline\n Q^{\\vee}). \n\\end{align*}\n\nUsing that\n\\begin{displaymath}\n \\pi _{1\\ast}(\\ch(\\Lambda ^{i}\\overline\n Q^{\\vee})\\Td(\\overline \\pi _{1}))=\n \\begin{cases}\n 1, &\\text{ if }i=0,\\\\\n 0, &\\text{ otherwise,} \n \\end{cases}\n\\end{displaymath}\nwe deduce\n\\begin{equation}\n \\label{eq:33}\n \\overline \\pi _{1\\flat}(T(\\overline p_{2},\\overline K, \\overline {\\mathcal O}_{{\\mathbb P}^{n}}))=t_{n,0}.\n\\end{equation}\n\nBy the arithmetic Riemann-Roch theorem for closed immersions\n\\begin{displaymath}\n \\amap(T(\\overline \\Delta\n ,\\overline {\\mathcal O}_{{\\mathbb P}^{n}},\\overline\n K))=\n \\sum_{i=0}^{n}(-1)^{i}p_{2}^{\\ast}\\chh(\\Lambda ^{i}\\overline\n Q^{\\vee})\\cdot\n p_{1}^{\\ast}\\chh(\\overline{{\\mathcal O}(-i)})-\\Delta _{\\ast}(\\chh(\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\Delta )).\n\\end{displaymath}\nFor $i>0$\n\\begin{multline*}\n \\pi _{1\\ast}\\left(p_{2\\ast}\\left(p_{2}^{\\ast}\\chh(\\Lambda ^{i}\\overline\n Q^{\\vee})p_{1}^{\\ast}\\chh(\\overline {{\\mathcal O}(-i)})\\widehat {\\Td}(\\overline{p_{2}})\\right)\n \\widehat {\\Td}(\\overline{\\pi} _{1})\\right)\\\\\n =\\pi _{1\\ast}(\\chh(\\Lambda ^{i}\\overline\n Q^{\\vee})\\widehat {\\Td}(\\overline{\\pi} _{1}))\\cdot \\pi _{\\ast}(\\chh(\\overline\n {{\\mathcal O}(-i)})\\widehat{\\Td}(\\overline \\pi )).\n\\end{multline*}\nSince\n\\begin{displaymath}\n \\zeta (\\pi _{1\\ast}(\\chh(\\Lambda ^{i}\\overline\n Q^{\\vee})\\widehat {\\Td}(\\overline \\pi _{1})))=\n \\zeta(\\pi _{\\ast}(\\chh(\\overline\n {{\\mathcal O}(-i)})\\widehat{\\Td}(\\overline \\pi )))=0\n\\end{displaymath}\nand $\\Ker \\zeta $ is a square zero ideal of the arithmetic Chow ring,\nwe deduce that, for $i>0$\n\\begin{equation}\\label{eq:36}\n \\pi _{1\\ast}\\left(p_{2\\ast}\\left(p_{2}^{\\ast}\\chh(\\Lambda ^{i}\\overline\n Q^{\\vee})p_{1}^{\\ast}\\chh(\\overline {{\\mathcal O}(-i)})\\widehat {\\Td}(\\overline{p_{2}})\\right)\n \\widehat {\\Td}(\\overline \\pi _{1})\\right)=0.\n\\end{equation}\nFor $i=0$ we compute\n\\begin{displaymath}\n \\pi _{1\\ast}\\left(p_{2\\ast}\\left(p_{2}^{\\ast}\\chh(\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}})p_{1}^{\\ast}\\chh(\\overline {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat {\\Td}(\\overline{p_{2}})\\right)\n \\widehat {\\Td}(\\overline \\pi _{1})\\right)=\\pi _{\\ast}(\\chh(\\overline\n{\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\pi))^{2}.\n\\end{displaymath}\nUsing that $\\chh(\\overline {\\mathcal O}_{{\\mathbb P}^{n}})=1$ and that $\\pi\n_{\\ast}(\\widehat{\\Td}(\\overline \\pi))-1\\in \\Ker \\zeta $ we obtain that\n\\begin{equation}\\label{eq:35}\n \\pi _{\\ast}(\\chh(\\overline\n{\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\pi))^{2}=-1+2\\pi\n_{\\ast}(\\widehat{\\Td}(\\overline \\pi )).\n\\end{equation}\nFurthermore, by the choice of metrics on the relative tangent complexes\n\\begin{equation}\\label{eq:34}\n \\pi _{1\\ast}p_{2\\ast}\\Delta _{\\ast}(\\chh(\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline \\Delta )\\widehat{\\Td}(\\overline\n p_{2})\\widehat{\\Td}(\\overline\\pi _{1}))=\n\\pi _{\\ast}(\\chh(\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline\\pi )).\n\\end{equation}\nUsing equations \\eqref{eq:36}, \\eqref{eq:35} and \\eqref{eq:34} we\ndeduce that\n\\begin{equation}\n \\label{eq:37}\n \\amap(\\overline\\pi _{1\\flat}(\\overline p_{2\\flat}(T(\\overline \\Delta ,\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}},\\overline K))))=\\pi _{\\ast}(\\chh(\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline\\pi ))-\\chh(\\overline {\\mathcal O}_{{\\mathbb Z}}).\n\\end{equation}\nBy equations \\eqref{eq:25}, \\eqref{eq:33} and \\eqref{eq:37} we conclude\n\\begin{displaymath}\n \\amap(t'_{n,0})=\\chh(\\overline {\\mathcal O}_{{\\mathbb Z}})-\\pi _{\\ast}(\\chh(\\overline\n {\\mathcal O}_{{\\mathbb P}^{n}})\\widehat{\\Td}(\\overline\\pi ))\n\\end{displaymath}\nproving the proposition.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm:7}] We now proceed with the\n induction step. We assume that equation \\eqref{eq:32} holds for some\n $k\\ge 0$ and all $n\\ge k$. Fix now $n\\ge k+1$. Consider the diagram \n\\begin{displaymath}\n \\xymatrix{\n {\\mathbb P}^{n-1}_{{\\mathbb Z}} \\ar[rd]_{\\pi _{n-1}} \\ar[r]^{s} & {\\mathbb P}^n_{{\\mathbb Z}}\n \\ar[d]^{\\pi _{n}}\\\\ \n & \\Spec {\\mathbb Z},}\n\\end{displaymath}\n where $s$ is the closed immersion induced by a section $s$ of\n ${\\mathcal O}_{{\\mathbb P}^{n}}(1)$. As in the proof of Proposition \\ref{prop:7}, we\n consider the relative hermitian \n structures defined by the Fubini-Study metric on the tangent\n bundles. In this way $\\overline \\pi _{n-1}=\\overline \\pi _{n}\\circ \\overline s$.\n\n We consider the Koszul complex\n \\begin{displaymath}\n K_{n,k}\\colon {\\mathcal O}_{{\\mathbb P}^{n}}(-k-1)\\overset{s}{\\longrightarrow }\n {\\mathcal O}_{{\\mathbb P}^{n}}(-k)\n \\end{displaymath}\n and we denote by $\\overline K_{n,k}$ the same complex provided with the\n Fubini-Study metrics.\n\n By the transitivity \\cite[Definition\n 7.1]{BurgosFreixasLitcanu:GenAnTor} of the homogeneous theory\n of analytic torsion, the equation\n \\begin{multline*}\n T^{h}(\\overline \\pi _{n-1},\\overline {{\\mathcal O}(-k)},\\overline {\\pi _{n-1\\ast}{\\mathcal O}(-k)}')=\\\\\n T^{h}(\\overline \\pi _{n},\\overline K_{n,k},\\overline {\\pi _{n-1\\ast}{\\mathcal O}(-k)}')+\n \\overline\\pi _{n\\flat}(T^{h}(\\overline s,\\overline {{\\mathcal O}(-k)},\\overline K_{n,k})).\n \\end{multline*}\n By definition $T^{h}(\\overline \\pi _{n-1},\\overline {{\\mathcal O}(-k)},\\overline {\\pi\n _{n-1\\ast}{\\mathcal O}(-k)}')=t'_{n-1,-k}$. By the choice of metrics, the\n exact sequence\n \\begin{equation} \\label{eq:38}\n 0\\to \\overline {\\pi _{n\\ast}{\\mathcal O}(-k-1)}'\n \\to \\overline {\\pi _{n\\ast}{\\mathcal O}(-k)}' \\to \\overline {\\pi\n _{n-1\\ast}{\\mathcal O}(-k)}'\\to 0\n \\end{equation}\n is orthogonal split (all the terms appearing in this short exact\n sequence are either $\\overline 0$ or $\\overline\n {\\mathcal O}_{{\\mathbb Z}}$). \n This implies that\n \\begin{multline*}\n T^{h}(\\overline \\pi _{n},\\overline K_{n,k},\\overline {\\pi _{n-1\\ast}{\\mathcal O}(-k)}')=\\\\\n T^{h}(\\overline \\pi _{n},\\overline{{\\mathcal O}(-k)},\\overline {\\pi _{n\\ast}{\\mathcal O}(-k)}')\n -T^{h}(\\overline \\pi _{n},\\overline{{\\mathcal O}(-k-1)},\\overline {\\pi\n _{n\\ast}{\\mathcal O}(-k-1)}')=\\\\\n t'_{n,-k}-t'_{n,-k-1}.\n \\end{multline*}\n By Lemma \\ref{lemm:4} (the case of closed immersions)\n \\begin{displaymath}\n T^{h}(\\overline s,\\overline {{\\mathcal O}(-k)},\\overline K_{n,k})=\\chh(\\overline {{\\mathcal O}(-k)})-\n \\chh(\\overline{{\\mathcal O}}(-k-1))-\\overline{s}_{\\flat}(\\chh(\\overline{{\\mathcal O}(-k)})).\n \\end{displaymath}\nThis implies that\n\\begin{multline*}\n \\overline{\\pi} _{n\\flat}(T^{h}(\\overline s,\\overline {{\\mathcal O}(-k)},\\overline K_{n,k}))=\\\\ \\overline{\\pi}\n _{n\\flat}\\chh(\\overline {{\\mathcal O}(-k)})- \n \\overline{\\pi} _{n\\flat} \\chh(\\overline{{\\mathcal O}}(-k-1))\n -\\overline{\\pi} _{n-1\\flat}\\chh(\\overline{{\\mathcal O}(-k)}).\n\\end{multline*}\nSumming up, we deduce\n\\begin{multline}\\label{eq:39}\n \\amap(t'_{n-1,-k}-t'_{n,-k}+t'_{n,-k-1})=\\\\\n -\\overline{\\pi} _{n-1\\flat}\\chh(\\overline{{\\mathcal O}(-k)})+\n \\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k)})+\n -\\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k-1)}).\n\\end{multline}\nApplying the induction hypothesis we get\n\\begin{displaymath}\n t'_{n,-k-1}=-\\chh(\\overline{\\pi _{n-1\\ast}{\\mathcal O}(-k)}')+\n \\chh(\\overline{\\pi _{n\\ast}{\\mathcal O}(-k)}')-\\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k-1)}). \n\\end{displaymath}\nUsing again that the exact sequence \\eqref{eq:38} is orthogonally\nsplit, we deduce that\n\\begin{displaymath}\n t'_{n,-k-1}=\\chh(\\overline{\\pi _{n\\ast}{\\mathcal O}(-k-1)}')-\\overline{\\pi} _{n\\flat}\\chh(\\overline{{\\mathcal O}(-k-1)}). \n\\end{displaymath}\ncompleting the inductive step and proving the Theorem \\ref{thm:7} and\ntherefore Theorem \\ref{thm:5}.\n\\end{proof}\n\nOnce we have proved that the formal properties of an analytic torsion\ntheory imply the arithmetic Riemann-Roch theorem we can compute\neasily the characteristic numbers $t_{n,k}$ for the homogeneous\nanalytic torsion and all $n\\ge 0$ and $k\\in {\\mathbb Z}$. By the Theorem\n\\ref{thm:5} they satisfy\n\\begin{displaymath}\n \\amap(t_{n,k})=\\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})-\\\n \\pi_{n\\ast}(\\chh(\\overline{{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})\\widehat{\\Td}(\\overline{\\pi}_{n})).\n\\end{displaymath}\nObserve it is enough to compute $t_{n,k}$ for $k\\geq -n$, since the self-duality of the homogenous analytic torsion \\cite[Thm. 9.12]{BurgosFreixasLitcanu:GenAnTor} immediately yields the relation\n\\begin{displaymath}\n\tt_{n,k}=(-1)^{n}t_{n,-k-n-1}.\n\\end{displaymath}\nTherefore, from now on we restrict to this range of values of $k$.\n\nWe first compute\n$\\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})^{(1)}$. For $-n\\leq k\\leq -1$ this quantity vanishes:\n\\begin{displaymath}\n\t \\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})^{(1)}=0,\\quad -n\\leq k\\leq -1.\n\\end{displaymath}\nSuppose now $k\\geq 0$. Using that the volume\nform $1\/n!\\omega _{FS}^{n}$ is given, in a coordinate patch, by\n\\begin{displaymath}\n \\mu =\\left(\\frac{i}{2\\pi }\\right)^{n}\\frac{\\dd z_{1}\\land \\dd \\overline\n z_{1}\\land\\dots \\land \\dd z_{n}\\land \\dd \\overline\n z_{n}}{(1+\\sum_{i=1}^{n}z_{i}\\overline z_{i})^{n+1}},\n\\end{displaymath}\nit is easy to see that the basis $\\{x_{0}^{a_{0}}\\dots\nx_{n}^{a_{n}}\\}_{a_{0}+\\dots+a_{n}=k}$ is orthonormal and satisfies\n\\begin{displaymath}\n \\|x_{0}^{a_{0}}\\dots\nx_{n}^{a_{n}}\\|^{2}_{L^{2}}=\\frac{a_{0}!\\dots a_{n}!}{(k+n)!}.\n\\end{displaymath}\nTherefore\n\\begin{displaymath}\n \\chh(\\overline{\\pi_{n\\ast}{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})^{(1)}=\n \\sum_{a_{0}+\\dots+a_{n}=k}-\\left(\\frac{1}{2}\\right)\n \\log\\left(\\frac{a_{0}!\\dots a_{n}!}{(k+n)!}\\right).\n\\end{displaymath}\n\n\n\nTo compute\n$\\pi_{n\\ast}(\\chh(\\overline{{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})\\widehat{\\Td}(\\overline{\\pi}_{n}))$\nwe follow \\cite{GilletSoule:ATAT}, where the case $k=0$ is\nconsidered. Let $\\alpha _{n,k}$ be the coefficient of $x^{n+1}$\nin the power series\n\\begin{math}\ne^{kx}\\left(\\frac{x}{1-e^{-x}}\\right)^{n+1} \n\\end{math}\nand let $\\beta _{n,k}$ be the coefficient of $x^{n}$ in the power series\n\\begin{math}\n \\int_{0}^{1}\\frac{\\phi(t)-\\phi(0)}{t}\\dd t, \n\\end{math}\nwhere\n\\begin{displaymath}\n \\phi (t)=e^{kx}\\left(\\frac{1}{tx}-\\frac{e^{-tx}}{1-e^{-tx}}\\right)\n \\left(\\frac{x}{1-e^{-x}}\\right)^{n+1}.\n\\end{displaymath}\nThen, by a slight modification of the argument in \\cite [Proposition\n2.2.2 \\& 2.2.3]{GilletSoule:ATAT}, we derive\n\\begin{displaymath}\n \\pi\n _{n\\ast}(\\chh(\\overline{{\\mathcal O}_{{\\mathbb P}^{n}_{{\\mathbb Z}}}(k)})\n \\widehat{\\Td}(\\overline{\\pi}_{n}))^{(1)}=\\frac{1}{2} \n \\amap\\left(\\alpha _{n,k}\\sum_{p=1}^{n}\\sum_{j=1}^{p}\\frac{1}{j}+\\beta\n _{n,k}\\right).\n\\end{displaymath}\nThe factor $(1\/2)$ appears from the different normalization used here (see\n\\cite[Theorem 3.33]{BurgosKramerKuehn:accavb}). We thus have to determine the coefficients $\\alpha_{n,k}$ and $\\beta_{n,k}$. The numbers $\\alpha_{n,k}$ can be obtained similarly to \\cite[Eq. (27)]{GilletSoule:ATAT}. We obtain\n\\begin{displaymath}\n\t\\alpha_{n,k}=\\begin{cases}\n\t\t0\t&\\text{for } -n\\leq k\\leq -1,\\\\\n\t\t\\binom{k+n}{n}\t&\\text{for }k\\geq 0.\n\t\\end{cases}\n\\end{displaymath}\nThe values of $\\beta_{n,k}$ are expressed in terms of some secondary Todd numbers, that in turn are determined by a generating series:\n\\begin{displaymath}\n\t\\beta_{n,k}=\\sum_{j=0}^{n}\\widetilde{\\Td}_{n-j}\\frac{k^{j}}{j!},\n\\end{displaymath}\nwhere the $\\widetilde{\\Td}_{m}$ are given by the equality of generating series\n\\begin{equation}\\label{eq:todd_numbers}\n\t\\sum_{m\\geq 0}\\frac{\\widetilde{\\Td}_{m}}{m+1}T^{m+1}=\n\t\\sum_{m\\geq 1}\\frac{\\zeta(-(2m-1))}{2m-1}\\frac{y^{2m}}{(2m)!},\\quad T=1-e^{-y}.\n\\end{equation}\nHere $\\zeta$ stands for the Riemann zeta function. The result is a direct consequence of the definition of $\\beta_{n,k}$ and the computation of the numbers $\\beta_{n,0}$ in \\cite[Prop. 2.2.3 \\& Lemma 2.4.3]{GilletSoule:ATAT}. \n\nWe summarize these computations for the principal characteristic numbers $t_{n,k}$, $-n\\leq k\\leq 0$, since it is particularly pleasant.\n\\begin{proposition}\nThe principal characteristic numbers $t_{n,k}$ are given by\n\\begin{displaymath}\n\tt_{n,k}=\\begin{cases}\n\t\t-\\frac{1}{2}\\sum_{p=1}^{n}\\sum_{j=1}^{p}\\frac{1}{j},\t&\\text{for }k=0,\\\\\t\n\t\t\\\\\n\t-\\frac{1}{2}\\sum_{j=0}^{n}\\widetilde{\\Td}_{n-j}\\frac{k^{j}}{j!},\t\t&\\text{for } -n\\leq k\\leq -1,\n\\end{cases}\n\\end{displaymath}\nwhere the sequence of numbers $\\widetilde{\\Td}_{m}$, $m\\geq 0$, is determined by \\eqref{eq:todd_numbers}. \n\\end{proposition}\n\n\n\\newcommand{\\noopsort}[1]{} \\newcommand{\\printfirst}[2]{#1}\n \\newcommand{\\singleletter}[1]{#1} \\newcommand{\\switchargs}[2]{#2#1}\n \\def$'${$'$}\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"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzqzsu b/data_all_eng_slimpj/shuffled/split2/finalzzqzsu new file mode 100644 index 0000000000000000000000000000000000000000..477255b8c16e7fd5cda1d5ebc61f9069829219c3 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzqzsu @@ -0,0 +1,5 @@ +{"text":"\n# Cover Copy\n\nAfter a passionless marriage under the scrutiny of high society, Avery Stowe is taking back her life. All she wants is a little privacy and a quiet place to raise her autistic daughter, Hailey. Redwood Ridge, Oregon, seems to offer all the right ingredients. Except for the problem of the local sexy veterinarian. The last thing she needs in her life is to fall for his irresistible allure, even if he is a nice guy who keeps doing her favors. But the well-meaning patrons of her new hometown have other ideas, and it appears playing Cupid is one of them.\n\nCade O'Grady has never met a woman he couldn't handle, but when Avery Stowe walks into his office late one night cradling an injured puppy, he's struck stupid. Which might explain her total lack of interest in him. But now that she's working for his family's clinic, he doesn't have to lust from a distance. He might just have a chance at convincing Avery\u2014and her too-guarded heart\u2014that falling for the right man isn't a mistake . . .\n\nVisit us at www.kensingtonbooks.com\n\n# Books by Kelly Moran\n\nRedwood Ridge\n\nPuppy Love\n\nPublished by Kensington Publishing Corporation\n\n# Puppy Love\n\nRedwood Ridge\n\nKelly Moran\n\nLYRICAL PRESS\n\nKensington Publishing Corp.\n\nwww.kensingtonbooks.com\n\n# Copyright\n\nLyrical Press books are published by\n\nKensington Publishing Corp. 119 West 40th Street New York, NY 10018\n\nCopyright \u00a9 2016 by Kelly Moran\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.\n\nAll Kensington titles, imprints, and distributed lines are available at special quantity discounts for bulk purchases for sales promotion, premiums, fund- raising, and educational or institutional use.\n\nTo the extent that the image or images on the cover of this book depict a person or persons, such person or persons are merely models, and are not intended to portray any character or characters featured in the book.\n\nSpecial book excerpts or customized printings can also be created to fit specific needs. For details, write or phone the office of the Kensington Special Sales Manager:\n\nKensington Publishing Corp.\n\n119 West 40th Street\n\nNew York, NY 10018\n\nAttn. Special Sales Department. Phone: 1-800-221-2647.\n\nKensington and the K logo Reg. U.S. Pat. & TM Off.\n\nLYRICAL PRESS Reg. U.S. Pat. & TM Off.\n\nLyrical Press and the L logo are trademarks of Kensington Publishing Corp.\n\nFirst Electronic Edition: February 2017\n\neISBN-13: 978-1-5161-0273-0\n\neISBN-10: 1-5161-0273-8\n\nFirst Print Edition: February 2017\n\nISBN-13: 978-1-5161-0276-1\n\nISBN-10: 1-5161-0276-2\n\nPrinted in the United States of America\n\n# Dedication\n\nThis book is whole-heartedly dedicated to my writing group, The Floozies. Vonnie, Alison, Angel, Auria, Sarah, Dixie, AJ, Mac, Amy & Arial...I couldn't do this without you.\n\nA very special note to my nephew Connor, who lives with Autism and teaches me strength, humor and resolve continually. \n\n# Acknowledgements\n\nA big thank you to the team at Brentwood Animal Hospital, especially Lynn, who made sure I got the veterinarian thing down right. Any errors are my own. And a gracious thanks to my street team member, Raquel, for answering aviary questions and sharing stories.\n\n# Chapter 1\n\nAvery Stowe squinted and leaned closer to the steering wheel, trying to see past the fat white snowflakes blanketing a dark, quiet Redwood Ridge. They sure didn't make storms like this in San Francisco. She supposed her mom's insight on trading in her Camry for a SUV before the move had been the right call. Her sedan never would've gotten them through an Oregon winter. Even her flighty mother had to be right once and awhile. Once, being the operative word.\n\nGrateful to almost be at their destination after two days of travel, Avery chanced a peek at Hailey in the backseat and breathed a sigh. Assured her daughter was still asleep in her booster, she directed her attention to the road.\n\nFour inches had already fallen since they hit the state border. It was insane. Pretty, but insane. Having never been outside of warm, sunny California, this was a culture shock. But...new year, new start. Both she and Hailey needed this.\n\nEven if the new town did look like _Silent Hill_. She checked for creepy zombie things, but found none.\n\nThe sidewalks had been rolled up for the night, the only illumination coming from the old world lampposts lining the two-lane cobblestone street. Avery thought her mom had fallen out of the crazy tree\u2014and hit every branch on the way down\u2014when she moved here ten years ago after inheriting a string of cabins from an aunt they never knew existed. Her mom had been happy, though, and figured Avery and Hailey would be, too.\n\nIt sure seemed ideal on paper. \"Not _Silent Hill_ , not _Silent Hill_.\" Seriously. Where was everybody?\n\nPocketed between the coast and the foothills of the Klamath Mountains, Redwood Ridge was both a tourist hotspot and a charming small town of fifteen hundred residents. It had to have merit if it was able to hold her mother's attention this long. Row after row of small, independent shops lined both sides of the street. It was like stepping back in time to simpler, sweeter days. If only it had people.\n\nTen minutes later, the snow was dwindling and they were on a private road cocooned by cypress, pine, and redwood. It was pretty freakin' amazing, but she'd have to appreciate it more later. In daylight. Right now it looked like the set for _Friday the 13th_.\n\nMaybe she should stop watching horror movies altogether.\n\nThey passed a few larger homes still decorated for Christmas and, five miles down, made the turnoff to the rentals. She pulled up to the first cabin and parked, eying her surroundings. _Not Friday the 13th_.\n\nThat was a yes to no more scary movies. Definitely.\n\nThere were five log cabins in total, evenly spaced and single story, dusted with snow. Cookie cutter in design, each had a small porch and an A-frame slant roof. The first had a warm, yellow glow illuminating the windows and smoke billowing from the chimney. Her mother's car was parked off to the side.\n\nFor the first time in what seemed like ten years, Avery breathed and closed her eyes. No sirens or horns. No conversation or rush. No ex or in-laws to fight. Just...peace.\n\nUntil Jason arrives in his white hockey mask...\n\nOkay, that's it. Nothing but comedy movies from now on.\n\nIt would take some getting used to, but the move would be worth it for Hailey. With her daughter's condition, too much stimulation equaled tantrums and fits. The city life wasn't for them. Maybe Hailey would excel better in this environment. Having her mother close by was a bonus, too. During her childhood, Avery often felt more like the parent in their two-person family as her mother was always off in la-la land. She never lacked for love, though, and right now, she desperately needed support.\n\nIt had been so long since she had someone to lean on.\n\nAvery glanced in the backseat and reached around to tap Hailey's knee. \"Hey, sweetie. We're here.\"\n\nLike a switch, her daughter's dark eyelashes flickered to reveal blue eyes the same color as her dad's. All her other traits had come from Avery. Thick, brown hair and a curvy, lithe frame. Even at seven years old, Hailey was the spitting image of her.\n\nHailey took in her surroundings in the seemingly distracted manner Avery had grown used to since her daughter's Autism diagnosis. Her gaze darted everywhere at once, never landing on one spot for more than a millisecond. After a moment, she squealed and flapped her hands.\n\n_Well done, Mother_ , Avery imagined her saying.\n\nSince Hailey was a nonverbal Autistic, at least so far, Avery often choreographed her own dialogue in her head. It had helped her cope.\n\nShe smiled, pleased Hailey liked what she saw. \"Grandma's inside waiting for us. Would you like to check out our new home?\" At least it would be home until Avery could find them an apartment or small house to rent. Perhaps _not_ next to Camp Crystal Lake.\n\nHailey squealed again and fumbled with the seatbelt, clumsily getting it undone. Avery quickly climbed out of the car and met her daughter by the rear passenger door before she could take off. Deciding to wait until after they saw the place, she left their bags in the car and directed Hailey up the porch steps, careful not to touch her more than necessary.\n\nThe door swung inward before they could knock, and the sight of her mother standing there had tears clogging Avery's throat. Justine Berry might be capricious and unpredictable, but she'd always been there. After everything her and her daughter had been through, Avery just needed...her mom.\n\n\"I'm so excited you're here!\" She bent down at eye level with Hailey, bursting at the seams to hug her granddaughter.\n\nIt went against her mother's instincts not to smother, but Avery knew Hailey's limitations. She had warned her a million times before the drive, just in case she forgot. Her mother and memory lapses were BFFs.\n\nHailey nudged her grandmother aside and rushed into the cabin. _Outta the way, G-ma. Better things to do_.\n\nAvery shrugged. \"She's excited. That's a good thing.\"\n\nIn the next instant, she was wrapped in her mom's arms and squeezed until breathing became impossible. The familiar scent of patchouli filled her nose. She bit back the tears threatening to fall and smiled. \"Hi, Mom.\"\n\n\"At least I can still hug you.\" Mom stepped back and patted her wild, shoulder-length brown hair. She preferred the natural approach to everything, so it probably hadn't seen conditioner or product. In decades. The fine lines around her mouth and eyes had grown deeper in the year since she'd last visited Avery in San Fran, which only added to her mother's charm. She was a woman who laughed often and loved hard. Four ex-husbands were proof. \"How were the roads?\"\n\nAvery closed the door behind her. \"A little slick, but not too bad. Wow, Mom. This place is great.\"\n\nNot B horror movie at all.\n\nThe whole cabin was natural wood and stone and glass. Shiny, clean, and rustic. A floor to ceiling redbrick fireplace created warmth from a corner. Plaid print couches were nestled over the bare wood floor with scarred pine tables. The living room was spacious and separated from the kitchen by an island. Large picture windows encased the back of the open room, where moonlight caught the water trickling in a thin riverbed.\n\nHailey disappeared down a short hallway and squealed. Avery went to follow, but her mom stopped her with a hand on her arm.\n\n\"There's no exit back there and nothing to get into. These cabins are rentals, so they're pretty bare. I did stock the kitchen for you, though.\" She smiled and hugged her again. \"I'm so glad you're here. Ten years and you just now get to see my town.\"\n\nAvery pushed down the guilt and nodded. The past couldn't be helped. \"You did a lot of work on the property, I remember you saying. This is lovely.\"\n\nMom sighed. \"Good contractors and money left in the will took care of it. We're pretty busy year round. I live above my shop in town, but I'll stay with you tonight, if that's all right?\"\n\nHer mother owned a secondhand clothing store she'd named \"Thrifty\" in addition to the rental cabins. As far as Avery was aware, she had someone else manage the businesses. Finance was not her mother's strong suit. She had idealistic dreams and ideas aplenty, but the numbers and details she wisely delegated to others.\n\n\"I'd love it if you stayed the night. It's getting late, anyway.\"\n\nAnd speaking of, Hailey had been pretty quiet the past few minutes. Worry and suspicion mounting, Avery made her way down the hall and found her daughter asleep in one of the twin beds, curled in a ball with her hat and coat still on.\n\nAffection and love so deep it hurt punched her chest. Carefully, she unzipped Hailey's coat, leaving it on so she didn't wake her. Removing her hat, Avery ran her fingers through her daughter's dark hair. The only time she got to openly touch Hailey was when she was asleep, otherwise contact bothered Hailey to the point she screamed. But she had these few quiet moments of the early evening to watch her and stroke her perfect little pale cheeks.\n\nBone tired, Avery barely took stock of the bare wood furniture or bay window before finding her mom in the kitchen stirring a pot at the stove. Stunned still, she waited in the doorway.\n\nMom turned and grinned. \"I made hot chocolate. Go have a seat in the living room. Take a load off. You look wiped out. I'll bring you a cup.\"\n\n\"Is it...edible?\"\n\nMom shook her head, then said, \"I think so.\"\n\n\"Remember that time\u2014\"\n\n\"One fire. That was one fire, Avery!\"\n\nToo exhausted to argue, she grinned, settled in a chair, and closed her eyes, amazed it was so comfortable. The crackling fire and scent of chocolate soothed her, allowing her to sink a bit deeper into her head. Darkness pulled heavy at her consciousness.\n\nThe next thing she knew she was shivering and a cooling cup of cocoa was next to her on the end table. There were mysterious chunks floating in it. Blinking, she straightened. Her mother was asleep in the chair beside her.\n\nWow. How long had they been out?\n\nTaking a moment to stretch before checking on Hailey, she stood and glanced across the room to make sure her mother had turned off the stove. It wouldn't be the first time she'd gotten distracted and forgot. Assured it was off, Avery tried to find the source of the draft circling the room and froze when she spotted the open back door.\n\nNo. God, no.\n\n\"Mom!\" Avery screamed, already halfway down the hall. Her heart hammered in sickening panic.\n\nHailey's bed was empty.\n\nNo, no, no, no, no...\n\nBolting back into the living room, she slammed into her mother and quickly moved around her.\n\n\"What is it?\"\n\nAvery shoved her feet into boots and grabbed her coat. \"Hailey's gone. We fell asleep. I didn't secure the door.\" A mistake she knew better than to make. Hailey took off way too often. Not to run away, but because she lived inside her head and had no concept of danger.\n\nOh, God. Her daughter was out there in the cold, in the middle of nowhere, at night. The region had mountain lions not to mention...\n\n\"Call the police.\"\n\nShe ran for the back door and rounded the house, but Hailey wasn't in the car or on the porch. Avery circled back around, fear clawing her throat as she slammed into her mom again.\n\n\"There's footprints.\" Mom tied a scarf around her neck. \"She went straight into the woods.\"\n\nAvery looked down. A small set of tracks her daughter's size led away from the cabin and deeper into the dense trees. She took off, following the tracks. Cold air rasped her lungs, and her fingers were numb by the time they reached the copse of pine.\n\nHailey was so small. She wouldn't make it long exposed to these temperatures. It had to be in the twenties. Hailey couldn't talk, either. If she needed help, she couldn't ask for it. Avery had done her research before the move. She knew the vegetation and wildlife, knew her daughter was at risk for an animal attack and from what animals. Black bears, mountain lions, and bobcats sprang to mind. Hailey wouldn't know how to defend herself.\n\nTears blurred her eyes. She quickened her pace to a sprint, kicking up snow in her wake.\n\nBe okay, sweetie. Be okay.\n\nThe footprints made a hard right and, as they rounded a bend, the breath left her lungs in a _whoosh_.\n\nHailey was sitting on a stump, her back to them. Her pink coat was still on, but she didn't have her hat. The relief was dizzying.\n\n\"Hailey.\" Avery circled the stump and squatted. \"We talked about this, sweetie. You cannot go running off\u2014\"\n\nThere was blood. Lots of blood. At Hailey's feet. On the front of her coat.\n\n\"Where are you hurt? Where are you hurt, sweetie?\" She ran her trembling, frozen fingers over Hailey's head, down her neck, to her chest, and stopped.\n\nA furry, warm head poked out of Hailey's partially opened coat.\n\nA scream wedged in Avery's throat until she realized it was a dog. No, a puppy. A little, tan-colored fluffy thing. Hailey was rocking it, stroking its head, and jerking her gaze around.\n\nRecognizing the motions as nervous and scared, Avery kept her voice quiet. Hailey would never hurt a living thing, so she had to have found the animal out here. \"You found a doggy. It's okay, Hailey. Is the doggy hurt? Is that where the blood is coming from? Can Mommy see?\"\n\nGently, she lifted the trembling ball of fur from her daughter's grasp and the poor thing yelped. Surprised by the noise in the quiet night, she fell back on her butt in the snow. It couldn't have been more than six weeks old. Seven pounds, max. Sad, scared brown eyes looked into hers and Avery melted.\n\n\"Well, crap on a cracker. You're adorable.\"\n\n\"Avery...its leg.\" Mom jerked her chin toward the dog and shoved her hands into her pockets.\n\nAvery's gaze swept over the animal in the moonlight and eyed what her mother was talking about. The lower half of one of its front legs was severed clean. Blood saturated the fur. Her stomach churned. What could've happened to it?\n\nNausea swirled in her gut. \"You poor thing.\"\n\nHailey began rocking in earnest.\n\nAvery reached over and clasped a hand on her daughter's arm. \"It's going to be okay, sweetie.\"\n\nShe looked at her mom, at a loss. She'd never had a pet before. It was freezing outside and who knew how long the little guy had been hurt out in the woods or how much blood it had lost. By all the red marring the snow, it seemed like a lot for such a small thing. It didn't have tags or a collar. It was doing little more than whimpering and trembling. She needed to get Hailey out of the elements, too.\n\nHer mom unwrapped the scarf from her neck and passed it to her. \"I'll call the O'Gradys. They own the vet clinic in town. Go. I'll take Hailey back to the cabin\u2014\"\n\nHailey shot to her feet and grabbed Avery's jacket, a sound of duress escaping her throat even as her gaze darted elsewhere.\n\n\"She wants to come.\" She looked at her mom. \"Zip her coat, would you? Call the vet. We need to go. This little guy doesn't have long.\"\n\n* * * *\n\nCade O'Grady stared at the tiny gray kitten as it sucked milk from the bottle he held. The furball was small enough to fit in one of his hands. Anger surged again, so he blew out a breath and glanced around the small confines of his clinic office.\n\nIt was late, and he'd decided to stay to catch up on some charts. That was two hours ago, and said charts were still in need of dictations. Good thing he'd stayed or the tiny kitten in his hand would've died, just like its mother and siblings.\n\nWhat the hell kind of person left a box of kittens outside a clinic door in the snow? Cade had no idea how long they'd been in the elements\u2014someone had dropped them by the kennels outside the back door\u2014but the kitten he was currently feeding was the only survivor. He gnashed his teeth, mad enough to kill the SOB if he ever found him.\n\nLuckily, the kitten, which looked like a Brazilian Shorthair, was made of strong stuff. She'd taken to feeding right away and didn't need an IV. In his exam, the temp and BP had been good, all things considered, and he found no outward signs of trouble.\n\nClosing his eyes, he listened to the answering machine kick on from the reception desk outside his office. If it were an emergency, he'd be paged, since it was his week on call. Animal Instincts was a small clinic, started by his father forty years ago and run by Cade and his two brothers since the old man had passed away. Almost nine years now. Hard to believe.\n\nThe bottle empty, he set it on his desk and eyed the kitten. \"You are a cute little bugger.\"\n\nIt mewed in agreement.\n\nHe laughed for the first time all day and rubbed its head. \"Modest, too. I think I shall call you Cutin. Get it? Combining cute and kitten? Cutin.\"\n\nMew.\n\n\"You're right. They should take my man card on the spot.\"\n\nPaws kneading, it settled into the crook of his arm and fell asleep.\n\n\"I'll take that as a yes. Can I get you anything else? A beer perhaps?\"\n\nIt didn't respond. She. _She_ didn't respond. He should stop calling her it.\n\nShaking his head, he pulled a chart closer. His pager went off. He cursed. Just as he reached for it in his scrubs pocket, a pounding rattled the front door.\n\nHe glanced at the kitten. \"Today sucked.\"\n\nShe mewed sleepily to second the thought. _Word! Try having my day_.\n\nRising, he settled Cutin on a pillow in a box on an office chair and checked the pager as he walked to the front door. The banging intensified. He didn't recognize the number, but the woman standing outside the clinic was Justine something-or-other, who owned a resale shop down the street.\n\nHe unlocked the door and held it open. At least the snow had stopped. \"Was it you who paged me?\"\n\nShe rushed inside, followed by a woman he didn't know, and a little girl about eight. \"Yes, I paged.\" Justine pushed dark, windblown hair out of her face.\n\nHe closed and relocked the door against the biting airstream coming down off the mountain.\n\nThe other woman held out her arms. Inside a knit scarf was a puppy.\n\nCade looked down at all the blood on the girl's pink coat. Shit. Kicked into action, he motioned for them to follow. \"This way.\"\n\n\"I'm going to wait out here in the lobby,\" Justine said, her face a suspicious shade of green. \"All that medical stuff... I'll be good out here.\" She sat down hard to emphasize her point.\n\nInside an exam room, he donned gloves, turned, and reached for the animal. \"What happened?\"\n\n\"I'm not sure. Hailey found it in the snow about thirty minutes ago.\" Her voice was hurried, but calm, not showing any signs she was unnerved by all the blood.\n\nGingerly, he set the puppy on the table and unwrapped the scarf, taking stock of the situation. Yellow lab. Male. Eyes barely open. Lethargic. Underweight. About five or six weeks old. Trembling. Leg severed below the knee. Wound clotting and not actively bleeding.\n\nSon of a bitch.\n\nBiting the inside of his cheek, he forced himself to look at the woman. \"Come over here and stand by the table while I get supplies please.\"\n\nDark, cocoa eyes widened at his harsh tone. She turned to the girl. \"Can you sit in the chair, sweetie? I'm going to be right over here.\"\n\nThe girl didn't answer, instead fidgeting and avoiding eye contact. After a moment, she settled into a corner chair. She was probably in shock, too.\n\nWhen the woman sidled up to the exam table, Cade moved away, grabbing a saline IV bag and heating it in the microwave. Pulling out a warming blanket, he plugged it in and slid it under the puppy, then wrapped it around his back. Snatching an otoscope from his pocket, he leaned over to look in the dog's ears.\n\n\"How long was he outside?\"\n\nShe shifted on her feet, and the scent of something fruity wafted in the space between them. Berries of some kind. \"I don't know. We\u2014\"\n\n\"You don't know,\" he repeated dully and checked the puppy's mouth. Gums were pale, but the teeth were fine.\n\nHe inserted a thermometer for a rectal temp and eyed the woman. Though he didn't know everyone in Redwood Ridge, she certainly wasn't familiar. She had a pretty, chubby face, and wavy brown hair that trailed to her shoulders under a knit hat. They were close in age, perhaps late twenties if he had to guess. She was biting her red lips to the point of swelling.\n\nGood. She should feel guilty. Leaving a new pet outside, unattended, was reprehensible. Plus, the puppy showed signs of neglect. By the look of his leg, he had probably been caught in a bear trap. Since they'd come in with Justine, he could only assume she was an idiot tourist renting one of the cabins who didn't know how dangerous the mountain\u2014or what wildlife that came down the mountain\u2014could be.\n\n\"Is he up to date on shots? Any other conditions?\"\n\n\"I'm not sure. It's not\u2014\"\n\n\"Is there anything you _do_ know?\" he barked.\n\nShe closed her mouth and turned her head to check on her daughter, who was now staring at the ceiling.\n\nGuilt for his tone rose in his gut but, dammit, neglecting a dog was lower than pond scum. He'd seen it so many damn times. People got a pet because it was adorable or they were lonely and had no idea how much responsibility went into having one. Then they'd go abandoned or to a shelter, forgotten.\n\nHe was sick of tourists, too. One of the clinic pets was a result of an idiot tourist who didn't bother to return to pick up their one-year-old Great Dane after he'd broken a leg.\n\nPeople really sucked.\n\nAs the youngest of the three O'Grady men, Cade was the affable one. He was generally good with pet owners and could laugh himself out of a bad mood. Today, not so much. He'd had to euthanize old man Kiser's hound dog, the first client he ever had as a vet, lost a two-year-old retriever to a bowel obstruction, and found a box of dead kittens by the back door. Well, all dead but one.\n\nSo today? Today blew. He had no patience left in reserve, especially not for a woman who probably just broke her daughter's heart by putting their pet at risk.\n\nRemoving the thermometer, he noted the temp was low, but not nearly as down as he'd figured. He shoved the buds of his stethoscope in his ears and listened to his heart, lungs and abdomen. Nothing worrisome. BP was good, too. The little guy lifted his head and whimpered when he tried to better examine the leg.\n\n\"I know, little guy. It hurts, huh? I'll getcha something for that in a moment.\"\n\nThe wound had clotted, and though it didn't look infected, the puppy would need surgery to amputate the rest of the leg at the hip joint. First, he had to make sure the animal stayed stable to get an IV into him for fluids and antibiotics.\n\nStraightening, Cade crossed his arms and faced the woman. \"He's in decent condition for now, which is shocking. Vitals are a little low, but good. If he's doing this well in a couple hours, we'll do surgery to remove the rest of the leg. He'll need to stay a few nights for monitoring.\"\n\nCade paused, waiting for her to say something. When she just stared at him with a cross between worry and confusion, he shook his head. \"This is going to be expensive, ma'am.\"\n\nNot that he cared. If she walked out and left the dog, he'd still do what he could to save him. He'd eat the cost. Once the puppy was well again, Cade would try to adopt him out or they'd keep him for a clinic pet. Either way, he wasn't euthanizing an animal just because the leg was shot. Not unless there was no other choice.\n\nShe rubbed her forehead. \"Can he live with three legs? Get around, I mean? I don't know much about\u2014\"\n\nHe ground his teeth. \"People manage just fine with one or both legs amputated, don't they? He's young. He'll adjust. Yes or no, ma'am?\"\n\nStartled, her gaze whipped to his. She'd be a stunner with her huge eyes, pouty mouth, and button nose if he hadn't disliked her on the spot. \"I... I'm not sure I know what you're asking.\"\n\nHe briefly closed his eyes and prayed for patience. \"There's a healing period involved. Money to pay for his care. Are you going to be up for that? Because if not, just leave. Either he's your dog or he's not.\"\n\n\"He's...\" Her brown gaze drifted to the exam table. Surprise flitted in her eyes. She covered her mouth with her hand as tears welled.\n\nNarrowing his eyes, he turned.\n\nThe little girl stroked the puppy's head, her face pressed to the fur on his neck. She didn't say anything, nor did she appear upset, but the puppy sure liked what she was doing. His tail limply thumped the table and his trusting eyes were watching her. They seemed to have a bond already in place, which would make it even crappier if the woman walked out and never came back.\n\nTurning to face her, he raised his brows in question, but she paid him no mind. She watched the girl and the puppy with a tentative smile and shock clearly evident. Why the shock? Kids loved animals. This was their dog, after all. It stood to reason the girl would be upset the little guy got hurt.\n\nEven though she seemed to be having some kind of moment, it was getting late. He needed to get going on an IV. \"Ma'am?\"\n\nShe flinched and looked between him and the table. After a second, she collected herself and wiped her eyes. \"Yes. He's our dog. Do whatever you need to do to help him.\" Her gaze softened once more as she stepped closer to the table and tapped the girl on the shoulder. \"Time to go, sweetie. The doctor here is going to help...him. We'll come check on him tomorrow, okay?\"\n\nHow odd she didn't try to comfort the child. Put her arm around her or offer a hug. Something. The girl had blood down the front of her and had been, no doubt, traumatized by seeing her pet injured. Yet the woman stood there as if she didn't have a heart beating or a compassionate bone in her body.\n\nThis was why he preferred animals to people. \"What's the dog's name?\"\n\n\"Oh. Um...\"\n\nHe sighed. \"Let me guess. You don't know.\"\n\nA sliver of irritation flicked in her eyes before it was masked and she looked to her daughter once more. \"His name is...\" She tilted her head, chocolate eyes glazing over. \"Seraph. His name is Seraph.\"\n\n\"Seraph?\" It was rare someone surprised him anymore. He would've guessed Lucky or Champ or some other clich\u00e9.\n\n\"It's another name for angel\u2014\"\n\n\"I know.\" Still, she just got bonus points. \"Head home. I'll get your info from Justine in the morning. You can come visit during clinic hours.\"\n\nShe nodded and kneeled next to her daughter. \"Come on, sweetie. We'll come see him tomorrow.\"\n\nAfter they'd gone, Cade inserted an IV filled with warm saline to bring the dog's temperature up a little and started antibiotics. He drew some blood and ran a CBC to check platelets, then did vitals again. The little guy took it in stride. Pleased the puppy was doing well, he yanked a chair up to the exam table and pulled out his cell.\n\nDrake was going to be pissed, but this couldn't wait until morning. The dog needed surgery, and his oldest brother was their clinic surgeon. Cade could do it himself, but he didn't want to call in a tech and Drake was better.\n\n\"I'm not on call.\"\n\nCade grinned. \"Maybe I just missed you.\"\n\nThere was a lengthy pause. \"What do you want? And by that, I mean you better be up to your ass in greyhounds needing my cutting expertise. It's almost midnight.\"\n\n\"Got a yellow lab puppy requiring an amputation. Does that qualify?\"\n\nDrake groaned. \"Is it stable?\"\n\nCade bit back the string of insults he wanted to let loose. It wasn't as if he wasn't used to being underestimated. \"I'm not an idiot, you know. I have a degree and everything. I'm even sure I can spell veterinarian if I try real hard\u2014\"\n\n\"I'll be there in ten. Prep the room.\"\n\nCade pocketed his phone and rubbed the puppy behind his ears, earning two thumps of a tail.\n\n\"Seraph.\" He shook his head. \"Your owner is really something. Kinda pretty, too. She gave you a great name, even if she did leave you out in the snow, the big meanie.\"\n\nTwo more tail thumps.\n\n\"I have to go prepare the surgery room for my ornery brother, but I'll be right back. You just hang out here for a minute.\" He stroked the puppy's back. \"I promise we'll get you fixed up. You'll be good as new soon.\"\n\n_Thump, thump_.\n\n# Chapter 2\n\nAfter two anal gland appointments and a lethargic guinea pig rounding out his patients for the morning, Cade walked up to the front desk and eyed his aunt, who was baby talking their clinic dog, Thor. The Great Dane hid under the desk, cowering from She-rah, their evil cat. And evil she was. Charts teetered precariously from the long counter. At least the lobby was empty. It had been a damn crazy morning.\n\n_Squawk._ \"You spin me right round.\" Gossip, the cockatoo\u2014yet another abandoned animal\u2014bobbed his head. One of these days, Cade would have to teach him to say something other than song titles or lyrics. As it was, that's all the bird did thanks to his former owner. That, and tease the cat.\n\nCade scratched his jaw. \"Am I good to go for lunch?\"\n\nAunt Rosa sighed dramatically. Everything was dramatic with Rosa from her red spiky hair to her cheetah print shirt. \"Will you look at this?\" She narrowed her eyes at Thor. \"Grow a pair and come out of there. It's just a cat.\"\n\nShe-rah licked her paw and meowed from the top of the printer, bored by the events. _I bet I can get the dog to lose his bladder. Wanna see?_\n\nThor didn't move.\n\nShaking his head, Cade picked up She-rah, much to her disdain\u2014 _put me down you insolent peasant_ \u2014and set her in the back room. Returning to the lobby, he called for Thor. The hundred and ten pound dog commando crawled from under the desk and hid behind Rosa's chair.\n\nCade lifted his brows. \"Now may I go to lunch?\"\n\nNot that Rosa was their boss or anything, but he and his brothers knew not to rock the boat. For twenty years, their aunt had managed the clinic and acted as receptionist. Poorly, but beggars couldn't be choosers. Rosa was their mother's sister and one of what Cade liked to refer to as The Battleaxes. Their mother, Gayle, Aunt Rosa, and other sister, Marie\u2014also town mayor\u2014ruled Redwood Ridge with an iron fist and oatmeal cookies. They were crazy, meddling women who he loved and feared. Mostly feared.\n\n\"Have you found my replacement yet?\"\n\nHe bit back a groan. Rosa had announced six weeks ago she wanted to retire to do...whatever it was the Battleaxes did. Eat small children, update Redwood Ridge's Twitter page with town gossip, matchmaking...\n\nHe batted his eyelashes. \"How could we ever replace you, Aunt Rosa?\"\n\n\"Can the cuteness. Save it for the ladies.\"\n\nRight. \"No, we haven't replaced you yet.\" He'd have to get an ad in the paper, which would bring out all the crazies or every single woman in a thirty-mile radius. Damn it. He should make Flynn handle the hiring. As the middle brother, he was the most organized. Except he was their traveling vet and wasn't in the office much. \"I'll get on it right away.\"\n\nShe narrowed her hazel eyes and tilted her head, her unnaturally red hair not moving with all the shellac she'd sprayed on it. \"You said that more than a month ago.\"\n\nWell, how was he to know she was serious? It was hard to tell with her. \"I mean it this time. Now can I go to lunch? Pretty please?\" There were only three people on earth he'd bust out the \"pretty please\" for, and one of them was right in front of him.\n\n\"Brent's already at lunch. Go ahead.\" She smacked his ass and waved him off.\n\nBrent being his vet tech. Why didn't Rosa smack _his_ ass? Brent would enjoy it. \"It's really pervy when you do that.\"\n\nShe feigned innocence. \"Love tap your rear end? It's a nice one.\"\n\nHe bit back a sigh. \"You're relation. It's pervy.\" He was two steps from freedom when he remembered something. \"Can you get a hold of Justine from that clothing store down the way? She came in with a tourist who had the injured lab\u2014\"\n\n\"You mean Avery Stowe? She's not a tourist. She's Justine's daughter. Just moved to town with her daughter. Bad divorce. They're staying up at one of Justine's rentals until they can find a place.\"\n\nPerhaps he should've been nicer to her, but she'd caught him at the tail end of a crappy day and the fact remained, she hadn't taken very good care of her pet. Even without the injury, the puppy was malnourished.\n\nPulling on his coat, he reached for the door. \"Can you get a number for her and let her know Seraph is recuperating nicely?\" Not that she seemed to care. She hadn't visited the puppy.\n\n\"No need. Avery called three times this morning to say she'd be in after lunch. Apparently, there was a problem with their moving truck getting lost. Poor woman. No good luck. Plus, the daughter, Hailey, was pretty upset over finding the stray, so it took Avery awhile to calm her down after they left here. They slept in.\"\n\nCade locked in on one word in that whole rant. \"What do you mean, _stray_? It's their dog, isn't it?\"\n\nShe offered him her classical _duh_ look. \"It is now, but not when they found it. Can you imagine that poor little girl coming across a scene like that?\"\n\nHe called to mind the exhaustion edging the woman's chocolate eyes, the way the daughter never spoke, and the way he'd all but jumped down her throat. He'd assumed the worst, which wasn't like him. The woman\u2014Avery?\u2014had rescued Seraph from bleeding out into the snow, alone and scared.\n\nAnd he'd been curt with her for doing the right thing.\n\nShit. He was an asshole.\n\n* * * *\n\nAvery wiped her hands on a dish towel and went to answer the incessant knocking. Hoping it was the moving van, she pulled the door open to find Seraph's vet. \"Oh.\" She took a step back and blinked. \"It's you.\"\n\nThe Jerk, she'd begun to call him in her head.\n\nLooking just as attractive as he had the night before, sans the irritation, his hands framed the doorway as he leaned into them, taking up the whole space. Sandy blond hair, a little on the longish side, curled around his ears and nape. Blue eyes damn near the color of the Pacific in June warily stared at her. A little gray mixed in to keep them from being too potent. His jaw had a day's worth of scruff and the man rocked a set of powder blue scrubs under an open leather coat.\n\nGod. He was an eye-gasm if she ever saw one.\n\nWhen he didn't say anything, her heart started to pound. \"Oh, no. Is...Seraph all right?\" She turned to peek at Hailey, who was doing a numbers app game on the iPad at the kitchen table.\n\n\"He's doing fine. Recuperating very well.\"\n\nHis voice had her pausing, just like the night before, but today it wasn't as angry. It wasn't quite coarse or too deep, but there was a melodic rhythm when he spoke. Great. So his voice was an ear-gasm on top of his too handsome looks. Crap on a cracker.\n\nShe caught herself wrapping the towel around her hand in nervousness and stopped. \"Then why are you here?\"\n\nPushing off the frame, he towered over her five-six height to what had to be six feet. \"I came to apologize. May I come in?\"\n\n\"Um, sure.\" She held the door open wider and glanced at Hailey. If he got upset again or had a naturally loud voice, it could upset her. \"Sweetie, why don't you do that in your bedroom for a few minutes? I'll be in soon.\"\n\nHailey grabbed her device and headed down the hall.\n\n\"She doesn't talk much, does she? For a girl, I mean. Thought they were all chatter boxes.\" He laughed uncomfortably and rubbed his neck.\n\nThe guy had adorable charm in spades.\n\nHis nervousness calmed her a bit. \"She's a nonverbal autistic. Pretty high functioning in other areas, though.\"\n\nHe stilled and pinned her with round eyes. \"I'm fu...mucking this up.\"\n\nShe smiled, more at ease with this version of him. Pretty sweet how he'd caught himself upright before cursing. \"You didn't know. It's okay.\"\n\nGazing heavenward, he crossed his arms. His leather jacket strained against his muscles and movement. \"Look, about last night, I'm sorry. I was rude. I thought you were a negligent owner. I'd had a bad day and took it out on you.\"\n\nAt the risk of sounding like a parrot, she said the only thing that came to mind. \"It's okay. You didn't know.\"\n\nThe flutter in her belly hadn't happened in so long, she was unnerved. Plus, people didn't often apologize to her, and she'd spent so long in the shadow of her ex-husband that she didn't know what to do with Cade's intense stare.\n\nOne corner of his mouth quirked. Too cute to be sexy, but head-tripping just the same. \"You never corrected me. You could've put me in my rightful place in the exam room. Why didn't you?\"\n\nShe glanced at the hallway and back to him. He was still staring as if trying to figure her out. \"Hailey gets nervous around raised voices. I figured we'd sort it out later.\"\n\nHe nodded slowly, a wrinkle forming between his brows. \"Did I upset her?\" He took a step forward and paused as if wanting to reach for her and thinking better of it. He seemed genuinely concerned about Hailey, the look in his eyes sincere.\n\n\"I don't believe so. She was worried about the dog, but I think she filtered everything else out.\" They stood awkwardly for a beat. Now what? \"Can I get you something to drink?\"\n\n\"No. I have to get back to the clinic, but thanks. I wanted to catch you before you came in so I could eat crow.\" One shoulder lifted.\n\nGod. The female species must flock to him. He didn't have a wedding band.\n\nShe tried for upbeat to put them on even footing. \"Without witnesses? It's like the apology never happened.\"\n\nHis face grew a little impish, which made her laugh. Been awhile since she did that, too. \"You haven't met my brothers yet. Or my aunt. She runs the front desk. For now. I'd never live it down if they got to see me removing my foot from my mouth.\"\n\nIt was a nice mouth, too. Full lips, firm. She shook herself just before a tremble could tear through her body.\n\nHe turned for the door. \"I'll see you when you come visit Seraph. He really is doing great. Nipped my stethoscope this morning trying to play.\"\n\nRelief settled deep. \"Okay. I'll be in soon. I had...stuff come up this morning.\" Like the moving van not following her directions and getting lost on the wrong side of the Klamath. They claimed it would be a few days before they arrived.\n\nThey reached for the doorknob at the same time. Their fingers brushed, and it seemed so intimate, she froze. He had big hands. Warm. Then she caught a whiff of something...gastric emanating from him.\n\nShe must've wrinkled her nose or something because he laughed nervously. \"Hazard of the job. That's Ode de Anal Gland you smell.\"\n\nShe pressed her lips together to hide a grin.\n\nHe closed his eyes and shook his head. \"I just keep sticking my foot...\" He sighed. \"I'll see you soon.\"\n\nShe waited until his car was gone before turning toward the hall. The muscles of her cheeks ached and she realized she was smiling. When was the last time that happened?\n\nAfter getting Hailey ready, they drove to the clinic. Hailey was a bundle of excited energy bouncing in her seat. Avery had never thought about getting a pet. Her ex, Richard, wouldn't have allowed one in the pristine house anyway, but Hailey had really seemed to connect with Seraph. They'd have to stop by a store to pick up supplies on the way back. What did a puppy need?\n\nShe opened the door to Animal Instincts and found...utter chaos. Unlike the night before, people filled the waiting room with varying breeds of dogs and cats. And...a snake? Yes, a big, big...\n\nShe chanced a peek at Hailey to see if the loud ruckus bothered her. It was hard sometimes to find her triggers. Car horns and loud music upset her the most. Shouting or raised voices, too. She seemed unfazed by the barking now.\n\nThe clinic was more spacious than it looked from the outside. To the left, the waiting room walls were painted in one giant mural of animals doing human things like cooking or reading. Two large windows flanked both sides to allow light inside. Slate tile was laid throughout, creating a natural feel to the place. She remembered from the night before that the hallway leading to the patient rooms had another mural of dogs walking their owners or cats petting their humans in a cute role reversal.\n\n\"Can I help you?\"\n\nAvery turned to the front desk. The middle-aged receptionist had unusually red hair and her penciled-in eyebrows pinged to her hairline. Her short, round frame leaned over to better see them.\n\n\"Um... We're here to visit our dog. We brought him in last night.\"\n\nThe receptionist's demeanor changed from irritated inquisitiveness to cajoling so fast Avery got whiplash. \"I'm Rosa. I'm the O'Grady boys' aunt. You must be Avery. Your mom and I are good friends. It's so nice to finally meet you.\" She came around the desk. \"And you must be Hailey.\" She squatted in front of her, but Hailey's attention was elsewhere.\n\nAvery didn't like walking around telling people why her daughter was seemingly so rude. In fact, she hated putting a label on her at all, but the only way for others to understand was to explain. Before she could open her mouth, Rosa stood.\n\n\"The rec center has a few other autistic kids, too. Miles and Anya run the place. You'll have to check it out.\" Glancing at Hailey, she spoke to her again. \"You'll like it there, hon. Plenty to do. Make new friends.\"\n\nSadly, Hailey didn't have any old friends. After her diagnosis at age two, their lives had been a steady stream of therapists.\n\nRosa waved her hand and assessed Avery as if plotting. Eyes narrowed, she nodded her head in approval. \"I'll get Cade to take you back to see Seraph.\"\n\nAvery eyed the waiting room. \"You guys are busy. Can we maybe just visit on our own or we can come back...?\"\n\n\"Nah. We're pretty slow today.\"\n\nSlow? Patients were packed like sardines. She hated to see what busy looked like. A cat hissed behind them and batted its paw at a poodle wearing a red bow and an incredulous expression. Two dogs were vying for butt-sniffing rights while two others cowered under chairs. The snake had slowly wound itself around its owner's arm and was climbing the wall near a framed picture of dogs playing poker.\n\nShe shuddered and turned back, but Rosa was already behind the desk and Cade was walking toward them. He'd changed into a darker set of blue scrubs. God. Just...he was all masculine grace as he ate up the distance. Sinewy muscle, wide shoulders, narrow waist. Avery bit her tongue so she didn't swallow it.\n\nHe grinned and, damn, there went her air. \"Come on back.\" He glanced at the waiting room and did a double take. \"George, get that reptile in a cage.\"\n\nWordlessly, they followed him down a long hall and into a back room where crates lined the wall with recovering animals. Yips and feral cries echoed off the walls.\n\n\"This is where we board.\"\n\nShe nodded. The twelve-by-twelve room smelled like wet fur and antiseptic. The walls had a continuation mural of grassy fields and blue sky. Several red fire hydrants were painted among the landscape. Pretty clever.\n\nHailey took off and knelt by one of the crates.\n\nBefore Avery could correct her, Cade strode over and squatted beside her. \"Hold on, squirt. I'll get him out for you.\" After opening the cage, he reached in and removed Seraph, nuzzling the puppy to his broad chest. A cone had been placed around the dog's neck, but Seraph nudged Cade's hand for petting. \"Who's a good boy?\"\n\nOh. Oh, swoon.\n\nShe hadn't had so much as a flicker of attraction in so long she almost forgot the sensation. Her cheeks heated and her belly quivered. A flare of jealousy sparked when Seraph licked Cade's face. The rumble of Cade's laugh had her biting back a moan.\n\n\"Hey, squirt. Can you sit on the floor? Your little buddy might not be up to running yet.\"\n\nAs usual, Hailey took a few seconds to process his words, then dutifully sat on the tile floor. Even her daughter seemed smitten. She squealed and flapped her hands. _Yes, cute doctor man. Whatever you say._\n\nCade gently set Seraph in her lap and, keeping one hand on the puppy, he lifted the other to Hailey. \"I'm going to show you how to pet him.\" He took her hand and used it to stroke the dog's back. \"Just like that,\" he said as if coaxing calm. \"Good. Just stay away from the ouchie on his leg, yeah?\"\n\nAvery made a conscious effort to shut her mouth. Good with animals _and_ kids. Not just kids, but disabled ones. He'd told Hailey what he was going to do before doing it and used a soothing voice for both her and Seraph.\n\nDouble swoon.\n\nCade grinned as Hailey giggled, and then he looked at Avery. \"Best friends already.\"\n\n\"Yeah,\" she whispered. Was she drooling? She cleared her throat and stepped closer. Her heart swelled at Hailey's complete adoration of the dog. \"When can we take him home?\"\n\n\"Tomorrow, if he's still looking good.\"\n\n\"At the risk of you getting upset, I need to know what he requires. I've never had a pet before.\"\n\nA flash of irritation lit his eyes before he blinked it away. He stood and dusted fur from his pants. \"I'm sorry about last night. How I behaved wasn't me. Don't be afraid to ask me questions.\" As if in afterthought, he added, \"I won't bite.\"\n\nWhat. A. Shame. _Bad, Avery_!\n\nStaring at her with intense, unwavering eyes, he sighed. \"You'll need a leash until you can train him to mind commands. A couple of dog bowls for food and water. Toys, especially ones he can chew on until...\" He rubbed the back of his neck. \"Know what? I'll take you out to the supply place and steer you in the right direction.\"\n\nHer jaw dropped. Again. \"You don't have to do that. If you just make a list\u2014\"\n\n\"How's seven tonight? We close at six. That'll give me time to shower.\" Though his tone was genial, it brooked no argument. He held her gaze, patiently waiting for an answer. When she didn't say anything\u2014because how long had it been since someone offered to go out of their way to help?\u2014he jerked a thumb at the door. \"I've got to get back to the patients, but meet me here at seven, yeah? I'll send Rosa in to help Seraph back into the kennel when you're done.\"\n\nWith that, he strode out, all alpha-male goodness.\n\nThe move and pressure he'd put on her reminded her of Richard. But Cade wasn't like her ex. Where Richard was controlling and cold, Cade was confident and warm. His mood today was a vast cry from the brash guy she'd met last night. She wondered which version was the real man, and then remembered the self-depreciating humor and the way he'd smiled with even his eyes.\n\nRosa opened the door and the barking started anew. \"Hush,\" she called and went right to Hailey. \"Time to put him back, but you get to take him home tomorrow. How exciting.\"\n\nHailey jerked her gaze around the room, upset.\n\nAvery patted her arm through the coat. \"They'll take good care of Seraph, sweetie. We'll see him tomorrow.\"\n\nRosa eyed Avery skeptically. She got the suspicion she was being dissected again, but had no clue why. \"How are you settling in? Redwood Ridge is quite the difference from a big city.\"\n\nAvery nodded. \"It is, but it's very pretty. It'll just take some getting used to.\"\n\n\"Have you found a job yet? Your mom said you were looking.\"\n\nShifting her gaze from Hailey\u2014who walked past the kennels\u2014to Rosa, she shrugged. \"I haven't had a chance yet.\" She'd been out of the workforce so long she didn't think she'd be hirable. Her savings would only stretch so far. \"Do you have a medical clinic in town?\"\n\nRosa pursed her lips. \"We've got Dr. Brad Crest at the edge of town. Otherwise you'd have to travel forty miles north. Why?\"\n\n\"Well, I've been a stay at home mom since Hailey was born, but before that I managed a cardiology practice.\" In fact, Richard had insisted she resign to remain at his beck and call for fundraisers or business dinners. He'd just used Hailey as an excuse to isolate her even more from society.\n\nShe squared her shoulders. Richard was a jerk of epic proportions, but she couldn't lay all the blame on him. She hadn't stood up to him or challenged his orders. But she'd gotten Hailey because of the sham of a marriage, and they were starting fresh. No sense dwelling on the past.\n\n\"Dr. Crest has a nurse and receptionist, so you're out of luck there.\" Rosa raked her gaze over Avery and nodded. \"You'll do. You're hired.\"\n\n\"What?\"\n\nRosa shrugged as if she hadn't just knocked Avery on her ass. \"I'm retiring. Cade's been dragging his heels finding a replacement, and you have experience.\"\n\nShe rubbed her forehead. \"I have experience in healthcare, not veterinary medicine. You haven't even checked my references.\"\n\n\"Don't need to. You're Justine's daughter, which is good enough for me. Besides, you're not going to be treating the animals, just running the office.\"\n\n\"True, but\u2014\"\n\n\"When can you start?\"\n\nAvery opened and closed her mouth several times. She did need a job and, though she hadn't been employed in a long time, it was work she could handle. It seemed a little too good to be true. She glanced at Hailey as she sat in front of Seraph's crate. Avery needed to get her enrolled at the elementary first, plus find someone for after school care.\n\n\"Um, Monday?\" She looked at Rosa, who was grinning like a Cheshire cat. \"What about my hours or salary or insurance?\"\n\n\"We're open from eight to six, Monday through Friday, and eight to noon on Saturdays. The office is closed on Sundays. You'd have Saturdays off, as that's just an emergency day for on-call. We'll get you enrolled in insurance right away, but you'll have to wait sixty days for it to be active. What are your salary requirements?\"\n\nHead spinning, she tried to recall what she made at the cardiology practice and quoted that number. \"Are you sure about this? It's awfully fast and you just met me.\"\n\nRosa slapped her shoulder. Avery nearly toppled with the force. \"Welcome to Redwood Ridge. Small town life is different than where you came from. We look out for each other around here. Word of mouth is grail and everyone knows your business.\"\n\nRight. Right, okay. Wow. She had a job.\n\n\"Come on. I'll show you around and introduce you to everybody.\"\n\nAvery called for Hailey and followed Rosa to the front desk. Charts were piled on every available surface, the phone rang off the hook, the waiting room spilled with patients, and a giant dog was cowering under the desk.\n\nA white bird with pretty yellow feathers standing straight up on its head ruffled its feathers from a stoop near the window. _Squawk._ \"Welcome to the jungle.\"\n\n# Chapter 3\n\nCade stood next to Flynn in his brother's office in front of the lightbox, staring at the x-ray for a Yorkie who'd eaten a ball of yarn. \"Crazy, yeah?\" His hands moved quickly to sign the words.\n\nFlynn nodded. \" _Wrapped completely in the intestinal track. Drake's got another surgery on his hands.\"_\n\nCade turned at the knock on the door. Because Flynn was deaf, most people didn't bother knocking. Aunt Rosa walked in, followed by Avery and Hailey.\n\n\"Meet Avery Stowe, your new office manager. She starts Monday, so consider this my notice.\"\n\nCade's gaze darted between Avery and his meddling aunt. Avery shuffled her feet, her cheeks pink as she stared at the floor. He and Flynn exchanged a look. Cade signed, \"She's the one who found the stray last night.\"\n\nFlynn grinned, the asshole. \" _The one that had you bumbling like a hormonal teenager? Nice. You're right. She's pretty.\"_\n\nHe sighed. This is what he got for talking to his brother. About anything. \"Shut up.\"\n\nAvery cleared her throat, then she...well, shit. She signed and spoke simultaneously. \"You're deaf?\"\n\nFlynn nodded as Cade reared. Not a lot of people knew sign language. Flynn could read lips if he was facing the person, which was how he typically got around the language barrier in the practice.\n\n\"You can sign?\" Which meant she'd understood the brief exchange between him and his brother. Wonderful. He just kept shooting himself in the foot.\n\n\"Yes. I learned how to communicate with Hailey, since she's nonverbal. Sometimes she signs when she needs to say something.\"\n\nCade was floored she was courteous enough to sign and speak at the same time for Flynn's benefit. And then he processed what she was doing here. \"You hired someone without asking?\"\n\nRosa crossed her arms. \"You weren't gonna do it.\"\n\nFlynn was more amused than Cade. \" _Does she have any experience?\"_\n\n\"I worked in a medical practice before staying home with Hailey. I'm a quick study.\" She was biting that pouty lower lip of hers. Her dark eyes bespoke more than simple nervousness, though. Like she had something to prove.\n\nFlynn nodded. \" _Good enough for me. Welcome_.\" He offered Cade a raised brow that said, _this should be fun_.\n\nCade returned it with his own _shut-up_ - _or-die_ scowl. He scratched his jaw, knowing he had no say in the matter, anyway. It wasn't as if he was against hiring Avery, but it would've been nice to be consulted.\n\nRosa turned for the door. \"I'm going back up front. Take her to meet Drake, would you?\"\n\nThough spoken as a question, Cade took it for the order it was intended. What in the hell was his aunt up to? Before he could comment, his tech, Brent, strode in. Hell, now it was a party. He pinched the bridge of his nose.\n\n\"You are all caught up.\" Brent sized up Avery with the kind of flare only an openly gay man could pull off. \"And who is this?\"\n\n\"Avery, meet Brent, my tech. Brent, Avery is our new office manager. Apparently.\"\n\nBrent's eyes lit up. \"Fresh meat. Hurray. Have you met Gabby yet? She's the other tech. You'll love her!\" He took Avery's hand and started to drag her from the room.\n\nAvery laughed and dug her heels in. \"I have my daughter with me. Hold on.\"\n\nThat laugh made Cade's gut do something twisty. \"I can watch Hailey for a couple minutes.\" Her _are-you-serious_ expression matched his internal _what-the-fuck_. Like he knew anything about kids. Still, he nodded. \"Drake's in surgery, so Hailey can't go back there. Gabby's assisting.\"\n\nBrent, as if just noticing the tiny human in the room, clapped his hands and knelt in front of Hailey near the corner. \"Hello. Aren't you a cutie pie? Gorgeous hair, too.\"\n\nThe only sign of acknowledgement Hailey offered was a half wave as she stared at the ceiling. It was more than Cade got.\n\nMore hot as hell lip chewing from Avery. \"Maybe I should just wait until Monday to meet everyone else.\"\n\nHell, she'd only be gone a few minutes. How much damage could he do to Hailey's little psyche in that time? \"Brent says I'm caught up, so go ahead.\" He looked at his tech. \"Tell Drake about the Yorkie. He needs surgery.\" Then he thought it over and decided to tell Drake himself. The case was complicated.\n\nFlynn, as if sensing the thought, tapped his shoulder. \" _Go ahead. I got the kid_.\"\n\n\"Are you sure?\" Avery eyed Flynn as if begging him to change his mind, giving Cade the impression she didn't leave Hailey in others' hands often.\n\n_\"I promise to not let her play with syringes_.\"\n\nAvery laughed, which was twice now, and he hadn't recovered from the first. She told Hailey she'd be back in a minute and allowed Brent to take her by the arm down the hall. Cade followed.\n\nBrent gave her a mask. \"Just hold this over your face and don't touch anything. And no matter what Dr. Drake says, he's happy to have you working here.\" He leaned in conspiratorially. \"He's broody.\"\n\n\"Got it.\" Amusement lit her dark eyes and, for the first time, Cade noticed the small hazel flecks mixed in with all that chocolate.\n\nBrent knocked once and strode in with a flourish. \"Dr. Drake, Gabby, this lovely doll is Avery, our new office manager. Gabby, we must do drinks tonight.\"\n\nGabby didn't look up. \"Can't. Got plans with my cousin. And I told you I'm never drinking with you again.\"\n\n\"Never say never.\"\n\n\"I mean, _never_.\" She peeked over her mask. \"I still can't eat nachos.\"\n\nAvery's hand shook where she held the mask. Her olive skin tone had lost some of its hue. As office staff, she probably wasn't used to seeing an open operation and had a weak stomach for blood. His aunt was the same way. Cade hoped she didn't yak all over the room.\n\nCade held the mask over his mouth and leaned on the doorframe. It looked like Drake was ready to close up on the Bullmastiff who had a tumor.\n\nDrake never lifted his dark head from his task. \"I'm in surgery. Go away.\"\n\nBrent sighed. \"Translation: Avery, it's nice to meet you.\"\n\n\"That's what I said.\"\n\nGabby's blue eyes slid to Avery, a smile in them. \"Hey there. We're almost done. Are you new to...? Damn. Cade, she's gonna\u2014\"\n\nPass out cold on the floor. Down went Avery in a heap. _Shit._\n\nBrent blinked in awe. \"How southern belle of her.\"\n\nDrake growled. \"Damn it, Cade. Get her out of here.\"\n\nAs if he was the one who'd brought her in the first place, but he was already at her side and cradling her head. Silky brown hair spilled over his hand and her berry scent rose up over the antiseptic. He lifted her in his arms. \"Get the door, Brent.\"\n\nCarrying her down the hall to his office, he tried to ignore the softness of her body against the hard planes of his. Dark lashes fluttered and lifted. He set her on the couch and asked Brent to get a cool cloth, never taking his eyes from her.\n\nHe lifted a strand of hair from her face as her eyes cleared of confusion. \"Hey. Lie still for just a second. You fainted.\"\n\nShe gasped. \"I did not.\"\n\nHe fought a grin. \"Afraid you did.\"\n\nShe tried to sit up and he let her, removing his hand from her shoulder. Glancing around the confines of his cluttered office, she winced. \"How humiliating.\"\n\nBrent strode in and offered her a wet cloth. \"Personally, I thought it was entertaining. I mean, swoosh. Like a feather, you fluttered down. Don't like blood then, doll?\"\n\nShe dropped her head in her hands and sighed. \"I guess not. I don't have much experience with it. I'm so embarrassed. I was fine last night with Seraph, but I guess I was too panicked to notice.\"\n\nCade looked at Brent. \"Tell Flynn we'll be a few more minutes.\"\n\n\"Oh God. Hailey\u2014\" She tried to stand.\n\n\"Uh-uh.\" With his hands around her wrists, he coaxed her butt back on the couch. \"She's fine. Sit for a few.\" He used his soothing tone reserved for cray-cray animals because her eyes were wide and the pulse in her throat beat double time. He casually pressed two fingers to her inner wrist and took her pulse.\n\nWhen he looked up, satisfied her heart rhythm was normal, her round cheeks were infused with red and she avoided his gaze like he was the second coming of the bubonic plague. He kept his hands where they were, in hers, rubbing his thumbs over her palms. Her skin was soft in comparison to his rough calluses. His own heart rate sped at the contact.\n\n\"Does your head hurt at all? I don't think you bumped it.\"\n\nShe shook her head, looking at her lap. Not shy, but obviously mortified.\n\nHe got the impression she hated\u2014or was unused to\u2014attention. With great regret, he let her go and stood as he was only adding to her embarrassment. \"I'll get you some juice. Stand by.\"\n\nHe poked his head in Flynn's office. Hailey was sitting cross-legged on the floor, arranging his brother's paperclips into neat rows.\n\nFlynn looked up from next to her. \" _I think we need to hire her, too. Great attention to detail.\"_\n\nCade laughed. \"You can send her back to my office. Cover my patients for a few, would you?\"\n\nHe strode to the break room, snatched a small bottle of OJ, and went back into his office. Avery had zipped Hailey into her coat and was tying a scarf around her neck.\n\nHe handed her the juice. \"Take a few sips before driving.\"\n\nShe nodded and uncapped the bottle, obliging him. The column of her throat worked a swallow and his gaze landed there. He wondered if she'd tasted as good as she smelled. Strawberries or melon. Something summery.\n\nRecapping the bottle, she finally met his eyes. \"Thank you. I'm sorry about...\" She waved her hand.\n\n\"Happens to everyone. Now we know not to put you in surgery.\"\n\nShe tapped Hailey's shoulder. \"Let's go, sweetie.\" Her gaze landed on his and away, so unlike the confident woman he'd met at the cabin today. \"I'll see you on Monday.\"\n\n\"Tonight.\" At her raised brows, he elaborated. \"Pet supplies. Instead of meeting here, I'll pick you up. Still seven?\"\n\n\"Yes, that's fine. Thank you.\"\n\nFrom his position in the doorway, he could just make out her retreating form as she passed the front desk.\n\n_Squawk._ \"Pretty woman.\"\n\nIndeed. She wasn't a sultry bombshell and not quite the girl next door, either. A conundrum. One minute she was all mama bear, the next she was tender, watching others with her kid. She had a sense of humor, displayed confidence, and yet he sensed a vulnerability she'd probably deny. Fascinating.\n\nFlynn walked by and slapped a chart against Cade's chest. \" _My Precious is acting 'off,' room five. Brent's in surgery with Drake. Gabby and I are hitting the road.\"_\n\nCade nodded. Flynn and Gabby traveled most of the day to local farms or elsewhere on home visits. They were getting a late start today.\n\nHe eyed the chart. Owners who brought in animals claiming to be \"off\" was code for: single woman alert. They only booked an appointment and used their pets as an excuse to make a play for the vet. He was used to it. A piece of meat. Ten grand said there was a plate of cookies or a casserole up front waiting for him. Just about every available female\u2014and some unavailable\u2014seemed to think the way to his heart was through his blood sugar levels.\n\nDrake had been a widow for almost four years now and, once most women figured out the broody fa\u00e7ade wasn't, in fact, a fa\u00e7ade, they moved on. With Flynn being deaf, a lot of potential dates passed him by because some people were just that damn shallow. Cade wasn't vapid enough to deny he and his brothers were attractive, or so Redwood Ridge's Twitter profile claimed, but it was Cade that got the most attention.\n\nThe funny one. The easy guy.\n\nThe good time.\n\nScrubbing his hands over his face, he sighed. Eying room five, he strode down the hall. _Slab of beef, coming right up_.\n\n* * * *\n\nAfter Avery got Hailey registered for second grade and talked with the special needs teacher on staff, she drove across town to her mom's clothing store to figure out after school care. Back in San Francisco, Hailey attended classes in the mornings and had therapists come to the house in the afternoons. Avery was always there. The only babysitter her daughter ever knew was a nanny they'd used when they'd had to attend a function for Richard.\n\nShe rubbed her forehead. This was a lot of change for Hailey all at once. New town, new home, new school, and now Avery wouldn't be with her as often as she was accustomed. For the longest time, it had been just her and Hailey. She'd been in Redwood Ridge less than twenty-four hours and she'd had more offers of help and welcome than the span of Hailey's life combined.\n\nThat was a tough thing to get used to\u2014ceding control. Hailey wasn't like other kids. Even leaving her with a sitter just to run to the store was a project. Plus, Avery feared she'd lost a lot of her social skills in the past few years, first with Richard being embarrassed by Hailey and wanting to keep her hidden, and then by the eighteen months it had taken for the divorce to go through. Rejected and emotionally drained, Avery hadn't had many friends during her marriage and had even less when she'd left Richard.\n\nShe pulled into a parking space in front of her mother's building and cut the engine. Pocketed among a row of other stores, it was similar in appearance. Small, brick, and two-story, it was square in design with a dark green awning over the front door. People milled about at caf\u00e9 tables or strolled the cobblestone walk, chatting, despite the chill in the air.\n\nExiting the car, she inhaled pine and snow while getting Hailey unbuckled. A thick fog blanketed the mountains in the distance, bringing a slightly humid quality to the air. A tang of brine hinted from the nearby ocean. It was another world from what she'd been used to, but she found herself liking it, the freshness.\n\nHand in Hailey's, she strode into the store and sought out her mom at the front desk. Scarves and hats lined the back wall in front of ten or so round racks with vintage clothes. It smelled old, like musk and time. Two other women hovered near the register. The chatter died when she stepped over.\n\n\"Oh, Avery.\" Mom pressed a hand to her chest. \"We just heard. Are you okay?\" She came around the counter and squeezed the breath from Avery's lungs, bathing her in patchouli.\n\n\"I'm fine. Heard what?\"\n\n\"About you fainting, of course. Rosa tweeted about it and I was just about to call.\"\n\nHer cheeks heated. She still couldn't believe she'd passed out right in front of everyone. And what did she mean Rosa had tweeted about it? Avery looked at the other two women close to her mother's age, a combination of curiosity and avid interest in their eyes.\n\nOne of them patted her arm as if to say, _poor dear_. \"It's all over the thread. How are you feeling? I'd drop like a fly, too, if they made me watch a surgery.\"\n\nThe vets hadn't made her do anything, and Avery had no way of knowing she'd respond that way. Still, mortification scorched her cheeks. \"Um, I'm fine. Why did Rosa tweet about me? I don't understand.\"\n\nThe woman laughed as if to say, _silly, silly girl_. \"Why, everything is news around here. Rosa's my sister. She manages our Pinterest boards and Twitter account. I'm Marie, the mayor of Redwood Ridge. Welcome to our great town.\"\n\nCrap on a cracker. Avery darted a glance at her mom, but no. She hadn't stepped into the Twilight Zone. She was really standing in her mother's store talking to the mayor about passing out not two hours before. Because it was on Twitter.\n\nShe cleared her throat. \"Thank you.\"\n\nThe other woman extended her hand. \"I'm Gayle, Rosa and Marie's sister. My sons are your new bosses.\"\n\nOh. Oh God. She saw the resemblance to Cade and Flynn now. Flynn had her coloring with reddish hair and pale skin. Cade had her blue eyes, gray flecks included, and her impossibly long eyelashes. \"Very nice to meet you.\"\n\n\"What brings you by, honey?\" Mom scooted back behind the counter.\n\nWould this conversation go on Twitter, too? She eyed the other two women, but they seemed in no hurry to leave. Was nothing private in this town?\n\n\"I just got Hailey registered for school, but I need to figure out care afterward. I'll still be at the clinic when she gets out.\"\n\nMom fluffed her already wild brown hair. \"That's easy enough. I can pick her up and bring her back here.\"\n\nAvery opened her mouth to protest when Marie inserted, \"Or you could have her attend the recreation center afterward. Some of the kids take the bus right there.\"\n\n\"Oh, I don't know.\" Avery glanced at Hailey. \"She's...\" Not like other kids. Different. She didn't want to say that aloud to strangers or make Hailey feel like an outcast. \"Her needs are pretty particular. Everyone right now is unfamiliar to her.\"\n\nMarie waved her hand. \"Pah. Strangers are friends we've yet to meet. Miles and Anya have other special needs kids at the center, too. She'd be in good hands.\" She whipped out a cell phone from her Samsonite-sized purse and thumbed a text. Her dark brown bob didn't move as she bent her head. \"There. All taken care of. They'll expect her on Monday after school. You should stop by today to introduce her, though.\"\n\nAvery forced her jaw shut, teetering between anger and shock. A pounding thumped her temples. This was her life, her daughter, and she'd spent enough time under her ex's rules to ever go back to someone else running her life.\n\nBoth Marie and Gayle knelt in front of Hailey and started talking to her as if they were old friends. Even though Hailey didn't converse, the ladies weren't deterred. In honesty, Richard's family and friends ignored Hailey, so the fact that anyone acknowledged her took the anger right out of Avery's sails.\n\nThis wasn't the first time, either. Flynn and Brent had made it a point to address her. They all seemed to know Hailey was autistic, probably because of her mom\u2014or Twitter\u2014but they didn't act uncomfortable. They... _included_.\n\nHailey ate up the attention, flapping her hands and letting out one of her rare barks of laughter. Avery's chest swelled, her throat going tight.\n\n\"We'll...\" Avery cleared her throat. \"We'll stop by the rec center to check it out. Thank you.\"\n\nMarie rose. \"No problem at all. You pop by my office or drop me a call should you need anything. Ta, Justine.\"\n\n\"Bye-bye.\" Mom sighed dreamily as they disappeared down the walk. \"I told you this place was great, didn't I?'\n\nAvery shook her head, not sure if she was being Punk'd. \"Twitter? I was on Twitter?\"\n\nMom looked at her as if she were the crazy one. She tapped out a few keys on the computer and waved her over.\n\nAvery made her way around the counter and looked at the screen. Sure enough, there was the @RedwoodRidge account. The banner was a wide pan of the main strip, and the profile picture was a sketch-drawn map with a tiny dot indicating the town. The most recent tweets were like that from a gossip rag.\n\nWord is our sexy fire dept will be doing another car wash this spring. Rawr, ladies!\n\nDr. Cade says the PB butter cookies are his fave. Dr. Flynn says choc chip. Dr. Drake had no comment. Psst, it's snickerdoodles!\n\nThe RR marching band needs new uniforms. Buy candy bars, peeps!\n\nWildlife is so gorgeous!\n\nAttached to that last tweet was a picture of what looked like a ranger leaning against a cypress and wiping his sweaty brow with his forearm. Lordy. Were all the men in Redwood Ridge hot? The veterinarians were.\n\nNew gal in town, Avery, fainted right in our OR. Poor thing!\n\nAvery rubbed her forehead, her gut churning. There were fifty-three @ responses ranging from _oh no_ to _the_ _dear lass_ to _hope she's okay_. She pulled out her phone, followed Redwood Ridge, and tweeted: _Doing great. Thanks for the concern!_ Jeez. She had one hundred and twenty-one new followers.\n\nThis _was_ The Twilight Zone. In Mayberry.\n\nShe needed to get out of here. Stimulation overload.\n\nAfter giving her mom a hug good-bye, she drove to the rec center to check it out. It was everything the mayor said it would be. They had several after school programs, and the woman who ran the center, Anya, had special needs experience. Figuring she'd give it a try, Avery signed Hailey up and checked the time.\n\nThey only had an hour before Cade was supposed to pick them up to get pet supplies. They'd been living on fast food the past two days while driving, but there wasn't time to make a decent meal and still be ready.\n\nShe drove through a hamburger joint and parked outside an ice cream shop so they could eat in the car. Avery mentally added another twenty minutes to her yoga tomorrow morning to make up for the food. She'd always been just north of chubby, but it seemed after she'd delivered Hailey, the pounds just kept tacking on.\n\nRichard hadn't wanted to have sex until she'd gotten back to her pre-pregnancy weight. He'd also attended all functions alone until the result of nearly starving herself and a rigorous workout routine got her back into a size twelve. Richard might be gone from her life, but the rejection and ugly feeling remained.\n\nShe longingly looked at the ice cream parlor and decided to go back to her healthy lifestyle tomorrow. A few days' setback wasn't going to kill her. Turning in her seat, she grinned at Hailey. \"How about dessert, sweetie?\"\n\nHailey squealed. _Sugar! Sugar, now!_\n\nThe kid at the counter was about sixteen and in that awkward stage of gangly limbs before his body filled out. He looked bored to be there, but welcomed them with a drone greeting.\n\n\"I'll have a scoop of...\" Oh God. They had rocky road. Rocky road was kryptonite. \"Uh, vanilla.\" Sigh. Cut at least a couple calories. \"And do you have anything nondairy?\"\n\nThe teenager rolled his eyes. \"Dad!\"\n\nA portly man in his fifties strolled out of the back and wiped his hands on an apron. \"Well, hello. I'm Hank. New in town? Just visiting?\"\n\nAvery went through the spiel about being Justine's daughter and just arriving.\n\nHe rubbed his bald head, a habit he seemed unaware of doing. \"Heard you took a tumble at the animal doc. How are you?\"\n\nShe resisted her own eye roll. Had anyone _not_ heard about her embarrassing intro to her new job? \"Great, thanks. I was wondering if you had anything nondairy? Hailey here gets rather sick from milk products.\"\n\nThe teenager walked to the freezer and scooped vanilla into a dish.\n\n\"Nothing frozen, I'm afraid, but I've got some cookies from Sweet Tooth down the road.\" He set a couple cookies in a bag and handed it to Hailey over the counter. \"On the house.\"\n\n\"Oh no. I want\u2014\"\n\n\"Nonsense. On us. Next time, we'll have something she can eat. How's that?\"\n\nGod. Everyone was so... _so nice_. \"Thank you.\"\n\nStill shell-shocked and contemplating whether the town was really inhabited by _Invasion of the Body Snatchers_ , she edged her car to the outskirts. Dusk was descending, bringing a nip to the air and making the shadows longer. As she pulled up to the row of rental cabins, Cade unwound himself from the porch steps and rose.\n\n\"Crap on a cracker, that man is sexy.\" He had on a pair of well-worn jeans, snug in all the right areas, and a T-shirt under a waist-length leather jacket. His dark blond hair was windblown and brushed his brow. She shook her head. \"He's your boss, you nutcase. Get a grip.\"\n\nHailey squealed _. I heard that, Mommy_.\n\nAs she unsnapped Hailey from the backseat and turned to face him, he...grinned. At Avery, then at Hailey. The breath left her lungs, and she got a little lightheaded again.\n\n# Chapter 4\n\nCade used his rearview mirror to glance at Hailey in her booster, then turned his attention back to the road. The pet supply place was forty minutes north, but Hailey seemed content with an iPad and headphones. It was odd seeing a little person in his backseat.\n\nHe cleared his throat. \"If you don't mind me asking, where's her dad? You're from San Diego, right?\"\n\n\"San Francisco.\" Avery turned to check on Hailey and resumed her spot in his passenger seat. \"I got full custody in the divorce.\" She went quiet for a beat. \"My ex didn't contest it.\"\n\n\"The divorce or custody?\"\n\n\"Either.\" She turned her focus to the side window while Cade's head swam. Some people were too stupid to know good when they had it.\n\nThey drove alongside the Klamouth Mountains, heading toward the Southern Coast Range in a contented quiet. With the occasional curve, glimpses of the Pacific came into view, the shoreline and cliffs rocky. The fog was heavy near the bank in the late dusk, but not nearly as much as his thoughts. Her words slammed around in his skull.\n\nGranted, he'd never been married, nor was he a father, but he couldn't see himself giving up rights to his kid. He'd probably learn more from Redwood Ridge's Twitter page than from Avery, but he was curious about her. \"Was he a total prick, your ex? Pardon my language.\"\n\nShe breathed a laugh. \"Doesn't matter. He's not in our lives anymore.\"\n\nGood point. \"What's his name?\"\n\nFor that, she turned to study him with intense brown eyes. There was so much hiding in her gaze that he had to force his away or wreck the car trying to figure her out.\n\n\"Richard. That's his name.\"\n\nHe grunted. \"So he _is_ a dick.\"\n\nCovering her face with her hand, she laughed. It sounded rusty at first, but it gained momentum. She had the kind of laugh that drifted into corners and made a person stop to listen. His chest did that funny twisty thing again at the sound, but damn, it felt good to make her laugh. He hadn't known her long, or at all, but it seemed to him she didn't do it often.\n\nShe sighed and laid her head against the seat. \"I suppose it's no secret, anyway. Yeah, he's a jerk. I think he just wanted a trophy wife, which was fine, I guess, until he started treating Hailey like she wasn't there. I filed for divorce two years ago and just got it granted last month. He stretched out the proceedings that long.\" She tilted her head toward him. \"Word to the wise, don't marry an attorney.\"\n\nHe fisted the wheel. She didn't strike him as a trophy wife, and the whole ignoring Hailey thing pissed him off. There was a wealth of information she wasn't telling him, but he let it go. \"What do you call fifty lawyers chained together at the bottom of the ocean?\"\n\nHer grin stole his air. \"A good start.\"\n\nHe laughed. \"A woman who knows her lawyer jokes. Impressive.\"\n\n\"Why don't snakes bite lawyers?\"\n\nHe shook his head. \"Why?\"\n\n\"Professional courtesy.\"\n\nLaughing, he scratched his jaw and made a couple of turns before speaking again. \"So you left the big city to come home to Redwood Ridge.\"\n\nShe appeared to mull that over. \"I don't know if it's home, but my mom's here and I think it's a better environment for Hailey.\"\n\nIf he traveled a thousand places, Redwood Ridge would always be home. Crazy and annoying and irritating as it could be sometimes, it was home. \"You'll settle in. Might take awhile to get used to things, but the people care about each other, take care of one another.\"\n\nShe nodded. \"It only took an hour for half the town to know I'd passed out at the clinic.\"\n\n\"Only an hour?\" he joked. \"Aunt Rosa's usually faster than that.\"\n\nShe grinned, but it fell flat. \"I met your mom today, and your other aunt. The mayor?\"\n\nHe made a dismissive grunt. \"Aunt Marie. My brothers and I call the three of them the Battleaxes. An impenetrable force of evil wrapped in good intentions. They meddle. A lot. Kind of frightening, actually.\"\n\nAnother laugh. He was on a roll, even if he was only half kidding.\n\n\"Your mom seems nice.\"\n\n\"She's certainly the tamest of the three. Still, don't stand too close or look in their eyes. It's a trick.\"\n\n\"Noted.\" She paused with a lazy smile. \"What about your dad? What does he do?\"\n\nDear old Dad. A pang of longing hit his gut. \"He passed away from a heart attack nine years ago.\"\n\n\"Oh wow. So young. I'm sorry. That must've been hard.\"\n\nNo sugarcoating that. \"Took us all by surprise. He started the clinic thirty years ago. My brothers and I never thought of doing anything else but following in his shoes.\"\n\nShe nodded. \"Has Flynn always been deaf?\" Her cheeks flamed. \"Is that too personal?\"\n\n\"Nothing's too personal around here. And yes, he was born deaf. Fluke of nature.\"\n\n\"And Drake? I didn't get to see much of him, what with my eyes closed and me on the floor.\"\n\nHe barked out a surprised laugh. \"Drake. What to say about him?\" Nothing she wouldn't find out via town gossip. \"He's...mourning. He married Heather, his high school sweetheart, right out of vet school. She died from an aggressive form of ovarian cancer three and a half years ago.\"\n\nAvery was silent as she stared out the window, rubbing circles over her collarbone. As he was about to ask if she was all right, she cleared her throat. \"I would be devastated.\"\n\nDrake had been beyond devastated. He'd been wrecked. He was getting better, but it had taken Cade and Flynn a year to even get him to go anywhere besides the clinic, and another year to pack up Heather's things to send for charity. Seeing his brother like that made Cade never want to fall that hard for someone, never sink that deep.\n\n\"He took it rough. You haven't met her yet, but Zoe was Heather's best friend. Zoe's our groomer. She has a little workshop attached to the back of our building.\"\n\nShe seemed lost in thought as he made the turnoff for the store. \"Who did the murals in the clinic? They're wonderful.\"\n\nHe grinned, happy for the topic change. \"That would be Zoe. She paints when she's not wrestling dogs into bathtime submission.\" He parked and cut the engine before turning to look at her head on, his arm on the back of her seat, hand inches from her soft brown waves.\n\n\"Are you really okay with hiring me?\"\n\nSomething told him not to brush off her question too casually. And hmm. She had a light dusting of freckles on her nose he hadn't noticed before. Her berry scent rose to claim him. He never thought fruit could be such a turn on.\n\n\"I'm very okay with hiring you.\" Suddenly, he had the strongest urge to prove to her not all men were dicks. But he'd been an ass the night they'd met, so her opinion of him couldn't be that high. It didn't sit right in his gut. \"I'm sorry for the way I behaved when you brought in Seraph. Truly, I am.\"\n\nHer lips parted and her breathing grew deeper. \"You said that.\"\n\nForcing his gaze to hers, and not dropping it to her mouth like he wanted, he swallowed. \"Bears repeating. I'm sorry.\" He studied her another moment. \"Can you do the job?\"\n\nShe blinked. \"Yes.\"\n\nOne corner of his mouth quirked in a grin, the one he knew drove women crazy. Charming her had just become his mission. Damned if he knew why. \"Then stop worrying about it.\"\n\n* * * *\n\nTwo hours into her new job on Monday, Avery knew she'd been handpicked for the position by the divinity himself. To use the term clusterfuck would be putting too much of a positive spin on the lack of organization.\n\nThey were going by a paper chart system, and there was no rhyme or reason to where they were stored. Some were in the back room, some on the front desk, others in the doctors' offices. It made her brain hurt. There was a small storage closet off the patient room hallway that wasn't in use.\n\nAfter she'd finished her new employee paperwork, she turned to Rosa. \"Can I do some organizing?\" She didn't want to overstep her boundaries, especially on the first day, but to continue this Dr. Seuss system would waste patient time. Rosa would only be training her for two weeks before she retired, so now would be the best time to get anything done while someone was around to man the desk.\n\nA slow grin spread over Rosa's face. \"Organize, you say?\"\n\n\"Um, yes.\" Why was she grinning like that?\n\n_Squawk_. \"Crazy.\"\n\nAvery eyed the cockatoo on the perch by the window. She didn't know if the bird was calling Rosa crazy, Avery's attempt to organize crazy, or if it meant in a general sense. Either way, the feathered beauty was growing on her. It said the most random things and only spoke in song lyrics. She'd laugh if she could breathe among the clutter.\n\n\"You go right ahead, my dear. Organize to your heart's content.\" Rosa's grin was calculating, and after what Cade had told her a few days ago, Avery figured she'd best not ask.\n\nWithout a word, she made her way down the hall and propped open the storage closet, deciding to start in Cade's office. She eyed the two tall filing cabinets before chancing a peek inside the drawers. Empty. Shaking her head, she moved them into the storage closet along the wall, and proceeded to do the same thing with the empty filing cabinets from Drake and Flynn's offices.\n\nShe went back up front. \"Which charts are for deceased patients?\"\n\nRosa waved her hand behind her to the stack teetering by the printer.\n\nAvery found a tote in the back room and dropped those charts inside before dragging it to the storage room in a corner. With one wall lined with filing cabinets, the other was bare, so she moved a few filing cabinets from the front desk area into the storage room and got to work putting charts away and labeling the drawers. By the time lunch rolled around, she was to the M's.\n\nCade walked past the room, stopped, and turned. He eyed her handiwork and put his hands on the top of the doorframe, stretching his light blue scrub top over his muscles. \"Whatcha doing?\"\n\nCaught between filing cabinets and, well...a hard place, she pressed her lips together, trying not to stare at his yummy body. \"Charting.\" She paused. She had asked Rosa first, but Animal Instincts belonged to Cade, Flynn, and Drake. \"Are you mad I moved things?\"\n\nHumor infused his eyes, igniting all that blue. \"Nope. Why don't you go to lunch? Or better yet, head over to the deli with me.\"\n\nShe bit her lip. \"I was going to pop over to Hailey's school. You know, stalk her to see how she's doing.\"\n\nHis grin was slow and knowing. \"Nervous, Mom? I'm sure she's doing fine.\"\n\nShe rubbed her forehead. \"I know. It's just, she's...\"\n\n\"Never been away from you this long?\" He dropped his hands from the doorframe, still smiling. \"Go on then, mama bear. We'll get lunch another time.\"\n\nMama bear? His tone was amused, low, raking over her skin. She shivered. _Shivered_ , damn it! And why was he asking her to lunch? Before she could say more, he stepped away, leaving her to fan herself with a chart.\n\nBrent walked past, chuckled as if the tech knew she was having a hot flash, and sashayed away.\n\nLike it was her fault Cade was so lick-able.\n\nDonning her coat, she walked the few blocks to Avery's school to get some air and chewed on a granola bar. It tasted like cardboard with chocolate chips, but she swallowed it to get something in her stomach.\n\nBreathing deep, she inhaled humid air infused with pine and salt. The temperature remained in the upper thirties, but the stiff breeze was chilling. A low fog hovered in the distance, and Avery was learning it never really dissipated. Through rays of sun or storm-drenched clouds, it was always there, like a protective bubble for Redwood Ridge.\n\nShe passed many of the storefronts, figuring she'd make some time over the weekend to swing into them and check things out. The town square, set up more like an I-shape, was perhaps two miles long, with the vet office being near the southern end. The town catered to the tourism market with a caf\u00e9, bakery, bookstore, herbal cooking, and a candle shop, but there were also accounting offices, an attorney, and a dentist.\n\nAt the end of the street, she cut left and strode to the chain link fence encompassing the playground. She searched for Hailey, and found her off to the side with another little girl perhaps a year or two older. A teacher was helping Hailey bounce a rubber ball to the girl in a game of catch.\n\nShe stilled, fingers gripping the cold metal fence. Tears sprang to her eyes at the grin on Hailey's face and the bark of laughter that floated across the playground. Her chest swelled. Hailey had made a friend. On her first day! She wasn't distressed by the commotion of the other kids, but instead she...played.\n\n\"Is she yours?\"\n\nAvery turned to the woman next to her she hadn't noticed and swiped her eyes. She cleared the emotion from her throat. \"Yes. We just moved here.\"\n\nThe woman nodded, tucking a stray piece of reddish hair behind her ear. Her gaze trained back to the girls. \"That's my daughter, Jenny. Grew up my whole life here, but I still come by every day at recess to check on her. I can't help it. I work at the pharmacy. I'm April, by the way.\"\n\n\"Avery, and that's my daughter, Hailey.\" She glanced at the girls again, noticing the characteristics of Down syndrome in Jenny.\n\n\"Heard you fainted at the\u2014\"\n\nAvery groaned, earning a laugh from April. \"Who hasn't heard? I'm so embarrassed.\"\n\nApril's smile transformed her thin, regal face into something more approachable and friendly. \"Did you faint because of the hot docs or something gory?\"\n\nShe breathed a laugh. \"Gory. I walked into the surgery room and down I went. Though the vets are attractive, aren't they?\" She immediately bit her tongue at the unprofessionalism, her cheeks heating.\n\n\"Yep, all three of them. Smokin'. You'll learn soon enough the tactics some women will go to just to get their attention.\" April tilted her head. \"Not many single options here in Redwood Ridge, never mind selections that delicious. You're a lucky woman, getting to work with them.\"\n\nShe shook her head at the tease.\n\nApril shoulder bumped her. \"Oh, come on. You wouldn't be admitting anything the rest of us don't know.\"\n\n\"True. So what tactics have you used?\" If this wasn't the oddest conversation...\n\n\"Nah. I'm happily married. My husband's a truck driver, so he's gone a lot.\" April shifted on her feet. \"You're staying up at the rental cabins, right?\" When Avery nodded, April said, \"We're right down the road in the apartments. We should get the girls together sometime. They seem to be clicking.\"\n\nThey pulled out their phones and exchanged numbers before April headed back to work. Avery needed to take off, too, but she glanced at Hailey one more time. Sighing in contentment, she walked to the clinic, her heart so happy it hurt.\n\nUntil she walked in the door and found Drake leaning against the front desk, arms crossed and a surgical cap covering most of his dark hair. Flynn and Cade stood off to the side, watching her entry.\n\nHer steps slowed as she glanced at the clock, wondering if she was late. But no, she still had five more minutes. Dread pitted her stomach as she unbuttoned her coat with shaking fingers. \"Is everything okay?\"\n\n\"You,\" Drake said, pointing a finger at her, his face an unreadable mask. \"Did you do this?\" He jerked his chin at the lack of charts and newly available counter space.\n\nThere was still a lot to be done, but not if they were angry. She'd asked Rosa first.\n\nAvery slowly walked to the desk and edged around Drake, the granola bar she'd eaten sitting heavy in her stomach. \"The charts for today's appointments are here in this basket. When you're done with them, I figured you could just set them back here, and I'll file them away.\"\n\nCade dropped his chin to his chest, lips quirking as if fighting a grin. Flynn stood next to him, eyeing the ceiling. Both men's expressions were in direct conflict with Drake's.\n\nSilence stretched, but she kept her chin up. She'd done nothing wrong, had asked permission first, and heck, the place needed organizing. How had they found anything before she came along? And it had only been one morning.\n\nDrake straightened and stepped into her space. \"You,\" he said, stretching the word out, \"are a keeper. Well done.\" With that, he strode down the hall to his office.\n\nAvery's jaw dropped. She forced it closed.\n\nCade chuckled and swiped a hand down his face.\n\nFlynn signed, \" _Thank you_ ,\" and followed Drake.\n\nAt Cade's silent laugh, his shoulders bouncing, Avery narrowed her eyes. \"Was it necessary to freak me out like that? You guys just can't pat me on the back like normal people?\"\n\n_Squawk_. \"Don't fear the reaper.\"\n\nCade laughed harder.\n\nBrent and Gabby walked in from the back. Brent lifted his brows at Avery's irritation and Cade's hysterics. \"What'd we miss? Dish, doll.\"\n\n_Squawk._ \"Laughing on the outside.\"\n\nShe eyed the bird. \"Be quiet.\"\n\n\"Don't go breaking my heart.\" _Squawk._\n\nApparently, that had been the last thread of composure for Cade. He wiped his eyes and groaned in distress when he couldn't stop laughing. Walking past her, lips pressed together, he patted her on the back and followed his brothers.\n\nAvery rolled her eyes and went to finish charting.\n\n_Squawk_. \"Don't go away mad.\"\n\n# Chapter 5\n\nAfter Cade and his brothers had mercilessly teased Avery for the systematic changes she'd made to the clinic, she'd stopped asking to do things and just did them. Her incentive was admirable, if not scary. It's not that the office was disorganized. It's just that it was...well, yeah. Disorganized. It had been so much easier just to succumb to the chaos than it had been to take the time to change.\n\nCade had dubbed Avery the Nazi but, damn, the clinic was running smoother than ever. So much so that Aunt Rosa decided not to finish out her two weeks, making today her last shift. Avery had hardly needed any training at all. She jumped right in with fervor and retained knowledge with frightening skill.\n\nOnce Avery had a chart system in place, she'd moved on to the storeroom. As in they had one now instead of boxes of supplies in various spots throughout the clinic. She'd cleaned out the janitor's closet, found some old shelving units, and unpacked. There were even\u2014 _gulp_ \u2014labels.\n\nIt actually had taken Brent twenty minutes to find twenty-two gauge syringes because they weren't buried in a box in the boarding room. They were on a shelf now. In a closet. With a label. Cade blew out a breath and shook his head in awe.\n\nFinished with a round of patients, he headed up front where Aunt Rosa was reading a romance book and Avery was typing away on the computer. \"Whatcha doing now?\"\n\nAvery didn't glance up from the screen. \"Making a supply list in Excel.\"\n\nHell. Why was that hot? She wasn't his usual type.\n\nOkay, to be honest, any female with too many brain cells wasn't his type. It wasn't out of a sense of shallowness he sought that variety, but preservation. Until he found someone who made his heart beat like Heather used to do for Drake, he'd stick with superficial. No point in getting hurt or hurting someone else. But ever since Heather died, Cade tried less and less to settle down or find that person. He was aware of it, aware he was doing it, and damn if he ever questioned his actions until now. People rarely recovered from that kind of love. Why search for it?\n\nPerhaps it was Avery's long legs in those black leggings, or her pink sweater the same shade as her cheeks when she blushed, or her brown hair\u2014more chestnut really in the sun\u2014piled on top of her head and held in place with a pencil that seized his interest. He bit back a sigh. Nope. It was her brain, her humor, and her strength.\n\nHot.\n\nAunt Rosa glanced at him over the top of her book, a knowing smirk on her face.\n\nBusted.\n\n\"A supply list?\" Bully for him. His voice sounded normal. He still had no freaking clue why, but if he wasn't bumbling like a moron around Avery, he was saying moronic things. Where the hell was his swagger?\n\n\"Mmhm.\" _Type_. \"So we don't over order things or run out. Gabby and Brent can just check off what they need and I can order from the supplier.\" _Type, type_. \"Did you know you had ten cases of cat litter? Insane.\"\n\nHe scratched his jaw. \"Uh. No.\"\n\n\"Putting Zoe's stuff on here, too. Her shampoos and whatnot for grooming.\" _Type._\n\n\"She's making a supply list,\" Rosa beamed, waving her hand like this was an epiphany. _I told you I'm all-wise. I hired her!_\n\nCade frowned at his aunt. She'd been the office manager, for Christ's sake. She should've been doing this.\n\nFlynn came up to the desk and tapped Avery's shoulder. \" _Have you seen my backup bag? Gabby and I need to head out to Miller's farm.\"_\n\nAvery nodded. \"In the supply room, stocked for you. Your new bag should be here Monday.\"\n\nAt Cade's questioning glance, Flynn signed, \" _A goat chewed my other one when Gabby was busy chasing a barn cat.\"_\n\nAnd this was why he wasn't the house call vet.\n\nFlynn tapped Avery's shoulder to get her attention again. \" _Marry me?\"_\n\nShe laughed. \"Not today, but you're welcome. Shoo, now.\"\n\nWhat. The. Hell.\n\nAs if sensing Cade's thought train and derailing it, Avery said, \"He asks me to marry him daily. Hourly, depending on what I've done.\" _Type, type_. \"Relax, Dr. Cade. He's joking. Besides, I don't do office romance.\" _Right, Flynn?_ she signed.\n\nFlynn grinned, the asshole, and strode into his office, only to emerge moments later with Gabby and his travel bag.\n\nCade flipped him off behind Avery's back as he was walking out the door.\n\n\"Saw that.\" _Type._\n\nOf course she did. All moms had eyes in the backs of their heads. And that was the other thing. She had a kid. Not that he didn't like kids. He did. Maybe even wanted a couple of his own someday. But it wasn't just one person involved when dating someone like Avery. It was two.\n\nAnd they weren't dating. Not even a little. He hadn't asked her out and, aside from that first day when her gaze had shown interest, she'd seemed immune to him.\n\nThat hadn't happened in...ever. He found it oddly refreshing.\n\nWith Flynn and Gabby gone, Cade glanced around. \"Where's She-rah?\" The cat was usually perched on top of the printer, plotting world domination.\n\n\"Avery put her on time-out. She's in the back room.\" Rosa's grin grew to _oh shit_ size, and he figured his aunt had read all his previous thoughts. Like a Vulcan mind meld. Which was never a good thing.\n\nWait. A time-out? He looked at Avery. \"You put a cat on time-out?\" He didn't know whether that was cute or genius.\n\nAvery never stopped typing, which was beginning to infuriate him. \"Yes, she was scaring Thor.\"\n\nCade looked down, just noticing Thor's head in Avery's lap from the other side of her chair. The Great Dane looked at him as if to say, _Neener, neener_. \"Dust bunnies scare that dog.\"\n\n\"Well, the cat needs manners. Plus, Thor and I are working on his courage. Aren't we, boy?\"\n\nThor barked. And not in fear. _Yes, my liege_.\n\nUnsure what to make of Avery, he scratched his jaw. She'd done more in one week on the job than Aunt Rosa had in twenty years. She was prettier to look at, too. Aunt Rosa caught him staring at Avery again, so he shook his head.\n\n\"Do I have any more patients this afternoon?\"\n\n\"Drake has one more surgery and you have two patients\u2014a terrier who chewed its way through a crate, thus cutting his gums, and strangely...\" She trailed off and brought up her schedule. \"A cat that's feeling 'off.'\"\n\nCade groaned. He was so not in the mood for\u2014\n\nRosa set her book down. \"It's Jeffery Harrison's cat.\"\n\nWell, that made no sense. Jeffrey had gone to high school with Cade, did not want to date Cade, and he didn't have a cat. Why make an appointment at all?\n\nRosa's brows lifted. \"It's not just the pretty females worming their way in for appointments anymore. Avery's getting popular.\" She tilted her head, her expression saying, _whatcha gonna do about it, boy?_\n\nSo men were making fake visits to his clinic just to see his new office manager? Hell, did Jeffrey buy a damn cat just to ask her out? This wasn't a dating service. Had people never heard of a cell phone? When his temples started to throb, he closed his eyes and took a deep breath.\n\n_Type, type_. \"Maybe Jeffrey is batting for the other team now. Or the cat really is feeling off?\"\n\nRosa laughed, high-fiving Avery, who never looked up from her computer.\n\nScrew this. Cade turned on his heels and went into his office.\n\n* * * *\n\nAvery looked over the front desk at her mom and Rosa\u2014who were scrolling through Pinterest pictures on Rosa's phone\u2014when they hemmed a third time. They'd been trying to goad her into looking, too, but Avery was on to them. They weren't even trying to be suave.\n\nThey were matchmaking, so she ignored them. Mostly. She did not need to be set up on a date. Especially not with her new boss. Any of them, for that matter. Avery had sworn off men, and besides, hello awkward? And Cade had more women hanging on his arm than the desert had sand granules. If she did date, it wouldn't be a man who loved women _that_ much.\n\nWhat had Cade called these ladies? The Battleaxes? Suiting. Seemed his mother and aunts were trying to recruit Avery's mother into their muah-ha-ha fold. The memory of one picture of Cade from last summer in board shorts and nothing else had her clenching her thighs together.\n\n\"Look at those biceps. Oh, he's cuddling a kitten. Avery, have you seen this?\"\n\nAvery closed her eyes and covered her ears. _Lead me not into temptation_...\n\nWho was she kidding? She hadn't been in church since her wedding nine years before.\n\nFinally, Cade's last patient of the day came out of an exam room. She blew out a breath.\n\nJeffrey seemed like a nice enough guy, if not a little obvious in stripping her naked with his eyes. He set the cat crate on the floor by his feet as Cade walked up front, followed by Brent.\n\nReady to cut Jeffrey off at the pass\u2014literally, he'd made several passes at her\u2014she pasted on a smile and stood. \"That was so nice of you to bring your mother's cat in for her.\"\n\nCade snorted and handed her a list of services. _Depression?_\n\nAvery gave Jeffrey the total and he squared the bill. \"You have a great weekend. Thanks for coming in.\"\n\nJeffrey shoved a baseball cap over his receding brown hair. \"About that. What are you up to tonight? Maybe we could get some dinner?\"\n\n\"Oh, that's so nice of you, but\u2014\"\n\n\"She's got plans.\"\n\nNo one moved for a fraction of a second. She was pretty sure the earth stopped rotating. They all slowly turned to face Cade. Yeah, that proverbial pin drop? They'd all heard it.\n\nCade froze, his eyes wide as if shocked he'd said anything, pen poised over the open chart in his hand.\n\nBrent took in Cade's distressed face and straightened. \"Um, yes. We're all headed to Shooters tonight. Avery's...spoken for?\" He looked at her as if to confirm.\n\n\"I am?\" Not that she was going to take Jeffrey up on the offer. She had no interest in dating, but still. Couldn't she just nicely turn him down? Why lie?\n\nBrent fixed her with a _go with it_ glare and then grinned at Jeffrey. \"Sorry, big guy.\"\n\nRosa and her mom's head whipped back and forth between the patient, Cade, and Brent. Their avid interest in Pinterest was gone and replaced by the live floor show.\n\nCade muttered a quiet curse.\n\nJeffrey's confused gaze scanned the room. \"Okay. Maybe I'll meet up with you later.\" He picked up the crate and left. Fast.\n\nAll eyes slid back to Cade. He opened and closed his mouth several times before tossing the chart aside, running a hand through his thick blond hair, and glancing heavenward. \"Right.\"\n\nFlynn and Gabby walked in the door at the same time Drake chose that moment to come up front from the back.\n\nDrake's gaze swept over each of them in a millimeter of a second. \"Are we having a moment of silence?\"\n\nRosa's eyes narrowed. \"Actually, we're trying to figure out why Cade just blocked Avery from accepting a date.\"\n\n\"I did not\u2014\"\n\n\"Ooh, a date?\" Gabby stepped forward and set her bag down before adjusting her long, blond ponytail. \"With who?\"\n\n\"Jeffrey Harrison.\" Brent waved his hand dismissively. \"Dodged a bullet with that one, doll. I'd thank Cade.\"\n\n\"I did not\u2014\"\n\n\"What time should I be over to watch Hailey?\" Mom interrupted, earning a nod of approval from Rosa. \"I'm glad you have plans to go out. You need to have a little fun.\"\n\nAvery shook her head. \"When did fake plans become real plans?\"\n\nFlynn's eyebrows shot to his hairline.\n\n\"Does seven o'clock work for everyone?\" Brent cocked a hip.\n\n\"I'm in,\" Gabby said. She picked up her bag and headed for the hall. \"Shooters?\"\n\nAvery rubbed her forehead. \"I didn't agree to\u2014\"\n\nRosa huffed. \"Now, now. Cade asked you out, and you're new in town, so this'll be a good opportunity to meet a lot of people.\"\n\n\"I did not ask her\u2014\"\n\n\"Semantics,\" Rosa said. \"Justine, dear, I'll see you tomorrow.\"\n\n\"I'll walk out with you. Avery, I'll be over at six thirty to watch Hailey.\"\n\nFlynn shrugged. \" _I'm in_.\" He looked at Avery. \" _Drinks on me for your awesome first week.\"_ Deserting her, too, he went into his office.\n\nDrake crossed his arms, looking like he wanted to kill Cade for some reason or erase the past five minutes from his memory. \"Have fun tonight, kids.\"\n\n\"Wait.\" Brent pouted. \"You have to come, too.\"\n\n\"I don't.\" Drake turned to go.\n\nThe tech was undeterred, using a teasing singsong voice. \"Avery cleaned out and organized the surgical room.\"\n\nDrake paused mid-step, back still to them, and sighed. \"One drink and I'm out. And...thanks, Avery.\"\n\nBrent clapped his hands like a giddy child and sauntered off, leaving Avery and Cade alone up front.\n\nSilence stretched for one minute. Two.\n\nSlowly, she turned in her chair to face him. \"What just happened?\"\n\nHe was leaning on the back counter with his chin down and fingers pinching the bridge of his nose. He didn't look up when he answered. \"We just got railroaded.\"\n\nYeah, she got that part. \"For the record, I didn't have any plans for tonight.\"\n\n\"Okay.\"\n\n\"I was going to take a hot bath and read.\"\n\nHe didn't move. \"Got it.\"\n\n\"Now I do have plans.\"\n\n\"Yep,\" he said tightly.\n\n\"Because of you. I believe your exact words were, 'She has plans.'\"\n\nHis shoulders tensed. \"I remember.\"\n\nShe strummed her fingernails on the desk. \"What if I wanted to go out with Jeffrey?\"\n\nFor that, he lifted his head. \"Do you?\"\n\nHe pinned her with that blue gaze of his, rife with interest, curiosity, and something that made her belly heat...then her chest...and her cheeks. All flaming. There were some serious mixed signals or crossed wires in the two feet between them. Why would he step in like that if he wasn't...jealous? But that was ridiculous. He was a Trojan god, and she was a divorcee with a kid who'd gotten way too soft around the middle.\n\nExcept he was still staring, waiting for an answer as if it were of utmost importance. She hadn't done this dance in so long she was certain she was reading him wrong.\n\nHe shoved off the counter and made his way over. Backing her chair up to the desk, he leaned down, placed his palms on either side of her to cage her in, and brought his face within inches of hers. The scent of fabric softener and animal fur followed him as he looked in her eyes.\n\nShe stopped breathing. Was pretty sure he did, too.\n\n\"Do you want to go out with him?\" he asked, his voice a low, dangerous rumble.\n\nHe had the start of a five o'clock shadow, which added to his devil-may-care attitude. The gray flecks in his eyes were more noticeable this close to him and the wide, almost bow shape of his mouth was only inches from hers. Sexual tension coiled in her gut, made drawing in air impossible. The heat from his body engulfed.\n\nCrap on a cracker. \"No.\"\n\nHe didn't move, but his lids lowered to half-mast when his gaze dipped to her mouth. Lazily, he brought it back up to her eyes. One corner of his lips quirked. \"Good.\"\n\nHe straightened and headed toward his office. \"I'll pick you up at seven.\" He was halfway there before she blinked and recovered.\n\n\"I can drive myself.\"\n\n\"Never said you couldn't.\" He kept right on going.\n\nShe bit her lip. \"This isn't a date.\"\n\n\"Never said it was.\" He stopped in his office doorway and turned to face her. His gaze leveled on her and held her captive. For a second, he seemed uncertain, but it was gone before she could react. Slowly, he drew in air and palmed the doorknob. \"If we do ever go on a date, you'll know it's a date. And we'll be alone.\"\n\nShe shivered.\n\nHe shut his office door.\n\n# Chapter 6\n\n\"Just what in the hell do you think you're doing?\"\n\nCade tore his gaze away from where Avery was playing darts with Brent, Zoe, and Gabby across the room to look at Drake. His older brother was in a mood. When wasn't he? \"I don't know what you're talking about.\"\n\nExcept he did know. Playing dumb seemed like a better option.\n\nThe jukebox inside Shooters blared hard rock, the peanuts on their table were stale, and there wasn't a face in the crowd Cade didn't know. Same ole, same ole. Since when was he restless? Typically he rolled with the punches, let the night play out how fate intended. He shrugged, suddenly tired of the mirror scene, the same cheese-ass d\u00e9cor and come-on lines.\n\nChrist. Was he getting...bored?\n\nFlynn paused with his beer halfway to his mouth, watching them closely to read their lips. Normally Cade wouldn't exclude him from conversation and would use sign language, but the bar was busy and Cade had a suspicion this was a conversation he didn't want the gang listening to.\n\nDrake crossed his arms and leaned back in the chair at their high-top table. \"You cock-blocked Avery today when someone asked her out.\"\n\nAs if he didn't feel like a big enough ass. \"I'm pretty sure she doesn't have a cock.\"\n\nFlynn snorted.\n\nDrake's brows lowered. \"This isn't a game. Whatever you're thinking, unthink it. She's not your usual good time.\"\n\nTranslation: She's too good for you.\n\nCade ground his jaw and slammed the rest of his pint. Never mind he knew that. He didn't need his brother pointing it out. Besides, who said he couldn't offer someone more than a little fun? He was getting sick of people underestimating him and putting him in a box. The label was getting old quick.\n\nAnd just when, exactly, had that happened?\n\nFlynn must've picked up on Cade's tension. \" _Maybe he really likes this one_.\"\n\nCade chanced a peek across the crowded bar, but the group wasn't paying attention and Flynn's back was to them.\n\nDrake shot out a laugh that had little to do with humor. \"He likes all of them. That's not the issue.\" He leaned forward, glare digging into Cade's patience. \"She's the best thing to happen to our clinic since before Dad died. If your dick screws that up\u2014\"\n\nCade slammed his glass down. \"I haven't done anything. And if I do, it wouldn't be before properly thinking it through.\" Except Avery had a way of shutting down thought.\n\nDrake shook his head, a sound of disgust rolling from his throat.\n\n\"You know, I graduated from college, never miss a day of work, own my own home\u2014\"\n\n\"And never sleep with the same woman twice.\" Drake shifted his glass on the table, studying him. \"There are levels and types of responsibilities, little brother. She has a kid. She doesn't need to date one.\"\n\nHe was sick of this conversation, too. Cade didn't know what more he needed to do to prove himself. What the hell difference did it make who got between his sheets? His sex life had nothing to do with his family or Animal Instincts.\n\nYet his gut twisted because he'd basically told himself the same thing Drake had not so charmingly spewed, and he was still thinking of making a play for Avery. She tied him up in knots. He wanted to dissect that, find out why.\n\n_\"They're coming back to the table_.\" Flynn sipped his beer.\n\nCade's gaze tracked Avery until she climbed onto the stool next to him. \"So who won?\"\n\nZoe grinned, the curve of her lips reminding him of her gypsy heritage, seductive and mysterious. \"Gabby, of course. When doesn't she win?\"\n\nZoe's once light brown hair was dyed blue this week. For more than a year, she'd colored it a different unnatural shade, changing it every week or two. No one knew why. She'd walked into the clinic one day with bright orange hair and never said a word as she'd set up in the back for her grooming appointments.\n\nThey'd all wondered. No one had asked.\n\nGabby took a healthy gulp of her ale. \"Gotta be good at something. At least Brent didn't maim anyone.\"\n\nBrent's aim was notoriously bad when it came to darts. Cade had a scar on his bicep as proof.\n\nBrent waved off her comment. \"Only because everyone scurried away, refusing to play with us.\" He turned in his seat and yelled to the bar over the noise. \"All pussies!\"\n\nGabby shoulder-bumped him. \"How would you know a pussy if you saw one?\"\n\nFlynn choked on his beer.\n\nCade looked at Avery to see how she was taking the light banter at Brent's expense, but the curve of her lips said she was amused by the conversation. She was quiet, but comfortable. He guessed he just wasn't used to quiet women. Come to think of it, if not for Avery, he'd swear they didn't exist.\n\nAvery dropped her chin in her hand. \"I'll bet he's been with at least one woman.\"\n\nBrent's eyes narrowed to catty, but he said nothing.\n\nShe shifted in her seat, crossing her long legs wrapped in skinny jeans that Cade would love nothing more than to peel off her. The knee-high black boots could stay.\n\n\"Are all of you from Redwood Ridge? You seem to know each other very well.\"\n\nMost of them had grown up together. Brent was a recent Seattle transplant from five years ago. Gabby had been in Flynn's graduating class, Zoe in Drake's. Cade and his brothers were only a year and a half apart, Drake being the oldest, but they were close. At the clinic there was a respectful working relationship, but after hours it was like being back in the sandbox again. A guy couldn't ask for better friends.\n\nGabby readjusted her ponytail. \"I can give you all the dirt.\"\n\nZoe laughed, a rich smoky sound that seemed to wind Drake tighter. His shoulders tensed as he swallowed the last of his whiskey neat and stood. \"I'm out. Avery, thank you for everything. Seriously.\"\n\nThey all lifted their glasses. \"To Avery!\"\n\nHer cheeks flushed as she stared at the table. \"Thanks.\"\n\nGabby winced and set down her glass. \"Damn, Cade. Cougar at twelve o'clock.\"\n\nJudging by her gaze directed over his shoulder, he assumed she meant her twelve o'clock, not his. He turned in just enough time to catch Cynthia's perfume before she sidled up next to him, plastering her double D's in his face. Cynthia had been a lapse in judgment about four years back, but she didn't seem to take that personally.\n\nHe looked to Flynn for help, but his jerkface brother grinned. \" _Just put her over your knee and she'll leave you alone.\"_\n\nWhy did he tell his brother anything? Yes, it was true Cynthia liked to be spanked, and yes, it was true she'd liked to scream out _Daddy_ when in the throes. To each her own. It just wasn't his thing. In fact, he was pretty sure he was still traumatized.\n\nAvery lifted her brows. \"Do I want to know?\" she signed to Flynn.\n\nGabby shook her head, eyes wide. \"Not unless you've got daddy issues. Keep your eyes down. Don't engage.\"\n\nFlynn laughed, wiping a hand down his face.\n\nCynthia shoved her tongue in Cade's ear and purred. Except she was so drunk it came out more like a tin drum roll. \"Let's go back to my place.\"\n\nHe resisted a shudder and set his hands on her waist. \"Ah, thanks for the offer, but I've got an early morning tomorrow. Besides, you know you're too much...woman for me to handle.\" Her curly red hair, courtesy of Clairol, tickled his face and barred him from seeing his friends laughing at his expense.\n\nCynthia licked his throat.\n\nAvery made a strangled sound of shock.\n\nCade always found it difficult rejecting women. One would think he'd become a Master Jedi at by now, but alas, \"no\" was not in his Webster. He'd beaten around the word, teased at it, and siphoned his way out of messy breakups, but rarely had to hurt feelings in his diversionary tactics. On the off chance Cynthia would remember this moment after she sobered, he'd hate for her to be humiliated in front of everyone at Shooters because the town manwhore had turned her down.\n\nAnother freaking term he couldn't stand. It wasn't as if he'd slept with every woman. He was actually more selective than people gave him credit. His reputation had been embellished greatly through the years, through no fault of his own. Not that he'd minded much. Not until recently. Hell, he'd been on a dry streak long enough to briefly\u2014perhaps punishingly\u2014consider Cynthia again.\n\nNope. Scratch that. He'd stay celibate.\n\nBrent, bless him, decided to save his ass from being swallowed whole. \"Cynthia, honey, retract your claws. Cade's with someone else tonight.\"\n\nCade pinched his eyes shut as Cynthia stilled halfway into her climb up his body. He held his hands out, not touching her, hoping the lack of contact wouldn't set her off. She'd been known to be a hothead.\n\nSlowly, she eased back to look at his face. Mascara smears shadowed the puffiness under her eyes, and her mouth hung open in shock, breathing stale beer in his face. She looked from him to Avery and back again. \"Who's she?\"\n\n\"Cynthia, meet Avery, our new office manager.\" Thinking fast, he tapped Cynthia's ass to cue her to get down. \"I think I saw Jared staring at you from the pool tables earlier.\" Jared being the ex high school football star whose life peaked ten years ago. No harm, no foul sending her Jared's way. \"I think he's interested. I'd hate to come between you two.\"\n\n\"Yeah?\" Her unfocused gaze scanned the room. \"Thanks, Cade. You're a helluva guy.\"\n\nAs she sauntered away, and the entirety of their table busted out into hysterics, he blew out a breath. \"And you people call yourselves friends.\"\n\nGabby swiped her eyes. \"I warned you she was coming.\"\n\nBrent held out his hand, wiggling his fingers. \"Zoe, hand over your purse. Quick.\" She passed it over and Brent dug through the contents, removed a small package of antibacterial wipes, and tossed it on the table in front of Cade. \"Use them. Hurry, before infection sets in.\"\n\nWith air being in such short supply, Gabby thunked her head on the table and wheezed through a round of giggles.\n\nAvery set her glass down. \"You knew she had wipes in her purse?\"\n\nAfter what just went down over the past five minutes, that was what shocked her?\n\nZoe patted Gabby's back and slid a glass of water in front of her. \"He's in my purse more than I am.\"\n\nBrent nodded in mock seriousness. \"True, dat.\"\n\nCade sighed and turned to Avery. \"You ready to go?\" He was so over this night.\n\nAmusement still shone in her eyes as she nodded.\n\nThey drove back to her cabin in silence, but a comfortable one. He parked in front of the porch and got out to open her door. She looked up with round eyes as if no one had been so courteous before and then avoided his gaze. As they climbed the porch steps and he wondered how to end the night, Justine opened the front door, rushed past them, and headed for her car.\n\n\"Hailey's asleep. There were no issues. You two go on inside. Together. Alone. I'll see you tomorrow, Avery.\" With that, she got in her car, sped out of the drive, and disappeared onto the main road.\n\nThey stood staring after the taillights had faded and Avery pressed a hand to her forehead. \"What is wrong with everybody?\"\n\n\"Subtlety is not her strong suit, I take it?\" He was used to his meddling family, but being the victim of matchmaking seemed new to her.\n\nHer gaze landed on his chest and suddenly he was aware of her closeness, of the summer and berries scent of her rising over the pine and snow. Her breath hitched, her lips parted, and her lids grew heavy.\n\nOne step, one step and his body would be flush with hers and he'd know what she tasted like. He'd thought of little else this week, and he was starting to forget the reasons why he shouldn't make a move.\n\n\"Um...\" She tucked a piece of hair behind her ear. \"The woman at the bar, you're not...together?\"\n\n\"No.\"\n\n\"But you used to be?\"\n\n\"Yes. One night, years ago.\"\n\nShe nodded. \"You were really nice about letting her down tonight.\"\n\n\"No sense in being an asshole about it.\" He shrugged.\n\nFinally, her gaze lifted to his. Part shock, part admiration lit the brown depths. He couldn't recall when, if ever, someone had looked at him and seen respect. His chest swelled, giving him the false impression he could do anything.\n\nThe porch light caught the golden flecks in her irises to add to the punch of dark brown. She had clever eyes, expressive, with long lashes that framed all that pretty color. Avery wasn't a stunner, but she had an unmistakable natural beauty not found often anymore. The freckles lightly dusting her nose only added to her charm.\n\nHer gaze dropped to his mouth and back up, telling him she was thinking of being naughty, too. \"You should stop looking at me like that. I'm not dessert. I'm not even an appetizer.\"\n\nHe stepped into her space and cupped her cheek, cool from the air. \"I disagree.\"\n\nCade wondered how long she'd been married to the dick who put those ideas in her head and how long it would take to undo the damage. She hadn't outright said anything, but he knew insecurity when he saw it. She was confident in work, with her daughter, in everything else, until he acted on impulse and got closer. Then her oh-shit meter went off.\n\nShe wasn't ready to be kissed, but he was going to pursue this chemistry to see just how combustible they were together. His mind was made up, if for no other reason than to show her just how desirable she was.\n\n\"I'll see you on Monday.\" Leaning in, he kissed her cheek, letting his lips drag across her smooth skin and graze a path to her temple. Her breath shuddered out, expelling puffs of air in the cool, damp night. \"For the record,\" he said quietly into her hair, \"I think you're sweeter than any dessert I've had.\"\n\nAnd he'd had a lot of dessert.\n\n* * * *\n\n_Squawk_. \"Highway to hell.\"\n\nCade muttered a curse.\n\nAvery looked up from the billing statement she'd been going over with him, glancing first at the cockatoo and then Cade. The muscles in his forearms flexed as he leaned into his hands on the desk next to her. Large hands. They'd felt good cupping her cheeks the other night.\n\nShe shook her head. _Focus_. \"What's wrong?\"\n\n\"The Battleaxes are here.\" He jerked his chin at the front door. \"All three of them at once. This can't be good.\"\n\nCade's mother and two aunts strode into the clinic. The three women couldn't be any more different. Rosa was in jeans and a T-shirt with a leather bomber jacket, her unnatural red hair plastered in place and not a stitch of makeup. Gayle's light blond bob moved freely as she walked. She was wearing a pair of khakis and a light blue sweater under a peacoat, sporting a natural look with a swipe of gloss and a little mascara. The mayor, Marie, was in professional mode in a tweed suit, her dark brown hair swept up in a twist and war paint in full operatus.\n\nAvery's gut twisted. Cade turned to leave, but she grabbed his wrist. \"Don't you dare leave me alone with them.\" She'd only met them a handful of times since moving, but they were intimidating as individuals. Together they could take down the Taliban.\n\n\"Mom, Aunties, what brings you out?\" He leaned over the desk to kiss their cheeks. \"Taking candy from babies? Solving the budget deficit and telling no one?\"\n\nAvery's pulse thrummed as heat pooled in her belly. There was nothing sexier than a good sense of humor.\n\n\"Har, har. Avery, dear. So good to see you.\" Marie took off her gloves and unbuttoned her coat.\n\n\"You, too, Mayor.\"\n\n\"Oh, please. Just Marie is fine.\"\n\nAvery chanced a peek at Cade, but he was pretending to read the billing statement in his hands. Wimp. She cleared her throat and smiled. \"What can I help you gals with today?\"\n\n\"Well,\" Gayle started, but was interrupted by Gabby who walked up front with her attention on a chart.\n\n\"Avery, have you seen the...\" Gabby's eyes rounded when they landed on the visitors. \"Hey there.\" She moved around the desk, hugging each in turn, not the least bit afraid of the trio. \"What a great surprise. Have you met Avery? Look at the wonderful job she's done in organizing us.\"\n\nCade tried to step behind Avery and leave, but she moved her chair back, blocking his exit. Stuck, he pasted on an affable grin. \"I've got patients, so...\"\n\nHe tried stepping in front of Avery. She stretched out her legs. \"No, you don't. Your schedule's clear for twenty minutes.\"\n\n\"I have paperwork.\"\n\n\"You don't.\"\n\n\"I do. I'm sure of it.\" His jaw ground, but amusement lit his eyes. Obviously he was enjoying the game. His expression implored her to release him.\n\n\"Nope,\" she sang.\n\nRosa cleared her throat and looked at her sisters. \"Told you.\"\n\nGayle's smile could've melted the arctic. \"I do see. You were right.\"\n\nCade tensed. \"See what? Right about what?\"\n\nThe trio, complete with Gabby, stared between her and Cade, a mischievous gleam in their eyes over an inside joke.\n\nInstead of answering, Marie nodded. \"We've heard about the wonderful job you've been doing, Avery. That's why we're here, actually. We have a proposition for you.\"\n\nOh God. \"Um...okay?\"\n\nCade chuckled. She kicked him in the ankle.\n\nEver since last Friday night, when they'd all gone out to Shooters and Cade almost kissed her, there'd been a jovial atmosphere at work. Like crossing the line between strangers and friends. They'd bantered, joked, teased. Even Drake had cracked a smile or...well, just the one time. But it had been fun. She loved working here, and she was beginning to gain a sense of purpose again.\n\nGayle took the lead, leaning on the counter with her big doe eyes. \"We're in need of someone with your organizational skills. You see, Jessica was put on bed rest and she normally runs the event committee\u2014\"\n\nCade barked out a laugh. Paused. Looked at each of the ladies. Held up a hand and laughed again. \"You want Avery to head the Redwood Ridge event committee?\"\n\nRosa nodded. \"Exactly.\" She looked at Avery, who was too confused to move. \"For now, it would just be the Valentine's dance, but we can see how that goes and then you could chair more things if all goes well.\"\n\nMarie nodded. \"The St. Patrick's pot luck, Easter egg hunt...\"\n\nGabby bounced on her toes. \"She'd be perfect. I mean, Avery's got great ideas, and you know she's super organized.\"\n\nThey kept chattering about her as if she wasn't there. She pressed a palm to her forehead.\n\nCade leaned down to whisper in her ear. \"This is how Jessica got roped into the job. I'd run. Run far. Run fast.\"\n\n\"Who is Jessica?\" She didn't even know this person and they were\u2014\n\n\"Oh, Jessica runs the nursery.\" Marie waved her hand. \"That's not important. She's stepped down and we need someone to replace her. We pick you.\"\n\n\"I...um.\" Avery sighed. God. \"That's very flattering, but I don't know the town very well, and I've never done anything like that before.\" There. That should get them to\u2014\n\n\"Nonsense. The meeting is tonight at the rec center. Be there at seven.\" Marie donned her gloves.\n\n\"Wait. I can't.\" Avery stood. \"My mom has book club and there's no one to watch Hailey. Besides that, I'm just too busy.\"\n\nGayle smiled as if this had all gone according to their crazy plan. \"The meetings are only ninety minutes once a week. Every two weeks if there's not an event coming up. Cade can watch Hailey while you're away.\"\n\nCade flinched. \"I what?\"\n\nAvery's temples started to throb. \"No. Hailey has special needs, and I'd need someone responsible who knows\u2014\"\n\n\"Are you saying I'm not responsible?\" Cade crossed his arms, his brows lifting.\n\nDang it. \"No, of course not. It's just\u2014\"\n\n\"I _am_ responsible. And for your information, I could handle a kid for a couple of hours.\"\n\n\"Perfect.\" Marie moved to the door before Avery could even blink, her sisters following. \"Cade will babysit Hailey, and we'll see you at the meeting. I knew this would work nicely.\"\n\nThey walked out with the same flair as when they'd strode in, leaving Avery's mouth agape. She stared at the door, trying to figure out if she was pissed off or freaked out.\n\nGabby grinned and slung an arm around her shoulders. \"You'll do a great job.\" With that complete statement of non-help, she went back into Flynn's office.\n\nSlowly, Avery turned to Cade.\n\n\"I walked right into that.\" He shook his head. \"Twenty-eight years and you'd think I'd know when I was being trapped. Don't look at me like that. They're sneaky. Tricky. You were standing right here, too.\"\n\nShe sighed. \"What did I just get myself into? And don't think you're off the hook. I can double book you on 'depressed' cats for the next three months.\"\n\nCade sprawled into the extra office chair and pinched the bridge of his nose. His dark blue scrubs stretched over the lean muscles of his thighs and biceps, momentarily distracting her.\n\nHe slapped his hand on his leg. \"Redwood Ridge does several community events throughout the year. Mostly holidays, but the funds raised go to causes like the rec center, the fire department, the library. The event coordinator kinda oversees all of that.\"\n\nAvery had spent a good amount of her time planning parties for Richard and his law firm. But it had been more than two years since they'd separated and she'd done nothing on this grand a scale. For a whole town? She was just beginning to get settled into the new job, new home. And Hailey wasn't accustomed to this many strangers around. She'd been doing really well in school and the rec center, but to leave her one-on-one with a man she barely knew was not something Avery was comfortable with, even if Cade was trustworthy.\n\nHeck, none of this was what Avery was used to. For ten years, she'd been isolated from others in the Stowe mansion, raising Hailey with no assistance and little more than herself for company.\n\nAvery sat in the chair across from Cade. She needed to give him an out. \"About you babysitting, I'll just take Hailey with me, or find someone...\"\n\nShe trailed off as his gaze met hers. Something like hurt flashed in his eyes, twisting his mouth, but then it was gone. He stared at her for a beat as if trying to calculate her reaction or his own. All she could think was, why didn't he want an out? Single, attractive man, Friday night\u2014surely he didn't want to babysit.\n\nIt was hard to get a read on him. Usually he was charming and playful. A few times, he'd teetered on irritation, but always he was good with the animals. Patient. Kind.\n\nHer breath caught. She could've sworn it was interest looking back at her, like the other night on her porch, but without the heat, the desire. Swallowing, she forced herself to look away when her cheeks flamed.\n\nScratching his jaw, he leaned forward, resting his forearms on his thighs. \"I can stay with Hailey for a couple of hours.\" His gaze bore into hers, not a trace of humor in them. \"I get the impression you don't lean on people often. It may seem like I got roped into it, and maybe I did, but that doesn't mean I don't want to do it.\" A moment passed, then two. \"She's a great kid, Avery.\"\n\nCade had spent a few hours with Hailey and thought she was a great kid. Richard had been her father and never saw it. Coming from any other guy, it might've sounded like a line. But no. Cade was genuine, from the inflection in his low voice to his direct gaze to his actions when around Hailey. A person couldn't fake sincerity.\n\nShe closed her eyes and leaned her head back. \"I'll drive you batshit with instructions. Blow up your phone with texts.\"\n\nHe rose and walked past her. \"I can take it.\"\n\n# Chapter 7\n\nCade had just enough time to head home and shower after work before needing to be at Avery's. Figuring he'd bring along his black lab, Freeman, so he could play with Seraph, he snagged the extra leash from the kitchen on his way in.\n\nStripping off his scrub top, he turned to head upstairs to the bathroom and paused at the living room threshold. Cutin was hanging by all fours from the front bay window drapes.\n\nStoically sitting on the floor was Freeman, one eyebrow quirked as if to say, _it was your idea to bring home a cat._\n\nCade cleared his throat.\n\nCutin squeaked out a meow and turned her head upside down to address him. _There was a wrinkle. I was just straightening it for you._\n\nSighing, he picked up the kitten and set her on his shoulder\u2014since that seemed to be her favorite spot when not clutching the drapes, shower curtain, or kitchen towel rack\u2014and walked upstairs to the bathroom.\n\n\"You're getting declawed on Monday. Just so you know.\"\n\nMeow.\n\n\"Don't take that tone with me.\"\n\nSetting her on the vanity, he stripped the rest of the way and showered off the clinic aroma. After toweling dry, he put Cutin back on his shoulder and headed into his bedroom.\n\n\"I'm going to be gone for a bit tonight. You will not destroy anything in my absence. Got it?\"\n\n_Meow_.\n\n\"Don't be sassy.\" He set her on the bed, dressed, and went to fetch Freeman. \"Who wants to go for a ride?\"\n\nFreeman, not one to bark, lifted his paw.\n\nTwenty minutes later, Cade found himself leaning against Avery's kitchen counter with his arms crossed as the dogs got to know each other and Avery fluttered around like the end of days was looming. She'd been talking nonstop for fifteen minutes while making PB&J for dinner, and if Hailey weren't sitting at the kitchen table, he'd be tempted to shut Avery up. With his mouth.\n\nShe was adorable all worked up. Flushed cheeks, hands flailing. Admittedly, he'd only caught about half the things she'd said because he found looking at her much more interesting. Those jeans molding her perfect ass weren't helping matters any. She had a great body. Curves and soft edges, not a rail with nothing to hold.\n\nShe huffed a breath. \"Are you listening?\"\n\nNot really. \"I should only let Hailey play with matches after dark, feed her candy bars and soda for snack, and scary movies are acceptable if she won't go to sleep. Got it.\" He sat next to Hailey at the table and held up a fist for her to bump. \"Oh, come on. You can't leave a guy hanging.\"\n\nHe was never really sure if Hailey was hearing him, as she never made eye contact and she always seemed to be deep in her head. Ducking his face close to hers, he grinned. \"Make a fist, squirt.\" When she did after a brief delay, he gently bumped his fist with hers. \"There you go. Don't leave me hanging again or no sugar high for you later.\"\n\nHailey whooped out a laugh, rough and short, but Cade's chest swelled at the small connection. The girl's round cheeks split in a smile as she glanced somewhere over his head. Sometimes she looked so much like Avery.\n\n\"Would you please be serious?\"\n\nCade reined in his grin. Avery was worried about leaving her daughter and he got that. All attempts to pacify that anxiety hadn't worked, so he met her gaze from across the room. \"She gets sick from dairy, she's prone to wandering, she's already in PJs, bedtime is eight, brush teeth first, cue her to potty, watch thirty minutes of her sleep video, and try not to touch her, especially on her head.\"\n\nAvery's head whipped back, gorgeous brown eyes blinking rapidly. \"Um...yes.\" She glanced around, clearly uneasy. \"You have my cell number?\"\n\n\"The first twenty times you asked. I got this. Go.\"\n\nShe sighed and looked down, where Seraph was doing his best to chase Freeman's tail. Finally, without anything else to nag about, she tapped Hailey's arm. \"Pretty please be good for Cade. I'll be back after you go to sleep, but I'll check in on you. Okay?\"\n\nHailey dug into her sandwich as an answer.\n\nAfter Avery left, Cade eyed his own sandwich and glass of milk Avery had set out. Wiping a hand down his face, he ate in silence, watching the dogs. Seraph was still wearing the cone around his neck to keep him from chewing off the bandages from his amputation. He was due in Monday for suture removal and a follow-up. Figuring he'd save Avery a trip, he looked at Hailey.\n\n\"Hey, squirt. Wanna be my assistant? Seraph's all better now, so we can get all that junk off him. What do you do say?\"\n\nShe squealed and flapped her hands, which he took as a yes.\n\n\"Awesome. Finish your...rice milk\"\u2014he shuddered\u2014\"and we'll hop to it.\"\n\nSince she was prone to wandering, he brought Hailey outside with him to the car to get supplies and headed back in. After tossing their paper plates away, he rinsed out Hailey's empty cup and set his on the counter so he could put Seraph on the table.\n\n\"Okay, squirt. Just stand next to me and pet your puppy while I check his ouchie.\" He turned when she didn't move and found her drinking his milk. \"Shoot, no. That's regular milk.\"\n\nTaking the glass from her, he eyed the contents and tried to determine how much she drank, but it couldn't have been much. He dumped the rest, filled a cup of water, and gave it to her. \"Just thirsty, eh? I hope that little bit didn't make you sick.\"\n\nShe seemed no worse for wear after a minute, so he pointed to the table and repeated his instructions. After a small delay, she did as he asked and petted Seraph's back while he removed the bandages.\n\nThe wound was closed and there were no signs of infection, so he opened a sterile disposable suture removal kit. \"Doing a great job, squirt.\" He bent and quickly snipped the sutures before feeling around the area. The puppy wiggled at all the attention, but Cade got it done without too much headache. \"I think we can take the cone off now.\"\n\nHailey squealed. Seraph barked.\n\nHe laughed and removed the cone, then tossed the stuff into the garbage. \"How about we give this guy a bath?\" Because of the injury, Cade had instructed Avery not to bathe Seraph, even though he needed one.\n\nFreeman took off for parts unknown at the word \"bath.\" For a lab, he hated water immensely. Hopefully Seraph was more cooperative than his own dog.\n\nHe rummaged around the bathroom until finding kid shampoo, then made his way back to the kitchen. He pulled a chair up to the counter, filled the sink with warm soapy water, and encouraged Hailey to climb on the chair before setting Seraph in the sink.\n\n\"I'm going to stand behind you, squirt. If I'm too close, just use your elbow on me.\" Straddling the chair, he caged Hailey in front of him and brought his arms around her. She didn't seem bothered, so he passed her a cup. \"Go ahead and pour water over the doggy.\"\n\nShe didn't seem to understand, so he gently wrapped his fingers around where hers held the cup and dipped it into the water, then poured it over Seraph.\n\nThe puppy yipped and shook his head, flinging suds and water everywhere. Hailey whooped out another laugh, which pulled one of his own from deep in his chest.\n\nBy the time they were done there was more water on them and the floor than in the sink. He towel-dried Seraph and set him on the floor, then cleaned up the mess. His pocket buzzed with two incoming texts as he dropped the towels into the washing machine in the room off the kitchen. But Hailey needed to change, so he fished around her dresser drawers and came up with a new set of pajamas.\n\n\"Can you dress by yourself?\" He made a show of covering his eyes and hoped the girl got the hint. When the rustle of clothing quieted, he peeked. \"Good job, squirt.\"\n\nHailey settled on the couch with her sleepy-time show. He tossed her wet PJs into the washer and checked his cell.\n\nAvery: Everything okay?\n\nAvery: Why aren't u answering?\n\nGrinning, he shook his head while thinking of how to answer. Deciding to goad her, he thumbed out his response.\n\nYep. Two corpses. Everything's fine.\n\nJust as he wondered if she'd get the reference, her response came back.\n\nAvery: Did you just quote the movie Clue to me?\n\nYes, ma'am. Impressed you knew that.\n\nAvery: Love that movie. How's Hailey?\n\nShe's fine. Stop texting and pay attention to ur meeting.\n\nHailey grabbed the hem of his damp shirt and tugged. He pocketed his phone and realized she looked a little pale. \"You all right, squirt? You\u2014\"\n\nWithout any preamble, she bent over and yakked down the front of him. Not just threw up\u2014no-no\u2014but projectile. White, milky vomit with chunks of PB&J. Oh, the stench. He'd never eat it again. Never.\n\nHe stilled, palms up, waiting to see if she was finished, unsure of how to handle this. He'd been yakked on a hundred times by animals, but never a tiny human. \"Wow. So that just happened.\"\n\nShe wailed, flapping her hands and jumping in place. Completely unlike her happy-flapping, the distress from her had panic rising in his chest.\n\nHe squatted. \"Hey, hey,\" he cooed. \"No biggie. We'll just...uh.\" Hissing a breath, he stood. \"Okay, wait right here.\"\n\nHe went to turn for the bathroom when he realized his clothes had to go or he'd drip the contents of Hailey's stomach all over the house. A quick survey showed she didn't get any on herself. At least there was that.\n\nSetting his phone on the counter, he quickly stripped down to his boxers and tossed the soiled clothes into the washer, added extra detergent, then started a load. He used some disinfectant wipes he found under the sink to wipe down the floor. Hailey never moved, but at least she had her color back.\n\n\"This would look so bad if someone were to walk in right now.\" Grown man in his skivvies with an eight-year-old girl.\n\nHe washed his hands and walked Hailey to the living room to finish her show. \"Be right back, squirt.\"\n\nHopping from foot to foot\u2014because the temp had dropped to twenty degrees\u2014he ran like a lunatic to his car, grabbed his extra set of scrubs, and thanked God that Avery didn't have neighbors. Once inside, he dressed and eased down next to Hailey on the couch.\n\nShould he call his mom? She'd know what to do. Except it would be all over Twitter in under an hour. Or call Avery? But she'd freak.\n\nThe kid seemed okay now. Her complexion looked normal, at least.\n\n\"Hey, squirt. I know you don't like it, but I'm just gonna feel your forehead.\" Slowly, he lifted his hand and pressed his palm to her cheek, then her forehead. Didn't feel hot or anything.\n\nHailey shoved his hand away, never taking her gaze from the TV.\n\n\"Okay, okay. All done.\" He blew out a breath and closed his eyes for a moment. \"You really do get sick on dairy, yeah?\"\n\nThe dogs circled the floor and laid down next to each other. Seraph was getting around really well with just the three legs. He'd grown a bit in the week since his surgery and the tan cotton puppy fuzz was beginning to look more like fur. Another few weeks and he'd resemble more dog than puppy.\n\nThe credits rolled on Hailey's show. He cut the power and followed her into the bathroom, watching as she brushed her teeth.\n\n\"So, um... Your mom says to go potty, too, yeah? I'm just going to...you know...step out.\" He waited outside the door, back turned, while Hailey hopefully had peeing by herself mastered.\n\nHe breathed when the distinct tinkle hit water. Waiting a few moments until a flush sounded, he helped her wash her hands and nudged her toward her bedroom.\n\nShe climbed onto the bed and pulled the covers up. Cade realized he should've paid more attention to Avery because now he wasn't sure what to do. Turn on a nightlight? Kiss her on the forehead good night?\n\nHe switched off the overhead light. Not finding a nightlight, he flicked on a lamp on the dresser in case she got scared and eyed Hailey's adorable little body, her heavy lids. She was an effing cutie. Dark hair just like Avery's and round cheeks. Hailey's eyes were blue, though, unlike Avery's brown.\n\nDeciding to leave the door open, just in case, he lifted his hand. \"Night, squirt. I'm right out here if you feel sick again or need anything.\"\n\nHailey sat up, the covers falling to her waist.\n\nCade stilled, wracking his brain for instructions he might've forgotten. \"Want me to stay in here?\"\n\nShe didn't move, so he turned off the lamp and sat next to her hip until she lay down again. The dogs came in, Freeman flopping on the floor by the foot of the bed and Seraph doing his damnedest to climb up by Hailey. Perhaps that was why she'd sat up. Used to sleeping with the dog?\n\nFiguring it wouldn't hurt, he picked up Seraph and set him on the bed. Both puppy and child sighed, closing their eyes. Double damn. That was adorable.\n\nExhausted, he stretched out crossways on the foot of her bed, head and feet hanging off, but at least he was right next to her if she needed anything.\n\n\"This babysitting thing is a lot of work, yeah?\"\n\nNo one answered, of course.\n\nAlong with complete muscle annihilation from tension, a fullness rose in his chest, a peace of sorts, at the night's events. A lot of work, yeah, but kind of rewarding in a way. The kid was pretty great when she wasn't puking on him, and getting her to laugh had felt damn good.\n\nThey didn't even burn the place down. Go them.\n\n* * * *\n\nHating to admit it, Avery was glad she'd gone to the meeting. Once she got past the questions of how her date with Cade went\u2014and repeatedly relaying it wasn't a date, just drinks with friends\u2014the night wasn't so bad.\n\nBesides Cade's mom and two aunts, there were five other women on the event committee. All of them were very nice and appreciative she'd taken the position. Not that she'd had a choice in the matter. Or a say.\n\nRedwood Ridge apparently celebrated most holidays as a community. It was sweet in a small town way. Tonight's meeting had been to discuss the Valentine's dance next month. In year's past, they'd done it in the high school gym, but the other ladies claimed it had been just blah, not romantic, and wanted more pizzazz. Their words, not hers.\n\nAvery had suggested a change of venue, easy enough, and they'd gone bonkers in glee. The dance would now be held at the botanical gardens, since they had a hall attached to the nursery dome. To boost interest and numbers, Avery had offered up the idea for a note exchange in the week before the dance.\n\nMarie, having loved the idea, would use her mayoral power to have the rec center kids make heart-shaped notes and pass them to the post office for delivery. People would place an order, with phrasing, and the whole thing would go down like a secret admirer swap. Best of all, the money went to charity. Each event the town held would. The Valentine's dance funds were going to the art program at the high school.\n\nPulling up to the cabin, she grabbed her purse and went inside, anxious to see how Cade had done watching Hailey. After his teasing text, she hadn't heard anything from him. It had taken a lot of willpower not to call fifty times.\n\nPausing by the couch, she set her purse down on a table and glanced around. The kitchen light was on, the washing machine in spin cycle, and the house was quiet. Too quiet. Cade was nowhere in sight, nor were the dogs. Beginning to panic, she rushed down the hall to Hailey's room and stopped dead in her tracks.\n\nCade's dog, Freeman, lifted his head from where he was sleeping on the floor and set it back down, uninterested in her arrival. Hailey was tucked in, sound asleep, her dark lashes shadowing her round cheeks.\n\nBut Cade\u2014God, her heart squeezed\u2014was draped over the foot of the bed, face down, legs and head dangling off the mattress, lightly snoring with Seraph curled in a fuzzy ball on his butt.\n\nShe clutched her chest at the... _adorableness_. Her throat suddenly got tight.\n\nCarefully stepping deeper into the room, she gingerly picked up Seraph off Cade's very tight, sexy backside and nuzzled the warm, sleepy puppy before setting him next to Hailey. Cade must've removed Seraph's cone, and the bandages were gone, too. He'd saved her an extra trip to his office by doing that.\n\nCade's head jerked up, his worried gaze falling on Hailey before relief filled his eyes. He rubbed his neck and spotted Avery. \"Hey, you're home.\" The sleepy, hoarse baritone pulled a tiny shiver from her.\n\nHe rolled over and rose slowly, all sinew and grace, his gaze on Hailey to make sure he didn't wake her. Her chest pinched again. At full height, he stretched and scrubbed his hands through his hair, making it stand on end.\n\nIt seemed too intimate, standing and staring, but damn he was something to look at. And here she thought her libido had been permanently broken. He wasn't wearing the same clothes from earlier. Instead he had on dark blue scrubs, his feet bare. Noticing her questioning glance, he tilted his head toward the doorway.\n\nShe nodded and bent down to kiss Hailey's cheek, taking a moment to run her fingers through the silkiness of her hair before following Cade into the kitchen.\n\nHe looked around as if confused, then glanced at the clock. \"It's still early. Guess she wiped me out.\" He offered a nervous laugh.\n\nIgnoring the urge to\u2014she didn't know, but it would be stupid\u2014she rubbed her arms. Scratch that. She knew exactly what she had in mind. She wanted to climb his body, kiss him to the point of breathlessness, and snuggle in his warmth.\n\nShe cleared her throat. \"How did it go?\"\n\n\"Uh...yeah.\" He scratched his jaw. \"Don't get mad, but when my back was turned, Hailey took a sip of my milk. For future reference, I prefer beer. Anyway, it wasn't much, but she kinda got sick. All over me.\"\n\nShe pressed her lips together to try not to smile, but it was pointless. He was so flustered the grin escaped. Obviously Hailey was no worse for wear, so everything was fine. He'd handled it. \"So those are your clothes in the washer?\"\n\n\"Yeah. And some towels. We gave Seraph a bath after I took his stitches out.\" He looked behind him at the laundry room and then at her grin. \"You can just give me back the clothes on Monday at work\u2014\" He crossed his arms suddenly. \"Stop smiling like that. You're really not mad?\"\n\nShe pressed a hand to her lips, shoulders shaking. \"No, I'm not mad. I tried to tell you not to leave a cup around. She picks up anything\u2014\"\n\n\"I wasn't listening that closely. Lesson learned.\"\n\nShe sobered a little. \"I'm sorry she got sick on you, though. Not fun. And thanks for doing all that for Seraph. It was really nice.\"\n\n\"Not a problem.\" He looked at her through blue eyes, humor gone and replaced with a soft affection. One corner of his mouth quirked as if undecided if his thoughts were funny or stupid.\n\nStatic charged the few feet between them. Pulling. His gaze never left hers as his throat worked a swallow. She'd kill to know what he was thinking, but something told her his thoughts weren't far from her own, that he was feeling the slow heat.\n\n\"Do you want something to drink?\" she asked, and her voice came out more sultry than normal.\n\nHe cleared his throat twice before answering. \"Yeah, sure. Whatever you have.\"\n\nStarting a pot of coffee\u2014decaf or she'd never sleep\u2014she focused on her task so she wasn't tempted to look at him. Eye. Gasm. \"Everything else went okay?\"\n\n\"Kosher. How was the meeting?\"\n\nShe turned. \"You know, it was actually kind of fun. I don't know if it was the adult interaction or just getting away from the house on my own, but I liked it. The ladies were nice. Gossips, but nice. They acted as if everything I said was an epiphany. I think they just needed a fresh set of eyes and ears. Anyway, the committee isn't too demanding, so I think I can handle it. Don't tell them I said that.\"\n\nHe'd taken a seat at the table while she was gabbing, his chin now on his palm and a lazy grin tipping his lips. \"Our secret.\"\n\nCrap on a cracker, he was yummy. Her pulse sped. She turned away. \"I'll have to get a sitter for Friday nights. Do you know anyone? One of the high school students, perhaps?\"\n\n\"I can do it.\"\n\nShe gripped the counter. Hard. \"An attractive single guy like you, occupied every Friday night? Can't ask you to do that.\" At the bar the other night, that woman had draped herself all over him. It wasn't the first instance, either. Since then, three new pictures with clingy females had all popped up on new boards.\n\n\"You think I'm attractive?\"\n\nOrdering her knees to lock and hold her upright, she faced him. \"Coy is unbecoming.\" Not really, because her cheeks were flaming, but it seemed like a decent thing to say.\n\n\"I'm not fishing for compliments. I'm merely interested that you find me attractive.\" He leaned back in his chair, stretching his legs out. \"And you didn't ask me to do anything. I offered. I'll watch Hailey during your meetings.\"\n\nUnsure, she poured them each a cup of coffee and brought them to the table. Grabbing cream and sugar, she set those down, too. She took a chair next to him, her leg brushing his until she shifted. The intimacy wasn't lost on her. The two of them sitting in the dim kitchen, drinking coffee together. Richard, when they'd still been together, hadn't been home much, and when he had been, he certainly hadn't sat at the table and chatted. Back then, it was usually just her and Hailey eating meals together. It was odd having a man beside her, in her home.\n\n\"She's a great kid, Avery.\"\n\nHer gaze shot to Cade's. And there he went again, being sincere and sweet. Most people had a hard time seeing past Hailey's disability to the girl inside. Cade didn't seem to see a handicap at all. He treated her like he did just about everyone else.\n\nShe took a sip of coffee before answering. \"Thank you.\"\n\nLeaning forward, he crossed his arms on the table. \"Is it hard for you with her not being able to talk?\"\n\nChewing her lip, she thought that over. No one had ever asked her before. \"Sometimes. She uses sign language when really frustrated, and she has a speaking app on the iPad, which has pictures and such to show what she needs if she has to. I...\"\n\nShe shook her head, her internal filter kicking in, not wanting to get too personal. He made it easy to talk to him, but he was still her boss and she didn't know what boundaries there were between them.\n\n\"You what?\"\n\nStaring at his large hands wrapped around his mug, she decided it didn't matter. Cade genuinely seemed interested. \"I guess the only time her being nonverbal bothers me is...in the little things I miss out on. The inane chatter from normal little girls, hearing a giggle.\" She paused. \"I'll never get to hear her say, 'I love you, Mommy,' like other parents take for granted.\"\n\nHe didn't say a word, but she could feel his gaze on her, quiet, intent, as she looked into her coffee. She closed her eyes briefly and shook her head, willing the flush to leave her cheeks.\n\n\"Anyway, I got used to the quiet a long time ago. At least I don't have to yell over noise, right?\" She forced a breathless laugh. God, she'd give anything to have the normal sounds of kids playing, fighting. Most parents wanted quiet. She desired the opposite.\n\nIn the silence that hung, she couldn't take it anymore and lifted her gaze to his. What she found in his eyes she hadn't seen in too many years, if ever. Not pity, but sympathy. Respect. Understanding.\n\nAwareness hummed. It had been years since someone had looked at her instead of through her. Her fingers tightened around her mug, and when he opened his mouth to speak, air trapped in her throat as she wondered if he'd shrug off the moment or embrace it.\n\nShe had no idea what she'd do if it was the latter.\n\n# Chapter 8\n\n\"Silence isn't always what it's cracked up to be,\" Cade said, and scratched his jaw at his witty, brilliant insight.\n\nHell, he'd rarely been shocked by anything. Growing up in Redwood Ridge, where everyone knew everything about you and secret was a word in the dictionary, he'd seen the best and worst people had to offer.\n\nBut sitting across from Avery at her kitchen table and listening to her share something so private his chest ached left him reeling like the floor had dropped out. He wondered how many times in his childhood he'd told his parents _I love you,_ and then tried to imagine his own hypothetical kids and how he'd feel if he never got to hear those three words.\n\nIt was more than that. The way she'd brushed off the comment and claimed to be used to quiet had him guessing she wasn't just referring to Hailey. He didn't know anything about her ex, but if the guy had let Avery go, he couldn't be too right in the head.\n\nHer cheeks were crimson. Even in the dim kitchen, he could make out the blush, and he was sorry to have embarrassed her. She met his gaze briefly, those cocoa eyes mixing with honey, and then skittered away.\n\nAnd wasn't it a crazy bitch, but he had the urge to reach across the table and pull her into his lap. Talk to her until the night waned and sunrise peeked over the mountains. Suddenly, he wanted to know everything about her, and that sent a ripple of shock and need through his gut.\n\nMostly shock. Because unless he was referring to his friends, talking wasn't what he typically sought from females.\n\nHe took a sip of his cooling coffee. \"Your ex, how is he with Hailey? I figure he's got holidays and every other weekend or something?\" And if the guy showed up in Redwood Ridge, Cade might just rearrange his face on principle.\n\nAvery shook her head, gaze trained on her cup, but weary exhaustion flitted in her eyes and was gone in a flash. \"He hasn't seen her in two years, not since I filed for divorce.\" She looked up at him and sighed. \"I have full custody. That was part of what took so long for the divorce to go through. The only thing I asked for was a small trust fund for Hailey, in case something happened to me. He didn't believe child support or a trust was pertinent if he signed parental rights away.\"\n\nForget rearranging his face. He'd break every bone in the guy's body. Instead of the plethora of choice terms that came to mind, Cade reined in his temper. \"She's better off without him. So are you.\"\n\nShe nodded. \"My sentiments exactly.\"\n\nWhen he'd asked about her ex before, she'd been tactful. Not that she wasn't now, but she wasn't choosing her words as carefully. \"You don't speak ill of him. Why?\" She had every right to, by the sound of it.\n\n\"I don't like painting him in a terrible light in front of Hailey. He might change his mind one day about seeing her, and I wouldn't want to scare her off. He's her dad.\" She sipped her coffee. \"The way I look at it, we both brought her into this world. By me bad-mouthing him, it gives her the impression half of her is bad, too.\"\n\nChrist. Her selflessness knew no bounds. Had it been him, he didn't think he'd be so mature. Anger and betrayal and hurt did things to people. It was normal to lash out and react. But it was as if her feelings didn't matter, putting only her daughter first. He shook his head in wonder, his respect for her notching near pedestal height.\n\nSince it seemed like she needed a topic change, he went back to his reason for being in her cabin in the first place. \"So what else went down at the meeting tonight? They rope you into doing anything else?\" He grinned to lighten the grief in her eyes, which he supposed was more for him than her.\n\nShe told him about moving the venue and the secret admirer exchange, which he found quite clever. \"Online invitations and flyers will go out on Monday. I guess the dance was pretty casual before, but the ladies thought something formal would be more romantic. To be allowed inside, guests must dress in red or pink to attend. And it's suit and tie.\"\n\nHe laughed. \"Like prom.\"\n\n\"Sort of.\" She shrugged, but her smile was back. \"I think it'll be a nice change of pace, at least by the sound of it.\" She gestured to his cup. \"You want me to heat that up?\"\n\n\"No, thanks. I should get going.\" Not that he wanted to.\n\nDisappointment settled in her smile, and he knew, right then and there, that she was feeling it, too. The attraction had been obvious, at least to him, from the minute she'd carried an injured Seraph into the clinic. It had only been amplifying since. He'd been taking things slow, biding his time for the moment to act instead of standing still.\n\nShe rose and rinsed their cups out in the sink. \"Thanks so much again for watching Hailey. We'll see about next week\u2014\"\n\nShe stopped the pleasantries when he came up behind her and moved her soft brown waves off her neck to expose her nape. A breath shuddered out of her mouth, and then she seemed to stop breathing altogether.\n\nLeaning in, he nuzzled the soft skin below her ear, keeping his body mere inches away, lest he jar her with his arousal. Her fruity scent encompassed him, filled him, and he fought the urge to nibble on her neck to find out if she tasted just as sweet. She didn't move, not away or closer, and he took that as a good sign.\n\nHe brought his lips to the shell of her ear. \"You didn't ask me why I don't mind babysitting your daughter on Friday nights.\"\n\nShe shivered. He grinned in satisfaction. Oh yeah, she was right there with him. And unlike before, when he'd driven her home from Shooters, she was ready to be kissed.\n\n\"This isn't a good idea, Cade.\" Her voice, a breathy whisper so unlike her usual calm, was beginning to tear his composure to shreds.\n\n\"Good ideas are rarely fun.\" He knew it was the wrong thing to say when she turned to face him, and he found her walls back up, her spine stiff.\n\nShe wasn't the type of woman to go looking for fun, playful, or short-lived, but damn if he hadn't met anyone who needed fun more than Avery. And at the moment, he was feeling the opposite of playful. Not for one second did he think this was a game.\n\n\"You're my boss, and I'm not in the market for romance.\"\n\nYep. Walls. Funny, he usually respected them. He had some of his own, for that matter.\n\nHe placed his hands on the counter behind her, caging her in and taking a step deeper into her gravitational pull. \"People don't usually shop for romance. It just happens.\" Like right now, for instance.\n\n\"You're still my boss. I need this job so I don't have to dip into Hailey's trust fund. People will think... It's not a good idea.\"\n\nPeople were going to think whatever they wanted, regardless, and her job wasn't at stake. \"Then how about we keep things just like this, until you're ready for more.\" He could do secret and he could do slow. Right? \"No one has to know unless you want them to.\"\n\nShe started to shake her head\u2014and damn, he'd never had to work this hard\u2014but he leaned in until his mouth was a whisper from hers. Her eyes widened.\n\n\"Since you refuse to ask me why I don't mind the babysitting gig, I'll tell you. I like your kid and I really like you. Your organizational mind is so hot, and I've been itching to unwind the tension from you since you alphabetized the magazines in my office. Yes, I noticed that,\" he added when she opened her mouth. \"I enjoy talking to you outside of work, like we did tonight, because you relax more and say things you normally wouldn't. Thus, the idea of seeing you every Friday night when you come home is a hell yes in my book.\"\n\nThe tiniest of wrinkles formed between her eyebrows as if she'd never heard something so ludicrous, but he could tell she was damn tempted to believe him. Wary hope and interest sparked in her eyes. \"Um...\"\n\n\"That. That right there nails home my point. You fluster the hell out of me.\" He'd put his foot in his mouth more times in the two weeks since she'd come to town than the whole of his teen years. \"And it's too much fun watching your mind go blank when I say something you're not expecting. Such as, I find you very attractive.\"\n\nShe let out an uneven breath that skated across his jaw. He went from semi-hard to _down, boy_.\n\n\"Here's the part where you say you find me attractive, too.\" He grinned for good measure, and was rewarded with the clouding of her eyes.\n\n\"You know you're attractive. I could cut my teeth on your abs, but...\"\n\nKilling him dead. \"You haven't seen my abs. Would you like to?\"\n\nThere went the eyes again, wide. A sexy as hell blush tinged her cheeks. He could all but feel the heat.\n\nFun as this was, he was sweating with the resistance. \"Going to kiss you now, Avery. Three, two, one...\"\n\nShe sucked in a breath, dragging his with her. He brushed his lips across hers, testing, getting her used to him. He hovered there, barely touching, until she took the initiative and added more pressure. Letting her test the waters, he followed her command.\n\nAs if unsure, she fumbled, tensing against him. His gut clenched with the realization she probably hadn't been kissed since filing for divorce. And hadn't she said they'd been separated before that? From the little she spoke of her marriage and from what he'd gleaned, Cade doubted the ass had paid any attention to her needs.\n\nShoving the prick from his mind, Cade cupped her jaw and took over. Confidence was a fragile thing, and he wasn't going to be the one to break hers. He tilted his head, parting his lips to take more of her. Her lids drifted shut and he was lost.\n\nOne hand at the small of her back, he eased the other into her hair to gently hold her to him. After that, he lost track of what the hell was happening. His heart jacked against his ribs, his mind vanishing in a fog of sensation.\n\nShe was soft. Everywhere. The crush of her breasts against his chest, the strands of her hair, her lips as they moved against his. So damn soft. In the space between newness and uncertainty, she found a rhythm and opened to him. Her hands fell to his shoulders and fisted. A holy shit female moan vibrated from her mouth to his. He was losing a battle with sanity.\n\n_Don't push her. Get a grip_.\n\nHe eased his mouth away and brushed a kiss to her temple, keeping her body flush with his, struggling to draw air. Her breaths panted against his neck, hot and damp. Giving himself just a moment, he stilled until the spots before his vision cleared.\n\nShe muttered something that sounded like _crap on a cracker_ , and he laughed. Looking in her lust-induced eyes, he fought the strange pleasure\/pain in his chest.\n\n\"I need to get home.\" No, he really didn't, but staying any longer would kill him.\n\nHe kissed her forehead and stepped away, not missing in the slightest that she grabbed the counter behind her as if needing it to stay upright. He whistled for his dog while dropping into a chair to shove his shoes on. Freeman strode into the room and sat by the back door as Cade shrugged into his coat.\n\n\"Why'd you name him Freeman?\"\n\nHe glanced at his dog and back to her. \"He has light markings under his eyes and the stoicism of Morgan Freeman.\" He shrugged.\n\nA slow smile spread over her face, and he had to remind himself he was leaving for a reason. \"You named your dog after an actor?\"\n\n\"Yep.\" He paused. \"Why Seraph?\" He jerked his chin toward the hallway where her puppy and child slept.\n\nHer smile turned wistful. \"When I saw Hailey snuggle up to him in the exam room, he was like an angel in that moment. I'd never seen her connect with something like she did with him.\" She rubbed her forehead and laughed. \"It's a hassle getting her to leave him to go to school. And when we get home, the dog follows her everywhere, even to the bathroom.\"\n\nThe awed expression on her face did him in. She was the type of woman to appreciate the little things, accumulate moments instead of seeing the grand spectrum of it all. Rare, indeed.\n\nHe grabbed the knob and opened the door. \"To him, you were the angels. 'Night, Avery.\"\n\n* * * *\n\nFor some strange reason, Animal Instincts was hopping the next week, more so than typical. And then on Friday, Avery checked her Twitter phone app and discovered why.\n\nCade's Aunt Rosa had tweeted about Avery taking the event committee deal and that Cade had watched Hailey while she'd been out. One of Redwood Ridge's most eligible bachelors had shown he liked kids and he could do the responsible thing, which made him even more attractive. To women, it meant he had the potential to be tied down.\n\nWhich explained the sudden crush of female owners bringing in their pets for anything from \"her coat isn't as shiny\" to \"it slept a lot yesterday\" to \"he looked at me funny.\"\n\nShe needed a bottle of wine and twelve hours of uninterrupted sleep. She'd barely tolerated the complaining clients when she'd had to shift some off of Cade's schedule to Flynn's. Even Drake had taken a couple, much to his dismay. People and Drake were not cozy bedfellows, which was why he mostly did surgeries. But she appreciated the help just the same. The patients? Not so much.\n\nFlynn came in from the back, glanced at the waiting room, and shook his head. \" _What the hell? They're still coming?\"_\n\nInstead of signing a response, she showed him her Twitter feed.\n\nHe leaned against the desk. \" _That explains it. They're vying for Cade's attention and checking out the competition.\"_\n\n\"I'm not competition. There's nothing going on between us _.\"_ Even though Cade had kissed her in her kitchen and her girly parts were still weeping with joy. The man knew how to kiss. But no one else knew that juicy tidbit.\n\n_\"Liar, liar_.\"\n\nShe shook her head. \"Whatever. I'm just glad you're able to pick up some of the slack.\" __ Luckily, Flynn and Gabby didn't have many home visits this week, so they'd helped with most of the add-on appointments.\n\nNow if she could just figure out what to do with the six casseroles in the break room fridge. Added to that, there were so many cookies and brownies that she'd gained twenty pounds just looking.\n\nCade and his recent appointment came out of an exam room. Avery couldn't remember the woman's name, but her white Persian was Fifi. No lie. Fifi. And Fifi's owner was a stacked blonde roughly in her mid-thirties who was severely underdressed for an Oregon winter.\n\nWhen they stopped by the front desk to talk, Avery turned to Flynn and signed. \"Watch this. Hair flip in point five seconds.\"\n\nOn cue, the woman let out a glass-shattering giggle and flipped her mane over her shoulder.\n\nFlynn covered his face and tried not to laugh, but his shoulders shook and he emitted an awkward snort\/groan combination.\n\nCade looked at them, narrowed his eyes, and tuned back into the senseless flirt.\n\n\"Wait for it,\" she signed. \"Casual arm touch...\"\n\nFifi's owner dropped her slender, manicured fingers to Cade's forearm and leaned in as if Cade were reciting Shakespeare naked.\n\nFlynn hunched over, face red in hysterics.\n\nCade finished the chat, sent the woman on her way, and turned to them, shoulders tense. \"What's so funny?\" he signed and spoke simultaneously, pissy eyes narrowed.\n\nFlynn sobered. Or tried to. \" _Your harem is growing by the minute.\"_\n\n\"You're hilarious.\"\n\nAvery sighed. \"They're an embarrassment to my entire gender.\"\n\nCade glanced at Avery. \"It's not like I ask for it. Seriously, what am I supposed to do with all that food? My freezer's full. So are Drake and Flynn's. And I don't encourage them, for the record.\"\n\nHe didn't discourage them, either. Which just gave her another reason not to get involved with him.\n\n\"Never said you did.\"\n\nThe look on his face dried her hilarity. He was irritated, edgy, and if she wasn't mistaken, embarrassed. His gaze dropped away from her and he closed his eyes, face tilted heavenward.\n\nGuilt churned. She didn't realize it actually bothered him\u2014her and his brother poking fun or the women treating him like a new toy. He was so affable and outgoing, letting things roll off his back. Not once, no matter how ridiculous the appointment or how busy they got, did he complain. His full charm had been cranked. Every patient was seen, and he didn't make them feel stupid for coming.\n\nFlynn strode off as they stared at one another.\n\nCade sighed. \"How's the afternoon looking? Is it just as crazy?\" He eyed the semi-full waiting room.\n\n\"Hey,\" she said softly, and waited until he met her gaze. \"Do you want me to handle this? The casseroles and desserts? I can screen the appointments better, too. Not book so full.\"\n\nHe opened his mouth as if to say something, but Brent strode up, handed him a chart, and took the next patient to exam. Cade scanned the file, breezed through what she figured was Brent's notes, and shook his head.\n\n\"Christ,\" he muttered. With his brow furrowed, he looked at her as if trying to formulate words, his gaze impenetrable. Frustration rolled off him in waves. Jaw locked, he glanced down the hallway, still making no attempt to move.\n\nAnd something clicked into place. This was all her fault. The crazy week, the women, all due to a few tweets because he'd been nice enough to help her. Her stomach sank. \"I'm sorry about all this. I'll get someone else to babysit Hailey for my meetings. It'll die down around here once the word gets out\u2014\"\n\nHe strode to her side of the desk so fast her breath seized. Gripping the arms of her chair, he put his face inches from hers. \"Know what else will calm things down? Agreeing to go out with me.\"\n\nOh. Wow. Not expected.\n\nThey hadn't officially discussed the situation after their mind-melding kiss, but he'd offered to keep things under the radar if she said yes. She hadn't said yes. She hadn't been able to say much of anything after he'd knocked her into next week. With his lips.\n\nBusiness and pleasure were not a good mix. Plus, people would think she'd gotten the job or preferential treatment because of their supposed romance. After Richard, she'd sworn off men. Two years strong. She'd been with him so long she didn't know how to do this anymore and, in honesty, Cade was way out of her league. They'd been flirty and cordial and mostly professional in the office. They hadn't seen each other out of the clinic. Things hadn't begun to get complicated yet.\n\nForcing back a shiver, she took in the perpetual shadow on his jaw, his direct blue eyes, and the scent of fabric softener. The muscles in his shoulders and forearms flexed, all predator-like. His full mouth was dialed to grim.\n\nWow. This alpha thing he had going on sometimes was just as earth-shattering as his sweet side. She forgot her own name. Her mouth dried to dust. Her heart pounded.\n\n\"Fine,\" he said at length. \"Have it your way. Our secret. I can handle the fallout. I'll be over tonight to watch Hailey.\"\n\nWith that, he shoved off her chair, straightened, and headed into an exam room.\n\nIt took her a full five seconds to remember breathing was a requirement of life.\n\n_Squawk_. \"Blurred lines.\"\n\nShe eyed the cockatoo with a frown and jumped when someone cleared their throat to get her attention.\n\nThor startled underneath the desk, hauled ass to his feet, and rammed into the counter, knocking over a few charts. Before Avery could calm the giant Great Dane, he lunged at her for protection from the big bad five-foot blonde woman holding a...turtle. Yep, a turtle.\n\nThat was the last thing she saw as her chair tipped backward, hitting the floor with a thud, leaving one hundred and fifty pounds of trembling dog sprawled over her and her feet up in the air. Flat on her back, she calculated on a one to ten scale of how embarrassing this was, and came up with an eleven. She blinked at the florescent lights overhead.\n\nShe swore She-rah laughed at her from her cat perch on the printer.\n\nFootsteps squeaked over the tile and moved closer, stopping beside her and Thor.\n\nDrake's face appeared above hers. \"He's not a lap dog.\"\n\nHe had a funny bone after all.\n\nAfter two more grueling hours, she went to pick up Hailey at the rec center, finding April and her daughter, Jenny, talking to Miles at the front desk. They'd spoken nearly every day on her lunch break when she went to visit Hailey at school, but this was the first time she'd encountered her at the center.\n\n\"Oh, hey.\" April turned Avery's way. \"I'm glad I caught you. How would you feel about getting the girls together for a sleepover at our place? They get along pretty great and it would give you a night off. I was thinking the night of the Valentine's dance, since we're not going. Plus, you'll be super busy with the event committee setting up things.\"\n\nJoy hit her first, her chest swelling and her throat tight. Hailey had an honest to God friend. But apprehension soon followed, twisting in her stomach. The girls did get along well, not that Hailey engaged much on the surface, but Avery was hesitant to agree.\n\n\"She's never been to a sleepover before.\" Hailey had never been away from Avery for an entire evening, come to think of it. She also had a lot of quirks and special needs. April was trustworthy and Jenny was a sweet girl, though. Why not let her try? \"I will be really busy that night. Could we test it out before then, that way if I need to come get her it won't be an issue?\"\n\nApril waved her hand. \"Sure. How about next Friday night? I could take them both home right from school.\"\n\nAvery wouldn't get to see Cade that night, which should've warned her the sexy vet was on her mind too much. She shook her head. \"That sounds great. I'll pack some overnight things in her schoolbag. If it goes well, you'll have to let Jenny stay with us next time.\"\n\nThey chatted for a couple minutes, and Avery collected Hailey to head home.\n\nLater, when Cade arrived at the cabin at five-thirty on the dot, Avery was still staring off into space. She looked up at him, blinking furiously to hold back tears. \"Hailey's having a sleepover next weekend. Can you believe it? She's made a friend.\"\n\n# Chapter 9\n\nCade stared at Hailey from the couch as she played a game of tic-tac-toe on her device. She'd been at it for fifteen minutes. At least she always won. \"Can I play?\"\n\nIn answer, she set the iPad down on the cushion between them and went first. Her hands flapped as her gaze darted over his shoulder.\n\nHe tapped the screen to place his X.\n\nWithout even seeming to look, she set her O.\n\nThis went on through three rounds, where she blocked his every move and won all three games. \"You're ruthless. Good game, squirt.\"\n\nShe scrolled out of the app and swiped through others. The language one Avery mentioned popped up, and Cade remembered what she'd said last week about never hearing the words _I love you_. What he knew about autism could be put in a shoebox, but maybe he could... He didn't know. Maybe teach Hailey?\n\nHe rubbed the back of his neck and figured it wouldn't hurt to try. \"What do you say we work on a project, just you and me?\" He tapped the app and fished through some of the pictures. \"Can you show me the I?\"\n\nShe didn't seem to be listening, but after a few seconds she tapped the photo of the eye and a robotic voice said \"eye.\"\n\n\"Rock on, squirt.\"\n\nShe flapped her hands and squealed.\n\nHe grinned. \"Okay, show me a heart.\"\n\nShe repeated the process, taking her time, and tapped the picture, earning a voice that said \"heart.\"\n\nNot exactly \"love,\" but close enough. \"What about the letter U? Do you know your alphabet?\" This one seemed to trip her up, so he swiped through the letters. \"It's tricky, I know. Think of a smile. That's what the letter U looks like.\" When she didn't move, he hovered over the letter.\n\nShe pushed his hand away and tapped the U, the voice calling out her comment.\n\n\"You're a pretty smart cookie.\" He shifted a little closer, still respecting her space. \"So that's how you tell your mom you love her. Eye, heart, you. Let's try it again.\"\n\nHailey backed out of the app and pulled up what looked like a neon coloring book. She dragged her finger over the screen, setting a bright pink line across the black background.\n\nClearly done with his lesson, he grinned and rolled with it. \"You like to color?\" He glanced at the fridge on the other side of the counter, but there were no pictures there. \"Be right back.\"\n\nHe stepped down the hall into Hailey's bedroom, but after sifting through her toys, he couldn't find any crayons or paper. They'd just moved, so perhaps Avery hadn't unpacked them yet. Undeterred, he went to his car and fished in his glove box for the rainbow Sharpies he kept for charting and a legal pad he had in his trunk. Items in hand, he strode back into the cabin.\n\n\"Hey, squirt. Come in the kitchen for a sec.\"\n\nDutifully, she turned off her iPad after the slightest delay and seated herself at the table.\n\nSetting the items in front of her, he pulled up a chair beside hers. \"Want to color?\"\n\nShe made no attempt to reach for the markers, so he grabbed one and uncapped it. She never glanced at him or the pad of paper as he drew a really terrible cartoon dog. When finished, he held it out for her, but she pushed it away.\n\n\"Not into coloring. Got it.\"\n\nHe glanced at the fridge again, remembering the one in his parents' kitchen growing up. It had been littered with drawings, report cards and, later, team schedules. Avery's was blank. Not even a grocery list.\n\nSighing, he eyed the girl again and her small hands planted on the table. \"Can I trace your hands? Would you let me do that?\" She didn't confirm nor deny. \"I'm going to touch your wrist to move your hand, yeah? If you don't like it, let me know.\"\n\nWatching her closely, he lifted her wrist, pushed the pad of paper over, and set her hand on top. When this didn't faze her, he bit the cap off a marker and leaned over. \"Hold real still, squirt. I'm going to make a drawing of your hand.\"\n\nQuickly, in case it bothered her, he traced an outline around her tiny fingers and sat back. To his utter shock, she set the other hand on the paper, too. He did that one as well.\n\n\"Should we have a look? Lift 'em up.\"\n\nHailey was neither ecstatic nor bored by what he'd done. She simply got up from the table unenthusiastically and went back into the living room, turning the device back on and settling on the couch.\n\n\"Right. Scratch coloring from the agenda.\"\n\nHe scrawled her name on the paper and the date. Since Avery didn't have any magnets, he'd hunted up some tape from a drawer and secured the drawing to the fridge. He glanced at the clock.\n\n\"Hey, squirt. Time to turn on your sleepy show.\"\n\nAfter poking his head over the back of the couch to make sure she complied, he let the dogs out and sat next to her. No wonder this was a bedtime ritual. The video was dragging his eyes closed. Cartoon sketches danced and swirled to drone, soothing music until finally, blessedly, the credits rolled.\n\nHe cued her to go potty, covering his eyes while she did, and then waited for her to brush her teeth. Leaning against the doorway, he couldn't get his mind off how the kid was so well behaved. Unless she didn't understand his directions, she pretty much did whatever was asked.\n\nCade and his brothers had been hellions at her age, always getting into mischief, to which his parents only knew the half. Girls must be tamer by nature. He figured even if Hailey were verbal, she'd still be quiet and cooperative.\n\nLike her mom.\n\nThere were pitfalls, too. He had to refrain from tucking her hair behind her ear when it fell in her face, since she didn't like touch, and more than once he'd had to fist his hands to avoid pulling her in for a hug. The lack of eye contact and seemingly distracted nature of her condition made her hard to read. He was beginning to catch on to her slight mannerisms, though.\n\nAfter an hour of just one-on-one with the girl, he was beginning to get sick of the sound of his own voice. He wondered how Avery had done this for eight straight years. To never have a conversation, a verbal response most take for granted, could be disheartening. But when he could pull a laugh from Hailey, awkward as the sound was, or when she flapped her hands and squealed in delight, that was really something. He'd had to work hard at it, but it was all the more rewarding.\n\nShe dutifully went to her bedroom after rinsing her mouth. Tucking her in, he set Seraph on the bed beside her and called Freeman out with a whistle. Unlike last week, Hailey closed her eyes and didn't fight him.\n\n\"'Night, squirt.\" Leaving the door ajar, he made his way down the hall, grabbed some charts he brought in need of dictation, and plopped on the couch.\n\nBut after ten minutes, the words blurred in front of him as he kept remembering the expression on Avery's face when he'd first arrived. She looked like she'd been slapped upside the head, her pretty lips parted and her unfocused brown eyes wide. A sleepover for Hailey, a normal girly ritual, a right of passage, had put that awe on her face.\n\nGiving up on the charts, he made his way into the kitchen to start a pot of decaf, since Avery would be back any minute. When it was brewing, he opened the fridge to grab creamer, and stilled.\n\nShe had a six-pack of beer in the door. He stared at the bottles of his favorite brand, trying to remember if she drank the stuff. The only time he'd seen her drink had been the one night at Shooters, and she'd had wine.\n\nHe closed the fridge and rubbed his neck. Jokingly last week, after discussing the Hailey milk incident, he'd told Avery he preferred beer. Had she taken him literally? His gaze landed on the door again, and he knew she had. Hell, she'd bought him beer.\n\nUnsure what to make of this as it seemed such an intimate act, he went back to his charts until she came in the kitchen door and set her purse down on the table.\n\nHe rose from the couch and walked closer. \"Hey. How was your meeting?\"\n\nShe'd changed out of her professional work clothes before she left, but he got a better look now. Her jeans were faded at the stress points, molding her curves, and the plain blue tee barely skimmed below the waistband. He'd bet if he asked her to reach high for something, he'd catch a peek of bare skin.\n\nShe laughed, drawing him back to her face. \"I'll give your aunt this, when she wants something, she's tenacious. Marie not only got the post office onboard for the admirer note exchange, she's got the rec center kids thinking it's a top secret mission.\"\n\nLeaning against the counter, he crossed his arms. \"Sounds like it was productive then.\"\n\nShe hummed in her throat and glanced at the coffeepot. After staring at it for a few moments, an unreadable expression on her face, she blinked. \"How was Hailey?\" She glanced down the length of him. \"I see she didn't get sick this time.\"\n\n\"Ha. No, we survived. She kicked my ass in tic-tac-toe.\"\n\nHer grin stopped his heart. \"She's quite good at that game. Let me just peek in on her. I'll be right back.\"\n\nAs she disappeared, he poured them each a cup of coffee and settled at the table to wait. When she came back, she eyed the cups and turned for the fridge, freezing in place with her hand mid-air over the handle.\n\nAnd he found himself holding his own breath.\n\n* * * *\n\nAvery swallowed hard, unable to tear her gaze from the paper on her fridge with the outline of her daughter's tiny hands. She recognized Cade's distinctive handwriting below them from his charts, the block letters and scratch.\n\n\"Tried to get her to color, but she wouldn't.\" The rough, quiet timbre of his voice held an edge of uncertainty.\n\n\"No,\" she breathed. \"She doesn't like it. The therapists tried getting her to attempt it several times. She'll do it on the iPad sometimes.\"\n\nHe grunted. \"Are you going to stare at that all night?\"\n\nConcern laced his tone, so she turned. He studied her with solemn eyes, his brow furrowed just enough to make out his uneasiness. Was he worried she'd be mad?\n\nShe pointed to the picture. \"That's her first art project.\"\n\nHe opened and closed his mouth, but ultimately said nothing.\n\n\"I have a few stored away from teachers and therapists, but they were wielded with them holding her hand. Nothing just from her.\"\n\nHe shifted uncomfortably in his seat. \"That's not original, either. I traced her hands.\"\n\nSince he wouldn't understand this _was_ unique, that he'd put her daughter's hands to something in the only way she'd allow, she opened the fridge and grabbed the creamer. She walked to the table and sat next to him, thinking first thing tomorrow, she'd buy a frame for that simple, endearing paper taped to her fridge.\n\nSilence stretched as they sipped their coffee, until he finally cleared his throat. \"So Hailey has a sleepover next Friday, yeah? Are you going to get a wink of sleep?\"\n\nThis dragged a laugh from her, which she assumed was his goal. \"Most likely not. She's never been away from home. I'll probably need a sedative not to call a hundred times.\"\n\nHis smile was sigh-worthy. \"April is good people. Hailey will be safe with her.\" He leaned forward and scratched his jaw. \"She and her husband moved here something like ten years ago. Nice family.\"\n\nAvery got the same impression, but said nothing.\n\nHe shook his head, grin widening into swoon-worthy territory. \"You'll still pace the floors with worry.\"\n\nShe nodded and looked away before she climbed over the table to lick him. \"I'll still pace the floor with worry,\" she confirmed. Then, she placed her palm to her forehead and laughed at how well he knew her after such a short time. Crazy, that. Even more insane was how easy he was to be around.\n\n\"Avery.\"\n\nShe glanced up.\n\nHis smile slipped, his gaze falling to her mouth. He muttered something unintelligible, kicked his foot out, and wrapped his leg around the rung of her chair. Slowly, he dragged her chair in front of his until their knees bumped, his gaze never leaving hers. He leaned forward, not touching her, but the heated desire in his eyes was an instant bolt of contact.\n\nShe let out an uneven breath. \"What are you doing?\"\n\n\"Honest to God, I don't know.\" After his whispered confession, his gaze took her in\u2014her hair, her eyes, her mouth\u2014as if trying to figure it out. \"I never know what the hell I'm doing around you.\" His lips parted like he wanted to say more, but he just shook his head and pressed his mouth to hers.\n\nLike the first time he'd kissed her, he brushed her lips with a tender caress, cajoling her into joining him. The unfamiliar contact was a mere blink of uncertainty before fire flared in her belly. Spread. Consumed. She breathed in his scent of male and fabric softener, but it didn't ground her as she'd hoped.\n\n\"Been wanting to do that all week,\" he said against her mouth. \"Kiss me back, Avery. Like you mean it. Give me some idea you're as\u2014\"\n\nShe sealed her lips to his, tilting her head to probe at a different angle. Parting her lips, she tentatively licked his lower lip, hoping he'd open. On a groan, he complied, and when their tongues finally met for the first time, something inside her snapped. Control and reason broke free of their leash and disappeared. Her hands fisted in the thick softness of his hair.\n\nHe sucked air through his nose and, never breaking the kiss, dropped his hands to her thighs and squeezed. A bolt of need shot straight to her core, causing an ache she hadn't experienced in too long, if ever.\n\nOh... His hands were on the move, sliding under her thighs. He lifted her from her chair as if she weighed nothing and deposited her in his lap to straddle him, bringing their chests flush. His hard to her soft.\n\nHe tore away to press his mouth to her throat, gulping air in tandem with her attempts to do the same. \"Slow down,\" he murmured against her skin, even as his tongue darted out to lick her pounding pulse.\n\nA shiver ripped through her body, his touch lighting her nerves, but she unclenched her fingers from his hair and eased back a smidgen to adhere to his request.\n\n\"Not you.\" He held her hips. \"I was talking to myself.\"\n\nHe lifted his head to look in her eyes, his drugged with the same fever. Then his mouth was pressed to hers again, kissing her blind. Desperate. Her breasts grew heavy, aching to be touched, so she crushed them to the hard wall of his chest to alleviate the throbbing. But then other parts began to throb, and the urge to grind into the thick bulge between them was almost feral.\n\nShe whimpered into his mouth, needing...something. Him? He groaned in response, his hands moving north to play with the hem of her shirt. His warm fingers dipped beneath on their way to her ribs, and she tensed.\n\nHe stopped on a dime, sensing her shift. Slowly, he drew back far enough to look in her eyes. \"Sorry. Too fast. Thought and reason aren't really processing just now.\" Though coarse with need, his voice was quiet, apologetic, making her fumble to formulate an explanation.\n\n\"It's just...\" She hadn't been touched in God knew how long and the hard pecs below her palms were a direct contrast to her rounded curves. He was all edges and yum and she...wasn't.\n\n\"Just what?\"\n\n\"I haven't had the time to do a lot of exercise or my usual yoga, and I might not be in the best shape, or what you're used to.\" She pinched her eyes closed as heat flared in her cheeks.\n\nWhen he didn't say anything for several erratic beats, she peeked. His jaw was clenched and his blue eyes glacial. Her heart stuttered to a halt.\n\n\"I swear, if you tell me that asshole ex of yours called you fat, or so much as implied it, I'm driving to where he is to beat the shit out of him. Tonight.\"\n\nAir seeped from her lungs. She'd had a hard time losing the baby fat after delivering Hailey, but she'd worked hard because the disgust in Richard's eyes had been palpable. But getting back to her target weight hadn't mattered in the long run because he'd sought someone much thinner and prettier from his office for his gratification. The betrayal and hurt, even after all this time, cut deep.\n\nIf she hadn't been enough to satisfy Richard, and they'd been college sweethearts bringing little experience to the relationship, how was she supposed to believe she could do so for Cade, who had worlds of knowledge and a trail of women in his wake?\n\n\"What's his address?\" Cade ground, drawing her gaze back to his.\n\nHer stomach rolled, and she looked away. \"It wasn't him.\"\n\nRichard wasn't to blame. No matter how he'd treated her, she was responsible for believing him or not. It was her old ghosts screwing with her head, and she thought she'd gotten over them.\n\nCade had been into it moments ago. His heart was still pounding beneath her palm, though she suspected that was in anger now and not lust. But his erection said he'd been turned on, too. Yet she'd doused that flame with reality, and the moment was lost.\n\n\"Avery\u2014\"\n\n\"No, it's okay.\" She climbed off his lap and he winced. From the loss of contact or something else, she couldn't tell. \"It's getting late, anyway.\"\n\nHe stared at her a few more seconds and rose, whistling for Freeman. Cade stepped into his shoes and walked into the living room. The dog strode into the kitchen and waited by the back door while she made herself busy by rinsing out their cups at the sink.\n\nShe didn't turn when Cade came back, but his gaze bore holes into her. She gripped the counter as he sidled up behind her, setting a small stack of charts down by her hand.\n\n\"Look at me.\"\n\nShe shook her head, her stomach knotting. This was a bad idea to get involved with him. He was a playboy who probably just saw the challenge in her. If things at the clinic became uncomfortable because of this flash romance, she'd have a hard time finding another position. People already figured she got her current job by sleeping with Cade. Plus, she had enough on her plate, had been burned enough to know better.\n\nYet she couldn't deny the pull, the flutter in her stomach when she was around Cade. He managed to make her feel giddy again. Hopeful.\n\nCade ignored her denial and turned her to face him. His eyes sharpened in understanding, his mouth a thin line. \"I'm not him.\"\n\nShe sighed. No, he wasn't anything like Richard, and to even compare the two was an insult to Cade. \"I know. I think I just need time to get used to having someone interested again. Been a long time.\" She laughed nervously and tried to step away.\n\nHe put an arm out to stop her. \"I'm interested. Make no mistake about that.\"\n\nShe lifted her gaze to his, and the breath punched from her chest. With dizzying absolution, she was positive no one had ever looked at her with the same desire and patience. And this thing between them had the potential to hurt her more than anything her ex had ever attempted.\n\nCade leaned in and kissed her. It was a gentle meeting, not passion-filled or hungry, and somehow that tripped her pulse more than what they'd done earlier.\n\nEasing back, he looked into her eyes. \"I wasn't going to take tonight any further than what we were doing. I'll go as slow as you want. Just do me a favor and push him from your head when we're together, yeah?\"\n\nBefore an intelligent response could squeak past her lips, he grabbed his charts and was gone.\n\n# Chapter 10\n\nNear lunchtime on Monday, Cade exited an exam room, sent Martha and her \"depressed\" hamster up front, rolled his eyes, and headed toward the break room to grab some water. He slammed into Flynn.\n\nHis brother grinned. \" _I want to show you something_.\" He crooked his finger, so Cade followed, since he was headed in that direction, anyway.\n\nFlynn paused at the fridge. \" _Take a look inside_.\"\n\nSighing\u2014because Cade knew there would be twenty more casseroles inside as they were back to back with mostly unnecessary appointments\u2014he opened the fridge. And found it empty. Or almost empty. There was their usual bottled water, a few brown bag lunches, and Drake's pudding stash, but not a casserole in sight. He still wondered why the single women of Redwood Ridge found it necessary to supply him with a year's worth of home cooking.\n\nFlynn lifted his brows. \" _Avery is telling the patients we're no longer allowed to accept food because it's against health regulations.\"_\n\nCade opened and closed his mouth. \"Why would she do that?\"\n\nNot that he was complaining. Half the time, Cade drove the stuff to the homeless shelter in the next town, since he couldn't eat it all. But it wasn't against health regulations as long as the food remained in the break room.\n\n_\"You're an idiot_.\" Flynn poked his chest. Hard. \" _She's doing it for you. She knows it bothers you to get all this unwanted attention_.\"\n\nThat gave him pause. It was true. He wasn't fond of all the flirting or subtle lean-ins or any other form of vying for his heart, much preferring to let things progress normally without a shove. But he'd never said so. Not once. In fact, to spare hurt feelings, he went out of his way to be gracious.\n\n_\"What did you have for dinner at her place on Friday_?\"\n\nCade blinked, sensing a trap. \"Pizza.\" Hailey didn't react violently to white cheeses, so that was one of the few dairy items she could eat, in moderation. He narrowed his eyes. \"Why?\"\n\nFlynn nodded knowingly. \" _The way I see it, if she's funny, kind, eats pizza, likes your sorry ass, and is good in bed, you should marry her. Yesterday.\"_\n\nChrist. \"We've known each other a month, man. I haven't slept with her.\" The rest of his brother's statement he wouldn't touch with a ten-foot pole.\n\nThat only made Flynn's grin widen, the asshole. \" _Thanks for proving my point. When's the last time you took a month to sleep with a woman?\"_\n\nCade scrubbed his hands over his face and headed up front to see if it was safe to take lunch. Alone.\n\nAvery was finishing up with Martha and the hamster as he approached. \"Thanks so much for coming in today. Please take a goodie on your way out.\" She pointed to several plates of cookies on the counter, which usually wound up in Cade's office, since they were brought for him. \"It's so nice of our patients to bring these for us, isn't it?\"\n\nMartha, a short brunette dental assistant from the office down the street, shoved a cookie into her mouth with a frown and left.\n\nCade would bet his right nut Martha had contributed to the sugar donation Avery was pawning off. Something pinched in his gut, which he attributed to hunger.\n\n_Squawk_. \"Sugar, sugar.\"\n\nAvery laughed, having not seen Cade yet, and was typing something into the computer schedule. \"Got that right, Gossip. She-rah, you leave that dog alone.\"\n\nCade glanced over, and sure enough, the cat had stretched from atop the printer and was eying a sleeping Thor by Avery's feet.\n\nThe cat paused, narrowed her evil eyes, and made to jump down.\n\n\"Ah-ah.\" Avery waggled her finger, never turning from the PC. \"You go near that dog and you'll get it.\"\n\nShe-rah sniffed.\n\nAvery reached over, snatched a spray bottle he hadn't noticed, and...squirted the cat in the face with a short stream of water. \"I said leave Thor alone.\"\n\nShe-rah hissed, but remained on the printer, stopping her planned assault on the dog. She swiped her face with a paw, sent Avery a warning glare, and laid back down.\n\nCade stood there for several moments as the pinch in his stomach moved into his chest, and he had to remind himself he was at work. They were taking things slow. But the pounding of his heart against his ribs wouldn't cease. She was... She was so damn hot in her bossy mode. And very different from the vulnerable woman on Friday night.\n\n\"Do you need something, Dr. Cade?\"\n\nHell. He wondered if he could get her to call him that when he kissed her again. Outside of the office.\n\nGiving his head a violent shake to clear it, he tossed a chart down next to Avery and eyed the empty waiting room. \"I'm taking twenty minutes for lunch.\"\n\nShe closed out her program and rose, preparing for her own break. \"That doesn't go there.\"\n\nHe glanced at the chart she referred to and then her, the urge to kiss her so damn jolting he almost flinched. Grabbing the chart, he set it in Avery's file basket, never taking his gaze from hers. The electricity between them could supply the town for a week.\n\nShe smiled, turned on her heel, and headed down the hall.\n\nHe caught up to her and pinned her to the wall, earning a surprised squeak from her. After a quick glance to make sure no one saw, he stepped closer, shoving his thigh between her legs and crushing her breasts to his chest.\n\nHer wide brown eyes didn't blink, and in this light, he could make out the honey flecks. \"What are you doing?\"\n\n\"This.\"\n\nWith his hands on her hips to hold her in place, he leaned in and kissed her. Not the careful enticement from her kitchen, but a full-on exploration to let her know how much she turned him on and just how much he appreciated what she'd done for him by micromanaging the plethora of female clients. And the cat. And damn it, the chickenshit dog.\n\nShe arched into him, fisting his scrubs, and moaned into his mouth like he made her forget their location, the day, and time space continuum, too. Her tongue stroked his, once, twice, rioting his thoughts right out of his head. Well, to the wrong head.\n\nShe smelled sweet, tasted sweeter, and if she kept grinding his thigh like that...\n\nHe tore away and rested his cheek to hers, breaths soughing. Holy hell, what she could do to him with a kiss. Her breathing wasn't stable either as she panted against his neck. His erection pressed into her hip, so he backed away, taking in her flushed cheeks and pert nipples.\n\nBiting back a groan, he said, \"Just wanted to tell you that. We can discuss it in great detail later, yeah?\"\n\nHer palms pressed to the wall behind her, those gorgeous eyes unfocused and her lips parted. \"Yeah.\"\n\nNodding, he headed out the back door instead of into the break room because suddenly he needed to cool off. However, several gulps of crisp, damp air did nothing to steady his heart rate.\n\n* * * *\n\nAvery snuck out of the clinic at five on the dot. If she did so because Cade was occupied in his office and wouldn't see her leave, well that was a coincidence.\n\nThis non-dating dance they were doing was heating up, and she didn't know how to feel about that, so she shoved it from her mind. Almost. Her lips were still swollen from his kiss, and every time she thought about how he pinned her to the wall and devoured her mouth like he had no choice, a blush flamed her cheeks.\n\nRichard had never pressed her up against the wall.\n\nStraightening her shoulders, she got in her car and drove the mile to get Hailey. Maybe they'd go to the diner tonight for dinner. Get out of the cabin. After she parked in the rec center lot, she pulled out her cell to text her mom to see if she could meet them.\n\nAnya manned the front desk and smiled as Avery opened the door. \"Hello. How's dating the sexy vet going? You're breaking hearts all over the Ridge.\"\n\nShe sighed. \"Cade and I aren't a thing.\" Despite what Twitter and the town's blog claimed. Despite the kissing. The really excellent kissing.\n\nA lot of women in town were under the impression Cade was theirs. Every minute a new picture of him popped up on Pinterest with a willing female. It made it hard to figure out how seriously to take him.\n\n\"Oh, I don't know. People are talking. Cade doesn't date, as in at all, so this is pretty big news around here.\"\n\nSo was watching paint dry on the new park bench, but she kept her mouth shut until Miles brought Hailey up front.\n\n\"I missed you.\" Avery bent to zip her daughter's coat and squeezed her shoulder when she really wanted to kiss her. \"How's she doing? Any problems?\"\n\n\"Nope.\" Anya tucked a piece of strawberry blond hair behind her ear. \"She's really great. The only thing that seems to upset her is the basketball games, so we keep her out of the gym.\"\n\nAvery nodded. \"Must be all the squeaky shoes.\"\n\nShe thanked them and led Hailey to the car. Once she had her belted in her booster, Avery checked her phone and noted her mom texted back with a confirmation of dinner.\n\nThey drove to the diner, which was surprisingly slow, but it was Monday. She noted a few men from the senior center were playing checkers at a table, but otherwise the place was empty. The fifties d\u00e9cor was retro-neat, she thought so when she first stopped in two weeks ago, but the neon signs and counter were under a layer of grease from years of fried food. The scent of French fries and hamburgers clung to the air, and Avery's stomach rumbled.\n\nShe spotted her mom in a booth and walked over, getting Hailey situated. \"Glad you could come, Mom.\"\n\n\"Well, I don't want to take up your time with Cade, so I just wait for you to call. A woman needs private time with her guy.\"\n\nAvery barely resisted an eye roll. \"He's not my guy. He's my boss.\"\n\nWho kisses her a lot. Up against the wall.\n\nBefore her mother could respond, a waitress made her way over, a woman Avery recognized from around town, but had never spoken to. Avery quickly scanned the menu for something Hailey could eat while her mom and the waitress made small talk. In her late sixties, the woman had wrinkles around her mouth and a craggy voice that bespoke years of smoking.\n\nHailey shrugged her shoulders as if bothered by her voice.\n\n\"And what can I get the cutie pie?\" The waitress, Mave, bent down at eye level with Hailey and ruffled her hair before Avery could stop her.\n\nHailey instantly stiffened and screeched, flailing her arms. Silverware and ice water went sailing. She slid to the floor under the table, earning gazes from the other customers, and continued the screeching at an insane decibel.\n\nMom laughed nervously and ducked her head.\n\nCrap on a cracker.\n\nHailey didn't act out very often, and when she did, it could be quite embarrassing, but her mother's reaction had fury building in Avery's chest. People would see her response and take their cues from her, thinking it was okay to stare at Hailey like a freak.\n\nMave's eyes popped. \"Oh dear. What's wrong with her?\"\n\nAvery tensed to the point of pain, her molars grinding. It always came down to that\u2014what was wrong with Hailey. She was different, so that made it okay to act like ignorant imbeciles, right?\n\nInstead of getting into it, Avery asked her for a few minutes and then slid out of the booth to squat by Hailey, who had stopped screeching. \"Hey, sweetie. She's gone. All better. Can you come out?\"\n\nA few moments passed and Hailey crawled back into the booth to stare out the window, all signs of distress vanishing except her all too common rocking.\n\nAvery reclaimed her seat, closed her eyes, and rubbed her forehead. \"If you're embarrassed by us, we can leave.\" She drilled her mom with an unforgiving glare.\n\nMom's eyes widened from where they were watching Hailey and shifted to Avery. \"I've never seen her do that. She's always so quiet.\"\n\nOf course she hadn't seen an outburst. For years, it had just been Avery and Hailey against the world. Mom had moved to Redwood Ridge before Hailey was born, and had only come to San Francisco a handful of times. Regret filled her because Avery hadn't made any attempt to visit her mom, either. Between Richard's schedule and Hailey's therapy, years snuck out from under them.\n\n\"I'm not embarrassed.\" Mom took her hand and squeezed. \"I'm sorry. I froze.\"\n\nThe tension drained from her shoulders. \"It's fine.\" Mom didn't know any better. Most people didn't, so Avery would educate where she could and shrug off when she couldn't.\n\nMave tentatively returned, a wary smile twisting her already wrinkled face. \"Everything okay?\"\n\nAvery forced a smile. \"This is my daughter, Hailey. She has autism and doesn't care to be touched, especially on her head. It's an over-stimulant for her, which is why she responded that way.\"\n\n\"Oh, I'm so sorry. I'll make sure not to do it again.\"\n\nSmiling, she tilted her head toward her daughter. \"Hailey, this is Mave. Can you say hello?\"\n\nA few moments passed as Hailey seemed to process the command, and then she signed, \" _Hello_.\"\n\nOnce they finished eating and waited for the check, Zoe strode in looking harried with her blue hair at haphazard angles and her button-down shirt half tucked into her jeans. No coat. She moved to the register and dropped some bills, tapping her foot while waiting. Since she seemed to be in a hurry, Avery didn't stop to talk as the groomer from the clinic grabbed two to-go boxes and darted outside.\n\n\"She's had a rough go of things lately.\"\n\nAvery glanced from Zoe's form disappearing in the fog back to her mom. \"How so?\"\n\nAside from her odd, unnatural choice in hair color, Zoe seemed okay. She'd gone out with them that one night at Shooters and she always came to work on time. Though not as chatty as Brent or Gabby, she was nice.\n\n\"Her mom was diagnosed with early onset dementia four years ago. Catherine's only fifty-one. There's no other family, so Zoe's taken it all on herself.\" Mom cleared her throat. \"We try to get over there and help out when we can. Cat used to be in our book club, but now she barely recognizes Zoe.\"\n\nGod. How awful. She rubbed at the sudden ache in her chest, unable to imagine it. \"That's rough.\"\n\nAfter parting ways with her mom, dark had descended, bringing a salt-scented brine to the humid, cold air. The fog had let up, but was still hovering in the distance near the banks. Avery and Hailey walked across the street to the ice cream shop to get Avery a fix for later before heading home.\n\nHank, the owner who'd greeted them the last time, grinned broadly. \"You've returned. Check it out.\" He pointed to the menu behind him. His round belly strained against a stained apron with the movement.\n\nAvery scanned the board, her gaze stopping at the new addition. She stilled, torn between a sudden bout of useless tears and gratitude. Surely he hadn't...\n\nHe had. He'd given Hailey her own special on the menu. Orange sherbet topped with marshmallows, kindly named \"The Hailey.\" He'd remembered she couldn't eat dairy. That was...dang. That was so nice.\n\nHank held up a Polaroid camera. She didn't even know those were still in existence.\n\n\"May I take a picture? For the board?\"\n\nAvery opened and closed her mouth. Swallowed hard. Her chest swelled, constricting her air. Dang it, she still couldn't talk, so she nodded.\n\nAs Hank took a shot of Hailey, her cheese face in place whenever someone said \"smile,\" a man's voice resounded behind them. Deep and low, it skimmed across her skin like a caress and set her nerves on fire.\n\n\"The Hailey. Sounds good.\"\n\nShe didn't have to turn to know who belonged to the voice. It had been the focal point of many of her recent fantasies as of late. She couldn't escape him, even when she should.\n\n\"This calls for a fist bump.\" Cade crouched down by Hailey, his grin huge. \"Remember what I said, you can't leave a guy hanging. Make a fist.\"\n\nHailey closed her fingers and awkwardly smacked Cade's fist, then squealed.\n\nAvery's chest tightened more. Even her daughter was enamored.\n\nHank stuck the picture of Hailey on the menu board right next to her option and turned. \"What'll it be, folks?\"\n\nCade stepped up next to her, pointing to the picture. \"We'll take three of those, Hank.\"\n\n# Chapter 11\n\nZoe leaned against the front desk near the end of the workday on Thursday where Brent and Gabby had congregated. \"Everyone's all aflutter with this Valentine's dance.\"\n\nBrent sighed dreamily. \"You, doll, are a genius.\"\n\nAvery's cheeks heated. \"I'm just glad people are interested. Job done.\"\n\nGabby shook her head. \"It's more than that. Seriously, you have no idea how boring the past years have been.\"\n\nAvery closed out her program and shut down the system, since Cade was on his last patient of the day and Drake had finished with surgery. \"You guys have a hot date picked out?\"\n\nThey all shook their heads.\n\n\"I say we go as a group, like in high school.\" Brent waved his hand at his brilliance. \"Power in numbers.\"\n\n_Squawk_. \"Rollin' with my homies.\"\n\nDrake strode in and handed Avery his billing form. \"We only did the front declaw, so don't charge for the back.\"\n\n\"You got it.\"\n\nHe went to leave, but Brent called his name. \"You're going to the Valentine's dance, right? Our Avery did a bang-up job planning.\"\n\nDrake paused and slowly turned, assessing Brent like he'd been dropped on his head one too many times as a child. \"No.\"\n\nZoe tucked a piece of blue hair behind her ear and sat on the desk. \"We're thinking of doing the group thing. No dates.\"\n\nDrake shifted his gaze to her and slowly shook his head.\n\nAvery recalled what Cade had told her awhile back about how hard it had been to get Drake out of the house and socializing since his wife had died. A dance was a huge social event. One nearly the whole town was planning to attend. She couldn't blame him for saying no.\n\nSince he looked uncomfortable, she tried to come up with something to lighten the mood. \"I could be your date, Drake. Platonic, of course. I'll beat the conversationalists off with a stick.\"\n\nHe tilted his head, studying her. And then it happened. He smiled. Not a full-on grin, but the corners of his lips quirked.\n\nZoe gasped. \"Did you... Did you just smile?\"\n\nIt fell, morphing into a scowl. \"No.\"\n\n\"I saw it.\" Gabby nodded. \"You did smile.\"\n\nThe frown deepened as he looked at Avery. \"Thought you said you'd beat them off.\"\n\nZoe grabbed her chest, faking heart failure. \"And he joked, too.\"\n\nPoor Drake. This was what he got for his trouble.\n\nAvery sighed. \"You've got more than a week to think about it. Open invitation to join us. We'll try to make it painless.\"\n\nSomething in his gaze softened, almost like a thank-you, before he nodded and strode away.\n\nA collective silence fell until Brent slapped her arm. \"Look at you, playing up the brothers. I'm telling Cade you did that.\" A sly grin split his face.\n\nAvery narrowed her eyes but, before she could retort, Cade and his last appointment came out of an exam room.\n\n\"Tell me what?\" He handed Avery the chart and looked at Mr. Townsend. \"Couple drops in each eye should clear it up. Give us a call if it doesn't.\" After the client left, he handed her the billing statement and glanced at the others. \"Tell me what?\" he repeated.\n\nBrent cocked a hip. \"Your girl just asked your brother to the Valentine's dance.\"\n\n\"I'm not his girl, and it was an invitation to join all of us.\" She stood. \"Stop instigating.\"\n\nCade's gaze landed on her. \"Which brother? I'll kill him.\" He grinned as if in afterthought.\n\n\"Yeah,\" Brent drolled. \"Not his girl.\"\n\nAvery shook her head and put the cockatoo back in his cage to cover him for the night. \"Good night, Gossip.\"\n\n_Squawk_. \"Enter sandman.\"\n\nZoe hopped down from the desk. \"Let's go dress shopping on Saturday. I can get someone to watch Mom.\" She looked at Gabby and then Avery.\n\n\"I'm in,\" Gabby said.\n\nCade deadpanned. \"I wouldn't look good in a dress.\"\n\nGabby laughed. \"Remember that Halloween in high school when you dressed as a cheerleader\u2014\"\n\n\"No. I don't remember, thus it never happened.\"\n\nAvery rolled her eyes and nodded at Zoe. \"I'll get my mom to watch Hailey. Barring a problem with that, I'm in, too.\" She didn't have a red or pink dress, so this solved her problem. She hadn't gone shopping with girlfriends since... Well, she couldn't remember when.\n\nThey made arrangements and the crew left for the day. All but Cade, who stood off to the side watching her. Since she couldn't read his expression, she called for Thor to put him away for the night.\n\nCade grabbed She-rah, much to the cat's dismay, and followed. \"It was a joke, the cheerleader thing.\"\n\nShe grinned. \"Okay.\"\n\n\"It was.\" He set the cat down and called Thor to follow. Once the animals were in the back room, he shut the door and eyed her.\n\nThe hallway suddenly seemed too small with her back pressed to the wall and him closing in like a predator, all fluid grace and delicious muscle.\n\nHe set his palms on either side of her head and leaned into them, his forearms flexing. \"I suddenly find myself thinking of little else than you in a dress, wondering if you'll go with red or pink, if you'll choose something that shows a lot of skin\u2014\" He frowned as he cut himself off and jerked his gaze from her breasts to her face. \"You asked my brother to the dance?\"\n\nA bubble of laughter rose in her throat, but she held it down. He made her feel young again. Between the kissing and his jokes and the giddy flutter in her belly. Whenever he was near, she wasn't a stressed out single mom with a disabled daughter and a useless ex-husband. Not with Cade. She was just...a woman again.\n\nAll of her reservations about them flittered away. She'd regret it later, but for now, this felt too good.\n\nHis blue-eyed gaze slid to her mouth and back up as if expecting an answer. The heat was unmistakable, but a flash of hurt twisted his mouth.\n\n\"Drake. I asked Drake to join us as a group, hoping he might say yes. You said it was hard to get him out since\u2014\"\n\nHe pressed a finger to her mouth and sucked in a harsh breath at the contact. \"What did he say?\"\n\n\"That he'd think about it.\" Or close enough, in Drake speak.\n\nSomething shifted in his expression as his eyebrows drew together, like he wasn't looking at her anymore. Grief warred with guilt. \"He hasn't attended many community functions since Heather died. Just the ones Mom makes him go to, like the Christmas parties.\" His gaze met hers again, tender and hollow all at once. \"He'll go. For you, he'll go.\"\n\nShe shook her head. It wasn't like that. Not how he was thinking.\n\nPart of her understood his brother, what Drake was going through. Her husband hadn't died, but the loss of comfort, of having someone there and then suddenly not, could gut a person. It made it hard to trust, to hold on to anything, and being around others only amplified the feeling that everything was slipping through her fingers. For Drake, it had to be especially difficult. Not only had someone he loved died, but everything around him had a memory tied to that loss.\n\n\"He's got a lot of respect for you.\" Cade cupped her cheek, his thumb tracing her lower lip, his steady gaze watching the movement. \"You're new. You don't remind him of her.\" He looked in her eyes, grateful and with relief, making her realize just how worried Cade had been for his brother. \"Thank you for that. For trying.\"\n\nHis hand slid from her jaw to her throat, his fingers brushing her collarbone under her blouse. \"The way he loved Heather, it was out of the storybooks. I used to watch them, wondering if anyone could live up to that. Then she got sick and...\" He shook his head. \"And I stopped dating altogether to avoid that kind of hurt. I did the random hook-ups and fun because there's no pain in not engaging.\"\n\nThey'd had quiet talks and they'd shared some heated kisses, but this was personal, on a different plane than where they'd ventured before. For the first time, it seemed like he was letting her see a piece of him no one else had, and it was costing him. Pleasure and pain lit his eyes, thinned his mouth. Whatever he was battling, he wasn't altogether happy about it.\n\nBecause her own heart was pounding, she cleared her throat and forced her lungs to take in air. \"Red.\" At his questioning glance, she elaborated. \"I'll probably pick a red dress for the dance. Pink's not a good shade on me.\" And since that admission wiped the regretful haze from his eyes, replacing it with something carnal, she kept flapping her mouth without the filter. \"I'll see what I can do about finding something that shows...skin.\"\n\nHe closed his eyes, his jaw clenched tight. Resting his forehead to hers, he let out a ragged breath. \"I can't wait to see you in it.\" He opened and closed his mouth several times before finally making up his mind. \"Go with me. Be _my_ date, Avery. No one has to know but us, as long as you...\"\n\nHer heart stuttered. \"As long as I what?\"\n\nHis breath skated across her cheek. He nuzzled her neck, behind her ear, sending shivers through her. The rasp of his growth echoed as it scraped her skin. \"As long as you come home with me afterward.\"\n\nHe tensed, waiting for her answer, motionless against her as if her response would make or break him.\n\nTheir breathing was the only noise, so loud it ricocheted off the walls. The heat from his body wrapped around her. The expectation of what she'd say kept them both taut, ready to snap. His heart pounded, the thump hitting her chest. And then he stopped breathing altogether, and there was no doubt in her mind that he was, somehow, against all reason, as nervous as she was to be here.\n\nThere were other doubts, reservations that pushed against her skull from the inside. Like would she measure up to his other conquests? Richard had cheated on her, and that had killed her confidence. She'd lost so much of herself in their marriage and she swore she'd never do it again. And there was the fact that Cade was her boss. She had Hailey to think about...\n\nHe cupped her cheek unexpectedly, the only movement he offered as if silently begging her to give him a chance.\n\nSome of the doubts were pushed aside. She wanted him. Wanted something just for herself that wasn't tainted or had ties to what she'd left behind. What would it hurt letting go for once? It didn't have to mean anything. Cade would make her feel good, could help her move past the stagnant halt her life had become.\n\nShe opened her mouth to agree to his request, but he kissed her forehead and stepped away. \"Think about it. There's no rush.\"\n\nShe had a sickening sensation in her gut that she'd just hurt him, rejected him when he'd laid himself out there for her. But he was out the door without a backward glance before her mouth caught up to her brain.\n\nAll through the next day her suspicions were confirmed. Cade barely spoke to her, and he didn't find a quiet moment to back her against a wall to kiss her stupid like he'd done several times throughout the week.\n\nThe strangest part was, the town\u2014aside from Cade's admirers\u2014all seemed to be gunning for her and Cade as a couple. And she had made sure they had no public displays or anything to egg it on. She just didn't get it.\n\nBy the time she got to her meeting, her brain was fried and her chest hurt.\n\nHailey also had her sleepover at her friend Jessica's tonight, so Avery's nerves were raw, her stomach twisted in knots. She'd texted April several times with instructions for Hailey. April responded to every instance, not making fun of her or telling her to calm down. Which was nice, having someone understand.\n\nThe event committee met at the Botanical Gardens in the hall to go over placement of tables and decorations instead of convening at the rec center. All she could think about, besides whether Hailey was doing okay or not, was how she wouldn't get to see Cade when she got home.\n\nShe'd begun to look forward to that, to having his presence in her house. Having his scent lingering in the rooms or to discover what activity he'd done with Hailey. Their quiet talks around her kitchen table, and the bone-melting kisses weren't a hardship, either.\n\nGod, he was so good with Hailey, too. Patient, never dominant, never pressuring her to be normal like other kids. If Richard had shown a smidgen of that understanding, Avery might not have left him.\n\n\"What do you think, Avery?\"\n\nShe shook her head and faced Marie. \"I'm sorry, I didn't hear you.\"\n\nThe other members looked at her with varying degrees of knowing smirks, and she wasn't in on the joke.\n\n\"About the tables?\" Marie pointed to the back wall, holding herself in complete professionalism. Being the mayor, Avery supposed she had to be, even though she was one of the Battleaxes and super sneaky. \"Line both sides of the room with tables? Bar at the back. DJ up front. You haven't listened to a thing, have you?\"\n\nHeat flared in her cheeks. \"I'm sorry. Hailey is having her first sleepover ever and my mind is elsewhere.\"\n\nThe women cooed, very sympathetic, which made Avery's face flame to scalding because Hailey had only been half of her distraction.\n\n\"I think that setup would be lovely.\" Avery glanced around, forcing herself to focus.\n\nThe room was long and open, with a rear wall of floor to ceiling glass that looked into the attached garden dome. Open wood beams slanted the rafters, giving a log cabin feel to the atmosphere. Two large chandeliers could be dimmed for mood lighting, and the floor was hardwood throughout.\n\n\"The streamers are going to be a pain to hang.\" Rosa directed her gaze up with a frown.\n\nAvery dialed back her wince. \"I think we should veto the streamers. The evening is for the adults, a romantic date night.\"\n\nShe eyed the tables along the wall, remembering Gayle had said they used this hall for a lot of weddings. \"White tablecloths with flower petals. White lights strung from the beams to look like starlight. We'll turn off all the overheads, all but the chandelier up front for the dance floor, and let the soft glow illuminate the rest.\"\n\nReally into the mood now, she glanced at Marie and did a double take. The other women were staring, slack-jawed.\n\nAvery rubbed her forehead. \"Or not. Whatever you think.\"\n\nMarie nodded. \"I think we were right to hire you.\"\n\nRosa slapped Avery's arm, nearly toppling her. \"I love it. Let's do it.\"\n\nBy the time Avery drove home, she was swamped with fatigue and wondered if she'd manage to get any sleep with Hailey gone. Oh, look at that, she'd checked her phone. Again. She was pathetic. When was the last time she'd had a night to herself? Never. And she couldn't stop looking at her cell.\n\nAs she made the turnoff for the cabin, she started to fantasize about a hot bath, a hotter book, and a glass of wine. But as she pulled up to the cabin and parked, an entirely different fantasy came to mind.\n\nOne that involved a sexy animal doctor who slowly stood to full height and grinned from where he'd been waiting on her front steps, his trusty, good-natured dog by his side.\n\nThe pull of desire in her belly was fierce and, unaccustomed to it, she stared for a few beats, giving herself an internal pep talk on what it meant that she felt anything at all. Years, _for years_ , she'd shut her needs down, and now it seemed they were making up for lost time.\n\nWith the worst possible candidate to snap the spell.\n\nThen again, Cade couldn't break her heart. Not only had it been broken before, but he wasn't the type to reach deep enough to attach. A steady calm washed over her, settling her tension.\n\nNodding, she grabbed the door handle.\n\n# Chapter 12\n\nCade watched Avery emerge from her car with a questioning glance. Hell, he didn't know why he was here either, other than it felt like he had something to prove. To her, that he wasn't just some playboy who sought fun. To himself, that this thing between them was different and he was capable of trying.\n\nBy showing up tonight, he wanted her aware that this wasn't about Hailey, nor work. And that he wanted to be here. Truth was, he thought of little else but Avery when she wasn't around.\n\n\"Hi,\" she said, coming closer. \"You get a Friday night free from babysitting and you come, anyway. Don't tell me we accidently domesticated you.\"\n\nHer grin was teasing, a little nervous, but his gut twisted. She was one of the few people to treat him like he wasn't a joke, like he had more inside him than he allowed to show. Yet her comment sliced, eradicating that.\n\nOr maybe she was just fishing.\n\nShe bent to pet Freeman, who ate up the attention.\n\nHe turned to retrieve the DVDs and microwave popcorn he'd brought, holding them up. \"To distract you from pacing the floors.\" After witnessing how she'd freaked out leaving Hailey with him for a couple hours, he figured she'd go catatonic with her daughter gone a whole night.\n\nShe froze, still bent to pet the dog, her gaze sliding to the movies and then locking on him. It seemed to take great effort, but she straightened and swallowed. Her breath fogged before her face, masking her eyes, but he caught the wide wonder in them first.\n\nThat was the thing about Avery. No one looked at him with a cross between hero worship and affection. And she did each time he offered to help, whenever he said something kind, like she'd yet to experience either. It made his chest expand with pride, made him want to do anything just to see it again.\n\nAnd made him want to kill her ex all the more.\n\nThe admiration was probably undeserving on his part. He had other, less honorable motives for being here. Like spending time alone with her. Because, yeah. The chemistry between them was a living thing. The heat in her eyes proved he wasn't alone.\n\nHe glanced at the movies in his hand just to break the intense moment. \"Horror or rom-com?\"\n\nHer brown eyes did this kind of shining in the moonlight thing that made him step closer just to see if the honey flecks would be there. Her eyes said so much about her, radiating the emotions she tried to quash.\n\n\"You picked out a romance?\"\n\nThe disbelieving tone had him shrugging. \"Chicks like them. And it's a comedy, too, lest you forget that.\" Why did he have the urge to shuffle his feet under her gaze?\n\n\"I like scary movies,\" she said at last, walking up the porch steps. She held the door open for him. \"Popcorn is my favorite food group.\"\n\nHe laughed and walked inside, calling Freeman. The dogs took turns sniffing each other's asses.\n\n\"I'm serious. Popcorn, chocolate, coffee, ice cream, and pizza. The five food groups. Health nuts are going to feel stupid one day, dying of nothing.\"\n\nOkay, that was it. She might possibly be the perfect woman.\n\nHe reached out and snagged her around the waist, hauling her to his chest. \"I have something I need to say.\" Leaning in, he kissed her soundly on the mouth and pulled away before things got too convoluted right off the bat.\n\nEyes wide, she reeled and pressed a hand to her forehead. \"I'm not sure I heard you. Could you repeat that?\"\n\nYep. The perfect woman.\n\n\"Maybe I should talk slower.\" Setting the movies aside, he wrapped both arms around her back, sliding his hands up the curve of her spine to tangle in her hair. \"Communication is very important.\"\n\nShe gasped as he took her mouth, at leisure this time, exploring her taste and breathing in her sweet scent. She was everywhere, all the time. Invading his Friday nights, working at the clinic. In his head. Somehow, he didn't mind.\n\nDespite their height difference, she fit against him, all soft and giving to his hard and unrelenting. And regardless of their other differences, they seemed to fit well in other areas, too. She was serious. He never tried to be. She was kind and generous. He typically gave what he got. She thought too much. He preferred action. But the oddity worked for them.\n\nWrapping her arms around his neck, she arched into him and raked her nails lightly against his scalp. Now he was hard. Painfully so. Hugest turn on? A woman who ran her fingers through his hair. Every time. He tore his mouth away before things went too far too fast and before she was ready. Not that she'd tried to tap the brakes.\n\n\"Grab the popcorn,\" he said against her lips, still holding her to him.\n\nShe reached behind him to the table. \"What are you carrying?\"\n\n\"You.\" Palming her ass, he lifted her and headed toward the kitchen, her toes dragging along the way. He set her on the counter and tossed the popcorn into the microwave. \"Stop trying to hit on me, by the way. A guy likes to play hard to get.\"\n\nShe threw her head back and laughed. For once, carefree, she covered her face and let go. It was the single most sexiest and sweetest sound he'd heard. Just like that, the air evaporated from the room.\n\nPushing her hair back, she sobered. \"I'll try to contain myself.\"\n\n\"Do that. Or don't.\" He stepped in front of her and set his hands on her hips. \"You don't laugh often enough. It's very hot.\"\n\nHer smile fell, but it stayed in her eyes. \"I'm trying.\"\n\n\"Fresh start and all that?\" She nodded, and he noticed her lack of tension. \"You don't stiffen when I touch you anymore.\"\n\nSetting her hands on his forearms, she stroked. \"I'm working on that, too. Haven't done this in awhile. In fact, I swore I wouldn't ever again.\" She looked away and shrugged.\n\n\"Wouldn't ever date again?\" She didn't respond, instead choosing to stare at his shirt, so he tilted her chin up until she met his gaze. \"How long's it been?\"\n\n\"Since I went out with someone? In secret or otherwise?\"\n\nHe didn't have the heart to tell her they weren't a secret and never would be in a town like Redwood Ridge. It was obvious to anyone they came in contact with that they were into each other, even if their every move wasn't tweeted and hashtagged. Yeah, they had their own hashtag. Damn Aunt Rosa.\n\nAnd Avery was deflecting. He couldn't have that. \"Either.\"\n\nShe let out a sigh infused with a groan. \"Richard is the only man I've been with. We met in college. He was two years ahead of me. We married right away. I think because he saw in me what he wanted from a wife, and he was laying the groundwork. He was on the fast track to partner in his dad's firm once he passed the bar.\"\n\nHe didn't care about what her ex did or didn't want, other than it directly resulted in how she saw herself. And she wasn't a house, damn it. \"What did he want in a wife?\"\n\n\"Arm candy. Quiet, not too pretty. Someone who was socially adept enough to carry conversation, but not challenge him.\" She set her hands on the counter behind her and leaned into them.\n\nAside from quiet, she was none of those things.\n\nHer shoulder lifted in a shrug. She did that a lot, shrugged things off. \"I lost too much of myself in him, until I didn't have a wish of my own or even trust my own tastes. He did it slowly. I didn't recognize what had happened and by then it was too late. Ten years too late. I won't ever get married again. Won't ever let someone break me down, mold me to their purpose.\"\n\nNot that he was an expert in relationships, but he always figured there should be give and take. No breaking or bending, just respect for the other person. Like his parents' marriage, like Drake and Heather's. No one should ever have to be someone they're not, and by the sound of it, that's what had happened to her. No wonder she was wary.\n\n\"I think that's why I'm getting to be okay with what's happening between us.\" She straightened, looking over his shoulder to avoid his gaze. \"There's no potential for more.\"\n\nTemples throbbing, he narrowed his eyes. He didn't know why that pissed him off, but it did. They were already \"more\" or he wouldn't be thinking about her day and night and in between. Yet he understood what she meant. His reputation wasn't to stick, and she had no desire to do so again. Why did his chest ache then?\n\n\"I'll be your date for the dance.\"\n\nHis gaze whipped to hers. Held.\n\n\"And...the afterward part.\" She bit her lip, her expression more open than he'd seen yet, her eyes unguarded.\n\nShe was trusting him with something she hadn't with anyone else. Not for a long time, anyway. The courage it must've taken just to say that wasn't lost on him. He'd do everything he could not to make her regret it.\n\n\"I didn't offer to pressure you. When I said come home with me, I didn't necessarily mean it literally.\" His gaze swept over her face. \"Not that I wouldn't love that, but I was serious when I said we don't need to rush.\"\n\nA small smile ghosted her lips. \"I wanted to tell you yes when you asked at the clinic, but...\" She tucked her hair behind her ear. \"You make me nervous.\"\n\n\"I make you nervous?\" She had his heart pounding so hard it was cracking ribs and he made her nervous? \"Likewise.\"\n\nOne of her perfectly arched brows rose, indicating she thought he was lying. \"You're too experienced and charming to be nervous.\"\n\nThat was where she was wrong. Oh, so wrong. He had no clue what the fuck he was doing with her. He had no game, nothing to fall back on. The women and his previous encounters weren't even in the same realm as this, as her.\n\nAnd this was...Not. Helping.\n\nHe pushed the button on the microwave, since he'd failed to do that before, and looked at her once more. She was grinning again, stopping his heart in the process as all his blood migrated south. \"What?\"\n\n\"You're cooking for me. It's sweet.\"\n\nKernels popped as the scent of butter wafted around the kitchen. She smelled better.\n\n\"I make a mean bag of popcorn. Nothing but the best for you. I make the finest mac and cheese this side of the fault line, too. Blue box, of course.\"\n\nShe patted her chest, her grin impish and adorable as hell. \"Be still my heart.\"\n\nNo kidding. His, too. Except she _was_ kidding.\n\nHe sighed. \"Let's start the movie. And if you find yourself scared, feel free to climb on my lap. I can take it.\"\n\nThis earned another laugh. She moved off the counter and, since he was still in front of her, every inch of that soft, lithe body slid against his in her descent.\n\nShe tilted her head to look up at him. \"What happens if you get scared?\"\n\nAlready there. \"You can hold me. With or without clothes.\"\n\n* * * *\n\nAvery put the movie into the disc player and checked her phone. Again. Cade served as a good distraction, but Hailey's bedtime had passed over an hour ago and April hadn't called. She hoped that meant Hailey was doing okay.\n\nShe sat next to Cade on the deep couch as the title screen popped up. \"I haven't seen _The Exorcist_ in years. Love this movie.\"\n\n\"Haven't seen it.\"\n\n\"What? Really? It's like a rite of passage.\"\n\n\"Will it frighten me?\" He grinned and waggled his brows.\n\n\"It scared me to death when I was younger.\" She checked her phone again.\n\nCade took it from her and held it out of reach. \"I will be the keeper of this until the movie's done.\"\n\n\"But\u2014\"\n\nHe shoved popcorn into her mouth and hit play. After setting the phone down on the table beside him, he threw his arm out, wrapped it around her waist, and pulled her closer until they were...snuggling.\n\nFor the first five minutes, she didn't move. Richard, even in their early years, didn't cuddle. She wasn't sure what to do. But eventually Cade turned off the lamp and she rested her head in the crook of his arm to tune into the film.\n\nSeraph paced the floor between Hailey's bedroom and the couch, obviously missing his friend. She called him over and set him in her lap, absently petting his back. Freeman eyed them and laid down by Cade's feet.\n\nIt was all so cozy.\n\nCade smelled too good. Fabric softener and warm male. Resisting the urge to bury her face in his rock hard chest and sniff, she assessed every twitch he made, every breath he took. His solid thighs under his well-worn jeans kept snagging her attention, and she itched to drop her hands there to stroke. Every inhale had her shoulder brushing his pec.\n\nSeraph moved to the floor by Freeman. Cade stretched his legs out, and she adjusted so she was reclining against his chest. His bulging, muscled bicep was right by her cheek, his thumb unconsciously caressing her shoulder.\n\nThis was crazy. A fire raged inside her, spreading and consuming. Her breasts ached. The apex of her thighs throbbed. She wanted...no, she _needed_ him to kiss her, to press his weight down on her and run his large hands over her body. Something, _anything_ to quell this fury of sudden lust.\n\nShe must've let out a frustrated moan because he laughed, gaze still on the screen.\n\n\"Yeah, I'll never eat pea soup again, either. Not that I cared for it much...\" He trailed off when she looked up at him, finally sensing her mood.\n\nHe stilled as if trying to comprehend the shift. His eyes clouded, closing part way. His lips opened to draw in a shallow breath. Cupping her jaw, a question in his brow, his gaze raked over her face\u2014her hair, her eyes, her cheeks...her mouth.\n\n\"Avery,\" he said quietly, but it sounded more like a prayer.\n\nShe was suddenly crawling out of her skin. If he didn't touch her soon, she was going to do something terribly embarrassing, like climb him and ride his thigh. Rip the T-shirt from his body so she could lick her way down to his jeans.\n\nHe gently brushed his nose with hers, the hesitation clearly costing him because the hand holding her jaw shook. Closing his eyes, he kissed her forehead, letting his lips linger on her temple. When he said her name again, she snapped.\n\nFisting her fingers in his thick, soft hair, she dragged his mouth to hers. A stunned beat passed before he became a willing participant and tilted his head. Taking the kiss deeper, he crushed her against him, stroking her tongue with his.\n\nYes. She all but heard the boom.\n\nShe moaned, and his hand traveled to her throat as if desiring proof the noise came from her. His shadow of a beard rasped her skin with the wild force of his mouth on hers, deliciously abrading.\n\nHe wrapped his arm around her back and eased her to the cushion, never breaking away. Lowering himself over her, he settled between her thighs and shoved his hands in her hair, holding her to his glorious assault as if he thought she'd evaporate. She wrapped her legs around his waist, and the hard bulge behind his fly ground into her heat, barely making a dent in her need.\n\nTearing his mouth away, he gulped air and made his way to her neck, licking and panting. She shivered from the nerve sensation overload and arched into him, his hard chest blocking the way, but her nipples cried _thank you_. He grabbed her hip, fingers clenching, before sliding his hand to her thigh to hold her leg in place.\n\nHe muttered something unintelligible and rocked his hips, pulling a feral sound from deep in her chest. Her breaths soughed until her lungs finally emptied. Cinching her shirt in his hands, he shoved the material past her breasts and stilled. His heated gaze swept over her blue bra, thumbs grazing the erect nipples straining against the lace.\n\n\"So beautiful,\" he murmured.\n\nDoubt crept in, a sickening swirl to mix with the lust. She tensed, and he looked up, his gaze soft and understanding compared to the rest of him. His hair stood at odd angles from her fingers in the strands.\n\nTrying to regulate her breathing, she fought to get the mood back, but it was lost. She didn't even know how to explain, but he deserved an answer. She pressed her hand to her forehead and swallowed. \"He cheated on me.\"\n\nDang it. That...was not what she meant to say. She closed her eyes to block out whatever ounce of pity she'd find on his face.\n\nCarefully, as if she might break, he tugged her shirt down and rose over her, bearing his weight on his arms. \"Look at me.\"\n\nShe blew out a breath and opened her eyes, not finding pity, but patience. Somehow, that seemed worse. And if she wasn't mistaken, there was fury there, too.\n\n\"I'm doing this wrong if you keep thinking about him, yeah?\" Frustration wrinkled his forehead, set a grim line to his mouth.\n\n\"What? No.\" She tried to sit up, but he didn't budge.\n\n\"Give me a second. I don't think I can walk.\" Tension stretched over his face when he finally looked back at her. \"I don't know what else to say, except he was wrong. In case it escaped your attention, I'm harder than steel and that's all your doing. He has the defect, not you.\"\n\nA rush of air whooshed from her lungs and the humiliation, the regret, seeped out of her pores. What was wrong with her? Hot guy making out with her on the couch, childfree for the night, and she couldn't get past first base. Or was petting second base? Didn't matter, she'd never score at this rate.\n\n\"I doubt it.\" He shook his head as if reading her mind.\n\n\"What?\"\n\n\"The movie. I doubt it.\"\n\nShe turned her head at the part where the characters were heavy into an exorcism. The priest kept chanting, \"The power of Christ compels you.\"\n\nA giddy bubble of laugher caught in her throat. Erupted. No, this lust-induced frenzy whenever they were around one another was not religion inspired. Though if they ever got to the actual sex, she suspected he could make it just as inspiring. In seconds, her body shook with hysterics, him joining her.\n\nThe tension drained away and she was so, so grateful he wasn't mad. Any other guy would be pissed or pressuring for more. Cade, completely attuned to her every mood and nuance, simply rolled with it.\n\nHe dropped his forehead to her chest and sighed. \"Such a shame. You really do have great breasts.\"\n\n# Chapter 13\n\nAfter Avery picked up Hailey from April's, she scrolled through the Internet on her laptop while Hailey played tug of war with Seraph. The puppy seemed to be winning because Hailey kept dropping the rope toy.\n\nMost of the articles she found on dog training were pretty straightforward. Now that Seraph was healed, she really should start training him better, but never having a pet before left her fumbling. He seemed to have potty training down. He'd only had one accident in the house, but she let him out routinely every hour and a half. Plus her mom came by on the days Avery was working to let him out.\n\nMaybe she should ask Cade, but she didn't want to bother him. Last night he'd gone home no doubt sore and seriously horny. She hadn't faired much better, spending half the night aching to finish what they started and too afraid to relax to get there. Eventually, his patience was going to wear thin.\n\nHailey whelped a laugh, which made Seraph yip in his puppy version of a bark.\n\nAvery smiled, setting the laptop aside. Two peas in a pod, just like Cade had said. The dog followed Hailey everywhere, and Avery latched on to the joy it brought seeing Hailey connect to something. Really connecting.\n\n\"You like your doggy a lot, sweetie?\"\n\nHailey squealed and flapped her hands. _Most entertaining, Mother!_\n\nShe laughed. \"And what about your new friend, Jenny? April said you had a great time at your sleepover. Would you like to do it again?\"\n\nOn repeat, she squealed and flapped her hands again.\n\nAvery's chest filled to capacity, causing her eyes to moisten. Aside from a slip-up or two, Hailey hadn't had any outbursts or shown any signs of distress since they'd landed in Redwood Ridge. The quiet, remote town was doing wonders for both of them. There was still an adjustment period, but they were getting there. She couldn't expect ten years to vanish with the fog.\n\nMom walked into the cabin and removed her coat. She eyed Hailey and Seraph and grinned before suffocating Avery in a hug. \"They're adorable. Look at them.\"\n\nMom had gone heavy on the patchouli today. Avery eased back before too much of the scent transferred.\n\nShe knelt down next to Hailey on the floor. \"Sweetie, I have to go meet some friends, but you be good for Grandma, okay?\"\n\nHer mother patted Avery's arm. \"What kind of dress are you going to get? Something sexy, I hope? You always dress too conservative.\"\n\nAvery resisted an eye roll. Mom's version of fashion was out of a hippie catalogue. \"I'll see what I can find. Thanks for watching her. She's not big into shopping, so she would've hated going.\"\n\nShe drove to the outer edge of town, past the mountains on her right and ocean to her left, navigating the winding roads. Zoe and Gabby said the outlet mall was in another county north of theirs, but right off the highway and easy to find. She hadn't ventured out much between work, Hailey, and the committee.\n\nFinding the shopping center, she parked and strode in. Zoe and Gabby were already there, browsing through the racks. Brent was there, too, wagging his finger in disapproval of one of their choices.\n\nHe grinned when he spotted her and made his way over. \"Let's find you something super sexy. Rawr.\"\n\nAvery laughed. \"Did your just rawr at me? And why is everyone so concerned about me finding a hot dress?\" She accepted a hug from Gabby and nodded a greeting to Zoe. \"Find anything good?\"\n\nGabby held up a pink sequin number and rolled her eyes. \"Didn't think finding pink and red would be so hard.\"\n\nZoe sighed. \"Let's try next door.\"\n\nAs they walked out of the shop and to the next stop, Brent shoulder-bumped Avery. \"Meant to offer before, doll. If you ever need a sitter, just call. My nephew's autistic, so I've got the basics down. Of course, I'm not a certain sexy animal doctor, but I've got a nice ass.\"\n\nGabby laughed and held the door open for them. \"I don't think a babysitting requirement is to have a fabulous ass.\" She slapped his butt when he strode past. \"It is nice, though.\"\n\nBrent rolled his eyes with dramatic flair. \"For reals, though. Just call.\"\n\nAvery grinned. \"I'd be afraid you'd turn Hailey into a diva.\"\n\nHe cocked a hip. \"And what, pray tell, is wrong with that?\"\n\n\"Got one,\" Zoe yelled from the back of the shop. She held up a short red spaghetti strap dress that darkened in color toward the hem.\n\n\"Love it.\" Brent fingered the satin material. \"Go try it on.\"\n\nWhen Zoe headed toward the dressing rooms, Gabby fished through a rack and sighed. \"I don't know why it matters so much. It's not like anyone's going to notice me.\"\n\nThe comment was unlike Gabby's typical enthusiasm. Avery frowned. Gabby's natural champagne blond hair was long and sleek. Her blue eyes had a lot of gray in them, adding a certain allure. She had a cute bow mouth and a nice, curvy body. Why wouldn't people notice her?\n\nWhen Avery said as much, Gabby shrugged. \"I'm the girl next door, the buddy, not the one men want to date. Especially around here, growing up with most of the guys.\"\n\nBrent and Avery shared a concerned look before Avery let the subject drop. She perused the rack and found a hideous dress with flowers sewn into the material. \"Brent, I found your dress.\"\n\nHis gaze raked the item. He shuddered. \"I'm here mainly to make sure you gals don't show up on Saturday looking hideous. Now put that back before someone sees you.\"\n\nShe laughed and moved on to another rack, spotting a light pink sheath dress with an empire waist. \"Gabby, for you?\" With her light complexion, the color would look great on her.\n\n\"Yes!\" She grabbed the dress, checked the size, and bounced toward the dressing room.\n\nA few minutes later, Brent made a noise of bottled excitement. \"Oh, Avery, doll? Check it out.\"\n\nThe red dress was a slinky slip style that came to a V in the front and had a swoop scarf back. If she didn't eat between now and Saturday, she might fit into it.\n\nShe pursed her lips. \"I'll try it on.\"\n\nMoments later, in the dressing room, her stomach fluttered as she checked herself out in the mirror. It was a perfect fit, even having a built in bra because of the low back and coming to a stop just above the curve of her rear end. It was racier than what she usually wore, but why not? Slimming, too.\n\nHanging up the dress, she changed and headed back out. \"Sold.\"\n\nBrent clapped. \"That was almost too easy. Let's grab some lunch and find shoes.\"\n\nBy the time Avery got home, she was exhausted but happy. She hadn't been shopping with girlfriends in ages, and the easy banter had lifted her spirits. It was nice for a change, not having to try so hard to say the perfect thing or act a certain way. And she was hereby taking Brent on all shopping excursions from now on.\n\nOn Monday, it snowed like the second coming. Eight inches by the time Avery was getting ready to leave for work and forcing a snow day from the school system. After dropping Hailey off at her mom's, Avery drove to the clinic, white-knuckling it the entire way.\n\nTheir little pocket between the ocean and the mountains didn't tend to get this much snow, so it was the topic of conversation. They'd gotten fourteen inches total by the end of the workday, and Avery stared at the accumulation through the front window. Everyone had taken off. All except Cade who was in his office finishing some charts. Avery's mom was keeping Hailey overnight, since school was cancelled tomorrow, too.\n\nHailey being away from home unnerved Avery, but Hailey had done well at Jenny's, and Avery needed to loosen her stranglehold on control. It took a lot of internal pep talk, but she conceded she wasn't alone anymore. She had help, she had friends, and letting them in was just one more step in moving on.\n\nEarlier in the day, she'd found a box with several tablets equipped with an electronic medical record system. Gabby told her they'd invested in the software two years ago, but no one was willing or had the time to scan charts and get them off the ground.\n\nDecision made, she went into the back room and grabbed the box with the tablets, figuring she'd get a jump on at least setting up a workstation to get the office digital ready. Maybe in an hour the roads would be clear and she wouldn't need heart medication to drive home. Until moving to Oregon, she'd only seen snow on TV.\n\nSettling in at the front desk, she connected the scanner and installed the program into the clinic computer system. She'd been working at the cardiology office when they'd switched from paper to EMR, but that had been before Hailey was born, so she was rusty. This system was a little different in that it was targeted more for veterinarian medicine than healthcare, but the operating procedures were similar.\n\nGlancing out the window, she noted the plows hadn't come through yet, so she checked next week's schedule and pulled the patient chart for first thing Monday to do a trial run on scanning the documents. After two attempts, she got the last few pages of dictation into the electronic chart. It would be a pain having to set up new accounts for the animals, and time consuming, but worth it.\n\nShe'd been so focused on her task that she hadn't heard Cade until he was right behind her. Jumping ten feet off her chair, she pressed a hand to her chest, heart thudding. \"Crap on a cracker, Cade. You scared me to death.\"\n\nHe grinned. \"Sorry. What are you still doing here? I thought I was alone.\"\n\nShe told him about Hailey and then the EMR project. Standing, she shut down the computer. \"It's getting late. I'll do more tomorrow. I was just installing for now while waiting for the roads to clear.\"\n\nThe grin slid from his face as he stared at her, gaze roaming her face in that unnerving way that made her flush in embarrassment. Richard had never looked at her like that, even when things were going well with their relationship. She knew she needed to stop comparing, but she'd only been with one man prior to Cade and rarely did she know what to expect. God, how she hated that, hated having no solid ground to stand on.\n\nHe stepped closer and pulled the pencil from her hair she'd used to pin the strands off her face. One corner of his mouth quirked in amusement. \"It's really hot when you do that, but I like your hair down.\" As if to emphasize his point, he combed his fingers through her hair and held the back of her head.\n\nHot. He'd just used her and hot in the same sentence.\n\nShe blinked, completely aware of him. Every single molecule and atom, aware. He made it impossible not to be. Cade filled a room, either with his laugh, his personality, or his blatant sex appeal. His hands were warm, almost as warm as his smile while he looked down at her, and she was helpless not to melt into him.\n\n\"I'm getting a few new fantasies about you and this desk.\" Leaning in, he brushed his lips over hers and sighed. \"It could be so good between us, Avery.\"\n\nHe kept his eyes open, blue gaze tracking her response, and she was putty. \"I want to.\"\n\n\"But?\"\n\nHow could she put into words her reservations when she didn't fully comprehend them herself? She wanted him more than she craved anything in too long. Yet old habits died hard, and Cade wouldn't be easy to walk away from afterward. She'd have to see him daily at work, around town. There would be no escaping.\n\nHe closed his eyes and dropped his forehead to hers. \"I hate what that prick did to you.\"\n\nShe wrapped her fingers around his wrists, him still cradling her head. \"He didn't do anything I didn't allow. I'm not a victim.\"\n\nHe laughed without mirth. \"You asked him to cheat on you? To ignore you and his own daughter? To kill your confidence one insult at a time?\"\n\nHer heart started to pound for an entirely different reason. They'd talked, but she hadn't delved that deeply into her past with Richard, not enough that Cade would know as much as he'd indicated. Which meant he'd read between the lines, he'd paid attention, and was figuring her out.\n\nTen plus years with Richard and he'd never remembered she hated peas. Six weeks and Cade could all but list her favorite foods. Angry at herself for comparing again, she shook it off.\n\n\"I didn't ask for him to do those things, but I allowed them. By not standing up for myself. For getting into a comfortable rut and not fighting, I allowed it.\" She sighed. \"I know you're frustrated with our pace. I'm sorry. If you want to stop, if you want to see someone else, I understand.\"\n\nNot that what they had was a relationship. She didn't know what label to slap on it, but relationship didn't fit. And the thought of ending things made her stomach twist in painful knots. But this would eventually end. His reputation proved he didn't have a long attention span.\n\nHe looked at her, a question in his eyes and fury closing ranks. \"I haven't looked at another woman since you swept into town. I won't cheat on you or make you feel like shit or ignore you. If we decide to end things down the road, then that's what we'll do, but for now, I'm here because I want to be.\"\n\nAnd there went the air from her lungs. He'd done nothing but prove himself, be completely honest with her, and she was acting like a skittish idiot. She just didn't know how to stop, to let go and...trust.\n\nShe threaded her fingers through his hair. He always initiated contact, and maybe by taking control of her actions she could take control of the doubt, too.\n\n\"I love it when you do that.\" His palms skimmed down her back and up again. \"Your hands in my hair? Major turn on.\"\n\nEmpowered, she lightly fisted the strands, earning a sharp inhale from him. This was what she needed. To know what she was doing right. \"What else do you like?\"\n\nHe seemed to be concentrating hard on breathing as he paused a split second before every inhale and exhale. His gaze never wavered from hers, but his eyes darkened with arousal.\n\nHis throat worked a swallow. \"I love it when your cheeks flush. It lets me know I've gotten under your skin.\"\n\nLike now? Because their open discussion was fueling her internal temperature to critical. She had no control over that, though. That was biology. \"What else?\"\n\n\"You make a needy little whimper in your throat when I kiss you. That? Extremely hot. And anything having to do with your breasts is good in my book. Like that green top you wore last week. It was this side of clingy and displayed those babies nicely.\"\n\n_Note to self: wear more formfitting shirts_. She could do that. The noise while kissing she'd been completely unaware she was doing, however.\n\nHe grabbed her hips, the motion seeming involuntary. \"When you get into your bossy mode, I go instantly hard watching you organize things.\"\n\nInteresting.\n\nJust to be a brat, she picked up the pencil he'd pulled from her hair and meticulously set it in the cup with her other pens. She lifted her brows in challenge.\n\nHe groaned and narrowed his eyes. \"Pushing your luck, sweetheart.\"\n\n_Sweetheart_. He'd never called her an endearment before. Heat flooded her core and spread. Her heartbeat thundered in her ears. Guess she liked it.\n\nSince it was fun watching him crack, she reached into his pocket and removed the paperclips he'd absently put there when she'd attach a note to his charts. Without taking her gaze from him, she dropped those into a tray with her other ones.\n\nIn the span of a second, she was pinned to his chest and held there by solid arms. His mouth crushed hers, needy, seeking. He shook against her, tension straining the muscles as he swept his tongue over hers. A mating. A claiming.\n\nHe backed her one step and lifted her onto the desk. She wrapped her legs around his waist and held him there. Driving her fingers into his hair\u2014because he said he liked that and she did, too\u2014she shifted to take the kiss deeper.\n\nThe phone rang.\n\nThey paused, still lip-locked. The machine kicked in, but no message was left. Cade patted his pockets out of habit, but she knew he didn't have the pager.\n\nShe tilted her head back. \"Drake's on call this week.\"\n\nHis gaze lifted from her throat to her mouth to her eyes. Need shifted into tenderness, and she realized he was surprised she'd known what he was looking for with his movements.\n\n\"I should get going. Early morning tomorrow.\"\n\nHe nodded, tucking her hair behind her ear. \"Drive careful, yeah? The roads look like a mess.\"\n\nShe smiled, pressed a quick kiss to his mouth, and hopped off the desk. He called her name when she was halfway to the break room to get her coat. She turned.\n\n\"Your laugh halts everything in my head. Forgot to mention that.\" He looked away as if...shy. He rubbed the back of his neck while she stood there, heart in her throat, tempted to jump his bones right there in the clinic.\n\nHe was good. Really good. A charmer without equal.\n\nBut he wasn't trying to persuade or lure or coax. He was being sincere, having picked up on her need to understand what he liked. He...saw her.\n\nSaw. Her.\n\n\"Good night, Avery.\" He stepped into his office and kicked the door shut.\n\n# Chapter 14\n\n\"Hey, squirt. Can I see your iPad?\"\n\nHailey pushed the device into Cade's hands and started to rock on the couch beside him. The action seemed restless and nervous to him.\n\n\"I'll give it right back. I promise.\"\n\nHe quickly connected the device to his laptop through an adaptor and pulled up Hailey's coloring app. After transferring several images she'd wielded, he unplugged and handed the device back to her.\n\n\"All done, squirt.\"\n\nShe didn't take it right away, leaving it to sit on the cushion between them. The rocking intensified and then worry slammed into his gut.\n\n\"What's wrong? Can you show me on your iPad? Or sign it to me?\"\n\nShe didn't so much as turn her head, but the rocking slowed.\n\nMaybe she needed space? He blew out a slow breath and walked to Avery's computer. After going into the control settings, he made her printer friendly with his laptop and ran off several of Hailey's pictures. Five should do it. They were nothing more than squiggly lines and mismatched shapes, but they were hers.\n\nWhen Cade had first arrived to babysit, he'd noticed Avery had taken the handprints he'd traced and framed them above the fireplace mantle. Moved, he'd decided to add to the fridge drawings, but since Hailey didn't like coloring, he was forced to pull up the ones from her device.\n\nHe printed a couple for his own fridge and turned back to the kid. She seemed off tonight for some reason. Avery was going to be later than usual because of the Valentine's dance tomorrow and getting last minute details squared away. He was on his own.\n\nCarefully, he sat next to Hailey and pulled up her language app. \"Do you feel sick, squirt?\" He'd watched her like a hawk, so he knew she hadn't gotten into anything. When she didn't respond right away, he guided her hand over the screen. \"Yes or no? Do you feel sick? Have an ouchie?\"\n\nShe pulled her hand away.\n\nAt a loss, he turned off the device and switched on the TV. Going into the DVR, he brought up her sleepy time show. Perhaps she was just tired. Being a kid was exhausting, after all.\n\nHis estimation was confirmed when she yawned and tuned into the program. Afterward, she wasted no time going potty and brushing her teeth.\n\nHe set Seraph in bed with her and cut the light. \"'Night, squirt. I'm right out here, yeah?\"\n\nMaking his way back into the living room, he put his printouts into his laptop bag and set it off to the side before heading into the kitchen and taping Hailey's pictures to the fridge. When he walked into the living room, Hailey was sitting on the couch.\n\nCade rubbed his neck. \"Not tired after all?\" He sighed, sitting next to her. How did Avery know what was up, since Hailey wasn't verbal? Something could really be wrong and he'd have no damn clue. \"Don't get mad, squirt. I'm going to touch your forehead.\"\n\nIt wasn't hot but, strangely, she didn't push his hand away.\n\n\"I'm at a loss. Can you point me in the right direction? Show me what's on your mind?\"\n\nJust as he was about to give up and try to coax her back to bed, Seraph jumped up on the couch and Hailey dropped her head in Cade's lap. He froze, hands up like a criminal. Seraph settled into the crook of Hailey's arm and dozed off. Hailey's eyes drooped.\n\nHell. Now what?\n\nHe waited several beats until her breathing was deep and even, then snatched a quilt from the back of the couch and covered her. She'd never tried to cuddle with him before. She'd never so much as touched him. If he moved, he'd wake her, so he settled back and rested on the couch. The weight of her tiny head on his thigh and smell of her kid shampoo was oddly endearing. Gently, he set a hand on her shoulder and closed his eyes.\n\nThe next thing he knew, he was jerked awake by someone tapping his knee and Avery was in front of him. Rubbing a hand down his face, he looked at his lap. Hailey hadn't moved. \"What time is it?\"\n\n\"Almost nine.\" Her gaze darted between him and her daughter, confusion parting her lips and a tugging softness in her eyes. \"What happened?\"\n\n\"She was kinda off tonight.\" He shrugged. \"Don't know what it was about. I tried to put her to bed, but she did...this.\"\n\nAvery pressed her fingers to her lips and cleared her throat. Twice. \"She missed you, I'll bet. She didn't see you last week because of her sleepover.\"\n\n\"You think?\" His heart somersaulted in his chest. Or maybe he had indigestion. He liked the idea of Hailey missing him. He didn't know shit about kids, but he liked this one. A lot.\n\nHer mom, too, if he was going there.\n\n\"I thought she didn't like touch.\"\n\nAvery tucked a piece of chestnut hair behind her ear, not meeting his gaze. \"She doesn't much. Sometimes she'll snuggle into my lap or play with my hair, but not often.\"\n\nShe closed her eyes and pulled in a breath as if centering herself. Slowly, she opened them and lifted Hailey from the couch. Since she seemed to need a minute, he didn't follow when she carried the girl into the bedroom.\n\nHe stood and stretched, then headed for the kitchen. Changing his mind about making coffee, he grabbed a beer and went back into the living room. He set the bottle aside and dropped onto the couch, head in his hands, wondering what the hell was going on. And why he didn't mind the new squishy feelings that seemed to be shoving around in his chest, taking up space.\n\nThe couch dipped beside him, but he didn't lift his head.\n\n\"Are you okay?\"\n\n\"Yeah.\" He sighed and sat back, staring at the ceiling. \"I guess it freaked me out a bit, her acting different. I thought something was wrong.\" He'd almost called Avery, but honestly, having Hailey fall asleep on him had been more jarring.\n\nShe studied him for a moment in that quiet, contemplative way of hers that made him want to crawl inside her head. Rattle out some words.\n\nSetting her elbow on the back of the couch, she rested her head in her hand. \"This one time, Hailey was maybe four years old, she wouldn't get out of bed. It scared me to death. She wouldn't use sign language, and she didn't have a device yet. This went on for two painful hours until a fever finally spiked and I realized she was sick. Nothing major, just the flu, but I was petrified.\"\n\nHe would've crawled out of his skin with worry. \"How do you do it? How do you not go nuts without a way to know what's wrong?\"\n\nShe shrugged. \"You just do. That's parenting. She has some tells, nonverbal cues. I guess I just know her so well I watch for the signs.\"\n\nHe was starting to pick up on them, too, but it was more than that. Hailey wasn't just silent. She didn't offer affection, either. Not in the typical way anyhow. How did Avery get by with no hugs, no laughter, no chatter?\n\nAnd then it dawned on him, a punch right in his gut, why Avery had such a hard time adapting to what was happening between them. She'd been separated from her dick of an ex for two years, which meant there were problems before that. She had a daughter who didn't speak, who she couldn't caress. She had a damn hard time trusting people and accepting help.\n\nBecause she'd been completely, irrevocably alone.\n\nHad he been the first man to touch her, offer comfort or see her vulnerable side in all that time? The first to want to? Hell, no wonder she had walls. Quiet house, quiet kid, quiet mind. Like she'd shut it all down to exist.\n\nHe looked in her eyes, at the strength and composure in the cocoa depths, and for once in his life, he wanted to be the hero in someone's tale. Wanted to be the guy she'd fall into. His heart stuttered behind his ribs, and a trickle of sweat beaded down his spine, but he held out his hand in the space between them.\n\n\"Come over here.\"\n\nShe offered him a quizzical look, but scooted closer. He grabbed her thigh and slid it over his so she straddled him. Then he removed his T-shirt and tossed it to the floor. Her lips parted as her gaze drifted over his bare chest, his abs\u2014too quick a perusal to really take him in\u2014before meeting his gaze with wide eyes.\n\n\"Don't freak out on me.\" Grabbing the hem of her sweater, he lifted it to expose her belly. \"We're just talking.\" With a couple swift flicks of his wrist, her shirt joined his on the floor.\n\nShe immediately covered herself, which was a crying shame. Those beautiful breasts were begging to be touched behind her yellow bra. But he set his hands on her thighs and left them there as he calmly looked in her eyes. It was the only idea he could conjure to get her used to him, comfortable in her own skin while in the same room with him. At some point, she had to stop living in her head and feel.\n\n\"How was the meeting?\" he asked.\n\nHer pink lips opened and closed. She cleared her throat. \"You needed my shirt off to ask me that?\"\n\nUnable to help it, he grinned. \"If I had my way, you'd do everything shirtless. Look at you. You're lovely.\" The subtle compliment wasn't lost on her as a disbelieving note traced her expression. He'd keep working her until that disappeared altogether. \"But no, that's not why I took your shirt off. I'm trying to drop a barrier. Now, how was the committee meeting?\"\n\n\"Um, good. We just...\"\n\nHe drew lazy circles over her leg, working upward to distract her, but she broke off, so he prompted her to continue. \"You just what?\"\n\n\"Details. We just worked out the details.\" Her breathing hitched as his hands inched higher, tracing over her outer thigh and shifting closer to her hips.\n\nHe was seconds from needing to adjust his pants, but he held still for the sake of his cause. \"What kind of details?\"\n\n\"Who will set up what and where. That...\" He brushed his fingertips up her sides, earning a shiver from her. \"Kind of thing.\"\n\nShe stopped covering herself to drop her hands to his forearms. He skimmed the soft, warm skin beneath her breasts, wanting to lean forward and kiss her there. Hell, kiss her everywhere. Her eyes glazed and he knew he had her out of her head and into the just-feel-me zone.\n\n\"And what are you in charge of setting up, Avery?\" He dipped his voice an octave, rumbling her name like a prayer. He'd need to start praying for patience soon. Watching her lose control, little by little, was the hottest thing he'd ever seen.\n\nHer breath caught and her eyes shut as he traced the curve of her breast above her bra. Slowly, he brought his mouth there, kissing and licking. Damn, she tasted as sweet as she smelled.\n\n\"Answer the question,\" he said against her skin, bringing his hands around her back and splaying them low enough to dip his pinkies into her jeans. \"What are you in charge of setting up?\"\n\n\"Ah...the lighting. I'm making starlight.\"\n\n\"Starlight?\" Her head rolled back and he kissed her exposed throat. Her pulse beat harder, faster beneath his lips. \"How does one make starlight?\"\n\nShe moaned. He nearly busted out of his inseam.\n\n\"I don't remember,\" she breathed, fisting her fingers on his shoulders as if unsure what to do with them.\n\n\"Touch me and tell me how you make lights into stars.\" He personally didn't give a shit how, but she was on the precipice of falling over the cliff of reason and, by God, he'd kill to get her there.\n\nWithout protest, her hands fell to his chest. Stroked. \"It's all about placement.\" Her statement ended on a moan, vibrating from her chest to his and back. Her fingers moved lower, over his abs and stopping at his fly.\n\nJesus. \"Placement?\" Panting against her throat, he ran his hands up her back and cupped her shoulders from behind. He thrust against her core, just enough for her to feel the hard ridge of his reaction to her. \"That kind of placement, sweetheart?\"\n\n\"Yes. No.\" She gripped his waistband.\n\nHis eyes rolled back in his head hard enough to thunk his skull. His lips grazed her cheek in a path to her mouth. \"Which is it? Yes or no?\"\n\nThey weren't talking lighting placement anymore and they both knew it. For once, she didn't hesitate. She turned her head to hover over his lips, stealing all his oxygen. It didn't matter what she said. He wouldn't take things beyond this point. They had all night alone tomorrow to explore. Tonight was about preparation, about the learning curve, and teaching her how to be touched again. A reminder she had needs and he'd be more than willing to sate them. Tomorrow.\n\nHis dick wasn't getting the memo, though, and when she brought her mouth to his in answer, he forgot his own damn name or why they needed caution.\n\nShe crushed her mouth to his, her breasts to his chest, and cupped his jaw in the most shocking display of initiative. He'd always had to come to her, and her going after him as if there was no stopping it knocked any sense out.\n\nHis hands tangled in her hair, holding her to him while he plunged. Pillaged. A groan rumbled deep in his chest, laying claim and surrendering all at once. Her tongue stroked his as her hips rocked against his erection, and he had to break away or die from suffocation.\n\nHe stared at her, breaths rasping in and out of his chest, her dark eyes dialed to take-me and lips swollen from his kiss. And fuck it. He didn't need to breathe. He pulled her to him again. Teeth scraped. Tongues warred. Her nails raked his scalp and he almost came. Right there. In his damn jeans.\n\nIt was her who tore away this time, pressing her face into his neck and fighting for air as badly as he was. \"Cade.\"\n\nHe closed his eyes to the breathy, needy tone of her usually calm voice and gripped her hips in an effort not to unzip her pants. \"Yeah?\"\n\nHer hot breath skated across his neck. \"This was a really good talk.\"\n\nTilting his head to look at her, he paused long enough to roll the words around until they sank in. He laughed and dropped his head back to the couch. \"I keep telling you communication is very important.\" Tucking a piece of hair behind her ear, he took in her flushed cheeks and swallowed hard.\n\nShe was so goddamn beautiful like this.\n\n\"I was right about your abs. I could cut my teeth on them.\"\n\nHe laughed again. \"A theory you can test another time.\" Like tomorrow, he hoped. He pressed his palms to his eyes to dispel the image of her nipping her way down his body. It didn't work. And he'd rather look at her, anyway.\n\n\"I'll meet you at the dance tomorrow instead of you picking me up. I have to be there early to set up.\"\n\nHe sighed, tracing his thumb over her bottom lip. \"To make starlight?\"\n\nShe smiled, and it hit her eyes like he hadn't seen often enough. \"Yes. To make starlight.\"\n\nIf he was a more poetic man, he'd find a way to say something romantic, but he wasn't, so he'd just wait until the dregs of arousal cooled enough for him to walk out without limping. It might take awhile.\n\n\"What does your dress look like? I should've snooped around while you were gone.\"\n\n\"I'm picturing you snooping around my closet.\" She shook her head as if pitying him. \"The dress isn't here. It's at Gabby's. Zoe and I are getting ready over there. Zoe's going to do my hair.\" She stopped and bit her lip. \"This does feel like prom all over again.\"\n\nHe wrapped a strand of her wavy hair around his finger. \"You're not going to let Zoe dye your hair blue like hers, are you?\"\n\n\"I think she's changing it to red to match her dress, but no. I'm not that adventurous.\"\n\nBefore Avery came along, he'd always been the adventurous one in his family, willing to take risks, however calculated. Aside from buying his house and working the clinic, he didn't often tie himself to one thing or person, happy to whittle away at life until he found something that stuck.\n\nHe had a suspicion Avery was more than a sticking point. She was a glue factory. Instead of getting the itch to bolt, he wanted to know more. Something besides her ex or the shitstorm of a marriage. Her life before that, maybe, and he refocused on what they'd been talking about.\n\n\"Who did you go to your prom with?\"\n\nShe bit her lip and narrowed her eyes. \"A boy from the baseball team. He played right field and kissed like he'd never made it to first base.\"\n\nHe laughed. \"Nice analogy. So no wild sex in the limo or after party?\"\n\n\"No, thank goodness.\"\n\nNodding, he realized the topic change worked and he figured he could stand without pain. The Redwood Ridge Valentine's dance wasn't prom, but he hoped to make a better impression afterward than her right fielder. Ending the night with a home run wouldn't suck, either.\n\n# Chapter 15\n\nAvery wrestled with a string of lights from the top rung of the ladder, blew a strand of hair out of her face, and cursed. The exposed beams in the botanical garden's hall were high. Too high for her to comfortably reach. Her and heights weren't exactly friends.\n\nNow what?\n\nFootsteps sounded from behind. She grabbed the ladder to turn, finding Drake and Flynn striding over. They tossed their coats by the front hall and made their way to the center of the room.\n\nFlynn assessed her in a swift glance and grinned. \" _I'm told you're making starlight. Need help?\"_\n\nHer conversation from last night with Cade came to mind and her cheeks heated. He'd obviously told his brothers some part of what had happened. Unsure how to feel about that, she resorted to sass to hide her embarrassment and grinned. \"Are you a starlight expert?\"\n\n_\"I can be_.\" Flynn winked.\n\nDrake rolled his eyes. \"Get down before you fall. Tell me what you need done.\"\n\nAvery climbed down and handed Drake the strand. \"What are you guys doing here, anyway?\"\n\nFlynn moved to the ladder and signed, \" _Cade called. He was going to come help you, but he got roped into doing something for Aunt Marie.\"_\n\nHe'd asked his brothers to help her? How...considerate.\n\nDrake looked down at her from the top rung. \"Lights? How do you want them?\"\n\nRight. She straightened and gave Drake directions, watching his forearms flex as his white T-shirt stretched across the muscles of his back. How was it all three of the O'Grady men were single? Even Flynn was attractive with his dark strawberry blond hair and lean body. Like a runner. Wide shoulders, narrow waist.\n\nBy the time they were done with the first beam, Avery was thinking of setting up the tables herself. It was a little useless to have her stand around while the guys worked.\n\nFlynn looked around. \" _It looks nice in here.\"_\n\n\"I'm not done yet, but thanks.\" She'd still be fighting with the first strand of lights if they hadn't shown up. It was looking pretty great. She had Drake bunching the strings together so they draped in clusters, resembling falling stars. \"How'd Cade talk you into this? And thank you, by the way.\"\n\nDrake looked down at her from the ladder, dark eyes intent. \"Help is around, Avery. You only need to ask.\"\n\nHer chest constricted at the words she'd heard before. Somehow, coming from Drake, the meaning dawned clearer. He may still be trying to get over his wife's death, and he may prefer his own company, but he was a friend, too. He'd come if she needed him. \"Thank you.\"\n\nHe nodded and went back to work.\n\nShe pulled out her cell and texted Rosa not to bother coming for the tables. She'd do them herself since she had time. She then texted the other ladies and told them the same thing with regards to their duties. Drake was nearly done, and she still had two hours before needing to be at Gabby's.\n\nWhen Drake set the ladder aside, Flynn turned to her. \" _So, you and my brother are a thing.\"_\n\nBeing new to Redwood Ridge and fresh off a divorce, the town gossip made her stomach twist. She'd led a sheltered, quiet life with Richard, and it was an adjustment having so many people interested in not only her love life, but everything else. She couldn't pass someone on the street without them asking how Hailey was or making idle chitchat about Cade. It was almost as if the whole town was trying to play matchmaker. Just this morning, the woman at the coffee shop had told her how Cade donated his time at a shelter every Thanksgiving before heading to his mom's. The woman followed this up by telling Avery what a \"catch\" he was.\n\nMaking friends and being social she could handle in moderation, but them delving into her personal life was hard to accept. She didn't know where this thing with Cade was headed. She wasn't interested in another long-term relationship or anything beyond what they were doing now. And he wasn't the husband and kids kind of guy. Why couldn't they just explore each other and have fun in private, without all the eyes?\n\nFlynn lifted his brows in question.\n\nShe opted for smartass again and pretended he meant the other brother. \"Drake might be shocked to learn we're a thing.\"\n\nFlynn's grin was infectious. _\"No, I think your type is someone a little younger, who's quicker to laugh. Am I right?\"_\n\n\"Are we going to paint each other's nails next? I didn't bring my manicure bag.\" Drake frowned, eying Flynn through narrow slits as if pissed he'd brought up the subject. Was he upset she was\u2014almost, kind of\u2014dating Cade or had Flynn said something he shouldn't have?\n\nFlynn shrugged. \" _Just making conversation_.\"\n\n\"Well, stop.\" Drake's features smoothed out before turning back to her. \"What else do you need done?\"\n\nShe bit her lip. \"I can get the rest.\"\n\n\"Or you could tell us what else needs to be done.\"\n\nRight. Help was around. Maybe by the time Hailey got to high school she'd have that mastered. \"The seating?\" She phrased it as a question, unsure if they really were going to stay.\n\nBoth Flynn and Drake waved her aside once she'd told them where they were supposed to arrange the round tables. Sighing at her uselessness once again, she went into the storage room to retrieve the d\u00e9cor and get a jump on that. By the front door, she assembled and decorated the check in area. Back in the hall, she placed a white tablecloth on each table as the guys got them set up, adding the flower petals and votives. Then she dragged the chairs away from the wall and arranged ten at each table.\n\nIt was early afternoon by the time they finished. The staff at the facility was supposed to let the DJ in and were handling the bar. She needed a shower before heading to Gabby's.\n\nFlynn waved. \" _See you tonight_.\"\n\n\"Thank you so much for helping.\"\n\nHe nodded and strode out, leaving her and Drake alone. She sighed and leaned against the wall, studying Drake's tense posture. His hard gaze traveled over the room as he raked a hand through his black hair.\n\n\"Heather and I had our wedding reception here.\"\n\nHer gaze flew to his as her heart squeezed. Being in the hall again had to be tough on him, yet he'd come to lend her a hand, anyway. Drake didn't talk much, but when he did, each word was infused with meaning. She didn't know why he'd mentioned it, but she was glad he had, glad he'd opened up a little.\n\n\"Cade showed me a few pictures of her. She was lovely.\"\n\nHe nodded. Slowly his gaze returned to her, pain and determination in his dark brown eyes. \"Did you need anything else?\"\n\n\"No, but thank you.\" She paused a beat. \"I understand it's difficult for you to be here, so it meant a lot you came.\"\n\nHis face was stony reserve, but his voice was gentle. \"We grew up together. Everywhere is hard to be. I make do. And you're welcome.\"\n\n\"Maybe one day you can tell me about her. I wish I could've met her.\" Her heart was breaking for what could have been. Drake obviously had loved Heather a whole lot.\n\nHis eyes glazed over as he stared at her feet. \"She would've liked you.\" He crossed his arms and blew out a breath.\n\nShe took mercy on him. \"Are you still coming tonight? I have my boxing gloves all ready to ward off any conversationalists.\"\n\nA ghost of a smile traced his lips. \"I think I'll pass. Appreciate the offer, though.\" He tucked his hands in his back pockets.\n\n\"It's because of the dress code, isn't it? You can tell me.\"\n\nHis smile widened. \"You got me. I'm not fond of neckties.\"\n\nShe eyed his jeans and T-shirt. \"How fond of that shirt are you?\"\n\nHe glanced down. \"I don't write it sonnets. Why?\"\n\nShe took a permanent marker from her back pocket and uncapped it with her teeth. \"Hold still.\" Securing the fabric in one hand, she drew a cartoon necktie on the front of the shirt, a couple buttons on the chest, and stepped back. \"I'm not as artistic as Zoe, but it'll do.\"\n\nHe looked at the shirt and then at her. Shaking his head in amusement, the heavy sorrow lifted from his eyes, degree by degree. \"You have me at a loss.\"\n\n\"Are we keeping tabs? I can write \"Caution: Will bite\" on your back.\"\n\nThe rusty sound of his short laugh echoed off the walls. \"You win. I'll come. One hour, then I'm out.\"\n\nShe grabbed her coat and purse from the front hall, trying her best not to show her excitement he'd changed his mind. \"I won't even make you dance with me. Brent, however, I can't control.\"\n\nHe shrugged into his coat and held the door, humor twisting his mouth. \"He's not my type.\"\n\nShe grinned all the way home, pleased she'd gotten Drake to come to the dance. She could only hope his grief would lesson a little more with each venture out of the house, replace old memories with new ones.\n\nMost of the snow had melted and the afternoon was mild, bringing in a thick fog. She breathed in the scent of pine and salt as she exited the car. Cade texted just as she unlocked the cabin door.\n\nI have Seraph at my house. That way your mom doesn't have to come by to let him out.\n\nA quick follow up came seconds later.\n\nThat wasn't meant to pressure you into staying. You can pick him up after the dance, if you like.\n\nShe smiled and shut the door. Cade hadn't pressured her at all, not even when most guys would've been pushed past the patience marker. She'd thought a lot about tonight and what might happen between them. Nerves pinged her belly, but it was mostly anticipation. Cade made her nervous, yet in a good way. It had just been so long since she'd had sex that she worried she'd let him down. Or embarrass herself. Richard hadn't exactly been experimental when it came to positions or trying new things.\n\nShaking her head, she thumbed a response. _Looking forward to tonight_.\n\nBefore heading to the bathroom, she called April to check on Hailey. Assured everything was fine, she stepped into the shower.\n\nWhen she got to Gabby's house\u2014a cute gingerbread tucked into the woods not far from Avery's rental\u2014everyone was waiting for her in the cozy kitchen. Scarred birch cabinets and green laminate floors\u2014very outdoorsy and rustic.\n\nBrent handed her a champagne glass filled with mimosa.\n\nShe muttered a thank you as another text pinged her phone. Digging in her purse while following them into the living room, she pulled out her cell. She grinned at Cade's name on the display and the response to her looking forward to tonight.\n\n_Not as much as I am_.\n\nBrent peeked over her shoulder. \"Oh, swoon. Doll, that man likes you bunches.\"\n\nZoe and Gabby started talking excitedly at once.\n\nAvery lifted her hand. \"Not talking about it, okay?\"\n\nShe quickly glanced around the living room in an effort to mute the subject, taking in the hardwood floors and colorful furniture. The fireplace mantle had more than a dozen photos in mismatched frames. The space suited Gabby, cheerful and open. Tall windows and decorative bottles. Magazines and plants. A fat orange tabby was curled in a corner, ignoring their existence.\n\nZoe plopped onto the arm of the navy sofa. She had, in fact, dyed her hair a bright, circus-fearing red. \"At least tell me you trimmed your naughty-zone. If you use the not-waxing excuse to back out of sex with Cade\u2014\"\n\n\"Not talking about it.\"\n\nHer face heated to inferno. Some things were too personal, and she'd only known these people for six weeks. Besides, she was good in that regard. Richard had insisted she have laser hair removal not only under her arms and on her legs, but in the bikini area, too. She'd huffed at the time. Now she was just grateful she didn't have to shave daily. Not that she'd tell her friends.\n\nGabby flopped on the couch, blond hair in rollers. \"Oh, come on. None of us have any juicy prospects. You could at least indulge us.\"\n\nAvery slammed half her mimosa and swallowed. \"Is nothing private?\"\n\nBrent laughed. \"You're so innocent it's adorable. Now, seriously. Dish.\"\n\n\"We haven't...\" She waved her hand and sighed. Giving in, she sat in a green armchair and rubbed her forehead. \"God, I'm so nervous. The only man I've ever slept with was my ex, and that was so long ago I'm not sure my parts still work.\"\n\nThey cooed and offered comfort.\n\nBrent squatted by her feet. \"It's just like riding a bicycle. A vibrating, naked, muscled bicycle.\"\n\nA giggle bubbled in her belly, morphing into a full-blown hysterical laugh. \"You're a nutcase.\"\n\n\"No question.\" He rose and looked her up and down. \"The first step is to make you look sexy. If you look sexy, you'll feel sexy. We need to get you some confidence.\"\n\n\"I'll do her hair,\" Zoe said.\n\n\"I'll apply makeup.\" Gabby stood. \"Let's do this.\"\n\nThey dragged her to a small vanity in Gabby's bedroom and shoved her onto the bench. Zoe got to work on her hair, pinning the curls up in a loose knot behind her head. Gabby worked on her face, using a smoky charcoal to lightly accentuate her eyes and a deep red for her lips. It was more war paint than she was used to, but she had to admit, it looked nice. The evening called for formality, so she supposed going all out was okay.\n\nWhat if Cade thought she was trying too hard? Would he like this look?\n\nRichard hated a lot of makeup. He'd wanted her to dress and look sophisticated at all times. A wallflower. Once, she'd gone to the grocery store just to pick up milk, deciding not to bother with cosmetics, and she'd run into his partner's wife. Richard had gotten on the phone within the hour to ask if she was sick and why she'd go out looking unpresentable.\n\n\"What's wrong, doll?\"\n\nShe glanced up at Brent and blinked the memory away. Cade wasn't Richard, and her ex no longer had a say in her life. She'd moved to Redwood Ridge to start fresh. Her plans never included getting mixed up in another romantic relationship, but she reminded herself Cade wasn't looking for one, either. This was just for fun until the attraction waned. No sense in freaking out over loss of control or worrying about someone trying to run her life.\n\nShe smiled. \"Nothing. I'm fine.\" She glanced at the sleigh-style bed where the dresses were laid out over a thick pink duvet. \"Should we get dressed now?\"\n\nZoe's cell phone rang. She glanced at the screen and cursed. \"That's Mrs. Tetherman. She's watching Mom for me.\" She took the call and walked into the hallway.\n\nGabby turned from the mirror. \"I hope she doesn't have to cancel. She's been looking forward to this for weeks. She hardly gets out anymore.\"\n\nAvery bit her lip, then remembered her lipstick and stopped. \"Her mom's dementia is really bad?\"\n\nGabby turned back to the mirror with a frown, applying eye shadow. \"Worse every day. People keep telling her to put her in a home, but Zoe won't do it.\"\n\n\"I'll go over there if there's a problem. That way she can still attend.\" Brent sat on the edge of the mattress. \"I don't have a date anyway, and her mother does better with men.\"\n\n\"That's strange, don't you think?\" Avery moved to the bed and sat next to him. \"I'd think females would be more comforting.\"\n\n\"Don't know, but Zoe's tried every female caregiver in three counties. None stick. I visit and her mom's fine.\"\n\nTheir conversation was cut short when Zoe came back into the room. \"False alarm. Mrs. Tetherman just wanted to know where Mom's bedtime dose of medicine was. I had to move it to a locked cabinet...\" She waved her hand. \"Let's get you dressed, Avery, since you have to leave early.\"\n\nShe did have to arrive in under an hour to see to any last minute details and be sure the ladies running the entry table had everything they needed. Stepping into the bathroom, she put her clothes into a small bag to wear tomorrow and carefully took the dress off the hanger.\n\nShe zipped the back and then looked in the mirror, hardly recognizing her reflection. The dress made her breasts more full, her waist thinner. The bright red matched her lipstick and brought out the natural chestnut highlights in her hair.\n\nPerhaps Brent was right. She looked good. Different, but good. Some of the nerves fled from her body. She was young, reasonably attractive, and had a man waiting for her tonight. She was going to have fun for once and not worry about the world closing in or controlling the outcome.\n\n# Chapter 16\n\nCade handed Aunt Rosa his coat in the entryway to the gardens and waited for his brothers to do the same. He endured a kiss from his mom, said hello to a few people, and headed toward the open ballroom.\n\nHalf the town was already in attendance, by the look of things. The lights were dimmed to nearly dark and candles flickered on the tables. To his right, the DJ played instrumental music while people mingled. To the left was the bar.\n\nBingo.\n\nHe ordered a beer and then turned to lean against the bar, his brothers flanking both sides. From here, he could take in the whole room. More importantly, scope out Avery. He knew she'd be fluttering about, and he wouldn't get to pin her down until the ball was rolling, so he'd wait here and hope to spot her in boss mode.\n\nSo hot.\n\nFlynn nudged his shoulder and pointed to the ceiling.\n\nA grin tugged at his lips. He'd be damned. It did look like starlight raining down from the rafters. In fact, the whole setup was so different from years past, he'd never know he wasn't at a wedding if not for all the pink and red.\n\nSpeaking of, Gabby and Zoe made their way over, Zoe decked out in a multi-tone red dress and Gabby in pale pink.\n\n\"You look lovely, ladies.\" Cade toasted them with his bottle.\n\nGabby lifted her champagne flute. \"Thank you. Avery outdid herself. Isn't this great?\"\n\nFlynn nodded, eyes full of mischief. \" _Drake hung the lights_.\"\n\nZoe's eyes rounded on Drake. \"You did?\"\n\nHis older brother shrugged, not meeting her gaze. \"Just did as Avery asked.\"\n\nShe stared at him a heartbeat too long and finally nodded. \"I'm glad you came.\"\n\nDrake cleared his throat. \"Let's get a table.\"\n\nThe ladies had already snagged one near the dance floor in a corner before they'd arrived. Cade followed them through the crowd, craning his neck to check for Avery. He was dying to see her, to find out if this mysterious dress would drop his jaw. The way he figured it, she could wear a paper bag and he'd want her anyway.\n\n\"She'll be here in a minute.\" Brent smiled as they sat down, reading Cade's intentions. \"She swore she'd stop working and have fun as soon as the music started.\"\n\nCade doubted it.\n\nThey made idle chitchat for a few minutes. Cade clenched his teeth and forced himself to sit still when all he wanted to do was prowl the room for Avery. His skin itched and his body tensed.\n\nHe didn't know if it was the anticipation for after the dance or just her not being within touching distance, but it wasn't like him to be so wound up over a woman. Sometime over the last six weeks, she'd crawled inside his head, his chest, and stayed there. Taking up space. Making him feel things. He rubbed the back of his neck and looked over the crowd.\n\nAnd saw her.\n\nThe deep red of her gown matched the shade of her lips. The material dipped into a V at her breasts, clung to the curves of her waist and hips, and flowed in a slip to her ankles. A slit up one side hinted at the long legs beneath, allowing nothing more than a peek. Her hair was pinned up, exposing her regal neck, and she'd done something to her eyes to make them a warmer shade of milk chocolate.\n\nWith slow grace, she drifted over in\u2014oh hell\u2014fuck me red heels, and sat between him and Drake.\n\n\"Hey, guys. You all look great.\"\n\nSince it was anatomically impossible to swallow his own tongue, he attributed the lump in his throat to nerves. Nerves? Him? He cleared his throat. \"So do you. Beautiful.\"\n\nThe smile hit her eyes, and him somewhere below the belt. \"Thank you.\"\n\nWhen she turned to Drake, Cade forgot to breathe. The back of her dress was open, dipping all the way to the small of her back. The pale smooth skin begged to be stroked, kissed. Slowly.\n\nHer calm voice brought his gaze up to his brother. \"After I went to all that trouble to make you a tie, and you busted out a suit, anyway.\"\n\nDrake ran a hand down his white shirt, necktie already loosened and first button undone. \"I'll save it for my next formal occasion.\" One corner of his mouth quirked in...damn. Drake was smiling. Avery had gotten him to come to the dance _and_ smile.\n\nThe rest of the table noticed, according to their wide eyes and parted lips, but no one said anything. Drake appeared oblivious to the attention. Cade met Flynn's gaze across the table. Flynn nodded in understanding.\n\nWait. She'd made his brother a tie?\n\nHe must've said the question aloud because Avery turned in her seat and breathed a laugh. \"Not really. I\u2014\"\n\n\"Drew on my T-shirt with a permanent marker.\" The other corner of Drake's mouth lifted, transforming his half-cocked smile into a grin.\n\nThe breath seeped from Cade's lips. He was immediately thrown back into childhood, of playing baseball and throwing snowballs. To Drake and Heather's wedding, where that grin never once left his brother's face. It had been so long, so damn long since he'd seen any sign of life in Drake. Flashes here and there, gone too fast to note they'd even manifested.\n\nThere was obviously an inside story about the shirt, but Cade didn't give a shit what it was because the only thought taking up residence in his mind was getting his mouth on Avery, every inch of her. For making Drake smile again. For organizing the clinic and this event. For being a great mom. For having the courage to leave a comfortable, privileged life and start over.\n\nFor making Cade's heart pound, stop, and pound again.\n\nThe squeal of a mic echoed through the room and he jumped. Shaking his head to clear it, he took a long pull from his beer and forced his gaze to the interruption. Aunt Marie was on the DJ's stage in full mayor mode. In year's past, she'd always said a few words before kicking off the event. He sat back and pretended he'd heard what she said.\n\nAnd then she called Avery's name.\n\nAvery muttered an \"oh crap\" in response while standing. The room cheered as she walked to Marie and took the place beside her. His aunt put Avery on display, a position she obviously was uncomfortable with if her blush and clenched fingers were any indication. Yet she plastered a smile on her face and waved.\n\n\"Are you having fun?\" Avery asked into the mic.\n\nThe patrons clapped and whistled.\n\n\"Good. I'm glad you all could make it. I'm getting a lot of compliments, but it wasn't just me who planned tonight. It was the effort of the committee, so please give them some love.\"\n\nThe room clapped. Avery rolled her eyes and waved her hand, indicating they weren't loud enough. The room erupted and she smiled.\n\n\"That's better. I'd also like to thank two special guys who helped me set up earlier and saved me from a fall off the ladder. Flynn O'Grady.\"\n\nGabby signed for Flynn, repeating Avery's words, and Flynn stood to take a mock bow, earning a laugh from Avery.\n\n\"And,\" she went on, dropping her voice to a conspiring tone, \"whatever you do, please do not tell the other guy I said he helped me. Drake O'Grady would be totally embarrassed to know I said anything at all.\"\n\nAll eyes shifted to Drake. And hell. His brother was still grinning, eyes on Avery and slowly shaking his head in amusement.\n\nTime seemed to stop as a collective pause filled the room. Then the applause erupted. Avery quickly quieted them down with a wave of her hand. She nodded to the DJ to go ahead and start, then bunched the front of her dress to walk down the stairs.\n\nCade couldn't testify to what song played or who did what as he tracked her hurrying back over to the table and reclaiming her seat.\n\n\"I can't believe Marie put me on the spot like that.\" Her cheeks were almost as red as her killer dress.\n\n\"Believe it, doll.\" Brent lifted his glass in mock toast. \"You did good.\"\n\nShe turned to Drake. \"Are you mad I called you out?\"\n\nDrake lifted one brow. \"Do I look mad?\"\n\nFlynn grinned. \" _How would she know the difference?\"_\n\nBrent rose from his seat and clasped Zoe's hand. \"Ladies, we're dancing now.\" He claimed Gabby's hand with his other. \"Avery?\"\n\n\"In a minute,\" she said, settling back in her chair. \"Go ahead.\"\n\nAnd that was pretty much how the next hour went. Flynn got up for a song or two to dance with the others, but the three of them remained seated, virtually not talking and watching the crowd.\n\nFinally, a slow song came on, and Cade shifted to Avery to ask her to dance, but she was conversing with Drake. Anyone who could get his brother to talk shouldn't be interrupted, so he draped his arm over the back of her chair and rubbed his thumb across the smooth, soft skin of her back.\n\nShe shivered. He grinned.\n\nFlynn stood to dance with Gabby. Brent already had Zoe locked in, twirling her around the floor like only a gay man could pull off. He made the others look bad.\n\n\"Don't look now, but you might turn into a pumpkin. It's been ninety minutes since you arrived.\" Avery laughed and sipped her champagne.\n\nDrake shrugged. \"I think it's my car that would turn into a pumpkin, not me.\" His gaze traveled the room. \"I'll take off in a bit.\"\n\n\"Is this bringing up bad memories?\" She asked so softly, Cade almost hadn't heard over the music.\n\nDrake swallowed, staring back at her with a blankness in his eyes. Just when Cade thought he wouldn't answer, Drake crossed his arms and shook his head. \"Brings up good memories, which is bad.\"\n\nCade stilled. In the four years since Heather had died, he'd tried everything except strapping Drake to a chair to get him to open up, to say anything about the grief so he could move past it. Avery seemed to pull words from him with ease as if they'd known each other a lifetime. Or maybe because they hadn't was exactly why Drake could talk to Avery.\n\n\"Where did you and Heather meet?\"\n\nHumor filled Drake's eyes. \"In the sandbox. I was four. She was three.\"\n\nAvery laughed in the lilting, silky way that wound around Cade's chest. \"Robbing the cradle. Was it love at first sight?\"\n\nDrake shook his head and looked down, but his lips were curved in an uncommitted version of a smile. \"No, that came later.\" He sucked a shallow breath and eyed the dance floor.\n\nCade followed his gaze. The others were making their way back to the table. A ballad was still playing, but Cade remained seated. He'd get her to dance with him at the next set.\n\nA wicked gleam lit Zoe's eyes when they sat down.\n\nBrent narrowed his eyes at her. \"What's that look about?\"\n\nZoe shrugged. \"Do you know how to do The Electric Slide, Avery?\"\n\n\"Uh...yes. Who doesn't?\"\n\nGabby threw her head back and laughed. \"Good. Because Zoe just put in a request. We're getting you out of that chair.\"\n\nAvery turned to Drake. \"This goes both ways. I said I'd protect you if you came. Help a girl out.\"\n\nDrake leaned his forearms on the table. \"I have no defense against The Slide. Sorry.\"\n\nThe slow set ended and the opening to Zoe's request filled the room. Cheers went up as others made their way to the dance floor.\n\nBrent gave Avery no warning. He stood, rounded the table, and wrapped an arm around her waist. \"Let's go.\"\n\nAvery shook her head.\n\nBrent dragged her to her feet. Zoe and Gabby got in on the action, tugging on each hand until she had no choice but to follow. On the floor, she stood awkwardly for a beat as the others line danced around her until Brent hip-bumped her for encouragement.\n\nThe moment she gave in, Cade's heart drummed a hard, thumping beat against his chest, stealing his oxygen. Face flushed, she moved across the floor with fluid grace and a grin that encompassed the entire state. For once, she let go of those reservations she'd clung to and relaxed.\n\nIt was like watching a shell crack and fall away. A piece of her wall crumbling. He wondered if this was the real Avery, if she'd just buried her so deep she'd forgotten that part of her was there. If that were the case, the rare glimpse was enough to make him promise to move the moon to get this side fully exposed.\n\nBecause damn. He couldn't breathe. Couldn't take his eyes from her.\n\n\"Be careful, little brother. That looks like more than a good time on your face.\"\n\nHe blinked and forced his lungs to accept air. \"So what if it is?\" When Drake didn't respond, Cade dragged his gaze away from Avery to find a hard, determined set to his brother's jaw and something like murder in his eyes. \"What?\"\n\n\"She's a good person. I like her.\"\n\nCade was feeling a little homicidal himself. He reined it in and took a deep pull of oxygen to stop the pounding in his temples. \"Maybe you should go out with her then.\"\n\nDrake went rigid, eyes narrowed to slits.\n\nWhat in the hell was wrong with him, to go after his brother like that? He knew that wasn't what Drake meant, and though Big Brother liked a lot of people, it had been too long since he'd remembered that fact.\n\n\"I'm sorry. That was uncalled for.\" Cade scratched his jaw and kicked his feet onto the chair beside him, crossing his ankles and his arms. \"It's just...\" Hell. Damned if he knew what the problem was, other than it involved a brunette about twenty feet away. \"It's just her ex was a real asshole. As in, I'd leave nothing left of him if we ever crossed paths.\"\n\nDrake's brows lifted. \"And what? You're going to be the guy who proves not all men are assholes? You're going to be the one to make it all better?\"\n\nHe scrubbed his hands over his face. \"Maybe.\"\n\nExcept there was no maybe about it. From day one, all he'd done was kill himself with patience. Not rushing her. Waiting her out. Living for the moment when he dragged a laugh from her belly or got her eyes to lose focus from his touch. Wanting nothing more than to show her someone enjoyed her company, her self-depreciating wit, and her inane capability of caring for others, even those who'd hurt her deeply.\n\nChrist. He'd been doing everything to get her to trust him, to fall into him. _Fall for him_. And he had a sinking suspicion he was getting way more out of his efforts than she was.\n\n\"I'll be damned.\"\n\nHe glared at Drake, at his quizzical brow and dropped jaw. \"What?\"\n\n\"You're into her. I mean, really into her.\"\n\nNo kidding. And to think he'd been happy Avery got his brother to talk again. \"Where have you been? I told you that weeks ago. You know, if for once in my life you gave me a little credit\u2014\"\n\n\"Shut up.\" Drake shook his head as if unable to make heads or tails of the situation. \"You're falling for her.\" When Cade opened his mouth, Drake lifted a finger. \"I said shut up. You haven't rolled over and exposed your underbelly yet, but you're sure as shit halfway there. Puppy love.\"\n\n\"Have you lost your goddamn mind?\" His heart started doing that jackhammer thing, and the edges of his vision grayed. That didn't mean his brother was right. Did it? \"I never said love, never implied it. It's too soon. I like her, yeah, but\u2014\"\n\n\"Puppy love. The beginning stages before you fall so hard the rest of your life before her is a blur.\" Drake stood.\n\n\"Where are you going?\"\n\n\"To see if my car's a pumpkin yet.\" The strangest smile twisted his mouth before he met Cade's eyes. \"Godspeed. It's all downhill from here.\"\n\nCade opened his mouth, but Drake was already halfway to the door. He ground his jaw, unsure why he was so pissed. Or perhaps that was fear shoving around in his chest, causing a cold sweat to break out over his back.\n\nHe didn't know how long he sat there, fuming. Panicking. No, it was fuming. But when he looked up, the DJ had changed the set to a slow mix and his friends were back in their seats staring at him with a cross between curiosity and trepidation.\n\nAll but Avery, who was at the next table talking to the fire department. Ignoring him. Laughing. Tucking a stray piece of hair behind her ear. Laughing some more. Not noticing him at all. Laughing...\n\nFlynn knocked on the table to get his attention. \" _You okay?\"_\n\n\"Define okay.\" Okay as in sunshine and rainbows? Was he still meandering through his life looking for the next good time? Was he still having fun and taking names?\n\nFlynn's smile fell. \" _Where's Drake? What's wrong?\"_\n\nShit. He sighed. \"He went home. Everything's fine. I'm fine.\" Or he would be as soon as he had Avery next to him again.\n\nHe shoved away from the table and walked over to where she was still talking to the firefighters. He used to like those guys.\n\nWithout giving her a chance to protest, he wrapped an arm around her waist, lifted her to the balls of her feet, and didn't deposit her again until they were in the middle of the dance floor. Then he put his arms around her and urged her closer, shifting into a slow glide. He breathed in her berry scent and closed his eyes.\n\nYes. That was much better.\n\n# Chapter 17\n\nAvery burrowed deeper into Cade's embrace as they danced, absorbing his warmth and getting more than a little excited being against the solid wall of him. Her pulse tripped at the caveman way he'd dragged her onto the floor. Primal. Territorial.\n\nTilting her head back to look at him, she noted the tension around his eyes and mouth. \"Something wrong?\"\n\nHe looked down his nose at her. \"Not anymore.\"\n\nBut there was something wrong before? \"I was having a nice conversation with the fire department.\"\n\n\"I noticed.\"\n\nWas he jealous? It was so absurd she almost laughed.\n\n\"Relax, Avery. I just wanted to dance with you. You spent most of the night talking to my brother, and we haven't had a second alone.\"\n\n\"I'm sorry. I was trying to make Drake more comfortable\u2014\"\n\n\"That's not what I meant.\" His chest expanded with an inhale. \"He came tonight, which was a miracle in itself. Don't ever be sorry for whatever it is you do to get him to open up.\" Resting his forehead to hers, he sighed. \"I just wanted to dance with you,\" he repeated.\n\n\"Well, I'm glad you almost asked me.\"\n\nHis grin turned her to mush. \"Almost?\"\n\n\"Actually, you didn't ask at all. You just kinda went all alpha.\"\n\n\"Alpha, huh?\" He kissed the tip of her nose. \"I'll show you how alpha I can be later.\"\n\nGod. Just...wow. Need and anticipation coursed through her veins. Her toes curled in her strappy shoes. \"Promises, promises.\"\n\nHe growled and brought his lips to her ear, his breath hot. \"I always keep my promises, sweetheart.\" His hands skimmed down her back and up again, making her shiver at his rough palms against her exposed skin.\n\nSuddenly, everything was on fire. Thrumming, burning. The breath caught in her throat. Her pulse went crazy. Sliding her hands from his chest to his neck, the erratic beat of his heart pounded beneath her palm. The bulge behind his zipper pressed into her hip as they swayed.\n\nHe was just as turned on.\n\nShe had a suspicion every eye in the joint was on them. There had been gossip for weeks about the two of them, but they hadn't given the town anything to fuel the speculation. Until now. The way he held her, intimately close, and the way he was looking at her, all hunger, would leave no doubt.\n\n\"They're playing your song,\" he said, his voice all gravel.\n\nTilting her head, she focused on the music. \" _Lady in Red_ is my song?\" She didn't know she had a song. Didn't know how to feel about him giving her one, either. That implied...more.\n\n\"You are wearing red, and last I checked, you are a lady.\" Humor blended with the heat in his eyes, turned the blue to a stormy gray. \"And you're dancing with me. Facts are facts.\"\n\nShaking her head, she breathed a laugh. \"Hard to argue with fact.\"\n\n\"I have more where that came from.\" He rested his cheek against hers. \"Fact: You look so beautiful I can't breathe if I stare too long. Fact: I was jealous of a gay man because Brent spent more time on the dance floor with you tonight than me.\" He turned his head, rubbing his jaw along her cheek. \"Fact: I want you so bad it hurts.\"\n\nHer legs threatened to give out. She whimpered, hypersensitive to the press of his chest to her breasts, the throbbing between her legs, the moist heat of his breath, the light linen scent of him. Everything inside her went haywire.\n\n\"How long do we have to stay? Are you required to help clean up afterward?\"\n\nShe blinked at his rapid topic change and stilled their motion. \"No, the committee is coming back tomorrow. They gave me the rest of tonight off, since I set up and...\"\n\nHe grabbed her hand and led her toward the back of the room, weaving through people at a clipped pace. In her heels, she struggled to keep up. He noticed and slowed his steps. A little.\n\nGuiding her out of the ballroom and to the coat check, they waited side by side in silence for his Aunt Rosa to collect their things. He tapped his foot and thrummed his fingers on his thigh.\n\nShe focused on breathing so her face wouldn't connect with the floor from lack of oxygen. This was it. They were going back to his place. They were going to have sex. Was it odd she was more nervous now with Cade than she had been to lose her virginity to Richard?\n\nWith a knowing smirk, Rosa handed over Avery's purse and coat. Cade shrugged into his jacket and then held hers out for her to step into. Wasting no time, he nodded to his aunt, placed a hand low on Avery's back, and guided her to the door.\n\nThe cool breeze hit her face and she gulped air as they walked across the parking lot. They were at Cade's car, him holding the passenger door open before she realized where they were. If she left with him, he'd need to drive her back in the morning in order for her to pick up Hailey. Plus, if her car stayed in the lot overnight, everyone would know she'd gone home with him.\n\n\"Avery?\"\n\nHer gaze flicked to his. \"My car's over there.\"\n\nHe stared at her a beat, a question in his open expression, hesitancy in his eyes. He opened his mouth as if to say something, but shut it again. Closing his eyes, he drew a slow breath, and her stomach twisted. He apparently interpreted her statement to mean she was turning him down.\n\n\"Can I follow you? So I have my car in the morning?\" Or later tonight. Should she sleep over? Did he want her to go home once they were...done? She didn't know the rules.\n\nHis gaze leveled on her, the relief in his eyes contrasting the muscles working his jaw. He nodded and shut the door. \"I live on a side street near Gabby's house.\"\n\n\"Okay. Just give me a minute and I'll follow.\"\n\nOn the short drive over, she checked her phone and was assured April hadn't called about Hailey. Relief filled her chest to mingle with nerves. She wove through town, passing the turn off for her cabins and made her way down Gabby's street. At the end of the road, she followed Cade onto a private drive that looped and curved through the dense woods. To the left was a modest two-story cabin, lights glowing from the interior to illuminate the pitch blackness.\n\nCade's car kept going another block until he turned onto a gravel driveway. She pulled in behind him and cut the engine, pausing for a moment to take in the house. Much like her rental, his home was a log cabin, but on a grander scale. Two stories and lots of windows. Pocketed between several pine, birch, and sequoia, it had a wraparound porch with bare wood rocking chairs.\n\nWhen he opened his car door, she blew out a breath and did the same. He waited for her to come to him, muscles tense and jaw set. She didn't know if he was as anxious as her or if something else had him on edge, but neither did anything to calm the riot inside her head.\n\nShe glanced at the cabin. \"I like your house.\"\n\nHe cleared his throat. \"Thank you. I need to let the dogs out, but how about a tour after?\"\n\nShe took the statement for what he'd intended\u2014a way to distract her and make her more comfortable in the transition. It shocked her all over again how well he knew her, how well he understood her quirks and nuances. \"Sure. I'd like that.\"\n\nShe followed him onto the porch and then stepped inside when he gestured for her to precede him. He'd left a few lamps on to guide her through a small foyer and into a wide open living room. Hardwood floors met a stone fireplace in the corner. Forest green leather couches made an L shape in the middle of the room, accompanied by bare wood accent tables. It was clean and tastefully decorated with forest prints on the walls and exposed beams.\n\n\"A man who knows how to decorate.\"\n\nHe laughed. \"Gabby and Zoe helped with that. The layout design was mine, though. Dad owned all this property. He was going to build the clinic here, but chose town instead. We passed Mom's house on the way in. Drake and Flynn's houses are just down the road.\"\n\nThat was really nice they lived so close together. The location was ideal as well, tucked into the woods, but not far from town. Peaceful, serene, without total isolation.\n\n\"Make yourself comfortable. I'm just going to let the dogs out.\"\n\nShe glanced over the room, remembering he had picked up Seraph earlier, but the dogs weren't around.\n\n\"I have them in the mudroom out back.\"\n\nShe nodded. When he was gone, she walked around, finding a flat screen TV in a corner bureau and photos of his family on the mantle. She grinned at a picture of Cade and his brothers from when he was about seven years old, standing by a riverbed, arms around each other. A photo of Heather and Drake from their wedding was next to it beside a picture of Cade and an older gentleman. Their eyes were so similar she knew it must be his father. Drake got his dark hair, and Flynn his smile.\n\nThe clattering of toenails hitting hardwood sounded behind her and she turned just in time to catch Seraph mid-leap. Kneeling, she gave the puppy some love, showing equal attention to Freeman. They licked her face and she laughed, looking up at Cade in the entryway. She paused.\n\nOn his shoulder was a small gray kitten, nuzzling his neck. Cade seemed unaware of the adorable furball, his attention on her and a smile curving his lips. He blinked and must've noticed her distraction because his hand came up to pet the kitten.\n\n\"This is Cutin. She was a stray left out in the snow. The evening we met, actually. She was one of the reasons for my sour mood that night. Her siblings didn't make it.\"\n\nOh God. As if he wasn't sexy as heck enough, he had to stand there with his delicious body, stroking a kitten, and saying something like that. Swoon.\n\nHe shrugged. \"She likes sitting on my shoulder for some reason. You want a tour?\"\n\nShe wanted to rip his clothes off and lick him from head to toe, but she stood, somehow not passing out from the lightness in her head. \"Yes.\"\n\nThey walked into a large kitchen off the living room, one she'd kill to have. About the only thing she missed about her time with Richard was their enormous kitchen. She'd loved sitting at the table early in the morning sipping coffee and watching the sunrise. When she'd gotten too lonely, she'd baked cookies or cooked a dinner he'd never seemed to make it home to eat. But the process had been the draw, the relaxation in the task.\n\nCade's design was much different, though. Black appliances, green marble countertops, and light oak cabinets. A patio door led to a deck next to a screened-in mudroom. An island had four stools tucked underneath, and off to the side, a nook held a table big enough to seat eight.\n\nShe imagined him and his brothers there sharing a beer. Or a gaggle of future kids fighting over the last drumstick while his wife laughed at something he said. Normal niceties, an everyday image. Things she'd never had because Richard had no interest in spending time with her and Hailey. A scene she'd never experience because she was done dreaming about such nonsense and had sworn off that kind of hurt again.\n\nStill, her chest ached a little.\n\n\"Where did you go just now?\" Cade stepped closer, running his fingers down one of her arms. \"You look miles away.\"\n\nShe smiled and reached up to pet the kitten on his shoulder. \"Nowhere. Just thinking.\" Cutin meowed and nuzzled her hand, making her smile widen. She was an adorable furball.\n\nHe nodded, looking like he didn't believe her, but he pointed toward the way they came. He gestured at a half bath near the base of the stairs and then started climbing.\n\nShe followed, her stomach flip-flopping with every step.\n\nHe led her to the right. \"Bathroom.\" He stepped into a bedroom with a simple queen bed and dresser, depositing the cat. \"Guest room. Though I never have guests, so I may need to rethink that.\" He scratched his jaw.\n\nShe laughed and trailed him past another bedroom.\n\n\"Room I don't know what to do with yet.\" He pivoted to the left, past the stairs, and into a tiny loft that held a computer desk and two bookshelves. \"That's the master bedroom.\" He stopped outside a doorway and met her gaze, brows drawn together as if waiting for direction from her on what to do next. He cleared his throat. \"My bedroom.\"\n\nWith more bravado than she felt, she walked into the room. Behind her, he let out a shallow breath that rasped between his teeth, and some of her uneasiness dissipated. If he was nervous, too, then she wasn't so alone.\n\nHis bedroom was amazing. Four poster mahogany bed in the middle of the room, with two matching dressers and two nightstands. The bedding was a slate gray pattern, complimenting the midnight blue walls. The colors might've made the space seem small if not for the patio doors letting in natural light and a window on the opposite wall. A porch had a grand view of the mountain base and forest.\n\nShe walked over to the sliding door and crossed her arms, taking in the moonlight hitting the trees and riverbed. A fog had rolled closer to the mountain base, thick and surreal as it hovered in the distance.\n\n\"This is really something, Cade.\"\n\nHe came up behind her, setting his hands on her shoulders and kissing her temple. \"I don't hate the view.\" His voice was low and coarse, riddled with need. His hands drifted over her collarbone, heating her skin. \"And I don't mean the view beyond the glass. I've pictured you here more times than I should admit.\"\n\nWhen he dropped a kiss to her neck, his lips lingering, she closed her eyes and leaned into him, his warmth. He traced a slow sensual path from her neck to behind her ear, cupping her jaw with his hand. His arousal pressed into her backside. Her breasts grew heavy and a shiver tore through her.\n\nBut then the shiver morphed into trembling, and she couldn't seem to stop. Turning in his arms, she pressed her face to his neck, breathing in the now familiar scent of him and trying to draw calm through the desire. A mind of its own, her body didn't comply. The trembling became almost violent.\n\nHe ran his hands up and down her arms and kissed her cheek. \"You're cold. I'll start a fire. Hang on a second.\"\n\nHe moved away from her, and it was all she could do to stay on her feet without his support. Tossing his suit coat on a chair, he loosened his tie, threw it aside, and rolled up his sleeves. He knelt in front of a small fireplace facing the foot of the bed and got to work building a fire, then he replaced the grate and disappeared into what she assumed was an adjoining bath to wash his hands.\n\nShe stood there like a nervous teenager, unsure of what to do. This was insane. She wanted him, wanted this, and she couldn't get her body to stop shaking. Her teeth gnashed, chin quivering. She fisted her hands, but that didn't cease the quaking, either.\n\nHe came back and stood in front of her, tenderness in his eyes, and reached up to hold her face. \"It'll warm up in a minute.\"\n\nFinding her voice, she blurted the first thing that came to mind. \"I'm not cold.\"\n\nHe froze halfway to her mouth, hovering inches from her lips. Tension coiled in the space between them, and still, the trembling wouldn't stop. His eyes slammed closed as a muscle worked his jaw. He stayed that way, suspended, as he obviously tried to work something out. A myriad of thoughts must've swirled in his head because his face twisted with too many expressions for her to catch up.\n\nSlowly, he straightened and opened his eyes, gaze blank and pinned to the glass behind her. She didn't know what conclusion he'd reached, but instinct said it wasn't in alignment with what she was thinking. Her stomach twisted into knots when he dropped his hands and backed away.\n\nIn her confusion, he moved to the dresser and pulled out a T-shirt, tossing it on the bed. After staring at it for a second, his chin dropped. \"The bathroom's through there. You can wear the shirt to bed. Just...\"\n\nHe sighed and finally met her gaze, patience and desire and guilt mirrored in his eyes. \"Just please stay. We'll sleep. I'll hold you, nothing more.\"\n\nAnd then she caught on. It took her a while, but she got it. Got him. Followed the dots to his conclusion. If she didn't set him straight, if she didn't make the move, they were never going to discover how explosive they could be together. He had too much honor to push, and she had little experience to guide her. But darn it, this was not how tonight was supposed to go.\n\nHis gaze never left hers. \"Avery?\" Barely whispered, her name was part question, part plea.\n\nForcing herself to move, she went to the bed and picked up the T-shirt, then walked to stand in front of him. His eyes tracked her movements, but he didn't seem to do so much as breathe. She passed him the shirt. He took it automatically, fisting it in his hands.\n\n\"I'm not cold and I'm not scared. Being with you doesn't frighten me, Cade.\" He seemed to need that reassurance, so she drove the point home. \"I'm not scared. I'm nervous. That's all. Just...nervous. I don't have the experience you do, and it's been a long time\u2014\"\n\n\"Avery, sweetheart.\" He dropped the shirt on the floor and hauled her against him, wrapping his arms behind her back. \"Don't you get it?\"\n\nNo, she didn't. But it didn't matter because he crashed his mouth to hers.\n\n# Chapter 18\n\nCade hauled Avery to him and crushed his mouth to hers. Heart pounding, he ran his hands up the smooth skin of her back exposed by the dress and cupped her neck, holding her to him.\n\nI'm not scared. I'm nervous.\n\nThat had been all he'd needed to hear. For a gut-dropping moment, he'd thought she'd put a stop to things. Thought she didn't want what he assumed was an inevitable progression. Didn't want him. He'd never forced himself on a woman, never would, and if he'd had any inkling she wasn't telling the truth, he would've been out the door and in his guest room.\n\nWhich reminded him how he was breaking his rules with Avery. With his previous lovers, if they hadn't done the deed at their place, they'd rarely made it past his living room. On the off chance that happened\u2014and he could count on one hand the number of times that had occurred\u2014they'd wound up in his guest room. Never his bedroom.\n\nAvery was the first. He didn't know why, knew he shouldn't analyze it, but there it was. She was in his bedroom where he'd envisioned her too many times to recant.\n\nAngling his head, he took the kiss deeper, gliding his hands into her hair and pulling the pins from her knot to release the silky brown strands. They fell around her shoulders, skimming her upper back, and he caught the unmistakable scent of her shampoo.\n\nNeed surged, but he kept his pace methodical, and he sensed the moment she let go. When she pressed her body to his, her trembling ceased, the muscles relaxing under his palms.\n\nNever taking his lips from hers, he unbuttoned his shirt and took her hands in his, encouraging her to strip it off, wanting her to have as much control as possible. It was a power play, him undressing first to give her a modicum of reassurance. He could only hope it worked.\n\nShe hesitated a moment, but slid the material from his shoulders, down his arms, where it pooled at their feet. Guiding her hands to his chest, he pressed his palms over her fingers and nudged them down. Her thumbs brushed his nipples and he tore his mouth away to suck in a breath.\n\nHer heavy lids lifted to stare at their joined hands, lips parting with a ragged inhale. On her own, her fingers descended past his pecs, down his abs to the button of his fly. Her knuckles brushed the skin below his naval, ripping a groan from his chest. There was something insanely arousing about her touching him with his hands covering hers. An intimacy, a joining.\n\nThat seemed to be all the encouragement she needed. Though her fingers trembled slightly, she unbuttoned his pants and eased his zipper down, her knuckles dragging along the underside of his straining erection.\n\nHe clenched his jaw to hold himself in check. Tonight was about her, not the end game, but restraint was nearing impossible with the scent of her in his orbit, with her touching him.\n\nShe tugged his pants down to his thighs, briefs along for the ride, and pressed a kiss to his chest. Emotion pinched behind his ribs, tenderness pulling at him from every direction. He kicked the clothing aside and wove his fingers through her hair. The satin from her dress caressed his shaft, silky, soft, and he pulsed against her.\n\nKeeping one hand in her hair, he wrapped his other arm around her waist and stepped backward until the mattress hit his knees. He yanked the comforter and sheets aside, gripped her thighs, and lifted her until she straddled him. Holding her, he leaned against the headboard, sitting her in his lap.\n\nThe position hiked her dress past her thighs, and her gaze met his with a mix of desire and question. With both hands, he traced a path from her knees to her ankles, stopping at the straps of her shoes. He unbuckled the clasps, keeping his gaze locked to hers, and removed her shoes. Another time, he'd ask her to keep them on while he made love to her because those heels invoked more than one fantasy.\n\nHe leaned forward to unfasten her dress and remembered the back was exposed. \"Where's the zipper?\"\n\nShe smiled, warming her brown eyes and igniting the golden flecks. \"On the side.\"\n\nBut before he could do the task himself, she brought her arm across her chest and slid the zipper down. The release of metal teeth mingled with their rapid breaths.\n\nCupping her elbows, he looked into her eyes. \"I've wanted to get you out of this dress since I first saw you in it.\"\n\nHer throat worked a swallow, but her gaze stayed on his, right there with him.\n\nSliding his hands up her arms, he slipped his fingers under the straps and slowly brought them down her slender shoulders. His gaze broke away to watch his movement, wanting to see how he exposed her inch by inch.\n\nThe dress bunched around her waist, baring her perfect breasts to him. She made an involuntary move to cover them, but he gripped her hands and kissed the inside of her wrists, meeting her gaze. \"Beautiful, Avery. Never hide.\"\n\nHe leaned forward and took one rose-tinted nipple into his mouth, and groaned at her immediate response. She fisted her hands in his hair, holding him to her and arching her back, thrusting herself into his mouth. He swirled his tongue around the pebbled bud, his erection pulsing when she raked her nails over his scalp.\n\nThat. Damn, he loved that. He went from hard to holy shit in one second flat.\n\nHe moved to the other breast to give it equal attention and figured if he died right now, he'd go out a happy man. She tasted as sweet as she smelled, was as soft as the dress fisted in his hands.\n\nTrailing kisses over her collarbone, he worked his way to her neck and licked the erratic pulse beating just for him. Her head fell back with a moan, and he grinned against her throat, inching her dress past her hips, up her body, and over her head.\n\nOh, hell. Black lace panties. He'd almost be sorry to see those go.\n\nHer gaze met his, desperate, nervous, and her need slammed into him with the force of a wrecking ball. She liked things orderly, done in her own way, and at whatever speed she saw fit. Except when they were together. Like this, she was silently pleading for him to take over.\n\nAnd that worked just fine. He'd give her anything she wanted, would enjoy it, and at the edges of his conscience, he feared that went beyond the bedroom, too.\n\nDidn't matter. He cupped the back of her head and flipped her beneath him. Her chestnut brown waves spread over his pillow, the sight making his chest ache. To relieve the pinch, he kissed her, softly at first until her hands tangled in his hair again and he saw stars.\n\n_Slow down, O'Grady_. She'd gone too long without a man. Taking her would be the equivalent of sinking into a virgin. God, he never wanted to cause her pain. He'd prepare her, get her ready so he wouldn't hurt her. If he could just get his damn body to read his head's memo, things would work out.\n\nHe kissed his way lower. Halfway down her soft, warm body, she called his name and he growled.\n\n\"Yes.\" _Me, not that asshole. Me_.\n\nDipping his tongue in her naval, he tugged her panties down her legs and tossed them over his shoulder. The breath stopped in his chest at finding her bare, just a small triangle of dark curls. So hot, so wet. Gently, he spread her legs and met resistance. She tensed under him and made a sound of protest.\n\nHe looked up and caught her wide gaze, hating the fucking apprehension he found in her eyes and the prick who'd put it there. He stilled, showing her he wouldn't do anything she didn't want him to, and gradually the hesitancy moved out. She bit her lip, clearly still uncertain, and he realized she may not have done this before. That perhaps in her entire short life, no one had thought to meet her needs.\n\nJesus.\n\nHe pressed his cheek to her inner thigh, biting back a sigh. \"Trust me.\" Watching her carefully, he turned his head and kissed the spot where his cheek had just been. \"Trust me.\"\n\nAnother moment passed before she nodded.\n\n\"Watch me, Avery.\" _See me, not him_.\n\nKeeping his gaze locked with hers, he spread her legs and dipped his head, lightly kissing her mound, then her core. She didn't move, but some of the tension uncoiled from the muscles under his hands. He licked a path from her opening to her clit and a gale force wind expelled from her lips.\n\nHer head flew back, her body bowing, breaking the spell. She was his again, in the room with him and not the uncertainty from before.\n\nCupping her bottom, he teased her clit with his tongue, working a fevered mewl from her throat before slipping a finger into her heat. Christ, she was tight. Her muscles clamped around him, urging more, so he inserted another finger and lightly bit her clit.\n\nShe cried out, her voice hoarse. Fisting the sheets at her hips, she chanted his name and, he swore to God, there was no sweeter sound on earth.\n\nBlood roared through his veins. His pulse pounded in his neck. His vision grayed until he forced himself to blink in order to watch her, not willing to miss this for anything.\n\nA few more strokes of his tongue, thrusts with his fingers, and she was there. Her mouth opened in a silent cry, brows drawn together in bliss. She stilled around him a heartbeat before her body quaked. Her pale skin flushed, rosy nipples peaked, breasts rising and falling as she gulped air.\n\nHe eased her down slowly, fighting the need to string her orgasm out, have her crash a second time just so he could watch. But he was too hard, too close to the edge himself.\n\nNext time. Next time, he'd expend more effort giving pleasure to them both. He needed her too much at the moment.\n\nWith her eyes closed, breaths rasping, he crawled up her body and reached into his nightstand for a condom. He rolled the sheath down his length, taking a second to stroke the ache before cupping her jaw.\n\nHe rubbed his thumb over her lower lip, swollen from his kiss and her biting. \"Open your eyes, sweetheart.\"\n\n* * * *\n\nAt his quiet command, Avery opened her eyes to his storm-tossed blue gaze looking intently down at her. Her body was still trembling in the aftershocks, the apex of her thighs pulsing.\n\nNo one had ever gone down on her before. Richard had been her only lover, and he didn't like oral. Once, he'd told her men didn't like the act. It was just something they did out of necessity.\n\nBut Cade hadn't made it seem like a hardship. He'd acted as if he'd wanted to, as if it pleasured him as much as it did her. And oh, pleasure her he had. Whatever nervousness she'd been feeling was gone, replaced with need and satisfaction. She stretched her languid muscles.\n\nCade shifted over her, settling his welcome weight between her thighs and crushing her breasts with his chest. The light dusting of his blond hair rubbed against her aching nipples, sending tingles through her, lighting her need anew.\n\n\"There you are,\" he whispered as if she'd gone somewhere else. As if that was even possible.\n\n\"Never left.\" She smiled and brushed her fingertips over the shadow on his jaw.\n\nHe swallowed, eyes darkening. A tiny wrinkle formed between his brows like he was trying to figure something out, but his features smoothed before she could question it.\n\nWithout taking his gaze from hers, he aligned his hips and reached between them to guide himself to her opening. His tip pushed against her, into her, and he stilled, staying just past the precipice of entering, waiting for permission.\n\nSince she'd been dying to explore him, she ran her hands over the knotted muscles of his shoulders, down his sculpted biceps. His lids lowered as if liking what she was doing. Directing her perusal to his pecs, she grazed his nipples with her thumbs, earning a sharp exhale from him before moving her hands lower, reveling in each indentation of his defined abs.\n\nWow, his body was really gorgeous. Corded and beautiful without the bodybuilder bulge. A delicious combination of hard and soft. She skimmed her fingers up his sides, around his back, and leaned in to kiss his throat.\n\nHe'd held still in her exploration, strain shadowing his face, and she had a suspicion he was nearing a breaking point. Wanting him inside her, she wrapped her legs around his waist, nudging him a little deeper.\n\nHe inhaled, hard, and closed his eyes, brows twisted as if in a painful-pleasure state of his own paradise. \"Avery.\" He kissed her forehead, her lips.\n\nEasing his hips forward, he pushed deeper, going so slow she could feel every ridge of him against her sensitive walls. He was thick and long, yet not too big for her to accept. Because it had been a while, the stretch was a noticeable pressure, but there was no pain.\n\nWhen she was full, so full, he stopped, completely rooted. She let out the breath she hadn't realized she'd been holding and wrapped her arms around his back, loving the shift of hard muscle under his skin.\n\nHe sighed against her lips, breath hot, body straining as he held himself motionless over her. \"God, Avery. I can't...\"\n\nShe hadn't been expecting tenderness, for desire to blend with sweet. The punch of her heart inside her chest was jarring. Her throat tightened, and she knew she'd underestimated this moment in her mind. There could be no casual with Cade, no simple act of fun and moving on. Somehow, she'd gotten invested. Not so deep she couldn't claw her way back out, but the cavern was there, calling from the depths.\n\nHe dropped his head and pressed his face into her neck. \"I can't...\" he said again, but she knew. She understood.\n\n\"Then don't wait.\"\n\nAs if that was all the permission he needed, he pulled out and thrust back inside, still so gentle, so careful. Remembering he liked her hands in his hair, she skimmed her fingers up his back and into his hair, fisting the strands.\n\nHe let out a low, guttural groan that rattled his chest. His breathing quickened, panting against her neck. Grabbing her thigh, he brought it higher on his hip, shifting the angle of his thrusts to go deeper, hitting her in a spot that made her eyes roll back in her skull. His other arm skated under her, lifting her hips.\n\nHe moved in earnest, their skin gliding against one another, their heartbeats pounding in unison. Her body hummed as she met him with every plunge. He held her tight, as if separating, as if allowing so much as air between them, would break the spell. They moved as a unit, as if they'd done this dance before, both new and old sensations taking hold.\n\nThey fit. Just...fit.\n\nA sharp tingle low in her spine was the only warning an orgasm loomed. The shock of it, the sheer improbability, nearly had her losing focus. She never came during sex. Never. She grasped at the thin tether, holding on. Fighting.\n\n\"Let go, sweetheart.\" He lifted his head, bringing his mouth to hers and crushing her in a kiss of pure annihilation. His tongue stroked, matching the thrusting of his hips. \"Let go,\" he said again, this time spoken through a groan as he dropped his forehead to hers. \"Come with me.\"\n\nNot _for me_ , but _with me_. Together. He wanted them to fall together.\n\nEven that one simple thing, a statement he probably didn't even realize the meaning for, demonstrated how different he was from anything she'd ever known. It wasn't about him or her, but them. He was selfless and giving. There was no defense against him.\n\n\"Avery, sweetheart.\" His tight, barely controlled voice filled her head.\n\nAnd the fight fled, becoming irrelevant as sensations took hold. The tight ball of need exploded. She shook in his arms. Grabbed his shoulders. Arched. Found oblivion.\n\nDied? Maybe. She couldn't tell.\n\nA primal noise, part scream, all male, emitted from his lips as he grew rigid. He pumped twice more and stilled, pulsing inside her as he released.\n\nBefore the effect wore off, before he was even finished, his handsome face twisted in wonder and determination in the tailspin of his orgasm. His blue gaze collided with hers. Accusing, accepting. There and gone in a flash.\n\nHe collapsed onto her and she held him as their breathing soughed, both of them fighting for air. After a moment, when they'd caught their wind, he raised onto his forearms and pressed his face between her breasts as if unable to look at her.\n\nHe stayed that way so long worry fluttered in her belly. \"Cade?\"\n\nHead down, he pulled out and rose from the bed, never meeting her gaze. \"I'll be right back.\"\n\nThen he went into the adjoining bathroom and shut the door.\n\n# Chapter 19\n\nCade tossed the condom in the bathroom trash and ran the faucet. Cold. Splashing water on his face, he looked at his reflection in the mirror, gritted his teeth, and threw more water on his cheeks.\n\nHe gripped the counter and leaned into his hands, forcing his lungs to take air before the black edging his vision had a chance to consume.\n\nDamn Drake and his cryptic shit. Spouting nonsense about puppy love and...well, crap. That's what this was. A panic attack brought on by his brother. The frantic, erratic beat of his heart and the vise around his throat had nothing to do with Avery and the fantastic, mind-blowing, holy shit sex they'd just had.\n\nSex that didn't feel like sex at all. Thus his panic. Brought on by his brother.\n\nOr not.\n\nIt had never been like that before. Sex was a physical, extremely enjoyable act. But that's all it was. Physical. There had never been any kind of emotional element. Never. He enjoyed sex, enjoyed women. He could sense their needs and desires and be anything they needed him to be, in any position, at any random time. He got his jollies and moved on.\n\nExcept what had just gone down in there was no act. Her needs before his own was paramount, making her comfortable with him a given. But afterward, when she'd relented to her hunger, he'd...relented something else. Like a chunk of himself. He didn't think, didn't need to. He'd just...felt. And lost himself. In her.\n\nHe blew out a long, unsteady breath. Okay, he'd known Avery was different. Check. He'd gotten to know her better than other lovers before hopping into bed. Check. There'd been an insane amount of buildup and foreplay in the previous weeks. Check. He'd had to work damn hard to get her to trust him and herself before ever getting to this stage. Check.\n\nThose things were rational. Explainable.\n\nHe was fine. Perfectly fine.\n\nYeah, he so wasn't. But he couldn't stay in his damn bathroom all night figuring it out and losing his shit in the process.\n\nStraightening, he splashed another stream of cold water on his face, toweled off, and avoided his reflection on his way to the door.\n\nWhen he emerged, Avery was on her knees on the floor, the bed sheet wrapped around her middle and her sweet round ass in the air as she looked under his bed.\n\nHe stopped and eyed her, not liking the tension building in his gut. But hell, she was such a sight if he ever saw one. Her hair was a tangled, wavy mess around her head, her cheeks were crimson, and her lips still swollen from his kiss. Her perfect breasts were playing peek-a-boo with the sheet and she looked adorable crawling around.\n\nSpeaking of...\"What are you doing?\"\n\nShe jerked up and blew a wayward strand of hair out of her face. \"I can't find my panties.\" Her hold on the sheet tightened. \"Or my other shoe. My extra set of clothes are in the car, and I can't go get them like this.\"\n\nBecause he had the violent urge to grip her by the shoulders and give her a firm shake, he leaned against the doorjamb and crossed his arms. \"Why do you need your extra clothes?\"\n\nOh, he knew. He wasn't an idiot, contrary to what his brothers seemed to think. But he'd be damned if he'd make this easy on her. He'd eaten the distance, had met her more than halfway, and she was still waiting for the other shoe to drop. Metaphorically speaking.\n\n\"To go home,\" she said slowly as if talking to a small child. \"I guess I could just put the dress back on,\" she mumbled to herself.\n\nEven that was adorable, which pissed him off to the point of no return. \"I said I'd be right back. What happened in the two minutes I was in the bathroom to make you bolt?\"\n\nShe sat on her haunches, confusion in her eyes, frustration wrinkling her brow. \"Fifteen minutes. You were in there fifteen minutes. I may not have your experience, but I can take a hint.\"\n\n\"I wasn't issuing any hints. I was...\" Freaking out. He sighed.\n\n\"Flipping out,\" she said, reading his mind. She went back to crawling around the floor. \"Look, I get it. No hard feelings.\"\n\n\"No hard...\" He pinched the bridge of his nose. \"You were in that bed, too. You were into it.\" _Into me_.\n\nIt was as if she hadn't heard him. Her head ducked under the bed. \"I warned you I wouldn't be any good. A husband doesn't cheat unless...\" She emerged with her shoe. \"There it is.\"\n\nHe remained where he was because if he moved he just might go postal. How many damn times did her ex need to whittle his way between them? \"Now who's freaking out?\"\n\n\"Cade.\" She sighed a weary moan and rested her head on the mattress, still on her knees. \"I'm sorry it sucked.\"\n\n\"Were you even in the same room?\" Yeah. Zero to postal. \"Whatever has you hunting for your clothes has nothing to do with you sucking and everything to do with the crap he put in your head. And you didn't. Suck, I mean.\"\n\nThere were many, many adjectives to attribute to what they'd done together, but suck wasn't even in the vicinity. In fact, he was pretty sure Webster hadn't invented nearly half the words coming to mind.\n\n\"Then why did you hide in the bathroom?\"\n\n\"I wasn't hiding.\" Much.\n\nHer brows hit her hairline, sardonic disbelief twisting her mouth.\n\nShe shook her head and grabbed her panties off the bed. With some kind of crazy female maneuver, she managed to step into them while maintaining a hold on the bed sheet. As if he hadn't just seen her naked, hadn't kissed every inch of her sweet, soft skin.\n\nPanic shoved at his chest. He crossed the room, tore the sheet out of her grasp, lifted her in his arms, and dropped her on the bed. Before she stopped bouncing, he crawled in after her and covered them with the blanket.\n\nShe turned on her side to face him, but blessedly made no attempt to leave.\n\nHe took in her mortified expression, the one she was obviously trying so damn hard to keep neutral, and struggled to find the right words. She could get him so rattled at the drop of a hat, he wound up eating crow more times than not.\n\nHe tucked a piece of hair behind her ear, letting his fingers linger by her jaw. \"It takes two people to have sex. If one of them isn't into it, isn't open to the experience or isn't willing to give as much pleasure as they're getting, then it will suck. You were right here with me, Avery. _We_ didn't suck.\" He swallowed hard. \"And when a man cheats on a woman, that's _his_ insecurity showing.\"\n\nShe wedged her hand under her cheek and stared at him. Her gaze softened, slipping back into the tender woman who he'd begun the night with and not the insecure person her ex had made of her. He was almost certain she didn't believe that dick's crap anymore, if she ever had. She was too strong in that regard. Yet he suspected no one had done anything to build up her confidence, either. The balance was askew.\n\nHer teeth sank into her lower lip and released. \"What happened in the bathroom?\"\n\n\"I threw out the condom and splashed cold water on my face.\" Went a little batshit and had an internal pep talk. He closed his eyes for a moment. She'd been nothing but honest with him and deserved to know the lack of post-coital glow wasn't on her.\n\nHe traced his thumb across her lip. \"I care about you. Sex has never been particularly emotional for me. I just needed a minute to get my head together. That's all.\" And if he could do it over differently, he would, if for no other reason than it made him sick he was another person to put doubt in her mind.\n\n\"This was just supposed to be fun.\" Concern showed in her eyes, but behind that he could've sworn there was hope.\n\nHe settled on humor as he didn't think either of them were ready for anything heavier. \"I don't know about you, but I'm having fun.\"\n\nA lazy smile curved her mouth. She brought her hand up to push away a strand of hair from his forehead, the move more maternal than sexual. \"Me, too.\"\n\nHe shifted his pillow a little closer to hers so that they were sharing air. \"Your marriage wasn't all bad, was it?\" Irritation and something he couldn't name battled in his gut. Empathy maybe. He didn't know. But the idea of her spending ten years with a man who didn't care enough about her to hold on to a good thing didn't sit well.\n\nHer smile fell a fraction. \"No, there were a lot of good things. We weren't big into anniversaries or Christmas presents, but he always remembered my birthday. We met on campus on my birthday, and he would send a bouquet of flowers to the house every year.\"\n\nFlowers weren't a gift. They were an accessory. And what kind of man wouldn't want to celebrate the day he got married?\n\nHe kept his thoughts to himself, though, as her expression drifted into memory. The moonlight streamed through the window at her back, casting her face mostly in shadow, but her scent mingled with his and settled his mood.\n\nHer gaze flickered to his. \"Richard was under a lot of pressure from his family even when we met. Rich relatives, big law firm. On paper, we fit. My mom was pretty flighty growing up. She loved me, does love me, but she's a dreamer. I saw in Richard the stability I'd been missing. At first, the friendship was enough to make up for the lack of chemistry. By the time we got married, it was like it was too late to back out. In time, his family changed him, made him cynical.\"\n\nShe sighed. \"I know Richard cheating wasn't my fault. But in a way, I can't blame him, either. Neither of us got to experience being young or carefree.\" Her gaze took in his face like she was comparing again. \"I guess I resorted back into the mind frame when you didn't say anything, when you...collected yourself in the bathroom.\"\n\nWhat she said gave Cade a clearer understanding of the dynamics, but it didn't change his view an iota. The guy still didn't appreciate or recognize a good thing when he had it. The prick's loss, Cade's gain.\n\nShe'd never once used the term \"love\" when it came to her ex, and by her explanation of the marriage, love had never come into play. For either side. He thought about his own romantic encounters. Love had never blindsided him, but he wasn't immune. No one was.\n\nDrake's words floated back to him, and he had to question whether his brother was right. His feelings for Avery, or the fact he had any at all, were unlike anything he'd come in contact with before. And what did it matter? Weeks ago, she'd told him she'd never lose herself in a relationship again.\n\nHe almost laughed. Denial aside, they were very much in a relationship.\n\nWrapping his arm around her waist, he tugged her closer, crushing her to his chest. He slid her panties down her legs and threw them over her shoulder. Then he settled on his back and drew her to his side, encouraging her to rest her cheek in the crook of his shoulder.\n\n\"Are you going to sleep?\" Her warm breath fanned his skin.\n\nWith his eyes closed, he smiled. \"Yep.\"\n\n\"Why did you take my panties off?\"\n\nHis grin widened against her temple. \"I'll demonstrate in a couple of hours after we recuperate.\"\n\nShe paused. He could all but hear the wheels turning in her head. Finally, she draped an arm over his chest and yawned. \"Promises, promises.\"\n\nHe laughed and kissed her hair. \"Oh, sweetheart. Challenge accepted.\"\n\nWhen Cade next opened his eyes, a muted dawn was peeking through the window, and he was spooning a warm, soft female. Avery had his hand tucked into hers, clasped against her breasts, her breathing deep and even. His thigh was wedged between her legs, his happiness to see her poking into her backside.\n\nTaking stock of the situation, he decided this sleeping with a woman thing wasn't half bad. The actual sleeping. Waking up to her scent and supple warmth could be quite addicting. Honestly, everything about Avery was an addiction.\n\nShe was all curves and no edges. He liked that, liked she wasn't a thin rail with nothing to hold on to. Both her body and her heart were giving, pliant\u2014a direct contrast to a lot of women he'd been with. She wasn't perfection, he wouldn't want that, but even her flaws were cute, making her...perfect _for him_. Hell.\n\nHe bent his head and kissed the curve of her shoulder where tiny freckles sprinkled her pale skin. In the darkness last night, he hadn't been able to take all of her in. This morning, he'd make up for that. Burying his face in her hair, he inhaled, unable to resist. She smelled so damn good.\n\nShe began to stir, arching her back and emitting the sexiest little moan. Her long, dark lashes fluttered open and she turned her head, offering a sleepy smile. \"Morning. What time is it?\"\n\n\"Early.\" He looked over his shoulder at the alarm clock. \"Not quite six.\" He pressed his face into her neck and licked her pulse, smiling when it kicked harder with his attention. \"I believe I promised you something last night. Remind me what that was again.\"\n\nShe brought her arm up and behind her to sink her fingers in his hair. \"I think it had something to do with missing panties.\"\n\nHis dick jumped. He swore to all that was holy, if she never took her fingers from his hair, he'd be a contented man. \"It's coming back to me now.\"\n\nHe cupped her breast, settling the small weight in his hand and brushing his thumb over the nipple. She inhaled sharply, pressing her bottom in much closer proximity to his straining erection. Her nails raked his scalp.\n\nThat was the other thing. She was so responsive. Whether it was his pathetic attempt at humor, the endless array of questions he asked, or her body to his touch. Didn't matter the scenario, she responded. He thrust against her, earning another moan. Like that. Action, reaction.\n\nSexy. As. Hell.\n\nBefore he completely lost his mind, he reached behind him into the nightstand and pulled out a condom, ripping it open with his teeth. Once protection was in place, he pressed his hand to her belly, reveling in the quiver beneath his palm.\n\nHe licked the shell of her ear. \"Place your hand over mine.\"\n\nShe did as he asked without issue, threading her fingers through his. He brought their joined hands lower, over her mound and between her folds. He groaned at how wet she was for him. Her head fell back as he spread her and dipped a finger inside. The motion was even more intimate with both their hands entwined as if they were stoking her pleasure together. He used his thumb to circle her clit and she bucked into his hand, emitting the hottest mewl from her throat.\n\nLast night, she hadn't made much noise. Aside from a gasp or moan, she came quietly. He wondered if that was restraint on her part or just her sexual makeup. \"You like that?\"\n\n\"Yes,\" she breathed, writhing under his hand.\n\nShe was more than ready for him, and he was throbbing. Withdrawing his hand from her heat, he aligned himself and, from behind, slowly sank into her. He slammed his eyes shut and waited a beat for her to adjust. And damn, she was so tight in this position breathing was impossible.\n\nWhen he could hold out no longer, he grabbed her thigh to lift it for a better angle and thrust. She cried out and buried her face in the pillow as if to smother the sound. He secured her leg in position with his own and cupped her cheek, finding it crimson. With embarrassment or desire, he wasn't sure.\n\n\"Don't hide from me. And be as loud as you want. We're alone.\" He didn't give her time to react. He thrust again, her inner walls like a glove milking his shaft. So hot, so tight. Perfection.\n\nHer head whipped back, exposing her throat. \"I don't... I'm not usually noisy. I'm sorry.\"\n\n\"Why the hell would you apologize for that? Let me hear you, sweetheart.\"\n\nAll she'd managed to do with her broken admission was prove she'd never been made love to properly. He was more than game for making it up to her. Repeatedly. He quickened his hips, pumping harder.\n\nBrushing her hair aside, he latched his mouth onto her neck and nipped. Thrusting furiously, he licked the sting away.\n\nA strangled moan tore from her again, and he knew she was close. He wouldn't hold out much longer, either. Sliding his hand between her legs, he circled her clit, working her into a fever.\n\nShe pressed back into him, tensing against his chest, her mouth open in a silent scream. Her hand fisted the sheet as her walls clenched him so tight he came undone. A tingle tracked from his lower back straight to his balls as she quaked in his arms.\n\nHe dropped his forehead to her neck, releasing hard and fast, until every last ounce of him had been wrenched dry.\n\nHe held her through the last of her aftershocks, depleted but determined to do better this morning than he had last night. The onslaught of foreign emotion was no different in the light of day. His chest cracked wide open and his throat seized. A painful, gratifying ache ripped through his gut and welled behind his lids, but he forced himself to breathe through it and figure out the cause later.\n\nWhen she went limp, he pulled out and dropped the condom in the bedside trash before rolling over and dragging her on top of him. Their arms and limbs tangled. Her hair spread over his chest, her breath warm and uneven on his skin where she rested her cheek. He wrapped his arms around her back, slipping one hand into her hair to hold her to him.\n\nHell. Better than any blanket.\n\n# Chapter 20\n\nAvery finger-combed her hair into what she hoped was submission, a little embarrassed she'd fallen back asleep. Especially since she woke up to find the bed beside her empty.\n\nAfter she used the bathroom, she shrugged into the T-shirt Cade had offered last night and followed the scent of bacon and coffee downstairs. She stopped short in the kitchen doorway at the sight of him in a loose pair of sweat pants low on his hips and nothing else. His back to her, he was frying bacon in a skillet at the stove. The shifting muscles in his back was a thing of beauty. His blond hair stood at odd angles, total bed head and rocking it. And him barefoot? In the kitchen?\n\nCrap on a cracker.\n\nThe dogs were sitting by his feet, tails wagging in a silent plea for food. Cade was having a conversation with them that entailed him saying \"bacon\" and the dogs tilting their heads, butts squirming in answer. The kitten was perched on his shoulder.\n\nGod. Clenching her thighs, she watched him another moment before the need for caffeine became astronomically pivotal.\n\nHe turned and grinned, his gaze sweeping down the length of her. His smile went from affectionate to predatory. Absently, he took Cutin from her perch and set her carefully on the floor, gaze trained on Avery. \"I like the look of you first thing in the morning.\"\n\nReally? She ran a hand through her hair. \"Animal Kingdom? I can see the appeal.\"\n\nHe laughed. Moving away from the stove, he walked over and set his hands on her waist, lifting her onto the island as if she weighed nothing. \"I'm instilling a new rule. If you're not naked, you can only wear my shirt.\"\n\nShe smiled as he stepped between her legs and wrapped his arms around her. \"That will go over real well at the clinic.\"\n\nPressing a kiss to her neck, he groaned. \"That reminds me of another fantasy.\" He licked her pulse and moved to the other side, nipping his way to her ear.\n\nGod, he was good at the nuzzling. Her nerves were on fire. Need zinged from where his mouth worked her over to straight between her thighs. No passing Go.\n\n\"What would it take to get you to call me Dr. Cade outside of work? Not all the time, mind you. Just once in a while. Something like, 'Oh, Dr. Cade, right there! Yes, that's so good.'\"\n\nWrapping her arms around his neck, she laughed until her cheeks hurt. \"Can I wear the white lab coat?\"\n\nHe groaned again, a deep rumble. \"Hell yes.\"\n\n\"I say the odds are good then.\"\n\n\"Definitely will remember that.\" He leaned back and kissed her forehead before stepping away. \"Do you like bacon?\" Back at the stove, he transferred the pieces to a plate to drain.\n\nShe reeled, the loss of his heat and arms a sudden vacancy. \"Are there people in existence who don't?\"\n\nHe laughed. \"None I'd want to know.\"\n\n\"Agreed.\" Her gaze drifted out the patio doors, where a thick fog blanketed the damp grass. The sun was trying to unsuccessfully break through the cloud cover. At least a lot of the snow had melted with the milder temperatures the past few days, but spring was a ways off yet.\n\nHe poured a cup of coffee and brought it over to her, holding it out of reach. \"Kiss and then coffee.\"\n\nBlackmail of the lowest form. Not that she minded.\n\nHe grazed her lips with his, gently at first, but it quickly gained momentum. Increasing the pressure, he opened and stroked her tongue, lazy, seductive. She melted and wrapped her legs around his waist.\n\n\"Nimble minx.\" He took her coffee to the table, picked her up again, and deposited her in a chair. \"We'll finish that thought after breakfast. Some brunette drained all my energy last night. And this morning. I need sustenance.\"\n\n\"Complain, complain.\"\n\nHe leaned close, brushed her nose with his, and busted out his panty-melting grin. \"Oh, sweetheart. Not even in the slightest.\"\n\nWhen his back was turned, she smiled at the easiness in being with him. Good conversation, humor, explosive in bed. He was good with Hailey and animals, cared about his family, and his attention to detail was staggering.\n\nIf she wasn't careful, she might break her rule of never getting involved again. She'd instilled it for a reason, good reasons, but Cade was slowly, increment by increment, wearing those rationales down to dust. Unsure what to think, or even if she should overanalyze what was intended to be simple, she sipped her coffee.\n\nHe brought the plate of bacon and a bowl of scrambled eggs to the table before digging into the fridge and removing a container of melon.\n\nHer gaze locked on the drawings tacked to the fridge she hadn't noticed before. Hailey's drawings. The ones from her app he'd printed for Avery's fridge, too.\n\nGod. He...crap. He'd put her daughter's pictures on his own fridge. She swallowed hard, torn between the urge to wrap her arms around him and the need to distance herself from the emotions swirling.\n\nSetting down two forks and plates, he took his seat next to her.\n\nShe shook her head to clear it.\n\nRichard had never cooked for her, even something as simple as scrambled eggs. They'd never made love in the morning or enjoyed breakfast while half dressed. Looking back on the past decade with an eagle eye, she realized it had been more a marriage of convenience than one of mutual love.\n\nShe hadn't been with Cade long, and she wasn't sure she could even deem what they were doing as dating, but the differences kept hitting her like a blow. Hard. His compassion to Richard's indifference. Cade's humor to Richard's sarcasm. Cade's attention and affection to Richard's disregard.\n\nWas this what it was like to matter to someone? Was her time with Cade and how they spent it normal? She swallowed with difficulty, wondering if she'd even recognize a regular relationship if it smacked her in the forehead.\n\nHe loaded eggs on her plate, looked up at her, and did a double take. \"Avery?\"\n\nThe concern in his blue eyes forced her to smile and shake her head. \"I thought you only cooked microwave popcorn or mac and cheese.\" She dug in, even though her appetite waned.\n\nMotionless for a moment, his gaze roamed her face as if trying to spot the lie. \"I have other basic culinary skills.\" He dished eggs and melon onto his plate, closed his eyes, and sighed. His brows creased in frustration before he lifted his lids to pin her with a wary look bordering on helplessness. \"One of these days, you'll be comfortable enough to tell me where your mind drifts off to. I can't stand not knowing what I did to cause such deep contemplation.\"\n\nDamn. How could he think he'd done anything wrong? And the way he made it sound implied they were long-term.\n\nMaybe she needed to be clear with him. Out of self-preservation and habit, she'd been holding a chunk of herself back. Openness wasn't something she was accustomed to, and for years she'd buried her thoughts and desires because the person she'd been with hadn't wanted to know them, hadn't cared.\n\nAfter they'd made love last night, Cade had claimed to care about her, that the experience had been emotional for him. It wasn't in his moral code to lie or mislead, which left her with more questions than answers about what they were doing. But if they were to keep at it\u2014and did she want to?\u2014then she had to make him understand he wasn't the one not functioning at capacity.\n\n\"Eat, sweetheart.\" Gaze trained on his plate, he shoveled eggs into his mouth with what seemed like effort.\n\nShe managed the bacon and took a bite of melon, forcing a swallow. \"If I told you it's me, not you, would you believe me?\"\n\nHe didn't so much as turn his head. \"No.\"\n\nShe rubbed her forehead. \"When we're together, you...\"\n\nHis body went rigid. \"I what?\"\n\nGod. She blew out an unsteady breath. \"You're...not what I was expecting, I guess. You're different than\u2014\"\n\nHis fork clattered to his plate. He glared at her with a look of pure pissed off male. \"Is that what this is about? Him?\" He turned to face her. \"I've got news for you, Avery. If you're waiting around for me to start acting like your dick of an ex, it's never going to happen.\"\n\n\"That's not what I\u2014\"\n\n\"At some point, you need to look at me and not see him.\" His voice steadily rose. \"I don't know if you feel the need to torture yourself out of some twisted form of guilt or if you think he's all you deserve, but I'm not going to do a one-eighty. I refuse to treat you like you don't matter.\" He stood and walked his plate to the sink, spine stiff and mood posting back off warnings.\n\n\"Cade...\"\n\nHe ran his hands through his hair and took more than a few ragged breaths before dropping his chin. He stood that way, back to her as if attempting to rein in his temper and get a hold of himself, while she could do little more than bite her lip trying to keep tears at bay. Not because he'd yelled at her and not because of the sudden distance between them, but because he was right.\n\nShe twisted her fingers together in her lap. \"I don't compare you and him to make you angry or to make you feel inferior.\" He didn't move, and her stomach cramped in dread. \"He's all I've ever known. So yes, I measure you against each other, to remind myself how different you are from him.\" She sighed and rubbed her forehead. \"Every other second you surprise me with a sweet phrase or a kind gesture or...\"\n\nThere was no point in explaining. He wouldn't understand. He didn't get that he was so much better a man than Richard could ever hope to be, and by her comparing, it forced her to allow their relationship to expand. Otherwise she'd never move past the hurt to see the good.\n\n\"Is that the truth?\"\n\nHer gaze landed on his tense back, and she wished so much she could rewind the past ten minutes and start this conversation over. \"Yes.\"\n\nHe was silent another moment, but she thought some of the stress left his shoulders. He turned and leaned against the counter, arms crossed over his bare chest.\n\n\"I don't want fun.\" His voice came out hoarse and quiet, but his words were a scream in her ears. \"You said you'd never let anyone break you like that again. I don't want to break you. I want to date you.\"\n\nDamn it. Tears welled in her eyes. She tried to blink them away, but it was fruitless. They fell, tracking hot paths down her cheeks. Her throat closed, but she refused to look away, refused to hide from his honesty.\n\nFor a moment, she thought he might cross the room, his expression was that tormented, but he only gripped the counter behind him until his knuckles turned white.\n\n\"You've been in Redwood Ridge long enough to know my reputation.\" He held her gaze, a determined set to his jaw. \"Ask me why a playboy like me is suddenly changing his tune.\"\n\nShaking, she wiped the wetness from her face with the back of her hands and crossed her arms. Unable to speak, she stared at him and shook her head, at a loss. The fact his admission had her in tears was answer enough in her book. She'd not shed one tear when her marriage ended. She suspected she felt way more intensely about Cade than she ever did her ex, which said a lot about her. And none of it good.\n\nHer heart beat double-time when he crossed the room and claimed the chair in front of her. He dropped his forearms on his thighs and leaned forward, dipping his head to look directly in her eyes. In her personal space, but not touching. The sympathy, the understanding in his gaze, brought a fresh wave of tears.\n\nHis throat worked a swallow, gaze suddenly unsure. \"None of my previous lovers made me want more.\" He offered a slight shake of his head, but it seemed unintentional, a battle waging in his mind. \"More time, more mornings like this or nights like last night. Just...more.\"\n\nStraightening, he cupped her cheeks with his warm, rough hands. \"Give me more. That's all I ask. A chance to figure this out and where it might go.\" His voice was pleading, but insistent. Determined. His blue gaze was kind, not a hint of the mischief or hunger she was used to finding.\n\nShe'd been weak with Richard. And though she told herself not to be weak in this moment, the truth was, Cade only made her stronger. For once, why not let herself feel good? She'd never find true happiness if she based every experience on the past.\n\nSniffing, she whispered, \"Okay.\"\n\nHis chin lifted in defiance. Or disbelief. \"Okay?\"\n\nShe dragged in a breath and smiled. \"Yes.\"\n\nHis eyes narrowed as he swiped the remnants of tears from her cheeks with his thumbs. \"That was almost too easy. And by almost, I mean you really know how to drag me across hot coals.\"\n\nBreathing out a laugh, she placed her hands over his, still cradling her face. \"There will be rules, though.\"\n\nHe groaned and closed his eyes, but he was smiling when he opened them. \"I knew it. Lay 'em on me. What are these...rules?\" His hands dropped to his lap and he gave a mock shudder.\n\nHailey had to come first. There was no one more important than her daughter. \"No sleepovers when Hailey's in the house. It might confuse her or give her the wrong impression. She's adjusted well to the move, but she needs a routine.\" Her daughter was already attached to him. If she got the impression they were a couple, Hailey might never recover from another uproot in her life.\n\nHe tilted his head in agreement. \"Not a problem. What else?\"\n\n\"No PDAs at work.\"\n\nFor this, his mouth twisted. \"Can I still sneak you off into a hallway and steal kisses?\"\n\nCheeks hot, stomach fluttering, she bit her lip. \"Yes.\"\n\n\"Then, I agree. Next.\" It was cute when he tried to maintain a serious face. He failed miserably, but his attempt was crushing her resolve.\n\n\"I'm not ready for the whole town to know about us.\" She loved the small community and the sense of home since moving here, but she needed privacy, space to breathe. At least for a little while longer.\n\n\"Are you ashamed to be with me?\" His tone was light, but something in his eyes told her the question wasn't all games.\n\n\"No, of course not. I'd rather take the time to see what happens before we go broadcasting things.\"\n\nHe scratched his jaw, bristled with morning whiskers. \"I don't know how to tell you this, but everyone knows about us. You can't hide anything around here.\"\n\nThat was one message she'd received loud and clear her first day. But there was a difference between rumor and truth. \"I'm more comfortable with it being gossip for now.\"\n\nA weary sigh fell from his lips. \"You're not my dirty little secret, and I won't treat you like one. But fine. Mum's the word.\" His gaze swept over her face. \"Anything else? Do I need to bark at the moon if I want to hold your hand or throw smoke signals if I want to see you?\"\n\nGod, he was a charmer. She was sunk. \"Nope.\"\n\n\"Excellent. I'm going to kiss you now, yeah?\"\n\nSunk deep. She even loved it when he did that, threw the word 'yeah' after a statement as if turning it into permission.\n\nHe didn't say anything more, just cupped her neck and pulled her to him. She went willingly, laying her hands on his shoulders and slanting her lips over his. The kiss wasn't seductive or infused with heat. He explored her mouth as if telling a story, their story, full of tenderness and longing and affection.\n\nBy the time he pulled away, she was shaking with unresolved apprehension for how much he could invoke inside her. He was the tingle down her spine, a pressure in her chest, a flutter in her belly, a force in her head. Everywhere, at once. Consuming and devouring. Eradicating and destructive.\n\nHe kissed her temple, his lips lingering as his jaw stroked her cheek. \"When do you have to pick up Hailey?\"\n\nShe blinked at the invasion and cleared her throat. \"Ten o'clock.\"\n\nHe glanced over his shoulder to the wall clock. \"Not much time. What are you two doing later?\"\n\nSuspicion mounted. \"Laundry.\"\n\nHe grunted. \"Can I convince you to do something more fun?\"\n\nNeeding some space, as they didn't have time to go at each other before she had to get Hailey, Avery collected her plate and brought it to the sink. \"Laundry can be fun.\"\n\nHe made a sound of dismissal. \"Do tell.\"\n\nWith her back to him, she grinned and rinsed off her plate. She wasn't flirty by nature, but he brought out her lighter side, the part that liked the teasing banter sometimes between them. \"For starters, it means I'll be wearing my nightgown while I wash and fold. Without panties. Seems to me that might be easy access if\u2014\"\n\nHe was behind her with his arms banding her waist and mouth on her neck before she could so much as laugh. \"I nominate laundry day every day. Twice on Sundays.\"\n\nShe laid her head back, the smile slipping as he worked his tongue and teeth over the sensitive spot on her neck. A tremble tore through her. Her legs gave out, but he caught her against his hard body.\n\n\"Hell.\" He lifted his head. Kissed her hair. \"We don't have time.\" Turning her in his arms, he caged her against the counter. \"I'd like to take you and Hailey somewhere this afternoon. A surprise,\" he amended before she could ask. \"But we most definitely have to do laundry afterward. Many loads. I insist.\"\n\nHow he could take her from powerful, raging lust to laughing in the course of seconds she'd never know. \"You insist? I guess I can't argue with that.\"\n\n# Chapter 21\n\nCade sucked in a cold blast of fresh air and glanced around the snow-covered hill. To the west was the steep incline of the Klamouth Mountains. To the east was the Pacific Ocean, which wasn't visible from their location, but the air was brine-scented and the crash of waves wasn't far off in the distance, reminding people it was there.\n\nThis was the perfect day weather-wise for their trip. The sky was a light gray overcast, so the reflection from the sun off the snow wouldn't be blinding and the temperature was in the lower thirties with heavy humidity.\n\nHis surprise was to drive Avery and Hailey an hour up the mountain to this sweet spot known mostly to the locals. The snow in Redwood Ridge had mostly melted, but up here this far north and seated next to the mountain there was powder aplenty. An inn catered to tourists was on the other side of the ridge, but this area was tucked away, semiprivate.\n\nKids were sledding the bunny slope at breakneck speed, and a shock of nostalgia hit him in the gut. \"My dad used to bring me and my brothers here as boys. I haven't been up this way in years.\"\n\nHe remembered his dad, grinning while watching him and his brothers play, biding his time to nail them with the perfect snowball. He'd laugh in that rough, rustic way of his and grab his stomach.\n\nJesus, he missed him so damn bad. Nine years and the sudden assault of memories still hurt realizing Dad was really gone.\n\nAvery smiled, cheeks pink from the wind. \"How many broken bones came afterward? I typically don't participate in activities requiring an ambulance.\"\n\nHe laughed. \"Such a mom thing to say.\"\n\nPeople were off to the side near the crest of pines where a flat area held a hot cocoa stand and some tables. Taking advantage of the wet snow, many were building forts and snowmen.\n\nHe watched the families for a moment, struck by how deeply he wanted his own family one day. Content to enjoy things as they were, he hadn't thought a lot about marriage or children, only that he thought he wanted them. Someday.\n\nBoys to play ball with like his dad used to, or girls to build a dollhouse for and pretend tea parties were the rage. Or the other way around, since he realized that was pretty sexist of him. Zoe would knock him upside the head for that, seeing as she was the best player on their summer baseball league.\n\nBut the warmth filling his chest meant the time to think about the future was present. He attributed the emotional shift to Avery and their new relationship. Perhaps because she already had a daughter, but he didn't think that was the reason. For the first time in his life, he didn't have the urge to move on, to find the next adventure or cheap thrill. He wasn't bored or itchy for action.\n\nAvery settled something inside him. It didn't matter if they were watching a movie on her couch, making love in his bedroom, or cooking dinner at her place. None of it was tedious. Hell, he couldn't stop thinking about her or when he could next see her. Couldn't stop wondering when he'd get to do the simple, mundane things with her again.\n\nWhen Mom lost Dad, and so suddenly, he'd had to witness just how hard it was for her to survive without him. They'd been a solid unit all his life. It was as if she suddenly didn't know how to function. And when Drake had lost Heather, well... Drake still wasn't himself almost four years later. It had gutted him. It made Cade wonder, and not for the first time, if getting this deep in a relationship was wise. All that pain awaited. But he couldn't live his life expecting the worst, could he?\n\nA small hand latched on to his and he glanced down. Hailey swung their arms while looking everywhere at once in that adorable, distracted way of hers.\n\nThat was the other thing. Children didn't typically frighten him, but they didn't give him the warm fuzzies, either. This one? This one brought out protective instincts he didn't know existed and made his throat close when she reached across the barriers of her handicap to form a connection.\n\nLike holding his hand. Or cuddling on the couch.\n\n\"I don't know if she'll like snow,\" Avery said, glancing over the horizon and back to him. \"Might be a sensory overload for her, like sand. She can't stand it.\" If she noticed what Hailey had done, she didn't say so. Her smile was warm, lighting her eyes as she shrugged. \"We can try.\"\n\nHe looked down at the tiny person next to him. \"Whatcha think, squirt? Want to try sledding or building a snowman?\" He knew she wouldn't answer, but sometimes she gave off subtle signals to tell him what she wanted. Nada this time.\n\nUndeterred, he led them closer to the pine trees to a quieter area and scooped up a handful of snow. He turned Hailey's hand over and set the fresh powder in her palm. She immediately dropped it and started rocking. Not good.\n\nAvery pulled the girl's mittens from her pockets and put them on Hailey. \"Try now.\"\n\nHe repeated the process with the same result. \"Guess not.\"\n\nPlopping his ass in the snow, he gathered enough powder to compact together for a snowman base, watching Hailey out of the corner of his eye. She didn't appear interested, but she didn't wander off, either. By the time he got to the head, Hailey had moved closer and Avery was snapping pictures on her phone.\n\nCade made a tiny snowball and set it in Hailey's hand. \"Don't drop it this time. Throw it.\" When she did neither, he gently grabbed her wrist and made the motion for her. The snow hit Avery square in the cheek.\n\nPhone poised in her outstretched hand, she froze and narrowed her eyes at him. \"Not funny.\"\n\nHe disagreed. Gathering another snowball, he helped Hailey throw again. This time, the girl barked a laugh. \"She thinks it's funny.\"\n\nAvery didn't retaliate, but her smile was reward enough.\n\nThey tried sledding, but Hailey never made it past sitting down. After an hour, they called it quits and packed it in.\n\nOn the drive back, Avery turned to him. \"I'm sorry she wasn't into it.\"\n\nHe shrugged. \"No worries.\" He had fun, nonetheless.\n\nHis cell went off in his pocket and he cursed for forgetting to attach Bluetooth. He dug the phone out and answered without glancing at the ID, keeping his eyes on the road.\n\nDrake's voice filled his ear. \"Family dinner has been bumped up an hour.\"\n\nCade glanced at the dashboard. \"We're on our way back now. We'll make it.\"\n\nHe glanced at Avery, but she was looking out the side window. He hadn't told her about dinner plans, figuring he'd work up to it. Every other Sunday, he and his brothers went over to Mom's. This would be the first time he'd brought someone along.\n\nDrake grunted his acknowledgement and disconnected.\n\nCade set the phone in his lap and wound his way over the curvy highway, wondering if she'd flip out about having dinner with his family. It wasn't as if she didn't know them, but working together or hanging out at the bar was different than sharing a meal in someone's home. The last thing he wanted to do was scare her off after they'd finally seemed to break ground this morning.\n\nHe drummed his thumbs on the steering wheel. \"We're heading to my mom's house. For family dinner. If that's okay with you.\" He ground his jaw at his brilliance in blurting it out in rapid-fire succession. \"Mom invited you. Hailey, too. It's just us and my brothers. Nothing fancy or anything.\"\n\nJesus, O'Grady. Shut it.\n\nShe looked at him, brows arched and lips hinting at a smile. \"Nervous?\"\n\nHell yes. Seemed like gnomes were doing cartwheels in his stomach, and he didn't like the foreign feeling one bit. Why he was nervous, he had no clue. This was Avery. She fit in anywhere and his brothers already loved her. His mom took a liking to her the first time they'd met and brought her up in conversation too often to ignore the hint.\n\nHe shrugged. \"A little.\"\n\nShe grinned.\n\n\"Fine. A lot.\" He sighed. \"Are you upset I didn't talk to you beforehand? We can do it another time, yeah?\"\n\n\"I prefer you ask me first next time, but I'm good with it.\"\n\nThey were on the steeper decline off the mountain road. One patch of black ice or a curve too fast and it would be lights out. He had precious cargo to think about. He couldn't chance a look to verify she really was okay. The inflection in her tone, one he was growing to know quite well, indicated she was fine. He, however, was not.\n\n\"You're the first woman I've brought home. Not including my prom date.\" Which had been Zoe on a friends-only basis, so that didn't count. \"Technically, you know my family, so this is more of a formality, yeah? I mean, it's just dinner.\"\n\nChrist. He'd need a straightjacket in about ninety seconds.\n\n\"Breathe,\" she said in her calm voice, amusement tinting her tone.\n\n\"I'm breathing.\" For the most part.\n\nHer hand came down on his thigh, and even through his jeans, her heat seeped into him, calming him in an instant. She stroked his leg with her thumb, meant to be a soothing action, but it was having the opposite effect with a certain part of his anatomy.\n\n\"Just remember, we can do laundry later.\"\n\nHis gaze whipped to hers and back to the road, long enough to catch a glance of her hand covering her mouth, attempting to contain a grin.\n\nHe was nervous and she threw humor at him. _At him_. The guy who typically made light of everything. Who didn't do the dating thing or bring home a date.\n\nYet Avery, who'd been burned by exactly what he was trying to attempt, opened herself to him and had the strength to go for it, no matter how much a relationship probably scared her. If anyone deserved to be cynical, it was her. Tenfold. But she wasn't. She was kind and giving and funny and quirky and organized and...\n\n_Perfect_.\n\nJust like that, every emotion he'd been battling with regards to her slammed into his chest with full seismic force.\n\nSurprise...happiness...nerves...worry...protectiveness...compassion...respect... lust.\n\nUtterly blinding. Deafening. Excruciating.\n\nOne after another after another until he thought he'd need to pull over or risk crashing the car. His hands shook against the wheel, forcing him to clench his fingers in a death grip. He was pretty sure this was what a heart attack felt like. No air. Crippling pain in his chest.\n\nThe worst of it all was, Drake had been right. At the forefront of the emotional upheaval and shoving its way to the surface was love. Yeah. _Love_. And not the cuddly puppy kind.\n\nHe should've seen this coming. And what in the hell was he supposed to do about it?\n\nFocusing on the road, on her soothing touch, he forced oxygen into and out of his lungs. By the time they'd pulled up to his mother's house, the answer hadn't been forthcoming, but he was measurably calmer. Figuring he'd corner Drake later, he slapped on his game face.\n\nShe didn't attempt to exit the car, instead shifting in her seat to face him. One look and he knew she saw right through him. Her eyebrows rose and her eyes took on a serious, compassionate softness that pummeled him.\n\nSlowly, she nodded in an ah-ha. \"If you're this upset, we don't have to go in. We don't have to do laundry, either. We can just call it a day.\"\n\nHe was wrong. She didn't get it. Or she was giving him an out.\n\nScrew that. He didn't want an out. \"I want to go inside and enjoy the time with my family. With you and Hailey right there.\"\n\nShe stared at him for an unnervingly long beat. He couldn't read her expression to save his life, but he suspected she was doing more of her mind juju. Then, without a word, she opened the door and slipped out of the car. Taking Hailey's hand, she walked up the porch steps.\n\nHe followed and nodded at Flynn when his brother opened the door. Cade hung up their coats in the hall closet, breathing in the scent of Mom's home cooking, hoping his Spidey senses were right and that was meatloaf he smelled.\n\nAs Avery and Hailey walked deeper into the house and said hello to Drake, Flynn turned to Cade.\n\n_\"Bringing home a girl to meet the family_ ,\" the asshole signed. \" _How the mighty have fallen. When's the wedding?\"_\n\nIn answer, Cade showed his brother his favorite digit, the one between his pointer and ring finger.\n\nStepping into the family room, he half expected to see Dad there. His father's medical journals were on the bookshelves right where he'd left them and his favorite chair was in the corner, empty. Nostalgia lanced through Cade's gut and he shook it off.\n\nBesides Avery and Hailey, Drake was the only one present, sitting on one of the plaid couches his parents had since 'Nam, and watching a nature show on Discovery Channel. Closed captioning was on for Flynn and a fire was roaring in the corner hearth. Drake's German Shepherd, Moses, and Flynn's golden retriever, Fetch, were sleeping on the thick beige carpet under the bay window. Neither moved to acknowledge the newcomers.\n\n\"Cade? Is that you?\" A clattering followed Mom's question and Cade imagined her fluttering around the kitchen, mashing potatoes and taking the meatloaf from the oven. Mom, the multi-tasker.\n\n\"Yeah,\" he hollered back. \"Avery and Hailey, too.\"\n\n\"Oh good. Avery, dear. Come in here with me. There's too much testosterone in there.\"\n\nSeeing her in his parents' house caused a cozy warmth to fill his core. He stayed in the doorway to take the moment in, but she was on her feet and motioning for Hailey to follow her.\n\n\"We're needed in the kitchen.\"\n\nFlynn shook his head. \" _I've got her. Go ahead_.\"\n\nCade looked down at Hailey, who hadn't moved too far from him and was watching Flynn's hands with more focus than he'd seen from her to date.\n\n\"Are you sure?\" Avery bit her lip, and he had to wonder when the day would come she'd realize her and her daughter weren't a hardship.\n\nFlynn nodded, redirecting his attention on the girl. \" _Remember me?\"_\n\nHailey didn't respond, but she was watching his brother intently, as was Avery with a worried wrinkle between her brows.\n\nFlynn squatted in front of Hailey. \" _I don't talk, either.\"_ He looked at Avery. \" _Does she play checkers?\"_\n\nAvery nodded. \"She prefers tic-tac-toe, though.\"\n\nFlynn grinned at the girl. \" _Think you can beat me?\"_ He didn't wait for an answer. Rising, he headed down the hallway toward their old bedrooms, Hailey in his wake.\n\nAfter a moment, Avery walked around the corner and disappeared into the kitchen. When his mother's voice drifted into the room, he knew Avery would be occupied for a while.\n\nCade looked at Drake and jerked his chin toward the door. \"Take a ride with me to get the dogs.\" They needed to be let out and Freeman usually came with him to Mom's.\n\nIt wasn't a request, and Drake didn't take it as one. He rose from the couch and slipped into his coat, then held the door open for Cade.\n\nIn silence, they drove the short distance to Cade's cabin and parked. He didn't have much time and wanted to get back before Avery knew he was gone. Yet he fisted the wheel and made no attempt to exit the vehicle. He wished his head could make sense of the emotional riot.\n\nDrake gave him an assessing side-glance. \"Are we going to make out?\"\n\nCade couldn't help it. He dropped his forehead to the steering wheel and laughed. The entire day, if not the past few weeks, had been surreal. \"Hell, Drake. It's good to have you back. One more thing I can thank her for.\"\n\nHis brother didn't respond, but tension filled the car, crackled in the air. Cade lifted his head and stared out the windshield, unsure what to do or say.\n\nAfter several long, awkward minutes, Drake sighed. \"I never went anywhere.\"\n\nNot physically, but ever since Heather died, his big brother had mentally and emotionally checked out. Maybe it was Avery's doing or maybe enough time had passed since Heather's death for Drake to move on. Perhaps a combination. Either way, his brother was acting less like a ghost and more like part of the human race again.\n\nDrake crossed his arms. \"Since we're not here for incestuous reasons, and thanks for that, I can only assume this is about a certain woman?\"\n\nHe leaned his head back and closed his eyes. \"All the great tragedies start with the phrase 'There's this girl...'\"\n\n\"As do the epic love stories. Not all end in tragedy.\"\n\nCade opened his eyes and stared at his brother. \"You look like Drake, sound like Drake, but\u2014\"\n\n\"You brought me out here. If you didn't want my opinion, you should've brooded in silence.\" One corner of his mouth quirked. \"It has its merits. Silence.\"\n\nHe returned his focus to the windshield, rubbing the back of his neck. \"How did you know it was love with Heather?\"\n\nDrake laughed without mirth. \"You just know. And the fact you're asking means you're there.\"\n\nCade had pretty much reached the same conclusion, but what he really wanted to say was wedged in his throat. Words that, if spoken aloud, couldn't be taken back, and he wasn't sure he wanted the truth.\n\nHe'd watched Drake grieve for his wife, stood there as his brother's life had crashed down around him, and he'd been powerless to do a thing. All that pain, all that misery. Cade was only hovering on the outskirts of that kind of love, and he couldn't fathom how Drake put one foot in front of the other after his world had been stolen.\n\n\"I don't...\" Cade blew out a ragged breath. \"I never wanted to fall for her, for anyone. Why set yourself up for pain when you can see it coming?\"\n\nDrake's gaze pinned him to the seat. Direct, swift, and more than a little irritated. \"So this man-to-man isn't about advice or confirmation. You're asking for permission to walk away. Because what? You could get hurt?\"\n\nCade shook his head, his unfocused gaze drifting off. He didn't think he could walk away from Avery, even if he should.\n\nHonestly, he didn't know why he'd brought Drake out here, other than his brother might very well be the only one who understood what he was dealing with. And the only person who could get him past his reservations about plunging altogether. When push came to shove, if Heather hadn't died, Cade didn't think they'd be having this conversation. Love would just be another grand adventure and not something to fear.\n\n\"You're not asking the right question, little brother.\" Drake's solemn gaze met his, his tone mild, but not brooking argument. \"You need to ask if it's worth it, if I'd do it again knowing the result.\" He leaned closer. \"Hell yes. I would.\"\n\nDrake grabbed the handle and opened the door, sliding one foot down to the ground. \"I'll get the dogs. You get your shit together.\"\n\n# Chapter 22\n\nAvery and Gayle had just brought the food out to the large kitchen table when Drake and Cade strode in, the dogs at their heels. Drake seemed no different, but Cade's gaze wouldn't meet hers and his mouth twisted in what seemed like unpleasant thoughts. She had no clue why it had taken thirty minutes to drive a block to get the dogs, but her stomach cranked in worry.\n\nCade rubbed the back of his neck and came deeper into the room, eying the table. \"Looks good, Mom.\"\n\n\"Thank you. Sit down, everyone.\"\n\nShe liked Gayle's home. Unlike Cade's two-story, his mom's was a ranch and not nearly as updated. The place had a lived-in feel. Photos were everywhere she turned\u2014the mantle, the walls, the fridge\u2014and went as far as his parents' wedding. The furniture and appliances seemed dated to the boys' youth. Even the kitchen table had nicks and scars. The house was simple and friendly, like Gayle.\n\nAvery hung back, waiting for everyone to take a seat in order to assess where they wanted her to sit. In the end, she wound up next to Flynn and across from Cade. Instead of taking the open spot by Cade, Hailey plopped down at the head of the table where no place setting had been laid out.\n\nAvery tapped Hailey's arm. \"Sweetie, there's a plate by Cade for you.\"\n\nHailey didn't move, and nerves pinged in Avery's belly, especially when everyone seemed to be sharing uncertain glances with each other.\n\n\"She's fine,\" Cade said at length, eyes on Hailey. \"Only someone as awesome as my dad gets to sit in his spot.\"\n\nOh God. That had been his dad's place at the head of the table. She rose. \"Hailey, please move to the other chair.\"\n\nDrake cleared his throat. \"Sit, Avery. Like Cade said, she's fine.\"\n\nAvery looked at Gayle just to be certain, still uneasy.\n\nShe smiled and nodded her agreement with her sons, waving her hand in dismissal of the situation.\n\nBlowing out a breath, Avery reclaimed her seat. She leveled a smile in Drake's direction. \"You're so bossy.\"\n\nHe grinned right back. \"This from the woman who alphabetized my surgical vials. Twice.\"\n\n\"Well, I don't know how you found anything in that mess of a cabinet. Besides, that's organizing, not bossing.\"\n\nHis brows lifted, grin widening. Who knew Drake had dimples? \"Okay, you told Missy Hamilton not to come back to the clinic unless she was wearing a bra. That's bossy.\"\n\nFlynn choked on his drink. \" _You did not. Really?\"_\n\nShe huffed. \"I did so. You can't walk into a place of business with all your...goodies hanging out. And who brings a goldfish in for a checkup? Come on.\"\n\nFlynn shook his head. \" _Dump my brother and marry me_.\"\n\nShe laughed. \"I'll consider it.\" Glancing around, she realized heads were pivoting between the conversation and she was holding up dinner. Her cheeks heated. \"Um... This looks really delicious, Gayle. Thanks for having us.\"\n\nGayle picked up a bowl of peas and passed it to Drake with a motherly smile. \"Oh no, dear. The pleasure is all mine.\"\n\nAvery went to load Hailey's plate only to find Cade had already done it. He also had a cup of milk in front of him, having switched out his lemonade with Hailey's glass.\n\nFlynn zeroed in on that, too. \" _Stealing milk from kids? Not nice_.\"\n\n\"She can't have dairy, jerkface.\"\n\n\"Language, Cade.\" Gayle tsked and forked a slice of meatloaf onto her plate. She looked at Avery. \"I'll remember the no-dairy for next time.\"\n\nCade's brows lowered. \"Jerkface is not a curse.\" He turned to Hailey. \"Right, squirt? Just don't go repeating it at school. Deal?\" He held up his fist and, after her typical delay, Hailey bumped it back.\n\nWhen he looked up at Avery, a half grin cranked in satisfaction, her heart outright stopped pumping. She couldn't look away from that intense shade of blue\u2014a force of nature when combined with amusement and a clear case of love. He wasn't trying to win her over with her daughter's affection or spending time with Hailey to get to Avery. He did all these sweet, simple things because he...legitimately loved Hailey.\n\nIt was blatantly apparent in the care he took, his patience, and the way he looked at her. He pushed her boundaries, but only to get a read on her, and never so far as to upset her mood. Able to read her mannerisms, he headed off problems before they began and was learning her likes.\n\nMost of all, he spent time with Hailey. Today's snow adventure was the truest test of his character. Instead of getting angry or disappointed Hailey hadn't been an excited participant, he'd shrugged it off.\n\nAvery's heart lodged in her throat, her eyes suspiciously close to watering. She stared down at her plate to get a hold of herself.\n\nRichard had never tried to bond with their daughter. Not truly, and definitely not after her autism diagnosis. He hadn't seen the point. He thought she was broken. In his eyes, Hailey had a major defect and since she didn't respond in the normal way to situations, he'd stopped trying at all. And though Avery swore to herself she'd strive to quit comparing Richard and Cade, in this one blaring instance, she couldn't help it.\n\nRichard was Hailey's father, but love never came into play. Cade had no blood tie, no familial connection, yet he adored Hailey. There was no question. That was the kind of man he was. Real. Genuine.\n\nShe wasn't sure if she should be jealous someone else was in Hailey's life. Stupid as it sounded, Avery had never needed to vie for her daughter's love. And as much as her head told her to wave a caution flag that Hailey was attached, she just couldn't. Hailey deserved to be adored.\n\nCade swallowed, his eyebrows drawn in a silent question, asking her what was wrong. That right there got her every time. His intuition.\n\nShe offered a smile and broke the potent contact to take a bite of mashed potatoes. Now was not the time to dissect feelings. Not surrounded by his family. She'd analyze her thoughts later and figure out what to do, if anything.\n\nBy the time they made it back to the cabin, it was pushing Hailey's bedtime. She seemed tired anyway, so Avery skipped the usual sleepy time show and had Hailey brush her teeth. Her eyes were drooping before her head hit the pillow.\n\nAvery sat on the edge of the mattress and tucked the sheet up to Hailey's chin. Seraph bounded into the room with his three-legged hobble and sat at Avery's feet to be picked up. He was getting bigger everyday, but the little pup hadn't mastered jumping quite yet.\n\nShe bent to pick him up, scratched behind his ears, and settled him next to an out cold Hailey. He circled the mattress twice before curling up to her chest. Since she was asleep, Avery took a moment to run her fingers through Hailey's dark hair. She let Avery hold her hand and occasionally cuddle, but sometimes Avery just craved touch. She loved these stolen moments, held them close to her chest when Hailey's limitations went against Avery's needs.\n\nCade was waiting in the other room. She pressed a kiss to Hailey's forehead and made her way out, finding him in the kitchen, leaning against the counter and staring off into space. That introspective expression was on his face, but he smiled when he noticed her walk in. Affection lit his eyes, warming his smile as she drew closer.\n\nHe kissed her temple as she stepped into his arms. \"Aunt Marie called your cell while you were in with Hailey.\"\n\nShe glanced over his shoulder to the counter. \"Probably wants to discuss the St. Patrick's Day potluck. She's been hinting about new ideas since we finalized the Valentine's dance.\"\n\nHe laughed. \"Luring you to the dark side, one event at a time.\" His grin was in full charm mode. \"The O'Gradys are the biggest contributors to the potluck, being Irish on Dad's side and all. You haven't tasted anything until you try his beef stew recipe.\"\n\nSettling closer, she wrapped her arms around his waist and got a little thrill having his hard body aligned perfectly to hers. \"Your dad cooked?\"\n\nHe made a sound of agreement. \"Sometimes. When he did, he was damn good.\" His gaze swept her face. \"My mom likes you.\"\n\nShe pressed her face to his neck, enjoying their few minutes alone. She didn't realize how much she'd liked to be held, to snuggle, until Cade. \"The feeling's mutual. Even if she is one of the... What do you call them? The Battleaxes?\"\n\nHis chest shook in a laugh. \"Yes. She's tame compared to her sisters, but she can be just as conniving if she's invested. Aunt Rosa's the instigator and Aunt Marie's the enforcer.\"\n\n\"They seem harmless.\" Mostly. Avery had been the target of their focus a few times, but none of it had been malicious.\n\n\"They're not. They're capable of mass manipulation of epic proportions. Match making, swaying public opinion to their cause, butting into personal business.\" He wrapped his arms tighter, giving her a squeeze. \"You're right, though. They mean well.\"\n\nShe let out a contented sigh as her muscles relaxed. His linen scent mixed with husky male was becoming her new drug. They stayed like that for a few minutes as if satisfied by nothing more than holding each other.\n\nThe washing machine went into spin cycle, the whirring sound filling the room, and she lifted her head. \"You started laundry?\" How...domestic.\n\nHe brushed a strand of hair away from her face. \"Hailey's load. And speaking of, you're not wearing the aforementioned laundry outfit.\"\n\nGod, he could melt the panties off with just one heated look, one word in that rough timbre tone. \"I guess we should remedy that.\" She went to step away, but he grabbed her waist and lifted her onto the counter.\n\nHands on her hips, he moved between her thighs. \"I like you best in nothing at all.\" Leaning in, he brought his mouth to hers, kissing her lower lip and sliding his tongue across it for entry.\n\nWhen she opened, he stroked her in a sensual, languid exploration that had her pulse tripping and her breasts heavy. Too soon, he eased back a margin to look in her eyes, and what stared back at her made her heartbeat speed for an entirely different reason. Behind the need, the heat, was adoration. She recognized the expression from movies, books, or other couples she'd been near, but not once had she ever been the recipient.\n\nMaybe she was reading too much into the moment. It was just a look after all, but her skin flushed under his study and, as they stared at each other, sharing air, the need in his eyes moved out to make room for...\n\nNo. Now she knew she was crazy. After less than two months, Cade couldn't possibly love her. Would he even know romantic love if it parachuted down in front of him?\n\nHeck, would she?\n\nThough he never abused his charm, he had every single woman in the county eating out of his palm. They sashayed into the clinic, assaulted him in the bar, and came on to him just walking the street. By her observation, he didn't lead them on and was always up front with them. But the fact remained, he'd never fallen for any of them.\n\nPerhaps he felt more deeply about her, but this wasn't love. He was confusing affection and caring for the real deal. Had to be.\n\nHis jaw clenched in what she'd come to understand was a nervous or angry gesture as he cupped her cheek. But his touch was a caress, stroking her with his thumb, his gaze tender. Total surrender. That's what his expression relayed.\n\nTotal surrender.\n\n\"Avery...\" His low, hoarse voice washed over her. A plea. An admission.\n\nShe couldn't do this, didn't want to. No, he wasn't Richard. Yes, Cade was a good man. But she'd done this merry-go-round before and it had been disastrous. Love stole everything\u2014personality, will, reason. Independence. She'd fought too hard to fall again. She was growing to depend on his attention, his kiss, his kindness, his sweet words. Hailey was already attached.\n\nHe tensed against her, and she opened her eyes to look into his. Infused in all that amazing blue was distressed alarm. Desperation.\n\n\"Don't,\" he said through gritted teeth. He sucked in a breath, speaking again in a softer voice. \"Don't put the wall back up.\"\n\nBefore she could wrap her head around the situation, he reached behind his neck and pulled off his shirt, revealing his ripped muscles and smooth skin lightly dusted with hair. He fumbled in his pocket, then withdrew a condom and set it next to her. Keeping his gaze locked to hers, he unbuttoned his fly and shucked his pants, standing before her in all his naked, yummy glory.\n\nThe man was built. Beautiful. Already hard, he stroked his length once, twice, his gaze open and devastatingly revealing. Hunger and heart.\n\nMouth watering, she remained still, welcoming the throb between her legs and the ache which followed. This she could do, and he seemed to sense that, like he always did. He'd closed the door on emotion to keep her grounded, keep her with him.\n\nBut at what cost? In time, maybe panic wouldn't clutch her chest, and she could grow to accept another pass at something more. But not now. And she would only hurt him in the process if his heart was engaged.\n\n\"I said don't.\" He closed the distance and yanked her shirt over her head. Unfastening her pants, he encouraged her to lift her hips, then tossed those aside, too. He ripped the condom wrapper with his teeth and rolled it down his length.\n\nAll the while, he kept her gaze held captive, unable to look away.\n\nThen his arms were around her and he was lifting her from the counter. She gasped, wrapping her legs around his hips and grabbing his shoulders as he backed them to the table. Her spine pressed against the cool, smooth surface, and he followed her down. His length rocked against her folds, testing her readiness as his mouth latched on to her neck.\n\nThe sensation stimulus was blinding. Her nipples rasped his chest hair, the contrast a burn. His erection teased her clit, sending sparks through her body. His tongue and teeth on her neck lit her nerves, drew a violent tremble from deep within.\n\nOh God. They were going to have sex in the kitchen, right there on her kitchen table.\n\nHis finger trailed from her mound to her opening, and at discovering how wet she was, he groaned. Fisting his base, he aligned his shaft and eased inside.\n\nShe moaned as he stretched her walls, filled her with hard heat, and kept going until he was rooted and she could do no more than pant against his neck. Every time they came together it was old and new, familiar and fresh, comforting and energizing. A collision of contrast and need and sensation.\n\nHe laced his fingers with hers and brought them over her head, his forearms on the table caging her in. The new position nudged him deeper, had more skin to skin than ever before. He looked into her eyes, not only trapping her with his body, but his gaze.\n\nTrapping her to him. All of him.\n\n\"Do you see how good we are, sweetheart?\" His throat worked a swallow. \"Do you feel it?\"\n\nThat was the problem in a nutshell, wasn't it? It was too good. And things that were too good to be true usually were. It just made the crash more painful.\n\nDespite the warning in her head and the lump in her throat, she nodded.\n\nHe brought his mouth to hers and kissed her so thoroughly she couldn't remember what planet she was on and didn't care. She just wanted him, for however long it lasted.\n\nReading her need, he began to move inside her, shallow thrusts at first and then with deeper strokes. She dug her heels into his backside, urging more, her body straining for the oblivion only he could give her. He rolled his hips with each drive, his pelvis grinding over her clit and bringing her closer to shattering.\n\nHis fingers squeezed hers, silently telling her he was close, asking her to let go. He kissed her again, frantic, searching. He must've found what he was seeking because his groan vibrated her chest and his pace quickened.\n\nShe threw her head back. Arched. Already so close, she tumbled over. Wave after wave, the force hit. She quaked underneath him, her breath trapped in her chest and her head reeling.\n\nHe pressed his cheek to hers and rasped her name as he came. Rigid, he dropped his forehead to her brow, eyes pinched tight, his handsome face twisted in desire. He let out the breath he'd been holding, his hips still thrusting with shallow strokes as if seeking to prolong the orgasm.\n\nSlowly, he opened his eyes, released her fingers and wrapped his arms around her back, providing a cushion from the hard table. So many emotions traveled in and out of his eyes, passing too quickly for her to read.\n\nMouth open, brows drawn together, he seemed like he wanted to say something and changed his mind. He eased out and dropped the condom in the trash, coming right back to pick her up.\n\nIt stopped being awkward after the first couple times, and she still had to bite back the urge to inform him she could walk, but he seemed to enjoy carting her places. Not often. A boost out of the car, a lift from one room to the next. Like a Cade O'Grady version of sweeping her off her feet. She could do without the shyness that rose or the damsel-in-distress impression, but Cade obviously got enjoyment from it, so she rested her head on his chest and went with it.\n\nCarrying her down the hall, he walked into her bedroom and gently set her on the bed, climbing in beside her.\n\nIt dawned on her as she turned on her side to face him that he'd always made it a point to care for her after making love. Even that first time, once he'd eventually emerged from the bathroom, he'd found a way to pull himself together. Whether it was cuddling, skimming a reassuring hand over her back, or just lying next to her, he'd find something touching to do to ride out the aftereffects.\n\nIn all her life, from her mom to her ex to Hailey, she'd always been the one taking care of others. Part of that was her need for control and order. But mostly, she'd been playing the hand she was dealt and had done what was needed. The role reversal with Cade only served to add another element to the heady mix.\n\nHe took her hand in his and kissed her palm. \"Relax. I know rule numero uno is no sleepovers with Hailey in the house. I'll leave in a little while.\" His thumb stroked her inner wrist. \"I'm not taking off seconds after being with you, though. Get used to it.\"\n\nGod. He was killing her. As Brent would say...swoon, doll, swoon.\n\nCade brushed her nose with his. \"Not that I'm complaining, but what's that smile for?\"\n\nShe made a sleepy noise in her throat. \"Just thought of a Brent-ism for this exact moment.\"\n\n\"Brent-ism? Should I be concerned you're thinking of my tech right now?\"\n\n\"Nope.\" Letting out a sigh, her lids drooped as exhaustion claimed her. What was she going to do with him?\n\n# Chapter 23\n\nIt was barely lunchtime and this was quickly turning into the day from hell. In the clinic hallway, Cade swiped the device screen to pull up his next patient's chart, hoping to hell for a depressed hamster or routine vaccination.\n\nHe'd had to euthanize Mrs. Frederick's eighteen-year-old cat when she'd brought it in for weight loss and he'd discovered a pancreatic tumor. Shit way to start the day. Especially because Mrs. Frederick was pushing ninety and was recently widowed. She loved that damn cat more than her deceased husband. Not that Cade blamed her. Mr. Frederick had been a crotchety old fart who spent most of Cade's youth bitching they played ball too loud. Like it was Cade's problem the Frederick's lived across the street from the little league park.\n\nThen they had to put down a German short hair for age-related issues, and Christ if watching ten-year-old Andy Diedry cry buckets over his dog didn't nearly kill Cade. Brent was still misty over that one.\n\nBetween those appointments, he'd had to pry two blondes and one brunette off him when the three of them cornered him in an exam room. The two blondes were retrievers in heat and their brunette owner was...well, in heat, too, he guessed.\n\nTo add insult to injury, something was off about Avery today. She'd barely spoken three words to him all morning and, judging by her expression, she was teetering between livid female and basket case, depending on when he caught her amid patients. He hadn't had five seconds to himself to ask her what was wrong, but he hoped to hell he hadn't done something.\n\nFor the past couple weeks, ever since he realized he was batshit crazy in love with her, things had been pretty good between them. They'd settled into a routine of complete denial where he didn't tell her his feelings and she pretended she didn't know he had any. No L words had been exchanged, but the look in her eyes that night after dinner with his mom had been like a billboard pronouncing her holy-hell-in-a-handbasket reaction.\n\nHe'd been dodging proposals and willing females for years who hadn't taken the hint when he'd ended things for getting too serious. And Avery wasn't ready to hear him say the three words. Oh, the irony.\n\nWhatever. He'd do denial for as long as it took for her be comfortable with the idea. He was in no hurry. He'd continue watching Hailey on Friday nights for Avery's committee meetings, have dinner at her house a couple times a week as they'd been doing, and sneak out her door after making love.\n\nHell, though. Would it kill anyone if they knew about their relationship? Why couldn't he take her out to dinner? Just once. Or hold her hand in public and not have her pull away as if he had leprosy. And though he understood the no sleepover thing, it was gutting him how badly he wanted to wake up next to her.\n\nFocusing on his workload, he skimmed through his next patient's chart in Avery's recently installed EMR system. Thank Almighty. A routine appointment. Knocking on exam room two's door, he strode in to shake Mr. Weaver's hand.\n\nA three-year-old Westie immediately attacked Cade's shoelaces with reckless abandon. Cade ignored mini-Cujo and looked into Mr. Weaver's rheumy eyes. \"What can I do for you today, sir?\"\n\n\"Well, Snowball here has something wrong with him.\" Mr. Weaver shook his head. \"He rubs himself all over the floor and does this shaking thing like he's got the devil in him.\"\n\nWithout even looking at the dog, Cade knew the issue, but he picked up the Westie and set him on the table. He took out an otoscope and checked Snowball's ears. \"Any change in diet?\"\n\n\"No, I feed him that expensive crap you told me to.\"\n\nBiting back a grin, Cade felt the dog's abdomen. \"He's sleeping okay? Playing normally?\"\n\n\"Yes.\"\n\nCade listened to the lungs and heart. \"He's getting the heartworm medication monthly?\" He checked the dog's gums and teeth, cooing to the terrier breed before it could get snippy. As a pup, Snowball had taken a good chunk out of Brent's finger. In the dog's defense, Brent had been taking a rectal temperature at the time.\n\n\"Yes. He gets that tick stuff, too.\"\n\nCade found the telltale skin irritation on the dog's back and paws, but chose his words carefully. Mr. Weaver was a Vietnam vet who never married and was pretty set in his ways. Though he only made appointments when necessary, he was pretty needy when he came in. Cade figured it was loneliness.\n\nHe rubbed Snowball's ears and gave him a treat for being good. \"He's got some skin allergies, I'm afraid. It's fairly common, but treatable. We can start him off on an oral medication, and if that doesn't work, we can combine meds or try injections.\"\n\n\"No. That's not the problem. He shakes. It has to be seizures.\" Mr. Weaver crossed his arms, and Cade mentally checked himself in for the long haul.\n\n\"Dogs often roll on the floor to scratch an itch and when they use their hind legs, it can appear as if they're shaking. I can completely understand where you got the notion.\"\n\n\"Then it's his heart. There's something wrong.\"\n\nAfter twenty minutes, Cade couldn't spare any more time and not piss off the other patients waiting, so he reiterated the allergy diagnosis, handed Mr. Weaver the pills, and chatted with him about the weather to distract him as they strode out.\n\nThe rest of the day was more shit. By the time he'd seen the last appointment of the day, he was ready for a hot shower and to sink into a hotter woman.\n\nClosing his office door, he made his way to the lobby to see if Avery was up for Chinese tonight when he found her leaning against the front desk staring at her phone. That had to be the fifth time he'd caught her today with the color drained from her face.\n\nHeart pounding and worry eating away the lining of his gut, he dipped his head to put his face level with hers. \"What's going on? Is Hailey all right?\"\n\nShe glanced up with wide eyes as if not realizing he was there. The hand holding her phone shook as she pocketed the cell. \"She's fine.\"\n\nHe opened his mouth to bombard her with more questions because something wasn't right, but Brent chose that moment to make an appearance.\n\n\"Hey, doll.\" His tech glanced between the two of them, brows steadily moving north. \"Ruh roh. Trouble in paradise?\"\n\nAvery quickly shook her head and straightened. \"Everything's fine. Something's come up, though. Would you mind closing shop for me?\"\n\nScrew all. That was the straw that broke the veterinarian's back. Avery didn't ask for help often, and she took her responsibilities very seriously. Not that he gave a rat's ass she'd asked Brent to close the clinic. Why she had asked bothered him. He took a step forward, but Brent was there again.\n\n\"You got it, doll. See you tomorrow.\"\n\nShe surged forward and kissed his cheek. \"Thank you.\"\n\n\"No problemo. Get your cute rear outta here.\" To put a period to his point, Brent swatted her \"cute rear\" as she passed him.\n\n_Squawk._ \"I like big butts.\"\n\nShe pointed a warning finger at the cockatoo and disappeared down the hall to retrieve her things.\n\nCade shifted his attention to Brent. \"Did you just smack my woman's ass?\"\n\nHe didn't know whether to be pissed or amused, but he was crawling out of his skin to find out Avery's issues and needed a distraction. He wouldn't get anything out of her until she was at home and had time to process. He was in for at least a couple more hours of worry.\n\nFrustrating woman. Frustrating, beautiful woman.\n\nBrent cocked a hip. \"Aw. Jealous much?\"\n\n\"I'm not jealous. Your door doesn't swing her way.\" Otherwise Cade might introduce Brent to his fist.\n\n\"I meant that I'm smacking her ass and not yours.\" Brent tsked and covered Gossip's cage.\n\n_Squawk._ \"Enter Sandman.\"\n\n\"That's right. Night, night.\" Brent turned to face Cade, picking up She-rah from the printer. \"What's up with our girl, anyway? She's been pouty all day.\" He stroked the cat's back like an evil lord, one pinky up in the air.\n\nCade didn't think Avery was pouty so much as...rattled. He had no clue what or who could do that. \"Hell if I know. I'll head over to her place later and find out.\"\n\nAvery rushed in from the back, waved to them, and used her hip to open the door. \"Night, guys. Thanks again, Brent.\"\n\nCade fisted his hands on the counter. \"I'll be over in an hour.\"\n\nHer gaze whipped to his, wide and alarmed. She ran a shaking hand over her forehead and looked down. \"Not tonight, okay? I'll call you later.\"\n\nThen she was gone, leaving Cade to stand there like an idiot. A confused, pissed off, scared to death idiot.\n\n\"Ouch. Dismissed. What did you do?\"\n\nCade raked a hand through his hair, staring after her. The urge to follow her and demand she tell him what the fuck was going on was so fierce his head pounded. Nauseous, he closed his eyes for a beat.\n\nHe could either respect her wishes and pace his house in worry tonight, or he could ignore her request and try to offer the comfort she needed. Despite her walls and roadblocks, she did need him. Not in a clingy, suffocating way, and not because she was weak. When they were alone, her eyes told him she was falling for him right back. So, yeah. She needed him.\n\nTogether, they had trust and respect and mutual admiration. They talked and laughed. And had really, really hot sex.\n\nTogether, they were stronger.\n\nHell. Decision made.\n\n* * * *\n\nAvery had forced herself to sit at the kitchen table after work and eat the stuffed peppers she'd put in the Crockpot this morning. It exhausted what little energy she had left. Time with Hailey was precious. Between Hailey's school and Avery's work schedule, she only had a couple hours at the end of the day to spend with her little girl.\n\nBut she hadn't been able to eat much, and instead of chattering to Hailey about her day, Avery kept replaying the events in her head. Eventually, she'd directed Hailey to the couch to play on her iPad so Avery could do the dishes.\n\nWhat she needed was a monster glass of wine and to forget today ever happened. A pint of Ben & Jerry's would have to suffice. And the day wasn't over yet.\n\nThe morning had started with the woman at the coffee counter badgering Avery about her relationship with Cade. It seemed innocent enough, and the woman probably meant no harm, but she wouldn't take the hint. Her parting shot had been telling Avery to check the town's Pinterest boards.\n\nNot liking the sound of that, she'd headed for the safety of her car to scroll the site on her phone. There had been an entire board created just for her and Cade. Pictures from the Valentine's dance, a couple from the ice cream shop, and the kicker\u2014a photo of Cade kissing her in the clinic's hallway. She had no clue when it was taken or by whom, but she knew it was not someone from Animal Instincts. They'd never do that. Yet it was a blatant disregard for her privacy and had her stomach cramping.\n\nThen she'd stumbled upon two clients outside the bathroom discussing none other than her. Avery flinched as she washed a plate and set it in the dish rack, remembering the hushed conversation between the two young women.\n\n\"She's not even that pretty. What's he doing with her?\"\n\n\"He'll get bored and move on to greener pastures in under a month.\"\n\nThe cattiness had stung and played on her insecurities. Though the barbs had hurt, she tried not to let them bother her. She was secure in the fact that Cade wanted her. Only her. She saw it in his eyes, felt it in his touch. For now, anyway.\n\nPlus, there were new pictures of Cade with different women on the Pinterest boards. Some as recent as two days ago. They were obviously non-sexual in nature and mostly selfies on the fly, but it made her question, all over again, if she could trust what was between them. How long could his interest last when he was used to...choices?\n\nYet it was just another thing to screw with her head today.\n\nTurning off the sink, she dried her hands and checked the clock again. Twenty minutes and she could put Hailey to bed. Normally, Avery cherished every minute with Hailey. Loneliness seeped through the fissures once her little girl was asleep, and in those quiet hours, Avery found herself hoping.\n\nFor someone. For normal. For an end to having to do it all alone.\n\nDangerous thinking, that. And yet something told her she had, in fact, found what she'd secretly been wanting, even if she couldn't admit it aloud. A heart's desire. Cade was coming hazardously close to proving he was The One, and she feared, more than anything, that it was all an illusion.\n\nShe blew out a breath and shook her head.\n\nYes. Bedtime was dangerous to her heart. Too much time to think. But tonight was different. The queasiness turning her stomach and abject uncertainty squeezing her chest wouldn't abate until she could be alone.\n\nBecause she needed to find out why Richard had called her today. Three times. He'd left a general message the first instance, claiming he needed to speak with her, but he hadn't bothered with voicemail after that.\n\nThousands of scenarios shoved around in her head. Did her ex suddenly want visitation rights? Did he want to fight custody? Had she left something when they'd moved? Was there an issue with the trust fund?\n\nThe acid in her stomach burned a path to her chest. She was going mad with anxiety. Fisting her shaking hands, she took a calming breath.\n\nShe sat next to Hailey on the couch and was glad her sleepy time show was almost through. \"I'm sorry I wasn't very with it today, sweetie. Mommy just had a bad day. But that's not your fault. You didn't do anything wrong.\"\n\nWhen the program was over, Avery shut off the TV and was about to direct Hailey to the bathroom when a knock sounded on the front door. Cade strode in without waiting for an answer.\n\nHer first reaction was how glad she was to see him. Her pulse fluttered and her belly quivered. Having no idea when she'd grown to rely on him, on looking forward to their time together, relief deflated her chest that he was here. Nothing sounded better than curling up on the couch with him, leaning on him for the support he was so willing to give. When had her happiness become so dependent on someone else?\n\nBut she'd asked him not to come tonight and he ignored that request. He didn't typically disregard her feelings, and on the day she really needed him to listen, he hadn't.\n\nHis gaze seemed tentative as he looked at her. He hadn't moved from just inside the doorway. Shoving his hands in his pockets, he implored her to say something with a pleading intensity to his eyes.\n\n\"Hailey, go brush your teeth. I'll be right there.\" Avery waited until the water ran in the faucet before addressing Cade. \"I asked you not to come tonight.\"\n\nNot one muscle moved. \"I know. I...\"\n\n\"What?\"\n\nHe sighed, his shoulders sagging. \"I was worried. You've been off all day and\u2014\"\n\nHer cell rang, reminding her why she was so \"off\" today, as he put it. She pulled it from her pocket and tensed at Richard's name on the screen. Sending the call to voicemail, she pressed a hand to her stomach to keep the contents inside and closed her eyes.\n\n\"Do you need to get that? I can put Hailey to bed.\"\n\nIrritation\u2014for Richard, for the bitchy clients talking about her in the hallway, for Pinterest\u2014surged in her chest until she nearly exploded. Her head pounding, her breathing short, she pinned Cade with a glare. \"She's my daughter. I got this.\"\n\nHe flinched as if she'd slapped him. His lips parted while he stepped forward. \"Avery.\"\n\nDamn. Regret added to the churn in her stomach. Her mood wasn't his fault. \"I'm sorry. I didn't mean that.\" She sighed. \"You should go. I'll talk to you tomorrow.\"\n\n\"Like hell. Talk to me now.\"\n\nShe was nearing the end of her rope. \"Damn it, Cade. I can't tonight.\"\n\nUndeterred, he jerked his chin toward the hallway. \"Do what you need to. I'll wait.\"\n\nShe was too darn exhausted to argue. Deciding this battle wasn't worth winning, she nodded and stepped out of the room.\n\nBut by the time she got Hailey settled in bed, Avery was shaking with fury. She was directing it at the wrong person. The events of her day weren't Cade's responsibility, but she didn't want him around for this. To see her cracked open and...exposed, two minutes from falling apart.\n\nHe hadn't moved from the spot she'd left him, hands in his pockets and standing in the middle of the living room. His shoulders were tense beneath his coat, his jaw working a grind.\n\nHis anger was masking concern, and something about him losing his cool was sexy beyond measure. Fierce blue eyes met hers, and all she could think about was closing the distance and letting him fix the mess. Kiss away the uncertainty and fear.\n\nFor the first time in her life, someone wanted to hold _her_ up. It was so damn tempting. Against her ingrained nature and experiences, yet tempting nonetheless.\n\nBut Richard needed a call back, and she'd never settle down until she found out what he wanted.\n\nShe pulled out her phone. \"I have to make a call. Wait if you want.\"\n\nHe made a motion like he was unhinging his jaw. \"I want.\" In emphasis, he shrugged out of his coat and tossed it on the couch.\n\nWith more bravado than she felt, she walked through the kitchen to the back door and stepped outside. Letting the cool, humid air wash over her heated skin, she squared her shoulders and connected the call.\n\n# Chapter 24\n\nCade swallowed past the boulder in his throat and rubbed a hand over his stomach. Acid was slowly burning a hole through his gut as he peeked around the living room corner a third time.\n\nAvery was still standing just outside her back door, illuminated in moonlight and motionless. With the phone pressed to her ear, she occasionally nodded or responded to the other person, seemingly as calm as the shallow riverbed snaking through the yard. But her body language screamed with tension. From the wrinkle on her forehead to the stiff way she held her shoulders, she was wrung tight.\n\nHe ran a hand over his neck and resisted the urge to storm through the door and find out who was making his sweet Avery a wreck. And why. All day, she'd been killing him. Her mood, the distance, was so unlike her, and it was still assembling. She'd always been reserved, a little aloof, but she'd never been abrasive. Hell, she'd actually snapped at him when he'd tried to help earlier.\n\nSomething was wrong, and the walls he'd meticulously torn down were rebuilding around her as he watched, helpless to do a damn thing. The urge to howl rose in his chest. He paced instead, wearing her hardwood floors to dust.\n\nFifteen grueling minutes later, the door latch clicked as she stepped through. Face pale, hands shaking, she walked into the living room, her gaze dazedly staring at her cell phone. She stopped near the couch and looked around as if unaware of her surroundings.\n\nShit, shit, shit. More shit. \"Avery, sweetheart. What is it? Talk to me.\" At this point, he wouldn't put it against him to shake an answer out of her.\n\nShe dropped to the couch and rubbed her forehead. \"My...\" She cleared her throat. \"My ex is getting remarried.\"\n\nHe stilled. The hurricane force of air he'd been holding demanded an escape from his lungs. Was he the biggest asshole in existence to be relieved? This wasn't exactly bad news, at least not to him, but the shell-shock in Avery's eyes wouldn't allow his pulse to settle.\n\nNot sure what to do, to say, he walked to the coffee table in front of her and perched at the edge. She wouldn't look him in the eye. An entirely different kind of anxiety raked his chest.\n\nShe shook her head and laughed without mirth. \"I was worried he was going to try to take Hailey away from me, and he called to say he's headed down the aisle.\" She looked at him with a bite in her eyes. \"He's not even marrying the assistant he cheated on me with. It's a socialite friend of the family.\" She rose so fast he got whiplash. \"He didn't ask about his daughter once. Never gave any indication he wanted her there for the nuptials. Can you believe him?\"\n\nCade stood and faced her. She had every right to be angry, and he understood her behavior now, but the fact she didn't lean on him, or attempt to tell him what had been going on, had his own fury battling for dominance. They'd been together long enough he should've been the one she went to when scared.\n\nShe still didn't get she wasn't alone anymore.\n\nMaybe he was itching for a fight, or perhaps he'd just reached the end of his patience, but his brain disconnected from his mouth. \"I'm more interested in why you didn't tell me he'd called.\"\n\nShe whirled to face him, brows arched. \"Excuse me?\"\n\n\"I was with you all day. Hell, I've been with you all along, trying my damnedest to get you to let me in.\" He paused. Did he just...Yes. He did.\n\nThe other side of the fence wasn't looking too great. Always careful, he'd never led his previous lovers on and was out the door the minute things even tilted in this particular direction. And though he'd never promised them anything, had been brutally honest, he was getting a glimpse of how they must've felt.\n\nIt sucked.\n\nRegardless, he slammed the door on his past and every other woman who'd tried to do this exact thing to him. Because Avery was it for him. He'd met his match. And she couldn't even open her mouth to tell him something was wrong.\n\nTo make this shitstorm complete, he had the worst sensation that it didn't matter how much time he gave her. She would never meet him halfway.\n\nHe closed his eyes and sucked in some much needed oxygen. The denial of not admitting how deep he'd sunk threatened to disintegrate. Words he'd never said to another woman, never wanted to. This was unchartered territory for him, and he didn't know what the hell he was doing. For once, he was pretty damn sure he wasn't the one fucking this up.\n\nSlowly, he opened his eyes to meet her unapologetic chocolate gaze. Eyes he'd gladly get lost in, and had. She had no idea how much she said with one look. If he were locked in a room with two terrorists, her ex, and a gun, and only had two bullets, he'd shoot her ex twice. He loved the guy that much. And she was still letting him control her life.\n\n\"Do you still love him?\"\n\n\"What kind of question is that?\" She crossed her arms over her chest with defiance, but the look in her eyes told him the bravado was false. She was just as scared.\n\n\"An honest one.\" He stepped closer until he was a heartbeat away and could breathe in her berry scent, which did nothing to calm him. \"Answer me.\"\n\nThe edges of anger in her eyes gradually softened to understanding. She clenched her fingers as if restraining herself from reaching out. \"No. I don't still love him.\"\n\nOkay. Hell. That was good. \"Then why couldn't you lean on me? Have I ever given you the impression I wouldn't understand?\"\n\nInstead of answering, she pulled out her phone, swiped the screen, and held it out to show him. \"I don't love my ex, but do you still love this lifestyle? Because these pictures indicate you do.\"\n\nPinterest. He fucking hated the site. And most of those pictures were taken out of context. A woman in Shooters who'd climbed in his lap. Etcetera. He ground his teeth. \"I\u2014\"\n\nSomething tugged at his shirt. He glanced down to find Hailey out of bed, holding out her iPad for him. \"Just a second, squirt.\" He looked back at Avery, but her focus was on her daughter, and Cade had lost his train of thought. A common occurrence these days.\n\nHailey tugged again.\n\nHe pinched the bridge of his nose. \"I said just a minute, Hailey.\"\n\nAvery gasped.\n\nAt his own harsh tone, he winced. Dropping to his knees in front of the girl, he barely resisted the urge to pull her in for a hug for being an asshole. \"I'm so sorry, squirt. I shouldn't have raised my voice. What do you need?\"\n\n\"I think you should leave.\"\n\nHis gaze cut to Avery's. Tension knotted his shoulders. Fear clawed his stomach. \"Don't do this.\" He rose.\n\nShe let out a long-winded sigh and closed her eyes. When she opened them again, his Avery was back. A tired, beaten version of her, but Avery just the same. \"It's been a really long, really bad day. Too much for me to take in. Please understand. The fighting is upsetting her. Just go, Cade.\"\n\n\"That's it?\" Just go. As if it were easy as that?\n\n\"No.\" She stepped closer and cupped his jaw, her touch tender. \"I just need some time to acclimate.\"\n\nHe understood she was used to doing things all by herself, and changing that pattern would take time, but he wanted to be around when she...acclimated. Stepping back, he dropped his hands on his hips, fighting the need to roar. Again. Head pounding, the pressure in his chest grew to unbearable.\n\nHe stared at her for he didn't know how long, attempting to get past her armor and remembering how to breathe. It was no use. She was trying, but she'd shut herself off. Not that she'd ever really opened in the first place. Not completely.\n\nHe should walk away. Cut his losses and move on. She obviously didn't give a good goddamn about the potential between them. At the very least, she wasn't as invested as him. Except he couldn't walk. She was in his head and heart and blood. In his every thought and action.\n\nAgony. Bliss.\n\nAnd no one had ever shown her she mattered, had put her first. He sure wasn't going to cement that proof in her mind by doing the same. Because...shit. He loved her even when she was trying not to do the same.\n\nAvery broke the epic stare down when a chill raced through the room and she shivered. She glanced toward the kitchen and froze. \"Oh God. _No_.\"\n\nIce crystallized in his gut as he followed her to the open back door. The room was chilly enough to indicate it had been open at least a few minutes.\n\nShe ran outside and screamed for Hailey, eyes wide, wind whipping her chestnut strands. \"I got distracted by Richard's call. I didn't lock the door after I came in.\" She stumbled down the back deck stairs. \"Hailey?\"\n\nThe hair on his nape stood erect. Abject terror, unlike anything he'd ever known, shoved through him, stealing the beat from his heart. He ran inside and down the hall, but the girl wasn't in her room. Seraph was gone, too. Pulling his cell from his pocket, he dialed Drake.\n\n\"It's almost ten o'clock\u2014\"\n\nCade grabbed his coat and walked back outside as he spoke. \"Hailey's missing. Get Flynn and meet me at Avery's. First rental cabin in the row on Justine's property. I'm going out to look for her. Have Flynn wait with Avery.\"\n\n\"On it.\"\n\nHe disconnected and gripped Avery's shoulders before she could take off alone into the night. \"Call 911 and wait here. My brothers are on their way.\"\n\n\"No, I need to go\u2014\"\n\n\"Stay here in case she comes back.\" He gave her a little shake to ensure she'd heard him.\n\nHer watery, panicked gaze met his. Her breath hitched. \"Cade?\"\n\n\"I'll find her. I promise.\" He didn't give her time to argue. He took off for the cluster of trees and didn't stop. Hailey couldn't be that far ahead of him, but he had no idea which way she'd gone.\n\nEngaging the flashlight app on his phone, he shone the beam on the ground as he ran. Most of the snow had melted, but after a few yards, he spotted footprints in the mud heading east. Shit. The closer Hailey got to the mountain base the more danger of encountering a wild animal.\n\nHis phone rang and he connected before looking at the screen.\n\nDrake's voice was short. \"Which direction did you take?\"\n\n\"East. I found footprints. Is Flynn with Avery?\" She needed someone with her, and Flynn wouldn't be able to hear anything if he was involved with the search.\n\n\"Yes. The sheriff is one minute out. I'm two minutes from you.\"\n\n\"Break northeast. I'm heading southeast.\" He went to disconnect, but thought of something. \"Call for the dog. Hailey won't answer, but Seraph might bark.\"\n\n\"Got it.\"\n\nThe foliage crunched under his boots as he wove through braches and over exposed roots. Every few seconds, he whistled for Seraph and called Hailey's name, stopping only long enough to listen for sounds. The wind was humid, but biting, and his fear notched to cataclysmic when he realized Hailey probably didn't have her coat.\n\nFootsteps pounded to his left, but they were too heavy to be Hailey's. Drake's voice rang through the still night, yards from his location. \"Hailey? Seraph? Come on, boy. Where are you?\"\n\nCade glanced down, but he couldn't make out her tracks with the dried leaves. Tears stung his eyes and clogged his throat. \"Hailey! Seraph!\"\n\nThis was all his goddamn fault. If he hadn't started the fight with Avery, if he'd just kept his mouth shut, he wouldn't have scared the girl and she wouldn't have bolted. He swore to all that was holy, if something happened to her, he didn't think he'd survive it. Would never forgive himself. Just like Avery, Hailey had woven herself around his heart. So embedded, they were an extension of him.\n\n\"Hailey! Seraph!\" He tried to whistle, but his jaw trembled too much to form the motion.\n\nHelpless, he stopped and tugged at his hair in frustration.\n\nA whine pierced the air, so faint he thought it might be himself making the noise out of desperation. He froze, eyes scanning, breath clouding in front of his face. It was too damn dark.\n\nThere. It came again. Due east.\n\nHe ran in that direction. \"Hailey! Seraph!\" He broke through a cluster of trees and came to a dead stop near a hollowed log.\n\nSeraph sat on his hind legs, tail wagging. He barked, and Cade could've sworn the damn puppy smiled.\n\nHailey was next to him on the ground, rocking in place, staring heavenward.\n\nRelief sagged his shoulders as he knelt in front of her, gaze scanning for injuries. His heart nearly cracked ribs with the force of each beat.\n\n\"Hey, squirt. Are you hurt?\" His voice was unsteady, but he did his best to keep it calm, not illustrating the sheer panic when he couldn't find her. He'd freaked her out enough for one night.\n\nHailey, of course, didn't respond, didn't so much as look at him, but she was rocking, indicating she was unnerved. Her face was pale, aside from her bright pink cheeks. Her dark hair was windblown and knotted, but otherwise she seemed all right.\n\nAlive. Thank Christ in heaven.\n\n\"I'm going to touch you just to be sure you have no ouchies, yeah?\" He didn't give her time to react. He ran his shaking hands over her head, down her arms, across her torso, and finished with her legs, carefully watching her face. She didn't flinch or show distress.\n\nShrugging out of his coat, he wrapped Hailey inside and dialed Drake. \"I've got her. She's okay.\" His voice broke. He bit his tongue and cleared his throat. \"Head back and tell Avery. I'm on my way.\"\n\nDisconnecting, he slammed his eyes closed as hot tears leaked down his cheeks. He attempted to draw in a steady breath, but damn it... His chest cracked open, spilling the fear and guilt. His shoulders shook as he swiped a hand down his face, hunched over, and gave himself a minute to collect his shit.\n\nHailey might not be his own, but this glimpse into fatherhood, into what it was like to love someone more than anything else, was gutting. The determination to protect her surged deep, a call unlike anything he'd known. Hell, he'd do anything not to go through something like this again.\n\nNo. She wasn't his daughter. Not by blood. But somewhere in the space of a couple months, she'd become his in every way that mattered.\n\nHe shook his head, noticing her tiny pink socks were wet and soiled. He plopped on his ass to remove his shoes. \"I'm so sorry I scared you, squirt. We weren't fighting about you. Sometimes... Sometimes adults fight. But everything's all right now.\"\n\nRemoving her socks, he put them in his pockets and worked his own on over her little feet. The sight of her swimming in his coat and wearing his socks brought a fresh wave of tears. Shoving his bare feet back into his shoes, he stood.\n\n\"Please, _please,_ don't ever scare me like this again.\" He picked up Seraph and set him in her lap, then lifted both the dog and the girl into his arms. He stared down at her a moment, caught between affection and apprehension. \"I love you, squirt. Just so you know.\" He sighed. \"Let's get you home.\"\n\nShe didn't fight him, and most of her shivering had stopped by the time he reached the clearing near the cabin.\n\nRed and blue lights from the police cruiser were a beacon in the black inkiness. Drake and Flynn stood with Avery on the back deck, the sheriff off to the side talking to Justine and Gabby. They all looked up when he approached.\n\nAvery wailed and ran toward him, launching into his arms. Fresh tears trekked down her pale cheeks as she wrapped her arms around his neck, squeezing Hailey between them. She took the girl from him, leaving his arms unbearably empty, and fell to her knees.\n\nSeraph loped off toward Gabby.\n\nCade glanced away and eyed his brothers, the relief evident in their eyes. If only Avery could see just how much these people, his family, loved her, too.\n\nThe energy drained out of him in a blink.\n\nGlancing back at her, he breathed for what felt like the first time in hours and cleared his throat. \"She's okay, sweetheart.\"\n\nHe wasn't okay, but at least Hailey was home safe. Everything else was a clusterfuck, but all's well and all that.\n\nAvery nodded repeatedly, running her hands over Hailey to check for injuries. Hailey was unusually complacent with the touch.\n\nNeeding a finger of whiskey and a hot shower, perhaps some bleach to scrub this night from his memory, he nodded to his brothers and walked around the cabin to his SUV. He was seconds from the worst adrenaline crash of his life. Hell if he was sticking around here for that.\n\nHe didn't answer the knock on his door at home an hour later when his brothers came calling. And he didn't respond to Avery's text, which only read: _Thank you._ He'd never made it to the liquor cabinet to pour whiskey, nor had he showered. Instead, he sat in his dark living room for hours, with Freeman lying across his feet and Cutin perched on his shoulder, while he tried to embrace the numbness.\n\nHe found himself missing his dad with a burning agony as if the loss was a fresh wound, and wishing the old man were here to tell him what to do. About Avery. About Hailey. About his feelings and future and how to handle this overwhelming need to take care of them both. Because Cade had no fucking clue.\n\nAll he did know was that despite all the crap that could go wrong, all the ways he could lose them, taking the risk was worth it. No matter the outcome.\n\nHe just had to prove that to Avery.\n\n# Chapter 25\n\nFrom the back deck steps, Avery smiled at Brent's attempts to play with Hailey by the riverbank. They were sitting on a log, Brent making girly sounds on behalf of the doll in his hand, and Hailey not taking the bait. She did laugh at Brent's antics a time or two, though. He'd come by after his shift at the clinic with the excuse to share his niece's Barbies with Hailey, but Avery suspected it was really a ploy to check up on her.\n\nWhich was fine. She'd needed the distraction from her heart-stopping scare. After the medics had checked out Hailey and Avery had gotten her settled in bed last night, she'd walked down the hall to find only Drake had remained behind. He'd insisted she take the rest of the week off, sat with her for an hour in quiet reassurance, and then left without another word.\n\nShe sighed. Strong, silent Drake. Flynn had been a godsend while the men were out searching. He'd paced with her, rubbed her back, paced some more. Both were so unlike their brother Cade it was hard to imagine they were related. They all had different degrees of humor and protectiveness, but Cade just...well, was different.\n\nCade hadn't responded to her text. She'd tried to call him today during his lunch hour, but it had gone right to voicemail. Their argument sat like lead in her belly. The fear in his eyes when they'd discovered Hailey gone tore at her heart. He loved her daughter. And he'd found her when Avery had been frozen in shock, too numb to move.\n\nFor years, it had just been her and Hailey. Even when Richard had still been in the picture, it was just them against the world. Cade had broken through the barrier she'd created to protect them both. He hadn't been afraid to let them in, to love her daughter, and she'd pushed him away with both hands.\n\nBefore Hailey had gone missing, Avery had intended to hash out their issues later, in the light of day and with a fresh mind. She was too raw from Richard's call, too overwhelmed at the time to think straight. When Cade had gotten short with Hailey, something inside her snapped.\n\nHad she been thinking rationally, she would have recognized Cade's apology immediately after he'd realized what he'd done. And he hadn't said sorry to Avery. He'd addressed Hailey himself. Something Richard would never have done.\n\nHeck, he was getting married and never thought to include his daughter.\n\nShe blinked and brought her surroundings into focus. The late afternoon sun was dipping west behind the tree line, creating long shadows and a chill in the air. The pink and purple hues of sunset reflected off the river behind Brent and Hailey. An owl hooted from that direction, and she loved the inquisitive noise. It beat the woodpecker driving her nuts in the mornings.\n\nDespite her initial reservations, and Camp Crystal Lake aside, it really was quite beautiful out here. It was getting to be peak season for tourism, though. Her mom would need the cabin back soon. Avery needed to amp up her apartment search. She'd sure miss this view.\n\nAs if reading her thoughts, her mom moved from a deck chair to sit beside Avery on the stairs. Her scent of patchouli rose over the pine and rain-dampened grass. She bumped Avery's shoulder with her own. \"I'm so glad you both are making friends. I knew transferring here would be good for you.\"\n\nAvery watched Brent and Hailey, unable to argue that point. Hailey was forming attachments in her own way, and was having fewer tantrums. In fact, Avery could count on one hand the number of fits since they'd moved to Redwood Ridge. Hailey also seemed more content, more...happy.\n\nShe sighed. \"Cade loves her an awful lot. You only have to look in his eyes to see it.\" Or note his actions when he thought no one was watching. That was the true test, the greatest illustration of his character. Heck, even his brothers were good with her.\n\n\"She's pretty hard not to love. You did a great job with her, honey.\" She paused a beat. \"I'm pretty sure Cade loves you as well.\"\n\nAvery closed her eyes. She knew that, too. \"I wasn't looking for love, Mom.\" She didn't want it, at least not initially. She'd barely survived the first time, and she hadn't loved Richard half as much or in the same way as she'd grown to love Cade.\n\n\"Well, take it from someone who knows. Whether you look for it or not, love finds you.\"\n\nSince Mom had been married four times before, Avery figured her mother was an expert in all things romance. Or was really, really bad at it. \"I don't know how you do it. How do you leave yourself vulnerable time and time again?\"\n\nHer mother looked out over the expanse of land and breathed deep. \"There's no greater feeling. The flutters when you first kiss, the putter of your heart when you know it's real. I live for that.\"\n\nAvery tried to remember a time when she'd felt those things with Richard, and came up blank. Being with Cade, on the other hand, was like riding a rollercoaster set for derailment.\n\n\"I've been chasing that feeling for almost twenty-eight years.\" Mom looked at her, and missing from her gaze was her typical airiness. \"I've loved other men since your father, but not as hard. So, I keep chasing it, hoping I'll find it again.\"\n\nThey never talked about her dad. Growing up, Avery had rarely asked and her mom wasn't forthcoming. Most of the time, she figured her mom didn't know who Avery's real father had been. \"What was he like?\"\n\nHer mother's smile grew wistful and distant. \"Like you, actually. You got my physical characteristics, but you inherited a lot of his personality.\" She leaned back on her elbows. \"He was...cautious. He never met a problem he couldn't solve or an emotion he couldn't master. Until I swept into his life. We had one wild, wonderful summer, and you were the result.\"\n\nPain filled her eyes, wrinkling her brow. \"I regretted not telling him about you, but he was destined for great things, and I didn't want to weigh him down. Then, this one day, you must've been three or four years old, I burned your grilled cheese sandwich. You put your little hands on your little hips and looked over your nose at me. I swear, you looked just like him.\" She sat forward, dropping her gaze. \"I decided to get in touch with him, only to find out he'd died in a boating accident the spring before.\"\n\nTears threatened, not for her own grief, she'd obviously never known her dad, but for her mother who clearly loved him even after all these years. Avery swallowed the tears and took her mother's hand. \"I'm so sorry, Mom.\"\n\n\"That's okay, honey.\" She kissed Avery's temple. \"I get a piece of him every day in you.\" She sighed, her breath fogging in the chilly air before her face.\n\nSuddenly, her mother looked her age. Years of life and laugh lines fanned around her eyes and mouth, punctuating the no-holds-barred way she lived. Not present on her face was regret, which made Avery wonder if, in twenty years, she'd have the same to say for herself.\n\nMom patted her hand. \"I don't want you to end up like me, loving too many people in a vain search for the one. But I don't want you to wind up like him either, so guarded about opening your heart that you miss love when it comes around.\"\n\nAvery inhaled slowly, filling her lungs. She had been doing that very thing her whole life, and she had to wonder now if it was genetic or taught. From day one, she'd been comparing Richard and Cade, weighing the similarities and differences. Not to measure them, but because the person she'd been with Richard was a vast cry from the person Cade brought out in her.\n\n\"You never loved Richard. I think you know that now since real love has entered your life.\" Mom grinned. \"He wears blue scrubs and a stethoscope, in case you were wondering who I meant.\"\n\nAvery laughed and dropped her head to her mom's shoulder. Maybe she should've taken more cues from her growing up. Sure, Mom forgot to charge her cell phone most days and she couldn't make a lasagna to save the homeless, but she was wiser than Avery had given her credit. \"Love you.\"\n\n\"Love you more.\"\n\n* * * *\n\nBy the time Friday rolled around, Avery was sick to death of her own company. She appreciated Drake giving her the time off, but she was crawling out of her skin. Which was interesting because before moving to Redwood Ridge, all she'd had was silence and Hailey. No friends, no job, no life.\n\nCade had texted this morning before his shift, letting her know he'd be over tonight to watch Hailey during her meeting. That's all the contact she'd had with him. In five days. One text. She was doing her best to give him space and let him think things through, but how was he supposed to do that without all the facts? The way they'd left things seemed like...an end.\n\nShe forced herself not to pick up the phone and call him, or snatch her keys to drive to the clinic. He said he'd be over tonight. She'd talk to him then.\n\nWhat other choice was there? Despite her protective stance and determination not to shift into her old pattern, she'd gone and fallen in love with Cade. Fallen hard for his humor and heart and heat. And because he was who he was, she knew she wouldn't regret it, even if they were over.\n\nUntil now, she hadn't been close to love. Not the real kind that hollowed her core when they weren't together, or made every stray thought drift to him, or made her laugh and cry and worry and hope.\n\nHer mother was right. She hadn't loved Richard. He had been a safety net. Settlement. The easy route she'd been too weak not to take.\n\nThere was no settling for Cade and no safety. With him, she didn't need those things. Because true security came from not being able to live without him and knowing, without any doubt, he felt the same. He'd put her first and do anything to make her happy. Without conditions and obligations. He'd been selflessly putting her back together from the beginning, and he hadn't been the one to break her in the first place. Men like Cade loved because they wanted to, because they had no choice. He was capable of the unguarded emotion in ways she'd never encountered or fully allowed of herself.\n\nHe'd catch her. Every time.\n\nIt made her sick to her stomach that he thought she didn't feel the same. In the past, she hadn't loved with bone-deep intensity, but she was smart enough to recognize it when it slammed into her chest.\n\nBlowing out a breath, she donned her coat and picked up Hailey from school. Back at the cabin, she settled Hailey at the table with her device and kept herself busy by making chili, lest she go insane waiting for Cade to arrive.\n\nHe walked in the front door just as Avery set two bowls on the table with a basket of rolls. Without glancing up, he stripped off his coat and hung it neatly on the peg. His boots were next, which he set by the door. He stepped into the kitchen and all the air evaporated.\n\nStill wearing his scrubs, the muscles of his arms and chest filled the dark blue material. Her heart fluttered and desire pooled in her belly. Her mom had been right about that, too. Any lingering doubt about loving him was erased. No wonder Mom chased this feeling. There was nothing like it.\n\nTaking a seat next to Hailey, Cade peeked over her shoulder at the device. \"What's up, squirt? Can I play?\"\n\nHailey squealed and flapped her hands.\n\nA grin split his face and he tapped the screen, placing an X on the tic-tac-toe app.\n\nAvery's stomach twisted into knots, her throat tight. He hadn't looked at her once. He smiled at Hailey, but there were shadows under his eyes as if he hadn't slept, and his shoulders sagged.\n\n\"Hi,\" she breathed. \"I made chili.\" She pinched her eyes closed at her lame attempt to start conversation.\n\nHis glance darted to the bowls. \"Smells good. We'll finish this game and dig in.\"\n\nStill no eye contact and his tone was as flat as the table.\n\n\"Can we talk?\" She twisted her fingers together and fisted them to stop. All she wanted was to run her hands through his hair and force him to look at her. Pull him in for a kiss and tell him how stupid she'd been.\n\nHis back tensed as he ran his hand over his neck. Several beats passed before he turned his head in her direction, but the blue of his eyes didn't lift to hers. \"Later, when you get home.\"\n\nDid he say that because he didn't want to discuss things in front of Hailey? Worried he'd raise his voice again and scare her? Did he not want to dump Avery with her daughter around?\n\nHeck, for all she knew, he was concerned about global warming or crop circles for all the indication he was giving.\n\nHer stomach threatened to revolt. This shielded, defensive man before her wasn't her Cade, wasn't the man she fell in love with.\n\n\"Cade...\"\n\nHis eyes closed for a moment, brows pinched as if in pain before he shook his head. When he spoke, his voice was weary and soft. \"Later, sweetheart. Go to your meeting.\"\n\n_Sweetheart._ If he used his endearment, that had to be a good sign, right?\n\nThey were going to be discussing the St. Patrick's potluck at the meeting. Mostly groundwork and ideas. The sooner she got there, the sooner she could get back and put herself out of this limbo misery. Perhaps she could even feign exhaustion and cut the pow-wow short a half hour early.\n\nShrugging into her coat, she kissed Hailey and said good-bye.\n\nShe couldn't attest to anything discussed at the meeting because she'd spent the entire time fidgeting in her seat, checking her phone, and dissecting every syllable that had escaped Cade's mouth. Since he'd only said a grand total of three sentences to her, the reliving was brief. Yet she'd scrutinized and dismembered those three sentences, analyzing the inflection of his voice until she'd nearly pulled her hair out with frustration.\n\nBlessedly, no one stopped her to chat on the way out, and she was able to drive home without wrapping the car around a tree. A miracle, considering her distracted state.\n\nCade's car wasn't parked out front when she arrived.\n\nPanic squeezed her throat as she fumbled with the seatbelt and exited the car. The lights were on, casting a warm, yellow glow from the cabin as she barreled up the porch steps. She was about to shove through the door when she noticed a small piece of paper taped to the glass above the knob.\n\nWe're at my house. Meet you there. ~Cade\n\nHis house? What were they doing there? It was past Hailey's bedtime. Plus, he'd never taken her anywhere while he'd watched her in the past. She checked her phone, but she hadn't missed a call.\n\nShe froze. Something was wrong.\n\nTrembling, she raced back to the car and sped down the tree-lined road toward Cade's place.\n\n# Chapter 26\n\nCade sat across from Hailey on the bed and practiced their project one last time on her iPad before Avery arrived. He suspected this was what the girl had been trying to show him on the device the night she'd found him and Avery in an argument, which only made him love Hailey even more.\n\nChecking his watch again, the acid in his gut became volcanic. Avery would be here any minute.\n\nAll week it had been a battle not to go over to her house, not to call or answer her texts. He missed her so damn much he'd gotten a collective eight hours of sleep between the time he'd walked out of her cabin and last night. He'd forced food down and went through the motions at work, all to bide his time to get to tonight.\n\nHe'd wanted her to have time to think, to get settled after their argument and reach her own conclusions without his interference. He could only hope she missed him half as much as he did her. If she didn't, all his plans and arrangements would be for naught.\n\nTruth was, he could wait her out. Yeah, he'd been impatient as of late, but when he picked apart her actions, her reactions, he knew their relationship would eventually lead where he wanted. She loved him. She might not know it yet or trust herself with the knowledge, but she loved him.\n\nExcept sitting on his hands and abiding by patience was not what she'd walk into any minute, nor was it working. And what he'd done would either throw her into his arms or have her bolting. He couldn't be sure which, but he had to do something. This holding pattern was killing him dead.\n\nThe front door slammed and Cade's heart tripped at the thud.\n\n\"Cade?\" Avery's voice floated up to him.\n\nHe sucked a harsh inhale and looked at Hailey. \"Showtime, squirt. You ready?\"\n\nShe squealed and flapped her hands.\n\n\"Cade?\"\n\nAvery's tone was panicked, so he rose quickly. \"Wait here. Be right back.\"\n\nLeaving the bedroom, he rounded the corner and stopped at the top of the stairs. \"Up here.\"\n\nShe darted to the base of the steps and stopped. \"What's wrong? Where's Hailey?\"\n\nDamn. Scaring her to death wasn't in the plan. \"She's upstairs. She's fine.\"\n\nHunching over, she grabbed her chest and exhaled.\n\nHe met her halfway up the steps and ran his hands down her arms. \"I'm sorry. I didn't mean to worry you.\"\n\nShe closed her eyes and nodded, her shoulders deflating.\n\nCompletely unprepared for what seeing her would do to him, he drank in the sight of her. It had only been a few days, but it had been the longest few days of his damn life. Before she'd left for her meeting, he couldn't look at her or he would've lost it.\n\nHis chest filled to capacity, and his throat suddenly grew tight. She had on a minimal amount of makeup, her chestnut hair was up in a hazard ponytail, and she wore nothing more fancy than jeans with a T-shirt under her coat, but she was the most beautiful thing he'd ever laid eyes on.\n\nThat was the thing that tugged at his gut the most. She didn't have to do or be anything but herself and he wanted her. No frills or fluff. She didn't cling or pretend or coax. Avery was real and genuine and simple.\n\nHer chocolate gaze met his, flecks of honey swimming in all that brown, and he pretty much stopped breathing. If eyes were the window to the soul, as the saying went, then her windows had just been wiped clean. Finally, there was no hesitation or restraint. She was wide open, looking at him as if her heart was there for the taking.\n\nLifting his arms, he stripped her coat off and tossed it down the stairs, then cupped her cheeks in his hands. \"I missed you.\"\n\nRelief filled her eyes. When, _when_ would she stop doubting him?\n\n\"I missed you, too. About what I said earlier this week, I'm sorry.\" A pained expression twisted her face. \"I don't want to hurt you for anything. You should know, I don't think I ever loved him, and I'm done letting the past cloud the present. I\u2014\"\n\nHe kissed her softly, not to shut her up, but to stop the excuse. He didn't need an explanation. He just needed her. He got why she'd said those things. Honestly, he did. He couldn't even blame her, but if it was the last thing he ever did, he was going to make her understand that he'd never treat her the way that bastard had.\n\nShe sighed when he pulled away, her warm breath skating across his jaw. Slowly, her heavy lids lifted and her gaze darted around. The hazy seduction began to clear. \"Why are you guys at your place, anyway? You've never taken Hailey to your\u2014\"\n\nHer gaze landed on the pictures he'd hung on the wall in alignment with the slope of the staircase. Five in total. One of him with both his parents after graduation. Another with him and his brothers outside the clinic last fall. A cute shot of Avery and Hailey he'd snapped on his phone during their snow excursion. Hailey and Seraph on the floor in front of the fireplace, playing tug-of-war, was closest to his favorite.\n\nThat honor belonged to the photo he'd stolen off Pinterest...him and Avery dancing. In her knockout red dress and him in his suit, she was smiling up at him. One of his hands cradled her face and the other was low on her back. Lights twinkled behind them. Cade had no idea who'd taken the picture, but he was grateful. One look at them and it was obvious she was his whole world.\n\nMaybe she'd finally get that.\n\nHer voice was thick when she broke the silence. \"When did you hang these?\" She was still staring at the pictures, so he couldn't read her reaction, but her body didn't stiffen against his.\n\n\"This week.\"\n\nHer gaze darted to his, and something battled in her eyes. Hope? \"After our fight?\"\n\nUnderstanding dawned. The frustrating woman thought he was done with their relationship and hanging pictures meant he wasn't. \"Yes, after our fight.\" He pulled her closer and kissed her forehead, the knot in his chest loosening as he inhaled her scent and touched her soft skin. \"I have something to show you. Come upstairs.\"\n\nShe nodded and he took her hand, leading her to what was once an empty bedroom he'd had no clue what to do with. He knew now, though.\n\nSwallowing hard, he pushed the door open and stepped inside, dropping Avery's hand to give her breathing room. She halted over the threshold and gasped.\n\nThe dogs were sitting on the bed, battling for Hailey's attention. She was busy playing on her device, paying them no mind.\n\n\"Oh, my God.\" Avery pressed a hand to her forehead.\n\nWas that a _this-is-awesome_ Oh my God or a _holy-shit_ Oh my God? He couldn't tell and, at the moment, she only offered her profile.\n\nDeciding that keeping his mouth shut was the best recourse until she gave him some direction to her thoughts, he took in the bedroom again as if trying to see it through her eyes.\n\nThe room was painted a light pink with a mural of a life-sized dollhouse on one wall. The lacey curtains over the bay window were white to match the new furniture\u2014a full-sized bed and dresser. There were throw pillows all over the floor because Hailey liked pillows, and a small table in the corner.\n\nAvery made a choking sound, and he feared he'd gone too far.\n\nNeeding to explain, he rubbed his neck. \"You said you needed to move out of the cabin before tourist season started. Instead of getting an apartment, I thought maybe you could move in here. With me.\"\n\nHer wide gaze whipped to his, and he snapped his jaw closed. Why the hell could he never say the right thing when it concerned her? The point was to try and ease her into things, and he was mucking it up.\n\n\"This would be Hailey's room. Obviously. Zoe painted the mural for her. Gabby picked out the furniture. The pillows were all me.\" Hell. He pinched the bridge of his nose and closed his eyes.\n\nWhen he opened them, she hadn't moved one iota and looked like he could knock her over with a feather. Then\u2014 _damn it_ \u2014tears started pooling in her eyes and panic took on a new name.\n\n\"You don't have to decide right away. Or do anything now. The offer's there, if you want it. Open for as long as you need.\" _Christ. Shut it, O'Grady._\n\nSilence hung.\n\nHailey squealed, drawing their attention to her. She bounced on the mattress, gaze trained up and hands flapping, making the dogs bark. At least one of his girls liked the idea of staying with him.\n\nHe couldn't take the silence. Walking to the bed, he sat on the edge and turned on the iPad. If he looked at Avery and found her closed up tight, he'd lose it, so he kept his gaze down.\n\n\"Cade...\"\n\nHe shook his head at her quivering, quiet voice. _Don't say no, sweetheart. Please._\n\nScrolling through the apps, he attempted one more time to explain. \"I know we haven't been together long, and I know you're scared. But this thing between us? It's real. I think about you all the time, and I want you in my life. Both of you.\"\n\nHell. He wanted to be a family.\n\nFinding the app he needed, he called Hailey to him and had her stand between his knees. \"Ready, squirt? Show your mom what we've been working on.\"\n\nHailey's finger hovered over the button a few seconds before she tapped the icon and a robotic voice said, \"Eye.\" Moving on to the next key, she tapped \"Heart\" with no hesitation.\n\nCade's gut clenched. This next one always stumped Hailey when they'd worked on their project.\n\nHailey paused, but after a moment, she tapped... \"U.\"\n\nCade threw his hands up and barked a laugh. \"Yeah! That's my girl. Knuckles.\" He held out his hand and she bumped her fist to his. \"She kept getting stuck on that last part, but...\" He looked up at Avery and did a double take.\n\nTears streamed down her pale cheeks, eyes red-rimmed and puffy. Her forehead was scrunched, her lips parted, and her breathing labored like she was on the verge of hyperventilating. With her hand fisting her shirt over her chest, she uttered some form of a startled noise as her gaze darted between him and Hailey.\n\n\"You...\" She pressed her lips together and wheezed. \"You taught her to say _I love you_?\"\n\n\"Uh...not exactly.\" He shot to his feet, worried she was going to collapse and wanting to be ready to catch her. \"We couldn't find a \"love\" button, so we had to go with heart.\" He shrugged.\n\nDamn if her breathing didn't grow more labored.\n\nShit. It dawned on him he hadn't said the words to her. He'd planned to start off this whole crazy tirade with that and got lost in his frayed nerves instead.\n\nGrabbing the device from the bed, he tapped out, \"Eye. Heart. You.\" Then he added his own ending. \"Two.\"\n\nThere it was. She could take it or leave it. He should probably warn her that if she chose the latter, he'd chase her to the ends of the earth and back.\n\nOr maybe she knew because finally\u2014thank Almighty, _finally_ \u2014a grin broke out on her face and she sucked in an uneven breath.\n\nJust in case he wasn't clear enough...\"I love you. Move in with me today, or next week, or next year. I don't care when, as long as you promise to do it eventually. When you get used to the idea of us living together, and my socks on the floor or my snoring doesn't bother you, then I'll ask you to marry me. I think it might be too soon for a proposal today, but you're it for me. Just know it's coming. Baby steps.\"\n\nHe had point five seconds to take in her watery, ragged laugh before she launched herself at him and pressed her lips to his. He stumbled backward and grabbed her ass, righting them before they fell.\n\nHell. Yes. Right where she belonged.\n\nSliding one hand into her hair, he deepened the kiss, stroking his tongue with hers, tasting the salty saline of her tears. She kissed him with no restraint, meeting him in the middle and pouring so much emotion into the act he almost got weepy himself. His heart was so happy it turned over behind his ribs.\n\nWhen oxygen became vital again, he eased back to look in her eyes. \"I'll have you know, this is very inappropriate behavior to do in front of a child.\"\n\nShe grinned and glanced at the bed. \"She's asleep.\"\n\nWith Avery still in his arms, he turned. Sure enough, Hailey was out cold. \"My plan worked. You have to spend the night with me.\" He pointed to the dogs. \"Stay.\"\n\nThen he cinched Avery higher so she could wrap her legs around his waist. He shut off the light before heading down the hallway to his bedroom. _Their_ bedroom.\n\n\"I heart you, too, Cade.\"\n\nHe came to an abrupt halt beside the bed. \"Is that a yes?\"\n\n\"To which part?\"\n\n\"Any of it. All of it. Take your pick. In fact\u2014\"\n\nShe laughed, the smoky sound sealing all the empty spaces in his soul. \"Never mind. I love you. A yes to everything. Now say it back and then take me to bed.\"\n\n\" _It back_ ,\" he teased, then laid her out on the sheets and followed her down. He pressed a kiss to her cheek. \"I do love you back.\" He kissed her other cheek. \"And front.\" Kissing her forehead, he whispered, \"And top.\" His lips trailed to her chin. \"And bottom. I love you everywhere in between.\"\n\nWrapping her arms around his neck, she laughed and nuzzled his jaw. \"Okay, okay. I get it.\"\n\nHe stared down at her, at everything he never knew he wanted but somehow got anyway. \"There was never a doubt in my mind, sweetheart.\"\nCheck out a preview for the next Redwood Ridge book _,_\n\n# Tracking You...\n\n# Chapter 1\n\nGabby Cosette smoothed her hand down the simple baby blue sundress she meticulously picked out for this evening and tried not to look too eager. Or throw up. That wouldn't do, either.\n\nFrom a back booth, she glanced around the only Italian restaurant in Redwood Ridge, comforted by the fact it was still early yet for the dinner rush. The place was a good choice. Right? Not as casual as Shooters\u2014the bar her and her friends frequented\u2014but not as formal as one of the seafood restaurants that dotted their Oregon coastal town. A step above grabbing coffee or a beer, yet it didn't scream desperation.\n\nWas a booth in the back too obvious? Had she overdone it with her makeup? Maybe she should've put her hair up instead of down?\n\nNo, no. She went for light and natural on purpose. The patrons of Redwood Ridge had known her all her life. It wasn't far out of the realm of ordinary for her to wear a dress and light cosmetics. She was being a basket case.\n\nIt's just... Well, she hadn't had a date in a year. A year!\n\nTo calm her nerves, she drew in a deep breath and focused on the red checkered tablecloth. A votive candle flickered on the windowsill to her right, the flame reflecting off the tinted glass. The parking lot stretched beyond, where her date's car was not in one of the available spots.\n\nIt was silly to get this worked up over a first date, especially with Tom. She'd gone to elementary and high school with him. His parents still lived down the street from hers. Strange how he'd never shown any interest in her romantically, yet out of the blue, he'd asked her out this week.\n\nThen again, most everyone in town viewed her as the sweet Cosette girl, everyone's friend. Thus the no date in a year. It was hard to get a guy to think about kissing her, never mind imagining her naked, when she had platonic all but tattooed on her forehead.\n\nThe waitress strolled over in her apron, holding a notepad in her hand. \"Are you waiting on someone, sweetie pie?\"\n\n\"Yes.\" She smiled and grabbed her cell on the table. Tom was five minutes late. \"He should be here any minute.\"\n\n\"Ooh. Is it a date?\" Mavis planted a hand on her plump waist and grinned, the wrinkles around her eyes growing to crevices. Gabby wasn't sure how old Mavis was, no one really knew, but she never seemed to age past the state from when Gabby was a child.\n\nGabby opened her mouth to answer, but Tom strode toward her, weaving around tables and plopping in the seat across the booth.\n\n\"Couldn't find ya at the bar. I wasn't expecting a table.\"\n\nIt was still early, and Le Italy didn't get that crowded even on a Friday night. How hard could it possibly have been to locate her? \"Give us a sec,\" she told Mavis and waited for her to step away.\n\nTom had blond hair too short for her preference and a thin mouth. His unremarkable brown gaze darted around the restaurant and back to her. He made no attempt to apologize for being late, and it appeared as if he'd just come from work. His jeans and T-shirt were paint-splattered. The hazard of working for his dad's commercial painting and roofing company.\n\n\"Thanks for meeting me.\" He took off his ball cap and scratched his head.\n\nWhy did that sound un-date-like? \"Um...sure thing. How's work going?\" Her gaze dipped to his hands, no better off than his clothes. Maybe she should've picked Shooters after all.\n\nSomething felt very, very off as her belly twisted. Not with nerves this time. Confused, Gabby's mind scrolled through their conversation from earlier in the week when he'd brought his dog into the vet clinic where she worked. As he was checking out, he'd anxiously spun around to face her and asked if she could meet him tonight.\n\n\"Good. Work's good.\" He put his hat back on and glanced outside. \"Getting to be warmer out, so the jobs are picking up.\"\n\nPerhaps he was just nervous, too. Her tension drained a degree.\n\nMavis returned and asked for a drink order.\n\nTom lifted his hand to wave her off. \"Nothing for me, thanks. I can't stay long. Got a poker game with the guys tonight. I need to shower before they show up.\"\n\nThe forced smile Gabby had plastered on her face began to wilt like her mom's petunias in August. What did he mean he couldn't stay long? And why would he ask her out and schedule a card game on the same night? Plus, he could shower for his friends, but not her?\n\nMavis divided her gaze between them, a mix of bewilderment and irritation lifting her brows. She tapped her pen to her pad as the silence hung. \"Can I get _you_ something?\" She focused on Gabby, her tone indicating she should order something.\n\n\"I'll have a sweet tea. Thank you.\" When the waitress walked away, Gabby looked at Tom. He'd thrown his arm over the back of the booth and had stretched his legs out. The aroma of Ode de Paint Thinner wafted across the table. \"So...?\"\n\n\"Right, right.\" Tom leaned forward and crossed his arms. \"I appreciate you letting me do this in person.\"\n\nShe stilled. \"Do what?\" Because she was definitely getting the this-is-not-a-date vibe now.\n\nA warring shift in contradiction took over her body. Everything inside grew rapidly chilly while her skin heated in what she hoped wasn't a blush. Her pale complexion always gave away her emotions, and she hated that more than she'd hated freshman algebra. Math was evil.\n\nHe let out a tense laugh, which sounded more like a guffaw, and drew several heads from other diners. \"Not exactly a conversation you want to have over the phone or somethin', ya know?\"\n\nNo. She didn't know. \"Maybe if you just tell me?\"\n\nHe played with the parmesan shaker, not meeting her gaze. \"Well, the whole town's buzzing about Rachel and Jeff's split.\"\n\nShe frowned, not connecting the dots on his crazy pattern. Her older sister had only dated Jeff for a few weeks which, per Rachel standards, might as well have been marriage. Rachel liked to keep her options\u2014and legs\u2014open.\n\nGuilt immediately consumed her for the crass thought, but it didn't make it any less true. She and Rachel couldn't be any more different. Rachel was aloof and sexy. Gabby was the girl next door. Men desired Rachel. The only thing they desired from Gabby was a shoulder to cry on after her sister shot them down.\n\nShe twirled a strand of hair around her finger to keep from fidgeting. \"I don't understand what Rachel and Jeff have to do with...\" Unable to finish the sentence\u2014because she had no idea anymore what \"this\" was\u2014she waved her hand between them.\n\n\"Well,\" he said in an aw-shucks kind of way that made her want to grind her teeth, \"now that Rachel's available, I thought maybe you could put in a good word for me?\" He blinked up at her hopefully.\n\nShe stared at him for several stunned beats.\n\nThe reality of the situation slowly crept into her head and shoved around her skull. Her stomach dropped somewhere near her ankles. When he'd asked her out at the clinic earlier this week, she supposed he hadn't actually asked her \"out.\" The phrasing he'd used had been something more like, _Can you meet up with me on Friday?_\n\nAnd stupid, stupid her had taken that to mean he wanted a date.\n\nAs if. Like anyone would ever be interested in her when her sister had gotten all the good genes and didn't have the reputation of being everyone's pal. Good ole Gabby.\n\n\"This wasn't a date,\" she muttered to herself, more to ground herself to the situation than for confirmation.\n\n\"Huh?\"\n\nClosing her eyes, she shook her head to let Tom know her utterance wasn't important. To him, it wasn't. Because she wasn't the one he wanted, and there was no sense in amplifying her mortification. It wasn't his fault she'd brainlessly gotten excited.\n\nGod, she was an idiot.\n\nHer heart sank a little as hope withered a painful death. She shouldn't be surprised, really. It wasn't like this was the first time someone had tried to use her as a go-between. If not with her sister, then her friends. Still, she'd been looking forward to tonight, had thought it was a blessed break in her dry spell.\n\nA lump formed in her throat as tears threatened. She looked around the room until she could get the pathetic emotions under control. Many of the tables had filled since she'd arrived. Man, if she started crying now...\n\n\"So what do you say?\" Tom set the shaker aside. \"Could you help a friend out?\"\n\nFriend. She nearly choked on the word. Instead, she cleared her throat and forced a smile. Who was she to stand in the way of potential true love? \"Of course. I'll talk to her tomorrow.\"\n\nHis nervous grin widened into something more genuine, drawing attention to the slight crookedness of his two front teeth. \"You're the best, Gabby.\"\n\nYep. That was her. She resisted patting her own back in a sarcastic response.\n\nDid she really want a relationship with him, anyway? Probably not. He wasn't classically handsome, but he had his charms. His looks didn't matter to her as long as he had a good heart or sense of humor. It was more the idea of having someone that appealed.\n\nWhich was not going to happen. Not tonight.\n\nTom rose from his seat and tipped his ball cap as if it were a Stetson. \"Thanks so much. I gotta go.\"\n\nOf course. Alone again. Maybe she should become a poet. It had worked for Hemmingway.\n\nShe nodded. Her gaze followed him to the front door, and then over to the bar where she was thinking of doing a little Cuervo therapy.\n\nFlynn was leaning against the bar, his direct sights on her. Still wearing his dark blue scrubs, his posture resonated his typical laid-back demeanor. Now there was an attractive man. Tall enough for the top of her head to reach his chin and ropey muscle on an athletic build. Wide shoulders, narrow waist.\n\nAll three of the O'Grady brothers were sexy in their own unique way. But they'd grown up together, and there had never been any chemistry between her and them. Cade, the youngest brother, was engaged to their office manager as of a couple months ago, and Drake, the eldest brother, was a widow. Gabby couldn't envision him dating again, at least not anytime soon. Flynn wasn't seeing anyone.\n\nNot that it mattered. She worked for Flynn and his brothers at their vet clinic, so that was an automatic hand-slap.\n\nFrom across the room, Flynn's eyes narrowed as he tilted his head in question toward the door. _Where'd your date go?_\n\nFlynn was deaf, and through the years she'd grown to read him easily. They always had a strong connection, being able to understand one another without words. Part of that was being good friends and part was due to working closely together for many years.\n\nShe shrugged in answer, keeping her disappointed expression open for him. Sucked to be her.\n\nHis brows lowered and he straightened from the bar, poised to head over until the bartender tapped his shoulder. Flynn signed for his takeout and carried it over to her booth, setting the bag down on the table before sitting.\n\nHis hazel eyes, framed by criminally long lashes, swept her face. \" _What happened?\"_ he signed with his hands. \" _I thought you had a date?\"_\n\nPer their routine, she signed and spoke simultaneously. \"Me, too. Turns out he wanted my help getting in with my sister.\" At his scowl, she shrugged, embarrassed enough without the urge to discuss it. \"My own fault. I read too much into the initial conversation.\"\n\nHe stared at her with disbelief and shook his head. His handsome, angular face was dialed to irritated and his full lips were twisted. He ran a hand through dark strawberry blond hair just this side of wavy. Flynn had a tendency to forget routine trims.\n\nMavis made her way back to the booth. Her gaze zeroed in on Flynn. \"Decided to eat in?\"\n\nHabit had him turning to Gabby. He could read lips, but sometimes people spoke too quickly or didn't face him fully so he couldn't see what they were saying. Gabby signed Mavis' question.\n\nHe grinned, back to his usual glower-free self, and nodded.\n\nWell, it wasn't a date, but Flynn was better company, anyway. Gabby looked at the waitress. \"He'll have a beer, whatever's on tap, and can I get the largest piece of tiramisu you can find?\"\n\n\"You got it, sweetie pie.\"\n\nGabby watched her walk away before letting out a sigh, chest deflating. When she looked at Flynn, his expression indicated he was patiently waiting for her attention again.\n\nHe leaned forward as if to punctuate a point. _\"He's an asshole_.\"\n\nShe laughed. \"Aren't they all?\"\n\n_\"Not all.\"_ He pulled a Styrofoam container of lasagna from the to-go bag, opened it, and grabbed her fork from her place setting. He waited for her to take it from him before signing, \" _Dig in.\"_\n\nHe picked up his fork and took a bite, then did a double take when she just poked at his lasagna. \" _Hey. You all right?\"_\n\n\"I'll be okay. Just not today. Today, I mope.\" He was one of the few people she'd admit that to, and since his gaze had softened and worry wrinkled his brow, she forced herself to take a bite. \"Thanks, Flynn.\"\n\nHe nodded, watching her intently. \" _Movie night. My house. I'll even let you pick_.\"\n\nWhy the hell wasn't he dating someone? Seriously.\n\nSad truth was, women tended to overlook Flynn because of his disability, just like they overlooked her for being in the friend zone. People sucked. \"Maybe we should make one of those pacts. You know, the one where if neither of us is married by the time we're thirty we marry each other.\"\n\nOne eyebrow quirked in his custom you-done-gone-crazy. \" _I'm thirty-one and you turn the same in a couple weeks. That ship has sailed.\"_\n\nYeah. \"Fine. Throw logic into my delusions.\"\n\nHis shoulders bounced in a silent laugh.\n\nShe smiled. \"Okay, hot date. What if I pick a sappy movie?\"\n\nHe shrugged. \" _I'll hide my man card. Tell no one_.\"\n\nCovering her face, she laughed until her chest ached. When she sobered, her mood was irrevocably lighter. Praise God for good friends. \"Just for that, I'll share my tiramisu.\"\n\n_\"Deal.\"_ He ate a few more forkfuls before his smile slipped a fraction, the hint of seriousness reflecting in his eyes. \" _For the record, I would've taken the pact_.\"\n\nShe dropped her chin in her hand before moving to sign. \"We would've had such cute babies, too.\"\n\n_\"Word. Now eat or I'll make you watch_ Die Hard _again_.\"\n\nShe scooped a bite of cheesy carb goodness. Calories didn't count on crappy days. \"Which one?\"\n\nHe whipped her a \"duh\" look. \" _All of them_.\"\n\nDeath by Bruce Willis. Could be worse things.\n\n# Meet the Author\n\nBestselling author Kelly Moran says she gets her ideas from everyone and everything around her and there's always a book playing out in her head. No one who knows her bats an eyelash when she talks to herself, and no one is safe from becoming her next fictional character. She is a Catherine Award winner, Readers Choice finalist, Holt Medallion Finalist, and earned one of the 10 Best Reads by _USA Today's_ HEA. She is also a Romance Writers of America member. Her interests include: sappy movies, MLB, NFL, driving others insane, and sleeping when she can. She is a closet caffeine junkie and chocoholic, but don't tell anyone. She resides in Wisconsin with her husband, three sons, and two dogs. Most of her family lives in the Carolinas, so she spends a lot of time there as well. She loves hearing from her readers. Please visit her at authorkellymoran.com, twitter.com\/authorkmoran, or facebook.com\/authorkellymoran.\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n# \n\n# \nCopyright \u00a9 2013 by Dean R. Owen\n\nAll rights reserved. No part of this book may be reproduced in any manner without the express written consent of the publisher, except in the case of brief excerpts in critical reviews or articles. All inquiries should be addressed to Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018.\n\nSkyhorse Publishing books may be purchased in bulk at special discounts for sales promotion, corporate gifts, fund-raising, or educational purposes. Special editions can also be created to specifications. For details, contact the Special Sales Department, Skyhorse Publishing, 307 West 36th Street, 11th Floor, New York, NY 10018 or info@skyhorsepublishing.com.\n\nSkyhorse\u00ae and Skyhorse Publishing\u00ae are registered trademarks of Skyhorse Publishing, Inc.\u00ae, a Delaware corporation.\n\nVisit our website at www.skyhorsepublishing.com.\n\n10 9 8 7 6 5 4 3 2 1\n\nLibrary of Congress Cataloging-in-Publication Data is available on file.\n\nCover design by Brian Peterson\n\nCover photo credit \u00a9 Tom Dillard Collection, Dallas Morning News, Sixth Floor Museum at Dealey Plaza\n\nPaperback ISBN: 978-1-63450-272-6\n\nHardcover ISBN: 978-1-62636-034-1\n\nEbook ISBN: 978-1-5107-0739-9\n\nPrinted in the United States of America\nTo my other and more important legacy: my wife of thirty years, Janet B. Owen; our older daughter and son-in-law, Whitney, and Josh Hammar; our younger daughter, Embry Wood Owen; and my parents, Edward and Elvira Owen, who taught me life's most important lesson, reflected in Luke: 6:31.\nTABLE OF CONTENTS\n\nForeword\n\nPreface\n\nAcknowledgements\n\nIntroduction\n\nSECTION ONE\n\nMembers of the Kennedy Administration and White House staff\n\nLetitia Baldrige Hollensteiner\u2014White House Social Secretary, expert on etiquette\n\nJoseph A. Califano Jr.\u2014Senior official in the Pentagon, Carter administration cabinet member\n\nMortimer M. Caplin\u2014Commissioner of the Internal Revenue Service, founder of Caplin & Drysdale\n\nCharles U. Daly\u2014White House congressional liaison, writer\n\nJohn Gunther Dean\u2014Senior State Department diplomat\n\nClint Hill\u2014U.S. Secret Service agent\n\nHarold Hill\u2014White House communications officer\n\nLem Johns\u2014U.S. Secret Service agent\n\nNicholas deB. Katzenbach\u2014Deputy Attorney General, Attorney General in the Johnson administration\n\nNewton N. Minow\u2014Chairman of the Federal Communications Commission\n\nNelson C. Pierce\u2014White House usher\n\nSue Vogelsinger\u2014White House press aide\n\nLee C. White\u2014Special Counsel to the President\n\nHarris Wofford\u2014Special Assistant to the President on Civil Rights, Associate Director of the Peace Corps, Peace Corps director in Ethiopia, U.S. senator\n\nSECTION TWO\n\nCivil Rights Leaders\n\nJulian Bond\u2014Founding member of the Student Non-Violent Coordinating Committee, chairman of the National Association for Advancement of Colored People, professor\n\nDolores Huerta\u2014Labor leader, advocate for rights of workers, immigrants, and women, co-founder of the National Farm Workers Association\n\nJohn Lewis\u2014Chairman of the Student Nonviolent Coordinating Committee, Freedom Rider, author, member of United States House of Representatives\n\nRoger Wilkins\u2014Assistant to the Administrator of the United States Agency for International Development, Assistant Attorney General in Johnson administration, Pulitzer Prize\u2013winning journalist, professor\n\nAndrew Young\u2014Executive Director of the Southern Christian Leadership Conference, mayor of Atlanta, U.S. Ambassador to the United Nations during Carter administration\n\nSECTION THREE\n\nChildren of Kennedy Advisors and Others\n\nDavid Acheson\u2014Attorney, scholar, son of former Secretary of State Dean Acheson\n\nFernando Chavez\u2014Attorney, son of Cesar Chavez, co-founder of the National Farm Workers Association\n\nNancy, Kate, and Walter Jr. (\"Chip\") Cronkite\u2014Children of Walter Cronkite, former CBS News anchor\n\nCasey Murrow\u2014Educator, son of Edward R. Murrow, journalist, director of United States Information Agency\n\nCraig McNamara\u2014Organic farmer, son of Robert McNamara, Secretary of Defense\n\nDavid Rusk\u2014Urban policy expert, scholar, son of Dean Rusk, Secretary of State\n\nSteven Schlesinger\u2014United Nations expert, scholar, son of Arthur Schlesinger Jr., Special Assistant to the President\n\nJim Swindal\u2014Physicist, son of James Swindal, pilot of Air Force One\n\nSECTION FOUR\n\nCelebrities\n\nVincent Bugliosi\u2014Attorney, author\n\nRev. Billy Graham\u2014Christian evangelist\n\nLee Iacocca\u2014America industrialist\n\nJimmy Piersall\u2014Major League Baseball player, author\n\nCliff Robertson\u2014Actor\n\nAlex Trebek\u2014Television personality\n\nSECTION FIVE\n\nFamily and Friends\n\nDr. Bob Arnot\u2014Physician, journalist, author\n\nCharles Bartlett\u2014Journalist, introduced John Kennedy to Jacqueline Bouvier\n\nBenjamin C. Bradlee\u2014Journalist\n\nRobert F. Kennedy\u2014Brother, Attorney General, United States Senator, Democratic Presidential Candidate\n\nRobert F. Kennedy Jr.\u2014Nephew, environmental activist\n\nChristopher Kennedy Lawford\u2014Nephew, author, filmmaker\n\nFrank Mankiewicz\u2014Peace Corps director in Peru, president of NPR, press secretary to Robert F. Kennedy\n\nJohn Seigenthaler Sr.\u2014Journalist, author, First Amendment historian\n\nSECTION SIX\n\nJournalists and Commentators\n\nEddie Barker\u2014KRLD Dallas, first journalist to confirm Kennedy's death\n\nTom Brokaw\u2014NBC News\n\nBob Clark\u2014ABC News, only journalist to cover assassinations of both John and Robert Kennedy\n\nSid Davis\u2014Westinghouse Broadcasting Company, NBC News, witnessed swearing in of Lyndon Johnson\n\nNicholas Gage\u2014Writer for The New York Times and other publications, author\n\nBob Jackson\u2014Dallas Times-Herald, captured Pulitzer Prize\u2013winning image of Jack Ruby shooting Lee Harvey Oswald\n\nMarianne Means\u2014Writer for Hearst newspapers\n\nMichael Medved\u2014Author, syndicated radio show host, commentator\n\nAl Neuharth\u2014Author, founder of USA Today, the Freedom Forum, and the Newseum\n\nBob Schieffer\u2014CBS News\n\nSanford Socolow\u2014CBS News\n\nAl Spivak\u2014United Press International, participated in the Kennedy-Nixon debates in 1960\n\nStan Stearns\u2014United Press International, captured iconic photo of John Kennedy Jr. saluting his father's casket\n\nRichard Stolley\u2014LIFE magazine, negotiated the rights to the only film of the assassination\n\nSander Vanocur\u2014NBC News and ABC News\n\nSECTION SEVEN\n\nPolitical Figures\n\nBarry Goldwater Jr.\u2014Member of the United States House of Representatives, businessman, son of U.S. Senator and 1964 Republican presidential nominee Barry Goldwater\n\nGary Hart\u2014Member of the United States Senate, Democratic presidential candidate\n\nKen Hechler\u2014Member of the United States House of Representatives, state official from West Virginia\n\nPete McCloskey\u2014Member of the United States House of Representatives, attorney\n\nWalter Mondale\u2014Vice President of the United States, member of the United States Senate, 1984 Democratic presidential nominee\n\nJim Wright\u2014Speaker of the United States House of Representatives\n\nSECTION EIGHT\n\nThose with Humorous, Poignant, Quirky, and Tragic Encounters and Connections\n\nPatricia Baillargeon\u2014Assistant to Eleanor Roosevelt\n\nDr. William Bernhard\u2014Surgeon who fought for thirty hours to save the life of Patrick Bouvier Kennedy\n\nJim Boyd\u2014Medical student working at Parkland Memorial Hospital\n\nLillian Brown\u2014Make-up artist for JFK and eight other presidents\n\nJosiah Bunting III\u2014Military officer, educator, foundation executive\n\nArthur \"Andy\" Carlson\u2014Army officer who led riderless horse \"Black Jack\" in funeral\n\nRobert Dellwo\u2014Attorney and county Democratic chairman\n\nPhyllis Elkins\u2014Waitress at Jim's Steak and Spaghetti House, Huntington, West Virginia\n\nRichard Gaudreau\u2014Military pallbearer\n\nMike Gefroh\u2014Little Leaguer who caught a ceremonial first pitch\n\nFrank Greer\u2014Senate page, media advisor to U.S. Presidents William Jefferson Clinton and Barack Obama\n\nRon Hall\u2014Artist, art dealer, author, coincidental bystander to assassination\n\nCatharine Hamm\u2014Travel writer and editor, Catholic who still grieves\n\nTerri Hazeleur\u2014Led thirty-one girls walking all night to meet the president\n\nFather Oscar Huber\u2014Catholic priest who administered the Last Rites\n\nDavid W. Knowlton\u2014West Point graduate with a foreboding premonition\n\nJames Leavelle\u2014Dallas police detective handcuffed to Lee Harvey Oswald\n\nPriscilla Johnson McMillan\u2014Knew both John Kennedy and Lee Harvey Oswald\n\nAnn Owens\u2014Inspired by \"Ask Not...\" to join the Peace Corps\n\nRuth Paine\u2014Living with Marina Oswald at the time of the assassination\n\nDr. Ira Seiler\u2014Pediatric resident who saved the life of John Kennedy Jr., born not breathing\n\nSamuel Stern\u2014Staff member of the Warren Commission\n FOREWORD\n\nIT is hard to believe fifty years have passed since John Fitzgerald Kennedy, the most inspired and inspiring of any president I covered, was assassinated in Dallas.\n\nFor journalists, his election three years earlier ushered in a new era of excitement and anticipation, marked by the memorable reference in his inaugural address: \"Let the word go forth from this time and place, to friend and foe alike, that the torch has been passed to a new generation of Americans.\" At age forty-three, he was the youngest president ever elected. (Theodore Roosevelt ascended to the presidency at age forty-two after William McKinley's assassination.) Kennedy was the first president born in the twentieth century. He had an abundance of charm, elegance, wit, and humor. Other presidents enjoyed some of those qualities, but John Kennedy had them all, and he used them effectively.\n\nFor example, responding to allegations of nepotism with the appointment of his brother Robert as attorney general, the President, speaking at Washington's Gridiron Dinner six weeks after taking office, remarked: \"Bobby has just received his law degree and we thought he should have some experience before he goes into private practice.\" Or at the conclusion of a state visit to France accompanied by his wife, he quipped, \"I am the man who accompanied Jacqueline Kennedy to Paris and I have enjoyed it.\"\n\nWhen Dean Owen contacted me in March of 2011, he already had been working on this book for four months. In November 22, 1963 he has compiled a series of reflections from people that reveal diverse perspectives and engaging anecdotes. Many people in this book are well known: Kennedy family members, civil rights leaders, journalists, and people who served in JFK's administration. Others are unknown to the general public, but provide interesting and poignant stories, some never before told: a second-year pediatric resident who saved John Kennedy Jr.'s life when he was born not breathing; children of Walter Cronkite and Edward R. Murrow, as well as those of Kennedy's advisors, such as Robert McNamara, Dean Acheson, and Arthur Schlesinger Jr.; the surgeon who tried desperately for thirty hours to save the life of the Kennedys' second son, Patrick; and one of, if not the only, person who knew both John Kennedy and Lee Harvey Oswald.\n\nI believe people from all walks of life, including those for whom John Kennedy's life and legacy comprise a footnote in a history book, will find in November 22, 1963 compelling commentaries about an engaging leader whose presidency remains one of the most studied in our nation's history.\n\nLike anyone old enough to remember that tragic day, the events of November 22, 1963 and the following weekend are etched indelibly in my mind. The day before I watched the president and Jackie walk across the South Lawn of the White House and board the presidential helicopter. He was holding his son's hand. All three flew to Andrews Air Force Base; John Jr. loved helicopters and airplanes and the president probably wanted to indulge him. The little boy, who would turn three the following Monday, the day his dad was buried, was driven back to the White House later that morning.\n\nThe next day, I was having lunch with Fran Lewine of the Associated Press and Pierrette Spiegler, a member of Mrs. Kennedy's staff. I was preparing to leave Saturday for Detroit to visit my family. Suddenly, we heard on a transistor radio that Kennedy had been shot. All of us bolted out of the restaurant, hailed cabs, and went to our offices.\n\nMy boss, Julius Frandsen, saw me and said, \"You're on vacation.\"\n\nI said, \"No, I'm not.\"\n\nLater I was told, \"OK, go to Andrews Air Force Base. You're going to Dallas.\"\n\nA plane was standing by to take journalists to Dallas, and en route to the air base in a cab, I heard the official announcement: The president was dead.\n\nI was stunned. And I did not want to believe it. I soon learned Lyndon Johnson had taken the oath of office and that he, Jackie, and John Kennedy's body were returning to Washington on Air Force One. Upon their arrival at Andrews, I was struck by the fact Jackie still was wearing the bloodstained suit. That took a lot of bravery on her part.\n\nOver the next three days, I covered every event at the White House involving the funeral. I stood on the steps of St. Matthew's Cathedral when Jackie and the family entered. It is unusual and uncharacteristic for reporters to cry while they are working; we're not supposed to show emotion. But I was dictating to UPI with tears running from my eyes.\n\nJohn Kennedy understood his time was short, that he had a rendezvous with death. The deaths of a sister and brother during World War II, as well as the horrors of war he experienced in the South Pacific, may have given him a sense of melancholy and foreboding.\n\nHe also had an infectious sense of hope and optimism for the future. The sky was not the limit. He believed in setting goals and taking steps to achieve them. John Kennedy instilled in young people the belief that public service was honorable and could be the capstones of their careers. He also wanted to demonstrate the American promise of excellence to other countries and so created the Peace Corps. He had his eyes on the stars and challenged the nation to send a man to the moon and bring him back to earth safely before the end of the 1960s, an historic achievement he did not live to witness.\n\nOne of his campaign themes in 1960 was \"A time for greatness.\" That greatness he envisioned is reflected in the remarks he would have delivered at a luncheon that fateful day in Dallas:\n\n\"Finally, it should be clear by now that a nation can be no stronger abroad than she is at home. Only an America which practices what it preaches about equal rights and social justice will be respected by those whose choice affects our future. Only an America which has fully educated its citizens is fully capable of tackling the complex problems and perceiving the hidden dangers of the world in which we live. And only an America which is growing and prospering economically can sustain the worldwide defenses of freedom, while demonstrating to all concerned the opportunities of our system and society.\"\n\nAs I have said so often, \"Thank you, Mr. President.\"\n\nHelen Thomas\n\nWashington, D.C.\n\nFebruary, 2013\n PREFACE\n\nMAJOR, sudden, and tragic events create unforgettable images, mental snapshots that enable people, decades later, to recall where they were when they heard the news: the September 11 attacks; the deaths of John Lennon and Diana, princess of Wales; Pearl Harbor.\n\nFor me and millions of others age fifty-five and over, it was the assassination of President John F. Kennedy. November 22, 1963 will forever be known for the shock and horror when the charismatic leader of the free world was killed with a gunshot to the head in Dealey Plaza in Dallas.\n\nI was seven years old in Mrs. Gurley's second grade class in Burbank School in Hayward, California. The older sister of a friend stopped by the classroom and told me the president had been shot. Like many of those whose reflections are in this book, I greeted the news with disbelief: \"Oh, yeah, sure.\"\n\nLater that afternoon and over the next three days, I was riveted to the television and read everything in the local newspapers about the details of Kennedy's assassination. Fifty years later, I can still hear the solemn funeral dirges and muffled drumbeats of the funeral procession from the White House to the Capitol. Then, Sunday morning, I watched Jack Ruby shoot the accused assassin, Lee Harvey Oswald, live on television.\n\nAs Tom Brokaw remarks in section six of this book, \"The television set was, if you will, the centrifuge for the country. Everybody drew from it in some fashion.\" For me, that weekend was the catalyst for my fascination with journalism and how the news media shape public opinion.\n\nIn December of 2010, I turned fifty-five, wondering about my legacy, and I realized everyone has two legacies\u2014a personal one and a professional one. The former is solid, as reflected in this book's dedication. For the latter, I decided to go back to that weekend in 1963, made a list of more than one hundred people I wanted to interview, and started making phone calls and writing letters.\n\nMy pitch was simple: Each person was offered the opportunity to review and edit a transcript of the interview, and I would not ask any questions about assassination conspiracy theories or John Kennedy's sex life. More than enough words have been written about both topics. After nearly two and a half years of early mornings, late nights, and long weekends, November 22, 1963 is presented for your interest and enjoyment. I intentionally omitted any summation or analysis of the contributors' reflections, and prefer to allow readers to draw their own conclusions about the life, assassination, and legacy of America's 35th president.\n\nDean R. Owen\n\nSeattle, Washington\n\nMarch, 2013\n ACKNOWLEDGEMENTS\n\nCREATING November 22, 1963 has been a labor of love, but could not have been accomplished without those I interviewed, for whom I am deeply grateful, as well the encouragement of family and friends. Many thanks to all of you, including: Gardi Wilks, my literary agent Marilyn Allen, Roger Flessing, Tom Parks, Ken Churchill, Manen Clements, Debbie Durham, Jeff Peabody, Steve McFarland, my brothers Dennis and Don Owen, Jeff Wright, Jeff Warnke, John Leckie, M. Joseph Sloan, Eddy Rodriguez, my mother-in-law Shirley Biggerstaff, Kari Costanza, Mike McLeod, John Yeager, Paul Nordlund, Betsy Grabinski, Bill Wolfsthal and colleagues at Skyhorse Publishing, Doug Boyles, Bill Dotterwick, Hope Swecker, Rachel Cota Hochstetler, Brian Vasey, Katy Krippaehne, John Jensen, Dave Wittenberg, Margaret Larson, Char Haake, Steve Matthews, John Clum, and Laurie Austin at the John F. Kennedy Library.\n INTRODUCTION\n\nTHEIR names read like a \"Who's Who\" of the past six decades: White House reporter Helen Thomas, House Speaker Jim Wright, civil rights leader Julian Bond, journalist Benjamin Bradlee, publisher and author Al Neuharth, Attorney General Nicholas deB. Katzenbach, White House social secretary Letitia Baldrige, actor Cliff Robertson, Peace Corps director Frank Mankiewicz, and journalist John Seigenthaler, Sr.\n\nThey all knew John Fitzgerald Kennedy. More importantly, their observations and insights into America's 35th president have helped paint a portrait of a man whose legacy will outlive all of us.\n\nSadly, they and many others whose reflections you will read in this book have passed away, their voices and memories never again to be heard.\n\nWith each passing year, people who worked for JFK, journalists who covered him, family members, close friends, and others are leaving us. Their recollections are now in the archives of the John F. Kennedy Presidential Library and Museum, the Sixth Floor Museum at Dealey Plaza, and elsewhere around the world. All are invaluable resources for anyone seeking to understand him.\n\n\"The preservation of these firsthand recollections is a crucial tool to understanding the broader story of the 1960s, including the frustration and skepticism that punctuated that turbulent decade,\" says Stephen Fagin, Associate Curator of the Sixth Floor Museum at Dealey Plaza.\n\nThat broader story of the era began with a slogan from John Kennedy's 1960 presidential campaign, \"Leadership for the 60s.\" For him, it continued over the next thousand days through the clarion call of his inaugural address, the debacle of the Bay of Pigs, a contentious meeting in Vienna with Soviet Premier Nikita Khrushchev, his challenge to send a man to the moon, the Cuban Missile Crisis, the inspirational speech in Berlin, and the Nuclear Test Ban Treaty.\n\nAnd then... Dallas.\n\n\"The assassination of President Kennedy touched the lives of people around the world,\" Mr. Fagin says. \"Today, memories of that moment provide emotional, insightful, and often poignant links to the past which enable younger generations to invest themselves in the experience and better appreciate the global significance of this tragic event and its vast and complex aftermath.... Collections like [this book] remind us that those few seconds in Dealey Plaza continue to impact our lives in so many ways.\"\n\nAnyone who wants to know more about John Kennedy\u2014beyond the lurid allegations of his extramarital affairs or the complex and convoluted assassination conspiracy theories\u2014should visit the museums in Boston and Dallas. They should watch his speeches and press conferences on YouTube and read some of the well-researched and documented books on his life. He was brilliant but flawed. He was vigorous but suffered great physical pain. He was a visionary but also a pragmatist.\n\nAs a former journalist, John Kennedy knew the power of words. As president, he gave some of the most powerful and poignant speeches of the twentieth century. His words and the words of those interviewed in this book will live on for generations.\n\nDean R. Owen\n\nSeattle, July 2015\nSECTION ONE:\n\nMembers of the Kennedy Administration and White House Staff\n Letitia Baldrige Hollensteiner\n\nWhite House Social Secretary, expert on etiquette\n\n\"When the president has people sticking to him like parasites and not letting go, I managed to get in there and uproot both of them, and let him escape.\"\n\nTHE first time I met him was on a train going to Newport, Rhode Island for a weekend in the summer. He had been elected to the Senate, and he was already being talked about as a handsome young man. Very, very ambitious. Member of a large political family. I already had heard all the gossip about him. And he personified it all.\n\nHe was so good looking and had a great sense of humor. And he had been everywhere in the world. It was an enthusiastic first sighting [laughter].\n\nWe were both going to stay at Jackie's mother's house, Hammersmith Farm. I would have loved to have spoken with him more that weekend, but there were too many people at him\u2014or after him. I could hardly get in a word.\n\nWhen the president has people sticking to him like parasites and not letting go, I managed to get in there and uproot both of them, and let him escape. That's where I was needed and that's one of the things a social secretary does. She watches over the boss, the host, and never lets him out of her sight, and removes anything that's too abrasive, too forceful, too boring. There are so many reasons to get rid of people who are surrounding your president, or your ambassador, whoever it is. You must get rid of those people, and let the ambassador or the president do his job. You take all the criticism and let people hate you. But you're doing it for the boss.\n\nExcerpt from A Lady First: My Life in the Kennedy White House and the American Embassies of Paris and Rome, by Letitia Baldrige \u00a9 2001 (Used with permission of author):\n\nI had just sat down to lunch with Town & Country magazine editor-in-chief Henry Sell at the Mid-America Club atop the Prudential skyscraper. In the middle of our chicken crepes, a waiter brought us some more wine and said, 'I can't believe it. Someone has shot the president.'\n\nIt was an extraordinary moment in my life. I couldn't believe it. In fact, I'm shaking my head right now as I'm saying this. It shouldn't have happened.\n\nI called the White House and spoke with Clark Clifford. He said, \"Get on a plane and come down here.\"\n\nI arrived in Washington, went to the White House, and stayed there for the next seven or eight days. I was involved in a little bit of everything: protocols, greeting people, calming people down, getting information for people. It was a time none of us will ever forget.\n\nJackie was completely stunned. She was appreciative of my help later. But while it was going on, she was stunned. And she handled herself with beauty and grace. She knew it was the funeral of the century. Everything was planned. The words she used. It was all carefully thought out. It was a script on how to plan a state funeral. She would think, \"This would mean so much to the children\"\u2014some particular phrase. She was incredible.\n\nYou just adored the man! He would not dwell on anything that was sad or gloomy. He was funny about anything that he could be funny about, and he taught us all how to do that. He would say, \"Don't dwell on that, that sad stuff. Start talking about what happened at the such and such.\"\n\nHe taught us public relations in many ways. He knew about public relations and he knew how to use it, how to control it. He was a real teacher. He was so naturally aware of the world. I've never known anyone like that. That was one of his major talents that most people don't think about, because they're so into the one talent or skill being discussed at the moment. But people should look at him with a wide lens. He had a wonderful team of people around him, but he would not have had them if he had not been an extraordinary person. The team devotes loyalty to its leader and grows more loyal when they see it in the leader.\n\nI was in contact with Jackie every once in a while after she left the White House. It was always in conjunction with some meeting. She worked very hard. She always had something on her plate. She would call me and ask me, and I would come spend a few days working with her on the event or activity. She was always grateful.\n\nThe main part of his legacy is to reach out to the young\u2014ages eight to eighty. He would have something to say to each group. To motivate them, to make them feel they have a real place in the government, in our country, and in future aspirations of this country. He was always thinking of the future, and how one issue or event would affect another.\n Joseph A. Califano Jr.\n\nSenior official in the Pentagon, Carter administration cabinet member\n\n\"(Robert Kennedy) whispered to me: 'This is where we'll bury the President.' I will never forget the words.\"\n\nI first noticed John Kennedy watching the 1956 Democratic convention on television. I was twenty-seven years old and working in the Office of the Judge Advocate General in Washington. Adlai Stevenson had thrown open the nomination for vice president to the delegates and Kennedy mounted a strong\u2014though ultimately unsuccessful\u2014effort against Estes Kefauver, the Democratic senator from Tennessee.\n\nAt that time, Kennedy had little going for him besides the glamour and wealth of his family and his socially prominent wife, Jacqueline Bouvier. But he came astonishingly close and lost to Kefauver by fewer than two hundred votes. I was impressed watching him make a politically adept and gracious appeal to the delegates to unanimously nominate his rival.\n\nFast-forward three and a half years. I woke up on February 20, 1960 ready to enjoy a long weekend for the George Washington birthday holiday. I had recently purchased James MacGregor Burns's campaign biography, John F. Kennedy: A Political Profile, after seeing so many news stories about John Kennedy's campaign for the presidency\u2014the first Catholic to seek the presidency since Al Smith in 1928. I spent most of the weekend reading that book.\n\nI identified with Kennedy as a strong anti-Communist Catholic, a cold warrior committed to defeating the atheistic Soviet Union\u2014a true struggle of good against evil. He also expressed concern for the poor and underprivileged, which was in line with the commitment to social justice that the Jesuits at Holy Cross and Brooklyn Prep had instilled in me. Burns wrote that Kennedy believed it was essential to \"track down the best talent\" to lead the United States in this \"revolutionary time.\" As a former editor of the Harvard Law Review, I believed I swam in that talent pool.\n\nA month later, I contacted John Stillman, a college friend of JFK who was chair of a New York area chapter of the Democratic Party, and enclosed my resume, indicating I was \"quite anxious to do anything that will further Senator Kennedy's candidacy.\" He wrote back and we later had lunch in Manhattan. I came away inspired. And three days later, I received a letter stating:\n\n\"I am very glad to know that you are in contact with our mutual friend John Stillman and you may be sure that I am most appreciative of your interest and support. I hope you will keep in close touch with John, as he will be aware of our plans in New York.\"\n\nIt was signed \"Jack Kennedy.\" And I immediately felt part of the campaign.\n\nFour days before the election in November, I saw John Kennedy for the first time\u2014standing in the rain at a rally in Manhattan. He was on stage with Lyndon Johnson and Johnson's wife and two daughters. The rain intensified our excitement as it drenched our clothes (and Kennedy's and Johnson's). He gave an ardent and amusing campaign speech: \"You have seen these elephants in the circus. They have their heads of ivory, thick skins, no vision, long memory, and when they move around the ring in the circus, they grab the tail of the elephant in front of them. Well, Dick (Nixon) grabbed that tail in 1952 and 1956, but in 1960 he is running, not the president.\"\n\nFive months later, I took my first steps as a member of Kennedy's \"New Frontier.\" My position? Special assistant to Cyrus Vance, the Defense Department's General Counsel. Over the next two and a half years, I served as Special Assistant to the Secretary of the Army and later General Counsel of the Army.\n\nOne of my proudest moments, indeed, one of the greatest moments of Kennedy's presidency, was his televised address to the nation on civil rights in June of 1963. He made an impressive argument: \"We are confronted with a moral issue. It is as old as Scriptures and as clear as the American Constitution. The heart of the question is whether all Americans will be afforded equal rights and equal opportunities, whether we are going to treat our fellow Americans as we want to be treated.\"\n\nThat statement came out of a meeting in the White House at which there was a discussion among several of his advisors, then-Vice President Johnson, and JFK regarding their concern about the politics of the whole civil rights movement, because of the damage it was doing among white voters, including white Democrats. In the course of that meeting, there was a wonderful exchange in which one of Kennedy's political advisors raised all these issues and Lyndon Johnson said, \"Wait a minute. This is a moral issue; this is not a political issue.\" There was silence for a minute. Everyone looked at Kennedy and the president said, \"Lyndon's right.\"\n\nOn November 22, 1963, I was in West Virginia inspecting a dam built by the U.S. Corps of Engineers. When I heard the tragic news out of Dallas, I flew immediately back to Washington and hurried into Vance's office and promptly informed him I was leaving the administration. He suggested I reconsider and, in the meantime, I had a new and urgent project: locate the burial place for John F. Kennedy.\n\nThe next day, I met a devastated, zombie-like Robert F. Kennedy at Arlington National Cemetery. In the pouring rain, we walked the perimeter of a 3.2-acre parcel of land above the Memorial Bridge and below the Lee-Custis Mansion. He whispered to me: \"This is where we'll bury the President.\" I will never forget the words.\n\nJohn Kennedy inspired me to enter public service. From ringing doorbells in the 1960 campaign to my work today in association with the National Center on Addiction and Substance Abuse at Columbia University, I have felt this calling. By playing a role in the struggle against substance abuse, I believe I am fulfilling his admonition, \"To those whom much is given, much is expected.\"\n\nKennedy's legacy is one of courage, hope, and determination. He was courageous during the Cuban Missile Crisis, standing up not only to Soviet Premier Nikita Khrushchev, but also to the American generals anxious to launch attacks on Cuba. He inspired hope in millions of young Americans that our nation and the world could flourish by their service to the less fortunate, both here and abroad. And he was determined to expand our exploration of space and land a man on the moon.\n\nThose profound words in his inaugural address, \"Ask not what your country can do for you. Ask what you can do for your country,\" are still a clarion call a half century later.\n Mortimer M. Caplin\n\nCommissioner of the Internal Revenue Service, founder of Caplin & Drysdale\n\n\"He was like a caged lion\u2014walking up and down the room, smoking the small cigarillos he enjoyed so much.\"\n\nI first met John F. Kennedy in 1958 at the University of Virginia where I was teaching at the law school. JFK was the honored speaker at Memorial Gymnasium celebrating the first \"Law Day\" set by President Dwight Eisenhower to mark United States' commitment to the rule of law. John S. Battle, former Governor of Virginia, introduced him as, \"The next president of the United States!\" You could hear the crowd gasp. A Virginia audience. And here, the son of a high Episcopalian minister was presenting a Roman Catholic as the next president of the United States.\n\nWe had a small gathering that evening at the home of William C. Battle, who had served in the Navy with JFK during World War II. Bill was the son of the former governor and one of his law partners. It was a great party\u2014hardly more than a dozen guests. Jackie was there, too. She was lovely.\n\nI taught two of JFK's brothers at UVA law, Robert and Edward, and also spent part of my time as counsel to the Battle firm. JFK knew I was teaching and asked about my recent testimony on tax reform before the House Ways and Means Committee. He asked that I send him a copy, which I was very happy to do.\n\nAs a professor at a state university, I found it prudent to publicly keep out of the 1960 political campaign, although I did write a brief piece in support of JFK in the Charlottesville Daily Progress. Sure enough, JFK won the election and in early December I received a telephone call from his principal aide, Ted Sorensen, asking me to serve on the President's Task Force on Taxation. The group met over a period of roughly a month, putting together a report on reforming our tax laws and stimulating our economy. On January 9, 1961, we delivered the report to the president-elect at Arthur Schlesinger's home in Cambridge.\n\nThe president-elect soon joined us. He was something to see: The pressures and strains of a close campaign had stretched him beyond belief. He just oozed confidence. Now, back from a brief vacation with his father in Florida, he was all tanned and lean. He was like a caged lion\u2014walking up and down the room, smoking the small cigarillos he enjoyed so much. We summarized our report and discussed his reactions and suggestions. He was well aware of our key issues and had already considered them with outside economists and others. We had a lively meeting.\n\nAfter we finished, JFK took me aside for a moment and said, \"Now, I want to talk to you. You'll be hearing from me or from Bobby in a week or so.\" That's all I wanted to know. While it certainly was on my mind, this was the first meaningful signal I'd had that a presidential appointment might indeed take place.\n\nI'd previously spoken by telephone with Robert Kennedy. He asked me to fly up and visit with him in Washington at the Democratic National Committee headquarters where he was quarterbacking appointments for posts in the new administration. I thought my meeting with Bob would be rather simple\u2014his former professor, brother Ted's professor, Bill Battle's law partner\u2014what could be better! And yet, while Bob was very cordial, he was also very business-like. We went through the interview in some detail and he then asked me to write him a letter on my views on tax administration in general and what I thought it would be like to head the IRS. I later wrote the letter, and it ultimately became my proxy statement and commitment on what I'd emphasize and how I'd run the IRS. It was sent to him December 5, 1960.\n\nNot much later, I received a telephone call from a local IRS revenue agent who said he was coming to my office to audit my tax returns. While the agent was still there, we were interrupted by a telephone call from the White House. It was the president's assistant press secretary, Andy Hatcher, who said, \"I'm going to announce this afternoon that you're the president's nominee for commissioner of the IRS.\"\n\nAfter the phone call from the White House, I immediately went to Washington and waited for the formal nomination and hearings. Matters moved far more quickly in those days, and the IRS commissioner was given a good deal more attention. In fact, my nomination was referred to the committee on January 30, confirmed by the committee on February l, and confirmed by the Senate itself on February 6. My official certificate of appointment had President Kennedy's signature on it the following day, February 7, 1961. Very hard to believe!\n\nMy first meeting with all my regional commissioners and district directors at the IRS was set for May 1, 1961. In making arrangements in March, someone in my office suggested, \"Wouldn't it be great if the president could come?\"\n\nI hadn't fully taken into account the magnitude of what was said. But I did call Kenny O'Donnell, the de facto chief of staff at the White House. I told him how important this would be to the IRS and to me. Kenny immediately replied, \"Are you kidding? Come on, there are no votes at the IRS!\"\n\nI called Bob Kennedy after that and raised the same question. He said he didn't know but would see. A week or so later, we were talking about something else and at the close he said, \"Oh, incidentally, Jack will be over.\"\n\nThat was the last time I heard him say, \"Jack.\" It was all so new. After that, it was always, \"the president.\"\n\nThe president did come over on May 1, 1961. It is the only time in history\u2014to this very day\u2014that any president has ever visited the IRS. The president's visit had me walking on water with the whole organization.\n\nPresident Kennedy was very supportive of the IRS, but totally hands off. All the way. There were very few times he called for any special treatment. On one occasion very early in the administration, he called to talk about some IRS related matter and then asked, \"Have you found a house yet, commissioner?\" I had been out house-hunting. You could see the smile on his face.\n\nDuring the Bay of Pigs crisis, I did hear from JFK through Ted Sorensen. The message was for a tax exemption ruling for a private group seeking release of the Cuban prisoners captured on the beach. Supported by the White House, certain prominent citizens\u2014Eleanor Roosevelt, Walter Reuther, Milton Eisenhower\u2014organized \"Tractors for Freedom\" to raise funds to finance exchanges of farm machinery for the prisoners. Castro had agreed to the exchange, but it was important to assure tax deductions for donors. At the same time, an uproar took place in Congress criticizing what some said was equivalent to the U.S. government paying ransom directly.\n\nI was very hesitant about the phone call. I didn't want to issue the ruling unless we had time to study the law. But fortunately, our lawyers saved the day when they uncovered, among other things, an old English statute, the Statute of Elizabeth I, which clearly treated payment of ransom as a charitable act. A favorable IRS ruling was quickly issued.\n\nLater, the agreement with Fidel Castro was changed making food and medicine, not tractors, the swap. It was much easier for the public to accept; but we did have a number of valuation battles with donor companies seeking to increase tax deductions.\n\nOn November 22, 1963, in the middle of an IRS business meeting with my top people, we were suddenly interrupted by the head of our inspection service, Mike Acree. Mike rushed in and whispered in my ear that he'd just heard the president had been shot. In a minute or so it was announced on the general news. Then other people started coming into the office saying, \"We just heard it over the radio.\" The room was hushed. The meeting had just begun; I looked around and said right away, \"We have to call this off. We better go home.\" The room cleared immediately.\n\nNext day, we\u2014all the Cabinet officers and presidential appointees\u2014were invited to the White House to view the casket. I later went downstairs to get my automobile and saw, standing against the wall waiting for his car, President Lyndon Johnson. He saw me, immediately stepped forward and put those big hands of his on my shoulders saying, \"Mortimer, I need you, Mortimer. I need you.\" He was trying to get as many Kennedy people as he could to stay in the administration with him.\n\nJohn F. Kennedy was an inspiration to all of us. He was a vibrant man with extraordinary intelligence and vitality. When he came to office after President Eisenhower, there was an unbelievable excitement throughout the country.\n\nHis inaugural address stressed the importance of public service, the theme he enlarged upon later at Vanderbilt University when he urged all, particularly the student body, \"to act, to enter the lists of public service.\" I had former students writing and calling\u2014and young people throughout the country\u2014who wanted to be a part of this, this special period in U.S. history. I put JFK very high on the list of our presidents.\n\nJFK was a devoted student of Thomas Jefferson, who in writing to an old friend said, \"There is a debt of service due from every man to his country, proportioned to the bounties which nature and fortune have measured to him.\"\n\nJefferson wrote this in 1796.\n\nSome 165 years later, in his inaugural address, JFK underscored this same call to service with the message, \"And so, my fellow Americans, ask not what your country can do for you. Ask what you can do for your country.\"\n\nIt was a privilege for me to serve, and I look back on this time as the best years of my life.\n Charles U. Daly\n\nWhite House congressional liaison, writer\n\n\"I feel fortunate to have been associated with this person who really did have vision. He was not perfect. From the waist down he had the same habits of his father. So what?\"\n\nI went to the Columbia Graduate School of Journalism and was awarded an American Political Science Association fellowship under which I spent eight months working with congressmen\u2014three in the House and three in the Senate. I chose Stewart Udall because he was from the West. In California, I had become interested in conservation. He (Udall) had a very open office. He was one of Kennedy's early supporters, so he suggested I take the other half of my fellowship with Kennedy.\n\nI had not been active in politics at all. By instinct, I was an FDR-Stevenson Democrat. I didn't think much of old Joe Kennedy, due to his record as weak on Hitler, and I didn't know a whole lot about his kids. But I thought, \"This could be interesting.\" So I went over to JFK's office, where I was welcomed due to Udall's good boost.\n\nDuring my fellowship with Kennedy, the best relationship I had was with Ted Sorensen. I had combined my fellowship with work for the Democratic Study Group which provided campaign information for House members more liberal than their national party. It became hectic. I could sense too many campaign stooges only focusing on positions in what they hoped would be a new administration. I decided to opt out.\n\n(A few years later) I was in the shower one day in our home in Menlo Park, California when my wife said, \"The White House is on the phone.\" It was Larry O'Brien, Kennedy's special assistant for congressional relations. He said many of the northern and western liberal members of Congress were being overlooked. He asked if I would come back to help out. We agreed I would call back ASAP.\n\nI (traveled to Washington) and went down to the Executive Office Building. They had no record of me! A guard called to a sentry at the northwest gate of the White House who said I was expected there. I went over and was ushered in and met Larry and his staff. He showed me my office. I was amazed: a big corner office on the second floor of the West Wing. \"That's yours.\" It was huge.\n\nJohn Kennedy's crew was very compartmentalized. Staff in the West Wing apparatus had little contact with staff in the East Wing. The exception was Arthur Schlesinger Jr. I think Kennedy figured him to be what he was\u2014a most able historian who worshipped at the Kennedy altar. Members of Congress had little direct contact with the president, so it was up to me to make my charges feel close to him.\n\nA good example of Kennedy's personal touch involves Dick Donahue, who was a vital member of O'Brien's staff. There was a congressman from Toledo, Ohio, Lud Ashley, whose dad had busted his hip during celebrations just before the inauguration.(The congressman's father) Old Meredith, who enjoyed a drink as much as the rest of us, ended up in Bethesda Naval Hospital. The night before the inauguration, at a party put on by Joe Kennedy, Congressman Ashley ran into Dick, whom he had never met before. Dick, as irreverent then as he is now, said, \"I'm sorry your father got drunk and broke his ass.\" Lud responded, \"Well, fuck you, pal.\" A great start to a new president's key staff man to start a relationship with an important member of Congress!\n\nOn the morning of the inauguration, Meredith Ashley was in in his bed of pain at the hospital. A Marine sergeant, his uniform still wet from the blizzard outside, handed him a big brown envelope. \"I've been ordered to deliver this to you.\" Inside was an advance copy of the inaugural address. At the top, John Kennedy had written: \"Dear Mr. Ashley, I am sorry to hear from Dick Donahue of your unfortunate accident. Hope you will enjoy reading parts of this address I'm going to give later on today.\"\n\nLater, when I introduced myself to Congressman Ashley as the new presidential assistant responsible for White House relations, he told me the story, adding, \"You've got an easy job with me.\" Lud and I became very close and remained so until the day he died.\n\nNovember 22, 1963, started as a slow day. It was a Friday and most of the Congress usually worked Tuesday through Thursday. I was sitting at round table in the White House mess, at the members-only table, that most exclusive of restaurants operated by the Navy in the West Wing's basement.\n\nAround 1:30, (Presidential Assistant Jack) McNally came in and just said very quietly, \"The president's been shot.\"\n\n\"How bad?\" I asked.\n\n\"I don't know,\" he replied.\n\nI darted up to my office and called (my wife) Mary.\n\nShe said, \"They've killed him.\"\n\nI responded, \"No they haven't. I've been shot and I'm not dead.\"\n\nI went down to the press office. No one was there except for (Special Counsel to the President) Lee White. (Assistant Press Secretary Malcolm) Killduf was in Dallas. I read the first AP bulletin, which said, \"President Kennedy was shot today as his motorcade left downtown Dallas.\"\n\nThen came the second bulletin. He was dead.\n\nThe president was going to be taken to Andrews Air Force Base. I thought about going there, but decided I didn't want to go out there\u2014or anywhere. So I sat down on my old brown leather sofa for a while, watching television as the president, inside a box, was being lowered on a forklift and then into an ambulance headed for Bethesda and an autopsy.\n\nAround midnight, I walked out on the little patio behind the president's office. It was a beautiful night. Henry Wilson (White House Liaison with the House of Representatives) came by, followed by Evelyn Lincoln, the President's secretary, hugging herself.\n\nAround four o'clock someone from the northwest gate of the White House called. \"The president's being brought in.\" I went down. There were floodlights focused on the road and the shadowy mass of a crowd beyond. I went to the East Wing. The windows were covered with black drapes. I left. Sometime later I wandered back. The casket was atop an elevated black catafalque. Now there was an honor guard and priests. Fortunately, the lid was closed. (During the Korean War) I had seen heads shattered by bullets, so I tried not to envision the mess under that lid. Then I thought of Mrs. Kennedy in that car, her husband's brains on her outfit. That was tough.\n\nOn Saturday morning, the mail arrived. Most of it was sent before the assassination. But there was one letter\u2014I don't know how it arrived so fast, perhaps hand-delivered\u2014sent by a congressman from California. I read it then threw it down on the sofa. About that time, a guard at the northwest gate called, \"Mr. Daly, Jimmy Breslin's out here. He says he's your pal and he won't go away. Do you want me to run him off?\" Breslin was working for the New York Herald Tribune at that time; I had known him for a long while. I said, \"Have him come up.\"\n\nNow Breslin is in my office the day after the assassination, saying, \"Well, I just want you to know that right now the whole world feels just the way you do.\"\n\nI said, \"Oh yeah? Take a look at this letter.\" And I handed him the letter from the congressman.\n\nJimmy said, \"Geez. Now I've seen it all.\"\n\nI said, \"You can't do anything with it.\"\n\nHe said, \"That guy crawls.\"\n\nOn Sunday, (Press Secretary Pierre) Salinger came in and handed me a copy of the Herald Tribune and said, \"I don't know where Breslin got this goddamn letter.\"\n\nThe column read:\n\nOn Saturday, almost an entire 24 hours after the murder of the President of the United States, a member of Congress who represents one of the great centers of culture and decency for which California is so famous, sat at his desk on Capitol Hill, dictated a letter to President Lyndon Johnson.\n\nDear Mr. President:\n\nJust to say that I and, I hope, all Members of the Congress believe it is our duty to support you and stand behind you at this time. I would also like to say that in 1960, at the Democratic National Convention in Los Angeles, I voted for you against John F. Kennedy.\n\nI tried to call Jimmy's house. He knew I was going to scream at him. So Rosemary (his wife) told me he was in the shower. When I finally reached him, he said, essentially, \"That freaking congressman is as dead as Kennedy.\" I hung up.\n\nOn Monday, we went to St. Matthew's Cathedral. I watched everyone hustling for aisle seats. I thought, \"Are they praying for their own survival or a promotion by LBJ, or are they praying for Jackie and the country?\" I still wonder.\n\nGod did not enter my mind during the service, and if he had, I would have been pissed off at him for fucking up this whole deal.\n\nWe came out of the church and there were two long lines of limos. Two or three single cars right behind the hearse then double rows of cars farther back. First, of course, the singles carried Jackie, the Kennedy clan, then Johnson. The rest of us got into the other cars. One row of VIPs was scheduled to follow the singles. The head cars of that group included this midget Haile Selassie (Ethiopian emperor) with all his medals, towered over by (French President Charles) De Gaulle. All of a sudden, Jack McNally jumps in front of the VIP row, raises out a mighty arm, and yells, \"Hold it!\" He directed all the staff cars to proceed. Then, and only then, he allowed the VIPs to move.\n\nI think John Kennedy's legacy is that he inspired millions. I feel fortunate to have been associated with this person who really did have vision. He was not perfect. From the waist down he had the same habits of his father. So what?\n\nHe really did make a difference in the world. I think that is where legacy and inspiration meld. Another part of the legacy is Robert Kennedy. Ethel and company cannot get over the fact that Bobby was never president. Again, so what? The fact is that much of his devotion to civil rights, and it was devotion, came from watching his brother learn about civil rights.\n\nPossibly the biggest part of John Kennedy's legacy was Ted Kennedy. He was a spoiled guy. You talk about nepotism with Bobby? Ted was handed a seat in the United States Senate. He got into the bottle of despair and couldn't get out. He left a girl to die at Chappaquiddick. Then he went on to become the most significant senator for shaping our present and, I hope, our future as a country. A woman who was at Chappaquiddick that night told me she said to Ted, not long before his death, \"I am so proud of what you have become.\"\n\nOverall, we are left a lasting legacy of this forever young president who, today, would be ninety-five years old.\n John Gunther Dean\n\nSenior State Department diplomat\n\n\"Kennedy was interested in moving the world forward.\"\n\nAS you know, 1960 was the big year\u2014when African nations gained independence. In 1962, Kennedy invited the President of Togo, Sylvanus Olympio, to come as an official guest of the United States. He was among the first African presidents to come to the White House. Olympio came away feeling that Kennedy was interested in helping the newly independent African nations. I had been, in 1960, the diplomatic adviser to Olympio in Togo. The meeting with Kennedy was very cordial. Kennedy was so nice.\n\nHe said, \"Here's my friend John, who works with you.\" He had a way of doing things that put people at ease.\n\nKennedy was interested in moving the world forward. That was well understood by the newly independent African countries and also by the rest of the world. Eisenhower, of course, had been the commander of the U.S. and Allied forces during World War II, and he knew North Africa well. He showed an interest in North African affairs. Kennedy sought to establish relations with newly independent nations in sub-Saharan Africa.\n\nIn November 1963, I served as an advisor to the U.S. delegation to the United Nations General Assembly in New York. They needed French speakers, such as myself. I was assigned to the head of the American delegation, former governor Adlai Stevenson. When we heard about the assassination of President Kennedy, he called for a formal meeting of all the U.N. delegates. Stevenson told them what had happened. Nearly everyone, and there were hundreds of people, came forward to shake hands with Stevenson to express their dismay at the loss of Kennedy. It was interpreted in a very personal way, as if they all had lost a great friend. All the delegates were affected by Kennedy's tragic disappearance from the scene.\n\nEven after Kennedy's death, many people around the world associated the Kennedy name with efforts to avoid war. You must remember the Kennedy-Khrushchev confrontation over Cuba. For many, that's when this association started.\n\nI was definitely influenced by Kennedy and what he stood for. In 1961, he issued a personal decree that every U.S. ambassador to a foreign country was his personal representative to that country, and therefore, each ambassador became responsible for all the services in the embassy. Very often, the various services in each embassy\u2014the military people, the State Department people, the antidrug people, the intelligence people, the political people\u2014were at odds with each other. They would send messages, and sometimes complaints, to the president. Kennedy's order made each ambassador the coordinator of all U.S. policies and interests, thereby enabling the embassy personnel to speak with one voice. This was a major contribution. I later found out how important this was when I served as ambassador. Unfortunately, this practice has been abandoned for the last thirty years.\n\nThe great legacy of the Kennedys is giving people hope. Hope is what religion is based on.\n Clint Hill\n\nU.S. Secret Service agent\n\n\"My objective was to get up on the back of the car and form a shield for President and Mrs. Kennedy to prevent any more damage from being done. Any agent would have done the same thing. I happened to be one.... I did have that opportunity, but it was too late.\"\n\nKENNEDY called each and every one of us agents by our first names. Johnson, when I first met him, was not too pleased to meet me, because I had been with the Kennedy detail. Jerry Ford was extremely friendly. His youngest son and my oldest son were high school classmates. He lived not far from where I was living in Alexandria, Virginia.\n\nOne of those situations where I learned of balancing my obligations and loyalty to Mrs. Kennedy involved an obnoxious UPI photographer, Roddy Mims, who once breached security at National Airport to take photos of her. As the agent responsible for the first lady's protection, I grabbed him, took his camera, and took all his film. I turned him over to the police because of his activities.\n\nWhen I got to the White House, I was advised that the president wanted to see me. So I went in to the Oval Office and there was the president and Pierre Salinger, his press secretary. I explained to them what happened. The president looked at me and said \"Unfortunately, you're going to be the scapegoat in this situation. We can't afford to have the press angry at us for what happened at National Airport and so you are going to get the blame. And we're going to return the film.\"\n\nWhen things like this happened, I had to weigh the risk of being chastised by the president himself. Eventually, the first lady gave me more and more responsibility for many things.\n\nWhen we traveled\u2014she often traveled alone overseas, or with Lee, her sister\u2014she no longer brought staff, except for a personal assistant. I became social secretary, press secretary, personal shopper. Whatever was necessary. I was trying to establish an environment in which she could function and do those things she wanted to do. And do them safely and with as much privacy as possible. It was difficult, but that's what I tried to do.\n\nNeither he nor Mrs. Kennedy ever carried cash. If she did, I was not aware of it. We'd go places and the children were along and if she wanted them to have ice cream or something, I would buy and later submit a personal expense request to the president's secretary, Mrs. Lincoln, and I'd be reimbursed. I'd buy Mrs. Kennedy magazines, newspapers, whatever was necessary.\n\nI remember when she was giving birth to John Jr. at Georgetown Hospital. After the baby was born, she was in her room and asked for the agent to come in. She wanted to borrow twenty dollars because she needed to tip her hair dresser. The agent didn't have any money. So he got the money from me. He gave it to her, and I got it back from Evelyn Lincoln.\n\nPrivacy for Mrs. Kennedy was difficult. She was well recognized, respected, and loved by people. They always wanted to be near her or associated with her in some way. That made it difficult.\n\nAfter leaving in 1964, yes, she called occasionally with questions about the children, recommendations about travel. In 1966, she sent me a Christmas gift, a book the president had written, and she inscribed it to me. That inscription is on the back of my book, Mrs. Kennedy and Me.\n\nThe last time I spoke with her was in 1968 at Robert Kennedy's funeral. It was very brief because, at that time, I was the agent in charge of presidential protection. I was there with President Johnson. It wasn't conducive to having any kind of conversation. I just expressed my sympathy and deep condolences.\n\nOn November 22, when I first heard the gunshots in Dealey Plaza, there was no hesitation at all. It's a reaction. And you immediately react. You do what we are taught to do and that's provide cover and hope we can evacuate before any damage is done.\n\nIn this case, the damage was already done by the time I got to them. In Dallas, my objective was to get up on the back of the car and form a shield for President and Mrs. Kennedy to prevent any more damage from being done. Any agent would have done the same thing. I happened to be one. The only one to be in a position to do anything because of the way it developed that day.\n\nNone of the other agents had a chance. I did have that opportunity, but it was too late.\n\nI wedged myself up on top of the back seat of the presidential limousine as high as possible to form a shield.\n\nPresident Kennedy has a mixed legacy. His speech at the inauguration\u2014\"Ask not what your country can do for you. Ask what you can do for your country\"\u2014that resonates still today. Some people go into public service and try to serve their country. He will always be known for establishing the Peace Corps and going into space and landing a man on the moon. That was his long term goal and it was accomplished. There were some bad moments like the Bay of Pigs and some good moments, such as when he prevented the nuclear holocaust with the Soviet Union during the Cuban Missile Crisis.\n Harold Hill\n\nWhite House communications officer\n\n\"My boss responded, 'What are you going to do? The president wants it.'\"\n\nI saw John Kennedy in passing a few times in Washington while I was still working with President Eisenhower. The first time I really met John Kennedy was in February of 1961, shortly after his inauguration. Within short notice he knew me by name. My job and the others' meant we put mikes in front of the president all the time. When you're face-to-face with him, you get acquainted pretty quickly.\n\nHe was very direct. If he asked you something, he wanted an answer. I saw him angry or unhappy only once\u2014when somebody's stupidity got him behind schedule. It was OK for him to get behind on his own, but not if it involved someone else's mistake.\n\nThe mission of our office was to record all public utterances of the president\u2014no matter where or when\u2014even though they might be social or political in nature, such as toasts at state dinners and the responding toasts. In Berlin, when he did the famous \"Ich bin ein Berliner\" speech, I was in the plaza setting up for his address to the crowd. Following the speech there was to be a luncheon. I had plenty of time to handle both, I thought.\n\nAll of a sudden a Secret Service agent came over to me and said, \"The president wants to see you. Now.\"\n\nSo I left my position in the adjacent room and walked up the president. He leaned over to me and said, \"Harold, they do the toasts here before we eat the meal, not afterward. Did you know that?\"\n\nI replied, \"No, Mr. President,\" and did what I needed to do to get the recording.\n\nDuring the speech in the plaza, I've never seen anything so well organized. They had runways five to six feet wide every thirty to forty feet. It was humid and the crowd was so large and crammed that when someone fainted from the heat, the person would be handed overhead by other people to one of the openings for first aid.\n\nOne of my fondest memories of President Kennedy involved his secretary, Evelyn Lincoln. She always kept bags and bags of PT 109 tie clips in her desk to give as gifts. When we traveled with the president, I always had a pocketful of them. They would help open doors. One day, I was at her desk and she said, \"Here, you deserve this.\" She handed me a white bound copy of the president's inaugural address. There were about a thousand of them made. All of a sudden the president walked out of his office and she said, \"Here, sign this for Harold.\" So he stopped, and I watched as he inscribed it for me and handed it to me.\n\nHe was a voracious reader. One day, we were recording his voice for a film celebrating the Chinese Year of the Tiger. So he came down and asked me, \"This is for the thing for China, right? What am I supposed to say?\"\n\nJust then, a guy came in with the script\u2014three or four pages, double-spaced. I watched him glance at each page for just a moment. He never looked at the script again and we did the recording. I later absconded with the script and read it while playing the tape. Except for a few \"ums\" and \"ahhs,\" it was verbatim. One reading.\n\nIn October of 1962, I was with the President Kennedy in Chicago. He was supposed to fly to Seattle to close the World's Fair. Air Force One stayed on the runway for a long time in Chicago. I couldn't figure out why. Finally, the plane took off, and they announced to the press that the president had a high temperature and was going back to D.C., rather than Seattle. Three days later, on Monday night, JFK made a nationally televised speech about \"Cuber\"\u2014the \"Cuber missiles.\"\n\nI flew on a red-eye that night, got to the White House, and was told the president is going to speak with the prime minister of England, Harold Macmillan. They were going to use a secure line with a scrambling device on both ends in the Situation Room, and they want it recorded.\n\nI said, \"Ohhhhh no. It's against the law to record a call on a scrambling device.\"\n\nI knew what the law was. Recording a call like that, you are subject to five years in prison and a fifty-thousand-dollar fine.\n\nMy boss responded, \"What are you going to do? The president wants it.\"\n\nI replied, \"I'm not doing it unless the president himself tells me he's authorizing the recording.\"\n\nSo JFK came down to the Situation Room and the duty officer was standing tall and straight; so was my boss, Colonel McNally.\n\nThe president said, \"Hi Harold. Are we all set to go?\"\n\nThe duty officer responded, \"Yes and no.\"\n\nThe president was quite taken aback. \"What do you mean, 'Yes and no'?\"\n\nThe officer replied, \"There's some question about using recording devices on cryptographic equipment.\"\n\nThe president turned to me. \"You can do it, can't you Harold?\"\n\n\"Yes sir.\"\n\n\"Then let's do it.\"\n\nI took that as a direct order. So I recorded their conversation, and they got into the nitty-gritty. Which platoon was going to hit which beach and what time. It was a long and elaborate phone call.\n\nOn Friday, November 22, another member of the communications staff and I were sitting in a rental car outside the hotel in Austin waiting for the third member of our crew. We were about to drive out to Bergstrom Air Force Base and prepare phone lines. At that time, the president always had to be within two minutes of a secure phone, and the LBJ ranch then was on a phone system with twenty-three other people run by a man with a hand-cranked phone, so we were working with the local phone company to set up a secure line.\n\nIt was just after lunch, and a couple of people walked by and said, \"Did you hear? The president's been shot.\"\n\n\"Oh sure,\" I replied.\n\nWe didn't believe it. We had heard similar things before, and of course, they were always hoaxes. Within a minute or two, two businessmen in suits came out of the hotel, and I overheard them talking to each other about the president being shot in Dallas.\n\nI said, \"Turn on the radio,\" and we heard the news. Indeed, he had been shot.\n\nI thought, \"That's awful. But as hard as this is, we now have to serve the new man.\"\n\nI ran into the hotel, up to the White House communications center, a special room set up with a secure line and a secure teletype. I sent a message on the teletype to our office in Dallas: \"What's the story?\"\n\nA moment later I received the reply: \"JFK dead. LBJ to be sworn in. More later.\"\n\nSo I went to my room, turned on the TV, and watched Walter Cronkite deliver the news of the president's death. A little while later, I got a call from Colonel McNally. He said, \"You're the most experienced person I've got. We need you to help put in a communications system at the ranch.\"\n\nI drove out to the ranch to begin the plans for laying cable\u2014telephone lines\u2014underground for several miles. Then I got a call from the White House. \"Mrs. Kennedy is asking for you. When can you get to a phone?\" I drove back to Austin\u2014about an hour. It was now Saturday. I spoke to Tish Baldrige, the White House social secretary who worked with Mrs. Kennedy.\n\nTish asked, \"Can you get back to town overnight? Mrs. Kennedy is asking for you to record the entire funeral.\"\n\nSo I flew from Austin to Dallas and then to Washington, and upon landing, the pilot came on the sound system and said, \"Will Mr. Harold Hill identify himself?\" I was the first one off the plane, and a White House car was waiting for me. It was Sunday the twenty-fourth, between two and three in the afternoon. We had only six people in the office, and we arranged to have police escorts to enable having an open mike at the Capitol, at St. Matthew's Cathedral, and then at the Arlington National Cemetery.\n\nTish came to see me two weeks later and gave me an invitation to and program for President Kennedy's service at St. Matthew's, with a handwritten thank you note from Mrs. Kennedy. To me, John F. Kennedy was a young, vital newcomer who led us out of the desert. He was the first really vibrant president we had had for some time, probably since Theodore Roosevelt.\n Lem Johns\n\nU.S. Secret Service agent\n\n\"I got off and walked right by President Kennedy's car. I saw the bouquet of flowers and blood on the upholstery. My first thought was, 'This is serious.' There was a clear indication someone had been hit.\"\n\nDURING the inauguration of President Kennedy, I was given the assignment of protecting family members. I was up at the top of the stairs at the Capitol for the inauguration. I met President Kennedy briefly that day.\n\nI got to see President Kennedy more regularly when I came back to the White House with the security detail for Vice President Johnson. I saw him as a very energetic young man, concerned about this country. He was very competent, and I thought he would carry out his plans, but he did not have enough time.\n\nDuring the Cuban Missile Crisis, we expanded security for the vice president to twenty-four-seven. There were three shifts. I observed President Kennedy coming to the meetings from the mansion. I knew it was extremely serious. He seemed very concerned, very worried. It occupied him totally. And as the tension mounted, it showed on his face.\n\nOn November 21, 1963, I accompanied Vice President and Mrs. Johnson via helicopter from the LBJ Ranch to San Antonio and met Rufus Youngblood, the senior agent in charge of the vice presidential detail. In a few minutes, the vice president and Mrs. Johnson met and joined the president and Mrs. Kennedy on their arrival.\n\nWe visited the Space Center in Houston, Texas, and then went on to Fort Worth. We arrived after dinner and stayed at the Texas Hotel. The next morning, the president spoke at a rally across the street, then at the chamber of commerce breakfast in the hotel. We then departed for the airport and flew to Love Field in Dallas.\n\nWe got the vice president and Mrs. Johnson into Love Field first. They greeted the Kennedys upon their arrival. The President and Mrs. Kennedy worked the lines of people by the fence, and then got into the cars for the motorcade. The president's limousine was an open convertible, although it had a clear bubble top in two pieces. It was kept in the trunk of the car. Immediately behind the president's car was a Secret Service car, then a Lincoln convertible that contained the vice president and Mrs. Johnson and Senator Ralph Yarborough. Behind the vice president's car, I was in a four-door sedan with two other agents. All three of us were assigned to LBJ. We had gone through downtown Dallas at what I call a parade speed, and as the cars slowed down, the agents were running behind the car with the president.\n\nI was sitting on the right side in the rear of the car with the window down, and was watching the right side. As we started down the incline on Elm Street, I heard the shots. I saw people on the grassy knoll hitting the ground. I started opening the door and jumping out to run to the vice president's car, which had slowed. Because of the way the door of my car was hinged, I had to jump backward. By the time I got out to start running forward, my car was speeding up. They left me standing in the street. Behind me were two convertibles with photographers, and the second one contained one of the photographers assigned to the White House. He saw me and said, \"Hey. That's Lem! Wait, pick him up.\" I jumped in the back of their car, and we went underneath the overpass.\n\nWe pulled up to the Trade Mart. There was a police officer with a three-wheeled motorcycle. I identified myself and asked, \"Where have they taken President Kennedy?\"\n\nHe responded, \"Right up the street to Parkland Hospital.\"\n\nI said, \"Would you mind taking me up there?\"\n\nHe said, \"No,\" and I jumped onto the padded seat behind his seat.\n\nThe officer took me to the emergency entrance. I got off and walked right by President Kennedy's car. I saw the bouquet of flowers and blood on the upholstery. My first thought was, \"This is serious. There was a clear indication someone had been hit.\"\n\nI walked into the hospital. I ran into an agent and asked, \"Where have they taken the vice president?\" He said they had separated him from Mrs. Kennedy and the others in the group. I did not see Mrs. Kennedy. I went into a room where the vice president was waiting.\n\nSo the vice president and his staff talked. \"Do we keep Air Force One at Love Field or send it to the air force base in Fort Worth?\"\n\nThey decided they would keep it at Love Field, close the airport, and place a security contingent there. Then it became, \"How do we get Vice President Johnson to the airport?\" I was asked to find an exit we could leave from without going through the press area. I went looking for a hospital administrator. We got the police cars lined up and they left the hospital. There were no photos of Vice President Johnson leaving Parkland.\n\nI saw the \"bag man\" in civilian clothes\u2014the guy who carries the briefcase with all the military codes. He was walking in the hallway. I told him to come with us. Then I saw (LBJ aide) Jack Valenti and pulled him into the car to go with us to Love Field. We got to the airport and went aboard the plane.\n\nThe vice president had called Bobby Kennedy and the president's mother, Rose Kennedy. The big question was whether he gets sworn in as president on the plane, or whether we wait until we're in Washington.\n\nMrs. Kennedy and the president's casket arrived. Some of the seats in the rear of the plan were removed to make room for the casket and to give Mrs. Kennedy some privacy.\n\n(United States District Court) Judge Sarah Hughes arrived. The mood was very solemn. When they got ready for the ceremony, the vice president asked someone to request Mrs. Kennedy to witness the swearing-in. She came forward, still in the same dress. Her leg was covered with blood. The vice president and Mrs. Johnson greeted her warmly. After the swearing in, the judge got off the plane and we took off for Washington.\n\nOver the next few days, I stayed on duty for the funeral events. We had two groups of agents. It did create some uncertainty. The guys on the Kennedy detail, of course, were very upset. The loss they felt was so great. Rufus later became the agent-in-charge for President Johnson, and I became the assistant agent-in-charge.\n\nJohn Kennedy's legacy is extremely good. I'm not really aware of all the bills or programs, but just as a representative of the country, he did a good job. One of the most telling things was when he went to West Berlin and made the famous remark, \"Ich ben ein Berliner.\" It was a city and a country divided. That was quite good.\n Nicholas deB. Katzenbach\n\nDeputy Attorney General, Attorney General in the Johnson administration\n\n\"President Kennedy asked, 'Where's the vice president?' Someone said, 'I don't think he was invited.' And the president replied, 'Go get him. We need him.'\"\n\nI first met John Kennedy a few days after he took office as president. He had a reception for all the new appointees for the government. I remember the receiving line with him and Jackie. And he said to me, \"Mr. Katzenbach. You're going to be dealing with Bobby and the Justice Department. I'm so glad.\"\n\nAnd I looked around at this enormously crowded room and thought, \"How did he know my name?\"\n\nHe certainly had done a lot of homework; although he likely received my name on a card from the aide who was introducing the guests, he knew I would be in the Justice Department. It was very flattering.\n\nSo my first impression of him was one of awe (laughter).\n\nMy more substantive impressions of him came later in meetings with him, as things went along, and how he reacted to problems and people. The first meeting that I recall having with him, which I describe in the book, had to do with the Federal Power Commission. Kennedy wanted to appoint Joseph Swindler as head, and the existing head, Jerome Kuykendall, did not want to resign. There was a U.S. Supreme Court opinion on the independence of the commissions that said you couldn't fire people because they were of a different political persuasion.\n\nMy assistant Harold Reis said, \"I think that's wrong. I think the president probably could fire the chairman as chairman, but not from the commission.\" Later, I went over to the White House on the issue to talk with Ralph Dungan, who was handling appointments for the president. I knew that Bobby was in with the president, and so I got in touch with Bobby and asked if I could come in to join him on that matter. So he said, \"Come on in.\"\n\nSo I came into the Oval Office. The president was there. And Ralph came in the other door. He said, \"I agree with Nick. He (chair of the commission) either ought to take this, or we litigate it.\"\n\nSo, the president took one look at it and said, \"I agree with Nick, too.\" That was very nice. The first time he called me by my first name. So my first two meetings with him were very nice. I came to admire his judgment (laughter).\n\nPresident Kennedy had all of the attributes one can reasonably attribute to a Harvard graduate. He was personable, well-mannered. He was intelligent and he was articulate. And he got along very well with people. He had all of the social graces that you have with that kind of an education and money as well.\n\nBut he had something more, which I think he acquired in World War II. The thing that always interested me about his naval career and his service as the captain of PT 109 was not the mission or the dangers, but the responsibility he felt for his crew. He risked his own life to save his crew. That was an attribute I admire greatly. He demonstrated loyalty to people, and he had responsibilities from his position, and he had to meet those responsibilities.\n\nHe took that sense of responsibility into the presidency.\n\nI have no doubt that President Kennedy and his brother Bobby, despite not knowing much about civil rights, did not approve of the second class citizenship that was evident and the very nearly caste system that existed in the South at that time. I think the problem was not whether you wanted it or liked it; the problem was what could be done about it.\n\nSo in that sense, the moral problem was, \"What are my obligations? My obligation is to make it better. But how in the name of God do I do that?\" And that was the most difficult problem we faced during the Kennedy presidency.\n\nThat was true of both the integration of the University of Alabama and the release of the Cuban prisoners. Allow me to explain.\n\nPresident Kennedy felt that he had made a bad mistake on the Bay of Pigs, which he had. All these Cuban prisoners were being held, and I think he felt a strong obligation for them. As long as they were there and being held hostage, he had a moral responsibility for them.\n\nBoth incidents, I think, reflected his position, essentially, \"I have a position of authority and I have a responsibility for these issues, and therefore, I have a moral obligation to do something.\"\n\nAnd the seed that grew into that moral obligation was planted in the Solomon Islands with PT 109, as well as earlier, during his time in England when his father was ambassador to the Court of St. James. He saw Hitler rising to power. He saw that morality and politics are not necessarily different.\n\nIn regard to the introduction of civil rights legislation, which the president did right after the integration of the University of Alabama, he and Bobby and (Assistant Attorney General for Civil Rights) Burke Marshall were probably the only people who were really for it in the White House. Nobody, maybe not even those three, thought it was possible. And my guess is that Bobby was urging it, but I don't think Bobby thought he was going to succeed.\n\nI think the squabble between Bobby and Johnson, which President Kennedy did not share\u2014but a lot of his staff did\u2014was harmful not simply to Johnson, but also to Kennedy. And one of the ways it was harmful was to civil rights. I say that not because I think Johnson should have or could have done anything particularly outside of what he did in his role on the Federal Employment Commission regarding government contracts for minorities. He did not have much to do during the Kennedy presidency, which was a mistake.\n\nI can remember on one occasion, it was a meeting on legislative tactics on the first civil rights bill that we had drafted. All the interested people were there. But no Lyndon Johnson. And President Kennedy asked, \"Where's the vice president?\" And someone said, \"I don't think he was invited.\" And the president replied, \"Go get him. We need him.\"\n\nIt was sad. And I'm sure Johnson knew what had happened. He came to the meeting and pointed out a section on fair employment was essential for the bill, and I think he was right.\n\nExcerpt from Some of it was Fun\u2014Working with RFK and LBJ by Nicholas deB. Katzenbach \u00a9 2008 (Used with the author's permission):\n\nOn November 22, I went to lunch at a restaurant near the department with Joe Dolan (colleague from the Department of Justice Anti-Trust Division). We had just been seated when we heard the radio behind the cashier's desk announce that the president had been shot in Dallas. We jumped up and ran back to the department, where we went to the attorney general's office. His secretary, Angie Novello, was there. Bobby was at Hickory Hill. He had had a meeting of his organized crime task force that morning and had taken Bob Morgenthau, the U.S. attorney for the Southern District of New York, home for lunch. Angie told us that it looked as though it might be fatal and that the president had been rushed to the hospital. The FBI was keeping her and Bobby informed.\n\nIt is impossible to describe the feelings all of us in the department had. It seemed as if all the lights had gone out, all senses down. I went to my office and after a few minutes called Bobby, feeling I should, but having no idea what to say. He answered the phone.\n\n\"Hoover just called me,\" he said. \"The president is dead. I think Hoover enjoyed giving me the news.\"\n\nI don't know what I said. There was nothing adequate to say. I turned on the TV and just sat. People started to come into my office. Mostly we were just silent. The phone rang. It was Bobby.\n\n\"They want to swear him in right away, in Texas. That's not necessary, is it?\"\n\nHe could not bear the swearing in. In some mythical sense, I think he wanted President Kennedy to return to Washington on Air Force One as if he were still president. I think, too, that swearing in JFK's successor in Texas was an offensive idea to him.\n\n\"No,\" I said. \"Not necessary.\"\n\n\"They want to know who can swear in the president. Does it have to be the chief justice?\"\n\n\"No,\" I replied. \"Anyone who can administer an oath\u2014a federal judge, for example.\"\n\nHe rang off. I was, frankly, appalled that Johnson's people were seeking legal advice from Bobby at this time. I could understand, however, and even sympathize for a prompt swearing in as a demonstration to the world that the government was intact and functioning.\n\nA few minutes later the phone rang again. It was (LBJ aide) Jack Valenti, calling from Air Force One.\n\n\"We want to swear Vice President Johnson in as president. The attorney general said you would have a copy of the oath.\"\n\nI got up and found a copy of the Constitution and read him the oath...\n\nThose days between the assassination and President Kennedy's funeral are something of a blur in my memory. I do remember a group of us going to the White House to pay our last respects to our fallen leader, a meeting arranged by Bobby to include an unusually large number of career attorneys. I can remember watching the procession to the Capitol from the department as it passed directly below us; the funeral with its foreign dignitaries and Bobby in close attendance on Jackie; and finally Bobby returning to the department the day after the funeral and trying his best to pretend things were normal. He looked like the ghost of his former self, and his efforts to tell humorous stories about events at the funeral were brave, but flat.\n\nI view this period as eight years of Kennedy and Johnson, since I served both presidents. Those eight years are remembered for both\u2014some of the successes on foreign policy going to Kennedy and other successes on domestic issues going to Johnson. Kennedy handled the economy much better than Johnson, as well as foreign affairs. I may have a bias because I served in both. And maybe I'm protecting myself (laughter).\n Newton N. Minow\n\nChairman of the Federal Communications Commission\n\n\"'Mr. President, I think you'll be pleased to know that Jackie's program had a higher rating than your press conferences.'\"\n\nI first met John Kennedy in 1954 or 1955. I was a friend of his sister and brother-in-law, Eunice and Sargent Shriver, and John Kennedy came to Chicago to give a speech. We (my wife and I) were invited by the Shrivers to accompany them to the speech, and then later there was a small reception in the hotel. I had followed his career before, and I was very taken with his intelligence, his charm, and his wit; I liked him immensely.\n\nI had decided prior to the 1960 campaign that I hoped Adlai Stevenson would not run again and I urged him to support John Kennedy. But when he (Governor Stevenson) didn't do that, I stayed supporting Adlai Stevenson all the way until Kennedy was nominated at the Democratic convention in Los Angeles. Then after that, we all supported Kennedy.\n\nThe day after the 1960 election, Sargent Shriver called me and said President-elect Kennedy asked him to be a recruiter and talent scout for the new administration. And he asked me if I was interested in joining the new administration in Washington. I said, no, I couldn't because I had a young family and had little financial resources. But he asked me if I would help him in the recruiting process for others, and I said, \"Of course.\" I worked with Sarge over the next few months helping him\u2014interviewing people, recommending people. Someone in that process knew how interested I was in television. They were looking for a new chairman of the FCC (Federal Communications Commission). They talked to Bob Kennedy and to Sarge and they both thought I would be good for that. So when they asked me to be chairman of the FCC, I couldn't resist, even though it was financially a disaster. It was the only job in the administration that I would undertake because I felt so strongly that I could accomplish something there.\n\nI had no interview with President-elect Kennedy. He knew me. I had seen him a number of times over the years, but he did not discuss the job with me.\n\nPresident Kennedy would call me occasionally, sometimes on a serious subject, sometimes not. He once called wanting to know what the ratings were when Jackie's tour of the White House was on television.\n\nIt was on CBS and I called Frank Stanton, then the president of CBS, and he said, \"We don't get those ratings for another couple of weeks.\"\n\nI said, \"I've got to have it today.\"\n\nHe later called me back with what were called the \"overnight ratings.\"\n\nI called the president back and said, \"Mr. President, I think you'll be pleased to know that Jackie's program had a higher rating than your press conferences.\"\n\nI would see him occasionally. The most interesting experience was when he invited me to accompany him in early May of 1961, when he was to speak to the National Association of Broadcasters. He asked me to meet him at the White House. So I was waiting outside the Oval Office and he came out. He had with him Alan Shepard, the astronaut who had just come back from his space flight, Mrs. Shepard, and the vice president.\n\nThe president said to me, \"What do you think about taking the Shepards to the National Association of Broadcasters?\"\n\nI said, \"That would be perfect. Very exciting.\"\n\nSo he said, \"Just wait here a minute\" and he went back into his office.\n\nHe came out a moment later and said, \"We're all going to go together. Come with me. I'm going to change my shirt and I want to talk to you.\"\n\nSo he took me up to the living quarters. He said, \"What do you think I should talk about to the broadcasters?\"\n\nAnd I said, \"I think you ought to say that in our country, when we have a thing like a space shot, we open it up to the press and radio and television, so that all Americans could see and share the experience. Whereas in the Soviet Union, everything was done behind closed doors because it is a closed society. You ought to contrast the difference between our system, an open society, and a closed society.\"\n\nThe president didn't say anything. He didn't say, \"That's good,\" or \"That's bad.\"\n\nHe finished changing his shirt and we went down and got in the car. The president and the Shepards got in the back seat. Lyndon Johnson, the vice president, and I got in the jump seats and drove to what was then the Wardman Park Hotel. It is now the Sheraton. The president got up and gave a perfect speech about the differences between an open society and a closed society, and in an open society why we have the broadcasters there. He had no notes; it was all extemporaneous.\n\nI left Washington in the summer of 1963 and went home to Chicago. On November 22, I was working in my new job at the Encyclopedia Britannica and was having lunch in the cafeteria, when my assistant came to get me and said, \"The news is on the radio and television. The president's been shot.\" So I immediately went to my office, heard what I could, and went home to be with my wife and children. And then we went to the funeral; we were invited to participate. It was so sad and so shocking. All of us were numb; literally speechless.\n\nLooking back, I felt that there was a five-year period in which four people died: Jack Kennedy, Martin Luther King Jr., Pope John the 23rd, and Robert Kennedy. In that terrible five-year period, the loss of those four great leaders moved all the institutions they had led backwards. The United States went backwards. The civil rights movement went backwards. And the Catholic Church went backwards. That Pope had convened Vatican II. The church was moving into the contemporary world.\n\nTelevision news came of age that weekend in November of 1963. The fact that they (the networks) cancelled everything else. They cancelled all the commercials. They covered everything live, including the murder of Lee Harvey Oswald. It (the news coverage) held the country together in a way that was extremely valuable and important. When television is live, it is especially important because it enables every viewer to feel that he or she is there.\n\nJohn Kennedy, particularly during the Cuban Missile Crisis in which I had a small role, saved our country from what could have been the end, with destruction from a nuclear exchange. I think he ushered in a whole generation of younger people who believed, as he did, that politics could be an honorable profession. He stimulated interest in public affairs and politics.\n\nUnfortunately for all of us, his life was too short. But I think his legacy will endure forever.\n Nelson C. Pierce\n\nWhite House usher\n\n\"I wondered what I would say to the first lady. As she came around the corner... she was still in her pink suit with bloodstains and I knew immediately. Our eyes met.\"\n\nPRESIDENT Kennedy was a wonderful man. When he smiled, you smiled. You couldn't help it. And he smiled quite often. For a luncheon, he had the guests in the Red Room, and when he came over from the office, I announced him to the group in the Red Room. I was shaking like a leaf, because that was my first time doing it. It was a real thrill.\n\nOne evening, I had a message for Maud Shaw, the children's governess, and I took it to the family dining room. The president and first lady were out of the city. And I gave the note to Miss Shaw. And Caroline said, \"Mr. Pierce, I have a terrible time doing somersaults, my legs either go to the right, or they go to the left.\"\n\nSo I said, \"Caroline, concentrate, now think hard about making your feet go straight over your head.\" And she did a couple, much improved, and I complimented her on them, but she said, \"Mr. Pierce, do somersaults with me.\" Miss Shaw came to my rescue, and I did not have to do somersaults on the dining room floor.\n\nOne day, Mrs. Kennedy was having trouble with her stereo in the West Sitting Hall\u2014that was their living room. And I had to escort the signal corps man upstairs to work on it. And while we were there, little John-John came over and said, \"Read a story.\"\n\nHe brought a book over to me, and I sat on the edge of the couch thinking he would be standing in front of the couch and leaning in, so I just sat on the edge. And he jumped up and sat back and then jumped down from the couch and pushed me in the chest and said, \"Sit back, sit back.\"\n\nSo I sat back and put my arm around him and read the story to him, and as soon as the story was over, he got down, put the book back, and went on his way.\n\nMrs. Kennedy, every once in a while, would call down to whoever was on duty and say, \"I need some help; could you come up and help me, please?\" So I went up and she said, \"Mr. Pierce, I would like to move the sofa from here to over there and move that chair somewhere else.\" And I asked, \"Do you want me to get the doorman to come up to help?\" And she said, \"No. I'll pick up one end and you pick up the other.\" It was a quite light sofa. So we moved it. And she didn't like it in that spot, so we moved it again.\n\nShe was wonderful to work for\u2014but you always knew your place. And you never went out of that place.\n\nThe night before the president left for Texas, they had a reception, with many members of the judiciary, including members of the Supreme Court. And the president had invited all of the ushers' wives to come, so Mr. (J. B.) West (head usher) arranged for us to have dinner served in our dressing room down in the basement, and then we all went up and he (Mr. West) escorted all the (ushers' wives) into the judicial reception.\n\nAfter a while, they all came back into the office. And when we heard that the president was coming, they all stood in our doorway. The president came around the corner to the elevator, and started in the elevator, and then backed up, and with a big, broad smile on his face said, \"Oh, you girls checkin' up to make sure your husbands are working tonight?\"\n\nAnd that big smile is what stayed with my wife months after that.\n\nOn November 22, 1963, I was working on an off-set press for a customer across the street on Fifteenth Street. It was just a little over a block to the East Gate of the White House. When I got to the gate, the police officer said, \"Pierce, hurry and get to the office. The boss has just been shot.\" And that started the day.\n\nI took the call from Texas that the president's death was official. So I called the engineers and had them lower the flag at half-staff at the White House. And I called the General Services Administration Control Center which notified all the embassies and all the ships at sea. Right after they lowered the flag to half-staff, there was a TV cameraman on the north grounds and he panned up to the flag, and that really shook me up because I knew I was the one who gave the order to put it down. And for all thirty days after the assassination, the flags were at half-staff. It got to me emotionally.\n\nAnd within ten minutes after Air Force One left Texas, Mrs. Kennedy's Secretary, Mary Gallagher, got on the phone and we got a radio patch from Air Force One that Mrs. Kennedy wanted the funeral as much like Lincoln's as possible. And so the curator's office and the U.S. Library of Congress did a remarkable research job in a very short period of time to find out the details of the Lincoln funeral.\n\nI was the \"day man.\" I relayed messages to the carpenter's shop, to the other shops that were involved, telling them what we needed, how soon we needed it. Basically, when the president's body arrived at Andrews and went to Bethesda Naval Hospital, they said the casket probably would arrive at the White House around ten o'clock (that evening). But that didn't happen. They called back and said it would be longer. And it was. It was about 4:20 am when he arrived at the White House.\n\nThe First Lady and Bobby Kennedy came in together and went to the East Room to open the casket. I was still in the office.\n\nWhen I heard they were coming down the hall, my doormen were busy with other members of the party. I locked the elevator and I wondered what I would say to the first lady. As she came around the corner, and of course, she was still in her pink suit with bloodstains and I knew immediately. Our eyes met. And we had a rapport and I knew that I didn't need to say a thing. She realized how I felt. We were all silent. I took them up to the second floor, the living quarters on the second floor.\n\nThat first night, we didn't get any sleep at all. It was the second night that I slept in the chair, Saturday night. Taking care of whatever needed to be done that I was asked to do.\n\nPresident Kennedy was a wonderful person. The first time I announced him, he turned to me and mouthed the words, \"Thank you.\" That made me feel better, but I got back in the office and my knees were still shaking.\n\nI'm asked time after time after time, \"Who was your favorite?\" I didn't have a favorite. I worked for the president of the United States, regardless of party affiliation. It was thrilling to work for six presidents. That's how I describe it: You have six friends. You like them all equally.\n Sue Vogelsinger\n\nWhite House press aide\n\n\"One of the stewards came through and said, 'We've got to pack up and get ready to move. Somebody's been shot.'\"\n\nI was hired in the fall of 1958, the year I graduated from college, and his entire staff was in Massachusetts, with the exception of Ralph Dungan, who was his counsel on the Senate Labor Committee. Ralph had hired me. So nobody was there when I was hired. I did not meet anybody until after they returned from the November election, an election Kennedy won by a huge vote.\n\nThe senator did not come back to the office for a while. I did not meet John Kennedy until early the following year, or maybe briefly in late December of 1958. I was struck by his good looks and vitality, but I had no particularly outstanding impression at that stage of the game.\n\nWhat was \"exhilarating\" about Kennedy? It was partially his youth. He was a presence. He seemed to emanate intelligence, humor, curiosity. Extremely out of the ordinary.\n\nMy first assignment was to work on what we called, \"political cards.\" The cards were designed to keep track of people the senator had met, to collect information on potential volunteers, potential contributors, and potential voters who would support him. Needless to say, that was a fairly boring operation. This was before the campaign started; even before he announced, actually. I was quickly bored with typing names on index cards and decided I was going to leave. Senator Kennedy heard about it for some reason.\n\nHe took an interest and called me in and asked, \"Why are you leaving?\"\n\nI really hardly knew him at this point. I said, \"Well, quite honestly, I don't like what I'm doing. I'm bored.\"\n\nHe said, \"Give me a little bit of time. I'll see what I can do about it.\"\n\nIndeed, a couple days later, I found myself working on the legislative staff, which means I was doing a bit more stimulating work than typing cards. He also started asking me to take dictation\u2014that was exciting. He was so fast. It was a challenge to be able to do that.\n\nEverybody was working on the campaign one way or the other. There weren't that many of us. It wasn't too long before he did announce, and I was asked at one point, whether I wanted to work with Ted Sorensen or Pierre Salinger.\n\nI said, being twenty-three years old, \"Which one will involve the most travel?\"\n\nThe answer was, \"Pierre probably.\"\n\nSo I said, \"OK, that's what I want to do.\"\n\nI did both. I did Senate work, and then I would be involved in the campaign. I was in West Virginia for the West Virginia primary. Then I'd come back and work on legislative stuff some more. Then, I became campaign staff.\n\nSenator Kennedy had won the Wisconsin primary just prior to West Virginia. Wisconsin had a fairly large Catholic population. He needed to win in a state that didn't have many Catholics. West Virginia did not, of course. So it was vital for him to win there. West Virginia was also real Hubert Humphrey country. He was beloved there.\n\nThe Kennedy campaign did a number of interesting things. They brought in FDR Jr. Of course, his father had been beloved in West Virginia. He campaigned with Kennedy and on his own. The Kennedy organization was a good one. They worked hard. They went to every door they could get to. They did a fabulous job.\n\nBob Kennedy, of course, was there. My roommate, a very good friend, had also been sent to West Virginia to work on the campaign. We did not have a big staff at the headquarters in Charleston. We all would go out at night and have a drink and dinner. And if Bob was around, he would go with us.\n\nHe said one night to Susan and me, \"You aren't Catholic, are you?\"\n\nAnd neither one of us were.\n\nHe said, \"How about if you come down the street with me tomorrow on your way to some non-Catholic church, wearing your Kennedy buttons?\"\n\nHe was only half-kidding. It was a good idea. So we did.\n\nAfter the primary, I went back to Washington. The labor bill was still being debated. I joined the presidential campaign around Labor Day. I was in Hyannis on election night. There were only three of us setting up in there. I was working with the press. My job was helping set up for media to be able to cover election night. The morning after the election, I was told I would be going to Palm Beach. That's where the president-elect was going. He needed some staff down there. So I pleaded with the pilot of the plane to stop in Washington, so I could get some warm weather clothes.\n\nIt's interesting, the differences between then and now. The fact was, if reporters asked and we could tell them something, we did. If we said, \"We can't tell you,\" we didn't. If we said, \"We'll find out,\" we might have to come back and say, \"We can't tell you.\" Reporters trusted that you were not lying. And we never were. It was really a very good relationship.\n\nThe night before the inauguration, I went to the Mayflower Hotel to give President Truman a copy of Kennedy's inaugural address. He had only one Secret Service agent. I was able to just take the elevator up to his floor and go knock on the door. He greeted me wearing slippers and invited me in to meet Bess.\n\nI was on the trip to Berlin and Ireland in 1963. Berlin was fascinating and a little frightening. I was with the crowd out in the square. It was an incredibly huge crowd, people pressing from all directions. Happily so, but it was a little frightening.\n\nOn November 22, we, of course, were working with the press. But part of my job, always, was getting press copy\u2014press releases and schedules\u2014ready, so the media had the speech and the schedule and knew what was going on, as well as getting the speech copy ready for the president. In Dallas, that's what I was doing. Chris Camp, another member of the White House Press Office, and I were on Air Force One in the president's cabin working on the president's speech copy for the next stop.\n\nOne of the stewards came through and said, \"We've got to pack up and get ready to move. Somebody's been shot.\"\n\nSo we started putting everything together to leave. We still did not know who had been shot or what had happened. There was a small television set on the plane. We turned it on and found out what had happened. Shortly thereafter, we found out he had died.\n\n\"Shocked\" is the only word you can use, but it doesn't even come close to saying what you were feeling.\n\nWe took our things and went to the back-up plane. We were not there when Johnson was sworn in. At that point, we did not know where the president's body would be taken. But we figured Air Force One would be needed for Johnson and his staff. The back-up plane held off-duty Secret Service agents and members of the Texas delegation. There were some Texans who were not unhappy with what was going on. I still remember it. At one point the Secret Service agents, hearing the same thing, came and sat with us, so we weren't too close to the Texans.\n\nBack in Washington, we were frantically busy for days on end. The role of the White House Press Office was to make it possible, as best we could, for the press to cover the funeral. Journalists came from all over the world, and the logistics involved to help them cover the funeral were extremely complicated.\n\nIt's astonishing that John Kennedy still generates the interest that he does. Part of it was him, of course, young, handsome, funny, smart, romantic, an attractive and exciting young family. Part of it, too, was that he followed eight years of Eisenhower, who was much older with a much older staff and a much older wife. So that, consciously or unconsciously, was the impact of a lot of the press stories. The feeling of excitement with Kennedy permeated everything.\n Lee C. White\n\nSpecial Counsel to the President\n\n\"After I was hired, I told my then-boss where I was going. He commented about John Kennedy, 'He's just a dilettante. He'll never amount to anything.'\"\n\nTHE first time I met John Kennedy was in his office, May or June of 1953. I had come up from Tennessee from my job with the Tennessee Valley Authority to interview with him.\n\nExcerpt from Oral history for the John F. Kennedy Library, volume 1, page 2, recorded May 25th, 1964 (copyright retained by Mr. White; used with his permission):\n\nHe was a little bit disorganized in the sense that his suit was rumpled and I recall he was not wearing any trousers. Mrs. Lincoln (Evelyn Lincoln, Kennedy's secretary) was trying to get some ink out of his trousers that he had spilled. He was going someplace or another and he just looked a mess. But he took Sorensen (Ted Sorensen, JFK's assistant) and me to lunch in the Capitol in the senators' dining room. Ted told me later this was fairly unusual. He didn't frequently have lunch with staff members.\n\nI went home thinking, \"Nice to meet that young senator.\" He was a lively guy, though at that time I had not realized his physical condition. He was spry and zippy.\n\nLater, after I was hired, I told my then-boss at the Tennessee Valley Authority where I was going. He commented about John Kennedy, \"He's just a dilettante. He'll never amount to anything.\"\n\nWorking for John Kennedy, you learned how to juggle. You do what you have to do and you get stretched a bit. As a boss, there were two qualities I admired about him. First, he expected staff to perform miracles, and the staff we had were pretty good, especially the indomitable Ted Sorensen. He was not easy, but he was fair, and he never asked or expected us to do anything that was impossible or degrading. On that front, he gets a good grade.\n\nWhen I left him as a senator, it was to go to the staff of the U.S. Senate Small Business Committee. It really was because I wanted to earn more money. It was funny. I was leaving at a time when everyone wanted to join his staff. But he didn't think I was nuts. He encouraged me and when I told him I was not going outside to make a fortune, but rather to work as a top dog for Republican Senator John Cooper from Kentucky, Kennedy said, \"Oh, that's different. When you want to get rid of the stigma (of working for a Republican), you come back to work for me.\" And that's how it worked out.\n\nJohn Kennedy did not go peacefully over that little hump of turning forty. I remember the event even today. Sometimes we joke or make gags about turning forty, but he was not joking. I don't quite know why, but I know damn well that he was not happy. I was not aware, as others were, about his medical and physical problems. He might have thought, \"Wow, I've been going uphill and now I'm going downhill.\" But it was a rude awakening, especially for a hot dog like he was.\n\nAfter his election to the presidency, I contacted Gloria Sitrin, Sorensen's assistant, and later Ted contacted me. About a week later, the word came back, \"You can go down and get your identification.\" I was in. It was either the last week of November, or the first week of December of 1961.\n\nExcerpt from oral history for the John F. Kennedy Library, volume 7, pages 382-384, recorded May 11, 1970 (copyright retained by Mr. White; used with his permission):\n\nOn the day of the assassination, I was having lunch in the White House mess when Jack McNally came in\u2014about one-forty in the afternoon\u2014and said that the president had been shot. My first instinct was that it was some sort of bizarre sense of humor... but all of a sudden, it was evident to me that nobody could have joked at that and his face made it clear.\n\nSo I quickly left and went upstairs to Pierre Salinger's office where the only (news wire service) ticker was located.... The tickers had the story.... Then, in a matter of just minutes, that room began to fill up with reporters. By default, I became the acting press secretary at the White House, although obviously (Malcolm) Kilduff, who was with the president, was the acting press secretary. So for those couple of hours, I had the responsibility\u2014self-designated\u2014to kind of work with these reporters and keep in touch with what was going on with Sorensen.\n\nWhat is John Kennedy's legacy? He and his wife Jackie created tremendous interest in the presidency. Some of us had an old saying: \"Franklin Roosevelt demonstrated that a man with a physical handicap could be president. Harry Truman demonstrated that a man with a mental handicap could be president, and Eisenhower demonstrated we didn't need a president.\"\n\nSo, here comes this young, good-looking whippersnapper, who was charming with a God-given smile that was irresistible. And suddenly, he's in the Oval Office. Boy, did that make a difference.\n\nIt really was the passing of the torch to a new generation.\n Harris Wofford\n\nSpecial Assistant to the President on Civil Rights, Associate Director of the Peace Corps, Peace Corps director in Ethiopia, U.S. senator\n\n\"The Bay of Pigs was a huge disappointment to the president. He was angered at himself and those who had planned it. So he asked his brother Robert from then on to keep a special eye on the CIA...\"\n\n(JOHN Kennedy) came across to me with a personality that would be able to win the election of 1960. I thought, and do think, that Kennedy had more of what we call charisma than anyone since Roosevelt and before Obama. A combination of seriousness and humor, I thought was appealing; a sort of political gaiety, like Roosevelt, \"The Happy Warrior.\"\n\nBeyond that, Kennedy represented what the world, as I knew it, had hoped for America. That we were young, optimistic, can-do, and full of energy and idealism.\n\n(B)y the time he left for Dallas, he had lived up to what he had promised in the campaign to try to do. Now I think that the revisionist writers about Kennedy, or at least the first wave of them that still ripples, view him as making low-level political judgment that would get him in trouble if he moved too fast (on civil rights and other issues). That is what I would call a low way of stating why Kennedy was very careful and, some would say, slow in acting. I would say there are maybe three key factors in measuring what he did.\n\nFirst, he, his father, his family, his \"Irish Mafia,\" as we jokingly called the staff and some of his closest gang, they very well knew the prejudice among whites in the North, epitomized by South Boston and what was also already evident in terms of the reaction of the whites to school integration in Boston, and their own Irish Catholic Americans. These were not the priests and the bishops and the serious devoted Catholics like Sargent Shriver, but the plain, American Roman Catholics who shared in their own way their own form of discrimination or prejudice racially.\n\nThey (JFK, his family, and the \"Irish Mafia\") genuinely, from the time they started paying attention to civil rights, worried that it was going to be counter-productive and that the backlash of moving too fast would set civil rights and the country back\u2014not just that they might lose an election. Most of the time they could not really tell whether action would win more votes from African Americans or lose more votes from prejudiced people. And they weren't just thinking only of the South, but very much they were thinking of the North.\n\nThe second thing that is a factor in measuring Kennedy is his overwhelming personal interest, that from the time he was doing anything public, it was international affairs. He wrote the book Why England Slept. He was forever interested in the world and for me, the chief appeal of Kennedy, aside from his charisma, was the fact that for him the world was the big question. And that was my own view\u2014the biggest question of all was the world.\n\nI came back from India thinking that civil rights was the great stain on the American soul and for us in our relations with the world. Beyond the reason of living up to ideals of the Declaration of Independence, it was crucial for our leadership in the world that we remove that stain, And John Kennedy not only thought that racial discrimination and prejudice were irrational and were problems that ought to be solved, but I think his biggest impetus in trying to get them solved was his sense of America and the world.\n\nNow remember, those I call the revisionists, they like to say that there wasn't a message on the inaugural address about civil rights which had been such a vital part of the campaign. The inaugural address had no domestic issue at all\u2014civil rights, or the economy\u2014aside from the call to service, \"ask not.\" No issue of domestic importance was pointed out, though Louis Martin (publisher, civil rights leader) and I got in those two words, \"'at home' and around the world.\"\n\nNow, the last thing that occurs to me. Beyond charisma. Beyond any particular issue, Kennedy's great promise that was beginning to be fulfilled (at the time of the assassination) was his political skill. He was forever trying to see what would really work. He was genuinely balancing his best judgment on that\u2014the way Lincoln did on when to sign the Emancipation Proclamation and how far-reaching that proclamation should be. In his first year, Lincoln was denounced by most of the abolitionist leaders for not signing it, and when he finally did, he was denounced for applying it initially only to the states in rebellion.\n\nThe main disappointment for me with Kennedy was the delay in signing the executive order (ending housing discrimination). Not because I thought that was an earth-changing action\u2014it was always necessarily limited in terms of how much it would end discrimination in federally assisted housing. It was frustrating that twice it was about to be signed and postponed. It came about because a few words he picked up from me: \"One stroke of a pen and you can do it.\" I was completely underestimating the ability of Congress and the political forces to make it dangerous to stroke that pen.\n\nIn retrospect, if he had done it right away, it would have caused less trouble than all the delay. It was not the main issue that people had in the civil rights movement. The issue going forward was what the platform promised and what he had espoused during the campaign: how to go forward with the most far-reaching Democratic platform\u2014any party's platform\u2014on civil rights. He was weighing whether the promise to do it in the first session of the first Congress was a wise thing to do. I happened to agree with him once we faced the fact that his margin of victory was one hundred thousand-plus (votes), and that his majority in Congress, in both houses, was dependent on Southern segregation-committed people. I think if he had gone forth with the civil rights legislation, it would have had a catastrophic effect. It would have fizzled out, and\/or been viewed as a major defeat. It would have reduced his capital.\n\nI would add that (President Lyndon) Johnson was able to get both civil rights acts\u2014the first and then the Voting Rights [Act] passed in the name of fulfilling Kennedy's promise. Public opinion was stirred in part on Kennedy's support for civil rights and the platform.\n\nThe Bay of Pigs was a huge disappointment to the president. He was angered at himself and those who had planned it. So he asked his brother Robert from then on to keep a special eye on the CIA, and Robert got increasingly important to Kennedy on foreign policy. Robert had come from being an arch cold warrior, starting with his own first passion which was (his opposition to) organized crime and the Communist infiltration of some labor unions. Robert Kennedy, by the time of the Cuban Missile Crisis, was profoundly shaken, as was (President) Kennedy, by the responsibility of dealing with nuclear war.\n\nThe first proposition of the inaugural address was \"man holds in his mortal hands the power to abolish all forms of human poverty and all forms of human life.\" Having that button to push and knowing the Russians had theirs was the great sobering thing.\n\nI believe that because John Kennedy was so rational and so cool and increasingly concerned about the violence of war, he would have been able to do what Johnson emotionally and irrationally could not do: Find a way to get out of Vietnam. He was on the way to resolving that. Remember, Kennedy's other great speech was the peace speech at American University\u2014the same week that he gave the civil rights speech. The civil rights and the peace speeches and the inaugural address are his greatest words.\n\nIn November of 1963, I was the director of the Peace Corps in Ethiopia. I'm sure by cable I was communicating with my boss and friend Sargent Shriver. I think he said that, \"If you can find a way to get here (for the president's funeral), we'd love to have you.\"\n\nI gave up the thought of getting there (to Washington). Then, suddenly my administrative officer came and interrupted my wife and me playing tennis to tell me that the (Ethiopian) Emperor Haile Selassie was going to the funeral. His big jet was landing in Addis Ababa to pick up some members of his party or his family or whoever. My administrative officer Ed Corboy said he was given the understanding that they would be quite happy if I could get there in time to join the plane.\n\n(My wife) Clare said, \"You certainly can't do that, they said the plane is about to land and would only be there a short time.\" And I was in shorts and tennis shoes. I looked at my administrative assistant. Ed was about the same size and he had been dressed for church\u2014it was a Sunday morning. Clare said, \"You can't do it.\"\n\nWe said, \"If you drive the Land Rover, we can change.\" So on the way I changed into Ed's business suit. We got there and Ed knew all the airport people. They said, \"We're waiting, hoping you'll make it. They're just about to leave so you can drive onto the field.\" And they opened the gates. And as we sped along toward the plane, it zoomed forward.\n\nWhere does Kennedy fit in the whole list (of presidential rankings)? Because he was killed after one thousand days, rationally, he can't be at the top with Lincoln or the greatest presidents. But he left an image of the kind of America that, at our best, we like to think of ourselves as\u2014that which the world has yearned for and believed in. That has to do with hope, optimism, can-do spirit. That image became more deeply etched in political memory because of Robert Kennedy's transformation, or change, or growth or transfiguration\u2014that's a romantic way of putting it. The two of them together was a high point in American politics and the promise.\n\nWhat if we had had thirty more years of John Kennedy, Robert Kennedy, and Martin Luther King? Who knows? No one knows. I like to think of those three figures as a whole.\nSECTION TWO:\n\nCivil Rights Leaders\n Julian Bond\n\nFounding member of the Student Non-Violent Coordinating Committee, chairman of the National Association for Advancement of Colored People, professor\n\n\"Kennedy watched the pictures of young people being fire-hosed in the streets of Birmingham, and he told onlookers, 'This makes me sick.'... For him to feel that way and to say that, means he was moved by it, and it shifted him.\"\n\n(ON November 22, 1963) I was having lunch and somebody came into the restaurant and said, \"The president's been shot.\" My first reaction was, because they did not say, \"The president's been killed,\" I thought, \"Gee, what a terrible thing. But he's not dead.\"\n\nWe continued with our meal. I was having lunch with a journalist, and she did not get up and say, \"I've got to go and take care of this.\" It did not seem that pressing to me.\n\nI went back, after lunch, to the office of the Student Non-Violent Coordinating Committee and talked to some other people. And I think, by then, we knew that he was dead.\n\nWe were a civil rights organization. We had been contending with President Kennedy. He had not done most of the things we had wanted him to do. He seemed disinterested in civil rights. He was not our favorite politician.\n\nAnd when we found out he was dead, we knew that meant Lyndon Johnson would be president. We began to call people we knew in Texas who could tell us something about him\u2014progressives, liberals in Texas, who could tell us what kind of person he was, what kind of president he might be.\n\nThey were not encouraging. They thought he would be a step backward. If Kennedy had been, pretty much, a do-nothing president, Johnson would be worse. And we discussed it. We were the kind of people who talked all the time.\n\nLike everybody else, I was glued to the TV. What's going on? What are people doing? What are they talking about? All those iconic scenes that everybody else was drawn to, they drew me, too. Television brought this catastrophic news event into our homes in ways we never thought it would. I never imagined it would.\n\nWhen he ran for office, we were inspired by him. But after he took office, it became fairly clear to us, at least we thought so, that he would not be the flaming liberal we had hoped he would be. Instead, he was cautious. He said early on that he saw no changes coming in the filibuster rule. That was just an awful thing for him to say from our point of view.\n\nSo we thought it was just going to be business as usual. And we were not enamored of him. We were of mixed emotions. On one hand, he and his wife were such attractive figures. Everybody was drawn to them. On the other hand, good looks alone wouldn't do it. It was deeds we were interested in, and there did not seem to be any good deeds forthcoming.\n\nAs time went on, in the months\u2014not the years, but the months\u2014before he was killed, he had begun to shift direction a bit. And you could see him becoming a different person. He introduced a civil rights bill, which became the civil rights bill the next year\u2014the Civil Rights Act of 1964. And that was a wonderful act, but could not have been done without Lyndon Johnson. If Kennedy had lived, it would not have been the bill that Johnson put through Congress. It would not have been as strong.\n\nIt seemed that he (Kennedy) was coming our way, though not as quickly as we wanted him to. Of course, we were young and impatient, angry and so on. When he was killed, you thought something good could happen here. But it's not going to happen now because this guy's been killed. And Lyndon Johnson is going to take his place and God knows who he'll be.\n\nKennedy watched the pictures of young people being fire-hosed in the streets of Birmingham, and he told onlookers, \"This makes me sick.\" And it must have had that effect on most people who saw it. But for him to feel that way and to say that, means he was moved by it, and it shifted him.\n\nI think both John and Robert Kennedy were na\u00efve about civil rights. They knew almost no black people and had few associations with them. This was alien territory to them. And so the Birmingham movement was an eye-opener for John F. Kennedy. And it was an eye-opener for Robert Kennedy, too.\n Dolores Huerta\n\nLabor leader, advocate for rights of workers, immigrants, and women, co-founder of the National Farm Workers Association\n\n\"One of my friends used to call John Kennedy 'our first Mexican president.' He stood for economic rights, as well as civil rights.\"\n\nI had the chance to meet John Kennedy and his brother, Ted, and turned it down. I was leading a non-partisan voter registration drive in San Diego in 1960 when he was running for the presidency. Being very innocent\u2014running the non-partisan Community Service Organization\u2014I felt I could not do anything partisan. I sent some of my volunteers.\n\nLater, I did see him because he had a parade in San Diego. But it was just a fleeting glimpse.\n\nI also met Robert Kennedy for the first time in San Diego. It was also during the campaign of 1960, during the voter registration drive. TIME magazine had published an article saying that a group of Latinos who were political and who called themselves \"Kennedy Clubs\" had organized the voter registration drive. We had asked some of these people to help us with the drive in San Diego, but they told us they were too busy to walk the streets with us. But then, they took credit for our work, saying they had registered all these people.\n\nRobert Kennedy was going to be speaking at the Hotel Del Coronado in San Diego. With the TIME magazine in hand, I went to see him. He saw me walking up the aisle, and he saw my copy of the magazine, and before I could even speak, he stopped me and said, \"Oh, I know it was your CSO (Community Service Organization) that did the voter registration. We'll correct that.\" And he did. He wrote a letter to TIME magazine, giving credit to our organization.\n\nBy 1963, Cesar Chavez and I had started the United Farm Workers Union. I was living in Stockton, California. On November 22, I was driving my young daughters to school. I had the radio on in my car, and the news came on that the president had been shot. And then they announced that two Latinos were seen in the grassy knoll (in Dallas's Dealey Plaza) and that these were the people believed to have fired the shots. Being a Latina, I just felt horrible about it.\n\nI was devastated.\n\nThe first instinct I had\u2014and many other people had\u2014was to go to church immediately. I remember the churches were packed with people. They were absolutely full. We prayed for him and our country.\n\nI was influenced by the 1960 debates (between Kennedy and then-Vice President Richard Nixon). And, being from California, Nixon was not a favorite of ours.\n\nThe very, very famous statement he (Kennedy) made in his inaugural address, \"Ask not what your country can do for you. Ask what you can do for your country,\" I still believe that now, especially with young people. We live in an age when some people believe that they are so entitled. Being a community organizer, this is basically what we say to people: \"Think about what you can do to help others.\"\n\nOne of my friends used to call John Kennedy \"our first Mexican president.\" He stood for economic rights, as well as civil rights. When the steel workers went on strike, and the steel companies were refusing to settle, he made them (company owners) bring the books in. He said, \"I want to see your books.\" I always thought that the reason he could do that was because he was one of their peers. He could say to them, \"I want to see your books.\"\n\nComparing that with what President Obama had to go through when he was trying to get Congress to approve the health care bill\u2014everything that the (health care companies) put him through, he did not have that peer authenticity that Kennedy or (President Franklin) Roosevelt had with the wealthy, when he (FDR) got the New Deal legislation passed. It was much harder for President Obama.\n\nJohn Kennedy had that kind of authority and he wasn't afraid to use it.\n John Lewis\n\nChairman of the Student Nonviolent Coordinating Committee, Freedom Rider, author, member of United States House of Representatives\n\n\"After the March on Washington...the moment we walked into the door of the Oval Office, he greeted us saying, 'You did a good job.' He was beaming like a proud father...When he got to Dr. King, he said, 'And you have a dream.' That was my last time seeing President Kennedy alive.\"\n\nTHAT day we met with President Kennedy (in June 1963) was my first opportunity to meet him, to literally be in his presence (two months before the March on Washington). I had heard him speak from a distance during the campaign in 1960, but this was my first time standing in the same space with him. I was so moved, honored, and blessed to be in his presence. I admired President Kennedy. The man gave me a great sense of hope and optimism, like he gave the entire country. Even as a candidate for president, his demeanor and style inspired many of us as young people during the sixties.\n\nI did not have much to say during that meeting, and I do not regret not speaking up. I was the youngest person in the room. A. Philip Randolph, who was considerably older, was the dean of our leadership, the dean of the group. He was our spokesperson. It was he who spoke up in his baritone voice, as I recall, and said, \"Mister President, the black masses are restless.\" And he went on to describe what was happening in America and to say that we were going to march on Washington.\n\nWhen President Kennedy heard that, you could tell from his body language that he was not pleased. He began rocking a little more vigorously in his rocking chair. He responded briskly and said something like, \"Mr. Randolph, if you bring all those people to Washington, won't there be violence and chaos and disorder?\"\n\nAnd Mr. Randolph responded and said, \"This will be a peaceful, orderly, non-violent protest.\" The others said a few other words, and before we got up to leave, the President said, \"We have problems, man-made problems that must be solved by men and women of goodwill.\"\n\nJohn Kennedy listened. He listened very well. He listened intently. I just felt that my presence, being a young guy, that I didn't have to say much of anything. After the meeting, we went out to the lawn of the White House to talk with the press and said, \"We had a productive and meaningful meeting with the president of the United States.\"\n\nEven before we met with President Kennedy that first time, he had made, in my estimation, one of the most meaningful speeches that any president had ever made at that time on the question of civil rights. I think that was the speech he gave on June 11 when he said that the question of civil rights and the question of race are moral issues. That confirmed in my own mind, his heart, my sense of his commitment.\n\nAlong the way, I think that he became reluctant to take action because he was concerned about the next election, about holding on to some of the Southern states. And he didn't want to do anything that would make it difficult or impossible to get some of his nominees through, especially in the judiciary. Senator (James) Eastland was the chairman of the Judiciary (Committee), and he (Kennedy) was very concerned about that. He had to be very cautious.\n\nAfter the March on Washington (August 28, 1963), we met with President Kennedy again. He had invited each of the speakers to the White House. The moment we walked into the door of the Oval Office, he greeted us saying, \"You did a good job.\" He was beaming like a proud father. He was glad that everything had gone off so well. He said to each one of us, \"You did a good job, you did a good job.\" And when he got to Dr. King, he said, \"And you have a dream.\" That was my last time seeing President Kennedy alive.\n\n(On November 22, 1963) I was in Nashville, getting into a car on my way to Detroit, and I was going to make a stop in Urbana at the University of Illinois at Champaign-Urbana to recruit students for the movement. Then I was planning to go on to Detroit to speak at a union event. When we heard the president had been shot, we cancelled the meeting in Illinois and we drove straight to Detroit. That meeting turned into a memorial service for President Kennedy.\n\nIt was just a very sad time. I had had family members pass, my great-grandfather, great-grandmother, other relatives, but never someone so prominent whom I had met, someone whom I admired and identified with. I felt that President Kennedy, even during his campaign, and then in his election, he created such a sense of hope and optimism. I've said that when President Kennedy was assassinated, and I've said the same thing about Dr. King and Bobby, something died in us. I know something died in me. I think something died in America.\n\nThose of us who lived through that period can never, ever forget it. I sometimes wonder what would our country be like, what would our world be like, if President Kennedy had lived.\n\nThe place for him (in my heart) is still real, still strong and deep. His personality, his voice, his sense that we as a nation can do almost anything. That drive, energy, the commitment, the dedication. The idea that we can go places as a nation and as a world community that we have never gone before. It is still there.\n\nSometimes when I'm working in politics, I often think, \"What would President Kennedy do? What would President Kennedy say?\" The same can be applied to Dr. King and Bobby Kennedy. I think these three men helped influence and mold my life. I'm eternally grateful that somehow, some way, they impacted me.\n\nDuring the weekend leading up to the funeral service for President Kennedy, we were holding a meeting in Washington at Howard University, and I proposed at the Student Nonviolent Coordinating Committee (SNCC) conference that we go out to the John F. Kennedy grave and pay tribute to him. But my fellow members of the SNCC staff did not want to be part of that. For many reasons, they vetoed the idea. Much later, I had the opportunity to go (to the gravesite). From time to time, I go out and (pause)...\n\nThere are ups and downs in life and in a nation's history, but you can never lose hope. I think that's what the lives of Dr. King and Robert Kennedy and President Kennedy say to us. These three young men inspired us to hold on and keep the faith, to be hopeful, be optimistic about the future.\n\nPresident John F. Kennedy's commitment, his dedication, his inspiration, his leadership continue to inspire people today. Many young people and many people not so young, many my age, people younger, people older still say, \"I got involved in public service, I got involved working for a non-profit, got involved with the Peace Corps because of President Kennedy.\"\n\nAnd several members of Congress say, \"It was John F. Kennedy who inspired me to do something with my life.\" People became teachers, they became lawyers and doctors. When I went to Africa for the first time in 1964, young Africans in West Africa, Central and East Africa, were talking about President Kennedy. And still, all these years after his death, he is a great source of inspiration. All over America and around the world.\n Roger Wilkins\n\nAssistant to the Administrator of the United States Agency for International Development, Assistant Attorney General in Johnson administration, Pulitzer Prize\u2013winning journalist, professor\n\n\"He said, 'Keep up the good work. We're lucky to have you.' I floated for a week. He was really interested in what this young, Negro man was doing and how it felt to be working in his administration.... It was no b.s.\"\n\nI knocked on doors for John Kennedy in 1960. We said, \"We're from the Democratic youth group and we'd like you to vote for Kennedy.\" We handed people pamphlets. Sometimes, we got the door slammed on us. But I would say more than 50 percent of the time people were decent. I was twenty-eight years old.\n\nI met John Kennedy after he was elected president. It was in the basement parking garage of the State Department. My boss, the USAID Administrator, was driven to and from the office in a limousine, so he allowed me to use his parking spot. As you may know, Kennedy did many of his press conferences in a theater in the State Department building, and one day, I came driving in and there were four or five cars lined up, including the presidential limousine. I got out of my car and the Secret Service agents looked at me, then all of a sudden, there he was. He was smiling and we walked toward each other.\n\nI said, \"Hello, my name is Roger Wilkins.\"\n\nHe remarked, \"Oh yes, Mr. Wilkins, very nice to meet you.\"\n\nAnd I thought to myself, \"He's heard of me? He knows me?\"\n\nI was three degrees from peeing in my pants.\n\nHe asked me, \"Well, how do you like it here at USAID?\"\n\nI responded, \"I'm very pleased to be here.\"\n\nThe president went on. \"What are you working on?\" he asked.\n\nI told him I was trying to fashion programs for Africa and also trying to make some sense out of the war in Vietnam.\n\n\"What else do you do?\" Kennedy asked me.\n\n\"Whatever the administrator wants,\" I responded. \"I deal with the inspector general. In fact, I meet with him every two weeks.\"\n\nThe president said, \"You're doing good work.\"\n\nAt this point, I'm giddy.\n\nThen, it clicked: He asked if I was related to Roy Wilkins, and I said, \"Yes, he's my uncle.\"\n\n\"Oh,\" the president responded, \"He's a fine man. We rely on him a lot.\"\n\nHe said, \"Keep up the good work. We're lucky to have you.\" And then he got into his car.\n\nI floated for a week. There was no question John Kennedy was a nice and polite man, as well as a good politician. He was really interested in what this young, Negro man was doing and how it felt to be working in his administration. I think we chatted for five to ten minutes. It was no b.s. I had heard from my uncle that Kennedy was like that, very engaging.\n\nOn November 22, 1963, there was an enormous explosion in my life and in my spirit. I was in Puerto Rico looking at possible locations for a police training center as part of USAID's work in Central America. We recognized that if you do not have good police, you cannot have an enriched economy. We were staying in the San Juan Hilton and I walked into the lobby. It was very quiet. It was weird. People looked shell-shocked. People were moving quietly. There was a kind of eerie silence.\n\nI said to myself, \"Something's wrong.\" I asked a busboy, and he looked at us and said, \"Kennedy el morte,\" and he made a fist and bounced it on the side of his head.\n\nI thought, \"What?! The president of the United States killed? No way. It didn't happen!\"\n\nWe soon learned it indeed had happened and we cut our trip short and flew back to Washington.\n\nJohn Kennedy helped us believe in ourselves, that we, as Americans, could do big things. He lifted our spirits and made us proud of ourselves. His youth, his attractiveness, his elegance of phrase. And, of course, Jackie. He was Mr. Cool. His self-effacing humor, as demonstrated by his comment when he visited France, \"I am the man who escorted Jacqueline Kennedy to Paris and I enjoyed it.\"\n\n(Senator) Pat Moynihan said it right: \"We may smile again, but we will never be young again.\"\n Andrew Young\n\nExecutive Director of the Southern Christian Leadership Conference, mayor of Atlanta, U.S. Ambassador to the United Nations during Carter administration\n\n\"His legacy really is hope in the American people. He put the burden of responsibility to maintain this vision on the people.\"\n\n(IN 1960) there was almost no attention paid to the black vote, and the fact that he showed up in a black community in Chicago\u2014I think the name of the place was Park Meadows. I had an aunt who was living there. It was a middle class high-rise on the south side. They had a rally there in the parking lot. We weren't used to seeing presidential candidates come into the heart of a black community that way. He stood on a truck and addressed the crowd directly.\n\nHe had a very interesting speaking style. You felt like he was talking to you. And he wasn't making promises. He was, by and large, reaffirming his faith in the country and his faith in us as a people. And I think part of what inspired me\u2014and this was before he asked, \"Ask not what your country can do for you. Ask what you can do for your country.\" That message was: \"We've got to pull together to straighten this world out, and we can do it.\"\n\nI knew Hubert Humphrey. And I liked Hubert Humphrey. I was always a Hubert Humphrey fan. It was like Obama and Hillary (in 2008). I knew Hillary (Clinton) much better than I knew Obama. And I was a supporter of Hillary. I didn't give up on Hillary as quickly as I gave up on Humphrey. I just felt that Kennedy was a younger clarion voice that the country would respond to.\n\n(On November 22, 1963) we were working with the citizenship education program, where we brought local leaders from across the South together, to train them in non-violence and citizenship education. To organize their neighbors to pass literacy tests and to get the right to vote. We had about fifty people at Penn Community Center in South Carolina. And these were poor black people for the most part from Mississippi, Louisiana, Alabama, South Carolina, and Georgia. And they were all very religious.\n\nNot only did we cry, but immediately we ended the conference, and the whole thing turned into a prayer meeting. People got down on their knees and prayed for the life of the president.\n\nDr. Martin Luther King Jr. saw President Kennedy's death as his own death. At the time he said, \"If they can't protect the president, there's nothing we can do. When they're ready for you, you're gone. And you don't have anything to say about when you go or how you go. You're gonna die. We're all gonna die. That's the one universal for all mankind. And we have no choice about when or how. The only choice we have is 'What is it that we are willing to die for?'\"\n\n(I was inspired by John Kennedy) in his willingness to insist that America be a global leader and I think that's what is required now. He was leading the nation and the world politically and in the area of atomic weapons and outer space. But what's needed now is someone who can create an economy that works well for everybody. And the strength that he brought to the office was that he was not an ideologue. He was a visionary.\n\nHis legacy really is hope in the American people. He built on the early constitutional vision and the government of the people, by the people, for the people of Lincoln. He put the burden of responsibility to maintain this vision on the people. The term usually used is \"American exceptionalism.\" I think what John Kennedy was saying is, \"We have to own up to the privileges and blessings that God has allowed this nation to develop.\"\n\nAnd I would say that my version of his legacy would be that \"to whom much has been given, of them much will be required.\" That was true of his brothers, Robert and Ted. They believed not only in the exceptionalism of the nation, but of their family and of the people. Those who are blessed must choose to serve others.\nSECTION THREE:\n\nChildren of Kennedy Advisors and Others\n David Acheson\n\nAttorney, scholar, son of former Secretary of State Dean Acheson\n\n\"I think Jack was a first rate public figure, that is, he knew how to handle himself in public. He was not, I think, a distinguished public servant. I was always a little disappointed in Jack because I thought he did not come up to the glamour he inspired.\"\n\nI first met Jack Kennedy in late September of 1960. I and a couple of others, because of our experience in previous campaigns, had been asked to go to New York City and run the five borough area for his campaign. The first time Jack came through, we were billeted at the old Biltmore Hotel\u2014that was the location of both the statewide and local Democratic headquarters. Mike Prendergast was the state chairman. Carmine DeSapio was the New York chairman, so we brought Jack into the hotel for a meeting with them.\n\nHe looked at me and asked, \"Do you mind if I ask you a question?\"\n\nI said, \"Of course not.\"\n\nHe said, \"What am I doing in this crummy hotel?\"\n\nWe explained why we were there. He was used to the Carlyle. The Biltmore was a few notches below his standards.\n\nThen he asked, \"So, what are we up to?\"\n\n\"The first thing on your calendar tomorrow is to meet the cardinal,\" I said. \"And the problem you have, of course, is that the Southern Baptists don't like you being a Catholic. And if you offend the cardinal, your Catholic supporters will take it amiss. So the question is, 'Do you kiss his ring when you are introduced to him, or do you just shake his hand?'\"\n\nJack, without any hesitation, said, \"Most of my Catholic friends and supporters think I'm not a very good Catholic, so I don't have much to fear from them. I have more to fear from the Southerners. So I think it's pretty obvious. I'll just shake his hand.\"\n\nI said, \"Well, that's our recommendation.\"\n\nHe said, \"OK, what's next on the agenda?\"\n\nSo we organized his moves for the next couple of days and then he left town. He came back several more times and the last time was the debate with Nixon. We took him over to the studio. But we were not permitted to be visible. We were very confident that he was doing well in the campaign. He was almost idolized by the press. Everything in his nature seemed to be in contrast with President Eisenhower.\n\nAnd he was quite the opposite of Nixon in his manner, appearance, and personality\u2014all of that was helping him immensely. Nixon regarded reporters as a risk and wanted to minimize his risk by not talking to them any more than he had to. Jack, like FDR, saw them as a medium, a medium that was perfectly prepared to cooperate.\n\nWe didn't see a lot of him during the campaign. We organized his moves in New York, communicated them to his campaign staff, and they would take off and handle it. We were not very visible and we did not want to be.\n\nAfter he was elected, I only saw him socially once or twice.\n\nI was taken with his personality. The thing that worried me was whether he was just a glib, attractive, superficial phenomenon. And close to the time he was assassinated, I began to realize that he was not really happy about being a leader in the civil rights campaign, because that threatened his position in the South.\n\nOn the other hand, Bobby Kennedy was attorney general when I became U.S. Attorney for the District of Columbia. I reported to Bobby, and I became convinced that Bobby, in spite of his reputation as being ruthless and cynical, really wasn't. He made some bad alliances early in his life through the political connections of his father.\n\nBobby was really interested in public service, including promoting civil rights. I completely changed my opinion about Bobby. We were never intimate friends, but we got to the point where, since I had a place on Cape Cod, we would fly up together and talk about various issues. Toward the end of his tenure as attorney general, some were pushing him to be LBJ's running mate in 1964. We would joke about this on the plane. When Johnson announced that no one in his Cabinet\u2014or sub-Cabinet\u2014would be considered to be his running mate, this was obviously a transparent formula for dismissing Bobby.\n\nJack seemed always preoccupied by his image, by the publicity about him. He was very sensitive to that. I got the impression he was a little too self-obsessed. And a little too cynical. He was worried about the civil rights movement because he did not want to alienate the South. He was worried about national security issues and worried about risking war with the Soviets over Berlin. All those things were worth worrying about, of course. I felt he was sort of paralyzed by the size of those problems and the difficulty of those decisions. I don't think he handled foreign affairs very well. He was risk-averse, and he was inexperienced. He did not have people around who were very good. Dean Rusk, who had worked for my father, was very hesitant, very timid, very poor at clear communication about what he thought the president should do.\n\nOn November 22, I had been working in my office and walked up half a block to a sandwich shop to get a quick lunch, and when I came back, my secretary said, \"Oh, Mr. Acheson, have you heard the awful news?\"\n\nAnd I hadn't. There was no television in the place where I was having my sandwich. But everyone in the office had heard it by then. It was, of course, a very, very sad day.\n\nI think Jack was a first rate public figure, that is, he knew how to handle himself in public. He was not, I think, a distinguished public servant. I was always a little disappointed in Jack because I thought he did not come up to the glamour he inspired. He had a potential that was greater than what he realized.\n\nJack had a breezy manner. His mind was quick; his responses were quick. He had a sardonic sense of humor. He liked ironies and gallows humor. I was attracted to him immediately for those reasons. My father enjoyed him as a person, but was more skeptical.\n Fernando Chavez\n\nAttorney, son of Cesar Chavez, co-founder of the National Farm Workers Association\n\n\"(L)ook at that time period, in the homes of Latinos, you would see calendars, posters, and photographs of President Kennedy. And you would see images of the Virgin Mary. That's the stature, that's the reverence with which he was perceived in Latino homes.\"\n\nI saw and heard him at a rally in Los Angeles in 1960. I was probably eleven years old. We were living then in East Los Angeles, and my dad was running the Community Service Organization. For its time, it was quite a militant organization, advocating things that were just not kosher: trying to register people, especially Hispanics, to vote; and forming what were called \"Viva Kennedy\" clubs.\n\nIt may have been the first time there was an organized, concerted effort on the part of Latinos to form a campaign for a presidential candidate. A lot of it had to do with the fact John Kennedy inspired a sense of hope and aspiration in people like my dad and Dolores Huerta, who were young, idealistic, and who were looking for a better way, not just for their children, but for their community as well.\n\nDuring the campaign, my dad was doing the \"get out the vote.\" One day, he recruited about thirty kids to distribute leaflets. It was a Saturday and it was raining like hell\u2014a miserable day. We made our own leaflets on an old mimeograph machine. We made thousands and thousands of flyers, and we kids were distributing the flyers to homes in Los Angeles. It was all day. We each were to be paid five dollars.\n\nAt the end of the day, someone from the Kennedy campaign gave a speech. It may have been a family member. My dad then asked everybody to come up onto the stage. He gave each one of the kids five dollars and then the campaign representative shook the hand of each kid. My name was called last. I had earmarked my five dollars and planned to spend it on something special. I walked onto the stage, and as I am about to get my five dollars, my dad turns to the audience and says, \"And Fernando is going to donate his five dollars to the campaign!\"\n\nKeep in mind, I am eleven years old. I don't know anything about the political process. But I'm determined to buy something very special with this money. And, all of a sudden, I have it taken from me. I wanted to die. My dad never thought how impactful that would be. When we got home, I told my mother and she could not believe it. She wound up giving me the five dollars.\n\nMy dad said, \"I'm sorry. I didn't mean it. I just wanted people to know how much we believe in this and how important this campaign is. How important this candidate (John Kennedy) is. How his election will make life better for us.\" He started giving me the political speech.\n\nSo I was baptized very early.\n\nIn 1962, we moved from Los Angeles to Delano, California. At that time, my father was starting to attract some national notoriety for his work creating the National Farm Workers Association. There were not a lot of Hispanics who were particularly prominent. That's when he was offered a job with the Kennedy administration running the Peace Corps in Latin America.\n\nThere are maybe a half dozen events that are imprinted and etched in my memory. This is one of them. My father had this habit\u2014with important decisions\u2014to gather our family for a family meeting, a discussion, and a vote. I don't know if this was to get us to endorse a decision he already had made, or in this case, to accept the fact that we were taking a vow of poverty. We were a family of ten living in a two-bedroom home. We all sat down at the dinner table and he said, \"Listen, I have been given this offer. It's a good offer. It's more money than I've ever made.\"\n\nWhatever the salary was\u2014twenty or thirty thousand dollars a year\u2014in the 1960s, that was pretty significant money.\n\nHe went on, \"We won't have to live in a small house. We'll have a big house with plenty of bedrooms. We'll probably have a maid, who will cook. We'll have people who will clean the house. And people to help your mom.\"\n\nHe went on and on about all the positive things. All these things are impressionable for children who don't have anything. Who are struggling and just trying to get by day-to-day.\n\nThen he says, \"But I won't be able to help farm workers.\" And he had a tear in his eye. For me and the older children it was very impactful.\n\nHe said, \"OK, let's vote.\"\n\nHe cut up pieces of paper and wrote on them, \"Yes to go,\" and \"No to stay and help farm workers.\"\n\nWe all voted and put our pieces of paper in a little plastic bowl. He mixed them up. He pulled out the first piece and read it: \"No to stay and help farm workers.\" All but one were the same, \"Stay and help farm workers.\" One of the votes said, \"I want to go and have a big house.\" It was my little brother, who afterward, said, \"I want to go. I'm tired of eating potatoes all the time!\"\n\nI don't tell that story very often. As I got older, I came to realize how important, how committed he was to what he was doing. He had a lifelong commitment, a commitment beyond the human norm. For people committed to causes, it is beyond just a commitment, it is your essence. It is who you are.\n\nOn November 22, 1963, I was fourteen years old and in middle school in Delano. President Kennedy's assassination was announced over the loudspeaker. My dad was devastated. I remember watching the (funeral) procession on our little black and white television with my father and my mother. And seeing my dad's tears coming down from his eyes. He said, \"This was a special person. He was going to make a difference for the country.\"\n\nKennedy's election triggered a sense of idealism, of hope, or anticipation. The Kennedy aura was elevated and grew out of the 1960s. A lot of the legislation that impacted minorities came afterward, from the Lyndon Johnson administration. But if you look at that time period, in the homes of Latinos, you would see calendars, posters, and photographs of President Kennedy. And you would see images of the Virgin Mary. That's the stature, that's the reverence with which he was perceived in Latino homes.\n Nancy, Kate, and Walter Jr. (\"Chip\") Cronkite\n\nChildren of Walter Cronkite, former CBS News anchor\n\n\"Once Dad told me he thought his breaking down on air was unprofessional, but apparently he didn't say this to anyone else and it may have just been a passing thought.\"\n\nNancy Cronkite\n\nON November 22, 1963, I was on the bus after school\u2014the Brearley School in New York\u2014and the flags were at half-mast. The rumor spread on the bus that the president had been killed. I got off the bus and went to my friend's house on East Seventy-Eighth Street. Her mother answered the door, and I pushed past her.\n\nShe was a friend of my mother and quite pretentious; she looked shocked and astonished. I said, \"I have to watch TV right now.\" She must have thought I was quite mad, but she soon realized the catastrophe that had happened.\n\nI probably turned on CBS to watch my father, but I truly cannot remember. When I first heard the news, I don't believe I thought of my father necessarily. It's hard for me to watch the news footage of my father announcing the president's death. I feel nostalgia when I see those images of him.\n\nGrowing up, the conversation at our dinner table was always about world affairs and politics. I've kind of rebelled against that. I don't follow politics in any great depth.\n\nHow would I describe John Kennedy's legacy? I was twelve years old when he was elected. He was a charming man, the first handsome president I ever saw, the kind a girl could have a crush on. I can't imagine anyone having a crush on Mr. Johnson or Mr. Nixon. Presidents Clinton and Obama also are charming; they were so gracious to come to my dad's memorial service.\n\nOf course, one of Kennedy's most important legacies is the Peace Corps.\n\nKate Cronkite\n\nI was thirteen years old on November 22 and was on the bus after school with a friend on the way to Woolworth's to buy cheap makeup. It was announced on the bus that the President had been killed and we thought it was a joke. I remember saying, \"Yeah, sure\u2014Lincoln!\" It was announced again in the store and we headed home. Like other people, I watched Dad on TV the rest of the day. It was comforting to have Dad to watch.\n\nOnce Dad told me he thought his breaking down on air was unprofessional, but apparently he didn't say this to anyone else and it may have just been a passing thought.\n\nWhat is John Kennedy's legacy? To me, he was the first youthful president who seemed to look to an optimistic and exciting future, with a vision of how great a nation we could be, revolutionary in its inclusion of social justice and science in that vision. That was the impression of a teenager who previously hadn't paid much attention to politics.\n\nThere was a lovely book I still remember called That Special Grace by Benjamin Bradlee. It summarized the John Kennedy that I like to think of.\n\nWalter Jr. (\"Chip\") Cronkite\n\nToday, watching my father's announcement of President Kennedy's death, I think of my father, not necessarily President Kennedy. I feel a bit emotional when I see him emotional. We're not used to seeing our fathers get emotional.\n\nI've also watched with some emotion the interview my father did with President Kennedy in September of 1963, the one that inaugurated the thirty-minute news broadcast for CBS. And during the 1960 campaign, my father called John Kennedy on his political sportsmanship after Kennedy wanted to re-shoot some portions of an interview.\n\nMy dad generally never discussed the assassination coverage at the dinner table. Rather, he often talked about minutiae from that evening's broadcast, not necessarily the big policy issues or stories of the day.\n\nKennedy's legacy, at least one aspect of it, is a celebrity legacy. How that affects the president and his own self-image can become a trap. You sometimes do what you think is expected of you because you want to look good.\n\nAnd, of course, one wonders whether he would have pulled us out of Vietnam. Going back, a few years after the end of World War II, we got engaged in Vietnam. Kennedy was just a middle layer in that whole process.\n Casey Murrow\n\nEducator, son of Edward R. Murrow, journalist, director of United States Information Agency\n\n\"It was a brave step to give that speech during the campaign in relation to the Catholic Church. It also was a great message for people of my age at the time\u2014that we need to separate religion and politics.\"\n\nI met John Kennedy for the first time when my father was sworn in as the Director of the United States Information Agency. I was sixteen years old and, for a kid that age, just going to the White House was exciting. I was, of course, excited that President Kennedy had been elected. He was interested in everyone around him, including me. He didn't focus on me to any great degree, but he was welcoming of someone of my age and not just looking to the \"more important\" people who were there.\n\nIn addition to my parents, the president, and me\u2014those in the photograph\u2014there were others there. It may be that Dean Rusk, the Secretary of State, was there. And I'm sure other White House staff members were there. I was very proud of my father. Of course, I didn't fully understand the job he was taking.\n\nOn November 22, 1963, I was a senior at school at Milton Academy outside of Boston. I was dumbfounded, like all my other classmates, when we heard the news. It was private school with children of many Republicans. They weren't all Kennedy supporters, but they were all shocked. They were all stunned. I went downstairs in my dormitory into a room with a television.\n\nMy father also was stunned and overcome. Because he was recuperating from cancer surgery, he couldn't do much for the administration. He was in the hospital; he wasn't at his post. He was quite upset that he could not be part of the USIA effort in reporting the story.\n\nMy parents later told me that Washington felt so subdued. A lot of people had questions on how the Johnson administration would function, especially in the first few days. No one really knew and my dad was somewhat out of the loop. My father's reflections would have been completely different if he had been able to participate.\n\nYes, I was influenced by John Kennedy, especially in his call to public service. Within two years of the president's death, I was in college at Yale and volunteering for a program tutoring inner-city kids in New Haven. I have worked in or with public schools ever since. In retrospect, I was very influenced by John Kennedy, but I'm not sure at the time I realized it.\n\nOne of his legacies was the focus on the separation of church and state. It was a brave step to give that speech during the campaign in relation to the Catholic Church. It also was a great message for people of my age at the time\u2014that we need to separate religion and politics. I have often wished that part of his legacy could have been to see a way through the quagmire of Vietnam that Johnson was unable to avoid.\n\nHe (Kennedy) was one of the most influential figures of the middle of the twentieth century. He touched the lives of so many\u2014even those who did not agree with him. He reached out to a variety of constituencies and pursued what, at the time, seemed like forward looking progressive ideas in so many realms.\n\nHe really was the first young, energetic president since Teddy Roosevelt.\n Craig McNamara\n\nOrganic farmer, son of Robert McNamara, Secretary of Defense\n\n\"'Dad, if you felt our strategic involvement in Vietnam was wrong, why could you not have spoken out on that earlier?' He believed it was his role to serve those presidents and not to be a public voice against foreign policy. His concept of service was different than mine, as was his concept of allegiance.\"\n\nI have several fond memories of meeting both President Kennedy and Mrs. Kennedy on many occasions. One of the earliest memories is when Mrs. Kennedy invited several of us to the screening of the film PT 109. I recall sitting in the White House private screening room. It was warm, relaxed, and very kid-friendly. All of Bobby and Ethel's children were there and the adults were happy. I remember sitting on the floor watching the film.\n\nOur family's friendship with Bobby and Ethel Kennedy's family was very strong. Kathleen, Joe, and Bobby Jr. were all my friends so we spent a lot of time together. The screening in 1963 included the Shrivers and other Kennedy family members.\n\nI also remember some other occasions: flying off from the South Lawn in a helicopter with the president and Mrs. Kennedy, and my mom and dad going to Camp David. I knew at certain times in my life, some things were very unique, very special. But I had no understanding of just how unique and special this particular event was. The fact is that this would happen only twice in my life and it would be a memory I would have forever. Landing at Camp David, I remember the natural beauty of the forests and the rustic cabins. President Kennedy was in his rocking chair in his cabin, sitting around the fireplace; this, too, left a lasting impression. I also recall there were no other kids there at that particular time, so I was encouraged to head off to the bowling alley. \"Go try some bowling,\" said one of the Secret Service agents.\n\nI have many memories of my father's work with President Kennedy: the allegiance, the pride, the call to service that my dad felt to this remarkable human being. Jackie wrote to my dad throughout the sixties, seventies, and eighties. The letters demonstrated a deep respect and friendship between the president and my father, and among my father, Jackie, and my mother. The letters were very supportive and complimentary, thanking him repeatedly over these decades for his service to the president and the country. Jacqueline Kennedy was one of my father's closest confidants after my mother's death.\n\nAnd even years later, when I asked him, \"Dad, if you felt our strategic involvement in Vietnam was wrong, why could you not have spoken out on that earlier? And why did you wait so many years to reveal these beliefs after you were Secretary of Defense?\"\n\nHe responded that he was appointed by two presidents as a cabinet member and believed it was his role to serve those presidents and not to be a public voice against foreign policy. His concept of service was different than mine, as was his concept of allegiance.\n\nMy father would come home from the Pentagon, sometimes at eight or nine o'clock in the evening, and he would rest his hand on the Cuban Missile Crisis calendar that President Kennedy gave him. This is a silver calendar mounted on black walnut with the month of October of 1962. The thirteen days of the crisis are embossed. Dad kept it on the coffee table in the living room. He would put his three middle fingers on the indented dates of the calendar, similar to reading braille, and remember those fateful days. I didn't understand it at the time, but I could see him so deeply moved. He lived with that memory for the rest of his life.\n\nThe letters from Jackie to him at that time are incredible. One of them, dated October 25, 1962, reads in part:\n\nDear Bob,\n\nYou must be very tired. I cannot find the right words to say all the things I feel. I am so proud of you and so grateful you are here. That sounds so pompous and sentimental. Please know, this is not said as Jack's wife, but just as a plain person...\n\nShe went on to say that she felt the whole world might be coming to an end.\n\nOn November 22, 1963, I was a student at Sidwell Friends School in Washington. I received the news of the assassination over the school PA system; the principal requested that I come to the office during his announcement to the student body. I remember walking out, and my mom was in our blue Ford Galaxy with our dog, picking me up at three o'clock that afternoon. I was absolutely crushed. Driving home, I was wondering how this could have happened; then I think I just fell into my bed sobbing, with my dog as my companion, for the next several hours.\n\nThat night, Mom and Dad and I drove that same Galaxy out to Bethesda Naval Hospital where the president's body had been driven. I was waiting in the car while Dad was in with Mrs. Kennedy. These are very profound memories. I have photographs of that evening that are rather haunting, to say the least.\n\nDad was very involved in selecting the president's gravesite. Along with Bobby, he chose one of the most strategic and beautiful locations overlooking the Lincoln Memorial and the Washington Monument.\n\nAfter the assassination, when Mrs. Kennedy moved out of the White House and into her home in Georgetown, I went over multiple times after school to play with Caroline and John. I would have been thirteen and I cherished these times with them. Mrs. Kennedy sent me some of John-John's drawings and his little notes to me. As tragic as it may sound, they were all of airplanes and helicopters, and him flying. From age three.\n\nThe early sixties were a magnificent time in our collective history. I'm not certain that had the president lived, his vision of Camelot, the New Frontier, and the goals of his Cabinet members would have been fulfilled. He was taken away from us in such a violent and critical fashion. This vision is frozen in time. But nothing really freezes in time; everything effects change. And I think life for that presidency would have changed significantly. I can't predict. It's like predicting where we would have been in the seventies, eighties, and nineties if we had not struggled so desperately in Vietnam. How would our lives have changed today if we had not invaded Iraq?\n\nThese are issues one cannot fathom.\n\nMy own personal feeling is that Camelot would have changed and that the realities of Vietnam and civil rights would have significantly altered his presidency. But he was successful in bringing together a remarkable group of people.\n\nThere was a tremendous amount of humor and goodwill. For these young Cabinet members and their families, can you imagine what it was like to be in that era? There was tremendous hopefulness coming out of the fifties. We had a president and first lady who demonstrated grace and had an appreciation for culture, worldliness, and justice. It must have been pretty damn exciting.\n David Rusk\n\nUrban policy expert, scholar, son of Dean Rusk, Secretary of State\n\n\"I began to wander around the family quarters all alone. I blundered into the Lincoln Bedroom and there, on the Lincoln night-table next to the Lincoln Bed, was a hardcover copy of Ian Fleming's Thunderball.\"\n\nI saw President Kennedy twice. The first time was from afar. He came to the campus of the University of California at Berkeley in March, 1962 to speak in Memorial Stadium for Cal's Charter Day. There were fifty thousand to sixty thousand people there.\n\nThe second time was up close. It was October 22, 1963. The night before, my dad called me and said, \"Tomorrow is the one-year anniversary of the height of the Cuban Missile Crisis, and the president has invited all the members of the ExCom to meet with him. He wants to give each of us a memento. Why don't you come by the house tomorrow morning, and we'll drive down together and I'll introduce you to the president?\"\n\nSo I went with my dad to the White House. President Kennedy did indeed have all the members of the ExCom there and I got to meet all of them. Kennedy gave each an engraved calendar of the month of October, 1962, with all the crucial days outlined, and with October 22 having a double outline.\n\nThe group had initially convened on the second floor of the White House in the family quarters. My first impression of President Kennedy was that he was standing very straight, almost as if he was suspended from a coat hanger. He must have had his back brace on.\n\nHe was quite cordial. He asked me, \"Have you ever seen the family quarters?\"\n\nI said, \"No.\"\n\nHe said, \"Well, while you're up here, why don't you have a look around?\"\n\nSo when shortly thereafter all the ExCom members went downstairs, either to the Cabinet Room or the Oval Office, I began to wander around the family quarters all alone. I blundered into the Lincoln Bedroom and there, on the Lincoln night-table next to the Lincoln Bed, was a hardcover copy of Ian Fleming's Thunderball. You may not be aware, but it was President Kennedy who helped launch the James Bond mania in the United States when it became known in 1962 that he was a great fan of the James Bond novels. Just after I stepped out of the Lincoln Bedroom, a White House usher caught up with me. I'm quite sure that he did not want strangers running around the family quarters so he gave me a hurried and perfunctory tour and I was on my way.\n\nA month later to the day, President Kennedy was assassinated.\n\n(On November 22, 1963) there was no school as the D.C. public schools were holding training sessions for teachers. As a local Urban League staffer, I was asked to speak on the civil rights movement to a group of teachers at an elementary school on North Capitol Street, a few blocks from the Urban League offices. The classroom where the session was being conducted must have been next to the school's office, and at one point, I heard a loud gasp through the wall. A few moments later, a secretary came in, clearly shaken. I was the panelist sitting closest to the door, and she handed me a note saying that President Kennedy had been shot and killed in Dallas. At that moment, her principal was speaking. I waited until she finished and then I got up and made the announcement.\n\nIt was like a thunderbolt. Most of the teachers in the room were black women and they were stricken, tears running down their faces. But we finished the workshop because the world's work must go on.\n\nAfter leaving the school, I walked down New York Avenue toward the White House. People on the street were grey. I took a bus home to (my wife) Delcia. Watching the TV news, we talked. We commiserated. And I have a photo taken that weekend, lying on a sofa with my eyes shut and with my week-old son Gregory lying on my chest. Having this new life at a time of a sudden and tragic loss of such an inspiring figure gave me great comfort.\n\nMy mom and dad that day were on an airplane over the Pacific heading for Japan with other members of the president's administration. They turned the plane back to Washington and I spoke with my dad on Saturday. I saw him the next day, Sunday, when I rode with him in a State Department car to the Treasury Department next to the White House. Treasury Secretary C. Douglas Dillon had set aside a corner office in the building for Cabinet members and their families to view the procession transferring the president's coffin from the White House down Pennsylvania Avenue to the Capitol for public viewing.\n\nOn the drive to the Treasury Department, my dad remarked that, \"What really worries me is that I hope they've got tight security around Lee Harvey Oswald.\" No sooner had we arrived in the garage of the Treasury Department then a security man opened the door to the car and said, \"Well, they got Oswald.\" That was the first we had heard that Jack Ruby had shot Oswald, which had just happened on national TV. My dad really wanted to get to the bottom of the assassination, and of course, Oswald's murder helped generate an industry of conspiracy theories that, for some people, haven't ended yet.\n\nI would characterize John F. Kennedy's legacy by the title of Robert Dallek's book, An Unfinished Life. I was an admirer of Kennedy, but I was savvy enough not to be starry-eyed. In the summer of 1963, I volunteered through the Urban League (which subsequently offered me a staff job) to help organize the great civil rights March on Washington for Jobs and Freedom. I remember full well that the march was launched, in part, to motivate President Kennedy and his brother Robert, the attorney general, to do more about civil rights. Ultimately, the president embraced the event, hosting some of the civil rights leaders at the White House afterwards.\n\nPresident Kennedy faced great difficulty moving his proposal on civil rights through Congress because of the alliance between southern Democrats and conservative Republicans. My dad said that President Kennedy approached civil rights issues in a cool, dispassionate manner as contrasted with Lyndon Johnson, who felt racial injustice in every bone in his body.\n Steven Schlesinger\n\nUnited Nations expert, scholar, son of Arthur Schlesinger Jr., Special Assistant to the President\n\n\"On Vietnam, which is the one issue where there's the ambiguity that hangs like a black cloud over Kennedy's presidency... I think he was desperately looking for an exit.\"\n\nMY father taught at Harvard from 1946 up until the time he joined the Kennedy Administration in 1961. He used to give a lawn party at the end of the college year, following the university's commencement ceremony. He would always have an afternoon reception for people who got honorary degrees, or who were guests at Harvard. Then-Senator Kennedy and Jackie Kennedy came occasionally to those parties, and I think that was the first time I ever met them. Even though, then, he was not yet president and did not have a national reputation, he and his wife were still a rather glamorous couple. I don't recall necessarily talking to either one of them.\n\nThe first encounter I had with him that was of any real value was when then-President-elect Kennedy came to my father's house in December of 1960 to use my father's house as a meeting place to talk to a number of different people\u2014professors at MIT and at Harvard\u2014to ask if they would join his administration. He came with a police escort and stopped in front of our house which was on Irving Street in Cambridge. I recall a lot of photographers and Secret Service agents around our house. When he came into our home, he sat in our living room. The door was closed when he was interviewing people. At one point, my father opened the door and brought me into the living room and Kennedy got up and shook my hand. And that was my first real encounter with him as president-elect and also as somebody who recognized me as an individual, as opposed to just being one of the children in the backyard during the time my father is giving commencement parties.\n\nThe second time\u2014and the one time I had a really vivid encounter with him\u2014was when my father was in the White House as a Special Assistant to the President. We (my father and I) were wandering down one of the outside porticos leading to the Rose Garden and we bumped into the president. He was then on crutches because his back was acting up. But he was in an ebullient mood. He was swinging his legs back and forth while leaning on the crutches. He must have just returned from Palm Beach because he had a tan. He honestly looked like a Roman god in my young eyes. He was such an attractive-looking guy anyway, but that tan and the vitality that exuded from his persona were overwhelming. It was almost like he was an immortal figure, in the classic sense of the word.\n\nHe and my father were talking about Latin America. He (the president) was kind of half-looking at me and saying, \"Should we intervene? Should we send down the troops?\" I was amazed that he was even talking to me about what I expect were national security secrets, although I couldn't tell if he was joking, or whether he was actually serious.\n\nHe asked me what I was doing, and I told him at that time I was working for Senator Paul Douglas from Illinois as an intern. It was the summer of 1962. And he (the president) said, \"If you can help us on the Hill, we'd be delighted.\" And I thought, \"He's asking me to help him? I'm just this lowly intern.\" But I was also kind of flattered that he said that to me.\n\nAt one point, I thought, I'd love to ask him the question that was raised by (author Leo) Tolstoy in War and Peace, which is whether one person can make any difference in history or whether history consumes everything, and people who are in power actually are instead driven by the dictates of history, and not able to have any impact themselves. But I could never summon up my courage to ask that question, and the moment passed.\n\nMy father was standing there, and I did not want to appear to be taking advantage of my father's presence and be some idiot son trying to horn in.\n\nThat was the last time I saw President Kennedy.\n\nOn November 22, 1963, I was a senior at Harvard and I was in Leverett House (a student residence) when somebody told me the news. And like many others, I thought, \"Oh, they're just joking. They can't be serious.\" When I realized they were serious, I called my home\u2014the rest of the family were living in Washington\u2014and said, \"I want to come home.\" My father wanted all of his family to be surrounding him when he was going through this ordeal. We obviously wanted to be there, too. Two sisters and a brother of mine were all there.\n\nI cannot remember if I flew that day or the next day. Everything was in an uproar. Nobody really knew what had happened and why. Of course, I remember the television flickering in the background. And then, of course, (Lee Harvey) Oswald was killed by Jack Ruby. Everything seemed haywire. The world seemed to be imploding. Everything that was certain was no longer certain, and in the midst of this, my father was directly involved, and as he portrays in his book A Thousand Days, in greeting the plane when it came back from Dallas. Later, Bobby (Kennedy, the attorney general) asked him to look at JFK's body to see whether the casket should be open or shut. My father saw Kennedy's disfigured face and said, \"No, it should be shut.\" He was not the only person asked to look at the body. It was eventually shut.\n\nMy father and I talked about JFK and his presidency a lot. I was an intern for the Democratic delegation from Massachusetts in 1960 in Los Angeles when JFK was nominated. I had been privy to a lot of conversations my father was having with key people. He tended to bring me in on a lot of this stuff. We (my father and I) had been talking about Kennedy for a number of years before the assassination. I adored Kennedy. He was such a charismatic figure that it was very difficult to go through those years and not be absolutely smitten by the guy.\n\nI think if Kennedy had lived, he would have fulfilled the agenda which he had campaigned on in 1960 and advanced as president\u2014including civil rights legislation, Medicare, the war on poverty, environmental protection, help for the arts, arms control treaties\u2014all of the things for which LBJ later gets credit for in history. And he would have not have prolonged and enlarged the U.S. involvement in the Vietnam War. In many ways, because he was only there for about one thousand days, he has remained more of a romantic figure than a substantive figure. He got a bum deal. A lot of the policies that Lyndon Johnson accomplished in his term actually began in the Kennedy years.\n\nOn Vietnam, which is the one issue where there's the ambiguity that hangs like a black cloud over Kennedy's presidency, it is my belief that he would not have gone down the path Johnson did, which was to make it an American war and pour in three hundred thousand or four hundred thousand troops. I think he (Kennedy) was desperately looking for an exit. And even though he did increase troop levels in a small way, I think he was buying time in order to get a negotiated peace. So I don't think Vietnam would have been the albatross for him that it became for Johnson.\n\nWe certainly remember Kennedy for the Peace Corps. We remember him for initiating the race to put a man on the moon, which happened as he predicted, by the end of that decade of the 1960s.\n\nThe Cuban Missile Crisis is another memorable moment in the Kennedy presidency because it was the closest the world has ever come to a nuclear showdown. JFK handled the showdown in a way that got us out of the crisis, and we won the confrontation with Khrushchev\u2014there was brilliance in the way he managed it. Everybody who looks at how each day went by during the thirteen days of that crisis remarks on how adeptly he was able to come to the right conclusion and produce a settlement that favored what the U.S. wanted. Observers looking back at the Cuban Missile Crisis are reminded of the brilliance of the Kennedy presidency.\n\nI am a specialist on the United Nations. JFK was very much a proponent of the United Nations. He was the only president in his inaugural address who ever mentioned the UN. No other president has ever done that\u2014before or since. I think he would have made the United Nations very much part of American security policy, which, from my point of view, is a very important thing.\n\nOne of the reasons people still have this memory of Kennedy, despite the shortness of his time in office, is the feeling that he was this charismatic, inspirational figure. And he still remains that today. When there are public measurements taken of presidencies in the twentieth century, he's always up near the top. The only person, I think, who obscures him today is Ronald Reagan, even though I think Reagan was a below average president\u2014except for his working out with (Soviet leader Mikhail) Gorbachev some sort of nuclear settlement. Reagan was able to glamorize the presidency the way Kennedy had. He gave it the same upbeat, exciting profile that Kennedy achieved. My own thinking is that the Reagan aura will fade over the years because he really didn't accomplish very much. Whereas, at least with Kennedy, you can point to specific things he did or that he initiated that soon came into fruition.\n\nKennedy is still an inspirational figure. Even with Vietnam, which, unfairly in my view, has become a blemish on his record. And his sexual escapades which have periodically become a matter of headlines. But all that is brushed aside when you look his overall presidency and\u2014from that point of view\u2014he certainly inspired me.\n Jim Swindal\n\nPhysicist, son of James Swindal, pilot of Air Force One\n\n\"(Flying back from Dallas) was 'the most awful mission I've ever been on...'\"\n\nMY father was in the right place at the right time. Just before he was assigned to be the pilot of Air Force One, he was the commander of the Special Missions Group at National Airport. He flew many important government officials and Washington dignitaries around the country and around the world. He also had an outstanding career as a pilot in the Air Force, starting with World War II. He also had a perfect safety record. All the important dignitaries liked him.\n\nI remember my father saying he had been invited to have lunch with the president and Jackie in the cabin of Air Force One while they were on a trip. But he turned the president down. He's probably one of the few people to turn down the opportunity to have lunch with the president. He did it because he had been talking to a press person on the plane and he told the reporter, \"If you guys found out that something bad had happened while I wasn't in the cockpit, you'd have a real good time at my expense.\"\n\nMy father thought very highly of the president and the Kennedy family. Kennedy loved Air Force One, as did my father. Actually, my father went to Seattle to pick up the plane when the Air Force bought it from Boeing. My father respected JFK, and JFK respected my father and the whole crew of Air Force One.\n\nHe visited my father from time to time in the cockpit, and Jackie frequently would sit in the cockpit on the console between the pilot's and copilot's seats. She would talk to the crew, look out the window, and watch them fly the airplane.\n\nOn November 22, 1963, I was walking down the hall at my office at United Aircraft Corporation, which later became United Technologies\u2014I was an engineer there\u2014and it was announced on the public address system. At the time, we only heard that he had been shot. I did not hear that he had been killed until later in the afternoon. I was twenty-six and living in Westport, Connecticut.\n\nI knew my father was in Dallas. I frequently went home to my parents' house on weekends, and I talked to my father a good bit on the phone. He would tell me about trips he was taking with the president, unless of course they were secret. My father saluted the president's coffin as it was brought onto the plane at Love Field in Dallas. He was quoted in the New York Times as saying, \"I didn't belong to the Lyndon Johnson team. My president was in that box.\"\n\nWhile they were still in Dallas, nobody knew what was going on. As far as they knew, the president's murder could have been the beginning of something much more serious\u2014possibly an effort by the Soviets to disrupt the U.S. government. So when they took off, they few as high as they could go safely and as fast as they could go. He said he pushed the throttle to the max.\n\nThe next time I spoke with my father, he told me it was \"the most awful mission I've ever been on\" even compared to those missions in World War II in the China-Burma Theater.\n\nThree days later, my father flew Air Force One over the gravesite during the funeral. He flew very low and dipped the wings of the aircraft left and right in tribute. There were a lot of tears evident.\nSECTION FOUR:\n\nCelebrities\n Vincent Bugliosi\n\nAttorney, author\n\n\"(T)here is little comparison between the nation's response to Kennedy's death and its response to the World Trade Center catastrophe on September 11, 2001... The World Trade Center victims were known only to their loved ones. But the dazzling first couple of JFK and Jackie, and their two children, Caroline and John-John, were perceived by many as the closest to royalty this nation has ever seen.\"\n\nON November 22, 1963, I was twenty-nine years old and in law school at UCLA. I was between classes and walking past the front desk of the law school office. One of the assistants said to me (and I'm paraphrasing), \"Mr. Bugliosi, the president has just been shot in Dallas, Texas.\"\n\nI happened to be president of my senior class at the time, and I took it upon myself to go to the two classes that were in session. I told the professors what had happened and asked if I could address the classes. I announced to both classes that President Kennedy had been shot in Dallas, and that I did not have any additional information at the moment. The professors immediately excused the classes as well as the rest of their classes for the day.\n\nAlthough there was the inevitable, immediate chatter around campus, one thing I remember is that there was a sense of numbness and there was mostly stunned silence. There was no loud chattering, as one might expect with a cataclysmic event.\n\nOver the weekend, the main recollection I have is that there was absolutely nothing on television except that which pertained to the assassination. I had never seen anything like this before\u2014or since. And there was no advertising.\n\nThe most indelible image from that weekend is TV showing throughout the night the silent, slow, endless progression of everyday Americans walking past the president's coffin in the Capitol rotunda. I've been told that there was some symphonic music playing, but I don't recall it. I don't recall any sound at all. The silent progression sticks in my mind.\n\nTelevision made the tragedy so much more real and palpable, much more than just reading about it. Or listening to the radio. Television kept the extremely penetrating shock of the tragedy alive longer. If the horror started to recede in one's mind, it was not allowed to do so. Television kept covering it. It kept the feeling of sorrow alive. It was the biggest thing television had ever covered. You felt the horror. You could see the tears in people's eyes. For the first time in the history of the medium, television was used to give America a new dimension to grieve.\n\nKennedy's popularity, next to that of Lincoln, remains above all other presidents from most polls I've seen. Also, whenever one speaks of a presidential candidate, in terms of hope or inspiration or idolization, JFK, invariably, is the ideal. He is the model used for comparison. People often say, \"He reminds one of JFK... \"\n\nExcerpt from Reclaiming History: The Assassination of President John F. Kennedy, by Vincent Bugliosi, \u00a9 2007 (Used with author's permission):\n\n(T)here is little comparison between the nation's response to Kennedy's death and its response to the World Trade Center catastrophe on September 11, 2001, even though the response to the latter was enormous. Just two indications among many of the differences. On the day of Kennedy's assassination and for three consecutive days thereafter, all three national television networks suspended all their commercial shows and advertising. And while only a relatively small number of books have been written about 9-11, far more books continue to be written to this very day, over forty years later, about Kennedy's assassination. How could one death cause greater personal anguish to more people than three thousand deaths? The World Trade Center victims were known only to their loved ones, entirely unknown to the rest of the country. But the dazzling First Couple of JFK and Jackie, and their two children, Caroline and John-John, were perceived by many as the closest to royalty this nation has ever seen. Nearly all Americans felt they knew JFK intimately, his charm and wit regularly lighting up the television screen at home. This is why polls showed that millions of Americans took his assassination like a \"death in the family.\"\n\nPerhaps the most impressive testament of what Kennedy possessed was the way his death was greeted by the tens of millions of people behind the Iron Curtain during the very height of the cold war. The masses behind the curtain were only fed Soviet propaganda, the level of censorship being virtually complete, isolating people from the outside world. Yet, when Kennedy died, the evidence is overwhelming that millions of Soviet citizens and those in the Soviet bloc satellite countries in Eastern Europe took his death almost as hard as we did in America, these adversary countries being immediately swept up in national mourning and tears. Under the prevailing censorship, how much could the people of these nations have possibly been exposed to Kennedy, the tiniest snippet of his person or his words reaching them during his three years in power? In addition to the youthful vigor and the indefinable charisma that he projected, my sense is that these masses, who heard so few of his words and could understand none of them, picked up in the sound of his voice and the inherent decency and sincerity in his always pleasant face and smile, that he was different, and it was these additional critical components that enabled the essence of the man, from just a glance, to pierce the curtain of iron that had descended upon these countries and to touch the hearts of its citizens.\n\nSecond only to Lincoln and rivaled only by Franklin Roosevelt, Kennedy rose above the national constraints of politics. Even with his own patrician pedigree, he tried to do what was right for the average American.\n\nJohn Kennedy was the most charismatic, attractive, and inspirational president in our history. He was a genuine American war hero. He truly was the Camelot figure who rekindled the notion that public service was a noble calling.\n\nDuring his presidency, idealism was in the air, and the nation's capital had never seen such an invasion of young people who wanted to change the world for the better. Also, no one before or since JFK instilled in Americans the sense of hope and promise for a better America. People spoke of JFK's \"irresistible charm;\" it was well known that he was not vindictive toward his political opponents, and unlike his brother Bobby, he had few, if any, bitter enemies. Except for segregationists and the militant right, as Look magazine said, \"Even his political opponents liked and respected him.\" There can be no question that there was something very special about John F. Kennedy.\n Rev. Billy Graham\n\nChristian evangelist\n\n\"During our conversations I became aware that he was concerned about the moral and spiritual condition of the nation. He was especially concerned about the scars which might have been left by the intense religious issue during the presidential campaign.\"\n\nEXCERPTS from \"Hour of Decision\" Sermon \"Our Fleeting Lives,\" broadcast on November 24, 1963, by Billy Graham \u00a9 1963, used by permission, all rights reserved:\n\nMany memories flooded in my mind and heart this week as I thought of President Kennedy. I first met him when he was a congressman. Then I got to know him better as a senator. Four days before he was inaugurated as president he invited me to Palm Beach, Florida, to spend the day with him. I had two meals with him, played fourteen holes of golf, and then he took me to a little party in the evening. That day he drove me around in his white Lincoln convertible.\n\nDuring our conversations I became aware that he was concerned about the moral and spiritual condition of the nation. He was especially concerned about the scars which might have been left by the intense religious issue during the presidential campaign. He asked me a number of questions about the Bible. One of his questions was, \"Where do you think history is going and what is the objective of history?\" I told him that the Bible teaches that history would someday come to a dramatic conclusion with the second coming of Jesus Christ and the setting up of God's kingdom on earth.\n\nHe replied, \"I'm interested in that.\"\n\n(On November 22) I thought of the president's elderly father and mother who had received the news from a workman who heard it on the radio. I thought of the shock and emotions that must have run through Senator Ted Kennedy's mind and heart as he heard the news while presiding over the Senate. Attorney General Robert Kennedy was having lunch in Virginia when the news came to him. The president's younger sister learned of the assassination as she watched a television program from Dallas.\n\nThe immediate sense of personal loss is much greater for the American people than it was at the death of McKinley, Garfield, and even Lincoln because by means of television the president was almost a daily visitor in our homes. A modern American president becomes a family friend and a familiar figure. Thus the sense of loss on the part of the peoples of the world is very deep.\n\nThis weekend the radio and television networks are playing religious music. The entire nation is thinking more about death and eternity than at any time since the war. If ever there was a time when Christians have an opportunity to witness to the saving grace and power of Christ, it is at this hour.\n\nLongfellow once said, \"It is not till time, with reckless hand, has torn out half the leaves from the book of human life to light the fires of passion... that man begins to see that the leaves which remain are few in number.\"\n\nPresident Kennedy never dreamed last Friday morning when he had his breakfast that by two o'clock he would be in eternity. We never know when our moment is coming. Certainly in this tragedy of the last few hours we should learn a lesson of the brevity of time and our need for preparation to meet God at any moment.\n Lee Iacocca\n\nAmerican industrialist\n\nI met John Kennedy when he was a senator. Ford Motor Company had an apartment in Washington, and Kennedy had an apartment nearby. I frequently went to Washington for meetings and sometimes saw Senator Kennedy in the morning wearing his Chesterfield coat. We exchanged pleasantries, but nothing more than that.\n\nOn November 22, 1963, I was crossing the street outside the world headquarters for Ford in Dearborn, Michigan, and someone came running up to the corner screaming the news that President Kennedy had been shot. I was stunned. That weekend, I read, watched, and listened to everything that was available. The poignant family images made the biggest impression on the enormous impact of the loss of President Kennedy.\n\nJohn Kennedy was a fascinating man who was great with the press. But he didn't live or was in office long enough to judge him fairly on any accomplishments. He personified charisma and spirit. Kennedy had a twinkle in his eye that left a lasting impression. He knew where he was headed.\n Jimmy Piersall\n\nMajor League Baseball player, author\n\n\"I was playing golf in Cape Cod. Just ahead of us on the course was Rose Kennedy playing alone... a Secret Service agent drove by us in a golf cart and said, 'The president's been shot and he's probably going to die.'\"\n\nEVELYN Lincoln, President Kennedy's secretary, was a big fan of mine and of the Red Sox, so she invited me to the White House in 1961. A few minutes after I arrived at her desk outside the Oval Office, I saw the president approaching, and it was evident he was quite upset. Somebody hadn't done something he had wanted. He said a hello and gave me a PT 109 bracelet and a pen and pencil set.\n\nI have a photo of him and me taken on opening day one year in Washington. He was standing behind the dugout, and these Secret Service agents came over and asked whether I wanted to have my picture taken with him. I said I'd be happy to.\n\nMy impression was that he seemed to enjoy people he wanted to know. And that was me at the time. He always had a big smile. I think being a Catholic with that smile helped get him into the White House. The next time I saw him, we were in Cape Cod. He loved to drive around in a golf cart and take his children and nieces and nephews into town to get ice cream.\n\nOn November 22, 1963, I was playing golf with three doctors in Cape Cod. Just ahead of us on the course was Rose Kennedy playing alone. The doctors and I were walking up a fairway and a Secret Service agent drove by us in a golf cart and said, \"The president's been shot and he's probably going to die.\" I was really, really shocked. Of course, we stopped playing. I had tears in my eyes. I felt we had lost a great man who had done a great job.\n\nI went to Washington for the funeral. I knew the man and he treated me very nicely. The fact he got shot broke my heart. I watched as the funeral procession went by the White House. He was a big baseball fan. And being from Boston, he was always there\u2014whether in Boston or Washington, or other cities, on opening day. And after a while, he used to bring his kids to the game.\n\nWhat's his legacy? Because he was Catholic, many people did not respect him. But I did. He had a happy way about him. Most presidents are ornery, but he wasn't; he got along with pretty much everybody.\n Cliff Robertson\n\nActor\n\n\"I arrived at the White House two hours early.... All of a sudden the door opened behind me, and I heard the voice, 'Hi, Cliff.'\"\n\nTHERE had been a lot of talk at that time. (Studios) were playing with the idea of a movie about Kennedy's experiences in the South Pacific in World War II, the subsequent crash (with the Japanese destroyer) and the heroic efforts to save some of his men. There was a lot of talk in Hollywood about who was going to play the part. The two who were suggested among others were Peter Fonda and Edd \"Kookie\" Byrnes, the fellow who had the hair. But I don't think he had as much as Kennedy.\n\nI had no concept in my wildest imagination that I would be considered. You don't identify yourself with presidents. And I didn't know him. I had never met him, though I had met Patricia, his sister, and I knew some of the Kennedys socially.\n\nNo one was more surprised than I, and when I was told I would be receiving a proposed script for PT 109, I was filming another picture at Paramount. They brought me this very impressive folder with all this information about Kennedy and PT boats. I was told to go home and learn some of the scenes. And I said, \"I'm working on this picture.\"\n\nAnd they said, \"This is all right. It's been arranged.\"\n\nI was surprised. But in this business, there are always conjectures. Rumors are always rife. So I went to Warner Brothers and shot several scenes and they treated me very nicely. Then I went back to work on the picture I had been doing previously.\n\nI got a phone call a few days later from a friend of mine in New York, and she said, \"Guess what?\"\n\nShe said, \"Your name has been printed in the columns and also in the New York Times, and they say you're going to play it.\" So she was calling me to congratulate me.\n\n(President Kennedy) had seen the different scenes that were shot for tests. (Press Secretary Pierre) Salinger told me that they were in the screening room. He (JFK) was a very busy man who wouldn't take a long time. He'd look at an actor's scenes and he'd comment one way or the other.\n\nSalinger said, \"As soon as you came on and you did your scene, he (JFK) said, 'That's it.'\" So, he was in a hurry. And I think the president said to himself, \"I'm so bored with these actors,\" and \"I'm going to pick the next one, even if it's a girl.\"\n\nSo there were pictures of me and the president in the papers. I finished the picture at Paramount, and then I started reading up as much as I could about John Kennedy. I made a habit of absorbing even more about Kennedy, anything I could get, whether from people who knew him or information I had read. I had an impression of him, and needless to say, I tried to enhance that by seeing as much and reading as much as I could about him.\n\nLike so many Americans, I was a big fan of his, and I felt that maybe this was kind of a dream I'm having. But I thought, \"I'll ride it out.\" I was delighted. I did not feel I was the best person for the part, but I certainly was not going to turn it down. I had such high regard for the president.\n\nI began to study more closely. He'd be on TV, and I would read anything I could about him, as I would with any role, particularly a biographical role. I was told that he didn't want me to try to do the accent. He indicated that any nightclub comic could do \"Ask not\" (in Bostonian accent). That would have been a distraction.\n\nSo, I received more information from Washington. And then I was told we were going to shoot it in Florida. And we shot some at Warner Brothers. They actually had a PT boat on the stage, in the water. So we shot all that stuff, when we were cut in half by the (Japanese) destroyer. We had the fire on the water. And I swam through that; it wasn't a big deal. You have all kinds of protection. But I did a little bit of my own stunts.\n\nAfter we finished, I got a call from the White House asking me would I like to come back and meet the president. And I said, \"You bet.\"\n\nSo I went to Washington and I arrived at the White House two hours early. I wasn't about to be late. (He) had an innate ability to make you feel very comfortable, and I did immediately, because of his graciousness and ease, regardless of having waited for two hours and nervously checking my watch every five minutes.\n\nAll of a sudden the door opened behind me, and I heard the voice, \"Hi, Cliff.\"\n\nIt was just as casual as that. He knew people that I knew, so it wasn't like we were complete strangers, and he said very nice things. I was very impressed. He could not have been more generous. And more helpful. We exchanged letters after that, for the short time left in his life. He also gave me a present for my daughter, Stephanie, because we talked a lot about our two daughters who were the same age.\n\nI admired the qualities of his character. Even though he was brought up, some might say, with a silver spoon, he had parents who were strict as far as scholarship, ambition, energy, and working for the country went. He was a public servant.\n\nI know that, in his younger days, he flirted with the idea of going into journalism, and he was a damn good writer. Those, too, had been my interests. I once worked for about twenty minutes for a daily paper, and I had visions of becoming a journalist, sort of jaded and cynical.\n\nPT 109 was a seminal film for me, because of the attention it received, as well as the respect that President Kennedy had garnered from the American public.\n\nWe have lived in such a turbulent time. I think about how some people, who were very prominent, faded away so quickly in history. With all the troubles we've had since Kennedy was killed, I think people respect him all the more. So often, people say, \"Well, Kennedy would not have done that,\" or \"That doesn't sound like Kennedy.\" He was truly presidential.\n Alex Trebek\n\nTelevision personality\n\n\"However, even though I was the most junior announcer on staff, I saw this for the huge news story that it was...\"\n\nLIKE most foreigners, I knew him primarily from news stories and from his televised press conferences, and like most foreigners, I was captivated by his intelligence and his great sense of humor, especially when he was dealing with the reporters. He and Jacqueline were so very different from their predecessors in the White House. They were much younger and so brought an element of youth and vigor to the capital as well as a certain style and elegance that were unprecedented.\n\nIn those days, I was working as a staff announcer for the Canadian Broadcasting Corporation at its English language headquarters in Toronto and hosting my first television show, Music Hop, a sort of Canadian version of American Bandstand.\n\nOn November 22, 1963, I was on booth duty doing news, weather, sports, and station breaks for the national radio network when the news came in from Dallas. I was the one who read the UPI wire that confirmed the death of the president.\n\nAt that time, the CBC had a standing protocol that with the death of her majesty the queen, the prime minister, or another major world figure, our regular programming would cease, and we would start playing classical music that was to be interrupted by announcements every fifteen minutes or so. That's what we did.\n\nHowever, even though I was the most junior announcer on staff, I saw this for the huge news story that it was, and one that was ongoing, so I went to see the news director on duty and made the case that the tragedy of the American president's assassination was far too important to cease all broadcasting, and that the CBC had a responsibility to our listeners to keep them informed as to what was happening. Fortunately he agreed, and for the rest of the day and into the weekend, we aired news feeds from all the American radio and TV networks.\n\nLetitia Baldrige, Jacqueline Kennedy's Social Secretary, gives remarks at an event at the White House in 1963, just a few months before President Kennedy's assassination. Joining the festivities are Mrs. Kennedy and her two children, Caroline and John. President Kennedy, Baldrige said, did not \"dwell on anything that was sad or gloomy. He was funny about anything that he could be funny about, and he taught us all how to do that.\" (Photo: \u00a9 Robert Knudsen, White House Collection, John F. Kennedy Presidential Library)\n\nDr. William F. Bernhard, left with his back to the camera, performs surgery in the hyperbaric chamber of the Industrial Medicine Department of the School of Public Health at Harvard. In 1963, Bernhard labored for thirty hours in the chamber to try to save the life of the Kennedys' second son Patrick. (Photo courtesy Dr. Bernhard)\n\nPresident Kennedy and Senator Frank Carlson (Republican of Kansas) look on as the Rev. Billy Graham delivers remarks at the National Prayer Breakfast in February, 1963. It would be the last time Rev. Graham saw John Kennedy. (Photo courtesy the Billy Graham Evangelistic Association)\n\nCivil rights activists Julian Bond, right with light-colored shirt, and Andrew Young, center in dark suit, holding hands with others in unity during the 1963 March on Washington. Bond remarks that \"Kennedy watched the pictures of young people being fire-hosed in the streets of Birmingham, and he told onlookers, 'This makes me sick'\u2014he was moved by it, and it shifted him.\" (Photo: \u00a9 Francis Miller\/Time Life Pictures\/Getty Images)\n\nBenjamin C. Bradlee, then a correspondent with Newsweek and later editor of the Washington Post, appears engrossed in conversation with John Kennedy, Jr. at the Kennedy estate near Middleburg, Virginia. The photo was taken less than two weeks before Bradlee's close friend, President John Kennedy, was assassinated. (Photo: \u00a9 Cecil W. Stoughton, White House Collection, John F. Kennedy Presidential Library)\n\nFuture NBC Anchor Tom Brokaw sharpens his broadcasting skills as a student on KUSD radio at South Dakota University in 1960. He later covered President Kennedy's 1962 visit to Offutt Air Force Base near Omaha, Nebraska. Regarding the news coverage of JFK's assassination and funeral, he remarked: \"The television set was, if you will, the centrifuge for the country. Everybody drew from it in some fashion.\" (Photo: \u00a9 South Dakota Public Radio Collection, University of South Dakota Archives and Special Collections)\n\nCharles U. Daly, a congressional liaison in Kennedy's White House, smiles alongside his boss. \"He really did make a difference in the world. I think that is where legacy and inspiration meld.\" (Photo courtesy Mr. Daly)\n\nRobert Dellwo, attorney and Spokane County Democratic Party chair, right, joins then-Senator John Kennedy at a downtown rally during the 1960 presidential campaign. \"Bobby and Ted were 'doers,' and were expected to take care of their older, prominent brother.... 'the boys,' as we called them, were always busy attending to details.\" (Photo: \u00a9 Argenta Images)\n\nBarry Goldwater Jr. photographs then-Senator John Kennedy at the U.S. Capitol. In 1963, Goldwater said his father, Senator Barry Goldwater, and Kennedy talked about running against each other (in 1964) and decided they would travel around the country together, stop at different cities, and debate each other. \"It would have been very entertaining and would have set a high standard for future campaigns.\" (Photo courtesy Mr. Goldwater)\n\nThen-Senator John Kennedy greets waitress Phyllis Elkins while meeting with Congressman Ken Hechler, right, and David Fox of the local Democratic Party during the 1960 primary in West Virginia. The brief encounter at Jim's Steak and Spaghetti House in Huntington is memorialized with a photo in the restaurant. (Photo courtesy Edward H. Dawson Jr., Editor and Publisher, The Herald-Dispatch, Huntington, West Virginia)\n\nPresident-elect John Kennedy comments to United Press International White House correspondent Helen Thomas after visiting his wife and newborn son John Kennedy, Jr. at Georgetown Hospital in Washington, D.C. In her foreword, Thomas states that Kennedy was \"the most inspired and inspiring of any president I covered.\" (Photo: \u00a9Bettmann\/Corbis Images)\n\nSenator John Kennedy, campaigning for the presidency in Portland, Oregon in May, 1960, addresses the opening ceremonies of the Riverside Little League. Mike Gefroh, who caught Kennedy's ceremonial first pitch, remarked that Kennedy made him more aware of politics and that, later that year as he watched on television the Democratic Convention in Los Angeles, \"the suspense of the balloting was a terrific initiation into the political process.\" (Photo: \u00a9 Argenta Images)\n\nA drawing for and a letter to eleven-year-old Craig McNamara, son of Defense Secretary Robert McNamara, from John F. Kennedy Jr., age three, less than a month after President Kennedy's assassination. The letter, presumably dictated by the young boy, was written by his mother Jacqueline Kennedy. Mr. McNamara remarks: \"I'm not certain that had the president lived, his vision of Camelot, the New Frontier, and the goals of his Cabinet members would have been fulfilled. He was taken away from us in such a violent and critical fashion.\" (Images courtesy Mr. McNamara)\n\nJohn Kennedy, following his nomination for the presidency by the Democratic Party in 1960, greets his nephew Christopher Kennedy Lawford and his sister Sydney in Los Angeles. \"(M)y uncle sat down and said he had just been nominated to be president of the United States, that it was going to be a very hard job, and that he wanted my help.\" (Photo: \u00a9 Argenta Images)\n\nJohn Lewis, then-chairman of the Student Non-violent Coordinating Committee, addresses thousands of civil rights supporters at the March on Washington, August 28, 1963. He and five other civil rights leaders, including the Rev. Dr. Martin Luther King, Jr., were congratulated later that day by President Kennedy in the Oval Office. (Photo: \u00a9 Bettmann\/Corbis)\n\nHearst reporter Marianne Means greets President Kennedy at a diplomatic dinner in Washington in 1962. Means first met then-Senator John Kennedy when she was a college student at the University of Nebraska and mentioned she wanted to be a journalist. \"He said, 'Well, when you come to Washington, look me up.' Being na\u00efve, I took him at his word.\" (Photo: \u00a9 Bettmann\/Corbis)\n\nFifteen-year-old Michael Medved, the future author, film critic, commentator, and syndicated radio show host, beams for the camera after the self-proclaimed \"political geek\" won an essay contest on American history. \"I recently found my notebook from that period. It has a very carefully hand-lettered 'Kennedy for President' message on it. Of course, I spelled his name wrong\u2014'Kenedy.'\"(Photo: \u00a9 Los Angeles Public Library Photo Collection)\n\nThen-Minnesota Attorney General Walter Mondale meets with John Kennedy at the White House in 1963. Reflecting on JFK's assassination, Mondale remarks, \"There still is a horrible sense of loss nearly fifty years later. We were robbed.\"(Photo: \u00a9 Minnesota Historical Society)\n\nCasey Murrow, left, observes his father, broadcast journalist Edward R. Murrow, and his mother Janet Murrow, joking with President Kennedy at the White House in March of 1961, on the day his father was sworn in as the director of the United States Information Agency. (Photo: \u00a9 Cecil W. Stoughton, White House Collection, John F. Kennedy Presidential Library)\n\nAl Neuharth, shown here in 1950 as the editor of the Volante, the student newspaper at the University of South Dakota in Vermillion, later founded USA Today, the Freedom Forum, and the Newseum, in Washington, D.C. He last saw John Kennedy at a speech in Miami, just a few days before the president's assassination. (Photo: \u00a9 South Dakota Public Radio Collection, University of South Dakota Archives and Special Collections)\n\nFather Oscar Huber walked three blocks from his parish to wave to President and Mrs. Kennedy as they drove through Dallas. Less than ninety minutes later, he administered the Last Rites of the Catholic Church to John Kennedy at Parkland Memorial Hospital. \"During this most trying ordeal, the perfect composure maintained by Mrs. Kennedy was beyond comprehension. I will never forget the blank stare in her eyes and the signs of agony on her face.\" (Photo: \u00a9 DeAndreis-Rosati Memorial Archives, DePaul University Archives, Chicago, Illinois, DePaul University)\n\nRobert F. Kennedy and his wife Ethel wave to well-wishers during the inaugural parade in Washington on January 20, 1961. Following the assassination of President Kennedy less than three years later, he wrote in a eulogy, \"President Kennedy was fond of quoting Dante that 'the hottest places in Hell are reserved for those who, in time of great moral crisis, maintain their neutrality.'\" (Photo courtesy of Ira Seiler, M.D.)\n\nFuture environmental lawyer and activist Robert F. Kennedy Jr. presents his uncle with a salamander in the Oval Office in 1961. He later writes, \"(T)here had been a sense of foreboding even within our family as he and Aunt Jackie prepared for the trip (to Dallas).\" (Photo: \u00a9 Abbie Rowe, White House Collection, John F. Kennedy Presidential Library)\n\nActor Cliff Robertson meets with John Kennedy in the Oval Office shortly after completing the film PT 109 about Kennedy's naval career. The movie, Robertson said, \"was a seminal film for me, because of the attention it received, as well as the respect that President Kennedy had garnered from the American public.\" (Photo: \u00a9 Cecil W. Stoughton, White House Collection; courtesy Mr. Robertson's daughter, Stephanie Saunders)\n\nAt a meeting during the Cuban Missile Crisis in October 1962, Secretary of State Dean Rusk looks on as President Kennedy examines a document in the White House Cabinet Room. Secretary of Defense Robert McNamara is seated to the right; Press Secretary Pierre Salinger stands at the rear. (Photo: \u00a9 Cecil W. Stoughton, White House Collection, John F. Kennedy Presidential Library)\n\nJohn Seigenthaler Sr. with Robert Kennedy. Seigenthaler first met John Kennedy when he covered the Senate for the Tennessean and later worked in the Department of Justice under Robert Kennedy. He was present when President-elect Kennedy convinced his younger brother to serve as attorney general, remarking, \"So that's it, General. Let's grab our balls and go.\" (Photo courtesy Mr. Seigenthaler)\n\nIra Seiler, M.D., examines an infant as part of his pediatric practice. In 1960, as a second-year resident at Georgetown Hospital in Washington, D.C., Dr. Seiler saved the life of John Kennedy Jr. who was born not breathing. He later received a personal note of thanks from then-President-elect Kennedy, and three years later, upon learning of JFK's assassination, wrote a letter of condolence to Jacqueline Kennedy. (Photo courtesy Dr. Seiler)\n\nCBS News Producer Sanford (\"Sandy\") Socolow, left, works alongside Anchor Walter Cronkite. Socolow ended his planned vacation abruptly on November 22, 1963, to work four days with little sleep \"looking for people to interview and feeding Cronkite and then later Charles Collingwood who spelled him.\" (Photo courtesy Mr. Socolow)\n\nAl Spivak, reporter for United Press International, far left, joins President Kennedy and other journalists saying farewell to a visitor leaving by helicopter from the Kennedy compound in Hyannis Port, Massachusetts. \"JFK felt that it was his job to protect the country and the world against Khrushchev. That was a great achievement. It kept the country out of a Third World War.\" (Photo courtesy Mr. Spivak)\n\nPresident Kennedy greets Air Force One Pilot Colonel James B. Swindal and his wife, Emily, at a reception on the South Lawn of the White House. Swindal's son, James L. Swindal, said his father, who flew missions in the China-Burma-India Theater during World War II, referred to the flight back to Washington from Dallas on November 22, 1963 as \"the most awful mission I've ever been on.\" (Photo courtesy Mr. Swindal)\n\nNBC White House Correspondent Sander Vanocur, center, and other journalists corner John Kennedy at the 1960 Democratic National Convention the day after JFK secured the presidential nomination. The senator had just left a meeting with Senator Lyndon B. Johnson to discuss the vice presidential position on the ticket. (Photo: \u00a9 Associated Press)\n\nMortimer Caplin, Commissioner of the Internal Revenue Service, second from right, comments to Kennedy aide, Kennedy O'Donnell, far right, after the president met with staff at the IRS offices. Kennedy, seen joking with Treasury C. Douglas Dillon, was the first and remains the only president to visit the IRS offices in Washington, D.C. (Photo: \u00a9 Abbie Rowe, White House Collection, John F. Kennedy Presidential Library)\n\nAssistant Attorney General Nicholas deB. Katzenbach, left, confers with Attorney General Robert F. Kennedy and Herbert J. Miller, another Department of Justice official. The artwork was a creation of Kennedy's son David. Katzenbach said he believed the issue of civil rights was \"the most difficult problem we faced during the Kennedy presidency.\" (Photo: \u00a9Bettmann\/Corbis)\n\nBob Schieffer, who looks more like a detective than a reporter with the Fort Worth Star-Telegram, wore his \"Dick Tracy hat\" and trench coat while escorting Lee Harvey Oswald's mother Marguerite into the Dallas Police Department on the afternoon of November 22, 1963. After a few hours, an FBI agent called his bluff, moments before he was to question the accused assassin. \"It would be the biggest story I almost got but didn't,\" he says. (Photo courtesy Mr. Schieffer)\n\nHarris Wofford, associate director of the Peace Corps, chats with President Kennedy during an event with Peace Corps volunteers on the South Lawn of the White House in August of 1962. Wofford previously served as JFK's special assistant on civil rights and later represented Pennsylvania in the U.S. Senate. (Photo: \u00a9 Rowland Scherman, Peace Corps\/John F. Kennedy Presidential Library and Museum)\n\nFort Worth congressman Jim Wright, far left, the future Speaker of the House of Representatives, addresses a crowd outside the Texas Hotel, just prior to President Kennedy's remarks on the morning of November 22, 1963. Vice President Lyndon Johnson, who would take the presidential oath of office just a few hours later, and Texas Governor John Connally (waving) are to the left of Kennedy. (Photo : \u00a9 Cecil W. Stoughton, White House Collection, John F. Kennedy Presidential Library)\nSECTION FIVE:\n\nFamily and Friends\n Dr. Bob Arnot\n\nPhysician, journalist, author\n\n\"For all of us, it was the end of hope. It's one of the few moments that I remember\u2014a particular time, a particular day.\"\n\nHE was the only hero I ever had.\n\nI first saw him while he was giving a speech at Boston College. He was running for president. I was twelve years old. I was mesmerized. I had never met anybody so electric, so charismatic. It was the most memorable event of my whole youth. I had that famous picture of JFK in his pinstriped suit right up above my bed, a three-by two-foot poster.\n\nWe lived, dreamed, and breathed John F. Kennedy. We knew members of the administration. I've been down to the Kennedy compound, playing tennis with Joe Kennedy (eldest son of Robert Kennedy). I was there the day Arnold Schwarzenegger first arrived and was down there for his wedding to Maria Shriver. It was spectacular.\n\nMy father had been a lifelong Republican. But he listened to one of the Kennedy-Nixon debates in 1960. He took his Nixon poster down and ripped it up and said, \"I've had it. I'm voting for JFK.\" I think it was the only time he voted for a Democrat. We lived in the most Republican town in the world, Wellesley Hills, Massachusetts.\n\nWhen he was president, there were a couple of occasions when I might have seen him from a distance. There was a motorcade in Boston where I saw him. Our best friends growing up were a family called Fulham. The father, Jack Fulham, was chairman of Kennedy's campaign organization in Massachusetts. One time, he was invited to a White House dinner and sat with Marjorie Merriweather Post. He was dying to turn the china over to see where it was made, but of course, he didn't want to sitting next to the etiquette lady. But then she turns the plate over to look!\n\nOn November 22, 1963, I was in Mr. Leyden's chemistry class in the tenth grade. There was a TV set in the room and it was announced the president had been shot. And we all went home, and I learned later he died. I never cried so hard in my life. Not when my dog died. We just sat by the television for days afterward. For all of us, it was the end of hope. It's one of the few moments that I remember\u2014a particular time, a particular day.\n\nThe most devastating image was when Air Force One came in at Andrews and the ambulance came by. We finally knew it was true.\n\nMy mother always told me, from the first minute I could understand language, \"To whom much is given, much is expected.\" She said that all the time to all of us. It was an interesting message with no way to execute it. Then JFK came along with the Peace Corps.\n\nJFK was pure inspiration. It wasn't necessarily inspiration just to join the Peace Corps or to do anything in particular. We all loved the Kennedy lifestyle. We spent summers down at Martha's Vineyard and Cape Cod. It was the same circle, all the same people, and so it could not be any closer to home.\n\nAnd then to have a Texan come in, especially LBJ, after all that glamour. It was devastating. Not that I have anything against Texans.\n\nKennedy's legacy is inspiration and youth. Rather than anything specific, it was pure, intoxicating exhilaration and inspiration.\n Charles Bartlett\n\nJournalist, introduced John Kennedy to Jacqueline Bouvier\n\n\"I talked to him the night before he left for Texas. Something nudged me to call him to say goodbye. I didn't normally do that.\"\n\nI met him in 1946 in Palm Beach, Florida. It was just before he announced his plans to run for Congress. Jack Kennedy and I got along immediately because he had a very, very good sense of humor which came through quickly. We nursed a mutual interest in politics and we knew many of the same people. There was a lot of common ground.\n\nHe was very aware that he was being pulled into politics because his brother (Joseph Kennedy, Jr.) had been killed. Someone in the family had to take on public life. It was up to him. Jack was ready to leave journalism. He felt that as a journalist one could not do much to help the country. He had been writing during the war for Hearst, but he was ready to give up writing because he believed he could do more as an elected official. He was ready to go into politics.\n\nI watched JFK's adjustment to the Washington scene. He was quickly immersed in the House because he was an instinctive politician with a good feel for the game. He had a quick mind, and his thinking went to the heart of a problem, avoiding the frills in a situation. He grabbed hard onto the issues he cared about and quickly matured as a legislator.\n\nHe told me he was going to run for president about four years before 1960. I was surprised, and I said, \"Jack, you've never been around except to Palm Beach and Hyannis Port. You don't really know this country at all. Why not wait eight years?\"\n\nAnd he said, \"We'll see. I'm going to get around.\"\n\nAnd he did so intently that I felt by the time he was nominated that he was thoroughly immersed in the nation's political geography and its disparities.\n\nI recall that in 1952 he was invited to address the Chattanooga Chamber of Commerce. This was his first exposure to a Southern audience. He did well, riding on the wave of his considerable charm.\n\nWhen Jack came back to Washington from Boston in 1953 (after being elected to the Senate), it was clear he wanted to find a wife. He knew it could be awkward to campaign as a bachelor even though he was known by friends to have a serious interest in women. So he was looking, really looking. Jackie Bouvier was a great friend of (my wife) Martha's and mine. We immediately thought of her. She was very good looking, very bright, and rather elegant. We thought she might be the one and we were right.\n\nThey met at a very small dinner in our house in Georgetown. The spark was immediately struck, and it kept the romance alive through months in which JFK was away tending to his base in Massachusetts.\n\nCredit for the next phase really belongs to Martha, because Jackie was already engaged to a nice fellow from Manhattan. This was not, it appeared, a hot romance, so Martha urged Jackie to invite JFK to escort her to the Dancing Class, a Washington social ritual. The evening clearly went well and this was the beginning of a courtship that went on to the alter. That was in December (of 1952) and they were married in September.\n\nI talked to him the night before he left for Texas. Something nudged me to call him to say goodbye. I didn't normally do that. I called about nine o'clock at night and found him in a very, very good mood. He gave no hint of foreboding. He had often talked to me about how easy it would be to kill a president. He believed an assassin would have no problem if he (the assassin) really wanted to do it. But this fatalistic sense was not on his mind the night before he left for Texas.\n\nWe talked of the political scene. His old friend Abe Ribicoff in the Senate had made a rather sharp criticism of him in a speech on the Senate floor. It was reported in the press and I saw it.\n\nAnd I asked Jack in our last talk, \"Why is Abe knocking you?\"\n\n\"That's easy,\" he laughed. \"Abe wants to be the first Jewish president and that's a long alley with no cans to kick.\"\n\n(On November 22, 1963) I was in my office banging out a story on a typewriter. The news came over the wire service. It was a huge shock. My first thought was to go see Bobby. But then I thought he would have all kinds of people swarming around him. So I just sat there and thought about all that Jack would have liked to have done and about Jackie.\n\nI didn't cover the funeral. In fact, I did not cover anything involving Kennedy. That was a problem for a newspaper man\u2014a very personal friend was the president of the United States. I made a decision: Nothing mattered except his success in the White House. I don't think I wrote about anything that was not already made public. I had no desire to exploit the friendship.\n\nThat night, we all went out to Bethesda Naval Hospital while doctors did the post-mortem. There were many friends and family there, and all seemed weighted with a glum sense of shock. JFK's death was a huge loss for me and for his friends. In all my conversations with JFK, he had never dwelled on negative gossip regarding individuals. He seemed to me in his private moments to nurse no animosities. In dealing with almost everyone, JFK was fun. He stood for politics at its best.\n\nKennedy was a model for politicians. In terms of issues, I don't think he really had the time to do a lot of things he wanted.\n\nIn the days before he was killed, he was in very good spirits. But in our conversations, there was one thing really on his mind\u2014the fact that he had been betrayed by Israel on nuclear weapons. He had just learned that one of the people who worked in the development of the Israeli bomb came to the CIA and told them. And the CIA told the president. The issue never blew up and he never talked about it publicly. But Kennedy was absolutely furious. He was raging mad. He felt it was a personal betrayal.\n\nHe said, \"I've done everything I can to help these people, and this government gave me no hint about what they were up to. Not one hint. And I want to tell you one thing. As long as I am president, they will never build another bomb, because I'm going to have inspectors in their laboratories all the time to insure they're not producing another bomb. There will be no second Israeli bomb.\" He was really firm, really angry. It was the first time I ever saw him this angry.\n\nOur conversations centered on this development for much of the last two or three weeks of his life. For him, it was a big issue. But it was the kind of issue a politician facing an election would not take to the public.\n\nHe talked about it (death) several times with me. I suppose any president would because assassination is an obvious hazard. But we had a talk about it in September before he was shot, I remember, up on the boat off Hyannis Port. It was just interesting that we would discuss it at some length. I mean, you know, subjects like what Lyndon would be like as president. He, by the way, told me that he liked LBJ very much.\n\nHe had that kind of mind that he was always talking about all the eventualities. I mean, on that same afternoon he talked about what he was going to do when he got out of the presidency. He said he thought he'd like to be ambassador to Italy if a friendly regime was in power. He thought that would be a good place because Jackie would like it, because he would be out of the country, and therefore, the man who took the presidency wouldn't be in his way.\n\nI do remember one time in Middleburg he was driving along a back country road, and a car sped by the Secret Service car and by us. And he was for a moment shaken and he said, \"The Secret Service should have blocked that car.\" But to cover his concern, he joked, \"Charley, that man could have shot you.\"\n Benjamin C. Bradlee\n\nJournalist\n\n\"We all rose, stricken and uncomfortable, as (Jacqueline Kennedy) came slowly toward us. When she got to Tony and me her eyes brightened a little, she raised her arms and then lurched into my arms, and sobbed.Then she turned to me and said, 'Oh Benny, do you want to hear what happened?' And then quickly added, 'But not as a reporter for Newsweek, okay?'\"\n\nI have to confess, I'm not a great historian. And John Kennedy and I did not discuss substantive issues of foreign policy.\n\nI knew John Kennedy accidentally and, with a few exceptions when it was just the two of us playing golf, we got together as a foursome socially\u2014he and his wife and me and my wife. I rarely saw him alone.\n\nExcerpt from Conversations with Kennedy by Benjamin C. Bradlee, copyright \u00a9 1975 (Used with author's permission):\n\nAs best I can now remember, that first moment came on the afternoon of a sparking late summer day in 1958. My wife and I bumped into the Kennedys during an afternoon walk in Georgetown. Already casual acquaintances, they now were new neighbors. We talked then for an hour or so in their garden, a few doors from our house, and we drew each other as dinner partners later that night. The subject of those conversations\u2014like so many that followed\u2014was the private lives and public postures of politicians, reporters, and friends.\n\nOur friendship just happened. It was never calculated. I didn't follow any script. We had a great time together. If I had any contribution to his life, it was that we did not have to worry about newspapers or issues. It was a friendship.\n\nExcerpt from Conversations with Kennedy by Benjamin C. Bradlee, copyright \u00a9 1975 (Used with author's permission):\n\nThe sledgehammer news that President Kennedy had been shot came to me while I was browsing through Brentano's book store on my lunch hour. First it was the barely distinguishable whispers of incredulous pedestrians, then all too finally it was the AP ticker in the Newsweek bureau on the twelfth floor of the National Press Building around the corner. It's not enough to say that you can't believe these terrible things as they happen.\n\nLate in the afternoon, Nancy Tuckerman, who was Jackie's social secretary, called to ask Tony and me to come to the White House.... About 9:00 Nancy Tuckerman told us that Jackie\u2014and everyone else connected with the slain president\u2014were going directly from Andrews Air Force Base, where Air Force One with the new president aboard would land, to Bethesda Naval Hospital. A final autopsy on the president's body would be performed there, and Jackie was apparently determined to stay as close as she could for as long as she could...\n\nThe next seven hours have stayed blurred in my mind for all these years, except for the moments right after Jackie entered the hospital suite on the arms of the Irish Mafia in the form of Larry O'Brien and Kenny O'Donnell, men she had never really understood or appreciated, but to whom she turned and clung now, strong men from the Irish political side of the dichotomous Kennedy, whom Jackie had never met on equal terms, but who now seemed to comfort her more than any of the rest of us.\n\nHer entrance, announced to us by the flashes of photographers' bulbs many stories below us, into that dreary hospital green room is scarred on my soul for the rest of my life. Her pink wool suit was copiously spattered with the blood of her dead husband, when she had cradled his shattered head in her lap. She looked so lovely, I remember thinking incongruously. But a closer look showed her to be dazed, moving ever so slowly, with eyes apparently not taking all of us in.\n\nWe all rose, stricken and uncomfortable, as she came slowly toward us. When she got to Tony and me her eyes brightened a little, she raised her arms and then lurched into my arms, and sobbed. After a minute or so she pulled back and greeted Tony in the same way. Then she turned to me and said, \"Oh Benny, do you want to hear what happened?\" And then quickly added, \"But not as a reporter for Newsweek, okay?\"\n\nI felt badly that she obviously felt she had to be that careful in that awful moment about the old problem and was about to proclaim my innocence as usual, when she started telling us about the actual shooting. I can remember now only the strangely graceful arc she described with her right hand as she told us that part of the president's head had been blown away by one bullet. She moved in a trance to talk to each of us there and to new friends as they arrived, ignoring the advice of friends and doctors to get some sleep and to change out of her bloody clothes. Those were some kind of dreadful badge of the disaster she had been through, and no one could persuade her to remove them.\n\nI was asked to be an usher, and as a result, I walked from the White House to St. Matthew's Cathedral as part of the procession. I remember that walk very, very well.\n\nExcerpt from A Good Life: Newspapering and Other Adventures, by Benjamin C. Bradlee, Copyright \u00a9 1995 (Used with author's permission):\n\nIn the weeks after Kennedy was assassinated, Tony and I spent a couple of emotional weekends at Atoka, the Kennedys' country house in Middleburg, Virginia, with Jackie, trying with no success to talk about something else, or someone else. Too soon and too emotional for healing, we proved only that the three of us had very little in common without the essential fourth.\n\nIf people read the book Conversations with Kennedy, I think they will realize that I'm not a great historian. Quite simply, a friend of mine got elected president of the United States.\n Robert F. Kennedy\n\nBrother, Attorney General, United States Senator, Democratic Presidential Candidate\n\n\"President Kennedy was fond of quoting Dante that 'the hottest places in Hell are reserved for those who, in time of great moral crisis, maintain their neutrality.'\"\n\nCourage is the virtue that President Kennedy most admired. He sought out those people who had demonstrated in some way, whether it was on a battlefield, or a baseball diamond, in a speech, or fighting for a cause, that they had courage, that they would stand up, that they could be counted on...\n\nPresident Kennedy would have been forty-seven in May of 1964. At least half of the days he spent on this earth were days of intense physical pain. He had diphtheria when he was very young and serious back trouble when he was older. In between, he had almost every conceivable ailment. When we were growing up together, we used to laugh about the great risk a mosquito took in biting Jack Kennedy\u2014 with some of his blood, the mosquito was sure to die... But during all this time, I never heard him complain. I never heard him say anything which would indicate that he felt God had dealt with him unjustly. Those who knew him well would know he was suffering only because his face was a little whiter, the lines around his eyes were a little deeper, his words a little sharper. Those who did not know him well detected nothing.\n\nHe didn't complain about his problem, so why should I complain about mine\u2014that is how one always felt.\n\nWhen he battled against against illness, when he fought in the war, when he ran for the Senate, when he stood against the powerful interests in Massachusetts to fight for the St. Lawrence Seaway, when he fought for a labor reform act in 1959, when he entered the West Virginia primary in 1960, when he debated Lyndon Johnson at the Democratic Convention in Los Angeles with no preparation, when he took the blame completely on himself for the failure at the Bay of Pigs, when he fought the steel companies, when he stood up at Berlin in 1961 and then again in 1962 for the freedom of that city, when he forced the withdrawal of the Soviet missiles from Cuba, when he spoke and fought for equal rights for all our citizens, and hundreds of other things both big and small, he was reflecting what is the best in human beings.\n\nHe was demonstrating conviction, courage, a desire to help those who needed help, and true and genuine love for his country....\n\nPresident Kennedy had only a thousand days in the White House instead of 3,000 days, yet so much was accomplished. Still so much needs to be done.\n\nPresident Kenedy was fond of quoting Dante that \"the hottest places in Hell are reserved for those who, in time of great moral crisis, maintain their neutrality.\"\n\nIf there is a lesson from the life of John F. Kennedy and from his death, it is that in this world of ours, none of us can afford to be lookers-on, the critics on the sidelines....\n\nPresident Kennedy's grandmother was living in Boston when he was assassinated. She was also alive the year President Lincoln was shot. We are a young country. We are growing and expanding until it appears that this planet will no longer contain us. We have problems now that people fifty, even ten years ago, would not have dreamed would have to be faced....\n\nIt has been fashionable in many places to look down on politics, on those in government. President Kennedy, I think, changed that and altered the public perception of government. He certainly did for those who participated. But, however we feel about politics, the arena of government is where decisions will be made affecting not only all our destinies, but the future of our children born and unborn....\n\nDecember 18, 1963\n\nReprinted with permission from A Tribute to John F. Kennedy,\n\n\u00a9 1964 by Encyclop\u00e6dia Britannica, Inc.\n Robert F. Kennedy Jr.\n\nNephew, environmental activist\n\n\"(T)here had been a sense of foreboding even within our family as he and Aunt Jackie prepared for the trip. Jack made an unscheduled trip to Cape Cod to say goodbye to my ailing grandfather. The night before the trip, Mummy found Jack distant and brooding at a dinner for the Supreme Court justices.\"\n\n(NOTE: This reflection was published previously by The Huffington Post. Used with author's permission)\n\nOn November 22, 1963, Mummy picked me up early from Sidwell Friends School in Washington, D.C. Driving home to Hickory Hill in Northern Virginia, I noticed that all the district flags were at half-staff. Mummy told us that a bad man had shot Uncle Jack and that he was in heaven.\n\nDaddy's friend and former football teammate, Dean Markham, a Justice Department Rackets Division Attorney, picked up my little brother David at Our Lady of Victory. \"Why did they kill Uncle Jack?\" David asked him. Dean, an ex-marine, combat veteran, known as the toughest linesmen on the \"GI-Bill Squad\"\u2014the toughest football team in Harvard University's history\u2014wasn't tough enough to field that question. He wept silently all the way to our driveway.\n\nWhen I got home, Daddy was walking in the yard with Brumus, our giant black Newfoundland, and Rusty, the Irish Setter. We ran and hugged him. We were all crying. He told us, \"He had the most wonderful life, and he never had a sad day.\"\n\nNeither Beck, Hannity, nor Savage nor the hate merchants at Fox News and talk radio can claim to have invented their genre. Toxic right-wing vitriol so dominated the public airwaves from the McCarthy era until 1963 that President Kennedy, that year, launched a citizen's campaign to enforce the Fairness Doctrine, which required accuracy and balance in the broadcast media. Students and civic and religious groups filed more than five hundred complaints against right-wing extremists and hate-mongering commentators before the Federal Communications Commission.\n\nThe Dallas, Texas, airwaves were particularly radioactive; preachers and political leaders and local businessmen spewed extremist vitriol on the city's radio and TV stations, inflaming the passions of the city's legions of unhinged fanatics. There was something about the city\u2014a rage or craziness, that, whether sensible or not, seemed to have set the stage for Jack's murder. The Voice of America, half an hour after the assassination, described Dallas as \"the center of extreme right wing.\" The Texas town was such a seething cauldron of right-wing depravity that historian William Manchester portrayed it as recalling the final days of the Weimar Republic.\n\n\"Mad things happened,\" reported Manchester. \"Huge billboards screamed 'Impeach Earl Warren.'\" Jewish stores were smeared with crude swastikas. Fanatical young matrons swayed in public to the chant, \"Stevenson's going to die\u2014his heart will stop, stop, stop and he will burn, burn, burn!\"\n\nThe mercantile elite that ruled the city carefully cultivated the seeds of hate. Radical-right broadsides were distributed in public schools; the Kennedy name was booed in classrooms; junior executives who refused to attend radical seminars were blackballed and fired.\n\nManchester continued: \"Dallas had become the mecca for medicine show evangelists of the National Independence Convention, the Christian Crusades, the Minutemen, the John Birch and Patrick Henry Societies and the headquarters of right wing oil man, H.L. Hunt, and his dubious activities...The city's mayor, Earl Carroll, a right wing co-founder of the John Birch Society, was known as \"the socialist mayor of Dallas\" because he maintained his affiliation with the Democratic Party.\"\n\nDallas's oil and gas barons who routinely denounced JFK as a \"comsymp\" (Communist sympathizer) had unbottled the genie of populist rage and harnessed it to the cause of radical ideology, anti-government fervor, and corporate dominion.\n\nUncle Jack's speech in Dallas was to have been an explosive broadside against the right wing. He found Dallas's streets packed five deep with Kennedy Democrats, but among them were the familiar ornaments of presidential hatred; high-flying confederate flags and hundreds of posters adorning the walls and streets of Dallas showing Jack's picture inscribed with, \"Wanted for Treason.\" One man held a posterboard saying, \"you a traitor [sic].\" Other placards accused him of being a Communist. When public school PA systems announced Jack's assassination, Dallas school children as young as the fourth grade applauded. A Birmingham radio caller declared that, \"Any white man who did what he did for niggers should be shot.\"\n\nAs my siblings and I visited the White House to console my cousins John and Caroline, a picket paraded out front with a sign \"God punished JFK.\"\n\nJack had received myriad warnings against visiting the right-wing Texas city. Indeed, there had been a sense of foreboding even within our family as he and Aunt Jackie prepared for the trip. Jack made an unscheduled trip to Cape Cod to say goodbye to my ailing grandfather. The night before the trip, Mummy found Jack distant and brooding at a dinner for the Supreme Court justices. He was very fond of Mummy, but for the first time ever, he looked right through her.\n\nJack's death forced a national bout of self-examination.\n\nIn 1964, Americans repudiated the forces of right-wing hatred and violence with an historic landslide in the presidential election between LBJ and Goldwater. For a while, the advocates of right-wing extremism receded from the public forum. Now they have returned with a vengeance\u2014to the broadcast media and to prominent positions in the political landscape.\n Christopher Kennedy Lawford\n\nNephew, author, filmmaker\n\n\"Jack Kennedy overcame serious health problems and that's in part what made him great. We often forget that.\"\n\nI was very young when my uncle was around, but I have recollections of him coming to visit my parents in Los Angeles when I was growing up on the beach here. I do remember the 1960 Democratic convention. I went to the convention when I was five years old. My mom was a delegate. I remember being amazed by this enormous spectacle in the LA Sports Arena.\n\nI remember sitting in a chair and then later being awakened in my uncle's hotel room. I was sitting on his bed and my uncle sat down and said he had just been nominated to be president of the United States, that it was going to be a very hard job, and that he wanted my help. I responded that I would help him, but that I was kind of tired right then, and I wanted my mom to take me home.\n\nMy uncle had a lot of energy obviously. We were a big Irish family; there was a lot of laughter. When he came to my home, we would go swimming in the ocean. I got to raise the presidential flag when he came to our home. People would gather outside our fence and try to look into our home. There was always a sense that he was very important and that people were very attentive to him. Everyone was always energized when he was around.\n\nI remember going to Washington and being in motorcades with my uncle. I got carsick once and threw up on his assistant, Kenny O'Donnell.\n\nI remember November 22 very vividly. I was in school\u2014St. Martin's Roman Catholic School. I was treated very well by the nuns there; it had nothing to do with me and everything to do with my family. I was in class and these women came to the door and they looked distraught. There was obviously something going on. They kept looking at me. I didn't know why. Then they motioned me to come over. They said something about my family. I don't remember if they said the family member was hurt or dead. But whatever it was, it was very bad.\n\nI thought they might have been referring to my grandfather, Joseph Kennedy. He was old and I remember, as a kid, you think those are the people who are going to die. My uncle was way too energetic, way too young, and way too full of life to be dead. And, at that age, I did not understand shootings.\n\nObviously, this was a big deal. They asked me if I wanted to go home. I said \"No,\" because it did not occur to me that I would want to go home. Later, at recess, we played stickball, and I remember being very prolific in my hitting that day. There was a certain energy. Up until then, nobody really focused on me. That kind of attention, that energy, is powerful. People were paying attention to me in a way that energized me. There was a level of concern, and in my family, that pity, any self-pity we never really went down that road. That sense of wanting to take care of me, I didn't want any part of.\n\nI later went home that day, and my mother was obviously unbelievably affected by this. I don't remember my father's reaction. There were a lot of people in front of my house. I went outside and walked among them and began to get a feeling of tension, which I did not particularly like. Then later, my mother asked me if I wanted to go to the funeral. I was having a friend spend the weekend and I did not want to disappoint him, and so I said, \"No,\" not really grasping what I was saying \"no\" to.\n\nI understood there was something big going on, but at the time, I thought it didn't really have anything to do with me. Also, there was a certain amount of denial about death and what death really was. It was the first death I ever experienced. I must say that for a long, long time, I really regretted that decision, in the sense I had really let my family down, my mother down particularly. I felt bad for many, many years. In hindsight, it was selfish. I was thinking how I did not want to disappoint my friend. That gives you a sense of a kid's viewpoint. My sister Sydney went to the funeral with my mother.\n\nMy uncle was very smart. I later went to Cuba and produced a film on the Cuban Missile Crisis called Thirteen Days. Obviously, Cuba played a big part in my uncle's presidency and in this country's more recent history. There were weapons of mass destruction in Cuba. The Soviets would have attacked. (Then-Cuban President Fidel) Castro told me that he had given orders to his field commanders that if American planes came, they were to rush to the missile sites and yell to the Soviet troops, \"Fire the missiles! Fire the missiles!\" The Soviets, Castro said, in the panic of it would have fired the missiles.\n\nHis legacy\u2014the promise of it, the youth of it, the vigor of it\u2014was something people respond to. But substantively, his legacy includes his commitment to the arts, to mental health issues, to those things I care about. In terms of the civil rights movement, poverty\u2014all the things that Lyndon Johnson was able to create in the Great Society\u2014were legacies of my uncle. Those things are missing today. We have a country today where people are so concerned about their own dwindling assets, they can't seem to get out of their own way and think about anyone but themselves. It's a shame. My uncle's legacy of \"Ask not what your country can do for you. Ask what you can do for your country,\" is a theme that has gone by the wayside. It's really unfortunate. Unless we get back to the things that made us great, our country will die on the vine.\n\nMy uncle struggled with a lot of illness and a lot of issues with regard to those illnesses. We grew up with this ethic and it came from him: in order to achieve great things in life, you have to overcome something great. He ascribed to that and he passed that on to us. There are millions of people in our society who have struggled with great things and they are marginalized. They've dealt with poverty, or mental health issues, or addiction issues, and they are not considered assets to our society. Jack Kennedy overcame serious health problems and that's in part what made him great.\n\nWe often forget that.\n Frank Mankiewicz\n\nPeace Corps director in Peru, president of NPR, press secretary to Robert F. Kennedy\n\n\"We've got a president who stays up until midnight watching television.\"\n\nI was about to go to Peru as the first director of the Peace Corps in that country. That would have been 1962, and I was waiting in Washington because there had been an armed overthrow of the elected government in Peru, and we were withholding the volunteers until democracy was restored. Meanwhile, President Kennedy was in the process of visiting every government agency just to say \"hello\" and be seen. And he got around to the Peace Corps, which would have been January or February of '62.\n\nWe all gathered in the auditorium, across the street from the White House in the Chamber of Commerce building. And (Peace Corps Director) Sarge Shriver asked me if I would stay on the platform and sort of entertain the troops while he went across the street to the White House to escort the president.\n\nSo I was telling some stories about my early visits to Latin America, and pretty soon there's a big commotion at the back of the hall, and here comes the president with Sarge Shriver. Photographers. Spotlights. He comes down the aisle, comes up on the stage. And Sarge is introducing him to people, and he says to the president, \"This is Frank Mankiewicz. He's our director in Peru.\"\n\nThe president shook my hand. Then he stopped and looked at me and said, \"I saw you on Mike Wallace's show last year talking about Peru.\"\n\nAnd I thought, Jesus, huh? We've got a president who stays up until midnight watching television.\n\nBut I was very impressed with him. He was very tan, very fit, very energetic. He had that quality of talking to someone or listening to them as though they were the only other person in the room, or indeed the country. A very serious listener.\n\n(On November 22) I was then in the second city in Peru, called Arequipa. We had some Peace Corps volunteers there. I was walking on the street, probably a block or two from the Peace Corps office, headed somewhere. And I noticed a big commotion. People talking. A man came toward me weeping. Physically crying.\n\nAnd I stopped him, and I said in Spanish, \"What's happened? What's going on?\"\n\nAnd he said to me, \"The president's been shot.\"\n\nAnd I said (and this is all in Spanish), \"You mean President Bela\u00fande?\" who was, of course, the president of Peru.\n\nAnd he said to me (in Spanish), \"I wish it were.\" A Peruvian. I always found that a very, very impressive response. Not because he wished ill of their president.\n\nAnd so I then rushed back to the Peace Corps office, and we started hearing reports on short-wave radio... I told the rest of the volunteers. I checked with Washington\u2014how serious this was. And we simply called off whatever activities we had, and I made an effort to try to get the volunteers together. We missed the funeral, and of course, there was no TV.\n\nI went back to Lima to go to the (U.S.) embassy as quickly as I could. I remember spending time with the ambassador. On the following Sunday morning, I was at the embassy and the ambassador came in, maybe from church service or something, and he said, \"I've heard the most horrible news. Someone had killed Lee Harvey Oswald in the Dallas police station.\" That was the first I had heard of that.\n\nI was inspired by President Kennedy then and I think I remain so. He was our president. He was from our generation, which was quite a change. Presidents up to then had always been father figures. Here was a man who was a few years older than I. He fought in \"our\" war. Like me, he was a World War II veteran. Young, vigorous. Thinking about the same things we were, and I think that was what contributed largely to my\u2014and I think my cohorts'\u2014inspiration by President Kennedy. Here was a guy who had been to school when I was. He left school, went into the service. Fought in the service. Came out; graduated. He was one of our guys.\n\nI think his legacy is pretty good. I know there's a lot of preoccupation with scandal now. A lot of things have happened in his family. But on the other hand, here's an extraordinary family. I'm not sure anyone (in the family) since Joe Kennedy has devoted more than an hour to personal aggrandizement\u2014to making money, to the American dream of becoming a rich person. I think they all have a strong social instinct. I know Robert Kennedy's kids all do. It's quite an extraordinary family.\n\nWe (Robert Kennedy and I) met once again in Panama. In those days, when you flew from Washington to Lima, you had to stop in Panama for re-fueling. The stop in Panama was usually for an hour and usually between two, or three, or four o'clock in the morning. I was having a meeting of all my country directors in Panama at the time, and I happened to read in the local paper that Kennedy was going to be coming through on this flight early the next morning.\n\nSo, I thought, I'll go down to the airport and say, \"hello.\"\n\nSo, at three o'clock, maybe two forty-five, I showed up at the airport, and there's the airplane. And, of course, there was no security, I just walked over. And there were a couple of newspaper reporters there.\n\nThey said, \"When's the press conference?\"\n\nI said, \"I don't know about any press conference. But I'll find out.\"\n\nSo, I went up on board to the plane. (The airline) had fixed up two of the overhead compartments into berths. Robert Kennedy was in one and Ethel was in the other\u2014in pajamas. So I came on board and I told him about these reporters, and he started to get out of bed and get dressed. And somebody said, \"There's no press conference scheduled. They just showed up.\"\n\nAnd he said to me, \"If I don't come down to the press conference, who gets hurt the most, those reporters or their publishers?\"\n\nI said, \"Undoubtedly, the reporters will get hurt.\"\n\nHe said, \"That's what I thought.\"\n\nHe started to get dressed. And then one of his staffers said to him, \"Senator, why don't you stay here? Stay in your bed, and we'll bring the reporters on board. And you can have the press conference right here. And that way, you won't have to get up and get dressed.\"\n\nRobert Kennedy looked at me and said, \"What do you think of that?\"\n\nAnd I thought, \"There's a key question here.\"\n\nI didn't have a lot of time to think, but I'm very happy to say that I said to him, \"Senator, I think that's what President De Gaulle would do.\"\n\nHe laughed. And said, \"Yeah, I agree. C'mon.\"\n\nSo he got up, got dressed, and we went down and had a little press conference. If he had done the press conference in his pajamas, I don't think we ever would have stopped hearing about it.\n John Seigenthaler Sr.\n\nJournalist, author, First Amendment historian\n\n\"(John Kennedy) poured me coffee, and then he was standing with the pot in one hand and his cup in the other, and said (to his brother Robert), 'So that's it, General. Let's grab our balls and go.'\"\n\nMY first encounter with President John F. Kennedy was in the spring of 1957 when I was a reporter for the Nashville Tennessean assigned to Washington. My assignment included covering the hearings of what was called the Senate Select Committee on Improper Activities in Labor and Management.\n\nI thought (and my view was shared by many correspondents covering the nation's capital) that Jack was in a class by himself\u2014an intellectual, who also was a street-wise politician, having been elected to the U. S. House of Representatives, and subsequently to the Senate\u2014and who was ambitious to run to succeed President Eisenhower in the White House.\n\nI was not alone among the reporters who put him in a class by himself. Thinking back on it, I realize that while my sense of that was a \"first impression,\" it did not come with an initial handshake or as a result of a few casual conversations. It was, really, an impression, strengthened and confirmed gradually, as I got to interact with him while he was in the Senate.\n\nLater, in 1958, I was a Nieman fellow at Harvard, and by the time I arrived in Boston in September, Jack was running for reelection to the Senate. Already the press was speculating that this race for the Senate was a prelude to a 1960 run for the White House.\n\nOur Nieman class\u2014twelve journalists from across the nation and five from abroad\u2014urged me from the first day to try to get Jack to come sit in on one of our weekly afternoon wine-beer-and-cheese seminars. There was some hesitation about it. It took a couple of phone calls to Ted Reardon, Jack's administrative assistant, but finally Ted called back. Jack accepted the invitation. And he came.\n\nIn fact, he showed up a bit early that afternoon, and it was a mark of the man that he was interested in knowing about my classmates. Where were they from? What news organizations did they represent? What was he or she like? As the session started, and we began to put questions to them, I noted that he would begin his responses by addressing the questioner as \"Mister Turner\" or \"Mister Morgan\" or \"Mister Parasuram.\" That kept the session rather formal, but still, his humor sometimes self-effacing, was calculated to set a tone that resulted in our questions beginning, \"Senator\" or \"Mister Kennedy.\" Perhaps half the class was his age, a couple perhaps his senior, and some of their questions were pointed. But I thought his demeanor set the tone throughout: formal, respectful, friendly.\n\nA few years later, I watched him during presidential press conferences, and thought back to that session with my Nieman class. He knew many of the journalists popping questions in those press conferences on a first name basis\u2014but still it always was \"Mister.\" And he had that same self-effacing sense of humor that let him laugh at himself as well as with others. He would take a tough question, respond with a wry comment, and, as the wheels turned, he would craft a substantive answer. It was a natural gift\u2014this \"think while you talk\"\u2014one shared by few politicians I have observed.\n\nThe Nieman Fellows seminar that afternoon was terrific for all of us. We were happy that he did not seem at all pressured to leave. When it was over, I escorted him out to a waiting car, and he said, \"I'm going to be moving around the greater Boston area during my campaign, and if you have any additional time while you are here at Harvard, any time you would like to ride with me, I'd be glad to have you along.\"\n\nI took him up on it. On one occasion, we were stuck in traffic in Boston, and Johnny Powers, who was a member of the Massachusetts legislature and quite an astute fellow, told the driver, \"When you get to the next corner, pull over.\" Powers tapped Jack on the shoulder and said, \"Jack, look up ahead. That's Mr. McCaffrey painting his house, and he is the key to this neighborhood. You need to have a conversation with him. Just a quick hello.\"\n\nThe senator got out of the car, went to the ladder, and called up to the fellow. The old man looked down, at first seeming puzzled, or harassed. Sort of irritated, he called down, \"What is it? What do you want?\" Maybe the sun was in his eyes. He didn't recognize him. (This was the first time I ever noticed Jack had any problems with his back. He started up the ladder with great care and took those steps one at a time and very slowly, under some stress\u2014one foot up, and then the other; one foot up, and then the other.)\n\nHe had taken a few steps up when the old fellow finally recognized him and said, \"Oh, Jack, Jack Kennedy! I'm coming down. I want to see you.\" And down he comes, Jack now backing down, still feeling some stress. Then they were on the sidewalk, shaking hands and into a conversation about how the campaign was going. Jack says something like, \"You ask me? I'm asking you. How are we doing?\"\n\nWithin minutes, passersby and people in the neighborhood started flocking around them, and it turned into a street corner rally. The U.S. marshal got out of the car and had to direct traffic. Jack had a ball. Mr. McCaffrey would introduce him to people and say, \"This is so and so, and his brother, he was in the Navy like you.\" Or he would say, \"This is Pat Egan, and his brother Joe or Uncle Charlie, you remember, is the head waiter (or maybe dishwasher) at such and such restaurant.\" And Jack would say, \"Oh yes, I've eaten there.\" Or, \"I remember him.\" It was warm and funny to watch. As (House Speaker) Tip O'Neill later said, \"All politics is local.\"\n\nI returned to the newspaper in late October, 1959. During this period, Jack Kennedy's plans to run for president were rapidly moving forward. Jack's formal announcement came in early January, 1960\u2014no surprise to anyone since the media had been reporting it for weeks in advance.\n\nBy spring, Bob (Robert Kennedy) was calling, asking if I would be interested in coming to work with him in the campaign as his administrative assistant. There finally came a point when I felt I needed a change, and in late June, I called Bob to ask if he still was looking for an administrative assistant. He was. The new editor, knowing that I was grousing (as were several of my colleagues), granted me a leave from August through the November election.\n\nImmediately after the election, Jack took some vacation time in Florida, and one day while playing golf, ran Bob's name up the flag pole as a possible candidate for attorney general. William Lawrence, the New York Times reporter in the golf foursome, wrote a story about the possibility, and it drew negative editorial responses including one from the Times that complained he had never practiced law as a litigator and had limited experience, except for his work as an attorney in the Senate. It was also reported\u2014accurately\u2014that Joseph P. Kennedy favored the idea, which did not contribute to any groundswell in its favor.\n\nJack was in the process of naming members of his cabinet and did so every couple of days from the small elevated front porch of his house in Georgetown. One afternoon, Bob asked me if I could devote the next day to driving him to several appointments he had made with various veterans of Washington political life. It would be a tight schedule and take the entire day.\n\nI was staying with Bob at Hickory Hill, and we left the house early. His first appointment was with Harry Truman at the Mayflower Hotel on Connecticut Avenue. I circled the block for half an hour while he talked about his future with the former president. When he got back in the car, it was apparent that it had not gone the way he had hoped. Next, we went to the Supreme Court building where he spent almost an hour with his friend, Associate Justice William O. Douglass. It was clear that Bill Douglass had made him no happier than Truman had.\n\nThen, to the Senate office building where he saw, separately, John McClellan and Bill Fulbright. He seemed no happier. Next, we stopped at the Justice Department (in those days the FBI headquarters was there), and he spent a half-hour with J. Edgar Hoover. Finally, he stopped at the law office of Clark Clifford, and then we headed home. The meetings had not gone well. On the way home, his spirits were down. He was disconsolate. The \"wise old heads\" he talked with that day (with one exception) did not tell him what he had hoped to hear\u2014that he should become attorney general, or even go to work in the Defense Department. The exception came from Hoover. \"He wasn't sincere about it,\" said Bob. \"I could tell. He just wanted me to tell Dad and Jack that he was for it. He was just being political.\"\n\nFinally, he decided that he would not do it, and that he would call his brother and tell him. Just before he picked up the phone, he said, \"This will kill my father.\" And, so he called his brother and told him he wouldn't do it. He recited the whole series of visits to his brother, Jack, who then said to him, \"Well, don't tell me now. I want to have breakfast with you in the morning. Come to the house.\"\n\nBob invited me to accompany him next morning. When we walked in, the president-elect was coming down the stairs, smiling. We went in toward the back of the house to the breakfast table. When we were done with our biscuits, eggs, and bacon, Bob says, \"Now, can we talk about my situation?\"\n\nThey were facing each other across the table; I was sitting on Jack's left, Bob's right. The president-elect reached over and grabbed my wrist, and for ten seconds he had eye contact with me and said, \"Have you heard the arguments for and against this thing?\" And he began to tell me\u2014but now he was talking directly to Bob, their eyes locked. Then he looked at Bob and said, and I'm paraphrasing, \"I have named the Cabinet, and I know some of the members. Some I do not know. I have known Rusk casually, but never had any intimate dealings with him. I know Stu Udall\u2014we served in the House together.\" He went down the list. Then he said to Bob, \"There will be times when I need the unvarnished sense of someone who is not interested in anything except the success of my presidency. There will be difficult times.\" He mentioned civil rights and organized crime. He ran through a list of critical issues and again added, \"I need someone who has an interest in the success of my administration.\"\n\nHe went on for about ten minutes. He was direct and to the point\u2014and he made the point in strong terms directly to his brother. It was clear. He needed Bob.\n\nHe then said to me, \"Do you want more coffee?\" He went into the kitchen and came back with the coffee pot. He poured me coffee, and then he was standing with the pot in one hand and his cup in the other, and said, \"So that's it, General. Let's grab our balls and go.\" We got up, and as we walked back out to the front porch, the president-elect told him to come back the next day and they would announce it.\n\nI had told Bob I did not want to work in the administration but wanted to return to journalism. Now he asked me to rethink that decision and come to work for at least a year as his administrative assistant in the Department of Justice. I agreed that very night to do it. I never regretted it.\n\nThe oral history projects I have participated in over the years since their deaths have included my reflections on that breakfast meeting at the president-elect's house. One of the revisionist historians on the Kennedy family, C. David Heyman, grossly distorted the account as I recorded it. He wrote that the meeting with me had been contrived by the two brothers, and that they were play-acting, assured that as a journalist I would write it for publication and repeat it to other journalists.\n\nTo believe him, you have to accept that Robert Kennedy took those trips to President Truman and the rest on the previous day, and that they \"arranged\" the phone call the night before\u2014just for my benefit. Both of them knew, of course, that I was not a journalist at the time, had not worked as one for many months\u2014and the fact is that I would not do so for at least 18 months more. Had I leaked what had transpired to some journalist (and I could have done so) both brothers would have considered it a breach of friendship, and I would have forfeited the opportunity to share future confidences with either of them.\n\nOn November 22, I was in my office at the Tennessean. A reporter from the AP came in. He had a torn strip from the AP wire machine and said, \"John, you should read this.\" I read it and flared with anger and said, \"If this is somebody's idea of a joke, it is sick.\" The reporter burst into tears and said, \"I wish it was a joke.\" I remember it vividly.\n\nThat night, the night of the autopsy, I got a call from I believe it was John Nolan at the hospital (Bethesda Naval Hospital), and he said Ethel would like to talk to me, and he put her on the line. Ethel said, \"Could you come up?\" And I said, \"Well sure.\" I planned to come, but I just didn't want to get in the way. (The next day) I went to the White House. It seems to be there were all sort of strange people in line as I was going through: Southern governors who really had been knocking Jack Kennedy's ass off for years. I went through the line and I was coming out the back of the foyer and I ran into Ethel. She told me where Bob was. He was in another part of the White House, upstairs. I remember my first impression was that he looked to me like a man who is just in intense pain; he looked like a man who hurt, I mean, you know, just physically hurt. He was glad I was there. He wanted to know if there was anything he could do for me, and I wanted to know if there was anything I could do for him. It was largely small talk; there wasn't really anything to say.\n\nBy this time, I was the editor of the newspaper, and we had a suite of offices on Connecticut Avenue. I was on the office's balcony with three members of my staff as the funeral procession came by. I did not march in the procession to St. Matthew's Cathedral. I did go to the cemetery and was on the periphery of the crowd. Almost everyone there was in tears. Looking back, the images are so vivid on television and in the newspapers. It is difficult to know what I took in visually and what became the lore of that funeral.\n\nHaving worked in that administration and then having the opportunity to be there, and living with the memories of some of the encounters like a visit to the White House for a birthday party for Caroline, you live with those memories. After all these years, the New Frontier is almost a clich\u00e9. But for those of us who worked in the 1960 campaign, the whole concept was very real to us. It was not political rhetoric. We knew quite well the civil rights challenges. The economy was sluggish, and internationally, there was a sense of lethargy. We took it very seriously. We felt deeply that it was vitally important to \"get the country moving again.\"\n\nTo me, Jack Kennedy's legacy is that his life and his administration served as a model for governance by politics\u2014in the best definition of the word. He looked at politics as an honorable profession. Any journalist who lived during that time or who has watched videos of his press conferences cannot come away with any other sense. He unconditionally loved the role of public service. For me, Jack will always be the happy warrior.\nSECTION SIX:\n\nJournalists and Commentators\n Eddie Barker\n\nKRLD Dallas, first journalist to confirm Kennedy's death\n\n\"John Kennedy was the darling of the press. To an extent that no other man who has held that office has enjoyed that...Let your own imagination create good or evil.\"\n\nON November 22, 1963, I was assigned to cover President Kennedy's speech at the Trade Mart for both KRLD radio and television. I realized something was wrong when the people supposedly coming in for the luncheon didn't come in. Up at the head table everything was very much in order, but no one was coming in to sit.\n\nExcerpt from Eddie Barker's Notebook by Eddie Barker and John Mark Dempsey \u00a9 2006 (Used with author's permission):\n\nIt was shortly before one o'clock central time. President Kennedy's arrival at Dallas's Trade Mart had been delayed.... Suddenly sirens were audible as the president's car and a parade of accompanying vehicles sped by the Trade Mart. We knew where they were headed: Parkland Hospital, where the most serious trauma cases were treated.\n\nI knew a doctor who was a member of the staff at Parkland Hospital and who happened to be at the Trade Mart for the president's speech. Like the rest of us, the doctor realized something wasn't right, went to a pay phone, and calmly placed a call at 12:35. An acquaintance in the emergency room matter-of-factly told the doctor that the president was dead when they brought him through the door. Of course, the president was not officially pronounced dead until 1:00 p.m., but the information my doctor acquaintance received more closely reflected the reality of the situation.\n\nMy friend saw me struggling to maintain a coherent broadcast with the limited information available, walked over, and whispered in my ear words that would shock the world and stay with me for a lifetime:\n\n\"Eddie, he's dead.\"\n\nThose words sent a cold chill running down my spine. I didn't want to believe them, but the source was too good.\n\nI then made a decision that has caused a lot of comment in the years since that strangely brilliant Friday afternoon. I told an audience that included the whole CBS network that a reliable source had confirmed to me that President Kennedy was dead. What I didn't know was that my shocking report caused a lot of anxiety as CBS News headquarters in New York. Walter Cronkite, holding the story at arm's length, repeatedly gave me, the local newsman, credit for saying the president was dead. As the minutes went by, Walter was generous in crediting me with the story, making sure the local reporter was given the credit he deserved. And the history books take note of the fact that CBS News stuck with the story of a reporter they knew and trusted.\n\nOnce a reporter, you're always a reporter. My God, I was told this, and I had to do something with it. Because it was pretty much straight from the horse's mouth, I went ahead and used it. I had been so close to CBS with various stories whose origins were here in Dallas. I was a known entity and the people in New York, including Walter, would say, \"Well, if Barker says it, it's correct.\"\n\nJohn Kennedy was the darling of the press. To an extent that no other man who has held that office has enjoyed that. I think you have to look at the time he served. Let your own imagination create good or evil. He was so popular\u2014even in Dallas\u2014so many thought the world of him. He missed the mark, but had he had the full run at it, he would have ranked up there with the most important presidents in the history of the United States.\n\nBut he was cut short.\n Tom Brokaw\n\nNBC News\n\n\"And I suppose I matured in that twenty-four hours, over those couple of days. I think I became more realistic as a journalist and saw the larger screen. And the consequences of daring action and the evil that can come, even to America. I do think it was a seminal time for me.\"\n\n(JOHN Kennedy) matched all my impressions because he was very cool and elegant. When his brother (Robert) came (to Omaha), it was different. He was rumpled, and his shirt was frayed, and his hair was messed, and he was very combative in an exchange with reporters.\n\nThere was this enormous generational shift. This was the World War II generation that had been forged by that war. They represented such a profound change from the generation of FDR, Harry Truman, and Dwight Eisenhower.\n\nThere was youthfulness about them that we could identify. They were in their forties\u2014Kennedy and Nixon for example.\n\nFor most of us in working class America, we had our nose pressed up against the glass of the Kennedys. They had a lifestyle that we could not imagine. It was F. Scott Fitzgeraldian in its own way. And they brought enormous energy to the campaign obviously. I remember first being aware of them when Bobby was counsel to the Senate Racketeering Committee and hearing that distinctive New England accent. We didn't hear that on a regular basis because television was just coming of age. And wherever we lived in the country, we could hear the different dialects now because they played out on that little screen every night. So the Kennedys brought dash, distinctive accent, and enormous wealth and lifestyle to parts of the country that we could not imagine.\n\nIn photography, they talk about a first and second generation print. I think we're in about the fifth generation now. And there are still people who were inspired by John Kennedy, who, even if they were not alive when he was president, know the ringing words, \"Ask not...\" and the whole \"Camelot\" concept still plays, I think, for a lot of people across the country.\n\nBut we've moved on from there. And there have been many people since then. Ronald Reagan came along and inspired an entirely different generation in an entirely different fashion. There's a Reagan generation as there was a Kennedy generation. We keep evolving in our politics and our reaction to it. There is an entire generation of social conservatives and entrepreneurial people in this country who were inspired by Ronald Reagan. They were in college when he was the president. And now they are in their late thirties, early to mid-forties and they make up a lot of this country. They got interested in politics because of him.\n\n(Regarding the impact of television coverage of the assassination) I don't think I turned to friends and said, \"This is a new era.\" But we were bound together by that screen. And we could watch the state burial and common grieving that was going on across the country on that screen. We had not seen anything quite like that. When FDR died, and he made that long train ride back across the South, that was primarily in Movietone News and still pictures. But this played out live. You were wedded to that screen.\n\nI remember being in church on Sunday morning when Lee Harvey Oswald was shot. The priest came out and said, \"What have we come to? They've now shot Lee Harvey Oswald.\" The church pretty well vacated, and we all ran home to watch the replay of it. And then I went to the office. The television set was, if you will, the centrifuge for the country. Everybody drew from it in some fashion.\n\nI was just a bit player. I was in Omaha, and I was racing around the city getting reactions. I did broadcast it for the NBC audience in Omaha because the network was dark for local programming. I also remember the Midwestern governors were meeting in Omaha at the time. Mitt Romney's father, George Romney, was there. And he was a Republican. Big strong executive. And we went down to catch his reaction coming out of a conference. And my cameraman said to him, \"Governor, I'm just so sorry.\" And Romney walked over and put his arms around him and said, \"We're all so sorry.\" And I thought, at that time, \"This goes up and down the generational ladder and across the country.\"\n\nAnd I suppose I matured in that twenty-four hours, over those couple of days. I think I became more realistic as a journalist and saw the larger screen. And the consequences of daring action and the evil that can come, even to America. I do think it was a seminal time for me.\n\nI think (John Kennedy's) legacy is in the boldness of his rhetoric and declaration, and the transition from one generation to another. Legislatively, there's not a lot you can look at. He wasn't there that long. I really do believe he brought younger people into the arena. And made the public arena a really attractive place for the adventurous, for the young. His offspring, politically speaking, were front and center of the anti-war movement of the sixties. Al Lowenstein and people like that who stepped up.\n\nSo I think his legacy is mostly in the memory of the dynamism he seemed to bring to the office. And the people that he would attract to high office. He made it honorable to go to Washington and a calling, if you will, for people of all stations of life.\n Bob Clark\n\nABC News, only journalist to cover assassinations of both John and Robert Kennedy\n\n\"We had no idea the president was going to the hospital. When we got there, we were permitted to run up next to his car. He was lying in the back seat with his head in Jackie's lap. That's the first we knew he had been shot. It was a horrific sight. I will never forget it.\"\n\nI started covering politics in Washington in 1948, and presidents, beginning with Eisenhower's first day in office in 1953. Both President Eisenhower and President Kennedy were very good at handling the press. Eisenhower gave direct, responsive answers; Kennedy made a little more of a game of it. He gave clever answers. I would put Eisenhower at least as equal with Kennedy.\n\nOn November 22, 1963, I was the back-up ABC reporter in Dallas, back up to Bill Lawrence, who was on vacation. I was selected as the network pool correspondent and was placed in the pool car for the motorcade. That car was just a few vehicles behind the limousine Kennedy was in. I was directly under the window where Oswald was shooting from when the shots rang out.\n\nThe shots were loud and clear, though we had no idea Kennedy had been hit. Merriman Smith of UPI, who was the senior wire service reporter, was in the car. He was a gun fancier and said, \"Those are shots.\"\n\nHe was permitted to sit in the front seat of the car and grabbed the only phone in the vehicle. Jack Bell, the AP reporter, didn't make an objection, but soon realized \"Smitty\" was going on and on and keeping him from filing (a bulletin). So, as we were nearing the hospital, Bell was reaching forward trying desperately to grab the phone out of Smitty's hand.\n\nWe followed the motorcade as fast as we could. We had no idea the president was going to the hospital. When we got there, we were permitted to run up next to his car. He was lying in the back seat with his head in Jackie's lap. That's the first we knew he had been shot. It was a horrific sight. I will never forget it. Moments later, they (hospital staff) brought out gurneys for Kennedy and the Texas governor John Connally.\n\nI went back to Love Field and witnessed the swearing in of Lyndon Johnson, but that weekend, did not cover the funeral. I was working the story from ABC's offices on Connecticut Avenue in Washington and watched the procession as it went by on the way to St. Matthew's Cathedral.\n Sid Davis\n\nWestinghouse Broadcasting Company, NBC News, witnessed swearing in of Lyndon Johnson\n\n\"What I reported in the pool report were the first words flashed to the world that Lyndon Johnson was sworn in as the thirty-sixth President of the United States.\"\n\nI first met John Kennedy when I began covering the 1960 campaign. I don't recall exactly my first trip out with Kennedy, but I covered Kennedy for most of the 1960 campaign, and I covered Nixon for eight weeks.\n\nThe campaign for the nomination started a few months into 1960. I would follow Kennedy here and there, not on a regular everyday basis. But as the year progressed, I went out more and more with the candidates. And he was one of them. I went to West Virginia to cover the primary. It was key to getting Kennedy the nomination.\n\nBoth the Bay of Pigs and the Cuban Missile Crisis were important to Kennedy's career, important to making him the president he eventually came to be. The attack on the Bay of Pigs was a dreadful mistake. A serious, grievous mistake. An embarrassment to the United States, and a lesson for John F. Kennedy. That Cuban operation took place within a few months after Kennedy became president. It was April of 1961. Kennedy had no role in planning the Bay of Pigs. It was a holdover from Eisenhower. It had been planned by the CIA, and poorly planned. No air support. No ground troops from the United States. Independent observers were very nervous about it, but the CIA assured Kennedy that they could pull it off.\n\nWhat's interesting is that later, President Eisenhower was astonished that Kennedy would undertake the thing. Immediately after the Bay of Pigs disaster, Kennedy looked for cover. He was embarrassed, and of course, the papers were all critical of what he did. So he met with President Eisenhower at Camp David. I went up for the meeting. They allowed us about twenty feet from the two of them as they took a walk. There's a great picture of Eisenhower and Kennedy taking a walk on one of the beautiful paths.\n\nEisenhower questioned Kennedy about how he went into the operation. \"Did you ask how many troops there were? Did you ask about air support? Did you ask that? Did you do this?\" And Kennedy said, \"No\" to every one of the Eisenhower inquiries. Mr. Eisenhower said, in effect, \"How could you have started this thing if you had not asked these questions?\"\n\nWe did not know of this conversation until years later. So, that led me to believe that Eisenhower was not in on the planning for the Bay of Pigs. It was a CIA operation. They obviously were doing it in secret and might have been afraid to bring it before Eisenhower, so they waited until '61 when Eisenhower was gone.\n\nThe Cuban Missile Crisis came a year later and was probably the most fascinating thing I covered, besides the assassination. My recollection of the night JFK addressed the nation, Monday, October 22, is that I was in the Oval Office behind one of the areas where the cameras were when Kennedy walked in to deliver his crisis speech that there were Soviet missiles in Cuba. He entered the room holding a sheaf of papers. He seemed a bit nervous before he went on the air, but as soon as the red camera light flashed, he was very cool, very tough. After the speech, he was surrounded by Ted Sorensen and others. My recollection is that I was the only pool reporter in the room. Pierre Salinger chose me, and I thought, \"Gee, look how fast I've risen.\" Well, the reason they chose me was that every other reporter who had any sense did not want to watch the speech at the White House. They wouldn't get much. They wanted to be at their office typewriters and teletypes, so they could file stories immediately.\n\nThe brilliance of John F. Kennedy was evident and was displayed in its finest detail during the thirteen days of the crisis. His wisdom and his handling of it, a potential conflagration, was brilliant, despite being this young, forty-five year old facing the generals at the National Security Agency and General Curtis LeMay, who was a World War II hero, sitting across the table from JFK. Of course, Kennedy had been a hero in the Navy with his PT boat. The other fortunate thing for Kennedy was that he had his brother in the room. When they say blood is thicker than water, they mean it. Kennedy looked around the room to see who was with him and who was not, and they\u2014the National Security Council\u2014were not entirely with him. But every time JFK looked at his brother Bobby when a question came up, Bobby would shake his head one way or another, usually in disagreement with most of the generals.\n\nThe options were: bomb, invade, blockade. The military people wanted to bomb, and then invade. And Bobby was urging to take the lesser of the more dangerous options. Kennedy held his ground and said, \"No, we'll go for the blockade. It won't be World War Three.\" And when he made his speech on Monday night, the twenty-second of October, the most important paragraph in that speech that caused Khrushchev to cave in was something Khrushchev apparently had never figured on putting missiles in Cuba. In JFK's speech you'll find a paragraph that says, \"It shall be the policy of this nation to regard any nuclear missile launched from Cuba against any nation in the Western Hemisphere as an attack on the United States, requiring a full retaliatory response upon the Soviet Union.\" Brilliant, is the only way I can describe the moment.\n\nThe U.S. would not only target Cuba, it was the entire Soviet Union open to a nuclear attack. At that point Khrushchev is said to have panicked: \"Why didn't we know that this is what they would do?\" But Kennedy put Khrushchev on notice that any action against the United States would be considered an action by the Soviet Union and not by Fidel Castro. I felt that was the most important part of the speech. And then, of course, the Russians started treading water with their ships heading toward Cuba, or turning back, leading to a removal of the missiles in Cuba. The boil had been pierced. The crisis was over.\n\nIn November of 1963, we first went to San Antonio\u2014I still have the press badge\u2014then we went to Houston, then to Fort Worth, then to Dallas and from Dallas, President Kennedy was to go to the LBJ Ranch for a barbeque. The Kennedys were going to spend the night at the ranch, and come back to Washington on Saturday.\n\nI was seven or eight cars behind the presidential limousine, in press bus number one. It was a city bus and probably carried thirty or more people. All reporters. This was mostly the national press. The crowds at the airport were warm and enthusiastic. Very admiring. That's the way it was all the way into town. It was a beautiful day. Jackie had on that beautiful, they called it raspberry-colored, wool suit with the pillbox hat. She was radiant. She was enjoying herself. She had had some serious personal problems in August of that year. They lost a child, two or three days after he was born, Patrick Bouvier Kennedy. So, she was coming out for the first time. We were rounding Elm and Houston streets and that's when the shots rang out. Our press bus, being seven or eight car lengths behind the presidential limousine, would have meant that we were about eighty feet behind JFK. With the presidential limousine being ahead of us that meant that Oswald, leaning out the window of the School Book Depository Building had to fire at an angle. It just so happened that the reason we heard the shots so clearly was because we were right under the window.\n\nSo I got to the hospital and went toward the emergency room area. The top had been put back onto the presidential limousine. But I could see blood on the back seat. I could see the flowers Mrs. Kennedy had carried lying on the back seat. And I could see the suit jacket of Clint Hill, the Secret Service agent, on the seat. He had taken his jacket off to cover the president's head.\n\nBy that time, the Kennedys had been removed and taken inside the hospital. I was told to get upstairs. I tried to get into the emergency room and was thrown out. So I went upstairs, found a telephone, and I called my office. I told them what was going on and filed my story. I had attended the news conference at the hospital at approximately ten after one, Central Standard Time. Kennedy was shot at 12:30. Malcolm Kilduff, the deputy press secretary, who deserves a lot of credit for the way he handled himself that day, told us at a brief news conference that, and I'll never forget the words, \"John F. Kennedy died at one o'clock, Central Standard Time today here in Dallas. He died of a gunshot wound in the brain. I have nothing further to say.\"\n\nSo I ran to the phones, started filing. I filed the story, did some Q & A's and color, and then a White House staffer grabbed me by the shoulder and said, \"We need you immediately. We need a pool.\"\n\nI said, \"I haven't got time. I'm on the air.\"\n\nAnd he said, \"I'm telling you. We're desperate. We need you right now. Tell your office you're coming with me.\"\n\nSo I said, \"Bye.\"\n\nThey raced us out to Love Field in an unmarked police car. About eighty miles an hour. No siren, no nothing. We got to Love Field, and we were told the Secret Service was bringing the casket aboard the airplane. They had removed four seats to get the casket into the rear of the plane. It was a huge casket. It weighed about eight hundred pounds.\n\nWe were brought into the cabin used by JFK as a conference room. Vice President Johnson was already in the compartment. There were a lot of people. The room was humid, sweltering hot. The scene was somber. LBJ was calm, reserved, deliberate, asking people to do certain things as he prepared to take the oath. They brought a federal judge from downtown Dallas, Sarah Hughes. Johnson turned to his secretary, Marie Fehmer, and said, \"Would you ask Mrs. Kennedy if she would like to stand with us?\"\n\nMarie went back to the rear of the plane where the casket was located with Mrs. Kennedy beside it. Marie asked Mrs. Kennedy what the Vice President asked her to do, and Mrs. Kennedy said she would like to attend, but that she needed a few minutes to compose herself.\n\nWe waited a few minutes. We were anxious, because the Secret Service wanted to get the president and the plane out of Dallas immediately. No one knew whether it was a conspiracy or not. In a few minutes, Mrs. Kennedy came into the cabin, and of course, everybody was awestruck. Women and men burst into tears. Jackie came into the room still wearing the same suit with blood on her skirt and stockings. Blood covered her right wrist, where she had held JFK's head. She was obviously in grief, but knew what was going on. She wished to be at the swearing-in ceremony. I think it was very important for her to be at the ceremony. Her presence was and remains a symbol of the nation's unity at this sad time. It took a great deal of courage for her to leave the casket, even if it was for a few minutes.\n\nJohnson placed Mrs. Kennedy to his left, his wife to his right, and told the judge to proceed. The oath took twenty-eight seconds. Only forty-five words.\n\nOne of the three of us had to brief the rest of the press still in downtown Dallas. I was told there were only two seats on the airplane back to Washington. And that I would have to flip a coin to see who would stay\u2014me or Chuck Roberts, the Newsweek reporter. Merriman Smith, because he was a wire service reporter, was entitled to one of the two seats. And the other seat could go to Chuck or myself. And I said, \"Chuck, I'm not going to flip for the seat, I'm going to get off and give the pool report.\"\n\nIn all selfishness, if I gave the pool report at Love Field, I could then file my own story with my office, and have an eyewitness account of the swearing in of the new president. If I had stayed on the airplane unable to file, I would have been airborne for almost three hours, and my office would have been waiting. And would not have heard from me until I had arrived back in Washington. It served me well in that I got to file my own eyewitness account that no one else had. What I reported in the pool report were the first words flashed to the world that Lyndon Johnson was sworn in as the thirty-sixth president of the United States.\n\nLBJ's behavior was magnificent. There have been stories to the contrary. It's baloney. Not true at all. Lyndon Johnson was calm. He was respectful to the Kennedy family. He understood the gravity of the situation. And his behavior for the next few months and years on behalf of the country and the Kennedys was magnificent. It was among LBJ's finest hours. He knew what to do. He brought a grieving nation together. And he would be exemplary in the civil rights bills and social welfare legislation he got approved, including voting rights and Medicare.\n\nThe Pan Am press charter took off about 9 or 10 o'clock. We arrived in Washington at about one in the morning, I went right to the White House, and I was asked to broadcast the arrival of the casket following an autopsy at Bethesda Naval Hospital. It took what seemed forever. I went on the air somewhere around four o'clock, Saturday morning, November 23. There were about three hundred or four hundred diligent, curious people outside the White House gates peering in. The body was brought to the North Portico in a Navy ambulance. Mrs. Kennedy did not want the body in a hearse.\n\nSmall kerosene lanterns were placed along either side of the driveway. It was an eerie and sad scene, but so appropriate. The leaves had fallen off the trees. It was late November. A chill cooled the night; temperature somewhere in the forties. The ambulance moved very slowly. An honor guard marched ahead of the ambulance in funeral cadence, slowly stepping up the driveway. I could see Mrs. Kennedy through the ambulance's window, sitting with the casket. Stoic, somber. She had helped make some of these arrangements. Closing the broadcast, I made the mistake of trying to quote poet Robert Frost. Frost was Kennedy's favorite poet. He had spoken at Kennedy's inauguration. And Kennedy used to quote from Robert Frost during the 1960 campaign, \"Walking By Woods On a Snowy Evening.\" It closes with, \"The woods are lovely, dark and deep, but I have miles to go before I sleep. And miles to go before I sleep.\"\n\nI tried to repeat those words as a tribute, and I didn't get through it. I cracked up on the air. What amazes me is how many people across the country heard me at four-thirty in the morning breaking down and wrote me letters. Very kind people. Some of them were college classmates I hadn't seen for years.\n\nFor the weekend, I covered everything that had to do with the White House. I stayed at the White House through the following Monday.\n\nI didn't see animosity between the Kennedy and LBJ staffs. You could understand the sadness of the Kennedy people. They had just lost their hero. They campaigned with him. They took him to the presidency. And they had such an exciting future to look forward to in running the country. But it all went away. It was destroyed by one guy with a $14.95 rifle.\n\nJFK left a promising legacy. In education, the arts, in Head Start for children. He was planning civil rights legislation that probably would have been tough to get approved were it not for Lyndon Johnson. It would have been tougher for Kennedy. Johnson, a Southerner, was able to break down barriers. He was a master parliamentarian, probably the best among all our presidents.\n\nI think Kennedy's presidency left a remarkable legacy. The Cuban Missile Crisis changed our relationship with the Soviet Union, leading to a nuclear test ban treaty. JFK's remarks in his inaugural speech, \"Ask not what your country can do for you. Ask what you can do for your country,\" still reverberate.\n\nKennedy was intelligent, witty, and sensitive. He and Mrs. Kennedy were as glamorous as the famous of Hollywood. Because he was in office only two and a half years, he did not have time to fulfill the great agenda he had hoped for. But his vision of America's greatness is lasting.\n Nicholas Gage\n\nWriter for The New York Times and other publications, author\n\n\"Kennedy was a groundbreaker in many ways. First, he was the first non-WASP to be elected president.... I don't think Obama would be there today if Kennedy had not opened the way.\"\n\nI met John Kennedy in 1963 when I won the Hearst Foundation National Journalism Award. It was a tremendously inspiring experience for me, because I had come to the United States fourteen years earlier without knowing a word of English and eventually won this national prize for the best published writing by a college student. It was presented at the White House by the president of the United States. It made me feel that anything was possible in this great country.\n\nExcerpt from A Place for Us by Nicholas Gage \u00a9 1989 (Used with author's permission):\n\nWe were ushered into a richly decorated waiting room and then joined by Kenneth O'Donnell, the president's director of appointments. Finally he said, \"The president is ready to see you now,\" and led us into the Oval Office.\n\nI blinked in the bright sunlight and saw the imposing figure of John Kennedy standing in front of his desk, holding out his hands in greeting. Then I heard a commotion, and turned around to discover several dozen reporters with cameras and microphones pointed toward us. They were all shouting, flashing light bulbs, waving, and calling to me as if I were Cary Grant disembarking from the Queen Elizabeth II: \"Look this way, Mr. Gage\"; \"Smile, please\"; \"How do you feel, meeting the president?\"\n\nAfter posing until the photographers were satisfied, Kennedy shooed most of them out of the room and spoke to each of us individually. He was relaxed and seemed to be enjoying himself, perhaps because his first job after leaving the Navy had been as a reporter for the Hearst Corporation, and he enjoyed the company of journalists and Randolph Hearst. He obviously had been briefed on our backgrounds, and made much of the fact that he and I were both from Massachusetts. \"You're getting a great start with this award,\" he said to me. \"You're obviously a much better reporter than I was. I was pretty bad, wasn't I, Randolph?\"\n\nHearst told him briefly how I had come to America as a war refugee and managed to learn English, enter journalism school, and work my way through college with a combination of academic scholarships and part-time jobs.\n\n\"And you're doing all this so you can become a reporter?\" Kennedy asked. \"Why do you want to be a newspaperman?\"\n\n\"Just a masochist, I guess,\" I answered, because I couldn't think of any way to encapsulate all my reasons in a few words.\n\nEveryone laughed as if this was a pearl of wit, and it was widely quoted in the newspapers the next day. The quarter-hour we spent in the Oval Office passed in an instant.\n\nOn November 22, 1963, only six months after my visit to the White House, I was studying for my master's degree at Columbia University's Graduate School of Journalism. We were in a class and one of the students came in and said, \"President Kennedy's been shot and they've taken him to the hospital.\"\n\nWe all got up and rushed to the Teletype machine. We read with great horror that he had been killed. All of us were stunned. We stopped whatever we were doing. A group of us went to a local bar to try to drink, but I couldn't even swallow. We thought the whole world had collapsed. There is no tragedy that does not produce something positive. I wound up marrying (one girl with us)\u2014Joan Paulson\u2014and we are still married forty-two years later. When the news came over the teletype machine, she tore off the story and still has it.\n\nI watched the news that weekend with Cronkite and read Tom Wicker's stories in The Times. Later, Wicker came and spoke to our class, and subsequently, I worked with him at The Times as an investigative reporter.\n\nKennedy was a groundbreaker in many ways. First, he was the first non-WASP to be elected president. He opened up the possibilities for other Americans to become president. I don't think Obama would be there today if Kennedy had not opened the way. Second, he generated optimism. Historians would say that other presidents accomplished more in similar periods, but I don't think anyone created such a feeling of optimism and well-being in the country as Kennedy, at least in the sixty years I've lived in this country.\n\nAnd the way he stood up to the Soviet Union showed his fierce opposition to authoritarianism. I was a refugee from the Communist civil war in Greece, where the Communists executed my mother for arranging the escape of her children so they wouldn't be taken to indoctrination camps behind the Iron Curtain. Kennedy was someone who had the right values, and he projected those onto the nation and world.\n Bob Jackson\n\nDallas Times-Herald, captured Pulitzer Prize\u2013winning image of Jack Ruby shooting Lee Harvey Oswald\n\n\"All of a sudden someone (Jack Ruby) stepped out from the right, and the thought flashed in my mind, 'He's blocking my view!'\"\n\nON November 22, my assignment was to go with our chief photographer, John Mazziotta, to photograph the arrival (of the president and Mrs. Kennedy at Love Field). I got some good pictures of them there. It was kind of a struggle. He decided to go over to the fence and greet people. It gave us a good photo op. We (myself and the other photographers) were kind of jammed in there, and as he moved along the fence, we had to back up and step on each other.\n\nI was in the eighth car behind the president. It was a convertible, and we were sitting up on the back of the back seat. Tom Dillard, chief photographer of the Dallas News, was sitting next to me. I had an engraved invitation to his luncheon (at the Dallas Trade Mart) where he was to speak. That event was the rest of my assignment.\n\n(Later) we had gotten around the corner and we heard the first shot. Then two more shots closer together. We were facing the Texas School Book Depository Building. I looked up there and there was a rifle on the window ledge. And I could see it being drawn in. And I said, \"There's a rifle!\"\n\nDillard said, \"Where?\"\n\nAnd I pointed out the empty window to him. Of course, he took a picture of that. I have an empty camera in my lap and wondering when I should try to reload it. On the Nikons in those days, the back wasn't hinged. You had to take the back off the camera, put the film in, and stick the back of the camera back on. That made it hard. The car was moving, and I didn't want to drop the back off the camera and lose it. So I wasn't able to do anything just then. So as we got to the corner (in front of the building), I just looked at the scene off to my left: the grassy knoll, people covering up their kids; people running. I saw the president's car disappear under the overpass and then Johnson's car. A cop rode his motorcycle up the slope of the grassy knoll; he jumped off, and the motorcycle kept going until it fell over. And he ran to the door of the building and another policeman joined him there. They went inside.\n\nI soon found out the president had been hit.\n\nI saw that the traffic, which had been held up at the corner of Houston and Elm, the police are letting the drivers go. So the traffic is coming toward me, over the evidence, any evidence that happened to be there. And I thought, \"Why are they (police) letting these cars down there?\"\n\nTwo other photographers and I flagged down a car. We told this lady we needed to go to the hospital. We got into her car\u2014just about the point where Kennedy had been hit. She pulled onto the on-ramp on the Stemmons Freeway, and a motorcycle cop was there, stopping traffic and checking everyone. We explained who we were and that we needed to get to the hospital. At that point, I overheard a conversation on the officer's radio, and someone said, \"But we don't know where the shots came from.\" And I said, \"Hey, I know where the shots came from.\" So he took my name and my phone number at the newspaper and let us go.\n\n(The driver) took us to the hospital. But, because I had not stayed in the motorcade, I did not have the access I could have had. I was kept back with the crowd behind the hospital's emergency entrance. In the meantime, I heard on another police radio that an officer is down in Oak Cliff. I remember commenting to someone, \"I wonder if that has anything to do with what just happened?\" But then I thought, \"Surely not, because whoever did that (the shooting of the president) is in custody.\" Sure enough, that's what it turned out to be.\n\nSometime between one o'clock and one-thirty, we got the announcement the president had died. The word spread outside soon after it was announced inside. I never got inside the hospital; never needed to, really\n\nEventually I got back to the newspaper office. To this day, I don't remember how I got there. I ran my film, and eventually I went to the Dallas Police Station, and I was there into the night. I was able to shoot images of the police bringing (Lee Harvey) Oswald down the hall. I also shot images of his wife Marina and his mother Marguerite. If I had known then what was going to happen Sunday morning, I would have shot images of everything I possibly could in the hallway. If nothing else, for the record. Jack Ruby was there as well.\n\nAt one point on Friday night, I wormed my way between Marina and Marguerite and the elevator door and I got several frames, including one really good close-up of their faces, showing the strain on them. I was tempted to just back into the elevator when the doors opened, but I realized there was no way the police would allow me to do that. But I thought about it.\n\nOn Sunday morning, November 24, we had one photographer stationed at the county jail awaiting Oswald's arrival there. I was the \"Sunday man\" on duty and the other photographer called me on Saturday night asking if I would cover the transfer of Oswald at the Dallas Police Department. I said, \"Sure.\" I knew I would be at the police department and thought, \"What if somebody makes an attempt on his life? What if something happens? What if he (Oswald) never gets to the county jail? What if... ?\"\n\nThe other photographer wanted to be at the county jail, where people had been putting hundreds of flowers at the entrance, and he probably was thinking in the back of his mind, \"That's the place for the picture.\" And it was! If Oswald had made it there.\n\nSo finally, the police said, \"We're bringing him (Oswald) down in about five minutes, so you have five minutes to get in position. We don't want anybody moving around when he appears.\" So I picked my spot which gave me a clear shot. I pre-focused the camera at about eleven feet. I must have checked it ten times! I expected it would be a pretty routine shot. It's a matter of the police escorting the prisoner toward you. Except in this case, it was a little more historic.\n\nAll of a sudden someone (Jack Ruby) stepped out from the right, and the thought flashed in my mind, \"He's blocking my view!\" At the same time, the car to take Oswald was backing up. It already should have been in place. I remember leaning against the fender at the time it (the car) stopped. I was leaning a bit to my left because this person (Ruby) on the right was coming out. All of this, of course, was quick. Instantaneous. He (Jack Ruby) took a few steps and \"bang.\"\n\nAnd I punched the shutter at the right time. It could not have come together any better. I could not have planned it any better. It just happened and I guess it was meant to be.\n\nAfter the shooting it was pandemonium. One cop ran over the car\u2014from the hood to the trunk\u2014and jumped into the pile. The next thing I know is that I am being shoved back by a plainclothes detective and I remember saying, \"Get your hand off my camera!\" He was really upset, and another policeman next to him had to calm him down.\n\nWe waited until they brought the ambulance in\u2014about five minutes. I was able to get a lucky shot when they put him (Oswald) into the back of the ambulance. I was holding my camera in the air and pointing it down. His shirt was pulled up and there was no blood. The bullet was lodged inside him.\n\nIn the spring of 1964, when the Pulitzer Prizes were announced no one in the newsroom broke out the champagne. It was the newspaper's first Pulitzer. The photo didn't really change my life. I got a twenty-five dollar a week raise. And it made getting another job easier. I never had to show a portfolio again.\n\nJohn Kennedy's legacy? I guess it's the space program and how he pushed for that. I was not a Kennedy lover. My family were Eisenhower Democrats. I was not into politics in 1963, and so I can't rave and rant about what a wonderful person he was. He was certainly charismatic and he brought the presidency closer to the people.\n\nHe was tough. Look at what he went through in the Pacific. I've read the account of the PT 109 sinking and that certainly prepared him for politics and for running the country, for going through crises, like the Vietnam War. He was an interesting character and I certainly think a lot more of him than I do of his younger brother Teddy.\n Marianne Means\n\nWriter for Hearst newspapers\n\n\"He wasn't simple. You couldn't see through him. He had his own thoughts in his own head and he had his own stuff.\"\n\nMY husband (then boyfriend) was the head of the student foreign affairs group (at University of Nebraska). We got ourselves assigned to go down and pick up (Senator) Kennedy and Ted Sorensen at the airport, because we had arranged for him (JFK) to come. He was a hot ticket then. He was (being considered as) a vice presidential candidate. So, we sat around at the airport, picked him up, talked to him on the way to school, and then, when we got back to the airport (after JFK's speech), the plane was late, of course, so we sat around and talked some more. We talked about how we, and I in particular, wanted to go into journalism. And he (JFK) said, \"Well, when you come to Washington, look me up.\"\n\nBeing na\u00efve, I took him at his word.\n\nMy boyfriend (and later husband) and I came to Washington. I knew three people on Capitol Hill who turned out to be very useful. One was John Kennedy, one was Ted Sorenson, and one was Evelyn Lincoln, who was his (JFK's) secretary. (Sorenson and Lincoln) were from Nebraska and being from Nebraska was a big asset.\n\nJohn Kennedy was engaging. He was charming. He didn't know what to do with me, so he transferred me to Sorenson. And then to Evelyn. I ended up as an executive assistant to the editor of Broadcasting magazine. It turns out I was his secretary and I hated it. But one of the guys there told me about an opening at the Northern Virginia Sun, and I worked there as the woman's editor. That was the only job they were willing to offer me at the time. Kennedy got elected and (several staff members) left and went into the (new) administration.\n\nI met the bureau chief of the Hearst papers. He was allowed to hire two reporters, and he picked two non-threatening people\u2014the son of a friend of his and me. So that's when I started working at the Hearst papers and that's where I felt comfortable for the rest of my career, almost fifty years with Hearst.\n\n(Kennedy) was the first president I had ever covered, and I had nobody to judge him against. I watched him. He muffed the Bay of Pigs, obviously. He had to live with that; it was his big mistake. But the rest, he did very well. Of course, he really didn't have much experience. He had been in the Senate, but nobody really knows until you're there (in the presidency). He was always so self-confident, in public, and when I was around. He was always genial, confident, and comfortable.\n\nThere was the trip to Vienna to meet Khrushchev which turned out to be a disaster. (Press Secretary Pierre) Salinger lied to us about the meeting, claiming it went well. I also went to Germany when he made his \"Ich bin ein Berliner\" speech. It was very emotional. Huge crowds had gathered to see him, waving hankies and flags. Then we went to Latin America for a conference. Huge crowds.\n\nAnd I went to Dallas.\n\nI was in the first press bus. Smitty (UPI's Merriman Smith) and Jack Bell of the Associated Press were in the car behind him (and Vice President Johnson). I was in the front seat of that press bus. And I remember hearing the shots, but I didn't know what a rifle shot sounded like.\n\nBut one of the guys, I think it was Bob Pierpoint (of CBS News), said, \"Those are rifle shots. I'm leaving.\" All the motion was on the grassy knoll. People were falling on their children because everybody thought the shots were aimed at the grassy knoll because they came from the (Texas School Book Depository) Building. Another thing we did not know was whether this was a mass conspiracy by the Soviets. We were deep in the Cold War then, and the Secret Service had to get Johnson out of there.\n\nThe second car, which had Johnson and Ralph Yarborough, sped ahead. So the cars sped ahead rapidly, and we could see what was happening, but we didn't know a damn thing. I remember everyone else in the press corps turning and running toward whatever transportation they could find. We had proceeded on to the Trade Mart because we didn't know what else to do. Of course we got there and there was not a sign of Kennedy.\n\nI got into a car with a couple of other guys. I was one of the first to get (to Parkland Hospital). Everybody was milling around, not knowing anything. But Yarborough and (Congressman Jack) Brooks were standing outside the hospital crying. We knew this was bad.\n\nSo we had to hang around until finally Mac (Malcolm) Kilduff (Deputy Press Secretary) came out and told us what to do and where we were to go and to sit in a certain room, and we went like the sheep we were. Kilduff came in and told us (JFK was dead), then we all had to go find phones. And I was fighting over a phone with some nurse at a desk. And I said, unfortunately, \"This is all Goldwater's fault.\" And she got very mad at me. We were all so damn emotional.\n\nWe were gathered there (at Parkland) and they wouldn't let us out until Johnson and the casket had gone to the airport. All because they thought it might be a Soviet plot.\n\nThe press plane was sitting at the airport, and Air Force One was sitting at the airport\u2014both were highly guarded. So, with all our press credentials and so forth, they wanted us out of the way. They hustled us to the (press) plane. And gave us Bloody Marys and things to calm us down.\n\nSome people were crying. Mostly me.\n\nWhen they went (the passengers on Air Force One), then we went. We all went to Andrews (Air Force Base) and went our merry ways. My boss said, \"I'll see you at the office.\" Click. (Over the next three days) it was very traumatic. It's all kind of a blur.\n\nHis brothers knew him well. And maybe a few of his old buddies, like Red Faye (friend from PT boat training in WWII; later appointed by JFK to serve as Undersecretary of the Navy). Certainly nobody like me. Presidents become very private. Once they become elected, they (presidents) don't know whom to trust. So they tend to trust the people they knew before they became presidents.\n\nHe (JFK) wasn't simple. You couldn't see through him. He had his own thoughts in his own head and he had his own stuff. He wasn't simple. No president is, including Barack Obama.\n\nHe lifted the spirits of the country. He came in after Eisenhower, who had been going along, treading along in the old, tired ways, rehashing World War II, and so forth. He was young and vibrant. And Jackie was gorgeous. He filled the city (Washington) and the country with new hope. And that's why it was so devastating when he died.\n\nCovering JFK consolidated my reputation in the Hearst Corporation\u2014that a young woman could actually handle all this. And that was important. There were no examples to follow; we had to cut our own way. It's inspiring to cover a president as a young journalist. You know his inner circle. You get all this inside information. And you can handle it well. It was inspiring and it was much fun.\n Michael Medved\n\nAuthor, syndicated radio show host, commentator\n\n\"Everyone was screaming and you didn't know why. But you did know why... It was an echo of November 22, 1963. The screams that night seemed to have started in Dallas.\"\n\nI first saw John Kennedy in October of 1960. I was twelve years old, and my parents allowed me to cut school from Dana Junior High School in San Diego where we lived. Senator Kennedy came to San Diego for a midday rally at Horton Plaza. I got there early, so I was up close, the front row, maybe one row back. When he came out to speak it was very exciting. There are, of course, two things you remember. First, his hair. At the time, his hair was famous. It was much fairer than what you saw on black and white TV. On TV, his hair looked dark. It was actually a sandy, light brown color. And second, he was slender, really slender. As we all know, television adds weight to a person.\n\nI saw him again in 1962 in San Diego, when he delivered a speech at San Diego State College. My dad taught at San Diego State. I was a huge Kennedy enthusiast. And I recently found my notebook from that period. It has a very carefully hand-lettered \"Kennedy for President\" message on it. Of course, I spelled his name wrong\u2014\"Kenedy.\" I was a political geek as a kid.\n\nIn my book, The Shadow Presidents, my chapter on Jack Kennedy was based on the wonderful cooperation of Ted Sorensen. Writing that book\u2014and the interviews I did with extraordinary people like Sherman Adams, Clark Clifford, and Dick Cheney\u2014made me a Republican. Sorensen, for the rest of his life, continued to feel hero worship for John Kennedy. I'm very positive toward Sorensen; he was a very effective aide to JFK. But the Jack Kennedy whom Ted Sorensen described did not come off well at all. I was struck by the detail that I had never heard before. Like the fact that Kennedy never carried money, or a wallet for that matter, so when it came to meals, drinks, and other expenses, Sorensen and other staff members were expected to shell out for everything.\n\nI don't think Jack Kennedy deserved the adulation he received\u2014and still receives\u2014from the American public. It always stuns me that polls of our greatest presidents rank Kennedy quite high. In some of the recent polling, Kennedy is placed at the top of the list as the greatest president we've ever had.\n\nKennedy relied on Sorensen in a different way than other presidents have relied on their chief aides. The funny thing about the Kennedy Administration is that so much was based on imposture. I learned this for the first time from Sorensen. Part of my youthful hero worship of John Kennedy had been based on the idea that, \"Oh, the Eisenhower Administration was that of a really old, dumb guy.\" Rather, with Kennedy, there was this image that he loved culture, as demonstrated by having cellist Pablo Casals perform at the White House. I learned from Sorensen that John Kennedy did not like classical music. In fact, he fell asleep during the concert. He liked Sinatra. There's nothing wrong with that. But that was part of the imposture.\n\nMost striking about the Kennedy inner-circle is that everybody was part of this grand imposture, such as conveying the idea that Kennedy was a devout Catholic. Not. And that he was mad crazy in love with his beautiful wife. And was an ideal family man. And was the picture of health. It just wasn't true, but the American people were led to believe it all was.\n\nOn November 22, 1963, I was a junior at Palisades High School in class and an announcement came on the PA system. The initial announcement by the principal was that the president had been shot and wounded, and he had been taken to a hospital. The implication was that he was going to be OK. For the rest of the day, there was not a lot of school work getting done. It was shortly afterward that the announcement came through that the president had died\u2014and it was that announcement that produced gasps, tears, and hysteria.\n\nI remember I got a ride home that day from Jean Hernandez, the English teacher who was closest to me at the time. She was compassionate because I was so upset; she commented that it was a terrible time for our country. Like a lot of people, my parents initially thought that because it occurred in Dallas, the assassination must have been associated with the far right and segregationists. The assumption was that there were dark forces at work in our country.\n\nMy fascination with the presidency\u2014what presidents could do, presidential greatness\u2014was all tied up with John Kennedy. I have one of those flashing campaign buttons with that fabulous profile of him, and on the other side, it reads, \"A time for greatness.\" I used to do Kennedy impersonations. I saved for and bought Vaughn Meader's album \"The First Family.\" It was a huge bestseller. I tried to emulate John Kennedy, but of course, I could not. I admired him, and the whole idea of the Peace Corps, public service. The line \"Ask not what your country can do for you...\" in the inaugural address... I was a textbook example of someone who was completely taken up with that.\n\nToday, I'm torn about John Kennedy's legacy. In the conservative world in which I function, it's become commonplace to suggest that John Kennedy would be a Republican today. He was pro-tax cuts. He clearly was pro-business. And he supported a strong defense, and he had a strong foreign policy. But it's tough to ignore his support for New Deal notions and his advocacy of Medicare and other dramatic expansions of government.\n\nThe question of his legacy is complicated. On one hand, part of his legacy was reinvigorating the competitive juices in the United States, making the nation feel more zesty and youthful, his call to American greatness. I think that was positive for the country, but it unfortunately involved a denigration of the previous two terms of Ike. In truth, any contrast between Kennedy and Eisenhower\u2014except the ability to impress people on TV\u2014shows Eisenhower as a greater, more successful president. Of course, Eisenhower never had the favorable congressional dispensation that Kennedy had. Kennedy was not in office long enough. But the Kennedy-Eisenhower contrast between style and substance as a national leader changes my outlook substantially. Eisenhower clearly wins over Kennedy.\n\nI was a junior in college during Robert Kennedy's presidential campaign. I got a leave of absence from Yale and went to work in the California primary. I was there at the Ambassador Hotel and was right in front of the podium. I'm heard at the end of his speech yelling, \"We love you, Bobby!\"\n\nThen, of course, he went into the kitchen, and you couldn't see anything and you really couldn't hear. The kitchen was too far away. Then, there was a scream. It started in the kitchen, moved into the front of the hall, then to the back of the hall, then back again. It was one, long wail.\n\nEveryone was screaming and you didn't know why. But you did know why. Everyone knew what had happened. You didn't have to be told. It was an echo of November 22, 1963. The screams that night seemed to have started in Dallas.\n Al Neuharth\n\nAuthor, founder of USA Today, the Freedom Forum, and the Newseum\n\n\"Kennedy congratulated (the Mercury Seven astronauts) and then said, 'Do you guys think we can go to the moon and back in this decade?' (Alan) Shepard said they all looked at each other and exclaimed, 'Sure!'... Then they left and got together for a beer and said to each other, 'Is he nuts?'\"\n\nI first met John Kennedy when he was in the Senate in the mid-fifties. I was in the Washington bureau for the Miami Herald, part of the Knight News organization. I covered him, though I did not get to know him very well. I thought he was a smart, cocky, young guy. When we met with senators\u2014either individually or in groups\u2014some stood out and some didn't. He was certainly one who stood out because he was a bright guy.\n\nOn November 22, 1963, I was in Palm Beach Florida attending the Inter-American Press Association convention. He had spoken to that group earlier that week before he went to Dallas. In that address, he did a good job of relating to the members of the association, which included leading people from Latin America, as well as the United States. I don't recall any specifics of the speech, but I do know that he was a hit.\n\nWe all just kept our eyes and ears on Walter Cronkite. My recollection of Cronkite and the TV coverage is much clearer than any newspaper coverage of the assassination. It was all intrigue and sticking to the tube until Cronkite told us he was dead.\n\nJohn Kennedy was a visionary and he was politically astute. My clearest memory of him is in connection with the space program. I was friends with all the original astronauts. I had started a magazine called Florida Today and was up at Cape Canaveral quite a bit for the launches, including the first one Alan Shepard rode up in. Later, when John Glenn made his orbit, Kennedy had the astronauts to the White House. Shepard often told the story that Kennedy congratulated them for what already had been done and then he said, \"Do you guys think we can go to the moon and back in this decade?\"\n\nShepard said they all looked at each other and exclaimed, \"Sure!\" So they talked about it some more and assured him of their interest in that program. Then they left and got together for a beer and said to each other, \"Is he nuts?\" Because none of them believed we would be in a position to do that. They hoped it could happen, but none of them thought it could.\n\nI don't know that I could say I was influenced by John Kennedy. I was certainly impressed by him. I like to think I was not too influenced by any public official. But I was impressed by him as much\u2014if not more\u2014than any other president. Maybe Eisenhower might be slightly ahead of Kennedy.\n\nWhat is John Kennedy's legacy? I am a bit prejudiced, being a Space Coast guy. His space program impressed me more than anything else he did. But today, decades later, it sure as hell is being compromised significantly.\n Bob Schieffer\n\nCBS News\n\n\"I've always had this theory that the most successful politicians are the ones who master the dominant medium of their time.... He changed the presidency forever. I still think nobody has mastered it quite the way he did.\"\n\nI never met John Kennedy, but my brother, who was in high school at the time, went down to the Hotel Texas (on the morning of November 22) and stood outside. When (Kennedy) came out to greet the crowds, he actually shook hands with the president. He was one of the last people in Fort Worth to do so, just because he happened to be standing close to the limousine.\n\nMy wife\u2014we were not married at the time\u2014and her father attended the breakfast that morning (where JFK spoke). She heard the president, she heard Lyndon Johnson, and she saw Jackie Kennedy when she came in and the president remarked, \"Nobody wonders what Lyndon and I wear.\"\n\nMy wife's father was a big Democrat. He had been a big supporter of John Connally. In fact, when she was growing up, John Connally lived across the street from her family.\n\nThe night before was such a big night in Fort Worth, but I was in a bad humor. I was the night police reporter at the Star-Telegram. There's nothing like having a big story in your town, and you're not a part of it. I had asked if I could cover it (JFK's visit) and the editors said, \"No,\" that the political reporters and the chief of our Austin bureau would be doing the story. Kennedy came to town and got this overwhelming welcome. He landed at Carswell Air Force Base, and ten thousand people showed up that night just to see him and Air Force One, and then the president motored into town to the Texas Hotel.\n\nWe had decided to keep the press club open beyond closing time for all the members of the White House Press Corps. To us local reporters, these were celebrities. Presidents didn't travel that much in those days. There was this \"coffee house\"\u2014an after-hours joint. I didn't get off work until three in the morning and I would often go down there. I was a well-known figure there. A lot of evenings, the night city editor and I would go there. They served coffee, but if you were a friend of the owner, they served you this Kool-Aid spiked with grain alcohol. They gave it away. It's a wonder it didn't kill us all. His place was well known. This was in the days when San Francisco was the headquarters for the beatniks\u2014Allan Ginsberg and the counter-culture. Fort Worth felt like it was on the cutting edge because we had our own \"coffee house.\" The waitresses wore underwear; that was their costumes. By today's standards, their outfits would be seen as bikinis.\n\nIt had quite a reputation, and some of these White House reporters had heard of it and told us, \"What is this thing? We've got to see this!\"\n\nSo Phil Record, the city editor, and I, we led a delegation down to this place from the press club a couple of blocks away. There were also some Secret Service guys there, too. I will preface this: If they were drinking, I never saw it. I don't think they were, quite frankly. But, of course, they were all off duty. My group\u2014the White House Press Corps\u2014stayed until after the sun rose. And we did have a few drinks. And we all had a fine time and then we all went home (or back to hotels).\n\nThat's why I was sound asleep the next day when my brother woke me up. I had not been asleep all that long, I was in a total fog, and he told me, \"You'd better get up. The president's been shot.\" I didn't believe it. At that time, we did not have all these horrible school shootings and these other violent incidents. We didn't know what had happened.\n\nExcerpt from This Just In: What I Couldn't Tell You on TV, by Bob Schieffer, \u00a9 2003 (Used with author's permission):\n\nI got dressed as fast as I could, grabbed my black felt snap-brim Dick Tracy hat, and roared off in my two-seater Triumph TR-4 sports car. As I parked in the lot of the Star-Telegram office, the radio confirmed the worst: The president was dead. It was as if someone had hit me with a hammer. At once, I was stunned, hurt and embarrassed. Stunned, because such violence was unthinkable in those days; hurt and embarrassed, because it had happed in our home state. Why did something like this have to happen and why did it have to happen in Texas?\n\n...When I got back to the city room, the confusion was worse than ever.... I hadn't even removed my hat when I settled behind a typewriter and picked up one of the ringing telephones. In all my years as a reporter, I would never again receive a call like that one. A woman's voice asked if we could spare anyone to give her a ride to Dallas.\n\n\"Lady,\" I said, \"this is not a taxi, and besides, the president has been shot.\"\n\n\"I know,\" she said. \"They think my son is the one who shot him.\"\n\nIt was the mother of Lee Harvey Oswald, and she had heard on the radio of her son's arrest.\n\n\"Where do you live?\" I blurted out.\"I'll be right over to get you...\"\n\nThe fact that Lee Harvey Oswald's mother called, everybody thought it was unusual, but not that unusual. The local newspaper was so much a part of the community.\n\nExcerpt from This Just In: What I Couldn't Tell You on TV, by Bob Schieffer, \u00a9 2003 (Used with author's permission):\n\nThe drive to Dallas took about an hour, and when we reached the police station, Bill (Foster, the newspaper's automotive editor) let us out and said he would join us later once he parked the car. Hundreds of reporters had converged on the station, most of them in a hallway where the detective offices were located. Since I was wearing the Dick Tracy hat, it was easy for me to pass for a plainclothesman. There was a uniformed cop behind a counter in one of the offices so I approached him and said, \"I'm the one who brought Oswald's mother over from Fort Worth. Is there someplace she can stay where she won't be bothered by all these reporters?\"\n\nThe officer guided us to a small space that seemed to be some kind of interrogation room and said, \"How's this?\" I said thanks, and settled Mrs. Oswald in and went into the hallway to see if I could help our guys.... Never once did anyone ask who I was. As the evening wore on, Oswald's wife was brought to the police station and an officer asked me if we would mind if they let her share the room. I told them I saw no problem...\n\nToward dark, Oswald's mother asked Detective Captain Will Fritz if they could visit Oswald. Fritz agreed, and led us into a holding room below the jail. The group included Oswald's wife, his mother, an FBI agent and me. I would soon be face-to-face with the man who was being charged with killing our president. Whatever Oswald said, this would have to be the story of a lifetime... We had only been there a few minutes, but to me it seemed like an eternity and I could feel my heart beginning to beat faster, when the FBI agent casually asked, \"And who are you with?\"\n\nI had watched veteran interrogators bluff their way with a suspect by answering a question with a question, and with my best imitation I sort of half snarled, \"Well, who are you with?\"\n\nThe agent seemed a little edgy now. \"Are you a reporter?\"\n\nNow I was really pushing it: \"Well, aren't you?\"\n\nIt was at this point that I believe I received my first official death threat. The embarrassed agent said he would kill me if he ever saw me again. Or at least that seemed to be what he was saying.\n\nI was already leaving as he said it.\n\nIt would be the biggest story I almost got but didn't, and I went back to the crowded corridor and blended with the rest of the reporters. For the next two days, I would just be part of the crowd.\n\nTruth be told, covering this story did not really boost my career. I was in Dallas. I wrote some stories about it. What really changed my career was when I went to Vietnam in 1965. That was the turning point.\n\nJack Kennedy figured out how to use television. I've always had this theory that the most successful politicians are the ones who master the dominant medium of their time. The founders were all great writers and that's how people got their news in those days. They got it in the written word. Franklin Roosevelt was the first politician to really understand radio. Before he came along, when a politician would be on the radio, he would speak as if he were addressing a crowd of forty thousand people at Madison Square Garden. Roosevelt figured out there would be three or four people sitting around the kitchen table or the living room listening to the radio and he spoke directly to them. He was so effective.\n\nTelevision came along and Eisenhower didn't understand it, didn't know what to do with it. He held news conferences, but they were so afraid he might say something untoward or start World War III that they never broadcast his news conferences until his press secretary, Jim Haggerty, reviewed the transcript.\n\nKennedy came along and he had this great wit and verve. He was so good. He changed the presidency forever. His mastery of television. The way he could answer questions off the cuff. They were broadcast live. I still think nobody has mastered it quite the way he did. It's hard to say anyone was better on television than Ronald Reagan. But Kennedy could do the interview. He could do the press conference. He could make a speech. He was just better at all of it.\n\nTelevision is no longer the dominant medium. We don't know where people get their news now. Television is still a powerful force, there's no doubt about that. When you add up the total number of people who watch the (network) evening news every night, it is still a very large group\u2014probably thirty million people. There were nearly sixty million people who watched the debate I moderated last year. We do three or three and a half million every Sunday morning on Face the Nation. Television is still influential, but when we start our evening news now, we assume that most of the people turning it on already know what the news is. They've heard it on the radio; they've gotten it on the Internet; they've gotten it on cable. So you can't say anymore that television is the dominant medium. Everything has changed. The whole communications revolution. We don't know where it's going, but we're dealing with a totally different world. It's 24\/7.\n\nThe good news about Twitter and other social media is that you no longer need a charismatic leader to lead a revolution, as we saw in Egypt. The bad news is that all the nuts now know where other nuts are, and they can find validation in their cause, no matter how bizarre.\n\nKennedy set a tone and a style for the presidency. He brought glamour to the presidency. He made a lot of young people want to take part in public service and serve their country. But the fact is, he was not there for very long, and as far as his accomplishments, they were not extensive. The person who got the most done was Lyndon Johnson. Perhaps, Johnson would not have been able to do it if it had not been for Kennedy's death. He (Johnson) got the civil rights programs passed. Had it not been for Vietnam where he made a mistake, he would have been remembered as one of our great presidents.\n\nKennedy's brother Bobby really disliked Lyndon Johnson. He didn't want (Johnson) on the ticket in the first place. They (RFK and LBJ) never healed that breach. They (JFK and RFK) didn't treat Lyndon Johnson very well as vice president for whatever the reasons, but it was Johnson who got done what Kennedy proposed.\n Sanford Socolow\n\nCBS News\n\n\"The press corps knew he was a flagrant womanizer, and the American people never got a hint of that.... The theory of the day was, which sounds specious when you think about it now, if it did not interfere with his professional or civic duties, his (the president's) private life was out of bounds for us.\"\n\nI shook John Kennedy's hand once. It was an event at the White House. It was brief\u2014\"Hello, glad to meet you.\" Something like that. I was enthralled, as most people were, by his youth, his vigor, his handsomeness. He seduced a lot of us (members of the press) of whom I am one.\n\nFor instance, one of the first things he did, which caused quite a stir in my memory, was allowed live coverage of his news conferences, which up to that point had been a no-no. Eisenhower allowed television news coverage of his press conferences after they had been vetted. But with Kennedy's live press conference there was an air of excitement and thrill to them. People anticipated he might goof. Don't forget, this was the height of the Cold War. The thrill that went through the press corps, I just can't dramatize enough. And, of course, that thrill influenced news coverage.\n\nIf you examine it, you'll find the coverage was unbelievably positive. From all corners. Beyond what would be expected. Let's get down to the down and dirty of it: The press corps knew he was a flagrant womanizer, and the American people never got a hint of that. That's one example of how his image was distorted by the press.\n\nThe Carlyle Hotel was the hotel Kennedy used when he came to New York. I was very friendly with Bob Pierpoint, now deceased, who was the White House correspondent for CBS, so when they came up (to New York), I would go to the press room at the Carlyle to meet Bob for drinks and dinner.\n\nI remember Pierre Salinger, the news secretary of the day, would come into the room and say, \"The lid is on,\" which was jargon for there would be no new developments for the rest of the day or the evening. \"You guys can go out and get drunk. I promise you nothing's going to happen while you're gone. The president has gone to visit his friend Mrs. Smith over on Fifth Avenue.\"\n\nSo we (journalists) would go on to have drinks. We knew at the time that Mrs. Smith was one of his amours.\n\nThe theory of the day was, which sounds specious when you think about it now, if it did not interfere with his professional or civic duties, his (the president's) private life was out of bounds for us.\n\nWhat changed was the so-called credibility gap (term referring to public skepticism first used to describe LBJ's policies on Vietnam). And Nixon with Watergate. People discovered the government was lying to them. This was, of course, post-Kennedy. That changed everything.\n\nOn November 22, 1963, I was on vacation in upstate New York with my then-pregnant wife. We had just arrived in Dutchess County and were having lunch at a down market diner. Somebody burst into the diner and yelled, \"The president's been shot.\" I didn't believe it. It was beyond comprehension to me. But I decided I ought to check, so I went to a nearby pay phone, and I couldn't get through to CBS. The CBS switchboard was dead in the water because of the overload. I explained to my wife; she was very understanding. We went back to the house we were staying in, packed up, and drove back to New York. Our vacation was maybe six or seven hours.\n\nI drove her home and went to the office. By the time I got there, it must have been seven or eight in the evening. And I was there for the next four days doing all sorts of odd jobs. I got only a few hours of sleep over the next four days.\n\nThose of us in the trenches did not reflect on the momentousness of the events. We were so busy doing the nitty-gritty of getting information out and researching. Looking for people to interview. Feeding Cronkite, and then later Charles Collingwood, who spelled him. Things were so frantic and so out of control. If you're doing your job, there is no time to reflect.\n\nI remember the big disappointment\u2014I cannot dramatize it enough\u2014that we were not on the air when Oswald got shot Sunday morning. We were in some form of ordinary, routine programming. I cannot remember what it was. This was an era with only three networks. NBC, which carried it live, their regular programming was The Today Show. They were live from Dallas. They lucked out and we at CBS were just beside ourselves. I was in the newsroom and within seconds of it happening (shooting of Oswald), we were on to it.\n\nAt the convention center to which Kennedy was headed (the Dallas Trade Mart) there was a fellow named Eddie Barker, then the news director of the CBS news station in Dallas and who owned Dallas, and Dan Rather, then the most junior member of the CBS news staff. He had been assigned to work with Barker at the convention center and be Barker's \"bat boy,\" doing whatever Barker needed. Eddie was the first news man to learn that Kennedy was dead (from a physician standing near him who had called Parkland Hospital). Eddie told Dan Rather. Rather called in to the Cronkite news room and told them Kennedy was dead. Cronkite rejected that as not being substantial enough to put on the air. Barker and Cronkite were close friends, and Cronkite later that day put Barker on the air.\n\nWe came out of this exhausted. Totally drained. Journalists at that time, especially the good ones, were not very reflective. Cronkite, in his heyday, was not a reflective person.\n\nSeveral years after the assassination, Cronkite published his memoir, A Reporter's Life, which was a huge bestseller in its day. Sometime after the book came out, I got a call from Bob Pierpoint who, as I said, was a very close friend of mine. He was beside himself, really upset.\n\nI said, \"Bob, what's the problem?\"\n\nBob said, \"I just read Cronkite's book.\"\n\nI said, \"So what?\"I personally didn't think much of Cronkite's book.\n\nHe said, \"Sandy, on the day of Kennedy's assassination, he's got Dan Rather at Parkland Hospital. Sandy, I was at Parkland Hospital. Dan Rather was at the convention center.\"\n\nWe're talking about the first edition of the Cronkite autobiography. Sure enough. There were repeated references of Rather at Parkland Hospital.\n\nSo I went into Cronkite's office, and I said, \"Walter, we've got a problem. In your book, you've got Rather at Parkland Hospital and it was Pierpoint.\"\n\nAnd he said, \"Get out of here. What are you talking about?\"\n\n\"What I'm talking about,\" I replied, \"is that on the day Kennedy was shot, Pierpoint was part of the White House entourage. He was at Parkland Hospital. Dan Rather was at the convention center.\"\n\nIt turned into a very heated conversation, with Cronkite finally dismissing me, shouting at me.\n\n\"I don't want to talk about it anymore, goddamn it, it was Rather at the hospital, not Pierpoint.\"\n\nAt one point, Cronkite said, \"I can see it in my mind's eye.\"\n\nI said, \"Walter, we didn't have the technology in those days to have a remote at the hospital in such quick time.\"\n\nHe dismissed me. We had a very angry display.\n\nI retreated. CBS had published a four-volume text of everything that was said over the four days of coverage, and what was seen on television. I proudly went home and pulled up volume one, day one. Sure enough, you go through it and there it is: Pierpoint, Pierpont, Pierpoint. At Parkland Hospital. There is hardly any reference to Rather. So I brought it into Cronkite the next day, and I laid it on his desk and said, \"You've got to read this.\"\n\nAnd he looked at it. And I must say, his lips paled right in front of me. He shouted at his assistant to get Pierpoint on the phone. He spoke to Bob and he groveled. He was dismayed. He promised Pierpoint it would be changed in subsequent editions. He apologized to Pierpoint over and over again for the horrendous mistake.\n\nAs I was leaving the office, Cronkite said to me, \"You know, in a courtroom under oath I would have sworn it was Dan Rather.\"\n\nThe moral of that story? I no longer believe any memoir I read. If the most trusted man in America could make that mistake of memory, you can't trust anybody's memory.\n\nWhat is John Kennedy's legacy? He saved us from nuclear disaster. We should all be grateful that we did not have a nuclear exchange with Russia over Cuba, though he did start our bad investment in Vietnam.\n Al Spivak\n\nUnited Press International, participated in the Kennedy-Nixon debates in 1960\n\n\"(During the campaign, there) were young teeny-boppers who would go running after his open convertible in the motorcades. And they would leap up and down as he drove past. Somebody in the Kennedy entourage named them 'the jumpers.'\"\n\nI first met John Kennedy when he was in the Senate. I covered the Labor Committee. I also covered the campaign of 1960 and got to know him well, because I was traveling with him on his family's airplane. It was a twin-engine Convair, \"The Caroline.\"\n\nI was a \"press pooler\" as a wire service reporter. I thought he was bright, capable, and charming, and as the campaign wore on, I thought he was \"out-campaigning\" Richard Nixon. I also had met Nixon in 1951, shortly after I arrived in Washington, and got to know him during his Senate days, and occasionally covered him when he was vice president.\n\nI watched the first debate on television, and I reacted the way most people did. On television, Kennedy came across much better than Nixon. There was, of course, the contrasting opinion that on radio, Nixon came through much better.\n\nAfter the first debate, the public response to Kennedy on the campaign leaped up by quite a bit. There were larger, more enthusiastic crowds. And people eager to see him; to touch his hand. That's when the so called \"jumpers\" came into existence. These were young teeny-boppers who would go running after his open convertible in the motorcades. And they would leap up and down as he drove past. And somebody in the Kennedy entourage named them \"the jumpers.\" It was like the reaction young girls gave to Frank Sinatra in his heyday. A lot of that for Sinatra may have been staged, but I don't think it was staged in the Kennedy campaign. It didn't have to be. The crowds for Nixon were more subdued. But then, he ran a more subdued campaign.\n\nAnd Kennedy was on the attack. He was attacking an incumbent vice president and an incumbent administration. If you review the transcripts of those campaign speeches, he hit it hard. My favorite line from his stump speech was the characterization of the Republicans: \"They remind me of the elephants in a circus parade. They lock their trunks onto the tails of the other elephants and they all march around in lock-step.\"\n\nOn November 22, 1963, I had just returned to my apartment in Arlington, Virginia, after an extended vacation in Europe. I had been in my apartment for only ten or fifteen minutes when the phone rang, and it was the number two man in our Washington bureau, Grant Dillman. He said, \"Thank goodness you're back. Don't unpack. Kennedy has been shot and we want you to go to Dallas.\" Not too long after that, Grant called me back and said, \"You're not going to Dallas. Kennedy's dead. But you are going to Andrews to cover the arrival of his body on Air Force One.\"\n\nAt Andrews, there was an eerie scene. It was dark and there was a rope stretched across a press area so reporters could see the plane well as it taxied up. Before the casket was removed, Merriman (Smith), who had been on Air Force One as a pooler, got off and rushed over, looking for whoever was there from UPI. He saw me and he handed me his copy and said, \"Phone this into the office.\" He had written a story, a follow-up to the assassination. He had been the only wire service reporter on Air Force One. He covered the swearing-in of Johnson, and he was able to interview people on the staff.\n\nDuring the funeral, my job was to cover the cemetery. I was dictating the graveside ceremony to a typist, play-by-play, and at a certain point there was a flyover of Air Force jets. Several years later, the young lady to whom I had been dictating said to me, \"Do you remember breaking down when the airplanes flew over?\"\n\nI responded, \"I did nothing of the sort.\"\n\nShe said, \"Yes you did. You choked up. And you had to say, 'Wait a minute.' And it took several minutes before you got back and started dictating some more.\"\n\nI said, \"I have no memory of it.\"\n\nBut thinking about it more and more, I am convinced that I did choke up. I choke up very easily. I also recall bagpipers performing. Anyone who would not choke up at that is made of iron.\n\nKennedy's legacy is one of inspiration. He inspired the country through his speeches\u2014not only his inaugural address, but other speeches as well. He came in as a younger president than the nation was accustomed to. And he loved to use the word \"vigor.\"\n\nHe performed magnificently during the Cuban Missile Crisis, and I think he started then to get his stride. Prior to that he felt, I am convinced, that Khrushchev \"did him dirt\" to coin an old phrase, and JFK felt that it was his job to protect the country and the world against Khrushchev. That was a great achievement. It kept the country out of a Third World War.\n Stan Stearns\n\nUnited Press International, captured iconic photo of John Kennedy Jr. saluting his father's casket\n\n\"One exposure. A roll of thirty-six exposures, only one exposure, that's it.\"\n\nI was a pool photographer for a state dinner one evening at the White House. And they put me in some anteroom, waiting for the entertainment, the famous cello player, Pablo Casals. So I'm in this ante waiting and President Kennedy walks in there. And there's nobody in there but me.\n\nI'm like about to pee in my pants. I mean, I'm a twenty-year-old kid.\n\nAnd he takes his coat off, and he calls me by my first name. I don't know how he remembered that. I was shocked. And he says, \"Stan, would you mind adjusting my corset?\" And I'm in the back, tightening up the laces on his corset.\n\nHe leaves, and I said, \"God, I need a drink.\" So then, I guess it was a half hour or so later, the entertainment started. And they brought me into the ballroom.\n\nOn November 22, 1963, I was out at the Bowie (Maryland) golf course. I was on the seventeenth green, putting for birdie. And the pro came out there and he says, \"Stan, I think you need to come with me, the president's been shot.\"\n\nI said, \"I'm sitting here putting for birdie.\"\n\nI thought he was pulling my leg, because he knew the guy I was with, we had a heavy bet going on. Well, I went back with him to the pro shop, called my boss George Grayling in Washington, and he says, \"Thank God you called. Get out to Andrews Air Force base as soon as you can. The plane left Dallas and they're en route.\" So that's when I started with the four-day funeral.\n\nThe most memorable image is Jackie with the blood on her legs, the big martyr, to prove to the American people. Jackie was in another world. Jackie set the motive. Hairstyles, fashion. She redecorated the White House. She was like the first queen, the only queen we ever had. And Jack was like a prince. It was different times.\n\nShe was a promoter. She used to hide the kids all the time, didn't want their publicity, didn't want them in front of photographers. But she would always bring them out at a strategic time. Her timing was unbelievable. And then nothing would happen for a couple months. But she timed it.\n\nSo I'm looking at the blood on her legs, and I'm making all the pictures. This is front page all over the world. It might have been fifty to sixty feet. It was quiet; you could drop a pin on that runway. Nobody said anything. And you could see Bobby look like he was almost in tears, Lyndon got out of there real quick. But I'll never forget the blood-soaked Jackie. In fact, one of the magazines in France, or somewhere else, I forget, they blew up her leg, just from the skirt down.\n\nAs far as the funeral, in my estimation, she would outdo Cecil B. DeMille. She copied all this stuff from the Lincoln funeral, I mean everything. And then the flame at the grave site had to be blue, because that was Jack's favorite color. The whole world stopped. Everybody was glued to that TV set. And I worked the full four days. I think I caught a two-hour nap once at the White House.\n\nIt was myself and an AP photographer who were chosen to walk from the White House with the entourage, with Jackie and all the world leaders, who walked from the White House to the church. And we walked out in front, to the side, and made pictures along this route. So when we got to the church, there were seventy or eighty other photographers. And I got under the rope, and there's no place to go. It's like sardines. And I wind up right next to a fellow UPI photographer Frank Cancellare.\n\nSo we agreed that I was just going to concentrate on Jackie, and he was going to photograph everything like I wasn't even standing there. So they come out, I'm looking through the finder at her, her expressions. Next thing you know, she reaches down, whispers in (John Jr.'s) ear, the hand goes up, click, and it's over.\n\nOne exposure. A roll of thirty-six exposures, only one exposure, that's it.\n\nSo I made that one exposure, and I'm supposed to go with the entourage for the funeral. And I ask Frank, I said, \"Did you get the salute?\"\n\nAnd he says, \"What are you talking about?\"\n\nI started asking every photographer I could get my hands on, did you see the salute? They all said, \"What are you talking about?\" They didn't see it. So I'm supposed to give my film to the motorcycle courier to go back to the office. And I said, \"Man, I know how good this picture is.\" So, I decided, I'm going to walk this back to the office myself. So I walked off the job.\n\nThe office was on Thirteenth and L streets. Well, I was just talking to myself, there's nobody there. And I'm talking to myself about how much this picture would be worth. And if I walked off the job, and if I sold it to LIFE, TIME, Newsweek, Paris Match. Then I said, \"Well, if I did that, they would sue me and get all the money back, and then I'd be in the headlines and nobody would hire me.\" It was just a passing thought that went through my head.\n\nSo I get in the office, and my boss sees me. I thought he was gonna pass out. I never saw a man that angry, and that wounded.\n\nI said, \"George, I've got the picture of the funeral.\"\n\nHe says, \"Are you shitting me? Because we've got twenty-five photographers. Look at this film right here. It looks like confetti.\" I never saw a guy that mad. And about that time, the big boss from New York walks over and he grabs me by the shirt collar and he says, \"You son of a bitch, you better have the picture or you're fired on the spot.\"\n\nAnd I said, \"Frank, take your hands off of me.\"\n\nThen George says, \"Give the film to the kid and let him develop it.\"\n\nI said, \"No, not in that crap that we develop it in.\" I said, \"This is a big enlargement, we've got to have fine grain developer.\" They all look at me like I'm a crazy man. So I go outside, I get in a cab, and I go down to the camera store, and I buy fine grain developer. I come back. I've got to fix up the chemicals. It takes time. I've got to cool it down, because you mix it up with hot water. So I go in the darkroom, and I develop this film.\n\nOne image. I look at it with a magnifying glass on the light box, and I'm gleaming from ear to ear, because I know I've got it. So I wash it properly, I dry it properly, I put it in a little plastic sleeve, and I walk outside. And they start to grab this thing.\n\nI said, \"Just everybody relax.\" And I walked over to my boss and I handed it to him.\n\nAnd he takes a magnifying glass, and he looks at it and he says, \"Holy shit, he does have the picture of the funeral.\"\n Richard Stolley\n\nLIFE magazine, negotiated the rights to the only film of the assassination\n\n\"Then we see frame 313.... I've been in journalism almost seventy years, and that is the most dramatic moment I can recall.\"\n\nMY connection to John Kennedy before his death was quite limited; my connection after his death was quite strong.\n\nI remember November 22 as if it were yesterday. I was in the LA bureau when I heard that John Kennedy had been shot. We got the news over the AP teletype. The first bulletins were not stories\u2014just fragments of sentences: \"shots heard in Dealey Plaza;\" \"presidential limo speeds up.\" I'm not quoting here, just trying to give a sense of what we were reading on the wire. Soon, it was clear that something terrible had happened: \"Kennedy believed wounded,\" or something like that.\n\nI called New York instantly and they were watching the same thing. I asked, \"What can we do?\" My editor in New York replied, \"How fast can you get to Dallas?\" The answer was that there was a National Airlines plane in an hour. I grabbed (correspondent) Tommy (Thompson) and two photographers and said, \"Let's go.\" We heard on the car radio just outside LA airport that Kennedy had died.\n\nThe plane was crowded with journalists. The pilot kept us informed on the flight about developments in Dallas and told us the police had arrested someone named Lee Harvey Oswald from Irving, Texas. Tommy turned to me and said, \"I know the cops in Irving, let me go after that part of the story.\" And I said, \"Godspeed.\"\n\nOnce we landed, Tommy took off for Irving with one of the photographers. The other photographer and I went to the Adolphus Hotel, where our bureau receptionist had booked us a suite. I had been there maybe two hours when I got a phone call from our stringer, Patsy Swank, who had been at the Dallas Police Department all afternoon. Another reporter had called her and said, \"Patsy, I just picked up a report from a cop that a Dallas businessman was in Dealey Plaza with a movie camera, and I'm told he photographed the assassination.\"\n\nI asked, \"What do we know about the photographer?\"\n\nShe replied, \"My friend just pronounced the name to me. He didn't know how to spell it. Zapruder.\"\n\nI had never been in Dallas before, so I picked up the Dallas phone book\u2014a phone book at that time was a very important research tool. I ran my finger down the Z's until I found the name, spelled exactly as she had pronounced it to me. Z-A-P-R-U-D-E-R, Abraham.\n\nSo I started calling. No answer. It was about six o'clock in the evening. At eleven o'clock I was still calling, and finally, this weary voice answered. I identified myself and asked if other reporters had contacted him. He said, \"No.\" He had been out having the film processed all evening long. I asked if I could come to his house and see it. At that point, he was exhausted and full of grief.\n\nHe said, \"Come to my office at nine a.m.\"\n\nI got there at eight. His office was just a few steps from Dealey Plaza. Zapruder looked slightly annoyed that I was so early. He was just about to show the film to two local Secret Service agents. But he said I could join them. He showed it to us in this small windowless room. Eight-millimeter film is tiny\u2014only about a quarter of an inch wide. The image we saw was not larger than a small television screen. He beamed it onto a white wall. You see the motorcade turning right into Dealey Plaza, and then it takes a sharp left. There were some cops on motorcycles out in front, then the limo disappears behind a big road sign. When it comes out, Kennedy's fists are at his throat, and you can see Governor John Connolly starting to fall off the jump seat, his mouth opening up in pain. The president is still clutching his throat. Jackie is looking at him quizzically.\n\nThen we see frame 313. The whole right rear of the president's skull explodes, a plume of blood and brain matter.\n\nAll three of us grunted when frame 313 came into view. I've been in journalism almost seventy years, and that is the most dramatic moment I can recall. I was overwhelmed, and immediately understood what an important piece of photography this was. I was determined not to leave that office without that film for LIFE magazine.\n\nOther reporters showed up. He showed them the film. I stayed away from them. Then Zapruder called us all into the hall outside his office. He said he realized they all wanted to talk to him about purchasing film rights, but, \"Mr. Stolley was the first person to contact me, so I feel obliged to talk to him first.\" He invited me into his office, and of course, the other reporters went nuts. There were no TV people among them. There was Saturday Evening Post, AP, several newspapers, and a newsreel. This was less than twenty-four hours after the president's death. Dan Rather and the other TV people didn't appear until Monday. That may seem astonishing, but TV news was only beginning; it had gone from fifteen to thirty-minute nightly broadcasts just a few months earlier.\n\nZapruder and I sat down and negotiated. He was a canny businessman, and he knew the value of what he had. I determined that very early on. We didn't have to bullshit after that. But he was also deeply concerned about the possible exploitation of the film. He realized how incredibly grisly and sensational it was.\n\nHe told me he had had a nightmare the night before, after the assassination. In it, he was walking through Times Square in New York, back when Times Square was particularly sleazy, and a guy in a pinstriped, double-breasted suit was on the sidewalk urging people, \"C'mon in, folks, and see the president get shot on the big screen!\" Zapruder said he woke up almost in tears. It haunted him.\n\nI was able to assure him that LIFE would not exploit this film. I said, \"You know LIFE magazine. LIFE has a reputation for graphic photography presented with taste.\" In the end, that factor was very important.\n\nWe kept raising the ante, and I got up to fifty thousand dollars. I said, \"Mr. Zapruder, that's as high as I am authorized to go. You'll have to excuse me so I can call New York.\"\n\nThe other reporters were banging on the door and shouting. I could see this was making him exceedingly nervous. The thought of going through this negotiation process with representatives of half a dozen other publications was weighing heavily on him.\n\nHe looked up at me and said, \"Let's do it.\"\n\nI had spoken with one of the TIME-LIFE lawyers the night before, and he told me how to compose a basic contract. I sat down at a desk in Mr. Z's office and typed out a six line contract for print rights only. We both signed it, and his business partner signed it as a witness. He gave me the original film and one copy. He had made three copies, and had already sent one to the FBI and another to the Dallas police department.\n\nIt was amazing the Secret Service agents didn't confiscate the original that morning. Astonishing.\n\nI got those two little reels and said, \"Do you have a back door to this place?\" Thank God he did, so I was able to sneak out and leave poor Mr. Z to go out and tell that hallway full of extremely angry journalists that he had just sold the film to LIFE magazine.\n\nI sent the original to Chicago by courier. We were on a tight deadline, and LIFE had set up a makeshift office at the plant where the magazine was printed. Normally, we would assemble all the editorial material for an issue in New York and put it on a train for Chicago and Donnelley, the printing company. The one copy of the film went to New York. The editors there watched it, called me Sunday, and said, \"This is an incredible piece of film. We now want all the rights.\" I called Zapruder, and he seemed so relieved to hear from me. He said, \"Meet me in my lawyer's office tomorrow.\"\n\nMeanwhile, a small staff of editors in Chicago was putting the film's images into the magazine in black and white because the plant could not print color that fast. And they did not use frame 313 out of deference to the Kennedy family.\n\nIn that frame, the brain matter is spilling forward, which proves fairly conclusively to me that there was no shot from the grassy knoll because it would then be going in the opposite direction. The editors were all set to close the magazine Sunday morning when Oswald was shot. So they had to rip open that issue and start all over again.\n\nTommy Thompson meanwhile had found the Oswald family, and it was an exclusive. He called me on Saturday and said, \"I got them. Nobody else has. We've got to hide them.\"\n\nWe did. We put them in a suite under an assumed name at the Adolphus Hotel. It was Marina, the two babies, Lee's brother Robert, their mother Marguerite, and a Russian interpreter. The Secret Service called us and demanded to know where they were, claiming, \"We've got to protect them.\" I denied any knowledge of their whereabouts. The agent got extremely profane with me, and I finally said, \"If I tell you where they are, will you promise me you won't tell any other reporters. I'm very serious about this.\" He said, \"I promise.\"And he kept his promise.\n\nI think they moved agents into the hotel near the family immediately, and the following day, Sunday, after Oswald was killed, they whisked the family to a motel on the outskirts of Dallas.\n\nThe day after that, Monday, I went to the office of Mr. Z's attorney, Sam Passman, and that's where Dan Rather comes in. He was there and saw the footage for the first time. I knew Dan. I had covered the South for four years and had seen him on a few racial stories working for CBS. He later wrote in The Camera Never Blinks that he was dismayed when LIFE magazine walked into the office because he knew we had \"deep pockets.\" He saw the film and left. He had been authorized to offer ten thousand dollars for film and broadcast rights.\n\nI went into Passman's office with him and Zapruder. I offered another hundred thousand dollars for the film and TV rights, and we ended our negotiations in about fifteen minutes. Zapruder asked that the amount not be revealed. I promised I would not.\n\nAfter that meeting, I never saw Abe Zapruder again, although we did speak on the phone once or twice. Time Inc. had agreed to split whatever revenue it made selling rights to the film after recouping the one hundred fifty thousand dollars, and he called to ask how sales were going. At the time, I realized that he had given all his copies of the film to law enforcement agencies, and I asked if I could send him a good first-generation copy. He said no, he did not want the film in his home.\n\nThe film almost didn't get made. Zapruder had not brought his camera to work that day. His secretary badgered him, saying, \"Mr. Z, it's not every day that the president comes through the neighborhood.\" All Dallas, including Oswald, knew the motorcade route because it had been published in the local papers. It was raining that morning, so Zapruder left his camera at home. But he drove back home and got it. Then he went to Dealey Plaza and found the one spot where he could film the whole thing from beginning to end.\n\nYou can see an almost imperceptible jerk three times in the film\u2014three shots. The first shot missed. It was fired when the limousine was the closest to the sixth floor of the Texas School Book Depository Building where Oswald was hiding. The second shot was fired, from Zapruder's vantage point, when the limousine was behind the road sign. And the fatal third shot came two or three seconds after that.\n\nEver since, I am frequently asked how I managed to obtain the film. Part of the answer came years later when I was speaking with Zapruder's business partner, whom I had no recollection of meeting even though he signed the first contract as a witness. It was that tense a situation. He asked me, \"Do you know why you got that film?\"\n\nI said, \"What do you mean?\"\n\n\"Do you know why you got it instead of all those other people in the hall?\"\n\nAnd I said, \"Well, we paid him fifty thousand dollars.\"\n\nHe said, \"No, he could have gotten that and maybe more from somebody out in the hall.\"\n\nI said, \"Well, LIFE's commitment not to exploit the film was important.\"\n\nHe replied, \"No question that was important to Abe. But do you know why YOU got it?\"\n\nI said, \"I give up.\"\n\nAnd he said, \"Because you were a gentleman.\"\n\nOn a lot of assignments you have to be a son of a bitch. But on this one, my instinct was not to go out to his house the night before. I considered it, because, in the end, there were not a lot of people who would say no to LIFE magazine. But I realized this was something none of us had ever been involved in before.\n\nJohn Kennedy's legacy is mostly spiritual, not substantial. Those of us who were around remember him with fondness and such \"what if\" longing. He never really got a chance to do very much. But there is one exception. I covered the South during four years of school desegregation. Kennedy tried to get civil rights bills through Congress, but there was no way in hell those bills were going to be passed. Lyndon Johnson, who was the most fascinating human being I ever met, did get them through.\n\nKennedy knew what the country needed and was blessed by having a vice president, whom he probably didn't like very much, and of course, Bobby Kennedy hated, who had great influence in Congress. I was in the Senate gallery when the 1964 Civil Rights Act was passed, a truly historic moment. Senator Everett Dirksen, who was from my hometown in Illinois, rallied the moderate Republicans, of whom there were still some then, to vote for the bill. Of course all the Southern Democrats voted no. It was an amazing example of how politics works. Kennedy realized that the civil rights bills were essential and just, and he was their spiritual leader. But the credit for their passage belongs to his successor.\n Sander Vanocur\n\nNBC News and ABC News\n\n\"During the Cuban Missile Crisis, we were very close (to a nuclear war). I had been told that in case of an evacuation of the president and his family to wherever they were going, I was to be the TV pool correspondent and go with them. And I didn't know how to explain this to my wife.\"\n\nI first met John Kennedy at Midway Airport in Chicago, and as luck would have it that day, Nixon and Kennedy were both there changing planes. It was 1958 or 1959. I was based in Chicago and one of them had a book, it might have been Kennedy's Profiles in Courage, and I was sent to the airport to get an interview with him.\n\nWhat I knew about him at the time I had learned from my college roommate at Northwestern University, Newton Minow, an Adlai Stevenson law partner. He was urging Adlai not to run again for the presidency in 1960, but rather to give his support to Kennedy. That was kind of difficult because Stevenson was a good friend of (Eleanor) Roosevelt. She didn't like Kennedy because she didn't like his father. She once said, I think, \"I do wish the senator from Massachusetts would, from time to time, show a bit more courage and a little less profile,\" referring, of course, to Kennedy's book.\n\nCatholicism was the most important issue in the 1960 election. In 1928, Al Smith (a Catholic) had run and lost to (Herbert) Hoover. The joke was, at the time, when the election was over, Al Smith sent a cable to the Pope, and all it said was \"Unpack.\" But (in 1960) the Catholic issue was still very strong and not entirely in the South. Kennedy made the decision to speak to the Greater Houston Ministerial Association one week before the (first) Kennedy-Nixon debate in Chicago. And he did very well. Later, Kennedy and Nixon spoke at the Al Smith Memorial Dinner in New York. I can't even say it was a bipartisan event because Cardinal Francis Spellman was in charge and Spellman didn't really like the Kennedys. The audience was mostly Republican, anyway.\n\nAfter the first debate in Chicago in 1960, in which I was one of the journalists asking questions of the candidates, I went up to the podium and took a yellow legal pad on which Kennedy had made notes. It's now in the Newseum in Washington, DC. Both Kennedy and Nixon left almost immediately after the debate. During the broadcast, Nixon had a little sweat on his upper lip, and he would glance once in a while at Kennedy as if\u2014and I am being a bit subjective here\u2014seeking approval. Kennedy hardly ever looked at Nixon; he looked straight at the camera. We later learned Kennedy had a perpetual tan to hide the pallor from taking prescription drugs for Addison's disease. We didn't know it at the time and had we known it, I'm not sure we would have reported it. Different kind of ethics then.\n\nLater, in 1962, at the end of the Cuban Missile Crisis, Kennedy had a press conference in the White House Situation Room and after he spoke, I went to where he had been standing. There was a yellow legal pad on which he had written one word five times and had circled it: \"Berlin.\" I went to grab it, and Kennedy's military aide, Major General Ted Clifton, put his hand on my wrist and said, \"Sandy, that's for the archives.\"\n\nDuring the Cuban Missile Crisis, we were very close (to a nuclear war). I came home one night and my late wife asked, \"Are we going to war?\" I said, \"I don't know. I need a drink.\" I had been told on Tuesday, after Kennedy's address to the nation on Monday night, that in case of an evacuation of the president and his family to wherever they were going, I was to be the TV pool correspondent and go with them. And I didn't know how to explain this to my wife. Anyone who says they knew there would not be a war, don't believe it. I was scared. I didn't know what to do. I probably would have stayed with my family.\n\nLater, when I visited Havana for an event commemorating the crisis, I came away with how angry the Soviets were at the Cubans for getting them into this, and how angry the Cubans\u2014at least Fidel Castro and his brother\u2014were because they thought the Soviets had collapsed at the strong stance Kennedy had taken.\n\nNewt Minow was a friend of Sargent Shriver, who married Kennedy's sister Eunice, and one night I attended a cocktail party at the Shrivers' apartment. The old man (Joseph P. Kennedy) came up to me and introduced himself. That is where he said, \"I keep telling Jack that television is very important and that (he) ought to pay more attention to you guys in television, than in the press.\"\n\nThe old man was more alert to the possibilities of television than Jack Kennedy was. Jack eased into it. He really was our first television president.\n\nThis continued with Kennedy after he was elected president. We in television thought the guys who had the best entry into the White House were Benjamin Bradlee at Newsweek and Hugh Sidey at TIME magazine. Also, Kennedy was a close friend of Bradlee. They were neighbors. Hugh Sidey had an in because TIME was very influential, terribly powerful at the time. And LIFE magazine, too.\n\nCovering the Kennedy White House was not a day at the beach because television had not come into the dominant position as it has today. After Kennedy and Jackie arrived at the White House, I think it was the day after the inauguration, he took her on a tour, and in the press room, he introduced her to Merriman Smith of UPI. He said, \"Jackie, this is Merriman Smith of the United Press. He owns the place.\"\n\nOn November 22, 1963, I was in Los Angeles. I had been asked by Robert Sarnoff, David's son, who was running the West Coast office of NBC, to be the speaker at a dinner. The next day I was playing tennis, and someone leaned out an apartment window and said, \"Kennedy's been shot.\"\n\nI took the first flight I could to Dallas. Air Force One had already left and so I flew then to Washington.\n\nOn the next day, Saturday, I was at the White House standing on the small street between the West Wing and what was then referred to as the \"Old State Department Building,\" now called the Eisenhower Executive Office Building. I was facing down toward the Potomac outside the office of (Kennedy's National Security Advisor) McGeorge Bundy and out came Kennedy's rocking chair. And about ten minutes later, in came a saddle on a stand. Visually, nothing could illustrate more forcibly the change.\n\nNow I have to say, President Johnson was very gracious and kind to Mrs. Kennedy in getting her through this.\n\nThe Kennedys were made not just for television but for still pictures as well: the image of Kennedy clapping while his children are jumping in the Oval Office; the image of John-John peeking out from the desk. The Kennedys were a good story.\n\nKennedy's legacy is generational. He and Nixon, even though they lived during Roosevelt's presidency and the New Deal, represented the post-World War II generation, symbolized by the GI Bill of Rights.\n\nIt was almost like central casting gave us Jack and Jackie. And their beautiful children.\nSECTION SEVEN:\n\nPolitical Figures\n Barry Goldwater Jr.\n\nMember of the United States House of Representatives, businessman, son of U.S. Senator and 1964 Republican presidential nominee Barry Goldwater\n\n\"They talked about running against each other, and instead of wasting a lot of money, decided to travel around the country together, stop at different towns and cities, and debate each other.\"\n\nI met John Kennedy as a young high school student. I was visiting my father at the United States Capitol. He and my father were talking, and a photographer took a picture of me taking a picture of the two of them. That was my only personal connection with the president. It was probably 1956.\n\nMy father and I talked about his expected campaign against John Kennedy in 1964, and I overheard many of my father's conversations about it. My father and JFK were good friends, having served together on the Senate Labor Committee (Senate Select Committee on Improper Activities in Labor and Management, also known as the \"McClellan Committee\"). He would spend time at the White House from time to time, sitting in the president's rocking chair, talking politics and sipping whiskey.\n\nThey talked about running against each other, and instead of wasting a lot of money, decided to travel around the country together, stop at different towns and cities, and debate each other. It would have been very entertaining and would have set a high standard for future campaigns. They ought to do that today.\n\nExcerpt from Barry M. Goldwater oral history interview for the JFK Library and Museum, recorded January 24, 1965 (Pages 16 and 23):\n\n\"... And I never will forget this\u2014and God, I hope they don't play this for a long, long time\u2014he came in and he had a little cigar in his mouth, you know, those little cigarette cigars. He looked down and he said, \"Do you want this (deleted) job?\" I said, \"Hell, no.\"\n\n(On whether Goldwater believed he could have defeated JFK in 1964): \"Well, I felt that I could, frankly. And I think that he felt that I had a chance. The tide was turning against him at the time he passed away. Certainly he couldn't have carried the South. The business fraternity was against him. I felt that I had a fair to middling chance of defeating him. I wouldn't have bet a lot of money on it... Kennedy would not have been afraid to debate as Johnson was... I imagine he (JFK) would have agreed to go along with me in town after town and have personal debates on the subjects and the issues. Johnson was afraid to get out of his shell. In fact, he never talked about an issue in his life. Kennedy enjoyed that kind of thing and I really looked forward to a campaign against Jack.\n\nOn November 22, 1963, I was on the floor of the Pacific Coast Stock Exchange, as a broker and trader, with two telephones to my ears. I was watching the ticker tape. In those days, stock brokers learned to read the ticker tape, so when the market opened, you never took your eyes off it. And it (the shooting of the president) was announced over the ticker tape. I saw it just as it came across. I was in the middle of a trade which I had to complete, but then I got a call from my dad. He was actually calling about the funeral of my grandmother. And he told me he had heard the president had been shot. We didn't talk much about the president; we were focused more on my grandmother's funeral.\n\nI think the difference between Kennedy and my father was their attitudes toward a strong defense and to react to situations the United States was confronted with in a strong and decisive manner. He was critical of Kennedy for not being more reactive and forthcoming on stopping the missiles in Cuba. Kennedy had an indecisive factor that my father did not appreciate.\n\nExcerpt from Barry M. Goldwater oral history interview for the JFK Library and Museum, recorded January 24, 1965 (Pages 25 and 26):\n\n(New York Senator Kenneth B. Keating) and I together had been pointing out on the floor of the Senate since June of that year that there were missiles in Cuba, that we should be doing something about it. In fact, I wrote down on a piece of paper flying back from Arizona after Labor Day that year that on or about the middle of October Kennedy would do something about Cuba and I gave it to Sam Shafer of Newsweek. And Sam, like you guys do, he printed it.\n\nBut Kennedy could have pulled this confrontation off at any time and he should have done it earlier. I don't hesitate a moment to say, as I've said before, I feel it was a politically motivated operation and it certainly cost the Republican Party a lot of seats in the House of Representatives. There's no question of that. I'll never forget for four or five days I was just like a fish out of water\u2014what the hell could you say? You had to back your president\u2014it was the right act. He didn't go far enough or long enough with it. I think at that time we could have completely toppled Castro and we wouldn't have had to have any bloodshed on our part about it.\n\nA lot of people judge presidents incorrectly. The most important attribute of a president is to be a leader, and through communications to the nation, move us in certain directions and rally the spirit of the American people to get together and solve problems, to get together and support initiatives. I was not around then, but Franklin Roosevelt had that same knack. I'm not sure if he was on the same level of a JFK or a Ronald Reagan, but he certainly made strong efforts to communicate to the American people in a non-partisan way.\n\nIn regard to Kennedy's legacy, the only thing that stands out is the Cuban Missile Crisis and that's a mixed bag. As a result, you wouldn't necessarily give him high marks. He wasn't in office long enough to do much else. The other thing I recall\u2014and Republicans like this\u2014he advocated reducing taxes. A main point of the Republican philosophy\u2014get the government off people's backs and out of their wallets. I respect that and JFK's message\u2014\"A rising tide lifts all boats.\" He was successful in building the nation's economy.\n\nHe stood out\u2014and he still stands out\u2014as an admired and respected president. More so than the others who came after him.\n Gary Hart\n\nMember of the United States Senate, Democratic presidential candidate\n\n\"He was going in to see the president with two guns!\"\n\nI saw John Kennedy twice in campaign appearances in the fall of 1960. I was a graduate student at the time and happened to be with some friends in New York City. We were in midtown Manhattan. There was a lot of excitement: campaign signs, leaflets being handed out. And he was making one of his first appearances after the Democratic convention in the fall campaign.\n\nIn those days, my recollection is that the conventions were held, but people didn't do very much in August, unlike now. The official campaign started after Labor Day. This would have been the first ten days or so in September. The event was a rally of supporters at the Waldorf Astoria Hotel ballroom, and we walked into the hotel, went up to the second or third floor to the ballroom, and more or less just walked right in. You obviously could never do that today. We were midway back in the crowd over on the edge. I remember it vividly. At this event and with others in the campaign, he usually ran an hour or two late. It was supposed to start at three or four in the afternoon. There were some preliminary speeches; a lot of standing around; a band playing.\n\nFinally, the curtains parted and out he came. Of course, he was an extraordinary looking man. Not someone you would mistake for anyone else; very tanned and Hollywood looking in a way.\n\nHe had a manner about him that was almost shy. He always looked a bit startled when he would walk out and see a bunch of people yelling and screaming. He had a seven-minute campaign speech: \"Let's get this country moving again.\" It was kind of standard rhetoric; all the sayings people wanted to hear. And that was it and he was gone. People stayed and cheered and clapped.\n\nThe next time I saw him was the Sunday morning before the election. He had a big rally at Madison Square Garden on Saturday evening. Then he made four or five stops through Westchester County at rallies, then two or three stops in Connecticut, well into the night. He was due on the New Haven Connecticut green at about nine o'clock Sunday morning. My wife and I went down there fairly early, around 8:30, and within minutes the green filled up. It was solidly packed. People were on every side street you could see. I think the crowd estimate was seventy-five thousand to one hundred thousand people, which was monstrous for a campaign at that time.\n\nMy principal impression was the degree to which he inspired my generation toward public service. A lot of people thought it was politics, but he never said, \"Run for office.\" He always said, \"Give something back to the country.\"\n\nIt was a broad challenge toward public service. I entered law school the next year and graduated in '64 and went into three years of government service in Washington. For my class, the class before us, and maybe one or two others, the thing was not to go to a Wall Street firm, but to go to Washington and work in government. I attribute that to John Kennedy. It traces back to the ancient Republic. He read a lot of literature from the Greek and Roman eras. He knew the theory of the Republic was citizen involvement and civic virtue. He conveyed that and I think was the last president to really do so.\n\nOf course, I recall vividly where I was on November 22. That was the fall of my senior year of law school, and so we were interviewing for positions for when we would be graduating the following spring. Law firms were coming to the campus to interview people. Ironically, I had an interview at two o'clock that afternoon with a partner from a Denver law firm, Holme, Roberts, and Owen, a very well-established firm.\n\nAt the time, my wife and I managed one of the international houses on Prospect Street up on the hill, and it was a fifteen minute walk downtown. I had the television on and was getting dressed up for the interview. The news came on, and of course, later the famous Walter Cronkite announcement. So I just went through the motions. I walked down the hill and was in a state of \"auto-pilot.\" It was unbelievable. I went to the room where the interviews were being held. The lawyer, who subsequently became a very good friend of mine, had just heard the news. I introduced myself and sat down.\n\nNeither one of us said a word for fifteen or twenty minutes. Then, I thanked him for his time and left.\n\nThe most important lessons (JFK) learned in office dealt with the Cuban Missile Crisis, as well as the confrontation with Soviet leader Khrushchev during their first summit meeting. The general consensus was that he (Kennedy) got overpowered. He was stunned at how blunt and demanding Khrushchev was and how little respect he paid to Kennedy. Kennedy came out of that saying to his close friends that he realized the seriousness of the Cold War.\n\nHe was very much an international president, as demonstrated by the famous trip to France and meetings with De Gaulle. Given who he was, and his background, and his family, he was by no means a novice in the ways of the world and international relations. He had traveled.\n\nHe had met a lot of important people, so it wasn't like this was a new set of experiences for him. But when he walked into the Elysee Palace, he was President of the United States, so that had to have made a big difference. I would imagine that the two shaping experiences of his presidency were the meetings with international leaders on the international stage and the civil rights struggle, how bitter it was given Southern sentiments.\n\nRegarding the Cuban Missile Crisis, he was a war veteran and had enormous respect for senior commanders, but not necessarily the hawkish ones, such as Curtis LeMay\u2014not only their willingness, but their eagerness to go to war in Cuba, really sobered him. And their underestimation of what it would take to overthrow Castro, I think it caused him to understand that he couldn't take somebody else's word for events on the ground, whether they were in intelligence or the military.\n\nThe concept of public service, as Kennedy spoke of it, is not prevalent today. The theme needs be re-echoed by leaders and particularly new leaders\u2014young leaders who can inspire idealism. Idealism is an interesting phenomenon.\n\nI've described myself as the oldest and last living American idealist. I tell student audiences to take a look because when I'm gone, there won't be any more. Idealism is something that, if it occurs to people at all, it usually occurs in their late teens to mid-twenties. Why that period? Because if you're going to have any spirit or sense of idealism\u2014meaning that we can make improvements to make our society better\u2014it's going to be in the period when you're old enough to understand what that takes\u2014a commitment to service, a belief that things can be improved, but also before you get life's burdens, meaning families, mortgages, and careers. It really is an appeal to a window of young people\u2014that's what the Peace Corps was about. It resonated with a few people over that age, but by and large, in that younger group.\n\nYou can't be against government and then challenge people to be involved in government. And that's an appeal of Republican ideology.\n\nWhat will it take to inspire people into public service? Dynamic young political leadership that issues the challenge. There's no magic to it; there's no secret formula. It's the theme and the reiteration of the theme and providing opportunities to fulfill it. One thing I tried to do in the late seventies and early eighties was to promote different forms of national service. This was a kind of update of the Peace Corps concept. Kennedy not only said, \"Give something back,\" but \"Here are a variety of ways you can do it, including internationally.\" And I think you have to do both; you have to say, \"Give something back and here are some ways that it can be done.\" By the way, not just in the national arena, but at the state and local levels as well.\n\nThe question of John Kennedy's legacy is so complex. I would rank him very high because of that intangible of appealing to idealism which cannot be qualified, cannot be measured, by any normal political science measurements. He had a profound impact on this country, at least in my generation. It hasn't survived much beyond that, and my generation is moving on. So I worry about that for future generations.\n\nHis legacy is not only the tangible events like the Cuban Missile Crisis and his very important American University speech, probably the most important and historic speech he gave. It was visionary. But it is also\u2014in those brief three years\u2014his inspirational contribution. And it was unique. I don't know American history well enough to know whether a Teddy Roosevelt sixty years earlier inspired that kind of activism. Of course, there was the Progressive Movement at the time. As historian Arthur Schlesinger commented, there are \"waves of reform\" and those, generally, are periods of idealism and social improvement.\n\nI would like to believe that if he had a chance to see what some of the people he inspired went on to do, that he would be very pleased. That is to say, a lot of people did go into the Peace Corps, the military, and had brilliant careers in government and diplomacy. A few of us got ourselves elected to office. The greatest rewards of my service are the people who have helped me, and then have gone on to do great things themselves. And some in political life and some, quite often, in other humanitarian causes. There have been an extraordinary number of them.\n\nOne last story. I got to the Senate in 1975 and was immediately appointed to the Select Committee to Investigate the Intelligence Agencies of the U.S. Government. It came to be called the \"Church Committee,\" after Senator Frank Church of Idaho. Shortly after we began our work in the spring of 1975, the CIA started to turn over a lot of files having to do with attempts to assassinate foreign leaders, particularly Fidel Castro. We got deeply involved in that\u2014how decisions were made and who made them. Was it John Kennedy? Was it Robert Kennedy? We spent endless hours on that. But in the process, we talked with a CIA deputy director of operations, who later became the director of the CIA, Richard Helms. We were trying to find out who had access to John Kennedy during that period, 1961 and 1962 principally, because the efforts to assassinate Castro ended with the Missile Crisis resolution in October of 1962.\n\nIn a meeting, Helms said, \"On my way out of the Oval Office in a planning session, the president asked me to stay behind and I did. He said, 'Dick, I've been reading these Ian Fleming books on 007,' which Kennedy made famous. And he said, 'Do we have a 007?'\"\n\nHelms said he had to think a while and he responded, \"'Well, not in the way those books portray, but one of our principal agents has a lot of exciting stories to tell.' And Kennedy said, 'Bring him around. I'd like to meet him.'\"\n\nHis name was William Harvey. Harvey become famous for digging the tunnel from West Berlin to East Berlin under the East Berlin phone exchange and tapping into Soviet telephone conversations for quite a while. And when the Soviets found out about it, they got tipped off and sent a whole bunch of people down the tunnel to West Berlin. Harvey was manning a machine gun at the other end of the tunnel ready to kill them if they came over.\n\nPhysically, however, he looked like the opposite of Sean Connery.\n\nSo we\u2014the members of the select committee\u2014called Harvey in to ask him if he ever met Kennedy. He said \"Yes.\" And we said, \"Tell us the circumstances.\"\n\nHe said, \"Dick Helms took me to the White House and we were waiting outside the Oval Office. The door was guarded by a Secret Service man. We waited a few minutes, and Helms came over to me and whispered in my ear, 'Bill, you're not carrying a weapon, are you?'\"\n\nAnd Harvey whispered back, \"Of course I'm carrying a weapon. I always carry a weapon.\"\n\nSo Helms goes to the Secret Service man and says, \"This man with me is one of our top agents at the agency, and he tells me he is armed.\"\n\nThe Secret Service guy turns pale and goes over to Harvey and says, \"Mr. Harvey, you're not going in to see the president with a weapon.\"\n\nHarvey reaches under his arm, pulls out a forty-five and a shoulder holster, and gives it to the Secret Service agent. So they sit for two or three more minutes. And Helms had a terrible idea. He went over and whispered to Harvey, \"Bill, you don't have any other weapons, do you?\"\n\nAnd Harvey whispered back, \"Of course I do. I always have a back-up weapon.\" And the Secret Service guy took a derringer off of Harvey's ankle holster.\n\nHe was going in to see the president with two guns!\n Ken Hechler\n\nMember of the United States House of Representatives, state official from West Virginia\n\n\"JFK and his campaign family spent so much time in West Virginia (during the 1960 primary) that West Virginia became a household name. (He) said that 'West Virginia' was the third word that his daughter Caroline learned to pronounce.\"\n\nI first met John F. Kennedy in 1947. I was teaching at Princeton University and took a group of my students to Washington. He talked to us in his congressional office. The students and I were not impressed with the off-hand manner in which he answered questions. Believe it or not, shortly thereafter, those same students interviewed Richard Nixon and were impressed with the directness of his answers.\n\nIn 1960, Senator Kennedy was trailing Hubert Humphrey midway in West Virginia's primary presidential campaign because of his Catholic religion. Ironically, Kennedy overcame this deficit with his crisp and direct answers to the media's questions, in contrast to Humphrey's over-expansive and drawn-out detail. Because Humphrey talked so much, it was often said of him: If you asked him what time it was, he would tell you how to build a watch!\n\nWhen I was a congressman, I once was flying into Charleston on the same plane with Senator Kennedy. The county Charleston is in is called Kanawha which is pronounced \"Ka-naw-wa.\" As I walked up the aisle of the plane passing by Senator Kennedy, I heard him softly repeating \"Ka-naw-wa\" over and over. Kennedy wasn't going to mispronounce the name as so many others often did.\n\nUltimately, Kennedy was able to articulate that he supported the Constitution of the United States and convinced West Virginians that he would adhere to the principles of peace and justice and would never let his religion interfere with his Americanism.\n\nDuring JFK's campaign he visited Huntington, West Virginia, home of then-Marshall College, which would not allow him to campaign on their campus. So I let him stand on the hood of my Chevy. JFK and his campaign family spent so much time in West Virginia that West Virginia became a household name. He said that \"West Virginia\" was the third word that his daughter Caroline learned to pronounce.\n\nThe head of the Logan County State Democratic Executive Committee was Raymond Chafin. For years he had been a machine boss who dictated the way elections would swing. Raymond told me a story about meeting with Kennedy's national organizer, Larry O'Brien, and being asked what it would take to get Raymond on board, so to speak. Raymond told me he said, \"Oh, about thirty-four.\" Later a suitcase was delivered that contained thirty-four thousand dollars, not the thirty-four hundred that Raymond had meant!\n\nOn the day that President Kennedy was assassinated, I was in my congressional office in Huntington preparing remarks for an address I was scheduled to make later that day in the town of Spencer. My address was on the leadership and influence of President Kennedy. During the preparation of my remarks, I received a phone call from the chairman of the Roane County meeting stating, \"Turn on your television set. President Kennedy has been assassinated in Dallas.\"\n\nWe went ahead with the meeting as scheduled, and I talked about the president's legacy. I said that \"A compassionate heart and courageous spirit have been stilled.\"\n\nAlong with millions of Americans, I was inspired with John F. Kennedy's brilliant inaugural address: \"Ask not what your country can do for you. Ask what you can do for your country.\" These stirring remarks inspired millions of Americans to join the Peace Corp and dedicate their lives toward helping their fellow Americans in areas like civil rights, health care, and programs to assist poor or underprivileged people.\n Pete McCloskey\n\nMember of the United States House of Representatives, attorney\n\n\"I said, 'Chuck, your boss is very impressive. If he can inspire a Republican to help him in a national cause, that's real leadership.'\"\n\nIN June of 1963, I was thirty-five and sitting in my little Palo Alto office and I get a telegram. It says, \"Would you please join me and other distinguished leaders of the bar in the East Room of the White House at three o'clock Thursday? Signed, John F. Kennedy.\"\n\nOf course I got on a red-eye flight and was the first one in the room at two-thirty in the afternoon. The room soon filled up with older lawyers. I recognized some of these older California lawyers.\n\nThey were with the big firms. There were, maybe, twenty I knew. They seemed surprised to see me, because I sure as hell did not rate with these fifty- and sixty-year-old distinguished lawyers.\n\nWe sat in a semicircle, and I believe there were about three hundred lawyers. There was a rostrum with the great seal of the presidency. Pretty soon, in comes the president with Lyndon Johnson on one side and Bobby Kennedy on the other. I didn't think much of Kennedy up to that point. I was a Republican. I didn't like the idea of appointing your brother as attorney general. That's nepotism.\n\nThe president opened with a very humble statement. He said, in close to the following words:\n\n\"Gentlemen, people think the president of the United States has great power. But there are things the president can't do. And there are things he can't do that you men can do... What I'd like to ask you to do is to help me. To do something I cannot do. I'd like you to go back to your communities and form committees of black and white lawyers. I'm not asking you to represent black people, but to act in a capacity so that when blacks have a legitimate grievance in your community, you make sure they can communicate with the power structure: the mayors, the chambers of commerce, the bankers. So that they can get redress or at least seek a means of redress for their grievances. If you can do that, you can help remove a problem which is going to occur in every city in this country, as the blacks move into the inner-cities and the whites move into the suburbs.\"\n\nAnd then we all went into the Rose Garden and shook hands with the president and went away. I walked out of the Rose Garden about three feet in the air and went to see my friend Chuck Daly in his office in the West Wing.\n\nI said, \"Chuck, your boss is very impressive. If he can inspire a Republican to help him in a national cause, that's real leadership. I've never thought about politics, but I think someday I might want to run for Congress. Who should I talk to?\"\n\nHe said, \"Nobody will talk to you around here. You're a goddamn Republican. But there's one guy who may give us some problems around here someday.\"\n\nHe picked up the phone and his secretary got this person on the phone for Chuck. Daly talked to the guy, saying, \"I have this guy who's got a crazy idea to get into politics.\" He handed me a slip of paper and said, \"He'll see you if you stick around until tomorrow at four o'clock.\"\n\nThe man who was advising me was John Lindsay, who later became mayor of New York City. It was clear the White House thought John Lindsay might be a threat to the Kennedy dynasty\u2014Jack, Bobby, and Teddy in that order. I tried to talk him out of becoming a Democrat when he was mayor of New York. John and I stayed friends until the day he died.\n\nHere's how I was invited to the meeting at the White House. Chuck is one of my closest friends in the world. We served together in the Marines in Korea. He worked in the Kennedy White House. It turns out Chuck had sat in on a meeting about this forthcoming conference on civil rights and Bobby Kennedy was present. The attorney general remarked, \"We have a lot of lawyers coming, but they all seem to be Democrats. Doesn't anybody know any Republican civil rights lawyers?\"\n\nDaly piped up and said, \"I know two.\" That was me and my then partner Lewis Butler, who later became an assistant secretary at the Department of Health, Education, and Welfare under Nixon, and the first person to quit the Nixon administration over the invasion of Cambodia in 1971.\n\nOn November 22, 1963, I was in San Mateo County Superior Court arguing an important case and the big utility company Pacific Gas & Electric was on the other side. There were four or five lawyers at the counsel table at one end. We had a visiting judge from one of the mountain counties. He was about seventy-five or eighty years old. At about eleven forty-five, I was in the midst of the final argument and the bailiff, sitting at the rear of the courtroom, came up to the judge and handed him a slip of paper. The judge said, \"Hold up your argument, young man. Gentlemen, I've just received news here that President Kennedy has been shot. Now, please continue your argument.\"\n\nI lamely finished the argument. At noon, I ran out of the courtroom, got in my car to go back to the office, turned on the radio, and by that time, it was clear Kennedy had died. I couldn't face it. I really couldn't face it. So, instead of going back to the office, I drove home and went to sleep.\n\nI felt a tremendous sadness when Kennedy was assassinated. He had given the nation great hope. He inspired not just me. He inspired a whole generation of people in his inaugural address. Here was a young guy coming out of the war, having been wounded in the war. He was a war hero. I had the feeling he never would have sent troops to Vietnam. Lyndon Johnson came in with this Texas, \"Bring the coonskin home\" attitude.\n Walter Mondale\n\nVice President of the United States, member of the United States Senate, 1984 Democratic presidential nominee\n\n\"There still is a horrible sense of loss nearly fifty years later. We were robbed.\"\n\nI first met John Kennedy in 1958. He was coming through Minnesota in his first effort to seek the presidency. He spoke at the annual DFL (Democratic-Farmer-Labor Party) dinner. I was awed by him. He was eloquent, although he was still learning how to speak with presidential timber. For young people especially, he was inspiring: his strength, youth, and his embrace of the need for change. It all hit me where I lived; most people in the room were impressed. I was affected because, at that time, I was a young man and thinking about going into politics.\n\nOf course, I met him many times after that.\n\nExcerpt from The Good Fight: A Life in Liberal Politics, by Walter F. Mondale \u00a9 2010 (Used with author's permission):\n\n\"I thought his inaugural address (in January 1961) was one of the finest political speeches of our generation. I believe that the stirring phrase he used\u2014that America would 'pay any price, bear any burden' to advance the cause of liberty\u2014was an enduring source of inspiration to the country.\"\n\nThat (statement) really hit me and encouraged my interest in politics. The one caveat or adjustment I would make more than fifty years later, after Vietnam, Iraq, and Afghanistan, is \"be careful.\" We've gotten into some bad wars and we've been ignorant about what we were getting into. These were toxic environments, and they've been tragic for us.\n\nKennedy had come to Duluth, Minnesota to speak at a conference on forestry in late September of 1963, a few weeks before the assassination. He stayed overnight at the Duluth Hotel. A bunch of us were sitting in his room, including Hubert (Senator Hubert Humphrey) and McCarthy (Senator Eugene McCarthy). All of a sudden, a loud foghorn went off. And the president said with a big smile, \"Will this be running all night?\" The answer, of course, was \"Yes.\" I don't think he got a lot of sleep that night.\n\nOn November 22, 1963, I was serving as Minnesota's attorney general and sitting in my office in the State Capitol with Nick Coleman, a leader in the state legislature and an old friend. I had agreed to be Kennedy's re-election campaign chairman for the state. Nick and I were working on the campaign when my secretary came in and said, \"I don't know how to say this, but I just heard the president's been shot. You'd better listen to the radio.\"\n\nSo we turned on the radio and heard the news that he was dead. It was one of the saddest moments of my life. The man we were working for was just gone.\n\nExcerpt from The Good Fight: A Life in Liberal Politics, by Walter F. Mondale \u00a9 2010 (Used with author's permission)\n\n\"After Kennedy's assassination, I believe we had moral momentum. And in the White House we had Johnson and Humphrey\u2014masters of Senate procedure and brilliant legislators. We could have passed nearly anything we wanted. The Senate has never moved so fast in its history. The Elementary and Secondary Education Act, which established the first federal aid to poorer children, was reported to the Senate floor on April 6 and passed on April 9. The Voting Rights Act\u2014reported on April 9 and passed on May 26. Medicare, reported to the Senate on June 30 and passed nine days later. The Immigration Act of 1965\u2014a landmark bill that did away with the old country-of-origin quotas\u2014was reported on September 15 and passed seven days later. It was like driving in a hailstorm\u2014Johnson kept hitting us with one bill after another.\n\n\"Some years later, in a talk at the University of Minnesota, I called that period of my career the 'high tide' for American liberalism. It was as if we took the intellectual heritage of Franklin Roosevelt, the moral inspiration of John Kennedy, and a decade of pent-up demand for social change and converted them in a social reality.\"\n\nI think we had the moral urgency of personal involvement for social change in the United States. There was a generational restlessness. Things had been sluggish for a long time. I think of Kennedy's line from the inaugural address, \"God's work must truly be our own.\"\n\nI think John Kennedy was denied a complete legacy. His commitment to civil rights, to Medicare, to health care, and even to environmental causes, were all part of his vision for America. And then, he was killed. I still think about him and that sense of urgency, the way he was able to take what we consider to be the noble causes of social justice and make them appealing by the way he carried himself, by the way he made compelling arguments for social causes.\n\nThere still is a horrible sense of loss nearly fifty years later. We were robbed.\n Jim Wright\n\nSpeaker of the United States House of Representatives\n\n\"It was the strongest emotional roller coaster I've ever experienced. We had been so happy that morning.\"\n\nFROM my first encounter with then-Senator John Kennedy, I was very favorably impressed. I was a member of the House of Representatives and hosted a fifteen-minute TV program in my community, Fort Worth. His was one of my earlier interviews. I had just finished reading his book Profiles in Courage and had been impressed with it. We did a thirty-minute interview.\n\nAfter the interview, I made a pledge: Anyone who was a senior in high school, or a student in a college and university, who would write a two-page or five-hundred word essay on his or her favorite character in American history, I would send them an autographed copy of John Kennedy's book.\n\nIn those days, the author's price for a book was about two dollars; brand new on the shelf, the book was about five dollars.\n\nI was able to give away a bunch of those books, and John Kennedy autographed each one personally. The books were first editions. Later, I gave more of them to other people\u2014faculty members of schools, students, and others I knew would appreciate them. This was long before I dreamed he would be a candidate for president. But you can image the value attached today to those books.\n\nI was impressed with his willingness to sit down for the interview. The spontaneity of the man, his engaging personality, and his ability to speak freely and with a strong measure of conviction on important issues. He was very much impressed with Sam Houston, who was a hero of mine. I developed an attachment to him and a friendship for him long before I ever imagined he would be a candidate for president.\n\nKennedy's magic was spontaneous and inspired people. He had the ability to say something that would be lasting, to grasp an idea or something that had been said, and to make a comment that would be memorable. He had a peculiar capacity for humor\u2014especially self-deprecating humor. He was the first man I ever knew who actually seemed to enjoy jokes made upon himself. He didn't just tolerate it; he made something out of it.\n\nEarly in 1960, after he was an announced candidate for the nomination, I attended the annual Gridiron Dinner in Washington. As was the custom in those days, they had professional actors portraying various characters in skits calculated to embarrass. Just before Kennedy was to appear, (they had chosen him to be the spokesman for the Democrats) was a skit capturing the most vulnerable spot in Kennedy's armor. The actors portrayed Kennedy out among crowds with a huge sack of money, and loosely scattering the money around, purchasing votes.\n\nKennedy had no idea this was coming. When the skit ended, the spotlight came upon him, and he was expected to make some response to this \"vote buying.\" He reached in his pocket, pulled out a piece of paper, looked at it, and said, \"I have just received a telegram from my wealthy father. It says, 'Dear Jack, Don't buy one vote more than necessary. I'll be damned if I'm going to pay for a landslide.'\" You can imagine how that captivated the audience.\n\nIn 1960, when he was campaigning in West Virginia, there was talk that Kennedy, never having been poor a day in his life, would not have any way to understand the problems of poor people. Not sure if this really happened, but he told me that after a campaign speech, a grizzled old coal miner came up to him and said, \"Senator, there's just one thing I want to tell you. That stuff those have said about you about never being poor, don't let that bother you none. I've been awfully poor, and I can say, you've not missed a damn thing.\"\n\nKennedy's Catholicism did not match in intensity the Cold War with the Soviet Union or other tensions in the world as an issue. And, thank God, it was not nearly as pivotal an issue as it had been in 1928. In the intervening years, I believe there had become a growing recognition that people had over-reacted and over-played their hands.\n\nKennedy did a great deal to help cultivate that calm among some poorly educated Protestants. I admired so much his courage, standing there in Houston before those prominent Protestant ministers from around the country. Bearing his breast, he told them exactly how he felt. It was a beautiful speech. I could sense a sensible mood among these people. They seemed content.\n\nThey seemed to recognize that some people of their faith had over-reacted. And that hostility had no place in this campaign. It still lingered, though, among some people. I made a statement afterward that Kennedy reminded me of Daniel in the lion's den. A few of my constituents got angry at me, accusing me of comparing Protestant ministers to lions. They were just looking for something to get shook up about.\n\nI'm glad I lived during the presidency of John F. Kennedy. It was a time when a man could have heroes and be unapologetic.\n\nExcerpt from Balance of Power: President and Congress from the Era of McCarthy to the Age of Gingrich, by Jim Wright \u00a9 1996 (Used with author's permission):\n\nOn that Friday morning, which today seems like an eternity ago, it would have been hard not to feel that the nation and this man who so perfectly symbolized it was their finest hour.... After his speech to the Fort Worth Chamber of Commerce, we flew to Dallas.... En route from Carswell Air Force Base to Dallas Love Field aboard Air Force One, I talked with President Kennedy. Someone had shown him a copy of the Dallas Morning News which contained a scurrilous display advertisement accusing the president of treason. What a boorishly rude and inhospitable thing! Any self-respecting newspaper, I thought, would have rejected such a vulgar advertisement on the day the president of the United States was visiting the city. Kennedy was puzzled by the extreme right-wing fanaticism that seemed to pervade the upper echelons of Dallas. What made the twin cities, Fort Worth and Dallas, so different, he asked.\n\nI was in the motorcade\u2014probably six cars behind the president. I heard the shots ring out. I thought at first it was a twenty-one gun salute. We were heading north toward the Texas School Book Depository Building where Oswald leaned out the window and shot him. We turned the corner and I saw the fallen president's body slump over. Suddenly, Jackie was on her knees crawling on the back of the car. Then the Secret Service agent jumped onto the car, pushing her down and throwing himself over both of them. Then the car rushed onto the highway toward the hospital. We followed it. I arrived as they were taking the president in on a stretcher. I looked into the back of the limousine and saw pools of blood and thought it must be fatal.\n\nIt was the strongest emotional roller coaster I've ever experienced. We had been so happy that morning. It was so triumphant to have the president appear in our town. He didn't ask for money. He didn't ask for votes. He spent his time thanking the citizens of Fort Worth for their service helping the nation's defenses over the years from the time of our municipal birth as a territory on the prairie saving the settlers from the ravages of the savage Comanches.\n\nIt was a marvelous speech. He said kind things about me, and his words impressed this young congressman. To have his president, in his town, talking to his people.\n\nThen, of course, at noon time we plunged into the depths of despondency. I don't know of anything career-wise that has rocked me as much as that tragedy. At the time his death was confirmed, we didn't know that Oswald had been arrested. I was worried it might be a conspiracy. My place was to stay in my home town until I found out what was going on. Of course, I flew back to Washington later that weekend for the funeral and burial.\n\nExcerpt from Balance of Power: President and Congress from the Era of McCarthy to the Age of Gingrich, by Jim Wright \u00a9 1996 (Used with author's permission):\n\nFor several days most of us moved about in a cloud. The trip back to Washington.... long queues of people, which streamed through the Capitol all Saturday night and into Sunday to pay respect at the casket as it lay in state in the rotunda.... the little boy, John, unaware of the finality of what had happened to his father, saluting the flag... the long, slow march to Arlington Cemetery... all this burned itself into our memories.\n\nJohn Kennedy was a great inspiration. He gave us hope, hope in which Americans can still take pride. I will always be grateful to have had the privilege of knowing him.\nSECTION EIGHT:\n\nThose with Humorous, Poignant, Quirky, and Tragic Encounters and Connections\n Patricia Baillargeon\n\nAssistant to Eleanor Roosevelt\n\n\"After almost an hour, he obviously just couldn't bear being a bystander any longer. He threw down his cane and went running onto the field. My first thought was, 'If your back is that bad, how can you do that?'\"\n\nIN mid-1953, I started working for Eleanor Roosevelt at the United Nations Association in New York City, and a few months later, the weekend of January 30-31, 1954, I was visiting a friend in Washington, D.C. That Sunday morning, longtime friend from Washington State, U.S. Senator Henry (\"Scoop\") Jackson, took me to breakfast at the Mayflower Hotel and then to the \"Red Mass\" held annually for government leaders at St. Matthew's Cathedral when Congress convenes.\n\nPresident and Mrs. Eisenhower were there, as well as many judges, senators, and representatives\u2014including Senator John Kennedy and his wife Jacqueline. Although there was no opportunity to meet the senator after Mass, I was to have such an occasion later that day.\n\nIt was the custom for Senator Jackson and other Senate colleagues and staff members to gather in Georgetown on Sunday afternoons to play touch football. So a couple hours after the cathedral service, we joined his group at a Georgetown playfield. Jack Kennedy and his brother Robert had just arrived and \"Scoop\" introduced me to them before he (\"Scoop\") and Robert went into the game. I was more formally dressed for traveling on the train, which I did later that day, back to New York.\n\nSenator Kennedy, who was walking with a cane, was not dressed for football either. So he and I stood on the sidelines and visited. We talked about current issues, the United Nations, and touched on various lighter subjects and I greatly enjoyed our conversation. After almost an hour, he obviously just couldn't bear being a bystander any longer. He threw down his cane and went running onto the field.\n\nMy first thought was, \"If your back is that bad, how can you do that?\"\n\nBut, of course, we know now that it was that bad. He was to have serious surgery later that year. After about ten minutes, he returned to the sidelines and quickly grabbed his cane. He obviously was in pain.\n\nIn a letter to my parents in Seattle dated February 2, 1954, all I said of John and Robert Kennedy was, \"They are most attractive.\" Rather an understatement. My impressions of Jack Kennedy that afternoon were that he was extremely vigorous and delightfully bright, energetically involved in the issues of the day. He connected with people individually; he had a sparkle. To use today's vernacular, he had charisma. To a young woman in her twenties, he was very handsome and I enjoyed his sense of humor. And he laughed a lot.\n\nIt was in the Senate dining room a few weeks later that I again saw Jack Kennedy. I was in Washington for a day, and Kennedy said he remembered me. We had a very friendly chat and my first impressions were reinforced. I saw Senator Kennedy and later President Kennedy several more times, but always with groups of various sizes.\n\nIn 1960, after winning the Democratic nomination, Senator Kennedy continued his efforts to acquire Mrs. Roosevelt's endorsement. He knew how invaluable it would be in the tight race against Richard Nixon. Finally, Mrs. Roosevelt was ready to express her support for the senator and a date was set for a private meeting the second week of August in Hyde Park. The day before the meeting, Mrs. Roosevelt's granddaughter tragically was killed falling off a horse. Senator Kennedy offered to postpone, but Mrs. Roosevelt kept the appointment. Although I was not in Hyde Park that day, I can envision the two of them having luncheon alone at a table by the corner window in Mrs. Roosevelt's living room at Val-Kill Cottage.\n\nThe very last time I saw him was at a distance during his visit to Seattle in November of 1961, when he delivered a speech on foreign policy. President Kennedy was here to commemorate the one-hundredth anniversary of the University of Washington. His remarks, more than a half century ago, are of special interest in today's world:\n\n\"We must face problems which do not lend themselves to easy or quick or permanent solutions. And we must face the fact that the United States is neither omnipotent nor omniscient\u2014that we are only 6 percent of the world's population and that we cannot impose our will upon the other 94 percent of mankind\u2014that we cannot right every wrong or reverse each adversity, and that, therefore, there cannot be an American solution to every world problem.\"\n\nLate in the evening of November 21, 1963, I returned to Seattle from New York City after attending a meeting of the Rockefeller Panel on the Performing Arts, a group of national leaders preparing a report assessing the arts in America. I had been asked to issue a press release in Seattle about the panel's work. So, on the morning of the twenty-second, I called a friend who was a producer at KING-5 television, the NBC affiliate in Seattle. She informed me that the network had just received a news bulletin that President Kennedy had been shot. \"Turn on your TV,\" she said, and we both instantly rang off. I turned to a friend who was with me and said, \"Thank God Mrs. Roosevelt isn't here to see this.\"\n\nOf course, I was stunned and shocked like everyone else. Americans of my generation had never experienced the assassination of a president. That weekend, I attended a memorial Mass where about one thousand people of all faiths were pressed into the church and more were lining the street outside. Television scenes became lasting images in the minds of people throughout the United States and the world: the dramatic and surreal sight of Jacqueline Kennedy in her pink suit climbing on the back of the presidential limousine as the Secret Service agent rushes toward her. And, of course, the solemn riderless horse in the funeral procession.\n\nJohn Kennedy's presidency offered many lighter moments: his self-deprecating humor when, following Jackie's thunderous welcome in France, he introduced himself, saying, \"I am the man who accompanied Jacqueline Kennedy to Paris, and I have enjoyed it.\"\n\nWhat is JFK's legacy? Perhaps it was the new focus on outer space or maybe his foreign aid program. But I think rather, most importantly, it was that he inspired several generations of Americans by his call to service: \"Ask not what your country can do for you. Ask what you can do for your country.\" Certainly the Peace Corps is one remarkable legacy, not only for its peaceful involvement with recipient countries, but for providing many mainstream American adults with an in-depth experience and an understanding of world problems and cultures.\n Dr. William Bernhard\n\nSurgeon who fought for thirty hours to save the life of Patrick Bouvier Kennedy\n\n\"He was a quick study... In my work, there usually is a lot of family emotion involved. There was no emotion in this. He was cool, but concerned. I've never talked about this experience publicly. It was all too sad.\"\n\nTHE hyperbaric tank belongs to the Industrial Medicine Department of the School of Public Health at Harvard, but it was accessible to Children's Hospital in Boston. And inside was a dreary, grubby room with a tank at one end, and in front of the tank was a relic rocking chair with some spokes missing. It also had some cracks in it. I remember sending a couple of my technicians to talk to the Secret Service agents and to tell them, \"Don't sit in the rocking chair.\"\n\nSo as the afternoon went on, President Kennedy came in. We had one of our chats (about his newborn son's condition), and immediately afterward, he leaped on to the rocking chair. He was rocking back and forth and really enjoying it. About the fourth or fifth rock, the whole thing came apart, and he went head over heels and landed on the cement floor. The Secret Service agents ran around frantically trying to pick him up, because he had back problems.\n\nAnd he got up laughing. He had a good sense of humor. Life was not all grim and gloomy to him.\n\nThat story sets the tone for what happened over the next two days.\n\nWe were doing research in this facility. I had some ideas that I wanted to increase the oxygenation of these very blue infants with congenital heart disease we were dealing with. If there was some way we could increase their arterial oxygen saturation, they might be a little more stable during anesthesia.\n\nThere was a second smaller chamber, a decompression chamber. Everyone who was in there for an hour or so had to go through decompression, so they would not have a nitrogen embolism. That second chamber was only about six or eight feet long. The only time I remember the president came in the decompression chamber was to get some food; he popped in to have a close look at the baby.\n\nDr. James Drorbaugh was selected to check this baby out in Hyannis. We were doing some research together on another project, and we knew each other quite well. Drorbaugh was the one they brought in, and they did not know what to do with this little kid. The child was obviously dying. He was a three-pound preemie. So the hyperbaric chamber was sort of the last resort.\n\nI was not paying attention to all the commotion outside, the police helicopters. Then I got a late phone call from Dr. Drorbaugh. He said, \"Things are not going well. Why don't you come up? Maybe we can do something for this little baby. \"\n\nI said, \"I'll come over. But I have avoided patients with lung disease in the past. I'm afraid I would damage their lungs with one hundred percent oxygen. I don't like to take any chances.\"\n\nHe said, \"Well, we don't have a lot of choices here, so if you're willing to come over, pop over and see him.\"\n\nSo I zipped over. I knew it was the son of the president, but I did not get involved with that. It was just another very sick baby. You can't be star-gazing. I was trying to figure out what I might be able to do. I had never treated anybody like this. The baby didn't have enough lung capacity to boost up his saturation. I listened to him and did not hear a lot of breath sounds, except in his upper chest. I wondered whether he had hypoplastic lungs, lungs that have lot of bronchioles that end up blunted. I didn't know whether he had hyaline membrane disease or hypoplastic lungs.\n\nSo we brought the infant down. We pumped the chamber up to three atmospheres absolute. Everything was all right, so we remained at three atmospheres absolute, about equivalent of the depth of sixty-six feet of sea water. Afterwards, we pressurized and checked the baby out. He was a little three-pounder, about as big as a small doll. And his chest was pumping away; his respirations were at least at one hundred. He was trying to stay alive, breathing one hundred percent oxygen.\n\nAfter about five or ten minutes, Drorbaugh, who was outside the chamber, said, \"His EKG looks better!\"\n\nThe baby's heart was getting a little more oxygen. We checked his saturation, and it had increased from about forty-five to fifty and then almost sixty. So that was a little improvement. Drorbaugh was overjoyed. So we continued at that pace and everything was very stable.\n\nThe baby was struggling, breathing about one hundred times per minute, despite a small improvement in oxygen saturation. He was still very short of breath.\n\nSo the president came in again and we had a little chat.\n\nI said, \"Things look improved on paper, but the baby doesn't appear any better, and I'm concerned we're not going to make out very well here.\"\n\nThe president said, \"Yeah, he looks like he's having a really hard time.\"\n\nI said, \"The deal is, he has to have enough energy to keep moving his diaphragm up and down. So he's burning fat.\"\n\nWe were giving him some glucose and water by vein, but we could never make up for the caloric loss he was sustaining by working so hard to breathe.\n\nSo I told the president, \"He's going to get tired if he keeps this up. And if we have to pick a moment as he gets tired to put an endotracheal tube in, get an anesthesiologist in here, and 'bag' him with one hundred percent oxygen to take over the load of his breathing, the chances of his survival are very, very slim.\"\n\nSo the president left again, and we continued on for a number of hours. I stayed the whole time. I couldn't bear to think of leaving the place, with nobody else knowing anything about it (function of the hyperbaric chamber), and no one there but technicians. There was no way I was going anywhere.\n\nAfter many more hours, it was sometime during the night, I got the feeling the baby was about ready to stop breathing. I watched the corners of his mouth. He seemed to be using accessory muscles plus his diaphragm.\n\nI said, \"Better get ahold of the anesthesia department.\" They sent a very good anesthesiologist over. He had not seen the baby before. He was pretty impressed that things were not looking very good, so he popped a tube in the trachea, taped it up and \"bagged\" him (assisted ventilation).\n\nAs soon as he \"bagged\" him, the baby stopped struggling. The baby looked absolutely peaceful because the anesthesiologist's hand was doing the work, rather than this infant.\n\nSo we continued on for another number of hours, but things basically did not improve.\n\nSaturation dropped back; his arterial saturation was in the upper fifties. All this time, he had hypertension in his right ventricle, that is, he had fetal circulation. This is the way the baby exists in the uterus. The lungs obviously are deflated\u2014there's no gas exchanged\u2014using the mother's arterial blood to keep the baby alive. The kid was having all these problems because he was receiving unsaturated blood.\n\nThere wasn't enough functioning lung to sustain life. And the baby died.\n\nThe president had been in three or four hours earlier, but he knew the baby probably wouldn't make it. So it was no shock to him.\n\nI never saw the president again. I had seen him four times over the previous several hours and in some cases we spent a fair amount of time together talking about what we\u2014the doctors\u2014were doing. He had been on the board of overseers at Harvard, and I did not want him to get too depressed about the School of Public Health tank facility. I did not want to say anything that would disturb the university's opinion of this facility used to instruct students.\n\nWe talked about a number of subjects. I had been a naval officer about two years after him. We talked quite a bit about the Navy.\n\nI never heard from him or anyone else in the family again. And I've never talked about this experience publicly. It was all too sad.\n\nJohn Kennedy was an amazing character. He was a diligent, caring parent, but he didn't bleed all over the floor. He kept his emotions to himself, made very intelligent inquiries, and was just the nicest person. He was very engaged. He wanted the best for his child, but he knew from the beginning it (the child's prognosis) was very bad.\n\nOn November 22, I was making rounds in the hospital seeing post-operative cardiac patients and the news of the assassination knocked me over. Someone had a radio in a patient's room. I just walked into it and got whacked in the face with this bad news. I remember blessing myself and saying a little prayer about it, because it really didn't fit. I couldn't put the whole thing together. I never gave the whole scenario any thought, that anybody would do anything like that.\n\nHe was inexperienced, but he was somebody who could make decisions. He was a typical young naval officer who had to make decisions all the time that could cause his death or the death of other people. When you're in that position, you have to quickly size things up and call some shots. He did that pretty well. The problem with him was that he was not experienced enough to know that other people were snookering him, like the Russians, for instance. They made light of him because he was young, and he was not sophisticated enough to know what they were doing, like loading missiles in Cuba. And the Bay of Pigs was the same thing; he got bad intelligence. Whether he should have been able to figure that out, I don't know.\n\nHe was a quick study; just the facts. You gave him the straight stuff, he got it, and that was it. In my work, there usually is a lot of family emotion involved. There was no emotion in this. He was cool, but concerned. Not many people can fall head over heels with a bad back onto a cement floor and come up laughing. I watched this and almost had a heart attack. I saw this man flip into the air, and I could not believe it.\n\nHe was a good person. He cared about other people, his family, and his children. He was a straight shooter. He was a good leader, a good man. I say a prayer for him. You should too.\n Jim Boyd\n\nMedical student working at Parkland Memorial Hospital\n\n\"When I arrived home that night I told my wife, 'These socks have Lee Harvey Oswald's blood on them!' My wife replied, 'Do you want me to wash them or frame them?'\n\nON November 22, 1963, I was a student at The University of Texas Southwestern Medical School. My wife worked near the medical school, and she came to work early that day so she could take some time off to watch the motorcade in which President John Kennedy would be riding on his visit to Dallas. I recall her saying, \"I'll be delighted when he's out of town. The climate for the president in this city is not good.\"\n\nI was in school that day. But I also had a dental appointment at Baylor Dental School which was across town from the school. When I arrived in the waiting room, I noticed a crowd around a small television on the wall. I recall my friend, the dental student with whom I had an appointment, coming up to me and saying, \"The president's been shot. I don't think we'll be doing anything here today.\"\n\nI left the dental school and drove quickly to the student parking lot at Parkland Memorial Hospital which had not yet been cordoned off. I walked onto the floor that led to the emergency room. The blood bank was on the right and trauma room one, where the president lay, was directly across the hall. Somebody saw me and grabbed me, saying, \"We need your blood. Governor Connally is B-positive and we need your blood.\" So I lay down on the table and gave a pint of blood.\n\nAt about that time, Kennedy was being pronounced dead right across the hall.\n\nThe next day, I was wearing my surgical scrubs and went to the surgical area which was on the second floor. I went in through the back stairs since the ER area was secured. In fact, everything was pretty well locked down. There were men in uniform on the roof with submachine guns. I was wearing my white coat, and with some of my fellow medical students, went into the hospital library, where the news people had been sequestered. As soon as we walked in, the reporters descended on my colleagues and me, thinking we were big shots and could offer an update on Governor Connally's condition.\n\nOn Sunday morning, the twenty-fourth, I was on call with other medical students. We were sitting in the TV room adjacent to the operative suites and were watching as Lee Harvey Oswald was being transferred in the basement of the city jail. We watched Jack Ruby shoot him.\n\nOne of our colleagues said to us, \"You'd better get up and help the surgery room personnel get ready. There's our next patient.\"\n\nWe ran down there. Because of all the various gases used at that time in ER, we usually put on special shoes. I kicked off my street shoes and went into the surgery room in my stocking feet. A couple of operating room nurses were scrambling to get instruments. It took about fifteen minutes for Oswald to arrive, and by then, four of the surgeons who had tried to save the president's life two days earlier were washing up, ready to go to work. They, too, had been watching television and saw Oswald get shot.\n\nI did a cut down in his ankle to get an IV going as I stood there in my stocking feet. As you can imagine they got a bit bloody. Later that day, I took my socks off before going home. When I arrived home that night I told my wife, \"These socks have Lee Harvey Oswald's blood on them!\" My wife replied, \"Do you want me to wash them or frame them?\"\n\nOswald was shot in the perfect spot to do the most damage. The bullet went into the wall of his abdomen at an angle. It hit his liver, the aorta, and one of his kidneys. It disintegrated a complex series of blood vessels where the small intestine goes into the stomach. All of that was blown out. We pumped thirty units of O-negative blood into him as fast we could go, and it was all coming out of the holes inside him just as fast.\n\nFour excellent trauma surgeons worked on Oswald for half an hour. Later, Dr. Mac Perry, a great surgeon, was stepping back and stripping off his surgical gloves, remarked, \"I wish we had had that chance with the president. We had a shot at saving Oswald.\"\n\nKennedy's legacy is a family legacy\u2014bright people, charismatic. But flawed and nobody seemed to care. There remains a great deal of fascination with the Kennedys\u2014books, movies. John Kennedy brought youth and a certain energy to our country.\n Lillian Brown\n\nMakeup artist for JFK and eight other presidents\n\n\"When you touch somebody's face you're getting pretty close to them. And either they were going to invite you back, or they would never invite you again.\"\n\nI first met John Kennedy as a senator on Face the Nation. He did eleven appearances on that program. He would call the producers and say, \"Do you want to make some news?\" And of course, they could not resist that. He read piles of newspapers every day; they were always scattered around him. And he was fascinated with cameras. He loved cameras and the cameras loved him.\n\nHe would ask questions of everybody in the studio. He would go to the lighting man and ask, \"How come I get those circles under my eyes?\" He would go to the audio man and say, \"How do you do this and that?\" And he would go to the camera man and say, \"How come these monitors don't match?\" He would ask dozens of questions, and I'm sure that he was determined to be the first television president, because we were just going from radio into television.\n\nHe was fascinated with the media and the media loved him back. No doubt about it.\n\nAt one point I suggested he get voice lessons. He had a strong Massachusetts or New England accent. The other thing he did, which I kidded him about, was during the Cuban Missile Crisis, he would pronounce it \"Cu-ber.\" I said, \"Jack, you should bring your speaking skills up to the level of some of your other skills. And bring your speaking voice to a more neutral tone.\" And he did it.\n\nAt that time, I was also teaching at Georgetown University. He would kid me and say, \"Why is a university professor doing makeup for a politician?\" And I said, \"Sir, I have three daughters to put through college by myself.\"\n\nJohn Kennedy was such a normal person. And yet, he was extraordinarily intelligent, and he would just milk everybody's brains. Everyone who came around him, if he thought they knew something he wanted to know, he would ask questions and try to find out everything he could. He thought that the people in television could guide him to become the kind of television personality he wanted to be. That's one of the reasons he was as successful as he was. And when you realize he made more than fifty appearances on television. And I said to him, \"Just make love to the camera.\" I would say things like that to him because he didn't care.\n\nOn television, he was superior as compared to all the other presidents since him and Eisenhower before him. It was because he worked so hard to study for it. He would say, \"Do you like this tie? How does this tie look on camera?\" And he would ask, \"Should I cross my legs?\" He would ask all these questions.\n\nAnd, of course, there were the debates in 1960. The first debate was in Chicago, not Washington. I couldn't go because of my classes, though I gave John Kennedy a compact, and I said, \"Before you go on the air, you go in and powder your nose.\" I did makeup for him before each of his nationally televised addresses. He did an unbelievable amount of advanced preparation. Reams of information. The point is: He always appeared knowledgeable about the subject.\n\nOn November 22, 1963, I was at the Washington Golf and Country Club. I had just finished giving a speech and came down, walked across the lobby, and was about to say good-bye to the girl at the front desk. She was white as a sheet. And I said, \"What's the matter?\" She could hardly speak the words. She was overcome. And I was overcome, too. I couldn't believe it and to this day, I don't know how I got home. I know I drove my car, but the whole time I was overcome with this terrible news.\n\nI was fascinated with this man, and I was, in a sense, a friend of his. Once we realized we worked so well together, he would call me to the White House frequently to prepare him for special appearances.\n\nI stayed home the whole weekend. I did not participate in the funeral, but I sure was impressed with what Jackie did. She was such a cultural person. He was so proud of her. Not just because she was so beautiful, but so brilliant. And when we did the White House tour in 1962 with (television newscaster) Charles Collingwood, the president was proud of her. They had a wonderful relationship.\n\nThe desk that he used? She found that in the White House basement where all this stuff was stored. The desk was first used by Rutherford B. Hayes and many other presidents after him, including Franklin Roosevelt. He loved the desk. I always, of course, carried a case with my makeup. And I would never, ever put that case on his desk. He always appreciated that. Other makeup people would set themselves up all over the place and put all this junk on his desk.\n\nHe carried a little bit of makeup I gave him in his pocket, and if he was somewhere else and about to go on television, he would use it. I've worked with nine presidents. When you touch somebody's face you're getting pretty close to them. And either they were going to invite you back, or they would never invite you again. Because of the accident with the PT boat, he was in constant pain. Excruciating pain. And when no one was around, he often walked with crutches. He never did it in public or when he thought he was being observed. But I sort of got closer to him than most people. What I would do is help him straighten up, because he would be leaning over in agony. He had that brace in his back. I would pretend to straighten his jacket at the shoulders, and I would push the palm of my hand into the small of his back, and just hold it there until he could straighten up, and look the way he ought to look on camera. Most people did not know the extent of the agony he was in all the time. It was unremitting.\n\nHe had numerous advisors and was always asking as nearest as he could get to the truth. It was incredible to walk into the Oval Office when he was by himself at the desk and didn't care how things looked. There would be newspapers everywhere.\n\nHis legacy? John Kennedy was not only handsome, well-educated, but he was a president who really cared for his country and really followed his ideals, responded to the needs of the people.\n Josiah Bunting III\n\nMilitary officer, educator, foundation executive\n\n\"(His administration) may have been on the verge of committing an overwhelming sin of intellectual and strategic arrogance with regard to Vietnam.\"\n\nI met President Kennedy approximately one month before graduation from VMI (Virginia Military Institute). It was May of 1963. We were in the middle of a series of centennial Civil War celebrations. The VMI Corps had fought in a battle quite famous at New Market, Virginia. The senior class at VMI typically in the spring went on a two or three-day visit to Washington, visiting various military posts and seeing the sights.\n\nHe was wearing a navy blue suit, and of course, he had that famous wide grin. The aide introduced him to our commandant, and then the commandant introduced me to the president. I was what was called the first captain, which is the senior cadet commander. I was very full of myself. The president came right over and stuck out his hand, saying \"It is a pleasure to meet you.\" Then he said in his distinct accent, \"Mr. Bunting, I understand you are a Rhodes Scholar.\"\n\nI replied, \"Yes, sir.\"\n\nAnd he said, \"Well that's wonderful. You have a wonderful time in England and at Oxford.\"\n\nThen he stood back and said, \"Well, it's a pleasure to have all of you here.\" And then he gave us a talk about the Green Berets, about unconventional warfare, and he mentioned Southeast Asia. At that time we already had advisors there.\n\nWhen he finished, there was applause; there may have been a question or two. And then I went up to thank him, and I said, \"Mr. President, you may not know this, but as our Commander-in-Chief, it is well within your authority to grant unconditional amnesty to all the cadets who are in trouble back at VMI. This is a traditional thing.\" And he laughed. When I say \"in trouble,\" typically it is what is called \"room confinement\" or \"marching penalty tours\" where you march up and down the courtyard with a rifle for an hour because of some relatively minor problem.\n\nHe said, \"Of course, that should be done.\"\n\nI can see it still. He waved his hand in the air and made everybody very happy. Then he went back inside and Major General Clifton (JFK's military aide) spoke to us for a moment or two and then we got back on our busses and on to our next appointment.\n\nWe just thought this man was remarkable. We wanted to follow him. I don't remember even being conscious of whether he was a Republican or a Democrat. He was a war hero. He was a strong guy. He is what we once had in political leaders. The only thing I can compare it to is that period when we had the Marshall Plan. That was a period comparable in its achievement to and in America's stature in the world. You had the sense that America was still looked at as a fount of goodwill and generosity. In the early days of JFK, really the whole time of JFK, we had that feeling. Idealism and unabashed patriotism. And optimism. The idea, \"We'll get it right.\"\n\nIn those days nobody cared about People magazine. Nobody cared about owning a house in the Hamptons. Nobody had ever heard of a Lexus. Celebrity was not a big thing. We wanted to be in the Peace Corps. Or the Marine Corps. Or we wanted to be priests. Or get PhDs. This American fixation on running a hedge fund\u2014we didn't care about that. I'm not sure we'll ever recapture that time or that feeling.\n\nOn November 22, 1963, I was in the middle of dinner at Christ Church Hall at Oxford. Suddenly, the dean of the college stood up and said, \"I must tell you the awful news that the American president has been felled by an assassin at Dallas, in Texas.\"\n\nI went back to my room to listen to the Armed Forces European Network. I got the news broadcast for the next couple of hours. All of the coverage was on the assassination and of Kennedy.\n\nI still have an achingly vivid memory of it. The cathedral was packed. Oxford is very international and many of the kids there were not American. But the one memory I have which is so vivid is that I was at the back of the cathedral. The pews in front of me were all crowded. And virtually everybody was holding onto each other; rows of people with their arms around each other's waists and shoulders. I can hardly describe the pathos of that moment.\n\nI thought of a famous line of William Wordsworth's, who said of being young during the time of the French Revolution: \"Bliss was it in that dawn to be alive, but to be young was very heaven.\" For my generation\u2014people who were eighteen, twenty, or twenty-two years old at that time, the early 1960s\u2014it was heaven.\n\nI was probably not in a completely rational state. The bereavement of others was for the country, or what would happen to the world. I thought of Mrs. Kennedy; I'm not sure why. She was very much a part of JFK.\n\nWe have to remind ourselves that his presidency was cut short. It was an unfinished work. I think of the fact that he and Senator (Barry) Goldwater had discussed flying together from city to city in the '64 campaign and debating. Kennedy was someone with whom everything was comfortable: exchanging strategic, political, and intellectual ideas at a time when partisanship did not pollute; a relatively easy, comfortable, and friendly intercourse among members of the two parties.\n\nMost of that generation of people highly placed in politics were veterans of the Second World War. They had that commonality of experience. When someone tells you that George McGovern flew thirty-five missions in a B-24 Liberator over Germany, it's hard to hate him and castigate him as a member of a far out party. Rather, he's an American patriot with whom you might disagree.\n\nI think John Kennedy's presidency was extraordinary and extraordinarily successful. It wasn't perfect. And it may have been on the verge of committing an overwhelming sin of intellectual and strategic arrogance with regard to Vietnam. It's a fairly short distance from the remark in his inaugural address, \".... that we shall pay any price, bear any burden.... to assure the survival and the success of liberty,\" to actually implementing a foreign policy which turns out to be based on faulty premises. Again, we just don't know. It was an unfinished presidency.\n\nAs a result, I rank his presidency very high, but with the provision that what followed from it was quite ugly. When you look at what people call \"the sixties,\" they're not talking about 1960 to 1964. They're talking about 1967 to 1973. This was a very different time. Young people today have no understanding of the turmoil that occurred.\n\nAnd JFK lived like what I thought was an American patriot of his time. Now we are losing one thousand members a day of the World War II generation\u2014people in their late eighties and early nineties\u2014and one of the things everybody says about these people is that it's hard to get them to talk. The phrase they use\u2014and you do not hear this much anymore\u2014is that \"I was in the service.\" And they leave it at that. It was something you just did. Kennedy was part of that generation.\n Arthur \"Andy\" Carlson\n\nArmy officer who led riderless horse \"Black Jack\" in funeral\n\n\"...(T)hat misbehaving horse, knew he was doing wrong, and I was furious with him. But I was told that some of the TV commentators talked about the horse as a symbol of the deceased president.\"\n\nON November 22, I was doing laundry at a laundromat across the street from the stables at Fort Meyer, Virginia. I saw people clustered around a car, listening intently to the car radio. So I walked over there to see what they were listening to. And when I heard the news, I said to myself, \"Well, I'd better get back to the stables. My day off is over.\"\n\nSince the Army's Third Infantry Regiment, \"The Old Guard,\" is the president's escort, I knew we'd have a very large role in his funeral. We had no idea what arrangements had been\u2014or would be\u2014made, but we knew we'd be involved it.\n\nI had a lot more esteem for the role of accompanying the riderless horse than most did. And I thought it should be done well because it obviously affected the family.\n\nYou see, the family is riding right behind the caisson and the horse. They are looking at this horse throughout the funeral procession. It's important to give a good appearance\u2014just in front of the deceased President's family.\n\nI was the first one who started spit-shining the boots in the stirrups. Previously, others would just brush shine them. I thought, \"People are going to be looking at these boots; they ought to have their best appearance.\"\n\nBlack Jack, that misbehaving horse, knew he was doing wrong, and I was furious with him. But I was told that some of the TV commentators talked about the horse as a symbol of the deceased president.\n\nBut Black Jack had one bad trait that always bothered me: a runny nose. He would wipe it on my back, or he would blow it on me. And I'd come in from a funeral at the cemetery with horse snot all down my right side. So I started before every funeral to take out my handkerchief and swab out his nostrils. Not during the funeral or out in public, of course, but before the funeral started.\n\nI was mission-oriented; this is my job. I wanted to do it the very best I could. Emotions are for later. On the flyover at the gravesite service, when Air Force One flew over alone, I allowed myself to get choked up then. After that, it was just exhaustion.\n\nThe next day, we were back to our regular routine. People were still dying, people needed to be buried.\n\nThere were things I would have done differently. I would have asked on the second day if I could put a martingale on him. That's a strap from the bit of the bridle to the girth around the horse's bridle that limits how high a horse can raise its head. Black Jack was raising his head as high as he could. The martingale would have taken a lot of strain off my arm.\n\nI also would have asked if I could ride across the Potomac River in a vehicle on the second day, rather than walk him to save energy. I walked from the Fort Meyer stables to the Capitol, then to the White House, then to St. Matthew's Cathedral, then back across the river to Arlington National Cemetery. At the time, I thought, \"I'll walk with him. I'll talk with him. I'll make sure he was settled down and OK.\"\n\nI learned later the horse's behavior was not a fluke. He was just going back to his early days. I got a call a few years ago from a man who said, as a soldier, he was the first to walk Black Jack. He said when Black Jack got to Fort Myer, where the stables are located, this horse was wild. He said they had to work with the horse for six months before Black Jack could be calm enough to be used in a funeral. And, even then, he would dance around a lot.\n\nAt an Old Guard reunion, one gentleman older than me said he had walked Black Jack one time. His instructions were: \"The horse knows its job: get behind the caisson, follow the caisson, but don't get too close to the caisson, or he'll eat all the felt off the back.\" But Tom failed to mind the other end, and Black Jack once kicked in the door of a car.\n\nI received about one hundred cards and letters after the funeral. They were all complimentary. People wrote me about what they thought of my performance and what it did for them. And I received an Army commendation medal for my participation in the president's funeral.\n\nAs a nation, we were sad and angry. Most people alive then, if you ask them where they were when they heard the news of Kennedy's assassination, they can tell you. To this day.\n\nIt was a traumatic experience for people old enough to understand it. Whether you voted for Kennedy, or you didn't vote for him. Or weren't old enough to vote. The nation was in shock and that misbehaving horse gave them something to focus on. A military funeral is very somber. But Black Jack gave the public some emotional relief.\n Robert Dellwo\n\nAttorney and county Democratic chairman\n\n\"Bobby and Ted were 'doers,'... and were expected to take care of their older, prominent brother... 'the boys,' as we called them, were always busy attending to details.\"\n\nIN 1960, I was the Democratic Party Chairman for Spokane County, Washington, as well as the county chair for the Kennedy for President Campaign. It was at this time that I first met Jack Kennedy. He came to Spokane to give the keynote speech at the Jefferson-Jackson Day banquet at the Davenport Hotel. He arrived early that day and we spent the day together.\n\nFirst, he drove with me and members of my committee to Whitworth College, where he spoke in the campus auditorium. It was here that I first realized what an attraction he was. The auditorium was packed and everyone cheering. Then we hastened back to Spokane for an event at Gonzaga University gymnasium. I remember walking up and seeing a line of people waiting to get in.\n\nOf course, that evening was the big speech. The Davenport Hotel ballroom was packed and after his remarks, the reception line was long\u2014about four people wide and stretching for what appeared to be at least one hundred yards. I was bird-dogging it, to make sure everything was going well.\n\nKennedy started at the head of the line and was greeting and shaking hands with everyone.\n\nAfter a while, he took me aside and said, \"Robert, I've got a problem. I've got to be back in Boston tomorrow morning.\"\n\nSo I took him through the hotel's kitchen and out a rear exit where I had a car parked. We were driven out to the airport and he got into his private plane. We had him there within fifteen or twenty minutes. I almost expected him to toot the horn as he flew over. All those people still in line at the hotel thought they'd be shaking Jack's hand by the time they got to front of the line.\n\nAfter he was elected president, I made many trips to Washington, both as a representative of the Amateur Athletic Union and as a representative of several Indian tribes. I would often go by the White House, and if he was not available, would just slip a note to his secretary, a note of greetings.\n\nBobby and Ted were \"doers,\" from my perspective, and were expected to take care of their older, prominent brother. They showed Jack respect and dignity. Jack had a dignity about him, and \"the boys,\" as we called them, were always busy attending to details.\n\nOn November 22, 1963, I was driving east of Spokane to a meeting of one of my legal clients\u2014a local water district board. I stopped at a restaurant, which had rooms in the back, where the meeting was to take place. As I was walking through the restaurant, someone, in an insipid, antagonistic voice said, \"Did you hear about the death of Jack Kennedy?\" And another person replied, \"Yes, I heard about it and we're rid of the guy now.\"\n\nI was shocked and got back into my car immediately and drove to the county courthouse. There were tears in my eyes. I walked into the courtroom of the presiding judge, who happened to be Ralph Foley, the father of the future Speaker of the House Tom Foley. Judge Foley, who had aspirations to be a federal judge, was informing people in his courtroom that the president had been assassinated. He asked everyone to sit side-by-side in a kind of semicircle and he led a discussion about what people thought would be the impact of the president's death on the nation.\n\nAt one point, he called on me. I couldn't speak; I just cried.\n Phyllis Elkins\n\nWaitress at Jim's Steak and Spaghetti House, Huntington, West Virginia\n\n\"He looked up at me, and he said, 'Do you mind having your picture made with me?'... And when I said, 'My pleasure,' he just busted out laughing because we talk kind of South-like here in West Virginia.\"\n\nIT was in the morning, and it was really good weather. It was a nice day. He came in and sat in my station. There were three of them. Senator Kennedy, Congressman Hechler, and the other gentleman, David Fox, who has passed away. They were having coffee.\n\nHe looked up at me, and he said, \"Do you mind having your picture made with me?\"\n\nI was pouring coffee, and I said, \"My pleasure.\"\n\nAnd when I said, \"My pleasure,\" he just busted out laughing because we talk kind of South-like here in West Virginia. And that's when they snapped the picture.\n\nHe didn't walk around to meet people. He just sat at the table and talked to the other two gentlemen. They only had coffee. And I don't remember if he gave me a tip.\n\nHe was such a gentleman. He was wonderful. And good looking! He stayed about an hour. He had a great smile, he really did.\n\nOn November 22, I was at work at Jimmie's. Someone ran into the restaurant and told us that President Kennedy had been shot. And I just bawled. I cried and I cried. And I still cry when I think about it.\n\nJimmie closed the restaurant for the weekend. She had a television for us. And she let us\u2014the employees\u2014stay there. We didn't do anything. We just sat at the counter and watched. The whole weekend. And it broke our hearts. And I'm crying right now. It was horrible.\n\nI will never forget him. I felt close to him. I don't know why. But I did\u2014and I still do. He had a big impact on me.\n\nThat picture has been up there\u2014in the restaurant\u2014forever. The customers who come in know that was me in the photo, even though sometimes I was working the grill, not handling tables. I was really good on the grill.\n\nI wish he hadn't been assassinated, because I think he would have really done this country, this world, great. He was the only president I really loved. I really cared about him. He's always been close to my heart. And always will be.\n Richard Gaudreau\n\nMilitary pallbearer\n\n\"Under my breath, I was saying to the cardinal, 'Come on, just get it over with!' He took the holy water and he said some words in Latin and then he blessed it. Finally, he finished, after six or seven minutes. We, the casket team, all drew from within and said, 'We'll get through this.' And we did.\"\n\nON November 22, I was stunned initially to hear over the radio that the president had been shot and possibility killed. Upon notifying the master sergeant, I was instructed to conduct a recall of all USAF Honor Guard personnel. All leaves were cancelled. I later learned the president's body was being flown back to Andrews and was instructed to assemble a team of Air Force pallbearers and to proceed to Andrews and stand by.\n\nAt Andrews, I was informed that the Joint Service Casket Team, composed of representatives from all the armed forces, would not handle the president's casket. The Secret Service personnel would do that. I felt that was odd and that we should have taken it from the lift truck. Air Force One arrived at about six o'clock, and I wound up helping place the casket into the ambulance. Like many Americans watching this live on television, the most vivid memory I have is seeing the bloodstains and other matter on Mrs. Kennedy's dress and stockings.\n\nThen I flew with the other members of the casket team by helicopter from Andrews to Bethesda Naval Hospital. There was some confusion regarding the arrival of a hearse at our location only to have it speed away without reason. After some ten or fifteen minutes waiting, the ambulance with Mrs. Kennedy and the president's body arrived. We were instructed at that time that Brigadier General Godfrey McHugh, the president's Air Force aide, would be helping our casket team to carry the casket. I thought, \"Why is this?\" We could handle this without his help.\n\nLater we were told to proceed to the hospital loading dock to bring in a new casket and place it in the autopsy room and to take the old casket and place on the loading dock. This was done because the first casket was damaged loading it onto Air Force One in Dallas. I later learned the old casket was dumped somewhere in the Atlantic Ocean.\n\nAfter the autopsy, we placed the president's casket into a hearse, and were directed to the vehicles that would transport us (the casket team) to the White House. Upon arrival at the White House, we formed up and proceeded to carry the casket inside the East Room where it was placed on the catafalque.\n\nLater, the casket team was expanded to eight members because of the weight of the coffin and the ceremony that would have the team carry it up the steps of the Capitol on Sunday. They didn't want it tilted coming down or going up, so we practiced carrying a casket at the Tomb of the Unknown Solider for six or seven hours. We used a \"dummy casket\" weighing only about four hundred pounds. The president's weighed more than seven hundred pounds. As a result, we had a guard at the Tomb of the Unknown Soldier lie in the casket and carried him.\n\nAll of the ceremonies went well except at the end of the service at St. Matthew's Cathedral. Cardinal Cushing wanted to bless the casket. The casket team came down a set of stairs. The flag was raised in front, and we stood at attention holding this very heavy casket. Under my breath, I was saying to the cardinal, \"Come on, just get it over with!\" He took the holy water and he said some words in Latin and then he blessed it. Finally, he finished, after six or seven minutes. That may not sound like a long time, but it really was. Now we had to carry the president's body to the middle of the street, raise the casket up again, place it onto the horse-drawn caisson. We, the casket team, all drew from within and said, \"We'll get through this.\" And we did. We were physically exhausted.\n\nWhat is John Kennedy's legacy? That is so hard to put into words. I would have liked to see him finish what he started, that is to complete the four years and be re-elected to a second term. The one big item that stands out in my mind was his commitment to place a man on the moon. It happened six years after his death. It was such an achievement for the United States.\n Mike Gefroh\n\nLittle Leaguer who caught a ceremonial first pitch\n\n\"The Kennedys were a catalyst for changing the American way of life for the next generation.\"\n\nIN 1960, I was twelve years old at the opening day of the Riverside Little League in Portland, Oregon. A huge crowd was present, maybe a thousand people. The parking lot was jam packed; everybody's family was there as well as all the media people. Senator John Kennedy was scheduled to throw out the first pitch.\n\nThere was an opening ceremony with a color guard and the national anthem. I was one of three players chosen to participate\u2014my role was to catch the first pitch. I'm not sure why I was selected, maybe because I was a catcher and Catholic. The whole nation\u2014particularly Catholics\u2014was abuzz with Kennedy's candidacy.\n\nHe stood on the mound, threw the pitch, and I caught it. No one explained to me that when I caught it, I was supposed to walk it back out to him. So I threw it back to him. He wasn't ready for it, and the ball flew between him and (the Little League President) as they were shaking hands. Another player brought the ball back to me, and I walked it out to Senator Kennedy. He was very cordial, very dignified, and very well dressed. He said in his Boston accent, \"You've got quite an arm there, young man.\" He autographed the ball, handed it back to me, and shook my hand.\n\nOn November 22, I was a junior at Central Catholic High School in Portland. During third period American history class, the principal came on the PA system. He spoke very slowly, trying to remain composed. He announced that President Kennedy had been shot in Dallas, Texas.\n\nWe all said a prayer and later, after lunch, it was announced that he had died. I was sad, of course, and the whole nation, especially the Catholic community, was distraught. School continued that day. The next day, or maybe the day after, there was a special memorial Mass at school. I remember the nuns crying.\n\nKennedy made me more aware of politics and current events. I remember watching the 1960 Democratic Convention with him running against Lyndon Johnson and Stuart Symington for the nomination. The suspense of the balloting was a terrific initiation into the political process.\n\nWhat's his legacy? He and his family were kind of like American royalty. I remember watching Jackie Kennedy's tour of the White House on TV and the Hyannis Port crowd playing touch football on Thanksgiving. President Kennedy got us to the moon and started many innovative programs, including the Peace Corps. We went from the post-war recovery days of the fifties to something much more progressive.\n\nThe Kennedys were a catalyst for changing the American way of life for the next generation.\n Frank Greer\n\nSenate page, media advisor to U.S. Presidents William Jefferson Clinton and Barack Obama\n\n\"It's like I was Forrest Gump\u2014a witness to history.\"\n\nI was born and raised in Alabama, and in 1963, at age fifteen, I went to Washington as a page for Senator John Sparkman. I was able to attend a few social events at the White House and met John Kennedy and his wife. I was in awe of him, and like so many people of that era, I admired and deeply respected him as a political force.\n\nHis presidential campaign was a major inspiration in my life. In my ninth grade social studies class at Tuscaloosa High School, we followed the campaign, and as an assignment, had to study and document one of the campaigns. Of course, I was the only one who followed Kennedy. Mine was an all-white school and there was another all-black high school.\n\nAs a kid growing up in the segregated South, I admired his courage on civil rights. There were other major figures who were inspiring at that time, such as Dr. Martin Luther King and A. Philip Randolph. John Kennedy also looked outward toward others and toward the world. He wanted America to be respected around the world.\n\nIn June of 1963, I heard there was going to be a big confrontation at the University of Alabama, which was three blocks from my house: The university was going to be integrated. And Governor George Wallace was going to \"stand in the schoolhouse door\" to prevent the integration. So I got on my bicycle, rode down to Foster Auditorium, and crawled under a table. I was ten or fifteen feet away when (Deputy Attorney General) Nicholas Katzenbach arrived to face George Wallace, the little bantam rooster that he was. Then Mr. Katzenbach read a federal court order. The next day, a young black woman, Vivian Malone, was admitted to the University of Alabama. I watched that unfold.\n\nIn August of 1963, Senator John Sparkman's office offered me a page appointment. I took the train\u2014the Southern Crescent\u2014to Washington, got off at Union Station, walked out the front door, and looked up and saw the Capitol. I went to the Senator's office and said, \"I'm Frank Greer and I'm here to go to work.\" A few weeks later, I heard there was going to be a civil rights march on Washington. Having gone through the searing experience of the struggle for civil rights in the South, I walked out of my rooming house, and headed down the Mall in Washington. I was astounded to see other white people, including many from labor unions\u2014the United Auto Workers, the steel workers, and the AFL-CIO, who were supporting civil rights.\n\nI had my page identification, this official-looking badge, and I convinced the security people that they should let me up close to the stage. I talked my way up onto the platform and sat down. Later that day, I was ten feet away when the Rev. King gave his famous \"I Have a Dream\" speech.\n\nIt's like I was Forrest Gump\u2014a witness to history.\n\nOn November 22, 1963, I was on the floor of the Senate working as a page and was called to the cloakroom. I was handed a note and told to give it to Senator Wayne Morse of Oregon. I ran into the chamber with a note saying, \"The president has been shot.\" Senator Morse then walked over and informed Senator Ted Kennedy, who was presiding over the Senate.\n\nOn the following Monday, many of us congressional pages volunteered to assist in the funeral. Several senators and we were bussed to Arlington National Cemetery and watched as the procession crossed the Memorial Bridge and came up the hill. At the gravesite, I stood near Haile Selassie and Charles De Gaulle.\n\nNineteen sixty-three was a searing year and an amazing experience. The events of that year were fundamental in my commitment to pursue a career in politics and non-profit service. I was, in many ways, a witness to history, and John Kennedy was a major positive force in my life and the life of our nation. He truly inspired a generation of young Americans, including Bill Clinton.\n\nI was inspired by John Kennedy's call to public service, his commitment to make politics an honorable profession, and, of course, his famous line in the inaugural address, \"Ask not...\" He was an outward-looking, a forward-looking leader, who inspired the nation and the world. Of course, I also admired his book Profiles in Courage. Regrettably, today, we have few profiles in courage. In this age of partisanship and selfish parochial politics, John Kennedy was unselfish. He inspired us to think about others, to think beyond our own self-interests. He called us to think about what was best for the nation and the people of the world.\n\nJohn Kennedy also transformed American politics. He demonstrated the power of television and, since his death, he has taught generations how to inspire with one's words and one's style.\n Ron Hall\n\nArtist, art dealer, author, coincidental bystander to assassination\n\n\"I said, 'The president's been shot? My goodness, he's right in front of us!'\"\n\nON November 22, 1963, I was eighteen years old, and we (my two friends and I) were sitting on the hood of my car. We were parked right directly below the Texas School Book Depository Building. Earlier we had been stopped by the police. We were not aware that Kennedy was coming downtown. It was just a coincidence\u2014a way of God remaining anonymous. My friends and I were trying to get to TCU (Texas Christian University)\u2014I was a freshman at East Texas State University, and my friends and I had dates with some rich sorority girls at TCU. We were just country boys thinking we were in high cotton because we were going to TCU and had dates for their homecoming. So we were in a hurry to get to TCU.\n\nA policeman stood in front of us as we were going through the downtown on Elm Street and were stopped. Ours was the last car that did not get stopped.\n\nWe asked him, \"What's the problem?\"\n\nThe officer said, \"The presidential motorcade is getting ready to pass. You'll have to wait until they pass before you can continue on.\"\n\nI said, \"OK, we've got a good seat!\"\n\nSo we got out of the car and sat on the hood and waited about five minutes. We were facing west on Elm Street, probably twenty-five or thirty feet away from the Book Depository building. While we were waiting, the crowd began to fill in around our car and in the street around us.\n\nAll of a sudden, we saw the presidential motorcade coming, and just in front of us, the motorcade had to take a hard left turn. We saw the president. We waved. They waved. Everybody waved. As the president passed in front of us and made the turn, we were not interested in anything else except getting into our car and busting through the crowd as soon as it eased up so we could get on to TCU.\n\nSo we jumped into the car to drive away and that's when the crowd started going everywhere. We didn't hear any gunshots, but we had the radio on. At that time, the limousine was probably fifty yards in front of us. The street in front of us was clear, so we drove off.\n\nAt this point, the president's limousine was about two hundred yards in front of us, just passed the grassy knoll area. So we were one of the first cars on the road behind the presidential car. At the time, people were running, but we didn't have a clue as to what was going on. So fifteen or thirty seconds later, we were at the grassy knoll, and a radio announcer said the president's been shot.\n\nI said, \"The president's been shot? My goodness, he's right in front of us!\"\n\nWe just kept chasing them. They turned onto the Stemmons Freeway going north, and we followed them as fast as we could to catch up to them. We probably were no more than three hundred or four hundred yards behind. And only one or two police cars were between us. Ours was the first civilian vehicle behind them. We passed the Trade Mart, where he was supposed to be having lunch. The radio was announcing the president's been shot. We kept following and pulled into Parkland Memorial Hospital, into the parking lot.\n\nThey had pulled up unto a carport. We did not see them take the president out of the car into the hospital. It was kind of eerie. There were not a lot of people in the parking lot. We stayed in the car, and we were about a hundred feet from the limousine at that time. We knew not to get out because we already were treading on thin ice just being there. We had no agenda; we just happened to end up there. Just to be there was incredible enough. We were listening to the radio and sat there for about thirty minutes, thinking he might come back out, and we would get to see him again.\n\nFinally, a Secret Service-looking guy with dark glasses said, \"What are you boys doin'?\"\n\nI said, \"We were just waiting here, maybe to see the president.\"\n\nHe said, \"You all better get on out of here.\"\n\nWe said, \"Yes sir.\"\n\nBy the time we got back on the freeway, they were saying (on the radio) that the president was dead.\n\nI thought, \"Wow. I just cannot believe he's dead.\"\n\nWe went to TCU, and of course, all the homecoming events were cancelled. I left my friends there\u2014they were going to stay at the dorm\u2014and went to my parents' house and stayed there. It was somber and everybody just watched the television. We were glued to the TV.\n\nOn Sunday, as we were going to church, they started saying, \"Oswald's been shot.\" And we thought, \"Oh my gosh.\"\n\nIt was a feeling of sadness and heaviness. You didn't know really what was going to happen in the world. I had never seen a funeral for a president or a fallen leader before: the horse-drawn carriage carrying the coffin, the president's widow in black and wearing a black veil, their son saluting the casket. These images are burned on my visual memory.\n\nWhenever I think of Kennedy, of course, I think of the assassination. I don't really think of him having accomplished so much as a president; he wasn't in office that long. He was the president associated with civil rights and integration. I remember the Cuban Missile Crisis and was worried we night have a nuclear war with the Soviets.\n\nI think of him as the assassinated president and me being one of the last of a few hundred people who saw him alive. Within five seconds after I took my eyes off of him, he was dead.\n Catharine Hamm\n\nTravel writer and editor, Catholic who still grieves\n\n\"I felt my world tilt. She asked us to observe silence. I don't remember whether she told us to pray. In my case, she didn't have to; I knew what our priests and nuns would have said.\"\n\nWEEKS before President Kennedy was killed, my father and mother had taken my middle sister and me to a parade honoring the president in downtown Honolulu. Kennedy, it seemed to me, had a sort of aura around him, and that smile was enough to melt even a nine-year-old's heart.\n\nI might have been predisposed to be enamored of this Irish Catholic president. My grandfather had once likened my mother's countenance to the \"map of Ireland.\" My grandfather died before I was born so I never got to know him, except through her. I knew she adored her father, who was as Irish as Paddy's pig.\n\nAnd I also knew that being Irish and Catholic wasn't necessarily something one flaunted in those days. There were still people who believed there was some kind of papal conspiracy and that Catholics hid guns in preparation for the overthrow. My mother never denied her roots, but it wasn't until Kennedy was elected that she fully embraced her heritage. He was like a gift from heaven to her and her family, and from then on, she instilled a sense of pride about both Irishness and Catholicism in us, her three daughters.\n\nMy dad didn't have quite the same fascination and might actually have voted for Nixon, which would explain why things were a little tense during the campaign. But by 1963, we had moved from Washington, D.C., to Hawaii, a heavily Democratic state. My father, ever the dedicated civil servant, finally acknowledged that a torch had, indeed, passed to a new generation\u2014his. Like Kennedy, he had been \"tempered by war.\" So complicated, so far above my understanding.\n\nAs a kid living in Hawaii, life was one giant playground. We worked hard\u2014fourth grade was demanding at Aikahi Elementary School in Kailua\u2014and we played hard. Few days were not play days and many days were beach days. It doesn't get better than that.\n\nThat day in November started like many others\u2014leave for school, take off shoes out of mom's sight, walk the rest of the way, and scurry into class, where Mrs. Uehara kept us under control with a firm, but not unkind hand. She returned to the classroom that morning tear-stained and told us the president had been shot and was dead.\n\nI felt my world tilt.\n\nShe asked us to observe silence. I don't remember whether she told us to pray. In my case, she didn't have to; I knew what our priests and nuns would have said.\n\nReleased from school, we gathered in the corridors. It was quiet, except for an outburst by a kid named Cindy who said she was glad that Kennedy was dead. We shushed her, and in typical elementary school fashion, told her she was stupid and offered to beat her senseless.\n\nWhen we got home, my mother had the TV on and we watched, as we did for the next three days, the scenes play out over and over. Johnson being sworn in. Jackie Kennedy near him in her bloodstained suit. Lee Harvey Oswald and the improbability of Jack Ruby stepping forward and turning a tragedy into a sickening mystery.\n\nOver and over and over again. We couldn't stop watching.\n\nOur house was quiet for days. It's too simple to say that life was never the same again, even for a kid. Our icon was gone, but his message to us\u2014be proud of who and what you are\u2014has suffused our lives.\n Terri Hazeleur\n\nLed thirty-one girls walking all night to meet the president\n\n\"He asked me how far we had walked. I told him thirty-five miles. He looked down at my bare feet and grinned and said, 'Well, next time, make it fifty.'\"\n\nJOHN Kennedy was very charismatic and youthful. Young people really liked him. My dad was involved in his election campaign. He was a staunch Democrat and chairman of the Democratic Party for Trinity County (California). In fact, he received an invitation to the inaugural ball in 1961, but we didn't have a lot of money. My father was a logger; he worked in the summer, but in the winter he didn't work at all. So, we could never have afforded to go. But it was quite an honor to be invited.\n\nI was fifteen years old and wanted to meet the president (who was coming to dedicate a nearby dam). But I thought the chances were really slim. So, we organized this thirty-five mile hike and thought we would get attention that way to meet him. We started the night before the president's visit; thirty-two girls hiking with flashlights over Buckhorn Mountain. My mom and dad and aunt supervised; they drove their car back and forth and brought us drinks and food. Our flashlights died out in the middle of the night.\n\nWe arrived at about five in the morning. President Kennedy arrived around ten. There were a bunch of chairs laid out for the audience, so we plopped ourselves on the chairs. We were tired. The reporters, who were already there, said, \"Oh you girls, these chairs are for the dignitaries. You get out of here.\"\n\nLater as the president was delivering his speech, someone said, \"Let all these girls in here.\" So we were allowed to sit with the dignitaries. After his speech, he started walking down a fenced-off area to meet members of the crowd. All of a sudden, one of the Secret Service agents said, \"The President of the United States would like to meet Terri Hodgetts.\"\n\nMy dad said, \"That's you!\"\n\nThey shoved me over the other side of the fenced off area\u2014I caught my britches on the fence and ripped them. So I tied a sweatshirt around my waist and walked up to him. He asked me how far we had walked. I told him thirty-five miles.\n\nHe looked down at my bare feet and grinned and said, \"Well, next time, make it fifty.\"\n\nIt was the short conversation that lasted a lifetime.\n\nHe was shorter than I thought he would be. He smelled really good. He had a beautiful smile. He really made an impression on me, and I think some of the other girls may have met him. The photographer just happened to be there and took the picture that went all around the country.\n\nOn November 22, 1963, I was in Spanish class at Hayfork High School. We had about one hundred and sixty in the school and twenty-five in my class. The teacher sent me to the office to get some paper. I got to the office and everyone in there was crying, even the principal. I said, \"What's the matter?\" They told me what had happened, and I started crying too.\n\nI went back to my class and told the teacher and the teacher said, \"That's a terrible joke to play on people, Terri, that's a sick joke.\"\n\nAnd I replied, \"You know me. I don't play sick jokes.\"\n\nThe teacher realized I was telling the truth. He left the classroom and went to the office. School was let out early.\n\nHis death was traumatic. Unbelievable. Back in those days, you really didn't hear about a lot of bad things happening like you do today. I recall the television coverage, especially the drums beating during the funeral.\n\nI always had the impression that John Kennedy wanted the nation to strive to do better. He instilled that in people, like his comment to me, \"Next time make it fifty.\"\n\nHe was a compassionate man. He would be finding ways for people to be proud of themselves. Like me. Some little nobody in some little town no one's ever heard of can do something and be successful. It's an encouraging thought.\n Father Oscar Huber\n\nCatholic priest who administered the Last Rites\n\n\"I will never forget the blank stare in her eyes and the signs of agony on her face.\"\n\n(ON November 22, 1963) on TV at 11:30 am, I saw his arrival at Love Field and heard the enthusiastic welcome given him. Then I walked down to Lemmon and Reagan Streets, about three blocks from Holy Trinity Church, to await the motorcade that would bring the president along the planned route that would end at the Dallas Trade Mart.\n\nThere both sides of the street were lined with people eagerly awaiting the president\u2014there also were the children of Holy Trinity School, their teachers, and lay teachers. Soon the car carrying the members of the presidential party passed by. The president and Mrs. Kennedy were waving and smiling to everyone, and these gestures of goodwill were enthusiastically returned by the happy onlookers along the way. It was a thrilling moment for me as I had never before seen a president of the United States.\n\nI returned to the rectory\u2014ate a brief lunch\u2014and had just finished when Father James N. Thompson, CM, one of my assistants, who had finished lunch previously and was watching TV\u2014came to the rectory and announced that the president had been shot. We went to the recreation room where we heard, over TV, the president had been taken to Parkland Memorial Hospital. This hospital is within the confines of Holy Trinity parish. Within a short time we were on our way to the hospital. Shortly after we left the rectory, a telephone call came from someone at Parkland Hospital saying Mrs. Kennedy was requesting a priest to administer to the spiritual needs of the president.\n\nWithin ten or fifteen minutes we were at the hospital. Father Thompson parked the car while I was escorted by a policeman to an emergency room where I found the fatally wounded president lying on a portable table. He was covered with a sheet that I removed from over his forehead before administering conditionally the Last Rites of the Catholic Church. These Rites are administered conditionally when a priest has no way of knowing the person's mind or whether the soul has yet left the body. In Latin, I said: \"I absolve you from your sins in the name of the Father, and of the Son, and of the Holy Ghost. Amen.\"\n\nDuring these ceremonies, Mrs. Kennedy was standing beside the president. She and others in the emergency room answered the prayers with which they were familiar. Mrs. Kennedy bent over and seemed to kiss the president. Soon after this, followed by Mrs. Kennedy and the others who were present, I walked from the emergency room into the adjoining corridor. Sorrow and consternation bowed the heads of everyone present. The silence that pervaded the corridor was mute evidence that another president of the United States had died at the hands of an assassin.\n\nDuring this most trying ordeal, the perfect composure maintained by Mrs. Kennedy was beyond comprehension. I will never forget the blank stare in her eyes and the signs of agony on her face. I extended my heartfelt sympathy and that of my parishioners to her. In a low tone of voice she thanked me graciously and asked me to pray for the president. I assured her I would do so. Shortly after this Father Thompson and I returned to Holy Trinity rectory.\n\nThe agonizing countenance of Mrs. Kennedy, her clear answer to the prayers recited, her gentle expression of thanks to me for administering the Last Rites and expressing my sympathy, the difficulty I had in controlling my emotions, the expression of shock written on the faces of the Secret Service men, the anxiety expressed by the people who crowded the corridors of Parkland Hospital waiting for a word of hope about the president's condition, the large attendance at Masses offered for the repose of his soul\u2014all these and many more incidents shall never be erased from my memory.\n\nCredit: DeAndreis-Rosati Memorial Archives, DePaul University Archives, Chicago, Illinois, DePaul University\n David W. Knowlton\n\nWest Point graduate with a foreboding premonition\n\n\"He spent ten minutes with us.... He was personable and interested in us. It was a wonderful experience. But I felt such terrible anxiety, and I think he could tell I was stressed.\"\n\nI was editor of the yearbook at West Point and went to Washington to present copies of the yearbook to all the top dignitaries\u2014the Secretary of Defense, the Chairman of the Joint Chiefs of Staff, and others. Normally, the president did not receive the yearbook directly; he had his military aide do it.\n\nBut John Kennedy always wanted to spend time with the cadets. That's how I ended up in the Oval office. Earlier, we had presented yearbooks to former President Eisenhower and former General of the Army Douglas McArthur. Eisenhower was just a regular guy\u2014like a man on the street. McArthur was a bit of a stuffed shirt; I don't think he even opened the book.\n\nI felt quite comfortable though with those two. But not when I met President Kennedy in the Oval Office.\n\nFor the second time in my life, the hackles went up on my neck, and I sensed danger, evil. It had happened to me a year earlier in 1962, when I decided I couldn't get on an airplane that later crashed in Georgia and killed everyone on board. I had terrible anxiety, and was afraid for my life. It cost me seventy-five dollars to change tickets\u2014a lot of money in 1962\u2014and when I arrived at my destination, Los Angeles, I learned that the other plane had crashed.\n\nWhen I met the president, he could tell I was distressed. He spent ten minutes with us. He said it was the best academy yearbook he'd ever seen, and I thanked him. He was personable and interested in us. It was a wonderful experience. But I felt such terrible anxiety, and I think he could tell I was stressed. I verbalized my feelings to my fellow graduate after we left the White House. I felt such a terrible foreboding.\n\nOn November 21, 1963, the night before the assassination, I arrived at my first duty station at Fort Carson, Colorado. On the morning of the 22nd, I was assigned to B Company, Seventh Engineer Battalion. I was the only officer except for the captain, so I was immediately the executive officer, which also meant I was the mess officer. The captain and I went over to the mess to get acquainted. All of a sudden, a soldier burst in and said, \"Sir, they've just shot President Kennedy. We're all on alert.\"\n\nI think John Kennedy was the greatest president we've ever had. But his legacy was unfinished. He was killed before he could have a legacy. I'm an Independent, leading toward conservatism. He had tremendous leadership. I've also admired Truman and Reagan. Of course, we know he was a womanizer. In fact, the White House assistant assigned to show us around Washington was gorgeous. She looked just like Marilyn Monroe. And I don't think she could tell the difference between a telephone and a typewriter.\n James Leavelle\n\nDallas police detective handcuffed to Lee Harvey Oswald\n\n\"If he'd just followed an old detective's advice one time, we would have had him (Oswald) at the county jail, got him to court, got him tried, got him convicted, got the death penalty on him.... But that didn't happen.\"\n\nON November 22, 1963, all the officers in the homicide office of the Dallas Police Department had assignments. We had two officers riding in the motorcade. But my partner was on vacation, so Captain (J. W.) Fritz told me to stick around the office and take care of anything that might come in. They furnished a uniformed patrolman to work with me, but he came up in plain clothes, and he and I went out and arrested a man for armed robbery whom I had a warrant for. On the way back from arresting him, we listened to the dispatcher keeping track of where the motorcade was from Love Field.\n\nAs we pulled in to the basement where Oswald was later shot, they were approaching Houston Street, where they had turned right for about a half block to get on Elm Street in order to make an exit onto the Stemmons (Freeway). We parked and turned the radio off. We went up the elevator from the basement to the third floor office, and I was told by Lt. Ted Wells that they had shot the president.\n\nAnd I said, \"Oh, yeah, yeah.\"\n\nBetween the time it took me to go from the car to the third floor office, the President was shot. I was not on the grassy knoll with ten thousand other people who told me they were there.\n\n(Later that day) they had Oswald in an interrogation room\u2014just a desk and a couple of chairs. No phone or anything, so there are no distractions. No one was with him. I went in and started talking to him about the shooting of Officer (J. D.) Tippit. At that time, I did not have a clue that he would be a suspect in the presidential shooting. There was nothing to indicate at that time that he was involved in that. He was two or three miles away from that scene (at the Texas School Book Depository Building). He denied shooting Officer Tippit.\n\nHe said, \"I didn't shoot anybody.\"\n\nI didn't think too much about it when he told me that, but a day or so later, I got to thinking about that: \"I didn't shoot anybody.\" Because I've worked through other officers' murders, suspects usually deny their involvement by saying, \"I didn't shoot the cop,\" or \"I didn't shoot the policemen.\"\n\nBut he didn't say that; he knew we were going to rap him for the president's shooting later, and he's getting his denial in to begin with, that he \"didn't shoot anybody.\"\n\nCaptain Fritz came back (to station headquarters) from the Book Depository building. They had found the rifle. They had found the window where he (Oswald) had shot from, and they picked up the empty hulls (shells) from underneath the window. And before he (Captain Fritz) left the building, he asked Mr. (Roy) Truly (warehouse manager) to do a head count on all of his employees. Oswald was the only one missing, and he hadn't had permission to leave. So the captain said to Truly, \"Give me that man's address.\" He got it and he sent people out there, but the address was wrong; Oswald had given Truly a bad address. When the captain walked in (to the Dallas Police headquarters), he started sending five or six detectives in different directions to look for Oswald.\n\nSomebody told him, \"Cap, the man Leavelle's talking to has got a name similar to that.\" So the captain came over, opened the door, and asked me his (Oswald's) name. I said, \"Lee Oswald.\" He looked at Oswald and asked, \"Where do you work?\" Oswald replied, \"The Texas School Book Depository Building.\" And the captain said, \"You're the man I want to talk to.\"\n\nSo I lost my prisoner and never questioned him again about any of the shootings because Cap had him on the presidential murder.\n\nThe first thing I regret is that the president was shot. And even worse that it happened here in my city. We got condolences from many other cities, telling us, \"It could have happened just as easily in our city when he was visiting here.\" I have become good friends with Clint Hill, the Secret Service agent who jumped up on the back of the car. He took it really bad that the president got shot.\n\nI've told him, \"Clint, you don't need to feel bad about it because you could have been handcuffed to him and he still could have been shot. I'm a perfect example of that. I handcuffed myself to the prisoner because we had so many threats against him.\"\n\nThe captain and I talked it over, and I said, \"I'll handcuff myself so if they take him, they've got to take me. And I'm not going to go peacefully.\"\n\nAnd that's how come I was handcuffed to him (Oswald).\n\nIf I hadn't had any more information than I had that day, I would not change anything I did (about the transfer of Oswald). But, if I had an inkling something like that might be happening, I would have done what I wanted to do and what I asked the Chief (Dallas Police Chief Jesse Curry) to do about an hour earlier. I asked him to let us take him out on the first floor.\n\nBeing it was a Sunday, there was nobody there. During the week the city courts are on that floor, and the elevator stops on that floor. They take the prisoners, drunks, and ne'er-do-wells down to the city court to be sentenced. But being a Sunday, there was no one there. We could have taken him out on that first floor, put him in a car on Main Street, and been at the county jail before anyone knew we had left the city jail with him.\n\nBut the chief said, \"Leavelle, I have given my word that they (members of the news media) can film the transfer. I want them to know that we didn't abuse him or mistreat him in any way. And the best way for them to know that is to let them film it, so we're going to take him out there (through the basement garage) and they can film it.\"\n\nThat's something else I've thought about years later. If he'd just followed an old detective's advice one time, we would have had him at the county jail, got him to court, got him tried, got him convicted, got the death penalty on him. And who knows? He could still be down there with appeals pending the way courts are handling things nowadays. But that didn't happen.\n Priscilla Johnson McMillan\n\nKnew both John Kennedy and Lee Harvey Oswald\n\n\"That afternoon, in Harvard Square, someone told me that President Kennedy had been murdered in Dallas.... I felt strangled. 'My God,' I said, 'I know that boy!'\"\n\nI first met John Kennedy in May, 1953, soon after he was sworn in for his first term in the U.S. Senate. A couple of months earlier I had finished work at Harvard on a Master's degree in Russian Studies and was fairly fluent in Russian. I went to Washington and applied for jobs at the offices of several congressmen and senators.\n\nA friend who worked for a senator on Capitol Hill mentioned that the new senator from Massachusetts was hiring, so I left my resume at his office, too. But when, after a couple of weeks, I had not found anything, I took a job translating Russian political articles for the Current Digest of the Soviet Press at Columbia University in New York.\n\nBefore the job was due to begin, I received an unexpected call from Senator Kennedy's office. I was told that the senator wanted me to work for him on a short-term project in Washington but wished to meet me first. When I went to see him at his office on the Hill, the first thing I noticed was how thin he was, so thin, in fact, that he looked concave. He apologized for not having a full-time position to offer me, and I said not to worry, I had accepted a job translating Russian, to start in a few weeks.\n\nWhat Senator Kennedy wanted was for me to do research on a subject he evidently had a special interest in: Should the U.S. pressure France, a country to which we gave financial assistance, to abandon its war in Indochina? I was to work for him for two weeks. I would not be on his office payroll; I was to be paid directly by his father. I would receive seventy-five dollars a week and work out of a small office in the Library of Congress Annex. He wanted his first Senate speech to be on foreign policy, and my research was to serve as its basis. The senator did not tell me why he was interested in Indochina. He did not mention that he had already been there two years before, that he knew quite a bit about the place, and that he held strong views about the French and their colonies. Nor did I ask his views, which I was not to learn for many, many years.\n\nOn two successive Friday afternoons in May, I briefed Senator Kennedy on why we should use our clout with the French to make them leave Indochina. After that, my work was done. I went to New York and my new job translating Russian newspapers. Over the next couple of years, I heard from time to time from the senator. He took me out for breakfast once after he had visited his doctor. Another time, after lunch in New York, he said he needed to see his tailor. The tailor's business was just down the street, but he insisted on taking a cab. When he got out of the cab, he quickly stepped into the tailor's shop, and I was left to pay the driver.\n\nI paid my first visit to the Soviet Union during the winter of 1955-56. Two years later, I got a job in Moscow as a rookie reporter for the North American Newspaper Alliance. It was a chance to learn about the country and improve my fluency in the language, and I struggled to extend my visas and stay as long as I could. In November of 1959, a consul at the American embassy in Moscow told me about a twenty-year-old Marine who wanted to defect to the Soviet Union. He was staying at the same hotel I was, the Hotel Metropole, but was angry for some reason and refused to speak with anyone at the embassy. Maybe he would speak to me since I was a woman.\n\nThe Marine's name was Lee Harvey Oswald.\n\nA bit to my surprise, he agreed to an interview. That evening we sat in my room for several hours while he told me his story. He seemed lonely and very, very young\u2014lost in a situation he did not understand. I felt sorry for him. A day or so later, I filed a story. Here is an excerpt:\n\nWith his suit of charcoal gray flannel, dark tie, and tan cashmere sweater, Lee looks, and sounds, like Joe College with a slight Southern drawl. But his life hasn't been that of a typical college boy.... Even though Russian officials have warned him Soviet citizenship is not easy to obtain, Lee already refers to the Soviet government as \"my government.\" \"But,\" says Lee, \"even if I am not accepted, on no account will I go back to the United States. I shall remain here, if necessary, as a resident alien.\"\n\nThe article appeared in newspapers back in the U.S., but I did not see Lee Oswald or hear of him again\u2014until November 22, 1963.\n\nThat afternoon, in Harvard Square, someone told me that President Kennedy had been murdered in Dallas. Later I ran into a friend and asked whether there was a suspect. \"Yes,\" she said, \"Someone named Lee H... Harvey something\"\u2014she could not remember the name\u2014had been arrested.\n\nI felt strangled. \"My God,\" I said, \"I know that boy!\"\n\nThat evening and the next day, because of my interview with Oswald and my article about him, FBI agents came to question me. I spoke with them, of course, and the following year testified before the Warren Commission. In the summer of 1964, in Texas, I interviewed Lee's widow, Marina, for many, many hours in Russian. My conversations with her and interviews with acquaintances of the Oswalds, along with essays and letters by Lee that were published in the Warren Commission Report, led to my book, Marina and Lee, which appeared in 1977.\n\nAs for the unfinished story of John Kennedy, I sometimes wonder whether the harsh edges of the Cold War might have been softened earlier, as early as the 1960s, had Kennedy still been President. And there is the question of Vietnam. In 1953, Kennedy hired me to do research on Indochina because he felt that the French should abandon their stake there. A decade later, in 1963 and afterward, would he, as president, have stood up to the pressure from Congress and the generals to raise our stake in that very same Vietnam?\n Ann Owens\n\nInspired by \"Ask Not...\" to join the Peace Corps\n\n\"There was a wonderful naivet\u00e9 about it. The idea that you could go to another country and show a side of America that we really do care. That we really are nice people. We're here to help you.\"\n\nI was in Peace Corps training in the summer of 1962 in Washington, D.C. at Georgetown University. Much to the surprise and delight of all of us, we were invited at the very end of our training to the White House to meet President Kennedy. There were two hundred of us being trained to go to Ethiopia. We were the first group to go to Ethiopia. President Kennedy spoke to us on the South Lawn of the White House on August 8, 1962.\n\nThen he went along the line and shook people's hands. Seeing and hearing him reconfirmed my great passion for him as our president. I was thrilled to be able to meet him. I was one of the people who got to shake his hand. He was very, very special. Very charismatic. Young, good looking. Strong.\n\nI was twenty-four years old. I had taught third grade for two years and lived at home with my mom. It was a good life. I had a nice boyfriend.\n\nBut when he gave his inaugural speech, the challenge to Americans, \"Ask not what your country can do for you. Ask what you can do for your country,\" it just really struck a chord with me. It appealed to my sense of service, and he called us to a higher calling to do something outside yourself, outside your country. It resonated with me. I had two years of experience. I was ready to do something different.\n\nThere was a wonderful naivet\u00e9 about it. The idea that you could go to another country and show a side of America that we really do care. That we really are nice people. We're here to help you. Of course, with the Peace Corps, they invite you. You have no link to foreign policy. And it's not like you're going as a missionary.\n\nI was with a group of Peace Corps volunteers teaching English and science and history in Tigray, a province north of the capital, Addis Ababa. On November 22, 1963, a doctor in the Peace Corps came into a house where I was visiting. He had a shortwave radio and was looking very gaunt. And he told us that Kennedy had been shot. And we didn't believe it. He was really ticked. He said, \"How can you not believe me? Would I come in and say this if it weren't true?\"\n\nKennedy was a hero in the best sense of the word. I looked up to him. He had fresh, new ideas. A new outlook on the world. That maybe we could get this right this time. We can go to the moon.\n\nWe can go to other countries and make a difference. We can help people out. He had new, profound thoughts on how we could get along in the world. The call to something bigger was a great motivator. He was smart. Very clever. A wonderful way with words. There was a cultural aspect to him that I appreciated. That was important to me. To have a president for whom life was not just politics.\n Ruth Paine\n\nLiving with Marina Oswald at the time of the assassination\n\n\"I have not been able to look in the face the idea that if I led my life differently President Kennedy might be alive. Perhaps most people whose lives touch the matter have a host of 'if only' thoughts. Mine will be with me forever.\"\n\nI was very impressed with him in office, and of course, Mrs. Kennedy. We felt like we were really being drawn into the activities of the country in a way I hadn't felt before. I feel that the things that happened informed him and made him a stronger president. In April of 1961, there was the invasion of the Bay of Pigs. That was a disaster. And I hoped that Kennedy would listen to his own wisdom more than the advice and counsel of his experts in the CIA. He was almost inoculated by that event to refer to his own understanding and wisdom.\n\nOn November 22, I had gotten my children up early to go to the dentist. I turned on the TV set to watch the Kennedys in Fort Worth for a breakfast speech. The Kennedys were a little late for arrival at the breakfast, and the news anchor had to fill time. And he talked about the assassination of President McKinley. There was something in the air and people were worried about his coming.\n\nOn that morning, I left with my kids for the dentist with the TV on because I knew Marina would want to see the motorcade and what was happening. But she wasn't up yet. She was feeding the baby in the night, and she slept when she could. When I got back from the dental appointment, she thanked me for having the TV on. She had watched him arrive at Love Field. Then we sat and watched the TV coverage.\n\nExcerpt from Exhibit 460, \"Testimony of Ruth Paine,\" from The Report of the President's Commission on the Assassination of President Kennedy (The Warren Commission):\n\nWe were on the sofa in the living room watching the television set when they announced that the president had been shot. I translated to her that the president had been wounded in the head. We waited for further word and the lunch I had prepared sat on the table untouched. I lit some plain candles. She asked if that was a way of praying and I told her yes, it was my private way. When the news came that the president was dead, I told her and we wept together. She said what a terrible thing it was for Mrs. Kennedy, how sad for her two children to grow up without a father.\n\nI first heard about Oswald being in custody when police arrived at my door and told me so. You have to understand, everybody was terribly upset. The police were. I was. Marina was. We didn't know what was going on. The police asked if Oswald had a gun, and I said, \"No.\" But I translated for Marina, who said, \"Yes, he did.\" She led them into the garage to show them the blanket that she believed the gun was in. It was not there.\n\nAnd it was at that moment, that I felt, \"Oh... it could have been Lee! He came out the night before. He could have gotten the gun.\"\n\nI was feeling like whatever these policemen and the sheriffs want to know, and if there's any way I can help, I'll help. They needed a little bit better practice in terms of what they took and what they told me they took. But I wanted to help in whatever way I could.\n\nThey put stuff in the trunk of the car and they said, \"We'd better go, you need to make a statement at the police station.\" My son was asleep. I was trying to think what to do in terms of a babysitter. One of the police thought I was taking too long. He grabbed me by the arm and said, \"We've got to get down to the police station right away.\"\n\nThey didn't know who they were dealing with. Another one turned to me and said, \"If you don't hurry up we'll just take the children and have them stay with juvenile.\"\n\nThat was a threat. I don't respond well to threats. So I turned to my daughter and said, \"Lynn, you may come with us.\" I was going to be holding her hand the whole time.\n\nI walked over to the home of a babysitter who regularly came to see if they could come and stay with my son while he was still sleeping and they did come. So we all piled into a couple of cars and went down to the police station.\n\nI've had to say, \"I could have done things differently. But that doesn't mean that the total outcome of Kennedy's life would have been different.\"\n\nExcerpt from Exhibit 460, \"Testimony of Ruth Paine,\" from The Report of the President's Commission on the Assassination of President Kennedy (The Warren Commission):\n\nI have not been able to look in the face the idea that if I led my life differently President Kennedy might be alive. Perhaps most people whose lives touch the matter have a host of \"if only\" thoughts. Mine will be with me forever.\n\nIf only I had known that Lee Oswald had hidden a rifle in my garage. If only I had apprised this man as someone able to do such terrible violence. If only the job that I helped him find hadn't put him in a building along the President's route. If only, quite by accident, I had done or not done a dozen things, the country might have been spared the tragedy, and Marina Oswald, whom I love as if she were a sister, would not have turned into an assassin's wife.\n\nGrief does not go away. That's how grief is. I've been upheld by a sense of the ancestors. There are folks who care who are on the other side of the veil of life and death, who have really helped me.\n\nI was inspired by the grace with which President Kennedy held that office, with his humor, and with his comfort talking to news people. His real interest in international affairs. He represented us so wonderfully. You miss him all the more for knowing how good he was and perhaps could have been in a second term.\n Dr. Ira Seiler\n\nPediatric resident who saved the life of John Kennedy Jr., born not breathing\n\n\"I then grabbed the baby back, reinserted the tube, and for about six minutes, breathed air into the lungs of the baby.\"\n\nI was a young, twenty-nine-year-old, second-year pediatric resident at Georgetown Hospital and had been given the morning off on Thanksgiving, November 25, 1960. I arrived at the hospital at about twelve noon and met Mrs. Kennedy's obstetrician in the hall. He informed me that Mrs. Kennedy was being admitted for an emergency C-section because she had placentia previa which causes blockage of the outlet from the uterus. I asked if he wanted me in the delivery room and was informed that he did. I remember thinking that if there was a problem with the delivery or with the baby, I would probably be blamed since I was the low man on the totem pole.\n\nIn 1960, we had no neonatologist or intensive care physician to assist in the care of a baby in distress. When JFK Jr. was born, the chief of the anesthesia department held him up by the ankles and slapped his buttocks. After doing this for several minutes, the infant became very cyanotic (blue in color), and I told him the baby needed to be intubated.\n\nHe handed the baby to me, and I passed a tube into the trachea of the baby. I then handed the infant back to him to breathe into the baby, since he was the chief of anesthesia. However, he was a bit nervous and knocked the tube out. I then grabbed the baby back, reinserted the tube, and for about six minutes, breathed air into the lungs of the baby. The baby was then transferred to the intensive care nursery, where I cared for him until his discharge on December 9, 1960. The discharge diagnosis was \"Respiratory distress syndrome of the newborn.\" In later years, it has been called \"Hyaline Membrane Disease.\"\n\nAfter I left the delivery room one of the reporters asked me the sex of the baby, and I replied that I could not give out any information and that he would have to speak to Pierre Salinger, President Kennedy's press secretary. It was later written in the paper that a young doctor came out of the delivery room and did not know the sex of the baby. I laughed.\n\nAbout a week later I was on the OB floor when President-elect Kennedy was visiting his wife, and I was introduced to him as the doctor who had done the initial resuscitation on the baby. I shook his hand, and he impressed me as being very sincere and a great man. Shortly afterward, on December 9, 1960, I received a letter from him in a plain envelope. He wrote:\n\nDear Dr. Seiler:\n\nI want to take this opportunity to thank you and the other members of your efficient hospital staff for the many kindnesses shown to Mrs. Kennedy during her stay at the hospital. Your wonderful care has contributed greatly for her and my new son to leave the hospital in the very best of health.\n\nWith every good wish.\n\nSincerely,\n\nJohn F. Kennedy\n\nOn January 10, 1961, I received a registered letter from the Presidential Inaugural Committee addressed to Dr. and Mrs. Seiler containing a card stating, \"President-elect Kennedy has requested the Inaugural Committee to forward to you the enclosed tickets for inaugural events.\" Enclosed were tickets to the ball at the armory, the governors' reception, the distinguished ladies' reception, the inaugural concert, and first row seats opposite the reviewing stand for the parade. My wife and I had been sent tickets to all the functions with the exception of the inauguration.\n\nHowever, three days later on January 13, 1961, I received a special delivery letter from the United State Senate which contained only a card asking me to go to the Old Senate Building to pick up my tickets for the inauguration. When I was given the tickets and was told that we were on the platform, I asked if President-elect Kennedy had actually asked that I be given these tickets. She checked and stated that I was on his private select list and should feel honored, which indeed, I was. We sat on the platform at the Capitol in front of where Kennedy and the dignitaries were assembled.\n\nSeveral weeks later, I was seeing a newborn at Georgetown Hospital and mentioned to one of the nurses how impressed I was in receiving all the inaugural invitations. She informed me that she had written to the inaugural committee that if I had not been there John Kennedy Jr. would have died.\n\nI wrote (Mrs. Kennedy) a letter after President Kennedy was assassinated and received a reply thanking me for my letter. We had just returned from vacation, and my wife was in the hospital with pneumonia. I had a home office at the time and was very depressed when I learned that he had been assassinated.\n\nHis death affected so many people. I thought he would have made a truly great president. His life was cut short; you really don't know how great a president he might have been. That has always bothered me. I considered him a brilliant and honest man. I didn't fully agree with his private life, but that's another story. His legacy is unfinished. He had great potential and made some errors, such as the Bay of Pigs fiasco, but getting the missiles out of Cuba was a great accomplishment. I feel it prevented a possible world war. Recently in the local newspaper, it was mentioned that he had some doubts about the value of the space program since it cost so much to fund. However, the standing of the United States was greatly enhanced as we became the leader in space exploration and science.\n Samuel Stern\n\nStaff member of the Warren Commission\n\n\"I think had he lived and had a second term, the world might have been a lot different in many favorable ways. Particularly I'm thinking of Vietnam. I think he would have found his way through that and dealt with the hawks who were really selling a preposterous notion.\"\n\nI met John Kennedy several times. The first time was at the Democratic Convention in 1956, when he was competing to be (Adlai) Stevenson's vice presidential running mate. (Senator Estes) Kefauver eventually beat him out for that assignment. I was a runner for the platform committee at the convention. I would sit outside and listen to them debate a plank. And when they had something to be printed, I would rush it down to the print shop in the hotel in Chicago, rush upstairs again, and listen some more.\n\nBut, more interestingly, several years after my clerkship with U.S. Supreme Court Chief Justice Earl Warren, in 1961 or '62\u2014or conceivably in early '63\u2014while Kennedy was president, I helped organize a dinner of the Warren clerks for him at the Metropolitan Club here in Washington. One of Warren's clerks, Jon Newman, who is now a Second Circuit judge, was on the White House staff at the time and invited President Kennedy. He was there for the cocktail reception, not the dinner. He was shaking hands with everybody and making small talk and Warren was overjoyed. Warren had been very, very moved by Kennedy and Mrs. Kennedy.\n\nHe was certainly charismatic, but I was less than overwhelmed by his presidency and substantially less than overwhelmed by his brother and his appointment of his brother as attorney general. There was the Bay of Pigs and a lot of on-the-job training, catching up to fill the office. So, at the time, I was reasonably critical, despite being a lifelong Democrat. I was much more a Stevenson Democrat than a Kennedy Democrat. In retrospect, considering everything that's come after him, I'm much more inclined to view his presidency favorably.\n\nOn November 22, I had just come back from lunch at the Metropolitan Club and got a phone call from a client and friend who was the president of Industrias Kaiser Argentina, the Kaiser subsidiary that manufactured Jeeps and Willys cars in Argentina. He was a devout Catholic and very devoted to Kennedy. He had just heard the news over the radio in Buenos Aires. He called to see what had happened and how serious it was. Initially, the news was that the president had been shot, but not that he had been killed. I remember it vividly\u2014everybody stopped everything.\n\nWe were very skeptical of Lyndon Johnson and what that meant to the country. Completely flummoxed by the notion that one insignificant human being took down the most powerful man in the world. It didn't compute.\n\nAt the time, it helped motivate me to become actively involved in the civil rights movement, and particularly, in the Lawyers' Committee for Civil Rights Under Law, led by Lloyd Cutler. When Johnson announced the creation of the Warren Commission, I wrote to the chief justice and said, \"If I can help, let me know.\" He passed that letter on to Howard Willens, a good friend who was then liaison between the Justice Department and the commission. I got a call from Willens and was invited to join the commission staff as an assistant counsel.\n\nJohn Kennedy has become something of a myth. He is larger than life. The most dramatized part of his presidency, aside from the assassination, was the Cuban Missile Crisis. If he did nothing else, that was of gigantic importance to the world, not just the United States. He handled that well. I think had he lived and had a second term, the world might have been a lot different in many favorable ways. Particularly I'm thinking of Vietnam.\n\nI think he would have found his way through that and dealt with the hawks who were really selling a preposterous notion. At the time, I was very opposed to the war. And in retrospect, it becomes even less sensible and supportable. And it skewed this country and the world in major ways. I think he was just catching on how to handle the LeMays (reference to Air Force General Curtis LeMay), and not just the military, but the civilian hawks. His own advisors were very much in favor of going to war and sustaining the war. The seeds of that terrible event were sown in Kennedy's Administration. There was very little in Eisenhower, and it could have been stopped immediately.\n\nPart of the difficulty for Kennedy is the same difficulty that President Obama has had, but in even greater terms. Suddenly you're elevated to this enormously powerful position without much real preparation for it, catching up as it unfolds. You have to respond immediately to huge crises after crises.\n\nHis personal life is part of that legacy. His relationship with his wife and all these beautiful things who were around the White House. There's still a sense of recklessness and a sense of being beyond apprehension and consequences. It was part of his psyche. He probably inspired Clinton. And that's too bad. I think when you take that job you have to stop being a sophomore.\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nEditors\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel\n\nRegulatory Pathways For Smart Grid Development in China\n\nEditors\n\nGert Brunekreeft\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\nTill Luhmann\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\nTobias Menz\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\nSven-Uwe M\u00fcller\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\nPaul Recknagel\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\nISBN 978-3-658-08462-2e-ISBN 978-3-658-08463-9\n\nDOI 10.1007\/978-3-658-08463-9\n\nSpringer Heidelberg Dordrecht London New York\n\nLibrary of Congress Control Number: 2015941028\n\n\u00a9 The Editor(s) (if applicable) and the Author(s) 2015\n\nThe book is published with open access at SpringerLink.\n\nOpen Access This book is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited. This book is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.\n\nThis work is subject to copyright. All commercial rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.\n\nThe use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.\n\nThe publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.\n\nPrinted on acid-free paper\n\nSpringer is part of Springer Science+Business Media (www.springer.com)\n\nProject information\n\nEditors\n\n * Prof. Dr. Gert BRUNEKREEFT, Jacobs University Bremen gGmbH\n\n * Dr. Till LUHMANN, BTC Business Technology Consulting AG\n\n * Dr. Tobias MENZ, BTC Business Technology Consulting AG\n\n * Dr. Sven-Uwe M\u00dcLLER, Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH\n\n * Paul RECKNAGEL, Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH\n\nKey authors\n\n * Prof. Dr. Gert BRUNEKREEFT, Jacobs University Bremen gGmbH\n\n * Marius BUCHMANN, Jacobs University Bremen gGmbH\n\n * Christian D\u00c4NEKAS, OFFIS Institut f\u00fcr Informatik e. V.\n\n * Dr. Xin GUO, BTC Business Technology Consulting AG\n\n * Dr. Till LUHMANN, BTC Business Technology Consulting AG\n\n * Dr. Christoph MAYER, OFFIS Institut f\u00fcr Informatik e. V.\n\n * Dr. Tobias MENZ, BTC Business Technology Consulting AG\n\n * Marcus MERKEL, EWE NETZ GmbH\n\n * Prof. Dr. Christian REHTANZ, ef.Ruhr Forschungs-GmbH\n\nContributing authors\n\n * Andr\u00e9 G\u00d6RING, OFFIS Institut f\u00fcr Informatik e. V.\n\n * Andre HERRMANN, BTC Business Technology Consulting AG\n\n * Ray KODALI, BTC Business Technology Consulting AG\n\n * Paul RECKNAGEL, Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH\n\n * Dr. Michael STADLER, BTC Business Technology Consulting AG\n\n * Dr. Mathias USLAR, OFFIS Institut f\u00fcr Informatik e. V.\n\n * Nils VOGEL, BTC Business Technology Consulting AG\n\nProject steering group in China\n\n * CHEN Tao, National Energy Administration of the P.R. China\n\n * Dr. TONG Guangyi, National Energy Administration of the P.R. China\n\n * GUO Tao, National Energy Administration of the P.R. China\n\n * BU Hongfang, National Energy Administration of the P.R. China\n\nProject group of Chinese experts\n\n * Dr. DONG Rick, China Southern Power Grid Electric Power Research Institute\n\n * Dr. HUANG Han, State Grid Energy Research Institute\n\n * Dr. JIA Bin, ENN Energy Holdings Limited\n\n * Dr. SHI Yaodong, Development Research Center of the State Council\n\n * Prof. Dr. WANG Shouxiang, Tianjin University\n\n * Academician, YU Yixin, Chinese Academy of Engineering, Prof., Tianjin University\n\nProject management\n\n * Dr. Tobias MENZ, BTC Business Technology Consulting AG\n\nProject coordination in China\n\n * Paul RECKNAGEL, Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH\n\nConsortium Members\n\n * BTC Business Technology Consulting AG, Oldenburg\n\n * Jacobs University Bremen gGmbH, Bremen\n\n * OFFIS Institut f\u00fcr Informatik e. V., Oldenburg\n\n * ef.Ruhr Forschungs-GmbH, Dortmund\n\n * EWE NETZ GmbH, Oldenburg\n\nProject Initiation & Funding\n\nThis study was conducted as part of the \"Sino-German Climate Change Programme\" implemented by Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH on behalf of the German Federal Ministry of Economic Cooperation and Development (BMZ).\n\nProject Term\n\nJanuary 2013 to June 2014\n\nNotice\n\nThe content of this study does not reflect the official opinion of the National Energy Administration of the P. R. China. Responsibility for the information and views expressed in the study lies entirely with the author(s). Neither the National Energy Administration of the P. R. China nor any person acting on its behalf may be held responsible for the use which may be made of the information contained therein.\n\nPreface\n\nby Prof. Dr. Dr. Klaus T\u00f6pfer , Executive Director of the Institute for Advanced Sustainability Studies, Potsdam\n\nSource: Schulzendorff\n\nUntil 2025, about 250 million people will move to cities in China, where by then there will be 221 cities with more than 1 million inhabitants. They will be clustered in about 11 regions with more than 60 million people each. In order to ensure energy supply with reduced impact on health and the environment a central element of the Chinese governments' five year plan is developing smart eco-cities. With this, the conservation of energy, water, land and materials, reducing pollution, optimizing the transportation, protecting the environment, and improving building comfort, health and safety should be maximized. This will be a critical moment in the acceleration of industrialization, urbanization and rural development providing great opportunities in many respects, business opportunities for the economic sector and saving potential for millions of households.\n\nAs China increasingly embraces clean energy, with newly set renewable energy targets and energy efficiency, smart grid capabilities are crucial for achieving and driving the low-carbon transition. With increasing shares of power from renewable energy facilities with their specifics of intermittency, the transformation of the demand and supply sides towards more flexibility is vital. This is recognized in the 12 th Five-Year Plan for National Economic and Social Development, where the People's National Congress has set up a goal for the acceleration of smart grid developments. Additionally to balancing the demand and supply sides smart grids can potentially contribute to a reduction in overall need of energy of up to 25 % and reduce the costs of integrating renewable energy into the power system. Doing this will both be useful for the economy, opening up new business models and be a challenge for the regulators, who need to create a supportive framework. Such a regulatory basis should provide incentives and ensure an acceleration of the smart grid development also encouraging competition in order to fuel innovation. Supporting the development of industrial clusters, by creating special industry funds can be one option to fully realize the potential of the economic opportunity of the smart grid development. Considering that China in 2013 for the first time surpassed the spending of the United States on smart grid technologies, accounting for more than a quarter of the worldwide smart grid spending, it is clear that a massive transformation of the country's energy landscape is underway. The potential of being a global leader in the technology development and serving as a role model in smart energy system development however still implies regulatory challenges in order to balance the energy policy goals of reliability, affordability and sustainability \u2013 keeping in mind that the social dimension of energy is central to sustainable energy systems.\n\nSincerely,\n\nKlaus T\u00f6pfer\n\nPreface\n\nby Dr. Werner Brinker, Chairman of the Board of the EWE AG, Oldenburg\n\nEurope is committed to the decarbonization of its economy, driven by the European Union (EU) climate and energy policies on renewable energy, low carbon emissions, energy savings and energy efficiency. The further development of the energy sector is pivotal to meeting these objectives, ensuring the transition towards a more sustainable energy system and driving innovation in the energy sector.\n\nWhile the German energy sector is already well prepared for a successful integration of decentralized power generation from renewable energy sources, Germany is currently heading the necessary legal and regulatory steps to build future-oriented electricity networks, complete the market integration of renewables and ensure at the same time the functioning of electricity markets.\n\nAutomation as well as information and communication technology (ICT) are playing an important role in this context. So called smart grids are deemed to improve the efficiency, reliability, and sustainability of the production and distribution of electricity. They are able to collect, transmit and use information about the behaviors of electricity producers and consumers in an automated fashion by means of automation and ICT.\n\nIn Germany, the transition towards smart grids is driven by a large variety of different institutions and companies interacting on well-functioning markets for electricity and associated products. However, despite all the valuable experiences with regard to the build-up of smart grids, Germany's regulatory framework has not yet been adapted completely to the vision of smart grids. In China, the transition towards smart grids is mainly pushed forward by the government and the politically powerful and vertically integrated grid operators due to the absence of competition in many parts of the energy sector and the non-existence of markets for electricity.\n\nThe present study aims to give regulatory recommendations for the deployment of smart grids in China based on German and international experiences and ongoing discussions. I am convinced that the results can help Chinese policy makers to optimize smart grid regulation in China. I am even more convinced that, based on this study, China and Germany have the unique opportunity to link their strengths, overcome weaknesses and withstand threats to maximize overall benefits for the society during the build-up of smart grids. As Germany is already a very important partner for China in Europe and China is of upmost importance for Germany in Asia, it could be important to align and ensure the right legal and regulatory framework as a precondition for a more sustainable energy sector in China.\n\nWe as the management of the EWE Group are honored that Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH chose experts from our companies to support the further development of smart grids in China. It was also a pleasure for us to welcome a group of Chinese experts in Oldenburg in April 2013.\n\nFinally, I would like to wish you many new insights during the reading of this study.\n\nYour\n\nWerner Brinker\n\nPreface\n\nby Dr. TONG Guangyi, Deputy Director General of the Electricity Department\n\nNational Energy Administration of the P.R. China, Beijing\n\nIn light of today's environmental challenges, in order to meet the requirements of sustainable development, economic restructuring as well as flexible transmission, distribution and utilization of electricity, optimizing the way we operate our power systems by building a smart grid has become an inevitable trend. Therefore, how to rationally and scientifically lay out a roadmap and at the same time design a sound legal and regulatory framework to promote smart grid development is a major question. This study gives important insights on modern smart grid concepts, policy frameworks and avenues for development in Germany and elsewhere providing us with a wealth of ideas and pathways to choose from.\n\nAccording to an old Chinese saying \"advice from others may help one overcome one's shortcomings\". To study the advanced experiences of other countries and learn from their example is of great significance to the development of China's smart grid. This study on \"Regulatory Pathways for Smart Grid Development in China\" includes an analysis of the current state of smart grid development in China and Germany, summarizes the latest discussion on the regulatory environment for smart grids in Germany and compares it with the current situation in China. Based on this analysis and in accordance with China's overall energy development targets, the study provides seven policy recommendations and three regulatory roadmaps to promote smart grid development in China.\n\nI believe this study provides a useful reference for the healthy development, effective guidance and supervision of China's smart grid. We want to take this opportunity and express our appreciation to Deutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH and the team of experts led by the German side. We thank all the contributors for their rigorous and prudent research approach as well as their professionalism, hard work and dedication in the whole process of writing this report.\n\nAt last, I sincerely wish all the best for the future development of smart grids in Germany and China.\n\nSincerely,\n\nTONG Guangyi\n\nPreface\n\nby Bernhard Zymla, Head of the Energy Department\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn\n\nThe primary driver for smart grid development in Germany is the integration of a rising share of electricity generated from variable renewable energies into the power system. The German Federal Government's Energy concept for an environmentally friendly, reliable and affordable energy supply of September 2010 and the Package of Energy Laws enacted in July 2011 contain guidelines and objectives related to Germany's future energy system. The government plans express a commitment to sustainable development and environmental protection by setting a target to reduce CO 2 emissions by 40 % compared to 1990 by 2020 and by 80 to 95 % by 2050.\n\nWith the German energy transition, the Energiewende , the German government has taken ambitious steps and action to tackle the problems related to fossil fuel combustion. To achieve the ambitious targets for reducing greenhouse gas emissions while also gradually phasing out nuclear power until 2022, a rapid expansion of renewable energy is essential. In 2025, 40 to 45 % of gross electricity consumption is to be covered by renewables, with the share planned to rise to 80 % by 2050. An increasing share of intermittent renewables requires a fundamental restructuring of our electricity system, allowing for an effective synchronization of demand with increasingly variable supply, while at the same time maintaining system reliability and stability. To achieve this objective, electricity markets have to be reformed to set the right incentives for an efficient utilization of the system's flexibility resources on the supply and demand side. For this purpose the development of a future-oriented electric power network infrastructure \u2013 or smart grid \u2013 is indispensable.\n\nSmart grids enable real-time communication between electricity suppliers, grid operators and consumers with the help of modern information and communication technologies facilitating an intelligent grid operation as well as the efficient utilization of all power system components through supply- and demand-side management. Thus, it is possible to integrate a high share of variable renewable energies without compromising the reliability of supply, while cutting peak loads and reducing the need for so-called baseload power plants. At the same time, the upgrade of the electric power grid with the help of innovative grid technologies helps to reduce the need for traditional grid expansion avoiding substantial costs. Smart grids lead to the emergence of new business models and new market players together with an increasing spectrum of energy services in the years to come.\n\nThe development of smart grids requires a sound legal and regulatory basis that sets the right incentives and clearly defines the roles of different power system actors, the interaction between them and enables a smooth communication between its components. This study presents an overview of China's and Germany's power system as well as each country's view on smart grid development. Built on this foundation recommendations for the adaptation of the policy and regulatory framework were developed aimed at facilitating the development of smart grids in China in order to allow the integration of a rising share of renewable energy in its power system.\n\nWe trust you will have an interesting and informative read.\n\nSincerely,\n\nBernhard Zymla\n\nAbbreviations\n\n3GThird\n\nThird Generation of Mobile Telecommunications Technology\n\nAbLaV\n\nVerordnung \u00fcber Vereinbarungen zu abschaltbaren Lasten (Deutschland)\/Ordinance on Disconnectable Loads (Germany)\n\nAC\n\nAlternating Current\n\nACATECH\n\nDeutsche Akademie der Technikwissenschaften (Deutschland)\/German Academy of Science and Engineering (Germany)\n\nACER\n\nAgency for the Cooperation of Energy Regulators\n\nAMI\n\nAdvanced Metering Infrastructure\n\nAMS\n\nAsset Management Systems\n\nARegV\n\nAnreizregulierungsverodnung (Deutschland)\/Incentive Regulation Ordinance (Germany)\n\nBDI\n\nBundesverband der Deutschen Industrie (Deutschland)\/Federation of German Industries (Germany)\n\nBImSchG\n\nBundesimmissionsschutzgesetz (Deutschland)\/Federal Pollution Control Act (Germany)\n\nBITKOM\n\nBundesverband Informationswirtschaft, Telekommunikation und neue Medien e. V. (Deutschland)\/Federal Association for Information Technology, Telecommunications and New Media (Germany)\n\nBMAS\n\nBundesministerium f\u00fcr Arbeit und Soziales (Deutschland)\/Federal Ministry of Labor, Social Affairs and Consumer Protection (Germany)\n\nBMUB\n\nBundesministerium f\u00fcr Umwelt, Naturschutz, Bau und Reaktorsicherheit (Deutschland)\/Federal Ministry for the Environment, Nature Conservation, Building and Nuclear Safety (Germany)\n\nBMVI\n\nBundesministerium f\u00fcr Verkehr und digitale Infrastruktur (Deutschland)\/Federal Ministry of Transport and Digital Infrastructure (Germany)\n\nBMWi\n\nBundesministerium f\u00fcr Wirtschaft und Energie (Deutschland)\/Federal Ministry for Economic Affairs and Energy (Germany)\n\nBNetzA\n\nBundesnetzagentur f\u00fcr Elektrizit\u00e4t, Gas, Telekommunikation, Post und Eisenbahnen (Deutschland)\/Federal Network Agency for Electricity, Gas, Telecommunications, Post and Railway (Germany)\n\nCDH\n\nCentral Data Hub\n\nCDMA\n\nCode Division Multiple Access\n\nCEN\n\nEuropean Committee for Standardization\n\nCENELEC\n\nEuropean Committee for Electrotechnical Organization\n\nCHP\n\nCombined Heat and Power Plants\n\nCO2\n\nCarbon Dioxide\n\nCSG\n\nChina Southern Power Grid Company Limited\n\nDAM\n\nData Access Point Manager\n\nDC\n\nDirect Current\n\nDIN\n\nDeutsches Institut f\u00fcr Normung (Deutschland)\/German Institute for Standardization (Germany)\n\nDSL\n\nDigital Subscriber Line\n\nDSLAM\n\nDigital Subscriber Line Access Multiplexer\n\nDSM\n\nDemand Side Management\n\nDSO\n\nDistribution System Operator(s)\n\nEC\n\nEuropean Commission\n\nEEG\n\nErneuerbare-Energien-Gesetz (Deutschland)\/Renewable Energy Act (Germany)\n\nEEX\n\nEuropean Energy Exchange\n\nEnLAG\n\nEnergieleitungsausbaugesetz (Deutschland)\/Energy Network Development Act (Germany)\n\nENTSO-E\n\nEuropean Network of Transmission System Operators for Electricity\n\nEnWG\n\nEnergiewirtschaftsgesetz (Deutschland)\/Energy Industry Act (Germany)\n\nETSI\n\nEuropean Telecommunications Standards Institute\n\nEU\n\nEuropean Union\n\nFACTS\n\nFlexible Control and AC Transmission Systems\n\nFEG\n\nFuture Energy Grid\n\nFINSENY\n\nFuture Internet for Smart Energy\n\nFNN\n\nForum Netztechnik\/Netzbetrieb im Verband der Elektrotechnik Elektronik Informationstechnik e. V. (Deutschland)\/Forum Network Technology\/Network Operation in the Association of the Electrical, Electronic and Information Technology (Germany)\n\nGDP\n\nGross Domestic Product\n\nGW\n\nGigawatt\n\nGWh\n\nGigawatt Hour\n\nGSM\n\nGlobal System for Mobile Communications\n\nGPKE\n\nGesch\u00e4ftsprozesse zur Belieferung von Kunden mit Elektrizit\u00e4t (Deutschland)\/Business Processes for Delivery of Electricity to Customers (Germany)\n\nGPRS\n\nGeneral Packet Radio Service\n\nGWAC\n\nGridWise Architecture Council\n\nGWB\n\nGesetz gegen Wettbewerbsbeschr\u00e4nkungen (Deutschland)\/Act Against Restraints of Competition (Germany)\n\nHV\n\nHigh Voltage\n\nHVDC\n\nHigh Voltage Direct Current\n\nICT\n\nInformation and Communication Technology\n\nIEC\n\nInternational Electrotechnical Commission\n\nIEEE\n\nInstitute of Electrical and Electronics Engineers\n\nISO\n\nInternational Organization for Standardization\n\nITO\n\nIndependent Transmission Operator\n\nITU\n\nInternational Telecommunication Union\n\nkV\n\nKilovolt\n\nkVA\n\nKilovolt Ampere\n\nkWh\n\nKilowatt Hour\n\nLV\n\nLow Voltage\n\nLTE\n\nLong Term Evolution\n\nMessZV\n\nMesszugangsverordnung (Deutschland)\/Metering Access Ordinance (Germany)\n\nMOF\n\nMinistry of Finance of the People's Republic of China\n\nMV\n\nMedium Voltage\n\nMWh\n\nMegawatt Hour\n\nNABEG\n\nNetzausbaubeschleunigungsgesetz \u00dcbertragungsnetz (Deutschland)\/Grid Expansion Acceleration Act for Transmission Networks (Germany)\n\nNDP\n\nNetwork Development Plan\n\nNDRC\n\nNational Development and Reform Commission of the People's Republic of China\n\nNEA\n\nNational Energy Administration of the People's Republic of China\n\nNOx\n\nNitrogen Oxides\n\nnTPA\n\nNegotiated Third Party Access\n\nOECD\n\nOrganisation for Economic Co-operation and Development\n\nOTC\n\nOver-The-Counter\n\nPAS\n\nPublicly Available Specification\n\nPLC\n\nPower-Line Communications\n\nPV\n\nPhotovoltaic\n\nR&D\n\nResearch and Development\n\nRES\n\nRenewable Energy Sources\n\nRES-E\n\nElectricity generated by means of RES\n\nROE\n\nReturn on Equity\n\nRPI\n\nRetail Price Index\n\nrTPA\n\nRegulated Third Party Access\n\nSCADA\n\nSupervisory Control and Data Acquisition\n\nSDO\n\nStandards Developing Organizations\n\nSERC\n\nState Electricity Regulatory Commission of the People's Republic of China\n\nSGAM\n\nSmart Grid Architecture Model\n\nSGCC\n\nState Grid Corporation of China\n\nSG-CG\n\nSmart Grid Coordination Group\n\nSGIS\n\nSmart Grid Information Security\n\nSGTF\n\nSmart Grids Task Force\n\nSO2\n\nSulfur Dioxide\n\nSOE\n\nState-Owned Enterprise(s)\n\nSSM\n\nSupply Side Management\n\nStromStG\n\nStromsteuergesetz (Deutschland)\/Electricity Tax Act (Germany)\n\nTSO\n\nTransmission System Operator(s)\n\nTWh\n\nTerawatt Hour\n\nUHV\n\nUltra High Voltage\n\nUMTS\n\nUniversal Mobile Telecommunications System\n\nUN\n\nUnited Nations\n\nUN\/EDIFACT\n\nUnited Nations Electronic Data Interchange for Administration, Commerce and Transport\n\nUSB\n\nUniversal Serial Bus\n\nUSDOD\n\nUnited States Department of Defense\n\nUSDOE\n\nUnited States Department of Energy\n\nVPP\n\nVirtual Power Plant(s)\n\nWAMS\n\nWide Area Monitoring Systems\n\nWG FSS\n\nWorking Group First Set of Standards\n\nWG RA\n\nWorking Group Reference Architecture\n\nWG SGIS\n\nWorking Group Smart Grid Information Security\n\nWG SP\n\nWorking Group Sustainable Processes\n\nWHO\n\nWorld Health Organization\n\nTable of Contents\n\nList of Figuresxxix\n\nList of Tablesxxxi\n\n1 Introduction 1\n\n2 Conceptual framework and background 7\n\n2.\u200b1 The power sector supply chain and regulatory environment of smart grids 8\n\n2.\u200b2 The role of regulation and technological progress for the development of electric power systems 10\n\n2.\u200b3 Smart grids \u2013 promising technological innovations 13\n\n3 China's way from conventional power grids towards smart grids 19\n\n3.\u200b1 Historical perspective 20\n\n3.\u200b2 Today's power system and its most pressing challenges 22\n\n3.\u200b2.\u200b1 Power generation 22\n\n3.\u200b2.\u200b2 Power consumption 25\n\n3.\u200b2.\u200b3 Power logistics 26\n\n3.\u200b3 Smart grid development in China 27\n\n3.\u200b3.\u200b1 Motivation for smart grids in China 27\n\n3.\u200b3.\u200b2 China's technological view of the smart grid 28\n\n3.\u200b3.\u200b3 This study's view on smart grids in China 30\n\n3.\u200b4 The regulation of China's electric power system 31\n\n3.\u200b4.\u200b1 Policy setting and fundamental institutions 31\n\n3.\u200b4.\u200b2 Market structure 34\n\n3.\u200b4.\u200b3 Market design and RES integration 35\n\n3.\u200b4.\u200b4 Development of infrastructure and network regulation 37\n\n3.\u200b4.\u200b5 Coordination of generation and consumption 38\n\n3.\u200b4.\u200b6 The role of information and communication 40\n\n4 Germany's way from conventional power grids towards smart grids 45\n\n4.\u200b1 Historical perspective 46\n\n4.\u200b2 Today's power system and its most pressing challenges 49\n\n4.\u200b2.\u200b1 Power generation 49\n\n4.\u200b2.\u200b2 Power consumption 50\n\n4.\u200b2.\u200b3 Power logistics 50\n\n4.\u200b3 Smart grid development in Germany 52\n\n4.\u200b3.\u200b1 Motivation for smart grids in Germany 52\n\n4.\u200b3.\u200b2 Germany's technological view of the smart grid 53\n\n4.\u200b4 The regulation of Germany's electric power system 57\n\n4.\u200b4.\u200b1 Policy setting and fundamental institutions 57\n\n4.\u200b4.\u200b2 Market structure 61\n\n4.\u200b4.\u200b3 Market design and RES integration 66\n\n4.\u200b4.\u200b4 Development of infrastructure and network regulation 70\n\n4.\u200b4.\u200b5 Coordination of generation and consumption 71\n\n4.\u200b4.\u200b6 The role of information and communication 72\n\n5 Recommended approaches for smart grid development in China 79\n\n5.\u200b1 Define a long-term strategy for the electricity sector and establish an independent and powerful regulator 81\n\n5.\u200b1.\u200b1 Background 81\n\n5.\u200b1.\u200b2 International practice 83\n\n5.\u200b1.\u200b3 Recommended approach for China 84\n\n5.\u200b2 Create level playing fields for access to power system infrastructure and information 85\n\n5.\u200b2.\u200b1 Background 85\n\n5.\u200b2.\u200b2 International practice 87\n\n5.\u200b2.\u200b3 Recommended approach for China 90\n\n5.\u200b3 Introduce network regulation for efficient investment incentives for electricity grid expansion and upgrade 90\n\n5.\u200b3.\u200b1 Background 91\n\n5.\u200b3.\u200b2 International practice 92\n\n5.\u200b3.\u200b3 Recommended approach for China 93\n\n5.\u200b4 Coordinate network expansion planning for electricity grid expansion and upgrade 94\n\n5.\u200b4.\u200b1 Background 94\n\n5.\u200b4.\u200b2 International practice 95\n\n5.\u200b4.\u200b3 Recommended approach for China 97\n\n5.\u200b5 Improve grid integration of RES 98\n\n5.\u200b5.\u200b1 Background 99\n\n5.\u200b5.\u200b2 International practice 100\n\n5.\u200b5.\u200b3 Recommended approach for China 103\n\n5.\u200b6 Optimize the balancing of electricity generation and consumption 104\n\n5.\u200b6.\u200b1 Background 104\n\n5.\u200b6.\u200b2 International practice 106\n\n5.\u200b6.\u200b3 Recommended approach for China 107\n\n5.\u200b7 Facilitate the development of a unified view of smart grids 109\n\n5.\u200b7.\u200b1 Background 109\n\n5.\u200b7.\u200b2 International practice 110\n\n5.\u200b7.\u200b3 Recommended approach for China 113\n\n6 Regulatory pathways for smart grid development in China 119\n\n6.\u200b1 Government targets for China's future electric power system 120\n\n6.\u200b2 Underlying scenarios 122\n\n6.\u200b3 Drawing the roadmaps 122\n\n6.\u200b3.\u200b1 Reliability\/\u200bSecurity of Supply scenario 123\n\n6.\u200b3.\u200b2 Ecological Sustainability scenario 128\n\n6.\u200b3.\u200b3 Affordability\/\u200bCompetition scenario 132\n\n6.\u200b4 Discussion of the three roadmaps 136\n\nBackmatter139\n\nAppendix A \u2013 Tables and Figures140\n\nAppendix B \u2013 Bottom-up view on China's technological smart grid vision146\n\nAppendix C \u2013 Integration levels of China's power system components in 2012 and 2010150\n\nAppendix D \u2013 Germany's smart grid vision according to the study Future Energy Grid154\n\nAppendix E \u2013 Extracts from specific laws157\n\nAppendix F \u2013 Further results from the European Mandate M\/161\n\nAppendix D \u2013 Germany's smart grid vision according to the study Future Energy Grid154\n\nList of Figures\n\nFig. Summary 1 Overview of this study's recommendations xvii\n\nFig. Summary 2 Overview of proposed implementation sequences for all three scenarios xxi\n\n Fig.\u200b 1.\u200b1 Structure of the study 4\n\n Fig.\u200b 2.\u200b1 Electric power sector supply chain model 9\n\n Fig.\u200b 2.\u200b2 Energy policy triangle 12\n\n Fig.\u200b 2.\u200b3 Development of smarter power systems 14\n\n Fig.\u200b 3.\u200b1 China's power generation capacities from 1980 to 2012 22\n\n Fig.\u200b 3.\u200b2 Standard coal consumption (2002\u20132012) 23\n\n Fig.\u200b 3.\u200b3 Installed capacities of renewable energies (2005\u20132012) 24\n\n Fig.\u200b 3.\u200b4 Energy intensity per unit of GDP (1991\u20132011) 25\n\n Fig.\u200b 3.\u200b5 Annual overall smart grid revenue of main smart grid technologies in China 29\n\n Fig.\u200b 3.\u200b6 Key Actors of China's Power Sector Governance 33\n\n Fig.\u200b 4.\u200b1 Electricity generation in Germany from 1993 to 2013 in TWh 48\n\n Fig.\u200b 4.\u200b2 Composition of the RES generation mix from 1993 to 2013 49\n\n Fig.\u200b 4.\u200b3 Abstract smart grid system model regarding the application of ICT within three distinct layers 56\n\n Fig.\u200b 4.\u200b4 Technology areas regarding ICT aspects of smart grid implementation in Germany 57\n\n Fig.\u200b 4.\u200b5 Long-term targets for Germany's energy sector 58\n\n Fig.\u200b 4.\u200b6 Responsibilities of BNetzA and of the Federal Cartel Office 59\n\n Fig.\u200b 4.\u200b7 BNetzA organization chart 60\n\n Fig.\u200b 4.\u200b8 Number of companies active in the German energy sector 62\n\n Fig.\u200b 4.\u200b9 Examples of established and new market actors in smart grids in Germany 63\n\n Fig.\u200b 4.\u200b10 Electricity wholesale markets in Germany 66\n\n Fig.\u200b 4.\u200b11 Effects of RES supply on the wholesale electricity prices 68\n\n Fig.\u200b 4.\u200b12 Development of the electricity price for private households in Germany 70\n\n Fig.\u200b 5.\u200b1 Process for the network development plan in Germany 96\n\n Fig.\u200b 5.\u200b2 Structure of the Future Oriented Energy Grids Platform 97\n\n Fig.\u200b 5.\u200b3 Organizational structure of M\/490 SG-CG 111\n\n Fig.\u200b 5.\u200b4 SGAM \u2013 The Smart Grid Architecture Model 112\n\n Fig.\u200b 6.\u200b1 Scenarios and their associations with energy policy goals 123\n\n Fig.\u200b 6.\u200b2 Priority of recommendations in the Reliability\/Security of Supply scenario 124\n\n Fig.\u200b 6.\u200b3 Starting points, timeslots of main activities, and dependencies between main aspects of recommendations in the scenario focusing on reliability and security of supply 127\n\n Fig.\u200b 6.\u200b4 Priority of recommendations in the Ecological Sustainability scenario 128\n\n Fig.\u200b 6.\u200b5 Starting points, timeslots of main activities, and dependencies between main aspects of recommendations in the scenario focusing on ecological sustainability 131\n\n Fig.\u200b 6.\u200b6 Priority of recommendations in the Affordability\/Competition scenario 132\n\n Fig.\u200b 6.\u200b7 Starting points, timeslots of main activities, and dependencies between main aspects of recommendations in the scenario focusing on competition and innovation 135\n\n Fig.\u200b 6.\u200b8 Overview of proposed implementation sequences for all three scenarios 137\n\nFig. A.1 Germany's power grids in 2012144 144\n\nFig. C.1 Power system components and market elements in the Chinese power system model 151\n\nFig. C.2 System integration levels in China in 2012 152\n\nFig. C.3 System integration levels in China 2020 153\n\nFig. F.1 Process to apply use cases as the basis for a standardization gap analysis under consideration of functional and security-related requirements 161\n\nFig. F.2 Overview on the smart grid security assessment proposed by WG SGIS162 162\n\nList of Tables\n\n Table 4.\u200b1 Electricity consumption in Germany in 2013 50\n\n Table 4.\u200b2 Demand side management potential according to German studies and sector 51\n\n Table 4.\u200b3 Frequently used measures to maintain supply security in the presence of RES 52\n\nTable A.1 Electricity consumption in China in 2011140 140\n\nTable A.2 Circuit length of transmission lines with 35-kV and above and installed capacity of transformers by the end of 2010 140\n\nTable A.3 Reliability rate of power supply for users in cities at the level of 1000-KV during the 11 th Five-Year Plan (2006\u20132010) 141\n\nTable A.4 Administrative regime of the power sector in China 141\n\nTable A.5 Regional and provincial grid operators in China 142\n\nTable A.6 Share of current application of ICT and challenges structured by voltage levels 143\n\nTable C.1 Definition of system integration levels of power system components151 151\n\nTable C.2 Definition of the prevalence of market elements 152\n\u00a9 The Author(s) 2015\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel (eds.)Regulatory Pathways For Smart Grid Development in China10.1007\/978-3-658-08463-9_1\n\n# 1. Introduction\n\nGert Brunekreeft1 , Marius Buchmann1, Christian D\u00e4nekas4, Xin Guo2, Christoph Mayer4, Marcus Merkel5, Christian Rehtanz6, Andr\u00e9 G\u00f6ring4, Andre Herrmann2, Ray Kodali2, Michael Stadler2, Mathias Uslar4, Nils Vogel2, Till Luhmann2 , Tobias Menz2 , Sven-Uwe M\u00fcller3 and Paul Recknagel3\n\n(1)\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\n(2)\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\n(3)\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\n(4)\n\nOFFIS Institut f\u00fcr Informatik e.V., Oldenburg, Germany\n\n(5)\n\nEWE NETZ GmbH, Oldenburg, Germany\n\n(6)\n\nef.Ruhr Forschungs-GmbH, Dortmund, Germany\n\nGert Brunekreeft\n\nEmail: g.brunekreeft@jacobs-university.de\n\nTill Luhmann\n\nEmail: Till.Luhmann@btc-ag.com\n\nTobias Menz (Corresponding author)\n\nEmail: Tobias.Menz@btc-ag.com\n\nSven-Uwe M\u00fcller\n\nEmail: sven-uwe.mueller@giz.de\n\nPaul Recknagel\n\nEmail: paul.recknagel@giz.de\n\nSmart grids \u2013 an innovative solution for upgrading the conventional power grid infrastructure\n\nAround the world, power grid operators have to cope with several technological challenges ranging from aging grid infrastructures and fluctuating load levels to the integration of intermittent1 renewable energy sources (RES). These challenges tend to result in supply security problems. The traditional way to deal with such problems is to invest heavily in the conventional (primary) grid infrastructure by installing components like new cables, overhead lines, transformers with higher capacities, additional switch panels, or converter stations.2\n\nSeveral modern technologies have been or are currently being developed which are aimed at reducing investments in the traditional grid infrastructure by increasing operational efficiency, making it possible to fully utilize the capacity of the existing power grid infrastructure and leveraging flexibilities in electricity generation and consumption.\n\nExamples of such innovations designed to upgrade electric power grids3 are modern information and communication technologies as well as advanced power grid components for metering, measurement and control, and automation. Power grids incorporating these new technologies are commonly referred to as smart grids [1].\n\nSmart grid technologies open up many new possibilities in power grid management and control, e. g. the status of distribution grids can be monitored in real-time when sensoring and communication networks are added. With the help of the information gathered, modern automation and control technologies can be employed to supervise and actively control grids increasing their utilization rate and preventing overloads via grid capacity management [1]. Thus, the need for costly grid expansion may be reduced while the stability of the grid and security of supply can be improved [1].\n\nChallenges to China's current electric power system\n\nChina faces tremendous challenges with regard to the development of its electric power system. According to China's 12 th Five-Year Plan for Energy Development, massive investments in power generation and grid capacities have to be made to cope with the steadily rising power consumption [2].\n\nDifferent sources point to a low level of equipment sophistication in parts of the electric power grid in China, especially at the distribution grid level. In some instances, outdated and inefficient distribution transformers are still in operation [3], [4]. This contributes to a considerable number of supply interruptions, despite the fact that significant advancements have been made in recent years. At the same time, requirements of China's quickly modernizing industry with regard to power quality are on the rise and cannot be met at all times [5].\n\nChina has to cope with increasing air pollution problems [6] and is the world's largest emitter of carbon dioxide (CO2) [7]. Its power generation sector, which is mainly relying on coal as an energy resource, is responsible for a large share of CO2 emissions.\n\nGovernment targets for China's future electric power system\n\nIn light of these challenges, the Chinese government issued specific targets for the physical and technological development of the power system up to 2015: power generation capacities are to increase from 970 gigawatts (GW) in 2010 to 1,490 GW in 2015 [2] and they are projected to attain 1,935 GW in 2020 [8]. Also, the share of power generation from RES and nuclear energy is planned to be increased significantly, with an official target of 11.4 % of total primary energy coming from non-fossil sources by 2015 and 15 % by 2020 [9]. In China, power generation from RES is considered a key instrument to reduce the dependency on coal power, stop the deterioration of air quality, and reduce the growth rate of CO2 emissions [9]. The State Council also stipulated that long-distance, inter-regional and inter-provincial transmission of electricity shall be promoted in order to build a nationally integrated backbone grid. In addition, urban and rural distribution grids are planned to be expanded and renovated [2]. Investments in the grid infrastructure are projected to reduce power supply interruptions and to increase power quality. Considering the power consumption side, there are plans to significantly increase the efficiency of power use and to provide access to electric power even in very remote regions of China by 2015 [2], [10].\n\nChina's government also plans to make considerable advancements on the organizational level: a modern energy market system is to be established, market-related reforms in key energy sectors are projected to take place, energy pricing mechanisms are to be improved, and the development towards an internationally competitive environmental and energy industry is to be accelerated [2].\n\nTo support accomplishment of the government targets for China's future electric power system, the government aims at starting to build up smart grids during the next several years. As has been formulated in the 12 th Five-Year Plan for Energy Development, China's government aims at promoting the development of smart grids [2]. Also, the Chinese Ministry of Science and Technology (MOST) released a special plan for technological smart grid improvements in 2012 [11].\n\nBarriers to the implementation of smart grids in China\n\nSome aspects of China's technological and regulatory framework may present obstacles to an effective and efficient smart grid development:\n\n * Overlapping responsibilities between different government institutions affect the government's ability to guide and facilitate the development of smart grids [5] [12].\n\n * Compared to the electricity sectors in other countries, the information and communication technology (ICT) industry, small-sized power generation companies, and other non-established players like prosumers4 are underrepresented in (smart) grid development planning in China.\n\n * Due to rather low electricity prices, the current tariff system may not offer sufficient incentives for saving electricity or for shifting electricity demands according to available generation and grid capacities [12].\n\n * A lack of sufficient incentives for grid integration of RES within the existing regulatory framework persists [13].\n\n * Similar to all other countries aiming at developing smart grids, some of the key smart grid technologies in China are not yet fully developed and some equipment specifications and standards are still inconsistent [3], [4], [14].\n\nIdea of the study\n\nThe aim of the present study is to analyze and discuss regulatory policies supporting the build-up of smart grids in China. The work is based on experiences gathered in Germany and other countries. In the context of the study, the term regulation is not restricted to purely regulatory issues. Rather, government policies such as the promotion of research and development (R&D) or standardization issues are also included.\n\nSmart grids follow an evolutionary pathway and their realization depends on the status quo of the existing grid infrastructure. Therefore, this study contains a detailed description of China's and Germany's electric power systems, their most pressing technological challenges, and their regulatory environments. Based on these descriptions, both countries' specific technological views on smart grids are presented.\n\nRegulatory smart grid pathways designed to meet the specific challenges in China are presented subsequently. The pathways include dedicated recommendations that are based on regulatory best practices from Germany and other countries. The recommendations build upon the current situation in China proposing achievable changes to the regulatory framework and relevant policies to promote smart grid development in China.\n\nThe structure of the study\n\nThe structure of the study is visualized in Fig. 1.1 and briefly outlined below:\n\nFig. 1.1\n\nStructure of the study\n\n * Chapter 2 presents the conceptual framework of the study. The chapter also introduces the so-called energy policy triangle covering the three main energy policy goals reliability, affordability, and sustainability. Fundamental premises highlighting the importance of smart grids and explaining the role of the government in the smart grid development process are presented as well. The chapter also discusses the importance of electric power markets and third parties, i. e. new participants in the value chain of the electric power sector, for smart grids.\n\n * Chapter 3 presents China's electric power system, its recent historical development, its regulation, government targets for China's future electric power system, and the role of smart grids in this context. A clear focus is placed on technological and regulatory challenges for China's electric power system.\n\n * Chapter 4 contains a description of the German situation focusing on lessons learned and sharing the German experience (corresponding to challenges presented in Chap.\u200b 3).\n\n * Chapter 5 presents recommendations designed to meet the specific challenges in China. Where appropriate, the recommendations refer to best practices from Germany.\n\n * Chapter 6 presents three different regulatory pathways (roadmaps) each focusing on a different objective of the energy policy triangle. This offers policy makers an insight of the effects different policy priorities may have on the implementation sequence of the study's recommendations.\n\nOpen Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.\n\nReferences\n\n1.\n\nBundesnetzagentur f\u00fcr Elektrizit\u00e4t, Gas, Telekommunikation, Post und Eisenbahnen (BNetzA), \"Smart Grid and Smart Market \u2013 Summary of the BNetzA Position Paper,\" November 2012. Online]. Available: [http:\/\/\u200bwww.\u200bbundesnetzagentu\u200br.\u200bde\/\u200bSharedDocs\/\u200bDownloads\/\u200bDE\/\u200bSachgebiete\/\u200bEnergie\/\u200bUnternehmen_\u200bInstitutionen\/\u200bNetzzugangUndMes\u200bswesen\/\u200bSmartGridEckpunk\u200btepapier\/\u200bSmartGridPapier_\u200bEN.\u200bpdf?\u200b_\u200b_\u200bblob=\u200bpublicationFile&\u200bv=\u200b3. [Accessed November 7, 2013].\n\n2.\n\nState Council of the People's Republic of China, \"12th Five-Year Plan for Energy Development,\" Guofa, Beijing, 2013.\n\n3.\n\nThe World Bank, \"China \u2013 Power Sector Transformer Efficiency Program Project,\" 2012. Online]. Available: [http:\/\/\u200bdocuments.\u200bworldbank.\u200borg\/\u200bcurated\/\u200ben\/\u200b2012\/\u200b01\/\u200b15641795\/\u200bchina-power-sector-transformer-efficiency-program-project. [Accessed February 4, 2014].\n\n4.\n\nH. Sun and Y. Zhang, \"Research on and Design of Intelligence Distribution Grid System,\" China Rural Water and Hydropower, no. 2, 2012.\n\n5.\n\nF. Kahrl, J. Williams, D. Jianhua and H. Junfeng, \"Challenges to China's Transition to a Low Carbon Electricity System,\" Energy Policy, vol. 39, no. 7, pp. 4032\u20134041, 2011. CrossRef\n\n6.\n\nThe China Greentech Initiative, \"The China Greentech Report 2013 \u2013 China at a Crossroads,\" 2013. Online]. Available: [http:\/\/\u200breport.\u200bchina-greentech.\u200bcom\/\u200b. [Accessed February 4, 2014].\n\n7.\n\nThe World Bank, \"CO2 Emissions (kt),\" The World Bank, 2013. Online]. Available: [http:\/\/\u200bdata.\u200bworldbank.\u200borg\/\u200bindicator\/\u200bEN.\u200bATM.\u200bCO2E.\u200bKT\/\u200bcountries\/\u200b1W?\u200bdisplay=\u200bdefault. [Accessed August 27, 2013].\n\n8.\n\nG. Tong, \"Status Quo of the Smart Grid Development in China and Its Driving Forces,\" National Energy Administration (NEA), Oldenburg, 2013.\n\n9.\n\nNational Energy Administration (NEA), \"Key Information at a Glance \u2013 China 12th Five-Year Plan for Renewable Energy Development,\" China National Renewable Energy Center (CNREC), Beijing, 2012.\n\n10.\n\nState Council Information Office of the People's Republic of China (SCIO), \"China's Energy Policy 2012,\" SCIO, Beijing, 2012.\n\n11.\n\nMinistry of Science and Technology of the People's Republic of China (MOST), \"Special Planning for Industrialization Program of S&T Related to Smart Grid During the 12th Five-Year Plan Period,\" MOST, Beijing, 2012.\n\n12.\n\nX. Qiu and H. Li, \"Energy Regulation and Legislation in China,\" Environmental Law Reporter, no. 7, pp. 10678\u201310693, 2012.\n\n13.\n\nC. Garcia, \"Policies and Institutions for Grid-Connected Renewable Energy: Best Practice and the Case of China,\" Governance: An International Journal of Policy, Governance, and Institutions, vol. 26, no. 1, pp. 119\u2013146, 2012. CrossRef\n\n14.\n\nY. Yu, J. Yang and B. Chen, \"The Smart Grids in China \u2013 A Review,\" Energies, vol. 5, pp. 1321\u20131338, 2012.CrossRef\n\nFootnotes\n\n1\n\nElectricity from an intermittent energy source is not continuously available due to factors outside direct control and cannot be dispatched to meet the demand of a power system. Intermittent RES are tidal power, wind power, or solar power, while biomass, geothermal, and hydro power are dispatchable and non-intermittent RES. Note that the term variable RES is used as a synonym for intermittent RES in the present study.\n\n2\n\nThe term conventional grid infrastructure refers to passive electronic components necessary for establishing an electric connection between electricity generators and consumers [1]. Note that this perspective mainly serves an explanatory purpose, because currently no grid is built or operated exclusively with passive components [1].\n\n3\n\nThe specific term electric power as well as the general term power are both used as a synonym for electricity in this study. Only if the term energy is used, non-electric power forms such as heat or kinetic energies are referred to.\n\n4\n\nProsumers are end consumers of electricity which also generate electricity, for example by means of rooftop photovoltaic installations.\n\u00a9 The Author(s) 2015\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel (eds.)Regulatory Pathways For Smart Grid Development in China10.1007\/978-3-658-08463-9_2\n\n# 2. Conceptual framework and background\n\nGert Brunekreeft1 , Marius Buchmann1, Christian D\u00e4nekas4, Xin Guo2, Christoph Mayer4, Marcus Merkel5, Christian Rehtanz6, Andr\u00e9 G\u00f6ring4, Andre Herrmann2, Ray Kodali2, Michael Stadler2, Mathias Uslar4, Nils Vogel2, Till Luhmann2 , Tobias Menz2 , Sven-Uwe M\u00fcller3 and Paul Recknagel3\n\n(1)\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\n(2)\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\n(3)\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\n(4)\n\nOFFIS Institut f\u00fcr Informatik e.V., Oldenburg, Germany\n\n(5)\n\nEWE NETZ GmbH, Oldenburg, Germany\n\n(6)\n\nef.Ruhr Forschungs-GmbH, Dortmund, Germany\n\nGert Brunekreeft\n\nEmail: g.brunekreeft@jacobs-university.de\n\nTill Luhmann\n\nEmail: Till.Luhmann@btc-ag.com\n\nTobias Menz (Corresponding author)\n\nEmail: Tobias.Menz@btc-ag.com\n\nSven-Uwe M\u00fcller\n\nEmail: sven-uwe.mueller@giz.de\n\nPaul Recknagel\n\nEmail: paul.recknagel@giz.de\n\nChapter at a glance\n\n * The study's conceptual framework is presented.\n\n * This chapter illustrates the role of regulation and technological progress for the development of the electric power system and introduces the energy policy triangle covering the three main energy policy goals reliability, affordability, and sustainability.\n\n * The concept of and the motivation for smart grids are introduced.\n\n * Fundamental premises highlighting the role of the government and new market actors in the smart grid development process are presented.\n\n## 2.1 The power sector supply chain and regulatory environment of smart grids\n\nThe supply chain model of the electric power sector\n\nThe delivery of electricity from generation sources to end consumers involves a multitude of technologies, actors, and processes, especially in the context of the development of an intelligent future-oriented power grid infrastructure. Therefore, it is useful to employ a supply chain model of the electric power sector (or electricity system) to structure the debate on smart grids.\n\nIn this study, a smart grid supply chain model (see Fig. 2.1) is used to structure the discussion on China's and Germany's power systems along with their most pressing problems and challenges as well as their particular approach of how to promote the development of smart grids. The model subdivides all activities\/processes, technologies, and actors of the power sector supply (or value) chain into four basic elements:1\n\nFig. 2.1\n\nElectric power sector supply chain model\n\n * Power Generation: the technical generation of electricity using various kinds of primary energy sources (such as coal, gas, nuclear, hydro, wind, and solar energy) and corresponding ancillary services.2\n\n * Power Logistics: the transmission, distribution, storage, and metering of electricity.\n\n * Power Trade & Retail: purchasing, trading, and selling electricity, as well as retail and billing services for end consumers.\n\n * Power Consumption: the end use of electricity by different end-user segments, e. g. rural and urban households, industrial and commercial consumers.\n\nTo capture the importance of ICT for transforming the traditional power sector into a smart grid, a corresponding fifth element is added to the model:\n\n * Information & Communication: operating, monitoring, and controlling power system components, as well as power system-related information exchange between various market actors with the help of ICT along all four stages of the power sector supply chain.\n\nThe government plays a pivotal role in the smart grid development process (see Sect. 2.3) as a guiding and supervising key player affecting processes and businesses across all five supply chain elements.\n\nThe regulatory environment of smart grids\n\nIn the remainder of this study, six regulatory areas will be used to describe the regulatory environments of China's and Germany's power systems.\n\nThe six regulatory areas are:\n\n * Policy Setting and Fundamental Institutions: Government leadership in form of policies, laws, and regulation is essential to promote smart grid development. To practice such a leadership, a governance structure with clearly assigned roles and responsibilities for the field of (smart) grid regulation is of crucial importance.\n\n * Market Structure: The (market) structure of the electric power sector is characterized by all the companies involved in the various stages of the power sector supply chain from Power Generation and Power Logistics to Power Trade & Retail (vertical market structure), the market composition and competitive conditions at different stages of the supply chain (horizontal market structure), and the roles and responsibilities of the market actors.\n\n * Market Design and RES Integration: Electricity prices (for generation and retail) are of crucial importance with regard to the utilization of different sources of power generation, investment decisions for new generation capacities, power consumption patterns, and investments in power saving technologies. By setting monetary incentives for investments into power generation capacities, generation electricity prices contribute particularly to the long-term coordination of power generation and consumption. At the same time, retail electricity prices have the power to directly affect the patterns of electricity consumption of end users. The formation of electricity prices is heavily affected by the market structure and the underlying market design, the latter of which is generally defined by the government. Feed-in tariffs for RES and associated regulations are also part of the market design because they represent nothing less than prices for electricity generation fixed by the government.\n\n * Development of Infrastructure and Network Regulation: The regulatory domain explains how investments in the (smart) grid infrastructure can be incentivized through power grid or network regulation. These regulatory practices directly impact the stable and affordable operation of the current grid infrastructure and the investments in the future grid infrastructure.\n\n * Coordination of Generation and Consumption: In this regulatory area, government policies for balancing electricity generation and consumption in the short term are described. Many smart grid-related technologies facilitate the balancing of generation and consumption.\n\n * The Role of Information and Communication: Integration of ICT with power system components across the supply chain is a key factor in smart grid development. This regulatory area covers the relevance of ICT companies, standardization issues, cyber security, and funding of smart grid research.\n\n## 2.2 The role of regulation and technological progress for the development of electric power systems\n\nRegulation in the electric power sector\n\nCompetition, where feasible free from government intervention, is generally considered as a very effective way of reducing production costs, improving quality of supply, and increasing product diversity [1], [2]. Yet, competition is not always feasible. Especially where natural monopolies prevail, competition cannot exist and government regulation is necessary [3].\n\nTransmission and distribution grids are such natural monopolies. Having an infrastructure of overlapping power grids from different competitive suppliers would be economically inefficient. Due to economies of scale, a single supplier (the monopolist) is able to offer the services of the electric power grid at lower costs. However, the monopolistic grid operator has a strong incentive to set high prices for its services or to scrimp on quality. Thus, the price and quality of service of the monopolistic grid operator must be regulated to protect grid users against the monopolist's market power [3].3\n\nIt is common to distinguish two related goals of monopoly regulation [3] which are both typically tasks for the regulator, like the Federal Network Agency for Electricity, Gas, Telecommunications, Post and Railway (BNetzA) in Germany:\n\n * Control of revenues and prices of the networks (i. e. network charges)\n\n * Secure access to the monopolistic infrastructure (i. e. the network) to facilitate competition among market actors in the commercial parts of the electricity value chain (e. g. generation, trade and retail).\n\nMoreover, the internalization of so-called externalities is a frequently cited argument in favor of government intervention even in the non-monopolistic parts of the electric power system:\n\n * Negative externalities: the generation of electricity is often connected to emissions of environmental pollutants. Without government intervention, neither producers nor consumers of electricity are sufficiently incentivized to behave in an environmentally friendly manner. Therefore, actions of the government such as pollution control laws, energy taxes, and subsidies for environmentally friendly technologies are used to reign in commercial activities and advance the overall societal objectives like environment protection and sustainable development [1].\n\n * Positive externalities: technological progress critically depends on the creation of new knowledge. After its creation, knowledge can be used over and over by various actors with almost no extra costs. From the perspective of the whole society, companies often do not invest enough in knowledge creation because a significant portion of the benefits related to knowledge creation accrues to outside stakeholders, e. g. other companies, institutions, or citizens. In this light, government actions such as a patent law, a stringent protection of intellectual property rights, or the subsidization of R&D can be employed to incentivize companies to invest in knowledge creation and innovation.\n\nThis study adopts a broad understanding of regulation and does not only focus on network regulation but also upon aspects such as RES integration, innovation policies, and standardization.\n\nA stylized history of power system development\n\nThe historical development of electric power systems dates back to the last decades of the 19th century in the world's most advanced countries of that era. The electrification process in these countries was mostly accomplished by the mid-20th century, when almost the whole population had access to electric power. In less developed regions, the electrification process started somewhat later and progressed at a considerably slower pace. Differences in onset and pace of the electrification process between countries can be explained mainly with technological, economic, political, and regulatory reasons [4].\n\nIn developing their electric power systems, many countries followed a similar stylized pathway. At the beginning of the electrification process, the emergence of new technologies and appliances consuming electric power led to a rapidly increasing demand for electricity. Thus, the main objective was to connect a growing number of consumers as quickly as possible to the electric power grid. The build-up process took place in a rather uncoordinated manner with only little government intervention. Thus, various small-sized companies set up overlapping infrastructures such as small-sized power plants and distribution lines. This uncoordinated set-up phase was often associated with low power quality and reliability as well as high costs. Therefore, governments started to regulate the electricity sector with the intention of making power supply more reliable and affordable. Formally assigning monopoly rights to power generation companies and grid operators created economies of scale, which reduced the overall production costs of electric power generation and distribution. After this assignment of monopoly rights, electricity market regulation became a decisive factor to protect consumers against the market power of the newly installed monopolies.\n\nWhen a certain level of wealth and economic development had been achieved, the awareness of environmental protection increased in many societies.4 Therefore, more emphasis was put on ecological sustainability of the electric power system and many governments introduced environmental protection policies.\n\nThe energy policy triangle\n\nThe schematic history presented above reveals that different objectives and regulations were prioritized at different development stages of the power system. Whereas the provision of widespread and reliable access to electricity was often the initial motivation, affordability of electric power emerged as a second driver of energy policy after the initial build-up phase. Finally, the ecological sustainability of the electric power system has often been formulated as a third important priority.\n\nToday, governments strive to incorporate all three objectives in their power sector regulation. They (1) want to provide a reliable and secure electricity supply for their economies, (2) simultaneously promote economic development by ensuring affordable electricity prices while (3) also achieving ecological sustainability. The three goals reliability, affordability, and sustainability are commonly referred to as the energy policy triangle (see Fig. 2.2).\n\nFig. 2.2\n\nEnergy policy triangle\n\nConflicting nature of the energy policy triangle\n\nThere are inherent conflicts between the goals of the energy policy triangle. A government forcing owners of fossil-fuel power plants to install filter technologies for the sake of improving air quality, for instance, is accepting that costs of power generation increase. Another government using expensive domestic resources for electricity generation instead of lower-priced resources available on world markets accepts higher generation costs for the sake of independent and predictable energy supply.\n\nThe complete and simultaneous achievement of all three policy goals is virtually impossible. Therefore, most countries place a priority on one or sometimes two policy goals, trying to achieve or maintain acceptable levels with regard to the non-prioritized policy goals. In recent years, the Chinese government, for example, prioritized (1) providing a reliable and secure electricity supply and (2) keeping electricity prices for the population at an affordable level. On the other hand, it has accepted high levels of environmental pollution. The German government, in contrast, has been placing a strong emphasis on ecological sustainability since the end of the 20th century. A prominent strategic decision in this context was the build-up of RES generation capacities so that currently approximately one fourth of Germany's electricity comes from RES [5]. The high share of RES in Germany currently leads to increasing challenges with regard to the policy goals of reliability and affordability. However, both conflicts may be mitigated by means of technological progress as well as smart design of policies and regulatory framework:\n\n * In most cases, electricity generated by means of RES, so-called RES-E, is still more expensive than electricity generated in fossil fuel-fired power plants. This difference in generation costs has contributed significantly to climbing end consumer prices for electricity in recent years. Nonetheless, RES-E is often already cheaper than electricity retail prices (known as grid parity) depending on the characteristics of location and end-user segment. The costs of RES-E have decreased considerably during the last few decades and are projected to decrease even more with further technological advancement and economies of scale through worldwide deployment [6]. Someday, the costs of renewably generated electricity will consequently fall below those of electricity generated in conventional power plants. From that time on, investments in RES will increase both sustainability and affordability of the power system provided that the power system and market design allow for a minimizing of grid integration costs.\n\n * With a rising of electric power generated by variable RES, more and more efforts have to be undertaken to guarantee the high stability and reliability of the power system. Below, two examples of temporary RES penetration in Germany, a power system with electricity loads commonly ranging from 40 to 80 GW, are presented: on December 5th 2013, a particularly windy day, roughly 27 GW of feed-in was generated by means of wind turbines in Germany [7]. On March 9th 2014, a very sunny day, photovoltaic (PV) installations alone fed a peak power of almost 22 GW into the grid [7]. Technological progress as well as adaptations to the existing design of the electricity market in Germany are essential to limit the cost of the ongoing transition towards a low carbon economy.\n\n## 2.3 Smart grids \u2013 promising technological innovations\n\nThe concept of smart grids\n\nThe capabilities of conventional power grids to cope with many of today's technological challenges, e. g. the integration of a large share of electricity generated by means of intermittent RES, are limited or can only be maintained with significant investments. In this light, smart grids represent a promising new technological concept. Today, the general understanding regarding the concept of smart grids seems to converge on a global level. A definition of smart grids which has found widespread acclaim in the professional community comes from the International Energy Agency (IEA) [8]:\n\n> An electricity network that uses digital and other advanced technologies to monitor and manage the transport of electricity from all generation sources to meet the varying electricity demands of end users. Smart grids co-ordinate the needs and capabilities of all generators, grid operators, end users and electricity market stakeholders to operate all parts of the system as efficiently as possible, minimizing costs and environmental impacts while maximizing system reliability, resilience and stability.\n\nAnother explanation of the smart grid concept comes from the International Electrotechnical Commission (IEC) [9]:\n\n> The general understanding is that the Smart Grid is the concept of modernizing the electric grid. The Smart Grid comprises everything related to the electric system in between any point of generation and any point of consumption. Through the addition of Smart Grid technologies the grid becomes more flexible, interactive and is able to provide real time feedback.\n> \n> It is an electricity network that can intelligently integrate the actions of all users connected to it \u2013 generators, consumers and those that do both \u2013 in order to efficiently deliver sustainable, economic and secure electricity supplies.\n\nIrrespective of the common general understanding of smart grids, the specific vision of smart grids differs substantially from case to case and from country to country. Technologies that are included in one smart grid concept are not necessarily included in another.\n\nMotivation for smart grids\n\nSmart grids are generally expected to cope with the most pressing problems and challenges of many power systems more effectively and efficiently than conventional grid technologies. Smart grids primarily aim at delivering enhanced levels of reliability and security of supply, facilitating the balancing of electricity generation and consumption, increasing the utilization rate of power system components, and reducing investments in the conventional (primary) grid infrastructure. Smart grid technologies can be used, for example, to\n\n * monitor and control electric power grids more effectively,\n\n * facilitate the grid connection and operation of generators of all sizes and technologies,\n\n * help to integrate electricity generated from intermittent RES,\n\n * allow industrial, commercial, and residential consumers to play a part in optimizing the operation of the system by adjusting electricity consumption behavior according to supply, and\n\n * provide consumers with greater information on their electricity consumption enabling electricity conservation [9].\n\nThe evolutionary character of smart grids\n\nSmart grids are not built from scratch. Rather, they evolve from conventional power grids which are upgraded with different kinds of innovative technologies and components. These technologies and components will be deployed at different periods in time depending on commercial attractiveness, compatibility with existing technologies, regulatory policies, and investment frameworks. The evolutionary pathway of smart grid technologies is similar to that of other innovations: after major R&D efforts, smart grid technologies are developed further in specific demonstration projects, tested in model cities or regions, and finally deployed on a larger scale.\n\nToday's smart grid technology landscape is highly diverse with some technologies reaching already high levels of maturity and others still in a development process. Smart grid-related technologies with high levels of maturity are, for example, power plant communication and control modules, grid control systems, and advanced metering infrastructures (AMI). On the other hand, technologies such as wide area monitoring and control, distribution management, or electric vehicle charging infrastructure require more progress before they are mature enough for large-scale deployment [8].\n\nFigure 2.3 visualizes a stylized evolutionary pathway of smart grids: in the past (left part of the illustration), electric power was mainly generated in thermal power plants. The status of high, medium, and low voltage grids was not remotely monitored. System operators were able to supervise substations at different voltage levels via rudimentary telecontrol or by means of manual patrol teams.\n\nFig. 2.3\n\nDevelopment of smarter power systems (\u00a9 OECD\/IEA [8])\n\nIn the present (middle part of the illustration), large wind farms complement thermal power plants feeding in electricity at the high voltage (HV) level. The status of transmission grids and the associated substations is remotely monitored. The resulting status information is received in the transmission control center and can be used to monitor and control the grid. Medium voltage (MV) and low voltage (LV) grids, however, are not yet equipped with sophisticated monitoring technologies. The distribution control center only monitors the associated substations via telecontrol or by means of manual patrol teams. For the first time, small-sized wind farms and PV installations are being connected to distribution grids.\n\nIn the future (right part of the illustration), power grids at all voltage levels are expected to be equipped with sophisticated monitoring and control technologies. Status information, e. g. on electricity generation and consumption, will also be available for distribution grids. RES, energy storage installations, and electric vehicles will become an integral part of the future electric power system and will be able to exchange status information with the grid.\n\nThe role of governments in the smart grid development process\n\nGiven the evolutionary character of smart grids and the low maturity levels of some smart grid technologies, both a systematic guidance and specific government support policies are essential to promote the development of smart grids. Exemplary government policies in this context are:\n\n * Setting up a government long-term strategy for the development of the future electricity system to reduce investment risks for potential investors in R&D and deployment of smart grid technologies.\n\n * Establishing network regulation for transmission and distribution grid operators which incentivizes them to invest in smart grid technologies.\n\n * Promoting a non-discriminatory and technology-neutral management of and access to power system data to enable companies to develop innovative business models making use of this data.\n\n * Partial government funding of R&D to increase the maturity level of smart grid technologies.\n\n * Supporting smart grid-related standardization, defining technical guidelines and regulations to reduce the costs for deployment and integration of different smart grid components by ensuring interoperability.\n\n * Promoting the exchange and collaboration between different government organizations, power sector companies, academia, associations, and other relevant actors to create a common understanding of smart grids.\n\nThe role of new market actors in the smart grid development process\n\nIn Germany and many other countries, one important trend associated with the migration towards smart grids and the transition to an electricity system relying to an increasing extent on RES-E is a rise in the number of market actors in the electricity sector. New market actors (third parties or non-incumbents) can expand the horizontal market structure by entering into competition with established companies (incumbents) with regard to existing products or services (e. g. operators of renewable energy plants or new power retail companies). New market actors can also expand the vertical market structure by being pioneers offering products and\/or services in new market sectors or niches (e. g. energy service companies or virtual power plant operators5) or in using existing knowledge and\/or infrastructure from other sectors in an innovative way upgrading products and services in the electric power sector (e. g. ICT companies).\n\nNew market actors offer innovative products and services that were not supplied by established market actors before. In a smart grid context, non-incumbents create new business models and offer new products and services by making use of available power system information and infrastructure in an innovative way. To give some tangible examples for the power sector innovations that may be driven by third parties, Sect.\u200b 4.\u200b4.\u200b2 highlights how non-incumbents in Germany create new business models in the smart grid development process. Note that the German evidence elaborated in Sect.\u200b 4.\u200b4.\u200b2 also demonstrates that new market actors might emerge due to government policies aimed explicitly at the promotion of competition.\n\nIn expanding the horizontal market structure, third parties contribute to an increased level of competition. Higher competition levels are usually considered to drive innovation, enable greater efficiency in the allocation of resources, increase cost efficiency of power sector enterprises with regard to their operations and investments, contribute to lower retail price levels, and stimulate innovation resulting in a higher variety of competitively priced products and services offered to the customer:\n\n * With regard to the electric power sector, IEA analyzed the market liberalization process in several of its member countries and came to the conclusion that higher competition levels in the electric power sector contributed to a reduction of electricity prices for industrial consumers [10].\n\n * With respect to the telecommunication sector, various empirical studies (including data from industrial and developing countries) found that high competition levels significantly increased the overall sector performance and led to more landline and mobile telephone connections, lower tariffs, and more connection capacity [11], [12], [13]\n\n * In the general economic literature, the relationship between competition and innovation has been examined in many theoretical and empirical studies. It is usually found that there is a kind of optimal market structure for R&D-spending, innovation and diffusion. Neither very low nor very high competition levels are an innovation-friendly environment. A competitive market with a limited number of companies seems to promote innovation best [14].\n\nEconomic efficiency of smart grids\n\nThe economic evaluation of smart grids is still ongoing. However, early evidence suggests that smart grids are an attractive solution compared to conventional grid expansion measures. A recent study summarizes the results of twelve smart grid cost-benefit analyses published between 2004 and 2012 in the United States, the UK, the Netherlands, Denmark, and the Czech Republic [15]. This overview shows that benefits of the investigated smart grid concepts outweigh costs in ten of twelve cases (with costs outweighing benefits in Denmark and the Czech Republic). Only four of these studies compared their results to costs and benefits of conventional grid expansion measures (increasing the grid capacity by traditional network expansion). These studies come to the conclusion that, from an economic point of view, investments in smart grid technologies are preferable to conventional expansion measures.\n\n\u25b6 Key findings\n\n### Key findings\n\n * Governments strive to provide a reliable, affordable, and sustainable electricity supply for their economies. Yet, there are inherent conflicts between these three energy policy goals.\n\n * Smart grids can be seen as an advanced way of operating grids supported by a broad set of new technologies in the areas of communication, metering, control, and automation. Smart grid-related operation concepts, technologies, and components can be used to upgrade the existing grid infrastructure and offer many new possibilities in grid management and control. Smart grids will be deployed at different periods in time depending on requirements of the particular power system, compatibility with existing technologies, regulatory policies, and investment frameworks.\n\n * Recent evidence has shown that investments in smart grid technologies are economically preferable to conventional grid expansion measures. Moreover, smart grids represent a promising concept that could increase reliability and security of supply, facilitate the balancing of electricity generation and consumption, ease the grid integration of electricity generated from variable RES, and increase the utilization rate of power grid assets.\n\n * Given the evolutionary character of smart grids and the still low maturity levels of some smart grid technologies, both systematic guidance and specific government support policies are essential to promote the development of smart grids.\n\n * New market actors in the electric power sector are of crucial importance to promote innovation in the electric power sector and speed-up the development of smart grids.\n\nOpen Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.\n\nReferences\n\n1.\n\nR. Perman, Y. Ma, J. McGilvray and M. Common, Natural Resource and Environmental Economics, Harlow: Pearson Education Limited, 2003.\n\n2.\n\nP. Krugman and R. Wells, Microeconomics, New York: Palgrave Macmillan, 2012.\n\n3.\n\nS. Stoft, Power System Economics, Piscataway: The Institute of Electrical and Electronics Engineers, 2002. CrossRef\n\n4.\n\nS. C. Bhattacharyya, Energy Economics \u2013 Concepts, Issues, Markets and Governance, Heidelberg: Springer, 2011.\n\n5.\n\nBundesverband der Energie- und Wasserwirtschaft (BDEW), \"Energie-Info Erneuerbare Energien und das EEG: Zahlen, Fakten, Grafiken (2013),\" BDEW, Berlin, 2013.\n\n6.\n\nC. Kost, J. N. Mayer, J. Thomsen, N. Hartmann, C. Senkpiel, S. Phillips, S. Nold, S. Lude and T. Schlegl, \"Stromgestehungskosten Erneuerbare Energien,\" Fraunhofer-Institut f\u00fcr Solare Energiesysteme, Freiburg, 2013.\n\n7.\n\nEuropean Energy Exchange (EEX), \"EEX Transparency in Energy Markets,\" EEX, November 2013. Online]. Available: [http:\/\/\u200bwww.\u200btransparency.\u200beex.\u200bcom\/\u200bde\/\u200b. [Accessed May 15, 2014].\n\n8.\n\nInternational Energy Agency (IEA), \"Technology Roadmap: Smart Grids,\" IEA, Paris, 2011.CrossRef\n\n9.\n\nInternational Electrotechnical Commission (IEC), \"What is a Smart Grid?,\" IEC, November 2013. Online]. Available: [http:\/\/\u200bwww.\u200biec.\u200bch\/\u200bsmartgrid\/\u200bbackground\/\u200bexplained.\u200bhtm. [Accessed November 15, 2013].\n\n10.\n\nInternational Energy Agency (IEA), \"Lessons from Liberalized Electricity Markets,\" IEA, Paris, 2005.\n\n11.\n\nA. J. Ros, \"Does Ownership or Competition Matter? The Effects of Telecommunications Reform on Network Expansion and Efficiency,\" Journal of Regulatory Economics, vol. 15, no. 1, pp. 65\u201392, 1999. CrossRef\n\n12.\n\nS. J. Wallsten, \"An Econometric Analysis of Telecom Competition, Privatization, and Regulation in Africa and Latin America,\" Journal of Industrial Economics, vol. 49, no. 1, pp. 1\u201319, 2001.MathSciNetCrossRef\n\n13.\n\nC. Fink, M. Aaditya and R. Randeep, \"An Assessment of Telecommunications Reform in Developing Countries,\" Information Economics & Policy, vol. 15, no. 4, pp. 443\u2013466, 2003. CrossRef00007-6)\n\n14.\n\nP. Aghion, N. Bloom, R. Blundell, R. Griffith and P. Howitt, \"Competition and Innovation: an Inverted-U Relationship,\" The Quarterly Journal of Economics, vol. 120, no. 2, pp. 701\u2013728, 2005.\n\n15.\n\nC. Mahlstedt, \"Bewertung international vorliegender Kosten-Nutzen-Analysen zur Errichtung eines intelligenten Stromnetzes (Smart Grid),\" Carl von Ossietzky Universit\u00e4t Oldenburg, Oldenburg, 2013.\n\n16.\n\nT. Luhmann and N. Vogel, \"EPM-Referenzmodell \u2013 Grundlage f\u00fcr ein marktgerechtes Leistungsportfolio f\u00fcr die Energiewirtschaft,\" Unpublished Manuscript, Oldenburg, 2010.\n\n17.\n\nE. Hirst and B. Kirby, \"Electric Power Ancillary Services,\" Oak Ridge National Laboratory, Oak Ridge, 1996.CrossRef\n\n18.\n\nS. Dinda, \"Environmental Kuznets Curve Hypothesis: A Survey,\" Ecological Economics, vol. 49, no. 4, pp. 431\u2013455, 2004.CrossRef\n\nFootnotes\n\n1\n\nThe smart grid supply chain model being referred to in this study is based on the so-called Energy Process Management (EPM) model [16].\n\n2\n\nAncillary services are functions performed by the equipment and people generating, controlling, transmitting, and distributing electricity with the intention of supporting basic services of generating capacity, energy supply, and power delivery [17]. Examples are load following, scheduling and dispatch, real-power loss replacement, and voltage control (e. g. by means of reactive power supply).\n\n3\n\nNote that, at the beginning of the electrification process, the power generation sector was also recognized as a natural monopoly: in the absence of power transmission lines, competition between power generation units from different cities or regions was not possible. With the rather small electricity demand in separated cities or regions, one larger power plant was able to provide the electric power at lower costs than many small competitive power plants due to economies of scale. However, the gradual deployment of transmission lines allowed competition between large power plants to evolve in such a manner that power generation is no longer considered to be a natural monopoly [3].\n\n4\n\nThe positive correlation between economic development and environmental preferences has been well documented during the last two decades [18].\n\n5\n\nBoth new market actors are introduced in more detail in Sect.\u200b 4.\u200b4.\u200b2.\n\u00a9 The Author(s) 2015\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel (eds.)Regulatory Pathways For Smart Grid Development in China10.1007\/978-3-658-08463-9_3\n\n# 3. China's way from conventional power grids towards smart grids\n\nGert Brunekreeft1 , Marius Buchmann1, Christian D\u00e4nekas4, Xin Guo2, Christoph Mayer4, Marcus Merkel5, Christian Rehtanz6, Andr\u00e9 G\u00f6ring4, Andre Herrmann2, Ray Kodali2, Michael Stadler2, Mathias Uslar4, Nils Vogel2, Till Luhmann2 , Tobias Menz2 , Sven-Uwe M\u00fcller3 and Paul Recknagel3\n\n(1)\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\n(2)\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\n(3)\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\n(4)\n\nOFFIS Institut f\u00fcr Informatik e.V., Oldenburg, Germany\n\n(5)\n\nEWE NETZ GmbH, Oldenburg, Germany\n\n(6)\n\nef.Ruhr Forschungs-GmbH, Dortmund, Germany\n\nGert Brunekreeft\n\nEmail: g.brunekreeft@jacobs-university.de\n\nTill Luhmann\n\nEmail: Till.Luhmann@btc-ag.com\n\nTobias Menz (Corresponding author)\n\nEmail: Tobias.Menz@btc-ag.com\n\nSven-Uwe M\u00fcller\n\nEmail: sven-uwe.mueller@giz.de\n\nPaul Recknagel\n\nEmail: paul.recknagel@giz.de\n\nChapter at a glance\n\n * The chapter gives an overview of the existing electric power system, the envisaged development of smart grids, and the regulatory environment in China. The main aspects emphasized in this chapter serve as a guideline for the description of the German background in Chap.\u200b 4 and the recommendations and regulatory pathways in Chaps.\u200b 5 and .\n\n * Section 3.1 gives an overview of the more recent development of the power sector to lay a solid foundation for identifying the challenges currently facing the sector.\n\n * Section 3.2 presents an overview of the physical structure of China's current electric power system and discusses in detail the main technological challenges in this context.\n\n * Section 3.3 introduces the Chinese smart grid vision resulting from the specific technological challenges. In this respect, the role of the government and the ICT industry in developing the Chinese smart grid vision is especially emphasized.\n\n * Section 3.4 describes the regulatory framework of the electric power system in China on the basis of the regulatory areas defined in Sect.\u200b 2.\u200b3. A clear focus is thereby on regulatory circumstances representing a barrier to smart grid development in China.\n\n## 3.1 Historical perspective\n\nTowards a reform of the power sector\n\nIn the past 15 years, a series of reforms have greatly improved the efficiency, reliability, and environmental performance of the Chinese power sector. However, significant challenges remain: rapidly rising electricity demand, concerns about power system reliability and energy security, environmental degradation and climate change [1].\n\nHistorically, all stages of China's power sector value chain \u2013 from generation, transmission and distribution to retail \u2013 were owned and operated directly by the central government. By the mid-1980 s, in the wake of China's opening up policy, a first set of reforms allowed new market actors, mostly provincial and local governments, to invest in power generation. This created a boom of so-called independent power producers (IPP), thereby alleviating power shortages. Today, IPP make up more than half of China's power generation assets.\n\nIn 1997, the former Ministry of Electric Power (MoEP) was dissolved and the generation, transmission and distribution assets previously under direct government control were transferred to the newly formed State Electric Power Corporation (SEPC). This marked a significant step towards the separation of market functions fulfilled by state-owned enterprises (SOE) and government regulatory authority.\n\nIn 2002, the State Council of the People's Republic of China promulgated a landmark reform for restructuring the Chinese power sector by separating power generation from grid operation (transmission & distribution). In the course of the reorganization, the vertically integrated SEPC was dismantled and its assets divided into eleven new SOE: two grid operators, five power generation companies, also known as Big Five (China Huaneng Group, China Datang Corporation, China Huadian Group, China Guodian Corporation, China Power Investment Corporation), and four power service companies providing advisory and ancillary services. At the same time, the State Council established a ministerial regulatory authority, the State Electricity Regulatory Commission (SERC), to oversee the developing competitive market structure and further push power sector liberalization and market-based reform. The aim of this major power sector reform was to:\n\n * break-up the power sector monopoly and introduce fair competition (mainly on the generation side) within the framework set by the regulator,\n\n * improve economic efficiency and reduce costs,\n\n * rationalize the electricity tariff system and optimize resource allocation,\n\n * promote the development of the power industry and push ahead nationwide interconnection,\n\n * set up an open, orderly and well- developed power market based upon the principles of separation of governmental oversight and power sector enterprises.\n\nIn retrospect, the reforms are considered a significant step towards the diversification of China's power sector market structure. However, the market-oriented reform of the power sector \u2013 as it had originally been envisaged by the State Council \u2013 stalled after these initial steps. No progress was made in unbundling the grid operators' transmission and distribution assets or in introducing more market-based electricity prices. This lack of progress despite ambitious plans for further reforms may be attributed in part to a lack of power and independent decision-making authority of the newly established regulator SERC, which suffered from very limited jurisdiction, capacity and resources. The National Development and Reform Commission (NDRC) is a successor to the State Planning Commission (SPC), which was renamed to State Development Planning Commission (SDPC) in 1998. In March 2003 the SDPC merged with the State Council Office for Restructuring the Economic System and parts of the State Economic and Trade Commission to form NDRC, which remained the most powerful policymaker for the power sector, retaining competence over electricity pricing and major energy project approval. The emergence of the newly formed State Grid Corporation of China (SGCC, in charge of 80 % of China's electricity grid) as a powerful influence in national energy policies further hampered the stride towards liberalization of the sector. In the most recent government restructuring in March 2013, the SERC was integrated into the National Energy Administration (NEA), an institution responsible for energy planning under the NDRC.\n\nThe trend towards low carbon development\n\nFor a long time, progress in the electrification process together with the provision of a reliable and affordable power supply were the main priorities of China's government in its efforts to promote industrial and economic development. In the last two decades, questions relating to the sustainability of the electric power system have steadily gained increasing importance in China.\n\nEnergy efficiency and environmental protection were first put forward as a prominent policy objective for power sector development in China's general energy law \u2013 the 1995 Electric Power Law [2]. On the one hand, the law aimed at legalizing the status of power generation companies as commercial entities and at establishing the legal basis for private ownership [3]. On the other hand, the law stressed the importance of the environmental sustainability of the development of the power system by stipulating that\n\n> the construction, production, supply and utilization of electric power shall protect the environment according to law, adopt new technologies, minimize discharge of poisonous waste, and prevent pollution and other public hazards. The state encourages and supports electricity generation by using renewable and clean energy resources. [4]\n\nThe shift towards environmental protection was reiterated in the 1998 Energy Conservation Law and the 2006 Renewable Energy Law, which respectively aim at promoting energy efficiency and deploying RES. The Renewable Energy Law has laid a solid legal foundation that has since been followed by a number of key implementation guidelines detailing national renewable energy targets, a mandatory connection and purchase policy, a national feed-in tariff system for wind and solar energy as well as arrangements for cost-sharing and funding of renewable energy incentives.\n\nChina put forward aggressive measures to reduce the energy intensity of its economy: in its 11th (2006\u20132010) Five-Year plan, China set a target of reducing its energy intensity, measured as energy consumption per unit of gross domestic product (GDP), by 20 %. At the world climate conference in Copenhagen in 2009, China complemented this target with an overarching goal to reduce carbon intensity, the measure of carbon dioxide emissions per unit of GDP, by 40 % to 45 % by 2020 compared to 2005 levels. The targets are backed up by comprehensive plans featuring numerous measures to facilitate increased energy efficiency and reduced emissions across different sectors. One of these efforts is the Small Plant Closure Program established in the 11th Five-Year Plan. It focused on closing down small and inefficient power plants and replacing them with larger state-of-the-art facilities. In the period from 2006 to 2010, more than 70 GW of coal-fired power generation capacity was shut down in the context of this program [5]. The program not only targeted power plants, but also facilitated the shut-down and replacement of a large number of outdated small factories of energy-intensive industries, e. g. in the iron and steel and cement sectors. More recently, small coal mines and inefficient pulp and paper mills have been subject to this government policy promoting consolidation and modernization of its heavy industry.\n\nThe Top 1000 Enterprises Energy Saving Program was another measure of the 11th Five-Year Plan to improve industrial energy efficiency by targeting China's largest energy consuming companies, which accounted for almost half of total industrial energy consumption and one third of total primary energy consumption in China. The program reportedly realized total energy savings of 150 million tons of coal equivalent, 50 % more than originally planned [6]. The ambitious targets and extensive efforts to promote energy efficiency and adjust China's energy mix through the deployment of renewable energies are evidence of China's commitment to decouple its economic growth from the growth of energy demand and emissions \u2013 to reduce the carbon intensity of its economy and fight pollution.\n\nAn increasing factor influencing China's energy and industrial policy is the increasing public awareness of environmental pollution and its detrimental effects on the ecosystem and on people's health. In particular, the extreme levels of air pollution in major Chinese cities may be seen as a potential source of social unrest and has thus been given great attention by the Chinese government, especially after the serious pollution events in Beijing in January 2013 saw particulate matter air pollution levels rise to figures exceeding the 24-hour mean level recommended by the World Health Organization (WHO) by more than 30 times.\n\n## 3.2 Today's power system and its most pressing challenges\n\n### 3.2.1 Power generation\n\nCompared to international averages, China's per capita reserves of coal, petroleum, and natural gas are rather low, a situation which in recent years has resulted in an increasing dependence on the import of fossil fuels [7]. In contrast, China is endowed with large renewable energy resources \u2013 estimated at 250 GW and 750 GW for onshore and offshore wind respectively, plus significant solar energy resources [8]. Since the 1980 s, China's power generation capacity has been steadily expanding (see Fig. 3.1). From 2006 to 2012, total installed capacity almost doubled, increasing from 621 GW to 1,147 GW [9].\n\nFig. 3.1\n\nChina's power generation capacities from 1980 to 2012, data from [10]\n\nCoal is currently the primary source of electricity generation in China; in 2012 it accounted for close to 80 % of total electricity generation [9]. This massive reliance on coal for power generation and industry makes China the biggest CO2 emitter worldwide [11]. The second pillar of China's electricity generation mix is hydro power, which has a share of roughly 17 % of total electricity generation. Nuclear and wind power contribute a share of roughly 2 % each. Other sources, such as gas, solar, and biomass power, with shares of less than 1 % each, currently play no more than a minor part in China's power mix [9].\n\nAlthough China's generation mix has been relatively stable over the past two decades, the composition of the country's coal-fired power plants has undergone a significant shift toward larger and more efficient units, especially during the 11th Five-Year Plan. By implementing the policy of replacing small-size units by large and efficient plants, the share of units with 300 MW and above rose from 42.67 % of total thermal generating capacity in 2000 to 89.1 % by the end of 2010. As a result, the standard coal consumption per kWh generated has been reduced significantly (see Fig. 3.2).\n\nFig. 3.2\n\nStandard coal consumption (2002\u20132012), data from [10]\n\nWith the introduction of the Renewable Energy Law in 2006, China triggered a boom in the expansion of renewable energy development (Fig. 3.3). China has since become the world's largest producer of wind power, with approximately 13 GW installed in 2012 reaching a total installed and grid-connected capacity of 61.4 GW. With the introduction of financially attractive feed-in-tariffs in 2013, China has also recently become the world's biggest market for solar PV with record installations of around 14 GW, more than 11 GW of which is grid-connected [9].\n\nFig. 3.3\n\nInstalled capacities of renewable energies (2005\u20132012), data from [10]\n\nChina's electricity system, however, is not well-prepared for such a rapid increase of intermittent wind and solar generation units. Due to the lack of gas-fired power plants in China, coal-fired power plants are mainly employed for load-following and peak generation. These activities require a significant cycling reducing the operational efficiency of coal-fired power plants [12]. Also, limited interregional transmission capacities often complicate the usage of hydro power plants for load-following and peak generation. This contributes to high grid integration costs for wind and PV power in China [12].\n\nAnother challenge of the rapid build-up of renewables is that grid connection is lagging behind. Wind turbines are erected much faster than power lines are built. Realizing this problem, the government has made efforts to slow wind power expansion in order to reduce the share of wind power capacity which is installed but not grid-connected (by year-end) from 30 to 20 % in 2012. Due to the heavy concentration of wind power in remote regions with rich wind resources and in so-called wind power bases with 10 GW and more capacity, curtailment due to grid congestion has become an important issue: 20 TWh of wind power had to be curtailed in 2012 [13], representing about 20 % of total electricity generation from wind [14]. Since financial losses due to curtailment are not reimbursed to RES owners, wind power developers exposed to the negative impact of curtailment are increasingly building wind farms in lower wind speed regions with less network congestions.\n\nChallenges in China's power generation sector\n\n * China is not particularly well endowed with fossil energy resources in per capita terms. A system relying on fossil-fueled generators may create an increasing dependence on foreign energy sources.\n\n * China's heavy reliance on coal for power generation and industrial processes is contributing to high air pollution levels and CO2 emissions.\n\n * Employing coal-fired power plants for load-following and peak generation reduces the efficiency of their operation.\n\n * Renewable energy deployment is poorly coordinated with grid development, so that grid connection is lagging behind the construction of renewable power plants and significant quantities of wind power are being curtailed due to grid congestion.\n\n * The high proportion of large coal-fired power plants and lack of gas-fired generators makes the Chinese power system less flexible. At the same time, an increasing share of electricity generated from intermittent renewable energy is increasing the demand for system flexibility.\n\n### 3.2.2 Power consumption\n\nSince the 1980 s, the increase in China's power consumption has been mainly driven by industrial growth. In 2011, more than 70 % of China's electricity was consumed by the industrial sector [15]. The residential sector accounted for approximately 12 % and the commercial sector for 3 % (see Table A.1 in the appendix). One reason for the high proportion of industrial loads in China is the focus on producing and exporting energy-intensive materials and goods [7]. While it is still relatively high in comparison to countries with a focus on less energy-intensive sectors [16], China's overall energy intensity of production (defined as energy consumption per unit of GDP) has been decreasing consistently due to a number of government measures to promote energy efficiency (see Fig. 3.4).\n\nFig. 3.4\n\nEnergy intensity per unit of GDP (1991\u20132011), data from [10]\n\nChina's economy is gradually shifting away from heavy industry towards high value-added industries, such as information technologies and the service sector [12]. The composition of the electricity consumption is projected to change in the coming decades. Shares of residential and commercial electricity consumption are expected to increase. As a consequence, a predicted increase in load variability and range of fluctuation will soon require a higher flexibility of electric power generation [12]. In addition, industrial demand for reliable and high-quality electricity is expected to increase due to the growing importance of information technology in all economic sectors.\n\nChallenges in China's power consumption sector\n\n * The efficiency of energy and electricity use is low.\n\n * Future energy demand from high value-added industries will require high power quality and reliability. Also, future demand will be more variable in nature, thus requiring a higher degree of flexibility in the Chinese power system.\n\n * There is a mismatch between the current power generation system with its limited flexibility and an increasing demand for flexible generation due to a projected shift in electricity consumption patterns towards more variability in demand.\n\n### 3.2.3 Power logistics\n\nDisparity between power generation and consumption\n\nCoal supplies are mainly located in the northwestern, northern, and northeastern parts of China (sometimes referred to as \"Three Norths\"). Similarly, wind resources are concentrated in the three northern regions as well as along China's coast. Solar energy resources are abundant in the west and north of China. Hydro power is concentrated in the southwest and in the upstream areas of the Yellow River. While energy resources are most abundant in China's north and west, regions that are typically remote and less economically developed, the load is concentrated in the economic and industrial centers along China's eastern coast. This discrepancy presents a major challenge for power logistics (transmission) as well as for the physical transportation of fossil resources and puts a heavy strain on power grid and road\/railway infrastructure.\n\nGrid infrastructure\n\nThe regional disparity between energy resource distribution and load profile on the one hand and between the geographical location of electricity generation facilities and major centers of consumption on the other means that electricity has to be transported from north to south and from west to east. However, China does not have a nationally integrated electricity network. Its network is fragmented into six regional grids with limited interconnection operated by three grid companies: SGCC, China Southern Power Grid (CSG) and the Inner Mongolia Grid Company (for a more comprehensive overview of China's power lines please refer to Table A.2 in the appendix).\n\nAside from the limited capacity of interconnectors between the regional networks, interregional electricity trade is also heavily impeded by administrative barriers. Regional grids are made up of provincial grids, where dispatch decisions are made with the aim of balancing supply and demand within the boundaries. For cross-border power trading, provinces have to conclude bilateral contracts specifying the annual amount of electricity transmitted in each direction typically netting close to zero [17]. This way of restricting interregional trade makes it difficult to leverage the power system's inherent flexibility potential across regional borders, e. g. the use of dispatchable hydro power capacities for peak generation and ancillary services in other regions [12].\n\nAround the turn of the century, in an effort to speed up nationwide grid integration and to connect regions with significant hydro power, solar, and wind capacities with the load centers on the east coast, China started to construct Ultra High Voltage (UHV) power transmission lines. Today, China is considered a global leader in UHV transmission and transformation technology [18]. Specifically, UHV alternating current (AC) lines are used for transmitting electricity generated in coal-fired power plants or by means of RES from China's northern and western regions to the load centers. UHV direct current (DC) lines are used for transmitting hydro power from South and Central China to the east coast.\n\nAsset utilization and supply security\n\nCompared to other countries, for instance the United States, average utilization rates of the grid infrastructure in China are low in spite of the rather flat load curve [19]. Major transmission lines, for example, seldom reach a high utilization rate. In 2011, for instance, two 800-kV lines for which data are available reached utilization rates of less than 35 %. Only five of eleven trans-regional lines with 500- and 660-kV achieved utilization rates above 50 % [20]. At the level of distribution grids, average utilization rates of 10-kV lines and transformers are even lower and only seldom reach values above 30 % [19].\n\nIn principle, low utilization rates point to a large margin for grid operation often resulting in a high level of system stability and security. In the specific case of China, average annual outage times of urban users during the last years exceeded seven hours (see Table A.3 in the appendix) and those of rural users even ten hours [19]. In Germany, to give a reference value from an industrialized country, average annual outage times were only approximately 15 minutes in 2010. This evidence points to low levels of equipment maturity in parts of the electric power grid (see also [21], [22]). In recent years, investments in distribution grids have been considerably smaller than those in transmission grids [12]. Most line losses and power outages in China occur in the distribution grid [19] pointing to less sophisticated equipment in the distribution grid compared to the transmission grid (see also [21]).\n\nInvestment needs in the grid infrastructure\n\nIn the coming years, massive investments will have to be made in the electric power grid in order to cope with the steadily rising power consumption, increase supply security, and facilitate the integration of wind and PV power. It is projected that, in 2020, the total length of power lines of 110 kV and above will reach 1.76 million kilometers with an associated transformer capacity of 7.9 billion kVA [23]. This represents a considerable increase compared to line length and transformer capacities in 2010 (see Table A.2 in the appendix).\n\nChallenges in China's power logistics sector:\n\n * The regional disparity between power generation and consumption in combination with barriers to interregional electricity exchange imposes a constraint on the efficient use and delivery of energy resources.\n\n * The barriers to interregional electricity exchange make it difficult to fully exploit the flexibility potential inherent in the power system.\n\n * Asset utilization rates of China's electric power grids are below those of highly industrialized countries such as the United States in spite of China's rather flat load curve.\n\n * Supply security in China is considerably lower than in highly industrialized countries. There is evidence that a large part of power outages and line losses in China originate from distribution grids. This indicates that distribution grid equipment is less mature and sophisticated than transmission grid equipment.\n\n## 3.3 Smart grid development in China\n\n### 3.3.1 Motivation for smart grids in China\n\nSmart grids for increasing supply security\n\nA core motivation for smart grids in China is their suspected positive impact on supply security. In China, especially distribution grids with voltage levels of 10-kV and less are limiting reliability for urban end-user's, causing roughly 80 % of all power blackouts [19]. Due to missing sensoring and monitoring technologies, it often takes a long time to analyze the respective line, locate and isolate the fault, and re-establish electricity supply [19].\n\nEnhancing distribution grids with smart grid technologies such as advanced sensoring and control technologies is often seen as a means to improve the reliability of the whole electric power system in China (see e. g. [19]).\n\nSmart grids for facilitating peak shaving\n\nThe rapid growth of electricity consumption reflected in very high peak loads may cause shortages on the generation side and network congestions. Also, the rising importance of residential electricity consumption compared to industrial electricity consumption will lead to a more pronounced load curve. This trend could be further accentuated by China's continuing urbanization process [24]. With low gas generation capacities and limited transmission capacities, coal-fired power plants are mainly employed for load-following and peak generation in China. This reduces their operational efficiency. Peak shaving is of critical importance as it contributes to a reduction of peak loads and thus helps to avoid peak generation. Peak shaving might also reduce potential network congestions when demand peaks occurring in periods of network congestions are reduced via demand side management. As such, it should be seen as an important factor towards a more reliable power system in China. In addition, by reducing the maximum grid load, peak shaving helps to increase average utilization rates of the grid infrastructure, thus reducing investment needs and increasing the affordability of electricity supply.\n\nTo enable peak shaving and demand side management it is necessary to deploy an ICT infrastructure for two-way communications between end-users \u2013 or so-called prosumers if they also generate electricity \u2013 and grid operators. Together with other smart grid technologies facilitating real-time data exchange, visualization of information as well as control of devices, this infrastructure enables the grid operator to have a clear picture of electricity consumption and generation at any given time, while allowing end-users to receive price signals to adapt their electricity consumption to the variable supply.\n\nSmart grids for preparing extensive integration of RES\n\nChina has aggressively expanded RES generation capacities within an extremely short time frame. Grid connection of RES is currently lagging behind and a considerable amount of RES-E is curtailed. In this context, smart grid technologies are often seen as a means to reduce RES integration costs:\n\n * Large-capacity battery storage systems facilitate the integration of centralized large-scale intermittent RES generation capacities of the type that are currently focused upon in China.1\n\n * Due to frequent network congestions at the local level, a considerable quantity of electricity generation from RES has to be curtailed. More effective grid capacity management using smart grid technologies could reduce curtailment of RES.\n\n * Smart grid technologies such as microgrids and virtual power plants ease the integration of RES at the local level.\n\n * The grid connection itself might be more effective with smart grid technologies. One aspiration expressed in this area is that RES might be integrated into grid operation via standardized ICT connectors so that the integration of standard RES plants might become as easy as connecting external devices with computers via universal serial bus (USB) interfaces (plug-and-play integration) [25].\n\n### 3.3.2 China's technological view of the smart grid\n\nThe development of smart grids in China\n\nAn early form of smart grids was proposed in 1999 by Lu and Mei [26] in their article Basic research on vital scientific problem with collapse prevention and optimal operation of large scale power systems [27]. Five years later, distributed computing was applied in a research project for a real-time simulation of the entire power grid [27]. In 2007, the East China Grid Company, a subsidiary of SGCC, carried out a feasibility study on smart grids that examined the promotion of digital substations and build-up of a unified enterprise platform for power system data [27].\n\nThe grid operator's view on smart grids\n\nSince then, China's smart grid development has mainly been pushed forward by China's grid operators, SGCC and CSG. In 2009, SGCC proposed the strategic goal of building strong and smart grids with Chinese characteristics. SGCC focuses on the nationwide integration of provincial and regional grids by means of a strong UHV AC backbone. The first objective of this backbone grid is to ease the balancing and power exchange between different regions, the second to integrate large-scale generation from RES. CSG, on the other hand, envisages a smart grid with a more decentralized focus using primarily high voltage direct current (HVDC) links as backbone systems. CSG intends to integrate remote energy sources, especially hydro power. HVDC development will lead to separated asynchronous provincial grids. The nationwide synchronization of China's power grids is not one of CSG's main goals. Comparing both strategies reveals that the two grid operators pursue different approaches with regard to the type of transmission technology and the nationwide synchronization of China's power grids.\n\nIn spite of this disagreement, the smart grid visions of both grid operators concentrate mainly on the upgrade of transmission grids: a look at the total annual smart grid revenues from the most important smart grid technologies in China reveals that currently more than 50 % of revenue is related to upgrades of transmission grids. Substation and distribution automation technologies are responsible for no more than a low share of overall smart grid revenues (see Fig. 3.5). According to markets forecasts for 2020, the absolute increase in smart grid revenues related to transmission upgrades is significantly higher than the increases of revenues from substation automation, distribution automation, and AMI (see Fig. 3.5).\n\nFig. 3.5\n\nAnnual overall smart grid revenue of main smart grid technologies in China (\u00a9 Navigant Research [28])\n\nThe government's view on smart grids\n\nChina's government has already acknowledged the importance of smart grids for China's future energy system. In its 12 th Five-Year Plan for National Economic and Social Development, the People's National Congress explicitly set the goal of accelerating smart grid developments in China (see [27]). In the Decision of the State Council on Accelerating the Fostering and Development of Strategic Emerging Industries, the State Council also underlined the importance of speeding up the development of smart grids (see [27]).\n\nIn addition to the general commitment to smart grids, NEA also issued a general definition of smart grids:\n\n> Smart grid technologies have the purpose to integrate new energy, materials and equipment as well as advanced technologies in information, automatic control and energy storage for realizing digital management, intelligent decision-making and interactive transaction in power generation, transmission, distribution, consumption and storage. Furthermore, smart grid assets optimize the resource allocation and satisfy diverse needs of customers as well as ensure the safety, reliability and cost-efficiency of power supply. Finally, the new technology [in the sense of smart technology] bridges the constraint of environmental protection and the development of the power market.\n\nMoreover, the Chinese government supports the technological development of main smart grid technologies by means of innovation policies such as standardization and R&D funding (see Sect. 3.4.6). However, in contrast to both grid operators, the Chinese government has not yet developed a perspective of its own on the technological and organizational design of China's smart grids. As a result, the discussion of such aspects is still dominated by the grid operators and there is currently no unanimously accepted comprehensive view of smart grids in China.\n\nThis lack of common understanding among the main stakeholders is the cause of many controversies and basic disagreements on the key aspects of smart grid development [27]. The absence of such a common vision increases uncertainty for potential smart grid investors because the profitability of their investments critically depends on whether a strong and smart or a decentralized smart grid will be realized. Thus, the lack of a comprehensive smart grid vision has to be viewed as an obstacle towards the development of smart grids in China [27].\n\nMain challenges with regard to the technological view of smart grids\n\n * Due to the contrary strategies of China's grid operators on the subject of smart grid development and the absence of a Chinese government view on smart grids, there is still no unanimously accepted vision on the technological and organizational design of smart grids in China. As a result, there is much uncertainty among potential smart grid investors regarding the future development.\n\n### 3.3.3 This study's view on smart grids in China\n\nThe present study has a clear focus on proposing regulatory policies supporting the evolution of smart grids in China. The creation of a widely accepted technological smart grid vision is beyond the study's scope. Nonetheless, a common understanding of desirable smart grid developments during the next years is necessary to determine the general direction of the regulatory recommendations. Given the missing unanimously accepted smart grid vision in China, a pragmatic three-sided approach was employed to develop such a common understanding:\n\n * Following a bottom-up approach, China's future smart grid is considered to comprise a broad portfolio of ICT together with various modern technologies for power generation, transmission, distribution, storage, and consumption.2 This also includes modern grid technologies such as UHV transmission grids or heat-resistant wires. A recent literature review supports this view by underlining that smart grids in China focus on all sections of the power system, including smart power generation, transmission, deployment, usage and storage. Specifically, the integration of RES should also be understood as part of the topic of smart grids in China [27].\n\n * The smartness of the current electric power grid as well as the desirable smartness of the power grid in 2020 was assessed. In particular, the current levels of system integration of single power system components as well as the projected levels in 2020 were described. In this context, system integration refers to the extent to which power plants, wind farms, transmission grids, distribution grids, and power consumers are expected to be remotely monitorable, controllable, or even autonomously controllable (self-healing).3\n\n * Given the government's will to establish energy markets, the study compares market elements used in China today with those projected for 2020.4\n\nThis three sided-approach has led to the following conclusions:\n\n * China has in recent years made important breakthroughs in the development of smart grid technologies. Examples for such technologies are UHV transmission grids and large-capacity battery energy storage technology.5\n\n * On the other hand, a big leap forward with regard to many smart grid technologies is indispensable at all stages of the smart grid supply chain.6 Like in many other countries engaged in the development of smart grids, some of China's key smart grid technologies are still immature and have somewhat inconsistent component specifications and standards [21], [22], [27].\n\n * In 2020, market elements such as regional energy marketplaces and virtual power plants are intended to be used much more intensively than today. They will be introduced at least on provincial level.\n\nWith the study's focus on regulatory issues in mind and in light of the conclusions described above, the regulatory pathways presented in this study aim specifically at:\n\n * facilitating the widespread deployment in 2020 of those smart grid technologies which have already achieved high maturity levels today. In this context, smart grid technologies also include modern grid technologies which are not necessarily included in smart grid concepts of other countries;\n\n * promoting technological innovations of rather immature smart grid technologies and increase their maturity;\n\n * promoting the development of so-called smart markets (see Sect.\u200b 4.\u200b3.\u200b2 for more details on smart markets). In this context, an important prerequisite is the integration of third parties. They are seen as key players in smart markets.\n\n## 3.4 The regulation of China's electric power system\n\n### 3.4.1 Policy setting and fundamental institutions\n\n#### 3.4.1.1 Policy Setting\n\nGovernment leadership is essential to promote smart grid development. The general strategy of policy-makers towards the development of the future power system, often containing quite specific targets for short-term and long-term development, is important for companies and other stakeholders in the electric power sector: indeed, this government strategy serves as an important basis for smart grid investment decisions of both companies and households.\n\nThe Chinese government has set quite specific targets with regard to the development of the energy system until 2015. However, there are no explicit targets beyond this point in time. The most important government targets for 2015 are briefly summarized below:\n\n * From 2010 to 2015, generation and grid capacities are planned to increase by roughly 50 % in order to cope with the steadily growing demand.\n\n * RES generation capacities are expected to increase out of proportion \u2013 their share in the electricity mix will increase significantly.\n\n * Average utilization rates of the grid infrastructure and supply security are targeted to increase. Specifically, power outages on the level of distribution grids, are expected to be reduced.\n\n * The efficiency of energy use and particularly of electricity use is planned to improve considerably.\n\nA detailed overview of government targets for China's future electric power sector is given in Sect.\u200b 6.\u200b1 in the context of the regulatory pathways.\n\n#### 3.4.1.2 Fundamental institutions\n\nTo realize the government agenda, ministries and other government institutions issue laws and ordinances, monitor compliance with these laws and ordinances, propose major technological standards, and promote innovations. The question regarding whether the different government practices can be executed effectively largely depends on the governance structure of ministries and government institutions.\n\nGovernance structure\n\nThe governance structure of China's energy system, and in particular the power sector, has been subject to frequent reorganization and currently comprises a broad variety of ministries and institutions. The fragmentation of responsibilities among a multiplicity of different stakeholders makes conflicts of competence inevitable and a coherent and continuous governance of the power sector difficult [2].\n\nFigure 3.6 presents a graphical overview of the main authorities of China's power sector governance (Table A.4 in the appendix further specifies influences, roles, and responsibilities of main governmental institutions). There are four main government authorities involved in China's power sector policy:\n\nFig. 3.6\n\nKey Actors of China's Power Sector Governance, (\u00a9 GIZ [29])\n\n * The State Council (SC) is the highest executive organ of the People's Republic of China. With regard to the electricity sector it sets the political agenda, takes the lead for major reforms and is in charge of promulgating major plans like the Five-Year Plan for energy development.\n\n * The NDRC is an agency under SC exerting broad administrative and planning control over the Chinese economy. It is the most important government authority for power sector regulation. Its powers include the regulation and setting of energy and electricity prices, the approval of major power sector projects, as well as energy efficiency policy.\n\n * The NEA proposes the energy development strategy, drafts energy development plans as well as energy-related policies, provisions and laws to be adopted by NDRC or SC. NEA also advises on power system reform and market regulation. In addition to these political functions, NEA is the regulatory authority for the power sector in charge of regulating power system construction, power safety, power supply and service, as well as tariff and information disclosure.\n\n * The State-owned Asset Supervision and Administration Commission of the State Council (SASAC) supervises the performance of SOE such as the grid operators and the major power generation companies. SASAC exerts its power through the right to appoint, dismiss and evaluate the performance of executives, the right to audit as well as to approve key decisions.\n\nThe existing literature on China's governance structure in the electric power sector (see for example [2], [3], and [30]) often stresses several regulatory challenges potentially preventing an effective development towards smart grids in China:\n\n * There is a significant fragmentation as well as overlap of responsibilities of the various government bodies involved in power system regulation, negatively affecting the efficiency of the sector's governance.\n\n * In international comparison, China's major institutions governing the energy sector have a rather low number of employees, staff numbers not always being adequate to fulfill the responsibility of regulating an electricity system the size of China.\n\n * The importance of the previous aspect even increases when the size and the power of China's major grid operators are taken into account. The grid operators' stakes are so high that it always pays off for them to hire consultants, lawyers, and lobbyists to argue their case. To address these claims, the regulator needs to be equipped with a sufficient number of highly qualified employees.\n\n * Essential instruments to steer power sector development, like electricity pricing, remain under the authority of NDRC. The concentration of power within NDRC limits the ability of NEA to drive power sector reform.\n\nRegulatory challenges in the area of policy setting and fundamental institutions\n\n * Efficient governance of the power sector is impeded by overlapping responsibilities and conflicts of interests between different government authorities.\n\n * In international comparison, NEA is understaffed, underfunded and without sufficient power to make independent decisions. The lack of independence and law enforcement authority has a negative effect on regulatory performance.\n\n * The concentration of power within NDRC limits the ability of NEA to drive power sector reform.\n\n### 3.4.2 Market structure\n\nThe market structure and especially the involvement of new market actors is a critical factor with regard to the innovation potential in the smart grid development process. The following paragraphs will describe the governance structures between the companies at the various stages of the supply chain (vertical market structure) and the competitive conditions for the companies in each part of the supply chain (horizontal market structure).\n\nVertical market structure\n\nThe competences and responsibilities of China's power system supply chain have been tightly bundled over a long period of time. The main electricity sector reform of 2002 mandated the separation (or unbundling) of the state-owned vertically integrated utility into five big power generation companies, two major grid operators handling transmission, distribution and retail as well as four power service corporations.\n\nAll of them are SOE. Under this market structure, the grid operators are assigned regional monopolies acting as single buyers from generation side as well as being the only seller with the electricity retail monopoly within their geographic area.\n\nHorizontal market structure\n\nIn the field of power generation, the five major power generation companies, the so-called Big Five are responsible for roughly 50 % of China's electricity generation. By means of its project approval process, NDRC tries to balance generation capacities between them [31]. Due to their size, the Big Five have a significant lobbying force contributing to China's power system governance together with the government and the grid operators. The remaining part of China's power is generated by thousands of smaller local and regional generation companies. In addition, there are specialized generation companies mainly focusing on power generation from one energy source, e. g. hydro or nuclear power. China's power generation sector can be described as liberalized, as it potentially allows competition between the different companies. However, compared to local governments or state-owned companies, private and foreign investors still face significant legal and administrative barriers restricting the development of a diverse ownership structure [2].\n\nPower transmission, distribution and retail are currently vertically integrated and operated by two major SOE with geographical monopolies including electricity retail: SGCC controls the eastern, central, northwestern, northern and northeastern grids; while CSG is in charge of the southern grid. In the sparsely populated province of Inner Mongolia, an independent grid operator, the Inner Mongolia Electric Power Corporation, controls the western part of the grid while SGCC controls the eastern part. Due to their importance for the development of China's power system, both SGCC and CSG are briefly introduced below:\n\n * State Grid Corporation of China (SGCC), an SOE, is the 7th biggest company in the world according to the 2012 Fortune Global 500 list, with almost 1.6 million employees [32]. SGCC is responsible for power transmission, distribution, and retail in all five major regions of China with the exception of South China. Its operations cover 26 provinces, autonomous regions and municipalities \u2013 88 % of the national territory \u2013 and 83 % of the national power consumption [33] (see Table A.5 in the appendix for a list of affiliated grid operators). The company has ministry-like status and is a powerful force in power sector governance [34].\n\n * China Southern Power Grid (CSG), also an SOE, ranks at position 152 on the Fortune Global 500 list and has roughly 300,000 employees. CSG is responsible for power transmission, distribution, and retail in the five provinces of South China, covering 12 % of the national territory and 17 % of the national power consumption [33].\n\nBoth grid operators are currently expanding their business portfolios across the value chain: SGCC has taken over domestic engineering firms and leading electric power equipment manufacturers, raising concerns about conflicts of interests and a threat to competition due to the concentration of multiple roles within one company (i. e. roles standard setting, manufacturing equipment, transmitting and selling electricity).\n\nAt the same time, SGCC is pursuing a \"going global\" strategy targeting the acquisition of assets abroad [35]. The advancing vertical integration of SGCC has caused discussion as to whether its mergers and acquisitions go against China's power sector reform policies of downsizing and unbundling grid operators. Recently a potential separation of grid operators into transmission and distribution companies or into smaller, regional businesses has been subject of debate [34].\n\nRegulatory challenges in the area of market structure\n\n * Transmission, distribution and retail of China's electricity are in the hands of two grid operators. There is no competition in the power retail sector.\n\n * The acquisitions of grid operators in other segments of the value chain (e. g. equipment manufacturing and services) threaten fair competition in these sectors.\n\n * The size and power of grid operators makes it difficult to regulate them.\n\n### 3.4.3 Market design and RES integration\n\nThe question regarding how electricity markets are designed is highly important for the development of smart grids because market design affects electricity pricing. Pricing mechanisms, including feed-in tariffs for RES, are of crucial importance with regard to the utilization of different generation sources, investment decisions for new generation and grid capacities, day-to-day power consumption patterns, and investments in power saving technologies.\n\nGeneral market design\n\nPrices for energy resources, power system equipment and electricity were fixed by the government for a long period of time. In the 1980 s, the government started to gradually liberalize coal prices. Coal is currently at least partly traded at variable market prices, either depending on spot market rates or on individually negotiated contracts [31].\n\nIrrespective of market liberalization tendencies, China still lacks a formal and transparent mechanism for linking real costs and prices of electricity [12]. Electricity markets with prices based on supply and demand do not exist in China. Two types of electricity prices exist in China:\n\n * On-grid prices are the prices power generators receive from their grid operator for each kWh generated.\n\n * Retail prices are the prices grid operators (which are also responsible for retail) charge to end consumers for each kWh consumed.\n\nBoth prices are fixed by the price department of NDRC and adjusted every 18\u201324 months.7\n\nSetting on-grid prices\n\nNDRC determines on-grid prices for power generation using two different approaches:\n\n * On-grid prices for thermal generators are set using a price benchmark for generators within the same technology class. Each power plant is contractually guaranteed a certain capacity factor (number of full load hours) equal to comparable facilities. The equal capacity factor is based on an estimate of annual power output as well as on average fixed and variable costs.\n\n * On-grid prices assigned to owners of hydro power and nuclear power plants differ from facility to facility. They are the sum of the plant's generation costs, governmental taxes and surcharges, and profits attributed to the plant owner [2] [12].\n\nTo give a short overview of the results of these price-setting mechanisms, the following list shows on-grid prices for electricity stemming from different conventional generation technologies in 2012 (prices from [31]):\n\n * Gas-fired power plants:\n\n0.395\u20130.710 RMB\/kWh\n\n * Coal-fired power plants:\n\n0.311\u20130.520 RMB\/kWh\n\n * Nuclear power plants:\n\n0.414\u20130.471 RMB\/kWh\n\n * Hydro power plants:\n\n0.145\u20130.411 RMB\/kWh\n\nConsequences of on-grid price setting\n\nBoth price setting approaches ensure that plant operators usually earn revenues sufficient for the repayment of their investment costs within one decade [2]. That is, in principle, both approaches are sufficient to incentivize investments in power generation capacities. However, such price setting approaches based on costs are often viewed quite critically in the economic literature (see e. g. [36], [37], and [38]): for example, they are quite time-consuming for the regulator. In China, the following aspects have to be emphasized with regard to the setting of on-grid prices:\n\n * Due to the fact that coal prices are flexible while on-grid electricity tariffs for coal-fired power plants are strictly regulated and adjusted rather infrequently, power companies are not able to directly pass on changes in fuel costs. During periods of high coal prices, power companies have seen their revenues squeezed to such an extent that they have restricted generation in order to limit their losses. This behavior further aggravates the problem of missing generation capacities and contributes to the emergence of power outages.\n\n * The approach used to set on-grid prices of hydro power and nuclear power plants gives few incentives to save costs, because potential cost savings in one specific power plant would lead to lower on-grid prices for the plant within 18\u201324 months. Even more, such approaches give an incentive to exaggerate generation costs by means of creative financial accounting to receive higher on-grid prices [38].\n\nSetting of retail prices\n\nRetail prices are amended from province to province to account for policy goals and status of economic development. They are also differentiated according to end user groups. The current national averages of electricity retail prices for different consumer groups are:\n\n * Commercial: 0.863 RMB\/kWh,\n\n * Non-residential lighting: 0.791 RMB\/kWh,\n\n * Industrial use: 0.698 RMB\/kWh,\n\n * Residential: 0.498 RMB\/kWh,\n\n * Agriculture: 0.419 RMB\/kWh,\n\n * Irrigation in poor areas: 0.201 RMB\/kWh.\n\nRetail tariffs are subject to cross-subsidization in certain market segments to account for differences in income and to ensure social stability. Tariffs for residential, agriculture, irrigation and drainage use, together reflecting about 20 % of the Chinese power consumption, are lower than average generation costs. It is notable that industrial users generally pay higher tariffs than households.\n\nConsequences of retail price setting\n\nThe main disadvantage of low retail prices is that a low price level does not incentivize consumers to use power more efficiently. There is evidence that residential electricity consumers in China are quite sensitive to changes in electricity price levels [39]. Thus, increasing electricity prices would probably lead to significant reductions of electricity consumption among Chinese households. However, increasing electricity prices for residential consumers is not a political option in China given the government's intention to keep prices for residential consumers low. Therefore, China currently employs a number of other retail pricing policies that serve to align economic and environmental targets of the electricity sector [1]:\n\n * A three-tiered pricing system has been applied to residential consumers since mid-2012. Electricity prices are set in blocks with higher rates for customers with a large electricity consumption (referred to as inclining block pricing).\n\n * In addition, China has implemented a policy differentiating the electricity tariffs based on the energy efficiency of industrial processes. The policy, which is applied to eight energy-intensive industries, aims at phasing out outdated industrial capacities or incentivizing upgrades.\n\nPromotion and integration of RES\n\nFeed-in tariffs have proven to be an efficient instrument to promote RES in China. They are paid to RES generation installations depending on the type of renewable energy and location of the installations. Feed-in tariffs for RES generation are considerably higher than on-grid prices for hydro power and coal-fired power plants. In 2012, the ranges for feed-in tariffs of different RES generation types in 2012 were [31]:\n\n * Solar: 1.00 RMB\/kWh\n\n * Biomass: 0.56\u20130.77 RMB\/kWh\n\n * Wind: 0.51\u20130.61 RMB\/kWh\n\nThe Renewable Energy Law obliges grid operators to feed in renewably generated electricity. The resulting costs are socialized through a renewable energy surcharge levied on all electricity users in China [31].8 Since the initial publication of the 2006 Renewable Energy Law, the renewable energy surcharge has been adjusted three times, with the last increase from 0.008 to 0.015 RMB\/kWh taking place in September 2013. To ensure that the grid companies do not misuse the surcharge, it is turned over to the Ministry of Finance (MOF) in a first step. The grid companies then apply to MOF and NEA for the disbursement of these funds with supporting documentation [40]. This procedure causes a delay in the reimbursement of grid operators' expenses for purchasing power generated from renewables, which in turn results in a lack of operating capital with serious effects on the whole supply chain. Grid operators delay payments to wind power developers, who in turn are unable to pay renewable energy equipment manufacturers (e. g. wind turbines), resulting in delayed payment of component suppliers. This situation is expected to be alleviated with the recent doubling of the renewable energy surcharge. From the grid operator's perspective, delays in reimbursement may be seen as a reason to delay the grid connection of renewable energy generators.\n\nRegulatory challenges in the area of market design and RES integration\n\n * On-grid and retail electricity prices do not reflect actual costs and are adjusted only infrequently.\n\n * The approach used for setting on-grid prices of hydro and nuclear power plants gives only few incentives for cost savings. Rather it gives incentives to exaggerate costs to achieve a higher on-grid price.\n\n * Although a system of tiered prices has been introduced for residential consumers, the low overall level of retail electricity prices does not provide sufficient incentives for consumers to use power more efficiently.\n\n * Delays in the disbursement of RES funds leads to a lack of operating capital along the renewable energy value chain and provides a disincentive for grid operators to connect RES plants to the grid.\n\n### 3.4.4 Development of infrastructure and network regulation\n\nThis section explains how grid planning is done in China and how investments in the (smart) grid infrastructure are incentivized. The pertinent regulatory practices directly impact the stable and affordable operation of the current grid infrastructure and the investments in the prospective smart grid infrastructure.\n\nNetwork expansion planning\n\nThe government's key target is cost-efficient extension and development of the power grid in order to cope with the steadily increasing electricity consumption and minimize supply interruptions.\n\nTargets on the development of grid infrastructure and grid operators' performance are specified by the government in a top-down process. However, little information is publicly available on official grid expansion plans [41]. As a result, the document Framework and Roadmap for Strong and Smart Grid Standards, published by SGCC in 2010 [42], serves as an unofficial (smart) grid development plan in China [41]. With respect to grid expansion planning, there is little guidance from the government and little coordination among existing power system stakeholders.\n\nNetwork regulation\n\nAt present, there is no explicit price for transmission and distribution (network charge) based on actual costs. The source of grid operators' income is the difference between the on-grid and the retail price for electricity. This amount covers all costs of the grid operators: grid operation and maintenance, grid upgrade and expansion, management, metering and billing, etc. The remainder makes up the profits of the grid operator. The government targets a rate-of-return for the grid operators in the range of about 8 to 10 % [43]. This practice, in combination with the rather infrequent adjustments of the retail electricity tariffs, has an impact on the ability of grid companies to quickly and predictably recoup costs. In turn, the grid companies may exhibit little motivation to invest and assume costs related to the implementation of government policies and regulations, e. g. for investments related to renewable grid connection and integration, investments in end-use energy efficiency or distributed generation [1]. Moreover, the cost structure of power sector companies lacks transparency and power sector data and information available to the public is very limited. Opaque costs affect the government's ability to regulate and inform the setting of electricity prices.\n\nThere are no clear rules for accounting of costs, revenues, and profits and no transparent administrative process for setting allowed revenues. Without this foundation the government is unable to audit companies' accounting records or exact penalties for noncompliance with reporting obligations, information requests or other government requirements [1].\n\nRegulatory challenges in the area of development of infrastructure and network regulation\n\n * Grid expansion planning is organized in a top-down process with low transparency and little involvement of players other than government authorities and grid operators.\n\n * The price-setting mechanism with respect to wholesale and retail prices together with the rather infrequent adjustments of retail prices reduces the ability of grid companies to quickly recoup costs. This might reduce their willingness to incur costs related to other government policies (for instance related to RES integration).\n\n * A lack of transparency of costs impedes the efficient and informed regulation of power sector companies.\n\n### 3.4.5 Coordination of generation and consumption\n\nThe following paragraphs describe general responsibilities for coordinating electricity generation and consumption. Government measures implemented in this context are also presented.\n\nLong-term coordination of generation and consumption\n\nChina's rapid economic growth has resulted in a steadily increasing electricity consumption. The Chinese government promotes the expansion of generation and grid capacities. In addition, the role of energy efficiency as an instrument to decouple the rise of power consumption from economic growth has been recognized. Numerous policies to promote energy efficiency have been promulgated: for instance, national targets for energy intensity, differential pricing for energy-intensive industries and energy efficiency obligations requiring Chinese grid companies to realize energy conservation targets (e. g. through end-user energy efficiency programs).\n\nShort-term coordination of generation\n\nHistorically, dispatch of power plants in China is organized by means of a so-called equal shares dispatch or generation quota system. China allocates operating hours equally among the coal-fired generators that constitute the bulk of China's generating capacity. This system is intended to give each generator an equal chance to recover capital costs and achieve a reasonable return-on-investment, but it largely ignores the fact that plants within the coal fleet vary significantly in terms of efficiency and environmental performance [44]. Annual operating hours for generators are set administratively by Provincial Economic and Trade Commissions (PETC) and approved nationally by NDRC. As a result, the overall performance of the power system has suffered significantly in terms of cost, environmental performance, and distorted investment decisions. The dependence of each coal-fired generator on running a similar guaranteed number of hours has become a major barrier to reforming dispatch in China.\n\nIn 2007, China started to pilot a so-called energy efficient dispatch system. This system sets a dispatch order prioritizing generators on the basis of heat and emissions rates favoring renewable and low carbon generation sources, with coal-fired power plants dispatched according to their thermal efficiency. This order is based on the priority order table created by PETC and is updated quarterly based on changes in generator parameters and the addition of new units [45]. Hence, the power generation quotas of power generation facilities are no longer guaranteed [46]. Since on-grid prices are calculated on the basis of a fixed estimate of annual operating hours, generators will face a revenue shortfall if average operating hours fall below the projected level [44]. One of the main drawbacks of the policy was a lack of compensation of power and grid companies for lost revenues due to the changes in dispatch. The pilot has proved to be difficult to implement and has not spread to the whole country [47].\n\nRegardless of whether a generation quota or an energy-efficient dispatch system is used, the specific day-to-day dispatching is in the hand of dispatch centers under the authority of national, regional, provincial, or local grid companies. The dispatch centers take into account factors like load forecasts, the availability of power plants, and constraints for system reliability. On this basis, the dispatch centers set day-ahead commitment plans for the power plants. Power generation companies are subsequently obliged to supply power as required [44].\n\nShort-term coordination of consumption\n\nLike in all other regions of the world, China's intraday power generation follows intraday power demand. However, early attempts have been made to influence the hourly pattern of China's power consumption and to shift power consumption from peak times to off-peak times: since the 1990 s, many provinces have started to coordinate power consumption by means of interruptible loads and time-of-use prices:\n\n * Interruptible loads refer to a pricing mechanism in which large industrial consumers are paid for curtailing their loads in times of network congestions. This demand response (DR) mechanism has been piloted and regionally applied in China [12].\n\n * Time-of-use pricing means that the electricity price varies depending on the time-of-day when electricity is provided. In times of network congestions prices tend to be high to incentivize electricity consumers to reduce their consumption. With the 2005 Interim Provisions for the Administration of Power Selling Prices, the Chinese government stressed the role of time-of-use prices to reflect real costs at different day times [2]. Time-of-use pricing has been implemented mainly for industrial and commercial users [48]. In some regions, time-of-use pricing is also applied to residential customers. The Chinese government has recently announced that it will introduce time-of-use pricing for residential consumers on a national level by the end of 2015 [49]. Overall, it is still questionable whether the existing incentives are actually sufficient to induce customers to shift electricity consumption to off-peak times [2].\n\nRegulatory challenges in the area of coordination of generation and consumption\n\n * The generation quota system ignores the fact that power plants within the coal fleet vary significantly in terms of efficiency and environmental performance. As a result, the overall performance of the system has suffered significantly in terms of cost, environmental performance and distorted investment decisions.\n\n * The dispatch model is closely linked to the electricity pricing mechanism; no dispatch reform can be introduced without a reform of electricity pricing, and such a reform would have to compensate coal power plants whose revenue is calculated on the basis of the existing dispatch model.\n\n * China has made remarkable progress with regard to the introduction of time-of-use pricing. Such pricing mechanisms are planned to be applied to all categories of customers in 2015. However, it is unclear whether the specific design of time-of-use prices actually sets sufficient incentives for shifting electricity use among all groups of customers.\n\n### 3.4.6 The role of information and communication\n\nThe government's role in promoting smart grid-related ICT\n\nThe Chinese government promotes innovations surrounding smart grids mainly by focusing on the supply of smart grid technologies: on the one hand, the Chinese government counts on public enterprises as main drivers of smart grid innovations. On the other hand, it allocates significant financial funds to promote R&D activities or to build up demonstration sites [50]. Specifically, in 2012 the MOST released a special plan for smart grids focusing on R&D and considering the 12th five-year period (from 2011 to 2015). The smart grid special plan focuses on three main strategic goals:\n\n * It is forward-looking and sets the agenda for the Chinese smart grid development after 2015. It promotes the development of so-called cutting-edge smart grid technologies which might be deployed after 2015.\n\n * It aims at ensuring that existing modern technologies such as those listed in appendix B are introduced very fast into the Chinese energy system.\n\n * It highlights that China must keep up with world-class smart grid research developments and should carry out cutting-edge smart grid technology research.\n\nThe general idea behind the smart grid special plan is that the development of smart grids should be business-orientated and primarily rely on the innovation capacity of domestic companies, which are intended to achieve international technical dominance by 2020. This will require a more profound and more effective involvement in international research cooperation and standardization processes.\n\nThe promotion of smart grid standards is another important channel for the government to influence the evolution of Chinese smart grids. One example for such activities is the work of the State Council, which has issued a plan for modifying and promoting the power equipment manufacturing industry [50]. Another example is the China Electric Power Research Institute directly emitting standards on Low Voltage Power Line Carrier Communication or Intelligent Control Network Data Terminal [50].\n\nPolicies focusing on the demand for smart grid technologies are of rather low importance for the Chinese government in its efforts to promote smart grids [50]. Note that this prevalence of supply-side policies together with the rather low importance of demand-side policies in China are a sign that, much like the situation in other countries, China's smart grid industry is still in the initial phase of its development [50].\n\nThe role of the ICT industry in promoting smart grids\n\nChina's ICT sector is representative of the massive changes in China's industry and economy. Since the economic reforms in 1978, it has been growing rapidly with large inward and outward foreign direct investment flows and export-led activities [51]. Today, most of China's ICT companies are private companies that are not owned by the state. Many of them have been founded by foreign investors or companies. The Chinese strategy of building national champions has already yielded the creation of several ambitious companies which became global players [51]: Huawei Technologies, Lenovo, and ZTE are good examples of such companies. Huawei and ZTE are major players in the Global System for Mobile Communications (GSM), Code Division Multiple Access (CDMA), Optical and Digital Subscriber Line Access Multiplexer (DSLAM) equipment markets. Huawei Technologies in particular has emerged as a leading provider of telecommunications networks that increasingly challenges established competitors like Siemens, Cisco, and Alcatel [51].\n\nIrrespective of the increasing importance of China's ICT sector, most ICT companies have few stakes in the strategic development of China's smart grid vision. Even though these companies act as component and technology suppliers, they are less engaged in the strategic development process, which is mainly pushed forward by the government, the grid companies, and the power generation companies.9\n\nRegulatory challenges in the area of the role of information and communication\n\n * There is a need for cutting-edge smart grid technology research to keep up with international level smart grid research developments.\n\n * China's ICT industry consists of many ambitious and competitive companies, some of them global players. However, ICT industry is currently underrepresented in the strategic development of smart grids in China. Therefore, the large innovation potential of the ICT sector risks not being fully integrated in the Chinese smart grid development.\n\n\u25b6 Key findings\n\n#### Key findings\n\n * In the past 15 years, a series of reforms have greatly improved the efficiency, reliability, and environmental performance of the Chinese power sector. However, significant challenges remain: rapidly rising electricity demand, concerns about power system reliability and energy security, low average utilization rates of the grid infrastructure, environmental degradation and climate change.\n\n * A core motivation for smart grids in China is their suspected positive impact on supply security and operational efficiency, especially on the distribution grid level. Peak shaving enabled by means of smart grid technologies plays an important role to increase supply security and operational efficiency. Smart grid technologies are also seen as a means to reduce RES integration costs, which is of critical importance given the Chinese government's aggressive RES expansion targets.\n\n * Due to the absence of a common smart grid view in China and the contrary strategies on smart grid development among China's grid operators, there is still no unanimously accepted technological and organizational concept of smart grids in China. As a result, the uncertainty of potential smart grid investors regarding the future technological development is high.\n\n * Some regulatory aspects of China's electric power system represent barriers for the effective and efficient development of smart grids in China:\n\n * The absence of government guidelines for the long-term development of the electric power sector, overlapping responsibilities and conflicts of interests between different government authorities, and the lack of independence and law enforcement of the regulatory authority point to insufficient government leadership with regard to smart grid development.\n\n * The market structure is dominated by China's grid operators, who are responsible for transmission, distribution, and retail. Innovative and new market actors, and specifically the ICT industry, are hardly involved in the smart grid development process.\n\n * On-grid prices, retail prices, and the operating hours of power plants are fixed by government authorities. Grid operators' income depends on the difference between retail and on-grid prices \u2013 network charges are not explicitly calculated. 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[Accessed June 7, 2014].\n\nFootnotes\n\n1\n\nOne example for the trend towards large-capacity battery storage are the activities of BYD. The Chinese manufacturer of automobiles and rechargeable batteries built one of the world's largest lithium-ion battery energy storage systems, a 36-megawatt-hour system, located in Zhangbei, Hebei [52].\n\n2\n\nPlease refer to appendix B for a complete overview of all modern technologies that are subsumed under the smart grid label in this study.\n\n3\n\nPlease refer to appendix C for a complete overview of the results.\n\n4\n\nPlease refer to appendix C in for a complete overview of the results of this discussion.\n\n5\n\nPlease refer to appendix B for a complete list and description of these technologies.\n\n6\n\nPlease refer to appendix C for an overview of the necessary technological advancement during the next years.\n\n7\n\nJointly with local and provincial price bureaus, the price department of NDRC is responsible for the formulation, inspection, approval and establishment of the tariff, as well as for the supervision and inspection of the tariff implementation. In addition to NDRC, the NEA may put forward suggestions to the price department of NDRC.\n\n8\n\nNote that agricultural uses and power consumers in Tibet are excepted from the surcharge [43].\n\n9\n\nThey seem to show even less engagement in China's smart grid development than international players such as Cisco or IBM.\n\u00a9 The Author(s) 2015\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel (eds.)Regulatory Pathways For Smart Grid Development in China10.1007\/978-3-658-08463-9_4\n\n# 4. Germany's way from conventional power grids towards smart grids\n\nGert Brunekreeft1 , Marius Buchmann1, Christian D\u00e4nekas4, Xin Guo2, Christoph Mayer4, Marcus Merkel5, Christian Rehtanz6, Andr\u00e9 G\u00f6ring4, Andre Herrmann2, Ray Kodali2, Michael Stadler2, Mathias Uslar4, Nils Vogel2, Till Luhmann2 , Tobias Menz2 , Sven-Uwe M\u00fcller3 and Paul Recknagel3\n\n(1)\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\n(2)\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\n(3)\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\n(4)\n\nOFFIS Institut f\u00fcr Informatik e.V., Oldenburg, Germany\n\n(5)\n\nEWE NETZ GmbH, Oldenburg, Germany\n\n(6)\n\nef.Ruhr Forschungs-GmbH, Dortmund, Germany\n\nGert Brunekreeft\n\nEmail: g.brunekreeft@jacobs-university.de\n\nTill Luhmann\n\nEmail: Till.Luhmann@btc-ag.com\n\nTobias Menz (Corresponding author)\n\nEmail: Tobias.Menz@btc-ag.com\n\nSven-Uwe M\u00fcller\n\nEmail: sven-uwe.mueller@giz.de\n\nPaul Recknagel\n\nEmail: paul.recknagel@giz.de\n\nChapter at a glance\n\n * This chapter gives an overview of Germany's electric power system, its physical infrastructure, the regulatory environment, and the vision for smart grid development. The main topics presented were selected with the intention of providing examples of lessons learned and of sharing the German experience in the area of the main technological and regulatory challenges presented in the previous chapter.\n\n * The chapter contains a detailed description of the historical development and current design of German electricity markets with a special emphasis on market liberalization policies. It also focuses on the effects of aggressively expanding RES generation capacities in the context of such markets. The evidence presented here might be insightful for Chinese policy-makers given their will to promote the establishment of electricity markets and to increase RES generation capacities.\n\n## 4.1 Historical perspective\n\nReliability and affordability as the first policy goals\n\nIn the first decades of the electrification process, Germany's electricity system developed rather independently from governmental regulation. Power generation units and electric power grids were built up in a decentralized manner and operated by a variety of local and regional companies. After World War I, 220-kV transmission grids were constructed to interconnect local and regional power grids. The trend towards a nationally integrated electric power grid contributed to increasing competition between companies from different regions which in turn resulted in a pronounced market consolidation.\n\nThe time of little government interference ended in 1935, when the German government issued the Energy Industry Act (EnWG). The main objective of this law was to pave the way for the effective and efficient development of a nationally integrated and reliable electricity grid. This goal was supposed to be achieved by incentivizing investments in generation units and in the grid infrastructure by formally assigning monopoly rights to predominant companies. Monopolistic structures were deemed more suitable to guarantee a reliable power grid and to operate the grid in a cost-efficient manner taking advantage of economies of scale.\n\nAs a result of regional monopoly rights, EnWG created an electricity system with a high degree of vertical integration and a low degree of competition. Electricity generation and transmission assets were owned and operated by integrated utilities, while electricity distribution and retail was in the hand of integrated municipal utilities. The municipal utilities were owned either by local governments or by the integrated utilities responsible for generation and transmission, which then combined all stages of the electricity supply chain into a single company. To protect consumers against the market power of the newly installed monopolies, EnWG obliged the companies to provide electricity to every end consumer; the Act also regulated construction and expansion of power plants in order to ensure system stability.\n\nSustainability as a more recent policy goal\n\nOne important shift in Germany's electricity market regulation regime had its origin in the 1970 s, when environmental protection gained momentum as a new policy goal [1]. Due to high levels of local and regional air pollution caused by the combustion of fossil fuels for electricity generation, the German government issued the Federal Pollution Control Act (BIMSchG) in 1974. This law and its ordinances obliged power plants to install filter technology in order to reduce, for instance, sulfur dioxide (SO2) or nitrogen oxides (NOx) emissions. In the 1990 s, the German government further strengthened the role of environmental protection:\n\n * The 1991 Act on the Feed-In of Electricity from Renewable Sources into the Public Grid and its more prominent successor, the Renewable Energy Act (EEG) of 2000, had the objective to incentivize investments in renewable energies by guaranteeing investors financially attractive feed-in tariffs.\n\n * Another example of Germany's regime shift to environmental protection is the Electricity Tax Act (StromStG) of 1999 which, amongst other objectives, had the aim of inducing consumers to consume less electricity by raising electricity prices.\n\nPublic acceptance \u2013 towards a fourth energy policy goal\n\nThe increasing importance of sustainability is generally supported by the German population. However, citizens are more frequently opposed to new energy infrastructures near residential areas if these infrastructures are related to visible, audible, or olfactory effects. In the light of Germany's rather high population density, the build-up of distributed and renewable energy sources has of late entailed rising public opposition. An increasing number of citizens disapprove of investments in new wind farms, biomass power plants or transmission lines [2]. During the last few years, a certain number of energy projects \u2013 for instance new transmission lines or demonstration sites for carbon capture and storage \u2013 have failed to be realized owing to public opposition against them [3], [4]. As a consequence, public acceptance has recently gained prominence in the discussion as a fourth general energy policy goal in Germany, since it is only with a high level of public acceptance that the government and the companies are able to realize their investment plans [2]. Experiences in Germany reveal that three elements are important to ensure the support of the population for investments in energy infrastructure:\n\n * There has to be transparency on costs, benefits, and risks of new investments and technologies while the underlying motivations of the stakeholders involved in a project have to be communicated to the public.\n\n * The public has to be included in the entire planning process of new projects. Private citizens and other public stakeholders must be able to communicate their position and may also be allowed to invest financial funds of their own in the project.\n\n * Given that some conflicts cannot be solved unanimously, specific institutions or procedures for mediation and reconcialiation of interest are necessary to reduce number of court-cases [2].\n\nA short summary of market liberalization tendencies since 1996\n\nFor a long time, Germany's electric power system was characterized by a high degree of vertical integration and a low degree of competition. Today, the different stages in the supply chain are in a state of far-reaching unbundling, and competition has been established in the generation and retail sectors.\n\nThe market liberalization process on the European level began in 1996 with the First Electricity Directive [5], which was issued by the European Union (EU) and motivated by two main objectives [6]:\n\n * To open the electric power sector for third parties and to prevent discriminatory behavior towards generation companies by grid operators.\n\n * To allow end consumers to choose their retailer in an effort to increase the affordability of electricity through more competition. Thus, the protected supply areas (regional monopolies) of the incumbent retail companies were abandoned in favor of retail competition.\n\nBased on this directive, the German government in 1998 revised EnWG and started to liberalize Germany's electricity sector. After a short period of promising results with market entries of independent retail companies and decreasing retail prices, retail prices increased again. In addition, the market concentration did not decline significantly. Questions emerged regarding whether competition in generation and retail could be achieved as long as grid operators still had ownership in generation. Accordingly, the Second Electricity Directive issued by the European Union in 2003 contained a package of requirements to achieve legal unbundling. Legal unbundling can be described as an unbundling of accounts, operations, and information. It requires that transmission and distribution grid operators are independent from each other, as well as from generation and retail. In practice, legal unbundling requires a functional unbundling by guaranteeing independence in terms of legal form, organization\/management and decision-making.\n\nBased on the 2007 inquiry into the energy sector, the European Commission (EC) stated that, in spite of legal unbundling, the level of competition in the European energy market was still too low [7]. Major challenges were identified with respect to market concentration and vertical foreclosure.1 This document criticized the fact that with legal unbundling, a utility might still be able to discriminate against competitors or even restrict access of new market actors to the infrastructure. In addition, the Commission Paper stated that a grid operator involved in competitive sectors might be able to cross-subsidize its activities in the market with the revenues generated from the monopoly part of its business [8]. Furthermore, the European Commission was concerned about insufficient incentives for network investments, especially across borders. Generally speaking, markets for electricity were organized on a national basis and there was only a weak relation between the various national markets, as shown by grid congestions at most borders. The Commission argued that incumbents might postpone investments into interconnector capacities in order to protect their own market against cheaper electricity imports. This behavior is known as strategic investment withholding by locally integrated utilities [9].\n\nIn 2009, the EU's Third Electricity Directive introduced a compromise with three different options for unbundling on the transmission level. Basically, the aim of this rule was to separate the transmission grid from the other stages of the supply chain. The three options were:\n\n * Full ownership unbundling prohibits ownership of network and generation or retail assets by one and the same firm.\n\n * A model based on an Independent System Operator requires that an entity independent from the transmission grid owner takes over grid operation. With an independent system operator, network ownership can remain within an integrated company which also owns generation assets.\n\n * A model based on an Independent Transmission Operator (ITO) allows companies to retain both network ownership and management, but it puts strong limitations on cross involvement of employees in order to ensure network independence (please refer to [10] for further explanations on this model). In effect, the ITO model is similar to legal unbundling, though in a stronger form.\n\nIn Germany, the ITO model was applied. It had to be ensured that the transmission system was owned and operated by the ITO, which is legally independent from the commercial businesses of electricity generation and retail. Currently three out of the four transmission system operators (TSO) in Germany apply full ownership unbundling; the fourth is a genuine ITO. Distribution grids are currently subject to legal unbundling requiring administrative separation similar to the ITO model though in a less restrictive form. The objective is to ensure that no commercially sensitive information is exchanged between the power grid and other parts of the supply chain within one integrated company. Note that administrative unbundling is only applied for distribution system operators (DSO) with more than 100,000 customers. DSO with fewer customers do not have to unbundle and can remain an integrated part of a utility. This exception is known as the de-minimis rule.\n\n## 4.2 Today's power system and its most pressing challenges\n\n### 4.2.1 Power generation\n\nIn 2013, Germany's gross electricity generation amounted to roughly 634 TWh. Coal is currently Germany's predominant primary energy source, accounting for more than 45 % of total electricity generation.2 Nuclear power and gas are the second and third most important generation sources, accounting for approximately 15 % and 11 % of overall electricity generation respectively. Roughly 24 % of total electricity generation comes from RES, with wind accounting for 8.4 %, biomass for 6.7 %, solar for 4.7 %, hydro for 3.2 %, and household waste for 0.8 % [11].\n\nDuring the last 20 years, overall electricity generation increased only slightly (see Fig. 4.1). However, the composition of the electricity mix has changed significantly owing to two specific governmental policies: the promotion of RES initiated in the 1990 s and the nuclear phase-out promulgated in 2002. As a consequence, there has been a steady decline in the proportion of electricity generated by means of nuclear power from 29.2 % in 1993 to 15.4 % in 2013, and coal, from 55.7 % in 1993 to 45.2 % in 2013, while the share of RES in the electricity mix has increased from 4.0 % in 1993 to 23.9 % in 2013 [11].\n\nFig. 4.1\n\nElectricity generation in Germany from 1993 to 2013 in TWh, data from [11]\n\nThe rise in the share of RES generation went along with a considerable shift of the importance of different RES generation sources. While hydro power was by far the most important RES generation source in 1993, it plays no more than a minor part in 2013. Wind, biomass, and solar power, virtually non-existent in 1993, are the most important RES generation sources in 2013 (see Fig. 4.2). Electricity generation from PV has seen large growth in recent years (from 4.4 TWh in 2008 to 30 TWh in 2013).\n\nFig. 4.2\n\nComposition of the RES generation mix from 1993 to 2013, data from [11]\n\nA large part of Germany's RES installations are distributed generation sources such as small rooftop PV installations, single wind turbines, or biomass plants. A look at PV installations, for example, reveals that more than 60 % of all installations feed in electricity in a decentralized manner at the level of low voltage grids [12]. The focus on distributed energy sources is reflected in a diverse ownership structure. More than 40 % of all RES installations in Germany are owned by private investors, with project developers and financial institutions following with 14 % and 13 % respectively [13]. Note that only roughly 12 % of RES installations are owned by power generation companies [13].\n\n### 4.2.2 Power consumption\n\nGermany's electric power consumption amounted to about 528 TWh in 2013 [14]. The difference between gross electricity generation and consumption results from power plants' own consumption, from electricity exports to other countries, and from line losses. Industry is the main consumer of electricity and is responsible for approximately 46 % of national electricity consumption (see Table 4.1). The residential sector follows with 26 % while the commercial and public sectors consume about 14 % and 10 % respectively. The transport and agricultural sectors play no more than minor roles with shares of roughly 2 % [14].\n\nTable 4.1\n\nElectricity consumption in Germany in 2013, data from [14]\n\nConsumer | Electricity Consumption in TWh in 2013 | in % of Total\n\n---|---|---\n\nMining and Industry | 241 | 45.6\n\nResidential Sector | 139 | 26.3\n\nWholesale, Retail, Trade | 76 | 14.4\n\nPublic Sector | 51 | 9.7\n\nTransport | 12 | 2.3\n\nAgriculture | 9 | 1.7\n\nIn comparison to China, the shares of residential and commercial loads are significantly higher in Germany. This results in a load curve with more pronounced peaks and valleys. The ancillary services necessary to cope with this pronounced load curve are mainly offered by gas-fired power plants in Germany. Neither total electricity consumption nor the relative importance of different types of consumers has changed significantly in recent years.\n\nThe increasing share of electricity generated from intermittent sources like wind and PV led to the question of how power consumption can adapt to fluctuating generation. The potential for load shifting, which is relatively easily accessible at reasonable costs, lies in Germany's industrial sector with its large electricity consumers. Table 4.2 presents the maximum power which can be disconnected (neg.) or connected (pos.) for a short period of time in the residential, commercial, or industrial sectors according to different studies. The numbers have to been seen in relation to the German overall peak load of 80 GW.\n\nTable 4.2\n\nDemand side management potential according to German studies and sector\n\nStudy | Residential | Commercial | Industrial\n\n---|---|---|---\n\nStadler [15] | \u221268 GW (pos.)\/28 GW (neg.)\n\nKlobasa [16] | 20 GW | \u2013 | 2.8 GW\n\nDena II [17] | 7\u201332 GW | 2.4 GW (pos.)\n\n14.3 GW (neg.) | 3.9 GW (pos.)\n\n6.5 GW (neg.)\n\nTU Dortmund University [18] | 5.0 GW (pos.)\n\n6.3 GW (neg.) | 3.0 GW (pos.)\n\n18.1 GW (neg.) | 0.9 GW (pos.)\n\n11.2 GW (neg.)\n\n### 4.2.3 Power logistics\n\nDisparity between generation and consumption\n\nPower generation and consumption are not equally distributed in Germany. The load centers are situated in western and southern Germany, both regions with strong industrial bases. Since the amount of electricity generated in nuclear and coal-fired power plants in these regions is generally not sufficient, they often have to import electricity from other parts of Germany or from neighboring countries. In contrast, Germany's north and east, with their significant wind capacities, quite regularly generate more electricity than they consume. Thus, both regions frequently transfer electricity to southern and western Germany.\n\nGrid infrastructure\n\nGermany's electric power grids can be classified into four different categories:\n\n * Extra high voltage grids (220-kV to 380-kV) form the German transmission grids. In addition to the transmission of electricity, they are responsible for the electricity feed-in of large generators such as nuclear and coal-fired power plants, or offshore wind farms. The transmission grid is mainly characterized by suspended above-surface cables with visible electricity pylons. There are currently approximately 35,000 km of transmission grids with 1,100 electricity transformers in Germany [19], [20].\n\n * High voltage grids (35-kV to 110-kV) are the highest voltage level of distribution grids. They act as a redistribution system at the regional level. Furthermore, high voltage grids provide electricity to large industrial consumers and are also employed to feed in electricity from smaller power plants, wind farms, and large PV parks. There are approximately 95,000 km of high voltage grids and 7,500 electricity transformers at this level [19] [20].\n\n * Medium voltage grids (10-kV to 30-kV) represent the subordinate level of distribution grids. They distribute electricity to the connected low voltage levels, provide electricity to connected bulk consumers, and feed in electricity from small PV parks or single wind turbines. The medium voltage level is characterized by underground cables; it is roughly 507,000 km in length and contains 560,000 local substations [19] [20].\n\n * Low voltage grids (230-V to 400-V) are typically also characterized by underground cables and distribute electricity from local substations to households and collect electricity from rooftop PV modules. It has an approximate length of 1,150,000 km [19].\n\nAn increasing amount of network congestion at times of peak generation is caused by small distributed rooftop PV installations on the low voltage level and rising feed-in from large wind farms at the high voltage level [21]. Due to the rapid build-up of RES generation capacities, grid capacities are not always sufficient to absorb RES-E. As a result, grid curtailment rates of solar and wind power have increased significantly within the last few years. In Schleswig-Holstein, a windy region in the north of Germany, 3.5 % of the total wind generation had to be curtailed in 2012 [22].\n\nThe curtailment of RES-E at times of peak generation can reduce the need for network investments. A recent study suggests that curtailing 30 % of PV peak production and 20 % of wind peak production could reduce infrastructure investments by 10 % between now and 2030 while a total of only 2 % of the annual electricity production from RES would be curtailed [23].\n\nThrough the transmission grid, Germany's electric power system is well interconnected with those of neighboring countries (please refer to Fig. A.1 in the appendix for a snapshot of Germany's transmission grids). All German TSO are members of the European Network of Transmission System Operators for Electricity (ENTSO-E), which was established in 2011 in order to\n\n> promote the completion and functioning of the internal market in electricity and cross-border trade and to ensure the optimal management, coordinated operation and sound technical evolution of the European electricity transmission network [24].\n\nSupply security in Germany\n\nIn comparison with other European countries, Germany's electric power system is characterized by a very high level of security of supply with, on average, only about 15 minutes of annual interruptions on the household level [25] [26].\n\nThe increasing feed-in of RES generation imposes challenges for the stability and reliability of Germany's distribution grids. Three technical challenges for network stability caused by RES integration into distribution grids in Germany are presented in Table 4.3, together with the measures most frequently used to overcome them.\n\nTable 4.3\n\nFrequently used measures to maintain supply security in the presence of RES, data from [27] | Supply Security Issue\n\n---|---\n\nMeasure | Grid Overload | Critical Voltage Variation | Power Quality\n\nDirect connection of RES to a substation\n\n| |\n\nX | X\n\nUpgrade of grid circuit conductors | X | X | X\n\nUpgrade of upstream transformer capacity | X | X | X\n\nReduction of the grid circuit length\n\n| |\n\nX | X\n\nSet point adjustment of transformer automatic voltage control\n\n| |\n\nX\n\n|\n\nUsing reactive power capabilities of RES\n\n| |\n\nX\n\n|\n\nConstruction of a new substation | X\n\n| |\n\nInvestment needs in the grid infrastructure\n\nSecuring a high level of supply security in spite of the increasing share of electricity generated by variable RES requires significant investments in transmission and distribution grids. On the transmission grid level, it is estimated that roughly 3,600 km of 380-KV AC overhead lines will have to be installed between now and 2023 [23]. This represents a total investment of EUR 21 billion [28]. On the distribution grid level, the pressure is even higher: between 135,000 and 193,000 km will have to be added to the existing network by 2030. In addition, between 21,000 and 25,000 km of the existing distribution grid will have to be modernized in the same period of time. According to a recent study, these numbers add up to a total investment need of roughly EUR 42.5 billion on the distribution grid level [18].\n\n## 4.3 Smart grid development in Germany\n\n### 4.3.1 Motivation for smart grids in Germany\n\nThe rising importance of intermittent RES generation is the main smart grid driver in Germany. Today, the general opinion of most energy market experts in Germany is that building a smart grid, especially a smart distribution grid, is a cost-efficient way of ensuring security of supply in the presence of large-scale integration of intermittent RES [29], [30].\n\nThe challenge of fluctuating RES in extra high voltage grids\n\nGermany's transmission grids (380-kV\/220-kV grid) have already achieved a high degree of smartness and are equipped with sophisticated real-time monitoring and control technologies. The increasing amount of wind power from large wind farms creates a need for more grid control. Sophisticated generation forecasts, for example, are needed to adequately react to the pools of fluctuating generators and maintain the 50 Hz grid frequency within its narrow tolerance range of \u00b1 0.2 Hz.\n\nThe challenge of fluctuating RES in high voltage grids\n\nThe 110-kV high voltage grid also requires high availability and near-real-time monitoring and control. The main challenge within high voltage grids is to maintain voltage levels and loads within a technically viable band. In the event of overloads, for example arising from a high volume of RES-E, electricity has to be transferred to the higher voltage level. Bidirectional flows of electrical power are an additional challenge at the level of 110-kV high voltage grids. If overloads cannot be transferred to the higher voltage level, generation has to be curtailed or additional loads have to be activiated.\n\nThe challenge of fluctuating RES in medium voltage grids\n\nSupply quality, specifically with regard to voltage maintenance, constitutes a major technical challenge in medium voltage grids due to the fluctuating and distributed generation from RES. The degree of utilization of ICT in medium voltage grids is limited. Continuous load measurement, for example, is used only for customers with consumption levels exceeding 100 MWh\/a. As prescribed by the Electricity Network Access Ordinance (StromNZV), these customers' average power consumption must be measured in periods of 15 minutes and this information delivered to the distribution grid operator which then uses the measurement data to compute a specific load profile. The measurement equipment is operated by the DSO or by the metering system operator. Like at the 110-kV level, wind and PV plants may result in inverted flows of electricity to the higher voltage level in order to avoid an overload of grid assets, especially in rural areas with a more limited infrastructure.\n\nThe challenge of fluctuating RES in low voltage grids\n\nToday, ICT-based grid operation is very rarely installed at the level of low voltage grids, where rooftop PV represents a major challenge in terms of voltage maintenance and can cause a more rapid aging of grid assets. Grid operators currently handle these challenges by expanding the grid infrastructure with new cables or local substations. In the future, electric mobility may further increase the necessity for active control of low voltage grids. It should be noted that the control of assets in low voltage grids is especially difficult due to the large number and high heterogeneity of the connected assets (e. g. households, rooftop PV modules, local substations, electric vehicles). Thus, standardization of control interfaces is viewed as one of the key issues for assets being installed in low voltage grids [31].\n\n### 4.3.2 Germany's technological view of the smart grid\n\nThe development of smart grids in Germany\n\nIn Germany, smart grid technologies have been described, combined, tested, and implemented in a bottom-up process by research institutions, companies from the electric power sector, component suppliers, and ICT companies.\n\nThe primary driver for smart grid development was the integration of RES into the operational environments of grid operators. Their integration mainly relies on large monolithic supervisory control and data acquisition (SCADA) systems. Small amounts of renewables were controlled in parallel to the overall grid operations, often in so-called distributed energy management systems (DEMS). In terms of communications, the systems used existing communication infrastructure and heterogeneous proprietary data models and protocols. The need to integrate RES in daily grid operations led to a change in the paradigms on how to design and control RES. Aspects relating to the connection between different assets were the first to be focused upon \u2013 general packet radio service (GPRS), GSM, universal mobile telecommunications system (UMTS), and currently long term evolution (LTE) or IP-based open networks such as the internet have been used. After this initial focus on connectivity, more emphasis was put on the semantics and syntactical aspects of communication.\n\nThe government's view on smart grids\n\nAs in China, different stakeholders in Germany have developed different views on smart grids. The primary goal of the German government, especially via the Federal Ministry for Economic Affairs and Energy (BMWi) and BNetzA, is to guide the debate and support convergence of the various stakeholders' smart grid visions. BNetzA, in late 2011, published a position paper called Smart Grid and Smart Market [32] (see [33] for an English summary of this document).\n\nThe main objective of this document was to introduce a clear-cut criterion on how smart grids and so-called smart markets can be differentiated and to discuss the regulatory consequences. BNetzA points out that electricity volumes and related services have traditionally been traded on electricity markets independently from the available grid capacity.3 In a power system based on smart grids, however, information on current grid status can be taken into account in market transactions. Markets allowing the trade of electricity volumes and related services based on available grid capacities are referred to as smart markets. Depending on the available grid capacity, smart markets can either operate without restriction \u2013 in case of sufficient grid capacity, or \u2013 in case of grid congestion \u2013 the grid operator has the right to intervene in the market to ensure grid stability and e. g. shut down power plants or cut off consumers [33]. One example for smart markets are regional energy market places.4 Within a specific region, industrial, commercial, and domestic customers are given the option of trading electricity volumes and\/or ancillary services in a market place. By trading ancillary services, power consumption schedules, and power generation (feed-in) schedules, market participants are exposed to price signals serving as an economic incentive to balance electricity supply and demand and thus stabilize the grid.\n\nThe position paper Smart Grid and Smart Market discusses relevant topics along six key concepts:\n\n * The first key concept, named Grid capacity and energy volumes as distinguishing criteria for grid and market, explains how grids and markets can be separated by identifying the main topics involved. All aspects relating to grid capacity (as measured in kW, MW, GW, etc.) refer to the grid whereas all topics relating to energy volumes (as measured in kWh, MWh, GWh, etc.) refer to the market.\n\n * The second key concept, Clarification of the discussion about the energy future through the terms of smart grid and smart market, follows-up on the first key concept. It clarifies that the term smart grid can be related to network issues while the term smart market can be related to energy volume issues.\n\n * The third key concept has the somewhat cumbersome title The energy future requires more responsibility on the market and more negotiated solutions. The grid should play a predominantly service role and should be separated from competitive activities as far as possible. It discusses the importance of new market actors in smart markets and underlines that competitive functions, especially those in smart markets, should not be attributed to grid operators. Grid operators are considered responsible only for the (smart) grid itself. Smart grids are seen as a platform for smart markets. Grid operators are consequently viewed as playing a supporting role for smart markets.\n\n * The fourth key concept, entitled Smart meters are part of, but not an absolute prerequisite for, the energy future, states that grids can be made smart without a widespread rollout of smart meters. The main argument is that it is sufficient to measure data on grid conditions in local substations or to install only some smart meters at potentially critical junctures in the grid.\n\n * The fifth key concept, named The smart grid is a part of an evolutionary, not a revolutionary, process, emphasizes that smart grids are not built from scratch but evolve in a gradual process. In the light of the heterogeneity of the various grid operators in Germany, BNetzA consequently stresses that a kind of uniform smart grid concept applicable to every grid operator does not exist and should not be promoted by means of regulation.\n\n * The sixth key concept is named If targets for the use of renewable energy are to be met it is essential that these producers, too, respond to market signals and grid exigencies. It underlines the importance of integrating RES more effectively in wholesale markets, potentially by redesigning the feed-in priority for RES.\n\nSmart grids according to a recent study by the German Academy of Science and Engineering\n\nIn 2012, under the guidance of the German Academy of Science and Engineering (acatech), representatives of the electric power sector, equipment manufacturers sector, ICT sector, and from academia and research institutions developed a smart grid model for Germany: the Future Energy Grid (FEG) model [30]. The model complements the BNetzA view on smart grids by developing a conceptual and technological foundation for the separation of smart grids and smart markets. In particular, FEG can serve as a best practice example of how to develop and formulate a comprehensive smart grid vision. FEG is a systematic and comprehensive top-down approach that can be used to evaluate the current smartness of grids and to define a smart grid vision. It systematically addresses specific problems and challenges in Germany's electric power system and introduces a model of system layers (see Fig. 4.3) and technology areas (see Fig. 4.4). The system layers represent different functions and requirements regarding the application of ICT in the power system. They were chosen in reference to a model adapted by the European Electricity Grid Initiative (see [34]).\n\nFig. 4.3\n\nAbstract smart grid system model regarding the application of ICT within three distinct layers, translated from [30]\n\nFig. 4.4\n\nTechnology areas regarding ICT aspects of smart grid implementation in Germany, translated from [30]\n\nIn total, FEG comprises the following three system layers (see Fig. 4.3):\n\n * The innermost layer, referred to as the closed system layer, contains the critical infrastructure and power system equipment that serves as the backbone of the system and requires a high level of security and safety. Therefore, external access to the resources within this layer is restricted and may be limited to the grid operator or to an equivalent actor. Central (bulk) power generation, transmission and distribution grids, and the corresponding ICT-based control systems are components of this layer.\n\n * The outermost layer is referred to as the networked system layer. It contains heterogeneous power system components (distributed power generators, power storage units, consumers, marketplaces, meters, control applications, etc.) which are characterized by a high level of communication and information exchange. In contrast to the closed system layer, much of the value within this layer is created by interactions between the different participants on smart markets. As the exchange of sensitive power system information, e. g. real-time data on power generation and consumption, is of particular importance in this context, strict ICT and data security protocols have to be applied to ensure individual privacy rights are respected and overall power system security is guaranteed.\n\n * The ICT infrastructure layer enables communication within and between the two other layers. It contains the communication networks and associated components that provide ICT interface functionalities. In order to ensure that different components of each layer can communicate with each other, interoperability is a key factor. Interoperability is achieved with the help of standardization of system interfaces and communication protocols.\n\nIn the study Future Energy Grid, a smart grid vision based on the three system layers described above and nineteen technology areas is outlined (see Fig. 4.4, for a detailed description refer to appendix D).\n\n## 4.4 The regulation of Germany's electric power system\n\n### 4.4.1 Policy setting and fundamental institutions\n\nPolicy setting\n\nThe Federal Government's Energy concept for an environmentally friendly, reliable and affordable energy supply of September 2010 and The road to the energy of the future \u2013 safe, affordable and environmentally friendly (Key Elements of an energy policy concept) of June 2011 [35] contain guidelines and objectives relating to Germany's future energy system. In particular, the trend towards more environmental protection is explicitly expressed by government plans to reduce CO2 emissions to 60 % of the 1990-level by 2020. It is planned to further reduce emissions to 20 % of the level of 1990 until 2050 [35].\n\nThese cuts in CO2 emissions are to be achieved by reduced energy use for transport and heating (see Fig. 4.5): e. g. energy consumption for room heating purposes should be reduced by 20 % between 2008 and 2020 and 80 % by 2050. For the power sector, the government's objective is to generate 35 % of electricity with RES in 2020 and to increase the share to 80 % by 2050 [35] as shown in Fig. 4.5. At the same time, in the aftermath of the nuclear disaster in Fukushima, the German government decided to completely phase out nuclear power generation by 2022 [35].\n\nFig. 4.5\n\nLong-term targets for Germany's energy sector (\u00a9 Heinrich-B\u00f6ll-Stiftung e. V. [36])\n\nMany specific objectives with regard to the development of Germany's power system are subordinated to the general goal of achieving more sustainability and the specific goal of increasing the importance of RES: for instance, the German government wants to expand transmission grids in the north-south direction, thus allowing a more effective transport of wind power from the north to the load centers in the south of the country. Other government goals such as improving energy efficiency and promoting energy storage technologies and electric vehicles are also related to the broad government plan of increasing the sustainability of Germany's power system.\n\nGeneral governance structure\n\nThe governance structure of Germany's energy system comprises several ministries and independent institutions. The ministries are responsible for enacting laws and ordinances that then have to be applied by independent institutions. This means that the ministry concerned can neither interfere in day-to-day business nor expand or restrict the competences of the institutions. Nonetheless, these institutions and the ministries cooperate closely.\n\nMinistries responsible for Germany's energy policy\n\nThere are currently two ministries at the core of the governance structure of the German electricity system:\n\n * The Federal Ministry for Economic Affairs and Energy (BMWi) has the main responsibility for formulating and implementing energy policy, including renewable energy, and is responsible for issues related to security of supply and competition policy.\n\n * The Federal Ministry for the Environment, Nature Conservation, Building and Nuclear Safety (BMUB) is responsible for those energy policy issues which are directly related to environmental protection, e. g. CO2 reduction, and energy efficiency in the building sector.\n\nThe market design of the electricity sector is a responsibility shared by BMWi and BMUB. Other relevant ministries in the context of energy and electricity sector policy and smart grids are:\n\n * The Federal Ministry of Transport and Digital Infrastructure (BMVI) takes responsibility for transportation and mobility issues as well as for the expansion of digital communication infrastructure, which is especially important as a backbone for smart grids.\n\n * The Federal Ministry of Labor, Social Affairs and Consumer Protection (BMAS) focuses on social issues related to energy.\n\nInstitutions responsible for Germany's energy policy\n\nThe following three government authorities are of particular relevance with regard to the regulation of Germany's electric power system:\n\n * The Federal Network Agency for Electricity, Gas, Telecommunications, Post and Railway (BNetzA) is responsible for regulation of the networks which are natural monopolies, including the electricity grid (see Fig. 4.6). The existence and competences of BNetzA are laid down in laws such as EnWG and the Grid Expansion Acceleration Act for Transmission Networks (NABEG).5 While BNetzA is in charge of national and interstate regulation it cooperates closely with regulatory counterparts on the level of the federal states. State regulators are responsible for DSO with less than 100,000 customers and BNetzA for all TSO and for DSO with more than 100,000 customers or with operations in more than one state.\n\nFig. 4.6\n\nResponsibilities of BNetzA and of the Federal Cartel Office\n\n * The Federal Cartel Office is responsible for general competition matters (see Fig. 4.6). If competition problems are related to natural monopoly networks, the Federal Cartel Office can authorize BNetzA to handle the issue. The existence and competences of the Federal Cartel Office are laid down in the Act Against Restraints of Competition (GWB).\n\n * The Monopoly Commission advises on competition and monopoly issues. Its advice is non-binding and it does not have decision-making powers. Nonetheless, the Monopoly Commission plays a vital role in checking and evaluating the regulator's work. The tasks of the Monopoly Commission are also laid down in GWB.\n\nA brief history of BNetzA\n\nThe liberalization of European electricity markets began with the EU's First Electricity Directive of 1996. A so-called negotiated Third Party Access (nTPA) was allowed as an option alongside regulated Third Party Access (rTPA). nTPA meant that access to the electricity networks, including network charges, had to be negotiated between network owners (grid operators) and network users (power companies). The directive did not explicitly prescribe a regulator and ultimately this approach failed to secure non-discriminatory network access and to deliver fair and reasonable network charges (cf. e. g. [37], [38] for an analysis and further literature).\n\nThe EU's Second Electricity Directive of 2003 contained significant changes: rTPA became the only option making non-discriminatory network access conditions a requirement by law. The Directive also demanded the establishment of an electricity sector regulator and the creation of a regulatory framework for fair and reasonable network charges. In Germany, EnWG was amended to satisfy these demands, resulting in the establishment of BNetzA as a federal regulator for monopolistic networks and the development of incentive regulation (see Sect. 4.4.4 for more information on incentive regulation). The regulatory competences of BNetzA were based on the competences of its predecessor, which was the Regulatory Agency for Telecommunications and Post Services. Competences relating to electricity and gas were added, and the agency was renamed to BNetzA. Later on, the task of regulating the railway infrastructure was added as well.\n\nThe EU's Third Electricity Directive of 2009 did not change the arrangements on rTPA or regulation. Instead, it strengthened the arrangements relating to unbundling rules. For the TSO, this led to the ITO approach while for DSO the unbundling rules stayed as they were in the Second Electricity Directive (see Sect. 4.1). BNetzA is also responsible for implementing the unbundling rules and monitoring compliance with them. Lastly, the Third Electricity Directive led to the creation of an Agency for the Cooperation of Energy Regulators (ACER). In a nutshell, ACER is responsible for cross-border issues and provides a platform for cooperation between various European regulators.\n\nMain tasks and competences of BNetzA\n\nThe mission of BNetzA is to regulate the monopolistic part of the supply chain \u2013 the grid or network infrastructure by:\n\n * guaranteeing an affordable, consumer-friendly, efficient and environmentally friendly supply of electricity and gas,\n\n * ensuring an effective and undistorted competition in the supply of electricity and gas as well as securing a reliable operation of electricity and gas grids,\n\n * transposing and implement EU law in the field of grid-bound energy supply and\n\n * facilitating efficient approval processes to adapt the German high-voltage transmission grid to the needs of a rising share of renewable energy [39].\n\nIn this context, the two main tasks of the BNetzA are:\n\n * to secure non-discriminatory access to the network and\n\n * to regulate network charges.\n\nThis is reflected by the organizational structure of BNetzA (see Fig. 4.7). BNetzA consists of several departments. Two of them (Department 6 and Department N) focus on energy and network regulation. The decision process within BNetzA takes place within so-called ruling chambers. BNetzA has nine ruling chambers with decision-making powers, with five of these relating to electricity and gas:\n\nFig. 4.7\n\nBNetzA organization chart, adapted from [40]\n\n * network development and approval of individual network charges (ruling chamber 4 in Fig. 4.7),\n\n * access to electricity networks (ruling chamber 6 in Fig. 4.7),\n\n * access to gas networks (ruling chamber 7 in Fig. 4.7),\n\n * regulation of electricity networks (ruling chamber 8 in Fig. 4.7), and\n\n * regulation of gas networks (ruling chamber 9 in Fig. 4.7).\n\nNote the focus on and restriction to networks as the core monopoly part of the supply chain. BNetzA is not responsible for the markets, where these are not related to the networks. Strictly speaking, BNetzA is not responsible for general competitive conditions, for example merger policy, which is one of the tasks of the Federal Cartel Office. In practice, however, the Federal Cartel Office and BNetzA cooperate closely. Moreover, BNetzA monitors market development in a so-called Monitoring Report, which is published on an annual basis.\n\nAdditional tasks of BNetzA\n\nIn addition to securing non-discriminatory access to the network and regulating network charges, further BNetzA tasks are:\n\n * ensuring consumer protection in retail issues (e. g. rules for switching the power retail company),\n\n * implementing and monitoring unbundling rules,\n\n * evaluating the network development plan (NDP),\n\n * approving network expansion plans and helping to accelerate licensing procedures for network expansion, as arranged by NABEG,\n\n * exchanging information with other European regulators, formally or informally, and cross-border issues (e. g. the interconnectors),\n\n * providing support for technical standards, and\n\n * providing data on power plants and electricity networks to the public.\n\nCompetences of BNetzA\n\nIt is of critical importance for the regulator to be powerful enough to impose sanctions on the grid operators. In Germany, this is regulated in \u00a7 29 to \u00a7 33 EnWG, which define the competences of BNetzA and the possible range of penalties it can impose:\n\n * \u00a7 29 EnWG lists all discriminating behaviors of grid operators which can be penalized by BNetzA.\n\n * According to \u00a7 30 EnWG, BNetzA can force grid operators to stop any discriminating behavior against other market participants.\n\n * According to \u00a7 31 EnWG, information on discriminatory behavior of a grid operator can be provided to BNetzA by any legal or natural person.\n\n * \u00a7 32 and \u00a7 33 EnWG specify how fines and compensation payments are to be settled in case of misconduct by a grid operator.\n\n### 4.4.2 Market structure\n\nVertical and horizontal market structure\n\nAs described in Sect. 4.1, the stages in the supply chain of Germany's electric power sector are in a state of far-reaching unbundling: transmission grids, for example, are owned and operated by fully unbundled companies that are independent from other parts of the supply chain. Distribution grid operators are legally unbundled from generation and retail companies so as to ensure that, within the same utility, no commercially sensitive information is exchanged between the power grid and other parts of the supply chain.\n\nCompetition in power generation has been increasing significantly in Germany since the EU's First Electricity Directive. Before 1996, generation was monopolized by four major companies (RWE, E.ON, Vattenfall Europe, and EnBW). Meanwhile, these four companies together represent a market share of no more than roughly 44 % of total installed electricity generation capacities [41]. The decreasing market share of the former monopolists is also a result of the nuclear phase-out and the increasing share of distributed generation from RES. The growing importance of RES in particular has served as a key driver for competition in the generation sector. While investments into conventional power plants are a capital-intensive business, investments into RES have become profitable for small investors due to the guaranteed feed-in tariffs for renewables. As a result, there are currently some 300 smaller generation companies with capacities starting at 1 MW up to hundreds of MW.\n\nThe situation is similar in the retail sector. The market share of the four former monopolists has been continuously decreasing from 50 % in 2008 to 45 % in 2011 [19]. Most German retail companies have a regional focus with a high market share within their established service areas. Consumer switching rates to other retailers are still quite low due to the end consumers' tendency to remain with the incumbent regional suppliers. In 2012, for example, only about 7.8 % of all households in Germany changed their electricity supplier [19].\n\nThe ownership structure on the transmission and distribution level is as follows: on the transmission level, four TSO own the infrastructure while roughly 900 DSO own parts of the distribution grid. Since electricity networks are a natural monopoly with network charges regulated by BNetzA, there is no competition for markets and customers between the different grid operators.\n\nThe emergence of new market actors\n\nThe unbundling process, the legally enforced trend towards more competition, as well as the migration towards smart grids and RES have contributed to the emergence of new market actors in Germany. Their growing importance can be considered as the most profound change in Germany's electricity market structure during the last few years. A considerable number of new players have entered the supply chain of the electric power system: Figure 4.8 shows that the number of companies active in the German energy sector (including electricity, gas, heat, etc.) increased from 15,666 in 2006 to 48,292 in 2011 [42]. This represents an increase of more than 200 % within five years. Especially companies with less than nine employees, often innovative start-ups and energy service providers, have contributed to this increase. Their number increased from 14,545 in 2006 to 46,967 in 2013 [43].\n\nFig. 4.8\n\nNumber of companies active in the German energy sector, data from [43]\n\nFigure 4.9 presents an exemplary overview of established and new market actors in smart grids, as they are present or emerging in Germany, classified into the different smart grid supply chain areas Power Generation, Power Logistics, Power Trade and Retail, Power Consumption, and Information and Communication. In the following, some examples of new market actors depicted in Fig. 4.9 are described together with a brief explanation on their role in the smart grid development process:\n\nFig. 4.9\n\nExamples of established and new market actors in smart grids in Germany\n\n#### Power Generation:\n\n * RES operators:\n\nTraditionally, power plants in Germany were exclusively owned and operated by large utilities. Due to the financial support codified in EEG (see Sect. 4.4.3), a number of more than 1,500,000 RES plants, especially onshore wind, PV and biomass plants, has been installed so far. The largest part of these plants is operated by households as well as small and medium-sized companies: in 2013 for example, 6 % of all German households had their own RES generation units, especially small rooftop PV installations [44]. Companies in the manufacturing industry have also long since started to build their own RES generation units. By 2005, roughly 5 % of all German manufacturing companies owned RES. This number has more than tripled since, reaching roughly 18 % in 2012 [45].\n\n * New energy cooperatives:\n\nIn the tradition of cooperatives founded in Germany in the beginning of the 20th century to develop the first power supply systems, new energy cooperatives have emerged in recent years. These associations allow individual citizens or civil society to pool their financial resources and jointly invest in power system components otherwise exceeding the financial resources of their individual members. In Germany, 650 energy cooperatives with roughly 130,000 members invested more than one billion euros in power plants based on RES until 2012 [46].\n\n#### Power Logistics:\n\n * Grid operations service providers:\n\nThis type of company specializes in offering services to operate smart grids for small-sized or municipally owned German DSO. The business model of grid operation service providers works out, since the small DSO often do not have the highly specialized personnel required for smart grid operation (i. e. with deep knowledge of ICT capabilities and with the required level of grid automation knowledge). A single grid operation service provider may operate the smart grids of several small DSO.\n\n#### Power Trade & Retail:\n\n * VPP operators:\n\nA virtual power plant is a network of decentralized, small to medium-scale power generating units such as biomass plants, combined heat and power (CHP) units, wind farms and solar parks. The interconnected units are partly operated through central control of the virtual power plant but nevertheless remain independent in their operation and ownership. Virtual power plants (VPP) deliver electricity products, such as balancing power, that can be traded on electricity market places. Product requirements, e. g. the minimum volume of the delivered power, are restrictive and usually cannot be met by single small scale power plants, like e. g. a single wind farm. VPP therefore bundle (aggregate) several small scale power plants and often even add other generation capacities and\/or flexible loads, to fulfil the product requirements of the energy market places. Thus, the power generation of the units in the virtual power plant is bundled \u2013 or aggregated \u2013 and sold by a single trader on the energy exchange or other energy market places (e. g. market for balancing power). As a result, VPP can gradually take over the role of traditional power plants \u2013 selling their output in the wholesale markets. Today, in Germany, about 20 medium sized companies operate VPP.\n\n * Specialized marketplace operators:\n\nThese market actors operate market places e. g. for ancillary services or for electricity from well-defined sources. The concept of specialized market places has been piloted in several research projects of the German E-Energy program (see Sect. 4.4.6 for more information on the E-Energy program).\n\n * Power traders:\n\nA person or entity that buys and sells energy goods and services in an organized electricity market (electricity or power exchange) or over-the-counter (OTC). Power traders offer dedicated electricity wholesale services to other market actors, e. g. industry companies or power retailers companies or larger end-users (like energy-intensive industry). Due to the complex nature of electricity markets, trading requires specialist knowledge and expertise, comparable to financial service providers. In Germany, power trading services are offered by some 50 companies [19].\n\n * Independent retailers:\n\nLiberalization of the energy market in Europe led to the establishment of mostly medium-sized power retail companies that are independent from the established utilities. These companies offer their customers heterogeneous energy-based retail products, e. g. regional tariffs, time-of-use pricing or electricity with a low CO2-footprint. These products are widely accepted both by the population and by enterprises.\n\n#### Power Consumption:\n\n * Smart appliance contractors:\n\nHouseholds as well as enterprises operate a growing multitude of power-consuming appliances like heating equipment, cooling devices or home electricity storage (so-called smart appliances). For these clients, smart appliance contractors offer individual services such as financing, installation, operation, maintenance, support and appliance replacement. Other contractors act as full-service providers and offer volume-based heating, cooling or load management services.\n\n * Prosumers:\n\nThe term prosumer is merged from the terms producer and consumer. Besides consuming power, these new market actors deliver surplus power to the grid, e. g. through small-scale rooftop PV or combined heat and power (CHP) plants.\n\n * Energy management service providers:\n\nEnergy management service providers deliver energy monitoring and controlling services to industry and large commercial companies. With their service portfolio they contribute to continuous improvement of energy procurement and use in smart grids.\n\n * Energy efficiency consultants:\n\nThese typically small-sized companies analyze the energy consumption of private households, enterprises, industry and municipalities in order to identify potentials for energy savings and energy efficiency improvements and consult the clients in efficient power usage. In a typical business model the advisory is paid for with a share of the savings generated from energy efficiency improvements. In Germany, a number of nearly 12,500 companies carried out more than 400,000 consulting projects in 2011 [47].\n\n * E-Vehicle infrastructure operators:\n\nElectric vehicles need charging stations. These are built and\/or operated by a growing number of infrastructure operators.\n\n * E-Vehicle service providers:\n\nThese new market actors are typically big-sized or mid-sized companies. E-Vehicle service providers operate pools of electric vehicles and rent them to companies and private consumers.\n\n#### Information & Communication:\n\n * Metering system operators:\n\nThese companies install and operate electricity metering equipment. Metering system operators are an example for a new market role that has been created by the German government. Their role is described by EnWG (\u00a7 21) and the Metering Access Ordinance (MessZV).\n\n * Metering service providers:\n\nMetering service providers offer the service of reading out meter systems and delivering the gathered data to power retailers as a basis for billing. Their role is also described by EnWG (\u00a7 21) and the Metering Access Ordinance.\n\n * Energy information service providers:\n\nAll market actors in smart grids require energy-related information to carry out their tasks and businesses, e. g. current or historical grid status data, metering data or weather data. Energy information service providers collect raw data from multiple sources, analyze and refine the data and then offer specialized information services to their customers. One example of an energy information service is wind and PV power generation forecasts, which are typically derived from a multitude of different weather data sources.\n\n * Energy system integrators:\n\nEnergy system integrators are established or new companies which develop ICT-based system solutions in all segments of the smart grid supply chain for their customers, e. g. solutions for advanced distribution system management and grid maintenance solutions for DSO, smart metering solutions for metering service providers or virtual power plant management solutions for VPP operators. The ICT sector in Europe has increasingly been participating in the development of smart grids and is involved in approximately 60 % of all related research projects [48]. The ICT-related smart grid concepts developed by energy system integrators contribute to the general understanding of smart grids among established and new market actors, public decision-makers, and the general public.\n\n### 4.4.3 Market design and RES integration\n\nGeneral market design\n\nGerman electricity wholesale markets bring together roughly 300 power generation companies, about 50 power trading companies, and approximately 1,110 power retail companies [19]. A high level of liquidity indicates that electricity wholesale markets are functioning well [19]. The German wholesale market is currently separated into two major energy-only markets (see Fig. 4.10):\n\nFig. 4.10\n\nElectricity wholesale markets in Germany\n\n * The European Energy Exchange (EEX) with two products: spot (short-term) and future (long-term) markets for electricity. In contrast to China, there is only one uniform wholesale price for electricity in Germany irrespective of the power source, production technology, or age of the power plant under consideration. The market price \u2013 for all generators \u2013 at any given time is determined by the marginal costs of the last power plant required to satisfy total electricity demand. This nationally integrated market leads to a situation in which, at any point in time, only those power plants with the lowest marginal costs of production are able to sell their electricity on the market.\n\n * The over-the-counter (OTC) market gives suppliers and buyers of electricity the opportunity to bilaterally trade electricity and to negotiate contracts and prices irrespective of standardized contracts or prices at the power exchange. Like the EEX, OTC contracts offer the possibility for spot and future trades. Products on both markets can be the same, e. g. short-term contracts with direct physical fulfillment can be either traded via the exchange or negotiated directly with another party on the OTC spot market.\n\nMost of Germany's electricity is traded bilaterally between generation and retail companies. In 2012 for example, 7,000 TWh of electricity were traded in OTC transactions, whereas only approximately 1,200 TWh were traded at the EEX [19].6 The attractiveness of OTC trading results from the fact that OTC products can be designed more flexibly according to the specific needs of the parties involved. Nonetheless, EEX prices are very important because they serve as a reference value for OTC trading.\n\nWhile generation and retail companies use the power exchange to trade electricity especially for short-term contracts (physical fulfillment), most of the trade at the power exchange is focused on the exchange of futures. Here electricity traders focus on financial exchanges. Traders expect to gain benefits through the arbitrage between different future periods. Retail companies have to pay the generators for the electricity produced and the grid operator for the transport of the electricity. The generation company needs to inform TSO in advance about the exact electricity volume that its facility will produce within a certain period of time and to which customer (e. g. power retail companies) the electricity needs to be transported.\n\nPromotion and integration of RES\n\nTo subsidize the development of RES, a fixed feed-in tariff which is significantly above market prices is paid to RES owners. The EEG, which regulates the promotion of RES, was enacted in 2000 on the basis of the former Act on the Feed-In of Electricity from RES into the Public Grid, itself enacted in December 1990. The EEG regulates a feed-in system that comprises four key elements:\n\n * Fixed feed-in tariff: for each kWh produced and fed into the grid, a fixed price is paid which is higher than the wholesale market price for electricity.\n\n * Take-up obligation: grid operators must buy the electricity from RES at all times and pay the feed-in tariff independently from current market prices.\n\n * RES priority: RES has priority over non-RES in case of network congestion.\n\n * RES curtailment in last resort: in case of network congestion, conventional power supply needs to be curtailed as much as possible before RES can be curtailed as well.\n\nFeed-in tariffs at a glance\n\nThe feed-in tariffs are usually paid for electricity stemming from hydro power, landfill gas, gas from purification plants, mine gas, biomass, biogas, geothermal power, onshore wind, offshore wind, small-sized rooftop PV installations, and large-scale PV parks. With regard to the specific design of the feed-in tariffs, three aspects must be considered:\n\n * First, feed-in tariffs differ depending on the power source under consideration.\n\n * Second, feed-in tariffs for installations using the same power source often depend on the installed capacity with higher feed-in tariffs applying to smaller installations.\n\n * Third, feed-in tariffs are paid for a period of 20 years and the feed-in tariff paid for each installation at the moment of its commissioning is guaranteed over the whole period.7\n\nFeed-in tariffs for new installations have been steadily adjusted downwards since the implementation of the EEG in 2000, reflecting technical progress and the declining costs of RES. However, feed-in tariffs for installations that went into service before the adjustments remain at their originally guaranteed level. To facilitate planning for RES investors, future reductions of the feed-in tariffs are already known today and recorded in specific reduction schemes that are part of governmental supplements to the EEG. Depending on the capacity and some other characteristics of the installations, the following ranges of feed-in tariffs for the power sources with the highest relevance were paid in 2012 [49]:\n\n * Hydro: 0.034 \u20ac\/kWh\u20130.127 \u20ac\/kWh\n\n * Onshore wind: 0.0893 \u20ac\/kWh\u20130.0991 \u20ac\/kWh\n\n * Offshore wind: 0.15 \u20ac\/kWh\u20130.19 \u20ac\/kWh\n\n * Biomass: 0.06 \u20ac\/kWh\u20130.143 \u20ac\/kWh\n\n * PV: 0.1794 \u20ac\/kWh\u20130.2443 \u20ac\/kWh\n\nFinancial burden caused by feed-in tariffs\n\nElectricity generated by means of RES (RES-E) is traded on wholesale markets irrespectively of the feed-in tariffs. RES-E enters Germany's wholesale markets in the following way:\n\n * Generators of RES-E receive the feed-in tariff from their respective distribution grid operator, who in turn gets an equivalent compensation from the transmission grid operator.\n\n * The transmission grid operator sells RES-E on a wholesale market, frequently receiving a price considerably lower than the governmentally fixed feed-in tariff.\n\n * To avoid financial burdens for transmission grid operators as a result of this practice, the difference between the fixed feed-in tariffs and the market prices for electricity is refunded in full to the transmission grid operator.\n\n * The financial capital for this compensation stems from the electricity consumers, who have to pay a surcharge for the promotion of RES on their electricity bill (renewable energy surcharge). The amount of the surcharge depends on the type of consumer (with high discounts for industrial consumers) but does not depend on the consumer's geographic location.\n\nThe financial burden caused by this compensation has increased significantly in the course of the past years. In 2000, approximately one billion euros was necessary to cover the difference costs of RES feed-in tariffs.8 This figure increased to approximately EUR 16 billion in 2012 and is projected to amount to roughly EUR 20 billion in 2014 [50]. Owing to the increasing share of RES in Germany's electricity mix, the renewable energy surcharge rose from 0.0008 \u20ac\/kWh in 2000 to 0.0528 \u20ac\/kWh in 2013 [51]. Germany has made the experience that setting up a system with feed-in tariffs financed by means of a surcharge that does not vary in different regions redirects purchasing power from regions with high loads towards regions with high RES capacities. Berlin, with its more than 3 million inhabitants (roughly 4.1 % of Germany's total population), received only 0.1 % of all RES connected payments, whereas Schleswig-Holstein, a federal state in Northern Germany with less than 3 million inhabitants (about 3.5 % of the population), received 7.0 % of all RES connected payments [51]. However, Berlin's population did not pay less than the population in Schleswig-Holstein to finance the RES funds. This means that purchasing power was implicitly redirected from Berlin to Schleswig-Holstein owing to the RES financing mechanism.\n\nThe effects of RES on wholesale electricity prices\n\nThe price on the wholesale electricity market is determined by the marginal costs of the last power plant required to satisfy total electricity demand setting the price which is applied to all generators at that point in time. The power plants are ranked according to their marginal costs of electricity generation (merit order), with the plants with the lowest marginal costs necessary to meet demand dispatched first and the ones with the highest marginal costs brought online last.\n\nTSO are mandated by law to prioritize the feed-in of RES before other conventional generation technologies. Once installed and connected to the grid, wind and PV installations can produce electricity with almost zero marginal costs, while costs of electricity generation from fossil fuel-fired power plants depends on the price of the combustibles used (fuel costs). Thus, electricity generated from RES enters the wholesale markets at the beginning of the merit order (at zero marginal costs) and is dispatched first. As a consequence, average wholesale prices decrease as the generation technologies with higher marginal costs are displaced by an increasing volume of RES-E. Thus, large-scale integration of RES-E suppresses wholesale electricity prices. This is known as the so-called merit order effect (see Fig. 4.11). With large amounts of RES-E traded on the wholesale markets (on windy and sunny days), wholesale prices are rather low. When high feed-in of RES-E corresponds to low demand on the consumption side (typically on Sundays), prices for electricity can even reach negative values. On these days, Germany sometimes exports electricity to foreign countries and has to remunerate these countries for absorbing the German electricity. There were negative spot market prices for almost 80 hours in 2013. Such negative prices occurred in ten of twelve months [52]. In conclusion, it can be said that the increasing share of RES leads to decreasing but much more volatile prices on the wholesale markets.\n\nFig. 4.11\n\nEffects of RES supply on the wholesale electricity prices\n\nAs wholesale market prices decrease, gas-fired power plants, which have high marginal costs, are dispatched less and less frequently making an economically viable operation difficult and deterring investors. However, with their flexibility and fast ramp times gas-fired power plants are considered a necessary part of a power system with a high share of variable RES. Due to these developments, discussions on a revision of the EEG and alternative support schemes and incentive mechanisms for investments in conventional power plants are currently taking place in Germany.\n\nElectricity retail markets and prices\n\nElectricity retail markets are based on bilateral standardized contracts without any interactions on marketplaces. They are less complex than wholesale markets. In both Germany and China, households and industrial consumers pay different retail prices. In contrast to China, German households have to pay significantly more than industrial consumers. In 2012, the price amounted to roughly 0.13 \u20ac\/kWh for industrial consumers, whereas the price for household consumers amounted to approximately 0.26 \u20ac\/kWh [53]. These privileges for industrial consumers were introduced to increase the competitiveness of Germany's industry on world markets.\n\nThe retail price for electricity can be subdivided into three main categories:\n\n * Taxes (electricity tax and value-added-tax) and fees (mainly concessional duties and the renewable energy surcharge) currently make up approximately 50 % of the electricity price.\n\n * Costs of power generation and retail amount to approximately 30 % of the price. Between 1998 and 2000, these costs decreased from 0.1291 \u20ac\/kWh to 0.0858 \u20ac\/kWh as a result of the market liberalization of 1998, which created more market competition in all areas of the power sector supply chain. In the following years, the size of this price component increased slowly but steadily until 2009 and has remained rather stable since then [53].\n\n * Governmentally regulated network charges compensating grid operators for electricity transmission and distribution. Network charges make up roughly 20 % of the retail price paid by household consumers [3]\n\nFigure 4.12 illustrates the development of electricity prices for private households and its composition in Germany since 2006. The electricity retail price has increased due to rising costs of power generation and retail as well as rising taxes and surcharges (fees), which increased from 0.0714 \u20ac\/kWh in 2006 to 0.1163 \u20ac\/kWh by 2012 [53]. The increase of the renewable energy surcharge from 0.008 \u20ac\/kWh in 2006 to 0.0528 \u20ac\/kWh in 2013 contributed to this development. In the same timeframe, the network charges decreased slightly.\n\nFig. 4.12\n\nDevelopment of the electricity price for private households in Germany, adapted from [3]\n\n### 4.4.4 Development of infrastructure and network regulation\n\nCoordination of network expansion\n\nIn Germany, many different stakeholders are involved in grid expansion planning. Even if planning activity is mainly in the hands of TSO and BNetzA, other established power sector companies, third parties and the public can also influence network expansion planning. From a legal point of view, the expansion of the electric power grid is mainly regulated by EnWG, by the Energy Network Development Act (EnLAG), and by NABEG:\n\n * \u00a7 12 EnWG states that transmission grid operators are responsible for elaborating and issuing a coordinated network expansion plan each year. This plan is supposed to describe which upgrades of the transmission grids will be necessary during the following ten years. The process of network expansion planning is monitored by BNetzA. It allows for public participation and is open to comments from various stakeholders.\n\n * EnLAG defines specific investment projects in single transmission lines with the intention of facilitating the integration of RES, improving the interconnection with neighboring countries, easing the connection of new power plants, and reducing network congestions.\n\n * NABEG further specifies procedures relating to the network expansion plan. Its main motivation is to accelerate the planning and approval procedures of network expansion.\n\nCost pass-through regulation until 2009\n\nThe costs of investments in the grid infrastructure are shared by all electricity consumers via network charges. Until 2009, investment into the grid infrastructure was regulated using a so-called cost pass-through regulation which was also applied in many European countries and the United States. Cost pass-through regulation adjusts permissible revenues according to the grid operator's accounting and capital costs. The primary advantage of this system is that it lowers investment risks as practically all costs can be passed on to the end-user (via network charges), thus encouraging investment in the infrastructure. However, this regulation does not set incentives for efficient grid operation especially important in power systems with a limited need for grid expansion and upgrade.\n\nIncentive-based regulation after 2009\n\nToday, network charges in Germany are regulated using incentive-based regulation in the form of a so-called revenue cap. This solution relates to a model proposed by the former UK Treasury economist Stephen Littlechild in 1983. He criticized the lack of efficiency incentives of cost pass-through regulation and proposed the price-based regulation, which is known as RPI-X [54]. Apart from Germany, similar systems exist across Europe (e. g. the UK) and in some areas of the United States as well.\n\nFor Germany, the details of revenue cap regulation are defined in the Incentive Regulation Ordinance (ARegV). With price-based regulation, the future revenue cap is defined ex-ante for the coming regulation period (five years in Germany). Within the regulation period, the formula used to calculate the precise level of the revenue cap remains unchanged. Permissible revenues therefore follow a predetermined path during the regulation period. The revenue cap is mainly based on previous-year revenues minus the so-called RPI-X Factor. This factor consists of the retail price index (RPI) and an anticipated increase in productivity (the so-called X-Factor). The X-Factor is an important element of incentive-based regulation. It is determined individually for each grid operator. If a grid operator reaches a higher increase in productivity than anticipated by the regulator, additional cost savings need not be passed through to the consumer and thus remain as additional profit for the company. This mechanism therefore represents an incentive to improve efficiency. The disadvantage of incentive-based regulation is that cost-saving pressure may be at the expense of network investment. In Germany, with its large network investment requirements, a reform of the regulatory system to facilitate efficient investment is therefore currently being discussed.\n\nRegulation of supply security\n\nNetwork regulation relates not only to network charges but also to monitoring supply security. EnWG contains several paragraphs on this aspect. \u00a7 13 and \u00a7 14 EnWG assign responsibility for stable grid operation to transmission grid operators and distribution grid operators respectively. In urgent situations with a national relevance (for example situations of network congestions), grid operators must contact BNetzA without any delays (\u00a7 13, section 6, EnWG). With regard to less urgent and more local situations, grid operators are obliged to issue a yearly report listing all supply interruptions within their respective grid area (\u00a7 52 EnWG). This report must be submitted to BNetzA every year by the end of April via an internet-based process (see [55]).\n\nThe description of each supply interruption must include the time, duration, scope, and cause of the interruption. Grid operators are also obliged to describe the preventive measures taken to avoid such interruptions in the future. A document entitled Guidelines of BNetzA concerning reporting duties for supply interruptions in electric power grids according to \u00a7 52 EnWG (see [56]) specifies the information to be transmitted to the regulator.\n\n### 4.4.5 Coordination of generation and consumption\n\nLong-term coordination vs. short-term balancing of generation and consumption\n\nNeither electricity generation nor electricity consumption has changed dramatically in Germany during the last two decades. Thus, policies focusing on facilitating the long-term coordination of electricity generation capacities with the development of electricity consumption are not a primary concern in Germany. However, due to the increasing intermittency of Germany's electricity generation caused by RES integration, policies aiming at balancing electricity generation and consumption in the short-term have become more and more important.\n\nThe role of TSO in balancing generation and consumption in the short-term\n\nBefore the beginning of the unbundling process, decisions such as the dispatching of power plants were coordinated within the firms themselves. Today, these decisions are coordinated in the wholesale and retail markets described in Sect. 4.4.3. In some cases, however, the balancing of generation and consumption and respective dispatching of power plants as determined by the market cannot be realized due to physical restrictions with regard to power grid infrastructure capacities. In these cases, the TSO are responsible for balancing generation and consumption in order to secure system stability. Specifically, German TSO are allowed to take the following measures and make the following adjustments:\n\n * So-called balancing markets are independent from EEX and OTC trading and allow generation and consumption to be adjusted in the very short term: according to \u00a7 12 EnWG, TSO can tender the required balancing power through a common internet platform.9 Three different reserves are tendered: primary, secondary, and tertiary reserve. Primary reserve needs to be available within 30 seconds and is tendered on a monthly basis. Secondary reserve must be available within 5 minutes and is also tendered on a monthly basis. Tertiary reserve has to replace the secondary reserve after 15 minutes. Tertiary reserve is tendered on a daily basis.\n\n * An Ordinance on Disconnectable Loads (AbLaV) was issued in 2013. It allows TSO to tender, on a monthly basis, loads of up to 3 GW that can be disconnected within 15 minutes if there is an urgent need to adjust consumption downwards. Like in the balancing markets, the loads are tendered on an internet platform.\n\n * If generation and consumption cannot be balanced in balancing markets or by disconnecting loads according to AbLaV, TSO are entitled to overrule market outcomes by forcing power generators to adjust their generation. In that case, the affected generation companies have to be compensated for financial losses and BNetzA has to be informed immediately on such measures.\n\nThe role of electricity tariffs in balancing generation and consumption in the short term\n\nTraditionally, the main function of electricity prices was the coordination of electricity generation and consumption in the long-run by incentivizing investments in generation capacities, grid capacities, and end-use-devices. Currently, there is a trend towards tariff structures on retail markets being designed to take over the balancing of generation and consumption even in the short term. The widespread introduction of such time-of-use pricing critically depends on a successful rollout of smart meters. Currently, only few households in Germany are equipped with smart meters. According to a recent survey among German energy market experts, the widespread rollout of smart meters is not expected to be completed before 2029 [57].\n\nAs a consequence, the German tariff system currently has less time-of-use pricing elements than the Chinese tariff system. The following elements provide examples for time-of-use pricing elements included in the German tariff system:\n\n * For more than 20 years, electricity prices for industrial consumers have been separated into a peak load price and a base load price. This offers users an incentive to keep peak demand as low as possible. Technically, energy management systems within factories supervise and control the processes within certain ranges to effectively reduce peak demand. In recent times, more differentiated time-of-use pricing has been introduced to take advantage of the flexibility within the industrial production process for load shifting.\n\n * Since 2011, EnWG has obliged each power retail company to offer at least one electricity tariff for residential consumers with price levels differentiating at least between times of peak and base load. However, only few German households have chosen such a tariff because the potential financial savings it offers are rather low [58].\n\n### 4.4.6 The role of information and communication\n\nThe role of government in promoting smart grid-related ICT\n\nIn addition to guiding the German debate on smart grid developments and including third parties in the smart grid development process, the German government promotes the development of smart grid technologies by means of innovation policies. The smart grid innovation policies of the German government currently focus on the promotion of R&D and are embedded into the government's broader energy research policy.\n\n * The first objective of Germany's energy research policy is to contribute to achieving the targets set by the government in relation to the energy sector and climate policy by supporting the early-stage development of new technologies, concepts and business models.\n\n * The second target is to enhance the position of German companies in the field of modern energy technologies.\n\n * The third objective is to secure and enhance technological options. This objective seeks to help improve the flexibility of Germany's energy supply and is consequently directly related to smart grid technologies.\n\nIn general, smart grid research projects are co-funded by the German government with a government grant amounting to 50 % of the total project costs being paid to industrial project members. Public research institutes and universities often get 100 % government funding. Mainly large consortia of industrial companies (utilities, manufacturers, telcos, innovative small and medium-sized enterprises, and energy service companies) and R&D institutions such as universities or independent institutes compete among each other for government funds. Their research proposals are evaluated by independent evaluators or government bodies and the best concepts are recommended for funding.\n\nSome results from early R&D projects\n\nGermany's main funding program for smart grid and smart market policies so far was the so-called E-Energy funding scheme set up by BMWi and supported by BMUB. Extended demonstration projects were carried out in six German regions to validate the integration and balancing of renewables and the inclusion of third parties and smart markets such as regional energy marketplaces. The development of new ICT solutions for smart grids and smart markets was an additional key target. The overall volume of this program was roughly EUR 140 million [59].\n\nThe main motivations behind the E-Energy funding scheme were\n\n * to establish a lead market in developing smart grid technologies,\n\n * to integrate smart grid developments into the European context, and\n\n * to guarantee the security of supply in the future power system.\n\nThe results of one of the E-Energy projects called E-DeMa show that in today's market conditions there are not enough incentives for residential consumers to apply DSM or demand response (DR) [60]. However, it is expected that the projected expansion of RES generation capacities will increase the demand and the corresponding business opportunity for DSM. Therefore, numerous German retailers have projects promoting consumers' commitment to shift electricity consumption to off-peak times and to use electricity more efficiently (e. g. by visualizing end users' electricity consumption). One important result in this context is that new market concepts are necessary to efficiently explore the load shifting potential of customers. An electronic marketplace developed in the framework of the E-Energy projects could, for example, serve as a communication and interaction platform for residential consumers.\n\nSelected findings and lessons learned from E-Energy\n\n * Household customers with detailed information on their load behavior are able to reduce electricity consumption by roughly 5 %.\n\n * Saving potentials are higher for commercial and industrial enterprises. These consumers were able to save up to 20 % with detailed information on their electricity consumption.\n\n * Electricity consumption needs to be made transparent with feedback instruments indicating current and historical consumption.\n\n * An illustration of historical consumption provided with the monthly electricity bill can sometimes constitute sufficient feedback for household consumers. In general, more sophisticated feedback instruments are necessary, especially for companies.\n\n * Transparent electricity consumption patterns are not sufficient to save on electric power. Consumers must also be empowered to assess the relevant information and decide on possible options. Advisory measures, efficiency indicators, and analytic tools are necessary in this context.\n\nSource: B.A.U.M. Consult G.m.b.H. [59]\n\nIn the wake of the E-Energy funding scheme, the call for project proposals for the Future Proof Power Grids research program took place in early 2013. The aim of the program is to improve cooperation between industry and academia throughout the value chain and facilitate international research cooperation. Another goal is to improve the environmental, economic, and resource efficiency of electricity networks as well as the security of electricity supply research under this program is supported with a total of EUR 150 million provided by three different ministries [61]. More than 400 companies and 300 academic and research institutions formed research consortia and submitted 171 project proposals. The large majority of project proposals deals with issues related to distribution grids, with proposed research on transmission grids also attracting a significant amount of proposals and wind power integration trailing behind [61]. The focus of most proposals is on the management of grid operations, followed by technical challenges of transmission and distribution grids as well as network planning.\n\nThe role of the ICT industry in promoting smart grids\n\nThe ICT industry has developed a prominent view on Germany's smart grid issues. Representatives of the ICT industry contributed to the creation of the comprehensive German smart grid vision elaborated in FEG. Germany's ICT industry is focusing less on basic aspects of communications but more on general services to end consumers, e. g. value-added services at residential level, apps for energy efficiency, and big data aspects relating to power grid data exchange, data processing, and archiving. While smart grid funding and lobbying is strongly influenced by the ICT industry, aspects related to distribution grids are still dominated by companies from the energy sector.\n\nThe Federation of German Industries (BDI) and the Federal Association for Information Technology, Telecommunications and New Media (BITKOM), which focus on the ICT point of view, can be regarded as important players to address the view of the ICT industry in the smart grid debate: the former is a large general industrial association communicating the interests of German industry to those in positions of political responsibility. The latter is a large association dedicated to information technology, telecommunications, and new media industry. In addition to the promotion of the business development, these associations focus on the aspect of data privacy. Therefore, important legislation and regulatory topics covered by both associations are smart metering (private consumption data), smart home gateways (also private data and service interruptions), and certain aspects of controllable local systems and communication requirements of the German Forum Network Technology\/Network Operation in the VDE (FNN).\n\nInformation security in Germany's smart grid environment\n\nBesides system operation, information security in Germany strongly focuses on the aspect of user acceptance, e. g. in the domain of smart metering. In that context, data privacy is a very important issue. The standards discussed in the context of information security in Germany include the IEC Technical Committee (TC) 57 family, ISA 99 and the North American Electric Reliability Corporation's (NERC) Critical Infrastructure Protection Committee (CIPC). Furthermore, studies for the BMWi have been carried out to provide an overview of previous attacks in the energy domain, existing solutions and security standards, and also insights on security metrics and patterns [63].\n\nStandardization in the information security sector seeks to unify the implementation of ICT security measures. The ultimate aim is to improve the common security level in the power system.\n\nAn overview of common security standards in Germany is given by BITKOM and DIN, although it does not cover the energy domain directly [64]. An evaluation of security standards and guidelines for the energy domain was conducted in the European project European Network for the Security of Control and Real Time Systems (ESCoRTS). This topic is also addressed by the Smart Grid Information Security (SGIS) working group, which is partly responsible for carrying out the European Mandate M\/490 as well as the corresponding DKE Group STD 1911.11 in Germany (see [65]).\n\nThe requirements stated in the white paper on Requirements for Secure Control and Telecommunication Systems [66] by the German Association of Energy and Water Industries (BDEW) aim to support the acquisition, development, and revision of control and telecommunications systems in the energy sector to minimize the consequences of threats.\n\n\u25b6 Key findings\n\n#### Key findings\n\n * Germany has a stable and nationwide integrated electric power system. The power sector is in an advanced state of unbundling, featuring widely used markets for power exchange. Retail prices are rather high in Germany due to taxes and levies imposed to finance the modernization of the power system.\n\n * An important goal of the German government is to increase the sustainability of the electric power system. RES generation capacities have therefore been strongly built up in recent years and are likely to further increase in importance during the next decades. The increasing feed-in of intermittent RES generation puts more and more stress on grid operation in Germany. In this context, Germany has gathered significant experience on topics relating to grid integration and curtailment of RES.\n\n * In Germany, smart grids are seen as a means to enhance the electric power grid so that it can cope with the increasing feed-in of RES and to avoid investments in the conventional (primary) grid infrastructure. New market concepts such as regional energy marketplaces, business services, and VPP also play an important role in the German smart grid concept. They are expected to increase business activities, integrate new market actors in the power sector, and facilitate the involvement of power consumers.\n\n * Representatives from the electric power sector, manufacturing sector, ICT sector, and from the science and research community recently developed a comprehensive smart grid vision for Germany that uses a systematic and comprehensive top-down approach. This approach can serve as a best practice example of how to develop and formulate a smart grid vision.\n\n * The government plays a strong role in Germany's energy sector regulation: it published credible long-term goals for the development of the power sector until 2050. There is also strong coordination between the different governmental institutions involved in energy policy. Finally, the regulatory authority is independent from the government and can be seen as a powerful player in Germany's power system. The government is very active in the smart grid development process as well: BNetzA has issued a widely acknowledged government position on smart grids and smart markets while BMWi aims at including new market actors in the smart grid development process.\n\n * The unbundling process, the legally enforced trend towards more competition, as well as the migration towards smart grids and RES have contributed to the emergence of new market actors in Germany. Their growing importance can be considered as the most profound change in Germany's market structure during the last few years: new market actors introduced innovative products and services and contributed to the modernization of Germany's energy power sector. 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Consult GmbH, \"Smart Energy Made in Germany \u2013 Interim Results of the E-Energy Pilot Projects towards the Internet of Energy,\" 2012. Online]. Available: [http:\/\/\u200bwww.\u200be-energy.\u200bde\/\u200bdocuments\/\u200bE-Energy_\u200bInterim_\u200bresults_\u200bFeb_\u200b2012.\u200bpdf. [Accessed November 25, 2013].\n\n60.\n\nH. J. Belitz, S. Winter, C. M\u00fcller , N. Langhammer, R. Kays, C. Wietfeld and C. Rehtanz, \"Technical and Economic Analysis of Future Smart Grid Applications in the E-DeMa Project,\" in Innovative Smart Grid Technologies 2012, Berlin, 2012.\n\n61.\n\nBundesministerium f\u00fcr Bildung und Forschung (BMBF), \"F\u00f6rderinitiative Zukunftsf\u00e4hige Stromnetze,\" 2013. Online]. Available: [http:\/\/\u200bwww.\u200bfona.\u200bde\/\u200bde\/\u200b16538. [Accessed November 25, 2013].\n\n62.\n\nM. Agsten, D. Bauknecht, A. Becker, W. Brinker, R. Conrads, V. Diebels, T. Erge, S. Feuerhahn, C. Heinemann, J. Hermsmeier, R. Hollinger, T. Klose, M. Koch, C. Mayer, G. Pistoor, C. Rosinger, H. R\u00fcttinger, T. Schmedes and M. Stadler, \"eTelligence final report,\" 2011. Online]. Available: [http:\/\/\u200bwww.\u200betelligence.\u200bde\/\u200bfeldtest\/\u200bfile\/\u200bEWE%20\u200b102189%20\u200bEVE%20\u200beTelligence%20\u200bAbschlussbericht\u200b%20\u200bInhalt%20\u200bGB%20\u200bInternet_\u200bsc.\u200bpdf. [Accessed February 11, 2014].\n\n63.\n\nC. Rosinger and M. Uslar, \"Smart Grid Security: IEC 62351 and Other Relevant Standards,\" in Standardization in Smart Grids. Introduction to IT-Related Methodologies, Architectures and Standards, Berlin, Heidelberg, Springer Verlag, 2013.\n\n64.\n\nBundesverband Informationswirtschaft, Telekommunikation und neue Medien e. V. (BITKOM)\/Deutsches Institut der Normung (DIN), \"Kompass der IT-Sicherheitsstandards \u2013 Leitfaden und Nachschlagewerk,\" BITKOM\/DIN, Berlin, 2007.\n\n65.\n\nNational Institute of Standards and Technology (NIST), NIST Framework and Roadmap for Smart Grid Interoperability Standards, US Department of Commerce, 2010.\n\n66.\n\nBundesverband der Energie- und Wasserwirtschaft (BDEW), Requirements for Secure Control and Telecommunication Systems, Berlin: BDEW, 2008.\n\nFootnotes\n\n1\n\nVertical foreclosure refers to a situation in which a company buys a supplier that supplies both the company and its competitors in order to discriminate against the competitors.\n\n2\n\nIn the German context, coal refers to both hard coal and lignite. These two fuels are used in roughly equal amounts.\n\n3\n\nHowever, system operators have the possibility of correcting market outcomes in the case of insufficient grid capacities. Nonetheless, grid capacity itself is not taken into account in the decisions of the market participants.\n\n4\n\nRegional energy markets have been tested in several demonstration projects in Germany, e. g. in the eTelligence project. A detailed overview of the results from eTelligence can be found in [62].\n\n5\n\nThe NABEG describes the precise steps and more importantly the timing of these steps to be fulfilled after a connection request. With this, it avoids unjustified delays with network connection. The role of BNetzA within this process is specified, for example, in paragraphs 5, 7, 8, and 9 of this law.\n\n6\n\nNote that the quantity of electricity virtually traded either via OTC or EEX is considerably higher than the physical quantity of electricity generation and consumption. This is due to hedging or arbitrage activities of market participants.\n\n7\n\nWind power represents an exception to this general framework: tariffs for wind farms are not constant over the whole period but are slightly elevated during the first years of the operation. Also, feed-in tariffs for wind farms do not decrease with the size of the wind farm.\n\n8\n\nDifference costs refer to the total amount of feed-in tariffs paid to investors minus the revenues from RES-E on wholesale markets.\n\n9\n\nPlease refer to http:\/\/\u200bwww.\u200bregelleistung.\u200bnet for more information on this internet platform.\n\u00a9 The Author(s) 2015\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel (eds.)Regulatory Pathways For Smart Grid Development in China10.1007\/978-3-658-08463-9_5\n\n# 5. Recommended approaches for smart grid development in China\n\nGert Brunekreeft1 , Marius Buchmann1, Christian D\u00e4nekas4, Xin Guo2, Christoph Mayer4, Marcus Merkel5, Christian Rehtanz6, Andr\u00e9 G\u00f6ring4, Andre Herrmann2, Ray Kodali2, Michael Stadler2, Mathias Uslar4, Nils Vogel2, Till Luhmann2 , Tobias Menz2 , Sven-Uwe M\u00fcller3 and Paul Recknagel3\n\n(1)\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\n(2)\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\n(3)\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\n(4)\n\nOFFIS Institut f\u00fcr Informatik e.V., Oldenburg, Germany\n\n(5)\n\nEWE NETZ GmbH, Oldenburg, Germany\n\n(6)\n\nef.Ruhr Forschungs-GmbH, Dortmund, Germany\n\nGert Brunekreeft\n\nEmail: g.brunekreeft@jacobs-university.de\n\nTill Luhmann\n\nEmail: Till.Luhmann@btc-ag.com\n\nTobias Menz (Corresponding author)\n\nEmail: Tobias.Menz@btc-ag.com\n\nSven-Uwe M\u00fcller\n\nEmail: sven-uwe.mueller@giz.de\n\nPaul Recknagel\n\nEmail: paul.recknagel@giz.de\n\n## 5.1 Define a long-term strategy for the electricity sector and establish an independent and powerful regulator\n\nChinese context\n\n * China has formulated explicit targets for the short-term future of its electric power system in its 12 th Five-Year Plan for Energy Development. However, there are no official goals for the period after 2015. Also, there is as yet no comprehensive vision of China's smart grid, especially with regard to distribution grids and the period after 2020.\n\n * The governance structure of China's energy system regulation comprises a broad variety of ministries and institutions. The multiplicity of these stakeholders as well as the frequent reorganizations that have been taking place lead to difficulties in coordination and increased overlap of competences between the various government institutions involved in the power system regulation.\n\n * China's earlier regulatory authority, the State Electricity Regulatory Commission (SERC), was merged with the National Energy Administration (NEA) in 2013. The National Development and Reform Commission (NDRC) and NEA are currently the responsible regulatory institutions for the electricity sector. Both share regulatory responsibilities and are subject to political and industrial influence.\n\n### 5.1.1 Background\n\nSmart grids require strong government leadership\n\nSmart grids are not an objective in its own right, but a new approach to meet the challenges that will arise with future electricity systems, for instance with an increasing share of renewable energies. Specifically, smart grids aim at achieving economic efficiency by combining the strengths of conventional grids with the new capabilities of ICT. In Germany, ICT requirements in electricity grids are driven mainly by intermittent supply from RES. Without RES, the conventional grid infrastructure would be sufficient to maintain high reliability levels in Germany's electric power system.\n\nThe evolution towards smart grids necessitates strong vision and leadership on the part of the government, because smart grids are not a market-driven concept:\n\n * A clearly defined long-term strategy for the electric power sector reduces uncertainty for smart grid investors and manufacturers. A lower level of uncertainty on future developments reduces the risks with respect to the investment's future cash flows. As such, smart grid investors and manufacturers are more willing to invest in smart grid technologies. A long-term strategy includes government targets with regard to the development of electricity generation capacities of different technologies (the so-called generation mix) and targets for energy efficiency.\n\n * Clearly defined roles and responsibilities of government bodies are a fundamental requirement for the development of smart grids. Especially, a clear delineation between the competences of the ministries and the regulator, and possibly other supervisory bodies is associated with a high degree of credibility and assertiveness of government policy.\n\n * An independent and powerful regulator functions as a coordinating institution between all market participants (incumbents and new market entrants). The coordination specifically focuses on the development of equal and non-discriminatory conditions of market entry where all market parties are treated equally and market power of single market actors is limited. The regulator should be sufficiently powerful to impose the measures necessary to manage the development of the electricity sector and the development of smart grids.\n\nCharacteristics of an independent regulator\n\nThere are two aspects to a regulator's independence:\n\n * independence from the regulated industry and\n\n * independence from politics.\n\nIt is important for the regulator to be independent from the industry that is being regulated. If independence is not guaranteed, the danger of so-called regulatory capture (i. e. the regulator rather favors commercial and industrial interests instead of public interests) is high [1]. On the other hand, a regulator needs to cooperate closely with the industry. Therefore, a relation of mutual trust is important. In some cases, the regulatory office is actually financed by the regulated industry. Note that this is not necessarily a contradiction to the requirement of independence. Following the rules set by the government, the firms are actually obliged to finance the work of the regulator on the basis of a pre-defined payment plan which is not based on a firm's satisfaction with the regulator's work. The risk of regulatory capture is therefore small.\n\nThe independence of the regulator from the regulated industry can be ensured in several ways:\n\n * The processes, decisions, and procedures of the regulator have to be transparent so that they can be controlled.\n\n * The regulator should make use of public consultations in which not only the regulated industry but also other stakeholders or the general public can express their views. Again, this avoids opaque agreements between the regulator and regulated industries.\n\n * The regulator should be monitored at regular intervals and should have to justify its work, for example, in an annual report.\n\n * Personal financial interests between the regulator and the regulated industry must be avoided. Two rules are specifically important in this context: first, senior regulator management should not be allowed to have financial stakes in the regulated industry. Second, a moratorium period during which senior regulator management cannot accept a job in the regulated industry is very useful.\n\nWhy the regulator should be independent from politics or from the ministry is a less straightforward question. Ultimately, the regulator is a government institution that is governed by the ministry. The relevant concern is the precise legal delineation of authority: who decides on what? In the following, four main arguments for independence of the regulator are listed:\n\n * Following the classical pattern of the separation of powers, the legislator, the body that sets the rules, needs to be separated from the regulator, who applies the rules, as a system of checks and balances.\n\n * The state may be the owner of the regulated industry, which causes an obvious conflict of interest.\n\n * Ministries are often very close to political decision-makers and base their decisions on criteria that are different from those of regulators. Politicians have to mediate and choose between various diverging preferences in their society. Regulators of a monopoly industry, in contrast, try to improve overall efficiency by applying legal and economic tools.\n\n * Regulation requires a long-term perspective. Grid operators for example, are incentivized to continuously invest in the grid infrastructure only with a high predictability and stability of legal and regulatory decisions. Whereas politicians tend to be subject to so-called short-termism (i. e. they often ignore long-term issues), an independent regulator is less vulnerable to short-termism as he is not elected by the public.\n\nHow can independence of the regulator from politics be achieved?\n\n * In the governance structure, the regulator can be an independent chamber of the ministry instead of an integrated department.\n\n * The appointment and especially the dismissal of the chief regulator(s) should be a transparent and well-defined process.\n\n * The regulatory office should have a budget that is largely independent from daily government business.\n\n * The duties and powers of the regulator should be laid down in a law. Specifically, the delineation of competence between the regulator and the ministry should be clearly described. In many countries, a general energy law contains a description of the duties and powers of the regulator.\n\n * There should be a system of checks and balances. Specifically, it has to be clear who is responsible for regulating and controlling the regulator. For example, this may be a judicial court system, or another regulatory authority with an equal level of substance (e. g. a competition commission). A system of checks and balances allows greater regulator independence.\n\nIt is important to note that the precise details of independence from politics or ministries depend strongly on the wider political and governmental structure in a country.\n\nCharacteristics of a powerful regulator\n\nThe need for a powerful regulator is more obvious:\n\n * The competences of the regulator need to be laid down in a law so that decisions can be enforced and challenged in court. In Europe, it has been extensively debated whether the liberalized parts of the energy sector could be regulated under a general competition law or whether sector-specific legislation is necessary to ensure the development of competition. The test of practical experience, for example in Germany, has shown convincingly that sector-specific regulation, executed by a sector-specific regulator, is necessary (e. g. [2]).\n\n * The regulator needs to have an adequate budget. The stakes in industry are so high that it always pays off for the industry to hire consultants, lawyers, and lobbyists to argue their case; the regulator needs to address these claims. Moreover, good regulation is difficult and requires highly qualified staff; the regulator will have to compete with the industry for qualified employees.\n\n * Electric power companies and other stakeholders should have the right to appeal the regulator's decisions before a court or other tribunals empowered to conduct judicial reviews. Such a system of checks and balances makes the regulator more powerful because knowing that another institution may check and correct the regulator allows him to take more risk.\n\n### 5.1.2 International practice\n\nLong-term strategies for the electric power sector\n\nMedium-term to long-term plans with concrete goals regarding electricity consumption, energy efficiency, and\/or renewables have been published in many industrialized and emerging countries. These plans often cover a period of at least 25 years (see [3], [4], and [5] for examples from UK, India, and Brazil). In its Federal Energy Concept, Germany has defined binding political goals for renewables until 2050. The government has specified the future share of RES-E in gross electricity production with four goals for different time periods: by 2020 renewables are to have a share of at least 35 % in gross electricity consumption, a 50 % share by 2030, 65 % by 2040 and 80 % by 2050 [6]. Similar targets have been defined with respect to gross electricity consumption that is planned to decrease by 50 % until 2050.\n\nDifferent studies commissioned by the German government analyze how to achieve the various government targets. At the same time, the scenarios for energy and power sector development put forward in these studies serve as a foundation for policy formulation and a point of reference for further studies on more particular topics such as smart grid development. A study entitled Long-term scenarios and strategies for the expansion of renewable energies in Germany, taking account of developments in Europe and across the world investigated potential scenarios with respect to the development of renewable energies in Germany [7]. The results of this study are used to define strategies for the development of smart grids in Germany (e. g. how to finance the large investment requirements for smart grids).\n\nThe role of an independent and powerful regulator\n\nThe OECD points at the importance of a clear definition of roles and responsibilities with respect to regulation. Within the OECD's recommendations on regulation, the necessity of a common government policy defining clear goals for the regulation process is specified. According to OECD, strategies shall be set for the implementation of these goals to give regulation a clearly defined framework [8]. Furthermore, OECD emphasized the importance of an independent and powerful regulator for the efficient development of the electricity sector. The government has the task to clearly define the duties and the power of the regulator [8]. OECD also points out that independence of the regulator is essential since its decisions can have serious economic effects on the regulated parties. This is especially the case for smart grids, where the introduction of competition is directly related to economic effects for the incumbents.\n\nThe US American way: FERC\n\nThe Federal Energy Regulatory Commission (FERC) is the regulatory authority in the United States. It is an independent regulator responsible for the regulation of the interstate transmission of electricity, natural gas, and oil. In addition, FERC reviews proposals to build liquefied natural gas (LNG) terminals and interstate natural gas pipelines, and it licenses hydro power projects. Further responsibilities of FERC outlined in the Energy Policy Act of 2005 include, amongst others:\n\n * review of mergers and acquisitions as well as corporate transactions by electricity companies,\n\n * approval of siting and abandonment of interstate natural gas pipelines and storage facilities,\n\n * licensing and inspection of hydro power projects,\n\n * protection of the reliability of the high voltage interstate transmission system through mandatory reliability standards,\n\n * monitoring and investigation of energy markets, and\n\n * administration of accounting and financial reporting regulations and conduction of regulated companies [58].\n\nThe independence and power of FERC are specified in 42 USC section 7172 g \u2013 Jurisdiction of the Commission [59]:\n\n> \"The decision of the Commission involving any function within its jurisdiction, other than action by it on a matter referred to it pursuant to section 7174 of this title, shall be final agency action within the meaning of section 704 of title 5 and shall not be subject to further review by the Secretary or any officer or employee of the Department.\"\n\nThe European Commission (EC) describes the characteristics of a powerful regulator in articles 37 et seq. of Directive 2009\/72\/EC (see appendix E). It states, for instance, that a powerful regulator should fulfill the following tasks and requirements:\n\n * issue decisions that are binding for electric power companies,\n\n * impose effective, proportionate, and dissuasive penalties on electric power companies,\n\n * ensure high standards of universal and public service,\n\n * protect vulnerable customers,\n\n * contribute to the effectiveness of consumer protection measures, and\n\n * promote effective competition and the proper functioning of the electricity market.\n\nGermany has established BNetzA, a regulator independent from the industry and the government. The powers and duties of BNetzA are recorded in EnWG. Neither the ministry nor the industry can overrule the decisions taken by BNetzA. The regulator's decisions can only be challenged before the court. The German regulator makes use of its power to supervise the network charges of the grid operators, to prevent or remove obstacles in access to energy supply networks for suppliers and consumers, to standardize processes for switching the power retail company, and to improve the conditions for connecting new generators to the grid. Driven by the growing share of renewables and the resulting need to expand the grid infrastructure in Germany, the regulator also has the task of supervising the network expansion process (see Sect.\u200b 4.\u200b4.\u200b1 for more detailed information on the German regulator).\n\n### 5.1.3 Recommended approach for China\n\nChina has not yet formulated explicit targets for the long-term future of its electric power system. This situation risks creating uncertainty among smart grid investors and manufacturers which might consequently postpone smart grid investments. Their uncertainty could be reduced by means of a clearly defined and committed long-term strategy for the electric power sector. Such a strategy should include government targets with regard to future generation capacities, shares of different generation technologies (generation mix), and targets for energy efficiency. In Germany, the long-term energy strategy contains binding government goals for a period of roughly 40 years, whereas long-term energy strategies in countries with higher economic growth rates and accordingly more dynamic energy sectors (e. g. India and Brazil) cover periods of approximately 25 years.\n\nThe governance structure of China's energy system comprises a broad variety of ministries and institutions. The multiplicity of these stakeholders as well as the frequent reorganizations that have been taking place lead to a rather low degree of coordination and to a rather high degree of overlaps of competence between the various government actors involved in the power system regulation. Based on OECD recommendations and on the German experiences, clearly defined roles and responsibilities concerning the regulation of the electricity sector and the development of smart grids would accelerate and ease the smart grid development in China. The existence of an independent and powerful regulator is by far the most important regulatory issue in this context. China's earlier regulatory authority, SERC, was recently merged with NEA. NDRC and NEA are currently the most relevant regulatory institutions for the electricity sector. As they share regulatory competences and are subject to political and industrial influence, regulation in China is less powerful and independent than in countries such as Germany and the United States. Thus, specific attention should be paid to the development of an independent and powerful regulator in China.\n\nThe recommended approach at a glance\n\n * A long-term strategy for the electric power system serves as a basis for more specific smart grid development strategies and objectives and is important for investors to gain investment security.\n\n * It is beneficial to centralize responsibilities for the regulation of the electricity sector in the hands of a single independent and powerful institution (regulator) that supervises the efficient development of the electric power system in general and smart grids in particular.\n\n## 5.2 Create level playing fields for access to power system infrastructure and information\n\nChinese context\n\n * Chinese grid operators are still integrated as they own and operate the electric power grids, are responsible for power retail, and invest in RES generation capacities. Also, power system data management (i. e. data collection and provision on grid status as well as generation and consumption quantities) is their task. Both major grid operators are currently developing systems for data management in smart grids. However, these systems focus on information collection by the grid operator for their own operation management, but not on information provision to other market actors.\n\n * New market actors are rarely participating in the development process of smart grids in China. Therefore, the innovation potential which could come from these new stakeholders, for instance from the ICT sector, currently remains untapped.\n\n * The Chinese government plans to establish a modern energy market system to increase competition and affordability. In particular, the reform of state-owned enterprises and the introduction of more market-related elements are envisaged.\n\n### 5.2.1 Background\n\nBenefits of integrating new market actors\n\nSmart grids are a relatively new concept aiming at making grid operation more reliable and efficient and accelerating the emergence of new energy-related products and services. Due to the novelty of the smart grid approach, many technological advances and ideas are necessary for smart grids to be developed in an effective and efficient way. Experiences from other sectors and countries suggest that new market actors, i. e. new competitors in the electric power sector or companies from other sectors such as the ICT industry, are key drivers of innovations: on the one hand, they offer innovative products and services that were not supplied by established market actors before. In a smart grid context, non-incumbents create new business models and offer new products and services by making use of available power system information and infrastructure in an innovative way (see Sect.\u200b 4.\u200b4.\u200b2 for examples of new market actors in the German electricity sector). On the other hand, new market actors contribute to an increased level of competition which is usually considered to drive innovation, enable greater cost efficiency in production, lower retail price levels, and provide a higher variety of products and services (see Sect.\u200b 2.\u200b3).\n\nA fair access to essential facilities is a prerequisite for new market actor integration\n\nNew market actors can only enter the markets if equal and non-discriminatory access to essential facilities (i. e. a level playing field) is guaranteed [9] [10]. In smart grids, level playing fields should be secured in two respects:\n\n * First, equal access to the physical grid infrastructure is important. Since connecting new power generation and consumption units to the power grid is laborious and often associated with high costs, grid operators might have a tendency to discriminate in favor of affiliated companies or against generators and consumers in remote regions. Regulation must prevent such a discriminatory behavior.\n\n * Second, access to power system information and data is required. Power system information and data are getting increasingly important and are a prerequisite for new market actors to offer new and innovative products and services [11]. This does not mean that each established or new market actor has access to all power system information, or that information is even open to the public; instead it means that each eligible market actor has equal and fair access to the information relevant for his business model. This is the task of power system data management, which should be organized in a way that both established and new market actors can participate in the rule-making process and can trust on non-discriminatory access.\n\nConcepts for a non-discriminatory access to power system information\n\nTwo concepts should ensure a non-discriminatory access to power system information and data in smart grids:\n\n * Technology neutrality is a regulatory concept for the telecommunications sector that was introduced by the European Commission in Directive 2002\/21\/EG. In this context, technology neutrality means that the regulator does not impose or discriminate in favor of a particular type of technology. This concept has been applied to several technological issues in Europe, for example with respect to the development of broadband internet, where the regulator left it open to the market to decide between the deployment of different technologies such as digital subscriber line (DSL), power-line communications (PLC), cable modem, or satellite. Technology neutrality is also important for smart grids, e. g. with respect to the question regarding which infrastructure should be used for the power system information exchange. This issue is currently being discussed under the headline of advanced metering infrastructure (AMI).1 The development of AMI is in its early introduction phase. Therefore, what specific technology should be used for building up the necessary infrastructure has not yet been settled. Different technologies could be applied, e. g. PLC, wireless, or fiber technologies. In this context, regulation needs to ensure that the most efficient technology will be applied, independently of which company supplies the technology.\n\n * Provider neutrality is another general regulatory concept that is also currently applied in the telecommunications sector in Europe. It basically states that the regulator should ensure that regulation does not discriminate against particular service providers. Applied to smart grids, this means that the regulator should ensure that established and new market actors are treated equally and allowed to offer services on an equal footing in smart grids. On the one hand, services could be provided by established players (e. g. DSO), on the other hand new market actors could provide complementary services or even substitute services of established players for the costumer. Provider neutrality should consequently ensure that competitive advantages (e. g. best technologies or low costs\/prices) and not the provider's market power affects consumer's choice.\n\nDefining roles and responsibilities of all market actors eases new market actor integration\n\nLiberalized energy markets for energy resources, electricity, capacity, or ancillary services necessitate the exchange of large amounts of information and data between different market actors. A mounting number of market actors leads to a more intensive inter-company exchange of operational and business-related information and data. Against this background, it is of critical importance that all market actors get assigned their respective roles and responsibilities [12]: on the one hand, they have to know the stakes and information requirements of their business partners and other actors in the electric power sector to better understand their business opportunities and their own contribution towards smart grid development. On the other hand, specific rules and data standards for inter-company exchange of business-related information help to reduce the transaction costs among all market actors.\n\n### 5.2.2 International practice\n\nEnsuring technology neutrality\n\nFor the development of smart grids, the concept of technology neutrality is particularly important with respect to the development of the ICT infrastructure. In principle, different communication technologies could be applied (PLC, 3G, etc.) and different data storages types could be used. Eurelectric, the association of the electricity industry in Europe, has defined basic requirements for the ICT infrastructures in smart grids. The German regulator has addressed this issue as well and specified that, currently, a medium-scale broadband connection should fulfill the necessary requirements to build up the AMI for smart grids. However, the German regulator stresses that with the use of real-time data the quality requirements concerning availability and latency cannot be fulfilled by all existing ICT solutions on the market [13].\n\nBasic requirements for ICT infrastructures in smart grids\n\nAccording to Eurelectric, the most important requirements for ICT in the context of smart grids are to:\n\n * \"Ensure that telecoms infrastructure and links are absolutely reliable, robust, meet operational requirements in terms of speed, capacity and latency and will be available at all times, particularly at times of critical incidents (e. g. be resilient to power outages for several hours because they are needed to rebuild the grid.\n\n * Provide well manageable and robust access control and user privileges mechanisms to the smart grid components and systems.\n\n * Guarantee the confidentiality, integrity and authentication of all smart grid-related communication events.\n\n * Guarantee a robust physical protection for the smart grid components as well as for the whole communication network.\n\n * Ensure that mission-critical telecommunications services are still alive during and up to the end of a wide area 72 hours blackout.\n\n * Implement strong monitoring systems to keep track of all the smart grid activity, implementing Security Information and Event Management (SIEM) systems for security related incidents analysis and maintain well trained security response teams to have a strong and quick response in the case of any security violation.\n\n * Warrant a true real time transfer of information: for a part the smart grid can be seen as an extension of the current SCADA systems; fully available at any time and the guaranteeing the perfect transfer of commands and feedback confirmation of the system operations.\n\n * Have an end-to-end security approach to guarantee a transversal security layer on the smart grid.\"\n\nSource: Wording from Eurelectric [61]\n\nEnsuring provider neutrality\n\nProvider neutrality is a key requirement for third-party access in general and is currently being discussed within the evaluation of different governance models for power system information management in smart grids. In this context, the European Commission (EC) has been trying to define which actor should be responsible for the data handling in smart grids. The models under discussion are based on unbundled companies or new market actors. The discussion on the different governance models is ongoing and no decision has been taken yet. Three potential models are currently being discussed (for more details see [11]):\n\n * DSO as market facilitator: This model allocates responsibility for power system data management, including collection of data and construction of the necessary information infrastructure to the DSO. The concentration of responsibilities within one institution has the advantage of centralized internal coordination and management of the ICT infrastructure. The main disadvantage of a DSO-centered solution addresses discrimination concerns. In principle, unbundling prescriptions require the neutrality of DSO in Europe. Yet, it is doubtful whether full neutrality can be guaranteed because of asymmetric information to the disadvantage of the regulator. This concept requires significant regulatory oversight. However, it is not new to the regulator, as the DSO are already regulated.\n\n * Independent central data hub (CDH): The CDH would be responsible for power system data management in smart grids and for central data storage under the supervision of the regulator. A key advantage of this approach is the neutrality of the market facilitator and the non-discriminatory access to information for third parties. The key challenge for this concept is the need to establish coordination mechanisms between the parties involved, e. g. the network owners\/system operators and the CDH. It should be noted that the regulator needs to ensure that the provider of the CDH does not discriminate against other parties or abuse its market power.\n\n * Data access point manager (DAM): The DAM concept focuses on a competitive market for power system information management and proposes to establish independent and unregulated service providers that consumers can choose from. Each DAM offers to build up the necessary information infrastructure for the consumer. Importantly, the DAM does not store the data centrally. Storage remains decentralized with the users, giving consumers full control of their own data. The DAM only acts as an interface which allows each consumer to decide which commercial party gets access to its main data. Such a decentralized approach requires a high degree of standardization to ensure flawless system integration. The DSO would have to control the quality of services provided by the DAM and each regulator would need to define the basic principles of the DAM to integrate them into the national electricity system.\n\nDefining roles and responsibilities of all market actors\n\nAt the European level, the Smart Grids Task Force (SGTF) of the European Commission broadly defined roles and responsibilities of various market actors in smart grids [12]: in a first step, all relevant smart grid actors (including companies from all supply chain stages of the electric power sector, end-users as well as influencing actors such as regulators, legislation authorities and standardization bodies) have been defined (see appendix E for an overview of all smart grid actors defined). In a second step, current as well as future responsibilities related to the smart grid development have been described for all actors. The work of SGTF should be understood as a practical toolset and guideline for further developments and business models for use by grid operators and grid users [12].\n\nFuture responsibilities of grid operators in smart grids\n\n\"[...] it appears that it is the DSOs who will have to face the biggest challenges so that Smart Grids will become a reality. The reasons include;\n\n * Growing distributed generation, active management of demand, local storage and electric vehicles (EV) will impact the DSO infrastructure. Thus the DSO will have to be an active participant in all such projects along with the actors implementing these projects as these projects will fundamentally change today's relatively static distribution system to a much more dynamic distribution system.\n\n * As more fluctuating distributed generation will feed into the distribution system, gathering and handling the data about the state of the distribution system will be one key issue for the DSO.\n\n * Attention will need to be paid to ensure that all privacy and system security recommendations (in line with the provisions defined by EG2) will be adhered to. Ownership of the data, length of time data is stored etc. will all need to be addressed in an appropriate way.\n\n * The data collected will enable the DSOs to fulfil their duty in relation to the overall grid stability and operational security, given that more and more distributed generation will be connected to the distribution grid.\n\nIn order to resolve the above challenges, the DSOs will have to continue upgrading their grid infrastructure, control centres and educating their employees accordingly.\n\nThe TSOs will have to provide more support & communication of data to the DSOs, but will also require more specific information from the DSOs, especially with more distributed generation coming from the distribution grids. In order to achieve this, both TSOs and DSOs need to ensure that the standards they implement for communication and data exchange are compatible. It also follows that the TSOs will have to gradually redesign power system control as well as market information management relating to forecasting the overall system load in conjunction with the DSOs. At the same time, the DSOs will have to strengthen their role in providing the required data relating to the distributed generation, local storage and electric vehicles within the distribution grid.\n\nBoth TSO and DSO should be able to execute their active role in Smart Grid management by ensuring more sophisticated legal provisions for system security management under increased uncertainty. Following the analysis about funding [...], these mechanisms should include the ability to interfere with the planned market activities in case of disturbed or emergency operational conditions, without \"automatic\" socialization of the related costs to other grid users.\n\nFinally, the role of grid communications will significantly increase as much more data will have to be gathered and exchanged frequently, which will be in turn used for different purposes by the grid operators and other service providers. As stated above, the standardization of communication protocols as well as clear rules for the handling and the security of this data will have to be developed and enforced. The security of the grid and supply systems as well as the privacy of customer data must remain the top priority.\"\n\nSource: Wording from Smart Grids Task Force [11]\n\nAt the German level, roles and responsibilities of companies in the electric power sector are further specified in EnWG, with an emphasis on specific conditions in Germany. This definition of roles and responsibilities in Germany's most prominent energy law contributed to a high understanding on business opportunities and legal obligations among affected companies in Germany.\n\nWith respect to the inter-company exchange of business-related information, BNetzA issued the so-called Business Processes for Delivery of Electricity to Customers (GPKE) based on the United Nations Electronic Data Interchange for Administration, Commerce and Transport (UN\/EDIFACT) in 2006 (see [14] for more information). GPKE standardize inter-company communication and data exchange in the case of typical business processes such as billing of customers or customers changing their power retail company. In reducing transaction costs of typical business processes, GPKE ease the emergence of new market actors.\n\n### 5.2.3 Recommended approach for China\n\nThe main electricity sector reform of 2002 mandated the separation (or unbundling) of the state-owned vertically integrated utility responsible for all supply chain stages across China into five big power generation companies, two major grid operators handling transmission, distribution and retail as well as four power service corporations. China's power generation sector can be described as liberalized, as it potentially allows competition between the major five generation companies and the thousands of smaller local and regional generation companies. Chinese grid operators are not completely unbundled as they own and operate the electric power grids, are responsible for retail, and also invest into RES generation capacities. Recently a potential separation of grid operators into transmission and distribution companies or into smaller, regional businesses has been subject of debate [15]. With respect to smart grid developments, non-incumbents (e. g. from the ICT sector) are not yet actively participating in the development process. Therefore, the innovation potential which could come from these new stakeholders, for instance the ICT sector, currently remains untapped.\n\nExperiences from Europe show that defining roles and responsibilities of established and new market actors (including specific rules and data standards for inter-company exchange of business-related information) leads to a better understanding of business opportunities and helps to reduce the transaction costs among all market actors. As such, the definition of roles and responsibilities contributed to the emergence of new market actors in the European electricity sector.\n\nCurrently, the management of power system data (e. g. grid status information or metering data on electricity generation and consumption) is in the hands of China's grid operators. The concepts of technology and provider neutrality are not applied. As soon as new market actors are to be integrated in the electric power system, power system data management will become more relevant on a broader scale. The regulator should develop a governance system that will ensure provider and technology neutrality and a level playing field for all stakeholders. Non-discriminatory access to information in smart grids is of particular importance for third parties to be able to develop their business plans. Neutral information management is therefore a key issue. There are various models currently being discussed in Europe. However, there is not yet one preferred solution fitting all possible contexts. Therefore, recommending a best practice approach to China in this context is not yet advisable. A better approach would be to evaluate what governance model best suits China to ensure non-discriminatory access to information for third parties in the near future.\n\nThe recommended approach at a glance\n\n * Defining roles and responsibilities of established and non-established market actors in the smart grid development facilitates the emergence of new market actors and helps to make the exchange of business-related information and data more efficient.\n\n * It should be evaluated which framework for smart grid data management is able to ensure provider and technology neutrality in China. On this basis, a suitable framework needs to be established.\n\n## 5.3 Introduce network regulation for efficient investment incentives for electricity grid expansion and upgrade\n\nChinese context\n\n * China's main challenge in the electricity sector is how to handle the massive electricity grid expansion necessary to facilitate the country's rapid economic growth and to integrate the increasing number of RES. A network regulation system that focuses on facilitating investments is lacking.\n\n * At present, there is no explicit price for power transmission and distribution (network charge) based on actual costs. The source of grid operators' income is the difference between the on-grid and the retail price for electricity, which are both fixed by the government.\n\n### 5.3.1 Background\n\nNetwork regulation for smart grids\n\nSmart grids require substantial investment and innovation, especially from the grid operators. In setting network charges (i. e. prices for power transmission and distribution), regulating institutions have an important influence on the investment behavior of grid operators. Which regulatory scheme (i. e. which specific method used for calculating network charges) is best suited to set incentives for efficient investment is far from settled: in general, regulation should facilitate necessary investment and avoid unnecessary investment at the same time.\n\nIn a smart grid context, incentivizing efficient investment into the infrastructure is becoming even more important because benefits of innovation and investment in smart grid technologies may not always accrue to the investor (more information on this issue is presented below). In these cases, additional incentive schemes are required to encourage grid operators to invest specifically in smart grid technologies.\n\nThe benefits of regulating only the monopolistic bottlenecks\n\nEven if it is far from settled which regulatory scheme is best suited to set incentives for efficient investment, it is generally accepted that network regulation should focus on the monopolistic bottlenecks (transmission and distribution grids), leaving the commercial businesses (generation, trade and retail) to competitive forces and monitoring by competition law [16], [17]. This approach is referred to as disaggregated regulation. Also with respect to the smart grid development, it has been emphasized that regulation should only focus on network charges and network planning while all other aspects should be subject to the market and competition law [13].\n\nDisaggregated regulation has the following advantages:\n\n * Regulation inevitably provides misdirected incentives: only competitive markets are able to provide incentives to hold prices down to marginal costs and to minimize long-run costs. Regulated markets can only do one or the other but not both [18]. In this light, a regulatory focus on the natural monopoly part of the supply chain avoids misdirected incentives at the competitive parts of the supply chain.\n\n * Leaving a stage of the supply chain unregulated eases the market entry of new market actors [19] because requirements for licenses, permits and monitoring obligations in regulated markets raise the investment needed to enter a market.\n\n * A focus on regulation of the natural monopoly may improve the quality of regulation because the regulator's most skilled employees can more easily focus their efforts on the natural monopoly part of the supply chain. It will then be easier for them to get a grip on the regulated firms' cost-developments and investment requirements.\n\n * Different stages of the electric power system may require different types of regulation. In case the regulator decides to regulate the retail stage of the electricity value chain, it is important to set incentives for quality of service and cost efficiency of power retail companies and to allow differentiated tariffs for different consumer groups. Regulation for networks, in turn, needs to set incentives for efficient investment, as network expansion has become so urgent.\n\nNetwork regulation and investment incentives\n\nDisaggregated regulation entails the necessity to apply a specific network regulation scheme. The challenge of network regulation is setting efficient investment incentives (i. e. allowing necessary investment while avoiding unnecessary investment) because the regulator is confronted with the following dilemma: the regulated companies know their own cost structures and market opportunities better than the regulator. As a consequence, regulation cannot provide full-powered incentives to incentivize necessary investment and to avoid unnecessary investment at the same time [18]. Regulators must always accept a trade-off between both goals.2 The regulator's challenge is to achieve acceptable levels of both goals at the same time. In accordance with the trade-off described above, two polar cases are commonly distinguished in regulation theory:\n\n * Rate-of-return (also known as cost-pass-through) regulation fixes the rate of return and requires revenues to adjust according to underlying costs. If costs go down, revenues should go down and if costs go up, revenues may go up as well. Therefore, the incentives to reduce costs are low. In fact, firms make profits by inflating the capital base as this is the basis for the rate of return. With cost-pass-through regulation, firms may actually have strong incentives to overinvest (i. e. they do not avoid unnecessary investment).\n\n * Price-cap regulation, also called revenue-cap regulation or RPI-X regulation, tries to avoid these very incentives [20]. Price-cap regulation fixes the price or revenue path ex-ante for the next regulatory period, irrespective of the actual cost development during the regulatory period. If the firm succeeds in reducing its costs more than anticipated by the regulator, the firm can keep the additional profits; this is an incentive to minimize costs. Therefore, this type of regulation is often called incentive regulation.3 A counterargument for price-cap regulation is a situation in which costs do not go down, but tend upwards. This typically happens if networks need to be expanded. Under the typical RPI-X regulation, firms then have an incentive to avoid necessary investments [21].\n\nThe necessity of additional incentive schemes in a smart grid context\n\nAn additional challenge is starting to emerge given the decentralization of decisions in the smart grid value chain. This challenge can be illustrated through the following example: suppose a new market actor, e. g. the operator of a wind farm, wants to invest in storage capacities close to a wind farm as the facility can store electricity from the wind farm at times of congestion in the grid. Imagine that this investment would be economically more efficient than expanding the distribution grid. From the perspective of the grid operator, only the costs associated with the investment in the grid infrastructure are usually taken into account for calculating the network charges. Thus, the grid operator has no incentive to support the investment in the economically more efficient storage facility. The regulatory framework needs to take account of the spill-over effects and allow cost- and revenue-sharing models to incentivize investment in smart grid solutions, like the storage facility in the example above.\n\n### 5.3.2 International practice\n\nRegulation of monopolistic bottlenecks and network regulation schemes\n\nIn Europe and many other countries with liberalized electricity markets, regulation focuses on the natural monopoly part of the supply chain, i. e. power transmission and distribution grids. The other elements of the supply chain, i. e. generation and retail, are liberalized and governed by general competition law only. As such, European power sector regulators focus primarily on the regulation of network charges for transmission and distribution grids. In fact, most of them are not even authorized to intervene in the competitive parts of the electricity sector.\n\nWith respect to specific network regulation schemes, the cost-pass-through regulation was traditionally applied in many European countries and the United States. Since this regulation scheme does not set incentives for an efficient grid operation, which is especially important in power systems with a limited need for grid expansion and upgrade, most European countries and some parts of the United States abandoned cost-pass-through regulation in favor of different variants of price cap regulation.4 Germany for example, applies an RPI-X regulation scheme since 2009 (see Sect.\u200b 4.\u200b4.\u200b4 for more details). Currently, only Belgium and most parts of the United States [22], [23] still apply cost-pass-through regulation. Owing to the large network investment requirements associated with the transition towards more RES and smart grids, a reform of the RPI-X system to facilitate efficient investment is currently being discussed in Germany and other parts of Europe. In the light of these developments, it can be seen that no single regulatory scheme is preferable in every situation. In the end, regulatory schemes have to take into account the current needs of each country under consideration.\n\nRegulatory approaches setting incentives for efficient investments\n\nNetwork regulation should set incentives for network operators to develop a secure and stable network at the lowest cost. Also, network regulation should set incentives to invest in smart grid solutions, especially in solutions that defer costly investments in grid expansion. Three possible incentive instruments have been applied in countries such as Italy, the United States, and the United Kingdom. These instruments are described below. However, note that this list is not comprehensive:\n\n * Explicit investment incentives could be applied. Such incentives could be so-called rate-of-return adders. The idea behind rate-of-return adders is that network operators can earn additional return on equity (ROE) for specific projects selected by the regulator. The rate-of-return adder (usually between 2 % and 3 %) increases the incentive for the network operator to build this specific line or to invest in the respective project. This concept has proven to be successful in Italy. Here, roughly 71 % of all investments made by the Italian TSO Terna in 2009 were priority projects with a rate-of-return adder. In Italy, this adder is provided for twelve years after the investment [24]. In Italy's case, it can be observed that overall investments, not only those with the rate-of-return adder, increased since the introduction of the adder. Similar effects were observed in the United States [24].\n\n * Explicit profit-sharing mechanisms, or sliding scales, are currently applied in the UK [25]. They can contribute to strengthening the incentives for investment in smart grid technologies. The idea behind the profit-sharing mechanism is that the grid operator is allowed to keep a share of a cost reduction achieved through the application of a smart application (e. g. a storage facility) as a substitute for grid investments. If the costs for the smart application are lower than the investment in the grid infrastructure, but both measures result in a more stable grid, then the grid operator has an incentive to invest in the smart application. Note that the sliding scales have to be granted by the regulator and that calculating their size is a complex task.\n\n * The innovation bonus is an additional instrument motivating the grid operator to invest in smart grid technologies. The innovation bonus is determined by the regulator and grants funds for R&D activities of the grid operators. The additional costs for the R&D activity are thereby at least partially refunded to the operator and do not reduce the operator's revenues. The innovation bonus is currently applied in the UK as well and has proved to be an efficient instrument [25]. Similar to rate-of-return adders, innovation bonuses can be designed and granted rather easily by the regulator.\n\n### 5.3.3 Recommended approach for China\n\nChina does currently not apply disaggregated regulation, but electricity wholesale and retail prices are subject to regulation. A basic step in the regulatory environment would be to focus on the monopolistic networks alone. This would avoid misdirected incentives at the generation and retail sectors, ease the emergence of new market actors, and improve the quality of regulation. In applying a specific regulation scheme, the regulator should focus on setting efficient investment incentives. As there is no regulation scheme suitable to all countries in all contexts, the specific design of a regulation scheme for China has to be elaborated with great rigor. The considerable investment needs in China's electric power grid and the recent experiences with RPI-X regulation in Germany should be taken into account.\n\nChina faces the primary challenge of having to increase security of supply. In addition, the potential that comes with smart grids can only be realized if there is an incentive for grid operators to invest into smart solutions (e. g. storage facilities or DSM). Thus, network regulation in China should specifically focus on incentives for security of supply and smart solutions. The application of rate-of-return adders for high priority projects for security of supply and for investment in smart solutions should be evaluated.\n\nThe recommended approach at a glance\n\n * Regulating only the natural monopoly part of the electric power sector (transmission and distribution grids) improves the opportunities for market entry of new market actors, reduces misdirected incentives, and may increase quality of regulation.\n\n * Network regulation should focus on setting efficient investment incentives balancing between network expansion and smart grid applications in an economically efficient way. Specifically, the application of rate-of-return adders for projects with a high priority for security of supply might be interesting for China. Furthermore, profit-sharing mechanisms or innovation bonuses could be applied to increase the diffusion of innovative technologies in China's electricity sector.\n\n## 5.4 Coordinate network expansion planning for electricity grid expansion and upgrade\n\nChinese context\n\n * From 2010 to 2015, generation and grid capacities are planned to increase by roughly 50 % in order to cope with the steadily growing demand. Their further expansion beyond that point in time is inevitable.\n\n * RES generation capacities are expected to increase out of proportion \u2013 their share in the electricity mix will increase significantly.\n\n * Electricity grid expansion planning is currently organized in a top-down process with low transparency and little involvement of actors other than government authorities and grid operators.\n\n### 5.4.1 Background\n\nTowards a more decentralized system\n\nSo far, electric power grids have been designed to transport the electricity generated in central power stations (e. g. coal-fired power plants) to industrial or residential load centers. The subordinated distribution grids have only been used to redistribute the electricity towards end-consumers. Single distribution grids have been operated quite independently from those in other areas. In this setting, a coordination of the necessary grid expansion measures has been important mainly at the level of transmission grids.\n\nThe current developments towards more intermittent RES and actively involved end-consumers result in a more decentralized system with bidirectional flows of electricity and information. Massive investments in the power grid infrastructure have to be undertaken to cope with these developments, especially at the distribution grid level: in Germany for example, the investment needs in the distribution grid infrastructure are considerably higher than in the transmission grid infrastructure [21], [26] because roughly 97 % of RES in Germany are connected to the distribution grids [27]. A coordination of the necessary grid expansion measures is especially important at the level of distribution grids to facilitate an efficient allocation of investments and reduce economic inefficiencies as much as possible.\n\nFragmentation of interests\n\nUp to now, investments in generation capacities were made mainly by operators of large-scale centralized power plants, a situation that allows grid operators to collect information on where and when a new power plant is planned for construction rather easily. Network expansion planning was a task with a moderate complexity organized by grid operators, operators of large-scale centralized power plants, and coordinating government institutions.\n\nCurrently, the number and the heterogeneity of stakeholders with interests in network expansion planning increases:\n\n * Investments into distributed RES are made by a larger number of more heterogeneous companies or even by private investors in many countries [28]. Therefore, advanced planning on how much new grid capacity need to be built in what area becomes more difficult for the network operators.\n\n * The trend towards smart grids is associated with an increasing amount of new market actors with diverse interests. As the business models of new market actors often depend on available grid capacities, their interests are also relevant for long-term electricity network planning alongside the stakes of established market actors making the process of stakeholder consultation more complex. If the interests of stakeholders such as grid operators on different voltage levels, power plant operators, industrial consumers, representatives of small end-users, environmental groups, local governments and central government ministries are not coordinated effectively, network expansion risks being economically inefficient [29].\n\nAgainst this background, it becomes of vital importance to develop a long-term centrally coordinated NDP that includes plans and interests of all relevant market actors and network users at an early stage.\n\n### 5.4.2 International practice\n\nNetwork expansion planning in Europe\n\nThe EU requires each member state to develop a 10-year NDP currently focusing on transmission grid expansion but not distribution grid expansion [30]. Given the high investment and coordination needs on the distribution grid level, the non-coordination of distribution grid expansion unnecessarily inflates costs of network expansion. In Germany, a group of geographically adjacent DSO consequently started initiatives to coordinate distribution grid expansion planning [31].\n\nThe network development plans are based on several scenarios (three scenarios in the case of Germany) concerning the future development of RES and the corresponding electricity system in each member state. The national planning processes are accompanied by 10-year network development plans of the ENTSO-E. Starting in 2010, these plans are to be issued every two years. The main results of the version published in 2012 are:\n\n * One third of investments planned in the first network development plan of 2010 are experiencing implementation delays owing to long approval processes.\n\n * 52,300 km of extra high voltage transmission lines clustered in 100 projects have to be modernized or constructed, mainly due to bottlenecks related to RES integration.\n\n * 20 megatons CO2 can be saved in the period up to 2022 due to further market integration in Europe.\n\n * Extending the grid by about 1.3 % enables the addition of 3 % more generation capacities [32].\n\nNetwork expansion planning in Germany\n\nIn Germany, the rules for developing the national NDP are defined in \u00a7 12 EnWG and the NABEG. Figure 5.1 illustrates the different steps towards the network development plan in Germany. These are [33]:\n\nFig. 5.1\n\nProcess for the network development plan in Germany, adapted from [34]\n\n * The starting point in the network development process is the scenario framework. It is drafted by the four German TSO and includes forecasts relating to the development of electricity generation, the shares of different generation technologies, and power consumption for the next ten years. To achieve a realistic forecast, three different scenarios based on different assumptions regarding the increase in generation capacities, RES expansion, and CO2 abatement are considered. The scenarios are submitted to BNetzA, which approves them after a public consultation process. During this process, distribution grid operators and other interested parties have the opportunity to comment on the different scenarios.\n\n * The four TSO subsequently employ the approved scenarios for calculating network expansion requirements. Selecting the necessary expansion measures and assigning a timeframe (two to ten years depending on the respective project) for their realization leads to a first draft of the respective network development plan. BNetzA again assesses this first draft within a public consultation process. During this process, the drafted network development plan is published on a public website and can be commented from various stakeholders and citizens. The network development plan is then revised by the TSO and once again assessed by BNetzA. Furthermore, BNetzA conducts the environmental impact assessment.\n\n * Based on the final version of the network development plan and the environmental impact report published at the same time, BNetzA has the responsibility to draft the so-called Federal Requirements Plan, which is finally brought in the parliamentary legislative procedure.\n\nWhile this process currently focuses on transmission grids only, it might serve as a best-practice example of how to organize the grid expansion planning on the distribution grid level as well.\n\nThe role of stakeholder platforms\n\nOne of the main goals of BMWi with respect to grid development is to involve new market actors in the strategic smart grid planning process. One prominent example of the ministry's activities is the Future-oriented Energy Grids Platform, which was founded in 2010. In February 2011, the platform was transformed into a permanent dialogue forum [35]. The platform involves all relevant parties of the electricity sector: grid operators, ICT-companies, environmental and consumer associations, research institutions, and representatives of several ministries and governmental institutions. One of the platform's main tasks is to discuss the draft documents of the network development plan. The platform is therefore an important instrument for the regulator to get feedback on NDP. Besides this task, the grid platform serves as a discussion platform for planning and approval procedures, the regulatory framework for investments into networks, grid connection of offshore wind farms, funding and testing of new technologies, development of smart meters, system stability, and applications of storage systems for network stability.\n\nFigure 5.2 shows the organizational structure of the platform. The plenary which adopts recommendations on actions is hosted by BMWi. The platform is subdivided in four different working groups, one of them directly responsible for smart grid issues. The different working groups provide consulting services for BMWi or other government institutions, make publicly available the results of their meetings, draft documents and recommendations, and support other publicly funded initiatives for smart grid development. The working group responsible for network development planning for example, proposed a joint grid connection process for offshore wind farms as well as an offshore master plan. In the meantime, both propositions have been included in NABEG [36].\n\nFig. 5.2\n\nStructure of the Future Oriented Energy Grids Platform, adapted from [35]\n\n### 5.4.3 Recommended approach for China\n\nChina faces a high need to expand the existing electricity network within the next few years. On both levels of the grid, transmission and distribution, this need is driven by growing consumer demand for electricity and the integration of RES. Supporting and guiding the formulation of a consistent and comprehensive concept regarding the set-up of China's future electric power grid is a key task for Chinese energy sector regulation. Government guidance is particularly important in the context of the build-up of RES generation capacities and smart grids both resulting in an increasing number of stakeholders involved in the electric power system.\n\nToday, there are no formalized institutions that acknowledge stakeholder interests in grid expansion planning in China. Therefore, stakeholder involvement currently focuses mainly on the grid operators, the government, and the China Electric Power Planning Institute. Other stakeholders are not integrated into the process in a formal way.\n\nThree aspects are especially important with respect to the coordination of network expansion in China:\n\n * First, the institution that has the responsibility to supervise the network planning process should be specified. Experience in Germany illustrates that a government institution such as the regulator should take this responsibility. This has the primary advantage that results from the planning process can be transferred directly into the government policy-making process.\n\n * Second, an evaluation should be made of which stakeholders are relevant for network expansion planning. Specifically, formal coordination between grid operators and power generation companies seems to be of great importance to align grid and generation capacity expansion processes. A platform such as the Future Oriented Energy Grids Platform in Germany has been proven effective in integrating new stakeholders in the strategic network development process. The evaluation of which stakeholders are relevant for network expansion planning could also happen within such a stakeholder platform.\n\n * Third, it is beneficial to specify procedures for the development of network plans. In this context, China could take advantage of German experience with the network development planning process. Making mandatory a NDP for both transmission and distribution grids, including smart grids, helps to make network expansion as efficient as possible. Clear scenarios about the development of RES, the general electricity mix, and electricity demand in China are needed as a basis for the definition of a network development plan in China. Such scenarios could then serve as a common basis for the network development process in China. Stakeholder involvement should be a key element of this procedure because the number of stakeholders is likely to increase in the near future due to the importance of non-incumbent market actors for smart grids. These new stakeholders should be integrated into the strategic network development process in a formal way.\n\nThe recommended approach at a glance\n\n * A stakeholder platform on the topic of the future electricity network involving all relevant market actors of the electricity sector (e. g. similar to the Future Oriented Energy Grids Platform in Germany) could help to evaluate which stakeholders are relevant for network expansion planning, support the network planning process, enhance mutual understanding among different stakeholders and provide recommendations to the government.\n\n * Make mandatory a network development plan (NDP) for transmission and distribution grids, including a procedure for consulting the NDP with third parties and the public. The NDP should reflect fundamental government policy, future energy policy goals, smart grid goals as well as reliability and security issues. The regulator should supervise and confirm the final NDP, which is then signed into law by the legislature.\n\n## 5.5 Improve grid integration of RES\n\nChinese context\n\n * Despite the fact that China has aggressively expanded solar and wind generation capacities since 2006, their shares in the electricity generation mix remain small. However, shares of both generation sources are planned to increase significantly according to government plans. Today, new market actors are only partly involved in the build-up of RES generation capacities.\n\n * Within the existing regulatory framework a lack of sufficient incentives for grid integration of RES persists even though significant improvements have been made in recent years. A considerable number of wind farms is only connected to the electric power grid with delay.\n\n * Often, wind farms can also not be connected due to the lack of a Low Voltage Ride Through Function. At the same time substantial quantities of wind power still have to be curtailed in order to increase grid stability.\n\n### 5.5.1 Background\n\nNew market actors require a physical connection to the power grid\n\nIn many countries, non-incumbent stakeholders have been investing in RES generation capacities and contributing to the increase of RES in the electricity mix: project developers, industrial and commercial companies, prosumers, and even venture capital and private equity companies [28]. These non-incumbent stakeholders need clearly defined conditions for the physical connection of RES installations to the power grid. Their RES plants need to be technically connected to the grid in such a manner that electricity generated from RES can be fed in. Without an equal and fair access, they abstain from investing in RES. As a result, a huge potential of capital resources for RES investment remains untapped.\n\nThe three elements of physical grid integration of RES\n\nGrid integration of RES, i. e. the issue of how effective, rapid, and fair the access of RES generation capacities to the electric power grid is organized, consists of three different aspects:\n\n * Grid expansion, necessary due to the connection of RES, is a prerequisite for the large-scale integration of RES (compare Sects. 5.3 and 5.4).\n\n * Grid connection of RES means technically interconnecting RES to the grid in such a manner that electricity generated from RES can be fed in.\n\n * Specific grid operation issues with large amounts of RES implies assuring that the generated electricity has access to the grid so that it can be transported to end consumers [37].\n\nGrid connection\n\nImportant issues in many countries are so-called long lead times (e. g. delays). This subject groups all aspects associated with the time the investor of a renewable energy generation unit has to wait before connecting to the grid and feeding of electricity in the grid is allowed. Long lead times increase the financial risk for investors, creating less favorable credit conditions for them, and thus reducing the incentives for investments in RES.\n\nThe effectiveness of grid connection is also affected by how grid connection costs are distributed among the parties involved in the process. In this context, shallow or deep cost approaches can be applied:\n\n * In a deep cost approach, the RES investors requesting connection not only cover the costs of grid connection but also further costs related to grid reinforcement and extension beyond the connection point (i. e. deeper into the network).\n\n * In a shallow cost approach, the RES investor only pays for grid connection costs to the connection point, but not for reinforcement and extension costs. Generally, the deep cost approach creates higher costs and risks for the RES investor. It is therefore more often viewed as an obstacle to RES deployment [37].\n\nGrid operation\n\nIn countries with a low share of intermittent RES-E in the energy mix, grid operation is usually not adversely affected by RES installations. However, European countries like Germany or Denmark have made the experience that, with a share of approximately 10 % to 20 % of intermittent RES-E, grid operation becomes more and more difficult because regional feed-in often exceeds regional loads and the grids do not have sufficient capacities to absorb and transport the complete surplus of RES-E. One straight-forward remedy for these situations is to expand grid capacities. However, curtailing specific RES generators in times of peak production might be economically more interesting: a study suggests that curtailing only 2 % of the annual electricity production from RES (in times of peak production) could reduce infrastructure investments by 10 % between now and 2030 [26].\n\nGrid curtailment is currently a critical issue in a number of European countries, especially due to the lack of specific curtailment rules and compensation issues [37]. Clear and well-defined grid codes for the interconnection of RES on all grid and voltage levels are a decisive factor to ease grid operation with RES [38]. These rules define the interaction of RES and the electric power grid in terms of grid operation. This also includes the handling of congestions and disturbances.\n\nThe ICT-integration of RES\n\nRES integration issues go beyond the physical connection of RES to the power grid. The basic idea of smart grids is to make all generators and consumers of electricity monitorable and in part remotely controllable by means of ICT. Only if RES plants are remotely controllable, RES generation can be coordinated with grid capacities at any time. In this light, the equipment of RES with communication interfaces is an essential part of smart grids. Also with respect to an economically efficient grid curtailment of RES, the capability of grid operators to monitor and control installations remotely is necessary.\n\nEspecially distributed RES (e. g. small rooftop PV installations or single wind turbines) are currently only rarely equipped with technology allowing the grid operator to remotely monitor or control the installations. Given that owners of RES, especially those of small-scale RES, have usually few incentives in investing in communication interfaces, the widespread deployment and usage of such interfaces depends on appropriate regulatory guidance.\n\n### 5.5.2 International practice\n\nThe European perspective\n\nIn its Directive 2001\/77\/EC2 [39], the EU laid down a common regulatory basis for the promotion of electricity generated from RES and a framework for the integration of RES into the grid [40]. Paragraphs 1\u20133 of article 7 contain general references to grid connection and operation issues demanding that EU member states\n\n> take the necessary measures to ensure that transmission system operators and DSOs in their territory guarantee the transmission and distribution of electricity produced from renewable energy sources.\n\nAlso, the directive proposes priority access to the grid for electricity from RES and favors a cost-sharing of grid expansion measures between grid operators and RES operators to pure deep cost and pure shallow costs approaches [39].\n\nPublished eight years later, Directive 2009\/28\/EC3 further regulates grid connection and operation issues. In paragraphs 2\u20134 of article 16, the EU makes a clear case for priority access for electricity from RES. The Directive recommends that such electricity be fed in with priority, that grid curtailment measures for RES be minimized and justified by the regulator, and that grid expansion costs related to renewable energies be transparent and born in full or in part by the grid operators.\n\nSince 2009, ENTSO-E has been working on a network code for grid connection requirements which aims at\n\n> setting out clear and objective requirements for generators for grid connection in order to contribute to non-discrimination, effective competition and the efficient functioning of the internal electricity market and to ensure system security [41].\n\nThe network code\n\n> defines a common set of requirements for power generation facilities, including synchronous generation units, power park modules and offshore generation facilities, to be connected to the network and sets up a common framework for grid connection agreements between grid operators and the power generation facility operators [41].\n\nIn defining these requirements, the network code clearly reduces conflicts between RES investors and grid operators and contributes to a better communication between them and to shorter lead times in RES connection as a result of more standardized procedures.\n\nGrid connection of RES in Germany\n\nPart 2, chapters 1\u20133 of the EEG contain regulations that are of critical importance with respect to the effectiveness of grid connection in Germany:\n\n * Chapter 1 section 5 emphasizes that grid operators must immediately and as a priority connect all RES to the grid even if this connection necessitates optimizing, boosting, or expanding the grid.\n\n * Paragraph 1 contains a definition of the so-called grid connection point. It is the point in the electricity grid which is at the nearest linear distance from the location of the RES installation. The generation capacity of the RES unit to be connected determines the voltage level of the nearest grid connection point.\n\n * The responsibility of building a connection between the RES installation and the grid connection point is assigned to the grid operator. However, the costs for building this direct connection are passed on to the RES investor as specified in chapter 3, section 13, and paragraph 1 of the EEG. Note that due to the rather close meshed electricity grids in Germany, the distances between the installations and the grid connection points are often rather short. Thus, connection costs to be paid by the RES investors are rather low in Germany.\n\n * As specified in chapter 3, section 14, however, the share of the costs within the entire electric power grid that is related to optimizing, boosting, and expanding the grid system is allocated to the grid operator (shallow cost approach). This cost-sharing mechanism prevents the investors in RES projects from installing capacities at great distances from the existing power grid, as they are the ones who have to cover the costs for the direct line to the nearest connection point.\n\n * Chapter 2, section 9 specifies further that the grid operator can only be forced to optimize or expand the electricity grid if this is not economically unreasonable.\n\nEEG \u2013 Paragraphs 1\u20134 of Part 2, Chapter 1, Section 5\n\n * Grid system operators shall immediately and as a priority connect installations generating electricity from renewable energy sources and from mine gas to that point in their grid system (grid connection point) which is suitable in terms of the voltage and which is at the shortest linear distance from the location of the installation if no other grid system has a technically and economically more favorable grid connection point. [...]\n\n * Installation operators shall be entitled to choose another grid connection point in this grid system or in another grid system which is suitable with regard to the voltage.\n\n * In derogation of subsections (1) and (2) above, the grid system operator shall be entitled to assign the installation a different grid connection point. This shall not apply where the purchase of electricity from the installation concerned would not be guaranteed in accordance with section 8(1).\n\n * The obligation to connect the installation to the grid system shall also apply where the purchase of the electricity is only made possible by optimizing, boosting or expanding the grid system in accordance with section 9.\n\nSource: Wording from the BMUB [62]\n\nThe combination of a shallow cost approach and an obligation to connect RES with priority is one important factor with regard to grid connection in Germany. Another important element of the EEG (with regard to grid connection) is the definition of what happens if the grid operator fails to connect RES. Part 2, chapter 1, section 10 of the EEG is based on the notion that grid connection rules can be effective only if non-compliance of the grid operator leads to financial losses.\n\nEEG \u2013 Paragraphs 1\u20132 of Part 2, Chapter 1, Section 10\n\n * In the event that the grid system operator violates his obligations under section 9(1), those interested in feeding in electricity may demand compensation for the damage incurred. The liability to pay compensation shall not apply if the grid system operator was not responsible for the violation of the obligation.\n\n * Where there are facts to substantiate the assumption that the grid system operator did not fulfill his obligation under section 9(1), installation operators may require the grid system operator to submit information concerning whether and to what extent the grid system operator did not meet his obligation to optimize, boost and expand his grid system. This information may be withheld if it is not necessary in order to establish whether the entitlement in accordance with subsection (1) above exists.\n\nSource: Wording from the BMUB [62]\n\nGrid operation in the presence of RES at the German level\n\nIn spite of the increasing share of electricity from wind and PV power, Germany's electric power grids are still among the most secure and reliable grids worldwide. Germany has made the experience that effective grid curtailment rules are necessary to sustain reliability and security of supply in times of rising feed-in from RES. To allow for an effective grid operation in spite of the presence of RES, the EEG distinguishes between normal and critical grid conditions:5\n\n * In normal grid conditions, electricity produced in RES installations has to be fed in at any time \u2013 irrespective of electricity consumption. This is regulated in part 2, chapter 1, section 8 of the EEG, which states that \"[transmission] grid system operators shall immediately and as a priority purchase, transmit and distribute the entire available quantity of electricity from renewable energy sources\".\n\n * During critical grid conditions, however, grid operators are allowed to curtail electricity from RES by taking technical control over installations connected to their grid system. Part 2, chapter 2, section 6 of the EEG specifies that all RES plants have to be equipped with a communication interface allowing the grid operator to take over technical control over installations or to limit their effective capacity to 70 % of the installed capacity instead. Those RES with a capacity exceeding 100 kW have to be unconditionally enhanced with a communication interface allowing the grid operator to remotely monitor the electricity feed-in in real time and to remotely control installations. To prevent a misuse of grid curtailment, grid operators are obliged to immediately report such measures to the BNetzA. This grid curtailment of RES is not to the detriment of RES investors because they receive equivalent compensations to the extent that they incur no financial losses as a result of the grid curtailment (see appendix E for extracts from EEG, chapter 2, part 2, sections 11 and 12.).\n\nEEG \u2013 Paragraphs 1\u20132 of Part 2, Chapter 1, Section 6\n\n * Installation operators and operators of CHP installations shall provide installations with an installed capacity exceeding100 kilowatts with technical facilities with which the grid system operator can, at any time:\n\n1.\n\nreduce output by remote means in the event of grid overload; and\n\n2.\n\ncall up the current electricity feed-in at any given point in time.\n\n * Operators of installations generating electricity from solar radiation:\n\n1.\n\nwith an installed capacity between 30 kilowatts and 100 kilowatts shall fulfil the obligation pursuant to subsection (1) no. 1 above;\n\n2.\n\nwith a maximum installed capacity of 30 kilowatts shall:\n\na)\n\nfulfil the obligation pursuant to subsection (1) no. 1 above; or\n\nb)\n\nlimit the maximum effective capacity fed in at the grid connection point with the grid system to 70 percent of the installed capacity.\n\nSource: Wording from the BMUB [62]\n\nEnsuring compliance with the EEG regulation\n\nWell specified laws alone might not be sufficient to ensure an effective grid integration of RES if there is no institution with a distinct judiciary function to monitor compliance with the rules and settle any dispute between grid operators and power plant owners. In this light, BMUB commissioned and funded the so-called Clearingstelle EEG in 2007. Since then the Clearingstelle EEG serves as a neutral and independent institution with the intention to settle any dispute between grid operators and power plant owners with regard to topics surrounding EEG. The staff of the Clearingstelle EEG consists of an interdisciplinary team of lawyers and engineers with expertise in the field of renewable energies [42].\n\nThe general idea is that ordinary courts settle potential disputes rather through costly and time consuming litigation while the Clearingstelle EEG can offer alternative dispute resolution options such as\n\n * mediation,\n\n * joint dispute resolution, and\n\n * arbitration [42].\n\nIn addition, the Clearingstelle EEG reports the results of past legal disputes such that a high transparency on the jurisdiction exists among grid operators and plant owners. In sum, the Clearingstelle EEG clearly contributes to a high compliance with the EEG regulation in Germany.6\n\n### 5.5.3 Recommended approach for China\n\nThe physical grid integration of RES in China currently lacks efficiency. This refers to both grid connection and grid operation. Specifically, a considerable part of wind power is currently being curtailed. There is also evidence that only a very small part of investments are being made by non-incumbent players in China: investments from venture capital and private equity companies for example, are quite low compared to many other countries [28]. For the government's ambitious RES expansion targets to be achieved and in order to incentivize new market actors to invest in RES, the physical grid integration of RES has to become more effective.\n\nTransparency and a clear division of responsibilities between grid operators and RES investors would encourage the deployment of RES and incentivize new market actors to invest in RES. The following recommendations are made based on best practices from Europe and especially Germany:\n\n * The grid operator is a monopolistic company with special responsibility. He has to define clear, transparent, and technically sound grid codes for the integration of generation units at all voltage levels. These codes have to be defined properly and made binding.\n\n * For an efficient and fast connection, grid connection points have to be defined properly for all kinds of RES on all voltage levels of the grid. RES need to be assigned a grid connection point on request, so that the interconnection point can be installed without delay and according to well-defined technical standards.\n\n * Grid operators have to bear liability for the grid connection of RES. This is very important, because liability puts a high priority on the establishment of the grid connection and avoids delays on the grid operator's side. In this context, definition of financial incentives (e. g. in form of penalties) is an effective measure to quicken the grid connection process and to reduce the risk for RES investors.\n\nThe same arguments which hold for the grid connection are equally valid for grid operation issues. To ensure grid stability, it is necessary to curtail RES in times of critical grid conditions. To provide transparent procedures to the RES operators, detailed processes for curtailment of RES (including documentation, transparency rules, timeframes, involved parties, etc.) have to be defined and the requirements for information exchange within these measures have to be specified. To ensure an economically efficient grid curtailment, the installation of communication interfaces at RES installations should be promoted by appropriate regulation (such as part 2, chapter 1, section 6 of EEG in Germany). Moreover, it is important that grid curtailment of RES only takes place if this is urgently needed to stabilize grid operation and that the compensation to the RES investor is calculated transparently.\n\nThe experiences from Germany further show that the establishment of an independent institution offering mediation, joint dispute resolution, and arbitration services contributes to a high compliance with the regulation.\n\nThe recommended approach at a glance\n\n * Grid connection points and binding grid codes specifying respective responsibilities of grid operators and power generators have to be defined.\n\n * Grid operators have to bear liability regarding the grid connection of RES.\n\n * Binding procedures for curtailment of RES have to be defined. To ensure an economically efficient grid curtailment, the installation of communication interfaces at RES installations should be promoted by appropriate regulation.\n\n * An institution offering mediation, joint dispute resolution, and arbitration services in the context of grid connection issues should be assigned.\n\n## 5.6 Optimize the balancing of electricity generation and consumption\n\nChinese context\n\n * Due to the high share of industrial load, China's overall electricity load curve is currently rather flat. In the future, China's electricity generation might become considerably more fluctuating owing to the further build-up of RES generation capacities planned by the government. In addition, shares of more intermittent residential and commercial electricity consumption are expected to increase. Both factors will lead to regionally higher load variability in China.\n\n * Higher load variability necessitates a more effective coordination of electricity generation and consumption. Specifically, with growing shares of intermittent RES it will become indispensable that electricity consumption will be, at least partly, able to follow electricity generation. This necessitates the use of economic incentives as well as sophisticated technologies facilitating the coordination of electricity generation and consumption. All types of consumers, i. e. industrial, commercial, and residential, have to be involved in these activities.\n\n * Time-of-use pricing (i. e. electricity prices depending on the time when electricity is provided) aims at incentivizing electricity consumers to shift their consumption according to generation and grid capacities. It has gradually been introduced in China to all categories of users except residential consumers and irrigational users. The Chinese government recently announced that it would introduce time-of-use pricing also for residential consumers by the end of 2015. While this is certainly a step in the right direction, it is still questionable whether the present tariff system offers sufficient price incentives for a pronounced intraday shifting of power demand.\n\n### 5.6.1 Background\n\nPeak shaving and residual peak load shaving\n\nThe balancing of generation and consumption of electricity is of crucial importance for the stable operation of electric power systems. In a power system with 100 % generation from large central (bulk) power plants, electricity generation follows the load. Sometimes, peak loads may cause shortages on the generation side. Peak shaving, i. e. reducing the electrical power consumption during periods of maximum electricity demand, is then an important factor to stabilize system operation. In power systems with a high share of RES, however, the motivation for peak shaving changes somewhat: in such power systems, the residual load, i. e. the difference between load and renewable generation, is highly volatile due to the intermittent nature of renewable generation. In these cases, large central power plants, usually conventional fossil fuel-driven power plants, have to cover the residual load \u2013 also known as residual peak load shaving (or residual load balancing).\n\nBalancing mechanisms and technologies like demand side management (DSM), supply side management (SSM), microgrids, virtual power plants (VPP), and energy storage can be employed to facilitate the coordination of electricity generation and consumption. All these mechanisms and technologies require an ICT infrastructure for measurement, control, and billing of various loads and\/or generation units at local, regional, or distributed sites.\n\nDemand side management\n\nDSM is an important mechanism for peak shaving and residual peak shaving for different types of loads. DSM means incentivizing electricity consumers to adapt their consumption to the availability of electricity generation. Principally, the loads can be influenced by means of two different mechanisms: on the one hand, end customers can react manually to suitable incentives such as price signals. On the other hand, automated load control is conceivable for devices where deferred use and a modulated operation mode entail no loss of comfort, economic consequences, or other restrictions in everyday household\/business life (dispensable loads). With respect to the potential of DSM on low voltage level, the necessary IT infrastructure and the relation between DSM and AMI have been discussed widely [43].\n\nSupply side management\n\nSupply side management (SSM) works similarly to DSM but refers to local or distributed generation facilities. SSM means incentivizing electricity producers to adapt their generation to the demand of electricity. The generation units are typically small-scale residential units like CHP plants or larger CHP units located at industrial facilities.\n\nVirtual power plants\n\nVirtual power plants (VPP) aggregate power generation, storage, or consumption units into one balancing unit by means of ICT metering and control technologies. These units may be spread out over the grid and belong to separate owners. VPP are especially useful in the context of electricity markets: the VPP operator controls or manages the operation of the generation units to follow a joint schedule or to offer balancing power to the system operators.\n\nMicrogrids\n\nMicrogrids are a specific solution to manage the intermittent character of RES on a regional level. They are grid areas in which generation from local RES or other distributed generation and consumption is balanced by local control mechanisms by means of information exchange between the devices through local ICT. As such, the higher-level distribution grid to which the microgrid is connected is not necessarily and immediately exposed to the intermittence of RES connected to the microgrid. Island grids without any connections to distribution grids are an extreme form of this approach. Note that microgrids are geographically connected grid regions, while VPP are virtually aggregated units which are distributed throughout larger grid areas.\n\nEnergy storage\n\nRequirements for power storage differ according to field of application. Some storage technologies are used to balance fluctuations within a very short timeframe (e. g. some milliseconds) and others are used for longer timeframes (e. g. days or weeks):\n\n * Hydrogen generated by means of electrolysis can be used to store electric power for timeframes lasting up to several months.\n\n * Pumped-storage and compressed air energy storage power stations are suitable for an intraday balancing of generation and consumption. On a global level, pumped-storage power plants are the most important technology to store electric power.\n\n * Electrochemical storage mediums like batteries are employed to store electricity for timeframes ranging from one hour to several days.\n\n * Centrifugal mass storage, super caps, and superconductive inductors can be charged and unloaded within an extremely short timeframe. They are used for grid stabilization services and voltage maintenance [44].\n\nBarriers to the development of balancing mechanisms and technologies\n\nThere are many different barriers to the development of balancing mechanisms and technologies as described above. The elimination of such barriers calls for suitable regulatory measures. In principal, these barriers can be classified into three different categories [45]:\n\n * Technological barriers refer to the fact that the maturity of some technologies that might be important for the balancing of generation and consumption is still rather low [46]. Electrochemical storage mediums such as batteries, for example, are in many cases still too expensive to be used in a widespread manner [44].\n\n * Economic barriers exist, because for many stakeholders in the electricity sector the costs of investing in balancing mechanisms and technologies are still higher than the benefits. For potential investors, especially for small-sized third parties, the administrative overhead and the necessary investments in ICT for controlling, balancing, and billing are relatively high. Many of them consequently refrain from investing in balancing technologies. Specifically, there are many open questions related to billing, balancing, and accounting, in the case that several market actors coordinate power generation and load for different business purposes.\n\n * Institutional barriers may arise if new technologies have to adapt to practices and codes developed in a context in which these new technologies were not yet known. For example, microgrids require specific connection codes that are different from small conventional distribution grids. The costs of deploying such balancing mechanisms and technologies critically depend on connection codes, legal reporting obligations, and application processes [45].\n\n### 5.6.2 International practice\n\nReducing technological barriers\n\nIn order to promote the development of technologies and business models to balance electricity generation and consumption, government-supported R&D programs are of fundamental importance (Germany's R&D program E-Energy is presented in Sect.\u200b 4.\u200b4.\u200b6). Many countries have established funding schemes to reduce existing technological barriers:\n\n * The United States for example, claim to invest at least $3 billion into smart grid projects [47]. Specifically, they have funded two important R&D programs for microgrid and DSM demonstration projects run by the United States Department of Defense (USDOD) and the United States Department of Energy (USDOE). The USDOD provided $38.5 million for three different military base microgrid demonstrations, with a focus on reliability and energy security [45]. The USDOE spent over $50 million for nine projects having the concrete goal of achieving a 15 % peak load reduction in the local distribution feeder [45].\n\n * In recent years, the EU has also invested significant amounts in smart grid-related R&D and devoted several major research efforts exclusively to DSM, SSM, VPP, microgrids, and energy storage. One promising example is the Future Internet for Smart Energy (FINSENY) project which was conducted as part of the private public partnership Future Internet from 2010 to 2013.7\n\nIn the United States and in Europe, the experience was made that two factors are especially important with respect to the success of R&D funding (cf. e. g. [45]):\n\n * Clear targets on what should be achieved by means of the research program have to be defined in advance by the funding organization. Only if such targets exist can the success of the research project be measured and controlled during and after the project.\n\n * The recipient of R&D subsidies often has to co-finance the research program with its own capital. Financial participation is an important additional incentive to ensure that the recipient of the subsidies will efficiently and effectively carry out the research project.\n\nReducing economic barriers\n\nAll regulatory approaches aiming at decreasing investment costs in balancing mechanisms or technologies or increasing profits related to their usage can be seen as an effective measure to promote their development:\n\n * A possible solution aiming at reducing investment costs is to directly subsidize the usage of balancing mechanisms or technologies. To promote electricity storage, for example, the German government recently established a co-funding scheme for electric battery storages newly installed in private houses in combination with PV. The purpose of this measure is to promote local consumption of electricity generated by PV, thereby limiting the PV feed-in to the distribution grid.\n\n * Time-of-use pricing is an important option to increase profits related to the usage of DSM, SSM, and energy storage. The higher the price differences between peak and base load prices, the higher the rate of return of DSM, SSM, and energy storage.\n\n * The attractiveness of microgrids critically depends on the difference of electricity generation costs within the microgrid and general retail prices for electricity. If a microgrid is able to produce its own electricity, consumers within the microgrid are independent from general electricity retail prices. However, it is only if general retail prices for electricity are higher than electricity generation costs within the microgrid that investments in these technologies might be potentially amortized after some years. In this light, increasing electricity retail tariffs can be seen as an important driver for investments in microgrids. Depending on the regulatory environment, microgrid operators might also be able to export the electricity to the distribution grid. In this case, the question to be asked is whether the operators receive payments for the electricity they export and what rate these payments depend on: wholesale, retail, or potentially a feed-in tariff. Feed-in tariffs that are higher than the general retail price for electricity and higher than electricity generation costs inside the microgrid can be seen as an effective measure to promote the development of microgrids [48].\n\nReducing institutional barriers\n\nStandards for regulating the general grid connection of RES are common in most countries. Like Europe, the United States, for example, have a standard for grid connection: IEEE 1547 was established in 2003. These standards often do not contain specific rules for VPP or microgrids. However, some countries have started to issue special regulation for VPP and\/or microgrids:\n\n * To ease the grid connection of microgrids, the United States issued IEEE 1547.4: Guide for Design, Operation, and Integration of Distributed Resource Island Systems with Electric Power Systems. This standard presents alternative approaches and good practices for the design and operation of microgrids and their integration with distribution grids. For instance, this includes the ability to separate from and reconnect to the distribution grid while providing power to the islanded microgrid. The IEEE 1547.4 standard is currently gaining approval on the international level.\n\n * Another example of institutional barriers for microgrids being reduced is Germany, where microgrid operators have been relieved of several legal requirements that have applied to distribution grid operators in Germany since 2005. Special rules applying to so-called site networks have been introduced by EnWG for grids covering campuses and enterprises. Such site networks can be designed as microgrids. For example, operators of site networks are not subject to the general connection obligation (\u00a7 18 EnWG): they are permitted to publish less reports on grid conditions (\u00a7 14b EnWG), have less monitoring obligations (\u00a7 35 EnWG), and less obligations to report service disruptions (\u00a7 52 EnWG).\n\n### 5.6.3 Recommended approach for China\n\nThe key driver for the usage of balancing mechanisms and technologies in China is peak shaving due to the high growth rate of power consumption. The balancing mechanisms and technologies described above, especially DSM and energy storage, will contribute to a more effective peak shaving and to an improved utilization rate of the grid infrastructure. In the long term, residual peak shaving \u2013 necessary due to the increasing expansion of RES generation capacities \u2013 will be an additional aspect in China. Balancing mechanisms and technologies like SSM, VPP, and microgrids will become more important in this context.\n\nIn 2011, more than 70 % of China's electricity was consumed by the industrial sector [49]. In general, the technical potential for peak shaving in this sector is comparatively high and can be realized with dedicated ICT solutions. Usually, the potential for peak shaving is smaller and more distributed in the commercial sector and especially in the residential sector. ICT requirements and costs of implementing balancing mechanisms and technologies in these sectors are therefore comparatively higher. In this light, balancing between generation and consumption in China should be optimized with a focus on industrial and commercial consumers in the short- to medium-term and on residential consumers in the long term.\n\nWith this general background in mind, several specific policies may foster the development and usage of specific balancing mechanisms and technologies:\n\n * Time-of-use pricing to incentivize the usage of DSM and energy storage: Time-of-use pricing has gradually been introduced to all categories of users except residential consumers and irrigational users. With regard to commercial and industrial consumers, China already has a very high time-of-use adoption rate, with roughly two thirds of large-scale customers using time-based electricity tariffs [50]. Even if the difference between peak and off-peak prices has increased in recent times, it is still questionable whether the current tariff system offers sufficient incentives for a pronounced intraday shifting of power demand [51]. The Chinese government recently announced that it would also introduce time-of-use pricing for residential consumers by the end of 2015 [50]. In designing this new tariff system, sufficient differences between peak and off-peak prices have to be considered as a key success factor. The same key success factor applies to time-of-use pricing in non-residential sectors.\n\n * Additional R&D funds for SSM, VPP, microgrid, and energy storage demonstration projects: To help abolish technological barriers, the Chinese government has decided to promote the development of different balancing mechanisms and technologies and it has already started to foster R&D in these areas. From an institutional point of view, the NEA has played the most active role in promoting balancing mechanisms [45]. Other institutions that are interested in the promotion of such technologies are the NDRC, the Ministry of Housing and Urban Rural Development, and the MOF [45]. However, the pertinent research has just started in China and consequently needs to be intensified in the future to catch up with international best practice technologies. Thus, additional R&D funds should be set up for SSM, VPP, microgrid, or energy storage demonstration projects. In this context, it is specifically important to set up concrete performance targets for funded demonstration projects and require a co-financing of the subsidies' recipients.\n\n * Feed-in tariffs and interconnection standards for microgrids: compared to European countries and many other countries, retail prices for electricity are very low in China. As a consequence, prices for locally generated electricity (e. g. in microgrids) are often significantly higher than local retail prices for electricity. This has already led individual investors to abandon microgrids demonstration projects [46]. However, increasing electricity retail prices to promote the development of microgrids would directly contrast one of China's primary energy policy goals: affordable retail prices. Therefore, feed-in tariffs for local RES or microgrids could be an option to incentivize investments in microgrids. Also, interconnection standards for microgrids (such as IEEE 1547.4), currently not existing in China, should be issued and made legally binding.\n\nThe recommended approach at a glance\n\n * Coordination between generation and consumption should be optimized with a focus on industrial and commercial consumers in the short- to medium-term and on residential consumers in the long term.\n\n * Peak shaving is of crucial importance in the short to medium term. To promote peak shaving, it is specifically recommended to refine time-of-use pricing in China in order to set sufficient incentives (by means of high differences between peak and off-peak prices) for investments in DSM and energy storage for all categories of consumers.\n\n * Residual peak shaving is relevant in the long term. Additional R&D funds for SSM, VPP, and microgrids as well as interconnection standards and feed-in tariffs specifically for microgrids can be employed to reduce technological, economic, and institutional barriers to the development of SSM, VPP, and microgrids.\n\n## 5.7 Facilitate the development of a unified view of smart grids\n\nChinese context\n\n * Due to the different strategies of China's grid operators with regard to smart grid development and the fact that the Chinese government has not yet publicly defined its view on smart grids, there is still no unanimously accepted vision on the technological and organizational design of smart grids in China. As a result, there is much uncertainty among potential smart grid investors regarding the future development.\n\n * The diversity of stakeholders interested in standardization in China has increased in recent years: in addition to the central government, research institutes, universities, civilian and defense industries, and regional governments, all have their own interests with regard to standardization and are increasingly willing to contribute to the standardization process [63].\n\n### 5.7.1 Background\n\nSmart grids require a common understanding of all stakeholders\n\nSmart grids represent a concept aiming at the integration of information and communication among market actors from various sectors as well as a multitude of power system components. In a smart grid, data on the grid status is exchanged as well as data related to services, products, and reporting obligations. A key challenge of smart grids is to integrate its different components and ensure that they can communicate with the help of ICT. The definition of common standards for interfaces and communication protocols is therefore of utmost importance in order to ensure interoperability and a smooth exchange of information between the different elements of a smart grid. The costs of connecting smart grid technologies through ICT (integration costs) significantly affect the overall costs of deploying smart grid technologies and are thus one key success factor for smart grids. A prerequisite for reducing integration costs is a unified view on smart grid technologies, business processes, and procedures. Furthermore, smart grid-related technologies, products, and services can only be developed by the variety of stakeholders in a cost-efficient manner if requirements for smart grid solutions are accepted by all smart grid stakeholders.\n\nStandardization as a means to create a common understanding\n\nStandardization can serve to create a common understanding between participating stakeholders, thus increasing interoperability, and reducing smart grid integration costs. According to the German Institute for Standardization (DIN), standardization usually has at least five major goals:\n\n * securing the competiveness of domestic industries in the international context of a broad ensemble of diverging technologies and procedures,\n\n * providing a strategic instrument to foster economic and social success,\n\n * supporting and relieving government regulation,\n\n * fostering technological convergence, and\n\n * creating efficient processes and instruments [52].\n\nThese goals illustrate that the current scope of standardization goes well beyond the integration of two or more individual systems. The following aspects should be focused on in the context of smart grid-related standardization issues:\n\n * How can the development of standards support the common understanding of future smart grids?\n\n * How can the process of national and international standardization be optimized for faster time to markets of necessary standards?\n\n * How can the heterogeneous requirements and viewpoints of different stakeholders with regard to smart grid architecture be expressed?\n\nThe government's role in promoting standardization\n\nIn some cases, governments are quite actively involved in standardization processes. By issuing laws and regulations providing minimum standards for certain goods, services, or technologies, governments are even able to legally enforce standards. In many cases, however, standardization is mainly pushed forward by the private industry in cooperation with accredited standards developing organizations (SDO).8 Note that, due to the involvement of many different stakeholders in SDO, a high level of coordination between the various stakeholders is necessary.\n\nEven if standardization is left to the private sector, many governments acknowledge its importance in today's quickly developing technological environment. In 2009, for instance, the German government issued a Standardization Policy Concept of the Federal Government [53]. This policy paper presents the goals and expectations of the government with regard to standardization as well as specific measures of different ministries. In general, governments have many options to influence standardization, and consequently they play an important role in this context. To give just a few examples, possible governmental measures to promote standardization are:\n\n * hosting conferences and symposia on standardization issues,\n\n * strengthening the role and (financial) power of SDO,\n\n * actively participating in the work of SDO, and\n\n * considering standardization issues in the educational system [53].\n\n### 5.7.2 International practice\n\nEuropean Mandate M\/490\n\nThe EU specifically acknowledges the importance of standardization in smart grids: the European Commission issued the European Mandate M\/490 with the intention to promote the development of a unified and commonly accepted view on smart grids to increase interoperability and reduce integration costs. To execute Mandate M\/490, the Smart Grid Coordination Group (SG-CG) was founded by the major European standardization organizations European Committee for Standardization (CEN), European Committee for Electrotechnical Organization (CENELEC), and European Telecommunications Standards Institute (ETSI). All of them are mirror-organizations of the main international standardization organizations International Organization for Standardization (ISO), International Electrotechnical Commission (IEC), and International Telecommunication Union (ITU).\n\nScope and objective of Mandate M\/490\n\n\"The challenge of Smart Grids deployment will require changes to existing standards, industry rules and processes.\n\nThis mandate is to address such a challenge in the field of standardization. The expected long term duration of Smart Grid deployment suggests the need for a framework that is:\n\n * Comprehensive and integrated enough to embrace the whole variety of Smart Grid actors and ensure communications between them\n\n * In-depth enough to guarantee interoperability of Smart Grids from basic connectivity to complex distributed business applications, including a unified set of definitions so that all Members States have a common understanding of the various components of the Smart Grid.\n\n * Flexible and fast enough to take advantage of the existing telecommunications infrastructure and services as well as the emergence of new technologies while enhancing competitiveness of the markets.\n\n * Flexible enough to accommodate some differences between EU Member States approaches to Smart Grids deployment [...].\n\nThe expected framework will consist of the following deliverables:\n\n * A technical reference architecture, which will represent the functional information data flows between the main domains and integrate many systems and subsystems architectures.\n\n * A set of consistent standards, which will support the information exchange (communication protocols and data models) and the integration of all users into the electric system operation.\n\n * Sustainable standardization processes and collaborative tools to enable stakeholder interactions, to improve the two above and adapt them to new requirements based on gap analysis, while ensuring the fit to high level system constraints such as interoperability, security, and privacy, etc.\"\n\nSource: Wording from the European Commission [65]\n\nOrganizational aspects of Mandate M\/490\n\nThe SG-CG consists of four working groups that are synchronized with those of mandates M\/441 (smart metering) and M\/468 (electric mobility). The following four working groups are organized under a joint steering committee (see Fig. 5.3):\n\nFig. 5.3\n\nOrganizational structure of M\/490 SG-CG, taken from [55]\n\n * First Set of Standards (WG FSS): Responsible for compiling a consistent set of smart grid standards based on existing standards by applying the concepts and processes from the other working groups.\n\n * Reference Architecture (WG RA): Responsible for the design of a technical reference architecture to be used in the other groups in order to consistently and comprehensively describe smart grids. The technical reference architecture can be thought of as a map showing the boundaries as well as different areas of smart grids. It can be used to increase the understanding of who does what with whom in which manner in smart grids.\n\n * Sustainable Processes (WG SP): Responsible for the design of processes for the identification and application of smart grid use cases. In the smart grid context, use cases describe specific smart grid applications and define the important actors, systems and technologies and their requirements and functions contributing to the development of a common understanding of smart grids [54]. Use cases are neutral with regard to specific projects, products, and vendors and can also be applied within a gap analysis revealing the need for future smart grid standardization.\n\n * Smart Grid Information Security (WG SGIS): Responsible for identifying the guidelines to achieve information security and privacy in the context of the application of current smart grid standards.\n\nMain results of European Mandate M\/490\n\nAs it provides the structure referenced in the results of the other SG-CG working groups, the so-called Smart Grid Architecture Model (SGAM), a technical reference architecture developed within WG RA, can be described as the first deliverable of the Mandate M\/490. SGAM (see Fig. 5.4) is a three-dimensional model of the European smart grid environment. It distinguishes between physical domains (generation, transmission, distribution, distributed energy resources, and customer premises), management zones (process, field, station, operation, enterprise, and market) and interoperability dimensions (component, communication, information, function, business).\n\nFig. 5.4\n\nSGAM \u2013 The Smart Grid Architecture Model, taken from [56]\n\nThe different interoperability dimensions are based on the GridWise Architecture Council's (GWAC) Inter-operability Context-Setting Framework [57]: the component layer represents the physical\/technical aspect of the system and is therefore used to model physical equipment and infrastructure. The communication layer further specifies protocols and procedures of data exchanges between the components, while the information layer outlines the information models used in the context of the components and information exchange. Above these solution-oriented dimensions, the function layer specifies the logical and thus technology-independent viewpoint in terms of the functions and services realized by the implementation, while the business layer finally represents the objectives as well as regulatory and legal requirements connected to these functions.\n\nWG SP applied the SGAM to identify smart grid use cases. The use case methodology is widely based on the Publicly Available Specification (PAS) IEC 62559 and includes a template for the unified description of use cases as well as a process to identify and manage use cases (subject to standardization as IEC 62559). WG FSS compiled a smart grid standards list based on SGAM and the smart grid use cases. The list structures the multitude of standards within a common framework reducing the complexity of the standardization landscape. Users of the list may search for appropriate smart grid standards based on their classification by domains, zones and layers (see appendix F for further information on the work of WG SP and WG SGIS as well as the next steps of Mandate M\/490).\n\n### 5.7.3 Recommended approach for China\n\nThe diversity of stakeholders interested in standardization issues in China has generally increased in recent years. The trend towards RES and more actively involved end consumers of electric power additionally tends to increase the number of stakeholders engaged in smart grid standardization. In particular, new market actors such as the ICT industry are expected to participate actively in order to unfold their innovative potential.\n\nExamples for current standardization activities in China and possible connections to the European Mandate M\/490\n\nThe SGCC Framework and Roadmap for Strong & Smart Grid Standards [66] stresses the importance of standardization for the smart grid development. Analyzing this document reveals the intention to identify gaps and thereby support the planning and implementation process. The mechanisms and concepts proposed in the context of the Mandate M\/490 are appropriate to be taken into consideration for the standardization in China and the coordination and harmonization of activities on the international level.\n\nThe issues and aims discussed in the context of the standardization process in China may benefit from coordination with similar activities in the international context. Within the organizational structure of the M\/490 SG-CG, possible liaisons with standardization activities in other countries including China are mentioned as future tasks within the scope of Mandate M\/490. Therefore a joint discussion on the issues of smart grid standardization may support the design of the smart grid architecture based on a unified planning process and a set of consistent standards.\n\nAs the goal of a consistent standardization framework is expressed in context of the SGCC Framework and Roadmap for Strong & Smart Grid Standards, its structure regarding the standardization system in China may represent a suitable starting point to integrate the European and Chinese viewpoints in context of a common framework. Furthermore, the national smart grid standardization task force established by the Standardization Administration of China (SAC) and NEA about two years ago may also be considered as an organizational platform to address these issues.\n\nThe further promotion of smart grid standardization activities would accelerate the development of an unanimously accepted vision on the technological and organizational design of smart grids in China. Specifically, the integration of new market actors in the standardization process would increase the innovation potential in the smart grid development. Based on the activities of the European Commission, the following policies are worth considering:\n\n * The establishment of an organizational arrangement for smart grid standardization: the Chinese government, for example via MOST, could promote the foundation of an organizational arrangement such as SG-CG to coordinate standardization issues and integrate new market actors in the standardization process. Within this organizational arrangement, clear structures and processes to foster the understanding of the smart grid concept and to provide the means to model and implement smart grid solutions should be defined and implemented. The requirements of smart grids should be analyzed and current and future stakeholders identified.\n\n * The commissioning of a reference architecture framework: the government could engage this organizational arrangement to develop architectural concepts such as a technical reference architecture (i. e. a Chinese Smart Grid Architecture Model). The models applied in this context should be able to describe interoperability aspects of the systems, ranging from business objectives to technical connectivity of the components involved in a solution.\n\n * The commissioning of smart grid use cases and standards: use case descriptions and a thorough process to address governance and quality aspects are beneficial to develop a set of consistent and complimentary smart grid standards. The work of the SG-CG may be regarded as a reference in this area. Moreover, adapting the results of SG-CG to the Chinese context or creating a compatible (mirror-like) approach could serve as the basis to initiate a joint discussion on the issues of smart grid standardization.\n\nFollowing these recommendations, the key quality requirements of interoperability, data management, and cyber security can then be thoroughly analyzed and managed to create appropriate architecture models and identify supporting smart grid standards. This way, Chinese standardization processes could be synchronized more with international standardization. For example, Chinese standards could be promoted to the international level. Currently, the joint work in standardizing information exchange for demand response and in connecting demand side equipment and\/or systems into the smart grid (in IEC PC 118) is a good example for the benefits of such cooperation.\n\nThe recommended approach at a glance\n\n * Promote the establishment of an organizational arrangement (e. g. similar to SG-CG) to coordinate smart grid standardization.\n\n * Initiate the development of a reference architecture framework taking into account technical and organizational aspects which are unique to China (i. e. a Chinese Smart Grid Architecture Model).\n\n * Ensure that effective and efficient standardization processes exist and promote the development of smart grid use cases and standards.\n\nOpen Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.\n\nReferences\n\n1.\n\nS. 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Updegrove, \"The Role of Government in ICT Standardization,\" Consortium Standards Bulletin, vol. 6, no. 2, 2007.\n\n65.\n\nThe European Commission, \"M\/490 Standardization Mandate to European Standardisation Organisations (ESOs) to Support European Smart Grid Deployment,\" The European Commission, Brussels, 2011.\n\n66.\n\nState Grid Corporation of China (SGCC), Framework and Roadmap for Strong & Smart Grid Standards, Beijing: SGCC, 2010.\n\nFootnotes\n\n1\n\nAMI is defined as \"systems [that] are comprised of state-of-the-art electronic\/digital hardware and software, which combine interval data measurement with continuously available remote communications. These systems enable measurement of detailed, time-based information and frequent collection and transmittal of such information to various parties.\" [60].\n\n2\n\nNote that the inevitability of this trade-off has been established repeatedly and with great rigor [18]. However, the extent of the regulator's dilemma can be reduced if the regulator has a fair amount of information on the cost structures and market opportunities of the regulated companies.\n\n3\n\nThe term incentive regulation is unfortunately somewhat misleading. Regulation always sets incentives: the question is merely what incentives and whether they are good or bad.\n\n4\n\nEven if the general idea behind the price cap regulation scheme is identical in the different countries, specific formula and parameters (e. g. the length of the regulation period or the allowed return of grid operators) differ from country to country. A more specific and detailed overview of network regulation in Europe can be found in [22].\n\n5\n\nCritical conditions occur when operational limits, according to common technical guidelines, are violated.\n\n6\n\nFurther information on the work of the Clearingstelle EEG can be found at https:\/\/\u200bwww.\u200bclearingstelle-eeg.\u200bde\/\u200benglish.\n\n7\n\nSee http:\/\/\u200bwww.\u200bfi-ppp-finseny.\u200beu for more information on this project.\n\n8\n\nEspecially in those areas where potential harm to citizens resulting from non-compliance with the standard is low, standardization is mainly left to private industry [64].\n\u00a9 The Author(s) 2015\n\nGert Brunekreeft, Till Luhmann, Tobias Menz, Sven-Uwe M\u00fcller and Paul Recknagel (eds.)Regulatory Pathways For Smart Grid Development in China10.1007\/978-3-658-08463-9_6\n\n# 6. Regulatory pathways for smart grid development in China\n\nGert Brunekreeft1 , Marius Buchmann1, Christian D\u00e4nekas4, Xin Guo2, Christoph Mayer4, Marcus Merkel5, Christian Rehtanz6, Andr\u00e9 G\u00f6ring4, Andre Herrmann2, Ray Kodali2, Michael Stadler2, Mathias Uslar4, Nils Vogel2, Till Luhmann2 , Tobias Menz2 , Sven-Uwe M\u00fcller3 and Paul Recknagel3\n\n(1)\n\nJacobs University Bremen gGmbH, Bremen, Germany\n\n(2)\n\nBTC Business Technology Consulting AG, Oldenburg, Germany\n\n(3)\n\nDeutsche Gesellschaft f\u00fcr Internationale Zusammenarbeit (GIZ) GmbH, Eschborn, Germany\n\n(4)\n\nOFFIS Institut f\u00fcr Informatik e.V., Oldenburg, Germany\n\n(5)\n\nEWE NETZ GmbH, Oldenburg, Germany\n\n(6)\n\nef.Ruhr Forschungs-GmbH, Dortmund, Germany\n\nGert Brunekreeft\n\nEmail: g.brunekreeft@jacobs-university.de\n\nTill Luhmann\n\nEmail: Till.Luhmann@btc-ag.com\n\nTobias Menz (Corresponding author)\n\nEmail: Tobias.Menz@btc-ag.com\n\nSven-Uwe M\u00fcller\n\nEmail: sven-uwe.mueller@giz.de\n\nPaul Recknagel\n\nEmail: paul.recknagel@giz.de\n\nChapter at a glance\n\n * There are inherent conflicts between the fundamental energy policy goals of reliability, affordability, and sustainability. Government priorities on the energy policy goals are often reflected in energy sector regulation.\n\n * Seven recommendations have been presented in support of smart grid development in China. In principle, each recommendation is intended as support for the fundamental energy policy goals. Nevertheless, the implementation sequence of the recommendations is not arbitrary with respect to the energy policy goals.\n\n * In order to give policy makers an impression of how policy goal prioritization influences the timeline in which the recommendations should be implemented, the present chapter will outline three possible regulatory pathways. Each of these pathways prioritizes one specific goal of the energy policy triangle and develops a suitable implementation roadmap. These roadmaps are intended to serve as blueprints for policy makers, who have to decide about proper regulation based on the individual Chinese prioritization of energy policy goals.\n\n## 6.1 Government targets for China's future electric power system\n\nThe following chapter will present the main targets of China's government for each stage of the electric power system. These targets are the basis for a comprehensive description of the scenarios underlying the regulatory pathways. The targets are based on the following documents:\n\n * 12 th Five-Year Plan for Energy Development, issued by China's State Council in 2013 [1],\n\n * Opinions on Accelerating the Development of Environmental Industry, issued by China's State Council in 2013 [2],\n\n * Key Information at a glance \u2013 China's 12 th Five-Year Plan for Renewable Energy Development, issued by NEA in 2012 [3],\n\n * China's 2012 Energy Policy, issued by the Information Office of the State Council in 2012 [4].\n\n * Understanding China's 12 th Five-Year Energy Plan, issued by IEA in 2013 [5].\n\nTargets for the power generation sector\n\nIn order to cope with rapidly increasing power consumption and to ensure power system reliability, China's government plans to increase total electricity generation capacities during the next years from 970 GW in 2010 to 1490 GW in 2015. It is projected that generation capacities will increase further to 1935 GW in 2020 [6].\n\nMore specifically, China's government aims at vigorously developing distributed and renewable energy sources. From 880 TWh in 2011, electricity generated solely from renewable sources is projected to increase to more than 1,200 TWh in 2015. The projected increases in generation capacities for solar power, wind power, and biomass power are impressive. If these expansion targets are to be achieved, RES must be integrated in the grid more effectively than today. The Chinese government additionally aims at stopping the deterioration of air quality as well as the increase of CO2 emissions. The target is therefore for the total amount of discharge of SO2 and NOx to remain constant between 2015 and 2020. Also, the increase in CO2 emissions will be slowed in spite of the planned coal power generation growth.\n\nGovernment targets for the power generation sector at a glance\n\n * Roughly double generation capacities from 2010 to 2020\n\n * Vigorously develop RES\n\n * Stop the deterioration of air quality.\n\n * Reduce the growth of CO2 emissions\n\nTargets for the power logistics sector\n\nChina's government plans to significantly expand transmission and distribution grids. The capacity of China's transmission grids is planned to double by 2020 (compared to 2010), creating a national backbone grid, interconnecting different Chinese regions and improving electricity transmission from the energy bases to the load centers. The plan also include upgrading urban and rural distribution grids, promoting distribution intelligence, and improving the capacity and reliability of the overall power supply. This enhancement of distribution grids primarily aims at reducing blackout times by roughly 40 % from 2012 to 2020. The government also aims at improving asset utilization of the grid infrastructure by peak shaving and by reducing the line loss rate from 6.5 % in 2010 to 6.3 % in 2015.\n\nWith the goal of facilitating the integration of large-scale and intermittent RES, the Chinese government aims at promoting energy storage technologies by funding R&D on energy storage, energy conversion and related key technologies.\n\nGovernment targets for the power logistics sector at a glance\n\n * Build up a backbone network and roughly double grid capacities from 2010 to 2020\n\n * Upgrade distribution grids and reduce blackout times significantly\n\n * Improve asset utilization of grid infrastructure\n\n * Promote energy storage\n\nTargets for the power trade and retail sector\n\nThe Chinese government plans to establish a more market-oriented energy system to increase competition and affordability. Administrative reforms of SOE are planned to take place: transmission and distribution will be separated in a number of pilot projects and independent power trading institutions will be established. Electricity markets and price-based coordination mechanisms are planned to be introduced: the government strives to form a pricing mechanism where markets set on-grid and off-grid prices and the government only sets the price for electricity transmission and distribution (network charge), the economic dispatch for power generation will be increased, and time-of-use prices, seasonal tariffs and interruptible load electricity tariffs are planned for more widespread introduction.\n\nGovernment targets for the power trade and retail sector at a glance\n\n * Establish a modern energy market system\n\n * Promote reforms in key energy sectors\n\n * Improve energy pricing mechanisms\n\nTargets for the power consumption sector\n\nOne of the government's major targets is to increase the efficiency of power use in order to reduce growth in total electricity consumption. Specifically, the government intends to reduce energy consumption per unit of GDP by roughly 16 % from 2010 to 2015. Given that some of China's most remote regions are still not connected to the electric power grid, China's government also wants to provide universal access to electric power even in most remote regions of China by 2015.\n\nGovernment targets for the power consumption sector at a glance\n\n * Increase efficiency of power use\n\n * Provide universal access to electric power even in most remote regions of China by 2015\n\nCross-sector targets\n\nWith regard to power system governance, the government aims at improving the planning and implementation of power sector development: targets and responsibilities of different governmental institutions will be clarified and coordinated; local governments and the pertinent departments of the State Council will be made responsible for binding indicators such as the share of non-fossil fuels in energy consumption or the intensity of energy consumption.\n\nSimultaneously the government is moving forward to strengthen energy industry management by accelerating the introduction of an Energy Law and by enacting amendments to the Coal Law and the Electric Power Law. Also, technology and equipment in the energy sector will be further standardized and capacities of statistics, monitoring, prediction, and warning will be built up.\n\nThe Chinese government envisages rationalizing energy investment and the management system of SOE. The government wants to explicitly adhere to the dominance of its state-owned economy in key energy sectors related to national security and economy but, at the same time, it wants to improve the management and assessment mechanisms of SOE.\n\nRegardless of the adherence to the dominant position of SOE in China's power system, the government also has the objective of diversifying investments in the energy sector. Approval for energy infrastructure investments is intended to become less restrictive so that the participation of private and foreign capital is encouraged. Single energy markets, like the oil market, will be opened up to competition. The government wants to strengthen the innovation potential of China's environmental industry in order to enable the industry to increase the supply of green products, expand market demand, and enhance export prospects.\n\nAnother goal is to accelerate the development of the environmental industry in order to promote the development of technologies and equipment that save energy, protect the environment, and encourage reuse of resources.\n\nGovernmental cross-sector targets at a glance\n\n * Improve planning and implementation\n\n * Strengthen the management of the energy industry\n\n * Rationalize energy investment and the management system of SOE\n\n * Diversify energy investment\n\n * Accelerate the development of the environmental industry\n\n## 6.2 Underlying scenarios\n\nIn the following chapter, three smart grid development scenarios will be presented, based on different energy policy goals. For each scenario, the recommendations made in Chap.\u200b 5 will be prioritized and their interdependencies analyzed. The results of this analysis are scenario-based propositions for the implementation sequences of the recommendations. They will be presented in the form of a roadmap. Each roadmap is intended to support China's smart grid development with a focus on one particular scenario. The smart grid scenarios presented in the chapter are:\n\n * Reliability\/Security of Supply: In this scenario, the goal is to maximize electricity availability and to minimize outages and disruptions of the electricity supply.\n\n * Ecological Sustainability: The second scenario focuses on improving air quality as well as on decreasing and potentially stopping the growth of CO2 emissions in order to contribute to climate protection. Amongst others, this is achieved by an accelerated installation of RES capacities.\n\n * Affordability\/Competition: Competition is a key factor for innovation and affordability. This scenario will consequently focus on the creation of level playing fields and of incentives for market participation of new market actors through appropriate market design.\n\nThe government targets presented in Sect. 6.1 are associated to the three scenarios (see Fig. 6.1). In some cases, a specific target contributes to more than one energy policy goal. Yet, in these cases, in an effort to increase clarity, only the most obvious association to one specific policy goal will be presented. Two targets are general smart grid development targets as recorded in the 12 th Five-Year Plan for Energy Development. They are placed in the center of the triangle, as smart grids are seen as a means to contribute to all energy policy goals.\n\nFig. 6.1\n\nScenarios and their associations with energy policy goals\n\n## 6.3 Drawing the roadmaps\n\nFor migrating from today's electric power system towards smart grids based on one of the given scenarios, it is favorable to implement the recommendations in a specific timeline. For deriving the implementation sequences, the recommendations were sorted by priority in a first step. In a second step, main aspects of the recommendations in each scenario were collected and allocated to the timeframes short term (2015\u20132016), medium term (2017\u20132020) and long term (2021\u20132030). Interdependencies between main aspects were explored additionally. Afterwards the recommendations were arranged chronologically.\n\nThe following three sections show recommended priorities, their main aspects, interdependencies, and the sequences worked out for the implementation for each scenario. Recommendations and dependencies are highlighted within the order of their respective importance for the scenario. In addition, a visualization pertaining to the scenario and summarizing the timeframes of the respective measures of each recommendation is also presented. Furthermore, a possible sequence of starting points for work on the recommendations has been worked out. The justifications for the prioritization of the recommendations are discussed together with each chart. Moreover, the specific measures to be executed for each recommendation, their respective interdependencies, and their timeframes are explained.\n\n### 6.3.1 Reliability\/Security of Supply scenario\n\nLooking at the overall prioritization with respect to the Reliability\/Security of Supply scenario, it can be seen that the recommendations to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator, Coordinate network expansion planning for electricity grid expansion and upgrade, and to Improve grid integration of RES have a high priority (Fig. 6.2). Remember that these priorities do not necessarily correspond to the proposed implementation sequence, because interdependencies within the given scenario have to be analyzed before recommendations are implemented.\n\nFig. 6.2\n\nPriority of recommendations in the Reliability\/Security of Supply scenario\n\nDefine a long-term strategy for the electricity sector and establish an independent and powerful regulator\n\nEffective government work is a precondition for the efficient development of smart grids. Therefore, roles and responsibilities for government administration and regulation have to be set from the start. As a consequence, it is important to implement in the short term all the measures proposed in the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator:\n\n * A clearly defined long-term strategy for electricity system development has to be specified by the government. The strategy should include long-term government targets with regard to the development of generation capacities of different generation technologies, and targets for energy efficiency. Such a strategy is a prerequisite for coordinated expansion of the electric power system as well as smart grid development. It might also serve as a point of reference for related government policies.\n\n * The centralization of all regulatory aspects in one independent and powerful regulatory institution is of critical importance to increase the effectiveness of regulation. The main tasks of the regulator are coordination of network expansion planning and network regulation. In the Reliability\/Security of Supply scenario, the regulator should specifically monitor power disruptions and outage times on national and local levels to provide an overview of the quality of electricity supply. Much like in Germany, grid operators should be obliged to regularly report data on power disruptions and outage times to the regulator (see Sect.\u200b 4.\u200b4.\u200b4 for related regulation in Germany). Pertinent laws (for example, a general energy law) should empower the Chinese regulator to effectively monitor power disruptions and outage times.\n\nCoordinate network expansion planning for electricity grid expansion and upgrade\n\nNetwork expansion planning has a high priority in this scenario because the electric power grid is the backbone of the power system. Coordinated and efficient network expansion is the basis of a power infrastructure that can be adapted to consumer needs in a timely manner. This maximizes energy availability as the reliability of the future energy system essentially depends on the grid and generation capacities being matched to the consumption side.\n\nWithin this scenario, the main measures of the recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade are:\n\n * In the short term, the main task to be accomplished for successful network expansion planning is to improve coordination between power system stakeholders and to define a common procedure for network expansion planning. Specifically, it is beneficial to coordinate network expansion planning and the construction of new generation capacities. The Future-oriented Energy Grids Platform can serve as a role model to encourage the establishment of a similar grid platform in China.\n\n * In the medium term, stakeholders involved in the grid planning process have to elaborate a network development plan that will be mandatory for TSO and DSO under supervision of the regulator.\n\nThe measures to be implemented within the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator should precede the measures within the recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade: a government strategy for the electric power system including long-term targets for generation capacities is a prerequisite for the network development process. An independent and powerful regulatory institution is supposed to supervise the network development process.\n\nImprove grid integration of RES\n\nIncreasing RES capacities contribute to supply the quickly increasing demand for electric power. Therefore, they support reliability and security of supply if the technical integration of RES is clearly defined. However, RES that have not been properly integrated may even endanger security of supply and stability.\n\nWithin this scenario, the main measures of the recommendation to Improve grid integration of RES are:\n\n * In the short term, grid connection points must be defined for all kinds of RES at all grid levels.\n\n * Defining a binding network code for grid connection that specifies responsibilities of grid operators as well as power generators is also a task in short term.\n\n * RES generation capacities are needed in order to maximize available electricity and thereby reduce load curtailments. However, for grid stability and security of supply to be ensured, RES generation has to be curtailed under specific conditions. Therefore, procedures and documentation for curtailment of RES and the requirements in terms of IT support must be specified in the medium term.\n\nEven if the recommendation to Improve grid integration of RES has high priority, measures to be implemented within several other recommendations serve as an input and should be implemented early: along with the need for an independent and powerful regulator to supervise curtailment procedures and define network codes, the clear specification of targeted RES generation capacities and well-defined network development procedures for network enhancement are preconditions for more effective RES integration. Also, a unified view of smart grids and technical standards contribute to a more effective integration of RES [7].\n\nOptimize the balancing of electricity generation and consumption\n\nWhen electricity consumption increases so quickly that the growth of generation capacities can hardly follow, balancing and especially peak shaving are measures that support security of supply. Grid operation with higher safety margins eases grid control. Therefore, reduced peak loads resulting from peak shaving help to avoid system instabilities. Coordination of electricity generation and consumption is therefore of considerable priority in the scenario of reliability and security of supply. This includes load management, time-of-use pricing, and smart meter rollout.\n\nWithin this scenario, the main measures of the recommendation to Optimize the balancing of electricity generation and consumption are:\n\n * In the short term, the main aspect is the further refinement of peak shaving mechanisms and technologies for system stabilization with focus on industrial and commercial consumers. One interesting option in this context is to tender the disconnectable loads on an internet platform (see Sect.\u200b 4.\u200b4.\u200b5 for related regulation in Germany).\n\n * As already planned by the Chinese government, the time-of-use pricing system has to be refined (sufficient price differences between peak and off- peak prices for all categories of consumers) in the short term as well. A more sophisticated time-of-use pricing would incentivize investments in balancing mechanisms and technologies \u2013 for instance DSM, SSM, and energy storage.\n\nAn additional interesting measure originally not included in the recommendation to Optimize the balancing of electricity generation and consumption is locational pricing that can be used to signal network congestion. Locational pricing refers to power prices which are calculated for a number of locations \u2013 called nodes \u2013 on the transmission grid. Each node represents a physical location where power is injected by generators or withdrawn by loads [8]. Locational pricing reduces network congestion and may also set incentives for network expansion in regions with many network congestions [9]. It is an option for the long term and might become necessary in future in order to efficiently integrate the projected RES generation capacities. The regulator is supposed to supervise such locational pricing by regulating network charges with a clear scheme for the interaction between network operators and market players.\n\nThe discussion reveals that measures implemented within the recommendations to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator and to Improve grid integration of RES serve as an input for the recommendation to Optimize the balancing of electricity generation and consumption.\n\nFacilitate the development of a unified view of smart grids\n\nA technical reference architecture helps better understand smart grids and allows holistic analyses of security aspects relating to their implementation. Interoperability minimizes individual integration efforts and reduces the probability of interface failures. Such a reference architecture will therefore increase reliability and security of supply.\n\nWithin this scenario, the main measures of the recommendation to Facilitate the development of a unified view of smart grids are:\n\n * In the short term, the main aspect is the creation and adoption of an organizational arrangement to model smart grid reference architectures and to coordinate smart grid standardization.\n\n * Defining effective and efficient standardization processes (e. g. based on smart grid use cases) as well as coordinating the work of SDO to establish effective and efficient standardization processes are two further important aspects that should take place in the same time frame.\n\n * In the medium term, a set of consistent smart grid standards should be created using the technical reference architecture.\n\nIntroduce network regulation for efficient investment incentives for electricity grid expansion and upgrade\n\nIn the scenario focusing on reliability and security of supply, network regulation is somewhat less important but not irrelevant. It contributes indirectly to this scenario, as it sets economic incentives for the build-up and maintenance of the grid infrastructure. Incentives for efficient network investments and investments in R&D and innovation are important in the context of the measures within this recommendation. These incentives, by contributing to the introduction of smart grid technologies, will directly ensure security of supply and reduce costs.\n\nWithin this scenario, the main measures of the recommendation to Introduce network regulation for efficient investment incentives for electricity grid expansion and upgrade are:\n\n * In the short term, the use of incentive instruments like rate-of-return adder or innovation bonus should be assessed for application in China.\n\n * A decision about the use of disaggregated regulation (i. e. regulating only the monopolistic bottlenecks) should be made in the short to medium term. If applicable, disaggregated regulation should be implemented in the medium term.\n\n * Also in the medium term, incentives should be defined for investment in innovation and smart applications. In this context, the network development plan should be aligned and critical network connections for system stability which necessitate stronger investment incentives should be identified. At this point in time, a rate-of-return adder can be especially beneficial.\n\nAs an independent and powerful regulator is a prerequisite for network regulation, the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator should be realized before network regulation is focused upon.\n\nCreate level playing fields for access to power system infrastructure and information\n\nAs new market actors are necessary more for innovation than for stability of the energy supply, the measures proposed in the recommendation to Create level playing fields for access to power system infrastructure and information are not considered as important in this scenario with a focus on reliability and security of supply.\n\nSequence of implementation\n\nIn the light of the interdependencies described above, the following implementation sequence is proposed in the scenario focusing on reliability\/security of supply:\n\n * Define a long-term strategy for the electricity sector and establish an independent and powerful regulator.\n\n * Coordinate network expansion planning for electricity grid expansion and upgrade.\n\n * Facilitate the development of a unified view of smart grids.\n\n * Improve grid integration of RES.\n\n * Optimize coordination of electricity generation and consumption.\n\n * Introduce network regulation for efficient investment incentives for electricity grid expansion and upgrade.\n\nFigure 6.3 depicts the points in time for undertaking the main activities and shows dependencies between the key aspects of recommendations for the scenario focusing on reliability and security of supply. Each recommendation is represented by a bar partitioned into timeframes from left to right. The starting point of each bar depicts the proposed beginning of work on each recommendation. Recommendations that are unimportant to the scenario are shown in gray. Proposed time points for the main measures are represented by the opaque blue regions on each bar. At the end of each bar, color gradients have been used to show that work on the recommendations continues even after the main aspects have been taken care of. The main measures are shown as boxes. The proposed starting time of each measure is shown by a diagonal line connecting the box to the bar. Dependencies between measures are shown by arrows while preconditions are marked with a white line crossing a bar at the beginning of the arrow. The end of the arrow points to the dependent measure.\n\nFig. 6.3\n\nStarting points, timeslots of main activities, and dependencies between main aspects of recommendations in the scenario focusing on reliability and security of supply\n\n### 6.3.2 Ecological Sustainability scenario\n\nAn introductory overview of the overall prioritization with respect to this scenario is presented in Fig. 6.4. It can be seen that the recommendations Define a long-term strategy for the electricity sector and establish an independent and powerful regulator, Improve grid integration of RES, and Create level playing fields for access to power system infrastructure and information have a high priority in the Ecological Sustainability scenario. Again, remember that these priorities do not necessarily correspond to the proposed implementation sequence because the interdependencies within the given scenario have to be analyzed before.\n\nFig. 6.4\n\nPriority of recommendations in the Ecological Sustainability scenario\n\nDefine a long-term strategy for the electricity sector and establish an independent and powerful regulator\n\nAs in the previous scenario, the commitment to a long-term strategy and the creation of an independent and powerful regulator has the highest priority in the scenario for ecological sustainability because a government-defined long-term strategy for the future electricity sector including specific RES expansion targets is a prerequisite for effective and efficient expansion of RES generation capacities.\n\nWithin this scenario, the main measures of the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator are:\n\n * Effective regulation is required to ensure RES integration. Thus, the centralization of regulatory aspects in one independent and powerful regulatory institution should be addressed in the short term. Responsibilities for grid integration of RES and for power system information management should be assigned in this context.\n\n * Official government targets for RES expansion are currently defined only until 2015. Within the forthcoming Five-Year Plan, the planning period will be extended to 2020. Defining a long-term government strategy concerning energy mix (including RES share), and energy efficiency indicators beyond 2020 is consequently a task for the medium term.\n\nImprove grid integration of RES\n\nTimely connection and low curtailment of RES will speed up the reduction of greenhouse gas emissions as well as emissions of local and regional air pollutants. RES therefore need to be integrated into the grid with a very high priority and all of the following tasks need to be implemented in the short term:\n\n * Proper definitions of grid connection points are necessary for all kinds of RES on all grid levels.\n\n * A binding network code specifying responsibilities of both grid operators and power generators needs to be defined, as investors in RES need clear time limits for grid connection of RES in order to deliver generated electricity to the market at the right time. Grid operators should have to bear liability for grid connection of RES \u2013 resulting in a shallow cost approach for interconnection, whereby grid operators have both the responsibility and the opportunity for efficient overall grid development.\n\n * Detailed procedures and documentation for curtailment of RES must be specified, and the requirements for IT support of these procedures must be clear. The implementation of this measure can be seen as a technical and economic framework to encourage investments into RES.\n\nMeasures related to the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator positively impact the effectiveness of the recommendation to Improve grid integration of RES: credible long-term government targets for RES generation capacities reduce risks related to investments in RES units. Another prerequisite is a powerful and independent regulator to supervise curtailment procedures and to define network codes.\n\nCreate level playing fields for access to power system infrastructure and information\n\nIn principle, new market actors in China can already invest in RES. However, they need to be integrated into electricity sector governance more effectively in order to ensure efficient integration of RES. Non-incumbent market actors will invest in RES generation capacities only if the necessary information for competition and market access is available to them. Methods for the integration of the innovation potential of third parties have to be institutionalized and non-discriminatory information sharing has to be guaranteed.\n\nWithin this scenario, the main measures of the recommendation to Create level playing fields for access to power system infrastructure and information are:\n\n * In the short term, the main measure of this recommendation is the definition of minimum necessary requirements for power system information management. Note that this task is related to the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator in Fig. 6.5 due to the pivotal role of the government in this context.\n\nFig. 6.5\n\nStarting points, timeslots of main activities, and dependencies between main aspects of recommendations in the scenario focusing on ecological sustainability\n\n * In the medium term, the role of new market actors in smart grids should be defined and how these actors can be integrated should be clarified.\n\n * An institution for power system information management should be specified, and this institution should set up a platform. It should then organize access to and exchange of information between all eligible parties.\n\nBefore the role of new market actors in smart grids can be defined, a long-term strategy for the electric power sector should be developed. Therefore, the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator should precede the recommendation to Create level playing fields for access to power system infrastructure and information. An architecture and standardization framework encourages the development of a unified smart grid model containing an overview of relevant use cases and business processes. Such a model, representing a form of Chinese SGAM, significantly eases the organization of power system information management. The platform for information management should therefore be implemented after the measures relating to the recommendation to Facilitate the development of a unified view of smart grids have been applied.\n\nOptimize the balancing of electricity generation and consumption\n\nWhen consumption is not coordinated with generation, significant curtailments of RES generation are necessary to ensure grid stability. This reduces the attractiveness of investments in RES generation capacities. The usage of VPP, microgrids, or energy storage units facilitates the integration of distributed RES.\n\nWithin the scenario, the main measures of the recommendation to Optimize the balancing of electricity generation and consumption are:\n\n * The first task in coordinating electricity generation and consumption is to promote VPP, microgrids, and energy storage technology for local integration in the short term by increasing R&D funding in this area.\n\n * The time-of-use pricing system, which depends on a successful smart meter rollout, also has to be refined (sufficient price differences between peak and off peak prices for all categories of consumers) in the short term. This measure is already planned by the Chinese government. The first focus should be set on industrial and commercial consumers and then on residential consumers.\n\n * Another interesting measure, originally not included in the recommendation to Optimize the balancing of electricity generation and consumption, is locational pricing. It might be interesting in the long term and contribute to RES integration (see Sect. 6.3.1 for a brief description of locational pricing).\n\nCoordinate network expansion planning for electricity grid expansion and upgrade\n\nNetwork development is necessary to connect new consumers and generators to the network, including generation units belonging to third parties. The network development plan allows planning of RES installation and thereby increases investments in RES. As a consequence, network expansion planning is relevant in this scenario. The main measures of the recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade should be taken in the short term:\n\n * Increased coordination between established stakeholders and new market actors should be promoted by the government.\n\n * A coordinated and mandatory network development plan including the aspect of RES integration should be established.\n\nBased on experiences in Germany, an independent and powerful regulator is the best-suited organization for organizing and implementing these measures. Therefore, the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator should be implemented before the recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade.\n\nFacilitate the development of a unified view of smart grids\n\nAs mentioned above, distributed energy resources are important and in this scenario their number is expected to rise. This implies an increase in system scale and complexity and calls for a smart grid architecture and standardization framework (i. e. an organizational arrangement to coordinate and promote smart grid standardization and the development of a reference architecture). A unified view of smart grids should exist to elicit the resulting requirements and to compare different architectural solutions. The smart grid architecture and standardization framework should focus on interoperability between actors and systems, which is a key aspect in the context of decentralized system architectures. The high number of interfaces between systems implies the need for security analyses, which in turn requires solid models of systems architectures [7]. Clear structures and processes should be defined in order to coordinate the work of SDO.\n\nWithin this scenario, the main measures of the recommendation to Facilitate the development of a unified view of smart grids are:\n\n * In the short term, an organizational arrangement to coordinate and promote smart grid standardization should be created and the development of a technical reference architecture framework should be assigned.\n\n * In the medium term, effective and efficient standardization processes (e. g. based on a collection of smart grid use cases) should be established.\n\nIntroduce network regulation for efficient investment incentives for electricity grid expansion and upgrade\n\nFor ecological sustainability, an electric power grid does not necessarily have to be developed at lowest costs. As network regulation focuses on efficient network investments, it is not an important regulation measure for this scenario.\n\nSequence of implementation\n\nFigure 6.5 shows the timeframes of the main activities and interdependencies involved in this scenario. For a detailed description of the layout of the visualization, please check back to the previous section. In the light of interdependencies described above, the following implementation sequence is proposed in the scenario focusing on ecological sustainability:\n\n * Define a long-term strategy for the electricity sector and establish an independent and powerful regulator.\n\n * Improve grid integration of RES.\n\n * Coordinate network expansion planning for electricity grid expansion and upgrade.\n\n * Create level playing fields for access to power system infrastructure and information.\n\n * Optimize the balancing of electricity generation and consumption.\n\n * Facilitate the development of a unified view of smart grids.\n\n### 6.3.3 Affordability\/Competition scenario\n\nAn introductory overview of the overall prioritization with respect to this scenario is presented in Fig. 6.6. It can be seen that the recommendations Create level playing fields for access to power system infrastructure and information, Coordinate network expansion planning, and Define a long-term strategy for the electricity sector and establish an independent and powerful regulator have a high priority in the Affordability\/Competition scenario.\n\nFig. 6.6\n\nPriority of recommendations in the Affordability\/Competition scenario\n\nCreate level playing fields for access to power system infrastructure and information\n\nIn the scenario focusing on competition and innovation, two very important aspects are power system information management and the inclusion of new market actors. As non-incumbent market actors are main drivers for competition and innovation, non-discriminatory access to relevant information is crucial for them to encourage investments in the electric power system. China's smart grid development is dominated by SGCC and CSG, whereas the ICT industry hardly participates in strategic smart grid developments. It would be highly beneficial to change this situation with regard to the non-involvement of the ICT sector and to create business opportunities for new market actors.\n\nWithin this scenario, the main measures of the recommendation to Create level playing fields for access to power system infrastructure and information are:\n\n * In the short term, the definition of the role of new market actors in smart grids and how these actors can be integrated is one of the main aspects.\n\n * Defining minimum requirements for a data platform to organize access to and exchange of power system information between all eligible parties has to be accomplished in the short term.\n\n * Also, the establishment of such a data platform is another task to be accomplished in this timeframe.\n\nA common understanding of the structural aspects of smart grids would encourage the identification and description of the roles of new market actors as well as their need for information and interfaces both on the business and technical level. An institutionalized smart grid architecture and standardization framework might therefore serve as an inter-company interaction mechanism to strengthen the role of third parties even beyond standardization.\n\nCoordinate network expansion planning for electricity grid expansion and upgrade\n\nThe recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade proposes that a stakeholder platform for network development should be established. This platform would mean more involvement of all market actors. All measures associated to this recommendation should be implemented in the short term:\n\n * Coordination between established power system stakeholders and new market actors has to be strengthened. For this purpose, a stakeholder platform similar to the Future-oriented Energy Grids Platform in Germany could be established and managed by a government institution.\n\n * Common procedures for network expansion planning should be defined. A network development plan should be elaborated within a coordinated process and shared responsibilities and made mandatory afterwards.\n\n * Responsibilities for network planning should be identified, thereby clarifying which new market actors should participate in this process. This task shall be accomplished by a government organization, but it can be supported by grid operators.\n\nBefore the start of network expansion planning including new market actors, the role of new market actors in China's electric power system must be defined. Therefore, the recommendation to Create level playing fields for access to power system infrastructure and information should precede the recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade.\n\nDefine a long-term strategy for the electricity sector and establish an independent and powerful regulator\n\nAs government agencies have to supervise and guide grid operation, an effective definition of the government's roles and responsibilities is important. This ensures favorable conditions for the development and rollout of innovations.\n\nWithin this scenario, the main measures of the recommendation to Define a long-term strategy for the electricity sector and establish an independent and powerful regulator are:\n\n * If the responsibilities for network planning have not already been defined within the previous recommendation, the government has to define these responsibilities in the short term.\n\n * An independent and powerful regulator should be established in the medium term.\n\n * A long-term strategy for the future electric power sector beyond 2020 must be defined in the same time frame.\n\nIntroduce network regulation for efficient investment incentives for electricity grid expansion and upgrade\n\nIn the scenario focusing on affordability and competition, disaggregated regulation should ensure that only monopolistic bottlenecks (transmission and distribution grids) are regulated. Consequently, competition can evolve in all other stages of the supply chain and incentives are created for a cost-efficient build-up of the grid and for technical innovations.\n\nWithin this scenario, the main measures of the recommendation to Introduce network regulation for efficient investment incentives for electricity grid expansion and upgrade within this scenario are:\n\n * Network charges offering efficient investment incentives in smart grids have to be introduced in the medium term.\n\n * After network charges have been fixed, disaggregated regulation can be introduced. Due to the high impact of disaggregated regulation on competition in the retail sector, this measure should also be implemented in the medium term.\n\nBefore network regulation can be established, the responsibility for network regulation has to be defined and an independent and powerful regulator is necessary.\n\nFacilitate the development of a unified view of smart grids\n\nAs mentioned above, a smart grid architecture and standardization framework (i. e. an organizational arrangement to coordinate smart grid standardization and the development of a reference architecture) can be institutionalized as an inter-company interaction mechanism to strengthen the role of new market actors. In addition, it ensures interoperability between solutions coming from different vendors, prevents vendor lock-in and makes Chinese smart grid technologies applicable in international markets as well as vice versa [10], [11]. Furthermore, the framework supports the standardization and engineering process and thereby the diffusion of knowledge.\n\nWithin this scenario the main measures of the recommendation to Facilitate the development of a unified view of smart grids are:\n\n * In the short term, the smart grid architecture and standardization framework has to be created or adopted to enable modeling of smart grid solution architectures.\n\n * Coordinating the work of organizations involved in the development of standards needs clear structures and incentives for new market actors to participate in the process. The definition of these structures and incentives is also a short-term task.\n\n * In addition, standardization processes need to be coordinated with international standardization in the short term in order to establish effective and efficient standardization processes (e. g. based on a collection of smart grid use cases) in the medium term.\n\nImprove grid integration of RES\n\nGrid integration of RES is important in this scenario. The stricter the conditions for grid connection of RES are formulated, the less smaller power generation companies are dependent on the grid operators. Rules favorable for the integration of RES attract new investors, i. e. third parties that need both stable investment and technical conditions. The following measures should therefore be implemented in the short term in order to allow competition in the field of RES generation:\n\n * Grid connection points need proper definition for all kinds of RES on all grid levels.\n\n * A binding network code for grid connection specifying the respective responsibilities of grid operators and power generators needs to be specified. In this context, grid operators should bear liability for grid connection of RES.\n\n * A detailed procedure and documentation for curtailment of RES including, for example, document exchange and transparency rules must be defined. Additionally, the requirements for IT support for these procedures need to be specified.\n\nWell-defined network development procedures for network enhancement involving RES will positively impact the effectiveness of these measures. The recommendation to Coordinate network expansion planning for electricity grid expansion and upgrade should therefore start before the recommendation to Improve grid integration of RES.\n\nOptimize the balancing of electricity generation and consumption\n\nThe recommendation with the lowest priority in this scenario is the balancing of electricity generation and consumption, which is relevant for integrating prosumers in the electric power system. Balancing of electricity generation and consumption includes load management, time-of-use pricing, smart meter rollout, and the promotion of VPP, microgrids, and energy storage by reducing barriers to their implementation.\n\nWithin this scenario, the main measures of the recommendation to Optimize the balancing of electricity generation and consumption are:\n\n * In the medium term, coordination between electricity generation and consumption can be optimized by peak shaving and a further refinement of time-of-use pricing. Such a time-of-use based framework for pricing with sufficient price differences will incentivize the usage of balancing mechanisms and technologies such as DSM, SSM, and energy storage.\n\nBalancing of electricity generation and consumption potentially depends on the grid integration of RES. However, power system information management with smart meter infrastructure and information interchange between all parties as well as an independent and powerful regulator for regulation of network charges are also necessary.\n\nSequence of Implementation\n\nFigure 6.7 shows the timeframes of main activities and interdependencies. In the light of the interdependencies described above, the following implementation sequence is proposed in the scenario focusing on ecological sustainability:\n\nFig. 6.7\n\nStarting points, timeslots of main activities, and dependencies between main aspects of recommendations in the scenario focusing on competition and innovation\n\n * Create level playing fields for access to power system infrastructure and information.\n\n * Coordinate network expansion planning for electricity grid expansion and upgrade.\n\n * Define a long-term strategy for the electricity sector and establish an independent and powerful regulator.\n\n * Introduce network regulation for efficient investment incentives for electricity grid expansion and upgrade.\n\n * Facilitate the development of a unified view of smart grids.\n\n * Improve grid integration of RES.\n\n * Optimize coordination of electricity generation and consumption.\n\n## 6.4 Discussion of the three roadmaps\n\nIn addition to the specific timeline of each scenario, the three different roadmaps presented above give policy makers the following general hints:\n\n * The priority and relevance of each recommendation can be assessed on the basis of the underlying scenario. For example, it can be seen that the measures of the recommendation to Create level playing fields for access to power system infrastructure and information have the highest priority if the government focuses on fostering competition and innovation but are virtually irrelevant if the government focuses only on reliability issues.\n\n * Comparing the proposed implementation sequences in all three scenarios shows which recommendations have high priorities in all three scenarios. Such recommendations can be considered as political imperatives and should be implemented irrespective of the underlying policy goals of the Chinese government.\n\nFigure 6.8 summarizes the proposed implementation sequences for all three scenarios. Those recommendations to be implemented in the beginning are presented in the left. For example, the Ecological Sustainability scenario starts with Define a long-term strategy for the electricity sector and establish an independent and powerful regulator, followed by Improve grid integration of RES and then by the remaining recommendations.\n\nFig. 6.8\n\nOverview of proposed implementation sequences for all three scenarios\n\nComparing the implementation sequences in all three scenarios reveals that there are two recommended approaches with the highest overall priority. The measures subsumed within these recommendations shall be implemented independently of the underlying scenario:\n\n * Define a long-term strategy for the electricity sector and establish an independent and powerful regulator, and\n\n * Coordinate network expansion planning for electricity grid expansion and upgrade.\n\nThree of the remaining five recommendations are relevant in each scenario, though with a lower priority:\n\n * The Improvement of the grid integration of RES has a very high priority under the Ecological Sustainability scenario and is also important for the Reliability\/Security of Supply scenario. It is somewhat less important in the Affordability\/Competition scenario.\n\n * The Facilitation of the development of a unified view of smart grids is especially important with respect to the Reliability\/Security of Supply scenario. It is somewhat less important with regard to the Affordability\/Competition and Ecological Sustainability scenarios.\n\n * The Optimization of the balancing of electricity generation and consumption is particularly relevant for the Reliability\/Security of Supply scenario. The recommendation has a lower relevance in the Ecological Sustainability scenario and is ranked last in the Affordability\/Competition scenario.\n\nThe two remaining recommendations are not relevant in every scenario. Rather, they contribute to single energy policy goals. In particular, both of these recommendations are essential for migrating towards smart grids focusing on affordability and competition:\n\n * The Creation of level playing fields for access to power system infrastructure and information is the first recommendation that should be implemented if the government strives to develop smart grids focusing on affordability and competition. It is also important if the government chooses to focus on ecological sustainability.\n\n * The Introduction of network regulation for electricity grid expansion and upgrade is at position four in the implementation sequence of the Affordability\/Competition scenario. It is also relevant, though at a later stage, in the Reliability\/Security of Supply scenario.\n\nOpen Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.\n\nReferences\n\n1.\n\nState Council of the People's Republic of China, \"12th Five-Year Plan for Energy Development,\" Guofa, Beijing, 2013.\n\n2.\n\nState Council of the People's Republic of China, \"Opinions on Accelerating the Development of Environmental Industry,\" Beijing, 2013.\n\n3.\n\nNational Energy Administration (NEA), \"Key Information at a Glance \u2013 China 12th Five-Year Plan for Renewable Energy Development,\" China National Renewable Energy Center (CNREC), Beijing, 2012.\n\n4.\n\nState Council Information Office of the People's Republic of China (SCIO), \"China's Energy Policy 2012,\" SCIO, Beijing, 2012.\n\n5.\n\nInternational Energy Agency (IEA), \"Understanding China's 12th Five-Year Energy Plan,\" IEA, Paris, 2013.\n\n6.\n\nG. Tong, \"Status Quo of the Smart Grid Development in China and Its Driving Forces,\" National Energy Administration (NEA), Oldenburg, 2013.\n\n7.\n\nBundesverband der Energie- und Wasserwirtschaft (BDEW), \"BDEW-Roadmap: Realistische Ziele zur Umsetzung von Smart Grids in Deutschland,\" BDEW, Berlin, 2013.\n\n8.\n\nD. Philliips, \"Nodal Pricing Basics,\" Online]. Available: [http:\/\/\u200bwww.\u200biemo.\u200bcom\/\u200bimoweb\/\u200bpubs\/\u200bconsult\/\u200bmep\/\u200bLMP_\u200bNodalBasics_\u200b2004jan14.\u200bpdf. [Accessed February 21, 2014].\n\n9.\n\nS. Stoft, Power System Economics, Piscataway: The Institute of Electrical and Electronics Engineers, 2002.CrossRef\n\n10.\n\nB. Qu\u00e9lin, T. Abdessemed, J.-P. Bonardi and R. Durand, \"Standardization of Network Technologies: Market Processes or the Result of Inter-Firm Co-Operation?,\" Journal of Economic Surveys, vol. 15, no. 4, pp. 543\u2013569, 2001.CrossRef\n\n11.\n\nG. Tassey, \"Standardization in Technology-Based Markets,\" Research Policy, vol. 29, no. 4\u20135, pp. 587\u2013602, 2000.CrossRef00091-8)\n\n## Appendix A \u2013 Tables and Figures\n\nTable A.1\n\nElectricity consumption in China in 2011, data from [1]\n\nConsumer | Electricity Consumption in TWh in 2011 | in % of Total\n\n---|---|---\n\nIndustry | 3,469 | 73.8\n\nResidential Sector | 562 | 12.0\n\nWholesale, Retail, Trade, and Hotels\/Restaurants | 150 | 3.1\n\nAgriculture, Forestry, Animal Husbandry, Fishery, and Water Conservancy | 101 | 2.2\n\nTransport, Storage, and Postal Service | 85 | 1.8\n\nConstruction | 57 | 1.2\n\nOthers | 275 | 5.9\n\nTable A.2\n\nCircuit length of transmission lines with 35-kV and above and installed capacity of transformers by the end of 2010\n\nVoltage level | Circuit length of transmission line\/km | Capacity of transformers\/ 10,000-kVA\n\n---|---|---\n\nTotal 35-kV and above | 1,336,772 | 361,742\n\n1000-kV AC | 1,006 | 600\n\n+\/\u2212 800-kV DC | 3,334 | 593\n\n750-kV AC | 6,685 | 3,870\n\n+\/\u2212 660-kV DC | 1,095\n\n|\n\n500-kV AC | 135,180 | 69,843\n\n+\/\u2212 500-kV DC | 8,081 | 4,031\n\n330-kV | 20,338 | 6,457\n\n220-kV | 277,988 | 118,247\n\n66-\/110-kV | 458,477 | 125,224\n\n35-kV | 432,668 | 37,501\n\nTable A.3\n\nReliability rate of power supply for users in cities at the level of 1000-KV during the 11 th Five-Year Plan (2006\u20132010)\n\nYear | Reliability ratio of power supply\/% | Average blackout time\/hours\/household\n\n---|---|---\n\n2010 | 99.923 | 6.722\n\n2009 | 99.896 | 9.111\n\n2008 | 98.863 | 12.071\n\n2007 | 99.882 | 10.360\n\n2006 | 99.849 | 13.191\n\nTable A.4\n\nAdministrative regime of the power sector in China\n\nOrganization | Description\n\n---|---\n\nPower sector regime reform working group | \u2013 Theoretically enjoying top decision-making rights regarding the power regime reform\n\nNational Development and Reform Commission (NDRC) | \u2013 Responsible for long-term planning and for the issuance of energy-related Five-Year Plans\n\n\u2013 Approving important investment projects\n\n\u2013 Regulating energy and electricity prices\n\n\u2013 Proposing energy conservation and new energy development policies\n\nNational Energy Administration (NEA) | \u2013 Proposing the energy development strategy\n\n\u2013 Drafting energy development plans and industrial policies and organizing their implementation\n\n\u2013 Drafting all energy-related provision and rules\n\n\u2013 Advancing energy regime reforms\n\n\u2013 Coordinating key issues emerging in the process of energy development and reforms\n\n\u2013 Exercising regulations on power system construction, power safety, power supply and service, as well tariff and information disclosure\n\nMinistry of Finance (MOF) | \u2013 Responsible for final decisions on some matters concerning the financial code and financial costs standards\n\nProvincial Economic and Trade Commission (PETC) | \u2013 Acting as the regulator at local level and serving as coordinators; in practice, they also perform the functions of local regulation institutions\n\nState-owned Assets Supervision and Administration Commission of the State Council (SASAC) | \u2013 Supervising the performance of SOE\n\n\u2013 Appointing and dismissing SOE executives\n\n\u2013 Formulating auditing requests and approving key decisions\n\nMinistry of Environmental Protection (MEP) | \u2013 Responsible for tasks related to overall environmental protection\n\n\u2013 Cooperating with NEA in the performance of its duties\n\nState Administration of Work Safety (SAWS) | \u2013 Giving guidance to safety inspection and being responsible for industrial safety regulation\n\nTable A.5\n\nRegional and provincial grid operators in China\n\nSGCC\n\n---\n\nRegional subsidiaries | Provincial subsidiaries | Province\n\nNorth China Grid Ltd. | Beijing Electric Power Corporation | Beijing\n\nTianjin Electric Power Corporation | Tianjin\n\nHebei Electric Power Corporation | Hebei\n\nShanxi Electric Power Corporation | Shanxi\n\nInner Mongolia Autonomous Region Electric Power (Group) Co. Ltd. | Inner Mongolia\n\nShangdong Electric Power Corporation | Shangdong\n\nNortheast China Grid Company Ltd. | Liaoning Electric Power Corporation | Liaoning\n\nJilin Electric Power Corporation | Jilin\n\nHeilongjiang Electric Power Corporation | Heilongjiang\n\nEast China Grid Company Ltd. | Shanghai Electric Power Corporation | Shanghai\n\nJiangsu Electric Power Corporation | Jiangsu\n\nZhejiang Electric Power Corporation | Zhejiang\n\nAnhui Electric Power Corporation | Anhui\n\nFujian Electric Power Corporation | Fujian\n\nCentral China Grid Company Ltd. | Henan Electric Power Corporation | Henan\n\nHubei Electric Power Corporation | Hubei\n\nHunan Electric Power Corporation | Hunan\n\nJiangxi Electric Power Corporation | Jiangxi\n\nSichuan Electric Power Corporation | Sichuan\n\nChongqing Electric Power Corporation | Chongqing\n\nNorthwest China Grid Company Ltd. | Shaanxi Electric Power Corporation | Shaanxi\n\nGansu Electric Power Corporation | Gansu\n\nQinghai Electric Power Corporation | Qinghai\n\nNingxia Electric Power Corporation | Ningxia\n\nXinjiang Electric Power Corporation | Xinjiang\n\nCSG\n\n|\n\nProvincial subsidiaries | Province\n\n\u2013 | Guangdong Power Grid Corporation | Guangdong\n\nGuangxi Power Grid Corporation | Guangxi\n\nYunnan Power Grid Corporation | Yunnan\n\nGuizhou Power Grid Corporation | Guizhou\n\nHainan Power Grid Corporation | Hainan\n\nGrid operators non-affiliated to SGCC or CSG\n\n\u2013 | East Inner Mongolia Electric Power Company Ltd. | Inner Mongolia\n\nTibet Electric Power Company Ltd. | Tibet\n\nCLP Holdings Ltd. | Hongkong\n\nTable A.6\n\nShare of current application of ICT and challenges structured by voltage levels, adapted from [2]\n\nVoltage Level | ICT in use | Character | Requirement | Control Functions | Challenges\n\n---|---|---|---|---|---\n\nExtra-high-voltage\/High-voltage\n\n380-kV\/\n\n220-kV | Control systems\n\nFurther development | Quasi-real-time | Data security\n\nAvailability (24\/365)\n\nActive data management | f (Hz) | Wind\n\nTrade | The grids are increasingly being operated at the limit of their capacity\n\nHigh-voltage\n\n110-kV | Control systems\n\nTelecontrol systems | Quasi-real-time | Data security\n\nAvailability (24\/365)\n\nActive data management | Voltage | Wind, large PV plants\n\nBidirectional load flow\n\nDemand response\n\nMedium voltage\n\n10-kV\/20-kV | Tele-control systems\n\nConsumption metering | Not time-critical | Data collection and processing | In part voltage | Wind, PV\n\nRural areas\n\nOver 800 system operators\n\nBidirectional load flow\n\nDemand response\n\nReactive power provision\n\nLow voltage\n\n0.4-kV | \u2013 | \u2013 | \u2013 | \u2013 | Distributed feed-in especially PV\n\nRural areas\n\nActive customers (prosumers)\n\nVirtual power plant\n\nConsumption follows generation\n\nElectric mobility\n\nFig. A.1\n\nGermany's power grids in 2012 (\u00a9 VDE e. V. [3])\n\n## Appendix B \u2013 Bottom-up view on China's technological smart grid vision\n\nThe following descriptions and lists present a full picture of the modern grid technologies that are being deployed and tested in demonstration projects, individual cities, or individual provinces. These descriptions and lists have been provided by the Chinese expert team, and they represent the state-of-the-art of China's smart grid technologies.\n\nTransmission grids\n\n * Control systems: SGCC has successfully developed a smart grid dispatching support system. In order to improve safe and stable operation, CSG carried out a series of advanced projects to improve the dispatching automation. An automatic dispatching system covering all dispatching areas has recently been completed; it is still based on a manual dispatcher but supplemented by automatic decisions. An integrated dispatching system is under construction; it will include strong simulation capability, off-line analysis, and decision-making capacity. Furthermore, research and pre-testing of intelligent system integration is in process.\n\n * Power lines: Several UHV transmission lines constructed by SGCC and CSG have gone into normal operation in recent years. This technology is perceived as an important component of the Chinese smart grid vision.\n\n * Transformers: At the end of December 2008, the 1,000-kV transformers manufactured by Baoding Tianwei Baobian Electrical Co., Ltd and TBEA Shenyang Transformer Co., Ltd all passed the pre-delivery test and on-site delivery test. On January 6 th 2009, they passed the 168-h assessment. The UHV transformer with single-unit capacity of 1,000 MVA and single-column capacity of 334 MVA is the largest in the world. Moreover, three of the largest transformer manufacturers in China have enjoyed enormous development in \u00b1 500-kV converter transformer by absorbing foreign technologies. Xian XD Transformer Co., Ltd has mastered the core technology for manufacturing \u00b1 500-kV DC power transmission devices with the capabilities of independent design and manufacturing. Now it has set up annual production of 18\u201350 \u00b1 500-kV converter transformers and smoothing reactors. Shenyang Transformer and Baobian Transformer have all introduced Siemens technologies. They have developed \u00b1 500-kV converter transformers jointly with Siemens and mastered the basic technology for designing and manufacturing converter transformers.\n\n * Fixed series compensation: This technology is seen as one of the most important technologies for improving transmission and distribution grid capacities. Since 2000, China has witnessed rapid development and satisfactory results in the 500-kV fixed serial compensation technology. One of the important technical issues in the promotion of fixed series compensation technology is the stability of sub-synchronous resonance induced by the interaction between fixed series compensation and large-scale turbo-generators. Recently, domestic companies have achieved technical breakthroughs in this area. For example, Inner Mongolia Shangdu Power Plant has successfully resolved the problem of sub-synchronous resonance by using supplementary excitation damping control and shaft torsion. Their achievements have created a basis for further promotion of the technical application of fixed series compensation.\n\n * Wires: CSG launched a pilot application of high-strength heat-resistant wires, carbon fiber wires, compact lines, and a helicopter patrolling line.\n\n * Substations: SGCC has developed intelligent components of high-voltage (HV) switchgears, intelligent transformer components, modular intelligent HV vacuum circuit breakers with phase selection function, integrated monitoring systems based on a unified information platform, a time synchronization system, grid security and grid status monitoring equipment, data and event logging equipment, polymorphic remote viewing inspection and firefighting system, secondary equipment for online automatic calibration and early warning systems as well as other critical equipment and systems. Turning to CSG, they launched the promotion and application of digital substations. One 220-kV and seven 110-kV digital substations have been built. Furthermore, online equipment monitoring technology has been applied. Equipment condition monitoring centers and technical supervision centers are under construction. These centers will provide monitoring and early warning capability, fault diagnosis, status evaluation, risk assessment, maintenance strategies as well as asset management and maintenance decision support. In 2009, the company completed the formulation of the relevant technical specifications and acceptance codes, and in 2010 it built the provincial host station system.\n\n * Gas insulated substations: Gas insulated substations have entered the application stage. Online ice monitoring and lightning detection monitoring tools have been gradually applied.\n\nDistribution grids\n\n * Distribution grids with ring structures: SGCC has developed an intelligent environmentally friendly ring main unit, and an automatic distribution system. CSG has continued to expand the distribution grid structure of urban and rural distribution grids. The current distribution grid is still weak and most of the loads have single-supply access with T connections without ring structures.\n\n * Troubleshooting and diagnosis: Currently, the troubleshooting and diagnostic work needs to be done manually. Distribution automation pilot installations exist only in large and medium-sized cities and mainly cover the telemetry data acquisition and monitoring, fault signal sending, problem solving and alarm, fault location and troubleshooting functions. The systems have information distribution and partition management functions via an integrated bus, in accordance with IEC61970 and IEC61968 public information model standards and distribution grid management system interface standard. They provide real-time information exchange and synchronization of data and models among distribution automation system and related automation systems.\n\nPower consumption\n\n * Smart home and consumption systems: SGCC exploits smart technologies such as a chip dedicated to information collection, smart meters, smart appliances, smart sockets, smart interactive terminals, smart energy storage systems, power grid management terminals and systems, smart energy-using service systems, electric vehicle charging equipment, self-service electricity service terminals, information collection systems, and smart interactive terminal detection devices. CSG has built a market automation system, standardized market businesses, carried out research on the power supply customer service standardization system, and strengthened demand side management activities of large consumers.\n\n * Market management systems: Through the construction of market and management systems in accordance with uniform technical specifications, Yunnan Power Grid Company and Hainan Power Grid Corporation have realized centralized provincial deployment systems, while other branch and subsidiary companies have deployed such systems at city level. In an effort to meet the actual needs of the power market, measurement automation systems have mainly encouraged off-peak power consumption and introduced electricity metering and monitoring, remote meter reading, and other functions.\n\n * Electric vehicle charging: Various Chinese companies have started research on the key technology for electric vehicle charging. Electricity charging stations and poles for electric vehicles have been constructed as trial points in Shenzhen, and exchange electricity Experience Centers in Guangzhou.\n\nInformation and communication technology (ICT)\n\n * ICT systems: SGCC has developed an optical phase conductor and the pertinent supporting equipment, information security technology inspection tools, and analysis platforms. It has also developed smart grid key equipment including intelligent switch combinations and electrical and optical fiber composite low-voltage cables, which have successfully been put into operation. Smart grid dispatching technology support systems, intelligent primary equipment and facilities for electric vehicle charging have achieved a major technological breakthrough. CSG has developed a common information model, enterprise data resource planning, SOA technical specifications as well as a series of technical standards. It has also undertaken the construction of a unified data center, an integrated distribution system, an enterprise cockpit, and a massive real-time data platform. In 2006\u20132010, the company launched the \"CSG 123 Plan\", which promotes unified business standards for the entire grid and unified information standards by building an integration system that includes IT infrastructure, technology architecture, and IT services to realize digitized grid support, informational business management as well as intelligent analysis and decision. As a result of this plan, the company has made remarkable achievements in terms of information technology, and information levels have been improved significantly: optical fiber communications now basically cover substations and generation stations for 110-kV and above. Emergency communication grids have been built. Distribution grid communication gives priority to optical fiber communication, public grid wireless communication, and medium voltage power line carrier communication. From an information security aspect, a series of technical standards and specifications involving information security protection technology have been worked out. System-level protection measures are being implemented in order to achieve information resource partition and hierarchical security protection. Information security technology and management systems have been developed, combining a variety of technical means including firewalls, intrusion detection systems, anti-virus systems, public key infrastructure, risk assessment, and other features.\n\nEnergy storage\n\n * Pumped-storage power: Energy storage is of great significance for the development of smart grids. It can be applied in the case of peak shaving in electrical systems to solve the power utilization problem or to improve power supply quality and grid reliability. Meanwhile, it can also be used to control the fluctuation of the electricity system in order to enhance the safety of the grid and to foster optimal utilization of renewable energy and promote further developments in this area. By 2011, 11 pumped storage power stations had already been constructed with total installed capacity accounting for 1.8 % of overall installed power. With the construction of pumped storage power stations during the 11 th Five-Year Period, the dispatchable hydro power including that of pumped storage power stations in the whole country except for Hubei and Lasha will be 3\u20137 % of China's total installed capacity once they have been put into normal operation.\n\n * New energy storage technologies: Significant progress has been made in research into new types of energy storage. In the past decades, organizations such as the Electrical Engineering Institute of the Chinese Academy of Sciences, the Beijing Feilun Energy Storage Flexible Research as well as some academic institutions have all been engaged in studying fly wheel energy storage. At present, important breakthroughs have been obtained in key technical fields such as superconductive magnetic levitation, high-speed motors, and power conversion. A research group at the Advanced Material Lab and Macromolecular Science Department of Fudan University has successfully developed a new kind of type of energy element-oriented carbon nano fiber. Based on this technology, a new type of solar fiber battery with a diameter of only 60\u2013100 micrometers can be manufactured, allowing PV conversion and energy storage in the same piece of fiber. In addition, China has also achieved lasting breakthroughs in research into energy storage using storage batteries. Since the research into battery supported energy storage with vanadium redox flow in 1995, China has developed a 10-kW-battery energy storage system and established battery energy storage lab modules. In 2008, China Electric Power Research Institute developed a 100-kW battery energy storage system applicable to wind power plants. In January 2011, China achieved an important breakthrough in the construction of large-capacity battery energy storage technology through the synchronization of a megawatt battery energy storage station of Southern Grid, a development representing material progress in the integrated application technology of Chinese large-capacity battery energy storage. The final construction scale of Southern Grid's 10-MW battery energy storage station is 10 MW for four hours with initial project construction of 5 MW for four hours. Currently, a 1-MW four-hour battery energy storage system has been successfully synchronized and put into normal operation.\n\n## Appendix C \u2013 Integration levels of China's power system components in 2012 and 2020\n\nThe system integration of China's power system components in 2012 and 2020 was discussed among the experts involved in the study generation process in the course of workshops carried out in Beijing in June 2013. In addition, the prevalence of market elements in 2012 and the targeted prevalence of market elements in 2020 were also discussed. In a first step, the experts agreed on the power system components and market elements that are relevant in China's current electric power system as well as on the corresponding components and elements in 2020 (see Fig. C.1 for a definition of the power system components and market elements).\n\nIn a second step, the experts developed a common understanding of the extent to which each single power system component is currently integrated in China's electric power grid. In other words, they examined whether the components are physically connected to the grid, remotely monitorable, remotely controllable, or autonomously controllable\/self-healing (see Table C.1 for a definition of integration levels). The targeted integration levels of the power systems components for 2020 were also discussed.\n\nAfter the Beijing workshops, an evaluation was made of the extent to which the market elements are currently used and will be used in 2020, ranging from demonstrational level to national level (see Table C.2 for a definition of these values). Figure C.2 shows China's current power grid integration as of 2012. The main findings are:\n\n * In the area of power generation, it can be seen that monitoring and control technologies are common in conventional power plants but are rather seldom used with RES. More particularly, currently installed rooftop PV installations and biomass power plants are only physically connected to the grid, without the possibility of remote monitoring or any control functionalities.\n\n * In the area of power logistics, China's transmission grids, the associated transformer substations, and the pumped storage power plants are currently already remotely controllable by the grid operator. On the distribution grid level, however, the various components' system integration levels are significantly lower \u2013 most of the components are not equipped with monitoring and control technologies.\n\n * China's end consumers of electricity are for the most part physically connected to the power grid. Early examples of remote monitoring technologies are used only at industrial and commercial levels and for large-scale urban buildings.\n\n * With respect to the prevalence of market elements, it can be stated that only few of these elements are currently used in China's power system: early demonstration projects for spot markets have been implemented by the grid operators and time-of-use pricing for residential consumers is being tested in the provinces. For the most part, China's electricity is still being traded at governmentally fixed prices. Elements such as futures markets, balancing markets, or capacity markets used in other countries have not yet been implemented in China.\n\nThe following advances are planned until 2020 (see Fig. C.3):\n\n * In the power generation area, various energy sources are projected to have a place in the Chinese power system of 2020. All of them will be fully integrated in the power grid and most of them will reach the highest integration level so that they can autonomously react to grid and\/or market conditions. A comparison with the present situation reveals that especially RES are likely to make a big leap forward. While a considerable part of RES is currently not connected to the grid, grid access of RES will have become plug-and-play by 2020.\n\n * In the area of power logistics, all voltage levels above 35-kV will have reached the highest level of system integration by 2020. Power grids and transformer substations at these voltage levels are projected to be enhanced with a large amount of control and monitoring devices so that they can react almost autonomously to unforeseen events. The distribution grids (voltage levels below 35-kV) and their associated local substations are projected to be either remotely monitorable or remotely controllable (depending on their location). This implies a significant improvement compared to their current system integration.\n\n * With regard to power storage, hydro-pumped storage plants will be autonomously controllable and, compared to the situation of today, new power storage technologies will be developed and integrated in the grid. Smart meters will be widely used and reach higher integration values than today.\n\n * In the power consumption area, industrial and commercial users \u2013 China's major power consumers \u2013 will be either remotely monitorable or remotely controllable in 2020. This means that grid operators and other players will be well informed with regard to their power consumption and have the opportunity to directly control some customers' loads. City dwellers residing in large buildings are expected to be fully integrated into the power grid with remote control activities by the grid operator. However, remote control activities are not planned for smaller, rural households. But even these customers will be remotely monitorable by means of smart meters. Advances are also projected to be made with regard to smart home technologies such as intelligent household devices and electric mobility being monitorable for grid operators.\n\n * In the power trade and retail area many new types of markets are envisaged for 2020. Whereas today governmental price regulation leads to low penetration with market elements, spot markets (including futures markets), OTC, and balancing markets prominent in other countries will be used in China's power system of 2020. Also, carbon markets, capacity markets, and market-related elements such as virtual power plants (VPP), time-of-use tariffs, and smart products are expected to be part of China's power grid in 2020.\n\n## Appendix D \u2013 Germany's smart grid vision according to the study Future Energy Grid\n\nUsing the three system layers presented in Fig.\u200b 4.\u200b3 and eight different domains of the power sector representing particular sub-fields of the power sector supply chain, nineteen relevant smart grid technology areas were identified 1], [2]. The set of these technology areas, their individual maturity levels, and development stages describe Germany's smart grid vision. The respective maturity levels help to assess the speed of development of different smart grid technologies in Germany. The different technology areas are shown in [ Fig.\u200b 4.\u200b4 and are described in more detail in the remainder of this section.\n\nSmart grid technologies of the closed system layer\n\nThe closed system layer requires a high level of security as well as the capability of real-time communication since it is supposed to ensure the stability of the power system. Only few actors are granted with controlling access of this system layer. ICT components used within this system layer address the operation of the grid infrastructure and large-scale power stations. These components are classified into the following technology areas:\n\n * Asset management systems (AMS) for grid components are employed to plan the usage of electrical equipment following technical and commercial constraints. Their focus is on grid infrastructure components and components of large conventional power stations. Today, AMS are mostly used for central assets managing static data. Automation of this process is only done for core components.\n\n * Grid control systems are used for monitoring and controlling electrical transmission and distribution grids, or bulk generation units. Grid control systems are mainly deployed in transmission grids and distribution grids of 20-kV or 10-kV but rather seldom below.\n\n * Wide area monitoring systems (WAMS) subsume technologies to measure, transfer, store, and visualize time-synchronized values with high temporal resolution. WAMS are currently used in some selected areas of transmission grids. The application within the distribution grids is evaluated in single pilot projects.\n\n * Grid automation refers to ICT components at the substation or field level processing data from grid components, measuring transducers, and giving control signals. Germany's high voltage segment is currently already automated. In medium and low voltage grids, automation solutions are subject to evaluation in a variety of field tests.\n\n * Flexible control and AC transmission systems (FACTS) represent power electronic control systems which are used to affect the power flows or voltage levels in the electricity grids. Currently, FACTS are occasionally used only in the transmission grids.\n\nSmart grid technologies of the ICT infrastructure layer\n\nThe ICT infrastructure layer accounts for the communication between the closed and the networked system layer. The technology area associated with this layer therefore deals with the interface functionality required for the realization of smart grids:\n\n * ICT connectivity refers to the prerequisites regarding the communication infrastructure and the amount and scope of the information exchange used in the power system. Thereby, the ICT connectivity accounts for the quality-of-service level regarding the discovery and accessibility of services. The transmission grids are currently equipped with ICT infrastructures using point-to-point connections and proprietary interfaces. In contrast to that, the distribution grid is normally not yet equipped with ICT infrastructures.\n\nSmart grid technologies of the networked system layer\n\nThe networked system layer is distinct regarding the high amount of heterogeneous stakeholders and power system components it is supposed to connect. It is expected that this system layer will gain more and more importance in the future, especially with respect to the rising amount of renewable generation. Therefore, this layer is expected to become system-critical in the process of developing robust smart grids and smart markets. The layer subsumes the following technology areas:\n\n * Asset management for distributed energy resources refers to the processing of operational and commercial data regarding power system assets. Distributed energy resources are referred separately because currently no end-to-end asset management systems exist for medium and small generation units which are expected to be of high importance for the development of smart grids in Germany.\n\n * Regional energy marketplaces are established to enable market participation for industrial, commercial and domestic customers, thereby taking requirements regarding grid operation, especially for the distribution grids, into account. First demonstrators of regional energy market places were evaluated in pilot projects, e. g. the E-Energy model regions.\n\n * Trade control systems support the analysis of relevant data and the subsequent planning and execution of energy trading activities. Current systems support trading on the spot and day-ahead markets, as well as energy portfolio management.\n\n * Forecasting systems compute state estimations of various measurement parameters. Examples for this are forecasts regarding electricity generation subject to weather conditions or the price elasticity of electricity demand.\n\n * Business services comprise technologies concerning the efficient use of resources, low costs, high availability and reliability in the context of important business processes of the energy sector. Main challenges in this area address the extension of service offerings for customers, handling of big data, cloud computing and the optimization of the supply chain across enterprise boundaries. Currently, processes prescribed by laws and regulations are fully supported. Additional services address basic customer management functions and the billing of electricity consumption.\n\n * Virtual power plant (VPP) systems aggregate power generation, storage and consumption units. This way the provision of effective and balancing power as well as system services is optimized. Today, VPP are used to provide balancing power. The installations are heterogeneous.\n\n * Plant communication and control modules refer to embedded systems, similar to grid automation components, providing access to distributed generation, storage or consumption units for data reading and control signal processing. Today, the communication interfaces of distributed energy resources are mostly proprietary. Their main function is disconnection from the grid.\n\n * The advanced metering infrastructure (AMI) provides remote access to energy consumption data. In later development stages, this shall lead to advanced smart meter processes, e. g. for calculation of grid conditions. Standardization, security, and mass data processing are key capabilities in this area. Today, there are no large-scale installations in Germany due to missing regulatory action. Functionalities and interfaces are proprietary.\n\n * Smart appliances refer to equipment in households, buildings or small enterprises possessing communication and control capabilities. The provision of a standardized infrastructure is currently one of the main challenges within this area. While smart appliances may not be essential regarding the technical smart grid infrastructure itself, they are expected to have a high market potential in the future. Currently, energy management systems (e. g. in the areas of lighting and heating) operate independently from the grid. Energy-related apps (e. g. for usage on smart phones or tablet computers) are developed and offered in the context of pilot projects.\n\n * Industrial demand side management\/demand response subsumes the integration between the requirements of the electrical infrastructure with those of industrial processes. This includes the possibility to take influence on industrial assets from outside the enterprises. DSM is motivated by pricing of electrical energy: the industrial enterprise tries to match its operations to the development of market prices. The enterprise may also agree to manually reduce their load on request of the grid operator. Automatic load shedding possibilities are partially realized in the context of industrial operations.\n\nCross-cutting technologies\n\nBeyond the sixteen functional technology areas introduced above, there are three key issues regarding quality aspects of the future smart grid vision in Germany. They are regarded as cross-cutting issues which have to be separately considered in context of each functional area.\n\n * Integration technologies address the aspect of interoperability being of critical importance for smart grids as ICT-based systems.\n\n * Management of data refers to procedures regarding structuring, aggregation, analysis, processing, and storage of high amounts of distributed and heterogeneous data.\n\n * Security, in terms of information security, provides procedures to protect the stakeholders of the power systems as well as the infrastructure. Key requirements refer to the aspects of availability, confidentiality, and integrity.\n\n## Appendix E \u2013 Extracts from specific laws\n\nDirective 2009\/72\/EC, Article 37\n\n\"Energy regulators should have the power to issue binding decisions in relation to electricity undertakings and to impose effective, proportionate and dissuasive penalties on electricity undertakings which fail to comply with their obligations or to propose that a competent court impose such penalties on them. Energy regulators should also be granted the power to decide, irrespective of the application of competition rules, on appropriate measures ensuring customer benefits through the promotion of effective competition necessary for the proper functioning of the internal market in electricity. [...]\n\nEnergy regulators should also be granted the power to contribute to ensuring high standards of universal and public service in compliance with market opening, to the protection of vulnerable customers, and to the full effectiveness of consumer protection measures. Those provisions should be without prejudice to both the Commission's powers concerning the application of competition rules including the examination of mergers with a Community dimension, and the rules on the internal market such as the free movement of capital. The independent body to which a party affected by the decision of a national regulator has a right to appeal could be a court or other tribunal empowered to conduct a judicial review.\"\n\nSource: Wording from The European Parliament and the Council of the European Union [1]\n\nRoles and Responsibilities of smart grid actors according to SGTF\n\n * \" Transmission System Operator (TSO): according to the Article 2.4 of the Electricity Directive 2009\/72\/EC (Directive): \"a natural or legal person responsible for operating, ensuring the maintenance of and, if necessary, developing the transmission system in a given area and, where applicable, its interconnections with other systems, and for ensuring the long-term ability of the system to meet reasonable demands for the transmission of electricity\". Moreover, the TSO is responsible for connection of all grid users at the transmission level and connection of the DSOs within the TSO control area.\n\n * Distribution System Operator (DSO): according to the Article 2.6 of the Directive: \"a natural or legal person responsible for operating, ensuring the maintenance of and, if necessary, developing the distribution system in a given area and, where applicable, its interconnections with other systems and for ensuring the long-term ability of the system to meet reasonable demands for the distribution of electricity\". Moreover, the DSO is responsible for regional grid access and grid stability, integration of renewables at the distribution level and regional load balancing.\n\n * Generator: Generating electricity, contributing actively to voltage and reactive power control, required to provide the relevant data (information on outages, forecast, actual production) to the energy marketplace (see also the Articles 2.1 and 2.2 of the Directive).\n\n * Electricity Installer\/Contractor: Electrical contractors design, install and maintain intelligent systems for all kinds of industrial, commercial and domestic purposes. Alongside the power and lighting applications, they equally install ICT and telecommunications, public street lighting, high medium and low voltage lines, control and energy management systems, access, fire and security control equipment, lightning protection systems, advertising and identification signs, emergency power generating systems and renewable energy systems.\n\n * Industrial customer: A large consumer of electricity in an industrial\/manufacturing industry. May be involved in contract based Demand\/Response.\n\n * Transportation customer: A consumer of electricity providing transport systems. May be involved in contract based Demand\/Response.\n\n * Buildings: A consumer of electricity which is a private or business building, may also be involved in contract-based Demand\/Response.\n\n * Home customer: A residential consumer of electricity (including also agriculture users) may also be involved in contract-based Demand\/Response.\n\n * Supplier [in the area of grid users] : A grid user who has a grid connection and access contract with the TSO or DSO. Moreover, suppliers are those actors which will provide new services, real-time information, energy efficiency services and dynamic energy pricing concepts with Time-of-Use (ToU). The suppliers also provide local aggregation of demand and supply, in order to increase the effectiveness and efficiency of the electricity supply at all voltage levels (including low\/medium voltage levels).\n\n * Retailer: Entity selling electrical energy to consumers \u2013 could also be a grid user who has a grid connection and access contract with the TSO or DSO.\n\n * Power Exchange: Provides a market place for trading physical and financial (capacity\/energy and derivates) contracts for capacity allocation by implicit auctions within the defined country, region or cross border.\n\n * Balance Responsible Party: Ensures that the supply of electricity corresponds to the anticipated consumption of electricity during a given time period and financially regulates for any imbalance that arises.\n\n * Clearing & Settlement Agent: Assumes liability for clearing and\/or settlement of contracts and provides contractual counterparty within a Power Exchange and for over-the-counter (OTC) contracts.\n\n * Trader: A person or entity that buys and sells energy goods and services in an organized electricity market (Power Exchange) or over-the-counter.\n\n * Supplier [in the areas of energy market places]: Has a contractual agreement with end customer relating to the supply of electricity.\n\n * Aggregator: offers services to aggregate energy production from different sources (generators) and acts towards the grid as one entity, including local aggregation of demand (Demand Response management) and supply (generation management). In cases where the aggregator is not a supplier, it maintains a contract with the supplier.\n\n * Electric Power Grid Equipment vendors\n\n * Ancillary Services providers\n\n * Metering operator: the entity which offers services to provide, install and maintain metering equipment related to a supply. In most EU Member States the DSO is also metering operator. In case of a specific contractual basis, the contract is mostly with the network operator, or may be with the customer or the supplier. The meter may be rented to, or exceptionally owned by, the customer.\n\n * Information & Communication Technology (ICT) service providers\n\n * Grid communications network providers Plan, build and maintain the communications systems that enable the data communication required to maintain grid stability, load balancing and fault protection systems by a TSO or DSO. This function is mostly executed by the TSO or the DSO, or may be performed by an independent actor but the overall responsibility and ownership of information remains with TSO and DSO. Grid communications network provider ensures compliance with the agreed service levels (Service Level Agreements including quality of service, data security and privacy) and compliance with any national and\/or international regulations as necessary.\n\n * Home Appliances vendors\n\n * Building Energy Management Systems (BEMS) providers , delivering the systems which facilitate management and control of building facilities, realizing energy saving and increasing comfortability of users of buildings and making full use of the state-of-the-art Information Technology.\n\n * Electric Transportation\/Vehicle Solutions providers\n\n * Grid User\/Customer\/Consumer: Entity or person being delivered electricity. How a customer perceives the value received from other actors in the electricity supply chain has a substantial influence on the economic viability of the grid in general and on the overall acceptance of how the electricity supply chain performs.\n\n * Regulator: Independent body responsible for the definition of framework (market rules), for setting up of system charges (tariffs), monitoring of the functioning and performance of energy markets and undertaking any necessary measures to ensure effective and efficient market, non-discriminative treatment of all actors and transparency and involvement of all affected stakeholders.\n\n * Standardization bodies: Responsible for standardization of all relevant elements and components within the electricity supply chain, which in turn leads to harmonization of relevant services, support towards removing barriers to trade, creating new market opportunities and reducing manufacturing costs.\n\n * EU and national legislation authorities: Entities are in charge of defining legislation and metrics for areas such as environmental policy, social policy, energy policy and economic policy. They are also responsible for the authorization needed to develop the electricity grid infrastructure.\n\n * Financial Sector undertakings: Provide capital to other actors or invest themselves into the projects within the electricity supply chain (grid, generation, etc.).\"\n\nSource: Wording from the EU Commission Task Force for Smart Grids, Expert Group 3 [2]\n\nEEG \u2013 paragraphs 1\u20133 of part 2, chapter 2, section 11\n\n1)\n\nNotwithstanding their obligation in accordance with section 9, grid system operators shall be entitled, by way of exception, to take technical control over installations connected to their grid system with a capacity of over 100 kilowatts for the generation of electricity from renewable energy sources, combined heat and power generation or mine gas, if\n\na)\n\nthe grid capacity in the respective grid system area would otherwise be overloaded on account of that electricity,\n\nb)\n\nthey have ensured that the largest possible quantity of electricity from renewable energy sources and from combined heat and power generation is being purchased, and\n\nc)\n\nthey have called up the data on the current feed-in situation in the relevant region of the grid system.\n\nTaking technical control over installations in accordance with the first sentence above shall only be permitted for a transitional period until measures referred to in section 9 are concluded.\n\n2)\n\nThe rights under section 13(1) and section 14(1) of the Energy Industry Act of 7 July 2005 shall continue to apply vis-\u00e0-vis the operators of installations for the generation of electricity from renewable energy sources, combined heat and power generation or from mine gas where the measures in accordance with subsection (1)\n\n3)\n\nGrid system operators shall, upon the request of those installation operators whose installations were affected by measures referred to in subsection (1) above, provide verification, within four weeks, for the need for the measure. The verification must enable a qualified third party to fully understand the need for the measures without any additional information; particularly the data ascertained in accordance with subsection (1) first sentence no. 3 above shall serve that purpose.\n\nSource: Wording from the BMUB [3]\n\nEEG \u2013 paragraphs 1\u20133 of part 2, chapter 2, section 12\n\n1)\n\nThe grid system operator whose grid system gives rise to the need for the assumption of technical control under section 11(1) shall compensate those installation operators who, on account of the measures under section 11(1), were not able to feed-in electricity to the extent agreed upon. Where no agreement has been reached, the lost tariffs and revenues from the use of heat less the expenses saved shall be paid.\n\n2)\n\nThe grid system operator may, when determining the charges for use of the grid system, add any charges arising on account of subsection (1) above if the measure was necessary and he bears no responsibility for it. The grid system operator shall, in particular, bear responsibility if he did not exhaust all the options for optimizing, boosting and expanding the grid system.\n\n3)\n\nClaims for compensation made by installation operators against the grid system operator shall remain unaffected.\n\nSource: Wording from the BMUB [3]\n\n## Appendix F \u2013 Further results from the European Mandate M\/490\n\nFigure F.1 outlines the application process of these use cases with reference to the SGAM (WG RA), the results of the WG SGIS and the set of standards compiled by the WG FSS.\n\nThe objective of WG SGIS was to support the analyses regarding information security for smart grids (see [1]). The group identified pertinent standards and assigned them to the needs of the stakeholders. Confidentiality, integrity, and availability were considered as the key requirements regarding information security in this context. The SGIS methodology supports the assignment of weights regarding these aspects.\n\nStructural elements of the approach were used in form of a security view assigned to the SGAM, the SGIS security levels as well as the smart grid data protection classes. Starting with the collection of use cases, the assessment process (referenced as the SGIS toolbox shown in Fig. F.2) includes the assessment of risk impact levels (1\u20135) for the assets in context of scenarios addressing the key requirements of confidentiality, integrity, and availability. The combination of these impact levels with their likelihood leads to the assignment of an overall security level. Appropriate standards are afterwards assigned as countermeasures appropriate to the individual security level.\n\nThe first iteration of Mandate M\/490 was completed in November 2012. A second iteration for 2013 to 2014 is currently ongoing, its main focus being the implementation and further refinement of the methodologies of the first iteration and development of a second set of standards as well as system interoperability testing methods and a conformance testing map.\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nMore than just make-believe\n\nLibrarian Phoebe O'Dunn deals in stories, but her passion for history has taught her that happy endings are rare. Her life in Knights Bridge, Massachusetts, is safe and uneventful...until she discovers the hidden room.\n\nAmong its secrets is a cache of vintage clothing, including a spectacular gown\u2014perfect for a gala masquerade in Boston. In the guise of a princess, Phoebe is captivated by a handsome swashbuckler who's also adopted a more daring persona. Noah Kendrick's wealth has made him wary, especially of women: everybody wants something.\n\nWhen Noah and Phoebe meet again in Knights Bridge, at first neither recognizes the other. And neither one is sure they can trust the magic of the night they shared\u2014until an unexpected threat prompts them to unmask their truest selves.\n\nAfter all, it takes more than just the right costume to live out your personal fairy tale. It takes heart...and the courage to be more than you ever dreamed.\nPraise for\n\nand her novels\n\n\"[A] beautifully described tale that rewards readers with an intriguing mystery as well as a deliciously satisfying romance.\" \n\u2014Library Journal on Secrets of the Lost Summer\n\n\"Neggers captures readers' attention with her usual flair and brilliance and gives us a romance, a mystery and a lesson in history.\" \n\u2014Top Pick, RT Book Reviews on Secrets of the Lost Summer\n\n\"Only a writer as gifted as Carla Neggers could use so few words to convey so much action and emotional depth.\" \n\u2014Sandra Brown\n\n\"With a great plot and excellent character development, Neggers' thriller, Saint's Gate, the first in a new series, is a fast-paced, action-packed tale of romantic suspense that will appeal to fans of Lisa Jackson and Lisa Gardner.\" \n\u2014Library Journal\n\n\"Saint's Gate is the best book yet from a writer at the absolute top of her craft.\" \n\u2014Providence Journal\n\n\"Cold Pursuit is the perfect name for this riveting read. Neggers' passages are so descriptive that one almost finds one's teeth chattering from fear and anticipation.\" \n\u2014Bookreporter.com\n\n\"[Neggers] forces her characters to confront issues of humanity, integrity and the multifaceted aspects of love without slowing the ever-quickening pace.\" \n\u2014Publishers Weekly\nAlso by Carla Neggers\n\nHERON'S COVE \nSECRETS OF THE LOST SUMMER \nSAINT'S GATE \nTHE WHISPER \nCOLD DAWN \nCOLD RIVER \nTHE MIST \nBETRAYALS \nCOLD PURSUIT \nTEMPTING FATE \nTHE ANGEL \nABANDON \nCUT AND RUN \nTHE WIDOW \nBREAKWATER \nDARK SKY \nTHE RAPIDS \nNIGHT'S LANDING \nCOLD RIDGE \nTHE HARBOR \nSTONEBROOK COTTAGE \nTHE CABIN \nTHE CARRIAGE HOUSE \nTHE WATERFALL \nON FIRE \nKISS THE MOON \nCLAIM THE CROWN\n\nTo my three sisters, Bonnie, Hilda and Gretchen\u2014nothing like jumping into a cold brook on a hot summer day!\nContents\n\nOne\n\nTwo\n\nThree\n\nFour\n\nFive\n\nSix\n\nSeven\n\nEight\n\nNine\n\nTen\n\nEleven\n\nTwelve\n\nThirteen\n\nFourteen\n\nFifteen\n\nSixteen\n\nSeventeen\n\nEighteen\n\nNineteen\n\nTwenty\n\nTwenty-One\n\nTwenty-Two\n\nTwenty-Three\n\nDear Reader\nOne\n\nBumblebees hummed in the frothy catmint on the edge of the stone terrace, the only sound to disturb the hot New England summer afternoon. Phoebe O'Dunn watched a solo bee hover above a purple blossom, as if debating what to do, then dart past the green-painted bench where she was seated and disappear across the herb and flower gardens. None of its fellow bumblebees followed.\n\nPhoebe had met on the terrace with her sister Maggie and her friend Olivia Frost to discuss the upcoming vintage fashion show at their small-town library, but inevitably talk had turned to the charity masquerade ball tomorrow night in Boston, two hours away. Maggie and Olivia were going. Phoebe wasn't, but she just might be able to help with costumes.\n\nThe dresses would be perfect.\n\nIf she'd had any doubts, they'd been dispelled when Maggie and Olivia sank into their chairs at the round, natural-wood table across the terrace and said they were stumped. With just twenty-four hours before they had to leave Knights Bridge for Boston, they had no idea what to wear.\n\nPhoebe did. She'd already had the dresses cleaned and now they were hanging in the back room at her little house on Thistle Lane, just off the Knights Bridge common. She hadn't mentioned them yet because\u2014well, she didn't know why, except that she couldn't help feeling as if she were handling someone else's secrets. She'd discovered the dresses two weeks ago in a mysterious hidden room in the library attic. So far she hadn't told anyone about them or the room.\n\n\"We should have figured this out sooner,\" Maggie said from the shaded table. Like Phoebe, Maggie had wild strawberry-blond hair, hers a tone darker and four inches shorter. And they had freckles. Lots of freckles, Maggie especially.\n\n\"Dylan didn't give us much notice,\" Olivia said without a hint of criticism. Her fianc\u00e9, Dylan McCaffrey, had purchased tickets to the masquerade ball to support the cause, a neonatal intensive care unit at a Boston hospital. He'd handed them to Olivia just before he and several friends took off to the White Mountains for a few days of hiking. She added with a sigh, \"I've never been to a masquerade.\"\n\n\"Neither have I,\" Maggie said. \"We must know someone in Boston who can help with costumes.\"\n\nPhoebe listened to the bumblebees hard at work in the catmint. She and Maggie had been friends with Olivia since preschool. They were gathered in Olivia's backyard. Fair-haired and pretty, she'd returned to Knights Bridge in the spring to convert her classic 1803 center-chimney house into The Farm at Carriage Hill. In the process, she'd met and fallen in love with Dylan, a former hockey player, now a wealthy San Diego businessman. His arrival in Knights Bridge had turned the out-of-the-way rural Massachusetts town on its head.\n\nPushing back stray curls, Phoebe got to her feet. She and Maggie both wore sundresses and sandals, but Olivia had on shorts and an old T-shirt after spending the morning in her gardens. When she'd left Boston, she'd put her graphic design skills and boundless energy to work in transforming her historic house into an idyllic spot for showers, meetings, girlfriend weekends and the occasional wedding\u2014including her sister's upcoming wedding in September and her own in December.\n\n\"You've been awfully quiet, Phoebe,\" Olivia said. \"Any ideas what we could wear?\"\n\n\"I was just thinking...\" Phoebe tried to sound casual. \"What if you two dressed up as Audrey Hepburn and Grace Kelly?\"\n\nOlivia pushed back her chair and eyed Phoebe with obvious interest. \"How would we pull off Audrey Hepburn and Grace Kelly? Do you have something in mind?\"\n\nMaggie, a caterer with two young sons, stood with her iced tea, the sprig of peppermint and wedge of lemon that she'd artfully hooked onto the glass now floating among the ice cubes. She cast Phoebe an amused look. \"Do you see me as Audrey Hepburn or Grace Kelly? Either one?\"\n\nPhoebe smiled at her sister. \"Sure, why not?\"\n\n\"You really do have an imagination,\" Maggie said. \"What are they, dresses that came in for the vintage fashion show?\"\n\nPhoebe hesitated, framing her explanation. As director of the Knights Bridge Free Public Library, the vintage fashion show was her brainchild, an end-of-the-summer event that would involve the entire community. It would showcase clothing from 1900 to 1975. The various library reading groups were focusing on twentieth-century books, the historical society was helping out, local businesses were donating food and staging materials\u2014it was an all-consuming project that now, finally, was well in hand.\n\nPhoebe had discovered the tiny hidden room while looking through the library attic for anything she could use for the show. It was as if she'd stepped into a time capsule, a secret hideaway. The room was filled with reproductions of dresses from movies up through the 1960s and from different historical periods\u2014Medieval, Regency, Victorian, Edwardian, Roaring Twenties.\n\nWho could have predicted such a thing?\n\nShe wanted to know more before she told anyone. Who had set up the room? Who had worked there, left everything behind? Why?\n\nDid anyone else know about it?\n\nShe'd started volunteering at the library as a teenager and working there in college, and she'd never heard a word about a hidden attic room.\n\nFinally she said, \"Everyone's been going through trunks and boxes in closets and attics for the fashion show. It's been loads of fun so far.\"\n\nOlivia nodded. \"I helped Gran load up her car trunk with old clothes from her and her friends. They're all getting a kick out of the idea.\"\n\n\"I can think of several dresses that would be perfect for a costume ball,\" Phoebe said. \"Two in particular. I'm not positive about sizes, but we can alter them if we need to.\"\n\n\"Easier to take in a seam than let one out,\" Maggie muttered.\n\n\"If we need to let out seams, we could add a strip of similar or contrasting fabric,\" Olivia said. \"It's a costume ball. No one's going to kick us out if our costumes are a little quirky.\"\n\n\"You'll be wearing masks, too,\" Phoebe said.\n\n\"Ah, yes. Plausible deniability.\" Olivia grinned, obviously liking that idea. \"No one else has to know it's me trying to pass myself off as Audrey Hepburn.\"\n\n\"Not as Audrey Hepburn herself,\" Phoebe amended. \"As one of the characters she played.\"\n\nOlivia laughed. \"Well, that just makes all the difference, doesn't it? Hey, if one of these outfits works, I'm all for it.\"\n\n\"Me, too,\" Maggie said, with somewhat less confidence. \"You're sure it's all right? We won't be stepping on anyone's toes borrowing a couple of the dresses?\"\n\n\"It'll be fine,\" Phoebe said, leaving it at that. \"Why don't you come by my cottage later? We can open a bottle of wine and you can see if the dresses work for you.\"\n\n\"What about you, Phoebe?\" Olivia asked. \"You have to come with us now. We can't go off to the ball like the wicked stepsisters and leave you sweeping the ashes out of the fireplace. Dylan left a half-dozen tickets. No one will use them if we don't.\"\n\nWhenever Olivia mentioned Dylan, Phoebe could see how very much her friend was in love with him.\n\nA happy ending.\n\nPhoebe's favorite books and movies were ones with happy endings, and she welcomed a real-life romantic happy ending, as rare as it could be.\n\nShe waved off a bee that had found its way to her. \"It's very generous of Dylan. A neonatal ICU is a great cause, and it'll be a wonderful night for everyone, I'm sure, but I can't go.\"\n\n\"Why not?\" Maggie asked, obviously skeptical.\n\n\"I have things to do.\" Phoebe glanced at her watch and winced. It really was later than she expected. \"I have to get back to the library. I have story hour, but I'll be home by six if you want to come by then.\"\n\n\"We'll be there,\" Olivia said, then turned to Maggie. \"I guess I shouldn't speak for you.\"\n\n\"Wouldn't miss it,\" Maggie said. \"I want to see these dresses and I need a costume.\"\n\nAware of her sister's eyes on her, Phoebe offered to help clear the table of the iced-tea glasses and plates of tarts they had sampled for possible addition to the Carriage Hill catering menu, but Olivia shooed her away. \"You need to get to story hour. The kids will get restless if you're late.\"\n\n\"An understatement,\" Phoebe said with a smile as she snatched a tiny apple-pear tart. \"This one's my favorite, but they're all fabulous. I'm off. I'll see you later.\"\n\nInstead of going back through the house and disturbing Olivia's dog, Buster, asleep in the mudroom, Phoebe followed a bark-mulch path through basil, oregano and dill plants soaking up the summer sun, then crossed a patch of shaded lawn and went around the side of the house to the front yard.\n\nShe had the door open to her Subaru, which she'd owned since she'd started commuting to the University of Massachusetts in nearby Amherst, when her sister burst out from the kitchen ell, a later addition to Olivia's old house. Phoebe didn't have a chance to get into the car before Maggie flew down the front walk and caught up with her.\n\n\"Phoebe, what on earth is wrong with you?\"\n\nShe knew exactly what her sister was getting at. \"I pay my own way, Maggie. You know that.\"\n\n\"It's not as if Dylan offered to pay off your mortgage for you. The tickets are his donation to a worthy cause. It looks good if the ball is well attended. It's great publicity for the neonatal ICU and what it does, and it gets other people thinking about giving. Everyone wins.\" Maggie sighed at her older sister. \"We can't be grinds all the time.\"\n\n\"I'm not a grind,\" Phoebe said. \"I love what I do. I have fun\u2014\"\n\n\"And you live within your means and never take a false step,\" Maggie finished for her, then winced. \"Sorry. That came out wrong.\"\n\n\"It's okay. It's just...\" Phoebe stared at the tiny tart in her hand, suddenly wishing she had left it on the table. \"It's a slippery slope, wanting more than you can have, but I take lots of false steps. We all do. I'm not being morally superior.\"\n\n\"Oh, Phoebe, I know. You're the kindest person in Knights Bridge. Probably in all of New England.\" Maggie's rich turquoise eyes shone with emotion. \"I just don't want you to miss out on a good time.\"\n\n\"Dressing you and Olivia for a masquerade will be a great time,\" Phoebe said, smiling at her sister. \"I have to go. Those boys of yours will be tearing up the library.\"\n\nMaggie groaned. \"They'll be full of energy after spending the day with Mom. She lets them do whatever they want. When I picked them up yesterday, they were helping her muck out the goat barn. Knee-deep in you-know-what. I wouldn't care except they didn't have a change of clothes. I'll never get the smell out of my van.\"\n\n\"It'll wear off in time but you probably don't want catering clients to get a whiff.\"\n\n\"No kidding.\"\n\nPhoebe commiserated even as she was amused at the image of her mother and her young nephews. \"I'll see you in a little while. I hope I wasn't rude to Olivia\u2014\"\n\n\"You weren't, and she'd understand if you were. Don't worry. She's trying to figure out things herself. This is new territory for her. For Dylan, too. He never pictured himself living in a little town like Knights Bridge until he met Olivia. He obviously still loves San Diego, too.\" Maggie stepped back from Phoebe's car and waved a hand. \"They'll figure it out. I should have such problems.\"\n\n\"What would you do with a fortune like Dylan has?\"\n\nHer eyes flashed with humor. \"Get someone to paint my house. I hate ladders.\"\n\nPhoebe laughed as she climbed into her car, but she also felt a pang of uncertainty about what was next for Dylan and his millions, and what it would mean for quiet, picturesque Knights Bridge.\n\nShe left her car windows open and drove back toward town. She could smell the clean, cool water of a stream that ran along the edge of the narrow road. Carriage Hill was the last house on the dead-end road, two miles from the Knights Bridge village center. The road hadn't always been a dead end. Once it had wound into the heart of the picturesque Swift River Valley. That was before four small valley towns were depopulated in the 1930s and deliberately flooded to create Quabbin Reservoir. The reservoir now provided pure drinking water for metropolitan Boston.\n\nBoston must have seemed so far away back then.\n\nIt seemed far enough away now. Barely two hours, but so different from her life in Knights Bridge. She'd never lived anywhere else. Olivia and Maggie had both lived in Boston for a few years before returning to their hometown, Olivia in March, Maggie last fall. Phoebe's biggest move had been from her mother's house\u2014or madhouse, as she and her sisters would say fondly\u2014to her own place on Thistle Lane. It was a cottage, really. Perfect for just her. She could even walk to her job at the library.\n\nPhoebe appreciated the peaceful back road as she pulled her thoughts together. Story hours were a favorite part of her job, but her visit with Maggie and Olivia had left her feeling edgy and frayed.\n\nQuestioning, wanting...dreaming.\n\nI like my life, she reminded herself as she came into Knights Bridge center, known as one of the prettiest villages in New England with its shaded town common surrounded by classic houses, a town hall, library, church and country store.\n\nPhoebe parked in front of the library, a solid, rambling brick building filled with endless nooks and crannies. Persistent stories said the library was haunted, to the point that the producers of a television series about ghosts had considered it for a show before choosing another supposedly haunted New England library. Phoebe often heard creaks, groans, moans, whistles and\u2014once\u2014what she would have sworn were whispers. But she'd never considered she might encounter an actual ghost.\n\nSpecifically, George Sanderson, founder of the library in 1872.\n\nUpon his death in 1904, he left the library his extensive collection of books and archives, a Steinway baby grand piano and a dozen straw hats made at one of the small mills he'd owned in the valley. The last Sanderson had vacated Knights Bridge during the Depression, when the family mills were demolished ahead of the damming of the Swift River for Quabbin. Homes, businesses, barns, fences, trees\u2014everything in the valley went. Even graves were moved to a new cemetery on the southern end of the reservoir.\n\nOld George's portrait still hung above the fireplace in the library's main room. He was handsome and stern-looking, not exactly the sort Phoebe imagined would encourage story hours for small children. As she headed up the sunlit brick walk, she heard squeals of laughter through the open front window, where the children's section was located.\n\nHer five-year-old nephew, Aidan, Maggie's younger son, pressed his face against the window screen. \"Hey, Aunt Phoebe!\"\n\n\"Aidan Sloan, do not poke that screen,\" she said firmly, picking up her pace.\n\nHe giggled and disappeared from sight.\n\nPhoebe ran up the steps and went inside, welcoming the cool, solid wood-paneled interior, hardly changed since the library was built to George Sanderson's specifications. The main room included a small stage, the piano tucked on one end. Before Phoebe's arrival as director, the library had seldom used the stage and the trustees had complained about the \"wasted space.\" With careful planning, she'd gained their support and found the money to launch a modest concert series, with musicians who didn't expect more than a few dozen in the audience, and opened up the stage for art and garden shows. It was where the library would hold its vintage fashion show in less than two weeks.\n\nWe make use of all that we have.\n\nThat was Phoebe's motto for the library as well as her own life. Why moan about what she didn't have when so much was right within her grasp?\n\nHer older nephew, Tyler, almost seven, was sitting cross-legged on the hardwood floor in front of the stage with a book about raptors in his lap. \"Aunt Phoebe, did you know that raptors have three eyelids?\"\n\n\"In fact, I did, Tyler.\" She laughed. \"You'd be surprised at what a librarian knows. Would you like to see a raptor's eyelids sometime?\"\n\nHe nodded eagerly.\n\n\"We'll have to figure that out, then. Right now, though, let's go in with the other kids.\"\n\n\"I want to stay here.\"\n\nTyler\u2014as redheaded as his mother\u2014preferred to read a book on his own than to be read to, especially with his squirming younger brother. Phoebe put out a hand, but he ignored it and stood up on his own. He shuffled past her into the children's section, his head down, shoulders slumped, as if she'd asked him to walk the plank.\n\nHe and Aidan would be tired after spending most of the day with Elly O'Dunn, their energetic maternal grandmother. She'd taken the afternoon off from her job at the town offices to look after the boys while Maggie catered a lunch and then met with Olivia at Carriage Hill. Phoebe, her mother and her two youngest sisters were doing what they could to help Maggie as she managed two young boys and a catering business on her own, without Brandon Sloan, her adrenaline-junkie carpenter husband. Phoebe didn't have all the details, but she knew Brandon's construction work in Boston had been on-and-off at best the past year or so. It had to have put a strain on his marriage. He had a tendency to take off into the mountains or up the coast when things got tough, instead of talking.\n\nBrandon was the third of six Sloan siblings\u2014five brothers and a sister. His family owned a successful construction business in Knights Bridge and would welcome him back, but returning to his hometown would signal defeat in his eyes. Phoebe had known him since nursery school. He'd wanted out of Knights Bridge at ten. Then he and Maggie fell for each other as teenagers and married in college. Almost no one in town had believed their marriage would last. Phoebe had hoped it would, because they were so much in love.\n\nShe sighed. She could be such an idiotic romantic. Hadn't she learned by now?\n\nShe gathered the dozen boys and girls onto a round, dark red rug. They came quickly to order, even her nephews. They were reading Beatrix Potter and had just started The Tale of Peter Rabbit, their last book of the summer, and they couldn't wait to find out what happened next.\n\n* * *\n\nWith Peter Rabbit and Knights Bridge's little ones safely back with their families, Phoebe locked up the library and walked across South Main Street and through the common to Main Street and the Swift River Country Store, a town fixture for the past hundred years. It sold everything from galoshes to canned goods and fresh vegetables to a decent selection of wine. The afternoon heat had eased but it was still warm when she headed back to the library with two bottles of pinot grigio, already chilled. Olivia would bring a bottle of some kind of red from a California winery owned by Noah Kendrick, Dylan's best friend and founder of NAK, Inc., the high-tech entertainment business that had made them both fortunes. The only thing Phoebe knew for sure was that her choice of white wine wouldn't be nearly as pricey as whatever red Olivia brought.\n\nHaving a friend fall for a wealthy Californian had its unexpected advantages.\n\nNormally she'd have walked home but her visit with Maggie and Olivia meant she had her car. She got in, set her wine on the front seat next to her and shut her eyes a moment, listening to the rustle of leaves as a gentle breeze floated through the shade trees on the wide library lawn.\n\nFinally she started her Subaru and turned off South Main onto Thistle Lane. The library stood on the corner. Thistle Lane led away from the common, connecting to a back road with views of the reservoir in the distance. On her trips to the library as a girl, Phoebe had dreamed of living on the quiet, tree-lined street, away from the chaotic life she had out in the country with her parents and younger sisters. Thistle Lane represented order, independence and, at least to a degree, prosperity.\n\nIn less than five minutes, she turned into her short paved driveway. An old American elm graced the corner of the yard next to hers, holding on against the ravages of Dutch elm disease, in part due to intervention by the town. It was a beautiful tree, a symbol of the past and yet very much part of her everyday world. When she bought her house eighteen months ago, she'd thought she was being practical, never mind that she was the only one to make an offer. The house was built in 1912 by one of the early directors of the library, then sold to a series of owners, until, finally, the town was forced to take possession when the heir to the last owner couldn't be located and property taxes went into arrears.\n\nPhoebe rolled up her car windows, shut off the engine and collected her wine bottles as she stepped out into the shade. With its new roof, furnace, windows, wiring and plumbing, the house was no longer a notch above a tear-down. It still needed a new kitchen and bathroom, but she had to save up before she tackled any more big projects. Right now, she was concentrating on some of the fun cosmetic work\u2014paint, wallpaper, gardens and restoring flea-market and yard-sale finds.\n\nWith her painting skills and eye for color, Olivia had been a huge help, but The Farm at Carriage Hill and her new life with Dylan were creating uncertainties for her. Phoebe had welcomed having Maggie and then Olivia move back to Knights Bridge, but that didn't mean more changes weren't coming. Change was inevitable, Phoebe thought. Her own life was more settled than the lives of her sisters and most of her friends. Her job at the library was secure. She had no plans to move, go into business for herself or get involved with a man.\n\nFive years from now, her life would likely look more or less as it did now.\n\n\"Just without an avocado-green refrigerator in my kitchen,\" she muttered happily as she headed down the curving stone walk with her wine.\n\nThe narrow clapboards of her small house were painted classic white. At Olivia's suggestion, Phoebe had chosen a warm, welcoming green for the front door. It was framed by pink roses that she'd pruned and trained to climb up the white-painted trellis by the porch steps. When she'd moved in, the yard was an overgrown mess. She didn't have Olivia's green thumb, but she'd nonetheless managed to save many of the shrubs and perennials that had come with the property.\n\nAs she started up the steps to the small, covered porch, she saw that her twin sisters had arrived ahead of her. They were seated on wicker chairs that Phoebe had reclaimed and painted white, adding cushions in a mix of pink, blue and white flowers. Ava and Ruby, at twenty-three the youngest of the O'Dunn sisters, were fraternal twins, but they were so much alike that people often assumed they were identical. In both appearance and temperament, they took after their late father, Patrick O'Dunn, an auburn-haired, green-eyed, gorgeous-looking dreamer, as hopelessly impractical as the widow he'd left behind almost ten years ago.\n\n\"Thanks for coming,\" Phoebe said as she unlocked the front door. \"Olivia and Maggie will be here any minute.\"\n\n\"This is going to be so much fun,\" Ruby said, tucking a pink rose blossom behind her ear. She had on a long black skirt and a white tank top, her short, wavy hair dyed a purple-black that made her skin seem even paler, more translucent. \"We brought all our goodies. Makeup, wigs, hairpieces, curling iron, needles and thread. We've already done up a half-dozen masks. Three are simple. You'd be able to recognize whoever's wearing them. Three are more elaborate. It'd be tougher to recognize who's wearing them.\"\n\nAva smiled. \"We will not fail you.\" She twirled a rose stem in her fingertips. Her hair was its natural reddish brown, trailing down her back in a loose ponytail. Her skirt, which came to just above her knees, was a deep, warm red that worked surprisingly well with her turquoise lace top. \"A masquerade ball in Boston. It doesn't get much fancier than that.\"\n\nPhoebe pushed open the door. \"Dylan has extra tickets if you want to go.\"\n\n\"I wish,\" Ava said wistfully, tossing her rose over the porch rail into the grass. \"We have to work, and classes start again next week.\"\n\n\"Otherwise we'd go in a heartbeat,\" Ruby added.\n\nNo doubt they would, Phoebe thought. \"It does sound grand,\" she said as she led them inside. \"Maggie and Olivia are counting on your theatrical flair. What do you think of Maggie in the blue gown Grace Kelly wore in To Catch a Thief and Olivia in Audrey Hepburn's black dress from Breakfast at Tiffany's?\"\n\nAva turned, intrigued. \"Do you have the dresses?\"\n\nPhoebe nodded. \"I have the dresses.\"\n\n\"Oh, wow. Excellent. Ruby?\"\n\n\"Grace's icy-blue chiffon gown? Audrey's little black dress?\" Ruby laughed. \"That's fantastic.\"\n\n\"I even have pearls and a cigarette holder,\" Phoebe said.\n\n\"Where did you get them?\" Ava asked.\n\n\"I'm thinking of including them in our vintage fashion show,\" Phoebe said evasively. Her sisters followed her into the kitchen, where she put the wine in the refrigerator, a relic that, somehow, still worked.\n\nAva leaned against the counter, a cheap wood that Phoebe had painted creamy white, her first renovation when she'd moved in. \"So, Phoebe,\" Ava said, crossing her arms on her chest. \"Have you decided what you're wearing?\"\n\nPhoebe got out wineglasses and set them on the cracked Formica counter, sidestepping her sister's question. The twins were in graduate school\u2014Ava in New York, Ruby in Boston\u2014but they were spending the summer in Knights Bridge, living at home to save money. They had student loans that would take years if not decades to pay off, and big dreams that might never pay off, but Phoebe hoped everything would work out for them, believed in them. She knew they felt the same way about her but suspected they had their doubts about her choices. Not her library work. Her solitary life\u2014or what to her sisters seemed like a solitary life. Meaning she didn't have a man.\n\nShe'd had one, once. She'd been on the road to marriage and a happy ending of her own, but it hadn't worked out that way.\n\nEveryone in town knew her story\u2014Phoebe O'Dunn, jilted at twenty, within forty-eight hours of finding her father dead from a tree-trimming accident. She'd shielded her mother and sisters from the depth of her pain, but the shock had taken its toll. Broken hearts healed but that didn't mean life was ever the same. Phoebe had deliberately shut the door on romance, at least for herself.\n\nBut it was fine, all fine, because she was fine. She loved her work, her family, her friends, her town. She couldn't be more content than she was right now.\n\nAva looked out the window over the sink at the backyard flower garden, dominated now, in mid-August, by hollyhocks that ranged from soft white through three shades of pink to deep maroon. \"You're not going to the ball, are you, Phoebe?\"\n\nPhoebe changed her mind and decided to pour the wine now. She grabbed the pinot grigio out of the refrigerator and set it on the counter. \"No, I'm not going,\" she said matter-of-factly as she rummaged in the utensil drawer for a corkscrew. \"Do you both want wine?\"\n\nRuby plopped her tote bag onto a chair at the table. \"Phoebe, you know you'd have a great time. You never go anywhere\u2014\"\n\n\"I have so much to do here. I'm taking vacation days before the end of the summer. I'll go someplace then.\"\n\n\"Where?\"\n\n\"I don't know. Someplace.\" Phoebe held up a glass. \"Wine?\"\n\n\"Sure,\" Ruby said with a sigh. \"Just don't think I've given up.\"\n\n\"Me, either,\" Ava said. \"You should go to this ball tomorrow, Phoebe. Maybe you'll change your mind when you see the masks Ruby and I made. Anyway, wine for me, too. I'll get our goodies out of the car.\"\n\n\"Hang on, I'll help.\" Ruby withdrew a square of golden-colored soap from her tote bag and tossed it to Phoebe. \"Check it out while we're setting up. It's a new soap Mom, Olivia and Maggie are trying out. Mom wants your opinion.\"\n\nOlivia and Maggie were experimenting with making their own artisan goat's milk soaps to sell at The Farm at Carriage Hill. If it worked, Elly O'Dunn's goats could go from being an expensive and impractical hobby to earning their own keep. Phoebe was happy to do what she could to help and knew Ava and Ruby were, too, although Ava in particular wasn't crazy about their mother's goats\u2014especially when she had to clean up after them. They all appreciated the mildness and purity of the soaps.\n\nPhoebe took in the gentle lavender scent of the bar Ruby had tossed her. \"It really is lovely, isn't it?\"\n\n\"Olivia's already designed the labels,\" Ruby said. \"Dreams do come true, Phoebe. Olivia's are.\"\n\n\"I know. I want yours to come true, too.\"\n\nAva stopped in the hall doorway. \"What about your dreams?\"\n\n\"My dream,\" Phoebe said lightly, abandoning the soap for her wine, \"is to see Maggie and Olivia all set for their charity ball. Go grab your stuff. I'll get the dresses.\"\n\n* * *\n\nThree hours, two and a half bottles of wine, a pot of vegetable curry and much laughter later, Phoebe was again alone in her kitchen. Olivia and Maggie had precise instructions, beautiful handmade masks and everything else they needed to transform themselves into their own versions of Audrey Hepburn and Grace Kelly.\n\nThe dresses had worked out even better than Phoebe had imagined.\n\nThe dresses.\n\nAva had recognized them first. \"Phoebe, these aren't like the dresses Audrey and Grace wore in Breakfast at Tiffany's and To Catch a Thief. They are the dresses.\"\n\n\"Close copies,\" Phoebe had said, then again deflected questions about where she'd gotten them.\n\nShe turned out the light in the kitchen and walked down a short hall to a small back room. For most of the past eighteen months, she'd used it to store paint supplies, tools and junk she'd collected from the rest of the house but wasn't sure what to do with. Then, on a rainy night earlier that summer, she'd cleaned everything out, wiped down the walls, mopped the floor and considered the possibilities. A guestroom? A study? A spa bathroom?\n\nIn another life, it would have made a great baby's room.\n\nShe felt the same pang of regret she'd felt that night, but it was ridiculous. If her father hadn't died and her steady college boyfriend hadn't given her an impossible ultimatum, she wouldn't have ended up on Thistle Lane at all, with or without babies.\n\nFlorida.\n\nShe'd have ended up in Florida.\n\nShe tore off the dry-cleaning plastic to a third dress she'd had cleaned along with the Grace Kelly and Audrey Hepburn dresses. It hung on a hook in the back room.\n\nShe stepped back, marveling at the creativity and the workmanship of the gown. It was Edwardian, one of the period pieces in the hidden room. Its creator had chosen a warm, rich brown silk satin, decorated it with sparkling beads, lace and embroidery, all in a matching brown. It had an empire waist, a deep square neckline and loose, belled lacy sleeves.\n\nAnd there was a matching hat.\n\nIt was as romantic and beautiful a dress as any Phoebe had ever seen.\n\nA gown for a princess.\n\nShe tried to shake off the thought. She'd had too much wine. Just two glasses, but she felt...well, a little reckless.\n\nAnd why not?\n\nAfter all, what could be more perfect for a masquerade ball than a gorgeous, mysterious dress from a secret attic room?\nTwo\n\n\"I could pass for a swashbuckler right now,\" Noah Kendrick said as he stretched out on an expanse of granite near the base of Mount Washington, the tallest peak in the White Mountains of northern New England. \"If I don't shave or shower before tonight, I'll be all set.\"\n\nDylan McCaffrey shrugged off his pack and sat on another boulder. Noah saw no sign that four days of hiking had had any effect on his friend beyond sweat, stubble and a certain grubbiness. Two of Dylan's hockey player friends had joined them but had split off that morning for several more days of tramping in the mountains. It was Dylan's and Noah's first time hiking in the White Mountains. They were in good shape, but Mount Washington was a hell of a climb, their last summit before heading back to civilization.\n\nAnd a charity ball.\n\nGreat, Noah thought without enthusiasm.\n\nHe doubted that anyone at NAK, Inc., had needed to reach him in the past three nights and four days. He was the founder of the high-tech entertainment company that bore his initials\u2014NAK, for Noah Andrew Kendrick. The convergence of technology and entertainment had fascinated him for as long as he could remember, and he'd managed to turn it into a profitable business. NAK was just four years old but had gone public last fall, a grueling process that had consumed him and his senior managers.\n\nHe'd stepped down as CEO in June. His idea.\n\nOne of his smarter moves had been to get Dylan, fresh out of the NHL and looking for something new to do, to help with NAK. He'd eased back from day-to-day involvement now, too.\n\nNAK would have gone bust within months without Dylan's help. Dylan knew how to read people. He knew how to fight in a way Noah didn't.\n\nThey were both keenly aware that a central challenge for a newly public company was to figure out what to do with the founder. Sometimes the best thing for the company was for the founder to stay on as CEO, or at least remain deeply involved in the stewardship of his or her creation.\n\nSometimes the best thing was for the founder to find something else to do.\n\nLike spend a few days hiking on the other side of the continent.\n\nNoah decided to focus on that problem another time. \"I promise I won't step foot in that ballroom until I've had a shower,\" he said. \"I wouldn't want to scare the ladies.\"\n\nDylan grunted. \"More like turn everyone off their hors d'oeuvres.\"\n\nNoah grinned, leaning back on one arm as he surveyed the view of the mixed hardwood forest they were about to enter, a relief after the rugged, open terrain above the tree line. At over 6000 feet, Mount Washington was the highest peak in the east and one of the deadliest mountains in the world, in part because of its proximity to a large and mobile population, in part because of its changeable and often extreme weather conditions. Noah liked it because unlike the other mountains in what was known as the Presidential Range\u2014a series of high peaks named after U.S. presidents\u2014Mount Washington had a weather observatory and a full caf\u00e9 with hot dogs at the top.\n\nHe couldn't remember the last time he'd had hot dogs, but he'd helped himself to two on his brief stay on the summit.\n\n\"It's a beautiful spot, Dylan,\" Noah said, meaning it, \"but the same mosquito that bit me yesterday at the Lake in the Clouds has found me again. I think it followed me up and down this mountain.\"\n\n\"It's not the same mosquito, Noah.\"\n\n\"I hate mosquitoes.\"\n\n\"At least it's only one. It could be a hundred.\"\n\n\"Maybe my lack of showering discouraged reinforcements.\"\n\nDylan grinned at him. \"You and mosquitoes. Imagine if you didn't have bug repellant.\"\n\n\"No, thanks.\"\n\n\"You never hiked up Mount Washington while you were at MIT?\"\n\nNoah shook his head. \"Never even considered it.\"\n\n\"Too busy doing math problems,\" Dylan said, amused.\n\nMath problems. Noah sighed. He had explained countless times in his long friendship with Dylan\u2014practically since first grade\u2014that \"math problems\" was too simplistic. It didn't explain how his mind worked.\n\n\"I'm not good at math,\" Dylan added.\n\n\"You don't like math. There's a difference. And your idea of 'math' is arithmetic. Adding fractions.\"\n\n\"I can add fractions. It's multiplying them that does me in.\"\n\nNoah glanced at Dylan but couldn't tell if he was serious.\n\n\"We shouldn't sit too long,\" Dylan said. \"We don't have much farther to go, but we want to make it down the mountain in time to get to Boston and turn into swashbucklers.\"\n\nFor a split second, Noah imagined himself lying back on the boulder and taking a nap. They'd encountered high winds, fog and temperatures in the low fifties on the last thousand feet or so to the summit. He appreciated the clear, quiet weather and relative warmth lower on the mountain. It was even sunny. By the time they reached the trailhead at Pinkham Notch, it would be in the seventies. He'd peeled off his jacket on the descent and continued in his special moisture-wicking Patagonia T-shirt and hiking pants. Dylan, who was built like a bull, was in Carhartt. Noah was fair and lean, more one for sessions in the gym or dojo than treks in the wilderness. Dylan had decided a few days in the White Mountains would be good for Noah.\n\nSame with the masquerade ball tonight.\n\nGood for him.\n\nNoah had gone along. Why not? It wasn't as if he had a whole lot else to do. Not like even just a couple of months ago. A year, two years, ten years ago, he'd navigated a hectic schedule that would have flattened most people he knew. So had Dylan.\n\n\"You couldn't sign me up for a simple black-tie ball,\" Noah said, sitting up straight on the New England granite. \"No. No way. My best friend since first grade has to sign me up for a masquerade. I have to wear a costume.\"\n\n\"More or less. It's not like Halloween.\" Dylan was clearly unmoved by Noah's complaints. \"All in the name of fun and a good cause.\"\n\n\"Right.\" Noah drank some water from his water bottle, relieved that he didn't see any mosquitoes. \"I've agreed to dress up in whatever swashbuckler outfit you've managed to find for me, but I'm skipping the long-haired wig and funny beard.\"\n\n\"Just not the sword,\" Dylan said.\n\nNoah grinned. \"Never the sword.\"\n\n\"A reenactment musketeer rapier is waiting for you in Boston. No one needs to know it's you behind the black mask. I understand you don't want your photo turning up on some gossip website asking if the most eligible bachelor in San Diego has lost his mind.\"\n\n\"Dylan, why do I have the feeling you aren't taking my concerns seriously?\"\n\n\"Because I'm not. You'd have even more women flocking to you if they could see you in your sword-fighting duds.\"\n\nSword-fighting duds. Noah shook his head. Expecting Dylan to appreciate proper fencing terminology was a waste of time. No doubt he felt the same when it came to Noah and the nuances of hockey.\n\n\"The costume has a cape, too,\" Dylan added.\n\n\"There's no hope for you, my friend.\"\n\nDylan shrugged as he drank some of his own water.\n\n\"You used to be the most eligible bachelor in San Diego,\" Noah said.\n\n\"Best-looking. You were always more eligible. You just have a habit of choosing the wrong women.\"\n\nNoah tucked his water bottle into the side mesh pocket on his pack and got to his feet, lifting the pack onto one shoulder. \"What wrong women?\"\n\n\"Hollywood babes for starters,\" Dylan said, standing with his pack.\n\n\"Only recently. I haven't been the same since I got dumped by that computation engineer my senior year at MIT. She was brilliant, cute\u2014\"\n\n\"Not that cute. I remember her.\" Dylan jumped onto the trail. He didn't seem to consider that he might slip and hit his head, twist an ankle or fall off the damn mountain. Of course, he landed lightly on his feet. \"She wasn't as cute as your latest actress.\"\n\n\"Her show just got canceled, and she's not cute. She's gorgeous.\"\n\n\"Smart?\"\n\n\"Yes, I guess so. We didn't get that far before we went our separate ways.\"\n\n\"Not many people are smart compared to you. It's a relative term.\"\n\nAlso one Noah seldom considered, but he had learned through hard experience that not everyone thought the way he did. And what did he know about relationships? His latest \"relationship,\" with the cute\/gorgeous actress of the canceled Sunday-night show, had lasted three weeks and ended that spring. He'd known from the start it wasn't an until-death-do-us-part match, but he'd thought it would last at least through the summer.\n\nHe was the one who had ended it. Just had to be done. Expensive dinners, gifts and such were one thing. Manipulating him to bankroll a movie she could star in was another.\n\n\"It's good you had this time to enjoy nature,\" Dylan said without any evidence of sarcasm.\n\n\"Right. Sure. I didn't even bring a cell phone.\"\n\nWaving off a mosquito that seemed to have singled him out, Noah joined Dylan in heading down the mountain. In a few minutes, they were in dappled shade, and he could hear water tumbling down a rock-strewn stream. Several hikers passed them, ascending the rugged, steep trail. There were no guaranteed safe trails up Mount Washington, but thousands climbed it without incident every year. Preparation and the right equipment were key, but so was the right mindset\u2014a clear understanding of one's abilities and a willingness to turn back if conditions warranted. A foolish risk on Mount Washington could prove dangerous, even deadly.\n\nWhen he'd decided to start his own business, Noah had assessed his situation with the same clarity and objectivity as he had when he agreed to join Dylan and his hockey friends hiking in the White Mountains. He'd realized within weeks of forming NAK that he needed Dylan McCaffrey on his team. They'd grown up together in suburban Los Angeles, but Noah had gone on to MIT and Dylan into the NHL. After a series of injuries ended Dylan's hockey career, he had blown most of his money and was sleeping in his car when Noah knocked on his window asking for his friend's help.\n\nDylan's instincts and no-nonsense view of people and business helped Noah get NAK going and keep it going. Its success had exceeded their dreams. Now Dylan was marrying a woman from a small New England town and reinventing his life.\n\nNoah had no idea what he was doing beyond taking a hot shower when he was back in civilization.\n\nMore mosquitoes descended on him when he rounded the next bend in the trail, but by then he didn't care. He could hear cars. After three nights sleeping in a tent, he was ready to check into a five-star Boston hotel, even if a B-movie swashbuckler costume was waiting for him.\n\n* * *\n\nDylan had booked a room at the sprawling Mount Washington Hotel, a National Historic Landmark that opened in Bretton Woods in 1902. Noah would have happily stayed there for several days and enjoyed the resort amenities and the spectacular views of the surrounding mountains, but he and Dylan had to get to Boston.\n\nThey took turns in the shower and changed into fresh clothes.\n\nNoah didn't shave. Dylan grinned at him. \"Four days' beard growth is essential for a swashbuckler, I take it.\"\n\nNoah shrugged. \"I'm just hoping it will help keep anyone from recognizing me.\"\n\nHe slipped into a black sport coat, which he wore over a silky black T-shirt and black trousers\u2014the uniform he'd adopted after graduating from MIT. He didn't remember why, except it had seemed like a good idea at the time. Dylan insisted it was because he wanted to appear older. Maybe it had been. Whatever the reason, now people expected him to show up head-to-toe in black.\n\nHe cupped his iPhone in one hand and started out into the hall.\n\n\"How many messages did you have?\" Dylan asked as they left the room.\n\n\"What makes you think I looked?\" At his friend's roll of the eyes, Noah answered with an exaggeration. \"Ten thousand.\"\n\n\"You mean ten, and one you answered.\"\n\nIt was close. That was Dylan. He could read people.\n\nThey headed down wide, elegant stairs to the main lobby, then outside onto a sweeping porch overlooking expansive lawns and the stunning mountains where they'd spent the past four days and three nights.\n\nAs they walked to Dylan's car, he frowned at Noah. \"Everything okay?\"\n\n\"I got bit by mosquitoes. Do you worry about West Nile virus?\"\n\n\"No, and you don't, either. What's up?\"\n\nNoah shook his head as he climbed into the passenger seat of Dylan's Audi. He'd bought the car for his Knights Bridge residence now that he was spending most of his time on the East Coast. Noah didn't offer to drive.\n\nHe needed to think.\n\nIn fact, he'd had one call from San Diego that made him uneasy. He would have to return it once they arrived in Boston. He had no choice.\n\nHe could see that Dylan was on alert. He would help in a heartbeat if Noah was in trouble. NAK trouble, personal trouble. It didn't matter.\n\nThis time, Noah didn't want Dylan to get involved.\n\nThe San Diego call was his problem.\n\nDylan seemed to guess that asking more questions would get him nowhere. His years on the ice, practicing, playing with a team, had honed his natural instincts about when to make a move, when to hold back. Noah had always been more of a solo operator.\n\nAs he started the car, Dylan took a breath, obviously reining in an urge to interrogate Noah. Finally he said, \"Olivia's done a lot of work on her house since you were there in April.\"\n\n\"That's good,\" Noah said neutrally. Olivia's house had needed a lot of work.\n\n\"We're tearing down my place,\" Dylan added.\n\n\"Ah.\"\n\nAs far as Noah was concerned, it was the only sensible option. He'd been to Knights Bridge just that one time, in early spring, not long after Dylan had received a handwritten note from Olivia Frost demanding he clean up his property, an eyesore for potential visitors to the getaway she was opening down the road from him.\n\nExcept her note was the first Dylan had heard of her, Knights Bridge or his ownership of a house there. He'd had no idea his treasure-hunter father had bought the house and left it to him upon his death two years ago. It was built in the 1840s but wasn't the architectural gem that Olivia's home was. In fact, it was a rundown wreck.\n\nDylan hadn't expected to discover that he had roots of his own in the out-of-the-way Swift River Valley, and he certainly hadn't expected to fall in love with Olivia Frost.\n\nDespite the miles he had hiked over the past few days, Noah felt restless, frustrated with his situation, even trapped, but at least he didn't have to keep the players straight in Knights Bridge, Massachusetts. He stuck out enough in Southern California but he enjoyed relative anonymity there compared to what he would endure in a small town straight out of Norman Rockwell. Dylan had tried to explain to him that, despite appearances to the contrary, time hadn't stopped in Knights Bridge.\n\nMaybe it hadn't, but it was still small.\n\nReally small.\n\nNoah stared out the window as the mountains and woodlands of northern New England gave way to the suburbs of metropolitan Boston. Dylan drove with occasional suspicious glances at him, but Noah didn't budge. He wasn't talking.\n\nWhen the Boston skyline came into view and traffic picked up, he sat up straight, wide-awake.\n\nThis was familiar territory.\n\nDylan valet-parked at the same five-star hotel in Copley Square where the charity event was being held and they each had booked a suite for the night. Their costumes for the evening would be delivered to their rooms.\n\n\"Noah,\" Dylan said as he climbed out of the car.\n\nNoah knew there was no point denying there was a problem. He shook his head. \"Later.\"\n\n\"Anytime. You know that.\"\n\n\"I do. Thanks.\"\n\nWhen he reached his suite, Noah dug out his iPhone and stood in the window overlooking the familiar city streets as he dialed Loretta Wrentham's number in San Diego. Loretta was Dylan's personal lawyer and friend, a striking woman in her early fifties who recently admitted she'd been his father's lover, at least briefly. According to Loretta, Duncan McCaffrey had never told her why he'd bought a house in Knights Bridge, either, but it had changed his son's life.\n\nThat was Duncan, Noah thought. He'd been a restless soul, divorcing Dylan's mother, traveling the world, having adventures. Fifteen years ago, he'd turned up in Boston when Noah was a freshman at MIT. Noah had been homesick, feeling like a misfit even among people just as dedicated to math and science as he was. Duncan McCaffrey had suggested Noah take up a martial art. \"Karate, tae kwon do, tai chi, fencing. Something.\" Noah had signed up for his first fencing lesson that week. Duncan had already gone off on some expedition.\n\nNoah had known Loretta since she'd started working with Dylan during his early years with the NHL and considered her a friend.\n\nShe answered on the first ring. She must have pounced on the phone. \"I haven't found out a thing,\" she said. \"Not. A. Thing.\"\n\nThat wasn't good. Loretta was a hound. One sniff, and she pinned her nose to the trail straight to the end. This one had her stumped.\n\nA few days before Noah flew to Boston for his hike in the White Mountains, he'd spotted a mystery man on his tail in San Diego. Or what he thought was a mystery man on his tail. He'd first noticed the man outside a waterfront restaurant, then at his fencing studio and finally outside the NAK offices in downtown San Diego.\n\nOn that third sighting, Noah had raced outside but got there too late. The man was gone. Loretta was on her way into the lobby of NAK's stylish high-rise. Noah asked her if she'd seen anyone. She said she hadn't, but offered to find out what she could. As a friend.\n\n\"It could just be my imagination that this guy's following me,\" Noah said, as he had a little over a week ago when he'd explained the situation to Loretta in San Diego.\n\n\"Do you have an imagination?\" She caught her breath. \"That didn't come out right. I don't mean it as an insult. You're just so...evidence-oriented. I'm a lawyer. I can relate.\"\n\nNoah had learned not to dwell on people's stereotypes about him but he was tempted to tell Loretta that if he didn't have an imagination, there would be no NAK, Inc.\n\nNor would there be a fortune for anyone to scheme and fight over.\n\nIf that was what was happening.\n\nHe didn't know if the man's reasons for tailing him were personal, professional or money related\u2014or even involved him.\n\n\"This guy could be a reporter,\" she said.\n\n\"I suppose,\" Noah said, unconvinced. So far, most journalistic interest in him since NAK had taken off had been legitimate, professional. No sneaking around, no following him.\n\n\"I wish you'd gotten a better look at him. Tall, gray hair, trim, wearing a dark gray suit. That's not much. You're sure you'd recognize him again if you saw him?\"\n\n\"Yes.\"\n\nLoretta sighed. \"Maybe he's looking into one of your Hollywood ex-girlfriends. A paparazzi type.\"\n\nNoah grimaced as he watched a young couple run across Boylston Street hand in hand. \"All I need is some idiot with a camera popping up out of nowhere and snapping shots of me dressed as a swashbuckler.\"\n\n\"A swashbuckler?\" Loretta gave a soft chuckle on the other end of the line. \"That I'd like to see. Dylan says you're damn good with a sword. Master fencer, right?\"\n\n\"Something like that,\" Noah said. The couple disappeared from his sight. He drew back from the window. \"I should get ready for this thing tonight. I really appreciate your help with this situation, Loretta.\"\n\n\"Happy to do what I can. I dealt with a few crazy stalker fans back in Dylan's hockey days. I'm not saying that's what's going on here, but you might send me a list of your ex-girlfriends.\"\n\nIt would be quite a list. \"I'm not going that far, Loretta. Not without more to go on.\"\n\n\"See? I said you were evidence-oriented. I'll pick at a few more possible leads, but I'm not optimistic. Keep your eyes open. If this thing gets serious and I think you have a real threat on your hands, I'll take additional steps.\"\n\n\"Such as?\"\n\n\"Calling the police. Recommending a bodyguard.\"\n\nNoah shook his head as if she were in the room with him. \"No bodyguard. Not without an actual threat.\"\n\n\"Have you told Dylan about this guy?\" Loretta asked.\n\n\"No. I don't want to distract him. He's moving on from NAK, as he should.\"\n\n\"He's still your friend. What if your mystery man is on your tail because of Dylan? Have you considered that?\"\n\nHe had. \"Now we're speculating. First things first. If there's a reason, I'll talk to Dylan. Right now there isn't.\"\n\n\"All right. Fair enough. How was your hike?\"\n\n\"There were mosquitoes,\" Noah said with a smile, then assured Loretta he'd keep his eyes open and let her know if there were any new developments.\n\nAfter they disconnected, he did a series of stretches. In addition to a master fencer, he was a brown belt in karate. He'd concentrate on advancing to black belt once he got over the nonstop work and pressure of taking NAK public\u2014and the loss of his best friend and closest business ally to New England.\n\nAnd to pretty, talented Olivia Frost.\n\nShe was the love of Dylan's life. And he of hers.\n\nNo question.\n\nNoah centered his mind, focused on his movements, the rhythm, the technique. Everything else\u2014doubts, questions, fears, noise\u2014fell away as he did his basic shorin ryu karate warm-up routine of calisthenics, blocks, punches and kicks, then eased into a series of simple katas.\n\nWhen he finished, he was sweating and loose, and he felt grounded, aware, in the moment.\n\nHis costume arrived. He laid it on the bed as if it were a dead musketeer and took another shower. He debated tripling his donation to the neonatal ICU and bowing out of tonight's festivities. He could stay in his room and watch movies.\n\nNo point. Dylan would just hunt him down. Might as well get on with it.\n\nStill damp from his shower, Noah donned the all-black costume, including the cape and the fake sword. He winced at his reflection. It wasn't so much that he looked bad or foolish. He just didn't look like himself.\n\nAt least there was a mask. It, too, was black, but fortunately it covered most of his face.\n\nIn San Diego, someone might recognize him even with the mask. In Boston?\n\nUnlikely.\n\n\"Good,\" he muttered, and headed down to the ballroom.\nThree\n\nPhoebe couldn't take her eyes off the man coming toward her as if they were the only two people in the crowded, glittering ballroom. As if nothing could stop him and he was determined to reach her.\n\nShe was standing by a pillar, next to a table of empty champagne glasses. She'd arrived twenty minutes ago, wanting just to watch the festivities with a glass of champagne. Olivia had left one of Dylan's extra tickets behind in case Phoebe decided to go after all, but she'd been so adamant about not going that now she didn't want to have to explain why she'd changed her mind. Because she was captivated by a dress, by the fantasy of an elegant masquerade ball?\n\nBest just to be the proverbial fly on the wall, then go back home with no one being the wiser. Let Olivia and Maggie enjoy their evening without worrying about her.\n\nShe adjusted her mask. Of the half-dozen masks Ava and Ruby had made for tonight, this one provided the most coverage. Her eyes and the line of her jaw were all that anyone could see of her face.\n\nPerfect.\n\nWith this swordfighter gliding toward her, Phoebe appreciated the anonymity.\n\nAnd he really was gliding. He moved with such smoothness, such an air of masculine purpose and self-control. He didn't pull away to the bar or meet up with another woman. His mask covered most of his face, as hers did, and he was tall and lean, wearing a black cape over sleek black trousers and shirt, with a sheathed costume sword at his side. He looked as if he could handle the sword, fake or not.\n\nHis eyes locked with hers.\n\nPhoebe started to duck away, but she was transfixed.\n\nWhy not stay?\n\nThere was a lull in the live music provided by a small, eclectic band near the separate dance floor. Her swordfighter continued toward her, his eyes still on her. She stared right back at him, ignoring the quickening of her heartbeat, the rush of self-consciousness.\n\nDo I know him?\n\nShe shook her head. Impossible.\n\nSo far she'd managed to avoid running into Maggie and Olivia. It definitely helped that she knew what they were wearing. Even so, she'd almost turned back several times before arriving at her pillar. First, when she'd started onto Storrow Drive into the heart of Boston. Then when she'd eased her car into a tight space in the parking garage. Finally on the escalator up to the ballroom. She'd glanced down at the hotel lobby, full of giant urns of fresh flowers and artfully arranged sofas and chairs. Above her, she could hear people gathering outside the ballroom.\n\nIf she hadn't been on an escalator, she'd have bolted then, for sure.\n\nOnce she reached the ballroom, she got caught up in the crowd, the music, the lights, the laughter and especially the costumes. Her mysterious Edwardian dress passed muster\u2014she'd known it would\u2014striking just the right note of elegance and daring.\n\nThe swashbuckler stopped a few yards from her. His eyes were a clear, striking blue, sexy and captivating. It wasn't just the contrast with his black mask or the glow of the chandeliers or even her few sips of champagne at work. They were great eyes. Fantastic eyes.\n\nShe held her glass motionless in one hand as a couple passed in front of her, blocking her swashbuckler from her view. When they were gone, he was right in front of her.\n\nPhoebe didn't breathe.\n\nI don't belong here.\n\nThen she remembered she was alone, anonymous and dressed as an Edwardian princess. Why not play the part? Why not be a little bold, even a little reckless?\n\nWith a deliberate smile, she raised her champagne glass in a flirtatious toast, hoping the man couldn't tell that her heart was hammering in her chest.\n\nNext thing she knew, he was at her side, an arm around her waist. \"Dance with me,\" he said, his voice low, deep and impossibly sexy.\n\nPhoebe nodded without saying a word. He took her glass and set it on the table, then swept her onto the dance floor. His movements were sure, fluid and strong. He'd obviously known what he'd do the second he reached her.\n\nShe stifled a jolt of panic. A real princess would know how to dance better than she did. At least she had on strappy sandals that had seen her through several weddings and library events, and she managed not to stumble.\n\n\"Just follow my lead,\" he whispered into her ear.\n\nShe licked her lips. \"All right.\"\n\nSomehow he got her arm in position on his shoulder before she realized she had moved. She felt the ripple of lean muscle under his black cape and noticed the stubble of tawny beard around the edges of his mask. She had no idea who he was and expected it was the same for him with her. She'd followed the instructions her younger sisters had given to Olivia and Maggie in applying her makeup, but she'd had to figure out her hat and wig on her own. They felt secure, and she refused to consider what would happen if they flew off, revealing her pinned-up strawberry-blond curls.\n\nThe room spun as her dance partner whirled her among the hundreds of guests in costumes and masks in various shapes and colors. The feel of his palm on her lower back, the way he held her right arm\u2014the way he moved with her\u2014made dancing easy. He was confident, physical and strong, and Phoebe let herself pretend that he really could fight off bandits and scoundrels.\n\n\"Do you know how to use that sword?\" she asked.\n\n\"I do, but it's a fake.\"\n\n\"You're a fencer?\"\n\nHe smiled but didn't answer. The music switched to a faster tune. Phoebe barely paid attention to the actual music as her swashbuckler spun her across the dance floor. She was glad her dress was a good fit. If not, she'd have been bursting buttons and hooks-and-eyes. As it was, the dress revealed more cleavage than was her custom.\n\nShe felt sexy, lithe, wanted.\n\nNot herself at all.\n\nWhen the music ended, Phoebe realized they were on the opposite side of the ballroom. She gave her hat and mask a quick, subtle check to make sure they weren't about to fall off while her dance partner accepted two fresh glasses of champagne from a passing waiter, handing one to her.\n\n\"Nice dancing with you, Princess,\" he said, clicking his glass against hers.\n\n\"That was wonderful. Thank you. You're quite a dancer.\"\n\nHe laughed. \"I watch a lot of movies. You're not so bad yourself.\"\n\n\"That's kind of you to say. What should I call you? D'Artagnan? Are you a king's musketeer?\"\n\n\"That works for me.\"\n\nPhoebe sipped her champagne, wondering if their dancing had loosened a strawberry curl or two from under her wig. Would her musketeer care that she didn't really have raven-black hair?\n\nWhat does it matter? None of this is real.\n\nShe shut her eyes a moment, bringing herself back to reality. This was her secret night out on the town. She would be Phoebe O'Dunn again before dawn. Probably before the stroke of midnight.\n\n\"What brought you here tonight?\" her swashbuckler asked.\n\nPhoebe quickly reminded herself she was playing a part. Flirtatious, confident, rich. An Edwardian princess could afford to pay her own way to a charity masquerade ball and wouldn't feel bad if she hadn't. \"It's a great cause,\" she said, settling on a vague answer.\n\n\"That it is.\"\n\n\"And you? What brings you here?\"\n\nHe shrugged. \"I owed a friend a favor.\" His so-blue eyes narrowed on her as he drank some of his champagne. \"And it's a good cause.\"\n\nThe music started again, a slow, romantic song. He took her champagne glass from her and set it and his glass on a small table, then drew her into his arms and back onto the dance floor.\n\nPhoebe laughed, feeling light-headed and free. She didn't want the night to end and yet she knew it would. Her swordfighter would go back to being whatever he was\u2014a pediatrician, a hospital administrator, a lawyer, a Boston businessman, a professor at one of the local colleges. She would go back to Knights Bridge. They lived in the real world. He wasn't a musketeer and she wasn't a princess.\n\nJust for tonight...\n\nHis hand eased lower, subtly, over the curve of her hip. \"Close your eyes,\" he whispered. \"Trust me.\"\n\nPhoebe did as he asked as he held her even closer. She had one arm around his middle and one on his shoulder, could feel the warmth of his skin through the black fabric of his costume. He wasn't a man she'd conjured up on a lazy, hot, quiet afternoon at the library. He wasn't a figment of her imagination.\n\nAs they danced, she heard only the music, felt as if they were floating together, as one. When the music finally stopped, he kept her close as she caught her breath and opened her eyes. \"That was amazing,\" she said with a smile.\n\nHis lips brushed hers. \"You're amazing, Princess.\"\n\nPhoebe started to tell him that she was no princess, but the words stuck in her throat. She didn't want the fantasy to end. For a while longer she wanted to be a princess. She lowered her hand from his shoulder and opened her palm on his chest. Who was he, really? Did she even want to know?\n\nThen she saw Dylan, dressed as a cross between Zorro and the Scarlet Pimpernel, standing with Olivia in her Audrey Hepburn dress. They gave no indication they recognized her or even were moving toward her. Phoebe glanced around for Maggie but didn't see her.\n\nHer swashbuckler released her and stood back a few inches, the muscles in his jaw visibly tensed as his eyes narrowed on something\u2014or someone\u2014behind Phoebe. \"Excuse me, I have something I need to do,\" he said, shifting back to her. He was enigmatic, decisive. \"Will you wait for me?\"\n\n\"I will. Yes, of course.\"\n\n\"Do you have friends with you?\"\n\n\"I'll be fine. Please, do what you have to do.\"\n\nHe touched a fingertip to her lips, then was gone in an instant. Phoebe watched him as he headed quickly through the crowd, his black cape flowing, his movements smooth and controlled.\n\nShe hoped he would come back but wasn't at all sure what she would do if he did.\n\nShe dipped out of Dylan and Olivia's line of sight and stopped at an hors d'oeuvres table. Out of the corner of her eye, she spotted her sister at the far end of the table in her gorgeous Grace Kelly gown. As a professional caterer, Maggie always liked to check out the food offerings at an event. Before Phoebe could decide what to do, her sister abruptly abandoned the hors d'oeuvres and whirled back toward Olivia and Dylan. At first Phoebe had no idea why. Then she saw a man dressed as a rogue of a pirate and she knew.\n\nBrandon.\n\nPhoebe immediately recognized her brother-in-law\u2014Maggie's soon-to-be-ex-husband\u2014as he stopped at a tray piled high with miniature brownies. She tried not to react to his unexpected presence or call attention to herself in any way, but she was too late. His eyes met hers and then he grinned that grin that Phoebe had first seen in nursery school and her sister had fallen for at fifteen.\n\nShe groaned inwardly. It just figured Brandon Sloan would turn up as a pirate, and that he would have no trouble recognizing her in her Edwardian costume.\n\nPhoebe didn't dare bolt. That would only draw more attention to her. Instead, pretending to be casual, she helped herself to a bit of apple and cheese and moved down the table to him.\n\n\"Oh, this is too good,\" Brandon said. \"Phoebe O'Dunn in sequins and a feathered hat.\"\n\n\"Maggie and Olivia don't know I'm here,\" Phoebe said through her clenched teeth.\n\n\"Dylan?\"\n\n\"No.\"\n\nBrandon polished off a tiny brownie in one bite. \"I didn't think you were the type to sneak into a charity ball. I'm proud of you, Phoebe.\"\n\n\"Do not make fun of me, Brandon.\"\n\nHis dark eyes softened behind his mask. \"Okay, I won't. You're shaking. Is everything all right? I saw you dancing\u2014\"\n\n\"I don't want to talk about it.\"\n\n\"All right. We won't talk about it. Why are you here on the sly?\"\n\n\"Just because.\"\n\n\"You've been doing too many kids' story hours. You sound like Aidan and Tyler.\"\n\nPhoebe ignored his teasing her and peered into the crowd. She didn't see her swashbuckler. Everything she hadn't noticed while she was dancing she noticed now. A cluster of people here. Another one there. A woman shrieking with laughter. A man spilling a drink down his front.\n\nClinking glasses.\n\nWaiters with trays of drinks and hors d'oeuvres.\n\nReading materials and displays about the neonatal ICU.\n\nWhat was I thinking, coming here tonight?\n\nHow had she let herself get caught up in dancing with a perfect stranger?\n\nThey were both playing a role.\n\n\"Phoebe?\" Brandon took her by the elbow. \"You look wobbly. Do you need to get out of here?\"\n\nShe nodded. \"Yes.\"\n\n\"How are you getting home?\"\n\n\"I have my car.\"\n\nHe grinned. \"You drove? Good for you.\"\n\nShe glared at him. \"Brandon\u2014\"\n\n\"I'm not patronizing you. I meant it. Driving in Boston is no picnic even for someone used to it. Do you have your cell phone on you? Call me if you need help. Got that? Maggie would kill me if I knew you were sneaking out of here alone and didn't look after you.\"\n\n\"I don't need looking after. Really. I'll be fine. Thank you.\" Phoebe started to leave, but stopped and turned back to him. \"Brandon, if you see the man I danced with...\" Was she completely mad? \"Never mind.\"\n\nShe spun into the crowd before he could respond. As she came to the large exit doors, she scanned a knot of people gathered there but didn't see her swashbuckler. When she reached the relative quiet of the ballroom lobby, she hesitated instead of plunging straight onto the escalators. Maybe she should go back to the ballroom and find him. Olivia and Maggie would understand that the only way she could have come tonight was exactly the way she had\u2014on her own, without telling anyone.\n\nIf she hadn't been on her own, hadn't been anonymous, she never would have danced with her swashbuckler. He might never have noticed her\u2014or she him\u2014if she'd been hanging out with Maggie, Olivia and Dylan.\n\nSuddenly her head itched under the raven-colored wig, her makeup felt like paste and her feet hurt in her strappy sandals. She turned away from the escalator. She'd freshen up, get her bearings, before heading to her car.\n\nAs she started down a carpeted corridor to the restrooms, she heard a man's voice and realized it was coming from a coatroom. \"He's here,\" the man said. \"I saw him with my own eyes. He's dressed head to toe in black as a swordfighter or some damn thing.\"\n\nPhoebe held her breath. Was he talking about her swordfighter? She edged to the wide-open doorway and peeked into the coatroom. A man was there, alone, his back to her as he spoke into a cell phone. He had short, dark hair with gray streaks and wore a black suit. He wasn't wearing a mask and he wasn't in costume.\n\n\"The bastard spotted me,\" he said. \"He's looking for me now. We don't have enough time to take action. We need more.\"\n\nPhoebe stiffened but didn't move from her position by the door.\n\nWho's we? What kind of action?\n\n\"You should have seen him dancing. The guy can move. He was with some woman dressed up like she was about to board the Titanic.\" Another pause, then a sigh. \"No, I don't know who she is. I'll find out. It shouldn't be hard.\"\n\nHe snapped his phone shut.\n\nPhoebe bolted down the hall and into the ladies' room, the door still swinging behind her as she ducked into a stall. She let out a breath. Should she try to find her swashbuckler and tell him what she'd just overheard?\n\nWhat had she just overheard?\n\nShe wasn't used to this kind of night. The crowds, the glitter, the elegance. She was out of her element. How could she judge the snippet of one-sided conversation with any clarity? For all she knew, her swashbuckler was in the middle of a divorce and tonight was his night to cut loose with a perfect stranger.\n\nIn which case it really was time to get back to Knights Bridge.\n\nPhoebe left the stall and washed her hands at the sink, avoiding her reflection in the mirror, grateful she was alone in the ladies' room. Should she peel off as much of her costume as possible before venturing back into the corridor?\n\nNo.\n\nShe didn't have another outfit to change into, and if the man she'd eavesdropped on saw her, he could recognize her dress, snap a picture of her with his phone and there she'd be, strawberry curls, freckles and all. He'd have her name and address in a heartbeat.\n\nBest just to make her exit now.\n\nShe'd planned to drive home tonight, anyway. She'd only had a few sips of champagne and was wide-awake. Dylan and Olivia were staying at the hotel, Maggie at Olivia's small apartment in town. Phoebe could join her sister, but that would mean telling her what she'd done.\n\nWhat I've done is gone completely mad.\n\nEasier just to stick to her plan and stay anonymous.\n\nThe dress had come with a tiny matching purse that hooked onto the waist. She pulled out the bright red lipstick that she had chosen from Ava and Ruby's theatrical makeup kit and reapplied it, noticing that her hand was shaking. What a night. She could be home with a nice cup of lemon-chamomile tea and a good book, or tucked on her couch watching a summer rerun of a favorite television show. Instead she was in Boston, dodging a stranger, her friends, her own sister.\n\nDancing with another stranger.\n\nA sexy stranger at that.\n\nHad he spotted the man in the coatroom? Was that why he'd left her so abruptly?\n\nWhat was he hiding?\n\nPhoebe tucked her lipstick back in her purse and pulled out her car keys as she finally took in her reflection. Her cheeks were flushed. Brandon hadn't been lying about that.\n\nThe dress and the hat and the elegant mask really were amazing.\n\nShe had no regrets, she realized. Even if someone recognized her now, as she made her exit, the night was worth any possible embarrassment. So what if her friends and sister discovered she was the one who'd danced with the swashbuckler?\n\nShe'd had a blast.\n\nPhoebe returned to the corridor and made it to the escalators without running into the man in the black suit, or anyone else.\n\nAs she stepped off the escalator, she glanced around the hotel lobby, half wishing that her dance partner would appear and sweep her into his arms again.\n\nMaybe more than half wishing.\n\nShe kept putting one foot in front of the other until she was in the parking garage unlocking her car door. She kicked off her sandals and threw them in the back. She'd tossed gym socks and a pair of sneakers onto the passenger seat. She slipped them on, feeling more normal as she settled behind the wheel and pulled off her wig. It wouldn't fool anyone now, anyway. At this point, if the man she'd overheard spotted her, he had only to jot down her license plate to find out who she was.\n\nThe same with her swordfighter.\n\nHer car started without any trouble. She'd visited Maggie, and even Olivia, often enough during their time in Boston that she had no trouble finding her way back to Storrow Drive. When she reached Route 2, she finally let out a long, cathartic breath.\n\nShe'd done it.\n\nNow her coach could turn back into a pumpkin and she could get back to her life in Knights Bridge.\nFour\n\nMaggie would have sworn the woman who had danced with Noah Kendrick was her sister Phoebe, but that just wasn't possible. It was wishful O'Dunn thinking at its craziest.\n\nEven crazier was thinking the pirate sauntering through the crowd was her husband.\n\nShe gulped more champagne than she should have. She was letting herself get upset over nothing. No way would any Sloan, and especially Brandon Sloan, show up for a masquerade ball.\n\nOf course, if Brandon did show up, it would be dressed as a pirate. She needed to get a better look at him.\n\n\"It can't be Brandon,\" Maggie said under her breath. \"It just can't be.\"\n\nOlivia eased next to her. \"The pirate, you mean?\"\n\nShe was stunning in her black Audrey Hepburn dress, complete with a revealing slit up one leg and multi-strand pearls. Maggie didn't feel nearly as elegant in her blue chiffon Grace Kelly dress.\n\n\"He reminds me of Brandon.\" She tried to sound dismissive. \"I must have had too much champagne.\"\n\n\"Ah.\"\n\nMaggie gave her friend a sharp look. \"Olivia? Is it Brandon?\"\n\n\"I don't know but I had the same thought when I saw the pirate. Dylan gave away so many tickets but he didn't mention Brandon. Several of his hockey buddies are here. Maybe the pirate's one of them.\"\n\n\"That must be it. He's one of Dylan's NHL friends.\"\n\n\"Do you want me to find out?\" Olivia asked.\n\n\"No! Not when I'm dressed up as Grace Kelly. Brandon would suck all the fun out of the experience.\" Maggie polished off the last of her champagne. It wasn't the reason her head was spinning. That pirate was. She forced herself to smile at Olivia. \"Several people have recognized my dress. I'm enjoying the fantasy, personally. The whole evening has been perfect.\"\n\n\"I'm glad. You deserve this break, Maggie.\"\n\n\"I do, don't I?\" She laughed, but she was on the verge of tears again. She had to put Brandon out of her mind. \"But I wouldn't change a thing about my life right now. I love my work, and the boys are the best\u2014I'd walk on hot coals for them. You're happy being back in our little hometown, aren't you?\"\n\n\"I don't miss Boston as much as I thought I might,\" Olivia said.\n\n\"Having Dylan up the road helps. Where is he, by the way, and when do I get to meet Noah? I'm glad you pointed him out to me. I've seen pictures of him but I'd never have recognized him in his costume.\"\n\n\"It's a great costume, isn't it? Dylan's with a couple of his NHL friends. I haven't seen Noah but I want to introduce you to him.\"\n\nGiven her relationship with Dylan, Olivia was naturally more attuned to the other attendees at the ball. Masks or not, most people had obviously recognized Dylan and were intensely curious about her. Maggie liked being able to enjoy the festivities with a measure of anonymity.\n\n\"Are you going to see Brandon while you're in town?\" Olivia asked.\n\n\"No,\" Maggie said without hesitation. \"I'm heading back home first thing in the morning and I'm Grace Kelly tonight. I'm not Mrs. Brandon Sloan. I won't be for much longer, anyway. Might as well get used to it.\"\n\n\"Maggie...\"\n\n\"I'm sorry. I don't know why I brought him up.\"\n\nOlivia hesitated, then smiled. \"Would you like more champagne? And have you tried the mini frittatas? They're great.\"\n\nMaggie frowned at her friend. They'd known each other since they were tots and Olivia was clearly not telling her something. They'd driven to Boston together, taking Maggie's car. They'd dropped off Maggie's things at the small apartment Olivia still had from her days with a Boston design studio and then walked over to the hotel. Dylan was already there, in costume, with Noah and his NHL friends.\n\nOlivia drank some of her champagne. Her behavior was definitely awkward, Maggie thought. \"Olivia? What's going on?\"\n\n\"I wouldn't bet good money that Brandon's at a sports bar watching the Red Sox tonight.\"\n\n\"What? Olivia\u2014is the pirate Brandon?\"\n\n\"I told you I don't know for sure.\" Olivia again hesitated. \"I think Brandon may have been in touch with Dylan.\"\n\nMaggie felt her mouth drop open but she quickly snapped it shut again. \"In touch how? Why?\"\n\n\"I don't know. The Sloans are working on Dylan's place. Maybe Brandon stopped by.\"\n\n\"He doesn't live in Knights Bridge. That's my life.\"\n\n\"His family's there. The boys.\"\n\n\"Believe me, I know.\"\n\nMaggie heard the bitterness in her own voice and regretted it. Don't do this tonight, she told herself. Let Brandon live his own life. That was what he wanted, wasn't it? She shook off her confusion, her sense of violation\u2014as if he had deliberately inserted himself into the new life she was building for herself, without him. She took in a deep breath. She prided herself on staying calm amid the chaos that her life sometimes threw at her as she juggled the multiple demands of her busy catering schedule, her two young sons, her three sisters, her widowed mother.\n\nHer estranged husband.\n\nShe looked into the crowd to see if she could spot the pirate. It had to be Brandon.\n\nShe forced another smile at her friend. \"We'll sort everything out later. We're Grace Kelly and Audrey Hepburn tonight, right?\"\n\nOlivia looked visibly relieved at Maggie's cheerful tone. \"Come on. Let's go find Dylan. Noah's around here somewhere, too. You'll have to meet him.\"\n\nMaggie spotted Dylan alone by the doors out to the ballroom lobby. \"He's on his way over here now. Why don't you two dance?\"\n\n\"I'm not going to abandon you if you're upset about Brandon\u2014\"\n\n\"Nothing I'm not used to. Don't worry about me. If Brandon is the pirate, he had his chance to annoy me and resisted. I'm fine, honestly. Go.\"\n\nWhen Olivia turned, Dylan was already in front of her. He took her in his arms and whisked her onto the dance floor. He moved like a hockey player on ice, Maggie thought, smiling as her friend snuggled close to her fianc\u00e9. Olivia had reinvented her life, too. She was doing well, and Maggie was glad to see her so happy.\n\nA thick arm went around her waist. \"No wallflowers allowed,\" the pirate whispered into her ear.\n\nBrandon.\n\nMaggie recognized his deep voice, his touch, but she pretended not to know it was him as she put a hand on his shoulder and let him spin her onto the dance floor. She'd be Grace Kelly in her flowing blue dress. Cool, calm, controlled, as if she were dancing with Cary Grant. But why was Brandon here? She let her questions die on her lips as he pulled her close to him. Did he know he was dancing with her\u2014with his wife, the mother of his children?\n\nOf course he knew.\n\nHe settled a hand on the curve of her hip. \"Shh. Let's just dance.\"\n\nIt was what she wanted, too. Just to dance. To pretend he was about to lift her into his arms and carry her off as he had so many times in the past.\n\nHow long had it been since he had held her like this?\n\nThere had been only one man in her life. Brandon Sloan. They had been so right together...and then so wrong. Money, pride, dreams, the busyness of life. They'd let them all erode what they'd had together.\n\nShe had so many questions. So much she wanted to say to him.\n\n\"Do you know Dylan McCaffrey?\"\n\nBrandon didn't hear her, or pretended not to as he held her close.\n\nMaggie almost didn't notice when the music stopped. He released her and smiled that rogue's smile of his. \"You're beautiful, Maggie O'Dunn Sloan.\"\n\nThen he was gone, and by the time Maggie pulled herself together, she was standing alone on the edge of the dance floor, wondering if she'd imagined him. Tears burned in her eyes but she hoped, with the mask, that no one would notice.\n\nSuddenly she felt hot in her Grace Kelly dress, ridiculous.\n\nNoah Kendrick eased in next to her in his swashbuckler costume and slipped off his mask. \"Maggie, I'm Noah Kendrick. Dylan's friend. Can I get you a glass of water?\"\n\n\"I'm all right, thanks. Just...just hot.\"\n\n\"The costumes have their drawbacks.\"\n\n\"They get us thinking about fantasies, don't they?\" Maggie appreciated his diplomacy. As Phoebe had said yesterday, it was a slippery slope to want what you knew you couldn't have. Maggie cleared her throat, straightened her shoulders, wondering if it really could have been her sister who had danced with the California billionaire. Dylan's friend, and now Olivia's friend. \"It's a pleasure to meet you, Noah.\"\n\nHe gave a mock bow. \"A pleasure to meet you, too, Maggie.\"\n\n\"Have you enjoyed your evening? Who was that you were dancing with?\"\n\nHe frowned. \"I never got her name. Did you see where she went?\"\n\nMaggie shook her head. \"Sorry, no.\"\n\nHis eyes settled on her. \"Do you know her?\"\n\n\"I never talked to her,\" she answered carefully. What if it was Phoebe who had danced with Noah? Had she realized who he was? Would she want Maggie blabbing her identity to him?\n\nShe gave herself a mental shake. She was being crazy. It wasn't Phoebe.\n\n\"I was just impressed with how you two danced together,\" Maggie said.\n\n\"I was distracted for a moment.\" He seemed to want to go on but sighed. \"Well, it's nothing for you to worry about. You and Olivia look lovely. Great costumes.\"\n\n\"Thanks. They're fun. My sisters helped. I should call it an evening. My mother has my sons for the night, and I want to check in. They're live wires as it is, and she tends to\u2014\" Maggie broke off. She wasn't going to criticize her mother in front of a man she barely knew. \"They always have a great time together.\"\n\n\"You're feeling better, then?\"\n\nShe nodded. \"It was just one of those things. I've learned to have my moments and move on. Are Dylan and Olivia still dancing?\"\n\n\"They're good together,\" Noah said simply.\n\n\"Yes, they are,\" Maggie said without hesitation. \"I hope you enjoyed your hike in the White Mountains.\"\n\n\"It was an experience,\" he said with a smile.\n\nNoah was quiet, but he radiated a confidence that Maggie hadn't noticed at first, probably due more to her preset ideas about him than anything else. She hadn't expected a high-tech genius, a hard-driving entrepreneur, to be so self-possessed. \"A good swordfight more to your taste than mountain climbing?\"\n\n\"Than staying a step ahead of a cloud of mosquitoes, for sure.\"\n\nMaggie laughed, feeling more herself again. \"Brandon and I climbed Mount Washington before the boys came along. What an experience. The views stay with you forever, don't they?\"\n\n\"I liked the hot dogs at the top,\" Noah said with a wink.\n\nDylan and Olivia joined them, and Maggie pulled off her mask. The evening was winding down and she wanted to change back into her regular clothes, forget any wild fantasies she'd had. She glanced around for her pirate husband, but he had disappeared. She expected to feel relief but she had to acknowledge a pang of disappointment, too.\n\nAnd of loneliness, she thought. For so long, she and Brandon had been at each other's sides. Lovers, best friends, parents to their two little boys.\n\nHow had they let that get away from them?\n\nMaggie pasted a smile on her face. She wasn't going to think about what had gone wrong between her and Brandon right now. She turned down Olivia and Dylan's offer to walk with her back to Olivia's apartment and instead headed out alone.\n\nThe night was warm and still, Copley Square filled with people. Maggie told herself she needed this time on her own. She'd loved living in Boston, but she didn't miss city life as much as she thought she would when she'd packed up herself and the boys and returned to Knights Bridge. Her hometown had plenty to offer, and it was a great place for Tyler and Aidan. They'd made new friends, loved being close to family. It was the same for Maggie. Even her work was better in Knights Bridge. In Boston, she'd worked part-time for different caterers. Now she had her own catering business, and it was getting off the ground faster than she'd anticipated or even had hoped it would.\n\nShe cut down to Commonwealth Avenue and continued on to attractive, residential Marlborough Street. She'd always wanted to live in Back Bay, but she and Brandon had rented a series of apartments in less expensive parts of the city. It wasn't just a question of finances, she'd finally realized. It was what he wanted, where he was comfortable. Back Bay wouldn't suit him.\n\nShe used Olivia's keys to get into the apartment. Olivia planned to give it up, but it definitely came in handy tonight. Maggie wouldn't have wanted to drive home after her evening as Grace Kelly.\n\nShe caught her reflection in the entry mirror. She'd managed to avoid mirrors all evening and was a little shocked at how she looked. Sexy, a little devil-may-care. Leave it to Ava and Ruby to get creative and theatrical. Phoebe's discovery of the look-alike dress from To Catch a Thief was perfect, but the twins were responsible for the subtle Grace Kelly makeup, the push-up bra, the blond wig and the glittery mask.\n\nMaggie pulled off the wig, then unpinned her hair and let it fall to her shoulders.\n\nAlready she looked and felt more like Maggie O'Dunn, mom to two young boys, second of four sisters, caterer to showers, weddings, meetings, reunions, fundraisers and even the occasional wake.\n\nIf not always the most practical person, she was at least able to manage on her own.\n\nDid she look and feel like the wife of Brandon Sloan anymore?\n\nHad he left the hotel and found his way to a sports bar?\n\nEveryone liked Brandon. He was easy to like since he didn't have to deal with the details of paying bills, raising their sons, figuring out their future. When faced with unemployment, he'd taken off for the mountains with a backpack and his dreams. He'd never meant to be a carpenter forever. He was good at it, he even liked it\u2014but he thought he should be doing something else. Maggie didn't even know what anymore. She doubted he did, either.\n\nShe put him out of her mind and dialed her mother's house. Tyler picked up. \"Gran's making hot chocolate.\"\n\n\"Hey, Tyler. Why are you still up?\"\n\n\"The bat woke us up.\"\n\n\"I see.\" Bats weren't unheard of at her mother's farmhouse, especially in summer. \"Where's the bat now?\"\n\n\"Gran shooed it outside with a broom. I helped.\"\n\n\"Good for you. What about your brother?\"\n\n\"He hid under his blanket. He's having hot chocolate, too.\"\n\n\"All right. Well, you two be good and help Gran. Tell her I called, okay?\"\n\n\"I will, Mom. When are you coming home?\"\n\n\"Tomorrow.\"\n\n\"Did you see Dad?\"\n\nShe couldn't lie to her son. \"I did, but just for a few minutes.\"\n\n\"He's taking me and Aidan camping.\"\n\nMaggie heard the questioning note in Tyler's voice and responded without hesitation. \"Yes, absolutely, he's taking you and Aidan camping.\" That was one thing she knew for certain: Brandon would keep his promise to his sons. \"Go enjoy your hot chocolate. I'll see you tomorrow.\"\n\nWhen she disconnected, she threw her phone onto the entry table and sank onto the sofa. It opened out into a bed. She would sleep there.\n\nShe kicked off her shoes and noticed a side seam in her flowing dress had split an inch, probably from dancing with her husband.\n\n\"Why aren't you here with me, Brandon?\"\n\nShe hugged her arms around herself and burst into tears.\n\n* * *\n\nPhoebe could hear the pitter-pat of rain on the library roof as she sat cross-legged on the wood floor of the hidden attic room. Too wired to sleep after the masquerade ball and the drive back to Knights Bridge, she'd changed into yoga pants and a lightweight fleece tunic, intending to do a few stretches on the living room floor, but she'd ended up grabbing a flashlight and heading out into what was then a light drizzle. As she'd breathed in the damp night air, she imagined her swashbuckler's arms around her.\n\nWhat a night it had been.\n\nShe'd walked down Thistle Lane to the library, letting herself in through the side door. Putting aside thoughts of ghosts, she'd debated a moment before starting up the back stairs. A more formal set of stairs in the main room led just to the second floor. In her five years with the library, she'd seldom ventured up to the attic. One of those rare times was two weeks ago, and it had resulted in the discovery of the dresses that she, Olivia and Maggie had worn tonight.\n\nIt was pouring rain now, pitch-dark outside. Phoebe had never been up to the attic at night. She half expected a bat to fly out from its dark recesses, crowded with cast-off library furnishings, archives, books and everything her waste-not, want-not predecessors over the past century-plus had thought might come in handy someday.\n\nShe'd come upon the hidden room accidentally, when she'd lifted a small paper bag sitting on top of an old filing cabinet and a dozen antique marbles broke out of the bottom. They dropped onto the floor, rolling every which way. Several rolled under two tin closets standing side by side, filled with more junk and treasures. She'd edged between the closets, determined to collect the marbles.\n\nAs she'd bent down to retrieve a colorful swirled boulder, she noticed a door behind the free-standing closets. She'd had no idea it was there. Madly curious, she'd tucked the marble in her dress pocket and shoved the closets back just enough to give her room to get at the door. It was unlocked but obviously hadn't been opened in a while. It hadn't given way easily.\n\nShe'd expected to find that it was a closet, probably stuffed with more of the mishmash of materials in the rest of the attic. Instead the door opened into a small room that she hadn't even realized existed. It was lined with shelves and cupboards neatly arranged with fabric, patterns, buttons, zippers, needles, thread, notions, buttons\u2014everything an avid seamstress might need.\n\nA secret sewing room.\n\nIt felt like a hideaway, a tiny retreat where someone could sit and work in peace and quiet. Another door opened onto a remote corner of the sprawling attic, by a small window that overlooked the town common. A dusty sewing table was positioned so that a seamstress could work with a pleasant view and a bit of natural light.\n\nThe only thing that seemed to be missing for a fully equipped sewing room was an actual sewing machine.\n\nPhoebe had done a quick survey of the contents of the room and discovered the Hollywood-inspired and period dresses in two matching cedar-lined trunks and several hanging garment bags. Leaving everything undisturbed, she'd replaced the tin closets in front of the door and decided to keep the room her secret for the time being.\n\nA few days later, she'd gone back and picked out the three dresses to be cleaned.\n\nNow, tired, a little spooked with the dark night and rain, she raised the lid on a sewing basket. Given the conditions, she was ever-watchful for mice and spiders but the sewing kit yielded only pins, needles, thread, embroidery floss, a tracing wheel, cards of zigzag and seam binding.\n\nWho had sewn up here? Why leave so much behind?\n\nPhoebe took a sharp breath. Had the sewer of all these clothes died? Was that why the incredible dresses were still here?\n\nI have to know.\n\nShe pulled all the notions and other items out of the sewing kit and laid them on the floor, looking for any clues that would help identify who had sewn the dresses she, Maggie and Olivia had worn to Boston tonight.\n\nHer Edwardian gown had attracted her swashbuckler and hidden her from the scrutiny of the mystery man in the coatroom.\n\nA night of mysteries, she thought, untangling several zippers.\n\nA browned sheet of paper was matted to the bottom of the sewing basket. Phoebe carefully peeled it off and saw that it was a practice sheet of the conjugation of the French verb to be in a neat, feminine handwriting: Je suis, tu es, il\/elle est, nous sommes, vous \u00eates, ils\/elles sont.\n\nPhoebe had taken French in high school and college but she was rusty and wasn't sure she could have managed to conjugate even a simple verb. Had the seamstress gone to high school in Knights Bridge? Had she been a student when she'd set up this room?\n\nSo many questions.\n\nPhoebe returned the sheet of French verbs to the sewing kit and carefully replaced all the supplies. She stood, finally feeling the effects of her long day. She grabbed her flashlight and shut the door, moved the closets back into place, then headed back down the steep, dark stairs. The creaks and groans of the old building normally didn't faze her, but the hidden room had her thinking about ghosts as she locked up.\n\nIt was still raining when she started back down Thistle Lane. She'd gone out without a raincoat or umbrella, but it was a warm, gentle rain, as if to remind her what was real and what wasn't real.\n\nPretending to be a princess and dancing with a mysterious swashbuckler at a Boston charity ball had been a fleeting fantasy, a peek into another kind of life.\n\nSomeone else's life. Not hers.\nFive\n\nNoah slept fitfully and awoke wishing he had sent a check for the neonatal ICU instead of attending the masquerade ball. He could have gone straight back to California after hiking in the White Mountains or stayed in California altogether. Either way, he'd have spared himself meeting the potential love of his life and letting her slip through his fingers.\n\nIt was his own fault. He never should have left his princess and chased after his mystery man, if, indeed, that was who he'd spotted.\n\nThere had to be a way to find her.\n\nHe decided he didn't want to deal with email and voice mail and \"accidentally\" dropped his iPhone in the water-filled bathroom sink.\n\nThe people who truly needed to reach him would figure it out.\n\nHe got dressed, appreciating his normal black trousers and black shirt. No more hiking clothes, no more swashbuckler cape. He went down for breakfast and tried to act as if he'd had a good night.\n\nOnce he had coffee, he decided he probably shouldn't have tossed his phone into the sink.\n\nHe'd run into people last night from his MIT days. Rumors were circulating about what was next for him now that NAK had gone public. One account had him staying on as CEO, another shifting into research and development. Focusing on his Central Coast winery. Getting deeper into venture capital, starting a new business, devoting himself to philanthropy, moving into academia.\n\nNone of the rumors were true, if only because Noah had no idea what was next for him beyond whole-wheat pancakes and warm Vermont maple syrup for breakfast.\n\nHe'd finished his pancakes when Dylan and Olivia wandered into the restaurant and joined him at his table. Waiters quickly brought out fresh place settings. Olivia had on lightweight jeans and a green linen top that matched her eyes. Dylan was in jeans and a hiking shirt, as if he hadn't thought about being at the Boston hotel this morning. Noah hadn't, either. He just generally wore the same thing.\n\nOlivia sat next to Dylan. She looked radiant, comfortable in her own skin in a way she hadn't on Noah's brief trip east in early spring.\n\nHe'd been assaulted by black flies then, he remembered.\n\n\"Loretta called,\" Dylan said. \"She said she emailed you and left you a voice mail and thought she'd hear back by now.\"\n\n\"Phone's broken.\"\n\n\"Dropped it again?\"\n\n\"In the sink. Water damage.\"\n\n\"Ah.\" Dylan shifted his gaze to Olivia. \"Noah breaks a lot of phones.\"\n\n\"You get distracted and drop them?\" she asked.\n\nNoah ignored Dylan's obvious amusement at her question. \"You could say that.\"\n\nDylan grinned. \"He gets pissed off and kills his phone.\"\n\n\"Not often,\" Noah said, keeping his attention on Olivia. \"Dropping my phone in water is an indulgence but the alternative is to get distracted by the thing.\"\n\n\"It's how his mind works,\" Dylan said, leaving it at that. \"What does Loretta want with you?\"\n\nNoah glanced past him and looked out tall windows as Boston slowly came to life on a quiet, sunny Saturday morning. \"I guess I'll call her and find out.\"\n\n\"Noah?\"\n\nHe heard the concern in his friend's voice and shifted back to him. \"It's all good, Dylan. No worries.\"\n\nDylan was clearly unconvinced. \"Loretta has no official role with NAK. She's my lawyer and business manager, and my friend. You two aren't cooking up a surprise party. Something's wrong. What?\"\n\n\"You're moving on. I don't want you to worry about this stuff.\"\n\n\"What stuff?\" Dylan asked, eyes narrowed, alert. \"What's going on, Noah? You might as well tell me. I'm not going to quit badgering you until you do.\"\n\nBadgering was one of Dylan's qualities that had been most helpful and necessary during the past four years. It also could be inconvenient and, occasionally, annoying. But Noah saw that he had to tell him.\n\nHe shrugged. \"Some guy's on my tail. He was here last night. At least I think it was him. I could be wrong\u2014\"\n\n\"But you're not.\" Dylan took a breath. \"Early fifties. Six feet tall. Dark hair with a lot of gray. Black suit. No costume.\"\n\nNoah wasn't surprised Dylan could describe the man. Between his years on the ice and at NAK, he missed nothing. He'd honed his natural instincts about people, their motives and character. He'd turned down a larger role with NAK, but he'd been indispensable in transforming Noah's ideas and technical skills into a viable\u2014and ultimately highly successful\u2014company.\n\n\"Did you talk to this man you saw?\" Noah asked.\n\nDylan shook his head. \"He was watching you dance with your princess. Was she with him?\"\n\n\"Why would you think that?\"\n\n\"She left the ballroom right after he did. I tried to follow her but she disappeared before I could catch up with her. I didn't see the older gentleman.\"\n\n\"Did you recognize him?\"\n\n\"No,\" Dylan said without hesitation, then turned to Olivia. \"What about you? Did you see this man?\"\n\nShe set down her coffee cup. \"I spoke with him briefly. I think he's the one you're talking about. He asked what I knew about the woman Noah was dancing with.\"\n\n\"How did you respond?\" Noah asked, keeping his tone neutral.\n\n\"I didn't, really. I just said I was there to enjoy the evening. I had the feeling he knew you, Noah. I didn't think anything of it. We spoke for less than a minute. Then he moved on. Is he a problem?\"\n\n\"He's an unknown.\" Noah poured himself more coffee from a small silver pot. \"He might not be a problem at all. I spotted him a few times in San Diego.\"\n\n\"How many is a 'few'?\" Dylan asked.\n\n\"Three. At a restaurant where I was enjoying a nice fish dinner with a friend.\"\n\n\"One of your actresses?\"\n\nNoah ignored him. \"Then at the fencing studio. Third time was outside our offices. I ran into Loretta and we agreed she'd see if she could find out who he is and what he wants.\"\n\n\"Why not use one of your own people?\"\n\n\"Who are my people nowadays, Dylan?\"\n\nDylan tapped his fingers on the white tablecloth. \"Noah, is there any reason this guy would bird-dog you? Personal, professional\u2014anything?\"\n\nNoah pushed away his untouched coffee refill. \"Not everyone needs a reason.\"\n\n\"What does Loretta say?\"\n\n\"She's stumped. I hoped it'd turn out to be a case of too much time on my hands. Then I saw this strange man again last night. It's too big a coincidence for me to spot him in San Diego and then in Boston.\"\n\nDylan sat back. \"I'll talk to Loretta and take care of this.\"\n\nNoah shook his head. \"No, Dylan. Thank you, but Loretta and I are handling this on our own.\"\n\n\"Any ideas who he is, what he wants?\" Dylan asked.\n\n\"No.\"\n\n\"Is he stalking you or what?\"\n\n\"I wouldn't say stalking.\"\n\nDylan took in a sharp breath. \"Maybe you should involve NAK security. You're worth a lot of money. Your company recently went public. You've made a few enemies in the process.\"\n\n\"I don't think this is about money, enemies or power. It feels different.\"\n\n\"Personal?\"\n\n\"Maybe.\"\n\n\"An ex-girlfriend's father?\" Olivia asked. \"Something like that?\"\n\nNoah smiled at her. \"You're assuming I have an ex-girlfriend.\"\n\n\"More like a legion of them,\" Dylan muttered. When Olivia raised her eyebrows, he added, \"Noah's high-profile. A lot of women want to have a night on the town with him, at his expense. Deep down, though, he's still the high school geek who was better at math than most of his teachers. I wasn't, in case you were wondering.\"\n\n\"In other words,\" Noah said, his eyes on Olivia, \"I have a low threshold of trust where women are concerned.\" He sat back, wishing now he'd waited longer to have his pancakes. \"I also get dumped a lot.\"\n\n\"Because you don't like being used,\" Dylan said. \"Maybe you flipped the switch of one of your actress's crazy uncles, or someone is seizing the moment to see what they can get off you. We can speculate all morning. It won't get us anywhere.\"\n\n\"And it's not a problem until it's a problem,\" Noah said.\n\n\"This man hasn't made direct contact with you?\"\n\n\"Not yet, no.\"\n\n\"Maybe he sent you one of the emails you didn't want to read this morning.\"\n\n\"I'm not worried, Dylan,\" Noah said truthfully. \"If he wanted to physically harm me, he's had several chances.\"\n\n\"He could know you're a master fencer and a brown belt in karate.\"\n\n\"I hope he does.\"\n\n\"What if he's looking up dirt on you so that he can harm NAK?\" Olivia asked. \"What if he wants to harm you\u2014your reputation?\"\n\n\"Let him try. I have no skeletons in the closet.\" Noah gave her a slight smile. \"I'm not that interesting, Olivia. More than likely this man is just angling for money.\"\n\nDylan eyed Noah. \"Any chance there's a connection to me?\"\n\n\"I have no reason to think so, or that there's a connection to Knights Bridge.\"\n\n\"Knights Bridge?\" Olivia sat forward. \"Why would there be a connection to Knights Bridge?\"\n\nNoah regretted his offhand comment and tried to reassure her. \"I'm sure there isn't one.\" He decided to change the subject. \"Unless my princess is hiding there. Are you positive you two didn't recognize her? She had quite arresting eyes. Almost turquoise. They reminded me of your friend Maggie's eyes but the color was deeper.\"\n\nOlivia reached suddenly for the cream pitcher. \"Really? I wonder who she could be.\"\n\nShe greeted the waiter a little too cheerfully when he arrived with her and Dylan's breakfasts. Noah glanced at Dylan and saw that he noticed her reaction, too.\n\nThe description of his dance partner had obviously struck a nerve with Olivia.\n\nNoah smiled. His princess might not be so lost, after all.\n\n* * *\n\nKnights Bridge was even prettier than Noah remembered from his visit in early April. Having leaves on the trees helped. He sat up front with Dylan while Olivia pointed out various landmarks from the backseat. She explained that the building of the Quabbin Reservoir and the subsequent flooding of much of the Swift River Valley had changed the development of the town, putting it off the beaten track and giving it a \"time has stopped here\" feel that was, both Olivia and Dylan again insisted, deceptive.\n\nMaybe so, Noah thought, but that didn't mean he wanted to do more than float in and out again. He had a chartered jet scheduled to meet him at a nearby private airport that evening.\n\nOf course, his princess could change everything. He'd hang out for a day or two in Knights Bridge and brave mosquitoes and its one restaurant if there was a chance he'd find out more about her.\n\nDylan turned onto a back road that wound toward Quabbin, his ease with the twists and turns suggesting a familiarity that reminded Noah that his best friend was, without a doubt, moving on from NAK. Less certain was whether he and Olivia planned to keep a home in San Diego. Noah would. Four New England winters during his years at MIT were enough for him.\n\nNot that he had any reason to move to Knights Bridge or anywhere else in New England.\n\nThe Farm at Carriage Hill was located in a picturesque mix of meadows, woods and stone walls. Its hand-painted sign, decorated with a cluster of chives, worked with the 1803 house with its cream-colored clapboards and rich blue front door. As he followed Olivia through her kitchen out to the stone terrace, Noah could see that she was turning her vision for her historic house into a reality. Even subtle changes were infused with her sense of color and design, and her love for her hometown. According to Dylan, she'd always planned on returning to Knights Bridge to open her own version of a bed-and-breakfast, even if her departure from Boston hadn't been entirely on her terms.\n\n\"Dylan and I will make lunch,\" she said. \"You can wait out here and familiarize yourself with New England herbs and flowers.\"\n\n\"You're assuming I want to know New England herbs and flowers.\"\n\nShe laughed. \"Yes, I am.\"\n\nShe went back inside, and Noah sat at the round table and observed the backyard. It really was attractive. Small-town life suited Olivia. He hadn't known her when she lived in Boston and worked at a prestigious design studio, but he knew from Dylan that she'd lost a major client in an underhanded way to a friend whose career Olivia had helped revive. The experience had served as a catalyst for her to transform her life.\n\nOne could only move forward from where one was standing, Noah thought as he stretched out his legs and tried to relax. Pretending otherwise was a fast way into trouble. He knew from hard experience that where he was standing at any given moment wasn't always where he wanted to be, or should be. That was just life. Not everything was under his control. Mistakes, incompetence, good intentions, bad intentions, good luck, bad luck, human nature\u2014lots of things beyond his control played a role.\n\nOf course, a lot under his control played a role in determining where he was, too. His own screwups, his own limitations, his own lack of vision and purpose.\n\nWere they what had this mystery man on his tail?\n\nNoah sank back in his chair, appreciating the quiet surroundings. Olivia certainly did have a knack with flowers and herbs. She came through the back door with a tray of sandwiches, her big, ugly dog trailing behind her.\n\nHe looked up at her as the dog, a German shepherd with a healthy mix of black Lab and probably several other breeds, promptly flopped down under the table, his big black-and-brown head on Noah's feet. \"What's his name again?\"\n\n\"Buster,\" Olivia said, placing the tray on the table. \"He adopted me when I first moved back here.\"\n\nDylan followed her onto the terrace, carrying two glasses of iced tea. He set one in front of Noah. \"Maybe you should get a dog, Noah.\"\n\nHe eased his foot out from under the dog's head. \"Does Buster have a brother?\"\n\n\"I hope not,\" Dylan said with a mock shudder.\n\nOlivia grinned at him. \"I thought you and Buster had bonded.\"\n\n\"We have, but one Buster is enough.\" He winked at her as he handed her the second glass of tea and sat across from Noah. \"All the world needs.\"\n\nBuster gave a deep, satisfied sigh from under the table. The dog was visibly calmer than when Noah had met him in April. A few months in Olivia's care no doubt had helped. Buster had clearly endeared himself to Dylan, despite an inauspicious meeting.\n\nNow here they all were\u2014Olivia Frost, Dylan McCaffrey and Buster.\n\nNoah smiled at what a great family they made. He'd never seen Dylan happier, and Olivia was fast becoming a friend herself. Noah helped himself to a chicken salad sandwich. It had some kind of herb in it. Fresh tarragon, he thought. If his princess was in Knights Bridge, was she into herbs, too?\n\n\"Who'll be minding Buster while you two are in San Diego?\" he asked casually.\n\n\"Maggie will be in every day,\" Olivia said. \"She and I are basically business partners. We're thinking about doing the paperwork to make it official. We work so well together.\"\n\n\"And she lives in Knights Bridge and likes herbs,\" Noah said.\n\n\"She also likes her mother's goats,\" Dylan added, his tone neutral. As he'd explained to Noah, the bonds between the people of Knights Bridge were sometimes tricky to navigate. The Frosts had been in the Swift River Valley and surrounding hills for generations. Despite Dylan's newly discovered roots in the region, he was still an outsider.\n\n\"Maggie loves herbs and goat's milk,\" Olivia said with a laugh. \"I don't know that much about goats, but the milk is perfect for the artisan soaps Maggie and I are making.\"\n\nNoah tried to keep any reaction to himself as it sank in that he was talking goats and soap at a two-hundred-year-old house on a dead-end road, surrounded by meadows, shade trees, green grass and a lot of flowers and herbs. It was a first.\n\nThe goats, he'd learned, belonged to Maggie's widowed mother and were a source of both tension and enjoyment within the O'Dunn family.\n\nObviously in a happy mood, Olivia sat between him and Dylan. \"I'll give you some samples of our goat's milk soap. We're still tinkering before we test-market it here. Maggie's on top of all the regulations.\"\n\n\"Complicated?\"\n\n\"Not too bad unless we make actual medicinal claims.\"\n\n\"Which you won't?\"\n\nShe shook her head. Noah saw that his interest surprised her, but she was the love of his best friend's life and he wanted to know about her and what she enjoyed. With Carriage Hill getting off the ground and the betrayal of her friend over stealing a client behind her, Olivia's natural optimism had clearly returned.\n\nFalling in love didn't hurt, either.\n\nNoah thought of his princess. He could feel the curve of her hip, see the warmth in her eyes, the soft swell of her creamy breasts. Why had he left her? Why hadn't he let the mystery man come to him?\n\nBecause he hadn't wanted his life in San Diego\u2014who he really was\u2014to intrude on the moment. The fantasy they both were enjoying.\n\nEither that, or he hadn't known what the hell he was thinking.\n\nHe wasn't thinking she'd disappear, that was for sure.\n\n\"Noah?\" Dylan asked with a frown.\n\nHe sighed. \"Mind drifting. Thinking about hiking in the mountains, then playing a swashbuckler at a ball\u2014I've got mental whiplash.\"\n\n\"Not a chance,\" his friend said without hesitation. \"You never have mental whiplash, whatever that is.\"\n\n\"It's a big change to go from waking up in a sleeping bag on a mountain to dancing at a charity ball that night.\"\n\nDylan was still obviously unconvinced. \"You knew the deal. There were no surprises.\" He shifted, then smiled. \"Except for your princess. I guess she could have you whiplashed in a number of ways.\"\n\n\"Funny, Dylan,\" Noah said.\n\nHe grinned. \"I thought so.\"\n\nAs they finished their simple lunch, Noah noticed a woman come out of a small shed at the far end of the yard. She had a cobalt-blue scarf tied around her head and long, dark strawberry curls trailing down her back. She started up a bark-mulch path, and Noah saw she wore a deep red top that accentuated her breasts and shorts that shaped slim hips. Her sport sandals, though, looked as if they'd gone up and down Mt. Washington a time or two.\n\nWhen she reached the terrace, she stayed on the path and motioned toward a raised flower bed as she addressed Olivia. \"The slugs got to the miniature dahlias, Liv. They're so gross. I put out slug bait and trimmed back the worst of the damage.\" She shuddered, then smiled brightly. \"I was admiring the gardens and couldn't resist going on slug patrol when I saw the carnage.\"\n\n\"Yuck,\" Olivia said. \"I hate slugs. Only thing worse are ticks.\"\n\nNoah glanced at Dylan. Slugs? Ticks? What had happened to bucolic small-town New England?\n\nDylan seemed to read his mind, with obvious amusement. \"Ticks suck blood and can be hard to see,\" he explained, not that an explanation was necessary or desired.\n\n\"Oh, sorry,\" Olivia said. \"Noah, this is my friend Phoebe O'Dunn. Maggie's sister. Phoebe, this is\u2014\"\n\n\"Noah. Noah Kendrick.\" He got to his feet and put out a hand. \"A pleasure, Phoebe.\"\n\nShe wiped a palm on her hip and smiled as she shook his hand, her skin warm, soft, her fingers long and slender. \"I wore garden gloves when I took on the slugs, but you never know. It's nice to meet you, Noah. I hope you're enjoying Knights Bridge.\"\n\n\"Hard not to on such a beautiful day, despite images of slugs and ticks.\"\n\n\"Sorry about that,\" she said, the twinkle in her eyes belying her words. \"Are you here for long?\"\n\n\"That's not the plan.\"\n\nNoah saw that her eyes were a similar turquoise to her sister's but shook off any comparison with his princess from last night. The false eyelashes, the heavy makeup\u2014how would he be able to tell for sure? He doubted he'd recognize her voice. He wasn't good at that sort of thing.\n\nNow if he could touch her hips...\n\nHe shook off that thought, too. Whatever Olivia knew about his dance partner and wasn't saying, it didn't involve this attractive slug-hunter in scarf and muddy clothes.\n\nDefinitely not the same turquoise eyes.\n\nWith one smooth movement, Phoebe pulled off her scarf and gave her curls a shake once they were free. She seemed natural, unselfconscious. In her element, he thought.\n\n\"Well, if you do decide to stay on,\" she said, \"we'll make sure you're not bored.\"\n\nNoah felt his eyebrows go up and heard Dylan give a little cough behind him.\n\n\"Phoebe's the town librarian,\" Olivia added quickly.\n\n\"She can keep me in reading material, then.\" Noah smiled at Knights Bridge's redheaded librarian. \"Nothing like a good book.\"\n\n\"That's right. I love to read. I meant to tie a hammock to two shade trees in my backyard this summer but I haven't gotten to it. Reading a good book in a hammock in the shade\u2014doesn't that sound like the perfect summer afternoon?\"\n\n\"Provided the hammock is tick- and slug-free,\" Noah said mildly.\n\nPhoebe laughed. \"Definitely.\" Her gaze steadied on him, her face angled so that the sunlight brought out the gold highlights in her hair and the spray of freckles on her nose and cheeks. \"Were you at the masquerade ball last night?\"\n\n\"Yes, I was. Were you?\"\n\nShe waved a hand. \"My youngest sisters and I helped Olivia and Maggie with their costumes.\"\n\n\"More O'Dunn sisters?\" Noah asked.\n\n\"There are four of us. Ava and Ruby are twins. They're home for the summer but they're in graduate school. They're studying theater. They'd have gone last night if they'd had the time.\"\n\n\"Maggie seemed to enjoy herself,\" Olivia said.\n\nPhoebe smiled. \"Oh, good. She'll be back home soon. I can't wait to hear how her night on the town went. I'm sure it was a fantastic evening.\"\n\nNoah sat back on his chair. There. He could rule out Phoebe O'Dunn and her turquoise eyes. Not that he'd seriously considered her, given her penchant for gardening and hammocks and her willingness to take on slugs. That didn't fit with his princess.\n\nSo what did Olivia know that she wasn't saying?\n\nNoah ground his teeth. What was he doing? The woman he'd danced with and then abandoned last night was as much a fantasy for him as her swashbuckler was for her.\n\nHe was no swashbuckler.\n\nRight now he was just bored. After the months of tension and the grueling pace of taking his company public, he finally had time to come up for air. He'd even looked forward to a few days hiking in the White Mountains.\n\nHis princess was a mirage, and maybe so was his mystery man.\n\nHe needed to head back to San Diego before he conjured up real trouble.\n\nExcept he didn't want to go back.\n\nHe felt his spine stiffen at the thought of staying in Knights Bridge. On the one hand, it felt like the right thing to do. A fun distraction. On the other hand...it was totally insane. The only person he really knew there was Dylan, and Dylan and Olivia were joining him on the flight back to San Diego tonight. She wanted to see NAK and her fianc\u00e9's home.\n\nNoah watched Phoebe O'Dunn kick a chunk of dried mud off one sandal and found that he wouldn't mind getting to know Knights Bridge's slug-hunting librarian.\n\nThree nights in the White Mountains must have affected his brain cells.\n\nWhen he tuned back into the conversation, Olivia and Phoebe had started to cross the terrace, chatting as they disappeared through the screen door into the kitchen.\n\nNoah frowned at Dylan. \"What did I miss?\"\n\nDylan sighed. \"Phoebe rode her bike here. That's why we didn't see her car.\"\n\n\"Ah.\"\n\n\"This is why you study fencing and karate. They force you to stay focused. If your mind wanders, you get stabbed or punched.\"\n\n\"It's not that simple.\"\n\n\"Nothing is with you,\" Dylan said without any hint of criticism.\n\nNoah stood. Except for a few holes in their waxy-green leaves, the flowers where Phoebe O'Dunn had done her slug work appeared fine to him. He wasn't sure he'd ever seen a slug. He stepped off the terrace onto the soft grass. Buster rolled out from under the table and followed him, then settled onto his stomach in the shade.\n\n\"I could always watch Buster while you and Olivia are in San Diego,\" Noah said.\n\nDylan scratched the side of his mouth. \"Dog sit? You're kidding, right? You've never even owned a dog.\"\n\n\"What difference does that make? How hard could it be to keep an eye on Buster here? Feed him, water him, walk him. Done.\"\n\nDylan didn't bother to hide his skepticism. \"What are you thinking, Noah?\"\n\nHe didn't really know what he was thinking. Sometimes he came up with a solution before he had fully, consciously grasped the problem. Buster yawned, then stretched as he relaxed completely. \"What if I told you I want to make sure our mystery man isn't here?\" Noah asked.\n\n\"In Knights Bridge, you mean?\"\n\n\"Correct.\"\n\n\"Is that a hunch, or do you have evidence he could be here?\"\n\n\"It's not even good enough to be a hunch.\"\n\nDylan didn't respond for a moment. They'd had similar conversations many times over their long friendship. \"Is there any possibility this guy is connected to my father?\"\n\n\"For all I know he's connected to the man on the moon.\" Noah reined in his frustration. \"I have zero to go on. I only know what I've told you.\"\n\n\"It's not enough.\"\n\n\"No, it's not.\"\n\nOlivia came out the back door. \"Phoebe just left. Did you guys say anything to her?\"\n\nDylan shook his head. \"I didn't. Noah?\"\n\n\"We talked slugs,\" Noah said. \"Why?\"\n\n\"She just seemed quieter than usual. I'm sure it's nothing. Noah, did I hear you say you want to dog sit?\"\n\n\"Sure.\" He realized he was at least semi-serious. He squatted down and patted Buster. \"Your pup and I have bonded.\"\n\n\"But you're...\" Olivia looked at Dylan, then back at Noah as if she hadn't heard him right. \"Of course you're welcome to stay here, but Maggie can look after Buster. I mean...\" She seemed at a loss for words.\n\n\"You mean you don't understand why a billionaire would offer to dog sit,\" Noah finished for her, matter-of-factly.\n\nShe blushed. \"I guess that's what I mean.\"\n\nHe suspected Olivia was eager to change the subject. \"Your friend Phoebe seems very nice. I hope I wasn't rude.\"\n\n\"Not at all. Phoebe's cool. She doesn't ruffle easily. Maggie's like that, too, at least when she's working. She can be hotheaded otherwise. I wish Phoebe had come with us last night but she wouldn't. Long story.\" Olivia glanced back at the kitchen, then shifted again to Noah. \"If you're not ready to go back to San Diego, Dylan and I can stay here with you. We're flexible. I want to see Southern California, but it's not going anywhere.\"\n\nNoah stood up. \"Are you nervous about flying?\"\n\n\"Nothing I can't handle.\" She spoke half under her breath, as if reminding herself that she had her fear of flying under control and wouldn't let it stop her from doing what she wanted to do. \"If you really do want to stay, you're welcome to use one of the guestrooms here.\"\n\n\"Complete with vintage linens, from what Dylan has told me.\" Noah considered the graceful center-chimney house and surrounding acreage that comprised The Farm at Carriage Hill. \"It sounds perfect, Olivia. I actually could use some time just to hang out on my own.\"\n\nDylan grunted. \"No one would look for you here, dog sitting, that's for damn sure.\"\n\n\"The plane's gassed up and ready,\" Noah said. \"Go on. I'll join you as soon as I've had enough of country life.\"\n\n\"Before we've reached altitude, then,\" Dylan joked. His good humor evaporated quickly and he narrowed his eyes again on Noah. \"I don't like the idea of leaving you here alone with this guy on your tail.\"\n\n\"I won't be alone. I'll have Buster.\"\n\n\"There'll be guys working up the road at my place, too. The Sloans. Maggie's in-laws. They're handling the construction.\" Dylan grimaced, shook his head. \"I still don't like it.\"\n\n\"Amazingly, Dylan, my friend, I have managed not just whole hours and days but whole weeks and now even months without you down the hall from me. I'll be fine.\"\n\nBuster roused himself and stood by Olivia. He seemed aware on some level that she was about to leave him with a stranger who knew next to nothing about dogs. She scratched his big head. \"Staying here won't be what you're used to, Noah. Not that I know what you're used to, but I'm not...\" She made a face but smiled through her discomfort. \"Do people often get tongue-tied when they try to talk to you?\"\n\nDylan answered before Noah could. \"Only until they learn he got perfect scores on his SATs and graduated a year early from MIT. Then it gets interesting. A billion in the bank's nothing compared to being good at math.\"\n\nNoah smiled at Olivia. \"Ignore him. He's been like that since kindergarten.\"\n\n\"This place is coming together,\" she said, \"but it's not done yet. There's so much more to do. So many possibilities.\"\n\n\"You love it here,\" Noah said.\n\n\"I do. As I mentioned, Maggie will be in and out, if you don't mind.\"\n\n\"It's your house.\"\n\n\"She can help you with anything you need. Phoebe can, too. She's always willing to help people, and she knows at least as much about what's going on in town as her mother does. Elly works at the town offices. She's a ball of fire, if you run into her. She'll tell you anything you want to know about goats and then some.\"\n\n\"Good to know,\" Noah said with a sideways glance at Dylan. Goats. They were discussing goats again.\n\n\"Sure you want to stay?\" Dylan asked with a grin.\n\nOlivia ignored them both. \"Noah, if you want to keep your presence here a secret, Maggie and Phoebe won't say anything. People in town are aware of who you are, because of my relationship with Dylan, but they'll assume you're in San Diego. They won't recognize you. For one thing, they'd expect you to have an entourage.\"\n\nNoah was amused. \"An entourage,\" he said.\n\n\"You know.\" Olivia shrugged. \"People. Bodyguards and chauffeurs and valets. That sort of thing.\"\n\nHe'd had a chauffeur and a bodyguard from time to time, but never a valet. He'd always managed to get himself dressed. He might have danced with a woman pretending to be a princess, but he wasn't a prince.\n\n\"The women he dates have entourages,\" Dylan said. \"Noah's just a guy with a lot of zeroes in his net worth.\"\n\nNoah rolled his eyes but addressed Olivia. \"Your fianc\u00e9 is dangerous when he starts talking numbers.\"\n\nShe laughed. \"I can see how you two were a good pair at NAK and have stayed friends for so long. I need to run into town. I want to see Maggie before we leave for the airport. Noah, you can hang out with Buster and see how it goes. There's still time to change your mind.\"\n\nAs she went back through the kitchen, Noah stepped back onto the terrace and sat at the table. He heard bumblebees buzzing in purple flowers behind a green-painted bench. They weren't chives. Some kind of mint, he thought. If he stayed in Knights Bridge for more than forty-eight hours, he probably would learn all about New England flowers and herbs.\n\n\"Do you think Olivia will fall in love with San Diego?\" he asked.\n\nDylan's eyes darkened. \"I hope so, but Knights Bridge is where she belongs. It's home for her. It always was, even when she was in Boston.\"\n\n\"You belong here, too,\" Noah said. \"You know there's ice here in the winter, right? Also in the fall and spring, and probably at times in the dead of summer, too.\"\n\n\"You're a riot, Noah,\" Dylan said.\n\n\"Think Buster needs a walk?\" Noah asked.\n\n\"You're the dog sitter. You tell me.\"\n\nNoah frowned at the big dog. Hell if he could tell. \"Think I've bitten off more than I can chew?\"\n\n\"Way more.\"\n\n\"Well. I need a walk, and I'll bet I can talk Buster into coming along. What about you?\"\n\nDylan shook his head in bemusement. \"I'll get Buster's leash.\"\nSix\n\nPhoebe had her gorgeous brown silk gown zipped into a garment bag and hung in a closet before Maggie, wearing a black sundress that drained what color there was in her face, arrived with the outfits she and Olivia had worn.\n\n\"Olivia's getting cold feet about going to San Diego,\" Maggie said as she entered Phoebe's small living room. \"I told her she has to go so she can bring the boys stuffed giraffes from the zoo.\"\n\n\"She'll go,\" Phoebe said. \"It's just preflight jitters.\"\n\n\"I hope so. The ball was incredible last night. You'd have loved the costumes.\"\n\nMaggie laid the Audrey Hepburn and Grace Kelly dresses on a loveseat angled against the front windows, the afternoon summer sun streaming through filmy curtains.\n\n\"Then you and Olivia had a good time?\" Phoebe asked.\n\n\"Olivia especially did. She's so much more comfortable in her own skin than she was when she moved back to town. Your dresses were a hit.\" Maggie gave her Grace Kelly gown a lingering look, as if she was thinking about last night and what might have been. She smiled stiffly at Phoebe. \"What have you been up to?\"\n\n\"I just came from Olivia's. The basil needs to be picked. She won't have time before she goes to San Diego. I thought maybe you and I could make pesto or something.\"\n\n\"Sure. That'd be great.\"\n\n\"She and Dylan arrived with Noah Kendrick,\" Phoebe said, wanting to do something\u2014say something\u2014to penetrate her sister's pensive mood. \"Did you meet him last night? He's not what I expected.\"\n\n\"In what way?\"\n\n\"I don't know. I guess I thought he'd be fidgety but he's not. He's...\" She thought a moment. \"Calm, I suppose.\"\n\n\"He and Dylan went straight from hiking in the White Mountains to the masquerade last night. He's probably tired, but it wouldn't matter. He strikes me as calm, too. Centered. I expected such a genius tech type to be a little weird, but he doesn't come across that way.\" Maggie smoothed a few wrinkles in the dress with her fingertips then stood back, marginally more cheerful. \"I should go. I promised the boys I'd ride bikes with them. I missed them last night.\"\n\nPhoebe followed her out to the porch, the afternoon still and warm. \"Maggie, are you okay?\"\n\n\"Just tired. I'm not used to nights on the town. What are you doing the rest of the day?\"\n\n\"I'll stop at Mom's later. Right now I'm going to put my feet up in the shade and read a book. Maybe a good swashbuckler tale.\"\n\nMaggie seemed to pull herself out of her own private thoughts and focus more on the conversation. \"There were swashbucklers at the ball last night.\"\n\n\"I'm sure there were,\" Phoebe said. She didn't want to lie outright but skirting the truth seemed just as duplicitous. She changed the subject. \"I'll have the dresses cleaned. We're going to use them in the fashion show. Maybe you and Olivia can model them.\"\n\n\"Only if you insist,\" Maggie said with a welcome laugh. \"Olivia and I will pay for the cleaning. We just didn't want to do anything without checking with you first. It was quite an experience last night. You'll get the Edwardian gown cleaned at the same time?\"\n\nPhoebe felt her heartbeat quicken. \"What Edwardian gown?\"\n\nHer sister paused on the top porch step and turned to her. \"The deep brown sequined Edwardian gown you wore last night. It was you, wasn't it?\" When Phoebe didn't respond, Maggie nearly choked. \"Phoebe! It was you! I was just testing. I wasn't really serious.\"\n\nPhoebe groaned. \"Maggie...\"\n\n\"I didn't know you could dance like that.\"\n\n\"I can't. The man I danced with can. I have no idea who he is. Please don't say anything, Maggie. I decided to go at the last minute. I know I could have called you and Olivia but if I had, I'd have chickened out and stayed here, or turned around halfway to Boston.\" Phoebe took a breath, trying to control the tumble of words, the feeling that she'd just been caught doing something embarrassing and dumb. \"I'd have been too self-conscious if you two knew. I'd never have gone through with it.\"\n\n\"Why? You were gorgeous. Really. People couldn't take their eyes off you.\"\n\n\"That's kind of you to say\u2014\"\n\n\"It's true. I saw you and wondered if it was you. Then you disappeared, and I was distracted.\"\n\nProbably by Brandon, Phoebe thought. \"You can't tell anyone I was there last night. I'd just die if anyone knew, and sneaking in like that just makes it worse. I wasn't myself.\" Phoebe looked at the pink roses trailing up her white-painted trellis. This is my life. She got her breathing and heartbeat under control and turned back to her sister. \"There is one thing you could do for me. I'd like to get a message to the man I danced with. I wonder if Dylan might know him.\"\n\nMaggie frowned. \"What kind of message, Phoebe?\"\n\n\"As I was leaving, I overheard another man talking on the phone about him. Not by name but by his description...\" She paused, removing a folded piece of paper from her dress pocket. \"After I got home last night, I typed up everything the man said.\" Except, she thought, what he'd said about finding out who her swashbuckler's dance partner was. She didn't want any reason for anyone\u2014Dylan, Olivia, her sister\u2014to worry about her. She thrust the note at Maggie. \"I was going to give it to Dylan when he got back from Boston, but I chickened out.\"\n\n\"Because you don't want him to know you were there last night,\" Maggie said with some sympathy.\n\nPhoebe nodded. \"I didn't tell Olivia, either. I got so carried away dancing. If I'd just slipped in, had a few hors d'oeuvres, checked out the costumes and left, that'd be one thing. But I didn't.\"\n\n\"You had a good time, Phoebe. You didn't make a spectacle of yourself, if that's what you're thinking.\"\n\nIt was exactly what she was thinking. \"I just don't want to have to explain. It was a moment, and it's over.\" She smiled. \"I'm back home where I belong.\"\n\n\"Right. I understand, Phoebe.\" Maggie held up the folded paper, a spark in her eyes now. \"What's in the note? Anything juicy?\"\n\n\"I only heard one end of the conversation so I don't have the context for what was said. It struck me as provocative. Like this guy had an ax to grind with my swashbuckler. He'd already disappeared on me, so I just got out of there. Then I thought about it and realized I probably should have found him and told him what I overheard. I hadn't wanted to chase him down if he didn't want...you know.\"\n\n\"You mean you didn't want to go find him if he'd ditched you,\" her sister said.\n\nPhoebe felt her cheeks flame. \"That's what I mean, yes.\"\n\nMaggie sighed. \"I can't say I blame you.\"\n\n\"When I got home I decided I should write everything down.\"\n\n\"Was there an implied or direct threat in what you overheard? Were you afraid? We can always talk to Eric.\"\n\nEric Sloan was Brandon's eldest brother, a town police officer. \"I'd have grabbed a security guard if I'd felt threatened. It wasn't anything that overt. Really, for all I know my swashbuckler was stepping out on his wife last night and she sicced this guy on him.\" Phoebe gave a small, thin laugh. \"That'd be right up my alley, wouldn't it? Not that I'll ever see him again.\"\n\n\"Phoebe\u2014\"\n\nShe held up a hand, stopping her sister. \"No, don't. Don't tell me anything you know, anything you suspect. I want to forget last night. I've assuaged my conscience by writing the note and giving it to you to give to Dylan. Let him think someone tucked it in your dress and you only just found it.\"\n\n\"You mean I lie to him instead of you lying to him?\" Maggie grinned suddenly, tucking the note in her pocket. \"I'm proud of you, Phoebe. I didn't think you had it in you to be a little devious.\"\n\n\"I just don't want this to become a thing. You can read the note. If you think I'm overreacting and it's not worth giving to Dylan, just toss it out your window.\"\n\n\"And risk having some big gossip in town find it? No way. Did you include a description of this man you overheard?\"\n\n\"I did, yes.\"\n\n\"You're a regular James Bond.\"\n\nPhoebe was relieved to see Maggie more animated, even if it was at her own expense. \"Whatever is going on, it has nothing to do with me. I was just a curiosity.\"\n\n\"Because of that killer dress, not to mention a couple of dips you took while dancing\u2014\"\n\n\"Maggie, please. Don't.\"\n\nHer sister tilted her head back, studying Phoebe in the late-afternoon light. \"This swashbuckler of yours really got to you, didn't he?\"\n\n\"We had a moment and now it's over, as well it should be. It wasn't real. For me, or for him.\"\n\nMaggie nodded. \"Okay. If you say so. Meet you at Mom's in a bit?\"\n\n\"Sure. Maggie, last night\u2014\"\n\n\"It was a strange night for both of us, Phoebe. Let's just leave it at that.\"\n\nBrandon, Phoebe thought, but she said nothing as Maggie trotted down the porch steps and back out to her car, at least in a better mood than when she'd arrived. Brandon had to have been the reason for her melancholy. Phoebe tensed, wishing she could pick up the phone and give her brother-in-law a piece of her mind. She might not always speak up for herself\u2014she wasn't by nature a bold person\u2014but she would defend her sisters.\n\nOnly Maggie had told Phoebe, Ava, Ruby and their mother to stay out of the problems between her and Brandon, and she was totally right to do so. They'd never really discussed the details of what had driven Maggie back to Knights Bridge without her husband.\n\nPhoebe sat on a wicker chair and breathed in the warm summer air, scented with roses. Why hadn't she wanted Maggie to tell her what she might know\u2014any suspicions she might have\u2014about her dance partner?\n\nMaybe, Phoebe thought, it wasn't just that she was worried that, unmasked, she'd be a disappointment in real life. Maybe she was worried he would be, too.\n\nShe liked the fantasy of last night, she realized.\n\nShe didn't want it to end.\n\n* * *\n\nMaggie was close to hyperventilating as she arrived at Carriage Hill, all but screeching to a stop on the side of the narrow back road.\n\nWhat was she going to do? That was Phoebe last night. With Noah Kendrick.\n\n\"My sister,\" she said aloud.\n\nShe had to calm down. It was one night. That was all there was to it. There was no budding relationship between her sister and Noah Kendrick.\n\nMaggie pushed open her van door with a groan. There'd been sparks between them, though.\n\nA lot of sparks.\n\nShe jumped out and headed for the kitchen ell, a newer addition to the pretty antique house. She peered through the screen door but didn't see anyone and let herself in. Her breathing more or less back under control, she went through the mudroom out to the terrace.\n\nNoah Kendrick sat alone at the round table. He was as still as a statue, dressed head-to-toe in black.\n\n\"Where's Olivia?\" Maggie asked before he could say a word.\n\n\"Upstairs packing, I believe.\" He turned to her with an enigmatic smile. \"Hello, Maggie.\"\n\n\"Hello. Sorry. I'm in a whirlwind.\"\n\n\"You enjoyed your night out in Boston?\"\n\n\"I did, yes.\" She didn't dare ask him if he'd enjoyed his. He was smart, rich, experienced. He'd see right through her. \"Olivia's packed for San Diego? She and Dylan are going back with you? She hasn't bailed on you, has she?\"\n\n\"They're flying to San Diego together. I'm staying here.\" Noah's expression didn't change. \"I'm dog sitting.\"\n\nMaggie gulped in air. Dog sitting? Was he serious? She really was going to pass out if her family and friends kept throwing curveballs at her. Did Noah know about Phoebe? Was that why he was staying? Had to be, Maggie thought. Why else would a billionaire dog sit in little Knights Bridge, especially with his best friend in San Diego?\n\nShe gave herself a mental shake. Maybe Noah was staying because of the mystery man Phoebe had overheard. Of course Maggie had read the note. Phoebe had typed up her transcript and printed it, probably so no one could recognize her handwriting. She was thorough like that. Maggie would have thought of such a cover-up only after the fact.\n\n\"You're staying here alone?\" she asked Noah.\n\n\"I'll have Buster for company.\"\n\nBuster was a great dog but he didn't qualify as company for a billionaire. But what did she know about billionaires? \"Do you think Olivia will love it in San Diego?\"\n\nNoah seemed surprised by her question, as if it would never occur to him to ask. \"Dylan's place on Coronado is very nice. It's a great location. You can see the Pacific from almost every window.\"\n\n\"Sounds lovely. Brandon and I went to California right after we got married. He's always loved to travel. Well, I can't stay long. I promised the boys I'd ride bikes with them. We live in the village.\" Why couldn't she calm down? Noah was already looking suspicious, as if he could read her mind and knew she wasn't being entirely straight with him. Not straight at all, in fact. \"I should go find Olivia.\"\n\n\"She said you'd be working here some of the time.\"\n\n\"That's right. Do you like herbs, Noah? I'm thinking about trying some new recipes.\"\n\nHis gaze, already steady, leveled on her. \"Just be sure no slugs end up in the pot with them. I met your sister earlier. She'd been on slug patrol.\"\n\n\"My sister? I have three sisters\u2014\"\n\n\"Phoebe.\"\n\nHe gave no hint of recognition but he struck Maggie as a man of supreme self-control. She tried not to choke. \"Phoebe loves to dig in the dirt. When she's not reading a book in the shade, that is. I'll...um...\" She cleared her throat, wondering if she was purple from not breathing properly. \"I'll watch for slugs. See you soon.\"\n\nShe welcomed the cooler air inside the house and ran through the living room to the stairs that led to the second floor. She paused to catch her breath and look where she was going. She tripped up Olivia's narrow, steep stairs half the time, even on a good day.\n\nDid Phoebe have any idea that it was Noah last night? Noah clearly had no idea it had been her. Maggie got that. Phoebe in shorts and a T-shirt, poking around for slugs. Not exactly the image she'd presented at the masquerade. He'd be hoping for...well, not for Phoebe O'Dunn. Maggie felt a surge of resentment that anyone could reject her older sister, but she also had to admit that Phoebe at home in Knights Bridge wasn't at all like the woman in the Edwardian gown last night.\n\nBrandon Sloan as a pirate, however\u2014that matched who he really was.\n\nMaggie mounted the stairs carefully, using the handrail. She'd gotten caught up in the fantasy of the charity ball herself. She'd danced with Brandon, pretended that all the frustration and pain of the past year didn't exist. It was a moment, no more real and lasting than what Phoebe had experienced.\n\nShe found Olivia zipping up her suitcase in her bedroom at the front of the house. \"We have to talk,\" Maggie said in a low voice.\n\nOlivia stood, pushing her fair hair out of her face. Maggie shut the door behind her. \"It was Phoebe last night,\" she said without further preamble.\n\n\"In the Edwardian dress?\"\n\n\"The woman who danced with Noah Kendrick, Olivia. It was Phoebe.\"\n\nOlivia sank onto the edge of her bed. \"I was afraid of that.\"\n\n\"We can't tell anyone. I'm only telling you because Phoebe doesn't know it was Noah and Noah doesn't know it was Phoebe, and I'm not going to be the one to spill the beans, accidentally or on purpose. So you have to give Noah the note.\"\n\n\"What note?\"\n\nLeave it to Olivia to cut through everything else and focus on the crux of the matter at hand. Maggie pulled the note out of her dress pocket and handed it over. \"You can read it but I wouldn't if I were you. It's the transcript of one end of a phone call of some guy talking about Noah.\"\n\nOlivia winced. \"Middle-aged? No costume?\"\n\n\"Yeah. He had on an expensive suit. Why?\"\n\n\"Noah spotted him a few times in San Diego and now he seems to have followed him east. Maggie, he and Dylan will want to know who wrote this note.\"\n\nMaggie stiffened. \"I don't know who wrote it.\"\n\n\"You do, too,\" Olivia said, as if they were seven again. \"It was Phoebe, wasn't it?\"\n\n\"I didn't say that. I just said it's a transcript.\" Maggie waved a hand. \"Tell Noah and Dylan I found the note on the ballroom floor, or someone tucked it in my pocket while I was sipping champagne. I don't care. Just don't mention Phoebe.\"\n\n\"But, Maggie\u2014\"\n\n\"Olivia. I mean it. You have to promise. I talked to Phoebe. She doesn't know anything.\"\n\nOlivia sighed, clearly pained. \"I can't keep things from Dylan.\"\n\n\"You're not keeping anything from Dylan. I'm telling you right now that I don't know who wrote that note.\" Unlike her eldest sister, Maggie had no compunctions about a strategic white lie. \"You just can't say that Phoebe was Noah's princess. That shouldn't be hard. It's not as if Dylan would ask or it would make any difference in finding this guy tailing Noah.\"\n\nOlivia leaned back on her elbows, frowning, obviously contemplating the knotty situation into which Maggie had just thrust her. \"You, Phoebe and I are smart. We've slayed our share of dragons. But Dylan and Noah...\"\n\n\"I know. They're in a different league when it comes to dragon-slaying. If they think we're hiding something about this guy, they have a right to be annoyed. But we're really not.\"\n\n\"I'm not saying they'll be annoyed. I'm saying they'll find out who wrote that note.\"\n\nMaggie suspected her friend had a point and wished Phoebe had danced with someone else last night, or simply hadn't overheard the man in the coatroom.\n\nShe stood by a window, its simple curtain billowing slightly in the afternoon breeze. \"Olivia, do you think Noah would hire someone to find out the identity of his princess from last night?\"\n\n\"Hire someone? Who?\"\n\n\"I don't know. A private investigator or something.\"\n\nOlivia sat up straight. \"Now you're getting carried away. I guess we both are. It's probably because we didn't expect...\" She sighed, shaking her head, then fastened her green eyes on Maggie. \"It really was Phoebe last night?\"\n\n\"It really was.\" Maggie glanced out the window, down at the shaded front yard and quiet back road. \"Why do you think Noah is staying here? It's not to dog sit. Do you think he suspects his princess is in Knights Bridge?\"\n\nOlivia eased up off the bed and stood her suitcase on end on the polished wide-board pine floor.\n\nMaggie narrowed her eyes on her friend. \"Olivia?\"\n\n\"I think I may have given him the impression that she's here. He mentioned that his princess's eyes reminded him of the color of your eyes. Not that he thought it was you he danced with. I don't think he assumes she was related to you. Just that I know more than I've let on, which now I do, for sure. I thought maybe it was Ava or Ruby, only because Phoebe was so adamant about not going to the ball.\"\n\n\"She prides herself on paying her own way,\" Maggie said.\n\nOlivia nodded. \"I can appreciate that.\"\n\n\"Phoebe feels she always has to be the sensible one. It's because the rest of us are all nuts. My mother and the twins and me. We're dreamers. We always have a bunch of different things going on at once. We've never been good with money. It's bad enough that I'm like that, but then I married a guy who's like that, too.\"\n\n\"You've done right by yourself and the boys,\" Olivia said. \"That's what matters.\"\n\n\"I'm always afraid I'll go just that one step too far in my catering business, and it'll all come crashing down on my head. I tell myself that I'd work it out if I did make a mess of things.\" Maggie forced a smile. \"We'd just move in with my mother and the goats.\"\n\nOlivia groaned, but her mood was noticeably lighter. She fingered the handle of her suitcase. \"I can postpone this trip to San Diego.\"\n\n\"No, that would only look suspicious, and you need to get out there and see where Dylan's from, what his life before he met you was like. You'll love it. I remember Brandon and I had pi\u00f1a coladas at the Hotel del Coronado. We didn't stay there. Too expensive. Noah said Dylan's house is just up the street.\"\n\n\"Dylan warned me it has no color. Everything in it is white, cream or cappuccino.\" Olivia laughed, only a slight strain in her voice as she continued, \"I'm sure it's grand, but Knights Bridge has its charms.\"\n\n\"You two can always have a bicoastal life if you want to. You'll figure it out. No angsting, okay? Just go and enjoy yourself.\"\n\n\"I will. Promise. You'll keep an eye on Noah and Phoebe?\"\n\n\"There is no Noah and Phoebe, Olivia.\"\n\n\"You know what I mean.\"\n\nMaggie blew out a breath, calmer if no more certain of what was going on. \"Noah must have all kinds of help in San Diego. Think he'll manage here on his own?\"\n\n\"He's a genius. He'll manage.\" With a welcome smile, Olivia added, \"Besides he'll have Buster.\"\n\n\"Maybe he can keep that mutt of yours from digging in the garden.\" Maggie wasn't one of Buster's big fans, but she noticed Olivia's smile fade and knew she was thinking about the long flight across the country. \"Liv, come on. You flew to England and back with Dylan not that long ago. That went fine. This will, too.\"\n\n\"I know it will but I still...\" Olivia didn't finish, just stared at her suitcase.\n\n\"I know. It's okay.\"\n\n\"Yeah.\" She looked up, smiled. \"Noah already thinks I'm hiding something. I don't want to end up drawing attention to Phoebe by suddenly bailing on the trip. Does she regret last night?\"\n\nMaggie thought a moment, then shook her head. \"No, I don't think she regrets it one little bit. She just doesn't want to get caught.\"\n\n\"If she knew it was Noah Kendrick\u2014\"\n\n\"I'm not telling her and neither are you, because you, my friend, are going to be in San Diego. Bring me back two stuffed giraffes from the zoo.\" She grinned at her longtime friend. \"Not life-size ones.\"\n\nThey chatted a few more minutes, deciding on a plan of action to deal with Phoebe's note, and when Maggie headed back downstairs, she noticed that Dylan had joined Noah on the terrace. They seemed relaxed. She left them to their drinks in the shade and went back to her van.\n\nShe rolled down the windows and listened to the water flowing over rocks in the brook across the road, the twitter of birds hidden in the trees. Was she focusing on Phoebe's situation with Noah as a distraction from thinking about Brandon, or did she have good reason to worry about her sister?\n\n\"Time will tell,\" she muttered, starting her van.\n\nShe felt only mildly guilty for not mentioning her encounter with Brandon last night. Olivia would want to know but she was just getting a handle on her anxiety over flying, and she didn't need more excuses to keep her from making this trip to San Diego.\n\nSo it was good, Maggie decided, that she'd kept quiet about that dance with her pirate of a husband.\n\nAfter all, what were friends for?\n\nShe sniffled, suddenly wanting nothing more than to go bike-riding with her sons, listen to their tales of their overnight with their maternal grandmother. When Maggie had checked in that morning, her mother had them picking tomatoes for fresh homemade sauce. Nothing like the prospect of spaghetti for supper to motivate them.\n\nAidan had found a spider. Tyler was looking for snakes.\n\nThey were having the time of their lives, all that saved Maggie from wishing she'd done anything last night besides dance with their father, as sexy and irresistible as ever.\nSeven\n\n\"Dog sitting, Noah?\" Dylan sat at the round terrace table with a glass of spiced iced tea, complete with an orange wedge and cinnamon stick. \"Are you sure about this?\"\n\nNoah, seated on a green-painted bench, ran a palm over the minty-looking plants with the purple flowers. He wondered if they had slugs. He noticed the bumblebees were back and withdrew his hand. \"It'll be easier on everyone if I stay,\" he said, although he wasn't precisely sure what he meant.\n\nObviously neither was Dylan, who gave Noah a skeptical look. \"Having you here on your own will be easier on everyone? How?\"\n\n\"I can tend to the gardens, as well as Buster.\"\n\n\"Have you ever done any gardening?\"\n\nIn fact, Noah had not. Even as a kid, he'd never done more than water the flowerpots on the deck of his parents' townhouse in suburban Los Angeles, under his mother's supervision.\n\nHe stood up with his own iced tea. Plain. No spiced tea for him. \"It'll be fine. Olivia will leave instructions. Remember, I recognized the chives on her note card when she wrote to you about your house. You thought they were clover. Anyway, you don't need me underfoot while you're showing her San Diego.\"\n\n\"I'm not buying it, Noah,\" Dylan said, shaking his head. \"You're not staying here because you've suddenly turned into Farmer Kendrick. It's that note Olivia gave you.\"\n\nSuddenly restless, Noah walked over to the table but didn't sit down. The afternoon had turned hazy and humid. Olivia and Dylan would be on their way to the airport soon. Noah wouldn't be going with them. He'd come close to changing his mind about staying. Then Olivia had handed him the note, telling him that he wasn't to ask where she got it.\n\n\"It was given to me in confidence,\" she'd said.\n\nMostly likely that meant her friend Maggie O'Dunn had given the note to her. But where had Maggie gotten it? Had she written it herself? Based on the contents, Noah doubted it. Maggie would simply have pulled him aside and told him what she'd overheard.\n\nThe mysteries of little Knights Bridge.\n\nThe note sealed his decision to stay on at least for a few days. He had no intention of pushing Olivia to break a confidence or telling her what he suspected. Nor would he involve Dylan, given that he was engaged to Olivia.\n\n\"I read the note, Noah,\" Dylan said.\n\nNoah sighed. \"When I was walking Buster up the road, testing whether he and I would get along for a few days?\"\n\n\"That's right. I figured you didn't want to hand the note to me outright but you wanted me to read it.\"\n\n\"Actually, I was just focused on keeping Buster from slobbering on me and didn't think about the note. I didn't want to involve you.\"\n\nDylan set his iced tea on the table. \"Noah, Olivia gave you that note. I am involved.\"\n\n\"That doesn't change anything,\" Noah said.\n\n\"The note was printed off a computer. Whoever wrote it didn't want you to see his or her handwriting.\"\n\n\"Could it have been Brandon Sloan?\"\n\n\"No,\" Dylan said without hesitation. \"He'd just have told you on the spot.\"\n\nNoah agreed. He sank onto a chair at the table, half wishing he were back in the White Mountains with nothing more pressing on his mind than survival. This was much more complicated. He looked over at his friend. \"You know it would take two seconds to find out what Olivia and Maggie have been up to,\" he said.\n\nDylan grimaced as he nodded. \"What makes you think Maggie's involved?\"\n\n\"She's always involved, isn't she?\"\n\n\"True. She and Olivia go back at least as far as you and I do.\" Dylan picked the cinnamon stick out of his tea and flicked it into the grass. \"I'm not going to spend a lot of energy guessing. The note says what it says. The person who overheard the man in question provided a thorough account and description. There's nothing more to add.\"\n\n\"What about my princess?\" Noah asked, his tone more serious than he'd intended.\n\n\"You mean is she a bystander? A potential victim? A potential accomplice?\"\n\n\"The note doesn't mention her, and yet this man described my presence at the ball in detail. If that had been you, don't you think you'd have mentioned my dance partner in the Queen Victoria dress?\"\n\n\"It was Edwardian,\" Olivia said, coming out from the kitchen. \"Titanic, Downton Abbey.\" She swallowed as if she might have gone too far. \"I noticed her dress. It was gorgeous.\" She added quickly, \"There were a lot of gorgeous dresses there last night.\"\n\nAs tempted as he was, Noah reminded himself that he wasn't going to grill his best friend's fianc\u00e9e about what she was holding back.\n\nDylan watched her walk past them into her gardens, mumbling about dealing with the basil before it went to seed. He glanced at Noah. \"Not a word.\"\n\n\"Nope. Not me.\" Noah put his feet up on another chair and settled back in the late-afternoon warmth. \"I don't know which I want more\u2014the identity of this stalker or of my princess. Wouldn't you have said that was a Victorian dress?\"\n\n\"I'd have said it was a dark blue dress.\"\n\n\"It was dark brown, Dylan.\"\n\nHe shrugged. \"I'm not big on colors.\"\n\n\"And you're engaged to a graphic designer who loves color?\"\n\n\"A case of opposites attracting, at least on that one. We have other things in common.\" His gaze was fixed on Olivia, kneeling in a sunny herb patch, checking what Noah assumed was basil. Finally Dylan added, \"Olivia and I are good together, Noah.\"\n\n\"No question about it. I'm happy for you.\"\n\n\"So, do you think she knows who your princess is?\"\n\nNoah debated answering, then said, \"Yes, I think she does.\"\n\nDylan sighed. \"I'm betting she does, too.\"\n\n\"A friend from Boston, maybe?\"\n\nHe shifted his gaze to Noah. \"I doubt it.\"\n\n\"Then a friend from Knights Bridge?\"\n\n\"It doesn't have to be a friend. She knows everyone in town.\"\n\nNoah looked up at the sky and contemplated the cloud formations. \"If my princess is from Knights Bridge, and Olivia and Maggie don't want to tell me\u2014\"\n\n\"Then you need to forget about her,\" Dylan said.\n\n\"Meaning they'll never give her up and they'll never forgive me if I find her and she doesn't want to be found.\" Noah dropped his feet onto the stone terrace and sat up straight. \"She wrote the note.\"\n\n\"Who? Olivia?\"\n\n\"My princess.\"\n\nDylan got to his feet. He looked pensive, tight.\n\n\"I'm not speculating, Dylan. I'm as certain about this as I was about starting my own company\u2014about knocking on your window when you were sleeping in your car. She wrote that note and got it to you because she thought you might know who her swashbuckler was and could get it to him.\"\n\n\"If that's the case, she took great pains to conceal her identity.\"\n\n\"Otherwise she would have just handed you the note herself, or Maggie and Olivia would have told you who she is.\"\n\n\"Maybe she doesn't want her swashbuckler to know who she is.\"\n\nNoah ignored the amusement in Dylan's voice. \"Olivia and Maggie know it was me dressed up like a musketeer last night. I don't think they've told her. Olivia said she was to give you the note because you might know the identity of the swashbuckler mentioned in the conversation. You invited a fair number of the guests, after all.\"\n\n\"She wasn't asked to get it specifically to Noah Kendrick. You, in other words.\"\n\n\"Right. No name.\"\n\nOlivia moved to another cluster of herbs. Noah didn't think she could hear the discussion between him and Dylan but suspected she had a fair idea of what was on their minds. He was rarely confident of his ability to read body language with any accuracy. He really didn't know what Olivia was thinking, or even his best friend of nearly thirty years. In contentious board meetings, dealing with the occasional backstabber, the ever-present sharks in the water, Dylan was better at getting at what was going on beneath the surface. Noah tended to focus on what he wanted. For the past four years, what he wanted had centered on business.\n\nNot right now. Right now, what he wanted was his princess.\n\nHe got to his feet and stood next to Dylan. \"My princess last night doesn't know it was me behind the mask.\"\n\n\"Do you think she'd be disappointed to find out she danced with the founder of NAK?\"\n\nDylan spoke as if disappointment was unimaginable. Noah remembered the persona he'd adopted last night. \"I'm no D'Artagnan,\" he said.\n\n\"You're as good with a sword as any musketeer.\"\n\n\"That's different. Anyway, if this woman has information on my mystery man, then it could help that I'm just...you know. Me.\"\n\n\"Your average, garden-variety California billionaire,\" Dylan said with some humor.\n\n\"All right, maybe it won't help.\"\n\nHis friend groaned suddenly. \"Are you confused at all? A mystery woman, a mystery stalker, small-town loyalties...\" He held up a hand before Noah could answer. \"Never mind. I know you're not confused.\"\n\n\"Do you have a short list of possibilities of who my Edwardian princess might be?\"\n\nDylan looked uncomfortable. \"Noah...\"\n\n\"Ah. A very short list. You don't have to tell me. I won't compromise you with your new friends here. I like a good challenge.\"\n\n\"Then you still plan to stay?\"\n\nNoah hadn't changed his mind. Not even close. \"It'll be fine. Nobody knows me in Knights Bridge except Maggie and her sister.\"\n\n\"But you're determined to find out who your princess is,\" Dylan said.\n\n\"As much as ever. I'll just keep it to myself when I do. Relax, Dylan. Think of me as taking a few days to enjoy the bucolic surroundings.\"\n\n\"You don't like bucolic surroundings.\"\n\n\"I do. I just don't like mosquitoes. I'll wear bug spray.\"\n\nDylan was still obviously skeptical. \"You're sure you're not just bored?\"\n\n\"I was bored. I'm not now.\"\n\nOlivia started up a path toward the terrace. Dylan kept his eyes on her as he continued, \"Are you avoiding San Diego? It's an adjustment, going from controlling everything to do with NAK to\u2014\"\n\n\"Controlling nothing?\" Noah gave a small smile. \"No one's going to feel sorry for me, Dylan, and I don't feel sorry for myself. You and I are both moving on. We made our choices about how we'd take NAK public. We still have a strong interest in the company, but we wanted fresh blood. The last thing the new people need is the founders skulking around.\"\n\n\"Founder,\" Dylan corrected. \"Singular.\"\n\nNoah didn't argue with him. They'd had this argument countless times in the past four years. \"We're both pivoting to what's next for us. I just didn't expect you to fall for someone from a little town on the other side of the country.\"\n\nOlivia joined them on the terrace and smiled at Noah as if she'd told him all she knew about the identity of his princess. \"I love it here but you'll be roughing it by your standards.\"\n\nNoah returned her smile. \"Like I just told Dylan. I'll wear bug spray.\"\n\n* * *\n\nOn his first evening alone in Knights Bridge, Noah listened to an owl in the woods behind the house and chased a mosquito out of the kitchen. He'd have killed it if he'd had the opportunity but it followed him outside when he walked Buster.\n\nHe had no problem killing mosquitoes.\n\nHis assistant in San Diego had arranged for a messenger to deliver a new phone to Carriage Hill. Noah scanned his messages. Loretta Wrentham had called seven times. When he called her back, she was annoyed with him for not responding sooner but she had no news.\n\nHe sat down at Olivia's white-painted kitchen table. \"If you don't know anything, why did you call me seven times?\"\n\n\"Because you didn't return my first two calls.\"\n\n\"That makes no sense, Loretta.\"\n\n\"It makes perfect sense to me. I hate being ignored.\"\n\n\"I wasn't ignoring you.\"\n\n\"What were you doing?\" she asked.\n\n\"I didn't have a phone.\"\n\nA half-beat's silence. \"You didn't have a phone? Really?\"\n\nHe heard the skepticism in her voice. \"Really,\" he said. \"I only got your voice mails just now.\"\n\n\"Don't you have an assistant who checks your messages?\"\n\n\"Not one who checks my private number.\"\n\n\"So are you and Dylan about to head back out here?\"\n\n\"Dylan is already en route,\" Noah said.\n\n\"Noah...\" Loretta took in an audible breath. \"Where are you?\"\n\n\"Knights Bridge. I'm staying at Olivia's place. I'm dog sitting.\"\n\nSilence on the other end of the connection.\n\n\"I almost wish it were cool enough tonight for a fire,\" Noah added. \"I like Olivia's fireplace. Fireplaces, actually. There are five or six. They all share one chimney. It's in the center of the house.\"\n\n\"Dear God.\"\n\nHe smiled into the phone. \"Have you ever been to New England, Loretta?\"\n\n\"Boston. Knights Bridge isn't Boston.\"\n\n\"Not even close.\" But he didn't mind that, he realized. At least for now. In another few days, it might make all the difference in the world. \"Dylan and Olivia will be arriving in San Diego soon. Don't let him get involved in this thing. He needs to show Olivia around town, take her to the zoo. Stuff like that.\"\n\n\"You're a romantic at heart,\" Loretta said.\n\nNoah laughed. \"Yeah, right.\"\n\n\"Are you sure it's wise to stay out there by yourself? You have a higher profile right now with NAK going public. You need to take your safety seriously. You've had corporate security training and you know fencing and karate, but if this guy's actually stalking you\u2014\"\n\n\"I'm not worried, Loretta. I don't want you to worry, either. Let's just identify this man and figure out what he wants.\"\n\n\"That high-IQ mind of yours is working the problem. I can feel it all the way out here on the other side of the continent.\"\n\n\"The problem has gotten a bit more complicated since we last talked.\"\n\nShe sighed. \"Of course it has. Tell me.\"\n\nNoah explained about his princess and the note Olivia had handed him. \"I'm positive my princess overheard our guy and wrote that note but I can't prove it.\"\n\n\"She doesn't know who you are and you don't know who she is,\" Loretta said.\n\n\"But Olivia and her friend Maggie know both\u2014who I am and who she is.\"\n\n\"I get it, I get it. You want to find this woman and your stalker, and you think staying in Knights Bridge will help. How, I don't know, but you're the genius. What are you doing tonight, since it's too warm for a fire?\"\n\n\"I'm listening to an owl right now.\"\n\nMuttering, Loretta disconnected.\n\nNoah got up from the table and stepped past a slumbering Buster onto the terrace, the early-evening air still and warm, fragrant with flowers and herbs. He looked out at the stone walls, fields and hills silhouetted against the darkening summer sky. He'd never been to this part of Massachusetts during his college days. On breaks, he'd gravitated to the beaches or gone home to Los Angeles. Not ever\u2014not once\u2014had he considered that Dylan might end up in a small New England town. He'd discovered that he had roots in the Swift River Valley\u2014a grandmother he'd never known, a woman now in her nineties who'd given up his father at birth.\n\nHence Duncan McCaffrey's purchase of the house up the road and Dylan's presence in Knights Bridge.\n\nAs much as Dylan appreciated the answers he'd discovered last spring, Noah knew they weren't why his friend was still here. Dylan was in Knights Bridge because of Olivia Frost. If she fell in love with San Diego and wanted to live there part-time, he would do that. He had the freedom to make whatever came next for him work for her, too.\n\nThe Farm at Carriage Hill was charming and sophisticated, and Olivia had every reason to be proud of what she'd accomplished in such a short time. It wasn't a traditional bed-and-breakfast that took in the odd overnight guest, and there were no events scheduled during his stay. Maggie O'Dunn would stop by during the day but for the most part Noah would have the place to himself.\n\nWell, he and Buster would.\n\nOlivia had lined up several painting projects in case he got bored.\n\nShe had a sense of humor. Noah did a lot of things but he didn't paint.\n\nHe headed upstairs to choose a bedroom for his New England sojourn. Only one, a small bedroom overlooking the side yard, didn't involve antique lace.\n\nThat was the one he chose.\nEight\n\nPhoebe took the call from Maggie in her back garden. They'd planned to head over to Carriage Hill and deal with Olivia's basil\u2014make a nice Sunday afternoon of it\u2014but Maggie couldn't. \"Ava and Ruby got their wires crossed and neither one will be around today,\" Maggie said. \"Mom needs help with the goats, although, of course, she insists she doesn't. The boys and I will go over there and do what we can.\"\n\n\"Nineteen goats are too many for her,\" Phoebe said.\n\n\"One goat is too many,\" Maggie added in exasperation, then sighed. \"I know she loves the goats. She's never asked for any of us to help take care of them, but you know she'd never manage without us.\"\n\nPhoebe didn't disagree. \"Getting into goat's milk soaps could make a difference.\"\n\n\"She says she's looking into selling a few of the goats. She knows she has to. We don't need nineteen, even if the soaps do well.\"\n\n\"Let me know if I can do anything to help,\" Phoebe said.\n\n\"Oh, we'll manage. The boys are still young enough to think mucking out the stalls is fun. Enjoy your quiet afternoon. We'll make the pesto later this week.\"\n\n\"I have all the ingredients. I can head over to Olivia's and see how much I can get done on my own this afternoon.\"\n\nHer younger sister took in a sharp breath. \"Phoebe...\"\n\n\"It'll be okay, Maggie. I can follow a recipe. If I screw up the pesto, there'll be more basil.\"\n\n\"What about Buster?\"\n\n\"He and I get along just fine.\"\n\nMaggie started to say something else, but Phoebe assured her she'd manage the basil on her own and got off the phone, eager to be on her way on what was turning into a hot, humid afternoon. Perfect for making pesto, she thought as she went back inside.\n\nNot that she'd ever made pesto.\n\nGiven the heat, she pinned up her hair and changed into shorts, a sleeveless linen top and flip-flops.\n\nFifteen minutes later, she parked at Carriage Hill, grabbed her canvas bag of pesto-making ingredients and headed up the stone walk to the kitchen ell. Maggie would have been by early to see to Buster, and Phoebe expected to have to use the extra key Olivia kept hidden behind a gutter. Instead she found the main door to the kitchen open and Buster nosing the screen door.\n\n\"Hey, Buster, did Maggie forget to lock up?\" Phoebe pulled open the door and stepped past the big, rambunctious dog into the country kitchen. Buster went from nosing the screen to nosing her as she set the bag on the counter. \"Easy. You remember me. I'm Phoebe. Olivia's friend.\"\n\n\"I do remember you.\"\n\nPhoebe jumped, startled at the sound of a man's voice, coming from the adjoining living room.\n\nNoah Kendrick appeared in the doorway. \"Phoebe O'Dunn, the slug-hunter,\" he said with an enigmatic smile. \"Hello, Phoebe.\"\n\nShe subtly breathed out in relief. \"Noah\u2014hi. I didn't realize anyone was here. I thought Dylan and Olivia were on their way to San Diego.\"\n\n\"They are. I stayed behind.\"\n\n\"But you'll be joining them?\"\n\n\"Eventually,\" he said.\n\nHe wore a black T-shirt over dark jeans, and as he entered the kitchen Phoebe saw he was barefoot. He didn't make a sound, his movements smooth, controlled. She'd noticed that about him during their brief meeting yesterday. She could see him glued to a computer but at the same time she could see him\u2014what? Doing yoga, maybe. She did yoga herself, at least sort of, and always felt more physically in control, poised, after a session.\n\nBuster followed Noah to the white porcelain sink and plopped down at his feet.\n\n\"Buster seems to like you,\" Phoebe said.\n\n\"I'm the man with the food. I think he misses Olivia. Maybe Dylan, too.\"\n\n\"You're taking care of Buster?\"\n\nHe leaned back against the sink. \"You seem surprised.\"\n\nNo kidding. \"I guess I am. Olivia didn't mention you'd be staying.\"\n\n\"It was a spur-of-the-moment decision.\"\n\nPhoebe wondered what had prompted it but shook off her questions. \"I'm here to make pesto.\" She pointed toward the mudroom and the back door out to the terrace and gardens. \"With the basil. For Olivia.\"\n\n\"Ah. Yes. Before it goes to seed.\"\n\n\"My sister Maggie was supposed to join me but she got called away.\"\n\n\"Anything I can do to help?\"\n\nHelp? Phoebe didn't know why she was so flustered, then realized she had every reason to be, with a house-sitting, dog-sitting Noah Kendrick a few yards from her. He had to be used to a different lifestyle than what he'd find in Knights Bridge. He also had to be used to having more to do\u2014or at least other things to do\u2014than what Carriage Hill offered.\n\nShe unloaded her canvas bag. \"I brought pine nuts, garlic, parmesan and olive oil. I think that's all I need.\" She didn't want him hanging around, watching her, bored, and quickly tried to think of something he could do. \"Olivia said she has a mortar and pestle. Do you think you could find them?\"\n\n\"A mortar and pestle,\" Noah said, his tone unreadable.\n\n\"They should be in a cupboard. You know what they are?\"\n\n\"Mmm.\"\n\nHe hadn't moved but she was intensely aware of his scrutiny as she set the bottle of virgin olive oil she'd brought on the counter. \"I've never actually made pesto but Maggie emailed me a recipe. I assume you've never...\" She stopped herself, rephrased. \"Have you ever made pesto?\"\n\nHe smiled that smile again. \"I haven't.\"\n\n\"It doesn't look hard.\"\n\nWhy was her heart beating so rapidly? Just because he was even richer than Dylan didn't mean she had to get crazy. But it wasn't just that. It was the way he looked at her, his air of self-control and calm. Those eyes. That smile. She hadn't noticed them yesterday the way she did now, perhaps because she'd been preoccupied with getting the transcript of the conversation she'd overheard to Dylan, so that he could get it to her swashbuckler. She'd planned to ask Maggie how that had gone when they were making pesto. She hadn't thought to ask her on the phone.\n\nPhoebe cleared her throat. \"How did your first night in Knights Bridge go?\"\n\n\"Quiet,\" Noah said. \"Just Buster, an owl and me.\"\n\nThe twitch of a smile, that spark of humor in his deep blue eyes\u2014Phoebe felt a rush of heat that she couldn't define or understand. She blamed Friday night. Sneaking past her sister and friends into the masquerade, dancing with a stranger and overhearing an alarming conversation from another stranger had kicked her adrenaline into high gear. Even venturing up to the hidden room in the library attic had taken a toll on her normally calm, sensible nature. If she hadn't found that room, she realized, she'd never have gone to the masquerade.\n\nShe turned to Noah with a pleasant smile, the sort that she often used when she was at a loss at the library. \"How long will you be here?\"\n\n\"I'm not sure. We'll see.\"\n\n\"I don't want to disturb you. I can pick the basil and then make the pesto back at my house.\" She gestured vaguely with one hand. \"I live in the village.\"\n\n\"You won't be disturbing me,\" he said. \"It's not as if I have a lot to do.\"\n\nA bored high-tech billionaire. Just what she needed. \"So you think pesto-making has possibilities?\"\n\nHe laughed. \"I wouldn't go that far, but I'm happy to help.\"\n\n\"Great,\" Phoebe said, half meaning it, half not. \"Why don't you look for that mortar and pestle while I start on the basil?\"\n\n\"Sounds good.\"\n\nHe seemed genuinely willing to help, but Phoebe wondered how long his interest would last before he got restless. If staying behind at Carriage Hill really was a spur-of-the-moment decision, then he wouldn't have any of his regular amusements and diversions with him. She supposed he could be working on a new business project. Something that required some quiet time to think.\n\nShe couldn't get out of the kitchen fast enough. She didn't even know why. Noah hadn't made any sarcastic remarks. He hadn't been condescending in any way toward her. He just put her on edge. She hated to think it had to do with his financial status. She wasn't the type to judge people by their net worth.\n\nNot that she'd met many billionaires, she thought as she made her way through Olivia's backyard to the garden shed. But it wasn't that. It wasn't money. It was...\n\n\"I just don't know,\" she said to herself, grabbing small clippers off a hook. Her swashbuckler Friday night and now Noah Kendrick Sunday afternoon. Maybe she was the one who was bored and restless.\n\nShe ducked out of the shed and up the path to the basil patch.\n\nNoah and Buster wandered out to the terrace. \"I found the mortar and pestle,\" Noah said.\n\n\"Excellent. We're in business.\"\n\nGiven past experience, Phoebe expected Buster to barrel to her and tear into the basil, but he stretched, yawned and lay down in a shady spot by the bench.\n\n\"Good dog,\" Noah said, obviously as surprised as Phoebe was. \"It must be Olivia's influence, or perhaps the heat. I haven't had him long enough to have an influence. How's the basil?\"\n\n\"It smells wonderful.\"\n\nHe stepped off the terrace into the grass. He was still barefoot. Phoebe noticed the muscles in his bare arms and, under his T-shirt, his shoulders. He was lean but clearly strong, far more fit than she'd have expected. His eyes settled on her and he smiled without saying a word, as if he knew she'd been appraising him.\n\nWith a flush that had nothing to do with the summer heat, she snipped a healthy hunk of basil and realized she hadn't brought anything to put it in. As she considered what she could use, Noah leaned over and took the basil from her. \"I'll get a colander,\" he said, then headed back to the terrace and into the kitchen.\n\nPhoebe took a breath, hoping to calm her racing heart. Maybe she should have rescheduled the pesto-making, after all.\n\nNoah returned with a colander. She laid more fresh-picked basil in it and thanked him. If he stayed this close to her, it was going to be a long afternoon. \"You don't have to do this,\" she said. \"If you want to take Buster for a walk in the woods, feel free.\"\n\n\"We already hiked up Carriage Hill this morning.\"\n\nCarriage Hill rose up beyond the open fields behind the house. \"I see.\" She snipped another basil plant and asked casually, \"How was hiking in the White Mountains?\"\n\n\"We went at hockey-player pace,\" he said with a wry smile.\n\n\"Is that faster or slower than your pace?\"\n\n\"Faster. Much faster. I prefer to savor each step up a mountain. I tend to be very deliberate about what I do.\" He reached down and brushed her bare shoulder with his fingertips, then smiled as he stood straight again. \"Bumblebee.\"\n\nPhoebe's mouth had gone dry at his touch. \"The bees like the catmint,\" she said, nodding to the frothy purple-flowered border. \"Olivia plans to move it to a less-trafficked area.\"\n\n\"Bumblebees have a natural preference for purple flowers, which tend to have more nectar than flowers of other colors.\"\n\n\"I didn't know that.\"\n\nHe shrugged. \"I read it in an article somewhere.\"\n\nAs smart as he was, she thought, he probably remembered everything he read. She tackled more basil, leaving enough for regrowth. Noah waited, then carried the overloaded colander to the terrace, Buster stirring enough to follow him inside.\n\nPhoebe returned the clippers to the shed. After sneaking into the charity ball on Friday and dealing with Maggie's suspicions yesterday, she'd wanted a quiet Sunday. Needed a quiet Sunday to get her bearings.\n\nAnd here she was, picking basil and making pesto with Noah Kendrick.\n\nWhen she returned to the kitchen, Buster was lapping water out of his bowl in the mudroom and Noah was sipping a glass of water at the table. The basil was in the sink. \"I rinsed it,\" he said. \"I didn't see any ants, spiders, worms or slugs. Just dirt.\"\n\n\"That's good. I'll do a second rinse. I always do with anything fresh out of the garden. It's not that I don't trust you.\"\n\nHe picked up his water glass. \"Of course not.\"\n\nAs she approached the sink, she noticed that one of the flyers Olivia had designed for the fashion show was on the table. It hadn't been there before. It announced the show and called for donations of pre-1975 vintage clothing in good condition.\n\nNoah tapped one finger on the flyer. \"I saw this on Olivia's bulletin board in the mudroom. A vintage fashion show at the local library. Your idea?\"\n\nPhoebe nodded. There'd been a change in him since he'd taken the colander inside. She couldn't put her finger on what it was, except that she was feeling caught, trapped\u2014as if he knew something that she didn't know.\n\nShe kept her tone even, professional, as she answered him. \"It came together fast and the response has been tremendous.\"\n\n\"And you're holding the show at the library?\"\n\n\"That's right. It has a stage. The founder, George Sanderson, insisted the design for the library include one. He envisioned lectures and concerts.\"\n\n\"Have you received many donations?\"\n\n\"Far more than I anticipated. It's been fun so far.\"\n\nNoah drank more of his water, then got to his feet in one smooth movement. \"Is that where Olivia and Maggie got their dresses for the other night?\" he asked as he walked over to the sink. \"Did they come in with a donation?\"\n\nPhoebe plunged one hand into the cold water. \"It's a bit more complicated than that, but the short answer is yes.\"\n\n\"And the masks?\"\n\n\"My youngest sisters made those. Ava and Ruby\u2014\"\n\n\"The theater majors.\"\n\n\"That's right.\" Phoebe tried to sound casual. \"So how did you enjoy the ball?\"\n\nHe leaned back against the sink and crossed his arms on his chest. \"It was quite a night.\"\n\nYes, it was, Phoebe thought. She hadn't noticed Noah in the ballroom, but she'd been too caught up in avoiding Olivia and Maggie and dancing with her swashbuckler to notice much else. Then there'd been Brandon, and the man she'd overheard. She hadn't even thought about Dylan's best friend, although she knew they'd been hiking in the White Mountains.\n\nNoah turned and got a stainless-steel grater out from a lower cupboard. \"I can grate the parmesan,\" he said.\n\nPhoebe had the feeling his mind wasn't on pesto but she smiled. \"That'd be good, thanks.\"\n\nShe laid the basil leaves on paper towels, watching him as he placed the grater and the hunk of parmesan on a wood cutting board. He glanced at her, and this time she paid close attention to the line of his jaw, the color and shape of his eyes. His smile was confident, knowing, but at the same time not at all easy to read, deliberately so, as if the man behind it guarded against letting anyone in.\n\nShe remembered her swashbuckler moving through the crowd to get to her, every movement precise, smooth, controlled.\n\nIt was all she could do not to gasp.\n\nIt's him.\n\nHer swashbuckler was Noah Kendrick.\n\nIf she'd been the one grating parmesan, she'd have cut herself. As it was, her hands shook. She tried to focus on blotting the basil dry but her mind was spinning. She'd danced with a billionaire. With Dylan McCaffrey's best friend. She'd let him kiss her.\n\nAnd he'd disappeared on her. Had he really meant to come back? Had he lost her? Had she left the ballroom too soon?\n\nDoes he know it was me?\n\nWhy hadn't she recognized him sooner? His voice, his eyes, his lean build\u2014so what if he'd shaved and wasn't wearing a mask and cape?\n\nShe hadn't expected that her swashbuckler would be Noah Kendrick. It was just that simple.\n\nShe blotted the basil, her heart hammering. Noah continued to grate the cheese for the pesto. It was all she could do not to think up an excuse and get out of there but she knew that would only draw more attention to her discomfort. He was a smart man. He'd figure out she'd asked him about the masquerade ball right before she unraveled.\n\nMaggie had to know it'd been Noah in the swashbuckler costume. Why hadn't she said so? Because I told her I didn't want to know. No doubt Maggie had assumed Noah would never recognize her sister as his princess.\n\nPhoebe didn't understand the intensity of her reaction. Why not just admit she recognized him? That it was her in the Edwardian dress?\n\nBecause it hadn't been her. Not really.\n\nShe should have just gone to the ball openly, with Maggie and Olivia. Then Noah would have known who she was. Probably he never would have danced with her\u2014or if he had, they wouldn't have gotten so carried away.\n\nShe glanced at him. He had a healthy mound of parmesan grated onto a cutting board. He gave no indication he thought of her as anything but the librarian friend of his best friend's fianc\u00e9e.\n\nOf course, that was what she was.\n\nPhoebe sighed and stood back from the sink. A slight breeze floated through the open window, calming her. Maggie would have given her note to Olivia, who would have given it to Dylan or even directly to Noah. That was why Noah had stayed behind in Knights Bridge. He wanted to figure out what the story was with this man in the coatroom. Dancing with a woman at a masquerade ball was probably par for the course for him, fun while it lasted but not particularly memorable.\n\nLet us both pretend that night never happened.\n\nAs matter-of-factly as she could, Phoebe nodded to the clean, reasonably dry basil. \"If you can chop the basil, I'll roast the pine nuts and mince the garlic.\"\n\n\"Then they all get pounded into a paste with the mortar and pestle.\" Nothing in his smile suggested he knew that she could hardly get a decent breath. \"I'm guessing, because I've had pesto.\"\n\n\"We pound the basil and garlic first. Then add the nuts. Then the parmesan and olive oil.\"\n\n\"And what do we do with all this pesto?\"\n\n\"Freeze it in ice-cube trays. Olivia and Maggie will use it all winter. They might use it at Olivia's wedding in December.\" Phoebe managed a smile. \"It'll remind everyone of summer.\"\n\n\"I'm sure it will.\"\n\n\"Will you be back for the wedding?\"\n\n\"Yes, absolutely.\"\n\nPhoebe looked at the parmesan, basil, garlic and pine nuts and thought about the work ahead to turn them into pesto. How would she be able to stand it, knowing what she did? She gathered up the damp paper towels from the basil and tossed them in the trash. She tried to appear casual as she turned back to Noah. \"You know, if there's somewhere else you'd rather be\u2014\"\n\n\"There isn't. I'm exactly where I want to be.\" He opened a drawer and removed a knife. \"I'll chop. You mince and roast.\"\n\n* * *\n\nOnce the pesto was in the freezer, Noah saw there was no keeping Phoebe at Carriage Hill. She was out of there, tucking her empty canvas bag under one arm and all but racing out the door. Although he wasn't by nature a patient man, years of martial arts practice and running a successful company had taught him that sometimes the best course of action was just to bide his time.\n\nHe followed her to her car. The afternoon sunlight caught the streaks of gold in her dark strawberry hair as she yanked open her car door. She turned to him with a quick smile. \"Thank you for your help with the pesto. Enjoy your stay.\"\n\n\"Anytime.\"\n\nShe climbed behind the wheel, and he shut the door for her. With another quick smile, she had the car started and was on her way.\n\nShe'd recognized him as her swashbuckler, obviously, but she still believed\u2014or was telling herself she believed\u2014that he hadn't recognized her.\n\nWell, he had.\n\nIt was the fashion show flyer on the bulletin board that had finally done the trick. He'd started to suspect when he'd found her in the kitchen. The way she'd licked her lips, smiled, moved. The line of her jaw, the deep turquoise of her eyes, the sound of her voice. The shape of her hips, the curve of her breasts. They'd all come together when he saw the flyer, and he'd known.\n\nPhoebe O'Dunn was his princess.\n\nNoah walked back through the house and liberated Buster from the mudroom. They went out to the quiet terrace, but the big dog looked as restless as he was. \"If you run off,\" Noah told him, \"I'll find you and I won't be happy about it. So spare both of us and stay put.\"\n\nBuster sat, panting, his dark eyes focused on Noah as if he'd gone crazy.\n\nNoah laughed. \"I just might have, my friend.\"\n\nThe pesto was in the freezer and the kitchen cleaned up, but even out on the terrace, he could smell the mix of basil, garlic, roasted pine nuts and pure virgin olive oil.\n\nVirgin olive oil. A Freudian slip, there. Dancing with his princess, he'd imagined her a virgin, as bold and as daring as she was when he'd swept her into his arms.\n\nWas Phoebe O'Dunn a virgin?\n\nNoah grimaced. Dylan would kill him dead for even letting such a question cross his mind. Dylan still had to tread carefully in Knights Bridge. Phoebe O'Dunn, her sister Maggie\u2014these were Olivia's people.\n\nTelling Phoebe that he knew she was his princess was out of the question until he'd had a chance to think. He could act quickly, decisively, but not when he didn't have a clue what in blazes to do. As he'd watched her pound the basil and garlic into a thick paste, he didn't know why he hadn't recognized her sooner. He hadn't been thrown off by her dark strawberry hair and freckles as much as the fact that she was from Knights Bridge and Olivia Frost's friend.\n\nThe note about his mystery man further complicated the situation.\n\nBuster stirred, and Noah noticed a thickset man hopping over the low stone wall from the field behind the house. \"Brandon Sloan,\" the man said, stepping over knee-high herbs onto a path. \"You must be Noah Kendrick. Dylan mentioned you'd be here for a few days. I'm working on his place up the road.\"\n\n\"You're one of the carpenters?\"\n\n\"Sloan & Sons. I'm one of the sons. There's a sister, too, but she showed up after the company was named. Sore subject.\" He polished off an energy bar and dusted his hands as he stepped onto the terrace. \"What do I smell?\"\n\n\"Pesto.\" Noah pointed to the patch of trimmed basil. \"Phoebe O'Dunn was here.\"\n\n\"Maggie, too?\"\n\n\"Not Maggie, no. You two are...\"\n\n\"Married.\" Brandon pulled out a chair at the table and sat down. \"I saw you the other night in Boston. You'd just come from hiking in the White Mountains. One of my favorite things to do.\"\n\n\"It was an experience,\" Noah said. \"You were at the masquerade ball?\"\n\nBrandon grimaced. \"I decided to go at the last minute. I'd told Dylan I'd rather have burning bamboo shoots shoved up my fingernails than go to a masquerade ball.\"\n\n\"What changed your mind?\"\n\n\"I found out Maggie was turning up\u2014I have my sources.\"\n\nThe other Sloans of Sloan & Sons, Noah suspected. He wondered if Brandon's presence at the ball explained why Maggie had been so upset. Noah decided his and Dylan's lives in San Diego, running NAK, were simple compared to the lives of the people he'd met so far in Knights Bridge.\n\n\"Is Maggie aware you're working on Dylan's place?\" Noah asked.\n\n\"Not yet, no. Olivia doesn't know, either. I asked Dylan to let me tell Maggie first.\" Brandon stretched out his thick legs. \"I'm camping up there. We start demolition on the house soon. I figure I can use the facilities here if need be. Olivia won't mind.\"\n\nHe seemed confident, even matter-of-fact, not at all presumptuous. He'd probably known Olivia\u2014and Maggie, his wife\u2014most if not all his life. Noah's one near-lifelong friend was Dylan.\n\n\"When did you arrive?\" he asked.\n\n\"This morning. I pitched my tent out of sight of the road. I'm glad Phoebe didn't see me. She's protective of her sisters. They stick together, those four.\" Brandon settled back in his chair, obviously not concerned about the O'Dunn sisters or anyone else. \"How do you like Carriage Hill?\"\n\n\"It's not as quiet as I thought it would be.\"\n\nBrandon grinned, then glanced around at the lawn and gardens, the fields, the hills in the distance. \"I want to get my two boys out here to help with the work on Dylan's place.\"\n\n\"How old are they?\"\n\n\"Five and six. Tyler's almost seven, though. Don't worry, I'm not talking about real work on the site. Just get them started on learning how to use a hammer and screwdriver. Maggie's got them baking tarts and peeling artichokes. I don't object, but they need this, too.\"\n\nNoah would guess that Brandon had learned to say he didn't object to his sons learning to bake tarts and peel artichokes. What he meant instead was that he was afraid his young sons were missing the influence of their rough-and-tumble father. Noah didn't have the particulars on Brandon Sloan's troubled relationship with his wife but could see that he loved his sons.\n\nBrandon stood abruptly, as if he wanted to escape wherever his thoughts had just taken him. \"Dylan offered me a ticket to this masquerade ball but I didn't take it. I wanted to pay my own way. I went as a pirate. Maggie made me faster than I thought she would. Maybe I should have gone as a banker instead.\" He paused, then added wryly, \"She'd never have recognized me as a banker.\"\n\nNoah made no comment but he thought that Brandon had a point.\n\nBrandon turned, his expression serious as he narrowed his dark eyes. \"Don't tell her that I'm camping out up at Dylan's. Leave that to me.\"\n\n\"No problem.\"\n\n\"I don't mind camping. I'm back on my feet financially but I want Maggie to take me as I am. With or without money.\"\n\n\"For better or worse,\" Noah said.\n\n\"That's right.\"\n\n\"Why are you sleeping in a tent?\"\n\n\"It beats staying with my folks or one of my siblings.\" Brandon gave a mock shudder. \"Trust me.\"\n\n\"Then you don't have your own place?\"\n\n\"I gave up my apartment in Boston the first of the month. I'm saving every dime I can. I was in and out of work for a while, but I've been working nonstop for the past six months. It's good. No complaints.\" He paused, looked at Noah. \"I won't drag this thing out. I just have to do this in my own time. Understood?\"\n\n\"Of course,\" Noah said. \"I'll respect your wishes.\"\n\nHe thought that Maggie O'Dunn Sloan\u2014or any woman in her position\u2014would appreciate knowing that her estranged husband was sleeping in a tent a few miles from her, but he wasn't one to offer advice on relationships.\n\n\"I can help out with anything you might run into here,\" Brandon said.\n\n\"I just made pesto with the town librarian. What could I run into?\"\n\nBrandon grinned. \"Snakes in the stone walls. 'Course, where you're from, you have poisonous snakes. A little old garter snake probably won't bother you, right?\"\n\n\"Probably not,\" Noah said.\n\n\"Bats?\"\n\nHe hadn't considered bats. He smiled. \"The hazards of country life.\"\n\nBrandon tilted his head back, eyeing Noah with an intensity that other people might find intimidating. \"You're not up to anything here, are you? Why didn't you go to San Diego with Dylan and Olivia?\"\n\n\"I'm dog sitting.\"\n\nAt first Brandon didn't respond. Then he laughed. \"Right. Dog sitting. Enjoy your pesto, or did Phoebe take it all back with her?\"\n\n\"It's in Olivia's freezer.\"\n\n\"I'd never had pesto until I met Maggie. I've known Phoebe since nursery school. We're the same age. She's a special person in Knights Bridge. She looks after all of us.\" He settled his gaze again on Noah. \"And we all look after her.\"\n\n\"Good to know,\" Noah said mildly.\n\nIt was as clear a warning between two men as one could get without Brandon Sloan coming right out and saying that he'd be watching and Noah had best behave himself with Phoebe O'Dunn.\n\nAnd why would Brandon think that Noah might not behave himself?\n\nBecause he knew that his sister-in-law had dressed up as an Edwardian princess the other night and had seen her dancing with her swashbuckler, who was now dog sitting in Knights Bridge.\n\nNoah assumed that Olivia and Maggie, who also had to know about Phoebe, didn't realize that Brandon was in on the secret, too.\n\nComplicated, complicated.\n\nBrandon headed off, back over the stone wall and through the field up to the house\u2014or what was left of it\u2014that Dylan had inherited from his father.\n\nNoah went inside. It was five o'clock in the afternoon. Now what was he supposed to do?\n\nHe'd take Buster for another walk, then see what Olivia had in terms of movies.\n\nAnd tomorrow?\n\nTomorrow was supposed to be another hot day.\n\nPerfect for a trip to the Knights Bridge Free Public Library.\nNine\n\nLoretta Wrentham parked in the driveway at Dylan's stucco house on Coronado. He'd left her three messages while she was sweating through a horrid exercise dance class. She'd finally texted him that she'd be right over, then showered, reapplied her makeup, put on slim jeans, a white shirt and red heels and, feeling energized if not any happier about exercise, headed across the San Diego\u2013Coronado Bay Bridge to the upscale island town where Dylan lived.\n\nHe'd told her that Coronado wasn't home for him like Knights Bridge was home for Olivia.\n\nLoretta believed him.\n\nHer cell phone trilled and she assumed it was Dylan again but saw Noah's name on her screen. This couldn't be good. Something clearly was up. She debated answering, but Noah was even worse about pestering her if he wanted a response. \"Isn't it the middle of the night on the East Coast?\" she asked him, knowing perfectly well what time it was in New England.\n\n\"It's midnight. I'm listening to my owl. I have all the windows in the house open. The stars are out. It's nice.\"\n\n\"I like stars. I heard an owl once on vacation in the mountains.\" Of course, she realized he hadn't called to talk about stars and owls. \"What can I do for you, Noah?\"\n\n\"Julius Hartley, Loretta. Who is he?\"\n\nShe was silent. Hartley. No wonder she had so many messages from Dylan and now Noah was on the phone with her.\n\n\"Loretta?\"\n\n\"He's your mystery man,\" she said.\n\n\"Is that a question or do you know?\"\n\n\"I know now that you've said his name. How did it pop up?\"\n\n\"Dylan checked the guest list at the masquerade ball. He couldn't resist. The name Julius Hartley stood out. He bought a ticket at the last minute, he came alone and he's from Los Angeles. He left his street address blank. Dylan doesn't know him.\"\n\nLoretta swore under her breath. \"I'll take care of this.\"\n\n\"Who is he, Loretta?\" Noah asked mildly.\n\nShe decided to tell him. \"Julius Hartley is a scumbag private investigator who won't return my calls.\"\n\n\"How do you know him?\"\n\n\"I don't. He showed up in my office a couple of weeks ago. I didn't think about him as a possibility for our mystery man until you told me you'd spotted your stalker in Boston. Something about your description this time finally clicked. I tried reaching Hartley. I only have his cell phone number and he didn't answer.\" She needed air and got out of her car. A cool evening breeze was blowing onshore off the Pacific. Damn. Had she screwed up this time? \"Where is Hartley now?\"\n\n\"I have no idea,\" Noah said, no hint of impatience or exasperation.\n\n\"All right. I'll see what I can do and call you when I know more.\"\n\n\"What did he want when he came to your office?\"\n\n\"He asked me about Duncan McCaffrey.\"\n\n\"Dylan's father? Why?\"\n\nLoretta had told Noah as well as Dylan about her brief affair with Duncan shortly before his death. At least she wouldn't have to rehash that indiscretion\u2014which was what it was, even if she didn't regret it.\n\nFinally she said, \"Hartley told me he was fascinated with treasure hunts and was curious about what would happen to Duncan's unfinished projects. Duncan's been gone for two years, so I figured it was a lame cover story for worming information out of me about Dylan, about you and your work together at NAK, what's next now that it's gone public.\"\n\nLoretta stood on the sidewalk in front of Dylan's house so that the breeze off the ocean caught her full in the face. She could see Julius Hartley in her office in La Jolla, a good-looking man around her own age, cocky, not really giving a damn that she didn't believe a word he was saying.\n\nShe should have pegged him as Noah's stalker from the get-go.\n\n\"I'll get to the bottom of this, Noah,\" she said.\n\n\"I know you will,\" he said, as calm as ever.\n\n\"He hasn't turned up in Knights Bridge, has he?\"\n\n\"Not that I know of.\"\n\nA stiff gust of wind brought with it the smell of saltwater. She could taste it as she tried to picture Noah alone in the out-of-the-way little New England town and found that she couldn't. Not that she'd ever been to Knights Bridge herself, but she'd never seen Noah outside of Southern California. Well, once at his winery on the Central Coast. He and Dylan had invited her up for a party celebrating NAK's fourth anniversary.\n\nOptimist that she was, when Dylan had told her he'd agreed to work with Noah, she'd figured NAK would go bust within months. But Dylan had been broke, going nowhere after ignoring one piece of good advice after another from her.\n\nA good thing she'd kept her mouth shut about NAK.\n\nDylan and Noah had done well in their work together, and Loretta had done well by Dylan and got to know and like Noah, even if she'd never understand him.\n\nWhat the hell did Julius Hartley want with him?\n\n\"Got anything more going on than listening to owls and watching the stars?\" she asked Noah, hoping she didn't sound as out of sorts as she felt.\n\n\"Olivia's dog keeps breaking out of the mudroom and getting up on the couch.\"\n\n\"The legendary Buster,\" Loretta said, then promised to keep Noah informed and disconnected.\n\nHer attempt at good humor didn't last. Dylan had come out onto his front porch and was waiting for her. She tossed her phone into her handbag and headed up the steps, the wind at her back now. Dylan had his eyes narrowed on her in that distinctly McCaffrey manner. She'd known him since his early days with the NHL. He was more aggressive than Noah. He'd pounce. With Noah, she thought, you could be bleeding on the floor before you knew he'd even come close to you.\n\n\"Does Julius Hartley work for you?\" Dylan asked.\n\n\"No. Never. He works up in L.A. He stopped by my office a couple of weeks ago.\"\n\n\"Why?\"\n\n\"I thought he was fishing for information on you and Noah. I kicked him out.\"\n\n\"As only you can,\" Dylan said.\n\nLoretta turned to look out at the darkening ocean and sky. The stars would be out here soon, too. She didn't know about owls.\n\nDylan stared at the Pacific, white caps visible as waves rolled onto the wide beach. \"I can't just step back from this, Loretta. Whatever Hartley is after involves Noah or me, or both of us.\"\n\n\"Could someone in Knights Bridge be a threat to Noah?\"\n\n\"Like who?\"\n\nLoretta waved a hand. \"I don't know. I'm just thinking out loud.\"\n\n\"Who was Hartley talking to on Friday?\"\n\nShe shook her head. \"No idea. No idea who his client is, either. I'll find out. You and Noah don't need to worry about this.\"\n\nDylan didn't look convinced but Loretta wasn't surprised. He'd had Noah's back for four years, allowing his friend to focus on his strengths in building NAK into a highly profitable company. It wouldn't be easy for him to give that up.\n\nA fair-haired woman Loretta took to be Olivia Frost, Dylan's fianc\u00e9e, stepped out of the house. Dylan introduced them, and any misgivings Loretta had about their sudden romance quickly disappeared. Olivia was smart and sophisticated, but also natural, and down-to-earth. She was perfect for Dylan. And Loretta saw that Dylan was perfect for Olivia, too.\n\n\"How do you like San Diego so far?\" Loretta asked.\n\nOlivia smiled. \"I absolutely love it. It's so different from Knights Bridge. That's where I've lived most of my life.\"\n\nOlivia's eyes lit up when she mentioned her hometown. Then Loretta saw it, too. What Dylan had been trying to tell her. That as much as Olivia might like other places, her town on the edge of the Quabbin Reservoir was home. That meant Dylan would make Knights Bridge his home.\n\nSo what were she and Noah supposed to do?\n\nMaybe that was why he was dog sitting, listening to owls and chasing a masked princess. In his own way, he was trying to figure out what he'd do with his best friend and business partner\u2014a man who was like a brother to him\u2014living on the other side of the continent.\n\nAnd if Noah and Dylan both ended up in Knights Bridge?\n\nLoretta didn't want to think about it.\n\nShe'd focus on tracking down Julius Hartley instead.\nTen\n\nPhoebe dressed for a hot August day but the library was cooler than she expected when she arrived early, well before eight o'clock. Mondays were generally quiet, and she was the only full-time employee. The part-time staff and volunteers wouldn't start arriving until after the library opened at nine-thirty but she appreciated the time to herself.\n\nWith a shiver, she grabbed an old sweater off one of the stacks of sorted vintage clothes on the stage. The sweater was several sizes too big and a dingy coral acrylic that didn't go at all with her sunflower-colored sundress but she wouldn't need it for long before the library warmed up.\n\nShe'd planned to shelve books and catch up on paperwork but instead went up onto the stage and sorted a box of clothes that had come in over the weekend. For the most part, donors had respected the specifications for the show and weren't just dropping off junk, although not everything could be used\u2014including the awful sweater she'd thrown on.\n\nThe front door creaked open at the stroke of nine-thirty. Phoebe looked up from her box, filled with a colorful collection of maxi skirts, fringed vests and headbands from the early 1970s. She expected to see her administrative assistant, but instead it was Noah Kendrick entering the library.\n\nPhoebe stood up, realized she still had on the old sweater. She wished she'd taken it off, then decided it was just as well. She hated the stereotype of the dowdy, introverted librarian and knew it didn't fit her or most professional librarians she knew, even if Noah was thinking exactly that right now. She was practical. She'd been cold and the sweater had been handy. She wasn't a princess.\n\nNot that Noah had recognized her as the woman he'd danced with.\n\nShe certainly had no intention of telling him.\n\nHe smiled, maintaining a stillness about him as he approached the stage. \"I half expected a ghost.\"\n\n\"Lots of people have said they've encountered ghosts in here, going back to when the library first opened in 1872.\"\n\n\"What a surprise,\" he said mildly.\n\nShe went to the edge of the stage. She expected to jump down to him on her own, but he caught her by the waist and lowered her to the hardwood floor with the same ease and sureness with which he'd swept her across the dance floor on Friday night.\n\nGlad for the dim light in the library, Phoebe pushed back strands of hair that had come out of its pins. Her oversize sweater had come off her shoulders. She let it drop to the floor and shoved it aside with one foot, then got control of herself.\n\n\"Good morning,\" she said politely, stepping back from Noah. \"What can I do for you?\"\n\nIt might have been her imagination, but she thought she saw a spark of pure male sexiness in his deep blue eyes, as if to say that she could do a hell of a lot for him. But he simply said, \"I was thinking I might borrow a book or two. Is that allowed?\"\n\n\"Sure. We'll figure it out. Wander around.\" Phoebe realized she wasn't cold anymore. \"Let me know if I can help you find anything.\"\n\nNoah peered up at the stage. \"The fashion show is shaping up well?\"\n\nShe nodded. \"Some of the clothes we've received are amazing. Others, not so much. The historical society is interested in checking out some of the unique items.\"\n\nHe shifted his gaze back to her. \"Are you involved in the historical society, too?\"\n\n\"Yes, as a matter of fact.\"\n\nShe didn't know why she felt defensive. There was nothing condescending in his tone or manner. His eyes were half closed, almost navy in the dim light by the stage. They lingered on her shoulders, then lifted to meet hers.\n\nHe knows.\n\nShe couldn't pinpoint what had tipped her off, but there was no question in her mind that Noah Kendrick had figured out that she was the woman in the Edwardian gown on Friday.\n\nHad he known yesterday?\n\nShe felt the heat of embarrassment but hoped he didn't notice her discomfort. \"The fashion show is turning out to be a lot of fun for everyone.\" Her throat was dry, tight, as she suddenly tingled with the memory of their brief kiss just two nights ago. They'd gotten carried away. No question about it. She added, \"Every donated garment has a story behind it.\"\n\n\"What's the story behind Maggie's and Olivia's dresses?\"\n\n\"And yours?\" Phoebe could almost hear him ask.\n\n\"They're copies from movies,\" she said. \"As I'm sure you know.\"\n\n\"To Catch a Thief and Breakfast at Tiffany's.\"\n\n\"That's right.\" She didn't explain further. She had no intention of telling him about the hidden room when she still hadn't told her sisters and Olivia\u2014anyone\u2014about it. \"Maggie and Olivia had a great time at the ball.\"\n\n\"So did I.\" His eyes held hers. \"More than I ever imagined.\"\n\nPhoebe reminded herself that she was a professional, experienced librarian, accustomed to dealing with tricky situations with the public. She would think of Noah as just that. A member of the public. She motioned toward the stacks. \"I'll let you get on with your browsing.\"\n\n\"Thanks.\" He walked over to the fireplace, then glanced back at her. \"Ever light a fire in here?\"\n\n\"Not in years.\"\n\n\"The library's centrally located. Do many people from out of town stop in to ask for information on residents?\"\n\n\"Some.\" She knew he was thinking about the note about the phone call she'd overheard. Did Noah realize she'd written the note? Was that why he was here? She pushed back her own questions and focused on what he'd asked her. \"We don't give out private information on anyone. That would include Olivia's guests, in case you're wondering.\"\n\n\"So you won't be spreading the word that I'm staying in town?\"\n\nPhoebe went behind the curved circulation desk and tried to act as if it was just another Monday morning. \"That's right.\"\n\nNoah glanced up at the oil portrait of an imposing George Sanderson. \"Has anyone been asking about Dylan or me?\"\n\n\"Not that I know of. Do you have anyone specific in mind?\"\n\nNoah moved back from the fireplace and scooped up the coral sweater she'd had on. He laid it on the stage. She had a feeling he knew it wasn't hers. \"What would you do if someone did ask about us?\" he asked.\n\n\"I might offer to take down a name, address and phone number and give them to Dylan, or to you if you're still in town.\" Phoebe shrugged, still containing her reaction to Noah's presence. \"Otherwise I stay out of personal business involving anyone in town.\"\n\n\"Smart. If someone does ask about either Dylan or me while I'm here, you'll let me know?\"\n\nShe nodded. \"Happy to.\"\n\n\"Thanks. I think I'll check out what you have on fencing. It's a hobby of mine. Classical fencing. In fact, Dylan couldn't resist having me dress up as a swashbuckler on Friday.\" Noah smiled. \"He has a sense of humor.\"\n\n\"So I've discovered.\"\n\nHe headed off to the stacks, but Phoebe knew he wasn't serious about checking out what the library had on fencing books or anything else. Once he was safely out of sight, she sat at her computer and let out a long, cathartic breath.\n\nVera Galeski, Phoebe's part-time assistant, arrived, cheerfully grumbling about the heat. In winter, she grumbled about the cold. She was a high-energy woman in her early sixties, devoted to books, married to a retired teacher, mother of four, grandmother of six and ever hopeful that Phoebe would find a man.\n\nVera nodded vaguely in the direction Noah had just gone. \"Who's that man dressed head-to-toe in black on a hot day like today?\"\n\n\"One of Dylan McCaffrey's friends,\" Phoebe said, hitting a few random keys on her keyboard to help herself look nonchalant.\n\nVera's pale blue eyes widened. \"Not Noah Kendrick,\" she whispered.\n\nPhoebe nodded, then added quickly, \"It's not something we're advertising.\"\n\n\"Of course not. I understand perfectly. Oh, my. I read an article in a magazine at the hairdresser's that mentioned him. It was about that actress...I can't think of her name. The one on that Sunday-night show that just got canceled. She played a lawyer.\"\n\n\"I should read more gossip magazines,\" Phoebe said with what she hoped was a credible laugh, then made an excuse to go upstairs.\n\nWithout so much as a glance in Noah's direction, she headed to the back stairs and ran all the way up to the attic without stopping. She switched on the dim overhead, then squeezed between the freestanding twin metal closets and entered the hidden sewing room. It was hot, airless. She opened the second door in the corner, letting in daylight from the small window overlooking the common. Children from a nearby daycare were sitting in a circle in the shade in front of the Civil War statue.\n\nNoah Kendrick's arrival notwithstanding, nothing in Knights Bridge had changed. This Monday was like last Monday.\n\nAnd next Monday?\n\nPhoebe pulled her gaze from the window and unzipped one of the garment bags. Inside were four dresses in various shades of red, as well as accessories carefully draped on hangers\u2014bright red scarves, sequined belts, gaudy costume jewelry.\n\nWho was the woman who'd sewn here, most likely in secret? Had she despaired that she was living in a small, out-of-the-way town? Had she wanted something that Knights Bridge just couldn't give her?\n\nPhoebe ran her fingertips over a scarlet crepe flapper dress. Would Noah have noticed her if she'd worn it the other night, or was there something about the Edwardian dress that had caught his eye, fired his imagination?\n\nShe glanced out the window again and saw him crossing South Main to the common. He didn't have any books with him. Had he returned to the circulation desk expecting to find her, or simply seized the moment to get out of there, leave her to her work? His princess, after all, had evaporated. She was a small-town librarian.\n\nAnd her swashbuckler?\n\nPhoebe watched him walk in the shade. He was one of the wealthiest people in the country, brilliant, successful, a master fencer, an expert in karate. As far as she could see, he was just missing the black cape, mask and sword at his side to be a real swashbuckler.\n\nHad her seamstress looked out at the window at a man crossing the common, yearned for him\u2014for a life other than the one she had, tucked up here with her fabrics and sewing notions?\n\nPhoebe's throat tightened at the thought of the carefully conjugated French verbs. A student doing homework, or a young woman dreaming of a different life?\n\nNoah disappeared from her view. Phoebe shut the corner door, the little children rolling in the grass now, playing some kind of game.\n\nWho was she kidding?\n\nThis Monday wasn't like last Monday. Last Monday, she hadn't met Noah Kendrick.\n\nShe left the hidden room, ran down the stairs and got back to work.\n\n* * *\n\nPhoebe left the library at four, as she did every Monday, and dropped off books at Rivendell, an assisted-living facility on a ridge just outside the village center. As she carried a box of fiction and nonfiction titles to the main entrance, she could see a peek of Quabbin in the distance, its pristine waters barely visible in the steamy haze. A number of the elderly residents remembered the Swift River Valley before it was flooded, and several were from the lost towns, including Grace Webster, Dylan's newly discovered grandmother.\n\nAfter leaving the books in the reading room, Phoebe went down a wide corridor to the sunroom, its tall windows overlooking the center's beautifully landscaped grounds. Grace was seated next to Audrey Frost, Olivia's grandmother, each with a set of binoculars. Grace, a retired teacher in her nineties, was an avid bird-watcher. Audrey, a former bookkeeper at the high school and a few years younger than Grace, was always up for a new hobby and was getting into the spirit of things.\n\n\"Phoebe, so good to see you, dear,\" Audrey said, lowering her binoculars. She had told Phoebe that she loved assisted living because she didn't have to cook every meal for herself, although she could if she wanted to, and she could still have her car. \"Did you bring us some good books? Your friend says he saw you at the library this morning.\"\n\nPhoebe frowned. \"My friend?\"\n\n\"Noah Kendrick,\" Grace said. \"You know him, don't you? He's Dylan's business associate and friend from San Diego.\"\n\n\"He was here?\"\n\n\"He still is here,\" Audrey said. \"He stepped outside for a minute to look at the view.\"\n\nPhoebe sank into a cushioned rocker and looked out at the array of bird feeders just outside the sunroom. They were empty now but would be kept filled over the winter. Bird-watching was a favorite activity for Rivendell residents.\n\nThe two older women eyed her. In addition to her work at the school, Audrey Frost had helped her late husband in his business specializing in custom reproduction millwork. Their son, Randy Frost, Olivia's father, now ran the mill with his wife. It was located behind a nineteenth-century sawmill the Frosts had converted into a residence.\n\nUntil recently, people in Knights Bridge would have said they knew all there was to know about Grace Webster's life. She'd moved to Knights Bridge as a teenager with her father and grandmother and became an English and Latin teacher. She'd lived out on Carriage Hill Road until two years ago, when she'd sold her house to Duncan McCaffrey and relocated to Rivendell. But Grace had her secrets. As a teenager, her family facing expulsion from their home ahead of the damming of the Swift River for Quabbin, she'd created a hideaway in a cabin on a small pond and met a British flyer on the run. They'd fallen in love, but he'd gone back to England, promising to return.\n\nWith a war on, starting her life over in a new town, Grace had discovered she was pregnant and realized her British flyer wasn't coming back to her. She gave birth to a baby boy in a Boston area hospital, put him up for adoption and went back home to Knights Bridge, only her grandmother and father ever aware of her secret.\n\nSeventy years later, Dylan's treasure-hunting father showed up in Knights Bridge and unearthed the story of his birth mother\u2014met her\u2014just before he died. Only he'd failed to tell his son, leaving Dylan to find out on his own. Even Grace hadn't realized that the handsome daredevil in his early seventies was the baby she'd never even held.\n\nPhoebe felt a rush of emotion, as she did whenever she thought of Grace's story. No one in town had ever guessed. People were still getting used to the idea that the starchy retired teacher was Dylan McCaffrey's grandmother.\n\nPhoebe realized that Audrey Frost was peering at her. \"What's on your mind, Phoebe?\" the older woman asked.\n\nShe collected her thoughts. \"I was wondering if you remember anyone from your days at the high school who was especially good at sewing, maybe took French and had an interest in Hollywood. She might have worked at the library, either as an employee or a volunteer.\"\n\nGrace frowned. \"What brought this up?\"\n\n\"I found some dresses that someone sewed and hid away. I'm not sure how long ago. Forty years, at least, I'd guess. I don't have enough information to talk about the details yet.\"\n\nBoth Grace and Audrey had no immediate memory of anyone who could fit Phoebe's vague description but promised to check their records.\n\nThe two elderly women smiled suddenly, pointing as Noah came into view out among the bird feeders. Audrey said, \"He's as good-looking in his own way as Dylan, don't you think?\"\n\nGrace concurred. \"You can tell he's a master fencer. He has the poise of a swordfighter.\" She glanced at Phoebe. \"He told us he helped you make pesto yesterday.\"\n\n\"We had to do something with Olivia's basil,\" she said, suppressing any reaction that might alert Grace or Audrey that something was up between her and Noah.\n\nHe entered the sunroom through a set of French doors and greeted the two Rivendell residents before turning to Phoebe. \"We do keep running into each other, don't we?\"\n\n\"It's a small town.\" She got to her feet, noticing that gray clouds were moving in from the west. \"I should get going. Grace, Audrey, good to see you.\"\n\nPhoebe quickly left the sunroom and got almost to her car when Noah caught up with her. \"Where's your car?\" she asked him.\n\n\"I don't have one. Olivia said I could borrow hers, but I walked into town and then out here.\"\n\nPhoebe stopped abruptly. \"You walked?\"\n\nHe smiled. \"It's something to do.\" He pointed up at the threatening sky. \"I didn't think about the fact that it rains more often here than in San Diego. Looks as if we're expecting a downpour.\"\n\n\"You don't worry about getting kidnapped walking around on your own? Not because Knights Bridge is a dangerous place, but given your\u2014\" She waved a hand. \"You know what I mean.\"\n\n\"I do, and I don't worry about kidnappers, no. I take reasonable precautions and I don't advertise my whereabouts. Besides, who would pay the ransom if I was the one snatched?\"\n\nHis tone was reassuring, not at all dismissive. Phoebe imagined he'd gotten over any surprise or disappointment he'd felt at discovering his Edwardian princess was one of Olivia's friends from Knights Bridge.\n\nFeeling more at ease, she opened her Subaru door. \"I can give you a ride back to Carriage Hill if you'd like. I'm done for the day. My assistant, Vera, will lock up the library.\"\n\n\"A ride would be great, thanks,\" he said.\n\nPhoebe got behind the wheel and waited as he came around to the passenger side. He settled next to her and snapped on his seat belt. She started the car, wondering if she'd flat-out lost her mind.\n\nShe tried to think of something innocuous to say. \"We're supposed to get scattered thunderstorms this evening. You don't have many thunderstorms in San Diego, do you?\"\n\n\"We don't.\"\n\n\"Have you heard from Dylan and Olivia? Do you know how she likes it out there?\"\n\n\"Dylan and I spoke last night. It was early evening in San Diego.\" Noah's tone was unreadable, as if deliberately so. \"He and Olivia had just come back from a walk on the beach.\"\n\n\"That sounds so romantic.\"\n\n\"Yes, it does,\" Noah said quietly.\n\nPhoebe felt his gaze on her and wished she hadn't mentioned romance. She drove out of the Rivendell parking lot, then turned toward the village. She heard a rumble of thunder in the distance. \"A front's moving through,\" she said. \"I won't mind saying goodbye to this heat and humidity.\"\n\nA fat raindrop splattered on her windshield, then another. Lightning flickered, followed immediately by a clap of thunder. By the time she pulled in front of Olivia's house, they were in the middle of a downpour.\n\nPhoebe looked at Noah next to her. \"You don't have an umbrella or raincoat. Anyway, you don't want to get struck by lightning. I don't mind waiting until the storm's passed. It'll probably be just a few minutes. Rain's one thing but lightning...\"\n\n\"Lightning can hurt,\" Noah finished for her, with a smile.\n\nShe switched off the engine and watched the rain stream down the windshield. It made the car feel even smaller. \"At least there's not much wind,\" she said. \"I'm glad you didn't walk back here in this weather.\"\n\n\"I'd have ended up playing Scrabble with Audrey and Grace until the storm passed. They'd have beaten me for sure. I've never had the patience for word games.\"\n\n\"For a lot of other things, though.\"\n\n\"Yes,\" he said.\n\nShe'd meant his patience for his work with NAK but something in his tone made her throat catch, made her think about his patience as a lover. She flashed on him bearing toward her at the masquerade. He'd moved with purpose and intention. Not patience. Patience, she thought, was something different.\n\nHe traced a raindrop as it slid down the other side of the passenger window. \"The rain's subsiding already,\" he said, then looked over at her. \"Thanks for the ride, Princess Phoebe.\"\n\nShe placed her hands on the steering wheel and stared straight out the windshield. \"How long have you known?\"\n\n\"Not long enough.\"\n\n\"Since yesterday?\"\n\n\"When I saw the flyer about the fashion show. I was suspicious before then, but not when we met here on Saturday.\" He paused, then added, with a hint of humor, \"The slugs threw me.\"\n\n\"I'm not fond of slugs,\" Phoebe said, then made herself breathe as she looked at him. \"I recognized you yesterday, too. I didn't want to say anything in case you didn't recognize me, or in case you just wanted to forget the masquerade. It's not as if it was that big a deal. We dressed up for a benefit. We danced.\" She pried her fingers from the steering wheel. \"I'm talking too much.\"\n\n\"You're fine,\" he said. \"Olivia and Maggie know?\"\n\n\"Yes. Well, I know Maggie does, and that means Olivia does.\"\n\n\"Anyone else?\"\n\n\"No.\" Phoebe remembered her brother-in-law and gave an inward groan. \"Wait. Brandon, Maggie's husband. He saw me and recognized me straight off.\"\n\nNoah smiled. \"I've only met him briefly but I can see that he would. Do you two get along well?\"\n\n\"We always have, but he knows Maggie's my sister.\"\n\n\"You're on her side.\"\n\n\"Always, but I hope there are no sides. He won't say anything about seeing me at the ball.\"\n\n\"Neither will I. I don't want to disrupt your life, Phoebe. I enjoyed our dance. I'm glad you were there that night.\"\n\nShe realized it was getting steamy in the car; the windows were fogging up. The worst of the brief downpour was over. She pushed open her door, welcoming the rush of air. She heard the water high in the stream across the road, tumbling toward the reservoir. Finally she said, \"We were both playing a part.\"\n\n\"What part were you playing?\"\n\n\"A bold, daring princess who'd let a swashbuckler sweep her off her feet.\"\n\n\"You played your part well, princess.\"\n\nShe jumped out of the car, gulping in a breath as she leaned against the hood, ignoring that it was wet. She was vaguely aware of Noah getting out, shutting his door, walking around the hood to her. This would all be easier, she thought, if he didn't play the role of a daring swashbuckler so well in real life.\n\nHe stood next to her but didn't lean against the car and get himself wet. \"Are you okay, Phoebe?\"\n\nShe nodded. \"I overheard that man talking about you.\" She could feel the cool rainwater soaking into the back of her sundress. \"I wrote the note Olivia gave you.\"\n\n\"I'm sorry you had to run into him. I'd spotted him. That's why I left you. I went to find him but I lost him. He must have slipped into the coatroom to make the call you overheard.\"\n\n\"You don't know who he is or what he wants?\"\n\nNoah hesitated, as if debating what to tell her. \"His name is Julius Hartley. He's a private investigator in Los Angeles. He does work for a prominent law firm there but that's not his only client. I saw him on my tail a few times in San Diego. I didn't expect him to show up in Boston.\"\n\n\"He wasn't in the mountains with you?\"\n\n\"No. Do you recognize his name?\"\n\nShe shook her head. \"It's new to me.\"\n\nShe was aware of Noah watching her as she stood straight, brushed her wet backside with one hand. \"Phoebe, tell me the rest. What wasn't in the note?\"\n\nShe didn't meet his eye. \"Hartley\u2014if that's who it was\u2014told whoever was on the other end that you were with, quote, some woman dressed up like she's about to board the Titanic, unquote. He said he didn't know who she was but would find out. That it shouldn't be hard.\"\n\nNoah was silent as she stepped into the middle of the quiet road and picked up a small stone. Everything was wet, dripping. Leaves, flowers, weeds, The Farm at Carriage Hill sign with its hand-painted signature clump of chives.\n\n\"It wasn't unnerving.\" Phoebe tossed the stone into the stream. \"I'm not worried or afraid Hartley will find me. He didn't see me, and he has no reason to think I was with Olivia or Maggie, or even that I'm from Knights Bridge. Only Brandon saw me and he was sneaking around himself.\"\n\nNoah stood with her on the edge of the road, in front of the stream. He brushed off a mosquito that had found him, then picked up another stone and pelted it into the water, the energy of his throw all that suggested he wasn't calm, wasn't unmoved by the prospect of an L.A. private investigator coming to find her. As he turned to her, Phoebe noticed the tension in the muscles in his forearms.\n\n\"I left right after I overheard the conversation,\" she said. \"I thought it could be a marital situation. I didn't know... I knew he was talking about the swashbuckler I'd danced with but I didn't know the swashbuckler was you. I was avoiding Dylan and Olivia. Maggie, too. Otherwise you and I might have met as\u2014\" As what? Phoebe smiled. \"As ourselves.\"\n\n\"I'm sorry about this, Phoebe.\"\n\n\"It's not your fault. If I remember anything else, I'll let you know, but I'm sure I've told you everything.\"\n\nHe came closer to her. \"If you see Hartley or hear from him\u2014\"\n\n\"I'll tell you right away.\"\n\n\"Call me. You have Olivia's landline, and I'll give you my private cell phone number.\" He gave a small smile. \"I had a new phone delivered. I was angry at myself for having lost you and threw my last one in the sink on Saturday morning. I tend to go through phones.\"\n\n\"You can only take so much intrusion.\" When he seemed surprised at her observation, she added, \"There's a core stillness about you, and you concentrate deeply, even if you have a number of thought threads going on at once. My nephew Tyler is a bit like that. Not Aidan.\" She paused. \"Of course, I don't know you well. I just observe a lot of people in my work.\"\n\n\"I imagine you do,\" he said. \"I want your number, too, Phoebe.\"\n\nShe saw the relentless entrepreneur in him. The drive. The self-possession. The focus on the next step he had to take\u2014on action. \"I understand. Noah, if this man is a threat to you\u2014\" She broke off, took a moment to collect her thoughts. \"A man in your position must have enemies. If Hartley thinks someone in Knights Bridge is involved in whatever bone he has to pick with you, that could be a problem.\"\n\nNoah winked at her. \"You have backbone, Princess Phoebe.\" He touched the wet bark of a tree. \"This is almost the same color as your dress the other night. It was a richer brown.\"\n\n\"Beautiful, wasn't it?\"\n\n\"Yes.\"\n\n\"I wish\u2014\"\n\nWhen she broke off, he fastened his blue eyes on her, his intensity almost palpable. \"You wish what?\"\n\n\"Nothing. I don't even know what I was going to say.\" Which was true, and not like her. \"Thank you for dancing with me. It was fun.\"\n\nNoah started to say something but she pulled away from him when she saw her brother-in-law ambling down the road. Thunder rumbled in the distance, to the east, but rays of late-afternoon sun were shining now on puddles and dripping leaves.\n\n\"Hey, Phoebe,\" Brandon said. \"Here for more basil?\"\n\nShe shook her head. \"I gave Noah a ride back from Rivendell. What are you doing here?\"\n\n\"I'm working up the road,\" he said, a tightness to his expression.\n\n\"At Dylan's place?\"\n\nHe hesitated, then nodded. \"That's right.\"\n\nPhoebe tried to contain her surprise. \"Does Maggie know? Where are you staying? Are you working for your family?\" The questions tumbled out, and she was aware once more of Noah watching her, tuning in to the dynamics between her and her sister's husband. \"Never mind. It's none of my business.\"\n\n\"I need to talk to Maggie first,\" Brandon said. \"She doesn't know I'm in town. I don't want to be a distraction. She's got a lot on her plate.\"\n\n\"You could be a help,\" Phoebe blurted.\n\n\"Maybe I could be, but it's not all up to me, is it? Maggie has a say.\" He turned to Noah with a wry smile. \"That was Phoebe's stern librarian voice you just heard. Can you imagine turning in a book late to her? I once had gum stuck in a book I returned. It wasn't my gum but that's another story. Phoebe was already volunteering at the library then. What were we, thirteen?\"\n\n\"About that.\" Still reeling from her conversation with Noah, she pointed a finger at her brother-in-law. \"You're putting me in a difficult position, Brandon. I'm not going to get caught between you and Maggie, but I can't not tell her that you're in town.\"\n\nBrandon was unperturbed. \"You don't have to tell her. I will in about five minutes. She's on her way over with the boys.\" He kept his gaze on Phoebe, the slightest hint of humor in his dark eyes. \"I have spies everywhere.\"\n\n\"Your brothers, you mean.\"\n\nHe shrugged, as if she'd stated the obvious, then glanced at Noah. \"Was I interrupting anything?\"\n\n\"Nothing,\" Phoebe said, answering for Noah. She crossed the road back to her car. \"I'll be on my way. Tell Maggie I said hi.\"\n\nNeither man stopped her as she climbed back behind the wheel of her old Subaru.\n\nAs soon as she arrived on Thistle Lane, the skies opened up again, but it was just a passing shower, no thunder and lightning. She parked in her driveway, then ran through the rain onto the porch. Soaked and shivering, she sat on a wicker chair, smelling roses and wet summer grass and thinking about Noah Kendrick's hand on her cheek.\nEleven\n\nMaggie fidgeted, grabbing a canvas bag of who-knew-what out of the back of her catering van just to give herself something to do, fussing at Aidan and Tyler when they jumped out of the back straight into a puddle. She didn't care one way or the other whether they got wet or muddied their shoes, and she wasn't at Carriage Hill to cook. She was just checking on the place.\n\nChecking on why she'd seen Phoebe driving in this direction with Noah Kendrick in the front seat of her car.\n\nMaggie shut the van door. She just needed to stay busy, give herself a chance to think.\n\nPhoebe, Noah.\n\nBrandon.\n\nIt was too much.\n\nBrandon scooped up his sons, one in each arm as if they weighed nothing. They got mud on his cargo pants but he didn't seem to notice as he set the boys on the grass in the front yard and turned to Noah. \"These are my sons, Tyler and Aidan Sloan. Guys, this is Noah Kendrick.\"\n\n\"Pleased to meet you,\" Noah said as the boys greeted him politely, then promptly went back to their puddle-stomping. He glanced at Brandon, then Maggie. \"I'll be out back.\"\n\nMaggie almost stopped him so that she wouldn't be alone with Brandon, but she kept quiet. She was being ridiculous. She'd known Brandon all her life. Even if he didn't want to live in Knights Bridge, his family was here. Now his sons were here. Whatever her relationship with him had become, he was still a part of her life.\n\nShe held her canvas bag against her hip, remembered that it contained different oils she and Olivia wanted to try out in their soap-making. Olive, almond, soy, coconut. \"Where's your truck?\" she asked Brandon.\n\n\"Up at Dylan's.\"\n\n\"I didn't see it there.\"\n\n\"It's out back.\"\n\n\"What did you do, park where I wouldn't see it?\"\n\n\"I didn't want you to run away or throw a brick through the windshield.\" When she glared at him, he held up a hand. \"It was a joke, Maggie. If you have things you need to do here, I can take the boys up to Dylan's and show them what's going on. Demolition starts this week.\"\n\n\"I'm not staying. I should get Aidan and Tyler home and into the tub.\"\n\n\"Why?\"\n\n\"Because they're dirty, Brandon.\"\n\n\"I mean why the rush?\"\n\n\"Schedules. Routine. Summer's winding down. School will be starting soon. Anyway, I don't have time to explain their day-to-day lives to you.\" She immediately regretted snapping at him. \"I'm sorry. I'll give them a few more minutes to get wet and muddy and then go.\"\n\nBrandon didn't fire back, as he would have six months ago. \"They can see Dylan's place another time.\"\n\n\"Another time? Aren't you going back to Boston?\"\n\n\"No. I'm working here, Maggie.\"\n\nShe was so shocked she gasped. \"With your family?\"\n\nHe nodded without hesitation, without any indication he was embarrassed, bitter, settling, anything.\n\n\"But why?\" she asked.\n\n\"It's a job.\" He spoke with a finality that shut down further questions.\n\nMaggie took a breath. \"Okay, then. Where are you staying? You're not commuting from Boston\u2014\"\n\n\"I pitched a tent at Dylan's.\" He gave her that devil-may-care Sloan smile. \"It's rent-free. I promised the boys I'd take them camping. I was thinking we'd just camp out here.\"\n\n\"They'd like that,\" Maggie said, her throat tight with emotion.\n\nHe started for the kitchen door. \"I'll head upstairs. I need to take a shower. Helps with the camping out.\"\n\n\"Helps with the smell, too.\"\n\nHe grinned at her. \"You never were one to beat around the bush.\"\n\n\"You're sweaty. It's not...\" Why had she brought up something as personal, as intimate, as that? \"Olivia and I made a spearmint-olive oil soap. I think there's some in the hall bathroom.\"\n\n\"No goat's milk?\"\n\nMaggie didn't know if he was making fun of her and Olivia's goat's milk soaps or if he was genuinely curious. She decided to give him a straight answer. \"It's the one soap we've tried so far that doesn't have goat's milk. Spearmint works for guys as well as women. You know. Instead of a lilac scent or something.\"\n\nHe scratched the side of his mouth and let his gaze linger on her. Maggie knew he was thinking about the two of them in the shower. She didn't know what had gotten into her, bringing up soaps. She wanted to blame him, because it'd be just like him to lead her down a dead-end road and let her figure out how to get out of there on her own.\n\nHe swaggered inside. He knew what she was thinking. He always knew. That was half their problem. She wished sometimes she wasn't so damn transparent.\n\nShe saw that the boys had settled into making mud pies and checking out worms. \"I'll be in the kitchen,\" she told them. \"I can see you from the window.\"\n\nIt was as much a warning to stay out of trouble as reassurance that she was near. They weren't toddlers anymore. She headed into the kitchen and set her bag on the island, then unloaded the oils. She might as well go ahead and leave them in Olivia's pantry.\n\nWhy was Brandon back working with his family?\n\nIt wouldn't last. He disliked construction on a good day. He did it to make a living. They'd married so young, had the boys so young. They'd both had to juggle dreams and practicalities. He was already restless, frayed at the edges, when his work slowed down in Boston last year and he got laid off. Brandon didn't do well being idle. No Sloan did. One day, Maggie came home to a note telling her he'd taken off on a canoe trip down the Moose River in Maine. He'd be back \"whenever.\"\n\nAnd that was that. She'd packed up herself and the boys and moved to Knights Bridge. She started her own catering business, and now she was working with Olivia, not just providing food for events but helping to shape The Farm at Carriage Hill.\n\nMaggie loved what she was doing, but her relationship with Brandon remained unresolved, in that no-man's-land between estrangement and divorce.\n\nIt wasn't as if there was another man in her life. It'd always been just the two of them.\n\nNoah came in from the mudroom, where Buster was sound asleep. In a combative mood, Maggie decided to confront him about Phoebe. \"My sister gave you a ride back here? Why?\"\n\n\"It was raining and I'd walked into town.\"\n\n\"Phoebe's...\" Maggie smacked a bottle of coconut oil onto the island with more force than was necessary. \"She loves books and history, and she knows everything that goes on in town. She's a good soul.\"\n\n\"She's a little quirky, too,\" Brandon said, entering the kitchen from the living room. He'd made fast work of his shower, the ends of his dark hair still damp.\n\nMaggie flashed him a look. \"Phoebe's reserved.\"\n\nHe shrugged. \"Compared to you and the twins, maybe. Ask the boys about her at story hour. She gets into it.\"\n\n\"I'm here every week for story hour. I have asked them.\" Maggie gritted her teeth, wishing she'd just gone straight home instead of coming out here, then smiled apologetically at Noah. \"I should go.\"\n\nNoah's interest clearly was piqued but he seemed to contain it. \"Was there a story hour when you all were Aidan and Tyler's age?\"\n\n\"There was,\" Maggie said. \"Brandon was disruptive.\"\n\nHe grinned at her. \"You remember.\"\n\nShe resisted comment.\n\n\"Phoebe and I know we danced with each other at the masquerade the other night,\" Noah said calmly.\n\nMaggie didn't bother hiding her relief. \"I'm glad that secret's out, at least among us. The whole town doesn't need to know.\" Then she remembered who she was talking to. \"The whole world in your case, I guess.\"\n\nShe wondered how Phoebe had taken discovering that Noah Kendrick was her swashbuckler but supposed her sister had more on her mind now. Maggie didn't want to get into the mystery man Phoebe had overheard. Let Noah explain to Brandon.\n\nSuddenly she just wanted to go home, walk over to Phoebe's house and talk about flowers and flea-market finds and never mind about men.\n\nBrandon eyed her but made no comment as he turned to Noah. \"Let's have a beer before I head back to my tent.\"\n\nNoah nodded. \"Sounds like a plan.\"\n\nMaggie gathered up the boys, who wanted to stay so they could see if the rain had brought out slugs, too. She told them they could check at home. She got them in the van with a promise they'd stop at Phoebe's house on the way back. They adored their aunt Phoebe.\n\nBut when Maggie pulled up in front of her sister's small house on Thistle Lane, no one was there. She drove past the library, noticed a light on in the attic. Phoebe was probably hunting for more clothes for the fashion show. Maggie almost stopped and joined her. She'd never been up to the library's attic. Given the stories of ghosts, she wasn't sure she ever wanted to, either.\n\n\"Oh, Phoebe,\" she whispered. \"Phoebe, Phoebe.\"\n\nHer sister, the romantic at heart. Her sister, whose heart was broken so long ago. For all her own heartbreak, Maggie would make the same stupid mistakes, fight the same useless battles and press the same empty arguments if that was what it took for her to have her sons. They were worth the agony of what she was going through with their father now.\n\nAnd she wouldn't give up the good years she'd had with him. Not for anything.\n\nShe looked up at the lighted attic. Was this life of her sister's worth what she'd endured? There'd been no full-on struggle when Phoebe's heart was broken.\n\nIt wasn't like Brandon and me.\n\nMaggie blinked back tears, remembering her sister's ashen face at twenty, their father not yet cold in his grave when her college boyfriend\u2014the man everyone expected her to marry\u2014had walked out on her. It hadn't been \"just like that.\" Nothing ever was. But it had been fast, permanent and devastating.\n\nBrandon and his brothers, ever Phoebe's champions, had wanted to chase down the weasel, but she'd stopped them. Maggie had looked him up on Facebook last year. He was a lawyer now, married and living in Orlando. She knew Phoebe would never look him up herself but hadn't told her that she had.\n\nKnowing Phoebe, she wished him well.\n\nMaggie didn't.\n\nHer ability to hold a grudge was something Brandon used to appreciate about her. He didn't anymore. \"Let it go, Maggie,\" he'd tell her. \"Just let it go.\"\n\n\"Mom,\" Aidan said from the back. \"When can we go camping with Daddy?\"\n\n\"He has a tent,\" Tyler said, sitting next to his brother.\n\nMaggie's idea of camping was a cabin with heat and indoor plumbing. \"We'll work out a date with your dad, okay?\"\n\nThey thought that'd be great and proceeded to regale her with all that they'd do with their father on their camping trip, even if it was just in Dylan McCaffrey's backyard.\n\nThat was Brandon, wasn't it? Always able to fire up their sons, fill them with can-do optimism. Even at his darkest moment, when he'd watched his dreams go up in smoke, when a temporary lay-off had dragged into flat-out unemployment, he hadn't taken his disappointments out on Tyler and Aidan. Maggie wasn't even sure if he'd taken them out on her, but she'd felt them, internalized them, let them make her bleed.\n\nShe hadn't wanted him to throw his dreams overboard and yet she'd known they were weighing them down, hurting their chances of creating a stable life for their sons. For themselves.\n\nNow here he was, in their hometown, working for Sloan & Sons.\n\n\"If I ever go back to Knights Bridge, Maggie, you'll know I've failed.\"\n\nHe'd been seventeen then.\n\nThings changed, she thought. People grew up.\n\nAnd yet, as she drove along the common and turned onto her own pretty little side street, she couldn't help but feel that Brandon had given up. That he did see himself as a failure...and maybe so did she.\n\n* * *\n\nAfter Maggie O'Dunn Sloan whirled out with her two sons, Noah got two beers out of the refrigerator, opened them and took them out to the terrace. He handed one to Brandon, who'd dried off a couple of chairs at Olivia's round table. He took a long drink. \"So, Noah. You may be good with a sword, but if I've misplaced my trust and you do anything to upset my wife or her sisters\u2014\"\n\n\"You'll key my car?\"\n\nBrandon grinned. \"Right. Key your car. You really are a trip. You don't even have a car here.\" He drank more of his beer. \"Why are you here? Really.\"\n\nNoah sat down. A cool breeze stirred. He swore he could smell pesto but assumed it was just rain-drenched basil. Finally he looked across the table at Brandon. \"A Los Angeles private investigator named Julius Hartley has been on my tail. I don't know why. I saw him several times in San Diego. Then I saw him in Boston.\"\n\n\"At the masquerade?\"\n\n\"That's right. I can't say for sure that he followed me east.\"\n\n\"But it's a safe bet,\" Brandon said.\n\nNoah didn't disagree. \"I stayed in Knights Bridge in part to make sure he's not hanging around here.\"\n\n\"Do you think he was hired by someone from here or from California?\"\n\nBrandon Sloan obviously had grasped the situation immediately. Noah drank some of his beer, appreciating the cooler, drier evening. How frank could he be with this man? \"It could be either one,\" he said finally.\n\n\"Explain.\"\n\nNoah told Brandon what he knew, but he left out his reaction to Phoebe\u2014and her reaction to him. The attraction they'd experienced at the ball hadn't been just a fleeting thing born of their anonymity and the roles they were playing.\n\nPhoebe O'Dunn, his princess.\n\nIf anything, he found her even more appealing with her wild red hair, in her element making pesto, working at the Knights Bridge Free Public Library, talking to Audrey Frost and Grace Webster at Rivendell.\n\nHer baggy sweater that morning at the library wasn't in the same league as her elegant Edwardian gown, but Noah didn't care. It'd been chilly in the library and Phoebe had obviously grabbed the sweater from the collection of vintage clothes for the upcoming fashion show. He appreciated her ease with herself and her surroundings.\n\nHe'd also noticed the swell of her breasts as the old sweater had slipped off her shoulders, but he blocked that image from his thoughts, in case Brandon Sloan could read minds and decide to throttle him.\n\nThese were treacherous waters he was navigating, Noah thought.\n\n\"Do you think Maggie and Phoebe have anything to do with this Hartley character?\" Brandon asked. \"Because if you do, you're wrong.\"\n\nNoah appreciated the other man's confidence. \"I understand your concern but I don't think anything. I'm trying not to speculate.\"\n\n\"You sound like my cop brother.\"\n\nNoah thought that was a compliment, or at least a neutral observation, but he couldn't be sure and therefore said nothing.\n\nBuster put his head on Brandon's lap. Brandon scratched behind the big dog's ears as he looked out at Olivia's garden. \"How much do you know about Phoebe?\"\n\n\"She's the director and sole full-time employee of the Knights Bridge library.\" Noah started to add that she could dance but reconsidered and said instead, \"She's the eldest of four sisters. Phoebe, Maggie, Ava and Ruby.\"\n\n\"Their mother is Elly. Elly O'Dunn. She's still around.\" Brandon patted Buster, then motioned for him to lie down on the terrace, which, miraculously, he did. \"Their father died when Phoebe was a junior in college. Maggie had just started her freshman year. The twins were still in school here in town. His death was sudden. An accident. He was trimming branches on a white pine and cut corners with safety. He fell and that was that. Broke his neck.\"\n\n\"I'm sorry to hear that,\" Noah said. \"It must have been a terrible shock.\"\n\n\"Patrick O'Dunn was a good guy but I don't think he ever figured he'd live a long life. He worked in forestry. He knew how to trim a tree. He made a mistake he shouldn't have made. I'm not saying he meant to die that day.\" Brandon drank more of his beer. \"It's all history now, anyway.\"\n\n\"Was Phoebe close to her father?\"\n\n\"They all were. He left Elly and the girls more or less penniless. Elly's managed to keep things together but I don't know that her daughters always see it that way, Phoebe especially. She likes to think of herself as the sensible O'Dunn.\"\n\n\"You've known them all a long time,\" Noah said.\n\nBrandon nodded thoughtfully, then grinned. \"As far back as I can remember, I've been arguing with one O'Dunn or another. Maggie and I have been together forever.\" He sighed, serious now. \"Were together forever, I guess I should say now.\"\n\n\"What happened between you? Do you mind if I ask?\"\n\nHe shrugged. \"I don't mind. You'll hear different stories around town. I was the dreamer she wanted but all my dreams went to hell. That's the short version.\" He held up his beer bottle. \"Now I'm having a beer with you instead of going home with my wife and sons.\"\n\n\"You're protective of all of them,\" Noah said, not certain Brandon Sloan would appreciate the observations of an outsider. \"Maggie, Phoebe, their twin sisters. Their mother.\"\n\n\"I guess I am. Just don't tell any one of them. I don't have anything against you or Dylan, Noah. In fact, so far, I like you both. Dylan's given my family work and therefore me work, and I'm happy to pitch a tent at his place. I hear talk about both of you\u2014you two are getting into venture capital, he's dabbling in adventure travel, finishing up some of his father's treasure hunts.\"\n\n\"Those things are true.\"\n\nBrandon shrugged. \"Some people thought his interest in Olivia would fizzle once he got used to the idea that his father had come here looking for his birth mother as well as a fortune in missing jewels. I can see that's not going to happen.\"\n\n\"He and Olivia love each other,\" Noah said simply.\n\n\"They do. I saw that for myself Friday night. You and Phoebe...\" Brandon grimaced as if he were questioning whether he should have begun his next thought that way. \"Phoebe's the sweetest person in Knights Bridge. She has a true heart of gold. Everyone here is protective of her.\"\n\n\"Point taken,\" Noah said. \"I gather there's no man in her life?\"\n\nBrandon looked straight at Noah and said, \"No. There's no man in her life.\"\n\nNoah wondered at the certainty in Brandon's tone. Also the finality. He wasn't saying anymore. Noah appreciated the history between the people in this little town, and he understood that he wasn't part of it.\n\nHe knew when he was the outsider.\n\nBrandon finished his beer and headed back to his tent.\n\nNoah hooked a leash on Buster and let the big dog lead the way down the road, in the opposite direction of Dylan's place\u2014Grace Webster's former home. He tried to picture the road before Quabbin, when it wound into a picturesque valley populated with small New England towns. Now Dana, Greenwich, Enfield and Prescott were gone.\n\nHe and Buster came to a yellow-painted gate that marked the border of the Quabbin watershed. The old road continued on the other side of the gate, eventually leading into the water's edge, as if the lost towns still were there.\n\n\"Sorry, Buster,\" Noah said. \"No dogs allowed. We have to turn around.\"\n\nThey walked back up the road to Carriage Hill. The dog pulled hard on the leash and Noah noticed a squirrel chattering at them from a pine branch. He could hear birds, but otherwise it was a cool, quiet summer evening, the daylight graying with the approach of dusk.\n\nBy the time he and Buster arrived in Olivia's kitchen, Noah was hungry. First, he'd feed Buster, then he'd heat up soup she'd frozen. He could always add some of Phoebe's pesto, or go pick a few herbs in the garden. He'd already learned that Knights Bridge had only one restaurant, so he better save that option. By the standards of the people who lived there, the town wasn't isolated\u2014they were used to driving to stores and restaurants in nearby towns.\n\nBy Noah's standards, it was the middle of nowhere.\n\nHe found a rag in the mudroom and wiped the dog's muddy paws. \"Well, Buster, my friend, there may not be a good Mexican restaurant within thirty miles, but we can consider ourselves lucky they take in strays around here.\"\n\nWhile his soup heated, he called Dylan but didn't reach him and hung up without leaving a voice mail.\n\nTwo minutes later, Dylan called back. \"You're bored,\" he said.\n\nNoah stirred his simmering soup with a wooden spoon. \"How could I be bored? There's always something to do here. If I'm not walking the dog, I'm giving him food and water, and if I'm not doing that, I'm dodging bees in the catmint.\"\n\n\"Catmint, Noah?\"\n\n\"It's the purple stuff by the terrace.\"\n\n\"I know what it is. Olivia told me. Who told you?\"\n\n\"Maybe I already knew.\"\n\n\"You didn't already know,\" Dylan said, confident.\n\nNoah wished he hadn't brought up catmint. \"How does Olivia like San Diego?\"\n\n\"Loves it. Who doesn't? We're out on my porch now looking at the ocean.\" Dylan paused. \"Anything new on Julius Hartley?\"\n\n\"Not on my end. I haven't talked to Loretta yet today. Why don't you forget about Hartley and enjoy the ocean breeze with Olivia?\"\n\n\"She got a good dose of what Loretta's like last night. We had martinis and talked about your stalker private investigator while we admired the sunset over the Pacific.\"\n\nNoah sighed. \"I miss the Pacific.\"\n\nDylan ignored him. \"Any sign of Hartley in Knights Bridge?\"\n\n\"No. I'm sorry you found out about him. Two years ago, we wouldn't have paid any attention. We'd have been too busy. Now you're busy and I'm...\" Noah frowned, noticing that Buster had wandered into the living room and jumped up on the couch. \"Does Olivia let Buster on the couch?\"\n\n\"No. Noah?\"\n\n\"I have to go. Buster and I need to straighten out who's boss.\"\n\n\"Good luck with that,\" Dylan muttered.\n\nNoah hung up and shooed Buster back onto his spot in front of the fireplace. The soup was bubbling on the kitchen stove. He found a pottery bowl and dumped in a healthy serving. The soup was orange and had a faint, pungent smell he couldn't identify. He checked the handwritten label on the freezer container.\n\nCarrot soup.\n\nNot much help. He knew carrots, but that wasn't what he smelled. He debated calling Dylan back to ask him. Or he could call Phoebe, eldest of the O'Dunn sisters. She'd probably know.\n\nInstead he brought his soup into the living room and sat with Buster in front of the cold fireplace. \"I'm a lonely man, Buster,\" he said with a laugh. \"A lonely, lonely man.\"\n\nAnd completely insane. All he had to do was dial his assistant, and he could have a car at The Farm at Carriage Hill in an hour and be on a flight somewhere\u2014anywhere\u2014before the sidewalks folded up in Knights Bridge.\n\nHe wondered if Brandon Sloan was managing to have a decent dinner up in his tent, but Brandon was a Knights Bridge native as well as a grown man. He could figure out what to eat for dinner.\n\nAnd Phoebe? What was she up to this quiet summer evening?\n\nWas she regretting that he hadn't kissed her when he'd had the chance during the storm, then again after the storm? Noah pictured her luminous turquoise eyes against the gray rain, and he could see her lick her lips. He squirmed as he felt pressure in his groin. Everyone in Knights Bridge could regard her as untouchable, but he didn't.\n\nAll he wanted to do was to touch her.\n\nTo make love to her.\n\nHe could see her wet skirt as she'd walked away from her car.\n\nHe took a long, slow, deep breath, held it, let it out again and tried his soup.\n\nGinger.\n\nThat was what he'd smelled. It was carrot-ginger soup, and it wasn't bad on a cool summer night on a dead-end road, with only a big, ugly dog for company.\nTwelve\n\nAfter a quiet, uneventful day at the library, Phoebe walked across the common to the Swift River Country Store and made her way back to the wine section. She was debating between two different brands of merlot when she heard a man talking up by the register. His voice sounded familiar but she couldn't quite place it. Abandoning the wine, she edged to the end of the aisle and peered past a display of homemade baked goods.\n\nThe man she'd overheard in Boston on Friday\u2014Julius Hartley, the private investigator tailing Noah\u2014was standing at the checkout counter, quizzing Greg Hughes, the teenage son of the owners.\n\nHartley had on a dark blue shirt, light khakis and light canvas shoes, as if he were about to step out onto a golf course.\n\nHe set a large coffee-to-go on the scarred wood counter. \"Sleepy Hollow here has one bed-and-breakfast,\" he said. \"I stopped by and guess what? The owners are in Montreal for the week. Doesn't New England have short summers? How can you run a bed-and-breakfast if you disappear for one whole week in August?\"\n\n\"It's kind of a hobby for them,\" Greg said from behind the register. He was an avid reader of science fiction and a recent high school graduate, on his way to Bowdoin College in Maine. \"They're professors at UMASS. They go to Montreal this time every year.\"\n\n\"Got it. I understand a new place has just opened up on some back road.\"\n\n\"Carriage Hill,\" Greg said, taking Hartley's money. \"It's not really a bed-and-breakfast. It caters to events. Weddings, showers. You know. Anyway, the owner's out of town right now, too.\"\n\n\"I see. Well, luckily I'm not staying. I just need directions to Elly O'Dunn's place. I understand she's selling some of her goats.\"\n\nPhoebe tensed. How did he know about her mother? What did he want with her?\n\n\"You're interested in buying goats?\" Greg asked, skeptical.\n\n\"Sure, why not? What's the O'Dunn farm like?\"\n\n\"Simple. A few acres, a couple of sheds, a house that has plumbing and electricity but not much else.\"\n\n\"A stove?\"\n\n\"Yeah, a stove. I guess.\"\n\n\"You guess?\"\n\n\"You haven't met Mrs. O'Dunn yet, have you?\"\n\n\"No, I have not,\" Hartley said. \"There are restaurants in town?\"\n\n\"One right now. Smith's. You can walk to it from here. There are more within easy driving distance. We have a good range of take-out food here at the store.\"\n\n\"Good to know,\" Hartley said without enthusiasm.\n\nHe left with his coffee, and Phoebe darted out of the store, giving Greg a quick wave. When she reached the sidewalk, Noah's mystery private investigator\/stalker had already crossed the street to the common and was making his way into the shade of a trio of sugar maples. He sat on a bench. He didn't look to be in a hurry.\n\nRemembering that he didn't know she'd overheard him or even had been in Boston, Phoebe took a breath and slowed down, crossing the street as she would if she had done what she'd planned to do\u2014buy a bottle of wine to go with a quiet dinner at home. No meetings, no family, no friends, no goings-on.\n\nNo Noah.\n\nShe'd dreamed about him. She didn't know what that meant but she'd awakened in a sweat and went out to the garden at dawn, calming herself by dead-heading her flowers. Noah Kendrick was off-limits. They'd gotten caught up in the drama of their night together, the thunderstorm, the moment of recognition that their identities were no longer secret.\n\nHis life was in San Diego. Hers was in Knights Bridge.\n\nHe was a billionaire with a fancy for Hollywood starlets, and she was a small-town librarian who loved her job and was devoted to her family. No Hollywood rakes for her. No men at all, lately.\n\nAnd Noah was Dylan McCaffrey's and now Olivia Frost's friend. Phoebe was, too, and she wasn't about to complicate their lives by getting involved with him.\n\nWhich was getting way ahead of herself but it'd been an intense dream.\n\nShe walked across the lawn, past the Civil War monument, her normal route back to the library, but instead of continuing to the opposite street, she paused in front of Julius Hartley. She couldn't let him drive out to her mother's place on the pretense of buying goats from her. That wasn't going to happen, Phoebe thought. It couldn't happen, and she wasn't waiting to get Noah out here to take care of it.\n\n\"Phoebe O'Dunn,\" Hartley said, looking up at her from the bench. He took a sip of his coffee. \"Town librarian and survivor of the Titanic. That was quite a dress the other night.\" He sat back and grinned at her. \"You can breathe, Phoebe. Your secret's safe with me.\"\n\nShe gave him what she hoped was a cool look. \"How did you know?\"\n\n\"I've been here in Sleepy Hollow for four hours. I know a lot about you and your little town.\" He pointed with his coffee. \"I even read the plaque on your Union soldier. It's my job to find out things. I'm good at it.\"\n\nPhoebe plucked a maple leaf off a low-hanging branch. He'd said he'd find out who she was, and he had. \"Your name's Julius Hartley. You're a private investigator from Los Angeles.\"\n\n\"So you've been busy, too. Who told you? Noah Kendrick? Dylan McCaffrey? Loretta Wrentham?\"\n\n\"It doesn't matter.\" She dropped the leaf into the grass. \"I want you to leave my mother alone.\"\n\n\"I can't go look at her goats?\"\n\n\"No, you can't.\"\n\nHartley got to his feet, casually, as if he didn't have a care in the world. \"You're tougher than you look, Phoebe O'Dunn.\" He drank more of his coffee. \"It's not apparent at first. You come across like a mild-mannered redhead in a pretty little sundress and sandals, but you're a pit bull when it comes to protecting your mother and your sisters. Who protects you?\"\n\n\"We all look after each other.\"\n\n\"What about Kendrick? Are you looking after him now, too?\"\n\n\"I'm not discussing Noah or anything else with you.\"\n\n\"Except for your mother's goats,\" Hartley said, clearly amused. \"Okay. I left my thumbscrews in California.\" He squinted toward the library, where two young children Phoebe recognized were running down the steps ahead of their very pregnant mother. \"I wonder what their favorite books are. I was a creepy little kid, I think. I liked Edgar Allan Poe.\"\n\n\"At four?\"\n\n\"I was a little older. Eight, maybe.\" He winked as he turned back to her. \"Consider that a telling clue, Phoebe. Are you going to tell Kendrick I'm in town?\"\n\n\"That assumes I'm in contact with him.\"\n\n\"Yes, it does.\"\n\nShe tilted her head back and eyed him. \"What are you doing in Knights Bridge, Mr. Hartley?\"\n\n\"Right now I'm drinking coffee and enjoying a pleasant summer afternoon.\"\n\n\"Are you here because of me\u2014because I danced with Noah?\"\n\n\"You two did steal the show the other night. Noah's friend Dylan is marrying a local girl. Olivia Frost. You know that, of course.\"\n\nPhoebe hadn't expected that response. Was he here because of Olivia? Because she was engaged to Dylan?\n\n\"Whoa. Easy there, Phoebe. No fainting.\"\n\n\"I'm not close to fainting.\" She straightened her spine. \"If I even think you're here to cause trouble, I'll notify the authorities.\"\n\nHartley laughed. \"Trouble? You have a good imagination, don't you? I guess being surrounded by books would fire up the creative juices.\" He paused, studied her again. \"You and Kendrick the other night. There was some kind of connection. Some sizzle between you two.\"\n\n\"We were in the land of make-believe.\" Phoebe immediately regretted her comment. She couldn't let this man get to her, couldn't engage him\u2014especially about Noah. \"I have to finish up at work. You'll leave my mother alone, right?\"\n\n\"Sure. No problem. Relax, Phoebe.\"\n\nShe didn't respond and ducked under the low maple branch.\n\n\"Does Kendrick know it was you the other night?\" Hartley called to her, his voice soft but no less cocky. \"A billionaire could solve all your problems.\"\n\nPhoebe spun around at him. \"I don't have any problems I can't solve on my own.\"\n\nHartley grinned at her. \"Sure you do. We all do. We all have dreams, too. I'll bet even you have dreams, Phoebe O'Dunn.\"\n\nEven you.\n\nHe crushed his coffee cup in one hand, kept his eyes on her. \"Noah Kendrick could make your dreams come true, don't you think? Then there's your sister the caterer, your twin sisters the theater majors and your eccentric mother. They all have big dreams. What about you, Phoebe? Do you have big dreams or little dreams?\"\n\nShe knew she should just walk away but didn't. \"There's no such thing as a little dream.\"\n\n\"Maybe so. A small-town New England librarian and a California billionaire. Dreams don't get bigger than that, do they?\"\n\nHe was overstepping, and Phoebe saw that he knew it. She met his gaze, drew on her experience with the public and her natural reserve to keep her emotions to herself. \"Why are you here, Mr. Hartley? Are you making sure we locals don't take advantage of Noah and Dylan? Are you looking for information on them for a client? For some blackmail scheme? Is that what this is about?\"\n\n\"How would you work your mother's Nigerian Dwarf goats into blackmail?\" Harley laughed, then waved a hand at her. \"Easy, Phoebe. I'm harmless. Go back to your musty books.\"\n\nShe didn't know if he was trying to be funny or deliberately insulting. She took a breath and watched him walk in the opposite direction across the common, back toward the country store. \"You still can't pretend you're interested in buying any of my mother's goats,\" she called to him.\n\nHe held up a hand without turning around, signaling acknowledgment of her statement more than acquiescence.\n\nShe headed back to the library, dialing the cell phone number Noah had given her. When he picked up on the second ring, she hardly waited for him to say hello. \"He's here,\" she said. \"Your guy. Julius Hartley. He's in Knights Bridge.\"\n\n\"Where are you?\"\n\n\"At the library. We close early on Tuesday.\"\n\n\"Wait for me there.\"\n\n* * *\n\nPhoebe headed up to the attic and into the hidden room, hot and stuffy even with the cooler, drier air. It'd be another ten minutes before Noah would get there from Carriage Hill. She opened the corner door, light streaming in from the window, no sign of Julius Hartley on the common. Did his reasons for following Noah east have anything to do with Knights Bridge? With Olivia or Dylan, or their upcoming marriage? Phoebe couldn't even guess. Noah and Dylan lived in such a different world than the one she knew.\n\nWhatever he wanted, she hoped Hartley was on his way back to San Diego.\n\nShe certainly didn't need him skulking around town ahead of the fashion show.\n\nShe would have to decide soon about using any of the dresses created by her mystery seamstress, exposing this room to town scrutiny.\n\nShe heard footsteps on the worn attic floor and assumed it was Vera, who was staying late to sort a new box of donated clothing for the fashion show. She'd been happily pulling out silk scarves when Phoebe had ventured up to the attic. \"I'll be right out,\" she called, quickly shutting the corner door by the window. She hadn't told Vera yet about the hidden room.\n\n\"It's me, Phoebe. Noah.\"\n\nOh, damn.\n\nHe was just outside the door that the free-standing closets had concealed. She'd pulled them away and left the hidden door\u2014now not-so-hidden door\u2014partly open. He had to have noticed. Not that he realized she was hiding anything. Which she wasn't, not really. She was just taking her time before she revealed her discovery.\n\nShe was suddenly sweating, her heart racing, as if the mysteries of her unknown seamstress somehow revealed deep, dark secrets of her own\u2014as if Noah would be able to see through her defenses.\n\nExcept she didn't have any deep, dark secrets.\n\n\"Phoebe,\" he said, his voice very near now. \"Are you\u2014\"\n\n\"I'm right here.\" She opened the door wide, not surprised to find Noah standing there. She smiled, feeling ridiculously nervous, even self-conscious. \"You're not a ghost.\"\n\n\"No, I'm not a ghost.\"\n\n\"What did you do, drive a hundred miles an hour to get here this fast?\"\n\n\"I was almost to town when you called. The library's closed but your assistant let me in. Can I come in?\"\n\nShe nodded. \"Of course. It's hot in here. I was just about to leave.\"\n\nNoah stepped into the tiny room. Since she'd closed off the window by shutting the corner door, the room was almost dark. She switched on an old lamp she'd dragged in from another part of the attic. It didn't offer much light as he glanced at the shelves and baskets of fabric and sewing supplies, the trunks and garment bags, the dresses she'd pulled out and left draped on a chair.\n\n\"I didn't expect this in a library attic,\" he said. \"The dresses you and Olivia and your sister wore to the masquerade came from here?\"\n\n\"This room's something of a mystery,\" Phoebe said, keeping her tone neutral. \"I didn't realize it existed until a few weeks ago. I haven't told anyone else about it. I stumbled on it when I was chasing marbles.\"\n\n\"Something for a librarian to do. Chase marbles.\"\n\nHis tone held a hint of humor, but she could see his focus and suspected he was trying to keep himself from pushing too hard, jumping on her for immediate answers about her encounter with Julius Hartley.\n\nShe ran a finger along the edge of the old sewing table. \"It's as if whoever created this room just stepped out, but I'm guessing it's been decades. I want to know who she was. Is, maybe. Someone in town, someone who left town?\"\n\n\"Any clues?\"\n\n\"Not many. She studied French and was clearly fascinated with Hollywood, and she could do anything with a needle and thread. She must have made her own patterns for most of these dresses.\"\n\n\"You found your Edwardian gown here?\"\n\nPhoebe stood back from the table. \"It's one of the first ones I came across. I had it and the dresses Olivia and Maggie wore cleaned\u2014\"\n\n\"But you didn't tell them about this room.\"\n\n\"Right. I will, though.\" She waved a hand. \"You're not here about an old sewing room, and it's not why I called you.\"\n\n\"You said Julius Hartley is here in town.\"\n\n\"I saw him at the country store. He took a coffee out to the common and I talked to him there.\"\n\nNoah steadied his gaze on her, his eyes a deep navy in the dim light. \"What did you two talk about?\"\n\nHis stillness, his control struck her as incredibly sexy, but she kept her reaction to herself and repeated her conversation with Hartley. She left out only his comments about dreams.\n\nWhen she finished, she picked up a sequined dress she'd draped on a rickety metal chair. \"You're sure you don't know Hartley or anything about why he's following you?\" she asked, folding the dress, a design of peach silk crepe, sequins and fringe.\n\n\"I'm sure,\" Noah said.\n\nHe'd remember, Phoebe thought. That was how his mind worked. \"What could he want with my mother? He's not interested in buying goats. I don't want that man in her life at all.\"\n\n\"I'll make sure he stays away from her. Where does she live?\"\n\nPhoebe shook her head. \"I appreciate the offer, Noah, but I'll see to my mother. She's off work now. She'll be at home. I think she was planning to can tomatoes tonight.\"\n\n\"Hartley's my problem, Phoebe.\"\n\n\"And my mother's mine,\" Phoebe said with a smile, then set the folded dress on the sewing table.\n\n\"Any idea how Hartley knew she keeps goats?\"\n\n\"Not specifically, no, but it's not a secret. If he asked around town, someone would have told him.\"\n\n\"Olivia and your sister Maggie are getting into goat's milk soaps. That might explain his interest in your mother.\" Noah touched the fringe on the folded dress. \"The workmanship is impressive.\"\n\n\"It is, isn't it? It's so precise. Whoever worked up here paid attention to every detail. Maybe it wasn't right to wear the dresses to the masquerade. I'd assumed we'd use some of the more fun and interesting outfits in here for the fashion show.\"\n\n\"That makes sense.\"\n\n\"When I came up here just now...\" Phoebe glanced around the small room. \"It felt as if I was intruding into someone's private life. Someone's hidden life.\"\n\nNoah turned from the sewing table. \"If no one was supposed to find this room, then why leave everything behind? Why create it in the first place?\"\n\n\"I should get Olivia up here when she's back home. She collects vintage linens. Maybe she can help. And Ava and Ruby know more about Hollywood-related fashions than I do. I recognized the dresses Maggie and Olivia wore, but who wouldn't?\"\n\n\"They're in good condition.\"\n\n\"They were stored properly. The fabrics have faded and yellowed over the years, but almost everything's in remarkably decent shape.\" Phoebe noticed that Noah's eyes were half closed as he watched her, and she forced a small laugh. \"Did I lose you at 'vintage'?\"\n\n\"You haven't lost me at all.\"\n\nShe believed him. She took in a breath. \"Who knows, maybe this room's the work of our resident ghost. Some people believe the library really is haunted.\"\n\n\"By a handsome sea captain?\"\n\n\"You never know.\"\n\n\"I don't see you sneaking up here to talk to ghosts.\"\n\n\"Not even a sea captain?\" She angled him a smile, ignored the sudden rush of emotion, of pure awareness at being alone with him in such a confined space, in the heat, the stillness and quiet of the hot summer afternoon. \"A swashbuckler, maybe?\"\n\nHe touched a finger to a long strand of her curls and tucked them behind her ear, where there was no chance they'd stay put. \"I'm sorry Hartley showed up in your town, Phoebe. He got under your skin. I won't let him hurt you or your family.\"\n\n\"You're more intuitive than you like to let on, I think.\" She realized he hadn't pulled back, hadn't tucked her hair behind her ear just because it was driving him crazy. \"You like people to think you're just a math genius who got lucky, but there's more to your success than that. You're always in control, aren't you?\"\n\n\"Not always,\" he said, then let his finger drift down to her cheek as if to give added meaning to his words.\n\nPhoebe licked her lips, cleared her throat. \"I've read several books and magazine articles on entrepreneurial thinking. You're a natural entrepreneur. You're young\u2014I wouldn't be surprised if you end up as a serial entrepreneur who is energized by starting new businesses, and helping other people start new businesses.\" She suddenly felt the bodice of her sundress sticking to her breasts. What was she saying? Just stop. Be quiet. \"Anyway. That's neither here nor there.\"\n\nHe smiled. \"I like it that you're trying to figure me out. Most people just give up.\"\n\n\"You're a good listener,\" she said, realizing she wasn't embarrassed at rattling on about entrepreneurship. \"I could talk to you all night.\" She winced at her own words, embarrassed now. \"Sorry. That didn't come out right.\"\n\n\"You can talk to me anytime, Phoebe.\"\n\nShe was used to talking to people in her work and around town, but not like this\u2014not with romantic undertones, with sexual undertones, and not to a man like Noah. Not to the best friend of her friend's fianc\u00e9. Phoebe couldn't deny she was attracted to him. She couldn't even pretend she wasn't. She just knew it didn't mean anything, and she wouldn't do anything to make things awkward between him and Dylan.\n\n\"We should go,\" she said. \"Do you have any idea what Julius Hartley wants with you? I imagine you have your share of troublemakers in your life. Is he one of them?\"\n\n\"He could be. I don't know. I'll figure it out.\" Noah traced a fingertip across her jaw, into her hair. \"I should have guessed you have red hair.\"\n\n\"Because of my freckles? I used Ava and Ruby's pancake theatrical makeup to cover them, but dancing with you took its toll. You put me through my paces on that dance floor.\"\n\nHe threaded his fingers through her hair. \"Not because of your freckles. Because of the color of your eyes.\"\n\nPhoebe could hardly breathe. It was as if being in the hidden room had returned them to the fantasy of the masquerade, to the intensity of their attraction to each other\u2014as swashbuckler and Edwardian princess.\n\n\"Phoebe.\" His voice was just above a whisper. \"I'm sorry Hartley interrupted us on Friday. I'm sorry he's in your town now. That I brought him here. You all don't have to worry about me.\" He curved his palm over her bare shoulder as he gave her one of his enigmatic smiles. \"I can take care of myself.\"\n\n\"You don't have a real sword with you, do you?\"\n\nShe was kidding, but he was amused. \"Only at home in San Diego.\"\n\n\"My guess is you don't do much by half measures.\"\n\nHe smoothed his hand down her upper arm. \"Hold that thought.\"\n\nWhen Olivia had first told her about Noah, Phoebe hadn't considered that he'd be this sexy, irresistible to the point that she was tingling, even burning. \"I think meeting me as an Edwardian princess colored your ideas about me.\"\n\n\"Nope. Meeting you hunting slugs in your scarf and shorts did.\"\n\nShe gave a small laugh. \"Are you making fun of me and my hometown, Noah?\"\n\n\"Never.\"\n\n\"Personally, I think you're just bored.\"\n\n\"Anything but bored, Phoebe.\"\n\nHe slipped his arm around her waist. She didn't protest, didn't pull back. It was as if they were dancing again. Their musty surroundings and the hot, still attic and the Hollywood dresses, the reality of their circumstances, didn't matter. She could almost hear the laughter of Friday night, the clinking of glasses, the music. She leaned into him more than she had then, in front of hundreds of people. He was all taut, lean muscle. That would be a surprise to most people, no doubt.\n\n\"I'm glad we danced,\" she whispered. \"It made the night special. Memorable.\"\n\nHe said her name again, then his mouth found hers, his lips grazing hers in the lightest of kisses. Yet it had her reeling, aching. He was so controlled, so deliberate. He knew exactly what he was doing. She put a palm on his arm, felt the hard muscle. In every way, he wasn't a man to underestimate.\n\n\"Something happened between us when we danced,\" he said, kissing her again. \"It's not going away.\"\n\nShe nodded. \"I know.\"\n\nHe looked into her eyes with an intensity she'd never experienced. \"It'll be okay, Princess Phoebe. I promise.\"\n\n\"You're more of a real swashbuckler than I am a real princess.\"\n\n\"I don't think so.\" He stood straight, glanced around the small room. \"Maybe this room was meant for you to discover.\"\n\n\"A custodian could have dusted and vacuumed in here and not realized no one knew about it. Anyway, I need to get out to my mother's and make sure Hartley didn't stop in, after all. She'd probably talk him into buying a goat. It'd serve him right.\" Phoebe inhaled, realized she was about to hyperventilate. \"I live just down Thistle Lane. I'll run and get my car. Did you walk?\"\n\n\"I have Olivia's car. Come on. I'll drive you out to your mother's place.\"\nThirteen\n\nNoah figured he could be Errol Flynn and it wouldn't make a difference to Phoebe O'Dunn.\n\nHe could be a real-life Zorro.\n\nA real-life musketeer.\n\nHe could be worth a trillion and she wouldn't be intimidated.\n\nHe wondered if she knew that about herself.\n\nElly O'Dunn lived a few miles out of the village center, on a winding dirt road that, according to a sign, led to a state forest. Somehow Noah didn't need Phoebe to point out that the mailbox topped with bluebirds marked her mother's driveway. He turned into it, pulling up behind a dark brown house nestled among shade trees, stone walls, gardens and a large yard that needed mowing. As he got out of the car, he noticed a couple of small, rough-wood outbuildings, a fenced-off pen, open fields and surrounding woods, still and green in the afternoon summer sun.\n\n\"My mother's and my sister Ava's cars are here,\" Phoebe said. \"I don't see any other cars. Maybe Hartley changed his mind.\"\n\nNoah recognized sunflowers standing tall in the late-day sun. A lot of sunflowers, he thought as Phoebe joined him on a flagstone walk.\n\nShe squinted at the flower and vegetable gardens. \"It's quite a bit for Mom to take care of on her own but she loves it.\"\n\n\"Does she have help?\"\n\n\"Ava and Ruby have been here this summer, and now Maggie lives in town. We all do what we can to help.\"\n\nHe smiled. \"I hope you like sunflowers.\"\n\n\"And goats,\" Phoebe added with a laugh.\n\n\"Are the goats much work?\"\n\n\"A ton, at least by my standards. They have to be milked twice a day, for starters. Mom never cared about having them earn their keep, but the artisan goat's milk soaps Maggie and Olivia are trying out could at least help with expenses.\"\n\n\"Do you have any pets?\" Noah asked as he followed Phoebe toward the screened-in back porch.\n\nShe shook her head. \"Not a one. I get my pet fix here. Mom has two cats and the laziest dog on the planet, too. She's probably lying in the sun somewhere. My mother's dog, not my mother.\" Phoebe laughed again, visibly more relaxed. \"Do you have pets?\"\n\n\"No pets.\" He heard wind chimes tinkling on the porch in the slight breeze. \"It's a lovely spot, Phoebe.\"\n\n\"My parents built the house together. It's got its quirks but it's warm in winter and the roof doesn't leak.\"\n\nA young woman with purple-black hair emerged from the six-foot-tall sunflowers. She carried a basket on one arm and waved as she stepped over a low wood-and-chicken-wire fence. \"I swear I never want to see another tomato, but I say the same thing at this time every year.\" She smiled at Noah. \"You must be Noah Kendrick. Hi, I'm Ruby O'Dunn, Phoebe's sister.\"\n\nNoah returned her smile. Her natural hair color, he guessed, was a shade of red. She wore a tank top, a long black skirt and sandals, and her basket was filled with fat, ripe tomatoes. \"Nice to meet you, Ruby.\"\n\n\"My sister Ava's in the kitchen. We're canning tomatoes. Mom's convinced we'll finish tonight but not a chance.\" Ruby set her basket on a porch step and stood up straight, frowning at Phoebe. \"You look awfully serious. What's up?\"\n\n\"Did a man name Julius Hartley show up here?\"\n\n\"No. Why? Do you know him? Who is he?\"\n\n\"I ran into him in town,\" Phoebe said. \"We need to talk about him.\"\n\n\"Whenever you're ready.\" Ruby was obviously intrigued but scooped up her basket again. \"I'll be in the kitchen dealing with these babies. They're only the beginning.\" She grinned at Noah. \"A friend of mine from New York was up here earlier this summer and said we reminded her of the Weasleys in Harry Potter. Red hair and cozy madhouse.\"\n\nPhoebe laughed. \"If only we were wizards, too.\"\n\n\"No kidding. Then Ava and I could wave our magic wands and these tomatoes would be canned and in the cupboard in a flash. See you guys later. Noah, nice meeting you.\"\n\nAs Phoebe led him across the lawn, past the fenced-in gardens, he thought Ruby's New York friend might have a point. The O'Dunn place was ramshackle and homey, radiating a chaotic warmth and good cheer that he suspected was more accidental than intentional.\n\nThey came to a small barn with an attached pen, miniature goats prancing in the grass. \"They're Nigerian Dwarf goats,\" Phoebe said. \"Cute, aren't they?\"\n\nNoah had never considered goats cute. In fact, he'd never considered goats at all.\n\nA woman came out of the shed, shutting the door behind her. Elly O'Dunn, he assumed. She was younger than he'd expected, barely in her fifties, with wild graying light red hair and warm blue eyes. \"Phoebe! I didn't know you were here. I was just tidying up the stalls before we start tackling the tomatoes.\"\n\nPhoebe introduced Noah to her mother, whose bubbly energy was a sharp contrast to her eldest daughter's quieter nature. \"Mom, has a man named Julius Hartley been in touch with you?\"\n\nElly nodded, no hint of alarm or suspicion. \"He stopped by the town offices and asked if I was interested in selling any of the goats. You know it's something I've been thinking about, since we have about three or four too many. So I told him, and he said he'd be by after I got out of work. I haven't seen him. I don't think he was really serious about the goats.\"\n\n\"Did he say why he was in town?\" Noah asked.\n\n\"No, but I was under the impression he knows people here.\" Elly came through a gate, shutting and latching it behind her as three goats with shiny brown coats nudged the fence. \"Don't let the size of my little friends here fool you. They're very adept at getting out of confinement. They'll eat anything. Well, I'll be up in the garden if you need me.\"\n\nElly headed up toward the house. Phoebe glanced at Noah. \"My mother and her goats and tomatoes and such are a bit different from what you're used to, I'm sure,\" she said.\n\nHe smiled. \"They don't allow goats where I live.\"\n\n\"My mother got two goats to keep her company after Dad died. I was commuting to college and Ava and Ruby were still at home, but they had their own things going on.\"\n\n\"Were the goats her idea?\"\n\n\"Yes, but we all encouraged her to find something just for herself.\" Phoebe smiled wistfully. \"Dad would never have had goats. Chickens, maybe, but he didn't really like having farm animals.\"\n\nThey walked past an old stone wall that marked a field that Phoebe told him the goats had cleared last summer. The tall grass was dotted with wildflowers\u2014yellow, white, deep orange. Noah had no idea what they were but suspected the O'Dunns would, Phoebe especially. She was the sort who soaked up knowledge, loved to learn new things.\n\nShe seemed pensive and yet also comfortable at her childhood home. She squinted back at the shed. \"My father built the shed when he first moved out here. He lived in it for a few years, before he met my mother. He always planned to convert it into a barn. There's a natural spring just into the woods, and a stream that he loved. There's a lake in the state forest farther down the road.\"\n\n\"This area seems to have a lot of water,\" Noah said.\n\n\"It does. Ponds, lakes, streams, rivers. That's why the Swift River Valley was chosen for a reservoir. Quabbin is a Native American word that translates as 'place of many waters.'\"\n\n\"It's a beautiful spot to have grown up, Phoebe.\"\n\n\"My father was twelve years older than my mother. He worked in forestry jobs. He wanted to make money as a farmer but it never happened.\" She took a breath. \"We still miss him. We always will.\"\n\n\"Brandon told me what happened. He didn't go into detail. Phoebe\u2014\"\n\n\"I found him,\" Phoebe said abruptly. \"I had to tell my mother, my sisters. It was a difficult time but we got through it. My mother has a full life but she's never remarried. Some days I don't know how she manages this place, even with our help. She has a live-for-today mentality that sometimes makes tomorrow a little sketchy.\"\n\n\"You worry about her,\" Noah said.\n\n\"It's hard not to sometimes. I can't imagine why Hartley went to the trouble of driving out here. The goats are providing milk for Maggie and Olivia's soaps. Olivia's the fianc\u00e9e of your best friend.\" Phoebe slowed her pace. \"Could Hartley be interested in Dylan and not you? Worried we're taking advantage of him?\"\n\n\"No one takes advantage of Dylan.\"\n\nPhoebe stopped, the sunlight deepening the rich turquoise of her eyes as she studied him a moment. \"Do people take advantage of you?\" she asked finally.\n\nNoah shrugged. \"They didn't when Dylan was working down the hall.\"\n\n\"Are you worried they will now?\"\n\n\"I'm making the transition from having total control over NAK. It's good for the company and its future. Regardless, I've always been better at creating ideas than watching for snakes in the grass.\"\n\n\"Fencing helps?\"\n\n\"It helps me focus on the present.\"\n\n\"Otherwise you get stabbed,\" Phoebe said. \"It must be a great motivator.\"\n\nHe leaned in close to her. \"In classical fencing, we try to touch without being touched. Any touch with a sharp sword can be deadly. Sport fencing is different. It's based on a point system.\"\n\n\"You get points for touching.\"\n\nNoah saw her amusement and smiled. \"It's a bit more complicated than that, but sports fencing isn't the same as a real fight. It's an athletic competition. Classical fencing isn't a real fight, either, since we're not out to kill each other and take safety seriously, but it simulates a real fight.\"\n\n\"Are you good?\" Phoebe asked.\n\n\"I'm not competitive in that way. I enjoy fencing and karate because they work with my schedule and personality.\" He heard the caw of a crow out in the field and suddenly realized how quiet it was. \"And you, Phoebe? Do you participate in a competitive sport?\"\n\nShe shook her head. \"Never have. I like to walk and work in the garden. I took a yoga class once. I enjoyed it, but yoga's hardly competitive.\" A breeze blew strands of red hair into her face as she gave him a sideways glance. \"So if you don't want to continue to run NAK or end up as the loose-end founder hanging out in the halls, you have to figure out what to do with the rest of your life.\"\n\n\"I could learn to make goat's milk soap.\"\n\n\"Maybe that's why Julius Hartley is following you,\" Phoebe said quietly. \"Maybe the uncertainty about what's next for you and even Dylan is causing problems in San Diego.\"\n\n\"Maybe it is,\" Noah said, slowing as they reached the shade of a birch tree, its trunk a bright white against the lush green leaves and grass, the clear blue of the sky. \"You're perceptive. Have you always been, or has your work at the library helped?\"\n\n\"Work at the library's helped me in many ways, but I don't think of myself as perceptive. I get surprised by people all the time.\"\n\n\"Who broke your heart, Phoebe?\"\n\nHer eyes widened but she smiled. \"See? I was just surprised by you. I didn't expect that.\"\n\nHe stood closer. \"Have you given up on love? Falling in love is fine for your sisters, your friends, but for you...\"\n\n\"I know everyone in Knights Bridge.\" She spoke lightly but the way she averted her eyes told a different story. \"I'd have to go looking out of town.\"\n\nNoah saw that she wasn't going any deeper, not here, not now. He smiled. \"You could always wait for a rich, good-looking swashbuckler to turn up.\"\n\nHe noticed her immediate relief at his teasing tone. \"My perfect fantasy,\" she said with a laugh. \"What's yours?\"\n\n\"Maybe it's a redheaded small-town librarian.\"\n\nAs they walked back to the house, Noah received a text message from Brandon Sloan that Julius Hartley had just taken a booth at Knights Bridge's one and only restaurant. Brandon was all set to meet Noah there, but Noah texted him back that he'd go alone.\n\nHe debated whether to tell Phoebe about the message from her brother-in-law and finally did.\n\n\"You can't move a muscle in Knights Bridge without a Sloan knowing,\" she said, then nodded up toward the driveway where he'd parked. \"Go on. I'll stay and help with the tomatoes. My mother or one of my sisters will give me a ride back to town. Smith's is the name of the restaurant. It's just down the street next to the country store.\"\n\n\"I'll find it.\"\n\nShe hesitated. \"Are you sure you want to meet Hartley on your own?\"\n\n\"I'm the one who brought him here,\" Noah said. \"I'll deal with him.\"\n\nPhoebe gave him the slightest of smiles. \"Then I recommend the turkey club,\" she said, heading off to her mother's vegetable garden.\n\n* * *\n\nJulius Hartley didn't seem surprised or nervous when Noah sat across from him at a dark-wood booth at Smith's, a small, family restaurant located in a 1920s house around the corner from the country store. Framed color photographs of the Quabbin Reservoir and wilderness hung on the walls. Noah studied a photograph of a bald eagle soaring above blue reservoir waters as he collected his thoughts.\n\n\"I'm having the turkey club,\" Hartley said. \"I hear it's an O'Dunn favorite.\"\n\nNoah knew it was meant as a provocative statement, a way to tell him that he was dealing with a private investigator, a man who was good at his job. Noah shifted his gaze from the eagle. Hartley already had a tall glass of iced tea on the table in front of him\n\n\"The meatloaf is supposed to be good, too,\" Hartley added. \"Homemade. Have you ever had meatloaf, Kendrick?\"\n\n\"Not in a while.\"\n\n\"Since the first million or the first billion?\"\n\nNoah didn't respond. \"What did you want with Elly O'Dunn?\"\n\n\"What, you don't believe I'm interested in buying goats, either?\" Hartley grinned and slung an arm over the back of the booth. \"I'm taking that as a positive. I assume you've been out to the O'Dunn place.\"\n\n\"I just came from there.\"\n\n\"With your princess?\" Hartley waited a half beat for a response and when he didn't get one, lowered his arm and helped himself to his tea. \"How much do you know about these people?\"\n\n\"That's not why I'm here.\"\n\nHartley raised his eyes. \"Here to kick my ass? No problem. I get that. Elly O'Dunn works in the town offices. She and your librarian princess are probably putting a dossier together on me.\"\n\n\"Good,\" Noah said.\n\n\"And if they aren't, you and your friend Loretta Wrentham are.\"\n\n\"You showed up in Loretta's office. Why? How do you know her?\"\n\nHartley shrugged, clearly unconcerned. \"It's my job. As you well know by now, I'm a private investigator in Los Angeles. I had a few questions, I got them answered and now I'm going home. It's all you need to know.\"\n\nNoah shook his head. \"I'll decide what I need to know.\"\n\nHartley was clearly not intimidated. \"Knock yourself out.\"\n\n\"Who are you working for?\"\n\n\"You know better than to ask.\" Hartley leaned back, glanced around the quiet restaurant. \"If you think what comes after NAK is here in Sleepy Hollow, you're kidding yourself.\"\n\nNoah said nothing. A waitress took his order, and he asked just for iced tea. He wasn't having dinner with Julius Hartley, L.A. private investigator. Noah was aware he could be naive about people and oblivious to obvious rogues and scoundrels, but he wasn't that naive and oblivious.\n\nHartley swirled the tea and ice in his glass. \"You like the action, the pressure of the work you do. Solving problems. Building something from scratch. Getting other people involved, like your hockey-player friend.\"\n\n\"Dylan was invaluable in building NAK.\"\n\n\"Invaluable is one of those words that sounds like it means the opposite of what it means. It doesn't matter to me what comes next for you. I'm not working for your board of directors, in case that's what you're thinking.\"\n\n\"It's not my board. It's the shareholders' board.\"\n\n\"Right. Things have changed for you.\"\n\n\"Is that why you're on my tail? To find out what I'm up to?\"\n\n\"I just told you I don't care what you're up to.\"\n\nTheir waitress delivered Hartley's club sandwich and Noah's iced tea.\n\nHartley helped himself to a golden-brown fry. \"I can't remember the last time I had fries and a club. I should have ordered a chocolate shake. What the hell, right?\" He reached for a bottle of ketchup on the side of the table. \"I'm not a threat, Kendrick. If there's a threat, it's these people and their ideas about you and yours about them.\"\n\n\"And you know what their ideas about me and my ideas about them are?\"\n\n\"They know you don't belong here. You're trying to pretend you do.\"\n\nNoah didn't rise to the bait. \"You were on my tail in San Diego and then you followed me out here. Why?\"\n\n\"I didn't 'follow' you. Careful with the language.\"\n\n\"I'm giving you a chance by meeting you here by myself. I could have called my security team at NAK and had them arrange for people to meet you here.\"\n\n\"That'd get this town talking. You don't want that, do you? You can fool them into thinking that you're normal until you summon a big black SUV filled with private security types.\"\n\nNoah hadn't tried to fool anyone, and even if he had, he hadn't met anyone in Knights Bridge who gave a damn that he was a billionaire. Except Phoebe, and then only because he'd kissed her.\n\n\"You don't think like normal people, Kendrick.\" Hartley dipped a fry into his mound of ketchup. \"You're a solo operator when it comes right down to it. Dylan McCaffrey got that about you in first grade. That's why you two get along.\"\n\nNoah drank some of his tea. He was used to people trying to figure him out. \"Does your presence here have anything to do with Dylan?\"\n\nHartley ate his fry in two bites. \"He and Olivia Frost are an interesting pair, aren't they? I drove out to the Frost sawmill. Pretty setting. Must be tough, your best friend, maybe your only real friend, falling for a woman on the other side of the country. Think that's making you vulnerable to Phoebe O'Dunn's charms?\"\n\n\"You're not answering my questions,\" Noah said.\n\n\"You date beautiful Hollywood actresses who want you to bankroll their chance at the big time. Phoebe's not in their league when it comes to being a good trophy.\" Hartley picked up a triangle of his club sandwich, then grinned at Noah. \"You're doing fencing breathing, slowing your heart rate, so you stay calm and don't go for my throat?\"\n\nIt wasn't that far from the truth. \"Did an actress I dated hire you?\" Noah asked calmly.\n\n\"You're assuming anyone hired me. Relax. I'm leaving Sleepy Hollow as soon as I finish my dinner. The club really is good. You should try it.\"\n\nNoah wasn't even close to being hungry. \"It's good you're leaving town but I still intend to find out why you're here.\"\n\n\"Go for it. The O'Dunns might grow tomatoes and raise goats and such, but it would be a mistake to think they're pushovers, or that they need you to protect them.\"\n\nNoah didn't want his presence in their town to harm them. Then why had he stayed? Why did he continue to stay?\n\nHartley wiped his fingers with a cloth napkin. \"One of the dessert specials is something called apple brown betty. It's made with local apples. First of the season, apparently. I don't think I can resist. I'll be in Boston tonight and back in sunny Southern California tomorrow.\"\n\nNoah got to his feet and tossed a few bills on the table for his tea. \"Then what?\"\n\n\"We'll see. Have fun in Sleepy Hollow, Noah. Every town has its secrets.\" Hartley grinned. \"So does every redhead. Ask your mild-mannered librarian how she got the scar on her knee.\"\n\n\"We're not done, Hartley.\"\n\n\"Nothing more dangerous than a bored billionaire,\" Hartley said, more amused than intimidated.\n\nNoah left the restaurant and walked up to Main Street where he'd parked. It was a pleasant summer evening. He was used to being a fish out of water wherever he was and Knights Bridge was no exception, but it was also different\u2014and not just because it was small. His friend had discovered a grandmother here and fallen in love here, and now he was making a home here.\n\nAnd there was Phoebe.\n\nShe was as intriguing as a small-town librarian as an Edwardian princess.\n\nAs he started to get into Olivia's car, he noticed Phoebe across the street on the common. She was with her sister Ruby and another young woman he took to be Ruby's twin, Ava.\n\nPhoebe waved to him. He thought she smiled.\n\nNoah was in no rush to get back to Buster. He crossed the quiet street.\n\nPhoebe broke away from her sisters and intercepted him by an old-fashioned gazebo. \"That was a fast dinner,\" she said. \"Was Hartley there?\"\n\n\"He was.\" Noah nodded toward her sisters. \"What happened to canning tomatoes?\"\n\n\"We got things rolling and then Mom ran us out so she could finish up in peace. She seemed preoccupied, which isn't like her, so we gave her some space.\" Phoebe pulled a lightweight shawl up over her bare shoulders. \"We're figuring out some of the staging for the fashion show. It's such a beautiful evening, we came out here.\"\n\n\"Sounds perfect.\" Noah suspected that she'd also recognized Olivia's car and hoped she'd catch him and get details on his meeting with Hartley.\n\n\"Then they're coming by my place to watch old movies,\" Phoebe added. \"Maggie might join us.\"\n\nPhoebe's shawl fell back off her shoulders, and she adjusted it again. Noah resisted an urge to help her. He knew what he wanted was to touch her creamy skin. Would it be cool now, in the evening air?\n\n\"Is this the life you always imagined for yourself?\" he asked her.\n\nHis question obviously caught her by surprise. \"It's the one I have.\"\n\n\"What about your sisters?\"\n\n\"Ava and Ruby are just twenty-three. We'll see what they end up doing after graduate school.\"\n\n\"Maggie?\"\n\n\"She got married so young. It was right after Dad died. Brandon had such big dreams. I don't know how she'll react to him giving them up.\"\n\n\"What makes you think he has?\"\n\n\"He's sleeping in a tent in Knights Bridge and working for his family.\"\n\nNoah shrugged. \"He's also two miles instead of seventy-five miles from his sons.\"\n\nPhoebe nodded, thoughtful. \"He's always been a good father. What are your plans for the rest of the evening?\"\n\nNoah didn't know what he was doing beyond not inviting himself to a girls' old movie night on Thistle Lane. \"I didn't gate Buster in the mudroom. I expect I'll be picking dog hair off the couch.\"\n\nHis touch of humor seemed to go right past her. She pulled up her shawl. \"And Hartley?\"\n\n\"On his way to Boston and then to California. I'm sorry if he caused you any alarm.\"\n\n\"Are you satisfied with what he told you about his reasons for following you?\"\n\n\"He didn't give me any reasons.\"\n\n\"So does that mean you're still on the case?\"\n\n\"It does, yes.\"\n\nPhoebe glanced back at her sisters, then shifted her gaze to him again. \"Ava and Ruby have so many great ideas for the fashion show. I think they're enjoying themselves.\"\n\n\"And you?\"\n\nShe smiled. \"Yes, me, too.\"\n\n\"You're an interesting family,\" Noah said. \"Any idea how Julius Hartley would know about a scar on your knee?\"\n\nHer eyes widened, and then she burst into laughter. \"Because everyone in town knows. I have no secrets, Noah. When I was twelve, I cut my knee at the Frosts' millpond. Maggie, Olivia and Jess Frost and I were swimming. The water was so cold\u2014not that it ever warms up\u2014but it was still early summer so it was especially frigid. They got out and I stayed in. Next thing I knew, one of Brandon's brothers was pulling me out of the water.\" Phoebe tilted her head back, the turquoise of her eyes rich and deep in the gray dusk. \"I had mild hypothermia, and I'd slipped on a rock. There was blood everywhere.\"\n\n\"Did you get stitches?\"\n\n\"No. We just bandaged up the cut. It's not that bad a scar. I don't know why people remember that story. It's not as if a cut knee is any big deal.\"\n\nIt wasn't just the cut knee, Noah realized. It was also staying in the cold millpond and ending up with hypothermia, to the point that she'd had to get pulled to safety by a Sloan.\n\nIt was a big deal because it was Phoebe O'Dunn.\n\nEveryone remembered her cut knee, hypothermia and rescue because they went against their ideas about her.\n\nHe touched a bit of fringe on her shawl and suspected it, too, was from the piles of old clothes donated for the fashion show. \"You're supposed to be the sensible O'Dunn.\"\n\nShe smiled at him. \"I am the sensible O'Dunn.\"\n\nNoah let her go and returned to Olivia's car. When he arrived at Carriage Hill, Buster had, in fact, liberated himself from the back room and was camped out on the sofa. Noah settled in next to him. He wasn't impulsive. He'd call Loretta. He'd stay focused.\n\nHe knew himself and he had clarity about what he wanted.\n\nAnd he wanted Phoebe O'Dunn.\nFourteen\n\nPhoebe opened a bottle of merlot and Maggie arrived with a variety of cheeses and crackers. They had fresh vegetables from their mother's garden, although Ava and Ruby insisted they couldn't look at another tomato right now.\n\nThey gathered in Phoebe's small living room and put on To Catch a Thief. They sighed over the beautiful Cote d'Azur scenery, took a poll about who thought Cary Grant was sexy\u2014all but Ava\u2014and argued about their favorite Alfred Hitchcock movies.\n\n\"That's my dress!\" Maggie pointed at the television with an olive-oil cracker topped with goat's cheese and cucumber. \"Oh, my. I'd forgotten how beautiful Grace Kelly was. I looked like a frump in comparison.\"\n\n\"You looked great,\" Phoebe said.\n\n\"At least I managed to get into the dress and we only had to let out a seam here and there.\" She popped her cracker into her mouth. \"Don't you think Cary Grant's kind of old for Grace?\"\n\n\"It's a movie,\" Ruby said with a roll of her eyes.\n\nThey settled down, enjoying the movie. Ava had the small sofa to herself, Ruby a chair, Maggie and Phoebe throw pillows on the floor. Phoebe hadn't confessed to Ava and Ruby that she'd ventured to Friday's charity masquerade, but, given their sideways glances at her, she suspected they had an inkling. Maggie slipped up a couple of times, and finally Phoebe just told them.\n\nThey loved the idea. \"Oh, excellent,\" Ava said. \"Now I wish I'd gone, too.\"\n\nRuby glanced at Maggie. \"Did Phoebe dance with anyone?\"\n\nPhoebe reached for a slice of cucumber. \"Why are you asking Maggie and not me?\"\n\n\"Because Maggie will tell us and you won't,\" Ruby said without hesitation.\n\nMaggie settled back on her stack of pillows. \"Phoebe danced with a dashing swordfighter.\"\n\n\"A swordfighter?\" Ruby peered down at Phoebe. \"Not Noah Kendrick. Olivia says he's a master fencer. You were out at Mom's with him, and then on the common\u2014Phoebe. It was Noah?\"\n\n\"Are you the reason he stayed at Olivia's?\" Ava asked.\n\nPhoebe shook her head. \"I'm not the reason he stayed.\"\n\n\"But he was your swordfighter,\" Ruby amended.\n\n\"Yes, but it's not...\" Phoebe jumped to her feet. How could she explain to her sisters what was going on between her and Noah when she didn't understand herself? \"I'll get more crackers.\"\n\nShe disappeared into the kitchen. She should have canceled tonight but she'd thought she needed it. A fun night with her sisters would remind her who she was\u2014and who Noah was.\n\nAt least now Ava and Ruby knew about Friday. Phoebe had thought about it during the day and decided she didn't like that she, Maggie and Olivia knew but the twins didn't. Of course, that was before she'd run into Julius Hartley at the country store and ended up kissing Noah in the library attic.\n\nPhoebe grabbed another box of crackers and headed back into her living room. Her sisters had turned their attention back to the movie, at least for the moment. She set the crackers on the coffee table and resumed her position on the floor, watching Grace Kelly and Cary Grant on their adventures in the south of France.\n\nWhen the movie ended, Ruby kicked out her legs and sighed. \"That was so much fun.\" She wiggled her bare toes. \"We'll have to watch Breakfast at Tiffany's next time.\"\n\n\"You still haven't told us where you got the dresses, Phoebe,\" Ava said.\n\nPhoebe helped herself to a slice of green pepper from her mother's garden. \"At the library,\" she said.\n\nRuby took a handful of cultivated blueberries, also courtesy of their mother. \"Someone donated them for the show?\"\n\n\"I found them in the library attic,\" Phoebe said, then told her sisters about the hidden sewing room.\n\nRuby sat up straight, tucking her knees under her chin. \"A secret room in the library attic? Oh, wow. I love it. You'll let us see it?\"\n\nPhoebe nodded. \"Maybe you can help figure out which outfits are copies from movies and which are original designs.\"\n\nAva was as enthusiastic as Ruby. Maggie was quiet, frowning at Phoebe. \"What about the man you overheard on Friday night?\"\n\n\"His name is Julius Hartley,\" Phoebe said.\n\n\"The guy who was interested in buying a goat?\" Ruby asked. \"He was at the masquerade ball, too?\"\n\nPhoebe nodded and brought them all up to speed about Noah and his mystery man from Friday night. \"Noah's still trying to figure out what Hartley wants with him,\" she said finally. \"I accidentally got in the middle of it. Hartley's going back to California and I imagine Noah won't be far behind.\"\n\nMaggie started to say something else but Ava stood up. Her hair was in a long, loose braid that hung down her back. \"When we were helping you clean up after you bought this place, I found a box of old books in a corner of that big hall closet upstairs. Novels, books on movies, biographies of stars and directors, a Vogue sewing book, a couple of books in French\u2014I remember one was Le Petit Prince. I wonder if it has anything to do with your hidden room.\"\n\n\"What did you do with the box?\" Phoebe asked.\n\n\"I left it where it was,\" Ava said. \"I figured we'd get to it eventually. It's in back of the closet with some old curtain rods and wool blankets. Phoebe, you know what this means, right?\"\n\nShe nodded as she got to her feet. \"It's possible that whoever created that hidden room used to live here.\"\n\n* * *\n\nAfter her sisters left, Phoebe sat outside on a wicker chair in the cool night air and called Olivia in San Diego. \"I'm watching fireflies,\" she said when her friend answered. \"What are you doing?\"\n\n\"I'm sitting on Dylan's porch looking out at the Pacific. It's quite a view.\" Olivia sounded content. \"How's everything in Knights Bridge?\"\n\n\"You know the mystery man from the ball is Julius Hartley, a California private investigator, right?\"\n\n\"Yes. Noah and Dylan have been in touch.\"\n\n\"He was out at my mother's place this afternoon. Hartley. It must be because I danced with Noah at the masquerade.\"\n\n\"Is there anything I can do, Phoebe?\"\n\n\"I don't know that there's anything to be done,\" she said, then told Olivia about the hidden room. \"Has anyone in your family ever mentioned a woman who loved Hollywood and was also an incredible seamstress?\"\n\n\"Not that I recall. I can ask.\"\n\n\"Maggie, the twins and I think she might have lived here\u2014in my house on Thistle Lane.\"\n\n\"Maybe my folks remember her. I can ask. Who else knows?\"\n\n\"Noah.\" Before Olivia could respond, Phoebe changed the subject. \"How do you like San Diego, Liv?\"\n\n\"I love it. I'm having a great time. Coronado is beautiful. I love Dylan's house. It's across from the beach.\" She added with a small laugh, \"The interior could use some color.\"\n\n\"It sounds idyllic.\"\n\n\"I don't see Dylan giving up San Diego entirely. I don't know that I want him to. We'll figure it out. It's not exactly a problem, you know?\"\n\nPhoebe stood up from her wicker chair and looked out at the dark lane, wondered if her seamstress had once done the same. \"Sometimes it's harder to open ourselves up to new possibilities than just to stay put where we are, emotionally, physically. You have so much going for you now. Dylan, too. If you hadn't left Boston when you did, the way you did...\"\n\n\"Phoebe? You don't sound like yourself. Is something wrong?\"\n\n\"No, nothing. It's good to talk to you. I hope you know I'll support whatever you and Dylan decide to do.\"\n\n\"We'll be flying back soon,\" Olivia said. \"We'll be there for the fashion show, for sure.\"\n\n\"I'm sure Noah will head back to San Diego now that he's found his mystery man. Julius Hartley's heading back to L.A., too.\" Phoebe tried to ignore a rush of emotion. \"Maggie and I can see to Buster and your place.\"\n\n\"Have you met Hartley?\"\n\n\"Just for a few minutes. He was pretending to be interested in buying goats from my mother. Noah and I went over there.\"\n\n\"What did he think of your mother's place?\" Olivia asked.\n\n\"He didn't say but it's obviously not what he's used to.\" Phoebe sat on the top porch step, next to the trellis and its tangle of roses, and smiled into the phone. \"Have you seen his place in San Diego?\"\n\n\"Just the NAK offices. They're incredible. It must be weird for him to go from working night and day to not knowing what's next.\"\n\n\"True.\" Phoebe wished she hadn't brought up Noah's life in San Diego. \"We missed you at movie night tonight. My head's spinning a little from the wine. We watched To Catch a Thief.\"\n\n\"And was Grace Kelly's dress as much like the one Maggie wore as we thought?\"\n\n\"It's identical. It's amazing.\"\n\n\"I understand Brandon's sleeping in a tent at Dylan's place,\" Olivia said. \"Dylan thinks he's there to get her back. What are the odds he does?\"\n\nPhoebe didn't hesitate. \"He will. No question. I'm just not telling her that's what I think.\"\n\nThey laughed, Phoebe finally relaxing as she and Olivia chatted for a few more minutes. When they hung up and she went back inside, she realized how quiet the house was, and how alone she was. She'd never felt alone before. How could she, here in the midst of her hometown? So, why did she now?\n\n\"It's Noah,\" she said aloud, knowing that no one was eavesdropping in the roses and hollyhocks outside her windows.\n\nShe and her sisters had gone upstairs and found the box in the closet. Phoebe carried it downstairs. They went through the contents for any obvious clues to their seamstress's identity. A name scrawled on the inside of a book, an old letter, an old bank statement. But there was nothing except the books themselves.\n\nPhoebe took a yellowed copy of The Moonspinners from the box and brought it up to bed with her. She'd lose herself in Mary Stewart's descriptions of Crete, and she wouldn't think about kissing Noah Kendrick in the library attic.\n\n* * *\n\nBrandon Sloan was a damn pirate at heart.\n\nMaggie held that thought as she watched him come out the front door of her Gothic Revival house. It was just the sort of place the Sloans loved: one in need of carpenters.\n\nHe sat next to her on the steps. The night was cool enough that he had on an old gray sweatshirt. She was chilly in her sundress but tried not to shiver. It was all she needed, him thinking he ought to put an arm around her to help keep her warm.\n\n\"The boys want hockey sticks,\" she said. \"Did they tell you?\"\n\n\"Hockey? I thought you had them baking cupcakes.\"\n\nShe resisted elbowing him because touching him in any way would just remind her of touching him in all the ways she had in the past.\n\nIt had turned into that kind of night.\n\nThey'd made such a damn mess of their marriage. It was all she'd thought about walking home from Thistle Lane. Maggie realized that something about the box of books they'd dragged out of Phoebe's upstairs closet had gotten to her. It was as if the books captured a moment in time of a woman's life, provided a window into her hopes and dreams. Sewing and Hollywood and adventures.\n\nWhat would the odds and ends of her own day-to-day life say about her, now, at this moment?\n\nMaggie pushed back the thought. \"The hockey is probably Dylan's influence,\" she said.\n\nIt was a gibe and Brandon obviously knew it, but he didn't jump up and storm off. He just stretched out his thick legs and shrugged. \"Maybe it is.\"\n\nMaggie regretted her crack. Whatever his faults\u2014whatever her own faults\u2014he'd always been there for the boys. Just not always for her. But she didn't need him, right? Wasn't that what she'd been telling herself for months? Telling him?\n\n\"I have ice skates in the budget for winter,\" she said. \"The town still does the small outdoor rink on the common.\" It was just a homemade rink done mostly with hoses and shovels, a Knights Bridge tradition going back at least to when her mother was a child. \"The boys and I will be able to walk over there so they can skate to their hearts' content. I think I even have my old skates.\"\n\n\"I remember when you and I would go ice-skating together,\" Brandon said.\n\nMaggie smiled despite a rush of emotion. \"You were a maniac. All that energy. What am I going to do if Tyler and Aidan have as much energy as teenagers as you did?\"\n\n\"Keep doing what you're doing.\"\n\n\"And watch them like a hawk,\" she muttered.\n\nBrandon grinned. \"Like I said, keep doing what you're doing.\" He looked out at the street, just one window lit in the saltbox house, one of the oldest houses in the village, opposite hers. \"You like living in town. The boys do, too. They like being able to walk to everything.\"\n\n\"You think I'm too soft,\" Maggie said, crossing her arms on her chest as she sat up straight. \"I'm not raising them to be tough Sloan men. I have them doing story hour at the library instead of roping a steer.\"\n\n\"Roping a steer? I guess they could rope goats at your mother's\u2014\"\n\n\"Goats are soft, too, right? She has a well-equipped toolshed, at least. It's got hammers, nails, drills, saws. No guns and fire hoses, though.\"\n\nBrandon sighed. \"What did I say, Maggie?\"\n\n\"Nothing. I can tell what you're thinking.\" She stood up, glaring down at him in the dark, on a roll now, her emotions boiling over. \"You and your family have always thought I was too soft, because my father was such a dreamer and then he died in a stupid accident and it's just been my sisters and my mother and me for so long.\"\n\n\"Your father was a good man, Maggie, but he still left you all with nothing.\"\n\n\"We have the land. We've all worked hard to help Mom hang on to it. Tyler and Aidan love it out there.\"\n\n\"I know they do.\" He stood, the light from the house casting dark shadows on his face. \"You're coping with a lot on your own, Maggie. It doesn't have to be that way. I can help.\"\n\n\"Help how?\"\n\n\"Any way you need.\"\n\nShe hadn't expected him to be so calm, not reacting to any of her barbs, deliberate or otherwise. She blinked back tears. \"I never should have had wine with my sisters. I'm sorry if I...\" If I what? She didn't even know. \"Never mind. Why are you back here, Brandon?\"\n\n\"My father needed the help.\"\n\n\"You're giving up on your dream of wandering the world?\"\n\nHis eyes held hers. \"I haven't given up on anything.\"\n\n\"What about Boston? You never wanted to live in a small town.\"\n\n\"If I hadn't gotten laid off, I'd have kept working in Boston.\" He spoke simply, without any obvious emotion. \"I do what I have to do. I always have.\"\n\n\"While wishing you were somewhere else.\"\n\n\"Not always,\" he said softly.\n\n\"Why are you living in a tent? I suppose it helps to have a temporary place, so you can pretend you're not really back in Knights Bridge.\"\n\n\"So I can save money.\"\n\n\"For a trip,\" she said. \"Not for ice skates.\"\n\nHe said nothing.\n\nMaggie regretted her sharp words. \"I'm sorry. I know you'd do anything for the boys. It's me...\" She stopped herself, cleared her throat. \"You won't be living in a tent once the snow flies. It's fine for now but...\" She left it at that. He knew what she was saying. What she was asking. Would he be staying in Knights Bridge?\n\n\"Don't worry about me, Maggie.\"\n\n\"I'm not worried about you. I'm trying to figure out what's next for you. For us.\" She waved a hand. \"Never mind. Let's talk about something else.\"\n\n\"All right. If that's what you want.\" He glanced back at her house with its \"gingerbread\" Gothic Revival details. \"The place is looking good. Going to paint it shades of pink? It's what you used to say when we walked past this place as kids. That you'd paint it shades of pink if you lived there.\"\n\n\"I'm actually thinking about a neutral color. Did the boys tell you they want to build a tree house out back? I can help but I'm not that great with hammers and nails. Mom's better but she's got her hands full.\"\n\n\"Their dad's a carpenter,\" Brandon said quietly. \"I can help my sons build a tree house, Maggie.\"\n\n\"They'd like that. They look up to you. I...\" Maggie sighed, her shoulders sagging as all the fight went out of her. \"When did it become so awkward between us? We used to be able to talk about anything. Not that you were ever a big talker but I never felt I couldn't speak my mind, that you couldn't speak yours. We were best friends.\"\n\nHe touched her cheek, her hair. \"You're tired. You're taking on a lot.\"\n\n\"I love what I'm doing. I love being back here. I wasn't sure I would but everything's turning out better than I anticipated. Don't worry, I still have my dreams.\"\n\n\"A gingerbread house in Knights Bridge village.\"\n\n\"Life could be worse, you know.\"\n\nHe smiled. \"You could be living in a tent.\"\n\n\"I remember some good nights with you in tents.\"\n\nHe winked. \"Damn straight.\"\n\nAfter he left and the boys were in bed, Maggie sat at her kitchen table with a stack of cookbooks. The kitchen was in good shape, with a relatively new gas stove and a decent refrigerator, but it still needed work. Buying the house hadn't felt as impulsive as it probably was. She'd been drawn to it since childhood, and she'd thought it'd be a great place for the boys. But it really was a fixer-upper, and here she was, the estranged wife of a carpenter who was related to all the other carpenters in town.\n\nShe pictured laughing with Brandon as they painted the kitchen together, but it wasn't going to happen. She was on her own. He would always be the father of their two young sons, but that was it.\n\n\"It can't be. It just can't be.\" Before she could burst into tears, she called Olivia as a distraction, as well as to check in on her friend. \"Am I catching you at a bad time?\"\n\n\"I'm not sure there is a bad time in San Diego. It's a stunningly beautiful day out here.\" Olivia sighed, obviously content. \"What're you up to?\"\n\nFrom the tone of her friend's voice, Maggie suspected Olivia knew that things were a bit complicated back home. \"I think we should try making our own essential oils for our soaps. They're so expensive to buy. You have some great herbs at your place. We'd have to dry them, and we'd need to buy equipment for distilling...\" She realized she was ready to burst into tears. \"What do you think?\"\n\n\"It's something I've been considering for a while,\" Olivia said. \"Will you have time?\"\n\n\"I'll make the time. Except for harvesting the herbs, we have flexibility. We can save up everything and make the soap during our down times. It should be quiet after foliage season, before the holidays. It'll be fun.\"\n\n\"Maggie? You sound upset. What's going on?\"\n\nMaggie immediately felt guilty for making the call when she was in such a down mood. \"Nothing I can't figure out. Tell me about California.\"\n\n\"If you tell me about Phoebe and Noah.\"\n\n\"Wait, what do you know about Phoebe and Noah?\"\n\n\"Not much except that something is going on between them. Phoebe's being tight-lipped, probably because Noah and Dylan are such close friends.\"\n\n\"Maybe she's afraid of mucking things up between you two.\"\n\n\"Not possible to muck things up between Dylan and me. So, what's going on? What am I missing?\"\n\nMaggie smiled through her tears and told her friend what she knew, which she realized wasn't everything, and what she surmised, which probably wasn't everything, either.\nFifteen\n\nCoronado was as beautiful as ever, the offshore night breeze prompting Loretta to grab a sweater out of her car as she and Olivia walked down to the historic Hotel del Coronado. Dylan had told them to go on ahead of him. He'd meet them shortly.\n\n\"He's worried about Noah,\" Olivia said, hugging her own sweater to her.\n\nLoretta shrugged. \"So far, Noah's managed just fine on his own in your little town.\"\n\n\"It's not just that.\" Olivia glanced out at the water, the lights of the sprawling hotel reflecting eerily in the white caps of the incoming tide. \"Dylan believes that none of this\u2014\" she waved a hand back toward Dylan's expensive house \"\u2014would have been possible if Noah hadn't knocked on his window when Dylan was sleeping in his car.\"\n\n\"Synergy,\" Loretta said. \"Noah knows he'd have crashed and burned without Dylan's help.\"\n\n\"And now what? What's next for both of them?\" Olivia took a deep breath. \"I don't want Dylan to move to Knights Bridge just for my sake. It's too much to ask, and I wouldn't. It's not that Knights Bridge doesn't measure up to San Diego. It does, at least for me. It's home. But this is home for him.\"\n\n\"You can do both, you know. Knights Bridge and San Diego.\" Loretta shivered in the gusty breeze, but she welcomed it at the same time, let it clear her head, keep back her own emotions. \"It's not like you two will be scrimping to pay for groceries.\"\n\nOlivia smiled. \"You're blunt, aren't you?\"\n\n\"You can't help either of those two if you're not. When Dylan was playing hockey, he could read the ice, read a defense, without thinking. He just knew. Same with NAK and his role there. Noah's smart, but he doesn't always pick up on what's going on around him. Karate and fencing help him tune in, I think.\"\n\n\"He gives people the benefit of the doubt until they give him reason not to.\"\n\n\"It's not a bad way to be. Dylan's not cynical but let's just say he gives people a shorter rope than Noah does.\" Loretta walked a few more steps as the tide came in on the wide sandy beach below them. \"Tell me about Phoebe O'Dunn.\"\n\n\"What about Phoebe?\"\n\nAs if Olivia didn't know what Loretta was asking. Loretta had already gathered that not much went on in Knights Bridge that Olivia and her family and friends didn't know about. That Grace Webster had managed to keep her affair with a British flyer and the birth of their son a secret for seventy years was a damn miracle as far as Loretta was concerned. Dylan said she'd understand when she met Grace. The assumption being that Loretta eventually would get to Knights Bridge.\n\nMaybe she would. She wondered if she'd understand the late Duncan McCaffrey any better when she did.\n\nProbably not.\n\nShe turned her attention back to the matter at hand. \"Phoebe was Noah's princess the other night. She overheard Julius Hartley talking on the phone to someone\u2014probably someone out here.\"\n\nOlivia seemed more amused than surprised. \"You do know everything, don't you?\"\n\nLoretta laughed. \"Not by half. Not when it comes to Noah and Dylan. So what about your friend Phoebe?\"\n\n\"We've been friends forever. My younger sister and I grew up with Phoebe and her sisters.\" Olivia glanced out at the Pacific, as if picturing her hometown in her mind. \"Jess and I grew up at an old sawmill and the O'Dunns grew up on a small farm. It was a great childhood.\"\n\nLoretta prodded her. \"And?\"\n\n\"Phoebe's the eldest. She's always felt responsible for the rest of us\u2014her sisters, and even Jess and me.\" Olivia hesitated, lowering her arms, letting her sweater flap in the breeze. She seemed to welcome the cooler air. \"Phoebe found her father after he died in a fall out of a tree he was trimming. His death was hard on all of them.\"\n\n\"Phoebe tried to fix things?\"\n\n\"I think she just tried to be there for everyone. Her mother was always a live-for-the-moment type but she became even more so after Patrick's death.\"\n\nLoretta imagined a woman facing early widowhood with four daughters and a farm. \"She can be impractical?\"\n\n\"That's one way of putting it. Phoebe commuted to college from home, so she's never lived anywhere but Knights Bridge. She loves her job at the library. She's good at it. She's smart and sophisticated, Loretta. Don't think just because she's from a small town that she's not.\"\n\n\"Whoa. Phoebe's not the only one who's protective.\"\n\nOlivia sighed as they crossed a driveway to the hotel, its distinctive red turrets and white exterior glowing in the night lights. \"Sorry,\" she said.\n\n\"Don't be sorry. It's good to have friends who worry about you.\" Loretta grinned, lightening the mood. \"I wish I had a few.\"\n\n\"You're like Phoebe. You do the worrying.\"\n\n\"Am I guessing right that something's going on between her and Noah?\"\n\nOlivia tightened her sweater around her again. \"I think so.\" She slowed her pace as they continued along a curving walk to the hotel. \"Will Noah hurt her, Loretta?\"\n\n\"Noah's more likely to get hurt himself than to hurt someone else.\"\n\n\"He dates Hollywood types\u2014\"\n\n\"Who are more interested in his money and his connections than in him.\"\n\n\"He's a very wealthy man, and he can't have taken his company to where it is without being driven, maybe even a little ruthless. Phoebe's a gentle soul, unless she thinks one of us is in trouble.\"\n\n\"Does she think one of you is in trouble now?\" Loretta asked.\n\nOlivia shrugged. \"I don't think so. I don't know. I'm not there.\"\n\n\"She can be tough, too, from the sounds of it. She kept her cool when she overheard Hartley in the coatroom. And didn't she show up at that masquerade on her own?\"\n\n\"You're right. I don't want to underestimate Phoebe. She can hold her own with a California billionaire.\" Olivia turned, the light from the hotel catching her green eyes as she smiled. \"I'd put running a small-town library on a shoestring right there next to running NAK.\"\n\nLoretta thought she heard a hint of homesickness in Olivia's voice. They found their way down to the waterfront behind the main hotel and sat at an outdoor table overlooking the wide beach and glittering ocean. They ordered pi\u00f1a coladas and watched the crowd. Loretta heard teenagers laughing, noticed a young couple holding hands, two older couples chatting quietly together over drinks.\n\n\"Do you come here often?\" Olivia asked.\n\n\"From time to time. It's a good place to take out-of-town guests.\"\n\n\"It's so romantic. Dylan says you two used to come here for a drink after you talked business.\"\n\n\"Money's never been his favorite topic,\" Loretta said with a smile. \"That's how he ended up sleeping in his car. As they say, you can lead a horse to water, but you can't make him drink. He wasn't as broke as he thought he was. I'd tucked some money away.\"\n\n\"But you didn't tell him,\" Olivia said.\n\n\"I told him when I did it. He just didn't pay attention.\"\n\nOlivia laughed, looking at ease with herself, her relationship with the man she would soon marry. Loretta felt a sudden sense of loss as she gazed out at the water. The wind had died down. She listened to the waves washing on the sand and wondered what her life would be like right now if Duncan McCaffrey had never gone to Knights Bridge, Massachusetts.\n\n\"I've known Noah and Dylan for a long time,\" she said finally. \"Dylan had just started with the NHL and Noah was still a student at MIT. I met Noah when he was out here on a break and went to one of Dylan's hockey games.\"\n\n\"They're like sons to you, aren't they?\"\n\nLoretta ignored a sudden tightness in her throat. \"Now you're making me feel old.\"\n\n\"I hope not.\" Olivia sat back with her pi\u00f1a colada and looked out at the dark ocean. \"What a beautiful spot.\"\n\n\"Some Like It Hot was filmed here.\"\n\nOlivia smiled. \"I'm glad Phoebe didn't try to put me in a Marilyn Monroe dress. It's so beautiful here, Loretta. Dylan's a very lucky man, and I love having him in my life. I love him. We never would have found each other without you.\"\n\nLoretta fought back tears that took her by surprise. She wasn't one for tears. As she studied the woman across from her, she was satisfied that Dylan had made the right choice in asking Olivia Frost to marry him\u2014as if choice had anything to do with it. The man was in love, and from what she'd heard in Noah's voice since he'd danced with Phoebe O'Dunn, he wasn't far behind. Loretta just wasn't sure he was in for as happy an ending as Dylan.\n\nShe kicked off her shoes and enjoyed her drink, subtly sniffling back any tears so Olivia wouldn't notice. Maybe Olivia had a point. Maybe in a way Dylan and Noah were like sons to her. She'd never regretted not having kids of her own.\n\nNoah had always struck her as a man looking for a real soul mate. A woman he loved, and who loved him, without condition. A woman he'd fight for, die for. It was the swordfighter in him, Loretta thought.\n\nShe'd never met two more decent men than Dylan McCaffrey and Noah Kendrick.\n\nWhat was she going to do if they both moved to Knights Bridge?\nSixteen\n\nNoah was in jeans\u2014no shirt, no shoes\u2014when Buster stirred and went to the kitchen door, barking through the screen as the two eldest O'Dunn sisters jumped out of Maggie's catering van. \"Company, Buster,\" Noah said, rising from the table with the last of his second cup of coffee. He figured he'd need a full pot of coffee before noon. It'd been a long night alone on the edge of Quabbin. Even Buster had been restless.\n\nMaggie and Phoebe approached the kitchen ell with an ease that suggested they'd forgotten he was dog sitting, which Noah doubted, or had heard rumors to the contrary. They were dressed in shorts and sport sandals. Maggie had on a Red Sox T-shirt, Phoebe a close-fitting tank top in a deep turquoise blue that matched her eyes.\n\nNoah finished his coffee as they entered the kitchen.\n\nAnd he'd told Buster it would be a boring morning.\n\nPhoebe all but gasped at seeing him. \"We didn't think anyone would be here.\" She kept one hand on the screen door, as if ready to bolt for the van. \"We thought\u2014we heard you'd left for California.\"\n\n\"Who told you that?\" Noah asked.\n\nPhoebe averted her eyes. \"It was the talk of the country store this morning.\"\n\nA vague answer at best but Noah let it go. \"I see.\"\n\n\"We're here to harvest mint,\" Maggie said, setting a basket on the kitchen island.\n\nNoah placed his mug in the sink. \"Dare I ask what you want to do with your mint harvest?\"\n\nMaggie turned to him. \"Olivia and I are having a go at making our own essential oils. She has several herbs that could work. We'll start with the orange mint.\"\n\n\"It'll have to dry first,\" Phoebe said. \"We won't actually be making essential oils today.\"\n\n\"You're off today?\" Noah asked her.\n\n\"Just this morning. I have an evening meeting.\"\n\nShe let the screen door shut behind her. Noah took that as a sign that she intended to stay for the mint-harvesting.\n\n\"Phoebe hasn't taken any vacation time this year,\" Maggie said. \"Right, Phoebe?\"\n\n\"I took time off in the spring to paint the porch.\"\n\n\"I rest my case,\" Maggie said, digging a pair of utility scissors out of an island drawer. \"If you want to help, Noah, that'd be great, but I suggest putting on a shirt. It'll be buggy in the mint patch.\"\n\nHe smiled. \"If you're warning me about insects, it means I should expect the worst.\"\n\nMaggie laughed and grabbed her empty basket. \"You're a riot, Noah,\" she said, heading for the mudroom and out the back door.\n\nPhoebe had her hair in a long, loose ponytail. She redid a clip that held stray curls off her face. Noah reached for a black shirt he'd brought downstairs with him. As he shrugged it on, he was aware of Phoebe watching him. He appreciated her reaction.\n\nHe fastened a few buttons. \"Did you sleep well last night?\" he asked her softly.\n\nBuster stirred, and she patted him. \"I read Mary Stewart until the wee hours. You?\"\n\n\"I was up late looking for things to do. Wood to chop, wild animals to slay. I did manage to chase Buster off the couch.\"\n\nShe laughed, visibly more relaxed. \"Olivia will appreciate that. Dylan won't care. Sorry we disturbed you. You don't have to help with the mint\u2014\"\n\n\"I don't mind.\"\n\nNoah had a feeling he'd be learning more about essential oils and soap-making than he ever thought of knowing before the morning was done.\n\nPhoebe followed her sister outside, and he put on shoes, grabbed the bug spray and Buster and joined them. The orange mint was at the end of the garden, almost at the shed where he'd first spotted Phoebe and hadn't even considered she might be his princess. He watched her squat down with a pair of clippers, snipping off the tops of the orange mint. He noticed the shape of her slim, bare legs.\n\nIt was definitely turning into a different morning than he'd planned. He was glad he hadn't called for a plane, after all.\n\nHe helped harvest the mint and bring it into a small room that shared the center chimney. Apparently Maggie and Olivia had conferred and decided the mint would dry there.\n\n\"What's it do for you?\" Noah asked as they spread the mint on a table, a flea-market find that Olivia had teased about putting on his painting list.\n\n\"Orange mint is supposed to be uplifting,\" Maggie said, then grinned at him. \"Aren't you uplifted?\"\n\n\"It'll be highly concentrated as an essential oil,\" Phoebe said. \"It's supposed to blend well with other essential oils.\"\n\n\"You never use essential oils directly on the skin,\" Maggie added. \"They're always diluted somehow.\"\n\nMaking an essential oil was a relatively complicated process that also involved a still, which Maggie said she had on order. Noah found the details surprisingly interesting. As they returned to the garden, she explained saponification, the chemical process that transformed a fat and alkali into soap and glycerin.\n\n\"We use only fresh goat's milk, not powdered,\" Maggie said. \"Soap making involves a range of my interests. Cooking, gardening, aromatherapy\u2014and my mother's goats, I guess. They've grown on me, finally. Each batch of our soap is handmade. I like that. We leave in the glycerin. A lot of commercial soap makers remove it because they can sell it.\"\n\n\"It's an ingredient in nitroglycerin,\" Noah said.\n\nShe wasn't that amused. \"Glycerin is a natural humectant. Goat's milk is very mild. A lot of people with sensitive skin swear by it because it has a pH level that's close to that of our skin.\"\n\n\"So that explains my baby-soft skin,\" Brandon Sloan said, climbing over a stone wall into the garden. \"I've been using the soap in Olivia's upstairs shower.\"\n\nMaggie wasn't that amused by her husband, either, but Noah could see Phoebe holding back a smile. He kept his mouth shut.\n\n\"Where are the boys?\" Brandon asked.\n\n\"With a couple of their friends. I'm picking them up for lunch.\" Maggie stepped onto the terrace, her enthusiasm for talking about mint harvesting and soap making on the wane.\n\nMaggie immediately headed back into the kitchen.\n\nPhoebe turned to Noah. \"Just leave the mint in the back room, out of any sunlight. It'll be fine.\" She smiled. \"Or it won't be fine and we'll toss it into the compost bin. Anyway, I should go. I have to be at the library soon.\"\n\n\"Weren't you two planning a picnic lunch?\" Brandon asked mildly. \"You know Maggie. She's always got food figured out.\"\n\nPhoebe scowled at him. \"Our morning didn't go quite as planned.\"\n\nHe shrugged. \"You could always leave the food for your poor starving brother-in-law.\"\n\n\"It's still in the van,\" Phoebe said, as if that explained everything.\n\nShe glanced at Noah, then left without another word.\n\nNoah stepped onto the stone terrace. He didn't know if he should follow Phoebe out to her sister's van and see them off\u2014or if he was supposed to take her retreat as her wish that he stay away. Dylan would know. Noah had no illusions that he was particularly good at figuring out what people were trying to say. Much easier if they just said it.\n\nBrandon picked up stray mint leaves off the terrace table. They heard the van start up out front. \"Fast exit,\" he said.\n\n\"Time got away.\"\n\n\"Yeah. That must be it. Did you just spend the morning picking mint?\"\n\n\"I walked Buster, too.\"\n\nThe big dog opened one eye from his spot under the table, as if he knew that life was rough for his dog sitter.\n\nBrandon grinned. \"Time to go back to San Diego?\"\n\nNoah didn't answer as he went into the kitchen, grabbed two beers out of the refrigerator and brought them outside. \"It's now officially after noon and I have nothing to do, so I can have a beer. If you're on the job\u2014\"\n\n\"I'm not. I'm taking the afternoon off. Maggie's dropping off Aidan and Tyler after lunch. We're hiking up Carriage Hill, then camping out at Dylan's place.\" Brandon uncapped his beer. \"Maggie's giving them instructions on spotting deer ticks. She's paranoid about Lyme disease. I guess that makes sense.\"\n\n\"I hadn't thought about Lyme disease,\" Noah said, then grinned. \"Now I will.\"\n\n\"Going out of your mind in our little town?\"\n\n\"It's only been a few days. I can do anything for a few days, but I've discovered that Knights Bridge is more complex than it might seem at first, despite the absence of traffic lights.\"\n\n\"I used to think it's isolated. It's really not. It's just small. It does help to have a driver's license if you're going to live here.\" Brandon dragged out a chair and sat down heavily, as if suddenly he had the weight of the world on his shoulders. \"You and NAK\u2014did you ever expect it to take off, get as big as it did?\"\n\n\"I worked toward that. It's the outcome I wanted.\"\n\n\"There were setbacks?\"\n\n\"Inevitably.\" When Brandon seemed to be looking for more, Noah added, \"We took steps each day, assessed, made adjustments, managed risk and learned to cope with uncertainty.\"\n\n\"No crystal ball?\"\n\nNoah smiled. \"No crystal ball.\"\n\n\"Maggie never used to mind taking a few risks. She jumped into catering with both feet, moved back here without a real plan, but she doesn't see it that way because it's her hometown. Her sisters are here. Her mother.\" Brandon drank some of his beer. \"It's me she wants to be practical.\"\n\n\"I think you can be practical and still take risks. You just want to be careful about not risking more than you can afford to lose, and you have to manage the uncertainties and unpredictability of the future.\"\n\nBrandon glanced back toward the kitchen, as if he were thinking about his estranged wife and their two young sons. He seemed to give himself a mental shake. \"Going public involved uncertainty, didn't it?\"\n\n\"It still does.\" Noah sat down, drank some of his beer. \"I didn't consider what I'd do after NAK went public as carefully as I could have.\"\n\n\"So that's why you're here dog sitting.\"\n\n\"Maybe so.\"\n\n\"Any paths not taken that you can take now that you can be free of the day-to-day running of your company?\" Brandon seemed to want to add something but was silent a moment. Finally he said, \"I suppose we all have paths not taken.\"\n\nNoah hadn't considered his situation in quite that way. \"I suppose so. What about Phoebe?\"\n\nBrandon narrowed his gaze on Noah. \"What about her?\"\n\n\"Her father died when she was in college and she stayed in Knights Bridge.\" Noah spoke carefully, aware that Phoebe was Brandon's sister-in-law, a woman he'd known since childhood versus a few days. \"Was that always her plan, or is there a path not taken?\"\n\n\"More like there's a guy who took off to Orlando without her. He wasn't from here,\" Brandon added quickly, as if that were a significant fact. \"They were at UMASS together. He was a senior and she was a junior when her father died. This guy didn't like sharing Phoebe with her mother and sisters on a good day.\"\n\n\"You met him?\"\n\n\"Yeah. Once. I was with Maggie at her mother's place. Those were tough days, right after their dad died. For a while they just didn't know...\" Brandon scowled as if irritated with himself. \"I'm talking too much. I never used to talk at all but I've been practicing.\"\n\nNoah hesitated but he knew he had to ask. \"Did Patrick O'Dunn commit suicide, Brandon?\"\n\nBrandon shook his head but the question clearly hadn't come as a shock. \"It crossed everyone's minds, but no, he didn't. It was just a stupid accident. Phoebe's guy\u2014he couldn't take it, having her in the middle of a family crisis. It was all about him. He gave her an ultimatum. Transfer out of UMASS and move with him to Florida or they were through.\"\n\n\"Phoebe's still here,\" Noah said.\n\n\"So she is. She hasn't been serious about a guy since then. Not that she'd tell me.\" Brandon settled back with his beer, no indication he had any bitterness toward Phoebe given his own troubled situation with her younger sister. \"You know sneaking into that ball the other night was a big deal for her, right?\"\n\nNoah nodded. \"She didn't tell anyone she was going.\"\n\n\"It was a last-minute decision. Phoebe's usually not impulsive. I ran into her. I kept my mouth shut but I guess the cat's out of the bag now.\" Brandon again narrowed his gaze on Noah. \"You two...\"\n\n\"I won't cause problems for her,\" Noah said quietly.\n\n\"As you say, not everything is predictable. It's up to us to respond to the unexpected. You didn't expect Phoebe. She didn't expect you.\" Brandon got to his feet. \"Life does have a say, doesn't it?\"\n\nNoah leaned back in his chair and thought he could smell orange mint in the warm air. \"You're not telling me all this as a friend. You're warning me.\"\n\n\"I guess you could look at it that way. I'm only telling you what everyone in town already knows.\"\n\n\"You don't want Phoebe hurt again.\"\n\n\"Let's just say I'm doing what I can to assess and manage risks.\"\n\nAnd I'm the risk, Noah thought. He was the stranger sweeping their Phoebe off her feet. Another man who could break her heart. She was happy with her life. No one wanted him to screw that up.\n\nNoah didn't want to, either.\n\nHe decided to shift the subject. \"What happened with you and Maggie?\"\n\n\"I'm in a tent for a reason.\" Brandon looked up at the sky. \"I'm not getting her back, Noah. It's not going to happen.\"\n\n\"Giving up easily, aren't you?\"\n\nBrandon sighed. \"Looking reality square in the eye. It's not something I always like to do, but I want Maggie to be happy. I know that much.\"\n\n\"Because you love her,\" Noah said.\n\n\"Always have, always will. That doesn't mean we can be together. Ack. I hate this kind of deep talk. I've been practicing, because she wants me to talk. Listening isn't enough. She says she has to hear my voice. I should practice talking to Buster. Hell of a lot easier to talk to a dog than to an O'Dunn.\"\n\nDespite Brandon's attempt to lighten his mood, Noah felt the other man's pain. \"I need to go back to San Diego to check on a few things,\" he said. \"You can see to Buster?\"\n\n\"Sure. I'll see to him.\" The big dog sat at Brandon's feet, obviously wanting to be petted. Brandon complied and grinned, his dark mood dissipating rapidly as some of his natural spark returned. \"The O'Dunn women are smart and quirky and pretty as hell, but damn, they're not easy.\"\n\nNoah smiled. \"What fun would easy be?\"\n\n\"Maggie's dress the other night makes me wonder if maybe she just wants a little old-fashioned romance in her life. What do you think?\"\n\n\"Like what?\"\n\n\"I don't know. Wooing.\"\n\nNoah stared at his new friend. \"Wooing?\"\n\nBrandon laughed. \"Yeah. I'll figure out some wooing options that won't break the bank. Meantime, I'll go up and take a shower with the goat's milk soap.\"\n\nBrandon seemed reenergized as he headed through the mudroom into his friend's house. Noah moved his chair into the sunlight and finished his beer. Bumblebees were again in the catmint.\n\nNo one in Knights Bridge had expected a man like him\u2014maybe any man\u2014to float into their librarian's life. It wasn't just his net worth. It was California. His work. His MIT background. His experience.\n\nHe was forbidden, he thought with a sigh.\n\nAt the same time, he liked the challenge, just as Brandon Sloan liked the challenge of \"wooing\" his wife back.\n\nBut what if Julius Hartley was right? Noah stood up in the sunshine, listened to the bees in the catmint, crows out in the fields. It didn't feel as if he'd seized on Phoebe because he was bored, but what if he had? What if he was drawn to her because she was so different, so out of reach? He wasn't playing games, and he was confident she was as attracted to him as he was to her.\n\nWell. Maybe not that confident. But confident that his interest wasn't one-way.\n\nHe'd had his share of Hollywood babes disappear on him. More who'd needed a push out of his life. He didn't want Phoebe to disappear and he didn't want to push her or cause her embarrassment, scrutiny or anything she'd live to regret.\n\nAfter his shower, Brandon walked up the road to Dylan's place instead of cutting through the field. Noah almost went with him, but there wasn't much to see. Dylan had shown him the plans for the new house and a barnlike building for his fledgling adventure travel business. He was also talking about finishing some of his father's treasure hunts. He and Olivia would live in the house, which would allow The Farm at Carriage Hill to function exclusively as a destination getaway. In addition to soap making, she and Maggie were talking about offering herbal lunches, tours and lectures at Carriage Hill.\n\nUnlike Noah, Dylan and Olivia and their friends in Knights Bridge didn't lack for ideas of what to do with themselves.\n\nIf Olivia hadn't met Dylan, she would have happily continued to live and work at her center-chimney house, with guests coming and going. Noah didn't see Dylan sharing a bedroom with her down the hall from strangers.\n\nHe'd shown Grace Webster the plans for her former property, too. She'd told Noah when he'd visited her that she couldn't wait to see the new house.\n\n\"I expect to live that long, you know,\" she'd said with a twinkle in her aged eyes.\n\nHe had no doubt.\n\nHe occupied himself with a few NAK-related calls, cleaning Buster's bowl, vacuuming Buster's hair off the couch and picturing Phoebe harvesting orange mint.\n\nThen he arranged for his flight back to San Diego himself. At six, he was scrounging in the freezer for something else to thaw for a quick dinner when Dylan called. Noah didn't bother hiding his relief. \"Someone to talk to who's not from Knights Bridge. At least not yet. What's up?\"\n\n\"You tell me,\" Dylan said. \"Is there anything else I need to know about you and Phoebe O'Dunn?\"\n\nHis friend might as well have been reading his mind. Noah was used to it. \"I won't screw things up for you here, Dylan.\"\n\n\"That's not an answer, is it?\"\n\n\"Why are you asking?\"\n\n\"I had a drink last night with Loretta and Olivia at the Hotel Del. We got to talking. It's been on my mind all day. Olivia asked how fencing has influenced you.\"\n\nNoah frowned as he dug out another container of frozen soup. Tomato-basil. Sounded good, and he didn't need much to eat before his flight. \"Influenced me how?\"\n\n\"In life. How you think, how you look at the world.\"\n\n\"Mostly in fencing I'm trying not to get a blade driven into my heart.\"\n\n\"Exactly Olivia's point. Loretta agrees. You should have seen her. It was as if she'd had this sudden epiphany about you, what makes you tick.\"\n\nNoah set the container on the counter. \"Dylan? Are you still jet lagged? You're not making any sense.\"\n\n\"You're skilled at avoiding the touch of a sword,\" Dylan said, apparently undeterred. \"Any touch, not just one that goes to the heart.\"\n\n\"That's because any touch can be fatal.\"\n\n\"Is that how you're thinking now, about Phoebe?\"\n\nNoah made a face. \"That she's\u2014what, a fencing partner?\"\n\n\"That in life as well as in fencing, you seek to avoid the blade.\"\n\n\"That's a tortured metaphor, Dylan.\"\n\nHis friend sighed. \"I had to try.\"\n\n\"Phoebe and Maggie were here earlier getting a start with making essential oils. That's all I know.\"\n\n\"Essential oils?\"\n\n\"For the goat's milk soaps. You know, this soap making is interesting.\"\n\nSilence on the other end of the phone.\n\nNoah grinned. \"You're interested. You're just surprised that I am, too. Never mind. I haven't talked to Loretta today. Anything to add about Julius Hartley?\"\n\n\"He does a lot of work in Hollywood,\" Dylan said. \"You have both business and personal connections there.\"\n\n\"I did have personal connections. I haven't in a while.\"\n\n\"Maybe that's why Hartley's on your tail. Maybe some pissed-off actress you dated sicced him on you when you didn't bankroll her in a movie.\"\n\n\"Maybe,\" Noah said. \"I need to know.\"\n\n\"I agree.\"\n\n\"I'm having a bowl of soup and then flying to San Diego later tonight.\"\n\n\"Good,\" Dylan said. \"You, Loretta and I need to put our heads together and see if we can figure out what's going on.\"\n\n\"Loretta and I can.\" Noah felt a light breeze through the window above the sink. \"You and Olivia will be walking on the beach.\"\n\n\"She wants to go to the zoo.\" Dylan sounded reasonably enthusiastic. \"She promised to bring back stuffed giraffes for Maggie's sons.\"\n\nNoah smiled. \"Then the zoo it is.\"\n\n\"Enjoy your soup. What kind?\"\n\n\"Tomato-basil. I might add some of the pesto Phoebe and I made, although that could be overkill.\"\n\n\"Noah...\" Dylan broke off. \"Never mind.\"\n\nAfter they disconnected, Noah peeled the top off the soup container. It was frozen solid. He heard an owl or a wild turkey or something in the woods and fields out back. Then he remembered the Sloan boys were camping with their father.\n\nHe left the soup to thaw on the counter and went into the living room. Buster had escaped from the mudroom and was back on the couch. Noah left him in peace and cleared a space in front of the fireplace. He eased into a series of basic fencing moves, then switched to karate and did several katas. He focused on his movements, his technique, his breathing. The positioning of a foot, a hand, a shoulder\u2014even a knuckle\u2014mattered. Every detail was important, worthy of his attention.\n\nWhen he finished, he took a shower in the upstairs hall bathroom, using a fresh bar of lemon-scented goat's milk soap. It was mild, soothing, reminded him of the beauty of the Swift River Valley and surrounding hills, of the sensibilities of the smart, kind and deceptively tough women who lived there.\n\nHe dried off and wrapped his towel around his waist as he went into one of Olivia's unused guestrooms. He noticed neatly ironed vintage pillowcases stacked at the foot of the queen-size bed. He looked out the window at the field behind the house, quiet in the early-evening light.\n\nThe library's fashion show was coming up soon. The Grace Kelly and Audrey Hepburn dresses Maggie and Olivia had worn in Boston weren't the only ones in Phoebe's hidden room copied from Hollywood movies.\n\nNoah turned from the window. Thoughts and possibilities\u2014odd connections\u2014came at him fast and furiously. They might amount to something, or they might amount to nothing, but he definitely had to go back to San Diego and talk to Loretta.\n\nAnd to Julius Hartley.\n\nHe walked down the hall to his bedroom and pulled on clean clothes, then headed back downstairs. Buster had vacated the couch and was sniffing at the counter.\n\n\"That's my soup, my friend,\" Noah said, getting out a bowl. He glanced at his watch. He had time to eat his soup before he had to be at the small private airport for his flight.\n\nHe didn't have time to eat his soup and stop to see Phoebe.\n\nSeeing Phoebe won over Olivia's soup, as good as it no doubt was.\n\n\"On second thought, Buster,\" Noah said, \"the soup is all yours.\"\n\nNot that the big dog was seriously interested in tomato-basil soup. Noah filled Buster's bowls with food and water, figuring he'd ask Phoebe to make sure someone looked after their friend's dog. It could be his excuse for stopping to see her, should the O'Dunns, the Frosts, the Sloans and the rest of little Knights Bridge be keeping an eye on Thistle Lane.\nSeventeen\n\nPhoebe had said goodnight to the last of a summer reading group that had met while she and the fashion show committee had gone over the last details of what promised to be a fun night. How profitable it would be was anyone's guess but at least they were managing to keep costs down.\n\nShe was tidying up the circulation desk, about ready to lock up and head home, when she heard the front door creak. She was surprised to see Noah enter the library. He moved with his usual smoothness, and he wore jeans and a black button-down shirt, his sleeves rolled up. She smiled to herself. He was even sexier than he'd been in his black cape and mask.\n\nHe pointed toward the children's section. \"I'll just be in here while you finish up,\" he said, then stepped into the empty alcove.\n\nPhoebe stifled images of him as a five-year-old\u2014then as a father, taking his children to the library. But would he? Had he ever gone to the library himself as a boy, picked out books, sat with other kids through a story hour? With her evening meeting, she'd had a long day and had spent much of it\u2014even while picking mint with him at Carriage Hill\u2014thinking about how little she really knew about Noah Kendrick.\n\nBeing a librarian, she'd searched out more information on him that afternoon, beyond what Vera had read at the hairdresser's or what everyone in town already knew since Dylan's arrival there in the spring. Phoebe had a few more facts at her fingertips. Noah was thirty-three, the only child of a structural engineer and a high-school chemistry teacher, both retired and living at their wealthy son's California Central Coast winery.\n\nIn addition to the winery, Noah owned a house in San Diego and a condo in Hawaii, and he collected antique swords.\n\nHe'd sailed through MIT. No surprise there.\n\nPhoebe thought of her avocado-colored refrigerator and her flea-market finds.\n\nA different world.\n\nThe women in Noah's life tended to be very attractive actresses, with or without talent.\n\nTalent, Phoebe suspected, wasn't that big an issue to him.\n\nShe glanced at her watch as if she had somewhere else she needed to be, but she didn't. And as Noah left the children's section and returned to the main room, book in hand, she realized she didn't want to be anywhere else.\n\n\"You read The Tale of Peter Rabbit to the kids?\" he asked her.\n\nShe nodded. \"They enjoyed it. It's hard not to identify with Peter.\"\n\n\"He drinks chamomile tea at the end.\" He smiled that enigmatic smile she'd noticed straight off at the masquerade in Boston. \"You and Olivia and your sisters must love chamomile tea.\"\n\n\"Especially with lemon,\" Phoebe said with a laugh.\n\n\"That Peter is a risk taker. Not all his risks work out that well, but his life is more exciting because he takes a few chances. At least he ends up with some great stories to tell.\" Noah set the book on the circulation desk. \"Okay if I leave it here? My mind was elsewhere and I don't remember which shelf it was on.\"\n\n\"We'll take care of it.\"\n\n\"I'm loquacious tonight. Must be that uplifting orange mint I inhaled all morning.\"\n\nNothing in her research had indicated that Noah Kendrick, founder and CEO of NAK, Inc., had a wry sense of humor, but Phoebe had discovered he did. He stood back, studied a series of framed photographs of Quabbin\u2014Winsor Dam, Goodnough Dike, the cemetery where graves and monuments from the lost towns were relocated and the beautiful, pristine waters of the reservoir itself.\n\nPhoebe eased in next to Noah and pointed to the steep, grassy hill formed by the dam on the southern end of the reservoir. \"My mother used to roll down the hill when she was a kid. Can you imagine? They don't allow that anymore, and they closed the road over the dam after 9\/11 for security reasons.\" She nodded to another photograph of an inundated section of what used to be Enfield, the largest of the towns that were depopulated, disincorporated and razed. \"Those islands were once hills in the valley towns.\"\n\nNoah gave her a sideways glance. \"You love this valley.\"\n\n\"It was flooded decades before I was born, but when I see people like Grace Webster, it doesn't feel that long ago. She loves that the protected wilderness has helped with the return of bald eagles to the area.\" Phoebe smiled. \"She's holding out for mountain lions.\"\n\n\"Was your father from around here?\"\n\n\"He grew up in a mill town just north of here. He came to Knights Bridge after he returned from Vietnam. He was drafted. He said all he ever wanted when he got back was to live in a small town and have a small farm.\" Phoebe went back behind her desk and grabbed her tote bag. \"I don't think he ever imagined meeting my mother and having four daughters.\"\n\nNoah turned from the photographs. \"How did they meet?\"\n\n\"My mother got lost in the woods by our house. There was just the shed then. Dad was living out there alone. He thought he'd be alone forever. Thought he wanted to be. Then Mom showed up, no idea where she was\u2014and she's lived in Knights Bridge all her life. She was dehydrated, covered in mosquito bites and singing Christmas carols. It was the dead of summer, but she says she could only remember the words to Christmas carols.\"\n\n\"What did your father do?\" Noah seemed genuinely interested. \"Did he have a phone? I think I'd have called the police, or at least an ambulance.\"\n\n\"No phone. He liked to tell us as kids that he tried to pretend the shed was abandoned, but my mother saw his truck and had heard stories about a Vietnam vet living out there, and she doesn't give up easily.\" Phoebe switched off her desk lamp, then started to the entrance, aware of Noah watching her, his stillness, his intensity. She glanced back at him. \"Coming?\"\n\n\"I want to hear the end of the story. What did your father do when your mother finally got him to come out of the shed?\"\n\n\"He gave her water and took her home. She said that was when she knew he was the man for her.\" Phoebe was silent a moment, picturing the two of them together. \"Dad always said Mom saved him from becoming a hermit. He really loved all of us so much.\"\n\nNoah joined her and they went out together. She locked up, then walked down the stairs with him. It was dark now. She'd noticed it becoming darker earlier, another sign the end of summer was near.\n\n\"Did your father retain some of his hermitlike ways?\" Noah asked as they descended the stairs.\n\nPhoebe noticed their shadows as they walked out to the street. Across the common, two teenagers were playing Frisbee. She took a breath, regaining control over her emotions. \"He never liked to go places,\" she said. \"He was content to stay in Knights Bridge.\"\n\n\"What about you, Phoebe?\" Noah asked softly. \"Do you want to travel?\"\n\n\"I'd love to. My mother's always wanted to do a walking tour in England.\"\n\n\"What do you want to do?\"\n\nShe smiled. \"A walking tour in England. Jane Austen country, I think.\"\n\n\"So you'd go with your mother.\"\n\n\"Yes, absolutely,\" Phoebe said without hesitation. \"We travel well together, not that we've ever gone that far. Olivia wants Maggie and me to see San Diego.\"\n\nHe glanced at her. \"I think you'd like it.\"\n\n\"There's so much I want to see.\" She expected him to get into Olivia's car but didn't see it parked on the street. Instead he stayed with her as she turned onto Thistle Lane. \"I have a travel savings account and I'm accumulating travel points on my credit card. Every little bit helps. What about you? Do you like to travel?\"\n\n\"When I have time. I haven't traveled for pleasure as much as you might think. Hiking last week with Dylan and his friends was enjoyable but a hotel room on business isn't the same as walking with you on a quiet summer evening.\"\n\nShe felt a rush of heat and was grateful for the darkness. She could feel Noah's eyes on her as they walked down the lane, close to each other but not quite touching.\n\n\"You're happy here, Phoebe.\"\n\n\"I'm happy with what I have. My life might be predictable compared to some people's lives, but that's okay. It's good.\" She looked up at the sky, a few stars glittering against the darkening sky. \"It's warm out tonight but I can feel summer's winding down. Can't you?\"\n\nHe laughed. \"Don't tell me it's going to snow.\"\n\n\"I suppose your idea of snow is a ski slope?\"\n\n\"More like watching other people ski while I sit by the fire with a good Scotch.\"\n\n\"What about your winery? Do you get up there often?\"\n\nHe shrugged. \"As often as I can. The people who run it are good friends, and my folks live up there now. They love it.\"\n\nPhoebe didn't tell him she'd read about his parents. \"Why did you buy a winery?\"\n\n\"It seemed like a good idea at the time. It's worked out okay.\"\n\n\"But it's not what's next for you,\" Phoebe said.\n\nHe shook his head. \"No.\"\n\n\"I'd love to see that part of the country.\"\n\n\"I'd love to show you.\"\n\nPhoebe felt a rush of panic that she was getting in too deep with him. Falling in love with him. Not the billionaire, she thought. The man.\n\nBut he was a billionaire.\n\nThey came to her small house, a light on above the front door. \"I think my mystery seamstress lived here,\" she said. \"We found a box of books and things that could have belonged to her.\"\n\nNoah turned to her, his face more angular in the dark shadows. \"Would you mind if I took a look?\"\n\nHer mouth went dry. His tone, his eyes, the way he stood. All she could think about was their kiss in the library attic. She had no idea what he was thinking about, except he did seem genuinely interested in the box, or was doing a good job faking it.\n\n\"It's in the kitchen,\" she said finally. \"Come on. I'll show you.\"\n\nShe led him onto the porch and through the front door, switching on lights as she showed him back to the kitchen. She knew he was taking in her simple furnishings. Since it was just her living there, she didn't have to worry about melding her taste with anyone else's.\n\nShe offered him wine but he shook his head. \"But don't let me stop you.\"\n\nIt would stop her. Definitely. \"I don't like to drink alone,\" she said, then added, \"Not that I drink that much.\"\n\nHe was already lifting up a copy of Assignment in Brittany. \"Have you ever read Helen MacInnes?\" he asked.\n\n\"Not yet, no, but I was looking at that book last night and would love to read it.\" Phoebe leaned against the counter, watching him, really wishing she'd had wine. \"Helen MacInnes, Mary Stewart, Victoria Holt, Daphne du Maurier\u2014they're all popular with the women at Rivendell. Younger women are reading them now, too. I started The Moonspinners last night.\"\n\n\"My mother's a Mary Stewart fan,\" Noah said.\n\nPhoebe walked over to the table and ran her fingers along the softened spine of Rebecca. \"I love a happy ending.\"\n\nHe raised his eyes to her. \"Do you believe a happy ending is in store for you?\"\n\nShe shrugged. \"I'm not a character in a novel.\"\n\nHe set the Helen MacInnes novel back in the box and picked up This Rough Magic, another by Mary Stewart. \"Do you think in real life we only get one chance at a happy ending, and if it doesn't happen, that's it? We've lost our chance for a happy ending?\"\n\nPhoebe stood back from the copy of Rebecca, her eyes narrowed on him. \"Someone told you about Richard.\"\n\nIf Noah felt guilty at all, he didn't show it. \"Richard's the name of the guy who moved to Orlando without you?\"\n\nOrlando. Noah even knew that much. Phoebe faked a laugh. \"You can see I wasn't kidding when I told you I have no secrets. Richard dumped me two days after my father's funeral. I'd thought...\" She took a breath. \"Well, I thought he wouldn't do that. Who told you? Not my sisters.\" She thought a moment. \"Brandon Sloan. Male solidarity at work.\"\n\nNoah smiled. \"Brandon is outnumbered by O'Dunn women.\"\n\n\"Maggie's outnumbered by Sloan men. No one would even remember Richard if I'd had a string of affairs or gotten married, but I haven't. I have no complaints, Noah. I like my life.\"\n\n\"You have a life that you think will never change. In five years, ten years, thirty years, you'll be the director of the Knights Bridge Free Library, living on Thistle Lane.\"\n\nPhoebe dug into the box for another book. \"And what's wrong with that?\"\n\nNoah remained intent on her. \"Nothing at all is wrong with that, except that I've found that the future is hard to predict.\"\n\n\"I get that, Noah. I get that anything is possible. I could get fired. The library could get shut down. I could decide to move in with my mother and raise goats. I could go into catering with Maggie. You see? I get it.\"\n\n\"Possible doesn't mean probable.\"\n\n\"Is that an MIT way of talking?\" With sudden energy\u2014a burst of defensiveness\u2014she lifted three books out of the box and set them on the table. \"The library's always been in excellent hands. People love it. It'll be okay regardless of what I do. I'm a temporary caretaker, whether I'm there for five years or fifty years.\"\n\n\"Phoebe...I'm not trying to upset you.\"\n\nShe nodded. \"I know.\" She opened the faded, yellowed paperback copy of Le Petit Prince. \"Do you speak French?\"\n\n\"Some.\"\n\n\"What does 'some' mean to you?\"\n\n\"It means I can get along okay in Paris,\" he said.\n\n\"Do you like Paris?\"\n\n\"Dylan and I were there on business a few times. It's such a romantic city, and there we were, a couple of straight guys on our own, working twelve-hour days. We both thought it was terribly unfair.\"\n\n\"You regretted not bringing one of your Hollywood babes,\" Phoebe said, then winced. \"I'm sorry. That was rude and uncalled for.\"\n\nNoah didn't seem to take offense. He tucked a finger under her chin. \"I regretted not being there with a woman I cared about. So did Dylan. At the time, though, we didn't have that kind of woman in our lives.\"\n\nPhoebe resisted an urge to grab his hand, thread her fingers into his. It seemed crazy and at the same time inevitable. \"You two worked hard,\" she said. \"You had no guarantees that any of the risks you took\u2014all your hard work\u2014would pay off, especially early on.\"\n\n\"Part of what made it fun did.\" Noah slid a hand along what was left of her ponytail, down her back. \"Phoebe...\"\n\nShe knew he was attracted to her. She could see it in his eyes, in the set of his jaw. She could feel it in the way his hand drifted lower on her back. \"My seamstress taught herself French and fashion design, and as I said, I'm guessing she lived here.\"\n\n\"You're starting to identify with her.\"\n\n\"I want to know her story. I wonder if she went to Hollywood. If it was her dream and she seized the moment and went. To act, to be a costume designer\u2014I don't know. If she wanted us to find her\u2014anyone in Knights Bridge, I mean\u2014she'd have been in touch in the past forty years. Maybe she doesn't want to be found.\"\n\n\"She could have changed her name.\"\n\n\"I don't even know her name when she was here.\"\n\n\"But you're on the case,\" Noah said.\n\nShe nodded. \"I love a good mystery, too.\"\n\n\"Nero Wolfe always gets his man.\"\n\n\"He never really changes. It's one of the things I enjoy about him. But that's not how life is, is it?\" She draped her arms over Noah's shoulders. \"I think I'd like you to kiss me again, Noah Kendrick.\"\n\nHe was already lowering his mouth to hers, slipping his arms around her waist. He drew her against him, lifting her off her feet, nothing gentle or tentative about him when his lips touched hers. She felt her dress ride up, her bodice go askew, but didn't care about her exposed skin.\n\nHe sat her on the table, old books falling onto the floor as he skimmed his hands up her sides, letting his thumbs ease under her breasts, find her nipples under the fabric.\n\nAwareness, the ache of desire, spread through her. She opened her mouth to his kiss, tasted him. She shut her eyes, gasped when she felt him ease her dress off her shoulders. The bodice fell to her waist.\n\nHardly aware of what was happening, she suddenly felt her bra come off, the cool night air on her breasts. She couldn't breathe. \"Noah,\" she whispered, kissing him again, holding on to his shoulders.\n\nHe trailed light kisses down her throat, slowing when he reached her breasts. She still had a grip on his shoulders as he took a nipple between his lips. When she felt the wet heat of his tongue, it was all she could do to stay upright on the table.\n\nThe crack of a heavy tome hitting the floor brought him to his senses.\n\nHe stood back, gently easing her dress up over her exposed breasts. His eyes were dusky, a rawness to his movements despite his impressive, never-faltering control. Phoebe tried not to let her gaze drift too low, but she knew he wasn't unaffected by their near lovemaking.\n\nShe held her dress to her and smiled. \"Well. I'm glad we're in the kitchen and don't have to worry about neighbors peeking in the windows.\"\n\n\"It's a small town.\"\n\n\"And you're leaving.\"\n\nHe didn't contradict her. \"I'll be back,\" he whispered. \"I promise.\"\nEighteen\n\nMaggie was glad when Phoebe knocked on her door at seven-thirty and dragged her to the library before it opened. That way she couldn't drive over to Carriage Hill too early and Brandon couldn't accuse her of hovering over Aidan and Tyler.\n\nShe hated an empty house. She'd never lived alone and she'd lain awake most of the night, hearing every creak and groan, imagining what she'd do if a bat got in, remembering having Brandon asleep next to her.\n\nA long damn night.\n\nShe met Phoebe on the library steps. It was a perfect August morning. The boys would be having a blast with their father.\n\nPhoebe frowned. \"You okay, Maggie?\"\n\n\"Cranky. I'll get coffee after we're done.\" She stood back, appraised her older sister. Phoebe wore a dull green sundress with no jewelry, her hair down, barely combed, as if she'd had a bad night herself. \"What about you? You okay?\"\n\n\"Just a lot to do before the fashion show.\"\n\nIt made sense but Maggie didn't think it was all. Phoebe was used to juggling her professional obligations. The fashion show had become personal, too, with the dresses\u2014with Noah Kendrick.\n\n\"Noah left a message that he's gone back to San Diego,\" Maggie said. \"I'll stop by Olivia's after we're done here and walk Buster. Did you see Noah before he left?\"\n\n\"Just for a minute.\" Phoebe waved a hand toward the street. \"Here are Ava and Ruby now.\"\n\nThe twins joined them, looking curious and sleepy but not as cranky as Maggie felt. Phoebe didn't waste any time and took them straight up to the attic.\n\nHer hidden room was even more amazing than Maggie had expected. She had zero interest in sewing and fashion design, but the dresses, the fabrics\u2014the atmosphere of the tiny, cramped room\u2014affected her. She could feel the talent, skill and obsession of whoever had created it.\n\nHow many years ago was it? Thirty? Forty?\n\n\"I can see why you didn't say anything right away,\" Maggie said, looking at the fabrics, the finished dresses, the simple shelves and sewing table. \"It feels as if we've walked into someone else's secrets.\"\n\n\"Someone who was vulnerable, maybe,\" Ava added, rubbing her fingertips over the rose-beige silk of a dress hanging in an open garment bag.\n\nRuby raised the lid on a cedar-lined trunk. \"Phoebe, did you bring Noah up here?\"\n\nPhoebe opened a creaky corner door, morning light streaming in from a small window. \"Why would I bring Noah up here?\" she asked casually. \"Actually, he came up here on his own. I was checking it out after I called him about running into Julius Hartley.\"\n\nMaggie stood back, reading Phoebe's expression. Ava did the same thing and emitted something between a groan and a squeal. \"Phoebe!\"\n\nRuby's eyes widened. \"Has Noah fallen for you?\"\n\n\"You've fallen for him.\" Ava raked a hand through her hair. \"Phoebe, you know he's a billionaire, right? He's not a regular guy. Dylan's rich but he's a hockey player at heart. Not that I have anything against Noah, but Phoebe...\"\n\n\"I know,\" she said. \"Don't worry. There's nothing between us. Really. He's gone back to San Diego.\"\n\nAva seemed to regret her words. \"I just want you to be happy, Phoebe. I don't know Noah. He seems nice.\"\n\n\"It was that dress you wore the other night,\" Maggie said. \"It sucked you into romantic fantasies. They'll bite you in the end. They always do.\"\n\nPhoebe gave her a knowing look, as if she suspected Maggie's remark had less to do with Noah than it did with one Brandon Sloan.\n\nMaybe it did, Maggie thought.\n\n\"What are you going to do now?\" Ruby asked.\n\n\"I have to work today,\" Phoebe said. \"We have a lot to do to get ready for the fashion show.\"\n\nRuby sighed. \"I meant about Noah.\"\n\nPhoebe turned from the open corner door. \"My life's here in Knights Bridge.\" She smiled. \"What would you all do without me?\"\n\n\"Visit you in San Diego in February,\" Ava said with a laugh.\n\nBut Ruby wasn't done yet. \"So you're not doing anything,\" she said, clearly frustrated. \"Noah Kendrick is interested in you, and now it's just business as usual?\"\n\n\"Maybe what's next isn't up to me,\" Phoebe said quietly.\n\nRuby groaned. \"Then who is it up to?\"\n\nPhoebe didn't answer as she ushered her sisters back downstairs. Ruby and Ava stayed behind to work on the fashion show, but Phoebe made them promise not to pester her about Noah.\n\nAs she drove out to Carriage Hill, Maggie remembered the look on Phoebe's face when they'd found her billionaire shirtless in Olivia's kitchen. Maggie didn't need confirmation. There was no question in her mind that her sister had fallen for Noah.\n\nIf he broke her heart, Maggie would fly to San Diego herself.\n\nAnd do what?\n\nNoah was Dylan's best friend, and Dylan was Olivia's fianc\u00e9. That wouldn't stop Maggie from giving Noah a piece of her mind, but it would Phoebe. She'd just keep her pain to herself and carry on.\n\nBut what if Noah had fallen for Phoebe, too? What if they were meant to be together, just like Olivia and Dylan?\n\n\"Then what?\" Maggie asked herself aloud as she climbed out of her catering van.\n\nShe ignored a pang of loneliness, loss\u2014she didn't know what it was. She wanted all three of her sisters and all of her friends to be happy and knew they wanted the same for her. Except she wasn't happy, she realized. Not romantically, anyway.\n\n\"What you are is discombobulated because the father of your sons is living in a damn tent.\"\n\nMuttering to herself couldn't be a good sign of anything, she thought as she headed into Olivia's kitchen. There was no sign of Buster. She assumed Noah had also let Brandon know he was leaving. She'd run into a couple of Sloans yesterday when she'd stopped at Dylan's place and tried to surreptitiously check out Brandon's tent, telling herself she just wanted to see where the boys would be camping. Sloan & Sons was working hard on carrying out the plans for the property. Brandon's uncle and his eldest brother had caught her peering into the tent and teased her. She'd found their underlying assumption that she and Brandon would get back together both annoying and comforting.\n\nThat kind of ambivalence couldn't last, she knew. She needed clarity in her life. Tyler and Aidan needed it, too.\n\nShe heard laughter and went into the mudroom, debated a moment before she stepped outside onto the terrace. It was a stunning morning, clear and dry, the sun shining on Olivia's flowers and herbs. Tyler and Aidan were charging up a path, laughing. Brandon ambled behind them with Buster on a leash.\n\nMaggie felt a jolt of awareness as her husband approached the terrace. He hadn't shaved, wore a black flannel shirt over jeans. She noticed the shape of his shoulders, his hips, his legs as he unclipped Buster's leash and warned him to stay out of the gardens. She blamed overwork and her sleepless night for her reaction and quickly looked away, although not before she saw Brandon grin. He'd noticed.\n\nOf course he had. Bastard.\n\nShe smiled at Tyler and Aidan as they jumped onto the terrace. \"Have you guys had breakfast?\"\n\n\"Cereal,\" Aidan said. \"I wanted pancakes.\"\n\n\"I bet I can find the ingredients for pancakes,\" Maggie said. \"I know Olivia has a griddle and maple syrup.\"\n\nTyler obviously liked that. \"Can we have blueberry pancakes?\"\n\n\"I don't know if there are blueberries\u2014\"\n\n\"We picked some with Dad,\" Aidan said.\n\nShe hadn't noticed the small covered plastic container that Brandon had in one hand. He set it on the table. \"Should be enough for pancakes.\"\n\nIt was what they'd done every summer since they were teenagers. Picked wild blueberries together. Made pancakes. Maggie fought back tears and grabbed the container. \"Why don't you guys burn off some energy out here and I'll see what I can do?\"\n\n\"Do you need any help?\" Tyler asked.\n\nHe was her budding chef, but she shook her head. \"You're on vacation today.\"\n\nShe returned to the kitchen. She'd catered a number of events for The Farm at Carriage Hill already and knew her way around the kitchen well. She quickly got out a pottery mixing bowl, measuring cups, measuring spoons and ingredients\u2014stone-ground cornmeal, baking powder, baking soda, salt, canola oil, buttermilk.\n\nBrandon came inside and dug out the electric griddle. Maggie tried not to think about how familiar it felt to have him there, working in the kitchen with her while the boys played outside. Familiarity was an illusion. They had gone their separate ways months ago.\n\n\"Noah's gone back to California,\" Brandon said.\n\n\"I know. He left me a message.\"\n\n\"We talked some about Dylan's plans to get into adventure travel. I could almost see the wheels turning in Noah's brain. Guy's smart. Always thinking.\"\n\n\"Does he want to get into adventure travel, too?\" Maggie asked, surprised.\n\nBrandon shook his head. \"No, but he thinks I should. He figured out I have a touch of wanderlust.\"\n\n\"More than a touch,\" Maggie said. \"You'd be good at adventure travel, Brandon. Do you think you'll talk to Dylan about it?\"\n\n\"I don't know.\"\n\nShe whisked together the dry ingredients for the pancakes. \"It'd be okay with me if you do. I'd like that. I never wanted you to give up your dreams. I just...\" She set aside the bowl and lifted the strainer of blueberries out of the sink. \"I was scared.\"\n\nBrandon opened his hand an inch above the griddle, testing the heat. \"I was in a dark place last year. Took it out on you.\"\n\n\"No, you didn't.\" Maggie dumped the blueberries onto paper towels. \"You took it out on yourself, and I reacted. I let my fear of going broke and all that went with it infect everything. What I did, how I felt, how I thought.\"\n\n\"My lack of faith in my future\u2014our future\u2014affected you. I didn't see that. I was caught up in my own stuff.\" He pulled his hand from the heat and turned to her, his eyes dark, filled with pain. \"I was out of work and I let pride get in the way of making good decisions. I didn't do right by you, Maggie.\"\n\nShe patted the blueberries dry, grateful that she had something to do. \"I don't want you coming back to Knights Bridge and working for your family just for me.\"\n\n\"There's no better reason, is there? Come on. Let's get these boys fed.\"\n\nMaggie mixed the dry and wet ingredients, and Brandon dropped the pancake batter onto the hot griddle while she sprinkled on a handful of the freshly picked blueberries. While they waited for the bottoms to brown, he leaned against the counter, holding a spatula. \"What are you going to do with your best friend marrying a multimillionaire and your sister marrying a billionaire?\"\n\nMaggie took a breath. What would she do? She scooped up another handful of berries. \"You're jumping the gun with Phoebe and Noah.\"\n\n\"Nope.\"\n\nIn spite of her concern for her sister and her emotional state\u2014the risks of getting involved with a man as complicated and intense as Noah\u2014Maggie liked hearing the confidence in Brandon's voice. She watched him flip the pancakes, smelled the sweetness of the heated, softened blueberries.\n\n\"What's on your mind, Maggie?\" he asked.\n\n\"Phoebe and Noah...I don't want him to break her heart.\"\n\nBrandon leveled a steady gaze on her. \"What if she breaks his heart?\"\n\nMaggie opened a jar of maple syrup that her mother had made in the spring from her own trees. Was he talking about Phoebe and Noah, or about himself and her? She didn't want to read too much into his words. Being physically close to him had her in a mess.\n\n\"Noah's the one who left Knights Bridge,\" she said. \"Phoebe's still here.\"\n\n\"Maybe that's part of the problem. Maybe Phoebe's got to give a little, too. See the possibilities.\"\n\n\"They've only known each other a few days.\"\n\n\"Instead of all their lives?\"\n\nBefore Maggie could respond, he grabbed a platter and flipped the pancakes onto it, then dropped more batter onto the griddle. She added the wild blueberries, wondering if he was right about Phoebe. About her. About them.\n\nHe went to the mudroom and called out the back door for the boys to come in for the pancakes. \"Get them while they're hot.\"\n\nMaggie watched him return to the griddle, wink at her as he picked up his spatula, and she realized she was in danger of falling in love with him all over again.\n\n* * *\n\nPhoebe worked in her backyard when she got home from the library. Pruning, weeding, checking for insect damage. It was a beautiful evening, and she appreciated having time to herself.\n\nNeeded time to herself.\n\nAfter her sisters had left that morning, she'd checked with the library's part-time custodian, a retired machinist and avid reader, but he didn't know a thing about any hidden room. His predecessor hadn't mentioned it. He'd died last year. Phoebe remembered him as a solid, unimaginative man. He could have cleaned the hidden room periodically and not thought twice about it.\n\nShe went back inside. She'd stacked up the books that she and Noah had knocked over. Her skin burned when she thought of sitting on the table with her dress half off.\n\nWhat was he doing now in San Diego?\n\nShe grabbed a Diet Coke out of her refrigerator and put the cold bottle against her cheek. She sat at the table. It stood to reason that someone who'd created a hidden room in a library would like to read. Seventy years ago, Grace Webster had buried herself in classic adventure tales while her world disappeared around her, literally scraped, burned, razed and carried off. She'd read The Three Musketeers, Scaramouche, The Scarlet Pimpernel.\n\nPhoebe wondered if Mary Stewart, Victoria Holt and Helen MacInnes had fired the imagination of her seamstress, diverted her on a bad day, entertained her on a good day.\n\nMaybe she'd collected the books for someone else or just hadn't gotten around to reading them.\n\nWho was she?\n\nAs Phoebe fingered one of the sewing books, she couldn't shake the feeling she'd had when she'd first discovered the attic room\u2014that somehow the woman who'd sewn there, dreamed there, had felt trapped by life in Knights Bridge.\n\nHad she abandoned the life\u2014escaped the life\u2014Phoebe was now living?\n\nShe gave an inward groan. So what if her mystery seamstress had hated Knights Bridge? Phoebe didn't. She loved her work. She loved her cottage. She loved her family and friends, being close to them, connected to her childhood.\n\nShe was just on edge and overthinking everything because of what had happened between her and Noah. She never should have let herself get so carried away with him. What had she been thinking?\n\nShe hadn't been thinking, obviously.\n\nOnce again she opened the yellowed copy of Le Petit Prince, its pages brittle with age.\n\nLorsque j'avais six ans j'ai vu, une fois, une magnifique image, dans un livre sur la For\u00eat Vierge qui s'appelait \"Histoires V\u00e9cues.\"\n\nCould her unknown seamstress have taken off to France? Phoebe flipped through the Antoine de Saint-Exup\u00e9ry novel. She didn't know what she hoped to find. An old letter tucked in the pages? A signature?\n\nShe opened her Diet Coke and methodically checked each of the books, looking for anything that could offer answers, even a clue that would point her in the right direction.\n\nThere was nothing.\n\nHad her mystery woman sat here, in this spot, listening to the crickets on a pleasant summer evening?\n\nShe heard a knock on her front door. \"Phoebe?\" It was her mother's voice. \"Can I come in?\"\n\n\"Of course,\" Phoebe said.\n\nShe started to get up but her mother was already through the living room. \"I worked late and thought I'd stop by,\" she said, getting a glass down and filling it with water from the tap. \"You don't mind, do you?\"\n\nPhoebe shook her head. \"It's water, Mom. Do you want anything else?\"\n\n\"No, this is fine.\"\n\nShe had on her work clothes, a flowered tunic over wide-legged white linen pants with neutral-colored slides. Phoebe had changed into shorts for her gardening. \"Mom?\"\n\nShe drank some of her water, then set her glass on the counter and walked over to the table. She patted the top book on the pile. \"The Vogue Sewing Book. My mother had a copy. She taught me to sew. I was never any good at it, but she did her best. I was always more interested in gardening and boys. My best subject in school was math. Isn't that funny? It didn't translate into being good with money, obviously.\"\n\n\"You've always managed to get by,\" Phoebe said.\n\n\"Barely.\" She tapped a finger on the front cover of the sewing book. \"Ava and Ruby told me about their visit to the library this morning. They said you gave them permission to tell me about the attic room.\"\n\n\"Did you know about it?\"\n\n\"No. I've only been in the attic once. A friend and I went up there in search of ghosts. We were in junior high...\" She sank into a chair, clearing her throat before she continued. \"We took French together. My friend was good at it but I just couldn't get the hang of it. There was a young woman who worked at the library who was fluent in French. She offered to tutor me.\"\n\nPhoebe stared at her mother. \"What was her name?\"\n\n\"Debbie Sanderson.\"\n\n\"Sanderson?\"\n\n\"She said she was George Sanderson's great-great-granddaughter but none of us ever believed her. She was here such a short time. It's been forty years, Phoebe. I was just a kid myself.\"\n\n\"What was her job at the library?\"\n\n\"I don't really know. An assistant, I think. She wasn't a librarian. I know that much. There was a bigger staff in those days.\"\n\n\"Four people instead of two,\" Phoebe said with a smile. \"How long was she here, do you know?\"\n\n\"It couldn't have been more than two years. She tutored me for half the school year. She didn't want any money, but my parents insisted on paying her.\"\n\n\"Did it work? Did you pass French?\"\n\n\"I most certainly did pass.\"\n\nAs she watched her mother pick up the copy of Le Petit Prince, Phoebe envisioned Elly O'Dunn\u2014then Elly Macintosh\u2014at twelve, conjugating French verbs. \"What was Debbie Sanderson like?\"\n\n\"She never wanted to be a librarian, but she had a fantastic imagination and loved to read. She loved to dress up in exotic clothes and speak French to us, and she loved movies, gothic novels and poetry.\"\n\n\"Poetry?\" That caught Phoebe by surprise since she hadn't discovered any poetry books in the box.\n\n\"That's right. I remember because...\" Elly set Le Petit Prince back on the table. \"Oh, Phoebe. I haven't thought about Debbie Sanderson in such a long time.\"\n\n\"Mom, you're about to cry. We don't have to talk about her\u2014\"\n\n\"It's okay. It's not that. I just tend not to let myself go back too far into the past. I didn't know your father yet when Debbie was tutoring me. He had just moved here. It wasn't long after he got back from Vietnam. He'd put enough money together to buy a few acres and was building his shed. He was a recluse, really.\"\n\nPhoebe pictured her father roaring with laughter when she and Maggie had told him about getting the better of the Sloan boys at the pond at the Frosts' sawmill. Ava and Ruby had been toddlers at the time, and he'd had them bouncing on his lap.\n\n\"He never liked being around a lot of people,\" Phoebe said. \"But he didn't stay a recluse.\"\n\nHer mother nodded, then said quietly, \"Because of Debbie.\"\n\n\"Did they\u2014were they an item, anything like that?\"\n\n\"It wasn't like that. He stopped at the library for a do-it-yourself book on plumbing, and she was there. She introduced him to poetry. He couldn't concentrate on a novel back then, but he could read a poem. He especially loved Robert Penn Warren's poetry.\"\n\n\"I remember,\" Phoebe said.\n\n\"He went on to reading novels. Robert Parker, Tom Clancy, Ross McDonald. He had so many favorites, but he continued to read poetry, too.\" She blinked back tears. \"He was a wonderful man, Phoebe. I had him\u2014we had him\u2014for the time we did because of Debbie Sanderson and the library. They helped him heal. They saved his life. There's no question in my mind.\"\n\nPhoebe felt her throat tighten with emotion. \"Do you know why Debbie came to Knights Bridge?\"\n\n\"She chose it because of her great-great-grandfather, but I think she came here to heal, too. I didn't realize she liked to sew as much as she must have, or that she was so good at it that she could copy Hollywood dresses. I knew very little about her. Just what I've told you.\"\n\n\"Did she say goodbye when she left town?\" Phoebe glanced around her small kitchen, wondered if it'd been much the same forty years ago. \"Did anyone notice she was gone?\"\n\nWhen her mother looked away, focused on the darkening night out the window by the table, Phoebe could see a glimpse of Elly Macintosh O'Dunn at twelve. \"It was summer,\" her mother said. \"I didn't even realize Debbie had left until I started school in September. I should have taken more of an interest. She was invisible, in a way.\"\n\n\"Mom, you were twelve.\"\n\n\"When I think back, I realize how young she was, too. Maybe twenty-one. She was such a dreamer. I could see it when she tutored me. She wanted a life that Knights Bridge couldn't give her.\"\n\n\"And all Dad wanted was Knights Bridge,\" Phoebe said quietly. \"He read poetry to us as kids.\"\n\n\"Poetry helped him cope with his experience in combat,\" her mother said. \"He didn't have a long life but he lived longer than many of the young men he served with. He took each day as it came and lived in the moment. Maybe that meant he wasn't as good with money and planning as some.\"\n\n\"But we have the land because of him.\"\n\nHer mother turned from the window. \"I have a good life, Phoebe. I like my job. It gets me out every day. What would I have done with a big insurance policy?\" She smiled, a spark coming back into her eyes. \"Blown it on horses instead of making do with goats.\"\n\nPhoebe smiled, too. \"Now the goats are coming in handy with Maggie and Olivia's soap making.\"\n\n\"Who'd have ever thought?\" Her mother laughed, but her lightheartedness didn't last. She leaned forward, took Phoebe's hand. \"Honey, I know you've helped me and I appreciate all you do, but I don't want you to worry about me. I don't want you not to live your life\u2014to feel tied to Knights Bridge\u2014because of me.\"\n\n\"I've never felt tied to Knights Bridge because of you or anyone else,\" Phoebe said. \"I like my life.\"\n\nHer mother didn't seem to hear her. \"Change is a part of life. Even if I knew deep down it was a delusion, I thought I'd grow old with your father. Instead I became a young widow with four teenage daughters. You have a big heart, Phoebe. Sometimes you ignore it so that you can be quote-unquote sensible. Don't ignore it now, okay? Not because you're worried about me, or your sisters. Open up your world if that's where your heart takes you.\"\n\nPhoebe shot to her feet, uncomfortable. She and her mother seldom had deep conversations and she didn't know what to make of this one.\n\n\"You have things to do,\" her mother said, rising. \"And I have more tomatoes to can tonight. I might turn them into sauce. I'll decide on the way home.\"\n\n\"Thanks, Mom,\" Phoebe said as she followed her through the living room. \"Thanks for stopping by.\"\n\n\"Phoebe...\"\n\n\"I'll be fine. Don't you worry about me, either, okay?\"\n\n\"I'm your mother. It's my job to worry.\" She laughed, and became her bouncy self again as she left.\n\nPhoebe returned to the kitchen and threw out the rest of her Diet Coke. She poured the last of a bottle of pinot grigio, took it out to the front porch and sat on the steps. So much for not drinking alone. It was quiet on Thistle Lane, but it always was. She sipped her chilled wine and smelled roses in the night air. A half moon created shadows that stirred in a gentle breeze.\n\nShe'd brought her cell phone out with her and stared at it in her palm. She had Noah's number memorized. That was a clue to her feelings, wasn't it? She debated just texting him but instead dialed the number.\n\nHe picked up right away. \"Phoebe.\"\n\nHis voice was calm, deep and made her heartbeat quicken. \"Hi. I hope I'm not catching you at a bad time\u2014\"\n\n\"Not possible.\"\n\nShe smiled and told him about Debbie Sanderson, her mother's French lessons, her father's poetry. He didn't interrupt. She could feel him listening to every word she said. She told him she'd brought her sisters up to the attic room that morning but didn't mention their questions about him.\n\nHe asked about the fashion show and what she was doing, where she was right now.\n\n\"It's a beautiful night,\" she said. \"It's nice after the heat.\"\n\n\"Helps since you have no air-conditioning.\"\n\n\"I only wish I had it a few nights a year. I have fans. I put one at the foot of my bed and...\" Phoebe stopped herself. \"I manage.\"\n\nNoah was silent, and she wondered if he was picturing her lying on her bed in next to nothing, or nothing at all, with a fan on her.\n\n\"Tell me about San Diego,\" she said.\n\n\"It's warm, sunny and not humid.\" He paused, and she could feel his smile. \"The same.\"\n\n\"Have you been to your office?\"\n\n\"Yes. I had pencils to sharpen.\"\n\nShe laughed. \"I love a good pencil.\"\n\n\"So do I, techie that I am. I'm meeting Loretta Wrentham tomorrow. She's getting Julius Hartley down here.\"\n\n\"I hope you get to the bottom of what he's up to.\"\n\n\"I'm sure we will,\" Noah said. \"Tell me again about Debbie Sanderson and her time in Knights Bridge.\"\nNineteen\n\nLoretta paced on Dylan's porch while she waited for Julius Hartley to park his BMW and join her. She could smell the ocean and taste it on the breeze, but she didn't care. That told her just how keyed up she was. She loved the ocean, the sand, the rocks, the birds, the colors of sky and water. Watching Navy SEALs run on the beach wasn't bad, either. She lived in La Jolla but she enjoyed coming out to Coronado.\n\nJust not today.\n\nDylan had disappeared with Olivia, saying something about stuffed giraffes from the zoo for kids back in Knights Bridge. Loretta knew what that meant: she was on her own. She'd helped make this mess with Hartley by trusting him, by not realizing sooner that he was Noah's mystery man.\n\nNow she could clean it up.\n\nNoah was back in San Diego, on his way to Coronado. He and Dylan would have already talked. She didn't know that for sure, but it was how the two of them operated. It was how it had always been and always would be. Friendships like theirs were rare. She'd seen that the first time she'd met them. Dylan and Noah had each other's backs. Dylan had a woman in his life now who understood that. Loretta didn't know if Noah ever would.\n\nShe watched Hartley mount the steps to the porch. He had on an expensive pale blue polo shirt and dark tan trousers, and he looked more like a country-club type than a scumbag private investigator. She'd dressed in a crisp black suit with her red heels and hoped she looked like she not only wanted to kick him down the stairs but could do it.\n\nHe smiled at her, no sign he knew how mad she was. \"Hi, Loretta. Nice day, isn't it?\"\n\n\"It's Southern California. It's always nice.\" She let him get onto the porch before she glared at him. \"You took it upon yourself to snoop on Dylan McCaffrey and especially Noah Kendrick. You snooped on a little rural town in New England.\"\n\n\"Yeah. Sure. It's what I do.\"\n\nShe pointed a red-nailed finger at him. \"You're a son of a bitch, Hartley.\"\n\nHe shrugged. \"Okay.\"\n\n\"Who were you on the phone with on Friday when Phoebe O'Dunn overheard you?\"\n\n\"It doesn't matter.\"\n\n\"It does matter.\" She paused to catch her breath. He was as roguishly good-looking up close, out of the sunlight. She reminded herself to stay focused. The man couldn't be trusted. \"Was it your client? Are you working for an attorney, or is this one of your private clients?\"\n\nHe turned and faced the water. \"This is nice. McCaffrey's giving up this place for Knights Bridge, huh?\"\n\n\"For Olivia Frost, and I don't know that he's giving it up. He hasn't asked me to look into putting it onto the market. Not that he will.\" Loretta gritted her teeth. \"I'll probably be banished from Noah's and Dylan's sight before cocktail hour tonight, thanks to you.\"\n\n\"That's some drama going there.\" Hartley gave her a sideways glance. \"Am I supposed to\u2014what? Feel guilty?\"\n\n\"You're supposed to tell the truth. I can't believe I didn't see through you sooner, but I just didn't take you for a snake.\"\n\n\"That's because I'm not a snake.\" He grinned, his eyes crinkling at the corners. \"Well, I'm not always a snake. I can be when the situation calls for some slithering. I tried to be discreet with my scrutiny of Dylan and Noah, if that helps any.\"\n\nLoretta shook her head. \"It doesn't.\"\n\n\"It was easier in Boston. Any stranger would stand out in Sleepy Hollow, but I really did. I had this thing in my head that I was looking into rich guys and should therefore try to blend in with them.\"\n\n\"Should have left the Rolex at home, huh?\"\n\n\"I'm just saying that I can see how people thought I was sneaking around.\"\n\n\"You were sneaking around,\" Loretta said.\n\n\"If I'd been sneaking around, no one would have ever known.\" His dark eyes narrowed on her. No smile or hint of humor now. \"Trust me on that, Loretta.\"\n\n\"Am I supposed to be intimidated?\"\n\nHe surprised her by laughing. \"Damn, you're a pain in the ass. I wanted to know what was going on with your two pals and this Sleepy Hollow little town in Massachusetts. That's all.\"\n\n\"That's not all, and Dylan and Noah can take care of themselves.\" Loretta squinted out at the ocean, added without looking at him, \"Are you providing intel to NAK corporate enemies?\"\n\n\"Let's not play twenty questions, Loretta.\"\n\nShe ignored him. \"The NAK board? Do Noah and Dylan have personal enemies I don't know about?\"\n\nA breeze off the water lifted the ends of Hartley's gray-streaked dark hair but he didn't seem to notice. \"Noah needs to decide what's next for him. It's driving the NAK board crazy not to know if he's going to try to run things there or open a fencing studio. But he knows that. You know that.\"\n\nLoretta tightened her hands into fists at her sides. \"You spied on my friends and you used me to do it.\"\n\n\"No one uses you, Loretta. I outwitted you. There's a difference.\"\n\nShe raised her chin at him. Not all men were taller than she was. \"I'm going to find out what you're really up to.\"\n\nHe didn't seem that threatened. \"You understand I have to respect attorney-client privilege.\"\n\n\"You're not an attorney. You're a sleazy private investigator.\"\n\n\"Part right. I'm a private investigator. I'm not sleazy. Which you know. You're just irritated because you're not in control of what's going on. You haven't been since Duncan McCaffrey left Dylan that house in Knights Bridge and you didn't know the whole story.\" Hartley had some sympathy in his expression now. \"You didn't know Dylan would go out there and fall for Olivia Frost.\"\n\n\"I'm glad he did,\" Loretta said stubbornly.\n\n\"You're glad he has someone in his life. You wish it was a woman from La Jolla instead of an out-of-the-way little town on the other side of the country. Now Noah's falling for this redheaded librarian.\" Hartley's sympathy turned to a knowing grin. \"I can just see you at the Knights Bridge Free Public Library. It's haunted, you know.\"\n\nOf course it was haunted. Loretta forced her palms open, tried to release some of her pent-up tension as Hartley pivoted and walked back down the steps without another word.\n\nShe inhaled deeply. She'd never been good with men.\n\nShe called to him. \"Do you like Mexican food?\"\n\nHe glanced up at her. \"I hate it.\"\n\n\"I love it.\" She followed him down to the sidewalk. \"There's a great place down the street. You can have a salad. Let's go. We can walk.\"\n\n\"You're the most difficult woman I've ever met. I think that's why I like you.\" He angled her a knowing look. \"Is Kendrick meeting us at this restaurant?\"\n\nLoretta didn't bother hiding her surprise.\n\n\"This one wasn't even hard to figure out,\" Hartley said with a bark of a laugh. \"You'd have had me meet you in La Jolla at your office if you weren't involving Dylan and Noah.\"\n\n\"Mr. McCaffrey and Mr. Kendrick to you,\" she said, sounding petty even to herself.\n\n\"Sure thing, Loretta. I have a son their age. He probably wants me to call him Mister, too.\"\n\n\"A son?\"\n\nHe grinned at her. \"Relax. I'm divorced.\"\n\nShe didn't relax but she didn't want to kill him as much as she had twenty minutes ago. They walked to a cluster of shops and restaurants, and for seconds\u2014or maybe only one second\u2014she pictured them as one of the honeymooning couples at the Hotel del Coronado. She'd never been married. Never had kids. Dylan's engagement had her thinking about what might have been, even if she had a good life, even if she had no regrets.\n\nNo serious regrets, anyway.\n\nThey got a table in the courtyard of the bustling restaurant. Loretta ordered a margarita and guacamole made fresh at the table. Hartley ordered a beer.\n\nShe dipped a warm tortilla chip into chunky, spicy salsa. \"You're going to tell Noah and me everything.\"\n\n\"No, I'm not. You know better than to ask.\"\n\n\"Then you are working for an attorney.\"\n\n\"I'm not saying.\"\n\n\"Why did you come if you're going to stonewall?\"\n\nHis beer arrived. \"Because you asked nicely.\"\n\nShe hadn't, but whatever. \"What did you think of Knights Bridge?\"\n\n\"I suffered. Goats, Loretta.\" He drank some of his beer, helped himself to a chip and salsa. \"Olivia Frost and the O'Dunn sisters are making soap out of the goat's milk.\"\n\n\"Goat's milk soap is nice.\"\n\n\"I'm sure it is.\"\n\n\"Hartley...\"\n\n\"Julius, okay?\"\n\nLoretta helped herself to another chip. \"Julius, did you check out the goats?\"\n\n\"Almost. I pretended I was going to buy one.\"\n\n\"And they bought that?\"\n\n\"No.\" His tone was matter-of-fact. \"The O'Dunns and their allies were about to get out the hot tar and pitchforks, so I made my exit. They're little goats, by the way. Nigerian Dwarf goats.\"\n\nLoretta felt laughter bubbling up despite how mad she was. \"Have you ever seen a goat in real life?\"\n\n\"The zoo.\" He sat back, looking at ease, comfortable in his own skin. \"Then there are the herbs for the soaps. The nineteenth-century library. The town common with its Civil War statue. The country store.\"\n\n\"It sounds idyllic.\"\n\n\"It's pretty,\" Hartley said, making it sound like a concession.\n\n\"Dylan says not to be fooled,\" Loretta said. \"Despite any evidence to the contrary, time hasn't stopped in Knights Bridge.\"\n\n\"Time never stops, does it?\"\n\nLoretta heard a note of wistfulness in Julius's voice, or thought she did. Maybe she was projecting. Maybe that was why she hadn't figured out what was going on with him to begin with. She'd wanted him to be someone he wasn't because she herself was coming to terms with the changes in her life. She'd been Dylan's attorney and business manager for a long time. She'd loved his father, even if for a short time.\n\n\"I'll have to see Knights Bridge myself soon,\" she said, digging into the fresh salsa. \"Dylan and Olivia have invited me to their wedding.\"\n\n\"It's at Christmas, you know.\"\n\nShe nodded. \"So I get to go there when it's freezing. I'll have to find myself a cute country inn with a fireplace, flowered wallpaper and a decent liquor cabinet.\"\n\nTheir waiter whipped together their guacamole and set it on the table with fresh, warm chips. Julius helped himself. Their table was pleasant, shaded by potted trees. Deep pink bougainvillea cascaded over a wall.\n\nFinally he said, \"Dylan and Noah are decent guys.\"\n\n\"Yes, I know.\"\n\n\"They're independent. Defiant, even. They do things their own way.\"\n\n\"What's Phoebe O'Dunn like?\" Loretta asked.\n\n\"From what I saw and heard, she's smart, positive, encouraging and genuinely nice.\"\n\n\"And?\"\n\n\"And protective of her family. She looks out for them.\"\n\n\"Who looks out for her?\"\n\n\"She'd say they look out for each other. Maybe they do, but she's stuck there.\"\n\n\"Maybe your idea of 'stuck' is her idea of fulfilling herself.\"\n\n\"That was before Noah Kendrick spotted her at that masquerade ball. Getting involved with a billionaire...\" Julius shrugged. \"Easier to fall for one of the local firefighters but she hasn't. She'll sacrifice herself if she thinks her family needs her. She's done it before.\"\n\nBefore Loretta could ask more questions, Noah arrived and joined them at their table. He was centered, focused and clearly on a mission. He was a man, she knew, who did best when he had a purpose, a result he was going after. Aimlessness didn't suit him.\n\nJulius picked up his beer, took another deliberate sip. \"How's Sleepy Hollow and your librarian? Same as ever?\"\n\nNoah leveled a cool gaze on him. \"Phoebe is getting ready for the vintage fashion show at the library. You know about that, right, Julius?\"\n\n\"I saw something about it when I was in Knights Bridge.\"\n\n\"You knew about the fashion show before you arrived there. It's what prompted you to check me out in the first place.\"\n\nLoretta frowned and noticed that Julius had gone silent. She glanced at Noah, but his gaze was fixed on the older man across from him.\n\nNoah reached for his water glass. \"It was logical to assume that I was the one who brought you to Knights Bridge, but it's not that simple.\" He sat back, as in control as Loretta had ever seen him. \"You knew that Dylan and his work with NAK, his friendship with me, could bring scrutiny to Knights Bridge. Could change things there. You figured out Phoebe must have discovered the hidden room where your client sewed and designed dresses as a young woman.\"\n\nLoretta held on tight to her margarita. Olivia and Dylan had told her about the hidden sewing room in the Knights Bridge library attic. She'd figured that sort of thing happened in small-town New England. She'd liked the idea of the Hollywood-inspired dresses. She hadn't considered\u2014not even for a split-second\u2014that they had anything to do with Julius Hartley.\n\n\"That hidden room,\" Noah said, \"is why you checked me out here in San Diego and why you followed me to Boston. It's why you went to Knights Bridge. You weren't just checking on me there. You were checking on the O'Dunns. Specifically, on Phoebe.\"\n\n\"It didn't take long. She's what we call an open book.\" Hartley smiled, added, \"No pun intended.\"\n\n\"Debbie Sanderson is the given name of the woman who created the hidden room.\" Noah kept his eyes on the man across the table. \"She lived in Knights Bridge for a year. Then she took off for Hollywood and reinvented herself.\"\n\n\"Are you speculating, or do you have facts to back up this claim?\" Julius asked.\n\nNoah pointed his water at him. \"You're protecting Debbie Sanderson's new identity.\"\n\nJulius sighed. \"You MIT types are just so damn smart, aren't you?\"\n\nLoretta let all the pieces fall into place in her own mind. She set her margarita on the table and looked at Julius. \"What did she have you do, watch for Knights Bridge in the news?\"\n\n\"She does that herself,\" he said. \"She saw a gossip piece saying that Dylan was engaged to a graphic designer from Knights Bridge. She read about his hockey years, his treasure-hunter father, his best friend the billionaire founder of NAK.\"\n\n\"Did she have you investigate, or did you volunteer?\" Loretta asked.\n\n\"She's a very special lady.\"\n\nMeaning he'd volunteered, Loretta thought.\n\n\"I'm the one who told her about the fashion show,\" Julius said. \"She knew it meant her room had been discovered. With you and Dylan in the picture, I had to investigate.\"\n\n\"What's her name now?\" Noah asked quietly.\n\nHartley didn't answer.\n\nNoah leaned forward. \"It's Daphne Stewart, isn't it?\"\n\nJulius looked uncomfortable but said nothing, and Loretta decided to give up on her margarita. She frowned at the two men. \"Who's Daphne Stewart?\"\n\n\"She's a highly respected independent costume designer in Hollywood,\" Noah said, his eyes still on Julius. \"She's worked on any number of movies. Daphne Stewart is the name she gave herself after she left Knights Bridge forty years ago and headed west.\"\n\n\"She guards her privacy.\" Julius suddenly wasn't as cocky. \"You of all people can understand that.\"\n\nNoah's expression softened. \"I won't intrude on her privacy. Neither will Dylan. Is she a friend?\"\n\n\"She's become one, yes. She's a client with a law firm I do a lot of work for. She's got a keen sense of drama. She likes knowing a private investigator.\" Julius seemed slightly less shaken by Noah's knowledge. \"What about Olivia and Phoebe O'Dunn and her family?\"\n\n\"What do you think?\" Noah asked, his tone as steady and controlled as ever. \"Do you think they want to intrude on Miss Stewart's privacy?\"\n\nJulius sighed. \"I've already told her that it's my judgment that they don't and her privacy and anonymity are safe if that's what she wants. Is Phoebe using any of Daphne's dresses in the show?\"\n\n\"I spoke to her a little while ago,\" Noah said. \"The library will only use the dresses with Miss Stewart's permission, now that she knows her identity. Even if the library can claim the contents of the sewing room she created, they won't. Miss Stewart is free to reclaim anything she left behind.\"\n\nLoretta shifted her attention from Julius to Noah. \"Have you seen this secret sewing room?\" she asked.\n\nNoah nodded. \"It's just as Miss Stewart left it at twenty-one.\" He steadied his gaze again on Julius. \"Please let her know that the people of Knights Bridge would welcome her anytime she'd like to return.\"\n\nJulius raised an eyebrow. \"Phoebe told you that?\"\n\nNoah didn't hesitate. \"Yes.\"\n\nLoretta sat forward. \"Is she here in San Diego?\"\n\n\"Not now, no,\" Noah said, his tone unreadable.\n\n\"Phoebe figured out that Debbie Sanderson and Daphne Stewart are one and the same?\" Julius asked.\n\n\"With the help of her sisters.\" Noah drank more of his water before he continued. \"Miss Stewart left a number of books behind in the house she rented while she lived in Knights Bridge.\"\n\n\"Phoebe's house now,\" Julius added.\n\n\"That's right. Rebecca, The Moonspinners.\"\n\n\"Daphne du Maurier and Mary Stewart,\" Loretta said. \"Daphne Stewart.\"\n\nNoah smiled for the first time. \"Exactly. Phoebe and her sisters figured out that the Debbie Sanderson who worked at the Knights Bridge library forty years ago and sewed dresses secretly in the attic had to be Daphne Stewart of Hollywood.\"\n\n\"I'll be damned.\" Loretta grinned. \"I read Daphne du Maurier and Mary Stewart as a kid. I love their books.\"\n\n* * *\n\nNoah looked out at the San Diego skyline from his corner office suite at NAK, the company he'd founded, built and taken public. He'd had the kernel of the idea for it at twelve. It had grown from there, so that now NAK was a leader in the convergence of technology and entertainment. It was an exciting company, with people clamoring to work there.\n\n\"We did a good job,\" he said, knowing that Dylan had entered the office.\n\n\"You did.\" Dylan joined Noah at the windows. \"I just helped you so that you could put your skills to their best use.\"\n\n\"You had my back. Who has yours, Dylan?\"\n\nHe shrugged. \"You do. You tapped on my car window four years ago. Where would I be if you hadn't?\"\n\n\"You'd have figured something out. You could have gotten into adventure travel with your father, or gone on treasure hunts with him, instead of keeping the jerks at bay so that I could do my thing. Now your father's gone\u2014\"\n\nDylan interrupted him. \"Noah, my father and I had our chances to do things together. I have a chance now to fulfill some of his dreams, the dreams we shared. Think I'd have that if I hadn't thrown in with you and made all this money?\"\n\nNoah smiled. \"That is one way of putting it.\"\n\nDylan looked out at the city they both loved. \"Olivia's family and friends in Knights Bridge watch out for her. She didn't know that at first. When her friend in Boston betrayed her and she moved back home, she let her pride get in her way. Now she knows.\"\n\n\"She'll do anything for her family and friends, and they'll do anything for her.\"\n\nDylan glanced sideways at Noah. \"It's that way with the O'Dunn sisters, too.\"\n\nNoah watched a U.S. Navy ship far out on the Pacific, just a gray blip on the blue water. \"Phoebe doesn't realize that it's not just her protecting her family and friends. They're protecting her, too.\"\n\n\"Maybe too much so,\" Dylan said.\n\n\"They think what she wants is to live alone in her little house on Thistle Lane for the rest of her life.\"\n\n\"What if it is what she wants?\"\n\nNoah didn't take his eyes off the ship. \"They don't want to see her heart broken again, so they tell themselves she's given up on love.\"\n\nDylan was silent a moment. Then he blew out a breath. \"Noah, you and Phoebe\u2014\"\n\nHe turned sharply away from the view and smiled at his friend. \"Never mind. I'm talking out of my hat. I'm no good at figuring out people. I'm just glad we figured out that Julius Hartley was protecting Daphne Stewart and had no particular bone to pick with either of us.\"\n\nDylan seemed unwilling to change the subject, but finally he sighed. \"He and Loretta are going to L.A. to talk to Miss Stewart. You're going with them, aren't you?\"\n\nNoah shook his head. \"Loretta and Julius are already in L.A. You and I, Dylan my friend, are meeting them at Daphne's house in Hollywood Hills.\"\n\nDylan scratched the side of his mouth. \"We are, huh?\"\n\n\"And Olivia, too. I've already talked to her. She wants to be back home in time for the fashion show. I've arranged for a flight from L.A., after our visit.\"\n\n\"You never fail to surprise me.\"\n\n\"Likewise,\" Noah said. \"How long have you known it was Phoebe in that Edwardian dress at the charity masquerade?\"\n\n\"Olivia didn't recognize her\u2014\"\n\n\"Olivia doesn't have your objectivity or experience with people. Design and colors, yes.\" Noah followed Dylan out of the office. \"When, Dylan?\"\n\n\"When you danced with her,\" he said when they reached the hall. \"I didn't tell Olivia because she didn't ask. She didn't tell me when she found out because I didn't ask. Worked out just fine.\"\n\n\"You two have a great life together.\"\n\n\"I like how you put that. We have a great life together now. It's not just in the future.\" But Dylan's eyes were serious as he and Noah stopped at the elevators. \"Noah, I don't want you worrying about Olivia and me if you and Phoebe...if you two...\" He grimaced. \"Hell.\"\n\nNoah grinned. \"Not going to have my back with this one, are you?\"\n\n\"I don't need to. You're a smart guy. You know what you're getting into.\"\n\nAs the elevator dinged, Noah could see Phoebe's turquoise eyes, her smile, her dark strawberry hair against her creamy skin. Yes, indeed. He did know what he was getting into, but what was going on between Phoebe and him was for them to sort out on their own\u2014without Dylan McCaffrey, Olivia Frost, Phoebe's three younger sisters, her eccentric mother, her brother-in-law, the rest of the Frosts, the rest of the Sloans or the rest of Knights Bridge.\n\nThe elevator doors opened and he and Dylan got in. \"Sometimes we hold on to an image of ourselves because we're convinced it's what we're supposed to be. It's what we want, what other people want from us.\" He was hardly aware that he was speaking to Dylan. \"I was supposed to be the techie rich guy with a babe on each arm.\"\n\n\"Maybe Phoebe is just who she is, Noah.\"\n\n\"She has a great life and a great job. I don't intend to mess that up for her.\"\n\n\"From what I saw in Boston...\" Dylan paused, glanced at Noah with a smile. \"I'm not sure Phoebe O'Dunn's given up on having a man in her life as much as she wants everyone else to think.\"\n\n\"Come on. Let's go to Hollywood.\"\nTwenty\n\nPhoebe had been involved in countless library and other town events since she was a kid, but tonight she was nervous. The vintage fashion show was special. Different. Not just because it was a first for Knights Bridge but because of the hidden attic room and the woman who'd created it, and why. The lasting impact of Debbie Sanderson's brief time in Knights Bridge, on herself, on the people she'd encountered here.\n\nOn Phoebe's own family. Her mother, her father.\n\nShe slipped into the rich brown sequined Edwardian gown. She'd debated whether to wear it to host the fashion show or just to model it. Wearing it to host had won out. She'd walk to the library and meet her mother and sisters there. No word from Olivia yet, but she and Dylan had planned to be back from California in time for the show.\n\nAnd no word from Noah...\n\nPhoebe adjusted the dress. She'd skip the matching hat tonight. She wasn't trying to conceal her identity from anyone, as she had at the masquerade ball. Now that Noah was back on his home turf, did his short stay in Knights Bridge seem completely unreal to him?\n\nIt almost did to her, she realized. Tonight would help get her refocused. Then she planned to take a week off. She'd stay home, work on her garden, help Olivia and Maggie with Carriage Hill, her mother with the goats and her freezing and canning. After that, she'd be into her fall routines at the library. She couldn't wait, really.\n\nSo why do I feel out of sorts?\n\nShe made herself smile in her bedroom mirror. As host tonight, she had to be happy and cheerful.\n\nVivacious. That was the word she was looking for.\n\nShe didn't feel vivacious.\n\nShe carried her sandals by the straps in one hand and headed downstairs barefoot. No black wig, mask or heavy makeup tonight, either. She'd found instructions for an updo on the internet and managed to follow them, if imperfectly. A few wisps of hair were already out of the pins. The do just had to last a couple of hours. Maggie was catering the post-show wine and cheese party, but then she, the twins and Phoebe would meet back on Thistle Lane and celebrate with a couple of bottles of white wine already chilling in the refrigerator. Phoebe didn't know if Olivia and Dylan would join them.\n\nHow much had Noah told his best friend about his stay in Knights Bridge? How much had Dylan guessed?\n\nPhoebe shook off the question. She couldn't change what had gone on between her and Noah. She knew now that she wouldn't even if she could. Any self-consciousness, embarrassment or awkwardness she might feel had to be endured.\n\nWas worth it, she thought as she stepped out onto the porch. She didn't know what was next in their relationship but she had no regrets so far.\n\nShe set her sandals on the porch floor. The evening was warm and clear, perfect for the fashion show. She couldn't have ordered up one better. They'd have a good turnout. She looked forward to telling people how Daphne Stewart had gotten her start as a Hollywood costume designer in their town.\n\nA sleek black sedan eased to a stop on Thistle Lane. A uniformed driver stepped out, went around and opened the back door. Noah got out, and the driver returned to his position behind the wheel.\n\nAs the car turned around in Phoebe's driveway and headed back up Thistle Lane, Noah crossed the yard, his movements as smooth and purposeful as she remembered from the first time she'd seen him. He wore an obviously expensive black suit this time. No cape, no mask, no sword.\n\nPhoebe watched him, her heart hammering. Her attraction to him wasn't going away. If anything, it had deepened, becoming not just physical, not just a fleeting connection to liven up a quiet summer. She loved hearing his voice, loved talking to him, listening to him. It didn't matter that she'd known him such a short time. She'd never felt like this about anyone. Her sisters, her parents, her nephews. She loved them with all her heart, but this was different.\n\nIt's too fast. Too crazy.\n\nMaybe so, but she couldn't deny the rush of emotion as he mounted the porch steps.\n\n\"Olivia and Dylan are right behind me,\" he said. \"I wanted to see you first.\"\n\nPhoebe steadied herself. \"My sisters are at the library already. Ava and Ruby are helping Maggie set up for the party after the fashion show. I'm meeting them there.\" She realized Noah couldn't possibly care about these details but she couldn't stop herself. \"Maggie's serving a merlot from your winery. She thought that was fitting. She says it's excellent. She knows more about wine than I do.\"\n\n\"Probably more than I do, too. I bought the winery for the view.\"\n\n\"And to help a struggling friend,\" Phoebe added, then smiled. \"I read about it in an article.\"\n\nHe stepped up onto the porch and gave her one of his slight, enigmatic smiles. \"I also liked the idea of wine.\"\n\n\"That's how you operate, isn't it?\" She felt his gaze on her and remembered she was in her Edwardian dress. She ignored a tingle of awareness and continued. \"You get an idea. Then you take action.\"\n\n\"It's hard to get anything done if you just think about it.\" He touched an errant curl that had flopped from her updo onto her cheek. \"You look beautiful tonight, Phoebe. I like the dress even better with your natural hair.\"\n\n\"I'm hosting the fashion show tonight. It seemed fitting to wear this dress.\"\n\n\"It's perfect. Tonight's special in a number of ways.\"\n\n\"Yes, it is.\" She felt calmer, steadier, even as she recognized that her head was spinning at having him back in Knights Bridge. \"Thank you for being here. I haven't talked to Olivia, so I didn't know you were coming.\"\n\n\"I asked her not to say anything. Things have been happening fast.\" He glanced back at Thistle Lane, then turned to her again. \"I appreciate all you and your sisters did to help figure out what Julius Hartley wanted.\"\n\n\"You're the one who put us in touch with Daphne Stewart.\"\n\n\"Through Hartley.\"\n\nPhoebe nodded. \"It's really exciting. I knew the hidden sewing room was special but I had no idea the woman who created it ended up in Hollywood. I haven't told anyone. I've just said it's okay to use the dresses\u2014that won't give Miss Stewart away, if she wants to stay anonymous.\"\n\n\"I don't know that she does,\" Noah said quietly. \"Olivia and Dylan will be here soon. They have Loretta Wrentham and Julius Hartley with them, and a guest.\"\n\nPhoebe stood straight. \"Noah?\"\n\nHis eyes seemed so blue in the evening light. \"We met Daphne Stewart in Los Angeles. Hartley had told her about the fashion show, and how you'd discovered her old sewing room. She said she wanted to be here tonight, if it's all right with you.\"\n\n\"She's here in Knights Bridge? Noah...\"\n\n\"I'll let her explain everything.\"\n\n\"She really is George Sanderson's great-great-granddaughter, you know.\"\n\nNoah smiled, visibly more relaxed. \"George Sanderson, the intrepid founder of the Knights Bridge Free Public Library.\"\n\n\"I wonder if being here inspired her.\"\n\n\"Or maybe his ghost did.\"\n\nNoah tucked another curl back into Phoebe's pins, then stood back as Dylan's car pulled in front of her house. He and Olivia got out. She gave Phoebe a quick wave, as if to say everything would be all right. Then the sleek black sedan returned, parked behind Dylan's car.\n\nJulius Hartley emerged from the backseat while a tall, slender woman in tight black pants, a belted white top and red heels stepped out from the front passenger seat. Julius helped another woman out of the back. She was petite and elegant, with copper hair in a pixie cut. She wore a simple rust-colored dress with a teal-and-rust scarf and sparkly gold flats.\n\nThe tall woman glided up the walk to the front porch. \"I'm Loretta Wrentham,\" she said, coming up the steps. \"Phoebe, right?\"\n\nPhoebe nodded, aware of Noah standing even closer to her. \"It's nice to meet you,\" she said. \"I've heard a lot about you.\"\n\n\"I'll bet,\" Loretta said. \"Cute place. I like wicker.\"\n\n\"Me, too,\" Phoebe said with a smile.\n\nNoah frowned at Loretta. She shrugged. \"Well, I do.\"\n\nHe said nothing.\n\nThe copper-haired woman crossed the lawn with Julius Hartley at her side. Phoebe started down the porch steps. Noah took her hand and walked with her. Loretta followed them. Phoebe thought she heard the older woman sigh.\n\nThe copper-haired woman left Julius's side and went to the trellis, tears glistening in her deep green eyes as she touched a pink rose. She lowered her hand and turned to Phoebe and Noah. \"I never thought I'd ever see this place again.\"\n\nJulius cleared his throat. \"Daphne, this is Phoebe O'Dunn, director of the library here in Knights Bridge. Phoebe, this is Daphne Stewart, or, as she was known when she was living here, Debbie Sanderson.\"\n\n\"It's a pleasure to meet you, Miss Stewart,\" Phoebe said, aware of Noah, silent, still, next to her.\n\n\"Likewise, Phoebe, and you must call me Daphne\u2014although standing here, seeing this place...\" She smiled through unshed tears. \"I feel like Debbie Sanderson again.\"\n\nPhoebe returned her smile. \"I can imagine. I'm glad you're here.\"\n\nDaphne stood back, eyeing Phoebe with obvious emotion, then pointed. \"Your dress...\"\n\nPhoebe glanced down at the beautiful gown. \"I couldn't resist.\"\n\nDaphne laughed unexpectedly. \"That's good. Excellent, in fact. My heavens, that dress does look stunning on you.\"\n\n\"I agree,\" Noah said softly.\n\nPhoebe noticed Loretta all but roll her eyes, but Daphne continued, \"You're Patrick's daughter. I can see him in you.\" She breathed in deeply, looking at the rose trellis and the small house where she used to live. \"I have roses at my home in Hollywood Hills. It's a cute little bungalow not that different from this place. Oh, I had such big dreams when I was here.\"\n\n\"Looks as if some of them came true,\" Loretta said, her directness breaking through the nostalgic mood.\n\n\"A lot of them did,\" Daphne said.\n\nOlivia touched Dylan's arm. \"Let's take Loretta and Julius to the library and get settled. Noah, Phoebe and Daphne can meet us there.\"\n\nHartley looked reluctant but made no comment as he followed Olivia, Dylan and Loretta out to Thistle Lane. Daphne smiled, watching them. \"Julius is someone you want to have on your side,\" she said quietly.\n\n\"You're welcome to come inside and have a look around,\" Phoebe said.\n\n\"Thank you but you have a show to put on.\" Daphne brushed back a few tears. \"I wasn't sure I'd even get out of the car. I'm glad I did. I'd like to walk to the library, if you don't mind.\"\n\n\"That was my plan, too.\"\n\n\"I thought it might be.\" She looked out at the shaded yard, a breeze stirring in the trees. \"I assumed someone had discovered my old sewing room years ago and given everything to Goodwill. It never occurred to me that it was still intact. I feel a little like Sleeping Beauty.\"\n\n\"It's a big attic,\" Phoebe said with a smile.\n\n\"I was surprised when Julius told me about the dresses Olivia, her friend Maggie and Noah's mysterious dance partner wore.\" Daphne brushed back a few tears. \"It's funny how things work out sometimes, isn't it? I see an article about Dylan McCaffrey and Knights Bridge, and you discover my secret sewing room. Meant to be, maybe.\"\n\n\"Maybe so.\"\n\nDaphne fingered another rose, as if remembering herself at twenty-one. \"I came to Knights Bridge because of my family's roots in the Swift River Valley. I thought it would help me to be close to my great-great-grandfather's energy, his can-do spirit. And it did, just not in all the ways I imagined. Well. Shall we go? Can you walk all right in that long skirt?\"\n\n\"Not a problem. I even danced in it.\"\n\nDaphne gave her a knowing smile. \"So I hear.\"\n\nWith a sudden burst of energy, she headed down the front walk. Phoebe glanced at Noah. \"Go,\" he said. \"I'll be right behind you.\"\n\nShe crossed the lawn and met up with Daphne Stewart. Noah stayed a few feet behind them as they started down Thistle Lane toward the library.\n\nDaphne hooked Phoebe's arm into hers. \"My father was a troubled man and as a result I had a troubled childhood. I knew I had to make a clean break. I saved up and moved to Knights Bridge.\"\n\n\"And you got a job at the library your great-great-grandfather founded,\" Phoebe said.\n\n\"I always knew it wasn't permanent. I loved it, and I loved the little house I rented, that you now own.\" Daphne tightened her hold on Phoebe, as if she needed to steady herself. \"I knew that to save myself, I had to totally reinvent who I was\u2014even who I thought I was. I was invisible here, but that's what I wanted.\"\n\n\"Invisible, maybe, but not unappreciated.\"\n\nDaphne cut her a sideways look. \"Thank you for that.\"\n\n\"How did you end up creating your sewing room?\" Phoebe asked.\n\n\"I volunteered to tidy up the attic.\" Daphne's grip eased, and she sounded more at ease, more the confident, successful woman she'd become. \"I thought it'd be a good opportunity to think, perhaps to encounter a ghost or two. I cleaned out a storage room and decided to make it my own. I enjoyed sewing. I knew what I was doing with a needle and thread. My grandmother taught me the basics. Then I got books out of the library and learned more.\"\n\n\"Did you know you wanted to be a designer?\"\n\n\"I knew I didn't want to be me,\" Daphne said quietly. \"Having a secret room allowed me to take risks I otherwise might not have. Not all risks involve jumping out of airplanes and climbing tall mountains. Some involve daring to strip away the limiting beliefs about yourself and seeing what's left.\"\n\nPhoebe glanced back at Noah, knew he could hear them, but he just gave her a slight wink.\n\nDaphne lowered her arm from Phoebe's and inhaled the night air. \"I never imagined at twenty that I'd fly back here in a private jet, that much I can tell you.\" There was no hint of tears in her eyes now. \"I used to walk to the library with fabric tucked under my coat. I spent every dime I had on fabric, thread, notions. I'd haunt thrift stores for old clothes that came in that I could tear apart and use in my sewing. I followed patterns at first. Then I found myself adapting them, adding this or that. I had such a good time.\"\n\n\"You taught yourself dress design, then.\"\n\n\"I did. I loved old movies. Of course, some of them weren't so old back then. I learned and practiced by copying dresses I took a fancy to in the movies. Then I pretended I was in charge of costumes for various movies and created my own designs.\"\n\n\"The dress I'm wearing is one of your original designs, isn't it?\" Phoebe asked. \"It's lovely. It caught my eye right away.\"\n\n\"You're very kind, Phoebe. Yes, I pretended I was designing costumes for a movie about the Titanic. It's not all that different from one of the dresses Kate Winslet wore many years later.\" Daphne sighed heavily, but with none of the earlier raw emotion. \"There was something about being up in the library attic...\" She trailed off, then smiled as she glanced again at Phoebe. \"Maybe it was my great-great-grandfather's presence.\"\n\nFor all Phoebe knew, it was.\n\nDaphne took Phoebe's arm again as the library came into view. \"He's quite a man, your Noah.\"\n\n\"We're not... He isn't...\" Phoebe took a breath. \"Sorry. I'm not usually tongue-tied. I just don't want to give the wrong impression.\"\n\n\"But you aren't, are you? You don't have to answer. Your remind me so much of your father. He was new to town, too. We helped each other. Knights Bridge was exactly where he wanted to be.\"\n\n\"He was such a dreamer,\" Phoebe said with affection.\n\n\"The war took its toll on his natural spirit and optimism, but he finally decided that the best way for him to honor the friends he lost was to live each day.\"\n\n\"He did that. No question.\"\n\nDaphne squeezed Phoebe's hand, then let go as the library came into view. \"One night, he walked me home from the library. It was January, absolutely frigid.\" She gave a mock shiver. \"Believe me, I've never missed below-zero temperatures living in Southern California. That night was so clear. I've never seen stars glittering in a night sky like that. Patrick said it was a combination of the cold air and Quabbin, the absence of ambient light. Now that I do miss.\"\n\n\"I love the night sky,\" Phoebe said.\n\n\"That night...\" Daphne looked up at the early-evening sky, gray with dusk. \"It felt as if the stars were talking to me. I knew I had to leave. I got home and packed up and left Knights Bridge that night.\"\n\n\"My father\u2014\"\n\n\"He knew I wasn't meant to stay here. He was so comfortable that night, as cold as it was. It was as if he were back in his own skin again. He was like a brother to me, you know. I'd have done anything for him.\"\n\n\"Did you stay in touch?\"\n\n\"I didn't even say goodbye,\" Daphne said softly, almost to herself. \"It was so, so cold that night. So cold, Phoebe. I knew it was Patrick's destiny to stay here and be himself just as it was mine to leave and become myself. I don't know if that makes sense to you but it does to me.\"\n\n\"It makes perfect sense,\" Phoebe said, meaning it.\n\n\"My attic room tells my story in a way, doesn't it?\"\n\nPhoebe smiled. \"It certainly does. How did you get to Hollywood?\"\n\n\"I took buses across the country. I was flat broke by the time I hit Wilshire Boulevard, but I got a job waitressing, found a roommate. I started working on sets, doing whatever work someone would hire me to do, and eventually I got into costume design.\" She grinned suddenly at Phoebe. \"That's the short version.\"\n\n\"Did you ever marry, have kids?\"\n\n\"Marry, yes. More than once. Have kids, no. Patrick O'Dunn, though...\" She shook her head in obvious \namazement. \"Four daughters.\"\n\n\"With your former French student,\" Phoebe added.\n\n\"Will Elly be here tonight?\"\n\n\"She wouldn't miss it.\"\n\nThey walked under a sugar maple and across the library's side yard to the front entrance. Phoebe saw Ava and Ruby on the steps, in their flapper dresses from Daphne's hidden room.\n\n\"I'm honored, Phoebe,\" Daphne said in a hoarse whisper. \"I thought I needed to pretend my past here never existed. I didn't want any reminders, anyone else to know about Debbie Sanderson and her abusive, alcoholic childhood.\"\n\n\"No one needs to know who you are,\" Phoebe said. \"If it's what you want, you can attend the fashion show as one of Dylan's friends from California.\"\n\nDaphne paused as they came to the main entrance. \"Loretta says you should introduce me.\"\n\n\"That sounds like Loretta,\" Noah said, joining them, easing in next to Phoebe. \"We'll do whatever you want.\"\n\nDaphne didn't respond, and Noah took her by the arm and escorted her up the library steps. Phoebe saw Ava and Ruby staring at her and gave them a quick smile as she followed Noah and Daphne.\n\nDylan and Julius were waiting at the top of the steps as people gathered for the show. Olivia would be with Maggie, changing into their Audrey Hepburn and Grace Kelly dresses.\n\nDaphne stopped at the open front door. \"I don't know what I'm getting worked up about. No one will remember me or give a hoot about meeting a Hollywood costume designer.\" She peered inside at the rows of chairs, filling up with people from the town where she'd lived for such a short time. Her great-great-grandfather stared down at the audience from his portrait above the fireplace. \"I used to talk to old George's ghost.\"\n\n\"Did he talk back to you?\" Dylan asked, taking her other arm. Julius, obviously protective of Daphne, scowled, but Dylan just shrugged. \"Fair question.\"\n\nDaphne laughed, visibly more at ease. \"In his own way, he definitely talked back to me. My father wasn't like him at all. That much I know.\" She left it at that and pointed into the library at Grace Webster and Audrey Frost, sitting next to each other up front. \"They encouraged me when I started tutoring a few kids in French. I could speak the language, but I was no teacher.\"\n\nPhoebe said, \"I thought at first whoever created the room had gone to Paris.\"\n\n\"I love Paris but Southern California is home.\" More tears shone in Daphne's deep green eyes. \"And for a while, so was Knights Bridge. Now go, Phoebe. Do your thing and enjoy every minute.\"\n\nAs she excused herself and headed backstage, Noah and Dylan, with Julius right behind them, escorted Daphne into the library. Noah caught Phoebe's eye and she mouthed, \"Thank you.\" She looked around for Loretta Wrentham but didn't see her...until she arrived backstage. The California lawyer\u2014Dylan McCaffrey and Noah Kendrick's friend\u2014had on a tie-dyed shirt, a fringed vest and wide-legged turquoise pants.\n\n\"I think I wore this outfit in sixth grade,\" she said with a grin. \"It's those twin sisters of yours. They could talk a frog into camping out under a cactus.\"\n\nPhoebe burst into laughter, and then Loretta did, too. Daphne Stewart, aka Debbie Sanderson, was living the life she always wanted. Phoebe realized that her father had, too\u2014that his untimely death didn't change the fact that his life in Knights Bridge with his wife and four daughters was exactly what he'd wanted.\n\nShe was living the life she wanted, too.\n\nExcept everything had changed when Noah Kendrick swept her onto the dance floor in the Edwardian dress that Daphne Stewart had sewn in her attic room forty years ago.\n\n* * *\n\nDaphne Stewart\/Debbie Sanderson was greeted like the celebrity she was, on the stage her great-great-grandfather had insisted be included in the small-town library he founded. Noah thought Phoebe was even more beautiful in her princess dress without the mask, the heavy makeup, the black wig\u2014the pretending to be someone else. With her dark strawberry curls falling out of their pins, framing her face, she smiled and laughed among people she knew and loved.\n\nHe sat with Julius, Dylan and Daphne in the row behind Grace Webster. Grace would glance back at Dylan as if he she'd dreamed of having a grandson just like him.\n\nMaggie and Olivia wowed the crowd with their Grace Kelly and Audrey Hepburn dresses, and Ava and Ruby O'Dunn had everyone laughing when they did a short skit in their flapper dresses. Their mother modeled a simple, elegant dress that Grace Webster's mother\u2014Dylan's great-grandmother\u2014had worn long before the people of the Swift River Valley had realized their towns were doomed.\n\nLoretta obviously had a grand time showing off her hippie outfit.\n\nJulius leaned toward Noah and whispered, \"That Loretta's a stitch.\"\n\nDylan overheard him and just shook his head. Noah didn't try to hide his amusement. Brandon Sloan was across the aisle with his sons. He looked more comfortable in his own skin, if also more interested in the women in the dresses than the dresses themselves. They were his friends, and one was his wife. Even if he packed up his tent and moved to California, Brandon would still be a part of Knights Bridge.\n\nIt was a warm evening. Noah found himself alone as the after-show party spilled outside onto the lawn and across South Main onto the town common. Daphne had grabbed a glass of wine and was chatting comfortably with Elly O'Dunn.\n\nHe spotted Phoebe under a tall, graceful elm and grabbed two glasses of wine and joined her. She'd slipped off her sandals and was barefoot in the grass. She thanked him as he handed her a glass, but her mind was clearly elsewhere.\n\nShe sipped the wine, her gaze on her mother and her former French tutor. \"My mother stayed in Knights Bridge and Daphne took off for Hollywood\u2014changed her name, became someone else.\"\n\n\"Or thought she became someone else,\" Noah said.\n\n\"It's what she needed to believe at the time. She wasn't running from her life here. She was running from her past and its hold on her.\" Phoebe turned to him, her eyes almost emerald in the shadows. \"Knights Bridge was a stop on the way to becoming who she is now. I'm glad it worked out for her, and I'm glad she's here tonight. Thank you for making that happen.\"\n\n\"You did far more than I did.\"\n\nHe saw spots of color in her cheeks as she smiled. \"Ava and Ruby are beside themselves. They're trying not to get too crazy but they're so excited to have Daphne here.\"\n\n\"I think Daphne's excited, too.\"\n\n\"People seemed to have a good time tonight. I hope you and Dylan and Julius weren't too bored.\"\n\n\"Not bored at all,\" Noah said, letting his gaze settle just for a moment on the swell of her breasts in her elegant gown. The spots of color in her cheeks deepened. He smiled. \"You did well, Princess Phoebe, and you look beautiful.\"\n\nHe saw she hadn't expected his comment. \"Thank you,\" she said, then quickly drank more wine.\n\nSeveral people approached Phoebe to comment on the evening. Noah stood back, observing her as she interacted with her family and friends. Elly O'Dunn broke off from Daphne to chase her grandsons, who'd clearly had enough of vintage fashion and partying. Daphne met up with Dylan and Olivia. They got wine and approached him under the elm.\n\nHe sipped his wine, knowing that he stood apart from the people around him. Forty years ago Debbie Sanderson had come to Knights Bridge as an outsider. She'd appreciated the welcome she'd received, but ultimately she'd decided she didn't belong.\n\nPhoebe did, Noah realized. She always would.\n\nIn her mind, belonging in Knights Bridge meant living there, being the director of the library, fixing up her little house on Thistle Lane. Any change would seem to her like the drastic break that Daphne had made.\n\nAs Phoebe glanced at him with a smile, Noah also saw that she'd convinced herself that he would never want to stay in Knights Bridge and make a place for himself there.\n\nMaybe she was right.\n\nHe felt himself go very still as he watched her, and he thought...no. She wasn't right.\n\nBut he couldn't tell her.\n\nShe needed to see it for herself.\n\n* * *\n\n\"This could have all gone wrong but it didn't,\" Loretta told Julius as she helped herself to a third glass of wine. They were small glasses. Knights Bridge\u2013size glasses, she thought with a grin. She'd changed back into her regular clothes but kind of missed her hippie outfit. \"I had a great time. Were you ever a hippie?\"\n\nJulius looked at her as if she'd turned green. \"No.\"\n\n\"Always a button-down type?\"\n\n\"Always.\"\n\n\"Now you're a high-priced private investigator for a high-priced Los Angeles law firm. You went to law school yourself?\"\n\n\"UCLA. Never took the bar.\"\n\n\"You went into the military,\" Loretta said, because it was what immediately made sense to her.\n\nHe nodded. \"I lost an uncle in Vietnam. My mother's brother. Great guy. He was twenty. The baby of the family. It affects you forever, that kind of loss.\"\n\nJulius was pensive, no wine for him. They'd promised to take Daphne to Boston tonight and then fly back to L.A. in the morning. She had things to do, she'd said, but Loretta knew she'd needed an exit strategy before she could commit to returning to Knights Bridge. Now the Hollywood designer was mingling with people she'd known forty years ago and hadn't seen since.\n\nKnights Bridge was a pretty town, small and off the beaten track, but Loretta could see that Dylan was right. Time hadn't stopped here.\n\nFinally Julius said, \"Daphne's time here in Knights Bridge meant something. It wasn't just about hiding and then running away. It wasn't just about her. I hope she sees that now.\"\n\n\"You're thinking about Patrick O'Dunn,\" Loretta said.\n\n\"Yeah.\" Julius's gaze was fixed on Daphne as she approached Phoebe under a big shade tree. His expression tightened. \"She wanted me to do it. I told her I couldn't.\"\n\n\"Do what?\"\n\n\"Watch. It's not easy to tell Daphne no, but this was for her to do.\"\n\nHe was silent as they watched Daphne hand Phoebe an envelope.\n\n\"What is it?\" Loretta asked.\n\n\"It's the letter she wrote to Patrick O'Dunn when she got to California. She never mailed it.\"\n\nPhoebe opened the envelope and unfolded the note inside. Loretta bit back her impatience. \"What's it say? You read it, right?\"\n\nJulius scowled. \"You're such a know-it-all, Loretta. Yeah, I read it.\"\n\n\"Well?\"\n\nHe hesitated, obviously debating how much to say. \"Daphne tells him she made it to Hollywood and while she doesn't know what the future will bring, she knows she made the right decision. 'Thanks to you, Patrick, I've finally found my home.'\" Julius cleared his throat. \"She tells him she'd never have made it to California without him. He helped her find the courage to take the plunge, go after her dream, just because he got up every day and lived his life, did the best he could.\"\n\nLoretta choked back tears. \"Damn. I think I'm going to cry.\"\n\n\"Don't do that,\" Julius said with a sudden grin. \"The world as we know it really will come to an end.\"\n\nDylan and Olivia chatted with Noah, whose gaze was on Phoebe O'Dunn and only Phoebe O'Dunn. Loretta sighed. \"It already has, but it's okay.\"\n\nJulius slung an arm over her shoulders. \"It's more than okay. It's damn good.\"\nTwenty-One\n\nMaggie slammed the door behind her as she entered Olivia's kitchen. After last night's perfect weather for the fashion show, the temperature had spiked today. It was midafternoon now and over ninety. The humidity level and dew point made it feel even hotter. Because The Farm at Carriage Hill was a getaway, Olivia was installing air-conditioning, but it wasn't up and running just yet.\n\nOlivia glanced at her from the sink. \"You're all red, Maggie. What have you been doing in this heat?\"\n\n\"I'm broiling hot and broiling mad.\" She raked a hand through her hair, determined not to take her mood out on her friend. \"I need to cool off.\"\n\n\"Why don't you take the boys up to the mill and jump in the pond? The water's always cold there.\"\n\nMaggie grunted. \"I could just dump a tray of ice cubes on Brandon's head. That'd cool me off.\"\n\nOlivia stepped back from the sink, drying her hands with a white dish towel. \"I had a feeling your red face had something to do with him. Do you want to talk about it?\"\n\n\"No.\"\n\nOlivia raised her eyebrows.\n\n\"I don't,\" Maggie said, adamant. \"You deserve a quiet day. Did you see the orange mint Phoebe, Noah and I harvested?\"\n\n\"It's drying nicely,\" Olivia said, setting the towel on the island. \"I also saw the pesto in the freezer.\"\n\n\"And the dog hair on your couch?\" Maggie tried to smile. \"Noah's a pushover when it comes to Buster. You've got him back on the straight and narrow?\"\n\n\"Noah or Buster?\"\n\nMaggie saw the smile in Olivia's eyes but couldn't relax, couldn't let go of how mad she was\u2014how upset. She sighed. \"Yes. Yes, I want to talk. Or tell you, anyway. Maybe not talk, because there's nothing to talk about. It is what it is.\"\n\n\"Do you want to go outside and sit?\"\n\nMaggie shook her head and started to pace in the kitchen. It was such an ideal kitchen, in an ell by itself, with great light and lots of cabinets and counters and a warm, country feel. She loved working there.\n\n\"Maggie...\"\n\n\"I'm sorry. I was thinking about how much I love it here. I love what you're doing with Carriage Hill.\"\n\n\"What we're doing,\" Olivia said.\n\n\"We do make a good team. This place has such possibilities. I've started to dream again, Olivia. I've started to have hope in the future. I don't dwell on the past so much.\" She stopped abruptly, stared out the window over the sink and noticed the haze, felt the oppressive heat. \"Brandon's been working nonstop for the past six months. He didn't tell me.\"\n\n\"Working for his family?\" Olivia asked.\n\n\"Mostly. Exclusively now.\"\n\n\"Is that good or bad?\"\n\nMaggie leaned back against the island. \"Good, I guess. Good that he's working. He's cut expenses to the bone. He's not living in that damn tent because he's broke. He's living there because he's saving every dime he makes.\"\n\n\"For what?\"\n\n\"To prove himself to me.\"\n\n\"He told you this?\"\n\nMaggie snorted. \"Are you kidding? He never tells me anything. I had to ferret it out on my own.\"\n\n\"Meaning you got one of his brothers to tell you,\" Olivia said.\n\n\"Christopher.\" He was the youngest of the six Sloan siblings, one of two full-time firefighters in town. \"It didn't take much doing. He says I need to prove myself to Brandon, too. Me. Like I did something wrong.\"\n\n\"What are you supposed to prove?\"\n\n\"That I still want to be with him, regardless of money.\"\n\nOlivia picked up her towel again and polished the butcher-block counter. \"Christopher said that, huh?\"\n\n\"Not in those exact words but I got the message.\" Maggie suddenly wished she had something to cook. She held up a hand before her friend could interrupt. \"I'm not asking you to take a side. I know you have to live in this town and it's crawling with Sloans. Whatever made me think I could come back here?\"\n\n\"There are a few O'Dunns here, too,\" Olivia said, her tone neutral.\n\n\"For how long? Ava and Ruby are going back to school. They'll end up in Hollywood or New York. And Phoebe. I don't know where Phoebe will end up. I used to think she'd always be here in Knights Bridge.\" Maggie hesitated, glanced toward the mudroom and back door. \"Where are Noah and Dylan?\"\n\n\"They're up the road talking demolition.\"\n\nWith Brandon, no doubt. Maggie gritted her teeth. \"I don't know. Maybe I'm crazy and Phoebe's not going anywhere. She and Noah seemed so good together last night at the fashion show, but she was in that Edwardian dress again. Maybe that makes all the difference.\"\n\n\"Or getting her out of her Edwardian dress,\" Olivia said half under her breath.\n\n\"Olivia!\"\n\n\"Well. Come on. You saw the sparks between those two.\"\n\n\"Yes, but\u2014\" Maggie stopped herself. \"I can't talk about my sister that way. And this is Phoebe. It's not Ava or Ruby. You know what I mean?\"\n\nOlivia nodded. \"I do, and maybe that's part of the problem.\"\n\nMaggie suddenly wished she'd brought something to do. Her mother had invited the boys up to the house to make pickles and would drop them off at the library later. Maggie would pick them up there. She hadn't planned on a project with Olivia, figured it was too soon after her return from San Diego. Too complicated with Noah Kendrick as a houseguest.\n\nAnd she was too mad at Brandon.\n\nShe made herself smile as she changed the subject. \"Daphne Stewart is something, isn't she?\"\n\n\"She wants to come back to Knights Bridge for a proper stay,\" Olivia said. \"She only stayed last night because it was all so last-minute and she had to get back to L.A. I also think she wasn't sure she could handle being back here.\"\n\n\"It was emotional for her. You could tell. She lived in town for such a short time but it had an impact, on her and on the people she met. She needed a brother, a man she could trust\u2014who wouldn't beat her up.\"\n\n\"Your father,\" Olivia said.\n\nMaggie nodded, emotional herself. \"Daphne helped him but he helped her. He was such a good guy, Liv.\"\n\n\"I remember.\"\n\n\"We all lost out when he died, but Phoebe most of all, although I don't think she and that rat bastard from Orlando were meant for each other. But she's never really had anyone since then.\"\n\n\"Maybe that's because she was waiting for a swashbuckler to sweep her off her feet,\" Olivia said with a smile.\n\nMaggie felt her own mood lighten. Sweat trickled down the nape of her neck. \"Phoebe's used to having people need her. Noah doesn't need her. What does she have to offer a billionaire? She's not a starlet. She's...Phoebe.\"\n\n\"Maybe Noah's wondering what he has to offer her.\"\n\n\"A winery, for starters,\" Maggie said, partly serious, partly facetious.\n\nOlivia went to the refrigerator and helped herself to a handful of crushed ice. She offered some to Maggie, rubbed her arms with hers. \"Maybe Phoebe doesn't want a winery. Maybe she just wants what she has.\"\n\n\"That would be my sister,\" Maggie said with a sigh. \"She's stubborn, you know. She'll tell herself she doesn't want anything but what she already has.\"\n\n\"Noah took her by surprise, didn't he?\"\n\n\"They took each other by surprise. It's not like Brandon and me. No surprises. I've known him my whole life. I know how he thinks. Wouldn't you think he'd know how I think?\"\n\nOlivia watched the ice chips melt on her arm and didn't answer.\n\n\"He can be such an idiot,\" Maggie said, back on that train of thought. \"What went on between us was never about money.\"\n\n\"If Brandon wasn't so thickheaded sometimes, Maggie, you wouldn't love him so much.\" Olivia took her towel, blotted her arm dry. \"Maybe you two got into a pattern of thinking that you knew what was going on with the other person\u2014like two halves of one whole instead of two individuals.\"\n\nMaggie wasn't willing to go that far. \"I talk.\"\n\n\"Maybe you both should talk.\" Then Olivia added, \"To each other.\"\n\n\"I'm too hot to talk, and when did you get so wise, my friend?\" Maggie smiled. \"Why don't we let the guys talk demolition while we go swimming in the millpond?\"\n\nOlivia grinned. \"I'll grab my suit.\"\n\n\"I've got one in the van.\"\n\nThey drove out to the nineteenth-century sawmill the Frosts owned on a tributary to the Swift River. The small millpond and wood-sided mill, now converted into an apartment, were still intact. Maggie shivered just looking at the clear, clean, copper-tinted water. Tyler and Aidan hated swimming here. They preferred the warmer water of their friends' pools. Maggie thought they might change their minds when they were older, although she didn't want them to get the cuts and bruises that she, Phoebe and Olivia had swimming out here growing up. Like their nephews, Ava and Ruby had never been big on plunging themselves into ice-cold water.\n\nMaggie sat on a sun-warmed boulder and dipped her feet into the water, then pulled them right out. \"It didn't seem this cold when we were kids.\"\n\nOlivia laughed. \"It did, too. We were just oblivious.\"\n\nThey eased into the water, each finding a rock to stand on as they got used to the chilly temperature. Maggie shut her eyes, appreciating the contrast between the waist-down cold and the waist-up heat. She remembered sneaking out here on a moonlit night with Brandon, back in the days when it felt like anything was possible.\n\nShe opened her eyes and realized Olivia was looking at her with concern. \"I'm okay.\" She smiled before she could burst into tears. \"It's a perfect day to be out here, isn't it?\"\n\n\"Perfect.\"\n\nThe pond was only five feet at its deepest. As she lowered herself into the water, getting wet up to her neck, Maggie listened to the flow of the brook over the stone dam. Olivia splashed her, and Maggie splashed her back. They shrieked with laughter as if they were twelve again.\n\nMaggie ducked her head underwater but popped up almost immediately. \"Whoa, that's refreshing,\" she said. \"I think I have goose bumps.\"\n\nOlivia eased back up onto their sunny boulder. \"Don't get hypothermia,\" she said.\n\nMaggie splashed her. \"Just like you to bring up hypothermia.\"\n\nOlivia pointed at her. \"Purple lips, shivering, goose bumps. You tell me.\"\n\n\"All right, all right.\" Maggie climbed up onto another boulder and stretched out her legs. She had purple-blue knees, too. She reached for a towel on the stone wall\u2014of course Olivia had remembered towels\u2014and draped one over her legs. \"It's still hot as blazes.\"\n\n\"It won't be for long.\" Olivia nodded up at dark clouds looming above the trees to the west. \"Looks like a storm's headed our way.\"\n\n\"That does look nasty,\" Maggie said. \"I guess we're done playing hooky for the afternoon.\"\n\nShe dried off as best she could and slipped back into her shorts and T-shirt. Olivia did the same. Thunder rumbled in the distance, the kind of low, deep, rolling thunder that suggested a strong storm was bearing down on them.\n\nThey headed to the parking lot by the much newer Frost Millworks building. Dylan was there, getting out of his car. \"We're not going anywhere right now. Knights Bridge is under a severe thunderstorm warning.\"\n\nMaggie shook her head. \"I have to pick up the boys at the library. I've driven in loads of storms\u2014\"\n\n\"Not like this one. I've seen the radar.\"\n\n\"We can duck into the mill,\" Olivia said. \"Where's Noah?\"\n\n\"And Brandon,\" Maggie added. \"He's not in that stupid tent, is he?\"\n\n\"They're at Carriage Hill,\" Dylan said.\n\nThey hurried up to the mill. No one else was there late on a Sunday afternoon. Maggie used the phone in the small front office and tried calling the library but no one picked up. Her mother would have dropped off the boys by now but Maggie called her just to be sure.\n\n\"I dropped them off twenty minutes ago,\" her mother said. \"Phoebe's there alone. We get thunderstorm warnings all the time, Maggie. It'll be okay.\"\n\nBut Maggie heard the note of worry in her mother's voice. They promised to keep each other updated.\n\n\"Stay at the mill,\" her mother said. \"Promise me, Maggie.\"\n\nThe phone went dead.\n\nMaggie cradled the receiver and went into the outer room. She could see wind whipping through the trees on the other side of the pond. Small limbs fell into the water. The ground was quickly littered with leaves and twigs.\n\nThen came the hail.\n\nOlivia and Dylan held hands. Maggie wrapped her arms around her middle and watched the pebble-size hail hit the walk and the rock walls. It pelted into the brook and collected on the grass.\n\nShe jumped at a simultaneous flash of lightning and crack of thunder.\n\n\"It's just the edge of the storm,\" Dylan said.\n\nMaggie insisted he hand her his iPhone. The local weather radar was still up on the screen. Reds, yellows, purples. It was a dangerous, severe thunderstorm, and if it stayed on course, Knights Bridge center was taking a direct hit.\n\nMaggie's stomach lurched. She bolted for the door but Dylan grabbed her. \"I have to get to the library,\" she said. \"The boys\u2014Phoebe.\"\n\n\"Phoebe knows what to do in a storm,\" Olivia said, white-faced.\n\n\"If she knows it's this bad...\"\n\n\"We wait this out,\" Dylan said. \"Then we go.\"\n\n* * *\n\n\"Aunt Phoebe! Aunt Phoebe!\"\n\n\"I'm here,\" she said, sitting up, wincing in pain. It was Aidan screaming her name. She tried to keep from moaning and further scaring her nephews. \"It's okay...\"\n\n\"Listen,\" Tyler told his younger brother, his tone reassuring. \"You hear the sirens? Uncle Chris will get us out.\"\n\nThey were alone in the library attic. They'd lost power but that was the least of their troubles. Despite the heat, Tyler and Aidan had wanted to see the attic and Phoebe's secret room. They didn't care about sewing, but someone had mentioned there were ghosts in the attic. Phoebe told them about the antique marbles she'd found, and they'd charged up the stairs ahead of her.\n\nWhat harm was there in a spooky little adventure on a hot summer afternoon?\n\nShe'd been right behind them on the back stairs when she noticed the threatening sky, the greenish light in the window on the second-floor landing. She'd grabbed the boys and started downstairs to get to an interior room, but the storm hit suddenly. A fierce wind gust uprooted a sugar maple and sent it into the library. Branches broke the window on the landing below them, just missing them and blocking their route back downstairs.\n\nShe all but threw the boys up to the attic. They took cover in Daphne Stewart's windowless sewing room. Hail pounded on the roof. Wind howled and whistled. The tiny room seemed to rattle with the booming thunder.\n\nPhoebe had held on to her nephews, shielding them in case part of the roof blew off.\n\nThe storm finally passed, and now it was quiet except for the sirens.\n\n\"I need to let someone know where we are,\" she said, keeping her voice calm. \"That we're safe.\"\n\n\"We're not safe,\" Tyler said, the pragmatic Sloan.\n\nAidan sucked in a breath and pointed at her. \"Aunt Phoebe...you're bleeding.\"\n\nShe saw that she was, in fact, bleeding from a cut on her left arm. She didn't remember being hurt, hadn't felt any pain until now. She took in a shallow breath. \"It's not bad. Are you boys okay? Let me look at you.\"\n\n\"We're fine.\" Tyler stood up. \"I'm going to yell out a window.\"\n\n\"You won't be able to open any of the windows up here,\" Phoebe said.\n\n\"We can throw a brick and break the glass,\" Aidan said.\n\nTyler rolled his eyes. \"Where are we going to get a brick?\"\n\n\"Then use something else,\" his brother said, impatient, scared.\n\nPhoebe struggled to her feet. \"I'll do it,\" she said. \"You two stay right there where I can see you and don't move. Understood? Don't move.\"\n\nAs she crept to the corner door, she heard a creaking sound in the tiny room. A ghost after all, maybe. She opened the door, felt blood drip into her eyes. A cut on her scalp, too? At least the blood hadn't reached her face and the boys hadn't seen it.\n\nShe saw Christopher Sloan down on South Main, yelling past two uprooted trees to someone out of view. Olivia was there with her father, a volunteer firefighter, and Dylan. No Noah. Then she saw Maggie, looking stricken as she approached her brother-in-law, picking her way through fallen limbs and scattered leaves.\n\nPhoebe tried to open the window, but she couldn't get it to budge. What was wrong with her? Her head was spinning, aching. Her arm stung from the cut.\n\nShe glanced back at her nephews with a quick smile. \"Everything's fine. We just have to give your Uncle Chris time to get up here.\"\n\n\"Because of the broken glass,\" Aidan said.\n\n\"And the tree in the way,\" Tyler added sarcastically. Phoebe saw that his toughness was a pretense, his own eyes wide with fear.\n\n\"Aidan! Tyler!\"\n\nBrandon. Of course. He was close, probably by the freestanding closets.\n\n\"We're in here,\" Phoebe called. \"We're okay.\"\n\n\"Aunt Phoebe's not okay,\" Tyler yelled.\n\nBrandon burst into the sewing room. \"Aunt Phoebe saved us,\" Aidan said, sobbing as his father scooped him up.\n\nHer brother-in-law looked straight at her. \"Sit, Phoebe.\"\n\n\"I'm fine\u2014\"\n\n\"You're not fine. Sit.\"\n\nThen Noah swept in behind him. \"Phoebe.\" He seemed hardly able to speak. \"I know a bit about cuts.\"\n\n\"From your fencing,\" Phoebe said, then clutched his arm, steadying herself. \"Oh, hell, Noah. Damn. I think I'm going to faint.\"\n\n\"Then you're right where you need to be.\"\n\nAnd she knew she was, even as she passed out in his arms.\nTwenty-Two\n\nNoah stood on Thistle Lane thirty yards from an ambulance as Phoebe reassured the crew that she was just fine. He'd hated to leave her but all eyes were on her. She'd regained consciousness almost immediately after she'd passed out, probably as much from heat and dehydration as anything else. She'd refused to wait for a stretcher. After firefighters had cleared the tree out of the way, she'd walked down from the attic on her own, Noah at her side.\n\n\"Phoebe's right,\" Dylan said, approaching Noah. \"She will be fine. Her cuts are superficial. She doesn't even need stitches.\"\n\n\"You should know. You got cut in hockey all the time.\"\n\n\"Regularly. Not all the time.\"\n\nThey both grinned, but Noah could still feel the after-effects of the adrenaline rush. He and Brandon Sloan had arrived in the village center minutes after high winds had blown down trees and wires, ripped off parts of roofs. Brandon managed to park his truck on South Main, and he and Noah jumped into action, charging into the damaged library.\n\nFor a terrifying minute, they'd thought Phoebe and the Sloan boys were under the debris on the stairs.\n\nNoah shook off the memory. Olivia was with Phoebe. Police and firefighters had cordoned off the library's side yard where two trees had come down in what they believed was a microburst.\n\nMaggie paced on the narrow lane as Tyler and Aidan told their story to their firefighter uncle.\n\n\"She's blazing,\" Dylan said.\n\nShe certainly was, Noah thought. \"She's had a fright.\"\n\n\"She needs to vent,\" Brandon said as he joined them. \"She can cater a dinner for seventy-five people without breaking a sweat, but this is different. It's her kids. The boys are good, though. All's well that ends well.\"\n\nNoah had entered the library with Brandon and knew how terrified he'd been for his sons, and for his sister-in-law.\n\nMaggie stalked over to him, hands on her hips, some color returning to her cheeks. \"Why didn't you wait for the firefighters?\" she asked as if in midthought.\n\n\"Noah and I were right there, Maggie,\" Brandon said. \"What were we supposed to do, twiddle our thumbs?\"\n\nShe ignored him and glared at Noah. \"You, too. You both had to go tearing into the library on your own.\" She didn't give him a chance to respond and spun back to her husband. \"What if that tree had dislodged and fallen on you? Phoebe and the boys were safe.\"\n\n\"We wouldn't have gone up if it wasn't safe,\" Brandon said.\n\n\"You would have.\"\n\n\"I've worked construction since I could pick up a hammer. I knew it was safe. I didn't know Phoebe and the boys were okay.\" His tone was patient, unwavering. \"We had a good angle. We got through. Coming back down with Phoebe and the boys was risky, so we waited for the firefighters.\"\n\n\"Phoebe had everything under control.\" Maggie blinked back tears. \"But if she and Tyler and Aidan had gone up those stairs thirty seconds later, and that tree, the window...\"\n\n\"They'd have been in a world of hurt,\" Brandon said bluntly. \"Lucky that didn't happen. Don't think about what could have happened, Maggie. Think about what did happen.\"\n\nShe nodded, calmer. Her sons edged toward their parents. Brandon slung an arm over Aidan's small shoulders. \"Aunt Phoebe protected us,\" the boy said.\n\nTyler nodded. \"It was a scary storm.\"\n\n\"We don't get many storms like that,\" Maggie said, reassuring them despite her own lingering fear.\n\nTyler kicked a small stone in the lane, then looked up at the adults, his eyes still wide. \"Aunt Phoebe said we just had to wait for someone to come get us. We were trapped, weren't we?\"\n\nBrandon pointed at the fallen tree. \"The storm took down that tree. It was blocking the stairs. It wasn't even close to you guys in the attic. You can see that from here, can't you, Tyler?\"\n\n\"Yeah,\" the boy said.\n\nNext to him, his younger brother took his father's hand. \"I wasn't scared, Dad.\"\n\n\"It's okay to be scared. We all get scared.\" Brandon looked straight at his wife. \"It's what we do when we're scared that matters.\"\n\n\"Aunt Phoebe is brave,\" Aidan said.\n\nBrandon nodded. \"She did the right thing today.\"\n\nMaggie looked at Noah. \"That's Phoebe. It's who she is. She always tries to do the right thing, for all of us.\"\n\nIt would be like that, Noah knew. The O'Dunn sisters would stick together. An attack on one\u2014even a perceived attack\u2014was an attack on all four. Most of the time, their solidarity was probably a positive for all of them. He glanced at the ambulance. Phoebe was back on her feet, her arm bandaged as she stood next to Olivia. She looked steady, her hair shining in what was now a clear, cool afternoon. He wondered how much freedom she had to do what she wanted and not just what her family wanted, or what she thought they wanted.\n\nElly O'Dunn arrived with Ava and Ruby, and Brandon and Maggie and their sons joined them at the ambulance. Phoebe smiled at her family. Noah knew that it'd be her instinct to reassure them.\n\nDylan sighed. \"Damn. That was too close for comfort.\"\n\n\"As Brandon says, all's well that ends well,\" Noah said. \"Phoebe and the Sloan boys are in good shape. The damage to the library is repairable.\"\n\n\"What about you, Noah?\" his friend asked him. \"You got to be a swashbuckler without having to stab anyone. Feeling pretty good?\"\n\n\"You were annoying in kindergarten, too, Dylan,\" Noah said with a grin. \"I don't know how we've stayed friends.\"\n\n\"We didn't do the same things. You're a swordfighter and I'm a hockey player. You're good at math and I'm not.\" Dylan paused, his gaze on Phoebe and Olivia. \"You and Phoebe do different things, too.\"\n\n\"No kidding,\" Noah muttered.\n\n\"It's good. You get bored easily. In fact, you're a pain in the ass when you're bored.\" Dylan looked more relaxed than when he'd arrived at the library. \"Brandon Sloan's a natural to work in adventure travel. He could do it on the side and still keep up with his construction work.\"\n\nNoah nodded. \"You could use the help.\"\n\n\"I thought this adventure travel initiative might burn out fast, but it's not going to. It's already taking off and we're not even fully set up yet.\"\n\nNoah wasn't surprised. With Duncan McCaffrey's treasure hunting background and Dylan's interests and contacts, adventure travel was a perfect next step for him. But it wouldn't be the only one. After four years of hyperfocusing on NAK, he was as ready to explore multiple interests as Noah was.\n\n\"Maggie and Olivia are on their way to becoming full business partners,\" Dylan added. \"Maggie's taking over more of the day-to-day running of Carriage Hill. She loves it and she's good at it. Olivia loves the design work, planning, coming up with new ideas.\"\n\n\"Finding old pillowcases,\" Noah added with a smile.\n\n\"And soap making. They're having a great time.\"\n\nNoah could get used to mosquitoes, and he could learn to love small-town life and hiking in the woods, but he wasn't going to get into making soap. He suspected Dylan was of a like mind.\n\n\"NAK's a success,\" Noah said. \"It's in good hands, but we're not done.\"\n\nAcross South Main, people were gathering to check out the damage from the storm. The Civil War monument was intact, the stone Union soldier standing tall amid several downed trees. Several of the buildings on South Main had sustained damage. Across the common, the Knights Bridge Country Store was untouched.\n\nExcept for Phoebe's cuts, there'd been no injuries. Noah watched her kneel down at eye level with her young nephews as she spoke to them. After their scare, they'd need reassurance.\n\n\"We're friends, Noah,\" Dylan said next to him. \"That's not changing, whatever comes next.\"\n\nHe meant Phoebe, Noah knew.\n\nShe stood, Tyler and Aidan laughing now. Everyone looked more at ease. Noah felt his throat tighten. \"I'm as sure about Phoebe as I've ever been about anything.\"\n\n\"I know,\" Dylan said.\n\n\"That obvious?\"\n\n\"It is to me.\"\n\n\"Phoebe and I need time together, but it won't change how sure I am.\" Noah went very still, a cool, gentle breeze blowing now. \"I know it won't.\"\n\nDylan didn't argue with him.\n\nOlivia broke off from the Sloans and O'Dunns and Dylan met her on the corner of South Main and Thistle Lane. He put his arm around her as if he'd known her forever. Noah glanced at the library. Brandon had said he thought it would reopen quickly, within a few days, although the damage would take longer to repair. Would Phoebe lead that effort as library director?\n\nShe hugged her mother, then left her family and walked toward him. He wanted to go to her but waited for her to reach him.\n\n\"Thank you, Noah,\" she said, stopping next to a small downed tree limb. \"I've never passed out before.\"\n\n\"You're sure you don't need to get checked out at the hospital?\"\n\n\"I'm sure. I've been cleared to go home. My mother wants us all to come out to her place for hamburgers and hot dogs.\" Phoebe smiled, added, \"And tomatoes, of course.\"\n\n\"Fresh from the garden,\" Noah said.\n\n\"Absolutely.\"\n\n\"Phoebe...\" Then as he let his fingertips graze hers, out of view of her family and friends, he heard himself tell her that he was going back to San Diego. \"I'm not a guy from Knights Bridge. I never will be. That's a statement of fact, Phoebe. It's not an apology.\"\n\nHer turquoise eyes widened but she didn't speak.\n\n\"When we saw each other that night in Boston, it was like having a flash grenade go off in our lives. It changed everything. I had it happen when NAK took off and turned me into a very wealthy man.\" He resisted an urge to touch her hair, her lips. \"You need time, Phoebe. I'm giving it to you.\"\n\nIf she said anything as he left, he didn't hear it. He waved a hand at Dylan, who immediately joined him on South Main. \"Noah, what's going on?\"\n\n\"I need a ride to the airport.\"\n\n\"You're going back to San Diego?\"\n\n\"Tonight.\"\n\n\"For how long?\"\n\nNoah glanced back at Phoebe, walking alone down Thistle Lane. \"For as long as it takes.\"\n\n* * *\n\nWord got out that Noah Kendrick had left Knights Bridge for the second time in less than a week. Phoebe was pouring wine when Maggie, Ava, Ruby and Olivia descended on Thistle Lane to take her to her mother's house. Brandon was already there with Tyler and Aidan.\n\nMaggie, looking emotionally ragged, took the wine bottle out of Phoebe's hands. \"Are you out of your ever-loving mind, Phoebe? It's bad enough I've screwed up everything with Brandon, but you can't screw up things with Noah because you're afraid of not being there for us, for Mom, for Knights Bridge. You just can't. I won't let you.\"\n\nAva glanced at her watch. \"Noah doesn't have that much of a head start. You have time to get to the airport before his plane takes off.\"\n\n\"You're assuming I was invited,\" Phoebe said.\n\nHer sisters groaned in unison. \"Who cares? Go!\"\n\n\"I'll drive you,\" Olivia said. \"If we're wrong and he doesn't want you on his plane, I'll drive you back here\u2014\"\n\n\"And we'll drink a case of his wine,\" Maggie said.\n\n\"It's a risk but not a crazy risk,\" Ruby said. \"He wants you to take it. He wants to prove he's not too much for you. Too rich, too California, too smart, too\u2014you know. Too everything.\"\n\n\"And he needs to know you're falling in love with him,\" Maggie added quietly. \"He needs to know he has something to offer you.\"\n\nPhoebe took a breath. \"Something to offer me?\"\n\n\"You have everything you need right here in Knights Bridge,\" Olivia said. \"Your family, your job, your friends.\"\n\nBut not Noah, Phoebe thought. She didn't tell her sisters and friend that even before the storm\u2014fainting into Noah's arms\u2014she'd drafted her letter of resignation from the library, just to get a feel for what it might be like to try something new. She could volunteer. She could get into adventure travel, work on Carriage Hill soaps, learn about venture capital and serial entrepreneurs. She could read books and chase toddlers.\n\nThere was so much she could do.\n\nShe saw possibilities where before she had only seen the path she was on.\n\nBefore Noah.\n\n\"It'll take me two seconds to pack,\" she said, already heading out of the kitchen.\n\nShe didn't know what she threw in her suitcase. If she forgot anything, there were stores in San Diego. And maybe I won't need clothes, she thought with a jolt. She blamed her scare with the storm, her mad dash up to the attic with her nephews. She still ached from her cuts, but at least she hadn't required stitches and didn't have a concussion.\n\nIn other words, she could fly. She could see the sights in San Diego.\n\nMake love to Noah.\n\nShe let out a breath. Don't get ahead of yourself.\n\nOn the drive to the small airport barely twenty miles from Knights Bridge, Olivia gripped the steering wheel, her eyes on the road as she spoke. \"If things don't work out with you and Noah, you and I will still be friends. You know that, right, Phoebe? It won't change anything between us.\"\n\n\"But Dylan\u2014\"\n\n\"It's the same for him and Noah. They've been friends for almost as long as we have. We're all grownups now, Phoebe. You know? We'll figure it out. You and Noah need the space to be whatever you're meant to be to each other.\"\n\n\"I appreciate that, Olivia,\" Phoebe said. \"You and Noah seem to get along well.\"\n\n\"Noah is\u2014he's just Noah. Not everyone gets him.\"\n\n\"Women?\"\n\n\"I don't know much about his past relationships.\" Olivia smiled. \"Except that they're past.\"\n\n\"I've never been to San Diego,\" Phoebe said half under her breath.\n\n\"You won't be intimidated by Noah's life there. You're not the type. Just because you're quiet and kind doesn't mean you're a pushover. You'd never have managed Knights Bridge Free Public Library if you were a pushover.\"\n\nPhoebe laughed. \"That's for sure.\"\n\nOlivia slowed for a curve. \"That's what Noah sees, you know. He believes in you.\"\n\nThey arrived at the airport.\n\nNoah's plane hadn't taken off yet.\n\nPhoebe had a strong suspicion that Dylan had given his friend advance warning. The pilot greeted her by name and escorted her to the private jet himself.\n\nNoah was there, fresh out of the shower, in a clean black shirt and dark jeans. He looked every inch the billionaire he was. He had champagne waiting, and when she sat next to him on a leather seat, Phoebe knew there was nowhere else on the planet she'd rather be.\n\n* * *\n\nBrandon slept on the couch at Maggie's \"gingerbread\" house off Knights Bridge common. It hadn't sustained any damage, but he said he wanted to be close to the boys their first night after their scare. Maggie didn't mind. It made sense, she told herself. Tyler and Aidan needed both parents.\n\nBrandon had an early start at work. He hadn't even stayed for coffee.\n\nIt was almost as if he hadn't been there.\n\nThe second night, however, was a different story.\n\nHis folks took the boys for the night. Christopher had promised he'd show his nephews some basic search-and-rescue techniques. Tyler and Aidan were so excited, Maggie couldn't say no, although she was reluctant to be apart from them. Never in her life had she had such a scare as when she'd arrived at the library after the storm.\n\nShe still wasn't over it, she thought as she crossed the yard to Grace Webster's old house\u2014the one Dylan's father had bought and then left to him, a simple act that, ultimately, had changed all their lives.\n\nThe heat of two days ago eased with the storm; it was downright chilly. Maggie had promised her in-laws that she'd drop off a jug of corn chowder for Brandon. She and her mother and younger sisters had made up tons with fresh corn from the garden.\n\nHe stood by an open fire in front of his tent. \"It's still warm,\" she said, handing the chowder to him.\n\n\"Thanks, Maggie.\"\n\nShe heard something in his voice but couldn't figure out what it was. He had a blanket spread out on the grass between his tent and the crackling fire. She appreciated the warmth of the flames.\n\n\"Sit with me a minute?\" he asked her.\n\nShe shrugged. \"Sure, why not?\"\n\nShe sat cross-legged on the blanket. He sat next to her, his legs stretched out, the light from the flames flickering on his face. He smiled at her. \"Nice night. I don't miss the heat.\"\n\n\"Me, either, although I'll probably regret saying that in January.\" She fidgeted, uncrossed her legs. \"I shouldn't stay long. I have things to do at home.\"\n\n\"At least stay until the stars are out.\"\n\nShe went still, narrowed her eyes on him. That was it, she thought. That knowing tone. That Sloan smugness. \"I've been set up, haven't I?\"\n\nHe leaned toward her. \"You didn't stand a chance. All of us Sloans united to get you out here tonight.\"\n\n\"My sisters weren't involved, were they?\"\n\n\"Do you see them here with pitchforks?\" Brandon asked wryly.\n\n\"It's not that they're against you. They're just with me.\"\n\nHis gaze softened. \"So am I, Maggie.\"\n\nShe looked away from him and saw a star twinkling brightly in the darkening night sky. \"Brandon...\" She didn't go on. What else was there to say?\n\n\"If you want to leave, Maggie\u2014\"\n\n\"I don't.\"\n\nThe words were out before she'd realized she'd said them and that it really was what she wanted. Brandon edged closer to her, and she sank against him, felt his arm settle around her. It was so quiet, just an owl hooting across the field toward Carriage Hill and Quabbin.\n\n\"Ah, Maggie,\" Brandon said. \"Maggie, Maggie.\"\n\n\"We're not kids anymore, are we?\"\n\n\"Maybe not, but we have years of fun left in us.\" He kissed her on the top of her head. \"Decades.\"\n\nAs they watched the stars come out, he talked to her about the work he was doing with his family, and he asked her about her catering business and what was up with her and Olivia at Carriage Hill. They talked about adventure travel and treasure hunts left over from Dylan's father.\n\nThe night turned dark, stars glittering overhead. Maggie watched the fire die down, just glowing coals now. At least she'd had the sense to wear jeans and a sweatshirt given the cool temperature. The mosquitoes left them alone.\n\nFinally she said, \"I was so afraid of wanting to be back in Knights Bridge\u2014wanting to raise Aidan and Tyler here\u2014that I ended up blaming you. I had to come home because we were on the skids. It was an excuse.\" She picked at a loose thread on the blanket, then looked up at him. \"It was a bad excuse, and it hurt you and the boys. And me.\"\n\n\"I'd talked you into thinking it'd be a sign of failure to come back here, and that I didn't want to.\"\n\nShe grunted. \"You didn't want to, Brandon. I've been listening to you say you couldn't wait to get out of Knights Bridge and then that you never wanted to go back for years.\"\n\n\"Yeah. I know.\" He shrugged. \"But things change.\"\n\nMaggie sat up straight, shocked. \"You want to be here?\"\n\n\"Pretty much.\" He grinned that easy Sloan grin. \"My family's been waiting to hear me say that since I was just out of diapers. Maggie, I don't care if you needed someone to blame for wanting to come back here for the boys\u2014for yourself. I can take it.\"\n\n\"I was so afraid of being impractical and impulsive. Starting my own business, buying a fixer-upper.\" She pushed hair out of her face. \"Oh, Brandon. I've been such an idiot.\"\n\n\"No, you haven't. I shut down. I told myself you and the boys would be better off on your own.\"\n\n\"You were wrong,\" she said, more forcefully than she'd intended. \"I know you've been through a rough period and you're being responsible, but I don't want you to give up your dreams. Not for my sake.\"\n\n\"When I lost my job, I felt like my dreams were what got us into trouble, and I dug a hole deeper for you and the boys.\" He touched a finger to her chin. \"There's a lot of history between us, Maggie. When I lost my job, I felt like a failure. I felt like everything I'd told you for years about what I was going to do, how we were going to live, was just a lot of BS.\"\n\n\"We were teenagers, Brandon. I wasn't going to hold you to what you said when you were seventeen.\"\n\n\"Or twenty-five? Thirty? And there you were, still filled with such dreams yourself. I felt like mine had only caused trouble for you and the boys, dug us a deeper hole. I put myself and my pride before you.\"\n\n\"You put words in my mouth, especially about money.\"\n\n\"They were what I was telling myself.\"\n\n\"I didn't know what to do. You've always been there for me, Brandon. Then you weren't. Or you were, but you didn't believe that you were. I never saw you as a screwup. You always had such hope and optimism. I didn't realize how much that meant to me until they weren't there.\"\n\n\"We went through a hard year.\"\n\n\"Maybe, but I'm stronger because of it. I'd never lived on my own. I'm not saying I needed to, but it worked out.\"\n\n\"I know it did. I can see it in you. The confidence.\" He shifted, his eyes lost in the shadows. \"I never wanted you to trim your dreams to make me look good. I'm not that kind of guy. In fact, I think that kind of guy's a jerk.\"\n\nShe smiled, even as she wanted to cry. \"I don't want you to give up on your dreams. I couldn't stand it.\"\n\n\"I haven't. This adventure travel gig's right up my alley.\" He leaned back on his outstretched arms. \"We'll see what happens. I know now that the only dream that matters is being with you and the boys.\"\n\n\"I know,\" Maggie said, up on her knees now, at eye level with him as she touched his dark hair. \"Deep down, I've always known.\"\n\nHe flicked a mosquito off her shoulder. \"The bugs have found us.\"\n\n\"What do you say we could go into your tent now?\"\n\n\"I thought you hated camping.\"\n\n\"It's not the camping part I'm thinking about.\"\n\n\"We worry about Phoebe,\" he said, \"but it's Noah we should worry about. The guy has no idea what he's in for getting involved with an O'Dunn.\"\nTwenty-Three\n\nPhoebe was alone in a room dedicated to Noah's collection of antique swords. The lighting and climate controls were set to protect the contents of the room. It was at the back of the house, on the second floor above the pool. She'd already dipped her feet into its warm, silky water. Noah had watched her from the patio. She'd smiled at him, mumbled something about the Southern California sun and her freckles. He'd smiled back and said nothing.\n\nHe was letting her get acquainted with his world, she thought as she leaned in close and studied the ornate handle\u2014or whatever it was called\u2014of another sword, an eighteenth-century French rapier.\n\n\"Note the shape of the blade,\" he said, coming into the small room. \"It's specifically designed for thrusting.\"\n\nShe stood straight. \"Thrusting as in...\"\n\n\"Just what it sounds like.\" He pointed at another sword next to it. \"This blade has a double edge. It's a bit longer. It can be used for thrusting but it can also slash.\"\n\n\"It's a fascinating subject.\"\n\n\"There are a lot of technical terms but it's not as complicated as it might seem,\" He nodded to the sharp tip of the first rapier. \"One touch in the right place is all it takes to kill one's opponent.\"\n\n\"Are you a thruster or a slasher, or is that too simplistic?\" She smiled. \"I'm sure I have a lot to learn.\"\n\nHis eyes held hers. \"I'd like to teach you.\"\n\nPhoebe tried to ignore a flutter in the pit of her stomach as she moved to another display. \"You have quite a collection here. One antique sword led to another antique sword?\"\n\n\"It was something to do on quiet nights after work,\" he said. \"What do you do?\"\n\n\"Lately I've been fixing up my house.\"\n\n\"You have your family and friends, too.\"\n\n\"Don't you?\"\n\n\"I have a small family and a few good friends. I know and like a lot of people, mostly from work, fencing, karate.\"\n\n\"Hollywood,\" Phoebe added.\n\nHe shrugged. \"Some.\" He walked over to her. \"This one is nineteenth-century Persian. Eventually I'll donate most of this collection to charity, to help young martial art athletes.\"\n\n\"You'll still fence and do karate.\"\n\nIt wasn't really a question but he nodded.\n\n\"And NAK?\"\n\n\"We'll see.\"\n\nPhoebe pretended to study the ornate sword but was intensely aware of his presence. They were alone, unlikely to be interrupted. They'd flown overnight. Now...she had to consider where she'd sleep.\n\nShe cleared her throat. \"Olivia and I were talking on the way to the airport. I mentioned that I've been reading about intensive seminars in entrepreneurship. New entrepreneurs spend a long weekend or even two or three months immersed in how to set up their own company. She said you and Dylan would be naturals. You could use the adventure travel barn for classes. People could stay at Carriage Hill. It's a thought, anyway.\"\n\nNoah was so close now she could almost feel his breath. \"It's a good thought,\" he said. \"It would give me another reason to be in Knights Bridge.\"\n\nShe shifted her attention back to the sword but couldn't focus on the details. She saw Noah in Knights Bridge. Saw him in winter, skating with her on the little homemade outdoor rink on the common. Saw him careening down a snow-covered hill in a toboggan with her and her nephews.\n\nIt all felt so right when just a short time ago she couldn't have pictured him in her small town at all.\n\nOr herself in San Diego, with him, and yet here she was.\n\n\"Phoebe.\" He stood next to her and took her hands, kissed her lightly on the lips. \"Come on. I'll show you the rest of the upstairs.\"\n\nHe explained that the house was new. He'd only moved in six months ago. It wasn't massive, certainly not as massive as he could afford. He'd hired a decorator because he hadn't had the time or the inclination to figure out what to do with each room, never mind what furnishings to use. He'd been satisfied with the results, but Phoebe realized that he didn't care that much about the specifics\u2014things like whether a refrigerator was stainless-steel or avocado-green. He'd wanted comfort, soothing colors, space where he could move, relax, think and entertain.\n\n\"Not that I entertain that much,\" he said as they came to the master bedroom. \"You're the first person I've had in here except for the decorators. I gave Olivia the grand tour when she was here with Dylan but we skipped this room.\"\n\n\"It's beautiful,\" Phoebe said, trying to ignore the catch in her throat. The room was all grayed neutrals and sleek lines. She walked to the windows that looked out at the bay. \"The view is breathtaking.\"\n\n\"That's San Diego for you.\"\n\nHe sat on the king-size bed. It was simply made up, the sheets pulled back, white-cased pillows piled at the headboard. Phoebe felt a tingle of awareness as she looked at him. He leaned back on one arm, his eyes a deeper blue than the sky and ocean outside the windows. A nighttime blue. A blue as intense and enigmatic as he was.\n\nThere were several guestrooms. She had only to pick one. He'd told her so in that steady manner he had. But as he watched her from his bed, she knew what his preference was.\n\nShe walked over to him and sat next to him, not quite touching. \"Noah, there's something you need to know about me.\"\n\n\"I want to know everything about you.\"\n\nShe turned to him, placing one knee on the bed. \"Everyone thinks I've given up on love and romance. I thought I had, too.\" She realized she felt comfortable talking to him, and the tension went out of her. She smiled. \"Then I found Daphne Stewart's sewing room in the library attic, and I started to see that I hadn't given up. I argued with myself.\"\n\n\"You didn't want to expose yourself to being hurt again,\" he said. \"Or expose your family to your pain.\"\n\n\"And I didn't want things to change. I liked my life.\" She put her hand on his upper arm. \"I've been torn ever since I saw those dresses, felt the presence\u2014the dreams and hopes\u2014of the woman who created them. They connected with me on the deepest level. I didn't see that at first.\"\n\n\"Sneaking into the ball in your Edwardian gown was part of the war with yourself,\" he said, brushing a curl off her face. \"I think I saw that. It's part of why I noticed you.\" He smiled, letting his hand drift along the line of her jaw. \"Also because you were so damn beautiful.\"\n\nShe laughed. \"It was the black wig.\"\n\nHis eyes sparked with amusement, and something else. Awareness, she thought. Desire. He leaned closer to her. \"It wasn't the black wig,\" he said softly. \"Trust me.\"\n\n\"I do trust you. The moment you swept me onto that ballroom dance floor, I knew I could trust you.\"\n\n\"Sure that's not just jet lag talking?\"\n\nShe smiled. \"Very sure.\"\n\n\"I love you, Phoebe. I don't know if I've ever known what love could be until I spotted you in your princess dress. It got even better when I ran into you hunting slugs, making pesto, warding off a chilly morning in an old sweater at the library.\"\n\n\"I'm not what you're used to\u2014\"\n\n\"I love you, Phoebe. Not some idea of you.\"\n\nHis hand eased over her shoulder. She had on a sleeveless top, could feel the warmth of his skin on hers.\n\n\"Daphne has invited us to lunch at her home in Hollywood Hills,\" Noah said. \"Julius Hartley and Loretta will join us. That's on Thursday. Then I thought we could drive up the coast to my winery. When do you have to be back at work?\"\n\n\"Whenever I want.\" She placed her hand on his, on her shoulder. \"If I want to go back. My future's wide open, too.\"\n\nThey were lost then. She could see it in his eyes, feel it as he withdrew his hand from hers and skimmed it down her bare arm. He was so close to her. So impossibly sexy. She reeled with a desire that was scarily intense and unrelenting. It made her feel vulnerable and open, as if she couldn't hide anything from Noah even if she wanted to\u2014even if she tried.\n\nHe was deliberate, as smooth and centered as he had been when he'd taken her into his arms in Boston, when he'd helped her pick basil and mint and she'd watched him chase after Buster.\n\nWhen he'd pushed his way through the debris from the storm and found her and her nephews in the library attic.\n\nHe was careful of her cuts, asked her if she was okay. She knew what he meant. Her heart skipped as she looked at him in the milky light. Their clothes were already scattered. She took in the ripples of lean muscle on his chest and arms. Damn, she thought. She could have stitches and she still wouldn't want to stop now.\n\nShe nodded. \"Don't worry.\" She draped her arms over his shoulders, heard the raggedness in her voice. How was she supposed to talk when she could hardly breathe? When her body was tingling, quaking for him to touch her. She managed a smile. \"No holding back.\"\n\nHe responded immediately, lowering her onto the bed, easing on top of her. Every touch, every caress, every kiss made her ache with wanting him. He left no inch of her, no part of her, untouched. As tender as he was, he was hard all over. She touched him everywhere, explored his body, touching, tasting. He was so controlled, so focused...until he wasn't.\n\nWhen he sank into her, she cried out, felt him try to ease up, to slow down, but it didn't work\u2014and she didn't want it to. She grabbed hold of his hips and gave herself up to him and the sensations taking over her.\n\n\"Phoebe,\" he breathed. \"Phoebe...\"\n\nShe heard the concern in his voice even as he didn't slacken his pace, didn't stop. \"I'm okay.\" She clutched him. \"I love you, Noah.\"\n\nShe barely got the words out before they both lost control. She'd never experienced anything like it before. Time seemed suspended. It was as if they were the only two people in the universe. She cried out with abandon, wrapped her legs around him as he drove into her one last time.\n\nThey collapsed together. She could feel his heart racing, his skin warm under her palms. She'd dreamed of mind-blowing sex with a man she loved but had convinced herself it would never happen.\n\nAnd yet here she was, with Noah.\n\nHe turned onto his side and locked his eyes with hers. After a moment, he touched the curve of one finger to her cheek. \"Why didn't you tell me that you'd never\u2014\"\n\n\"Because it doesn't matter.\" She caught her breath. \"Didn't matter.\"\n\nHe kissed her softly. \"I'll be more careful next time.\"\n\nShe took his hand into hers and smiled. \"I hope not.\"\n\n* * *\n\nNoah enjoyed showing Phoebe the NAK offices. She wanted to see everything and talk to everyone, and she was interested in every detail of NAK's work. He'd already met with his executive team and set up a formal meeting to redefine his role with the company.\n\nThings were changing, because\u2014well, things always changed.\n\nAnd because of Phoebe. He loved talking to her. He loved listening to her, and making love to her. After two days together, he was as certain as ever that they belonged together. She was kind, intelligent, courageous and so damn beautiful.\n\nShe was also still at least a little afraid of what the changes loving him meant for her.\n\n\"I thought I could predict my future,\" she said as she looked out at the view from his office windows. \"But then you decided to dog sit Buster.\"\n\n\"I knew Olivia had an idea of who my princess was.\"\n\n\"If you hadn't stayed in Knights Bridge, my swashbuckler would have remained a mirage.\" She smiled at him. \"But you're no mirage.\"\n\nHe expected he'd proved that over the past two days. He winked at her. \"You've got that right, princess.\" He stood next to her, slipping an arm around her waist. \"My best friend is marrying one of your best friends. We'd have found each other.\"\n\nPhoebe leaned against his arm. \"I can't wait for Olivia and Dylan's wedding. I love a happy ending.\"\n\nNoah kissed her on the top of her head. They'd showered together that morning, and he could still smell the shampoo she'd used on her hair. \"Phoebe...\" For the first time since she'd arrived in San Diego, he found himself struggling for the right words. He'd had an engagement ring delivered to his house at the winery. The timing didn't feel too soon. It felt just right. Perfect, even. Finally smiled at her. \"We'll write our own happy ending.\"\n\nHer smile reached her eyes. \"We already are.\"\n\nThey headed down to the lobby together. They didn't want to keep Daphne Stewart waiting. They'd take their time driving up the coast to his winery and enjoying their stay there.\n\nThen they'd go back to Knights Bridge.\n\n* * * * *\nDear Reader,\n\nThank you for reading That Night on Thistle Lane. I hope you enjoyed the story! Since I grew up in a small town on the western edge of the Quabbin Reservoir, I'm having the best time writing books set in the Swift River Valley and surrounding hills. You can find scenes and even a video on my website depicting this beautiful part of rural New England.\n\nI've appreciated hearing from people with ties to the \"lost\" towns of Dana, Enfield, Greenwich and Prescott, all depopulated and razed to create the massive reservoir. Old roads and trails lace the protected wilderness, much of it open to the public. When I walk there, I look for cellar holes and other remnants of the people who used to live in this beautiful valley, but most of all I just enjoy the incredible scenery.\n\nIf you haven't read Dylan and Olivia's story yet, it's called Secrets of the Lost Summer. It actually starts in early spring, when those of us in northern climates are eager for March to turn into a lamb.\n\nI'm looking forward to whatever's next for fictional Knights Bridge and the Frosts, Sloans, O'Dunns...and, of course, our guys from San Diego, Dylan McCaffrey and Noah Kendrick. It's a lot of fun for me, and I'm so glad to have you join me!\n\nPlease visit www.carlaneggers.com for updates, links, goodies and photos, and join me on Facebook, Twitter and Pinterest. And write to me anytime at Carla@CarlaNeggers.com. I'd love to hear from you.\n\nHappy reading,\n\nWe hope you enjoyed this Harlequin ebook. Connect with us for info on our new releases, access to exclusive offers, free online reads and much more!\n\nSubscribe to our newsletter: Harlequin.com\/newsletters\n\nVisit Harlequin.com\n\nWe like you\u2014why not like us on Facebook: Facebook.com\/HarlequinBooks\n\nFollow us on Twitter: Twitter.com\/HarlequinBooks\n\nRead our blog for all the latest news on our authors and books: HarlequinBlog.com\nISBN: 9781460303283\n\nCopyright \u00a9 2013 by Carla Neggers\n\nAll rights reserved. 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, Harlequin Enterprises Limited, 225 Duncan Mill Road, Don Mills, Ontario, Canada M3B 3K9.\n\nThis is a work of fiction. Names, characters, places and incidents are either 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. This 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"}} +{"text":"\n\n\n\nProduced by Karin Spence, Juliet Sutherland and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n [Illustration: _From the painting by F. Brangwyn_\n\n \"THEIR LANTERN JUST BROUGHT OUT THE GHOSTLINESS OF GRAVESTONES LEANING\n BETWEEN THE COLUMNS OF THE CYPRESSES\"\n\n _See \"The Valley of Mills,\" page 659_]\n\n\n\n\n McCLURE'S MAGAZINE\n\n VOL. XXXI OCTOBER, 1908 No. 6\n\n\n\n\n [Illustration]\n\nFAMILIAR LETTERS OF AUGUSTUS SAINT-GAUDENS\n\nEDITED BY ROSE STANDISH NICHOLS\n\nILLUSTRATED WITH PHOTOGRAPHS\n\n_Copyright, 1908, by The S. S. McClure Co. All rights reserved_\n\n\nThese familiar letters from Augustus Saint-Gaudens show the artist as\nhis intimate friends knew him. They were written at odd moments, often\nin haste, and never with a shadow of self-consciousness. They are\ninteresting, not as literary productions, but as the simple record of a\ncritical period in his career.\n\n\"Le Coeur au Metier,\" the motto which he wished to place in his studio,\nwill be seen to express the spirit of his life. Other keen interests\nhe had, but they were never allowed to interfere with his work, and\nhe seldom felt the need of any recreation apart from it. One of his\nfriends used to complain that in the midst of their merrymaking an\nabstracted look would come into his eyes and his mind would hark\nback to sculpture. Although he was extremely modest and was given to\nunderrating his powers in other directions, from his childhood he\nconfidently expected to be a great artist. As a little school-boy, sent\nfrom his father's shop to do errands, he would sit in the omnibus and\nlook about at his well-dressed fellow-passengers, and wonder what they\nwould think if they realized what he was going to be some day. But even\nas a child he never dreamed of achieving his ambition without years of\nceaseless struggle.\n\nWhen the boy left school, at the age of thirteen, this struggle began.\nIn 1848 his father, a Frenchman, had brought his Irish wife and his\nbaby, Augustus, to New York, where he worked as a shoemaker. He was\npoor, and was anxious that his eldest son should become self-supporting\nas soon as possible; so at thirteen the boy was apprenticed to a\ncameo-cutter, whose trade he mastered with surprising readiness, at the\nsame time studying drawing at the Cooper Institute in the evenings. In\na little while he was not only earning his own living by cameo-cutting,\nbut excelled all his fellow-pupils at the night-school in talent and\nperseverance.\n\n [Illustration: AUGUSTUS SAINT-GAUDENS FROM A PHOTOGRAPH IN THE\n COLLECTION OF MRS. ROSE NICHOLS]\n\nSaint-Gaudens' artistic education was completed in Europe, where he\nwent at the age of eighteen and stayed almost continuously for nearly\nfourteen years. His father sent him first to Paris. There his progress\nin the art schools was marked, although he continued to support himself\nby his trade, and could give only half his time to sculpture. At the\noutbreak of the Franco-Prussian War he reluctantly refrained from\nenlisting in the French army and left for Italy. It was in Rome that\nhe first found sculpture remunerative, and finally was able to drop\ncameo-cutting. The years from 1866 to 1880, which he spent in Rome and\nParis, with only occasional visits to America, were singularly happy\nones, characterized by a capacity for continuous work at a high pitch\nof excellence.\n\nThe letters from Saint-Gaudens printed here were written eighteen\nyears later, when the sculptor had come into full possession of his\ngenius. They cover a most critical period in his career, and record\nhis greatest artistic triumph--his recognition in France as one of the\nforemost of modern sculptors. After he returned to the United States\nin 1880 he lived and worked in New York, and by 1897 had built up a\nnational reputation. His work was progressing under the most favorable\nconditions, with the encouragement of an ever-increasing circle of\nfriends and admirers. On the other hand, in France, his father's\ncountry, where he himself had been educated, his work was practically\nunknown. A few of his former comrades at the Beaux-Arts, judging his\nsculpture from photographs, did not hesitate to tell Saint-Gaudens\nthat it had been over-praised in America and would obtain no such\nappreciation in France. The sculptor felt that, in order to learn\nhis own deficiencies and to find out where he really stood among his\ncontemporaries, he must return to Paris, exhibit at the Salon, and run\nthe gauntlet of the best critics. All his friends on both sides of the\nwater discouraged him from taking this step, and he himself dreaded it;\nbut he believed that, in justice to himself and to his work, he must\nmake this venture.\n\nAfter his decision was made, however, his departure had to be postponed\nuntil various duties were fulfilled. The Shaw and Logan monuments had\nfirst to be completed and unveiled, and a number of smaller commissions\nhad to be executed. From the beginning of his work upon the Shaw\nmemorial there had been bitter opposition upon the part of his friends\nto the symbolical figure hovering above Colonel Shaw and his men, but\nthe sculptor clung to his original conception with great tenacity.\nSaint-Gaudens' best friend, Bion, a Parisian sculptor and critic, whose\nopinion he valued highly, had never liked the idea of this figure.\nJust before Bion's death he received a photograph of the monument as\nfinished in the clay, and he wrote a long letter to Saint-Gaudens,\ncomplaining that the angel was as superfluous as a figure of Simplicity\nwould be, floating in the air above the bent figures in Millet's\n\"Gleaners,\" and concluding: \"I had no need of your 'nom de Dieu'\nallegory on the ceiling. Your s marching in step and your Colonel\nleading them told me enough. Your priestess merely bores me as she\ntries to impress upon me the beauty of their action.\"\n\n [Illustration: CARYATID FOR THE VANDERBILT HOUSE]\n\nConcerning this letter of Bion's, Saint-Gaudens wrote:\n\n \"The Players, New York, Jan. 26th, 1897\n\n \"Dear ----\n\n \"I meant to write you at length tonight but I started with a\n letter to Bion which has kept me busy till now, 11 P.M. It is in\n reply to the one from him that I enclose, in which at the end he\n says a word of you.\n\n \"I am not disturbed by his dislike of my figure. It is because\n it does not look well in the photograph. If the figure in itself\n looked well, he would have liked it, I know, and notwithstanding\n his admirable comparison with the Millet I still think that a\n figure, if well done in that relation to the rest of the scheme,\n is a fine thing to do. The Greeks and Romans did it finely in\n their sculpture. After all it's the way the thing's done that\n makes it right or wrong, that's about the only creed I have in\n art. However his letter is interesting, although very sad, dear\n old boy.\n\n [Illustration: DETAIL FROM THE SHAW MEMORIAL, SHOWING THE ALLEGORICAL\n FIGURE FULL FACE, AS IN THE FIRST DESIGN]\n\n [Illustration: _Copley print, copyrighted by Curtis and Cameron_\n\n DETAIL FROM THE SHAW MEMORIAL, SHOWING THE ALLEGORICAL FIGURE WITH THE\n HEAD TURNED MORE IN PROFILE, AS IN THE FINAL EXECUTION]\n\n \"All of the Shaw is out of the studio. They cast the Logan on\n Monday and I am working like the devil on the Sherman. I've found\n precisely the model I wished, just his size, the same pose of the\n head and the same thinness; a Milanese peasant who poses like a\n rock. Next week I commence the nude of the Victory from a South\n Carolinian girl with a figure like a goddess.\n\n \"Affectionately yours\n A. ST.-G.\"\n\nBion died shortly after writing his objections to the allegorical\nfigure, and if anything could have changed Saint-Gaudens' decision\nregarding his composition of the Shaw monument, his friend's letter\nwould certainly have done so. Although Saint-Gaudens and Bion had\nstudied sculpture together at the Beaux-Arts in their youth, it was\nnot until years afterward that, through a constant interchange of\nletters, their relation became a close one. Bion gave up sculpture as\na profession, and devoted himself to friendship and philosophy. He\ndropped into the studios of a few intimates every day, frequented art\nexhibitions, and attended lectures upon philosophy and psychology at\nthe Sorbonne or the College de France; but the long letters which he\nused to write Saint-Gaudens every week became more and more the chief\nbusiness of his life. He kept his friend informed as to what was going\non in Paris; of the doings of their little circle of acquaintances; and\nwrote him detailed descriptions of all important events in the world\nof art, besides giving him a great deal of disinterested advice upon\nevery conceivable subject, including his work and the conduct of his\nlife. Saint-Gaudens used to reply at great length, but his letters were\ndestroyed, according to directions left in his friend's will. When the\nnews of Bion's death reached Saint-Gaudens, he wrote:\n\n \"148 W. 36th St., Feb. 17th, 1897\n\n \"Of course the one thing on my mind, the terrible spectre that\n looms up, is poor Bion's death; night and day, at all moments, it\n comes over me like a wave that overwhelms me, and it takes away\n all heart that I may have in anything. Today, however, I have had\n a kind of sad feeling of companionship with him, that seems to\n bring him to me, in working over the head of the flying figure of\n the Shaw. The bronze founders are not ready for it yet. I have had\n a stamp made of the figure and have helped it a great deal, I am\n sure you will think. You know that Thayer told me he thought an\n idea I once had of turning the head more profile, was a better one\n than that I had evolved, and I've always wished to do it. It is\n done, and it's the feeling of death and mystery and love in the\n making of it that brought my friend back to me so much today....\n But the young, thank Heaven, do not feel these blows so profoundly\n as do older people. In one of my blue fits the other day I felt\n the end of all things, and reasoning from one thing to the other\n and about the hopelessness of trying to fathom what it all means,\n I reached this: that we know nothing, (of course) but a deep\n conviction came over me like a flash that at the bottom of it all,\n whatever it is, the mystery must be beneficent. It does not seem\n as if the bottom of all were something malevolent; and the thought\n was a great comfort.\n\n [Illustration: _Copley print, copyrighted by Curtis and Cameron_\n\n MONUMENT TO COLONEL ROBERT GOULD SHAW, ERECTED AT BOSTON]\n\n \"I shall be all the week at the figure. I've made an olive branch\n instead of the palm,--it looks less 'Christian martyr'-like,--and\n I have lightened and simplified the drapery a great deal. I had\n not seen it for two or three months and I had a fresh impression.\n\n \"At 27th Street I've finished the nude of the Sherman and next\n week I begin to put his clothes on him. I had another day with the\n model for the Victory last Sunday, and that, too, is progressing\n rapidly. Zorn, the Swedish artist, was with me all day Sunday\n making an etching of me while the model rested; it is an admirable\n thing and I will send you a copy of it.\n\n \"The studio is once more in a fearful condition with the casting\n of the Logan, and the getting of the Puritan ready to photograph\n and cast for the Boston Museum and to send abroad to have the\n reductions made....\n\n \"This letter is no good, but it must go; the clatter of seven\n moulders and sculptors does not help to the expression or the\n development of thought, confusion only----\n\n \"Affectionately\n A. ST.-G.\"\n\n \"May 15th or 16th, 1897\n\n \"The Shaw goes to Boston on Thursday or Friday. I've done little\n else lately but run around about it until I am frantic. On the\n other hand, while waiting for some workmen yesterday, I had a\n great walk in the Babylonian East Side here. It was a beautiful\n day and one of great impressions.\n\n \"I have not commenced the Howells medallion yet, as I expected to\n be absent. I believe I told you I had a nice note from him.\n\n A. ST.-G.\"\n\n [Illustration: MURAL PLAQUE ERECTED IN MEMORY OF DR. JAMES McCOSH]\n\nThe Shaw memorial was unveiled in Boston, in the latter part of May,\n1897. The erection of the monument had been so long delayed that\nSaint-Gaudens feared that the public had lost interest in the work, or\nwould expect too much and be disappointed. On the contrary, its success\nwas immediate, and made him very happy. Its appeal was to men of every\ncondition, laymen as well as artists, and nothing ever pleased the\nsculptor more than the way it arrested the attention of almost every\npasser-by. In June, scarcely a month after the unveiling of the Shaw,\nanother soldier's monument, the equestrian statue of General Logan, was\nunveiled at Chicago, and Saint-Gaudens went there to be present at the\nceremony.\n\n [Illustration: STATUE OF PETER COOPER, NEW YORK]\n\n \"1142 The Rookery, Chicago, June 23, 1897\n\n \"I am again at the top of this big building here, and I will\n give you some description of the last 24 hours. At one o'clock\n yesterday Mrs. Deering, Mrs. French, Mr. French (brother and\n sister-in-law of Dan French) and I were placed in one carriage,\n Mr. Deering, Mrs. St. G. and the editor of the 'Chicago Tribune'\n in another, and in the wake of a lot of other carriages and\n followed by a procession of them, we drove to the big stand. A\n great day; with a high wind and glorious sun. I was put in one\n of the seats in the Holy of Holies alongside of Mrs. Logan,\n if you please, and the president of the ceremonies. A lot of\n speeches, one of which was very good, and at the right moment the\n complicated arrangement of flags dropped, the cannon fired, the\n band played, Mrs. Logan wept, and I posed for a thousand snap\n photographs, 'a gleam of triumph passed over my face,' think of\n that! (vide 'Chicago Tribune').\n\n [Illustration: THE LOGAN MONUMENT, ERECTED AT CHICAGO]\n\n \"However, the monument looks impressive as I see it this morning\n for the first time with much of the disfiguring scaffolding gone.\n I stay here until Sunday, when I take the 5.30 P.M. train and\n shall get to New York Monday at 6 or 7. Last night we went to a\n great golf place where high merriment prevailed. This afternoon to\n Fort Sheridan. Tonight a reception at the Art Institute; tomorrow\n a lawn party at Burnham's and Sunday a visit to the great dredging\n canal; on Monday the cars and rest.\"\n\nAfter the sculptor's return from Chicago, he continued his preparations\nfor departure in New York.\n\n \"The Players, August 7, 1897\n\n \"Brander Matthews has just come and interrupted this with a long\n and interesting talk on the conventional in art and an article he\n has written and sent to Scribner's on it. You have often wondered\n what I think about things--I wonder myself; I think anything and\n everything. This seeing a subject so that I can side with either\n side with equal sympathy and equal convictions I sometimes think a\n weakness. Then again I'm thinking it a strength.\n\n \"Last night I dined with X---- and Y---- and passed a delightful\n evening with them. X---- cracked his constructed jokes and\n manufactured his silversmith puns, and cackled over them. We\n talked literature, English, French, and Taine's great work on\n English literature. We afterward went to the open air concert\n at the Madison Square Garden, and when we were not talking of\n anything else we talked on that subject of eternal interest and\n mystery 'les femmes.'\"\n\nFinally, in the autumn of 1897, after both the Shaw and Logan monuments\nhad been unveiled, and various minor obstacles to his departure had\nbeen removed, Saint-Gaudens was ready to leave America. Opposition to\nhis plan still came from every side. Many of his friends in New York\nseemed to feel that he was casting a certain reproach upon his country\nby his desire to profit by foreign criticism and to measure his work\nby European standards. They prophesied that his work would deteriorate\nunder French influence. His few friends in Paris were equally\ndiscouraging. They did not hesitate to warn him that if he persisted\nin coming there he must be prepared to face indifference and failure.\nEven Bion, when Saint-Gaudens had asked him to get the opinions of a\nfew French artists upon photographs of the Shaw memorial, had refused\nto do so, saying: \"I shan't show your photographs to anyone. Shiff,\nMacMonnies, and Proctor have seen them, my poor old friend, and the\nothers do not know you. They are quite indifferent about what goes on\noutside their own little show.\"\n\nSaint-Gaudens himself feared that he might be making a serious mistake.\nThe ocean voyage in itself was an ordeal to him, and before leaving\nhe wrote: \"I continue fencing and am preparing for the voyage as one\nprepares for a fight. I go to the theatre and that tides over the\nblue hours which lie between dinner and bed-time.\" But he felt that\nhe must make the venture, whatever lay before him, and that he could\nnever be satisfied until he had stood the test of a comparison with\nhis chief contemporaries and until his work had been passed upon by\nthe most sophisticated and penetrating critics of art. At the end of\nSeptember, 1897, accompanied by his wife and his son, Homer, he sailed\nfor England. After crossing to France, he thus described his first\nimpressions:\n\n \"Hotel Normandy, Paris, Nov. 7th, 1897\n\n \"The beauty of the scenery and of the English homes and villages\n on the railroad from Southampton to London recalled the delightful\n impression of the last trip, when I was so light-hearted. The\n sense of order and thrift appealed to me strongly in comparison\n with the shiftlessness of America. Then London with its\n extraordinary impression of power and also of order. Homer and I\n went to see Hamlet. Read it, R----. As I grow older, the greatness\n of Shakspeare looms higher and higher; every line, every word is\n so deep, so true, 'never offending the modesty of nature withal,'\n as Hamlet himself advises the players.\n\n \"From London we came on the following day to Paris. The country\n between Calais and Dover seemed very grand; great rolling lands\n with immense fields being ploughed in the waning day. The peace,\n simplicity, and calm of it all was profoundly impressive. Just a\n ploughman and a boy, alone in the country on a hillside, following\n the horses and the plough along the deep, straight furrows; no\n fences, a clear sky with the half moon, and only a small clump or\n two of trees--all so orderly and grand.\"\n\nFor the first few weeks in Paris Saint-Gaudens was miserable. His\nstudio, on the Rue de Bagneux, in the Latin Quarter, was large and\ncheerful, with comfortable quarters adjoining for his assistants, and\nhe was extremely interested in his work upon the equestrian statue\nof General Sherman. But he missed his old friends and haunts in New\nYork, the weather was gloomy and depressing, and he felt enervated and\nhomesick. Almost none of the friends of his student days were there to\nwelcome him back to Paris, and he was not in the mood to make new ones.\nDr. Shiff, a retired physician with a philosophic turn of mind, and\nmany years the sculptor's senior, was the only man he could count upon\nfor regular companionship, though occasionally an old friend like Henry\nAdams, John Alexander, or Garnier would drop into the studio. John\nSargent was another warm friend who helped to keep up his spirits and\nwhom he admired intensely both as a man and as an artist. With Helleu,\nthe etcher, they enjoyed spending a day or two at Chartres and Rheims.\nIn the following letter he describes his first meeting with Whistler:\n\n \"Paris, Nov. 16th, 1897\n\n \"Mac and I made a short call on Whistler, whom I found much more\n human than I imagined him to be, and today I went to the Court of\n Appeals where a trial of his was to come off--it didn't,--but I\n had a delightful chat with him. He is a very attractive man with\n very queer clothes, a kind of 1830 coat with an enormous collar\n greater even than those of that period; a monocle, a strong jaw,\n very frizzly hair with a white mesh in it, and an extraordinary\n hat.\"\n\nThe brightest spot in Saint-Gaudens' winter was his visit to the south\nof France and to Italy, in the company of his friend Garnier, who, like\nBion, had been a fellow-student of his at the Ecole des Beaux-Arts\nyears before. They left Paris in December, and went almost directly to\nAspet and Salies du Salat, Gascon villages where Saint-Gaudens' father\nwas born and where he worked at his trade as a young man. This was the\nfirst time that Augustus Saint-Gaudens had visited that country on the\nSpanish frontier where his paternal ancestors had lived for centuries\nand where many of their name still survived.\n\n \"Aspet, December, 1897\n\n \"I write this in the village where my father was born and\n today has been one of the most delightful days of my life. I\n have invited my old friend Garnier (a dear friend and the most\n delightful of companions) to travel with me. We left Paris\n yesterday morning and slept at Toulouse last night. We left there\n this morning before dawn and saw the sun rise over the Pyrenees on\n our way to Salies du Salat, a most picturesque and dirty village\n at the foot of the beautiful mountains. I inquired at the station\n if any Saint-Gaudens lived there. 'Yes, opposite the mairie.' We\n walked up a narrow Spanish-looking street and there was a little\n shoe-store and on it the sign 'Saint-Gaudens.' I woke my cousin\n up. His is the very house where father passed his childhood. We\n three walked over the town up to the cradle of the 'Comminges'\n just back of father's house, and we went around on the sward and\n on the old moat where the children now play and where his father\n and my father played when children. I cannot describe to you how I\n was moved by it all.\n\n \"After a characteristic dejeuner with the cousin, a typical French\n peasant, and his typical wife, we hired a wagon with two horses\n and drove three hours into the mountains through a wonderfully\n beautiful country, very Spanish in character, to this delightful\n village. Here father was born, and baptized in the little church\n right at hand from where I write. There are delightful fountains\n at every corner and an air of thrift, order, and cleanliness that\n you cannot imagine. We are in a nice hotel, a homelike place, and\n tomorrow, after seeing Market Day, we walk to Saint-Gaudens, about\n 12 miles from here. It is a most romantic spot; all the country\n and the people here have a good deal of the Spanish dignity. We\n are 30 miles from the frontier of Spain. I must stop now because\n my third cousin (his grandfather and mine were brothers) is\n coming. He is the postman of the village and the surrounding\n country, a handsome young fellow who carries the mail around on\n horseback, and who between times makes shoes.\"\n\nLeaving this out-of-the-way corner of Gascony, under the shadow of the\nPyrenees, Saint-Gaudens and Garnier traveled by Toulouse to Marseilles.\nFrom this port the sculptor had sailed twenty-seven or eight years\nbefore, when he first went to study in Rome. Now, with his old friend,\nhe again climbed up to where the church of Notre-Dame de la Garde\noverlooks the Mediterranean, and was amused to remember the three\ndays he had spent upon that hill-top, with little to eat but figs and\nchocolate, while awaiting the departure of his ship for Italy.\n\nThe two artists went by train from Marseilles to Nice and Ventimiglia,\nand then walked along the superb Cornice road to San Remo, conscious\nthat every step brought them nearer to their beloved Italy. The hills,\ncovered with palms and orange-trees, the sacred-looking groves of\ngray-green olives detached against the deep blue of the sea, recalled\nto Saint-Gaudens a story by Anatole France describing some early\nChristians in an olive grove overlooking the Mediterranean.\n\nIn Italy they stopped first at Pisa, and did not reach Rome much before\nmidnight. Regardless of fatigue, Saint-Gaudens insisted upon starting\nout that night to revisit the favorite haunts of his student days,\ntaking the reluctant Garnier with him. At a late hour they ended their\nexcursion at the Cafe Greco, where the sculptor talked with a waiter\nwho had served him with coffee in 1871. The next morning they spent in\nthe gardens and the Bosco of the Villa Medici. Nothing seemed to them\nmuch changed, and their happiness was as great as if they had found\ntheir youth again in the land where they had left it. Saint-Gaudens\nafterward said that on the night of that arrival in Rome he felt as if\nhe were slaking a great thirst. Before their return they also visited\nthe Bay of Naples. Vivid memories of Italy were present with the\nsculptor until the end of his life, and during his last illness he said\nthat one thing he wished to live for was to take again the drive from\nSalerno to Amalfi: the vineyards clinging to the hillsides, the cliffs\nwith the blue waves breaking at their base, haunted him as a vision of\nexquisite beauty.\n\nLate in the winter Saint-Gaudens returned to Paris, and when spring\nand the pleasant weather came on he was working again with great\nenthusiasm, preparing for the Salon. His exhibit at the Champs de\nMars attracted much attention and elicited unexpected praise from the\nseverest French critics.\n\n \"3, rue de Bagneux, Paris, May 16th, 1898\n\n ... \"I must be brief today for Dr. Shiff is coming in to talk,\n and help me with his consoling philosophy as Bion did; and I must\n work, for the model leaves shortly, and I must use him every hour\n I can; so I will tell you briefly of what has happened.\n\n \"This Paris experience, as far as my art goes, has been a great\n thing for me. I never felt sure of myself before, I groped ahead.\n All blindness seems to have been washed away. I see my place\n clearly now, I know, or think I know, just where I stand. A great\n self-confidence has come over me and a tremendous desire and will\n to achieve high things, with a confidence that I shall, has taken\n possession of me. I exhibited at the Champs de Mars and the papers\n have spoken well and it seems as if I were having what they call\n a 'success' here. I send you some of the extracts from several of\n the principal artistic papers here, the 'Gazette des Beaux-Arts,'\n 'Art et Decoration,' and from the 'Dictionaire Encyclopedique\n Larousse'; four of these have asked permission to reproduce my\n work. The Director of the Luxembourg tells me he wishes something\n of mine, and other friends have asked that I be given the Legion\n of Honour. Of this latter you must say nothing, and I only speak\n of it to give you a true idea of what impressions I am undergoing.\n\n \"For four months it rained incessantly, but the great interest of\n preparing for the Salon has interested me. The sunshine has been a\n blessing, and Paris, with her smiles and green dress and the blue\n skies overhead captivates like a beautiful woman.\n\n \"There is something in the air here which pushes one to do\n beautiful things; it seems something actually atmospheric,\n something soft and gentle in the air.... Later Sargent came in\n very good spirits. We dined and went to the theatre together\n last night. He wished me to tell him when I go to London, as the\n fellows there wish to give me a great 'blow off.' And so it all\n goes; the sun is now pouring into the studio, and it all seems\n like a great dream.\"\n\nThe article in _Art et Decoration_ to which Saint-Gaudens refers was\nwritten by Paul Leprieur. After attacking with great severity Rodin's\n\"Balzac,\" the critic said:\n\n\"The more completely to forget this sinister vision, one may well\nlinger before the work of a great sculptor, almost unknown among\nus, who reveals himself to us, so to speak, for the first time,\nwith an altogether remarkable collection of monumental sculpture\nand photographs of monuments previously executed. We refer to M.\nSaint-Gaudens, an Irishman by birth, who has worked mainly for America,\nand who was, if I mistake not, the teacher of Mr. MacMonnies--a teacher\nfar superior to his pupil. His exhibit is one of the surprises and\ndelights of the Champs de Mars.\n\n\"Had we only the photographs which he shows us--whether of his Peter\nCooper, his President Lincoln, the noble and serious allegorical figure\nfor a tomb, called the Peace of God, or the charming caryatid for the\nVanderbilt house--we could already perceive the grasp of composition,\nthe decision of the contours, the depth of the sentiment expressed\nwithout any splurge or noise. This sculpture, in its acceptance, or\ningenious re-shaping, of traditions from ancient sources, as well as in\nits modern inventiveness, imparts a savor of intimate charm, of dignity\nwithout parade, which are rare indeed in our day.\n\n\"The actual work exhibited simply confirms the impression of the\nphotographs. To say nothing of the placques and medallions, models of\na fine funeral bas-relief, and the highly entertaining and picturesque\nstatue of a Puritan, the large high-relief dedicated to the memory\nof Colonel Robert Gould Shaw may well be esteemed as a model of\nintelligent decoration.\n\n\"The idea of representing, not the death scene itself, but the moment\npreceding it, and of showing the army of blacks, led by the white\nofficer, filing by as if in a march to death, grave of mien, solemn,\nand heroic, is as novel as it is boldly treated. While presenting\nprodigies of skill (absolutely without triviality or pettiness in\nmatters of detail), and modeled with a great freedom and understanding\nof how to arrange the various groups of lines in perspective,--which\nall men of his profession will admire,--everything is kept subordinate\nto the ensemble and to the predetermined unity of motion. Upon each\nof the faces one feels more or less the reflection of the motto of\nself-sacrifice and enthusiastic faith inscribed on a flat surface in\nthe background (Omnia relinquit servare rem publicam), and the superb\nfigure of a woman with flying drapery, symbolical of glory or of death,\ncomparable to the loveliest creations in this style by Watts or Gustave\nMoreau, succeeds in giving to this very sculpturesque composition a\ndistinguished moral significance.\"\n\nTwo months later the critic Leonce Benedite, in his article on the\nsalons of 1898, wrote, in the _Gazette des Beaux-Arts_:\n\n\"It is a foreign sculptor, an American artist whose name alone had\npreviously reached us, M. Augustus Saint-Gaudens, who affords us an\nexample of a commemorative monument composed of modern elements and\nbroadly executed in the simplest and purest sculptural spirit. Half\nFrench, not only by descent, but by his whole education, trained in our\nschool,--which he honors today,--the illustrious chief of the future\nAmerican school of sculpture has produced numerous beautiful works in\nhis own country. Photographic reproductions of these accompany his\nexhibited works and demonstrate their rare dignity and grandeur of\nstyle. His beautiful mortuary statues, one of which is on exhibition at\nthe Salon, together with the caryatid of the Vanderbilt house--long and\nslender, with beautiful, severe draperies--are figures of distinguished\nelegance, of austere grace.\n\n\"But above all, the statues of President Lincoln and Peter Cooper, the\nmural tablets of Dr. McCosh and Dr. Bellows, show us with how exalted\nan appreciation of his art the American master has succeeded in making\nthe most of the complete modernity of his subjects. To be sure, he has\nnot misrepresented the characteristic local physiognomy of his models,\nor the unique effect of the accessories of costume and furniture; far\nfrom it. But with what elegance and vigor he makes them all speak to\none, from the skirt of the coat to the slightest fold of the trousers!\n\n\"We find ourselves face to face with a powerful and self-restrained\nmaster, who is able to comprehend and to express emotion, who speaks a\nsimple but expressive language, and who has the power to convince and\nto fascinate. The monument to Colonel Robert Gould Shaw, erected at\nBoston, and exhibited in plaster at the Salon, affords us a striking\nproof of this. It is a large high-relief, set in a graceful and\nexceedingly simple architectural frame. In the center a young officer,\nmounted, sword in hand, is leading a company of black soldiers who\nare marching by his side, musket on shoulder, with a drummer at their\nhead. In the upper field floats a grave and melancholy figure, flying\nhorizontally; it is Duty, and with a sweeping and eloquently mournful\ngesture she points out to them the road leading to glory and to\ndeath. The measured march of the men, the expression of resigned and\nsubmissive gravity on the faces of those troops, contrasting\nwith the proud, absorbed energy of the young white man who leads\nthem, his beautiful young steed nervous and quivering, emphasizes\nyet more the restrained enthusiasm and patient determination of the\ncommander. All this, and even the sculptural comprehension of all this\nparaphernalia of war, impresses one simply yet powerfully, and holds\none enthralled by its genuine epic grandeur.\"\n\n \"June 14th, Paris\n\n \"I am going to stay alone in Paris and on Sundays go and see Brush\n and Garnier and the Proctors and go to St. Moritz for a week or\n ten days; further than that I have no plans.... I see Shiff every\n other night and dine with him then; occasionally I see F----,\n whom I rather like. I'm working hard but slowly. I want a little\n rest, so in two days I go to London to see the exhibit there;\n besides, Sargent gives me a dinner on the 20th. Paris is really\n a wonderfully attractive city and the 'cut' atmosphere, to use\n a very unpleasant phrase, is clearly a great thing. There can\n never be more than a few big men that one respects, but there are\n so many people deeply interested in art, literature and music,\n so many that are working hard, that you feel a great deal of\n intelligence around you in the direction in which you are working,\n beside the unusual amount of general intelligence which surrounds\n one.\"\n\nToward the end of June Saint-Gaudens and his family went to England.\nIn London, Sargent, always hospitable, gave a dinner to introduce\nSaint-Gaudens to many distinguished sculptors and painters.\nBurne-Jones, unfortunately, had died a few days before. Saint-Gaudens\nhad always admired his work greatly, and treasured photographs of his\npictures.\n\nAfter two days at Broadway with Edwin Abbey, the family separated.\nSaint-Gaudens and his son Homer then returned to Paris for the summer,\nwhile Mrs. Saint-Gaudens went to take a cure at Vichy and St. Moritz.\nDuring that summer in Paris Saint-Gaudens saw as much as possible\nof George De Forest Brush and his family, who were then living near\nFontainebleau. His intimacy with the Brushes dated back to his student\ndays in Paris, and had been kept up in America. The two families had\noften been neighbors at Cornish, New Hampshire. Indeed, the Brushes had\nspent their first summer there encamped in an Indian \"tepee,\" which was\npitched on the edge of a field in front of the Saint-Gaudens' house.\nTheir life always impressed every one as singularly beautiful and\nhappy, and their presence so near Paris helped Saint-Gaudens to get\nthrough the long, dull weeks of the summer.\n\n \"Paris, July 10th or 11th\n\n \"Lately I have had a great time with X----, driving and lunching\n with him and sometimes with the ladies, going to Versailles and\n the museums. Next Sunday we go to Chantilly, another day to\n Dampierre where Rude's great statue of Louis (XIII, I think) is.\n We go to the Cluny, to the Louvre, and sit sipping in front of\n cafes, X---- telling me how much the woman question from one point\n of view troubles him and I doing the same from another, and the\n big world turns round, and we all suffer, and men fight, and women\n mourn. Courage and love is what we all need, isn't it?\n\n \"Yesterday I went with Homer to Fontainebleau to see Brush and\n Proctor who live near there at 'Marlotte Montigny.' The day was\n fine, and I enjoyed it greatly, particularly the walk with Brush\n and his two lovely eldest children. How remarkable Brush is! All\n the children are so beautiful and nice-mannered. He has commenced\n another picture of his wife, this time with all the children and\n himself, and it is already a stimulating thing, the composition is\n so fine and what there is of it that is drawn, is so splendidly\n drawn.\"\n\n \"Paris, July 14th\n\n \"It is the third or fourth really fine day that we have had\n since coming to France eight months ago. The whole city is alive\n with sunshine, a sky with white floating clouds, and every place\n brilliant with flags, and there is an unusual feeling of peace in\n this big studio as I sit alone in it and write to you.\n\n \"I have your letter with the enclosure from the _Transcript_.\n 'That's the way things is,' as Bryant said to me. I send you some\n more Hosannahs in my honour by this mail, and there is going to be\n more still in the 'Gazette des Beaux-Arts,' as I judge from the\n way Ary Renan talked to me the other night. He is son of the great\n Renan and is one of the editors of the 'Gazette des Beaux-Arts'\n and wished to meet me so much that Pallier, another critic, asked\n us to dine with him night before last. Pallier is the one who\n wrote the long article in the Liberte about me.\n\n \"You speak of Browning--I shall read the 'Ring and the Book,' but\n unless a man's style is clear I am too lazy and I have too little\n time to devote to digging gold out of the rocks, fine as it may\n be. On the other hand I got the Schopenhauer that Shiff spoke\n about with the intention of sending it to you, but it is so deadly\n in its pessimism, judging from the ten or eleven lines that I\n read, that I flung it away. It was so terribly true from his point\n of view, but what's the use of taking that point of view? We can't\n remedy matters by weeping and gnashing our teeth over the misery\n of things. 'That's the way things is' again, and although I have\n been told all my life it's best to put on a brave face and bear\n all cheerfully, it's only lately that it is really coming into my\n philosophy.\n\n \"It seems as if we are all in one open boat on the ocean,\n abandoned and drifting no one knows where, and while doing all\n we can to get somewhere, it is better to be cheerful than to be\n melancholy; the latter does not help the situation, and the former\n cheers up one's comrades.\n\n \"Michel, a friend of mine, had a beautiful nude marble bought for\n the Luxembourg, a pure noble chaste figure. There was a remarkable\n statuette by Gerome, two or three other good things in sculpture\n and the same among the objets d'art, and one swell thing in\n painting, the Puvis de Chavannes. _That_ appealed to me, but of\n course there were a lot of other very fine things, by Aman Jean,\n Henri Martin, Besnard and others. I send you some publications\n with the good things marked. I think if the Champs-Elysees were\n sifted there would be more good work found in it or as much as at\n the Champs de Mars. It is remarkable how much good work is done in\n Paris, but the first impression is bad, as the good is concealed\n in such a mountain of trash; but it's like gold in a mountain.\"\n\n \"Paris, July 24th\n\n \"Last night I dined with an old 'camarade d'atelier' at his home\n in the Cite Boileau at Passy and it was a great pleasure to be\n with him, one of the nicest kind of Frenchmen, a sculptor who\n is doing admirable work, a man of calm manners and large views,\n intensely interested in his work. His wife and three children\n are by the seaside, and on their return, if Homer does not go to\n America and I remain too, I'm looking forward to Homer's meeting\n his children. His boy, who is seventeen, is going to work in his\n atelier with him. It was delightful, as he took one through the\n rooms of his three children, to see the photographs of admirable\n works of art they had selected to hang on the walls. He has a\n house with a garden and we dined outside. (His name is) Lenoir\n and he is the son of a distinguished architect and grandson of a\n Lenoir whose bust is erected in the Cour des Beaux-Arts, a man\n of great distinction here on account of his love of art and his\n efforts to prevent the Revolutionists in 1795 from destroying the\n public monuments.\"\n\nEarly in August, while his wife was still away, Saint-Gaudens took his\nson Homer to Holland, where they had a delightful trip, extending to\nthe quaint dead cities of the north. Ten days or so after their return\nto Paris they made another successful expedition together to join some\nfriends at the sea-shore.\n\n \"3 rue de Bagneux, Paris, Aug. 26.\n\n \"It was intensely hot in Paris. I discovered that the Brushes were\n at Boulogne as well as the Proctors, so off we packed and we have\n had a great time, what with bathing and lolling all day on the\n cliffs, which I adore doing. The two Mears sisters followed us\n down there, and we, the Brushes, Proctors, Mears, babies, and all\n started off in the mornings, and, with the luncheon mixed up with\n the babies in the carriage, passed most delightful days, either on\n the cliffs or by the shore.\"\n\nSaint-Gaudens, however, could never be happy long away from his work,\nand he was soon writing from his studio again.\n\n \"Paris, Sept. 2d\n\n \"A Russian professor at one of the Universities here has sent me\n his translation of Tolstoi's last work 'What is Art?' and has\n asked me (with highly eulogistic terms about what I have done, in\n an inscription on the fly leaf) to give him my opinion, which he\n wishes to publish with those of other men of note. So I am in for\n reading it. You read it too, please, and tell me what you think\n of it, then I'll sign it and send it as my opinion! For I have\n no opinion, or so many that trying to put them into shape would\n result in driving me into the mad-house sooner than I am naturally\n destined to be there. Yes, 5000 different points of view that are\n possible. After all, we are like lots of microscopical microbes on\n this infinitesimal ball in space, and all these discussions seem\n humourous at times. I suppose that every earnest effort toward\n great sincerity or honesty or beauty in one's production is a drop\n added to the ocean of evolution, to the Something higher that I\n suppose we are rising slowly (d----d slowly) to, and all the other\n discussions upon the subject seem simply one way of helping the\n seriousness of it all.\n\n \"Shiff's letter that I enclose is in reply to one asking whether\n the professor's request was all right and whether I should bother\n about it. In answer he wrote that the Russian was a very serious\n man who had done admirable work. I once told Shiff that at times\n I thought that 'beauty must mean at least some goodness'--that\n explains part of his letter to me.\"\n\nTO BE CONCLUDED IN NOVEMBER\n\n [Illustration]\n\n\n\n\n [Illustration]\n\nTHURNLEY ABBEY\n\nBY PERCEVAL LANDON\n\n\nThree years ago I was on my way out to the East, and as an extra day\nin London was of some importance, I took the Friday evening mail train\nto Brindisi instead of the usual Thursday morning Marseilles express.\nMany people shrink from the long forty-eight-hour train journey\nthrough Europe, and the subsequent rush across the Mediterranean on\nthe nineteen-knot _Isis_ or the _Osiris_; but there is really very\nlittle discomfort on either the train or the mail-boat, and unless\nthere is actually nothing for me to do, I always like to save the extra\nday and a half in London before I say good-bye to her for one of my\nlonger tramps. This time--it was early, I remember, in the shipping\nseason, probably about the beginning of September--there were few\npassengers, and I had a compartment in the P. and O. Indian express to\nmyself all the way from Calais. All Sunday I watched the blue waves\ndimpling the Adriatic, and the pale rosemary along the cuttings; the\nplain white towns, with their flat roofs and their bold \"duomos,\" and\nthe gray-green gnarled olive orchards of Apulia. The journey was just\nlike any other. We ate in the dining-car as often and as long as we\ndecently could. We slept after luncheon; we dawdled the afternoon away\nwith yellow-backed novels; sometimes we exchanged platitudes in the\nsmoking-room, and it was there that I met Alistair Colvin.\n\nColvin was a man of middle height, with a resolute, well-cut jaw; his\nhair was turning gray; his mustache was sun-whitened, otherwise he was\nclean-shaven--obviously a gentleman, and obviously also a preoccupied\nman. He had no great wit. When spoken to, he made the usual remarks in\nthe right way, and I dare say he refrained from banalities only because\nhe spoke less than the rest of us; most of the time he buried himself\nin the Wagonlit Company's Time-table, but seemed unable to concentrate\nhis attention on any one page of it. He found that I had been over the\nSiberian railway, and for a quarter of an hour he discussed it with\nme. Then he lost interest in it, and rose to go to his compartment.\nBut he came back again very soon, and seemed glad to pick up the\nconversation again.\n\nOf course this did not seem to me to be of any importance. Most\ntravelers by train become a trifle infirm of purpose after thirty-six\nhours' rattling. But Colvin's restless way I noticed in somewhat marked\ncontrast with the man's personal importance and dignity; especially\nill suited was it to his finely made large hand with strong, broad,\nregular nails and its few lines. As I looked at his hand I noticed a\nlong, deep, and recent scar of ragged shape. However, it is absurd to\npretend that I thought anything was unusual. I went off at five o'clock\non Sunday afternoon to sleep away the hour or two that had still to be\ngot through before we arrived at Brindisi.\n\nOnce there, we few passengers transhipped our hand baggage, verified\nour berths--there were only a score of us in all--and then, after an\naimless ramble of half an hour in Brindisi, we returned to dinner at\nthe Hotel International, not wholly surprised that the town had been\nthe death of Virgil. If I remember rightly, there is a gaily painted\nhall at the International--I do not wish to advertise anything, but\nthere is no other place in Brindisi at which to await the coming of the\nmails--and after dinner I was looking with awe at a trellis overgrown\nwith blue vines, when Colvin moved across the room to my table. He\npicked up _Il Secolo_, but almost immediately gave up the pretense of\nreading it. He turned squarely to me and said:\n\n\"Would you do me a favor?\"\n\nOne doesn't do favors to stray acquaintances on Continental expresses\nwithout knowing something more of them than I knew of Colvin. But I\nsmiled in a noncommittal way, and asked him what he wanted. I wasn't\nwrong in part of my estimate of him; he said bluntly:\n\n\"Will you let me sleep in your cabin on the _Osiris_?\" And he a\nlittle as he said it.\n\nNow, there is nothing more tiresome than having to put up with a\nstable-companion at sea, and I asked him rather pointedly:\n\n\"Surely there is room for all of us?\" I thought that perhaps he had\nbeen partnered off with some mangy Levantine, and wanted to escape from\nhim at all hazards.\n\nColvin, still somewhat confused, said: \"Yes; I am in a cabin by myself.\nBut you would do me the greatest favor if you would allow me to share\nyours.\"\n\nThis was all very well, but, besides the fact that I always sleep\nbetter when alone, there had been some recent thefts on board these\nboats, and I hesitated, frank and honest and self-conscious as Colvin\nwas. Just then the mail-train came in with a clatter and a rush of\nescaping steam, and I asked him to see me again about it on the boat\nwhen we started. He answered me curtly--I suppose he saw the mistrust\nin my manner--\"I am a member of White's and the Beefsteak.\" I smiled\nto myself as he said it, but I remembered in a moment that the man--if\nhe were really what he claimed to be, and I make no doubt that he\nwas--must have been sorely put to it before he urged the fact as a\nguarantee of his respectability to a total stranger at a Brindisi hotel.\n\nThat evening, as we cleared the red and green harbor-lights of\nBrindisi, Colvin explained. This is his story in his own words:\n\n\"When I was traveling in India some years ago, I made the acquaintance\nof a youngish man in the Woods and Forests. We camped out together for\na week, and I found him a pleasant companion. John Broughton was a\nlight-hearted soul when off duty, but a steady and capable man in any\nof the small emergencies that continually arise in that department. He\nwas liked and trusted by the natives, and his future was well assured\nin Government service, when a fair-sized estate was unexpectedly left\nto him, and he joyfully shook the dust of the Indian plains from his\nfeet and returned to England. For five years he drifted about London.\nI saw him now and then. We dined together about every eighteen months,\nand I could trace pretty exactly the gradual sickening of Broughton\nwith a merely idle life. He then set out on a couple of long voyages,\nreturned as restless as before, and at last told me that he had decided\nto marry and settle down at his place, Thurnley Abbey, which had long\nbeen empty. He spoke about looking after the property and standing\nfor his constituency in the usual way. He was quite happy and full of\ninformation about his future.\n\n\"Among other things, I asked him about Thurnley Abbey. He confessed\nthat he hardly knew the place. The last tenant, a man called Clarke,\nhad lived in one wing for fifteen years and seen no one. He had been\na miser and a hermit. It was the rarest thing for a light to be seen\nat the Abbey after dark. Only the barest necessities of life were\nordered, and the tenant himself received them at the side-door. His\none half-caste man-servant, after a month's stay in the house, had\nabruptly left without warning, and had returned to the Southern States.\nOne thing Broughton complained bitterly about: Clarke had wilfully\nspread the rumor among the villagers that the Abbey was haunted, and\nhad even condescended to play childish tricks with spirit-lamps and\nsalt in order to scare trespassers away at night. He had been detected\nin the act of this tomfoolery, but the story spread, and no one, said\nBroughton, would venture near the house except in broad daylight. The\nhauntedness of Thurnley Abbey was now, he said with a grin, part of\nthe gospel of the countryside, but he and his young wife were going to\nchange all that. Would I propose myself any time I liked? I, of course,\nsaid I would, and equally, of course, intended to do nothing of the\nsort without a definite invitation.\n\n\"The house was put in thorough repair, though not a stick of the old\nfurniture and tapestry were removed. Floors and ceilings were relaid;\nthe roof was made watertight again, and the dust of half a century was\nscoured out. He showed me some photographs of the place. It was called\nan Abbey, though as a matter of fact it had been only the infirmary of\nthe long-vanished Abbey of Closter some five miles away. The larger\npart of this building remained as it had been in pre-Reformation days,\nbut a wing had been added in Jacobean times, and that part of the\nhouse had been kept in something like repair by Mr. Clarke. He had in\nboth the ground and the first floors set a heavy timber door, strongly\nbarred with iron, in the passage between the earlier and the Jacobean\nparts of the house, and had entirely neglected the former. So there had\nbeen a good deal of work to be done.\n\n\"Broughton, whom I saw in London two or three times about this time,\nmade a deal of fun over the positive refusal of the workmen to remain\nafter sundown. Even after the electric light had been put into every\nroom, nothing would induce them to remain, though, as Broughton\nobserved, electric light was death on ghosts. The legend of the Abbey's\nghosts had gone far and wide, and the men would take no risks. On the\nwhole, though nothing of any sort or kind had been conjured up even\nby their heated imaginations during their five months' work upon the\nAbbey, the belief in the ghosts was rather strengthened than otherwise\nin Thurnley because of the men's confessed nervousness, and local\ntradition declared itself in favor of the ghost of an immured nun.\n\n\"'Good old nun!' said Broughton.\n\n\"I asked him whether in general he believed in the possibility of\nghosts, and, rather to my surprise, he said that he couldn't say he\nentirely disbelieved in them. A man in India had told him one morning\nin camp that he believed that his mother was dead in England, as her\nvision had come to his tent the night before. He had not been alarmed,\nbut had said nothing, and the figure vanished again. As a matter of\nfact, the next possible dak-walla brought on a telegram announcing the\nmother's death. 'There the thing was,' said Broughton.\n\n\"'My own idea,' said he, 'is that if a ghost ever does come in one's\nway, one ought to speak to it.'\n\n\"I agreed. Little as I knew of the ghost world and its conventions,\nI had already remembered that a spook was in honor bound to wait to\nbe spoken to. It didn't seem much to do, and I felt that the sound\nof one's own voice would at any rate reassure oneself as to one's\nwakefulness. But there are few ghosts outside Europe--few, that is,\nthat a white man can see--and I had never been troubled with any.\nHowever, as I have said, I told Broughton that I agreed.\n\n\"So the wedding took place and I went to it in a tall hat which I\nbought for the occasion, and the new Mrs. Broughton smiled very nicely\nat me afterwards. As it had to happen, I took the Orient Express that\nevening and was not in England again for nearly six months. Just before\nI came back I got a letter from Broughton. He asked if I could see him\nin London or come to Thurnley, as he thought I should be better able\nto help him than any one else he knew. His wife sent a nice message to\nme at the end, so I was reassured about at least one thing. I wrote\nfrom Budapest that I would come and see him at Thurnley two days after\nmy arrival in London, and as I sauntered out of the Pannonia into the\nKerepesi Ut to post my letters, I wondered of what earthly service I\ncould be to Broughton. I had been out with him after tiger on foot,\nand I could imagine few men better able at a pinch to manage their own\nbusiness. However, I had nothing to do, so after dealing with some\nsmall accumulations of business during my absence, I packed a kit-bag\nand departed to Euston.\n\n\"I was met by a trap at Thurnley Road station, and after a drive of\nnearly seven miles we echoed through the sleepy streets of Thurnley\nvillage, into which the main gates of the park thrust themselves,\nsplendid with pillars and spread-eagles and tom-cats rampant atop of\nthem. From the gates a quadruple avenue of beech-trees led inwards for\na quarter of a mile. Beneath them a neat strip of fine turf edged the\nroad and ran back until the poison of the dead beech-leaves had killed\nit under the trees. There were many wheel-tracks on the road, and a\ncomfortable little pony trap jogged past me laden with a country parson\nand his wife and daughter. Evidently there was some garden party going\non at the Abbey. The road dropped away to the right at the end of the\navenue, and I could see the Abbey across a wide pasturage and a broad\nlawn thickly dotted with guests.\n\n\"The end of the building was plain. It must have been almost\nmercilessly austere when it was first built, but time had crumbled the\nedges and toned the stone down to an orange-lichened gray wherever it\nshowed behind its curtain of magnolia, jasmine, and ivy. Farther on\nwas the three-storied Jacobean house, plain and handsome. There had\nnot been the slightest attempt to adapt the one to the other, but the\nkindly ivy had glossed over the touching-point. There was a tall fleche\nin the middle of the building, surmounting a small bell tower. Behind\nthe house there rose the mountainous verdure of Spanish chestnuts all\nthe way up the hill.\n\n\"Broughton had seen me coming from afar, and walked across from his\nother guests to welcome me before turning me over to the butler's care.\nThis man was sandy-haired and rather inclined to be talkative. He\ncould, however, answer hardly any questions about the house: he had, he\nsaid, only been there three weeks. Mindful of what Broughton had told\nme, I made no inquiries about ghosts, though the room into which I was\nshown might have justified anything. It was a very large low room with\noak beams projecting from the white ceiling. Every inch of the walls,\nincluding the doors, was covered with tapestry, and a remarkably fine\nItalian fourpost bedstead, heavily draped, added to the darkness and\ndignity of the place. All the furniture was old, well made, and dark.\nUnderfoot there was a plain green pile carpet, the only new thing about\nthe room except the electric light fittings and the jugs and basins.\nEven the looking-glass on the dressing-table was an old pyramidal\nVenetian glass set in heavy repousse frame of tarnished silver.\n\n\"After a few minutes cleaning up, I went downstairs and out upon the\nlawn, where I greeted my hostess. The people gathered there were of\nthe usual country type, all anxious to be pleased and roundly curious\nas to the new master of the Abbey. Rather to my surprise, and quite to\nmy pleasure, I rediscovered Glenham, whom I had known well in old days\nin Barotseland: he lived quite close, as, he remarked with a grin, I\nought to have known. 'But,' he added, 'I don't live in a place like\nthis.' He swept his hand to the long, low lines of the Abbey in obvious\nadmiration, and then, to my intense interest, muttered beneath his\nbreath, 'Thank God!' He saw that I had overheard him, and turning to me\nsaid decidedly, 'Yes, thank God I said, and I meant I wouldn't live at\nthe Abbey for all Broughton's money.'\n\n\"'But surely,' I demurred, 'you know that old Clarke was discovered in\nthe very act of setting light to his bug-a-boos?'\n\n\"Glenham shrugged his shoulders. 'Yes, I know about that. But there is\nsomething wrong with the place still. All I can say is that Broughton\nis a different man since he has lived here. I don't believe that he\nwill remain much longer. But--you're staying here?--Well, you'll\nhear all about it to-night. There's a big dinner, I understand.' The\nconversation turned off to old reminiscences, and Glenham soon after\nhad to go.\n\n\"Before I went to dress that evening I had twenty minutes' talk with\nBroughton in his library. There was no doubt that the man was altered,\ngravely altered. He was nervous and fidgety, and I found him looking at\nme only when my eye was off him. I naturally asked him what he wanted\nof me. I told him I would do anything I could, but that I couldn't\nconceive what he lacked that I could provide. He said with a lustreless\nsmile that there was, however, something, and that he would tell me the\nfollowing morning. It struck me that he was somehow ashamed of himself,\nand perhaps ashamed of the part he was asking me to play. However, I\ndismissed the subject from my mind and went up to dress in my palatial\nroom. As I shut the door a draught blew out the Queen of Sheba from the\nwall, and I noticed that the tapestries were not fastened to the wall\nat the bottom. I have always held very practical views about spooks,\nand it has often seemed to me that the slow waving in firelight of\nloose tapestry upon a wall would account for ninety-nine per cent of\nthe stories one hears, and certainly the dignified undulation of this\nlady with her attendants and huntsmen--one of whom was untidily cutting\nthe throat of a fallow deer upon the very steps on which King Solomon,\na gray-faced Flemish nobleman with the order of the Golden Fleece,\nawaited his fair visitor--gave color to my hypothesis.\n\n\"Nothing much happened at dinner. The people were very much like those\nof the garden party. After the ladies had gone, I found myself talking\nto the rural dean. He was a thin, earnest man, who at once turned the\nconversation to old Clarke's buffooneries. But, he said, Mr. Broughton\nhad introduced such a new and cheerful spirit, not only into the Abbey,\nbut, he might say, into the whole neighborhood, that he had great\nhopes that the ignorant superstitions of the past were from henceforth\ndestined to oblivion. Thereupon his other neighbor, a portly gentleman\nof independent means and position, audibly remarked 'Amen,' which\ndamped the rural dean, and we talked of partridges past, partridges\npresent, and pheasants to come. At the other end of the table Broughton\nsat with a couple of his friends, red-faced hunting men. Once I noticed\nthat they were discussing me, but I paid no attention to it at the\ntime. I remembered it a few hours later.\n\n\"By eleven all the guests were gone, and Broughton, his wife, and I\nwere alone together under the fine plaster ceiling of the Jacobean\ndrawing-room. Mrs. Broughton talked about one or two of the neighbors,\nand then, with a smile, said that she knew I would excuse her, shook\nhands with me, and went off to bed. I am not very good at analyzing\nthings, but I felt that she talked a little uncomfortably and with a\nsuspicion of effort, smiled rather conventionally, and was obviously\nglad to go. These things seem trifling enough to repeat, but I had\nthroughout the faint feeling that everything was not square. Under the\ncircumstances, this was enough to set me wondering what on earth the\nservice could be that I was to render--wondering also whether the whole\nbusiness were not some ill-advised jest in order to make me come down\nfrom London for a mere shooting party.\n\n\"Broughton said little after she had gone. But he was evidently\nlaboring to bring the conversation round to the so-called haunting\nof the Abbey. As soon as I saw this, of course I asked him directly\nabout it. He then seemed at once to lose interest in the matter. There\nwas no doubt about it: Broughton was somehow a changed man, and to my\nmind he had changed in no way for the better. Mrs. Broughton seemed no\nsufficient cause. He was clearly very fond of her, and she of him. I\nreminded him that he was going to tell me what I could do for him in\nthe morning, pleaded my journey, lighted a candle, and went upstairs\nwith him. At the end of the passage leading into the old house he\ngrinned weakly and said, 'Mind, if you see a ghost, do talk to it; you\nsaid you would,' He stood irresolutely a moment and then turned away.\nAt the door of his dressing-room he paused a moment: 'I'm here,' he\ncalled out, 'if you should want anything. Good-night,' and he shut his\ndoor.\n\n\"I went along the passage to my room, undressed, switched on a lamp\nbeside my bed, read a few pages of the _Jungle Book_, and then, more\nthan ready for sleep, switched the light off and went fast asleep.\n\n * * * * *\n\n\"Three hours later I woke up. There was not a breath of wind outside.\nIt was so silent that my ears found employment in listening for the\nthrobbing of the blood within them. There was not even a flicker of\nlight from the fireplace. As I lay there, an ash tinkled slightly\nas it cooled, but there was hardly a gleam of the dullest red in the\ngrate. An owl cried among the silent Spanish chestnuts on the \noutside. I idly reviewed the events of the day, hoping that I should\nfall off to sleep again before I reached dinner. But at the end I\nseemed as wakeful as ever. There was no help for it. I must read my\n_Jungle Book_ again till I felt ready to go off, so I fumbled for\nthe pear at the end of the cord that hung down inside the bed, and I\nswitched on the bedside lamp. The sudden glory dazzled me for a moment.\nI felt under my pillow for my book with half-shut eyes. Then, growing\nused to the light, I happened to look down to the foot of my bed.\n\n * * * * *\n\n\"I can never tell you really what happened then. Nothing I could ever\nconfess in the most abject words could even faintly picture to you\nwhat I felt. I know that my heart stopped dead, and my throat shut\nautomatically. In one instinctive movement I crouched back up against\nthe head-boards of the bed, staring at the horror. The movement set my\nheart going again, and the sweat dripped from every pore. I am not a\nparticularly religious man, but I had always believed that God would\nnever allow any supernatural appearance to present itself to man in\nsuch a guise and in such circumstances that harm, either bodily or\nmental, could result to him. I can only tell you that at that moment\nboth my life and my reason rocked unsteadily on their seats.\"\n\nThe other _Osiris_ passengers had gone to bed. Only he and I remained\nleaning over the starboard railing, which rattled uneasily now and then\nunder the fierce vibration of the over-engined mail-boat. Far over,\nthere were the lights of a few fishing-smacks riding out the night, and\na great rush of white combing and seething water fell out and away from\nus overside.\n\nAt last Colvin went on:\n\n\"Leaning over the foot of my bed, looking at me, was a figure swathed\nin a rotten and tattered veiling. This shroud passed over the head, but\nleft both eyes and the right side of the face bare. It then followed\nthe line of the arm down to where the hand grasped the bed-end. The\nface was not that entirely of a skull, though the eyes and the flesh of\nthe face were totally gone, There was a thin, dry skin drawn tightly\nover the features, and there was some skin left on the hand. One wisp\nof hair crossed the forehead. It was perfectly still. I looked at it,\nand it looked at me, and my brains turned dry and hot in my head. I\nhad still got the pear of the electric lamp in my hand, and I played\nidly with it; only I dared not turn the light out again. I shut my\neyes, only to open them in a hideous terror the same second. The thing\nhad not moved. My heart was thumping, and the sweat cooled me as it\nevaporated. Another cinder tinkled in the grate, and a panel creaked in\nthe wall.\n\n\"My reason failed me. For twenty minutes, or twenty seconds, I was\nable to think of nothing else but this awful figure, till there came,\nhurtling through the empty channels of my senses, the remembrance that\nBroughton and his friends had discussed me furtively at dinner. The dim\npossibility of it being a hoax stole gratefully into my unhappy mind,\nand once there, one's pluck came creeping back along a thousand tiny\nveins. My first sensation was one of blind unreasoning thankfulness\nthat my brain was going to stand the trial. I am not a timid man,\nbut the best of us needs some human handle to steady him in time of\nextremity, and in this faint but growing hope that after all it might\nbe only a brutal hoax, I found the fulcrum that I needed. At last I\nmoved.\n\n\"How I managed to do it, I cannot tell you, but with one spring\ntowards the foot of the bed I got within arm's length and struck out\none fearful blow with my fist at the thing. It crumbled under it, and\nmy hand was cut to the bone. With the sickening revulsion after my\nterror, I dropped half-fainting across the end of the bed. So it was\nmerely a foul trick after all. No doubt the trick had been played many\na time before: no doubt Broughton and his friends had had some bet\namong themselves as to what I should do when I discovered the gruesome\nthing. From my state of abject terror I found myself transported into\nan insensate anger. I shouted curses upon Broughton. I dived rather\nthan climbed over the bed-end on to the sofa. I tore at the robed\nskeleton--how well the whole thing had been carried out, I thought--I\nbroke the skull against the floor, and stamped upon its dry bones. I\nflung the head away under the bed, and rent the brittle bones of the\ntrunk in pieces. I snapped the thin thigh-bones across my knee, and\nflung them in different directions. The shin-bones I set up against\na stool and broke with my heel. I raged like a Berserker against the\nloathly thing, and stripped the ribs from the backbone and slung the\nbreastbone against the cupboard. My fury increased as the work of\ndestruction went on. I tore the frail rotten veil into twenty pieces,\nand the dust went up over everything, over the clean blotting-paper and\nthe silver inkstand. At last my work was done. There was but a raffle\nof broken bones and strips of parchment and crumbling wool. Then,\npicking up a piece of the skull--it was the cheek and temple bone of\nthe right side, I remember--I opened the door and went down the passage\nto Broughton's dressing-room. I remember still how my sweat-dripping\npajamas clung to me as I walked. At the door I kicked and entered.\n\n\"Broughton was in bed. He had already turned the light on and seemed\nshrunken and horrified. For a moment he could hardly pull himself\ntogether. Then I spoke. I don't know what I said. Only I know that from\na heart full and over-full with hatred and contempt, spurred on by\nshame of my own recent cowardice, I let my tongue run on. He answered\nnothing. I was amazed at my own fluency. My hair still clung lankily\nto my wet temples, my hand was bleeding profusely, and I must have\nlooked a strange sight. Broughton huddled himself up at the head of\nthe bed just as I had. Still he made no answer, no defence. He seemed\npreoccupied with something besides my reproaches, and once or twice\nmoistened his lips with his tongue. But he could say nothing, though he\nmoved his hands now and then, just as a baby who cannot speak moves his\nhands.\n\n\"At last the door into Mrs. Broughton's room opened and she came in,\nwhite and terrified. 'What is it? What is it? Oh, in God's name! what\nis it?' she cried again and again, and then she went up to her husband\nand sat on the bed; and the two faced me in speechless terror. I told\nher what the matter was. I spared her husband not a word for her\npresence there. Yet he seemed hardly to understand. I told the pair\nthat I had spoiled their cowardly joke for them. Broughton looked up.\n\n\"'I have smashed the foul thing into a hundred pieces,' I said.\nBroughton licked his lips again and his mouth worked. 'By God!' I\nshouted, 'it would serve you right if I thrashed you within an inch\nof your life. I will take care that not a decent man or woman of my\nacquaintance ever speaks to you again. And there,' I added, throwing\nthe broken piece of the skull upon the floor beside his bed, 'there is\na souvenir for you, of your damned work to-night!'\n\n\"Broughton saw the bone, and in a moment it was his turn to frighten\nme. He squealed like a hare caught in a trap. He screamed and screamed\ntill Mrs. Broughton, almost as terrified as I, held on to him and\ncoaxed him like a child to be quiet. But Broughton--and as he moved I\nthought that ten minutes ago I perhaps looked as terribly ill as he\ndid--thrust her from him, and scrambled out of the bed on to the floor,\nand still screaming put out his hand to the bone. It had blood on it\nfrom my hand. He paid no attention to me whatever. In truth I said\nnothing. This was a new turn indeed to the horrors of the evening. He\nrose from the floor with the bone in his hand, and stood silent. He\nseemed to be listening. 'Time, time, perhaps,' he muttered, and almost\nat the same moment fell at full length on the carpet, cutting his head\nagainst the fender. The bone flew from his hand and came to rest near\nthe door. I picked Broughton up, haggard and broken, with blood over\nhis face. He whispered hoarsely and quickly, 'Listen, listen!' We\nlistened.\n\n\"After ten seconds' utter quiet, I seemed to hear something. I could\nnot be sure, but at last there was no doubt. There was a quiet sound\nas of one moving along the passage. Little regular steps came towards\nus over the hard oak flooring. Broughton moved to where his wife\nsat, white and speechless, on the bed, and pressed her face into his\nshoulder.\n\n\"Then the last thing that I could see as he turned the light out, he\nfell forward with his own head pressed into the pillow of the bed.\nSomething in their company, something in their cowardice, helped me,\nand I faced the open doorway of the room, which was outlined fairly\nclearly against the dimly lighted passage. I put out one hand and\ntouched Mrs. Broughton's shoulder in the darkness. But at the last\nmoment I too failed. I sank on my knees and put my face in the bed.\nOnly, we all heard. The footsteps came to the door, and there they\nstopped. The piece of bone was lying a yard inside the door. There was\na rustle of moving stuff, and the thing was in the room. Mrs. Broughton\nwas silent: I could hear Broughton's voice praying, muffled in the\npillow: I was cursing my own cowardice. Then the steps moved out again\non the oak boards of the passage, and I heard the sounds dying away. In\na flash of remorse I went to the door and looked out. There at the end\nof the corridor was a small bowed figure in a gray veil--I knew it only\ntoo well. But this time there was a pathos in the drooped head that\nleft me standing with my forehead bowed in shame against the jamb of\nthe door.\n\n\"'You can turn the light on,' I said, and there was an answering flare.\nThere was no bone at my feet. Mrs. Broughton had fainted. Broughton was\nalmost useless, and it took me ten minutes to bring her to. Broughton\nonly said one thing worth remembering. For the most part he went on\nmuttering prayers. But I was glad afterwards to recollect that he had\nsaid that thing. He said in a colorless voice, half as a question,\nhalf as a reproach, 'You didn't speak to her.'\n\n\"We spent the remainder of the night together. Mrs. Broughton actually\nfell off into a kind of sleep before dawn, but she suffered so horribly\nin her dreams that I shook her into consciousness again. Never was dawn\nso long in coming. Three or four times Broughton spoke to himself. Mrs.\nBroughton would then just tighten her hold on his arm, but she could\nsay nothing. As for me, I can honestly say that I grew worse as the\nhours passed and the light strengthened. The two violent reactions had\nbattered down my steadiness of view, and I felt that the foundations of\nmy life had been built upon the sand. I said nothing, and after binding\nup my hand with a towel, I did not move. It was better so. They helped\nme and I helped them, and we all three knew that our reason had gone\nvery near to ruin that night. At last, when the light came in pretty\nstrongly, and the birds outside were chattering and singing, we felt\nthat we must do something. Yet we never moved. You might have thought\nthat we should particularly dislike being found as we were by the\nservants: yet nothing of the kind mattered a straw, and an overpowering\nlistlessness bound us as we sat, until Chapman, Broughton's man,\nactually knocked and opened the door. None of us moved. Broughton,\nspeaking hardly and stiffly, said: 'Chapman, you can come back in\nfive minutes.' Chapman was a discreet man, but it would have made no\ndifference if he had carried his news to the 'room' at once.\n\n\"We looked at each other and I said I must go back. I meant to wait\noutside till Chapman returned. I simply dared not re-enter my bedroom\nalone. Broughton roused himself and said that he would come with me.\nMrs. Broughton agreed to remain in her own room for five minutes if the\nblinds were drawn up and all the doors left open.\n\n\"So Broughton and I, leaning stiffly one against the other, went down\nto my room. By the morning light that filtered past the blinds we could\nsee our way, and I released the blinds. There was nothing wrong in the\nroom from end to end, except smears of my own blood on the bed, on the\nsofa, and on the carpet where I had torn the thing to pieces.\"\n\nColvin had finished his story. There was nothing to say. Seven bells\nstuttered out from the fo'c'sle, and the answering cry wailed through\nthe darkness. I took him downstairs.\n\n\"Of course I am much better now, but it is a kindness of you to let me\nsleep in your cabin.\"\n\n\n\n\nTHE TERROR\n\nBY A. E. THOMAS\n\nILLUSTRATIONS BY HERMAN C. WALL\n\n\nIt was a gray and bitter morning in January when Tim first saw The\nVale. For weeks winter had lain heavy upon the sunny South. A cold rain\nhad swept the countryside; then came zero weather for days, till the\nice lay inch-thick on all the broad pikes of Lexington County, and only\nthe firs were green.\n\nTim and his mother had left the little cabin they called home at the\nfirst crack of dawn and together had tramped the five miles that\nspelled the road to The Vale. All the way they spoke scarce a word, for\nthey knew that parting was near and that it had to be. Colonel Darnton\nwas to take the boy and make a jockey of him, if he could, and the\nstables of The Vale were to be his home thereafter.\n\nThe s were feeding the stallions when the boy and his mother\ntrudged up to the big barn. They sat on a feed-box until the Colonel\nhad finished his breakfast and come out from the big house under the\ntrees.\n\n\"Morning to you, Mrs. Doolin,\" said the Colonel. \"And so you've brought\nthe boy, eh?\"\n\n\"I have that,\" responded Mrs. Doolin, in her odd mixture of brogue and\nSouthern drawl. \"An' I beg ye t' be good tew him. Since Pete died, he's\nall I hov, an' it's the good lad he's been to me, an' phwat it is I'll\nbe doin' widout him whin he's gawn, I dinnaw. Will ye be afther lettin'\nhim come down t' see me wanst a fortnight, sor?\"\n\n\"Of course I will,\" smiled the Colonel, and then he turned to Tim,\nstanding there, so pale and little.\n\n\"And you, boy,\" he said, taking the lad's chin in his big hand and\nturning the blue eyes up to his gaze, \"how about you--strong for the\nhosses, eh?\"\n\nTim's lip quivered. He was only twelve. But he looked the Colonel\nbravely in the face.\n\n\"I reck'n,\" he said.\n\n\"Well, well, we'll see,\" said the Colonel, mercifully releasing the\nboy's chin. \"'Twould be odd if you weren't. Your father was mighty\nhandy with 'em all--mighty handy.\"\n\n\"Savin' yer prisince, Colonel, I'd hov jist wan wurrud wid th' boy,\"\nsaid the woman, and she drew Tim aside.\n\n\"Lookee yew here, yew Tim Doolin,\" she said, when she had him by\nhimself, \"don't yew niver fergit thet yew're up here tew The Vale tew\nlarn hosses. Raymimber thet.\" The boy drew one ragged sleeve across his\nblue eyes.\n\n\"All right, maw,\" he quavered.\n\n\"An' raymimber this, too,\" she went on. \"There niver yit was wan Doolin\nthet wasn't on the square. Hoss racin' ain't prayin', an' all them as\nraces hosses ain't like the Colonel. But there niver was wan Doolin yit\nthet wasn't on the level. Mind yew ain't the fust crook in the clan, er\nelse yew needn't niver come home t' the Blue Grass ter look yewr maw in\nthe face.\"\n\nThin and gaunt and gray-haired, she stood in the biting wind that\nfought to tear her shawl from her bony shoulders. For a moment she\nstared, stern and dry-eyed, at the boy. Somehow he had never seemed so\ntiny before.\n\n\"Will yew raymimber thet?\" she demanded at last. Tim dropped his eyes\nin boyish embarrassment.\n\n\"I reck'n,\" he said.\n\nHis mother drew her shawl tightly about her shoulders and departed\nwithout more ado.\n\nThe life of a stable-boy on a great breeding-farm is not all beer and\nskittles, whatever that may be. His principal business is to look sharp\nand do as he is told and never forget. It's always early to rise,\nbefore dawn in the winter time, and often late to bed, if some of the\npriceless thoroughbreds are ailing. Moreover, the tongues of stable\nforemen are sharp, and their hands are heavy.\n\nTim made his mistakes. Once, after they came to trust him at The Vale,\non a sharp morning when he was giving King Faraway, the head of the\nstud, his morning gallop on the pike, he fell to dreaming. A little\nbrook ran under a wooden bridge built for carriage use. But to one\nside there was a ford through which people drove in summer to give\ntheir horses drink. The brook was solid ice that morning, but Tim, not\nthinking, turned King Faraway into the ford. The great horse slipped\nand fell.\n\nTim sprang up from the far side of the brook with the blood gushing\nfrom a nasty cut on his forehead. But he didn't think of that. Was King\nFaraway hurt?\n\nHe walked the three miles back to The Vale, the stallion limping behind\nhim, and at the stable he told the truth and got a thrashing.\n\nKing Faraway was on three legs for a month. But he recovered. Every\nnight of that month the boy slept on a heap of straw in the stallion's\nbox stall, waking up half a dozen times a night to rub the injured\nstifle; and in the end the great horse was as good as new.\n\nAgain, one chilly November night Tim left one of his yearlings out in\nthe South Paddock. Late that night a cold, driving storm came up. In\nthe morning they found the yearling shivering by the paddock gate.\nThe Colonel himself worked his fingers off over that yearling colt,\nfor he was bred in the purple. The youngster had pneumonia, but they\nsaved him, and the Colonel said that Tim's nursing was what pulled him\nthrough.\n\nOn an April morning something over two years after the day Tim came\nto The Vale, he started with the season's two-year-olds for the big\ntracks at New York. He had helped break the youngsters to the saddle\nand to the track on the half-mile race-course on the farm, and he knew\nevery one of the lot as if he had been its mother. So when they rounded\nthem up to take them to the special box-cars that were waiting in the\nfreight yards, the Colonel took the lad aside.\n\n\"Really want to be a jockey, Tim?\" he asked.\n\n\"Sure,\" said Tim.\n\n\"Want to leave us, then, eh?\" The boy looked away, and the Colonel\nspared him.\n\n\"All right,\" he said with a laugh. \"To the races you go. You can come\nback if you don't like it.\"\n\nAll the broad acres of The Vale and the costly stallions and the brood\nmares belonged to David Holland, a captain of finance. He was too busy\nmanipulating the ticker to pay much attention to the stock-farm itself.\nHe knew nothing whatever about the breeding of horses and was clever\nenough to admit it. He paid the bills and got his fun out of \"seeing\n'em run.\"\n\nThe Holland stable was already quartered at Sheepshead Bay when the\nColonel and Tim arrived with the two-year-olds. Pat Faulkner, the\ntrainer, was there to meet them. He and the Colonel drew aside and\nleft the boy to himself. The hours for morning gallops were long since\nover, and when Tim climbed the white rail fence that enclosed the\nback-stretch, the big and beautiful track was absolutely deserted.\n\n\"Well,\" said Faulkner, \"what sort of a grist have you brought me this\ntrip? I've been bitin' me nails off to find out, but not a word would\nyou write.\"\n\nThey had out the chestnut colt with the one white foot, and the black\nwith the white blaze, and the bay filly by Checkers-Flighty, and a few\nother individuals, while the trainer felt them over and looked them up\nand down and round about, and had them walked and trotted and cantered\nthrough the stable yard.\n\nWhen it was all over, and he knew that here was material that would\nmake his rivals sit up, Faulkner's eyes fell upon a slim shape sitting\non the white rail fence.\n\n\"What's the kid?\" he demanded.\n\n\"That?\" said the Colonel, with a smile, \"why, that's Tim Doolin, a\nchampion jockey I've brought you.\" The trainer grunted.\n\n\"How old?\" he asked.\n\n\"Going on fifteen, weighs seventy-three pounds, is kind and clever,\nknows the hosses, and they'll do for him. Try him out at exercise work,\nand if he makes good, give him a chance to ride.\"\n\nThat same night the Colonel departed.\n\nAfter that Tim's work was cut out for him. There were twenty-six\ntwo-year-olds in the Holland stables, twelve three-year-olds, and six\nor eight thoroughbreds in the aged division. Faulkner kept a big staff\nof grooms and exercise boys, but there was always a day's work for each\nof them. Aside from the routine exercise for every horse in training,\nthe feeding, the grooming, and so on, all the youngsters had to be\nbroken to the starting barrier. Some trainers didn't pay much attention\nto that.\n\n\"Let 'em come to it in their races,\" said they. Not so, Faulkner. He\ndrilled every last one of his two-year-olds till the starting gate was\nno more to them than so much steel and wood and webbing.\n\nTim was not long in winning the trainer's confidence. The job of\nbreaking to the barrier was turned over to the stable foreman, under\nwhose eyes the grooms and exercise boys worked. But one afternoon\nFaulkner himself came out to see how things were going. He noticed that\nthe three two-year-olds that were Tim's especial care were already\nbarrier-broken. He cross-examined the lad. Tim was reticent.\n\n\"I--I--jest get 'em used to it,\" he faltered.\n\n\"How?\" demanded the trainer.\n\n\"I--I jest lead 'em up to it, first along, an' let 'em smell of it\nand look at it. Then I git one of the boys to spring it while I'm\na-standin' by at their heads. They git used to it pretty soon. Then I\nride 'em up to it.\"\n\n\"Humph!\" grunted the trainer; but later he said to the foreman: \"That\nkid's got sense.\"\n\nIt wasn't long before Tim was exercising three-year-olds, and one gray\nmorning when he turned out of the loft where he slept, the foreman\nshouted:\n\n\"Hurry up, you Tim, an' git yer breakfast.\"\n\nThe boy wondered and obeyed. He gulped down the last of his oatmeal,\nshot out of the training kitchen, and ran up to the stables, where a\n groom was holding a big bay horse, about which Faulkner himself\nwas busily working. The trainer arose as the boy ran up.\n\n\"Up you go, kid,\" he said and tossed Tim into the saddle.\n\nAnd Tim knew that he was to exercise Lear! And everybody knew that the\nHolland stable was pointing Lear for the Brooklyn Handicap! It was a\nproud moment for Tim. But his honors didn't sit too heavily on his\nsmall shoulders, for Faulkner was a hard task-master.\n\n\"Jog him to the mile post and send him the last half in .55 an' keep\nyer eye on the flag,\" the trainer would order.\n\nThen the boy would canter away through the gray light, and the trainer,\nhandkerchief in one hand and stop-watch in the other, would mount the\nfence. If the clock said .57 for that last half mile, or anything\nbetween that and .55, there was a slap on the back and a \"Good kid,\"\nfor Tim, but woe to him if the clicking hand cut it down to .53.\n\nMistakes he made, and many of them, but they grew fewer and fewer. Good\nhands he had (for they are born with a boy, if he's ever to have them)\nand an intuitive knowledge of the temper of a horse. A good seat they\nhad taught him at The Vale. And gradually, little by little and bit by\nbit, he came to be what only one jockey in fifty ever grows into--an\nunerring judge of pace.\n\nJust what it is that tells a boy whether the muscles of steel that he\nbestrides are shooting him rhythmically over a furlong of dull brown\nearth or black and slimy mud in .12-1\/2 or .13-1\/4, some person may\nperhaps be able to tell, but certain it is that no person ever has told\nit. Long after Tim had learned the secret as few boys have ever known\nit, I asked him.\n\n\"Why,\" said he, \"yew know your hoss, an' after thet, why, yew jest feel\nit.\"\n\nIt was not until the autumn meeting at Gravesend that Tim first wore\nthe colors. It was in an overnight selling race for two-year-olds, for\nwhich Faulkner had in despair named Gracious.\n\nGracious was a merry little short-bodied filly, who was bred as well as\nany of the Holland lot, but who hadn't done well. Out of six starts she\nhad never shown anything, and Faulkner had determined to start her once\nmore and then weed her out. The weight, eighty-seven pounds, was so\nlight that the stable jockey couldn't make it. Then Faulkner remembered\nthe Colonel's words: \"Give him a chance, if he makes good.\"\n\n\"I'll do it,\" he said, and told Tim.\n\nTim didn't sleep well that night, and with wide eyes he welcomed the\nfirst light of the great day. At last he was to wear the colors!\n\n\"Just get her off well and take your time,\" said Faulkner, as he put\nthe boy up. \"Rate her along to the stretch and then drive her.\"\n\nTim did all that. Coming into the stretch, there were four horses\nahead of him on the rail. But two of them were weakening. Then Tim\ncalled on the filly. She answered and went up. But the colt next her\nwas staggering. He swerved, and Tim had to pull out. He got Gracious\ngoing again and landed her third, only a head behind the second horse.\nFaulkner was radiant as Tim dismounted.\n\n\"Good kid,\" he said. He had backed the filly a bit to run third. But\nTim was almost weeping.\n\n\"I could have won,\" he moaned, \"if thet there Blinger hed kep'\nstraight.\"\n\nThe boy rode half a dozen races in the next month, all of them for\ntwo-year-olds. He won once and was second twice. Among the other\napprentice riders he was already a personage, although, of course, he\nscarcely dared speak to the full-fledged jockeys.\n\nAnd then the Terror came.\n\nIt was Gracious that brought it. There were eight two-years-olds in the\nseven-furlong sprint on the main track at Morris Park. The filly had\ngone slightly off her feed the night before the race, but she seemed\nperfectly fit otherwise, and Faulkner determined to start her.\n\n\"She won't finish as strong as she would a week ago,\" he told the boy,\nas the saddling bugle blew. \"So you send her along a bit at the start\nand get the rail. Keep her goin' an' let her die in front.\"\n\n\"I reck'n,\" said Tim confidently, and they swung him into the saddle.\n\nGracious, under Tim's riding, was a quick breaker. She leaped away the\ninstant the barrier rose, and from the middle of the track the boy took\nher to the rail before the run up the back-stretch was over. She held\nher lead till the field had rounded into the stretch, and then he felt\nher falter. In an instant he began to ride, first with hands, then with\nhands and feet, then with hands and feet and whip. But it was not in\nthe filly to answer. At the six-furlong pole she had gone stale--gone\nstale between two jumps. But the boy kept at her with might and main.\n\n [Illustration: \"TIM AND HIS MOTHER HAD LEFT THEIR LITTLE CABIN AT THE\n FIRST CRACK OF DAWN\"]\n\nIt was useless. In six strides a brown muzzle crept up to his saddle\ngirth. In two jumps more it reached the filly's shoulder. In three more\nstrides the two were head and head; and then the brown muzzle was in\nfront.\n\nSuddenly the brown muzzle drooped, and the colt faltered. Tim took\nheart again. Perhaps, perhaps he might still nurse the filly home in\nfront. He gripped her withers a bit tighter with his knees and spoke to\nher, softly and pleadingly, as was his wont, through his clenched teeth:\n\n\"Come on, yew gal--come on, yew baby--come jes' once mo'--jes'\nonce--we's mos' home now--come--come. Come, yew gal!\"\n\nBack to the boy's stirrup came the saddle girth of the brown colt, as\nhis stride shortened under the staggering drive. Tim's heart leaped in\nhis bosom, for there was the wire not ten jumps away and--he was going\nto win.\n\n\"Come--come, yew baby,\" he whispered almost into the filly's ear, as he\nleaned far over her nodding head. The ecstasy of victory thrilled his\nsmall body to his very toes.\n\nAt that instant the brown colt swerved against him. The pungent odor\nof sweating horseflesh smote his nostrils--the roar of a horrified\ncrowd filled his ears--the track rose up to meet him. A flash of red\nenveloped his brain--then came darkness and oblivion.\n\nWhen he came to himself, the first faint light of dawn was sifting in\nthrough a window somewhere. \"Time I was up fer exercisin',\" he thought,\nand he struggled to rise. A flash of pain in his left arm turned him\nfaint and sick. As he wondered over this, he became aware of a dull,\nsteady roar that filled the room.\n\nAgain he opened his eyes. Dimly he made out the form of a white-capped\nwoman standing over him. Then he knew that he was not lying in the\nloft at Sheepshead Bay.\n\n\"Are you awake, little boy?\" said a soft voice.\n\n\"I--I reck'n,\" said Tim faintly.\n\nThere came the rattle of a heavy vehicle pounding over pavements, the\nshrill shriek of a whistle, the roar of horses' hoofs.\n\nThen he remembered it all and turned his face to the wall.\n\nThat same evening Faulkner came in to see him.\n\n\"Well, Tim,\" he said, \"'twas a bad tumble, hey? How d'you feel? better?\"\n\n\"Sure,\" said the boy feebly.\n\n\"That's fine, that's fine,\" cried the trainer heartily. \"'Twa'n't your\nfault. You done fine. You'd 'a' won, sure, 'f that chump Reilly had\nkep' his colt straight. But don't you care. We'll have you out in a few\ndays, the Doc says. I telegraphed the Colonel you was all to the good,\nan' he'll tell yer ma, so don't you worry about that, kid.\" He leaned\nover, smiled kindly, and put a huge hand on the boy's head.\n\nIt smelled horribly of sweaty horseflesh. With a shudder Tim turned his\nhead away.\n\n\"You musn't mind a little thing like a tumble,\" said the trainer\nanxiously. \"They all get 'em. Why, I remember when I was ridin' a hoss\nnamed ----\"\n\nAnd the kindly horseman blundered on in an attempt to cheer the\nhelpless lad. It seemed to Tim that he simply must cry out to him to\nstop, when the nurse came swiftly up and warned the trainer not to stay\nany longer.\n\n\"Well, so long, kid,\" was Faulkner's parting word. \"Oh, 'course yer\nbusted arm won't let yer ride again this fall, but the season's most\nover anyway. Only two more days o' Morris Park, and y' know we ain't\ngot any cheap ones to start at Aqueduct. Anythin' I kin do f' you?\" Tim\nopened his eyes again.\n\n\"Filly hurted?\" he asked faintly.\n\nThe trainer laughed.\n\n\"Nothin' to hurt,\" he said. \"Skinned her knees a bit, but I was goin'\nto put her out o' trainin' anyhow. She's O.K.\"\n\nTo Tim's unspeakable relief he lumbered away.\n\nWith his arm in a sling, Tim was out again at the end of a week.\nMuch against the boy's will, Faulkner took him one day to the\nmeeting at Aqueduct. There the trainer was soon surrounded by\nprofessional colleagues, and Tim fled to a seat in the highest row\nof the grandstand. Thence he looked down upon the first stages of a\nsix-furlong sprint, but when three horses labored home in a tight-fit\nfinish he buried his face in his hands that he might not see them.\n\nWhen he lifted his face again, he glanced furtively about, thankful,\noh, so thankful, that nobody had noticed him.\n\nThen self-scorn descended upon him. If he could only go away somewhere\nand die! Furtively, he wept, wiping the tears away with one pudgy,\nbrown fist. For some minutes he stared, heavy-eyed and broken, at his\nfeet.\n\n\"Ta-ra-ta-ta-ta! Ta-ra-ta-ta!\"\n\nThe bugle spoke, calling the handicap horses to the post.\n\nTim started up and edged toward the aisle. His racing feet carried him\nin panic half way down to the lawn. One idea possessed him--to get\naway--to hide himself, he didn't care where--anywhere where he couldn't\nsee the horses run.\n\nA hand seized him by the shoulder and spun him around.\n\n\"Hey, kid,\" said a voice, \"how you feelin'? All to the mustard, hey?\"\n\nIt was Bud Noble, star jockey of the Holland stable, radiant with all\nthe prestige that comes with twenty thousand a year and the adulation\nof the racing public.\n\n\"I reck'n,\" said Tim, and fled again.\n\nHe had no notion of flight. His feet bore him along unsentiently.\nSuddenly they stopped. And then he knew that he couldn't run away.\nHe must see that race. Something within him that would not be denied\ncommanded it. Slowly he retraced his steps, muttering unconsciously: \"I\ngotter do it. I gotter do it.\"\n\nPresently he found himself back in the top row of the grandstand. As in\na dream, he watched the parade of brilliant colors to the post. As in\na dream, he saw the barrier flash up. The old-time roar \"They're off!\"\ncame faint and faraway to his ears. Dreamlike, the field drifted up the\nback stretch, rounded the turn, and straightened out for home. He dug\nthe fingers of his one good hand into the hard wooden bench and held\nhis eyes upon the horses.\n\n\"I gotter do it. I gotter do it,\" he muttered still.\n\nThey were years in reaching the wire. No mortal thoroughbreds ever\nran so slowly before since time began. But at last, at the end of the\nworld, they finished. And up on the highest bench of the grandstand a\nlittle boy, with white face and wide eyes, sat back, limp and still.\n\nTim's arm was still in a sling when he got back to Lexington, and it\nwas January before he could use it to any effect. The intervening weeks\nhe spent at home, helping his mother as best he could in the round of\nher hard life, running her errands and bearing to and fro the various\nwashings by which she lived. For the first time in his life it worried\nhim to see her work so hard.\n\n [Illustration: \"A GROOM WAS HOLDING A BIG BAY HORSE, ABOUT WHICH\n FALKNER WAS BUSILY WORKING\"]\n\n\"Nivver mind, Tim,\" she would say, lifting her bent back from the tub\nin the corner of the kitchen, \"soon you'll be the famous jockey wid\nthousands a year. Thin it's your ould mother that'll be wearin' the\nfine duds and wurruk no more.\"\n\nAnd then the boy, sick with shame and fear, would steal from the\nhouse--anywhere to be out of the sight of her and the sound of her\nvoice.\n\nSometimes the Terror would grip him in his sleep, in the middle of the\nwinter night, when the wind shrieked under the shingles on the cabin\nroof or the cold rain drove against the window-pane. More than once he\nstarted up, broad awake, with the smell of sweating horseflesh sharp\nand agonizing in his nostrils. Once it was the sound of his own voice\nthat woke him, and he was crying out:\n\n\"Come on, yew baby, come, come, yew gal!\"\n\nThen he sat on the edge of his cot, with the blanket over his\nshoulders, until daybreak, with such thoughts as a boy may know.\n\nBut on a sunny morning in February, it was Tim who stood in the great\ndoorway of the stallion stable at The Vale, saying to the Colonel:\n\n\"Thought mebbe I could help yew with the two-year-olds.\"\n\nDay by day he strove with himself. Little by little he fought the\nTerror down. The very smell of the stables turned him faint for a week.\nHe used to creep into King Faraway's box-stall when the big horse\nstood, wet under his blanket, after his morning gallop, and bury his\nface in the stallion's mane and rub his nose along the giant withers,\ntill at last the horrible smell of sweating horseflesh had power\nto terrify him no more. It was weeks before he could mount without\ntrembling, but at last he came to do it and--to hope.\n\nAt last came April, and one evening, as Tim was helping with the\nfeeding, he heard the Colonel's voice calling him. He trembled a\nlittle, for he knew what was coming.\n\n\"I've a letter from Faulkner,\" said the Colonel, \"and he's asking\nfor you, Tim. Shall I tell him you'll be up with the new batch of\nyoungsters?\" It was the cast of the die.\n\n\"I reck'n,\" said Tim stoutly.\n\nBut it wasn't quite the same old Sheepshead Bay that Tim went back to.\nHe did his work as faithfully and skilfully as ever. His hand was just\nas light and sure; he had not lost his sense of pace. But the first\npale light of day did not send him out to the stables with every nerve\nin his lithe body tingling for very joy of the work that was coming.\nAnd once, when he saw a stable-boy thrown--the Terror rose at him\nagain; not with the old terrible leap, to be sure, but he saw Its face\nfor an instant.\n\nHe will never forget his first race that spring. Again he rode a\ntwo-year-old, and he won without difficulty, nobody guessed at what\nexpense. As the season went on, he rode again and again, and sometimes\nhe won, and oftener not.\n\nBut Faulkner saw and shook his head. If Tim's horse won, it was because\nits own speed and the judgment of its rider did it. Nobody ever saw Tim\ntake a chance. Other boys might leave him space to squeeze through if\nthey liked. He never did it. It was the longest way 'round and plain\nsailing for Tim. No mad, brilliant rush for the rail. No fine finishes\nfrom unlucky beginnings.\n\nAnd Faulkner watched and saw it all. Once the boy caught the trainer\nlooking at him, thoughtful and puzzled. A big lump rose in his throat\nand strangled him, and he stumbled away with his grief. It seemed to\nhim that he could not live on any longer. He grew even more grave and\nsilent as the days went on, shunned the other stable-boys, and kept\nstolidly to himself.\n\nIt had to end sometime, somehow, and the ending of it was\nnotable--because Tim was Tim, I suppose.\n\nFor the Suburban Handicap, with the Brooklyn the greatest of the\nclassic races for the older horses, the Holland stable had two\ncandidates. The first was the five-year-old Gladstone, son of Juniper\nand winner of fifteen races, one of them a Metropolitan. The second\nwas Kate Greenaway, a three-year-old filly by King Faraway, whose only\nclaim to distinction was that she had won third place in the Futurity\nof the preceding year. But, though Gladstone was the stable's main\nreliance, the filly's work had been dazzling, and the shrewd Faulkner\nhad hopes of her.\n\nBud Noble, as stable jockey, was to ride Gladstone, while the trainer\nrelied on the light-weight Ban Johnson, on whom the stable had second\ncall, to handle Kate Greenaway. Tim knew the filly as no one else knew\nher or could know her. Down at The Vale, before ever he came to the\nraces, he had been the first to put halter and bridle on her; his small\nlegs were the first to bestride her; he had broken her to the barrier\nuntil she seemed actually to like the thing, and in her work she had\nbeen his especial charge. But he had never ridden her in a race.\n\nThe running of a big handicap at a Metropolitan track is an impressive\nevent, even to the man who knows nothing of horses. To him who loves\nthe thoroughbred it is inspiring. To Tim it was something more than\nthat--a thing to make you tremble.\n\nAll morning the boy hung uneasily about the stable. He ate scarcely\nany dinner and roved restlessly about until it was time to take the\nfilly to the paddock. He got her there just as the horses were going to\nthe post for the third race. The Suburban was the fourth. Up and down\nunder the great shed he walked his charge, blanketed and hooded, in the\nwake of towering, black Gladstone. Soon a shouting from the grandstand\nannounced that the third race was over.\n\nThen came a rush of hundreds to see the Suburban horses saddled. One\nby one, the candidates filed out to the track for their warming-up\ngallops--Boston, top-weight, favorite and winner of the Metropolitan,\nand second in the Brooklyn; Carley, winner of the Advance the season\nbefore; Catchall, the speedy Hastings mare; and all the rest--all save\nKate Greenaway. Once, in a warming-up gallop, she had run away, and\nFaulkner would never take chances with her after that. So Tim walked\nher up and down by herself, thankful, yet ashamed, that somebody else\nwas to ride her.\n\nSuddenly the stable foreman ran up.\n\n\"Hi, you Tim,\" he shouted, \"hustle over to the dressin' room an' git on\nyer duds. Skin along, now, no time to lose.\"\n\nTim stood gaping.\n\n\"Git a move on--git a move! My Gawd! You ain't got no time to lose.\nBan's fell down an' sprained his ankle.\"\n\nTim trudged over to the jockey's house, his eyes on the ground. Over in\nthe paddock, Faulkner listened stubbornly to the foreman.\n\n\"I tell you,\" the latter was saying, \"the kid's lost his nerve. Ain't\nyou seen it all along? He ain't took a chance sence his tumble. Why\ndontcher give the mount to Tyson or Biff Barry? They ain't neither of\n'em got a mount.\"\n\n\"Nothin' doin',\" rejoined the trainer. \"The kid knows the\nfilly--brought her up, almost. He can ride, too, if he don't get in\na tight place, an' that ain't likely. Tyson can't make the weight.\nB'sides, I told the Colonel I'd give the kid a chance. An',\" he\nconcluded, \"this is it.\"\n\n\"All right,\" said the foreman, \"but you'll see. He's lost his nerve.\nWhy, he got white eraoun' the gills when I tol' him.\"\n\n [Illustration: \"HE SAT ON THE EDGE OF HIS COT, WITH THE BLANKET OVER\n HIS SHOULDERS, UNTIL DAYBREAK\"]\n\nTim had grown like a weed since he first saw Sheepshead Bay, but it\nwas a slender, fragile figure that the trainer tossed into the chestnut\nfilly's saddle when the bugle blew.\n\n\"Now, kid,\" said Faulkner quietly, throwing one arm over the crupper,\n\"you're third from the rail. You know the filly as well as I do. She's\nfit to the minute. She'll run in 2.03, if she ain't rushed in the first\nhalf. Hold yer place an' let the sprinters do their sprintin'. They'll\ncome back. Keep her goin' her pace for a mile, an' if you have to ride\nher the last quarter, make her sweat for it. She's game fer a drive.\nThey don't make 'em no gamer.\"\n\nThe lad heard scarcely a word. He wasn't frightened. He was sullen,\nrebellious against--against everything. It was one more race to\nhim--commonplace, perfunctory, tiresome. He was going to get through\nwith it in the easiest way he could. He thought with relief of the wide\nspaces and easy turns of the great track.\n\n\"Keep up yer nerve, kid,\" said Bud Noble, turning in his saddle and\nlooking back at Tim as the field filed through the paddock gate.\n\nTim grinned scornfully. What a notion! Why should anybody need nerve\nto gallop a horse around a track? He had only one idea--to keep out\nof trouble. So, perfectly calm and very much bored, he danced to the\nstarting-gate on the chestnut filly. He paid little attention to the\nfretful doings there. He was haunted by no fear that he might be left.\nIt was a nuisance to have to keep an eye on the vicious heels of Baldy,\nthe swayback gelding at his left--that was all.\n\nBut Kate Greenaway had no intention of being left. She kept her dainty\nnose on the webbing from the instant she got it there, for hadn't\nTim taught her that? And when, at last, all the fussing and fuming\nwas over, and the whips of the starter's assistants had ceased their\nhissing, and the pleadings and threats of the starter himself were\ndone, and the gate swished up before the fourteen racers, the filly's\nfirst bound beat the gate by half a length.\n\nTim was a trifle disgusted. \"Blast the filly, anyhow!\" he thought. It\nwas no part of his plan to lead that roaring field. He took a double\nwrap on the reins, and his mount came back till two lithe, lean forms\nslid up abreast her on the rail, and a third on the outside. That\nwas better, thought Tim, and the sprinters drew out ahead of him.\nContentedly he fell in on the rail behind them.\n\nA storm of dirt clods smote the filly in the face. Another pelted\nTim on the forehead. He took a tighter hold on Kate Greenaway, and\nthe sprinters drew away another length. It would have been an easy\nthing for him to choke her back still further, but somehow a surge\nof generous feeling for the game creature beat down his sullen\nselfishness, and he hadn't the heart to strangle her.\n\n [Illustration: \"IN HIS EARS WAS THE ROAR FROM THIRTY THOUSAND THROATS\n IN THE GRANDSTAND\"]\n\nThe leaders had by this time swung around the first turn, and as they\npassed the half-mile mark two noses intruded themselves on Tim's vision\non the outside.\n\n\"Hello,\" he thought, \"old long-distance Boston is movin' up. An'\nCarley, to keep him from gettin' lonesome.\" But the track was wide,\nthey ran straight and true and kept their distance.\n\nSuddenly the sprinters began to come back. In five seconds Tim would\nhave to pull up behind them. This was disgusting! If only he were on\nthe outside! A clod of earth struck his breast. Instinctively he let\nout a wrap on the reins.\n\nThe filly went up to the sprinters in ten jumps. As he ranged\nalongside, Tim took another hold on her. No more front positions for\nhim. He was outside, and he meant to stay there and be derned to 'em!\n\nThen one of the sprinters fell back, beaten already, and as Boston\nsomehow sifted into the vacant place Tim noted with a gasp that here\nwas the far turn already, and he was with the leaders. This surprised\nhim so much that the last turn leaped past before he realised that\nthere were only two horses between him and the rail. One of them was\nblack Boston, top-weight at one hundred and twenty-nine; the other was\nCarley.\n\nHe was getting a bit interested in spite of himself. The boys on the\nolder horses began to urge them a bit, and as they swung around the\nturn and into the stretch they drew away a couple of lengths. Tim sat\nstill. He was in that delightful outside place, with acres of room. He\neven glanced over at the in-field where the patrol judge stood with his\nglasses to his eyes. He remembered afterward that that official's weird\nwhiskers amused him. Then something happened.\n\nKate Greenaway became mistress of herself. As she swung round the turn,\na wide space confronted her, left by the leaders between themselves and\nthe rail. Kate Greenaway had been taught to hunt that rail as a homing\npigeon its cote. She sought it now so sharply that Tim all but lost his\nseat.\n\nInstantly the boy awoke. He remembered the prize he was riding for--the\nSuburban! the Suburban! Straight before him for a quarter of a mile\ngleamed the track, yellow in the June sunlight. Nothing to do but\nride--straight--straight to the wire.\n\nAll the slumbering life in his body awoke from its sullen sleep. He\nblessed the splendid filly racing so true and so strong beneath him,\nand he sat down for the first time to help her with every ounce of his\npower and every trace of his skill.\n\nHe knew she could win. He knew she had been going well within herself,\nand still she was where she could strike. Now was the time to ride,\nand he rode as he had never ridden before, standing in the stirrups,\ncrouched over the gallant filly's neck, rising and falling in perfect\nrhythm with her every stride. And, bless her! that stride had not begun\nto shorten yet.\n\nSteadily she crept up on the older horses fighting their duel before\nher. Tim could see from the tail of his eye that both their riders were\nworking for dear life--and he had only just begun to ride. His heart\nbounded again beneath his brilliant jacket, and again he urged the\nfilly.\n\nBut what was that? Surely, surely his path was growing narrower. In six\nstrides more he was sure of it. Carley, on the outside, was boring in\nunder the drive, and Boston was pulling in to keep from fouling.\n\nThere's no time to pick daisies in the last furlong of the Suburban.\nAll the months of Tim's purgatory called to him to pull up before\nthey squeezed him against that deadly rail. He tried to do it, but\nhis wrists had gone limp. The next instant the bay and the black were\nrunning stride for stride half a length before the filly--and closing\nin.\n\nThen rose the Terror and gripped Tim by the throat. The moment had\ncome. They had pinned him on the rail.\n\nUnder the gruelling drive Carley staggered again. He bumped Boston. Tim\nfelt the big horse graze his boot as he wavered. Instantly that pungent\nsmell of sweating horseflesh stung his nostrils, and with it flashed\nthe memory of that awful day to smite him helpless.\n\nAgain he tried to pull up, and again he failed. His wrists were\npalsied. Why didn't he fall! Oh, why didn't he fall!\n\nUnder his quaking knees the withers of the gallant filly still rose and\nfell, mightily, rhythmically; her lean, beautiful neck stretched out\nas if to meet the goal, her nostrils wide and blood-red, through which\nthe air came and went, roaring, like the escape of steam from a mighty\nvalve, her eyeballs starting from their sockets.\n\nThen sickening shame smote him on his quivering lips. He seemed to\nrealise for the first time that the filly was waging her terrible fight\nalone.\n\nThe Terror dropped from the boy like a bad dream when one awakes. A\nfrenzy of pride and love for the filly swept over him. He had no hope.\nThe next instant he would hear that terrified roar of the crowd, the\ntrack would leap up to meet him, that flash of red would smite him,\nand blackness would fold him about. But the beautiful filly should not\ngo down with a coward astride her! He found himself talking to her as\nof old, crouching low till his lips all but brushed her fine, straight\nears:\n\n\"Come on, yew gal! Katie--yew Katie! Come on! Almos' home! Almos'!\nCome--come, yew darlin'!\"\n\nCloser pressed the driven Boston, till his rider's stirrup locked\nTim's. And then the boy knew that the last moment had come. It was fall\nor win and instantly. In his ears was the creak and protest of the\nstraining saddles and girths, the roar from thirty thousand throats in\nthe grandstand, the whistle of the breath of three great horses locked\nin a desperate struggle, the thunder of the flying hoofs behind him. He\nhad the right of way--let them unbar it, or crash to destruction--all\nthree!\n\nGripping the reins with his right hand, he raised his whip in his left\nand let it fall, once--twice--three times. Somewhere in her straining,\nbreathless, driven body the filly had one ounce more left. Gallantly,\ninstantly, she gave it. The rail grazed the boy's left boot. His right\nwas driven up to the filly's loins.\n\nShe faltered--but she was through--through that strangling pocket,\nreeling, staggering, half-blind and splendid, and the Suburban was hers\nby a nod.\n\nThey lifted Tim in the famous floral horse-shoe, and they cheered and\ncheered him again. \"Grandest finish I ever see,\" said Faulkner, and \"My\nGawd! what a drive!\" said the stable foreman, gaping.\n\nBut to little Tim it meant only one thing--the greatest, most beautiful\nthing that could be--the Terror was gone forever. He took a deep breath\nand looked about him on a new world.\n\n\n\n\n [Illustration]\n\nJAPAN'S STRENGTH IN WAR\n\nBY GENERAL KUROPATKIN\n\nTRANSLATED BY GEORGE KENNAN\n\nILLUSTRATED WITH PHOTOGRAPHS\n\n\nAlthough the trial of war through which our country and our army passed\nin 1904-5 is now a subject for history, the material thus far collected\nis not sufficiently abundant to enable the historian to estimate fairly\nthe events that preceded the war, nor to give a detailed explanation\nof the defeats that we sustained in the course of it. It is urgently\nnecessary, however, that we should make immediate use of our recent\nexperience, because by ascertaining the nature of our mistakes and the\nweaknesses of our troops we may learn what means should be adopted\nto increase, hereafter, the material and spiritual strength of our\nmilitary force.\n\nIn times past, when wars were carried on by small standing armies,\ndefeats did not affect the every-day interests of the whole nation\nso profoundly as they affect them now, when the obligation to render\nmilitary service is general, and when, in time of war, most of our\nsoldiers are drawn from the great body of the people. If a war is to\nbe successful, in these days, it must be carried on, not by an army,\nbut by an armed nation, and in such a contest all sides of the national\nlife are more seriously affected and all defeats are more acutely felt\nthan they were in times past.\n\nWhen the national pride has been humiliated by failure in war, attempts\nare usually made to ascertain what brought about the failure and who\nwas responsible for it. Some persons attribute it to general causes,\nothers to special causes. Some censure the system, or the regime,\nwhile others throw the blame on particular individuals. I have been\nso closely connected with immensely important events in the Far East,\nand have been responsible to such an extent for the failure of our\nmilitary operations there, that I can hardly hope to take an absolutely\ndispassionate and objective view of the persons and matters that I\nshall deal with in the present work; but my object is not so much\nto justify myself by replying to the charges that have been brought\nagainst me personally as to furnish material that will make it easier\nfor the future historian to state fairly the reasons for our defeat,\nand thus render possible the adoption of measures that will prevent\nsuch defeats hereafter. The army that Russia put into the field in\n1904-5 was unable, in the time allowed, to conquer the Japanese; and\nyet Japan, only a short time before the war began, had no regular army\nand was regarded by us as a second-class Power. How was she able to win\na complete victory over Russia at sea, and to defeat a powerful Russian\narmy on land? Many writers will study this question and, in time, they\nwill give us a comprehensive answer to it; but I shall confine myself,\nin the present work, to an enumeration of the most broad and general\nreasons for Japanese success. Among the most important of such reasons\nis the following:--we did not fully appreciate the material and moral\nstrength of Japan and did not regard a conflict with her seriously\nenough.[A]\n\n\n_The Secret Growth of Japan's Army_\n\nThe Japanese first became our neighbors when, in the reign of Peter\nthe Great, we acquired the peninsula of Kamchatka. In 1860, by virtue\nof the Treaty of Peking, we took peaceful possession of the extensive\nUsuri territory; moved down to the boundary of Korea; and obtained\nan outlet on the Sea of Japan. This sea, which is almost completely\nenclosed by Korea and the Japanese islands, was immensely important\nto the whole adjacent coast of the main land; but as the straits that\nconnected it with the ocean were in the hands of the Japanese, we might\neasily be prevented by them from getting free access to the Pacific.\nWhen we acquired the island of Sakhalin, we obtained an outlet through\nthe Tartar Strait; but that was all we had, and during a large part of\nthe time it was frozen over.\n\nFor a long time, Japan lived a life that was wholly apart from ours and\ndid not particularly attract our attention. We knew the Japanese as\nextremely skilful and patient artisans; we were fond of the things that\nthey made; and we were charmed with the delicacy and bright coloring of\ntheir artistic products; but, from a military point of view, we took no\ninterest in them and regarded them as a weak nation. Our sailors always\nspoke with sympathetic appreciation of the country and its inhabitants,\nand were delighted to stay in Japanese ports--especially Nagasaki,\nwhere they were liked and favorably remembered; but our travellers,\ndiplomats, and naval officers entirely overlooked the awakening of an\nenergetic, independent people.\n\nIn 1867, the army of Japan consisted of nine battalions of infantry,\ntwo squadrons of cavalry, and eight batteries, and numbered only 10,000\nmen. This force, which formed the _cadre_ of the present army, had French\nteachers and adopted from the latter the French uniform. After the\nFranco-German war of 1870-71, German officers took the places of the\nFrench instructors; military service was made a national obligation;\nand Japanese officers were sent to Europe, every year, for the purpose\nof study. At the time of her war with China, Japan had an army\nconsisting of seven infantry divisions; but finding herself unable, at\nthe end of that war, to retain the fruits of her victory, on account\nof her weakness both on land and at sea, she made every possible\neffort to create an army and a fleet that would be strong enough to\nprotect her interests. On the 19th of March, 1896, the Mikado issued\na decree providing for such a reorganization of the army as would\ndouble its strength in the course of seven years. This reorganization\nwas completed in 1903. Our military and naval authorities did not\noverlook the creation and development in Japan of a strong army and\nfleet; but they confined themselves to the collection and tabulation of\nstatistics. We kept an account of every ship built and every division\nof troops organized; but we did not estimate highly enough these\nbeginnings of Japan, and did not admit the possibility of measuring her\nfighting-power by European standards. The latest information that we\nhad with regard to her military strength, prior to the late war, was\ncompiled by our General Staff from the reports of Colonel Vannofski\nand other Russian military agents in Tokio. It showed that her army,\non a peace footing, numbered 8,116 officers and 133,457 men (not\nincluding the troops in Formosa); and on a war footing, 10,735 officers\n(not including reserve officers) and 348,074 men, with perhaps 50,000\nuntrained reserve recruits. There was no mention of additional reserve\nforces.\n\n\n_Russian Generals Pigeonhole Reports of Japan's Fighting Strength_\n\nIn 1903 Colonel Adabash, who had just visited Japan, gave to General\nZhilinski, of our General Staff, very important information with regard\nto new reserves which the Japanese were organizing for service in case\nof war. Inasmuch, however, as this information did not agree at all\nwith that previously furnished by Colonel Vannofski, General Zhilinski\ndid not give it credence. A few months later, Captain Rusine, a very\ntalented officer who was acting as naval observer in Japan, made a\nsimilar report upon Japanese reserves to his superiors, and extracts\nfrom it were furnished to General Sakharoff, Chief of Staff of the\narmy. Although the information contained in this report ultimately\nproved to be perfectly accurate, the report was pigeonholed, simply\nbecause Generals Zhilinski and Sakharoff did not believe it; and in our\ncompendium of data with regard to the military strength of Japan in\n1903-4, no reference whatever was made to additional reserve forces.\nAccording to the figures of our General Staff, therefore, the total\nnumber of available men in the standing army, the territorial army, and\nthe regular reserve of Japan, was a little more than 400,000.[B]\n\n [Illustration: _Stereograph copyright, 1904, by Underwood & Underwood_\n\n SCHOOL CHILDREN BEING DRILLED IN MILITARY TACTICS NEAR TOKIO, JAPAN]\n\nRecently published official reports of General Kipke, Chief Medical\nInspector of the Japanese army, show that the loss of the Japanese in\nkilled and wounded, in the course of the war, was as follows:\n\n Killed 47,387\n Wounded 173,425\n\n Total 220,812\n\nTheir loss in killed, wounded, and sick was 554,885--a number\nconsiderably greater than the whole force which, according to the\nfigures of our General Staff, they could put into the field. They sent\n320,000 sick and wounded back from Manchuria to Japan.\n\n [Illustration: VISCOUNT KATSURA\n\n PRIME MINISTER OF JAPAN DURING THE RUSSIAN-JAPANESE WAR]\n\nOther available information is to the effect that the bodies of 60,624\nkilled were buried in the cemetery of honor in Tokio, and that, in\naddition to these, 75,545 men died from wounds or disease. The Japanese\nthus admit the loss of 135,000 men by death.[C]\n\nTheir Chief Medical Inspector says that their killed and wounded\namounted to 14.58 per cent of their entire force, from which it would\nappear that they put into the field against us troops of various\ncategories to the number of 1,500,000--or more than three times the\nestimate of our General Staff. In view of these facts, it is evident\nthat our information with regard to their fighting strength was\ninsufficient. At the time when they had hundreds of avowed and secret\nagents in the Far East, studying the strength of our land and naval\nforces, we entrusted the collection of data with regard to their\nmilitary strength and resources to a single officer of the General\nStaff, and, unfortunately, our military observers were not always\nwell selected. One of these experts in Japanese affairs said, in\nVladivostok, before hostilities began, that, in the event of war, we\nmight count on one Russian soldier as equal to three Japanese. After\nthe first engagements he moderated his tone and admitted that it might\nbe necessary to put one Russian against every Japanese. At the end of\nanother month he declared that, in order to win victories, we must meet\nevery Japanese soldier in the field with three Russians. Another of\nour military agents, who had been in Japan, predicted authoritatively\nthat Port Arthur would fall in a very short time, and that immediately\nthereafter the same fate would overtake Vladivostok. I sharply\nreprimanded the faint-hearted babbler and threatened to dismiss him\nfrom the army if he continued to make such injurious and inopportune\nremarks.\n\n\n_Moral Superiority of the Japanese_\n\nBut it was not only with regard to Japan's material strength that\nour information was insufficient. We underestimated, or entirely\noverlooked, her moral strength. According to that great leader\nNapoleon, three fourths of an army's success in war is due to the moral\ncharacter of its soldiers. This relation of moral character to material\nsuccess still exists, although the conditions of battle, in these\ndays, are more trying than they were in the Napoleonic wars. And now,\nmore than ever before, the moral strength of the army depends upon the\ntemper of the nation. Armies are now so organized that, in case of war,\nsoldiers are drawn, for the most part, from the reserves. A successful\nwar, therefore, must be a popular war, and victory must be attained by\nthe hearty cooperation of the whole people with its Government.\nThe recent contest in Manchuria was a popular war for the Japanese, but\nnot for us. The Korean question, and the question of naval supremacy\non the waters of the Pacific, involved vital Japanese interests, and\nthe immense importance of these interests was so clearly understood\nand so fully appreciated by the Japanese people that the war for their\nprotection was a national war. Japan spent ten years in preparing\nfor it, and then the whole nation carried it on. Japanese soldiers,\ndeeply conscious of the bearing that their exploits might have on\nthe future of the country, fought with a self-sacrificing devotion\nand a stubbornness that we had never seen in any war in which we had\npreviously been engaged. Sometimes, in villages that we had taken by\nassault, a handful of Japanese soldiers would barricade themselves in\nnative houses and die there rather than retreat or surrender. Japanese\nofficers who fell into our hands--even wounded officers--generally\ncommitted suicide.\n\nIt is quite possible that when we have a true history of the war based\non Japanese sources of information, our pride may receive another blow.\nWe already know that in many cases we outnumbered the enemy, and still\nwe were not victorious. The explanation of this, however, is very\nsimple. The Japanese, in these cases, were inferior to us materially,\nbut they were stronger than we morally.[D] To this aspect of the\nstruggle we should give particular attention, because military history\nshows that, in all wars, the antagonist who is strongest morally wins\nthe victory. The only exceptions are such contests as that between the\nEnglish and the Boers in South Africa and that between the North and\nthe South in America. The English were weaker than the Boers morally,\nbut they put into the field an overwhelming force, and, in spite of\nmany defeats, they finally conquered. In the American war, the army\nof the South was in the same position that the Boer army was, and the\nNortherners had to put a superior force into the field in order to\novercome it.\n\n [Illustration: GENERAL TERAUCHI\n\n JAPANESE MINISTER OF WAR]\n\n\n_Extraordinary Popularity of the War in Japan_\n\nAmong the sources of moral strength that failed to attract our\nattention in Japan were the following: The training of her citizens had\nlong been patriotic and warlike in tendency; her educational system had\ninculcated an ardent love of country; and even in her primary schools\nchildren were prepared, from their earliest years, to be soldiers. The\npeople regarded the army with profound respect and trust, and young\nmen served in it with pride. All these things we failed to see, and we\noverlooked also the iron discipline enforced in the army and the role\nplayed in it by the samurai officers. We wholly failed to appreciate,\nmoreover, the vital importance of the Korean question to Japan, and the\nstrength of the hostile feeling that was raised against us when the\nJapanese were deprived of the fruits of their victory after their war\nwith China. The party of Young Japan had long insisted upon war with\nRussia and had been restrained only by a prudent Government.\n\nWhen the war began, we recovered our powers of perception, but it was\nthen too late. And at that time, when the war was not only unpopular\nin Russia but incomprehensible to the Russian people, the Japanese,\nwith a great outburst of enthusiastic patriotism, were responding,\nlike a single man, to the call to arms. In some cases Japanese mothers\neven killed themselves, when their sons, on account of weakness or ill\nhealth, were denied admission to the army. Hundreds of men volunteered\nto undertake the most desperate enterprises, in the face of certain\ndeath; and many officers and soldiers, before going to the front, had\nfuneral ceremonies performed over their bodies, in order to show that\nthey intended to die for their native land. The youth of the Empire\ncrowded into the army, and the heads of the most distinguished families\nsought to serve their country by enlisting themselves, by sending\ntheir sons to the front, or by helping to pay the expenses of the war.\nSome Japanese regiments, in attacking our positions, threw themselves\nwith the cry of \"Banzai!\" upon our obstructions, struggled over or\nthrough them, filled our ditches with the bodies of their dead, and\nthen, rushing across upon the corpses of their comrades, forced their\nway into our entrenchments. The army and the whole people appreciated\nthe importance of the war, understood the significance of the events\nthat were taking place, and were ready to make sacrifices in order to\nachieve success.\n\n [Illustration: _Copyrighted by Underwood & Underwood_\n\n GENERAL KODAMA\n\n CHIEF STAFF OFFICER OF THE JAPANESE ARMY IN MANCHURIA]\n\n\n_Military Training of Japanese Children_\n\nAfter the Japanese-Chinese war, of which I made a most careful and\ndetailed study, I myself was inspired with a feeling of respect for the\nJapanese army and watched its growth with anxiety. Then, in 1900, the\npart played by the Japanese troops that cooperated with ours in the\nprovince of Pechili confirmed me in the belief that they were excellent\nsoldiers. During my short stay in Japan, I was unable to acquaint\nmyself thoroughly with the country and its military forces, but what I\ndid learn was enough to convince me that the results attained by the\nJapanese in the course of twenty-five or thirty years were astounding.\nI saw a beautiful country, with a large and industrious population.\nIntense activity prevailed everywhere, and I was impressed by the\npeople's joy in life, their love of country, and their faith in\ntheir future. In their military school, where I saw a Spartan system\nof education, the exercises of the cadets with pikes, rifles, and\nbroadswords were not approached by anything of the kind that I had\nwitnessed in Europe,--it was fighting of the fiercest character. At\nthe end of the struggle there was a hand-to-hand combat, which lasted\nuntil the victors stood triumphant over the bodies of the vanquished\nand tore off their masks. In these exercises, which were very severe,\nthe cadets struck one another fiercely and with wild cries; but the\nmoment a prearranged signal was given, or the fight came to an end,\nthe combatants drew themselves up in a line and their faces assumed an\nexpression of wooden composure.\n\n [Illustration: _Stereograph copyrighted by the H. C. White Co._\n\n MARSHAL OYAMA\n\n COMMANDER-IN-CHIEF IN MANCHURIA OF THE JAPANESE ARMY DURING THE\n RUSSO-JAPANESE WAR]\n\nIn all the public schools prominence was given to military exercises,\nand the pupils took part in them with enthusiasm. Even in their walks\nthey practised running, flanking, and sudden, unexpected attacks of\none party on another. The history of Japan was everywhere made a means\nof strengthening the pupils' patriotism and their belief in Japan's\ninvincibility. Particular stress was laid upon the country's successful\nwars, the heroes of them were extolled, and the children were taught\nthat none of Japan's military enterprises had ever failed.\n\n\n_Japan's Material Resources_\n\nIn the manufactories of arms I saw the turning out of rifles in immense\nnumbers, and the work was being done swiftly, accurately, and cheaply.\nIn Kobe and Nagasaki I inspected attentively the ship-building yards,\nwhere they were constructing not only torpedo boats but armored\ncruisers, and where all the work was being done by their own mechanics\nand foremen under the direction of their own engineers. At the great\nnational exposition in Osaka there was a splendid and instructive\ndisplay of the country's manufactures, including textiles, products of\ncottage industry, complicated instruments, grand pianos, and guns of\nthe largest caliber--all made in Japan, by Japanese workmen, and out of\nJapanese materials. I saw nothing of foreign origin except raw cotton\nand iron, which were imported from China and Europe. And the products\ndisplayed at this exposition were not more worthy of attention than the\nobservant, courteous, and always dignified throng of Japanese visitors.\n\nIn the agriculture of Japan many of the methods were ancient, but the\nculture was unquestionably high. The fields were carefully worked,\nand the effort to make every foot of land yield all that it could,\nthe struggle to raise crops even on the mountain sides, and the\ninsufficiency of the country's food products despite this intensive\nculture, showed that the people were becoming overcrowded on their\nislands, and that the Korean question was for them a question of vital\nimportance. I lived ten days among the fishermen, and saw something\nof the reverse side of Japan's rapid development under European\nconditions. Many complaints were made to me of heavy taxes, which\nhad increased greatly in later years, and of the high cost of the\nnecessaries of life.\n\nI witnessed reviews of the Japanese troops, including the division\nof Guards, two regiments of the First Division, two regiments of\ncavalry, and many batteries. The marching was admirable, and the\ncommon soldiers appeared like our younkers. The officers and leaders\nof the Japanese army whom I saw and met made upon me a very favorable\nimpression. The culture and knowledge of military affairs that many\nof them possessed would have given them places of honor in any army.\nWith General Terauchi, the Japanese Minister of War, I had had\nfriendly relations ever since 1886, when we met in France at the\ngreat manoeuvers directed by General Levalle. Among others whose\nacquaintance I made were Generals Yamagata, Oyama, Kodama, Fukushima,\nNodzu, Hasegawa, and Murata, and the Imperial princes, Fushimi and\nKanin. In spite of a terrible war, which has separated by a barrier\nnations that were apparently created for union and friendship, I\nstill cherish a sympathetic feeling for my Tokio acquaintances.\nEspecially do I remember with respect their ardent love of country and\ntheir devotion to their Emperor--feelings that they have since made\nmanifest in deeds. I met also in Tokio many leaders in fields other\nthan that of war, among whom were Ito, Katsura, and Komura. In the\nreport that I made to the Emperor, after my return from Japan, I placed\nthe military power of the Japanese on a level with that of European\nnations. I regarded one of our battalions as equal to two battalions of\nJapanese in defence, but I estimated that in attack we should have two\nbattalions to their one. The test of war has shown that my conclusions\nwere correct. There were lamentable cases, of course, in which the\nJapanese, with a smaller number of battalions, drove our forces from\nthe positions that they occupied; but these results were due either to\nmistakes in the direction of our troops, or to numerical inferiority\nin the fighting strength of our battalions. In the last days of the\nbattle of Mukden, some of our brigades consisted of hardly more than a\nthousand bayonets. It is evident that the Japanese had to put into the\nfield only two or three battalions in order to deal successfully with a\nbrigade of such depleted strength.\n\nAll that I saw and learned of Japan, or her military strength, and of\nthe nature of her problems in the Far East, convinced me that it would\nbe necessary for us to come to a peaceable understanding with her, and\nthat we should have to make great concessions--concessions that, at\nfirst sight, might seem humiliating to our national pride--in order to\navoid war with her. As I have already said, I did not hesitate even\nto propose the return of Port Arthur and Kwang-tung to China and the\nsale of the southern branch of the Eastern Chinese Railway. I foresaw\nthat the threatened war would be extremely unpopular in Russia; that\nthere would be no manifestation of patriotic spirit, on account of the\npeople's ignorance of the objects of the war; and that the leaders of\nthe anti-Government party would avail themselves of the opportunity\nto increase domestic discontent and disorder. I did not, however,\nanticipate that the Japanese would display so much energy, activity,\ncourage, and lofty patriotism, and I therefore erred in my estimate of\nthe time that the struggle would require. In view of the insufficiency\nof our railroad transportation, we should have allowed three years for\nthe war, instead of the year and a half that I thought would be enough.\n\nWith all their strong points, the Japanese manifested weaknesses that\nmay be shown again in future wars. I shall not enumerate them, but I\nwill say that, in many cases, the outcome of the fight was in doubt,\nand that in other cases we escaped defeat only through the errors of\nthe Japanese commanders. There is a saying that \"the victor is not\njudged.\" I may add that to the victor is rendered homage, and this is\ntrue of the Japanese. The general tone of the whole press was favorable\nto them, and even their practical and well-balanced heads might well\nhave been turned by the praise that they received. No one went further\nin this direction than Count Leo Tolstoi. In an article published in\na foreign journal,[E] our gifted author and philosopher expressed\nthe conviction that the Japanese defeated us because, owing to their\nwarlike patriotism and the power of their ruling authorities, they are\nthe mightiest nation on earth, and are not to be conquered by any one,\neither at sea or on land.\n\n [Illustration: _Stereograph copyrighted by Underwood & Underwood_\n\n JAPANESE ARMY TRANSPORTATION CORPS MOVING ONE OF THE GREAT SIEGE GUNS\n WHICH WERE USED IN THE DESTRUCTION OF PORT ARTHUR]\n\nThe strength of Japan was in the complete union of her people, army,\nand government, and it was this union that gave her the victory. We\ncarried on the contest with our army alone, and even the army was\nweakened by the unfavorable disposition of the people toward all things\nmilitary. Our aims in the Far East were not understood by our officers\nand soldiers, and, furthermore, the general feeling of discontent\nwhich already prevailed in all classes of our population made the war\nso hateful that it aroused no patriotism whatever. Many good officers\nhastened to offer their services--a fact that is easily explained--but\nall ranks of society remained indifferent. A few hundreds of the common\npeople volunteered, but no eagerness to enter the army was shown by the\nsons of our high dignitaries, of our merchants, or of our scientific\nmen. Out of the tens of thousands of students who were then living\nin idleness,[F] many of them at the expense of the Empire, only a\nhandful volunteered,[G] while at that very time, in Japan, sons of the\nmost distinguished citizens--even boys fourteen and fifteen years of\nage--were striving for places in the ranks. Japanese mothers, as I have\nalready said, killed themselves through shame, when their sons were\nfound to be physically unfit for military service.\n\n\n_Russian Discipline Undermined by the Revolutionists_\n\nThe indifference of Russia to the bloody struggle which her sons were\ncarrying on--for little understood objects and in a foreign land--could\nnot fail to discourage even the best soldiers. Men are not inspired\nto deeds of heroism by such an attitude toward them on the part of\ntheir country. But Russia was not merely indifferent. Leaders of the\nrevolutionary party strove, with extraordinary energy, to multiply our\nchances of failure, hoping thus to facilitate the attainment of their\nown dark objects. There appeared a whole literature of clandestine\npublications, intended to lessen the confidence of officers in their\nsuperiors, to shake the trust of soldiers in their officers, and to\nundermine the faith of the whole army in the Government. In an \"Address\nto the Officers of the Russian Army,\" published and widely circulated\nby the Social Revolutionists, the main idea was expressed as follows:\n\n\"The worst and most dangerous enemy of the Russian people--in fact, its\nonly enemy--is the present Government. It is this Government that is\ncarrying on the war with Japan, and you are fighting under its banners\nin an unjust cause. Every victory that you win threatens Russia with\nthe calamity involved in the maintenance of what the Government calls\n'order,' and every defeat that you suffer brings nearer the hour of\ndeliverance. Is it surprising, therefore, that Russians rejoice when\nyour adversary is victorious?\"\n\nBut persons who had nothing in common with the Social Revolutionary\nparty, and who sincerely loved their country, gave aid to Russia's\nenemies by expressing the opinion, in the press, that the war was\nirrational, and by criticizing the mistakes of the Government that had\nfailed to prevent it. In a brochure entitled \"Thoughts Suggested by\nRecent Military Operations,\" M. Gorbatoff referred to such persons as\nfollows:\n\n\"But it is a still more grievous fact that while our heroic soldiers\nare carrying on a life-and-death struggle, these so-called friends of\nthe people whisper to them: 'Gentlemen, you are heroes, but you are\nfacing death without reason. You will die to pay for Russia's mistaken\npolicy, and not to defend Russia's vital interests.' What can be more\nterrible than the part played by these so-called friends of the people\nwhen they undermine in this way the intellectual faith of heroic men\nwho are going to their death? One can easily imagine the state of\nmind of an officer or soldier who goes into battle after reading,\nin newspapers or magazines, articles referring in this way to the\nirrationality and uselessness of the war. It is from these self-styled\nfriends of the people that the revolutionary party gets support in its\neffort to break down the discipline of our troops.\"\n\nSoldiers of the reserves, when called into active service, were\nfurnished by the anti-Government party with proclamations intended\nto prejudice them against their officers, and similar proclamations\nwere sent to the army in Manchuria. Troops in the field received\nletters apprising them of popular disorders in Russia, and men sick in\nhospitals, as well as men on duty in our advanced positions, read in\nthe newspapers articles that undermined their faith in their commanders\nand their leaders. The work of breaking down the discipline of the army\nwas carried on energetically, and, of course, it was not altogether\nfruitless. The leaders of the movement, in striving to attain their\nwell defined objects, took for their motto: \"The worse things are, the\nbetter\"; and the ideal at which they aimed was the state of affairs\nbrought about by the mutinous sailors on the armor-clad warship\n\"Potemkin.\" These enemies of the army and the country were aided by\ncertain other persons who were simply foolish and unreasonable. One can\nimagine the indignation that the Menchikoffs, the Kirilloffs and the\nKuprins would feel, if they were told that they played the same part in\nthe army that was played by the persons who incited the insubordination\non the \"Potemkin\"; yet such was the case. It would be difficult,\nindeed, to imagine anything that could have been said to the sailors of\nthe armor-clad for the purpose of exciting them against their officers\nthat would have been worse than the language of Menchikoff, when, in\nwriting of our army officers, he referred to their \"blunted conscience,\ntheir drunkenness, their moral looseness, and their inveterate\nlaziness.\" Firm in spirit though Russians might be, the indifference of\none class of the population, and the seditious incitement of another,\ncould hardly fail to have upon many of them an influence that was not\nfavorable to the successful prosecution of war.\n\n\n_Attacks of the Russian Press_\n\n [Illustration: _Copyrighted by Underwood & Underwood_\n\n SCENE IN SHIBA PARK, TOKIO, WHERE TOGO'S NAVAL VICTORY WAS CELEBRATED\n WITH WILD ENTHUSIASM]\n\nThe party opposed to the Government distributed among our troops,\nespecially in the West, hundreds of thousands of seditious\nproclamations exhorting the soldiers to work for defeat rather than\nfor victory. Writers for newspapers and magazines, even though they\ndid not belong to the anti-Government party, contributed to its\nsuccess by lavishing abuse upon the army and its representatives. War\ncorrespondents, who knew little about our operations and still less\nabout those of the Japanese, and who based their statements, not upon\nwhat they had seen, but upon what they had heard from untrustworthy\nsources, increased the disaffection of the people by exaggerating the\nseriousness of our failures. Even army officers, writing from the\ntheatre of war, or after returning to Russia for reasons that were\nnot always creditable to them, sought to gain reputation by means of\nhasty criticism which was often erroneous in its statements of fact and\ngenerally discouraging or complaining in tone. On the fighting line,\nheroic men without number faced and fought the enemy courageously for\nmonths, without ever losing their faith in ultimate victory; but from\nthat part of the field little trustworthy news came. Brave soldiers,\nmodest junior officers, and the commanders of regiments, companies,\nsquadrons, and batteries in our advanced positions, did not write and\nhad no time to write of their own labors and exploits, and few of the\ncorrespondents were willing to share their perils for the sake of being\nable to observe and describe their heroic deeds. There were among the\ncorrespondents some brave men who sincerely wished to be of use; but\ntheir lack of even elementary training in military science made it\nimpossible for them to understand the complicated problems of war, and\ntheir work therefore was comparatively unproductive. The persons best\nqualified to see and judge, and to give information to the reading\npublic, were the foreign military observers, who were attached to our\narmies in the field and who, in many cases, were extremely fortunate\nselections. These officers felt a brotherly affection for the soldiers\nwhose perils and hardships they shared, and were regarded by the latter\nwith love and esteem. Their reports, however, are very long in coming\nto us.\n\n [Illustration: _Stereograph copyrighted by the H. C. White Co._\n\n JAPANESE ARTILLERY TRANSPORTING A 7-1\/2 C. M. MOUNTAIN GUN ACROSS THE\n HILLS]\n\nSome of our correspondents, who lived in the rear of the army and saw\nthe seamy side of the war, wrote descriptions of drunkenness, revelry,\nand profligacy (at Kharbin, for example) which distressed our reading\npublic and gave a one-sided view of army life. Our press might have\nmade our first defeats a means of rousing the spirit of patriotism\nand self-sacrifice; it might have exhorted the people to redouble\ntheir efforts as the difficulties of the war increased; it might have\nhelped the Government to fill the gaps in our thinned ranks; it might\nhave encouraged the faint-hearted, called forth the country's noblest\nsons, and opened to the army new sources of material and spiritual\nstrength. But instead of doing any of these things, it played more or\nless into the hands of our foreign and domestic enemies; made the war\nhateful to the great mass of the population; depressed the spirits of\nsoldiers going to the front, and undermined, in every way, the latter's\nfaith in their officers and their rulers. This course of procedure\ndid not rouse in the nation a determination to increase its efforts\nand to win victory at last, in spite of all difficulties. Quite the\ncontrary! The soldiers who went to the front to fill up or reinforce\nour army carried with them seditious proclamations and the seeds of\nfuture defeats. Commanding officers in the Siberian military districts\nreported, as early as February, that detachments of supernumerary\ntroops and reservists had plundered several railway stations, and at\na later time regular troops, on their way to the front, were guilty\nof similar bad conduct. The drifting to the rear of large numbers\nof soldiers--especially the older reservists--while battles were in\nprogress, was due not so much to cowardice as to the unsettling of\nthe men's minds and to a disinclination on their part to continue the\nwar. I may add that the opening of peace negotiations in Portsmouth,\nat a time when we were preparing for decisive operations, affected\nunfavorably the morale of the army's strongest elements.\n\n\n_The Russian Army Cut Off from the Nation_\n\nMr. E. Martinoff, in an article entitled \"Spirit and Temper of the\nTwo Armies,\" points out that \"even in time of peace, the Japanese\npeople were so educated as to develop in them a patriotic and martial\nspirit. The very idea of war with Russia was generally popular, and\nthroughout the contest the army was supported by the sympathy of the\nnation. In Russia, the reverse was true. Patriotism was shaken by the\ndissemination of ideas of cosmopolitanism and disarmament, and in the\nmidst of a difficult campaign the attitude of the country toward the\narmy was one of indifference, if not of actual hostility.\"\n\nThis judgment is accurate, and it is evident, of course, that with\nsuch a relation between Russian society and the Manchurian army, it\nwas impossible to expect from the latter any patriotic spirit, or\nany readiness to sacrifice life for the sake of the fatherland. In\nan admirable article entitled \"The Feeling of Duty and the Love of\nCountry,\" published in the \"Russian Invalid\" in 1906, Mr. A. Bilderling\nexpressed certain profoundly true ideas as follows:\n\n\"Our lack of success may have been due, in part, to various and\ncomplicated causes; to the misconduct of particular persons, to bad\ngeneralship, to lack of preparation in the army and the navy, to\ninadequacy of material resources, and to misappropriations in the\ndepartments of equipment and supply; but the principal reason for our\ndefeat lies deeper, and is to be found in lack of patriotism, and in\nthe absence of a feeling of duty toward and love for the fatherland.\nIn a conflict between two peoples, the things of most importance are\nnot material resources, but moral strength, exaltation of spirit, and\npatriotism. Victory is most likely to be achieved by the nation in\nwhich these qualities are most highly developed. Japan had long been\npreparing for war with us; all of her people desired it; and a feeling\nof lofty patriotism pervaded the whole country. In her army and her\nfleet, therefore, every man, from the commander-in-chief to the last\nsoldier, not only knew what he was fighting for and what he might have\nto die for, but understood clearly that upon success in the struggle\ndepended the fate of Japan, her political importance, and her future in\nthe history of the world. Every soldier knew also that the whole nation\nstood behind him. With us, on the other hand, the war was unpopular\nfrom the very beginning. We neither desired it nor anticipated it, and,\nconsequently, we were not prepared for it. Soldiers were hastily put\ninto railway trains, and when, after a journey that lasted a month,\nthey alighted in Manchuria, they did not know in what country they\nwere, nor whom they were to fight, nor what the war was about. Even our\nhigher commanders went to the front unwillingly and from a mere sense\nof duty. The whole army, moreover, felt that it was regarded by the\ncountry with indifference; that its life was not shared by the people;\nand that it was a mere fragment, cut off from the nation, thrown to a\ndistance of nine thousand versts, and there abandoned to the caprice of\nfate. Before decisive fighting began, therefore, one of the contending\narmies advanced with the full expectation and confident belief that it\nwould be victorious, while the other went forward with a demoralizing\ndoubt of its own success.\"\n\nGenerally speaking, the man who conquers in war is the man who is\nleast afraid of death. We were unprepared in previous wars, as well\nas in this, and in previous wars we made mistakes; but when the\npreponderance of moral strength was on our side, as in the wars with\nthe Swedes, the French, the Turks, the Caucasian mountaineers and the\nnatives of Central Asia, we were victorious. In the late war, for\nreasons that are extremely complicated, our moral strength was less\nthan that of the Japanese; and it was this inferiority, rather than\nmistakes in generalship, that caused our defeats, and that forced us\nto make tremendous efforts in order to succeed at all. Our lack of\nmoral strength--as compared with the Japanese--affected all ranks of\nour army, from the highest to the lowest, and greatly reduced our\nfighting power. In a war waged under different conditions--a war in\nwhich the army had the confidence and encouragement of the country--the\nsame officers and the same troops would have accomplished far more\nthan they accomplished in Manchuria. The lack of martial spirit, of\nmoral exaltation, and of heroic impulse, affected particularly our\nstubbornness in battle. In many cases we did not have dogged resolution\nenough to conquer such antagonists as the Japanese. Instead of holding,\nwith unshakable tenacity, the positions assigned them, our troops often\nretreated, and, in such cases, our commanding officers of all ranks,\nwithout exception, lacked the power or the means to set things right.\nInstead of making renewed and extraordinary efforts to wrest victory\nfrom the enemy, they either permitted the retreat of the troops under\ntheir command, or themselves ordered such retreat. The army, however,\nnever lost its strong sense of duty; and it was this that enabled\nmany divisions, regiments, and battalions to increase their power\nof resistance with every battle. This peculiarity of the late war,\ntogether with our final acquirement of numerical preponderance and a\nnoticeable decline of Japanese ardor, gave us reason to regard the\nfuture with confidence, and left no room for doubt as to our ultimate\nvictory.\n\n\n_The Failure of the Russian Fleet_\n\nAmong other reasons for the success of the Japanese, I may mention the\nfollowing.\n\nThe leading part in the war was to have been taken by our fleet. In the\nGeneral Staff of the navy, as well as in that of the army, a detailed\naccount was kept of all Japan's ships of war; but the directors\nof naval affairs in the Far East reckoned only tonnage, guns, and\ncalibers, and when, in 1903, they found that the arithmetical totals\nof our Far Eastern fleet exceeded those of the entire Japanese fleet,\nthey adopted, as a basis for our plan of operations, the following\nconclusions:\n\n1. \"The relation that the strength of the Japanese fleet bears to the\nstrength of our fleet is such that the possibility of the defeat of the\nlatter is inadmissible.\"\n\n2. \"The landing of the Japanese at Yinkow, or in Korea Bay, is not to\nbe regarded as practicable.\"\n\nThe strength of the land force that a war with Japan would require\ndepended upon three things: (1) the strength of the army that the\nJapanese could put into Manchuria, or across our boundary; (2) the\nstrength of our fleet; and (3) the transporting capacity of the railway\nupon which our troops would have to depend in concentration. If our\nfleet could defeat the fleet of the Japanese, military operations on\nthe main land would be unnecessary. And even if the Japanese were\nnot defeated in a general naval engagement, they would either have\nto obtain complete mastery of the sea, or leave a considerable part\nof their army at home for the protection of their own coast. Without\ncommand of the sea, moreover, they could not risk a landing on the\nLiao-tung peninsula, but would have to march through Korea, and that\nwould give us time for concentration. By their desperate night attack\nupon our fleet at Port Arthur, before the declaration of war,[H] they\nobtained a temporary superiority in armored vessels, and made great\nuse of it in getting command of the sea. Our fleet--especially after\nthe death of Admiral Makaroff at the most critical moment in the\nexecution of the Japanese plan of campaign--offered no resistance to\nthe enemy whatever. Even when they landed in the immediate vicinity of\nPort Arthur, we did not make so much as an attempt to interfere with\nthem. The results of this inaction were very damaging to our army. The\nJapanese, instead of finding it impossible to land troops in Korea Bay,\nas our naval authorities anticipated, were able to threaten us with a\ndescent along the whole coast of the Liao-tung peninsula, beginning at\nKwang-tung. Notwithstanding our weakness on land, Admiral Alexeieff\nthought it necessary to authorize a wide scattering of our troops, so\nwe prepared to meet the Japanese on the Yalu, at Yinkow, and in the\nprovince of Kwang-tung. He had also permitted a dispersal of our naval\nforces, so that we were weak everywhere.\n\n\n_Advantages Secured by Japan's Naval Victory_\n\nInstead of making a landing in Korea only,--as was anticipated in the\nplan worked out at Port Arthur,--the Japanese, with their immense fleet\nof transports, landed three armies on the Liao-tung peninsula and a\nfourth in Korea. Then, leaving one army in front of Port Arthur, they\npushed the other three forward toward our forces, Which were slowly\nconcentrating on the Haicheng-Liaoyang line in southern Manchuria.\nThus, having taken the initiative at sea, they obtained the same\nadvantage on land. Their command of the sea enabled them to disregard\nthe defence of their own coast and move against us with their entire\nstrength. In this way--contrary to our anticipations--they were able,\nin the first stage of the war, to put into the field a force that was\nsuperior to ours. Command of the sea, moreover, made it possible for\nthem to supply their armies quickly with all necessary munitions, and\nto transport to the field, in a few days, masses of heavy supplies,\nwhich we, with our feeble railroad, were hardly able to get in months.\nBut command of the sea, and the almost complete inactivity of our\nfleet, gave them another advantage, not less important, and that\nwas the possibility of bringing safely to their ports and arsenals\nquantities of commissary and military stores, weapons, horses, and\ncattle, which had been ordered in Europe and America. Their line of\ncommunications, furthermore, was short and secure, while we were at\na distance of eight thousand versts from our base of supplies and\nwere connected with our country only by one weak line of railway. The\nadvantage that they had over us in this respect was immense. The slow\nconcentration of our army, which had to be brought eight thousand\nversts over a single-track railroad, gave them time, after the war\nbegan, to form new bodies of troops, in considerable numbers, and send\nthem to the front. They had time enough, also, to supply their army\nwith innumerable machine guns, after they had observed, in the early\nstages of the war, the importance of machine-gun fire.\n\nThe field of military operations in Manchuria had been familiar to the\nJapanese ever since their war with China. Its heat, its heavy rains,\nits mountains and its kiaoliang, were well known to them, because they\nhad seen them all in their own country. In the mountains, especially,\nthey felt perfectly at home, while a mountainous environment, to our\ntroops, was oppressive. The Japanese, moreover, in their ten years of\npreparation for war with us, had not only studied Manchuria, but had\nsecured there their own agents, who were of the greatest use to their\narmy. The Chinese, I may add, assisted the Japanese, notwithstanding\nthe severity and even cruelty with which the latter treated them.\n\nThe Japanese had a considerable advantage over us, also, in their\nhigh-powered ammunition, their machine guns, their innumerable mountain\nguns, their abundant supply of explosives, and their means of attack\nand defence in the shape of wire, mines, and hand grenades. Their\norganization, equipment, and transport carts were all better adapted\nto the field of operations than ours were, and their bodies of sappers\nwere more numerous than ours.\n\nThe Japanese soldiers had been so trained as to develop self-reliance\nand ability to take the lead, and they were credited by foreign\nmilitary observers with \"intelligence, initiative, and quickness,\" In\nthe fighting instructions that were given them, very material changes\nwere made after the war began. At the outset, for example, night\nattacks were not recommended; but they soon satisfied themselves that\nnight attacks were profitable and they afterward made great use of\nthem. Major von Luwitz, of the German army, in a brochure entitled \"The\nJapanese Attack in the War in Eastern Asia in 1904-05\" says that while\nthe Japanese did not neglect any means of making attacks effective, the\nsecret of their success lay in their determination to get close to the\nenemy, regardless of consequences.\n\n\n_The Intellectual Superiority of the Japanese Soldier_\n\nThe non-commissioned officers in the Japanese army were much superior\nto ours, on account of the better education and greater intellectual\ndevelopment of the Japanese common people. Many of them might have\ndischarged the duties of commissioned officers with perfect success.\nThe defects of our soldiers--both regulars and reservists--were the\ndefects of the population as a whole. The peasants were imperfectly\ndeveloped intellectually, and they made soldiers who had the same\nfailing. The intellectual backwardness of our soldiers was a great\ndisadvantage to us, because war now requires far more intelligence and\ninitiative, on the part of the individual soldier, than ever before.\nOur men fought heroically in compact masses, or in fairly close\nformation, but if deprived of their officers they were more likely to\nfall back than to advance. In the mass we had immense strength; but few\nof our soldiers were capable of fighting intelligently as individuals.\nIn this respect the Japanese were much superior to us. Their\nnon-commissioned officers were far better developed, intellectually,\nthan ours, and among such officers, as well as among many of the\ncommon soldiers, whom we took as prisoners, we found diaries which\nshowed not only good education but knowledge of what was happening and\nintelligent comprehension of the military problems to be solved. Many\nof them could draw maps skilfully, and one common soldier was able to\nshow accurately, by means of a plan sketched in the sand, the relative\npositions of the Japanese forces and ours.\n\nBut the qualities that contributed most to the triumph of the Japanese\nwere their high moral spirit, and the stubborn determination with which\nthe struggle for success was carried on by every man in their army,\nfrom the common soldier to the commander-in-chief. In many cases, their\nsituation was so distressing that it required extraordinary power\nof will on their part to stand fast or to advance. But the officers\nseemed to have resolution enough to call on their men for impossible\nefforts--not even hesitating to shoot those that fell back--and the\nsoldiers, rallying their last physical and spiritual strength, often\nwrested the victory away from us. One thing is certain: if the whole\nJapanese army had not been inspired with an ardent patriotism; if it\nhad not been sympathetically supported by the whole nation; and if all\nits officers and soldiers had not appreciated the immense importance\nof the struggle, even such resolution as that of the Japanese leaders\nwould have failed to achieve such results.\n\n [A] General Kuropatkin makes frequent use of the expression\n \"moral strength,\" or \"moral character,\" and often employs the\n English word \"moral\" instead of the corresponding Russian word.\n He evidently intends that the adjective shall be understood in\n its broadest signification, as a term covering patriotism, the\n sense of duty, capacity for self-sacrifice, and all the qualities\n that go to make up character as distinct from mere intellectual\n ability.--G. K.\n\n [B] Considerations of space have forced me to omit the greater\n part of General Kuropatkin's detailed and somewhat technical\n statement with regard to Japan's military strength and the extent\n to which it was underestimated by the Russian General Staff.--G. K.\n\n [C] According to information contained in Immanuel's work, \"The\n Russo-Japanese War,\" the Japanese lost 218,000 men in battle.\n\n [D] General Kuropatkin uses the English words \"materially\" and\n \"morally.\"--G. K.\n\n [E] _Fortnightly Review._\n\n [F] On account of student disorders that had led to the closing of\n the universities.--G. K.\n\n [G] Medical students excepted.\n\n [H] General Kuropatkin, it will be noticed, calls this night\n attack \"desperate,\" but does not characterize it as treacherous\n or unfair. At the time when it occurred, however, the Russian\n Government denounced it as a dishonorable violation of civilized\n usage, if not of international law, while the loyal Russian\n press held Japan up to the scorn of the world as a tricky and\n treacherous antagonist. It is an interesting but little known fact\n that the Tsar himself had ordered Admiral Alexeieff to attack\n the Japanese in the same way, without notice and before any\n declaration of war had been made. In the historically important\n series of official dispatches from the archives of Port Arthur,\n published in the liberal Russian review \"Osvobozhdenie\" at\n Stuttgart in 1905 appears the following telegram sent by the Tsar\n to the Viceroy just after the Japanese had broken off diplomatic\n relations.\n\n ST. PETERSBURG, JANUARY 26, 1904, O. S.\n\n ALEXEIEFF\n\n PORT ARTHUR.\n\n It is desirable that the Japanese, and not we, should begin\n military operations. If, therefore, they do not attack us, you\n must not oppose their landing in southern Korea, or on the eastern\n coast as far north as Gensan, inclusive. But if their fleet makes\n a descent upon the western coast, or, without making a descent,\n goes north of the 38th parallel, you are authorized to attack\n them, without waiting for the first shot from their side. I rely\n on you. May God assist you.\n\n (Signed)\n\n NICHOLAS\n\n (Signature in the Tsar's own hand)\n\n It thus appears that Russia intended to attack Japan without\n notice and without a declaration of war, but Alexeieff was not\n quick enough--G. K.\n\n\n\n\n [Illustration]\n\nTHE DEATH OF HENRY IRVING\n\nBY ELLEN TERRY\n\nILLUSTRATED WITH PHOTOGRAPHS\n\n_Copyright, 1908, by Ellen Terry (Mrs. Carew)_\n\n\nI have now nearly finished the history of my fifty years upon the stage.\n\nA good deal has been left out through want of skill in selection. Some\nthings have been included which perhaps it would have been wiser to\nomit. I have tried my best to tell \"all things faithfully,\" and it is\npossible that I have given offence where offence was not dreamed of;\nthat some people will think that I should not have said this, while\nothers, approving of \"this,\" will be quite certain that I ought not to\nhave said \"that.\"\n\n\"One said it thundered ... another that an angel spake----\"\n\nIt's the point of view.\n\nDuring my struggles with my refractory, fragmentary, and unsatisfactory\nmemories, I have realised that life itself is a point of view. So if\nany one said to me: \"And is this, then, what you call your life?\" I\nshould not resent the question one little bit.\n\n\"We have heard,\" continues my imaginary and disappointed interlocutor,\n\"a great deal about your life in the theatre. You have told us of plays\nand parts and rehearsals, of actors good and bad, of critics and of\nplaywrights, of success and failure, but after all your whole life has\nnot been lived in the theatre. Have you nothing to tell us about your\ndifferent homes, your family life, your social diversions, your friends\nand acquaintances? During your long life there have been great changes\nin manners and customs; political parties have altered; a great Queen\nhas died; your country has been engaged in two or three serious wars.\nDid all these things make no impression on you? Can you tell us nothing\nof your life in the world?\"\n\nAnd I have to answer that I have lived very little in the world. After\nall, the life of an actress belongs to the theatre, as the life of a\npolitician to the State.\n\n * * * * *\n\nThe recognition of my fifty years of stage life by the public and by\nmy profession was quite unexpected. Henry Irving said to me not long\nbefore his death in 1905 that he believed that they (the theatrical\nprofession) \"intended to celebrate our Jubilee.\" (If he had lived, he\nwould have completed his fifty years on the stage in the autumn of\n1906.) He said that there would be a monster performance at Drury Lane,\nand that already the profession were discussing what form it was to\ntake.\n\nAfter his death, I thought no more of the matter. Indeed, I did not\nwant to think about it, for any recognition of my Jubilee which did not\ninclude his seemed to me very unnecessary.\n\n [Illustration: SIR HENRY IRVING\n\n FROM A PHOTOGRAPH IN THE POSSESSION OF MISS EVELYN SMALLEY]\n\n [Illustration: _From the collection of Miss Evelyn Smalley_\n\n \"OLIVIA\"\n\n DRAWN BY SIR EDWIN HENRY FOR MISS HENRY'S JUBILEE PROGRAMME]\n\nOf course, I was pleased that others thought it necessary. I enjoyed\nall the celebrations. Even the speeches that I had to make did not\nspoil my enjoyment. The difficulty was to thank people as they deserved.\n\nI can never forget that London's youngest newspaper first conceived the\nidea of celebrating my stage Jubilee. Of course, the old-established\njournals didn't like it, but I suppose no scheme of this kind is ever\norganized without some people not liking something!\n\nThe matinee given in my honour at Drury Lane by the theatrical\nprofession was a wonderful sight. The two things about it which touched\nme most deeply were my visit the night before to the crowd who were\nwaiting to get into the gallery, and the presence of Eleonora Duse,\nwho came all the way from Florence just to honour me. I appreciated\nvery much, too, the kindness of Signor Caruso in singing for me. I did\nnot know him at all, and the gift of his service was essentially the\nimpersonal desire of an artist to honour another artist.\n\nWhen the details of my Jubilee performance at Drury Lane were being\narranged, the committee decided to ask certain distinguished artists\nto contribute to the programme. They were all delighted about it, and\nsuch busy men as Sir Lawrence Alma-Tadema, Mr. Abbey, Mr. Byam Shaw,\nMr. Walter Crane, Mr. Bernard Partridge, Mr. James Pryde, Mr. Orpen,\nand Mr. William Nicholson all gave some of their work to me. Mr.\nSargent was asked if he would allow the first Lady Macbeth sketch to\nbe reproduced. He found that it would not reproduce well, so in the\nheight of the season and of his work with fashionable sitters, he did\nan entirely new sketch, in black and white, of the same subject! This\nact of kind friendship I could never forget, even if the picture were\nnot in front of me at this minute to remind me of it! \"You must think\nof me as one of the people bowing down to you in the picture,\" he wrote\nto me when he sent the new version for the programme. Nothing during my\nJubilee celebrations touched me more than this wonderful kindness of\nMr. Sargent's.\n\nBurne-Jones would have done something for my Jubilee programme too,\nI think, had he lived. He was one of my kindest friends, and his\nletters--he was a heaven-born letter-writer--were like no one else's,\nfull of charm and humour and feeling. Once, when I sent him a trifle\nfor some charity, he wrote me this particularly charming letter:\n\n\"Dear Lady,\n\n\"This morning came the delightful crinkly paper that always means\nyou! If anybody else ever used it, I think I should assault them! I\ncertainly wouldn't read their letter or answer it.\n\n\"And I know the cheque will be very useful. If I thought much about\nthose wretched homes, or saw them often, I should do no more work, I\nknow. There is but one thing to do--to help with a little money if you\ncan manage it, and then try hard to forget. Yes, I am certain that I\nshould never paint again if I saw much of those hopeless lives that\nhave no remedy....\n\n\"You would always have been lovely and made some beauty about you if\nyou had been born there--but I should have got drunk and beaten my\nfamily and been altogether horrible! When everything goes just as I\nlike, and painting prospers a bit, and the air is warm, and friends\nwell, and everything perfectly comfortable, I can just manage to behave\ndecently, and a spoilt fool I am--that's the truth. But wherever you\nwere, some garden would grow.\n\n\"Yes, I know Winchelsea and Rye and Lynn and Hythe--all bonny places,\nand Hythe has a church it may be proud of. Under the sea is another\nWinchelsea, a poor drowned city--about a mile out at sea, I think,\nalways marked in old maps as 'Winchelsea Dround.' If ever the sea goes\nback on that changing coast, there may be great fun when the spires and\ntowers come up again. It's a pretty land to drive in.\n\n\"I am growing downright stupid--I can't work at all, nor think of\nanything. Will my wits ever come back to me?\n\n [Illustration: _From the collection of Miss Evelyn Smalley_\n\n _Copyright by Window & Grove_\n\n ELLEN TERRY AS HERMIONE IN \"THE WINTER'S TALE\"\n\n THE PART PLAYED BY MISS TERRY AT HIS MAJESTY'S THEATRE IN 1906]\n\n\"And when are you coming back--when will the Lyceum be in its rightful\nhands again? I refuse to go there till you come back....\"\n\n * * * * *\n\nOne of those little things almost too good to be true happened at\nthe close of the Drury Lane matinee. A four-wheeler was hailed for\nme by the stage-door keeper, and my daughter and I drove off to Lady\nBancroft's in Berkeley Square to leave some flowers. Outside the\nhouse, the cabman told my daughter that in old days he had often\ndriven Charles Kean from the Princess Theatre, and that sometimes the\nlittle Miss Terrys were put inside the cab too and given a lift! My\ndaughter thought it such an extraordinary coincidence that the old\nman should have come to the stage door of Drury Lane by a mere chance\non my Jubilee day, that she took his address, and I was to send him a\nphotograph and remuneration. But I promptly lost the address, and was\nnever able to trace the old man.\n\n [Illustration: 150 GRAFTON STREET\n\n THE HOUSE WHERE HENRY IRVING LIVED DURING THE PERIOD OF HIS LYCEUM\n MANAGEMENT]\n\nI was often asked during these Jubilee days, \"how I felt about it all,\"\nand I never could answer sensibly. The strange thing is that I don't\nknow even now what was in my heart. Perhaps it was one of my chief joys\nthat I had not to say good-bye at any of the celebrations. I could\nstill speak to my profession as a fellow-comrade on the active list and\nto the public as one still in their service.\n\nAll the time I knew perfectly well that the great show of honour and\n\"friending\" was not for me alone. Never for one instant did I forget\nthis, nor that the light of the great man by whose side I had worked\nfor a quarter of a century was still shining on me from his grave.\n\n * * * * *\n\nIt is commonly known, I think, that Henry Irving's health first began\nto fail in 1896.\n\nHe went home to Grafton Street after the first night of the revival\nof \"Richard III.\" and slipped on the stairs, injuring his knee. With\ncharacteristic fortitude, he struggled to his feet unassisted and\nwalked to his room. This made the consequences of the accident far more\nserious, and he was not able to act for weeks.\n\nIt was a bad year at the Lyceum.\n\nIn 1898, when we were on tour, he caught a chill. Inflammation of the\nlungs, bronchitis, pneumonia followed. His heart was affected. He was\nnever really well again.\n\nWhen I think of his work during the next seven years, I could weep!\nNever was there a more admirable, extraordinary worker; never was any\none more splendid-couraged and patient.\n\nThe seriousness of his illness in 1898 was never really known. He\nnearly died.\n\n\"I am still fearfully anxious about H,\" I wrote to my daughter at the\ntime. \"It will be a long time at the best before he gains strength....\nBut now I do hope for the best. I'm fairly well so far. All he wants is\nfor me to keep my health, not my head. He knows I'm doing that! Last\nnight I did three acts of Sans-Gene and Nance Oldfield thrown in! That\nis a bit too much--awful work--and I can't risk it again.\n\n\"A telegram just came: 'Steadily improving.' ... You should have seen\nNorman[I] as Shylock! It was not a bare 'get-through.' It was--the\nfirst night--an admirable performance, as well as a plucky one.... H.\nis more seriously ill than anyone dreams.... His look! Like the last\nact of Louis XI.\"\n\n [Illustration: HENRY IRVING AS BECKET\n\n THE PART IN WHICH IRVING MADE HIS LAST APPEARANCE ON OCTOBER 13, 1905,\n THE NIGHT OF HIS DEATH]\n\nIn 1902, on the last provincial tour that we ever went together, he\nwas ill again, but he did not give in. One night when his cough was\nrending him, and he could hardly stand up for weakness, he acted so\nbrilliantly and strongly that it was easy to believe in Christian\nScience \"treatment.\" Strange to say, a newspaper man noticed the\nsplendid power of his performance that night and wrote of it with\nuncommon discernment--a _provincial_ critic, by the way.\n\nIn London, at the time, they were always urging Henry Irving to produce\nnew plays by new playwrights! But in the face of the failure of most of\nthe new work, and of his departing strength--and of the extraordinary\nsupport given him in the old plays (during this 1902 tour we took\nL4,000 at Glasgow in one week!)--Henry took the wiser course in doing\nnothing but the old plays to the end of the chapter.\n\nI realised how near, not only the end of the chapter, but the end of\nthe book was when he was taken ill at Wolverhampton in the spring of\n1905.\n\nWe had not acted together for more than two years then, and times were\nchanged indeed.\n\nI went down to Wolverhampton when the news of his illness reached\nLondon. I arrived late and went to an hotel. It was not a good hotel,\nnor could I find a very good florist when I got up early the next\nday and went out with the intention of buying Henry some flowers. I\nwanted some bright- ones for him--he had always liked bright\nflowers--and this florist dealt chiefly in white flowers--_funeral_\nflowers.\n\nAt last I found some daffodils--my favourite flower. I bought a bunch,\nand the kind florist, whose heart was in the right place if his flowers\nwere not, found me a nice simple glass to put it in. I knew the sort of\nvase that I should find at Henry's hotel.\n\nI remembered, on my way to the doctor's--for I had decided to see the\ndoctor first--that in 1892, when my dear mother died, and I did not\nact for a few nights, when I came back, I found my room at the Lyceum\nfilled with daffodils. \"To make it look like sunshine,\" Henry said.\n\nThe doctor talked to me quite frankly.\n\n\"His heart is dangerously weak,\" he said.\n\n\"Have you told him?\" I asked.\n\n\"I had to, because, the heart being in that condition, he must be\ncareful.\"\n\n\"Did he understand _really_?\"\n\n\"Oh, yes. He said he quite understood.\"\n\n(Yet, a few minutes later when I saw Henry, and begged him to remember\nwhat the doctor had said about his heart, he exclaimed: \"Fiddle! It's\nnot my heart at all! It's my _breath_!\" Oh, the ignorance of great men!)\n\n\"I also told him,\" the Wolverhampton doctor went on, \"that he must not\nwork so hard in future.\"\n\nI said; \"He will, though,--and he's stronger than any one.\"\n\nThen I went round to the hotel.\n\nI found him sitting up in bed, drinking his coffee.\n\nHe looked like some beautiful gray tree that I have seen in Savannah.\nHis old dressing-gown hung about his frail yet majestic figure like\nsome mysterious gray drapery.\n\nWe were both very much moved and said little.\n\n\"I'm glad you've come. Two Queens have been kind to me this morning.\nQueen Alexandra telegraphed to say how sorry she was I was ill, and now\nyou----\"\n\nHe showed me the Queen's gracious message.\n\nI told him he looked thin and ill, but _rested_.\n\n\"Rested! I should think so. I have plenty of time to rest. They tell\nme I shall be here eight weeks. Of course I shan't, but still--It\nwas that rug in front of the door. I tripped over it. A commercial\ntraveller picked me up--a kind fellow, but damn him, he wouldn't leave\nme afterwards--wanted to talk to me all night.\"\n\nI remembered his having said this, when I was told by his servant,\nWalter Collinson, that on the night of his death at Bradford he\nstumbled over the rug when he walked into the hotel corridor.\n\nWe fell to talking about work. He said he hoped that I had a good\nmanager ... agreed very heartily with me about Frohman, saying he was\nalways so fair--more than fair.\n\n\"What a wonderful life you've had, haven't you?\" I exclaimed, thinking\nof it all in a flash.\n\n\"Oh, yes,\" he said quietly, ... \"a wonderful life--of work.\"\n\n [Illustration: _Copyright by the London Stereoscopic Co._\n\n HENRY IRVING AS MATTHIAS IN \"THE BELLS\"\n\n IRVING GAVE HIS LAST PERFORMANCE OF \"THE BELLS\" AT BRADFORD, ON\n OCTOBER 12, 1905, THE NIGHT BEFORE HIS DEATH ]\n\n [Illustration: _Copyright by the Topical Press Agency_\n\n IRVING'S DEATH MASK]\n\n\"And there's nothing better, after all, is there?\"\n\n\"Nothing.\"\n\n\"What have you got out of it all?... You and I are 'getting on,' as\nthey say. Do you ever think, as I do sometimes, what you have got out\nof life?\"\n\n\"What have I got out of it?\" said Henry, stroking his chin and\nsmiling slightly. \"Let me see.... Well, a good cigar, a good glass of\nwine--good friends--\" Here he kissed my hand with courtesy. Always he\nwas so courteous--always his actions, like this little one of kissing\nmy hand, were so beautifully timed. They came just before the spoken\nwords, and gave them peculiar value.\n\n\"That's not a bad summing up of it all,\" I said. \"And the end.... How\nwould you like that to come?\"\n\n\"How would I like that to come?\" He repeated my question, lightly, yet\nmeditatively too. Then he was silent for some thirty seconds before he\nsnapped his fingers--the action again before the words.\n\n\"Like that!\"\n\nI thought of the definition of inspiration--\"A calculation quickly\nmade.\" Perhaps he had never thought of the manner of his death before.\nNow he had an inspiration as to how it would come.\n\nWe were silent a long time, I thinking how like some splendid Doge of\nVenice he looked, sitting up in bed, his beautiful mobile hand stroking\nhis chin.\n\nI agreed, when I could speak, that to be snuffed out like a candle\nwould save a lot of trouble.\n\nAfter Henry Irving's death in October of the same year, some of his\nfriends protested against the statement that it was the kind of death\nhe desired--that they knew, on the contrary, that he thought sudden\ndeath inexpressibly sad.\n\nI can only say what he told me.\n\nI stayed with him about three hours at Wolverhampton. Before I left,\nI went back to see the doctor again--a very nice man, by the way, and\nclever. He told me that Henry ought never to play \"The Bells\" again,\neven if he acted again, which he said ought not to be.\n\nIt was clever of the doctor to see what a terrible emotional strain\n\"The Bells\" put upon Henry--how he never could play the part of\nMatthias \"on his head,\" as he could Louis XI., for example.\n\nEvery time he heard the sound of the bells, the throbbing of his heart\nmust have nearly killed him. He used always to turn quite white--there\nwas no trick about it. It was imagination acting physically on the body.\n\n [Illustration: _From the collection of Miss Evelyn Smalley_\n\n IRVING'S TOMB IN WESTMINSTER ABBEY]\n\nHis death as Matthias--the death of a strong, robust man--was different\nfrom all his other stage deaths. He did really almost die--he imagined\ndeath with such horrible intensity. His eyes would disappear upwards,\nhis face grow gray, his limbs cold.\n\nNo wonder, then, that the first time that the Wolverhampton doctor's\nwarning was disregarded, and Henry played \"The Bells\" at Bradford, his\nheart could not stand the strain. Within twenty-four hours of his last\ndeath as Matthias, he was dead.\n\nWhat a heroic thing was that last performance of Becket which came\nbetween! I am told by those who were in the company at the time that\nhe was obviously suffering and dazed this last night of life. But he\nwent through it all as usual. All that he had done for years, he did\nfaithfully for the last time.\n\nYes, I know it seems sad to the ordinary mind that he should have\ndied in the entrance to an hotel in a country town, with no friend,\nno relation near him; only his faithful and devoted servant, Walter\nCollinson, whom--as was not his usual custom--he had asked to drive\nback to the hotel with him that night, was there. Do I not feel the\ntragedy of the beautiful body, for so many years the house of a\nthousand souls, being laid out in death by hands faithful and devoted\nenough, but not the hands of his kindred either in blood or in\nsympathy?...\n\nI do feel it, yet I know it was more appropriate to such a man than\nthe deathbed where friends and relations weep. Henry Irving belonged\nto England, not to a family. England showed that she knew it when she\nburied him in Westminster Abbey.\n\nYears before I had discussed, half in joke, the possibility of this\nhonour. I remember his saying to me with great simplicity, when I asked\nhim what he expected of the public after his death: \"I should like them\nto do their duty by me. And they will--they will!\"\n\nThere was not a touch of arrogance in this, just as I hope there was\nno touch of heartlessness in me because my chief thought during the\nfuneral in Westminster Abbey was: \"How Henry would have liked it!\"\nThe right note was struck, as I think was not the case at Tennyson's\nfuneral thirteen years earlier.\n\n\"Tennyson is buried to-day in Westminster Abbey,\" I wrote in my diary\nOctober 12th, 1892. \"His majestic life and death spoke of him better\nthan the service.... The music was poor and dull and weak while he was\n_strong_. The triumphant should have been the sentiment expressed.\nFaces one knew everywhere. Lord Salisbury looked fine. His massive head\nand sad eyes were remarkable. No face there, however, looked anything\nby the side of Henry's.... He looked very pale and slim and wonderful!\"\n\nHow terribly I missed that face at Henry's own funeral! I kept on\nexpecting to see it, for indeed it seemed to me that he was directing\nthe whole most moving and impressive ceremony.... I could almost hear\nhim saying \"Get on! get on!\" in the parts of the service that dragged.\nWhen the sun, such a splendid tawny sun, burst across the solemn misty\ngray of the Abbey, at the very moment when the coffin, under its superb\npall of laurel leaves, was carried up to the choir, I felt that it was\nan effect which he would have loved.\n\nI can understand any one who was present at Henry Irving's funeral\nthinking that this was his best memorial, and that any attempt to\nhonour him afterwards would be superfluous and inadequate. But after\nall it was Henry Irving's commanding genius and his devotion of it\nto high objects, his personal influence on the English people, which\nsecured him burial among England's great dead. The petition for the\nburial, presented to the Dean of the Chapter, and signed on the\ninitiative of Henry Irving's leading fellow-actors by representative\npersonages of influence, succeeded only because of Henry's unique\nposition.\n\n\"We worked very hard to get it done,\" I heard said more than once. And\nI often longed to answer: \"Yes, you worked for it between Henry's death\nand his funeral. He worked for it all his life!\"\n\nI have always desired some other memorial to Henry Irving than his\nhonoured grave; not so much for his sake as for the sake of those who\nloved him, and would gladly welcome the opportunity of some great test\nof their devotion.\n\nTHE END\n\n [I] Mr. Norman Forbes-Robertson.\n\n\n\n\nTHE VALLEY OF MILLS\n\nBY H. G. DWIGHT\n\nWITH A PAINTING BY F. BRANGWYN\n\n_The American Dragoman narrates to the Second Secretary_\n\n\nI shall never forget the night I got there. The train went no farther\nthan Nicomedia in those days, and it took so long that you nearly died\nof old age on the way. But when the three red lights on the tail of\nit dwindled into the dark, I had the queerest sense of having been\ndropped into another world. It was the more so because one couldn't\nsee an earthly thing--not a star, not even the Gulf which we were to\ncross. I only heard the lapping of it, close by, when the rumble of the\ntrain died out of the stillness. That and the crunch of steps on the\nsand were all there was to hear, and an occasional word I didn't catch.\nThe men could hardly have been more silent if our lives had depended\non it. I had no idea how many of them there were, or what they looked\nlike--much less where they were taking me. They simply hoisted a sail\nand put off into the night. I would have sworn, too, that there was\nno wind. The sail filled, however: I could see the swaying pallor of\nit, and hear the ripple under the bow. And as my eyes got used to the\ndarkness, I discovered an irregular silhouette in front of us, and a\nfloating will-o'-the-wisp of a light. The silhouette grew taller and\nblacker till the boat grounded under it. Then, by the light of the\nwill-o'-the-wisp, which was a sputtering oil lantern on shore, I made\nout some immense cypresses. You have no idea how eerie that landing\nwas, in a waterside cemetery that was for all the world like Boecklin's\nIsland of Death. The men moved like shadows about their Flying\nDutchman of a boat, and their lantern just brought out the ghostliness\nof gravestones leaning between the columns of the cypresses. And I\nsuddenly became aware of the strangest sound. I had no idea what it was\nor where it came from, but it was a sort of low moaning that fairly\nwent into your bones. It grew louder when we started on again. We\nclimbed an invisible trail where branches slashed at us in the dark,\nand all kinds of sharp and sweet and queer smells came put of it in\nwaves. And nightingales began to sing like mad around us, and off in\nthe distance somewhere jackals were barking, and under it all that low\nmoaning went on and on and on. And at last we came out into an open\nspace on top of the hill, where a bonfire made a hole in the black, and\na couple of naked figures stood redly out in the penumbra of it, with a\nring of faces flickering around them....\n\nI found out afterwards that the bonfire business was nothing but a\nwrestling match--they had them almost every night on the _meidan_--and\nthe moaning came from the mill-wheels in the valley. But I never quite\ngot over that first impression--that sense of walking through all kinds\nof things without seeing them. No sooner would I begin to feel a bit at\nhome than something would bring me up with a jerk and remind me that\nI was a stranger in a strange land. I suppose it was natural enough,\nconsidering that I had only just come out then. The place was nothing\nbut a snarl of muddy lanes and mud shanties, tossed into a filbert\nvalley where water tumbled down to the Gulf. It was only about fifty\nmiles away from here, but it might have been five thousand and fifty.\nThere was none of the contrast with Europe that is always bothering you\nhere--though perhaps it really sets things off. The people were all\nTurks, and their village was Asia pure and simple. That extraordinary\njuxtaposition of care and neglect, of the exquisite and the nauseating,\nwhich begins to strike you in Italy, and which strikes you so much more\nhere, simply went to the top notch there. It was under your eyes--and\nnose--every minute. There were rugs and tiles and brasses that you\ncouldn't keep your hands off of, in houses plastered with cow-dung. And\nthe people used the gutters for drains, and their principal business\nwas making attar of rose. You should have seen what gardens there\nwere, hidden away behind mud-walls!\n\nWhat struck me most, though, was a something in it all which I never\ncould lay my finger on. It seemed incredible that a country inhabited\nso long should show so few signs of it. The people might have camped\nin a clearing over night, and the woods were just waiting to cover up\ntheir tracks. But the wildness was not the good blank, unconscious\nwildness we have at home. There was a melancholy about it. The silence\nthat hung over the place was really a little uncanny. The mills only\ncried it out, in that monotonous minor of theirs. They were picturesque\nold wooden things, all green with moss and maidenhair fern, that went\ngrinding and groaning on forever, and making you wonder what on earth\nit was all about. I can't say that I ever found out, either. But I\ncertainly got grist enough for my own mill.\n\nFor that matter, I don't imagine that I was precisely an open book\nmyself. In this part of the world they haven't got our passion for\npoking around where we don't belong: perhaps they've had more time to\nfind out how little there is in it. And for a mysterious individual\nfrom lands beyond the sea, whose servant can't be prevented from\nbragging of the splendor in which he lives at Constantinople, to bury\nhimself in a wild country village, must mean something queer. Does one\ngive up a _konak_ on the Bosphorus for a _khan_ in the Marmora? And\nare there no teachers of Turkish in Stamboul? I believe it didn't take\nlong for the _Moutessarif_ of Nicomedia to find out I was there, and\nfor him to ascertain in ways best known to himself what I was up to. I\nhave often wondered what his version of it was. At all events it didn't\nprevent the great men of the village from smoking cigarettes of peace\nwith me in a little vine-shaded coffee-house at the top of the hill.\nThere was the _Mudir_, a plump and harmless _effendi_ of a governor;\nand the _Naib_, who was some kind of country justice; and a charming\nold _Imam_ in a green turban and a white beard and a rose-\nrobe; and a _Tchaouche_, an officer of police, all done up in yellow\nbraid and brass whistles; and various other personages. And I couldn't\nimagine where in the world they had all picked up their courtliness and\nconversation. The _Mudir_ was from town, and one or two of the others\nhad been there; but if such things were to be had for a visit to town\nthey'd be a little more common at home. Of course, I was asked a good\nmany questions, and some of them were pretty personal. That is a part\nof Oriental etiquette, you will find. It was marvelous, though, what a\n_savoir faire_ they had, to say nothing of a sense of life and a few\nother things. I couldn't make them out--taken with their vile village\nand their half-tamed fields. The thing used to bother me half to death,\ntoo. I thought all I had to do was to sit down and look pleasant and\nturn them inside out at my leisure. Whereas more than once I had a\nvague feeling, after it was over, of having been turned inside out\nmyself. Altogether it makes me grin when I remember what an idiotic\nyoung ostrich I was. I have been at the business quite a while now, and\nto this day I am never sure of my man--how that Asiatic head of his\nwill work in any given case. I can only console myself by remembering\nthat I'm not the only one. In the last two generations I presume there\nmust have been as many as four Anglo-Saxons--and three of those,\nEnglishmen--who didn't more or less make jackasses of themselves when\nthey ran up against Asia. And I fancy it took them rather more than a\nyear to arrive at even that negative degree of comprehension.\n\nHowever, various things went into my hopper first and last, to the tune\nof the mill-wheels in the valley--particularly last.... It was lucky\nfor me that the wireless telegraphy I sometimes felt about me allowed\nthe _Mudir_ to cultivate his natural inclinations. He was bored enough\nin his exile, and I think he was genuinely glad that his advices from\nheadquarters made him free of my company. I certainly am. I have never\ncome into just such relations with any of the officials here. He was a\ngrave, mild, suave personage who might have made an excellent _Cadi_\nof tradition if he had never heard of Paris. As it was, I'm afraid he\ntook less thought for his peasants' troubles than of the extent to\nwhich they could be made to repay him for his own. He liked to practise\nhis French on me as much as I liked to practise my Turkish on him,\nand on such occasions as I had the honor of squatting at his little\nround board, his knowledge of the Occident would manifest itself in an\nincredible profusion of spoons. I also discovered that he was by no\nmeans averse to sampling my modest cellar. He didn't care so much about\nbeing found out, though. They are tremendous prohibitionists, you know,\nand while the pashas have accepted champagne with their tight trousers,\nthey're not so public about it. Just watch when you go to your first\ncourt dinner.\n\nA person of whom I thought more than the _Mudir_, and who interested\nme more as a type, was the _Imam_. A more kindly, honest, simple,\ndelightful old man it has seldom been my luck to meet. He was a Turk\nof the old school, without an atom of Europe in his composition. I\nwish they were not getting so confoundedly rare. They are worth a\nmillion times more than these Johnnies who pick up the Roman alphabet\nand a few half-baked ideas about what we are pleased to call progress.\nI took daily lessons from him. He was a mighty theologian--made me\nread the Koran, and all that, and was much interested in what I had to\ntell him of our own beliefs. He used to make me ashamed of knowing so\nlittle about them. Before he got through with me, he taught me rather\nmore than was in the bond, I fancy. I had always cherished a notion\nthat because a Turk could have four wives, and didn't think much of\nmy chances for the world to come, and was somewhat free in the use of\nantidotes to human life, his morality wasn't worth talking about. But I\ngot something of an eye-opener on that point.\n\nAltogether, I managed to have a very decent time of it. My pill of\nlearning the most of the language in the least possible time was so\ningeniously sugared that the business was one prolonged picnic. In\nfact, living in a _khan_, as I did at first, is nothing but camping.\nThey're all about the same, you know. You can see the model any day\nover in Stamboul--a rambling stack of galleries round a court of\ncattle and wheels, and big bare rooms where twenty people could live.\nThey often do, too. You spread your own bedding on the wooden divan\nsurrounding two or three sides of the room, and your servant cooks for\nyou in a series of little charcoal pits under the huge chimney. It's\nrather amusing for a while, if you're not too fussy about smells and\ncrawling things. I suppose I must have been, for the _Mudir_ eventually\npersuaded me to rent a house from an absentee rose-growing pasha. It\nwas about the only wooden one in the place--a huge rattlety-bang old\naffair that stood on the edge of the bluff, a little apart from the\ntown. It leaked so villainously that I had to sit under an umbrella\nevery time there was a shower, but the view and the garden made up for\nit. I used to prowl around the country a good deal, though. Everything\nwas so strange to me--the faces, the costumes, the curious implements,\nthe hairy black buffaloes, the fat-tailed sheep with their dabs of\nred dye, the solid-wheeled carts that lamented more loudly, if less\ncontinuously, than the water-wheels, the piratish-looking caravels\nstrutting up and down the Gulf under a balloon of a mainsail. I took\nthem by the day, sometimes, to go fishing or exploring. All of which\nmust have been highly incomprehensible to my astonished neighbors. I\nbelieve my man had to invent some legend of a doctor and a cure to\naccount for so eccentric a master. It was only when I came more and\nmore to spend my days among the cypresses on the edge of the beach\nthat I became less an object of suspicion; for while a Turk is little\nof a sportsman and less of mere aimless sight-seer, he likes nothing\nbetter than sitting philosophically under the greenwood tree.\n\nMy greenwood was, as I have said, a cemetery. Heaven knows how long it\nhad been there. The cypresses were enormously tall and thick and dark.\nAnd the stones under them--with their carved turbans and arabesques,\nand their holes and rain-hollows for restless or thirsty ghosts--were\nall gray and lichened with time, and pitched every which way between\nthe coiling roots. You may think it a queer kind of place to sit around\nin, but it took my fancy enormously. I don't know--there was something\nso still and old about it, and the spring had such a look between the\nblack trees. It wasn't quite still, either, for that strange, low minor\nof the water-wheels was always in your ears. It ran on and on, like the\nsound of the quiet and the sunshine and the cypresses and the ancient\nstones. And it made all sorts of things go through your head. I presume\nthat first impression had something to do with it. You wondered whether\nthe trees would have lived so long if so many dead people had not lain\namong their roots. You wondered--I don't know what you didn't wonder.\n\nAs hot weather came on, I used to pack a hammock and reading and\nwriting and cooking things on a donkey nearly every day, and drop down\nthrough the filberts to my cypresses. There was fairly decent bathing\nthere, over an outrageous bottom of stones and sea-urchins. What I\nliked best, though, was simply to lie around and watch the world go\nby. Not that much of it does go by the Gulf of Nicomedia. If it hadn't\nbeen for a sail every now and then, you would have supposed that people\nhad forgotten all about that little blue pocket of a firth leading\nnowhere between its antique hills. Then there were two or three trains\na day, whose black you could just make out, crawling through the green\nof the opposite shore. And there was a steamer a day each way that it\nwas as much as your life was worth to put your foot into. You wouldn't\nthink so, though, to see the people who packed the decks. Sometimes I\nused to go down to the landing for the pleasure of the contrast they\nmade, solemnly huddled up in their picturesque rags, with the noisy\nmodern steamer. It was a miracle where so many of them came from and\nwent to. That's the wildest part of the Marmora, you know, for all\ntheir railroad on the north shore. Some day, I suppose, when German\nexpresses go thundering through to the Persian Gulf, it'll be all\nfactory chimneys and summer hotels, like the rest of the world. But\nnow there's nothing worse than vineyards and tobacco plantations. On\nthe south coast there's hardly that. The hills stand up pretty straight\nout of the water, and they're wooded down to the rocks. You might\nthink it virgin forest if you didn't know the Nicene Creed came out of\nit--to say nothing of invisible villages, and eyes looking out at you\nwithout your knowing. It all gave one such an idea of the extraordinary\nwreckage that has been left on the shores of that old Greek Sea. Only\nyou don't get it as you do here, where races and creeds march past you\non the Bridge while you stand by and admire. There's something more\nsecret and ancient about it--more like Homer and the Bible and the\nArabian Nights.\n\nThe caravans gave the most telling touch. You don't often see camels\nup here any longer, but they're still common enough in the interior.\nI could hardly believe my eyes the first time a procession of them\nappeared on my beach. First came a man on horseback, with a couple\nof Persian saddle-bags to make your mouth water, and then the long\nstring of camels roped together like barges in a tow. What an air\nthey had--the fantastic tawny line of them swinging against the blue\nof the Gulf! And how softly they padded along the shingle, with the\npicturesque ruffians in charge of them throned high among their\nmysterious bales! They passed without so much as a turn of the eye, my\nWise Men of the East, and disappeared behind the point as silently as\nthey came. It gave me the strangest sensation. I had felt something of\nthe same before. I could scarcely help it, looking out between those\ntragic trees at the white strip of beach and the blue strip of sea and\nthe green strip of hills that were so much like other hills and seas\nand beaches and yet so different. But there had never come to me before\nquite such a sense of the strangeness of this world where so many\nthings had been buried from the time of Jason and the Argo--of this\nworld of which I knew nothing and to which I was nothing.\n\nYou may believe that I was delighted when I went back to the village\nthat night and found it full of camels. The air was sizzling with\nbonfires and _kebabs_--you know those bits of lamb they broil on a long\nwooden spit?--and strange faces were at every corner. They filled the\ncoffee-house, too, when I finally got there. By that time it was too\ndark to stare as hard as I would have liked. But perhaps the scene was\nall the more picturesque for the shadowy figures scattered under the\nvine in the dusk, and the bubble of nargilehs filling the intervals of\ntalk. A feature would come saliently out here and there in the red of\na cigarette--a shining eye, a hawk nose, a bronzed cheek-bone. And out\non the _meidan_ were groups around fires, with their little pipes that\nhave all the trouble of the East in them, and their little tomtoms of\nsuch inimitable rhythms.\n\nI found my friends established as usual in the seat of honor--an old\nsofa in the corner of the cafe--and as usual they made place for me\namongst them. When the ceremony of their welcome subsided, the _Mudir_\ntook occasion to whisper to me that the leader of the caravan, an\nexcellent fellow who had stopped there before, was telling stories. I\nthen recognized, in the light of the _cafedij's_ lamp, the man I had\nseen that afternoon on horseback. He sat on a stool in front of the\ndivan of honor, and behind him were crowded all the other stools and\nmats in the place. Although he had not deigned, before, to turn his\nhead toward me, he now testified by the depth of his salaam to the\nhonor he felt in such an addition to his circle. He was a curiously\nhandsome chap, burnt and bearded, with the high-hung jaw of his people,\nthe arched brow, the almost Roman nose. And, shaky as I still was in\nthe language, he didn't leave me long to wonder why he was the center\nof the circle. He was a born _raconteur_--one of those story-tellers\nwho in the East still carry on the tradition of the troubadours. Not\nthat he sang to us, or recited poetry--although the _Imam_ told me with\npride that the man was a dictionary of the Persian poets. But he went\non with a story he had begun before my entrance. It was one of those\nendless old eastern tales that are such a charming mixture of serpent\nwisdom and childish _naivete_. And he told it with a vividness of\ngesture and inflection that you never get from print.\n\nWell, you can imagine! I always had a fancy for that sort of thing,\nbut it's so deuced hard to get at--at least, for people like us. And\nafter that queer turn the first sight of the caravan gave me, down\nby the water, it made me feel as if I were really beginning to lay\nmy hand on things at last. So I was disappointed enough when at the\nend of the story the party began to break up. Upon my signifying as\nmuch to my neighbor, the _Mudir_, however, he said that nothing would\nbe easier than to summon the man to a private session. If I would do\nhim the honor to come to the _konak_--I was tickled enough to take\nup with the idea, provided the meeting should take place at my house\ninstead. I knew there would be bakshish, which I didn't like to put the\n_Mudir_ in for, after all he had done. Moreover, I had a whim to get\nthe camel-driver under my own roof--by way of nailing the East, so to\nspeak!\n\nSo the upshot of the business was that we made a night of it. Oh, I\ndon't mean any of your wild and woolly ones. To be sure, we did wet\nthings down a trifle more than is the custom of the country. There\nhappened to be a decanter on the table, which the camel-driver looked\nat as if he wouldn't mind knowing what it contained; and being a bit\nawkward at first, I knew no better than to trot it out. The _Mudir_, to\nwhom of course I offered it first, wouldn't have any. I suppose he had\nhis reputation to keep up before an inferior. I was rather surprised,\nall the same, for it was plain enough that the camel-driver was by\nno means the kind of man the name implies, and a little Greek wine\nwouldn't hurt a baby. Moreover, I had heard of this _raki_ of theirs,\nwhich is so much fire-water, and I didn't take their temperance very\nseriously. As for the camel-driver, he was rather amusing.\n\n\"You tempt me to my death!\" he laughed, taking the glass I poured\nout for him. \"Do you know that my men would kill me if they saw me\nnow? These country people have not the ideas of the _effendi_ and\nmyself. They follow blindly the Prophet, not realizing how many rooms\nthere are in the house of a wise man. They found out that I had been\naffording opportunity for the forgiveness of God, and they took it\nquite seriously. They threatened to kill me if I did not make a public\nconfession. And I had to do it, to please them. On the next Friday I\nmade a solemn confession of my sins in mosque, and swore never to smell\nanother drop.\"\n\nAt this I didn't know just what to do. I looked at the _Mudir_, and the\n_Mudir_ looked at the camel-driver. The latter, however, waved his hand\nwith a smile of goodfellowship.\n\n\"There is no harm now,\" he said. \"We break caravan to-morrow at\nNicomedia. Moreover, I do not drink saying it is right. I should\nblaspheme God, who has commanded me not to drink. But I acknowledge\nthat I sin. Great be the name of God!\" With which he tipped the glass\ninto his mouth. \"My soul!\" he exclaimed, \"That is better than a\ncucumber in August!\"\n\nThese people are democratic, you know, to a degree of which we haven't\nan idea--for all our declaration of independence. Yet there are certain\ninvisible lines which are sure to trip a foreigner up and which made\nme mighty uncertain what to do with the governor of a _mudirlik_ and\nthe leader of a caravan. But the latter proceeded to look out for that.\nSuch a jolly good fellow you never saw in your life, with his stories,\nand the way he had with him, and the things he had been up to. It\nturned out that he knew western Asia a good deal better than I know\nwestern Europe. Tabriz, Tashkend, Samarkand, Cabul, to say nothing of\nMecca and Cairo and Tripoli--such names dropped from him as Liverpool\nand Marseilles might from me. Where camel goes he had been, and for\nhim Asia Minor was no more than a sort of ironic tongue stuck out at\nEurope by the huge continent behind. It gave me my first inkling of how\nthis empire is tied up. It seems to hang so loosely together, without\nthe rails and wires that put Sitka and St. Augustine in easier reach\nof each other than Constantinople and Bagdad. I began to learn then\nthat wires and rails are not everything--that there are stronger nets\nthan those. Altogether it was a momentous occasion. To sit there in\nthat queer old house, in a wild hill village of the Marmora, and speak\nfamiliarly with that camel-driver who carried the secrets of Asia in\nhis pocket--it brought me nearer than I had ever dreamed to that life\nwhich was always so tantalizing me by my inability to get at it.\n\nWhen the man finally withdrew, and the _Mudir_ after him, I was in no\nmood to go to bed. They had opened to me their ancient world, with all\nits poetry and mystery, and I did not want to lose it again. I could\nsee it stretching dimly beyond the windows where the water-wheels went\nmoaning under the moon. I went out into it. The night was--you have no\nidea what those nights could be. They had such a way of swallowing up\nthe squalidness of things, and bringing out all their melancholy magic.\nThe rose season was at its height, and the air was one perfume from the\nhidden gardens. Then the nightingales were at that heart-breaking music\nof theirs. And the moon! It wasn't one of those glaring round things,\nlike a coachman's button or a butcher's boy with the mumps, by which\nyoung ladies are commonly put into spasms; but it was an old wasted\none, with such a light!\n\nIt was all the more extraordinary because not a creature was\nabout--except a man who lay asleep on the ground, not far from the\ndoor. Apparently they dropped off wherever they happened to be, down\nthere, and I used to envy them for it. I stood still for a while, in\nthe shadow of the house, taking it all in. Don't you know, it happens\nonce in a while that you have a mood, and that your surroundings come\nup to it? It doesn't happen very often, either--at least, to workaday\npeople like us. So I stood there, looking and listening and breathing.\nAnd when I saw the edge of the shadow of the house crumble up at one\nplace, without any visible cause, and creep out into the moonlight,\nI--I only looked at it. Nothing had any visible cause in that strange\nworld of mine, and I watched the slowly lengthening finger of shadow\nwith the passivity of a man who has seen too many wonders to wonder any\nmore. But then I made out a darker darkness winding back toward the\nhouse. And--I don't know--I thought of the man on the ground. I looked\nat him.\n\nIt was my camel-driver, dead as Darius, with the blood running out\nof a hole in his back like water out of a spout. For the moment I\nwas still too far away from every day to be startled, or even very\nmuch surprised. It was only a part of that mysterious world, with its\nmysterious people and mysterious ways that I never could understand.\nWhat was he doing there dead, who had been so full of life a little\nwhile before? Was it one of his jokes? The night was the most\nenchanting you could imagine, the air was heady with the breath of\nrose-gardens, the nightingales were singing in the trees (down in the\nvalley I heard, low, low, the weary water-wheels), and here was the\nprince of story-tellers with his tongue stopped forever, and the blood\nof him making a snaky black trail across the moonlight....\n\n * * * * *\n\nWhat happened next? My dear fellow, you remind me of these kids who\nwill never let you finish their story! Nothing happened next. That was\nthe beauty of it. I guess I got one pretty good case of the jim-jams\nafter a while, and when I got through wondering whether I was going\nto be elected next, I began to wonder whether they wouldn't think I'd\ndone it. Of course, I had done it, as a matter of fact, and that didn't\ntend to composure of mind. Neither did my speculations as to what the\n_Mudir_ might or might not have noticed when he left me that evening.\nBut, if you will believe it, nobody ever lifted a finger. The next\nmorning the caravan was gone and apparently everything was the same as\nbefore. If anything, they were more decent than before. That was the\nworst of it. I don't believe I'd have minded so much if they'd stoned\nme and ridden me out on a rail and set the Government after me and\nraised the devil generally. I should at least have felt less at sea. As\nit was--hello, there's Carmignani! Let's take him over to Tokatlian's.\n\n\n\n\nTHE UNREMEMBERED\n\nFRAGMENTS OF A LOST MEMORY\n\nBY FLORENCE WILKINSON\n\n\n Where have they gone, the unremembered things,\n The hours, the faces,\n The trumpet-call, the wild boughs of white spring?\n Would I might pluck you from forbidden spaces,\n All ye, the vanished tenants of my places!\n\n Stay but one moment, speak that I may hear,\n Swift passer-by!\n The wind of your strange garments in my ear\n Catches the heart like a beloved cry\n From lips, alas, forgotten utterly.\n\n An odour haunts, a colour in the mesh,\n A step that mounts the stair;\n Come to me, I would touch your living flesh--\n Look how they disappear, ah, where, ah, where?\n Because I name them not, deaf to my prayer.\n\n If I could only call them as I used,\n Each by his name!\n That violin--what ancient voice that mused!\n Yon is the hill, I see the beacon flame.\n My feet have found the road where once I came.\n Quick--but again the dark, darkness and shame.\n\n\n\n\nTHE BATTLE AGAINST THE SHERMAN LAW\n\nHOW CAPITAL AND LABOR COMBINE TO SAFEGUARD THE TRUST AND LEGALIZE THE\nBOYCOTT\n\nBY BURTON J. HENDRICK\n\nILLUSTRATED WITH PHOTOGRAPHS\n\n\nUnder the existing laws of the United States, it is a crime to\norganize a combination of individuals or corporations into a business\naggregation in restraint of trade. It is likewise a crime for labor\nmen or labor unions in different States to combine for the prosecution\nof certain aggressive enterprises popularly described as boycotts. Any\nperson convicted of engaging in either of these prohibited acts may be\nfined not more than $5,000 for each offense, or imprisoned for one year\nat hard labor, or both.\n\nAccording to reliable estimates, there are in the neighborhood of five\nhundred large trusts or combinations that daily violate this law.\nThere are many thousands of smaller corporations and business firms\nthat indulge in secret practices for which their officers may at any\ntime be lodged in jail. As for the national prohibition of boycotts,\nlabor organizations openly exist for the express purpose of conducting\nthem. The constitution of the most powerful labor organization in this\ncountry, the Federation of Labor, specifically provides for engaging in\nthis form of industrial warfare.\n\nThe statute that outlaws these combinations of both capital and labor\nis the famous Sherman Anti-trust Law. It is one of the briefest, most\npointed, and most comprehensive measures ever passed by Congress. It\ncontains only about seven hundred words and would fill less than a page\nof this magazine. In its first three lines, without any modifications\nor circumlocutions, it declares illegal \"every contract, combination in\nthe form of trust or otherwise, or conspiracy, in restraint of trade or\ncommerce among the several States or with foreign nations.\" The next\nfew lines provide the punishment, cited above, for breaking the law.\nThe Sherman Act does not say that \"some combinations\" are illegal and\ncriminal, but that \"every\" one is. It does not provide that certain\noffenders may be punished, but that \"every\" one \"shall be.\" It leaves\nabsolutely no discretion to prosecuting officers or to the courts.\nWithin its comprehensive folds are gathered, on the one hand, the most\ncommanding captains of industry and the greatest railroad magnates;\nand, on the other, the most insignificant puddlers in their furnaces\nand stokers on their trains.\n\nThe Sherman Act has thus established a community of interest between\nlabor and capital which has had important practical results. Both\ncapital and labor are openly evading the law. Both have many times been\nhaled into court, convicted of infringing this statute, and enjoined\nfrom continuing in their illegal combinations. Both consequently find\nit an irksome impediment to their present plans and ambitions. In their\nactive opposition to the law the two previously warring elements now\nmeet on common ground.\n\nThe platform of the Republican party calls for amendments which, to\nall practical purposes, will seriously weaken the law, so far as its\napplication to corporate combinations is concerned. The Democratic\nplatform demands such changes as will exempt labor unions from its\noperation,--which is virtually the same thing as demanding the\nlegalization of the boycott. At the last session of Congress the\nspectacle was presented of important labor unions and great corporation\nlawyers working hand in hand to this common end. Though this agitation\nfailed for the time being, it may safely be asserted that the repeal\nor modification of the Sherman Act will continue to be a fixed article\nof the policy both of large aggregations of wealth and of large\naggregations of labor. This fact makes important a study of its history\nand of its practical effects upon corporate and labor organizations.\n\n\n_The Sherman Law Not Rushed Through Congress_\n\nHardly any important legislation has been so imperfectly understood\nor more persistently misrepresented. Although the law was passed only\neighteen years ago, a large number of legends have already grown up\nabout it. According to popular belief, the Sherman Anti-trust Act is an\nimperfect piece of legislation; a measure which was drawn up hastily,\nwithout thorough study or knowledge of the economic and social problems\nwhich it was intended to solve. The corporations declare that it was\nnever intended to meet industrial conditions as they exist now: labor\nleaders have repeatedly asserted that the framers of the measure never\nintended that it should affect organizations of labor.\n\nA study of the congressional debates which preceded the passage of the\nSherman Act dissipates these misconceptions. The law was not rushed\nthrough Congress. It was seriously proposed as a carefully thought-out\nattempt to check great and clearly comprehended evils. In essence those\nevils did not differ from the ones which confront the American people\ntoday. In 1890 the trust, or the industrial combination, had almost\nreached its present state of development. Large aggregations of capital\nhad already secured a monopoly of many of the necessaries of life. The\nStandard Oil Trust was then, as it is now, the most conspicuous of\nthese combinations, and had already attained an unpopularity almost\nas great as it enjoys today; the Sugar Trust controlled practically\nthe whole output of refined sugar. The Steel Trust, it is true, did\nnot exist; but many combinations in steel products had already been\nformed. Combinations on steel rails dictated prices; nails, barbed\nfence wire, copper, lead, nickel, zinc, cordage, cottonseed oil,--all\nthese products had already been brought largely under trust control.\nThe Salt Trust and the Whiskey Trust had been organized. Combinations\nof railroads, for the purpose of fixing charges for transportation,\nhad existed for twenty-five years. In 1875 Commodore Vanderbilt\ncalled the first great meeting of railroad trunk lines at Saratoga;\nand this conference adopted a \"pooling\" arrangement. The accumulated\nrailroad abuses of a generation, especially this practice of \"pooling\"\nearnings, had led to the passage of the Interstate Commerce Act in\n1887--three years before the enactment of the Sherman Law.\n\nOther combinations, which disdained the name of trusts, but which had\nalready developed certain points in common with them, also flourished.\nThe labor union, for example, was in full flower. The Knights of\nLabor, under Powderly, had passed through many triumphant years; the\nFederation of Labor was firmly entrenched, and Samuel Gompers was its\nPresident then as he is today. The unions existed then, as they do\nnow, to secure higher wages and greater advantages of employment for\ntheir members; and one of their weapons then, as it is at present, was\nthe boycott. Organizations of farmers, which existed for a similar\npurpose--the Farmers' Alliance, the National League--had also reached a\nhigh state of development.\n\n\n_Statesmen who Framed the Sherman Law_\n\nNor were the framers of this law inexperienced legislators who hastily\nscrambled together a measure to meet certain political exigencies. The\nmen chiefly responsible for the anti-trust law were John Sherman of\nOhio, George F. Edmunds of Vermont, George F. Hoar of Massachusetts,\nGeorge Gray of Delaware, and James Z. George of Mississippi. Senator\nSpooner recently declared that no greater body of lawyers ever sat in\nCongress; no one would venture to contend that there is any similar\ngroup of five men in Washington today. John Sherman had served almost\ncontinuously in Congress since 1854; he had represented Ohio in\nthe Senate throughout the Civil War and the reconstruction period,\ndisplaying especial talent in dealing with questions of national\nfinance; and, as Secretary of the Treasury in President Hayes' cabinet,\nhad carried through with masterly success the resumption of specie\npayments. George F. Edmunds was generally regarded as the greatest\nlawyer then in the Senate. Starting his career in that body in 1866,\nwhen Congress had to handle the intricate constitutional problems\ninvolved in the readmission of the Southern States, he immediately\nbecame one of an influential group of which the other members were\nSumner, Fessenden, Trumbull, and Wade, and took an important part in\nframing the legislation of the reconstruction period. George F. Hoar\nhad, by 1890, represented Massachusetts in the Senate for thirteen\nyears; his great learning, his comprehensive knowledge of public\nquestions, his independence, his genuine devotion to the best public\ninterests had made him one of the most commanding figures in that body.\nGeorge Gray of Delaware, at present a judge of the United States\nCircuit Court, and for many years one of the most conservative forces\nin the Democratic party--the same George Gray upon whom many of Mr.\nBryan's opponents hoped to unite a few months ago as the Democratic\npresidential nominee--was also recognized as one of the Senate's\ngreatest authorities on the Constitution. Senator George had served for\nmany years as chief justice of the Supreme Court of Mississippi, and\nwas the author and compiler of many works on law which are still widely\nused.\n\nOver the question of federal control of large combinations these five\nmen and their colleagues debated for nearly two years. Senator Sherman\nintroduced his first anti-trust act August 14, 1888; the present\nstatute finally became a law on July 21, 1890. During this period six\nseparate trust bills, all modifications of that originally introduced\nby Mr. Sherman, were laid before the Senate. They were considered by\ntwo committees--the Finance and the Judiciary--and debated at great\nlength in the committee of the whole. The discussions occupy one\nhundred and fifty pages of the Congressional Record.\n\nA striking illustration of the general ignorance of the circumstances\nunder which the Sherman Act was passed is furnished by the present\nRepublican platform. This declares that \"the Republican party passed\nthe Sherman Anti-Trust Act over Democratic opposition.\" The records\nof Congress, however, show no indications of any opposition at all,\nDemocratic or other. Of the five men most conspicuous in framing the\nlaw, three were Republicans and two were Democrats. In the Senate only\none senator voted against the passage; in the House two hundred and\nforty-two votes were cast in favor of the act, and not a single one was\ncast against it. The whole debate was notable for its seriousness and\nits dignity; one or two Democrats did suggest that a revision of the\ntariff might help to curb the trusts; but that was the only partisan\nnote struck. Congress keenly appreciated the issues raised by the trust\nproblem and the necessity of taking action that would be beneficial and\npermanent. Everybody realized, also, the inherent difficulties of the\nsituation. The debates in the Senate on this issue, far from indicating\na scrappy investigation, furnish material for a liberal education in\nthe constitutional questions involved in dealing with monopolies.\nSenator Hoar, in preparation for the work, studied the history of\nlegislation concerning monopolies from the time of Zeno. One of the\nsections in the bill--that providing that a successful litigant against\na trust can recover three times the damages suffered from it--Mr. Hoar\nincorporated from a statute on monopolies passed in the reign of James\nI.\n\n [Illustration: SAMUEL GOMPERS, FOR TWENTY-FIVE YEARS PRESIDENT OF THE\n FEDERATION OF LABOR. MR. GOMPERS DEMANDS AN AMENDMENT OF THE SHERMAN\n ANTI-TRUST ACT THAT WOULD MAKE LEGAL THE INTERSTATE BOYCOTT]\n\n\n_Sherman Act Intended to Apply to Labor Unions_\n\nOf all the legends which have grown up about this law, perhaps the\nmost absurd is that it was never intended to apply to workingmen. \"As\na matter of fact,\" said Samuel Gompers before the Judiciary Committee\nof the House last winter, \"every man who now lives and is familiar\nwith the legislation of the day knows that the Sherman Anti-trust Law\nwas never intended to include organizations of labor,\" Chief Justice\nFuller, in a recent decision of the United States Supreme Court, flatly\ncontradicts Mr. Gompers' statement. \"The records of Congress show,\"\nsays Justice Fuller, \"that several efforts were made to exempt, by\nlegislation, organizations of farmers and laborers from the operation\nof the act and that all these efforts failed,\" In fact, the question\nof the relation of labor unions and the law occupied a conspicuous\nplace in the debates; it was almost as constantly in the minds of the\nSenators as the question of capitalistic combinations themselves.\nTo meet this situation, Senator Sherman introduced an amendment\nspecifically excepting labor unions and agricultural associations from\nthe operation of his statute. Mr. Gompers, according to his remarks\nbefore the Judiciary Committee last winter, was partly responsible for\nthe introduction of this amendment. Senator Edmunds opposed it on the\nground that it granted rights to labor which it withheld from capital,\nand he insisted that both sides should be treated upon an exact\nequality. In the following words he disposed for all time of Senator\nSherman's plea for preferential treatment of laboring men:\n\n The fact is that this matter of capital, as it is called, of\n business, and of labor, is an equation, and you cannot disturb one\n side of the equation without disturbing the other. If it costs\n for labor 50 per cent. more to produce a ton of iron, that 50 per\n cent. more goes into what that iron must sell for, or some part of\n it. I take it everybody will agree to that.\n\n Very well. Now, if you say to one side of that equation, \"You may\n make the value or the price of this iron by your combination for\n wages in the whole Republic or on the continent, but the man for\n whom you have made the iron shall not arrange with his neighbors\n as to the price they will sell it for, so as not to destroy each\n other,\" the whole business will certainly break, because the\n connection between the plant, as I will call it for short, and\n the labor that works that plant, is one that no legislation and\n no force in the world--and there is only one outside of the world\n that can do it--can possibly separate. They cannot be divorced.\n Neither speeches nor laws nor judgments of courts nor anything\n else can change it, and therefore I say that to provide on one\n side of that equation that there may be combination and on the\n other side that there shall not, is contrary to the very inherent\n principle upon which such business must depend. If we are to have\n equality, as we ought to have, if the combination on the one side\n is to be prohibited, the combination on the other side must be\n prohibited, or there will be certain destruction in the end....\n\n On the one side you say that it is a crime and on the other side\n you say it is a valuable and proper under-taking. That will\n not do, Mr. President. You can not get on in that way. It is\n impossible to separate them; and the principle of it therefore is\n that if one side, no matter which it is, is authorized to combine,\n the other side must be authorized to combine, or the thing will\n break and there will be universal bankruptcy. That is what it will\n come to.\n\n [Illustration: SENATOR GEORGE F. EDMUNDS, GENERALLY REGARDED AS ONE OF\n THE GREATEST CONSTITUTIONAL LAWYERS OF HIS TIME. THE SHERMAN ACT, AS\n IT STANDS AT PRESENT, IS VERY LARGELY HIS WORK]\n\nSenator Edmunds' logic absolutely killed any attempt to place capital\nand labor upon different footings, Instead of adopting this proposed\namendment, the Senate referred the whole question of trust legislation\nto the Judiciary Committee, of which Senator Edmunds was chairman. Mr.\nEdmunds and his colleagues threw into the waste basket all the pending\ntrust bills and their amendments and struck out on new lines. As a\nconsequence, Senator Edmunds became the chief author of the Sherman\nAnti-Trust Law. His most active associates, were Senator Hoar and\nSenator George. The one man who had practically nothing to do with the\nstatute as it stands to-day was Senator Sherman himself. He played\nan important part in the preliminary discussion and in framing the\nmeasures which served as a basis for this discussion; but the bill as\nit was finally adopted by Congress bore little resemblance to his.\nThe amendment upon which he laid especial stress--that of exempting\nlaboring and agricultural organizations from the operation of the\nAnti-trust Law--was absolutely ignored.\n\nAs finally adopted, the act did not prohibit labor unions per se or\ncombinations of labor unions formed to accomplish lawful ends; it\ndid, however, strike at certain labor union practices. That this was\nthe clear intention of the Senate is evident from a statement made by\nSenator Edmunds in a newspaper interview as far back as 1892. \"The\nSherman Law,\" said Mr. Edmunds, \"is intended to cover and I think will\ncover every form of combination that seeks in any way to interfere with\nor restrain free competition, whether it be capital in the form of\ntrusts, combinations, railroad pools, or agreements, or labor through\nthe form of boycotting organizations that say a man shall not earn his\nbread unless he joins this or that society. Both are wrong; both are\ncrimes and indictable under the Anti-trust Law.\"\n\n\n_Unsuccessful Efforts to Destroy the Law_\n\nFor eighteen years the anti-trust statute has represented American\npolicy and American law in federal regulation of combinations in\nrestraint of trade. In that period the act has been repeatedly\nassailed from many legal standpoints. It has been passed upon more\nthan two hundred and fifty times by the federal courts, and has been\nconsidered fifty-five times by the United States Supreme Court. The\ngreatest constitutional lawyers of this generation--such men as\nEdward J. Phelps, James C. Carter, John F. Dillon, and Francis Lynde\nStetson--have attempted to destroy it and have not succeeded. The\ngreatest railroads and corporations, on the one hand, and the largest\nand most influential labor unions, on the other, have both failed in\ntheir attempts to secure exemption from its operation.\n\n [Illustration: JUDGE GEORGE GRAY OF DELAWARE, WHO, AS UNITED STATES\n SENATOR, IN 1890, TOOK AN IMPORTANT PART IN FRAMING THE SHERMAN LAW]\n\nThe history of the Sherman Act has absolutely justified the wisdom\nand integrity of the Supreme Court. Scores of times the lower courts\nhave decided against the government; and the most important decisions\nhave been those in which the Supreme Court has reversed the inferior\ntribunals. The record of federal prosecutions under this law affords an\ninteresting insight into the attitude of the several administrations\ntoward trust regulation. President Harrison, under whose administration\nthe law was passed, accomplished little. His attorney-general brought\nseven actions--four bills in equity and three criminal indictments.\nUnder the equity proceedings, he obtained three injunctions; the\ncriminal proceedings all ended in failure. One of the cases instituted\nby President Harrison, however,--that against the Trans-Missouri\nFreight Association,--was afterward taken to the Supreme Court by\nPresident Cleveland's attorney-general, and resulted in securing one of\nthe most important decisions in the history of the law.\n\nPresident Cleveland showed considerably more activity than his\npredecessor. Though only eight proceedings stand to his credit,\nseveral of them were of the greatest importance. He used the Sherman\nLaw in fighting the Debs cases growing out of the Pullman strike; and\nin the well-known Addyston Pipe & Steel Company case he dissolved a\ncombination, formed by several manufacturers of gas and sewer pipe, to\nmonopolize the trade of most large American municipalities. President\nMcKinley apparently had little interest in the Sherman Law; throughout\nhis four and a half years only three cases were prosecuted, none of\nwhich were of much consequence. With the administration of President\nRoosevelt, however, the situation changed. Against the seven cases\ninstituted by Harrison, the eight by Cleveland, the three by McKinley,\nstand thirty-seven started by Roosevelt. That is, he has instituted\ntwice as many cases as all his predecessors combined, and many of\nthe Roosevelt prosecutions have proved successful. Nineteen of these\nthirty-seven cases have already been decided; the government has won\nseventeen and lost only two.\n\nAs a result of these many proceedings and interpretations, the Sherman\nAnti-trust Law is now fairly well understood. There has recently been\nmuch complaint that the law is not sufficiently \"specific\"; that\nbusiness men and labor leaders are groping very much in the dark;\nthat it is impossible to say what this statute prohibits and what it\npermits. From the judicial literature which has accumulated in the\nlast eighteen years, however, a fairly clear idea of its bearings\nupon large enterprises, both of labor and capital, can be obtained.\nSenator Hoar declared, when the bill came up for final passage, that\nit enunciated no new principle of law. It made illegal \"restraints of\ntrade\" and \"monopolies,\" but these had been for centuries unlawful in\nall Anglo-Saxon countries. As far back as the reign of Henry VI. in\nEngland, in 1436, a law was passed declaring that \"all agreements in\nrestraint of trade are illegal and voide.\" This principle has ever\nsince been part of the law of England, and is at present part of the\ncommon law of many States in the Union.\n\n [Illustration: FRANCIS LYNDE STETSON, CHIEF COUNSEL FOR THE UNITED\n STATES STEEL CORPORATION AND OTHER MORGAN ORGANIZATIONS. MR. STETSON\n WAS ONE OF THE DRAFTERS OF LAST WINTER'S TRUST BILL. IF IT HAD BECOME\n A LAW, THIS MEASURE WOULD HAVE MADE THE UNITED STATES STEEL COMPANY\n PRACTICALLY IMMUNE FROM FEDERAL PROSECUTION]\n\nIn the United States itself, however,--that is, in the federal\ncourts--there is no common law; everything must be fixed and regulated\nby statute. What the Sherman Act did was to make this common law on the\nsubjects of restraints and monopolies the statute law of the United\nStates. Under the common law of practically every State, monopolies\nand restraining combinations were illegal; Congress made these illegal\nwhen they involved inter-State trade. Under the common law boycotts\nwere illegal also; Congress made illegal the inter-State boycott.\nCongressional action on this subject was demanded, because the larger\nnumber of these unlawful combinations could be reached only by federal\naction, inasmuch as they usually involved more than one State.\n\nUnder the rulings of the Supreme Court, combinations and conspiracies\nwhich restrain trade and develop monopolies are those which, broadly\nspeaking, deprive the public of the benefits of free competition. This\nact recognizes the competitive system as the one industrial ideal,\nand outlaws anything that interferes with a free, unobstructed flow of\ntrade. A trust that gets control of the larger part of a particular\nproduct and manipulates the output so as to prevent trade from flowing\nin its natural course--that is an illegal restraint. Labor unions that\ncombine to divert artificially this same course of trade--as they\nunquestionably do when they persuade the public not to have business\nrelations with particular persons or corporations against which\nthey have declared a boycott--also engage in an illegal restraint.\nThe Sherman Law aims only to protect the public against these\nunnatural influences; to restore business to normal conditions. With\ncorporations, the final test as to whether they restrain trade or not\nis whether their effect is to increase prices. If they do not increase\nprices, then they do not restrain trade and consequently do not violate\nthe Sherman Act. The Supreme Court has insisted upon one important\nmodification of this principle. The effect upon prices must be\nimmediate and not remote. An arbitrary agreement that definitely fixes\nthe prices of a product is clearly illegal; an agreement which, in the\nlast analysis, might tend to influence prices, would not necessarily be\nso.\n\n [Illustration: SETH LOW, EX-MAYOR OF NEW YORK, WHO, AS PRESIDENT OF\n THE NATIONAL CIVIC FEDERATION, ADVOCATES THE AMENDMENT OF THE SHERMAN\n LAW]\n\n\n_Railroads Stopped from Making Rate Agreements_\n\nIn the first ten years after the passing of the Sherman Act, the\ngovernment attacked most successfully, not the great solidified\naggregations of capital popularly known as trusts, but the more or\nless loosely organized federations of corporations, formed chiefly for\nthe purpose of regulating and establishing prices. Trade agreements,\nnot monopolistic corporations, became its chief quarry. In proscribing\nthese agreements as illegal, the Sherman Act was found to be extremely\neffective. The very first case under this law was directed against a\ncombination of coal-mining companies in Kentucky and Tennessee, which\nexisted for the express purpose of regulating output and fixing prices.\nThe courts promptly decided that this agreement violated the Sherman\nAct. In 1892 eighteen railroads, nearly all operating west of the\nMissouri River, organized what they called the Trans-Missouri Freight\nAssociation. This association included many of the great Western roads,\ncompanies of the magnitude of the Santa Fe, the Missouri Pacific, and\nthe Rock Island. Its object, as clearly stated in the articles of\nassociation, was \"mutual protection by establishing and maintaining\nreasonable rates, rules, and regulations, in all freight traffic, both\nthrough and local.\" In other words, it proposed to fix arbitrarily\nthe price of transportation throughout the enormous territory covered\nby the eighteen railroads in question. The old \"pooling\" agreements,\nwhich had existed for many years, had been prohibited by the Interstate\nCommerce Law passed in 1887; and this Traffic Association was an\nattempt to accomplish the same end--that is, stop competition among the\nrailroads and maintain rates--in a different way. The Supreme Court,\nby a vote of five to four, decided that this agreement was prohibited\nby the Sherman Anti-trust Act, because, as an attempt to fix prices,\nit restrained trade. The famous Trans-Missouri decision, which settled\nthis case, made the Sherman Law an insurmountable bulwark against all\nrailroad combinations of this kind. Until this decision was finally\ngiven in 1897, this act had not been seriously regarded; after the\nSupreme Court had spoken, however, capitalists suddenly awoke to its\nsignificance. The decision settled many important points, which will be\nreferred to subsequently in this article, and it changed as well the\nwhole policy of railroad management.\n\nThe Sherman Act has stopped, not only railroad combinations, but\nsimilar agreements existing among manufacturers for the regulation\nof prices. The case of the Addyston Pipe & Steel Company is the most\ncelebrated of this kind. In 1894 a large number of manufacturers\nof sewer and gas pipe, the Addyston Company being one, formed a\ncombination to monopolize business and fix prices in thirty-six States\nand Territories. All companies which were parties to the agreement\nreserved the right to compete with each other outside of these\nthirty-six States as fiercely as before. They significantly called the\nsection in which there was to be no competition \"pay territory\"; and\nthe States outside of this section were known as \"free territory.\"\nThese manufacturers dealt chiefly with municipalities, which usually\nlet contracts for sewer and gas pipe by public bidding. Whenever such\na contract was offered, the Addyston combination would meet secretly,\ndecide upon the price they would charge, and then arrange a program\nof fictitious bids. They then divided the profits among themselves.\nIn this way they forced practically all purchasers in the sections in\nwhich they traded to pay exorbitant prices. Indeed, the subsequent\nhistory of this combination beautifully illustrates the practical\neffect upon the public of agreements of this kind. The Addyston and\nits associate members sold certain pipe in \"pay territory,\" where\nthe combination was enforced, at twenty-four dollars a ton; in \"free\nterritory,\" where they competed with each other, they frequently sold\nidentically the same product at fourteen dollars. The Supreme Court\ndecided that this agreement violated the Sherman Act--that it was a\ncombination or a conspiracy in restraint of trade. William H. Taft,\nthen United States Circuit Judge, wrote an opinion discussing the\nmerits of this dispute which has since become a legal classic. Mr. Taft\nspent six months in studying the questions involved.\n\nNearly all such cases, however, involved merely what may be called\ntrade agreements. In each case there were actual attempts to fix\nprices by compact, and these agreements were the only things in\ncommon among the different corporations that became parties to them.\nThe several corporations preserved their independent existence; they\nwere not trusts in the sense in which the Standard Oil Company, the\nAmerican Sugar Refining Company, the United States Steel Company, are\ntrusts--that is, single corporations, producing and distributing the\ngreater part of some particular product. Until President Roosevelt's\nadministration, these trusts had, for the larger part, escaped\nprosecution under the Sherman Law, the few attempts that had been made\nto assail them; having ingloriously failed.\n\n [Illustration: JOHN SHERMAN, A STATESMAN WHO SERVED THE GOVERNMENT\n FOR MORE THAN FORTY YEARS, AS SENATOR, SECRETARY OF THE TREASURY, AND\n SECRETARY OF STATE. BETWEEN 1888 AND 1890 HE INTRODUCED SIX SEPARATE\n BILLS FOR THE REGULATION OF TRUSTS, AND OUT OF THESE GREW THE MEASURE\n WHICH NOW BEARS HIS NAME]\n\nMeanwhile, in the first twelve years after the passage of the\nAnti-trust Act, and in the teeth of it, some of the largest\nmonopolistic corporations were formed. Many persons have maintained\nthat the Sherman Law, far from forestalling these corporations, has\nactually precipitated them. Their point is that, since this act\nclearly outlawed trade agreements among independent corporations, these\ncorporations, in order to get control of the situation, have been\ncompelled to amalgamate themselves under one ownership. The Sherman\nAct made illegal, for example, rate agreements among railroads; as a\nconsequence, in order to control railroad policy, the owners of the\ngreat trunk lines have purchased large blocks of stock in each other's\nproperty--on what is popularly known as the \"community of interest\"\nidea.\n\nPresident Roosevelt, however, has succeeded in applying the Sherman\nAct to the trusts, as that word is popularly understood. The famous\nNorthern Securities case is his greatest victory along that line. In\nthis instance, Mr. J. J. Hill and J. Pierpont Morgan formed a new\ncorporation, the Northern Securities Company, which acquired the actual\nstock ownership of nine-tenths of the stock of the Northern Pacific\nRailroad and three-fourths of that of the Great Northern. The Northern\nSecurities Company thus obtained a virtual monopoly of railroad\ntransportation from the Great Lakes to the Pacific Ocean in the\nnorthern section of the United States. The Roosevelt administration,\nrelying solely upon the Sherman Act, destroyed this corporation. The\nadministration has followed up this victory by instituting suits\nagainst the Standard Oil Company, the American Tobacco Company, and\nother powerful monopolies.\n\n [Illustration: GEORGE FRISBIE HOAR, UNITED STATES SENATOR FROM 1877 TO\n 1904, AND ONE OF THE AUTHORS OF THE SHERMAN ANTI-TRUST ACT]\n\n\n_Labor Unions, as Such, Not Prohibited_\n\nMeanwhile, the same law has proved an effective weapon in opposing\nthat other form of combination and restraint against which it was\nframed,--the labor trust. Under it a new code of federal laws affecting\nlabor unions has developed; and to a large extent it has strengthened\nthe cause of legitimate labor organization. No intelligent person now\ndisputes the right of workingmen to organize. A few labor leaders have\npublicly declared their apprehension that the Sherman Law prohibits\npeaceable labor organizations; no man, however, has thus far had the\nhardihood to raise this question legally; and, in the present state\nof public opinion as to the rights of labor, no one is likely to.\nThe United States Courts, in decisions defining the scope of the\nSherman Act, have specifically stated that it does not prohibit the\nordinary peaceful activities of labor unions. Justice White, in a\ndecision of the Supreme Court, has declared that an agreement among\n\"locomotive engineers, firemen, or trainmen engaged in the service\nof an inter-State railroad not to work for less than a certain named\ncompensation\" would not be illegal. William H. Taft, in one of the most\nimportant decisions affecting the rights of workmen under the Sherman\nAct, has defined the situation in words which are now widely accepted\nas a clear statement of what is not only good law but sound public\npolicy:\n\n The employees of the receiver had the right to organize into or\n join a labor union which would take action as to the terms of\n their employment. It is a benefit to them and to the public\n that laborers should unite for their common interest and for\n lawful purposes. They have labor to sell. If they stand together,\n they are often able, all of them, to obtain better prices for\n their labor than dealing singly with rich employers, because\n the necessities of the single employee may compel him to accept\n any price that is offered. The accumulation of a fund for those\n who feel that the wages offered are below the legitimate market\n value of such labor is desirable. They have the right to appoint\n officers, who shall advise them as to the course to be taken in\n relations with their employers. They may unite with other unions.\n The officers they appoint, or any other person they choose to\n listen to, may advise them as to the proper course to be taken in\n regard to their common employment; or if they choose to appoint\n any one, he may order them on pain of expulsion from the union\n peaceably to leave the employ of their employer because any of the\n terms of the employment are unsatisfactory.\n\nIt is clearly indicated, therefore, what labor leaders, under the\nSherman Act, can do. They have the right to organize, to combine--that\nis, to form unions; they have the right to refuse to work for wages or\nterms of employment unsatisfactory to themselves--that is, to strike.\nUnder the Sherman Act, indeed, mere organizations of laboring men are\nregarded as no more outlawed than ordinary social clubs or college\nfraternities.\n\n\n_How the Chicago Strike of 1894 Restrained Trade_\n\nOn the other hand, labor leaders know what, under the Sherman Act,\nthey can not do. They cannot enter into combinations that restrain\ntrade. This vital point has been settled in several important\nproceedings--those involving the Chicago disturbances in 1894, and,\nmore recently the decision just handed down in the matter of the\nDanbury Hatters. These cases so clearly show the bearing of the Sherman\nAct upon illegal labor practices, that they may profitably be reviewed\nhere.\n\n [Illustration: _Copyrighted by C. R. Buck_\n\n CONGRESSMAN CHARLES E. LITTLEFIELD OF MAINE, WHOSE KEEN ANALYSIS OF\n LAST WINTER'S CIVIC FEDERATION TRUST BILL WAS LARGELY RESPONSIBLE FOR\n ITS DEFEAT]\n\nIn 1894 the employees of the Pullman Palace Car Company of Chicago\nstruck for higher wages. These employees were not railway men; they\nwere workmen engaged in the manufacture of railway cars. In spite\nof this, about four thousand had been admitted to membership in the\nAmerican Railway Union, an organization of railroad operatives,\nwhich, under the vigorous management of Eugene V. Debs, had acquired\na membership of 250,000, and a correspondingly great power in the\nfield of railroad labor. In order to help the Pullman workmen in their\nstruggle with the Pullman Company, the American Railway Union declared\nwhat was in effect a boycott upon all railroads using Pullman cars.\nNearly all the larger American railroads had entered into contracts\nwith the Pullman Company, by which parlor and sleeping cars were to be\nused on their trains. Debs now demanded that these railroads should\nbreak their contracts, and thereby, of course, become responsible for\nheavy damages to the Pullman Company. In other words, he demanded that\nall American railroads cease patronizing the Pullman Company because of\nits \"unfair\" attitude toward union labor; that is, he started a boycott\nagainst the Pullman Company. When the railroad companies refused to\nmeet his demand, he ordered out all American Railway Union men employed\non these lines. He even declared war upon several of the Vanderbilt\nroads, which had no Pullman sleepers, operating instead the Wagner\ncars. In effect, in order that several thousand workmen in Chicago\nmight profitably settle their private grievances with their employers,\nDebs proposed, practically to end railroad communication in the larger\npart of the United States.\n\n\"The gigantic character of the conspiracy,\" said William H. Taft in\na well-known decision resulting from these proceedings, \"staggers\nthe imagination. The railroads have become as necessary to the life\nand health and comfort of the people of this country as are the\narteries to the human body.\" The larger part of our food supply,\nfor example, is furnished by means of the railway; the interruption\nof railroad transportation for any considerable period would, among\nother calamities, bring famine upon large sections of the country. In\nChicago, in Cincinnati, and in other large cities, Debs despatched his\nlieutenants with orders to tie up all railroads using Pullman cars.\nHe gave particular instructions to interfere with freight trains,\nsince freight was the main source of railroad revenue. In many places\nriots followed; in Chicago, strikers began wrecking trains, blowing up\nbridges, burning freight yards, tearing up tracks--indeed, nearly all\nthe twenty-three railroads centering in that city ceased operations.\nThe fundamental principles of the constitution, guaranteeing the\nsafety of life and property, had apparently given way to lawlessness\nand anarchy. In the opinion of Grover Cleveland, then President of\nthe United States, these proceedings constituted a \"conspiracy in\nrestraint of trade\" among the States, and as such were prohibited by\nthe Sherman Act. That the purpose and effect of Debs' proceedings was\nto restrain trade is sufficiently clear; indeed, no more complete\nrestraint than the cessation of railroad communication could be\nimagined. Trade in this case was not only restrained; it was entirely\nstopped. That the means by which this was to be accomplished had all\nthe essential elements of the inter-State boycott has also been shown.\nIn several cities, acting under the President's instructions, United\nStates district attorneys obtained injunctions on the ground that the\nstrike leaders were violating the Sherman Act, and also interfering\nwith the carriage of United States mails. In Chicago Eugene V. Debs\nwas enjoined, and, when he disobeyed the injunction, was arrested\nand afterward sentenced to six months' imprisonment. In Cincinnati\nhis associate, Frank W. Phelan, was likewise enjoined and likewise\nimprisoned for contempt. It was his act as judge in sending Phelan to\nprison for violating the Sherman Law that first made William H. Taft\na national figure. The circuit courts[J] decided, in several cases,\nthat the combination formed by Debs against nearly all the trunk lines\nwas a boycott, \"a conspiracy in restraint of trade,\" and punished the\nleaders, under the Sherman Act. William H. Taft declared that \"the\ncombination is in the teeth of the act of July 2, 1890.\"\n\n\n_The Danbury Hatters Attempt to \"Restrain Trade\"_\n\nThis boycott involved violence as an incident; the Supreme Court,\nhowever, has recently taken still more advanced ground, and decided\nthat a peaceable boycott also violates the Sherman Act. In the last\nfifteen years a terrific warfare has raged between the American\nFederation of Labor and nearly all American manufacturers of hats.\nThe American Federation has a membership variously estimated at from\n1,500,000 to 2,000,000, including workmen in practically every State\nand Territory. It is, as its name implies, a central association\norganized for the purpose of bringing into one effective machine all\nthe local labor organizations scattered throughout the country. It\nis an association of associations, and, as indicating its national\nscope, has its headquarters in Washington. It keeps constantly in touch\nwith its membership through its monthly publication, the American\nFederationist, as well as through the many journals of the unions\nwith which it is affiliated. It regularly employs nearly one thousand\nagents who continually push the interests of its members in the larger\npart of the United States and Canada. Mr. Samuel Gompers constantly\nuses this organization for the prosecution of inter-State boycotts.\nIn his petition to intervene in the Danbury Hatters case, Mr. Gompers\nstated, over his own signature, that \"the constitution of said American\nFederation of Labor makes special provision for the prosecution of\nboycotts, so-called, when instituted by a constituent or affiliated\norganization.\" In a public speech on May 1, 1908, Mr. Gompers declared\nthat the Supreme Court might \"as well dissolve and destroy the\norganization of labor as to enforce these decisions\"--that is, the\ndecisions against boycotts. Obviously, the Federation of Labor has an\nadvantageous organization for work of this kind. A local union, with\nmembership extending not beyond the limits of a town or State, could\nmake little headway against a manufacturer against whose goods it had\ndeclared a boycott, inasmuch as his trade usually extends over a large\narea. The American Federation of Labor, however, by embracing the local\nunions' cause can make the boycott effective in practically every part\nof the country. In the last twelve years, Mr. Gompers' organization has\ndeclared four hundred and eight boycotts.\n\nIn particular, it has prosecuted with considerable success boycotts\nagainst the manufacturers of fur hats. About ten years ago, Mr.\nGompers, working with the United Hatters of North America, inaugurated\nan elaborate program to compel all such manufacturers to unionize their\nshops. By using their well-known methods, they have brought to terms\nseventy out of the eighty-two manufacturers in this country. The firm\nof D. L. Loewe & Co. of Danbury, Connecticut, however, had persistently\nrefused to comply with these demands. Mr. Loewe was not a large\nmanufacturer; he had, however, built up a prosperous business, and,\nthough he had never shown any hostility to union labor, had insisted on\nmaintaining an open shop. In 1901 the United Hatters' Union practically\nordered him to discharge his non-union men and unionize his factory.\nMr. Loewe again refused to do this, and a strike immediately followed.\nMr. Loewe, however, promptly engaged new non-union men, and soon his\nfactory was running as busily and as profitably as before.\n\nMr. Gompers then brought the whole machinery of his organization to\nbear upon this recalcitrant hatter. On July 25, 1902, the Federation of\nLabor and the United Hatters declared a boycott against his products.\nThey denounced this concern in their several publications as \"unfair,\"\nand notified nearly all the wholesale and retail hat dealers throughout\nthe United States that they must not handle the Loewe goods, under\npain of being boycotted themselves. It is said that their agents kept\nespionage, in Danbury, over all freight consignments from the Loewe\nfactory, and thus obtained a fairly complete list of their customers;\ncommittees of labor men in many cities waited upon these customers,\nand, in several instances, persuaded them to drop the Loewe hats. Some\nfirms who refused to obey this dictation were themselves boycotted;\nand, in San Francisco, Philadelphia, Baltimore, and Richmond, the\nboycott was pursued with particular virulence. The Federation went so\nfar as to grant a special dispensation to its members to purchase hats\nmade by other non-union labor, rather than patronize the Loewe brand.\nMr. Loewe, though he suffered enormous loss as a result of these\nproceedings, pluckily kept up the fight. Under the Sherman Law, an\naggrieved citizen is authorized to bring private suit against persons\nengaged in a conspiracy to restrain his trade, and, if he successfully\nmaintains his case, may recover three-fold damages. Mr. Loewe quietly\nwent to work and had made an inventory of all property-holders actively\nengaged in boycotting his goods. He then brought suits for $340,000\ndamages against a large number of labor men, filing in the District\nCourt 240 separate attachments. The Supreme Court of the United States\nmade short work of this case. Chief Justice Fuller, who wrote the\ndecision, declared that \"the combination described in the declaration\nis a combination 'in restraint of trade or commerce among the several\nStates' in the sense in which these words are used in the act, and the\naction can be maintained accordingly.\" An interesting feature of the\ncase is that the decision of the Supreme Court was unanimous. In nearly\nall the other proceedings involving the Sherman Law--the Trans-Missouri\ncase, the Northern Securities--the government has won by a bare\nmajority; every member of the Supreme bench, however, at once concluded\nthat Mr. Gompers' activities against the firm of D. L. Loewe & Co.\nrestrained inter-State trade, and thus violated the Sherman Law.\n\nThus, in eighteen years, the Sherman Act has proved an effective weapon\nagainst the two forms of trust and conspiracy with which the public\nis most familiar--combinations of capitalists to restrain inter-State\ntrade and arbitrarily fix prices, and combinations of labor unions\norganized for the prosecution of inter-State boycotts. It strikes\nimpartially the Northern Securities Company and the American Federation\nof Labor; it does not discriminate between the activities of Mr. J.\nPierpont Morgan and of Mr. Samuel Gompers. At the last session of\nCongress, the two forces which it opposes bent all their energies to\ndestroy this law; in all probability they will renew and redouble their\nefforts this winter.\n\n\n_National Civic Federation Attempts to Amend the Law_\n\nFor many years the National Civic Federation has been collecting data\nbearing upon the trust and labor problem. In 1899 it held a trust\nconference; and again, in October, 1907, it called a large meeting\nat Chicago for the consideration of the trust situation. Delegates\nappointed by the governors of forty-two States and representatives of\nmore than ninety commercial, agricultural, and labor organizations\ncontributed to these discussions. Referring to these Chicago\nproceedings, Mr. Theodore Marburg, one of the participants, said\nbefore the Judiciary Committee in Washington last winter: \"Mr. Nicholas\nMurray Butler sounded the note of attack upon the Sherman Anti-trust\nLaw.... I take it that the gentlemen will agree with me that it was\na dominant note of that conference.\" As a result, a bill radically\namending the Sherman Anti-trust Act was introduced in Congress at the\nlast session. Its most active sponsors in Washington were Seth Low,\npresident of the National Civic Federation, Professor Jeremiah W. Jenks\nof Cornell, and Samuel Gompers, president of the Federation of Labor.\nWell-known men who had participated in the conference that preceded\nthe framing of the bill were E. H. Gary, chairman of the Board of the\nUnited States Steel Corporation, Henry L. Higginson, Isaac N. Seligman,\nand James Speyer and August Belmont, bankers. Francis Lynde Stetson,\nchief counsel for the United States Steel Corporation and other Morgan\ncorporations, and Victor Morawetz, counsel for the Santa Fe Railroad,\nwrote the drafts. This latter fact was publicly stated by Mr. Low and\nMr. Jenks in the course of the hearings before the Judiciary Committee.\nThe authorship of the bill was early brought out in the following\ncolloquy between Congressman Charles E. Littlefield and Mr. Low:\n\n MR. LITTLEFIELD: Right there, Mr. Low, if there is no objection,\n who are the people that actually participated in the preparation\n of the bill? Who are the men who actually drew it?\n\n MR. LOW: We conferred with Judge Gary, of the United States Steel\n Corporation.\n\n MR. LITTLEFIELD: E. H. Gary, president of their board of directors?\n\n MR. LOW: E. H. Gary. The lawyers actually engaged in the drafting\n of the bill were Mr. Stetson----\n\n MR. LITTLEFIELD: That is, Francis Lynde Stetson?\n\n MR. LOW: Francis Lynde Stetson; and Mr. Morawetz.\n\n MR. LITTLEFIELD: Victor Morawetz?\n\n MR. LOW: Victor Morawetz.\n\nAt another time, Mr. Low described Mr. Stetson and Mr. Morawetz as\n\"the drafters\" of the bill. Herbert Knox Smith, commissioner of\ncorporations, also had a hand in framing the measure. President\nRoosevelt openly indorsed it and sent in an emergency message urging,\namong other things, its passage. Extensive hearings, extending through\nseveral months, were held before the Judiciary Committee. Many\nrepresentatives of capital and labor appeared in favor of the measure.\nAlthough Congressman Littlefield, who presided over these hearings,\nmany times expressed his wish to examine Mr. Stetson and Mr. Morawetz,\nthese gentlemen never appeared. Although Mr. Low promised that they\nwould submit a brief, explaining several disputed legal points, they\nnever did so. The burden of discussing the many intricate legal points\nthat constantly arose rested entirely upon the shoulders of Mr. Low and\nProfessor Jenks, neither of whom had had any legal training. Through\nthe efforts of Congressman Littlefield, James A. Emery, counsel for\nthe National Association for Industrial Defense, and Daniel Davenport,\ncounsel for the Anti-Boycott Association, the proposed law was\ndefeated, but the proceedings are of great interest and importance\nas illustrating the changes desired by both labor and capital in the\npresent anti-trust law.\n\n\n_Gompers Asks that the Boycott be Legalized_\n\nMr. Gompers' demands were entirely simple and direct. He wished labor\nunions entirely exempted from the operations of the Sherman Act. That\nlaw, if properly respected and enforced, would practically put an end\nto Mr. Gompers' occupation. Referring lately in a public speech to\nthe effect of a recent court decision against inter-State boycotts,\nMr. Gompers quoted, as applicable to his own organization, Shylock's\nspeech in \"The Merchant of Venice,\" \"You might as well take from\nme my life as take from me the means whereby I live.\" Mr. Gompers'\nchief interest in the Civic Federation bill, therefore, was a clause\nwhich specifically declared that the Anti-trust Act should not be so\ninterpreted \"as to interfere with or restrict any right of employees\nto strike for any cause or to combine or to contract with each\nother or with employers for the purpose of peaceably obtaining from\nemployers satisfactory terms of their labor or satisfactory conditions\nof employment.\" Mr. Low and Mr. Jenks denied that this language\nlegalized the boycott; Congressman Littlefield, however, and many other\nopponents of the measure, emphatically asserted that it did. Such\nsweeping concessions as \"_to strike for any cause_\" and \"_to combine\nor to contract with each other or with employers for the purpose\nof peaceably obtaining from employers satisfactory terms_,\" it was\nmaintained, clearly authorized such boycotts as that prosecuted against\nthe Danbury Hatters. That proceeding, it was pointed out, was entirely\npeaceable--there was no law-breaking, no rioting, no bloodshed. It\nwould also legalize, it was said, many of those arrangements between\nlabor unions and employers--by which employers' associations contract\nto employ only members of certain labor unions, the latter, on their\npart, contracting to work only for certain employers--which were\nbrought to such perfection by the late Sam Parks. Mr. Gompers demanded\nthat, if the clause in question did not authorize boycotts, another\nshould be substituted which did; to make the case sure, therefore, he\nproposed an amendment which did so in no uncertain tone. The following\nextract from the record clearly defines Mr. Gompers' position:\n\n MR. LITTLEFIELD: Now, Mr. Gompers, a word. Would this amendment\n you suggest, if it became a law, authorize the prosecution of such\n a boycott as was attempted in the Danbury Hatters' case, which was\n in violation of the Sherman Anti-trust Law? Is that the purpose?\n\n MR. GOMPERS: One of the purposes; yes, sir. That case was brought\n under the Sherman Anti-trust Law.\n\n MR. LITTLEFIELD: Yes. And the purpose of the amendment you have\n offered is to relieve you from the operation of the Sherman\n Anti-trust Law as construed by the court in that case?\n\n MR. GOMPERS: Yes, sir.\n\n MR. LITTLEFIELD: And to authorize that kind of an inter-State\n boycott?\n\n MR. GOMPERS: Yes, sir.\n\n MR. LITTLEFIELD: Do you, as the representative of organized labor,\n favor the boycott, both as an inter-State and a local proposition?\n\n MR. GOMPERS: I do, sir.\n\n MR. LITTLEFIELD: And your organization stands for that?\n\n MR. GOMPERS: It does, sir.[K]\n\n\n_Government to Discriminate Between Good and Bad Trusts_\n\nAs to monopolistic corporations, the proposed act placed them entirely\nunder the supervision of the executive branch of the government. If\nyou wished to form a trust, or enter into a restraining contract,\nand, at the same time, to escape the prohibition of the Sherman\nAct, you would first, under the provision of this bill, submit the\nproposed arrangement to the Commissioner of Corporations and answer\nsuch questions as he saw fit to ask. If he gave approval, you could\ngo ahead and carry out the deal, practically secure against further\ninterference. If he disapproved, you would be liable to attack under\nthe Sherman Act. In fact, the administration was to be given arbitrary\npower to discriminate between good and bad trusts, to separate the\ncorporation sheep from the corporation goats. \"You are all right,\" it\ncould say to one combination; \"you are all wrong,\" it could say to\nanother. The federal government, in other words, was to rule absolutely\nthe business activities of nearly 80,000,000 of people; merely by a\nword it could authorize a gigantic combination like the United States\nSteel Company, and prohibit another like the Standard Oil.\n\n\n_\"Reasonable\" and \"Unreasonable\" Combinations_\n\nThe above statement gives the effect and not precisely the form\nof the proposed legislation. What its authors really hoped to\naccomplish was executive discrimination between those combinations\nand those restraints of trade which were reasonable and those which\nwere unreasonable. They based their measure upon the theory that\ncertain combinations, even many whose tendency is to restrain trade\nand increase prices to the consumer, may still work for the public\ninterest. The word \"reasonable\" has played an important part in the\nhistory of the Sherman Act. In several cases the corporations, in\ncontesting the law, have made the claim that this act did not prohibit\nall combinations in restraint of trade, but only those which were\n\"unreasonable.\" They set up this defense most strongly in the famous\nTrans-Missouri case, already described. Eighteen railroads, it may be\nrepeated, had formed an association for the purpose of fixing freight\nrates. James C. Carter, who argued the case, strongly asserted that\nsuch an agreement was beneficial both to the railroads and to the\npublic; the history of railroads having conclusively proved that\ncut-throat competition inevitably led to bankruptcy and demoralization\nin railroad service. He therefore claimed that the proposed restraint\nin trade was \"reasonable\" and consequently not prohibited by the\nSherman Act. The Supreme Court, by a majority of five to four, rejected\nthis theory. The Sherman Act, it pointed out, in express language\nmade illegal \"_every_ contract, combination in the form of trust or\notherwise, or conspiracy, in restraint of trade\"; and made \"_every_\nperson\" who was a party to such contract a criminal. It left absolutely\nno leeway--it did not discriminate in the remotest degree between those\nwhich were reasonable and those which were not. Since then all demands\nfor the modification of the act have hinged upon this one point.\n\n\n_Andrew Carnegie on Combinations_\n\nThis demand, of course, has precipitated a very nice problem in\ndefinition. What is a reasonable combination? What is an unreasonable\none? What is a good trust? What is a bad one? Upon this all-important\nquestion the many weary hearings extending through four months before\nthe Judiciary Committee last winter shed practically no light. The\nCivic Federation bill was based upon this fundamental distinction;\nand a large number of distinguished citizens appeared in favor of\nit. Congressman Littlefield, as each speaker appeared before the\nCommittee, asked him to give a concrete illustration of a combination,\nforbidden by the Sherman Act, which really promoted the public interest\nand was therefore \"reasonable.\" Mr. Seth Low frankly admitted that he\ncould name no concrete case of the kind. He caused some amusement,\nhowever, when he read a letter from Andrew Carnegie touching upon this\nvery subject. \"One point seems to me essential,\" wrote Mr. Carnegie,\n\"without it, little general progress can be made; namely, when new\ncombinations are proposed, the first question must always be 'what is\nthe object sought?' _In ninety-nine cases out of a hundred, it will\nundoubtedly be to rob the community of its right to the benefits of\nfree competition, disguise it as one may_; therefore the Commissioner's\nduty is to obtain satisfactory proof that the application is to\ncover an exceptional case. The conditions must be peculiar, as those\nof common carriers and steel-rail agreements are.\" Mr. Carnegie's\nstatement that ninety-nine per cent of trade agreements are made for\nthe purpose of \"robbing the community\" and his implication that the\nexceptional one per cent are the agreements involving the manufacturers\nof steel rails, naturally provoked much hilarity.\n\nOnly two other illustrations were furnished of benevolent combinations.\nMr. Herbert Knox Smith, commissioner of corporations, instanced a\nproposed agreement among lumber men to cut only a certain amount\nof timber each year, the ostensible purpose being to prevent the\nwanton destruction of the forests. It appeared, however, that the\nreal purpose of such an agreement was not to preserve the forests,\nbut to restrict the output, and increase prices, and consequently\nthe profits of the lumber men. Another illustration offered was the\ncombination of patent medicine dealers to fix prices and prohibit\nprice cutting--the object, it was said, being to prevent the unfair\ncompetition of large department stores with retail druggists. But this,\nin the last analysis, was generally believed to be a concerted attempt\nto destroy competition and enhance the profits of patent medicine\nmakers. Congressman Littlefield insisted, throughout the entire\nproceedings, that the fundamental purpose of forbidden combinations was\nto control the product and thereby increase the price to the consumer.\nIf there were any combinations that did not have that purpose or\nresult, then the Sherman Act, according to Mr. Littlefield's analysis,\ndid not prohibit them. Thus in all attempts to define practically\nreasonableness and unreasonableness, as applied to trade agreements,\nthe statement was repeatedly made that the large part of the business\nof this country was done in violation of law; that business men lived\nconstantly in a state of terror from the fear of its enforcement; that\nits presence on the statute books largely explained existing business\ndepression. When it came to defining precisely what they wished to do,\nhowever, none of those who favored the bill became specific. The thing\nfinally simmered down to a statement by Mr. Low that the law was \"a\nvery important element in the psychological condition of business men\nto-day.\"\n\n\n_Indulgences to be Granted to Corporations_\n\nThis particular power of defining reasonableness and unreasonableness,\nhowever, the proposed law centered in the President, acting through\nthe Commissioner of Corporations. It provided a limited system of\nfederal registration for corporations, and, in a modified form, for\nfederal license and publicity--the two circumstances which probably\nled President Roosevelt to support the measure. In effect it granted\nindulgences to corporations to combine, provided they would do certain\nthings. The Sherman Law, as it stands to-day, was not specifically to\nbe repealed; it was simply to be waived in favor of those combinations\nand trusts which paid the price of these indulgences. In order to\nobtain absolution, the offending corporation must do two things:\nregister with the Bureau of Corporations and answer such questions\nas might be propounded to it. The bill authorized the President to\ndetermine precisely what information should be exacted, and also to\nchange from time to time the requirements regarding data. That is, for\nregistered corporations, it gave the executive branch of the government\nabsolute inquisitorial power. Registered corporations had the right\nto file with the Bureau any agreement or contract or combination to\nwhich it became a party--the precise kind of transactions made illegal\nby the Sherman Act. The Commissioner had thirty days in which to\nexamine such contracts; if, within that period, he declared them in\nreasonable restraint of trade, then they became practically legal.[L]\nIf not, then they could be proceeded against under the Sherman Law.\nThe chief point of criticism in this arrangement was the stipulation\nfor a thirty-day period during which the Commissioner must pass upon\nthese contracts. This, it was asserted, was the loop-hole by which the\ncorporations were to secure immunity. The Commissioner must declare\nthese contracts reasonable or unreasonable within thirty days; if\nhe failed to act upon them in that time, they became reasonable,\nprecisely as if he had declared them to be so. How, it has been\nasked, could the Bureau possibly act intelligently within that period\nupon many of the exceedingly intricate questions which would come up\nfor judgment? Whether a contract is reasonable, of course, largely\ndepends upon the way it affects prices. An examination would therefore\nfrequently involve an economic study of the particular trade, as well\nas the organization of the particular corporation involved. It would\nbe necessary to go deeply into capitalization, values behind this\ncapitalization, cost of production, wages, transportation charges\nand so on. There are said to be more than 200,000 corporations in\nexistence. Supposing half or a quarter should register,--how could\nthe Bureau possible examine them within thirty days? Would it be\npossible to investigate the United States Steel Corporation within\nthat period? Under the suggested law, however, unless the Commissioner\npassed judgment within this time, all these contracts and combinations\nwould automatically receive a certificate of good character. In their\ninterest, the Sherman Act would practically be repealed.\n\nIn the main, this provision referred to contracts made and combinations\nto be formed in the future; another section practically extended\nimmunity to all contracts and combinations now in existence. Nearly\nall trusts organized in the last forty years, and all restraining\nagreements, were to become valid. The government was to have a year in\nwhich to institute proceedings against such corporations as declined\nto register. If it failed to do so within this time, then these\ncombinations could never be attacked on any ground whatever, and\nbecame regularly fixed institutions. As there are about five hundred\ncorporations popularly known as trusts and myriads of trade agreements\nnow forbidden, the law department, it was suggested, would have its\nhands full if it attempted to bring suit against them all within twelve\nmonths. Moreover, after the passage of the proposed act, the government\ncould not proceed against any combination except on one ground--that\nit was an unreasonable restraint of trade. Under the Sherman Act, it\nwill be remembered, it can prosecute without any reference to the\nquestion as to whether the restraint is reasonable or not. If the act\nhad passed, in other words, the position of the government would have\nbeen this: within a year it could have assailed the trusts only on\nthe grounds of unreasonableness; after the expiration of a year it\ncould assail them on no ground whatever. A saving clause, however,\nprovided that the government could prosecute all actions already\nbegun. That is, it could follow up to the end the pending cases against\nthe Standard Oil, the American Tobacco Company and other corporations\nagainst which it has already started suit. It could not prosecute,\nhowever, the United States Steel Corporation, for it has instituted no\nproceeding in that direction. It was the Attorney of the United States\nSteel Corporation, Mr. Francis Lynde Stetson, who had a large hand in\nframing the bill.\n\nThese facts have led many observers to believe that the bill in\nquestion represented an underhanded attempt, by large corporations,\nespecially the United States Steel, practically to remove the\nSherman Anti-trust Law from the statute book. Mr. E. H. Gary and\nMr. George W. Perkins spent many days in Congress while the bill\nwas under discussion, though they did not once openly appear before\nthe committee. No criticism affecting the good faith of Mr. Low and\nProfessor Jenks, the most active open advocates of the bill, was\nput forth. The discussion disclosed the fact, however, that the\nSherman Act, as it stands at present, has many friends. Organizations\ninterested in curbing the unlawful activities of labor unions insisted\nthat that law, as interpreted by the Supreme Court, is practically\nthe only protection American industry has against the boycott. Repeal\nor seriously modify it, they declared, and a regime of labor union\nterrorism far surpassing any hitherto known in any country, would\nat once begin. The plan of Mr. Gompers and his associates to shelve\nthis law, they insisted, was merely part of their general scheme to\nremove all legal restraints from the operations of labor unions.\nOpinions did not seem quite so unanimous as to the wisdom of the\nSherman Act in its bearings upon corporations. Though many declared\nthat this measure is too sweeping and drastic, and should be amended,\nno one has yet suggested any practical way of framing a new law. No\none who has studied the problem of trust regulation, it is believed,\nhas thus far hit upon a plan that, while it gives greater leeway to\nthe corporations, protects the public from arbitrarily high prices\nand other exactions. There is thus a growing conviction that the act\npassed by the great constitutional lawyers of 1890 represents the best\nattainable result in this direction. It has not stopped the growth of\ntrusts, it is true; but whether that is because it does not furnish the\nmeans or because it has not been sufficiently enforced, is the disputed\nquestion. \"What is needed,\" recently said ex-Senator Edmunds, the\nman who was the real author of the Sherman Act, \"is not so much more\nlegislation as competent and earnest administration of the laws that\nexist.\"\n\n [J] In the Debs case the Circuit Court based its decision almost\n entirely upon the Sherman Law. The Supreme Court of the United\n States, in affirming this decision, rested mainly on the broader\n question of the interference with the United States mails. Justice\n Brewer, however, who wrote the decision, specifically said that\n this fact did not mean that the Supreme Court dissented from the\n grounds on which the lower tribunal had decided the case.\n\n [K] In Justice to Mr. Low and Mr Jenks it should be said that they\n disclaimed any intention of indorsing a bill which authorized\n the boycott. They afterward amended the clause in question by\n authorizing employees \"to strike for any cause not unlawful at\n common law,\" which modification leads into many legal fogs which\n it is hardly worth while to enter in this place.\n\n [L] The bill provided, it is true, that the contracts might still\n be assailed on the ground of unreasonableness. The practical\n effect, however, it was generally conceded--virtually admitted by\n Herbert Knox Smith--would be to give them immunity for all time.\n\n\n\n\nTHE ETERNAL FEMININE\n\nBY TEMPLE BAILEY\n\n\nIf it had been any one but Anne Beaumont!\n\n\"I don't like turning conventionalities topsy-turvy, Sophie,\" she said,\nas we went downstairs; \"I don't believe I can ever ask a man to dance\nwith me.\"\n\n\"Other women do,\" I murmured.\n\n\"My husband would never have agreed to such a thing,\" Anne stated.\n\nThat is where Anne always had the advantage of me. Although she had\nbeen a widow for five years, she still quoted the authoritative\nmasculine point of view, while I, having in my teens chosen a career\ninstead of a husband, and never having rectified my mistake, was forced\nto fall back on the unsupported feminine.\n\n\"Perhaps you'd rather sit out the dances,\" was my somewhat malicious\nway of putting it.\n\nAnne, poised like a white butterfly on the landing, turned on me a\nreproachful glance.\n\n\"No woman would rather be a wallflower,\" she affirmed.\n\n\"Of course not,\" I returned promptly, \"and I don't believe it is going\nto be very bad after the first plunge.\"\n\nAnne leaned over the stair rail and surveyed the formidable group of\nmen in the lower hall. \"It's dreadful,\" she said. Then, gathering about\nher a scarf of silver tissue, she commanded, \"You go first, Sophie,\"\nand we descended together.\n\nAt the foot of the stairs, Charlemagne Dabney met us.\n\n\"Charlie, boy,\" Anne said plaintively, \"ask me to dance with you. I\nsimply can't get used to the leap-year idea----\"\n\nAnd I, having prepared to blunder into a formal, \"May I have the\npleasure?\" was so illumined by her method that I employed it with\nsuccess--for though I lacked Anne's appealing coquetry, I challenged\nold friends, and my card was soon filled.\n\nBut Anne did not depend on old friends. She danced with the count\nfrom Hungary, the multi-millionaire from the West, the Senator from\nKentucky, and to fill up spaces she fell back on Charlemagne Dabney.\n\n\"I think it was lovely of you,\" she told him at supper, \"to open the\nhouse for the week-end and the dance. Only, it's too bad that you\ninsist on the leap-year idea for the whole time.\"\n\nAcross the table Elizabeth Ames sparkled radiantly. \"I like it. I\ndidn't dance with a single bore, and before I go home I am going to ask\nall of the men to marry me!\"\n\nAnne's face wore its most gracious expression, but I knew how she felt.\nElizabeth is eighteen and pretty. Anne is twice eighteen and pretty.\nAnd there's a difference.\n\nAnne opened her eyes very wide and said to Charlemagne, \"You see what\nyou've done? Elizabeth is going to ask you to marry her.\"\n\nCharlemagne smiled at Elizabeth. \"No such good luck. There are too many\nyoung fellows who will accept her before she gives me the chance.\"\n\nElizabeth laughed back, \"Don't be too sure that you'll escape.\"\n\nAnne's delicate eyebrows were raised. \"Of course she is joking; no\nwoman would really ask a man----\"\n\nCharlemagne sighed. \"I wish one woman would.\"\n\nAnne's lashes fluttered. \"Why don't you ask her?\" she challenged.\n\nHe shrugged his shoulders. \"I feel weak in the knees when I think of\nit,\" he said, \"for fear she might say 'no'.\"\n\n\"Faint heart,\" I murmured, but no one paid any attention to me.\n\nIt seemed to me, after that, as if some of the brightness had gone\nout of Elizabeth's face. But Anne fairly scintillated. And she was\nexceedingly amiable to Elizabeth.\n\n\"Ask the count first,\" I heard her say, \"he's simply charming.\"\n\nElizabeth flung up her head in a quick way. She was all in sheer pale\nyellow, bordered with daffodils, and there was a twist of gold ribbon\nin her fair hair. Only extreme youth could have worn it, and, as she\nflashed her answer back to Anne, I had never seen her more beautiful.\n\n\"The count wouldn't have me as a precious gift,\" she said. \"I'm too\ncrude. He likes a more finished product--like you, dear Mrs. Beaumont.\"\n\n\"Now, what do you suppose she meant by that?\" said Anne that night,\nwhen we were in our kimonos and were comforting our complexions with\ncold cream. \"Do you think she meant it for a compliment, or was it a\nreflection on my age?\"\n\n\"No one can reflect on your age,\" I told her. \"Nobody knows it but\nCharlemagne and me, and we won't tell.\"\n\n\"That's the advantage of living on the other side and coming back to\nmeet the younger generation,\" said Anne; \"they haven't kept tab on the\nyears.\"\n\nShe got up and moved restlessly about the room. With the cream on her\nface and with her hair down, she looked old, and I had a vision of\nElizabeth in the yellow gown.\n\nPerhaps something of my thought showed, for Anne stopped suddenly and\ngazed into a long mirror set in the door. \"Oh, youth, youth, Sophie,\"\nshe cried.\n\n\"Anne,\" I said, \"come away from that mirror. No one can be beautiful\nwith her face full of cold cream.\"\n\nShe laughed and dropped down on the rug in front of me, and after a\nwhile she said, \"Did you hear what he said to-night?\"\n\n\"About wishing a certain woman would ask him?\"\n\n\"Yes. He will never ask me, Sophie. He thinks I am still mourning my\nhusband--he thinks I don't care----\"\n\nThere wasn't much to be said after that. But before I left her, I\nwhispered, \"Why don't you tell him, Anne?\"\n\nAnne's shocked eyes condemned me. \"Oh, Sophie, as if a woman _could_!\"\n\nI passed Elizabeth Ames' room on my way to my own, and she called to\nme, \"Come in, Miss Sophie.\"\n\n\"It's so late,\" I protested, standing on the threshold.\n\nBut she was insistent. \"Please come,\" she begged.\n\n\"You ought to be in bed,\" I scolded, \"getting your beauty sleep.\"\n\nBut even as I said it, I knew she didn't need it, for she was as\ndaintily fresh as a rose. Her fair hair hung down in two heavy braids\nover her white gown. She looked like a lovely child.\n\n\"Miss Sophie,\" she said abruptly, when she had put me into a big chair\nin front of the fire, \"tell me about Anne Beaumont and Mr. Dabney----\"\n\n\"What about them?\" I asked innocently.\n\n\"Were they in love with each other--years ago--before she married Mr.\nBeaumont?\"\n\nI nodded. \"They were engaged, and Anne was very young. She had never\nseen much of other men, and when Mr. Beaumont came along, with his\nair of foreign distinction, she was fascinated and broke off her\nengagement. But she never really cared for Mr. Beaumont----\"\n\n\"And you think Mr. Dabney has--has stayed single for her sake?\"\n\n\"I think so. Yes.\"\n\n\"And you think he loves her still----?\"\n\n\"You heard what he said to-night?\"\n\n\"I don't call that love,\" she cried. \"If he cared, he'd tell her. He\ncouldn't help it. It would just come--if he really loved her----\"\n\n\"He thinks that she has never cared--and he isn't an impetuous boy----\"\n\n\"I know--but he's a _man_.\" She was all aglow. \"And if he cared, his\nheart would say, 'I love you, I love you, I love you,' and then his\nlips would say it----\"\n\n\"You believe, then, that he doesn't care for her?\"\n\n\"His allegiance is a memory--an old dream--of the girl she was, not of\nthe woman she is. Isn't she older than he, Miss Sophie?\"\n\n\"She is younger,\" I said gravely.\n\n\"She seems older--and--it's spoiling his life. He--he won't look at\nanother woman--because in a way he feels bound to her. Some day I'm\ngoing to tell him.\"\n\nI stared at her. \"Tell him what, Elizabeth?\"\n\n\"That he is throwing away his happiness--that there are other women.\"\n\nShe had risen and stood in front of me with her hand on her heart. Her\neyes were like stars, and the radiance of youth shone from within and\nround about her. If Charlemagne should see her in such a mood----\n\nI thought of Anne, dear Anne.\n\n\"Elizabeth,\" I said sharply, \"if you should tell him that, he would\nthink--that you--cared.\"\n\nShe swept out her arms in a charming gesture of surrender.\n\n\"Well, if he did,\" she cried, defiantly, \"what then?\"\n\n * * * * *\n\nAll that night Elizabeth and Anne contended in my dreams, and in the\nmorning, worn to a frazzle, I went down to breakfast, to find that\nElizabeth had gone for a ride with Charlemagne, and that Anne was still\nin bed.\n\nI drifted into the library and found there a circle of somewhat\nfagged-out feminines. The men were riding or on the links.\n\nFrom the light bits of conversation that were wafted to me as I sat\nand read in the window-seat, I gathered that most of the women took\nCharlemagne's leap-year idea as a joke, but I knew that to Elizabeth\nand Anne the question presented itself seriously, and that each would\nsettle it in her own way, and according to the tradition of her own\ntime.\n\nFor that education and environment had made the difference, I did not\ndoubt. Had Elizabeth been born eighteen years earlier, when women were\ntaught the mysteries of advance and retreat, that coquetry was their\nbest weapon, and that man must always be the wooer, she might have felt\nall of Anne's shrinking from a revelation of herself; whereas had Anne\nbeen brought up in the later days when boys and girls mingle in close\ncomradeship, when plays and books subtly analyze the state of woman as\nthe pursuer and man as the pursued, she might have been as frank about\nher feelings as Elizabeth.\n\nHence, I argued, they were both of them what their generation had made\nthem, and I, who loved Anne, and adored her for her womanliness, was\nyet forced to admit the potency of Elizabeth's youth, and the charm of\nher complete surrender.\n\nAfter a time the men began to drift in, and I heard the\nmulti-millionaire from the West inquiring for Elizabeth. He was a big,\nbroad-shouldered fellow, sure of himself, but not unpleasantly so, and\nwhen he couldn't find the girl he wanted, he came over and talked to me.\n\n\"Say,\" he began at once, \"it's all tommyrot about this leap-year\nbusiness. When I want a girl to do anything, I want to ask her. It\nmakes me feel foolish to have to wait for her to come to me. I wish\nDabney would cut it out.\"\n\n\"But think what an opportunity for a girl to get what _she_ wants,\" I\nsaid.\n\n\"They don't know what they want,\" he stated dogmatically. \"The way to\nwin a woman is to pick her up and put her on a horse and run away with\nher----\"\n\n\"Suppose she doesn't care to be run away with?\" I asked.\n\n\"Oh, she'd settle down to it,\" he said securely; \"and besides that, I\ncan't really imagine a nice girl asking a man to marry her.\"\n\nI thought of Elizabeth as she had stood with her hand on her heart and\nhad hurled defiance at conventions.\n\n\"Girls are hard to understand,\" I murmured.\n\n\"Oh, I don't know,\" he contended. \"If a man gets right down to\nprimitive principles and keeps after her, he'll get her--and it makes\nme hot to think I am wasting valuable time trying to stick to Dabney's\nold rules, when I have to go back West again on Monday.\"\n\nI wanted to be sure, so I murmured, \"Of course it's Elizabeth Ames?\"\n\n\"Who else?\" he demanded. \"Oh, I'm going to jump over the traces, Miss\nSophie, and let her know I mean business. This thing of sitting around\nand letting her go off with another man--you know she's riding with\nDabney this morning?\"\n\nI nodded.\n\n\"He's twice her age, and she _thinks_ she likes him. Girls get romantic\nstreaks, and Dabney's the kind they put up on a pedestal, but he isn't\nany more suited to her than--a bunch of beets----\"\n\n\"I suppose not,\" was all the response I dared venture in the face of\nsuch an outpouring of eloquence.\n\n\"They are coming now,\" he said, and through the window I saw\nthem--Elizabeth, looking like a little girl in her three-cornered hat,\nwith her hair tied with a broad black ribbon, and Charlemagne sitting\nhis horse like a centaur.\n\nThe Westerner deserted me at once, and, the rest of the guests\nfollowing, I was left alone in the library.\n\nI curled up in the window-seat, drew the curtains to shield me from the\ngaze of those who might step within, and tried to take forty winks to\nmake up for the four hundred I had missed the night before.\n\nBut I couldn't sleep. Elizabeth and Anne--Anne and Elizabeth! I\ncouldn't get their affairs out of my mind. Would Elizabeth propose,\nwould Anne, would Charlemagne, would the multi-millionaire? Again and\nagain I tried to fit together their widely different theories, until in\ndespair I wished that Charlemagne and his leap-year week-end had not\ntempted me from my maidenly apartment in town, where the worries of\nlovers were confined to my manuscripts.\n\nAnd even as I pondered, I heard Elizabeth's voice saying, as she came\nin from the porch, \"I suppose you think I am awfully forward to make\nyou spend all your morning with me----\"\n\nAs he followed her into the library, Charlemagne laughed. \"I might\nfeel flattered,\" he said, \"if I didn't know you were doing it to make\nMcChesney furious.\"\n\nMcChesney was the multi-millionaire.\n\n\"McChesney?\" Elizabeth's tone was startled.\n\n\"Don't hedge,\" Charlemagne teased. \"He's bound to win out, Elizabeth.\nNo woman can escape a man when he goes for her like that. You might as\nwell give in.\"\n\n\"I shall never give in.\"\n\n\"He's a nice fellow.\"\n\n\"He's not my ideal----\" there was a pathetic note of appeal in her\nyoung voice.\n\n\"Ah--ideals----\" Charlemagne had dropped his banter. \"Don't spoil your\nhappiness looking for the ideal man--he's like the pot of gold at the\nend of the rainbow--something we hear of, but have never seen.\"\n\nThere was a heavy silence. Then Elizabeth said, catching her breath,\n\"But--but I have found my ideal, Mr. Dabney.\"\n\n\"You have? And it's not McChesney?\"\n\nI peeped at them through the curtain. They were in big wicker chairs\nin front of the door that led to the porch. Elizabeth had taken off\nher coat, showing her thin white blouse with its crisp frills. Her\ncheeks were as pink as the rose which she picked to pieces with nervous\nfingers.\n\n\"No,\" she said tremulously, \"it's--it's not Mr. McChesney.\"\n\nI held my breath. Would she dare?\n\n\"It's--it's a man much older than I am,\" she went on, \"and--and I don't\nknow that he has ever thought of me--in that way--perhaps if he had, he\nmight like me--a little----\"\n\nI am sure that Charlemagne felt the charm of her youth, as she made\nher little confession, and I am just as sure that he was absolutely\ninnocent that he was the object of it.\n\n\"He would undoubtedly love you more than a little,\" he said heartily.\n\"Look here, Elizabeth, you won't mind telling me who he is--will\nyou----?\"\n\nHere was an opportunity holding out open arms, and did Elizabeth\nembrace it as beseemed an advocate of woman's right to woo?\n\nNot she! She simply gasped in a panic-stricken way and stood up.\n\n\"Oh, _no_,\" she whispered, with her cheeks flaming, \"I couldn't--I\ncouldn't tell any one.\"\n\nBefore Charlemagne could answer, McChesney blundered in.\n\n\"Say----\" he stopped dead still on the threshold, \"I think this is a\ncase of monopoly. I'm tired of hanging around waiting for the girl I\nwant. I am going to break the rules, Dabney, and ask Miss Ames to take\nme for a walk in the rose garden.\"\n\nAnd Elizabeth actually turned to him with an air of relief.\n\n\"Oh, yes,\" she said breathlessly, \"I'd love it!\"\n\nAnd away they went. And Charlemagne, turning back into the library, met\nAnne Beaumont coming in at the other door.\n\nShe wore a thin, trailing white gown, and there were dark shadows under\nher eyes. She looked tired and fragile and every day of her thirty-six\nyears.\n\n\"Anne!\" Charlemagne said, as if for him all the morning stars sang\ntogether.\n\nAnne dropped into the chair where Elizabeth had been.\n\n\"I'm afraid I'm awfully late getting down,\" she faltered, \"but--but my\nhead ached.\"\n\nCharlemagne stood behind her chair, and there was a look on his face\nthat, for the first time, made me ashamed of my eavesdropping. The\nother had been comedy, but this was real.\n\n\"Poor little Anne,\" he said.\n\nAnne propped her chin on her hand and gazed out through the open door\nwith wide eyes.\n\n\"Yes,\" she said slowly, \"poor little Anne.\"\n\nHe came around and took the other chair. \"I wish--I knew how I might\ncomfort you,\" he said.\n\nFor a moment Anne looked at him with that wide stare, then, like a\nflash, it came. \"Oh, Charlie, Charlie boy,\" she cried, \"why don't you\nask me to marry you--I can't ask you, you know----\"\n\nBefore she had finished, he was on his knees beside her, and then I\nshut my eyes and put my fingers in my ears, for the time had come when\nI had no right to hear or see.\n\nBut as for theories--Oh, who knows _what_ a woman will do? There was\nElizabeth and there was Anne----\n\nBut I never would have believed it of Anne!\n\n [Illustration]\n\n\n\n\nTHE MOTHER OF ANGELA ANN\n\nBY CLARA E. LAUGHLIN\n\nILLUSTRATIONS BY ALICE BARBER STEPHENS\n\n\nI\n\nHenry Street, drowned in November murk, was black as Tartarus and\na shade more dreadful, as a heavily built man stumbled along its\nunfamiliar bumps and intermittent stretches of sidewalk, stopping now\nand then to peer vainly at doors for a number. Presently he encountered\na wisp of a girl with a jacket thrown about her head and shoulders.\n\n\"Where's twenty-one?\" he asked.\n\nShe pointed. \"Who d'ye want?\"\n\n\"Casey.\"\n\n\"In the rear--I'll show ye,\" and she led the way to a precipitous\nflight of steps. \"Ye go down, an' 'long 's far 's ye kin, thin turn t'\nth' right an' knock,\" she said, and disappeared in the mist.\n\nGroping his way, the man reached the end of a long passage between two\ntenements and knocked at a rear door. A woman opened it.\n\n\"Th' ditictive,\" she murmured, and let him in.\n\nThe kitchen was stifling close; a fire raged to the brim of the big,\nheavily nickeled stove which had cost the Caseys so dear in instalments\nand in worry. Casey had been working for two weeks, and the bin outside\nthe kitchen door had a ton of soft coal in it. In a bracket above the\nsink was a lamp whose tin reflector, instead of diffusing the light\nrays, seemed to concentrate them, like a feeble searchlight, so that\nthe corners of the kitchen were all in gloom, and half-lost in gloom\nwere the forms of the Caseys, whose pallid faces showed sharply against\nthe dusk.\n\n\"Had any word?\" said the detective, addressing Mrs. Casey. To the\nrelief of the parents and the bitter disappointment of the children, he\nwas a plain-clothes man.\n\n\"Niver a worrd.\"\n\nThe detective consulted a memorandum.\n\n\"You say she left home Monday morning, just as usual, to go to work?\"\n\n\"Yissir; she wint down th' alley here hummin' a chune an' as gay as a\nburrd.\"\n\n\"And you don't think she intended to stay away?\"\n\nMary Casey's eyes flashed. \"If I t'ought a gyurl o' mine could walk out\nan' l'ave me, intintional, wid a chune on her lyin' lips, I'd not ask\nye t' be findin' her,\" she said.\n\n\"Did she have a beau?\"\n\n\"None thot I iver see. She used t' be after talkin', sometoimes, 'bout\ngran' fellies she'd see downtown, an' I always sez to her, 'You mark\nme worrds an' l'ave gran' fellies be. They don't mane no good t' th'\nloikes o' you,' I sez. 'Thim fellies spinds ivry cint they git on their\ngold watches an' swallie-tails, an' whin they marry they got t' marry\na gyurl wid money t' support thim. Whin yer old enough t' take up wid\nanny wan,' I sez, 'yer pa or yer Uncle Tim'll introjuce ye t' some nice\nyoung lab'rin' man wid a good trade an' ambition t' git on, an' you\nwork fer him whoile he works fer you.' 'Ah, ye don' know nothin' 'bout\nit,' she'd say t' me, an' 'Don't you belave thot,' I'd say t' her, 'I'm\nnothin' t' look at, an' I ain't got mooch style about me, but I got\nsome knowlidge o' min,' I sez, 'an' they're a bad lot, aven th' bist o'\nthim. An' you git it out o' yer hid,' I sez, 'thot anny gran' felly's\ngoin' t' marry you, or th' loikes o' you. Ye may rade such foolishness\nin yer story paapers er see it at yer theayters, but ye kin mark me\nworrds thot love is fer tony folks thot kin afford it, an' not fer th'\nloikes o' you an' me.'\"\n\nUp to this time Casey had been conspicuously quiet. He had had his own\nexperiences with the Chicago police, who more than once had ordered\nhim to keep away from his abused family or go to the Bridewell. This\nwas buried deep in the voluminous records of the desk sergeant; but\nCasey had not the comfort of knowing that there were a thousand kindred\ncases piled a-top of his, so he kept discreetly in the shadow until\nthe detective asked, \"Was she gay at all?\" and Mrs. Casey replied:\n\n\"She be a little granehorn, wid no sinse yet. I'm after taalkin' t' her\nth' whole, blissed toime 'bout kapin' straight, an' not l'avin' her go\nby dances er stay out nights, but I dunno--ye can't kape thim in yer\npocket, an' whin a gyurl have her livin' t' earn anny place she kin\nfoind it, 't ain't her mother thot know fer sure wheer she is or what\nshe be.\"\n\nAt this Casey sat suddenly forward in his chair, and the streak of\nlight fell full across his face, swollen with tears and streaked with\nthe grime of three awful days. Despite the grime, however, despite the\nstubble of reddish beard, the unkempt hair and untidy clothes, there\nwas something singularly pathetic about him, with his great, Irish-blue\neyes and youthful, innocent-looking face. He had not been drinking for\nsome weeks, and he wore no air of sottishness, nor of vagrancy, nor of\nany of his other crimes against self and family and society.\n\n\"I dunno what I ever done,\" he had moaned for three days, rocking back\nand forth in his misery, the tears raining down his unwashed cheeks and\nsplashing from his stubbly chin, \"I dunno what I ever done that this\nthing should 'a' happened t' me! My gyurl! My Ang'la Ann!\"\n\n\"She were a good gyurl,\" he said to the detective, sitting suddenly\nforward.\n\n\"So far 's we know, she were,\" his wife amended, \"but she had no sinse\nyet, bein' so young, an' th' young niver belaves th' old. I don' see\nhow a gyurl o' mine could go wrong, an' me hatin' it th' way I do. But\nshe have more o' him in her nor o' me, down t' thim same shifty blue\neyes thot kin look so swate, an' God knows what divilment's behint\nthim!\"\n\nCasey smiled in wan coquetry at this charge against his fascinations,\nbut reiterated in defense of his daughter:\n\n\"She were a good gyurl. I seen a piece o' this world, of'cer, an' I kin\ntill--min like us, we kin till gyurls that's merely flightsome from\nthim that's gon' t' th' bad. If she's bad, I don' want ye t' find her.\nJes' show me th' felly thot lied t' her, an' I'll kill him--but I don'\nwant ye t' find her; I don' niver want t' set eyes on her ag'in, if\nshe've brought disgrace on me.\"\n\n\"Ye won't lit it git in th' paapers, will ye?\" Mary Casey pleaded\nfor the twentieth time in her brief communications with the police.\n\"Yell kape thim aff av her, won't ye--fer th' love o' Hiven? I'm after\ntellin' th' childern I'll kill th' first wan o' thim thot breathes t'\na soul we don' know wheer Ang'la Ann is. Ag'in' she be all right an'\ncome home some day, it'd go hard wid her if these Sheenies 'round here\nknew she was gon'--people do belave th' worst of a gyurl, always. I\ndunno what t' think o' my Ang'la Ann, but I don' want it to go haard\nwid her if she don' desarve it.\"\n\nThe detective promised about the papers and went his way. A missing\ngirl, with no probable complications of a horrible murder, excited\nonly the feeblest interest at Maxwell Street, and this visit would\ncomprehend the whole of the police activity expended in the case unless\nAngela Ann should happen to turn up under their incurious noses.\n\nThe facts of the case were these: Angela Ann Casey, a slim,\nunder-sized, pretty young thing just under eighteen, had left home\non Monday morning, November 7th, apparently to go to work, and had\nnot been seen since by her family or any one they knew. She was an\nunskilled worker, a bit of flotsam in the industrial whirlpool so cruel\nto her kind. In the summer she had worked for a few weeks in a cannery,\npasting labels on fruit cans. When the cannery shut down, she answered\nan \"ad\" for extra help in the rush season of a cap factory, which laid\nher off when work slackened. And after a fortnight's idleness she was\ntaken on as a bundle-wrapper in a cheap department store, where she met\na girl who told her of a place needing more girls for the manufacture\nof cheap finery for the \"levee\" trade. Angela Ann applied, and was\ngiven work at a knife-pleating machine, at four dollars and a half a\nweek. She was in this job, to the best of her mother's belief, when\nshe disappeared; but a visit to the place on Tuesday laid bare the\nstartling fact that she had \"give notice\" on Saturday night.\n\nAngela Ann had few intimates; her associates changed with her changes\nof occupation, and these were so many that she took root nowhere. A\ngirl on Blue Island Avenue, to whose house Angela Ann sometimes went,\ncalled at Henry Street Tuesday evening and was told that Angela was out.\n\n\"She's tellin' me she have a gran' fella,\" said the girl questioningly.\n\n\"She have,\" lied Mary promptly, \"did she iver tell ye his name?\"\n\nNo, she hadn't; so Mary said maybe Angela Ann wouldn't want her to tell\nit either.\n\nMary's sister, Maggie O'Connor, who was married to a \"will-t'-do\"\nblacksmith and lived but a few blocks away, had also heard of a stylish\nyoung man who could not be asked to the back cellar on Henry Street, or\neven allowed to suspect it. In family council Mrs. O'Connor testified\nthat she had offered her own \"parlie\" for the courting.\n\n\"'Bring him here an' l'ave us have a look at him,' I sez to her. 'Ye\nkin have th' parlie anny toime ye want it,' I sez, 'an' if yer 'shamed\no' yer Uncle Tim's brogue, he kin stay in th' shop, an' I'll talk t'\nhim mesilf,' I sez.\"\n\nBut Angela Ann had not accepted this handsome offer, nor had she\nconfided the name of the young man to Mrs. O'Connor, who only knew that\nAngela Ann had assured her he was a gentleman beyond a doubt, for he\nhad a gold watch and chain.\n\nFired by this information, which he considered an important clue,\nCasey was for carrying it at once to the police so that they might\ninvestigate all young men wearing gold watches and thereby in due\nprocess find the one who knew Angela Ann. But before he could get\naway to furnish the detectives with this important information, Mrs.\nO'Connor had made some further suggestions. The chief of these was\ntouching the advisability of consulting a fortune-teller.\n\n\"Thim coppers,\" she opined, \"is no good. Tim's after radin' a lot about\nthim in th' paapers, an' he sez they niver ketch nothin' 't all. He\nsint ye a dollar wid me and sez he, 'You till thim t' stop foolin' wid\ncoppers an' go t' th' forchune-teller,' sez he.\"\n\n\"I belave it have more t' do wid what th' forchune-teller know than\nwid what thim coppers kin foind out,\" reflected Mary Casey. It was the\nmorning after the detective's visit, and Mrs. O'Connor had come over\nto ask the news. \"Theer's somet'ing I didn't till th' ditictive,\" Mary\nconfessed, \"not knowin' how he'd take it--but the day befoore Ang'la\nAnn wint, a quare, wan-eyed cat kem here. Ivrywheer I wint thot day she\ntraipsed at me heels, an' all Monday noight whin I was up watchin' fer\nAng'la, th' cat was on th' windie-sill, howlin' what sounded joost like\nAan-gla, Aan-gla, Aan-gla. Now what d'ye make o' thot?\"\n\nMrs. O'Connor had been fumbling in her plush wrist-bag during this\nrecital. \"Say,\" she said presently, holding out a very dirty card, \"th'\nlas' noight Ang'la Ann was t' our house she was after l'avin' th' baby\nplay wid her purse, an' th' baby spilt all th' t'ings out av it. We\npicked thim up, an' I t'ought we got thim all, but whin I was clanein'\nyiste'day, I foun' this card. It mus' be hers, fer Tim say he niver see\nit, an' no more did I.\"\n\nThe card read:\n\n [Illustration: O. HALBERG,\n\n _Dramatic Agent--West Madison Street_.]\n\n\"That's him, I bet ye!\" cried Casey excitedly, \"that's th' felly wid\nth' gol' watch an' chain!\"\n\n\"Wait a minute!\" commanded Mrs. O'Connor impatiently, \"Tim sez thot\nhave somet'ing t' do wid a theayter.\"\n\n\"Sure,\" said Mary Casey, \"Ang'la Ann wouldn't be so grane as t' ixpict\nno theayter guy t' marry her! She'd ought t' know thim niver marries;\nor if they do, they have a woife in ivery town, loike soldiers an'\ntravelin'-min! I niver bin to no theayter in my loife, but I know that\nmooch!\"\n\nCasey, who had lost his job by default, and had sat apathetically by\nthe stove ever since gray morning dawned after the frantic vigil of\nMonday night, was struggling with the lacings of his shoes preparatory\nto setting forth to demolish O. Halberg if he proved his guilt by\nwearing a gold watch and chain.\n\n\"Ye kin spend yer dollar on yer wan-eyed cat,\" he said indulgently,\n\"but as fer me, I got t' foind thot felly thot lied t' me gyurl.\"\n\nSo the inaction of the past three days was over, temporarily at least.\nCasey was bound for O. Halberg's and Mrs. Casey and Mrs. O'Connor were\ngoing to approach some fortune-teller with the dollar and the tale of\nthe cat. But first of all Mary must go to the school and take Johnny\nout to mind Dewey and the baby in her absence.\n\n\"Now you be keerful,\" she adjured Casey as he made ready to go, \"an'\ndon' kill nobody be mistaake. Th' bist way is t' kill nobody at all,\"\nshe continued cautiously.\n\nIn spite of this caution, however, there would have been danger in\nprospect if Casey had owned a gun or if he had taken a few drinks. As\nit was, he was not a formidable figure when he presented himself at the\nnumber on West Madison Street, a few doors from Halsted.\n\nThere was a pawnshop on the first floor, and beside it a narrow door,\nwhich opened upon a long flight of wooden stairs rising steeply to a\ndark hall, where, by the light of a two-foot gas burner, Casey could\nmake out the name \"O. Halberg\" on one of the dozen doors. The name was\npainted on a black tin plate tacked to a rear door. Casey knocked.\n\n\"Come in,\" said a guttural voice.\n\nEntering, Casey saw a man sitting with his feet on a battered desk;\nhe was reading the morning paper and smoking a vile cigar. The\nwalls, calcimined a kind of ultramarine blue, but grimed and fouled\nunspeakably, were hung with theatrical lithographs depicting thrilling\nscenes from plays on the blood-and-thunder circuit. For the rest, the\nfurnishings were two wooden chairs, a giant cuspidor, and the desk,\nwhich looked as if it had never been new.\n\n\"Have I,\" said Casey in his grandest manner, \"th' honor t' addriss Mr.\nO. Halberg?\"\n\nO. Halberg grunted that he had. Then Casey advanced a step further\ninto the room and looked about for a sight or trace of Angela Ann.\nNothing could have been more damning than O. Halberg's gold chain, but\nin no likelihood would Angela Ann, by any stretch of courtesy, have\ncalled him young; he was probably fifty, and not prepossessing from any\npossible point of view.\n\n\"Me name is Casey,\" ventured the visitor, \"me gyurl is lost, an' I'm\nlookin' fer her. We found this,\" proffering the dirty card, \"an' we\nt'ought mebbe you'd know wheer she is.\"\n\nCasey was proud of the neatness and despatch of his \"ditictive\"\nmethods, but more than a little disappointed to find so soon that he\nwas on the wrong trail entirely. Mr. Halberg was truly surprised to be\napproached with any such query. A great many little silly, stage-struck\ngirls flocked to see him, of course, and no doubt some of them got\nhold of his cards \"in the hope of using them to impress managers,\" but\nhe had no recollection of any girl named Casey--none whatever. And he\nresumed the reading of his paper.\n\n\"I got th' coppers after her,\" murmured Casey apologetically, as he\ntook his leave, \"but thim coppers is no good. Ag'in' ye want ditictive\nwork done, ye better do it yersilf.\"\n\nO. Halberg did not deign to reply, but when Casey was safely outside he\nstepped to the door and locked it. In case the \"coppers\" came around,\nit would be just as well to be \"out\"--it would save the coppers some\ntroublesome pretense.\n\nIn his descent of the steep stairs Casey met two girls coming up. They\nwere about Angela Ann's age and were giggling nervously. One of them\nheld between thumb and finger a quarter-inch \"ad\" from a morning paper,\noffering:\n\n\"High-salaried positions in good road companies to young ladies of\npleasing appearance. O. Halberg, Dramatic Agent--West Madison Street.\"\n\n\"Ask him if this is the place,\" said the girl who appeared to be\nfollowing the other's lead. Casey directed them to O. Halberg's door,\nthen went on his way. A moment later, while he stood on the corner of\nHalsted Street waiting for a south-bound car, he saw the girls emerge\nfrom the door by the pawnshop. They passed him as they went to take an\neast-bound Madison Street car on the opposite corner.\n\n\"Did ye foind him?\" Casey asked.\n\n\"No, he wasn't in.\"\n\n\"That's quare,\" he said, startled, \"he was there wan minute before.\"\n\nOn his way home Casey dropped in at the Maxwell Street Station in a\nfree-and-easy manner he would not have dreamed possible two days ago.\nHe was so full of his \"ditictive\" experience that he felt he must have\nsome one, if only a copper, to talk it over with. The detective who had\ncalled the night before wasn't in, so Casey related his recent daring\nexploit to no less a personage than the desk sergeant himself.\n\nIt was well poor Casey could not hear the desk sergeant's account of\nthe call after the self-appointed sleuth had gone on his way.\n\nMrs. Casey was at home when her husband got there. Relating her\nadventures, after she had listened to his, she said that the\nfortune-teller, after accepting the dollar, had asked several searching\nquestions about the one-eyed cat.\n\n\"'Ag'in' th' cat come back, yer gyurl 'll come home,' she sez t' me.\"\n\n\nII\n\nThe days dragged by. There seemed to be a complete lapse of the\nstone-cutting industry, so Casey had nothing to take his mind from\nhis \"ditictive\" operations, which were interesting and unexhausting,\nthough expensive in car-fare and unproductive of results. Angela Ann's\nweekly wage, for many years the main dependence of the family, being\nlost to them, they were closer even than was their wont to starvation\nand eviction; and winter was beginning to snarl around their warped,\nill-fitting doors.\n\nAs time wore on, the poignant horror of Angela Ann's absence grew\nmercifully less for all but Mary Casey. Night after night she wept the\nlong hours through, until Casey complained of the depressing effect of\nher grief, and she felt constrained to hide it.\n\n\"If I could on'y know she were dacintly dead,\" was her heart's cry, as\nbetter hopes died in her, \"Ag'in' a bye l'ave home, he kin knock around\nan' pick up a bite here an' a lodgin' theer, an' be none th' worse fer\nit. But a gyurl bees diff'runt! Theer's always thim watchin' 'round\nthot's riddy t' do her harm.\"\n\nMeanwhile she lied bravely to the neighbors. \"Angela Ann bees livin'\nout an' have th' graandes' plaace,\" she told them impressively; \"th'\nlady she live wid 's after takin' her to Floridy fer to mind her little\nbye.\"\n\nMary's hope was strong that Christmas would see the wanderer's return,\nbut the holidays passed in unrewarded waiting. Casey had perforce\nabandoned his search, and worked a day or two now and then. Though the\ntraces of really terrible suffering were still in his weak, winsome\nface, he had long since forsaken all hope of Angela Ann's \"safety with\nhonor,\" and, when it had come to seem unlikely that she ever would do\nso, took comfort in vowing that she should never again darken the door\nof his outraged home.\n\nMary gave over pleading for her girl, in the interests of family peace,\nbut, more and more like a specter as the weeks wore away, she haunted\nlocalities where Angela Ann had been or might be. Sometimes she had\nthe baby in her arms, but oftener she left it with Dewey at their Aunt\nMaggie's, and roamed the streets unhampered in her never-ending quest.\n\nEvenings she would say, \"I'll be goin' t' yer aunt's a bit,\" and\nslip away into the engulfing dark, to reappear in the glare of light\nmarking the entrance to some cheap West Side theater or dance hall.\nGradually her excursions extended downtown, where she would take up her\nstation at the door of some place of amusement and stand watching the\npleasure-seekers pour in, then turn away and wander aimlessly up and\ndown the streets for an hour or so before facing homeward. In some way\nshe heard about stage doors, and took to haunting them. She saw many\ngirls of Angela's type, and wondered sadly if their mothers knew where\nthey were, but her own girl was not among them. In those nights on the\nflaming streets she learned more about vice than she had ever dreamed\nof in all her life, and the world came to seem to her a vast trap set\nby the bestial for the unwary.\n\nNot hunger, nor cold, nor abuse, nor sickness, nor death, as it came\nto five of her children, had driven Mary Casey to anything like the\npoignancy of feeling that was hers now. Heretofore she had been\npatiently dumb under affliction; now her spirit cried out in a passion\nof pain that called straight upon Almighty God for an answer to its\nanguished questionings.\n\nWith the aid of Casey, who was a \"scollard,\" and could \"r'ade 'n'\nwrite joost as aisy,\" she pored over the sensational papers in search\nof stories about girls in trouble, and never a horror happened to an\nunidentified girl anywhere but Mary was sure it was Angela Ann.\n\nOnce there was an account of an unknown young woman found dead on\nthe prairies near Dunning, the county institution. It was Johnnie\nwho laboriously spelled out this story for her--Casey having gone to\nthat club of congenial spirits, O'Shaughannessy's saloon--and at ten\no'clock, when the children were all abed, her anxieties could brook\nno more delay. Throwing a shawl about her head and shoulders, she\nstole along the pitchy passageway, up the long flight of steps to the\nsidewalk, clutching the torn fragment of newspaper in the hand that\nheld the shawl together beneath her chin.\n\nIt was Saturday night, and the avenue was still brightly lighted. One\nor two acquaintances greeted her, but she hurried by with only a nod\nand a word. At Harrison and Halsted and Blue Island Avenue, where three\nstreams of ceaseless activity converge, there is always a whirlpool\nrapids of traffic and humanity, and here, in a brilliant drug store,\nMary felt far enough from her own haunts and all who knew her and\nAngela Ann to venture on her errand.\n\n\"I want t' tillyphome,\" she whispered to the clerk, who pointed\nimpatiently to the booth.\n\n\"I dunno how,\" said Mary imploringly. \"I want ye t' do it fer me. R'ade\nthat.\" She thrust the dirty, crumpled fragment of the evening's yellow\njournal into his hand.\n\nThe young man glanced at it, and then curiously at her. \"I've read it,\"\nhe said.\n\n\"Down here, somewheers,\" said Mary, pointing vaguely towards the last\nparagraph, \"it till wheer she be, an' I want ye t' tillyphome that\nplace an' ask thim have she a laarge brown mole on her lift side. If\nshe have, I'm goin' out theer this night, fer 'tis my gyurl I t'ink she\nbe.\"\n\nThis was not as startling an episode to the young man addressed as it\nmight have been to one in a quieter locality. Nevertheless, it smacked\nof the dramatic sufficiently to interest him, and when Mary proffered\nher nickel he called up the Dunning morgue.\n\nAfter what seemed an interminable wait, while the sleepy morgue\nattendant at the county poor-house was being summoned by repeated\nrings, and the brief colloquy was in progress, the clerk emerged from\nthe booth.\n\n\"The girl has been identified this evening,\" he said.\n\nDisappointment mingled with relief in Mary's countenance: she had\nreached that stage where it would have been not altogether unendurable\nto look at Angela Ann's dead face, even in a morgue.\n\nAs she retraced her way home, the chill of the sharp February night\nstruck into her mercilessly. When she set forth, she had scarcely\nnoticed in it her preoccupation; but now that another expectation,\nhowever tragic, had proved false, and the situation stretched ahead\nof her indefinitely dull and despairing again, the abrupt relaxation\nleft her physically as well as mentally \"let down,\" and she shivered\nviolently as she hurried along.\n\n\"Mother o' God,\" she cried, the tears rolling swiftly down her shrunken\ncheeks, \"wheer is my gyurl this noight? If I could on'y know she had a\nroof over her head an' a fire t' kape her warrm!\"\n\nCasey was still out when she got back, and she was thankful, for the\nsight of her tears made him ugly these days. \"She've disgraaced us,\"\nhe said of Angela Ann, \"an' she be dead t' me, an' ought t' be t' you,\nif ye had proper shame.\"\n\nMary could give herself up to the luxury of grief, therefore, and\nshe did, until she fell asleep. The next morning she was up betimes,\nmeaning to go to early mass in the basement of the church before\n\"drissy folks\" were abroad in their Sunday finery. For more than one\nreason Mary avoided the later masses; her rags were small shame to her\ncompared with the more than half-suspicious inquiries of acquaintances\nas to the whereabouts of Angela Ann.\n\n\"'Tis more lies I'm after tellin',\" thought poor Mary, \"than th' praste\nkin iver take aft o' me. 'N' ag'in' I do pinance enough t' kape me\nbusy half me time, an' go t' git me holy c'munion, I'm not out o' th'\nprisence o' th' blissed Sacrament befoore I'm havin' t' lie ag'in t'\nsave that poor, silly gyurl's name!\"\n\nThis morning, however, in spite of her early rising and her efforts to\nget to seven o'clock mass, events conspired to thwart her intentions.\nMollie woke up with a headache, and Johnnie had to be despatched on a\nvinegar-borrowing expedition, so that the time-honored application of\nbrown paper soaked in vinegar might be made to the poor little head.\nThe baby cried lustily, with a colicky cry, and Mary had to hasten the\nboiling of tea, that wee Annie might have a good, hot cup to soothe\nher. Casey, complaining profanely of broken slumbers, was in no mood to\nbe left home with fretting children while Mary went to mass.\n\nIt was nine o'clock before she could get away; the last mass in the\nbasement was at nine o'clock. But the Elevation of the Host had been\ncelebrated before she got there, and she turned disappointedly to\nthe stairs; she would have to wait for half-past nine mass in the\nmain church. It seemed as if Providence were balking her, but on the\nstairway she learned the reason why.\n\n\"Ye mus' be sure t' say a spicial prayer on this mass,\" said one woman\nwho passed her to another, \"'tis the first mass this young praste have\niver said, an' a blissin' go wid it t' thim thot prays wid him.\"\n\nSaul on the Damascus road had no more overwhelming sense of arrest and\nredirection than Mary Casey had, as, trembling with excitement, she\nreached the top of the stairway.\n\n\"Think o' that now,\" she told herself, \"an' if I had come t' th' airly\nmass I'd niver 'a' known it!\"\n\nHardly would her knees uphold her until she could sink into an obscure\npew, far back under the gallery. And there, at the tense moment when\nthe silver-toned bell proclaimed commemoration of the great lifting-up\nin suffering, Mary raised her faith-full prayer: \"A'mighty God, sind\nme gyurl back t' me! But if it don' be in yer heart t' do thot mooch,\nmaake her a good gyurl wheeriver she be. Fer th' love av Christ, Amin.\"\n\nNot often in any lifetime, perhaps, does it come to pass that one prays\nwith such sublime assurance of crying straight into the listening\near of Omnipotence that will inevitably keep faith with poor flesh.\nFor nigh on to forty years Mary Casey had listened to reiterations\nof the old and new Covenants, but they had fallen on sterile ground\nin her soul. It was the little chance remark about the new priest's\nfirst mass, dropping into harrowed and watered soil, that flowered in\nimmediate faith.\n\n * * * * *\n\nThe mass ended and the throngs of worshipers passed out, but Mary sat\nunheeded and unheeding in her dim corner, her simple mind grappling\nwith the stupendous idea of its Covenant with Heaven.\n\nBefore she had any realizing sense of time, the church had filled again\nfor high mass. Then the lighting of the great white altar fascinated\nher, and she felt an intense desire to live again through such a moment\nof assurance as she had lately experienced--to hear that bell ring\nagain, to smell the incense, and to believe that in some wonderful,\nwonderful way it was all a part of that prayer of hers that Heaven was\nbound to answer.\n\nSo she stayed on, in her far-away pew, to the remotest corner of which\nshe was crowded as the enormous church filled to its capacity. With the\nentrance of the preacher into the pulpit, though, she was conscious of\na distinct \"let-down.\" She had never liked sermons; they dealt with\nthings so formally. Even when the priests made their greatest efforts\nto be plain-spoken and understandable, she seldom got any personal help\nfrom their discourse. They were prone to denunciations of adultery and\ndrunkenness and other sins of which she was innocent, and to vague\nexhortations looking toward a hereafter on which her imagination had\nnever taken any but the feeblest hold. But what was this priest saying?\nSomething about a little household that the Lord had loved, and one of\nits two sisters had gone astray!\n\nThe woman sitting next to Mary nudged her other neighbor and glanced in\nthe direction of Mary's face, thrust forward as if so as not to lose a\nsyllable, the tears chasing each other unheeded down its furrows. In\nher lap Mary's gnarled hands were clasped in painful intensity.\n\nOver and over, since she was a tiny child in Ireland, she had heard\nthis Catholic rendering, of Mary of Bethany's story, but it had never\nmeant anything to her. To-day it meant everything.\n\n [Illustration: \"MARY SAT UNHEEDED AND UNHEEDING IN HER DIM CORNER, HER\n MIND GRAPPLING WITH THE STUPENDOUS IDEA\"]\n\n\"An' I said I niver wanted t' see her ag'in if she'd disgraaced me,\"\nshe told herself, and was appalled at the remembrance.\n\nThat afternoon, toward the early dusk, she sat in the dark kitchen\nholding Annie in her lap; all the other children were out. Casey,\nwho had not left the house all day, was huddled up to the stove,\nsmoking his rank pipe; he was unshaven and unwashed, and wore a\ncoarse undershirt of a peculiar mustard color which lent his pallid,\ngrime-streaked face a ghastly hue. He had been talking about a \"gran'\njob\" of which a man had told him, and building large castles about\nmoving to a better street and a better house and buying a \"parlie suit\nbe aisy paymints.\"\n\nMary listened believingly; twenty years of listening to these dreams\nwhich never came true had not killed her hopefulness. As she listened,\nthough, her hopes outran Casey's, for she could conceive no possible\nfelicity without Angela Ann. How to introduce the now-forbidden subject\nof Angela was a problem, but clearly the only way was to plunge in.\n\n\"Yis,\" she assented, \"I t'ink we should have a parlie. It have always\nbeen my belafe thot if we'd had a parlie Ang'la wouldn't niver 'a' wint\naway. Ag'in' she come home, I'm goin' t' kape th' parlie noice fer her\nan' lave her have her beau ivry noight, an' no wan t' bother thim. An'\nI ain't goin' t' lave her go downtown t' work no more--theer's too\nmanny bad min. She kin stay home an' moind th' house, an' I'll git\nscrubbin' t' do t' th' Imporium. Wid what you earn an' what I earn, we\nkin give her mebbe a dollar a wake fer spindin' money.\"\n\nMary waxed excited as her dream unfolded, but Casey was ironical.\n\n\"Whin d'ye _ixpict_ her?\" he inquired, with pride in the sarcasm.\n\n\"I dunno,\" said Mary, undaunted, \"but I know she'll come. An' whin she\ndo, I'll not ask her anny quistions. I don' keer how she come t' me, so\nshe come. No matter what she've done, theer mus' be dipths she haven't\nr'ached yit, an' all I ask now is t' save her from gittin' anny worse\nthan she be. D'ye know what I prayed t' th' Mother o God befoore I\nlift th' church this mornin'? I prayed that our Ang'la Ann'd git in\ntrouble--in tur'ble trouble 'n' disgraace so thot thim thot's lid her\naway'd t'row her out, 'n' no wan but God 'n' her mother'd take her in!\"\n\nIn speechless astonishment Casey gazed at the vehement woman before\nhim. Some instinct made him hold his peace while she told about the\npriest's first mass, about the sermon, about the answer she confidently\nexpected to her prayer. While he listened, his easy Irish emotionalism\ncaught the contagion of her belief, and his tears flowed unchecked as\nhe alternately cursed the man that had led Angela away, and prophesied\nglowingly of the \"parlie\" that was to be.\n\nIt was pitchy dark in the kitchen now, and Mary got up to light the\nlamp. As she did so, a sound at the door caused her nearly to drop the\nlamp. Hurrying to the door, she threw it open, and with the light in\none hand peered out into the black yard.\n\n\"Here, pussy, pussy,\" she called. Then, as her call was answered, \"My\nGod! what did I tell ye? Tis the wan-eyed cat!\"\n\n\nIII\n\nThe next morning the postman brought a letter. Mary was not surprised\nto get it. Casey had gone to look for the \"gran' job,\" and the older\nchildren were in school, so the letter could not be read, but she could\nmake out the signature, written in the large, unformed hand where-with\nAngela had covered every available space in the days of her brief but\nlaborious apprenticeship to the art of writing.\n\nWith trembling hand Mary tucked the letter in her bosom, hastily got\nready herself and Dewey and the baby, and started for Maggie's. Maggie\nwas younger and had enjoyed more educational advantages. She could\n\"r'ade printin'\" easily, and \"writin\"' fairly well if it hadn't too\nmany flourishes.\n\n\"She says,\" spelled out Mrs. O'Connor, \"'Dear Ma, I'm at ---- West\nRandolph Street I'm sick I'm afraid to go home count of Pa Your Loving\ndaughter Angela Ann Casey.' I'll go wid ye,\" finished Mrs. O'Connor in\nthe same breath.\n\nOut of her small store of tawdry finery she lent several articles\nto make Mary \"look more drissy,\" and while they got ready for their\nmomentous journey, Mary related the events of the day before, and of\nSaturday night.\n\n\"Me an' Tim,\" said Maggie, when the tale had reached the stage of the\n\"parlie\" and Mary's earnings as a scrub-woman, \"was figgerin' how we\ncould help out a bit, ag'in' she come home, an' Tim have promised t'\ntake me 'n' her to th' theayter quite frayquint of a Sat'day noight,\nan' together we're goin' t' give her half a dollar ivry wake t' spind\non her clo'es.\"\n\nThe number they sought on West Randolph Street was not far from the\nfateful Haymarket Square. There was a store on the ground floor, with\nliving rooms behind. And above, a long flight of oilcloth-covered\nstairs led to a \"hotel.\"\n\nThey inquired first in the store, but no one there had ever heard of\nAngela Ann. Then, with fast-beating hearts, the women mounted to the\noffice of the hotel, an inside room facing the head of the first flight\nof stairs. The door stood open, and they looked, before entering, into\na gas-lighted room furnished with yellow-painted wooden arm-chairs\nranged along the walls and flanked by a sparser row of cuspidors; a\nbig sheet-iron stove on a square zinc plateau filled the middle of\nthe room, and near the door, behind a small desk like a butcher-store\ncashier's, sat the \"clerk,\" chewing vigorously and expectorating\nwithout accuracy.\n\n\"Yes, she has a room here,\" he answered to Mary's question, \"hall room,\nrear, third floor.\"\n\n\"In a minute!\" called Angela Ann's voice when Mary had knocked.\n\n\"My God, 'tis hersilf,\" sobbed Mary, and fell a-weeping violently.\n\n\"Ma!\" cried Angela Ann, and threw open the door. She had been in bed\nwhen they knocked, and had not waited to put on her clothes when she\nheard her mother's voice. At the touch of her, the clinging clasp of\nher poor, thin, cold little arms, Mary grew hysterical.\n\n\"Don't, Ma, don't,\" begged Angela.\n\n\"She've grieved hersilf sick over ye,\" said Maggie, unable to forbear\nthis much of a reprimand now that the sinner was found. \"Iver since ye\nwint she've been loike wan crazy. Come, Mary; now ye've got her, brace\nup!\"\n\n\"Sure, Ma,\" echoed the girl, \"now ye've got me, brace up, I ain't never\ngoin' t' lave ye no more, Ma--honest t' God, I ain't.\"\n\n\"Wheer ye been?\" Mary raised her head, and drawing back from the girl\npeered anxiously into her face. \"In God's name, Ang'la Ann, wheer you\nbeen? Tell me ye've kep' dacint, gyurl, tell me ye've kep' dacint!\"\n\nAngela sat down on the dingy, disordered bed and began to cry, hiding\nher face in her hands. For a long moment the silence, save for her soft\nsobbing, was profound. Then a low moan escaped Mary, a moan of anguish\ninexpressible, showing how deeply, notwithstanding her resolution of\nyesterday, she had cherished the hope of her daughter's safety.\n\n [Illustration: IN GOD'S NAME, ANG'LA ANN, WHEER YOU BEEN?]\n\nAngela raised her head. The pain in her mother's moan was beyond\nher comprehension, and she could only understand it as horror and\ncondemnation.\n\n\"Are ye--are ye--goin' t' t'row me off?\"' she asked.\n\n\"T'row ye off? Ah, me gyurl, if ye'll on'y stick t' me as long as\nI'll stick t' you, 'tis all I'll ask o' Hiven! Tis fer yer sake I was\nprayin' no harm had come t' ye--not fer mine. Whativer happen t' ye,\nye're me Ang'la Ann thot I nursed from yer first brith. An' ye don'\nknow all I'm fixin' t' do fer ye--me an' yer pa an' yer Aunt Maggie,\nhere, and yer Uncle Tim----\"\n\nAnd there followed a glowing account of the feast prepared for the\nprodigal's return.\n\n\"Th' idare o' you bein' afraid o' yer pa,\" chided Mary, \"an' him fixin'\nt' git a stiddy job an' not have ye go downtown no more.\"\n\nFar shrewder than her mother, Angela Ann did not overestimate\nthis excellent intention of her pa's, but she said nothing of the\nbitterness that was in her heart on account of his past crimes. It was\na long-standing grievance with her that her mother could never, for\nmore than a fleeting, irritated moment at a time, be made to see Casey\nas others saw him. Angela Ann had been working for him since she was\neleven (child-labor laws were lax, then) and giving up her every penny\nto pay rent and buy insufficient mites of coal and food--just enough to\nkeep them alive and no more--and it was starvation of many sorts that\nsent her at last into the clutches of them that prey. The girl was full\nof self-pity, and impatient with her mother because the older woman had\nforgotten how to rebel.\n\n\"Yer pa say, though,\" added Mary, \"thot he won't promise not i' kill\nthe felly thot lid ye away; he've got tur'ble wingeance on him--yer pa\nhave.\"\n\nAngela Ann smiled grimly. \"I guess theer's quite a few pa's lookin' fer\nhim,\" she said, \"but they don't ever seem t' find him.\"\n\n\"Did he prom'se t' marry ye?\" asked Mary anxiously.\n\n\"I should say not! He promised to make me a primmy donny.\"\n\n\"What's that?\" fearfully.\n\n\"'Tis a kind of actress that wear tights an' sings,\" explained Angela.\n\"I'm after r'adin' in books how gran' they be, an' in the papers\nit tell how the swell fellies do be runnin' after thim with diming\nnecklusses, an' marryin' of 'em. 'Tis all a lie!\" she cried shrilly.\n\n\"Ye see,\" Mary could not refrain from reminding her. \"I tol' ye thim\ntheayters was all wrong. We kind o' t'ought it might be thim thot got\nye, an' yer pa wint t' see this here Halberg, whin we foun' the caard\nout o' yer pocke'-book. But he said he niver hear tell o' ye.\"\n\n\"Did pa go there?\" questioned Angela eagerly. She was all interest to\nknow how the search for her had been carried on, and \"did th' p'lice\nknow?\" and \"how did ye kape it out o' th' papers?\"\n\nYes, it had been Halberg \"all the time,\" she admitted. She had answered\nhis advertisement, and after a week's drill he had sent her, true to\nhis published word, in a \"road company\" that mitigated the gloom of\ncoal miners' lives by singing and dancing--and carousing--in a circuit\nof saloons in the soft coal regions of Illinois. When she fell sick,\nthe company abandoned her without the formality of paying her any\nsalary, and a foul-tongued, soft-hearted landlady, whose own young\ndaughter was God knew where, had let Angela stay in her wretched hotel\nuntil she was able by dishwashing and lampfilling chores to earn the\nfew dollars to take her back to Chicago.\n\n\"But I couldn' get no stren'th back,\" the girl went on, \"an' that woman\nat th' hotel, Mis' Schlogel, she sez t' me, 'You better go home t' yer\nma, that's wheer you better go,' an' she bundled me off Friday mornin'.\nBut I was scairt t' go home right t' wunst till I seen how youse was\ngoin' t' be t' me, so I come here wheer I stayed whin I was studyin'\nwid O. Halberg, an' Friday night I got awful sick an' laid here all\nnight awake an' burnin' up an' my head achin' t' beat th' band. An' all\nday Sat'day an' Sunday I wasn't able to go out fer nothin' t' eat, an'\nth' propri'ter wouldn't order me nothin' sent in fer fear I wouldn't be\nable t' pay. A woman in the nex' room light-house-keeps, an' she made\nme tea a couple o' times after she heard I was sick an' alone.\"\n\n\"Why in Hivin's name,\" Maggie broke in, \"did ye niver drap yer ma a\nline t' say ye were aloive? Ye needn't 'a' tol' wheer ye was, but ye\ncould 'a' said ye were in the land o' th' livin', surely?\"\n\n\"I was 'shamed,\" whimpered Angela; \"I fought ye wouldn't keer wheer I\nwas if I wasn't doin' dacint.\"\n\n\"Think o' that, now!\" cried Mary. \"That's all a gyurl do know about\nher ma. Whin yer a ma yersilf ye'll know better, an' not till thin, I\nsuppose.\"\n\nThus was Angela Ann made sure of her welcome home.\n\n\"An' not wan but yer own kin know ye've been missin'\" said Mary, as she\nhelped the girl to get ready for the return, \"so ye kin hol' up yer hid\nan' look th' world in th' faace. An' may God fergive yer mother the\nloies she've tol' t' save yer name!\"\n\n\n\n\n [Illustration]\n\nBORDEN\n\nBY GEORGE C. SHEDD\n\nILLUSTRATIONS BY WALTER BIGGS\n\nOne rainy afternoon I was sitting with my friend Carter, in his log\nhouse. Through the open door we could see the road, all cut up by\nwagon-tracks, running with water; lumps of mud thrust their black heads\nup in it everywhere; the bordering grass was wet and heavy. And down by\nthe creek the fringe of trees made only a gray blur.\n\nWe had talked ourselves pretty near out when a rider splashed up to\nthe door. His ragged beard stuck out stiff, full of rain-drops, and\nhis slouch hat had an unpleasant tilt forward. To Carter's invitation\nto enter he shook his head, asked if such-and-such a person had passed\nwithin the hour, and, receiving an affirmative reply, pulled his hat\ndown tighter and galloped away west. \"Who is that?\" I inquired.\n\n\"That! Why, that's Borden. It's easy to see you're new out here. His\nhand holds the river from Saint Joe to Omaha, and men think twice\nbefore trying to break his grip.\" He drew out his pipe and tobacco,\nstuffed the bowl thoughtfully, and struck a match. \"If you want to hear\nabout the first time I saw him at work, I'll tell you.\"\n\nI nodded.\n\n\"Eh? Well, this was the way of it\":\n\n * * * * *\n\nAt the end of the war I settled here--that was five years ago. Borden\nlived a mile up the creek, and so, as times went, we were neighbors.\nBy the people yonder in Kinton he was not liked, being grim, rough,\nsavage, altogether unsociable and short of word. Besides, they\nremembered '57. In that year he appeared from no one knew where, took\nhis claim, and proceeded to live after his own fashion. Then the\nhigh-handed Claim Club of the village went about it to drive him \"in or\nover the river\"--a bad night for them. They rode back to Kinton with\nthree dead men laid across saddles. That was in the rough days of the\nTerritory, the days when men in the Nebraska hills along the Missouri\nwere a law unto themselves.\n\n [Illustration: \"THEY CROWDED HIS HORSE UNTIL IT HUNG BACK FROM THE\n OTHERS\"]\n\nOnce he tied up on his own deck a steamboat captain who was drunk\nand bent on murder; single-handed he ran down two horse-thieves; and\nanother time he choked the money out of a river-gambler who had robbed\na boy. Oh, they knew Borden up and down the river in those days! Then\nhe went to war as one of Thayer's sharpshooters, returning at the end\nof it to be appointed United States marshal. And he had been riding\nthat saddle six months when I came.\n\nOne day he and another pulled rein at my door.\n\n\"Come with me,\" he said abruptly. \"I want you to look after this\nfellow--you're my deputy till further notice.\" He did not waste time\nover oaths or official nonsense.\n\n\"Now, see here--\" the man started to say. But Borden cut him off with a\nscowl.\n\n\"Who is he?\" I asked.\n\n\"Him?--Fitch. You've heard of him, I guess.\"\n\nHeard of him, of course, as everyone had; of his sly, petty legal\ntricks by which he grabbed land here and land there until his titles\nspotted the country about Nebraska City; of his rent-squeezing that\nsmelled over the whole town; of these, and other things. He was a lean,\ndark, uneasy fellow, wearing a rumpled tile and a shiny coat, riding\nall crouched up, and pulling his horse away from everybody we met.\n\nAfter we started, Borden told me that Fitch had brought him notice to\nserve on Dempster--old John Dempster, his friend. Now, that made a bad\njob for the Marshal. I saw it from the way he answered not a word to\nFitch, who now and then pressed up--intent on the business--to make him\ntalk. Once Borden pulled out his heavy wrinkled boot from the stirrup\nand kicked the other's horse in the belly until it reared on its\nhaunches. For Borden was the law's officer, but no man's servant.\n\nOur way ran three miles up from Kinton. There was no road, and we\nfollowed along the edge of the bluffs as best we were able, until\nfinally we dipped down into a ravine and so came to our destination. It\nwas a wooded flat on the bank of the river, made by a sudden retreat\nof the hills--a sort of pocket. The space was not large, a handful of\nacres, and it looked smaller than it really was. The bluffs curved\naround it on three sides in a yellow, crumbling wall; on the fourth\nflowed the muddy waters of the Missouri. The house was in the center\nof a small clearing, and when we came in sight of it Fitch pulled up\nbehind a small thicket of scrub. Borden, as if he never saw the fellow\nhalt, rode straight up to the door where John Dempster sat shaping an\naxe-haft.\n\n\"Jack,\" said Borden, swinging down from his saddle, \"I've come to have\na talk with you.\"\n\nDempster shaved the haft a minute, laid it aside, and gazed off toward\nthe clump of scrub. The two men were something alike, though the man\nseated on the door-sill was the older, both past the prime, both spare\nof words, both come to the West in the same year. They had lain side by\nside behind a sleety log before Fort Donelson, and each in his three\nyears of service had felt the touch of hot lead.\n\n\"How d'you come--friend or enemy?\"\n\n\"The first, and always, I hope. It depends on you. Why did you kick him\noff of here yesterday, Jack? He's full of poison over it.\"\n\n\"Let him keep off then,\" was the gruff response.\n\nBoth looked again at the clump where Fitch could be seen through the\nthin screen of bushes. After a while Dempster took out his tobacco, cut\noff a piece, and passed the rest to us.\n\n\"You're in a dirty way of business when you're mixed up with him,\" he\nsaid slowly. \"An' I 'spose you've come to run me out.\"\n\n\"What's at the bottom of this trouble?\" returned Borden, evading the\npoint. \"'Tain't the land--what is it he's after?\"\n\nDempster spat. \"He's gettin' even. I knocked him down last spring when\nI was at Nebraska City, for lyin' about--never mind. That's all. So he\nsneaked around an' hunted out where I live an' filed on the land.\" A\ndull fire lighted up under his bushy eyebrows.\n\n\"Why didn't you file long ago?\"\n\n\"Does the gover'ment take away a man's home when he's fought in the\nwar?\"\n\n\"You know how I feel about it,\" replied Borden, and he laid his hand on\nthe other's shoulder. \"But it's too late for you to try to keep it now.\nYou'd better look up another place.\"\n\n\"No, I'm goin' to stay here, I guess, or nowhere.\"\n\nBorden knew that the decision was inflexible. As he rose, put his foot\nin the stirrup, and raised himself into the saddle, he determined,\nhowever, to have another try.\n\n\"Come and settle up along the creek by me. There's an open claim just\nbeyond mine, better than this piece.\"\n\n [Illustration: \"'YOU GOT THE BEST O' ME, DICK; I'LL GO'\"]\n\nDempster shook his head; maybe he was thinking of the clearing back in\nIndiana and the boughs under which he had drawn his first breath,\nmaybe this poor fringe of woods along the river was dearer to him than\nall the treeless prairie.\n\n\"We've lived here near ten years now,\" he said at last, \"the old woman\nan' Joe--an' me, 'ceptin' when I was at war. I guess if we go, you'll\nhave to use your gun.\"\n\n\"I'm sorry, Jack, but you've got to go. And I give you a week. It's not\nme that says so, it's the law.\"\n\n\"Law!\" answered Dempster, with sudden rising fierceness. \"Does the law\ndrive a man off his own?\"\n\nIt was the law, not justice, that was driving him. Without replying a\nword, Borden, and I by his side, rode away. When we reached the lean,\neager face behind the scrub, the Marshal broke out, \"You vulture, keep\nbehind us! If you try to ride even, I'll sink your carcass in the\nriver.\" And in that order, with him trailing us, we came back to Kinton.\n\nWell, during the next week the more I turned the thing over on my\ntongue the less I liked the taste of it, but Borden was not one to\nconsider dislikes--neither another man's nor his own--when he was\nriding the law's saddle. So I resolved to go through with it, and was\nready Thursday morning. He came out from Nebraska City, accompanied\nby six deputies, men he had tried, who would not back off from the\nmouth of a gun, for he knew the door he must enter that day. Fitch was\namong them; oh, he was yellow over it! Borden had dragged him along to\nthe whole end of the dirty business. The tale, too, was out among the\ndeputies, and Fitch saw plainly what rope they would have swung him\nby. Grim looks were his every mile; when he pushed up among them, they\ncrowded his horse to the withers until it hung back from the others;\none cursed him fully and foully. They intended that he should earn that\nbit of ground before the day was done.\n\nIn the ravine at the edge of the flat we tied our horses. The men\nunslung their rifles, hitched their revolvers about, and waited, while\nBorden went down the hollow to reconnoiter. Perhaps half an hour had\npassed when he climbed down the bank above our heads and dropped into\nour midst.\n\n\"Quick! The boy's gone for water to the spring. Straight ahead there.\nNo shooting till I give the word.\"\n\nThe men nodded, we filed down the ravine single-file, and the next\nminute were advancing noiselessly through the trees, spreading out\ngradually as we crossed the flat toward the clearing where stood the\nlog house. The deputies went ahead, alert, silent, with an eye on\nBorden, who walked a little before them, each keeping a tree in line\nwith the door.\n\nPerhaps things were no different that morning than they were at any\ntime; yet the little flat seemed possessed of a very great quiet,\nbroken only by the slight swish of our boots through the dry grass.\nAs we neared the cabin, we saw that its windows and door were shut.\nFitch, who clung to me as though he found more comfort in my company,\noccasionally wiped drops of sweat from his yellow forehead, and removed\nhis high hat to let the wind blow through his hair.\n\nThe other men went ahead unconcerned enough. One big fellow dropped\nhis gun into the crook of his arm, pulled out a piece of tobacco, and\ncarefully picked the lint off it. When he had had a bite, he tossed it\nto a comrade, who caught it handily, buried it for a moment under his\nmustache, and then held up the remnant to the other's sight, grinning.\nHe tossed it back; neither had lost his place in the advancing line.\n\nFifty yards from the house Borden signaled a halt. Rifle-butts slipped\nto the ground, and the men leaned with backs against their trees--all\nexcept two, who handed their guns to others and veered off towards\nthe bluffs, the direction Borden indicated, to the spring. A brown,\ngrizzled fellow, sheltered behind an elm a few feet from me, turned\nhis attention to Fitch, whom he examined curiously and at leisure,\nconcluding his inspection by spitting his way. Then his look strayed\nsouth. After a little he began to sing softly:\n\n The flat-boats 'r in an' the bull-boats 'r a-stoppin'\n An' licker runnin' free,--oh, hell is a-poppin'!\n Down on the river, down on----\n\nHe broke off suddenly, turning his head a little way towards where the\ntwo men had entered the bushes, listening. Directly he finished the\nlines:\n\n Down on the river, down on the river,\n Down on the Misser-ee when the boats come in.\n\nThe man must have had ears like an Indian's. He folded his arms across\nthe muzzle of his rifle and began watching the bushes that fringed the\nbase of the hill; the other men also were looking that way. A minute\npassed. All at once a young fellow slipped out from nowhere, running\nand carrying a full bucket. He was bare-headed, his sleeves rolled\nto the elbows. He ran a few steps toward the house, quickly slanted\noff, and kept going, while turning his head this way and that. I saw\nthe cause of his sudden change in direction, for there was one of the\ndeputies running parallel with him, but between him and the door. The\nsecond came in sight a minute later, farther down, and from behind a\nthicket, abreast of the other two. They had the young fellow between\nthem.\n\nThe rest of us were strung about before the house in a half-circle, the\nthree runners being on the outside of the circle. Everything was quiet,\nfor Borden's hounds don't hunt with their mouths open. Young Dempster\ncarried his bucket of water with scarcely a slop or a splash; the inner\ndeputy gradually moved out and behind him. Two men at the tail of the\nline fell away from their trees to meet him--and there he was in a\nring. The man nearest me, still leaning on his rifle, gave a cluck of\nhis tongue as if it were all over. But it was not. A shot cracked from\nthe door, and the deputy who was on the outside flipped his hand in\nthe air as if he had been stung. His fingers were all bloody. That was\na pretty shot, I tell you; old Jack Dempster ticked the button on his\nson's shirt to make it, for the men were running breast and breast from\nthe door.\n\nThe boy saw the trap he was in. Just as he came even with me, he\nwhirled and took his chance through the line. It was quick--oh,\nquick as a cat! Three of us met him. But he was in moccasins and\nlight-footed, jumping this way and that, and though my neighbor flung\nhis rifle between his legs, he skipped it and was nearly through. He\nsprang to one side, leaped at Fitch--the water was splashing now--and\nswerved past him. Maybe it was the nasty look on his face that made\nFitch shoot, anyway the fellow fired his revolver. It did not seem as\nif he could miss; Joe ran straight for the cabin. Half way there the\nbucket slipped from his hand; then he began to stagger a little. Near\nthe door he went to his knees and, with a look over his shoulder at us\nwhile fumbling for his revolver, crawled behind the chopping-log.\n\n\"I got him before he got me,\" said Fitch, fairly green about the mouth,\n\"He was going to kill me.\"\n\nBorden took a step toward him, paused for the time of a single breath,\nwhirled around, and was behind his tree. As for the other men, I never\nwant to see such faces as they wore.\n\nAfter that it seemed to me as if our business had come to a standstill.\nIt was little shelter we had, just a tree apiece. We might as well have\nbeen tied to them with cords, for the old man was watching from his\nlair, and that with his boy's blood red in his eyes, ready to catch us\neither advancing or retiring. Nor was the young fellow so badly hurt\nbut what he could pull a trigger. And Borden never retired that I ever\nheard of--that wasn't his way. Any instant I expected to hear a bullet\nsnip the bark on my tree. I never felt so big before or since, big as\na hill, and I drew myself together mighty small, I can tell you.\n\nWhile I was wondering what would come next, Borden stepped out into\nthe open. He walked toward the door, calm and steady, and without\nparticular haste, his revolver in its holster. It all happened so\nquickly it took me by surprise; the Dempsters, man and boy, must have\nbeen struck by it, for not a shot was fired. But to advance that way,\nto clasp hands with death! Maybe you've heard soldiers tell about\ncharging in the face of cannon, how they felt--I know I felt worse just\nto see him go straight toward the house. I got dizzy, dizzy sick. Then\nit had all fallen so still, the little wind in the trees and the leaves\nstirring over the ground. I looked at the other men, thinking they\ncould somehow change it; the grizzled old chap was chewing his tobacco\nas fast as he could, and the man with the bloody fingers had finished\ntying them up in his handkerchief. First thing I knew I was half out\nfrom behind my tree, watching him.\n\n\"Keep back, Dick Borden,\" warned the man in the house--I swear his\nvoice shook as he said it--\"keep back, or, by God, I'll shoot!\"\n\n\"I'm coming into that door, Jack Dempster,\" was Borden's reply.\n\nHe never flinched, never stopped. Then the rifle sounded, and, like an\necho, the boy's revolver echoed it. Borden was hit--how could they fail\nat that distance and such a mark? But he managed to win the log where\nyoung Dempster lay. He stood there an instant, then slowly sat down\nupon it. A second time the young fellow lifted his weapon, and every\nman of us could see the Marshal looking into the muzzle. Orders or no\norders, that was too much for even the deputies; the click of their\nrifle hammers ran along the trees. Borden heard it.\n\n\"Don't shoot, men!\"\n\nHis voice was not loud, but harsh, and keyed high, as if his throat was\ndry. I think the next sound was a groan from the boy, and his revolver\nwavered and slipped in his fingers.\n\n\"It's the gun you gave me,\" he said, \"an' I can't kill you with it.\"\n\nBorden turned his head painfully from side to side, saw a stick, bent\ndown laboriously, got it at last, and by its aid raised himself to\nhis feet. That seemed to exhaust him. He stood for a moment, inert\nand useless, like an old man. Then he began to hoist himself forward\nstep by step to the door. Iron will, just iron, it was. And it was\nterrible to see him--one shoulder and arm swinging low and limp, his\nknees lifting high as if knotted with stiffness, his head protruding in\nintense effort. The distance was short, but long, long for him.\n\n\"Keep back! keep back!\" cried Dempster. He himself was half out of the\ndoor, gripping his gun with one hand, warding the relentless Marshal\noff with the other.\n\nBorden answered nothing, another step.\n\n\"You've got to stop!\" begged Dempster. \"Don't make me kill you, an'\nI can't let you in. Go back, go back! We fought together, we marched\ntogether, we ate and slept together, Dick--for God's sake, don't come\nnearer!\"\n\nOne step at a time, putting his stick forward bit by bit and dragging\nhimself to it with his queer uplifting knees, Borden moved himself\nahead. There was something stern and inhuman in this persistence. So it\nwent to the last bitter inch. Then Borden's breast touched the rifle's\nmuzzle. The two men stood looking into each other's eyes, measuring\nlife and death.\n\nThat is a minute in my mind forever. The young fellow had dragged\nhimself a little way from behind his log--half-following, fascinated,\nsupporting himself by his two hands--and was staring at them. The empty\nbucket lay on its side in the sunshine. The wind whined and whined\nthrough the trees. And the wife's haggard face peered over Dempster's\nshoulder in the door.\n\n\"I arrest you!\"\n\nThe stick dropped from his fingers, he clutched at the man's sleeve and\nfell across the door-sill. All I remember is that we were all crowding\nabout the door, with the boy cursing from the ground behind us for\nsomeone to help him. Even Fitch had come, twisting and pushing among\nthe rest.\n\nBorden was white and still, but he came around directly and stared at\nus a little. We laid him on a blanket outside the door, along with\nJoe, who carried his lead just below the knee. The Marshal was pretty\nbad, having a bullet through his collar-bone and another through his\nside, this one a big ugly hole. There were plenty of us to help, some\nto cut and to strip their clothes, some to fetch water, some to wash\nthe wounds, some to tear bandages. One had already started south for a\ndoctor. Dempster was on his knees by his old comrade.\n\n\"You got the best o' me, Dick; I'll go.\"\n\nBorden smiled a little. It was good to look at their two faces then.\n\nFitch, who was rubbing his hands evilly, put in, \"Yes, you get off here\nwithin an hour. And I'll have the law on you, too, for the kicking you\ngave me.\"\n\nOne of the men struck him across the mouth.\n\n\"Tie him,\" said Borden, \"and hang him.\"\n\nWell, there was a noisy to-do, the fellow screeching that it was\nagainst the law, that he shot the boy for trying to kill him, that it\nwas on his own land, and the like. He kept it up until his screech fell\ninto a quaver, and terror came into his eyes. Borden smiled again at\nsight of him, this time with lips that made a straight white line.\n\n\"The law!\" he said, at last. \"I am the law.\"\n\nHe let the matter go as far as the rope around the wretch's neck; then\nit seemed as if Fitch was dead already. No, Borden didn't hang him;\nhe had another idea, the claim. He waited until Fitch had his senses\nonce more and told him he would be taken to Nebraska City and tried for\nattempted murder. Fitch began to beg, while Borden listened with grim\nsatisfaction. He would let the claim go, he would start down the river,\nquit the country. The rope was thrown off and Borden ordered him away;\nand with a sudden fierce oath that made him gasp from pain, Borden\nswore he would shoot him with his own hand if he caught sight of him\nagain.\n\nFitch knew that Borden meant what he said, and he wasn't seen again in\nNebraska. Six months or so fetched Borden round, and let him into the\nsaddle again. It must be lead in the heart or brain to kill men of his\nfiber--and Dempster had been shaky with his gun. Things got a little\nloose while the Marshal was on his back up there in the cabin, but he\ntightened them up again soon. We'll ride up there some day and see the\nspot. Yes, the Dempsters have the title to the place now.\n\n [Illustration]\n\n [Illustration]\n\n\n\n\n [Illustration]\n\nTHE GLOUCESTER MOTHER\n\nBY SARAH ORNE JEWETT\n\nDECORATION BY WLADYSLAW T. BENDA\n\n [Illustration]\n\n When Autumn winds are high\n They wake and trouble me,\n With thoughts of people lost\n A-coming on the coast,\n And all the ships at sea.\n\n How dark, how dark and cold.\n And fearful in the waves,\n Are tired folk who lie not still\n And quiet in their graves;--\n In moving waters deep,\n That will not let men sleep\n As they may sleep on any hill;\n May sleep ashore till time is old,\n And all the earth is frosty cold.--\n Under the flowers a thousand springs\n They sleep and dream of many things.\n\n God bless them all who die at sea!\n If they must sleep in restless waves,\n God make them dream they are ashore,\n With grass above their graves.\n\n\n\n\nALCOHOL AND THE INDIVIDUAL\n\nBY HENRY SMITH WILLIAMS, M.D., LL.D.\n\n\nSome very puzzling differences of opinion about the use of alcoholic\nbeverages find expression. This is natural enough, since alcohol is a\nvery curious drug, and the human organism a very complex mechanism. The\neffects of this drug upon this mechanism are often very mystifying. Not\nmany persons are competent to analyze these effects in their totality.\nStill fewer can examine any of them quite without prejudice. But in\nrecent years a large number of scientific investigators have attempted\nto substitute knowledge for guesswork as to the effects of alcohol,\nthrough the institution of definitive experiments. Some have tested its\neffects on the digestive apparatus; others, its power over the heart\nand voluntary muscles; still others, its influence upon the brain. On\nthe whole, the results of these experiments are singularly consistent.\nUndoubtedly they tend to upset a good many time-honored preconceptions.\nBut they give better grounds for judgment as to what is the rational\nattitude toward alcohol than have hitherto been available.\n\nThe traditional role of alcohol is that of a stimulant. It has been\nsupposed to stimulate digestion and assimilation; to stimulate the\nheart's action; to stimulate muscular activity and strength; to\nstimulate the mind. The new evidence seems to show that, in the final\nanalysis, alcohol stimulates none of these activities; that its final\neffect is everywhere depressive and inhibitory (at any rate, as regards\nhigher functions) rather than stimulative; that, in short, it is\nproperly to be classed with the anesthetics and narcotics. The grounds\nfor this view should be of interest to every user of alcohol; of\ninterest, for that matter, to every citizen, considering that more than\none thousand million gallons of alcoholic beverages are consumed in the\nUnited States each year.\n\nI should like to present the new evidence far more fully than space\nwill permit. I shall attempt, however, to describe some of the more\nsignificant observations and experiments in sufficient detail to enable\nthe reader to draw his own conclusions. To make room for this, I must\ndeal with other portions of the testimony in a very summary manner.\nAs regards digestion, for example, I must be content to note that\nthe experiments show that alcohol does indeed stimulate the flow of\ndigestive fluids, but that it also tends to interfere with their normal\naction; so that ordinarily one effect neutralizes the other. As regards\nthe action on the heart, I shall merely state that the ultimate effect\nof alcohol is to depress, in large doses to paralyze, that organ.\nThese, after all, are matters that concern the physician rather than\nthe general reader.\n\nThe effect of alcohol on muscular activity has a larger measure of\npopular interest; indeed, it is a question of the utmost practicality.\nThe experiments show that alcohol does not increase the capacity to do\nmuscular work, but distinctly decreases it. Doubtless this seems at\nvariance with many a man's observation of himself; but the explanation\nis found in the fact that alcohol blurs the judgment. As Voit remarks,\nit gives, not strength, but, at most, the feeling of strength. A man\nmay think he is working faster and better under the influence of\nalcohol than he would otherwise do; but rigidly conducted experiments\ndo not confirm this opinion. \"Both science and the experience of life,\"\nsays Dr. John J. Abel, of Johns Hopkins University, \"have exploded\nthe pernicious theory that alcohol gives any persistent increase of\nmuscular power. The disappearance of this universal error will greatly\nreduce the consumption of alcohol among laboring men. It is well\nunderstood by all who control large bodies of men engaged in physical\nlabor, that alcohol and effective work are incompatible.\"\n\nIt is even questionable whether the energy derived from the oxidation\nof alcohol in the body can be directly used at all as a source of\nmuscular energy. Such competent observers as Schumberg and Scheffer\nindependently reached the conclusion that it cannot. Dr. Abel inclines\nto the same opinion. He suggests that \"alcohol is not a food in the\nsense in which fats and carbohydrates are food; it should be defined\nas an easily oxidizable drug with numerous untoward effects which\ninevitably appear when a certain minimum dose is exceeded,\" He thinks\nthat alcohol should be classed \"with the more or less dangerous\nstimulants and narcotics, such as hasheesh, tobacco, etc., rather\nthan with truly sustaining foodstuffs,\" Some of the grounds for this\nview will appear presently, as we now turn to examine the alleged\nstimulating effects of alcohol upon the mental processes.\n\n\n_Alcohol as a Brain Stimulant_\n\nThe celebrated physicist Von Helmholtz, one of the foremost thinkers\nof the nineteenth century, declared that the very smallest quantity\nof alcohol served effectively, while its influence lasted, to banish\nfrom his mind all possibility of creative effort; all capacity to\nsolve an abstruse problem. The result of recent experiments in the\nfield of physiological psychology convince one that the same thing is\ntrue in some measure of every other mind capable of creative thinking.\nCertainly all the evidence goes to show that no mind is capable of\nits best efforts when influenced by even small quantities of alcohol.\nIf any reader of these words is disposed to challenge this statement,\non the strength of his own personal experience, I would ask him to\nreflect carefully as to whether what he has been disposed to regard as\na stimulant effect may not be better explained along lines suggested\nby these words of Professor James: \"The reason for craving alcohol is\nthat it is an anesthetic even in moderate quantities. It obliterates a\npart of the field of consciousness and abolishes collateral trains of\nthought.\"\n\nThe experimental evidence that tends to establish the position of\nalcohol as an inhibitor and disturber rather than a promoter of mental\nactivity has been gathered largely by German investigators. Many of\ntheir experiments are of a rather technical character, aiming to test\nthe basal operations of the mind. Others, however, are eminently\npractical, as we shall see. The earliest experiments, made by Exner in\nVienna so long ago as 1873, aimed to determine the effect of alcohol\nupon the so-called reaction-time. The subject of the experiment sits at\na table, with his finger upon a telegraph key. At a given signal--say\na flash of light--he releases the key. The time that elapses between\nsignal and response--measured electrically in fractions of a second--is\ncalled the simple or direct reaction-time. This varies for different\nindividuals, but is relatively constant, under given conditions, for\nthe same individual. Exner found, however, that when an individual had\nimbibed a small quantity of alcohol, his reaction-time was lengthened,\nthough the subject believed himself to be responding more promptly\nthan before.\n\nThese highly suggestive experiments attracted no very great amount of\nattention at the time. Some years later, however, they were repeated by\nseveral investigators, including Dietl, Vintschgau, and in particular\nKraepelin and his pupils. It was then discovered that, in the case\nof a robust young man, if the quantity of alcohol ingested was very\nsmall, and the tests were made immediately, the direct reaction-time\nwas not lengthened, but appreciably shortened instead. If, however, the\nquantity of alcohol was increased, or if the experiments were made at\na considerable interval of time after its ingestion, the reaction-time\nfell below the normal, as in Exner's experiments.\n\nSubsequent experiments tested mental processes of a somewhat more\ncomplicated character. For example, the subject would place, each\nhand on a telegraph key, at right and left. The signals would then\nbe varied, it being understood that one key or the other would be\npressed promptly accordingly as a red or a white light, appeared. It\nbecame necessary, therefore, to recognize the color of the light,\nand to recall which hand was to be moved at that particular, signal:\nin other words, to make a choice not unlike that which a locomotive\nengineer is required to make when he encounters an unexpected signal\nlight. The tests showed that after the ingestion of a small quantity\nof alcohol--say a glass of beer--there was a marked disturbance of\nthe mental processes involved in this reaction. On the average, the\nkeys were released more rapidly than before the alcohol was taken,\nbut the wrong key was much more frequently released than under normal\ncircumstances. Speed was attained at the cost of correct judgment.\nThus, as Dr. Stier remarks, the experiment shows the elements of two\nof the most significant and persistent effects of alcohol, namely, the\nvitiating of mental processes and the increased tendency to hasty or\nincoordinate movements. Stated otherwise, a levelling down process is\ninvolved, whereby the higher function is dulled, the lower function\naccentuated.\n\nEqually suggestive are the results of some experiments devised by Ach\nand Maljarewski to test the effects of alcohol upon the perception and\ncomprehension of printed symbols. The subject was required to read\naloud a continuous series of letters or meaningless syllables or short\nwords, as viewed through a small slit in a revolving cylinder. It was\nfound that after taking a small quantity of alcohol, the subject was\nnoticeably less able to read correctly. His capacity to repeat, after\na short interval, a number of letters correctly read, was also much\nimpaired. He made more omissions than before, and tended to substitute\nwords and syllables for those actually seen. It is especially\nnoteworthy that the largest number of mistakes were made in the reading\nof meaningless syllables,--that is to say, in the part of the task\ncalling for the highest or most complicated type of mental activity.\n\nAnother striking illustration of the tendency of alcohol to impair the\nhigher mental processes was given by some experiments instituted by\nKraepelin to test the association of ideas, In these experiments, a\nword is pronounced, and the subject is required to pronounce the first\nword that suggests itself in response. Some very interesting secrets\nof the subconscious personality are revealed thereby, as was shown,\nfor example, in a series of experiments conducted last year at Zuerich\nby Dr. Frederick Peterson of New York. But I cannot dwell on these\nhere. Suffice it for our purpose that the possible responses are of\ntwo general types. The suggested word being, let us say, \"book,\" the\nsubject may (1) think of some word associated logically with the idea\nof a book, such as \"read\" or \"leaves\"; or he may (2) think of some\nword associated merely through similarity of sound, such as \"cook\" or\n\"shook.\" In a large series of tests, any given individual tends to show\na tolerably uniform proportion between the two types of association;\nand this ratio is in a sense explicative of his type of mind. Generally\nspeaking, the higher the intelligence, the higher will be the ratio of\nlogical to merely rhymed associations. Moreover, the same individual\nwill exhibit more associations of the logical type when his mind is\nfresh than when it is exhausted, as after a hard day's work.\n\nIn Kraepelin's experiments it appeared that even the smallest quantity\nof alcohol had virtually the effect of fatiguing the mind of the\nsubject, so that the number of his rhymed responses rose far above the\nnormal. That is to say, the lower form of association of ideas was\naccentuated, at the expense of the higher. In effect, the particular\nmind experimented upon was always brought for the time being to a lower\nlevel by the alcohol.\n\n\n_The Effect of a Bottle of Wine a Day_\n\nWhen a single dose of alcohol is administered, its effects gradually\ndisappear, as a matter of course. But they are far more persistent than\nmight be supposed. Some experiments conducted by Fuerer are illuminative\nas to this. He tested a person for several days, at a given hour, as to\nreaction-time, the association of ideas, the capacity to memorize, and\nfacility in adding. The subject was then allowed to drink two litres of\nbeer in the course of a day. No intoxicating effects whatever were to\nbe discovered by ordinary methods. The psychological tests, however,\nshowed marked disturbance of all the reactions, a diminished capacity\nto memorize, decreased facility in adding, etc., not merely on the day\nwhen the alcohol was taken, but on succeeding days as well. Not until\nthe third day was there a gradual restoration to complete normality;\nalthough the subject himself--and this should be particularly\nnoted--felt absolutely fresh and free from after-effects of alcohol on\nthe day following that on which the beer was taken.\n\nSimilarly Ruedin found the effects of a single dose of alcohol to\npersist, as regards some forms of mental disturbance, for twelve hours,\nfor other forms twenty-four hours, and for yet others thirty-six hours\nand more. But Ruedin's experiments bring out another aspect of the\nsubject, which no one who considers the alcohol question in any of\nits phases should overlook: the fact, namely, that individuals differ\ngreatly in their response to a given quantity of the drug. Thus,\nof four healthy young students who formed the subjects of Ruedin's\nexperiment, two showed very marked disturbance of the mental functions\nfor more than forty-eight hours, whereas the third was influenced for a\nshorter time, and the fourth was scarcely affected at all. The student\nwho was least affected was not, as might be supposed, one who had been\naccustomed to take alcoholics habitually, but, on the contrary, one who\nfor six years had been a total abstainer.\n\nNoting thus that the effects of a single dose of alcohol may persist\nfor two or three days, one is led to inquire what the result will\nbe if the dose is repeated day after day. Will there then be a\ncumulative effect, or will the system become tolerant of the drug\nand hence unresponsive? Some experiments of Smith, and others of\nKuerz and Kraepelin have been directed toward the solution of this\nall-important question. The results of the experiments show a piling\nup of the disturbing effects of the alcohol. Kuerz and Kraepelin\nestimate that after giving eighty grams per day to an individual for\ntwelve successive days, the working capacity of that individual's mind\nwas lessened by from twenty-five to forty per cent. Smith found an\nimpairment of the power to add, after twelve days, amounting to forty\nper cent.; the power to memorize was reduced by about seventy per cent.\n\nForty to eighty grams of alcohol, the amounts used in producing these\nastounding results, is no more than the quantity contained in one to\ntwo litres of beer or in a half-bottle to a bottle of ordinary wine.\nProfessor Aschaffenburg, commenting on these experiments, points the\nobvious moral that the so-called moderate drinker, who consumes his\nbottle of wine as a matter of course each day with his dinner--and\nwho doubtless would declare that he is never under the influence of\nliquor--is in reality never actually sober from one week's end to\nanother. Neither in bodily nor in mental activity is he ever up to what\nshould be his normal level.\n\nThat this fair inference from laboratory experiments may be\ndemonstrated in a thoroughly practical field, has been shown by\nProfessor Aschaffenburg himself, through a series of tests made on\nfour professional typesetters. The tests were made with all the rigor\nof the psychological laboratory (the experimenter is a former pupil\nof Kraepelin), but they were conducted in a printing office, where\nthe subjects worked at their ordinary desks, and in precisely the\nordinary way, except that the copy from which the type was set was\nalways printed, to secure perfect uniformity. The author summarizes the\nresults of the experiment as follows:\n\n\n_A Loss of Ten Per Cent. in Working Efficiency_\n\n\"The experiment extended over four days. The first and third days were\nobserved as normal days, no alcohol being given. On the second and\nfourth days each worker received thirty-five grams (a little more than\none ounce) of alcohol, in the form of Greek wine. A comparison of the\nresults of work on normal and on alcoholic days showed, in the case\nof one of the workers, no difference. But the remaining three showed\ngreater or less retardation of work, amounting in the most pronounced\ncase to almost fourteen per cent. As typesetting is paid for by\nmeasure, such a worker would actually earn ten per cent. less on days\nwhen he consumed even this small quantity of alcohol.\"\n\nIn the light of such observations, a glass of beer or even the cheapest\nbottle of wine is seen to be an expensive luxury. To forfeit ten per\ncent. of one's working efficiency is no trifling matter in these days\nof strenuous competition. Perhaps it should be noted that the subjects\nof the experiment were all men habituated to the use of liquor, one\nof them being accustomed to take four glasses of beer each week day,\nand eight or ten on Sundays. This heaviest drinker was the one whose\nwork was most influenced in the experiment just related. The one whose\nwork was least influenced was the only one of the four who did not\nhabitually drink beer every day; and he drank regularly on Sundays. It\ngoes without saying that all abstained from beer during the experiment.\nWe may note, further, that all the men admitted that they habitually\nfound it more difficult to work on Mondays, after the over-indulgence\nof Sunday, than on other days, and that they made more mistakes on\nthat day. Aside from that, however, the men were by no means disposed\nto admit, before the experiment, that their habitual use of beer\ninterfered with their work. That it really did so could not well be\ndoubted after the experiment.\n\n\n_The Effect of Beer-drinking on German School-children_\n\nSome doubly significant observations as to the practical effects\nof beer and wine in dulling the faculties were made by Bayer, who\ninvestigated the habits of 591 children in a public school in Vienna.\nThese pupils were ranked by their teachers into three groups, denoting\nprogress as \"good,\" \"fair,\" or \"poor\" respectively. Bayer found, on\ninvestigation, that 134 of these pupils took no alcoholic drink; that\n164 drank alcoholics very seldom; but that 219 drank beer or wine once\ndaily; 71 drank it twice daily; and three drank it with every meal.\nOf the total abstainers, 42 per cent. ranked in the school as \"good,\"\n49 per cent. as \"fair,\" and 9 per cent. as \"poor.\" Of the occasional\ndrinkers, 34 per cent. ranked as \"good,\" 57 per cent. as \"fair,\" and\n9 per cent. as \"poor.\" Of the daily drinkers, 28 per cent. ranked as\n\"good,\" 58 per cent. as \"fair,\" and 14 per cent. as \"poor.\" Those\nwho drank twice daily ranked 25 per cent. \"good,\" 58% \"fair,\" and 18\nper cent. \"poor,\" Of the three who drank thrice daily, one ranked\nas \"fair,\" and the other two as \"poor.\" Statistics of this sort are\nrather tiresome; but these will repay a moment's examination. As\nAschaffenburg, from whom I quote them, remarks, detailed comment is\nsuperfluous: the figures speak for themselves.\n\nNeither in England nor America, fortunately, would it be possible to\ngather statistics comparable to these as to the effects of alcohol on\ngrowing children; for the Anglo-Saxon does not believe in alcohol for\nthe child, whatever his view as to its utility for adults. The effects\nof alcohol upon the growing organism have, however, been studied\nhere with the aid of subjects drawn from lower orders of the animal\nkingdom. Professor C. F. Hodge, of Clark University, gave alcohol to\ntwo kittens, with very striking results. \"In beginning the experiment,\"\nhe says, \"it was remarkable how quickly and completely all the higher\npsychic characteristics of both the kittens dropped out. Playfulness,\npurring, cleanliness and care of coat, interest in mice, fear of dogs,\nwhile normally developed before the experiment began, all disappeared\nso suddenly that it could hardly be explained otherwise than as a\ndirect influence of the alcohol upon the higher centers of the brain.\nThe kittens simply ate and slept, and could scarcely have been less\nactive had the greater part of their cerebral hemisphere been removed\nby the knife.\"\n\n\n_The Development of Fear in Alcoholized Dogs_\n\nProfessor Hodge's experiments extended also to dogs. He found that the\nalcoholized dogs in his kennel were lacking in spontaneous activity\nand in alertness in retrieving a ball. These defects must be in part\nexplained by lack of cerebral energy, in part by weakening of the\nmuscular system. Various other symptoms were presented that showed the\nlowered tone of the entire organism under the influence of alcohol;\nbut perhaps the most interesting phenomenon was the development of\nextreme timidity on the part of all the alcoholized dogs. The least\nthing out of the ordinary caused them to exhibit fear, while their\nkennel companions exhibited only curiosity or interest. \"Whistles and\nbells, in the distance, never ceased to throw them into a panic in\nwhich they howled and yelped while the normal dogs simply barked.\" One\nof the dogs even had \"paroxysms of causeless fear with some evidence of\nhallucination. He would apparently start at some imaginary object, and\ngo into fits of howling.\"\n\nThe characteristic timidity of the alcoholized dogs did not altogether\ndisappear even when they no longer received alcohol in their diet.\nTimidity had become with them a \"habit of life.\" As Professor Hodge\nsuggests, we are here apparently dealing with \"one of the profound\nphysiological causes of fear, having wide application to its phenomena\nin man. Fear is commonly recognized as a characteristic feature in\nalcoholic insanity, and delirium tremens is the most terrible form\nof fear psychosis known,\" The development of the same psychosis, in\na modified degree, through the continued use of small quantities of\nalcohol, emphasizes the causal relation between the use of alcohol\nand the genesis of timidity. It shows how pathetically mistaken is\nthe popular notion that alcohol inspires courage; and, to anyone who\nclearly appreciates the share courage plays in the battle of life, it\nsuggests yet another lamentable way in which alcohol handicaps its\ndevotees.\n\n\n_Is Alcohol A Poison?_\n\nIt is perhaps hardly necessary to cite further experiments directly\nshowing the depressing effects of alcohol, even in small quantities,\nupon the mental activities, Whoever examines the evidence in its\nentirety will scarcely avoid the conclusion reached by Smith, as\nthe result of his experiments already referred to, which Dr. Abel\nsummarizes thus: \"One half to one bottle of wine, or two to four\nglasses of beer a day, not only counteract the beneficial effects\nof 'practice' in any given occupation, but also depress every form\nof intellectual activity; therefore every man, who, according to\nhis own notions, is only a moderate drinker places himself by this\nindulgence on a lower intellectual level and opposes the full and\ncomplete utilization of his intellectual powers.\" I content myself with\nrepeating that, to the thoughtful man, the beer and the wine must seem\ndear at such a price.\n\nTo any one who may reply that he is willing to pay this price for\nthe sake of the pleasurable emotions and passions that are sometimes\npermitted to hold sway in the absence of those higher faculties of\nreason which alcohol tends to banish, I would suggest that there is\nstill another aspect of the account which we have not as yet examined.\nWe have seen that alcohol may be a potent disturber of the functions\nof digestion, of muscular activity, and of mental energizing. But we\nhave spoken all along of function and not of structure. We have not\neven raised a question as to what might be the tangible effects of\nthis disturber of functions upon the physical organism through which\nthese functions are manifested. We must complete our inquiry by asking\nwhether alcohol, in disturbing digestion, may not leave its mark upon\nthe digestive apparatus; whether in disturbing the circulation it may\nnot put its stamp upon heart and blood vessels; whether in disturbing\nthe mind it may not leave some indelible record on the tissues of the\nbrain.\n\nStated otherwise, the question is this: Is alcohol a poison to the\nanimal organism? A poison being, in the ordinary acceptance of the\nword, an agent that may injuriously affect the tissues of the body, and\ntend to shorten life.\n\nStudents of pathology answer this question with no uncertain voice.\nThe matter is presented in a nutshell by the Professor of Pathology at\nJohns Hopkins University, Dr. William H. Welch, when he says: \"Alcohol\nin sufficient quantities is a poison to all living organisms, both\nanimal and vegetable.\" To that unequivocal pronouncement there is, I\nbelieve, no dissenting voice, except that a word-quibble was at one\ntime raised over the claim that alcohol in exceedingly small doses\nmight be harmless. The obvious answer is that the same thing is true of\nany and every poison whatsoever. Arsenic and strychnine, in appropriate\ndoses, are recognized by all physicians as admirable tonics; but no one\nargues in consequence that they are not virulent poisons.\n\nOpen any work on the practice of medicine quite at random, and whether\nyou chance to read of diseased stomach or heart or blood-vessels\nor liver or kidneys or muscles or connective tissues or nerves or\nbrain--it is all one: in any case you will learn that alcohol may be\nan active factor in the causation, and a retarding factor in the cure,\nof some, at least, of the important diseases of the organ or set of\norgans about which you are reading. You will rise with the conviction\nthat alcohol is not merely a poison, but the most subtle, the most\nfar-reaching, and, judged by its ultimate effects, incomparably the\nmost virulent of all poisons.\n\n\n_Alcohol and Disease_\n\nHere are a few corroborative facts, stated baldly, almost at random:\nRauber found that a ten per cent. solution of alcohol \"acted as a\ndefinite protoplasmic poison to all forms of cell life with which he\nexperimented--including the hydra, tapeworms, earthworms, leeches,\ncrayfish, various species of fish, Mexican axolotl, and mammals,\nincluding the human subject.\" Berkely found, in four rabbits out\nof five in which he had induced chronic alcohol poisoning, fatty\ndegeneration of the heart muscle. This condition, he says, \"seems to be\npresent in all animals subject to a continual administration of alcohol\nin which sufficient time between the doses is not allowed for complete\nelimination.\" Cowan finds that alcoholic cases \"bear acute diseases\nbadly, failure of the heart always ensuing at an earlier period than\none would anticipate.\" Bollinger found the beer-drinkers of Munich\nso subject to hypertrophied or dilated hearts as to justify Liebe in\ndeclaring that \"one man in sixteen in Munich drinks himself to death.\"\n\nDr. Sims Woodhead, Professor of Pathology in the University of\nCambridge, says of the effect of alcohol on the heart: \"In addition\nto the fatty degeneration of the heart that is so frequently met with\nin chronic alcoholics, there appears in some cases to be an increase\nof fibrous tissue between the muscle fibers, accompanied by wasting\nof these tissues.... Heart failure, one of the most frequent causes\nof death in people of adult and advanced years, is often due to fatty\ndegeneration, and a patient who suffers from alcoholic degeneration\nnecessarily runs a much greater risk of heart failure during the\ncourse of acute fevers or from overwork, exhaustion, and an overloaded\nstomach, and the like, than does the man with a strong, healthy heart\nunaffected by alcohol or similar poisons.\"\n\nIt must be obvious that these words give a clue to the agency of\nalcohol in shortening the lives of tens of thousands of persons with\nwhose decease the name of alcohol is never associated in the minds of\ntheir friends or in the death certificates.\n\nDr. Woodhead has this to say about the blood-vessels: \"In chronic\nalcoholism in which the poison is acting continuously, over a long\nperiod, a peculiar fibrous condition of the vessels is met with; this,\napparently, is the result of a slight irritation of the connective\ntissue of the walls of these vessels. The wall of the vessel may become\nthickened throughout its whole extent or irregularly, and the muscular\ncoat may waste away as a new fibrous or scar-like tissue is formed. The\nwasting muscles may undergo fatty degeneration, and, in these, lime\nsalts may be deposited; the rigid, brittle, so-called pipestem vessels\nare the result.\" Referring to these degenerated arteries, Dr. Welch\nsays: \"In this way alcoholic excess may stand in a causative relation\nto cerebral disorders, such as apoplexy and paralysis, and also the\ndiseases of the heart and kidneys.\"\n\nFrom our present standpoint it is particularly worthy of remark that\nProfessor Woodhead states that this calcification of the blood-vessels\nis likely to occur in persons who have never been either habitual\nor occasional drunkards, but who have taken only \"what they are\npleased to call 'moderate' quantities of alcohol.\" Similarly, Dr.\nWelch declares that \"alcoholic diseases are certainly not limited\nto persons recognized as drunkards. Instances have been recorded in\nincreasing number in recent years of the occurrence of diseases of\nthe circulatory, renal, and nervous systems, reasonably or positively\nattributable to the use of alcoholic liquors, in persons who never\nbecame really intoxicated and were regarded by themselves and by others\nas 'moderate drinkers.'\"\n\n\"It is well established,\" adds Dr. Welch, \"that the general mortality\nfrom diseases of the liver, kidney, heart, blood-vessels, and nervous\nsystem is much higher in those following occupations which expose them\nto the temptation of drinking than in others.\" Strumpell declares that\nchronic inflammation of the stomach and bowels is almost exclusively\nof alcoholic origin; and that when a man in the prime of life dies of\ncertain chronic kidney affections, one may safely infer that he has\nbeen a lover of beer and other alcoholic drinks. Similarly, cirrhosis\nof the liver is universally recognized as being, nine times in ten, of\nalcoholic origin. The nervous affections of like origin are numerous\nand important, implicating both brain cells and peripheral fibres.\n\n\n_How the Poison Works_\n\nWithout going into further details as to the precise changes that\nalcohol may effect in the various organs of the body, we may note\nthat these pathological changes are everywhere of the same general\ntype. There is an ever-present tendency to destroy the higher form of\ncells--those that are directly concerned with the vital processes--and\nto replace them with useless or harmful connective tissue. \"Whether\nthis scar tissue formation goes on in the heart, in the kidneys, in\nthe liver, in the blood-vessels, or in the nerves,\" says Woodhead,\n\"the process is essentially the same, and it must be associated with\nthe accumulation of poisonous or waste products in the lymph spaces\nthrough which the nutrient fluids pass to the tissues. The contracting\nscar tissue of a wound has its exact homologue in the contracting scar\ntissue that is met with in the liver, in the kidney, and in the brain.\"\n\nIt is not altogether pleasant to think that one's bodily tissues--from\nthe brain to the remotest nerve fibril, from the heart to the minutest\narteriole--may perhaps be undergoing day by day such changes as these.\nYet that is the possibility which every habitual drinker of alcoholic\nbeverages--\"moderate drinker\" though he be--must face. This is an added\ntoll that does not appear in the first price of the glass of beer or\nbottle of wine, but it is a toll that may refuse to be overlooked in\nthe final accounting.\n\n\n_Alcohol and Acute Infections_\n\nIn connection with experiments in rendering animals and men immune\nfrom certain contagious diseases through inoculation with specific\nserums, Delearde, working in Calmette's laboratory in Lille, showed\nthat alcoholized rabbits are not protected by inoculation, as normal\nones are, against hydrophobia. Moreover, he reports the case of\nan intemperate man, bitten by a mad dog, who died notwithstanding\nanti-rabic treatment, whereas a boy of thirteen, much more severely\nbitten by the same dog on the same day, recovered under treatment.\nDelearde strongly advises any one bitten by a mad dog to abstain from\nalcohol, not only during the anti-rabic treatment but for some months\nthereafter, lest the alcohol counteract the effects of the protective\nserum.\n\nSimilar laboratory experiments have been made by Laitenan, who became\nfully convinced that alcohol increases the susceptibility of animals to\nsplenic fever, tuberculosis, and diphtheria. Dr. A. C. Abbott, of the\nUniversity of Pennsylvania, made an elaborate series of experiments to\ntest the susceptibility of rabbits to various micro-organisms causing\npus-formation and blood poisoning. He found that the normal resistance\nof rabbits to infection from this source was in most cases \"markedly\ndiminished through the influence of alcohol when given daily to a stage\nof acute intoxication.\" \"It is interesting to note,\" Dr. Abbott adds,\n\"that the results of inoculation of the alcoholized rabbits with the\nerysipelas coccus correspond in a way with clinical observations on\nhuman beings addicted to the excessive use of alcohol when infected by\nthis organism.\"\n\nAdditional confirmation of the deleterious effects of alcohol in this\nconnection was furnished by the cats and dogs of Professor Hodge's\nexperiments, already referred to. All of these showed peculiar\nsusceptibility to infectious diseases, not only being attacked earlier\nthan their normal companions, but also suffering more severely, This\naccords with numerous observations on the human subject; for example,\nwith the claim made some years ago by McCleod and Milles that Europeans\nin Shanghai who used alcohol showed increased susceptibility to Asiatic\ncholera, and suffered from a more virulent type of the disease.\nProfessor Woodhead points out that many of the foremost authorities now\nconcede the justice of this view, and unreservedly condemn the giving\nof alcohol, even in medicinal doses, to patients suffering from cholera\nor from various other acute diseases and intoxications, including\ndiphtheria, tetanus, snake-bite, and pneumonia, as being not merely\nuseless but positively harmful. Even when the patient has advanced far\ntoward recovery from an acute infectious disease, it is held still\nto be highly unwise to administer alcohol, since this may interfere\nwith the beneficent action of the anti-toxins that have developed in\nthe tissues of the body, and in virtue of which the disease has been\novercome.\n\n\n_The Ally of Tuberculosis_\n\nNot many physicians, perhaps, will go so far as Dr. Muirhead of\nEdinburgh, who at one time claimed that he had scarcely known of a\ndeath in a case of pneumonia uncomplicated by alcoholism; but almost\nevery physician will admit that he contemplates with increased\nsolicitude every case of pneumonia thus complicated. Equally potent,\nseemingly, is alcohol in complicating that other ever-menacing lung\ndisease, tuberculosis. Dr. Crothers long ago asserted that inebriety\nand tuberculosis are practically interconvertible conditions; a view\nthat may be interpreted in the words of Dr. Dickinson's Baillie\nLecture: \"We may conclude, and that confidently, that alcohol\npromotes tubercle, not because it begets the bacilli, but because it\nimpairs the tissues, and makes them ready to yield to the attacks\nof the parasites.\" Dr. Brouardel, at the Congress for the Study of\nTuberculosis, in London, was equally emphatic as to the influence of\nalcohol in preparing the way for tuberculosis, and increasing its\nvirulence; and this view has now become general--curiously reversing\nthe popular impression, once held by the medical profession as well,\nthat alcohol is antagonistic to consumption.\n\nCorroborative evidence of the baleful alliance between alcohol and\ntuberculosis is furnished by the fact that in France the regions\nwhere tuberculosis is most prevalent correspond with those in which\nthe consumption of alcohol is greatest. Where the average annual\nconsumption was 12.5 litres per person, the death rate from consumption\nwas found by Baudron to be 32.8 per thousand. Where alcoholic\nconsumption rose to 35.4 litres, the death rate from consumption\nincreased to 107.8 per thousand. Equally suggestive are facts put\nforward by Guttstadt in regard to the causes of death in the various\ncallings in Prussia. He found that tuberculosis claimed 160 victims\nin every thousand deaths of persons over twenty-five years of age.\nBut the number of deaths from this disease per thousand deaths among\ngymnasium teachers, physicians, and Protestant clergymen, for example,\namounted respectively to 126, 113, and 76 only; whereas the numbers\nrose, for hotelkeepers, to 237, for brewers, to 344, and for waiters,\nto 556. No doubt several factors complicate the problem here, but one\nhazards little in suggesting that a difference of habit as to the use\nof alcohol was the chief determinant in running up the death rate due\nto tuberculosis from 76 per thousand at one end of the scale to 556 at\nthe other.\n\nPneumonia and tuberculosis combined account for one-fifth of all deaths\nin the United States, year by year. In the light of what has just been\nshown, it would appear that alcohol here has a hand in the carrying\noff of other untold thousands with whose untimely demise its name is\nnot officially associated. I may add that certain German authorities,\nincluding, for example, Dr. Liebe, present evidence--not as yet\ndemonstrative--to show that cancer must also be added to the list of\ndiseases to which alcohol predisposes the organism.\n\n\n_Hereditary Effects of Alcohol_\n\nIf additional evidence of the all-pervading influence of alcohol is\nrequired, it may be found in the thought-compelling fact that the\neffects are not limited to the individual who imbibes the alcohol, but\nmay be passed on to his descendants. The offspring of alcoholics show\nimpaired vitality of the most deep-seated character. Sometimes this\nimpaired vitality is manifested in the non-viability of the offspring;\nsometimes in deformity; very frequently in neuroses, which may take the\nsevere forms of chorea, infantile convulsions, epilepsy, or idiocy. In\nexamining into the history of 2554 idiotic, epileptic, hysterical, or\nweak-minded children in the institution at Bicetre, France, Bourneville\nfound that over 41 per cent. had alcoholic parents. In more than 9 per\ncent. of the cases, it was ascertained that one or both parents were\nunder the influence of alcohol at the time of procreation,--a fact\nof positively terrifying significance, when we reflect how alcohol\ninflames the passions while subordinating the judgment and the ethical\nscruples by which these passions are normally held in check. Of\nsimilar import are the observations of Bezzola and of Hartmann that\na large proportion of the idiots and the criminals in Switzerland\nwere conceived during the season of the year when the customs of the\ncountry--\"May-fests,\" etc.--lead to the disproportionate consumption of\nalcohol.\n\nExperimental evidence of very striking character is furnished by the\nreproductive histories of Professor Hodge's alcoholized dogs. Of 23\nwhelps born in four litters to a pair of tipplers, 9 were born dead, 8\nwere deformed, and only 4 were viable and seemingly normal. Meantime, a\npair of normal kennel-companions produced 45 whelps, of which 41 were\nviable and normal--a percentage of 90.2 against the 17.4 per cent. of\nviable alcoholics. Professor Hodge points out that these results are\nstrikingly similar to the observations of Demme on the progeny of ten\nalcoholic as compared with ten normal families of human beings. The ten\nalcoholic families produced 57 children, of whom 10 were deformed, 6\nidiotic, 6 choreic or epileptic, 25 non-viable, and only 10, or 17 per\ncent, of the whole were normal. The ten normal families produced 61\nchildren, two of whom were deformed, 2 pronounced \"backward,\" though\nnot suffering from disease, and 3 non-viable, leaving 54, or 88.5 per\ncent., normal.\n\nAs I am writing this article, the latest report of the Craig Colony for\nEpileptics, at Sonyea, New York, chances to come to my desk. Glancing\nat the tables of statistics, I find that the superintendent, Dr.\nSpratling, reports a history of alcoholism in the parents of 313 out\nof 950 recent cases. More than 22 per cent. of these unfortunates are\nthus suffering from the mistakes of their parents. Nor does this by any\nmeans tell the whole story, for the report shows that 577 additional\ncases--more than 60 per cent, of the whole--suffer from \"neuropathic\nheredity\"; which means that their parents were themselves the victims\nof one or another of those neuroses that are peculiarly heritable, and\nthat unquestionably tell, in a large number of cases, of alcoholic\nindulgence on the part of their progenitors. \"Even to the third and\nfourth generation,\" said the wise Hebrew of old; and the laws of\nheredity have not changed since then.\n\nI cite the data from this report of the Epileptic Colony, not because\nits record is in any way exceptional, but because it is absolutely\ntypical. The mental image that it brings up is precisely comparable\nto that which would arise were we to examine the life histories of\nthe inmates of any institution whatever where dependent or delinquent\nchildren are cared for, be it idiot asylum, orphanage, hospital, or\nreformatory. The same picture, with the same insistent moral, would be\nbefore us could we visit a clinic where nervous diseases are treated;\nor--turning to the other end of the social scale--could we sit in the\noffice of a fashionable specialist in nervous diseases and behold the\nsuccession of neurotics, epileptics, paralytics, and degenerates that\ncome day by day under his observation. It is this picture, along with\nothers which the preceding pages may in some measure have suggested,\nthat comes to mind and will not readily be banished when one hears\nadvocated \"on physiological grounds\" the regular use of alcoholic\ndrinks, \"in moderation.\" A vast number of the misguided individuals\nwho were responsible for all this misery never did use alcohol except\nin what they believed to be strict \"moderation\"; and of those that did\nuse it to excess, there were few indeed who could not have restricted\ntheir use of alcohol to moderate quantities, or have abandoned its use\naltogether, had not the drug itself made them its slaves by depriving\nthem of all power of choice. Few men indeed are voluntary inebriates.\n\n\n_Alcohol and the \"Moderate\" Drinker_\n\nIt does not fall within the scope of my present purpose to dwell upon\nthe familiar aspect of the effects of alcohol suggested by the last\nsentence. It requires no scientific experiments to prove that one of\nthe subtlest effects of this many-sided drug is to produce a craving\nfor itself, while weakening the will that could resist that craving.\nBut beyond noting that this is precisely in line with what we have\neverywhere seen to be the typical effect of alcohol--the weakening of\nhigher functions and faculties, with corresponding exaggeration of\nlower ones--I shall not comment here upon this all too familiar phase\nof the alcohol problem. Throughout this paper I have had in mind the\nhidden cumulative effects of relatively small quantities of alcohol\nrather than the patent effects of excessive indulgence, I have had in\nmind the voluntary \"social\" drinker, rather than the drunkard. I have\nwished to raise a question in the mind of each and every habitual user\nof alcohol in \"moderation\" who chances to read this article, as to\nwhether he is acting wisely in using alcohol habitually in any quantity\nwhatever.\n\nIf in reply the reader shall say: \"There is some quantity of alcohol\nthat constitutes actual moderation; some quantity that will give me\npleasure and yet not menace me with these evils,\" I answer thus:\n\nConceivably that is true, though it is not proved. But in any event,\nno man can tell you what the safe quantity is--if safe quantity there\nbe--in any individual case. We have seen how widely individuals\ndiffer in susceptibility. In the laboratory some animals are killed\nby doses that seem harmless to their companions. These are matters of\ntemperament that as yet elude explanation. But this much I can predict\nwith confidence: whatever the \"safe\" quantity of alcohol for you to\ntake, you will unquestionably at times exceed it. In a tolerably wide\nexperience of men of many nations, I have never known an habitual\ndrinker who did not sometimes take more alcohol than even the most\nliberal scientific estimate could claim as harmless. Therefore I\nbelieve that you must do the same.\n\nSo I am bound to believe, on the evidence, that if you take alcohol\nhabitually, in any quantity whatever, it is to some extent a menace to\nyou. I am bound to believe, in the light of what science has revealed:\n(1) that you are tangibly threatening the physical structures of your\nstomach, your liver, your kidneys, your heart, your blood-vessels,\nyour nerves, your brain; (2) that you are unequivocally decreasing\nyour capacity for work in any field, be it physical, intellectual, or\nartistic; (3) that you are in some measure lowering the grade of your\nmind, dulling your higher esthetic sense, and taking the finer edge\noff your morals; (4) that you are distinctly lessening your chances\nof maintaining health and attaining longevity; and (5) that you may\nbe entailing upon your descendants yet unborn a bond of incalculable\nmisery.\n\nSuch, I am bound to believe, is the probable cost of your \"moderate\"\nindulgence in alcoholic beverages. Part of that cost you must pay in\nperson; the balance will be the heritage of future generations. As a\nmere business proposition: Is your glass of beer, your bottle of wine,\nyour high-ball, or your cocktail worth such a price?\n\n\n\n\n [Illustration]\n\nEDITORIALS\n\nTHE PEASANT SALOON-KEEPER--RULER OF AMERICAN CITIES\n\n [Illustration]\n\n\nThe great wave of temperance which is now sweeping Europe and America\nhas its chief impulse, no doubt, in ethical and religious sentiment.\nBut a new force is operative--the force of an exact knowledge of the\nevil physical effects of alcohol. It would be impossible to exaggerate\nthe importance of this new element in temperance reform.\n\nThe story of the modern series of scientific experiments with alcohol,\nbegun about twenty-five years ago and still in progress, is given by\nDr. Henry Smith Williams in this number of MCCLURE'S MAGAZINE.\nThese investigations, largely conducted in Continental Europe, include\nexperiments on the senses, upon the muscles, and upon the different\nhuman intellectual activities, from the simplest to the most complex.\nWithout exception they show that every function of the normal human\nbody is injured by the use of alcohol--even the moderate use; and that\nthe injury is both serious and permanent.\n\nThis knowledge is of concern to all the world. But there is in America\na particular and special concern over a condition which may be believed\nto be unparalleled in human history--certainly in modern civilization:\nthe power of the saloon in American government, especially the\ngovernment of cities.\n\nThe fact is notorious; yet the condition is not clearly understood.\nSixty years ago, with the first flood of European immigration, the\ncharacter of American city governments changed suddenly and entirely.\nA great proportion of the peasantry who arrived here from the farms of\nEurope stopped in our cities. They were isolated from the rest of the\npopulation; their one great social center was the saloon. And out of\nthis social center came their political leaders and the manipulators of\ntheir votes. The European peasant saloon-keeper, for more than half a\ncentury, has been the ruler of a great proportion of American cities.\n\nThe case of Tammany Hall, for so many years the real governing body of\nNew York, is most familiar. Its politicians for half a century have\ngraduated into public affairs through the common school of the saloon.\nIts leaders at the present time are perfect examples of the European\npeasant saloon-keeper type, which has come to govern us. The same\ncondition exists to a large extent in nearly every one of the larger\ncities in the country. An analysis of the member-ship of the boards of\naldermen in these cities for the past few decades shows a percentage of\nsaloon-keepers with foreign names which is astonishing.\n\nA government necessarily takes the character of those conducting it.\nThe business of saloon-keeping, which produced the present management\nof our cities, involves, from the conditions which surround it, a\ndisregard for both law and proper moral ideals. Ordinary commercial\nmotives urge the proprietors, as a class, to increase the sale of a\ncommodity which the State everywhere endeavors to restrict; and a\nsavage condition of competition drives them still further--till a\ngreat proportion break the provisions of the law in some way; while\na considerable number ally themselves with the most degraded and\ndangerous forms of vice.\n\nThe government by this class has been exactly what might have been\nexpected. A body of men--drawn from an ancestry which has never\npossessed any knowledge or traditions of free government; educated in\na business whose financial successes are made through the disregard of\nlaw--are elevated to the control of the machinery of law and order in\nthe great cities. Another type of citizen--men of force and enterprise\nunsurpassed in the history of the world--by adapting the discoveries of\nthe most inventive century of the world to the uses of commerce, have\nmassed together in the past half century a chain of great cities upon\nthe face of a half savage continent, and left them to the government\nof such people as these. The commercial enterprise of these cities has\nbeen the marvel of the world; their government has reached a point of\nmoral degradation and inefficiency scarcely less than Oriental.\n\nThe debauching of our city life by this kind of government has been\nfrequently pictured in this magazine. A government by saloon-keepers,\nand by dealers in flagrant immorality, finds both its power and profit\nin the establishment of vice by its official position. The progress of\nsuch a government is shown in George Kennan's description of the former\nregime in San Francisco, published in MCCLURE'S MAGAZINE of\nSeptember, 1907:\n\n\"Instead of protecting the public by enforcing the laws, it devoted\nitself mainly to making money by allowing gamblers, policy-sellers,\nbrothel keepers, and prostitutes to break the laws. Its honest\nofficers and men tried, at first, to do their duty; but the police\ncommissioners, under the influence or direction of Ruef, interfered\nwith their efforts to close illegal and immoral resorts; the police\ncourt judges, allowing themselves to be swayed by selfish political\nconsiderations, released the prisoners whom they arrested.\"\n\nConditions similar to this have been shown in this magazine to exist\nin New York, Chicago, St. Louis, Pittsburg, and other great cities\nof America. The results have been a general disintegration in the\nmoral fiber of cities. Life itself is much more unsafe than under the\nwell-ordered governments of European cities. The murder rate in Chicago\nand New York is six or eight times as great as in London and Berlin.\nEven such a primary necessity of civilization as the safety of women is\nlost sight of. A leading Chicago newspaper said in 1906:\n\n\"It has ever been our proudest boast as a people that in this country\nwoman is respected and protected as she is in no other. That boast is\nbecoming an empty one in Chicago. Women have not only been annoyed and\ninsulted in great numbers on the street within a very short time, but\nnot a few have been murdered. In the year before the Hollister tragedy,\nthere were seventeen murders of women in Chicago, which attracted the\nattention of the city.\"\n\nThe system of government which produces this result was well described\nsome years ago by the late Bishop Potter, speaking of conditions in New\nYork.\n\n\"A corrupt system,\" he said, \"whose infamous details have been steadily\nuncovered, to our increasing horror and humiliation, was brazenly\nignored by those who were fattening on its spoils, and the world was\npresented with the astounding spectacle of a great municipality, whose\ncivic mechanism was largely employed in trading in the bodies and\nsouls of the defenseless.\"\n\nAside from giving direct encouragement and propagation to the more\nterrible forms of vice, the European peasant saloon-keeper government\nof our cities furnishes a fitting field for so-called respectable\nmen--but really criminals of the worst type--who help organize and\nperpetuate saloon government for the purpose of securing, by bribery,\nfranchises for public utilities without paying therefor. Thus American\ncities have been robbed as well as badly governed.\n\nThere are signs of amelioration of these conditions in most of the\ngreat cities of the country. But every advance is made against the\nfierce antagonism of just such systems as Bishop Potter described;\nand those systems exist in every large American city to-day--either\nin direct control or ready to take control at the slightest sign of\nrelaxation by the forces which are opposing them. And the foundation of\nthis evil structure is the European peasant saloon-keeper.\n\nMCCLURE'S MAGAZINE, in the next year, will consider the\nhorrible influence of the saloon on American life. Dr. Williams will\nfollow his article in the present number by studies of the influence of\nalcohol upon society at large, upon racial development, and upon the\nState. The author is especially equipped for his work. He is in the\nfirst place perhaps the greatest living popularizer of national science\nand history in America; and he has himself made life-long observations\nupon the influence of alcohol--both physical and social--first as\na medical practitioner in the treatment of the insane at the great\nasylums at Bloomingdale and Randalls Island, and later by study and\nobservation in the chief capitals of Europe, where he has lived the\ngreater part of the last ten years. The sound judgment and impartial\ntemper which have characterized his work in other fields will be found\nin his treatment of this great subject.\n\n\n\n\nTHE ELDER STATESMEN\n\n\nSenators Sherman, Hoar, Edmunds, George, and Gray; these were the men\nwho made the present Sherman Anti-trust Law. They were the men who\nmade largely the financial and constitutional history of the United\nStates for the three decades following the Civil War. They brought\nto the consideration of the trust problem an intimate knowledge of\nconstitutional law, an open, unbiased attitude toward property rights,\nand a thorough devotion to the public interest. They gave long and\ncareful attention to the question, spending two years on this bill.\nThere was nothing hasty or ill-considered about their action. They\nsought to end special privilege and put all citizens on the same basis\nof free competition. Of all their great services to the nation none\nprobably equals in importance this bill, which may be called the Magna\nCharta of industrial and commercial liberty.\n\nThe amendment of the Sherman Act may be an important public issue for\nsome time to come. If it were possible to assemble for this work a body\nof men as able and as disinterested as the Elder Statesmen who framed\nthe original act, the interests of the public would be safe.\n\n\n\n\nTranscriber's Note\n\n\nHyphenated words have been retained as in the original text.\n\nTypographical errors have been silently corrected.\n\nOE ligatures have been expanded.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of McClure's Magazine, Vol. XXXI, No. 6,\nOctober, 1908, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzrdtb b/data_all_eng_slimpj/shuffled/split2/finalzzrdtb new file mode 100644 index 0000000000000000000000000000000000000000..9a84f1018940854c62a966a82b9b0e880a265fca --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzrdtb @@ -0,0 +1,5 @@ +{"text":"\n\nTable of Contents\n\nTitle Page\n\nForeword\n\nPreface\n\nChapter 1\n\nChapter 2\n\nHISTORY\n\nA CHANGING LANDSCAPE\n\nMEDICAL MARIJUANA, DOCTORS, AND YOU\n\nCONDITIONS MARIJUANA CAN HELP\n\nMARIJUANA VARIETIES\n\nMETHODS OF USING MARIJUANA\n\nHERBAL MEDICINE\n\nChapter 3\n\nSAFETY INSPECTING MARIJUANA\n\nMOLD AND MILDEW\n\nPESTICIDES AND FERTILIZERS\n\nDISEASES AND PATHOGENS\n\nBASIC COOKING SAFETY\n\nCLEANLINESS\n\nDEDICATED WORK AREA\n\nHYGIENE\n\nSAFETY EQUIPMENT\n\nEDUCATION\n\nSTORAGE\n\nKNOW FOR WHOM YOU COOK\u2014GET THE DOSAGE RIGHT\n\nTITRATION FOR OPTIMAL AFFECT\n\nSMOKING VS. EATING\n\nUSING CANNABIS RESPONSIBLY\n\nChapter 4\n\nCONTROLLING THE MEDICINAL POTENCY OF MARIJUANA FOODS\n\nADJUSTING FOR MARIJUANA QUALITY\n\nUSING BUDS, FLOWERS, LEAF TRIM, OR SHAKE TO CONTROL STRENGTH\n\nREDUCING THE PLANT MATERIAL FOR LOWER POTENCY\n\nADJUSTING THE STRENGTH OF CANNABUTTER, OIL, OR TINCTURE\n\nVARYING THE SERVING SIZE FOR OPTIMAL AFFECT\n\nAUNT SANDY'S FORMULAS FOR BUTTER, OIL, TINCTURE, AND FLOUR\n\nAunt Sandy's - 10x Cannabutter\n\nAunt Sandy's - Cannaflour\n\nAunt Sandy's - Cannabis Oil\n\nAunt Sandy's - Marijuana Tincture\n\nKNOW FOR WHOM YOU COOK\n\nChapter 5\n\nLOW CALORIE OPTIONS\n\nVEGETARIAN DIET\n\nVEGAN DIET\n\nSUGAR FREE, DIABETIC, AND HYPOGLYCEMIC DIETS\n\nGLUTEN FREE\n\nLOW SODIUM ALTERNATIVES\n\nChapter 6\n\nBananacanna Fofana - Nut Bars\n\nBerry Peachy - Cannabutter Cobbler\n\nBlue Sky - Lemon Bars\n\nChocolate Coconut - Pecan Pie\n\nCoconut Cannabis - Cookies\n\nLarry's - Munchie Bar\n\nKahlua - Cannabis Cake\n\nLemon - Poppy Seed Bars\n\nOatmeal - Raisin Cookies\n\nPeanut Butter - Bundt Cake\n\nPumpkin - Streusel Muffins\n\nCannabis Milk - & Bhang\n\nThree Cannabis - Frostings\n\nChapter 7\n\nRanch - Dressing\n\nBlue Cheese - Dressing\/Dip\n\nLemony - Hummus Dip\n\nMarshall's - Favorite Dip\n\nAunt Sandy's - 10x Hot Sauce\n\nChapter 8\n\nCannafire - Buffalo Wings\n\nCannaflour - Chicken Tenders\n\nCannabis - Gumbo\n\nCreamy - Zucchini Soup\n\nMinestrone\n\nPumpkin - Soup\n\nTomato - Bisque\n\nChapter 9\n\nAdobo - Pork Chops\n\nBraised Short Ribs & Egg Noodles\n\nCannamaca Roni & Cheese\n\nCannabis - Chicken Curry\n\nCioppino\n\nClassic - Cannabis Pesto\n\nDizzy Bird - Turkey with Stuffing\n\nStuffing - for Dizzy Bird\n\nGourmet Grilled Brie - Sandwiches with Apricot Jam\n\nJambalaya a la Cannabis\n\nChapter 10\n\nGreen - Rice\n\nScalloped - Potatoes\n\nRosie's - Risotto\n\nChapter 11\n\nPopcorn\n\nOlive - Medley\n\nSweet & Spicy - Roasted Nuts\n\nCheesy - Savory Nuts\n\nSpicy - Pepitas\n\nAppendix A - Equivalents and Conversions\n\nAcknowledgments\n\nIndex\n\nOAKSTERDAM UNIVERSITY\n\nCopyright Page\n\nForeword\n\nby Dennis Peron Medical Marijuana Pioneer\n\n**I** first met Aunt Sandy while teaching the History of Medical Marijuana at Oaksterdam University. Over the years we have fostered our friendship, and I am honored to be a part of Aunt Sandy's Medical Marijuana Cookbook.\n\nAunt Sandy has told me that she relies on my book all the time. She is referring to \"Brownie Mary's Marijuana Cookbook and Dennis Peron's Recipe for Social Change, co-authored with Mary Rathbun, in 1996. In the mid-70s, I met \"Brownie Mary\" Rathbun outside of a San Francisco caf\u00e9. A friend pointed her out to me and from then on we were very close friends.\n\nThe 1980s AIDS epidemic happened at the height of America's War on Drugs. We knew marijuana could relieve the pain and weight loss due to wasting syndrome in AIDS patients, but we were in the \"Belly of the Beast\". Brownie Mary made cannabis brownies in my kitchen and distributed them for free to street kids and AIDS patients in a blatant act of defiance to the arcane drug laws. The sweet taste of the brownies helped them consume the medicine, which in turn restored their appetites and improved their quality of life. She continued to help people in need, regardless of her life mission's conflict with our drug laws.\n\nBrownie Mary died in 2000, but her legacy lives on through Aunt Sandy's work. After suffering a stroke, I have been unable to smoke cannabis. Aunt Sandy brings me medicated foods every Tuesday. Her food is delicious, innovative, and nutritious, but more importantly Aunt Sandy's foods truly relieve pain in the body, mind and soul. Her warmth and compassion shines through in her recipes. I truly believe that the information in this book is invaluable and could potentially be life changing for you, your friends, or a loved one.\n\n**Preface**\n\n**by Richard Lee President, Oaksterdam University**\n\n**A** unt Sandy's edibles are a favorite around Blue Sky Coffeeshop, in Oakland, and in the Bay area. Since joining us, her recipes have also become popular with Oaksterdam University students, but more importantly Sandy herself has been part of the Oaksterdam family for years. Aunt Sandy is now taking cooking with cannabis to a higher level with this cookbook. We are proud to have Sandy enhance and contribute to our Methods of Ingestion curricululm. She is helping Oaksterdam University bring quality training to the cannabis and hospitality industry.\n\n_\"Keep the Faith!\"_\n\n**Chapter 1**\n\n**Welcome to Aunt Sandy's Kitchen**\n\n_Ever since I was a teenager I have loved cooking. My home-style cooking can best be described as \"comfort food.\" As I began to understand the therapeutic benefits of cooking with marijuana, I learned how these foods can also help to heal a person's mind as well as their body. I am delighted that_ Aunt Sandy's Medical Marijuana Cookbook\u2014Comfort Food for Mind and Body _can help you to learn the secrets of cooking with marijuana so that you too can provide relief and joy with foods from your kitchen._\n\n**O** ne of the greatest feelings in the world is to cook wonderful food that people love. When that food also has the power to help a person feel better both physically and emotionally, that feeling becomes even greater. Early on I realized the therapeutic value of the cannabis plant, long before the idea came back to the mainstream. I began cooking marijuana foods in 1974 to help my brother relieve his chronic pain. At the time, I was na\u00efve and just added a bag of marijuana\u2014sticks, seeds, stems\u2014and all to a cookie recipe and my poor brother had to eat it. Even though the cannabis did its job, the cookie was difficult to eat and had the bitter taste of chlorophyll from the plant material. Over the years I have perfected the process and have developed methods that enhance the taste of the food while lessening the harsh taste of marijuana.\n\nI've lived in California my whole life and have been a medical marijuana advocate for decades. In 1996 California passed the groundbreaking medical marijuana law Proposition 215, which allowed me to further explore the benefits of marijuana cooking in a safe and legal environment. I now have hundreds of patients that depend on my tasty, medicated foods to make their world a better place.\n\nCurrently I prepare my dishes for a select group of medical marijuana dispensing collectives in California. Being involved with these organizations has given me the opportunity to see first-hand how these foods have the power to change a person's life and bring them a level of comfort and healing that other medicines could not. When a patient comes up to me and tells me how one of my lemon bars helped her sleep through the night or manage his pain more regularly, it warms my heart. Those moments keep me working hard in the kitchen to provide the very best medicinal foods to the patients I serve. I now teach my craft to students at Oakland's revolutionary marijuana trade school, Oaksterdam University, so that others can learn to provide their own comfort foods for themselves or for the patients in their lives.\n\nI'm proud to be the mother of three successful daughters who support my vision to provide healing foods to patients. My daughters, a nurse, a chef, and a student, have experienced first-hand my passion for making home-cooked marijuana remedies. They have always been my buddies, and they totally understand the medical benefits of cannabis for patients in need. I thank them for their support, which has helped me to discover my calling\u2014creating flavorful and effective therapies for an increasingly wider patient base.\n\nMy kitchen is a place of compassion and healing. While there are still certain legal and societal hurdles to jump to provide medical marijuana, I do not worry because I know whole-heartedly that what I am doing is right. It is unacceptable for those who oppose marijuana to allow for patients to suffer because of lack of understanding or misinformation about this wonderful and healing plant. Patients deserve to have safe access to reliable and effective marijuana medicines, and I am honored to be able to do my part to help.\n\nBy creating _Aunt Sandy's Medical Marijuana Cookbook\u2014Comfort Food for Mind and Body_ , I can help others learn how to cook marijuana foods that help bring relief and wellness to their lives. Most of the recipes in this book are easy enough to make, no matter how experienced you are in the kitchen. By following the simple instructions you will be able to create delicious meals that can also treat and heal a plethora of illnesses and ailments. Always check with your physician to ensure that marijuana food, like any other medicine, is suitable for your condition and safe for you to take with or in lieu of other medications. Cannabis medicines are not for everyone, but many people report them being extremely beneficial.\n\nPlease take the time to read carefully the following chapters before you start cooking. They will inform and educate you about marijuana as a medicine, marijuana and cooking safety, and methods for cooking with marijuana. It is important to take the time to learn about marijuana and how to cook with it safely and responsibly. You will find your personal comfort zone as it relates to proper dosage and the different strengths and potency levels of marijuana medicines.\n\nThe recipes are divided into categories that make it easy to find a delicious dessert, starter, soup, main course, side dish or snack to suit your needs. I hope you enjoy making some of my favorite recipes. They are rich in flavor and created with a lot of love. After mastering my marijuana cooking methods and techniques you may also enjoy adding marijuana to your favorite recipe. Most importantly: Always be safe, always be happy, and always remember to make cooking fun. I do hope you find my book to be a reliable and trusted companion in your journey to preparing great foods with marijuana.\n\n_From my kitchen to yours\u2014Happy cooking!_ \n_\u2014Aunt Sandy_\n\n**Chapter 2**\n\n**Marijuana as Medicine**\n\n_Marijuana medicines have played a role in human culture for thousands of years. The active ingredients in the marijuana plant make it beneficial for many different medical conditions. It is a safe, effective, and most importantly, a natural remedy that has proven to be helpful in treating pain afflictions, combating nausea, assisting in digestion, limiting seizures, increasing appetites, and more._\n\n**T** HC is the most recognized in a group of cannabinoids and constituents found in marijuana. It has shown to be neuroprotective, reduce the aggressiveness of several cancers, and in its raw form is both antimicrobial and an antioxidant. Most of the other cannabinoids are found in smaller but varying quantities, and have virtually no effect upon the \"high\" or euphoric effect, other than perhaps CBD. Cannabidiol, or CBD, does mitigate the sometimes dysphoric (unwanted or uncomfortable) effects and can reduce the \"mind high\". CBD has shown to greatly improve the \"body effects\" by binding to CB2 receptors that are found predominately in body organs and muscle tissue, helping those patients that suffer from muscle spasms and disorders. CBD also has therapeutic value because of its anti-inflammatory activities, is neuroprotective, and reduces metastasis in breast cancer. THC and CBN bind to the CB1 receptors in the brain, which affect the uptake of serotonin. This is why marijuana can relieve many physical and environmental stresses, and provide a better quality of life.\n\nBecause the federal government has so drastically limited marijuana research, there are very few clinical trials on whole plant marijuana. What is known is that hundreds of thousands of patients have confirmed that this plant benefits their medical condition and many use it regularly to relieve symptoms.\n\n# **HISTORY**\n\nThe medicinal qualities of this plant trace back through recorded history. Ancient civilizations, including China, Greece, India, and medieval cultures documented cannabis as being a beneficial therapy for humans and even animals. It was a very common treatment throughout the eighteenth and nineteenth centuries in America. Cannabis was the physicians' primary pain reliever until the discovery of aspirin. Cannabis was a prominent part of the US Pharmacopoeia until 1942, and several major pharmaceutical corporations distributed it and held patents for medicines containing cannabis. It was outlawed by the Marijuana Tax Act of 1937 and less commonly used thereafter.\n\nIn 1996 California voters passed Proposition 215, which was the first state law to allow patients a legal right to use medical marijuana. Since then many other states have passed similar legislation by voter initiative or the legislative process. In 2009 the US Congress lifted a ban, allowing Washington DC residents legal access to medical marijuana.\n\nIn the same year, the United States Department of Justice under the Obama Administration issued a directive for federal enforcement to cease prosecution of those operating in compliance with state law. This was a sea change in policy, after more than a decade of controversial federal enforcement.\n\n# **A CHANGING LANDSCAPE**\n\nCurrently several countries, including the Netherlands, Canada, Spain, and Italy allow marijuana to be prescribed in its plant form. The American Medical Association (AMA) and the American College of Physicians (ACP), the world's largest organizations of physicians, have called for the rescheduling of marijuana studies to explore its medical benefits. Medical organizations across the globe have endorsed the medical benefits of marijuana, and it is once again emerging as a legitimate therapy in medical circles.\n\nCalifornia and other states have marijuana dispensing collectives, or dispensaries, where patients can access their medicines in a clean and safe environment. Patients continue to gain safe access to these medicines while medical professionals, patient activists, and concerned citizens continue to break down the barriers of decades of misinformation, and allow the truth about the benefits of marijuana to come to light. Legal, political, and social changes continue to evolve within the medical marijuana landscape.\n\n# **MEDICAL MARIJUANA, DOCTORS, AND YOU**\n\nHow does a patient know if medical marijuana is right for his condition? It is always best to consult with a doctor about taking any medication; cannabis is no different. Because of the limited research or negative bias, it may be difficult to find a doctor supportive of marijuana therapies. If your physician seems to have a knee-jerk negative response it may be worthwhile to get a second opinion. In states where medical marijuana is allowed there are doctors who specialize in this area of alternative medicine. Seeking out a doctor who is knowledgeable about the benefits of medical marijuana will help you clearly understand if this treatment may help with your condition.\n\nWhile marijuana is not beneficial for everyone, it has improved the quality of life for many patients. Marijuana is not a cure for any disease or illness. Like most medicines, it increases comfort levels and helps to regulate a condition so a patient can be more productive in daily life. Cannabis can benefit a number of afflictions because of its ability to decrease certain kinds of pain, stimulate appetite, decrease spasms and seizures, and normalize dietary regiments. Deciding whether to use marijuana medicines should be a decision based on whether the effect of the medicine helps you function more normally and helps you cope with the symptoms of the conditions.\n\n# **CONDITIONS MARIJUANA CAN HELP**\n\n\u2022 **Cancer and Chemotherapy Treatment:** Cannabis is effective in curbing nausea and increasing the appetite of patients experiencing the harsh side effects of chemotherapy and radiation. It can also reduce the pain associated with the disease.\n\n\u2022 **HIV\/AIDS:** HIV\/AIDS patients often experience wasting syndrome from the disease and the multitudes of medicines used to combat it. Marijuana stimulates their appetite allowing them to eat more regularly, as well as helping them to ease the pains associated with the disease itself.\n\n\u2022 **Pain Afflictions:** Research shows that cannabis is a safe and effective treatment for a variety of pain related afflictions, including deep tissue pains, muscle and back pain, and neuropathy.\n\n\u2022 **Multiple Sclerosis:** Marijuana improves spasticity and improves tremors in MS patients. It helps control involuntary muscle contraction, improves balance, and assists with bladder control, speech, and eyesight. Cannabis helps with the immune system, which is thought to be the underlying pathogenic process in MS patients.\n\n\u2022 **Gastrointestinal Disorders:** Marijuana has value as an anti-emetic and analgesic medication. It helps combat the symptoms brought on by disorders such as Crohn's Disease, irritable bowel syndrome, and ulcerative colitis. Cannabis interacts with the endogenous cannabinoid receptors in the digestive tract, which can result in calming spasms, assuaging pain, and improving motility. Marijuana has also been shown to have anti-inflammatory properties and recent research has demonstrated that cannabinoids are immune system modulators, either enhancing or suppressing immune response.\n\n\u2022 **Movement Disorders:** Cannabis is effective in treating muscular spasticity, a common condition affecting millions of people. This condition afflicts individuals who have suffered strokes, as well as those with multiple sclerosis, cerebral palsy, paraplegia, quadriplegia, and spinal cord injuries. Conventional medical therapy offers little to address spasticity problems. Because cannabinoids have anti-spasticity, analgesic, anti-tremor, and anti-ataxia properties, they can be helpful.\n\n\u2022 **Aging:** Marijuana has been found to help many patients suffering from conditions that afflict older patients, including arthritis, chronic pain, cancer, Alzheimer's disease, diabetes, and spasticity associated with such diseases as Parkinson's.\n\n\u2022 **Arthritis:** There are two common types of arthritis, rheumatoid arthritis and osteoarthritis, and each affects the joints, causing pain and swelling, and limiting movement. The analgesic properties of marijuana make it useful in treating the pain associated with arthritis, both on its own and as an adjunct therapy that enhances the efficacy of opioid painkillers. Cannabis has also been shown to have powerful immune-modulation and anti-inflammatory properties suggesting that it could play a role in treating arthritis, and not just in symptom management.\n\n _Source: Americans for Safe Access_ \n_Toll-free 888-929-4367_ \n_www.americansforsafeaccess.org_ \n_info@safeaccessnow.org_\n\n# **MARIJUANA VARIETIES**\n\nMarijuana is a medicinal plant with many different genetic varieties. Patients report that certain types of cannabis work better for specific afflictions than others. Medical marijuana is generally derived from three varieties: indica, sativa, and hybrids. An indica-dominant strain is commonly thought to have deeper body effects and helps with many physical afflictions. Sativa-dominant strains produce more of a cerebral, uplifting, and calming effect. A hybrid is a strain that has been developed by combining indica- and sativa-dominant plants to create marijuana medicine that has the traits and effects of both indica and sativa. Due to years of genetic modification and breeding, there are few pure strains of just sativa or indica; but the dominant genetics in most marijuana types generally lean one way or another. In each of these types of cannabis there are hundreds of different strains that have been bred to produce certain qualities. If possible, it may be necessary to try different varieties of marijuana to find which works best for you.\n\n# **METHODS OF USING MARIJUANA**\n\nThe raw plant material of marijuana can be ingested or used in other ways, such as smoking or vaporization. Vaporization heats the plant material to a temperature that activates the cannabinoids but does not burn the plant, so that the vapors can be inhaled. Cannabis can also be infused into butters, oils, and tinctures. The recipes in this book include these infusions as ingredients to provide the medicinal value to the foods.\n\n# **HERBAL MEDICINE**\n\nCannabis is one of the safest medicines on the planet. It is an herbal medicine that helps people find relief where traditional medicines have failed them. The World Health Organization estimates that 80% of the world population uses herbal remedies. This is not uncharted territory, as many who oppose marijuana may have you believe. Cannabis is not a magical plant that can heal any condition; but if used properly it can provide extended relief for many symptoms and drastically increase one's quality of life _if used safely_ , and that is the topic of the next chapter.\n\n**Chapter 3**\n\n**Cooking Safely with Marijuana**\n\n_Cooking with marijuana should always be done in a safe and responsible manner. Handling food in itself requires you to take precautions to ensure that you and other people do not become sick. Cooking with cannabis requires the same diligence in food preparation, while also considering the effects of the active ingredients on the user, and sometimes on the food handler. With proper plant inspection, strict safety standards, adequate education, keen awareness, and a conduct of responsibility, you can be sure to have a good experience and avoid problems. Unsafe usage of marijuana can result in unintended consequences, so it is important for you to be aware of basic safety standards for marijuana handling and cooking._\n\n# **SAFETY INSPECTING MARIJUANA**\n\n**W** hen beginning to cook with marijuana it is important to inspect the medicine being used for signs of contamination. Marijuana can contain biological contaminants, chemical contaminants, and even diseases. Plant material with measurable levels of pesticides or fungicides should not be used under any circumstances. Powdery mildew, while not shown to have any pathogenic qualities in humans, should also be avoided. Taking time to examine the marijuana before cooking with it can help to detect and avoid unwanted contamination of your medical foods.\n\nIt is best if you know the source of the medication to have a better idea of the cultivation process. Knowing the conditions of cultivation and storage helps you to know the sanitary standards, pesticides used, and if the medicine was properly stored to preserve active ingredients. Many times you do not know where the marijuana was grown, and that is okay. You can inspect the marijuana for problems and take certain safety precautions before cooking to help ensure it is safe for patient use. Here are some common marijuana contaminants and how to recognize them.\n\n# **MOLD AND MILDEW**\n\nFungal growths, such as molds and mildews, can be present on marijuana. In a perfect world all food and plants would be mold free, but when making medicinal foods it is extra important to be diligent. White powdery mildew is unfortunately very common. This manifests in a white powdery substance on the leaf and flower surfaces that feels chalky instead of sticky. You may be able to detect other fungal problems by smell; the plant material may smell musty. When the material is fresh, it can feel a little slimy, and that is better than dry and chalky. Dried plant material can contain mold spores. Dark discoloration can be a sign that your marijuana has been effected by mold contamination. Use a magnifying device or microscope to take a closer look if you suspect problems. Under a microscope molds will look like thin strands with distinct heads where you can see spore-like growth.\n\n_Healthy bud_\n\nThere are acceptable levels of molds in many herbal medications. However, patients with compromised immune systems should never take a chance. When in doubt, throw it out. The American Nursing Association recommends boiling any mixture containing plant material at 325\u00b0 for at least 30 minutes. THC evaporates at 380\u00b0, so you can boil marijuana at lower temperatures without losing its active ingredient. While this will kill certain kinds of molds, it will not kill them all; so never risk your health if your condition is worsened by molds and mildews.\n\n_Mildew on leaf_\n\n# **PESTICIDES AND FERTILIZERS**\n\nMarijuana cultivation has been under-regulated for a long time. As a result, people often use dangerous and harsh pesticides or chemical growth additives in their plants. Some of these chemicals are dangerous for patients and should be avoided. Identifying pesticides is more difficult, but sometimes you can smell or taste a small portion of leaf and sense if there is a detectable chemical residue. The plant material being damaged or discolored may be a sign of pesticide contamination, as well.\n\n_Mold on bud_\n\nIf pesticides are dangerous to your condition be sure to exercise caution if using marijuana from unknown sources. There are home testing kits that can identify common toxins on foods if you have serious concern. Plant material with measurable levels of pesticides or fungicides should not be used under any circumstances.\n\n# **DISEASES AND PATHOGENS**\n\nMarijuana is a plant that is susceptible to the same biological and bacterial problems faced by other plants. If not properly cultivated and handled, plants can be exposed to a number of diseases that pose risk to humans, making them unfit for human consumption. Some pathogens can be identified by a discoloration or change in surface texture of the plant material. Also, the normally pungent odor of the plant may smell stale. Having a reputable source for the medicine is the first choice. Always discard any questionable medicine.\n\n# **BASIC COOKING SAFETY**\n\nAfter you have inspected the marijuana for safety you can begin cooking. Safety in the kitchen is essential. Following protocol will help avoid contamination in the kitchen and ensure the foods you create are of the highest quality. When making medical marijuana foods, it is necessary to take extra care to ensure the food you prepare does no harm.\n\n# **CLEANLINESS**\n\nThe first thing is to maintain a clean and sterile kitchen. Always clean before beginning a project, during preparation, and upon finishing that project. Use safe and non-toxic cleaning solutions and prepare foods on nonporous surfaces. All surfaces, sink areas, and materials should be cleaned and equipment should be in good repair. The number one source of food contamination is failure to properly clean. It is your duty to make sure your food preparation and cooking areas are literally clean enough to eat off of.\n\n# **DEDICATED WORK AREA**\n\nWhile you are preparing medicinal foods, your food prep area should be dedicated only to making medicinal foods. People, pets, things, and food not related to the making of medical foods should be removed from the area. Making marijuana food requires focus, so it is important to remove any possible distractions prior to working with the medicine. While you want to have fun cooking, be sure to do that in a safe and responsible manner.\n\n# **HYGIENE**\n\nAlways wash your hands thoroughly with warm water and soap before handling medicine or food. Be sure to follow sanitation standards if hand washing your equipment and utensils. This will ensure no bacteria or biological contaminants are transferred on to the foods.\n\n# **SAFETY EQUIPMENT**\n\nWhen cooking for others outside the home, use safety equipment, like plastic or non-powdered latex food handling gloves and a head covering, to avoid contamination of the food. When cooking for others it is important to be cautious, as nobody wants to find a stray hair in their dish. A patient with severe allergies can become ill if precaution is not taken to avoid cross-contamination. Safety equipment can help avoid these problems by promoting sanitary conditions.\n\n# **EDUCATION**\n\nAll states offer food safety courses. If you are cooking for others it is beneficial to take a course to be educated on temperature matters, hygiene standards, food borne disease and allergens, and cross-contamination.\n\n# **STORAGE**\n\nAlways store marijuana in a cool, dry area. To avoid heat degradation, never store items in cupboards near the stove or over the refrigerator. You need to be sure that all cannabis extractions and foods containing cannabis are kept secured and away from unknowing or curious people. Clearly label the medicine, butters, oils, tinctures, and medicated foods to avoid confusion. Treat medicinal foods with the same care and diligence you would a bottle of pills and always keep them out of reach of kids. Check the recipe for proper storage conditions and be aware of the shelf life of the foods to ensure your products are always fresh.\n\n# **KNOW FOR WHOM YOU COOK\u2014GET THE DOSAGE RIGHT**\n\nDosage is one of the most important aspects of marijuana cooking and the subject many people are most curious about. Dosage is relevant to both the potency of the medicine and the condition and size of the patient. There are three ways to control the medicinal potency and the affect of the finished cannabis-infused foods: 1) adjust the amount of plant material used in the butter, oil, or tincture according to the quality of the marijuana and depending on the part of the plant you are using, bud, flower, leaf trim, or shake; 2) adjust the strength of the cannabis butter, oil, tincture, or flour used in the recipe; and 3) adjust the serving size of the food.\n\nIn Chapter 4 Aunt Sandy's Marijuana Cooking Secrets, we discuss more about controlling the potency of the marijuana. Right now, let's focus on determining the condition of you or the patient you are cooking for and how that condition effects your choices during the marijuana preparation process.\n\nThere are many personal factors that can go into establishing the right dosage for you or your patient including weight (generally a heavier person will need more than a light person), metabolism (how your body metabolizes foods and the chemicals in cannabis can change how it effects you), and eating habits (eating cannabis on an empty stomach will have increased effect). Properly managing your dosage can help you to have a better experience with your medicine. Every patient is different, so understanding your personal boundaries and needs helps to maintain a consistent dosage rate that provides adequate relief.\n\nEating too much marijuana can result in drowsiness, diminished focus, increased heart rate or blood pressure, and euphoric feelings. If you feel you may have ingested too much, remain calm. Marijuana is non-toxic, so it will not have any lasting or fatal effects. The symptoms will subside in a few hours. Be sure you drink plenty of water and eat non-medicated foods to help symptoms pass.\n\n# **TITRATION FOR OPTIMAL AFFECT**\n\nIn medicine, titration is the process of gradually adjusting the dose of a medication until the desired affect is achieved. Titration allows the patient to determine which amount of certain medicinal foods adequately medicate, without overwhelming and overdosing. When an inexperienced patient is confronted with a new medicated food, she should begin by eating a small portion of the food and waiting an hour to examine the effects. At that point you can assess the need for more or less medicine. If necessary, eat another portion of the food and wait another thirty minutes to an hour. If you require more medicine for your condition, you can consume more food. Over time you will begin to accurately judge what portion size of particular foods work well for your medical needs. Use good judgment and caution when trying new medicated foods.\n\n# **SMOKING VS. EATING**\n\nChances are you may have only experienced using marijuana through inhalation by smoking or vaporizing. Smoking has a rapid onset, and the effects can be felt almost immediately. Patients that inhale their medicine generally know right away if they need to take more or less to achieve their comfort level. Smoking can create a lighter more \"heady\" experience and often will not last as long as eating marijuana.\n\n_Digital vaporizer_\n\nEating cannabis is much like ingesting a pill. It takes a period of time for your digestive system to break down the medicine and get it into your blood stream. Ingesting cannabis in foods or drinks can often have longer lasting deep body effects. Eating cannabis causes the active ingredients to remain in your system longer, and some patients experience residual effects even after sleeping. It is difficult to inhale enough medicine through smoke to cause a hallucinogenic effect, but patients who have consumed too much cannabis have reported minor psychedelic hallucinations and extreme euphoria. Just because you may smoke larger quantities of marijuana does not mean you can eat more. It is a whole different experience and precaution should be taken.\n\n# **USING CANNABIS RESPONSIBLY**\n\nYou should always use cannabis in a safe and responsible manner. Like any drug, marijuana can be abused. It should never be your intention to use marijuana at extreme levels to increase the effect to a point of losing control. While there is little evidence of physical addiction to marijuana, some medical professionals have observed people suffering from mental addictions. If you feel your marijuana use is making your life unmanageable, interfering with your productivity, or straining your relationships, it may be necessary to reevaluate and consult your physician.\n\nNever use cannabis when driving a car or motor vehicle, operating heavy machinery, or dangerous equipment. Always be aware of your surroundings and make sure you avoid hazards while medicated. Be prepared for the effects of your medicine and do not take it if you are unable to do so in a responsible and safe setting. Be aware of how cannabis interacts with other substances you may be taking, including alcohol and pharmaceuticals.\n\nMarijuana should contribute to your health and well-being, not detract from it. Understand your limitations and exercise discretion. Never allow social pressure to influence your cannabis use practices. Be aware of the situation and determine what amount of cannabis, if any, is appropriate. It is your duty to ensure that your marijuana does not get into the hands of others, including young people.\n\nBe discreet if you encounter law enforcement, especially if you live in a state with limited legal protections. It is never recommended to flaunt your marijuana use in public, no matter if it is legal in your area or not.\n\nFollowing these safety protocols will make your experience using medical marijuana more enjoyable. It is up to you to control the situation and make sure that cannabis is used safely. Now let's move on to the methods of cooking with marijuana.\n\n**Chapter 4**\n\n**Aunt Sandy's Marijuana Cooking Secrets**\n\n_Because marijuana can taste bitter and plant material can be hard to digest, simply adding marijuana plant material to your favorite recipe is not ideal. The taste becomes much more palatable when you draw out the cannabinoids, the active ingredients in marijuana, by infusing them into substances like butter, oil, alcohol, or milk fats. The cannabinoids bond to these substances and the plant material can then be removed so that the infusion can be added to various recipes. My recipes call for adding **Aunt Sandy's 10x Cannabutter, Aunt Sandy's Cannabis Oil, Aunt Sandy's Marijuana Tincture, Aunt Sandy's Cannaflour** , or a combination of these. Because I have developed these methods over the years for my clients, you can be sure that each recipe has a solid foundation of medicinal properties so that you can create both delicious and effective medical marijuana foods._\n\n# **CONTROLLING THE MEDICINAL POTENCY OF MARIJUANA FOODS**\n\n**A** s we discussed earlier, there are three ways to control the medicinal potency and the affect of the finished marijuana foods:\n\n1. Adjust the amount of plant material used in the butter, oil, or tincture according to the quality of the marijuana and depending on the part of the plant you are using, bud, flower, leaf trim, or shake.\n\n2. Adjust the strength of the cannabis butter, oil, tincture, or flour used in the recipe.\n\n3. Adjust the serving size of the food.\n\n# **ADJUSTING FOR MARIJUANA QUALITY**\n\nThe quality of the marijuana you use to create the butter, oil, tincture, or flour effects the potency of the finished food. An ounce of high quality marijuana will have more active ingredients than an ounce of low-grade marijuana. I consistently use high-grade medical cannabis, but depending on the marijuana you have access to, the amount of cannabis can be increased or decreased accordingly.\n\nYou can use sensory methods to help better understand whether to use more or less marijuana when preparing cooking extracts. If the medicine looks mediocre, lacks smell, is not resinous to the touch, tastes stale, has limited effects, or lacks freshness, it is likely that using more marijuana will be more beneficial. If the marijuana meets all of the requirements and can be considered high-grade, then a normal or lower amount may be used to achieve optimal therapeutic results. Here are some easy ways to use your senses to distinguish between high- and low-grade marijuana.\n\n_Weighing marijuana_\n\n**Sight:** Are there visible trichomes (resinous crystal-like formations)? Does it look discolored or old? You can use a magnifying device to examine the material more closely for signs of quality.\n\n**Smell:** Is the plant material pungent and ripe, or does it lack aroma or smell somewhat stale? Is there a musty smell? Fresh high quality marijuana has a strong distinct odor.\n\n**Touch:** Does the material have a sticky resin when rubbed between your fingers? Is it a chalky feeling (powdery mildew), or does it have a coarse texture?\n\n**Taste:** When smoked or vaporized does the marijuana have a chemical or perfume taste? Does it taste fresh or stale? Is the taste powerful or weak?\n\n**Effect:** When smoked or vaporized does the medicine have a strong and immediate effect, or does it take more than normal to be adequately medicated?\n\n**Freshness:** Aged medicine or improperly stored medicine may experience a breakdown of cannabinoids. Crumbling, over-dried, or discolored marijuana could be a sign of lack of freshness.\n\n# **USING BUDS, FLOWERS, LEAF TRIM, OR SHAKE TO CONTROL STRENGTH**\n\nUnderstanding the ratio of marijuana buds to marijuana leaf trim or shake is essential to developing cooking extracts that produce desired effects consistently. While it is not an exact science because every cannabis plant is different, the folks at Steep Hill Labs in California who test medicine for patient collectives in the state verify that on average the leaves or trimmings from a cannabis plant are about 25% as strong as buds or flowering clusters from the same plant. This means that in general a 1:4 ratio of bud\/flower to leaf trim or shake will produce similar medicinal effects.\n\nIf you are using the buds\/flowers of the plant you should only use from \u00bc ounce buds\/ flowers to an ounce of leaf trim or shake; or on a larger scale of weaker marijuana, use one ounce of bud\/flowers to a quarter pound of leaf trim or shake. This 1:4 general ratio should provide a relatively consistent result. The \"1:4 Ratio Chart for Determining Potency\" on the next page provides guidelines for amounts to use when preparing butter, oils, and tinctures.\n\n_Marijuana shake_\n\n**1:4 RATIO CHART FOR INCREASING OR DECREASING POTENCY**\n\nSTRENGTH| BUDS\/FLOWERS| LEAF TRIM\/SHAKE \n---|---|--- \nAunt Sandy's 10x Maximum Strength Formula| _1 Ounce_ | _4 Ounces_ \nHigh Strength Formula| _\u00be Ounce_ | _3 Ounces_ \nElevated Strength Formula| _\u00bd Ounce_ | _2 Ounces_ \nLow Strength Formula| _\u00bc Ounce_ | _1 Ounce_\n\n# **REDUCING THE PLANT MATERIAL FOR LOWER POTENCY**\n\nIf you want lower potency, another option is to add less plant material than is called for in the extraction formula to create a less potent potion to cook with. This is a good method if you do not have access to large quantities of marijuana but still want to enjoy Aunt Sandy's marijuana cooking recipes. Making butter, oil, or tincture with lower quantities of marijuana may also be desirable if you want to create entire meals for a group of patients or enjoy larger portions or any combination of snacks, soups, starters, sides, entrees, or desserts.\n\nAdjusting the potency of the marijuana butter, oils, and tinctures, will help you to have a better experience, and in turn will help foster wellness. Always err on the side of caution. (Refer to Appendix A for measurement equivalents.)\n\n# **ADJUSTING THE STRENGTH OF CANNABUTTER, OIL, OR TINCTURE**\n\nMany of my recipes, especially the desserts, use my famous Aunt Sandy's 10x Cannabutter to provide the greatest value and maximum effect to the patients I work with. When made with high-quality buds and flowers, this is an extremely potent formula. **If you are unsure of your tolerance to marijuana as a medicine, do not ingest it at full strength. Also, if you are not sure of the strength of the marijuana you are using, be cautious. You may want to dilute the recipe with non-medicated butter starting at a 25% Aunt Sandy's 10x Cannabutter to 75% regular butter.** Dilute the butters by melting them together in a pot and gently stirring. After ingesting foods at this ratio and following the titration methods to determine the proper portion size (see page 16), you can decide whether to increase or decrease the ratio accordingly.\n\nThe same is true for oils; you can dilute any Aunt Sandy's Cannabis Oil by adding more of the oil you used for the infusion. The same ratio works: 25% Aunt Sandy's Cannabis Oil to 75% regular oil. You can manipulate the ratio from there on up to 10x strength.\n\nThis will help you to have a better experience, and in turn will help foster wellness. Always err on the side of caution. It is always easier to eat more if needed, but it is difficult to reduce the amount once you may have ingested too much.\n\n# **VARYING THE SERVING SIZE FOR OPTIMAL AFFECT**\n\nOnce the tasty Aunt Sandy's recipe is made, your last chance to control the potency of the medicine is by adjusting the size of the portion of food. This is where the titration method comes in: if you are unsure of how much medication you need, even though you have controlled the recipe by the amount of plant material added and the amount of Cannabutter or cannabis oil used, you can choose how much of the food to eat.\n\nWhen trying a new recipe, start with a small serving to be on the conservative side. This is very useful when you have prepared a dessert like my Blue Sky Lemon Bars at its highest possible potency. Try a small piece of one bar. Wait an hour to feel the results. You can always eat more if you need it. Once you have experienced your body's response to the medicated food, you will know how much of that particular food makes you feel better. You will feel the relief from this wonderful medicine at its optimal strength for you. And that is the point of my whole story. Bon app\u00e9tit!\n\n# **AUNT SANDY'S FORMULAS FOR BUTTER, OIL, TINCTURE, AND FLOUR**\n\nOn the following pages is the famous Aunt Sandy's 10x Cannabutter formula. Remember that I use my most potent butter for patients in need of high doses of medication, and at full strength it is a powerful medicine. Please adjust your recipe as needed by either diluting the butter or using less plant material if you do not feel as if you need a maximum strength dosage or if you plan on cooking a dish or meal with larger portions. While Aunt Sandy's recipe provides great value, at it highest strength it can be overwhelming if you have a lower threshold, reduced medical need, or low tolerance for the active ingredients of marijuana.\n\n# **Aunt Sandy's**\n\n## **10x Cannabutter**\n\nYield:\n\n_Approximately 2 cups_\n\nEquipment:\n\n_scale, 5-quart stock pot (no lid), strainer or colander, catch pot (large pot), potato masher or stiff spoon, cheesecloth, kitchen gloves (plastic or non-powdered latex), ice cream scoop or big spoon, food storage bags or containers_\n\nIngredients:\n\n1\u20134 _ounces of cannabis leaf trim or shake. 4 ounces results in maximum strength 10x Cannabutter. See the 1:4 Ratio Chart on page 22 for lower strengths._\n\n1 _pound of unsalted butter (salted or sweet cream butter if preferred)_\n\n2 _quarts of water_\n\nButter is a great way to add marijuana to any recipe that calls for butter or oil. Try it in your own recipes and learn to make your favorite foods into delicious medicines. It is important to note THC evaporates at 380\u00b0 so keep recipes at a safe 350\u00b0. Boiling point is 212\u00b0, so it's okay to cook at a rapid boil or a strong saut\u00e9.\n\nINSTRUCTIONS:\n\n\u2022 Weigh desired amount of plant material on scale.\n\n\u2022 Combine these ingredients in a 5-quart stock pot.\n\n\u2022 Bring the ingredients to a slow boil where they will begin to conjoin in love and happiness.\n\n\u2022 Reduce heat and let the mixture simmer with no lid for 3\u20134 hours. The marijuana mixture will cook down to a concentrated level and most of the water will dissipate. If water evaporates before 3 hours, add one cup more. The idea is too cook off as much liquid as possible without burning the plant material.\n\n\u2022 Turn off the stove.\n\n\u2022 In a second pot or catch pot you will strain the plant material from the butter and liquid mixture using a common kitchen strainer or colander.\n\n_1) Measuring marijuana using scale_\n\n_2) Pouring measured marijuana into cooking pot_\n\n_3) Putting butter into cooking pot_\n\n\u2022 Place the strainer over the pot and allow the liquids to drain into it.\n\n\u2022 Using a blunt, firm object such as a potato masher or stiff spoon apply pressure to the remaining plant material to extract the remaining cannabis butter.\n\n\u2022 Allow mixture to cool to where it can be handled.\n\n\u2022 Using cheesecloth to strain the remaining butter from the plant material is suggested. Wrap a piece of the cheesecloth around the remaining plant material until there are no areas that leaf or shake can escape. Squeeze the material to separate the remaining butter into the pot. Use gloves to avoid absorption into skin, which at concentrated levels can produce psychoactive effects.\n\n\u2022 If there is any noticeable plant material remaining in the butter-liquid mixture it may be necessary to restrain the mix.\n\n\u2022 Place the liquid butter mixture into the refrigerator to cool (normally overnight). The butter will congeal and separate from the remaining water mix.\n\n\u2022 Remove the solidified butter from the water with an ice cream scoop or big spoon. Be sure to remove all loose pieces from the water.\n\n _4) Marijuana mixed with butter and water in cooking pot_\n\n_5) Stirring mixture until it boils, then simmering_\n\n_6) Pouring cooked mixture into strainer and catch pot_\n\n\u2022 The beautiful amber-colored water can be saved for use in other recipes.\n\n\u2022 Let the butter dry or pat with a paper towel to remove leftover moisture.\n\n\u2022 Store the butter in appropriate food storage containers and\/or food storage bags. Butter can be melted briefly if needed to accommodate storage. You can measure it into one-cup containers for convenience.\n\n_The shelf life of Aunt Sandy's 10x Cannabutter is roughly three months refrigerated, six months in the freezer._\n\n_7) Mashing mixture to strain cannabis butter into catch pot_\n\n_8) Squeezing remaining liquid through cheesecloth into catch pot_\n\n_9) Removing (refrigerated) solidified butter from the water_\n\n# **Aunt Sandy's**\n\n## **Cannaflour**\n\nYield:\n\n_1\/3 of a cup_\n\nEquipment:\n\n_Blender or food processor, airtight jar or canister_\n\nIngredients\n\n1 _ounce of dry cannabis leaf trim or shake_\n\nThis cannabis flour is a great accent to normal flour for recipes that require breading, such as fried chicken. Or it can be a great garnish, as well. It has milder effects but can be a tasty and effective additive to a number of recipes.\n\nINSTRUCTIONS\n\n\u2022 Place cannabis in blender.\n\n\u2022 Blend on high until plant material is a fine powder.\n\n\u2022 Store in an airtight jar or canister. Store in a dark and cool cabinet or drawer.\n\n_Shelf life of powdered cannabis leaves is about a year if stored properly. Like other dried herbs, it will not spoil but will lose its strength, smell, and taste over time._\n\n1. _Pouring marijuana into blender_\n\n2. _Blending marijuana into a fine powder-like flour_\n\n3. _Storing powdered leaves in an air tight container_\n\n# **Aunt Sandy's**\n\n## **Cannabis Oil**\n\nYield:\n\n_2 cups_\n\nEquipment:\n\n_scale, saucepan or small stockpot, strainer, small catch pot, potato masher or stiff spoon, cheesecloth, kitchen gloves (plastic or non-powdered latex), funnel, airtight jar or bottle with lid_\n\nIngredients\n\n\u00bc _ounce to 1 ounce of ground marijuana buds; 1 ounce results in maximum strength marijuana oil. See the 1:4 Ratio Chart on page 22 for lower strengths. Maximum strength may not be possible if using leaf trim\/shake due to saturation volume. Only use as much leaf trim\/shake as can be saturated by the oil_\n\n2 _cups of cooking oil (olive, canola, peanut, vegetable, etc.)_\n\nTry using a combination of 1 cup of butter and 1 cup of oil in this saut\u00e9 method to give it a rich and hearty flavor. Try using it when saut\u00e9-ing shrimp and enjoy the caramelized cannabis flavor. It is so-oooo scrumptious.\n\nINSTRUCTIONS\n\n\u2022 Weigh desired amount of plant material on scale.\n\n\u2022 If using buds, break the buds up into small chunks.\n\n\u2022 Combine oil and plant material in a saucepan or small stockpot.\n\n\u2022 Saut\u00e9 over medium heat for 20 minutes.\n\n\u2022 Strain the oil from the plant material into a separate container using a medium to fine strainer or appropriate colander.\n\n\u2022 To maximize yield you may want to strain the remaining plant material again.\n\n\u2022 Be sure to allow the material to cool before handling. Use gloves to avoid possible psychoactive effects.\n\n\u2022 Wrap it in cheesecloth and squeeze the remaining oil from it.\n\n_1) Pouring oil into saut\u00e9 pan with buds_\n\n_2) Stirring bud and oil mixture_\n\n\u2022 Discard the leaf material.\n\n\u2022 The finished oil is a beautiful color green.\n\n\u2022 Using a funnel, pour the oil into a clean, airtight container.\n\n\u2022 Be sure the storage container is completely dry to avoid the growth of molds.\n\n\u2022 Store the oil in a container like a bottle, jar, or food storage container with a lid.\n\n_The shelf life of marijuana infused oils is 4\u20136 months in a dark and cool pantry away from heat and up to a year if refrigerated._\n\n_If your cannabis oil grows mold, there is either too much water content in the herb or moisture in the jar. Use dry buds or wilt and dry the leaf before using._\n\n_Always ensure oil is stored in an airtight container._\n\n_3) Pouring oil and bud mixture through strainer_\n\n_4) Pouring strained cannabis oil into bottle_\n\n# **Aunt Sandy's**\n\n## **Marijuana Tincture**\n\nYield:\n\n_Approximately 3 cups_\n\nEquipment:\n\n_scale, scissors (optional), fine strainer, funnel, 1 quart glass jar with an airtight lid_\n\nIngredients\n\n\u00bd _ounce of broken up marijuana flowers or buds. See the 1:4 Ratio Chart on page 22 if you are using leaf trim or shake._\n\n_A Fifth (750ml) of 100 proof or higher spirits (Vodka, Rum, Brandy, etc.) For an alcohol free tincture try using vegetable glycerin or apple cider in place of spirits._\n\nMarijuana tincture is a great additive to enhance your favorite beverage. Adding a dropper to a cup of tea can make for a relaxing and holistic experience.\n\nINSTRUCTIONS\n\n\u2022 Weigh the appropriate amount of plant material.\n\n\u2022 Break down the marijuana. If it is dry enough you may do it by hand. Otherwise use scissors. Break it down to pieces at least as small as your pinky nail to expose more of the active ingredients. If using shake or leaf trim, it is generally not necessary to break down further.\n\n\u2022 After breaking down the marijuana, place it in a large quart (32 oz.) glass jar (Mason jar or the like).\n\n\u2022 Add the spirits to the plant material, filling the jar approximately 3\/4 of the way or 3 inches over the plant material.\n\n\u2022 Place the lid tightly on the jar.\n\n\u2022 Shake well.\n\n\u2022 Store in a cool and dark cabinet for 4 weeks.\n\n1. _Selecting cannabis buds_\n\n2. _Shaking cannabis buds and alcohol prior to storing_\n\n3. _Straining (stored) material and pouring through funnel_ Aunt Sandy's Medical Marijuana\n\n\u2022 Shake the jar every day during the maceration process. This helps to simulate the transformation process.\n\n\u2022 After four weeks (one moon cycle) has passed remove the jar from the cabinet and take the lid off.\n\n\u2022 Remove the plant material from the jar using a strainer and reserve the liquid, which is now your potent tincture liquid.\n\n\u2022 Using a funnel, pour the strained liquid into a clean dark, medicine bottle.\n\n\u2022 Be sure the medicine bottle, or tincture container is dry.\n\n\u2022 Store your tincture in a cool dark area to slow down degradation.\n\n _The shelf life for an alcohol-based tincture is extremely long, at least two years._\n\n# **KNOW FOR WHOM YOU COOK**\n\nUnderstanding who you are cooking for is imperative for the marijuana chef. Knowing if the persons who will be consuming the foods you make with marijuana are \"seasoned veterans\" with a lot of experience eating marijuana foods, or if they are \"lightweights,\" or beginners, will help you to adjust and regulate dosage accordingly and avoid issues. While marijuana is a very safe substance in terms of toxicity, it can be a very powerful psychoactive substance, especially when ingested in foods. Consuming too much cannabis can lead you to experience feelings of panic, tension, confusion, and lethargy. This should never be the goal of marijuana ingestion. Be sure to be clear with those eating your marijuana delicacies about their potency levels.\n\nWhen a patient tries a new marijuana food, he should begin by eating a small serving and waiting an hour to examine the effects. At that point you can assess the need for more or less medicine. If necessary, eat another portion of the food and wait another thirty minutes to an hour. Shortly, you will know what size of particular foods work well for your medical needs.\n\n**If you are unsure of the potency level of the foods or the tolerance of the person eating it, always err on the side of caution. Less is more when you are considering the effects on others. Other things to consider when cooking for yourself or others are dietary preferences or needs. The following chapter will let you know how to best handle those issues.**\n\n**Chapter 5**\n\n**Dietary Considerations**\n\n_Many people have special dietary needs. This book is filled with home-cooked favorites that may not adhere to your particular diet. Don't worry. In most recipes you can substitute ingredients to make them conform. The good news is that marijuana is a vegetarian, gluten free, low calorie, low-sodium plant on its own. It is the ingredients in the recipes that you will need to alter to meet your needs. The following are some specialized diets and ideas on how to best use_ Aunt Sandy's Medical Marijuana Cookbook _under those conditions._\n\n# **LOW CALORIE OPTIONS**\n\n**Y** ou use margarine when making the marijuana extract if you want to try a low calorie option with butter-like qualities and flavoring. Use Aunt Sandy's Cannabis Oil formula (see page 28), melting the margarine first. Most recipes where the butter or oils are baked in or cooked thoroughly it is hard to tell that margarine was used instead of butter. The taste is still great and your waistline will not be affected.\n\n# **VEGETARIAN DIET**\n\nBeing vegetarian is fairly common these days. You may be a strict vegetarian that eats only plant-based foods including fruits, nuts, and grains. Some vegetarians do eat dairy products and eggs; others do not. Some may occasionally have a small piece of fish for protein and are often referred to as pescatar-ians. Whatever your vegetarian diet includes, it is a snap to adapt these recipes to meet your needs.\n\nIf you are a vegetarian that enjoys dairy then you may be able to use Aunt Sandy's 10x Cannabutter recipe, as is. If you do not eat dairy, you can always use a vegetable oil in place of butter. There are many different types of vegetable oils and choosing the right one to use depends on your taste preferences and price and availability of the oils. Olive oil is very common, but it can be expensive and often not have the taste you may want for a dessert dish. A plain canola or vegetable blend oil can often be better for baking and have a smoother taste for sweets. Some folks even add a dash of salt to emulate the taste of butter. Any vegetable-based oil will work fine, but deciding what effect the taste of the oil will have on your recipe is an important consideration. Of course, you may just want to use what you have available, and that is fine, too. Cannabis can be infused into any great vegetable oil to provide excellent medicinal effects.\n\nIn recipes that call for meat, you can often substitute for non-meat options. For example, you can still make a \"Dizzy Bird\" with a tofu turkey if you would like. In a dish like pot roast or pork chops, you may have a favorite meat substitute product that you can use. This book contains many great recipes that can be altered to accommodate any vegetarian diet. Also, veggies saut\u00e9ed in cannabis butter or oil are always a delicious and healthy way to eat your medicine.\n\n# **VEGAN DIET**\n\nWhile many of the same concepts from the vegetarian section above apply to vegans as well, there are some further considerations that can enhance the vegan diet. Being vegan means one adheres to a strict plant diet, eating nothing derived from a living being, including honey, gelatin, and whey. This means you will most likely be infusing recipes with a vegetable-based oil and substituting vegan products where needed. There may be some recipes that will not be able to be prepared vegan; but substituting alternatives, like soymilk, soy cheese, and egg substitute where applicable can make most of my recipes just fine. These types of products normally explain on the package how to integrate them into standard recipes. Also, making marijuana oil for use in other vegan recipes is a great way to make marijuana foods that meet your dietary norms. Nothing is more delicious than cannabis sesame oil drizzled over cold, crisp cucumbers on a warm summer day or a rich cannabis olive oil over sun-ripened tomatoes and fresh basil. Many oils go great with cannabis, including sesame oil and coconut oil. With a little ingenuity and common sense, you can very easily adapt most recipes to meet vegan standards and still enjoy the benefits of marijuana cooking.\n\n# **SUGAR FREE, DIABETIC, AND HYPOGLYCEMIC DIETS**\n\nMany people are unable to use sugar in their normal diets for any number of reasons. Some have medical conditions, like diabetes, while others may opt for lower calorie and low-carb alternatives. The good news is that using a sugar substitute called xylitol can make every recipe in this book that uses sugar. The ratio of xylitol to sugar is one to one, so there is no need to adjust the measurements. Simply use the xylitol in place of the sugar for delicious sweets that are safe for diabetic and hypoglycemic patients. Xylitol has about two-thirds the calories of standard sugar and a very low glycemic index. It is a better substitute than some on the market because it is naturally derived from fruits and vegetables and has virtually no aftertaste. So instead of skipping past the dessert chapter, use this great alternative and enjoy some sweet, delicious cannabis treats.\n\n# **GLUTEN FREE**\n\nThe gluten free diet is becoming more common with many modern eateries offering regular gluten alternatives. You may be allergic to gluten or simply find that your body does not process it well. Whatever the reason for avoiding gluten, there are many ways to enjoy my recipes substituting simple gluten-free alternatives. There are ample gluten-free flour mixes on the market, or you can make your own mix to accommodate the recipe. For breading, a single-grain, gluten-free flour may work, but for baking you will want to use a combination of these flours. Some gluten-free options that can be substituted are rice flour, potato and potato starch flour, quinoa flour, millet flour, tapioca flour, teff flour, soy flour, cornstarch, corn flour, and many more. Finding the right blend can make gluten-free baking very enjoyable.\n\nGluten is what makes dough sticky when baking. When using substitutes for normal flour, it is important to compensate for some of the properties that gluten provides. Bread will not hold its shape without gluten. Using pans with sidewalls, like Bundt pans or loaf pans, helps the bread retain its shape. Use muffin pans for rolls. You can add gums, such as guar or xanthan gums in small amounts (1\/8 to \u00bc teaspoon per cup of flour) to simulate the sticky consistency of gluten. Gluten is a protein, so you can try replacing a half a cup of water in a recipe with egg whites to retain the value of the protein lost. All of these methods can help to make your favorite recipes while maintaining your gluten free diet.\n\n# **LOW SODIUM ALTERNATIVES**\n\nIf you need to control your sodium content, you can look for ingredients that have low sodium options. Definitely use unsalted butter for your extract. You can also lower or eliminate salts in recipes where it is not an essential ingredient. There are salt substitutes on the market that can be used, normally made from potassium and lysine. These substitutes are usually used at the same rates as regular salt; follow the manufacturer's recommendations.\n\nNow that you have considered all of the dietary ins and outs, you are ready to make some delicious foods from the library of great recipes in the following chapters. Because it is the best part of the meal and everyone's favorite, we will begin with the desserts first. Happy cooking!\n\n**Chapter 6**\n\n**Desserts**\n\n_Dessert is the best part of the meal, so why save it for last? These delicious sweets are sure to make you smile. Cakes, pie, cookies, bars, cobblers, and muffins provide a range of tasty sweets you can make with medical marijuana. Many patients enjoy medicating with desserts because they are easy to portion into manageable sizes, and of course, because they are scrumptious. You, too, might find that a spoonful of sugar makes the medicine go down._\n\n_These are so yummy you will want more than one so cut the cannabutter dosage and enjoy more._\n\nYield:\n\n_24 pieces; serving size should be determined by titration to achieve desired affect_\n\nEquipment:\n\n_13 by 9-inch glass baking pan, small bowl, large bowl_\n\nIngredients:\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter** (see page 24)_ \n1 _tablespoon regular butter for_ \n_greasing pan_ \n1\u00bd _cups all-purpose flour_ \n2 _teaspoons baking powder_ \n\u00bc _teaspoon baking soda_ \n\u00bc _teaspoon salt_ \n1 _teaspoon ground cinnamon_ \n1\/8 _teaspoon ground nutmeg_ \n2\/3 _cup chopped walnuts_ \n\u00be _cup packed brown sugar_ \n1 _large egg_ \n1\u00bd _cup mashed bananas (2 to 3)_ \n1 _teaspoon vanilla_\n\n# **Bananacanna Fofana**\n\n## **Nut Bars**\n\nThis recipe is a hit! A favorite taste from childhood is now a medicinal treat.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Grease baking pan with regular butter.\n\n\u2022 In a small bowl, combine flour, baking powder, baking soda, salt, cinnamon, nutmeg and walnuts and gently mix together.\n\n\u2022 In a large mixing bowl combine cannabutter with sugar and blend until it is a creamy consistency.\n\n\u2022 Add the egg to the cannabutter and sugar mix and blend the mixture again.\n\n\u2022 After the eggs are blended completely in, add the mashed bananas and vanilla. Blend the mix again until smooth.\n\n\u2022 Gradually add the small bowl with the flour, nuts, and baking ingredients to the creamy mixture while continuing to blend until they are well mixed.\n\n\u2022 Evenly spread the batter into the greased pan.\n\n\u2022 Bake 30 minutes or until a toothpick inserted comes out clean.\n\n\u2022 After cooling, remove from pan.\n\n\u2022 Cut into 24 pieces (4 by 6 pattern).\n\n\u2022 Store bars in airtight food storage containers or appropriate food storage bags.\n\n_The shelf life of these bars is 5\u20137 days at room temperature, 7\u201310 days refrigerated, three plus months frozen._\n\nYield:\n\n_12 pieces_\n\nEquipment:\n\n_saucepan, oven-safe 12-inch skillet, bowl, stiff spoon or whisk, spatula_\n\nIngredients:\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n3 _cups sliced peaches_ \n1 _cup raspberries_ \n1 _cup sugar_ \n1 _cup flour_ \n2 _teaspoons baking powder_ \n_Pinch of salt_ \n1 _cup milk_ \n1 _teaspoon vanilla_\n\nOptional:\n\n_whipped cream or ice cream for topping._\n\n# Berry Peachy\n\n## Cannabutter Cobbler\n\nA California spin on one of Willie Nelson's favorite desserts, country-fresh peach cobbler.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Use one tablespoon of cannabutter to grease the skillet.\n\n\u2022 Melt the remaining butter in the saucepan over low heat.\n\n\u2022 Remove the greased skillet from the stove and set it aside.\n\n\u2022 Combine the peaches and raspberries in a bowl and sprinkle a tablespoon of sugar over them. Gently stir and set to the side.\n\n\u2022 In a separate bowl, add the flour, baking powder and salt. Mix them together.\n\n\u2022 Add the remaining sugar, milk and vanilla extract to the bowl. Mix with a stiff spoon or whisk until all of the ingredients are evenly blended (If using a mixer set speed to medium).\n\n\u2022 Pour the melted butter into the batter and continue to stir rapidly.\n\n\u2022 Put the skillet on medium heat and let it warm.\n\n\u2022 Pour the batter into the skillet evenly.\n\n\u2022 Add the fruit mixture with its natural juices on top of the batter, spooning it evenly. Press it into the batter lightly with a spatula.\n\n\u2022 Bake the cobbler until the top is golden brown, about an hour.\n\n\u2022 Serve warm with ice cream or whipped cream.\n\nYield:\n\n_18 bars_\n\nEquipment:\n\n_mixing bowls, bowl, 13 by 9-inch baking pan, pastry cutter or fork_\n\nCrust Ingredients:\n\n\u00be _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1\u00bd _cups unsifted all-purpose flour_ \n3 _cups sugar_ \n_Pinch of salt_\n\nFilling Ingredients:\n\n6 _large eggs_ \n3 _cups sugar_ \n_Grated zest of 1 lemon_ \n1 _cup fresh lemon juice_ \n\u00bd _cup sifted all-purpose flour_ \n\u00bc _cup powdered sugar_\n\n# Blue Sky\n\n## **Lemon** Bars\n\nLarry at Blue Sky Cafe challenged me to make something unique and different, my Blue Sky Lemon Bars are now the #1 selling edible there.\n\nINSTRUCTIONS:\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 In a mixing bowl, mix the 1\u00bd cups of flour and 3 cups sugar and salt.\n\n\u2022 Add the cannabutter to the mixing bowl with the dry ingredients.\n\n\u2022 Using a pastry cutter (or a fork pressed against the side of the bowl), cut the dough mixture into pea-sized pieces.\n\n\u2022 Press the dough into the bottom and up the sides of the 13 by 9-inch baking pan, ungreased.\n\n\u2022 Bake about 25 minutes or until golden brown.\n\n\u2022 Remove from oven and let cool.\n\n\u2022 Reduce the oven temperature to 300\u00b0.\n\n\u2022 In a bowl, whisk eggs and 3 cups of sugar together until smooth.\n\n\u2022 Stir in lemon zest and juice.\n\n\u2022 Add the sifted flour to the egg mixture and mix thoroughly.\n\n\u2022 Pour the lemony egg batter over the baked crust.\n\n\u2022 Bake another 35 minutes, until the topping is set.\n\n\u2022 Remove from oven and let cool.\n\n\u2022 Finish by sprinkling powdered sugar on top.\n\nYield:\n\n_8 to12 pieces_\n\nEquipment:\n\n_mixing bowl, kitchen spoon_\n\nIngredients:\n\n\u00bc _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n\u00be _cup sugar_ \n2\u00bc _teaspoons vanilla extract_ \n3 _eggs slightly beaten_ \n3 _tablespoons all-purpose flour_ \n6 _ounces sweetened dark_ \n_chocolate bar, finely chopped_ \n\u00bd _cup chopped pecans_ \n\u00bd _cup shredded unsweetened_ \n_coconut_ \n_9-inch prepared piecrust_\n\nOptional:\n\n_whipped cream topping_\n\n# Chocolate Coconut\n\n## Pecan Pie\n\nThis tasty pie's flavors combine to perfection and remind me of my favorite candy bar. Can you guess which one?\n\nINSTRUCTIONS:\n\n\u2022 Heat oven to 350\u00b0.\n\n\u2022 In a mixing bowl combine melted cannabutter, sugar, and vanilla extract. Stir well.\n\n\u2022 Add the eggs and the flour. Mix well.\n\n\u2022 Stir the chocolate, pecans and coconut into the mixture.\n\n\u2022 Pour the mixture into piecrust and bake for 30 minutes. Pie will rise during baking.\n\n\u2022 Let the pie cool. It may contract a bit.\n\n\u2022 Delicious topped with whipped cream.\n\nYield:\n\n_6 dozen cookies_\n\nEquipment:\n\n_baking sheet, mixing bowl, bowl, whisk or kitchen spoon_\n\nIngredients:\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _tablespoon regular butter for_ \n_greasing pan_ \n1 _cup sugar_ \n1 _cup packed brown sugar_ \n2 _eggs_ \n1 _teaspoon vanilla extract_ \n2 _cups all-purpose flour, sifted_ \n2 _cups old fashioned oats_ \n1 _teaspoon baking powder_ \n1 _teaspoon baking soda_ \n\u00bd _teaspoon salt_ \n2 _cups flaked coconut_ \n1 _cup chopped walnuts_\n\n# **Coconut Cannabis**\n\n## **Cookies**\n\nTurn your afternoon tea into an adventurous experiment with these light and tasty morsels.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 325\u00b0 and grease the baking sheet.\n\n\u2022 In a mixing bowl, cream the cannabutter and add the sugars.\n\n\u2022 Beat until the mixture is fluffy.\n\n\u2022 Add the eggs one at a time beating well after each one. Beat in the vanilla until the mixture is smooth.\n\n\u2022 In a smaller bowl, combine flour, oats, baking powder, baking soda and salt.\n\n\u2022 Gradually add to the egg mixture and blend well.\n\n\u2022 Stir in the coconut and walnuts and continue to blend until the batter is well mixed.\n\n\u2022 Drop by rounded teaspoonfuls 3\" apart onto the greased baking sheet. Flatten slightly.\n\n\u2022 Bake for 8\u201310 minutes until the cookies are golden brown.\n\n\u2022 Cool for 2 minutes before removing to wire racks.\n\n\u2022 Store in airtight container.\n\n_Shelf life is 5\u20137 days at room temperate, 7\u201310 days in the refrigerator, or 3 plus months in the freezer._\n\n1. _Scooping measured cookie dough on to baking sheet_\n\n2. _Placing baking sheet in oven_\n\n3. _Removing cooled batch of cookies from baking sheet_ Aunt Sandy's Medical Marijuana\n\nYield:\n\n_12\u20133 x 3-inch pieces_\n\nEquipment:\n\n_mixing bowl, small bowls, pastry cutter or fork, 13 by 9-inch baking pan_\n\nIngredients:\n\n\u00be _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1\u00bd _cups all-purpose flour_ \n\u00bc _cup powdered sugar_ \n_Pinch of salt_ \n3 _cups chocolate chips_ \n1 _cup butterscotch chips_ \n1 _cup flaked coconut_ \n1 _cup crunched walnuts or_ \n_pecans_\n\n# **Larry's**\n\n## Munchie Bar\n\nLarry Richards from Blue Sky Cafe in Oakland inspired me to follow my dreams. This recipe is made in honor of his dedication and love to the cause.\n\n**INSTRUCTIONS:**\n\n\u2022 In a mixing bowl, mix together the flour, sugar and pinch of salt.\n\n\u2022 Add the cannabutter to the bowl with the flour, sugar, and salt. Mix well.\n\n\u2022 Using a pastry cutter or a fork pressed against the side of the bowl, cut the dough into small pea-size pieces.\n\n\u2022 Press the dough into the bottom and sides of a 13 by 9-inch baking pan.\n\n\u2022 In bowl, combine the chips, coconut and nuts to make the munchie mixture.\n\n\u2022 Add munchie mixture to top of the dough in baking pan.\n\n\u2022 Bake for 25 minutes until golden brown.\n\nYield:\n\n_18 pieces_\n\nEquipment:\n\n_two 9-inch round cake tins, saucepan, bowls, mixing bowl_\n\nIngredients:\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter** (creamed or_ \n_melted or room temp)_ \n1 _tablespoon regular butter for_ \n_greasing pan_ \n1 _tablespoon instant coffee_ \n_powder_ \n\u00be _cup, boiling water_ \n\u00bd _cup Kahlua or coffee liqueur_ \n2\u00bd _cup all-purpose flour_ \n2 _teaspoons baking soda_ \n\u00be _teaspoon salt_ \n\u00bc _teaspoon cinnamon_ \n\u00bd _cup applesauce_ \n1 _teaspoon vanilla extract_ \n2 _teaspoons maple flavoring_ \n3 _eggs separated_ \n1\u00bd _cup sugar_\n\n# Kahlua\n\n## **Cannabis Cake**\n\nSomewhat of a Tiramisu spinoff, this light and fluffy cake soaked in coffee liqueur makes a wonderful after dinner delight.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Grease baking pans with butter.\n\n\u2022 In a bowl, dissolve coffee powder in boiling water. Stir Kahlua into the coffee powder mix and allow to cool.\n\n\u2022 In a separate bowl, combine flour, baking soda, salt and cinnamon. Mix gently and set aside.\n\n\u2022 In another bowl, combine the cannabutter, applesauce, vanilla extract, maple flavoring and egg yolks. Mix well. Stir in the cooled Kahlua mixture.\n\n\u2022 In a large mixing bowl, beat egg whites with a mixer until soft peaks form.\n\n\u2022 Gradually add sugar to the egg whites and beat until peaks are stiff but not dry.\n\n\u2022 Using a spatula gradually fold in the about half of the dry ingredients to the egg whites. Add about half of the Kahlua mixture and mix. Repeat with the remaining dry ingredients and Kahlua mixture and stir until the batter is a smooth consistency.\n\n\u2022 Pour the batter into the buttered baking dish.\n\n\u2022 Bake at 350\u00b0 for 30 minutes until a toothpick inserted comes out clean.\n\n\u2022 Remove from oven and let cool.\n\nOption:\n\n_Ice the bottom layer and then put top layer on and ice the whole cake with **Chocolate Frosting** (see page 57). Then place leaf or desired pattern on top and dust desired pattern for decoration. Then gently remove your \"pattern\"._\n\n\u2022 Decorate by placing \"leaf\" pattern (or pattern of your own choice) on top of cake and sprinkling powdered sugar or cannaflour over the entire top of the cake. Then remove the pattern.\n\n1. _Sifting cannaflour over marijuana leaf pattern on frosted cake_\n\n2. _Removing leaf pattern from frosted cake_\n\nYield:\n\n_24 bars_\n\nEquipment:\n\n_mixing bowl, bowl, 13 by 9-inch baking pan, stiff spoon or whisk_\n\nIngredients:\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n2 _cups all-purpose flour_ \n1 _tablespoon baking powder_ \n\u00bd _teaspoon salt_ \n1\u00bd _teaspoon poppy seeds_ \n2 _large eggs_ \n1 _cup milk_ \n2\/3 _cup packed brown sugar_ \n\u00bd _cup lemon curd (prepared and_ \n_sold in glass jars)_\n\nOption:\n\n_Drizzle lemon flavored **White Frosting** (see page 56) on each bar._\n\n# Lemon\n\n## Poppy Seed Bars\n\nThe best part about this recipe is it's simplicity. It is a zesty flavored dessert that can be whipped up in no time.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Grease 13 by 9-inch baking pan.\n\n\u2022 In a bowl, mix together flour, baking powder, salt and poppy seeds.\n\n\u2022 In a mixing bowl, beat the cannabutter and sugars until they are creamy.\n\n\u2022 Add the eggs one at a time beating well after each.\n\n\u2022 Gradually stir the dry ingredients into the egg mixture and mix well.\n\n\u2022 Add the lemon curd to the batter and mix well.\n\n\u2022 Spread evenly in the greased baking pan.\n\n\u2022 Bake for 30 minutes or until a toothpick inserted comes out clean.\n\n\u2022 Remove from oven and let it cool on a baking rack.\n\n\u2022 Cut into 24 pieces in a 4 by 6 pattern.\n\n\u2022 Store in airtight food storage container or food storage bags.\n\n_Shelf life is 5\u20137 days at room temperature, 7\u201310 days in the refrigerator, or three plus months in the freezer._\n\nYield:\n\n_4 dozen cookies_\n\nEquipment: :\n\n_mixing bowl, bowl, cookie sheet pan, stiff spoon or spatula_\n\nIngredients: **Ingredients:**\n\n1 _cup softened **Aunt Sandy's**_ \n_**10x Cannabutter**_ \n1 _cup sugar_ \n1 _cup packed brown sugar_ \n2 _eggs_ \n1 _teaspoon vanilla extract_ \n3 _cups old fashioned oats_ \n1\u00bd _cups all-purpose flour_ \n1 _tablespoon baking soda_ \n1 _teaspoon salt_ \n\u00bd _cup chopped walnuts_ \n\u00bd _cup golden raisins_\n\n# Oatmeal\n\n## Raisin Cookies\n\nWho doesn't love Oatmeal Raisin Cookies? A classic recipe that has withstood the test of time.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 In a mixing bowl, cream the softened cannabutter.\n\n\u2022 Add both sugars and beat the mixture until it is light and fluffy.\n\n\u2022 Mix in the eggs one at a time beating well after each one.\n\n\u2022 Beat in the vanilla.\n\n\u2022 In a separate bowl, combine the oats, flour, baking soda, and salt and blend together.\n\n\u2022 Gradually add the dry ingredients to the egg and butter mixture. Mix well.\n\n\u2022 Stir in the walnuts and raisins.\n\n\u2022 On a baking sheet (ungreased), drop tablespoon sized globs of batter two inches apart.\n\n\u2022 Bake for 10-12 minutes, or until golden brown.\n\n\u2022 Cool for two minutes before removing to wire racks.\n\n\u2022 Store in airtight container or food storage bags.\n\n_Shelf life is 5\u20137 days at room temperature, 7\u201310 days in the refrigerator, or three plus months in the freezer._\n\nYield:\n\n_1 Bundt cake or 12 pieces_\n\nEquipment:\n\n_Bundt cake pan, 2 mixing bowls, stiff spoon or spatula_\n\nIngredients:\n\n1 _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _tablespoon regular butter for_ \n_greasing pan_ \n1\u00bd _cups all-purpose flour_ \n\u00bd _teaspoon baking soda_ \n\u00bd _cup sugar_ \n\u00bd _cup brown sugar_ \n1 _large egg_ \n1 _cup peanut butter_ \n\u00bd _teaspoon vanilla_\n\n# Peanut Butter\n\n## Bundt Cake\n\nNotice the blending of the cannabis butter and peanut butter flavors. This is a marijuana food connoisseur's favorite.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Grease Bundt cake pan with regular butter.\n\n\u2022 In a mixing bowl, mix flour and baking soda and blend together.\n\n\u2022 In a separate mixing bowl, blend cannabutter, sugar, egg, peanut butter and vanilla. Mix well.\n\n\u2022 Add the flour and baking soda to the egg mixture and mix thoroughly.\n\n\u2022 Pour the batter into the Bundt cake pan.\n\n\u2022 Bake 30 minutes or until a toothpick inserted comes out clean.\n\n\u2022 Let cool and remove from pan.\n\n\u2022 Store in an airtight cake dish or wrapped tightly.\n\n_Shelf life is 5 days at room temperature or a week in the refrigerator. Frozen it will keep for 3 months._\n\nYield:\n\n_12 muffins_\n\nEquipment:\n\n_muffin tin, muffin paper or nonstick spray, bowls, mixing bowl, stiff spoon or spatula_\n\nPumpkin Muffins Ingredients:\n\n\u00bc _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _egg_ \n1 _tablespoon water_ \n\u00be _cup canned pumpkin puree_ \n1 _teaspoon vanilla extract_ \n1\u00be _cup all-purpose flour_ \n\u00bd _cup sugar_ \n\u00bd _teaspoon salt_ \n\u00bd _teaspoon nutmeg_ \n2\u00bd _teaspoons baking powder_ \n1 _teaspoon baking soda_ \n1 _teaspoon cinnamon_\n\nStreusel Topping Ingredients:\n\n\u00bc _cup sugar_ \n\u00bd _teaspoon cinnamon_ \n1 _tablespoon creamed butter_ \n\u00bc _cup finely chopped pecans_ \n3 _tablespoons quick cooking_ \n_oats, uncooked_\n\n# Pumpkin\n\n## **Streusel** Muffins\n\nA harvest season favorite of mine. The smell of it baking is invigorating.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Coat the inside of muffin tin with non-stick spray or line with muffin paper.\n\n\u2022 In a bowl, combine the egg, cannabutter, water, pumpkin and vanilla extract.\n\n\u2022 In a mixing bowl, stir together the flour, sugar, salt, nutmeg, baking powder, baking soda, and cinnamon.\n\n\u2022 Add pumpkin and egg mixture to the dry ingredients and stir until mixed well.\n\n\u2022 Spoon the mixture into the muffin tin.\n\n\u2022 In a separate bowl, combine the streusel topping ingredients and mix well.\n\n\u2022 Sprinkle the topping evenly over each muffin.\n\n\u2022 Bake the muffins for 18\u201320 minutes or until lightly browned on top.\n\n\u2022 Store in airtight containers.\n\n_Shelf life for muffins is 2\u20135 days at room temp, a week in the refrigerator, and up to 3 months frozen if packaged correctly._\n\nYield:\n\n_2 cups_\n\nEquipment:\n\n_saucepan, strainer, funnel_\n\nCannabis Milk Ingredients:\n\n1\/8 _ounce **cannabis bud** s_ \n2 _cups milk, cream or soymilk_\n\nYield:\n\n_1 cup_\n\nEquipment:\n\n_saucepan, strainer, funnel_\n\nBhang Ingredients:\n\n2 _teaspoons **Cannabis Milk**_ \n_(see page 55)_ \n1 _teaspoon sugar_ \n1 _cup of your favorite tea_\n\n# Cannabis Milk\n\n## & Bhang\n\nCannabis Milk is as versatile as milk itself. It can be used in a variety of ways, as easy as pouring it on your cereal.\n\n**INSTRUCTIONS:**\n\n\u2022 Over low heat, mix together buds and milk.\n\n\u2022 Bring to a rolling boil.\n\n\u2022 Reduce heat and simmer for 30 minutes.\n\n\u2022 Cool and strain the plant material from the milk.\n\n_Store in the refrigerator in a container with a lid._\n\n## Bhang\n\nSome say this is the second most popular way to medicate in the world.\n\n**INSTRUCTIONS:**\n\n\u2022 In a small pot, heat all ingredients to a desirable temperature.\n\n# Three Cannabis\n\n## Frostings\n\nWhat would a cake be without frosting? These creamy finishing touches of vanilla, chocolate, mocha, lemon, orange, rum, cream cheese, bourbon, or cinnamon are sure to make any cake great.\n\nYield:\n\n_4 cups_\n\nEquipment:\n\n_mixing bowl (mixer optional), stiff spoon or whisk_\n\nIngredients:\n\n\u00bd _cup **Aunt Sandy's 10x**_ \n_**Cannabutter** , softened_ \n4 _cups powdered sugar_\n\n4 _to 6 tablespoons milk_ \n2 _teaspoons vanilla extract_ \n\u00bc _teaspoon salt_\n\nOptions:\n\n_Flavored frosting: Substitute the milk with dry sherry, rum, coffee or bourbon._\n\n_Lemon frosting: Add grated zest of 1 lemon and 2 tablespoons fresh lemon juice. Mix with the milk._\n\n_Orange frosting: Add grated zest of 1 small orange and 4 to 6 tablespoons fresh orange juice as a substitute for milk._\n\n_Mocha frosting: Add 2 tablespoons cocoa powder and 1 teaspoon instant coffee._\n\n## White **Frosting**\n\n**INSTRUCTIONS:**\n\n\u2022 In a mixing bowl, combine all of the ingredients.\n\n\u2022 Mix the ingredients until smooth. If using a mixer set to medium speed.\n\nYield:\n\n_4 cups_\n\nEquipment:\n\n_saucepan, mixing bowl (mixer optional), stiff spoon or spatula_\n\nIngredients:\n\n6 _tablespoons **Aunt Sandy's**_ \n_**10x Cannabutter**_ \n6 _ounces chocolate chips_ \n\u00bd _cup milk_ \n2 _teaspoons vanilla_ \n4 _cups powdered sugar_\n\nYield:\n\n_4 cups_\n\nEquipment:\n\n_mixing bowl, stiff spoon or spatula_\n\nIngredients:\n\n6 _tablespoons softened **Aunt**_ \n_**Sandy's 10x Cannabutter**_ \n8 _ounces cream cheese_ \n2 _tablespoons vanilla_ \n3 _cups powdered sugar_\n\nOptions:\n\n_For additional variety of flavors, add grated lemon or orange zest, cinnamon, bourbon or rum._\n\n## **Chocolate Frosting**\n\n**INSTRUCTIONS:**\n\n\u2022 In a saucepan, melt chips and cannabutter.\n\n\u2022 In a mixing bowl, combine the cannabutter, milk, and vanilla. Mix thoroughly.\n\n\u2022 Gradually add sugar and beat until smooth (If using a mixer set to medium speed).\n\n## Cream **Cheese Frosting**\n\n**INSTRUCTIONS:**\n\n\u2022 In a mixing bowl, combine all of the ingredients except sugar and mix together.\n\n\u2022 Gradually add sugar and beat until smooth. (If using a mixer, set to medium speed).\n\n**Chapter 7**\n\n**Sauces, Dressings, and Dips**\n\n_A good sauce, dressing, or dip can make or break a meal. Enjoy these super duper cannabis versions of some condiment classics. I love a good dip in the afternoon, with some crisp and cool veggies. Let these recipes bring your meal to life. From spicy, to savory, to delicious\u2014this chapter is full of great ideas._\n\n_Remember to be careful with your dosage for these sauces, dressings, and dips._\n\nYield:\n\n_\u00be cup_\n\nEquipment:\n\n_saucepan, mixing bowl, whisk_\n\nIngredients:\n\n3 _tablespoons melted **Aunt**_ \n_**Sandy's 10x Cannabutter**_ \n1 _garlic clove_ \n3 _pinches of salt_ \n\u00be _cup buttermilk_ \n3 _tablespoons fresh lime juice_ \n1 _tablespoon minced parsley_ \n1 _tablespoon chives, snipped_ \n_Salt and pepper to taste_\n\n# Ranch\n\n## Dressing\n\nRanch dressing is America's favorite dip. It tastes great on so many things. Try it as a healthier alternative to mayonnaise.\n\n**INSTRUCTIONS:**\n\n\u2022 After melting cannabutter in a saucepan, add all of the ingredients, except for the salt and pepper, to a mixing bowl.\n\n\u2022 Whisk well until all ingredients are thoroughly blended.\n\n\u2022 Add salt and pepper to taste.\n\n_Refrigerate in an airtight jar or bottle._\n\n_Shake well if using after sitting for a period of time._\n\nYield:\n\n_2\u00bd cups_\n\nEquipment:\n\n_saucepan, mixing bowl, whisk_\n\nIngredients:\n\n\u00bd _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n\u00be _cups ranch dressing_ \n1 _tablespoon hot pepper sauce_ \n\u00bc _teaspoon cayenne_ \n\u00be _cup blue cheese, crumbled_ \n\u00bd _cup sour cream_ \n3 _tablespoons milk_\n\n# Blue Cheese\n\n## Dressing\/Dip\n\nOne person's dressing is another person's dip. I love Blue Cheese on fresh veggies, hot wings, or just a fresh garden salad.\n\n**INSTRUCTIONS:**\n\n\u2022 Combine all ingredients in a mixing bowl.\n\n\u2022 Whisk thoroughly until well blended and sour cream is a smooth consistency.\n\n_Refrigerate in an airtight jar or bottle._\n\n_Shake well if using after sitting for a period of time._\n\nYield:\n\n_5 cups_\n\nEquipment:\n\n_small saucepan, blender_\n\nIngredients:\n\n1 _cup **Aunt Sandy's Cannabis**_ \n_**Oil** (see page 28, olive oil_ \n_works best)_ \n4 _large cloves of garlic, thinly_ \n_sliced_ \n1 _teaspoon ground cumin_ \n2 _cans (15\u00bd ounce) chickpeas,_ \n_drained and rinsed_ \n3 _tablespoons tahini_ \n3 _tablespoons fresh lemon juice_ \n1 _tablespoon soy sauce_ \n\u00bd _teaspoon salt_ \n\u00bc _cup cold water_\n\n# Lemony\n\n## Hummus Dip\n\nAlso try this one with warm pita bread. It is a delicious and healthy choice for any afternoon.\n\n**INSTRUCTIONS:**\n\n\u2022 In a small saucepan, mix \u2153 cup of the cannabis oil with the garlic and cumin.\n\n\u2022 Simmer on low heat until the garlic softens. It takes about 3 minutes after you see it sizzling. Don't let the garlic brown.\n\n\u2022 Let the garlic and oil cool completely.\n\n\u2022 Put the chickpeas, tahini, lemon juice, soy sauce and salt in a blender. Add the cooled, softened garlic and blend well.\n\n\u2022 Slowly pour (infuse) the remaining oils into the mixture and continue to blend.\n\n\u2022 Add the water and blend again, until the mixture is creamy.\n\n\u2022 Add more salt and lemon juice to taste.\n\n\u2022 Let it stand for at least an hour so the flavors can congeal.\n\n\u2022 Serve at room temperature with a variety of raw vegetables: fennel, celery, carrots, green onions, cucumbers, peppers, cauliflower, broccoli, etc.\n\n_Store in an airtight container in the refrigerator._\n\nYield:\n\n_12 cups_\n\nEquipment:\n\n_13 by 9-inch glass or ceramic baking dish_\n\nIngredients:\n\n1 _cup **Ranch Dressing** with_ \n_cannabis (see page 60)_ \n4 _cans (14-ounce cans) of_ \n_artichoke hearts, cut into_ \n_small pieces_ \n3 _small cans (4-ounce cans) of_ \n_jalapeno peppers, diced_ \n4 _cups mozzarella cheese,_ \n_shredded_\n\n# Marshall's\n\n## Favorite Dip\n\nMy nephew Marshall loves this dip. He begs me to make it. This medicated version is a delight at any celebration or gathering.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven 350\u00b0.\n\n\u2022 Use about 25% of the artichokes making a thin layer in a 13 by 9-inch baking dish.\n\n\u2022 Do the same with 25% of the jalapenos.\n\n\u2022 Spread \u00bc cup of the cannabis Ranch Dressing over the artichokes\/jalapenos.\n\n\u2022 Top with 1 cup of the mozzarella cheese.\n\n\u2022 Repeat in the same order, making four identical layers.\n\n\u2022 Finish with a layer of cheese.\n\n\u2022 Bake for 45 minutes. Remove when the edges are slightly brown.\n\n\u2022 Serve warm with tortilla chips.\n\n_If storing, let it cool and place in an airtight container in the refrigerator._\n\nYield:\n\n_1 cup_\n\nEquipment:\n\n_medium saucepan, strainer, bowl, bottle with air tight lid_\n\nIngredients:\n\n\u00bd _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n\u00be _cup canned chopped tomatoes,_ \n_drained_ \n6 _tablespoons chili sauce_ \n6 _tablespoons cider vinegar_ \n3 _tablespoons prepared_ \n_horseradish_ \n2 _teaspoons minced onion_ \n\u00be _teaspoon curry powder_ \n\u00be _teaspoon sugar_ \n\u00bc _black pepper_ \n_Pinch of ground red pepper_ \n1 _garlic clove, sliced_\n\n# **Aunt Sandy's**\n\n## 10x Hot Sauce\n\nThis is a hot sauce that can turn any Super Sunday into a truly super experience. Hot wings and sports of any kind always go well together.\n\n**INSTRUCTIONS:**\n\n\u2022 In medium saucepan, combine all ingredients except cannabutter.\n\n\u2022 Simmer on low heat until mixture thickens, about 40 minutes.\n\n\u2022 Strain thickened mixture into bowl.\n\n\u2022 Add melted cannabutter to the bowl and mix well.\n\n\u2022 Serve as a condiment or as part of a meal (Great for wings).\n\n_Store in an airtight bottle._\n\n1. _Pouring tomatoes into cooking pot_\n\n2. _Adding spices into cooking pot_ Aunt Sandy's Medical Marijuana Cook\n\n**Chapter 8**\n\n**Soups and Starters**\n\n_Get your meal off to a fabulous start with any of these great soups and starters. Delicious finger foods are a hit for any party or to start a meal. A warm delicious soup can make the best of a gloomy day and when paired with a sandwich can make a sensible meal. Enjoy these great beginnings to a wonderful cannabis experience._\n\n_Adjust your dosage as needed._\n\nYield:\n\n_24 pieces_\n\nEquipment:\n\n_baking sheet pan with sidewalls, sauce pan, kitchen scissors or knife, mixing bowl_\n\nIngredients:\n\n\u00bd _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n\u00bd _teaspoon **Aunt Sandy's 10x**_ \n_**Hot Sauce** (see page 64)_ \n12 _whole chicken wings or 24_ \n_halves_ \n2 _tablespoons of water_ \n1 _small clove garlic, minced_ \n_Pinch of salt_\n\n# **Cannafire**\n\n## **Buffalo Wings**\n\nA tailgater's dream, these wings are sure to make any event special\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 If using whole wings, remove the tips of the wings and discard. Clip wings at the joint, making 24 halves.\n\n\u2022 Place the water in the bottom of the sheet pan.\n\n\u2022 Spread wings evenly over the sheet pan and bake for 30 minutes, partially cooking the wings.\n\n\u2022 Remove the wings from the baking sheet and pat dry. Let them cool 1 hour in the refrigerator.\n\n\u2022 Melt the cannabutter in a saucepan. Pour the warm sauce into a mixing bowl.\n\n\u2022 Remove the wings from the refrigerator and toss them in the mixing bowl with the warm sauce. Pour a drop (no more than \u00bd teaspoon) over the wings and mix well.\n\n\u2022 Place the wings coated in sauce on a baking sheet and bake for another 30 minutes or until golden brown.\n\n\u2022 Serve warm.\n\n1. _Cutting chicken wings_\n\n2. _Pouring hot sauce over baked wings_\n\nYield:\n\n_6 servings_\n\nEquipment:\n\n_shallow dish or bowl, skillet, baking pan, tongs_\n\nIngredients:\n\n2\u00bd _cups **Aunt Sandy's**_ \n_**Cannaflour** (see page 27)_ \n1 _tablespoon regular butter for_ \n_greasing the pan_ \n2\u00bd _cups bread crumbs (Try_ \n_Panko, a Japanese bread_ \n_crumb)_ \n_Salt and pepper to taste_ \n1\u00bd _pounds chicken tenders or 3_ \n_boneless chicken breasts, cut_ \n_into 1\u00bd inch strips_ \n\u00bd _cup canola oil_\n\n# Cannaflour\n\n## **Chicken** Tenders\n\nMixing the breading with Aunt Sandy's Cannaflour gives the chicken a wonderful hint of marijuana in a crisp golden breading. Delicious!\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 In a shallow dish or bowl combine the cannaflour, bread crumbs, salt and pepper, mixing them together.\n\n\u2022 Add the chicken tenders to the bread crumb mixture and coat each piece completely.\n\n\u2022 Add the canola oil to a skillet over medium heat.\n\n\u2022 After the oil has warmed, add the breaded chicken tenders and saut\u00e9 for 10 minutes.\n\n\u2022 Remove tenders from skillet and place on a paper towel to drain.\n\n\u2022 Grease baking pan.\n\n\u2022 Place on tenders on the greased baking pan and brown in the oven until golden brown, about 20 minutes.\n\n\u2022 Serve on a platter with Blue Cheese Dipping Sauce (see page 61).\n\nYield:\n\n_20 servings_\n\nEquipment:\n\n_bowl, mixing bowl, baking pan, 6-quart pot, mixing bowl, spoon_\n\nIngredients:\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _cup all-purpose flour_ \n1 _teaspoon paprika_ \n1 _teaspoon ground thyme_ \n1 _teaspoon Old Bay seasonings_ \n1 _teaspoon salt_ \n\u00bd _teaspoon cayenne pepper_ \n2 _quarts chicken stock_ \n\u00bd _pound smoked sausage, sliced_ \n1 _cup onions, chopped_ \n\u00bd _cup bell pepper, chopped_ \n1 _cup celery, chopped_ \n1 _pound boneless, skinless_ \n_chicken breasts, cubed_ \n1 _pound shrimp, peeled and_ \n_deveined_ \n1 _\u2013_ 2 _cups water_\n\n# Cannabis\n\n## Gumbo\n\nMy sister-in-law Leslie recommends my gumbo because it reminds her of my parents and all their stories of \"Na'Orleans.\"\n\n**INSTRUCTIONS:**\n\n\u2022 In a bowl, mix the flour, paprika, thyme, Old Bay seasonings, salt and pepper.\n\n\u2022 Place in a baking pan and bake about 30 minutes, stirring often, until it turns a brown color. Set to the side.\n\n\u2022 Add the chicken stock to a 6-quart pot with the smoked sausage, onions, bell peppers and celery. Bring to a boil. Reduce heat and simmer for 15 minutes.\n\n\u2022 Take the baked spice mixture and put it in a mixing bowl. Stir in a cup or more of water until it has the consistency of a smooth paste.\n\n\u2022 Slowly add the paste to the pot with chicken stock, stirring occasionally over low heat.\n\n\u2022 Add the chicken, shrimp and cannabutter.\n\n\u2022 Cook on a medium low heat for 2 hours. The sauce will thicken over time.\n\n\u2022 Serve over regular rice or try it with Green Rice (see page 94).\n\n1. _Chopping sausage_\n\n2. _Adding peppers_\n\n3. _Making the paste_\n\n4. _Adding liquid_\n\nYield:\n\n_6 servings_\n\nEquipment:\n\n_large saucepan, blender, fine mesh, large strainer, masher, large bowl_\n\nIngredients:\n\n\u00bc _cup loosely packed **cannabis**_ \n_**leaves** (about 20 leaves),_ \n_washed and stemmed, plus 6_ \n_more for a garnish_ \n1 _teaspoon olive oil_ \n1 _large yellow onion, chopped_ \n2 _pounds zucchini, sliced \u00bc'_ \n_thick_ \n4 _cups chicken broth_ \n2 _tablespoons cr\u00e8me fraiche or_ \n_sour cream, plus more for a_ \n_garnish_ \n\u00bc _teaspoon chili powder_ \n_Pinch of salt_\n\n# Creamy\n\n## Zucchini Soup\n\nThe beautiful green hue of this soup gives it an eye-catching appeal. Garnish it with a dollop of sour cream and put a fresh cannabis leaf on top.\n\n**INSTRUCTIONS:**\n\n\u2022 Heat the olive oil in a large saucepan over medium heat.\n\n\u2022 Add the onion and cook until it looks translucent, about 5 minutes.\n\n\u2022 Add the zucchini and cook another 2 minutes.\n\n\u2022 Then add chicken broth and fresh cannabis leaves.\n\n\u2022 Reduce heat and simmer for 20 minutes with the lid on.\n\n\u2022 Pour the soup into blender and puree on high.\n\n\u2022 Pour soup through a strainer into a bowl, using a masher or hard spoon to push any lumps through.\n\n\u2022 Add the cr\u00e8me fraiche and chili powder and stir.\n\n\u2022 Add salt to taste.\n\n\u2022 Divide soup into bowls and garnish with a dollop of cr\u00e8me fraiche, a sprinkle of chili powder and a cannabis leaf.\n\n\u2022 Serve warm.\n\n_Store in an airtight container. Freezing is fine._\n\nYield:\n\n_12 servings_\n\nEquipment:\n\n_large soup pot with lid, colander or strainer, large stiff spoon_\n\nIngredients:\n\n\u00bd _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1\u00bd _quarts chicken stock_ \n1 _medium white onion, chopped_ \n3 _cloves garlic, minced_ \n1 _leek, well rinsed and sliced_ \n2 _tablespoons minced parsley_ \n3 _tablespoons tomato paste_ \n3 _stalks celery, chopped_ \n2 _large carrots, peeled and sliced_ \n2 _cups cabbage, shredded_ \n2 _medium zucchini, sliced_ \n1 _can (28 ounces) chopped_ \n_tomatoes with juice_ \n\u00bc _teaspoon thyme_ \n\u00bc _teaspoon basil_ \n\u00bc _teaspoon sage_ \n\u00bc _teaspoon rosemary_ \n\u00bc _teaspoon marjoram_ \n\u00bc _teaspoon black pepper_ \n1 _can (28 ounces) red kidney_ \n_beans, drained_ \n\u2153 _cup uncooked noodles_\n\n# Minestrone\n\nI love the farmer's market. I hand pick the best freshest vegetables for this soup and turn them into a hearty pot of warm and comforting goodness.\n\n**INSTRUCTIONS:**\n\n\u2022 Put all the ingredients, except for the cannabutter, beans, and noodles, in a large soup pot and bring it to a boil.\n\n\u2022 Reduce the heat and cover. Simmer for 1 hour.\n\n\u2022 Rinse and drain the beans to remove canning solution. Stir them into the pot.\n\n\u2022 Add the noodles to the pot.\n\n\u2022 Then add the cannabutter and stir well.\n\n\u2022 Put lid on and cook on a very low heat for at least 2 hours and up to 7 hours for better results.\n\n\u2022 Serve warm.\n\n_Store in an airtight container. Freezing is fine._\n\nYield:\n\n_6 servings_\n\nEquipment:\n\n_baking sheet pan, scoop or large spoon, blender, large soup pot with lid_\n\nIngredients:\n\n\u00bc _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n2 _pounds pumpkin puree, fresh_ \n_or canned_ \n_Olive oil, if needed_ \n1 _yellow onion, finely chopped_ \n2 _tablespoons flour_ \n5 _cups chicken or vegetable stock_ \n\u00bc _cup vermouth or dry sherry_ \n_(optional)_ \n\u00be _teaspoon ginger_ \n\u00bd _teaspoon nutmeg_ \n\u00bd _teaspoon pepper_ \n2 _egg yolks_ \n1 _cup heavy cream_\n\n# **Pumpkin**\n\n## Soup\n\nTry this recipe with warm corn bread. In the harvest season there is nothing better. You can substitute acorn or butternut squash for the pumpkin.\n\n**INSTRUCTIONS:**\n\n\u2022 If using fresh pumpkin, rinse the shell well and scoop and slice it in half crosswise. Scoop out seeds and strings. Place the two halves flesh side down on a baking sheet and bake for one hour or until pumpkin is fork tender. Scoop out flesh and puree in a blender. You may need to add a little olive oil to help blend.\n\n\u2022 In a large soup pot, saut\u00e9 the onions in a tablespoon of the cannabutter over medium heat for three minutes.\n\n\u2022 Stir in the flour, mixing thoroughly.\n\n\u2022 Add the pumpkin puree and the vegetable stock. (Add optional vermouth or sherry at this time).\n\n\u2022 Cover the pot and reduce heat. Let simmer for 15 minutes. Cook a little longer if using canned puree.\n\n\u2022 Add ginger, nutmeg, and pepper and stir in.\n\n\u2022 Combine the egg yolks and the cream. Spoon a little hot soup into the egg-cream mixture to prevent curdling. Add this mixture to the soup and stir.\n\n\u2022 Continue heating the soup on low heat. Be sure to not let it boil.\n\n\u2022 Serve warm.\n\n_Store in an airtight container. Freezing is fine._\n\nYield:\n\n_6 servings_\n\nEquipment:\n\n_large saucepan with lid, blender, strainer, stiff spoon_\n\nIngredients:\n\n\u00bc _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n2 _medium onions, chopped_ \n2 _cloves garlic, minced_ \n2 _cups chicken stock_ \n1 _can (28 ounces) whole_ \n_tomatoes, drained_ \n\u00bc _cup brown rice, uncooked_ \n1 _pinch of basil and thyme_ \n2 _cups light cream_ \n_Dash of cayenne pepper_ \n2 _tablespoons chopped parsley_\n\n# Tomato\n\n## Bisque\n\nServe this savory and creamy soup with a warm grilled cheese sandwich. It is a taste sensation. Just ask Oprah....\n\n**INSTRUCTIONS:**\n\n\u2022 In a large saucepan saut\u00e9 the onions and garlic in a teaspoon of the cannabutter over medium heat until they are soft, but not brown.\n\n\u2022 Add the chicken stock, tomatoes, rice, basil, and thyme. Stir and cover.\n\n\u2022 Bring the ingredients to a boil. Reduce heat and let simmer for 45 minutes.\n\n\u2022 Puree the soup in a blender.\n\n\u2022 Pour the blended soup through a kitchen strainer and discard the solids.\n\n\u2022 Return the soup to the pot and add the cream. Blend and heat the soup\u2014do not boil.\n\n\u2022 Garnish with a sprinkle of chopped parsley.\n\n\u2022 Serve warm.\n\n_Store in an airtight container. Freezing is fine._\n\n**Chapter 9**\n\n**Main Dishes**\n\n_Why not just call it the main event? These recipes are not for the faint of heart. They are fabulous enough to wow your dinner guests or simple enough to prepare and freeze in portions for later use. This is a very comforting set of recipes that please your palate and leave your mind and body at ease. An entree is a commitment, so be sure to manage your dosage accordingly. As a guide, I have suggested small portions. I like to think of these recipes as a \"best of\" version of my life cooking._\n\n_Enjoy them, as I have over the years, but as main courses, the portions may be larger so you may want to reduce the dosage._\n\nYield:\n\n_6 chops_\n\nEquipment:\n\n_bowl, large plastic food storage bag, grill_\n\nIngredients:\n\n\u00bc _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _teaspoon **cannabis herb** ,_ \n_dried and crushed_ \n2 _tablespoons packed brown_ \n_sugar_ \n2 _tablespoons orange juice_ \n2 _tablespoons fresh cilantro_ \n1 _tablespoon red wine vinegar_ \n2 _teaspoons chili powder_ \n1 _teaspoon ground cumin_ \n\u00bc _teaspoon cayenne pepper_ \n\u00bc _teaspoon cinnamon_ \n3 _cloves garlic, minced_ \n\u00bd _teaspoon salt_ \n6 _boneless pork top loin chops_ \n_cut \u00be\" thick_\n\n# Adobo\n\n## Pork Chops\n\nI have fond memories of this recipe cooking at my grandparents and filling the house with its savory aroma before our traditional Sunday dinners.\n\n**INSTRUCTIONS:**\n\n\u2022 Combine all ingredients, except the pork chops, in a bowl. Stir thoroughly to create the marinade.\n\n\u2022 Put the pork chops in the plastic food storage bag and let it rest inside a shallow dish or pan.\n\n\u2022 Pour the marinade over the chops and seal the bag. Mix and turn to coat chops completely.\n\n\u2022 Marinate in refrigerator 2 to 24 hours.\n\n\u2022 Remove the chops and discard the marinade.\n\n\u2022 Grill on rack directly over medium heat for 12 to 15 minutes. Turn once. Chops are done when firm to the touch.\n\nYield:\n\n_4 ribs or 8 servings_\n\nEquipment:\n\n_shallow dish, large skillet, aluminum foil, plate, stock pot_\n\nIngredients:\n\n\u00bd _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1\u00bd _pounds trimmed short ribs (4_ \n_ribs)_ \n3\u00be _teaspoons salt, divided_ \n\u00bd _teaspoon black pepper_ \n\u00bd _cup all-purpose flour_ \n2 _tablespoons olive oil, divided_ \n\u00be _cup chopped carrot (1 large_ \n_carrot)_ \n\u00bd _cup chopped onion (1 small_ \n_onion)_ \n8 _ounces sliced Cremini_ \n_mushrooms_ \n3 _garlic cloves, minced_ \n1 _tablespoon tomato paste_ \n3 _quarts and 2\u00bd cups water_ \n8 _ounces medium egg noodles_\n\n# Braised **Short Ribs** & Egg Noodles\n\nThis classic warms the body and the soul on a cold winter night. It is a tender, hearty meal.\n\n**INSTRUCTIONS:**\n\n\u2022 Sprinkle beef evenly with \u00bc teaspoon of salt and pepper.\n\n\u2022 Coat the ribs with flour, rolling them in a shallow dish.\n\n\u2022 Place a large skillet over medium high heat.\n\n\u2022 Add 1 tablespoon of the olive oil to the pan and swirl to coat the skillet.\n\n\u2022 Place the beef ribs in the skillet and cook 4 minutes or until browned, turning occasionally.\n\n\u2022 Add 2\u00bd cups water and scrape the pan to loosen browned bits and bring it to a boil.\n\n\u2022 Cover, reduce heat, and let simmer for 1 hour and 45 minutes. Beef should be fork tender.\n\n\u2022 Remove beef from the pan to a plate and cover with foil to keep warm.\n\n\u2022 Remove cooking liquid, and save it for later.\n\n\u2022 Now heat the skillet over medium heat. Add the remaining tablespoon of olive oil and swirl to coat.\n\n\u2022 Add the carrots and onion and cook 4 minutes, stirring occasionally.\n\n\u2022 Add the mushrooms and \u00bd teaspoon salt and the cannabutter.\n\n\u2022 Saut\u00e9 for 5 minutes, stirring occasionally.\n\n\u2022 Add the garlic and cook for 30 seconds, stirring rapidly.\n\n\u2022 Then add the tomato paste and cook for another 30 seconds continuing to stir.\n\n\u2022 Stir in the reserved cooking liquid and bring to a boil.\n\n\u2022 Reduce heat and simmer for 6 more minutes or until slightly thickened.\n\n\u2022 Bring 3 quarts of water to a boil in a stockpot, add 1 tablespoon of salt.\n\n\u2022 Add the egg noodles and cook 5 minutes or until al dente.\n\n\u2022 Drain the noodles.\n\n\u2022 Serve the ribs and sauce over the egg noodles.\n\nYield:\n\n_8 to 12 servings_\n\nEquipment:\n\n_13 by 9-inch baking pan or large casserole dish, kitchen spoon_\n\nIngredients:\n\n1 _tablespoon of **Aunt Sandy's**_ \n_**10x Cannabutter** to grease_ \n_the pan_ \n1 _pound elbow cooked macaroni_ \n_(use package instructions)_ \n3\u00bc _cups half and half or whole_ \n_milk_ \n18 _\u2013_ 20 _slices of American or_ \n_Cheddar cheese_ \n4 _tablespoons butter_ \n12 _buttery crackers_ \n_Salt, pepper, and paprika to_ \n_taste_\n\nOption:\n\n_For more kick, use residual cannabutter water to cook macaroni. If you don't use the residual cannabutter water, add cannabutter to the macaroni cooking water._\n\n# Cannamaca Roni & Cheese\n\nThis is a cheesy classic that just screams comfort food. For more kick, use the residue liquid from Aunt Sandy's Cannabutter formula.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Grease a 13 by 9-inch baking pan or a large casserole dish with cannabutter.\n\n\u2022 Spoon about a third of the cooked macaroni into the pan.\n\n\u2022 Pour in 1 cup of the half and half or milk.\n\n\u2022 Cover the macaroni with 4 to 6 slices of cheese.\n\n\u2022 Add two more layers of pasta, milk, and cheese.\n\n\u2022 Add small pieces of butter across the top.\n\n\u2022 Place the crackers in a sealable plastic bag and crush them with a rolling pin for crumb topping.\n\n\u2022 Mix in the salt, pepper and paprika with the crumbs.\n\n\u2022 Sprinkle the crumbs on top of the pasta and cheese.\n\n\u2022 Bake for 45 minutes, until bubbly and browning on top.\n\n\u2022 Let stand 5 minutes before serving.\n\nYield:\n\n_6 servings_\n\nEquipment:\n\n_large non-stick frying pan, whisk_\n\nIngredients:\n\n\u00bd _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _tablespoon canola oil_ \n12 _ounces skinless boneless_ \n_chicken breast, cut into 1-inch_ \n_pieces_ \n\u00bd _teaspoon salt_ \n2 _cups green bell pepper, cut into_ \n_strips_ \n2 _tablespoons fresh lime juice_ \n2 _tablespoons red curry paste_ \n1 _teaspoon sugar_ \n1 _can (14-ounce) light coconut_ \n_milk_ \n3 _cups cooked long grain rice_ \n_Lime wedges_\n\nOption:\n\n_If you want some more flavors, add golden raisins, frozen peas, shredded coconut, nuts, and\/or dried cranberries._\n\n# **Cannabis**\n\n## Chicken Curry\n\nI love a good Thai dish. The condiment combinations allow for customizing the dish to one's preference and make for a culinary adventure.\n\n**INSTRUCTIONS:**\n\n\u2022 Heat a large nonstick frying pan over medium-high heat. Add oil to pan and swirl to coat.\n\n\u2022 Season the chicken evenly with salt.\n\n\u2022 Add chicken to pan and cook 6 minutes or until browned, turning once.\n\n\u2022 Then add the bell peppers and saut\u00e9 another 6 minutes, stirring occasionally.\n\n\u2022 Remove chicken and peppers from the pan.\n\n\u2022 Combine the cannabutter, juice, soy sauce, curry paste, and sugar in a small bowl, stirring with a whisk.\n\n\u2022 Add juice mixture and coconut milk to the pan and bring to a boil.\n\n\u2022 Simmer about 12 minutes to slightly thicken the sauce.\n\n\u2022 Return the chicken and peppers to the pan (add options here) and cook 2 minutes or until heated thoroughly.\n\n\u2022 Serve over warm rice or try it with Green Rice (see page 94).\n\n\u2022 Garnish with lime wedges.\n\nYield:\n\n_10 servings_\n\nEquipment:\n\n_large soup pot, steamer, soup tureen, ladle_\n\nIngredients:\n\n\u00bd _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n\u2153 _cup olive oil_ \n3 _large onions, chopped finely_ \n4 _cloves garlic, minced_ \n2 _cups dry white wine_ \n\u00bd _cup chopped parsley_ \n1\u00bd _teaspoons dried basil_ \n1\u00bd _teaspoons oregano_ \n\u00bc _teaspoon cayenne pepper_ \n1 _can (12 ounces) tomato sauce_ \n1 _can (28 ounces) peeled_ \n_tomatoes, drained and_ \n_chopped_ \n1 _pound large raw shrimp,_ \n_shelled and de-veined_ \n3 _pounds fillet sea bass_ \n3 _pounds halibut or cod_ \n1 _pound scallops_ \n2 _dozen clams or mussels in_ \n_shell_ \n2 _pounds cracked crab in shell_\n\n# Cioppino\n\nCioppino is a San Francisco classic. This hearty seafood stew goes great with garlic bread and is sure to thrill the seafood lover in you.\n\n**INSTRUCTIONS:**\n\n\u2022 Heat the oil in a large soup pot and saut\u00e9 the onions for ten minutes.\n\n\u2022 Add the garlic, wine, parsley, basil, oregano, cayenne pepper, and cannabutter and stir. Simmer for 10 minutes over low heat.\n\n\u2022 Add the tomatoes and the tomato sauce. Continue simmering over low heat for 30 minutes.\n\n\u2022 Cut all fish into bite size pieces.\n\n\u2022 Stir and add the fish pieces, shrimp, and scallops to the pot and continue to simmer for another 30 minutes. You may need to add additional wine or a bit of water if the sauce gets too thick.\n\n\u2022 In a steamer, let the clams and mussels steam for 5\u201310 minutes until shells open. Discard any that do not open. Add just before serving.\n\n\u2022 Stir soup well and ladle into warmed bowls, leaving the clams and mussels in their shells.\n\n1. _Breaking mushrooms apart_\n\n2. _Breaking crab legs apart_\n\n3. _Chopping fish into pieces_\n\n4. _Adding spices to Cioppino_\n\n5. _Ladling cooked Cioppino into bowls_\n\nYield:\n\n_1 cup pesto and serving bowl of pasta, serves 8_\n\nEquipment:\n\n_small saucepan, flat spoon, blender, 4\u2013quart pot_\n\nIngredients:\n\n\u00bc _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n\u00bd _cup fresh **cannabis leaves**_ \n_(about 20 leaves)_ \n4 _large or 6 medium, finely_ \n_chopped garlic cloves_ \n\u2153 _cup shredded Romano cheese_ \n3 _tablespoons pine nuts_ \n2 _teaspoons minced parsley_ \n\u00bd _teaspoon salt_ \n1 _pound cooked pasta_\n\nOption:\n\n_Instead of fresh cannabis leaves, you can use the leaf mixture left over from making the **Aunt Sandy's 10x Cannabutter** to make the pesto. This mixture will still have a lot of residual active ingredients._\n\n# Classic\n\n## Cannabis Pesto\n\nThis is a rich and flavorful sensation. It can be made with residual plant material from my cannabutter formula. Nothing goes to waste.\n\n**INSTRUCTIONS:**\n\n\u2022 Pound the cannabis leaves until smooth.\n\n\u2022 In a small saucepan, melt the cannabutter.\n\n\u2022 Add the cannabis leaves to the butter.\n\n\u2022 Put them in a blender and add the remaining ingredients.\n\n\u2022 Blend until smooth.\n\n\u2022 Serve warm over pasta. (You may want to add more cheese.)\n\nYield:\n\n_1 turkey, 12 cups of stuffing_\n\nDizzy Bird Equipment:\n\n_Roasting pan with rack, basting brush, meat thermometer_\n\nIngredients:\n\n6 _tablespoons melted **Aunt**_ \n_**Sandy's 10x Cannabutter**_ \n12 _to 16 pound turkey_\n\n# **Dizzy** Bird\n\n## Turkey with Stuffing\n\nMy brother Al recomends this recipe to share with your friends and family as he did on his vacation at Burning Man.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 325\u00b0.\n\n\u2022 Remove the turkey giblets and neck from the chest and neck cavities.\n\n\u2022 Loosely pack the body and neck cavities with stuffing (below) and skewer or sew the opening closed.\n\n\u2022 Brush the skin all over with 6 tablespoons of melted cannabutter.\n\n\u2022 Baste the turkey every 30 minutes with pan drippings.\n\n\u2022 Bake for 20 minutes per pound.\n\n\u2022 The Dizzy Bird is done when a thermometer in the thigh area registers 170\u00b0.\n\n\u2022 Remove the turkey to a platter and let rest for 20 minutes before carving.\n\nStuffing Equipment:\n\n_large skillet, meat thermometer, large bowl_\n\nIngredients\n\n1 _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n2 _cups finely chopped onion_ \n1 _cup finely chopped celery_ \n10 _cups, lightly packed small_ \n_bread cubes for stuffing_ \n\u00bd _cup fresh parsley, minced_ \n1 _tablespoon dried sage_ \n1 _tablespoon dried thyme_ \n\u00be _teaspoon salt_ \n\u00bd _teaspoon black pepper_ \n\u00bc _teaspoon nutmeg_ \n\u215b _teaspoon ground cloves_ \n1 _cup chicken stock_\n\n# **S** tuffing\n\n## for Dizzy Bird\n\n**INSTRUCTIONS:**\n\n\u2022 In a large skillet, saut\u00e9 the cannabutter, onion and celery over a medium heat until the onion looks translucent, about 10 minutes.\n\n\u2022 Remove the skillet from heat and stir in the bread cubes, parsley, sage, thyme, salt, black pepper, nutmeg and cloves. Stir well.\n\n\u2022 Put the mixture in a large bowl.\n\n\u2022 Add 1 cup of chicken stock, a little at a time until the stuffing is moist but not too wet.\n\n\u2022 Spoon the stuffing into the turkey.\n\n\u2022 Stuffing is done when the turkey is done.\n\nYield:\n\n_8 servings (half sandwich each)_\n\nEquipment:\n\n_Frying pan with lid, spatula_\n\nIngredients:\n\n\u00bc _cup **Aunt Sandy's 10x**_ \n_**Cannabutter** at room_ \n_temperature_ \n8 _slices French bread_ \n3 _tablespoons apricot jam_ \n5 _ounces Brie (or Camembert)_\n\nOption:\n\n_You can also make it with the traditional combos of ham & cheese\u2014or creamy Velveeta cheese and prosciutto\u2014or Fontina and sun dried tomatoes\u2014or your favorite grilled cheese combo. Be creative and enjoy!_\n\n# Gourmet Grilled Brie\n\n## Sandwiches with Apricot Jam\n\nA cannabis twist on a delicious classic. There has never been such an easy and delicious way to take your medicine. Try it with a cup of tomato bisque soup.\n\n**INSTRUCTIONS:**\n\n\u2022 Butter all bread slices on one side with cannabutter, placing them butter side down.\n\n\u2022 Spread half of the slices with apricot jam.\n\n\u2022 Remove rind from the brie, cutting while cold.\n\n\u2022 Spread room temperature brie on the other 4 slices.\n\n\u2022 Combine one apricot jam side with one side with the brie, butter side out. Repeat 3 more times to make 4 sandwiches.\n\n\u2022 Place the sandwiches in a preheated skillet, over medium heat.\n\n\u2022 Cover frying pan with lid so cheese melts.\n\n\u2022 Remove lid and turn sandwiches. Cook until they are golden brown on both sides.\n\n\u2022 Serve immediately.\n\n1. _Spreading cannabutter on first bread side (apricot on second half)_\n\n2. _Spreading cannabutter on second sandwich side_\n\n3. _Flipping grilled cheese sandwich_\n\nYield:\n\n_8 servings_\n\nEquipment:\n\n_large frying pan or pot with lid_\n\nIngredients:\n\n\u00be _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _teaspoon **Sandy's 10x Hot**_ \n_**Sauce** (see page 64)_ \n2 _tablespoons olive oil_ \n2 _boneless, skinless chicken_ \n_breasts, cut into 1\" chunks_ \n\u00bd _pound Andouille sausage,_ \n_cooked, and thinly sliced_ \n1 _medium onion, chopped_ \n2 _large celery stalks, chopped_ \n1 _small bell pepper, seeded and_ \n_chopped_ \n2 _cloves garlic, peeled and_ \n_chopped_ \n\u00bc _teaspoon salt_ \n\u215b _teaspoon ground black pepper_ \n1 _teaspoon ground black pepper_ \n1 _cup canned, chopped tomatoes,_ \n_not drained_ \n2 _cups uncooked white rice_ \n6 _cups chicken broth_ \n2 _tablespoons Worcestershire_ \n_sauce_\n\n# **Jambalaya** a la Cannabis\n\nAfter a family trip, I created this to remind me of the vibrant spirit that the people of New Orleans share with the world every Fat Tuesday.\n\n**INSTRUCTIONS:**\n\n\u2022 Combine the olive oil and 2 tablespoons of the cannabutter in a large frying pan or pot over medium to high heat.\n\n\u2022 Add the chicken and cook through, about 5 minutes.\n\n\u2022 Reduce the heat to medium and stir in the sausage, onion, celery, bell pepper and garlic.\n\n\u2022 Sprinkle in the salt and pepper and continue to stir.\n\n\u2022 Continue to cook the mixture for 5 minutes, stirring occasionally.\n\n\u2022 Then stir in the chopped tomatoes and the uncooked rice.\n\n\u2022 Add the chicken broth and the rest of the butter and bring the ingredients to a boil.\n\n\u2022 Reduce heat to low and cover the pan.\n\n\u2022 Simmer the jambalaya until the rice is tender, about 30 minutes.\n\n\u2022 Finish with the Worcestershire and Aunt Sandy's hot sauce, stirring them in.\n\n\u2022 Serve warm.\n\n**Chapter 10**\n\n**Sides**\n\n_A main dish always is better with a trusty sidekick. These easy and versatile sides will really tie the meal together. If cooking as a side to other cannabis dishes, be aware of the cumulative effect of the medicine. You can always make them without marijuana if need be. Either way, these sides are sure to make your meal into an event. These are some of my favorite leftovers, as well. Adding a cannabis side to any dinner makes it a fabulous and functional experience._\n\n_If you want larger portions or seconds, consider reducing the cannabutter dosage._\n\nYield:\n\n_2 cups_\n\nEquipment:\n\n_small pot_\n\nIngredients:\n\n2 _tablespoons **Aunt Sandy's**_ \n_**10x Cannabutter**_ \n1 _cup water_ \n1 _cup rice_ \n2 _cloves garlic, minced_ \n1 _teaspoon parsley_\n\n# Green\n\n## Rice\n\nA dish that resembles cannabis in color and flavor as the rice takes on all aspects of the cannabis that will compliment any entree.\n\n**INSTRUCTIONS:**\n\n\u2022 Cook the rice with all the ingredients for as long as the package suggests.\n\n\u2022 Serve as side dish.\n\nYield:\n\n_12 servings_\n\nEquipment:\n\n_13 by 9-inch glass baking dish, saucepan_\n\nIngredients:\n\n\u00bc _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _teaspoon oil_ \n5 _large potatoes, peeled and_ \n_sliced_ \n2 _medium onions, chopped_ \n_thinly_ \n\u00bd _cup flour_ \n2 _cups milk_\n\n# **Scalloped**\n\n## Potatoes\n\nThis is a buttery, creamy side dish that perfectly complements any meat dish, such as ham, roast beef, or chicken.\n\n**INSTRUCTIONS:**\n\n\u2022 Heat oven to 350\u00b0.\n\n\u2022 Spray 13 by 9-inch glass baking dish with oil to create crispy edges.\n\n\u2022 Layer one third of the potatoes in baking dish and sprinkle with one third of the chopped onions.\n\n\u2022 Keep layering until the dish is full.\n\n\u2022 To make the sauce (also called b\u00e9chamel) melt the cannabutter in a medium saucepan until it hot.\n\n\u2022 Add flour slowly stirring constantly until blended.\n\n\u2022 Continue to cook over medium to low heat, stirring constantly until flour and butter blend to form a brown color.\n\n\u2022 Add milk slowly, whisking the whole time until it is a smooth, slightly thickened texture.\n\n\u2022 Pour over the potato and onion mixture.\n\n\u2022 Cover with plastic wrap; then cover with tin foil.\n\n\u2022 Bake for 80 minutes until golden brown.\n\n\u2022 When knife easily cuts through the potatoes, they are done.\n\n\u2022 Serve immediately.\n\nYield:\n\n_12 servings_\n\nEquipment:\n\n_large pan, small pan_\n\nIngredients:\n\n\u215c _cup **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n2 _tablespoons olive oil_ \n1 _onion, chopped_ \n2 _cups uncooked, Arborio rice_ \n1 _cup Cremini mushrooms_ \n1 _cup white wine_ \n1 _cup chicken or vegetable broth_ \n\u00bd _teaspoon parsley_ \n\u00bd _teaspoon thyme_ \n1\u00bd _cup grated Parmesan or_ \n_Romano cheese_ \n_Salt and pepper to taste_\n\n# **Rosie's**\n\n## Risotto\n\nThis yummy creamy and cheesy dish is filling enough to stand on its own as a main course, or serve it as a great side to a chicken or fish entree.\n\n**INSTRUCTIONS:**\n\n\u2022 In a large pan, saut\u00e9 onions in olive oil until translucent, about 8 minutes.\n\n\u2022 Add rice and cook until rice turns plump, about 20 minutes.\n\n\u2022 In a separate pan, saut\u00e9 mushrooms in cannabutter and set aside.\n\n\u2022 In the large pan with the rice, very slowly add \u00bd cup chicken broth and 1 cup white wine, and stir constantly while maintaining a simmer.\n\n\u2022 Add saut\u00e9ed mushrooms.\n\n\u2022 Add the parsley and thyme.\n\n\u2022 The risotto is done when rice is tender, about 20 minutes.\n\n\u2022 Fold in grated Parmesan or Romano cheese and season to taste with salt and pepper.\n\n**Chapter 11**\n\n**Tasty Snacks**\n\n_Cannabis snacks make sense. Because of the individuality of popcorn, nuts, and seeds, you can regulate your dosage almost to a science. When you find a level that works for you, you can easily prepackage these snacks into adequate portions for home or on the go. Take three nuts and call me in the morning. Beware of the munchies. Prepare to have non-medicated alternative snacks for munching on if cannabis heavily stimulates your appetite._\n\n_These will sneak up on you, so watch your dosage!_\n\nYield:\n\n_12 cups_\n\nEquipment:\n\n_large saucepan or pot with lid_\n\nIngredients:\n\n\u00bc _cup melted **Aunt Sandy's**_ \n_**10x Cannabutter**_ \n\u00bd _cup of your favorite popcorn_ \n_kernels_ \n\u00bc _cup canola or vegetable oil_ \n_Salt to taste_\n\n# Popcorn\n\nTake your own popcorn to the movies. There is nothing like a tasty therapeutic snack.\n\n**INSTRUCTIONS:**\n\n\u2022 Heat the oil in the large saucepan or pot on medium-high heat.\n\n\u2022 Place 2\u20133 kernels in the oil and cover the pot.\n\n\u2022 When these kernels pop, add the rest of the kernels in an even layer.\n\n\u2022 Cover the pot and let pop, shaking gently to ensure all kernels pop.\n\n\u2022 When popping slows to several seconds between kernels, remove the pan from heat.\n\n\u2022 Add melted cannabutter and mix well.\n\n\u2022 Remove from pan and serve warm.\n\nYield:\n\n_3 cups_\n\nEquipment:\n\n_quart-size jar with lid, saucepan_\n\nIngredients:\n\n\u00bc _cup of **Aunt Sandy's 10x**_ \n_**Cannabis Oil** (use olive oil,_ \n_see page 28)_ \n3 _small sprigs of **cannabis bud**_ \n3 _cups of mixed olives, rinsed_ \n_and drained well_ \n\u00be _cup regular olive oil_ \n3 _sprigs of fresh thyme_ \n4 _sprigs of fresh rosemary_ \n1\u00bd _teaspoons whole fennel seeds_ \n2 _strips orange zest_ \n\u00bc _teaspoon crushed red pepper_ \n_flakes_ \n2 _bay leaves_ \n1 _clove garlic, slivered_ \n\u00bc _cup fresh lemon juice_\n\n# **Olive**\n\n## Medley\n\nI love olives. Simply infuse them with cannabis olive oil. It creates a quick, easy, and delicious way to take your medicine.\n\n**INSTRUCTIONS:**\n\n\u2022 Place the olives in a quart jar.\n\n\u2022 In a small sauce pan, combine oil, thyme, rosemary, cannabis sprigs, fennel seeds, orange zest, red pepper flakes, bay leaf, and garlic and simmer for 10 minutes.\n\n\u2022 Pour the oil and seasonings over the olives in the jar.\n\n\u2022 Add the lemon juice and put the lid tightly on the jar.\n\n\u2022 Turn the jar a few times to evenly distribute the seasonings.\n\n\u2022 Reopen the jar and let cool.\n\n\u2022 Replace lid and store in the refrigerator.\n\nYield:\n\n_3\u00bd cups_\n\nEquipment:\n\n_baking sheet pan, aluminum foil, bowl, mixing bowl_\n\nIngredients:\n\n\u00bc _cup melted **Aunt Sandy's 10x**_ \n_**Cannabutter**_ \n1 _pound (3\u00bd cups) mixed_ \n_unsalted almonds, pecans and_ \n_cashews_ \n1\u00bd _tablespoons light brown sugar_ \n2 _teaspoons fresh thyme_ \n\u00bd _teaspoon chipotle powder_ \n1 _teaspoon salt_\n\n# Sweet & Spicy\n\n## Roasted Nuts\n\nThis makes for a great addition to a sundae or a salad on a warm summer night.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 375\u00b0.\n\n\u2022 Line the baking sheet pan with aluminum foil.\n\n\u2022 Spread the nuts evenly on the baking sheet.\n\n\u2022 Roast about 15 minutes, until the nuts are lightly brown.\n\n\u2022 Combine sugar, thyme, chipotle powder and salt in a bowl. Mix together.\n\n\u2022 Place the warm nuts in a mixing bowl and add the melted cannabutter. Toss well, coating all of the nuts.\n\n\u2022 Add the brown sugar and spices to the mixing bowl and toss again to coat evenly.\n\n\u2022 Serve warm or cold.\n\n\u2022 Store in an airtight container.\n\nYield:\n\n_2 cups_\n\nEquipment:\n\n_baking sheet pan, aluminum foil, bowl, mixing bowl_\n\nIngredients:\n\n2 _tablespoons melted **Aunt**_ \n_**Sandy's 10x Cannabutter**_ \n1 _tablespoon **Aunt Sandy's**_ \n_**Cannaflour** (see page 27)_ \n2 _tablespoons olive oil_ \n2 _teaspoons salt_ \n1 _teaspoon fennel seed_ \n\u00bc _teaspoon pepper_ \n2 _egg whites_ \n\u00bd _cup finely grated Parmesan-_ \n_Reggiano cheese_ \n\u00bd _pound (1\u00bd cups) almonds_ \n_with skin on_\n\n# Cheesy\n\n## Savory Nuts\n\nThese are simply delightful on their own.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 300\u00b0.\n\n\u2022 Line the baking sheet with foil and coat evenly with olive oil.\n\n\u2022 In a bowl, combine the cannaflour, salt, pepper and fennel, and mix together.\n\n\u2022 In a mixing bowl, whisk the egg whites until they are foamy.\n\n\u2022 Add melted cannabutter, spices and cheese and mix again.\n\n\u2022 Put the nuts in the mixing bowl, tossing to thoroughly coat.\n\n\u2022 Spread the nuts onto the baking sheet.\n\n\u2022 Bake for 45 minutes, turning the nuts every 15 minutes.\n\n\u2022 Remove the nuts to a clean foil sheet to cool.\n\n\u2022 Store in an airtight food storage container or jar. To preserve crispness, do not store in plastic bags.\n\nYield:\n\n_2\u00bd cups_\n\nEquipment:\n\n_baking sheet pan, aluminum foil_\n\nIngredients:\n\n4 _tablespoons melted **Aunt**_ \n_**Sandy's 10x Cannabutter**_ \n2 _cups pepitas (raw hulled_ \n_pumpkin seeds)_ \n\u00bd _teaspoon pepper_ \n\u00bd _teaspoon cayenne_ \n1 _teaspoon cumin_ \n1 _teaspoon chili powder_ \n1 _teaspoon salt_ \n1 _tablespoon lime juice_\n\n# Spicy\n\n## Pepitas\n\nFor a medicinal treat, these are good for someone on the go.\n\n**INSTRUCTIONS:**\n\n\u2022 Preheat oven to 350\u00b0.\n\n\u2022 Line a baking sheet with aluminum foil.\n\n\u2022 Toss all ingredients together in a bowl. Mix well.\n\n\u2022 Spread the mixture evenly on the baking sheet.\n\n\u2022 Bake for 10 minutes, until golden brown.\n\n\u2022 Remove from baking sheet.\n\n\u2022 Serve warm or cool.\n\n**Appendix A**\n\n**Equivalents and Conversions**\n\nMeasurement Abbreviations\n\nt| =| teaspoon \n---|---|--- \ng| =| gram \nT| =| Tablespoon \nkg| =| kilogram \nc| =| cup \nmL| =| milliliter \npt.| =| pint \nL| =| liter \nqt.| =| quart \noz.| =| ounce \nlb.| =| pound\n\nFluid Equivalents\n\n3 t| =| 1 T| =| 15 mL \n---|---|---|---|--- \n4 T| =| 1\/4 c| =| 60 mL \n1 c| =| 8 oz.| =| 240 mL \n2 c| =| 1 pt.| =| 473 mL \n2 pts.| =| 1 qt.| =| 0.94 L\n\nDry Equivalents\n\n1\/4 oz.| =| 7 g| \n---|---|---|--- \n1\/2 oz.| =| 14 g| \n1 oz.| =| 28.4 g| \n8 oz.| =| 227 g| \n16 oz.| =| 454 g =| 1 lb. \n1000 g| =| 1 kg =| 2.2 lbs. \n100 kg| =| a federal rap\n\n **U.S.** Food Measures\n\nButter: 1 lb.| =| 2 cups \n---|---|--- \nFlour: 1 lb.| =| 4 cups \nSugar: 1 lb.| =| 2 cups\n\nTemperature Conversions\n\n**\u00b0Fahrenheit** | | **\u00b0Celsius** \n---|---|--- \n150| =| 66 \n212| =| 100 \n250| =| 120 \n275| =| 135 \n300| =| 149 \n325| =| 163 \n350| =| 177 \n375| =| 191 \n400| =| 204 \n425| =| 220 \n450| =| 230 \n475| =| 240 \n500| =| 260 \n550| =| 288 \n600| =| 316\n\nPowdered Marijuana **(Leaf** & Bud)\n\n1 t| =| 1.8 g| | \n---|---|---|---|--- \n1 T| =| 5.5 g| =| ~1\/4 oz. \n1\/4 c| =| 22.4 g| | \n1.3 c| =| 28 g| =| 1 oz. \n1\/2 c| =| 45 g| | \n2\/3 c| =| 56 g| =| 2 oz.\n**Acknowledgments**\n\n**W** riting a cookbook requires the concerted effort and synergy of many talented people\u2014everyone working toward a common goal.\n\nIt is the people mentioned here that made this cookbook possible.\n\nI want to express thanks to my publisher, Ed Rosenthal, for his lifelong commitment to cannabis, and for wanting to publish this cookbook.\n\nThanks to his wife Jane Klein, for her design and marketing expertise.\n\nI want to offer my special thanks to Oaksterdam University Staff for the stellar job they do educating and especially to Richard Lee, the founder, for his continuing efforts to legitimize the cannabis industry.\n\nI am deeply grateful to Dennis Peron for his lifelong dedication to making this plant available to all who need it.\n\nI want to especially thank Jack Jennings, my editor who was instrumental in helping to shape this book into one of the most authoritative medical marijuana cookbooks. He also assembled the greatest team anyone could hope for: Joe Burull, a superb photographer, for the most mouth-watering, beautiful pictures ever; Mickey Martin for editing and writing from his depth of understanding and wealth of knowledge about marijuana as medicine; Alvaro Villanueva for his artistic cover design concept and layout; Leslie Kwartin for her focus and friendship in coordinating all aspects of this project; Cindy Jennings, the fastest, funniest, best copy editor; and to the many testers and tasters and the staff at Quick Trading Publishing, especially Angela Bacca.\n\nMy special thanks to friends and family: my brother and good friend Al for being a good friend and teaching me the ways of our world and his wife Leslie for all the fun we had when figuring out our personal dosage; my special daughter Ashley, who graduated from the California Culinary Academy, for sharing her cooking knowledge and enriching my life with her love; my wonderful daughter Erin for her eye of the lens and her artistic compassion in everything and her dedication and help while creating this cookbook; my daughter Marci, who generously opened her kitchen to all of us; Chef Mike for his professional advise; Marshall and Krisianne for their unconditional support and love; my good buddy Rose for all the hard work and fun in the kitchen perfecting these recipes; in remembering my doctor Dr. Tod Mikuriya as a sincere pioneer and his son Sean for carrying on....\n\nSo many people are involved when a new cookbook gets published. It isn't just about the author. In the world of cannabis, we are a family with one common goal.\n**Index**\n\n## A\n\nacorn squash\n\nAdobo Pork Chops\n\naging\n\nalcohol-based tincture\n\nalcohol-free tincture\n\nallergies, food\n\nAlzheimer's disease\n\nAmerican College of Physicians (ACP)\n\nAmerican Medical Association (AMA)\n\nAmerican Nursing Association\n\nAmericans for Safe Access\n\nanalgesic, marijuana as\n\nanti-inflammatory, marijuana as\n\nappetite stimulation\n\napricot jam\n\narthritis\n\nartichoke hearts\n\naspirin\n\nAunt Sandy's 10x Cannabutter: adjusting potency of; as medicinal additive; recipe; residual leaf use; residual water use; shelf life; unsalted; for vegetarian diets\n\nAunt Sandy's 10x Hot Sauce\n\nAunt Sandy's Cannabis Oil\n\nAunt Sandy's Cannaflour\n\nAunt Sandy's Marijuana Tincture\n\n_Aunt Sandy's Medical Marijuana Cookbook\u2014Comfort Food for Mind and Body_\n\n## B\n\nbaking, gluten-free\n\nBananacanna Fofana Nut Bars\n\nbananas\n\nbars: Bananacanna Fofana Nut Bars; Blue Sky Lemon Bars; Lemon Poppy Seed Bars\n\nb\u00e9chamel sauce\n\nbeef\n\nBerry Peachy Cannabutter Cobbler\n\nbeverages: Bhang; Cannabis Milk; tincture as additive to\n\nBhang\n\nBlue Cheese Dressing\/Dip\n\nBlue Sky Cafe\n\nBlue Sky Lemon Bars\n\nbourbon\n\nbrain chemistry\n\nBraised Short Ribs with Egg Noodles\n\nbreading, gluten free\n\nbrie cheese\n\nbud: vs. leaf trim; to control strength; making cannabis oil; using sprigs of\n\nbuffalo wings\n\nBundt cakes\n\nBurning Man\n\nbutter, cannabis; see also Aunt Sandy's 10x Cannabutter\n\nbuttternut squash\n\n## C\n\ncakes: Kahlua Cannabis Cake; Peanut Butter Bundt Cake\n\nCalifornia legislation\n\ncancer\n\ncannabidiol (CBD)\n\ncannabinoids: evaporation point; extracting; therapeutic effects of; vaporization of\n\ncannabinol (CBN)\n\nCannabis Chicken Curry\n\nCannabis Gumbo\n\nCannabis Milk\n\ncannabis oil: Aunt Sandy's Cannabis Oil as ingestion method\n\nCannabutter, Aunt Sandy's 10x. see Aunt Sandy's 10x Cannabutter\n\nCannafire Buffalo Wings\n\nCannaflour Chicken Tenders\n\nCannamacca Roni & Cheese\n\ncanola oil\n\nCB1 receptors\n\ncheese: Blue Cheese Dressing\/ Dip; Cannamacca Roni & Cheese; Cheesy Savory Nuts; Gourmet Grilled Brie Sandwiches with Apricot Jam; mozzarella; in risotto; variants, for grilled cheese\n\nCheesy Savory Nuts\n\nchemotheraphy treatment\n\nchicken: Cannabis Chicken Curry; Cannabis Gumbo; Cannafire Buffalo Wings; Cannaflour Chicken Tenders; Jambalaya a la Cannabis\n\nchicken tenders\n\nchicken wings\n\nchickpeas\n\nchilies\n\nchlorophyll\n\nchocolate: Chocolate Coconut Pecan Pie; Chocolate Frosting; Larry's Munchie Bar\n\nChocolate Coconut Pecan Pie\n\nCioppino\n\nClassic Cannabis Pesto\n\ncleanliness\n\nclinical trials\n\ncobbler\n\ncoconut\n\nCoconut Cannabis cookies\n\ncoffee\n\ncoffee liqueur\n\ncomfort food, medicated\n\ncondiments\n\ncontaminants\n\ncookies: Coconut Cannabis cookies; Oatmeal Raisin Cookies\n\ncorn bread\n\ncrab legs, breaking\n\nCreamy Zucchini Soup\n\ncurried chicken\n\n## D\n\ndairy products: alternatives to; on vegetarian diets\n\ndiabetes\n\ndiabetic diets\n\ndietary restrictions\n\ndigestive issues\n\ndips\n\ndisease, plant\n\ndispensaries, medical marijuana\n\nDizzy Bird Turkey\n\ndosages: of entree plus side dish; for entrees; factors affecting; get it right; marijuana quality as influencing; in sauces; in snacks; titration\n\ndressings: blue cheese; ranch\n\n## E\n\neating, vs. smoking\n\neating habits\n\neffects, unintended\n\negg noodles\n\neggs\n\negg whites\n\nentrees\n\neuphoria\n\n## F\n\nFat Tuesday\n\nfederal law enforcement\n\nfish\n\nflour, cannabis\n\nfood safety\n\nFrench bread\n\nfrosting: chocolate; lemon; mocha; cream cheese; white\n\nfungicides\n\nfungus\n\n## G\n\ngastrointestinal disorders\n\ngenetic variants\n\ngluten\n\ngluten free diets\n\nGourmet Grilled Brie Sandwiches with Apricot Jam\n\nGreen Rice\n\ngrilled cheese sandwiches\n\ngumbo\n\n## H\n\nhallugenogenic effects\n\nhand washing\n\nherb, cannabis, in pork chop\n\nherbal medicine\n\nhigh-grade marijuana\n\nhistory, of mairjuana medicine\n\nHIV\/AIDS\n\nhot sauce\n\nhummus\n\nhybrid strains\n\nhygiene\n\nhypoglycemic diets\n\n## I\n\nimmune system: compromised; response of\n\nindicas\n\ninfusions, cannabis\n\ningestion methods\n\n## J\n\njalape\u00f1os\n\njam, apricot\n\nJambalaya a la Cannabis\n\n## K\n\nKahlua Cannabis Cake\n\n## L\n\nlabeling\n\nLarry's Munchie Bar\n\nleaf trim\n\nleaves, fresh cannabis\n\nlegislation\n\nlemon bars\n\nlemon frosting\n\nLemon Poppy Seed Bars\n\nLemony Hummus Dip\n\nlow calorie options\n\nlow-grade marijuana\n\nlow sodium diets\n\n## M\n\nmain courses\n\nmargarine\n\nMarijuana Tax Act (1937)\n\nMarshall's Favorite Dip\n\nmayonnaise\n\nmeat substitutes\n\nmedical system use\n\nmetabolism\n\nmethods of ingestion\n\nmildew, powdery\n\nmilk, cannabis\n\nMinestrone Soup\n\nmocha frosting\n\nmold\n\nmuffins\n\nmultiple sclerosis\n\nmunchies\n\nmushrooms\n\n## N\n\nNelson, Willie\n\nNew Orleans\n\nnuts: Bananacanna Fofana Nut Bars; Cheesy Savory Nuts; as curry additive; Larry's Munchie Bar; Pumpkin Streusel Muffins; Spicy Pepitas; Sweet and Spicy Roasted Nuts\n\n## O\n\nOakland\n\nOaksterdam University\n\nOatmeal Raisin Cookies\n\nObama Administration\n\noil: cannabis-infused; in vegan diets; vegetable oil choices\n\nolive oil\n\nolives\n\n1:4 Ratio Chart for Determining Potency\n\norange frosting\n\nover-dosage, symptoms of\n\n## P\n\npain relief\n\nParkinsons\n\npasta: Braised Short Ribs with Egg Noodles; Cannamacca Roni & Cheese; Classic Cannabis Pesto\n\npathogens\n\npatients, medical marijuana: access to medicine; adjusting dosage for; author's cooking for; legal rights\n\npeaches\n\nPeanut Butter Bundt Cake\n\npecan pie\n\npesticides\n\nPesto, Classic Cannabis\n\nphysicians, contacting\n\npies\/cobblers: Berry Peachy Cannabutter Cobbler; Chocolate Coconut Pecan Pie\n\npita bread\n\npopcorn\n\npoppy seeds\n\npork chops\n\npotatoes, scalloped\n\npotency: adjusting ratios for; caution in testing; chart for determining; factors affecting; and patient capacity; and raw material quality; reasons for reducing; reducing plant material and; and serving size\n\npowdery mildew\n\nPropositioon 215\n\npsychoactivity\n\npumpkins\n\nPumpkin Streusel Muffins\n\n## Q\n\nquality, plant\n\n## R\n\nraisins\n\nRanch Dressing\n\nRatio Chart for Determining Potency\n\nresearch, limited\n\nresidual effects\n\nresponsibility: in cannabis use; in food preparation\n\nribs\n\nrice: Green Rice; in jambalaya; Rosie's Risotto\n\nRichards, Larry\n\nroasted turkey\n\nRosie's Risotto\n\nrum\n\n## S\n\nsafety: dosage measurement. see dosages; equipment; food preparation; of herbal medicine; inspection, of marijuana; learning; marijuana as non-toxic\n\nSan Francisco\n\nsanitaiton\n\nsativas\n\nsauces\n\nsausage\n\nScalloped Potatoes\n\nseafood stew\n\nseizures\n\nserving size\n\nshake, marijuana\n\nsherry\n\nshrimp\n\nsmell\n\nsmoking\n\nsnacks\n\nsocial pressure\n\nsodium, controllling\n\nsoups: Cioppino; Creamy Zucchini Soup; Minestrone Soup; Pumpkin Soup; Tomato Bisque\n\nsource of medicine, knowing\n\nspasms\n\nspicey hot sauce\n\nSpicy Pepitas\n\nsquash\n\nstarters: Cannabis Gumbo; Cannafire Buffalo Wings; Cannaflour Chicken Tenders\n\nSteep Hill Labs\n\nsterilization\n\nstew\n\nstorage: cannabis oil; cannabis powder; Cannabutter; marijuana; tincture\n\nstreusel\n\nstuffing, turkey\n\nsuffering, allowing patient\n\nsugar free diets\n\nSunday dinner\n\nSweet and Spicy Roasted Nuts\n\nsymptoms served\n\n## T\n\ntaste of marijuana\n\ntea: cookies for; tincture as additive to\n\nteaching marijuana cooking\n\ntesting, for contaminants\n\ntexture: and marijuana content; and raw material quality\n\nThai curry\n\nTHC\n\ntherapeutic effects: controlling; by genetic strain; ingestion method as varying; list of; pain relief; and raw material quality\n\ntincture, cannabis: as ingestion method; recipe for making\n\nTiramisu\n\ntitration\n\nTomato Bisque\n\ntrichomes\n\nturkey\n\n## U\n\nUS Pharmacopoeia\n\n## V\n\nvaporization\n\nvegan diets\n\nvegetables: for minestrone; sauces for; in turkey stuffing\n\nvegetarian diets\n\n## W\n\nWashington DC legislation\n\nweight\n\nweight loss diets\n\nwhite frosting\n\nwings\n\nWorld Health Organizaion\n\n## X\n\nxylitol\n**OAKSTERDAM UNIVERSITY**\n\n**QUALITY TRAINING**\n\n**13 week semester courses** \n**Basic and Advanced weekend seminars** \n**Horticulture semester programs** \n**New 30,000 square foot campus** \n**Featuring the very best instructors that the cannabis** \n**industry has to offer** \n**Learn to grow your own cannabis, train to work in a** \n**collective, or open your own!**\n\nTo Enroll: **oaksterdam.com** (510) 251-1544 \n1600 Broadway, Oakland California 94612\nAUNT SANDY'S MEDICAL MARIJUANA COOKBOOK \nComfort Food for Mind and Body\n\nCopyright c 2010 Sandy Moriarty\n\neISBN : 978-1-936-80709-3\n\nPublished by Quick American Publishing \nA Division of Quick Trading Company \nPiedmont, California\n\nPublisher's Cataloging-in-Publication \n(Provided by Quality Books, Inc.)\n\nMoriarty, Sandy.\n\nAunt Sandy's medical marijuana cookbook : comfort food for body & mind \/ Sandy Moriarty.\n\np. cm.\n\nIncludes index.\n\n1. Marijuana\u2014Therapeutic use\u2014Recipes. 2. Diet therapy. 3. Cooking. 4. Cookbooks. I. Title. II. Title: Medical marijuana cookbook.\n\nRM666.C266M67 2010\n\n615'.7827 QBI10-600139\n\nThe material offered in this book is presented as information that should be available to the public. The Publisher does not advocate breaking the law. The Author and Publisher have tested the recipes included here, but no warranty is provided, nor results guaranteed for the use, or misuse of this information.\n\nAll rights reserved. No part of this book may be used or reproduced in any manner without written permission of the Publisher, except in the case of brief quotations embodied in critical articles or reviews.\n\nTry your bookstore first but you may order this book from our website-www.Quicktrading.com or by mail from the Publisher.\n\nQuick Trading\/Quick American Publishing \n9 Lake Ave \nPiedmont, CA 94611\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nE-text prepared by Mark C. Orton, Anita Hammond, and the Online\nDistributed Proofreading Team (http:\/\/www.pgdp.net)\n\n\n\nTranscriber's note:\n\n Text enclosed by underscores is in italics (_italics_).\n\n Text enclosed by plus signs is in bold face (+bold+).\n\n Text enclosed by tilde characters is underlined (~underlined~).\n\n Text enclosed by equal signs is double underlined\n (=double underlined=).\n\n The Key to Pronunciation, p. 37 (Spelling Lesson 3), contains\n characters with diacritical marks not available in Latin-1\n character encoding. Therefore, they have been transcribed as\n follows:\n [=x] character 'x' with macron (bar) above the letter\n [.x] character 'x' with dot above the letter\n [~x] character 'x' with tilde (curved bar) above the letter\n\n\n\n\n\nPLAIN ENGLISH\n\nby\n\nMARIAN WHARTON\n\n_For the Education of the Workers\nby the Workers_\n\n\n\n\n\n\n\nPublished by\nThe People's College\nFort Scott, Kansas\n1917\n\n\n\n\nHe who aspires to master the art of expression must first of all\nconsecrate himself completely to some great cause and the greatest cause\nof all is the cause of humanity. He must learn to feel deeply and think\nclearly, to express himself eloquently. He must be absolutely true to\nthe best there is in him, if he has to stand alone.\n\nSuch natural powers as he may have should be cultivated by the study\nof history, science and literature. He must not only keep close to the\npeople but remember that he is one of them, and not above the meanest.\nHe must feel the wrongs of others so keenly that he forgets his own, and\nresolve to combat these wrongs with all the power at his command.\n\nThe most thrilling, inspiring oratory, the most powerful and\nimpressive eloquence is the voice of the disinherited, the oppressed,\nthe suffering and submerged; it is the voice of poverty and misery, of\nrags and crusts, of wretchedness and despair; the voice of humanity\ncrying to the infinite; the voice that resounds throughout the earth and\nreaches Heaven; the voice that awakens the conscience of a race and\nproclaims the truths that fill the world with life and liberty and love.\n\n --EUGENE V. DEBS.\n\n\n\n\n FOREWORD\n\n\nEvery generation has added a little to the store of truth of which the\nhuman race has possessed itself throughout the long sweep of the\ncenturies. Every truth expressed and preserved by those who lived in the\npast, is a contribution which enriches the lives of those who live in\nthe present. We, as members of the human race, are not separate atoms\nindependent of the universe, but we are atoms of it. We are the product\nof all time, and partake of the truth of all preceding generations, in\nwhich the power to express ideas and preserve them has existed.\n\nOne reason why the race has not profited more largely by the discoveries\nof previous generations, is the fact that we feel so profoundly the\ndiscovery of a truth of any nature, that we are prone to dogmatize it by\na rule or set of rules.\n\nThis usually results in shutting away from us the real principle of\nwhich the rule is but an evidence. A mechanic may learn every detail of\nevery rule for the construction of a steam engine, but if he lacks the\nunderstanding of the principles which give rise to the rules, they will\navail nothing and his work must fail. If, however, he understands the\nprinciples involved, his work will stand the test, though he has no\nknowledge of rules as such.\n\nIn teaching the English language, the rules have been stressed, while\nthe principles have been submerged, so that the teaching of rules has\nnot resulted in the improvement of the student.\n\nThe People's College, realizing this, has, through the author of this\nwork, revolutionized the teaching of the fundamental principles that\nunderlie the use of language. The stress is laid upon principles instead\nof rules, so that the student, whether he remember a rule or not, will\nnever forget the application of these principles to the use of the\nwritten and spoken word.\n\nThe assertion is ventured that no more practical and effective method\ncan be devised for the rapid and thorough teaching of these principles.\nMoreover, the importance of this new departure in method cannot be\nover-estimated, when we consider that only through the use of language\ncan information be disseminated concerning other branches of learning.\nThis science, then, lies at the very base of all real education, and a\nmastery of it puts the student in possession of the only weapon by means\nof which he may master all other sciences.\n\nThe author has, with peculiar aptitude, grasped the fundamental\ncharacter of the foregoing facts and has adapted the study of language\nto the real principles involved. All the dry rules that are the\nwitnesses of principles in the ordinary text are done away, while the\nprinciples evidenced by those rules come forth to the light in practical\napplication, with a beauty of expression and a real utility that render\nthe mastery of the subject an entertaining excursion into the realms of\nlearning, rather than a dry imprisonment of the faculties in an effort\nto memorize misunderstood rules without apparent reason or real use.\n\nIt is the principle behind the rule that has power in it. When this is\nunderstood, the method pursued by the author in this course will be\nuniversally applied to all branches of learning, and will end forever\nthe imprisonment of children for the useless worship of rules.\n\nThe author's grasp of this fact and the exemplification of it, contained\nin this work are even more far-reaching than the foregoing would\nindicate. It really means the application of a new viewpoint to life\nitself. It means the questioning of the utility of authority; the\nquestioning of the utility of institutions; the application, we might\nsay, of such a test as this: Does any rule, does any authority, does any\nprinciple, conserve the interests of humanity? If not, away with it.\nThis means rationalism, the use of common sense. It means that at last\nthe race is beginning to consciously direct its own destiny.\n\nIt is with a profound sense of the necessity of education as a part of\nthe evolutionary process now in the conscious grasp of the race, and\nwith a conviction of the fundamental importance of the new viewpoint so\nably presented by the author that we dedicate this work \"To the\nEducation of the Workers by the Workers.\"\n\n THE PEOPLE'S COLLEGE.\n\n\n\n\n TABLE OF CONTENTS\n\n\n PLAIN ENGLISH\n\n I. Language Study 9\n II. Nouns and Verbs 18\n III. Parts of Speech 27\n IV. Nouns 38\n V. Verbs 50\n VI. Inflection of Verbs 58\n VII. Time Forms of Verbs 69\n VIII. Time Forms, Cont'd. 78\n IX. Participles and Infinitives 88\n X. Helping Verbs 97\n XI. Verbs--Common Errors 106\n XII. Pronouns 115\n XIII. Pronouns, Cont'd. 127\n XIV. Adjectives 138\n XV. Adjectives, Cont'd. 148\n XVI. Adverbs 160\n XVII. Adverbs, Cont'd. 169\n XVIII. Prepositions 179\n XIX. Prepositions, Cont'd. 189\n XX. Conjunctions 200\n XXI. Conjunctions, Cont'd. 212\n XXII. Adjective Clauses 222\n XXIII. Independent Constructions 232\n XXIV. Sentence Building 243\n XXV. Sentence Analysis 255\n XXVI. Sentence Building 267\n XXVII. Sentence Building 278\n XXVIII. The Use of Capitals 288\n XXIX. Punctuation 299\n XXX. Punctuation, Cont'd. 310\n\n\n SPELLING\n\n I. Definition 17\n II. Vowels and Consonants 26\n III. Diacritical Marks 36\n IV. Digraphs 49\n V. Diphthongs 57\n VI. Syllabification 68\n VII. Syllabification, Cont'd. 77\n VIII. Accent 87\n IX. Compound Words 96\n X. Prefixes and Suffixes 105\n XI. Derivatives 114\n XII. Derivatives, Cont'd. 126\n XIII. Silent E 137\n XIV. Words Ending in Y 146\n XV. Words with ei or ie 159\n XVI. Homonyms 168\n XVII. Derivative Nouns 178\n XVIII. Verbs with Prepositions 187\n XIX. Derivative Prepositions 199\n XX. Derivative Adverbs 211\n XXI. Derivative Adjectives 221\n XXII. Words in able and ible 231\n XXIII. Simplified Spelling 241\n XXIV. Verbs with Suffixes 254\n XXV. Cognate Sounds 265\n XXVI. Words beginning with dis 277\n XXVII. The prefixes in, un and mis 287\n XXVIII. Synonyms 297\n XXIX. Antonyms 308\n XXX. Common Errors 320\n\n\n\n\n PLAIN ENGLISH\n\n Lesson I\n\n\n Open Letter\n\nDear Comrade:\n\nYou are beginning a course of study in the use of Plain English. We do\nnot know what your previous study may have been, but the object of this\ncourse is to give the basic principles and practice of the use of the\nEnglish Language for the benefit of those who have not had the\nopportunity of a high school education and possibly have not finished in\nthe grade school.\n\nFor this reason we have avoided, as much as possible, the statement of\nrules and formulas to be learned by rote and have made the few rules\nwhich it is necessary to know, grow naturally out of the need for them\nin the development of expression in language.\n\nWe have taken for granted several things in the preparation of this\ncourse. First, we assume that you have never studied grammar, or if you\nhave, that you will be glad to review it in simplified form. This course\ndoes not follow the lines laid down by technical grammarians. It has\nbeen worked out on the basis of plain, common sense. Our purpose is not\nto make of you a grammarian, versed in the knowledge of rules and\nreasons, but to give you the power to express yourself more readily,\nfluently and correctly--in other words to speak and write _good_\nEnglish.\n\nSecond, we assume that you are interested and willing to work and eager\nto increase your store of knowledge. Your progress in this branch of\nknowledge will depend, to a large extent, upon your own efforts. We have\nendeavored to avoid unnecessary and uninteresting rules and make the\ncourse as simple, clear and plain as possible; but that does not mean\nyou will not have to work in order to master this study. We trust it\nwill be pleasant and interesting work, bringing you joy as it brings you\na growing sense of power.\n\nProbably no two people will use the same plan of work. Your work, to be\na pleasure, must express your own individuality. However, we want to\nmake a few suggestions which we know from experience you will find\nhelpful.\n\n+1st.+ +Be Systematic.+ Find some time each day which you can regularly\nspend in study. Do not be discouraged if it is only fifteen minutes each\nday. The student who will spend fifteen minutes every day regularly in\nintensive study can easily complete this course within the prescribed\ntime.\n\n+2d.+ +Concentrate.+ By this we mean that when you study, you should do\nit to the exclusion of everything else. Keep your mind upon the subject.\nYou may find this difficult at first. Your mind will wander; but you\nwill soon acquire the student habit if you persevere.\n\n+3d.+ +Have Faith in Yourself.+ Do not be easily discouraged. You have\nthe power to master this subject and _you will_. You will find it of\nimmeasurable value to you to be able to speak and write fluently and\ncorrectly. Those whom you admire for their ready use of good English\nwere not born with the \"gift of gab.\" They learned how to speak by\nstudying the rules of grammar, the meaning of words, just as you are\nstudying them. What they have done, _you will do_.\n\n+4th.+ +Go Slowly and Surely.+ Do not skim through these lessons. Be\nsure you understand thoroughly as you go along. Read carefully and\n_think_ for yourself. If there is anything you do not understand at any\ntime, write us and ask about it. These lessons have been carefully\nprepared and are for your benefit. Make them yours and call upon us\nfreely for help. This is your College and its only ideal is service.\n\n+5th.+ +Get a Note-Book.+ Make your note-book your work-shop. Write in\nit an outline of each lesson. Fill it with notes, examples, anything\nwhich is of interest on the subject. Note down your own frequent\nmistakes in the use of English. Watch the conversation of your friends;\nlisten to good speakers. Write down the mistakes you notice. Whenever\nyou hear a word which seems particularly good, or when you see one in\nyour reading, write it in your note-book and make it part of your\nvocabulary. You will find your interest continually growing and also\nyour ability to express the thoughts you yearn to express.\n\nIf we can bring to you an increasing joy in life because of a growing\npower of expression; if we can enlarge your ability to serve the world;\nif we can, through the study of this wonderful language of ours, open\nwider the door of opportunity for you,--our comrade,--The People's\nCollege will have served its purpose and realized its ideal.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n GOOD ENGLISH--WHAT IS IT?\n\n+1.+ People seem to differ in their idea as to what constitutes \"Good\nEnglish.\" Have you never seen a man suddenly called upon to make a\nformal speech or introduced into the company of distinguished men and\nwomen? Quite often, he will drop his simple every-day mode of speech and\nspeak in stilted, unnatural language, using all the \"big\" words he can\npossibly remember. He no doubt fondly imagines he is making an\nimpression and using \"good\" English.\n\nThe purpose of language is to make one's self understood, and, of\ncourse, this can be done in very simple and crude English. The man who\nbreaks every rule of grammar, intersperses his remarks with every\nvariety of slang phrase, may make himself understood, but he is not\nusing _good_ English.\n\n+2.+ +Good English is that which is good for its purpose and conforms to\nthe standards of usage.+\n\nWe have one purpose when we write a business letter and quite another\nwhen we are writing or speaking of the great issues of life. There is a\nplace for the simple, direct, plain, unadorned language of every-day\nbusiness life--the life of the work-a-day world--and there is a place\nalso for the beauty and charm of the language of poetry. If we are\ntalking with the man who works beside us of the work of the day, we will\nnaturally use plain, simple, forceful words. But, if we are speaking to\nour comrades, striving to arouse them out of their lethargy, to stir\nthem to action as men and women, we will just as naturally use the fine\nand noble words which touch the depths of human emotion--the heights of\nhuman endeavor.\n\n+3.+ There are certain rules for the use of English which have grown up\nthrough the years, to which we must conform. These are not arbitrary.\nThey have not been made by any man or any set of men. In fact, they are\nconstantly changing, as the common usage of the people forces the\nchanges. For these rules are only the expression of the common usage,\nand as usage changes, the rules change.\n\nBut these changes come slowly, so we can set down in a book the rules\nwhich express the established usage of today. The ability to use good\nEnglish does not mean the ability to use long, high-sounding words. To\nbe a master of good English means to be able to use the word that meets\nyour need and use it correctly.\n\nDo not strive for _effect_, strive for _effective expression_.\n\n\n USE YOUR DICTIONARY\n\n+4.+ Do you know that the average individual s through life with\na vocabulary of a few hundred words when he might easily have at his\ncommand as many thousands?\n\nWe are misers with our words. Here hid away in this book we call the\ndictionary is a wealth of words, a rich mine of expression, and yet in\nour every-day conversation we halt and stammer, using meaningless words\nand phrases largely made up of current slang.\n\nNever let a word pass by that you do not understand thoroughly. Look it\nup at once in your dictionary and master it then and there. Dollars may\nbe difficult to earn and more difficult to keep, but here is a wealth\neasily gained and the more you use it the more you possess it.\n\nYou will find your dictionary an exceedingly interesting book when you\nget acquainted with it.\n\nUse it constantly; make it your familiar companion.\n\n\n OUR LANGUAGE\n\n+5.+ Did you ever stop to think what the world would be if we had no way\nof communicating, one with another? Think of Helen Keller, shut up in\nher prison-house of silence. Her only mode of communication with her\nfellows is through the sense of touch.\n\nEvery form of life that has consciousness has some way of expressing its\nfeelings. Every animal, by the movements of its body or the tones of its\nvoice, expresses its emotions of pain, pleasure, rage, hate, joy, hunger\nand the many passions that sway its life. The child knows without being\ntaught how to express its wants. We understand its cry of hunger, its\nscream of pain, its laugh of delight. This is the natural language, the\nlanguage of feeling. It is the universal language that needs no rules\nand no interpreter. Life on every plane knows and understands it.\n\n\n WHEN WE BEGIN TO THINK\n\n+6.+ Our feelings and desires are not the only things we wish to\ncommunicate. The natural language satisfies a child for a time, but as\nthe child grows he begins to _think_, then he feels the need of a more\neffective means of expressing himself. You can express your feelings to\na certain extent by the natural language. You can make one know that you\nare glad by the expression of the face, the attitude of the body or the\ntone of the voice. But could you make anyone understand _why_ you are\nglad, by these signs and gestures?\n\n+7.+ To express thoughts and ideas, man had to devise another sort of\nlanguage. So the language of _words_ grew up out of the need to\ncommunicate ideas to other people. As man's ability to think grew, so\nhis language grew. At first, this language was only a spoken language.\nThe ideas of one generation were handed down to the next by the spoken\nword. Gradually a crude form of writing was invented from which our\nwritten language has developed. This has made it possible to put the\nwisdom of the ages into books for the benefit of the world.\n\n+8.+ +Hence, language is the means of expressing thought and feeling.+\nIt has grown out of our need for expression.\n\n+A word is a symbol of an idea.+ It is a sound or combination of sounds\nwhich we use to represent an idea. The use of words makes it possible\nfor us to readily convey our thoughts to other people.\n\nThrough the medium of words we are able to communicate to others our\nthoughts, not only of the external world about us, but also of the\nmental world in which we live. We can tell of our loves, our hates, our\ndreams and our ideals. Animals find the natural language of looks and\ntones and gestures sufficient because they live almost wholly upon the\nphysical plane. But man lives in a mental world as well as in a physical\none, and must have a spoken and written language by which to express his\nthoughts.\n\n\n Exercise 1\n\nSelect from the following sentences those which it is possible to\nexpress by a look or tone or gesture, and those which can not be\nexpressed without words:\n\n 1. I am glad.\n 2. I am glad because men are struggling for freedom.\n 3. I am hungry.\n 4. I am hungry for the chance for an education.\n 5. Come.\n 6. Come, let us reason together.\n 7. I am afraid.\n 8. I am afraid that we must wait long for peace.\n 9. Go.\n 10. Go, search the world over for the truth.\n 11. I am disgusted.\n 12. I am disgusted with those who will not think for themselves.\n 13. I am tired.\n 14. I am tired of these petty squabbles among comrades.\n\n\n OUR EXPRESSION\n\n+9.+ Our knowledge of language opens up a new world to us. We\ncan communicate with those about us; we can open the storehouse of\nthe knowledge of the past as recorded in books, or as two of our writers\nhave expressed it:\n\n Have you ever rightly considered what the mere ability to read\n means--that it is the key which admits to the world of thought and\n fancy and imagination--to the company of saint and sage, of the\n wisest and wittiest at their wisest and wittiest moments--that it\n enables us to see with the keenest eyes, hear with the finest ears\n and listen to the sweetest voices of all time?--_Lowell_.\n\n Strip man of his books and his papers, and he becomes a mere slave,\n ignorant of his own resources, ignorant of his rights and\n opportunities. The difference between the free citizen of today and\n the savage of yesterday is almost entirely a thing of books. The man\n who dislikes books can never be entirely happy, and he who loves a\n good book can never be wholly miserable.--_Hillis_.\n\nHave you never felt that struggle within and the sense of defeat when\nyou have tried to make some one feel as you feel, understand as you\nunderstand, see some great truth as you see it, and could not find the\nwords with which to express your ideas?\n\n+10.+ The mastery of words gives; first, _the ability to understand the\nspoken or written thoughts of others_; second, _the ability to\nadequately express our own thoughts_; and third, _the ability to think\nclearly and to grow in our intellectual life_.\n\nA connected chain of reasoning is impossible without the knowledge of\nthe words that express the development of the ideas and the varying\nshades of meaning. To gain this mastery, you must know the words of our\nlanguage and their use. Words are the symbols of ideas and perform\ncertain functions in expressing our thoughts. This, simply stated, is\nall that the study of English Grammar comprises--_the study of English\nwords and their use in the expression of thought and feeling_.\n\n\n THE THOUGHT AND THE WORD\n\n+11.+ We have found that the invention of words grew out of the ability\nto _think_ and the need for expression. But we first _thought_! So, in\norder to express yourself clearly you must first _think_ clearly. Any\nthought can be simply and clearly expressed. When you read something\ndifficult of understanding, where the thought is buried under an\navalanche of words, you can be assured the writer was not thinking\nclearly. He did not have the perfect mastery of his thought. On the\nother hand, one may have a valuable thought in mind and not be able to\nexpress it because he does not have the words at his command. In the one\ncase, we have words and no idea; in the other, the idea and no words.\n\nThis study is intended to enable you to master words, the tools of\nexpression. In whatever work you are engaged, it was first necessary to\nlearn to use the tools with which you work. So, you must master the use\nof English words, the tools of your expression. You can in that way\nlearn to express your thoughts clearly and exactly. You will not need to\nresort to slang, or to the tiresome repetition of a few words.\n\nThe best of everything is none too good for you. It is your right, your\nheritage, and the best in the English language will bring you into the\ncompany and comradeship of the men and women who have striven and toiled\nfor humanity, who will talk to you of dreams and deeds worth while, who\nwill place in your hands the key to a new world.\n\n\n A COMPLETE THOUGHT\n\n+12.+ When we want to express a thought we use more than one word. Words\nare the symbols of ideas, but a thought is the expression of the\nrelation between ideas. For example, I say _man_, and you get an idea or\nan image in your mind of a man, but I have not said anything about any\nman. But if I say, _Man works_, then I have expressed a thought. I have\nrelated the idea of a man and the idea of work and have expressed a\ncomplete thought.\n\nSo we express our thoughts by _groups of words_. The very smallest group\nof words which will express a complete thought must, therefore, contain\ntwo words. If I say _men_, _fire_, _flowers_, and stop, you wonder what\nI mean, for I have not expressed a thought. Or, I might say, _work_,\n_burns_, _bloom_, and you would still be in the dark as to my meaning;\nbut, when I say, _Men work_, _Fire burns_, _Flowers bloom_, you\nunderstand, for I have told you my complete thought. I have put two\nwords together in a way to make sense; I have formed a sentence.\n\n+13.+ If we say, _Go_ or _Wait_, in the form of a command or entreaty,\nthe single word seems to make complete sense and to form a sentence in\nitself. But this is only because _you_, who are to do the going or the\nwaiting, is clearly implied. The words _go_ or _wait_, by themselves, do\nnot make sense or form a sentence unless they are uttered in the\ncommanding or beseeching tone of voice which makes you understand that\n_You go_ or _You wait_ is the intended meaning. With the exception of\nwords used in this way as a command or entreaty, it is always necessary\nto use at least two words to express a complete thought.\n\nBut will any two words make a sentence--express a complete thought?\n\n+14.+ Which of these combinations of words are sentences and which are\nnot?\n\n Busy men.\n Men travel.\n Snow flies.\n Blue sky.\n Red flag.\n Rustling trees.\n Workers strike.\n Bees sting.\n Grass grows.\n Cold winds.\n Green fields.\n Happy children.\n\n_Busy men_ does not express a complete thought. We are wondering _busy\nmen do what?_ But, _men travel_ is a complete thought. It makes sense\nand forms a sentence, and tells us what men _do_. In the words, _busy\nmen_, we have spoken the name of something but have made no assertion\nconcerning it. In the two words, _men travel_, we have spoken the name\n_men_ and we have told what they _do_.\n\nIf we were walking down the street together we might say:\n\n The street is crowded to-day.\n Does the open road attract you?\n See the jostling crowds.\n\nOr if we were discussing the class struggle, we might say:\n\n Two classes have always existed.\n To which class do you belong?\n Join your class in the struggle.\n\nIn every one of these six groups of words we have a complete thought\nexpressed. Each of these groups of words we call a sentence.\n\n+15.+ +A sentence is a group of words expressing a complete thought.+\n\n\n Exercise 2\n\nWrite in each blank space the word necessary to express a complete\nthought.\n\n Men...... ......fade.\n\n Leaves...... ......bloom.\n\n Water...... ......run.\n\n Fire...... ......write.\n\n Women...... ......grow.\n\n Children...... ......speak.\n\n\n SUBJECT AND PREDICATE\n\n+16.+ We have found that every sentence must have at least two words,\none word to name that about which something is said and another word\nwhich does the saying or makes the assertion. In the sentence, _Men\nwork_, we have these two parts; _men_ which is the part about which\nsomething is said, and _work_ which tells what men do.\n\n+The part about which something is said is called the subject.+\n\nIn this sentence, _Men work_, _men_, therefore, is the subject, for it\nnames that about which something is said.\n\n+17.+ +The part that asserts or says something about the subject is\ncalled the predicate.+\n\nTherefore in this sentence, _Men work_, _work_ is the predicate. In the\nfollowing sentences draw a single line under the subject and a double\nline under the predicate, thus, _~Birds~ =fly=_.\n\n Ships sail.\n Soldiers fight.\n Flowers fade.\n Horses neigh.\n Flags wave.\n Snow comes.\n War rages.\n Winds blow.\n Fish swim.\n\n+18.+ We may add other words to the subject or the predicate and so\nenlarge their meaning, as for instance we may say:\n\n The stately ships sail proudly away.\n The war in Europe rages furiously.\n The soldiers in the army fight like men gone mad.\n\nYet in every one of these sentences you will find the subject and the\npredicate,--_Ships sail_, _War rages_, _Soldiers fight_.\n\nEvery sentence must have a subject and a predicate, and it is a very\nimportant part of the study of sentences to be able to distinguish\nquickly and readily the subject and the predicate. Find that about which\nsomething is said, and that will always be the subject. Find that which\nis said about the subject, and that will be the predicate.\n\n+Every sentence must contain a subject and a predicate.+\n\n+The subject of a sentence names that about which something is said.+\n\n+The predicate tells that which is said about the subject.+\n\n\n Exercise 3\n\nIn the following sentences add other words to the subject and to the\npredicate to enlarge their meaning, then draw a single line under the\nsubject and a double line under the predicate:\n\n Ships sail.\n Tides flow.\n Stars shine.\n Rain falls.\n Children play.\n Nature sleeps.\n Waves break.\n War rages.\n Birds sing.\n\n\n Exercise 4\n\nIn the following sentences the subject and the predicate have other\nwords added to enlarge their meaning. Find the subject and predicate and\ndraw a single line under the subject and a double line under the\npredicate, as in the sentence,\n\n _The ~workers~ of the world =build= palaces for other people._\n\n 1. Our success lies in solidarity.\n 2. New occasions teach new duties.\n 3. Two classes exist in the world.\n 4. Labor creates all wealth.\n 5. The workers fight all battles.\n 6. Our time calls for earnest deeds.\n 7. Knowledge unlocks the door of life.\n 8. Ignorance bars the path to progress.\n 9. Few people think for themselves.\n 10. Hope stirs us to action.\n\n\n\n\n SPELLING\n\n LESSON 1\n\n\n+Spelling is the process of naming or writing in proper order the\nletters of a word.+ There is nothing that marks us so quickly as lacking\nin the qualities that go to make up a good education as our inability to\nspell the words most commonly used.\n\nSpelling in English is rather difficult. If each letter represented but\none sound, spelling would be an easy matter. Every word would be spelled\njust as it sounds. This is the goal of those who advocate phonetic\nspelling. Phonetic spelling simply means spelling according to sound.\nBut our alphabet does not have a letter for every sound.\n\nThere are some forty-two different sounds used in English words and we\nhave only twenty-six letters in the alphabet. Therefore some letters\nmust do duty for several sounds. Then we have words which contain\nletters which are not sounded at all when the word is pronounced, so,\nall in all, spelling is a matter of memorizing.\n\nThe best way to become an accurate speller is to read much, to observe\nclosely the forms of words and to write frequently. Always spell any\nword of which you are uncertain aloud several times and write it out\nseveral times. In this way you have aided the memory both through the\neye and through the ear. If you are not sure of the spelling of a word\ndo not use it until you have looked it up in the dictionary and made\nsure.\n\nThe words in this lesson are taken out of Lesson 1, Plain English\nCourse. There are thirty in all, five for each day of the week. (1) Look\nup the meaning in the dictionary. (2) Learn the correct spelling. (3)\nLearn the correct pronunciation. (4) Use the word in a sentence of your\nown construction. (5) Use it during the day in your conversation; strive\nto make it a part of your working vocabulary.\n\n +Monday+\n\n Mode\n English\n Grammar\n Expression\n Complete\n\n +Tuesday+\n\n Language\n Emotion\n Group\n Mastery\n Dictionary\n\n +Wednesday+\n\n Thought\n Symbol\n Ability\n Idea\n Knowledge\n\n +Thursday+\n\n Subject\n Predicate\n Vocabulary\n Practice\n History\n\n +Friday+\n\n Memory\n Sentence\n Write\n Right\n Purpose\n\n +Saturday+\n\n Propose\n Growth\n Learn\n Teach\n Pronounce\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 2\n\n\nDear Comrade:\n\nReview Lesson 1 before taking up this lesson. Do not try\nto learn by rote the contents of these lessons. Our endeavor is to make\nyou see the reason for every rule and definition before they are given.\nWe want you to see unfolding before you the development of language and\nthrough this evolution you can catch a glimpse of the developing life of\nman. Language like customs, religion, government, has grown with the\neconomic advancement of man. As man has evolved on the economic plane,\nthe material plane, as he has improved his means of providing for\nhimself food and clothes and shelter, he has developed a language suited\nto his needs.\n\nSo we can trace the growth of the race as we study the development of\nlanguage from the sign language of the primitive savage to the language\nof the philosopher of today by which he makes known to us the story of\nthe stars, and the innermost secrets of our hearts and minds.\nCivilization began with the invention of the phonetic alphabet and the\nuse of writing. So the study of language becomes not a dull and stupid\nconning of useless rules and formulas, but an absorbing study of a\nliving, growing, changing thing that mirrors forth the very life of man.\n\nThink while you study. As you look for the definition of words in your\ndictionary and realize how many shades of meaning we can express in\nwords, remember that this power is a heritage that comes to us from a\nlong past of incessant struggle.\n\nWe of to-day are also writing history in words. By our efforts we are\nadding new words to the language and giving old words a richer meaning.\n_Brotherhood_, _justice_, for example! The world is coming to understand\nthese glorious words more fully and giving them a new interpretation.\n\nYou will see a new beauty and glory in words after you have finished\nthis course and you will have a mastery of this wonderful language of\nours.\n\nWatch carefully the use of words in your reading. Especially this week\ndistinguish the nouns and verbs. Use your dictionary constantly and add\na few words to your vocabulary every day.\n\nWhenever there is a word used in these lessons which you do not\nthoroughly understand, look it up at once in your dictionary and master\nit then and there. Make a list in your note book of the words you look\nup and at the end of the week go over them again and see if you have\nthem clearly in mind. Watch also the pronunciation of the words. Do not\ntry to do everything all at once, nor should you be discouraged if your\nprogress seems slow. We approach the goal one step at a time and each\nstep takes us nearer and nearer. Just keep steadily at it, Comrade.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n KINDS OF SENTENCES\n\n+19.+ We have found that we use sentences to express our thoughts. But\nwe also find that we use these sentences in different ways for different\npurposes. Can you notice any difference in the following sentences?\n\n Two classes have always existed.\n To which class do you belong?\n Join your class in the struggle.\n\nWhen I say, _Two classes have always existed_, I am making a simple\nassertion, stating what I know or believe to be true.\n\nWhen I say, _To which class do you belong?_ I am asking a question.\n\nWhen I say, _Join your class in the struggle_, I am giving a command or\nmaking a request.\n\n+20.+ +These three kinds of sentences are called assertive,\ninterrogative and imperative.+\n\n+An assertive sentence states a fact or an opinion.+\n\n+An interrogative sentence asks a question.+\n\n+An imperative sentence gives a command, makes a request or\nexpresses a wish.+\n\n+21.+ Any of these three kinds of sentences may be exclamatory; that is,\nit may express surprise, excitement, impatience, or some other emotion.\nFor example:\n\n Hurrah! Freedom is coming!\n\nThis is an assertion expressed as an exclamation.\n\n Oh! Why should war continue?\n\nHere we have a question in the form of an exclamation.\n\n Come! Keep your courage up.\n\nIn this, we have a command, an imperative sentence, expressed in the\nform of an exclamation.\n\n+An exclamatory sentence expresses surprise, excitement or some other\nemotion.+\n\nIn these three forms of sentences, the assertive, the interrogative and\nthe imperative, together with the exclamatory, we are able to express\nevery thought and feeling which demands expression, either for practical\nor artistic purposes.\n\nThe sentence is the basis of spoken and written language and as we trace\nits development we trace the history of the evolution of man and the\ngrowth of his power of expression, as he has developed his powers of\nmind.\n\n+22.+ +Every sentence must begin with a capital letter.+\n\n+Every assertive and imperative sentence should end with a period.+\n\n+Every interrogative sentence should end with a question mark.+\n\nThe word in an exclamatory sentence which expresses strong emotion is\nfollowed by an exclamation point. The sentence itself if in\ninterrogative form should be followed by a question mark; if in the\nassertive or the imperative form it may be followed either by an\nexclamation point or a period.\n\n\n Exercise 1\n\nMark the assertive sentences among the following with an _a_ in the\nblank space. Mark the interrogative sentences with a _q_ for question;\nthe imperative sentences with a _c_ for command; and the exclamatory\nwith an _e_ for exclamation.\n\n 1. ...... Books are the true levelers.\n 2. ...... Put not your trust in princes.\n 3. ...... To err is human; to forgive divine.\n 4. ...... What are the rights of a child?\n 5. ...... Seize common occasions and make them great.\n 6. ...... Not until all are free, is any free.\n 7. ...... Freemen! Shall not we demand our own?\n 8. ...... Is a world of happiness but a Utopian dream?\n 9. ...... He who will not work, shall not eat.\n 10. ...... Strike at the polls for freedom!\n 11. ...... Do the majority want social justice?\n 12. ...... A friend is the hope of the heart.\n 13. ...... How beautiful is the vision of peace!\n 14. ...... Acquire the thinking habit.\n 15. ...... Is it glorious to die for our country?\n 16. ...... Lo! Women are waking and claiming their own!\n 17. ...... Claim your right to the best.\n 18. ...... What is the highest good?\n 19. ...... Workers of the world, unite!\n 20. ...... To remain ignorant is to remain a slave.\n\n\n WORDS--THEIR USES\n\n+23.+ We have learned from our study that we use sentences to express\nour thoughts. These sentences are made up of words; therefore we call\nwords _parts of speech_. Words are only fractions or parts of speech,\nand it is by combining them into sentences that we are able to express\nour thoughts.\n\nThere are many thousands of words in the English language. It would be\nimpossible for us to study each word separately. But these words, like\npeople, are divided into classes, so we can study each class of words.\nThese thousands of words are divided into classes much as people are, or\nrather as people ought to be; for words are divided into classes\naccording to the work which they do. In the Industrial Commonwealth\nthere will be no upper or lower class, but men will be divided into\ngroups according to the work which they do. There will be various\nindustrial groups, groups of agricultural workers, groups of clerical\nworkers, etc. So words are divided into classes according to the work\nwhich they do in helping us to express our ideas.\n\n+24.+ +Words are divided into kinds or classes according to their use in\nsentences.+\n\n+There are eight of these classes of words, called parts of speech.+\n\n\n THE NAMES OF THINGS\n\n+25.+ What a word _does_ determines what part of speech it is. When\nprimitive man, long ago, first began to use words, in all probability\nthe first words which he invented were those used to name familiar\nobjects about him. He invented a word for _man_, _boy_, _tree_,\n_animal_, etc. Gradually, all the things he met in his daily life\nreceived a name. About one half of the words in our language are of this\nclass, the _names_ of things.\n\nEvery word which is used as a name of something is called a _noun_. This\nword _noun_ is derived from the Latin word which means _name_, so it is\nquite the same thing as saying _name_. Notice the following sentences:\n\n Boys run.\n Fish swim.\n Horses neigh.\n Soldiers march.\n Flags wave.\n Flowers fade.\n Girls study.\n Winds blow.\n Men work.\n\nAll of the words used like _boys_, _girls_, _fish_, _horses_,\n_soldiers_, _flag_, _winds_, _flowers_ and _men_, are the names of\nobjects, therefore all of these words are _nouns_. The subject of a\nsentence is always a noun or a word used as a noun. However, we may use\nin a sentence many nouns besides the noun which is used as the subject,\nthe noun about which the statement is made. We will study the use of\nthese nouns later in our lessons.\n\n_The famous palace of the kings of the Moors, at Granada, in Spain, was\ncalled the Alhambra._ We have six nouns in this sentence, _palace_,\n_kings_, _Moors_, _Granada_, _Spain_ and _Alhambra_, but the noun\n_palace_ is the noun which is the subject--the noun which is the name of\nthat about which something is said. _Palace_ is the subject; and _was\ncalled_ is the predicate in this sentence.\n\n+26.+ +A noun is a word used as the name of something.+\n\nNow we want to learn to distinguish every word that is used as a name.\nPick out the nouns as you read your books and papers until you are able\nto tell every word which is used as a noun, the name of something.\n\nIn the following paragraph, the nouns are printed in italics. Carefully\nstudy these nouns:\n\nThe _fire_ in the _grate_, the _lamp_ by the _bedside_, the _water_ in\nthe _tumbler_, the _fly_ on the _ceiling_ above, the _flower_ in the\n_vase_ on the _table_, all _things_ have their _history_ and can reveal\nto us _nature's_ invisible _forces_.\n\n\n Exercise 2\n\nUnderscore every noun in the following quotation:\n\n The whole history of the earth has been one of gradual development, of\n progress, of slow and painful climbing through the ages. Not only have\n the hills and the mountains, the rivers and the stars, the trees and\n the cattle, the beasts and the birds, been developing; but man\n himself--his mind and his body--has been developing. Men are marvelous\n little creatures; they have weighed the sun in their balances,\n measured the stars and analyzed the light and beauty of the rainbow;\n they have sounded the depths of the ocean; they have learned how the\n sun and the mountains were born and the rivers were laid in their\n mighty beds; they have learned how the seas became salt, what the\n stars are made of. They have learned so much, and yet when it comes to\n matters of time and space, and law and motion, they still know so\n little. The only man who is conscious of his ignorance is he who has\n learned a great deal.--_McMillan_.\n\n\n WORDS THAT ASSERT\n\n+27.+ After the primitive man had invented names for the things about\nhim, probably his next step was to invent words of action. He very\nnaturally wanted to tell what all of these various things _did_. So the\nwords that tell what things do, the words of action, the words that\nassert, came into the language. A child follows much the same\ndevelopment. As you can readily observe, it first names the objects\nabout it, then learns the words that tell what these objects do.\n\nSo the words that tell what things _do_, become the second class of\nwords. These words we call _verbs_. The word _verb_, like the word\n_noun_, is taken into our language from the Latin. In Latin, the word\n_verbum_ means _the word_; and the verb is practically _the_ word in a\nsentence, for we cannot have a sentence without a verb. You may string a\nnumber of words together, but if you do not have an asserting word, you\nwill not have a sentence.\n\nNotice the following sentences:\n\n Men work.\n Flowers fade.\n Snow flies.\n Winds blow.\n\nIn these sentences, the words _work_, _fade_, _flies_ and _blow_, are\nthe words used to assert or say something of the subject, hence they are\nthe verbs in these sentences.\n\n+28.+ Sometimes it takes more than one word to express the action or\nmake the assertion. Notice the following sentences:\n\n The men are working.\n The boy has been studying.\n\nIn the first sentence it takes two words, _are working_, to make the\nassertion; in the second, three are required, _has been studying_. These\ngroups of words are called _verb phrases_.\n\n+29.+ +A verb is a word that asserts.+\n\n+A verb phrase is a group of words used as a single verb.+\n\nThe verb is perhaps the most difficult part of speech to master. It is\nnot hard to find the verb in short sentences, but in longer sentences it\nis sometimes difficult.\n\nFor example:\n\n The sun shines.\n The man walks.\n The boys strike.\n\nWe very easily see that _shine_, _walk_ and _strike_ are the verbs in\nthese sentences. But let us add other words, as for example:\n\n The sun shines brightly.\n The man walks for his health.\n The boys strike the dog.\n\nNow we are very apt to confuse the verb with the words which state _how_\nand _why_ the action is performed, or the _object_ towards which the\naction is directed. But in these sentences, _shine_ and _walks_ and\n_strike_ are still the verbs, just as in the first sentences. The verb\nasserts the action; the other words merely give additional information\nabout _how_ or _why_ or _upon what_ the action is performed.\n\n+30.+ Another thing which makes it difficult for us to distinguish verbs\nin English is that the same word may be used both as a noun and as a\nverb; but always remember that words are separated into classes\naccording to the work which they do. When a word is used as a _name_ it\nis a _noun_; when it is used as an _asserting_ word it is a _verb_. Note\nthe following sentences:\n\n The _play_ made the child tired.\n The children _play_ in the yard.\n\nIn the first sentence _play_ is a noun, the subject of the verb _made_.\nIn the second sentence _play_ is the verb, telling what the children\n_do_. Always classify words according to the work which they perform in\nthe sentence. This will help you very much in finding your verb.\n\n+31.+ Then we have some verbs which do not assert action but express\nrather a connection or relation between the subject and some other word\nor words. For example:\n\n The dog belongs to the man.\n The girl is happy.\n\nIn these sentences _belongs_ and _is_ are the verbs. _Belongs_ asserts\nor shows the relation between _the dog_ and _the man_. _Is_ shows the\nrelation between _the girl_ and _happy_. If we simply say _girl_ and\n_happy_, we do not show any connection between them or make any\nstatement relating the two, but when we say, _The girl is happy_, we are\nasserting something, and the word _is_ makes the assertion.\n\nOr when we say, _The girl was happy_, or _The girl will be_ or _may be\nhappy_, in each of these cases, it is the verb or verb phrase _was_ or\n_will be_ or _may be_, that asserts or shows the relation between the\nsubject _girl_ and the descriptive word _happy_. You will observe that\nthe verbs _will be_ and _may be_ are composed of more than one word and\nare _verb phrases_.\n\nWe will study the verb in succeeding lessons, but let us remember from\nthis lesson that the word or group of words that makes the assertion in\nthe sentence is the verb. Remember too that every sentence must contain\na verb.\n\nGet this basic principle firmly fixed in mind that what a word _does_\ndecides what it _is_--to what part of speech it belongs, and that every\nclass of words fulfills its own function in sentence building.\n\n+32.+ Remember:--\n\n+Every sentence must have a subject and a predicate.+\n\n+Every sentence must express a complete thought.+\n\n+Every sentence must contain a verb.+\n\n+A noun is the name of something.+\n\n+A verb is a word that asserts.+\n\n+What a word does determines what it is.+\n\nStudy carefully the following quotation. The verbs are printed in\n_italics_.\n\n Slowly, painfully, _proceeds_ the struggle of man against the power of\n Mammon. The past _is written_ in tears and blood. The future _is_ dim\n and unknown, but the final outcome of this world-wide struggle _is_\n not in doubt. Freedom _will conquer_ slavery, truth _will prevail_\n over error, justice _will triumph_ over injustice, the light _will\n vanquish_ the darkness; and humanity _will rise_ in the glory of\n universal brotherhood.--_Warren_.\n\n\n Exercise 3\n\nUnderscore all verbs and verb phrases in the following quotation:\n\n+The Dream of Labor+: Ours is not the cause of one class, of one sex, of\none tribe, of one city, of one state, of one continent.\n\nIt is the wish for a better world where Man shall be Man; where the\nbeast shall become subdued; where everything shall lead to complete\ndevelopment; where the good of each shall be bound up in the good of\nall; where all shall feel the sorrows of each and shall run to his\nrescue.\n\nA glimpse of this ideal takes us into the Land of Promise, where peace\nand plenty shall reign supreme; where brothers shall no longer battle\namong themselves, but for one another; where the atmosphere shall be\nladen with love, the love that saves; where the hate that kills shall be\nunknown; where heart and brain shall work together and shall make life\nbetter and more complete; where the fullness of life shall be for all\nand where men and women shall be as happy at their work as little\nchildren at their play.\n\nThe mere glimpse into that land makes life worth living, makes work\nworth doing, makes dreams worth dreaming, gives us hope and faith--the\nfaith we need in the labor for our cause, the faith which shall help us\nwin.--_Oscar Leonard_.\n\n\n Exercise 4\n\nWe have found that there are a number of words in English which may be\nused either as nouns or verbs, depending upon the function they serve in\nthe sentence. In the following sentences underscore the nouns with a\nsingle line, the verbs with two lines:\n\n 1. They _man_ the boats.\n 2. The _man_ has a boat.\n 3. The women _pass_ this way.\n 4. They held the _pass_ for hours.\n 5. Little children _work_ in the mines.\n 6. The _work_ of the world is done by machinery today.\n 7. The armies will _cross_ the bridge.\n 8. He built a _cross_ of rude stones.\n 9. The leopard cannot _change_ its spots.\n 10. We will force a _change_ in the law.\n\n\n Exercise 5\n\nIn the following poem, mark every noun and every verb and verb phrase.\nYou will find the verb phrases in several places divided by the word\n_not_, as in _I do not obey_. _Do obey_ is the verb phrase. We will\nlearn to what part of speech _not_ belongs a little later.\n\n I DO NOT OBEY, I THINK.\n\n \"Captain, what do you think,\" I asked,\n \"Of the part your soldiers play?\"\n The Captain answered, \"I do not think--\n I do not think, I obey.\"\n\n \"Do you think your conscience was meant to die,\n And your brains to rot away?\"\n The Captain answered, \"I do not think--\n I do not think, I obey.\"\n\n \"Do you think you should shoot a patriot down,\n And help a tyrant slay?\"\n The Captain answered, \"I do not think--\n I do not think, I obey.\"\n\n \"Then if this is your soldier's code,\" I cried,\n \"You're a mean, unmanly crew;\n And with all your feathers and gilt and braid,\n I am more of a man than you;\n\n \"For whatever my lot on earth may be\n And whether I swim or sink,\n I can say with pride, 'I do not obey--\n I do not obey, I think.'\"\n\n --_Ernest Crosby_.\n\n\n\n\n SPELLING\n\n LESSON 2\n\n\nThe twenty-six letters in the English alphabet are divided into vowels\nand consonants. A vowel is a letter which represents a sound of the\nhuman voice but slightly interrupted by the vocal organs. The vowels are\n_a_, _e_, _i_, _o_ and _u_. All of the remaining letters of the alphabet\nare consonants. A consonant is a letter which represents a sound of the\nhuman voice greatly obstructed by the vocal organs. Consonant is from\nthe Latin _con_, meaning _with_, and _sono_--_I sound_. So it means\nliterally _I sound with_.\n\nThe consonants are produced by union of the breath with the vocal\norgans. The consonant sounds are so called because they are always\n\"sounded with\" a vowel; they are used only in combination with vowels in\nforming words or syllables.\n\nIn English a consonant alone never forms a word or a syllable. Sound the\ndifferent consonants _b_, _c_, _d_, _f_, _g_, _h_, _j_, _k_, _l_, _m_,\n_n_, _p_, _q_, _r_, _s_, _t_, _v_, _x_ and _z_, by themselves and you\nwill see how the sound of the breath is obstructed or changed by the use\nof the vocal organs--the lips, the tongue, the teeth, etc.--in making\nthese various sounds.\n\n_W_ and _y_ are sometimes vowels and sometimes consonants. _W_ and _y_\nare vowels when they are used with another vowel representing a vowel\nsound as in _awe_, _new_, _joy_, _eye_, etc. _Y_ is sometimes used as a\nvowel by itself as in _by_, _cry_, etc. _W_ and _y_ are consonants when\nthey are used at the beginning of a syllable or before a vowel in the\nsame syllable as in _wine_, _twine_, _yield_ and _year_.\n\nLook up the meaning of the words in this week's lesson. Master the\nspelling and use them in sentences of your own construction.\n\n +Monday+\n\n Reason\n Evolution\n Justice\n Thorough\n Beauty\n\n +Tuesday+\n\n Assertive\n Review\n Surprise\n Basis\n Separate\n\n +Wednesday+\n\n Interrogative\n Period\n Capital\n Capitol\n Function\n\n +Thursday+\n\n Example\n Contain\n Imperative\n Question\n Speech\n\n +Friday+\n\n Method\n Various\n Familiar\n Industry\n Alphabet\n\n +Saturday+\n\n Travel\n Sense\n Cents\n Sail\n Sale\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 3\n\n\nDear Comrade:\n\nIn this lesson we are taking up a short study of the different parts of\nspeech. In later lessons we will study each part of speech more\nthoroughly but this lesson covers the ground quickly and briefly. It is\nsufficient, however, to form a basis for our understanding of the\nevolution of language.\n\nYou will see, as you study this lesson, how each part of speech has been\nadded to meet a growing need. There are many, many thousand words in the\nEnglish language, but they can all be grouped under these eight parts of\nspeech, for they all answer in some way to one of these great needs.\n\nThe object in studying grammar, as in studying any other science, is not\nto fill one's mind with a great many unrelated facts--facts which may or\nmay not prove useful to one hereafter. The object of all study is to\ndevelop one's power of observation and one's ability to think. Added to\nthis must be the practical ability to make use of this knowledge. Here\nthe study of grammar has an advantage over the study of every other\nscience. It deals with words, something which we use every day.\n\nYou do not need any laboratory or expensive apparatus in order to study\ngrammar. All that you need lies ready to your hand. And in addition to\nthis the knowledge which you gain is something which is of practical use\nto every man and woman no matter what their work, no matter what their\nplace or position in life may be.\n\nRemember that dogmatism has no place in the study of grammar.\n\"Grammarians are the guardians, not the authors, of language.\" We do not\nsay, \"You should say this or that, or you violate a rule of grammar,\"\nbut we say \"The common usage among those who use good English is thus\nand so.\" If we do not believe that the common usage is the best usage,\nthen we follow the democratic method of seeking to change the common\nusage into that which we consider the more sensible way. Thus, those who\nadvocate simplified spelling have not sought to pass a law whereby every\none should be compelled to spell words exactly as they sound, but they\nhave striven to influence our writers and people in general to use this\nmore sensible way of spelling words.\n\nSo _think_ while you study. Do not try to learn rules and formulas. See\n_why_ the rules and formulas exist. Once having seen this you do not\nneed to learn them--you know them already. The study of any language is\nan intellectual discipline of the highest order.\n\nSo apply yourself diligently to this most interesting study and you will\nsee that the result of this application will affect your daily life in\nevery particular.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n WORDS ADDED TO NOUNS\n\n+33.+ When man began to invent words to express his ideas of the world\nin which he lived, we have found that probably the first need was that\nof names for the things about him. So we have nouns. The second need was\nof words to tell what these things _do_, and so we have verbs. But\nprimitive man soon felt the need of other classes of words.\n\nThe objects about us are not all alike. For example, we have a word for\nman, but when we say _man_ that is not sufficient to describe the many\ndifferent kinds of men. There are tall men, short men, white men, black\nmen, strong men, weak men, busy men, lazy men. There are all sorts of\nmen in the world, and we need words by which we can describe these\ndifferent types and also indicate which man we mean.\n\n+34.+ So we have a class of words which are called adjectives.\n_Adjective_ is a word derived from the Latin. It comes from the Latin\nword _ad_, meaning _to_, and the Latin word _jecto_, which means _to\nthrow_; hence an adjective is a word _thrown to_ or _added to_ a noun.\n\nIf you will stop to think for a moment, you will see that it is by their\nqualities that we know the things about us. Some men are strong, some\nare weak, some are tall, some are short. These qualities belong to\ndifferent men. And we separate or group them into classes as they\nresemble each other or differ from one another in these qualities.\nThings are alike which have the same qualities; things are unlike whose\nqualities are different. Apples and oranges are alike in the fact that\nboth are round, both are edible. They are unlike in the fact that one is\nred and one is yellow; one may be sour and the other sweet. So we\nseparate them in our minds because of their different qualities; and we\nhave a class of words, _adjectives_, which describe these various\nqualities.\n\n+35.+ We use adjectives for other purposes also. For example, when we\nsay _trees_, we are not speaking of any particular trees, but of trees\nin general. But we may add certain adjectives which point out particular\ntrees, as for example: _these_ trees, or _those_ trees, or _eight_ trees\nor _nine_ trees. These adjectives limit the trees of which we are\nspeaking to the particular trees pointed out. They do not express any\nparticular qualities of the trees like the adjectives _tall_ or\n_beautiful_ express, but they limit the use of the word _trees_ in its\napplication. So we have our definition of the adjective.\n\n+36.+ +An adjective is a word added to a noun to qualify or limit its\nmeaning.+\n\n\n Exercise 1\n\nUnderscore all of the adjectives in the following quotation. Notice also\nthe nouns and verbs in this quotation.\n\n Yet fearsome and terrible are all the footsteps of men upon the earth,\n for they either descend or climb.\n\n They descend from little mounds and high peaks and lofty altitudes,\n through wide roads and narrow paths, down noble marble stairs and\n creaky stairs of wood--and some go down to the cellar, and some to the\n grave, and some down to the pits of shame and infamy, and still some\n to the glory of an unfathomable abyss where there is nothing but the\n staring, white, stony eye-balls of Destiny.\n\n They descend and they climb, the fearful footsteps of men, and some\n limp, some drag, some speed, some trot, some run--they are quiet,\n slow, noisy, brisk, quick, feverish, mad, and most awful in their\n cadence to the ears of the one who stands still.\n\n But of all the footsteps of men that either descend or climb, no\n footsteps are so fearsome and terrible as those that go straight on\n the dead level of a prison floor, from a yellow stone wall to a red\n iron gate.--From _The Walker_. _Giovannitti_.\n\n\n WORDS ADDED TO VERBS\n\n+37.+ From our study, you see how our classes of words grew out of man's\nneed of them in expressing his thoughts. And notice also how the many\nthousands of words in our language can all be grouped under these few\nclasses. We _name_ the things about us; we invent words to tell what\nthese things _do_; we have another class of words which _describe_ the\nthings which we have named; and now we come to a fourth class of words\nfor which we also find great need.\n\nWhen we come to tell what things _do_, we find that we need words which\nwill tell us _how_ or _where_ or _when_ these things are done. Notice\nthe following sentences:\n\n The men work busily.\n The men work late.\n The men work now.\n The men work here.\n The men work hard.\n The men work well.\n The men work inside.\n The men work more.\n\nWe would have a complete sentence and express a complete thought if we\nsaid simply, _The men work_, but each of these words which we have\nadded, like _busily_, _hard_, _late_, etc., adds something to the\nmeaning of the verb. These words add something to the action which is\nasserted by the verb, for they show _how_ and _when_ and _where_ and\n_how much_ the men work.\n\n+38.+ We call this class of words _adverbs_, because they are added to\nverbs to make the meaning more definite, very much as adjectives are\nadded to nouns. Adverb means literally _to the verb_.\n\nAn adverb will always answer one of these questions: _how?_ _when?_ _how\nlong?_ _how often?_ _how much?_ _how far?_ or _how late?_ If you want to\nfind the adverbs in your sentences just ask one of these questions, and\nthe word that answers it will be the adverb.\n\n+39.+ An adverb may be used also with an adjective. Notice the following\nsentences:\n\n The book is _very_ long.\n _Too_ many people never think.\n\nNotice here that the adverbs _very_ and _too_ modify the adjectives\n_long_ and _many_.\n\n+40.+ Adverbs may also be used with other adverbs. Notice the following\nsentences:\n\n He speaks _very_ distinctly.\n He walks _too_ slowly.\n\nHere the adverbs _very_ and _too_ are used with the adverbs _distinctly_\nand _slowly_, and add to their meaning. We will study more fully in\nlater lessons concerning both the adjective and the adverb, but we can\nsee by this brief study why adverbs were added as a class of words, a\npart of speech, for they are absolutely necessary in order to describe\nthe action expressed by verbs, and also to add to the meaning of\nadjectives and other adverbs. Hence we have our definition of an adverb.\n\n+41.+ +An adverb is a word that modifies the meaning of a verb, an\nadjective, or another adverb.+\n\n\n Exercise 2\n\nUnderscore all adverbs in the following sentences:\n\n 1. He will not come today.\n 2. Here and now is the day of opportunity.\n 3. Very slowly, but even then entirely too rapidly, the fire crept\n forward.\n 4. The room was very quiet and still.\n 5. He was too weary to go farther.\n 6. One must learn to feel deeply and think clearly in order to\n express himself eloquently.\n 7. Ferrer stood there, so calmly and so bravely facing the firing\n squad.\n 8. He was condemned to death because he stood uncompromisingly and\n courageously for the education of the masses.\n 9. Ferrer understood thoroughly that the schools of today cleverly\n and effectively adapt their teaching to maintain the present\n system of society.\n 10. He said \"The school imprisons the children physically,\n intellectually and morally.\"\n\n\n WORDS USED IN PLACE OF NOUNS\n\n+42.+ Now we come to study another class of words which are also very\nnecessary in order to express our ideas. Suppose you had just arrived in\na strange town and you wanted to find the way to a friend's house. You\ninquire of a stranger, \"Can you tell me who lives in the house on the\ncorner?\"\n\nNotice the words _you_ and _me_ and _who_. You could not call the\nstranger by name for you do not know his name, and hence you say _you_.\nAnd if you used your own name instead of _me_, he would not recognize\nit, and you would both be puzzled to find a substitute for that little\nword _who_.\n\nIf you knew the stranger and he knew your name, you might say, \"Can Mr.\nSmith tell Mr. Jones what person lives in the house on the corner.\" But\nthis would sound very stilted and unnatural and awkward. So we have\nthese little words like _you_ and _me_ and _who_, which we use _in place\nof nouns_. These words are called pronouns. This word is taken from the\nLatin also. In the Latin the word _pro_ means _in place of_. So the word\npronoun means literally in place of a noun.\n\n+43.+ +A pronoun is a word that is used in place of a noun.+\n\nThese pronouns are very useful little words. They save us a great deal\nof tiresome repetition. Notice the awkwardness of the following:\n\n The workers will succeed in gaining the workers' freedom if the\n workers learn solidarity.\n\nAnd yet this would be the way we would have to express this idea\nif we did not have pronouns. Instead we say:\n\n The workers will succeed in gaining their freedom if they learn\n solidarity.\n\n+44.+ We will study the pronoun in detail in later lessons, but we can\nreadily recognize these words which are used in place of nouns. The most\ncommon pronouns are:\n\n I\n you\n he\n she\n it\n we\n they\n me\n him\n her\n us\n them\n my\n your\n his\n her\n its\n our\n their\n that\n which\n who\n whose\n whom\n what\n\n\n Exercise 3\n\nUnderscore the pronouns in the following story:\n\n A man in South Africa picked up a small piece of stone. It was dirty\n and Rough.\n\n \"Make me beautiful,\" said the stone.\n\n \"I shall have to hurt you,\" said the man.\n\n \"Well, if it hurts me, I will bear it,\" said the stone.\n\n So the man took it to a clever craftsman, who put it into a tight\n vise, and cut it with his sharp instrument.\n\n \"Oh!\" cried the stone.\n\n And he ground it till the dust fell all about it.\n\n \"Oh!\" cried the stone.\n\n And he polished it very hard.\n\n \"Oh!\" cried the stone.\n\n And then he set it in a crown and sent it to the Queen. On a sunny day\n she wore her crown, and the stone--it was a diamond--sparkled in long\n rays of crimson and green and yellow and silvery white. And all the\n people greeted their queen. She showed them her crown and they praised\n the beautiful stone.\n\n The training was hard, but the improvement was glorious.\n\n\n PREPOSITIONS\n\n+45.+ Notice the following sentences:\n\n I want the book _on_ the box.\n I want the book _under_ the box.\n I want the book _in_ the box.\n I want the book _beside_ the box.\n I want the book _behind_ the box.\n I want the book _beyond_ the box.\n\nDo you notice any word in these sentences which does not belong to any\nof the classes of words which we have studied? _I_ is a pronoun, _want_\nis a verb, _the_ is an adjective, _book_ is a noun, _the_ is an\nadjective, _box_ is a noun; but the words, _on_, _under_, _in_,\n_beside_, _behind_ and _beyond_ are not nouns, verbs, adjectives,\nadverbs or pronouns.\n\nYet would it be possible to express the meaning in these sentences\nwithout these words? Read the sentences without them, and you will see\nthat no one could tell the relation which you wish to express between\nthe _book_ and the _box_. And you will notice too that each word\nexpresses a different relation, for it means one thing to say _on the\nbox_ and another thing to say _in the box_, and so through the list.\n\n+46.+ The words which are used to show this relation are called\n_prepositions_. The groups of words introduced by the preposition, like\n_on the box_ and _in the box_, and so on, are called prepositional\nphrases. The noun which follows a preposition as _box_ follows the\nprepositions _in_, _on_, _beside_, _beyond_, etc., is called the\n_object_ of the preposition.\n\n_Preposition_ is a word which comes into our language from the Latin. It\nis formed from the Latin _pre_, which means _before_, and the Latin verb\nwhich means _to place_, so preposition means literally _to place\nbefore_. It is given this name because it is placed before the noun or\npronoun which is its object. Therefore our definition of a preposition\nis as follows:\n\n+47.+ +A preposition is a word that shows the relation of its object to\nsome other word.+\n\n+48.+ Either a noun or a pronoun may be the object of a preposition.\nNotice the following sentences:\n\n Bring the book to me.\n Lay the book on the table.\n He will speak to you.\n I will speak to the man.\n\nIn these sentences the noun _table_ is the object of the preposition\n_on_; the pronoun _me_ is the object of the preposition _to_; and in the\nlast two sentences the pronoun _you_ and the noun _man_ are the objects\nof the preposition _to_.\n\n+49.+ There are not many prepositions in the language and they are\neasily learned and easily distinguished. Here is a list of the most\ncommon and the most important prepositions. Use each one in a sentence.\n\n at\n across\n around\n about\n among\n above\n against\n along\n behind\n beside\n between\n below\n beyond\n by\n before\n beneath\n down\n for\n from\n in\n into\n off\n on\n over\n to\n toward\n under\n up\n upon\n with\n within\n without\n\n\n Exercise 4\n\nUnderscore the prepositions in the following sentences:\n\n He went to the door and looked out upon the field.\n Over the river and through the woods, to Grandfather's house we go.\n He saw them in the distance as they were coming toward him.\n They went along the road, across the bridge, and hid among the trees\n at the foot of the hill.\n They came from Minneapolis down the river by boat.\n The war between the classes is a struggle against exploitation.\n The army was intrenched behind the barricades before dawn.\n His claim was within the law but without justice.\n\n\n CONJUNCTIONS\n\n+50.+ We have found that the preposition is a very important connective\nword. It connects two words and shows what one of them has to do with\nthe other, but the preposition is not the only connective word which we\nuse in English. We have another part of speech which performs an\nimportant function as a connective word. Notice the following sentence:\n\n Men and women struggle for their rights.\n\nCan you find a word in this sentence which is a connective word besides\nthe preposition _for_? Did you notice that little word _and_? The noun\n_men_ and the noun _women_ are both subjects of the verb _struggle_, and\nthey are joined by this little connective word _and_. If we did not have\nthis word we would have to use two sentences to express our thought,\nthus:\n\n Men struggle for their rights.\n Women struggle for their rights.\n\nBut with the use of this connective word _and_ we can combine these\ntwo sentences and express it all in one sentence:\n\n Men _and_ women struggle for their rights.\n\nThis word is used in a different manner from the preposition. The\npreposition connects two words and makes one modify the other. When we\nsay, _Get the book on the table_, the phrase _on the table_ designates\nthe book just as much as if we had said, _Get the green book_. So the\nuse of the preposition enables us to show the relation between two words\nand to make one word describe or modify the other.\n\n+51.+ This little word _and_ in the sentence, _Men and women struggle\nfor their rights_, is a connective word also, but it connects two words\nthat are used in the same way, so it is a different sort of connective\nword from the preposition. Words used in this way are called\n_conjunctions_. Conjunction is a word which is taken from the Latin,\nbeing made up of the Latin word _con_, which means _together_, and the\nLatin verb _juncto_, which means _to join_. So conjunction means\nliterally _to join together_.\n\n+52.+ +A conjunction is a word that connects sentences or parts of\nsentences.+\n\nNotice the following sentence:\n\n The class struggle is waged on the political field and on the\n industrial field.\n\nHere we have the conjunction _and_ connecting the two phrases _on the_\n_political field_ and _on the industrial field_. Without the use of this\nconnective word, we would have to use two sentences to express these two\nthoughts:\n\n The class struggle is waged on the political field.\n The class struggle is waged on the industrial field.\n\n+53.+ So a conjunction may be used to connect phrases as well as words.\n\nNow notice the following sentences:\n\n He will speak. I will listen.\n He will speak, _and_ I will listen.\n He will speak, _but_ I will listen.\n He will speak, _if_ I will listen.\n He will speak, _therefore_ I will listen.\n He will speak, _because_ I will listen.\n He will speak, _until_ I will listen.\n\n+54.+ These _sentences_ are joined by different conjunctions, and the\nconjunction used alters the meaning of the sentence.\n\nThe conjunction is a very useful part of speech. Without it we would\nhave many disconnected sentences requiring tiresome repetition of the\nsame words. Like prepositions, there are not many conjunctions in\nEnglish and they are readily recognized.\n\n+55.+ We will study about these conjunctions at length in later lessons.\nIf you consult the following list of those most commonly used, you can\neasily pick out the conjunctions in your reading:\n\n and\n as\n as if\n after\n although\n as soon as\n because\n besides\n before\n but\n either\n for\n hence\n in order that\n lest\n neither\n nor\n or\n since\n still\n so\n then\n though\n that\n than\n therefore\n till\n until\n unless\n while\n whether\n yet\n\nThe seven classes of words which we have studied make up all of our\nsentences. The hundreds of words which we use in forming our sentences\nand expressing our thoughts belong to these seven classes. They are\neither nouns, pronouns, verbs, adverbs, adjectives, prepositions or\nconjunctions.\n\n\n Exercise 5\n\nUnderscore the conjunctions in the following sentences. Notice whether\nthey connect words or phrases or sentences.\n\n 1. We cannot win unless we are organized.\n 2. Books and music are true friends.\n 3. Men, women and children work under conditions neither proper nor\n just.\n 4. We must educate and organize.\n 5. The workers on the farms and in the factories must be united.\n 6. Winter has come and the birds are going South.\n 7. We have been ignorant, therefore we have been exploited.\n 8. We must learn before we can teach.\n 9. We do not understand the situation, because we do not know the\n facts.\n 10. Do you know whether these statements are true or false?\n\n\n IT CAN BE DONE\n\n Somebody said that it couldn't be done,\n But he, with a chuckle, replied\n That \"maybe it couldn't,\" but he would be one\n Who wouldn't say so till he tried.\n So he buckled right in, with a trace of a grin\n On his face. If he worried he hid it.\n He started to sing as he tackled the thing\n That couldn't be done--and he did it.\n\n Somebody scoffed, \"Oh, you'll never do that;\n At least no one ever has done it.\"\n But he took off his coat and he took off his hat,\n And the first thing we knew he'd begun it;\n With the lift of his chin, and a bit of a grin,\n Without any doubting or quiddit,\n He started to sing as he tackled the thing\n That couldn't be done--and he did it.\n\n There are thousands to tell you it can not be done;\n There are thousands to prophesy failure;\n There are thousands to point out to you, one by one,\n The dangers that wait to assail you.\n But buckle right in, with a lift of your chin,\n Then take off your coat and go to it;\n Just start in to sing as you tackle the thing\n That \"can not be done,\"--and you'll do it.\n\n\n INTERJECTIONS\n\n+56.+ There is another class of words which we use _with_ sentences, but\nwhich are really not _parts_ of the sentences. They are emotional\nexpressions which seem to belong more to the natural language than to\nthe invented language. For example:\n\n Oh! You hurt me!\n Aha! Now I have you.\n\n_Oh_, used in this way, is very apt to sound like a groan, and _aha_\nlike a shout of triumph. These words do not really belong in the\nconstruction of the sentence. The sentence would be complete without\nthem, but they are thrown in to express the emotion which accompanies\nthe thought. We call expressions such as these _interjections_.\nInterjection is from the Latin and means literally _thrown into the\nmidst of_. It comes from the Latin word _inter_, which means _between_,\nand the Latin verb _jecto_, _to throw_, so it literally means _to throw\nbetween_.\n\nSome of these words imitate sounds, as for example:\n\n Bang! There goes another shot.\n Ding-dong! There goes the first bell.\n\nWe do not use interjections very frequently in writing on scientific\nsubjects that express deep thought, but you will find them often used in\npoetry, fiction, oratory or any emotional writing. Therefore we have our\ndefinition of an interjection:\n\n+57.+ +An interjection is an exclamatory word or phrase used to express\nfeeling or to imitate some sound.+\n\n+58.+ Following is a list of commonly used interjections. Use them in\nsentences of your own.\n\n oh\n hello\n bravo\n ahoy\n aha\n hurrah\n bow wow\n ssh\n alas\n hist\n whirr\n pshaw\n fie\n whoa\n ding-dong\n rub-a-dub\n\n\n Exercise 6\n\nMark the interjections in the following sentences. Notice those which\nexpress emotion and those which imitate sound.\n\n 1. Oh! Is it possible.\n 2. Hurrah! We have good news at last.\n 3. Whirr! Whirr! goes the giant machine.\n 4. Come! Keep up your courage.\n 5. What! I cannot believe it.\n 6. Courage! We shall yet win.\n 7. Bravo! Let those words ring down the centuries.\n 8. Ding-dong! the bells ring out the hour!\n\n\n\n\n SPELLING\n\n LESSON 3\n\nSince there are forty-two elementary sounds used in the formation of our\nwords and only twenty-six letters to represent these sounds, some of\nthese letters must necessarily represent more than one sound.\n\nOf the forty-two elementary sounds, eighteen are vowel sounds, but we\nhave only five vowels with which to represent these sounds, so each\nvowel has several different sounds.\n\nTherefore we must have a key to pronunciation to indicate the various\nsounds which are represented by these letters used in forming the words.\nWhen you look up words in your dictionary you will find the vowels\nmarked by certain signs to indicate the pronunciation. These signs are\ncalled diacritical marks.\n\nThe following table gives the diacritical marks for the vowels. Study\nthis table and learn to pronounce the words you look up. When you have\ndetermined the correct pronunciation of the word, repeat it over to\nyourself aloud a number of times until you have accustomed your ear to\nthe correct pronunciation.\n\nDifferent dictionaries use different keys to pronunciation. This table\nis taken from the dictionary which we are using in connection with this\ncourse--Winston's New Universal Self-Pronouncing Dictionary.\n\n Key to Pronunciation\n\n [=a] as in _late_, _fade_.\n a-umlaut as in _mar_, _father_.\n [.a] as in _mask_, _dance_.\n a as in _cat_, _had_.\n aw as in _awl_, _fall_.\n [=e] as in _he_, _feet_.\n [~e] as in _her_, _verge_.\n e as in _let_, _men_.\n [=i] as in _line_, _time_.\n i as in _tin_, _little_.\n [=o] as in _vote_, _home_.\n o-circumflex as in _orb_, _form_.\n o as in _lot_, _odd_.\n oi as in _oil_, _join_.\n [=oo] as in _moon_, _school_.\n oo as in _cook_, _foot_.\n ou as in _out_, _house_.\n [=u] as in _mute_, _unit_.\n u as in _nut_, _drum_.\n\n\nThe spelling lesson for this week is composed of words containing the\ndifferent vowel sounds. Look up in your dictionary and mark all the\n_a's_ in Monday's lesson, all the _e's_ in Tuesday's lesson, all the\n_i's_ in Wednesday's lesson, all the _o's_ in Thursday's lesson, and all\nthe _u's_ in Friday's lesson. In Saturday's lesson note the use of _w_\nand _y_ as vowels.\n\n +Monday+\n\n Pause\n Adjective\n Lazy\n Quality\n Advance\n\n +Tuesday+\n\n Resemble\n Descend\n Adverb\n Interjection\n Complete\n\n +Wednesday+\n\n Limit\n Define\n Distinct\n Imprison\n Civilize\n\n +Thursday+\n\n Form\n Footsteps\n Proof\n Report\n Common\n\n +Friday+\n\n Union\n Under\n Unusual\n Summer\n Commune\n\n +Saturday+\n\n Comply\n Employ\n Vowel\n News\n Lawful\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 4\n\n\nDear Comrade:\n\nWe are studying in this lesson a most interesting part of our language,\nthe words that are the names of things. If we could trace these names of\nthings and the order and time of their coming into the language of men\nwe would have a progressive history of mankind. Way back yonder in the\ndim dawn of history, men lived upon fruit and nuts. They had no\nknowledge of the use of fire and could not use foods that required\ncooking. They communicated with one another by signs. Then they\ndiscovered fire and invented the bow and arrow. They could now use fish\nand flesh for food and they commenced to use articulate speech. This\nstage has been called the Middle Stage of Savagery. With the invention\nof the bow and arrow, began the third stage of savagery which merged\ninto the first stage of barbarism with the invention of pottery.\n\nThere are three stages of barbarism before we come to the beginning of\nthe era of civilization which begins with the use of the phonetic\nalphabet and the production of literary records. All tribes that have\nnever attained the art of pottery are classed as savages and those who\npossess this art but have never attained a phonetic alphabet and the use\nof writing are classed as barbarians. Civilization began with the spoken\nand written language and it has been well said that all that separates\nus from savagery is a wall of books. It is upon the accumulated wisdom\nof the past that we build. Without this we would be helpless.\n\nSo these various names of things have come to us with developing\nevolving life. As the men of the past gained a knowledge of the use of\nfire, as they learned to bake the clay and make various utensils; to\nheat and forge the iron into weapons; to conquer nature in all her\nphases, to feed the race, to clothe the race, to shelter the race more\nadequately, our language has grown in volume, strength and beauty.\n\nThe study of words and their uses is of great importance to you. Master\nthe few rules necessary and watch your words daily. We are living in an\nage full of wondrous things and yet many of us have almost as limited a\nvocabulary as the men of those bygone days, who had never dreamed of the\nmarvels that are commonplace to us.\n\nAs you use your dictionary watch closely the meaning of the words and\nchoose the words that most aptly express your ideas. Listen to good\nEnglish spoken as often as you can. _Read_ good English. Mark the\ndifference between good and bad English and gradually you will find\nyourself using good English naturally and continually.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n CLASSES OF NOUNS\n\n+59.+ We have learned that the words in a sentence are classified\naccording to the work which each word does. The words which assert are\ncalled verbs; the words which are the names of things are called nouns.\nBut now we shall see that these words are again divided into classes\naccording to the _special_ work which they perform. Just as we may\ngather the people of the world into one great class, the working class,\nthen classify them according to the industry in which they work, thus\nsome are farmers, some teachers, some factory workers; then each class\nmay be subdivided according to the special work which they perform, as\ntruck farmers, high school teachers, machinists, etc.\n\nSo we find that nouns are divided into classes according to their\nmeaning in the sentence.\n\nIn the sentence, _Lincoln was a man of the people_, we have two nouns\nreferring to the same person, _Lincoln_ and _man_, but they are\ndifferent kinds of names. The word _man_ is a name that may apply to any\none of a million persons but the name _Lincoln_ applies to one person\nonly. Some nouns, then, represent a thing as being of a certain kind or\nclass, without showing which particular one is meant. Other nouns are\nnames given to designate a particular individual. These are called\n_common_ and _proper_ nouns.\n\n+60.+ +A proper noun is a special name meant for only one person, place\nor thing.+\n\nAll other nouns are common nouns.\n\n+A common noun is a name which belongs to all things of a\nclass of objects.+\n\n+Every proper noun should begin with a capital letter.+\n\nIndicate the proper nouns in the following list by drawing a line under\nthe letters that ought to be capitals:\n\n king\n month\n city\n france\n dog\n virginia\n war\n wilson\n november\n doctor\n colonel\n napoleon\n chicago\n governor\n independence day\n freedom\n ocean\n atlantic ocean\n thanksgiving\n thanksgiving day\n uncle william\n thursday\n week\n general sherman\n karl marx\n union\n labor\n united mine workers\n newspaper\n the daily call\n\nWrite the special or _proper_ names of several individuals in each of\nthe following classes:--as city,--Chicago, New York, etc.\n\nRiver, king, author, country, state, inventor, martyr, month, book,\ncollege.\n\n\n COLLECTIVE NOUNS\n\n+61.+ Some nouns are the names of groups or collections of things and\nare called collective nouns.\n\nMany soldiers taken together form collectively an _army_--a number of\nsheep form a _drove_. Many of these group or collective nouns will\nreadily occur to your mind.\n\n+A collective noun is one that in the singular form, denotes a number of\nseparate persons or things.+\n\n\n Exercise 1\n\nOpposite each of the following collective nouns, write the name of the\nindividuals represented by the collection; as an army of _soldiers_; a\nswarm of _bees_; a flock of _birds_.\n\n A gang of......\n A committee of......\n A herd of......\n A drove of......\n A hive of......\n A corps of......\n A suite of......\n A group of......\n A class of......\n A multitude of......\n\nFill the following blanks with appropriate collective nouns.\n\n A......of horses.\n A......of sailors.\n A......of wolves.\n A......of savages.\n A......of singers.\n A......of girls.\n A......of ships.\n A......of quail.\n A......of birds.\n A......of workers.\n\n\n ABSTRACT NOUNS\n\n+62.+ When primitive man began to name the objects about him, doubtless\nhe first named the things which he could see, hear, taste, smell and\ntouch,--the objects which he could perceive by the five senses. Then\ngradually he came to understand that these objects had certain qualities\nwhich he could consider apart from the object itself.\n\nHe hunted among the stones to find those which were suitable for making\nhis arrow-heads. For this purpose he needed the hardest stone which he\ncould find, so _hardness_ became something which he could think of as\nsomething apart from the object itself.\n\nHe saw the men about him and found a name for them. Then he knew that\nsome men were stronger than others, so _strength_ was a quality which he\ncould consider apart from the man himself.\n\nThese men performed certain actions; they ran, they climbed,--so\n_running_ and _climbing_ became actions which he could think of as\nsomething apart from any individual.\n\nHe noted too that men lived in certain conditions; for example, some men\nwere free, some were slaves, so he came to think of _slavery_ and\n_freedom_ as conditions which could be thought of as something apart\nfrom the individual.\n\nSo we draw away, or separate certain ideas; the _quality_ from the thing\nwhich has it and the _action_ from the thing which does it and the\n_condition_ from the thing which is in it. These nouns which are used to\ndescribe these qualities, actions or conditions are called _abstract_\nnouns. Abstract is a word derived from the Latin _abs_, _away from_, and\n_tractus_, _drawn_, so it literally means _drawn away from_.\n\nThe nouns which are names of things which we can see, hear, taste, smell\nand touch or perceive by any of the five senses are called _concrete_\nnouns.\n\n+63.+ +A concrete noun is the name of an object which may be perceived\nby one or more of the five senses.+\n\n+An abstract noun is the name of a quality, a condition or an action.+\n\n+64.+ You remember we found in the study of adjectives that we have a\nclass of adjectives which are used to describe the qualities of objects,\nas for example--_good_, _noble_, _honest_, _true_, _wise_, etc. Since\nabstract nouns are the names of qualities, many of our abstract nouns\nare formed from adjectives. Study carefully the following list of\nadjectives and nouns. Note that the word is an _adjective_ when it is\nused with a noun to _describe_ certain qualities. It is a _noun_ when it\nis used by itself to _name_ that quality.\n\n +Adjectives+ +Abstract Nouns+\n\n 1. honest honesty\n 2. pure purity\n 3. true truth\n 4. strong strength\n 5. wise wisdom\n 6. good goodness\n 7. bold boldness\n 8. just justice\n 9. silent silence\n 10. wide width\n 11. patient patience\n 12. stupid stupidity\n\n+65.+ You will notice that another use of abstract nouns is to name\nactions. The verb is the part of speech which expresses action,\ntherefore many abstract nouns are formed from verbs. Notice the\nfollowing list:\n\n +Verbs+ +Abstract Nouns+\n\n 1. learn learning\n 2. invent invention\n 3. choose choice\n 4. defend defense\n 5. try trial\n 6. judge judgment\n 7. read reading\n 8. please pleasure\n 9. elect election\n 10. move motion\n\n+66.+ An abstract noun is also the name of a condition. These nouns are\nderived from the concrete noun which is the name of the person or thing\nwhich is _in_ the condition.\n\n +Concrete Nouns+ +Abstract Nouns+\n\n 1. slave slavery\n 2. friend friendship\n 3. thief theft\n 4. man manhood\n 5. child childhood\n 6. leader leadership\n 7. hero heroism\n 8. martyr martyrdom\n\n\n Exercise 2\n\nForm abstract nouns from the following adjectives, verbs and nouns.\n\n long\n simple\n rapid\n lovely\n loyal\n fresh\n prove\n sing\n run\n behave\n believe\n reflect\n write\n child\n agent\n infant\n rascal\n clerk\n president\n coward\n\n\n NUMBER FORM\n\n+67.+ So we find that we classify our nouns according to the special\nwork which they do. Now sometimes we find it necessary to change the\nform of the noun to make it express our thought. Thus we say, _book_,\n_man_, _boy_, _knife_, when we wish to express the idea of only one of\neach object mentioned. But when we wish to express the idea of more than\none of them, we say, _books_, _men_, _boys_, _knives_.\n\nWe say, _The boy calls_; _the boys call_. The form of the noun _boy_ is\nchanged by adding an _s_ to it. The meaning has also changed. _Boy_\ndenotes one lad; _boys_ denotes two or more lads. Any change in form and\nmeaning of words is called _inflection_. The change to denote more than\none object is called _number_. The word _boy_, denoting _one_ is in the\n_singular number_; the word _boys_, denoting _more than one_ is in the\n_plural number_.\n\n+68.+ +Inflection is a change in the form of a word to denote a\ndifferent application or use.+\n\n+Number is the form of a noun which shows whether it denotes one or more\nthan one.+\n\n+The singular number denotes one thing.+\n\n+The plural number denotes more than one thing.+\n\nThere are a few rules governing the formation of plurals which we must\nknow, and these rules are of great assistance in correct spelling.\n\n+69.+ Most nouns form their plural by adding _s_--thus:\n\n boat\n boats\n\n day\n days\n\n book\n books\n\n boy\n boys\n\nLong ago in early English all plurals were formed by adding _es_, and\nyou will read in the first translation of the Bible, for instance, such\nwords as _bird-es_, _cloud-es_. Later the _e_ was dropped and _s_ added\nto the singular without an increase of syllables. But when the singular\nends in an _s_ sound, the original syllable _es_ is retained, for two\nhissing sounds will not unite.\n\n+70.+ So nouns ending in _s_, _x_, _z_, _sh_ or soft _ch_, form the\nplural by adding _es_ to the singular. These words end with a sound so\nmuch like that of _s_ that we cannot pronounce the plural easily without\nmaking another syllable. Thus:\n\n class\n classes\n\n tax\n taxes\n\n topaz\n topazes\n\n wish\n wishes\n\n ditch\n ditches\n\n+71.+ In words ending with the _s_ sound but with a final _e_, only _s_\nis added to form the plural, but in pronouncing the word we then have\ntwo syllables, thus:\n\n house\n houses\n\n place\n places\n\n size\n sizes\n\n cage\n cages\n\n niche\n niches\n\n+72.+ Letters, figures, signs, etc., are made plural by adding an\napostrophe and the letter _s_ ('s), thus:\n\n Cross your t's and dot your i's.\n Do you know the table of 4's?\n\nWhile most of our nouns form their plural in this regular way by adding\n_s_ or _es_, there are some nouns that form their plural by some other\nchange in the form of the word.\n\n+73.+ Notice the following list of words and their plurals:\n\n fly\n flies\n city\n cities\n key\n keys\n day\n days\n story\n stories\n enemy\n enemies\n tray\n trays\n boy\n boys\n\nThese nouns all end in _y_, yet they form the plural differently. Some\nsimply add _s_ and the rest change the _y_ to _i_ and add _es_. Can you\ndiscover the reason?\n\nWherever the _y_ is preceded by a vowel, as _e_ in _key_, _a_ in _tray_,\n_o_ in _boy_, the plural is formed by adding _s_. But when the _y_ is\npreceded by a consonant, as _l_ in _fly_, _r_ in _story_, _t_ in _city_,\nand _m_ in _enemy_, the _y_ is changed to _i_ and _es_ added in forming\nthe plural.\n\n+If the singular ends in _y_ after a consonant, change _y_ to _i_\nand add _es_ in the plural.+\n\n+74.+ There are thirteen nouns ending in _f_ and three in _fe_ which\nform the plural in _ves_. They are:\n\n beef beeves\n calf calves\n elf elves\n half halves\n leaf leaves\n loaf loaves\n self selves\n sheaf sheaves\n shelf shelves\n staff staves\n thief thieves\n wharf wharves\n wolf wolves\n knife knives\n life lives\n wife wives\n\nAll other nouns in _f_ or _fe_ are regular; adding only _s_, to form the\nplural.\n\n+75.+ About forty nouns ending in _o_ after a consonant form the plural\nin _es_. The most common ones are:\n\n buffalo\n cargo\n potato\n tomato\n \n veto\n cargo\n echo\n calico\n embargo\n hero\n mulatto\n mosquito\n motto\n tornado\n volcano\n torpedo\n flamingo\n\nMost nouns ending in _o_ form the plural regularly, adding only _s_, as\n_pianos_, _banjos_, _cameos_, etc.\n\n+76.+ A few words form their plurals by a change in the word and without\nadding _s_ or _es_.\n\nThe most common of these words are:\n\n man men\n goose geese\n ox oxen\n woman women\n foot feet\n mouse mice\n brother brethren\n tooth teeth\n child children\n louse lice\n\n+77.+ Proper nouns, when made plural, generally follow the same rule as\ncommon nouns. Thus we write:\n\n All the Smiths, the Joneses, both the Miss Johnsons, one of the Dr.\n Davidsons, and the Mrs. Wilsons, were present.\n\nBut to prevent the confusion and misunderstanding which might arise in\nchanging the form of a proper noun, we do not change its form in writing\nthe plurals; for example:\n\n There were eight Henrys, kings of England.\n The two Marys reigned in the kingdom.\n\nIt would be confusing to say _eight Henries_, the _two Maries_.\n\nThe title is made plural when several are referred to, thus:\n\n Mr. Hayes The Messrs. Hayes\n Miss Smith The Misses Smith\n\n+78.+ The title is made plural when used with several names, thus:\n\n Messrs. Brown and White.\n Generals Lee and Grant.\n Drs. Long and Larson.\n\n+79.+ In the case of nouns formed of two or more words, when\nthe compound word is so familiar that the parts are not thought\nof separately the _s_ is added to the whole compound word, as\n_four-in-hands_; _forget-me-nots_; _court-yards_; _spoonfuls_;\n_green-houses_; etc. But when one of the parts is more important than\nthe others, the _s_ is added to the more important part, thus:\n\n mothers-in-law\n commanders-in-chief\n hangers-on\n men-of-war\n by-standers\n attorneys-at-law\n passers-by\n step-sons\n\n+80.+ We have many words in our language taken from other languages.\nThey do not form the plural in these languages as we do, and some of\nthese words retain their foreign plurals. Some of the most commonly used\nof these nouns are the following:\n\n +Singular+ +Plural+\n\n alumnus alumni\n analysis analyses\n axis axes\n datum data\n erratum errata\n ellipsis ellipses\n appendix appendices\n bacterium bacteria\n basis bases\n crisis crises\n parenthesis parentheses\n radius radii\n terminus termini\n hypothesis hypotheses\n larva larvae\n madame mesdames\n memorandum memoranda\n phenomenon phenomena\n stratum strata\n thesis theses\n\n+81.+ The following nouns are treated as singular: _news_, _pains_\n(meaning care), _acoustics_, _mathematics_, _economics_, _ethics_,\n_molasses_, _physics_, _politics_, and other nouns ending in _ics_\nexcept _athletics_. With these always use the s-form of the verb. For\nexample:\n\n The news _is_ distorted. Not, The news _are_ distorted.\n Economics _is_ an important study. Not, Economics _are_, etc.\n\n+82.+ The following nouns are always plural:\n\n alms\n annals\n amends\n antipodes\n bellows\n billiards\n clothes\n dregs\n eaves\n fireworks\n hysterics\n measles\n mumps\n matins\n nippers\n nuptials\n oats\n premises\n proceeds\n pincers\n riches\n rickets\n suds\n scissors\n thanks\n tidings\n tongs\n trousers\n vitals\n victuals\n vespers\n\nWith all these nouns always use the form of the verb which is used with\nthe plural subject. Thus:\n\n Alms are given.\n Riches are easily lost.\n\n+83.+ The following nouns have the same form for both plural and\nsingular, _corps_, _cannon_, _deer_, _grouse_, _heathen_, _hose_,\n_means_, _odds_, _series_, _sheep_, _species_, _swine_, _vermin_,\n_wages_. You can tell whether the singular or plural is meant by the\nmeaning of the sentence. For example:\n\n_The cannon is loaded._ Here we are speaking of _one_ cannon.\n\n_The cannon used in the war are of tremendous size._ Here we know are\nmeant all the big guns used in the war.\n\nWhen you say, _The sheep is lost_, we know you mean _one_ sheep, but\nwhen you say, _The sheep are in the pasture_, we know you mean the\nentire drove.\n\n+84.+ When preceded by a numeral, the following nouns have the same form\nfor both singular and plural. Without the numerals, the plural is formed\nby the adding of _s_; _brace_, _couple_, _dozen_, _hundred_, _pair_,\n_score_, _thousand_, _yoke_. For example:\n\n Thousands enlisted.\n Three thousand enlisted.\n Dozens came at my call.\n Two dozen came when I called.\n\n\n GENDER\n\n+85.+ All of the changes we have studied so far have been for the\npurpose of indicating number; but among the nouns that name living\nbeings, many change to show to which sex the object named belongs. These\nnouns change in form to distinguish between the masculine and the\nfeminine. This is called _gender_.\n\n +Gender is the distinction in words that denotes sex.+\n\n +The nouns that denote females are called feminine nouns.+\n\n +The nouns that denote males are called masculine nouns.+\n\n+86.+ The feminine form is generally made by the addition of _ess_\nto the masculine form. Thus:\n\n prince princess\n master mistress\n host hostess\n count countess\n tiger tigress\n lion lioness\n actor actress\n god goddess\n\n+87.+ Names of things without sex are, of course, of neither gender, and\nare called _neuter nouns_. Neuter means literally _neither_. Such nouns\nas _mountain_, _iron_, _river_, _chair_, are neuter.\n\nSometimes the feminine is an entirely different word from the masculine.\nThus:\n\n king queen\n lord lady\n man woman\n youth maiden\n sir madam\n stag hind\n\n+88.+ Many nouns that denote living beings apply alike to male and\nfemale, and are said to be of _common gender_. As woman enters more and\nmore into the business world and pursues the same occupations as man,\nthe change in form to denote the feminine is used less frequently, and\nwhat we have called the masculine form is used for both sexes, thus:\n\n_Poet_, _waiter_, _doctor_, _editor_--these nouns are used for both men\nand women.\n\n\n POSSESSIVE FORM\n\n+89.+ There is just one more change made in the form of a noun, and that\nis when we wish to show who or what owns or possesses a thing. Thus we\nwrite:\n\n John's book.\n The boy's hat.\n\nAnd since this form of the noun denotes possession, it is called the\n_possessive form_. Some grammarians call this the possessive case.\n\nThe possessive form of nouns is made by adding an apostrophe and _s_,\n('s); thus, _day's_, _lady's_, _girl's_, _clerk's_.\n\nTo plural nouns ending in _s_ add only an apostrophe; thus, _days'_,\n_ladies'_, _girls'_, _clerks'_.\n\nWhen plural nouns do not end in _s_, their possessive forms are made by\nadding the apostrophe and _s_, the same as singular nouns, thus:\n\n They make _men's_ and _women's_ shoes.\n\n+90.+ In words which end with a sound that resembles that of _s_, the\napostrophe with _s_ forms an additional syllable. Thus:\n\n James's (pronounced James-ez.)\n Mr. Lynch's (pronounced Lynch-ez.)\n\nThe only exception to the rule occurs when the addition of another _s_\nwould make too many hissing sounds, then we add the apostrophe alone.\nThus:\n\n For goodness' sake.\n In Jesus' name.\n\n+91.+ In forming the possessive of compound nouns, the possessive sign\nis always placed at the end, thus:\n\n My son-in-law's sister.\n The man-of-war's cannon.\n\n+92.+ When we wish to show that a thing belongs to two or more persons\nwho are joint owners of it, we add the possessive sign to the last word\nonly, thus:\n\n Carson, Price and Scott's store.\n Mason and Hamlin's pianos.\n\nIf it is a separate ownership that we wish to denote, we place the\npossessive sign after each name, thus:\n\n Bring me John's and Mary's books.\n Lee's and Grant's armies met in battle.\n\nRemember that the noun has just _three_ changes in form, one for the\nplural number, one to denote gender and one for the possessive form.\nWatch carefully your own language and that of your friends and note if\nthese changes are correctly made.\n\n\n Exercise 3\n\nWrite the plural form of each of the following:\n\n ax\n beef\n chief\n hero\n knife\n T\n hoof\n man-of-war\n axis\n basis\n cherry\n leaf\n son-in-law\n Mr. Smith\n thief\n Doctor Wood\n alley\n buffalo\n chimney\n staff\n Frenchman\n Miss Brown\n ox\n spoonful\n alto\n calf\n cargo\n two\n 3\n tooth\n foot\n turkey\n\n\n Exercise 4\n\nUnderscore the nouns in the following:\n\nHow many abstract nouns?\n\nHow many concrete?\n\nHow many singular?\n\nHow many plural?\n\n\n FIVE AND FIFTY\n\n _Charlotte Perkins Gilman_\n\n If fifty men did all the work\n And gave the price to five;\n And let those five make all the rules--\n You'd say the fifty men were fools,\n Unfit to be alive.\n\n And if you heard complaining cries\n From fifty brawny men,\n Blaming the five for graft and greed,\n Injustice, cruelty indeed--\n What would you call them then?\n\n Not by their own superior force\n Do five on fifty live,\n But by election and assent--\n And privilege of government--\n Powers that the fifty give.\n\n If fifty men are really fools--\n And five have all the brains--\n The five must rule as now we find;\n But if the fifty have the mind--\n Why don't they take the reins?\n\n\n Exercise 5\n\nSelect all the nouns in the following. Write their singular, plural and\npossessive forms. Decide whether they are abstract or concrete, common\nor proper or collective, masculine, feminine or neuter.\n\n Brother!\n\n Whoever you are, wherever you are on all the earth, I greet you.\n\n I extend to you my right hand.\n\n I make you a pledge.\n\n Here is my pledge to you:--\n\n I refuse to kill your father. I refuse to slay your mother's son. I\n refuse to plunge a bayonet into the breast of your sister's brother. I\n refuse to slaughter your sweetheart's lover. I refuse to murder your\n wife's husband. I refuse to butcher your little child's father. I\n refuse to wet the earth with blood and blind kind eyes with tears. I\n refuse to assassinate you and then hide my stained fists in the folds\n of _any_ flag.\n\n Will you thus pledge me and pledge all the members of our working\n class?--_Kirkpatrick._\n\n\n\n\n SPELLING\n\n LESSON 4\n\n\nSome of our consonants also have more than one sound. We have also\ncertain combinations of consonants which represent one sound. This\ncombination of two letters to represent one sound is called a digraph,\nas _gh_, in _cough_, _ch_ in _church_. A digraph may either be a\ncombination of two consonants or of two vowels or of a vowel and a\nconsonant. The following table contains the consonants which have more\nthan one sound:\n\n c--k as in _cat_\n c--s as in _vice_\n g--j as in _ginger_\n g--_hard_ as in _go_\n s--sh as in _sure_\n s--zh as in _usual_\n s--_soft_ as in _also_\n s--z as in _does_\n x--_soft_ as in _extra_\n x--gz as in _exist_\n\nThe following table gives the digraphs most commonly used:\n\n ng--as in _ring_, _tongue_\n ch--as in _church_ and _much_\n ch--k as in _chasm_\n ch--sh as in _chagrin_\n th--as in _then_, _those_\n th--as in _thin_ and _worth_\n ce--sh as in _ocean_\n ci--sh as in _special_\n dg--j as in _edge_\n gh--f as in _rough_\n ph--f as in _sylph_\n qu--kw as in _quart_\n qu--k as in _conquer_\n sh--as in _shall_\n si--sh as in _tension_\n si--zh as in _vision_\n ti--sh as in _motion_\n\n\nThe use of these digraphs gives us a number of additional sounds. Notice\nthe use of the consonants which have more than one sound and also the\ndigraphs in the spelling lesson for the week. Mark the consonants and\ndigraphs.\n\n +Monday+\n\n Commence\n Certain\n General\n Gradual\n Sugar\n\n +Tuesday+\n\n Soldier\n Season\n Pleasure\n Exact\n Exercise\n\n +Wednesday+\n\n Singular\n Chemistry\n Chapter\n Machine\n Changing\n\n +Thursday+\n\n Theory\n Thither\n Ocean\n Racial\n Budget\n\n +Friday+\n\n Philosophy\n Enough\n Quorum\n Bouquet\n Phonetic\n\n +Saturday+\n\n Permission\n Asia\n Attention\n Marshall\n Martial\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 5\n\n\nDear Comrade:\n\nWe want to say just a word about the lesson assignment. This has been\narranged on a schedule of days merely to assist you in systematizing\nyour time and making the most of the leisure at your disposal. It is not\nintended that you should slavishly follow it. We thoroughly believe in\nindividuality and all that contributes toward its development. But we\nare also confident that many foolish things are done in the name of\nliberty. Whenever we set ourselves to the performance of any task we\nnecessarily limit our activities in some other direction. Power comes by\nconcentration of force. Whenever we combine with others for the\naccomplishment of any purpose, it becomes necessary to have some plan of\naction and we give and take for the end which we have in view. The\nmusician because he follows the law of harmony in music has not given up\nhis liberty. He has only found a new freedom which enables him to make\nglorious music where only discord reigned before. System in our work\ndoes not mean loss of liberty or of individuality but only finding a\nchannel through which individuality can flow into the great ocean of\nreal freedom.\n\nSo use this suggestive lesson assignment to meet your own need and find\nexpression for your real individuality in full freedom.\n\nThis is the first of several lessons concerning verbs. The verb is\nperhaps the most difficult part of speech to thoroughly master, so do\nnot be discouraged if there are some parts of this lesson you do not\nunderstand. Succeeding lessons will clear up these difficult points.\nKeep your eyes open as you read every day, and be careful of your\nspelling and pronunciation.\n\nSome of us mis-spell the common words which we see and use every day. In\na student's letter we recently noted that, with our letter before him in\nwhich the word was printed in large type and correctly spelled, he\nspelled College, _Colledge_.\n\nDo not be satisfied with half-way things or less than that which is\nworthy of you. Demand the best for yourself. Read aloud this little\nverse from the Good Grey Poet, Walt Whitman:\n\n \"O, the joy of a manly self-hood;\n To be servile to none, to defer to none, not to any tyrant known or\n unknown,\n To walk with erect carriage, a step springy and elastic,\n To look with calm gaze or with a flashing eye,\n To speak with a full and sonorous voice out of a broad chest,\n To confront with your personality all the other personalities of the\n earth.\"\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n THE WORD THAT ASSERTS\n\n+93.+ You remember when we studied sentences we found that we could not\nhave a sentence without a verb or a word that asserts. The life of a\nsentence is the verb, for without the verb we cannot assert, question or\ncommand. It was on account of this importance that the Romans called the\nverb, _verbum_, which meant the word. Verbs, like nouns, are divided\ninto classes.\n\n+94.+ In some of our sentences the verb alone is enough to make a\ncomplete assertion, but in other sentences we use verbs that need to be\nfollowed by one or more words to complete the assertion. Notice the\nfollowing sentences:\n\n The boy ran.\n The boy found the ball.\n The earth revolves.\n The earth is round.\n\nDo you notice any difference in the verbs used in these sentences?\nNotice that the verbs _ran_ and _revolves_ make the complete assertion\nabout their subjects. Notice the verbs _found_ and _is_. These are not\ncomplete without the addition of the words _ball_ and _round_. If we say\n_The boy found_, _The earth is_, you at once ask, _The boy found WHAT?_\n_The earth is WHAT?_ The sense is incomplete without the addition of\nthese words _ball_ and _round_. A part of the thought is unexpressed;\nbut when we say _The boy found the ball_, _The earth is round_, the\nsense is complete.\n\nSo we have two classes of verbs, _COMPLETE AND INCOMPLETE VERBS_.\n\n+95.+ +An incomplete verb is one that requires the addition of one or\nmore words to complete its meaning.+\n\n+The word or words added to an incomplete verb to complete its meaning\nare called the complement.+\n\n+A complete verb is one that requires no complement to complete its\nmeaning.+\n\n+96.+ You can readily tell when a verb is complete and when it is\nincomplete by asking the question _What?_ If you put the question _what_\nafter the verb, and it makes a sensible question the verb is\n_incomplete_. For example:\n\n Farmers raise--_what?_\n The employer discharged--_what?_\n We were--_what?_\n The earth is--_what?_\n\nIf the question _what?_ does not make sense after the verb, then the\nverb is _complete_. For example:\n\n The sun shines.\n Water flows.\n Men work.\n\nThe question _what_ after these verbs would not make sense, as:\n\n The sun shines--_what?_\n Men work--_what?_\n Water flows--_what?_\n\nSo these verbs are _complete_ verbs.\n\n+97.+ The same verb, however, may be complete or incomplete, according\nto the way in which it is used. For example:\n\n The corn grows.\n The farmer grows corn.\n\nIn the sentence, _Corn grows_, _grows_ is a complete verb. You could not\nsay _The corn grows--what?_ for it does not grow anything. It merely\ngrows, and the verb _grows_ in this sense is a complete verb. But in the\nsentence, _The farmer grows corn_, you are using the verb _grows_ in a\nslightly different sense. It is an _incomplete verb_, for you do not\nmean, _The farmer grows_, but you mean that _the farmer grows CORN_.\n\n\n Exercise 1\n\nIn the following sentences, underscore the complete verbs with one line,\nthe incomplete with two lines. Ask the question _what?_ after each verb\nto determine whether it is complete or incomplete.\n\n He returned today.\n He returned the book.\n The rose smells sweet.\n He smelled the rose.\n The trees shake in the wind.\n The wind shakes the trees.\n The ship plows through the waves.\n The farmer plows the field.\n The birds sing sweetly.\n They sang the Marseillaise.\n He worries over the matter.\n The matters worry him.\n The table feels rough.\n He feels the rough surface.\n It tastes bitter.\n He tasted the bitter dregs.\n\n\n Exercise 2\n\nUse the following verbs in sentences as both complete and incomplete\nverbs, as for example, _The snow melts._ _The sun melts the snow._\n\n melts\n write\n stopped\n answer\n rings\n fall\n see\n strike\n\n\n INCOMPLETE VERBS\n\n+98.+ Do you notice any difference in the two verbs in the following\nsentences:\n\n The boy found the ball.\n The earth is round.\n\nIn the sentence, _The boy found the ball_, the word _ball_ tells _what_\nthe boy _found_. The verb _found_ expresses action; it tells what the\nboy _does_. _Boy_ is the subject of the action--the one who performs the\naction. The word _ball_ is the _object_ of the action. It shows the\nreceiver of the action. In the sentence, _The earth is round_, _is_ does\nnot express action. The earth is not doing anything, it simply _is_. The\nverb _is_ expresses a state or condition and is incomplete, for you do\nnot know what state or condition is expressed until we add the other\nword or words which describe the state or condition.\n\nNotice the following sentences:\n\n The earth is round.\n The earth is our home.\n The earth is a sphere.\n The earth is large.\n\nThe words _round_, _sphere_, _home_ and _large_, describe the earth\nwhich is the subject of the verb _is_.\n\n+99.+ So we have two classes of incomplete verbs, the verbs that express\naction and the verbs that express state or condition. The verbs which\nexpress action are called _transitive_ verbs. Transitive is a word\nderived from the Latin, and means literally _passing over_.\n\n+100.+ So a transitive verb describes an action which _passes over_\nfrom the subject to the object. As for example in the sentence, _The\nplayer struck the ball_, _struck_ is a transitive verb--a verb of\naction--describing the action of the subject, _player_, which passes\nover to the object, _ball_. Therefore we have our definition of a\ntransitive verb:\n\n+A transitive verb is one that has a complement showing who or what\nreceives the action expressed by the verb.+\n\n+The complement or word that denotes the receiver of the action\nexpressed by a transitive verb is called the object.+\n\nWhen you look up the meaning of verbs in your dictionary, you will find\nsome verbs marked _v.i._, and some verbs marked _v.t._ _V.t._ is the\nabbreviation for _verb transitive_. Whenever you find a verb marked\n_v.t._, you know that it is a transitive verb, a verb of action, one\nwhich requires an object to complete its meaning. _V.i._ is the\nabbreviation for _verb intransitive_. Some grammarians use the term\n_intransitive_ to include both _complete_ and _copulative_ verbs. We\nhave used the terms complete and incomplete because they are much\nsimpler and clearer in describing the two general classes of verbs, but\nyou will remember that when you find verbs marked _v.i._ in the\ndictionary that these include _complete_ and _copulative_ verbs.\n\n+101.+ Now notice these sentences:\n\n The earth is round.\n The earth is a sphere.\n\nIn these sentences the verb _is_ does not express action, but _connects_\nor _couples_ the complements _round_ and _sphere_ with the subject\n_earth_. Verbs used in this way are called _copulative_ verbs, from the\nword _copula_, which means to _complete_ or to _connect_. The words\n_round_ and _sphere_ are not the objects of the verb, for they do not\ndescribe the receiver of any action. They are the words which describe\nthe state or condition expressed in the verb _is_, and are called the\nattribute complement of the verb.\n\nYou note that this complement may be either an adjective or a noun. In\nthe sentence, _The earth is round_, the adjective, _round_, is used as\nthe complement; in the sentence, _The earth is a sphere_, the noun,\n_sphere_, is used as the complement. So we have our definition of\ncopulative verbs.\n\n+102.+ +Verbs that express state or condition are called copulative\nverbs.+\n\n+The word or words that complete the meaning of an incomplete verb\nexpressing state or condition, are called the complement, or attribute\ncomplement.+\n\nThere are only a few of these copulative verbs. All forms of the verb,\n_be_; like _am_, _is_, _are_, _was_ and _were_, and the verb phrases\nlike _must be_, _can be_, _will be_, _shall be_, _have been_, _had\nbeen_, etc.; and the verbs _seem_, _appear_, _become_, _look_, _feel_,\n_taste_, _sound_ and _smell_, are the principal copulative verbs.\n\n\n Exercise 3\n\nStudy carefully the following sentences. Note whether the complement of\nthe copulative verb is an adjective or a noun. Draw one line under each\n_adjective_ used as a complement and two lines under each _noun_ used as\na complement.\n\n The day is beautiful.\n I am weary and tired.\n The men were soldiers.\n The tasks seem endless.\n All men must be free.\n The workers have been slaves.\n The burden becomes heavier every day.\n The children feel happy and care-free.\n Evolution is the development of life.\n Grammar is the study of words and their use.\n Knowledge is freedom.\n The music sounds sweet on the midnight air.\n He looks well today.\n The dregs taste bitter.\n The incense smells sweet.\n\n\n Exercise 4\n\nComplete the following sentences by adding an object or a complement.\n\n\n 1. Perseverance in your study will bring.......\n 2. The great need of the working class is.......\n 3. We shall never acknowledge.......\n 4. By the sweat of no other's brow shalt thou eat.......\n 5. The Revolutionary fathers founded.......\n 6. The workers demand.......\n 7. Labor's only road to freedom is.......\n 8. Life, liberty and the pursuit of happiness are.......\n 9. If you struggle, you will gain.......\n 10. An incomplete verb requires.......\n 11. The complement of a transitive verb is called.......\n 12. The complement of a copulative verb may be either......or.......\n\n\n+103. There are two classes of verbs, complete and incomplete.+\n\n+A complete verb is one that requires no complement.+\n\n+An incomplete verb is one that requires a complement to complete its\nmeaning.+\n\n+Incomplete verbs are of two kinds: 1. Those that express action; 2.\nThose that express state or condition.+\n\n+Incomplete verbs that express action are called transitive verbs.+\n\n+Incomplete verbs that express state or condition are called copulative\nverbs.+\n\n+The complement or the word that denotes the receiver of the action\nexpressed in a transitive verb is called the object.+\n\n+The word or words that complete the meaning of a copulative verb are\ncalled the complement, or attribute complement.+\n\n+The same verb may be complete or incomplete, according to the way in\nwhich it is used.+\n\n\n Exercise 5\n\nIn the following sentences draw a single line under the complete verbs\nand a double line under the incomplete verbs. Then determine whether the\nincomplete verbs are transitive or copulative verbs, and draw a line\nthrough the object or the complement.\n\n 1. Some plants are poisonous.\n 2. A rolling stone gathers no moss.\n 3. Perseverance brings success.\n 4. Delays are dangerous.\n 5. A man's actions show his character.\n 6. He looks well and feels stronger.\n 7. The snows come and the flowers fade.\n 8. Labor creates all wealth.\n 9. Labor must be free.\n 10. The boy writes well.\n 11. The man wrote a letter.\n 12. The skies are clear.\n 13. The hail destroyed the wheat.\n 14. No man is ever too old to learn.\n 15. Competition makes enemies.\n 16. Co-operation makes friends.\n 17. Competition breeds hatred.\n 18. Co-operation breeds good will.\n 19. Competition ensures war.\n 20. Co-operation ensures peace.\n\n\n Exercise 6\n\nIn the following quotation all of the verbs are printed in _italics_.\nDetermine whether they are complete or incomplete verbs. If incomplete,\ndetermine whether they are transitive or copulative verbs. Draw a line\nunder the object of every transitive verb and two lines under the\ncomplement of every copulative verb. Remember that sometimes we have\nseveral words combined into a verb phrase and used as a single verb.\nWatch for the verb phrases in the following, as for example: _must be_,\nin the sentence, _Labor must be free_.\n\n\n The history of man _is_ simply the history of slavery. Slavery\n _includes_ all other crimes. It _degrades_ labor and _corrupts_\n leisure. With the idea that labor _is_ the basis of progress _goes_\n the truth that labor _must be_ free. The laborer _must be_ a free man.\n\n There _is_ something wrong in a government where honesty _wears_ a rag\n and rascality _dons_ a robe; where the loving _eat_ a crust while the\n infamous _sit_ at banquets.\n\n _Talk_ about equal opportunity! Capitalism _ties_ a balloon to the\n shoulders of the rich child; it _ties_ a ball and chain to the feet of\n the poor child; and _tells_ them that they _have_ an equal\n opportunity!\n\n Once the master _hunted_ for the slaves, now the slave _hunts_ for a\n master.\n\n\n Exercise 7\n\nMark the verbs in the following poem. Often in poetry words are omitted\nwhich in strict grammatical construction should be expressed. As for\nexample in the fourth line of this poem _which are_, is omitted before\nthe word _bought_. In prose this would read, _The pews which are bought\nby the profits_, etc. So the word _bought_ is a part of the verb phrase,\n_are bought_. In the last line of the third stanza there is another\nomission before the word _planning_. The meaning is, _while they are\nplanning slaughter_. _Planning_ is a part of the verb phrase _are\nplanning_. And in the last line _is_ is omitted before the word\n_beloved_. _Is beloved_ is the verb phrase. Determine whether the verbs\nin this poem are complete, transitive or copulative, and mark the\nobjects and the complements of the transitive and the copulative verbs.\n\n\n WHO IS A CHRISTIAN?\n\n _Ella Wheeler Wilcox_\n\n \"Who is a Christian in this Christian land\n Of many churches and of lofty spires?\n Not he who sits in soft, upholstered pews\n Bought by the profits of unholy greed,\n And looks devotion while he thinks of gain.\n\n Not he who sends petitions from the lips\n That lie to-morrow in the street and mart.\n Not he who fattens on another's toil,\n And flings his unearned riches to the poor\n Or aids the heathen with a lessened wage,\n And builds cathedrals with an increased rent.\n\n Christ, with Thy great, sweet, simple creed of love,\n How must Thou weary of earth's \"Christian\" clans,\n Who preach salvation through Thy saving blood\n While planning slaughter of their fellow men.\n\n Who is a Christian? It is one whose life\n Is built on love, on kindness and on faith;\n Who holds his brother as his other self;\n Who toils for justice, equity and peace,\n And hides no aim or purpose in his heart\n That will not chord with universal good.\n Though he be a pagan, heretic or Jew\n That man is Christian and beloved of Christ.\"\n\n\n\n\n SPELLING\n\n LESSON 5\n\n\nWe often have two vowels used in the same syllable as a single sound, as\n_ou_ in _round_, _oi_ in _oil_, etc.\n\n+A diphthong is a union of two vowels to represent a single\nsound different from that of either alone.+\n\nSometimes we have two vowels used together in a combination which is\nreally not a diphthong for they do not unite in a different sound. Only\none of the vowels is used and the other is silent as _ai_ in _rain_,\n_oa_ in _soap_, etc.\n\nThe most common diphthongs are:\n\n ou as in _sound_.\n ow as in _owl_.\n oi as in _oil_.\n oy as in _boy_.\n\nIn the spelling lesson for this week mark the words in which the\ncombination of vowels forms a diphthong. In some of the words the\ncombination of vowels does not form a diphthong for only one of the\nvowels is sounded. Draw a line through the silent letter.\n\n +Monday+\n\n Straight\n Aisle\n Search\n Breadth\n Defeat\n\n +Tuesday+\n\n Exploit\n Ceiling\n Height\n People\n Feudal\n\n +Wednesday+\n\n Brought\n Shoulder\n Group\n Compound\n Trouble\n\n +Thursday+\n\n Royal\n Coarse\n Course\n Broad\n Flower\n\n +Friday+\n\n Laughter\n Haunted\n Plaid\n Invoice\n Chair\n\n +Saturday+\n\n Guide\n Build\n Grieve\n Sieve\n Renown\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 6\n\n\nDear Comrade:\n\nWe have this week another lesson in verbs. Do not be discouraged if you\ndo not understand it all at once. Little by little, it will grow clearer\nand you will master this important word.\n\nThe verb may seem involved to you, but a little application will soon\nmake it clear. It is the most important word in the language to master.\nIt almost seems as though the verb were a living, thinking thing. It\nchanges outward form to accommodate itself to its subject in the number\nform and person form change. If it is entertaining a subject in the\nsingular it adopts one dress; if it is entertaining a plural subject,\nmore than one, the verb wears a different dress.\n\nSo also if the subject is the first person, the person speaking, or the\nsecond person, the person spoken to, or the third person, the person\nspoken of, the verb accommodates itself to the subject. The verb is the\nmost agreeable thing for it changes its form to agree with its subject!\nSo watch your verb and see that it agrees.\n\nRefer constantly to your list of irregular verbs given in this lesson\nfor we so often make mistakes in the use of these verb forms.\n\nThen, too, the verb kindly changes its form to accommodate itself to the\ntime of the action--action in the present, in the past, in the\nfuture--action completed before the present time--before some time\npast--or before some future time--and action progressing and not yet\ncompleted in the present, in the past or in the future. Then it can also\nchange to show whether its subject is acting or being acted upon. Isn't\nthe verb a wonderfully accommodating member of the co-operative\ncommonwealth of words?\n\nAnd can you not see hidden under all this, a marvelous development in\nthe intellectual needs of men from the day of the savage's signs and\ngrunts to the day when we can express such shades of meaning? This tool\nof expression, language, has had a wonderful evolution side by side with\nthe evolution of the other tools by which man expresses his creative\ngenius; from the forked stick with which man scratched the soil to the\ngreat machine-driven plow of today; from the simple threshing flail to\nthe monster threshing machine of modern times.\n\nThere is nothing so wonderful as man's ability to express himself. Add a\nlittle to your knowledge every day and the sum total will soon surprise\nyou.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n INFLECTION--CHANGES IN FORM\n\n+104.+ You remember that nouns have certain changes in form to indicate\nchanges in use. Verbs also have several changes in form to correspond\nwith changes in their use or meaning. Notice the following sentences:\n\n I think.\n I thought.\n I work.\n I worked.\n\nWhat is the difference in the meaning of _I think_ and _I thought_? of\n_I work_ and _I worked_? When we say, _I think_, or _I work_, we mean\nthat the action is now, to-day, in the present; but when we say, _I\nthought_, or _I worked_, we mean that _now_ is not the time of the\naction, but that the action was performed sometime in the past. So we\nhave a change in the verb form to denote _time_. The simple form of the\nverb, like _think_ or _work_, is used to denote _present time_. When we\nwish to express _past time_ we do it by changing the form of the verb.\nNow note the following:\n\n } call\n I, We, You, They, } send\n } fall\n The men } bring\n } hide\n\n } calls\n He, She, It, } sends\n } falls\n The man } brings\n } hides\n\nNow let us write this in another way.\n\n +Present Time+\n\n _Singular_ _Plural_\n\n 1st person--I call. We call.\n 2nd person--You call. You call.\n\n He }\n 3rd person She } calls. They, or } call.\n It } The men }\n The man }\n\n+105.+ You notice in this table we use the expressions _first person_,\n_second person_, and _third person_. _I_ and _we_ indicate the person or\npersons speaking and are called the first person. _You_ indicates the\nperson or persons spoken to and is called the second person. _He_,\n_she_, _it_, _they_, and the person or persons or things spoken of, are\ncalled the third person.\n\nWe use the word _you_ when speaking to one or more than one now-a-days.\nIt used to be that when speaking to a single person, people said _thou_,\nand in speaking to two or more they said _you_. But we today have\ndropped the old form _thou_, and use _you_ for both singular and plural.\n\n+106.+ Now note, in the above table, that there is only one form change\nin the verb, and this is in the _third person singular_. We say _I\ncall_, _You call_, _We call_, _They_, or _The men call_, but we say\n_He_, or _the man calls_, in speaking of one person or thing. So we\nchange the form of the verb with any subject which denotes the third\nperson and the singular number. This form is made by adding _s_ to the\nsimple form of the verb, therefore we may call it the _s-form_ because\nit always ends in _s_.\n\nRemember that this _s-form_ is used to express present time with a third\nperson, singular subject. _BE CAREFUL NOT TO USE THIS FORM WITH ANY\nPLURAL SUBJECT._ There is no other change in the verb form in expressing\nthe present time in any verb, except in the verb _be_.\n\n+107.+ This little verb _be_ is one of the most troublesome verbs in our\nlanguage, and since it is used in forming verb phrases, it will be well\nto commit the following table to memory. Watch closely your use of this\nbothersome little word. Note that it has a change in form for the _first\nperson singular_, as well as for the third person singular. All other\nverbs have just the one change, the _s-form_ for the third person\nsingular. The verb _be_ has a form also to use with the first person\nsingular, the pronoun _I_.\n\n +Present Time+ +Past Time+\n\n _Singular_ _Singular_\n\n 1. I am. 1. I was.\n 2. You are. 2. You were.\n 3. He is. 3. He was.\n\n _Plural_ _Plural_\n\n 1. We are. 1. We were.\n 2. You are. 2. You were.\n 3. They are. 3. They were.\n\n+108.+ +The present time form is the form which expresses present time.\nIt is expressed by the simple form of the verb with the exception of the\nthird person singular, which is expressed by the _s-form_.+\n\n\n PAST TIME\n\n+109.+ To express _past time_ we change the form of the verb. Notice the\nfollowing:\n\n I } called We } called\n She } sent You } sent\n He } fell They } fell\n It } brought The men } brought\n The man } hid } hid\n\nNotice that these various forms of the verb which express past time are\nall made by changes from the simple form, which expresses present time.\nYou will also notice that these five verbs used in the above table all\nform their past time form in different ways. For example, _call_ adds\n_ed_; _send_ changes the final letter from _d_ to _t_; _fall_ changes\nthe vowel in the middle of the word from _a_ to _e_; _bring_ changes\nboth the vowel and the final letter from _bring_ to _brought_; _hide_\ndrops the final letter _e_.\n\n+110.+ +Verbs whose past time forms are made by adding _d_ or _ed_\nto the simple form are called regular verbs.+\n\n+Verbs whose past time forms are made in some other way than by adding\n_d_ or _ed_ are called irregular verbs.+\n\n+111.+ There are about two hundred of these irregular verbs which form\ntheir past time in the following ways:\n\n1. By change in the vowel letter, as _fall_, _fell_; _write_, _wrote_;\n_see_, _saw_; _sing_, _sang_; _come_, _came_.\n\n2. By dropping the final vowel; as _hide_, _hid_; _slide_, _slid_;\n_bite_, _bit_.\n\n3. By dropping a vowel from the middle of the word; as _bleed_, _bled_;\n_feed_, _fed_; _lead_, _led_.\n\n4. By changing the final letter or letters; as _send_, _sent_; _lose_,\n_lost_; _spend_, _spent_.\n\n5. By changing the vowel and final letters; as _bring_, _brought_;\n_seek_, _sought_; _catch_, _caught_.\n\n6. By changing the vowel sound and adding _t_ or _d_; as _sleep_,\n_slept_; _feel_, _felt_; _flee_, _fled_.\n\nThere are some irregular forms which we must learn and be exceedingly\ncareful in their use. Study the list in this lesson.\n\n\n Exercise 1\n\nWrite the _present_ and _past_ time forms of the following verbs as the\nverb _think_ is written in the table given below.\n\n think\n ride\n have\n give\n write\n ask\n make\n try\n speak\n run\n see\n do\n\n +Present Time+ +Past Time+\n\n _Singular_ _Singular_\n\n 1. I think 1. I thought\n 2. You think 2. You thought\n 3. He thinks 3. He thought\n\n _Plural_ _Plural_\n\n 1. We think 1. We thought\n 2. You think 2. You thought\n 3. They think 3. They thought\n\n+112.+ Be very careful not to use the _s-form_ except for the third\nperson singular. Be especially careful in the use of different forms of\nthe verb _be_. It is in the use of this verb that we so frequently make\nmistakes. Watch your own language and the conversation of your friends\nand note these mistakes and correct them in your own mind. These common\nblunders in the use of English mark us as careless or uneducated by\neveryone who hears us speak. We have fallen into bad habits oftentimes\nand make these mistakes when we know better, and only constant\nwatchfulness for a time can overcome the habit. After a time we learn to\nspeak correctly without effort, and then these mistakes made by others\noffend the ear like a false note in music.\n\n\n Exercise 2\n\nCross out the wrong form in the following:\n\n They _was_--_were_ not here.\n The clouds _has_--_have_ gathered.\n People _is_--_are_ indifferent.\n The train _was_--_were_ on time.\n The men _was_--_were_ armed.\n Our school building _is_--_are_ inadequate.\n The workers _earn_--_earns_ their wages.\n The voters _elect_--_elects_ the President.\n They _do_--_does_ as they please.\n We _was_--_were_ there on time.\n\n\n DOING DOUBLE WORK\n\n+113.+ We have found now three forms of the verb, the _simple form_, the\n_s-form_, and the _past time form_, and, in addition, the _I-form_, or\nthe first person form of the verb _be_. There are no other real verb\nforms, but there are two other changes made in the form of the verb when\nit ceases to be used as the predicate, the asserting word of the\nsentence, and becomes, in part, another part of speech.\n\nNotice in the following sentences:\n\n Making shoes is his work.\n He enjoys making shoes.\n\nIn each of these sentences the word _making_, from the verb _make_, is\nused as a noun. In the first, _Making shoes is his work_, _making_ is\nused as the subject of the sentence. In the second, _He enjoys making\nshoes_, _making_ is used as the object of the verb _enjoys_. But\n_making_ is not like the ordinary noun, for it has an object\n_making_--_what?_--_making shoes_. _Shoes_ is the object of the action\nexpressed in _making_. A noun never takes an object; so while the word\n_making_ is used as a noun, it is also partly a verb. It is a form of\nthe verb used as a noun, but keeping in part its verb nature, partaking\nof the nature of two parts of speech at the same time.\n\nHence these forms of the verb are called _participles_. Participle means\n_partaker_.\n\nThe participle may also be used as an adjective. Notice the following:\n\n The _crying_ child came toward us.\n The _rescuing_ party arrived.\n\nIn these sentences _crying_ and _rescuing_ are formed from the verbs\n_cry_ and _rescue_, and are used as adjectives to describe the noun\n_child_ and the noun _party_. So a participle is a mixed part of speech.\nIt is partially a verb, but is not a true verb. A true verb is always\nused as the predicate, the asserting word in the sentence and _always_\nhas a subject. The participle _never_ has a subject; it may have an\nobject, but not a subject.\n\n+114.+ There are two forms of the participle. The active form or the\npresent form as it is sometimes called, ends in _ing_, as, _waiting_,\n_walking_, _saying_. It expresses action, existence, or possession as\ngoing on at the time mentioned in the sentence.\n\n+115.+ The other form of the participle is the passive form or the past\nform of the participle. This ends in _ed_ in the regular verbs, and has\nvarious forms in the irregular verbs. It is formed in regular verbs by\nadding _d_ or _ed_ to the simple form, hence has the same form as the\npast time form, as for example, present time form, _call_--past time\nform, _called_--past participle, _called_. You will find the past\nparticiple forms of irregular verbs in the list of irregular verbs given\nin this lesson, as for example--present time form, _go_--past time form,\n_went_--past participle, _gone_.\n\n+116.+ You will find as we study the verb phrases in later lessons that\nthese participles are used in forming verb phrases. As for example:\n\n He is coming.\n They are trying.\n He has gone.\n\n+A participle is a word derived from a verb, partaking of the nature of\na verb and also of an adjective or a noun.+\n\n\n LET US SUM UP\n\n+117.+ +Verbs have five form changes.+\n\n Simple S-Form Past Time Present Part. Past Part.\n\n call calls called calling called\n\n go goes went going gone\n\n\n Exercise 3\n\nWrite in columns like the above the five forms of the following verbs:\n\n do\n try\n give\n hope\n live\n rob\n have\n think\n sing\n get\n wave\n lose\n come\n make\n\n\n Exercise 4\n\nStudy carefully the following quotation. You will find in it all five of\nthe form changes of the verb--_the present time form_, _the s-form_,\n_the past time form_, _the present participle_ and _the past\nparticiple_. In the verb phrases _had been filled_, _has survived_, _has\ngone_, _has proved_ and _be dismayed_, you will find the past participle\nused in forming the verb phrase. We will study these verb phrases in\nlater lessons.\n\nIn the verb phrases, _was stumbling_, _was groping_, _is conquering_,\n_are carrying_, the present participle is used in forming the verb\nphrases. _Could reconcile_ is also a verb phrase. We will study these\nverb phrases also in later lessons.\n\nThe present participles, _struggling_, _persevering_ and _regaining_ are\nused as adjectives. Study them carefully and find the words which they\ndescribe. The present participles _imagining_, _learning_ and\n_suffering_ are used as nouns. Note their use.\n\nThe past participles _rebuffed_, _self-reproached_, _discouraged_ and\n_promised_ are used as adjectives. Find the words which they modify.\nThere are several _present time forms_, several _past time forms_, and\nseveral _s-forms_. Find them and study carefully their usage.\n\n\n OUT OF THE DARK\n\n _By Helen Keller_\n\n _America's famous blind girl, who has come to see more than most\n people with normal eyes._\n\n Step by step my investigation of blindness _led_ me into the\n industrial world. And what a world it _is_. I _faced_ unflinchingly a\n world of facts--a world of misery and degradation, of blindness,\n crookedness, and sin, a world _struggling_ against the elements,\n against the unknown, against itself. How _could_ I _reconcile_ this\n world of fact with the bright world of my _imagining_? My darkness\n _had been filled_ with the light of intelligence, and, _behold_, the\n outer day-lit world _was stumbling_, _was groping_ in social\n blindness. At first, I _was_ most unhappy, but deeper study _restored_\n my confidence. By _learning_ the _suffering_ and burdens of men, I\n _became_ aware as never before of the life-power which _has survived_\n the forces of darkness--the power which, though never completely\n victorious, _is_ continuously _conquering_. The very fact that we\n _are_ still carrying on the contest against the hosts of annihilation\n _proves_ that on the whole the battle _has gone_ for humanity. The\n world's great heart _has proved_ equal to the prodigious undertaking\n which God _set_ it. _Rebuffed_, but always _persevering_;\n _self-reproached_, but ever _regaining_ faith; undaunted, tenacious,\n the heart of man _labors_ towards immeasurably distant goals.\n _Discouraged_ not by difficulties without, or the anguish of ages\n within, the heart _listens_ to a secret voice that _whispers_: \"_Be_\n not _dismayed_; in the future _lies_ the _Promised_ Land.\"\n\n\nList of Irregular Verbs\n\nHere is a list of the principal irregular verbs--the present and past\ntime forms and the past participle are called the principal parts of a\nverb.\n\n(Those marked with an _r_ have also the regular form.)\n\n +Present T.+ +Past T.+ +Past Part.+\n\n abide abode abode\n arise arose arisen\n awake awoke, _r_ awaked\n be or am was been\n bear bore borne\n beat beat beaten\n begin began begun\n bend bent, _r_ bent, _r_\n bereave bereft, _r_ bereft, _r_\n beseech besought besought\n bet bet bet\n bid bid or bade bid (den)\n bind bound bound\n bite bit bit (ten)\n bleed bled bled\n blow blew blown\n break broke broken\n breed bred bred\n bring brought brought\n build built, _r_ built, _r_\n burn burnt, _r_ burnt, _r_\n burst burst burst\n buy bought bought\n cast cast cast\n catch caught caught\n chide chid chid (den)\n choose chose chosen\n cling clung clung\n clothe clad, _r_ clad, _r_\n come came come\n cost cost cost\n creep crept crept\n cut cut cut\n deal dealt, _r_ dealt, _r_\n dig dug, _r_ dug, _r_\n do did done\n draw drew drawn\n dream dreamt, _r_ dreamt, _r_\n drink drank drunk\n drive drove driven\n dwell dwelt, _r_ dwelt, _r_\n eat ate eaten\n fall fell fallen\n feed fed fed\n feel felt felt\n fight fought fought\n find found found\n flee fled fled\n fling flung flung\n fly flew flown\n forget forgot forgotten\n forgive forgave forgiven\n forsake forsook forsaken\n get got got (ten)\n give gave given\n go went gone\n grind ground ground\n grow grew grown\n hang hung, _r_ hung, _r_\n have had had\n hear heard heard\n hew hewed hewn, _r_\n hide hid hidden\n hit hit hit\n hold held held\n hurt hurt hurt\n keep kept kept\n kneel knelt, _r_ knelt, _r_\n knit knit, _r_ knit, _r_\n know knew known\n lay laid laid\n lead led led\n leave left left\n lend lent lent\n let let let\n lie lay lain\n light lit, _r_ lit, _r_\n lose lost lost\n make made made\n mean meant meant\n meet met met\n mistake mistook mistaken\n mow mowed mown, _r_\n pay paid paid\n plead pled, _r_ pled, _r_\n put put put\n quit quit, _r_ quit, _r_\n read read read\n rend rent rent\n rid rid rid\n ride rode ridden\n ring rang rung\n rise rose risen\n run ran run\n saw sawed sawn, _r_\n say said said\n see saw seen\n seek sought sought\n sell sold sold\n send sent sent\n set set set\n shake shook shaken\n shape shaped shapen, _r_\n shave shaved shaven, _r_\n shear sheared shorn, _r_\n shed shed shed\n shine shone, _r_ shone, _r_\n shoe shod shod\n shoot shot shot\n show showed shown, _r_\n shrink shrank shrunk (en)\n shut shut shut\n sing sang sung\n sink sank sunk\n sit sat sat\n slay slew slain\n sleep slept slept\n slide slid slid (en)\n sling slung slung\n slink slunk slunk\n slit slit slit\n smite smote smitten\n sow sowed sown, _r_\n speak spoke spoken\n speed sped sped\n spend spent spent\n spill spilt, _r_ spilt, _r_\n spin spun spun\n spit spit spit\n split split split\n spoil spoilt, _r_ spoilt, _r_\n spread spread spread\n spring sprang sprung\n stand stood stood\n stave stove, _r_ stove, _r_\n steal stole stolen\n stick stuck stuck\n sting stung stung\n stink stunk stunk\n strike struck struck\n strike struck stricken\n stride strode stridden\n string strung strung\n strive strove striven\n strew strewed strewn, _r_\n swear sworn sworn\n sweat sweat, _r_ sweat, _r_\n sweep swept swept\n swell swelled swollen, _r_\n swim swam swum\n swing swung swung\n take took taken\n teach taught taught\n tear tore torn\n tell told told\n think thought thought\n throw threw thrown\n thrust thrust thrust\n tread trod trod (den)\n wake woke, _r_ woke, _r_\n wear wore worn\n weave wove woven\n wed wed, _r_ wed, _r_\n weep wept wept\n wet wet, _r_ wet, _r_\n whet whet, _r_ whet, _r_\n win won won\n wind wound wound\n work wrought, _r_ wrought, _r_\n wring wrung wrung\n write wrote written\n\n\n\n\n SPELLING\n\n LESSON 6\n\n\nEvery vowel or every vowel combination pronounced as one vowel sound\nindicates a syllable (excepting final _e_ in such words as _fate_,\n_late_, _rode_, etc.) Take the word _combination_, for example. In this\nword we have four syllables, thus: _Com-bi-na-tion_.\n\n+A syllable is that part of a word which can be uttered distinctly by a\nsingle effort of the voice.+ Remember that each syllable must contain a\nvowel or a vowel combination like _oi_ or _ou_, which is pronounced as\none vowel. Sometimes the vowel alone makes the syllable as in _a-lone_,\n_e-qual_, etc. The final _e_ in words like _late_, and _fate_ is not\nsounded. It is silent, we say.\n\nAll words ending in silent _e_ have the long vowel sound, with a very\nfew exceptions. Words without the final _e_ have the short vowel sound\nas for example: _fate_, _fat_; _mate_, _mat_; _hide_, _hid_; _rode_,\n_rod_.\n\nIn dividing words into syllables the consonant is written with the\npreceding vowel when that vowel is short. If the vowel is long the\nconsonant is written with the next syllable, as for example, de-fine and\ndef-i-ni-tion. In de-fine the _e_ is long therefore _f_, the consonant\nfollowing, is written with the next syllable, _fine_. In def-i-ni-tion\nthe _e_ has the short sound, therefore the _f_ is written with the _e_\nin the syllable, _def_.\n\nWhen there are two consonants following the vowel, divide between the\nconsonants, as for example, _in-ven-tion_, _foun-da-tion_, etc. Never\ndivide a digraph, that is, two consonants which are sounded together as\none sound, as for example, _moth-er_, _catch-er_, _te-leg-ra-pher_, etc.\n\nIn writing words containing double consonants like _dd_, _ll_, _ss_,\ndivide the word into syllables between the double consonants, as for\nexample, _per-mit-ted_, _ad-mis-sion_, _sad-dest_, etc.\n\n +Monday+\n\n Important\n Accommodate\n Person\n Correspond\n Action\n\n +Tuesday+\n\n Difference\n Notice\n Indicate\n Remember\n Irregular\n\n +Wednesday+\n\n Mistake\n Conversation\n Correctly\n President\n Ordinary\n\n +Thursday+\n\n Participle\n Passive\n Various\n Phrase\n Quotation\n\n +Friday+\n\n Imagine\n Confidence\n Humanity\n Faith\n Future\n\n +Saturday+\n\n Whisper\n Thought\n Ability\n Knowledge\n Genius\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 7\n\n\nDear Comrade:\n\nI wonder if you have ever thought as to how our language grew.\n\nWe get the words in our language from many sources. The English language\ntoday is a development of the early Anglo-Saxon. England was called\noriginally Angle-land which was gradually shortened into England. So we\nhave in our language what are called pure English or Anglo-Saxon words.\nThese words form the bulk of our every day vocabulary, being simple,\nstrong, forceful words. Then we have in our English many foreign words\nwhich we have adopted from other languages. There are many Latin and\nGreek words; these we use in our more elegant speech or writing.\n\nThere is an interesting bit written by Sir Walter Scott in his novel of\nearly England, \"Ivanhoe,\" which illustrates the manner in which words\nhave come into our language and also the difference in speech which\nmarks the working class and the exploiting class. As those who do the\nwork of the world rid themselves of the parasites who have appropriated\nthe produce of their labor, through the ages, they will demand that\nwhich belongs to them--the best--the best in language as in everything\nelse.\n\n \"'... I advise thee to call off Fangs and leave the herd to their\n destiny, which, whether they meet with bands of traveling soldiers, or\n of outlaws, or of wandering pilgrims, can be little else than to be\n converted into Normans before morning, to thy no small ease and\n comfort.'\n\n \"'The swine turned into Normans to my comfort!' quoth Gurth. 'Expound\n that to me, Wamba, for my brain is too dull and my mind too vexed to\n read riddles.'\n\n \"'Why, how call you these grunting brutes running about on their four\n legs?' demanded Wamba.\n\n \"'Swine, fool, swine,' said the herd; 'every fool knows that.'\n\n \"'And swine is good Saxon,' said the jester; 'but how call you the sow\n when she is flayed, and drawn, and quartered, and hung by the heels\n like a traitor?'\n\n \"'Pork,' answered the swineherd.\n\n \"'I am glad every fool knows that too,' said Wamba; 'and pork, I\n think, is good Norman-French, and so when the brute lives and is in\n charge of a Saxon slave, she goes by her Saxon name; but becomes a\n Norman and is called pork when she is carried to the castle-hall to\n feast among the nobles. What dost thou think of that, friend Gurth,\n ha?' ...\"\n\nSo you see even in words the distinction is made between those who\nproduce and those who possess.\n\nBut the day is at hand when those who work shall also enjoy. We have\nfought for religious and political freedom. Today we are waging the\nbattle for industrial freedom. It is _your_ struggle. Study--prepare\nyourself to do battle for your rights.\n\n Yours for Freedom,\n\n THE PEOPLE'S COLLEGE.\n\n\n FUTURE TIME\n\n+118.+ We have learned how to express present time and past time, by\nchanges in the form of the verb. But we very often desire to make a\nstatement in which we do not express either present or past time, thus\nwe may say:\n\n We shall enjoy our rights some day.\n He will join us in the struggle.\n\nWe do not mean to say that we do enjoy our rights now, in the present,\nor that we did in the past, but that we _shall_ enjoy our rights some\ntime in the future. In the second sentence, _will join_ expresses the\nsame idea of future time. To indicate future time, we do not make a\nchange in the verb form, but we use _shall_ and _will_ with the simple\nform of the verb.\n\n+119.+ +We denote future time by use of a verb phrase made by placing\n_shall_ or _will_ before the simple form of the verb.+\n\n+120.+ The rule of some grammarians is to use _shall_ always in the\nfirst person, the person speaking, to denote future time, and _will_\nwith the second person, the person spoken to, and with the third person,\nthe person spoken of, to denote future time. But common usage does not\nalways follow the rules of the grammarians, and, in the course of time,\naffects and changes these rules. So our common usage of today uses\n_will_ in the first person to express future time, as well as _shall_.\n\nThis rule of grammarians marks a nicety of speech and conveys a\ndistinction of meaning which it really seems worth while to retain. The\nidea of the grammarians is that when we use _will_ with the first person\nand _shall_ with the second or third person, we express a _promise_ or\n_determination_. Thus if I say, _I shall go_, I simply mean that my\ngoing will be in the future. But if I say, _I will go_, I either mean\nthat I am promising to go or that I am expressing my determination to\ngo. So also if we use _shall_ in the second and third persons. If we\nsay, _You will go_ or _He will go_, we are simply stating that the going\nwill be in the future, but if we say, _You shall go_, or _He shall go_,\nwe mean that we promise or are determined that you or he shall go.\n\nTo be technically correct this distinction should be observed. _Shall_\nin the first person, and _will_ in the second and third express simple\nfuturity. _Will_ in the first person and _shall_ in the second and third\nexpress promise or determination. But in every day conversation this\ndistinction is not observed, and many of our best writers do not follow\nthis rule.\n\n\n Exercise 1\n\nMark the future time forms in the following sentences:\n\n 1. I shall speak of liberty.\n 2. I will never give up.\n 3. I shall write to him.\n 4. He shall not starve.\n 5. We shall expect you.\n 6. They shall suffer for this.\n 7. I shall go to New York.\n 8. He will call for me.\n 9. The hungry shall be fed.\n 10. You will soon see the reason.\n 11. You shall never want for a friend.\n 12. They shall some day see the truth.\n 13. We will not fight against our class.\n 14. We will stand together.\n\n\n PERFECT TIME\n\n+121.+ Past, present and future, being the three divisions of time, one\nwould naturally expect that when we had found how to express these three\nforms, we would be through, but if you stop to think, you will find that\nthere are other verb phrases of which we have need.\n\nWhen we wish to speak of action as completed at the present time, we do\nnot say:\n\n I study my lessons every day, _but_, I have studied my lessons every\n day.\n _Not_, You work for him every day, _but_, You have worked for him\n every day.\n _Not_, He sees her frequently, _but_, He has seen her frequently.\n\nCan you not readily see the difference in the meaning expressed in _I\nwork every day_, and _I have worked every day_? In the first sentence\nyou express a general truth, _I work every day_, a truth which has been\ntrue in the past, is true in the present, and the implication is that it\nwill continue to be true in the future. But when you say, _I have worked\nevery day_, you are saying nothing as to the future, but you are\ndescribing an action which is completed at the present time. This is\ncalled the _present complete_ or _present perfect_ time.\n\n+122.+ Perfect means complete, and present perfect describes an action\nperfected or completed at the present time. So it is possible for us to\nexpress a necessary shade of meaning by the present perfect time form.\n\n+123.+ +The present perfect time form describes an action completed at\nthe present time, and is formed by using the present time form of\n_have_ and the _past_ participle of the verb.+\n\n +Present Perfect Time+\n\n _Singular_ _Plural_\n\n 1st. I have seen. We have seen.\n 2d. You have seen. You have seen.\n 3d. He has seen. They have seen.\n\n+124.+ Review in the last lesson how to form the past participle.\nRemember that it is one of the principal parts of the verb. In regular\nverbs the past participle is the same form as the past time form. In\nirregular verbs the past participle is quite often different from the\npast time form, as for example: _go_, _went_, _gone_; _do_, _did_,\n_done_, etc.\n\nWatch closely your irregular verbs and see that you always use the past\n_participle_ with _have_ or _had_; never use the past _time_ form with\n_have_ or _had_.\n\n\n PAST PERFECT\n\n+125.+ When you desire to express an action complete at some definite\npast time, you do not say:\n\n We finished when they came, _but_, We had finished when they came.\n _Not_, They went when we arrived, _but_, They had gone when we\n arrived.\n _Not_, I worked six months when he began, _but_, I had worked six\n months when he began.\n\nCan you see a difference in the meaning expressed in these sentences: _I\nworked six months when he began_; and _I had worked six months when he\nbegan_? This last sentence describes an action completed or perfected\nbefore some definite past time.\n\n+126.+ +Past perfect time denotes an action perfected or completed at\nsome definite past time. It is formed by using _had_ and the past\nparticiple of the verb.+\n\nRemember always, with irregular verbs, to use the _past participle_.\nNever use the _past time form_ with _had_.\n\n +Past Perfect Time+\n\n _Singular_ _Plural_\n\n 1st. I had seen. We had seen.\n 2d. You had seen. You had seen.\n 3d. He had seen. They had seen.\n\n\n Exercise 2\n\nCorrect the following sentences in which the past time form is used\ninstead of the past participle. Look up the word in the list of\nirregular verbs and use the past participle instead of the past time\nform.\n\n 1. I have saw it often.\n 2. He had shook his fist.\n 3. She has sang for us.\n 4. The boat has sank here.\n 5. He has spoke the truth.\n 6. They had stole the books.\n 7. He has swore to the truth.\n 8. He had took the wrong road.\n 9. She has tore her dress.\n 10. He had threw the ball away.\n 11. The girl had wore the dress.\n 12. He had wrote the letters.\n 13. He had drank too much.\n 14. He had rode the horse.\n 15. The sun has rose.\n 16. He has bore his part.\n 17. They have began already.\n 18. The wind has blew all night.\n 19. It had broke when it fell.\n 20. He has chose the right.\n 21. You have did your duty.\n 22. He has ate his breakfast.\n 23. A heavy rain has fell.\n 24. They had gave it to me.\n 25. He has became rich.\n 26. It has grew rapidly.\n 27. He has knew it always.\n 28. He has mistook her for another.\n\n\n FUTURE PERFECT TIME\n\n+127.+ We find also that we need a verb phrase to express time _before_\nsome other future time, to describe an action that will be finished,\nperfected, or completed, before some other future action. Thus,\n\n I shall have gone before you arrive.\n You will have earned your money before you get it.\n I shall have worked thirty days when pay-day comes.\n\nCan you not see a difference in saying, _I shall work thirty days when\npay-day comes_, and _I shall have worked thirty days when pay-day\ncomes_? The first sentence expresses simple future time, or what you\nwill do when pay-day comes; the second describes an action which will be\ncompleted or perfected _before_ pay-day comes. So there is quite a\ndifference in the meaning of the future and the future perfect time.\n\n+128.+ +The future perfect time form expresses or describes an action\nthat will be perfected or completed before some other future time. It is\nformed by using _shall have_ or _will have_ with the past\nparticiple.+\n\nBe careful to use the past participle. Never use the past time form with\n_shall have_ or _will have_.\n\n +Future Perfect Time+\n\n _Singular_ _Plural_\n\n 1st. I shall have seen. We shall have seen.\n 2d. You will have seen. You will have seen.\n 3d. He will have seen. They will have seen.\n\n\n LET US SUM UP\n\n+129.+ We have three time forms, _present_, _past_, _future_.\n\n +Present+ +Past+ +Future+\n\n I see I saw I shall see.\n\nEach of these three time forms has a _perfect_ form; that is, a time\nform which expresses an action as completed or perfected at the present\ntime, or before some definite past or future time.\n\n +Present+ +Past+ +Future+\n +Perfect Time+ +Perfect Time+ +Perfect Time+\n\n I have seen I had seen I shall have seen\n\n+130.+ It is wonderful how a knowledge of words and their uses enables\nus to express so many shades of meaning. It is like our development in\nobserving colors. You know the savage always admires vivid reds and\ngreens and blues. He does not yet see the beautiful shades and\ngradations of color. We enjoy the delicate pinks and blues and all the\nvarying shades between the primal seven colors of the spectrum. And as\nwe develop our artistic ability we see and enjoy all the beauties of\ncolor.\n\nIn music too, we observe the same development. The barbarian enjoys\nloud, crashing, discordant sounds which he calls music, but which to the\neducated ear are only harsh noises. The trained musician catches the\ndelicate overtones and undertones and finds deepest ecstasy in sounds\nwhich the uneducated ear does not even catch. So as we study words and\ntheir uses, we find ourselves able to express shades of meaning, to\npaint our word pictures, not in gaudy, glaring chromo-tints, but in the\nwondrous blending of color that reveals the true artist.\n\nNow get these modes of expressing time firmly fixed in your mind.\n\n+131.+ +Let us get all we have learned about verbs into a summary and\nhave it clearly in mind.+\n\n\n VERBS--SUMMARY\n\n +Two Classes+\n\n _Complete_--Taking _no_ complement.\n\n _Incomplete_--{ Verbs of action requiring object.\n { Copulative verbs requiring complement.\n\n +Inflection--Changes of Form+\n\n _Simple Form_ _S-Form_ _Past Time_ _Present Part._ _Past Part._\n see sees saw seeing seen\n\n\n TIME FORMS\n\n Present\n\n _Singular_ _Plural_\n\n 1. I see. We see.\n 2. You see. You see.\n 3. He sees. They see.\n\n\n Past\n\n _Singular_ _Plural_\n\n 1. I saw. We saw.\n 2. You saw. You saw.\n 3. He saw. They saw.\n\n\n Future\n\n _Singular_ _Plural_\n\n 1. I shall see. We shall see.\n 2. You will see. You will see.\n 3. He will see. They will see.\n\n\n Present Perfect\n\n _Singular_ _Plural_\n\n 1. I have seen. We have seen.\n 2. You have seen. You have seen.\n 3. He has seen. They have seen.\n\n\n Past Perfect\n\n _Singular_ _Plural_\n\n 1. I had seen. We had seen.\n 2. You had seen. You had seen.\n 3. He had seen. They had seen.\n\n\n Future Perfect\n\n _Singular_ _Plural_\n\n 1. I shall have seen. We shall have seen.\n 2. You will have seen. You will have seen.\n 3. He will have seen. They will have seen.\n\n\n Exercise 3\n\nRead carefully the following quotation. All of the verbs and verb\nphrases are written in _italics_. Study these carefully and decide\nwhether they indicate present, past, future, present perfect, past\nperfect or future perfect time. The verb phrases--_is seizing_, _is\nput_, _is praised_, _is defended_, _can see_, _must have_, _are owned_,\nand _are conducted_, do not belong to any of these six forms. They are\nverb phrases used in ways which we shall study later. All of the other\nverbs or verb phrases belong to one of the six time forms which we have\nstudied. Classify them.\n\n\n The Working Class Must Strike the Blow\n\n You _remember_ Victor Hugo's story of the devil-fish; how the monster\n _put_ forth one tentacle after another and _coiled_ it around his\n victim; how the hero _recalled_ that there _was_ but one vulnerable\n spot in his brute enemy; how at the strategic moment he _struck_ a\n blow at that spot, and the terrible demon of the deep _shuddered_,\n _released_ his grasp and _fell_ dead.\n\n Capitalism _is_ a monster which _is seizing_ the body politic. One\n tentacle _is put_ forth to grasp the major part of the earnings of the\n working class; another _has seized_ the working-woman; another\n _reaches_ forth to the child; another _has fastened_ upon government\n and _has made_ that the instrument of the powerful classes; still\n another _has turned_ the pen of the journalist into a weapon by which\n the injustice of Capitalism _is praised_ and _is defended_; and still\n another _has seized_ the pulpit, _has silenced_ those who _profess_ to\n speak for God and man, or _has turned_ their phrases into open apology\n and defense for the crimes of Capitalism!\n\n But there _is_ one vulnerable spot in Capitalism. If the working class\n of the world _can see_ that spot and _will strike_, they _shall be_\n free.\n\n The fundamental wrong, the basic injustice of the Capitalist System,\n _is_ that the resources of land and machinery, to which all the people\n _must have_ access, in order to live and labor, _are owned_ by the few\n and _are conducted_ by the few for their private profit.\n\n This _is_ the social tragedy, the monstrous wrong of our time.--_J.\n Stitt Wilson_.\n\n\n Exercise 4\n\nSelect two verbs out of the following poem and write their six time\nforms, in the same manner as the time forms of the verb _see_ are given\nin section 131.\n\n\n A MAGIC WORD\n\n There's a little word below, with letters three,\n Which, if you only grasp its potency,\n Will send you higher\n Toward the goal where you aspire,\n Which, without its precious aid, you'll never see--\n _NOW!_\n\n Success attends the man who views it right.\n Its back and forward meanings differ quite;\n For this is how it reads\n To the man of ready deeds,\n Who spells it backwards from achievement's height--\n _WON!_\n\n\n TENSE\n\nThe grammatical term for the time form of the verb is _TENSE_, which is\nderived from a Latin word meaning _time_. The present time-form of the\nverb is called the _present tense_; the past time-form, the _past\ntense_; the future time-form, the _future tense_; the present perfect\ntime-form, the _present perfect tense_, etc.\n\n\n Exercise 5\n\nWrite each of the following four sentences in the six time-forms, or\ntenses,--present, past, future, present perfect, past perfect and future\nperfect, as follows:\n\n _Present_--Labor _creates_ all wealth.\n _Past_--Labor _created_ all wealth.\n _Future_--Labor _will create_ all wealth.\n _Present Perfect_--Labor _has created_ all wealth.\n _Past Perfect_--Labor _had created_ all wealth.\n _Future Perfect_--Labor _will have created_ all wealth.\n\n 1. Hope stirs us to action.\n 2. Human progress is our business.\n 3. The majority demand justice.\n 4. The workers fight all the battles.\n\n\n\n\n SPELLING\n\n LESSON 7\n\n\nThe division of words into syllables is quite important as an aid to\npronunciation. It is also a very important matter to understand in our\nwritten speech for it is often necessary to divide a word at the end of\na line. If the word is not properly divided, it is much more difficult\nto read and understand. The hyphen is used to divide words into\nsyllables when carrying a portion to the next line.\n\nWhen you must divide a word at the end of a line divide it only between\nsyllables. Never divide a word of one syllable, no matter how long it\nmay be. If you cannot get all of it on the line, write it all on the\nnext line. Do not divide a short word of two syllables if you can avoid\nit and never divide such a word when it leaves only one letter on the\nline or only one letter to be carried over to the next line, as for\nexample: _luck-y_, _a-loud_, etc.\n\nWhen two or more vowels are used together to make one sound they should\nnever be separated by the hyphen, as for example, joy-ous, anx-ious,\ntrail, dis-course, de-feat, boor-ish.\n\nWhen two or more vowels placed together are not used to form one sound\nthen these vowels may be divided, as for example, _tri-al_,\n_co-or-di-nate_, _he-ro-ic_.\n\nLook up the words in this week's lesson in the dictionary carefully and\ndivide into syllables. Notice specially the division of words into\nsyllables where the word contains a diphthong and where it contains two\nvowels written together which are not diphthongs. Notice also the words\nwhich have a single vowel as the first or last syllable.\n\n +Monday+\n\n Museum\n Creatures\n Peaceable\n Accruing\n Already\n\n +Tuesday+\n\n Persuade\n Trivial\n Plague\n Alert\n Inquiry\n\n +Wednesday+\n\n Piteous\n Patriot\n Poetry\n Evil\n Business\n\n +Thursday+\n\n Obey\n Breathe\n Society\n Ether\n Sociable\n\n +Friday+\n\n Idealism\n Pledge\n Ache\n Acre\n Pronunciation\n\n +Saturday+\n\n Idle\n Idol\n Mutual\n Wealthy\n Neighbors\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 8\n\n\nDear Comrade:\n\nYou have often read the words _organic_ and _inorganic_ but did you ever\nstop to think of the meaning of these words? We say a body is organic--a\nrock is inorganic; one grows from within, the other is built from\nwithout. A tree is organic; it grows. A house is inorganic; it is built.\nThe house was never a baby house, growing from a tiny house to a large\none. But the tree was once a baby tree, a sapling, and grew branch by\nbranch to its present height. So we have two classes of things--those\nwhich grow and those which are made.\n\nLanguage belongs to the class of things which grows. It is organic. We\nhave even used the same terms in speaking about language that we use in\ntalking of a tree. We use the words ROOT, STEM and BRANCH to describe\nits growth.\n\nLanguage, too, has its different terms of life like a tree, its youth,\nits maturity, its old age, its death.\n\nSo we have dead languages like Latin and Greek--languages which are no\nlonger living,--no longer serving mankind. But these dead languages have\nleft living children, languages that have descended from them.\n\nThe Italian language for example is the child, the descendant of the\nclassical Latin. We have many words in our English language from these\ndead languages. About five-sevenths of the words in our English are from\nthese classical languages. The remaining two-sevenths are from the\nAnglo-Saxon. We use the Anglo-Saxon words more frequently, however, in\nour every day speech.\n\nAnd it is interesting to note that our best poetry--that which stirs our\nblood and touches our hearts--is written in the strong forceful\nAnglo-Saxon words.\n\nThese words we are studying have been through some interesting\nexperiences as they have passed from race to race down to us and the\nhistory of life is mirrored in their changes. How much more interesting\nthey seem when we know something of their sources, just as we are more\ninterested in a man when we know something of his boyhood and youth and\nthe experiences through which he has passed.\n\nYou may think that the study of verbs is rather difficult and involved,\nbut it is more simple in English than in any other language. There are\nfewer changes in the verb form in order to express time and person. Do\nnot rely on the memorizing of the rules. Rules never made one a fluent\nspeaker. Write sentences in which the correct form is used. Read aloud\nfrom the best authors until the sound of the words is familiar and they\ncome readily to the tongue. We have used for the exercises in these\nlessons excerpts from the best authors.\n\nStudy these exercises carefully and note the use of the different verbs\nespecially, this week. Verbs, like all else, are yours to command.\nCommand them.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n PROGRESSIVE VERB PHRASES\n\n+132.+ We have learned how to form the three principal time forms,\n_present_, _past_ and _future_ and the perfect or completed form of each\nof the three, _present perfect_, _past perfect_ and _future perfect_.\nAnd still we have such a wonderful language that we can express other\nshades of meaning in _time_.\n\n+133.+ There is still another phase of action which we must have a verb\nphrase to express. Suppose you want to describe something you are now\ndoing and are continuing to do, something not yet completed. To say, _I\ndo it now_, is not satisfactory. Instead we say, _I am doing it now_.\n\nYou have by the verb phrase, _am doing_, described a progressive action,\nan action _going on_ in the present. You may also want to describe what\nyou were doing yesterday, an action that continued or _progressed_ in\nthe past. You would not say, _I built the house yesterday_ but, _I was\nbuilding the house yesterday_. Again you may want to describe an action\nwhich will be _progressing_ or going on in the future. You do not say,\n_I shall build the house next week_ but, _I shall be building the house\nnext week_.\n\nSo we have progressive verb phrases.\n\n+134.+ +The present progressive describes an action as continuing or\nprogressing in the present.+\n\n+It is formed by using the present time form of the verb _be_ and the\npresent participle.+\n\nYou remember that the present participle is formed by adding _ing_\nto the simple form of the verb.\n\n Present Progressive\n\n _Singular_ _Plural_\n\n 1st. I am seeing. We are seeing.\n 2d. You are seeing. You are seeing.\n 3d. He is seeing. They are seeing.\n\n+135.+ +The past progressive time form describes an action which was\ncontinuing or progressing in the past. It is formed by using the past\ntime form of the verb _be_ and the present participle.+\n\n Past Progressive\n\n _Singular_ _Plural_\n\n 1st. I was seeing. We were seeing.\n 2d. You were seeing. You were seeing.\n 3d. He was seeing. They were seeing.\n\n+136.+ +The future progressive describes an action which will be\nprogressing or going on in the future. It is formed by using the future\ntime form of the verb _be_ and the present participle.+\n\n Future Progressive\n\n _Singular_ _Plural_\n\n 1st. I shall be seeing. We shall be seeing.\n 2d. You will be seeing. You will be seeing.\n 3d. He will be seeing. They will be seeing.\n\n+137.+ The perfect time forms also have a progressive form. There is a\ndifference of meaning in the _present perfect_ and its progressive form.\nYou say for instance, _I have tried all my life to be free_. You mean\nyou have tried until the present time and the inference is that now you\nhave ceased to try. But, if you say, _I have been trying all my life to\nbe free_, we understand that you have tried and are _still_ trying.\n\n+138.+ +So we have the present perfect progressive which describes an\naction which progressed in the past and continued up to the present\ntime. It is formed by using the present perfect form of the verb _be_\nand the present participle.+\n\n Present Perfect Progressive\n\n _Singular_ _Plural_\n\n 1st. I have been seeing. We have been seeing.\n 2d. You have been seeing. You have been seeing.\n 3d. He has been seeing. They have been seeing.\n\n+139.+ +The past perfect progressive describes an action which was\ncontinuing or progressing at some past time. It is formed by using the\npast perfect time form of the verb _be_ and the present participle.+\n\n Past Perfect Progressive\n\n _Singular_ _Plural_\n\n 1st. I had been seeing. We had been seeing.\n 2d. You had been seeing. You had been seeing.\n 3d. He had been seeing. They had been seeing.\n\n+140.+ +The future perfect progressive describes an action which will be\nprogressing at some future time. It is formed by using the future\nperfect time form of the verb _be_ and the present participle.+\n\n Future Perfect Progressive\n\n _Singular_ _Plural_\n\n 1st. I shall have been seeing. We shall have been seeing.\n 2d. You will have been seeing. You will have been seeing.\n 3d. He will have been seeing. They will have been seeing.\n\n\n Exercise 1\n\nIn the following sentences mark all the progressive forms, and note\nwhether they are present, past, future, present perfect, past perfect or\nfuture perfect.\n\n 1. The old order is passing.\n 2. Men will be struggling for freedom so long as slavery exists.\n 3. The class struggle has been growing more intense as wealth has\n accumulated.\n 4. The workers are realizing their power.\n 5. He had been talking for an hour when we arrived.\n 6. Next Monday I shall have been working for one year.\n 7. The workers will be paying interest on war debts for generations\n to come unless they repudiate.\n 8. While Marx was writing his books, he lived in abject poverty.\n 9. The Industrial Relations Commission has been investigating\n industrial conditions.\n 10. Ferrer was martyred because the Modern Schools were educating the\n people.\n 11. The nations of Europe had been preparing for war for many years.\n\n\n ACTIVE AND PASSIVE\n\n+141.+ Notice carefully the following sentences; select the subjects in\nthese sentences which show _who_ or _what_ performed the action; select\nthe subjects that show _who_ or _what_ receives the action. Do you\nnotice any difference in the meaning of these sentences? Do you notice\nany difference in their form?\n\n The engine struck the man.\n The man was struck by the engine.\n\n The system enslaves men.\n Men are enslaved by the system.\n\n Leaders often betray the people.\n The people are often betrayed by leaders.\n\nLet us look carefully at the first two sentences. You remember when we\nstudied transitive verbs we found that every transitive verb had an\n_object_ which was the receiver of the action expressed in the verb. Now\nyou notice in this first sentence, _The engine struck the man_, we have\nthe transitive verb _struck_. _Engine_ is the subject of the verb and\n_man_ is the object of the verb, the receiver of the action expressed by\nthe verb _struck_.\n\nNow in the sentence, _The man was struck by the engine_, we have the\nsame thought expressed but in a different manner. The word _man_, which\nwas the object of the verb _struck_ in the first sentence, has now\nbecome the subject of the sentence, and we have changed our verb form\nfrom _struck_ to _was struck_. In the first sentence of the subject,\n_engine_ was the _actor_. In the second sentence, _The man was struck by\nthe engine_, the subject of the sentence, _man_, is the _receiver_ of\nthe action expressed in the verb.\n\n+142.+ So we have thus changed the verb form from _struck_ to _was\nstruck_ to indicate that the subject of the verb is the receiver of the\naction. _Struck_ is called the active form of the verb because the\nsubject of the verb is the actor. _Was struck_ is called the passive\nform of the verb because the subject receives the action. Passive means\n_receiving_. In the passive form the subject is the receiver of the\naction expressed in the verb.\n\n+143.+ You remember that complete verbs have no object or complement,\ntherefore it would follow that they cannot be put in the passive form\nfor there is no object to become the receiver of the action. Take the\ncomplete verb, _sleep_, for example. We do not _sleep_ anything, hence\n_sleep_ has no passive form for there is no object which can be used as\nthe subject, the receiver of the action.\n\n+Only transitive verbs can be put into the passive form.+ Remember that\na transitive verb in the passive form is one that represents its subject\nas receiving the action.\n\nThe present, past, future and all the perfect time forms of transitive\nverbs can be changed from active to passive. The progressive time forms\ncan be changed into the passive, but it makes an awkward construction\nand should be avoided as much as possible. Occasionally, however, we\nfind it worth our while to use these forms, as for example:\n\n The book is being written by the man.\n\nThis is the passive form of the present progressive, _The man is writing\na book_.\n\n The book was being written by the man.\n\nThis is the passive form of the past progressive, _The man was writing\nthe book_.\n\n+144.+ The future progressive passive is awkward, and the present and\npast progressive forms are the only forms we find used in the passive.\nThe best writers use them sparingly for we can usually say the same\nthing by using the active form of the verb and have a sentence which\nsounds much better.\n\n\n Exercise 2\n\nAll the verbs in the following sentences are _transitive_ verbs in the\n_active_ form. Rewrite each sentence, putting the verb into the\n_passive_ form and making the _object_ of the _active_ verb the\n_subject_ of the _passive_ verb; as, for example, the first sentence\nshould be rewritten as follows:\n\n _War on Russia was declared by Germany on August 1, 1914._\n\n 1. Germany declared war on Russia, August 1, 1914.\n 2. Who will sign the Emancipation Proclamation of the Proletariat?\n 3. Marx and Engels wrote the Communist Manifesto.\n 4. Spain murdered Francisco Ferrer, October 13, 1909.\n 5. We celebrate the first of May as International Labor Day.\n 6. The people of Paris stormed the Bastille, July 14, 1789.\n 7. Wat Tyler was leading the English workers in rebellion against\n the King when the Mayor of London stabbed him in 1381.\n 8. The Inquisition burned Bruno at the stake for heresy in 1600.\n 9. The Paris Commune followed the German siege of Paris in 1871.\n\n\n SUMMARY\n\n+145.+ Now let us take the verb _see_ and name all the time forms which\nwe can describe with the changes in the verb forms which we have learned\nto make and also with the verb phrases which we can construct with the\nhelp of the verbs, _be_, _have_, _shall_ and _will_.\n\nFirst, we want to express the present, what is happening now, and we\nwant to put it in both the active and passive forms, so we say:\n\n +PRESENT TIME+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I see. I am seen.\n You see. You are seen.\n He sees. He is seen.\n\n _Plural_ _Plural_\n\n We see. We are seen.\n You see. You are seen.\n They see. They are seen.\n\nNote that the only change in the verb form in the present ACTIVE is the\n_s-form_ for the third person singular. In the present passive the only\nchange is the special form of the verb _be_ for the first and third\npersons, singular.\n\nWhen we want to tell what occurred yesterday or some time in the past,\nstated in the active and passive form, we say:\n\n +PAST TIME+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I saw. I was seen.\n You saw. You were seen.\n He saw. He was seen.\n\n _Plural_ _Plural_\n\n We saw. We were seen.\n You saw. You were seen.\n They saw. They were seen.\n\nWe have one other division of time which we must express--the future.\nPrimitive man doubtless lived principally in the present, but with the\ndevelopment of memory and the means of recording events by a written\nlanguage, he was able to make the deeds and achievements of the past a\nvital part of his life. But not until the faculty of thinking developed\nwas the mind able to project itself into the future and make tomorrow\nthe hope of today. Future time expresses hope, desire, growth.\n\n +FUTURE TIME+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I shall see. I shall be seen.\n You will see. You will be seen.\n He will see. He will be seen.\n\n _Plural_ _Plural_\n\n We shall see. We shall be seen.\n You will see. You will be seen.\n They will see. They will be seen.\n\nThen you remember we had to devise a way of describing an action\nperfected or completed at the present or at some time in the past or at\nsome time in the future--so we have present perfect, past perfect and\nfuture perfect.\n\n +PRESENT PERFECT+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I have seen. I have been seen.\n You have seen. You have been seen.\n He has seen. He has been seen.\n\n _Plural_ _Plural_\n\n We have seen. We have been seen.\n You have seen. You have been seen.\n They have seen. They have been seen.\n\n +PAST PERFECT+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I had seen. I had been seen.\n You had seen. You had been seen.\n He had seen. He had been seen.\n\n _Plural_ _Plural_\n\n We had seen. We had been seen.\n You had seen. You had been seen.\n They had seen. They had been seen.\n\n +FUTURE PERFECT+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I shall have seen. I shall have been seen.\n You will have seen. You will have been seen.\n He will have seen. He will have been seen.\n\n _Plural_ _Plural_\n\n We shall have seen. We shall have been seen.\n You will have seen. You will have been seen.\n They will have seen. They will have been seen.\n\n+146.+ But these are not all the phases of time which we can express. We\nhave progressive, continuous action. So each of these six time forms has\na progressive form.\n\n +PRESENT PROGRESSIVE+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I am seeing. I am being seen.\n You are seeing. You are being seen.\n He is seeing. He is being seen.\n\n _Plural_ _Plural_\n\n We are seeing. We are being seen.\n You are seeing. You are being seen.\n They are seeing. They are being seen.\n\n +PAST PROGRESSIVE+\n\n +Active+ +Passive+\n\n _Singular_ _Singular_\n\n I was seeing. I was being seen.\n You were seeing. You were being seen.\n He was seeing. He was being seen.\n\n _Plural_ _Plural_\n\n We were seeing. We were being seen.\n You were seeing. You were being seen.\n They were seeing. They were being seen.\n\nOnly the Present and Past Progressive forms have a passive form. The\nrest of the Progressive forms are expressed in the active forms only.\n\n +FUTURE PROGRESSIVE+\n\n _Singular_ _Plural_\n\n I shall be seeing. We shall be seeing.\n You will be seeing. You will be seeing.\n He will be seeing. They will be seeing.\n\n +PRESENT PERFECT PROGRESSIVE+\n\n _Singular_ _Plural_\n\n I have been seeing. We have been seeing.\n You have been seeing. You have been seeing.\n He has been seeing. They have been seeing.\n\n +PAST PERFECT PROGRESSIVE+\n\n _Singular_ _Plural_\n\n I had been seeing. We had been seeing.\n You had been seeing. You had been seeing.\n He had been seeing. They had been seeing.\n\n\n +FUTURE PERFECT PROGRESSIVE+\n\n _Singular_ _Plural_\n\n I shall have been seeing. We shall have been seeing.\n You will have been seeing. You will have been seeing.\n He will have been seeing. They will have been seeing.\n\n\n Exercise 3\n\nWrite the four following sentences in their active and passive forms, as\nthe sentence, _War sweeps the earth_, is written.\n\n 1. Education gives power.\n 2. Knowledge frees men.\n 3. Labor unions help the workers.\n 4. The people seek justice.\n\n +Present+ _Active_ War sweeps the earth.\n _Passive_ The earth is swept by war.\n\n +Past+ _Active_ War swept the earth.\n _Passive_ The earth was swept by war.\n\n +Future+ _Active_ War shall sweep the earth.\n _Passive_ The earth shall be swept by war.\n\n +Pres. Per.+ _Active_ War has swept the earth.\n _Passive_ The earth has been swept by war.\n\n +Past Per.+ _Active_ War had swept the earth.\n _Passive_ The earth had been swept by war.\n\n +Fut. Per.+ _Active_ War shall have swept the earth.\n _Passive_ The earth shall have been swept by war.\n\n\n Exercise 4\n\nUnderscore all the verbs and verb phrases in the following quotation.\nWrite all the time forms of the transitive verb, _lose_, as the time\nforms of the verb _see_ are written in the foregoing table.\n\n When we study the animal world and try to explain to ourselves that\n struggle for existence which is maintained by each living being\n against adverse circumstances and against its enemies, we realize that\n the more the principles of solidarity and equality are developed in an\n animal society, and have become habitual to it, the more chance it has\n of surviving and coming triumphantly out of the struggle against\n hardships and foes. The more thoroughly each member of the society\n feels his solidarity with each other member of the society, the more\n completely are developed in all of them those two qualities which are\n the main factors of all progress; courage, on the one hand, and, on\n the other, free individual initiative. And, on the contrary, the more\n any animal society, or little group of animals, loses this feeling of\n solidarity--which may chance as the result of exceptional scarcity or\n else of exceptional plenty--the more the two other factors of\n progress, courage and individual initiative, diminish; in the end they\n disappear, and the society falls into decay and sinks before its foes.\n Without mutual confidence no struggle is possible; there is no\n courage, no initiative, no solidarity--and no victory!--_Kropotkin_.\n\n\n\n\n SPELLING\n\n LESSON 8\n\n\nIn pronouncing words of more than one syllable we always lay a little\ngreater stress upon one syllable of the word; that is, that syllable\nreceives the emphasis of the voice so as to make it more prominent than\nthe other syllables. This is called accent, and the syllable which\nreceives the special stress is called the accented syllable.\n\n+Accent is the stress of the voice upon one syllable of the word.+\n\nYou will notice when you look up the pronunciation of words in your\ndictionary that a little mark called the accent mark is placed after the\naccented syllable, as for example: di-vide'.\n\nMany words differ in meaning according to which syllable receives the\naccent. Our spelling lesson for this week contains a number of these\nwords.\n\nThese words, when accented on the first syllable, are nouns; when\naccented on the second syllable, they are verbs.\n\n +Monday+\n\n Con' tract Con tract'\n Pro' test Pro test'\n Rec' ord Re cord'\n Im' port Im port'\n De' tail De tail'\n\n +Tuesday+\n\n Con' vert Con vert'\n Con' flict Con flict'\n Prog' ress Pro gress'\n Im' press Im press'\n Ref' use Re fuse'\n\n +Wednesday+\n\n Con' test Con test'\n Con' duct Con duct'\n Proj' ect Pro ject'\n Des' ert De sert'\n Ex' tract Ex tract'\n\n +Thursday+\n\n Con' trast Con trast'\n Con' sort Con sort'\n Reb' el Re bel'\n Con' script Con script'\n Pres' ent Pre sent'\n\n +Friday+\n\n Com' pound Com pound'\n Re' tail Re tail'\n Com' press Com press'\n Im' print Im print'\n Com' bine Com bine'\n\n +Saturday+\n\n Con' fine Con fine'\n Sus' pect Sus pect'\n Com' mune Com mune'\n Ex' port Ex port'\n In' crease In crease'\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 9\n\n\nDear Comrade:\n\nYou have been studying several weeks now in this Plain English Course\nand we trust you are enjoying the unfolding of the powers of expression.\nWe have been necessarily studying rules to some extent but you have seen\nhow these grew out of the need for expression. We have been breaking the\nsentence up into its different parts. First we had the names of things\nand now we are studying the words used to tell what these things _do_\nand _are_--namely verbs. And as our life has grown complex and our\npowers of thinking diversified covering the whole range of time, past,\npresent and future, we have had to invent many forms of the verb to\nexpress it all.\n\nNow do not try to commit these facts concerning the verb to memory. You\nare not studying English in order to know rules. You are studying\nEnglish that you may be able to say and write the things you _think_. So\nfirst of all, _think_, _think_! That is your inalienable right! Do not\naccept anything just by blind belief. Think it out for yourself. Study\nuntil you see the '_why_' of it all. \"Independent thinking has given us\nthe present, and we will forever continue to make tomorrow better than\ntoday. The right to think is inalienable, or a man is a machine. Thought\nis life or a human soul is a thing.\"\n\nAnd do not lack the courage of your own thoughts. _You_ do not need to\ncringe or apologize to any man. \"Our life is not an apology but a life.\"\nDare to think and dare to express and live your thought.\n\nDid you ever read Emerson's definition of genius? \"To believe your own\nthought, to believe that what is true for you in your private heart is\ntrue for all men,--that is genius.\" Then he says, \"We dismiss without\nnotice our own thoughts, because they are ours. Tomorrow a stranger will\nsay with masterly good sense, precisely what we have thought and felt\nall along and we shall be forced to take with shame our own opinion from\nanother.\"\n\nHave you not experienced this? How often we hear some one express a\ntruth and we say to ourselves, \"That is just what I have long believed\nbut I have never dared say so.\" We have been so taught all our lives to\ndepend on some outside power and discredit the power within ourselves,\nthat we pay no attention to the thoughts that are ours for who are we\nthat we should dare to think and perchance disagree with those who have\nassumed authority over us! But that is precisely what we should dare to\ndo--to think and to do our own thinking always. Who dares place anything\nbefore a man!\n\nSo _think_ as you study these lessons and use these rules and formulas\nsimply as means to an end, as tools to aid you in expressing these\nthoughts.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n PARTICIPLES\n\n+147.+ We have found that the verb has five forms, made by internal\nchanges in the verb itself,--the present time form, the s-form, the past\ntime form, the present participle and the past participle.\n\nWe have also found that we can express various time forms by verb\nphrases formed by using the helping verbs, _shall_, _will_, _have_ and\n_be_ with one of the verb forms. All of these forms are used as the\nasserting word in the sentence. So long as the verb or verb phrase forms\nthe predicate--the word or words that assert something of the\nsubject--it still remains a verb. But we have found that the participle\nforms of the verb may be used as other parts of speech while still\nretaining some of the qualities of the verb.\n\n+148.+ You remember a sentence which we used when we studied\nparticiples, _Making shoes is his work_. Here we have the present\nparticiple _making_, with its object _shoes_, used as the subject of the\nverb _is_. Now a noun never takes an object, so _making_ in this\nsentence is partly a verb, partly a noun, and is called a participle,\nwhich means _partaker_.\n\nWe have studied and used two forms of participles, the present and the\npast participle. The present participle always ends in _ing_ and\nexpresses action or existence in the present, or at the time mentioned\nin the sentence. For example, _being_, _bringing_, _working_, _seeing_,\n_loving_, _hating_, etc.\n\nThe past participle we found to be one of the principal parts of the\nverb. It expresses action or existence which is past or completed, at\nthe time mentioned in the sentence. It is formed by adding _d_ or _ed_\nto the regular verbs and by a change in the form in irregular verbs. For\nexample, regular verbs: _learned_ from _learn_, _defeated_ from\n_defeat_, _watched_ from _watch_. Irregular verbs: _taught_ from\n_teach_, _seen_ from _see_, _won_ from _win_.\n\n+We have found that these participles may be used either as nouns or as\nadjectives.+ As for example:\n\n The _crying_ of the child annoyed the people.\n The _crying_ child ran to its mother.\n The _coming_ of the new day will bring peace.\n We await the _coming_ day of peace.\n\n\n PARTICIPLE PHRASES\n\n+149.+ The present and the past participles are each single words; but\nwe may also have participle phrases; that is, two or more words used as\na participle, as for example:\n\n His _having joined_ the strikers caused him to lose his job.\n The man, _having been discharged_, left the mill.\n\nIn these sentences we have the participle phrases, _having joined_ and\n_having been discharged_. _Having joined_ is a participle phrase used as\na noun, the subject of the verb _caused_. _Having been discharged_ is a\nparticiple phrase used as an adjective to modify the noun _man_. Notice\nthat _having joined_ is an active participle describing the action\nperformed by the man who is referred to by the pronoun _his_. _Having\nbeen discharged_ is a passive participle expressing an action of which\nthe subject of the sentence, _man_, is the receiver.\n\nThese are both perfect participles, expressing actions which are\ncomplete at the present time.\n\n+150.+ We have also progressive participles expressing action which is\ncontinuing or progressing. These progressive participles are also used\nin both the active and the passive forms. The progressive active\nparticiple is formed by using _having been_ with the present participle,\nas _having been working_. The progressive passive participle is formed\nby using _being_ with the past participle, as for example, _being\nwatched_, _being driven_, _being gone_, etc. So we have six participles,\nthree active and three passive.\n\nNote the following table:\n\n +Active+\n\n _Present._ Sending.\n\n _Perfect._ Having sent.\n\n _Progressive._ Having been sending.\n\n\n +Passive+\n\n _Past._ Sent.\n\n _Perfect._ Having been sent.\n\n _Progressive._ Being sent.\n\n+These participle phrases may be used either as nouns or as\nadjectives.+\n\n\n Exercise 1\n\nIn the following sentences mark the participles and the participle\nphrases. Underscore those used as _nouns_ with a single line; those used\nas _adjectives_ with two lines.\n\n 1. He denies having been hired by the employer.\n 2. Our friends, having arrived, joined us at dinner.\n 3. The rain, falling incessantly, kept us from going.\n 4. Having often seen him passing, I judged he lived near.\n 5. The man, being discouraged and ill, was unable to do his work well.\n 6. Happiness shared is happiness doubled.\n 7. Having finished his work, he rests at last.\n 8. The army, beaten but not vanquished, waited for the morrow.\n 9. The men, having been unemployed for months, were desperate.\n 10. Being prepared will not save us from war.\n 11. \"Rest is not quitting this busy career;\n Rest is the fitting of self to its sphere.\n It's loving and serving the highest and best;\n It's onward, not swerving; and that is true rest.\"\n\n\n Exercise 2\n\nWrite the six participle forms of the verbs _see_ and _teach_, and use\nin sentences of your own construction.\n\n\n INFINITIVES\n\n+151.+ We have found that the various forms of the participles may be\nused as other parts of speech. They partake of the nature of a verb and\neither of a noun or an adjective. Notice the following sentences:\n\n Traveling is pleasant.\n Eating is necessary.\n\nCan you think of any other way in which you could express the same\nthought? Do you not sometimes say,\n\n To travel is pleasant.\n To eat is necessary.\n\nWe have expressed practically the same thought in these two sentences,\nwhich is expressed in the sentences above, where we used the participle.\n_To travel_ and _to eat_ are used as nouns, subjects of the verb _is_\njust as _traveling_ and _eating_ are used as nouns, the subjects of the\nverb _is_.\n\nHere we have another form of the verb used as a noun. When we use the\nverb in this way, we are not speaking of the _traveling_ or _eating_ as\nbelonging to or being done by any particular person, nor do we indicate\nwhether one person or more than one is concerned in the action. It might\nbe anyone doing the traveling or eating, and it might be one person or a\nthousand. We are making a general statement of everybody in the world,\nso we call this form the _infinitive_.\n\n+152.+ Infinite means _unlimited_, without limit as to persons or\nnumber. Almost every verb in the language may be used in this way, and\nsince _to_ is generally used before the infinitive, _to_ is often called\nthe sign of the infinitive. For example:\n\n _To be_, or not _to be_, that is the question.\n _To have_ and _to hold_ is the problem.\n He likes _to travel_.\n\nYou note in all of these infinitives _to_ is used with the simple form\nof the verb.\n\n+153.+ _To_ is generally omitted after verbs like _help_, _hear_, _bid_,\n_feel_, _let_, _make_, _see_ and _have_, or words of similar meaning.\nFor example:\n\n Help me (to) find it.\n He bade me (to) stay.\n Feel it (to) shake.\n Make him (to) come.\n Hear me (to) sing.\n Let us (to) go.\n See him (to) run.\n Have him (to) copy this.\n\n+154.+ _To_ is also omitted after _need_ and _dare_ when _not_ is used.\n\n They need to work.\n They need not work.\n\n They dared to come.\n They dared not come.\n\n+155.+ _To_ is sometimes omitted after prepositions:\n\n He will do anything for his class, except (to) fight for it.\n He would do nothing but (to) go away.\n\n+156.+ We have a number of different forms of the infinitive, both\nactive and passive. Note the following table:\n\n +Active+\n\n _Present._ To love.\n _Perfect._ To have loved.\n _Present Prog._ To be loving.\n _Perfect Prog._ To have been loving.\n\n +Passive+\n\n _Present._ To be loved.\n _Perfect._ To have been loved.\n\n+157.+ Notice that only the _present_ and _perfect_ infinitives have the\n_passive_ form. The progressive infinitives cannot be used in the\npassive. Remember also that only _incomplete_ verbs, those which require\nan object to receive the action, can have a passive form.\n\nThe verb _loved_, which we have used in the above table, has a passive\nform because it is an incomplete verb, for there must be that which is\nthe object of our love.\n\n+158.+ The complete verbs,--verbs which require no object,--cannot have\na passive form for there is no object to become the receiver of the\naction. Take for example the verb _dwell_. This is a complete verb which\ncan have no passive form. You cannot dwell anything, therefore you\ncannot say _to be dwelt_ or _to have been dwelt_.\n\n+So complete verbs have only the four active forms+, as follows:\n\n +Active+\n\n _Present._ To dwell.\n _Perfect._ To have dwelt.\n _Present Prog._ To be dwelling.\n _Perfect Prog._ To have been dwelling.\n\n+159.+ Infinitives, like participles, may be used either as nouns or\nadjectives. When used as nouns, they are used in the various ways in\nwhich nouns are used. The infinitive may be the _subject_ of a sentence,\nthus:\n\n _To hesitate_ now will be fatal.\n _To be defeated_ is no crime.\n\n+160.+ The infinitive may be the _object_ or _complement_ of the verb.\nFor example:\n\n He wanted _to see_ you.\n His desire is _to learn_.\n\n+161.+ The infinitive may be used as the object of a _preposition_; as,\n\n He is about _to go_.\n They will do anything for the cause except _to live_ for it.\n\n+162.+ The infinitive may be used as an adjective to modify a noun. For\nexample:\n\n He showed me the way _to go_.\n We must have food _to eat_ and clothes _to wear_.\n The question _to be decided_ is before us.\n Claim your right _to live_.\n\n+163.+ The infinitive may also be used as an adverb to modify the\nmeaning of a verb, adjective or adverb, thus:\n\n He was forced _to go_.\n They are slow _to learn_.\n The fruit was not ripe enough _to eat_.\n\nNote that the infinitives in these sentences may all be changed into\nadverb phrases. As for example in the first sentence, He was forced _to\ngo_, the infinitive _to go_, which modifies the verb _forced_, may be\nchanged to the adverb phrase, _into going_, thus, _He was forced into\ngoing_. In the second sentence, _They are slow to learn_, the infinitive\n_to learn_ may be changed into the adverb phrase _in learning_, thus,\n_They are slow in learning_. In the last sentence, _The fruit is not\nripe enough to eat_, the infinitive _to eat_, which modifies the adverb\n_enough_, may be changed into the adverb phrase, _for eating_, as for\nexample, _The fruit was not ripe enough for eating_.\n\n+164.+ The infinitive is quite a useful form of the verb, and we will\nfind that we use it very frequently in expressing our ideas. While it is\nnot the asserting word in the sentence, it retains the nature of a verb\nand may have both an object and an adverb modifier. As for example, in\nthe sentence:\n\n I wish _to learn_ my lesson quickly.\n\n_To learn_ is the infinitive, used as a noun, the object of the verb\n_wish_. The infinitive also has an object, to learn--_what?_ _My lesson_\nis the object of the infinitive _to learn_. We also have an adverb\nmodifier in the adverb _quickly_, which tells _how_ I wish to learn my\nlesson. So the infinitive retains its verb nature, in that it may have\nan object and it may be modified by an adverb.\n\n\n Exercise 3\n\nNotice carefully the use of the infinitives in the following sentences.\nUnderscore all infinitives.\n\n 1. To remain ignorant is to remain a slave.\n 2. Teach us to think and give us courage to act.\n 3. Children love to be praised, but hate to be censured.\n 4. To obey is the creed taught the working class by the masters.\n 5. To be exploited has always been the fate of the workers.\n 6. Ferrer wrote on his prison wall, \"To love a woman passionately, to\n have an ideal which I can serve, to have the desire to fight\n until I win--what more can I wish or ask?\"\n 7. The people wish the man to be punished for the crime.\n 8. Primitive man found plenty of wood to burn.\n 9. We have learned to use coal and oil.\n 10. The lecture to have been given this evening has been postponed.\n 11. They are eager to hear the news.\n 12. He has failed to come.\n 13. We felt the house shake on its foundation.\n 14. Have him find the book for me.\n 15. To be defeated is no crime; never to have dared is the real crime.\n 16. The rich will do anything for the poor except to get off their\n backs.\n 17. To have slept while others fought is your shame.\n 18. Claim your right to do, to dream and to dare.\n\n\n Exercise 4\n\nWrite sentences containing the six infinitive forms of the verb _obey_.\n\n\n DON'TS FOR INFINITIVES\n\n+165.+ +Don't split your infinitives.+ Keep the _to_ and the infinitive\ntogether as much as possible. Don't say, _They intended to never come\nback_. Say rather, _They intended never to come back_. Sometimes,\nhowever, the meaning can be more aptly expressed by placing the adverb\nmodifier between the _to_ and the infinitive, as for example:\n\n To almost succeed is not enough.\n It will be found to far exceed our expectations.\n\nIn these sentences the adverbs _almost_ and _far_ express our meaning\nmore closely if they are placed between the _to_ and the infinitive.\nOrdinarily, however, do not split your infinitives, but place the adverb\nmodifier either before or after the infinitive.\n\n+166.+ +Don't use _to_ by itself without the rest of the infinitive.+\nDon't say, _Do as I tell you to_. Say instead, _Do as I tell you to do_;\nor, _Do as I tell you_. Don't say, _He deceived us once and he is likely\nto again_. Say rather, _He deceived us once and he is likely to deceive\nus again_, or _to do so again_.\n\n+167.+ +Don't use _and_ for _to_. Don't say, _Try and go if you can_.\nSay instead, _Try to go if you can_.\n\nCorrect the following sentences:\n\n We ought to bravely fight for our rights.\n I will do all my employer tells me to.\n We shall try and get our lessons.\n I ought to at least help my comrades but I am afraid to.\n\n\n Exercise 5\n\nStudy carefully the infinitives in the following quotation. Notice which\nare active and which are passive infinitives.\n\n The twenty thousand men prematurely slain on a field of battle, mean,\n to the women of their race, twenty thousand human creatures _to be\n borne_ within them for months, _to be given_ birth to in anguish, _to\n be fed_ from their breasts and _to be reared_ with toil, if the\n members of the tribe and the strength of the nation are _to be\n maintained_. In nations continually at war, incessant and unbroken\n child-bearing is by war imposed on all women if the state is _to\n survive_; and whenever war occurs, if numbers are _to be maintained_,\n there must be an increased child-bearing and rearing. This throws upon\n woman, as woman, a war tax, compared with which all that the male\n expends in military preparations is comparatively light.\n\n It is especially in the domain of war that we, the bearers of men's\n bodies, who supply its most valuable munition, who, not amid the\n clamor and ardor of battle, but singly, and alone, with a\n three-in-the-morning courage, shed our blood and face death that the\n battle-field might have its food, a food more precious to us than our\n heart's blood; it is we, especially, who, in the domain of war, have\n our word _to say_, a word no man can say for us. It is our intention\n _to enter_ into the domain of war and _to labor_ there till in the\n course of generations we have extinguished it.--_Olive Schreiner_.\n\n\n Exercise 6\n\nMark the participles and infinitives.\n\n Gold! Gold! Gold! Gold!\n Bright and yellow, hard and cold,\n Molten, graven, hammer'd and roll'd;\n Heavy to get, and light to hold;\n Hoarded, barter'd, bought, and sold,\n Stolen, borrow'd, squander'd, doled:\n Spurn'd by the young, but hugg'd by the old\n To the very verge of the churchyard mould;\n Price of many a crime untold:\n Gold! Gold! Gold! Gold!\n Good or bad a thousand-fold!\n How widely its agencies vary--\n To save--to ruin--to curse--to bless--\n As even its minted coins express,\n Now stamp'd with the image of Good Queen Bess,\n And now of a bloody Mary.--_Thos. Hood_.\n\n\n\n\n SPELLING\n\n LESSON 9\n\n\nIn our English lessons, we have been studying the division of words into\nparts of speech. We have been studying them as we use them in expressing\nour thoughts but we may study them in other ways also. We may study them\nas words alone.\n\nStudied in this way we find that we have simple, compound and derivative\nwords. For example, _man_, _man-slaughter_, _manly_. _Man_ is a simple\nword. _Man-slaughter_ is a compound word formed of two simple words.\n_Manly_ is a derivative word derived from _man_.\n\nWhen a compound word is first formed, it is usually written with a\nhyphen; but after the word has been used awhile the hyphen is often\ndropped and the two parts are written together as a simple word.\n\n+A simple word is a single word which cannot be divided into other words\nwithout changing its meaning.+\n\n+A compound word is composed of two or more simple words into which it\nmay be divided, each retaining its own meaning.+\n\n+A derivative word is one which is derived from a simple word by the\naddition of another syllable.+\n\nIn next week's lesson we will take up the study of these derivatives.\n\nDivide the compound words in this week's lesson into the simple words of\nwhich they are composed.\n\n +Monday+\n\n Birthday\n Coal-tar\n Craftsman\n Foreman\n Gunpowder\n\n +Tuesday+\n\n Handkerchief\n Headquarters\n Lawsuit\n Lockout\n Bookkeeper\n\n +Wednesday+\n\n Motorman\n Newspaper\n Pasteboard\n Postage-stamp\n Postmaster\n\n +Thursday+\n\n Salesman\n Second-hand\n Shirtwaist\n Sidewalk\n Staircase\n\n +Friday+\n\n Trademark\n Time-table\n Typewriter\n Tableware\n Sewing-machine\n\n +Saturday+\n\n Undergarment\n Underhand\n Water-mark\n Woodwork\n Workshop\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 10\n\n\nDear Comrade:\n\nWe have been studying this course in Plain English for some weeks now\nand I trust that you have been enjoying as well as benefiting by the\nstudy of our wonderful and expressive language. Did you ever stop to\nthink what a wonderful step it was in evolution when man first began to\nuse the spoken word? And yet it was a still more wonderful step in\nadvance when he began to use the written word for our highest evolution,\nand development would have been impossible without the help of written\nspeech. An illiterate man may be a good workman and prosperous so far as\nthe material things of life and his immediate contact with his fellow\nmen are concerned, but we have only to think for a moment of what this\nworld would be if we had no written language, to understand what a\nmighty power it has been in evolution.\n\nSuppose we had no way by which we could communicate with our friends at\na distance. Suppose there were no written words by which we could set\ndown the countless dealings between man and man. What a hopeless tangle\nthis social life of ours would soon become! Suppose also that we had no\nknowledge of the past, no knowledge of the discoveries and inventions of\npast generations except that which could be handed down to us through\noral speech. All our knowledge of history, of the deeds and development\nof the past, all the observations by which science has uncovered to us\nthe mysteries of nature would be largely lost to us. It was the\ninvention of writing alone which made possible man's growth from\nbarbarism to civilization, and it is more true than we oftentimes\nrealize, that it is \"only a wall of books that separates the civilized\nman of to-day from the savage of yesterday.\" And yet I wonder if we have\never stopped to think how this art of writing developed. Knowledge of\nthe alphabet and of the letters by which we form our words and hence are\nable to express our ideas, has become such a common-place thing to us\nthat we have forgotten what a wonder it is and how it has slowly grown\nand developed through the centuries. Yet there are races to-day that\nhave no written language such as we know and to whom our written\nlanguage seems truly a miracle.\n\nThe story is told of an Indian who was sent from one colony to another\nwith four loaves of bread accompanied by a letter stating their number.\nThe Indian ate one of the loaves and of course, was found out. The next\ntime when he was sent upon a similar errand he repeated the theft but he\ntook the precaution to hide the letter under a stone while he was eating\nthe bread so that it might not see him!\n\nBut it is only the things that we do not understand which we invest with\nmystery and as we study the story of the alphabet in this series of\nletters we find that it has been a natural development accomplished by\nthe growing powers of man. In succeeding letters we will trace this most\ninteresting story of the alphabet.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n HELPING VERBS\n\n+168.+ We have found that whenever a verb is used by itself in making an\nassertion it denotes either present or past time. When we use a verb\nphrase, it expresses some other time than the past or present. These\nverb phrases are formed by using _shall_, _will_, _have_, _had_, and the\nvarious forms of the verb _be_ with some form of the principal verb.\nThese verbs which help to form verb phrases are called _helping_, or\n_auxiliary verbs_. Auxiliary means helping.\n\nWe have used _have_ and _had_ with the past participle to form the\npresent perfect and past perfect time forms. We have used _shall_ and\n_will_ with different forms of the verb to denote future time, and we\nhave used different forms of the verb _be_ in making the various other\ntime forms. So _shall_, _will_, _have_, _had_ and the various forms of\nthe verb _be_ are _helping verbs_, which we use to help us in making\nverb phrases.\n\n+169.+ But these are not all of the helping verbs. There are other\nhelping verbs which we use in forming verb phrases to express different\nideas. These are such verbs as _should_ and _would_, _may_ and _might_,\n_can_ and _could_, _must_ and _ought_, _do_ and _did_.\n\n\n Exercise 1\n\nFill the blank spaces in the following sentences with the appropriate\nforms of the helping verbs, _shall_, _will_, _have_, _had_ and _be_.\n\n 1. When......the workers organize?\n 2. Education......help us win.\n 3. The world......had enough of war.\n 4. We......deceived by the masters.\n 5. The workers......organized into craft unions.\n 6. They......never ceased the struggle.\n 7. The state......founded on exploitation.\n 8. Mutual aid......been an important factor in evolution.\n 9. The truth......taught to the people.\n 10. The victory......gained by the proletariat.\n 11. The nations of Europe......preparing for war for years.\n 12. The International......recognized war for defense.\n 13. We......not made the class distinctions, but we......recognize\n them as long as they exist.\n 14. The evolution of animals and the evolution of\n plants......proceeded according to the same general laws.\n 15. We......never win while the majority remains ignorant.\n 16. The strikers......betrayed by their leaders.\n\n\n SHOULD AND WOULD\n\n+170.+ _Should_ and _would_ are the past-time forms of _shall_ and\n_will_. We use them to express action or existence dependent upon some\ncondition, thus:\n\n I should go if I were well enough.\n He should join us if you asked him.\n\nIn these sentences _should_ and _would_ express action which is possible\nnow or will be in the future, provided some other action takes place.\n\nThe same distinction which we found made in the use _shall_ and _will_\nhas been made with _should_ and _would_; that is, that _should_ used\nwith the first person, expresses action dependent upon condition; but\n_would_, used with the first person, implies exercise of the will. This\nrule is not closely followed, though it expresses a nice distinction in\nthe use of _should_ and _would_. In ordinary usage we use either\n_should_ or _would_ with the first person without any distinction of\nmeaning, as for example:\n\n I should struggle on even if it meant death.\n I would stand for my principles though I stood entirely alone.\n\nWe do not use _should_ however, with the second and third persons to\nexpress an action or existence dependent upon some condition. _Should_\nused with the second and third person implies obligation. _Would_ is\nused with the second or third person to express an action dependent upon\nsome condition, as for example:\n\n He would not go, even if you insisted.\n They would come if you invited them.\n You would believe him if you could hear him.\n You would be surprised if I should tell you the reason.\n\n+171.+ _Should_ and _would_ in all of the sentences which we have quoted\nare used to express action or existence dependent upon some condition\nwhich is expressed in that part of the sentence introduced by such\nconjunctions as _if_ and _though_.\n\nThe parts of the sentence introduced by these conjunctions express the\ncondition upon which the other action is dependent. When we use _should_\nin sentences without this condition, it means practically the same as\n_ought_, and implies an obligation. We use _should_ with the first and\nsecond and third persons when we use it with this meaning, as for\nexample:\n\n I should have gone yesterday.\n You should be with us in this fight.\n They should never fear defeat.\n\n+172.+ _Ought_ could be used in all these sentences and express\npractically the same meaning. _Should_ used in this way implies\nobligation.\n\n\n Exercise 2\n\nStudy carefully the following sentences. Write in the blank space\npreceding each sentence the number of the paragraph in the lesson which\ngoverns the use of the helping verb in that sentence.\n\n 1. ...... The workers should organize if they desire to control\n production.\n 2. ...... The proletariat would destroy this system if they\n understood their power.\n 3. ...... Every worker would join his fellows if he could but realize\n the class struggle.\n 4. ...... We would all enjoy plenty if we produced for use instead of\n for profit.\n 5. ...... The ruling class would not give up their privileges even\n though they knew that their cupidity endangers society.\n 6. ...... The injury of one should be the injury of all.\n 7. ...... The workers' International should stand for the\n international solidarity of the workers.\n 8. ...... You should never fear the ridicule of little minds.\n 9. ...... You would never fear ridicule if you were conscious of your\n own power.\n 10. ...... No man should fear to think for himself.\n 11. ...... No man would fear to think for himself if the world were\n truly free.\n 12. ...... Compromise now would mean defeat.\n\n\n MAY AND MIGHT\n\n+173.+ _May_ used as a helping verb means present permission in regard\nto an action or possession, as:\n\n You may come with us.\n He may have the money.\n\n+174.+ It may also mean a possible action or possession. _You may come\nwith us_, for example, might mean that some time in the future it is\npossible that you will come with us. _He may have the money_, might mean\neither _He is given permission to have the money_, or _It is possible\nthat he has it_.\n\n_May_, used with many verb forms, means _it is possible_. For example:\n_He may be hungry_, _He may have starved_. _He may have been starving_;\nthat is, it is possible that _he is hungry_; that _he has starved_; that\n_he was starving_.\n\n+175.+ _Might_ is the past form of _may_ and expresses past permission\nto do or to be and also possibility in the past. For example: _The\nofficer said he might go_. That is, he gave him permission to go. _You\nmight have helped your comrades_; that is, _you had the power to have\nhelped_.\n\n_Might_ is also used to express permission or the power to do in the\npresent and future, on condition. For example:\n\n He might find work if he were trained.\n The workers might destroy this insane system if they would.\n\n\n Exercise 3\n\nStudy carefully the following sentences. Write in the blank space\npreceding each sentence the number of the paragraph in the lesson which\ngoverns the use of the helping verbs _may_ or _might_ in that sentence.\n\n 1. ...... The solidarity of the workers might have averted this war.\n 2. ...... \"Of all sad words of tongue or pen,\n The saddest are these--'it might have been.'\"\n 3. ...... You might join us.\n 4. ...... The people struggle that they may live.\n 5. ...... Try; you might succeed.\n 6. ...... The day may come when this day's deeds shall be remembered.\n 7. ...... Victory might be ours if we dared to face the issue.\n 8. ...... \"Men may come and men may go;\n But I go on forever.\"\n 9. ...... It seemed possible that we might win.\n 10. ...... May we ever be loyal and true!\n 11. ...... It appeared for a time that we might be involved in war.\n 12. ...... Let come what may, we will not yield.\n\n\n CAN AND COULD\n\n+176.+ _Can_ is the present-time form and _could_ the past-time form,\nand both imply ability or power to do or to be. _You can go_ means _You\nare able to go_,--_You have the power to go_. _You may go_ means _You\nhave permission to go_. _Can_ is often used when we should use _may_,\nwhen we mean to give permission. Habit plays a great part in our life\nand knowledge of the right way does not always suffice. It is only\ncontinued effort that will establish correct habits of speech. Good\nEnglish would be easy of accomplishment if \"to do were as easy as to\nknow what it were good to do.\"\n\nWe are too often like the mother in the story. \"Can I have a piece of\npie?\" asked the child. \"May I?\" the mother corrected. Then the child\nasked, \"May I have a piece of pie?\" and the mother answered, \"Yes, you\ncan.\" Knowledge said, _may_; habit said _can_, and the ready tongue\nobeyed the force of habit.\n\nSay the correct word over and over aloud until it sounds right to your\near and flows readily to your tongue.\n\n+177.+ _Could_ is sometimes used in the present sense to denote power to\ndo, conditioned upon willingness, as:\n\n He could if he would.\n\n\n Exercise 4\n\nStudy carefully the following sentences. Write in the blank space\npreceding each sentence the number of the paragraph in the lesson which\ngoverns the use of the helping verbs _can_ or _could_ in that sentence.\n\n 1. ...... I can say love when others say hate;\n I can say every man when others say one man;\n What can I do? I can give myself to life,\n When other men refuse themselves to life.\n 2. ...... No one can be free till all are free.\n 3. ...... They could win their freedom if they would prepare\n themselves to be free.\n 4. ...... What can I do, being alone?\n 5. ...... If all men could catch the vision of freedom, wars would\n cease.\n 6. ...... Could you find a better way to spend your time than in\n study?\n 7. ...... Men would rise in revolt if they could know the facts.\n\n\n MUST AND OUGHT\n\n+178.+ _Must_ and _ought_ imply obligation. _Must_ conveys the idea of\nbeing obliged to do an action from necessity or compulsion, as,\n\n You must have known it.\n He must go.\n\n_Ought_ was originally the past time form of _owe_, hence means _to be\nindebted to_, _to owe_. It conveys the idea of a moral obligation, as,\n\n You ought to help the cause.\n You ought to understand.\n\n+179+. _Ought_ is always used with the infinitive, and the same form is\nused to express both the present and the past time. The difference in\ntime is expressed by a change in the infinitive instead of a change in\nthe form of the helping verb. With _may_ and _might_ and _can_ and\n_could_, present and past time are expressed by a change in the form of\nthe helping verb. With the helping verb _ought_, the difference in time\nis expressed in the infinitive. For example:\n\n He ought to pay us our wages.\n\nThis means, _He owes it to us to pay us our wages now_.\n\n He ought to have paid us our wages.\n\nThis means, _He owed it to us to pay us our wages some time in the\npast_.\n\n+180.+ The present infinitive is used with the helping verb _ought_ to\nexpress present time and the perfect infinitive is used with _ought_ to\nexpress past time.\n\n\n Exercise 5\n\nStudy carefully the following sentences. Write in the blank space\npreceding each sentence the number of the paragraph in the lesson which\ngoverns the use of the helping verb _must_ or _ought_ in that sentence.\n\n 1. ...... Service must be the key note of the future.\n 2. ...... Competition must give place to co-operation.\n 3. ...... Ought we to fear, who know the truth?\n 4. ...... Government ought to be the administration of things.\n 5. ...... No man ought to have the power of life and death over any\n other human being.\n 6. ...... It may cost much but humanity must be set free at any cost.\n 7. ...... What ought to be the attitude of the workers toward war?\n 8. ...... \"For man must work and woman must weep,\n For there is little to do and many to keep.\"\n 9. ...... The day must come when we can live the dream.\n\n\n DO AND DID\n\n+181.+ _Do_ and _did_ are used as helping verbs to give emphasis--to\nform emphatic verb phrases. _Do_ is the present time form and _did_ the\npast time form, as for example:\n\n I do wish you would come.\n I did hope he would win.\n\n+182.+ When we use the negative _not_ we use the helping verbs _do_ and\n_did_ to form our verb phrases. For example, we do not say:\n\n I obey not.\n I walked not.\n He comes not.\n They arrived not.\n\nBut in expressing the present and past time forms with the negative\n_not_, we say instead:\n\n I do not obey.\n I did not walk.\n He does not come.\n They did not arrive.\n\n+183.+ We also use _do_ and _did_ with the present and past time forms\nof the verb in writing interrogative sentences. For example, we do not\nsay:\n\n Comes he with them?\n Studied you yesterday?\n Found they the book?\n Think you it is true?\n\nBut we say instead:\n\n Does he come with them?\n Did you study yesterday?\n Did they find the book?\n Do you think it is true?\n\n\n Exercise 6\n\nWrite in the blank space before each sentence the number of the\nparagraph which governs the use of the helping verb _do_ or _did_ in\nthat sentence.\n\n 1. ...... Slaves do not think; they obey.\n 2. ...... Men do not obey; they think.\n 3. ...... Do you know that two per cent of the people own sixty per\n cent of the wealth?\n 4. ...... The children of the masses do not have the opportunity to\n attend school.\n 5. ...... Did not every nation claim a war for defense?\n 6. ...... \"We did not dare to breathe a prayer,\n Or give our anguish scope.\"\n 7. ...... We do desire the freedom of the people.\n 8. ...... We did hope that war might be averted.\n\n+Let us sum up the auxiliary or helping verbs.+\n\n+184.+ Helping verbs are used to express:\n\n +The different time forms+--_shall_, _will_, _have_, _had_, _be_.\n +Power to do or to be+--_can_, _could_, _might_.\n +Permission+--_may_ and _might_.\n +Possibility+--_may_ and _might_.\n +Obligation+--_must_, _ought_ and _should_.\n +Necessity+--_must_.\n +Condition+--_would_.\n\nMark the helping verbs in the following exercise:\n\n\n Exercise 7\n\n The earth shall rise on new foundations.\n We have been naught, we shall be all.\n No more tradition's chains shall bind us.\n Oh! Liberty! Can man resign thee?\n Can dungeon's bolts and bars confine thee?\n Capital could never have existed if labor had not first existed.\n What can I do? I can talk out when others are silent. I can say man\n when others say money.\n Do you hear the children weeping, O my brothers?\n Political freedom can exist only where there is industrial freedom.\n Political democracy can exist only where there is industrial\n democracy.\n Who would be free, themselves must strike the blow.\n If there is anything that cannot bear free thought, let it crack.\n No doctrine, however established, should be protected from discussion.\n Society can overlook murder, adultery or swindling; it never forgives\n the preaching of a new gospel.\n The tree of liberty must be refreshed from time to time with the\n blood of patriots and tyrants.\n Every man is a consumer and ought to be a producer.\n No picture of life can have any variety which does not admit the\n odious facts.\n I know not what course others may take, but as for me, give me liberty\n or give me death.\n\n\n Exercise 8\n\nNote the use of the helping verbs in the following quotation. Could you\nuse _might_ or _must_ or _ought_ anywhere and strengthen the emphasis?\n\n \"I have looked at this claim by the light of history and my own\n confidence, and it seems to me, so looked at, to be a most just claim,\n and that resistance to it means nothing short of a denial of the whole\n of civilization.\n\n This then is the claim:\n\n It is right and necessary that all men should have work to do which\n shall be worth doing and be of itself pleasant to do; and which should\n be done under such conditions as would make it neither over-wearisome\n nor over-anxious.\n\n Turn that claim about as I may, think of it as long as I can, I cannot\n find that it is an exorbitant claim; yet if society would or could\n admit it, the face of the earth would be changed; discontent and\n strife and dishonesty would be ended. To feel that we were doing work\n useful to others and pleasant to ourselves, and that such work and its\n due reward could not fail us! What serious harm could happen to us\n then? And the price to be paid for so making the world happy, must be\n revolution.\"--_William Morris_.\n\n\n\n\n SPELLING\n\n LESSON 10\n\n\nSimple words are sometimes spoken of as root words. _Root_ means that\nfrom which something grows. We know our language is a living, growing\nthing and these root words are the roots where the growth begins. One\nway in which this growth is accomplished and new words added to our\nlanguage is by placing syllables before or after the root word--the\nsimple word--as, for example: _unmanly_.\n\nIn this we have a syllable placed before and a syllable placed after the\nroot word _man_. The syllable placed before the root word is called the\nprefix from the Latin _pre_ meaning _before_ and the Latin word to\nplace. Therefore, prefix means literally _to place before_.\n\n+A prefix consists of one or more syllables placed before a word\nto qualify its meaning.+\n\nThe syllable placed after the root word, or simple word, is called the\nsuffix, from the Latin _sub_ meaning after and the Latin word to place.\n_Subfix_ the word should be literally, but for the sake of the\nsound--the euphony, the good sound--we say _suffix_.\n\n+A suffix consists of one or more syllables placed after a word to\nqualify its meaning.+\n\n+The words made by adding prefixes and suffixes are called derivative\nwords.+\n\nYou remember we used a suffix in forming participles. The present\nparticiple is formed by adding the suffix _ing_ to the simple form of\nthe verb. The past participle is formed by adding the suffix _ed_ to the\nsimple form of the verb.\n\nThe words in the spelling lesson for this week are derivative words\nformed by adding a prefix or suffix, or both, to the simple word. Draw a\nline through the prefix and the suffix and leave the simple or root\nword.\n\n +Monday+\n\n Wonderful\n Prosperous\n Disloyalty\n Uncovered\n Government\n\n +Tuesday+\n\n Memorize\n Unreality\n Co-operation\n Dependent\n Truly\n\n +Wednesday+\n\n Beautify\n Countless\n Uncomfortable\n Dishonesty\n Producer\n\n +Thursday+\n\n Existence\n Untruthfulness\n Discontentment\n Victory\n Removable\n\n +Friday+\n\n Impurity\n Unwillingness\n Indebted\n Overwearisome\n Enjoyable\n\n +Saturday+\n\n Obligation\n Hopeless\n Endanger\n Precaution\n Denial\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 11\n\n\nDear Comrade:\n\nAs we begin the study of the story of the alphabet and the evolution of\nwritten speech, we discover that primitive man imagined the art of\nwriting to have had divine origin, to have been handed down from the\npowers above.\n\nIt is natural for us to personify and envelop in mystery the things that\nwe do not understand. So these primitive people have attributed the\ndiscovery of the art of writing to the gods and have looked upon the\nparchment containing the written word which they cannot understand, as\npossessing magical power; but as we come to learn the origin and causes\nof things, they are divested of their mystery and become no longer gods\nand enslavers of men. We understand the laws that govern their action\nand they become our servants. Take lightning for example. Primitive\npeople personified the lightning or called it the thunder bolts of Jove\nor attributed it to an act of divine providence. We have learned the\nlaws that govern the action of electricity and so this mighty giant is\nno longer a god to whom we bow in submission, and who slays us at his\nwhim. He has become our most faithful servant who travels along the\nwires at our behest and obeys our every bidding. So in the early stages,\nthe art of writing belonged only to the favored few and was made the\nmeans of enslavement of the common people instead of the means of\nliberation.\n\nKnowledge has always been power and the ruling classes of the world,\ndesiring power over the people, have striven to keep knowledge within\ntheir own circle; so the art of writing was known only to the few. The\nfew books in circulation were laboriously written by hand and\ncirculated, largely among the clergy, who used it as priests have ever\nused their power--from medicine man to Pope,--for the enslavement of the\npeople and the protection of the privileges of a few. This is aptly\nillustrated in the law which was known as \"the benefit of clergy\" which\nwas not entirely repealed until the year 1827. Under this statute,\nexemption from trial for criminal offenses was given to the clergy and\nalso to any man who could read. If a person were sentenced to death for\nsome criminal offense, the bishop of that community might claim him as a\nclerk and if, when given a Latin book, he could read a verse or two, the\ncourt would declare \"he reads like a clerk\" and the offender was only\nburned in the hand and then set free.\n\nThe invention of the printing press in the fifteenth century which made\npossible the diffusion of knowledge among the people, was the beginning\nof the emancipation of the workers of the world. But while we realize,\nperhaps, what this art of writing means to us and by the knowledge of\nits growth and development no longer ascribe it to divine origin or\nconsider it a blessing designed by a supreme being for a favored few,\nstill most of us know very little of the interesting evolution which\nmade possible the alphabet which is the basis of our written and spoken\nlanguage of to-day. When we realize how through all these long centuries\nman has been struggling, striving, evolving, developing, reaching out\ntoward fuller, freer and richer life, it gives us courage in our\nstruggle and makes us see ourselves, not as individuals alone, but as\nlinks in a mighty chain clasping hands with that primitive man of the\npast, from whom we have inherited the power we now possess, and reaching\nforth also to clasp the hands of those who shall come and handing on to\nthem the things for which we have struggled and added to the inheritance\nof the past.\n\nNext week we will have the story of man's first beginning in the art of\nwriting.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n THE VERB \"BE\"\n\n+185.+ The verb is perhaps the most difficult part of speech to master\nbecause it has more form changes than any other part of speech.\n\nIn this lesson we are going to emphasize the most important things to\nremember in the study of the verb and also call attention to the most\ncommon mistakes.\n\n+186.+ First, master that little verb be in all its forms. The only way\nto do this is to commit to memory these forms. Say them over and over\nuntil any other form does not sound right.\n\n +Present+ +Past+ +Future+\n\n _Singular_ _Singular_ _Singular_\n\n 1. I am. I was. I shall be.\n 2. You are. You were. You will be.\n 3. He is. He was. He will be.\n\n _Plural_ _Plural_ _Plural_\n\n 1. We are. We were. We shall be.\n 2. You are. You were. You will be.\n 3. They are. They were. They will be.\n\n _Pres. Perf._ _Past Perf._ _Fut. Perf._\n\n Have been. Had been. Shall have been.\n\n+187.+ Do not use _aint_ for _is not_ or _am not_. Do not say, _He aint\nhere_, or _I aint going_. Say, _He isn't here_; _I am not going_.\n\n\n A FREQUENT MISTAKE\n\n+188.+ Perhaps one of the most frequent mistakes is the confusion in the\nuse of the past time form and the past participle. Remember that the\npast time form is never used except in expressing past time; never use\nit in forming a verb phrase. Take the verb _do_, for example--say, _He\ndid the work_, never, _He done the work_; but we should say, _He has\ndone the work_, never, _He has did the work_. _Say_ and _seen_ are\nconfused in the same manner. Watch this carefully.\n\n\n Exercise 1\n\nUnderline the correct word in the following:\n\n 1. Who did--done it?\n 2. He sung--sang well.\n 3. He sunk--sank before we could reach him.\n 4. She written--wrote him a letter.\n 5. He taken--took the book.\n 6. They swum--swam the river.\n 7. I saw--seen him do it.\n 8. They drank--drunk too much.\n 9. He soon began--begun to fail.\n 10. The lad ran--run home.\n 11. They come--came yesterday.\n\n\n WITH HELPING VERBS\n\n+189.+ Never use the past time form with the helping verbs _has_, _had_,\n_was_ and _were_. Always use the past participle. Watch this carefully.\nFor example, never say, _He has went_. _Went_ is the past time form.\nSay, _He has gone_.\n\n\n Exercise 2\n\nUnderscore the correct word in the following sentences:\n\n 1. He had tore--torn the book.\n 2. Have you ever sang--sung this tune?\n 3. They have showed--shown us how to win.\n 4. She has went--gone away.\n 5. The trees were shook--shaken by the wind.\n 6. He was chose--chosen for leader.\n 7. He has rose--risen from the ranks.\n 8. It was wrote--written by him.\n 9. He has took--taken the prize.\n 10. He was gave--given the money.\n 11. I have forgot--forgotten the rule.\n 12. The river was froze--frozen over.\n 13. The machine was broke--broken.\n 14. It was wore--worn out.\n 15. The meal was ate--eaten in silence.\n\n\n PAST TIME FORMS\n\n+190.+ Watch your speech to see if you use an incorrect verb form for\nthe past time form. Study the table of irregular verbs and refer to it\nfrequently. We often make the mistake of forming the past time form by\nadding _ed_ when properly it is formed irregularly. For example: we\noften say _drawed_ for _drew_, _throwed_ for _threw_, etc.\n\n\n Exercise 3\n\nDraw a line under the correct form in the following:\n\n 1. He grew--growed rapidly.\n 2. He knew--knowed better.\n 3. He catched--caught the ball.\n 4. He drew--drawed the water.\n 5. They threw--throwed him over.\n 6. I drinked--drank the water.\n 7. I climbed--clumb the tree.\n 8. I seed--saw him do it.\n 9. She teached--taught school.\n\n\n VERBS OF SIMILAR FORM\n\n+191.+ Do not use one verb for another of similar form but different\nmeaning. The following are the most common of these:\n\n+Lay+ (incomplete verb, requires an object) meaning to place or to put;\nas, _to lay the book down_. Principal parts: _Present_, lay; _Past_,\nlaid; _Past participle_, laid.\n\n+Lie+ (complete verb, takes no object) meaning to recline, to rest; as,\n_to lie in bed_. Principal parts: _Present_, lie; _Past_, lay; _Past\nparticiple_, lain.\n\n+Set+ (incomplete verb, requires an object) meaning to place or to put;\nas, _to set the table_. Principal parts: _Present_, set; _Past_, set;\n_Past participle_, set.\n\n+Sit+ (complete verb, takes no object) meaning to rest, as, _to sit in a\nchair_. Principal parts: _Present_, sit; _Past_, sat; _Past participle_,\nsat.\n\n+Raise+ (incomplete verb, requires an object) meaning to cause to rise,\nto lift up. Principal parts: _Present_, raise; _Past_, raised; _Past\nparticiple_, raised.\n\n+Rise+ (complete verb, takes no object) meaning to get up, to ascend.\nPrincipal parts: _Present_, rise; _Past_, rose; _Past participle_,\nrisen.\n\n+192.+ +NOTE--These three verbs need an object to complete their\nmeaning:+\n\n _Present_ _Past_ _Past Participle_\n\n set set set\n lay laid laid\n raise raised raised\n\n+193.+ +NOTE--These three verbs need no object:+\n\n _Present_ _Past_ _Past Participle_\n\n sit sat sat\n lie lay lain\n rise rose risen\n\n\n Exercise 4\n\nFill in the following blanks with the correct form of the verbs _sit_,\n_set_, _lay_, _lie_, _raise_ and _rise_:\n\n 1. I......it on the table and there it.......\n 2. They......the battle ship, Maine.\n 3. Where did you......it?\n 4. A mile of pipe has been.......\n 5. The miners......a large strike fund.\n 6. She......down to sleep.\n 7. The body......in state three days.\n 8. The farmers of the U. S.......an enormous wheat crop.\n 9. The city......on the right bank.\n 10. We have......the corner stone.\n 11. When wages are......, prices are......too.\n 12. He......in bed all morning.\n 13. ......down Fido.\n 14. The sun......at six this morning.\n 15. She has been......there all day.\n 16. The ship......to during the storm.\n 17. They have been......new tracks.\n 18. The hen is......on the eggs.\n 19. Somebody said, \"Early to bed and early to......,\n Makes a man healthy, wealthy and wise.\"\n 20. He......motionless for an hour.\n 21. He......out the trees in rows.\n 22. He will......in his position.\n 23. The court will......in May.\n 24. Where did he......?\n 25. She......the table while he......there.\n 26. He......the clock for six o'clock.\n 27. The water has......two feet since the rain.\n 28. He......the book down and......on it.\n 29. The hen has been......a week.\n 30. ......it on the table.\n 31. He......in the shade and watched her......the plants.\n\n\n COMMON ERRORS\n\n+194.+ Remember that in the present time form the third person singular\ntakes the s-form, but the s-form is never used _except_ with the _third\nperson singular_. We often make the mistake of using the _s-form_ with a\n_plural_ subject. Notice carefully the following sentences, and correct\nthe errors. All of the sentences are wrong.\n\n 1. The days is getting shorter.\n 2. The men has struck.\n 3. The trains was late.\n 4. These papers is written for you.\n 5. You was disappointed, wasn't you?\n 6. There is several coming.\n 7. The nights was dark and cloudy.\n 8. The clouds has gathered.\n 9. They was anxious to come.\n\n+195.+ +When two subjects are connected by _and_, the s-form of the\nverb must not be used+, unless both subjects refer to one person; as:\n\n The president and the secretary (two persons) were late.\n The president and secretary (one person) was elected.\n\n+196.+ +But when the two subjects are connected by _or_ or _nor_\nthen use the s-form of the verb+; as:\n\n Neither Germany nor Russia admits a war of offense.\n Either the House or the Senate rejects the bill.\n\n+197.+ +Never use the infinitive sign _to_ by itself+; as:\n\n I have not written and do not expect _to_.\n He has not gone nor does he intend _to_.\n\n+198.+ +Never use don't for doesn't.+ The use of _don't_ for _doesn't_\nis a very common mistake. _Don't_ is a contraction of _do not_ and\n_doesn't_ of _does not_. When you are in doubt as to which to use, think\nor speak the two words in full and see if the verb agrees with the\nsubject. _Do not_ is used with a plural subject, and _does not_ with a\nsingular subject. For example: _He don't believe me_. This sentence in\nfull would be, _He do not believe me_, which is incorrect. _He does not_\n(_doesn't_) _believe me_ is correct. Or, _They doesn't believe me_. This\nsentence in full would read, _They does not believe me_, which is\nincorrect. _They do not_ (_don't_) _believe me_ is correct.\n\n+199.+ +Do not use _has got_, or _have got_ for _must_.+ For\nexample, do not say, _We have got to go_. Say, _We must go_. Not, _He\nhas got to do what I say_; but, _He must do as I say_.\n\n+200.+ +Do not say _had ought_.+ For example: _You had ought to know\nbetter_. Omit the _had_; it is unnecessary and incorrect. Say, _You\nought to know better_.\n\n+201.+ +Do not say _says I_ or _thinks I_.+\n\n Says I, \"Will you go?\"\n Says he, \"That's what will happen.\"\n Thinks I to myself, \"I'll show you.\"\n\nThese are incorrect. Say instead:\n\n I said, \"Will you go?\"\n He said, \"That's what will happen.\"\n I thought, \"I'll show you.\"\n\n\n Exercise 5\n\nMark all the verbs in the following quotations and note carefully their\nuse.\n\n 1. Speak properly and in as few words as you can but always plainly;\n for the end of speech is not ostentation but to be\n understood.--_Penn_.\n\n 2. \"Freedom's battle, once begun,\n Bequeathed from bleeding sire to son,\n Though baffled oft, is ever won.\"\n\nNote the use of _may_ and _can_ in this quotation:\n\n 3. Knowledge cannot be stolen from us. It cannot be bought or sold. We\n may be poor, and the sheriff may come and sell our furniture, or\n drive away our cow, or take our pet lamb and leave us homeless and\n penniless; but he cannot lay the law's hand upon the jewelry of our\n minds.--_E. Burritt_.\n\nNote the use of _shall_ and _will_ and _would_ and _should_ in the\nfollowing. Richard Grant White says: \"I do not know in English\nliterature another passage in which the distinction between _shall_ and\n_will_ and _would_ and _should_ is at once so elegantly, so variously,\nso precisely, and so compactly illustrated.\"\n\n 4. \"How long I shall love him I can no more tell,\n Than, had I a fever, when I should be well.\n My passion shall kill me before I will show it,\n And yet I would give all the world he did know it;\n But oh how I sigh, when I think, should he woo me,\n I cannot refuse what I know would undo me.\"\n\n 5. I want it said of me by those who know me best that I always\n plucked a thistle and planted a flower where I thought a flower\n would grow.--_Abraham Lincoln_.\n\n\n Exercise 6\n\nNote the nouns as well as the verbs in the following quotation. Note\nalso the use of infinitives and participles. Mark every verb and use it\nin a sentence of your own.\n\n\n +Faith and Truth+\n\n You say \"Believe;\" I say \"Trust.\"\n\n Between those two words is a great gulf fixed.\n\n The idea that there can be a moral obligation to believe external\n facts is unworthy of a freeman, but to trust is as much the true\n nature of man as it is that of a babe to draw in its mother's\n milk.\n\n You say \"Creed;\" I say \"Faith.\"\n\n A creed at best is but a sorry caricature of a faith.\n\n Faith is the proper atmosphere of man, trust is his native buoyancy,\n and his only obligation is to follow the highest law of his being.\n\n You have one supreme duty above all creeds and conventions--namely,\n to think honestly, and say what you think.\n\n Have you doubts about your creed? say so; only thus has the true faith\n ever advanced.\n\n It is not God, but the devil, who whispers: \"Think at your peril!\"\n\n Do you see flaws in the ancient structure of respectability and law\n and order? Say so; only thus has the condition of man ever\n improved.\n\n Have courage to be the heretic and traitor that you are by nature, and\n do not worry about the consequences.\n\n Be a creator, as you were born to be, and spurn beyond all infamies\n the wretched role of a repeater and apologist.\n\n The world lives and grows by heresy and treason.\n\n It dies by conformity to error and loyalty to wrong.\n\n _Ernest Crosby_.\n\n\n Exercise 7\n\nIn the following paragraph, the predicates are printed in italics, and\nthe participles and infinitives in italic capitals. Study carefully.\n\n If it _were taught_ to every child, and in every school and college,\n that it _is_ morally wrong for anyone _TO LIVE_ upon the _COMBINED_\n labor of his fellowmen without _CONTRIBUTING_ an approximately equal\n amount of useful labor, whether physical or mental, in return, all\n kinds of _GAMBLING_, as well as many other kinds of useless\n occupations, _would be seen_ _TO BE_ of the same nature as direct\n dishonesty or fraud, and, therefore _would_ soon _come_ _TO BE\n CONSIDERED_ disgraceful as well as immoral.\n\n _Alfred Russel Wallace_.\n\n\n Exercise 8\n\nUnderscore all the verbs in the following and note the participles, the\ninfinitives and the various time forms; also the helping verbs:\n\n What, speaking in quite unofficial language, is the net purport of\n war? To my knowledge, for example, there dwell and toil, in the\n British village of Dumrudge, usually some five hundred souls. From\n these, by certain 'natural enemies' of the French, there are selected,\n say thirty able-bodied men; Dumrudge, at her own expense, has suckled\n and nursed them; she has, not without difficulty and sorrow, fed them\n up to manhood and trained them in the crafts, so that one can weave,\n another build and another hammer. Nevertheless, amidst much weeping\n and swearing, they are selected; all dressed in red and shipped away,\n at the public charges, some two thousand miles, or, say only to the\n south of Spain, and fed there till wanted. And now to that same spot\n in the south of Spain are thirty similar French artisans, in like\n manner, wending their ways; till at length the thirty stand facing the\n thirty, each with his gun in his hand. Straightway, the word 'Fire' is\n given, and they blow the souls out of one another; and in the place of\n the sixty brisk, useful craftsmen, the world has sixty dead carcasses,\n which it must bury and anew shed tears for.\n\n Had these men any quarrel? Busy as the devil is, not the smallest!\n They lived far enough apart; were the entirest strangers; nay, in so\n wide a universe, there was even, unconsciously, by commerce, some\n mutual helpfulness between them.\n\n How then?\n\n Simpleton! Their governors had fallen out; and instead of shooting one\n another, had these poor blockheads shoot.--_Carlyle_.\n\n\n\n\n SPELLING\n\n LESSON 11\n\n\nThere are but few rules which can be learned to aid in the spelling of\nEnglish words. The spelling of words must be largely mastered by\nconcentration and effort of the memory. It will help you to memorize the\ncorrect spelling if you will write each word a number of times. This\ngives you a visual image of the word. Then spell it aloud a number of\ntimes. This will give you an auditory image.\n\nWords which you find difficult to master, write in a list by themselves\nand review frequently. There are a few rules, however, which are helpful\nto know. There is one rule of spelling we want to learn this week\nconcerning words formed by adding a suffix.\n\n+A word of one syllable which ends in a single consonant before which\nstands a single vowel, doubles the final consonant when a suffix\nbeginning with a vowel is added.+\n\nFor example: _mat_, _matted_, _matting_; _sun_, _sunned_, _sunning_.\n\n_Mat_ ends in _t_, a single consonant which is preceded by the single\nvowel _a_,--so you double the _t_ when you add the suffix _ed_ or _ing_,\nwhich begin with a vowel.\n\nNotice these: _Blend_, _blended_, _blending_; _Help_, _helped_,\n_helping_.\n\nThese words do not end in a single consonant, so you do not double the\nconsonant.\n\nNotice also: _Lean_, _leaned_, _leaning_; _Rain_, _rained_, _raining_.\n\nThese words end in a single consonant, but before the consonant is a\ndouble vowel, _ea_ in _lean_ and _ai_ in _rain_. So we do not double the\nfinal consonant.\n\nThis same rule holds true of any suffix, beginning with a vowel, as _er_\nand _est_, for example: _sad_, _sadder_, _saddest_. _Slim_, _slimmer_,\n_slimmest_.\n\nLearn to spell the following words. Add the suffixes _ed_ and _ing_ to\nthe words for Monday, Tuesday and Wednesday. Add _er_ and _est_ to the\nwords for Thursday, Friday and Saturday.\n\n +Monday+\n\n Chat\n Cheat\n Grin\n Groan\n Suit\n\n +Tuesday+\n\n Sap\n Soap\n Bet\n Beat\n Rot\n\n +Wednesday+\n\n Talk\n Teach\n Gain\n Stir\n Plan\n\n +Thursday+\n\n Thin\n Dear\n Flat\n Cheap\n Straight\n\n +Friday+\n\n Clean\n Brief\n Fair\n Shrill\n Wet\n\n +Saturday+\n\n Strong\n Great\n Mad\n Fleet\n Fat\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 12\n\n\nDear Comrade:\n\nIn this lesson we are beginning the study of still another part of\nspeech. You will notice that in words, at least, we give credit and\nplace in society only because of _work performed_. In the society of\nmen, people are given place and position too often because of outward\ndress and form or because of some special privilege. They are not given\ntheir place in society because of the work which they do or because they\nperform any useful function. In fact, in our topsy-turvy world, those\nwho perform no work at all, but are simply parasites upon society, have\nclaimed for themselves the best of everything and the highest positions.\n\nSurely some time we shall see a society as successfully organized as our\nsociety of words, when men will be received, not because of that which\nthey possess, but because of that which they do and are. Man has really\nlaid the foundation for an ideal commonwealth in his organization of\nwords into a spoken and written language.\n\nWhen we think back across the centuries and think of the primitive man\nas he dwelt in trees to protect himself from the wild animals, we wonder\nwhat sort of speech he used then. Possibly it was only a little more\narticulate than the speech of some animals.\n\nBut man had within him the instinct to question, and this has been the\nroot of all his progress. We can imagine these primitive men witnessing\nthe wonder of fire, as the terrible unknown god of the lightning set\nfire to the forest in which they lived; but after the fear had subsided,\nsome adventurous, inquiring forefather of ours ventured near the ashes,\nand began to investigate concerning this fearful and wonderful thing.\n\nSo gradually they discovered the use of fire, and with it a wonderful\nnew future opened before the primitive man. With these great\ndiscoveries, he needed a better form of communication with his comrades,\nso articulate speech developed. But when we go back into the beginning\nof written speech, it is difficult for us to trace it to its beginning.\n\nThe first evidence we find was of man as a sign maker. On the walls of\ncaves in France and Belgium and here in America, we have found rude\nsketches which the scientists tell us date back to the Ice Age and the\nOld Stone Age. Here the primitive man has drawn for us crude pictures\ndescribing different phases of his life, the animals about him, the hunt\nand the chase, and in these pictures we find the very beginning of our\nalphabet of to-day.\n\nHow much more wonderful it makes our spoken and written language to know\nthat man has developed it himself. It has not been handed down by some\ngod or powers above; but the spirit of rebellion against the things that\nbe; the great desire to know more and to find out the reason _why_ of\nall the things around us,--these have been the forces that have led the\nrace from the animal-like beings that lived in trees to the race of\ntoday that understands in a large measure the laws that govern life.\n\nIt is only as we, through this spirit of rebellion, this same divine\ndiscontent with the things that are, seek to do our own thinking that we\ncan add our share to the heritage of the race. Let us have the same\ncourage that must have inspired the heart of that primitive man who\ndared to venture and inquire concerning the fearful things of nature\nround about him. Let us think for ourselves. Ask always the question\n\"why\" and demand the reason for all things. Thus we shall free ourselves\nand help to free the race.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n IN PLACE OF A NOUN\n\n+202.+ You remember in our study of the parts of speech we found that we\nhave one part of speech that can be used in place of a noun. This is a\nvery helpful part of speech for it saves us a great deal of tiresome\nrepetition. Notice the following sentences:\n\n John Smith is a machinist.\n John Smith works at the machine.\n The machine is John Smith's master.\n\nThis is awkward and the repetition is tiresome. So we say instead:\n\n John Smith is a machinist.\n He works at the machine.\n It is his master.\n\nYou readily understand who and what we mean by _he_ and _it_ and _his_,\nand we will all agree that the latter is a much better way of making the\nstatements. These words like _he_ and _his_ and _it_, which we use in\nplace of the noun, we call _pronouns_. _Pro_ means literally in the\nLatin, _for_ or _in place of_; so when we say pronoun we are practically\nsaying, in place of a noun.\n\n+A pronoun is a word that is used in place of a noun.+\n\n+203.+ The word for which a pronoun stands or the noun in whose place it\nis used is called its antecedent. _Ante_ means _before_ and _cedent_\ncomes from the Latin word meaning _go_, hence antecedent means\nliterally, _going before_.\n\nNotice this sentence: _The manager spoke to the men before he left and\ntold them to stop at the office_. _Manager_ is the antecedent of the\npronoun _he_, and _men_ is the antecedent of the pronoun _them_.\n\n+The word for which a pronoun stands is called its antecedent.+\n\n\n KINDS OF PRONOUNS\n\n+204.+ The Latin language has had a great deal of influence upon\nEnglish. Many of our words are taken from the Latin. You remember that\nall of the names of our parts of speech are derived from Latin words. We\nalso feel the influence of the Latin language in the way in which we\nnumber our personal pronouns. The Romans naturally thought that one\nwould think of one's self first, and so the pronouns referring to one's\nself, or the person speaking, are called the _first_ person pronouns.\nThey are, _I_, _my_, _mine_, _me_ and _we_, _our_, _ours_, and _us_.\n\nThen they naturally thought that one would think second of the person\nspoken to, so the pronouns referring to the person spoken to are called\nthe _second_ person pronouns. Formerly _thou_ was used in speaking to\none person. In German and many other languages this form is still used,\nbut in English we do not today use the singular form _thou_ with its\nvariations, _thy_, _thine_, and _thee_, except in poetry or poetic\nprose. In every-day speech we use _you_ and its forms, _your_ and\n_yours_, for both the singular and the plural.\n\nThen the Romans considered last the person or thing of whom they were\nspeaking; so pronouns referring to the person or thing spoken of are\ncalled the _third_ person pronouns. These are _he_, _she_, and _it_,\nwith their other forms, _his_, _him_, _her_, _hers_, _its_, in the\nsingular, and _they_, _their_, _theirs_ and _them_ in the plural.\n\n+A personal pronoun is one that denotes the speaker, the person spoken\nto, or the person or thing spoken of.+\n\n\n COMPOUND PERSONAL PRONOUNS\n\n+205.+ All of these forms of pronouns which we have named are simple\nforms; but we have several personal pronouns which have a compound form;\nthat is, a form made by the addition of _self_ or _selves_ to the simple\nforms.\n\nThese are called compound personal pronouns. They are, in the singular,\n_myself_, _thyself_, _yourself_, _himself_, _herself_, _itself_, and in\nthe plural, _ourselves_, _yourselves_ and _themselves_.\n\nThe compound personal pronouns have two uses, reflexive and emphatic.\n\n\n Reflexive\n\n+206.+ A compound personal pronoun has a reflexive use when the actor\nbecomes the object of its own action or in other words when the subject\nand the object refer to the same thing; as in this sentence, _He has\nhurt himself_, _himself_ is the object of the incomplete verb _has\nhurt_, but it refers to the subject _he_. Reflexive is from the Latin\n_re_ meaning _back_ and from the Latin verb meaning _throw_, so\nreflexive means literally _thrown_ back. These pronouns throw their\nmeaning back to the subject.\n\n\n Emphatic\n\n+207.+ A compound personal pronoun has also an emphatic use when it\ndirects especial attention to the noun or pronoun to which it refers.\nFor example in the sentence, _He did the work himself_, or, _He,\nhimself, did the work_, _himself_ gives emphasis or intensifies the\nmeaning of the pronoun _he_.\n\nRemember a compound personal pronoun is correctly used only in these two\nways, reflexive and emphatic. For example, the following sentences are\nincorrect:\n\n This is for yourself and your comrade.\n Ourselves will find out the reason.\n\nThe correct form would be:\n\n This is for you and your comrade.\n We, ourselves, will find out the reason.\n\n+208.+ You can readily distinguish between the reflexive and the\nemphatic use. In the reflexive, the compound personal pronoun is always\nthe _object_ of a verb or preposition, and the subject of the sentence\nis its antecedent. The subject and the object always refer to the same\nthing.\n\nIn the emphatic use, the compound personal pronoun is neither the\nsubject nor the object, but is thrown into the sentence simply to render\nit emphatic, and to call special attention to its antecedent.\n\n\n Exercise 1\n\nSupply the compound personal pronoun in the following blanks and tell\nwhether the use is reflexive or emphatic.\n\n 1. He discovered the truth.......\n 2. The workers have robbed......by their ignorance.\n 3. You must educate.......\n 4. You must do the work.......\n 5. He must defend.......\n 6. Capitalism overreaches.......\n 7. The people will rule.......\n 8. We will settle the question.......\n\n\nWrite six sentences in which the compound personal pronouns are\ncorrectly used.\n\n\n SINGULAR AND PLURAL\n\n+209.+ Personal pronouns, like nouns, have number form. Nouns simply add\n_s_ to the singular form to denote the plural, but in personal pronouns\nwe have different words which we use to express one or more than one\nperson or thing. In the first, second, and third person forms, personal\npronouns also have different forms for the object form, the possessive\nand the subject form. The following table gives the singular and plural\nof the subject form,--that is the form which is used as the subject of\nthe sentence.\n\n +Subject Form+\n\n _Singular_ _Plural_\n\n _First person._ I We\n _Second person._ You You\n _Third person._ He, she, it. They\n\n +Compound Personal Pronouns+\n\n _Singular_ _Plural_\n\n _First._ Myself Ourselves\n _Second._ Yourself Yourselves\n _Third._ Himself, herself, itself. Themselves\n\n+210.+ Remember that the first person refers to the person speaking, the\nsecond to the person spoken to, and the third person to the person or\nthings spoken of. When we speak of things, we never use the first or\nsecond person, unless we are speaking of them in a personified form. So\nin the third person singular, we have the pronoun _it_ which refers to\none thing. In the plural, we have no special pronoun referring to\nthings, but the pronoun _they_ is used to refer both to persons and\nthings.\n\n\n Exercise 2\n\nWhich of the following pronouns refer to the person speaking, which to\nthe person spoken to, and which to the person or thing spoken of? Which\nare singular, which plural?\n\n I will defend my principles.\n Give them to me for they are mine.\n Do you believe him to be your friend?\n We saw their mistake at once.\n They acknowledged it was their fault.\n Success will be your portion if you persevere.\n He struggles for his rights; she does not understand her rights.\n It forces us to struggle for our education.\n Woman craves her freedom.\n Workers of the world, unite; you have a world to gain and nothing to\n lose but your chains.\n\nForm sentences of your own containing all these pronouns.\n\n\n POSSESSIVE FORM\n\n+211.+ You will note in these sentences above that we have used the\npronoun _my_ and _your_ and _his_ and _her_ as _my principles_, _your\nfriend_, _his rights_, _her freedom_. This is the possessive form of\nthese personal pronouns, the form that denotes ownership or possession.\nYou remember that nouns had a possessive form, a form to denote\npossession or ownership, as, _The man's book._ _The boy's school._ _The\nworker's college._ So pronouns also have a possessive form which we use\nto show that an object belongs to such and such a person or thing. If I\nwant to tell you that I own or possess a home, I say, _I own my home_.\nEach personal pronoun has its possessive form, thus:\n\n +Singular+\n\n _Subject Form_ _Possessive_\n\n _First person._ I My, mine\n _Second person._ You Your, yours\n _Third person._ He, she, it His, her, hers, its\n\n +Plural+\n\n _Subject Form_ _Possessive_\n\n _First person._ We Our, ours\n _Second person._ You Your, yours\n _Third person._ They Their, theirs\n\n\n POSSESSIVE FORM\n\n+212.+ You will notice that the possessive forms, _my_, _our_, _her_,\n_your_, _its_, _his_ and _their_, are always used with the name of the\nobject possessed. As for example; _my work_, _our library_, _her\ndelight_, _your task_, _its purpose_, _his home_, _their mistake_.\n\n+213.+ The possessive forms, _mine_, _thine_, _hers_, _ours_, _yours_\nand _theirs_, are always used by themselves and are used either as\nsubject, object or complement. As for example:\n\n That letter is mine.\n The work is hers.\n Thine is the glory.\n Is that yours?\n Theirs not to reason why; theirs but to do and die.\n\nThe possessive form _his_ may be used either in connection with the name\nof the object possessed or by itself. For example:\n\n This is _his_ home.\n This home is _his_.\n\n\n OBJECT FORM\n\n+214.+ Pronouns have one form which nouns do not have. We use the same\nform for the noun no matter whether it is the subject or the object. For\nexample:\n\n The man saw me.\n I saw the man.\n\nIn the first sentence _man_ is the subject of the verb _saw_, and in the\nsecond sentence _man_ is the object of the verb _saw_. The same word is\nused; but you will notice that in the first sentence _me_ is the object\nof the verb _saw_, and in the second _I_ is the subject; yet both refer\nto the same person, the first person, the person speaking.\n\nSo we have a different form of the pronoun for the object, for example:\n_I saw him._ _He saw me._ _She watched us._ _We watched her._ _You found\nthem._ _Him_, _me_, _us_, _her_, and _them_ in these sentences are used\nas the objects of the verbs, _see_, _watch_ and _found_, and are called\nthe object forms of the pronouns. _You_ and _it_ have the same form for\nboth the subject and object; as, _You did it._ _It frightens you._ _Her_\nis used as both the possessive form and the object form, as, _Her work\ntires her._\n\n+215.+ The following table gives the subject and the object forms of the\npersonal pronouns, and these should never be confused in their usage. We\nmust not use the object form as the subject of the verb, nor the subject\nform as the object of the verb.\n\n +Singular+\n\n _Subject_ _Object_\n\n _First._ I Me\n _Second._ You You\n _Third._ He, she, it Him, her, it\n\n\n +Plural+\n\n _Subject_ _Object_\n\n _First._ We Us\n _Second._ You You\n _Third._ They Them\n\n\n GENDER\n\n+216.+ You notice in all of these tables that there are three forms\ngiven for the third person singular, _he_, _she_, and _it_. These are\nthe only forms in which pronouns express gender. In all other forms the\ngender can be determined only by the gender of the antecedent.\n\n+He, representing a male, is masculine.+\n\n+She, representing a female, is the feminine.+\n\n+It represents a sexless thing, and hence is said to be of the neuter\ngender.+\n\n\n THE LITTLE VERB _BE_\n\n+217.+ You remember when we studied verbs, we had the incomplete verb\nthat took an object; the complete verb that needed no object, since it\nwas complete in itself; and one other kind of a verb. Do you remember\nthis third kind of verb? This third kind is the copulative verb, and the\ncopulative verb which we use most frequently is the one in the use of\nwhich we make the most mistakes.\n\nIt is that troublesome, bothersome, little verb _be_, which is so\ndifficult to master. You remember it is an incomplete verb, but instead\nof taking an object, it takes a complement or completing word. So when\nyou see a pronoun with any form of this verb _be_, you must use the\n_subject_ form and not the _object_ form. This copulative verb _be_ is\nsimply a connecting word, not a verb that asserts action or takes an\nobject.\n\n+218.+ Here is where we make so many mistakes. We say, _It was me_, _It\nwas them_, _It was him_, _It wasn't her_; instead of, _It was I_, _It\nwas they_, _It was he_, _It wasn't she_. We have used the incorrect form\nin this particular so often that the correct form has a strange sound to\nour ears.\n\nThe only way to remedy this is to repeat over and over aloud the correct\nform until it has a familiar sound. Don't think this is putting words,\nas you should do in everything. We of the working class have built the\nworld in its beauty. Why should we live in shacks, dress in shoddy, talk\nin slang? There is no reason except that we endure it. When the united\nworking class demands its own, it will receive it. Demand yours and\narouse the stupid from their sleep as rapidly as you can.\n\nRepeat the following sentences aloud ten times every day this week and\nsee if the correct form does not come to your lips more readily. We can\nlearn the rule, but only continued practice and watchfulness can break\nus of our old habits.\n\n It is I who seek my own.\n It shall be they who are defeated.\n It was I who was ignorant.\n It is they who cause all wars.\n It is he who must be aroused.\n It is we who strive for freedom.\n It shall be I who shall win.\n It was she who was enslaved.\n It shall be we who shall demand equality.\n It shall be they who shall conquer.\n\n\n Agreement\n\n+219.+ Pronouns are very agreeable members of the co-operative\ncommonwealth of words. They strive to agree with their antecedents.\nSometimes we do not allow the pronoun to agree, and then our sentence is\nincorrect.\n\n+A pronoun must agree with its antecedent in number, gender and person.+\n\nFor example, if you are referring to one man, you must use a masculine\npronoun, singular, third person form, as _I saw the man but he did not\nsee me_. _Man_ is the antecedent. It is singular, masculine, third\nperson and so we use the pronoun _he_.\n\n_The girl came, but she could not stay._ In this sentence _girl_ is the\nantecedent; it is singular, feminine, third person, and so we use the\npronoun _she_.\n\n_The boys did not come when the teacher called them._ In this sentence\n_boys_ is the antecedent; it is plural, masculine, third person, and so\nwe use the pronoun _them_.\n\n+220.+ +Sometimes there are two words used as the antecedent, joined by\n_and_.+ We use a singular pronoun in referring to them if they denote the\nsame person or thing; as:\n\n The secretary and treasurer (one person) resigned _his_ position.\n My comrade and friend (one person) gave me _his_ help.\n\n+221.+ +But two nouns joined by _and_, that mean different persons or\nthings, must be represented by a plural pronoun, thus+:\n\n Marx and Engels (two persons) wrote _their_ call to liberty, the\n Communist-Manifesto.\n Men and women will struggle for _their_ freedom.\n Childhood and youth should have _their_ rightful joys.\n\n+222.+ +Use the singular pronoun when the nouns are kept separate by the\nuse of _each_, _every_, _many a_, or _no_.+\n\n Each man and boy must do _his_ part. (Not _their_ part.)\n Every soldier and every officer must do _his_ duty.\n Many a city and many a village gave _its_ best to the army.\n No comrade and no Socialist will give _his_ consent to war.\n\n+223.+ +If you have two singular nouns as antecedents, joined by _or_,\nor _nor_, use the singular pronoun+, thus:\n\n Either Germany or France must abandon _its_ position.\n Neither Wilson nor Bryan kept _his_ promise to the people.\n\n+224.+ +When you use a collective noun and are speaking of the\ncollection as a whole, use a singular pronoun+, as:\n\n The committee will make _its_ report.\n The audience was hearty in _its_ appreciation.\n The jury has returned _its_ verdict.\n\n+225.+ +But if you are referring to the individuals of the collection\nseparately, use a plural pronoun+; as:\n\n The committee adjourned for _their_ dinner.\n The audience kept _their_ seats until the close.\n The jury argued until _their_ nerves were on edge.\n\n\n PERSONIFICATION\n\n+226.+ We sometimes speak of things as if they were persons, and so use\neither masculine or feminine pronouns in referring to them. Such objects\nare said to be personified. Thus, we say:\n\n The sun his ceaseless course doth run.\n The moon sheds her silvery ray.\n Nature dons her robes of green.\n\nHere we speak of the sun as though it were a man or possessing the\nqualities of a man and use the pronoun _his_. Then we speak of the moon\nand nature as though they were women and use the pronoun in the feminine\nform.\n\n\n REMEMBER\n\n+227.+ +A pronoun must agree with its antecedent.+\n\n+Use the subject form of the pronoun if the pronoun is the subject of\nthe sentence.+\n\n+Use the object form when the pronoun is the object of a verb or a\npreposition.+\n\n+Use the compound personal pronouns only in their reflexive or emphatic\nuse.+\n\n+With all forms of the verb _be_, use the subject form of the\npronouns.+\n\n\n SUMMARY\n\n SUBJECT POSSESSIVE OBJECT\n First person (_Singular_ I my (mine) me\n (_Plural_ we our (ours) us\n\n Second person (_Singular_\n (_Plural_ you your (yours) you\n\n Third person (_Sing. Masc._ he his him\n (_Sing. Fem._ she her (hers) her\n (_Sing. Neut._ it its it\n (_Plural_ they their (theirs) them\n\n\n Exercise 3\n\nRead carefully the following beautiful dream of Olive Schreiner's. Mark\nall of the personal pronouns and note carefully their use and by\nreferring to the table above decide just what form each pronoun is.\nWatch carefully too for the antecedents of the pronouns and note the\nagreement of the pronoun with its antecedent.\n\n\n \"I THOUGHT I STOOD\"\n\n I.\n\n I thought I stood in Heaven before God's throne, and God asked me what\n I had come for. I said I had come to arraign my brother, Man.\n\n God said, \"What has he done?\"\n\n I said, \"He has taken my sister, Woman, and has stricken her and\n wounded her and thrust her out into the streets; she lies there\n prostrate. His hands are red with blood. I am here to arraign him;\n that the kingdom be taken from him, because he is not worthy, and\n given unto me. My hands are pure.\"\n\n I showed them.\n\n God said, \"Thy hands are pure. Lift up thy robe.\"\n\n I raised it; my feet were red, blood-red, as if I had trodden in wine.\n\n God said, \"How is this?\"\n\n I said, \"Dear Lord, the streets on earth are full of mire. If I should\n walk straight on in them my outer robe might be bespotted, you see how\n white it is! Therefore I pick my way.\"\n\n God said, \"_On what?_\"\n\n I was silent, and let my robe fall. I wrapped my mantle about my\n head. I went out softly. I was afraid that the angels would see me.\n\n\n II.\n\n Once more I stood at the gate of Heaven, I and another. We held fast\n by one another; We were very tired. We looked up at the great gates;\n angels opened them, and we went in. The mud was on our garments. We\n walked across the marble floor, and up to the great throne. Then the\n angels divided us. Her, they set upon the top step, but me, upon the\n bottom; for, they said, \"Last time this woman came here she left red\n foot-marks on the floor; we had to wash them out with our tears. Let\n her not go up.\"\n\n Then she with whom I came, looked back and stretched out her hands to\n me; and I went and stood beside her. And the angels, they, the shining\n ones who never sinned and never suffered, walked by us, to and fro, up\n and down; I think we should have felt a little lonely there if it had\n not been for one another, the angels were so bright.\n\n God asked me what I had come for; and I drew my sister forward a\n little that He might see her.\n\n God said, \"How is it you are here together today?\"\n\n I said, \"She was upon the ground in the street, and they passed over\n her; I lay down by her, and she put her arms around my neck, and so I\n lifted her, and we two rose together.\"\n\n God said, \"Whom are you now come to accuse before Me?\"\n\n I said, \"We are come to accuse no man.\"\n\n And God bent and said, \"My children--what is it that you seek?\"\n\n And she beside me drew my hand that I should speak for both.\n\n I said, \"We have come to ask that Thou shouldst speak to Man, our\n brother, and give us a message for him that he might understand, and\n that he might----\"\n\n God said, \"Go, take the message down to him!\"\n\n I said, \"But what _is_ the message?\"\n\n God said, \"Upon your hearts it is written; take it down to him.\"\n\n And we turned to go; the angels went with us to the door. They looked\n at us.\n\n And one said, \"Ah! but their dresses are beautiful!\"\n\n And the other said, \"I thought it was mire when they came in, but see,\n it is all golden!\"\n\n But another said, \"Hush, it is the light from their faces!\"\n\n And we went down to him.\n\n --_Olive Schreiner_.\n\n\n The Cry of the People\n\n Tremble before your chattels,\n Lords of the scheme of things!\n Fighters of all earth's battles,\n Ours is the might of kings!\n Guided by seers and sages,\n The world's heart-beat for a drum,\n Snapping the chains of ages,\n Out of the night we come!\n\n Lend us no ear that pities!\n Offer no almoner's hand!\n Alms for the builders of cities!\n When will you understand?\n Down with your pride of birth\n And your golden gods of trade!\n A man is worth to his mother, Earth,\n All that a man has made!\n\n We are the workers and makers!\n We are no longer dumb!\n Tremble, O Shirkers and Takers!\n Sweeping the earth--we come!\n Ranked in the world-wide dawn,\n Marching into the day!\n The night is gone and the sword is drawn\n And the scabbard is thrown away!\n\n --_Neihardt_.\n\n\n SPELLING\n\n LESSON 12\n\n\nLast week we learned the rule governing the spelling of derivatives of\n_one_ syllable ending in a single consonant preceded by a single vowel\nwhen we add a suffix beginning with a vowel.\n\nThe same rule applies to words of two or more syllables, accented on the\nlast syllable.\n\nFor example:\n\n _Compel_, compelled, compelling.\n _Prefer_, preferred, preferring.\n\n+Words accented on the last syllable, when they end in a single\nconsonant preceded by a single vowel, double the final consonant when\nyou add a suffix beginning with a vowel.+\n\nWhen these words take a suffix that begins with a _consonant_, they do\n_not_ double the final consonant; as, _preferment_.\n\nWords accented on any syllable but the last, do _not_ double the final\nconsonant; as, _offer_, _offered_, _offering_.\n\nWords that have two vowels before a single final consonant do not double\nthe final consonant; as, _reveal_, _revealed_, _revealing_.\n\nWords that end in a double consonant or any two consonants, keep the two\nconsonants, no matter what suffix they take; as, _indent_, _indented_;\n_skill_, _skilled_, _skillful_.\n\nThe only exception to this rule is when the addition of the suffix\nthrows the accent back to a preceding syllable. When this is the case,\nthe final consonant is not doubled. For example: _refer_, _referred_,\n_ref'erence_; _confer_, _conferring_, _con'ference_.\n\nLook up the following words in the dictionary, watch for the accent,\nmark and add the suffixes, _ed_, _ing_, _ence_ or _ance_, if possible.\n\n +Monday+\n\n Repel\n Alter\n Prefer\n Debar\n Answer\n\n +Tuesday+\n\n Inter\n Offer\n Demur\n Wonder\n Succeed\n\n +Wednesday+\n\n Detain\n Combat\n Compel\n Occur\n Cancel\n\n +Thursday+\n\n Permit\n Travel\n Repeal\n Control\n Profit\n\n +Friday+\n\n Forbid\n Neglect\n Expel\n Render\n Infer\n\n +Saturday+\n\n Benefit\n Retain\n Submit\n Reveal\n Limit\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 13\n\n\nDear Comrade:\n\nDid you ever tie a knot in your handkerchief to help you remember to get\nsomething you felt almost sure you would forget? Well, tying a knot in a\ncord was one of the first ways devised by our ancestors of long ago to\naid them to remember. They also used this plan to send word to those at\na distance or to keep track of things for succeeding generations. A\nrelic of this old device of our forefathers is also found in the rosary\non which the Roman Catholic counts his beads as an aid to memory.\n\nThere are some primitive tribes to-day who still use knotted strings as\nan aid to memory. These consist of a main cord, and fastened at given\ndistances are finer cords of different colors. Each cord is knotted in\ndifferent ways to mean different things and each color, too, has its own\nmeaning. A red string stands for soldiers, a yellow for gold, and a\ngreen for corn, and so on, while a single knot may mean ten, two single\nknots twenty, a double knot 100, two double knots 200. In this way, they\nkeep a record of things, transmit orders and use them for various\npurposes.\n\nOnly a generation ago the tax gatherers in the Island of Hawaii kept\naccount of the assessable property on lines of cordage knotted in this\nmanner, and these cords in some cases were three thousand feet long. The\nmethod of keeping track of things by means of a notched stick is easily\nwithin the memory of many people living today. For in England in the\nearly part of the last century, accounts of debts to the government were\nkept by means of tally sticks, which were merely notched sticks.\n\nSuch methods as these were the only ways primitive man had of keeping\ntrack of things before he had discovered the art of written speech. And\neven after written speech was known and used, these old methods\npersisted.\n\nGradually, step by step, man has come along the path of progress.\nAdventurous spirits, not satisfied with the old way of doing things,\nsought new ways. The conservatives of their day thought them dangerous\npeople, no doubt, and feared that they would destroy the very\nfoundations of society. And this they oft-times did, but only that there\nmight rise a more perfect form of society. It is the seeking,\nquestioning mind that demands the reason for all things, that seeks ever\nbetter ways of doing things. They have always throughout the ages\nrefused to bow to the authority of the past but have dared to live their\nown lives. To them we owe the progress of the world and we are the\ninheritors of their spirit.\n\nLet us prove our kinship by daring to live our own lives and think our\nown thoughts.\n\n Yours for Freedom,\n\n THE PEOPLE'S COLLEGE.\n\n\n INTERROGATIVE PRONOUNS\n\n+228+. You recall that in our first lesson we studied concerning the\nfour different kinds of sentences which we use in expressing our\nthoughts, the _assertive_, the _interrogative_, the _imperative_ and the\n_exclamatory_. The interrogative sentence is the form which we use in\nasking a question, _interrogative_ being derived from the Latin _inter_,\nmeaning _between_, and _rogare_, _to ask_, meaning literally _to ask\nbetween_. The interrogative sentence differs from the assertive sentence\nin the arrangement of the words; for in order to ask questions, we\nusually place the predicate, or part of it at least, before the subject,\nthus:\n\n _Can_ you _use_ good English?\n _Did_ you _spell_ the word correctly?\n _Has_ he _studied_ grammar?\n\nIn these sentences, you note that the helping verbs, _can_, _did_ and\n_has_, are placed first instead of the subject. It is by this\narrangement that we put the sentence in the interrogative form.\n\n+229.+ Frequently, however, in asking questions we wish to ask\nconcerning a person or thing whose name we do not know. So we need a\nword to refer to the unknown object. See how these uses of words grow\nout of our need! We have three interrogative pronouns, _who_ and _which_\nand _what_, that we use to meet this need. Notice the use of these three\npronouns in the following sentences:\n\n _Who_ wrote the Communist Manifesto?\n _Which_ of the two men is the better known?\n _What_ are the closing words of this famous document?\n\nIn these sentences, _who_ and _which_ and _what_ are the interrogative\npronouns, used to ask questions concerning the unknown persons or\nobjects.\n\n+230.+ +Who refers only to human beings or to personified objects.+\n\n+Which refers either to human beings, animals or things.+\n\n+What refers only to things.+\n\n_Which_ and _what_ have the same form for both the subject and the\nobject. _Who_ has a different form for all three forms, the subject\nform, the possessive form, and the object form. It uses the same form,\nhowever, both in singular and plural.\n\n _Subject form_ _Possessive form_ _Object form_\n\n Who Whose Whom\n\n+231.+ We often make mistakes in the use of the different forms of the\npronoun _who_. We often use the subject form for the object form, using\n_who_ where we should have used _whom_. For example:\n\n Who did you see?\n\nThe correct form is:\n\n Whom did you see?\n\nThe pronoun _whom_ is the object of the verb _see_, hence the object\nform should be used. However, the use of the subject form _who_ instead\nof _whom_ is coming into such general use today that some grammarians\naccept it as a permissible usage. The will of the people influences\nlanguage, as it does all other human institutions, and gradually creates\nnew rules.\n\nWrite three sentences, using _who_, _which_ and _what_ as interrogative\npronouns.\n\n+An interrogative pronoun is a pronoun used to ask a question.+\n\n\n RELATIVE PRONOUNS\n\n+232.+ There is one other class of pronouns which plays a great part in\nour speech and is a wonderful help to us. For example, suppose I want to\ntell you several things about this book. I say: _I am reading this book.\nIt interests me greatly._ Now it would be a great advantage to me if I\ncould put these two sentences together, and we have for this use a\npronoun which makes it possible for us to combine these sentences, and\nso I say:\n\n The book which I am reading interests me greatly.\n\nThus I am able to unite two short sentences into a long sentence, which\nconveys my meaning better than the two short sentences and gives a\nsmoother bit of reading. We have four pronouns which we use in this way,\n_who_, _which_, _that_ and _what_ and they are called relative pronouns\nbecause they refer or relate to some noun in the sentence and they also\nserve to connect two statements.\n\n+233.+ +A relative pronoun is a pronoun that relates to an antecedent\nand at the same time connects two statements.+\n\nA relative pronoun always relates to its antecedent and at the same time\nconnects the statement that it introduces with the one that contains the\nantecedent to which it relates, as in the sentence above, _The book\nwhich I am reading, interests me greatly._ _Which_ is the relative\npronoun; first, because it relates to the antecedent, _book_; and\nsecond, because it connects the statement, _I am reading_, with the rest\nof the sentence. Notice these sentences also:\n\n The man who thinks will not enlist in the army.\n We will destroy the system that enslaves us.\n\n_Who_ and _that_ are the relative pronouns in these two sentences and\ntheir antecedents are _man_ and _system_, and they connect the\nstatements, _who thinks_ and _that enslaves us_, with the rest of the\nsentence.\n\n+234.+ +Who is used to relate to persons.+\n\n+Which is used to relate only to animals and things.+\n\n+That may relate to either persons, animals or things.+\n\n+What relates to things.+\n\nNote that _which_, as an interrogative, may refer to persons as well as\nto animals and things; but as a relative, _which_ never refers to\npersons.\n\n+235+. Note that we use the same pronouns _who_, _which_ and _what_ as\nboth relative and interrogative pronouns. You will not be confused in\nthis matter if you will remember that they are called interrogative\npronouns only when they are used to ask questions. When they are used as\ninterrogative pronouns they never have an antecedent. _Who_ and _which_\nand _what_ are always relative pronouns when used in an assertive\nsentence and referring to an antecedent.\n\n_That_ and _what_ have the same form for both the subject and object\nforms. They have no possessive form. _Who_ has a different form for the\nsubject form and the possessive form and the object form. _Which_ has\nthe same form for subject and object forms, and a different form for the\npossessive form. Note the following:\n\n _Subject form_ _Possessive form_ _Object form_\n\n who whose whom\n which whose which\n\n I know the man _who_ called him.\n I know the man _whose_ voice I hear.\n I know the man _whom_ they called.\n\nIn these three sentences we have the pronoun _who_ used in its three\nforms, subject, possessive and object form. We should be very careful\nnot to confuse the subject and the object forms of the pronoun _who_.\n\n This is the book _which_ tells the truth.\n This is the book _whose_ author is in prison.\n This is the book _which_ I wanted.\n\nIn these three sentences we have the pronoun _which_ used in its three\nforms, _subject_ form, _possessive_ form and _object_ form. In the first\nsentence the pronoun is the subject of the verb _tells_; in the second\nsentence, it is used in the possessive form with the noun _author_; in\nthe third sentence, it is used as the object of the verb _wanted_.\n\n+236.+ _What_ differs from the other relative pronouns in that its\nantecedent is never expressed, for it is implied in the word itself.\n_What_ is always equivalent to _that which_, or _the thing which_. For\nexample, the sentence, _Do not tell what I have told you_, is equivalent\nto saying, _Do not tell that which I have told you_, or _the thing which\nI have told you_.\n\n+237+. Never use _what_ in a sentence as a _relative_ pronoun unless you\ncan replace it and make good sense by using _that which_, or _the thing\nwhich_ in place of _what_.\n\nFor example, do not say, _I know that what he would say_. This is\nincorrect. You should say, _I know that which he would say_, or _I know\nwhat he would say_, using _what_ in place of _that which_. Here is a\nsentence that occurred in an English examination recently, which\nillustrates most aptly this point. _A subject is that what something is\nsaid about._ Here _what_ is used incorrectly. _A subject is that about\nwhich something is said_, would have been the correct form.\n\nWatch for this in your speech for it is a most common error and to the\neducated ear is harsh and marks the speaker as uneducated. All of these\nmistakes which we make so commonly will require a considerable amount of\neffort to overcome, but the result is worth the effort, for even those\nabout us who will not take the pains or give the required time and\neffort to acquiring an education for themselves, will give greater heed\nto the speech of those who do speak correctly, and will readily\nacknowledge the leadership of those who have given the time and effort\nto self-development.\n\n+238.+ The antecedent of _who_ is sometimes omitted and understood; for\nexample, _Who follows the cause must endure hardship_, _He_, is\nunderstood and omitted. _He who follows the cause must endure hardship._\n\n+239.+ The relative pronoun itself is often omitted. For example:\n\n These are the men (whom) you must help.\n The words (that) you use and the deeds (that) you do, are your judges.\n\n+240.+ The relative pronouns have compound forms also, such as\n_whoever_, _whosoever_, _whichever_, _whichsoever_, _whatever_ and\n_whatsoever_, which are used in the same manner as the simple forms.\n\n\n COMMON ERRORS\n\n+241.+ Here are a number of common errors which only constant practice\nand watchfulness can overcome. Study these over and watch your\nconversation closely. Force yourself to speak correctly for a time, and\nsoon correct speech will become a habit.\n\n+1.+ +Do not use both a noun and a pronoun as the subject of a\nsentence+; as, _John, he waited for me._ _Mary, she refused to go._\nLeave out the pronouns _he_ and _she_ in these sentences. They are\nunnecessary and incorrect.\n\n+2.+ +Never use+ _hern_, _ourn_, _hisn_ or _yourn_ for _hers_, _ours_,\n_his_ and _yours_; as, _The book is hisn._ _Ourn stopped on the first._\n_Did you get yourn?_ Say: _This book is his._ _Ours stopped on the\nfirst._ _Did you get yours?_\n\n+3.+ +Never say+ _hisself_ for _himself_. There is no such word as\n_hisself_. Do not say, _He hurt hisself_. Say, _He hurt himself_.\n\n+4.+ +Do not say+ _them_ for _those_; as, _Did you bring them songs?_\n_Them things are not right._ Say, _Did you bring those songs?_ _Those\nthings are not right._\n\n+5.+ +Do not use an apostrophe in writing the possessive forms of\npronouns+, as _her's_, _our's_, _it's_. Leave out the apostrophe and\nwrite _hers_, _ours_, _its_.\n\n+6.+ +Do not use _who_ to relate to animals or things+; as, _The dog\nwho bit me was killed_. Say, _The dog that bit me was killed_.\n\n+7.+ +Do not use _myself_ as the subject+. It can be used only as an\nemphatic or reflexive pronoun. It is correct to say, _I found the book\nmyself_, and _I hurt myself_. But do not say, _They asked my friend and\nmyself_, or _Myself and my wife will go_. Say, _They asked my friend and\nme_. _My wife and I will go._\n\n+8.+ +Avoid the use of pronouns when the reference to the antecedent is\nnot clear.+ Better repeat the nouns or re-write the sentence. For\nexample:\n\n He said to his friend that if he did not feel better soon he thought\n he had better go home.\n\nNow you can interpret this in at least four different ways. No one but\nthe speaker can ever know to whom the pronouns _he_ refer, whether to\nthe speaker or to his friend. Or in the sentence,\n\n A tried to see B in the crowd, but could not because he was so short.\n\nWho was short, _A_ or _B_? _John's father died before he was born._ Did\nJohn's father die before John was born or did John's father die before\nJohn's father, himself, was born? Be careful in the use of pronouns in\nthis way.\n\n+9.+ +Remember that _I_, _we_, _he_, _she_, _they_ and _who_ are\nalways used as subject forms and also as the complement of all forms of\nthe verb _be_.+\n\n+10.+ +Remember that _me_, _him_, _her_, _them_, _us_ and _whom_ are\nalways object forms+. Never say, _They charged he and I too much_. Say,\n_They charged him and me too much_. In an attempt to speak correctly and\nfollow the niceties of English, this mistake is so often made. Always\nuse the object form as the object of a verb or preposition.\n\n+11.+ +When a participle is used as a _noun_, and a pronoun is used\nwith it, the pronoun should always be in the _possessive_ form+. We\nmake this mistake so frequently. For example, we say: _Us going there\nwas a mistake_. We should have used the possessive form, _Our going\nthere was a mistake_. _I have never known of him being absent from\nwork._ We should say: _I have never known of his being absent from\nwork_. _Did he tell you about me joining with them?_ This should be,\n_Did he tell you about my joining with them?_ _You talking to him set\nhim to thinking._ This should be, _Your talking to him set him to\nthinking_. Watch this and wherever you have used a participle as a\n_noun_, use the pronoun in the _possessive_ form, as you would with any\nother noun.\n\n+12.+ +Watch carefully that the number of the pronoun always agrees with\nthe number of its antecedent.+ If you are speaking of one person or\nthing use a singular pronoun. If you are speaking of more than one\nperson or thing in your antecedent, use the plural pronoun. For example:\n_Each man must do his own work._ _The soldiers fully understood their\ndanger._\n\n+13.+ +When a singular noun, in the common gender (this means that it\nmay name either a male or female being), is the antecedent of the\npronoun, it is customary for us to use the masculine pronoun.+ For\nexample:\n\n Every student should send in _his_ examination paper promptly.\n\n Every member of the class may select _his_ own subject.\n\nDo not use the pronoun _their_ when the antecedent is a singular noun.\n\n\n SUMMARY\n\n Pronoun--In Place of a Noun\n\n CLASSES\n\n _Personal_ {Simple-- {1st Person, _speaking_.\n {Compound-- {2nd Person, _spoken to_.\n {3rd Person, _spoken of_.\n\n _Interrogative_ {To ask questions.\n {_Who_, _which_ and _what_.\n\n _Relative_ {To refer to another word and connect two statements.\n {_Who_, _which_, _that_ and _what_.\n\n\n Exercise 1\n\nComplete the following sentences by using the correct form of _I_, _me_,\nor _myself_, in the blank spaces:\n\n 1. My partner and......joined the union.\n 2. They asked Henry and......to go.\n 3. May my friend and......call?\n 4. I will attend to that.......\n 5. Let my comrade and......go with you.\n 6. Are you sure it was......?\n 7. I blame......for joining with them.\n 8. They accused......of bothering them.\n 9. I am nearly beside......with grief.\n 10. The manager dismissed the men......among the rest.\n\n\n Exercise 2\n\nComplete the following sentences by using the correct form of _we_, _us_\nor _ourselves_ in the blank spaces:\n\n 1. They are better off than.......\n 2. The French as well as......claim a war of defense.\n 3. Can you blame......who have always stood by you?\n 4. We will do that for.......\n 5. Between......comrades there should be no differences.\n 6. They gave......men work.\n 7. Do not trouble;......will attend to this.......\n 8. They sent a special notice to our friends and.......\n\n\n Exercise 3\n\nComplete the following sentences by using the correct form of _thou_,\n_thee_, _thy_ or _thyself_ in the blank spaces:\n\n 1. To......be true, and it follows as the night the day......\n canst not then be false to any man.\n 2. Paul,......art beside......; much learning hath made ......mad.\n 3. ......shalt love......neighbor as.......\n 4. Trust....... Every heart vibrates to that iron string.\n\n\n Exercise 4\n\nComplete the following sentences by using the correct form of _he_,\n_him_, or _himself_ in the blank spaces:\n\n 1. ......and John are to blame.\n 2. I think it was.......\n 3. My friend and......called on you.\n 4. He blamed......for the accident.\n 5. You are no better than.......\n 6. I shall call for you and.......\n 7. You and......must come on time.\n 8. He found the place.......\n 9. There should be no quarrel between you and......who loves you.\n 10. If you were......would you go?\n\n\n Exercise 5\n\nComplete the following sentences by using the correct form of _she_,\n_her_, or _herself_ in the blank spaces:\n\n 1. They asked Mary and......to go.\n 2. Mary and......went.\n 3. May......and I go with you?\n 4. Let......and Harry go.\n 5. Is that Mary? Yes, it is.......\n 6. There are many points of difference between......and me.\n 7. You are more beautiful than.......\n 8. She brought it to me.......\n 9. If......and I join you, will you go?\n 10. They must not quarrel over......and me.\n\n\n Exercise 6\n\nComplete the following sentences using the correct form of _they_,\n_them_, or _themselves_ in the blank spaces:\n\n 1. They gave......up.\n 2. ......and I will finish the work.\n 3. I found......where......hath thrown......down to rest.\n 4. I am sure it was......for I saw......plainly.\n 5. The workers enslave......by their lack of solidarity.\n 6. ......must learn the lesson.......\n\n\n Exercise 7\n\nCross out the wrong word in the following sentences:\n\n 1. Everybody do--does as he pleases--they please.\n 2. No one should waste his--their opportunities.\n 3. The jury rendered its--their verdict.\n 4. If anyone wishes war, let him--them do the fighting.\n 5. The audience displayed its--their approval by its--their applause.\n 6. The audience remained quietly in its--their seats.\n 7. The jury adjourned for its--their dinner.\n 8. Nobody willingly gives up his--their rights.\n 9. Each one may express his--their opinion.\n 10. Every man received his--their wages.\n\n\n Exercise 8\n\nComplete the following sentences by using the correct form of the\npronouns _who_, _whose_, or _whom_:\n\n 1. ......do you think I am?\n 2. I am the man......you taught yesterday.\n 3. With......are you going?\n 4. The contract was let to a man......we are sure cannot fulfill it.\n 5. The contractor......wishes to bid will come tomorrow.\n 6. On......are you depending?\n 7. The friends......counsel I took, stood by me.\n 8. He is a man......I am sure will succeed.\n 9. We tried to talk to those......we thought would understand us.\n 10. For......did you work?\n\n\n Exercise 9\n\nInsert _who_, _whose_, _whom_, _which_, _that_ or _what_ in the blanks\nin the following sentences:\n\n 1. Man is the only animal......uses a written speech.\n 2. Can you save......you earn?\n 3. Ricardo's law was that the workers always receive a\n wage......permits them to produce and reproduce.\n 4. Have you read the book \"War, What For\"......Kirkpatrick wrote.\n 5. Newspapers......distort the news......they print to serve the\n ruling class are dangerous foes to the workers.\n 6. The massacre at Ludlow was an event......aroused the working\n class.\n 7. They......live by the labor of others are drones in society and\n should be given the fate......they deserve.\n 8. The big machine gun......will destroy slavery is the printing\n press.\n 9. The man......leadership we should follow is he......preaches\n social equality.\n 10. We know......we need and we will demand......is our right.\n\n\n Exercise 10\n\nIn the following quotations note the use of the pronouns and mark\nwhether they are _personal_, _relative_ or _interrogative_, whether they\nare used in the _subject_ form, _possessive_ form or _object_ form:\n\n 1. \"Camerado, I give you my hand,\n I give you my love more precious than money,\n I give you myself before preaching or law;\n Will you give me yourself, will you come travel with me,\n Shall we stick by each other as long as we live?\"\n\n 2. \"I think I could turn and live with animals they are so placid and\n self-contained,\n I stand and look at them long and long, they do not sweat and whine\n about their condition,\n They do not lie awake in the dark and weep for their sins,\n They do not make me sick discussing their duty to God;\n Not one is dis-satisfied, not one is demented with the mania of\n owning things.\n Not one kneels to another nor to his kind, that lived thousands of\n years ago,\n Not one is respectable or unhappy over the whole earth.\"\n --_Whitman_.\n\n\n Exercise 11\n\nNote the omission of the antecedent in the first sentence, also the use\nof the relative _what_ in the last sentence of the first paragraph:\n\n \"Whoso would be a man, must be nonconformist. He who would gather\n immortal palms must not be hindered by the name of goodness, but must\n explore if it be goodness. Nothing is at last sacred but the integrity\n of our own mind. Absolve you to yourself, and you shall have the\n suffrage of the world. I remember an answer which, when quite young, I\n was prompted to make to a valued adviser who was wont to importune me\n with the dear old doctrines of the church. On my saying, \"What have I\n to do with the sacredness of traditions, if I live wholly from\n within?\" my friend suggested--\"But these impulses may be from below,\n not from above.\" I replied, \"They do not seem to me to be such; but if\n I am the devil's child, I will live then from the devil.\" No law can\n be sacred to me but that of my nature. Good and bad are but names very\n readily transferable to that or this; the only right is what is after\n my constitution; the only wrong what is against it.\n\n A foolish consistency is the hobgoblin of little minds, adored by\n little statesmen and philosophers and divines. With consistency a\n great soul has simply nothing to do. He may as well concern himself\n with his shadow on the wall. Out upon your guarded lips! Sew them up\n with pack threads, do. Else, if you would be a man, speak what you\n think today in words as hard as cannon balls, and tomorrow speak what\n tomorrow thinks in hard words again, though you contradict everything\n you said today. Ah, then, exclaim the aged ladies, you shall be sure\n to be misunderstood. Misunderstood! It is a right fool's word. Is it\n so bad then to be misunderstood? Pythagoras was misunderstood, and\n Socrates, and Jesus, and Luther, and Copernicus, and Galileo, and\n Newton, and every pure and wise spirit that ever took flesh. To be\n great is to be misunderstood.\"--_Emerson_.\n\n\n\n\n SPELLING\n\n LESSON 13\n\n\nThere are a few more rules governing the spelling of derivative words.\nWords ending in silent _e_ keep the _e_ before the suffix beginning with\na consonant. Notice the following words:\n\n excite excitement\n like likeness\n force forceful\n shame shameless\n lone lonesome\n live lively\n\nWords ending in silent _e_ drop the _e_ before the suffix beginning with\na vowel, as:\n\n excite excitable\n live living\n grieve grievous\n force forcible\n\nSome words ending in silent _e_ retain the _e_ before the suffix\nbeginning with a vowel, to prevent a change in the pronunciation or to\npreserve the identity of the word. Notice the following words:\n\n peace peaceable\n courage courageous\n singe singeing\n change changeable\n shoe shoeing\n notice noticeable\n\nThese are words ending in the soft sound of _c_ and _g_, where the _e_\nis retained to preserve the correct pronunciation of the _c_ and _g_,\nand with some few words like _toe_, _dye_, etc., where the dropping of\nthe _e_ would lose the identity of the word.\n\nThe _e_ is dropped in a few words before the suffix beginning with a\nconsonant, as in _wholly_, _nursling_, _judgment_, _wisdom_, _lodgment_.\n\nAdd the suffixes _ment_ and _ing_ to the words in Monday's lesson; the\nsuffix _able_ to the words for Tuesday and Wednesday; the suffixes\n_some_ and _ous_ to the words for Thursday; the suffixes _ly_ or _ness_\nto the words for Friday and Saturday.\n\n +Monday+\n\n Excite\n Advise\n Chastise\n Disfranchise\n Enslave\n\n +Tuesday+\n\n Manage\n Receive\n Blame\n Exchange\n Imagine\n\n +Wednesday+\n\n Admire\n Service\n Desire\n Peace\n Pronounce\n\n +Thursday+\n\n Whole\n Meddle\n Courage\n Advantage\n Outrage\n\n +Friday+\n\n Accurate\n Positive\n False\n Definite\n Distinct\n\n +Saturday+\n\n Agreeable\n Careful\n Awful\n Sure\n Secure\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 14\n\n\nDear Comrade:\n\nYou remember our definition of a word; a word is the sign of an idea. In\nour lessons we have been studying the different kinds of words which we\nuse in the expression of our complete thoughts. Probably the first step\nin the development of language was to name the objects about us. Then\nthe next logical step would be to invent words which would tell what\nthese objects did. So we have our nouns, which are the names of things;\nour verbs, which tell what these things do; and in these we have the\nfoundation for spoken and written speech. We soon found, however, that\nthe constant repetition of a name was tiresome and annoying, so we\ninvented words which we could use in place of these nouns; and we have\npronouns.\n\nAll of the things about us possess certain qualities and our next great\nneed was for words to describe these qualities; so we have adjectives.\nEach adjective is a sign of an idea. It adds its part to the expression\nof our complete thought. So we find that each part of speech comes\nlogically in its place to fill a certain need. Without any one of them,\nwe would be crippled in our power of expression. Each different word is\nthe sign of an idea and the combination of these ideas as represented by\nthe various signs gives us the complete expression of our thought.\n\nSo primitive man in the development of written speech had signs to\nexpress the various things about him. Naturally his first sign was a\npicture, as nearly as he could draw it, of the object itself. If he\nwanted to tell you about a tree he drew a picture of the tree; the\npicture of a man represented a man, and so on. You will notice among\nchildren that this is the first development in their endeavor to express\ntheir thoughts in writing. They draw pictures. The average small child\ncannot understand why you read those strange marks on the page. They\nwant you to read the pictures. To their mind that is the only way to\ncommunicate ideas.\n\nThese early forefathers of ours grew to be very adept at this picture\nwriting. We have examples of this among the Indians of our own country.\nThere is a picture on the face of a big rock on the shores of Lake\nSuperior which records an expedition across the lake led by a noted\nIndian chief. Canoes are shown in the picture with the crew denoted by a\nseries of upright strokes and there is a picture of the chief on\nhorseback. You or I would have great difficulty in reading this picture\nwriting, but an Indian could read it right off just as we would read a\nwritten page. Aids to memory such as knotted strings and tally sticks\nwere the first step toward written speech. This picture writing was the\nsecond step toward the development of written speech.\n\nWe owe a great deal to the work which these primitive ancestors of ours\naccomplished. It took them years and years to develop through these\ndifferent stages and our rapid development of the last few centuries has\nonly been made possible because of this slow and patient building of the\nfoundation. An understanding of this helps us to appreciate the place we\noccupy in this great struggle of the ages. The power of written speech\nopens up to us such tremendous possibilities. Let us make the most of\nthem, that we too may hand on worth while things to those who follow us.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n CLASSES OF ADJECTIVES\n\n+242.+ Adjectives, like nouns and pronouns, are divided into classes.\nAdjectives are divided into two main classes, _qualifying_ and\n_limiting_.\n\n+243.+ An adjective which qualifies a noun is one which names some\nquality which is possessed by the word which it modifies. When we say,\n_Trees grow_, we are making a general statement; that is, we are saying\nsomething that is true of any kind of trees. We have not described any\nparticular tree. But when we say, _The tall trees grow_, _The old trees\ngrow_, _The young trees grow_, the words _tall_, _old_ and _young_\ndescribe certain qualities of the trees, which separate them into\nclasses. So these adjectives are _qualifying adjectives_.\n\nAn adjective qualifies a noun when it attributes some quality to the\nnoun, as, _The brave man_, _The sweet apple_, _The pretty girl_, _The\nlarge house_, etc.\n\n+244.+ But if we say, _this tree_, _that tree_, _some trees_, _many\ntrees_, _three trees_, or _four trees_, we are not giving any quality of\nthe tree, but are pointing out a particular tree or trees and limiting\nthe word to the ones pointed out. So such adjectives as _the_, _this_,\n_that_, _some_, _many_, _three_ and _four_ are limiting adjectives. An\nadjective limits a noun when it restricts or limits its meaning as to\nquantity or number.\n\n+245.+ So adjectives are divided into two classes, _qualifying_\nadjectives and _limiting_ adjectives.\n\n+Words that limit or qualify other words are called _modifiers_\nbecause they modify or affect the meaning of the words to which they\nare added.+ So adjectives are modifiers of the nouns and pronouns to\nwhich they are added because they modify or qualify or limit the meaning\nof the noun or pronoun.\n\nThe limiting adjectives answer the questions _which_ and _how many_. The\nqualifying adjectives answer the questions _which_ and _what kind_.\n\n+246.+ +A qualifying adjective is an adjective which describes the noun\nit modifies by attributing to it some quality.+\n\n+A limiting adjective is an adjective which merely shows which one or\nhow many, without describing the noun it modifies.+\n\n\n HOW TO DISCOVER AN ADJECTIVE\n\n+247.+ Sometimes the noun may have several adjectives qualifying or\nmodifying it; as,\n\n The beautiful, old elm tree shades the lawn.\n\n_The_, _beautiful_, _old_ and _elm_, all modify _tree_, telling\nsomething of the qualities or pointing out which tree we are speaking\nof. You can discover an adjective in a sentence by asking the questions,\n_which_, _what kind_, or _how many_; and the words that answer these\nquestions will be the adjectives in the sentence. For example in this\nsentence:\n\n Those three immense factories employ thousands of men.\n\n_Factories_ is the noun, subject of the sentence. _Which_ factory is\nindicated by the adjective _those_. _How many_ factories is indicated by\nthe adjective _three_. _What kind_ of factories is indicated by the\nadjective _immense_. So we have three adjectives answering the three\nquestions, _which_, _what kind_ and _how many_.\n\n\n Exercise 1\n\nIn the following sentences the adjectives are printed in _italics_.\nStudy them carefully and determine which are qualifying and which are\nlimiting adjectives. Note that the possessive nouns and possessive\npronouns are _not_ adjectives. _Its_ in the phrases _its cruel fangs_\nand _its savage claws_, is a possessive pronoun, third person singular.\nIn the last sentence _beggar's_, _miser's_, and _Ingersoll's_, are nouns\nin the possessive form.\n\n _This terrible_ war in Europe is slaughtering _the_ working-class.\n _Gaunt_ famine follows war.\n A _docile_, _meek_, _humble_, working-class makes war _possible_.\n _The shrieking_ shell snarls like a _living_ thing; like _some wild_\n beast in _ferocious_ glee it thrusts its _cruel_ fangs in earth and\n rock and rends _living_ flesh with its _savage_ claws.\n Its _fetid_ breath of _poison_ powder scorches in _the autumn_ winds.\n _Shattered_ bones, _torn_ flesh and _flowing_ blood were mingled on\n _the_ battlefield with _broken_ swords and _split_ rifles.\n _The best modern_ rifles will force _a_ bullet through _five human_\n bodies at _a_ range of _twelve hundred_ feet.\n _The pitiful_ dead, _slain_ in war, sleep under _the solemn_ pines,\n _the sad_ hemlock, _the tearful_ willow and _the embracing_ vines.\n A world without _the_ beggar's _outstretched_ palm, _the_ miser's\n _heartless_ _stony_ stare, _the piteous_ wail of want, _the livid_\n lips of lies, _the cruel_ eyes of scorn, was Ingersoll's vision of\n _the_ future.\n\n\n QUALIFYING ADJECTIVES\n\n+248.+ Qualifying adjectives are also called _descriptive_ adjectives\nbecause they describe the noun. They answer the questions _which_ and\n_what kind_.\n\nYou remember we found in the beginning of our study of English, that\nwords were grouped into classes according to the work which they do in\nthe sentence, not according to the form of the word itself. For\ninstance, we have already found that some words, without changing their\nform, may be used either as a noun or as a verb. Take the word _oil_,\nfor instance. I may say, _I oil the engine_. Here I have used the word\n_oil_ as a verb telling what I do. But I may say, _The oil is gone_.\nHere I have used the word _oil_ as a noun, subject of the sentence. The\npart of speech to which a word belongs in the English language, always\ndepends upon the work which it does in the sentence.\n\n+1.+ So we have nouns which are used as descriptive adjectives, for\nexample the word _oil_, which we have found we can use either as a noun\nor a verb, may also be used as an adjective. For example; I may say,\n_the oil tank_. Here I have used the word _oil_ as a descriptive\nadjective modifying the word _tank_. So also we may say, _the oak tree_,\n_the stone curb_, _the earth wall_. In these expressions _oak_, _stone_\nand _earth_ are nouns used as descriptive adjectives.\n\n+2.+ We have descriptive adjectives derived from proper nouns, as\nFrench, English, American. These are called proper adjectives; and since\nall proper nouns must begin with a capital letter, these proper\nadjectives, also, should always begin with a capital letter.\n\n+3.+ We have also descriptive adjectives derived from verbs as _active_,\n_talkative_, _movable_, _desirable_, derived by the addition of suffixes\nto the verbs _act_, _talk_, _move_ and _desire_.\n\n\n LIMITING ADJECTIVES\n\n+249.+ Limiting adjectives are also divided into classes, the\n_numerals_, the _demonstratives_ and the _articles_.\n\n\n Numeral Adjectives\n\n+250.+ Numeral adjectives are those which limit nouns as to number or\norder. They are such adjectives as _one_, _two_, _three_, _four_, etc.,\nand _first_, _second_ and _third_, etc., as for example:\n\n _Three_ men applied for work.\n The train ran at the rate of _forty_ miles an hour.\n There have always been _two_ classes in the world.\n The _first_ martyr to anti-militarism was Jaures.\n The _eighteenth_ day of March is the anniversary of the Paris Commune.\n\nIn these sentences the adjectives _three_, _forty_, _two_, _first_ and\n_eighteenth_ are all numeral adjectives. They limit the nouns which they\nmodify as to number or order.\n\n+Adjectives that limit nouns as to number or order are called numeral\nadjectives. Numeral adjectives answer the question how many or in what\norder.+\n\n\n Demonstratives\n\n+251.+ We have also a class of adjectives which are used to point out\nsome particular person or thing. These are called _demonstrative_\nadjectives. Demonstrate means literally _to point out_. So these\nadjectives point out from a number of things, one particular thing to\nour attention. These demonstrative adjectives are _this_, _that_,\n_those_, _these_, _yonder_, _former_, _latter_ and _same_.\n\n_These_ and _those_ are the plural forms of _this_ and _that_. _This_\nand _these_ are used to point out things near at hand. _That_ and\n_those_ are used to point out things more distant, as _This is my book_.\n_These are my papers_, meaning _this book_ or _these papers_, close to\nme. By, _That is my pencil_ and _Those are my letters_, I mean _that\npencil_, and _those letters_, which are farther away from me.\n\n_Former_ and _latter_ are used to show which of two things already\nmentioned is referred to, and to point out things in point of time, not\nof place. For example, we may say:\n\n We no longer observe the _former_ customs, but rather prefer the\n _latter_.\n He did not like his _former_ job but this _latter_ job pleases him.\n\nYou understand from this that we have been discussing and describing two\nkinds of work, and that the first in point of time was unpleasant and\nthe second pleasant.\n\nThe demonstrative adjective _same_ refers to something of which we have\njust spoken, as for example, _He has gone to work, I must do the same\nthing_. These demonstrative adjectives answer the question which, so\nwhen you wish to discover a demonstrative in a sentence, ask the\nquestion _which_, and the answer will be the demonstrative adjective.\n\n\n Exercise 2\n\n 1. _This_ study is very interesting.\n 2. _These_ comrades will stand by us.\n 3. _That_ solution will never deceive the people.\n 4. _Those_ books have opened our eyes.\n 5. _Yonder_ battle appals the world.\n 6. _Former_ investigations have had no results.\n 7. _This latter_ decision has reversed the _former_.\n 8. The class struggle has persisted through the centuries; we are\n engaged in the _same_ struggle.\n\nMake sentences of your own containing these demonstrative adjectives.\n\n\n ARTICLES\n\n+252.+ We have three adjectives which are used so commonly that we have\nput them in a class by themselves. These three little words are _a_,\n_an_ and _the_, and we call them articles. The word _article_ literally\nmeans a little joint or limb, and these three little words are so\nclosely connected with the nouns with which they are used that they seem\nto be a part or joint or limb of the noun itself, and so we have called\nthem articles.\n\n_A_ and _an_ are called the _indefinite_ articles because they point out\nan object in a very indefinite manner. _The_ is called the _definite_\narticle for it points out in a more definite way.\n\nWe use _a_ before words beginning with a consonant sound, as _a man_, _a\ntree_, _a book_; and we use _an_ before words beginning with a vowel\nsound, as _an apple_, _an editor_, _an orange_, _an heir_. In _heir_ the\n_h_ is silent, and we say _an_ because the word begins with a vowel\nsound. _A_ is used before words beginning with _u_ because long _u_ is\nequivalent in sound to a consonant, for the blending of the sounds of\nwhich long _u_ is composed produces the initial sound of _y_, which is a\nconsonant sound. For example, we say, _a university_, _a useful work_,\netc., and not _an university_. Before words beginning with short _u_,\nuse _an_, as, _an upstart_, etc.\n\nIn deciding whether to use _a_ or _an_, watch the initial _sound_ of the\nword, not the initial _letter_. If it is a vowel sound use _an_, if a\nconsonant sound, use _a_.\n\n\n Exercise 3\n\nUnderscore the correct article in the following sentences:\n\n 1. Bring me an--a apple.\n 2. He is a--an able orator.\n 3. A--an heir was born to the German King.\n 4. He built a--an house for his family.\n 5. He is an--a honest man.\n 6. He is a--an undertaker.\n 7. I had to take a--an upper berth.\n 8. He joined a--an union.\n 9. It is a--an unique book.\n 10. He is a--an unruly member of society.\n 11. He told a--an untruth.\n 12. He wears a--an uniform.\n 13. It is a--an honor to be chosen.\n\n+253.+ When a singular noun is modified by several adjectives, only one\nof the articles _an_ or _a_ must be used if the noun denotes but _one_\nobject; but if the noun denotes more than one object the article must be\nrepeated before each noun. For example, I say, _A red, white and blue\nflag_. You know I mean but one flag, containing the three colors, red,\nwhite and blue. But if I say, _A red, a white and a blue flag_, you know\nI mean three flags, one red, one white, and one blue.\n\nNote the use of the article in the following sentences:\n\n He wears a black and white suit.\n He wears a black and a white suit.\n He sold a red and white cow.\n He sold a red and a white cow.\n He bought a gas and coal stove.\n He bought a gas and a coal stove.\n\nThe first sentences in each of the above series refers to only one\nobject. The second sentences all refer to two objects.\n\n+254.+ There are some rules concerning the article _the_ that it is well\nto know because we do not always say what we wish to say, if we do not\nobserve these rules or customs of speech. For example, I say, _The\neditor and publisher of this book is unknown_. I have used the article\n_the_ but once, and I mean that the editor and publisher is one person.\nBut I may say, _The editor and the publisher of this book are well\nknown_. In this sentence I have used the article _the_ twice, _the_\neditor and _the_ publisher, and I mean that the editor and the publisher\nare two different persons.\n\nSo when two or more nouns following each other denote the same person or\nthing, the article is not repeated, but when the nouns denote different\npersons or things, the article must be repeated before each noun. Be\nsure to use the proper form of the verb.\n\nNote the following sentences and underscore the proper verb to complete\nthe meaning:\n\n The secretary and treasurer were--was here.\n The secretary and the treasurer were--was elected.\n The singer and artist were--was with me.\n The singer and the artist were--was on the program.\n\nSometimes we have two things so closely associated in use that they may\nbe considered as forming a single idea, so that we may use the article\nbefore the first one only. For example:\n\n The pen and ink is gone.\n He bought a horse and buggy.\n The bread and butter is on the plate.\n\n\n INTERROGATIVE ADJECTIVES\n\n+255.+ You remember we found in the study of pronouns that we have\ninterrogative pronouns which we use in asking questions when we do not\nknow the name of the object concerning which we are asking. We also have\nadjectives which we use in asking questions when we do not know the\nnumber or quality of the object concerning which we are asking. For\nexample:\n\n _Which_ book did you enjoy most?\n _What_ work are you doing now?\n _What_ machine did you order?\n\n_Which_ and _what_ are the interrogative adjectives in these sentences.\n\n+Interrogative adjectives are adjectives used in asking questions.+\n\n\n INDEFINITES\n\n+256.+ We have one more class of adjectives called indefinites.\n\n+An indefinite adjective is one that does not denote any particular\nperson or thing.+\n\nAll such adjectives as _each_, _every_, _either_, _neither_, _some_,\n_any_, _many_, _much_, _few_, _all_, _both_, _no_, _none_, _several_ and\n_certain_ are indefinite adjectives. We use them when we are not\nspeaking of any particular person or thing, but are speaking in a broad,\ngeneral sense and in an indefinite manner.\n\n+257.+ The interrogative adjectives are sometimes used in this\nindefinite way. They are sometimes used to modify nouns when a direct\nquestion is not asked, and they are then used, not as interrogative\nadjectives, but as indefinite adjectives. For example:\n\n He did not know which party to join.\n I have not learned what time he will go.\n\nIn these sentences _which_ and _what_ are not used to ask questions, but\nare used to describe an unknown object.\n\n\n Exercise 4\n\nAll the words in italics are adjectives. Decide to which class each\nadjective belongs.\n\nNote in this exercise the compound words used as adjectives, as:\n_earth-born_, _self-made_, _new-lit_, _blood-rusted_. Look up the\nmeaning of these adjectives and see if you can use other adjectives in\ntheir places and keep the same meaning. Note the use of _fellest_.\n\n Slavery, _the earth-born_ Cyclops, _fellest_ of _the giant_ brood,\n Sons of _brutish_ Force and Darkness, who have drenched _the_ earth\n with blood,\n _Famished_ in his _self-made_ desert, _blinded_ by our _purer_ day,\n Gropes in yet _unblasted_ regions for his _miserable_ prey;--\n Shall we guide his _gory_ fingers where our _helpless_ children play?\n They have rights who dare maintain them; we are traitors to our sires,\n _Smothering_ in their _holy_ ashes Freedom's _new-lit_ altar-fires;\n Shall we make their creed our jailer? Shall we, in our haste to slay,\n From the tombs of _the old_ prophets steal _the funeral_ lamps away\n To light up _the_ martyr-fagots round _the_ prophets of to-day?\n\n _New_ occasions teach _new_ duties; Time makes _ancient_ good,\n _uncouth_;\n They must upward still, and onward, who would keep _abreast_ of\n Truth;\n Lo, before us gleam her camp-fires! We ourselves must Pilgrims be,\n Launch our Mayflower, and steer boldly through _the desperate winter_\n sea,\n Nor attempt _the_ Future's portal with _the_ Past's _blood-rusted_\n key.\n --_Lowell_.\n\n\n Exercise 5\n\nThe following is from Oscar Wilde's story of _The Young King_. Oscar\nWilde was a master of English, and if you have the opportunity, read all\nof this beautiful story and watch his use of adjectives. Mark the\nadjectives in this excerpt and use them in sentences of your own.\n\n And as the young King slept he dreamed a dream, and this was his\n dream. He thought that he was standing in a long, low attic, amidst\n the whirr and clatter of many looms. The meager daylight peered in\n through the grated windows and showed him the gaunt figures of the\n weavers, bending over their cases. Pale, sickly-looking children were\n crouched on the huge crossbeams. As the shuttles dashed through the\n warp they lifted up the heavy battens, and when the shuttles stopped\n they let the battens fall and pressed the threads together. Their\n faces were pinched with famine, and their thin hands shook and\n trembled. Some haggard women were seated at a table, sewing. A\n horrible odor filled the place. The air was foul and heavy, and the\n walls dripped and streamed with damp.\n\n The young King went over to one of the weavers and stood by him and\n watched him.\n\n And the weaver looked at him angrily and said, \"Why art thou watching\n me? Art thou a spy set on us by our master?\"\n\n \"Who is thy master?\" asked the young King.\n\n \"Our master!\" cried the weaver, bitterly. \"He is a man like myself.\n Indeed, there is but this difference between us--that he wears fine\n clothes while I go in rags, and that while I am weak from hunger he\n suffers not a little from overfeeding.\"\n\n \"The land is free,\" said the young King, \"and thou art no man's\n slave.\"\n\n \"In war,\" answered the weaver, \"the strong make slaves of the weak,\n and in peace the rich make slaves of the poor. We must work to live,\n and they give us such mean wages that we die. We toil for them all day\n long, and they heap up gold in their coffers, and our children fade\n away before their time, and the faces of those we love become hard and\n evil. We tread out the grapes, another drinks the wine. We sow the\n corn, and our own board is empty. We have chains, though no eye\n beholds them; and are slaves, though men call us free.\"\n\n \"Is it so with all?\" he asked.\n\n \"It is so with all,\" answered the weaver, \"with the young as well as\n with the old, with the women as well as with the men, with the little\n children as well as with those who are stricken in years. The\n merchants grind us down, and we must needs do their bidding. The\n priest rides by and tells his beads, and no man has care of us.\n Through our sunless lanes creeps Poverty with her hungry eyes, and Sin\n with his sodden face follows close behind her. Misery wakes us in the\n morning, and Shame sits with us at night. But what are these things to\n thee? Thou art not one of us. Thy face is too happy.\" And he turned\n away scowling, and threw the shuttle across the loom, and the\n young King saw that it was threaded with a thread of gold.\n\n And a great terror seized upon him, and he said to the weaver, \"What\n robe is this that thou art weaving?\"\n\n \"It is the robe for the coronation of the young King,\" he answered;\n \"What is that to thee?\"\n\n And the young King gave a loud cry and woke and lo! he was in his own\n chamber, and through the window he saw the great honey- moon\n hanging in the dusky air.\n\n\n\n\n SPELLING\n\n LESSON 14\n\n\nYou remember in the formation of plurals, we learned that words ending\nin _y_ change _y_ to _i_ when _es_ is added; as, _lady, ladies_; _baby,\nbabies_; _dry, dries_, etc.\n\nThere are several rules concerning words ending in _y_, knowledge of\nwhich will aid us greatly in spelling.\n\n+1.+ +Words ending in _ie_ change the _ie_ to _y_ before _ing_\nto prevent a confusing number of vowels.+ For example, _die, dying_;\n_lie, lying_; _tie, tying_.\n\n+2.+ +Words of more than one syllable ending in _y_ preceded by a\nconsonant, change _y_ into _i_ before all suffixes except those\nbeginning with _i_.+ For example:\n\n happy, happily, happiness;\n witty, wittier, wittiest;\n satisfy, satisfied, satisfying;\n envy, enviable, envying.\n\nThis exception is made for suffixes beginning with _i_, the most common\nof which is _ing_, to avoid having a confusing number of _i's_.\n\n+3.+ +Most words ending in _y_ preceded by a vowel retain the _y_\nbefore a suffix.+ For example:\n\n destroy, destroyer, destroying;\n buy, buyer, buying;\n essay, essayed, essayist.\n\nThe following words are exception to this rule:\n\n laid,\n paid,\n said,\n daily,\n staid.\n\nMake as many words as you can out of the words given in this week's\nspelling lesson by adding one or more of the following suffixes: _er_,\n_est_, _ed_, _es_, _ing_, _ly_, _ness_, _ful_, _ment_, _al_.\n\n +Monday+\n\n Beauty\n Portray\n Deny\n Rare\n Multiply\n\n +Tuesday+\n\n Mercy\n Bury\n Obey\n Lovely\n Envy\n\n +Wednesday+\n\n Tie\n Defy\n Study\n Decry\n Crazy\n\n +Thursday+\n\n Merry\n Silly\n Lusty\n Imply\n Day\n\n +Friday+\n\n Dismay\n Duty\n Employ\n Satisfy\n Pretty\n\n +Saturday+\n\n Pay\n Joy\n Journey\n Qualify\n Sorry\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 15\n\n\nDear Comrade:\n\nIn this week's lesson we are finishing the study of adjectives, which\nadds another part of speech to those which we have studied. We can see\nin the study of each additional part of speech how each part has its\nplace in the expression of our ideas. We could not express ourselves\nfully if we lacked any of these parts of speech. Each one is not an\narbitrary addition to our language but has come to us out of the need\nfor it. We see that there are no arbitrary rules but in language, as in\nall things else, growing needs have developed more efficient tools. With\nthese have grown up certain rules of action so we can have a common\nusage and system in our use of these tools. It has taken years of effort\nto accomplish this. The changes have been slow and gradual, and this\nlanguage which we are studying is the finished product.\n\nThis slow development in the use of language, even in our own lives,\nmakes us realize how many thousands of years it must have taken our\nprimitive ancestors to reach a point where they could use the phonetic\nalphabet. We have found that at first they used simple aids to memory,\nas knotted strings and tally sticks. Then they began to draw pictures of\nthings about them and so were able to communicate with one another by\nmeans of these pictures. When a man was going away from his cave and\nwanted to leave word for those who might come, telling them where he had\ngone and how soon he would return, he drew a picture of a man over the\nentrance with the arm extended in the direction in which he had gone.\nThen he drew another picture of a man in a sleeping position and also\none of a man with both hands extended in the gesture which indicated\nmany. These two pictures showed that he would be away over many nights.\nIn some such rude manner as this, they were able to communicate with one\nanother.\n\nBut man soon began to _think_, and he needed to express ideas concerning\nthings of which he could not draw pictures. He could draw a picture of\nthe sun, but how could he indicate light? How could he indicate the\ndifferent professions in which men engaged, such as the farmer and\npriest, etc.?\n\nHe was forced to invent symbols or signs to express these ideas, so his\nwriting was no longer a picture of some object, but he added to it\nsymbols of abstract ideas. A circle which stood for the sun written with\nthe crescent which stood for the moon, indicated light. The bee became a\nsymbol of industry. An ostrich feather was a symbol of justice, because\nthese feathers were supposed to be of equal length. A picture of a woman\nstood simply for a woman, but a picture of two women stood for strife,\nand three women stood for intrigue. These old ancestors of ours became\nwise quite early concerning some things. The symbol for a priest in the\nearly Egyptian picture writing was a jackal. Perhaps not because he\n\"devoured widows' houses,\" but because the jackal was a very watchful\nanimal. The symbol for mother was a vulture because that bird was\nbelieved to nourish its young with its own blood.\n\nIt naturally required a good memory and a clear grasp of association to\nbe able to read this sort of writing. It required many centuries for\nthis slow development of written speech.\n\nThe development of language has been a marvelous growth and a wonderful\nheritage has come to us. Let us never be satisfied until we have a\nmastery of our language and find a way to express the ideas that surge\nwithin us. A mastery of these lessons will help us.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n ADJECTIVES AND PRONOUNS\n\n+258.+ From our study of the adjective, we know that it is a word used\nwith a noun to qualify or limit its meaning. But a great many times we\nfind these adjectives used without the noun which they modify. As, for\nexample, I may say, _This is mine_, and the adjective _this_ is used\nalone without the noun which it modifies, and you are able to tell only\nby what I have been saying or by some action of mine to what I am\nreferring when I say _this_.\n\nWhen adjectives are used in this manner, they are used like pronouns--in\nplace of a noun. So sometimes we find an adjective used with a noun, and\nsometimes used as a pronoun, in place of a noun; and since we name our\nparts of speech by the work which they do in the sentence, an adjective\nused in this way is not an adjective, but a pronoun or word used in\nplace of a noun.\n\nSo these words are pronouns when they stand alone to represent\nthings--when they are used in place of a noun. They are adjectives when\nthey are used _with_ a noun to limit or qualify the noun. For example, I\nmay say, _This tree is an elm, but that tree is an oak_. _This_ and\n_that_ in this sentence are adjectives used to modify the noun _tree_.\nBut I may say, _This is an oak and that is an elm_, and in this sentence\n_this_ and _that_ are used without a noun, they are used as pronouns.\n\n+259.+ Our being able to name every part of speech is not nearly so\nimportant as our being able to understand the functions of the different\nparts of speech and being able to use them correctly. But still it is\nwell for us to be able to take a sentence and point out its different\nparts and tell what each part is and the function which it serves in the\nsentence. So sometimes in doing this we may find it difficult to tell\nwhether certain words are adjectives or pronouns. We can distinguish\nbetween adjectives and pronouns by this rule:\n\nWhen you cannot supply the noun which the adjective modifies, from the\n_same_ sentence, then the word which takes the place of the noun is a\npronoun, but if you can supply the omitted noun from the same sentence,\nthen the word is used as an adjective. Thus, we do not say that the noun\nis understood unless it has already been used in the same sentence and\nis omitted to avoid repetition. We make each sentence a law unto itself\nand classify each word in the sentence according to what it does in its\nown sentence.\n\nSo if a noun does not occur in the same sentence with the word about\nwhich we are in doubt as to whether it is a pronoun or adjective, it is\na pronoun or word used in place of a noun. For example, in the sentence,\n_This book is good but that is better_; _book_ is understood after the\nword _that_ and left out to avoid tiresome repetition of the word\n_book_. Therefore _that_ is an adjective in this sentence. But if I say,\n_This is good, but that is better_; there is no noun understood, for\nthere is no noun in the sentence which we can supply with _this_ and\n_that_. Therefore in this sentence _this_ and _that_ are pronouns, used\nin place of the noun. And since _this_ and _that_, when used as\nadjectives, are called demonstrative adjectives; therefore when _this_\nand _that_, _these_ and _those_, and similar words, are used as pronouns\nthey are called demonstrative pronouns.\n\n+260.+ Be careful not to confuse the possessive pronouns with\nadjectives. Possessive pronouns modify the nouns with which they are\nused, but they are not adjectives, they are possessive pronouns. _My_,\n_his_, _her_, _its_, _our_, _your_ and _their_ are all possessive\npronouns, not adjectives. Also be careful not to confuse nouns in the\npossessive form with adjectives.\n\n\n ADJECTIVES AS NOUNS\n\n+261.+ Sometimes you will find words, which we are accustomed to look\nupon as adjectives, used alone in the sentence without a noun which they\nmodify. For example, we say, _The strong enslave the weak_. Here we have\nused the adjectives _strong_ and _weak_ without any accompanying noun.\nIn sentences like this, these adjectives, being used as nouns, are\nclassed as nouns. Remember, in your analysis of a sentence, that you\nname every word according to the work which it does in that sentence, so\nwhile these adjectives are doing the work of nouns, we will consider\nthem as nouns.\n\nThese words are not used in the same manner in which demonstrative\nadjectives are used as pronouns. There is no noun omitted which might be\ninserted, but these adjectives are used rather to name a class. As, for\nexample; when we say, _The strong_, _The weak_, we mean all those who\nare strong and all those who are weak, considered as a class. You will\nfind adjectives used in this way quite often in your reading, and you\nwill find that you use this construction very often in your ordinary\nspeech. As, for example:\n\n The rich look down upon the poor.\n The wise instruct the ignorant.\n\nMany examples will occur to you. Remember these adjectives are nouns\nwhen they do the work of nouns.\n\n\n ADJECTIVES WITH PRONOUNS\n\n+262.+ Since pronouns are used in place of nouns, they may have\nmodifiers, also, just as nouns do. So you will often find adjectives\nused to modify pronouns. As, for example; _He, tired, weak and ill, was\nunable to hold his position_. Here, _tired_, _weak_ and _ill_ are\nadjectives modifying the pronoun _he_.\n\n+263.+ We often find a participle used as an adjective with a pronoun.\nAs, for example:\n\n She, having finished her work, went home.\n They, having completed the organization, left the city.\n He, having been defeated, became discouraged.\n\nIn these sentences, the participles, _having finished_, _having\ncompleted_, and _having been defeated_, are used as adjectives to modify\nthe pronouns _she_, _they_ and _he_.\n\n\n COMPARISON\n\n+264.+ We have found that adjectives are a very important part of our\nspeech for without them we could not describe the various objects about\nus and make known to others our ideas concerning their various\nqualities. But with the addition of these helpful words we can describe\nvery fully the qualities of the things with which we come into contact.\nWe soon find, however, that there are varying degrees of these\nqualities. Some objects possess them in slight degree, some more fully\nand some in the highest degree. So we must have some way of expressing\nthese varying degrees in the use of our adjectives.\n\nThis brings us to the study of comparison of adjectives. Suppose I say:\n\n That orange is sweet, the one yonder is sweeter, but this one is\n sweetest.\n\nI have used the adjective _sweet_ expressing a quality possessed by\noranges in three different forms, _sweet_, _sweeter_ and _sweetest_.\nThis is the change in the form of adjectives to show different degrees\nof quality. This change is called comparison, because we use it when we\ncompare one thing with another in respect to some quality which they\npossess, but possess in different degrees.\n\nThe form of the adjective which expresses a simple quality, as _sweet_,\nis called the positive degree. That which expresses a quality in a\ngreater degree, as _sweeter_, is called the comparative degree. That\nwhich expresses a quality in the greatest degree, as _sweetest_, is\ncalled the superlative degree.\n\n+265.+ +Comparison is the change of form of an adjective to denote\ndifferent degrees of quality.+\n\n+There are three degrees of comparison, positive, comparative and\nsuperlative.+\n\n+The positive degree of an adjective denotes simple quality.+\n\n+The comparative degree denotes a higher degree of a quality.+\n\n+The superlative degree denotes the highest degree of a quality.+\n\n+266.+ Most adjectives of one syllable and many adjectives of two\nsyllables regularly add _er_ to the positive to form the comparative\ndegree, and _est_ to the positive to form the superlative degree, as:\n\n _Positive_ _Comparative_ _Superlative_\n\n sweet sweeter sweetest\n cold colder coldest\n soft softer softest\n brave braver bravest\n clear clearer clearest\n\n+267.+ Adjectives ending in _y_ change _y_ to _i_ and add _er_ and _est_\nto form the comparative and superlative degree, as:\n\n _Positive_ _Comparative_ _Superlative_\n\n busy busier busiest\n lazy lazier laziest\n sly slier sliest\n witty wittier wittiest\n\n+268.+ Many adjectives cannot be compared by this change in the word\nitself, since the addition of _er_ and _est_ would make awkward or\nill-sounding words. Hence we must employ another method to form the\ncomparison of this sort of words. To say, _beautiful_, _beautifuller_,\n_beautifullest_, is awkward and does not sound well. So we say\n_beautiful_, _more beautiful_, _most beautiful_.\n\nMany adjectives form the comparative and superlative degree by using\n_more_ and _most_ with the simple form of the adjective, as:\n\n _Positive_ _Comparative_ _Superlative_\n\n beautiful more beautiful most beautiful\n thankful more thankful most thankful\n sensitive more sensitive most sensitive\n wonderful more wonderful most wonderful\n\n+269.+ Adjectives of two syllables, to which _er_ and _est_ are added to\nform the comparison, are chiefly those ending in _y_ or _le_, such as:\n\n _Positive_ _Comparative_ _Superlative_\n\n happy happier happiest\n noble nobler noblest\n steady steadier steadiest\n feeble feebler feeblest\n able abler ablest\n witty wittier wittiest\n\n+270.+ Some adjectives, few in number, but which we use very often, are\nirregular in their comparison. The most important of these are as\nfollows: (It would be well to memorize these.)\n\n _Positive_ _Comparative_ _Superlative_\n\n good better best\n well \" \"\n bad worse worst\n ill \" \"\n much more most\n many \" \"\n little less least\n late later latest\n latter last\n far farther farthest\n (up) adv. upper uppermost\n (in) adv. inner innermost\n\n\n DESCENDING COMPARISON\n\n+271.+ The change in form of adjectives in the positive, comparative and\nsuperlative shows that one object has more of a quality than others with\nwhich it is compared. But we also wish at times to express the fact that\none object has less of the quality than is possessed by others with\nwhich it is compared; so we have what we may call the descending\ncomparison, by means of phrases formed by using _less_ and _least_\ninstead of _more_ and _most_. Using _less_ with the positive degree\nmeans a degree less than the positive, while using _least_ expresses the\nlowest degree. For example:\n\n Descending Comparison\n\n _Positive_ _Comparative_ _Superlative_\n\n beautiful less beautiful least beautiful\n intelligent less intelligent least intelligent\n sensitive less sensitive least sensitive\n thankful less thankful least thankful\n\n\n PARTICIPLES AS ADJECTIVES\n\n+272.+ You remember, when we studied the participle, that we found it\nwas called a participle because it partook of the nature of two or more\nparts of speech. For example; in the sentence, _The singing of the birds\ngreeted us_; _singing_ is a participle derived from the verb _sing_, and\nis used as a noun, the subject of the verb _greeted_.\n\nBut participles are used not only as nouns; they may also be used as\nadjectives. For example; we may say, _The singing birds greeted us_.\nHere the participle _singing_ describes the birds, telling what kind of\nbirds greeted us, and is used as an adjective modifying the noun\n_birds_.\n\nYou will recall that we found there were two forms of the participle,\nthe present participle and the past participle. The present participle\nis formed by adding _ing_ to the root form of the verb; and the past\nparticiple in regular verbs is formed by adding _d_ or _ed_ to the root\nform, and in irregular verbs by a change in the verb form itself. These\ntwo simple forms of participles are often used as adjectives.\n\n+273.+ The present participle is almost always active; that is, it\nrefers to the actor. As, for example; _Vessels, carrying soldiers, are\nconstantly arriving_. Here the present participle _carrying_ describes\nthe noun _vessels_, and yet retains its function as a verb and has an\nobject, _soldiers_. So it partakes of two parts of speech, the verb and\nthe adjective.\n\n+274.+ The past participle, when used alone, is almost always passive,\nfor it refers not to the actor, but to what is acted upon, thus:\n\n The army, beaten but not conquered, prepared for a siege.\n\nIn this sentence _beaten_ is the past participle of the irregular verb\n_beat_, and _conquered_ is the past participle of the regular verb\n_conquer_, and both modify the noun _army_, but refer to it, not as the\nactor, but as the receiver of the action. Hence, the past participle is\nalso the _passive_ participle.\n\nNote in the following sentences the use of the present and past\nparticiple as adjectives:\n\n A _refreshing_ breeze came from the hills.\n They escaped from the _burning_ building.\n _Toiling_, _rejoicing_, _sorrowing_, onward through life he goes.\n The man, _defeated_ in his purpose, gave up in despair.\n The child, _driven_ in its youth to work, is robbed of the joy of\n childhood.\n The army, _forced_ to retreat, destroyed all in its path.\n The children, _neglected_ by society, grow up without their rightful\n opportunities.\n\n\n Exercise 1\n\nThe adjectives and participles used as adjectives in the following\nsentences are printed in _italics_. Determine which adjectives are\ncapable of comparison, and whether they are compared by adding _er_ or\n_est_, or by the use of _more_ and _most_.\n\n In _a_ community _regulated_ by laws of demand and supply, but\n _protected_ from _open_ violence, _the_ persons who become _rich_ are,\n generally _speaking_, _industrious_, _resolute_, _proud_, _covetous_,\n _prompt_, _methodical_, _sensible_, _unimaginative_, _insensitive_ and\n _ignorant_. _The_ persons who remain _poor_ are _the_ entirely\n _foolish_, _the_ entirely _wise_, _the idle_, _the reckless_, _the\n humble_, _the thoughtful_, _the dull_, _the imaginative_, _the\n sensitive_, _the well-informed_, _the improvident_, _the_ irregularly\n and impulsively _wicked_, _the clumsy_ knave, _the open_ thief, and\n _the_ entirely _merciful_, _just_ and _godly_ persons.--_Ruskin_.\n\n\n PARTICIPLE PHRASES\n\n+275.+ If you will refer now to Lesson 9 you will find that we studied\nin that lesson concerning participle phrases; that is, several words\nused as a participle. We found that these participle phrases may also be\nused as nouns; as, for example:\n\n His having joined the union caused him to lose his position.\n\n_Having joined_ is here a participle phrase used as a noun, subject of\nthe verb _caused_. Participle phrases may also be used as adjectives.\n\nYou remember that we had four participle phrases, as follows:\n\n +Present perfect+, _active_, having called.\n +Present perfect+, _passive_, having been called.\n +Progressive+, _active_, having been calling.\n +Progressive+, _passive_, being called.\n\nThese participle phrases are used as adjectives to describe and modify\nnouns, thus:\n\n The soldier, _having joined_ his comrades, fought in the trenches.\n The nurse, _having been watching_ for days, was nearly exhausted.\n\nThe passive phrases also are used as adjectives, thus:\n\n The woman, _having been hired_ by the manager, went to work.\n The man, _being attacked_, fought bravely.\n\nHere the participle phrases _having been hired_ and _being attacked_ are\nused as adjectives to modify the nouns _woman_ and _man_.\n\nUse the participles and participle phrases of the verbs _see_ and _obey_\nin sentences of your own.\n\n\n USES OF ADJECTIVES\n\n+276.+ In our use of adjectives, we find it convenient to use them in\nseveral different ways. The most common use is closely connected with\nthe noun as a modifying word, seeming in a sense almost a part of the\nnoun; as in the sentence, _These brave men have bequeathed to us\nsplendid victories_. In this sentence _these_ and _brave_ are easily\ndiscovered to be adjectives, being used in such close connection with\nthe noun.\n\nBut sometimes we find the adjectives a little farther away from the noun\nwhich it describes, and then it becomes a little more difficult to find.\nYou will recall, in our study of the copulative verb _be_, that we found\nit was simply a connecting word, connecting that which followed the verb\nwith its subject. So we often find an adjective used in the predicate\nwith a copulative verb showing what is asserted of the subject. When an\nadjective is used in this way, it modifies the subject just as much as\nif it were directly connected by being placed immediately before the\nnoun. For example:\n\n The lesson was long and difficult.\n\n_Long_ and _difficult_ are used in the predicate after the copulative\nverb _was_, but are used to modify the subject _lesson_ just as much as\nthough we said instead, _It was a long and difficult lesson_. So watch\ncarefully for adjectives used with the copulative verb _be_ in all its\nforms, _am_, _is_, _are_, _was_, _were_; and the phrases, _has been_,\n_will be_, _must be_, etc.\n\n+277.+ You may find adjectives also used following the noun. As, for\nexample: _The man, cool and resolute, awaited the attack_. _Cool_ and\n_resolute_ are adjectives modifying the noun _man_, but they follow the\nnoun, instead of being placed before it.\n\n\n COMMON ERRORS\n\n+278.+ There are a number of common errors which we make in comparison,\nwhich we should be careful to avoid.\n\n1. A number of adjectives cannot be compared for they in themselves\nexpress the highest degree of quality, so they have no shades of meaning\nand will not admit of comparison. For example: _full_, _empty_, _level_,\n_round_, _square_. If a thing is full or empty or level or round or\nsquare, it cannot be more full, or more empty, or more level, or more\nround, or more square. So do not compare adjectives that already express\nthe highest degree of a quality. Also such words as _supreme_,\n_eternal_, and _infallible_, cannot be compared for they also express\nthe highest degree of quality.\n\n2. Do not use _more_ with the comparative form made by using _er_, or\n_most_ with the superlative form, made by using _est_. For example: do\nnot say, _They cannot be more happier than they are_. Say, _They cannot\nbe happier_; or _They cannot be more happy_. Use either form but never\nboth. Do not say, _That is the most wisest plan_. Say either, _That is\nthe wisest plan_; or _That is the most wise plan_, but never use both\nforms. Never use _most_ with a superlative form.\n\n3. Do not use the superlative form in comparing _two_ objects. The\nsuperlative form is used only when more than two are compared. For\nexample; do not say, _He is the smallest of the two_. Say, _He is the\nsmaller of the two_. _Which is the largest end?_ is incorrect. _Which is\nthe larger end?_ is correct. _Which is the oldest, John or Henry?_ is\nalso incorrect. This should be, _Which is the older, John or Henry?_ Use\nthe _comparative_ form always when comparing _two_ objects.\n\n4. In stating a comparison, avoid comparing a thing with itself. For\nexample; _New York is larger than any city in the United States_. In\nthis sentence, when you say _any_ city in the United States, you are\nincluding New York; so you are really comparing New York with itself,\nand you are saying that New York is larger than itself. You should have\nsaid, _New York is larger than any other city in the United States_; or,\n_New York is the largest city in the United States_. When you compare an\nobject with all others of its kind be sure that the word _other_ follows\nthe comparative word _than_.\n\n5. When an adjective denoting _one_ or _more than one_ modifies a noun,\nthe adjective and the noun must agree in number. For example; _The house\nis 30 foot square_. _Thirty_ denotes more than one, so a plural noun\nshould be used, and this sentence should be, _The house is 30 feet\nsquare_. _We are traveling at the rate of 40 mile an hour._ This should\nbe, _We are traveling at the rate of 40 miles an hour_.\n\n6. Only two adjectives, _this_ and _that_ change their form when\nmodifying a plural noun. _These_ and _those_ are the plural forms of\n_this_ and _that_. So remember always to use _this_ and _that_ with\nsingular nouns and _these_ and _those_ with plural nouns. For example;\ndo not say, _These kind of people will never join us_. You should say,\n_This kind of people will never join us_. Or, _Those sort of flowers\ngrows easily_. You should say, _That sort of flowers grows easily_.\n\n7. Place your adjectives where there can be no doubt as to what you\nintend them to modify. Put the adjective _with_ the noun which it\nmodifies. For example; do not say, _a fresh bunch of flowers_, _a new\npair of shoes_, _a salt barrel of pork_, _an old box of clothes_, _a\ncold cup of water_, _a new load of hay_. Put the adjective with the noun\nwhich it modifies, and say, _a bunch of fresh flowers_, _a pair of new\nshoes_, _a barrel of salt pork_, _a box of old clothes_, _a cup of cold\nwater_, _a load of new hay_.\n\n8. Adjectives are usually placed before the nouns they qualify, but\nsometimes, especially in poetry or in the use of participles, they\nfollow the nouns. They should not, however, be placed too far away from\nthe noun which they modify or be unnecessarily separated from the noun.\nWhere there are two or more adjectives used to qualify the same noun,\nplace nearest the noun the adjective most closely connected with the\nobject described and place farthest from the noun the adjective least\nclosely connected with the noun. If they are all of the same rank, place\nthem where they will sound best, usually according to their length,\nnaming the shortest adjective first.\n\nCorrect the following sentences by arranging the adjectives in the\nproper order:\n\n The summer sky was a blue, soft, beautiful sky.\n He bought a brown, fine, big horse.\n A gold, beautiful, expensive watch was given her.\n The new, beautiful apartment building is on the corner.\n He advertised for a young, intelligent, wide awake man.\n\n9. Never use _them_ as an adjective. _Them_ is a pronoun. One of the\nworst mistakes which we can make is to use such phrases as _them\nthings_, _them men_, _them books_. Say, _those things_, _those men_,\n_those books_.\n\n10. Do not use _less_ for the comparative form of _few_. The comparative\nform of _few_ is _fewer_. _Less_ refers only to quantity, _fewer_ to\nnumber. For example:\n\n He raised _less_ grain this year than last, because he has _fewer_\n horses now than he had then.\n He uses _fewer_ words because he has _less_ to say.\n There are but _few_ people here today; there were still _fewer_ (not\n less) yesterday.\n\n\n Exercise 2\n\nCorrect the adjectives in this exercise:\n\n 1. Hand me the little knife.\n 2. He claims to be more infallible than anyone else.\n 3. Mary is the oldest of the two.\n 4. He was the bestest boy in school.\n 5. The barn is forty foot long.\n 6. Yonder is a happy crowd of children.\n 7. Which is the largest end?\n 8. I found the bestest book.\n 9. This is the most principal rule.\n 10. Give me a cold cup of water.\n 11. These kind of books will not do.\n 12. Give me them books.\n 13. Who is the tallest, you or John?\n\n\n Exercise 3\n\nMark all the adjectives in this poem. Note especially the participles\nused as adjectives.\n\n THE COLLECTION\n\n I passed the plate in church.\n There was a little silver, but the crisp bank-notes heaped\n themselves up high before me;\n And ever as the pile grew, the plate became warmer and warmer, until\n it fairly burned my fingers, and a smell of scorching flesh rose\n from it, and I perceived that some of the notes were beginning\n to smolder and curl, half-browned, at the edges.\n And then I saw through the smoke into the very substance of the\n money, and I beheld what it really was:\n I saw the stolen earnings of the poor, the wide margin of wages\n pared down to starvation;\n I saw the underpaid factory girl eking out her living on the street,\n and the over-worked child, and the suicide of the discharged\n miner;\n I saw the poisonous gases from great manufactories, spreading\n disease and death;\n I saw despair and drudgery filling the dram-shop;\n I saw rents screwed out of brother men for permission to live on\n God's land;\n I saw men shut out from the bosom of the earth and begging for the\n poor privilege to work, in vain, and becoming tramps and paupers\n and drunkards and lunatics, and crowding into almshouses, insane\n asylums and prisons;\n I saw ignorance and vice and crime growing rank in stifling, filthy\n slums;\n I saw shoddy cloth and adulterated food and lying goods of all\n kinds, cheapening men and women, and vulgarizing the world;\n I saw hideousness extending itself from coal-mine and foundry over\n forest and river and field;\n I saw money grabbed from fellow grabbers and swindled from fellow\n swindlers, and underneath the workman forever spinning it out of\n his vitals;\n I saw the laboring world, thin and pale and bent and care-worn and\n driven, pouring out this tribute from its toil and sweat into\n the laps of the richly dressed men and women in the pews, who\n only glanced at them to shrink from them with disgust;\n I saw all this, and the plate burned my fingers so that I had to\n hold it first in one hand and then in the other; and I was glad\n when the parson in his white robes took the smoking pile from me\n on the chancel steps and, turning about, lifted it up and laid\n it on the altar.\n It was an old-time altar, indeed, for it bore a burnt offering of\n flesh and blood--a sweet savor unto the Moloch whom these people\n worship with their daily round of human sacrifices.\n The shambles are in the temple as of yore, and the tables of the\n money-changers waiting to be overturned.\n\n --_Ernest Crosby_.\n\n\n\n\n SPELLING\n\n LESSON 15\n\n\nThere is a class of words having the sound of long _e_, represented by\nthe diphthong _ie_, and another class having the same sound represented\nby _ei_. It is a matter of perplexity at times to determine whether one\nof these words should be spelled with _ie_ or _ei_. Here is a little\nrhyme which you will find a valuable aid to the memory in spelling these\nwords:\n\n When the letter _c_ you spy,\n Put the _e_ before the _i_.\n\nFor example, in such words as _deceit_, _receive_ and _ceiling_, the\nspelling is _ei_. On the other hand, when the diphthong is not preceded\nby the letter _c_, the spelling is _ie_, as in _grief_, _field_,\n_siege_, etc.\n\nThere are a few exceptions to this rule, such as _either_, _neither_,\n_leisure_, _seize_ and _weird_. Most words, however, conform to the\nrule--when preceded by _c_, _ei_ should be used; when preceded by any\nother letter, _ie_.\n\nObserve that this rule applies only when there is a diphthong having the\nsound of long _e_. When the two letters do not have the sound of long\n_e_, as in _ancient_, the rule does not apply.\n\n +Monday+\n\n Deceive\n Belief\n Conceive\n Brief\n Ceiling\n\n +Tuesday+\n\n Field\n Receive\n Piece\n Chief\n Leisure\n\n +Wednesday+\n\n Receipt\n Wield\n Weird\n Thief\n Perceive\n\n +Thursday+\n\n Deceit\n Yield\n Grief\n Seize\n Conceit\n\n +Friday+\n\n Relieve\n Neither\n Liege\n Shield\n Niece\n\n +Saturday+\n\n Relief\n Achievement\n Reprieve\n Lien\n Siege\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 16\n\n\nDear Comrade:\n\nWe have been tracing the development of written speech in order that we\nmight have a clearer understanding of our own language. We have found\nhow our earliest ancestors communicated with each other by signs and an\narticulate speech that was probably a little better than that of some\nanimals of today. They gradually developed this articulate speech and\nthen began to have need for some form of written speech. That which\ndistinguishes man from the animals primarily is his power to remember\nand to associate one idea with another. From this comes his ability to\nreason concerning the connection of these ideas. Without this power of\nassociative memory we would not be able to reason. If you could not\nrecall the things that happened yesterday and had not the power of\nimagination concerning the things that may happen tomorrow, your\nreasoning concerning today would not be above that of the animals.\n\nSo man soon found it necessary to have some way of recalling accurately,\nin a manner that he could depend upon, the things that happened\nyesterday and the day before and still farther back in time. So that his\nfirst step was the invention of simple aids to memory such as the\nknotted strings and tally sticks. Then he began to draw pictures of the\nobjects about him which he could perceive by the five senses, the things\nwhich he could see and hear and touch and taste and smell.\n\nBut man, the Thinker, began to develop and he began to have ideas about\nthings which he could not see and hear and touch and taste and smell. He\nbegan to think of abstract ideas such as light and darkness, love and\nhate, and if he was to have written speech he must have symbols which\nwould express these ideas. So we have found that he used pictures of the\nthings he perceived with his five senses to symbolize some of his\nabstract ideas, as for example; a picture of the sun and moon to\nrepresent light; the bee to symbolize industry; the ostrich feather to\nrepresent justice. But as his ideas began to develop you can readily see\nthat in the course of time there were not enough symbols to go around\nand this sort of written speech became very confusing and very difficult\nto read.\n\nNecessity is truly the mother of invention, and so this need of man\nforced him to invent something entirely new--something which had been\nundreamed of before. He began now to use pictures which were different\nin sense but the names of which had the same sound. You can find an\nexample of this same thing on the Children's Puzzle Page in the rebus\nwhich is given for the children to solve. As for example: A picture of\nan eye, a saw, a boy, a swallow, a goose and a berry, and this would\nstand for the sentence, I saw a boy swallow a gooseberry.\n\nPerhaps you have used the same idea in some guessing game where a mill,\na walk and a key stands for Milwaukee. And so we have a new form of\npicture writing. Notice in this that an entirely new idea has entered\nin, for the picture may not stand for the whole word but may stand for\none syllable of the word as in the example given above. The mill stands\nfor one syllable, walk for another and key for another. This was a great\nstep for it meant the division of the word into various sounds\nrepresented by the syllables.\n\nWhat a new insight it gives us into life when we realize that not only\nour bodies but the environment in which we live, the machines with which\nwe work and even the language which we use has been a product of man's\nown effort. Man has developed these things for himself through a\nconstant and steady evolution. It makes us feel that we are part of one\nstupendous whole; we belong to the class which has done the work of the\nworld and accomplished these mighty things. The same blood flows in us;\nthe same power belongs to us. Truly, with this idea, we can stand erect\nand look the whole world in the face and demand the opportunity to live\nour own lives to the full.\n\n Yours for Freedom,\n\n THE PEOPLE'S COLLEGE.\n\n\n WORDS ADDED TO VERBS\n\n+279.+ We have just finished the study of adjectives and we have found\nthat adjectives are words added to nouns to qualify or to limit their\nmeaning. Without this class of words it would be impossible for us to\nexpress all of our ideas, for we would be at a loss to describe the\nobjects about us. Adjectives enable us to name the qualities or tell the\nnumber of the objects with which we come in contact.\n\nThe verb, we have found, expresses the action of these objects; in other\nwords, the verb tells what things do. So with adjectives and verbs we\ncan describe the objects named by the nouns and tell what they do. For\nexample, I may say, _Men work_. Here I have used simply a noun and a\nverb; then I may add various adjectives to this and say, _Strong,\nindustrious, ambitious men work_. By the use of these adjectives, I have\ntold you about the kind of men who work; but I have said nothing about\nthe action expressed in the verb _work_. I may want to tell you _how_\nthey work and _when_ they work; _where_ they work and _how much_; in\nother words, describe fully the action expressed in the verb _work_, so\nI say:\n\n The men work busily.\n The men work late.\n The men work well.\n The men work inside.\n The men work hard.\n The men work here.\n The men work now.\n The men work more.\n\nWords like _busily_, _hard_, _late_, _here_, _well_, _now_, _inside_,\nand _more_, show _how_, _when_, _where_ and _how much_ the men work.\n\nWe could leave off these words and still have a sentence, since the\nother words make sense without them, but these words describe the action\nexpressed in the verb.\n\nWords used in this way are called adverbs because they are added to\nverbs to make our meaning more definite, very much as adjectives are\nadded to nouns.\n\n+280.+ The word adverb means, literally, _to the verb_, and one would\nsuppose from this name that the adverb was strictly a verb modifier, but\nan adverb is used to modify other words as well. An adverb may be used\nto modify an adjective; for example, we might say: _The man was very\nbusy_. _This lesson is too long._ Here _very_ and _too_ are added to the\nadjectives _busy_ and _long_ to qualify their meaning.\n\n+281.+ You remember in the comparison of adjectives, we used the words\n_more_ and _most_ to make the comparative and superlative degrees. Here\n_more_ and _most_ are adverbs used with the adjectives to qualify their\nmeaning. Adverbs used in this way will always answer the question, _how\nmuch_, _how long_, etc. In the sentence, _The man is very busy_, _very_\nis used to answer the question _how_ busy. And in the sentence, _The\nlesson is too long_, the adverb _too_ answers the question _how_ long.\n\nAn adverb is also added to another adverb sometimes to answer the\nquestion _how_. For example; we say, _The man works very hard_. Here the\nadverb _hard_ tells _how_ the man works and _very_ modifies the adverb\n_hard_, and answers the question _how hard_. So we have our definition\nof an adverb:\n\n+282.+ +An adverb is a word that modifies the meaning of a verb, an\nadjective or another adverb.+\n\nRemember that adjectives are used only with nouns or pronouns, but the\nadverb may be used with a verb or an adjective or another adverb. You\nremember that we had in our first lesson, as the definition of a word,\nthat, _a word is a sign of an idea_. The idea is a part of a complete\nthought. See how all of these various words represent ideas, and each\ndoes its part to help us express our thoughts.\n\n\n HOW TO TELL ADVERBS\n\n+283.+ We need not have much difficulty in always being able to tell\nwhich words in a sentence are adverbs, for they will always answer one\nof the following questions: _How?_ _When?_ _Where?_ _Why?_ _How long?_\n_How often?_ _How much?_ _How far?_ or _How little?_ etc. Just ask one\nof these questions and the word that answers it is the adverb in your\nsentence. Take the following sentence:\n\n He _always_ came _down too rapidly_.\n\nThe word _always_ answers the question _when_. So _always_ is an adverb,\ndescribing the time of the action expressed in the verb _came_--He\n_always_ came. _Down_ answers the question _where_. So _down_ is the\nadverb describing the _place_ of the action. _Rapidly_ answers the\nquestion _how_, and is the adverb describing the _manner_ of the action.\n_Too_ also answers the question _how_, and modifies the adverb\n_rapidly_.\n\n\n Exercise 1\n\nUnderscore the adverbs in the following sentences and tell which\nword they modify:\n\n 1. He writes correctly.\n 2. She answered quickly.\n 3. A very wonderful future awaits us.\n 4. You should not speak so hastily.\n 5. You can speak freely here.\n 6. He could never wait patiently.\n 7. We very often make mistakes.\n 8. She very seldom goes there.\n 9. He usually walks very rapidly.\n 10. I have read the lesson quite carefully.\n 11. We would willingly and cheerfully give our all for the cause.\n 12. He frequently comes here but I do not expect him today.\n 13. If we work diligently and faithfully we will soon learn to speak\n correctly and fluently.\n 14. I am almost sure I can go there tomorrow.\n 15. It was more beautifully painted than the other.\n 16. We eagerly await the news from the front.\n 17. He always gladly obeyed his father.\n 18. She spoke quite simply and met with a very enthusiastic reception.\n 19. The difficulty can be easily and readily adjusted.\n\n\n Exercise 2\n\nUse the following adverbs in sentences to modify verbs:\n\n slowly\n here\n now\n gently\n loudly\n never\n soon\n carefully\n nobly\n down\n seldom\n easily\n\nUse the following adverbs in sentences to modify adjectives:\n\n quite\n very\n more\n too\n most\n less\n nearly\n so\n\nUse the following adverbs in sentences to modify adverbs:\n\n too\n very\n quite\n less\n more\n most\n least\n so\n\n\n CLASSES OF ADVERBS\n\n+284.+ There are a good many adverbs in our language, yet they may be\ndivided, according to their meaning, into six principal classes:\n\n+1. Adverbs of time.+ These answer the question _when_, and are such\nadverbs as _now_, _then_, _soon_, _never_, _always_, etc.\n\n+2. Adverbs of place.+ These answer the question _where_, and are such\nadverbs as _here_, _there_, _yonder_, _down_, _above_, _below_, etc.\n\n+3. Adverbs of manner.+ These answer the question _how_, and are such\nadverbs as _well_, _ill_, _thus_, _so_, _slowly_, _hastily_, etc.\n\n+4. Adverbs of degree.+ These answer the questions _how much_, _how\nlittle_, _how far_, etc., and are such adverbs as _much_, _very_,\n_almost_, _scarcely_, _hardly_, _more_, _quite_, _little_, etc.\n\n+5. Adverbs of cause.+ These answer the question _why_, and are such\nadverbs as _therefore_, _accordingly_, _hence_, etc.\n\n+6. Adverbs of number.+ These are such adverbs as _first_, _second_,\n_third_, etc.\n\n\n Exercise 3\n\nIn the following sentences there are adverbs of each class used. Find\nthe adverbs of the different classes.\n\n 1. We shall always be found in the forefront of the struggle.\n 2. It is much more effective to train the young.\n 3. He came first and remained through the entire program.\n 4. It is pleasant to know that we have done well.\n 5. Our comrades are fighting yonder in the trenches.\n 6. Therefore we shall never acknowledge defeat.\n 7. Come down and discuss the matter with us.\n 8. We would soon be able to agree if we understood the facts.\n 9. Study your lessons slowly and carefully.\n 10. He was scarcely able to tell his story.\n 11. Accordingly I am sending you full particulars of the plan.\n 12. He came third in the ranks.\n\n\n INTERROGATIVE ADVERBS\n\n+285.+ The adverbs _how_, _when_, _where_, _why_, _whither_, _whence_,\netc., are used in asking questions, and when they are used in this way\nthey are called interrogative adverbs. For example:\n\n _How_ did it happen?\n _Where_ are you going?\n _Whence_ came he?\n _When_ did he come?\n _Why_ did you do it?\n _Whither_ are you going?\n\nThese adverbs, _how_, _when_, _where_, _why_, _whence_ and _whither_,\nare used in these sentences to modify the verbs and ask the questions\nconcerning the _time_ or _place_ or _manner_ of action expressed in the\nverb.\n\n_How_ may also be used as an interrogative adverb modifying an adjective\nor another adverb. For example:\n\n How late did he stay?\n How large is the house?\n\nIn the first sentence, the adverb _how_ modifies the adverb _late_, and\nintroduces the question. In the second sentence _how_ modifies the\nadjective _large_ and introduces the question.\n\n\n Exercise 4\n\nWrite sentences containing the interrogative adverbs _how_, _when_,\n_where_ and _why_, to modify verbs and ask simple questions.\n\nWrite sentences using the interrogative adverb _how_ to modify an\nadjective and an adverb and to introduce a question.\n\n\n ADVERBS OF MODE\n\n+286.+ There are some adverbs which scarcely fall into any of the above\nclasses and cannot be said to answer any of these questions. They are\nsuch adverbs as _indeed_, _certainly_, _fairly_, _truly_, _surely_,\n_perhaps_ and _possibly_. These adverbs really modify the entire\nsentence, in a way, and are used to show how the statement is\nmade,--whether in a positive or negative way or in a doubtful way. For\nexample:\n\n _Surely_ you will not leave me.\n _Truly_ I cannot understand the matter as you do.\n _Perhaps_ he knows no better.\n _Indeed_, I cannot go with you.\n\nHere, these adverbs, _truly_, _surely_, _perhaps_ and _indeed_, show the\nmanner in which the entire statement is made; so they have been put in a\nclass by themselves and called +adverbs of mode+. _Mode_ means literally\n_manner_, but these are not adverbs that express manner of action, like\n_slowly_ or _wisely_ or _well_ or _ill_. They express rather the manner\nin which the entire statement is made, and so really modify the whole\nsentence.\n\n\n PHRASE ADVERBS\n\n+287.+ We have certain little phrases which we have used so often that\nthey have come to be used and regarded as single adverbs. They are such\nphrases as _of course_, _of late_, _for good_, _of old_, _at all_, _at\nlength_, _by and by_, _over and over_, _again and again_, _through and\nthrough_, _hand in hand_, _ere long_, _in vain_, _to and fro_, _up and\ndown_, _as usual_, _by far_, _at last_, _at least_, _in general_, _in\nshort_, etc. These words which we find used so often in these phrases we\nmay count as single adverbs.\n\n\n ADJECTIVES AND ADVERBS\n\n+288.+ Sometimes the same word may be used either as an adjective or as\nan adverb, and you may have some difficulty in telling whether it is an\nadjective or an adverb. Some of these words are: _better_, _little_,\n_late_, _far_, _hard_, _further_, _first_, _last_, _long_, _short_,\n_much_, _more_ and _high_. For example:\n\n The _late_ news verifies our statement.\n The man came _late_ to his work.\n\nIn the first sentence, the word _late_ is used as an adjective modifying\nthe noun _news_. In the second sentence, the word _late_ is used as an\nadverb to modify the verb _came_.\n\n+289.+ You can always distinguish between adjectives and adverbs by\nthis rule: Adjectives modify _only nouns_ and _pronouns_, and the one\nessential characteristic of the adverb, as a limiting word, is that it\nis _always_ joined to some other part of speech than a noun. An adverb\nmay modify a verb, adjective or other adverb, but never a noun or\npronoun.\n\nYou recall the rule which we have made the very foundation of our study:\nnamely, that every word is classified in the sentence according to the\n_work_ which it does in that sentence. So a word is an adjective when it\nlimits or modifies or qualifies a noun or pronoun; a word is an adverb\nwhen it qualifies any part of speech other than a noun or pronoun,\neither a verb or an adjective or an adverb, or even an entire sentence,\nas is the case with adverbs of mode.\n\n+290.+ Many adverbs are regularly made from nouns and adjectives by\nprefixes and suffixes. Adverbs are made from adjectives chiefly by\nadding the suffix _ly_, or by changing _ble_ to _bly_. For example:\n_honestly_, _rarely_, _dearly_, _ably_, _nobly_, _feebly_. But all words\nthat end in _ly_ are not adverbs. Some adjectives end in _ly_ also, as,\n_kingly_, _courtly_, etc. The only way we can determine to which class a\nword belongs is by its use in the sentence.\n\n\n Exercise 5\n\nIn the following sentences, tell whether the words printed in italics\nare used as adjectives or as adverbs: also note the words ending in\n_ly_. Some are adverbs and some adjectives.\n\n 1. The boy was very _little_.\n 2. It was a _little_ early to arrive.\n 3. It was a _hard_ lesson.\n 4. She works _hard_ every day.\n 5. I read the _first_ book.\n 6. I read the book _first_ then gave it to him.\n 7. He went to a _high_ mountain.\n 8. The eagle flew _high_ in the air.\n 9. We saw clearly the lovely picture.\n 10. He is a wonderfully jolly man.\n 11. His courtly manner failed when he saw his homely bride.\n 12. He speaks slowly and clearly.\n 13. They are very cleanly in their habits.\n\n\n NOUNS AS ADVERBS\n\n+291.+ Words that are ordinarily used as nouns, are sometimes used as\nadverbs. These are the nouns that denote time, distance, measure of\nvalue or direction. They are added to verbs and adjectives to denote the\ndefinite time at which a thing took place, or to denote the extent of\ntime or distance and the measure of value, of weight, number or age.\nThey are sometimes used to indicate direction. For example:\n\n They were gone a _year_.\n He talked an _hour_.\n They will return next _week_.\n They went _south_ for the winter.\n They traveled 100 _miles_.\n The wheat is a _foot_ high.\n The man weighed 200 _pounds_.\n\nIn these sentences, the nouns, _year_, _miles_, _hour_, _foot_, _week_,\n_pounds_ and _south_ are used as adverbs. Remember every word is\nclassified according to the work which it does in the sentence.\n\n\n Exercise 6\n\nMark the adverbs in the following poem and determine what words they\nmodify:\n\n THE BRIDGE OF SIGHS\n\n One more Unfortunate\n Weary of breath,\n Rashly importunate,\n Gone to her death!\n\n Take her up tenderly,\n Lift her with care;\n Fashion'd so slenderly,\n Young, and so fair!\n\n Look at her garments\n Clinging like cerements;\n Whilst the wave constantly\n Drips from her clothing;\n Take her up instantly,\n Loving, not loathing.\n\n Touch her not scornfully;\n Think of her mournfully,\n Gently and humanly;\n Not of the stains of her--\n All that remains of her\n Now is pure womanly.\n\n Make no deep scrutiny\n Into her mutiny\n Rash and undutiful;\n Past all dishonor,\n Death has left on her\n Only the beautiful.\n\n * * * * * * *\n\n Alas! for the rarity\n Of Christian charity\n Under the sun!\n O! it was pitiful!\n Near a whole city full,\n Home, she had none.\n\n * * * * * * *\n\n\n The bleak wind of March\n Made her tremble and shiver;\n But not the dark arch,\n Or the black flowing river:\n Mad from life's history\n Glad to death's mystery\n Swift to be hurled--\n Anywhere, anywhere\n Out of the world!\n\n In she plunged boldly,\n No matter how coldly\n The rough river ran;\n Over the brink of it,--\n Picture it, think of it,\n Dissolute Man!\n Lave in it, drink of it,\n Then, if you can!\n\n Take her up tenderly,\n Lift her with care;\n Fashion'd so slenderly,\n Young and so fair!\n\n Ere her limbs frigidly\n Stiffen too rigidly,\n Decently, kindly,\n Smooth and compose them;\n And her eyes, close them,\n Staring so blindly!\n\n Dreadfully staring\n Thro' muddy impurity,\n As when with the daring\n Last look of despairing\n Fix'd on futurity.\n\n Perishing gloomily,\n Spurr'd by contumely,\n Cold inhumanity,\n Burning insanity,\n Into her rest.\n Cross her hands humbly\n As if praying dumbly,\n Over her breast!\n\n Owning her weakness,\n Her evil behavior,\n And leaving, with meekness,\n Her sins to her Saviour!\n\n --_Thomas Hood_.\n\n\n\n\n SPELLING\n\n LESSON 16\n\n\nThe English language is truly a melting pot, into which have been thrown\nwords from almost every language under the sun. This makes our spelling\nvery confusing many times. Because of this also, we have in our\nlanguage, words which have the same sound but different meaning, having\ncome into the language from different sources. These words are called\n_homonyms_.\n\n+Homonyms are words having the same sound but different meaning.+ For\nexample:\n\n Plane, plain;\n write, right.\n\nSynonyms are words which have the same meaning. For example:\n\n Allow, permit;\n lazy, idle.\n\nOur spelling lesson for this week contains a list of most of the\ncommonly used homonyms. Look up the meaning in the dictionary and use\nthem correctly in sentences. You will note that in some instances there\nare three different words which have the same sound, but different\nmeanings.\n\nNotice especially _principal_ and _principle_. Perhaps there are no two\nwords which we use frequently which are so confused in their spelling.\n_Principle_ is a noun. _Principal_ is an adjective. You can remember the\ncorrect spelling by remembering that _adjective_ begins with _a_.\n_Principal_, the adjective, is spelled with an _a_, _pal_.\n\nNotice also the distinction between _two_, _to_ and _too_. Look these up\ncarefully, for mistakes are very often made in the use of these three\nwords. Also notice the words _no_ and _know_ and _here_ and _hear_.\n\n +Monday+\n\n Buy--by\n Fair--fare\n Meat--meet\n Our--hour\n Pain--pane\n\n +Tuesday+\n\n Deer--dear\n Hear--here\n New--knew\n No--know\n Peace--piece\n\n +Wednesday+\n\n Two--to--too\n Pair--pare--pear\n Birth--berth\n Ore--oar\n Ought--aught\n\n +Thursday+\n\n Seen--scene\n Miner--minor\n Aloud--allowed\n Stare--stair\n Would--wood\n\n +Friday+\n\n Bear--bare\n Ascent--assent\n Sight--site--cite\n Rain--reign--rein\n Rote--wrote\n\n +Saturday+\n\n Great--grate\n Foul--fowl\n Least--leased\n Principle--principal\n Sale--sail\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 17\n\n\nDear Comrade:\n\nWe are finishing in this lesson the study of a very important part of\nspeech. Adverbs are a necessary part of our vocabulary, and most of us\nneed a greater supply than we at present possess. We usually have a few\nadverbs and adjectives in our vocabulary which are continually\noverworked. Add a few new ones to your vocabulary this week.\n\nDo not slight the exercises in these lessons. The study of the lesson is\nonly the beginning of the theoretical knowledge. You do not really know\na thing until you put it into practice. You may take a correspondence\ncourse on how to run an automobile but you can not really know how to\nrun a machine until you have had the practical experience. There is only\none way to become expert in the use of words and that is to use them.\nEvery day try to talk to some one who thinks and reads. While talking\nwatch their language and your own. When a word is used that you do not\nfully understand, look it up at your very first opportunity and if you\nlike the word use it a number of times until it has become your word.\n\nWe have been following in these letters, which are our weekly talks\ntogether, the development of the alphabet. It is really a wonderful\nstory. It brings to us most vividly the struggle of the men of the past.\nLast week we found how they began to use symbols to express syllables,\nparts of a word. We found that this was a great step in advance. Do you\nnot see that this was not an eye picture but an ear picture? The symbol\ndid not stand for the picture of the object it named but each symbol\nstood for the sound which composed part of the word.\n\nAfter a while it dawned upon some one that all the words which man used\nwere expressed by just a few sounds. We do not know just when this\nhappened but we do know that it was a wonderful step in advance.\nCumbersome pictures and symbols could be done away with now. The same\nidea could be expressed by a few signs which represented the few sounds\nwhich were used over and over again in all words. Let us not fail to\nrealize what a great step in advance this was. These symbols represented\nsounds. The appeal was through the _ear gate_ of man, not through the\n_eye gate_.\n\nThus came about the birth of the alphabet, one of the greatest and most\nmomentous triumphs of the human mind. Because of this discovery, we can\nnow form thousands of combinations expressing all our ideas with only\ntwenty-three or twenty-four symbols,--letters that represent sounds.\nSince we have at our command all of this rich fund of words, let us not\nbe content to possess only a few for ourselves. Add a word daily to your\nvocabulary and you will soon be surprised at the ease and fluency of\nyour spoken and written speech; and with this fluency in speech will\ncome added power in every part of your life.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n COMPARISON OF ADVERBS\n\n+292.+ You will recall that we found that adjectives change in form to\nshow different degrees of quality. A few adverbs are compared the same\nas adjectives. Some form the comparative and superlative degree in the\nregular way, just as adjectives, by adding _er_ and _est_; for example:\n\n _Positive_ _Comparative_ _Superlative_\n\n soon sooner soonest\n late later latest\n often oftener oftenest\n early earlier earliest\n fast faster fastest\n\n+293.+ Most adverbs form their comparative and superlative by the use of\n_more_ and _most_ or _less_ and _least_, just as adjectives do; for\nexample:\n\n _Positive_ _Comparative_ _Superlative_\n\n clearly more clearly most clearly\n nobly more nobly most nobly\n ably more ably most ably\n truly more truly most truly\n\nOr, in the descending comparison:\n\n clearly less clearly least clearly\n nobly less nobly least nobly\n ably less ably least ably\n truly less truly least truly\n\n+294.+ The following adverbs are compared irregularly. It would be well\nto memorize this list:\n\n _Positive_ _Comparative_ _Superlative_\n\n ill worse worst\n well better best\n badly worse worst\n far further (farther) furthest (farthest)\n little less least\n much more most\n\nSome adverbs are incapable of comparison, as _here_, _there_, _now_,\n_today_, _hence_, _therefore_, etc.\n\n\n Exercise 1\n\nIn the following sentences mark which adverbs are used in the positive,\nwhich in the comparative and which in the superlative degree:\n\n 1. He came too late to get his letter.\n 2. I can understand clearly since you have explained the matter to me.\n 3. He speaks most truly concerning a matter of which he is well\n informed.\n 4. If he comes quickly he will arrive in time.\n 5. I will be able to speak more effectively when I have studied the\n subject.\n 6. Those who argue most ably are those who are in complete possession\n of the facts.\n 7. He needs to take a course such as this very badly.\n 8. I am too weary to go farther today.\n 9. This is the least expensive of them all.\n 10. If he arrives later in the day I will not be able to see him.\n 11. I can understand him more clearly than I can his friend.\n 12. You must work more rapidly under the Taylor system of efficiency.\n 13. Those who are least trained lose their positions first.\n 14. Those who are best fitted for the positions do not always receive\n them.\n\n\n POSITION OF ADVERBS\n\n+295.+ When we use an adverb with an adjective or other adverb, we\nusually place the adverb before the adjective or adverb which it\nmodifies. For example:\n\n She is _very_ studious.\n Results come _rather_ slowly.\n It is _quite_ evident.\n He speaks _too_ rapidly.\n\nWhen we use an adverb with the simple form of the verb, (that is, either\nthe present or past time form or any time form in which we do not need\nto use a phrase), if the verb is a complete verb, we place the adverb\nafter the verb. For example:\n\n The boat arrived _safely_.\n The man came _quickly_.\n The boy ran _fast_.\n The teacher spoke _hastily_.\n\nBut when the verb is an incomplete verb used in the simple form, the\nadverb usually precedes it in order not to come between the verb and its\nobject. As, for example:\n\n He _willingly_ gave his consent to the proposition.\n She _gladly_ wrote the letter which we requested.\n A soldier _always_ obeys the command of a superior officer.\n\nWhen the object of the incomplete verb is short, then the adverb is\nsometimes placed after the object. As, for example:\n\n I study my lessons _carefully_.\n He wrote a letter _hastily_.\n\nThe object is more closely connected with the verb and so is placed\nnearer the verb. However, when the object is modified by a phrase the\nadverb is sometimes placed immediately after the verb, as:\n\n I studied _carefully_ the lessons given for this month.\n He wrote _hastily_ a short letter to his son.\n\nWhen we use an adverb with a verb phrase, we usually place the adverb\nafter the first word in the verb phrase. For example:\n\n The boy has _always_ worked.\n The workers will _then_ understand.\n He will _surely_ have arrived by that time.\n\nWhen the verb is in the passive form the adverb immediately precedes the\nprincipal verb, as for example:\n\n The work can be _quickly_ finished.\n The obstacles can be _readily_ overcome.\n The lesson must be _carefully_ prepared.\n The workers must be _thoroughly_ organized.\n\nWhen an adverb of time and an adverb of manner or place are used to\nmodify the same verb, the adverb of time is placed first and the adverb\nof manner or place second, as for example:\n\n I _often_ stop _there_.\n He _usually_ walks _very rapidly_.\n They _soon_ learn to work _rapidly_.\n\nIf the sentence contains adverbs of time, of place, and of manner; the\nadverb of time should come first; of place, second; and of manner,\nthird; as:\n\n He _usually_ comes _here quickly_.\n\n\n Exercise 2\n\nImprove the location of the adverbs in the following sentences and\nobserve how the change of place of the adverb may alter the meaning of\nthe sentence:\n\n 1. I _only_ saw the President once.\n 2. Such prices are _only_ paid in times of great scarcity.\n 3. No man has _ever_ so much wealth that he does not want more.\n 4. It seems that the workers can be _never_ aroused.\n 5. I want to _briefly_ state the reason for my action.\n 6. I shall be glad to help you _always_.\n 7. I _only_ mention a few of the facts.\n 8. He _nearly_ walked to town.\n 9. We are told that the Japanese _chiefly_ live upon rice.\n 10. They expected them to sign a treaty _daily_.\n 11. Having _nearly_ lost all his money he feared _again_ to venture.\n\n\n ADVERBS AND INFINITIVES\n\n+296.+ You remember when we studied the infinitive in Lesson 9, we found\nthat it was not good usage to split the infinitive; that is, to put the\nmodifying word between _to_ and the verb. For example: _We ought to\nbravely stand for our rights_. The correct form of this is: _We ought to\nstand bravely for our rights_.\n\nBut we have found, also, that common usage breaks down the old rules and\nmakes new rules and laws for itself, and so we frequently find the\nadverb placed between the infinitive and its sign.\n\nSometimes it seems difficult to express our meaning accurately in any\nother way; for example, when we say: _To almost succeed is not enough_,\nwe do not make the statement as forceful or as nearly expressive of our\nreal idea, if we try to put the adverb _almost_ in any other position.\nThis is also true in such phrases as _to far exceed_, _to more than\ncounterbalance_, _to fully appreciate_, and various other examples which\nyou will readily find in your reading. The purpose of written and spoken\nlanguage is to express our ideas adequately and accurately.\n\nSo we place our words in sentences to fulfill this purpose and not\naccording to any stereotyped rule of grammarians. Ordinarily, though, it\nwould be best not to place the adverb between the infinitive verb and\nits sign _to_. Do not split the infinitive unless by so doing you\nexpress your idea more accurately.\n\n\n COMMON ERRORS\n\n+297.+ The position in the sentence of such adverbs as, _only_, _also_\nand _merely_, depends upon the meaning to be conveyed. The place where\nthese adverbs occur in the sentences, may completely alter the meaning\nof the sentence. For example:\n\n_Only the address can be written on this side._ We mean that nothing but\nthe address can be written on this side.\n\n_The address can only be written on this side._ We mean that the address\ncannot be printed, but must be written.\n\n_The address can be written only on this side._ We mean that it cannot\nbe written on any other side, but on this side only.\n\nSo you see that the place in which the adverb appears in the sentence\ndepends upon the meaning to be conveyed and the adverb should be placed\nin the sentence so as to convey the meaning intended.\n\n+Never use an adjective for an adverb.+ One common error is using an\nadjective for an adverb. Remember that adjectives modify nouns only.\nWhenever you use a word to modify a verb, adjective or another adverb,\nuse an adverb. For example, _He speaks slow and plain_. This is\nincorrect. The sentence should be, _He speaks slowly and plainly_. Watch\nthis carefully. It is a very common error.\n\n+Another very common error is that of using an adverb instead of an\nadjective with the copulative verb.+ Never use an adverb in place of an\nadjective to complete a copulative verb. When a verb asserts an action\non the part of the subject, the qualifying word that follows the verb is\nan adverb. For example, you would say:\n\n The sea was calm.\n\nHere we use an _adjective_ in the predicate, for we are describing the\nappearance of the sea, no action is expressed. But if we say: _He spoke\ncalmly_, we use the adverb _calmly_, for the verb _spoke_ expresses an\naction on the part of the subject, and the adverb _calmly_ describes\nthat action, it tells how he spoke. So we say: _The water looks clear_,\nbut, _We see clearly_. _She appears truthful._ _They answered\ntruthfully._ _She looked sweet._ _She smiled sweetly._\n\nWith all forms of the verb _be_, as _am_, _is_, _are_, _was_, _were_,\n_have been_, _has been_, _will be_, etc., use an adjective in the\npredicate; as, _He is glad_. _I am happy._ _They were eager._ _They will\nbe sad._ Use an adjective in the predicate with verbs like _look_,\n_smell_, _taste_, _feel_, _appear_ and _seem_. For example: _He looks\nbad._ _It smells good._ _The candy tastes sweet_. _The man feels fine\ntoday._ _She appears anxious._ _He seems weary._\n\n+Never use two negative words in the same sentence.+ The second negative\ndestroys the first and we really make an affirmative statement. The two\nnegatives neutralize each other and spoil the meaning of the sentence.\nFor example, never say:\n\n I don't want no education.\n He didn't have no money.\n Don't say nothing to nobody.\n She never goes nowhere.\n He won't say nothing to you.\n He does not know nothing about it.\n He never stops for nothing.\n The stingy man gives nothing to nobody.\n\nIn all of these sentences we have used more than one negative; _not_ and\n_no_, or _not_ and _nothing_, or _never_ and _no_, or _never_ and\n_nothing_. Never use these double negatives. The correct forms of these\nsentences are:\n\n I don't want any education.\n He didn't have any money.\n Don't say anything to any one.\n She never goes anywhere.\n He won't say anything to you.\n He knows nothing about it.\n He never stops for anything.\n The stingy man gives nothing to any one.\n\n+Where to place the negative adverb, not.+ In English we do not use the\nnegative adverb _not_ with the common verb form, but when we use _not_\nin a sentence, we use the auxiliary _do_. For example, we do not say:\n\n I like it not.\n They think not so.\n He loves me not.\n We strive not to succeed.\n\nOnly in poetry do we use such expressions as these. In ordinary English,\nwe say:\n\n I do not like it.\n They do not think so.\n He does not love me.\n We do not strive to succeed.\n\n+We often use _here_ and _there_ incorrectly after the words _this_\nand _that_.+ For example, we say:\n\n This here lesson is shorter than that there one was.\n\nThis should be: _This lesson is shorter than that one_.\n\n Bring me that there book.\n This here man will not listen.\n\nThese sentences should read:\n\n Bring me that book.\n This man will not listen.\n\nNever use _here_ and _there_ in this manner.\n\n+Another common mistake is using _most_ for _almost_.+\n\nFor example, we say:\n\n We are most there.\n I see her most every day.\n\nThese sentences should read:\n\n We are almost there.\n I see her almost every day.\n\n_Most_ is the superlative degree of _much_, and should be used only in\nthat meaning.\n\n+We often use the adjective _real_ in place of _very_ or _quite_,\nto modify an adverb or an adjective.+\n\nFor example, we say:\n\n I was real glad to know it.\n She looked real nice.\n You must come real soon.\n\nSay instead:\n\n I am very glad to know it.\n She looked very nice.\n You must come quite soon.\n\n_Really_ is the adverb form of the adjective _real_. You might have\nsaid:\n\n I am really glad to know it.\n\nBut never use _real_ when you mean _very_ or _quite_ or _really_.\n\n+We use the adjective _some_ many times when we should use the adverb\n_somewhat_.+ For example, we say:\n\n I am some anxious to hear from him.\n I was some tired after my trip.\n\nWhat we intended to say was:\n\n I am somewhat anxious to hear from him.\n I was somewhat tired after my trip.\n\n+Do not use _what for_ when you mean _why_.+ Do not say:\n\n What did you do that for?\n\nOr worse still,\n\n What for did you do that?\n\nSay:\n\n Why did you do that?\n\n+Do not use _worse_ in place of _more_.+ Do not say:\n\n I want to go worse than I ever did.\n\nSay:\n\n I want to go _more_ than I ever did.\n\n+Observe the distinction between the words _further_ and _farther_.+\nFarther always refers to distance, or extent. For example:\n\n He could go no farther that day.\n We will go farther into the matter some other time.\n\nFurther means more. For example:\n\n He would say nothing further in regard to the subject.\n\n+Never use _good_ as an adverb+. _Good_ is always an adjective. _Well_\nis the adverb form. _Good_ and _well_ are compared in the same way,\n_good_, _better_, _best_, and _well_, _better_, _best_. So _better_ and\n_best_ can be used either as adjectives or adverbs; but _good_ is always\nan adjective. Do not say, _He talks good_. Say, _He talks well_. Note\nthat _ill_ is both an adjective and an adverb and that _illy_ is always\nincorrect.\n\n\n Exercise 3\n\nCorrect the adverbs in the following sentences. All but two of these\nsentences are wrong.\n\n 1. Come quick, I need you.\n 2. The boy feels badly.\n 3. Give me that there pencil.\n 4. I am some hungry.\n 5. The people learn slow.\n 6. He never stopped for nothing.\n 7. What did you say that for?\n 8. This here machine won't run.\n 9. I make a mistake most every time.\n 10. Watch careful every word.\n 11. The man works good.\n 12. The tone sounds harsh.\n 13. I don't want no dinner.\n 14. I hope it comes real soon.\n 15. I want to learn worse than ever.\n 16. She looked lovely.\n 17. She smiled sweet.\n 18. He sees good for one so old.\n 19. She answered correct.\n 20. He won't say nothing about it.\n 21. I will be real glad to see you.\n 22. That tastes sweetly.\n 23. The man acted too hasty.\n 24. We had most reached home.\n 25. They ride too rapid.\n\n\n DO NOT USE TOO MANY ADVERBS\n\n+298.+ Like adjectives it is better to use adverbs sparingly. This is\nespecially true of the adverbs used to intensify our meaning. Do not use\nthe adverbs, _very_, _awfully_, etc., with every other word. It makes\nour speech sound like that of a gushing school girl, to whom everything\nis _very, awfully sweet_. More than that, it does not leave us any words\nto use when we really want to be intense in speech. Save these words\nuntil the right occasion comes to use them.\n\n\n Exercise 4\n\nAdverbs should always be placed where there can be no doubt as to what\nthey are intended to modify. A mistake in placing the adverb in the\nsentence often alters the meaning of the sentence. Choose the right word\nin each of the following sentences:\n\n 1. He looked glad--gladly when I told him the news.\n 2. Slaves have always been treated harsh--harshly.\n 3. I prefer my eggs boiled soft--softly.\n 4. The lecturer was tolerable--tolerably well informed.\n 5. Speak slower--more slowly so I can understand you.\n 6. The evening bells sound sweet--sweetly.\n 7. The house appears comfortable--comfortably and\n pleasant--pleasantly.\n 8. If you will come quick--quickly you can hear the music.\n 9. I was exceeding--exceedingly glad to hear from you.\n 10. The bashful young man appeared very awkward--awkwardly.\n 11. The young lady looked beautiful--beautifully and she sang\n beautiful--beautifully.\n 12. I looked quick--quickly in the direction of the sound.\n 13. The sun is shining bright--brightly today and the grass looks\n green--greenly.\n\n\n\n\n SPELLING\n\n LESSON 17\n\n\nIn our study of adjectives we have found that we use them to express\nsome quality possessed by a noun or pronoun which they modify. You will\nrecall when we studied nouns, we had one class of nouns, called abstract\nnouns, which were the names of qualities. So we find that from these\nadjectives expressing quality we form nouns which we use as the name of\nthat quality.\n\nFor example from the adjective _happy_, we form the noun _happiness_,\nwhich is the name of the quality described by the adjective _happy_, by\nthe addition of the suffix _ness_. We use this suffix _ness_ quite often\nin forming these derivative nouns from adjectives but there are other\nsuffixes also which we use; as for example, the suffix _ty_ as in\n_security_, formed from the adjective _secure_, changing the _e_ to _i_\nand adding the suffix _ty_. When the word ends in _t_ we sometimes add\nonly _y_ as in _honesty_, derived from the adjective _honest_.\n\nYou remember that an abstract noun may express not only quality but also\naction, considered apart from the actor; so abstract nouns may be made\nfrom verbs. For example:\n\n_Running_, from the verb _run_; _settlement_, from the verb _settle_.\n\nIn our lesson for this week the list for Monday, Tuesday and Wednesday\ncontains adjectives of quality from which abstract nouns expressing\nquality can be made, by the addition of the proper suffix, either\n_ness_, _y_, _ty_ or _tion_. The list for Thursday, Friday and Saturday\nconsists of verbs from which abstract nouns can be made by the addition\nof the suffixes _ment_ and _ing_.\n\nMake from each adjective and verb in this week's lesson an abstract noun\nby the addition of the proper suffix. Be able to distinguish between the\nuse of the qualifying adjective and the noun expressing quality.\n\n +Monday+\n\n Stately\n Forgetful\n Real\n Concise\n Noble\n\n +Tuesday+\n\n Slender\n Empty\n Equal\n Righteous\n Deliberate\n\n +Wednesday+\n\n Submissive\n Dreadful\n Eager\n Sincere\n Resolute\n\n +Thursday+\n Enlist\n Defile\n Adorn\n Nourish\n Commence\n\n +Friday+\n\n Content\n Adjust\n Induce\n Indict\n Adjourn\n\n +Saturday+\n\n Discourage\n Refine\n Acquire\n Enrich\n Infringe\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 18\n\n\nDear Comrade:\n\nLast week we finished the study of adverbs and we found that they were a\nvery important part of our vocabulary, and that most of us needed a\ngreater supply than we at present possess. This is true of both adverbs\nand adjectives. While we do not use as many adverbs as adjectives in our\nordinary speech, nevertheless, adverbs are a very important factor in\nexpression. A great many adjectives can be readily turned into adverbs.\nThey are adjectives when they are used to describe a noun, but by the\naddition of a suffix, they become adverbs used to describe the action\nexpressed by the verb. So in adding to our stock of adjectives we also\nadd adverbs to our vocabulary as well.\n\nWatch your speech this week and make a list of the adverbs which you use\nmost commonly, then go to your dictionary and see if you cannot find\nsynonyms for these adverbs. Try using these synonyms for awhile and give\nthe adverbs which you have been using for so long, a well earned rest.\nRemember that our vocabulary, and the power to use it, is like our\nmuscles, it can only grow and develop by exercise.\n\nThe best exercise which you can possibly find for this purpose is\nconversation. We spend much more time in talking than in reading or in\nwriting. Conversation is an inexpensive pleasure and it does not even\nrequire leisure always, for we can talk as we work; yet our conversation\ncan become a great source of inspiration and of influence as well as a\npleasant pastime. But do not spend your time in vapid and unprofitable\nconversation. Surely there is some one in the list of your acquaintances\nwho would like to talk of things worth while. Hunt up this some one and\nspend some portion of your day in profitable conversation.\n\nRemember also that a limited vocabulary means also a limited mental\ndevelopment. Did you ever stop to think that when we think clearly we\nthink in words? Our thinking capacity is limited, unless we have the\nwords to follow our ideas out to their logical conclusions.\n\nThis matter of vocabulary is a matter, too, that is exceedingly\npractical. It means success or failure to us in the work which we would\nlike to do in the world. A command of words means added power and\nefficiency; it means the power to control, or at least affect, our\nenvironment; it means the power over men and things; it means the\ndifference between being people of ability and influence and being\nobscure, inefficient members of society.\n\nSo feel when you are spending your time in increasing your vocabulary\nthat you are not only adding to your enjoyment of life but that you are\ndoing yourself the best practical turn; you are increasing your\nefficiency in putting yourself in a position where you can make your\ninfluence felt upon the people and circumstances about you. This effort\nupon your part will bear practical fruit in your every day life.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n A GROUP OF WORDS\n\n+299.+ We have studied about the independent parts of speech, that is,\nthe nouns and pronouns and verbs. These are independent because with\nthem we can form sentences without the help of other words. And these\nare the only three parts of speech which are so independent--with which\nwe can form complete sentences. Then we have studied also the words that\nmodify,--that is, the words that are used with nouns and pronouns and\nverbs to describe and explain more fully the ideas which they express.\nSo we have studied adjectives, which modify nouns and pronouns; and\nadverbs, which modify verbs or adjectives or other adverbs.\n\n+300.+ The adjectives and adverbs which we have studied thus far are\nsingle words; but we find that we may use little groups of words in\nabout the same way, to express the same idea which we have expressed in\nthe single adjective or adverb. For example, we may say:\n\n Strong men, _or_, men of strength.\n City men, _or_, men from the city.\n Jobless men, _or_, men without jobs.\n Moneyed men, _or_, men with money.\n\nThese groups of words like, _of strength_, _from the city_, _without\njobs_, and _with money_, express the same ideas that are expressed in\nthe single adjectives, _strong_, _city_, _jobless_ and _moneyed_.\n\nYou recall that we defined any group of words used as a single word as a\n_phrase_; so these groups of words are phrases which are used as\nadjectives. The phrase, _of strength_, modifies the noun _men_, just as\nthe adjective _strong_ modifies the noun _men_. So we may call these\nphrases which modify nouns, or which may be used to modify pronouns\nalso, _adjective phrases_, for they are groups of words used as\nadjectives.\n\n\n Exercise 1\n\nChange the adjectives which are printed in italics in the following\nsentences into phrases:\n\n 1. _Strong_ men know no fear.\n 2. She bought a _Turkish_ rug.\n 3. He followed the _river_ bed.\n 4. _Fashionable_ women are parasites.\n 5. He left on his _homeward_ journey.\n 6. _Sensible_ men readily understand their economic slavery.\n 7. _Intelligent_ people will not always submit to robbery.\n 8. _Senseless_ arguments cannot convince us of the truth.\n\n\n USED AS ADVERBS\n\n+301.+ These phrases may be used in the place of single adverbs also.\nYou remember an adverb is a word that modifies a verb or an adjective or\nanother adverb. Let us see if we can not use a phrase or a group of\nwords in the place of a single adverb. For example:\n\n The man works rapidly, or, The man works with rapidity.\n The man works now, or, The man works at this time.\n The man works here, or, The man works at this place.\n\nIn these sentences _rapidly_, _now_ and _here_ are single adverbs\nmodifying the verb _work_. The phrases, _with rapidity_, _at this time_,\nand _at this place_, express practically the same ideas, conveyed by the\nsingle adverbs, _rapidly_, _now_ and _here_. These phrases modify the\nverb in exactly the same manner as the single adverbs. Therefore we call\nthese groups of words used as single adverbs, _adverb phrases_.\n\nWe also use adverbs to modify adjectives. Let us see if we can use\nadverb phrases in the same way:\n\n Rockefeller is _excessively_ rich; or, Rockefeller is rich _to\n excess_.\n He is _bodily_ perfect, but _mentally_ weak; or, He is perfect _in\n body_ but weak _in mind_.\n\nIn the sentences above, the adverb _excessively_ modifies the adjective\n_rich_; the same meaning is expressed in the adverb phrase, _to excess_.\nIn the sentence, _He is bodily perfect, but mentally weak_, the adverb\n_bodily_ modifies the adjective _perfect_ and the adverb _mentally_\nmodifies the adjective _weak_. In the last sentence, the same meaning is\nexpressed by the adverb phrases, _in body_ and _in mind_. These phrases\nmodify the adjectives _perfect_ and _weak_, just as do the single\nadverbs _bodily_ and _mentally_.\n\n+302.+ We can use a phrase in the place of almost any adverb or\nadjective. It very often happens, however, that there is no adjective or\nadverb which we can use to exactly express our meaning and we are forced\nto use a phrase. For example:\n\n He bought the large house _by the river_.\n The man _on the train_ is going _to the city_.\n He came _from the country_.\n\nIt is impossible to find single words that express the meaning of these\nphrases, _by the river_, _on the train_, _to the city_, and _from the\ncountry_. You could not say the _river house_; that is not what you\nmean. You mean the large house _by the river_, yet the phrase _by the\nriver_ modifies and describes the house quite as much as the adjective\n_large_. It is an adjective phrase used to modify the noun _house_, yet\nit would be impossible to express its meaning in a single word.\n\n\n Exercise 2\n\nWhich phrases in the following sentences are used as adjectives and\nwhich phrases are used as adverbs?\n\nChange these phrases to adjectives or adverbs, if you can think of any\nthat express the same meaning.\n\n 1. Men lived _in caves_ long ago.\n 2. Man's discovery _of fire_ was the beginning _of industry_.\n 3. _After this discovery_, men lived _in groups_.\n 4. The work _of the world_ is done _by machinery_.\n 5. The workers _of Europe_ were betrayed.\n 6. They are fighting _for their country_.\n 7. The struggle _for markets_ is the cause _of war_.\n 8. The history _of the world_ records the struggle _of the workers_.\n 9. The idea _of democracy_ is equal opportunity _for all_.\n 10. The invention _of the printing press_ placed knowledge _within the\n reach_ _of the masses_.\n 11. If you will study _with diligence_ you can learn _with ease_.\n 12. This knowledge will be _of great value_ _to you_.\n 13. Diplomacy means that the plans _of nations_ are made _in secret_.\n 14. The men _in the factory_ are all paid _by the month_.\n 15. They are afraid to take a trip _through Europe_ _at this time_.\n\n\n Exercise 3\n\nUse a phrase instead of the adjective or adverb in the following\nsentences:\n\n 1. The men in the trenches are fighting _bravely_.\n 2. An _uneducated_ man is _easily_ exploited.\n 3. Our _educational_ system is inadequate.\n 4. The _skilled_ workers must be organized.\n 5. _Careless_ men endanger the lives of others.\n 6. The plans have been _carefully_ laid.\n 7. _Ambitious_ men often trample on the rights of others.\n 8. Shall our education be controlled by _wealthy_ men?\n 9. We want to live _courageously_.\n 10. We want to face the future _fearlessly_.\n 11. We want to possess _peacefully_ the fruits of our labor.\n 12. By constant practice we can learn to speak _effectively_.\n 13. This book will be a _valuable_ addition to your library.\n 14. The number of _unemployed_ men _constantly_ increases.\n 15. The men mastered each step _thoroughly_ as they proceeded.\n 16. In order to express one's self _eloquently_ it is necessary to\n think _clearly_.\n 17. We must consecrate ourselves _completely_ to the cause of\n humanity.\n 18. A _kind_ act is its own reward.\n 19. _Experienced_ workers can _more easily_ secure positions.\n 20. He spoke _thoughtlessly_ but the people listened _eagerly_.\n 21. The soldier was rewarded for his _heroic_ deed.\n 22. He is an _honorable_ man and I am not surprised at this _brave_\n act.\n 23. A _prudent_ man should be chosen to fill that _important_ office.\n\n\n PREPOSITIONS\n\n+303.+ Have you noticed that all of these phrases, which we have been\nstudying and using as adjectives and adverbs, begin with a little word\nlike _of_, _with_, _from_, _in_, _at_ or _by_, which connects the phrase\nwith the word it modifies? We could scarcely express our meaning without\nthese little words. They are connecting words and fill an important\nfunction. These words usually come first in the phrase. For this reason,\nthey are called _prepositions_, which means _to place before_.\n\nLet us see what a useful place these little words fill in our language.\nSuppose we were watching the play of some boys outside our windows and\nwere reporting their hiding place. We might say:\n\n The boys are hiding _in_ the bushes.\n The boys are hiding _among_ the bushes.\n The boys are hiding _under_ the bushes.\n The boys are hiding _behind_ the bushes.\n The boys are hiding _beyond_ the bushes.\n\nThese sentences are all alike except the prepositions _in_, _among_,\n_under_, _behind_ and _beyond_. If you read the sentences and leave out\nthese prepositions entirely, you will see that nobody could possibly\ntell what connection the _bushes_ had with the rest of the sentence. The\nprepositions are necessary to express the relation of the word _bushes_\nto the rest of the sentence.\n\nBut this is not all. You can readily see that the use of a different\npreposition changes the meaning of the sentence. It means quite a\ndifferent thing to say, _The boys are hiding in the bushes_, and to say,\n_The boys are hiding beyond the bushes_. So the preposition has a great\ndeal to do with the true expression of our ideas.\n\nThe noun _bushes_ is used as the object of the preposition, and the\npreposition shows the relation of its object to the word which it\nmodifies. You remember that nouns have the same form whether they are\nused as subject or as object, but if you are using a pronoun after a\npreposition, always use the object form of the pronoun. For example:\n\n I bought the book from _him_.\n I took the message to _them_.\n I found the place for _her_.\n\nIn these sentences the pronouns, _him_, _them_, and _her_ are used as\nobjects of the prepositions _from_, _to_ and _for_. So we have used the\nobject forms of these pronouns.\n\n+304+. The noun or pronoun that follows the preposition, and is used\nwith it to make a phrase, is the object of the preposition. The\npreposition is used to show the relation that exists between its object\nand the word the object modifies. In the sentence above, _The boys are\nhiding in the bushes_, the preposition _in_ shows the relationship\nbetween the verb phrase, _are hiding_ and the object of the preposition,\n_bushes_.\n\nThe noun or pronoun which is the object of a preposition may also have\nits modifiers. In the sentences used about the noun _bush_, which is the\nobject of the prepositions used, is modified by the adjective _the_.\nOther modifiers might also be added, as for example:\n\n The boys are hiding in the tall, thick bushes.\n\nThe entire phrase, _in the tall, thick bushes_, is made up of the\npreposition _in_, its object _bushes_ and the modifiers of bushes,\n_the_, _tall_ and _thick_.\n\n+305+. The preposition, with its object and the modifiers of the object,\nforms a phrase which we call a _prepositional phrase_. These\nprepositional phrases may be used either as adjectives or as adverbs, so\nwe have our definitions:\n\n+A preposition is a word that shows the relation of its object to some\nother word.+\n\n+A phrase is a group of words used as a single word.+\n\n+A prepositional phrase is a phrase composed of a preposition and its\nobject and modifiers.+\n\n+An adjective phrase is a prepositional phrase used as an adjective.+\n\n+An adverb phrase is a prepositional phrase used as an adverb.+\n\n+306.+ Here is a list of the most common and most important\nprepositions. Use each one in a sentence:\n\n above\n about\n across\n after\n against\n along\n around\n among\n at\n before\n behind\n below\n beneath\n beside\n between\n beyond\n by\n down\n for\n from\n in\n into\n of\n off\n on\n over\n to\n toward\n through\n up\n upon\n under\n with\n within\n without\n\n\n ADVERBS AND PREPOSITIONS\n\n+307.+ Many of the words that are used as prepositions are used also as\nadverbs. It may be a little confusing to tell whether the word is an\nadverb or a preposition, but if you will remember this simple rule you\nwill have no trouble:\n\n+A preposition is always followed by either a noun or a pronoun as its\nobject, while an adverb never has an object.+\n\nSo when you find a word, that can be used either as a preposition or an\nadverb, used alone in a sentence without an object, it is an adverb; but\nif it is followed by an object, then it is a preposition. This brings\nagain to our minds the fundamental rule which we have laid down, that\nevery word is classified according to the work which it does in a\nsentence. The work of a preposition is to show the relation between its\nobject and the word which that object modifies. So whenever a word is\nused in this way it is a preposition. For example: _He went about his\nbusiness_.\n\nHere, _about_ is a preposition and _business_ is its object. But in the\nsentence, _He is able to be about_, _about_ is used as an adverb. It has\nno object.\n\n_He sailed before the mast._ Here, _before_ is a preposition introducing\nthe phrase _before the mast_, which modifies the verb _sailed_. But in\nthe sentence, _I told you that before_, _before_ is an adverb modifying\nthe verb _told_.\n\nBy applying this rule you can always readily determine whether the word\nin question is an adverb or a preposition.\n\n\n Exercise 4\n\nTell whether the words printed in italics in the following sentences,\nare prepositions or adverbs and the reason why:\n\n 1. He came _across_ the street.\n 2. He is _without_ work.\n 3. Come _in_.\n 4. He lives _near_.\n 5. He brought it _for_ me.\n 6. I cannot get _across_.\n 7. We will go _outside_.\n 8. This is _between_ you and me.\n 9. He can go _without_.\n 10. Stay _in_ the house.\n 11. Do not come _near_ me.\n 12. They all went _aboard_ at six o'clock.\n 13. He enlisted _in_ the navy and sailed _before_ the mast.\n 14. I do not know what lies _beyond_.\n 15. I will soon be _through_.\n 16. The aeroplane flew _above_ the city for hours.\n\n\n PHRASE PREPOSITIONS\n\n+308.+ Sometimes we have a preposition made up of several words which we\nhave used so commonly together that they are used as a single word and\nwe call the entire phrase a preposition. As, for example: _According\nto_--_on account of_--_by means of_, etc.\n\n 1. He answered _according to_ the rule.\n 2. I could not go _on account of_ illness.\n 3. He won the election _by means of_ fraud.\n 4. The strike was won _by help of_ all the comrades.\n 5. You can learn to spell only _by dint of_ memory.\n 6. We speak incorrectly _by force of_ habit.\n 7. He went to New York _by way of_ Chicago.\n 8. Ferrer died _for the sake of_ his ideals.\n 9. _In consideration of_ this payment, we will send you the set of\n books.\n 10. Germany issued her ultimatum _in defiance of_ the world.\n 11. _In view of_ all the facts, we are convinced of his innocence.\n 12. He will gladly suffer _in place of_ his comrade.\n 13. _In conformity with_ the information contained in your letter,\n I will join you on the 10th.\n\n\n Exercise 5\n\nMark the prepositions in the following quotation. In the first three\nparagraphs the prepositional phrases are printed in italics. Determine\nwhether they are used as adjective phrases or as adverb phrases.\nUnderscore the prepositional phrases in the remainder of the quotation\nand determine which word is used as the object of the preposition.\n\n\n THE SUNLIGHT LAY ACROSS MY BED\n\n_In the dark_ one night I lay _upon my bed_. And _in the dark_ I dreamed\na dream. I dreamed God took my soul _to Hell_.\n\nAnd we came where hell opened _into a plain_, and a great house stood\nthere. Marble pillars upheld the roof, and white marble steps led up _to\nit_. The wind _of heaven_ blew _through it_. Only _at the back_ hung a\nthick curtain. Fair men and women there feasted _at long tables_. They\ndanced, and I saw the robes _of women_ flutter _in the air_ and heard\nthe laugh _of strong men_. They feasted _with wine_; they drew it _from\nlarge jars_ which stood somewhat _in the background_, and I saw the wine\nsparkle as they drew it.\n\nAnd I said _to God_, \"I should like to go up and drink.\" And God said,\n\"Wait.\" And I saw men coming _into the banquet house_; they came in\n_from the back_ and lifted the corner _of the curtain_ _at the sides_\nand crept in quickly; and they let the curtain fall _behind them_; they\nbore great jars they could hardly carry. And the men and women crowded\n_round them_, and the newcomers opened their jars and gave them _of the\nwine_ to drink; and I saw that the women drank even more greedily than\nthe men. And when others had well drunken they set the jars _among the\nold ones_ _beside the wall_, and took their places _at the table_. And I\nsaw that some _of the jars_ were very old and mildewed and dusty, but\nothers had still drops _of new must_ _on them_ and shone _from the\nfurnace_.\n\nAnd I said to God, \"What is that?\" For amid the sounds of the singing,\nand over the dancing of feet, and over the laughing across the winecups,\nI heard a cry.\n\nAnd God said, \"Stand away off.\"\n\nAnd He took me where I saw both sides of the curtain. Behind the house\nwas a wine-press where the wine was made. I saw the grapes crushed, and\nI heard them cry. I said, \"Do not they on the other side hear it?\"\n\nGod said, \"The curtain is thick; they are feasting.\"\n\nAnd I said, \"But the men who came in last. They saw?\"\n\nGod said, \"They let the curtain fall behind them--and they forgot!\"\n\nI said, \"How came they by their jars of wine?\"\n\nGod said, \"In the treading of the press these are they who came to the\ntop; they have climbed out over the edge and filled their jars from\nbelow; and have gone into the house.\"\n\nAnd I said, \"And if they had fallen as they climbed--?\"\n\nGod said, \"They had been wine.\"\n\nI stood away off watching in the sunshine, and I shivered.\n\nAnd after a while I looked, and I saw the curtain that hung behind the\nhouse moving.\n\nI said to God, \"Is it a wind?\"\n\nGod said, \"A wind.\"\n\nAnd it seemed to me that against the curtain I saw pressed the forms of\nmen and women. And after a while, the feasters saw it move, and they\nwhispered one to another. Then some rose and gathered the most worn-out\ncups, and into them they put what was left at the bottom of other\nvessels. Mothers whispered to their children, \"Do not drink all, save a\nlittle drop when you have drunk.\" And when they had collected all the\ndregs they slipped the cups out under the bottom of the curtain without\nlifting it. After a while the curtain left off moving.\n\nI said to God, \"How is it so quiet?\"\n\nHe said, \"They have gone away to drink it.\"\n\nI said, \"They drink it--their own!\"\n\nGod said, \"It comes from this side of the curtain, and they are very\nthirsty.\"\n\nAnd still the feast went on.\n\nMen and women sat at the tables quaffing great bowls. Some rose, and\nthrew their arms about each other and danced and sang. They pledged each\nother in the wine, and kissed each other's blood-red lips.\n\nMen drank till they could drink no longer, and laid their heads upon the\ntable, sleeping heavily. Women who could dance no more leaned back on\nthe benches with their heads against their lovers' shoulders. Little\nchildren, sick with wine, lay down upon the edge of their mothers'\nrobes.\n\nI said, \"I cannot see more, I am afraid of Hell. When I see men dancing\nI hear the time beaten in with sobs; and their wine is living! Oh, I\ncannot bear Hell!\"\n\nGod said, \"Where will you go?\"\n\nI said, \"To the earth from which I came; it was better there.\"\n\nAnd God laughed at me; and I wondered why He laughed.\n\n --_Olive Schreiner_.\n\n\n\n\n SPELLING\n\n LESSON 18\n\n\nThere are a number of words that are ordinarily followed by a\npreposition with its phrase. We make a great many mistakes in the use of\nthe proper preposition with these words. Our spelling lesson this week\ncovers a number of these words with examples illustrating the\nappropriate preposition to be used with each word. Learn to spell these\nwords, look up their meaning in the dictionary and use each word with\nits proper preposition in sentences of your own construction.\n\n +MONDAY+\n\n +Abhorrence+, of; We have an abhorrence _of_ war.\n +Abhorrent+, to; War is abhorrent _to_ us.\n +Acquaint+, with; I will acquaint you _with_ the facts in the case.\n You will then be acquainted _with_ the facts.\n +Acquit+, of; The man was acquitted _of_ the charge.\n +Adequate+, to; Our resources are not adequate _to_ the demand.\n\n +TUESDAY+\n\n +Angry+, with, at; We are angry _with_ persons and angry _at_ things.\n +Astonished+, at or by; (Never with) I am astonished _at_ you, or\n _by_ you, not _with_ you.\n +Confer+; We confer _with_ people, _upon_ or _about_ matters.\n +Contrary+; A thing is contrary _to_ our ideas, (not _from_ or\n _than_).\n +Controversy+; with, between, or about, (not over). I had a\n controversy _with_ you. There is a controversy _between_ the two\n _about_ the result.\n\n +WEDNESDAY+\n\n +Convicted+, of (not for). He was convicted _of_ the crime.\n +Copy+; We copy _after_ people, _from_ things, and _out_ of books.\n +Deprive+, of, (not from). We are deprived _of_ an education.\n +Desire+, of and for; We may speak of the desire _of_ a man, meaning\n man's desire; but we should always say \"He has a desire _for_\n position, _for_ wealth,\" etc.\n +Die+, of, for and from; A person dies _of_, not _from_, a disease. He\n dies _from_ the effects of an injury. One person may die _with_\n another, but never _with_ a disease, for the disease does not die.\n\n +THURSDAY+\n\n +Differ+, from, among, about, concerning, with; Persons or things\n differ _from_ each other; that is, they are dissimilar in\n appearance. Two persons may differ _with_ each other; that is,\n contend or disagree. Several persons differ _among_ themselves\n _about_ or _concerning_ some matter.\n +Dissent+, from (not to). There was a general dissent _from_ that\n idea.\n +Guilty+, of (not for). He is guilty _of_ the crime.\n +Incentive+, to (not for). It is a great incentive _to_ action.\n +Receive+, from, (not of). Received _from_ John Smith, thirty dollars,\n etc.\n\n +FRIDAY+\n\n +Infer+, from, (not by). I infer this _from_ your remarks, not _by_\n your remarks.\n +Introduce+; A man is introduced _to_ a woman, a speaker _to_ an\n audience; _into_ society or _into_ new surroundings. We introduce\n a bill _in_ Congress or a resolution _in_ a committee.\n +Involved+, in (not with). We are involved _in_ difficulties.\n +Listen+; We listen +for+ the expected news; we listen +to+ our\n friends, not _at_.\n +Married+; One person is married +to+ another, not +with+ another.\n\n +SATURDAY+\n\n +Matter+, with, (not of). What is the matter _with_ this?\n +Opposition+, to (not against). There is opposition _to_ the motion.\n +Part+, to part _from_, means to leave. I will part _from_ my friends.\n To part _with_ means to give up. A fool soon parts _with_ his\n money.\n +Remedy+, for; We have a remedy _for_ the disease.\n +Preventive+, against; We have a preventive _against_ disease.\n\n * * * * *\n\n It is easy to sit in the sunshine\n And talk to the man in the shade;\n It is easy to float in a well-trimmed boat,\n And point out the places to wade.\n\n But once we pass into the shadows\n We murmur and fret and frown;\n At our length from the bank, we shout for a plank,\n Or throw up our hands and go down.\n\n It is easy to sit in a carriage\n And counsel the man on foot;\n But get down and walk and you'll change your talk,\n _As_ you feel the peg in your boot.\n\n It is easy to tell the toiler\n _How_ best he can carry his pack;\n But not one can rate a burden's weight\n _Until_ it has been on his back.\n\n The up-curled mouth of pleasure\n Can preach of sorrow's worth;\n But give it a sip, and a wryer lip,\n Was never made on earth.\n\n --_Ella Wheeler Wilcox_.\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 19\n\n\nDear Comrade:\n\nIn this lesson we are completing our study of the preposition. The\npreposition is one of the last parts of speech which we take up for\nstudy and it is also one of the last parts of speech to be added to our\nvocabulary. The child does not use the preposition when it first begins\nto talk. It uses the names of things; words of action; words that\ndescribe objects and actions. It does not begin to use prepositions\nuntil it begins to relate ideas.\n\nThe relation of ideas means that we are thinking; combining ideas into\nthoughts. Then we begin to need prepositions, which are words of\nrelation, connecting words, expressing the relationship between ideas.\nThe measure of the fullness and richness of our lives is the measure of\nour understanding of the world about us, of the relationship existing\nbetween the different phases of that world and of our relationship to it\nall.\n\nSo words do not mean much to us until we can relate them to our own\nlives and our own experiences. When you look up a word in the\ndictionary, do not study the word alone; study also the thing for which\nit stands. A person with a good memory might acquire a vocabulary by\nsheer feat of memory; but what good would it do unless each word could\nbe related to practical experience? It is only in this way that words\nbecome _alive_ to us. We must have an idea, a concept and knowledge of\nthe thing for which the word stands.\n\nSo let us use our dictionary in this way. Do not be satisfied when you\nhave looked up a word simply to know how to spell and pronounce the word\nand understand somewhat of its meaning. Do not be satisfied until it has\nbecome a live word to you. Have a clear image and understanding of just\nwhat each word stands for. Use the words in sentences of your own. Use\nthem in your conversation. Make them a part of your every-day life.\n\nDo not pass over any of the words in the lesson without understanding\ntheir meaning. Study the poem _Abou Ben Adhem_ in this week's lesson.\nAfter you have read it over a number of times, close the book and\nrewrite the poem in prose in your own language. Then compare your\nversion with the poem. Note where you have used different expressions\nand decide which is the better, the words used in the poem or your own\nwords. Rewrite it several times until you have a well-written version of\nthis poem.\n\nExercises such as this will increase your vocabulary and quickly develop\nthe power of expression. No power can come in any department of life\nwithout effort having been expended in its acquisition. Our great\nwriters have been careful students. Robert Louis Stevenson says that he\nhas often spent a half a day searching for the particular word which he\nneeded to express precisely the idea in mind. Stevenson is a master of\nthe English and this power came to him by this sort of studious and\nearnest work.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n AN IMPORTANT WORD\n\n+309.+ Things are not always to be judged by their size. Sometimes the\nmost important things are very small and unimportant in appearance. A\ngreat machine is before you. You see its giant wheels, its huge levers.\nThese may seem to be the most important parts of the machine, but here\nand there throughout this great machine are little screws and bolts.\nThese bind the giant parts together. Without these connecting links, the\ngreat wheels and levers and revolving belts could not work together. Let\na little bolt slip out of its place in the mechanism, and the great\nwheels stop, the throbbing machinery comes to a standstill. No work is\npossible until this little bolt has been replaced.\n\nSo in our sentence building, the _preposition_ is the bolt that joins\nwords together. The importance of the preposition in a language\nincreases just in proportion as the nation learns to think more exactly\nand express itself more accurately.\n\nWe found in our last lesson that by changing a preposition we can change\nthe entire meaning of the sentence. A man's life may depend upon the use\nof a certain preposition. You may swear his life away by bearing\ntestimony to the fact whether you saw him _within_ the house, or\n_without_ the house; or _before_ dark, or _after_ dark.\n\n+310.+ The preposition is an important word in the sentence. We can use\nit to serve our purpose in various ways. We have found, for instance,\nthat we can use it:\n\n_First_, to change an adjective into an adjective phrase. As, for\nexample:\n\n The _fearless_ man demands his rights.\n The man _without fear_ demands his rights.\n\n_Second_, to change an adverb into an adverb phrase. As, for example:\n\n We want to possess _peacefully_ the fruits of our labor.\n We want to possess _in peace_ the fruits of our labor.\n\n_Third_, to express a meaning which we can express in no other way; as,\nfor example, _They are fighting for their country_. There is no single\nword which we can use to express the meaning which we express in the\nphrase, _for their country_.\n\n+311.+ So the preposition has given us a new means of expression, _the\nprepositional phrase_. We can, by its help, use a phrase in place of an\nadjective to modify a noun or a pronoun, and in place of an adverb to\nmodify a verb or an adjective. And we can also use the prepositional\nphrase to express relationship which we cannot express by a single\nadjective or adverb.\n\nIf I want to tell you that I see a bird in yonder tree, such an\nexpression would be impossible without that little preposition _in_. By\nthe use of various prepositions, I can express to you the relationship\nbetween the bird and the tree. I can tell you whether it is _under_ the\ntree, or _in_ the tree, or _over_ the tree, or flying _around_ the tree,\nor _near_ the tree. By the use of the various prepositions, I can\nexpress accurately the relationship that exists between the _bird_ and\nthe _tree_.\n\n\n Exercise 1\n\nLook up the list of prepositions in Section 306, on page 184. Use the\nfollowing pairs of words in sentences and use as many different\nprepositions as you can to express the different relationships which may\nbe expressed between these words. For example, take the two words, _man_\nand _house_. You may say:\n\n The man went _around_ the house.\n The man went _about_ the house.\n The man went _over_ the house.\n The man went _under_ the house.\n The man went _without_ the house.\n The man went _into_ the house.\n The man went _by_ the house.\n The man went _beyond_ the house.\n The man went _to_ the house.\n\n enemy city\n soldiers cannon\n man machine\n woman factory\n children school\n government people\n\n\n A GOVERNING WORD\n\n+312.+ The preposition shows the relation between two words. In this way\nit enables us to use a noun or a pronoun as a modifying word. For\nexample, in the sentence given above, _I see the bird in the tree_, the\npreposition _in_ shows the relationship between _bird_ and _tree_, and\nmakes of _tree_ a modifying word. It expresses a different meaning than\nif we used the word _tree_ as an adjective. For we do not mean that we\nsee a tree bird, but a bird in a tree. So with the help of the\npreposition _in_, we have used _tree_ as a modifying word.\n\nBut the preposition _in_ also governs the form of the word that follows\nit. Since nouns have the same form whether they are used as subject or\nobject, this does not mean any change in the form of the nouns. But\npronouns have different forms for the subject and object, so when we use\na pronoun with a preposition, we must use the object form. There are\nseven object forms of the personal pronouns, and after a preposition,\nalways use one of these object forms.\n\n He gave it to _me_.\n Give it to _him_.\n Give it to _her_.\n Add this to _it_.\n Bring it to _us_.\n I will give it to _you_.\n He gave it to _them_.\n\n+313.+ Be careful to always use the object form of the pronoun following\na preposition. Observe this also in the use of the relative and\ninterrogative pronoun \"who.\" The object form is \"whom.\" For example:\n\n To whom will you go?\n This is the man to whom I wrote.\n For whom are you looking?\n Where is the woman for whom you would make such a sacrifice?\n\n\n Where to Put the Preposition\n\n+314.+ The preposition generally precedes its object. This is the reason\nit was given its name, _preposition_, meaning _to place before_.\nSometimes, however, the preposition is separated from its object. This\nis often true when it is used with an interrogative or relative pronoun.\nWith these pronouns, the preposition is often thrown to the end of the\nsentence. For example:\n\n This is the book about which I was speaking; _or_, This is the book\n which I was speaking about.\n To whom shall I give this letter; _or_, Who shall I give this\n letter to?\n\nThe sentence, _To whom shall I give this letter_, is grammatically\ncorrect; but in ordinary usage we use the form, _Who shall I give this\nletter to?_\n\nWhile the rule calls for the object form of the relative pronoun after a\npreposition--so that the use of _to whom_ is grammatically correct--in\ncommon usage we use the subject form of the pronoun when it is used so\nfar away from the preposition which governs it. So we find this use\ncommon. For example, instead of saying, _For whom is this letter?_ we\nsay, _Who is this letter for?_\n\n+315.+ In poetry also, we often find the object coming before the\npreposition. For example:\n\n \"The interlacing boughs between\n Shadows dark and sunlight sheen,\n Alternate, come and go.\"\n\n_Boughs_ is here the object of the preposition _between_, but in this\npoetic expression the object is placed before the preposition. Note also\nin the following:\n\n \"The unseen mermaid's pearly song,\n Comes bubbling up the weeds _among_.\"\n\n \"Forever panting and forever young,\n All breathing human passion far _beyond_.\"\n\n+316.+ After an interrogative adjective, the preposition is also often\nthrown to the end of the sentence. As, for example:\n\n What men are the people talking _about_?\n Which person did you write _to_?\n\nWith these few exceptions, however, the preposition usually precedes its\nobject, as:\n\n We were astonished _at_ the news.\n He arose _from_ his sleep.\n\n\n POSSESSIVE PHRASES\n\n+317.+ Review Lesson 4, in which we studied the possessive use of nouns.\nYou will recall that we make the possessive form of the nouns by the use\nof the apostrophe and _s_. But instead of using the possessive forms of\nthe name of inanimate things; that is, things without life, we generally\ndenote possession by the use of a phrase. Thus we would say, _The arm of\nthe chair_, instead of, _The chair's arm_; or, _The roof of the house_,\ninstead of, _The house's roof_.\n\n+318.+ We also use a possessive phrase when the use of a possessive form\nwould give an awkward construction. As, for example: _Jesus' sayings_.\nSo many hissing sounds are not pleasant to the ear and so, we say\ninstead, _The sayings of Jesus_.\n\n+319.+ We use a phrase also where both nouns are in the plural form. In\nmany words, there is no difference in the sound between a single noun in\nthe possessive form and a plural noun in the possessive form. We can\nreadily tell the meaning when it is written, because the place of the\napostrophe indicates the meaning, but when it is spoken the sound is\nexactly the same. As, for example:\n\n The lady's hats.\n The ladies' hats.\n\nWritten out in this way, you know that in the first instance I am\nspeaking of the hats belonging to one lady, but in the second instance\nof the hats belonging to two or more ladies. But when it is spoken, you\ncan not tell whether I mean one lady or a number of ladies. So we use a\nphrase and say, _The hats of the lady_; or, _The hats of the ladies_.\nThen the meaning is entirely clear.\n\n+320.+ Sometimes we want to use two possessives together, and in this\ncase it is better to change one of them into a phrase; for example,\n_This is my comrade's father's book_. This is an awkward construction.\nSay instead, _This is the book belonging to my comrade's father_.\n\n+321.+ Do not overlook the fact, however, that the phrase beginning with\n_of_ does not always mean possession. Consider the following examples\nand see if there is not a difference in meaning:\n\n The history of Wilson is interesting.\n Wilson's history is interesting.\n\nIn the first instance, I mean the history of Wilson's life is\ninteresting; in the second instance I mean the history belongs to or\nwritten by Wilson is interesting. So there is quite a difference in the\nmeaning. The phrase _of Wilson_ used in the first example does not\nindicate possession.\n\nNote the difference in meaning between the following sentences:\n\n The picture of Millet is good.\n Millet's picture is good.\n\n The statue of Rodin stands in the park.\n Rodin's statue stands in the park.\n\nWould you say:\n\n The invention of gunpowder, or gunpowder's invention?\n The destruction of Louvain, or Louvain's destruction?\n The siege of Antwerp, or Antwerp's siege?\n The boat's keel, or the keel of the boat?\n\n\n COMMON ERRORS\n\n+322.+ Prepositions are usually very small and seemingly unimportant\nwords, yet we make a great many mistakes in their use. It is these\nlittle mistakes that are most difficult to avoid.\n\nNotice carefully in your own speech this week, and in the conversation\nwhich you overhear, the use of the prepositions. Notice especially the\nfollowing cautions:\n\n+1. Do not use prepositions needlessly.+ We often throw a preposition in\nat the close of a sentence which we have already used in the sentence,\nand which we should not use again. The little preposition _at_ is most\nfrequently used in this way. See how many times this week you hear\npeople use such phrases as:\n\n At which store do you trade at?\n At what corner did you stop at?\n\nThe last _at_ is entirely unnecessary. It has already been used once and\nthat is enough. We also use _at_ and _to_ at the close of sentences\nbeginning with an interrogative adverb, where they are not necessary.\nFor example, we say:\n\n Where did you go to?\n Where did you stop at?\n Where am I at?\n\nThe correct form of these sentences is:\n\n Where did you go?\n Where did you stop?\n Where am I?\n\nDo not use _at_ and _to_ in this way, they are entirely superfluous and\ngive a most disagreeable sound to the sentence. Do not close a sentence\nwith a preposition in this way.\n\n+2. Do not omit the preposition where it properly belongs.+ For example,\nwe often say:\n\n The idea is no use to me.\n\nWe should say, _The idea is of no use to me_.\n\n I was home yesterday.\n\nWe should say, _I was at home_ yesterday.\n\n+3. Do not use the preposition _of_ with a verb that requires an\nobject.+ The noun cannot be the object of both the verb and the\npreposition. As, for example:\n\n He does not remember _of_ seeing you.\n Do you approve _of_ his action?\n\n_Remember_ and _approve_ are both incomplete verbs requiring an object,\nand the nouns _seeing_ and _action_ are the objects of the incomplete\nverbs _remember_ and _approve_. The preposition _of_ is entirely\nsuperfluous. The sentences should read:\n\n He does not remember seeing you.\n Do you approve his action?\n\nOther verbs with which we commonly use the preposition _of_ in this way\nare the verbs _accept_ and _recollect_. As, for example:\n\n Will you accept _of_ this kindness?\n Will you try to recollect _of_ it?\n\nThese sentences should read:\n\n Will you accept this kindness?\n Will you try to recollect it?\n\n\n The Correct Preposition\n\n+323.+ We make a great many mistakes also in the choice of\nprepositions. For example, the preposition _between_ refers to two\nobjects and should never be used when you are speaking of more than two,\nthus:\n\n We settled the quarrel _between_ the two men.\n\nThis is correct, but it is incorrect to say:\n\n We settled the quarrel _between_ the members of the Union.\n\nWe cannot settle a quarrel between a _dozen_ people. When there are more\nthan two, use the word _among_. We can perhaps attempt to settle a\nquarrel _among_ a dozen people. _Between_ refers to two objects, _among_\nrefers to more than two. For example:\n\n Divide the work _between_ the two men.\n Divide the work _among_ twenty men.\n\n+324.+ Do not confuse the use of _in_ and _into_. When entrance is\ndenoted use _into_. As, for example:\n\n He came into the room.\n He got into the auto.\n\nOften the use of _in_ will give an entirely different meaning to the\nsentence. For example:\n\n He ran _in_ the water.\n He ran _into_ the water.\n The man acted as our guide _in_ the city.\n The man acted as our guide _into_ the city.\n The horse ran _in_ the pasture.\n The horse ran _into_ the pasture.\n\n+325.+ Do not use _below_ and _under_ to mean _less_ or _fewer_ in\nregard to an amount or number. _Below_ and _under_ have reference to\nplace only. It is correct to say:\n\n He went _under_ the bridge.\n He came out _below_ the falls.\n\nBut it is incorrect to say:\n\n The price is _below_ cost.\n There were _under_ fifty present.\n\nSay instead:\n\n The price is _less_ than cost.\n There were _fewer_ than fifty present.\n\n+326.+ Do not misuse _over_ and _above_. These prepositions have\nreference only to _place_. They are incorrectly used to mean _more than_\nor _greater than_.\n\nIt is correct to say:\n\n The boat anchored above the landing.\n He flew over the city.\n\nIt is incorrect to say:\n\n He bought above a hundred acres.\n He lives over a mile from here.\n\nThese sentences should be:\n\n He bought more than a hundred acres.\n He lives more than a mile from here.\n\n\n THE PREPOSITION WITH VERBS\n\n+327.+ In our first lesson on prepositions, we had a list of verbs and\nthe correct preposition to use with these verbs. There are a few words\nwhich we use very commonly in which the meaning is slightly different\naccording to the preposition which we use in connection with the verb.\nForeigners especially who are learning the English language have great\ndifficulty with the prepositions. Here are a few of these common words:\n\n+Adapt.+ With _adapt_ we can use either the preposition _to_ or _for_.\nFor example; we adapt ourselves _to_ circumstances, that is, we\naccommodate or conform ourselves; but a thing can be adapted _for_ a\ncertain purpose.\n\n+Agree.+ We can use the prepositions _with_ and _to_ with the verb\n_agree_, but with different meanings. For example, we say, We agree\n_with_ you about a certain matter; and, We agree _to_ the proposal which\nyou make.\n\n+Ask.+ We ask a favor _of_ a person. We ask a friend _for_ a favor. We\nask _about_ some one or thing that we wish to hear about.\n\n+Charge.+ There are several prepositions we can use with the verb\n_charge_. Your grocer charges you _for_ the things that you buy. If you\nrun an account you are charged _with_ a certain amount. These things are\ncharged _to_ you; but in war the enemy charges _upon_ you.\n\n+Compare.+ One thing is compared _with_ another in quality, but it is\ncompared _to_ another when we are using the comparison for an\nillustration.\n\n+Complain.+ We make complaint _to_ the manager _of_ the things we do not\nlike.\n\n+Comply.+ We comply _with_ the request of another, but he does a thing\n_in_ compliance _with_ that request. Do not use the preposition _to_\nwith compliance.\n\n+Correspond.+ With correspond, we use either the preposition _with_ or\n_to_. For example, I may correspond _with_ you, meaning that I\ncommunicate with you by letter, but one thing corresponds _to_ another,\nmeaning that it is like the other.\n\n+Disgust.+ We are disgusted _with_ our friends sometimes _at_ the things\nwhich they do. We are disgusted _with_ people and _at_ things.\n\n+Reconcile.+ With reconcile, we use either the preposition _with_ or\n_to_. For example, I may become reconciled _with_ you; that is, I am\nrestored to friendship or favor after an estrangement. But we reconcile\none thing _to_ another; that is, we harmonize one thing with another.\n\n+Taste.+ We have a taste _for_ music, art or literature, but we enjoy\nthe taste _of_ good things to eat. When taste refers to one of the five\nsenses, use the preposition _of_, but when you use it to mean\nintellectual relish or enjoyment, use the preposition _for_.\n\n\n Exercise 2\n\nMark all of the prepositional phrases in the following poem:\n\n THE ANGEL OF DISCONTENT\n\n When the world was formed and the morning stars\n Upon their paths were sent,\n The loftiest-browed of the angels was made\n The Angel of Discontent.\n\n And he dwelt with man in the caves of the hills,\n Where the crested serpents sting,\n And the tiger tears and the she-wolf howls,\n And he told of better things.\n\n And he led them forth to the towered town,\n And forth to the fields of corn,\n And told of the ampler work ahead,\n For which his race was born.\n\n And he whispers to men of those hills he sees\n In the blush of the misty west;\n And they look to the heights of his lifted eye--\n And they hate the name of rest.\n\n In the light of that eye does the slave behold\n A hope that is high and brave;\n And the madness of war comes into his blood--\n For he knows himself a slave.\n\n The serfs of wrong by the light of that eye\n March with victorious songs;\n For the strength of the right comes into their hearts\n When they behold their wrongs.\n\n 'Tis by the light of that lifted eye\n That error's mists are rent;\n A guide to the table-lands of Truth\n Is the Angel of Discontent.\n\n And still he looks with his lifted eye,\n And his glance is far away,\n On a light that shines on the glimmering hills\n Of a diviner day.\n\n --_Sam Walter Foss_.\n\n\n Exercise 3\n\nMark all of the prepositions in the following poem. Write out the entire\nphrases and mark the word which is the object of the preposition. For\nexample, in the phrase in the second line; _from a rich dream_, _dream_\nis the object of the preposition _from_; and _a_ and _rich_ modify the\nnoun _dream_.\n\n Abou Ben Adhem (may his tribe increase)\n Awoke one night from a rich dream of peace,\n And saw, within the moonlight of his room,\n Making it rich, and like a lily in bloom,\n An angel, writing in a book of gold.\n Exceeding peace had made Ben Adhem bold,\n And to the Presence in the room he said,\n \"What writest thou?\" The Vision raised its head,\n And, with a look made of all sweet accord,\n Answered, \"The names of those who love the Lord.\"\n \"And is mine one?\" said Abou. \"Nay, not so,\"\n Replied the Angel. Abou spoke, more low,\n But cheerily still; and said, \"I pray thee, then,\n Write me as one that loves his fellow-men.\"\n The angel wrote and vanished. The next night\n It came again, with a great wakening light,\n And showed the names whom love of God had blessed,\n And lo! Ben Adhem's name led all the rest.\n\n --_Leigh Hunt_.\n\n\n\n\n SPELLING\n\n LESSON 19\n\n\nThere are a few prepositions which might really be called derivative\nprepositions.\n\n1. A few prepositions are formed from verbs. These are really participle\nprepositions, for they are the present participles of the verbs but have\ncome to be used like prepositions. These are such as _concerning_,\n_excepting_, _regarding_, _respecting_, _during_, _according_, etc.\nNearly all of these participle prepositions can be expressed by a\npreposition phrase, as for example, we can either say; I wrote\n_regarding_ these facts, or I wrote you _in regard to_ these facts. I\nmentioned them all _excepting_ the last, or, I mentioned them all _with\nthe exception of_ the last. I have gone _according_ to the directions,\nor, I have gone _in accord with_ the directions.\n\n2. Derivative prepositions are also formed by prefixing _a_ to other\nparts of speech, as _along_, _around_, _abroad_, etc. Strictly speaking\nthese might be called compound prepositions for the prefix _a_ is really\nfrom the preposition _on_.\n\n3. We have also compound prepositions formed:\n\nBy uniting two prepositions, as _into_, _within_, _throughout_, etc.\n\nBy uniting a preposition and some other part of speech, usually a noun\nor an adjective, as _beside_, _below_ and _beyond_.\n\nWe also have a number of compound verbs which are made by prefixing a\npreposition to a verb. Some of these compound words have quite a\ndifferent meaning from the meaning conveyed by the two words used\nseparately; as for example, the compound verb _withstand_, derived from\nthe preposition _with_ and the verb _stand_, has almost the opposite\nmeaning from _stand with_.\n\nOur spelling lesson this week includes a number of these compound verbs\nformed by the use of the verb and a preposition. Look up the meaning in\nthe dictionary. Use them in sentences in the compound form; then the two\nwords separately as a verb and a preposition and note the difference in\nthe meaning.\n\n +Monday+\n\n Upset\n Withdraw\n Outrun\n Overlook\n Understand\n\n +Tuesday+\n\n Oversee\n Undergo\n Outnumber\n Withhold\n Overcome\n\n +Wednesday+\n\n Overflow\n Undertake\n Overreach\n Overthrow\n Outshine\n\n +Thursday+\n\n Overhear\n Withstand\n Overgrow\n Overhaul\n Overrun\n\n +Friday+\n\n Concerning\n Regarding\n Respecting\n According\n Excepting\n\n +Saturday+\n\n Against\n Throughout\n Around\n Between\n Beneath\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 20\n\n\nDear Comrade:\n\nWe are taking up in this lesson the study of the last important part of\nspeech. We have spent some little time on the study of these parts of\nspeech, but it has been time well spent. We cannot use good English and\nconstruct sentences that express our thoughts without an adequate\nknowledge of the words we use in sentence building. As soon as we finish\nthe study of the parts of speech, we will spend several weeks in\nsentence building. This will give us a review of these lessons in which\nwe have studied separate words.\n\nThe English language is one of the most interesting of all to study. It\nis the most truly international of all languages, for the English\nlanguage contains words from almost every language in the world. Did you\never stop to think that we could have internationalism in language as\nwell as in other things? We can be as narrowly patriotic concerning\nwords as concerning anything else.\n\nNations have been prone to consider all those who do not speak their\nlanguage as barbarians. Germany, perhaps, possesses as strong a\nnationalistic spirit as any country, and in Germany this spirit has\nfound expression in a society formed for the purpose of keeping all\nforeign words out of the German language. They have published handbooks\nof native words for almost every department of modern life. They insist\nthat the people use these words, instead of foreign importations. The\nGerman State takes great pride in the German language and considers it\nthe most perfect of any spoken today. The rulers of Germany believe that\nit is a part of their duty to the world to see that all other nations\nspeak the German language. In conquered Poland, only German is permitted\nto be taught in the schools or to be spoken as the language of commerce.\nThe patriots in language seem to believe that there is some connection\nbetween purity of language and purity of race.\n\nIn English, however, we have the beginnings of an international speech.\nOur civilization is derived from various sources. Here in America we are\ntruly the melting-pot of the nations, and this is mirrored forth in our\nlanguage which is, in a way, a melting-pot also, in which have been\nthrown words from every tongue. Those for whom nationalism is an\nimportant thing will probably cling to the idea of a pure unmixed\nlanguage, but to those of us to whom Internationalism is not an empty\nword, but a living ideal, an international language becomes also part of\nthe ideal.\n\nThere is a wealth of wonderful literature open to us once we have gained\na command of the English language. Pay especial attention to the\nquotations given in each lesson. These are quotations from the very best\nliterature. If there are any of them that arouse your interest and you\nwould like to read more from the same author, write us and we shall be\nglad to furnish you full information concerning further reading.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n CONJUNCTIONS\n\n+328.+ You remember that in Lesson 3, where we studied the parts of\nspeech, we found that we had another connective word besides the\npreposition,--the conjunction.\n\nA preposition connects two words and shows what one of them has to do\nwith the other. The conjunction plays a different part as a connective,\nfor it connects not only words but also phrases and clauses. Note the\nfollowing sentences:\n\n Shall we be men _or_ machines?\n We must struggle for ourselves _and_ for our children.\n We pray for peace _but_ furnish ammunition for war.\n\nThe use of the conjunction saves a great deal of tiresome repetition,\nfor, by its use, where two subjects have the same predicate or two\npredicates have the same subject, we can combine it all into one\nsentence.\n\nYou will readily realize how important this part of speech is to us. If\nwe did not have conjunctions our speech would be cumbersome and we would\nhave to use a great many short sentences and a great deal of repetition.\nIf we wanted to make the same statement concerning a number of things,\nwithout conjunctions, we would have each time to repeat the entire\nstatement. Try to write a description of a scene and avoid the use of\nconjunctions and you will see what an important part these connective\nwords play in our power of expression.\n\nWithout the use of the conjunction, you would necessarily use a great\nmany short expressions and repeat the same words again and again, and\nyour description would be a jerky, tiresome, unsatisfactory piece of\nwriting.\n\n\n Exercise 1\n\nRewrite the following sentences, writing in separate sentences the\nclauses that are united by the conjunctions:\n\n\n 1. The birds are singing _and_ spring is here.\n 2. We talk of peace, _but_ war still rages.\n 3. The unemployed cannot find work _and_ they are dying of hunger.\n 4. We believed in war for defense _and_ every nation is now fighting\n for defense.\n 5. We believe in education _and_ we are struggling for universal\n education.\n 6. The old order is fast passing _and_ the new order is rapidly\n appearing.\n 7. Profit is the keynote of the present, _but_ service shall be the\n keynote of the future.\n 8. All children should be in school, _but_ thousands must earn their\n bread.\n\nNote that these sentences are made up of two or more simple sentences\ncombined; and each of these simple sentences is called a clause, and\neach clause must contain a subject and a predicate.\n\n\n Exercise 2\n\nRewrite the following simple sentences, using conjunctions to avoid a\nrepetition of the same subject and predicate. Rewrite these into a\nparagraph, making as well written a paragraph as you possibly can:\n\n One hundred years ago the workers fought for universal education.\n As a result we have our public schools of today.\n Our public schools have been our chief bulwark against oppression.\n Our public schools are our chief bulwark against oppression.\n Our public schools are our greatest safeguard for the protection of\n such liberty as we enjoy.\n Our public school system embodies a socialistic ideal.\n Our public school system is the most democratic of our institutions.\n There has been a subtle subversion of the ideal.\n The public school system has been made to serve the master class.\n We have spent millions to make the ideal a reality.\n Have we realized the ideal?\n Is there universal education?\n Is there education for every child beneath the flag?\n The grounds of our public schools have cost millions.\n The buildings have cost millions.\n The courses of study are many.\n They are varied.\n They are elaborate.\n But the workers of the world do not enjoy this feast.\n The children of the workers do not enjoy this feast.\n\n\n CLASSES OF CONJUNCTIONS\n\n+329.+ Conjunctions are divided into classes, as are other parts of\nspeech, according to the work which they do. Notice the following\nsentences and notice how the use of a different conjunction changes the\nmeaning of the sentence.\n\n We are united _and_ we shall win.\n _When_ we are united, we shall win.\n\nIn the first sentence the conjunction _and_ connects the two clauses,\n_we are united_ and _we shall win_. They are both independent clauses,\nneither is dependent upon the other, and both are of equal importance.\nBut by the use of the conjunction _when_, instead of the conjunction\n_and_, we have changed the meaning of the sentence. There is quite a\ndifference in saying, _We are united and we shall win_, and _When we are\nunited we shall win_.\n\nBy connecting these two statements with the conjunction _when_, we have\nmade of the clause, _we are united_, a dependent clause, it modifies the\nverb phrase _shall win_. It tells _when_ we shall win, just as much as\nif we had used an _adverb_ to modify the verb phrase, and had said, _We\nshall win tomorrow_, instead of, _We shall win when we are united_.\n\nSo in these two sentences we have two different kinds of conjunctions,\nthe conjunction _and_, which connects clauses of equal rank or order,\nand the conjunction _when_, which connects a dependent clause to the\nprincipal clause.\n\n+330.+ So the conjunctions like _and_ are called co-ordinate\nconjunctions. _Co-ordinate_ means literally of equal rank or order.\nConjunctions like _when_ are called sub-ordinate conjunctions.\n_Sub-ordinate_ means of inferior rank or order.\n\nSo we have our definitions:\n\n+331.+ +A conjunction is a word that connects words or phrases or\nclauses.+\n\n+A co-ordinate conjunction is one that joins words, phrases or clauses\nhaving the same rank.+\n\n+A subordinate conjunction is one that connects a dependent clause to\nthe principal clause.+\n\n\n CO-ORDINATE CONJUNCTIONS\n\n+332.+ +Co-ordinate conjunctions connect words, phrases or clauses of\nequal rank.+ The most commonly used co-ordinate conjunctions are; _and_,\n_but_, _or_, _nor_.\n\n+333.+ But there are a number of words which we often use as adverbs,\nwhich may also be used as co-ordinate conjunctions. These words are not\nalways conjunctions, for they are sometimes used as adverbs. When they\nare used as conjunctions they retain something of their adverbial\nmeaning; but still they are conjunctions, for they are used to show the\nconnection between two clauses of equal rank. Thus:\n\n I am not in favor of the motion, _nevertheless_ I shall vote for it.\n The deputies voted for the war appropriation, _notwithstanding_ they\n had carried on an extensive anti-war propaganda.\n I did not believe in the change, _however_ I did not oppose it.\n\n+334.+ The co-ordinate conjunctions which we use with this adverbial\nmeaning also, are; _therefore_, _hence_, _still_, _besides_,\n_consequently_, _yet_, _likewise_, _moreover_, _else_, _than_, _also_,\n_accordingly_, _nevertheless_, _notwithstanding_, _otherwise_,\n_however_, _so_ and _furthermore_.\n\nThese conjunctions always refer to what has been said before and serve\nto introduce and connect new statements.\n\n+335.+ We often use these conjunctions, and also, _and_, _but_, _or_,\nand _nor_, at the beginning of a separate sentence or paragraph to\nconnect it in meaning with that which has gone before. You will often\nsee the use of these conjunctions as the first word of a new paragraph,\nthus relating this paragraph to that which has preceded it.\n\n+336.+ +Co-ordinate conjunctions connect words of equal rank.+\n\n\n NOUNS\n\nCo-ordinate conjunctions may connect two or more _nouns_ used as the\nsubject of a verb. As:\n\n _Death_ and _disaster_ follow in the wake of war.\n\nIn this sentence, _death_ is just as much the subject of the verb\n_follow_ as is the word _disaster_, but no more so. You can omit either\nof these words and the other will make a subject for the sentence. They\nare both of equal importance, both of the same rank in the sentence, and\nneither depends upon the other. These two words taken together form the\nsubject of the sentence. This is called the _compound subject_, for it\nconsists of two simple subjects.\n\nCo-ordinate conjunctions may connect two or more nouns used as the\n_object_ of a verb.\n\n He studies history and science.\n\nIn this sentence the words _history_ and _science_ are both used as\nobjects of the verb _studies_.\n\nCo-ordinate conjunctions may connect two or more nouns used as the\nobject of a _preposition_.\n\n He called for the letters and the papers.\n\nIn this sentence _letters_ and _papers_ are both objects of the\npreposition _for_, connected by the co-ordinate conjunction _and_.\n\n\n Exercise 3\n\nNote in the following sentences the nouns which are connected by\nconjunctions and decide whether they are used as the subject of the\nsentences or the object of verbs or of prepositions. Draw a line under\ncompound subjects.\n\n 1. John and Henry are going home.\n 2. Music and painting are fine arts.\n 3. The grounds and buildings of our public schools have cost millions.\n 4. The time calls for brave men and women.\n 5. We struggle for truth and freedom.\n 6. Will you study English or arithmetic?\n 7. Education and organization are necessary for success.\n 8. We must learn the truth about production and distribution.\n 9. We demand justice and liberty.\n 10. The great struggle is between the working class and the ruling\n class.\n\n\n PRONOUNS\n\n+337.+ +Co-ordinate conjunctions may also connect pronouns.+\n\nThese are used in the same way as nouns,--either as subject or object.\nNouns have the same form whether used as subject or object. Pronouns,\nhowever, have different forms when used as the object. Here is where we\noften make mistakes in the use of pronouns. When the pronouns are\nconnected by co-ordinate conjunctions they are of the same rank and are\nused in the same construction;--if they are used as subjects both must\nbe used in the subject form;--if they are used as objects, both must be\nused in the object form. For example, it is incorrect to say, _He told\nthe story to her and I_. Here _her_ is properly used in the object form,\nfor it is the object of the preposition _to_; the pronoun _I_ connected\nwith _her_ by the use of the conjunction _and_ is also the object of the\npreposition _to_, and the object form should be used. You would not say,\n_He told the story to I_. The sentence should read, _He told the story\nto her and me_.\n\nCo-ordinate conjunctions may connect two pronouns used as the _subject_\nof a sentence, as for example:\n\n _She_ and _I_ arrived today.\n\nCo-ordinate conjunctions may connect two pronouns used as the _object_\nof the verb, as for example:\n\n Did you call _her_ or _me_?\n\nCo-ordinate conjunctions may connect two pronouns used as the object of\nthe _preposition_, as:\n\n He gave that to _you_ and _me_.\n\n\n Exercise 4\n\nStudy closely the following sentences and correct those in which the\nwrong form of the pronoun is used.\n\n 1. He and I are old friends.\n 2. Did you ask him or me?\n 3. They promised him and I that they would come.\n 4. Find the place for she and me.\n 5. Me and him will get it for you and she.\n 6. She and I will go with you.\n 7. You and I must decide matters for ourselves.\n 8. You will find him and her to be loyal comrades.\n\n\n VERBS\n\n+338.+ +Co-ordinate conjunctions are also used to connect verbs.+ Verbs\nconnected in this way have the same subject; and with the use of the\nconjunction to connect the verbs, we save repeating the subject.\n\n He _reads_ and _studies_ constantly.\n\nIn this sentence _reads_ and _studies_ are words of the same kind and of\nthe same rank; either could be omitted and the other would make a\npredicate for the sentence. They are of equal importance in the sentence\nand are connected by the conjunction _and_. They have a single subject,\nthe pronoun _he_.\n\nThis is called a compound predicate.\n\nIn the sentence, _He reads constantly_, we have a simple predicate, the\nsingle verb _reads_; but in the sentence, _He reads and studies\nconstantly_, we have a compound predicate, compound of the two verbs\n_reads_ and _studies_. A sentence may have both a compound subject and a\ncompound predicate. As, for example:\n\n John and James read and study constantly.\n\nIn this sentence _John_ and _James_ is the compound subject of both the\nverbs, _read_ and _study_. So we have a compound subject and a compound\npredicate.\n\n\n Exercise 5\n\nNotice the verbs in the following sentences connected by co-ordinate\nconjunctions. Draw lines under each compound predicate.\n\n 1. The days come _and_ go in a ceaseless round.\n 2. The brave man dreams _and_ dares to live the dream.\n 3. The coward dreams _but_ dares not live the dream.\n 4. We produce splendidly _but_ distribute miserably.\n 5. The bought press twists _and_ distorts the facts.\n 6. Only a traitor aids _or_ supports the enemy.\n 7. We agitate _and_ educate for the cause of liberty.\n\n\n ADJECTIVES\n\n+339.+ +Co-ordinate conjunctions are used to connect adjectives.+\n\nIn this way we use a number of adjectives to modify the same word\nwithout tiresome repetition. When several adjectives are used to modify\nthe same word, the conjunction is used only between the last two\nadjectives. As, for example:\n\nA _simple_, _clear_ and _concise_ course has been prepared.\n\n\n Exercise 6\n\nIn the following sentences, underscore the adjectives which are\nconnected by co-ordinate conjunctions.\n\n 1. The plains of France are covered with the dead and dying soldiers.\n 2. Education should be both universal and free.\n 3. They are faithful and loyal comrades.\n 4. This was only our just and legal right.\n 5. Old and hoary was the man who sat on the stool by the fireless and\n godless altar.\n 6. The service of humanity is a sweet and noble task.\n 7. We must be brave and true.\n 8. He lived a noble and courageous life.\n 9. All was old and cold and mournful.\n 10. Most powerful and eloquent is the voice of the disinherited.\n\n\n ADVERBS\n\n+340.+ +Co-ordinate conjunctions are also used to connect adverbs.+ This\ngives us the power to describe the action expressed in verbs without the\ntiresome repetition of the verb. For example:\n\n He spoke _fluently_ and _eloquently_.\n\n\n Exercise 7\n\nIn the following sentences underscore the adverbs which are connected\nby co-ordinate conjunctions:\n\n 1. Man selfishly and greedily prevents his fellow men from the\n enjoyment of nature's bounties.\n 2. She is wonderfully and gloriously brave.\n 3. He speaks eloquently and impressively, but very slowly.\n 4. Nature has provided lavishly and bountifully for her children.\n 5. Advice spoken truly and wisely is always in season.\n 6. We must resist injustice bravely and courageously.\n 7. He feels keenly and deeply the wrongs of his class.\n 8. He writes easily and rapidly.\n 9. The words, calmly and coolly spoken, were instantly opposed.\n 10. He reached that conclusion naturally and inevitably.\n 11. He was gently but unwaveringly firm.\n 12. The revolution comes slowly but surely.\n\n\n PHRASES\n\n+341.+ +Co-ordinate conjunctions are used, not only to connect words,\nbut also to connect phrases.+\n\n\n Verb Phrases\n\n+342.+ Verb phrases may be connected by conjunctions. For example:\n\n The People's College _is owned_ and _controlled_ by the working class.\n We _have made_ and _are making_ a fierce struggle for a free press.\n\n\nIn this last sentence the two verb phrases, _have made_ and _are making_\nare connected by the co-ordinate conjunction _and_. Often in using verb\nphrases, we use phrases in which the same helping verb occurs in both\nphrases. When this is the case the helping verb is quite often omitted\nin the second phrase and only the participle is connected by the\nconjunction. As, for example:\n\n The People's College is owned and controlled by the working class.\n\nIn this sentence the helping verb _is_ belongs in both the phrases but\nis omitted in the second phrase in order to make a smoother sounding\nsentence. In the second phrase, only the past participle _controlled_ is\nused. It is understood that we mean,\n\n The People's College _is owned_ and _is controlled_ by the working\n class.\n\n\n Exercise 8\n\nNote the use of the conjunction in the following sentences to connect\nthe verb phrases. Supply the helping verb where it is omitted.\n\n 1. Our system of education is rooted and grounded in outgrown dogmas.\n 2. We have written but have received no answer.\n 3. Will you come or stay?\n 4. Man must struggle or remain in slavery.\n 5. The workers are organizing and demanding their rights.\n 6. We must arouse and educate our comrades.\n 7. We have sought but have not found.\n\n\n Prepositional Phrases\n\n+343.+ +Co-ordinate conjunctions are used to connect prepositional\nphrases.+\n\nThese phrases may be used as adjective phrases. For example:\n\n The books _in the book case_ and _on the table_ belong to me.\n\nThese phrases may be used as adverb phrases. For example:\n\n He works _with speed_ and _with ease_.\n\n\n Exercise 9\n\nNote in the following sentences, the prepositional phrases which\nare connected by co-ordinate conjunctions. Mark which are used as\nadjective and which as adverb phrases.\n\n 1. Education is the road out of ignorance and into the light.\n 2. The army charged over the plain and up the hill.\n 3. The first men lived in groups and in clans.\n 4. Democracy means government of the people and by the people.\n 5. Shall we take the path toward progress or toward barbarism.\n 6. They are not fighting for their country but for their king.\n 7. Human rights are not protected by the law nor by the courts.\n 8. The problem of the working class and of society is the problem of\n equitable distribution.\n 9. They are deceived by their leaders and by their press.\n 10. You can pay either by the week or by the month.\n 11. Our government is not the rule of the majority but of the\n minority.\n\n\n Infinitives and Participles\n\n+344.+ +Co-ordinate conjunctions are also used to connect infinitives\nand participles.+\n\n\n Exercise 10\n\nIn the following sentences mark the infinitives and participles\nconnected by co-ordinate conjunctions.\n\n 1. Those words will inspire us to dream and to dare.\n 2. We shall learn to produce and to distribute.\n 3. To be or not to be, that is the question.\n 4. Puffing and panting, the great engine pulled up to the station.\n 5. A cringing and trembling coward fears to demand his own.\n 6. The warped and twisted facts in the daily press deceive the masses.\n 7. Singing and dancing should be enjoyed by all children.\n 8. The exploiting and robbing of the people is made a virtue in ruling\n class ethics.\n\n\n CLAUSES\n\n+345.+ +Co-ordinate conjunctions are also used to connect clauses of\nequal rank.+ For example:\n\n _The floods came and the winds blew._\n\nEach of these clauses is a complete sentence in itself, but they are\ncombined into one compound sentence by the use of the co-ordinate\nconjunction, _and_. Clauses united in this way may have a compound\nsubject and a compound predicate, but two complete clauses must be\nunited by a co-ordinate conjunction in order to form a compound\nsentence. For example:\n\nThe rain and snow fell, _and_ the wind blew a mighty gale.\n\nHere the first clause in the compound sentence, _the rain and snow\nfell_, contains a compound subject, _rain and snow_.\n\nThe boys are running and shouting, _and_ the girls are gathering\nflowers.\n\nHere the first clause has a compound predicate, _are running_ and\n_shouting_. The second _and_ connects the two clauses forming the\ncompound sentence.\n\n\n CORRELATIVES\n\n+346.+ Certain co-ordinate conjunctions are used in pairs, such as\n_both, and_; _either, or_; _neither, nor_; _whether, or_. These pairs\nare called correlatives. The first word in the pair, as, _both_,\n_either_, _neither_, or _whether_, is used as an assistant conjunction\nhelping the other to do the connecting. These are used in such sentences\nas:\n\n I have _both_ seen _and_ heard him.\n They will join us _either_ in April _or_ in May.\n Labor has received _neither_ liberty _nor_ justice.\n _Whether_ to go forward _or_ to retreat was the problem.\n\nNote that _nor_ is always the proper correlative to use with _neither_\nand also with the negatives _not_ and _never_ when they apply to what\nfollows as well as to what precedes. For example:\n\n There are thousands in this country who can _neither_ read _nor_\n write.\n _Neither_ you _nor_ I can foretell the future.\n He will _not_ write _nor_ should you.\n Capital punishment is _nothing_ more _nor_ less than legalized murder.\n We shall _never_ lower our colors _nor_ retreat.\n\n_Or_ is always used with the correlative _either_. For example:\n\n We will _either_ come _or_ write you.\n _Either_ he was mistaken _or_ he deliberately lied.\n\n\n Exercise 11\n\nNote the use of the co-ordinate conjunctions _and_, _but_, _or_ and\n_nor_, in the following quotation. Mark especially the use of _and_ as\nan introductory conjunction, introducing a new sentence, but connecting\nit with that which has gone before.\n\n In my judgment slavery is the child of ignorance. Liberty is born of\n intelligence. Only a few years ago there was a great awakening in the\n human mind. Men began to inquire, \"By what right does a crowned robber\n make me work for him?\" The man who asked this question was called a\n traitor.\n\n They said then, and they say now, that it is dangerous for the mind of\n man to be free. I deny it. Out on the intellectual sea there is room\n for every sail. In the intellectual air, there is space enough for\n every wing. And the man who does not do his own thinking is a slave,\n and does not do his duty to his fellow men. For one, I expect to do my\n own thinking. And I will take my oath this minute that I will express\n what thoughts I have, honestly and sincerely. I am the slave of no man\n and of no organization. I stand under the blue sky and the stars,\n under the infinite flag of nature, the peer of every human being.\n\n All I claim, all I plead is simple liberty of thought. That is all. I\n do not pretend to tell what is true nor all the truth. I do not claim\n that I have floated level with the heights of thought, nor that I have\n descended to the depths of things; I simply claim that what ideas I\n have, I have a right to express, and any man that denies it to me is\n an intellectual thief and robber.\n\n Every creed that we have today has upon it the mark of the whip or the\n chain or the fagot. I do not want it. Free labor will give us wealth,\n and has given us wealth, and why? Because a free brain goes into\n partnership with a free hand. That is why. And when a man works for\n his wife and children, the problem of liberty is, how to do the most\n work in the shortest space of time; but the problem of slavery is, how\n to do the least work in the longest space of time. Slavery is poverty;\n liberty is wealth.\n\n It is the same in thought. Free thought will give us truth; and the\n man who is not in favor of free thought occupies the same relation to\n those he can govern that the slaveholder occupied to his slaves,\n exactly. Free thought will give us wealth. There has not been a\n generation of free thought yet. It will be time to write a creed when\n there have been a few generations of free-brained men and splendid\n women in this world. I don't know what the future may bring forth; I\n don't know what inventions are in the brain of the future; I don't\n know what garments may be woven, with the years to come; but I do\n know, coming from the infinite sea of the future, there will never\n touch this \"bank and shoal of time\" a greater blessing nor a grander\n glory, than liberty for man, woman and child.\n\n Oh, liberty! Float not forever in the far horizon! Remain not forever\n in the dream of the enthusiast and the poet and the philanthropist.\n But come and take up thine abode with the children of men\n forever.--_Ingersoll_.\n\n\n\n\n SPELLING\n\n LESSON 20\n\n\nWe found that we often formed adjectives by adding suffixes to other\nwords. We also form many adverbs by the addition of suffixes to other\nwords. Derivative adverbs are formed in the following ways:\n\n1. By adding suffixes to adjectives, chiefly the suffix _ly_, as for\nexample; _chiefly_, _truly_, _really_, _lately_, etc.\n\n2. By changing _ble_ to _bly_, as in _ably_, _nobly_, etc.\n\n3. By adding the suffix _ward_, as in _forward_, _upward_, _skyward_,\n_downward_, _homeward_, etc.\n\n4. We have some adverbs formed by adding the prefix _a_ to adjectives\nand nouns, as _ahead_, _afoot_, _afresh_, also by adding the prefix\n_be_, as in _besides_, _beyond_.\n\nWe often misspell a number of adverbs by adding _s_ where it does not\nrightfully belong; as, _anywheres_, _everywheres_, _backwards_,\n_forwards_, _towards_, _upwards_, _downwards_, _afterwards_,\n_homewards_, etc. All of these words should be written without the _s_.\n\nWe also have a number of compound adverbs which are made by the union of\ntwo other parts of speech, such as _sometime_, _henceforth_, _forever_,\n_overheard_, _outside_, etc.\n\nIn the lesson for Monday, Tuesday and Wednesday, adjectives are given\nhaving opposite meanings. Make the proper adverbs from these adjectives\nby the addition of the suffix _ly_.\n\nThursday's and Friday's lessons are made up of both adjectives and\nadverbs that end in _ly_. Look up in your dictionary and be sure you\nknow which are adjectives and which are adverbs.\n\nSaturday's lesson is made up of compound adverbs.\n\n +Monday+\n\n Haughty--Humble\n Wise--Ignorant\n Careful--Careless\n Firm--Wavering\n\n +Tuesday+\n\n Honest--Deceitful\n Fearful--Fearless\n Punctual--Tardy\n Identical--Different\n\n +Wednesday+\n\n Thoughtful--Thoughtless\n Rich--Poor\n Attentive--Inattentive\n Industrious--Lazy\n\n +Thursday+\n\n Quickly\n Lovely\n Clearly\n Cleanly\n\n +Friday+\n\n Homely\n Truly\n Courtly\n Nearly\n\n +Saturday+\n\n Otherwise\n Herewith\n Sometime\n Always\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 21\n\n\nDear Comrade:\n\nIn this lesson we are completing the study of conjunctions. We have\nstudied the conjunction last among the parts of speech and in the order\nof the development of language, the conjunction naturally comes last.\nThe need of connective words does not come in any language until the\nlanguage is quite well developed. You will notice that the connective\nwords, such as prepositions and conjunctions are the last words the\nchild begins to use. The child first begins to use the names of the\nthings with which it comes in contact, then it learns the words that\nexpress what these things do. But it is not until the child begins to\nreason that it begins to use connective words. These become necessary\nwhen we have reached a stage of development where we can consider the\nrelationship existing between things.\n\nThe use of conjunctions, however, can be greatly overdone. The long and\ninvolved sentences are more difficult to understand. If you will note\nthe authors which you enjoy the most, it will probably be those who use\nshort and crisp sentences. We have some authors who by the use of\nconjunctions can string one sentence out over several pages. You wonder\nhow they manage to exist so long without stopping for breath. It is very\neasy for us to fall into this error when we are thinking rapidly and our\nthoughts all seem to be closely connected. But no mind can grasp many\nideas at one time. Break your sentences up and express your ideas\nconcisely and clearly. Use conjunctions rather sparingly, especially\nthese subordinate conjunctions. Do not have too many subordinate clauses\nin one sentence.\n\nNotice in your reading for this week those who use the short, crisp\nsentences and those who use the longer and more involved sentences.\nNotice which are understood more readily and which are more enjoyable to\nread. Take some of the paragraphs from those who write long and involved\nsentences and break them up into short sentences and see if these\nshorter sentences do not make the meaning simpler and clearer. This will\nbe excellent practice also in gaining the power of expression.\n\nEspecially in the class struggle do we need those who can write clearly\nand simply of the great problems of the day. As the work of the world is\nconducted today, the workers have too little time for reading. They are\napt, after a hard day's work, to be too tired to follow an author\nthrough long, winding, involved passages.\n\nIn the spoken word, this is also true. You will find your hearers much\nmore in sympathy with you if you will use short sentences. Break your\nthought up so they can readily grasp your meaning and follow you to your\nconclusion.\n\nConjunctions are very important to save us from tiresome repetitions and\nshort, jerky sentences, but we must avoid using them too frequently.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n SUBORDINATE CONJUNCTIONS\n\n+347.+ We have found that co-ordinate conjunctions connect words,\nphrases and also clauses that are entirely independent; that is, they do\nnot depend in the slightest degree upon any other word, phrase or\nclause. Subordinate conjunctions connect inferior clauses to the main\nclauses of the sentence. These inferior clauses are dependent clauses.\nSubordinate conjunctions never connect words or phrases; but only\ndependent clauses, to the rest of the sentence. Note the following\nsentences:\n\n He came _quickly_.\n He came _on time_.\n He came _when he was called_.\n\nIn the first sentence the word _quickly_ is an adverb modifying the verb\n_came_ and answers the question _when_. It tells _when_ he came. In the\nsecond sentence, the phrase _on time_ is an adverb phrase modifying the\nverb _came_, and answers the question _when_. It tells _when_ he came.\nIn the third sentence, the clause _when he was called_, also answers the\nquestion _when_, and tells _when_ he came. Therefore, it is a clause\nused as an adverb. It is different from the phrase _on time_, for the\nphrase _on time_ does not contain a subject and a predicate.\n\n+348.+ The difference between the phrase and the clause is that the\nphrase does not contain either a subject or a predicate, while the\nclause _always_ contains both a subject and a predicate. So in the\nclause, _when he was called_, _he_ is the subject and _was called_ is\nthe predicate, and _when_ is the subordinate conjunction, which connects\nthis adverb clause to the verb _came_, which it modifies. The clause _he\ncame_, and the clause _when he was called_, are not of equal rank and\nimportance, because the clause, _when he was called_, simply modifies\nthe verb contained in the clause _he came_, by describing the _time_ of\nthe action expressed in the verb _came_. So the clause, _when he was\ncalled_, is a subordinate or dependent clause, and the conjunctions\nwhich connect this class of clauses to the main clause are called\nsubordinate conjunctions.\n\n+349.+ +A subordinate conjunction is one that connects a dependent\nclause to the principal clause.+\n\n\n CLASSES OF SUBORDINATE CONJUNCTIONS\n\n+350.+ Most subordinate conjunctions are used to make adverb clauses.\nThese clauses will answer some one of the questions answered by adverbs.\nThey will tell _how_, _when_, _where_ or _why_ the action expressed in\nthe verb in the principal clause occurred. There are six classes of\nthese subordinate conjunctions which are used to introduce adverb\nclauses. They introduce:\n\n+351.+ +Adverb clause of time.+ These clauses will answer the question\n_when_ and are introduced by such subordinate conjunctions as, _before_,\n_since_, _as_, _while_, _until_, _when_, _after_ and _as soon as_.\nNotice in the following sentences the difference made in the meaning of\nthe sentences by the use of the different conjunctions:\n\n We waited _until_ you came.\n We waited _after_ you came.\n We waited _as_ you came.\n We waited _before_ you came.\n We waited _since_ you came.\n We left _while_ you were gone.\n We left _when_ you were gone.\n We left _as soon as_ you were gone.\n\n+352.+ +Adverb clause of place.+ These answer the question _where_, and\nare introduced by the conjunctions, _where_, _whence_, _whither_.\n\n I will go _where_ you go.\n The wind blows _whither_ it listeth.\n He went _whence_ he came.\n\n+353.+ +Adverb clauses expressing cause or reason.+ These will answer\nthe question _why_. They are introduced by such subordinate conjunctions\nas, _because_, _for_, _since_, _as_, _whereas_, _inasmuch as_, etc.\n\nNote the difference in the meaning of the following sentences expressed\nby the use of different conjunctions:\n\n I will come _because_ you expect me.\n I will come _since_ you expect me.\n I will come _as_ you expect me.\n I will come _for_ you expect me.\n I will come _inasmuch as_ you expect me.\n\n+354.+ +Adverb clauses of manner.+ These clauses will answer the\nquestion _how_, and are introduced by such subordinate conjunctions as,\n_as_, _as if_, _as though_, etc.\n\n Study _as though_ you were in earnest.\n Come _as if_ you had been called.\n Do _as_ I say, not _as_ I do.\n\nIn these clauses of _manner_, introduced by _as if_, and _as though_,\n_were_ is used in the present form with either singular or plural\nsubjects. For example:\n\n He writes as if he _were_ informed of the facts.\n They talk as though they _were_ confident of success.\n You act as though I _were_ your slave.\n\n+355.+ +Adverb clauses of comparison.+ These clauses are introduced by\nthe subordinate conjunctions _than_ and _as_. The verbs are often\nomitted in these dependent clauses introduced by _than_ and _as_. For\nexample: _He is taller than I_. The complete sentence would be: _He is\ntaller than I am_. _He is not so tall as I._ Here the sentence would be:\n_He is not so tall as I am_.\n\nWhen the pronoun occurs in these dependent clauses, be sure to use the\nproper form of the pronoun. It may be the subject or the object of the\nverb which is not expressed. For example; it is incorrect to say: _I am\nnot so tall as him_. The correct form is: _I am not so tall as he_. The\ncomplete sentence would be: _I am not so tall as he is_, and the pronoun\nshould be in the subject form, for it is the subject of the verb _is_,\nwhich is understood and omitted.\n\nThe use of the _subject_ or of the _object_ form may make a difference\nin the meaning of your sentence. For example, you say: _I admire them as\nmuch as he_. You mean that you admire them as much as he admires them.\nBut if you say, _I admire them as much as him_, you mean that you admire\nthem as much as you admire him. Quite a different meaning!\n\nBe careful in the use of your pronouns in this way, for you can express\nquite a different meaning. For example, if you say, _I care more for you\nthan he_, you mean, I care more for you than he cares for you. But if\nyou say, _I care more for you than him_, you mean, I care more for you\nthan I care for him. A mistake like this might mean a great deal to you\nsome time, if the one to whom you had been speaking had been studying a\ncourse in Plain English!\n\n+356.+ +Adverb clauses of condition.+ These clauses are introduced by\nsuch conjunctions as, _if_, _provided_, _supposing_, _unless_, _except_,\n_otherwise_, _though_, _notwithstanding_, _albeit_, and _whether_. For\nexample:\n\n I will come _if_ you need me.\n I will come _provided_ you need me.\n I will go _notwithstanding_ you need me.\n I will not go _unless_ I am called.\n He will not go _except_ he is called.\n He will not go _though_ he is called.\n He came, _otherwise_ I would go.\n He will go _whether_ you go or stay.\n\nWhen subordinate clauses beginning with _if_, _though_ or _unless_ are\njoined to clauses containing _might_, _could_, _would_ or _should_, the\nverb _were_ is sometimes used with a singular subject, in such sentences\nas:\n\n If this _were_ true, I should know it.\n Unless I _were_ positive, I would not say so.\n Though our leader _were_ lost, yet we would not despair.\n If he _were_ here, he would explain it himself.\n If I _were_ with you, I might make you understand.\n\nSometimes in sentences like these, _if_ is omitted in the clause, and\nthe verb placed first. For example:\n\n _Were_ he here, he would deny these slanders.\n _Were_ he truly class-conscious, he would oppose this war.\n _Were_ this fact known, the people would never submit.\n\nThese clauses express something which is uncertain, or which is to be\ndecided in the future; a supposition contrary to a fact or a wish.\nOccasionally you will find the verb _be_ used instead of _is_, in\nclauses of this kind introduced by _if_, _though_, _unless_, _except_,\n_lest_, etc. For example:\n\n If it _be_ true, I will hear it.\n Though he _be_ guilty, we will not desert him.\n\nIn subordinate clauses connected by _if_, _unless_, etc., with a\nprincipal clause which expresses future time, the present form of the\nverb is used in the subordinate clause. For example:\n\n If they are willing, we will join them.\n Unless he comes, I shall not leave.\n If it rains, we will not go.\n\n+357.+ +Adverb clauses expressing purpose.+ These are introduced by such\nsubordinate conjunctions as, _that_, _in order that_ and _lest_. For\nexample:\n\n Take good care _that_ you understand this lesson.\n I will go today _in order that_ I may meet him.\n Watch these carefully _lest_ they be stolen.\n Read the labor press _that_ you may know the truth.\n\nNotice that _that_, when used in this way, as a pure conjunction, means\n_in order that_. For example, the sentence above might read:\n\n Read the labor press _in order that_ you may know the truth.\n\n+358.+ +Adverb clauses expressing result.+ These are introduced by the\nsubordinate conjunction _that_, as for example:\n\n They were so late _that_ I could not go.\n\n\n SUMMARY\n\n+359.+ We have then adverb clauses introduced by subordinate\nconjunctions expressing:\n\n 1. +Time.+ Answer the question _when_.\n 2. +Place.+ Answer the question _where_.\n 3. +Cause or reason.+ Answer the question _why_.\n 4. +Manner.+ Answer the question _how_.\n 5. +Comparison.+ Used to compare.\n 6. +Condition.+ Answer the question _on what condition_.\n 7. +Purpose.+ Answer the question _for what purpose_.\n 8. +Result.+ Answer the question _to what result_.\n\n\n Exercise 1\n\nIn the following sentences, mark the conjunctions and tell to what class\nthey belong; ask the question _when_, _where_, _why_, _how_, _on what\ncondition_, _for what purpose_, _to what result_. Underscore the\nsubordinate clauses. The subjects of the subordinate clauses are printed\nin italics.\n\n 1. Speech was developed that _we_ might be able to communicate with\n one another.\n 2. The International failed in the crisis because _it_ had no\n definite war program.\n 3. We will fail if _we_ have no definite program.\n 4. If _labor_ were united, we could destroy wage slavery.\n 5. When the _people_ understand, they will no longer submit.\n 6. Labor cannot win until _it_ learns solidarity.\n 7. After the terrible _war_ is over, the workers in all countries may\n come closer together.\n 8. We are convinced of the folly of nationalism since the _war_ has\n been declared.\n 9. If _we_ knew the facts we could not be misled.\n 10. Inform yourself before _you_ seek to teach others.\n 11. We must unite in order that _we_ may possess power.\n 12. It is more than the _heart_ can bear.\n 13. May you have courage to dare ere _you_ have ceased to dream.\n 14. If _we_ remain ignorant, we shall remain enslaved.\n 15. We sometimes fear to trust our own thought because _it_ is our\n own.\n 16. Though _we_ should lose the strike we will not despair.\n 17. The battle waged so fiercely that _thousands_ were slain.\n\n\n PHRASE CONJUNCTIONS\n\n+360.+ There are certain phrases which have come to be used together as\nconjunctions so commonly that we may consider them as conjunctions. They\nare:\n\n_As if_, _as though_, _but also_, _but likewise_, _so that_, _except\nthat_, _inasmuch as_, _notwithstanding that_, _in order that_, _as well\nas_, _as far as_, _so far as_, _as little as_, _provided that_, _seeing\nthat_, etc.\n\n\n Exercise 2\n\nWrite sentences using these phrase conjunctions to introduce clauses.\n\n\n NOUN CLAUSES\n\n+361.+ We have found that there are two kinds of clauses, principal\nclauses and subordinate clauses.\n\n+A principal clause is one that does not depend on any word.+\n\n+A subordinate clause is one that depends upon some word or words in the\nprincipal clause.+\n\nWe have found, also, that these principal clauses are always connected\nby co-ordinate conjunctions, for they are of equal rank and importance;\nneither is dependent upon the other.\n\nSubordinate clauses are always connected with the principal clause by a\nsubordinate conjunction. The subordinate clauses which we have been\nstudying have all been adverb clauses which are used to describe the\naction expressed in the verb contained in the principal clauses.\n\nThe subordinate clause in a sentence may also be used as a noun. When\nthe subordinate clause is used as a noun it is called a noun clause.\n\n+362.+ +A noun clause is a clause used as a noun.+\n\nA noun clause may be used in any way in which a noun is used, except as\na possessive. It may be used as a subject, an object, a predicate\ncomplement, or in apposition with a noun. These noun clauses may be\nintroduced by either relative pronouns, interrogative pronouns or by\nconjunctions. For example:\n\n I know _who_ he is.\n He asked, \"_what_ do you want?\"\n I know _where_ it is.\n\nIn the first sentence, _who he is_, is a noun clause used as the object\nof the verb _know_. It tells _what_ I know, and is the object of the\nverb _know_,--just as if I had said; _I know the facts_. In this\nsentence the noun, _facts_, is the object of the verb _know_.\n\nIn the second sentence, _He asked, \"what do you want?_\" the noun clause\n_what do you want_ is the object of the verb _asked_, and is introduced\nby the interrogative pronoun _what_.\n\nWe will study in a subsequent lesson the use of noun clauses introduced\nby relative pronouns. In this lesson we are studying the conjunctions.\n\nIn the last sentence, _I know where it is_, the noun clause _where it\nis_, is the object of the verb _know_, and is introduced by the\nconjunction _where_.\n\n+363.+ Noun clauses are introduced by the subordinate conjunctions,\n_where_, _when_, _whence_, _whither_, _whether_, _how_, _why_, and also\nby the subordinate conjunction _that_. For example:\n\n I know _where_ I can find it.\n I inquired _when_ he would arrive.\n We do not know _whence_ it cometh nor whither it goeth.\n Ask _whether_ the train has gone.\n I don't know _how_ I can find you.\n I cannot understand _why_ he does so.\n I believe _that_ he is honest.\n\nIn all of these examples the noun clauses are used as the objects of the\nverb. Noun clauses may also be used as objects of prepositions. As, for\nexample:\n\n You do not listen to _what is said_.\n He talked to me about _what had happened_.\n He told me to come to _where he was_.\n\n+364.+ Noun clauses may also be used as the subject of a sentence. As\nfor example:\n\n _That he is innocent_ is admitted by all.\n _That he was guilty_ has been proven.\n _Why he should do this_ is very strange.\n _How we are to live_ is the great problem.\n\nIn all of these sentences, the noun clause is used as the subject of the\nverb. You will note that most frequently the noun clause used as subject\nof the verb is introduced by the subordinate conjunction _that_. But\nquite often we write these sentences in a somewhat different way. For\nexample:\n\n It is admitted by all _that he is innocent_.\n It has been proven _that he was guilty_.\n\nYou will notice in these sentences we have expressed practically the\nsame thought as in the sentences where the noun clause was used as the\nsubject of the verb.\n\nBut now we have this little pronoun _it_ used as the subject, instead of\nthe clause, which is the real subject of the sentence. _It_ is simply\nused as the introductory word in the sentence. The noun clause is in\nreality the subject of the sentence.\n\n+365.+ Noun clauses may also be used as the predicate complement with a\ncopulative verb. For example:\n\n The general opinion is _that he is innocent_.\n The problem is _how we may accomplish this quickly_.\n The question was _why any one should believe such statements_.\n\nIn all of these sentences the noun clause is used as the complement of\nthe incomplete verbs _is_ and _was_, to complete the meaning, just as we\nuse a noun as the predicate complement of a copulative verb in such\nsentences as, _Socialism is a science._ _War is murder._\n\n+366.+ A noun clause may also be used in apposition to a noun to explain\nits meaning. Apposition means to place alongside of. Note in the\nfollowing sentences:\n\n The fact, _that such a law had been passed_, alters the situation.\n His motion, _that the matter should be laid on the table_, was\n adopted.\n\nIn the first sentence, the clause, _that such a law had been passed_, is\nplaced beside the noun _fact_ and explains _what_ that fact is. The\nclause, _that the matter should be laid on the table_, is in apposition\nto and explains the noun _motion_.\n\nThese noun clauses are used in apposition.\n\n\n Exercise 3\n\nComplete the following sentences by inserting the appropriate\nconjunctions and pronouns in the blank spaces:\n\n 1. Can you tell......Germany has a million fighting men?\n 2. Would you be pleased......the United States should intervene in\n Mexico?\n 3. The Mexican revolution will continue......the people possess the\n land.\n 4. No one may vote in the convention......he has credentials.\n 5. ......Debs was in Woodstock jail, he became in Socialist.\n 6. ......the treaty was signed, hostilities ceased.\n 7. We shall win......we have the courage.\n 8. ......we have lost this battle we shall not cease to struggle.\n 9. All are enslaved......one is enslaved.\n 10. Humanity will be free......labor is free.\n 11. Let us do our duty......we understand it.\n 12. Man will never reach his best......he walks side by side with\n woman.\n 13. We must struggle......we would be free.\n 14. ......we shout for peace, we support war.\n 15. All our sympathies should be with the man......toils,......we\n know......labor is the foundation of all.\n 16. ......all have the right to think and to express their thoughts\n every brain will give to all the best......it has.\n 17. ......man develops he places greater value upon his own rights.\n 18. ......man values his own rights he begins to value the rights of\n others.\n 19. ......all men give to all others the rights......they claim for\n themselves this world will be civilized.\n\n\n Exercise 4\n\nNote all the co-ordinate and subordinate conjunctions in the following\nverses from \"The Ballad of Reading Gaol.\" Underscore the subordinate\nclauses. Are they adverb or noun clauses? Do the co-ordinate\nconjunctions connect words, phrases or clauses?\n\n I know not _whether_ Laws be right,\n Or _whether_ Laws be wrong;\n All that we know who lie in gaol\n Is _that_ the wall is strong;\n _And that_ each day is like a year,\n A year whose days are long.\n\n _But_ this I know, _that_ every Law\n That men have made for Man,\n _Since_ first Man took his brother's life,\n _And_ the sad world began,\n But straws the wheat _and_ saves the chaff\n With a most evil fan.\n\n This too I know--_and_ wise it were\n _If_ each could know the same--\n _That_ every prison that men build\n Is built with bricks of shame,\n _And_ bound with bars _lest_ Christ should see\n _How_ men their brothers maim.\n\n With bars they blur the gracious moon,\n _And_ blind the goodly sun:\n _And_ they do well to hide their Hell,\n _For_ in it things are done\n That son of God _nor_ son of Man\n Ever should look upon!\n\n In Reading gaol by Reading town\n There is a pit of shame,\n _And_ in it lies a wretched man\n Eaten by teeth of flame,\n In a burning winding sheet he lies,\n _And_ his grave has got no name.\n\n _And_ there, _till_ Christ call forth the dead,\n In silence let him lie:\n No need to waste the foolish tear,\n _Or_ heave the windy sigh:\n The man had killed the thing he loved,\n _And so_ he had to die.\n\n _And_ all men kill the thing they love,\n By all let this be heard,\n Some do it with a bitter look,\n Some with a flattering word,\n The coward does it with a kiss,\n The brave man with a sword.\n\n --_Oscar Wilde_.\n\n\n\n\n SPELLING\n\n LESSON 21\n\n\nIn Lesson No. 17 we studied concerning abstract nouns derived from\nqualifying adjectives. We found that we formed these nouns expressing\nquality from adjectives that describe quality by the addition of\nsuffixes.\n\nAdjectives may likewise be formed from nouns and also from verbs by the\naddition of suffixes. There are a number of suffixes which may be used\nto form adjectives in this way; as, _al_, _ous_, _ic_, _ful_, _less_,\n_able_, _ible_, _ary_ and _ory_. Notice the following words: nation,\n_national_; peril, _perilous_; reason, _reasonable_; sense, _sensible_;\ncustom, _customary_; advise, _advisory_; hero, _heroic_; care,\n_careful_, _careless_.\n\nTo some words, more than one suffix may be added and an adjective of\ndifferent meaning formed; for example, use, _useless_, _useful_; care,\n_careless_, _careful_.\n\nMake as many adjectives as you can from the nouns and verbs given in the\nspelling lesson for this week by the addition of one or more of the\nfollowing suffixes:\n\n_Al_, _less_, _ous_, _ic_, _ful_, _able_, _ible_, _ary_, _ory_, and\n_ly_.\n\n +Monday+\n\n Accident\n Danger\n Origin\n Commend\n Element\n\n +Tuesday+\n\n Critic\n Libel\n Attain\n Revolution\n Contradict\n\n +Wednesday+\n\n Cynic\n Injury\n Respect\n Station\n Migrate\n\n +Thursday+\n\n Event\n Parent\n Order\n Virtue\n Marvel\n\n +Friday+\n\n Second\n Fashion\n Consider\n Murder\n Incident\n\n +Saturday+\n\n Constitution\n Industry\n Vibrate\n Tribute\n Compliment\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 22\n\n\nDear Comrade:\n\nWe have practically finished the study of the different parts of speech.\nWe are now in possession of a knowledge of the tools which we need to\nuse in expressing ourselves. We are ready to make practical application\nof this knowledge in writing and speaking. We will find that with our\nincreasing ability to express ourselves there comes also the power to\nthink clearly. The analysis of language has meant a growing power to\n_think_ on the part of the people.\n\nWe sometimes imagine that simplicity of language was a part of primitive\nlife, but this is not true. Simplicity of language is the product of\nhigh civilization. Primitive life was marked, not by simplicity of\nlanguage, but by the scarcity of language. They made one word stand for\nan entire sentence, and if they wished to express a little different\nmeaning, an entirely different word had to be used, as for example, in\nthe primitive language: _I said to her_, would be one word, and _I said\nto him_, would be another, entirely different, word.\n\nBut as the power of thought began to develop, we began to analyze our\nmeaning and we found that this thought was identical except the _him_\nand the _her_. So as we analyzed our thought our expression of it became\nmore simple. In most languages, the different meaning of the verb, for\nexample, is expressed by an arbitrary change in the verb form. This is\ncalled the inflection of the verb. In English we would use several words\nto express the same thing. For example, the Latin word _Fuissem_\nrequires four English words to express the same meaning; _I should have\nbeen_, we say in English. So instead of having to learn a great number\nof different changes in the verb form, we, by the use of auxiliary\nverbs, _have_, _shall_, _do_, _be_, etc., are able to express all these\nshades of thought much more simply and clearly.\n\nMost other languages also have changes for gender. Every noun has a\ngender of its own and sometimes this form gives the wrong gender to\nliving beings and attributes sex to sexless objects and the only way to\nknow the gender of the noun is simply by memory. Then the adjectives,\npossessive pronouns and the articles _a_ and _the_ have gender also and\nhave to be changed to suit the gender of the noun; this involves a great\neffort of memory. So while the English may seem somewhat involved to\nyou, it is, after all, much simpler than other languages. It has been\nfreed from many superfluous endings and unnecessary complications.\n\nTake a little time each day to read something out of the best\nliterature. The quotations given in each of these lessons are from our\nvery best writers. A study of these will be a wonderful help and\ninspiration to you and bring you in touch with some of the great\nthinkers of the revolution. They are our comrades and are putting into\nwords the thoughts and hopes and dreams of our lives.\n\n Yours for the Revolution,\n\n THE PEOPLE'S COLLEGE.\n\n\n ADJECTIVE CLAUSES\n\n+367.+ In our study of subordinate clauses, we have studied subordinate\nclauses used as adverbs and as nouns. We have found that adverb clauses\ncan be used in the same way as adverbs, to describe the time, place,\nmanner, cause, condition or purpose of the action expressed in the verb.\nWe have found, also, that a noun clause may be used in any way in which\na noun can be used, as the subject of the sentence, the object of a verb\nor preposition or as the predicate complement. But these are not the\nonly uses to which the subordinate clause may be put. Note the following\nsentences:\n\n _Wealthy_ men desire to control the education of the people.\n Men _of wealth_ desire to control the education of the people.\n Men _who are wealthy_ desire to control the education of the people.\n\nDo you see any difference in the words which are used to modify the noun\n_men_? In the first sentence, _wealthy_ is an adjective, modifying the\nnoun _men_. In the second sentence, _of wealth_ is a prepositional\nphrase, used as an adjective modifying the noun _men_. In the last\nsentence, _who are wealthy_ is a clause used in exactly the same way\nthat the adjective _wealthy_ and the adjective phrase _of wealth_ are\nused, to modify the noun _men_.\n\nWe have expressed practically the same meaning in these three ways: by a\nword; by a phrase; by a clause.\n\n+368.+ +A word used to describe and modify a noun is an adjective.+\n\n+A phrase used to describe and modify a noun is an adjective phrase.+\n\n+A clause used to describe and modify a noun is an adjective clause.+\n\nNote the difference between a phrase and a clause.\n\n+369.+ A prepositional phrase, used as an adjective, consists of the\npreposition and the noun which is its object, together with its\nmodifiers. A phrase never has either a subject or a predicate. _Who are\nwealthy_, is a clause because it does contain a subject and a predicate.\nThe pronoun _who_ is the subject in the clause, and the predicate is the\ncopulative verb _are_ with the predicate complement, the adjective\n_wealthy_.\n\n\n Exercise 1\n\nIn the following sentences change the adjective into a phrase and also\ninto a clause, if possible. For example:\n\n A _fearless_ man always defends his rights.\n A man _without fear_ always defends his rights.\n A man _who is fearless_ always defends his rights.\n\n 1. The _unemployed_ men are becoming desperate.\n 2. The _uneducated_ masses are demanding equal opportunity.\n 3. The discovery of gold was an _important_ discovery.\n 4. _Unorganized_ labor is helpless.\n 5. The revolution needs _intelligent_ rebels.\n 6. A few _wealthy_ men are striving to control education.\n 7. This will be a _progressive_ movement.\n 8. _Labor-saving_ inventions throw men out of employment.\n 9. _Scientific_ men prophesy a great advance for the mass.\n\n\n THE INTRODUCING WORD\n\n+370.+ You will notice that these adjective clauses are introduced by\nthe relative pronouns _who_, _which_ and _that_. These relative pronouns\nfulfil something of the office of a conjunction, because they are\nserving as connecting elements; they join these subordinate clauses to\nthe words which they modify. But you will note, also, that these\nrelative pronouns not only serve as connecting elements, but they also\nplay a part in the subordinate clause, as either the subject or object.\nFor example:\n\n The man who has no education is handicapped in the struggle.\n Are these the books that you ordered?\n\nIn the first sentence, _who has an education_ is an adjective clause\nmodifying the noun _man_, introduced by the relative pronoun _who_,\nwhich is also the subject of the verb _has_.\n\nIn the second sentence, _that you ordered_ is an adjective clause,\nmodifying the noun _books_, introduced by the relative pronoun _that_,\nwhich is also the object of the verb _ordered_.\n\n+371.+ There is no need to be confused in this matter of clauses. If the\nclause is used as a noun, either as the subject or the object or in any\nother way in which a noun can be used, it is a noun clause. If it is\nused as an adverb and will answer any of the questions _why_, _when_,\n_where_, or _how_, etc., it is an adverb clause. If it is used as an\nadjective,--if it modifies a noun or pronoun,--it is an adjective\nclause.\n\nYou will note that the only way in which a noun is used that does not\nhave its corresponding clause is as a possessive. We do not have\npossessive clauses. The clause used as an adjective always modifies a\nnoun or pronoun.\n\n+372.+ +An adjective clause is a clause used as an adjective and hence\nalways modifies a noun or pronoun.+\n\nAn adjective clause may be introduced by the relative pronouns, _who_,\n_which_ or _that_. The use of this clause is a great help to us in the\nexpression of our ideas, for it enables us to combine several sentences\ncontaining related thoughts into one sentence so we have it all\npresented to the mind at once.\n\n\n Exercise 2\n\nIn the following sentences, note which are the noun clauses and which\nare the adjective clauses and which are the adverb clauses. The verb in\nthe subordinate clause is in italics.\n\n 1. Life is what we _make_ it.\n 2. We acquire the strength that we _overcome_.\n 3. While he _slept_ the enemy came.\n 4. All that he _does_ is to distribute what others _produce_.\n 5. When faith _is lost_, when honor _dies_, the man is dead.\n 6. Thrice is he armed who _hath_ his quarrel just; he is naked though\n he _be locked_ up in steel whose conscience with injustice is\n _corrupted_.\n 7. When strength and justice _are_ true yoke fellows, where can we\n find a mightier pair than they?\n 8. You will gain a good reputation if you _endeavor_ to be what you\n _desire_ to appear.\n 9. Live as though life _were_ earnest and life will be so.\n 10. He that _loveth_ makes his own the grandeur that he _loves_.\n 11. Who _does_ the best his circumstance _allows_ does well; angels\n could do no more.\n 12. He is not worthy of the honeycomb that _shuns_ the hive because\n the bees _have_ stings.\n 13. We always may be what we _might have been_.\n 14. Rich gifts wax poor when givers _prove_ unkind.\n 15. Let me make the songs of the people and I care not who _makes_ the\n laws.\n 16. Attention is the stuff that memory _is made_ of.\n 17. A great writer has said that grace _is_ beauty in action; I say\n that justice _is_ truth in action.\n 18. If we do not _plant_ knowledge when young it will give us no shade\n when we _are_ old.\n 19. You can no more exercise your reason if you _live_ in constant\n dread of laughter than you _can enjoy_ your life if you _live_ in\n constant dread and terror of death.\n\n\n WHICH RELATIVE PRONOUN TO USE\n\n+373.+ We are sometimes confused as to which relative pronoun to use in\nintroducing an adjective clause. We hesitate as to whether we should use\n_that_ or _who_ or _which_. Remember that _who_ always refers to\n_persons_, _which_ refers to _animals_ or _things_, and _that_ may refer\nto either _persons_, _animals_ or _things_.\n\nSo when referring to a _person_, we may use either _who_ or _that_, and\nwhen referring to _animals_ or _things_, we may use either _which_ or\n_that_. As, for example, we may say, either, _The man who was here\nyesterday came back today_, or _The man that was here yesterday came\nback today_. Either is correct, for _who_ and _that_ both refer to\npersons.\n\n+374.+ We may make a little distinction in the use of _who_ and _that_\nwhen referring to _persons_, however. A clause introduced by _that_ is\nusually a restrictive clause. It limits or restricts the meaning of the\nnoun which it modifies. When you say, _The man that was here yesterday_,\nyou mean _that_ man and no other, limiting your meaning to that\nparticular man. On the other hand, when you say, _The man who was here\nyesterday_, there is no restriction or limitation expressed in the use\nof the clause, but it is merely a descriptive clause, adding a new fact\nto our knowledge concerning that particular man.\n\nThe same is true when we are speaking of _things_ using either _that_ or\n_which_. The clause introduced by _which_ is presumably a descriptive\nclause. We do, however, often use _who_ or _which_ when the sense of the\nclause is restrictive, but we should never use _that_ to introduce an\nadjective clause, unless the sense is restrictive. When in your\nsentences you can use, instead of the relative pronoun _who_ or _which_,\nthe conjunction _and_, you can know that the use of the pronoun _who_ or\n_which_ is correct. As, for example:\n\n I have read the book, _which_ I found very interesting.\n\nYou could say instead:\n\n I have read the book _and_ I found it very interesting.\n\nThis would express the same meaning. But if you say: _I have read the\nbook that I found very interesting_, you mean that you limit your idea\nto this particular book.\n\n+375.+ We do not always observe these niceties in our spoken and written\nspeech, but it is interesting to know the shades of thought and meaning\nwhich you can express by the proper use of the language. The man who\nruns an engine and learns to know and love his machine almost as though\nit were a human being, can easily recognize the slightest change in the\naction of his machine. His ear catches the least difference in the sound\nof the running of the machine, a difference which we, who do not know\nand love the machine, would never notice.\n\nSo it is in language. Once we have sensed its beauty and its wondrous\npower of expression, we notice all these slight differences and shades\nof meaning which may be expressed by the use of words. In just the same\nmanner the musician catches the undertones and overtones of the music,\nwhich we, who possess an uneducated ear, cannot know; and the artist\nalso has a wondrous range of color, while we, who are not sensitive to\ncolor, know only a few of the primal colors.\n\n\n ADJECTIVE CLAUSES WITH CONJUNCTIONS\n\n+376.+ The adjective clauses which we have been studying so far have\nbeen introduced by relative pronouns. Adjective clauses may also be\nintroduced by conjunctions, such as, _where_, _when_, _whence_, or\n_why_. As, for example:\n\n Antwerp is the place where a terrible battle was fought.\n No man knows the hour when opportunity will be his.\n Each group has a different reason why this world-war was precipitated.\n\nNote in these sentences the clauses, _where a terrible battle was\nfought_, _when opportunity will be his_, _why this world-war was\nprecipitated_, are all adjective clauses modifying the nouns _place_,\n_hour_ and _reason_, and are introduced by the conjunctions _where_,\n_when_, and _why_. These are adjective clauses because they modify, by\neither limiting or describing, the nouns with which they are used. You\nwill note that we could omit the nouns in the first two of these\nsentences and these clauses would become noun clauses, for they would be\nused in the place of a noun. As, for example:\n\n Antwerp is where a terrible battle was fought.\n No man knows when opportunity will be his.\n\n+377.+ We determine whether a clause is an adjective or an adverb or a\nnoun clause just as we determine whether a word is an adjective, adverb\nor noun, by the work which it does in a sentence. Noun clauses are used\nin the place of a noun; adverb clauses modify verbs, adjectives, and\nadverbs; adjective clauses modify nouns and pronouns.\n\n\n THE LITTLE WORD \"AS\"\n\n+378.+ Adjective clauses may also be introduced by _as_. _As_ is a very\nconvenient word and may be used in several different ways; sometimes as\nan adverb, sometimes as a conjunction; and it may also be used as a\nrelative pronoun after _such_, _same_ and _many_. For example:\n\n Such books _as_ you should read are listed here.\n No such person _as_ he ever came here.\n We are facing the same crisis _as_ our comrades faced.\n This is the same _as_ you gave before.\n He has made as many mistakes _as_ you have.\n\nIn these sentences _as_ is really used as a relative pronoun, connecting\nthese adjective clauses to the words which they modify. _As_ may also be\nused as an adverb. _I am as tall as you are._\n\nHere the first _as_ modifies _tall_ and is used as an adverb; the second\n_as_ is a conjunction connecting the subordinate clause _you are_, with\nthe principal clause. Note that in making comparisons, _as_ is always\nused when the comparison is equal, _so_ when it is unequal, thus:\n\n I am _as_ tall as you are.\n She is not _so_ tall as you are.\n\nWe have found that _as_ is also used as a conjunction to introduce an\nadverb clause. For example:\n\n She is as beautiful _as_ she is good.\n\nThe clause, _as she is good_, is an adverb clause, modifying the\nadjective _beautiful_. In the sentence, _Do as I say_, _as I say_ is an\nadverb clause of manner, modifying the verb _do_.\n\n\n CONNECTIVE WORDS\n\n+379.+ Let us not be confused in this matter of connectives. There are\njust four classes of connective words:\n\n 1. +Copulative verbs.+\n 2. +Relative pronouns.+\n 3. +Prepositions.+\n 4. +Conjunctions.+\n\n+380.+ The copulative verb is not a pure connective, for it serves\nanother purpose in the sentence. For example, in the sentence, _The book\nis interesting_, the copulative verb _is_ connects the adjective\n_interesting_ with the noun _book_, which it modifies; but it also is\nthe asserting word in the sentence. So it fulfils a double function. It\nis an asserting word and also a connective word.\n\n+381.+ The relative pronoun also is not a pure connective, for it serves\ntwo purposes in the sentence. It not only connects the clause which it\nintroduces, with the word which it modifies, but it also serves as\neither the subject or object in the clause. For example: _The man who\nwas here has gone_. The clause, _who was here_, is introduced by the\nrelative pronoun _who_, which connects that clause with the noun _man_,\nwhich the clause modifies. _Who_ also serves as the subject of the verb\n_was_.\n\nIn the sentence, _The men whom we seek have gone_, the clause, _whom we\nseek_, is introduced by the relative pronoun _whom_, which connects the\nclause with the word _men_, which it modifies. _Whom_ also serves as the\nobject of the verb in the clause, the verb _seek_.\n\n+382.+ A preposition is not a pure connective, since it serves a double\nfunction. It shows the relation of its object to the rest of the\nsentence and also governs the form of its object. As, for example, in\nthe sentence: _The man before me is not the culprit_, the preposition\n_before_ connects its object _me_ with the noun _man_, which the\nprepositional phrase modifies, showing the relation between them; and it\ngoverns the form of its object, for the pronoun following a preposition\nmust be used in the _object_ form.\n\n+383.+ Even co-ordinate conjunctions can scarcely be considered pure\nconnectives unless it be the co-ordinate conjunction _and_. Co-ordinate\nconjunctions such as _but_, _yet_, _still_, _however_, etc., not only\nconnect words, phrases and clauses of equal rank, but in addition to\nconnecting the words and expressions they also indicate that they are\nopposite in thought.\n\n+384.+ Co-ordinate conjunctions like _therefore_, _hence_, _then_, etc.,\nconnect words, phrases and clauses of equal rank, and also introduce a\n_reason_ or _cause_. Co-ordinate conjunctions like _or_, _either_,\n_nor_, _neither_, _whether_, etc., connect words, phrases and clauses of\nequal rank, and also express the choice of an alternative. Thus these\nco-ordinate conjunctions can scarcely be considered as pure connectives.\n\n+385.+ Subordinate conjunctions are most frequently used to introduce\nadverb clauses and have an adverbial meaning. They express, as do\nadverbs, _place_, _time_, _manner_, _cause_, _reason_, _purpose_,\n_condition_ or _result_. Some authorities indicate this double function\nby calling such words as these conjunctive adverbs, because, even when\nthey are used as conjunctions, they retain some of their adverbial\nforce.\n\nBut according to our rule that every word in the sentence is classified\naccording to the function which it performs in that sentence, all words\nthat perform the function of a conjunction are called conjunctions,\nalthough we understand that these conjunctions which introduce dependent\nclauses do still retain some of their adverbial meaning.\n\n\n Exercise 3\n\nIn the following sentences the connectives are in italics. Determine\nwhether they are copulative verbs, relative pronouns, prepositions,\nco-ordinate conjunctions or subordinate conjunctions.\n\n 1. They _are_ slaves _who_ dare not be _in_ the right _with_ two\n _or_ three.\n 2. _In_ the twentieth century war _will be_ dead, dogmas _will be_\n dead, _but_ man will live.\n 3. The abuse _of_ free speech dies _in_ a day, _but_ its denial slays\n the life _of_ the people _and_ entombs the race.\n 4. Liberty _for_ the few _is_ not liberty.\n 5. Liberty _for_ me _and_ slavery _for_ you means slavery _for_ both.\n 6. The greatest thing _in_ the world _is for_ a man to know _that_ he\n _is_ his own.\n 7. Nothing can work me damage _except_ myself.\n 8. He _that_ loveth maketh his own the grandeur _which_ he loves.\n 9. My life _is_ not an apology, _but_ a life.\n 10. I cannot consent to pay _for_ a privilege _where_ I have intrinsic\n right.\n 11. It _is_ difficult to free fools _from_ the chains _which_ they\n revere.\n 12. Desire nothing _for_ yourself _which_ you do not desire _for\n others_.\n 13. All our liberties _are_ due _to_ men _who_, _when_ their\n conscience compelled them, have broken the laws _of_ the land.\n 14. \"It takes great strength to live _where_ you belong,\n _When_ other people think _that_ you _are_ wrong.\"\n 15. _If_ the truth shall make you free, ye _shall be_ free indeed.\n 16. He _is_ true _to_ God _who is_ true _to_ man.\n\n\n Exercise 4\n\nIn the following sentences underscore all the connectives--copulative\nverbs, prepositions, relative pronouns, co-ordinate and subordinate\nconjunctions.\n\n\"There was a bird's egg once, picked up by chance upon the ground, and\nthose who found it bore it home and placed it under a barn-yard fowl.\nAnd in time the chick bred out, and those who had found it chained it by\nthe leg to a log lest it should stray and be lost. And by and by they\ngathered round it, and speculated as to what the bird might be.\n\nOne said, \"It is surely a waterfowl, a duck, or it may be a goose; if we\ntook it to the water it would swim and gabble.\" But another said, \"It\nhas no webs to its feet; it is a barn-yard fowl; if you should let it\nloose it will scratch and cackle with the others on the dungheap.\" But a\nthird speculated, \"Look now at its curved beak; no doubt it is a parrot,\nand can crack nuts.\"\n\nBut a fourth said, \"No, but look at its wings; perhaps it is a bird of\ngreat flight.\" But several cried, \"Nonsense! No one has ever seen it\nfly! Why should it fly? Can you suppose that a thing can do a thing\nwhich no one has ever seen it do?\" And the bird, with its leg chained\nclose to the log, preened its wings.\n\nSo they say about it, speculating and discussing it: and one said this,\nand another that.\n\nAnd all the while, as they talked, the bird sat motionless, \"Suppose we\nlet the creature loose to see what it will do?\"--and the bird shivered.\nBut the others cried, \"It is too valuable; it might get lost. If it were\nto try to fly it might fall down and break its neck.\" And the bird, with\nits foot chained to the log, sat looking upward into the clear sky; the\nsky, in which it had never been--for the bird--the bird, knew what it\nwould do--because it was an eaglet!\"\n\n --_Olive Schreiner_.\n\n\n Exercise 5\n\nThese stirring lines are taken from Arturo Giovannitti's \"Arrows in the\nGale\" and are a part of the poem \"The Sermon on the Common.\" Note the\nuse of the conjunctions. Mark all of the clauses.\n\n Ye are the power of the earth, the foundations of society, the\n thinkers and the doers of all things good and all things fair and\n useful, the makers and dispensers of all the bounties and the joys and\n the happiness of the world, and if ye fold your mighty arms, all the\n life of the world stands still and death hovers on the darkened abodes\n of man.\n\n Ye are the light of the world. There was darkness in all the ages when\n the torch of your will did not blaze forth, and the past and the\n future are full of the radiance that cometh from your eyes.\n\n Ye are eternal, even as your father, labor, is eternal, and no power\n of time and dissolution can prevail against you.\n\n Ages have come and gone, kingdoms and powers and dynasties have risen\n and fallen, old glories and ancient wisdoms have been turned into\n dust, heroes and sages have been forgotten and many a mighty and\n fearsome god has been hurled into the lightless chasms of oblivion.\n\n But ye, Plebs, Populace, People, Rabble, Mob, Proletariat, live and\n abide forever.\n\n Therefore I say unto you, banish fear from your hearts, dispel the\n mists of ignorance from your minds, arm your yearning with your\n strength, your vision with your will, and open your eyes and behold.\n\n Do not moan, do not submit, do not kneel, do not pray, do not wait.\n\n Think, dare, do, rebel, fight--ARISE!\n\n It is not true that ye are condemned to serve and to suffer in shame\n forever.\n\n It is not true that injustice, iniquity, hunger, misery, abjection,\n depravity, hatred, theft, murder and fratricide are eternal.\n\n There is no destiny that the will of man cannot break.\n\n There are no chains of iron that other iron cannot destroy.\n\n There is nothing that the power of your arms, lighted by the power of\n your mind, cannot transform and reconstruct and remake.\n\n Arise, then, ye men of the plow and the hammer, the helm and the\n lever, and send forth to the four winds of the earth your new\n proclamation of freedom which shall be the last and shall abide\n forevermore.\n\n Through you, through your united, almighty strength, order shall\n become equity, law shall become liberty, duty shall become love and\n religion shall become truth.\n\n Through you, the man-beast shall die and the man be born.\n\n Through you, the dark and bloody chronicles of the brute shall cease\n and the story of man shall begin.\n\n Through you, by the power of your brain and hand,\n\n All the predictions of the prophets,\n\n All the wisdom of the sages,\n\n All the dreams of the poets,\n\n All the hopes of the heroes,\n\n All the visions of the martyrs,\n\n All the prayers of the saints,\n\n All the crushed, tortured, strangled, maimed and murdered ideals of\n the ages, and all the glorious destinies of mankind shall become a\n triumphant and everlasting reality in the name of labor and bread and\n love, the great threefold truth forever.\n\n And lo and behold, my brothers, this shall be called the revolution.\n\n\n\n\n SPELLING\n\n LESSON 22\n\n\nIn our study of the spelling of English words we have found that there\nare not many rules that apply. In fact, the only way to learn to spell\ncorrectly is by sheer dint of memory.\n\nIn last week's lesson we found that a number of adjectives can be formed\nfrom nouns or verbs by the addition of _able_ or _ible_, but we find it\ndifficult to determine whether to add _able_ or _ible_. The sound is\npractically the same and we are confused as to whether we should use _a_\nor _i_. There is no rule which applies in this case and there is nothing\nto do but to master the spelling of these words by memory. These are\nwords which we use a great deal and which are very helpful members of\nour working vocabulary.\n\nOur list of words in this week's lesson contain some of the most common\nwords which we use ending in _ible_ or _able_. The words for Monday,\nTuesday, and Wednesday all end in _able_; the words for Thursday,\nFriday, and Saturday will end in _ible_. Notice them carefully and get\nfixed firmly in mind the correct spelling. Notice also that most of\nthese adjectives can be changed into adverbs by changing _ble_ to _bly_.\nSo when you have added these adjectives to your vocabulary, you have\nalso added the adverbs as well.\n\n +Monday+\n\n Probable\n Capable\n Usable\n Considerable\n Respectable\n\n +Tuesday+\n\n Durable\n Salable\n Advisable\n Available\n Equitable\n\n +Wednesday+\n\n Tolerable\n Profitable\n Remarkable\n Valuable\n Comfortable\n\n +Thursday+\n\n Possible\n Horrible\n Plausible\n Intelligible\n Terrible\n\n +Friday+\n\n Credible\n Visible\n Infallible\n Responsible\n Sensible\n\n +Saturday+\n\n Forcible\n Permissible\n Feasible\n Corruptible\n Eligible\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 23\n\n\nDear Comrade:\n\nIn this lesson we are taking up the study of interjections.\nInterjections are the language of emotion. This was probably the\nearliest form of speech. You notice that children use these exclamations\noften, and the sounds which are imitations of the noises about them.\nThis language belongs also to the savage, whose peculiar and expressive\ngrunts contain whole areas of condensed thought. As we progress from\nfeeling to thinking, the use of the interjection diminishes.\n\nYou will not find interjections used in a book on mathematics or\nphysical science or history. To attempt to read one of these books may\nmake you use interjections and express your emotion in violent language,\nbut you will not find interjections in these books. These books of\nscience are books that express thought and not feeling. But if you turn\nto fiction and to oratory you will find the interjection used freely,\nfor these are the books which treat of the human emotions and feelings.\nEspecially in poetry will you find the interjection used, for poetry is\nthe language of feeling and the interjection is an important part of the\npoet's stock in trade.\n\nIn conversation, these exclamatory words are very useful. They fill the\ngaps in our conversation and they help to put the listener and the\nspeaker in touch with one another. They are usually accompanied by a\ngesture, which adds force to the word. The tone of the voice in which\nthey are expressed also means a great deal. You can say, Oh! in half a\ndozen different ways; you may express surprise, wonder, joy, sorrow,\npain, or disgust. A great many different and widely separated feelings\ncan be expressed simply by the tone in which you use the exclamatory\nwords. Some one has said that these words grease the wheels of talk.\nThey serve to help the timid, to give time to the unready and to keep up\na pleasant semblance of familiarity.\n\nWhen we use them in the stress of emotion to express deep feeling, their\nuse is perfectly justified. But one author has called these words \"the\nmiserable refuge of the speechless.\" We use them many times because we\nhave no words with which to express ourselves. This use is unjustified.\nBe careful that you do not use them in this way. It has been said that\nthe degree of a man's civilization can be pretty fairly judged by the\nexpletives which he uses. Do not sprinkle your conversation with\ninterjections and even stronger words because you are at a loss for\nother words.\n\nThere is a rich mine of words at your disposal. Do not be satisfied with\nbits of glass that have no value, when the rich diamonds of real\nexpression can be yours for just a little digging. Save your emotional\nlanguage for the time when you really need it to express deep emotion.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n INTERJECTIONS\n\n+386.+ We have been studying the parts of speech,--the elements of which\nsentences are composed. But we have another class of words which we call\nparts of speech because they are spoken and written as words, but which\nare really not parts of speech in the same sense as the words which we\nhave been discussing. These are words which we call interjections.\n\nInterjection means, literally, thrown between, from _jecto_, to throw,\nand _inter_, between. So interjections do not enter into the\nconstruction of sentences but are only thrown in between. Every word\nthat is really a part of the sentence is either a noun, a pronoun, a\nverb, an adjective, an adverb, a preposition or a conjunction.\n\nThere are words, however, that we use with sentences which do not enter\ninto the construction. For example, you say:\n\n Oh! I am wounded.\n Aha! I have conquered.\n Alas! He came too late.\n\n+387.+ Words which we use in these sentences, like, _oh_, _aha_, _alas_,\nare used to express the emotion which you feel in making the statement.\nYour _Oh!_ in a sentence like: _Oh! I am wounded_, would probably sound\nvery much like a groan. But your _Aha!_ in the, _Aha! I have conquered_,\nwill sound like a shout of victory, and your _Alas!_ in the sentence,\n_Alas! He came too late_, will express grief or regret over the fact\nthat he came too late.\n\nThese words do not assert anything and very much of the meaning which we\ngive them must come from the tone in which they are uttered. Every one,\nupon hearing them, knows at once whether they express grief or delight.\n\n+388.+ +An interjection is an exclamatory word or phrase used to express\nfeeling or to imitate some sound.+\n\n+389.+ Interjections may be divided into four classes:\n\n1. +Words which we use instead of an assertion to express feeling of\nvarious kinds+, as:\n\n (a) Surprise or wonder; as, _Oh_, _Aha_, _What_.\n (b) Pleasure, joy, or exaltation; as, _Hurrah_, _Ha, Ha_.\n (c) Pain, sadness or sorrow; as, _Alas_, _Alack_.\n (d) Contempt or disgust; as _Fie_, _Fudge_, _Ugh_, _Pshaw_.\n\n2. +Words used instead of a question+; as, _Eh?_ _Hey?_\n\n3. +Words used instead of a command+; as:\n\n (a) To call attention; as, _Hello_, _Ahoy_, _Whoa_.\n (b) To express silence; as, _Shh_, _Hush_, _Hist_.\n (c) To direct or drive out, etc., as, _Whoa_, _Gee_, _Haw_, _Scat_.\n\n4. +Words used to imitate sounds made by animals, machines, etc.+, as,\n_Bow-wow_, _Ding-dong_, _Bang_, _Rub-a-dub_.\n\nWhen we wish to imitate noises or sounds made by animals, machines,\netc., in writing, we spell out the words as nearly as we can, just as we\nwrite _ding-dong_ to represent the sound of the bell or _tick-tock_ to\nindicate the ticking of a clock.\n\nNote that a number of our verbs and nouns have been formed from\nimitating the sound which these nouns or verbs describe or express, as\nfor instance, _crash_, _roar_, _buzz_, _hush_, _groan_, _bang_, _puff_,\netc.\n\n\n Exercise 1\n\nMark the interjections in the following sentences. Which express\nsurprise? Which joy? Which sorrow? Which disgust?\n\n 1. Alas! We shall never meet again.\n 2. Bravo! You have done well.\n 3. Pshaw! Is that the best you can do?\n 4. Ship ahoy! All hands on deck.\n 5. Hello! When did you come?\n 6. Hurrah! We have won the victory.\n 7. Alas, alack! Those days will never come again.\n 8. Hist! You must be as still as mice.\n\n\n Exercise 2\n\nWrite sentences using an interjection to express: 1. Joy. 2. Surprise.\n3. Pain. 4. Sorrow. 5. Disgust. 6. To ask a question. 7. To call\nattention. 8. To silence. 9. To direct. 10. To imitate the sound made by\nan animal. 11. By a machine.\n\n\n EXCLAMATORY WORDS\n\n+390.+ Interjections express only emotion or feeling. They do not\nexpress ideas. However, we have a number of words which are used\nsomewhat as interjections are used, which we may class as exclamatory\nwords, but they express more than interjections, for they express ideas\nas well as emotions; but, like interjections, they are used\nindependently and have no part in the construction of the sentence.\n\n+391.+ Many ordinary words and phrases are used in this way as\nexclamations. When they are so used they have no place in the\nconstruction of the sentence; that is, they do not depend upon the\nsentence in which they are used, in any way. A noun used in this way is\nnot used as the subject or the object, but simply as an exclamation.\n\nFor example; the noun _nonsense_ may be used as an interjection, as in\nthe sentence; _Nonsense! I do not believe a word of it_. In this\nsentence, _nonsense_ is a noun used as an interjection and plays no part\nin the sentence, either as subject or object, but is an independent\nconstruction. There are a number of words used in this way:\n\n1. Nouns and pronouns, as _fire_, _mercy_, _shame_, _nonsense_, _the\nidea_, _what_.\n\n2. Verbs like, _help_, _look_, _see_, _listen_, _hark_, _behold_,\n_begone_.\n\n3. Adjectives like, _good_, _well_, _brave_, _welcome_, _strange_.\n\n4. Adverbs like, _out_, _indeed_, _how_, _why_, _back_, _forward_.\n\n5. Prepositions like, _on_, _up_, _down_.\n\n6. Phrases like, _Oh dear_, _dear me_, _good bye_.\n\nWords and phrases such as these, used as exclamations, are not true\ninterjections, for they express a little more than feeling. They express\nan idea which, in our haste, we do not completely express. The other\nwords necessary to the expression of the idea are omitted because of the\nstress of emotion. For example:\n\n Silence! I will hear no more.\n\nIn this sentence it is understood that we mean, _Let us have silence, I\nwill hear no more_. But in the stress of our emotion, we have omitted\nthe words, _Let us have_.\n\nIf we say, _Good! that will do splendidly_, you know that we mean, _That\nis good_, we have simply omitted _That is_, which is necessary to\ncomplete the sentence. Sometimes when we are greatly excited we abandon\nour sentence construction altogether and use only the most important\nwords. For example:\n\n A sail! a sail!\n\nThis is not a sentence, for it does not contain a verb, yet we know that\nwhat was meant was, _I see a sail, I see a sail_.\n\n\n Exercise 3\n\nWrite sentences using the words given in the foregoing list as\nexclamatory words, and add as many more to the list as you can think of.\n\n\n YES AND NO\n\n+392.+ The words _yes_ and _no_, which we use in reply to questions were\noriginally adverbs, but we no longer use them as adverbs. We no longer\ncombine them with other words as modifying or limiting words, but use\nthem independently. They are in themselves complete answers. Thus, if\nyou ask me the question, _Will you come?_ I may say _Yes_, meaning, _I\nwill come_; or, _No_, meaning, _I will not come_.\n\nThe responsives _yes_ and _no_ thus stand for whole sentences, so they\nare really independent words. We may use them in connection with other\nsentences. For example; I may say, _Yes, I will come_, or _No, I will\nnot come_. Used in this way, they still retain an independent\nconstruction in the sentence. We call them responsives because they are\nused in response to questions.\n\n\n OTHER INDEPENDENT EXPRESSIONS\n\n+393.+ Other words may be used in an independent construction in\nsentences, without depending upon the sentence in which they are used or\nwithout having the sentences depend upon them, such as:\n\n1. +A word used in address.+ For example:\n\n Mr. President, I move that a committee be now appointed.\n Fellow Workers, I rise to address you.\n\nIn these sentences, _Mr. President_ and _Fellow Workers_ are nouns used\nindependently; that is, they are neither the subject of the sentence nor\nused as object or predicate complement. They are independent of all\nother words in the sentence.\n\nThe most common use of words used independently in direct address occurs\nwith imperative sentences. For example:\n\n _Comrades_, rouse yourselves.\n _Men_, strike for freedom.\n\n2. +Exclamatory expressions.+ These are nouns used in the manner in\nwhich we have already discussed, as in the sentence:\n\n _Nonsense!_ I do not believe a word of it.\n Alas! poor _Yorick_! I knew him well.\n\n3. +Words and phrases used parenthetically+, as for example:\n\n _By the way_, I met a friend of yours today.\n We cannot, _however_, join you at once.\n He called, _it seems_, while we were gone.\n\nIn these sentences such words as, _however_, and such phrases as, _by\nthe way_, and, _it seems_, are used independently,--in parenthesis, as\nit were; that is, they are just thrown into the sentences in such a way\nthat they do not modify or depend upon any other word in the sentence.\nWhen we analyze our sentences, these independent words are not\nconsidered as elements of the sentences in which they are used. It is\nsufficient to say that they are independent words.\n\n4. +Conjunctions used as introductory words.+ We have noted the use of\nconjunctions like the co-ordinates _and_, _but_, etc., and the\nsubordinates _because_, _in order that_, _so_, _for_, _wherefore_,\n_how_, _whether_, etc., which are used to introduce sentences and\nconnect them in thought with sentences and paragraphs which have gone\nbefore.\n\n\n INTRODUCTORY WORDS\n\n+394.+ +We have a number of words which we use to introduce our\nsentences.+ They are such words as, _so_, _well_ and _why_. These are\nordinarily adverbs, but when they are used merely to introduce a\nsentence they retain little of their adverbial force. For example:\n\n _So_, that is your only excuse.\n _Well_, I cannot understand why you should accept it.\n _Why_, that is no reason at all.\n\nIn these sentences, _so_, _well_ and _why_ do not modify any of the\nwords in the sentences, but are used merely to introduce the sentences.\nThey serve in a measure to connect them with something which has gone\nbefore.\n\n+395.+ +The adverb _there_ is also used as an introductory word.+ When\nit is used in this manner, it loses its adverbial force. _There_, as\nordinarily used, is an adverb of place, but it is often used to\nintroduce a sentence. For example: _There is some mistake about it_. In\nthis sentence _there_ is not used as an adverb, but it is used simply as\nan introductory word. It is used to introduce a sentence in which the\nverb comes before the real subject. _Mistake_ is the real subject of the\nverb is, and _there_ is used simply as the introductory word.\n\n+396.+ +The indefinite pronoun _it_ is also used as an introductory\nword+, to introduce a sentence in much the same manner as _there_. The\nreal subject of the verb occurs later in the sentence. For example:\n\n It is best to know the truth.\n\nThis could be written, _To know the truth is best_, and the entire\nmeaning of the sentence would be conveyed.\n\n+397.+ +Adverbs of mode.+ You remember in our study of adverbs, we had\ncertain adverbs which were called adverbs of mode. These are used to\nmodify the entire sentence. They express the feeling in which the entire\nsentence is uttered. Adverbs of mode may be regarded also as independent\nwords. They are such words as, _indeed_, _surely_, _certainly_,\n_perhaps_, etc. For example:\n\n _Indeed_, I cannot tell you now.\n _Surely_, I will comply with your request.\n _Perhaps_ it may be true.\n I _certainly_ hope to do so before long.\n\n\n Exercise 4\n\nNote in the following sentences the words which are pure interjections,\nand those which are other parts of speech used as exclamatory words.\nMark those which are used in direct address, those which are used\nparenthetically, and those which are used as mere introductory words.\n\n 1. Oh, it seems impossible to believe it.\n 2. Surely, you will accept my word.\n 3. Nonsense, there is not the least truth in the story.\n 4. It will be impossible for us to join.\n 5. Therefore we urge you to join in this campaign.\n 6. There is only one solution to the problem.\n 7. It is difficult to discover the true facts.\n 8. Well, I have done my best to persuade you.\n 9. Mr. Chairman, I rise to a point of order.\n 10. Comrades, come and stand for your rights.\n 11. Yes, I have studied that philosophy.\n 12. Enough! we have been enslaved too long.\n 13. Hark! we hear the tramp of the army of labor.\n 14. Alas! that any should refuse to join in this battle.\n 15. You have not, it seems, understood the issue.\n 16. Indeed, solidarity is our only hope.\n 17. Br-r-r-r-r-r-r, thus whirl the machines that grind our children's\n lives.\n 18. Hush! Over the crash of the cannon sounds the wail of Europe's\n women and children.\n\n\n EXPLANATORY WORDS\n\n+398.+ We sometimes use words which do not belong in the construction of\na sentence to explain other words in the sentence. For example:\n\n We, _the undersigned_, subscribe as follows:\n Helen Keller, _the most wonderful woman of this age_, champions the\n cause of the working class.\n\nIn the first sentence, the words, _the undersigned_, are added to the\npronoun _we_ to explain who _we_ means. In the second sentence, the\nwords, _the most wonderful woman of this age_, are added to explain who\nHelen Keller is. Words added to other words in this way are called\nexplanatory words. They are placed in apposition to the noun which they\nexplain. Apposition means _by the side of_, or _in position near_. You\nremember that in clauses we found that a clause may be placed in\napposition with a noun to explain the meaning of that noun. For example:\n\n There is an old saying, _in union there is strength_.\n\nThese words in apposition may themselves be modified or limited by other\nwords or phrases or clauses. For example:\n\n Helen Keller, the most wonderful woman of this age, champions the\n cause of the working class.\n\nIn this sentence, _woman_ is the noun placed in apposition to the\nparticular name, Helen Keller, and the noun _woman_ is modified by the\nadjectives _the_, and _wonderful_, and by the phrase _of this age_.\n\nSometimes a second explanatory word is placed in apposition to the first\none. This is quite often the case in legal documents or resolutions,\nwhere the language is quite formal. For example:\n\n We, the undersigned, _members of Local No. 38_, do hereby move, etc.\n I, John Smith, _Notary Public_, in and for the county of Clay, etc.\n\nThese words, _undersigned_ and _members_, are both placed in apposition\nto the pronoun _We_, explaining to whom that pronoun refers.\n\n\n Exercise 5\n\nIn the following sentences note the explanatory words and their\nmodifiers:\n\n 1. Wendell Phillips, the great abolitionist, was a man of genius.\n 2. Buckle, the historian, writes from the view point of the\n materialistic conception of history.\n 3. Giovannitti, the poet, wrote \"Arrows in the Gale.\"\n 4. Helen Keller, champion of the working class, wrote the introduction\n to this book.\n 5. We, the workers of the world, will some day claim our own.\n 6. He was found guilty of treason, a crime punishable by death.\n 7. Ferrer, the martyr of the twentieth century, was put to death by\n the Spanish government.\n 8. Jaures, the great French socialist, was the first martyr to peace.\n 9. But ye, Plebs, Populace, People, Rabble, Mob, Proletariat, live and\n abide forever.\n 10. Ye are eternal, even as your father, labor, is eternal.\n 11. This document, the Constitution of the United States, hinders the\n progress of the people.\n 12. The memory of Guttenberg, the inventor of the printing press,\n should be reverenced by every class-conscious worker.\n 13. Wallace, the scientist and author, was co-discoverer with Darwin\n of the theory of evolution.\n 14. Karl Marx, the thinker, applied this theory to social forces.\n 15. Do you understand the three basic principles of Socialism--the\n class struggle, economic determinism and surplus value?\n\n\n Exercise 6\n\nRead the following list of words and note the ideas which they suggest\nto you, then make sentences containing these words, _modified by a word\nor group of words in apposition_, which explain more fully these words.\n\n Law, martyr, society, education, inventor, commander, freedom, Eugene\n V. Debs, Karl Marx, Kaiser Wilhelm, The Balkan, Lawrence, Colorado,\n Calumet.\n\n\n ABSOLUTE CONSTRUCTION\n\n+399.+ We have found that every word in a sentence bears some relation\nto every other word, except these words which we have been studying,\nwhich we use independently. These explanatory words which we have just\nbeen studying are not used independently, but do in a sense modify the\nnoun with which they are placed in apposition. Sometimes we place a noun\nor a pronoun and its modifiers alongside the whole sentence and it does\nnot really modify any part of the sentence, but modifies the whole\nsentence in a way, for it expresses an attendant thought or an\naccompanying circumstance. For example:\n\n The workers being unorganized, the strike was easily defeated.\n The strikers having won, work was resumed on their terms.\n\n_The workers being unorganised_ and _the strikers having won_ are not\nclauses for they do not contain a verb. _Being unorganized_ and _having\nwon_ are participles. Neither do they modify any word in the sentence.\nThey are not placed in apposition with any other word. While they do\nexpress a thought in connection with the sentence, in construction they\nseem to be cut loose from the rest of the sentence; that is, they are\nnot closely connected with the sentence, hence they are called absolute\nconstructions. _Ab_ means from, and _solute_, loose; so this means,\nliterally, loose from the rest of the sentence.\n\nWe speak of these as absolute constructions, instead of independent,\nbecause the thought expressed is connected with the main thought of the\nsentence and is really a part of it. Notice that the noun used in the\nabsolute construction is not the _subject_ of the sentence.\n\nTake the sentence, _The workers being unorganized, the strike was_\n_easily defeated_, the noun _strike_ is the subject of the sentence, and\nthe noun _workers_ is used in the absolute construction with the\nparticiple, _being unorganized_.\n\nThese absolute constructions can ordinarily be rewritten into adverb\nclauses. For example, this sentence might read: _The strike was easily\ndefeated because the workers were unorganized_. Do not make the mistake\nof rewriting your sentences and using the noun in the absolute\nconstruction as the subject of the sentence. For example:\n\n The workers, being unorganized, were easily defeated.\n\nThis is not the meaning of this sentence. The meaning of the sentence is\nthat the _strike_ was easily defeated _because_ the workers were\nunorganized. But the adverb clause, _because the workers were\nunorganized_, instead of being written as an adverb clause, has been\nwritten in the absolute construction, _the workers being unorganized_.\n\nWhile it is nearly always possible to change these absolute\nconstructions into adverb clauses the sentences are sometimes weakened\nby the change. These absolute constructions often enable us to make a\nstatement in a stronger manner than we could make it with a clause or in\nany other way.\n\n\n Exercise 7\n\nIn the following sentences, note the groups of words which are used in\nabsolute construction. Rewrite these sentences and if possible change\nthese words used in absolute construction into equivalent adverb phrases\nor clauses. Note how some of the sentences are weakened when you make\nthis change.\n\n 1. _Nationalism having been taught to generation after generation_,\n the workers obeyed the call of the master class to slaughter their\n fellow workers.\n 2. _The hour having arrived_, Ferrer was blindfolded and led forth to\n die.\n 3. _The mass being without education_, capitalism gains an easy\n victory.\n 4. _The class struggle being a fact_, why should we hesitate to join\n our class?\n 5. _These facts being true_, such a conclusion is inevitable.\n 6. _Darwin having stated the theory of evolution_, Marx applied its\n principles to social science.\n 7. _Chattel slavery having been destroyed_, wage-slavery became the\n corner stone of capitalism.\n 8. _The price having been paid_, we claim our own.\n 9. _The battle ended_, the army left the trenches.\n\n\n Exercise 8\n\nMark the interjections in the following quotations. Note the independent\nconstructions. Mark the words used as explanatory words in apposition.\n\n In the mind's eye, I see a wonderful building, something like the\n Coliseum of ancient Rome. The galleries are black with people; tier\n upon tier rise like waves the multitude of spectators who have come to\n see a great contest. A great contest, indeed! A contest in which all\n the world and all the centuries are interested. It is the contest--the\n fight to death--between Truth and Error.\n\n The door opens, and a slight, small, shy and insignificant looking\n thing steps into the arena. It is Truth. The vast audience bursts into\n hilarious and derisive laughter. What! Is this Truth? This shuddering\n thing in tattered clothes, and almost naked? And the house shakes\n again with mocking and hisses.\n\n The door opens again, and Error enters--clad in cloth of gold,\n imposing in appearance, tall of stature, glittering with gems, sleek\n and huge and ponderous, causing the building to tremble with the thud\n of its steps. The audience is for a moment dazzled into silence, then\n it breaks into applause, long and deafening. \"Welcome!\" \"Welcome!\" is\n the greeting from the multitude. \"Welcome!\" shout ten thousand\n throats.\n\n The two contestants face each other. Error, in full armor--backed by\n the sympathies of the audience, greeted by the clamorous cheering of\n the spectators; and Truth, scorned, scoffed at, and hated. \"The issue\n is a foregone conclusion,\" murmurs the vast audience. \"Error will\n trample Truth under its feet.\"\n\n The battle begins. The two clinch, separate, and clinch again. Truth\n holds its own. The spectators are alarmed. Anxiety appears in their\n faces. Their voices grow faint. Is it possible? Look! See! There!\n Error recedes! It fears the gaze of Truth! It shuns its beauteous\n eyes! Hear it shriek and scream as it feels Truth's squeeze upon its\n wrists. Error is trying to break away from Truth's grip. It is making\n for the door. It is gone!\n\n The spectators are mute. Every tongue is smitten with the palsy. The\n people bite their lips until they bleed. They cannot explain what they\n have seen. \"Oh! who would have believed it?\" \"Is it possible?\"--they\n exclaim. But they cannot doubt what their eyes have seen--that puny\n and insignificant looking thing called Truth has put ancient and\n entrenched Error, backed by the throne, the altar, the army, the\n press, the people and the gods--to rout.\n\n The pursuit of truth! Is it not worth living for? To seek the truth,\n to love the truth, to live the truth? Can any religion offer\n more?--_Mangasarian_.\n\n\n\n\n SPELLING\n\n LESSON 23\n\n\nMany words contain letters for which there are no corresponding sounds\nin the spoken words. Thus, in the spoken word _though_ there are only\ntwo sounds, the _th_ and the _o_; _u_ and _g_ and _h_ are silent. There\nare a great many words in the English language which contain these\nsilent letters. There has been a movement inaugurated for the purpose of\nsimplifying the spelling of these words, omitting these silent letters.\nSome writers have adopted this method of simplified spelling, and so in\nsome magazines and books which you read you will find these silent\nletters dropped; for example, you will find _though_ spelled _tho_,\n_through_ spelled _thru_.\n\nThis method of simplified spelling has not been universally adopted and\nwe have not followed it in these lessons because we feared that it would\nbe confusing. Probably in most of your reading you will find the old\nmethod of spelling followed and all of these silent letters included. No\ndoubt, as time goes on, we shall adopt this simplified method of\nspelling and drop all of these silent and useless letters.\n\nIn our spelling lesson for this week we have a number of words\ncontaining silent letters.\n\n\n +MONDAY+\n\nIn a number of words you will find _ea_ pronounced as short _e_. The\nboard of simplified spelling has suggested that we drop the _a_, which\nis a silent letter, from these words. If we adopted their suggestion,\nwords like _head_ would be spelled _hed_. Note the spelling of the\nfollowing words in which _ea_ is pronounced as short _e_ and the _a_ is\nsilent.\n\nSpread, stead, threat, meant, pleasant, stealth.\n\n\n +TUESDAY+\n\nWe have a number of words ending in _ough_ in which the _gh_ is silent.\n\n1. In some of these words the _ou_ is pronounced like _ow_. We have\nalready changed the spelling of a few of these words, for example, we no\nlonger use _plough_, but write it _plow_.\n\n2. In other words ending with _ough_ the _ugh_ is silent and the words\nend with a long _o_ sound, as in _though_. Many writers have dropped the\nsilent letters ugh and spell this simply _tho_.\n\n3. A few other words ending with _ough_ end with a _u_ sound and those\nwho adopt the simplified spelling have dropped the _ough_ and used\nsimply _u_, as in _through_; many writers spell it simply _thru_.\nObserve the spelling of the following words and mark the silent letters:\n\nBough, through, thorough, furlough, borough, though.\n\n\n +WEDNESDAY+\n\nWe have a number of words ending in _mn_ in which the _n_ is silent.\nNote the spelling of the following words:\n\nAutumn, solemn, column, kiln, hymn, condemn.\n\n\n +THURSDAY+\n\nWe have a number of words containing a silent _b_. Notice the spelling\nof the following words:\n\nDoubt, debt, dumb, limb, thumb, lamb.\n\n\n +FRIDAY+\n\nA number of words end with silent _ue_ after _g_. Some writers omit the\nue and probably after a while we will drop this silent _ue_, but you\nwill find it used now in most of your reading. These are such words as:\n\nCatalogue, demagogue, decalogue, tongue, league, harangue.\n\n\n +SATURDAY+\n\nWe have a number of words ending with _gh_ in which the _gh_ has the\nsound of _f_, as in the following words:\n\nTrough, rough, enough, laugh, tough, cough.\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 24\n\n\nDear Comrade:\n\nWe have finished our study of the different parts of speech and are\ngoing to enter upon the work of sentence building. In the next few\nlessons we will gather up all that we have been studying in these\nlessons so far. This is a good time to give this work a thorough review.\nPerhaps there have been a number of things in the lessons which you have\nnot thoroughly understood, or perhaps there have been some rules for\nwhich you have not seen the reason. Now as we begin to construct our\nsentences, all of this will fit into its place. We shall find the reason\nfor many of the things which may not have seemed thoroughly clear to us.\n\nThere _is_ a science in language as in everything else, and language,\nafter all, is governed by the will of the people. This has seemed so\nself-evident to those who make a special study of the language and its\ndevelopment that they have given this power a special name. They speak\nof the \"Genius of the Language\" as though there was some spirit guiding\nand directing the developing power of language.\n\nThere is a spirit guiding and directing the developing power of\nlanguage. That spirit is the creative genius of the people. It is the\nsame spirit that would guide and direct all phases of life into full and\nfree expression, if it were permitted to act. There being no private\nprofit connected with the control of the language, the creative genius\nof the people has had fuller sway.\n\nThe educator sitting in his study cannot make arbitrary rules to change\nor conserve the use of words. The people themselves are the final\narbiter in language. It is the current usage among the masses which puts\nthe final stamp upon any word. Think what this same creative genius\nmight do if it were set free in social life, in industrial life. It\nwould work out those principles which were best fitted to the advance of\nthe people themselves. But those who would profit by the enslavement of\nthe people have put stumbling blocks,--laws, conventions, morals,\ncustoms,--in the way of the people.\n\nTheir creative genius does not have full sway or free sweep, but let us\nrejoice that in language, at least, we are free. And let us, as we\nrealize the power of the people manifest in this phase of life,\ndetermine that the same power shall be set free to work out its will in\nall life. Some day the revolution will come. The people will be free to\nrule themselves, to express their will, not in the realms of words\nalone, but in their social and economic life; and as we become free\nwithin, dare to think for ourselves and to demand our own, we each\nbecome a torch of the revolution, a center of rebellion--one of those\nwho make straight the path for the future.\n\n Yours for the Revolution,\n\n THE PEOPLE'S COLLEGE.\n\n\n SENTENCE BUILDING\n\n+400.+ Every expression of a complete thought is a sentence. A sentence\nis the unit in language. Words are the material out of which we build\nour sentences, so we have been studying the various parts of speech that\nare used in sentence building. Now we are ready to use these parts of\nspeech in the building of sentences. We have found that there are eight\nparts of speech, though the interjection, which is termed the eighth\npart of speech, is not in reality a part of the sentence; but is a\ncomplete, independent construction. So in your sentences all of the many\nhundreds of words which we use can be grouped into seven divisions;\n_nouns_, _pronouns_, _adjectives_, _verbs_, _adverbs_, _prepositions_\nand _conjunctions_.\n\n+401.+ You remember in our first lesson we found that there were just\nthree kinds of sentences. The _assertive_, the _interrogative_ and the\n_imperative_; or in other words, sentences which state a _fact_, ask a\n_question_ or give a _command_. We also found that these three kinds of\nsentences could all be expressed in _exclamatory_ form.\n\n\n THREE KINDS OF SENTENCES\n\n +Assertive.+ Makes a statement.\n +Interrogative.+ Asks a question.\n +Imperative.+ Gives a command.\n\n +Assertive sentence;+ _I remember the day._\n +Interrogative sentence;+ _Do you not remember the day?_\n +Imperative sentence;+ _Remember the day._\n\n\n In Exclamatory Form\n\n +Assertive;+ _Nonsense! I remember the day._\n +Interrogative;+ _What! Do you not remember the day?_\n +Imperative;+ _Oh come! Remember the day._\n\n\n ANALYSIS--SIMPLE SENTENCES\n\n+402.+ Now that we have finished the study of the various parts of\nspeech, we are ready for sentence building and for sentence analysis.\nSentence analysis is the breaking up of the sentence into its different\nparts in order to find out how and why it is thus put together. To\nanalyze anything is to break it up or separate it into its different\nparts. We speak of analyzing a sentence when we pick out the subject and\nthe predicate and their modifiers, because we thus unloosen them or\nseparate them from one another.\n\nThese parts of the sentence are called the elements of the sentence. The\nelements of a sentence consist of the words, phrases and clauses used in\nforming the sentence.\n\n+403.+ Let us begin from the simplest beginning and build up our\nsentences, using the various parts of speech as we have studied them.\nLet us take the simplest form of sentence which we can consider. For\nexample:\n\n Men work.\n\nThere are only three parts of speech which can be used to make a simple\nsentence in this manner, and these are, either the noun and the verb, or\nthe pronoun and the verb. We might say instead of _Men work_, _They\nwork_, and have a complete sentence.\n\nIn the sentence _Men work_, _men_ is the subject and _work_ is the\npredicate. The subject and the predicate are the two principal elements\nin a sentence. No sentence can be formed without these two parts and\nthese two parts can express a thought without the help of other\nelements. Now we may begin to enlarge the subject by adding modifiers.\n\nYou remember we have found that a noun may be modified by an adjective.\nSo we add the adjective _busy_, and we have:\n\n Busy men work.\n\nOur simple subject is still the noun _men_, but the complete subject is\nthe noun with its modifier, _busy men_. We may add other adjectives and\nsay:\n\n The busy, industrious men with families work.\n\nHere we have our simple subject _men_ modified by the adjectives, _the_,\n_busy_ and _industrious_, and also by the adjective phrase, _with\nfamilies_. So the complete subject of the sentence now is, _the busy,\nindustrious men with families_.\n\nOur predicate is still the single verb _work_. Let us now enlarge the\npredicate. We have found that adverbs are used to modify verbs, and so\nwe may say:\n\n The busy, industrious men with families work hard.\n The busy, industrious men with families work hard in the factory.\n\nOur simple predicate, _work_ is now enlarged. It is modified by the\nadverb, _hard_ and the adverb phrase, _in the factory_. So our complete\npredicate is now, _work hard in the factory_.\n\n+404.+ These sentences with the simple subject and the simple predicate\nand their modifying words and phrases form simple sentences.\n\n+A simple sentence is one which expresses a single statement, question\nor command.+\n\n+405.+ A simple sentence, therefore, will contain but one subject and\none predicate. The subject may be a compound subject and the predicate\nmay be a compound predicate, but still the sentence expresses a single\nthought. For example: _The boys sing_. This is a simple statement with a\nsimple subject and a simple predicate. Then we may say: _The boys sing\nand play_. We still have a single statement, but a compound predicate,\n_sing and play_.\n\nNow we may make a compound subject, and say, _The boys and girls sing\nand play_, but we have still a single statement, for both predicates are\nasserted of both subjects. So, _The boys and girls sing and play_, is a\nsimple sentence.\n\nIf we say, _The boys sing and the girls play_, we have a compound\nsentence, composed of two simple sentences, _The boys sing_, _The girls\nplay_.\n\nIf we say, _The boys sing while the girls play_, we have a complex\nsentence formed of the simple sentence, _The boys sing_, and the\ndependent clause, _while the girls play_.\n\n+406.+ Now let us sum up our definitions:\n\n+Every sentence must contain two parts, a subject and a predicate.+\n\n+The subject of a sentence is that part about which something is said.+\n\n+The predicate is that part which asserts something of the subject.+\n\n+The simple subject of a sentence is a noun, or the word used in place\nof a noun, without modifiers.+\n\n+The simple predicate is the verb or verb phrase without its modifiers.+\n\n+The complete subject of a sentence is the simple subject with all of\nits modifiers.+\n\n+The complete predicate of a sentence is the simple predicate with all\nof its modifiers.+\n\n+A simple sentence is one which expresses a single statement, question\nor command.+\n\n+A complex sentence is one containing an independent clause and one or\nmore dependent clauses.+\n\n+A compound sentence is one containing two or more independent clauses.+\n\n+A clause is a part of a sentence containing a subject and a predicate.+\n\n\n Exercise 1\n\nIn the following sentences the simple subject and the simple predicate\nare printed in _italics_. Find all of the modifiers of the subject and\nall of the modifiers of the predicate, and draw a single line under the\ncomplete subject and two lines under the complete predicate.\n\n 1. Beautiful _pictures hang_ on the wall.\n 2. Those elm _trees grow_ rapidly every year.\n 3. A terrible _storm broke_ unexpectedly at sea.\n 4. The clear, crystal _water runs_ swiftly to the sea.\n 5. The beautiful _flowers fade_ quickly in the heat.\n 6. The happy, boisterous _children play_ at school every day.\n 7. The sturdy _oak_ in the forest _stands_ bravely through every\n storm.\n 8. Their arching _tops_ almost _speak_ to us.\n 9. A _cry_ of joy _rings_ through the land.\n 10. The _leaves_ of the trees _flutter_ in the wind.\n 11. Great _clouds_ of smoke _float_ in the air.\n\n\n Exercise 2\n\nNote carefully the following simple sentences. Each of these groups of\ntwo words will suggest ideas and pictures to you. Lengthen each sentence\nby adding modifiers to the simple subject and to the simple predicate so\nas to make a fuller and more definite statement. For example: _Ships\nsail_. This is a simple subject and simple predicate. We add adjectives\nand an adjective phrase and adverbs and an adverb phrase as modifiers\nand we have, as follows:\n\n The stately _ships_ in the bay _sail_ proudly away to foreign shores.\n\n Snow melts.\n Winds blow.\n House stands.\n Boys run.\n Soldiers fight.\n Tides flow.\n Children play.\n Ships sail.\n Guns boom.\n Women endure.\n\n\n ANOTHER ELEMENT\n\n+407.+ You will note that all of these verbs which we have used in these\nsentences have been complete verbs as _hang_, _grow_, _runs_, _fade_,\netc. A complete verb, you will remember, is a verb that does not need an\nobject or a complement. It is complete within itself. It may be modified\nby an adverb or an adverb phrase, but when you leave off these modifiers\nyou still have complete sense.\n\nIn any of the sentences above you may cross out the adverb or the adverb\nphrase which modifies the verb and you will still have complete\nsentences. For example:\n\n Great clouds of smoke float in the air.\n\nHere, the adverb phrase, _in the air_, may be omitted and still we have\ncomplete sense, thus:\n\n Great clouds of smoke float.\n\n+408.+ The incomplete verbs, however, require either an object or\ncomplement to complete their meaning.\n\nIncomplete verbs are of two kinds; those that express _action_ and those\nthat express _state_ or _condition_.\n\nAn incomplete verb that expresses action requires an object which is the\nreceiver of the action expressed in the verb, so we have another element\nwhich enters into the simple sentence, when we use an incomplete verb.\nFor example:\n\n The busy man makes shoes.\n\nIn order to complete the sentence, we must use an object with the\nincomplete verb _makes_. To say, _The busy man makes_, is not enough. We\nmust have an object which is the receiver of the action expressed in the\nverb, _makes_. Verbs of action often have two objects. One object names\nthe _thing_ that _receives_ the action and the other names the _thing_\nindirectly _affected_ by the action. For example:\n\n The tailor made him a coat.\n\n+409.+ _Coat_ is the _direct_ object of the verb _made_. But we have\nanother object in the pronoun _him_. We do not mean that the tailor made\n_him_, but that the tailor made him a _coat_. _Coat_ is the direct\nobject and _him_ is the indirect object. The indirect object is always\nplaced before the direct object. The indirect object may be used as the\nobject of the preposition _to_ or _for_. As for example, this sentence\nmight be rewritten to read, _The tailor made a coat for him_. In this\nsentence, _him_ is not the indirect object of the verb, but is the\nobject of the preposition _for_.\n\n+410.+ The direct object of the verb always answers the question _what?_\nAs for example, the tailor made _what?_--_a coat_. The indirect object\nof the verb names the person or thing _to_ or _for_ which the act is\ndone,--_the tailor made a coat for whom?_--for _him_.\n\nThe direct and indirect object become a part of the complete predicate\nof the sentence. There may be other modifiers also, as adverbs or adverb\nphrases, and all of these taken together form the complete predicate in\nthe sentences where you have used an incomplete verb. As for example:\n\n The tailor gladly made him a coat for the occasion.\n\nThe complete predicate is, _gladly made him a coat for the occasion_,\nformed of the verb _made_, the direct object, _coat_, the indirect\nobject _him_, the adverb modifier, _gladly_, and the phrase modifier,\n_for the occasion_.\n\n\n Exercise 3\n\nIn the following sentences, underscore the direct object with one line\nand the indirect object with two lines. The verb is in italics.\n\n 1. He _gave_ her a book.\n 2. He _wrote_ me a long letter.\n 3. Her father _bought_ her a watch.\n 4. The nurse _gave_ the patient his medicine.\n 5. The mother _gave_ her daughter a present.\n 6. _Give_ me time to think.\n 7. The clerk _sold_ her a dress.\n 8. The teacher _read_ the children a story.\n 9. The company _furnishes_ the men food and shelter.\n 10. The man _showed_ us his wounds.\n\n\n Exercise 4\n\nIn the following sentences underscore the complete subject and the\ncomplete predicate. Notice especially the direct and the indirect\nobjects of the incomplete verbs. The simple subjects and the direct\nobjects are in italics.\n\n 1. A great many _miles_ separate _us_ from our friends.\n 2. The merry _shouts_ of the children fill the _air_ with music.\n 3. A gentle _breeze_ brings us the _perfume_ of the flowers.\n 4. A careless _druggist_ gave the unfortunate man the wrong\n _medicine_.\n 5. His admiring _friends_ gave him a beautiful _ring_.\n 6. _Soldiers_ obey _orders_ from their superiors.\n 7. This terrible _war_ claims _thousands_ of victims.\n 8. The _power_ of hunger drives the _unemployed_ to rebellion.\n 9. The _workers_ of the world produce _enough_ for all.\n 10. The retiring _secretary_ showed us a _letter_ from the president.\n 11. The old sea _captain_ told them an interesting _story_ of life at\n sea.\n 12. _Labor_ produces all _wealth_.\n\n\n COPULATIVE VERBS\n\n+411.+ We have another class of incomplete verbs which require a\ncomplement to complete their meaning. These are the copulative verbs.\nThe number of copulative verbs is small. They are: all forms of the verb\n_be_; also, _like_, _appear_, _look_, _feel_, _sound_, _smell_,\n_become_, _seem_, etc. These verbs require a noun or an adjective or a\nphrase as a complement, to complete their meaning. They are really\nconnective words serving to connect the noun or adjective or phrase used\nin the predicate with the noun which they modify. The noun or adjective\nor phrase used to complete the meaning of the copulative verb is called\na predicate complement. For example:\n\n The man is a hero.\n\nHere we have a noun, _hero_, used as a predicate complement after the\ncopulative verb, _is_, to describe the noun _man_.\n\n The man is class-conscious.\n\nIn this sentence, we have an adjective, _class-conscious_, in the\npredicate to modify the subject, _man_. It is connected with the subject\nby the copulative verb _is_.\n\n The man is in earnest.\n\nHere we have a phrase, _in earnest_, used in the predicate to modify the\nnoun _man_, and connected with the subject by the copulative verb _is_.\n\n+412.+ So in the predicate with the copulative verbs--incomplete verbs\nwhich express state or condition--we may use a noun or an adjective or a\nphrase. A noun used as the predicate complement may have modifiers. It\nmay be modified by one or more adjectives or adjective phrases. These\nadjectives in turn may be modified by adverbs. The complete predicate,\nthen, is the copulative verb with its predicate complement and all its\nmodifiers. For example:\n\n Grant was the most famous general of the Civil war.\n\nIn this sentence, _Grant_ is the complete subject, _was the most famous\ngeneral of the Civil war_ is the complete predicate. _Was_ is the\ncopulative verb; _general_ is the noun used as the predicate complement;\n_the_ and _famous_ are adjectives modifying _general_; _most_ is an\nadverb modifying the adjective _famous_, and, _of the Civil war_ is an\nadjective phrase modifying _general_, so our complete predicate is, _was\nthe most famous general of the Civil war_.\n\nWhen an adjective is used in the predicate complement it, too, may have\nmodifiers and more than one adjective may be used. For example:\n\n The man is very brave and loyal to his class.\n\nHere we have two adjectives used in the predicate complement, _brave_\nand _loyal_. _Brave_ is modified by the adverb _very_, and _loyal_ is\nmodified by the adverb phrase, _to his class_. The complete predicate\nis, _is very brave and loyal to his class_.\n\nWhen we use a phrase as a predicate complement, it, too, may have\nmodifiers and more than one phrase may be used. For example:\n\n The man is in the fight and deeply in earnest.\n\nIn this sentence, two phrases are used in the predicate complement, _in\nthe fight_ and _in earnest_. The second phrase, _in earnest_ is modified\nby the adverb _deeply_. The complete predicate is, _is in the fight and\ndeeply in earnest_.\n\n\n Exercise 5\n\nFill the blanks in the following sentences with a noun and its modifiers\nused as predicate complement. Name all of the parts of speech which you\nhave used in the predicate complement as we have done in the sentences\nanalyzed above:\n\n The men are _loyal members of the Union_.\n Slavery is.......\n Liberty will be.......\n War is.......\n The machine is.......\n The children were.......\n\nFill the blanks in the following sentences with one or more adjectives\nand their modifiers used in the predicate complement.\n\n The work is _hard and destructive to the children_.\n The history will be.......\n Labor has been.......\n Peace will be.......\n Poverty is.......\n\nFill the blanks in the following sentences with a phrase used in the\npredicate complement.\n\n His service was _for his class_.\n Socialism is.......\n The workers are.......\n The message shall be.......\n The government is.......\n The opportunity is.......\n\n\n VERB PHRASES\n\n+413.+ Note that in most of the sentences which we have used, we have\nused the simple form of the verb, the form that is used to express\n_past_ and _present_ time. In expressing other time forms we use verb\nphrases. Note the summary given in section 145, which gives the\ndifferent time forms of the verb.\n\n+414.+ Sometimes in using the verb phrase you will find that other words\nmay separate the words forming the phrase. When you analyze your\nsentence this will not confuse you. You will easily be able to pick out\nthe verb phrase. For example:\n\n I shall very soon find out the trouble.\n\nHere the adverbs, _very_ and _soon_, separate _find_ from its auxiliary\n_shall_. The verb phrase is, _shall find_. The negative _not_ very often\nseparates the words forming a verb phrase. For example:\n\n I will not go.\n\nIn this sentence, _will go_ is the verb phrase.\n\nWhen we use the auxiliary verb _do_ to express emphasis, and also the\nnegative _not_, _not_ comes between the auxiliary verb _do_, and the\nprincipal verb. For example:\n\n I do not obey, I think.\n\nIn this sentence, _do obey_ is the verb phrase.\n\nIn interrogative sentences, the verb phrase is inverted and a part of\nthe verb phrase is placed first and the subject after. For example:\n\n Will you go with us?\n\n_You_ is the subject of this interrogative sentence and _will go_ is the\nverb phrase; but in order to ask the question, the order is inverted and\npart of the verb phrase placed first. In using interrogative adverbs in\nasking a question, the same inverted order is used. For example:\n\n When will this work be commenced?\n\nIn this sentence, _work_ is the subject of the sentence and _will be\ncommenced_ is the verb phrase. If you should write this in assertive\nform, it would be:\n\n This work will be commenced when?\n\nBy paying close attention we can easily distinguish the verb phrases\neven when they are used in the inverted form or when they are separated\nby other parts of speech.\n\n\n LET US SUM UP\n\n+415.+ The elements of a sentence are the words, phrases or clauses of\nwhich it is composed.\n\n+A simple sentence is one which contains a single statement, question or\ncommand.+\n\n+A simple sentence contains only words and phrases.+ It does not contain\ndependent clauses. The elements of a simple sentence are:\n\n {The simple subject--the noun, or the\n The complete subject { word used in place of the noun--and\n { all its modifiers.\n\n The complete predicate {The simple predicate--the verb, and\n { all its modifiers.\n\n\n Exercise 6\n\nIn the following sentences, the simple subjects and the simple\npredicates of the principal clauses are printed in italics. Locate all\nthe modifiers of the subjects and predicates, and determine the part of\nspeech of each word in the sentence.\n\nSentences Nos. 1, 5, 6, 14, 15, 16, 18, 30, 31, 32 and 37 are simple\nsentences.\n\nSentences Nos. 2, 4, 7, 11, 13, 17, 19, 22, 26, 28, 33, 34 and 36 are\ncomplex.\n\nSentences Nos. 3, 10, 12, 21, 23, 24, 25, 29 and 35 are compound.\n\nNo. 8 is incomplete, having neither subject nor predicate.\n\nNo. 9 is incomplete, there being no predicate in the principal clause.\n\nNo. 20 is a simple sentence, with a complex sentence in parenthesis.\n\nNo. 27 consists of two dependent clauses.\n\nIn the complex sentences, draw a line under the dependent clauses.\n\n \"Br--r--r--r--r--r--r--r--r--.\"\n\n 1. What _are_ the _machines saying_, a hundred of them in one long\n room?\n 2. _They must be talking_ to themselves, for I see no one else for\n them to talk to.\n 3. But yes, there _is_ a boy's red _head_ bending over one of them,\n and beyond _I see_ a pale face fringed with brown curly locks.\n 4. There _are_ only five _boys_ in all, on the floor, half-hidden by\n the clattering machines, for one bright lad can manage twenty-five\n of them.\n 5. Each _machine makes_ one cheap, stout sock in five minutes,\n without seam, complete from toe to ankle, cutting the thread at the\n end and beginning another of its own accord.\n 6. The _boys have_ nothing to do but to clean and burnish and oil the\n steel rods and replace the spools of yarn.\n 7. But how rapidly and nervously _they do_ it--the slower hands\n straining to accomplish as much as the fastest!\n 8. Working at high tension for ten hours a day in the close, greasy\n air and endless whirr----\n 9. _Boys_ who ought to be out playing ball in the fields or taking a\n swim in the river this fine summer afternoon.\n 10. And in these good times, the _machines go_ all night, and other\n _shifts_ of boys _are kept_ from their beds to watch them.\n 11. The young _girls_ in the mending and finishing rooms downstairs\n _are_ not so strong as the boys.\n 12. _They have_ an unaccountable way of fainting and collapsing in\n the noise and smell, and then _they are_ of no use for the rest of\n the day.\n 13. The kind _stockholders have had_ to provide a room for collapsed\n girls and to employ a doctor, who finds it expedient not to\n understand this strange new disease.\n 14. Perhaps their _children will be_ more stalwart in the next\n generation.\n 15. Yet this _factory is_ one of the triumphs of our civilization.\n 16. With only twenty boys at a time at the machines in all the rooms,\n _it produces_ five thousand dozen pairs of socks in twenty-four\n hours for the toilers of the land.\n 17. _It would take_ an army of fifty thousand hand-knitters to do what\n these small boys perform.\n\n \"Br--r--r--r--r--r--r--r--r--.\"\n\n 18. What _are_ the _machines saying_?\n 19. _They are saying_, \"We are hungry.\"\n 20. \"_We have eaten_ up the men and women. (There is no longer a\n market for men and women, they come too high)--\n 21. _We have eaten_ up the men and women, and now _we are devouring_\n the boys and girls.\n 22. How good _they taste_ as we suck the blood from their rounded\n cheeks and forms, and cast them aside sallow and thin and\n careworn, and then call for more.\n 23. Br--r--r--r--r--r--r--r! how good _they taste_; but _they give_\n us so few boys and girls to eat nowadays, although there are so\n many outside begging to come in--.\n 24. Only one _boy_ to twenty of us, and _we are_ nearly _famished_!\n 25. _We eat_ those they give us and _those_ outside _will starve_, and\n soon _we shall be left_ almost alone in the world with the\n stockholders.\n 26. Br--r--r--r--r--r--r--r! What shall we do then for our food?\" the\n _machines chatter_ on.\n 27. \"When we are piling up millions of socks a day for the toilers\n and then there are no toilers left to buy them and wear them.\n 28. Then perhaps we shall have to turn upon the kind stockholders and\n feast on them (how fat and tender and toothsome they will be!)\n until at last we alone remain, clattering and chattering in a\n desolate land,\" _growled the machines_.\n 29. While the _boys went_ on anxiously, hurriedly rubbing and\n polishing, and the _girls_ downstairs _went_ on collapsing.\n 30. \"Br--r--r--r--r--r--r--r!\" _growled_ the _machines_.\n 31. The _devil has_ somehow _got_ into the machines.\n 32. _They came_ like the good gnomes and fairies of old, to be our\n willing slaves and make our lives easy.\n 33. Now that, by their help, one man can do the work of a score, why\n _have we_ not plenty for all, with only enough work to keep us\n happy?\n 34. _Who could have foreseen_ all the ills of our factory workers and\n of those who are displaced and cast aside by factory work?\n 35. The good wood and iron _elves came_ to bless us all, but _some_ of\n us _have succeeded_ in bewitching them to our own ends and turning\n them against the rest of mankind.\n 36. _We must break_ the sinister charm and _win_ over the docile,\n tireless machines until they refuse to shut out a single human\n being from their benefits.\n 37. _We must cast_ the devil out of the machines.\n\n --_Ernest Crosby_.\n\n\n\n\n SPELLING\n\n LESSON 24\n\n\nAmong the common suffixes in English are the suffixes _or_ and _er_.\nThese suffixes mean _one who_ or _that which_. For example, _builder_,\none who builds; _actor_, one who acts; _heater_, that which heats. But\nwe are confused many times to know whether to add the suffix _or_ or\n_er_ to form these derivative words. There is no exact rule which can be\ngiven, but the following rule usually applies with but few exceptions:\n\nTo the shorter and commoner words in the language add the suffix _er_.\nFor example, _writer_, _boxer_, _singer_, etc. To the longer and less\ncommon words, usually those derived from the Latin or the Greek, add the\nsuffix _or_. For example, _legislator_, _conqueror_, etc.\n\nThere are a number of words in the English like _honor_, in which the\nlast syllable used to be spelled _our_ instead of _or_. You will\nprobably run across such words as these in your reading. This mode of\nspelling these words, however, is being rapidly dropped and the ending\n_or_ is being used instead of _our_. There are also a number of words in\nour language like _center_, which used to be spelled with _re_ instead\nof _er_. The _re_ ending is not used any more, although you may run\nacross it occasionally in your reading. The proper ending for all such\nwords as these is _er_. There are a few words, however, like _timbre_ (a\nmusical term) and _acre_, which are still properly spelled with the _re_\nending.\n\nThe spelling lessons for Monday, Tuesday, Wednesday and Thursday,\ncontain words from which derivatives can be formed by adding _er_ or\n_or_. Look these words up in the dictionary and be sure that you have\nadded the proper suffix. The list for Friday consists of words which you\nmay find in your reading spelled with the _our_ ending. The list for\nSaturday contains words which you may find spelled with the _re_ ending\ninstead of the _er_.\n\n +Monday+\n\n Create\n Produce\n Profess\n Debate\n Govern\n\n +Tuesday+\n\n Edit\n Consume\n Consign\n Legislate\n Design\n\n +Wednesday+\n\n Solicit\n Pay\n Success\n Observe\n Invent\n\n +Thursday+\n\n Vote\n Debt\n Organize\n Sail\n Strike\n\n +Friday+\n\n Labor\n Neighbor\n Rumor\n Valor\n Candor\n\n +Saturday+\n\n Theater\n Scepter\n Fiber\n Somber\n Meager\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 25\n\n\nDear Comrade:\n\nIn logic, we have two ways of reasoning, from the general to the\nparticular and from the particular to the general. In other words, we\nmay take a certain number of facts and reason to a conclusion; or we may\ngo the other way about and start with our conclusion and reason back to\nthe facts which produce the conclusion. Scientists use the former\nmethod. They gather together all the facts which they possibly can and\nfrom these facts they reach their conclusions.\n\nThis was what Karl Marx did for the social problems of his day. He\nanalyzed these problems. He gathered together all of the facts which he\ncould obtain concerning conditions of his day and from these facts he\nreached certain conclusions. He foretold the rise of capitalism and\noutlined present day conditions so perfectly that had he lived long ago\namong superstitious people, they would probably have called him a\nprophet.\n\nThis mastery of analysis, of marshaling our facts and from them reaching\nconclusions, is a wonderful power to possess, and this is exactly what\nwe are doing in our English work. We are analyzing our sentences,\nfinding the elements of which they are composed, and then building the\nsentence; and since neither the thought nor the sentence can be really\nstudied except in connection with each other, this analysis of sentences\ngives us an understanding of the thought. The effort to analyze a\ndifficult sentence leads to a fuller appreciation of the meaning of the\nsentence. This, in turn, cultivates accuracy in our own thought and in\nits expression.\n\nSo do not slight the analysis of the sentence or this work in sentence\nbuilding. You will find it will help you to a quicker understanding of\nthat which you are reading and it will also give you a logical habit of\nmind. You will be able to think more accurately and express yourself\nmore clearly. After a little practice in analysis you will find that in\nyour reading you will be able to grasp the author's meaning quickly. You\nwill see at a glance, without thinking about it consciously, the subject\nand the predicate and the modifiers in the sentence. Then you will not\nconfuse the meaning. You will not have to go back and reread the passage\nto find out just what the author was talking about; and when you come to\nwrite and speak yourself, you will have formed the habit of logical\nexpression. In this way you will be able to put your thought in such a\nmanner that your listener can make no mistake as to just what you mean.\n\nNow, no habit comes without practice. You cannot do a thing\nunconsciously until you have done it consciously a great many times. So\npractice this analysis of sentences over and over. It really is an\ninteresting game in itself, and the results which it will bring to you\nare tremendously worth while.\n\nNothing is too much trouble which will give us the power to think for\nourselves and to put that thought into words.\n\n Yours for Freedom,\n\n THE PEOPLE'S COLLEGE.\n\n\n THE SUBJECT OF A SIMPLE SENTENCE\n\n+416.+ We have found that the two parts of a simple sentence are the\ncomplete subject and the complete predicate. The noun is most often used\nas the subject of a sentence. It may have a number of modifiers, but\nwhen we strip away these modifiers we can usually find a noun which is\nthe subject of the sentence. Occasionally the subject is a pronoun or a\nparticiple or adjective used as a noun but most frequently the subject\nis a noun. As for example:\n\n A wild piercing _cry_ rang out.\n Hopeless, helpless _children_ work in the cotton mills.\n The golden _age_ of peace will come.\n Little child _lives_ are coined into money.\n Defenseless, helpless _children_ suffer most under capitalism.\n Every neglected _child_ smites my conscience in the name of humanity.\n The thrilling, far-sounding _battle-cry_ shall resound.\n\nNote that in all of these sentences the word in italics is a noun, which\nis the simple subject of the sentence. All of the other words which\ncomprise the complete subject are the modifiers of this noun, or\nmodifiers of its modifiers.\n\nBut in our study of words, we have found that there are a number of\nother words which can be used in place of a noun and these may all be\nused as the subject of a sentence.\n\n+417.+ +A pronoun may be used as the subject of a sentence+, for the\npronoun is a word used in place of the noun; and a pronoun used as the\nsubject of a sentence may have modifiers just as a noun. It may be\nmodified by adjectives or adjective phrases, as for example:\n\n _We_ are confident of success.\n _He_, worried and out of employment, committed suicide.\n _She_, heartsick and weary, waited for an answer.\n _She_, with her happy, watchful ways, blessed the household.\n _They_, victorious and triumphant, entered the city.\n How can _I_, without money or friends, succeed?\n\n \"Out of the night that covers me,\n Black as the pit from pole to pole,\n _I_ thank whatever Gods there be\n For my unconquerable soul.\"\n\nIn all of these sentences the pronoun is the simple subject of the\nsentence, and the pronoun with all of its modifiers is the complete\nsubject of the sentence.\n\n+418.+ +The participle may be used as a noun, the subject of the\nsentence.+ For example:\n\n _Traveling_ is pleasant.\n\nHere the present participle _traveling_ is used as a noun, subject of\nthe sentence.\n\nParticiple phrases may also be used as nouns, as for example:\n\n _Being prepared_ will not save us from war.\n His _having signed_ the note was the cause of the trouble.\n\nIn these sentences, _being prepared_ and _having signed_ are participle\nphrases used as nouns, the subjects of the verbs _will save_ and _was_.\nNote the use of the participle used as the subject in the following\nsentences:\n\n _Painting_ is an art.\n _Making_ shoes is his work.\n _Being discovered_ seems to be the real crime.\n His _having joined_ his comrades was a brave act.\n Your _remaining_ here will be dangerous.\n\nNote that when the participle is used as a noun, the possessive form\nof the pronoun is always used with it, as in the sentence above:\n\n _Your_ remaining here will be dangerous.\n\nNotice that in some of these sentences the participle has an object; as,\nmaking _shoes_, his having joined his _comrades_. The participle still\nretains some of its verb nature in that it may take an object. The\nentire phrases, _His having joined his comrades_, and, _Making shoes_,\nare the subjects of the sentences.\n\n+419.+ +The infinitive may also be used as a noun, the subject of the\nsentence.+ Note in the following sentences the use of the infinitive as\nthe subject of the sentence:\n\n _To err_ is human; _to forgive_ is divine.\n _To be_ or not _to be_ is the question.\n _To toil_ all day is wearisome.\n _To aim_ is one thing; _to hit_ the mark is another.\n _To remain_ ignorant is to remain a slave.\n\n+420.+ +An adjective can also be used as the subject.+ You remember in\nour study of adjectives we found that an adjective may be used as a\nnoun, as for example:\n\n The _strong_ enslave the weak.\n\nHere the adjective _strong_ is used as a noun, subject of the sentence.\nNote in the following sentences, the use of the adjectives as subjects:\n\n The _wise_ instruct the ignorant.\n The _dead_ were left upon the battlefields.\n The _rich_ look down upon the poor.\n The _mighty_ of the earth have forced this war upon us.\n The _poor_ are enslaved by their ignorance.\n The _wounded_ were carried to the hospitals.\n\n\n PLACE OF THE SUBJECT IN A SENTENCE\n\nThe subject usually comes first in the sentence. If it has any\nmodifiers, they alone precede the subject, as for example:\n\n A wonderful, inspiring _lecture_ was given.\n The weary _army_ slept in the trenches.\n\nBut occasionally we find the subject after the verb.\n\n+421.+ +By simple inversion.+\n\nWe will often find this use in poetry or in poetic prose, as for\nexample:\n\n Never have _I_ heard one word to the contrary.\n\nIn this sentence _I_ is the subject of the sentence, _have heard_ is the\nverb, and _never_ is an adverb modifying the verb phrase, _have heard_.\nBut in order to place emphasis upon the word _never_, which is the\nemphatic word in the sentence, _never_ is placed first, and the verb\nphrase inverted so that the subject _I_ comes in between the two words\nwhich form the verb phrase. The sentence expressed in its usual order\nwould be:\n\n I have never heard one word to the contrary.\n\nYou will note that this statement does not carry the same emphasis upon\nthe word _never_ as the inverted statement.\n\n+422.+ +In interrogative sentences, the subject comes after the helping\nverb or after the interrogative used to introduce the sentence.+ As for\nexample:\n\n Have _you_ heard the news?\n When will _we_ hear from you?\n How have the _people_ been managing?\n What will the _children_ do then?\n Will the _students_ come later?\n Can the _work_ be accomplished quickly?\n Must our _youth_ end so quickly?\n\n+423.+ +The real subject comes after the verb when we use the\nintroductory word it.+ As for example:\n\n It will not be safe _to go_.\n\n_To go_ is really the subject of the sentence. _To go will not be safe._\n\n_It_ is sometimes the real subject of a sentence, as in the sentence;\n_It is a wonderful story_.\n\nHere _it_ is the subject of the sentence and _a wonderful story_ is the\npredicate complement. But in the sentence:\n\n It is wonderful to hear him tell the story.\n\n_To hear him tell the story_ is the real subject of the sentence. The\nfirst sentence, _It is a wonderful story_, could not be rewritten, but\nthe second sentence could be rewritten, as follows:\n\n To hear him tell the story is wonderful.\n\n+424.+ +The introductory word there reverses the order of the sentence+,\njust as the introductory word _it_. The real subject is used later in\nthe sentence. As for example:\n\n There were a great many people present.\n\nThis could be rewritten, omitting the introductory word _there_. We\ncould say:\n\n A great many people were present.\n\nThe noun _people_ is the subject of the sentence.\n\n\n Exercise 1\n\nIn the following sentences, underscore the complete subject with one\nline, and the simple subject with two lines, and decide whether the\nsimple subject is a noun, pronoun, participle, infinitive or an\nadjective used as a noun:\n\n\n 1. A great man is universal and elemental.\n 2. To love justice was his creed.\n 3. A more inspiring and noble declaration of faith was never born of\n human heart.\n 4. The reading of good books should begin in childhood.\n 5. Dreaming of great things will not bring us to the goal.\n 6. The weary seek for rest.\n 7. To believe in yourself is the first essential.\n 8. He, speaking and writing constantly for the cause, has given his\n life to the movement.\n 9. To remain ignorant is to remain a slave.\n 10. A foolish consistency is the hobgoblin of little minds.\n 11. A great soul has simply nothing to do with consistency.\n 12. To be great is to be misunderstood.\n 13. Traveling is a fool's paradise.\n 14. It is not enough to be sincere.\n 15. We, seeking the truth, have found our own.\n 16. There are thousands of comrades with us.\n\n\n THE COMPLETE PREDICATE\n\n+425.+ Look first in the predicate for your verb. It will always be the\nprincipal part of your predicate. It may be a verb or a verb phrase, but\nthe first thing in analyzing the complete predicate of the sentence is\nto find the verb. The verb or verb phrase without any of its modifiers\nconstitutes the simple predicate. If the verb is a complete verb, its\nonly modifiers will be adverbs or adverb phrases. For example:\n\n A splendid statue of Lincoln stands yonder in the park.\n\nIn this sentence, _stands yonder in the park_ is the complete predicate.\n_Stands_ is a complete verb. It requires no object, but it is modified\nby the adverb _yonder_ and by the adverb phrase _in the park_.\n\n\n INCOMPLETE VERBS\n\n+426.+ If the verb in the predicate is an incomplete verb of action,\nthen the object of the verb is also part of the predicate. The complete\npredicate containing an incomplete verb of action may contain five\nparts; a verb, a direct object, an indirect object, an adverb and an\nadverb phrase. As for example:\n\n The tailor gladly made him a coat at that time.\n\nIn this sentence, the complete predicate is _gladly made him a coat at\nthat time_. _Made_ is the verb. It is an incomplete verb of action, and\n_coat_ is its direct object. _Him_ is the indirect object. _Made_ is\nalso modified by the adverb _gladly_, and the adverb phrase, _at that\ntime_.\n\nAll of these are not always used, of course, in every predicate; but\nthese are the elements which may occur in the predicate with an\nincomplete verb.\n\n\n THE OBJECT OF THE VERB\n\n+427.+ Words used as objects of a verb are practically the same as those\nwhich may be used for its subject.\n\n+We may have a noun used as the object of the verb.+ For example:\n\n Hail destroyed the _crops_.\n The banks rob the _farmers_.\n We must educate the _children_.\n Labor produces all _wealth_.\n\nIn these sentences, _crops_, _farmers_, _children_ and _wealth_ are\nnouns used as the object of the verb.\n\n+A pronoun may also be used as the object of a verb.+ For example:\n\n Will you not teach _me_?\n Send _them_ to her.\n They have invited _us_.\n The comrades will remember _him_.\n\nIn the above sentences, _me_, _them_, _us_ and _him_ are the objects of\nthe verbs, _will teach_, _send_, _have invited_ and _will remember_.\n\nRemember that in pronouns we have a different form for the object form,\nas, _me_, _her_, _him_, _us_ and _them_.\n\n+428.+ +An infinitive may also be used as the object of a verb+, thus:\n\n I like _to study_.\n He asked _to go_.\n I want _to learn_ all that I can.\n\nIn this last sentence, the infinitive, _to learn_, is the direct object\nof the verb _want_. The object of the infinitive, _to learn_, is _all\nthat I can_. All of this taken together with the verb _want_, forms the\ncomplete predicate, _want to learn all that I can_.\n\n+429.+ +The participle may also be used as the object of a verb+, thus:\n\n We heard the _thundering_ of the cannon.\n We enjoyed the _dancing_.\n Do you hear the _singing_ of the birds?\n\nIn these sentences, the participles _thundering_, _dancing_, and\n_singing_ are the objects of the verbs _heard_, _enjoyed_ and _do hear_.\n\n+430.+ +An adjective used as a noun may also be used as the object of a\nverb+, thus:\n\n I saw the _rich_ and the _poor_ struggling together.\n The struggle for existence crushes the _weak_.\n Seek the _good_ and the _true_.\n\nIn these sentences the adjectives _rich_, _poor_, _weak_, _good_ and\n_true_, are used as nouns and are the objects of the verbs _saw_,\n_crushes_ and _seek_.\n\n\n VERBS OF STATE OR CONDITION\n\nWe have found that with the incomplete verbs of state or condition, or\ncopulative verbs, the predicate complement may be either a noun, as,\n_The man is a hero_; or an adjective, as, _The man is class-conscious_;\nor a phrase, as, _The man is in earnest_.\n\nThe predicate complement may also be:\n\n+431.+ +A pronoun+; as,\n\n Who is she?\n That was he.\n This is I.\n\nIn these sentences the subjects of the verbs are _she_, _that_ and\n_this_, and the pronouns _who_, _he_ and _I_ are used as predicate\ncomplements.\n\n+432.+ +Infinitives may also be used as the predicate complement+, thus:\n\n To remain ignorant is _to remain_ a slave.\n\n_To remain ignorant_, is the subject of the copulative verb _is_, and\nthe infinitive, _to remain_, with its complement, _a slave_, is the\npredicate complement.\n\n+433.+ +A participle used as a noun may also be used as the predicate\ncomplement+, thus:\n\n Society is the mingling of many elements.\n\n_Mingling_, in this sentence is a participle of the verb _mingle_, but\nis used as a noun, the predicate complement of the verb _is_. _Society_\nis the subject of the verb.\n\nWhere the present participle is used to form a verb phrase, the\nparticiple is part of the verb phrase, thus:\n\n We are mingling in society.\n\nHere, _are mingling_, is the present progressive verb phrase, and the\nparticiple _mingling_ is not used as a noun or adjective, but is part of\nthe verb phrase _are mingling_.\n\nIf you will observe the different parts of speech carefully, you will\nnot be easily confused as to whether the participle is a noun or a part\nof the verb phrase.\n\n\n Exercise 2\n\nIn the following sentences the incomplete verbs, including infinitives\nand participles, are in italics. Mark the words, phrases or clauses\nwhich are used as objects or complements, to complete the meaning of\nthese verbs.\n\n There _is_ no such thing in America as an independent press, unless it\n _is_ in the country towns.\n\n You _have_ and I _know_ it. There _is_ not one of you who _dares to\n write_ his honest opinions. If you did, you _know_ beforehand that it\n would never appear in print.\n\n I _am paid_ $150.00 a week for _keeping_ my honest opinions out of the\n paper with which I am connected. Others of you _are paid_ similar\n salaries for similar things. Any one of you who _would be_ so foolish\n as _to write_ his honest opinions _would be_ out on the streets\n looking for another job.\n\n The business of the New York journalist _is to destroy_ the truth, to\n lie outright, to pervert, to villify, to fawn at the feet of Mammon,\n and _to sell_ his race and his country for his daily bread.\n\n You _know_ this and I _know_ it. So what folly _is_ this _to be\n toasting_ an \"Independent Press.\"\n\n We _are_ the tools and vassals of rich men behind the scenes. We _are_\n the jumping-jacks; they _pull_ the strings and we dance. Our talents,\n our possibilities and our lives _are_ all the property of other men.\n We _are_ intellectual prostitutes.--_John Swinton_.\n\n\n MODIFIERS OF THE SIMPLE SENTENCE\n\n+434.+ Remember that a simple sentence is one that contains a single\nstatement, question or command. It is a clause, for it contains a\nsubject and a predicate; but it contains only the one subject and the\none predicate. A sentence containing two principal clauses, or a\nprincipal clause and a subordinate clause, would contain two complete\nstatements, questions or commands, therefore it would not be a simple\nsentence, but compound or complex.\n\nRemember, however, that the simple sentences may contain two or more\nsubjects with the same predicate, or two or more predicates with the\nsame subject, or both a compound subject and a compound predicate.\n\n+435.+ The modifiers in a simple sentence are always words or phrases.\nThe modifiers of the subject are either adjectives or adjective phrases.\nThe modifiers of the predicate are either adverbs or adverb phrases. If\nan adjective or an adverb clause is used as a modifier, then the\nsentence is no longer a simple sentence, but becomes a _complex_\nsentence, for it now contains a dependent clause.\n\n\n ORDER OF ELEMENTS\n\n+436.+ The usual order of the principal elements in the sentence is the\nsubject, the predicate and the object or complement, thus:\n\n _Subject_ _Predicate_\n _Men_ _work_\n\n _Subject_ _Predicate_ _Object_\n _Men_ _build_ _houses_\n\n _Subject_ _Predicate_ _Complement_\n _Books_ _are_ _helpful_\n\nThis is called the natural or logical order. Logical means according to\nsense or reason.\n\nAdjectives usually stand before the nouns they modify, thus:\n\n _Good_ books are helpful.\n\nAdverbs may be placed either before or after the verbs they modify,\nthus:\n\n The men _then_ came _quickly_ to the rescue.\n\nThe adverb _then_ precedes the verb _came_, which it modifies; and the\nadverb _quickly_ is placed after the verb.\n\nAdverbs which modify adjectives or other adverbs are placed before the\nwords which they modify, thus:\n\n The _more_ industrious students learn _quite_ rapidly.\n\nIn this sentence, the adverb _more_ is placed before the adjective\n_industrious_, which it modifies; and the adverb _quite_ is placed\nbefore the adverb _rapidly_, which it modifies.\n\nAdjective and adverb phrases usually follow the words which they modify,\nthus:\n\n The men _in the car_ came quickly _to the rescue_.\n The manager _of the mine_ remained _with the men_.\n\nIn this last sentence, the adjective phrase, _of the mine_, is placed\nafter the noun _manager_, which it modifies, and the adverb phrase,\n_with the men_, is placed after the verb _remained_, which it modifies.\n\n+437.+ These sentences illustrate the logical order in which the\nelements of the sentence usually come. But this logical order is not\nstrictly adhered to. Many times, in order to place the emphasis upon\ncertain words, we reverse this order and place the emphasized words\nfirst, as:\n\n _Without your help_, we cannot win.\n\nThe logical order of this sentence is:\n\n We cannot win without your help.\n\nBut we want to place the emphasis upon _your help_, so we change the\norder of the words and place the phrase, _without your help_, first.\n\n+438.+ This inversion of the order helps us to express our thought with\nmore emphasis. Our language is so flexible that we can express the same\nthought in different ways by simply changing the order of the elements\nin the sentence. Notice in the following sentences, the inversion of the\nusual order, and see what difference this makes in the expression of the\nthought.\n\n Uneasy lies the head that wears a crown.\n A more terrible scene you cannot imagine.\n With the shrieking of shot and shell the battle raged.\n Louder and louder thundered the tempest.\n Silently and sadly the men returned to their homes.\n\nTo transpose these inverted sentences--that is to place the elements in\ntheir logical order, gives us an insight into the thought expressed in\nthe sentence. It is worth a great deal to us to be able in our reading\nto see the live elements in the sentence at a glance, and in this way we\ncan grasp at once the thought of the sentence. So you will find that\nthis analyzing of the sentences is very helpful to us in our reading.\n\n+439.+ When we have learned to analyze a sentence quickly we will not be\nlost in the maze of words. A paragraph is often like a string of pearls.\nThe author has a single thread of thought running through the different\nsentences which compose the paragraph and if we have trained ourselves\nwell in sentence analysis, we will never lose this thread. It will be\nlike a life line to which we cling while the breakers of thought and\nemotion roar about us.\n\n\n Exercise 3\n\nIn the following poem, study carefully the inverted order of the\nsentences. Rewrite them, placing the elements in their logical order. As\nfor example:\n\n To the poor man you've been true from of old.\n\nThe elements of the sentence are inverted in this quotation. Rewritten\nin their logical order this would read:\n\n You've been true to the poor man from of old.\n\nYou will note that this inversion is quite common in poetry.\n\n\n HUNGER AND COLD\n\n Sisters, two, all praise to you,\n With your faces pinched and blue;\n To the poor man you've been true,\n From of old;\n You can speak the keenest word,\n You are sure of being heard,\n From the point you're never stirred,\n Hunger and Cold!\n\n Let sleek statesmen temporize;\n Palsied are their shifts and lies\n When they meet your bloodshot eyes,\n Grim and bold;\n Policy you set at naught,\n In their traps you'll not be caught,\n You're too honest to be bought,\n Hunger and Cold!\n\n Let them guard both hall and bower;\n Through the window you will glower,\n Patient till your reckoning hour\n Shall be tolled;\n Cheeks are pale, but hands are red,\n Guiltless blood may chance be shed,\n But ye must and will be fed,\n Hunger and Cold!\n\n God has plans man must not spoil,\n Some were made to starve and toil,\n Some to share the wine and oil,\n We are told;\n Devil's theories are these,\n Stifling hope and love and peace,\n Framed your hideous lusts to please,\n Hunger and Cold!\n\n Scatter ashes on thy head,\n Tears of burning sorrow shed,\n Earth! and be by Pity led\n To love's fold;\n Ere they block the very door\n With lean corpses of the poor,\n And will hush for naught but gore,\n Hunger and Cold!\n\n --_Lowell_.\n\n\n\n\n SPELLING\n\n LESSON 25\n\n\nYou remember in our lesson in the study of consonants we found there\nwere a number of consonants in English which had more than one sound;\nfor example, _c_, _s_, _g_, _x_, etc.\n\nA number of other consonants have sounds which are similar; that is,\nthey are made with the organs of articulation in the same position, only\none is a soft, and the other a hard sound; for example, _p_ and _b_, _t_\nand _d_, _f_ and _v_, etc. These sounds are called cognate sounds.\nCognate means literally _of the same nature_, and so these sounds are of\nthe same nature, only in one the obstruction of the vocal organs is more\ncomplete than in the other.\n\nOur language contains a number of words in which there is a difference\nin the pronunciation of the final consonant when the word is used as a\nnoun and as a verb. The final consonants in these words are the cognate\nsounds, _f_, _v_; _t_, _d_; _th_ soft or _th_ hard, _s_ soft, or _s_\nhard. When the consonant sound is a soft sound, the word is a noun; and\nwhen the consonant sound is a hard sound the word is a verb. For\nexample; _use_ and _use_; _breath_ and _breathe_; _life_ and _live_,\netc.\n\nThe spelling lessons for Monday, Tuesday and Wednesday contain words\nending in cognate sounds, in which the words ending with a soft sound\nare nouns and the words ending in the hard sounds are verbs. Add others\nto this list as they occur to you.\n\nWe have a number of words in the English beginning with _ex_. In some of\nthese words, the _ex_ has the sound of _eks_, and in some of the words\nthe _ex_ has the sound of _egs_. It is not easy at times to know which\nsound to use.\n\nIn regard to the use of _ex_, follow this rule: When a word beginning\nwith _ex_ is followed by an accented syllable beginning with a vowel,\nthe _ex_ is pronounced _egs_; in all other words _ex_ is pronounced\n_eks_; for example, in _executor_, the _ex_ is followed by an accented\nsyllable beginning with a vowel, therefore, _ex_ is pronounced _egs_. In\n_execute_, the _ex_ is followed by an unaccented syllable beginning with\na vowel, and therefore _ex_ is pronounced _eks_. In _explain_, _ex_ is\nfollowed by a syllable beginning with a consonant, and it is therefore\npronounced _eks_.\n\nNote that in words like _exhibit_, _exhort_, etc., the _ex_ is followed\nby a vowel sound, the _h_ being silent, and it is therefore, pronounced\n_egs_, for it is followed by an accented syllable beginning with a vowel\nsound.\n\nThe spelling list for Thursday, Friday and Saturday contains words\nbeginning with _ex_. Watch carefully the pronunciation.\n\n +Monday+\n\n Excuse Excuse\n Abuse Abuse\n Grease Grease\n Sacrifice Sacrifice\n Device Devise\n\n +Tuesday+\n\n Intent Intend\n Advice Advise\n Relief Relieve\n Cloth Clothe\n Reproof Reprove\n\n +Wednesday+\n\n Ascent Ascend\n Strife Strive\n Mouth Mouth\n Grief Grieve\n Bath Bathe\n\n +Thursday+\n\n Exile\n Except\n Exhibit\n Expert\n Exempt\n\n +Friday+\n\n Example\n Excellent\n Exhaust\n Exit\n Expropriate\n\n +Saturday+\n\n Exercise\n Exist\n Experiment\n Exaggerate\n Explanation\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 26\n\n\nDear Comrade:\n\nThere are really two things which will come to us out of the study of\ngrammar. One of these, which we discussed in our letter last week, is\nthe power of logical thinking. The second is the ability to express our\nthoughts correctly; that is, according to accepted usage. So you can\nconsider your spoken and written speech from two viewpoints. First, you\ncan look to see if you have used the words correctly. We have noted\nthese common errors especially in our study of the various parts of\nspeech. There are certain errors we often make, as for example, using a\nplural noun with a singular verb, or using the past time form of the\nverb for the past participle.\n\nWe have noted a great many of these errors in our speech. We might make\nourselves understood and express ourselves fairly accurately and still\nmake these mistakes, but it is wise for us to try to eliminate them from\nour speech for several reasons. To those who understand the use of\ncorrect English, these mistakes mark us as ignorant and uneducated. No\nmatter how important and absolutely accurate the thought we are\nexpressing, if we make these grammatical errors, they very naturally\ndiscount our thought also. They feel that if we cannot speak correctly,\nin all probability we cannot think accurately, either.\n\nThen, too, these words in our speech distract the attention of our\nhearers from the things which we are saying. It is like the mannerism of\nan actor. If he has any peculiar manner of walking or of talking and\npersists in carrying that into whatever character he is interpreting, we\nalways see the actor himself, instead of the character which he is\nportraying. His mannerisms get in the way and interfere with our grasp\nof the idea.\n\nSo in music. You may be absorbed in a wonderful selection which some one\nis playing and if suddenly he strikes a wrong note, the discord\ndistracts your attention and perhaps you never get back into the spirit\nof the music again.\n\nSo we must watch these common errors in our speech, but we must not let\nour study of English be simply that alone. The greatest benefit which we\nare deriving from this study is the analytic method of thought and the\nlogical habit of mind, which the effort to express ourselves clearly and\naccurately and in well-chosen words will give us. Put as much time as\nyou can possibly spare into this analysis of sentences. Take your\nfavorite writer and analyze his sentences and find out what is his\nparticular charm for you. If there is any sentence which gives you a\nlittle trouble and you cannot analyze it properly, copy it in your next\nexamination paper and state where the difficulty lies. Rewrite the\npassages which please you most and then compare your version with the\nauthor's and see if you really grasped his meaning. In this way you will\nadd quickly to your enjoyment of the writing of others and to your power\nof expressing yourself.\n\n Yours for Freedom,\n\n THE PEOPLE'S COLLEGE.\n\n\n THE SIMPLE SENTENCE\n\n+440.+ We have been analyzing the simple sentence, which contains only\nwords and phrases. We have found that there may enter into the simple\nsentence, the following elements:\n\n 1. The simple subject.\n 2. The simple predicate.\n 3. The modifiers of the subject.\n 4. The object of the verb.\n 5. The predicate complement.\n 6. The modifiers of the predicate.\n\nThis is not the order in which the elements will appear in the sentence,\nbut this is the order of their importance. We first look for the simple\nsubject and the simple predicate; then we can determine which words are\nthe modifiers of the subject; then we find the object or predicate\ncomplement of the verb and the modifiers of the verb; and thus we have\nall of the elements which go into the construction of the simple\nsentence.\n\nWe may also have two nouns used as the subject or two verbs used in the\npredicate, connected by a co-ordinate conjunction, thus:\n\n Marx and Engels lived and worked together.\n\nHere we have two proper nouns used as the subject, _Marx_ and _Engels_.\nWe have also two verbs used as the predicate, _lived_ and _worked_. We\ncall this a compound subject and compound predicate.\n\nSo in one simple sentence, that is a sentence which makes a single\nassertion, we may have every part of speech. For example:\n\n The most intelligent men and women think for themselves.\n\nIn this sentence, we have a _noun_, _verb_, _pronoun_, _adjective_,\n_adverb_, _conjunction_ and _preposition_--every part of speech except\nthe _interjection_, which is an independent element and does not enter\ninto the construction of the sentence.\n\n\n Exercise 1\n\nWrite simple sentences of your own containing:\n\n 1. A compound subject.\n 2. A compound predicate.\n 3. A noun as subject modified by one or more adjectives.\n 4. A noun as subject modified by a phrase.\n 5. An incomplete verb with a direct and an indirect object.\n 6. An incomplete verb with a predicate complement.\n 7. A predicate modified by one or more adverbs.\n 8. A predicate modified by an adverb phrase.\n\n\n COMPLEX SENTENCES\n\n+441.+ The simple sentence is the unit of speech. It is a combination of\nwords which makes a single statement, question or command. But many\ntimes a constant repetition of these short sentences would become\ntiresome, and our written and spoken speech would not flow as smoothly\nand rapidly as we desire. So we have evolved a way in which we may\ncombine these sentences into longer statements. Let us take the two\n_simple_ sentences:\n\n\n We are united.\n We shall succeed.\n\nWe may combine these into a single sentence by using the co-ordinate\nconjunction _and_. Then our sentence reads:\n\n We are united and we shall succeed.\n\nThis is a _compound_ sentence, formed by uniting two simple sentences.\nBoth of the clauses are independent and are of equal rank. Neither\ndepends upon the other. They are united by the co-ordinate conjunction\n_and_. We can combine these sentences in a different way. For example,\nwe may say:\n\n If we are united, we shall succeed.\n\nNow we have a subordinate clause, _if we are united_, which is used to\nmodify the verb of the main clause, _succeed_. We have used the\nsubordinate conjunction _if_, and so we have a _complex_ sentence formed\nby uniting the principal clause and a dependent clause.\n\n+442.+ The next step in sentence building, after the simple sentence, is\nthe complex sentence. A complex sentence is a combination of two or more\nsimple sentences, which are so united that one sentence remains the main\nsentence--the backbone, as it were--and the other sentence becomes\nsubordinate or dependent upon it.\n\n+443.+ +A complex sentence is one containing a principal clause and one\nor more subordinate clauses.+\n\n+A principal clause is one which makes a complete statement without the\nhelp of any other clause or clauses.+\n\n+A subordinate or dependent clause is one which makes a statement\ndependent upon or modifying some word or words in the principal clause.+\n\n\n KINDS OF DEPENDENT CLAUSES\n\n+444.+ Dependent clauses are of three kinds. They may be used either as\n_nouns_, _adjectives_ or _adverbs_, and so are called _noun clauses_,\n_adjective clauses_ or _adverb clauses_.\n\n\n NOUN CLAUSES\n\n+445.+ +Noun clauses are those which are used in place of a noun.+ They\nmay be used in any way in which a noun may be used, except as a\npossessive.\n\n1. +The noun clause may be used as the subject of the sentence.+ For\nexample:\n\n _That he is innocent_ is admitted by all.\n\nThe clause, _that he is innocent_ is used as a _noun_, the subject of\nthe sentence.\n\n2. +The noun clause may be used as the object of a verb+, thus:\n\n I admit _that I cannot understand your argument_.\n\nThe clause, _that I cannot understand your argument_, is in this\nsentence the object of the verb _admit_.\n\n3. +The noun clause may be used as the predicate complement+, thus:\n\n The fact is _that this policy will never win_.\n\nThe clause, _that this policy will never win_, is here used in the\npredicate with the copulative verb _is_.\n\n4. +The noun clause may also be used in apposition, explaining the noun\nwith which it is used+, thus:\n\n The motion, _that the question should be reconsidered_, was carried.\n\n_That the question should be reconsidered_, is here a noun clause, used\nin apposition with the noun _motion_, and explains the meaning of the\nnoun.\n\n5. +The noun clause may also be used as the object of a preposition+,\nthus:\n\n I now refer to _what he claims_.\n\nThe noun clause, _what he claims_, is here the object of the\npreposition, _to_.\n\n\n Exercise 2\n\nIn the following sentences the noun clauses are printed in italics.\nDetermine whether they are used as the subject, or object of the verb,\nas predicate complement, in apposition, or as the object of a\npreposition.\n\n 1. The fact is _that I was not listening_.\n 2. _Whatever King Midas looked upon_ turned to gold.\n 3. He acknowledged _what we had suspected_.\n 4. We will never know _what the real situation was_.\n 5. The fact _that the wage is insufficient_ can be easily proved.\n 6. He replied to _what had been asked_.\n 7. The claim was _that he had made a speech inciting to riot_.\n 8. The law _that labor unions are in restraint of trade_ was upheld.\n 9. _That we cannot win by compromise_ is readily apparent.\n 10. Labor demands _that it shall have its full product_.\n 11. _Whoever controls education_ controls the future.\n 12. He came to _where the militia was in camp_.\n\n\n Exercise 3\n\nWrite sentences containing noun clauses used:\n\n 1. As the subject of a verb.\n 2. As the object of a verb.\n 3. As a predicate complement.\n 4. In apposition.\n 5. As the object of a preposition.\n\n\n ADJECTIVE CLAUSES\n\n+446.+ A dependent clause in a complex sentence may also be an adjective\nclause.\n\n+An adjective clause is a clause used as an adjective+, and, hence,\nalways modifies a noun or some word used as a noun, such as a pronoun or\na participle. In Lesson 22, we studied adjective clauses and found that\nthey could be introduced by the relative pronouns, _who_, _which_,\n_that_ and _as_, and also by conjunctions such as, _when_, _where_,\n_whither_, _whence_, etc. An adjective clause may modify any noun or any\nword used as a noun in the sentence.\n\n1. +An adjective clause may modify the subject+, thus:\n\n Men _who have become class-conscious_ do not make good soldiers.\n\nIn this sentence the clause, _who have become class-conscious_, modifies\nthe noun _men_, and is introduced by the relative pronoun _who_.\n\n2. +An adjective clause may modify the noun which is the object of the\nverb+, as:\n\n The men supported the party _which fought for their rights_.\n\nHere the clause, _which fought for their rights_, is an adjective clause\nintroduced by the pronoun _which_, and it modifies the noun _party_,\nwhich is the object of the verb _supported_.\n\n3. +An adjective clause may also be used to modify the noun which is\nused in the predicate complement+, as:\n\n That was the book _which I enjoyed_.\n\nIn this sentence the clause, _which I enjoyed_, is an adjective clause\nmodifying the noun _book_, which is used as the predicate complement\nwith the copulative verb _was_.\n\n4. +An adjective clause may also be used to modify the noun which is\nused as the object of a preposition+, as:\n\n He arrived on the train _which was late_.\n\nHere the adjective clause, _which was late_, modifies the noun _train_,\nwhich is the object of the preposition _on_.\n\nSometimes it is a little difficult to discover these adjective clauses,\nfor frequently the connecting word is omitted, as for example:\n\n I could not find the man _I wanted_.\n\nIn this sentence, the pronoun _whom_ is omitted; the complete sentence\nwould read:\n\n I could not find the man _whom I wanted_.\n\n_Whom I wanted_ is an adjective clause modifying the noun _man_.\n\n\n Exercise 4\n\nIn the following sentences the relative pronouns and the conjunctions\nintroducing adjective clauses are omitted. Rewrite the sentences using\nthe proper relative pronouns and conjunctions. The adjective clauses are\nin italics.\n\n 1. The people _you are seeking_ are not here.\n 2. I have read the book _you brought_.\n 3. The articles _you mentioned_ are not listed.\n 4. I will go to the place _you say_.\n 5. This is a book _you should read_.\n 6. Those are ideals _the people will readily grasp_.\n 7. We make Gods of the things _we fear_.\n 8. I listened to every word _he said_.\n 9. I should love the cause _you love_.\n 10. The things _the people demand_ are just and right.\n\n\n Exercise 5\n\nIn the following sentences the adjective clauses are all printed in\nitalics. Determine whether they modify the subject or the object, the\npredicate complement or the object of the preposition.\n\n 1. In that moment _when he saw the light_ he joined our cause.\n 2. Other men are lenses _through which we read our own minds_.\n 3. This is perhaps the reason _why we are unable to agree_.\n 4. He _that loveth_ maketh his own the grandeur _that he loves_.\n 5. The other terror _that scares us from self-trust_ is our\n consistency.\n 6. There is a popular fable of a sot _who was picked up dead drunk in\n the street, carried to the Duke's house, washed and dressed and\n laid in the Duke's bed, and, on his waking, treated with all\n ceremony like a duke and assured that he had been insane_.\n 7. He _who would gather immortal palms_ must not be hindered by the\n name of goodness, but must explore if it be goodness.\n 8. Superstition, _who is the mother of fear and faith_, still rules\n many people.\n 9. We are looking for the time _when the useful shall be the\n honorable_.\n 10. He _who enslaves another_ cannot be free.\n 11. He _who attacks the right_ assaults himself.\n 12. The force _that is in every atom and every star, in everything\n that grows and thinks, that hopes and suffers_, is the only\n possible God.\n 13. He _who adds to the sum of human misery_ is a blasphemer.\n 14. The grandest ambition _that can enter the soul_ is the desire to\n know the truth.\n\n\n ADVERB CLAUSES\n\n+447.+ The third kind of clause which we may use in a complex sentence\nis the adverb clause.\n\n+An adverb clause is a clause which takes the place of an adverb.+ It\nmay modify a _verb_, an _adjective_, or an _adverb_. We studied adverb\nclauses in lesson 21 and we found eight classes of adverb clauses,\nexpressing _time_, _place_, _cause_ or _reason_, _manner_, _comparison_,\n_condition_, _purpose_ and _result_. For example:\n\n 1. +Adverb clause of time:+\n No man is truly free _until all are free_.\n\n 2. +Adverb clause of place:+\n We must live _where we can find work_.\n\n 3. +Adverb clause expressing cause or reason:+\n We lost the strike _because the men were not class-conscious_.\n\n 4. +Adverb clause of manner:+\n We must work _as if the result depended entirely upon us_.\n\n 5. +Adverb clause of comparison:+\n The working class must become more class-conscious _than it is\n today_.\n\n 6. +Adverb clause of condition:+\n We will continue to be exploited _if we do not demand our rights_.\n\n 7. +Adverb clause expressing purpose:+\n We must read the labor press _in order that we may know the truth\n concerning conditions_.\n\n 8. +Adverb clause expressing result:+\n The battle raged so furiously _that thousands were slain_.\n\n\n ANALYZING COMPLEX SENTENCES\n\n+448.+ To analyze a complex sentence; that is, to break it up into its\ndifferent parts--treat the sentence first as a whole, then find the\nsimple subject and the simple predicate. If a noun clause is the\nsubject, treat it first as a noun. Treat adjective clauses as adjectives\nmodifying certain words and the adverb clauses as adverbs modifying\ncertain words.\n\nIn other words, analyze the sentence first as a simple sentence with\ndependent clauses considered as modifying words; then analyze each\ndependent clause as though it were a simple sentence. Make an outline\nlike the following and use it in your analysis of the sentence. Let us\ntake this sentence and analyze it:\n\n Conscious solidarity in the ranks would give the working class of the\n world, now, in our day, the freedom which they seek.\n\n +Simple subject+, _solidarity_.\n\n +Simple predicate+, _would give_.\n\n Modifiers of the subject:\n\n Adjective, _conscious_.\n Adjective phrase, _in the ranks_.\n Adjective clause, (_none_).\n\n +Complete subject+, _Conscious solidarity in the ranks_.\n\n Modifiers of the predicate:\n\n Adverb, _now_.\n Adverb phrase, _in our day_.\n Adverb clause, (_none_).\n\n +Direct object+, _freedom_.\n\n Modifiers of direct object:\n\n Adjective, _the_.\n Adjective phrase, (_none_).\n Adjective clause, _which they seek_,\n\n +Indirect object+, _class_.\n\n Modifiers of indirect object:\n\n Adjectives, _the_, _working_.\n Adjective phrase, _of the world_.\n Adjective clause, (_none_).\n\n +Complete predicate+, _would give the working class of the world,\n now, in our day, the freedom which they seek_.\n\nAnalyze the dependent clause, _which they seek_, just as a principal\nclause is analyzed. _They_ is the simple subject, _seek_ is the simple\npredicate, _which_ is the direct object. The complete predicate is _seek\nwhich_.\n\n+449.+ Notice that the first two sentences given in the exercise below\nare imperative sentences,--the subject, the pronoun _you_, being omitted\nso that the entire sentence is the complete predicate. As for example:\n_Take the place which belongs to you_. The omitted subject is the\npronoun _you_. _Take the place which belongs to you_ is the complete\npredicate, made up of the simple predicate _take_; its object, the noun\n_place_; the adjective _the_, and the adjective clause, _which belongs\nto you_, both of which modify the noun _place_.\n\n\n Exercise 6\n\nUsing the outline given above, analyze the following complex sentences.\n\n 1. Take the place which belongs to you.\n 2. Let us believe that brave deeds will never die.\n 3. The orator knows that the greatest ideas should be expressed in\n the simplest words.\n 4. Gratitude is the fairest flower that sheds its perfume in the\n human heart.\n 5. Children should be taught that it is their duty to think for\n themselves.\n 6. We will be slaves as long as we are ignorant.\n 7. We must teach our fellow men that honor comes from within.\n 8. Cause and effect cannot be severed for the effect already blooms\n in the cause.\n 9. Men measure their esteem of each other by what each has.\n 10. Our esteem should be measured by what each is.\n 11. What I must do is all that concerns me.\n 12. The great man is he who, in the midst of the crowd, keeps the\n independence of solitude.\n 13. The only right is what is after my constitution.\n 14. Whoso would be a man must be a non-conformist.\n 15. They who build on ideas build for eternity.\n\n\n Exercise 7\n\nWe have studied all the parts of speech, and now our work is to combine\nthese parts for the expression of thought. It will be good practice and\nvery helpful to us to mark these different parts of speech in our\nreading. This helps us to grow familiar with their use. It also helps us\nto add words to our vocabulary and to learn how to use them correctly.\nIn the following quotation, mark underneath each word, the name of every\npart of speech. Use _n._ for noun, _v._ for verb, _pro._ for pronoun,\n_adv._ for adverb, _adj._ for adjective, _p._ for preposition and _c._\nfor conjunction. Write _v. p._ under the verb phrases. For example:\n\n +The workers of the world do not have,\n _adj._ _n._ _p._ _adj._ _n._ _v.p._ _adv._ _v.p._\n\n under this system, very many opportunities\n _p._ _adj._ _n._ _adv._ _adj._ _n._\n\n for rest and pleasure for themselves.+\n _p._ _n._ _c._ _n._ _p._ _pro._\n\n\nMark in this manner every part of speech in the following quotation:\n\n The history of all hitherto existing society is the history of class\n struggles.\n\n Freeman and slave, patrician and plebeian, lord and serf, guildmaster\n and journeyman,--in a word, oppressor and oppressed,--stood in\n constant opposition to one another, carried on an uninterrupted, now\n hidden, now open fight, a fight that each time ended, either in a\n revolutionary re-constitution of society at large, or in the common\n ruin of the contending classes.\n\n In the earlier epochs of history, we find almost everywhere a\n complicated arrangement of society into various orders, a manifold\n gradation of social rank. In ancient Rome we have patricians, knights,\n plebeians, slaves; in the middle ages, feudal lords, vassals,\n guild-masters, journeymen, apprentices, serfs; in almost all of these\n classes, again, subordinate gradations.\n\n The modern bourgeois society, that has sprouted from the ruins of\n feudal society, has not done away with class antagonisms. It has but\n established new classes, new conditions of oppression, new forms of\n struggle in place of the old ones.\n\n --_Communist Manifesto_.\n\n\n Exercise 8\n\nIn the following quotation, mark all of the clauses and determine\nwhether they are dependent or independent clauses. If they are dependent\nclauses, determine whether they are noun, adjective or adverb clauses.\nMark all the sentences and tell whether they are simple or complex.\n\nI see in the near future a crisis approaching that unnerves me, and\ncauses me to tremble for the safety of my country. As a result of war,\ncorporations have been enthroned, and an era of corruption in high\nplaces will follow. The money power of the country will endeavor to\nprolong its reign by working upon the prejudices of the people, until\nall the wealth is aggregated in a few hands, and the republic is\ndestroyed. I feel at this moment more anxiety for the safety of our\ncountry than ever before, even in the midst of war. God grant that my\nforebodings may be groundless. Monarchy itself is sometimes hinted at as\na refuge from the power of the people. In my present position I could\nscarcely be justified were I to omit to raise a warning voice against\nthe approach of a returning despotism.... It is assumed that labor is\navailable only in connection with capital; that nobody labors unless\nsomebody else, owning capital, somehow, by the use of it, induces him to\nlabor. Labor is prior to and independent of capital. Capital is only the\nfruit of labor, and could not have existed if labor had not first\nexisted. Labor is the superior of capital, and deserves much the higher\nconsideration. I bid the laboring people beware of surrendering the\npower which they possess, and which, if surrendered, will surely be used\nto shut the door of advancement for such as they, and fix new\ndisabilities and burdens upon them until all of liberty shall be lost.\n\n * * * * *\n\nIn the early days of our race the Almighty said to the first of mankind,\n\"In the sweat of thy face shalt thou eat bread,\" and since then, if we\nexcept the light and air of Heaven, no good thing has been or can be\nenjoyed by us without first having cost labor. And inasmuch as most good\nthings have been produced by labor, it follows that all such things\nbelong of right to those whose labor has produced them. But it has so\nhappened, in all ages of the world, that some have labored and others\nhave without labor enjoyed a large portion of the fruits. This is wrong,\nand should not continue. To secure to each laborer the whole product of\nhis labor, as nearly as possible, is a worthy object of any government.\n\n * * * * *\n\nIt seems strange that any man should dare to ask a just God's assistance\nin wringing bread from the sweat of other men's faces.\n\n * * * * *\n\nThis country, with its institutions, belongs to the people who inhabit\nit.\n\n --_Lincoln_.\n\n\n Exercise 9\n\nIn the following poem find all of the assertive, interrogative and\nimperative sentences. Mark all of the simple sentences and all of the\ncomplex sentences. Mark all of the dependent clauses and determine\nwhether each is used as a noun, adjective or adverb clause. The verbs\nand the verb phrases are in italics.\n\n _Shall_ you _complain_ who _feed_ the world,\n Who _clothe_ the world,\n Who _house_ the world?\n _Shall_ you _complain_ who _are_ the world,\n Of what the world _may do_?\n As from this hour you _are_ the power,\n The world _must follow_ you.\n\n The world's life _hangs_ on your right hand,\n Your strong right hand,\n Your skilled right hand;\n You _hold_ the whole world in your hand;\n _See_ to it what you _do_!\n For dark or light or wrong or right,\n The world _is made_ by you.\n\n Then _rise_ as you never _rose_ before,\n Nor _hoped_ before,\n Nor _dared_ before;\n And _show_ as never _was shown_ before\n The power that _lies_ in you.\n _Stand_ all as one; _see_ justice done;\n _Believe_ and _dare_ and _do_.\n\n --_Charlotte Perkins Gilman_.\n\n\n\n\n SPELLING\n\n LESSON 26\n\n\nIn our last lesson we had examples of words in which the _s_ had the\nsoft sound, and also of words in which the _s_ had the sound of _z_. In\nsome English words, it is difficult to determine which sound to use.\nThere are a number of words in English beginning with _dis_. In a few of\nthe words, the _s_ has the sound of _z_, and in other words it has the\nsound of _s_. There are only a few words which are pronounced with the\n_diz_ sound. _Discern_, _dismal_ and _dissolve_ are always pronounced\nwith the _diz_ sound. _Disease_ and _disaster_ are pronounced both ways.\nSome dictionaries give the _diz_ sound and some give the _dis_ sound.\n\nThe spelling lesson for Monday, Tuesday and Wednesday contains a number\nof words beginning with _dis_. Be sure of the pronunciation. Run through\nthe words in the dictionary beginning with the _dis_ sound and mark\nthose in which the _dis_ has the sound of _diz_.\n\nWe have also a number of words in the English language which end in\n_ise_ or _ize_, and we are often confused to know which ending to use.\nThere is a rule, which has very few exceptions, which covers the use of\n_ise_ and _ize_. Words should be spelled with the _ize_ ending when the\n_ize_ can be cut off, and the word that is left can be used alone. For\nexample; _author_, _authorize_. In this word you can cut off the _ize_\nand the word _author_ can be used alone. But in the word _exercise_, if\nyou cut off the _ise_, the remaining portion cannot be used alone.\n\n_Recognize_ and _criticise_ are exceptions to this rule. When used as a\nsuffix added to a noun or adjective to form a verb, _ize_ is the proper\nending; as _theory_, _theorize_, _civil_, _civilize_, etc. Final _e_ or\n_y_ is dropped before _ize_, as in the words _memorize_, _sterilize_,\netc.\n\nThe spelling lesson for Thursday, Friday and Saturday contains a number\nof common words ending with _ize_ or _ise_. Study carefully this list\nand add as many words to it as you can.\n\n +Monday+\n\n Disappear\n Distress\n Discern\n Disburse\n Discipline\n\n +Tuesday+\n\n Discount\n Discredit\n Distribute\n Dismal\n Disseminate\n\n +Wednesday+\n\n Disguise\n Distance\n Dissolve\n Discontent\n Disposition\n\n +Thursday+\n\n Franchise\n Civilize\n Surprise\n Organize\n Compromise\n\n +Friday+\n\n Monopolize\n Revise\n Legalize\n Enterprise\n Capitalize\n\n +Saturday+\n\n Memorize\n Advertise\n Theorize\n Comprise\n Systematize\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 27\n\n\nDear Comrade:\n\nIngersoll said: \"Words are the garments of thought and the robes of\nideas.\" This is a beautiful and poetic way of expressing the\nrelationship between words and thoughts. Words are really the body which\nwe give to our thoughts. Until they are clothed in words, our thoughts\nare only ghosts of ideas. Other people cannot see or come into contact\nwith them, and they can have but little influence upon the world.\n\nWithout thought, no language is possible. It is equally true that\nwithout language, no growth of thought is possible. It is futile to try\nto determine which is first, language or thought. The two are entirely\nnecessary to each other and make possible social and individual\ndevelopment.\n\nEvery time that you add a word to your vocabulary, you have added to\nyour mental equipment. You have also added greatly to your power of\nenjoyment. Through these words you will come into a new relationship to\nyour fellow men. Each new word enlarges the circle of your acquaintance.\nA knowledge of language brings us into a circle of wonderful friends.\nWhen we have learned to read we need never more be lonely. Some one has\nwritten in a book somewhere just the thing we are hungry for at this\nmoment.\n\nIn the pages of a book we can meet and talk with the great souls who\nhave written in these pages their life's experience. No matter what mood\nyou are in, you can find a book to suit that mood. No matter what your\nneed, there is a book which meets that need. Form the habit of reading\nand you will find it a wonderful source of pleasure and of profit.\n\nNor do we need to be barred because of our lack of educational\nadvantages in our youth. Buckle, the author of the greatest history that\nhas ever been written, left school at the age of fourteen, and it is\nsaid that at that age, except a smattering of mathematics, he knew only\nhow to read; but when he died at the age of forty, this man, who did not\nknow his letters when he was eight years old, could read and write seven\nlanguages and was familiar with ten or twelve more. He had written a\nwonderful book and had become a teacher of teachers. Engraven upon his\nmarble altar tomb is the following couplet:\n\n \"The written word remains long after the writer.\n The writer is resting under the earth, but his words endure.\"\n\nGood books are so cheap nowadays that they are within the reach of every\none of us. Let us not be content to live in the narrow world of work and\nworry. Let us forget the struggle occasionally in the reading of books,\nand let us prepare ourselves, by reading and studying, for the battle\nfor the emancipation of the workers of the world.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n KINDS OF SENTENCES\n\n+450.+ +A simple sentence is a sentence which makes a single assertion,\nquestion or command.+\n\nThe simple sentence contains only words and phrases.\n\n+451.+ +A complex sentence is one which contains a principal statement\nand one or more modifying statements.+\n\nThe statements made in addition to the principal statement are made in\ndependent clauses. The complex sentence has only one main clause and one\nor more dependent clauses.\n\n+452.+ +A compound sentence is one which contains two or more\nindependent clauses.+\n\nThese compound sentences may contain any number of dependent clauses but\nthey must always have at least two independent or principal clauses.\nThese principal clauses are always connected by co-ordinate\nconjunctions, for the principal clauses in a compound sentence are\nalways of equal rank or order.\n\n\n Exercise 1\n\nReview the lesson on co-ordinate conjunctions and notice which\nconjunctions are used to unite principal clauses into single sentences.\nUse these co-ordinate conjunctions to unite the following pairs of\nsimple sentences into compound sentences. For example:\n\n The sun rises _and_ the day dawns.\n The men work _but_ the boys play.\n\nThe sun rises. The day dawns.\n\nHe studies diligently. He learns rapidly.\n\nHe came early. He could not stay.\n\nThe weather is cold. The plants are not growing.\n\nThe men work. The boys play.\n\nThe day is cold. The wind is blowing.\n\nTake the above sentences and use subordinate instead of co-ordinate\nconjunctions, and make complex sentences instead of compound out of each\npair of simple sentences. For example:\n\n _When_ the sun rises, the day dawns.\n The men work _while_ the boys play.\n\n\n KINDS OF COMPOUND SENTENCES\n\n+453.+ +Compound sentences may be made up of two simple sentences.+\n\nRewrite the following compound sentences, making of each sentence two\nsimple sentences:\n\n The birds are singing and spring is here.\n He believes in war but his brother is against it.\n We must arouse ourselves or we shall be involved.\n He will not study nor will he allow any one else to study.\n\n+454.+ +A compound sentence may be made up of a simple sentence and a\ncomplex sentence, joined by a co-ordinate conjunction.+ For example:\n\n John goes to school, but Mary stays at home in order that she may help\n her mother.\n\nThis compound sentence is made up of the simple sentence, _John goes to\nschool_, and the complex sentence, _Mary stays at home in order that she\nmay help her mother_.\n\n+455.+ +Both parts of the compound sentence may be complex; that is,\nboth principal clauses in a compound sentence may contain dependent\nclauses.+ For example:\n\n John goes to school where his brother goes, but Mary stays at home in\n order that she may help her mother.\n\nThis compound sentence is made up of two complex sentences. The\nsentence, _John goes to school where his brother goes_, is complex\nbecause it contains the dependent clause, _where his brother goes_; the\nsentence, _Mary stays at home in order that she may help her mother_, is\ncomplex because it contains the dependent clause, _in order that she may\nhelp her mother_.\n\n\n Exercise 2\n\nRead carefully the following sentences, determine which are simple\nsentences, which are complex and which are compound.\n\n 1. When the state is corrupt, then the laws are most multiplied.\n 2. To teach the alphabet is to inaugurate revolution.\n 3. Freedom degenerates unless it has to struggle in its own defense.\n 4. The destroyers have always been honored.\n 5. Liberty of thought is a mockery if liberty of speech is denied.\n 6. Where slavery is, there liberty cannot be; and where liberty is,\n there slavery cannot be.\n 7. All our greatness was born of liberty and we cannot strangle the\n mother without destroying her children.\n 8. In the twentieth century, war will be dead, but man will live.\n 9. The abuse of free speech dies in a day, but the denial entombs the\n hope of the race.\n\n\n SENTENCE ANALYSIS\n\n+456.+ There is no more important part of the study of English than the\nanalysis of sentences. The very best result that can come to one from\nthe study of grammar is the logical habit of mind. The effort to analyze\na difficult passage gives us a fuller appreciation of its meaning. This\ncultivates in us accuracy, both of thought and of expression. So, spend\nas much time as you can on the analysis of sentences.\n\nThe subject and the predicate are the very body of the sentence, upon\nwhich all the rest of the sentence is hung. The other parts of the\nsentence are but the drapery and the garments which clothe the body of\nthe sentence. Hence, the most important thing in sentence analysis is to\nbe able to discover the _subject_ and _predicate_.\n\nIn the expression of a thought, there are always two important\nessentials, that about which something is said,--which constitutes the\nsubject,--and that which is said about the subject, which constitutes\nthe predicate.\n\nThere may be a number of modifying words, phrases or subordinate\nclauses, but there is always a main clause which contains a simple\nsubject and a simple predicate. Find these first, and you can then fit\nthe modifying words and phrases and clauses into their proper places.\n\n+457.+ Let us take for study and analysis the following paragraph from\nJack London:\n\n Man's efficiency for food-getting and shelter-getting has not\n diminished since the day of the cave-man. It has increased a\n thousand-fold. Wonderful artifices and marvelous inventions have been\n made. Why then do millions of modern men live more miserably than the\n cave-man lived?\n\nLet us take the first sentence out of this paragraph and analyze it.\n_Man's efficiency for food-getting and shelter-getting has not\ndiminished since the day of the cave-man._ What is the main word in this\nsentence--the word about which the entire statement is made? Clearly it\nis the word _efficiency_. _Efficiency_ is the noun which is the subject\nof the sentence.\n\nThen you might ask _what sort of_ efficiency and _whose_ efficiency?\nWhat sort of efficiency is explained by the adjective phrase, _for\nfood-getting and shelter-getting_. Whose efficiency is explained by the\npossessive noun, _man's_. Therefore, the complete subject is, _Man's\nefficiency for food-getting and shelter-getting_.\n\nNow we are ready to consider the predicate. What has efficiency done? It\n_has not diminished_. _Has diminished_ is the verb phrase, which is the\nsimple predicate of this sentence. It is modified by the adverb _not_,\nso we have _Man's efficiency has not diminished_. Then we might ask,\n_when_ has it not diminished? And this is answered by the phrase, _since\nthe day of the cave-man_. So we have our complete predicate, _Has not\ndiminished since the day of the cave-man_.\n\nIn this way we can analyze or break up into its different parts, every\nsentence. First find the subject, then ask what that subject does, and\nthe answer will be the predicate or verb. Do not confuse the verb with\nthe words which state _how_ or _why_ the action is performed, and do not\nconfuse the verb with the _object_ of the action. The verb simply\nasserts the action. The other words will add the additional information\nas to how or why or when or upon whom the action was performed.\n\nLet us finish the analysis of the sentences in the paragraph quoted from\nJack London. In the second sentence, _It has increased a thousand-fold_,\nthe personal pronoun _it_, which refers to the noun _efficiency_, is the\nsubject of the sentence; and when you ask what _it_ has _done_, you find\nthat the question is answered by the verb, _has increased_. Therefore,\n_has increased_ is the verb in the sentence. The noun, _thousand-fold_\nis used as an adverb telling how much it has increased. It is an\nadverb-noun, which you will find explained in Section 291.\n\nIn the next sentence, _Wonderful artifices and marvelous inventions have\nbeen made_, we find two _nouns_ about which a statement is made.\n_Artifices_ have been made and _inventions_ have been made; so\n_artifices_ and _inventions_ are both the _subjects_ of the sentence.\nTherefore, we have a compound subject with a single verb, _have been\nmade_. _Artifices_ is modified by the adjective _wonderful_, and\n_inventions_ is modified by the adjective _marvelous_, so we have\n_wonderful artifices and marvelous inventions_, as the complete subject,\nand _have been made_, as the complete predicate.\n\nIn the last sentence, _Why then do millions of modern men live more\nmiserably than the cave-man lived?_, we find a sentence which is a\ntrifle more difficult of analysis. It is written in the interrogative\nform. If you find it difficult to determine the subject and the verb or\nverb phrase in an interrogative sentence, rewrite the sentence in the\nassertive form, and you will find it easier to analyze.\n\nWhen we rewrite this sentence we have, _Millions of modern men do live\nmore miserably than the cave-man lived_. Now it is evident that the noun\n_millions_ is the subject of the sentence. We see quickly that _men_\ncannot be the subject because it is the object of the preposition _of_,\nin the phrase, _of modern men_. So we decide that the noun _millions_ is\nthe simple subject.\n\nWhen we ask the question what millions _do_, our question is answered by\nthe verb phrase, _do live_. So _do live_ is the simple predicate, and\nthe skeleton of our sentence, the simple subject and the simple\npredicate, is _millions do live_. The subject _millions_ is modified by\nthe adjective phrase _of modern men_.\n\nThen we ask, _how_ do men live? And we find our question answered by\n_they live miserably_. But we are told _how_ miserably they live by the\nadverb _more_ and the adverb clause, _than the cave-man lived_, both\nmodifying the adverb _miserably_. So we have our complete predicate, _do\nlive more miserably than the cave-man lived_.\n\nThis interrogative sentence is introduced by the interrogative adverb\n_why_.\n\nDo not drop this subject until you are able to determine readily the\n_subject_ and _predicate_ in every sentence and properly place all\nmodifying words. There is nothing that will so increase your power of\nunderstanding what you read, and your ability to write clearly, as this\nfacility in analyzing sentences.\n\n\n Exercise 3\n\nThe following is Elbert Hubbard's description of the child-laborers of\nthe Southern cotton-mills. Read it carefully. Notice that the sentences\nare all short sentences, and the cumulative effect of these short\nsentences is a picture of the condition of these child-workers which one\ncan never forget. The subjects and predicates are in italics. When you\nhave finished your study of this question, rewrite it from memory and\nthen compare your version with the original version.\n\n _I thought_ that _I would lift_ one of the little toilers. _I wanted_\n to ascertain his weight. Straightway through his thirty-five pounds of\n skin and bone there _ran_ a _tremor_ of fear. _He struggled_ forward\n to tie a broken thread. _I attracted_ his attention by a touch. _I\n offered_ him a silver dime. _He looked_ at me dumbly from a face _that\n might have belonged_ to a man of sixty. _It was_ so furrowed, tightly\n drawn and full of pain. _He did_ not _reach_ for the money. _He did_\n not _know_ what _it was_. There _were dozens_ of such children in this\n particular mill. A _physician who was_ with me _said_ that _they\n would_ probably all _be_ dead in two years. Their _places would be_\n easily _filled_, however, for there _were_ plenty _more_. _Pneumonia\n carries_ off most of them. Their _systems are_ ripe for disease and\n when _it comes_ there _is_ no _rebound_. _Medicine_ simply _does_ not\n _act_. _Nature is whipped, beaten, discouraged._ _The child sinks_\n into a stupor and _dies_.\n\n\n Exercise 4\n\nIn the following sentences, mark the simple sentences, the complex\nsentences and the compound sentences, and analyze these sentences\naccording to the rules given for analyzing simple sentences, complex\nsentences and compound sentences:\n\n 1. Force is no remedy.\n 2. Law grinds the poor, and the rich men rule the law.\n 3. Force and fraud are in war the two cardinal virtues.\n 4. Freedom is a new religion, a religion of our time.\n 5. Desire nothing for yourself which you do not desire for others.\n 6. An ambassador is a man who goes abroad to lie for the good of his\n country.\n 7. A journalist is a man who stays at home to pursue the same\n vocation.\n 8. Without free speech no search for truth is possible.\n 9. Liberty for the few is not liberty.\n 10. Liberty for me and slavery for you mean slavery for both.\n 11. No revolution ever rises above the intellectual level of those who\n make it.\n 12. Men submit everywhere to oppression when they have only to lift\n their heads to throw off the yoke.\n 13. Many politicians of our time are in the habit of saying that no\n people ought to be free till they are fit to use freedom. The\n maxim is worthy of the fool in the old story who resolved not to\n go into the water till he had learned to swim. If men are to wait\n for liberty till they become wise and good in slavery they may\n indeed wait forever.\n\n\n SUMMARY\n\n+458.+ The following is a summary of that which we have learned in\nsentence building:\n\n { { Assertive\n { Use { Interrogative\n Sentences are { { Imperative\n classified { { Exclamatory\n according to {\n { { Simple\n { Form { Complex\n { { Compound\n\n Elements { +Words+, the eight parts of speech.\n of { +Phrases+, adjective, adverb and verb phrases.\n The Sentence. { +Clauses+, adjective, adverb and noun clauses.\n\n\n+459.+\n ESSENTIALS OF A SIMPLE SENTENCE\n\n +Subject+ +Predicate+\n\n Subject Complete Verb\n Subject Copulative Verb Predicate Complement\n Subject Transitive Verb Direct Object\n Subject Transitive Verb Direct Object Indirect Object\n\n+460.+\n THE SUBJECT\n\n { _Noun_--The _man_ came.\n { _Pronoun_--_He_ came.\n +The simple subject+ { _Adjective_--The _poor_ came.\n may be { _Infinitive_--_To find_ work is difficult.\n { _Participle_--_Walking_ is good exercise.\n { _Clause_--_What I learn_ cannot be lost.\n\n +Complete subject+--Simple subject and modifiers.\n\n Modifiers of the Subject\n\n { Word--_Wealthy_ men rule.\n +Adjective+ { Phrase--Men _of wealth_ rule.\n { Clause--Men _who are wealthy_ rule.\n\n +Possessive+--The _man's_ energy was great.\n\n { Word--The poet, _Lowell_, was the author.\n +Appositive+ { Clause--The fact, _that you came_, pleases me.\n\n { The soldiers, _wounded and dying_, were\n +Participle+ { left on the field\n\n +Infinitive+--A plan _to end the war_ was discussed.\n\n+461.+\n THE PREDICATE\n\n +The simple+ { _Verb_--The man _came_.\n +predicate+ { _Verb phrase_--The man _has been coming_ daily.\n\n { +Predicate Complement+--The man was a _hero_.\n A COMPLETE { +Direct Object+--The man brought the _book_.\n PREDICATE { +The Indirect Object+--The man brought _me_ the book.\n _equals a verb {\n or verb phrase { {_Word_--The man works _rapidly_.\n and_ { +Adverb+ { _Phrase_--The man works _in the factory_.\n { +Modifiers+ { _Clause_--The man works _whenever he\n { can_.\n\n { _Words_--The man works hard.\n SIMPLE SENTENCES { _Phrases_--The man _on your right_ works _in the\n CONTAIN ONLY { factory._\n\n { _Words_, The man works steadily\n +Complex sentences+ { _Phrases_ in the factory _whenever\n +contain+ { and there is work_.\n { _Dependent clauses._\n\n+Compound sentences contain+ two or more principal clauses, as:\n\n _The sun rises_ and _the day dawns_.\n\n+462.+ Take the simple subjects and simple predicates in Exercise 5, and\nbuild up sentences; first, by adding a word, then a phrase and then a\nclause to modify the subject; then add a word and a phrase and a clause\nto modify the predicate.\n\nSo long as you have only words and phrases you have simple sentences.\nWhen you add a dependent clause you have a complex sentence. When you\nunite two independent clauses in one sentence, then you have a compound\nsentence, and the connecting word will always be a co-ordinate\nconjunction. These will be readily distinguished for there are only a\nfew co-ordinate conjunctions.\n\nGo back to the lesson on co-ordinate conjunctions and find out what\nthese are, and whenever you find two clauses connected by these\nco-ordinate conjunctions you know that you have a compound sentence.\nRemember that each clause must contain a subject and predicate of its\nown. When you have two words connected by these co-ordinate conjunctions\nyou do not have a clause. Each clause must contain a subject and a\npredicate of its own.\n\n+463.+ Here is an example of a sentence built up from a simple subject\nand a simple predicate:\n\n\n SIMPLE SUBJECT ENLARGED\n\n+Simple Subject and Predicate+--_Soldiers obey._\n\n_Adjectives_ added--_The enlisted_ soldiers obey.\n\n_Phrase_ added--The enlisted soldiers _in the trenches_ obey.\n\n_Clause_ added--The enlisted soldiers in the trenches, _who are\ndoomed to die_, obey.\n\n\n SIMPLE PREDICATE ENLARGED\n\n+Simple Subject and Predicate+--_Soldiers obey._\n\n_Object_ added--Soldiers obey _orders_.\n\n_Adverb_ added--Soldiers obey orders _quickly_.\n\n_Phrase_ added--Soldiers obey orders quickly and _without\nquestion_.\n\n_Clause_ added--Soldiers obey orders quickly and without question\n_because they are taught to do so_.\n\nCombining our enlarged subject and predicate we have the sentence:\n\n The enlisted soldiers in the trenches, who are doomed to die, obey\n orders quickly and without question because they are taught to do so.\n\nThis is a complex sentence because it contains dependent clauses. We\nmight add another independent clause and make of this a compound\nsentence. For example:\n\n The enlisted soldiers in the trenches, who are doomed to die, obey\n orders quickly and without question because they are taught to do so,\n and _this is patriotism_.\n\n\n Exercise 5\n\nEnlarge the following simple subjects and simple predicates:\n\n\n Men write.\n Boys play.\n People study.\n The law rules.\n\n\n Exercise 6\n\n In the following poem underscore all of the dependent clauses.\n Determine whether they are noun, adjective or adverb clauses. Do you\n find any simple or compound sentences in this poem?\n\n MEN! whose\n boast it is that ye\n Come of fathers brave and free,\n If there breathe on earth a slave,\n Are you truly free and brave?\n If ye do not feel the chain,\n When it works a brother's pain,\n Are ye not base slaves indeed,\n Slaves unworthy to be freed?\n\n Women! who shall one day bear\n Sons to breathe New England air,\n If ye hear without a blush,\n Deeds to make the roused blood rush\n Like red lava through your veins,\n For your sisters now in chains,--\n Answer! are you fit to be\n Mothers of the brave and free?\n\n Is true Freedom but to break\n Fetters for our own dear sake,\n And, with leathern hearts, forget\n That we owe mankind a debt?\n No! true freedom is to share\n All the chains our brothers wear,\n And, with heart and hand, to be\n Earnest to make others free!\n\n They are slaves who fear to speak\n For the fallen and the weak;\n They are slaves who will not choose\n Hatred, scoffing, and abuse,\n Rather than in silence shrink\n From the truth they needs must think;\n They are slaves who dare not be\n In the right with two or three.\n\n --_Lowell_.\n\n\n\n\n SPELLING\n\n LESSON 27\n\n\nWe have studied concerning the formation of derivatives by the addition\nof suffixes. Derivatives are also formed by the addition of prefixes.\nYou remember that a prefix is a syllable which is placed before a simple\nword to form the derivative. Among the most common of these prefixes are\n_in_, _un_ and _mis_. The prefix _in_ used with an adjective or adverb\nmeans _not_; for example, _insane_ means _not_ sane; _incorrect_ means\n_not_ correct, etc.\n\nThe prefix _in_ used with a noun means _lack of_; for example,\n_inexperience_ means _lack of_ experience; _inability_ means _lack of_\nability, etc.\n\nIn words beginning with _m_ or _p_, _in_, meaning _not_ or _lack of_, is\nchanged to _im_. This is done for the sake of euphony. The _n_ does not\nunite readily with the sound of _m_ or _p_. So we do not say _inmodest_\nand _inpartial_, but _immodest_ and _impartial_.\n\nThe prefix _un_, used with participles, means _not_; for example,\n_unprepared_ means _not_ prepared; _unguarded_ means _not_ guarded, etc.\n\nThe prefix _un_ used with verbs, means to take off or to reverse; for\nexample, _uncover_ means to take off the cover; _untwist_ means to\nreverse the process of the twisting.\n\nThe prefix _un_ used with adjectives means _not_; for example,\n_uncertain_ means _not_ certain; _uncommon_ means _not_ common.\n\nThe prefix _mis_ used with nouns or verbs, means _wrong_. For example,\n_mistreatment_ means _wrong_ treatment; _to misspell_ means to spell\n_wrong_.\n\nAdd the prefix _in_ to the nouns given in Monday's list; add the prefix\n_in_ to the adjectives given in Tuesday's list; add the prefix _im_ to\nthe adjectives and nouns in Wednesday's lesson; add the prefix _un_ to\nthe participles and adjectives in Thursday's lesson; add the prefix _un_\nto the verbs in Friday's lesson, and add the prefix _mis_ to the nouns\nand verbs in Saturday's lesson.\n\n +Monday+\n\n Tolerance\n Frequency\n Competence\n Efficiency\n Coherence\n\n +Tuesday+\n\n Convenient\n Expedient\n Famous\n Adequate\n Solvent\n\n +Wednesday+\n\n Pertinent\n Morality\n Patience\n Moderate\n Pious\n\n +Thursday+\n\n Balanced\n Biased\n Gracious\n Stable\n Solicited\n\n +Friday+\n\n Burden\n Veil\n Fasten\n Screw\n Furl\n\n +Saturday+\n\n Construe\n Apprehension\n Inform\n Guide\n Judge\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 28\n\n\nDear Comrade:\n\nWe are beginning with this lesson the study of the use of capitals and\nof punctuation. The use of capitals as well as punctuation has nothing\nto do with our spoken words, but both are very important in our written\nlanguage.\n\nThere is nothing that will mark us as uneducated more quickly than bad\nspelling, faulty punctuation and the incorrect use of capitals.\n\nThe rules for the use of capitals may seem somewhat arbitrary. After an\nunderstanding of them, however, you will discover that they are not\narbitrary, but are based upon a single principle. The word which is of\nthe most importance, or which should receive the most emphasis is the\nword which is capitalized, as for example, the principal words in a\ntitle, the first words in a sentence, proper names, etc.\n\nStudy these rules carefully, note the use of the capitals in your\nreading and watch your written language carefully for a time. Soon the\nproper use of capitals will seem easy and most natural. In the meantime\ndo not fail to keep up your study of words. Add at least one word to\nyour vocabulary every day.\n\nDid you ever consider how we think in pictures? Nearly every word that\nwe use calls up a certain image or picture in our minds. The content of\nwords has grown and developed as our ability to think has developed.\n\nTake, for example, words like head or hand. Head originally referred to\na portion of the body of a living thing; then it was used to refer to\nsome part of an inanimate object which might resemble or call up a\npicture of an animal's head, for example, the head of a pin. Again, it\nwas used to refer to some part of an inanimate thing which was\nassociated with the head of a human being, as the head of the bed. Then,\nby the power of association, since the head was considered the most\nconspicuous and important part of the body, that which was most\nconspicuous and important was called the head, as the head of the army,\nthe head of the nation.\n\nThen, since the head was the seat of the brain and of the mental\nfaculties, the head was often used instead of the brain or mental\nfaculties. We speak of a clear head or a cool head. Thus we have a\nnumber of idiomatic expressions. We may speak of the head of the river;\nor the subject matter was divided under four heads; or again, the matter\ncame to a head; he is head and ears in debt; we cannot make head against\nthe opposition, etc.\n\nThis transfer of our ideas from the physical to the mental and spiritual\nmarks vividly the growth of the language and the development of thought.\nTrace the words like hands, arm, foot, eye, tongue, in their use, first\nas physical then as mental or spiritual.\n\nThis will be the most interesting pastime and will enlarge the content\nof the words which you use.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n CAPITAL LETTERS\n\n+464.+ In our written speech we often display our lack of education by\nour use of capital letters and punctuation. We may understand the use of\nwords and be able to speak fairly well, but if we do not understand the\nproper use of capitals and of punctuation marks, our written language\nreadily betrays our ignorance.\n\n+465.+ There are a number of rules for the use of capitals which we must\nobserve. Some of the writers in our magazines defy these rules of\ncapitalization, in an effort to seem different from other people,\nperhaps. These rules for the use of capital letters, like all other\nrules, are not arbitrary rules laid down by any body of men, but are\nsimply a statement of accepted usage among people. We should not feel\nthat we should say this or that or we are violating a rule of grammar.\nWe should feel rather that the majority of the people who speak and\nwrite good English do thus, and so, for this reason, I shall do it also.\n\nThis is simply obeying the standard of majority rule. If there is any\ngood and sufficient reason why we feel this should not be a rule, we may\nbe justified in breaking it and making a new rule. Many people feel that\nour spelling should be simplified and so they insist upon spelling\ncertain words in a more simple way. They feel that they have good and\nsufficient reason for insisting upon this change and gradually if these\nreasons appeal to the majority as being good and sufficient reasons,\nthen this simplified mode of spelling will become the accepted usage.\n\nBut there seems no good reason why any writer should scatter capital\nletters with a lavish hand throughout his writing. One feels as though a\nwriter in so doing is expressing his desire to be different, in a very\nsuperficial manner. Let us be unique and individual in our thought. If\nthis forces us to a different mode of living or of expression from the\nrest of the world, then we are justified in being different from the\nrest. We have thought and reason behind our action. This is far\ndifferent from the attitude of one who poses as a radical and whose only\nprotest is in the superficial external things. So let us learn and\nobserve these rules for the use of capital letters.\n\n\n RULES FOR THE USE OF CAPITAL LETTERS\n\n+466.+ +Use a capital for the first word of every sentence.+\n\nWhen you begin a new sentence always begin that sentence with a capital\nletter. Each sentence is a statement of a complete thought and is\nindependent of every other sentence. The use of the capital letter\nindicates this independence and calls attention to the fact that you are\nbeginning a new thought.\n\n+467.+ +Begin every line of poetry with a capital letter.+ Sometimes in\npoetry, the line is too long to be printed on a single line and must be\ncarried over into another line; in this case, the first word of the\nsecond line does _not_ begin with a capital letter.\n\n+468.+ +Use a capital for every proper noun.+ This includes names of\npersons, countries, states, towns, cities, streets and geographical\nnames, as the names of seas, lakes, mountains, rivers, etc.\n\n+469.+ +The words North, South, East and West are capitalized when they\nare used to refer to geographical divisions.+ When these words simply\nrefer to the points of the compass, they should not begin with a\ncapital.\n\n+470.+ +The pronoun _I_ and the interjection _O_ should always be\ncapitals.+ Never write the pronoun _I_ with a small _i_.\n\n+471.+ +Every proper adjective should begin with a capital letter.+\nProper adjectives are adjectives derived from proper nouns. For example:\nthe _Marxian_ philosophy, the _Darwinian_ theory, _Indian_ money,\n_Japanese_ labor, etc.\n\n+472.+ +Always begin the names of the months and the days of the week\nwith capital letters.+ For example: _January_, _February_, _August_,\n_Monday_, _Tuesday_, _Friday_, etc.\n\n+473.+ +Use a capital letter for every name or title of the Deity.+ For\nexample: _God_, _Jehovah_, _Christ_, _Jesus_, etc. It is also customary\nto capitalize all personal pronouns referring to God or Christ.\n\n+474.+ +Begin with a capital letter names of all religious sects and\npolitical parties, also all adjectives derived from them.+ As for\nexample: _Christian Church_, _Methodism_, _Republican Party_,\n_Mohammedan_, _Socialist_, etc.\n\n+475.+ +Begin the names of all things spoken of as persons with a\ncapital.+ In poetry or poetic prose we often speak of _war_, _fame_,\n_death_, _hope_, _fancy_, _liberty_, etc., as persons. Whenever these\nwords are used in this way they should begin with a capital letter.\n\n+476.+ +Use capital letters to begin important words in the title of a\nbook or the subject of a composition.+ In titles the nouns, pronouns,\nadjectives, verbs and adverbs should begin with a capital, while the\nprepositions and conjunctions should begin with small letters. The\narticles, _the_, _a_ and _an_ are not capitalized unless they are the\ninitial word in the title.\n\n+477.+ +Use a capital to begin every direct quotation.+ The first word\nof an indirect quotation should begin with a small letter. A direct\nquotation is one which uses the exact words of the speaker. For example:\n_He said_, \"_I will come_.\" This is a direct quotation, but _He said\nthat he would come_, is an indirect quotation.\n\n+478.+ +Use a capital to begin an important statement or to ask a\nquestion.+ For example: _Resolved; That the United States should\ndemocratize war. The question is, Shall the people determine the\nquestion of war?_\n\n+479.+ +Use capitals for the chief items of any enumeration of\nparticulars.+ For example;\n\n The bill is as follows:\n For Composition $20.00\n For Press Work 10.00\n Paper 25.00\n\n+480.+ +Begin the words indicating titles of offices and honor with a\ncapital.+ For example, _President Wilson_, _Doctor Smith_, _Professor\nLocke_. When you use a title of this kind as a general term, that is,\nnot indicating any particular person, do not use a capital. As for\nexample: _The society has had several presidents._ But if you use the\ntitle to take the place of the person's name, for example: _The\nPresident read the message to Congress_, always use a capital.\n\n+481.+ +Use capitals for the titles at the beginning of a letter or in\nwritten composition and in direct address.+ For example: _My dear\nFather_, _My dear Mother_, _My dear Comrade_, _Dear Aunt Emma_, _Dear\nFriend_, _Dear Fellow Workers_, etc. Also in conversation.\n\n Are you coming with me, Mother?\n What did the Doctor say, Comrade Smith?\n\nWhen these words are not used in direct address, however, they should\nnot be capitalized. For example, at the close of a letter you would\nwrite:\n\n Your sincere friend.\n Your loving brother.\n\nOr in conversation:\n\n I asked my mother to go with me.\n My brother wrote me concerning the matter.\n\n+482.+ +Begin the names of important buildings and localities with a\ncapital.+ For example:\n\n Public Library, High School, The East Side, The Union Square, Central\n Market, etc.\n\nThese words used in a general sense, however, should not begin with a\ncapital letter. For example:\n\nOur public libraries, our high schools, jails, prisons, post offices,\netc.\n\n+483.+ +The words state and territory, when they refer to particular\ndivisions of the country, should be capitalized.+ For example:\n\nThe State of New York, The Territory of Alaska, The French Government,\netc.\n\n_State_ and _government_ are also capitalized when they are used in\nplace of proper names. For example:\n\n The State is based on exploitation.\n The Government has issued an edict of war.\n\nWe do not use a capital in such expressions as:\n\n Church and state, state affairs; they occupy a large territory, etc.\n\n+484.+ +In directing letters or other matter for the mail, capitalize\nall words except prepositions, conjunctions or articles.+ These should\nbe capitalized only when they begin a line.\n\n\n Exercise 1\n\nDraw a line under each word in the following that should be begun with a\ncapital:\n\n john joffre, lake michigan, day, thursday, friday, spring, august,\n december, germany, country, france, man, jones, smith, doctor, doctor\n george, professor moore, girl, mary, susan, methodist, mohammedan,\n church, party, republican party, socialist, company, national electric\n light company, river, mississippi river, the red river, essex county,\n state of illinois, iowa, railway, new york, new york central railway,\n the french revolution, novel, the sea wolf, poem, arrows in the gale,\n american.\n\n\n Exercise 2\n\nNotice carefully the following quotations and sentences and capitalize\nevery word that should begin with a capital letter.\n\n 1. iron, the twin brother of fire, the first born out of the matrix\n of the earth, a witness everlasting to the glory of thy labor, am\n i, o man.\n 2. therefore i say unto you, banish fear from your hearts.\n 3. but ye, plebs, populists, people, rebels, mob, proletariat, live\n and abide forever.\n 4. and they came here from all parts of the earth, the syrians and\n the armenians, the thracians and the tartars, the jews, the greeks\n and the romans, the gauls and the angles and the huns and the\n hibernians, even from the deserts of the sands to the deserts of\n ice they came to listen unto his words.\n 5. marx and engels wrote the communist manifesto.\n 6. its closing words are; working men of all countries unite.\n 7. italy was the last of the great powers of europe to become\n involved in the war.\n 8. john randolph submitted an amendment to the constitution providing\n that the judges of the supreme court of the united states shall be\n removed by the president on the joint address of both houses of\n congress.\n 9. eugene v. debs spent six months in woodstock jail for exercising\n his right of free speech.\n 10. col. the abbreviation for colorado, is easily confused with cal.\n the abbreviation for california.\n 11. the people's college is a college maintained by the working\n class.\n 12. william jennings bryan won his first nomination for president of\n the united states by a very dramatic speech delivered in the\n national democratic convention.\n 13. marion craig wentworth, a socialist playwright, has written a play\n called \"war brides.\"\n 14. the play closes with these words; a message to the emperor: i\n refuse to bear my child until you promise there shall be no more\n war.\n 15. olive schreiner's \"woman and labor\" is full of fascinating\n thought.\n\n\n Exercise 3\n\nNotice carefully the use of capitals in the following quotations, and\ndetermine the reason for the use of every capital:\n\n As the nobles of England wrung their independence from King John, and\n as the tradesmen of France broke through the ring of privilege\n enclosing the Three Estates; so today the millions who serve society\n in arduous labor on the highways, and aloft on the scaffoldings, and\n by the sides of the whirring machines, are demanding that they, too,\n and their children, shall enjoy all of the blessings that justify and\n make beautiful this life.--_Frank Walsh_.\n\n \"The toad beneath the harrow knows\n Exactly where the tooth-point goes.\n The butterfly beside the road\n Doth preach contentment to that toad.\"\n\n \"When I came here, it was said that the Colorado Fuel and Iron Company\n voted every man and woman in their employ without any regard to their\n being naturalized or not; and even their mules, it used to be\n remarked, were registered if they were fortunate enough to possess\n names.\" _From a letter written by Mr. L. M. Bowers, Chairman of The\n Board of Directors of the Colorado Fuel and Iron Company, to the\n Secretary of Mr. John D. Rockefeller, Jr., under date of May 13,\n 1913._\n\n Master, I've done Thy bidding, wrought in Thy many lands.\n Not by my sins wilt Thou judge me, but by the work of my hands.\n Master, I've done Thy bidding, and the light is low in the west,\n And the long, long shift is over ... Master, I've earned it--Rest.\n\n --_Robert Service_.\n\n It's O! to be a slave\n Along with the barbarous Turk,\n Where woman has never a soul to save,\n If this is Christian work!\n\n --_Thos. Hood_.\n\n While there is a lower class, I am in it.\n While there is a criminal element, I am of it.\n While there is a soul in jail, I am not free.\n\n --_Eugene V. Debs_.\n\n When Adam delved and Eve span,\n Who was then the gentleman?\n\n The vilest deeds, like poison weeds,\n Bloom well in prison-air;\n It is only what is good in man\n That wastes and withers there:\n Pale Anguish keeps the heavy gate,\n And the Warder is Despair.\n\n --_Oscar Wilde_.\n\n\n ABBREVIATIONS AND CONTRACTIONS\n\n+485.+ There are a number of words which we abbreviate or contract, in\nour every-day use. A contraction is a shortened form of the word used to\nsave time or space and is made by omitting a letter or letters. The\napostrophe is used to indicate the omission in a contracted word. As,\nfor example:\n\n B'ld'g, B'l'v'd, M'f'g.\n\nWhen the word is contracted in this way and the apostrophe is used,\nthese contractions are not followed by the period but are used just as\nthe completely written word would be used. There is no accepted list of\nthese contractions. We devise them according to our need at the moment.\n\nAn abbreviation, however, is an authorized contraction of the word. It\nis the shortening of a term which is habitually used to save time and\nspace. The apostrophe is not used and the abbreviation should be\nfollowed by a period. As for example:\n\n Bldg. Blvd. Mfg.\n\nThese abbreviations and contractions are very helpful to us in saving\ntime and space but should not be used too frequently. Too many\ncontractions or abbreviations make writing ridiculous. Take time to\nwrite out the majority of words. Only use abbreviations or contractions\nfor certain accepted words. Avoid an excessive use of abbreviations.\n\n\n COMMONLY USED ABBREVIATIONS\n\n+486.+ We quite often abbreviate the names of the months, especially\nthose which have long names. Short names like _March_, _April_, _May_,\n_June_ and _July_, should never be abbreviated. For the other months we\nuse in correspondence the abbreviations, _Jan._, _Feb._, _Aug._,\n_Sept._, _Oct._, _Nov._, _Dec._ Days of the week are also sometimes\nabbreviated as follows: _Sun._, _Mon._, _Tues._, _Wed._, _Thur._,\n_Fri._, _Sat._ Do not use these abbreviations too often. Spell out the\nnames of the months and of the days of the week except in lists of dates\nor something that calls for abbreviations to save time or space.\n\n_Mr._, _Mrs._, _Messrs._, _Jr._, _Sr._, are never spelled out, but are\nalways written in the abbreviated form. You will often find _Doctor_ and\n_Professor_ abbreviated to _Dr._, _Prof._ This is permissible but it is\nalways good form to write them out in full.\n\n+487.+ We have abbreviated forms for a number of names; as for example:\n_Geo._, _Chas._, _Thos._, _Wm._, etc. But it is always much better to\nwrite these names out in full: _George_, _Charles_, _Thomas_, _William_,\netc.\n\nRemember that nicknames are not abbreviations and do not require a\nperiod after them. _Jim_, _Charley_, _Tom_, and _Bill_ are not\nabbreviations but nicknames.\n\nIn correspondence or in any circumstance that demands the saving of time\nor space, we abbreviate the names of states and territories, as follows:\n\n Alabama, Ala.\n Arizona, Ariz.\n Arkansas, Ark.\n California, Cal.\n Colorado, Colo.\n Connecticut, Conn.\n Delaware, Del.\n District of Columbia, D. C.\n Florida, Fla.\n Georgia, Ga.\n Idaho, Ida.\n Illinois, Ill.\n Indiana, Ind.\n Iowa, Ia.\n Kansas, Kan.\n Kentucky, Ky.\n Louisiana, La.\n Maine, Me.\n Maryland, Md.\n Massachusetts, Mass.\n Michigan, Mich.\n Minnesota, Minn.\n Mississippi, Miss.\n Missouri, Mo.\n Montana, Mont.\n Nebraska, Neb.\n Nevada, Nev.\n New Hampshire, N. H.\n New Jersey, N. J.\n New Mexico, N. M.\n New York, N. Y.\n North Carolina, N. C.\n North Dakota, N. D.\n Ohio, O.\n Oklahoma, Okla.\n Oregon, Ore.\n Pennsylvania, Pa. or Penna.\n Rhode Island, R. I.\n South Carolina, S. C.\n South Dakota, S. D.\n Tennessee, Tenn.\n Texas, Tex.\n Vermont, Vt.\n Virginia, Va.\n Washington, Wash.\n West Virginia, W. Va.\n Wisconsin, Wis.\n Wyoming, Wyo.\n\n+488.+ Use _a. m._ and _p. m._ after dates in lists of dates or\nschedules of trains or for any similar purpose, but in the text of a\nletter or manuscript it is better to write them out in full. As for\nexample, do not say:\n\n I will arrive tomorrow a. m., or, You may call about eight p. m.\n\nSay rather:\n\n I will arrive tomorrow morning. You may call at eight o'clock this\n evening.\n\nThe letters _a. m._ are the abbreviation for ante meridiem, Latin for\nbefore noon; and _p. m._ for post meridiem, meaning afternoon.\n\n+489.+ Two consecutive years may be written 1914-15, but use 1915 rather\nthan '15. In the heading of letters it is better to write the date out\nin full, as, _May 28, 1915_, instead of 5-28-15.\n\nIn the back of your dictionary you will find a complete list of accepted\nabbreviations used in writing and printing. The list that follows\ncontains abbreviations most commonly used, especially in business\ncorrespondence:\n\n @ for at\n acct. for account\n agt. for agent\n amt. for amount\n ans. for answer\n asst. for assistant\n atty. for attorney\n av. for average\n bal. for balance\n bbl. for barrel\n bdl. for bundle\n bro. for brother\n bros. for brothers\n blk. for black\n bls. for bales\n bu. or bush. for bushels\n Co. for company\n chgd. for charged\n C. O. D. for \"cash on delivery\"\n cr. creditor\n cts. cents\n cwt. for hundred weight\n cu. for cubic\n do. for the same\n dr. for debtor\n doz. for dozen\n ea. for \"each\"\n et al. for \"and others\"\n e. g. for example\n etc. for \"and so forth\"\n ft. for foot or feet\n frt. freight\n f. o. b. \"free on board\"\n gal. gallon\n guar. for guaranty\n hdkfs. for handkerchiefs\n h. p. horse power\n in. for inches\n ins. for insurance\n inst. for this month\n i. e. for \"that is\"\n Jr. for junior\n lb. for pound\n memo. for memorandum\n Mon. for Monday\n mo. for month\n mos. for months\n mdse. for merchandise\n mfg. for manufacturing\n Mss. for manuscript\n no. for number\n N. B. for take notice\n O. K. for \"all correct\"\n oz. for ounce\n % for per cent\n pp. pages\n pr. for pair\n pt. for pint\n pk. for peck\n prox. for next month\n qt. for quart\n recd. for received\n sec. for second\n Sec. for secretary\n Sr. for senior\n Supt. for superintendent\n ult. for last month\n via by way of\n viz. namely\n vol. for volume\n wt. for weight\n yd. for yard\n yds. for yards\n yr. for year\n\n\n Exercise 4\n\nWrite the proper abbreviations for the following words:\n\n Building\n Charles\n Boulevard\n Tuesday\n Arkansas\n Mississippi\n Foot\n Virginia\n Georgia\n Senior\n By way of\n Per cent\n Charged\n Avenue\n October\n Delaware\n Professor\n Thursday\n Colorado\n Kansas\n Handkerchiefs\n January\n Secretary\n Superintendent\n Received\n That is\n Free on board\n Monday\n Oklahoma\n July\n Thomas\n California\n Company\n Account\n Friday\n Merchandise\n Number\n All correct\n Cash on delivery\n And so forth\n Colonel\n Maine\n August\n William\n Missouri\n Brothers\n Amount\n Wyoming\n\n\n\n\n SPELLING\n\n LESSON 28\n\n\nThere is no way to learn to spell except by constant application. Watch\nin your reading the spelling of all words. Whenever you wish to add a\ncertain word to your vocabulary, master immediately the spelling as well\nas the meaning of that word. Keep your dictionary handy; use it\nconstantly in the study of your lessons. Do not guess at the spelling of\nthe word. You are not likely to forget quickly the spelling of any word\nwhich you have taken the trouble to look up.\n\nRead your examinations over carefully before sending them in, watching\nclosely for any error in spelling and in punctuation. When your papers\nare graded and returned you, make a list of all the words which are\nmisspelled and master then and there the spelling of these words. Do not\nbe guilty of the same error twice. Remember that correct spelling is a\nmark of intelligence and scholarship and that nothing will so detract\nfrom the influence of your written work as incorrect spelling.\n\nWhile there is always a certain word which more aptly expresses our\nmeaning than any other, we can usually find two or more words which\nexpress practically the same meaning.\n\n+Words which have nearly the same meaning are called synonyms.+\n\nIt is always an interesting exercise and will add greatly to your\nvocabulary to select a certain paragraph and go through it replacing\ncertain words with other words which have practically the same meaning.\nIt is this mastery of synonyms which gives the great writers and orators\ntheir power. They do not use the same word over and over again until our\nears have grown weary of it. With their wonderful mastery of language\nthey are never at a loss for words in which to re-clothe their meaning.\n\nFor the first three days of this week's work in spelling we have words\nand their synonyms. For the words given in the lessons for the last\nthree days, look up in your dictionary a suitable synonym.\n\n +Monday+\n\n Abundant\n Plenty\n\n Precarious\n Uncertain\n\n Behavior\n Conduct\n\n +Tuesday+\n\n Abuse\n Invective\n\n Hateful\n Odious\n\n Praise\n Applause\n\n +Wednesday+\n\n Sufficient\n Enough\n\n Refuge\n Asylum\n\n Achieve\n Attain\n\n +Thursday+\n\n Insolent\n Revenge\n Curb\n Repudiate\n Censure\n Regret\n\n +Friday+\n\n Prosperity\n Subterfuge\n Event\n Observe\n Portion\n Destroy\n\n +Saturday+\n\n Talkative\n Indolent\n Profit\n Volunteer\n Cordial\n Enormous\n\nThere are a number of nouns very similar in form, yet different in\nmeaning, which we very often use incorrectly.\n\nCross out in these sentences the incorrect word. Look them up in the\ndictionary and be sure of the exact meaning:\n\n Roger's _essay_--_assay_ won him praise.\n The _assay_--_essay_ indicated the quantity of gold in the metal.\n The _completion_--_completeness_ of the course entitled me to a\n Diploma.\n The _completion_--_completeness_ of the arrangements fills us with\n hope of success.\n _Confidants_--_confidence_ often betray us.\n The business world is built upon _confidants_--_confidence_.\n The _conscience_--_consciousness_ of a religious person is very\n sensitive.\n The class struggle develops class _conscience_--_consciousness_.\n The strikers listened to unwise _counsel_--_council_.\n The _council_--_counsel_ refused the franchise.\n You knew he was a _cultured_--_cultivated_ man, the moment you met\n him.\n It is a highly _cultured_--_cultivated_ plant.\n I asked her for the _recipe_--_receipt_ for making cake.\n He gave her a _receipt_--_recipe_ for the money.\n _Emigration_---_immigration_ has reduced the population of Servia.\n _Emigration_--_immigration_ is flooding the United States with cheap\n labor.\n Edison's _discovery_--_invention_ of the storage battery was a\n momentous event.\n The _discovery_--_invention_ of gold in Alaska attracted the attention\n of the world.\n The state placed a _limitation_--_limit_ upon the sale of liquor\n within certain _limits_--_limitations_.\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 29\n\n\nDear Comrade:\n\nThe spirit of self-help is the root of all genuine growth in any man or\nwoman. It is the things which we do for ourselves in any line of work\nthat count the most for us. The things which come to us without any\neffort on our part do not stay with us very long nor do us much good\nwhile we have them.\n\nSometimes we feel discouraged because we have not had the opportunity to\nattend school as much as we would like. There is no gainsaying but that\nthis is a tremendous handicap and yet, after all, it is not an\ninsurmountable obstacle. It is much better to have the appetite without\nthe food than to have the food without the appetite. There is always a\nchance of securing the food if we want it bad enough and will struggle\nhard enough. So in the matter of an education. Many a man who has never\nseen the inside of a college is better educated than those who have been\nthrough college.\n\nThese men have really wanted knowledge, have sought it early and late,\nand have found knowledge; and because they were in the work-a-day world,\nin constant contact with their fellow-men, they were able to relate the\nknowledge which they gained out of books to the world in which they\nlived and this is true education. This is, also, what many college-bred\npeople lack. A student is half made as soon as he seeks knowledge for\nits own sake. If you are striving to learn, not to make grades or to\npass examinations or to secure a degree, but simply for the sake of\nknowing things, then indeed you are on the way to become really\neducated.\n\nStimulate within yourself a desire for knowledge, observe the things\nabout you, add to your store of information daily; read a good book each\nday, even if you have time to read only a page or two, and you will be\nsurprised at the result in your life.\n\nTake, for example, our spelling. Why should we continually misspell the\nwords which we use every day and which we see every day on a printed\npage. If we are wide-awake and have our eyes open, we can soon learn to\nspell correctly all these common words, at least. Make a list this week\nof fifty things with which you come in contact in your daily work, then\nlook these words up in your dictionary and see how many of them you have\nmisspelled. There is no reason why we should not be learning constantly\nand the more we observe, the more acute becomes our power of\nobservation.\n\nLet us determine more than ever to feel that we are part of the great\nworld movement, that we belong in the ranks of those who have caught the\nvision of what the world might be, and that we belong to that glorious\narmy of those who are fighting for the dream; so we may take courage; so\nwe may find joy in the struggle, bitter as it may be, and so we may do\nour part in the fight.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n PUNCTUATION\n\n+490.+ Marks of punctuation are very important in our written language.\nThey take the place of the gesture and pause and inflection and\nintonation of the voice, by which we make our meaning clear in vocal\nspeech. So the marks of punctuation do not become mere mechanical\ndevices. They are marks full of meaning and necessary to express our\nthought.\n\nPunctuation is a word derived from the Latin word _punctum_ which means\n_a point_. We have other words from the same derivation, as puncture,\netc.\n\n+Punctuation is the art of pointing off our written language so as to\nmake its meaning clear.+\n\nSome very amusing errors have occurred because of the misplacing or the\nomission of punctuation marks. It is said, that a toast was one time\ngiven at a public dinner; \"Woman! without her, man would be a savage.\"\nThe next day it appeared in print; \"Woman, without her man, would be a\nsavage.\" You can readily see that the punctuation in this instance made\na very great difference in the meaning of the sentence.\n\n+491.+ In conversation, the tone of the voice which we use, has a great\neffect upon our meaning, for example I might say, _The International\nfailed_, in such a tone of voice, that it would express despair and\nchagrin, and indicate that the International was a thing of the past; or\nI might say, _The International failed_, with such an inflection, that\nyou would understand that even the suggestion was to be treated with\ncontempt, that the International was still powerful and its triumph\ninevitable. And in writing, the only way we have of expressing these\nshades of meaning is by means of punctuation marks.\n\nSo these marks of punctuation are not thrown upon a page haphazardly, or\nput there simply for decoration; they have a meaning and a very great\nmeaning. Those who use short, crisp sentences have less need for\npunctuation marks than those who use longer and more involved sentences.\nWhen we have learned to express ourselves directly and simply, we will\nnaturally use fewer marks of punctuation.\n\n+492.+ You will find that, in writing in connection with business, there\nis much less need of punctuation than in literary and philosophical\nwritings. Business writing is usually direct and simple in style. Its\npurpose is to state facts. The literary and philosophical writing,\nhowever, expresses more involved ideas and emotions, and in these, the\npunctuation is exceedingly important.\n\n+493.+ One of the great purposes served by punctuation is to indicate a\npause or break in the thought. A very good rule to go by in punctuating\nis to repeat the sentence aloud, and whenever you pause for breath or\nbecause of a break in the thought, it is a pretty safe indication that\nin that place, you should have a punctuation mark.\n\n+494.+ The following are the chief marks of punctuation:\n\n 1. The Comma ,\n 2. The Semi-colon ;\n 3. The Colon :\n 4. The Period .\n 5. The Interrogation Point ?\n 6. The Exclamation Point !\n 7. The Dash --\n 8. The Parenthesis ()\n 9. The Bracket []\n 10. The Quotation Marks \"\"\n 11. The Apostrophe '\n 12. The Hyphen -\n\n\n THE COMMA\n\n+495.+ The comma is the mark used to indicate a slight break in the\nthought.\n\nThere are a number of rules given for the use of commas. These rules,\nlike the rules for the use of capitals, you cannot commit to memory;\nbut, after repeated practice in your own writing and paying attention to\nyour reading, you will gradually develop an instinctive sense of the use\nof the comma. Select some book which you are reading and go through it,\nnoticing especially the use of the commas. See if you can determine the\nreason which prompted the author to place his commas where he did.\nNotice, also, what effect the placing or the omission of the comma would\nhave upon the meaning of the sentence.\n\n+496.+ +The Comma indicates the slightest degree of separation between\nthe parts of a sentence.+\n\n+RULE 1.+\n\n+497.+ +Words, phrases and clauses, forming a series and used in the\nsame construction, should be separated from each other by commas when\nthe conjunctions are omitted.+\n\n\n WORDS WHICH FORM A SERIES\n\n+498.+ The words which form a series, separated by a comma may\nbe either nouns, adjectives, adverbs or verbs. The comma is only used\nwhere the conjunction is omitted. Note carefully the following\nsentences:\n\n Love, laughter and happiness are the right of every child.\n He visited every city, town and village.\n The working class has been meek, humble, docile and gullible.\n All the crushed, tortured, strangled, maimed and murdered ideals of\n the ages shall become an everlasting reality.\n He struggled patiently, faithfully and fearlessly for the cause.\n If labor thinks, dares, rebels, fights, it will be victorious.\n\n\n PHRASES WHICH FORM SERIES\n\n+499.+ Phrases which are used in the same construction and form a series\nare separated by commas where the conjunction is omitted. For example:\n\n Day after day, year after year, century after century, the class\n struggle has proceeded.\n The struggle in the mines, in the fields, in the factories and in the\n shops, will go on until labor receives the product of its toil.\n\n\n CLAUSES USED IN A SERIES\n\n+500.+ Sometimes clauses are used without the co-ordinate conjunction\nand a comma is used to indicate the omission. For example:\n\n Do not moan, do not submit, do not kneel, do not pray, do not wait.\n Speak as you mean, do as you profess, perform what you promise.\n\n\n+RULE 2.+\n\n+501.+ +Explanatory and introductory expressions, words in direct\naddress, parenthetical words and phrases, are separated from the rest of\nthe sentence by commas.+\n\nNote carefully the following examples:\n\n Jaures, the great French Socialist, was the first martyr to peace.\n War having been declared, the troops were mobilized.\n No, I cannot believe you.\n Mr. Chairman, I desire to speak to the convention.\n We can, of course, give you the information you desire.\n\n\n+RULE 3.+\n\n+502.+ +Words, phrases or clauses written in the sentence out of their\nnatural order should be separated from the rest of the sentence by\ncommas.+\n\nThese words, phrases and clauses are often written at the beginning of\nthe sentences or at the end of the sentences, or in some place out of\ntheir natural order, for the sake of emphasis, instead of with the words\nthey modify.\n\nNotice in the following sentences how these words, phrases and clauses\nare separated from the rest of the sentence by commas. Rewrite these\nsentences, placing these words, phrases and clauses in their natural\norder and omit the commas.\n\n Longingly and anxiously, he waited.\n With this exception, the figures are correct.\n The music, sweet and dreamy, floated upon the air.\n The waves came rolling in, white with foam.\n To deceive the men, he resorted to shameful tricks.\n Before anyone else could speak, he was on his feet.\n\n\n+RULE 4.+\n\n+503.+ +Co-ordinate clauses, when closely related in meaning are\nseparated by commas. The comma should precede the co-ordinate\nconjunction.+ For example:\n\n I have not intended to detain you, but the matter required\n explanation.\n\n\n+RULE 5.+\n\n+504.+ +The omission of the verb in a sentence or a clause should be\nindicated by a comma.+ Sometimes in writing for effect or to give\nemphasis we omit the verb in the sentence; at other times we omit the\nverb when the same verb occurs in a series of brief sentences, and its\ncontinued use would mean a tiresome repetition. For example:\n\n Reading maketh a full man; conference, a ready man; writing, an\n exact man.\n\nHere the verb is omitted in the last two clauses and the omission is\nindicated by the use of the comma.\n\n\n+RULE 6.+\n\n+505.+ +Short, direct quotations should be preceded by a comma.+ For\nexample:\n\n Their slogan is, \"An injury to one is the concern of all.\"\n Ferrer's last words were, \"Long live the modern school.\"\n\n\n+RULE 7.+\n\n+506.+ +Separate the figures in large numbers into groups of three\nfigures each by the use of commas.+ For example:\n\n The population of the United States has now reached 100,000,000.\n According to the census of 1900, there are 29,073,233 people engaged\n in gainful occupations in the United States.\n\n\n Exercise 1\n\nSupply commas in the following sentences in the proper places:\n\n 1. Food clothes and shelter are the fundamental needs of life.\n 2. We believe in education free from theocracy aristocracy or\n plutocracy.\n 3. Man is the master of nature of law of life.\n 4. We shall struggle rebel arise and claim all being for our own.\n 5. Sickness and suffering sorrow and despair crime and war are the\n fruits of poverty.\n 6. You should seek after knowledge steadily faithfully and\n perseveringly.\n 7. The most inspiring powerful and impressive oratory is the voice of\n the disinherited.\n 8. Through your united almighty strength order shall become equity law\n shall become liberty duty shall become love and religion shall\n become truth.\n 9. First let us consider the main question.\n 10. Mr. President I rise to a point of order.\n 11. We the workers of the world must unite.\n 12. The class struggle being a fact why should we hesitate to join our\n class?\n 13. You have not it seems understood the issue.\n 14. Of all our needs education is the greatest.\n 15. Regularly and monotonously the machine whirs to and fro.\n 16. Before any one can take special training he must have a good\n knowledge of English.\n 17. We plead for education universal and free.\n 18. The first ingredient in conversation is truth the next good sense\n the third good humor and the fourth wit.\n 19. The slogan of the People's College is The education of the workers\n by the workers.\n 20. According to the last census the enrollment of the schools of the\n United States is 18521002.\n 21. There are 4611000 in the first grade and 155000 in the last year\n of high school.\n\n\n THE SEMI-COLON\n\n+507.+ The semi-colon indicates a break more complete than that of the\ncomma. The period indicates a complete break in the thought. So the\ncomma indicates a slight break, the semi-colon a greater break in the\nthought, and the period, the completion of the thought.\n\n\n RULES FOR THE USE OF THE SEMI-COLON\n\n+508.+ The semi-colon is often used instead of the comma where a longer\npause is desired or we wish to indicate a greater break in the thought.\nFor example:\n\n \"The wind is chill;\n But let it whistle as it will,\n We'll keep our Christmas merry still.\"\n\n+509.+ As a rule we separate by semi-colons those parts of the sentences\nthat are already punctuated by commas. For example:\n\n After considerable delay, he came back to look for his friends; but,\n though he looked diligently, he could not find them.\n\n+510.+ The semi-colon is used to separate closely connected simple\nsentences when the conjunction is omitted. The continual repetition of\nthe conjunction would become very tiresome and detract from the\nforcefulness of our sentences. So instead of continually repeating the\nconjunction we separate these simple sentences by semi-colons. For\nexample:\n\n Through the industrial revolution, the face of the earth is making\n over even as to its physical forms; political boundaries are wiped out\n and moved about as if they were indeed only lines on a paper map;\n population is hurriedly gathered into cities from the ends of the\n earth; habits of living are altered with startling abruptness; the\n search for the truths of nature is infinitely stimulated; and the\n application of these truths to life is made not only practicable, but\n commercially necessary.\n\n+511.+ The semi-colon should be used after each item in a series of\nspecific statements. For example:\n\n We quote you the following prices: Grade No. 1, $1.00; Grade No. 2,\n $2.90; poorer grades not in demand.\n\n\n RULES FOR THE USE OF THE COLON\n\n+512.+ The colon is not used as much as it formerly was. The comma and\nthe semi-colon and the period are now used in most of the places where\nolder writers used the colon.\n\nOne authority in English says that, \"in strict logic the colon is to the\nsentence in which it is used what the mark of equality is in\nmathematics.\"\n\n+513.+ The colon is used before a formal list of items. For example:\n\n Economics has three important divisions: production, distribution,\n consumption.\n\n+514.+ The colon is used after a salutation at the beginning of a\nletter. For example: _Dear Sir:_ _Gentlemen:_ _Comrades:_\n\nIn such cases the dash is also frequently used with the colon. For\nexample: _My dear Sir:--_ _Gentlemen:--_ _Comrades:--_\n\n+515.+ The colon is more often used instead of the semi-colon after such\nexpressions as, _thus:_ _as follows:_ _the following:_ _for example:_\netc.\n\nThe colon is also used to separate a series of sentences which are\nexplanatory of the main clause. For example:\n\n The People's College has two great aims: the first is to bring\n education within the reach of every worker; the second is to teach\n from the viewpoint of the working class.\n We were advised to proceed thus: first, to be systematic in our work;\n second, to concentrate; third, to go slowly and surely; and last of\n all, to think for ourselves.\n\n\n RULES FOR THE USE OF THE PERIOD\n\n+516.+ +The period is a mark of punctuation that denotes the completion\nof a sentence.+\n\n+517.+ The period is used at the close of all assertive and imperative\nsentences. For example:\n\n There is talk of peace but preparation for war.\n Claim your own at any hazard.\n\n+518.+ The period is used after all initials and all abbreviations, as\nfor example: E. V. Debs; T. P. O'Connor; Mr., Dr., Co., Mass., N. Y., C.\nO. D., F. O. B., U. S. A., etc.\n\n+519.+ The period is used to separate whole numbers and decimal numbers.\nFor example: 3.1416 9.342.\n\nA period is used for the decimal point between dollars and cents;\nas: $4.50, $2.25, $16.54, $35926.72.\n\nIt is also used to separate the various denominations of sterling money,\nas: L14. 15s. 6d.\n\n+520.+ The period is used after letters used as numerals or after\nfigures used to number paragraphs, notes, remarks, questions or any list\nof particulars. For example:\n\nThe letters which are used to denote sub-heads in the enumeration of\nrules as _a. b. c._, etc., also the numerals and letters marking\nsections or sub-sections in chapters, as _Chapter 8._ _Paragraph 1._\n_Rule 1._ _Page 4._ _Volume 2._ _Paragraph 3._ _P. 16._\n\n+521.+ The period is also used after headings and titles, after dates\nand signatures to letters and other documents; also at the close of the\naddress at the beginning of a letter, and of the name at the close of\nthe letter; also after the last item in the direction of an envelope or\npackage.\n\n\n Exercise 2\n\nIn the following quotations place the commas, semi-colons, colons and\nperiods in their proper places, and be able to give a reason for what\nyou do:\n\n The man who stabs his brother to death is a criminal and is hanged the\n general who under a flag slays a regiment is a hero and is decorated\n with a cross\n\n The most thrilling oratory the most powerful and impressive eloquence\n is the voice of the disinherited the oppressed the suffering and the\n submerged it is the voice of poverty and misery of wretchedness and\n despair it is the voice of humanity crying to the infinite it is the\n voice that resounds throughout the earth and reaches heaven it is the\n voice that wakens the conscience of the race and proclaims the truths\n that fill the world with life liberty and love\n\n The number of lives lost in the great wars of the world have been as\n follows Napoleonic wars 1900000 our Civil War 656000 Franco-German War\n 290000 Boer War 90898 Russo-Japanese 555900 and in the present\n world-war untold millions\n\n Walt Whitman who represents individualism at its best writes \"I sing\n the song of myself\" To this the Socialist replies \"Inasmuch as my\n redemption is bound up in that of my class I sing the song of my\n class\"\n\n We believe with John Ruskin \"whether there be one God or three no God\n or ten thousand children should be fed and their bodies should be kept\n clean\"\n\n My dear Mr Smith Your letter of the 15th has been received\n\n Through the dreams of all the ages rings the voice of labor beginning\n as a murmur growing in volume and grandeur as it rolls round the world\n And this is the burden of its message By the sweat of no other's brow\n shalt thou eat bread\n\n The sun of the new world is rising it is rising out of the solidarity\n of the working class Its rays of light are bursting through the dark\n horizon which ignorance and deceit have so long riveted upon us It is\n lighting up the faces of a new order of men and women supermen and\n women men and women not discouraged by defeat god-like men and women\n who have found the secret springs of life and are already drinking\n deep and glorious draughts men and women who are standing erect and\n whose joined hands encircle the world men and women who see the\n world's wretchedness and the world's poverty and are ready to throw\n away their lives with a song on their lips that such things shall not\n be\n\n\n Exercise 3\n\nNote the punctuation in the following poem and determine for yourself,\nin accordance with the rules we have studied, why the commas,\nsemi-colons, colons and periods are used as they are:\n\n JOHN BROWN\n\n States are not great\n Except as man may make them;\n Men are not great except they do and dare.\n But States, like men,\n Have destinies that take them--\n That bear them on, not knowing why or where.\n\n The _why_ repels\n The philosophic searcher--\n The _why_ and _where_ all questionings defy,\n Until we find,\n Far back in youthful nurture,\n Prophetic facts that constitute the _why_.\n\n All merit comes\n From braving the unequal;\n All glory comes from daring to begin.\n Fame loves the State\n That, reckless of the sequel,\n Fights long and well, whether it lose or win.\n\n * * * * *\n\n And there is one\n Whose faith, whose fight, whose failing,\n Fame shall placard upon the walls of time.\n He dared begin--\n Despite the unavailing,\n He dared begin, when failure was a crime.\n\n When over Africa\n Some future cycle\n Shall sweep the lake-gemmed uplands with its surge;\n When, as with trumpet\n Of Archangel Michael,\n Culture shall bid a race emerge;\n\n * * * * *\n\n From boulevards\n O'erlooking both Nyanzas,\n The statured bronze shall glitter in the sun,\n With rugged lettering:\n \"JOHN BROWN OF KANSAS:\n HE DARED BEGIN;\n HE LOST,\n BUT, LOSING, WON.\"\n\n --_Eugene Ware_.\n\n\n\n\n SPELLING\n\n LESSON 29\n\n\nLast week we studied words which had the same, or nearly the same,\nmeaning. There is always a slight distinction in the meaning of words,\nbut some of them are so nearly the same that it makes very little\ndifference which word we use. Some writers, however, are very careful\nand spend a great deal of time in the selection of just the right word\nto express their meaning.\n\nRobert Louis Stevenson once said a good writer would wait half a day in\norder to secure the best word to convey a certain idea.\n\nA very amusing story is told of Thomas Carlyle, who was very careful to\nuse words expressing just the shade of meaning which he desired to\nexpress. He had a habit of writing in a note book these words as they\noccurred to him, so he would have them for ready reference and use. One\nday he had searched all day for a certain word which eluded him.\nSuddenly in the middle of the night he wakened with the word flashing in\nhis mind. He wanted to write it down immediately lest he should forget\nit in the morning, but it was cold and he dreaded getting up in the cold\nto secure his note book so he nudged Jeanie, his wife, and said:\n\"Jeanie, Jeanie, get up! I have thought of a good word, and I want you\nto write it down.\" Now it was equally cold for Jeanie, so Jeanie nudged\nThomas and said: \"Thomas, Thomas, get up yourself. I have thought of a\nbad one!\"\n\nNevertheless, it is a good idea when these good words occur to you to\nwrite them down. Possibly to save trouble, you had better write them for\nyourself!\n\nBut in addition to words which have the same meaning, or almost the same\nmeaning, there are also words which express just the opposite meaning,\nand it is well for us to be master of these words also.\n\n+These words which express opposite meaning are called antonyms.+ Words\nand their antonyms are given in this week's spelling lesson in the words\nfor the first three days' study. For the last three days, words only are\ngiven. Look these words up in your dictionary and determine upon the\nmost suitable antonyms.\n\n +Monday+\n\n Legal\n Illegal\n\n Artificial\n Natural\n\n Assert\n Deny\n\n +Tuesday+\n\n Civilized\n Barbarous\n\n Courage\n Cowardice\n\n Active\n Passive\n\n +Wednesday+\n\n Initial\n Final\n\n Temporary\n Permanent\n\n History\n Legend\n\n +Thursday+\n\n Addition\n Cleverness\n Assured\n Genuine\n Acquit\n Increase\n\n +Friday+\n\n Affection\n Composure\n Enlarge\n Anxious\n Prompt\n Discord\n\n +Saturday+\n\n Succeed\n Describe\n Winning\n Wasteful\n Superficial\n Grieve\n\nWrite the proper word in the following blanks:\n\n PATIENTS or PATIENCE\n\n The Doctor has many.......\n We have no......with stupidity.\n\n NEGLIGENCE or NEGLECT\n\n The accident was due to the......of the employer.\n He has been guilty of......of his family for he was injured by the\n criminal......of the Railroad Company.\n\n OBSERVANCE or OBSERVATION\n\n The troops were concealed from.......\n Trade Unions never fail in the......of Labor Day.\n A man's own......will guide him in the......of all good customs.\n\n RELATIVES or RELATIONS\n\n Taft and Roosevelt did not always have pleasant......with each other.\n He has gone to visit his.......\n We do not always have pleasant......with our.......\n\n SECTS or SEX\n\n There are many religious.......\n Woman is refused the ballot because of her.......\n\n STATUE or STATUTE\n\n The law was placed upon the......books.\n The world will sometime erect a......to the man of the people.\n\nDo not fear to be thought a \"high-brow\" if you use these words in your\nevery day speech. The very people who may laugh are in their hearts\nadmiring you, and are, in all probability, envious. The man who has\naccused another of being a \"high-brow\" has by that very act, admitted\nhis own inferiority.\n\nDemand the best for yourself in words, as in everything else.\n\n\n\n\n PLAIN ENGLISH\n\n LESSON 30\n\n\nDear Comrade:\n\nWith this lesson we are finishing this course in Plain English. We have\ncovered a great deal of ground and have studied the essentials of\ngrammar. We have tried, as far as possible, to avoid the stupid conning\nof rules or learning by rote. We have attempted at least to make the\nreason and necessity for every rule apparent before the rule was stated.\n\nWe have also tried to weave into the lessons something of the romance of\nlanguage, for language is a romance; in its growth is written the epic\nof the race. Our words portray the struggle of man from savage to sage.\nSo, feeble as our efforts in this regard may have been, we trust that\nyou have enjoyed and profited by this course and have caught a new\nvision of life. Most of us are forced so inexorably into the bitter\nstruggle for existence that we have little time or opportunity to catch\nmuch of the beauty of life. That is the curse of a society that dooms\nits citizens to weary, toil-burdened lives, robbed of the joy and beauty\nof living.\n\nYet, if we know how to read we can always have access to books and\nthrough them we can escape the sordidness and ugliness of the life in\nwhich we are compelled to live and spend at least a little time each day\nin the company of great souls who speak to us from the printed page. The\nquotations in these lessons have been taken from these great writers.\n\nWill you not pursue the acquaintanceship and become real friends with\nthese men and women? Above all things they will bring you into the\natmosphere of liberty and of freedom. For throughout all the pain of the\nstruggle of the past and of the present, there has been the fight of man\nfor freedom. We have gained the mastery over nature. Wild animals, which\nwere a constant menace to savage man, have been destroyed. We have been\nfreed from fear and superstition by the discovery of the laws of nature.\nWith the invention of the machine, man has increased his ability to\nprovide the essentials of life,--food, clothing and shelter--a\nthousandfold. The past has seen revolution after revolution in the\nstruggle for mastery.\n\nWe now stand on the threshold of another great revolution when man shall\nmaster the machines which he has invented and shall cease serving them\nand make them serve him. His increased facilities for food-getting and\nshelter-getting shall be made to serve all mankind. We have a part to\nplay in that great revolution.\n\nWhatever you may have gained from the study of this course; what\nincreased facility of understanding or of expression may have come to\nyou; may it be not only for the service of yourself but also for the\nservice of the revolution that shall bring the worker into his own.\n\n Yours for Education,\n\n THE PEOPLE'S COLLEGE.\n\n\n THE ETERNAL WHY\n\n+522.+ There is no more important mark of punctuation than the\nInterrogation Point. Asking questions is the foundation and beginning of\nall wisdom. Progress is based upon the eternal _Why_. If men had always\nbeen satisfied with the knowledge of their age and had not continually\nasked questions which they set themselves to answer, we would still be\nliving in caves or dwelling in trees.\n\nThe natural child, that is, the child whose will has not been broken, is\nan animated Interrogation Point. He is full of questions. He wants to\nknow _why_ this and _why_ that. This is a most natural trait and one\nthat should not be destroyed. It may sadly interfere sometimes with the\nthings that we wish to do, to stop and answer the child's questions as\nto why cats have tails or who made the world and what did he stand on\nwhile he was doing it; but it is decidedly important that some one\nshould answer these questions which the child asks, in a manner to\nsatisfy its present craving for knowledge. The fact that this trait has\nbeen quenched in so many children by the impatient grown-ups explains\ntheir stupidity in later years. Encourage every child to ask questions.\nEncourage it also to be persistent until it finds somewhere the answer\nto its questions.\n\nCultivate also this trait yourself. Do not accept a thing simply because\nsome one says it is so. Insist upon knowing for yourself. This is the\nsecret of progress, that we should think for ourselves, investigate for\nourselves and not fear to face the facts of life or to express our own\nideas. The wise man does not accept a thing because it is old nor does\nhe reject it because it is new. He inquires, demands, reasons and\nsatisfies himself as to the merit of the question. So the Interrogation\nPoint in the written language of man has a tremendous meaning. It stands\nfor the open and inquiring mind; for the courage that dares question all\nthings and seek the truth.\n\n\n THE INTERROGATION POINT\n\n+523.+ An Interrogation Point should be placed after every direct\nquestion.\n\nA direct question is one that can be answered. An indirect question is\none that cannot be answered. If I say, _Why do you not study?_, I am\nasking a direct question to which you can give an answer; but if I say,\n_I wonder why you do not study_, I have asked an indirect question which\ndoes not require a direct answer.\n\n Why do you not go? (_Direct_)\n He asked why you did not go. (_Indirect_)\n\n+524.+ When an interrogative clause is repeated in the body of another\nsentence, use the interrogation point after the clause, and begin the\nclause with a capital letter. For example:\n\n The question, _Shall we be involved in war?_, should be settled by\n the people.\n\n\n THE EXCLAMATION POINT\n\n+525.+ The exclamation point should be placed after words, phrases or\nsentences that express strong emotion. For example:\n\n Oh! When shall peace reign again?\n Alas! I am undone!\n To the firing line! the battle rages!\n\n+526.+ Ordinarily the exclamation point is placed immediately after the\ninterjection or word used as an interjection, but frequently when the\nstrong emotion continues throughout the expression, the exclamation\npoint is placed at the close of the sentence instead of after the\ninterjection, even though the interjection comes first in the sentence.\nFor example:\n\n On, Comrades, on!\n Charge, Chester, charge!\n\n\n THE DASH\n\n+527.+ The dash is a much abused punctuation mark. A great many writers\nwho are not familiar with the rules of punctuation use a dash whenever\nthey feel the need of some sort of a punctuation mark. Their rule seems\nto be, \"whenever you pause make a dash.\" Punctuation marks indicate\npauses but a dash should not be used upon every occasion. The dash\nshould not be used as a substitute for the comma, semi-colon, colon,\netc. In reality, the dash should be used only when these marks cannot be\ncorrectly used.\n\n+528.+ The chief use of the dash is to indicate a sudden break in the\nthought or a sudden change in the construction of the sentence. For\nexample:\n\n In the next place--but I cannot discuss the matter further under the\n circumstances.\n\n+529.+ The dash is frequently used to set a parenthetical expression off\nfrom the rest of the sentence when it has not as close connection with\nthe sentence as would be indicated by commas. As for example:\n\n\n The contention may be true--although I do not believe it--that this\n sort of training is necessary.\n\n+530.+ The dash is also used in place of commas to denote a longer or\nmore expressive pause. For example:\n\n The man sank--then rose--then sank again.\n\n+531.+ The dash is often used after an enumeration of several items as a\nsumming up. For example:\n\n Production, distribution, consumption--all are a part of economics.\n\n+532.+ A dash is often used when a word or phrase is repeated for\nemphasis. For example:\n\n Is there universal education--education for every child beneath the\n flag? It is not for the masses of the children--not for the children\n of the masses.\n\n+533.+ If the parenthetical statements within dashes require punctuation\nmarks, this mark should be placed before the second dash. For example:\n\n War for defense--and was there ever a war that was not for\n defense?--was permitted by the International.\n This sight--what a wonderful sight it was!--greeted our eyes with the\n dawn.\n\n+534.+ The dash is also used to indicate the omission of a word,\nespecially such words as _as_, _namely_, _viz._, etc. For example:\n\n Society is divided into two classes--the exploited and the exploiting\n classes.\n\n+535.+ After a quotation, use the dash before the name of the author.\nFor example:\n\n Life only avails, not the having lived.--_Emerson_.\n\n+536.+ The dash is used to mark the omission of letters or figures. For\nexample:\n\n It happened in the city of M--.\n It was in the year 18--.\n\n\n PARENTHESIS\n\n+537.+ In our study of the comma and the dash we have found that\nparenthetical statements are set off from the rest of the sentence\nsometimes by a comma and sometimes by a dash. When the connection with\nthe rest of the sentence is close, and yet the words are thrown in in a\nparenthetical way, commas are used to separate the parenthetical\nstatement from the rest of the sentence.\n\n+538.+ When the connection is not quite so close, the dash is used\ninstead of the comma to indicate the fact that this statement is thrown\nin by way of explanation or additional statement. But when we use\nexplanatory words or parenthetical statements that have little or no\nconnection with the rest of the sentence, these phrases or clauses are\nseparated from the rest of the sentences by the parenthesis.\n\n+539.+ +GENERAL RULE:--Marks of parenthesis are used to set off\nexpressions that have no vital connection with the rest of the\nsentence.+ For example:\n\n Ignorance (and why should we hesitate to acknowledge it?) keeps us\n enslaved.\n Education (and this is a point that needs continual emphasis) is the\n foundation of all progress.\n\n\n THE PUNCTUATION OF THE PARENTHESIS\n\n+540.+ If the parenthetical statement asks a question or voices an\nexclamation, it should be followed by the interrogation point or the\nexclamation point, within the parenthesis. For example:\n\n We are all of us (who can deny it?) partial to our own failings.\n The lecturer (and what a marvelous orator he is!) held the audience\n spellbound for hours.\n\n\n OTHER USES OF THE PARENTHESIS\n\n+541.+ An Interrogation Point is oftentimes placed within a parenthesis\nin the body of a sentence to express doubt or uncertainty as to the\naccuracy of our statement. For example:\n\n In 1858 (?) this great movement was started.\n John (?) Smith was the next witness.\n\n+542.+ The parenthesis is used to include numerals or letters in the\nenumeration of particulars. For example:\n\n Economics deals with (1) production, (2) distribution,\n (3) consumption.\n There are three sub-heads; (a) grammar, (b) rhetoric, (c) composition.\n\n+543.+ Marks of parenthesis are used to inclose an amount or number\nwritten in figures when it is also written in words, as:\n\n We will need forty (40) machines in addition to those we now have.\n Enclosed find Forty Dollars ($40.00) to apply on account.\n\n\n THE BRACKET\n\n+544.+ The bracket [] indicates that the word or words included in the\nbracket are not in the original discourse.\n\n+545.+ The bracket is generally used by editors in supplying missing\nwords, dates and the like, and for corrections, additions and\nexplanations. For example:\n\n This rule usually applies though there are some exceptions. [See Note\n 3, Rule 1, Page 67].\n\n+546.+ All interpretations, notes, corrections and explanations, which\nintroduce words or phrases not used by the author himself, should be\nenclosed in brackets.\n\n+547.+ Brackets are also used for a parenthesis within a parenthesis. If\nwe wish to introduce a parenthetical statement within a parenthetical\nstatement this should be enclosed in a bracket. For example:\n\n He admits that this fact (the same fact which the previous witness\n [Mr. James E. Smith] had denied) was only partially true.\n\n\n QUOTATION MARKS\n\n+548.+ Quotation marks are used to show that the words enclosed by them\nare the exact words of the writer or speaker.\n\n+549.+ A direct quotation is always enclosed in quotation marks. For\nexample:\n\n He remarked, \"I believe it to be true.\"\n\nBut an indirect quotation is not enclosed in quotation marks. For\nexample:\n\n He remarked that he believed it was true.\n\n+550.+ When the name of an author is given at the close of a quotation\nit is not necessary to use the quotation marks. For example:\n\n All courage comes from braving the unequal.--_Eugene F. Ware_.\n\nWhen the name of the author precedes the quotation, the marks are used,\nas in the following:\n\n It was Eugene F. Ware who said, \"Men are not great except they do and\n dare.\"\n\n+551.+ When we are referring to titles of books, magazines or\nnewspapers, or words and phrases used in illustration, we enclose them\nin quotation marks, unless they are written in italics. For example:\n\n \"Whitman's Leaves of Grass\" or _Whitman's Leaves of Grass_. \"The New\n York Call\" or _The New York Call_. The word \"book\" is a noun, or, The\n word _book_ is a noun.\n\n\n THE QUOTATION WITHIN A QUOTATION\n\n+552.+ When a quotation is contained within another, the included\nquotation should be enclosed by single quotation marks and the entire\nquotation enclosed by the usual marks. For example:\n\n He began by saying, \"The last words of Ferrer, 'Long live the modern\n school' might serve as the text for this lecture.\"\n The speaker replied, \"It was Karl Marx who said, 'Government always\n belongs to those who control the wealth of the country.'\"\n\nYou will note in this sentence that the quotation within the quotation\noccurs at the end of the sentence so there are three apostrophes used\nafter it, the single apostrophe to indicate the included quotation and\nthe double apostrophe which follows the entire quotation.\n\n\n PUNCTUATION WITH QUOTATION MARKS\n\n+553.+ Marks of punctuation are (except the interrogation point and the\nexclamation point which are explained later) placed inside the quotation\nmarks. For example:\n\n A wise man said, \"Know thyself.\"\n\nNotice that the period is placed after the word _thyself_ and is\nfollowed by the quotation marks.\n\n \"We can easily rout the enemy,\" declared the speaker.\n\nNotice that the comma is placed after _enemy_, and before the quotation\nmarks.\n\n+554.+ The Interrogation Point and the Exclamation Point are placed\nwithin the quotation marks if they refer _only_ to the words quoted, but\nif they belong to the entire sentence they should be placed outside the\nquotation marks. For example:\n\n He said, \"Will you come now?\"\n Did he say, \"Will you come now\"?\n He said, \"What a beautiful night!\"\n How wonderfully inspiring is Walt Whitman's poem, \"The Song of the\n Open Road\"!\n\n+555.+ Sometimes parenthetical or explanatory words are inserted within\na quotation. These words should be set off by commas, and both parts of\nthe quotation enclosed in quotation marks. For example:\n\n \"I am aware,\" he said, \"that you do not agree with me.\"\n \"But why,\" the speaker was asked, \"should you make such a statement?\"\n \"I do not believe,\" he replied, \"that you have understood me.\"\n\n\n THE APOSTROPHE\n\n+556.+ The apostrophe is used to indicate the omission of letters or\nsyllables, as: _He doesn't_, instead of _does not_; _We're_, instead of\n_we are_; _I'm_, instead of _I am_; _ it's_, instead of _it is_;\n_ne'er_, instead of _never_; _they'll_, instead of _they will_, etc.\n\n+557.+ The apostrophe is also used to denote possession. In the single\nform of the nouns it precedes the _s_. In the plural form of nouns\nending in _s_ it follows the _s_. For example:\n\n Boy's, man's, girl's, king's, friend's, etc.\n Boys', men's, girls', kings', friends', etc.\n\nNote that the apostrophe is not used with the possessive pronouns\n_ours_, _yours_, _its_, _theirs_, _hers_.\n\n+558.+ The apostrophe is used to indicate the plural of letters, figures\nor signs. For example:\n\n Dot your _i's_ and cross your _t's_.\n He seems unable to learn the table of 8's and 9's.\n Do not make your _n's_ and _u's_ so much alike.\n\n+559.+ The apostrophe is used to mark the omission of the century in\ndates, as: '87 instead of 1887, '15 instead of 1915.\n\n\n THE HYPHEN\n\n+560.+ The hyphen is used between the parts of a compound word or at the\nend of a line to indicate that a word is divided. We have so many\ncompound words in our language which we have used so often that we have\nalmost forgotten that they were compound words so it is not always easy\nto decide whether the hyphen belongs in a word or not. As, for example;\nwe find such words as _schoolhouse_, _bookkeeper_, _railway_ and many\nothers which are, in reality, compound words and in the beginning were\nwritten with the hyphen. We have used them so frequently and their use\nas compound words has become so commonplace, that we no longer use the\nhyphen in writing them. Yet frequently you will find them written with\nthe hyphen by some careful writer.\n\n+561.+ As a general rule the parts of all words which are made by\nuniting two or more words into one should be joined by hyphens, as:\n\n Men-of-war, knee-deep, half-hearted, full-grown, mother-in-law, etc.\n\n+562.+ The numerals expressing a compound number should be united by a\nhyphen, as; _forty-two_, _twenty-seven_, _thirty-nine_, etc.\n\n+563.+ When the word _self_ is used with an adverb, a noun or an\nadjective, it is always connected by the hyphen, as; _self-confidence_,\n_self-confident_, _self-confidently_, _self-command_, _self-assertive_,\n_self-asserting_, etc.\n\n+564.+ When the word _fold_ is added to a number of more than one\nsyllable, the hyphen is always used, as; _thirty-fold_, _forty-fold_,\n_fifty-fold_, etc. If the numeral has but one syllable, do not use the\nhyphen, as; _twofold_, _threefold_, _fourfold_, etc.\n\n+565.+ When fractions are written in words instead of figures always use\nthe hyphen, as; _one-half_, _one-fourth_, _three-sevenths_,\n_nine-twelfths_, etc.\n\n+566.+ The words _half_ and _quarter_, when used with any word, should\nbe connected by a hyphen, as; _half-dollar_, _quarter-pound_,\n_half-skilled_, _half-barbaric_, _half-civilized_, _half-dead_,\n_half-spent_, etc.\n\n+567.+ Sometimes we coin a phrase for temporary use in which the words\nare connected by the hyphen. For example:\n\n It was a never-to-be-forgotten day.\n He wore a sort of I-told-you-so air.\n They were fresh-from-the-pen copies.\n\n\n ADDITIONAL MARKS OF PUNCTUATION\n\nThere are a few other marks of punctuation which we do not often use in\nwriting but which we find on the printed page. It is well for us to know\nthe meaning of these marks.\n\n+568.+ The caret (^) is used to mark the omission of a letter or word or\na number of words. The omitted part is generally written above, and the\ncaret shows where it should be inserted. For example:\n\n s\n I cannot give you this permis ion.\n ^\n received\n I have just a letter from him.\n ^\n\n Please write your matriculation number on all examination\n and all letters\n papers sent in to the College.\n ^\n\nThe above examples illustrate the use of the caret with the omission of\na letter, a word or phrase.\n\n+569.+ If a letter or manuscript is not too long, it should always be\nrewritten and the omissions properly inserted. Occasionally, however, we\nare in a hurry and our time is too limited to rewrite an entire letter\nbecause of the omission of a single letter or word so we can insert it\nby the use of the caret. If, however, there are many mistakes, the\nletter or paper should be rewritten, for the too frequent use of the\ncaret indicates carelessness in writing and does not produce a favorable\nimpression upon the recipient of your letter or manuscript.\n\n\n MARKS OF ELLIPSIS\n\n+570.+ Sometimes a long dash (--------) or succession of asterisks (* *\n* * * *) or of points (. . . . . .) is used to indicate the omission of\na portion of a sentence or a discourse. In printed matter usually the\nasterisks are used to indicate an omission. In typewritten matter\nusually a succession of points is used to indicate an omission. In\nwriting, these are difficult to make and the omission of the portion of\nmaterial is usually indicated by a succession of short dashes (-- -- --\n--).\n\n\n MARKS OF REFERENCE\n\n+571.+ On the printed page you will often find the asterisk (*), or the\ndagger, ([Symbol: dagger]), the section (symbol for Section), or parallel\nlines (||), used to call your attention to some note or remark written at\nthe close of the paragraph or on the margin, at the bottom of the page or\nthe end of the chapter. It is advisable to hunt these up as soon as you\ncome to the mark which indicates their presence, for they usually contain\nsome matter which explains or adds to the meaning of the sentence which\nyou have just finished reading.\n\n\n Exercise 1\n\nIn the following exercise, note the various marks of punctuation and\ndetermine why each one is used:\n\n\n THE MARSEILLAISE\n\n Ye sons of toil, awake to glory!\n Hark, hark, what myriads bid you rise;\n Your children, wives and grandsires hoary--\n Behold their tears and hear their cries!\n Shall hateful tyrants, mischief breeding,\n With hireling hosts, a ruffian band,--\n Affright and desolate the land,\n While peace and liberty lie bleeding?\n\n CHORUS\n\n To arms! to arms! ye brave!\n Th' avenging sword unsheathe!\n March on, march on, all hearts resolved\n On Victory or Death.\n\n With luxury and pride surrounded,\n The vile, insatiate despots dare,\n Their thirst for gold and power unbounded,\n To mete and vend the light and air;\n Like beasts of burden would they load us,\n Like gods would bid their slaves adore,\n But Man is Man, and who is more?\n Then shall they longer lash and goad us? (CHORUS)\n\n O Liberty! can man resign thee,\n Once having felt thy generous flame?\n Can dungeons' bolts and bars confine thee,\n Or whip thy noble spirit tame?\n Too long the world has wept bewailing,\n That Falsehood's dagger tyrants wield;\n But Freedom is our sword and shield,\n And all their arts are unavailing! (CHORUS)\n\n --_Rouget de Lisle_.\n\n\n THUS SPAKE ZARATHUSTRA\n\n I teach ye the Over-man. The man is something who shall be overcome.\n What have ye done to overcome him?\n\n All being before this made something beyond itself: and you will be\n the ebb of this great flood, and rather go back to the beast than\n overcome the man?\n\n What is the ape to the man? A mockery or a painful shame. And even so\n shall man be to the Over-man: a mockery or a painful shame.\n\n Man is a cord, tied between Beast and Over-man--a cord above an abyss.\n\n A perilous arriving, a perilous traveling, a perilous looking\n backward, a perilous trembling and standing still.\n\n What is great in man is that he is a bridge, and no goal; what can be\n loved in man is that he is a going-over and a going-under.\n\n I love them that know how to live, be it even as those going under,\n for such are those going across.\n\n I love them that are great in scorn, because these are they that are\n great in reverence, and arrows of longing toward the other\n shore!--_Nietzsche_.\n\n\n\n\n SPELLING\n\n LESSON 30\n\nThere are a great many words in English which are frequently\nmispronounced; the accent is placed upon the wrong syllable; for\nexample, _thea'ter_ instead of _the'ater_; the wrong sound is given to\nthe vowel, for example, _hearth_ is pronounced _hurth_. Sometimes, too,\nan extra letter is added in the pronunciation; for example, _once_ is\noften pronounced as though it were spelled _wunst_.\n\nThe following is a list of common words that are frequently\nmispronounced, and there are many others which you may add to this list\nas they occur to you. Look up the correct pronunciation in the\ndictionary and pronounce them many times aloud.\n\nIn the second column in this list is given the incorrect pronunciation,\nwhich we often hear.\n\n Acoustics a-cow-stics\n Aeroplane air-e-o-plane\n Apron a-pron\n Athlete ath-a-lete\n Autopsy au-top'-sy\n Awkward awk-ard\n Column col-yum\n Coupon coo-pon\n Deficit de-fic'it\n Diphtheria dip-ther-y\n Economic ee'co-nom-ic\n Errand ur-rant\n Faucet fos-set\n Figure fig-ger\n Film fill-um\n Finance fi'nance\n Guardian guar-deen'\n Height heighth\n Hostile hos-tile'\n Hundred hund'erd\n Idea i-dee'\n Inaugurate in-aug-er-ate\n Inquiry in'qui-ry\n Inventory in-ven'-to-ry\n Length lenth\n Magazine mag'-a'zinn\n Mischievous mis-chie'-vi-ous\n Municipal mu-ni-cip'-al\n Opponent op'-ponent\n Overalls over-hauls\n Rheumatism rheumatiz\n Stomach stum-ick\n Twice twict\n Vaudeville vaw'de-ville\n\nThere are a number of words in English which sound very much alike and\nwhich we are apt to confuse. For example, I heard a man recently say in\na speech that the party to which he belonged had taken slow poison and\nnow needed an anecdote. It is presumed that he meant that it needed an\nantidote. Some one else remarked that a certain individual had not been\nexpelled but simply expended. He undoubtedly meant that the individual\nhad been suspended.\n\nThis confusion in the use of words detracts from the influence which our\nstatements would otherwise have. There are a number of words which are\nso nearly alike that it is very easy to be confused in the use of them.\nIn our spelling lesson for this week we have a number of the most common\nof these easily confounded words. Add to the list as many others as you\ncan.\n\n +Monday+\n\n Lightening, _to make light_\n Lightning, _an electric flash_\n Prophesy, _to foretell_\n Prophecy, _a prediction_\n Accept, _to take_\n Except, _to leave out_\n\n +Tuesday+\n\n Advice, _counsel_\n Advise, _to give counsel_\n Attendants, _servants_\n Attendance, _those present_\n Stationary, _fixed_\n Stationery, _pens_, _paper_, _etc._\n\n +Wednesday+\n\n Formerly, _in the past_\n Formally, _in a formal way_\n Addition, _process of adding_\n Edition, _publication_\n Celery, _a vegetable_\n Salary, _wages_\n\n +Thursday+\n\n Series, _a succession_\n Serious, _solemn_\n Precedent, _an example_\n President, _chief or head_\n Partition, _a division_\n Petition, _a request_\n\n +Friday+\n\n Ingenious, _skillful_\n Ingenuous, _honest_\n Jester, _one who jests_\n Gesture, _action_\n Lose, _to suffer loss_\n Loose, _to untie_\n\n +Saturday+\n\n Presence, _nearness_\n Presents, _gifts_\n Veracity, _truthfulness_\n Voracity, _greediness_\n Disease, _illness_\n Decease, _death_\n\n\n THE END AND THE BEGINNING\n\nAs we look back over the study of these thirty lessons we find that we\nhave covered quite a little ground. We have covered the entire field of\nEnglish grammar including punctuation. But our study of English must not\nconclude with the study of this course. This is simply the foundation\nwhich we have laid for future work. You know when students graduate from\nhigh school or college the graduation is called the Commencement. That\nis a peculiarly fitting term, for the gaining of knowledge ought truly\nto be the commencement of life for us.\n\nSome one has said that the pursuit of knowledge might be compared to a\nman's marriage to a charming, wealthy woman. He pursued and married her\nbecause of her wealth but after marriage found her so charming that he\ngrew to love her for herself. So we ofttimes pursue wisdom for practical\nreasons because we expect it to serve us in the matter of making a\nliving; because we expect it to make us more efficient workers; to\nincrease our efficiency to such an extent that we may command a higher\nsalary, enter a better profession and be more certain of a job.\n\nAll this is well; but we often find that after we have pursued wisdom\nfor these reasons, practical as they are, we have fallen in love with\nher for her own sake. We begin to take pleasure in her society; we begin\nto want to know things for the sake of knowing them, for the pleasure\nthat it brings us, quite divorced from any idea of monetary gain.\n\nSo while we have urged upon you the study of English because of the\ngreat practical benefit that it will be to you, we trust that you have\nalso grown to love the study for its own sake.\n\nMake this but the beginning of your work in the study of English.\n\n\n\n\n INDEX\n (by Section No.)\n\n Abbreviations, 486-489\n\n Absolute Construction, 399\n\n Adjectives\n Defined, 36\n Classification of, 242-245\n Qualifying, 246\n Limiting, 246\n Descriptive, 248\n Numeral, 249-250\n Demonstrative, 251\n How to discover, 247\n Interrogative, 255\n Indefinite, 256-257\n Used as pronouns, 258-259\n Used as nouns, 261\n Comparison of, 264-271\n Participles used as, 272-274\n Participle phrases used as, 275\n\n Adverbs\n Defined, 41, 282\n Use of, 279-281\n How to tell, 283\n Classes of, 284\n Interrogative, 285\n Of mode, 286, 397\n Phrase Adverbs, 287\n To Distinguish from Adj, 288-289\n Derivation of, 290\n Nouns used as, 291\n Comparison of, 292-294\n Position of, 295\n With Infinitive, 296\n Common errors in use of, 297-298\n\n Articles\n A and An, use of, 252-253\n The, use of, 254\n\n Capital Letters\n Need of, 464\n Uses of, 22, 60, 465\n Rules for, 466-484\n\n Clauses\n Defined, 406\n Noun, 361-366, 371, 445\n Adjective, 367-372, 446\n With Conjunctions, 376\n Introduced by as, 378\n Adverb, 447\n Dependent, kinds of, 444-447\n\n Conjunctions\n Defined, 52, 331\n Uses of, 328\n Classes of, 329-330\n Co-ordinate, 332-334\n Uses of, 336-345\n Correlatives, 346\n Subordinate, defined, 349\n Use of, 347\n Classes of, 350-359\n Phrase Conjunctions, 360\n\n Connective Words\n Classes of, 379\n Uses of, 380-385\n\n Contractions, 485\n\n Dictionary, Use of, 4\n\n Exclamatory Words, 390-391\n\n Explanatory Words, 398\n\n Good English, defined, 2\n\n Grammar, English, defined, 10\n\n Independent Expressions, 393\n\n Infinitives\n Use of, 151-167\n To, omitted, 153-155\n Forms of, 156\n Passive, 156-157\n\n Interjections\n Defined, 57, 388\n Classes of, 389\n\n Introductory Words, 394-396\n\n Language\n Defined, 8\n Natural, 5\n Spoken, 6\n Written, 7\n\n Nouns\n Defined, 26\n Classification of, 59\n Proper, defined, 60\n Common, defined, 60\n Collective, defined, 61\n Abstract, 62-66\n Concrete, 63\n Number, defined, 68\n Number, Singular, 68\n Number, Plural, 68\n Formation of Plural, 69-84\n Formation of Possessive, 89-90, 92\n Compound, 91\n Gender, defined, 85\n Formation of Feminine, 86\n Neuter, 87\n Common, 88\n\n Object\n Direct, 100, 408-410, 427-430\n Indirect, 408-410\n\n Participle\n Defined, 116\n Active form, 114\n Present form, 114, 148\n Passive form, 115, 148\n Past form, 115\n Past irregular forms, 124\n Used as nouns, 148\n Used as adjective, 272-274\n Phrase, 149-150\n Phrase used as adjective, 275\n\n Parts of Speech, 24\n\n Phrases\n Verb, 29, 413\n Adverbs, 287\n Prepositional, 300-305, 317-321\n Prepositions, 308\n Conjunctions, 360\n\n Predicate\n Defined, 17\n Complete, 406, 425\n Simple, 406\n Simple Enlarged, 463\n Complement, 411-412\n Modifiers of, 461\n\n Prepositions\n Defined, 47, 305\n Use of, 309-312\n Object of, 304, 313\n List of, 306\n How to Distinguish from Adverbs, 307\n Phrase prepositions, 308\n Place of, 314-316\n Common errors in use of, 322\n With verbs, 327\n Choice of, 323-326\n\n Prepositional Phrases, 300-305\n Use of, 317-321\n\n Pronouns\n Defined, 43, 202\n Antecedent of, 203\n Personal, 204\n Compound personal, 205-208\n Number forms of, 209\n Object forms of, 214-215\n Possessive forms of, 211-213\n Gender forms of, 216\n With verb \"be\", 217-218\n Agreement of, 219-225\n Personification, 226\n Interrogative, 228-231\n Relative, 232-236\n What, 234, 236-240\n Who, 234, 235, 240\n Which, 234-236, 240\n That, 234-236\n Omitted, 239\n\n Punctuation\n Need of, 490-493\n Marks of, 494\n The Comma, 495-496\n Rules for use of, 497-506\n The Semi-colon, 407-511\n The Colon, 512-515\n The Period, 22, 516-521\n The Interrogation Point, 22, 523-524\n The Exclamation Point, 22, 525-526\n The Dash, 527-536\n The Parenthesis, 537-543\n The Bracket, 544-547\n The Quotation Marks, 548-555\n The Apostrophe, 556-559\n The Hyphen, 560-567\n The Caret, 568\n Marks of Ellipsis, 570\n Marks of Reference, 571\n\n Responsives, 392\n\n Sentence\n Defined, 15\n Essentials of, 18\n Use of, 19\n Assertive, 20\n Interrogative, 20\n Imperative, 20\n Exclamatory, 21\n Elements, order of, 436-438\n Analysis of, 456-457\n Simple, defined, 404-406\n Modifiers of, 434-435\n Essentials of, 459\n Analysis of, 402-405\n Complex, 406, 443, 451\n Analysis of, 448\n Compound, defined, 406, 452\n Kinds of, 453-455\n Building of, 400\n Classification of, 401\n Summary of, 458\n Subject of, 416-420\n\n Subject\n Defined, 16\n Complete, 406\n Simple, 406\n Simple, enlarged, 462\n Place of, 421-424\n\n Thought, Complete, 12-14\n\n Verb\n Defined, 29\n Complete, 95, 103, 131, 158\n Incomplete, 95, 103, 131, 426\n Classified, 99, 103\n Complement of, 95, 102\n Transitive, defined, 100, 103\n Object of, 100, 141\n Copulative, 102-103, 431-433\n Time forms\n Present, 104, 108, 111\n Past, 104, 109, 111\n Future, 118-120\n Pres. Perf., 121-123, 145\n Past Perf., 126\n Future Perf., 128\n Regular, 110\n Irregular, 110\n Progressive Form, 133\n Present, 134, 146\n Past, 135, 146\n Future, 136, 146\n Pres. Perf., 138, 146\n Past Perf., 139, 146\n Fut. Perf., 140, 146\n Active, 142\n Passive, 141-146\n Helping, 168-184\n Be, 186\n Lay, lie, set, sit, raise, rise, 191-193\n S-form, 106, 194-196\n Phrase, 29, 413-414\n\n Words\n Defined, 8\n Mastery of, 10\n Use of, 23\n\n\n\n\n * * * * *\n\n\n\n\nTranscriber's note:\n\n 1. Punctuation errors such as incorrect or missing end-of-sentence\n punctuation, period for comma in mid-sentence, and missing end\n quotation marks have been corrected without comment. Inconsistency\n in the author's spelling of certain words, such as today\/to-day have\n been retained.\n\n 2. The list of foreign words broken across pp. 44-45 (section 80.) and\n the list of abbreviations broken across pp. 295-296 (section 489.)\n were rearranged to preserve alphabetical order.\n\n 3. The numbering in Exercise 4 on p. 110 (section 193.) was corrected.\n\n 4. Added ditto marks (\") to the table on p. 153 (section 270).\n\n 5. Commas were added to the separate the abbreviations on p. 305\n (section 518).\n\n 6. The following typographical errors were corrected:\n\n Page\n 10 \"your vocabularly\" changed to \"your vocabulary\"\n 23 \"verb-phrase\" changed to \"verb phrase\"\n 38 \"as limited a vocabularly\" changed to \"as limited a vocabulary\"\n 41 \"the name of person\" changed to \"the name of the person\"\n 44 \"Mr. Hays\" changed to \"Mr. Hayes\"\n 82 \"the Bastile\" changed to \"the Bastille\"\n 143 \"publiher\" changed to \"publisher\"\n 157 \"than he had them\" changed to \"than he had then\"\n 180 \"the noun _man_\" changed to \"the noun _men_\" (two instances)\n 182 \"a little work\" changed to \"a little word\"\n 187 \"_of_ the desire of\" changed to \"of the desire _of_\"\n 191 \"expresed\" changed to \"expressed\"\n 207 \"He feels keenly and deeply and wrongs of his class.\" changed\n to \"He feels keenly and deeply the wrongs of his class.\"\n 222 \"our expression of it become more simple.\" changed to \"our\n expression of it became more simple.\"\n 238 \"in apposition to the pronoun I\" changed to \"in apposition to the\n pronoun We\".\n 252 \"_I_ see a pale face\" changed to \"_I see_ a pale face\"\n 265 \"With your faces pinches and blue\" changed to \"with your faces\n pinched and blue\"\n 271 \"the _party which fought for their rights_\" changed to \"the\n party _which fought for their rights_\"\n 277 \"Find _e_ or _y_\" changed to \"Final _e_ or _y_\"\n 287 \"The prefix _in_ used with adjectives\" changed to \"The prefix\n _un_ used with adjectives\"\n 312 \"The dash if often used\" changed to \"The dash is often used\"\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n_The Chalk Artist_ is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nCopyright \u00a9 2017 by Allegra Goodman\n\nAll rights reserved.\n\nPublished in the United States by The Dial Press, an imprint of Random House, a division of Penguin Random House LLC, New York.\n\nTHE DIAL PRESS and the HOUSE colophon are registered trademarks of Penguin Random House LLC.\n\nGrateful acknowledgment is made to the following for permission to reprint previously published material:\n\n**The Belknap Press of Harvard University Press:** Excerpt from \"There is a pain so utter\" from _The Poems of Emily Dickinson_ , edited by Thomas H. Johnson (Cambridge, MA: The Belknap Press of Harvard University Press), copyright \u00a9 1951, 1955 by the President and Fellows of Harvard College. Copyright \u00a9 renewed 1979, 1983 by the President and Fellows of Harvard College. Copyright \u00a9 1914, 1918, 1919, 1924, 1929, 1930, 1932, 1935, 1937, 1942 by Martha Dickinson Bianchi. Copyright \u00a9 1952, 1957, 1958, 1963, 1965 by Mary L. Hampson. Reprinted by permission of The Belknap Press of Harvard University Press.\n\n**New Directions Publishing Corporation:** Excerpt from \"A Virginal\" from _Personae_ by Ezra Pound, copyright \u00a9 1926 by Ezra Pound. Reprinted by permission of New Directions Publishing Corporation.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nNames: Goodman, Allegra, author.\n\nTitle: The chalk artist : a novel \/ Allegra Goodman.\n\nDescription: First edition. | New York : The Dial Press, [2017]\n\nIdentifiers: LCCN 2016001270| ISBN 9781400069873 | ISBN 9780679605041 (ebook)\n\nSubjects: LCSH: Computer games\u2014Social aspects\u2014Fiction. | Video games\u2014Social aspects\u2014Fiction. | Man-woman relationships\u2014Fiction. | Interpersonal relations\u2014Fiction. | GSAFD: Love stories.\n\nClassification: LCC PS3557.O5829 C47 2017 | DDC 813\/.54\u2014dc23 LC record available at https:\/\/lccn.loc.gov\/\u200b2016001270\n\nEbook ISBN 9780679605041\n\nrandomhousebooks.com\n\nBook design by Caroline Cunningham, adapted for ebook\n\nHand lettering on title page and leaf ornament by Rachel Willey\n\nCover design: Rachel Willey\n\nCover photograph: Rosemary Calvert\/Getty Images (leaves)\n\nv4.1\n\nep\n\n# Contents\n\nCover\n\nTitle Page\n\nCopyright\n\nChapter 1: Grendel's Den\n\nChapter 2: The Orchard\n\nChapter 3: Emerson\n\nChapter 4: Everwhen\n\nChapter 5: Wait for Me\n\nChapter 6: Snow Day\n\nChapter 7: Dream\n\nChapter 8: No Moon, No Stars\n\nChapter 9: Drink Me\n\nChapter 10: In Her Eyes\n\nChapter 11: Caution\n\nChapter 12: The Leopard\n\nChapter 13: Crossing Over\n\nChapter 14: The Visit\n\nChapter 15: Open Door\n\nChapter 16: The Interview\n\nChapter 17: Arkadia\n\nChapter 18: The Gates\n\nChapter 19: Deer\n\nChapter 20: Very Close\n\nChapter 21: Face-to-Face\n\nChapter 22: Pursuit\n\nChapter 23: Admission\n\nChapter 24: Discovery\n\nChapter 25: Lucky\n\nChapter 26: Walden Woods\n\nChapter 27: Half Magic\n\nChapter 28: The Question\n\nChapter 29: The Kiss\n\nChapter 30: Poetry Inaction\n\nChapter 31: Joy Street\n\nChapter 32: Two Rivers\n\nChapter 33: Busted\n\nChapter 34: Bird on a Wire\n\nChapter 35: In the Hall\n\nChapter 36: World-Jumping\n\nChapter 37: Contagion\n\nChapter 38: Win, Lose, or Draw\n\nDedication\n\nBy Allegra Goodman\n\nAbout the Author\n\nHer long hair curtained her face as she sat marking papers. Drunk graduate students surrounded her, but she didn't even look up. Rock pounding, dishes clattering, this was Grendel's in winter, the old Cambridge dive, loud, warm, and subterranean, half a flight down from Winthrop Street. A green lamp lit every table, a hundred mirrors hung on paneled walls. Collin watched her reflection from every angle. She looked so elegant and out of place.\n\nShe came on Tuesday nights, and sometimes Thursdays too. She would order a Mediterranean salad and start grading papers. She was slender, fair, her eyes dark and shining, as though she knew some secret\u2014she alone. Whenever he got close enough, he looked over her shoulder. Her handwriting was precise, her pen purple, extra fine. Once she glanced up and nearly smiled. You realize, he told her silently, if I drop something it's your fault. If I break a plate, it's all because of you.\n\nHe saw guys leering, even if she didn't. \"Everybody's looking at her,\" he told Samantha, the bartender.\n\nSam said, \"Yeah, but mostly you.\"\n\nCollin was twenty-three, bright, artistic, and unhappy. He had just left college for the second time, and although he had good reasons, his mother was upset with him. His ex-girlfriend Noelle was out of patience. His father was in the navy; he had not seen or even heard from the man in seven years. Collin had thought of enlisting, mostly to travel, but he had grown up on a street where signs in the front yards read WAR IS NOT THE ANSWER. He never did enlist. He didn't go anywhere.\n\nHe worked at a bar and went out drinking afterward. Even if he'd enjoyed college and respected his instructors, even if he had excelled at Web design and programming, he didn't have time to go to class. He was busy collecting tips and partying, waking up in other people's beds. Sometimes he despised himself; not often. Sometimes he decided to get serious, but he kept working nights and sleeping in, and hanging with his high school friends, and all of this became a full-time job; youth itself was his vocation.\n\nFor this reason, the girl's diligence fascinated him. She sat for hours grading at her table, and she was so young\u2014way too young to be a teacher. She should have known better than to sit alone down there. Few came to work at Grendel's, and those who tried, didn't get much done. They would open their computers and close them gratefully when drinks arrived. This girl did not respond to guys circling her table. She looked royal in her cardigans and trailing scarves and calfskin boots. He sketched her on his order pad. The princess of solitude, with a crown.\n\nOne Tuesday, when she started packing up, her coat slipped off the back of her chair, and Collin ran to catch it for her. She stood to go, and he realized how tall she was, almost his height. He was close enough to see the gold flecks in her eyes, the freckles dusting her face. He held his breath as she slipped her arms into the sleeves. Then she thanked him, and rushed off.\n\n\"Nice,\" teased a waitress named Kayte. \"Could you catch my coat too? Before it touches the ground?\"\n\nCollin watched for the girl on Thursday while he carried out chicken wings and plates of stuffed potato skins. He served foaming Guinness, caught bits of conversation: _Seriously? How much did that cost? I feel guilty but..._ The Who pounding. Students wailing, _\"The exodus is here.\"_ Busy night and no free time, but Collin kept watching until Sam started flicking ice at him from behind the bar. \"Who're you waiting for?\"\n\n\"Shut up.\"\n\n\"So you admit it.\" Sam was tiny but in your face. She was compiling a book of vintage cocktails.\n\n\"I'm not admitting anything.\"\n\nTrue, Collin wondered about the teacher. He speculated about her at Broadway Bicycle School, where he taught wheel changing, tire patching\u2014basic repair. She had sounded American, but he decided that she came from Paris. Or London. He said, \"Inflate the tube and listen.\" Maybe Barcelona.\n\nOn Monday he colored backdrops for the theater company he had founded with his roommate, Darius. Working with wet chalk on old-fashioned rolling blackboards, he drew slender trunks and arching branches, layered cherry blossoms, white and pink. The edge of his chalk crumbled. He rubbed white and red together with his thumb, and he thought and thought about her. Sometimes she glanced up and she was looking at him, he was sure of it. The next second he would think, No, that can't be true. Daydreaming about her, he felt lighthearted, amused. His fantasies were so chaste and so persistent. She was always sitting at her table, just out of reach, and he liked her there\u2014although he was intensely curious. What was she doing all alone? A girl like that would have a boyfriend. There had to be some story. A long-distance relationship\u2014but she didn't look lonely. He wanted to know her. Or at least to hear her name.\n\nThere were days she never even crossed his mind. He spent a weekend with Noelle. They went to a party and stayed out late dancing, and then they went to her place and he began undressing her. She laughed, and he knew why. Now that they'd sworn off each other their bodies were so eager.\n\nLate the next day they woke stale and headachy, annoyed with themselves. Even so, Darius's girlfriend, Emma, had four tickets to Lady Lamb the Beekeeper in Davis Square, and so they went. All that time, Collin didn't think about the girl, until Lady Lamb bent over her guitar, her long hair curtaining her face. Then suddenly he imagined the girl watching him. He saw himself through her eyes and he was cheap, and aimless. He felt poor, as well, although he didn't consider himself poor. He considered himself free.\n\n\u2014\n\nThe next week, he was taking orders for a party of six when she materialized again. He looked up, and there she was, already seated in Kayte's territory. He was not getting off early, but when he saw the huge stack of papers on her table, he made a secret deal with her. If you keep at it until eleven, I'll walk out with you.\n\nAll night he watched her table, willing her to stay. When she began to stir, he murmured: \"No, you don't. Keep working. You aren't going anywhere.\"\n\nTen forty-five, she pushed back her chair. From behind, he saw her shoulders shaking, and thought she must be sobbing, or choking. He rushed over. \"Are you all right?\"\n\nWhen she looked up, she was laughing, not crying, and she showed him an essay. Curvy handwriting on lined paper, the title in bigger script: _Juliet: Shakespeare's Heroin._ \"What do you think?\"\n\nA thousand ideas crowded his mind, none about her student's spelling, as he watched her add an _e._ \"Are you really a teacher?\"\n\nShe said, \"I keep asking myself.\"\n\n\"You don't look like one.\"\n\nShe shook back her long brown hair and glanced up at him, amused. \"What's a teacher supposed to look like?\"\n\n\"Old,\" he told her. \"Bitter.\"\n\n\"I'm bitter.\"\n\n\"How long have you been teaching?\"\n\n\"Three months.\"\n\n\"Your students are that bad?\"\n\nShe frowned as she looked down at her check. Annoyed? Or just figuring out the tip?\n\nHe said, \"My friend Darius was thinking of directing _Romeo and Juliet,_ but he couldn't find a church.\"\n\n\"He couldn't get permission?\" Already she was shouldering her bag, and standing up to go.\n\n\"He wanted to do it in a cathedral with stained glass and confessionals, but the only church interested was Unitarian.\"\n\n\"Are you an actor?\"\n\nJean-Philippe, the busboy, was trying to get by, and Collin stepped sideways. \"I'm an actor and an artist.\" He regretted the words as soon as he said them. He sounded pretentious. \"Mostly chalk.\"\n\nShe looked puzzled. \"On sidewalks?\"\n\n\"Yeah, but other places too. I do all the art for Theater Without Walls.\"\n\n\"I've heard of them!\"\n\n\"In the _Phoenix_?\" He turned, glancing backward at Kayte. Cover for me, he begged her silently. She was shaking her head, but he knew she liked him. Just five minutes. My tips are yours! \"Wait, let me walk you out.\" He handed the girl a leaflet for _The Cherry Orchard,_ a new production at the MIT tennis courts by Theater Without Walls. Art Director: Collin James.\n\n\"Tennis courts in December?\"\n\n\"They're indoor.\" He led the way upstairs and opened the door for her. The snow around them lit the darkness. \"I'm designing the lights...and the trees. I'm in it too.\"\n\n\"You perform in Sennott Park, right?\"\n\n\"We perform all over. We did _The Tempest_ on a traffic island.\"\n\n\"That's it! I read about the car accident.\"\n\n\"It was just one guy hitting a pole,\" he said. \"Nobody got hurt.\"\n\nShe smiled.\n\n\"Come to _The Cherry Orchard._ I'll get you a ticket. Give me your name and I'll put you on the list.\"\n\nShe didn't say yes, and she didn't say no. She just looked at him, and her eyes were so dark and bright that he drew closer, until she began to laugh.\n\n\"Or not.\" He took a full step back.\n\n\"I wasn't laughing at the play.\"\n\n\"Why, then?\" He had wild curly hair, black eyes, a quick, athletic body, a defensive look.\n\n\"I don't know,\" she said in some confusion.\n\n\"Come if you want,\" he said coolly.\n\n\"Okay,\" she said, automatically polite.\n\nHe didn't ask her name; he pretended she was just a customer. \"Have a good night.\"\n\nCollin could not remember why Darius had insisted on the MIT Bubble. Maybe he'd assumed he could reserve all the courts. As it happened, the actors got just one. All through _The Cherry Orchard_ they competed with the skid and squeak of athletic shoes, and the _thwack_ of tennis balls. Sometimes a stray ball flew over from guys rallying just steps away.\n\nThe idea was no chairs. Everyone could walk freely so that, as the play progressed, the audience followed the action, advancing, retreating, and approaching the net. There was no set except for Collin's chalk orchard, his blackboards filled with blooming trees.\n\nThe other idea\u2014all Darius\u2014was to stage the play as farce. They had fought about this. \"Dude,\" Collin had told Darius, \"the lady returns to where her kid drowned and then finds out she's going to lose her house.\"\n\n\"Yeah, so?\" Darius was a big guy, not as tall as Collin, but broader. Smart, and avant-garde, and something of a rainmaker, he came up with grants from the Cambridge Arts Council, permission to perform in public places.\n\n\"The play is sad,\" Collin said.\n\nDarius dismissed this. \"Yeah, the sad version is really overdone.\"\n\nIn performance, Darius's girlfriend, Emma, romped across the court, starring as the romantic Ranevskaya, who would lose her childhood home, her past, her everything. Emma was more folk-rock Mainer than Russian nobility to begin with, and when she spun around calling, \"Goodbye, old house, old grandfather house,\" you had the feeling she was excited to move on.\n\nPouring imaginary tea as the servant, Collin listened to the audience's cautious laughter, and he wanted to dash his imaginary teapot onto the court and smash all his imaginary cups and saucers too. Where was the darkness in the play? Where were the shadows? In rehearsal Darius had conceded that some moments were bittersweet. Now, under rented lights, nothing bittersweet came through. No darkness, except for Collin's seething servant, and Noelle, who played Ranevskaya's daughter Anya.\n\nEveryone wore street clothes instead of costumes, and Noelle dressed in a little undershirt and a pair of frayed jeans that fit her like a second skin. She was an ex-ballerina with spiky hair and a pierced tongue, and she was pissed. She had a lot of anger in general, but she was angry at Collin in particular. They'd had a huge fight the night before, during which she had said, among many other things, \"First of all, I hate you. Second of all, you are one hundred percent bad for me, because the only thing you care about is the beginning and the end. You can never be in the middle; you can never actually be with someone or learn something or get something done, because you're always starting and then leaving, which is why I hate myself after I've been with you.\"\n\nEven now, performing, he heard hostility in every one of Noelle's lines. At the end of the play, she was brutal when she announced in her husky smoker's voice, \"The cherry orchard is sold, it's gone, that's true, but don't cry, Mama.\" Noelle was hard-core for an ingenue.\n\nCollin was glad the girl from Grendel's hadn't come. Her laughter haunted him, because she had known then\u2014she'd known in advance exactly how the play would be. Pretentious and amateurish all at once. Everyone watching related to or sleeping with the cast. After Act II, a couple of strangers wandered in, but they were carrying their racquet bags.\n\nHe could see the audience tiring, clustering near the baseline. They perked up when the estate was sold and all the characters said their goodbyes. As soon as Chekhov's characters began talking about the future, everybody started folding camping stools and gathering bags. People were already heading out when Collin flipped his blackboards over, one by one. The audience froze for just a moment as he revealed his chalk drawings on the other side. Jagged stumps, fallen petals, broken branches, a holocaust of trees.\n\n\u2014\n\n\"Don't touch the art,\" Collin's mother, Maia, told him at the party afterward. \"Do you have pictures of that orchard?\"\n\n\"Darius had the camera.\"\n\n\"Hey,\" Maia called out to Darius, who sat on a couch overflowing with actors. \"Do you have photos of Collin's orchard?\" She was unhappy with Collin and also fiercely proud, saving every scrap of art, celebrating each performance, inviting the whole cast to her winter solstice celebration. \"That orchard was the best character in the play.\"\n\nDarius raised his beer bottle.\n\nCollin said, \"Good thing he can't hear you.\"\n\n\"I'm your mother,\" said Maia. She meant, You think I care?\n\nShe was tall and young for a mother, darker than Collin, so they didn't quite match. What am I? he used to ask when he was little. Black Irish, his mother said. His father was Irish and she was Italian, French Canadian, and a little bit Native American as well. She worked at the Fletcher Maynard Academy, in the basement therapy room with swings and finger paints, giant inflatable balls. The house was full of teaching prizes, clocks and plaques, two crystal apples from the district. She had posters too. WHAT DO TEACHERS MAKE? I MAKE KIDS WONDER. I MAKE THEM QUESTION...Collin had grown up with all these tributes, apples, and rainbows. He had been famous in school as Maia's son, and had played in her therapy room as a small child. There had never been money for babysitters, so he'd attended his mother's afterschool dance classes as well. Jazz, tap, and tango.\n\nNow Maia's colleagues were arriving, and she served them borscht in honor of the Russian play. She ladled the thick magenta soup into bowls, and when she ran out of bowls, she handed mugs to Mrs. McCabe, the librarian, and Ms. Jamil, the occupational therapist. She found a \"Mad Genius\" mug for Mr. Cooperman, the fifth-grade science teacher, also known as Scienceman.\n\nYoga friends tramped up the porch of the triple-decker and left their boots in the stairwell. In socks they padded into the dining room, where Maia had covered the Ping-Pong table with rich fabric and flatbreads, quiche, empanadas, latkes.\n\nUpstairs neighbors came in slippers. Lois, the art teacher from the second floor. Sage and her wife, Melissa, who grew tomatoes in window boxes on the third floor. Strawberry-blond Kerry O'Neil came from across the street, along with her twins, Aidan and Diana, suddenly sixteen.\n\n\"Can you believe it?\" Maia asked Collin.\n\n\"How'd you people get so big?\" said Collin. \"Did I ever babysit you?\"\n\nThey didn't answer. Aidan wolfed down his food and left, while Diana stayed, nibbling crispy spring rolls.\n\n\"Collin!\" Lois exclaimed, because she had seen the show. \"Your orchard!\"\n\nLois's white hair was short and spiky, her vest hand-quilted, her earrings fashioned of the most precious Scrabble tiles, Z and Q. \"When the production ends, what happens to your art?\"\n\n\"Well...\" Collin began.\n\n\"Don't say it.\"\n\nCollin enjoyed saying it. Gentle, sentimental, Lois and Maia were crushed when he explained, \"I'll wash the boards and start something new.\"\n\nLois tried arguing with him, but he couldn't hear. Everyone was laughing over in the living room. Darius laughed so hard that he had to lean over Emma to defog his black-framed glasses.\n\nThe actors were talking about putting on a druggier, _Big Chill_ Chekhov, like in a big house up in Maine. Meanwhile, the neighbors were discussing snow emergencies and parking bans, and how they were disappointed in the president. Troops still in Afghanistan, prisoners still rotting in Guant\u00e1namo. The economy still in the toilet. What good was he?\n\nThe battered oak floors thrummed with Crosby, Stills, and Nash. \"Judy Blue Eyes,\" \"Marrakesh Express,\" \"Guinnevere.\" Noelle was flirting with some woodworker named Austin. He was much taller than she was, so she rose up on her toes, _en pointe,_ to look into his eyes.\n\nCollin brushed past Noelle and said, \"How's the view?\"\n\nShe ignored him, and he hated her. No, he didn't hate her; he felt nothing. He felt dead.\n\nDarius and Emma started dancing. Maia and Mrs. McCabe joined in, flushed with laughter and with wine, but Collin turned toward the front window to watch the snow. The little street was melting away, all the houses turning to gingerbread, white drifts icing porches and peaked roofs. He wanted to be out in it, away from everyone, especially himself. He pulled his jacket from the pile in the entryway, sat on the stairs and laced his boots.\n\nWhen he stepped onto the porch he was so lonely he could almost see it like his breath. Impatient with himself, he seized the shovel propped up near the door, and began clearing the front steps. Working fast, he excavated a path to the street and started on the sidewalk. It was still snowing heavily, but he was hot, and unzipped his jacket.\n\nHis mother stepped outside and watched him from the porch. \"Oh, come on.\"\n\nHe didn't answer.\n\nShe sighed and disappeared into the house. A moment later, she returned in boots. She was holding a glass of wine.\n\n\"It's only natural to feel a letdown.\"\n\n\"Letdown from what? The performance wasn't any good.\"\n\n\"So then it's even more natural, because the play wasn't as great as the one in your head. That's art.\"\n\n\"That's life,\" he said.\n\n\"I've seen you and Darius do amazing stuff.\"\n\n\"It's _his_ stuff. It's all about him.\"\n\n\"You'll get your chance.\"\n\nCollin planted his shovel in the snow. \"I'm not working with him anymore.\"\n\nHer temper flared. \"Right, that's the answer. Just give up.\"\n\n\"Nobody's interested in theater on traffic islands and indoor tennis courts.\"\n\n\"Who's nobody?\"\n\n\"Nobody in the real world.\"\n\n\"The real world is overrated,\" Maia said.\n\n\"I wouldn't know.\"\n\n\"Your loss.\"\n\nThey stood in silence as the city plow thundered down the street, sparking tiny flames, metal scraping asphalt as it clattered past. Then Collin started shoveling again, heaping snow into the garden.\n\n\"Careful,\" Maia said. \"My hydrangeas are somewhere under there.\"\n\n\"They're okay.\"\n\n\"They'd better be.\" She watched him for a moment in silence and then tried again. \"Do you want my advice?\"\n\n\"No.\" He didn't want advice. He wanted to escape. He wanted to break away, but he kept coming home, working with Darius, returning to Noelle. What was wrong with him? He could draw, but he drew only for Darius. He was a hired hand. No, just a hand! It wasn't like Darius paid him.\n\nHe had cleared the sidewalk and now he carried his shovel up the steps. \"It's a play about regret.\"\n\n\"I know.\" Maia followed him.\n\n\"It's about wanting what you can't have,\" Collin declared under the porch light. \"Maybe that's too obvious for Darius. He always gets what he wants, so he can't even see it.\"\n\nMaia nodded. \"I understand.\"\n\n\"No, you don't.\"\n\n\"Mm-hm. What's her name?\"\n\nLoaded down with books and folders, Nina shouldered the heavy school door open. VISITORS REPORT TO THE OFFICE. THIS DOOR TO REMAIN LOCKED. OUR CHILDREN ARE OUR FUTURE. Dreading the day ahead, she backed her way into school.\n\nDread was not too strong a word for Wednesdays. She had to start with the wild ones, her American lit class. Students came unprepared. (They're kids, said Mrs. West, her department chair.) Some didn't come at all. (It's not about you, said Jeff, her TeacherCorps mentor.) Nothing worked with these eleventh graders. Her lessons were too difficult, her readings too long, her assignments way too complicated.\n\n\"Good morning, Miss Lazare,\" said Mrs. West as they stood in the glass front office, signing in. All the teachers spoke to one another like that, as if afraid a student might overhear their given names. \"Are you okay?\"\n\nNina could only imagine how she looked\u2014pale, sleep-deprived, floundering. Mrs. West, on the other hand, could walk the halls with total mastery, her midnight-blue manicure adorned with tiny crystal stars. Mrs. West was famous for her fingernails, for her languages, French and Haitian Creole, for her gorgeous singing voice, for her in-class performances of _Romeo and Juliet._ Above all, for her way with words. Nina had heard her harangue one boy into submission in the hall. \"You can stand there and tell me that's your best work, but you _know_ that's not true, so get your butt back in your chair and do it again. The end. That's all!\"\n\n\"I've got my eleventh graders,\" Nina said.\n\n\"Don't let up!\" Part coach, part cheerleader, part preacher, Mrs. West admonished Nina, \"If those kids test you\u2014then you test them back!\"\n\nA sudden buzz and pulsing orange lights. A couple of boys had set off the metal detector, and the school's police officers sprang into action. \"Over here, you guys. Bags on the table. Keep the line moving.\" Kids continued shuffling in with their backpacks, their headscarves, their puffy jackets, their attitudes. Nina joined them, climbing the chipped cement steps to her classroom, where she opened her desk and locked her purse inside.\n\nEmerson High School was small, diverse, and experimental. There were no exams, only year-end portfolios in which students collected what they considered their best work. The school was open to everyone by lottery, but had a reputation for \"out of the box\" kids\u2014those who were artistic, or autistic, those with learning differences, or special gifts, or both at once. Each student was required to keep a journal of personal discovery in a marbled composition notebook. Required personal discovery seemed like an oxymoron to Nina, but so did many other aspects of the school. \"A community of learners\" with metal detectors. A non-linear curriculum in a rectilinear 1930s building. The wiring was antique, the doorknobs brass. The basement flooded during rainstorms, and in places the roof leaked. There were computers in every classroom, but some rooms had no heat. There were no whiteboards, let alone SMART Boards. Blackboards were strictly black.\n\nNina took a deep breath and picked up her white chalk. _DO NOW,_ she wrote on the board. _List three adjectives Emerson uses to describe..._\n\n\"Miss?\" Students were starting to drift in. \"Miss?\"\n\n\"Get out your packets,\" Nina announced. There were no textbooks for Emerson's English classes. Humanities teachers had to develop their own materials. \"Turn to the essay 'Nature.' Where's your reading?\" she asked Xavier.\n\n\"I'm not exactly sure.\"\n\n\"Miss.\" Rakim approached the board. \"I think the page was missing or maybe\u2014\"\n\n\"Let me see,\" Nina said.\n\n\"I think my packet is defective or something like that.\"\n\nNina found \"Nature\" for him. \"Have a seat.\"\n\n\"I looked for it,\" he said.\n\n\"Rakim, please sit down.\"\n\nHe raised his hands in mock surrender as he backed away. Two girls started giggling, and Rakim played to the gallery. \"Okay. Okay!\"\n\nSome people had it\u2014that mysterious rapport, the ability to catch a student's gaze and hold it, to direct without seeming to direct, to take a joke and lob it back. Nina thought of her own high school teachers, charismatic Mr. Kincaid, witty Mr. Rousse. Was it a gift like perfect pitch, or something you could learn? She remembered the silence in Mr. Rousse's classroom, the deliberate way he spoke, the way he made you wait for the next word. You shivered when he called on you.\n\nShe had done well in training. For five weeks at Bowdoin College in blackfly season, Nina had learned, in theory, to teach high school. After long days of role-playing, mini lessons, and crash courses in curriculum development, she would sit with other TeacherCorps recruits in a darkened lecture hall to hear testimonies of transformation.\n\nAlumni perched on stools in front of the eight hundred trainees, and, one by one, those alumni stood and walked into the spotlight to tell of raising test scores, breaking through. \"Is there anything you can do in this world that's more important?\" asked one former fellow, now working at McKinsey. \"Is there anything more valuable than the life of a child?\"\n\nAll the alumni said that their students had taught them lessons they would carry with them their whole lives. Nina never doubted this, but as her own kids came in dancing, slouching, scuffling playfully, she hoped she could teach them something too.\n\n\"Miss, could you sign this?\"\n\nLeila was holding a pink slip.\n\n\"You're dropping language arts?\"\n\n\"Switching,\" said Leila, a picture of innocence, framed by her white headscarf. \"Mr. DeLaurentis thinks I'll learn better with Mrs. West.\"\n\nOf course he does, Nina thought miserably. Mrs. West had been teaching almost thirty years. Nina had been trying for thirteen weeks. Mrs. West invented her own acronyms: OWL (Own it, Work it, Learn it). QUACK (Question Underlying Assumptions Critically and Knowledgeably). Nina was still trying to keep her kids in chairs.\n\nShe bent to sign her name, and felt for a moment as though she were signing a confession. _I'm a fraud. I have no idea. I've failed to reach you._ Everyone spoke about epiphanies and transcendent moments, the _Miracle Worker_ of it all: teacher and student swept up in revelation, spelling into each other's palms\u2014touching the word, grasping the concept, feeling the rush of water. People didn't talk so much about students switching out of your class.\n\n\"Okay.\" She handed Leila the pink paper. \"Go for it.\"\n\n\"Thanks.\" Leila ran out, and Nina shut the door behind her.\n\n\"Take out your notebooks,\" she directed, as she took attendance. Sixteen students, eight absent, including one learning better with Mrs. West. \"Write your three adjectives.\"\n\n\"Three adjectives about what?\" Rakim asked, and Nina realized that she hadn't finished writing her _DO NOW_ on the board.\n\n\"Three adjectives Emerson uses to describe...nature.\" She scribbled the missing word.\n\n\"Can I borrow a\u2014\"\n\n\"Can I use a\u2014\"\n\nAlready two kids were up to sharpen pencils by the window.\n\nA pregnant girl named Brynna asked, \"Miss, where's the bathroom pass?\"\n\nYou just got here, Nina thought, as she searched her desk. \"Three adjectives. Faheen, I don't see you writing.\"\n\n\"I'm thinking!\"\n\nSqueaking chairs, rustling papers. Always moving, always whispering, the class never settled. As soon as Nina shushed one conversation, another started. She needed to be everywhere at once, but she tried to focus on one student at a time as she walked between desks, reading over shoulders.\n\n\"Beautiful, gorgeous, natural. No, that's not quite it. I want adjectives that _Emerson_ uses about nature,\" Nina told Diana, who looked up, insulted.\n\n\"Miss?\" Brynna asked again.\n\nNina handed over the restroom key with its big block of wood attached and she began writing on the board the adjectives the students had found in Emerson: _tranquil, perpetual, transparent._\n\n\"Is _theory_ an adjective?\" Chantal asked.\n\n\"Is it descriptive?\"\n\n\"Yeah.\"\n\n\"Give me an example of _theory_ as an adjective.\"\n\n\"Theory of relativity.\"\n\n\"Excuse me?\" Nina asked Marisol and Cierra. The girls looked up from their conversation as Nina began walking over, but she got distracted by Rakim writing furiously, filling in some worksheet for another class. Shit. Nina had forgotten to collect homework. \"Time out. Please hand in your short responses from last night. Rakim? Marisol? Pass them to the front. Cierra, did you hear me?\"\n\nNine homework sheets came in, accompanied by at least five excuses.\n\n\"I was absent yesterday.\"\n\n\"Miss, I didn't get the assignment.\"\n\n\"I never got the questions.\"\n\nNina struggled to assert herself. \"If you were absent, you're still responsible.\"\n\nUnfortunately, when she was frustrated, her voice got quiet. Her kids continued writing, ripping pages out of notebooks, sharpening pencils, and talking, talking.\n\nYou had to get louder in this profession, not softer. You couldn't just look disappointed. You were supposed to scream to show you cared. Mrs. West would cry out, \"Listen up. I'm talking to _you._ \" Across the hall, Ms. Powers stamped her foot. Mr. Allan could bugle like a moose. Nina's emotions were all wrong, if that was possible. She wasn't angry when kids didn't do the reading. She was crushed.\n\n\"If you don't understand, speak up!\" she pleaded. \"If you never got the questions, then come and get them from me.\" For a moment, her class looked at her, gauging her annoyance. \"I'm tired of excuses,\" she added, but the chatter had started up again.\n\n\"You should all be on page thirteen of your packets. Rosie. Page thirteen?\" Once more she walked up and down between the chairs. \"Page thirteen,\" she repeated. \" 'Nature.' Faheen, read the second paragraph for us, starting with _'Nature is...'_ I'm sorry, would you please sit down?\" she told Sevonna, who was standing by the windows. She was a big girl scribbling on a tiny piece of paper, which she stuffed into the back pocket of her jeans. \"Cierra? Marisol? Sevonna, would you please sit down? Please?\" Nina repeated, even as Faheen read slowly, _\"Nature is a setting that fits equally well a comic or a mourning piece. In good health, the air is a cordial of incredible virtue.\"_\n\nThe sky outside was bright. From the fourth-floor window Nina could see Lincoln Playground, framed by black trees striped with snow. \"Keep reading, Faheen.\"\n\n_\"Crossing a bare common, in snow puddles, at twilight, under a clouded sky, without having in my thoughts any occurrence of special good fortune, I have enjoyed a perfect exhilaration.\"_\n\n\"Marisol,\" Nina said. \"What does Emerson mean by _exhilaration_?\"\n\n\"Don't kick my chair, man,\" said Rakim.\n\n\"I never touched your chair.\" Xavier stretched out his long arms and legs and tilted back his own chair until he sat on a diagonal.\n\n\"It was on the homework. _Exhilaration._ \" The noise level was rising, but Nina risked turning her back to write the word on the board, along with the words _transport_ and _ecstasy._ She drew a line and then wrote SUBLIME as the header for these vocabulary words. \"The Transcendentalists were interested in the sublime.\" She spun around just in time to hear a crash and laughter, as Xavier and his chair slammed onto the floor.\n\n\"Ow! Fuck!\" Xavier moaned good-naturedly, and took his time getting up again.\n\n\"Not in my class,\" Nina said, and Xavier apologized for his language, but that took more time. Here she was, clock ticking, eight students absent, not to mention one in the bathroom, and she had covered three sentences.\n\n\"You deserve it.\" Diana was talking to Xavier, but Nina thought Diana might as well have been speaking to her.\n\nAs a child, Nina had wondered why some teachers were so boring. Now she understood. They were bad actors\u2014terrible at performing what they knew. Everything those teachers said fell flat, every lesson trailed off. First the class stopped listening. Then kids began whispering, laughing, talking openly. Finally, they seized power for themselves. As a student she had seen it happen. Now she watched her students turn on her. Xavier's attempts at comic relief, Marisol's and Cierra's and Sevonna's disrespect. Diana's scorn, Brynna's exit into the real drama of the girls' bathroom. Clear-eyed, Nina watched her class spin away from her.\n\nYou don't look like a teacher, Collin James had said at Grendel's. How does a teacher look? Old, he'd said, and bitter. He was a flirt\u2014she'd seen him with the waitresses\u2014but he was right: Teachers had to be hard and spiky, barbed in self-defense. You had to be bitter, to deter kids from eating you alive.\n\nAlready the bell was ringing, and she was trying to explain the homework, even though she hadn't finished her lesson. Her class was racing out the door and Nina wanted to run after them. _Standing on the bare ground,\u2014my head bathed by the blithe air, and uplifted into infinite space,\u2014all mean egotism vanishes._ She would carry Emerson to them. Wait! You forgot this! But they were gone. Too late, too late. She'd lost her chance.\n\n\"Now!\" Sword in hand, he ran through tall grass. \"Jump!\" Together they bounded over rushing water.\n\nThe sky was darkening. Smoke clouded the landscape, smudging hills and trees. Faster and faster he ran, and she flew after him, blue hair streaming. No time to talk, no time to breathe. They raced to the edge of the Trackless Wood, which crackled with fire. Danger never stopped them. They kept running, weaving through the burning trees.\n\nShe was a Tree Elf named Riyah. He was a Water Elf, Tildor. They came from different realms, but for the past three nights they'd qwested, traded, and killed together. They had hunted basilisks, slain dragons, and retrieved two diamonds, which Riyah carried in the bag hanging at her waist. She was an amazing marksman, and beautiful, even for an Elf, her eyes huge, her body supple. Her breasts swayed as she ran, her quiver bouncing behind her.\n\nFlaming branches crashed around them, the crack of falling trees\u2014then something else\u2014a ripping sound.\n\nThey whirled around to face a colossus with jagged teeth and claws, a shifting, seething monster, half man, half bear, tar black. He rose up on his hind legs to seize them with clawed hands. They slashed him with their swords, but he reconstituted, oozing and bubbling. He snatched up his left hand and jammed it on again. Screaming with pain, he retrieved his severed leg and screwed it onto his own bloody stump.\n\n\"Can't finish him with steel,\" Riyah gasped.\n\n\"Shoot.\"\n\n\"Arrows can't penetrate.\"\n\n\"Take a\u2014\"\n\n\"Let me!\" The beast heaved up roaring and she sent her silver dagger into his eye. The colossus melted like a pile of burning tires. \"Yes!\" Breathing hard, she raised her arms in celebration.\n\nA diamond glittered at the melting monster's core. As Tildor plucked the jewel, a white nimbus glowed around him.\n\nRiyah's voice was hushed. \"The third.\"\n\nThorny branches overhead turned into talons, flaming twigs to ashen feathers. The forest phoenix woke, and Riyah and Tildor threw themselves onto the bird's back. The landscape shifted under them as they soared into the air. With each wingbeat, the phoenix carried them over smoldering trees and moonlit fields, twisted sunflowers, stubble glowing with white frost. Wind whistled in their ears. \"That's the Keep.\" Riyah pointed to stone towers in the distance.\n\n\"And there's the\u2014\"\n\n\"Aidan?\"\n\n\"Wait for me,\" he told Riyah.\n\nHe hurried to his bedroom door.\n\n\"Aidan!\"\n\nHe opened the door. _\"What?\"_\n\n\"Take off that headset.\"\n\nHe obeyed and lost the music of the wind and air, the hoots of owls and beating wings, the sound of Riyah's voice. Onscreen the phoenix soared over tangled woods and frozen ponds in the silver winter night of EverWhen. Offscreen, only a computer on a desk, an unmade bed, a backpack open on the floor.\n\n\"Do you know what time it is?\" his mother demanded. She was home from the night shift at the hospital and he was supposed to be getting ready for school.\n\nKerry snatched the headset from his hands. \"Do you even know what day it is? Aidan? Look at me when I'm talking to you!\"\n\nHe looked at her. Unprotesting, he sat down on his bed. Lately when his mother screamed at him, he listened. When she said EverWhen was sucking the life out of him, he didn't argue. If she asked for remorse, he showed remorse. If she declared, You're sixteen years old, you're wasting your time, you're failing school, he said he was sorry and promised to do better. He said whatever she wanted him to say, and all the time he kept his computer screen in view. The landscape dark and hidden, stars spelling out the name ARKADIA.\n\n\u2014\n\nShe worked so hard. He said, I know. She loved him so much. He bent his head. She asked what he thought she should do. He couldn't think of anything. She asked whether he thought life was precious. He said yes.\n\nBut his mother defined life as singular. He rejected that. He didn't live one life. He lived two. Was it his fault that he preferred the second? In EverWhen he was a healer and an Elvish prince, a leader of his company. He had a pile of gold, and a sword worth eighty marks, a magic ring, a diamond flask filled with a hatchling dragon's blood. He had fashioned his own gear: chain mail, silver helmet, enchanted boots. He'd trained for transformation. If he wanted to run like a deer, he could become one. If he qwested underwater, he could be a beaver, otter, or eel. And now in EverWhen he fought with Riyah at his side. Meanwhile, in the outside world, he was just a skinny kid, blue-eyed, dirty-blond, ignored at school. His friends were equally unpopular. Jack was a Water Elf, but in real life he took college-level math. Liam was an amazing warrior, but in real life he smoked so much pot Kerry refused to let him in the door.\n\nAidan's mother worried about drugs, and games, and bad influences\u2014not to mention the transmission on the car, the second mortgage on the house. They lived in a two-family. Priscilla, the piano teacher, paid rent on the other side of the living room wall, but Kerry owned the place. If the roof leaked or the pipes burst, Kerry was responsible. She slept mornings and worked nights, and in her world every day was like the last. No qwests awaited, no treasure maps arrived, no burning trees turned into birds, no cities into stone.\n\nHe knew he could outlast his mother because she was so tired. He let her confiscate his headset and joystick. He promised that he'd stop playing; he would get to school. When Kerry dragged herself to bed, he hunted up his other joystick and slipped back into EverWhen.\n\nThe sun was rising. Not the sun outside his window, but the sun inside the game, blood orange, melting frost and warming icy air. The colors were so clean and bright it took him a moment to adjust his eyes. Riyah was waiting for him in a thicket. She spoke, but without his headset he couldn't hear.\n\nHe typed into the chat box: sorry\n\nAfter a moment, her answer appeared on the screen. you should be. But she turned toward him, beautiful as ever, in the glowing morning light. more?\n\ncant\n\ntoo bad.\n\nlater? he pleaded.\n\nmaybe\n\nwheres the bird?\n\nlook\n\nJoystick in hand, he turned his Water Elf around. Pivoting onscreen, Tildor searched the thicket.\n\nNow Aidan saw the phoenix in new form, long feathers changed to white birches marked with what had been the bird's black eyes and claws.\n\n!!!\n\ni know!\n\nHis sister's alarm was beeping in the next room. He could hear Diana in the bathroom, flushing the toilet, starting her shower, getting ready for school.\n\nI have togo\n\nRiyah answered, bfn.\n\nwait whats your name?\n\nHe waited and waited. He heard the water stop, the shower curtain slide on its metal loops, Diana thumping down the stairs, the kitchen cabinets, the jolt of silverware in the drawer.\n\nEven as he gave up hoping, a word materialized onscreen: daphne\n\nFor a moment he just stared. He had never met a Daphne, and assumed the name was another alias. no, he typed, your real name\n\nThat IS my real name! Riyah folded her arms across her chest.\n\nHe took a breath. i want to see you.\n\nWhat do you want to see?\n\nHe began to answer and then he stopped. Her question confused him. Did he want to see the girl who played with him at night? The one who said her name was Daphne? Or the archer, dagger-thrower, Riyah? It seemed spell-breaking to write, I want to know who you really are\u2014and he wasn't sure if that was true. He wanted to play with Riyah forever. Not his school friends, or his old company. He wanted his headset back. He wanted to talk to her: I need to stay in EverWhen with you.\n\n\"Hey, Aidan.\" Diana was knocking.\n\n\"What?\" Instinctively he shifted in his chair to block his screen.\n\nShe opened his door and stood before him in black jeans and a black sweatshirt. \"You never took the recycling out.\"\n\n\"Because it was your turn.\"\n\n\"It wasn't my turn. Check the calendar.\"\n\n\"You check.\"\n\n\"The truck's already coming up the street.\" She yanked his window shade, but it didn't roll up. She pulled again, harder, and the shade only grew longer.\n\n\"Let go. I believe you.\"\n\nToo late. One final tug and his shade came crashing down. \"Diana! God!\" he whispered fiercely. \"You're going to wake Mom.\"\n\nGray sky. Dirty snow. The orange recycling truck lumbered up Antrim Street under a small, cold winter sun.\n\nu there? Riyah's question floated onscreen.\n\n\"Are you taking it out?\" Diana pressed.\n\n\"No! Get out of my room.\"\n\nThe broken window shade billowed on the floor, but Diana did not apologize. She turned on him. \"I can't believe you're playing again right after you promised Mom not to.\"\n\n\"I said get out.\"\n\n\"I'm going to tell her,\" Diana said slowly as she backed away.\n\n\"Go ahead,\" he said, but he knew she wouldn't tell. They were close, or had been. They had a pact, and even now he trusted Diana. He knew his twin would not betray him.\n\nDiana slammed the door, and the house rattled. He could hear his mother's voice, \"Aidan!\"\n\nHe typed, Invaded. I'll comeback.\n\nWould you try another world?\n\nRL???\n\nanother game\n\nwhich one?\n\ncan you keep a secret?\n\n\"Aidan?\" his mother called.\n\n\"I'm getting dressed.\"\n\nhuge? Riyah asked.\n\nwhat????\n\nHis screen faded to gray. The network hung. No, it only blinked. He and Riyah stood together as before, surrounded by white birches.\n\nI have UnderWorld beta, she told him.\n\n!!!!!! ru shitting me?\n\nno I have it\n\nnoone has it\n\ni do\n\nhow???\n\nyou want it?\n\nYES\n\nwanna play?\n\nHOW WHERE WHEN?\n\nyou need black box\n\nsend!\n\nnot yet\n\nwhen??\n\nyouwant it?\n\nHe pounded out his answer in frustration: Comeon\n\nwhat can you do for me?\n\nWas he still offended? All evening Nina tried to catch his eye, but he was carrying armfuls of dishes, dashing between tables\u2014keeping his distance. She hadn't meant to hurt him. The play had sounded funny and he was charming and she laughed when she was nervous. She couldn't help it.\n\nShe tried to work. _In the play \"A Midsummer Night's Dream\" by William Shakespeare portrays many characters, all with distinguished characteristics._ She marked this sentence: _In the play \"A Midsummer Night's Dream,\"_ _by_ _William Shakespeare portrays many characters, all with_ _distinguished_ **do you mean distinguishing?** _characteristics._ Then, looking up, she saw him emerging from the kitchen.\n\nHe saw her too, and then avoided her.\n\nShe couldn't work at all. She waited for her chance when he passed by. \"Collin,\" she said.\n\nPivoting, he turned toward her.\n\n\"How was the play?\"\n\n\"I'd tell you, but I'd get in trouble standing here so long.\"\n\n\"Oh!\" Instantly she turned back to her work, like a kid caught talking during study hall. _Puck is one of the most unique characters who..._ But this was no study hall, and even with her head down, she felt the warmth of his smile.\n\n\"What's your name?\"\n\n\"Nina.\"\n\nHe bent over his order pad, scribbling, before he rushed away.\n\nA scrap of paper floated down over her student's essay. A drawing no bigger than a silver dollar. Lines quick and confident. Her own face, half hidden by her hair, her chin propped up on her hand. She looked up, trying to find him in the crowded room, but he had disappeared into the kitchen.\n\nHad he really drawn all this just now? His work looked finished, less a sketch than a miniature portrait in black pen. He couldn't have, but yes\u2014she'd told him her name a second ago, and he'd printed in rapid block letters: NINA WAIT FOR ME I'M OFF 11.\n\n\u2014\n\nThat winter was so cold that the drifts never melted, and each storm topped off the one before. Even now, fresh snow was falling, velveting the streets.\n\nThey stood in front of Grendel's, facing JFK Street. The park there, colorful in summer with pigeons, buskers, tourists, now pure and white.\n\n\"I know where to go,\" Collin said. \"I grew up here, so.\"\n\n\"So did I.\"\n\n\"You did not.\"\n\n\"I went to college here too.\"\n\nOh, great, he thought. \"Went to college\" meant Harvard. If she had gone to Lesley she would have come out with it. If she'd attended Boston University, she would have said BU. Anywhere else, she would have named the school.\n\n\"My dad and stepmother still live here,\" she told him.\n\n\"What street?\"\n\nShe hesitated for a fraction of a second. \"Highland.\"\n\nThere were two Highlands in Cambridge. One down near his mother's place, and one west of the Square with spreading trees and Victorian houses layered like wedding cakes. He looked at her and knew which one she meant. I was right the first time, he thought. You're not from here.\n\nEven so, he told her about the play, and he made it sound amazing. Deep instead of flippant, poignant where it had been absurd.\n\nNina said, \"I wish I'd seen it.\"\n\nYou have a boyfriend, Collin thought.\n\nBut Nina said, \"I'm always behind preparing class.\"\n\nHe led the way between snowbanks as they walked up Brattle. Without asking, he took her bags, two canvas totes filled with composition notebooks. \"Where do you teach?\"\n\n\"Emerson High School.\"\n\n\"No way.\" He turned around so fast she almost stepped on him. \"I went there!\"\n\n\"Really?\"\n\n\"Yeah. I know Mr. DeLaurentis.\"\n\nShe could picture Collin in the principal's office. Prankster, troublemaker, effortlessly popular.\n\n\"We had these same notebooks.\" Collin glanced down at her tote bags.\n\n\"You kept Discovery Journals too?\"\n\n\"Mostly I drew pictures of...\"\n\n\"What?\" she demanded playfully.\n\n\"What do you think?\" Collin said.\n\nThey went to Caf\u00e9 Algiers, with its polished samovars and copper bar, its Egyptian almond cookies behind glass. The red walls were hung with ceramic tiles and antique maps, a framed illustration of \"the oriental coffee shop.\" They sat at a wobbly table and Collin ordered hummus, pita triangles, bulgur salad, and Lebanese wine. They were the last customers in the door.\n\n\"Is this table okay?\" He sounded nervous. Actually he sounded like a waiter as he asked, \"Do you want to switch?\"\n\n\"I think they all wobble,\" Nina said. Her knee was almost touching his.\n\nWhen the food came she didn't eat anything. \"I already had dinner,\" she explained.\n\nOf course. He had forgotten that. She'd been nibbling at Grendel's all evening.\n\n\"Go ahead,\" she said.\n\n\"Okay, I'm starving,\" he confessed, but he tried not to eat too fast. He tried not to finish all his wine at once. She was just sipping hers. He could see she didn't really drink.\n\nHe asked Nina, \"Why did you decide to teach?\"\n\nShe parried, \"Why did you decide to draw?\"\n\n\"For the money.\"\n\n\"Oh, okay. That's why I teach too.\"\n\n\"No, really.\"\n\nStill, she didn't answer. She turned the conversation back to his own art. \"I've never seen anybody draw so fast.\"\n\n\"That's nothing,\" he said. In the summer he could chalk a whole Van Gogh painting in an afternoon. Sunflowers outside of Faneuil Hall. Irises. He laid down one wet color after another, deep purple, violet, scarlet, gold. Oh, he thought, you don't know what I can do.\n\n\"Are you in art school?\"\n\n\"Well...\" Collin hedged. \"I've been in art school.\"\n\nYou dropped out, she thought, surprised, confused.\n\nHe said, \"I think, in general, school is overrated. No offense!\"\n\n\"No problem.\"\n\n\"But you're an educator.\"\n\n\"Not a good one,\" Nina said.\n\n\"You look like the sort of person who's good at anything they try.\"\n\n\"Kids hate me.\"\n\nHis eyes were sparkling, full of fun. \"Why?\"\n\n\"My last name is Lazare, so they call me Laser Lips.\"\n\n\"Let me see.\" He got up and leaned over the table. Her smile was tender, rueful. How would you draw a mouth like that? Ever so slightly, she drew back.\n\nHe sat down. \"You're strict.\"\n\n\"Well, I want them to learn.\"\n\n\"Yeah, but I mean, do you want them to learn a _lot_?\"\n\n\"Of course.\"\n\n\"Then obviously they hate you. But that's not bad,\" he hastened to add. \"I mean, sometimes it's good to be feared.\"\n\n\"Nobody fears me. There's always some boy goofing off or fighting or falling out of his chair...\"\n\n\"You're making me feel guilty,\" Collin said. \"Is Mrs. West still there?\"\n\n\"You had Mrs. West?\" She lit up. \"Tell me what you did to Mrs. West.\"\n\n\"I loved her. Mrs. West was great.\"\n\n\"Oh.\"\n\nShe looked so disappointed, he wanted to wrap his arms around her. \"She's been teaching for like a hundred years. She has all that experience.\"\n\n\"I hate hearing how you need experience.\"\n\n\"Me too.\"\n\nHer left hand rested on the table, waiting for him. What else could her hand be doing there? He covered it with his. \"Real teachers are like fifty years old, and they have a wrinkle right between their eyes. Have you seen Miss Dorfman? Studio art?\"\n\nDistracted, she spoke slowly. \"Well, I haven't gone around looking at her eyes.\"\n\n\"You should.\"\n\n\"I'm trying to figure out...\" She wanted his hand back, as soon as he released hers.\n\n\"Kids can tell when teachers are trying,\" Collin said. \"It's like how dogs smell fear.\"\n\n\"I have five weeks of training. I'm totally unqualified.\"\n\nHe spoke with confidence. \"Real teachers don't worry about that.\"\n\n\"Yeah. Because they know what they're doing.\"\n\n\"No, because they leave all the actual learning to the kids. Like\u2014here's the French Revolution. Just putting it out there. Here's physics. What you do with it is up to you.\"\n\n\"No, I don't think that would work.\"\n\n\"What are you actually trying to do?\"\n\n\"I want to teach my students to read\u2014not just superficially, but from the inside.\"\n\n\"Kiss of death!\"\n\n\"I want them to make poetry their own.\"\n\n\"Never ever tell them that!\"\n\n\"I _know._ \"\n\n\"You told them.\"\n\n\"My students think I'm...\"\n\nGorgeous, Collin thought.\n\n\"Hopeless.\"\n\n\"They barely know you,\" Collin said. \"They're still watching you.\"\n\nThe restaurant was closing. It didn't feel late, but the whole city was shutting down. There wasn't anywhere to go.\n\nHe said he'd walk her home. She said no need, she lived really close. In the end he walked her halfway in the snow-lit night.\n\n\"You need a hood,\" Nina told him, as she pulled up hers.\n\n\"I like the snow.\" He ran his fingers through his curly hair until it looked wilder than ever. Then he shook his head like a dog so that the snow flew off.\n\n\"You're silly.\"\n\nHe nodded.\n\nWith a hint of regret, she confessed, \"I'm not.\"\n\nHe took her arm. \"I can help you with that.\"\n\nFresh snow buried Collin's small basement window. His bedroom was dark to begin with, and he'd painted the walls with blackboard paint. Now his room looked like an ice cave. In the stillness and the half-light he lay in bed remembering the night before. He wanted to hold on to every second as long as possible. And then, suddenly, he sprang out of bed and threw on his clothes.\n\nHe stepped into a living room full of salvaged furniture\u2014scarred old tables, mushy upholstered chairs. It was just seven. Darius and Emma were fast asleep, but Collin pulled on coat and boots, swiped a piece of leftover pizza from the fridge, and ran outside. Plunging knee-deep into the snow, he tripped on one of his landlady's scrap-metal sculptures, a sharp-eared cat. He swore and laughed, dusting himself off.\n\nThe house was narrow and Victorian, ocher-trimmed with cadmium red. Collin and Darius had painted it over the summer in exchange for rent, while the owner, Dawn, was visiting her daughter in Northampton. Dawn wasn't just a landlady, of course. She was one of the women Collin had always known, drinking coffee in his mother's kitchen, talking about the universe.\n\nPerfect powder, and still falling. Central Square was ghostly white. All but the coffee shops were closed and dark. Teddy Shoes, where you could buy patent-leather pumps with a kitten heel in men's thirteen. Classic Graphx, where you could print one hundred programs cheap for your stealth underground production of _A Streetcar Named Desire._ Collin passed the Seven Stars bookstore, then 1369 Coffee House, its steamy windows already filled with poets, students, therapists. On second thought, he turned back to 1369 and bought chocolate croissants and a gingerbread latte. Then, holding the warm paper cup, he tramped to Antrim Street to find a sled and ask his mother for the car.\n\n\u2014\n\n\"You'll have to shovel out Sage and Lois too,\" Maia told him, after he gave her the latte. During citywide snow emergencies, the women parked in the one driveway, herding together Maia's ancient Volvo wagon, Lois's Toyota bristling with bike racks, Sage's Subaru. Collin could see three humps of snow through the kitchen window.\n\n\"Just give me the keys so I can move them.\"\n\nMaia shook out the calabash by the front door and tossed each set of keys. He caught them in the air.\n\n\"I like you this way.\"\n\n\"Which way?\"\n\n\"Motivated.\"\n\nCollin flashed her a smile and headed to the cellar door.\n\n\"Hold the railing,\" she called out as he started down the creaky stairs.\n\nBy the light of a naked bulb, he could make out his mother's gardening tools, her leaf bags, damp gloves, and metal rakes, her extra cans of paint and varnish. He squeezed between the rolling blackboards she kept for him. Containers of broken chalk; string bags of sand toys; Cheyenne, the plastic rocking horse galloping on rusty springs and tossing its plastic mane. Lois kept her mountain bikes down here, and Sage had a dresser she was planning to refinish. He had to search behind a collapsed wading pool, a stack of cartons labeled CHRISTMAS, and a pair of ratty wicker chairs before he found the sleds. Two blue singles and a black plastic double. He dusted off the double and texted Nina, Found it.\n\n\u2014\n\nShe looked chilly waiting in the Square. He opened the car door for her and apologized for being late. \"I had to shovel out three cars to get here.\"\n\n\"The heat feels good,\" she said, basking in the car's hot air. She was wearing a white coat, dark velvety jeans, and black suede boots.\n\n\"Are you going to be warm enough?\" He kept looking at her as he drove, because he had never seen her in daylight. Her eyes were not dark as he'd thought, but clear gray, like water over river stones.\n\nThe streets were barely plowed, so they had to inch to Danehy Park. \"I'm sorry this is taking so long,\" he said, although he wasn't really. There she was, close enough to kiss\u2014but he would wait. He could be patient sometimes.\n\nAs he eased into the parking lot, she saw kids tramping up the big hill. \"I wonder if I'll see my students here.\"\n\n\"We'll mow 'em down.\" Collin took her gloved hand and led the way, breaking a path. Her kidskin gloves felt like nothing inside his, which were puffy, cheap, but insulated. Her clothes were way too delicate for real weather.\n\nA stand of dwarf pines covered the hill. Little trees bent down with drifts. Collin held the sled while Nina climbed on, bracing her feet in front of her. She looked out at red coats and purple mittens, a black Lab barking, a little girl crying that her sister never let her steer. All along the snowy slope, she could see children and their parents, every bright detail. \"My father took me here once, and we went down together.\"\n\nCollin sat behind her and wrapped his arms around her waist. \"Like this?\"\n\nHer heart jumped, but she had no time to answer. He pushed off with his feet and the sled flew.\n\nThe hill was faster than either of them remembered. Nina nearly slipped out of his arms, and he had to hold her tight against his chest.\n\nThey skimmed the slope, accelerating into a swoop of snow. Breathless at the bottom, they didn't speak. Her legs felt wobbly, and Collin helped her up, offering his hand.\n\nAgain and again they sped down, and trudged back up the hill, which seemed to grow taller as they climbed. She stepped into his footsteps as he led the way, pulling the sled. At the top he looked at her and saw her glowing cheeks, her trailing scarf, her thin gloves soaked through. \"Take mine,\" he said. \"They're better.\"\n\nHis bare hands stung with cold, but he didn't care.\n\nSledding down, wind scoured her nose. Her hood fell back; her hair whipped into his face. He closed his eyes, concentrating on her and not the hill. When they hit a bump they both went flying.\n\nThey lay together where they fell, and it was sweet and it was surprising, like waking up together in bed. She turned toward him, drawn by his dark eyes, his laughter, his question, \"Are you all right?\"\n\n\"I'm fine.\"\n\n\"You sure?\"\n\nHe brushed her hair from her face with his raw hands. \"No broken bones?\"\n\n\"No.\"\n\n\"No sprains?\"\n\nShe shook her head, even as sleds swerved all around them.\n\n\"Not even a scratch?\" His lips brushed hers so lightly that she wasn't sure he'd kissed her. Before she could decide, he kissed her for real.\n\nTrembling, she sat up. She wanted more, but he helped her to her feet instead.\n\nSlowly they began walking to his car. She was shivering, but she hardly noticed. He was hungry, but he forgot the croissants flattened in his coat pocket. Already it was afternoon, the sun no longer bright but softer, pale gold. Were the days so short now? Or had they been out so long?\n\nCollin drove with his left hand and took Nina's in his right. He said, \"I used to wait and wait for you.\"\n\nShe said, \"I know.\"\n\nThey drove to the Brattle Theatre, the classic-movie house, and bought tickets for _The Adventures of Robin Hood_ with Errol Flynn, the early show. They had the place to themselves, and they spread their wet coats over empty seats. The Brattle's clock glowed lavender, and they leaned back and ate all their buttered popcorn while they waited for the movie to begin.\n\n\"It's warm in here,\" Nina said drowsily.\n\n\"It's quiet.\"\n\n\"I'm starting to feel my toes again.\"\n\n\"Uh-oh.\" He reached down and slipped off her boots, feeling for her feet. \"You're soaked!\" He began pulling off her wet socks.\n\nShe was startled, and then she let him. \"I didn't know the snow would be so deep.\"\n\nThe cartoon was starting, a brief Bugs Bunny called _Rabbit Hood._\n\n\"Didn't your mom teach you to wear good boots for sledding?\" He rubbed her bare feet with his hands.\n\n\"I lived with my father.\" She kept her eyes on the screen.\n\n\"Divorce?\"\n\n\"No, they never married.\"\n\n\"So your dad...\"\n\n\"He doesn't really think about boots.\" Onscreen, with heroic music playing, Bugs Bunny stole a carrot from the King's Carrot Patch, despite warnings posted on stone walls: NO POACHING. NOT EVEN AN EGG.\n\n\"What does he do?\"\n\n\"What's up, Doc?\" Bugs Bunny asked the King's skinny sheriff.\n\n\"What's up, _forsooth_!\" the sheriff retorted, longbow poised to shoot, arrow pointed at Bugs Bunny's heart.\n\n\"Whatever he wants.\"\n\n\"Cool. Where does he work?\"\n\nShe hesitated, even as he rubbed her instep with his thumb.\n\n\"Around here?\" Collin asked.\n\n\"Arkadia.\"\n\n\"Seriously?\" As kids, Collin and Darius had haunted Arkadia's message boards for any hint of the next EverWhen expansion. Once they took the commuter rail out to Waltham and walked three miles from the station just to stand in the company parking lot. The two of them streaked with dirt and sweat, two pilgrims at the shrine. \"What does he work on?\"\n\nNow Nina tucked her bare feet under her. \"You're a gamer.\"\n\n\"I used to play EverWhen.\"\n\n\"What level were you?\"\n\n\"Well...\"\n\n\"Come on.\"\n\n\"Sixty.\"\n\n\"Sixty!\" He could hedge, but Nina had played that labyrinthine game. She had wandered the Trackless Wood of EverWhen and qwested in the caves of EverSea. She knew what level sixty meant. He'd claimed five kingdoms, and hunted all twelve dragons to their dens. He had raced from green to golden fields, and lived a thousand years in Arkadian worlds, before he'd met her. He was just getting to know Nina, but he'd already journeyed deep into her father's realm.\n\nOther parents practiced law or medicine; they traded stock, or ran for office, built houses or sold real estate. Nina's father ran a company that produced MMORPGs (massively multiplayer online role-playing games). Millions lived and dreamed inside his virtual worlds. His work was nowhere and everywhere, ephemeral and everlasting.\n\n\"I used to play all the time,\" Collin admitted. \"I can close my eyes and see the Keep.\"\n\n\"His Most Royal Majesty cometh! Sound the welcomes and blow the crumpets!\" cried Bugs Bunny.\n\nCollin didn't even glance at the cartoon. \"Is your dad a developer? Or is he like...I mean is he specialized, or more...\" He interrupted himself. \"Is he working on UnderWorld?\"\n\n\"He works on everything.\"\n\nIn flickering light, Nina watched Collin put the facts together. Her students called her Laser Lips. Her last name was Lazare. Her father was at Arkadia. He worked on everything, including the long-awaited UnderWorld, unreleased, but trending everywhere with its hash tag, #seeyouinhell.\n\n\"Is your father...?\"\n\n\"Viktor Lazare,\" she said.\n\n\"Nina!\" Playfully, reproachfully, Collin said, \"Your father doesn't work at Arkadia.\"\n\n\"No,\" she conceded.\n\nSoftly Collin said, \"He owns it.\"\n\n\"H _aving once this juice,\"_ Nina read to her sophomores, _\"I'll watch Titania when she is asleep, \/ And drop the liquor of it in her eyes.\"_ She looked up from her book and studied her best-behaved class. \"What's happening here?\"\n\n\"Oberon's going to drug Titania,\" Noemi said.\n\n\"With what kind of juice?\"\n\n\"Liquor.\"\n\n\"Right. Liquor\u2014which is the nectar of a magic flower. Sean, could you read the rest?\"\n\n_\"The next thing then she waking looks upon, \/ Be it on lion, bear, or wolf, or bull, \/ On meddling monkey, or on busy ape, \/ She shall pursue it with the soul of love.\"_\n\n\"So what's going to happen?\"\n\n\"You're asking me?\" Sean looked up with an easy smile.\n\n\"Yes.\"\n\n\"She's going to fall in love.\"\n\nFor a second the answer startled her. Then she said, \"Who or what will Titania love? Sasha?\"\n\n\"The first thing she sees.\"\n\n\"Yes!\" Silently, for just a moment, Nina celebrated. Her kids got it! Then she thought\u2014three kids got it. She looked at the other thirteen arrayed before her, some whispering, some passing notes. Isaiah sat in back, staring out the window. Anton was drawing Titania\u2014or was it Sasha?\u2014topless. He was smart and tough, green-eyed, Russian. Athletic, but he'd been kicked off the basketball team.\n\n\"Anton, why is Oberon going to play this trick on Titania?\"\n\nAnton kept drawing.\n\n\"Put the notebook away,\" Nina said.\n\nHe did not put the notebook away.\n\nShe repeated, \"Please put the notebook away so you can concentrate.\"\n\nAt this moment, Jeff, her TeacherCorps mentor, slipped into the room. He found an empty chair and took out his computer. Nina tensed as she stood her ground, confronting Anton.\n\nThe class rustled, as trees rustle when the breeze picks up. Jeff watched her, and the kids watched Jeff watching her. For the third time she repeated, \"Put the notebook away and tell me about these fairies\u2014Oberon and Titania.\"\n\nSlowly, Anton closed his notebook, and she could breathe again.\n\nMeanwhile, Jeff was typing rapidly. He had cropped blond hair, a runner's build, eyes Eagle Scout blue. He thought data was the answer. That was why he recorded every bent head in Nina's class.\n\n\u2014\n\n\"I noticed,\" he would tell Nina later, \"that two kids had their heads down on their desks when I came in, and three were missing books. Five minutes after that, we've got three kids with heads down on their desks. Everyone is sitting, which is great, but between the missing books and body language, you had six disengaged when you began your line of questioning.\" She couldn't stand him. He distracted the class with his tapping on the keyboard, and then he aggravated her afterward. He always said, \"Look, it's your call,\" before he told her what to do.\n\n\"What are the fairies like?\" she asked her class now. \"Are they kind and good-natured?\" No one answered. \"Are they mischievous? Chandra?\" She appealed to her smallest, quietest student.\n\n\"Mischievous,\" Chandra answered in a tiny voice.\n\n\"Why is that?\"\n\n\"Because they're powerful.\"\n\n\"Yes!\"\n\nHer response amused the class. Miss Lazare was so intense. She jumped if you guessed what she was thinking\u2014like she was playing Bingo in her mind.\n\nNina walked to the board. \"Does power make you mischievous? Is that true? Who thinks it's true?\" She found a broken piece of chalk. \"Isaiah, where does power come from in this play?\"\n\n\"Magic,\" Isaiah said.\n\n\"Good. Where else? Let's list some places power comes from.\"\n\n\"In the play,\" Isaiah asked, \"or in the real world?\"\n\n\"Both!\"\n\n\"Money,\" said Brittani.\n\nNina wrote that on the board.\n\n\"Etienne?\"\n\n\"Wealth.\"\n\n\"Okay.\" Nina wrote wealth next to money.\n\n\"Politics,\" said Sasha, who had been nibbling a cookie when she thought Nina wasn't looking.\n\n\"Jonee?\" Nina tried to call on Jonee every once in a while, but usually Jonee shook her head. Jonee had written permission to keep quiet in class because of a psychosomatic condition that caused her to faint when agitated. Nina found this confusing. She had assumed psychosomatic conditions weren't real. Apparently they were real enough. \"Where else does power come from?\" she asked.\n\n\"Technology,\" said Chandra.\n\n\"Interesting!\" Look at us, she thought, as she scribbled \"technology\" on the board. Hey, Jeff, we're springboarding from Shakespeare to technology. \"Where else does power come from?\"\n\nNo one answered, but she tried to coax the discussion, cup her hands and blow upon the spark. \"What about knowledge? Knowledge is power. Francis Bacon said that. What about language? And art? What about poetry?\"\n\nAnton was drawing again. Girls were whispering in the back.\n\n_\"I know a bank where the wild thyme blows,\"_ Nina said.\n\nSeveral kids looked up, startled by her sudden shift to Shakespeare's language.\n\n_\"Where oxlips and the nodding violet grows...\"_\n\nThe girls stopped talking. The three kids with heads down began to stir themselves.\n\n_\"Quite over-canopied with luscious woodbine, \/ With sweet musk-roses, and with eglantine: \/ There sleeps Titania some time of the night, \/ Lull'd in these flowers with dances and delight.\"_\n\nThey were all staring. Even Anton stared. Jeff himself stopped typing and gazed at Nina with a mixture of admiration and alarm. She was holding her book open in her hands, but she wasn't looking at it.\n\nStrange and still, the mood inside her room. She felt it happening\u2014she had the kids' attention. If only she could make it last. She wished she could enchant her students with the liquor of a magic flower. Cast a spell so that they would fall in love with poetry.\n\n\"How do you know all that by heart?\" Isaiah asked.\n\n\"Well...\" she began.\n\nWhat would she say if she were honest? I memorized almost the whole play in my father's library. _Your father has a library?_ her kids would say. I fell in love with Shakespeare when I was eleven. _When you were eleven? Whoa!_ I was so lonely. No, she wouldn't tell them that.\n\n\"They barely know you,\" Collin had said, and she felt a rush of pleasure, remembering his dark eyes. \"They're still watching you.\"\n\n\"I read the words over and over and over,\" she told the class. \"I kept repeating lines, just the way you learn a song.\"\n\n\"Yeah, but songs have music,\" Sasha pointed out, \"so that's much easier.\"\n\nNina said, \"Poetry has music too.\"\n\nShe lost some of them then. Once again, the girls started whispering, and Nina thought, as she did so many times a day, How do I lure them back? She opened her book. \"Everybody start from _And there the snake._ Read aloud together.\"\n\n_\"And there the snake throws her enamell'd skin,\"_ the kids read together shyly, some mumbling, some faking it. _\"Weed wide enough to wrap a fairy in.\"_\n\n\"Again.\"\n\n_\"And there the snake throws her enamell'd skin,\"_ they read in chorus. _\"Weed wide enough to wrap a fairy in.\"_\n\n\"Again.\"\n\nThey glanced up, wondering if she was serious. _\"And there the snake...\"_\n\nShe looked at her students murmuring together and imagined the conversation she couldn't have with them.\n\nI grew up in an enormous house. _What do you mean \"enormous\"?_ her kids would ask. Her father's house had carved woodwork and marble floors, a polished staircase, rising and turning like the vortex of a storm. There was a glass conservatory and a library with a gallery. _Awesome!_ her kids would shout. _Field trip! Par-tay!_\n\nShe hadn't always lived there. As a tiny girl, she'd lived with her grandparents on Evans Road in Brookline. Her grandmother had been pale, with powder in her creases. She set out crystal bowls filled with butterscotch candies. Her grandfather had deep, dark eyes, a raspy voice, a fearsome smile. When he walked, he dragged his feet, because they were so heavy for him. Her grandparents talked to her in Russian and read her Russian books. They beamed at Nina, spending all their warmth on her. Left to themselves, they sat for hours without speaking. They were a pair of armored lizards; they were stone.\n\nSlowly, Nina's grandfather climbed the stairs, and slowly he descended. The stairs were carpeted dark green like moss, the walls papered with lilacs, the soaps in the bathroom carved like cabbage roses. Everything in her grandparents' house looked like something else. The boot scraper took the shape of a hedgehog, the throw pillows were embroidered cats. Even Nina's grandmother began to look like something else, the Blue Fairy in Nina's book. She wore blue quilted slippers and a sky-blue robe. Her eyes grew paler, purer blue. When she left for the hospital, Viktor took Nina to live with him across the river. She was four years old.\n\n\"Can I go home?\" she would ask her father.\n\n\"Later,\" Viktor said, but she had sensed, even then, that he was powerful and he was mischievous. She saw her grandfather only a few times after that. She never saw her grandmother again.\n\nIn her father's house there were no candy dishes. Nina discovered a stone bird, instead, and a gold man sitting in a flower. A giant cartoon hung in the breakfast room. A naked lady lived in the library without arms or nose.\n\nThere were other toys, and other women. In the library, her father kept two globes, a green and blue globe of the Earth, and a black globe of the sky. In the hall he kept a grandfather clock. Nina sat on the stairs and studied it, imagining her grandfather rising like the sun painted on the round clock face. She saw her grandmother in robe and slippers, creeping across the sky over days and weeks, growing and then disappearing, like the silver moon.\n\nNow she walked between her students' desks and her class was chanting all around her. _\"And there the snake throws her enamell'd skin \/ Weed wide enough to wrap a fairy in.\"_\n\n\"Stand up,\" Nina said. The kids shuffled to their feet. Jeff stood too. \"Read it one more time _loudly._ \"\n\nThe kids were almost shouting, _\"And there the snake throws her enamell'd skin...\"_\n\n\"Read it one last time, softly.\"\n\n_\"And there the snake...\"_ her students whispered, rhythmically.\n\n\"Good. Now close your books. Everybody close your books.\"\n\nThe kids closed their books like hymnals. \"No, don't sit down,\" Nina said. \"Say the lines with me.\"\n\nTogether they recited from memory, _\"And there the snake throws her enamell'd skin \/ Weed wide enough to wrap a fairy in.\"_\n\n\"That's it,\" she said, even as the class applauded. A couple of kids bowed deeply. They were only joking, but even Jeff was smiling. Ha! Nina thought. She'd engaged every student for at least thirty seconds. Log that!\n\nSaturday night, while Kerry was at work, Diana lay in bed listening to her brother and his friends playing EverWhen. Their deep voices rumbled through the wall as they compared weapons, traded for gold arrows, debated their next move. Tunnel underground? Or fly?\n\n\"Hey, Aidan.\" Diana banged the wall above her head.\n\nThen her brother told Jack and Liam to keep it down, and the noise died away, at least for a little while.\n\nShe had no idea how long she had been sleeping when she woke to voices in the hall. \"It's still early,\" Liam was protesting.\n\nJack said, \"It's the weekend!\"\n\n\"Gotta sleep,\" Aidan insisted.\n\nHis friends clattered down the stairs and barreled through the living room to find their boots. The front door slammed. The storm door snapped shut like a trap. Silence, and then a tapping from Aidan's room. Diana sat up and listened. What was he doing? Texting. Playing with someone else, now that his friends were gone. You aren't sleepy at all, she thought. You lied to them.\n\nHe would rather be alone. He would rather play online with strangers than qwest with friends in the same room. He'd rather live in EverWhen than in this house.\n\nShe knew her brother. He might lie to other people, but he couldn't fool her. He was just two minutes older. When they were little they'd played every game together. Hide and seek, tag, and chase. They'd joined forces against big kids who wouldn't let them onto the tire swing at Sennott Park. They had shared a bunk bed at their father's place when he was still allowed to see them. If their dad tried to punish Aidan they would drag that metal bed to barricade the door. They had always protected each other. The year before, when Kerry banned Liam from the house, Diana had lied for Aidan, vouching that his friend no longer came over. In middle school, when someone called Diana fat, Aidan would chase him.\n\nIt was true, although her mother never used that word, even to describe other people. Kerry said \"plump,\" which sounded like pillows, or \"heavy,\" which sounded like uranium down at the bottom of the periodic table. Her mother always said, Stand up straight, you're a beautiful girl. She said all this, but even Diana's arms were fat. She hid behind her long black hair and her mother said, Don't do that! Why do you do that? Kerry had read _Reviving Ophelia_ and she was afraid Diana would end up starving or cutting herself, or dying, like the original Ophelia, who drowned herself with wildflowers in her hair.\n\nWhy do you wear black clothes all the time? her mother asked in her sad, pleading voice. Because I'm in mourning, Diana said. Because I'm Wiccan. I'm practicing black magic. She never admitted the real reason, which was simply that she was trying to disappear. After all, people called her a whale\u2014not to her face, but still. She felt guilty, because she was almost as big around as Brynna, who was six months pregnant. Diana had no excuse, because she had nobody inside her.\n\nShe burrowed down in bed. Hibernated in a nest of blankets and limp pillows, along with her math homework and her Discovery Journal with Miss Lazare's elaborate question. _Thoreau writes: \"I went to the woods because I wished to live deliberately, to front only the essential facts of life, and see if I could not learn what it had to teach, and not, when I came to die, discover that I had not lived.\" What does he mean by this? Why do you go into the woods?_\n\nTo this, Diana replied, _We don't go into the woods very much because my mom is afraid of ticks. Also we do not live near the woods, obviously._\n\nBut that night she dreamed of trees. It was spring, and she was running through green leaves. Branches brushed Diana's shoulders as she floated down the street. She was racing, flying to the Charles River, but she didn't stop there. She took the dirt path along the water. She was warm and sweaty as she ran fast and faster. She ran past rowers with long oars, and joyriders in snarling motorboats. Running east, she overtook the sun. She ran with its heat at her back and flung her clothes onto the grass.\n\nShe woke with a jolt, heart pounding as she fell to earth. Sunday morning. No oars, no river. No piano lessons next door. Priscilla played organ at church services on Sundays. This was as quiet as it ever got. Aidan sleeping. Their mother home from work, rustling _The Boston Globe._\n\nIn plaid flannel pajama bottoms and a giant T-shirt, Diana padded down to the kitchen, where she ate cereal and chocolate milk and two peanut butter cookies. And then a banana, which was healthy.\n\nHer mother said, \"Do you think you should have cookies for breakfast?\"\n\nDiana said, \"It's more like brunch.\"\n\nThe kitchen was chilly because the windows were old and cost a fortune to replace. Kerry had already bought a new boiler and rebuilt the double porch and patched the roof. Even after the repairs, there were squirrels up in the attic. Diana's mother said that couldn't be, but Priscilla heard them with her keen musician's ears. She insisted, \"I hear them trapped inside,\" and Diana pictured rodents crazed with hunger, eating their own children.\n\nShe wished Priscilla would move out. Then they'd have the double house all to themselves. Two living rooms, and two kitchens. Six bedrooms on the second floor! All that space, and no more sonatinas and minuets and little fugues. It was a never-ending guilt trip, sharing a two-family with your old piano teacher.\n\nDiana was not musical, but everyone had been heartbroken when Aidan quit in seventh grade. Priscilla still looked at him wistfully. She would catch him on the porch and say, \"I wish that you'd start up again.\" If Diana was around she'd add, \"You too!\"\n\nAt which point Diana would tell Priscilla, \"I heard the squirrels last night.\"\n\n\"Could you start the laundry?\" Kerry asked now.\n\n\"Could I have some money?\" Diana replied.\n\n\"How much?\"\n\n\"A hundred twenty dollars.\"\n\n\"Why? What do you need that for?\"\n\nDiana went to the front door and picked up her old silver Nikes.\n\n\"Didn't we just buy you shoes?\"\n\n\"Look.\" The uppers were splayed open, the rubber heels warped, the laces frayed and broken.\n\n\"What did you do to them?\" Kerry gazed at Diana's feet, afraid that they were widening with the rest of her. \"Let's go shopping this afternoon.\"\n\n\"Maybe.\" Diana didn't want to hurt her mother's feelings and say she'd already made plans.\n\nWhen her mother went upstairs to sleep, Diana dragged her laundry bag along with her mother's down to the basement. Two heavy sacks thumping down the stairs. She didn't collect Aidan's. He didn't need clothes, since he lived in EverWhen.\n\nShe was afraid of the dank smell and creepy toys. A sophomore at school had been raped in her own basement. A whole gang of boys\u2014guys the girl knew\u2014got in and forced her down. Fast as she could, Diana loaded the washing machine and ran upstairs. Breathing hard, she shut the door behind her.\n\nA heavy step in the living room. She whirled around and saw Brynna. \"How did you get in here?\"\n\n\"I'm doing well,\" said Brynna. \"How are you?\"\n\n\"Seriously, wasn't the door locked?\"\n\n\"No.\" Brynna squeezed herself into a kitchen chair. Pregnancy was good for her skin. Her forehead had cleared up completely. Her green eyes were beautiful to begin with, and she wore her wavy brown hair down over her shoulders, so she looked huge, but gorgeous too.\n\n\"I was having brunch,\" Diana said. \"Care to join me?\"\n\n\"No,\" Brynna said. \"I'm eating right.\"\n\n\"Good for you.\"\n\nBrynna sat on Diana's bed while she got dressed. Diana pulled an all-black sweater over her head, and when she emerged, Brynna was holding her journal.\n\n\"Hey, give that back.\"\n\nBrynna leafed through Diana's black composition notebook. _One single word to describe myself would be conspicuous. People in the halls are always trying to get around me. If I actually look back at them that's considered rude, like how dare you block my view? Last year I..._\n\n\"I said give it back.\" Diana snatched the book from Brynna's hands.\n\n\"Okay!\" Brynna said. \"Sorry! It's not like I was spying on you!\"\n\n\u2014\n\nThey took the bus through the slush to the CambridgeSide Galleria on the river at the edge of town. The Galleria would give anyone a headache, but it had a lot of stores.\n\n\"What are you going to name her?\" Diana asked, as they braved the shiny walkways, all glitter and glass and sleigh bells ringing.\n\n\"I like 'Tasha,' \" Brynna said.\n\n\"Tasha? Is that a name?\"\n\n\"Yes, it's a name.\"\n\n\"Maybe for a cat.\" Diana paused in front of Godiva to look at the truffles in the window. \" _Godiva_ is a good name.\"\n\n\" _What?_ My child is not a candy company!\"\n\n\"Does Anton get a vote?\"\n\nAnton was the baby's father, and Brynna didn't even answer that.\n\nThey fingered dresses at Motherhood Maternity, but they were so ridiculously expensive that Brynna didn't try on a single one. At Sears, they walked past the baby gear in the infant and toddler department. There were cribs and baby swings and play centers and mobiles with themes like rainbow pandas, or tropical islands. Everything was puffy, soft, and new. Brynna was curious, and at the same time afraid to look.\n\n\"Do you want to get something?\" Diana asked.\n\n\"No!\"\n\n\"Why not?\"\n\n\"Because!\"\n\n\"Okay, let's go home.\"\n\n\"Shoes,\" Brynna reminded her.\n\n\"Later.\"\n\n\"Just get it over with.\" Brynna steered her toward Lady Foot Locker.\n\n\"Don't make me go in there.\"\n\n\"Come on. You said you wanted shoes.\" It was ironic that Brynna was the pregnant one, because she was so responsible. Maybe it wasn't ironic. Brynna was already such a mom.\n\nBrynna scanned the walls of shoes arrayed for walking, running, cross-training. Diana sat on the polished blond-wood bench. \"What about these?\" Brynna held up a pair of Sauconys.\n\n\"They have green on them,\" Diana said.\n\n\"One little stripe!\"\n\n\"I don't wear green.\"\n\n\"How about these?\" Brynna held up a pair of silver Nikes.\n\nDiana shook her head.\n\n\"These are the same shoes you have on,\" Brynna said. \"Look. They're exactly the same, except they're not falling apart.\"\n\n\"No, I don't think so.\"\n\n\"You're not even going to buy an exact replica of the shoes you have?\"\n\nDiana looked down at her feet. \"I hate it here. Let's go.\"\n\nShe was almost out the door when a sales associate in a referee uniform flagged her down, asking to help. He was dark black, African, and the name on his badge was Joseph. He had an accent and a nervous look. It was probably his second day. He wanted to know what kind of shoe Diana was looking for and what sport she played.\n\n\"She doesn't play sports,\" Brynna said.\n\nJoseph didn't give up. \"A shoe for exercise?\"\n\n\"She doesn't\u2014\"\n\n\"I do exercise,\" Diana said.\n\nIncredulous, Brynna asked, \"Since when?\"\n\nThe question upset Diana. \"I _could_ exercise. I might.\"\n\nBrynna snorted.\n\n\"Cross-trainers?\" Joseph suggested.\n\n\"Something black. Something like this.\" She pointed to a black shoe on the wall.\n\n\"This one is for running,\" Joseph said.\n\n\u2014\n\nThat night in her room, Diana opened the cardboard box from Lady Foot Locker and took out a pair of pure black running shoes. Black uppers, black heels, black soles, black laces.\n\nOne shoe in each hand, she tapped the wall. She loved the new-shoe smell, clean leather and fresh rubber.\n\n\"Stop,\" Aidan said, after a few seconds.\n\nShe kept drumming her new shoes at the spot she hoped was just above his computer monitor.\n\n\"Stop or I'll kill you.\"\n\n\"He speaks,\" Diana said. \"Hey, Aidan. I got new shoes.\"\n\nNo answer.\n\nWhen she laced the shoes and walked around her room, she bounced. The floor felt like those giant inflatable birthday party castles. She jumped, and jumped again. Her dresser rattled when she landed. She could hear Aidan shuffling around, her fellow cell mate, self-incarcerated in his room. \"I'm going to do something,\" she told him. \"Do you dare me?\"\n\nNo answer.\n\n\"You dare me,\" she answered for him, but that sounded pathetic. \"Okay, I dare myself.\" That sounded even worse, like she was trying to be inspirational. Even when she was little she had hated anything inspirational, like books where kids saved the day or movies involving wildlife and parents getting back together and slow-motion horse races at the end. She sat on her bed and looked down at her feet in the new shoes. Who are you kidding? she thought. She embarrassed herself, even when she was alone.\n\nLong ago, when she was six and seven, she had swung bar by bar across the climbing structure. As a small girl, she'd gone to GAB, Gymnastics Academy of Boston, which was the only day camp open in the very last weeks of August. At the gym near Fresh Pond, she had practiced pikes and flips on the giant trampoline, hurled herself with all her force up and over the vault. She had been a little gymnast and Aidan had been a swimmer, and they had been a matched pair, wiry and strong. Then at about twelve, he grew tall, and she grew round. The weight came on in cookies and gumdrops, and late-night snacks. Aidan ate too, but it didn't show on him. He was over six feet and growing; she was done at five feet four. He paced the house, while Diana hunkered down. He refueled standing at the kitchen counter, while she curled up with goldfish crackers on the couch. He started killing monsters, and she built up her defenses, practiced her self-doubt.\n\nShe was not a small girl anymore, nor was she fast, nor was she flexible. She could barely remember hurtling over anything. For a while now, her tiny pediatrician had been talking about exercise and healthy nutrition. Diana was thirty pounds overweight.\n\nSometimes she felt doomed. Other times she felt as though she were carrying somebody's lost luggage. When would the real owner come to claim it? She felt disgust, resignation, surprise, but no sense of recognition. She avoided herself. Stayed away from scales, mirrors, bright lights, shorts, and bathing suits.\n\nKerry could talk all she wanted about standing up straight and being beautiful; she could say it a thousand times. Words could not change anything. \"You're a beautiful girl\" was like saying God is good. You didn't say these things because they were true, you said them because you hoped the universe would take pity on you.\n\nDiana pounded once more on the wall.\n\n\"What?\" Aidan shouted.\n\nThis time she didn't answer.\n\n\"What?\" he called again.\n\nWhen she spoke, she wasn't even talking to him anymore. Thumping down the stairs, she berated herself. \"Go. Go. Go. I'm tired of waiting for you.\"\n\nNear the front door at the bottom of the stairs, she found her broken Nikes, and picked them up by the laces. Outside she flung them in the garbage can and shut the lid.\n\n\"Where are you going?\" her mother called out from the couch.\n\n\"Nowhere,\" Diana said.\n\nThe night was mild, the snowbanks melting, no longer white, but newsprint gray, the sidewalks cracked, but clear of ice. Diana walked down Antrim Street to Broadway, and she took deep breaths, swinging her arms, speed-walking like the old ladies in the mall. They were probably in better shape than she was. Diana was already hungry after five minutes in the fresh air. At the corner of Prospect Street, she nearly stepped into Tedeschi's market for a bag of chips. The only thing stopping her was Aidan's friend Jack, walking out with two gallons of milk.\n\nHe wore glasses, but he had a way of squinting to look at you. He had been the small one. Tiny! Now he was all legs and bony shoulders, incredibly long arms. \"Diana.\" He couldn't wave, because he had a gallon weighing down each arm. \"What are you doing here?\"\n\n\"Nothing.\" She hated how he examined her.\n\nSince they had known each other since preschool, he thought he had a right to trail after her. He followed her to the traffic light. \"We ran out of milk.\"\n\n\"Yeah. I see that.\" He was heading home to Norfolk Street, where he lived in the Chocolate Factory apartments, so called because yes, the building had been a chocolate factory. She said, \"Okay, I cross here.\"\n\nHe knew she lived in the opposite direction. \"Why are you...?\"\n\nShe didn't wait for the light. She dodged cars, crossing Broadway to Sennott Park. Then she checked that he was gone. She didn't want anyone to see her, so she hid behind a tree to touch her toes.\n\nShe didn't know the real stretches, the kind they did on teams at school. She didn't know the right way or the fast way, so she just started walking the perimeter of the field. There was no moon; she saw no stars. She wasn't quick enough to pass anything moving, only houses and little stores, and the great silent trees. Her legs were heavy under her, and her sides ached. Breathing hard, she began to run.\n\nNow Nina came to Grendel's just to see him. She stacked her students' Discovery Journals on the table and looked up at Collin as he brushed past. He was always hovering near her, or scribbling little notes. His friends thought the situation was hilarious. Not just that Nina was a teacher, but that she'd turned him into such a courtier.\n\nAt closing time, Collin and Kayte cleaned up, while Nina waited at the bar.\n\n\"You're anomalous,\" Samantha told her.\n\n\"Don't listen to her,\" Collin said as he wiped tables.\n\n\"You aren't a teetotaler, are you?\" Sam asked Nina.\n\n\"No.\"\n\n\"Because you never even order beer.\"\n\n\"I drink other things.\"\n\n\"Like what?\"\n\nNina hesitated.\n\n\"Collin!\" Sam cried out as if to say, I can't believe her.\n\nHe threw his wet rag and Sam ducked behind the bar.\n\n\"Drink me! Drink me!\" Sam poured Nina a drop of cr\u00e8me de menthe. Nina shivered, tasting Sam's strange medicine. She really did look like Alice falling down the rabbit hole. Curiouser and curiouser. Collin had to kiss her.\n\n\u2014\n\nShe went with Collin to Charlie's Kitchen, the almost-all-night diner in the Square. They sat together in a red vinyl booth, and he told her of his days performing plays about nutrition at the Children's Museum, where Darius had worn a full tomato suit.\n\n\"Full tomato? Is that like full metal jacket? What were you?\"\n\n\"A loaf of bread.\"\n\nShe laughed.\n\n\"What?\"\n\n\"I'm trying to picture that,\" Nina said.\n\nThey discovered that they were both turning twenty-four in January. Their birthdays were just a week apart, and they had been born in the same hospital, Beth Israel Deaconess. Strange that they had never met.\n\nShe had attended Cambridge-Ellis as a toddler, while he'd spent his days at Aisha's Family Daycare. No chance of meeting there. Ice cream? She had walked to Lizzy's. He'd worked at Christina's. Pizza? She went to The Village Kitchen. He went to Angelo's. Summers she'd interned at CIRCLE, the Center for Information and Research on Civic Learning & Engagement at Tufts. He had taught swimming at the Y. They had grown up two miles apart, but it was as if they came from different cities. He said it was funny. She said, \"I'm not so sure.\"\n\nHe caught the guilty note. \"That's a West Cambridge thing to say.\"\n\nHe told her about the triple-deckers of Antrim Street. The back porches where you could hang in summers. He told her his old girlfriend had worked as a nanny for a baby named Moses. Noelle would lull Mo to sleep inside and then she and Collin would sit out on the porch and smoke weed until all the trees and green leaves shimmered. He told Nina this, but he downplayed the smoking part. He focused on the trees.\n\n\"They're huge,\" he said. \"And people worship them. There was an elm that died and my mom's friend Lois had a funeral before the city took it down.\"\n\n\"What's that like?\" Nina asked. \"A funeral for a tree?\"\n\n\"Pretty straightforward. Everybody gathered and Lois said a eulogy.\"\n\nHe described his mother's garden, tiny but so well planted that you couldn't set foot in it without stepping on a flower. There were pale-green hydrangeas, and purple irises, soft lamb's ears, creeping strawberry vines. \"You'll see,\" he told her. \"If this winter ever ends.\"\n\nNina's stepmother had a garden too. Helen had terraced lawns, and a swimming pool edged with bluestone, and a clay tennis court, but Nina didn't say all that at once. She started with the roses, and the moss on the stone walls.\n\nEach night they stayed out later. They walked to Broadway Bicycle after the shop had closed. Bike seats hung like hunting trophies, the size and shape of deer skulls on the wall. Behind the register, Nina saw hundreds of plastic drawers labeled like body parts: SPIKES, NUTS, SHINS, FANGS.\n\nHe took her to Christina's, where he had chocolate-orange and she had gingersnap ice cream. There were no open seats, so they walked up Cambridge Street with their cones past Boutique Fabulous. When they came upon Rosie's Bakery, Collin said, \"My father used to take me here before he joined the navy.\"\n\n\"That must have been hard,\" Nina said.\n\n\"What do you mean?\"\n\n\"Missing each other?\"\n\n\"They can't get you for child support if you enlist.\"\n\nThey walked down Fayette to the new coffee shop called Dwelltime. \"I liked the boyfriends better,\" Collin said, and he described Maia's main ex-boyfriends: the poet, Greg, who wrote obituaries until he got laid off from _The Boston Globe._ Tony, the chiropractor, who taught Collin how to drive. Best of all was Chris, the guy who'd lived with them until Collin was twelve. \"He gave me this,\" Collin said, showing off an old-fashioned watch with a worn leather band. Chris didn't really work, but he would take Maia and Collin to his parents' farm in Western Mass. They'd drive out in the fall to pick apples and press their own cider. All the ground was covered with peaches, plums, and pears, the fruit ripening, splitting open in the sun, fermenting, so the whole orchard smelled like wine.\n\nThey sipped their coffee and she told him about her stepmother. \"She's taller than my father,\" Nina said. \"She's tall and jealous.\"\n\n\"Without reason?\"\n\n\"No, she has reason.\"\n\nShe told Collin how Viktor traveled, and how she had waited up for him. She described his parties and his renovations, his fights with Peter, his younger brother and business partner.\n\n\"What do they fight about?\" Collin asked.\n\n\"Design, schedule, money,\" Nina said. \"But my father always wins.\"\n\n\"Why?\"\n\n\"Because he's the commercial one.\"\n\nShe told Collin of Viktor's dazzling inventions and the lawsuits afterward. His platforms MORPH and OVID (ocular-virtual integration device). Ideas like comets with long tails. Viktor had invented new ways to use aeroflakes, tiny sensor-receptors that filtered light to construct interactive, immersive fantasies. Aeroflakes drew power wirelessly through walls.\n\n\"I want to see that!\"\n\n\"You will,\" she told him. \"Everybody will.\"\n\nAt work, Viktor was charismatic and aggressive, at home, affectionate and preoccupied, by turns jovial, baffling, furious. Once she had seen her father fly into a rage and smash a table lamp. Another time she'd found him, early in the morning, kissing her au pair. Then Nina ran away to hide. She was the guilty one, terrified he'd punish her.\n\nWhen Nina told Collin this, he saw her all alone and small, and he wondered what else she'd seen with her gray eyes.\n\nThey saw each other almost every night, but she never let him walk her home. Did she think it was too soon? It didn't seem too soon to him. She was so soft, her mouth so sweet. They talked for hours, but she held back. He knew she wasn't teasing; she was a serious person. She didn't take relationships lightly, and that was fine\u2014except he wasn't used to it. He loved the rush, the free fall into intimacy. You had your whole life for conversation afterward.\n\nOne night he pulled her close and closer, swept her hair back from her face, and kissed her neck. Serious as you want, he promised silently. Anything you want. \"Let me take you home.\"\n\n\"No, that's okay.\"\n\n\"Don't walk by yourself.\"\n\nShe stood there in her white down jacket. \"I'm not afraid.\"\n\n\"Yes, you are,\" he said. \"You're afraid of me.\"\n\nShe didn't answer.\n\n\"What's wrong? Are you ashamed of where you live?\"\n\n\"A little bit,\" she confessed.\n\n\"Do you live in some big mansion too expensive for a teacher?\"\n\n\"No, just an apartment.\"\n\n\"Where?\"\n\n\"Mem Drive.\"\n\nShe lived in one of those buildings on the river. He had always wondered who lived there. \"Show me.\"\n\n\"It's not mine,\" she told him as they began walking. \"My father owns it.\"\n\n\"Okay.\" He was not surprised.\n\nShe looked at Collin earnestly. \"Do you know Arkadia's symbol?\"\n\n\"An ouroboros.\"\n\n\"Right. The dragon eating its own tail.\"\n\n\"And that's your father?\"\n\n\"That's my family.\"\n\n\"Lots of families eat their own tails,\" Collin reassured her, but even as he spoke, he realized that she was warning him. Hers was cruel.\n\nShe told Collin about how her father married Helen at the Cape. Nina was six and wore green silk, and she cried during pictures on the pier. She'd trailed her hand on the weathered railing and a splinter had pierced her palm.\n\nThe weather had been perfect, water glassy in the cove called Pleasant Bay. Barely a breeze ruffled the long sea grass, but Nina's tears ruined the photos and annoyed Helen. Nina's uncle Peter took tweezers and worked the splinter out. \"You hate her, don't you?\" Nina's uncle said. He was like a magician, drawing the idea out of her. As soon as he said the words, Nina knew that they were true.\n\nShe was walking slowly now. Collin waited, but for a long moment she didn't speak.\n\nThey were standing in front of Nina's redbrick building, with its bay windows, its faux balconies of fanciful wrought iron.\n\nShe said of her family, \"I love them. Unfortunately.\"\n\nHe slipped his hands into her coat pockets and felt the rounded corners of her phone, her jingling pocket change. He ran his thumb over the rough edges of her keys. \"Why unfortunately?\"\n\n\"I don't believe in them.\"\n\n\"What does that mean? You don't trust them?\"\n\n\"I don't trust them\u2014but I can't get away from them.\"\n\n\"Have you ever tried?\"\n\n\"I'm trying now.\"\n\n\"How? Standing in the cold with me?\"\n\nHe was impudent and funny, more straightforward than other guys she had known. He spoke without embarrassment about his talents and his difficulties. He loved performing, but he hated computer programming. He drew well but he had dyslexia and didn't like to read. He said that, but he read Nina. He listened intently, and he watched her face. Stories of her family didn't scare him. He kept his eyes on her. It was a simple thing, but it was rare. He really looked at her. \"Are you freezing?\" she asked.\n\n\"My hands are warm.\"\n\nShe felt his hands through the lining of her coat. She felt his warmth and she wanted to kiss him, but he must have known before she did, because he was already kissing her, his mouth softly on her mouth. He hadn't shaved, and his rough cheek scratched her face.\n\nWhen she tried to make sense of what was happening, she got scared. She'd known him for only two weeks. He had no career. He wasn't even a student. Once or twice she wondered what he might want from her, and then she felt dishonorable thinking that way. She had never known anyone so uncalculating.\n\nPlease, his body begged.\n\n\"Wait.\"\n\n\"What's wrong?\"\n\n\"Nothing.\"\n\n\"Should I go?\"\n\n\"No.\"\n\nHe murmured in her ear, \"It doesn't matter who your father is or where you live.\"\n\nShe pulled back, just enough to breathe. \"That's not been my experience.\"\n\n\"Try me.\" His hands closed inside her pockets.\n\n\"You have my keys,\" she said.\n\n\u2014\n\nFrom outside, the apartments looked like jewel boxes with their gilt-framed mirrors and carved furniture, their book-lined rooms. Inside, the lobby was dusty and old-fashioned. Chipped plasterwork and scuffed white marble stairs, a squeaky elevator trimmed with brass. The halls were hushed as libraries. Collin pictured old professors tucked away in bed. He smelled wood polish, noticed the umbrella stand inside Nina's door. Who had an umbrella stand?\n\nHer upholstered furniture, her kitchen big enough for chairs, her view, the shining river at her feet. All in an instant, she saw him take it in. Silently she dared him to speak.\n\nHe said nothing. He turned toward her instead, his expression rapt, his dark eyes bright. Even so, he waited. Though they stood just a few feet apart, the distance and the silence seemed dangerous to cross. \"I'll hang up your...\" she began, but the coat closet was too far away. When he pulled her in, she let his jacket fall.\n\nAs they unwound scarves, unlaced boots, she didn't offer him a drink or something to eat. Undressing, they tasted nothing but each other. They lay down on the couch and then on the carpet. And then they were so warm that they forgot the time, the view, the world outside. They forgot that it was winter.\n\nNina still assigned too much homework and popped too many quizzes. She was just as serious in January as she'd been in September\u2014and yet she had changed. She was more relaxed, sitting on her desk or leaning back against the board. Less fearful, less self-conscious, she smiled as she brushed chalk dust from her clothes.\n\nAs soon as he walked into her classroom, Jeff noticed her new confidence. Just as he'd predicted, after weathering the first three months, Nina had returned from winter break with fresh purpose. She struggled, but she wasn't nearly so bewildered. At last she understood what she could cover in one period, and arrived at class with two or three main questions instead of an entire lecture.\n\n\"Sevonna,\" Nina said. \"Sevonna. Cierra...\" Nina walked over to the pencil sharpener, where the girls were whispering, and escorted them to their seats. \"We were talking about the way Puritans policed one another. Xavier?\"\n\n\"Courts.\"\n\n\"Peer pressure,\" said Rakim.\n\n8:20 good intervention \/ continuity with scarlet letter...Jeff typed into his computer log.\n\n\"In those days a lot of morality came from peer pressure,\" Rakim said.\n\n\"Say more!\"\n\n\"Like the stocks,\" Xavier said. \"When you were publicly humiliated.\"\n\n\"But did it work?\" Nina asked.\n\n8:25 avoid jumping in too fast\n\n\"Obviously it didn't work for Hester Prynne,\" said Diana, \"because...\"\n\n\"Because what?\"\n\nJeff surveyed the class. Only two heads down. Open books on half the desks. One hand raised. Nina waited for Diana, even as she shook her head slightly at Rakim, who was leaning back in his chair again. He landed with a thud, but only a few people laughed.\n\n\"Because she had sex anyway, so the peer pressure wasn't working on her,\" said Diana.\n\n\"Exactly.\" Nina tried not to look at Brynna, who was examining her own long hair, holding up strands as she looked for split ends. Probably lots of pregnant high school girls studied _The Scarlet Letter,_ but Brynna was Nina's first, and she couldn't help worrying about what a sixteen-year-old in her third trimester might make of this.\n\n\"It's so ironic,\" Xavier said, without raising his hand, \"that usually peer pressure is for bad things, but in this book it's all about morality.\"\n\nNina smiled.\n\nWas she really smiling at Xavier? The eleventh graders shuffled in their seats. Oh, my God, Xavier was such a player. The word _ironic_ was like crack to Miss Lazare.\n\ngood pause, Jeff typed. He assumed Nina was counting silently to ten, as he had suggested at their last meeting, to allow her students more time to answer.\n\nShe was not counting to ten, or any other number. She felt delicious, strangely alert, then suddenly sleepy. She and Collin had been together three weeks.\n\nAs soon as the bell rang, she rushed off with the students. Jeff tried to catch her with his notes, but she slipped into the windowless, overheated photocopy room and hid behind the supply shelves stacked with paper and toner cartridges. There were several old wooden chairs behind the shelves, and she sat there for five minutes, just to close her eyes. She had to think, she had to dream, but the bell was ringing again. How did it ring so loud? So fast?\n\nAt lunchtime, she escaped to the basement, threading her way through tiled corridors, past the cafeteria smelling of disinfectant and steamed broccoli, to an abandoned resource room filled with giant therapy balls. As she leaned against the biggest, the ball deflated slightly, cushioning her body and her sleepy head. There she could rest and feel his hands. Remember him kissing her bare shoulders, burying himself inside her, breath quickening, fingers knit into hers.\n\nShe had been in love before. Away at school there had been a boy named Emmett, a runner with long dirty-blond hair, always in his eyes. She would sneak out early before class to find him coming back from morning practice and they would walk together through the Hill School's misty playing fields, to lie down in the wet grass. She had ruined a coat that way, spreading it like a blanket over sticks and stones. Emmett was already warm in his running shirt and shorts; he was wet anyway, his body sleek with sweat. Nina was the one who got suspicious looks at breakfast. Leaves caught in her long hair. She had to carry her black coat to math.\n\nIn college she had loved a scruffy literary guy named Jonah who concentrated in philosophy and wrote for the _Lampoon._ Theirs was almost a shipboard relationship; they had lived in such close quarters, studying and sleeping in his narrow bed in Adams House, editing each other's papers, reading poetry.\n\nJonah had curly hair and wore faded cords and raggedy old sweaters. He had theories about religion and politics and the frayed dynamic between love and friendship. He was interested in transcendental meditation and tech design and stand-up comedy. He wanted to be rich, but, like a juggler tossing knife, tennis ball, and frying pan, he debated management consulting, Hollywood, and graduate school. He never tired of perseverating about his future, or pondering the world. He'd hurt Nina when he began tweeting bits of news she'd told him about Viktor and OVID.\n\nCollin came as a relief. He didn't ask about her father, nor did he talk about the future. He brought takeout to her apartment and he spent the night and they laughed about her students. After bad days he comforted her.\n\n\"I could not get them to listen,\" she said.\n\n\"Buy a police whistle,\" he suggested. \"Bring free food.\"\n\nShe leaned against him on the couch. \"My students deserve a better teacher.\"\n\n\"Do something else, then. You could do so many other things.\"\n\n\"But this is what I want to do.\"\n\n\"Why?\"\n\n\"I want to give back.\"\n\nHe looked at her and said in all seriousness, \"Why? What did you take?\"\n\nShe shook her head. He knew that she was rich, but didn't see the rest of it. Her father produced mind-blowing, immersive entertainment. She wanted to separate herself from that. She dreamed of enchanting kids with words instead of optics.\n\n\"There are lots of other ways to give back,\" Collin pointed out. \"Homeless people, clean water, the environment.\"\n\nShe was almost too shy to look at him. \"I want to teach because that's the real magic.\"\n\nHe nodded, because, of course he'd heard this language before. He'd grown up with his mother's golden apples, her #1 TEACHER paperweights. \"Transforming lives.\"\n\n\"I want to give at least a little bit of what my teachers gave to me\u2014but my kids don't even listen.\"\n\n\"I guess you have to be patient.\"\n\nShe tucked her legs under her and considered him. If he'd been patient he would have stayed in college. \"Why did you quit?\"\n\nHe thought of his mother, always hoping he would learn marketable skills. \"I hate Web design.\"\n\n\"But you could do studio art.\"\n\n\"Nah.\"\n\n\"Why not?\n\n\"I'm not conceptual.\"\n\n\"What does that mean?\"\n\n\"I'm not big-picture. My art's not deep.\"\n\n\"You're just being modest!\" She was thinking of the tiny line drawing he'd given her in Grendel's.\n\n\"No, seriously. I have nothing to say. I like to draw. That's all.\"\n\n\"What?\" She had never heard anyone admit to such a thing. Jonah had been all ideas; he'd never stopped talking. \"That can't be true.\"\n\nCollin teased, \"You think I'm tragic!\"\n\nGuilty again. \"No, that's not what I meant!\"\n\n\"I don't need ideas,\" Collin declared. \"I don't need theories in my life.\"\n\n\"What do you need, then?\" she asked, partly curious, partly fishing.\n\nWas he supposed to say you? All I need is you? He answered, \"Just a box of chalk.\"\n\n\"You're funny.\"\n\n\"I'm serious,\" he said, as he caressed her hand.\n\n\u2014\n\nThe world was brighter now, and strange. She saw rabbit prints on the clean snow, and trees of diamonds glittering. When the last bell rang, kids flooded the staircases, and she could lose herself in the crowd, forgetting books and lessons as her students surged around her. Thrilled to leave, she was becoming just like them.\n\nShe took Collin to Burdick's in the Square. He had never had such dark hot chocolate. She took him to Upstairs on the Square just two flights up, but a world away from Grendel's. The dining room all pink and gold, with marble-topped tables and gilt fireplaces.\n\nEvery object in her apartment had a history. Her furniture came from the 1950s. The atomic clock on the kitchen wall came from Finland. She had a Narnia chess set\u2014Aslan and his fauns carved of ebony, arrayed against the White Queen and her henchmen cut in alabaster. When Nina was ten, her father had promised her the set if she could beat him, and then relented when she fought him to a draw.\n\nOn her bookcase she kept a framed drawing of her father, a pen-and-ink caricature by Al Hirschfeld, an artist Collin didn't know. Collin studied her father's cartoon face, his dark eyes, his curly hair, his exaggerated nose. She told him, \"If you look here, you can find my name.\"\n\nSure enough, Hirschfeld had hidden the name _Nina_ in Viktor's bushy brow.\n\nIn the moment, none of this seemed strange. She had beautiful things, but she piled her dishes in the sink like everybody else. When they were together he felt at home. White kitchen, river view, clean sheets. Then he got back to his own place, and he felt like a hobbit living underground. He stood in disbelief on his own threshold, taking in the mousetraps in the kitchen and the dank, shared bathroom. Reentry required several beers. He would sit on Darius's salvaged couch and he would blast Bent Shapes, and draw until he collapsed into his unmade bed.\n\nNever in his life had he devoted so much time to anyone; he barely saw his friends; he abandoned his old haunts\u2014but his behavior didn't seem unusual to Nina. Always, in her quiet way, she wanted more. When she couldn't reach him, she texted, IMY.\n\nAlarmed, he typed, Dont do that!\n\nThe next time they saw each other she asked, \"What's wrong with saying I miss you? It's just a fact.\"\n\nBut it wasn't a fact for him. It was a demand. He read IMY as \"I want more of you.\" He told her, \"This is all the time I have.\"\n\n\"I understand,\" she said, but she didn't, really.\n\n\"I can't be with you every minute of the day,\" he told her.\n\nShe shot back, \"I know! I never asked you to.\"\n\nShe got skittish. She needed reassurance\u2014not just words but hours, entire afternoons. Day to day they held each other in suspense. He had to back off and breathe. She wanted to know him better, to unfold their friendship like a map. What else could he give her? Sometimes the question scared him. Sometimes the answer came easily. He would give himself.\n\nOn a slushy day in late January he met her after school and said, \"I want to show you something.\"\n\nIt was drizzly cold as he hurried her up Cambridge Street, past the fabric store, and the senior center, the Portuguese savings and loan. They dashed across Hampshire in the rain. The trees and bus shelters were dripping. Even the birds hunched up, wet and miserable, on telephone wires.\n\n\"Where are we going?\" Nina asked.\n\n\"You'll see.\"\n\n\"Your apartment?\"\n\n\"That needs fumigating first.\"\n\n\"You always say that.\"\n\n\"Because it's always true.\"\n\n\"I still want to see it.\"\n\n\"No,\" he told her. \"It's embarrassing. Emma labels all her food with skull and crossbones, and Darius forgets to flush.\"\n\n\"I wouldn't mind.\"\n\nThat angered him. Of course she didn't mind. She didn't live there. \"It's a pit.\"\n\n\"But it's yours.\"\n\n\"Exactly,\" he exploded. \"It's _my_ pit of an apartment, and I promise you won't like it there.\"\n\nShe said, \"But it doesn't matter what\u2014\"\n\nHe cut her off. \"Don't tell me it doesn't matter and you wouldn't mind. Don't be so fucking condescending.\"\n\nHe had never spoken so harshly to her. Maybe nobody had spoken to her that way before. He watched her turn and walk away toward Kendall Square.\n\nShe provoked him with her eagerness, her gentleness, her noblesse oblige. After all, what was she doing, spending time with him? He had nothing, he'd done nothing; and when Nina said she didn't mind, she acknowledged it was true.\n\nHe watched her figure receding, and he was furious with her and with himself. He'd been planning to surprise her.\n\n\"Nina, wait.\"\n\nShe didn't turn.\n\n\"Don't go,\" he called out, as he sprinted down the street, splashing through slush puddles. His shoes were soaked when he finally caught up to her.\n\nShe spun around to face him. \"I was telling the truth,\" she said. \"I wouldn't mind. And didn't you say the same thing to me about my apartment? It didn't matter where I live?\" She had been continuing the argument in her head.\n\nBreathing hard, he took her hands in his. \"Let me take you somewhere.\"\n\nShe shook her head.\n\n\"I didn't mean to hurt you.\"\n\n\"Yes, you did.\"\n\n\"I'm an idiot.\"\n\nShe didn't contradict him.\n\n\"Let me start over. There's something I have to show you.\"\n\nIt was as if they'd never quarreled; his mood changed that fast. She was the one who lagged behind. His anger flared and burned out fast; hers smoldered.\n\nHe said, \"I'll take you to my place when Darius and Emma are away in Maine.\"\n\n\"No, that's okay.\"\n\n\"I'll wash everything down.\"\n\nSilence.\n\nHe almost coaxed a smile when he said, \"I'll cook.\"\n\nThey retraced their steps to Antrim Street, and he led her to a dark-green triple-decker, three Victorian apartments stacked one atop the other with three porches.\n\nNina turned to him in surprise. \"Did you tell her that...Shouldn't you call to warn her first?\"\n\n\"Why would I warn my own mother?\"\n\nShe hung back. \"What if it's a bad time?\"\n\n\"Sh.\" He ushered her into the entrance hall and knocked on Maia's door. \"There's no bad time.\"\n\n\"Hey,\" a voice called out. \"Come in.\"\n\nThey slipped off their wet coats and shoes and walked into a half-painted living room with all the furniture piled in the center of the floor. \"Who is it?\"\n\nCollin called back, \"Me.\"\n\n\"Hello, you,\" Maia said, as Collin led Nina into the kitchen.\n\n\"This is Nina.\"\n\nNina felt flustered, entirely unready. It was just like Collin to spring this on her. Even so, she gazed in fascination as Maia took her hands. Collin's mother was tall, and muscular, a dancer wearing plaid pajamas. She had close-cropped hair, showing off her small ears, her beautifully shaped head. Her almond eyes were dark, and she had a birthmark in one, a sepia ink blot in the white. Nina saw the mark, and immediately forgot it, as everybody did.\n\nMaia ushered Nina in with such warmth that Collin shook his head in silent warning.\n\n\"She's so pretty,\" Maia whispered when he followed her to the hall closet with the coats.\n\nHe shot her a look. \"Don't screw up.\"\n\n\"That's just what I was going to say to you.\"\n\nIn the kitchen Maia poured them each a glass of wine.\n\nCollin asked his mother, \"What are you doing with the living room?\"\n\n\"You don't like the green?\"\n\n\"I think it looks like mushy guacamole.\"\n\n\"Okay, thank you, sweetheart.\" Maia turned to Nina. \"I'm not listening to him. Peanut brittle? Caramel corn? Fruit cake?\" She covered the kitchen table with Christmas gifts parents had brought her. \"Try this.\" She sliced homemade fruit cake, dark-spiced and walnut-studded, jeweled with candied cherries, carbuncled with pineapple.\n\n\"Wow,\" said Nina. \"I got candy canes.\"\n\nMaia waved her hand over her cards and deluxe candy apples, her mugs filled with gift certificates, and she told Nina, \"Someday all this will be yours.\"\n\nNina looked doubtful, and Maia laughed.\n\n\"Tell her your secrets,\" Collin said.\n\n\"I don't have secrets.\"\n\n\"You know what I mean. Secrets of teaching.\"\n\n\"Well, the first ten years are the hardest,\" Maia told Nina.\n\n\"Come on, Mom.\"\n\n\"It's humbling. I'm not gonna lie.\"\n\n\"Give her something she can use.\"\n\n\"Hmmm.\" Maia took a long sip and set her wineglass down. \"There is such a thing as reincarnation. If you teach long enough, the same kids keep coming back again.\"\n\nNina said, \"In different forms?\"\n\n\"Yeah, but they're recognizable\u2014like, Oh, _yes._ I remember you. The ones who can't keep still. The ones who don't listen. The ones who fall in love with you.\"\n\n\"That's not advice,\" Collin pointed out.\n\n\"Advice.\" Maia pondered. \"Be funny.\"\n\n\"I'm not,\" said Nina.\n\n\"Be desperate.\"\n\n\"Okay, I'm good at that.\"\n\n\"See, you are funny,\" Maia said. \"Be surprising. For example, I came in one weekend and painted my classroom purple. It's good to blow their minds.\"\n\n\"She likes paint,\" Collin said.\n\n\"Yeah. If you're wondering where he gets the art from, it's from me.\"\n\nMaia pulled down Collin's paintings from the refrigerator, his splotchy grade-school pictures of the river. She spread his drawings of ducks across the kitchen table. \"The early years. See how he did the webbed feet?\"\n\n\"I should have warned you,\" Collin told Nina. \"She keeps all my old stuff.\"\n\n\"I'd keep the new stuff too,\" Maia said serenely. \"Oh, wait. You work in _chalk._ \"\n\n\"I like chalk.\"\n\n\"I like a green living room.\"\n\n\"That green is way too blue.\"\n\nMaia watched Collin pace the kitchen and drink another glass of wine. \"Come back and see it in the light.\"\n\n\"It won't work in daylight either.\" Collin swiped a kitchen towel, soaking it in water.\n\nMaia said, \"If you want a rag, look under the sink.\"\n\nCollin ignored this and disappeared into his old bedroom for socks and shoes. \"Come on.\" He ushered Nina to the entry, where he opened the cellar door and pulled the light string.\n\nMaia called after them, \"Careful on those stairs.\"\n\nCollin ran down quickly and then held out his hand, helping Nina with the last rickety steps.\n\n\"Over here.\" Guiding Nina past the toys and rusty bicycles, he led her to seven rolling blackboards, dark, old-fashioned, like the ones at school. Rising from the basement clutter, they stood mounted in wooden frames with casters underneath.\n\nQuickly, he took the wet towel and wiped the board in front, rubbing the surface clean of every bit of dust and dirt so that it gleamed in the dim light. Then he reached down to excavate a box brimming with chalk. \"What would you like?\"\n\n\"What do you mean?\"\n\n\"What do you want?\" Focused, almost fierce, he picked up a piece of chalk.\n\nShe hesitated. \"Well...what do you like to draw?\"\n\n\"Everything.\" Already his hand was moving across the board. \"Horses.\" He found a clear space and drew a horse with mane and tail streaming. \"Birds.\" A pair of white swans glided from his hand, their long necks curving, reflections rippling in a glassy lake. \"When I was a kid I liked castles.\" In seconds he drew crenellated towers and pennants flying, drawbridge opening. \"Now I draw people.\" With one fluid line, he drew a woman lying on a beach\u2014no, on a bed. It was Nina, leaning back, supporting herself with her elbows. That was her hair falling over her shoulders. Those were her arms, her breasts...\n\n\"Collin!\" She had never seen anyone draw so well, so fast. His horses turned their heads to look at each other, his castle stood on foundations of rough stone, his sketch of Nina captured her tender expression, her soft, mussed hair.\n\nHe stood back for just a moment to admire his work, and then he wiped it all away. Castle, swans, horses, Nina's portrait vanished, and in their place he drew the hemlock in front of his mother's house. He worked with thick and thin edges, smudging snow onto the branches with his hand, outlining telephone wires in white. \"What else?\"\n\nWhat else? She'd like to know how he drew swans so easily. How did he toss off castles in five seconds? He was so quick! He'd told her about his asphalt water lilies and his sidewalk Van Goghs, but she'd been to street fairs, glanced at sidewalk art on Church Street. His work was something else.\n\nHis art was quick but never crude or facile. His drawings were lively, streamlined, beautifully observed. As mimics capture gestures in performance, he drew essential details, the curve of a neck, the soft dent in a pillow, the arc of a careening sled. With each sketch, Nina felt a shock of recognition. She forgot the fight, entirely forgot her funk, lost all consciousness of place and time and Maia upstairs. He was drawing the hill at Danehy Park, the dwarf pines weighted down with snow. How did he do it? He seemed to steal from the world.\n\nBy now he held four pieces of chalk in his hand. \"Tell me what you want.\"\n\n\"A cat.\"\n\nA white cat capered atop a brick wall, and then a calico on a windowsill, and a dark cat with white feet and white nose crouching in tall grass. \"What else?\" He never hesitated as his chalk moved across the board. Sometimes he erased a line or smudged it with his hand, but he seemed to have a picture in his mind. Another cat emerged, a slinky, half-grown animal, ginger with green eyes.\n\n\"You couldn't learn this in school,\" Nina said.\n\n\"No, but I drew all the teachers.\"\n\n\"You drew the wrinkle between Miss Dorfman's eyes.\"\n\nHis right hand traveled over the board. \"Go crazy,\" he told Nina. \"Ask for furniture.\" He drew his mother's kitchen table with six chairs. \"Or tennis.\" He sketched himself, long, sinuous, jumping up to serve. He looked at her. \"I'm showing off.\"\n\n\"Don't stop.\" She remembered his words\u2014\"I have nothing to say\"\u2014but it seemed to her the opposite was true. He had everything to say.\n\n\"What else?\" he pressed.\n\n\"Wait,\" she said. \"Just let me look.\" He had charmed her, delighted her before. She had enjoyed him for himself. He didn't need to impress her, but to her chagrin, she saw him differently now. It should not have mattered, but it did; she saw his gift. \"No!\" she begged. \"Don't erase your pictures.\"\n\nWith a defiant smile he took his rag and erased his art to start again with a sailboat skimming an imaginary sea.\n\nHer heart was racing, because she could help him if he let her. She didn't tell him, because she was afraid once more he'd take offense. To tell him was to flaunt her own position. Worse than that, to judge him. You're so good. What are you doing with your time? She watched in silence as he drew a sail swelling with the wind. She did not speak, even as the words rushed in her ears. Let me do something for you.\n\nOn Sunday Nina met her friends Julianne and Lily at Henrietta's Table. She decided not to mention Collin, because she didn't need her friends passing judgment. The relationship was so new, she wanted to protect it\u2014but they guessed. They knew that she'd been seeing someone. After all, she had been ignoring them.\n\n\"He's an artist,\" Nina said at last. \"Sometimes an actor.\"\n\n\"Oh, no,\" said Julianne.\n\n\"What do you mean, 'Oh, no'?\" This vehemence surprised Nina, because Julianne was studying to be an opera singer.\n\n\"Sometimes an actor?\" Lily asked.\n\n\"I mean, he's not just an actor,\" Nina said.\n\n\"Uh-huh,\" said Lily.\n\n\"He's smart, and he works hard.\"\n\n\"You're blushing!\" Julianne said.\n\n\"I'm not.\"\n\n\"Yes, you are.\"\n\n\"Only because you're staring at me,\" Nina protested, but Julianne had known her since first grade. The girls had grown up ice-skating, scribbling their names inside Nina's closet, pitching tents in the music room of Julianne's stone manor in Milton. This was a house so grand it had its own agent for photo shoots and concerts and commercials. Servants' bells hung in the kitchen. The window seats were deep enough to put on plays.\n\nNina had loved Julianne's house because it made hers seem more ordinary. There had been something last days of Versailles about the empty ballroom and the vaulted dining room. Gold silverware and crystal filled the butler's pantry, but it was hard to find anything to eat. Hundreds of gilt chairs arrived by truck for chamber music concerts, but couches were few and far between. At night, on the third floor, the girls lay in Julianne's four-poster bed, and they heard distant music downstairs and far away. The house was so big that when Julianne's pet tortoise escaped, he was never seen again.\n\n\"Look at yourself.\" Julianne held up her phone now.\n\n\"No, don't!\"\n\nJulianne took a picture of Nina laughing and covering her face.\n\n\"Is he cute?\" said Julianne.\n\n\"Is he an egomaniac?\" said Lily.\n\n\"He has an ego, but he's not a maniac.\"\n\n\"So he's just self-involved.\"\n\n\"No! The opposite.\"\n\n\"But he doesn't have a job.\"\n\n\"He has at least two jobs,\" Nina said. \"Maybe three.\"\n\n\"Hold on.\" Lily held her hand up. \"At _least_ two jobs, or at _most_ two jobs?\"\n\nLily, Nina's college roommate, had been adopted from China but raised a Klein in Santa Barbara. Her mother was a pediatrician, her father an endocrinologist. She had begun college as a writer and a humanist, comping the poetry board of _The Advocate,_ studying folklore and mythology\u2014but within the year, she'd dropped Old Norse and reverted to organic chemistry, dumping her boyfriend from Vermont, along with all his poet friends. In their matchbox room in Thayer, she had confessed to Nina, \"Student poets suck.\"\n\n\"Didn't e. e. cummings go here?\" Nina asked. \"And T. S. Eliot?\"\n\nLily said, \"Yeah, but he didn't write _The Waste Land_ as an undergraduate.\" Lily had lost all sympathy for literary guys. After graduation, she had crossed the river for Harvard Medical School, leaving Asgard far behind.\n\n\"He's talented,\" said Nina.\n\nLily said, \"Apparently.\"\n\n\"He's a chalk artist.\"\n\n\"Like sidewalk chalk?\"\n\n\"He can draw anything.\"\n\n\"Okay.\" Julianne considered this. \"He's good with his hands.\"\n\n\"Be serious!\"\n\n\"I am serious.\" In her low-cut shirt, Julianne looked like a Renaissance goddess of spring, a buxom mezzo with strawberry-blond hair. \"This is your transitional guy.\"\n\n\"What are you reading?\" Nina said.\n\n\"I'm not reading anything!\" The language of recovery came naturally to Julianne, who had been in therapy since she was a child. \"I'm just so glad.\" She'd always said Nina needed to fool around.\n\nBut Nina wasn't fooling. \"I think I can get him a job.\"\n\nLily pounced. \"So he isn't actually working.\"\n\n\"He's working, but I have an idea he could do better.\"\n\n\"Doing what?\" Julianne asked, because she kept an open mind.\n\n\"I thought he could work at Arkadia.\"\n\nSilence.\n\n\"You haven't seen him draw.\"\n\nJulianne and Lily looked at each other. They knew Nina's enthusiasms and her stubborn resolve. They had seen her disappointed tears.\n\n\"Don't you remember Jonah?\" Lily said.\n\nNina quieted a little, even as she said, \"He's nothing like Jonah.\"\n\nLily said, \"You felt used.\"\n\nNina told them, \"He's an artist, and he's incredible.\"\n\n\"Oh, my God,\" Julianne told Lily. \"She's completely in love with him.\"\n\nHer friends didn't understand. They assumed Nina was slumming, although they would never use the word. Of course sidewalk art was just as valid as anything else, and joining TeacherCorps was meaningful. It was the kind of thing everyone should try. They believed all this, and then they were startled when Nina actually followed through. She didn't mean it, right? She was just rebelling. This was her version of sex, drugs, and rock 'n' roll\u2014dating a chalk artist, and teaching school.\n\nThey were prejudiced. No, that wasn't fair. They just wanted to protect her. They were glad she'd found Collin; relieved that she had slept with him. What alarmed them was her urge to help him. She'd been seeing him for what? Six weeks? Obviously he cared about her, but he knew who her father was.\n\nJulianne said, \"I wouldn't help him get a job.\"\n\n\"Why not?\"\n\n\"You aren't thinking,\" Lily said.\n\n\"I am thinking. I haven't decided anything.\"\n\n\"Good,\" said Julianne. \"Don't!\"\n\n\"Don't introduce them?\"\n\n\"Just don't decide.\"\n\nNina's friends hugged her after dinner. The three of them stood outside the Charles Hotel and said goodbye and love you. Julianne and Lily were driving back to Brookline. They offered to drop off Nina, but she said no. Are you upset? they asked. Of course not, she lied, and she walked home alone.\n\nShe wished she'd never told them. Her desire to help Collin tarnished in the open air. She would not admit that they were right, but their doubts awakened hers. Once she introduced him to her father, there would be no going back. She would be pulling strings, and he would always see her that way. She would be manipulating the situation\u2014giving up any pretense of equality. What would that be like? Risky, Julianne said. Big mistake, said Lily. How can you tell what he really wants? Julianne had asked gently, and of course Nina knew what that meant. You have too many things he needs.\n\nThey didn't understand, because they didn't know him. She thought they might change their minds once they met Collin. Then she admitted to herself she wasn't sure.\n\nHe had laughed when she showed him her father's house. It was at night and everyone had been out of town. He'd stepped into the hushed foyer and he'd laughed and laughed. \"Is this for real?\" It wasn't the grand Victorian that flummoxed him, not the gables and turrets and tall windows, but the art inside, the white-flower painting by Georgia O'Keeffe, the eye-popping Lichtenstein of a woman crying on the telephone, the gold Buddha in the dining room.\n\nWhen Collin saw the statue of Venus standing in the library, he caressed the goddess from her broken shoulder over the curve of her breast and through the folds of her drapery to the hem of her robe. Exultant, he said, \"It's like a gallery, but you can touch, and no one's watching!\" He seemed to forget that Nina was watching him. He seemed to forget everything.\n\nSleepless that night, she sat up, marking Discovery Journals.\n\n> _Q. Why does Hester Prynne conceal the identity of her child? Does her decision make sense to you?_\n\nIn large, round handwriting, Marisol wrote: _Hester Prynne not revealing the father of her child isn't so impossible to understand. 1. It's embarrassing. 2. In those days it was not appropriate._\n\nTiara wrote: _It is dangerous to reveal anything when your a Puritan._\n\nDangerous, Nina thought. That was the word. Not the games at Arkadia. The people there.\n\n_To me the whole book is ironic,_ wrote Xavier.\n\nSevonna said, _Without secrets life would be so blatantly obvious it would be ridiculous._\n\nCollin might enjoy her help, and he might resent it too. He might be grateful, but she wasn't sure she wanted gratitude. She imagined his mixed feelings, his guilty delight. She considered how he might treat her, as if he owed her something, or as if she'd trapped him in some grand plan.\n\nDiana's entry was an inky thicket. _Everybody's got secrets. Whats more interesting is when you find out other peoples. Then the question is do you tell on THEM? For example my twin and I were like blood brothers only moreso. Now its like he moved away. I hear him whispering daphne daphne._\n\nDaphne? Nina stopped there, puzzled. Then the word receded into the tangle on the page. _Maybe because he has something to hide, Nathaniel Hawthorne is trying to write in a confusing way. Sometimes_ _its like Nathaniel Hawthorne is trying to be deep._\n\nNina imagined speaking to her father. I have to ask you something. It was a big thing to ask\u2014a serious and revealing question.\n\nGo ahead, Viktor would answer, and as he listened, he would jump three steps ahead of her, and he would laugh.\n\nShe turned off the light and lay awake in bed. She couldn't keep Collin from her father. Knowing her, he must know Viktor\u2014but not so soon! Coming to Arkadia, Collin would know her uncle Peter too. This gave her pause. Her father was scientific and jovial, devoted to technology. Peter was artistic and perverse.\n\nArkadia was harsh, fantastical, a tricky labyrinth. She wanted to shield Collin, but that seemed wrong, discounting his independence and his gift. She remembered just a week before, she and Collin had celebrated their birthdays at a little place called Carmen, in the North End. Brown kraft paper covered all the tables, and as they ate, Collin covered the paper in black pen, sketching places they had been\u2014the bike shop, the bar at Grendel's, the stone castle atop Prospect Hill. When the waitress arrived with their check, she looked wide-eyed at the illustrations and said, \"Do you want me to wrap that up for you?\"\n\nI have to help him, she thought the next morning, as she drove to school. But was there some way she could do it indirectly? She wove through winter streets, looping around the Cambridge Common, dipping into the underpass, and she wished she could help him secretly.\n\nShe was not surprised to see police when she arrived at Emerson. She assumed patrol cars meant a safety drill, but then she saw security blocking off the sidewalk, a traffic jam of cars and school buses redirected to the back entrance. She had to circle the block to find a parking place.\n\nInside the school building, yellow tape cordoned off the lobby, and police officers were directing students downstairs to the gym. CAUTION POLICE LINE DO NOT CROSS. Vandals had attacked the building, spray-painting the entryway.\n\n\"Oh, wonderful,\" said Mr. Allan.\n\nMrs. West was demanding, \"How did they get in?\"\n\nThe black graffiti extended across the lobby wall in a series of initials over a foot high. The other teachers didn't know yet what the letters meant, nor did Mr. DeLaurentis, who stood outside his office, talking on his phone. Was it a new gang? Or some random prank?\n\nOnly Nina understood. She knew instantly, and her face burned. There she'd been, plotting to find Collin a job. Now she wished Arkadia away, along with its obsessive fans. They had tagged her own school with UnderWorld's catchphrase, See You In Hell.\n\nCUCUCUCUCU\u200bCUCUCUCUCU\u200bCUCUCUCUCU\u200bCUCUCUCUCU\u200bCUCUCUCUCU\n\nAcross the river, Kerry was catching up on paperwork when she received a text from Emerson. Partial lockdown, vandalism with violent videogame content, classes continuing, increased security, no imminent danger, please wait until dismissal before coming to school. She read all this at once, and then twice, three times, but the only words she saw were \"violent videogame content.\" She hardly noticed signing out, zipping up her coat, shouldering her bag to leave the ICU.\n\nShe had no idea what was going on, but her thoughts were all for Aidan, even as she glanced back at patients and parents in their glass-walled rooms. She left\u2014although you could never leave entirely\u2014and took the elevator down to the lobby, which was teeming with families and strollers, musicians toting their guitars, pet therapists leading wise-eyed dogs, hospital ambassadors, costumed clowns.\n\nKerry hurried past bright murals and saltwater aquariums. Reaching back, she retied her thick blond hair, stuffing the ponytail into her hood. She passed a girl in a wheelchair with head support, then a man-size boy, clinging to his mother. \"Bless you,\" Kerry said, as the boy sneezed loudly, and then sneezed again.\n\nShe was a believer. She believed in four-leaf clovers and shooting stars. She believed in Jesus and in angels, although they worked mysteriously. She had seen more of death than most, and she believed, to some extent, in ghosts\u2014not the ghoulish kind, but the quiet ones who come to you in dreams.\n\nShe had her superstitions about shoes on tables, and open umbrellas in the house. She had her rituals, and she was not ashamed of them. After all, she worked in a place where knowledge ended and belief began. She got out of bed on the right side, and when she had a chance to swim in the echoing War Memorial pool, she used only a red kickboard\u2014red for happiness\u2014never blue. She didn't think that she could sway the universe, but she hoped that you could nudge it with a prayer.\n\nShe prayed now, as she drove through the maze of hospitals\u2014Children's, Beth Israel Deaconess. Her hands were cold inside her gloves, her car's heater was still warming up, and she shivered as she prayed for her own children.\n\nDiana was secretive, but she followed rules; she did her homework and her chores. Trusting Aidan was an act of faith. Some days were easier. He said hello, or washed the dishes. He came downstairs and looked awake. Other days he didn't even glance at her. His games consumed him and he had nothing left. What would become of him? How would he get into college? She berated him and he listened in silence, waiting for her to leave. At those moments she hated home as much as he did. She would drive to the Star Market on Sidney Street, and wheel her cart through the white aisles, and cry.\n\nDriving across the BU Bridge now, she felt a rising dread. The river spread before her, icy near the banks, but lively in the center, gold water dancing in the morning light. She had loved the drive on other mornings. Today her back tightened and her shoulders ached. She tried to breathe and fogged the windows.\n\nPlease, she prayed as she wiped the glass with her knit glove. I know you work in your own ways\u2014but could you send me a sign that this year will get better? She didn't ask for a miracle. She'd nearly given up on those, but please, she thought. Just something small?\n\nSlowly, she eased her car between the snowbanks in her shared driveway and picked her way up icy steps. Anxiously, she stepped inside the door. All was quiet; the twins were at school. Of course they were. Even so, she peeked into their rooms. She gazed at Diana's rumpled bed. Briefly, she ventured into Aidan's cave. He had covered his window with aluminum foil so that not a crack of light could penetrate. He'd stripped the walls of posters, and wedged his computer desk into the corner. Kerry reached out to touch the monitor and then drew back. He would know.\n\nTiptoeing through her own house, she retreated to wash dishes, to take down a load of laundry, to sort the mail, to read _The Boston Globe._ Finally, she took her little photo album to bed with her and flipped through photos of the twins when they were small. She looked at one picture of her children coming down a slide. It was a little slide and they were wearing overalls. She had forgotten about those overalls. They couldn't have been more than two, laughing Aidan first, his hair white gold. Dark-haired Diana peeking over his shoulder, apprehensive, as she slid down after him. Those days had not been easy, but they had been happier. For one thing, Kerry's parents had been alive. She had moved in with them after her ex left. Her parents had watched the children in North Cambridge while she worked night shifts. Her father had built a sandbox for the children, and a little table with matching stools. Later, her parents left her a small inheritance, which she'd used for the down payment on the house.\n\nNow she wished that she could travel back in time. She wouldn't go far. She had no interest in history or adventure\u2014the recent past was all she wanted. Her mother's voice, her father's patient carpentry, the playground with the green slide, the twins at six, learning in school that vitamin A was good for your eyesight. Venturing down to the basement, they held carrots like torches. She had found Aidan and Diana standing in the dark, nibbling the tips.\n\nHours later, Kerry woke in pale winter light. A creaking, clicking sound, the tick of the gas burner on the stove. \"Diana?\" Kerry called.\n\nNo answer.\n\n\"Diana?\" Kerry descended to find her daughter in the kitchen. \"Diana!\"\n\n_\"What?\"_ Diana shouted. \"Stop calling me over and over.\"\n\n\"Start answering!\"\n\n\"I did answer.\"\n\n\"I couldn't hear you.\"\n\nDiana opened three packets of instant oatmeal.\n\n\"It's only twelve-twenty.\"\n\n\"Early dismissal.\"\n\n\"Because of the lockdown?\"\n\n\"I guess.\" Diana felt for her mother. Kerry's face looked pallid; even her blond hair was dull and fuzzy, not gold, as it had been. Stand up straight, you're a beautiful girl, Diana thought. It was no use. Exhaustion beat Kerry down. Her delicate features had faded, her hands were raw from washing at the hospital.\n\n\"Where's your brother?\" Kerry asked.\n\n\"Where he always is.\"\n\n\u2014\n\nUpstairs, in his room, Aidan was turning a BoX over in his hands. The BoX was black and beautifully smooth, a perfect cube of plastic. He knew what was inside, but he could not find a way to open it.\n\nThe cube was compact, small enough to fit into a backpack, but heavy enough to strain the straps. Kneeling, he ran his fingers over the surface, pressing for a secret spring. Gently, he tapped each side. Nothing happened. He shook the BoX, and shook it harder. \"What's wrong with you?\" He could not open this inert console, although he'd been trying since he got home from school. He'd snatched up the parcel, marked PRIORITY MAIL, and now, like a prisoner, he pried its edges, paced the floor, threw himself down onto his bed, dreamed and despaired of his escape\u2014except that most prisoners imagined getting out. Aidan wanted to get in.\n\nHe'd checked the package a hundred times and found no instructions, no note from Daphne, no code, no Web link. He'd searched online, typing \"UnderWorld,\" \"black BoX,\" \"new platform.\" He'd only turned up articles he'd seen before. Now he sat with the BoX on his lap and entered EverWhen, roaming across the screen, sending Tildor over snowfields to search for Daphne.\n\n???, he typed into the chat box.\n\nNothing.\n\nGotit now what?\n\nNothing.\n\nhow does it open???\n\nNothing.\n\nComeon\n\nNothing.\n\nHelp me open it.\n\nNothing.\n\nIts fake, he typed in rage and in frustration. soru. bitch.\n\nWatch your language.\n\nThe answer came so fast he jumped, and the BoX slid off his lap and crashed onto the floor.\n\nHe was afraid he'd broken it. Kneeling, he found a hairline crack, but as he turned the BoX, he saw the surface wasn't cracked at all, but subtly divided. As with a Rubik's Cube, you could twist the top half of the BoX away from the bottom. As with a child-safe medicine bottle, you twisted while pressing down. Oh, I get it, he thought, even as the top popped off in his hands.\n\nThe room went dark.\n\nFrantic, Aidan gasped for air in what looked like smoke. In fact, he could breathe easily. The air in his room was just the same; only his perception of the atmosphere had changed. He was crouching in a stream of dust motes. These were aeroflakes, imperceptible on their own, flying together in a cloud.\n\nHis room was not his room. His ceiling was dissolving, his walls warping, rippling. A mist rose up around him. Fog that wasn't wet; dry ice that wasn't cold. For a long time he was afraid to stand. Bed, desk, and chair had disappeared, the floor seemed to slide beneath him. Faintly he heard the trickle of water, the rustle of leaves. He imagined a deep forest, but he could barely see. He reached for the light switch on what had been his wall. Aeroflakes shifted and resolved themselves, illuminated by Aidan's ceiling light. He felt for the door of his closet and opened it, groping for the light switch inside. Once more the accumulating mist began to shift and change. Filtering and reflecting light, the particles responded to Aidan's movements and to one another, projecting a multidimensional landscape, deceiving and delighting the eye, coloring the air, even transmitting sound.\n\nNow Aidan perceived bare branches and jagged pieces of blue sky. A forest floor carpeted with leaves, bracken crackling underfoot, trees looming overhead. This was no tableau framed by a computer screen. Without glasses, headset, or joystick, he was standing in a world expanding and deepening every moment.\n\nHe tried to take it all in, the piles of leaves and patches of snow, the ancient trees, the bright sky where his ceiling had been. A sunny afternoon, a winter wood. This was where he found himself. Literally found himself, his avatar, a knight in chain mail, taking shape before his eyes, no flat cartoon, but a shifting, sculptural figure cast from his own body, conjured like the woodland from a cloud of dust.\n\nHe raised his arm, and the knight raised his in turn. He pivoted, and the knight pivoted as well, so that Aidan couldn't see his alter ego's face. He took a step in place, and the knight began walking through the rustling leaves. As in a dream, Aidan watched himself, his motions fluid, his body long and strong.\n\nPlayfully, motes mapped themselves onto the ordinary features of his room. As his knight walked on, Aidan saw his bed take the shape of a great boulder and then a fallen log. Veiled in cloud, illuminating a vaulting winter sky, Aidan's ceiling light shone with the complexity and brilliance of the sun. The knight was just Aidan's height and build, carrying himself as Aidan did. Light-headed, Aidan watched his knight venture deeper. The trees grew closer, stockading against the sky.\n\nSnap of a twig. An animal. No. Something else. He sensed some creature stalking him. Heard it breathing behind him. No, above him. Something in the trees. He wanted to stop, but his knight kept walking. He sensed the creature coming closer. \"Stop,\" he whispered. Then he stamped his foot. His alter ego stopped immediately.\n\nAidan lifted his arm, and his knight drew his sword. He heard the creature hiss. Snake? Dragon? Spitting monster? He lunged, but he guessed wrong. The thing pounced, screaming, tackling him from behind. Whirling, he fought a leopard, sinuous and dark. He slashed, but could not wound the massive cat. He attacked again, but didn't hit. The leopard sank her teeth into his shoulder, and he saw his own blood showering, drenching his tunic and his arm. Shocked, he fell to his knees and his avatar plunged to the forest floor. The leopard came in for the kill, gold eyes shining, long body undulating. She bit his neck and pinned him down, drinking his blood. She was gentle now, teeth no longer penetrating, claws no longer bared, her tongue almost caressing his raw wound.\n\nAs the leopard lapped him up, he felt himself unmoored. He shed his sword and shield, and then he shed his body, legs, arms, head, torso. Whitening like toppled statuary, the corpse of Aidan's knight lay on the forest floor, but the knight's ghost floated free, a weightless spirit-version of his venturing self.\n\nNow the leopard released him. She seemed to purr as she drew back into herself, and he saw that she'd begun to change as well, black velvet lightening to a tawny glow. At first her spots stood out boldly; then they too began to shift and fade; the animal's gold eyes darkened, her head and body turned elfin, white and delicate, claws changing into tapered fingers, great cat changing to a girl in transparent silken robes.\n\nDaphne's voice. \"Let's go.\"\n\nHe forgot he'd ever called her fake. He forgot his anger altogether. She was not an elf, nor was she a warrior. She seemed herself, luscious and three-dimensional. She had never seemed so real. \"This is the most amazing place I've ever been.\"\n\n\"You haven't even seen the Gates.\"\n\n\"Take me.\"\n\n\"You have to cross the river first.\" She brushed his phantom body with her hand, and he had to imagine what he could not feel. This world could represent the subtlest exchange, a word, a sigh, even a breath, the smallest gesture, the quirk of an eyebrow, the tremble of a lip\u2014nothing was lost, except for touch.\n\nHe couldn't touch her, so he followed her instead. He took the first step, and his knight broke through bracken and forded streams, clambering after Daphne.\n\nGradually, the river revealed itself. At first he saw nothing more than a shimmer between trees. The shining water unfolded like a ribbon, then a banner. As Daphne led him from the forest, the river opened further, a watery valley, a realm unto itself between steep banks.\n\nSilver, heavy, vast, the river looked like liquid mercury, so slow it scarcely seemed to move. Aidan picked up a pebble and tried to skip it across the surface. The rock sank without a ripple. No birds flew overhead, no fish surfaced, no reeds or plants grew on the dull clay bank. Aidan threw a bigger rock. In this water nothing splashed. Absorbing each stone, the sluggish flow healed itself.\n\nWeird river. Amazing place. Aidan drew his sword and dipped it in the water.\n\nEven Daphne gasped as heavy silver wicked up the blade and continued up his arm as well, cloaking him in metal to his shoulder. He dropped his sword on the riverbank and the liquid metal stopped rising. Ghostly still, he faced Daphne with a silver arm. He flexed his silver fingers, clenched a gleaming fist.\n\nDrumbeats. Thunder. An incessant pounding in the distance.\n\n\"What's that?\" It took him a moment to realize the pounding was his mother at the door.\n\n\"Do you have to go?\"\n\n\"No,\" he whispered.\n\nOutside, Kerry watched the strip of light beneath the locked bedroom door. She saw the light brighten, shifting from gray-green to silver, heard her son's whisper and his shuffling feet. No tapping noise. No typing at all. She stopped knocking and stood still, straining to listen.\n\n\"Why not?\" she heard Aidan say, and then in a louder voice, \"I did everything.\"\n\nKerry held her breath.\n\nA long pause, and then he said, \"I can't.\"\n\n\"Aidan!\" Kerry called out. She started banging all over again.\n\nOn the other side of the door, Aidan saw a black speck on the water. Slowly, slowly, a boat emerged, an ancient ship with long oars and a black sail. The oars rowed themselves across the heavy river, but hovered mid-stroke just before they reached the bank.\n\n\"Stay,\" begged Aidan.\n\nLightly Daphne jumped into the boat, which began rowing her away.\n\n\"Come back!\"\n\n\"Let me in!\" Kerry called outside his door.\n\n\"I need more,\" Aidan called out.\n\nNow Kerry couldn't make out the words. She heard her own heart beating faster, but she tried to calm herself. Aidan was in his room. He'd gone to school, as usual. Surely the vandalism was some other gamer's work. Aidan had never defaced anything. She took a deep breath and sat down on the stairs.\n\nOn the other side of the door, on the far banks of the silver river, Daphne told Aidan, \"This time, make it silver.\"\n\n\"I can't!\" he answered, but his knight held out his arms to her.\n\nThe pull and slap of heavy water, the rhythmic stroke of oars. _Make it silver._ Kerry heard and yet she didn't hear. Sitting on the top stair, Kerry leaned her head against the wall.\n\nAt the hospital she had seen parents waiting for diagnoses. Cancer, tumor, genetic disease. When doctors came to speak to them, the parents knew what was coming, but couldn't bring themselves to ask. Kerry had seen mothers do this, holding still, afraid to speak. They weren't cowardly; they weren't lying to themselves. They clung to uncertainty in order to survive. She held still now. She would speak to Aidan; she knew she had to speak to him\u2014but not today. She closed her eyes.\n\nLike a dreamer who didn't want to wake, Aidan played and replayed UnderWorld's opening. The nights his mother slept at home, he slept. The nights she worked, he took the black BoX from his desk drawer under a pile of old school papers. He unscrewed the top, and his room filled with aeroflakes, those lively flecks scattering and gathering into UnderWorld's barren landscape. He played through the night, his movements increasingly fluid, his reflexes faster, his consciousness expanding so that his knight no longer seemed a projection, but a real person. Aidan's ordinary body seemed a dim reflection of his gaming self.\n\nEvery morning before dawn, he closed the BoX. He had to put his whole weight into it when he turned the lid. Yielding slowly, the lid attracted aeroflakes. The silver river faded, its barren cliffs collapsed into themselves, as, like metal shavings aligned by a magnetic force, the particles flew in.\n\nHe closed the BoX, but he could not shut UnderWorld away for long. He explored the river's edge in every light. Darkness, and pale morning, cloudy afternoon.\n\nAidan collected stones and threw them in the heavy water. He tried throwing several at once to watch them sink together. Then, with a stick, he dug a shallow trench in the clay at water's edge. Silver oozed up to fill the hole. He was still trying to find a way across, even though he knew there was no way, unless he obeyed Daphne.\n\nHe decided to do the thing she asked. Sometimes fear caught him by surprise\u2014a falling sensation, just as he was drifting off to sleep. A sudden chill walking to school. He did not change his mind. Instead, he tried to judge his dread dispassionately. His anxiety, he thought, was superficial, like a nosebleed. He looked down and saw his fear, but hardly felt it. Hour after hour, the river in his room worked its strange magic, inciting him to cross.\n\nWonderful to await the next installment of his secret life. He became calm, efficient, pleasant, sitting at the kitchen table, catching up on geometry, glancing discreetly at the clock, which looked like a red apple cut in half, with seeds marking the hours. He solved one problem after another, writing out his answers with a sharp pencil. You're so bright, his mother always said. \"Now is the time,\" Mr. Allan had told him at his college counseling meeting. \"Your test scores are outstanding. If you do your work, you'll have options. You could compete for scholarships.\"\n\nAs Kerry cooked spaghetti and meat sauce, she turned to look at Aidan. \"You see.\"\n\n\"See what?\" Aidan asked.\n\nShe didn't answer. She was thinking, Here you are, working at the table. Here you are, returned to me. Wasn't that exactly what she'd prayed for? She wished. She hoped\u2014and doubted.\n\nAidan kept working, and Kerry talked about how he only had to try, and said the thing she always said about how more than half of life was showing up.\n\n\"How much more than half?\" Aidan asked and she pretended to whack him with her spoon. He looked at his mother with that mixture of love and pity he felt when they were closest, and he had no idea she suspected him. In fact she had spent all afternoon online on gaming-addiction message boards. She had Googled UnderWorld-related vandalism, and found one case in Seattle, one in Austin, in addition to the one in Cambridge.\n\nNow Kerry drained the pasta, and the room filled with steam. She made everyone hold hands to say grace, and that meant Aidan had to reach across to Diana. Kerry bowed her head, but Diana and Aidan kept their eyes on the counter, where the brownies were cooling, half with pecans and half without. \"Thank you for this food,\" Kerry said. \"Thank you for our family,\" and then she added silently, Help me figure out what Aidan plans to do.\n\n\"Mom?\" Diana asked.\n\nKerry looked up, and everybody said amen.\n\nDiana had seconds and Aidan had thirds of the spaghetti, although he picked out all the onion from the glistening sauce. Kerry asked what was going on in school, and the twins said nothing, and they said, stuff. It was an ordinary dinner, with Aidan clearing and Diana taking out the recycling, even as Kerry held up Diana's copy of _The Scarlet Letter,_ with its cover torn away. \"Why do you mutilate your books like that?\"\n\nDiana stood in the doorway holding an overflowing bag of newspapers. \"They're all online anyway.\"\n\nKerry said, \"You're mumbling. I can't understand a word you're saying.\"\n\nAidan said, \"Yeah, Diana,\" who shot back, \"Shut up, Aidan,\" and their mother looked nostalgic because they were bickering again.\n\nBut that night, Aidan lay awake, listening to the sleeping house. He heard his sister snoring in her room next door, pictured his mother buried deep in bed. Fully dressed, he heard dark old beams settling, snow dripping, tiny paws thumping as animals ran across the roof.\n\nHe crept downstairs in his socks, and stepped into his waiting boots. Slipped on coat and gloves, collected the bag he'd hidden behind Priscilla's fermenting compost bin. He felt nothing as he walked to the corner, not the slightest apprehension. Like his avatar in UnderWorld, he moved weightlessly, covering the ground with his long strides. Supernatural, he glided across Broadway, ignoring traffic lights and passing cars. Lightly he hopped the low fence and dropped into the field where he'd played soccer years ago.\n\nExcept for a few patches, the snow had melted, exposing tangled old grass and spongy ground. He unzipped his jacket and crouched low, shaking his paint can. A metallic clicking like ball bearings rolling around. The can was full, and, finger to the nozzle, he scarcely had to press. A silver stream hissed straight from his hand.\n\nQuickly now, he did his work, keeping head and shoulders down, shielding the spray can with his body as he crossed the field. Looping over the patchy field, his letters gleamed like a skater's figure eights. He didn't stop to watch for passersby, nor did he pause to consider his own writing. He felt, rather than saw, that he wrote elegantly, his CUs smooth.\n\nAny second, a patrol car could expose him with white lights. He was just blocks from the police station, around the corner from City Hall, but he felt invincible, as though he drew his own luck, and painted his own rules. He carried nothing in his pockets. He had no backup plans, or explanations. All he knew was that no one could catch him; he was too fast.\n\nHe ran past basketball courts, past the children's playground with its corkscrew slides, all the way to the Koreana restaurant, where he threw the paint can into the garbage bin in back. In the morning everyone would see what he had done, but no would find him. No one could pin it on him. He was long gone. Gloved hands in his pockets, he slipped inside the kitchen door. Athlete, artist, ghost.\n\n\"Aidan.\"\n\nHe froze.\n\n\"Come here.\"\n\nKerry was sitting with her coffee at the table.\n\n\"What's going on?\"\n\n\"Nothing.\" He glanced at the apple clock above her head.\n\n\"What am I going to find out in the morning?\"\n\n\"I don't know.\"\n\nShe forced herself to ask, \"Were you at school tonight?\"\n\nHe told the truth. \"No!\"\n\n\"Are you involved in this new game CU?\"\n\n\"There's no game CU.\"\n\n\"UnderWorld, then. Are you playing?\" He heard the dread in her voice. She might as well have asked him: Are you using?\n\n\"It's not even out yet.\"\n\n\"Don't lie to me.\"\n\nHe looked her in the eye. \"I'm not.\"\n\nAidan saw her tension ease, and knew he'd won. Exultant and forgiving all at once, he felt a wave of love for his mother. Kerry was too tired to fight on, but he would be merciful. He would protect her, even as he deceived her.\n\n\"Don't worry. I'm fine,\" he told Kerry softly. This was an understatement. He was strong, and he was young. Immortal. He feared nothing as he stood on the threshold of his other life.\n\n\u2014\n\nJust before dawn, he opened his BoX and saw the river glimmering in darkness. He had already texted Daphne to tell her what he'd done. Now he waited on the bank, expecting his reward. A million stars appeared, more than he had ever seen on Earth. The night opened, unfolding and expanding all around him, a pocket universe.\n\nThe sun rose in muted colors and the stars began to fade. A milky fog obscured the water, neither day nor night. What did it mean? Gradually, Aidan realized that his closet light had burned out.\n\nIn the half-light at the glowing river's edge, his knight dug up rocks and turned them over, looking for a clue. Some were unyielding; others shattered in his hand. One cracked slowly like an egg. Slime trickled out, and then a ragged tooth emerged, prying the rock open from inside. The hatchling looked like a tiny coiled crocodile, its thorny skin the clay color of the riverbank. Hissing, it uncurled and sprang up snapping, tearing and biting Aidan's spectral self. The monster's little teeth snapped right through Aidan's ghostly form, but could not penetrate his silver arm at all; its teeth clanged instead, metal on metal.\n\nHe slashed the creature's throat with his sword, and a red-black ooze pooled under its stony head. Carefully, Aidan laid out the body full-length on the ground. Like a dancer in front of a mirror, he watched himself as he slit the monster lengthwise and peered inside, searching entrails for some clue or sign. He found something hard, a forked rod, a wishbone. Even as he knelt on the riverbank, he felt the bone drawing him toward water with magnetic force. Then again, he really had to pee! He'd put it off as long as possible, but he had to go, and there was no way to pause.\n\nHe dropped the divining rod and dashed to the bathroom, but the door was locked. Pounding, he called to Diana, and heard her answer, \"Wait.\" Running down to the bathroom near the kitchen, he tripped and caught himself, cursing the narrow stairs and his own feet. It wasn't just depressing; it was disorienting, emerging from the game to clomp down the stairs or stand at the toilet with no defense against the peeling paint, the sharp corners of the real world.\n\nHe dashed back again and saw the upstairs bathroom open and empty. He had left his own door ajar, and on reentry caught Diana standing there, gazing at his river and his sky.\n\n\"Out.\"\n\n\"This is so weird,\" she said. The world was beginning to work on her, the mist of particles, the strange half-light, the slow pull of the river. He closed his door and the game closed around them, the river coming into focus, moving even slower than before, like cooling glass.\n\n\"Out,\" he repeated.\n\nDiana didn't move. She stood there staring at the barren riverbank and shattered dragon egg, and for just a moment he let the game wait. He stood there with her, and they were brother and sister, exploring the basement. They were a pair of eight-year-olds walking to the library. They were the woodcutter's children, and their father had abandoned them.\n\n\"Who are you?\" Diana asked, as she gazed at Aidan's ghostly avatar.\n\n\"I don't have a name yet.\"\n\n\"I thought you pick a name at the beginning when you design your body and all that.\"\n\n\"It's not that kind of game.\"\n\n\"What kind of...\" Diana began, and then finished her own sentence. \"This is UnderWorld.\"\n\n\"Just the demo.\"\n\n\"What do you mean? What demo? Where did you get it?\"\n\n\"A girl I know.\"\n\nDiana folded her arms across her chest, and just like that, the spell was broken. She stood there in her black clothes, a wall of skepticism, a frown on her round face. \"Daphne.\"\n\n\"You're spying!\"\n\n\"I'm not! I overheard.\"\n\nHe grabbed her by the shoulder. \"What do you mean you overheard? You never heard anything.\"\n\nShe wrenched away. \"Let me go. I never said anything. I never did anything to you.\" Diana didn't care anymore about the river in his room; she didn't look twice at Aidan's ghost, or silver sword, or his divining rod, quivering on the bank. \"There is no girl. None of this is real.\"\n\n\"It's real,\" Aidan told her. \"Obviously the game is real. She gave it to me.\"\n\n\"You want that to be true.\"\n\n\"How do you know what's true or not?\" Aidan asked his twin, his doubter.\n\n\"How do you?\"\n\n\"Get out.\"\n\n\"Fine.\" Diana turned to leave.\n\n\"You were never here.\"\n\nDiana shot back, \"Neither were you!\"\n\nAt first he drew Nina from memory. After seeing her, he came back to his room and drew her on the walls, chalking her shoulders, her breasts, her slender waist. He grew so intent on capturing her that sometimes he felt he had to get away from her.\n\nOver days and weeks, imagination yielded to experience. In February Collin began bringing her to his apartment. He stopped worrying about what she would think, and slept with her in his own bed. Now his blackboard-painted walls were covered with Nina asleep, awake, half dressed, turning her head. She was brushing her hair. She was sitting in a chair with the ribbon strap of her slip looped over her bare arm.\n\nHe hardly went to parties anymore. He had stopped drawing backdrops for Theater Without Walls. Darius said, \"Who do you think you are, Edgar fucking Degas?\" Noelle said\u2014\"What? Are you too good for us?\" He spent all his time with Nina. Drew her and erased her, studying her every mood.\n\nBut Nina studied Collin too. She watched him focus on one aspect of her body and chalk it over and over until he could re-create it without a second glance. He learned to draw her profile, and then he used that view next time, generalizing with slight variations. He was amazing and relentless, developing a shorthand for her body and her face.\n\nHe studied shapes and colors the way poets studied words. He kept a bird's nest in his room and a collection of old tools. He had a broken ceiling fan on the floor, a bouquet of shriveled flowers, a crumpled chrome fender, because he liked the way it reflected light. He had a whole library of curling art posters, old masters and fantasy art all mixed together. She saw no books, no tablets, no computers.\n\nOne night as he drew her, Nina asked, \"How do you learn your parts when you perform?\"\n\nWith his foot, Collin nudged a tangle of headphones on the floor.\n\n\"You just listen?\"\n\n\"Sometimes I slow down the audio and write the lines.\" As so often when he was holding chalk, he illustrated his point, writing in fluid script, _We are such stuff as dreams are made on..._\n\n\"How do you do that?\" His words looked like calligraphy. \"My printing is so bad half my students say they can't read the homework on the board.\"\n\n\"You're very trusting.\"\n\nShe said quite seriously, \"I want to be.\"\n\nThen he stood back from the board and wondered at her. She was so earnest. \"What were you like when you were little?\"\n\n\"Sad,\" she told him. \"Worried.\"\n\n\"Why?\" He sat next to her on the bed.\n\n\"I worried I was adopted, and my father was going to give me back. Then I worried I wasn't adopted. I was afraid suicide ran in families, so I worried I would die like my mother did.\"\n\nHe was a little shocked, so he asked lightly, \"Oh, is that all?\"\n\nShe didn't mention that her father had lied about her mother, substituting physical for mental illness. Nina had been almost twelve when she found out the truth. She just said, \"I worried about other things too.\"\n\nShe had been afraid of fire, afraid of thunderstorms, afraid of the ocean, afraid of dark, shadowy sharks that could come ripping at her through the water. She had worried about burglars. Distrusting the security system, she had double-checked the doors\u2014all six regular doors and the French doors in her father's house. \"I was afraid of my father leaving.\"\n\n\"Poor Nina!\"\n\n\"Well, I was lucky too.\"\n\nMoney, Collin thought.\n\nBut Nina said, \"I played my dad's games first.\"\n\n\"Before release?\"\n\n\"Before everything.\"\n\n\"Which ones?\"\n\n\"EverRest. EverSea. That's my friend Julianne. That's her voice when all the mermaids sing.\"\n\n\"Seriously? So when you hear the mermaids you're thinking of her.\"\n\nNina shrugged. The memory was strange, pleasure mixed with anger. Julianne singing in the studio, Nina aware of Peter watching.\n\n\"What's it like to be you?\" Collin asked, playfully.\n\n\"This is me when I was eleven.\" Nina reached for her phone and searched for the image from the game. \"The girl collecting sand dollars in the sea cave.\"\n\nCollin looked at a waif of a girl in a green kelp dress. Her hair was long and shimmering, her face serious, her eyes clear gray. \"You should tell your students!\"\n\nNina looked at him in disbelief.\n\n\"You're like a celebrity!\"\n\n\"No way!\"\n\n\"You're no fun.\" Then again, Collin had always preferred the teachers you could sidetrack: Mr. Dillingham, who talked about running during biology; Mrs. Giannetti, who paused while solving equations to reminisce about her summers with Earthwatch as a volunteer archaeologist. His best teachers had been the most distractible. \"When we did Shakespeare, Mrs. West had us perform scenes.\"\n\n\"She still does.\"\n\n\"You should do that! Come dressed as something. Titania.\" He scrambled to his feet and drew Nina as the fairy queen. His sketch was part Pre-Raphaelite, part 1920s _Vogue,_ white chalk and lavender, deep purple, and long trailing wings. He drew a gossamer dress, draped over Nina's naked body.\n\n\"I don't think so!\"\n\n\"Bring props.\"\n\n\"Like ass ears?\"\n\n\"Let me visit,\" Collin told her. \"I'm a veteran.\"\n\n\"Of what?\"\n\n\"Veteran dramatic educator!\"\n\nShe teased. \"Full tomato.\"\n\n\"Hey, I played Edward Winslow for two seasons!\" He knew his living history. He'd done his time as an interpreter at Plimoth Plantation. Walked up and down in wool breeches and white linen, doffing his hat, cleaning his musket, tending goats, thatching roofs, mending woven eel traps while telling the story of the _Mayflower_ to schoolchildren, old couples, tourists from Singapore, Germany, Japan. \"We used to do class visits all the time.\"\n\nNina tried to picture Collin as a seventeenth-century colonist.\n\n\"The voyage was long and stormy,\" he intoned. \"Yet none but a blaspheming sailor was lost at sea...\"\n\n\"But I'm not teaching Pilgrims.\"\n\n\"I could do Shakespeare, no problem.\"\n\n\"My kids are tough,\" said Nina.\n\nHe scoffed. \"I work in Harvard Square. My art's been peed on.\"\n\n\"Are you serious?\"\n\n\"Twice.\"\n\n\"Ohh.\" She sounded so dismayed. He had to laugh. It wasn't just that she'd never seen such a thing. She had never even thought of such a thing.\n\n\u2014\n\nThe last day before February vacation, he bounded into Emerson, setting off the metal detector. Framed by pulsing orange lights, Collin stood resplendent in plumed hat, brown jerkin and slashed doublet, white shirt with pointed lace collar, sweeping velvet cloak lined with tawny satin, thick brown hose.\n\n\"Good morrow, fair lady.\" He doffed his hat to Nina, as security came running.\n\n\"Okay, let's have the sword,\" the guard said.\n\nCollin's hand tightened on the hilt. \"What, surrender my weapon?\"\n\n\"Yup.\"\n\n\"In death alone,\" said Collin. \"Else forfeit sacred honor.\"\n\nNina was laughing. \"Stop!\"\n\n\"Let's have the sword.\"\n\n\"Officer,\" Nina said, \"it isn't real.\"\n\n\"Zero tolerance,\" he replied.\n\nNina watched the guard march Collin into the office. \"My good Lord DeLaurentis,\" Collin announced himself to Mrs. Solomon, the principal's secretary. \"Is he within?\"\n\n\"Collin?\" Mrs. Solomon strained to recognize the young Emerson alumnus in his Jacobean clothes.\n\nAfter some discussion, he surrendered his weapon to DeLaurentis himself, and signed the school guest book with a wonderful crabbed hand. Name: _Wm. Shakespeare._ Organization: _The King's Men._\n\nThen Collin led the way up the stairs of his old school, striding through corridors where crowds parted before him, kids whistling, laughing, calling to one another, \"Oh, my God, check this out. Check the boots!\" Some thought he was from King Richard's Faire. Others said he was doing the LGBTQ assembly, since he looked like such an amazing queen. Gawkers peeked into Nina's room even when passing time was over. Her students loved it when she closed the door, and William Shakespeare belonged to them alone.\n\n\"Today we have a special guest,\" Nina began, as Collin paced behind her, examining the poster of the old Globe Theatre, experimenting with the light switches, gazing at the ceiling, marveling at the fluorescent glow. \"He is an actor, director, playwright, poet...the envy of his peers\u2014\"\n\n\"I have no peers,\" Shakespeare interrupted.\n\n\"Ahem!\" Nina shot him a look and the class laughed. \"His peers, including...Ben Jonson\u2014\"\n\n\"Prithee,\" Shakespeare asked the class, \"hath anybody seen his work?\"\n\nNobody had.\n\n\"Christopher Marlowe,\" Nina continued.\n\nNone of the kids had heard of him. Shakespeare gave Marlowe the thumbs-down.\n\n\"Beaumont and Fletcher.\"\n\n\"Boo!\" Shakespeare cried, and then to Nina, \"Fair lady, I'm the last man standing! They've heard only of me!\" Rhythmically, he clapped his hands and gestured for the students to join in.\n\n\"He was born in Stratford, and he worked in London,\" Nina said as her kids clapped along with Collin. \"He founded his own theater, the Globe.\" She looked down at her notes. \"He invented the words _radiance, equivocal, lustrous, amazement._ Phrases such as _sea change, all of a sudden, dead as a doornail, in stitches, the game is up, fancy free..._ \"\n\nShakespeare gestured for Miss Lazare to move along, and the class laughed as she cut her introduction short. \"His plays are still performed all over the world. I give you William Shakespeare.\"\n\n\"Can I call you Will?\" Isaiah called out, as the applause died.\n\nShakespeare raised an eyebrow.\n\n\"What do we call you, then?\" Isaiah asked.\n\n\"The Bard of Avon.\"\n\nAfter this answer, bold and pretentious, the class drew back a little. Shakespeare didn't seem to mind. He walked up and down, perfectly possessed, actually enjoying the growing distance between him and his audience.\n\nNina tried to catch Collin's eye, to remind him of the Q&A they'd planned. But Collin ignored her. He gazed coolly at the class, and, fascinated, they stared back at him. Without a word, he attracted their attention, offering no greeting, no information, nothing but suspense.\n\n\"We prepared some questions.\" Nina spoke as much to her students as to Shakespeare. \"Who'd like to start?\" She looked out at the students, issuing a silent appeal. Anyone?\n\nFinally, Chandra asked in her small voice, \"When did you start writing?\"\n\n\"When my players needed parts.\"\n\n\"What's your favorite play?\" asked Isaiah.\n\n\"The one that pays.\"\n\nAnother pause, and Nina gestured to Jonee, who read from her open notebook: \"Who did you write the sonnets for?\"\n\n\"None but you,\" Shakespeare cried gallantly, and took her by the hand!\n\nThe class sniggered. Holding hands with Jonee? She was this large, pale, seriously depressing girl, and there she was, with Shakespeare escorting her to the front of the room, seating her on Miss Lazare's own desk.\n\nKhalil laughed so hard he started coughing. \"And you!\" Shakespeare added, offering his hand to Khalil as well.\n\nThe class exploded with whoops and catcalls, as Khalil shrank back.\n\nShakespeare knelt down on the spot. _\"Shall I compare thee to a summer's day?\"_ He looked adoringly at Khalil's stubble and his dreadlocks, then gazed into the boy's dark eyes.\n\nNo! Nina thought. She hovered behind Khalil's chair, trying to catch Collin's attention.\n\n_\"Thou art more lovely and more temperate...\"_\n\n\"Holy shit!\" Khalil interjected, even as Nina gestured for Collin to stop.\n\n_\"Rough winds do shake the darling buds of May.\"_\n\nThe class was in hysterics.\n\n\"Back off!\" Khalil snarled.\n\nIn vain, Nina told her students to settle down. Unheard, her explanation that actually this sonnet was written for a young man. Doubled over, clutching themselves, kids were falling out of their chairs\u2014all but Khalil, who lost his cool completely, and shoved Shakespeare with two hands.\n\nThe Bard lost his balance, since he had been down on one knee, but he recovered fast, springing to his feet.\n\n\"Let's take a moment here,\" Nina said, trying to intercede, but Shakespeare waved her off and turned toward Jonee, still sitting where he had left her on Nina's gray Steelcase desk.\n\n_\"Sometime too hot the eye of heaven shines...\"_ he told Jonee. You could see all the blood rush to her face as everyone's attention turned toward her. _\"And often is his gold complexion dimmed.\"_\n\nThe class watched in shock. How could the Bard have known that Jonee panicked if you even looked at her? She had a thing where she could hyperventilate at any moment. She could do it now, even as Shakespeare knelt down, velvet cloak streaming around him. He spoke to her as though they were alone. Jonee's eyes widened as he declared, _\"And every fair from fair sometime declines...\"_\n\nJonee gasped, breathing faster and then faster. Her classmates watched Miss Lazare pluck at Shakespeare's sleeve\u2014to no avail. Jonee's cheeks burned redder than her limp strawberry-blond hair. She swayed.\n\nMiss Lazare tried to stand between Shakespeare and his victim. He waved Lazare away. The guy was hard-core.\n\nEven Anton, who usually kept his head down, was on the edge of his seat. Fully recovered from the assault on his sexuality, Khalil grinned in disbelief. This was going to be the most amazing class in the history of the school; Shakespeare could be causing a medical emergency.\n\nMeanwhile, Shakespeare sounded blissed. _\"But thy eternal summer shall not fade,\"_ he intoned, looking deep into Jonee's eyes.\n\nThe Bard got a standing ovation, especially when Jonee flumped off the desk. She would have fallen in a heap, if not for the Bard's steadying hand. Her classmates shouted, \"Encore! Encore!\" Everybody saw the faint smile on Jonee's face as Shakespeare returned her to her seat.\n\nNina saw Collin raise a frenzy and then with a flick of his hand, silence the whole class. She watched him and she thought, You should have it all\u2014money, fame, every success.\n\nCollin himself broke into her reverie. He whipped around and bowed to her. Then, right in front of all her students, he took her by the hand. There she'd been, enjoying the show. Now suddenly she was in it. \"No. No!\" she whispered. Unfair to ambush her like that! \"Collin,\" she begged under her breath.\n\nHer kids thought this was hilarious. \"Pray be seated,\" Shakespeare said, and Lazare was shaking her head, but she had to listen. The guy was a rock star. She took the hot seat atop her desk.\n\nShakespeare didn't kneel this time. Everybody watched as he looked Lazare up and down. She was so embarrassed she couldn't even meet his eye. Whooo! The guy was her boyfriend. The more she tried to hide it, the more it showed.\n\nWith a wave of his hand, Shakespeare signaled his next speech.\n\n_\"My mistress' eyes,\"_ he said, _\"are nothing like the sun; \/ Coral is far more red than her lips' red.\"_\n\nOh God, Nina thought, knowing what was coming, but he recited so well. The kids loved him, and loved her embarrassment even more.\n\nShakespeare took a step back to view her in profile. _\"If snow be white, why then her breasts are dun.\"_\n\n\"Owwww!\"\n\n\"Her breasts are _dung_?\"\n\n_\"If hairs be wires...\"_ \u2014Shakespeare lifted a lock of Nina's hair\u2014 _\"black wires grow on her head.\"_\n\n\"Hey,\" Nina whispered. She had planned a discussion of playwriting and acting, costuming, performing at the Globe Theatre. \"Collin.\"\n\nEither he ignored her or he couldn't hear. He had galvanized the class, now cheering.\n\n_\"I have seen roses damasked, red and white, \/ But no such roses see I in her cheeks; And in some perfumes is there more delight\"_ \u2014the Bard turned away\u2014 _\"Than in the breath that from my mistress reeks...\"_\n\n\"Ewww!\"\n\n\"Death match! Death match!\" the students screamed.\n\nIsaiah called, \"Miss, don't let him tear you down!\"\n\n_\"I love to hear her speak, yet well I know \/ That music hath a far more pleasing sound.\"_\n\n\"Kill! Kill!\"\n\n_\"I grant I never saw a goddess go; \/ My mistress when she walks treads on the ground. \/ And yet...\"_ Shakespeare paused and looked out at the class and they hushed, wondering what he would do.\n\nHe fell to his knees and pulled a single red plastic rose from his sleeve. \"Awwww,\" chorused the class, amused, but also disappointed to see him cave.\n\nSean said, \"Miss? You gonna _accept_ that from him?\"\n\n\"Reject! Reject!\" the class thundered.\n\nNina shook her head. Her kids cheered as she turned the flower down.\n\nDespite this, Shakespeare had the last word: _\"And yet,_ _by heaven, I think my love as rare \/ As any she belied with false compare.\"_\n\nApplause and cheers and school bells ringing. Wait. Those were alarms pulsing in the halls. Kids groaned and started shuffling for the door. As usual, whenever anything fun happened, DeLaurentis hit them with a fire drill.\n\nCollin was still carrying the plastic rose when everybody trooped outside. Students and staff gathered in the designated area behind the school, where Mr. DeLaurentis waited, furious, in his charcoal-gray suit. This was not a scheduled fire drill, so either a real fire was raging somewhere in the building, or some kid had pulled the alarm.\n\n\"Line up! Line up!\" DeLaurentis barked into his electric bullhorn, as students poured out of the building, but no one paid attention. Fire trucks had already arrived, and firemen were tromping into the school with empty hoses trailing behind them. Meanwhile, it was so close to dismissal that the students figured February break had started. _Yes!_ The problem was their teachers had rushed them out the door without coats. They had to huddle and hug each other to stay warm.\n\nLike temporary parents, Collin and Nina shepherded their class to the school's spiked black iron fence.\n\n\"Yo, Shakespeare,\" one of the boys called out, but Collin wasn't Shakespeare anymore.\n\n\"No reentry,\" blared Mr. DeLaurentis. \"No reentry.\"\n\n\"Bard of Avon,\" Isaiah asked, \"you coming back?\"\n\nCollin stole a look at Nina, but she glanced away.\n\n\"Did I say you could leave?\" Mrs. West was confronting a tall blond kid edging toward the street. \"Did I dismiss you, Aidan?\"\n\n\"Come on, Nina,\" Collin said, but she wouldn't speak to him where her students could overhear.\n\n\"Down. No climbing!\" DeLaurentis crackled, even as the student body surged.\n\nNina saw Aidan close his eyes and lean against the fence. There was something grand about his silent resignation, a royal insolence.\n\nKids and teachers waited for what felt like hours. In fact, after twenty minutes, the returning firefighters reported no fire, no smoke, no evidence of faulty wiring. The official cause for evacuation was some student, and DeLaurentis was talking about consequences, but no one stuck around to listen, because the fire marshal had just given the all-clear.\n\nIn the throng of students racing to empty lockers for vacation week, Collin cleared a path, sheltering Nina with his arm. At last he opened the door for Nina and they took refuge in her empty classroom.\n\n\"Phew.\" Collin tried to keep it light. \"Next time we'll do the theater games.\"\n\nShe was picking up stray papers, rescuing paperbacks splayed open on the floor.\n\n\"Nina?\"\n\nShe turned to face him. \"I never said you could recite sonnets to me in front of my class.\"\n\n\"I know, but\u2014\"\n\n\"I asked you to stop!\"\n\n\"You were laughing.\" He crossed the room to make his appeal.\n\n\"Why didn't you listen?\"\n\n\"Sorry. I gave you away. The secret's out. You have a crazy boyfriend.\"\n\n\"I wanted this lesson to be about Shakespeare, not you.\"\n\n\"It _was_ about Shakespeare,\" he retorted. \"And what's the big deal? The kids already know that you're in love with him.\"\n\n\"You don't understand. They hardly listen to me. I've barely got them sitting down.\"\n\nHe tossed hat and plastic rose onto her desk. \"I wasn't trying to embarrass you.\"\n\n\"No. Just having fun at my expense.\"\n\n\"That's not fair.\"\n\n\"Well, what's fair, Collin?\"\n\n\"I don't know.\" He took her hands.\n\nHe was hard to resist; he was so warm. \"I was embarrassed,\" she admitted. \"and I was...\"\n\n\"What?\"\n\n\"Jealous,\" she confessed. \"You're just...surprising. I don't know what to do with you.\"\n\nHe forgot the classroom door was open. He cupped her face in his hands and kissed her.\n\nAs if on cue, some kid whistled in the hall.\n\nThey sprang apart.\n\n\"No more sonnets,\" he assured her. \"No more school visits. I don't want to mess up your life.\"\n\nHer response surprised him. \"I don't want to mess up yours.\"\n\n\"Yeah, I don't think there's any danger.\"\n\n\"I don't want to do the wrong thing.\"\n\n\"Go ahead,\" he said, not knowing what she meant. \"Do the wrong thing. You need the practice.\"\n\n\"You're so talented...\" she began.\n\nNow his eyes hardened. His body tensed as he guessed where this was going. He had listened to this speech from his mother, from Noelle, from every other serious girlfriend, and he felt a kind of grief to hear Nina start in on him now. Two months before, they had been sledding in Danehy Park. Now the long talks had begun.\n\nShe said, \"You could do a lot more, and earn a lot more, and have a career if you want.\"\n\nHis voice was cold. \"I'm not going back to school.\"\n\n\"It's just an idea\u2014and I keep debating whether to tell you.\" She took a breath. \"I could take you to Arkadia.\"\n\n\"Take me?\"\n\n\"Well...introduce you...\"\n\n\"That's what's on your mind. You want to introduce me to Arkadia.\"\n\n\"I do and I don't,\" she confessed.\n\n\"That's what you're debating. Whether to leave me in my poor miserable life, or invite me to the big leagues?\"\n\n\"I didn't mean it to come out that way.\"\n\n\"I know!\" He almost laughed. \"That's the arrogant part. It comes naturally.\"\n\n\"Wait. Let me explain.\"\n\n\"You don't have to explain. I understand.\"\n\nShe persisted. \"Listen.\"\n\n\"I don't need you to find me a job. I'm not your charity. I'm not your good deed, okay? I don't want help.\"\n\nHands on her hips, she protested, \"I let you help me! I let you embarrass me in front of my whole class.\"\n\n\"That was nothing. You're talking about a job. You've got my whole career planned out.\"\n\n\"I don't,\" she said. \"I don't want to change you or mess up your art.\"\n\n\"You've planned so far ahead you're already up to feeling guilty about it.\"\n\n\"Don't tell me what I'm planning.\"\n\n\"Don't tell me what to do.\"\n\n\"I'm not! And don't assume you'd get a job.\"\n\nShe began rubbing out vocabulary words with her felt eraser. \"You're the arrogant one.\"\n\nHe didn't answer.\n\n\"You don't even listen.\"\n\n\"I'm listening,\" Collin retorted.\n\n\"Is it that hard for you to accept a favor?\"\n\n\"Stop.\" He took the eraser from her hand and drenched it with water from the bottle on her desk. Then he wiped her cloudy board until it was sleek and black again.\n\n\"You would blow them all away.\"\n\n\"Let's go.\"\n\n\"Okay.\" She picked up her bag, but then she said, \"Just let me introduce you to my father.\"\n\n\"He won't meet with me.\"\n\n\"Yes, he will.\"\n\n\"Because of you.\"\n\n\"Well, that has to be enough.\"\n\nHe shot her a look.\n\n\"Can't you just take a chance?\"\n\n\"You're not offering me a chance,\" he said. \"You're offering a gift, out of the goodness of your privileged heart.\"\n\nFor a long moment she didn't answer.\n\nYoung as she was, she understood her position. She could teach for two years or she could quit tomorrow. She could travel, study abroad, go to law school, or do nothing at all. She didn't need to earn a living. Nothing kept her at Emerson but idealism and interest. \"It's not a gift,\" she said at last. \"It's not a job. It's just an open door.\"\n\n\"Thanks. I'll open my own doors.\"\n\nShe leaned against her desk. \"And how will you do that?\"\n\n\"None of your business.\"\n\nNina thought of Collin's theatrics, and her students' laughter\u2014his crazy visit. \"That's what I should have told you!\"\n\n\"You're right,\" he said. \"I'm sorry. Let's get out of here and have a drink.\"\n\nEven then, she persisted. \"Just think about it.\"\n\n\"I told you I don't want help.\"\n\n\"I know,\" she said, \"but you deserve it.\"\n\nThose words pierced him. His mother and his girlfriends and his instructors always accused him. He was wasting his time. He wasn't living up to his potential. Nina said the thing he hardly dared to tell himself.\n\n\"What's wrong?\" she asked, searching his bright eyes.\n\n\"Nothing.\" He wrapped her in his cape.\n\nHe held her in his arms and the velvet cape trailed all around her shoulders. Yes, she thought. Yes he would listen to her, but she was half afraid of what she'd done.\n\nIn silence they headed out together. Seriously, almost ceremonially, he took his plumed hat from her desk. She turned off the lights and he followed her out into the hall, where janitors roared up and down with vacuum canisters strapped onto their backs.\n\n\"I can't program,\" he reminded Nina.\n\n\"That wouldn't matter.\"\n\n\"Yes, it would.\"\n\n\"No, no, no. They'll know what to do with you.\"\n\nCollin wore a button-down shirt, blue-striped, clean but wrinkled and papery. Where had he found it? Of course Nina couldn't ask. Her father was sitting next to her, ordering wine, and Collin had been exiled across the table.\n\nThey were eating dinner at Harvest, and Collin was trying not to stare. Viktor was fifty-four, but he didn't look old. He looked like a guy who woke at dawn to bike up mountains and ford icy streams. His eyes were black, his nose craggy. His dark hair and bushy eyebrows stood up as if to say, Who're you calling short? He smiled to himself as though delighted with his own ideas\u2014and why not? They were worth a ton of money. Nina's father set Collin on edge immediately. Collin saw the laughing ferocity in Viktor's eyes.\n\n\"Nina tells me you act and sing and dance and draw, and I don't know what else,\" Nina's father said. \"Jack of all trades.\"\n\nFuck you, Collin thought. \"I draw,\" he said.\n\n\"Nina says you're very good.\"\n\n\"Dad!\" Nina exclaimed. Clearly he hadn't even opened Collin's portfolio.\n\nViktor heard the reproach. Nina loved him, but she judged him. Even as a small girl, she'd studied him, until he'd had to look away.\n\nOnce, when he was leaving for the airport, he'd knelt down to apologize. \"I wish I could stay.\"\n\n\"Why are you going, then?\" she had demanded.\n\nRecently, in the heat of argument, she had accused him of ignoring her. \"You see,\" he'd declared. \"That proves it. You're my conscience.\" She had been his family before he had a family. (It did not occur to him to count her mother.) Once it had been the two of them\u2014Viktor and his gray-eyed child, his smaller, better self.\n\nNow Nina drew herself up, as if to say, This is your big gesture? Agreeing to have dinner? And Viktor was sorry. For a moment he felt guilty, but the moment passed.\n\n\"Where did you go to school?\" he asked Collin.\n\n\"MassArt. But I'm still there. I mean technically I'm\u2014\"\n\n\"Enrolled?\"\n\n\"Well, not currently, but\u2014\"\n\n\"The best ones leave,\" Viktor said. His words were conciliatory, his tone half-mocking. \"The best ones teach themselves.\"\n\nThey ate steaks and drank a dark Bordeaux, and Viktor watched Collin. \"Did you study game design?\"\n\nSlow down, Collin told himself, as he drank his second glass of wine. \"Not exactly,\" he told Viktor, \"but I have a long-standing interest in EverWhen, so...\" The more he drank, the more he found in the Bordeaux. Autumn and dusky stairwells, and dark old jewels and soft lead pencils sinking into blotting paper. He began to feel warm. Viktor was drinking too, but the wine didn't change him; it was Collin who felt overheated, dangerously glib. Pace yourself, he thought, but he kept talking fast. \"I was a gamer when I was younger, and I know the monsters. For example, I can draw every major dragon in EverWhen.\"\n\n\"But not the minor ones?\" Viktor asked lightly.\n\n\"I can draw them too,\" said Collin. \"And the serpents, basilisks, and hydras, Gnomes, Elves, mermaids, bears, wolves...\" Nina was looking at him anxiously, but he didn't stop. \"Birds of prey, owls, phoenixes _..._ \"\n\n\"In other words\u2014\" Viktor began.\n\n\"Everything,\" Collin cut him off boldly.\n\n\"Excellent,\" Viktor said, without a flicker of surprise. Some fans were like that. Everheads programmed their own game mods, copied screenshots, directed their own films. Arkadia kept online galleries of their art, thousands of drawings and paintings. Viktor appreciated these tributes, but they didn't excite him. He studied optics, graphics, vision, the interplay of imagination and perception. He lived for innovation, not obsessive imitation. \"I'll tell you where to send your stuff.\"\n\nCollin reminded Nina's father, \"I've already sent my stuff to you.\"\n\nNina, Viktor thought. What have you been promising this kid?\n\nShe shot him a look that meant, Stop! You're overbearing and dismissive.\n\nViktor was hurt. He wasn't dismissive; he saw through people quickly. He wasn't overbearing; he was busy.\n\n\"Can I get you some dessert?\" the waiter asked.\n\nViktor said, \"No, thank you.\"\n\nCollin glanced at Nina, who sat with hands clasped together on the table. So much for her sweet, arrogant idea. She knew her father, but Collin knew something about sending unsolicited portfolios. He might have reached across or smiled to reassure her. I don't care; it doesn't matter. The wine was good and I forgive you for the rest. But at that moment Collin cared greatly. He felt an intense desire to prove himself, as soon as he realized he didn't stand a chance.\n\n\"Coffee?\" the waiter suggested.\n\n\"Just the check,\" said Viktor.\n\nBut Collin told the waiter, \"Just a pen.\"\n\nWhen the check arrived, Nina's heart stopped as Collin snatched it out of Viktor's hands. Without even glancing at the numbers, he flipped the little paper over and began drawing on the back, working over the entire surface with the restaurant's black ballpoint. The waiter returned long before he finished, and Viktor had to request another check. He said, \"We're having this one embellished.\"\n\nCollin kept his head down, scribing the paper, so small and flimsy, cross-hatching his shadows, exhausting the pen's ink supply. Viktor watched with mild interest. Nina held her breath.\n\n\"Here.\" Unsmiling, Collin handed Viktor his drawing.\n\nViktor squinted at the drawing, holding it close and then farther away.\n\nNina plucked Viktor's reading glasses from his breast pocket. \"Put these on.\"\n\nViktor sensed his daughter's eagerness, her tremendous hope as the drawing came into focus. \"Look at that,\" he said gently. \"It's the ouroboros.\"\n\nCollin had drawn the dragon exactly as he appeared in EverWhen. Slinky, snarky, with evil needle-teeth. Somehow, even in ballpoint, the ouroboros took on a silvery sheen, scales delicately rendered, claws distinct, serpentine body curled around a treasure chest overflowing with gold coins. The dragon's head was long and vicious, jaws bloodied, eyes rolling backward in ecstatic pain as it devoured its own tail. Collin had used every millimeter, puncturing the paper more than once.\n\nNow, as Viktor held the drawing between his thumb and forefinger, he saw Collin for the first time. Here was a young man who could dash off a perfect dragon in five minutes, drunk. This was a prodigious act of illustration\u2014not only lively but anatomically correct. Did Collin have total recall of the dragon's five claws? Had he studied the beast's twenty-one spikes descending in size down his back? Viktor looked over the top of his reading glasses at Collin. \"What other monsters did you say you draw?\"\n\n\"All of them.\"\n\nViktor considered his own empire and the myriad creatures in it. No one could remember all of them, let alone draw each one from memory.\n\nEven so, Collin boasted, \"I can draw anything.\"\n\nViktor smiled as he studied the fierce dragon in his hands.\n\nCome on, Dad, thought Nina. Say it. He's an artist.\n\nViktor made her wait. He loved pleasing Nina. Pretending he cared about her friends was much less satisfying. Even so, he liked the dragon's looping body\u2014back arching, scales spiking. Collin had caught the monster's self-destroying spirit. \"It's good,\" he said, at last.\n\nNina was almost too glad. \"It would have been even better with a decent pen.\"\n\n\"I don't like pen in general.\" Collin's face was flushed. \"I like to work in chalk. Chalk is pretty much my forte, because you can do so much with dust.\"\n\n\"I can understand that,\" Viktor said. \"I do a lot of work with dust.\"\n\nCollin nodded. \"I like smudging colors, and layering.\"\n\n\"I want to show you something,\" Viktor said.\n\n\u2014\n\nThey drove out to Waltham in Viktor's little BMW, Nina and her father in the front, Collin folded queasily in back. The car was hardly meant for passengers. Hostages, maybe, with their legs trussed up around their ears.\n\nIt was a wet February night, as they sped past glass hotels and low-slung office parks. Viktor was talking to Nina in a low voice and Collin couldn't hear the conversation. He felt like cargo, until Nina turned around to look at him. She looked excited as a child.\n\nWhen they arrived, Collin saw that Arkadia had grown since he and Darius had visited as kids. Like a space colony, its polygonal buildings extended on and on into the night.\n\nThere were the usual glass doors and guards. There was a guest book, and Collin got a sticky name tag printed VISITOR. There were desks and workstations. There was a glass atrium set up as a caf\u00e9. However, on examination, every ordinary feature seemed a little strange. Glass doors darkened as visitors passed through. A life-size sculpture of Toth, the mountain king, loomed over the salad bar, commanding attention with his bear's head and great clawed paws.\n\nViktor led the way through clusters of cubicles. Corporate enough, but as they walked, the cubicles grew larger, and their gray walls taller. It was like entering a forest. With each step, Arkadia grew darker. Programmers clustered at monitors like moths to flames. As his eyes adjusted, Collin saw bits of EverWhen on each monitor, fragments of the Trackless Wood, Elves battling tarry monsters.\n\n\"Where's Peter?\" Viktor asked. Despite the late hour, Arkadia was full of people. It might have been the middle of the day. \"Anyone seen Peter?\"\n\nEmployees looked up, startled by the sudden visit. They seemed almost afraid to answer. \"I saw him heading over there,\" one woman ventured.\n\n\"Hello?\" Viktor was standing before a self-contained room, a windowless cabin in the darkness. He opened the door to a cube insulated and baffled with wavy black foam, a music studio dominated by huge black speakers. There were multiple electric guitars, black piano keyboards, giant computer screens. A burly, bearded man was sitting there, reading a printed book to a little boy in pajamas, visible on his monitor.\n\n\"That's Nicholas,\" Nina whispered to Collin. \"He's a sound engineer.\"\n\nNicholas spun around in his swivel chair to greet them. He wore a Jerry Garcia T-shirt and his voice was husky. He looked like a retired football player, and he sounded like a rocker before his first cup of coffee. \"Bedtime story for my kid.\"\n\n\"Go on, go on!\" Viktor encouraged him.\n\n_\"Of course the Neverland had been make-believe in those days,\"_ read Nicholas, _\"but it was real now...\"_ He waved as his visitors backed out, shutting the heavy door behind them.\n\nStranger and stranger, Arkadia glowed with Whennish light. Viktor led Collin and Nina into a misty corridor. Ethereal shapes appeared, insubstantial from a distance, overwhelming up close. Collin flinched as a dark leopard approached with golden eyes. A few steps farther, he nearly tripped over a bloody giant lying at his feet. Craggy mountains rose up in the distance. He could see an EXIT sign and then the outlines of a door embedded in the rock face, but the door itself was bursting into flame.\n\n\"Nina,\" Collin whispered. Dazzled, he was looking everywhere at once, but she looked at him alone.\n\n\"There's so much more,\" Nina told him.\n\nCollin watched in awe as Nina walked through each illusion. She was the magician's daughter, and mountains shattered, dragons shrank before her. Monsters turned to dust motes in her wake.\n\nThey walked into another dark space, a labyrinth of tall black walls, each workstation a peephole, a glimpse of river, a dark cavern, or white bats. Curious, the nearest animators glanced up at Viktor. \"These are our hellions,\" Viktor said. \"And this is Peter, their developer.\"\n\nA tall man pushed his swivel chair away from his desk.\n\nPeter was ten years younger, his brother's partner, but not quite his equal. Viktor's technology drove the company and much of the marketplace as well. Peter's role as developer, scheduler, manager, and coordinator of every team at UnderWorld was secondary. Nevertheless, Peter had powers of his own. Storyteller, Gorey winner, he was Arkadia's chief geographer, historian, world-builder. The press called him the Dark Lord, while Viktor was simply CEO.\n\nCollin was amazed by Peter's height, his leonine body, his long dark hair and glowing eyes, more gold than brown, like liquid amber. Collin couldn't help staring, but Peter looked at his niece alone.\n\nViktor said, \"Here's our artist, Nina's friend.\"\n\n\"Any friend of Nina's,\" Peter said.\n\nNervous, Nina tried to read her uncle's face, and he gazed back, amused. He was mesmerizing, strange. He had performed card tricks for Nina when she was small. Coin tricks, sleight of hand. He had told stories of blood and magic, seven brothers sewing wings into their flesh as they turned into swans, princesses dismembering frogs. He had taught her the violence in fairy tales, and the cruelty of dragonflies. If he built a sand castle with Nina, he'd show her how to build a siege ramp too. And then there came a time when she rejected him. She had watched him enchant young artists and then throw their work away. She had seen him in the recording booth with Julianne. He had never touched her friend, but she'd caught his predatory gaze.\n\n\"He's not just any artist,\" Nina said, and Peter heard the warning in her voice: Don't hurt him. Don't dismiss him.\n\nPeter turned to Collin. \"What kind of artist are you?\"\n\n\"Any kind you want.\"\n\nYou're cute, Peter thought. Eager, insubstantial. \"What if I want Rembrandt?\"\n\nHis questions nettled Collin. They were insistent, but idle as well. Peter leaned back against the edge of a cubicle and his whole body seemed to say, I'm already bored with you. \"I couldn't _be_ Rembrandt,\" Collin said, \"but I could draw a Rembrandt.\"\n\n\"So you're not an artist. You're a copyist.\"\n\n\"I'm both.\" Asshole, Collin added silently.\n\nPeter led the way to a huge whiteboard. Covering half a wall, its surface was adorned with diagrams and flow charts, doodles and sketches.\n\nCollin frowned. He hated dry-erase markers, their scanty ink, their faded colors and anemic lines. He was up for anything\u2014but how could he show Peter what he could do?\n\nPeter seemed to read his mind. He dipped his finger in the aluminum chalk tray at the bottom of the board, and the white surface, along with all its graffiti, disappeared, changing to pure glossy black. Then he handed Collin a pair of plastic styluses, one thick, one thin. \"You've got your colors here.\" Peter showed Collin the array of colored squares in the chalk tray. \"Just dip the stylus.\" Peter demonstrated with a quick sketch of the girl from EverSea. Collin recognized Nina immediately, ten, maybe eleven, with her hair falling over her shoulders.\n\nNina had never told Collin that Peter could draw. The sketch was simple and unshaded, just a line drawing in silver, and yet it conveyed a kind of magic. Peter's hand was so light, the expression on Nina's face so tender.\n\nPeter cleared the board with a brush of his hand. \"Okay, let's see what you can do.\"\n\nTentatively, Collin touched the electronic chalk tray with the thin stylus and saw the tip turn green. He touched the board and left a dot. As he applied pressure his dot expanded into a green pool, a lake.\n\nHe had never played with an electronic board like this. He drew one line and then another. He scribbled with the wireless chalk, and the board picked up his slightest gesture, responding to his every touch. He could dip into any of a hundred colors, and try a thousand shades. He met with no resistance, and no crumbling. With ordinary chalk he would layer, fuss, and wet his sticks to produce saturated color; here his tints were luminous each time.\n\nLines came fast; color flowed endlessly. The trouble was the surface felt so slick. He was like a runner trying ice skates for the first time. He could not control his strokes. Every time he tried, his stylus glided out from under him. He had to erase, brushing away his blunders with his hand. \"Shit. Sorry!\" he murmured. \"I'm not...\" He tried again, and then again, and all the time he sensed Peter growing colder.\n\nNina turned on her uncle. \"Why can't he have pen and paper?\"\n\n\"What are those?\" Peter replied.\n\n\"Take your time,\" said Viktor, enjoying the sport.\n\nCollin knew he couldn't take his time. He had to figure this out _now._ He had about half a minute before Peter lost interest altogether.\n\nShut them out, he thought, as he glared at the glossy board. Viktor, Peter. Even Nina. He had to forget her hopes for him. Lighten your strokes. Limit yourself. Keep the stylus under you. Don't overdraw.\n\nHe began an easy dragon, an ordinary fire-eater with rolling eyes and iridescent wings. He drew the dragon big, working its undulating body across the length of the board. Then he drew a dragon's nest. He didn't try for every detail; he practiced with the thick stylus until he had the nest just right. Now he drew a dragon with its breath aflame. With his thin stylus he added scales and claws, jagged, sooty teeth.\n\nBuilding confidence, he drew faster. A phoenix swooping through the air. A silver falcon. Then, tiring of birds and flying monsters, he brushed them away with his arm, erasing with his shirtsleeve.\n\n\"Oh.\" Nina sighed. Collin's drawings bloomed like fireworks, dazzling and brief.\n\n\"Don't worry,\" Viktor said, because the board saved each image, captured every stroke.\n\nCollin drew Gnomes and Fire Elves, forest creatures, deer camouflaged in trees. He still slipped, but he corrected quickly. He understood the surface now. With each drawing, Collin grew bolder; his work grew more precise. The board changed into a shimmering landscape and the room began to change as well. He sensed hellions gathering to watch.\n\nPeter stood among them and he felt a rush of jealous pleasure\u2014surprise at Collin's skill, admiration of his line. When Peter glanced at Nina she met his eye as if to say, You see?\n\nHellions stood in silence as Collin drew a riderless horse, a stallion tossing its long mane and tail. He didn't know there was a horse in UnderWorld. He only knew he had an audience.\n\nHe drew the horse huge, devouring the wall. He was working freely now with his whole arm. Galloping across the blackboard, Collin's horse was fearsome, and also strangely beautiful. In silver lines alone, in two dimensions only, Collin animated the horse's corded muscles, its powerful legs, and flying feet. Intent on his work, he couldn't see Nina's rapt expression or Viktor's triumphant smile. He saw none of this, but he sensed Peter drawing closer. Now I have you, Collin thought.\n\nWhen Collin finished he stepped back amid a rustling, a whispering from the crowd. A mix of admiration and foreboding, because Collin's horse was better than the one in UnderWorld. More powerful, more dynamic, and subtler too. The horse they had appeared cartoonish in comparison. The hellions knew, before Peter said a word. They saw fresh art coming, long days and sleepless nights ahead. They saw it all, even before Peter pointed to Collin's stallion and said, \"I want that.\"\n\nWhen Collin started working at Arkadia, he collected enough company T-shirts, caps, ear warmers, and fleece vests to outfit everyone in Theater Without Walls. He got his own Arkadian backpack and water bottle and high-fidelity headphones for blasting music late at night. He had never seen so many gaming toys\u2014not just electronics, but miniatures of every beast and warrior. He could have played for hours with the Elves, no bigger than toy soldiers, but far more beautiful, with their blue hair and meticulously painted clothes. Some carried longbows, some knelt to shoot atop computer monitors, some guarded keyboards, brandishing their needle swords. Once, at night, he stumbled upon an entire squadron lined up in formation on the floor.\n\nEach day, Collin discovered something new, a funhouse mirror, a hall wallpapered in a skull print. Ping-Pong, foosball, mini basketball. He found a cache of real skeletons, an art book on Michelangelo, a war room with a map of the world projected on the wall. In this map's glow, a handful of troubleshooters manned workstations, watching for power outages and natural disasters that could short-circuit Arkadia's network and interrupt players' negotiations and alliances, their qwests and feuds.\n\nThe place was scientific and theatrical. Meetings addressed the physics of an avalanche, the look and feel of UnderWorld's caverns, the speed of flaming arrows, the way a castle might explode. Testers reported back on lags. For example, when you were mowing down your enemies in battle, you wanted them to fall before you instantly. Death throes were fun to watch when you were fighting one-on-one, but in the aggregate they dragged. Speed, fluidity, efficiency: These weren't just computational problems, they were artistic problems too. You could accelerate a war with visual shortcuts. \"Know your history,\" Peter said, and he showed classic sequences from World of Warcraft and Call of Duty and Grand Theft Auto.\n\nA long-haired archivist named Robbie presided over a cache of antique Game Boys, X-Boxes, and arcade consoles. He told Collin, \"I played Pong before you were born.\"\n\nTempted, Collin studied the library of boxed classics, everything from American McGee's Alice to the Bitmap Brothers' Z. Myst, with its subtle island veiled in cloud. Reluctantly, he turned away. He saw how people worked, watched Peter deliver schedules and agendas. UnderWorld was a vast construction project, a virtual cathedral. Animators, modelers, and programmers all scrummed together in small pods, and together the pods built up the game.\n\nHe had signed and initialed a thirty-page contract filled with references to company ownership, licensing, and intellectual property. He had taught his final class at Broadway Bicycle, and waited his last tables at Grendel's. Samantha presented him with her bartender's business card. \"Hey, think of me for parties.\"\n\nParties! He wasn't thinking about parties. He was living, breathing, dreaming horses. He drew them in silhouette, in small black thumbnails, racing, turning, leaping on his slate. Peter judged each variant\u2014one small and muscular like a mustang, one massive, one slender, built to run. He chose elements he liked and Collin worked up a detailed study, a horse noble but also wild, with dark, rolling eyes, broad shoulders, nervous ears, rough mane and tail.\n\nIf Peter approved an image you were golden. More often, he dismissed his artists' work. These were not quiet critiques, nor were they cushioned with encouragement. Peter used group meetings to tear into artists, ripping them apart. Hellions called it getting drawn and quartered.\n\n\"I didn't think this could get worse,\" Peter told an artist named Akosh. \"Somehow you found a way.\"\n\nCollin watched with twenty others as Peter stood at the whiteboard and destroyed an entire winter landscape, gorgeous trees knee-deep in snow. He slashed the forest through with one black stroke.\n\n\"It's pale, it's soft, it's _pretty,_ \" Peter said, and Collin saw that \"pretty\" was a felony.\n\n\"As for you...\" Peter cleared the board and brought up a sketch by an artist named Obi. \"You call this a knight? I asked for rust. I told you his sword is filthy. What is this shit?\" With his stylus Peter roughened armor, bloodied the knight's gleaming weapon.\n\n\"I'll change it,\" Obi said.\n\n\"Don't change it. Start over.\"\n\n\"Okay.\"\n\nObi's conciliatory tone only irritated Peter further. \"Okay. Okay, I'll change it. _No._ Don't tell me what you think I want to hear. Do it right the first time.\"\n\nObi ventured, \"I thought you wanted the knight first and then we'd modify it.\"\n\nPeter didn't answer this. He stood there staring at Obi until the hapless artist had to look away. \"I'm not interested in what you thought,\" Peter said at last. He seemed at that moment a monarch, denying all history and memory. Nothing mattered but his current inclination.\n\n\"He's harsh,\" Collin told Nina.\n\nHer reply surprised him. \"That's good. You know where you stand.\"\n\n\"Have you ever heard him?\" Collin asked in disbelief.\n\nShe nodded. \"Praise is worse.\"\n\n\"Why?\"\n\n\"That's where he messes with your mind.\"\n\nIn fact Peter had singled Collin out for commendation. \"Okay,\" he said when Collin showed him his new horses. Collin could scarcely believe it. Compared to what he'd heard, \"okay\" seemed a precious gem. He treasured that single word for days. \"Could be interesting,\" Peter added on another occasion, and Collin wanted to leap with joy. He remembered Nina's words, but Peter didn't lavish praise on Collin. Instead, he lavished time.\n\nPeter began watching Collin work, and even guiding Collin as he drew. \"More sinew. Ears back. Neck outstretched,\" he directed, as Collin made mid-course corrections.\n\nOther hellions watched Peter work with Collin. True, Collin stood out as a draftsman. Some artists could talk about procedural shaders and facial animation systems. Some could sculpt a gorgeous scene with aeroflakes. But nobody could draw like Collin. As Peter's favorite, he didn't win friends, but gradually he earned respect. Arkadians would crowd around at lunch to see his horses, and he began posting sketches on his cubicle walls.\n\nIn those early days he took his slate and stylus everywhere. He worked at home and at lunch, and even on the company shuttle bus. In the seat next to him, Collin's Brazilian scrum master, Tomas, watched a horse emerge white against the black surface of the slate. Behind them, Akosh leaned over the back of Collin's seat, as did Obi, whose chestnut hair flowed almost to his waist, in the style of the Elvish kings. Only one person on the shuttle held herself apart. It seemed a matter of pride. She wouldn't let Collin catch her watching.\n\nHer name was Daphne. She had just graduated from Full Sail University, and she worked in marketing. Her face was impudent, her dark hair cropped short. Her eyes were pure blue, her expression mocking. She wore black jeans and a CU sweatshirt, so she was all covered up, whether at her desk, or gaming late at night. Only her hands showed, and Collin found himself gazing at the flowers tattooed on her wrists. The petals were finely drawn, as if they'd been scribed in India ink. He wondered who had done the work, and if the design continued up her arms.\n\nDaphne wasn't a programmer, and she wasn't an artist, and she wasn't really a tester, but she was a brilliant gamer. Like a chess master, she could play multiple games at once. She would stand in a pool of light with her avatars lit up around her as she showed off unreleased UnderWorld to the select few\u2014those beta gamers she found most extreme, all kids, all boys.\n\nFluid as a dancer, she darted in and out of worlds. No hellion could beat her in a duel. Nobody even came close, and Collin loved to watch her take her colleagues down. She took such pleasure in it; you couldn't begrudge her gloating. Sometimes she would take a bow. Peter liked to watch as well, and this was the other thing that fascinated Collin. Peter was the only one who rattled Daphne. In his presence her whole body tensed. She shot her virtual arrows and she missed. No one was immune to Nina's uncle. No one except Nina herself\u2014but then she didn't work for him.\n\nHellions feared Peter, but they also worshipped him, waiting for him as he strode the halls, following him with questions. They competed for answers and decisions, but most of all for Peter's sketches. His vision was dark and strange and always new. He never developed drawings; he left his sketches bare and suggestive, a landscape ever so subtly wrong, an imminent nightmare. These were the challenges he set his artists, and then he wondered why they couldn't read his mind.\n\nCollin began to see what Peter was about. Nina's uncle detested all things shiny, perfect, new. He was building a world of darkness, decay, infection, and Collin adapted quickly to this vision. He drew Peter to him with his facility and speed, and soon Peter was thinking out loud while Collin took down ideas on his slate. \"A white dragon,\" Peter said. As Collin sketched, Peter ordered, \"Keep the wings and lose the scales. No. Scales on the body. Open wings. Keep opening.\" Collin drew wings opening big enough to fill a room. \"Bigger,\" Peter said. \"Wings big enough to block the sun.\" An hour passed, but it felt like just a minute. Collin's drawing evolved that fast.\n\nThen Peter lost interest and demanded something new. \"Two horses,\" he said, as though he were ordering at a restaurant. When Collin drew the horses, Peter rejected them instantly. \"What are they, twins? Nobody can tell them apart. Again.\"\n\nCollin looked up, startled.\n\n\"I said draw them again.\"\n\nPeter never let up, but Collin realized this was a sign of favor. He had won Peter's attention.\n\nAll the others noticed. Jealous, they watched Collin draw for Peter. Even Daphne gave up acting cool and hovered, challenging Collin for Peter's time. She caught Peter's attention when she reported news of her campaign. \"We've got CU trending now.\" She held up a tablet, offering him fresh numbers.\n\nWhenever Peter worked with Collin, she waited on the periphery. Eventually she began looking at Collin's work. She started glancing and then she began watching him at night when she thought he wasn't looking. At last she scooted her swivel chair to his workstation to gaze at the horses on his monitor.\n\n\"Seriously?\" she said under her breath.\n\n\"What?\"\n\nHe watched her struggle with herself. She wouldn't compliment him. No, she would not give him the satisfaction. \"Is that all you can do?\"\n\n\"Jealous?\" Collin teased.\n\n\"You wish!\"\n\nHe didn't know Daphne, but her trash talk amused him. He felt chosen by them both\u2014Peter the master, and Daphne the marketer. But they were different. Peter demanded while Daphne teased. Peter pushed while Daphne bantered. There was something hard about Daphne, and also sweet. He felt at ease with her, and confused by her as well. He couldn't tell what she really thought of him.\n\nAlmost unconsciously he began sketching Daphne on his electronic slate, covering the surface with her clever face, her cropped hair and wide blue eyes. He drew Daphne working at her terminal with her legs tucked under her. He drew her in Elvish guise, delicate in thin draperies, and then he sketched her gaming like a boy, brandishing an imaginary sword.\n\n\"What are you doing?\" she said.\n\n\"Nothing. Drawing you.\"\n\nShe drew close and closer. He could feel her quick breath. Then she brushed his slate clean with the cuff of her sleeve. \"I never said you could.\"\n\n\"I didn't know I had to ask!\"\n\n\"Yeah, I'm copyright.\"\n\n\"You can't copyright yourself.\"\n\n\"Well, you have to ask permission.\"\n\n\"Why? I don't need permission to look at you,\" Collin pointed out.\n\n\"Yes, you do.\" For just a moment, she covered his eyes with her small hand.\n\nHe laughed.\n\n\"What?\" she demanded, mock seriously.\n\n\"You really want me to stop drawing you?\"\n\nShe had a smile too quick to capture. Her eyes shone with fun. \"I want to play with you.\"\n\nThe first time they fought in EverWhen, she was a Fire Elf and Collin was a Forest Elf. They fought with spears, and in two blows Daphne cut Collin down, dismembered him, and left him for dead.\n\n\"Once more,\" Collin said.\n\nThey fought with cudgels and he landed only one glancing blow before she knocked Collin to the ground and brained him.\n\nThe third time they clashed swords. They stood on the banks of a crystal stream, and Collin forced Daphne into the water, but once again she was too fast for him. In one swift move, she slashed his arm and then drove her blade through his Elf's chest. The purling water turned blood red.\n\n\"Just one more time,\" he said.\n\n\"Go practice.\"\n\nHe felt a flash of anger, but it lasted only for an instant. Luminous in the Arkadian darkness, she disarmed him.\n\nDaphne was like girls he had once known. A little selfish, a little dangerous. She liked to play; she liked to drink. After long days, Obi and Akosh and Collin and Daphne would go out drinking after work. \"The bars of Waltham,\" Daphne intoned. They would sit together at O'Riley's, joking, and it was like the old days, when he could leave his work behind, and no one wondered where he was at night. Not that he wanted his old life. He knew the difference between odd jobs and full employment. He understood the difference between lust and love, and he wanted what lasted. Of course he did. But the \"of course\" part rankled. How strict and narrow real life turned out to be. Of course you wanted a career. Of course you wanted to be with the girl so much better than you it wasn't even funny. But he missed being funny and stupid and irresponsible.\n\nHe was Collin, even in Arkadia. Especially in Arkadia. He got drunk enough to flirt with Daphne. He drank enough to be himself, pushing her sleeves up off her wrists, catching a glimpse of ink. Most people flaunted their tattoos. Strangely modest, or perversely teasing, Daphne kept hers covered up.\n\n\"What kind of leaves are these?\" He studied her forearms.\n\n\"Laurels,\" she said. \"Duh.\"\n\n\"Show me.\"\n\nShe slipped off her barstool and said, \"You'd have to beat me first.\"\n\nThen Obi and Akosh were laughing as though she'd said something witty, but she meant it, and handed Collin a pool cue. He wasn't bad, but she was lethal with her angles, maddening, and also cute, undeniably funny when she triumphed over him. \"Ha!\" She raised her cue like a Whennish spear. \"I'm so much better than you!\"\n\n\u2014\n\nOver days and weeks, Collin drew a hundred horses. Hoofbeats thundered in his ears. Even when he closed his eyes for a few minutes, he dreamed of horses. He slept on the hellions' black leather couch and saw his horses racing on the beach, kicking sand in the salt air. Then he dreamed of Daphne\u2014a strange Arkadian dream. Leaves unfolded on her slender arms, stems and tendrils crept over her neck. Shocked, he watched her disappear. Her ears changed to budding twigs, her nipples hardened into berries. Her limbs were smooth and silvery, her toes rooted to the ground. Those were not her eyes anymore, but birds sheltering in her branches. That was not her mouth, but a dark nest with fledglings where her tongue had been.\n\n\u2014\n\nGradually Nina saw a haze come over Collin. That spring he was exhausted. Not just a little sleepy, spent. Often when they met for dinner he was too tired to talk, certainly too tired to draw on kraft paper. Sometimes he was too tired to eat. They met at Grafton Street and ordered a Margherita pizza, but his slices grew cold.\n\n\"How's your hand?\" Nina said.\n\n\"It's fine. It hurts.\"\n\n\"Is it hurting now?\"\n\n\"No. It's not too bad.\"\n\nShe said, \"We've got spring fever at school.\" She had been teaching William Carlos Williams to her eleventh graders. \"This Is Just to Say\" and \"Burning the Christmas Greens\" and \"Spring and All.\" \"My student had a baby,\" she told Collin. \"Brynna had a baby girl.\"\n\nCollin smiled. Nina was so beautiful and he was so relieved to see her. She was like home; she took him back.\n\n\"Collin?\"\n\nHe'd drifted off only for a moment.\n\nNina said, \"You can't work all day and all night too.\"\n\n\"I don't.\"\n\nShe leaned across the table. \"Don't let them wear you down.\"\n\n\"Don't lecture me.\"\n\nThat offended her, not just his words, but his cold tone.\n\nIn silence, he paid for dinner. In silence, Nina handed him the leftover pizza in a box. When they walked outside the night was misty, fogging the windows of Harvard Book Store. The spring air woke him, but Nina was angry. He knew it, even though she didn't say it. He knew because she didn't speak.\n\n\"I wasn't lecturing,\" Nina said at last.\n\n\"Yes, you were. That's what you do.\" He had to get away; he had to rest. He would have canceled dinner, except that he'd have disappointed her.\n\nShe looked at him and sensed something was wrong. \"You never draw anymore.\"\n\nHe said, \"I draw all day.\"\n\n\"And you're tired of it now.\"\n\n\"No, I'm just tired.\"\n\n\"You should have said you were too tired to meet me.\"\n\n\"You'd be upset.\"\n\n\"Like I am now?\"\n\n\"Yeah, because I never see you anymore, or I'm working so much that I ignore you.\"\n\n\"I never said that.\"\n\n\"That's what you were thinking.\"\n\n\"Don't tell me what I've been thinking.\"\n\n\"Don't make such a big deal out of everything.\"\n\nTears started in her eyes.\n\nWhat am I doing? Collin thought. Why did he want to hurt her? \"No,\" he said softly.\n\n\"No what?\" Nina asked.\n\n\"Just don't cry.\"\n\nShe took a long breath.\n\n\"I don't forget you. I could close my eyes and draw your face.\" With his finger he traced her profile on the frosty bookstore window. With one curving line he captured her forehead, her straight nose, her decided chin, her long neck. In one more stroke he drew her hair flowing down over her shoulders. \"You're the one who can't remember.\"\n\nShe gazed at her image, a portrait in two lines, the work of a moment, subtle, ephemeral. Collin had caught her likeness, but she found him in the glass as well. She saw his easy grace, his quicksilver imagination. \"I do remember,\" she said. \"That's why I miss you.\"\n\nNow he was creeping closer to the Gates. Digging into sludge with dragon's bone, Aidan's knight caught a glint of gold. He knelt to unearth a token for the ferry, which began gliding toward him, drawn by the bright coin. At last he boarded, crossing to the darker shore.\n\nDaphne didn't show herself, but he heard her voice as he jumped off the boat into thick mud. \"Look down,\" she called. \"Look down.\"\n\nAidan stared at his knight's legs. Leeches, long, black, and gelatinous, had torn his leggings and his boots away. He pulled them off, one after another, flung them away, but they snapped back, wrapping themselves around his arms and neck. Each place they sucked, an ooze of silver flowed and blackened. He dripped with tarnished phantom blood.\n\nInvisible, Daphne laughed as he finally threw the suckers off. He slashed them with his sword and they fell, curling like dark ribbons at the mucky river's edge.\n\nShe taunted him, but he was strong. Furious, he charged ahead and destroyed the next monster he saw, slaughtered a double-headed dog, slashing two throats with one stroke.\n\n\"Where are you?\" he called out to Daphne.\n\n\"Inside.\"\n\nHe stumbled through a marshy swamp, tripping over rocks, and roots, and fallen trees. He cut a path before him, but his sword dulled and slowed as he fought onward. In EverWhen you got stronger, earning more power and collecting more weapons in a qwest. In UnderWorld, you struggled as you advanced; your weapon failed you. The silver blackened on his blade, and his silvered arm tarnished as well.\n\nLight-headed, he played on, late into the night. His schoolbooks lay forgotten in the mud. Jack and Liam wanted to come over, but he told them no. Messages from his company in EverWhen remained unread.\n\nSmaller and smaller in the distance, the people he had known in real life. It was as if he'd left them on the riverbank. Their faces became indistinct and their voices died away. Diana was the last one he could hear. She threatened to tell their mother that he played all night, but he knew she wouldn't. Week after week she kept his secret. Their pact held, even as the wall between them turned to stone.\n\nHe battled demons while his sister slept and his mother worked her night shift at the hospital. He traveled leagues, and it was morning when at last he saw UnderWorld's horizon glowing red, a smudge of fire in the distance. Then, with new strength, he rushed through bog and bracken to arrive stunned, breathless, at massive gates with runic messages forged in iron. ABANDON HOPE he read, and pumped his fist. Yes! He had reached the gates of hell.\n\nSpringtime was an old movie outside. Loud daffodils and bright birds singing. Junior year was a recurring dream. For months he had done the minimum to avoid calls home. Dashed off lab reports, polished off math problems, filled bubbles on answer sheets. He learned the way he ate\u2014gulping down enough to get through the day. School was a holding pen, home a portal for the game.\n\n\"When can I see you?\" he asked Daphne the next night.\n\n\"Inside.\"\n\nHe pulled at the Gates' iron bars with no success. The metal warped and bent, refashioning itself into a massive Iron Man with slits for eyes. The Iron Man was twice the size of Aidan's knight. Under iron feet, the earth resounded like a drum. Aidan tried to fight, but with one blow, the Iron Man knocked him down and snapped his sword.\n\n\"Daphne!\" Aidan dropped his stump of a sword and struggled to his feet.\n\nThe Iron Man had no weapons. He killed with his body and his claws. Without even bending, he kicked Aidan's chest in.\n\nAidan could not recover from that blow. He watched the Iron Man pound his gaming body again and again. Black blood trickled from Aidan's mouth. \"Daphne,\" he called again. As if to silence him, the giant kicked Aidan's head off, and with a surge of nausea, Aidan watched his avatar's staring eyes freeze and his hair blacken in a fountain of blood.\n\nThe world went dark. \"Daphne!\"\n\nNo answer.\n\n\"Where am I?\"\n\n\"Inside.\" Her voice was hollow, echoey, as though she were hiding in a deep cave underground. Then he saw that the game was not altogether dark, but shadowy. He found himself in a new place, vast and wet. He heard the drip of water, and he began to make out the contours of cavern walls. He was ashen, from his bloody hair to his torn leather boots. With his two hands he tried to lift his head and screw it back in place. He got it on backward at first. His gaming vision blurred, and he almost lost his balance. He reached out with an arm to steady himself, and the stone walls buckled for a queasy moment before he got his head on straight.\n\n\"I'm over here.\"\n\nHe whipped around, starting a small avalanche of pebbles. He heard them pinging far below. Looking down he could not see the cavern floor, only ledges upon ledges, piled with guano.\n\n\"Careful.\"\n\nAt last. She leaned against the cavern wall. Her hair was white blond in the darkness, her leather bodice half unlaced.\n\n\"Who are you, really?\" he demanded.\n\n\"Daphne,\" she told him, for the thousandth time.\n\n\"Where do you live?\"\n\n\"Close.\"\n\nFor a moment he couldn't speak; he could scarcely breathe. He played with people from all over the world, and took huge distances for granted. He had never pictured her nearby. \"Please.\" He had to see her; he had to touch her; he couldn't wait.\n\nSlowly he stepped closer. Ever so slowly, he pulled the laces on her bodice, so that for an instant he saw her breasts, or at least the swollen image of her breasts exposed.\n\nHe heard a car and froze. Footsteps on the porch, a door opening. No, that was the other door. Priscilla, not his mother.\n\nIn that instant, Daphne turned away and laced herself up again.\n\nHe lifted his hand and watched himself touch her shoulder. She almost smiled as she looked back at him. He reached, but she escaped again. Sure-footed, she ran down the narrow ledge and vanished.\n\nScrambling after her, he found a long fissure in the rock, an entrance to another cave, not vast, but intimate, a cavern flickering with candles set into the hollows of the walls. He edged inside. \"Which way?\"\n\n\"Right in front of you.\"\n\nHe found her blocking the entrance to a tunnel absolutely dark.\n\n\"This is the way to the First Circle.\"\n\n\"Let me in.\"\n\n\"Why?\" she asked lightly. \"What's in it for me?\" Her teasing voice hurt after all this time qwesting.\n\n\"Nothing,\" he said.\n\n\"Then you can't go on.\" She sealed the entrance with a boulder, easily rolling the gigantic rock. \"You can wait for the release like everybody else.\"\n\n\"Don't go.\"\n\nIn his dreams she struggled, but she couldn't get away. Then he devoured her, ripping off her clothes, pushing himself inside her, biting her nipples until they bled. His dreams throbbed fast and hard as he licked her white skin and tasted her blood, sweet as metal. In his dreams she belonged to him, but in the game, he couldn't catch her.\n\nCaverns and candlelight disappeared. When she materialized again she was standing on a stone bridge, still out of reach. In EverWhen they'd played together. Now she played against him, toying with him, and he was confused by how much he wanted her, and how much he hated her. He sensed that she was using him, but he could not break free.\n\n\"Fight me.\" He bounded onto the bridge, brandishing his broken sword.\n\n\"You know what I want.\" Her voice was close, and slightly agitated, tempted by his intensity.\n\n\"You want me to paint.\" He watched her materialize, holding two weapons.\n\nShe tossed him a new sword, but before he could get a grip, she lunged and sliced his shoulder.\n\n\"See you in hell,\" he said.\n\n\"Write it out.\"\n\n\"If you win,\" he said, parrying her silver blade, \"I'll write it out. But if I win...\"\n\nShe grazed his ear with the knife edge of her blade.\n\n\"I get to see _you._ \" He slashed and severed her wrist. Her left hand fell to the cavern floor.\n\nShe kept her eyes on him as she bent to pick it up, but he was quicker, and he got there first, pocketing her white hand as his prize.\n\nStroke and counterstroke, they watched themselves bleed, clanging swords and breathing hard. They couldn't feel a thing, but they were gasping with each blow. \"See you...\" He pushed hard, driving her back against the wall. \"In real life.\"\n\nShe parried, but he cut her thigh in the gap between her leggings and her tall boots. Then in one fluid motion, he knocked her sword from her hands. \"I get to meet you,\" he said, drawing his dagger, plunging the point deep into her heart.\n\nIn the next room, Diana typed with books and papers spread out on her bed. _In my opinion Huckleberry Finn is a character whose morality is different from society but in a good way._ She picked up her paperback and leafed through it, looking for a quote. Lazare loved quotes, so Diana used as many as possible. The trouble was Lazare expected a lot of other things too. She had written out a rubric, also a series of step-by-step instructions, starting _1. Write with a sense of purpose._\n\nAs usual, her mother was knocking on her door. \"What?\"\n\n\"Hey, it's Aidan.\"\n\nDiana was so surprised she lost her place.\n\n\"Can I come in?\"\n\n\"I guess.\"\n\nHe closed the door behind him, even though their mother wasn't home. She stared as he took a seat on her desk chair, straddling it backward. \"I have to ask a favor.\"\n\n\"No.\"\n\n\"Wait, listen to me first.\"\n\n\"No,\" she repeated, just to annoy him. She missed him so much.\n\nThere were days Diana didn't talk to anyone at all. She woke and ate breakfast alone, walked to school alone, kept her head down in the halls. Brynna was still home with her new baby, and she was coming back to school at some point, but even when she talked about returning, she seemed far away, like a girl who'd died, promising to visit Earth again. Brynna had changed forever, no matter what she said. She was living this strange afterlife, sequestered with her parents and her newborn and a million stuffed animals and white china picture frames with wings and halos. She'd even named the baby Angela.\n\n\"Are you writing your paper?\" Aidan cast his eyes over the papers on Diana's bed.\n\n\"Obviously.\"\n\n\"When you finish, can I copy you?\"\n\nHe spoke without hesitation, but when Diana looked at him, he quavered just a moment. He'd left her far behind, but he still needed her. He ignored her, but she knew his secrets. She'd heard him whispering Daphne's name and talking about tagging walls at night. She'd listened as he crept downstairs. She'd seen his graffiti, the long chain of letters CUCU tagging the lobby of the school.\n\nNaturally, Diana had heard DeLaurentis lecture about defacing public property. She'd scrunched down in the auditorium, knees up on the seat back in front of her, and watched the principal exhort students to come forward with information. Diana had not come forward. That wasn't even a question in her mind. As far as property went, she hated school. As far as Aidan went, he was no criminal; or if he was a criminal on occasion, he was also a genius, mastering everything he tried. He had been the musical twin, the academic twin, the fantastic test-taking twin. He got the highest scores and he got into the worst trouble. When teachers talked about not fulfilling potential, they were just a little worried about Diana; Aidan was the one who really scared them. The gap between his performance and abilities was so huge.\n\n\"Can I?\" Aidan asked again.\n\n\"No! We'll get in trouble.\"\n\n\"Why? We're not even in the same class.\" Diana had Miss Lazare and he had Mrs. West.\n\n\"What are you talking about? We'll get caught and fail English. I'm not taking summer school.\"\n\n\"We won't get caught,\" Aidan insisted. \"And if we do, I'll take the blame.\"\n\n\"Don't you think Lazare and West compare their students' work?\"\n\n\"How would they have time for that?\"\n\n\"What do you mean? That's like their job. That's what teachers do.\"\n\n\"I always let you copy me,\" Aidan said.\n\nThis was true. More than once in elementary school, when Aidan had finished homework first, he'd handed over his word searches or long-division worksheets to Diana. \"Why do you want my paper now?\" she asked. \"You never asked before.\"\n\n\"I have to turn something in tomorrow.\"\n\n\"Just do it.\"\n\n\"I haven't read the book. I don't have time.\"\n\nDiana thought about the silver river in her brother's room. \"You've been playing that imaginary girl.\"\n\n\"She's not imaginary.\"\n\n\"You have an imaginary friend in a totally imaginary place.\" She spoke mockingly, but she also envied him. She wanted what Aidan had, crazy as that seemed. All through the winter she'd walked and run along the Charles. Panting, she had slogged through snow and ice, and now, in spring, through mud. She ran until her breath came hard and her feet ached, and sometimes even then, she could not outrun her loneliness. \"Let me play,\" she said.\n\n\"No way.\"\n\n\"Let me try.\"\n\n\"You've already tried EverWhen.\"\n\n\"No, the new one.\"\n\n\"I can't. It's secret. It's not even on sale.\"\n\n\"I want to see it.\"\n\n\"Why?\"\n\nShe sprang off her bed. \"I want to see where you live. Come on.\"\n\n\"I can't.\"\n\nShe loved that she could make him nervous. \"I'll let you copy, if you let me play.\"\n\n\u2014\n\n\"You can't tell anyone about it,\" he warned, as he led the way into his room. \"And you can only play a little bit. You'll have to use my avatar. I can't give you a new one. Stand here. Just wait. Stand still.\"\n\nDiana wasn't listening. She was watching the game rise up around her. Great caves shadowed Aidan's walls, dark passageways came into focus, and suddenly a flight of tiny animals. She sprang back as a thousand white bats swooped down upon her.\n\n\"That's you,\" said Aidan, pointing to a ghostly knight, ducking and weaving in the onslaught. \"You can fight off these bats with your sword. Lift your right hand.\"\n\nShe lifted her right hand and saw her weapon. When she slashed her sword, the bats screamed around her, and their red blood spattered. Startled, she stopped moving, and the creatures swarmed her ghostlike body, biting and ripping at her neck, arms, and face. She covered her eyes as Aidan warned, \"Don't drop the sword.\"\n\nToo late. More and more bats attacked her, flying mice with tiny vampire fangs. They covered her entire body. Whenever she moved, the creatures moved with her, a mass of squirming bodies and red eyes. She swatted at her face, and watched herself knocking bats away.\n\nHer stomach lurched when she saw what was left of her avatar. The bats had eaten half the knight's flesh away, but they had not exposed muscle or bone. No, their attack revealed something else, another creature, an elongated nose, black eyes, wide-set, rolling independently. An ear, unfolding like a leaf from the raw patch where the knight's ear had been, and from his forehead, nubs of horns. Doubled over now, she heard Aidan calling to her.\n\nHer image doubled over too, and shook the bats away as human limbs morphed into four legs. She was changing into a deer. She could reach out and almost touch her other self, the doe hovering before her, pale flanks foaming, ears twitching, body quivering with a strange, borrowed life.\n\nAidan said, \"Don't throw up in my room.\"\n\n\"I won't!\" She felt hot and tearful all the same. As she tried to catch her breath, she lifted her head, and saw the deer prick up her ears. Her head was small, her neck long and delicate, her legs slender. How beautiful she was. Heart pounding, adrenaline racing through her body, she couldn't take her eyes off her deer-self.\n\nA clanging echo in the cavern, great footsteps like giants walking. A jolt of fear. How could she escape?\n\n\"Aidan!\" she pleaded.\n\nHe knelt and closed a black box, a console without buttons. As he screwed top to bottom, the deer vanished, along with swirling bats. The bloodstained cavern melted, and Aidan's walls emerged again, his foil-covered window, his bookcase, his floor strewn with dirty clothes.\n\nDiana sank onto the bed.\n\nHe looked at her anxiously. He couldn't copy her essay if she couldn't write. \"Are you okay?\"\n\n\"That was sick.\"\n\n\"I know.\"\n\nShe closed her eyes and leaned back against the wall. \"I'm seeing spots.\"\n\n\"They go away.\"\n\n\"I can see them with my eyes open.\" The spots were small and bright like fireflies, but they didn't last.\n\nShe had a strong stomach. She could read on long car rides. She did fine in boats. After a few minutes, Diana sat up on the edge of his bed. \"That was seriously the most nauseating thing I've ever seen. Can I play again?\"\n\n\"You promised. You can't change your mind now.\"\n\n\"I was kidding. God!\"\n\nHe didn't see the joke. She didn't care about the game, she didn't care about the copying; she cared about him.\n\n\"It's late,\" he said.\n\nShe asked, \"Is this like the first thing you're going to turn in all year?\"\n\nHe opened the door. \"Get started.\"\n\n\"Stop panicking!\"\n\nShe returned to her room, gathered up computer, rubric, paperback, and descended to the kitchen. There, beneath the apple clock, she ate half a bag of pretzels as she pounded out three pages. She was writing with a sense of purpose now, and words came easily. She laid on the quotations. She expatiated on Huck Finn's personal morality. She even threw in Lazare's favorite word, _ironically._ _Huck Finn decided to go to Hell but for a good cause which ironically shows some things are more important than what you believe society wants you to do._\n\nThat year school went on almost forever. To make up for snow days, the district mandated extra class time straight through the end of June. Kids pulled together their portfolios, and, without warning, the weather changed from cold to scorching. Bees flew in through the open windows of Emerson's un-air-conditioned classrooms. Kids brought miniature spray-bottle fans, and Mr. DeLaurentis had to announce that these devices were for personal use only, which caused some snickering. You could spray yourself, but not your girlfriend.\n\nThen, just when it got too hot to do anything, it started pouring. It rained so hard the morning of the annual Antrim Street Block Party that the yard sale had to move indoors to the Presbyterian church. It was still drizzling a couple of hours later, and people carried umbrellas for the garden tour. Maia led neighbors through Antrim's secret gardens, lush oases behind closely built houses.\n\nGazing at her neighbors' crimson roses, their hidden lawns, and flowering ginkgos, Kerry wished that she had time to tend her own overgrown patch, or money to afford a gardener. Long ago she had imagined that her children would help her. Together as a family they would clear away the dead branches and the big weeds and sow new grass. \"Apart from seeing friends, no teenager will go outside,\" Maia had warned her, and Kerry understood that now, as she did so many other things.\n\nWhen evening came and neighbors cordoned Antrim off with orange traffic cones, the street began its transformation. Lois strung sparkling lights through oak branches. Sage set up camping tables. Neighbors carried out their lawn chairs and their salad bowls piled high with fruit or pasta, watermelon slices, bulgur wheat. There were casseroles, and bags of pretzels, and roasting pans filled with deep-fried cauliflower. Greg played his banjo, and his new girlfriend, Nella, joined him on her flute. Preschoolers ran in a pack and were thrilled to draw chalk pictures where they couldn't play on any other day, the middle of the street.\n\nLike moths Aidan and Diana materialized, pale, in the fading light. Diana hovered near the grilled portobello mushrooms and listened to her mother and Maia go on about the year the whole street flooded. Remember that?\n\n\"You and Aidan were just two years old,\" said Kerry. \"The basement filled with water.\"\n\n\"That was the worst flood I've ever seen,\" said Maia. \"That was build-an-ark-type rain. Collin and Darius took a kayak and paddled down the street.\"\n\nLois was testing out her photos for the slideshow, even as Greg fixed the screen, a white sheet strung from the great branches of a maple. \"Oh, God,\" said Diana, because there she was at four, riding her tricycle in a purple satin cape and a gold crown. \"I had to run behind you,\" Kerry said. \"I had to lift your cape, or it would get caught up in the wheels.\"\n\n\"And there's your brother, and there's Liam, and who's that? Jack?\" The boys must have been in kindergarten. They looked so small and delicate, standing shirtless, holding water balloons.\n\n\"Yeah, that was me,\" said Jack. He was helping his father, aka Scienceman, set up a giant gyroscope.\n\n\"I can't believe you still do that,\" Diana said.\n\n\"This is my community,\" Jack told her. \"There aren't a lot of block parties anymore.\"\n\nShe just stared. There was something so horribly sincere about him. She remembered her mother dragging her along with Aidan to watch Jack and his mom perform in the North Cambridge Family Opera as singing insects. What had he been? A dung beetle? Cricket? She concluded, \"You're just weird.\"\n\nJack told her, \"You're just mean.\"\n\nHis directness startled her. When he narrowed his blue eyes to look at her, she wanted to hurt him. \"My brother is completely bored with you,\" she blurted out.\n\nCoolly Jack said, \"Then you and I have something in common,\" and he left her standing there.\n\nPictures in the trees of Lois and her godchild from Uganda, the water fight when five boys got ahold of the Mednicks' garden hose.\n\n\"Hey, baby! Hey, Nina,\" Maia called out.\n\nDiana turned to look, and then she looked again at Nina. Of course teachers had first names and didn't live at school. Miss Lazare didn't sleep under her desk. Diana knew all this in theory, but it was a shock to see her teacher there.\n\nInstinctively, Diana shrank back as Nina and Collin approached the table for their drinks.\n\n\"Hi.\" Collin breezed by, but Miss Lazare gazed into Diana's eyes as though she could see inside of her.\n\n_What?_ Diana demanded silently. Instantly she felt huge and guilty.\n\nShe knows, Diana thought, even as Lazare walked on. She knows!\n\nDiana scanned the crowd. Old guys, couples drinking hard cider, mothers nursing babies in lawn chairs. Kids rumbling up and down in Big Wheels. There he was, sitting alone on their front steps.\n\n\"Aidan.\" She ran up to him.\n\n\"What?\"\n\n\"Lazare knows we cheated.\"\n\n\"No, she doesn't.\"\n\n\"Yes, she does.\"\n\n\"What did she say?\"\n\n\"Nothing. She just looked at me. It's obvious. They're all sitting at school comparing end-of-year portfolios.\"\n\nAidan dismissed this. \"Yeah, I don't think so.\"\n\n\"I'm serious.\"\n\nAidan stood up. \"You don't have to do anything. I take the blame and say I copied you.\"\n\n\"They're not going to believe that.\"\n\n\"They will if you let them.\"\n\nWalking down the street with Maia, Kerry saw her children standing in the soft light of her porch. \"They're really very close,\" she told Maia. She saw the conversation, but she couldn't hear the words.\n\nMaia said, \"And there you were, worrying all winter.\"\n\nQuietly Kerry said, \"You would have worried too.\"\n\n\"They come out of it. They start growing up eventually. Look at Collin!\"\n\nKerry flushed under her freckles. She could not share Maia's joy\u2014not while Collin worked at Arkadia. Yes, it was a full-time job. Yes, he had benefits. Yes, he could earn a living making art, but the thought sickened her. \"It's like working at a munitions factory.\"\n\n\"Kerry.\"\n\n\"I'm sorry. It's true.\" Kerry had known her neighbor almost fifteen years. Maia was just about her closest friend, but she spoke out anyway. \"It's like building bombs.\"\n\nMaia's temper flared. \"First of all, as you'll find out, you don't tell your twenty-four-year-old son where he can and cannot work. Second of all, games aren't bombs.\"\n\n\"Yes, they are. They are! They're weapons of mass destruction,\" Kerry burst out. In cloud, in smoke, in myth, Arkadian games were detonating in a million minds. She had been following the news. She'd read about the kid in Austin caught defacing public property, the kid in Seattle charged with hacking his school website so the banner read CU.\n\n\"Hey, are we fighting about this now?\" Maia asked gently.\n\nKerry was too upset to speak.\n\n\"They're pastimes,\" Maia said. \"They're part of life.\"\n\nDeath is part of life, thought Kerry. Maia's words reminded her of chaplains and hospice nurses at the hospital.\n\nMaia said, \"Games are just like music and art and dreaming.\"\n\n\"Whose dreams?\" Kerry demanded. \"Not _my_ dreams for my children!\"\n\nMeanwhile, on the porch, Diana asked Aidan, \"What are you going to do when they say they want to talk to you?\"\n\n\"If that ever happens I'll just sit there,\" Aidan said.\n\n\"And you'll admit you cheated?\"\n\n\"Yeah.\"\n\nHe opened the glass storm door.\n\n\"But what about Mom?\"\n\nThe question pierced him. If Diana was right and they got disciplined, Kerry would search his room.\n\n\"We have to think.\" Diana followed him inside the house.\n\nBut he was thinking of himself. He had to hide his BoX. \"I want you to do something,\" he told his sister.\n\n\"No,\" Diana said, but she trailed him upstairs to his room, where he rooted in the closet under laundry, worn-out shoes, old schoolbooks, a pair of hockey skates. From the depths he pulled out the scuffed black BoX.\n\n\"It's heavy,\" Aidan warned.\n\nShe started back as if he'd handed her a loaded gun.\n\n\"Don't drop it.\"\n\n\"Your game? You're giving me your game?\" she asked, incredulous.\n\n\"Just hide it for a little while.\"\n\n\"I'm not keeping this in my room.\"\n\n\"Hide it somewhere else, then.\"\n\n\"Why?\"\n\n\"So I won't play.\"\n\n\"You're going to stop?\"\n\n\"Just hide it and don't tell me where it is.\"\n\nHe saw her wavering.\n\n\"I'm taking the blame,\" he reminded her. \"Just keep it somewhere.\"\n\nThe BoX was cold and smooth. She was afraid of it, and at the same time she thought, But he won't play. He's going to stop.\n\nHe looked at her with trust, with urgency. \"Hide it.\"\n\nDon't even touch it, she thought, but she took the BoX.\n\nHe promised, \"You don't ever have to tell me where.\"\n\nAnd she accepted this fiction; she took this lie to heart, even as she said, \"Yeah, so you won't know where it is when Mom comes after you.\"\n\n\u2014\n\nOn the Monday after the block party, Aidan and Diana faced Miss Lazare and Mrs. West and Mr. DeLaurentis and their mother. It was two-fifty in the afternoon, and the last bells had rung. DeLaurentis's first-floor office vibrated with students' feet.\n\nMr. DeLaurentis hung his suit jacket on the back of his chair, and all Diana could think was it took a lot of cloth to sew that blue dress shirt.\n\nThe principal had a whiteboard covered with a grid for days of the week, a poster that said BE THE CHANGE YOU WANT TO SEE. Binders filled his bookshelves, and his phone lay on the desk, along with twin essays on morality in _The Adventures of Huckleberry Finn._\n\n\"It's my fault,\" Aidan confessed immediately. \"I stole her paper without her knowledge.\"\n\n\"You expect us to believe that?\" DeLaurentis asked, and he turned to Diana. \"You had no idea this was happening?\"\n\nShe shook her head.\n\n\"How do you steal a paper from your sister? You found it? You just saw it lying around? What?\"\n\n\"I took it from her computer,\" Aidan said.\n\n\"You just left it there on your computer?\" DeLaurentis asked Diana.\n\nShe nodded. \"That's where I wrote it.\"\n\n\"Don't you have a password?\"\n\n\"Yes.\"\n\n\"But I know it.\" Aidan was way too calm, taking all the blame.\n\nMrs. West said, \"Aidan, this was cheating. You get that, right?\"\n\nHe didn't even blink.\n\nMiss Lazare started talking about what would happen if they were at college, but DeLaurentis cut her off and said, \"Excuse me, one step at a time. The goal is getting into college. Let's get there first.\"\n\nLazare looked hot and flushed, all You can do better, and you know it in your heart. Kerry's eyes filled with tears, and DeLaurentis reached behind him for his box of tissues, but even then\u2014especially then\u2014Aidan sat unmoved.\n\nHe betrayed no emotion as DeLaurentis talked about the Honor Code and pride in end-of-year portfolios and summer school. I knew it, Diana thought; and she was scared, not so much by the situation as by her own psychic powers, the whole thing playing out as she'd imagined.\n\nShe held still, afraid of crying, but Aidan radiated confidence. He was a beautiful liar, his voice unwavering, his details bold and magical, conjured up as though he were remembering. _She was sleeping with her head down on the kitchen table and her computer open._\n\nDiana listened in awe. Her brother was so smart. Nobody, not DeLaurentis, not Lazare, not even their mother could trip him up or force him into inconsistencies.\n\nExcept that his teachers had caught him copying. DeLaurentis would not allow Aidan to forget that.\n\n\"So you're admitting that you plagiarized,\" DeLaurentis said.\n\n\"Pretty much.\"\n\n\"Excuse me?\"\n\n\"I did it.\"\n\nBy the time the meeting ended, the building was deserted. The twins followed Kerry through dank tiled halls. Outside, the field was empty. Kerry led the way and unlocked the car. She said nothing. She let herself in and just stared out the dirty windshield. She wasn't crying anymore. Her eyes were blank, and that was almost worse.\n\nThe twins exchanged glances, and for the first time Diana saw regret in Aidan's eyes. A touch of sadness and embarrassment. Poor Mom. Poor us. Wordless, like little children, they piled in back with all their stuff. Neither would brave the front seat. Diana felt miserable, but more than that, a sense of solidarity, as Aidan untangled his seatbelt from hers.\n\nHow angry was their mother? White-hot. Incinerating. She marched up the porch steps, unlocked the door, and threw her keys down on the kitchen table. In silence, Diana and Aidan watched Kerry open mail, ripping envelopes, tossing junk into the brown paper recycling bag. In silence, they saw her pile up breakfast dishes in the sink, scummy cereal bowls sitting out since morning. Then up the stairs she went, but not to her own room. They heard her slam Aidan's door behind her.\n\nDiana retreated to the living room couch and considered her brother leaning in the kitchen doorway. \"What now?\" she said.\n\n\"What now?\" Aidan echoed, mockingly. He could be so warm. He was brilliant as the sun, and then the next moment he turned his back on you.\n\n\"I feel like we should do something.\"\n\nAidan lashed out. \"You mean like make a card?\"\n\nThere had been a time when the two of them gave Kerry cards after they did something wrong. A drawing of a heart, a portrait of the stray cat they'd adopted. A rainbow and the single word SORY. That wasn't happening now. Kerry was ransacking Aidan's room. Aidan had predicted this, but he was furious at the invasion. He had never been sorry for what he'd done. Now he wasn't even sorry for his mother.\n\n\u2014\n\nUpstairs Kerry collected two joysticks and a headset. Pawing through his laundry, she searched for more. She would do better. She would be stronger. Take Aidan's games away, cut him off from his computer. She had to find a way to punish him\u2014and not just for cheating. He'd dragged Diana into trouble. This was the hold Arkadia had on him.\n\nShe emptied Aidan's closet, piling the floor with childish things, old Legos, outgrown clothes, broken toys. She would fumigate. Burn all his stuff. Attack his computer with a baseball bat. If only it would make a difference. If only she could drown his phone.\n\nStripping his bed and peering under it, she excavated dirty T-shirts and twisted jeans. She dumped all the old papers and candy wrappers from his desk drawers, riffled through the binders piled on the floor. Just two years before, Aidan had blazed through homework every afternoon. He'd aced his tests and writing assignments too. Didn't she have his A paper on the battle of Vicksburg?\n\nJostling his monitor, Kerry saw his screen glow and darken. A silvery pattern, the sheen of water, shimmered and rippled eerily, but she didn't know how to break into his machine.\n\nShe knelt down, peering under Aidan's desk and touched a scar on the wood floor, a deeper gash than other scratches. For a moment she paused, tracing the raw place, but she didn't know the cause.\n\nFeeling for his surge protector, Kerry unplugged the black cord. She had done it before. Now, once again, she would pull up her son's computer by the roots, discontinue cable service, disable the house router. She would stanch the electric river through which games flowed. None of this would frighten Aidan. Last time, he'd simply walked away, surfacing days later at Liam's house. Panicked, Kerry had come running after him. This time she would not negotiate. Don't give in, she told herself, although her child was almost seventeen, and more than six feet tall.\n\nShe sat back on her heels, and sadness overcame her as the monitor went dark. Other kids enjoyed games for a weekend. Jack would play, but he went to Math Circle and competed on the robotics team. Even Liam had his band. Aidan was the one who couldn't stop. For this, she blamed herself.\n\nShe had a good job, but it wasn't good enough. She earned decent money, but not enough. She loved him, but that was not enough. She could not afford a mountain program; she didn't have the cash to send him to some snow-capped wilderness where he might waken from his soul-sucking dream. If she took him to the police, maybe that would scare him. If she found explosives in his closet, if she discovered drugs, or caught him dealing, then she'd have some leverage. What could she do with a son whose drug of choice was legal? Whose weapon was his own imagination?\n\nJust as Kerry despaired of Aidan and his future, he grew quiet, almost docile. He accepted the loss of his computer, along with being grounded every afternoon and weekend. He accepted that he had failed English and would repeat the class next year. He accepted that he had failed biology as well. After the term ended he would return for summer school. When Kerry asked him to wash dishes or take out garbage or bring up the laundry, he did it instantly.\n\nAt first, his compliance made her nervous. She had prayed, but this was more than she had hoped. Aidan returned from school to do his homework. He ate dinner. He even slept at night\u2014he really slept. When Kerry returned from her night shift, she didn't hear a sound. Aidan's behavior seemed to her too good too fast, and yet she wanted to believe in him. Maybe it was true, as she had read, that deep down teens craved structure and authority. Confiscating Aidan's electronics may have been his secret wish!\n\nAidan never asked for his computer and his games. Nor did he run away to play with friends. Day by day, he worked to earn his mother's trust. He held out his open palm for Kerry, and, like a hungry woodland creature, she watched him from a distance. Steadily he made his offering. No sudden movements, no threatening gestures, as Kerry crept closer. Trembling, she circled hungrily. She knew better. Experience told her otherwise, but need trumped fear as she began grazing from his hand.\n\nKerry re-established Sunday breakfast, which she prepared after her Saturday-night shift. She made the children French toast as soon as she came home, and they ate together at the kitchen table, their plates drenched with syrup, their glasses filled with fresh-squeezed orange juice. Aidan was quiet, but his mother didn't mind. Just two weeks after she had confiscated his computer, she saw a calmer manner, a steadier gaze. No longer did he fidget at the table and race away to play. To her mind, he'd hit rock bottom, and now he was rebuilding. She rejoiced, but tried to temper her excitement. On Friday morning, when he said he wasn't feeling well, she took his temperature. Trust but verify. He had a fever of 101, and she allowed him to stay home from school.\n\n\"You rest,\" she told him as she hunted for ibuprofen. \"Take these, and I'll check on you when I wake up.\" Then gratefully, she lay down in her own bed.\n\nComing down with a cold that morning wasn't planned. Aidan's body ached, and his throat hurt when he swallowed, but he didn't care. His heart pounded as he listened to Diana leave for school. Then, when she was gone and all was quiet, he slipped downstairs, supporting himself with his arms on wall and banister, so that his feet scarcely touched the creaky treads. Softly he escaped through the kitchen. Gently, he shut the back door.\n\nGreat trees canopied his street and Maia's roses were all blooming luscious red, but Aidan didn't stop to look. He ran to Central Square to catch the T inbound. Plunged, with his student CharlieCard into dank tunnels and took the rattling train to Boston.\n\nHe had never been to the Seaport World Trade Center. He had hardly been to Boston Harbor except on trips to the Children's Museum when he was small. Now he saw the ferries and the sailboats, the rusty fishing boats, and all the people swarming the gray convention center on the water. A friendly mob had gathered even before doors opened, gamers in costume, entire companies brandishing their spears like shaggy Vikings at the harbor's edge. Hand-sewn Elvish shoes with turned-up toes, custom knives and swords, the glint of chain mail in the sun\u2014this was EverCon.\n\n\"Yo, why aren't you dressed?\" demanded a blue-haired Fire Elf.\n\nShe was a girl from Kansas City traveling with three friends from her company, all dressed in black leather bustiers and thigh-high boots.\n\nAidan did feel undressed in ordinary clothes. He was relieved to see that some Everheads had come in jeans\u2014the buskers selling ouroboros T-shirts, the guys scalping tickets to the evening show. There were jugglers and musicians crowding the door. A man entertained the crowd with a Whennish lute he'd built himself, a seven-stringed instrument inlaid with mother-of-pearl and a twangy sound like a medieval banjo.\n\nThe crowd was joyous. Everheads were eating breakfast harborside in folding beach chairs, trading tips and telling war stories, reuniting with their companies, donning matching T-shirts. After campaigning together for months and years online, qwesters embraced, many meeting in person for the first time. They were college kids and couples, and hordes of hard-core single gaming geeks. They were men and women, Elves, and Gnomes, masters and journeymen. My people, Aidan thought. Arkadian nation.\n\nEven so, he hesitated at the Trade Center doors. At least twenty protesters stood right in front, flanked by police. Christians Against Gaming Exploitation wore matching CAGE T-shirts, and brandished big hand-painted signs for the cameras of Channel 7 Eyewitness News.\n\n> CU IN HEAVEN!\n> \n> BE A PRAY-ER, NOT A PLAYER\n> \n> HELL NO: DON'T GO\n\n\"What are you doing here, son?\" one Christian asked. The man had a round, friendly face, and he wore round, friendly glasses. \"Shouldn't you be at school?\"\n\nAidan flinched, nervous about the cops.\n\n\"Won't you join us?\" a second Christian asked. She had long blond hair, and she wore a little gold cross on a chain. \"Won't you consider reading this?\" She held out a leaflet printed:\n\n> YOU AND CHRIST: WIN WIN!\n\nKeep walking, Aidan thought. He wouldn't take the leaflets offered him.\n\n\"God has other plans for you!\" the picketers called after Aidan as he hurried toward the World Trade Center doors. \"Jesus is waiting, if you let him in!\"\n\nWhat a relief to dash inside without anyone chasing or reporting him. Aidan presented his school ID at the registration table and no one questioned him. His name was on the master list, and he received an all-access pass to wear on a chain around his neck. He fingered that strand of tiny metal beads as though it were a chain of Elvish gold.\n\n\"Straight ahead,\" one of the EverCon staffers said.\n\nAidan entered the blue-carpeted Commonwealth Hall, and it was dark after the June sun. Vast as a theater, dim-lit, draped in black. For a moment Aidan stopped in awe. Thousands of gamers had brought their home computers and powered up, logging in to play EverWhen, together and apart. Rustling, shuffling, clicking, the assembled armchair heroes sounded like cicadas on a summer night as they slayed dragons with trackballs and joysticks. Above them, banners hung from the vaulted ceiling, pennants floating over the virtual fair. There were monitors pimped out with flashing police lights, computers bedazzled with crystals. Aidan saw a PC transformed into a steampunk masterpiece of cherrywood, antique typewriter buttons, and polished brass.\n\nThere were girls dressed like cheerleaders walking the aisles to toss out T-shirts, bumper stickers, free download codes. There was the blue-haired Elf from Kansas City, who had followed him inside. She told him she had legally changed her name to Kalinda, but he had someone else in mind.\n\nDaphne was here somewhere in this mass of people, and she'd promised, after his long days and dangerous nights, after a million refusals, that he could meet her face-to-face. She had arranged his all-access pass.\n\nCurtained off from the gaming hall, vendors sold costumes, crystals, dragon masks, silver ouroboros bracelets, and pendants of amethyst. Some wares were silly, like needlepoint phoenix pillows, or hand-painted eyeglass cases with your company's insignia. Some exquisite, like the swords of tempered steel. You could buy anything, from Elf-inspired candles to prosthetic noses, humps, and wings. At the Arkadian brokerage, you could sit at a bank of computers, log in, and bid real money for an imaginary jewel or weapon or gold ring. You could buy an enchanted sword that would take months to earn in EverWhen\u2014or you could sell your hand-forged virtual armor in an instant.\n\nDaphne had promised Aidan that he'd find her, but the convention was even bigger than he'd imagined. He watched early-round tournament play on giant screens. He saw Viktor Lazare speaking to a thousand Everheads about UnderWorld and its new platform. \"This is official,\" Lazare announced. \"Everything you've heard is true. We're launching in December.\" But even Lazare's keynote couldn't hold Aidan for long. He searched every hall, texting, where ru?\n\nDaphne did not reply, and yet he scanned the crowds. Every blond girl in black leather startled him. He texted again, but his phone didn't even blink. He remembered Diana's words\u2014\"There is no girl.\"\n\nBut his sister was wrong. Daphne was real. She had to be. He had traveled with Daphne; he had fought with her and against her. She had been his guide, his companion, and his closest friend. Even without his headset, he heard her throaty voice, her whisper, and her laughter in his ears.\n\nHe hadn't eaten, and after several hours his body ached in the highly air-conditioned hall. The place began to look fake, with its black drapes and kitschy painted booths. He kept looking for Daphne, but he moved slowly now. He saw spectacular costumes, but others seemed hokey and embarrassed him. So many girls fell short compared to Elvish women. Their arms were flabby, and they were always tugging at their bodices, afraid they would fall down. So many guys were old and bald. They didn't have the physique or hair of warriors. Everheads could not live up to the real thing.\n\nDiscouraged, he drifted through the demo booths, with their banks of computers set up for free trials. EverHeart. EverFlight. He sank into a swivel chair, and a black-shirted Arkadian staffer offered him a headset to play EverSea.\n\nAt first he said no, because he wanted to keep searching, but after a few minutes he slipped the headset on. Logging in as his old Elf, Tildor, he found himself in the Golden Islands, where he came upon a skiff beached on the shore. A fair wind was blowing, and the tide began to rise. The golden ocean swelled, murmuring around him, and he leapt into the skiff and began to play.\n\nThe wind whipped up. Salt spray flew into his face and water roared in his ears, along with the pure voices of mermaids singing on outlying rocks. The mermaids' breasts were full and lovely, their hidden tails serpentine, deadly underneath.\n\n\"Hey, buddy.\" A male voice broke in. \"Five-minute warning before we have to shut this thing off.\"\n\n\"Okay.\" Aidan kept his eyes on the screen.\n\nQuickly, nimbly, he tapped and clicked, approaching the mermaids, but keeping his skiff just out of reach. He kept his distance as each mermaid questioned him in turn.\n\n\"Who are you?\"\n\n\"What are you doing?\"\n\n\"Where have you been?\"\n\nHe froze when he heard the last mermaid. Daphne.\n\n\"Listen, I've really gotta shut you down,\" the EverSea demonstration supervisor said apologetically.\n\n\"Just one second,\" Aidan pleaded as he scanned waves and rocks. Then into his headset, \"What do you mean where have I been? I've been looking for you everywhere.\"\n\nDaphne laughed.\n\n\"You said you'd come here as yourself,\" he said.\n\n\"I _am_ here as myself.\"\n\n\"Yeah, right.\"\n\nThe islands vanished; the singing ended.\n\n\"Sorry.\" The supervisor started packing up the monitor and CPU.\n\nAidan took off his headset, but, strangely, he could still hear Daphne's mocking voice. It took him a moment to realize she wasn't speaking to him from inside the game. \"I'm standing right behind you. Turn around.\"\n\nHe spun around in the swivel chair and there she was, looking nothing like Riyah. No flowing locks, no leather bodice, no heaving breasts. She wore black jeans and a black sweatshirt. She was so covered up, he could barely even see her hands. When she pulled up her sleeves, he saw indigo flowers tattooed on her wrists.\n\n\"What?\" she asked playfully.\n\n\"Nothing. You look different.\"\n\n\"From what you expected?\"\n\nHe looked down, embarrassed. \"I probably look different too.\"\n\n\"No,\" she told him. \"You look exactly the way I imagined you.\"\n\nHer voice was just as he remembered\u2014knowing, teasing. ARKADIA was the word on her hooded sweatshirt.\n\n\"You work here.\"\n\n\"Yeah,\" she said. \"Of course I do.\"\n\nShe wasn't Elvish but elfin, with her slight frame, short hair, and huge blue eyes. The object of his obsession was not an object at all. She was bright, her expression lively, her smile incandescent. He had no idea how old she was, but he figured at least twenty, college age or more. Her voice was condescending, her expression curious.\n\n\"What do you want to play?\" she asked.\n\n\"Nothing. I want to talk to you.\"\n\n\"I don't like talking.\"\n\nHis throat was dry. \"What do you like instead?\"\n\nShe took him upstairs to the Harborview Ballroom, where several hundred champion qwesters were playing EverWhen.\n\nAidan said, \"I thought you have to register for the tournament to play.\"\n\n\"Not if you're with me.\"\n\nShe commandeered two Arkadian workstations, and they sat in swivel chairs across the table from each other. A strange place for gaming, with its sea-green corporate carpeting, but once you started, you didn't notice anything.\n\nCharging on horseback, fighting with broadswords, shooting arrows, casting spells that rained down sparks like fireworks, she played well, but he played better. Trembling, feverish, his reflexes were faster. Jousting, he unhorsed her. He thrust and parried, forcing her down on the leafy forest floor. Again and again he vanquished Daphne. He frustrated and dazzled her. He fought brilliantly, but she would not admit defeat. She sprang up, challenging him in archery. Together they sent arrows flying after moving targets, gold birds darting and wheeling in the bright sky.\n\n\"Once more, once more,\" she urged him on.\n\n\"I won,\" he said. \"Admit it.\"\n\n\"One more round.\"\n\n\"No.\" He scooted his swivel chair to her side of the table.\n\nReluctantly, Daphne turned away from the game.\n\nOnce again he said, \"I need to talk to you.\"\n\nShe felt a prick of fear when she saw Aidan's face, because he looked so serious. To tell the truth, she had considered standing him up. All day she had ignored his messages, and when he texted that he was playing EverSea, she had circled for a long time, watching.\n\nHe was too involved with her. He had revealed way too much about himself. His name, his school, his loneliness. Sad Aidan was sadly predictable, because, of course, Daphne knew exactly what he was going through. Once upon a time she'd been a miserable teenager, shutting herself up and gaming for days. She didn't need to hear about it when she'd lived it. Even so, he'd won the chance to see her. He'd stabbed her through the heart, and she played fair. \"What do you want to talk about?\"\n\nHe was practically shaking, meeting her like this. He had no weapons; he wore no armor. Even so, he forced himself to speak. \"I want to know you.\"\n\n\"You're a kid,\" she said, as much to reassure herself as to remind him.\n\nHe wouldn't let her put him off. \"Who are you?\"\n\n\"I'm exactly what you see.\"\n\n\"No, you're not.\"\n\n\"Stop!\" She brushed his hand with hers. The brief contact, meant as friendly and dismissive, shocked them both. Her touch surged through his body. She felt his heat. \"What are you on?\" She was only half joking. \"I want some.\"\n\n\"I'm not on anything.\"\n\n\"I was kidding!\" Then she added, \"You don't get to know me.\"\n\n\"Why?\"\n\n\"Because it's not good for you.\"\n\n\"How do you know what's good for me?\"\n\n\"Okay, we're done now.\" She sprang out of her chair and began walking.\n\nHe hurried after her. \"Why are you, of all people, talking about what's good for me?\"\n\nShe walked faster. She was nearly running to the mezzanine.\n\n\"Daphne!\"\n\nShe slipped into a crowded elevator going down.\n\n\"Wait!\" Aidan called out as the doors were closing.\n\nShe might have escaped then, but a friendly Water Elf held the door, and Aidan squeezed in with a crowd of costumed Everheads.\n\n\"Going to the show?\" the Water Elf asked Daphne.\n\nShe answered, \"Absolutely.\"\n\nAidan tried to edge closer, but too many sweaty bodies blocked him. Last in, first out when the doors opened, Aidan tried to catch Daphne leaving, but other elevators were opening too. Streams of Everheads were exiting, and she slipped into the crowd, ignoring him, as she race-walked to Commonwealth Hall.\n\nHe followed Daphne into a conference venue transformed into a rock concert. Sparkling light rained down, as Aidan plunged into a sea of bodies. For a moment he couldn't even see. He almost lost Daphne in the darkness and the virtual fireworks, the huge reverb, the thunder of Arkadia's house band, the Velvet Pixels, rocking out onstage, the roar of a thousand Everheads partying. Half blind, he reached out and caught the hood of her sweatshirt.\n\nShe glanced over her shoulder, amazed at his tenacity. Her eyes were mocking; her whole body told him, Catch me\u2014I dare you. Her hood ripped as she tore away.\n\n\"Stop,\" he called. She paused for just a second, and then she raced off again.\n\nHe pursued her from the dance floor to the great hall of gamers, with their glowing monitors. Security tried to block him, but he hardly noticed. If Daphne had stopped here, guards would have caught him, but she kept running through the hall. She tried one door, and then another. When she hit an unlocked exit, he was right there after her.\n\nGasping, laughing, she sprinted down two flights of stairs. His speed was thrilling, actually Elvish. His legs were longer than hers and he jumped the last four stairs at the exit to the parking lot.\n\n\"Okay,\" she said.\n\n\"Okay what?\"\n\nShe kissed him on the lips.\n\nHe was already light-headed, and now he felt like he was floating. For a long moment he was dreaming, flying! But no, he was really standing there with Daphne. That was her tongue pressing against his.\n\nHis hands spread over her shoulders. Still kissing, he unzipped her sweatshirt. He touched her through her ribbed undershirt as he kissed her neck, her collarbone. Then, opening his eyes, he saw her ink. Thick twigs and flowers covered her arms and shoulders, climbing over her shoulder blades. For such a long time he'd fantasized about her Elvish flesh, her pure white character in game. He'd imagined her softness and her nakedness, but she had clothed herself in intricate designs. Black branches, berries, thick leaves tattooed her entire torso.\n\nThe instant he paused to look, she pulled away.\n\nHe reached for her. Just wait, he pleaded with his eyes.\n\nShe glanced at him with a hint of sympathy. \"Ohh,\" she said, as she might speak to a small child. Then she ran upstairs.\n\nHe couldn't follow. His head was pounding. His muscles throbbed. With sheer adrenaline, he'd chased her down, but he had nothing left, and he sank back against the wall. He was so sick, his mind and body racing. He thought his heart would burst. It wasn't fair. He'd caught her, touched her, half undressed her, and still she wasn't his.\n\n\u2014\n\nUpstairs, the Velvet Pixels were performing and the audience was belting out the chorus. _\"Reality! Reality!\"_ Grateful, Daphne slipped into the crowd. People were singing, and they were swaying, and they were holding light sticks. The darkness and the mayhem comforted her. She loved crowds because you had strangers to fall back on when you were bruised and rattled, out of breath.\n\nShe hung back when the Velvet Pixels took their bows and walked offstage, changing from rock stars to programmers and engineers. She did not join her colleagues as they met the band outside to head to Viktor's after-party at the Institute of Contemporary Art.\n\nObi saw her and called, \"Come on.\" Arkadians had been working the conference all day, answering questions, demonstrating features, mingling with fans, and they were ready to escape and celebrate.\n\n\"Later,\" Daphne said.\n\nHer colleagues left without her. All the Arkadians had changed their clothes, because the invitation had suggested evening dress and vintage gowns. They stuffed their jeans and T-shirts into backpacks or trunks of people's cars, and trooped outside into the warm summer night.\n\n\u2014\n\nCollin wondered where Daphne had gone, and then he thought, She's way too cool for this. Leaving with the others, he thought the hellions betrayed their name, dressed up like a bunch of kids at prom. Even so, his heart beat faster as the group crossed parking lots and wove between construction sites to reach the ICA, glowing like a square-cut diamond on the water.\n\nHow strange, coming to a museum as an invited guest. Collin wore a used tux from Keezer's in Central Square, and his black dress shoes echoed on wood planks so that arriving felt like boarding ship.\n\nArkadians had gathered on the deck outside with their guests. Scanning the crowd, Collin saw Akosh's wife, Meta, in a sari of gold silk and Obi with a Russian graduate student dressed in a ballroom dancer's plunging evening gown. There was Viktor, holding forth with his six-foot wife, Helen. There was Peter, with his amber eyes. But where was Nina? It took Collin a moment to recognize her, even as he kissed her. She was wearing a silvery-blue sheath, the lightest, smoothest fabric rippling over her body. A diamond clip was shining in her hair. \"They're just rhinestones,\" she told Collin, but she looked so elegant, he only half believed her.\n\nFor a moment he felt like a stranger. He thought, No way; I don't belong here with Nina and her uncle and her father. Martinis helped. He finished his first and had another. The evening shimmered; the warm summer night caressed him. When Nina introduced him to her stepmother, the museum levitated on the water.\n\nHelen stared at Collin as she offered her cold hand.\n\nCollin gazed back at the dark-eyed, dark-haired woman in her tight dress. Tall as she was, she wore stiletto heels. She enjoyed her height.\n\n\"Good evening,\" Helen said in accented English.\n\n\"Where are you from?\" Collin asked, a little louder than he'd intended.\n\n\"I come from Greece,\" said Helen. Her voice was courtesy itself. Her eyes said, Who the hell are you?\n\nViktor told Collin, \"Now you know what I mean when I say that I'm a Hellenized Jew.\"\n\nCollin smiled, pretending he knew what Viktor meant.\n\n\"Collin draws horses,\" Peter was telling Helen.\n\n\"Oh, good,\" she said.\n\n\"He's better than good,\" Viktor said. \"He's insanely great.\" Showman, technocrat, Gatsby without tears, he told Collin, \"Someday you'll develop games for us.\"\n\nHelen warned, \"My husband likes making promises.\"\n\n\"Predictions!\" Viktor corrected.\n\n\"They're easier,\" Peter pointed out.\n\nViktor waved all this away. Partly flattering, partly self-satirizing, he told Collin, \"We'll make you a star.\"\n\n\"Don't believe him,\" Helen said. \"Don't trust anything he says.\"\n\n\"You're beautiful tonight,\" Viktor told his wife.\n\nHelen was not amused, but Viktor didn't mind. He entertained himself. He was fierce and jovial, voracious and self-satisfied. Announcing UnderWorld's release date in December, he had chosen an audacious path, accelerating development to the deadline. Peter had worried about rushing, cutting corners, cheapening the game. He had protested, \"Don't sacrifice the story for effects. Give us time to build the myth.\" He'd made his case, but Viktor chose December anyway, high risk and high reward. OVID couldn't wait. Arkadia's rival, Urania, was building its own platform for gaming in the round.\n\nOthers called him ruthless, but disruption, quick reversals, and bold decisions spelled leadership for Nina's father. By the same token, escapism and delusion were not problems but products in Viktor's lexicon. He never doubted his profession the way some of his colleagues did. There were those at Arkadia who had trained as researchers\u2014not just computer scientists, but biologists and physicists. There were those nostalgic for their old disciplines and their youthful goals. Once they had studied structures of living things, and pondered laws of the actual universe. Now they spoke wistfully of science as their homeland, and their old religion. Viktor was not among them. He had been and remained a scientist, investigating vision\u2014fundamental questions of perception. Applying his research to gaming did not disappoint him. Consummate player, joyful inventor, he never apologized for the diversions he marketed. Not at EverCon, where players thanked him for bringing them together and granting them such pleasure and such power. Not after that morning's symposium on role-playing as therapy for autism, not after the Q&A, where a gamer in a wheelchair told Viktor, In EverWhen I run, I swim, I fly.\n\n\"Am I glib?\" Viktor asked Nina.\n\n\"Sometimes.\"\n\n\"I thought so.\" Viktor spoke with such satisfaction that she laughed and Collin laughed with her. Even Helen smiled. With few exceptions, Viktor captivated people. When he was happy the whole world glowed. That very moment the sky was deepening from lavender to violet. He might have cast that sunset over his shoulder; it would have been just like him.\n\nPeter was one of the exceptions, and he kept quiet as he watched Nina and Collin basking in his older brother's praise.\n\n\"The point is, you'll go far,\" Viktor told Collin.\n\n\"Promise or prediction?\" Peter asked.\n\nThe question startled Collin, but Viktor was whispering to Nina. No one heard, but she was glowing with her father's words.\n\nShe knew flattery came easily to Viktor. He didn't just say what you wanted to hear. He said what you hardly dared to hope. Even so, he loved talent. He loved youth, and he appreciated art. Nina could have kissed him when he praised Collin.\n\nShe had been to parties at the ICA before. She had watched the waves framed by the glass window on the mezzanine, but when she stood with Collin, it was like holding the ocean in your hands, studying the weave up close, the warp and weft of water. The glass galleries seemed stranger, funnier, far more beautiful. A sofa sank under papier-m\u00e2ch\u00e9 lovers. A woman in a video installation stood gazing out at shimmering water from her balcony. Collin sat on a marble bench, chaste white, funereal, carved with the words PROTECT ME FROM WHAT I WANT. When he opened his arms for Nina, a guard hurried over. \"Sir! I'm sorry\u2014you can't sit there!\"\n\n\"See, that's the big-picture stuff,\" Collin told Nina. \"Guards coming over and acting out the message of the piece! I would never think of that.\"\n\nThey took the glass freight elevator to view a three-story paper pagoda from above. They ate oysters. They drank Champagne. They listened to the jazz trio and they danced together. She didn't know the fox trot or salsa or tango, but Collin whispered instructions in her ear. \"One, two, onetwo, one two onetwo. Look up. Just look at me.\"\n\nHe had told her that his mother taught dancing, but Nina had always imagined him watching the class. Now she realized that he had learned as well.\n\nHe said, \"I had to dance with all the girls who didn't have a partner. I'd be the youngest one, and I was like a foot shorter, so I was staring straight at all these eighth-grade breasts.\"\n\n\"Sh!\"\n\n\"What? It was like the highlight of my childhood.\"\n\nShe was laughing as he spun her out. \"How do you think those girls felt?\"\n\n\"You need more Champagne,\" said Collin.\n\n\"Oh, I don't think so.\"\n\n\"I think you do.\"\n\nThey walked outside to the bar and got two glasses, but she didn't drink from hers.\n\n\"Do you think there's such a thing as too much fun?\" He set his empty Champagne flute on the railing.\n\n\"Well, it depends...\" she began.\n\n\"Okay, just by that answer I can tell you aren't drunk enough.\"\n\nAlready he was more at home than she was. He seemed born for music and Champagne and black-tie parties. He looked dangerously handsome, not tamed, but liberated in formal clothes.\n\nHe was leaning back against the railing and Nina was facing him. This was how Daphne saw them as she walked up the stairs to the deck. Her first thought was, Look at you, Collin, with your tuxedo and your girlfriend. Her second thought: You know I'm here but you're pretending you can't see me.\n\nShe had come only for a quick drink. That's what she explained to everyone who asked. She was still wearing her black Arkadia sweatshirt with the ripped hood.\n\n\"What happened to you?\" Tomas asked her at the bar.\n\nShe said, \"I got held up at EverCon. I had to tear myself away.\"\n\n\"It's almost ten,\" Peter pointed out.\n\nShe shrugged, trying to seem nonchalant, even as he studied her. \"I wasn't going to come at all.\"\n\nDrink in hand, Daphne made her way to the dessert table, where she surveyed lemon tarts the size of silver dollars, chocolate mousse in shot glasses, marzipan fruit, petits fours adorned with candied violets. Collecting one of each, she looked up to see Collin and Nina just ahead of her in line.\n\n\"Is that your dinner?\" Collin asked, and he was not avoiding her. He was expansive, and a little drunk.\n\n\"Definitely.\"\n\nIt was dark now, and they stood in candlelight. Collin with his princess at his side. All around them Arkadians were plucking petits fours like flowers. \"Nina, this is Daphne.\"\n\nDaphne lifted her glass in greeting. \"Hey.\"\n\nNina was confused. She'd heard Daphne's name. She was sure she'd met Daphne somewhere before\u2014but Collin was the one who knew her, demanding playfully, \"Where were you?\"\n\n\"Working! Did you like our protesters?\"\n\n\"What do you mean?\"\n\nDaphne smiled.\n\nNina said, \"You hired them.\"\n\n\"They got great coverage! They were on the news at six, and again at ten. Wait.\" Daphne checked her phone. \"They're on _now._ Look.\"\n\nNina watched Daphne and Collin exclaim over the unfolding newscast on the phone, and the whole evening began to change. She watched the two of them together, and their conversation, spoken and unspoken, seemed familiar. Collin had never mentioned Daphne, but she was not simply an acquaintance. She was his close friend.\n\nDaphne was explaining, \"I had to coordinate the protest, and then I did the giveaway at eight, but the worst was I got chased!\" She rolled up her sleeve and showed off the purpling bruise on her wrist.\n\n\"Who did that?\"\n\n\"A rabid fanboy ran me down. He had me cornered in a stairwell.\" Daphne spoke ruefully, but her eyes were mischievous as Collin examined her. \"And he was _big._ At least sixteen.\"\n\n\"Why would he chase you?\" Nina asked.\n\n\"You need ice.\" Collin pressed his cold glass to Daphne's wrist.\n\nSurprised and hurt, Nina turned to look at him, although he was just being Collin: friendly, flirtatious, drinking too much.\n\n\"Occupational hazard,\" Daphne said.\n\n\"What does that mean?\"\n\n\"It's like sometimes you get bitten. I feed new games to hungry kids.\"\n\n\"Why do they bite?\"\n\n\"Because they get wild.\"\n\n\"Is that the plan?\"\n\n\"No!\" Daphne protested. \"I just want them a little bit obsessed.\"\n\nCollin chided, \"You can't just tell people where to find you.\"\n\nHe was so warm, so playful with Daphne, that Nina took a full step back. He didn't even notice. His concern was all for Daphne as Nina walked away.\n\nShe drifted to the dessert table, then to the bar. The glass freight elevator descended gently, and all the music and all the art and all the laughter of the evening seemed to sink slowly, noiselessly. All of it seemed joyless now.\n\nIdly she watched the waiters carrying their trays. Champagne flutes by the dozen. Chocolates decorated with gold leaf. Nina wanted nothing but to get away. She had nearly disappeared down the wood stairs when Collin caught up with her.\n\n\"Stop and tell me what's wrong.\"\n\nShe kept walking to the parking lot. \"You have to ask. That's what's wrong.\"\n\n\"Daphne offended you.\"\n\n\"No, you did.\"\n\n\"Why?\"\n\n\"Because you were defending her!\"\n\n\"She's the one who got hurt.\"\n\n\"Do you know that?\" she demanded. \"How do you know that? You aren't thinking about that kid.\"\n\n\"This isn't like you,\" he said.\n\n\"Really? How am I supposed to be?\"\n\n\"I don't know. I don't go around telling people how they should be. I know how you are.\"\n\n\"You don't know anything.\"\n\nHe looked into her hurt eyes. \"You're jealous.\"\n\n\"Not true!\"\n\nNina turned her back on him, unlocking her car. She slipped inside and slammed the door. Then she opened the door again because she'd caught her dress.\n\nCollin said, \"Wait\u2014let me come with you.\"\n\n\"No.\"\n\nShe could be so lovely and so delighted with the world, and then she doubted everything. All he wanted was to see her happy, to make her laugh, but she had to find some catch. He could be moody, he had a temper, but hers had this weight, this crushing moral force. Nina wasn't just angry\u2014she had to drag in the whole universe. \"Don't be like this.\"\n\nWhat did that mean? Was she too serious? She couldn't help it. She was serious the way she was left-handed, and she couldn't change. She had been raised on lies and fairy tales, and she hated deception and excuses. She had grown up with games, and she craved truth.\n\nHe repeated, \"Let me come with you.\"\n\n\"No, stay. You want to.\" She nearly caught his fingers as she closed the car door.\n\n\"You know what your problem is?\" he called out as she started the engine.\n\n\"My problem is you,\" she called back. \"My problem is that I'm in love with you.\"\n\nHe could hardly see when he got home that night. Body shaking, teeth chattering with fever, Aidan collapsed inside the door. His mother said, Oh, my God, what happened to you, even as she asked, Where have you been, what did you do? At first he couldn't answer. He was throbbing with pain. He lay on the floor, and when his mother tried to help him up, he vomited. Oh, my God, his mother kept saying. I'm taking you in.\n\n\"No,\" Aidan groaned.\n\n\"Diana!\"\n\n\"I'm right here.\" Diana hovered near the kitchen door. Whenever Aidan got sick, she had a sinking feeling she would be next.\n\n\"Get me the thermometer from my bathroom.\"\n\nDiana raced up the stairs.\n\n\"Don't bite,\" Kerry told Aidan. \"Don't break the glass.\" A moment later, she turned around and told Diana, \"Help me get him to the car.\"\n\n\"Isn't the doctor closed?\"\n\n\"We aren't going to the doctor's office.\"\n\nHe thrashed and fought as they tried to move him.\n\n\"Ow!\" Diana turned to her mother. \"He scratched me!\"\n\n\"Come nicely or I'm calling an ambulance,\" Kerry said.\n\nDiana turned on Aidan. \"What is wrong with you?\"\n\nThey were trying to pull him through the front door, but he clung to the doorframe. \"Let me go! Let go!\" he screamed, and it took all their strength to hold him, even in his weakened state. He demanded, \"Give me my sword.\"\n\n\"Is he high?\" his sister whispered. His eyes were glassy, his face dead white.\n\n\"Hold his arm back,\" Kerry ordered. \"His left arm. I've got his right.\" Pinning his arms back, holding him on either side, they got him through the door, and all the time Kerry spoke to Aidan. Her voice was stern and quiet in his ear. \"You're sick. You're delirious. You're fighting a hundred and four fever and you might have a bacterial infection. I'm taking you to the ER.\"\n\nStep by step, she talked him down the stairs into the humid night. She never took her eyes off him, so he felt her gaze, even though he wouldn't look at her. \"Now we're going to the car,\" she said, for Diana's benefit as much as for Aidan's. \"We're going to the car. I'm going to strap you in.\"\n\nHe stopped fighting. They couldn't tell whether he went limp from exhaustion or in protest, but they knew better than to let go. Half carrying him, they edged into the driveway.\n\n\"Get my purse,\" Kerry told Diana.\n\nNow Aidan cried out and tried to wrest himself away again.\n\nKerry pinned him against the old Subaru. Diana opened the passenger-side door, and he closed his eyes as Kerry strapped him in.\n\n\"Am I coming?\" Diana asked.\n\n\"No, wait here.\"\n\nKerry was already starting the car. She had to concentrate on Aidan. \"You'll be all right,\" she told him as she reached over to adjust his seat, sliding and tilting so he could lie back.\n\nShe touched his burning forehead, and he couldn't move his head away. His neck was like a superheated steel rod. He couldn't bend; he couldn't turn without bursting into flame.\n\nYou can do this, Kerry kept telling herself, as she drove through the night. You can do this. But when she began threading through the maze of Longwood's labs and hospitals, Beth Israel, Joslin, Dana-Farber, she began to cry.\n\nShe pulled up in front of Children's, her own hospital, and she got help with a wheelchair and rushed Aidan inside.\n\nHow many times had she met parents as they rushed in hyperventilating, sick with fear? Take a breath. Sit for just a second. Let me get you a cup of juice. Try to drink this. Just take a little sip. All those years, all the waking hours of her working life, Kerry had nursed other people's children. Now she was the one wheeling her son past fish tanks and patient art, doves and rainbows, painted words: DREAM! HOPE! LOVE! She was the mother printing Aidan's birth date, affirming no known allergies, as children's cartoons laughed, wheezed, screamed. She was like everybody else who came in with a sick baby. The only difference was that Kerry didn't have to wait.\n\n\"I'm taking you to see the doctors,\" she told Aidan, unconsciously simplifying her language. \"They're going to look at you.\"\n\nHe was unresponsive when the nurse spoke to him. He knew vaguely that he was in the hospital, but the voices were too loud, the walls too bright. Docile, he gave over his body, submitting to the doctors' tests. He wasn't trying to cooperate; he was too sick to object.\n\nEveryone was explaining what would happen. Some tests, some blood, some time to rest, some spinal fluid to see if he was cloudy. He knew the answer. He was so cloudy that he couldn't see. A dark moon blocked his vision, but he could hear and he could feel. Jackhammer pain, sharp needles in his neck, and in his head, behind his eyes. Dry heaves and distant voices. A lurching, sickening puncture in his back.\n\n\"They're going to admit you,\" his mother told him, and he thought, Where did I get in? \"They're going to help you heal,\" she said. \"That's why you've got the IV. That's for your antibiotics. Don't pull it out.\" He had not been aware that he was pulling. Those were just hands. Nothing to do with him.\n\nKerry called Diana and told her that Aidan had to stay the night.\n\n\"Okay,\" Diana said.\n\n\"They're doing some tests,\" Kerry said. \"He has an infection.\" She couldn't bring herself to say \"meningitis.\"\n\n\"Okay.\"\n\n\"I have to stay.\"\n\n\"When's he coming home?\"\n\n\"I don't know.\"\n\nShe took this in. \"Could he die?\"\n\nFor a split second, Kerry hesitated. Then she said, \"No!\" as if the very question were offensive, as if from that day forward nobody would ever die again.\n\nA muffled sound at the other end of the line.\n\n\"I need you to stay calm.\"\n\nLong pause. \"Okay,\" Diana said.\n\nAll that night, Kerry stayed with Aidan in the ICU. She envisioned his recovery. She had seen it happen. At the same time, she could imagine losing him; she'd seen that too.\n\nShe watched him in the darkened room; she just sat and watched. She tried to pray, but fear silenced her. The enormity of the situation smothered her. Every thought flew to Aidan. He looked so still, so straight, so narrow, like a beautiful felled tree. His hair stuck out every which way, and Kerry thought, You need a haircut. What a strange idea, when he needed so many other things first. His hands lay quiet on his blanket. Beautiful fingers, long and tapered. Piano fingers, Priscilla always said. Piano fingers, gaming fingers, lightning quick.\n\n\"You were hardly ever sick,\" she told him. \"When you were little, you barely had an ear infection. If you and Diana came down with something you'd be sick for just a day. That's all.\"\n\nHe breathed evenly, but his eyes were only partly closed. She always told parents to assume their child could hear them.\n\n\"You're strong,\" she told him. \"You're very strong, and you're in the right place. This is the place...\"\n\nHe stirred and cried out softly.\n\n\"It's me. It's just me,\" she told him in her low, urgent voice. \"I know we've fought. It doesn't matter now. It didn't even matter then. I love you just the same as I did when you were born. I'm with you now, just the way I was then.\"\n\nNurses came to check on Kerry and Aidan. Robyn, the duty nurse, came in to work, but she and others also came to clasp Kerry's hand, to bring her tea, to promise, I'll watch him if you need to sleep. I'll sit here for you. No, no, Kerry said. I'll stay. But all that long night, as Kerry watched over Aidan, her friends watched over her.\n\nWhile Kerry sat with Aidan in the hospital, Nina paced her apartment. She had changed from her silk dress into a cotton nightshirt, tossed her hair clips onto the dresser. She tried watching a movie on her computer. Then she tried to read her battered Thoreau. She leafed through her students' papers, but all she saw was Collin holding his cold glass to Daphne's wrist.\n\nShe tried to stifle jealousy, or at least outwork it. Sitting on the couch, she stacked and restacked her students' end-of-year portfolios. One by one, she opened composition books. She stared at Brynna's perfect, even printing, Marisol's huge letters, ballooning in blue ballpoint, every _s_ a sail, every dot a circle, round as a full moon. She opened Anton's journal and saw anime drawings, spiky-haired punks with evil grins and inky eyes.\n\nThe sun was rising when Nina fell asleep with her computer and her headphones and her students' journals on the couch. She woke hours later in a jumble of composition books and wires. Bleary, barely conscious, she thought of Collin and she missed him, even as the events of the evening came rushing back to her. Half dreading, half hoping for a message, she checked her phone and found a medical alert from school: ONE OF OUR STUDENTS WAS ADMITTED LAST NIGHT TO CHILDREN'S HOSPITAL. AIDAN O'NEIL IS IN SERIOUS BUT STABLE CONDITION WITH A DIAGNOSIS OF MENINGITIS.\n\nNow she was awake, startled from her own unhappiness. Aidan? She thought about the disciplinary meeting\u2014Aidan's performance in DeLaurentis's office. SOME FORMS OF MENINGITIS ARE CONTAGIOUS. WE ARE WAITING TO LEARN WHETHER...What was it about him? A kind of chivalry as he took the blame, a strange protectiveness, although he had dragged his sister into trouble, copying her work. There was something brilliant and dangerous about Aidan. \"I've had kids like him before,\" Mrs. West had told Nina after the plagiarism meeting. \"I taught a kid named Daniel with an attitude like that. You know where he is now? In jail.\"\n\nAlarmed, uncertain, Nina sat up. She remembered that Diana had written about her brother. He was a liar or a stoner. Diana had written unhappily about him.\n\nNina began searching for Diana's Journal of Discovery. She sifted through her composition books, and eventually she found it in an unread pile in her bedroom. Nina sat in bed and returned to the black-ink thicket of Diana's writing. Entries on _The Scarlet Letter,_ entries on Thoreau. When she'd marked the journal the first time, Nina had been checking to see that Diana had done the reading. Now Nina skipped over Diana's analysis, such as it was, and searched out the places where Diana digressed or misinterpreted or ignored the question.\n\n_Q. How would you characterize Hawthorne's ghost stories? How does he use ghosts to represent the past?_\n\n_Living with my brother its like living with a ghost_. _If you leave out a sandwich it might not be there later but you won't see him, only his shoes. When I see him_\n\nNina turned the page.\n\n_...I'm like who are you because he is inside his game. He is obsessed with this game like he prefers to be there\u2014even though he is not allowed. Hes not supposed to play but he always finds a way to get inside._\n\n_Q. How would you characterize the mood in Whitman's opening line: \"I sing the body electric\"? What do you think about when you read these words?_\n\n_Danger. High voltage! People turning into machines. Instead of blood vessels wires instead of brains circuits chips etc. Or machines turning into people inhabiting people infecting their brains, controlling them so they do whatever the machine wants them to._\n\n_Or what would it be like if a machine controlled the world instead? Like the world is a game and you just live in it? You think you're a player but actually you're getting played?_\n\nOnce again, Nina remembered Aidan in DeLaurentis's office. Clear-eyed, he had countered every question. Such was his confidence. He might have been a patriot captured and questioned by the enemy, or a young saint who heard angelic voices.\n\n_Q. Consider Bartleby's famous line: \"I would prefer not to.\" What is he rejecting? What do those words mean to you?_\n\n_I would prefer not to wake up in the dark that really sucks. I would prefer not to go to school. I would prefer not to be seen. I would prefer not to enter the cafeteria. I would prefer not to eat lunch. I would prefer not to care what anybody thinks. I would prefer not to do my homework. I would prefer not to take out the recycling. I would prefer not to hear squirrels in the attic. I would prefer not to go down to the basement (mice). I would prefer no games, no paint, no lies. I would prefer you to leave since your gone already. CU Aidan. CUCU_\n\nNina stared at that chain of letters linked together. No paint. No lies. He had tagged the school. Diana had confessed it.\n\n_Everybody's got secrets. Whats more interesting is when you find out other peoples. Then the question is do you tell on THEM? For example my twin and I were like blood brothers only moreso. Now its like he moved away. I hear him whispering daphne daphne._\n\nNo, it couldn't be. It was too strange. She reread Diana's words, expecting them to change\u2014but they did not. This was where she'd seen Daphne's name\u2014in the Discovery Journal. And this was how Daphne did her work. Aidan was Daphne's rabid fanboy. Nina could see it even now. Daphne had led him on, although she had not mentioned where she'd led him.\n\nDon't panic. Think. There were rules. There must be rules for this. The school had guidelines for red flags and danger signs. In her capacity as Language Arts Team Leader, Mrs. West had briefed new teachers on calls for help. I think I'm going to drown myself. I fantasize about shooting my parents or my teachers or my classmates. I'm going to set myself on fire in the gym. That kind of thing. \"You'll know it when you see it. And if you see it, report it,\" Mrs. West told the assembled faculty at orientation. \"Do not hesitate!\"\n\nWhat should she do? Go to Mrs. West? Run to Miss Sorentino, the school psychologist? Diana's journal was neither hate speech nor suicide note, but suggested Aidan had committed a crime. The diary revealed this\u2014but could anybody apart from Nina see it?\n\nHer mind raced on, and she imagined Aidan with a new can of paint, then with a dagger, then a gun. Millions play, her father always told reporters. In every population of this size you'll find a couple of crazies. That was what her father said after the mall shooting in Connecticut, the massacre in Norway. Gaming is like the world. No one can prove gaming causes violent crime. It's not what gaming does to you. Gaming is what you bring. Despite this, she imagined Daphne enticing Aidan. She imagined EverCon, and beyond that a bigger marketplace, and she felt responsible\u2014not theoretically, but personally, responsible. Arkadia was not simply property of her father and her uncle. Viktor had given her a share of the company when she was a little child. Nina was a silent partner in more ways than one; she never spoke of this.\n\nShe was implicated. She had not divested herself, but had maintained her position. She had been practical, imagining she could use her money to do good work. She had been loyal to her father's enterprise, despite her doubts. What a hypocrite she'd been.\n\nShe threw off the covers and snatched up Diana's journal. Took the composition book to her desk in the living room and began scanning pages, one after another.\n\nNina? Collin texted Nina on her phone. u there?\n\nno, she typed back.\n\ncan you talk?\n\ncan't.\n\ncant or wont?\n\nwe'll just fight.\n\nA moment later, phone in hand, he was standing in her doorway. \"Let's fight, then.\"\n\n\"You scared me!\" She was pale, exhausted, her hair rough, unbrushed.\n\n\"What are you doing?\"\n\nShe cut him off. \"There's just one thing I want to say to you.\"\n\n\"Say whatever you want.\" He meant, Say you hate me. Say I drink too much. Say that I was out of line.\n\nBut she said none of those things. She stood in front of her scanner and told Collin, \"Daphne can't play with kids. It's wrong, and if you don't see why, then I can never, ever be with you.\"\n\n\"Why don't you just admit you're angry at me?\"\n\n\"You're not thinking what Daphne does to people.\"\n\n\"She's never done anything to me.\"\n\n\"She plays kids. She played Aidan and he chased her and now he's in the hospital.\"\n\n\"What? Slow down! Aidan O'Neil? Kerry's kid?\"\n\nAt last Nina had startled him. She showed Collin the medical alert from school and he said, \"What if he got Daphne sick?\"\n\n\"That's what you're worried about?\"\n\nIt seemed obvious to Nina that nothing bad would ever happen to Daphne. She was the perpetrator, not the victim. \"She led him on. That's her test marketing.\"\n\n\"With one sixteen-year-old. Who could be anyone!\"\n\n\"She uses him to get the word out.\"\n\n\"How would Aidan get the word out?\"\n\nShe spoke with fevered intensity. \"In pranks. On walls.\"\n\n\"What are you talking about?\"\n\n\"He tagged Emerson.\"\n\n\"Prove it.\"\n\nShe almost showed him Diana's journal. She nearly opened to the page\u2014and then she held back, afraid of making a mistake, betraying student-teacher confidentiality, exposing Aidan, helpless in the hospital.\n\n\"You're imagining what _could_ happen,\" Collin told her.\n\nShe took a deep breath. \"I'm not imagining. I know.\"\n\n\"You know for sure? You can say without a doubt?\"\n\n\"Yes.\"\n\n\"How?\"\n\n\"I can't tell you.\"\n\n\"So I just have to trust you.\"\n\nShe nodded.\n\n\"Then can I ask you something?\"\n\n\"What?\"\n\n\"Why can't you trust me?\"\n\nDiana slept on the couch. The air conditioners were too heavy to carry from the basement, and without them, it was too hot to sleep upstairs. Monday morning, she padded to the kitchen and listened to the house creak. The phone rang, and people left messages. Mrs. Solomon called to inform Kerry that Diana was absent from school.\n\nMaia came over with a casserole of vegetarian enchiladas, and Diana said thank you very much, but as soon as Maia left, she slid the whole thing into the freezer. She was too hot to eat. Too hot for anything. She took off her shirt and lay on the living room rug. Then she took off her shorts. Her long hair smothered her neck and shoulders.\n\nWhen her mother phoned that afternoon, Diana roused herself to answer. \"Hello.\"\n\n\"Diana?\"\n\n\"No, this is a robber,\" Diana said.\n\n\"What?\"\n\n\"Yes, Mom. It's Diana. Who else would it be?\"\n\n\"How are you, sweetheart?\"\n\n\"I'm fine,\" Diana said.\n\n\"He's sleeping now. He's stable,\" said Kerry, although Diana hadn't asked.\n\n\"Okay.\"\n\n\"You can see him. I can't leave, but Priscilla says she'll drive you.\"\n\nDiana didn't answer.\n\n\"Are you there? Diana?\"\n\n\"What?\"\n\n\"Just go next door and ask. She's glad to take you. She has to finish a lesson this afternoon but then...\"\n\nEven as her mother spoke, Diana climbed the stairs to the stifling second floor. Her entire body was slippery with sweat.\n\n\"You're breaking up,\" Kerry said.\n\nDiana dropped the receiver into the laundry hamper in the bathroom. Then she peeled off her underwear.\n\nThe bathroom had white penny tiles on the floor. There was a shower curtain printed with yellow rubber duckies, a small sink with a square mirror over it, a glass shelf cluttered with acne wash, toothpaste, and a jar of toothbrushes, combs, and scissors. Diana's blue bathrobe hung on a hook on the door, but it was too hot to wear. For a long time, Diana stood in the cold shower. She stood there until she couldn't stand it any longer. Then she stepped out, shivering.\n\nEven she was surprised by the girl in the mirror. She wasn't thin, but she was sturdy, muscular from so much running, her breasts pink-tipped, her eyes fierce, her hair dripping down her back. She didn't recognize herself at all.\n\nShe took a comb and raked all her hair forward so that it covered her face. She could barely see herself in the mirror through black strands. She took the scissors and held it open in her hand, cool blades against her palm. For a moment she wanted to feel that blade slicing her skin. She imagined her own blood, red-black, spotting the tile floor. She thought about it, but she cut her hair instead.\n\nDeliciously cool to shear off all that heavy hair. She didn't try to cut evenly, just hacked away until she could run her hand through the mop she had left. Her eyes were darker in the mirror, her face paler. She looked almost like a boy.\n\nShe dressed in basketball shorts, a sports bra, and a black mesh shirt, but couldn't find clean socks. She glanced at her bed covered with laundry, her dresser piled high with crap. Pencils and rubber bands, marbles and beads, staplers and the wrong-size staples, a broken alarm clock covered in thick dust. She said aloud, \"I feel bad for whoever lives here.\"\n\nIn the kitchen she found a stack of paper grocery bags. She took them to her room and threw them on the floor. She announced, \"Let's recycle, shall we?\" and swept her dresser clean. She picked up all the stray school assignments and receipts and cardboard boxes from her floor and stuffed them into bags. Stripped her bed of its faded princess sheets and stuffed them into a bag along with her pillowcases. She pulled plush animals from her closet. Duck. Rainbow fish. Black horse with a mane matted from the dryer. There was a small bison, which she had called her \"dison.\" \"Time to get a life,\" she told her childhood elephant.\n\nThe doorbell startled her. At first she didn't answer. Then she heard the bell again. Her mother? No, stupid, your mother has the key. Except she might have lost it! She rushed downstairs.\n\nMiss Lazare.\n\nDiana held the glass storm door open with her body.\n\n\"Diana?\"\n\n\"Yes?\" Forgetting her new look, unconscious of the stuffed elephant in her hands, Diana wondered why Lazare seemed so confused.\n\n\"I was so sorry to hear about your brother,\" Lazare began.\n\n\"What do you mean?\" Diana felt a surge of dread.\n\n\"I heard he's in the hospital.\"\n\n\"Oh.\" Diana could breathe again.\n\n\"I'm sorry,\" Lazare repeated.\n\n\"Thanks.\"\n\n\"I wanted to ask if you're okay.\"\n\nDiana gazed into Lazare's anxious eyes. \"I'm good.\"\n\n\"Because you weren't in school.\"\n\n\"So I hear,\" Diana said.\n\n\"And I have your Discovery Journal.\" Lazare held out the composition notebook. \"And I was wondering...I wanted to know if we could...You wrote some things that made me think you might want to talk, either to me or to someone at school.\"\n\nDiana took the composition notebook.\n\n\"There were some things you wrote about your brother,\" her teacher said in that very gentle voice adults used when they wanted to pry something out of you.\n\nJust a step back, and Diana could retreat inside. The glass door would snap shut again.\n\n\"You seemed worried,\" Lazare said.\n\n\"Not really.\"\n\nDiana's teacher took a deep breath. \"You say he's playing nonstop, and you don't know if you should tell on him.\"\n\n\"Wow,\" Diana said slowly, amazed that her teacher would come out and talk to her like that, not only rereading but repeating written words\u2014dragging Diana's dark, unformed thoughts into the light.\n\n\"I thought you might want to\u2014\"\n\n\"Hey!\" Diana cut her off. \"Are you talking about my family?\" Her face was burning. She had never spoken to a teacher in this way.\n\nLazare said, \"I was afraid you wrote about your brother for a reason.\"\n\n\"No, not at all,\" Diana said. \"Only for the assignment. Educational purposes only!\"\n\nShe stepped inside and the glass door snapped shut. Then she closed the wood door, and locked it too. She was the only one left, and it was up to her to defend the house. Go away, Diana thought. Get the hell away from me.\n\nShe ran down to the basement, which was piled with cartons and moldering window shades. She pushed hard and opened a door to the utility room. There, behind the water heater, she had hidden her doll carriage. Inside the carriage under blankets lay Aidan's black BoX. \"Stay there,\" she said, and stuffed her journal deep inside. \"Sit tight.\" She left her elephant on top.\n\n\u2014\n\nHour after hour, Aidan slept. The sun had set, but he didn't know the difference. He had been moved from the ICU to a regular room, and in two days he had two different roommates, a little boy and then a baby in a hospital crib. Doctors rounded, nurses came to change his bags and check his drips. Janitors emptied receptacles of trash and sharps and linens. At night they buffed and polished the smooth floors. Children tottered outside his door and their parents supported them, one step at a time. A chaplain came and blessed the baby on the other side of the curtain. Aidan slept on. He saw none of this.\n\nSitting at his side, Kerry watched pain, fear, longing flash across his face, and she spoke to him. Sometimes she prayed, \"Please.\" Sometimes she whispered, \"Aidan. Where are you?\"\n\nEven as she sat with Aidan, he fought on. He killed a thousand bats, yet every moment, more attacked. With huge effort he shook them off, and still they flew at him from their caverns until at last he could do no more than kneel, feeling for some tunnel, or some hole in which to hide. If he could get to the river, the great silver river, if he could immerse himself, that dead water would cover him entirely. Nothing could hurt him then.\n\nKerry roused herself to cheer him on. \"You'll be all right. You're breathing on your own. Your body is resting.\" She asked and answered her own questions. \"Tell me, Aidan. Do you know where you are? You're at the hospital. Do you know why you're here? You have meningitis. Do you know what's going to happen? You're going to get better. Can you hear me? Can you squeeze my hand?\"\n\nHe did hear his mother, not all the words, but most of them. Do you know you're at the hospital? Her voice was both close and far away, like the rush when you pressed a seashell to your ear. Can you hear my hand?\n\nTen o'clock at night, Kerry dozed in her chair, and Aidan clawed for some way out, trying to escape the bats feasting on his flesh.\n\n\"Mom?\"\n\nKerry jumped at the light touch on her shoulder. \"Aidan!\" For a split second, she thought he'd risen from his bed. She saw him standing before her in his black shirt, basketball shorts.\n\n\"It's just me,\" Diana said.\n\n\"What did you do to yourself?\" Kerry cried out. \"What happened to your hair?\"\n\n\"I cut it.\" Diana was gazing at Aidan. Her mother was still talking, but Diana could barely hear.\n\nSleep-deprived and overwrought, Kerry felt possessed by her twins' strange reciprocity. Even as she'd told herself Aidan needed a haircut, Diana had chopped off her own hair.\n\n\"You just went at it with a scissors? Diana. _Why?_ \"\n\n\"I was hot,\" Diana answered automatically.\n\n\"I left you all alone.\"\n\nDiana just stared at Aidan, with the needles in his hand and tubes and bags.\n\n\"Where's Priscilla?\" Kerry asked her daughter. \"Did she drive you?\"\n\n\"No.\" Aidan was shrouded in his sheets, his faintly printed hospital gown. His chest seemed empty as he breathed in and out.\n\n\"I don't think you should take the T at this hour of night.\"\n\n\"I didn't take the T. I just ran over. Hey, Aidan.\"\n\n\"He needs to rest.\"\n\n\"It's me,\" Diana told her brother.\n\nKerry broke in, \"Wait. What do you mean you just ran over?\"\n\n\"Look at me,\" Diana said, but Aidan's eyes remained closed. There was something strange about the lids, as though they'd been sealed, gold lashes glued together. His skin was strange as well, almost translucent. Yes, he was breathing, but he wasn't sleeping normally. He was becoming a statue, an icon of himself. She could see the metal in his cheek. \"Open your eyes.\"\n\nHe didn't move.\n\nDiana's voice trembled. \"Open your eyes because you're scaring me.\"\n\nHe stirred, turning ever so slightly toward her.\n\nKerry stood next to Diana. \"Keep talking.\"\n\nDiana's voice sounded hollow, as though someone else were speaking. \"So now that you're here, I figured it was a good time to cut my hair, flunk my exams, and sell your stuff.\"\n\nHis eyelids twitched.\n\n\"Kidding!\"\n\n\"Go on,\" Kerry whispered.\n\nDiana told Aidan whatever came into her head. \"Remember in fifth grade when we had that field trip and all the mummies kept beeping when we stepped too close?\"\n\nNo response.\n\nRemember the bodies in their coffins? Diana thought. Remember their gold masks? She could see Aidan's body; she could see his mask.\n\n\"Remember when Jack caught that bird?\" If Aidan remembered anything, he would remember this. On a different field trip, in Copley Square, Jack had actually caught an obese pigeon with his bare hands. Teachers started screaming, Oh, my God, what are you doing? Don't touch that thing\u2014it's filthy! Drop it! Drop it! The whole time the pigeon kept flapping in Jack's hands.\n\nNothing.\n\n\"Remember when we went on the T into Boston by ourselves and we thought someone was going to kidnap us? Remember when we thought if we clasped our hands together we could cast spells? Remember how I turned into a deer? I ran three miles tonight. I ran across the BU Bridge and followed the signs for Longwood Medical Area. I was like that crazed deer in UnderWorld.\"\n\nWhen she said \"UnderWorld,\" Aidan opened his eyes.\n\n\"Don't stop,\" Kerry whispered.\n\nAidan's irises were bright as coins, expressionless. Looking at him hurt, like staring at the sun.\n\n\"Keep talking,\" Kerry said.\n\nDiana repeated the one word that was working. \"I've been playing UnderWorld on your computer. I've been using your account,\" she added, just to get a rise out of him. \"No, actually, I can't log in.\" She paused. Then asked tauntingly, \"Could I have your password?\"\n\nAidan's voice was hoarse but distinct, as he spoke for the first time in three days. \"No!\"\n\nKerry was crying. She was so relieved. Her tears were falling all over Aidan's pillow. Diana was embarrassed, because her mother always made such a big deal about everything, and because when she started crying she made Diana cry. \"Aidan, make her stop,\" Diana said, brushing away her own tears, but Aidan did nothing. He had no energy to tell his mother what to do. To tell the truth, if he'd tried a stunt like that, Kerry would have cried even harder.\n\nThe nurse came in. Everyone was talking, but Aidan looked up at Diana. The gold was fading from his cheek, and his eyes were human, soft again, slightly amused. Can you believe this? he was asking silently.\n\n\"You're awake!\" Kerry told him, as if he didn't know.\n\nThen Aidan smiled at Diana and she knew exactly what he meant. Their mother was so crazy. Dying was so boring. As soon as I can walk, I'm outta here. He said all this without words, and Diana understood. Kerry could read her children, but only haltingly. They were her second language. Diana was a native speaker; she came from Aidan's country. When he closed his eyes again, she knew he wasn't going anywhere.\n\nNow the air-conditioning rushed over her, and her sweaty T-shirt chilled her skin. The hairs were standing up on her arms, and she could feel each one individually. She could feel everything from the most enormous, overarching joy down to the jagged middle toenail stabbing the toe next to it inside her right shoe. She wasn't just happy, she was thirsty. Actually, she was starving. She realized all this as she watched Aidan drift off to sleep again. Kerry was still tiptoeing around the bed, but Diana said, \"Mom, could we order pizza?\" because everything was good now. Aidan had decided to come back to life.\n\n\u2014\n\nThe next day, he began to look about him. He saw his IV, his plastic bracelet, his scanty gown, his uneaten dinner on the swing arm tray. He began to see the colors of the hospital, grays and muted pinks, the red sharps container on the wall.\n\nHis eyes were fierce, even as he lay crumpled in his bed. He seemed to Kerry like a rescued bird of prey, one of those injured hawks caged with a few dead mice for dinner. He was quiet and obedient and\u2014he even let her touch him\u2014almost tame. Stay like this, she thought, even as she prayed for his recovery. Stay gentle. Please don't fly away.\n\nSlowly his strength returned. He lost track of days, but every morning he limped along, leaning on his mother in the hall. Through open doors he saw other mothers in other patients' rooms. Beds decorated with helium balloons, windows covered with cards and paper flowers. Wasn't he too big for this? At the end of the hall they stopped at a lounge filled with toys and dolls and children's books. There were shelves of children's movies. \"People donate their old stuff,\" his mother said. There was a play operating table as well, where a large teddy bear awaited surgery. Or was it a rabbit? He couldn't tell. He had to rest, and, sinking into a tiny armchair, he felt childish and unsteady. He was growing younger and more wobbly by the minute, even as his mother glowed, telling all her nurse friends, We're so lucky. Look, we're up and around!\n\nHe knew where his mother was going with all this. In her mind, if you were lucky, you were also blessed, and once you were blessed, it was just a tiny jump to everything happening for a reason, and God putting you on this earth for a purpose. He didn't buy any of it, but he didn't argue either. Leaning on Kerry as they walked back to his room, he was simply grateful for his mother, and happy that she didn't have his BoX.\n\nHe was thankful in small ways as well. Glad to wear his own pajamas, grateful to lose his catheter. He felt something else too, a strange curiosity as he observed his own recovery. After four days at the hospital, he closed his eyes and he saw nothing, no spots or sparks, no monsters approaching, just darkness. On the fifth morning, he woke empty, as though he'd run out of dreams. Could that happen? Could your imagination actually run out of things to see? Fascinated, he tried to remember where he'd been last. Caverns? Tunnels? Crevasses? Which circle? Seven or Eight? Briefly his gaming history vanished, and he could not remember where he had left off.\n\nOnly Daphne stayed with him. The kiss, the shock of meeting her, his intense desire, and his humiliation. He wanted to speak to her, wanted very much to hurt her. He imagined ripping her open with his sword. But his fantasy had little heat. His mind drifted away.\n\nHis fingers began tapping. Softly they drummed the edge of the mattress, and he thought he would be a drummer. He would play drums as he had before, in Liam's band. Or he would play keyboard. Scales and sonatinas returned to him as he lay in bed, and he thought, I'll play again.\n\nThe next day this conviction faded. He watched the young resident listening to his heartbeat with a stethoscope. For the first time it occurred to Aidan how strange that was, to listen to another life. The doctor was listening intently, as though he could hear distant hoofbeats, and Aidan thought, I will study medicine.\n\nBy evening this vocation floated off as well. All his ideas were abstract, all his desires theoretical. He observed, he admired, he imagined, but he wanted nothing\u2014not even UnderWorld.\n\nMaybe meningitis had wiped out his gaming life. He knew that this was what his mother hoped. This was what she really meant by lucky. When she said everything happened for a reason, she was praying that fever burned the games right out of you.\n\n\"Aidan.\" She was sitting on his bed, and the sky was dark outside his window. \"I want to ask you something.\"\n\n\"I know,\" he said. She wanted him to start over, to return from the hospital cleansed of all his sins. Then this ordeal would be a blessing, his meningitis actually an act of God.\n\n\"Could you promise?\" She looked so thin. Her hair and hands, her arms, were light as straw. All the color had leached out of her, as though she had taken on his illness for herself. Even Kerry's eyes were ghostly blue. \"Promise you won't play again.\"\n\nHe hadn't touched a computer in what seemed like years. Looking down upon himself, as from a great distance, his own thoughts seemed strange, oddly colorless, like clouds. Not even clouds, but the shadows of clouds on empty valleys. How still it was, how slow. Delicious to lie back powerless, to drop the threads of all your gaming lives. It was a kind of death, an abdication. His kingdom would carry on without him as he lay in state. He would sleep and sleep. It was easy to say yes.\n\nThe heat smothered Collin outside Arkadia's air-conditioned halls. He had been working such long hours, drawing such beautiful and terrible things, that he felt a kind of grief to leave, as though he were giving up his wings to walk the streets. Summer days were white, overexposed. Trees dusty, and all the flowers overblown. Old houses flaked, bricks needed repointing, and you couldn't do the job in one stroke either. You couldn't change colors in an instant, or render different trees, or refresh roses dry and withered in the sun.\n\nNina was waiting for him in her VW bug, and he eased himself in. \"Thank you.\"\n\n\"You're welcome,\" she said, shyly.\n\nQuiet, careful with each other, they drove past Waltham's office parks. Were those tiny birds filling the air? Like aeroflakes, they scattered and then drew together.\n\n\"Where do you want to go?\" she said.\n\n\"Anywhere.\"\n\n\"Let's get lunch.\"\n\n\"No,\" he told her. \"Let's get out of here.\"\n\nThey drove to Walden Woods, just a few minutes away. She wasn't dressed for hiking, and they had no food or water, but they parked at the trailhead and they went walking anyway, taking the dirt path into the shady trees.\n\nThe light was green, the boulders massive. Chartreuse silkworms swayed on invisible threads. Some leaves were bright, some dark and glossy, some dull, some pale, some olive, and all around them, pines grew straight up to the sun.\n\nIt was amazing how fast Arkadia dematerialized in this green light. Mountain ranges and wild horses, brilliant sunsets, Whennish alliances, even UnderWorld's silver river faded away. The lowliest leaf, brown and shriveled, brittle-veined, showed more life and detail than anything in EverWhen. The humblest rock displayed more intricate patterns, lichened, mica-flecked, cool and wet on its underside, sheltering a thousand ants.\n\nCollin helped Nina when the trail got steep. He offered her his hand as they crossed a shallow stream. His shoes squelched, his jeans, her skirt wicked water. Mossy stones slipped under them. No flying here, no bounding over rivers. The stream was slippery and refreshing, but their feet were slow, the whole forest heavy with summer heat.\n\nHer sandals weren't good for climbing, so they didn't get far. They found a leafy knoll, and turned off the trail to rest. Higher and drier than the rocky path, carpeted with pine needles, the knoll was shaded by young trees. Collin sat down, leaning back against a maple. Nina sat on his lap, leaning back against him.\n\n\"What if we were the size of tiny insects\u2014like mayflies?\" he asked.\n\n\"I wouldn't want to live for just one day.\"\n\n\"But it wouldn't seem like just a day to us.\" Collin closed his eyes.\n\n\"I would rather be a bird,\" Nina said.\n\n\"What kind? A little one, or something like an ostrich?\"\n\nShe answered slowly. \"Maybe an owl.\"\n\n\"Are you sleepy?\"\n\n\"Are you happy?\" Nina was so quiet, and so close, the question might have been his own.\n\n\"I am now.\"\n\n\"I went to see Diana,\" Nina told him, \"but she wouldn't talk to me.\"\n\n\"Of course not. You're a teacher,\" Collin said.\n\nShe turned to face him and she wasn't thinking about Diana. She was thinking about Daphne. \"I do jump to conclusions.\"\n\nHe pushed back her long heavy hair. \"It's okay.\"\n\n\"Not really.\"\n\n\"Nina,\" he said, \"it was a million years ago.\"\n\n\"A week.\"\n\n\"That's like a million years in EverWhen.\"\n\n\"I don't like her.\"\n\nHe tried to keep it light. \"She's not your type.\"\n\nShe settled down again, leaning against him. As far as Collin was concerned, their fight was history, and so Nina's next question startled him. \"Do you ever draw her?\"\n\n\"What?\"\n\n\"Do you draw Daphne?\"\n\n\"No!\" Collin lied reflexively. Why was she still thinking that way? How could she mention Daphne in this place, under these trees? \"She's obsessed with Peter,\" Collin added.\n\n\"So are you.\"\n\n\"Not like that.\"\n\n\"He's got you drawing nonstop. He'll get you to the point where you _can't_ stop.\"\n\nCollin frowned.\n\nNina said, \"He's hard to resist.\" She was thinking of the way Peter had driven Julianne, recording her for hours. Julianne had been sixteen when she sang those mermaid voices. She had wilted with exhaustion, but Peter would not let up. He kept her working and he kept watching her. \"I'm afraid he'll crush you.\"\n\n\"Give me a little credit!\"\n\n\"Sorry.\" Nina's face was pink, flushed with heat.\n\n\"You're way too serious,\" Collin said. \"What's the opposite of a martyr?\"\n\n\"A tormentor?\"\n\n\"No. Someone too good.\"\n\n\"A savior?\"\n\n\"Yeah, you think you have to be a savior.\"\n\nShe bent her head. \"I was out of line with Diana.\"\n\n\"And that's the other thing. You think everything is set in stone. You screwed up with Diana. So what? We had a fight. It's over. Just erase and start again.\"\n\nShe almost laughed.\n\n\"You see?\" He caressed her until she licked his neck, salty with sweat, rolled up his shirt, so that his bare skin slid against hers. He had her to himself again.\n\n\"What was that?\" She raised her head.\n\nChildren calling to each other on the trail below. The panting, jingling sound of dogs. \"Let's go back.\" She meant to her apartment, where they could be alone.\n\nThe children's voices faded away. Someone called the dogs, and they ran off.\n\nThey turned toward each other, tousled, pine needles in their hair. \"Do you really want to go home?\"\n\n\"No.\"\n\n\"Then stay.\"\n\nBut already mosquitoes were hovering, grazing their wrists. Nina sprang to her feet and shook them off.\n\nHe wished he could draw Nina then. Her white arms. Her waist, where T-shirt and skirt didn't quite meet. He wished he could do justice to her eyes, not gray but silver in the dappled light. Instinctively he felt in his pockets for a bit of chalk, but he had none. He picked up a stick. \"Pick a tree, any tree.\"\n\n\"What do you mean?\"\n\n\"Just choose one.\"\n\nShe pointed to the pine in front of her.\n\n\"Too close.\"\n\nShe pointed to a bigger pine farther away.\n\nHe flung his stick like a machete, end over end, and hit the pine square on the trunk. \"Another.\"\n\nShe pointed to a maple, even farther off.\n\n\"That's Peter.\" He picked up a second stick and she watched it cartwheeling through the air. It struck the trunk dead center and bounced backward.\n\nShe chose a dogwood, and an oak, another sticky pine, some trees close, and others far away. He threw and threw, and he hit every single mark.\n\n\"How do you do that?\"\n\n\"Pure talent,\" he teased, but it was true. He was outstanding at stick throwing. Fence climbing. Stone skipping. \"I'm good at all the sports nobody plays.\"\n\nShe kissed him and she felt his warmth.\n\nHe cupped his hands around her face, and he admonished himself as well as Nina. \"Remember this.\"\n\nIf Walden Woods banished Arkadia from your mind, the opposite was also true. At work, the outside world felt like a dream, far away and insubstantial. You earned real money, but you barely had the time to spend it. Collin bought himself his dream bike, an Eddy Merckx AX, but he was too busy to shop for other toys. He could afford a new apartment\u2014except he didn't have a day to look. Maybe it was for the best. He didn't want to be materialistic\u2014and he loved having money in the bank. He could pay for dinner. He could go to a movie without thinking.\n\nLong summer days turned into nights, and he slept at Arkadia on the hellions' black couch. He closed his eyes and imagined his hands on Nina's waist, his fingers unbuttoning her shirt, her hair tumbling down her bare back. He wanted her, but woke to work and play again. He drew horses for hours, and then he turned to EverWhen to stoke his imagination.\n\nThere was a cove near the Whennish shore where waves washed into tide pools, and you could search for pearls. There was another place he liked to wander, beyond the tree line on EverRest. The world was white, with craggy peaks of ice, the only shelter crystal caves between the rocks. You had to break your way inside, cutting down icicles, which shattered at your feet. In the deepest cave, a race of Dwarves served a white-haired, white-eyed queen. Her eyes were frosty, her hands translucent, her behavior secretive. You knew that she was hiding a rich jewel. Other gamers told you where. She had a ruby for a heart.\n\nHe played so late and worked so long that his own art raced away from him. His horses ran untamed through the Trackless Wood. Lovely onscreen, they overwhelmed you in the round. First you heard them. Hoofbeats coming closer. Then the sound of branches breaking as they tore the underbrush. Magical, unearthly, they surrounded you with tangled manes and flying legs and dark eyes flashing through the trees. Drawing them apart, he developed each horse as a character, the gentle dapple-gray, the chestnut mare with the scar on its right flank, the black high-kicking stallion. No detail was too small to realize, none too difficult to render. Any line he drew might live and breathe in game.\n\nHe sat with Nicholas in his black padded office, and they tried out sound prints for each animal. Collin had no control over UnderWorld's soundtrack, but it was magic listening to the possibilities.\n\n\"These are for the chestnut.\" Nicholas clicked a file on his monitor and Collin heard a quick light step, a sleek rustling, a whinnying, a single twig breaking. \"These are for the black.\" Nicholas opened another file, and Collin heard the stallion's pounding hooves, its heavy breath, the rip and tear of brambles in its path.\n\n\"What about when they run through water?\"\n\n\"Yeah, I've got this muddy sucking sound, and splashing.\"\n\nEven without visuals, Collin could hear the animals fording a stream. \"What if one of them slips? One of them could slip scrambling up the bank.\"\n\n\"Yeah!\" Nicholas said. \"You hear them slide and then they're struggling. You hear them flailing. And then they scream.\"\n\nNicholas was at least fifty, a veteran of three major games and countless game expansions. His beard was gray and his hair was thin on top, and a lot of times he had to wear reading glasses, but he played all kinds of guitars and he could sing. He was a vocalist of the Bob Dylan variety, singer-screamer-songwriter. \"How's this?\" He opened new files: whinnying, running, and footfalls. Clicking on his computer, he adjusted the volume, and suddenly a horse was screaming, thrashing, dying in a ditch.\n\nCollin flinched, and Nicholas said, \"Okay. We're doing something right.\"\n\n\"That was like the worst thing I've ever heard.\"\n\nNicholas leaned back his swivel chair. \"Thank you.\"\n\nHe looked so supremely satisfied that Collin said, \"You love it here.\"\n\nNicholas thought about this. \"Sometimes I hate it,\" he said at last. \"Sometimes I'm burned out\u2014but I try to stay inspired.\" He held up his phone, flashing a picture of his son.\n\nOf course, Collin thought. You have to support your kid. That changes everything.\n\n\"Sometimes I get bored,\" Nicholas admitted.\n\nUnconsciously, Collin massaged his right hand with his left. \"Then what?\"\n\n\"I keep going anyway.\"\n\n\"You force yourself?\"\n\nNicholas kept clicking at his keyboard even as he spoke. \"Yeah, but you keep going with your own shit too. You keep your little projects going on the side.\"\n\nA sound of bells, no, faint wind chimes, filled the padded room. The sound of wind rustling in the trees, and then soft chords on a guitar, music frayed around the edges, a wordless kind of blues. All this emerged from Nicholas's speakers.\n\n\"What's that?\"\n\n\"That's me,\" said Nicholas. \"That's my stuff.\"\n\nThe music stopped instantly when he closed the file. \"Okay, what else you got?\"\n\n\"I'm drawing the stallion. Again.\"\n\n\u2014\n\nThat night in the conference room called the Keep, Collin presented his latest stallion to the hellions. He touched his slate and transferred his small drawings to the big electronic board.\n\nIn silence, Peter gazed at Collin's creation, a horse scarier than any other, a biter and a kicker with a bold head and hard black eyes. It was crueler than Collin's previous attempts. Each of those stallions had caught Peter's attention briefly, and then, one by one, they wouldn't suit. They had been too heroic or too noble, or simply too beautiful for UnderWorld. Nobody could say that now. The new horse looked like death, with its gaunt body and vicious mouth, its ragged coat, translucent, dirty white.\n\nPeter frowned, and Collin steeled himself.\n\nPeter examined Collin's work in silence. Come on, thought Collin, but Peter didn't speak.\n\nHe took the stylus from Collin's hand. Then, wielding the slender rod like his own wand, Peter touched the stallion's eye. It took only a moment, but the effect was brilliant, ghastly. The horse's bright eye filled with blood. Now the stallion looked possessed. Now Collin's drawing became a character, riveting, repellent.\n\nThe whole room hushed as Peter stepped back from the board. One touch and Peter had transformed Collin's horse entirely.\n\n\"That's sick,\" said Peter, contemplating his own work.\n\nA team player would have rejoiced. A humbler artist might have thanked Peter for the lesson. Collin did neither. His feelings were confused. Awe mixed with anger, a sense that Peter had destroyed something. And yet Peter had not destroyed the horse. He had revealed its full potential\u2014and made it his own. All eyes were on Peter, all praise to him. Even Daphne clapped her hands. It's true\u2014you're obsessed with him, thought Collin. You can't take your eyes off him.\n\nBut the moment passed and Daphne was herself again. She was warmer to Collin than before, staying late, after everybody left, laughing, commiserating, showing off her work. She drew up a chair and revealed her new project, posting on fan forums as an anti-gaming activist named Christian Wench.\n\n\"Listen to this.\" Daphne read from her tablet, _\"Rise up against UnderWorld! It is the work of Satan. Destroy this game before it destroys our children!\"_\n\n\"What's the point of that?\"\n\n\"Rallying the troops!\"\n\n\"More like aggravating them.\"\n\n\"That's publicity,\" she said. \"We got a hundred responses in the first minute.\" Daphne scrolled through furious comments. \"Oh, look. My second death threat. _I'm cuming for u cutting off your..._ \" She seemed amused, if slightly rattled, by the hatred Christian Wench inspired.\n\n\"That's not funny.\" He was reading over her shoulder.\n\n\"Oh, come on. I'm trying to cheer you up.\"\n\nHer words startled him. She had never before acknowledged what he might feel, or admitted any impulse beyond the desire to win.\n\n\"I don't need cheering up.\"\n\n\"You looked crushed in there.\"\n\n\"Fuck you.\"\n\n\"Play with me.\"\n\n\"Why?\"\n\n\"Because no one's here.\" Her tone was light, but her breath was soft and eager. The building couldn't be empty, and yet for now they were alone. Other hellions had gone home. Their idle workstations had passed into dream mode, monitors displaying ARKADIA in stars.\n\nShe took both his hands and he let her pull him to his feet. She reminded him of Noelle when she was high. Eager, sweetly wild. In the pale starlight, Daphne's eyes took on a strange sheen from too much gaming, too much everything. \"Come on.\"\n\nHe wanted her then. Hands on her hips, he pulled her in, but she sprang back, teasing, \"Duel in EverWhen.\"\n\nInsulted, he shot back, \"No.\"\n\n\"What, then?\"\n\nOf course she knew exactly what he wanted. He stepped closer, but, like a boxer in her hooded sweatshirt, she danced out of reach.\n\nHis heart was pounding, but he kept his voice steady. \"I want to draw you.\"\n\nFor once he surprised her. \"You've drawn me before.\"\n\n\"I want to draw you like this.\"\n\nShe held still for just a moment. His gaze unnerved and attracted her. She followed as he wheeled a swivel chair into the conference room where he had presented the ghost horse that afternoon. \"Here. Now.\" With a swipe of his hand, he erased the red-eyed stallion covering the board. \"Sit there.\" He pointed to the chair.\n\n\"What if I say no?\"\n\nHands on her shoulders, he seated her.\n\nNow she looked up at him, curious, mischievous, seriously tempted, but he wouldn't touch her again. He wasn't a child for her to tease. He backed away to draw her on his slate.\n\nHe sketched her short dark hair, her bright face, the soft folds of her sweatshirt. Fierce, efficient, he drew Daphne's picture, and she was amused, slightly disbelieving, as she sat for this formal portrait. Boldly she gazed back at Collin, but he controlled the situation. Her expression, sweet and dangerous, belonged to him.\n\nFinishing his first sketch, he showed her the slate, and she was startled by the likeness. Wonderingly she gazed at her own blue eyes, her parted lips, her body floating in the dark. They stood together looking at her picture and he never touched her, but he had left his mark on her. She was impressed.\n\n\"Once more,\" he said.\n\nHe never asked, but she saw the question in his eyes. She unzipped her sweatshirt.\n\nHe drew her with the sweatshirt open to reveal her undershirt, and the leaves inked on her bare collarbone.\n\nDaphne was serious now, conscious of her emerging image. Her body stilled and her eyes darkened as Collin drew her. Her mood began to match his.\n\nAs before, he showed her the slate. \"Again?\" She let him slip the sweatshirt off her shoulders.\n\nNow he drew her neck, with its intricate design of leaves. He outlined her bare arms inked with stems. With total concentration, he drew her tattooed shoulders and the ribbed cloth of her clinging shirt. He maintained his distance, but he had her in his hands. For her part, she never moved. She never flinched as she gazed back at him.\n\nCollin showed Daphne the slate and then erased it. He erased each drawing and deleted every image for good measure.\n\nThey didn't speak. He didn't need to ask. In silence, she locked the door and took off her undershirt. They were alone, and no one could walk in on them. She sat for Collin and her breasts were white, her nipples small and hard. He stepped back and took a breath, but even then, he didn't think that anyone would ever know.\n\nNo one walked in, but one person saw. Invisible to them, Peter sat in his office, watching Collin's sketch emerge on his own monitor. He could not see Collin, but he knew Collin's line. Only Collin drew like that, with such detail and speed.\n\n\"Careful, Collin,\" Peter said.\n\nEven as Collin deleted Daphne's image, his drawing remained on Peter's workstation. Line by line, Peter watched the next drawing develop. Daphne's face, her short, tousled hair, her patterned arms and shoulders, her bare torso. \"Careful,\" Peter murmured, but Collin couldn't hear.\n\nAidan sat at the kitchen table with his thousand-page textbook spread before him. He was sweating it out in summer school, learning cellular respiration. Oxidative phosphorylation. Cast away in the kitchen, without computer, without weapons, without any other voices in his ears, he gazed at the page, and the diagrams meant nothing to him. He waited, closed his eyes, and looked again. It was easy once he got his mind inside the picture. Once there, he could inhabit each cell's tiny factory. The hard part was getting in. Only sheer boredom did the trick. The table was bare except for a bag of pretzels. When those were gone, when he was left with salt dust at the bottom of the bag, he glared at his biology text until, like a sulky cat, his imagination came around again.\n\nHis mother said that he had changed, and she was right. She said life was a miracle, and he could see it. Little things like ice cream, or summer cloudbursts. He would think, Here I am, tasting Toscanini's cocoa pudding. Here I am, with rain streaming down my face. I might have missed all this. He watched sparrows hopping into puddles to wash their dusty feathers, and he thought, How sweet they are; I never noticed.\n\nHe was hungry again. He devoured dinner and then prowled the kitchen late at night, finishing whatever leftovers he could find. His desire to play grew stronger too. As he woke from sickness, regaining stamina and appetite, he dreamed about his spectral life. Even now, under the table, his right foot tapped nervously\u2014but he held back. Thrill seeker, storm chaser, he couldn't qwest a little bit.\n\nHe sat in class three hours a day. He met with the school psychologist, Miss Sorentino, who was pale and strange, with huge eyeglasses and a metallic voice. \"Who's better than you?\" Sorentino would say whenever something pleased her. Patiently he sat at her desk and stared at her collection of miniature turtles. \"This one's soapstone,\" she told Aidan. \"This teeny one's made out of an acorn. This one's marble. This one's glass.\"\n\nHe did his time. He listened to Sorentino talk about his uniqueness and his future. \"What do you think?\" she asked. Sometimes she said, \"It's up to you.\"\n\n\"I get it,\" he told her, and he wasn't lying. When he thought of gaming, he wrote in the margins of his bio notes, _Not now._ He knew how fortunate he was. Plagiarism was minor. As of yet, no one had accused him of his real crimes, and he knew better than to press his luck.\n\nEven so, he felt the tidal pull of UnderWorld. Like water flowing underground, UnderWorld seeped into his ordinary life. How beautiful, how strange, to see dragons in the shadows, and silver in the Charles River. Phantom visions. He could drown in his own thoughts.\n\nDiana was running down the stairs.\n\n\"Where are you going?\" he called out to her.\n\n\"Where I always go.\" She had a summer job across the street. Every afternoon she took care of Sage and Melissa's one-year-old, Henry.\n\n\"Where's the BoX?\"\n\n\"What?\" She thought that she had misheard him, he asked so casually.\n\n\"Where did you put it?\"\n\nShe stood there in shorts and T-shirt and dusty running shoes. \"You said you wouldn't.\"\n\n\"I won't!\"\n\n\"You said you wouldn't even ask.\"\n\n\"I was just curious.\" He turned back to his diagram of leaf and tree. \"You don't have to tell me anything.\"\n\n\u2014\n\nAll afternoon, Diana felt uneasy. She took Henry to Hancock Park with a net bag of pails and shovels dangling from the stroller. She helped the toddler into the sandbox and tried to keep his sun hat on his head, and the whole time she was thinking, \"You don't have to tell me anything.\" Was that supposed to reassure her? Or was it some kind of threat? Did Aidan plan to find the BoX himself?\n\nShe told Henry, \"People as bald as you are should wear hats, or they'll get cancer.\"\n\n\"No,\" said Henry. It was one of his best words, along with \"my.\"\n\n\"He'll never find it,\" she told Henry.\n\n\"No.\"\n\n\"He can't find it.\"\n\n\"No.\" Henry took off his hat.\n\n_\"Yes.\"_ She put his hat back on again.\n\nHe smiled, showing his two teeth, and took off the hat.\n\nStop panicking, she thought. He can't find it, because I don't have it. \"He can look and look,\" she told Henry, as she pushed the stroller back to Antrim Street, \"but I won't tell, and you won't tell.\"\n\nHenry laughed because the ride was bumpy. He loved the jolts where the sidewalk buckled over tree roots.\n\n\u2014\n\n\"It's the little things, right?\" Diana told Brynna that evening, as she laced her shoes. \"Henry gets that.\"\n\n\"You like babies!\"\n\n\"I like money.\"\n\nThey ran together along the river. At first Brynna complained she couldn't keep up, but six weeks into the summer she had lost what was left of her pregnancy weight. Diana burned\u2014she was fair and freckled\u2014but Brynna's body loved the sun. She tanned and her thick hair flew out behind her, streaked with gold.\n\n\"I think I might switch religions,\" Diana said.\n\n\"You can't switch,\" Brynna panted. \"You're Catholic.\"\n\n\"I know, but...not really.\"\n\n\"You go to church,\" Brynna pointed out.\n\n\"My mom goes.\"\n\n\"What would you switch to?\"\n\nDiana glanced at the wind-ruffled river. \"I'd be pagan.\"\n\nBrynna snorted. \"No, you wouldn't.\"\n\n\"I already am.\"\n\n\"Last year you were Wiccan,\" Brynna reminded her.\n\n\"That was different. I was younger then!\"\n\nThey stopped at the light at the Eliot Bridge, and Brynna bent over, laughing. \"Yeah, and now you're almost seventeen.\"\n\n\"I hate you,\" Diana said, although the opposite was true. When she was with Brynna, her uneasy feeling disappeared. When the light changed Diana took off again and her friend followed. Diana said, \"I would be a pagan god.\"\n\nBrynna didn't have the breath to answer.\n\n\"I'd be the god of secrecy,\" Diana said. \"No one could get anything out of me.\"\n\n\"How much farther?\"\n\n\"Just Fresh Pond.\"\n\n\"No way,\" protested Brynna.\n\nStill, Diana pressed on, and Brynna followed.\n\nThey cut away from the river and ran up Brattle Street, and then up Sparks to Huron Avenue, past fancy stores like Graymist, which sold Nantucket baskets, and Marimekko, which sold umbrellas covered with enormous poppies. \"You got this far,\" Diana encouraged her, but when they got to Formaggio Kitchen, Brynna sank down on the old-fashioned park bench out front.\n\n\"Please!\" Diana begged. \"We're almost there. I'll let you take the bus home.\"\n\nDiana knew her friend could run all the way, but Brynna didn't. Brynna's cheeks were scarlet, and her shirt clung to her, soaking wet. Diana had to coax her into the store for air-conditioning and water.\n\n\"Wow.\" Cool air prickled the hair on their arms as they stared at all the treats. Ripe apricots, pluots, and nectarines. Strange pears. Tart cherry scones. Marble tables displaying slabs of sheep cheese, goat cheese, runny cheese, unpasteurized cheese. Brynna said, \"Isn't that against the law?\"\n\nThere were boxes of Belgian chocolate and Italian nougat. Salted caramels from Seattle. Candied violets. Amaretto truffles. You could buy chocolate-covered grapefruit peel. \"Oh, my God, look,\" Brynna said. \"Wild strawberries.\"\n\nTiny berries trailed over green pint-size containers. \"This place is incredible,\" Diana said, in a hushed voice.\n\nBrynna checked the price per pint. \"And everything costs a bazillion dollars.\"\n\nYou couldn't even buy regular water. You had to buy spring water untouched by man. They burst out laughing as they escaped with their single purchase.\n\nDiana took a long swig and handed the bottle to Brynna. \"This water bubbled up through natural aquifers in a volcano.\"\n\nBrynna splashed water on her own face. \"How does that work?\"\n\n\"Come on.\" Diana pulled her by the hand. \"This is the best part of the day. See? The sun is going down.\"\n\n\"Next time.\" Brynna untied and retied her hair. \"I have to take care of Angela.\"\n\n\"Please.\"\n\n\"Come back with me,\" Brynna offered.\n\nDiana shook her head.\n\n\u2014\n\nAlone, Diana ran up Huron Avenue to Fresh Pond, the reservoir, hidden in trees. There was a good trail there all around the pond, and a water fountain where you could splash yourself and dunk your head. Older ladies strolled along, deep in conversation, and every once in a while one of them would look at the water and say, It's so beautiful\u2014look how still the water is. It's like glass; it's like a mirror. Look at the mist! And for a split second they would stop and look, and then they'd go right on talking. Joggers brought their terriers, and big black Labs and goldens, and Irish setters straining at the leash. But if you knew where to go, you didn't run into people so much. There was another pond, a smaller one, all choked with water lilies. The big pond was fenced, but at Little Fresh Pond, you could run right to the edge.\n\nDiana knew a fallen tree, an oak that looked like it had taken its own life, keeling over and trailing its branches in the water. She ran to this tree, pulled off her shoes without unlacing them, and peeled off her sweaty socks. Broken bottles lay scattered on the ground. Frowning, she collected a few of the big shards, but the shore was littered with little jagged pieces. She gave up and climbed the toppled trunk to dangle her legs and soak her feet.\n\nThe sun had set. The first stars began to show themselves, but it wasn't dark. The evening was bright, and the trees stood black against the sky. Diana stretched herself out the length of the tree trunk, her back against the bark. It was good to rest; it was good to be alone, to test the edge of your own loneliness. When she looked up at the night sky she saw how impersonal it was, how big, how changeable. Everything was moving, trees, moon, stars. You tried to keep up, but you couldn't run fast enough.\n\nA scuffling, growling sound.\n\nShe started up and then held still as possible, as she saw two pit bulls, racing through the trees off-leash. Their master called to them, but they scented her immediately and ran for her, barking, red-eyed.\n\n\"Anton!\" she screamed when the dogs' master appeared. \"Get your fucking dogs away!\"\n\nHe ordered them to stay and then to sit. He clipped leashes to his dogs' collars, but they never took their eyes off her, and they growled deep in their throats, even as they obeyed. Anton didn't speak, but he stared as she scrambled off the tree trunk and retrieved her shoes and socks. His eyes traveled down her shoulders, and over her breasts, down the front of her wet shirt to the waistband of her shorts, then lightly over her bare legs to her feet.\n\n\"I didn't recognize you,\" he said.\n\n\"I'm the same,\" she told him flatly.\n\n\"No, you're not.\"\n\n\"You wouldn't know.\" Probably she should have been afraid of him, but she felt a confused kind of power.\n\n\"Did you run here?\"\n\nShe didn't answer.\n\n\"I'll drive you back.\"\n\n\"Yeah, right,\" Diana said. \"Is that what you told Brynna last year?\"\n\n\u2014\n\n\"What happened to you?\" Aidan asked when she got home. She was dirt-streaked, blistered, carrying her shoes.\n\n\"Are you playing?\" Diana demanded. Aidan was sitting just where she'd left him, with the open biology textbook, but she saw that he'd got ahold of their mother's clunky old desktop computer.\n\n\"No,\" Aidan said. \"Why're you in such a bad mood?\"\n\nHow to answer? My feet are bleeding. I had to fight off wild dogs. I'm scared for you.\n\nShe limped upstairs and saw that he had searched her room. Her piles on the floor had toppled. The clean clothes on her chair had shifted. Her closet door was ever so slightly askew. \"Aidan!\" she screamed. She wanted to run downstairs and knock the chair out from under him, but her legs were so sore that she couldn't move. _\"Aidan.\"_\n\nNo answer.\n\nShe threw herself onto her bed and opened her phone.\n\nWTF, she texted him.\n\n?? he replied, as if anybody else would have searched her room.\n\nYou know what u did dont deny it. what is wrong with you??\n\ncalm down, he wrote.\n\nNO\n\nIts been 6 weeks\n\nIdc\n\nI have like one week of summer. By this he meant one week between the end of his summer course and the first day of school. He had seven days of freedom, and he wanted to spend it with his BoX. That would be Aidan's vacation. A week of caves and red-eyed vampire bats. You promised you wont play, she typed.\n\nInstantly his answer appeared. I promised mom not you.\n\nIn the last days of summer Aidan got his class assignments for senior year. Physics, calculus, Spanish, European history, American literature again because he'd plagiarized. He talked his mother into computer access too. Kerry had given him her ancient desktop\u2014safe, she thought, because it was too primitive and slow for gaming. Aidan couldn't play, but he scanned fan forums all day long, following the news. He couldn't enter games, but he could dream about them. He stood in the center of his room and closed his eyes and he could see the leaden river, and the Gates, the dark caverns all around him. He clenched his fist and saw his arm changing to silver. He drew his sword\u2014and yet no caverns rose up around him, no tunnels opened up. He had to live outside of UnderWorld.\n\nOfficial beta testing was beginning, but he had not been chosen. The BoX was to go on sale in the winter, but he couldn't wait until December. The situation maddened him because he had a BoX, but couldn't get his hands on it.\n\nHe searched every room, every closet and cabinet. On Labor Day weekend Diana found him hunting in the basement. Carrying her laundry bag downstairs, she found him rummaging in dusty cartons labeled CHILDREN'S BOOKS; HUMIDIFIER; TENNIS; WINTER CLOTHES.\n\n\"Just stop!\" Diana said, acutely aware of the doll carriage in the boiler room.\n\n\"Tell me where it is.\"\n\n\"No.\"\n\n\"I finished summer school,\" he said.\n\n\"So what?\"\n\n\"I did what I was supposed to do.\"\n\n\"You gave it to me,\" Diana said. \"You said keep this for me.\"\n\n\"Not forever.\"\n\nShe set down her laundry bag as if to stake her claim. \"Yes! Forever. You gave it to me when you were getting sick.\"\n\n\"I wasn't getting sick! I got sick later.\"\n\nHe was always correcting her on that, but in her mind, his illness and the black BoX went together. \"You said I could have it, and now, guess what, Aidan? You don't get it back again. It wasn't like a long-term loan.\"\n\n\"I want it.\"\n\nDiana snapped, \"What makes you think I kept it?\"\n\nHe dipped into the tennis box and hurled a yellow ball at her.\n\nShielding her face, she caught it. \"What the hell is wrong with you?\"\n\nAidan didn't answer.\n\n\u2014\n\nJust an hour later, Diana sneaked into the boiler room, retrieved the BoX, and stuffed it into her backpack. Then she ran across the street.\n\nFor a moment she stood looking up at the third floor of Maia's triple-decker. All she could see were the tomato plants in Sage's window boxes. She had the key from babysitting, but she felt some trepidation as she slipped inside and climbed the stairs. Sage and Melissa had taken Henry to the Cape, and she felt like a trespasser.\n\nStifling hot without AC, airless, dusty, deathly clean, Sage's apartment scared Diana. No, that wasn't true. She brought her anxiety with her, and she was afraid to touch anything, almost afraid to touch the floor.\n\nShe set her backpack on the couch and took out the BoX, so heavy and compact. No lights, no buttons, no electric ports. Not even the words MADE IN CHINA. Just a perfect cube, smooth and mute. Diana felt it now, the strange desire to touch this thing, to open it. She opened the hall closet instead, rooted behind ice skates and hockey sticks, and heaved the BoX in.\n\nDone. She tried to leave, and yet she felt no relief. Hiding the thing was not enough. She took up her burden again, carrying the BoX out in her backpack. Slowly, she walked around the block. Once around, and she felt the BoX hard against her back. A second time, and she felt the weight between her shoulder blades. The third time around, she met Lois returning from the gym with her furled yoga mat. \"Excited about senior year?\" Lois asked.\n\n\"Not really.\" Diana was watching for her mother's car.\n\n\"It's a transition,\" Lois said, all teacherly.\n\n\"Mom!\" Diana saw her mother drive up. \"Mom!\" Diana sprinted across the street.\n\n\"What's wrong?\" Kerry called through the open car window.\n\n\"I have to give you something.\"\n\n\"Okay.\" Kerry was drenched in sweat. Her car's air-conditioning had broken long ago. \"Just let me get inside.\"\n\n\"No. Not inside. Here.\" Diana blocked her mother's way. \"This is Aidan's.\" She unzipped her backpack and unwrapped the BoX.\n\nKerry flinched. \"What is that?\" she asked reflexively, but she knew. She seized the BoX, even before Diana finished explaining.\n\n\"Don't bring it in,\" Diana pleaded, as Kerry marched up the steps. \"Don't let him see.\"\n\nKerry took the BoX into the kitchen and set it on the table. \"Throw it away! Get that thing out of here,\" Diana begged her mother. She was afraid to see it in the open.\n\nKerry was rummaging in the broom closet, but Diana hovered over the black BoX. She had done the thing she'd sworn she'd never do. She had betrayed her twin, and her younger self as well\u2014the girl who'd promised, chanting along with Aidan, We won't tell no matter what.\n\n\"What are you doing?\" Diana cried out when her mother returned with a hammer in her hand. \"Wait, not in the house.\"\n\nKerry didn't hesitate. She took her hammer and she struck again and then again. She hit so hard the salt and pepper shakers jumped. She struck a third time and a jagged hole opened in the smooth black cube.\n\nA cloud of aeroflakes flew up to the ceiling, shrouding the light fixture, clouding the air. Kerry didn't even look. Harder and harder, she kept hammering down blows. This was for UnderWorld. This was for Arkadia. This was for night shifts. For lack of time. For lack of money. For unanswered prayers.\n\nKerry smashed the BoX to pieces, even as Aidan ran downstairs. He watched in silence as the walls of the black BoX shattered. Disoriented, unfocused, the little motes faded, faint as watermarks on the kitchen walls.\n\n\"Stop,\" Diana said, as Kerry hammered pieces into powder.\n\nWith a long, shuddering sigh, Kerry set down her hammer and sank into a chair. Diana was the one who sponged the plastic shards into the trash.\n\n\u2014\n\nAll that afternoon, Aidan kept himself locked upstairs. Diana heard him pacing. She heard him moving furniture. His desk? His bed?\n\nShe knew better than to knock. She texted him just once. I hadto.\n\nHe did not respond.\n\nHer mother tried to comfort her. She sat with Diana on the couch and she kept murmuring, \"You did the right thing.\"\n\n\"Yeah, I can tell,\" said Diana, \"because I feel like shit.\"\n\n\"Don't punish yourself like that.\"\n\nDiana buried her head in the cushions. \"Just get away from me, okay? The only one punishing me is you.\"\n\nIn the heat of the evening, the afterburn of the long summer day, Diana ran all the way down Magazine Street to Cambridgeport to bang on Brynna's door. \"No!\" Brynna protested. \"My feet still hurt. I can't.\"\n\n\"Please, please, please,\" Diana begged, and even as she asked, she kept moving, jogging in place, hopping from one foot to another. \"I need you.\"\n\n\"What happened?\" Brynna asked.\n\n\"Nothing, if you'll come with me,\" Diana said. Her face was tearstained, her body eager, strong.\n\nTogether, they ran along the river. Diana charged ahead, and the sullen breeze riffled her dark hair. Brynna followed, protesting, \"Wait up. You're way too fast.\"\n\nDiana was too upset to wait. She was still too close to home; she had to get away. All she wanted was to run faster, to outstrip the setting sun and plunge into darkness. She wanted to feel nothing, remember nothing, be nothing.\n\nBut even she couldn't keep up this furious pace. Gradually the girls found their rhythm, running up Huron Avenue. Diana wiped the tears from her eyes, and Brynna stopped complaining, gathering her strength, saving her breath until they reached Fresh Pond, the reservoir, fringed with trees.\n\n\"Rest,\" Diana said.\n\nThey stopped at the fountain near the water treatment plant and splashed their faces.\n\nBrynna sighed. \"That feels so good.\"\n\n\"Wait. I'll show you something better,\" Diana said.\n\nShe led Brynna to Little Fresh Pond and showed her the half-drowned tree. \"Watch out. There's glass.\" Together they picked their way around the broken bottles, and climbed up on the fallen trunk. They pulled off their running shoes, peeled off their sweaty socks.\n\nNight fell, sky and water deepening. Barefoot, Diana walked along the fallen tree to the point where the trunk dipped into the pond. She sat there trailing her legs in the waterlogged branches. She reached for Brynna, who hesitated. \"No one's here.\" Diana stretched out her white arms. She was persuasive, confident. Even her voice seemed lower, whispering. She was a different creature in the dark.\n\nBrynna followed Diana and sat next to her. Water lapped the backs of her knees, green-black leaves felt like seaweed on their feet. \"Are you still hot?\" Diana's voice was so soft, Brynna wasn't sure whether Diana was asking her, or talking to herself.\n\nDiana took off her sweat-soaked shirt, balled it up, and tossed it onto the bank. Brynna hesitated a moment and then took off hers. They faced each other in shorts and jogging bras. Fair Diana, and darker Brynna with her hair tied back. The air was heavy, the summer weighed upon them, humming with mosquitoes. They could hear the frogs and the cicadas, the scrabbling claws of unseen animals, the rustling trees.\n\nA branch snapped.\n\nBrynna started. \"What's that?\"\n\n\"Nothing.\" With her thumb, Diana smudged out a mosquito on Brynna's neck.\n\n\"We'll get eaten alive,\" Brynna said.\n\n\"Come swimming then.\"\n\nBrynna glanced at the dark trees. \"We're not supposed to.\"\n\n\"So what? I swim here.\"\n\n\"When?\"\n\n\"I run up here and swim at night.\"\n\nAs they sat side by side, their bare arms were almost touching. Brynna said, \"Isn't that dangerous?\"\n\n\"When you were younger\u2014\" Diana began.\n\n\"You're always saying, 'When you were younger,' \" Brynna said.\n\n\"Okay, before you had the baby, you wouldn't ask.\"\n\n\"I'm not so different.\"\n\n\"Yes, you are.\"\n\n\"Because of Angela?\"\n\n\"Because of Anton, obviously.\"\n\nBrynna said, \"I never even see him anymore.\"\n\n\"Good.\"\n\nBrynna sighed. Her breath brushed against Diana's ear. \"Do you swim in your clothes?\"\n\n\"No,\" Diana whispered. \"I take them off.\"\n\n\"Would you now?\" Brynna asked.\n\n\"If you come with me.\"\n\nBrynna held still, absorbing the suggestion.\n\n\"Do you want to?\" Diana asked, as she unhooked the back of Brynna's bra.\n\n\"Yes.\"\n\nBrynna's breasts were full, her nipples dark. Brynna's heart was racing underneath Diana's hand. \"I just want to touch you,\" Diana whispered, and Brynna didn't pull away. Her skin was soft under Diana's tongue.\n\nThey slipped into the water, and Diana had no twin; she had no mother. She had Brynna alone. They were standing waist-deep, and Brynna glanced back at the water's edge. \"What if someone sees us?\"\n\nFor a moment Diana thought of Anton and his dogs\u2014but no one had followed them. \"No one sees us,\" Diana murmured, as she stroked Brynna. \"No one's here.\" The darkness and the water concealed them.\n\nShe touched Brynna's face. She kissed Brynna's lips and she forgot everything but Brynna's soft mouth. She forgot Aidan and she forgot herself. It was happening. She was becoming someone else.\n\nAidan would not forgive his sister. He wouldn't even look at her. Exiled in the hospital, he had been too weak to care. Now he was strong and restless, and resentful. He had endured days and weeks of summer school, and this was his reward. He had trusted Diana, or so he told himself. He had counted on her to keep his black BoX safe, and then just when he needed it\u2014just when he'd earned it\u2014she turned on him and gave it to their mother.\n\nHis mother had taken back her desktop, and so he had rehabilitated Liam's old laptop to play while she was at work. The laptop was easy to hide but almost useless, because it was so flat and slow. His BoX had spoiled him. As color eclipsed black and white, as computers supplanted typewriters, so OVID overshadowed all the gaming platforms that had come before. Onscreen graphics seemed old as picture books to Aidan. All he wanted now was gaming in the round.\n\nHe tortured himself, inhaling articles on UnderWorld's paradigm-shifting technology. He read about the millions waiting for BoXes in December, and it killed him to think he'd held his own BoX in his hands. i've been inside, he typed on UnderWorld's fan forum. no way, the other Everheads shot back. whats it like? your full of shit. prove it. pix! But he had no proof.\n\nThose last days of summer he swallowed his pride and texted Daphne, he messaged her on fan forums, he emailed her at her official Arkadia address, but she did not reply. At last he did what he should have done first. He logged in to EverWhen and searched for Riyah. He wandered the Trackless Wood and hunted by the shore, he journeyed past the tree line on the slopes of EverRest. The night before school started, he sat in bed wearing his old headset and fell asleep trying to find her.\n\nThe crunch of autumn leaves, the rush of water.\n\n\"What's up, Tildor?\"\n\nHe opened his eyes and heard Daphne speaking to him, although at first all he could see was his screen saver.\n\nHe touched his trackpad, and there she stood on a rough boulder before a thundering waterfall. She was wearing her thigh-high boots and scant bodice of black leather. Her eyes were huge, her blue hair flowed down her back, but her avatar was just that to him, because he'd seen her for real.\n\n\"Sleeping?\" She touched his Elf lightly on the shoulder.\n\nHe shook himself awake. \"I need another BoX.\"\n\n\"Sorry. One per person!\" Riyah folded her arms across her chest.\n\n\"Wait. Listen. My mom took a hammer and destroyed mine.\"\n\nShe threw back her head and laughed.\n\n\"Go to hell,\" breathed Aidan.\n\n\"Excuse me?\" said Daphne. \"Which of us can't get there?\"\n\nShe repulsed him. Much as he'd admired her when they'd met, honored as he had been, qwesting in her company, he hated her now. He knew Daphne was a schemer and a marketer, neither Elf nor ordinary gamer, and he didn't want her anymore. He had one desire. \"I have to play.\"\n\n\"Oh, well.\" Riyah jumped down from her rock and began searching for jewels in the crystalline pool beneath the waterfall.\n\n\"I'll make a deal with you,\" Aidan said.\n\n\"I can't.\" Riyah waded deeper into the water. She looked like a tiny dominatrix, but she sounded like an irritated teacher. \"You have to wait like everybody else.\"\n\nTildor splashed after her. \"I won't wait.\"\n\n\"You have to.\"\n\n\"I'll tell everybody you're breaking the law.\"\n\n\"Vandalizing property?\" Riyah said. \"That would be you.\"\n\n\"You made me.\"\n\n\"Hey. Stop right there.\" She drew her sword.\n\nHe drew his shining weapon and advanced. Steel on steel, the two of them were fighting, waist-deep in the water. \"You tricked me!\"\n\nShe struck and slashed his arm. \"I never tricked you. Think about it.\"\n\nIn a fury, he forced her back toward the roaring waterfall. White water reddened with his blood, but he would not let up. \"You said I could play.\"\n\n\"You did play.\"\n\n\"You promised.\"\n\n\"I kept my promises.\"\n\n\"I need to play now.\"\n\n\"Grow up.\"\n\n\"You owe me!\" He knocked her down into the water and with one massive stroke sliced off her head. The pool was black now with her blood, the water churning with her headless body.\n\n\"Okay, game over.\" Still bleeding, Riyah picked up her head and leapt onto a rock.\n\nHe panicked. \"Please. Just give me a new BoX and I promise you'll never see me again.\"\n\nShe stood there holding her head as casually as a fencer holds her helmet after a bout. \"You won't see me either.\" Riyah replaced her head and turned her back on him.\n\n\"Come back,\" Aidan called out.\n\nAlready she was out of reach, leaping from rock to rock.\n\n\"I did everything you wanted.\"\n\n\"You did what _you_ wanted.\" Those were the last words Daphne ever said to him.\n\n\u2014\n\nHe was in a killing mood that night. He rampaged through the Trackless Wood, murdering animals, one by one. He slew a wild boar, hacking the beast until he was knee-deep in blood. He tracked a golden fox and shot it through the eye. Charged a two-headed dog and killed it twice. A few people from his old company texted, Tildor u back? Or Qwest now? He hunted on, alone and furious.\n\nIn the starlight his Elf shot down a phoenix with a gold arrow. The bird flamed like a falling star, but all Aidan could think was how much better the fire would have been in UnderWorld, with sparks showering down upon him.\n\nIn the distance he could hear Diana rustling across the hall. Far away he heard her knocking at the door. \"You know what day it is, right?\"\n\nHe took off his headphones but he didn't answer. He waited until she left the house to log off and walk to Emerson. Let her wonder whether he'd show up the first day. Let her think he had forgotten. Angry as he was at Daphne, he was angrier at his sister.\n\n\u2014\n\nSchool smelled like paint. A few days before, a troop of City Corps kids had descended on the building to spruce it up. Scuffed floors were lemon-fresh, walls bright with new bulletin boards. A purple banner hung in the lobby blazoned POETRY IN ACTION.\n\nPeople were embracing. Girls were calling to one another like long-lost sisters. Everyone was supercharged with energy. The shoes alone were blinding. Aidan wanted to turn around and leave.\n\n\"Good morning, Aidan.\" Miss Lazare pursued him up the stairs. \"Welcome back!\"\n\nWhat was that about? Aidan ducked into his math class and out of range. Lazare was such a lethal mix of concern and hopefulness.\n\n\u2014\n\nIf I were Mrs. West, thought Nina, I would have demanded a response. \"Excuse me, young man. You look me in the eye and say, 'Good morning,' when I greet you.\" Next time, Nina thought, even as she added a new resolution to her mental list: Don't compare yourself to Mrs. West.\n\nHer list was long. Start off strict and set expectations high. Mean 'til Halloween, as Mr. Allan said. Insist on homework. No lost books, no missing papers, no unexcused absences. Keep track of time. Finish lessons before the bell. She was wearing Collin's old-fashioned watch.\n\nThis year will be different, she thought, as she stepped into her classroom. I'll get it right. I won't fail you, she promised her kids silently, as she shut the door.\n\n\"Good morning,\" she said.\n\nA few kids answered, \"Good morning.\"\n\n\"I'm Miss Lazare, and this is American Literature,\" she announced, just in case someone had wandered into the wrong room. \"I'll take attendance before we get started, and I'll just say,\" she added as a couple of boys walked in, \"I need everybody here on time. Colleen. Matisse. Jared. Australia...\" She tagged each student in her mind with a mnemonic. Colleen, with the eyebrow piercing. Rachelle, with hennaed hair. Candace was chewing gum. \"Take out the gum, Candace. Take these poems. Pass them down.\"\n\nShe watched her photocopied handout ripple from hand to hand. \"As a class we'll study American poets and writers. As a school, we have a new initiative this year. You may have seen the banner.\" She paused and felt a little dumb. The banner was a good twelve feet long and hard to miss. \"Our school will be participating in the national recitation competition.\"\n\nGroans and laughter. Confusion about what recitation might entail. Scuffling in the back.\n\n\"Miss?\" Candace held up the extra photocopies.\n\nKeep it moving, Nina thought. \"So we'll start the year with two American poems. Do I have a volunteer to read the first?\"\n\nSilence.\n\n\"Anyone?\" Nina volunteered a girl named Zena, who had been whispering. \"Right here.\" Nina interposed herself between Zena and Colleen.\n\n_\"I hear America singing,\"_ Zena began, _\"the varied carols I hear, \/ Those of mechanics, each one singing his as it should be blithe and strong...\"_\n\nAs Zena read aloud: _\"The carpenter singing his as he measures his plank or beam...\"_ Nina opened a fresh box of chalk. Was there anything more perfect than new chalk? _\"The mason singing his as he makes ready for work, or leaves off work...\"_\n\n\"How would you describe the mood here?\"\n\n\"Ummm,\" Zena said.\n\nNina waited. Unconsciously, she played with Collin's watch. It was so big, it kept flipping over, buckle up, facedown, rubbing the inside of her wrist.\n\n\"Mmmm.\" Zena considered the words before her.\n\n\"Happy,\" called out a boy named Trey.\n\n\"Okay!\" Nina wrote \"happy\" on the board.\n\n\"Cheerful.\"\n\n\"Yes.\" Nina wrote that too.\n\n\"Full of it,\" Australia suggested, and Nina heard the class cackling behind her.\n\n\"No, that's good. Why do you say that?\"\n\nAustralia did not explain.\n\n\"Does anybody else think Whitman's full of it?\"\n\nNatalie and Mikayla raised their hands.\n\n\"He's too happy about everything,\" said Mikayla, but Natalie forgot what she was going to say.\n\n\"It will come back to you,\" said Nina. \"Let's see how Langston Hughes responds to Whitman. Trey, read the second poem aloud for us.\"\n\nTrey began reading, _\"I, too, sing America...\"_ but a snickering undercurrent accompanied him.\n\n\"Hold on,\" Nina interrupted, and she waited for silence. \"Okay, go ahead.\"\n\n_\"I am the darker brother,\"_ he read, and the other kids burst out laughing, because Trey's skin was darkest in the class.\n\nIn second period Nina repeated the two views of America, Walt Whitman's and Langston Hughes's. Once again she mentioned the recitation contest, which kids were already calling Poetry Inaction. Every student had to choose and memorize a poem from the contest website to recite in class. Then the class would vote for a winner to compete in front of the whole school.\n\nStudents shifted in their seats.\n\n\"What if I have no memory?\"\n\n\"Is it graded?\"\n\n\"Whose idea was that?\"\n\nNina almost said, \"You can thank Mrs. West.\"\n\nInstead she held still, as she'd seen Collin do when he visited as Shakespeare, cool and distant.\n\nKids paused in their conversations when they realized she was no longer talking. They looked up, curious.\n\nLazare had a reputation now. She had a mad-hot, cross-dressing boyfriend, but she was so strict she corrected grammar when you were only talking. Total mind reader, she could tell whether you'd plagiarized just by looking at you. She was confusing. One minute she was all intimidating, and then when people didn't listen, she got emotional, so you couldn't hate her without feeling bad. That was the worst! On the other hand, she was fun to watch. She had a photographic memory, and if you were lucky she'd stop teaching and show what she could do.\n\nAt the end of each class, Nina closed her attendance book and left it on the desk. Then she looked each student in the eye and came up with the right name. \"Darsy, Lalitha, Jean-Albert, Theresa, Cameo, Susannah, Tyrell, Joanna, Sebastian, Yasmin, Aria...\" Without a single mistake, she had identified each kid in her first class, but she faltered now in second period. \"Becca, Jameson, Nico,\" she began. \"Shana, Rafael, Siddhartha, Miles.\" She named each kid correctly until she got to the back of the room. \"Aidan.\"\n\nThe boy shook his head.\n\n\"I'm sorry. I mean Ethan!\" Aidan was missing, although she'd just seen him on the stairs.\n\nAll this happened in a moment, but when Nina turned to the next student, her memory failed her. She looked at the girl's round face. Black eyes, smooth hair, gold hoop earrings, tight shirt. Nina looked into the girl's dark eyes.\n\nAnxious\u2014was she unmemorable?\u2014the girl stared back.\n\nSiddhartha called out, \"It starts with an _S_...\"\n\n\"Sofia,\" Lazare said at last. Gold hoops quivering, Sofia sank back with relief.\n\nNina texted Collin after the bell. I did it! Then, a little later, when he hadn't answered, Are you there??\n\nBlood in his eye, the Ghost Horse raced through caverns, wheeling, screaming, rising up on his hind legs, then crashing down, a beast possessed.\n\n\"Okay,\" Peter said.\n\nThey were sitting in the sound booth. When Nicholas stopped the demo, Peter rose to leave.\n\n\"What about the rest?\" Collin asked.\n\nPeter glanced at the second monitor, where Collin's wild horses streaked across the screen. The Ghost Horse belonged to Peter now, but the herd was Collin's joy. The gray, the chestnut, the palomino, shining pale gold. They stretched their necks and ran together. You could see them close, coming down upon you. On a third monitor you could see them from above, and their tails streamed behind them so they looked like shooting stars. All the time you heard the hoofbeats and the hot breath of those horses.\n\n\"No,\" Peter said.\n\nNicholas clicked once and the room was silent. The horses froze.\n\nPeter said, \"We won't use these.\"\n\nAt first Collin couldn't even hear. He could not absorb the words. He had worked for months. He had spent the entire summer on these horses.\n\nPeter turned back to the horses caught in midair. \"They're pretty.\" Peter bent and clicked on the palomino, highlighting the horse in blue, isolating the creature's body, its slender neck, its little ears and feet. Suddenly the horse _was_ pretty. Peter clicked again and vaporized the animal. Now he touched the chestnut, with its soft eyes and flowing tail. Clicked once, clicked twice. It disappeared.\n\nShocking how cold he was as he killed each of Collin's steeds: the velvet black, the gray.\n\nCollin said, \"You wanted them before.\"\n\n\"No, they aren't right.\"\n\n\"What isn't right about them?\"\n\nPeter looked at Collin, and in that moment he smothered Collin's protest. \"They're like illustrations in a children's book.\"\n\nHe spoke with experience and authority. He spoke with his uncanny insight. The chestnut, the palomino, and the gray were nothing more than wild ponies, lovely, gentle, sweet. They had no edge, none of the Ghost Horse's snarling cruelty. How had Collin been such a fool? He had been too in love with his own work to understand. These horses did not belong in UnderWorld.\n\n\"They're just not interesting.\"\n\nCollin nodded; he could not argue.\n\n\"You'll be okay,\" Nicholas counseled after Peter left. \"It happens all the time. Not everything gets used.\"\n\nCollin nodded, but the verdict cut deep. He had never been sentimental about his art. He had erased his own work with freedom and with joy, but this was different. He had worked so long on these bright horses, invested so much of himself\u2014and he had not been the one to wipe the board.\n\nPhone in hand, he stood that evening in the parking lot and texted Nina, Im fine working late. He was okay, just as Nicholas had said, but he couldn't talk to her. He had to absorb the shock, along with his new assignment.\n\n\u2014\n\nHe would work on UnderWorld's Flamethrowers, evil Amazonian women guarding treasure in UnderWorld's Sixth Circle. Testers had reported that these characters looked too much like elves. Collin had to make them scarier.\n\n\"Condolences,\" Daphne told him the next day. It was as if he'd been demoted to a janitor, cleaning up other people's work.\n\n\"I don't care.\"\n\nShe pretended to believe him.\n\nWith numb determination he drew the Flamethrowers and their fiery arrows. His heart wasn't in it, but he forced himself, working for weeks to make the women nastier, sharpening features, drawing snarling faces. He even tried Peter's trick of bloody eyes. Peter rejected every iteration, and responded to the red eyes with contempt. \"You're painting by number now, and that's just desperate.\"\n\nDiscouraged, Collin took his slate and scrolled through his old work. He saw all the faults that Peter found and more. Crude Flamethrowers and pretty horses, thoughtless cartoons. He was a copyist, just as Peter had predicted. His art was superficial, glib.\n\nNina said, Don't let Peter mess with you. Her uncle was harsh, unreasonable. She pleaded, Don't believe him. None of this meant anything to Collin. In fact, her exhortations made him feel worse. He would rather talk to Nicholas, who said, Shit happens. He would rather play Daphne, dueling late at night with broadswords. For the first time, Collin avoided Nina. He used work as an excuse, and when she asked about Arkadia, he picked a fight or pushed her away. He said, Don't tell me what to do.\n\nOffended, Nina drew back. Her own days weren't easy. Her kids were rowdy, and she had her evaluation coming\u2014observations and online surveys of her students. She wanted to confide in Collin as she had before, but she held back because he had so much going on. This too will pass, she told herself. She would teach and he would find his way. Therefore, she didn't offer help, nor did she come with Collin to the hellions' Halloween party. She didn't trust herself to see Peter without losing her temper, so she stayed home.\n\nCollin's roommates were not so reticent.\n\n\"Take us with you!\" Emma said, as Collin set out for Peter's house.\n\n\"It's hellions only.\"\n\nDarius handed him a stack of Theater Without Walls postcards. \"Disseminate!\" The company was staging _The Importance of Being Earnest_ at South Station, with the audience convening on one platform and then departing on two different trains.\n\nCollin shoved the postcards into his jacket pocket.\n\n\"Seriously. Pass those around.\"\n\n\"It's the least you can do,\" Emma pointed out.\n\nCollin said, \"People at this party don't care about plays.\"\n\n\"What are you talking about?\" Darius exclaimed. \"Videogames are us. Games _are_ plays.\"\n\nCollin began wheeling his bike out the door.\n\n\"Don't be jaded,\" Darius said.\n\n\"I'm not jaded. I'm late.\"\n\nDarius called after Collin, \"If they need actors for UnderWorld, we're available!\"\n\nCollin jumped on his bike. \"They already recorded all the voices.\"\n\nAcross the shining river, through the looking glass to Boston, Collin sped toward Joy Street. The wind was harsh, but it felt good to travel to a new place, neither Cambridge nor Arkadia. He was nervous but curious as well, crossing the bridge into Peter's city, with its gas lamps and cobblestone streets.\n\nPeter lived in a pair of townhouses, catercornered. One faced Joy, and the other faced Myrtle, and they shared a basement, like conjoined twins. There were two sets of windows, and two front doors, but the windows were all curtained, and both doors closed. Locking his bike to a lamppost, Collin eyed the Myrtle Street entrance. A bouncer stood on Peter's steps. He was a massive man with short arms and a little derby hat.\n\n\"Name?\"\n\n\"Collin James.\"\n\nOfficiously, the bouncer scrolled down on his tablet. \"You're not on the list.\"\n\n\"Yes I am. I'm from Arkadia.\"\n\n\"You aren't here.\"\n\nCollin felt a prick of anxiety. Then he said, \"Were you looking under _C_ for _Collin,_ or _J_ for _James_?\"\n\nThe bouncer opened the door, releasing a tidal wave of sound.\n\nWalls pulsed with laughter and with music, and it was dark, the entryway cavernous, shrouded in mist.\n\nAs his eyes adjusted, Collin found himself at the foot of a grand staircase with a body splayed across the bottom. A woman in a ripped ball gown lay quite still, her body painted white, blood trickling from the corner of her mouth. Nimbly, caterers stepped around her. In the next room, guests posed for pictures with another dead body languishing on Peter's velvet couch.\n\nThe furniture was dark and battered, fantastic, unrestored. Velvet upholstery split at the seams, leather crazed and cracked, rough to the touch. A pair of gilt clocks on Peter's mantelpiece stood motionless, hands stopped, but the effect was lively, nothing like a morgue. The whole place thrummed with song and shouted conversation, bartenders in every corner. He saw the Dresden Dolls, the actual Dolls, in matching corsets, pounding drums and electric keyboard.\n\nAnimators crowded the dance floor\u2014colleagues Collin saw each day at meetings, solving problems, scrumming together. They looked ghostly now, in slippery black gowns and feathered masks, black lipstick, fangs. The pudgiest programmers seemed ethereal in flickering candlelight. Like night-blooming flowers, they came into their own, singing together\u2014 _\"Coin. Operated boy. Coin. Operated boy\"_ \u2014louder and louder, with pedantic glee.\n\nDrink in hand, Collin scanned the crowd for Peter, but couldn't find him. Maybe he didn't come to his own parties. Maybe he just wound them up and hid somewhere to watch.\n\nA tarnished mirror leaned against the wall. Collin recognized that mirror, the model or dilapidated cousin of the Magic Glass in EverWhen. When Collin touched the surface he half expected it to dimple and then melt, a watery portal to the Trackless Wood.\n\nPeter was everywhere, even when you couldn't find him. He had changed each room into a theater. Aeroflakes transformed the paneled library into a forest of shifting autumn leaves, and Obi and Akosh were gaming there. Obi was a Fire Elf breaking a path, cutting through underbrush with his ax. Akosh was a falcon. Like a dancer, he gestured with his hands and wrists, flying his avatar above his head. At a little distance, Collin discovered Peter watching every move.\n\n\"Welcome.\" Peter turned to Collin as if he were sharing a particularly lovely view. \"What do you think of falcons in the round?\"\n\nTogether they watched the falcon soaring and dipping between trees. Collin kept thinking Peter would leave, but he was in a rare mood, gazing at the colors haloing Akosh and Obi.\n\nCollin ventured, \"What if Akosh could do more than fly? What if he could see like a falcon too? He'd have this incredible vision where suddenly every detail was magnified twenty times. The whole game would shift to his point of view.\"\n\nIntrigued, Peter glanced at Collin, who seized the moment. \"You'd be swooping through the air but you could see your prey down on the forest floor and every clue, and every crevice. You'd see it all so fast.\"\n\n\"That would be terrifying,\" Peter said.\n\nHe could be gracious, acknowledging your contribution. He could be generous. He had opened his grand home, draping windows with the softest, darkest velvet, adorning his marble mantelpieces with prosthetic limbs, filling each room with a different food or drink, a raw bar in the parlor, a dim sum station in the breakfast room. He had staged this party with corpses that looked real, and flowers that seemed artificial. He'd filled great urns with protea\u2014blossoms spiky green, and curling coral, and deep pink fringed with feathery black.\n\n\"That's beautiful,\" Collin murmured as he watched night falling in the game.\n\n\"You think so?\" Peter pounced. \"What's beautiful about it?\"\n\nCollin gazed at the deepening shadows. Lavender, lilac, indigo. \"The colors.\"\n\n\"They're boring,\" Peter said. \"They're just what you'd expect.\"\n\nOf course Collin saw his mistake. Once again, he'd made the easy, sentimental choice.\n\nPeter kept his eyes fixed on the game. \"Let me give you some advice.\"\n\nCollin waited.\n\n\"Don't rely on clich\u00e9s. You aren't good at them.\"\n\nDappled forest light played on Peter's face, but he looked at Collin now. \"That's why you can't get the Flamethrowers right. Are you tired of them?\"\n\nYou know I am, Collin shouted inwardly, but he said, \"What would you suggest?\"\n\n\"Use your sketches of Daphne.\"\n\nIn all that noise and all that shifting light, Collin stood perfectly still. Peter spoke without rancor. He made his suggestion without heat, but Collin didn't for an instant take Peter's words as artistic advice. He heard them as a casual display of power: I know about your drawings; I know everything about you.\n\n\"You do better when you draw from life,\" Peter concluded as he walked away.\n\nThe party was raging all around, but Collin heard only his own heartbeat. I know you, Peter had told him without words. I know you and I know that you've been drawing Daphne.\n\nBut how did Peter know? Gazing through the doorway where Peter had gone, Collin saw a flash of white. Three steps and he had caught Daphne in the hall.\n\nShe had no idea what was wrong as she laughed up at him. Her blue eyes were dark, almost black in the black shifting light. She was barefoot and she wore a sheer slip of a dress.\n\n\"Come here,\" he said.\n\n\"Why?\"\n\nHand on her back, he steered her away from the dance floor.\n\n\"I'm not drunk enough for this.\"\n\nHe hurried her out to the dark entryway.\n\n\"It's cold.\" The stone floor chilled her feet.\n\nCollin didn't listen. \"You told Peter.\"\n\n\"What?\"\n\n\"You told Peter I drew you.\"\n\n\"I did not.\"\n\n\"Of course you did.\"\n\n\"I never told him anything!\"\n\n\"Tell the truth.\"\n\nPlayground-sincere, she said, \"Cross my heart and hope to die.\"\n\nHer mockery infuriated him. Either she had told Peter, or she had let him watch. He took her by the shoulders and demanded, \"What is wrong with you?\"\n\n\"Nothing!\"\n\nHis hand tightened on her shoulder. \"He saw us and you knew the whole time, and you never warned me.\"\n\n\"You're hurting me.\"\n\nHe heard the fear in her voice and he let go. She saw her chance and slipped into the dark.\n\nThe party burst apart. The corpse in the ripped ball gown sat on the stairs with a plate of calamari. The man impaled near the bar took to the dance floor with his ax protruding from his chest. Other cadavers mingled, dancing, laughing, rolling cigarettes.\n\nHellions crowded the dining room with its massive table and carved chairs. As Peter spoke, those material objects began to change. Above the table a Japanese lantern waxed into a full moon in the gathering mist. Dead water pooled where Peter's table had been. Stalagmites replaced chairs, and UnderWorld's white bats swarmed overhead. In the confusion, it took a moment for the assembled animators to perceive a dead knight lying in the water.\n\nCollin edged closer to hear.\n\n\"Those would have been our maggots.\" Like a surgeon demonstrating in the operating theater, Peter pointed to the corpse with a red-tipped laser. Tiny silver worms were writhing in every open wound and orifice, consuming flesh inside ears and eye sockets, underneath the skin. \"These would have been our rats.\" Peter announced with grim satisfaction. He pointed to a crevice where a blind rat devoured her own pups.\n\nAt first Collin didn't understand. Then, gradually, he saw that Peter was displaying outtakes, gorgeous horrors he had cut from UnderWorld before the launch.\n\nA thundering of hooves, a crashing avalanche of stone, and all bats scattered as Collin's horses broke through virtual stone. Watching his horses run together, Collin felt his heart shift. He was watching his horses and they weren't pretty. They were earth shakers, storm raisers. How had Peter stolen Collin's confidence? He had messed with Collin's mind. He was messing with Collin even now.\n\n\"These would have been our horses. We had to choose just one.\"\n\nThe herd raced past, tails streaming.\n\n\"Look,\" Akosh exclaimed. \"There must be some way to use them.\"\n\n\"Murder your darlings,\" Peter said.\n\n\"Asshole,\" Collin whispered. Fucking vampire. He pushed his way out of the dining room, past hellions deep in conversation. He skirted the dance floor and crossed virtual fields, stubble white with frost. He dodged dart games and turned away from tables of good food. He would not touch any of it. He took narrow stairs into a dark passageway. He kept moving away from the music. Suddenly, he realized he'd passed through to Peter's other house.\n\nThis twin house was grand, but empty and undecorated, dimly lit, with plaster peeling, windows covered with brown paper, floors bare and scuffed. In one room Collin saw mattresses upended and leaning against a wall, in another an old couch draped with a white drop cloth. Doors opened into empty rooms and narrow passageways. Desolate, confused, half dreaming, Collin imagined snow behind those doors, a lamppost in a wintry wood. Instead, he found dark closets and brick walls.\n\n\"Are you looking for the way out?\"\n\nHe turned and saw Daphne sitting all alone in a small straight-backed chair. He had never seen her so quiet or so still. \"I didn't tell him,\" she added.\n\n\"How does he know, then?\"\n\n\"He has the sketches. He has everything.\"\n\n\"I deleted them. They don't exist.\"\n\n\"Deleting doesn't help. The system backs up every image. You know that!\"\n\nHer words chilled him. He had known, but he had imagined some vast dump, electronic compost no one sorted. He had not pictured Peter collecting his discarded work. Every image, every version, every line.\n\n\"I thought you didn't care.\"\n\nPeter hadn't just heard about the drawings. He hadn't just glanced at them. He owned them.\n\n\"What's going on?\"\n\nShe wouldn't even look at him.\n\nHe knelt down at her feet so he could see her face. \"Tell me.\"\n\n\"Nothing to tell.\"\n\n\"Then why are you crying?\"\n\nShe glanced down at her own elaborately inked arms, and it seemed to him that her tears magnified and blurred her leaves.\n\n\"Come on.\" He took her hand. \"It's late.\"\n\nHer words were lively, but her voice was sad. \"No, it's not. It's early.\"\n\n\"You know what I mean. Let's go.\"\n\nShe shook her head.\n\n\"I'll take you home.\"\n\n\"That's just it,\" she said. \"I live here.\"\n\nFear mixed with anger as Collin fought his way through passages and crowded rooms to push open the front door. Confused images flashed before his eyes. His horses and Peter's face, and Daphne's tears. She lived with Peter. She was no longer obsessed or tangentially involved. Nor was she wild or independent, as she pretended. She lived in Peter's house. She belonged to him.\n\nHow different Peter's past behavior seemed. Not just critical, but jealous. Not just impossible to please, but punishing. He had punished Collin with advice. I've got you. That's what he'd been saying. I know what you have and I know what you want.\n\nThere he was even now. Standing outside on the steps, Collin opened a message from Peter on his phone. Subject: Flamethrowers. Attachment, a huge cache of sketches. Not just three or four, but every sketch of Daphne. Dozens of drawings, quick and slow, dressed, undressed. Peter had them all. He had been collecting them.\n\nCollin whirled around to stare at the closed door on Joy Street. This was a threat. This was Peter saying, I have your art\u2014not just your art, but your ideas\u2014and I can send them anywhere. He could send the sketches to Nina. He might have sent the file already.\n\nCollin sprang onto his bike and raced away. It did not occur to him that Peter would protect his niece from these pictures. Collin sped across the bridge with just one idea. He had to get to Nina first.\n\nCold knifing his throat, legs burning, Collin didn't stop to rest. He had to catch her before she left for school.\n\nHe used his own key, took the elevator up, and rushed into her dark apartment.\n\nThe rooms were still. At first he thought he was too late. Then he heard her stirring in the bedroom. \"Collin?\" She appeared in her big Hill School shirt. \"What's wrong?\"\n\n\"I thought I'd missed you.\"\n\n\"Why?\"\n\n\"Aren't you supposed to be at school?\"\n\n\"It's Saturday.\"\n\nHe sank onto the couch, and pulled her down next to him. It was the weekend and he hadn't even realized. They had the whole morning, but she was wide awake now, and her laptop lay there on the table like a bomb.\n\n\"What is it?\"\n\nHe wanted to say, Nothing\u2014I just missed you; instead he forced himself to tell the truth. \"I drew some sketches and I never showed them to you.\"\n\n\"Sketches of what?\"\n\n\"Peter got them, and he says that I should use them.\"\n\nNina looked at him and said, \"You drew her, didn't you?\"\n\n\"The thing you have to understand\u2014\"\n\n\"Show me.\"\n\n\"I just wanted to tell you...they're rough sketches. They weren't...\"\n\nShe wasn't having that. He didn't get to talk about his drawings in the abstract. She handed him her laptop.\n\nGuilty and indignant, hating himself, but hating Peter more, he logged in and opened Peter's message.\n\nThe first image emerged, a line drawing of Daphne in silver. Next, a sketch of Daphne slouched down on the shuttle bus. Daphne leaning over, playing pool. Daphne drinking at a bar. Daphne gaming like a dervish with her arms outstretched. One by one, each image filled Nina's field of vision, and she saw Daphne with her wide eyes and her laughing mouth and her cropped hair. She saw all this and she saw the time Collin and Daphne spent together. Daphne drunk and funny, messed up, impudent.\n\nThen she saw Daphne sitting just for him. Full color, fully shaded, Daphne in her hooded sweatshirt, Daphne unzipped, Daphne in her undershirt, Daphne undressed.\n\n\"How could you draw her like that?\" Nina gazed at Daphne's arched back, her breasts, soft and white, in contrast to her patterned torso, her black-inked collarbone.\n\n\"They're just studies.\"\n\n\"Studies of what?\"\n\n\"They're not important,\" Collin said. \"That's what you have to understand.\"\n\nNina said nothing.\n\n\"I didn't even keep them!\"\n\nShe was remembering the summer day they'd spent together. The pine trees and the heat, the taste of salt. She had asked him, \"Do you draw Daphne?\" And he had lied to her, even at that moment. He'd lied to her then.\n\nNow he told Nina, \"I was practicing. I was just experimenting, and I erased them all.\"\n\n\"I can see why.\"\n\n\"No, it wasn't like that! They were just like chalk drawings. I wasn't trying to keep them from you. I didn't even keep them for myself.\"\n\n\"Is that why you said you didn't draw her?\"\n\n\"I said it because...it wasn't important. And it didn't matter.\"\n\nShe closed the laptop and hugged it to her chest.\n\nHe struggled to explain. \"I want to be open with you. That's why I'm showing you the sketches.\"\n\n\"You're showing them because you're afraid of Peter.\"\n\n\"No! I'm showing them because I won't let him hold this over me\u2014or us.\"\n\nOver days and weeks Nina had dismissed Daphne from her mind; she had fought against distrust. Now all she saw was Daphne's body and her inked arms and her kissed mouth. All this had happened. It was still happening. Collin spent every day with her. \"You see her all the time. You'll see her today.\"\n\n\"No, I won't. I don't want to see her.\"\n\nHe could talk as much as he liked. His drawings drowned him out. They weren't ordinary. They weren't occasional. There were too many.\n\n\"I'm being honest,\" he protested.\n\n\"You _have to be_ honest!\" she shot back. Peter had Collin's art, and Peter had Daphne, and now Peter had Collin too. Because of this, Collin had confessed what he'd been keeping from her. He could say that these were only sketches. He could insist he didn't take them seriously. She knew better. After all, he'd drawn her too. She read a whole relationship in these studies, a second life entirely, overlaying theirs.\n\n\"Nina,\" he said softly.\n\n\"Just get away from me,\" she said. \"Just leave.\"\n\n\u2014\n\nHe left in frustration, but arrived home guilty. There were no little lies for Nina. There was no action without meaning. Wasn't that what he loved about her? Now he had hurt her; he'd misjudged her. He reached for his phone without a plan, without any motive but apology.\n\nToo late. Nina didn't want to talk. She didn't want to see him. When he texted and he called, she didn't answer. All Saturday, he left her voicemails. He wrote emails. He kept starting them, anyway. _Dear Nina, This is such a mess...; Dear Nina, I realize..._ They weren't any good.\n\nDarius said he had to stop. On Saturday night, he took Collin to The Plough & Stars and said, \"Believe me, the only thing worse than cheating is going on and on about it.\"\n\n\"I didn't cheat!\" Collin burst out.\n\n\"Whatever,\" Darius said. \"Stop talking about it.\"\n\nHe began writing on a yellow legal pad. _Nina, I shouldn't have kept those drawings from you. Please believe me when I say they didn't matter. The sketches aren't important. I don't think about them. I think about you._\n\nHis words were colorless. They could never capture what he felt. After all, what could he say? I lied. I should have told the truth when you asked. But also\u2014you're different from anyone I know. A bunch of drawings, a few late nights, a girl taking off her clothes. What did any of it matter? Noelle had worked as a model at the Museum School. Collin never took it personally\u2014but Nina looked inside art, uncovering intentions. She had seen his curiosity, his pleasure, his intense attraction.\n\nLate at night, Collin anguished over Darius's words. He considered his own denials and felt guiltier than before. To be honest, he would have cheated. He would have slept with Daphne, but she didn't let him, so he drew her instead.\n\nLoving Nina didn't mean he'd changed; he was the same guy as before. The same except for his remorse, his growing understanding as the hours passed. She had trusted him. She had risked her heart with him. Why had he taken it so lightly?\n\nHow could you? he kept asking himself. And at the same time that other voice grew stronger. The voice demanding, Why are you surprised? She's way too good for you. You knew it all along and now you've proved it. This was inevitable, his conscience told him. Give up. You don't belong together. He told himself all this, and yet his heart jumped every time he got a message. If only he could reach her.\n\n\u2014\n\nNina turned off her phone. She left her computer closed on the table. She couldn't bear his explanations and apologies, abject but self-serving. His images possessed her, multiplying in her mind, and she filled in the blanks, imagining where his art would lead. Collin was undressing Daphne, touching her, caressing her.\n\nShe didn't eat or drink. She had lessons to prepare, but when she looked at Emily Dickinson's poetry, the words seemed foreign. _It was not Night, for all the Bells \/ Put out their Tongues..._ What did that mean? She had no idea. Only when she was calm and happy had she understood those lines about despair.\n\nAt some point Sunday night, Nina's alarm clock began beeping. She struggled to open her eyes, and did not remember falling into bed. She did not remember anything in those first moments between sleep and waking. Then her disappointment came crashing down upon her. Pinned, she looked out at her shadowy room and saw her laptop blinking, her stacks of papers, Emily Dickinson facedown on the floor.\n\nMonday, she thought, and then, I can't. She had nothing to say, and nothing to give. _I believe that each of you has a unique contribution to make,_ she had typed on her syllabus. _Therefore, I expect you to come to class prepared. I understand that you are very busy, but I am asking you to make this class a priority, as it is a priority for me..._ Had she really written that? God, how insufferable she had been.\n\nShe forced herself to shower. Threw on some clothes, gathered papers, remembered to run a comb through her wet hair. When she took the elevator down, she was surprised to find her car keys in her hand.\n\nIt was street-cleaning day. When she arrived at school she saw the trucks, Phil's Towing\u2014WE MEET BY ACCIDENT\u2014hitching up and pulling the parked cars away. Raindrops beaded on her windows as she drove past the school, looking for a legal space. Then her windows fogged. She was driving in a cloud.\n\nSlowly, she climbed the stairs with the last stragglers. She had had no coffee; she hadn't even brushed her teeth. She hugged her thick copy of Dickinson to her chest. The bell was ringing, but a crowd stood outside her classroom door.\n\n\"Miss! Miss! Check this out,\" her students called to her. Her class was standing room only. As soon as she saw her blackboards, she knew why.\n\nHer double boards had been transformed. Black no longer, they had changed into a pair of landscapes\u2014two views of the Eliot Bridge over the Charles, one in winter, one in summer. The winter river glimmered white. Snow outlined the bridge, bare bushes, and park benches on the icy bank. The world was cold, the sky pale, with just a hint of red suggesting the early-setting sun. Next to this winter scene, the summer river showed the same bridge and trees, but here the dark water danced under a bright sky. The bridge was ruddy, the bank thick with grass and tender leaves. The pictures were huge, but also detailed. In winter you could see a lost mitten on one snowy bench. In summer, a family of ducks clambered up the riverbank.\n\n\"Who _did_ this?\" Trey asked.\n\nRachelle wondered aloud, \"How'd they get in?\"\n\n\"Is this, like, vandalism?\" Tentatively, Candace touched the summer riverbank, smudging the grass. \"It comes off.\"\n\n\"Stupid,\" said Trey. \"You know it's chalk.\"\n\n\"All right, everybody in my class sit down,\" Nina said. \"Everybody else\u2014go where you belong.\"\n\nReluctantly, students from other classes backed out the door. They couldn't take their eyes off those pictures. No one could. Nina gazed at the blackboards and saw the Charles sparkling in white stillness, as it had that first snowy winter night. It hurt, but she could not look away. She saw the scene as if no time had passed, and she was there with Collin and without him, looking at the same river twice.\n\nMeanwhile her kids were taking pictures with their phones. \"Okay,\" Nina said softly. \"You know the rule.\"\n\nReluctantly her students shoved their phones into their pockets and their backpacks. They found their seats, but couldn't settle down.\n\nZachary echoed Trey. \"Who did this?\"\n\nNina took a quick attendance. Colleen, Matisse, Jared, Australia, Candace, Rachelle...\n\n\"Because he's, like, a genius,\" Zachary said.\n\n\"How do you know it's a guy?\" Australia demanded.\n\nFrom force of habit, Nina turned to write, _DO NOW,_ on the board. She stopped. There was no room, and the rivers were too beautiful to erase. Setting down her chalk, she began her lesson. \"Dickinson leaves space for your own imagination. She leaves a space around each word so you can think about it. \" _Memory is a strange Bell..._ What do you associate with the word _bell_? _Memory is a strange Bell\u2014Jubilee, and Knell._ What's a _jubilee_? What's a _knell_?\"\n\nThe kids shifted restlessly. She called on Sebastian, and he just stared at her. She was heartless in her students' eyes, standing with her back to this amazing art.\n\nAustralia was pointing to the footings of the chalk bridge and asking Trey, \"That's the place where the geese live, right?\"\n\n\"No!\"\n\nSeveral kids corrected her at once. \"The geese live at the BU Bridge.\"\n\n\"We're looking at _Memory is a strange Bell,_ \" Nina said, but she was distracted too. How had he gotten in? How many hours had he spent on this? Amazed, she thought, You must have worked all night. Affronted, she thought, And you think that you can color over everything. \"All right, listen up.\"\n\nNobody paid the least attention.\n\n\"I'll wait,\" Nina announced, but she thought\u2014how? How could she get through this lesson and four more classes as well? \"I'm ready.\"\n\n\"Shut up, Liam,\" Tanya responded to some unseen slight.\n\nThat did it. Nina yanked open the file drawer of her Steelcase desk and found her water bottle. At last her students hushed as she unscrewed the cap.\n\nSilence as Nina poured out the water, wetting her industrial-size eraser just as she had seen Collin do. A long horrified sigh as she swiped the center of Collin's winter river. \"Miss,\" they murmured. \"Ohhh,\" they exhaled, as she ruined Collin's beautiful illusion, sweeping it clean. She felt almost criminal, but she didn't stop.\n\nWhen she turned around to face her kids, they sat chastened in their metal chairs. For the first time, they were afraid of her, because her eyes were filled with tears.\n\n\u2014\n\nShe was faster than her students when the bell rang. First out and down the stairs. She would have fifteen minutes to wash her face, to soak paper towels and press them like a rough compress against her closed eyes. She got to the staff restroom, and realized that in her rush, she had forgotten the key.\n\nShe glanced behind her, but she saw Jeff coming. She turned and ran down the hall, took the back stairs to the basement.\n\nIn the therapy room she could close her eyes. She could shut herself inside\u2014but someone else had come down here as well. She saw him at the end of the dark corridor, typing into the keypad on the wall, disabling the alarm.\n\n\"Stop!\"\n\nThe boy shrank back, trying to disappear. Too late. Nina threw herself between Aidan and the door. This is how you do it, she thought. This is how you skip out in the middle of the day. What do I tell him? How do I keep him here? What can I do? The questions tumbled over one another, as her mind woke. Now. Now. Today of all days. \"Aidan!\"\n\nHe turned back, trying to avoid her. He wanted to walk back to class, pretend nothing had happened, but she had caught him and she wouldn't let him go. He knew he was in trouble. She knew this was her chance.\n\nHe was unreachably tall, eyes fierce, head crowned with tangled golden hair.\n\n\"You know you can't leave school.\"\n\n\"I'm not leaving.\"\n\n\"You have class,\" she said. \"You've got _my_ class now.\"\n\nSilence. He didn't move, but she braced herself against the door. \"I know what's going on.\"\n\nThat irritated him, her sad-and-disappointed look.\n\n\"I know you're smart. I know you're capable...\" Standing with her back against the door, she remembered Maia's advice. Be funny. _But I'm not funny._ Be desperate. Now Nina told Aidan, \"I've seen your work.\"\n\nThat surprised him. He had not turned in many assignments.\n\n\"On walls.\"\n\nHe stared in disbelief. She didn't know. She couldn't know. Her accusation made no sense. He searched her pale, tear-streaked face. What was wrong with her? Lazare wasn't an adult anymore; she looked like a kid his age.\n\nEven so, she spoke as his teacher, with authority. \"I know what you've been doing.\"\n\n\"What have I been doing?\" he asked coolly.\n\n\"Playing UnderWorld with Daphne. Tagging the school.\"\n\nThen he went cold. Lazare was a mind reader, just like people said. No, she'd seen him. She'd watched him somehow\u2014at school or in the park. How did she know? He understood, of course, that she was Viktor Lazare's daughter. Neither in school nor in the neighborhood had she flaunted that connection, but it was creepy to consider now. \"You talked to my sister,\" he said in a hushed voice.\n\n\"She didn't let me.\"\n\n\"You came to my house.\"\n\nThe bell was ringing. She was going to be late. \"I haven't told anyone else yet.\"\n\n\"Wait.\"\n\nShe interrupted. \"You have a choice. Meet me in my classroom after school and we'll start working. Or meet me in the office with Mr. DeLaurentis.\" She turned to go.\n\n\"What do you mean\u2014with DeLaurentis?\" Now he was following her through the basement, up the stairs.\n\n\"You know what I mean.\"\n\nHe trailed her to the classroom, where the other kids were waiting. Head down, he slipped inside and sat in back.\n\nAll through her lesson, she sensed Aidan slouching in his chair, but she never spoke to him. She taught around him, calling on Becca to his left and Siddhartha to his right. Becca, could you read aloud? Siddhartha, why does Dickinson use the word _abyss_? Aidan looked up once, but she was careful not to catch his eye. She almost had him, and she wouldn't press her point.\n\n\"Look at Dickinson's dashes,\" she told her class. \"What does she use them for?\"\n\n\"Punctuation,\" said Miles.\n\n\"Yes,\" Nina said. \"What else?\"\n\n\"Instead of commas?\" Miles asked.\n\n\"Look at these lines.\" Nina wrote them on her chalk-smeared board.\n\n> _There is a pain\u2014so utter\u2014_\n> \n> _It swallows substance up\u2014_\n> \n> _Then covers the Abyss with Trance\u2014_\n\nShe repeated her question. \"Why do you think she uses dashes?\"\n\nShana said, \"To show where you should breathe?\"\n\n\"Good! Say more.\"\n\nShana hesitated and Nina had to resist the impulse to rush in and flood the room with questions. Did the dashes allow space for imagination? Could they be like rests in music? A nod to ambiguity? A way to honor the unsaid?\n\nShe said none of this. She held back, calm and quiet. When kids talked in class or rocked their desk chairs back, when they flirted or they fought, she stood between them and pointed silently to the correct line on the page. After her disastrous morning she began to right herself, slowing down and teaching her lesson, class by class. She had thought the day would be impossible, her lessons incoherent. In fact, she made a lot of sense. She saw it in her kids' faces. Less prepared than usual, she didn't try to cover so much ground. She kept it simple, giving students time to think.\n\n\"Thank you,\" she told her fifth-period seniors, right before the bell. Her students looked puzzled, but Nina couldn't tell them what she really meant. Thank you for crowding into my room. Thanks for whispering behind my back. Thanks for reading with me. She was grateful even when her kids refused her. I don't get poetry. I still don't understand. They took all her energy, and all her heart.\n\nWhen the last bell rang, her students thundered down the stairs, and Nina stood alone. Her windows were dusty, but her room glowed in the gold November sun. Washing down the last of Collin's rivers, she thought, I taught more than one hundred kids today\u2014and some of them were listening. She thought, I'm getting better at this. At the same time, she was deeply sad.\n\n\"How do you know Daphne?\"\n\nAidan was standing at her desk.\n\n\"Have a seat.\"\n\nHe remained standing. \"How did you...?\"\n\nShe dragged over a chair. \"I don't want to talk about her. You're here so that we don't have to talk about her. Please.\" She gestured to the chair.\n\nWary, Aidan sat down.\n\nNina took her grade book out, along with her weekly planner. \"Okay. You owe me six vocabulary sheets, three journals, and two expositions. Then for next week you have your analysis, and your poem.\"\n\n\"What poem?\"\n\n\"For Poetry in Action.\" She studied the planner. \"Start with the exposition. You can choose a Dickinson poem to analyze and memorize it at the same time.\" She looked up. \"You'll need to come in every day.\"\n\nHe stared at her in disbelief. First of all, he had never heard of meeting with a teacher every day. Detention, yes. Private tutoring? Nobody did that. And he had work. He was bagging leaves after school. He had begun with his mother's yard. Then he'd cleaned up Maia's place. His mother said that she was proud of him. His sister said nothing. She suspected him, but he ignored her. He wasn't angry with Diana anymore. He barely thought about her.\n\nHe lived alone, he worked alone, raking yards on Fayette and Amory and even as far away as Kirkland Street. He had filled forty giant bags with the fallen leaves of a giant copper beech. His arms were hard, his hands rough and cracked, because he didn't wear gloves, but he was more than halfway to a new BoX.\n\n\"I have a job.\"\n\nNina met his gaze with her gray eyes. \"Adjust your schedule.\"\n\nHe said, \"I have to do my\u2014\"\n\n\"Sorry.\"\n\nHe persisted. \"I have a lot of\u2014\"\n\nShe interrupted, \"I don't care.\"\n\n\u2014\n\nNina wasn't sure that Aidan would return the next day. Even when he showed up in class, she doubted he would meet her after school again. He looked so distant leaning back, his legs stretched out before him. His body told her, I'm here temporarily.\n\nAfter school, she regretted telling him everything at once. She had surprised him, but he wouldn't stay surprised for long. She had presented a stark choice, but on consideration he might not care. If he called her bluff, she would have to go to DeLaurentis. A whole investigation would follow. She would have to explain why she had never mentioned Diana's journal before, and she would have to answer questions about her own behavior, her impulsive visit to Diana, her threat to Aidan, her flouting student privacy. There would be consequences for her as well as him. She stood in her empty classroom and she waited, but Aidan didn't come.\n\nAt last, Nina started loading her bags with student papers. She was about to leave when she saw Aidan's face in the window of her classroom door. Quickly she dropped her bags behind her desk and ushered him inside.\n\n\"You're late.\"\n\nHe didn't answer.\n\n\"Take a seat.\"\n\nThey sat together at two student desks, and she handed him a vocabulary worksheet, the first assignment that he'd missed.\n\nHe looked at her uncertainly. Was she planning to sit there while he did these? That seemed to be her intention. She actually watched him fill in blanks with _misanthrope, jocular, protean, lassitude, inchoate._\n\n\"You know the words.\"\n\n\"Why wouldn't I?\"\n\nShe heard the challenge in his voice and countered, \"I didn't know that you had time to read.\"\n\nHe looked up and she saw a flash of fear in his blue eyes.\n\n\"Just keep working,\" Nina said.\n\nEvery day after that, they met in her empty classroom after school. They sat together at her desk and read Dickinson. She had him type his analysis on the classroom computer, where she could see him. _\"I heard a Fly buzz\u2014when I died\" is an eerily calm poem about death. The narrator is detached from the situation..._\n\n\"What are you saying here?\" She pointed to his opening paragraph.\n\nAidan searched Nina's face for clues, trying to find the words she wanted, hoping to fulfill his obligation fast.\n\n\"You're not interested in any of this.\"\n\n\"Not really.\"\n\nShe was sitting with her elbows on her desk, her chin resting in her hand. She was very serious, and very beautiful. \"What if I say you have to be?\"\n\n\"You can't make someone interested,\" he pointed out.\n\nShe ignored this and handed him his analysis. \"Go ahead and read it to me.\"\n\n\"Aloud?\"\n\nHe caught the hint of a smile and he thought, Oh yeah. Otherwise she couldn't hear me. But when he started reading, she challenged him.\n\n\"Are you sure you want to say death doesn't bother her?\" she asked. \"How do you know? How can anybody know?\"\n\n\"True.\" He considered his own words. \"I'll say it seems like death doesn't bother her.\"\n\nHe kept reading, and she bent her head to listen. It amazed her that he had plagiarized a paper. He wrote clearly, choosing his words well. He was more than capable. What was it then that had compelled him? Gaming in itself, or Daphne?\n\nTears welled up, but she didn't think that Aidan noticed.\n\nHe saw more than she realized. He saw her sadness. He saw her ink-smudged hand, her forearm where her sleeve fell back, her small wrist. He saw her head dip slightly. She was exhausted.\n\nAt first he watched for those moments when she let her guard down, the split second when she almost fell asleep. He willed her to lay down her head so he could rush outside.\n\nGradually, however, Nina drew him in. Shaking off sleep, she had such an edge about her, an intense concentration. She reminded him of someone, and he kept searching his memory. Who was it?\n\n\"Remember the speaker,\" she said of the Dickinson poem. \"Who is speaking?\"\n\n\"Someone dead.\"\n\n\"And what does that sound like?\"\n\n\"Calm,\" Aidan said. \"Numb.\"\n\nNina was leaning forward, intent on every word. When she pulled back, he caught the resemblance. She was like the doctor at the hospital, listening to his heart.\n\n\u2014\n\nDay by day, Aidan began to catch her moods. Sometimes she was sleepy, sometimes miserable, sometimes bored. He forgot about escape. Her determination fascinated him. Her eyes closed for just a moment, but she never put her head down. She sat up straight and she was Miss Lazare again. \"Here,\" she said, \"I think you overstate your case.\"\n\nHer knowledge scared him. Not her knowledge of literature, but her knowledge of his secret life.\n\n\"You need to memorize the poem.\" She passed him her thick book, the complete Dickinson, its paper jacket printed with blue and green leaves.\n\n\"I can memorize it now.\" He read the poem rapidly to himself and then turned the volume facedown on the desk.\n\n\"Don't do that! It's old. You'll break the spine.\" Protectively, she gathered her book up again.\n\nStartled, he blurted, \"Sorry.\" Then, trying to appease her, he recited the poem rapidly all in one breath, _\"I heard a Fly buzz\u2014when I died\u2014 \/ The Stillness in the Room \/ Was like the Stillness in the Air\u2014 \/ Between the Heaves of Storm\u2014 \/ The Eyes around...\"_\n\nShe waited until he'd finished and then she said, \"No. Say it like you mean it.\"\n\n\"But I don't mean it,\" he answered, sullen, because she was not impressed.\n\nShe looked at him steadily, without anger, without reproach. She could have snapped at him. She could have walked out, but she said nothing, and she stayed.\n\nGazing back, he saw his teacher's patience, vast and still, spread out before him like an inland sea.\n\nAt last she ventured, \"You're a good student when you're in school.\"\n\n\"How do you know?\"\n\n\"I've read your transcript, obviously.\"\n\nShe wanted more than anything to bring Aidan down to earth, but that was not the effect she had on him. She seemed to him enchanted, dangerous, a mermaid in the Whennish Sea.\n\nWhen she was dissatisfied, she barely looked at him. When he worked harder, she nodded, as though she'd promised herself she wouldn't praise him. Her determination drew him to her. Lovely and difficult to please, she sent him off to prove himself in thorny fields, and set new tasks as soon as he was done.\n\nShe said, \"When you recite, you have to slow down and think about the meaning of the words.\" But when he recited again, _\"I heard a Fly buzz...\"_ he was thinking about her. He was imagining her elusive smile.\n\n_\" 'I heard a Fly buzz\u2014when I died' is an eerily calm poem about death,\"_ she said, quoting his essay back at him. _\"The narrator is detached from the situation.\"_\n\nBriefly, he felt it. Detached from the situation, he watched Miss Lazare sitting at her desk. He saw himself, looking into his teacher's eyes.\n\nCollin felt as though his world had changed to black and white. He had known magic. Nina had touched his life with gold. Now he could not reach her. He could not get back to where he'd been.\n\nIn games you could redeem yourself with points or jewels. You could trade gold for freedom, or even give an extra life away. In EverWhen you always had some recourse. Collin envied Everheads counting down to Launch Day in their ordered companies, their banners rippling overhead. He wished that he could run away and take his place among them. He wished that he could quit his job and disappear. Such was his guilt, his increasing sense of shame. Then he thought, No. Quitting was just what Peter wanted.\n\nMonday morning, Collin walked to the bus stop, although he thought more than once of turning back. Waiting, he remained uncertain, but when Arkadia's shuttle arrived, he boarded. He would not give up. He had seen his horses thundering through the dining room. He had seen them with his own eyes, and they weren't pretty. They had silenced everybody.\n\nThe other hellions on the bus were quiet. How much did they know? Had they seen him confront Daphne at the party? She was not riding the bus, but had they spoken to her? Maybe they thought he was a coward, crawling back. I can't win, Collin thought suddenly. He's got me to the point where I look like a quitter if I leave and a punching bag if I return.\n\nCollin hunched down alone in a window seat. Just do your job. Just do your work and you'll get paid. He told himself this, but he thought of Peter. His harsh words, his frown, his cold dissatisfaction.\n\nWhen the bus stopped, he roused himself to follow the others, stepped off with furious determination. To do what? He hardly knew. He wanted the impossible, the life he'd had before\u2014his work, his love, his happiness.\n\nExiled from Nina, Collin found himself alone at Arkadia as well. In the next days and weeks, Peter reorganized pods so that Collin worked with all new people on background\u2014texture and detail, condensation on rock faces, mists and shadows, damp stains on stone.\n\nCollin was not the only artist reassigned as the launch approached. Even so, Collin sensed that Peter was isolating him. No one disabused him of the notion. Obi, Akosh, Tomas, and Daphne kept a friendly distance. They smiled and even played Ping-Pong with him on occasion, but they kept the conversation light. No gossip, no commiseration.\n\nDaphne approached with cheerful caution, never once alluding to the drawings or the party or her tears. Only once was she at all real with him.\n\nCollin passed her desk and startled her. She cleared her screen with a quick swipe of the hand.\n\n\"What was that?\" Collin said. \"Was that your new marketing plan?\"\n\n\"New project,\" she admitted.\n\n\"For UnderWorld or something else?\"\n\n\"You don't want to know.\"\n\nOf course new projects cropped up all the time. Stealth mode was common, but Collin understood that he had been excluded once again.\n\n\"You'll get through this,\" Nicholas told Collin in the sound booth.\n\n\"Maybe.\" Collin had been rendering stalagmites in UnderWorld's caverns, and dragon bones, and mountains of slick guano. He was beginning to wonder if he would ever draw a face again, and yet he toiled on, sometimes bitter, sometimes laughing at himself. He was drawing bat shit, after all.\n\nWork got even worse. Peter moved him to color correction. For days he didn't draw at all. He studied sequences at his workstation, analyzing the tint and continuity of dragon scales in the dim light, underground. His hands hovered over the keyboard, and, like a caged panther, his imagination turned and turned upon itself.\n\nHe stood and stretched his arms. He spun around and he saw dragon scales on every surface. Every cubicle glowed silver. He had never been so tired or so bored, and he wanted to bust out of there. Pride prevented him. Don't give up now, he told himself. Don't let Peter win. He forced himself to work, imprisoned in his chair. The days stretched out, hour after hour.\n\n\u2014\n\nOn Launch Day Collin watched news reporters arrive, trailing long black cables. He watched the hallways fill with virtual mist, an atmosphere of mystery advertising aeroflakes. From a distance, he watched Viktor speak.\n\n\"Today we are announcing that OVID will support all Arkadian games. Not just UnderWorld, but EverWhen as well. We are opening portals for world-jumping so that players can move seamlessly from one game to the next.\"\n\nThe metallic kiss of cameras.\n\n\"A world without edges,\" Viktor said. \"No screens, no frames. You go where your imagination takes you.\"\n\nI wish, thought Collin.\n\nHe wandered through the building, and there was food, and there were kegs. There were demos. Reporters standing at the Gates and dueling in the halls. Bright as a harvest moon, a virtual clock was counting down to midnight. The illusion caught Collin's attention, but he could not forget the maze of cubicles beneath.\n\nHe paused for a moment at his desk. On impulse, he pocketed a black pen and five packs of sticky notes. Then he headed outside.\n\nThe wind hit hard. The evening was bitter, and he was glad. It was quiet outdoors. No speeches, no demos, but as he rounded the corner of the building, he heard voices.\n\n\"Hey, hey, ho, ho. UnderWorld has got to go.\" Shoulder to shoulder, members of Christians Against Gaming Exploitation were marching in the west parking lot, a dozen young men and women carrying hand-painted signs. CU IN HEAVEN! JESUS PLAYS FOR KEEPS.\n\nCollin stared at the picketers bundled up like carolers in their knit hats and Christmas sweaters. One in particular, a paunchy guy in rectangular glasses, called out to Collin. \"Hey, man, listen!\"\n\nCollin studied the protestor's round face.\n\n\"The Lord tests the righteous, but His soul hates the wicked and the one who loves violence.\"\n\nWearily, Collin said, \"Hi, Darius.\"\n\n\"Repent!\" cried a young woman in a reindeer sweater. Her long hair spilled out under the brim of her candy-striped hat. Her eyes were bright, her nose and lips and eyebrows ringless.\n\n\"Hey, Emma,\" Collin said.\n\nHe took the early shuttle back to Cambridge, but it was dark when he got off in Harvard Square. He stood for a moment on Mt. Auburn Street, just watching the students, and the construction workers, and the homeless veteran, and the people carrying rolled yoga mats. He glanced at Grafton Street and the Harvard Book Store, where Nina used to meet him.\n\nHis heart rebelled against his loneliness. His body ached from hunching at his desk, and now his fingers flexed. He walked across the Square to Grendel's. He hurried down the stairs to the old dive, and all the green lamps on the tables welcomed him.\n\n\"Collin!\" Sam cried out as he approached the bar. Tiny, sharp-eyed, hospitable, she brought a pint without his asking. \"You never call, you never write.\"\n\n\"Hey.\" Kayte was still waiting tables. \"How's Nina?\"\n\nCollin didn't answer, and she didn't press.\n\nHe sat at the bar and watched Sam mixing drinks as Grendel's music pounded all around him.\n\n\"What's it like there?\" Sam asked.\n\n\"It's just work.\"\n\n\"Come on, give us some dirt,\" Kayte called out as she passed through to the kitchen.\n\n\"I got nothing,\" Collin said.\n\n\"Empty-handed,\" Sam said.\n\n\"No.\" He felt in his jacket pockets and pulled out the sticky notes. He lined them up on the bar along with the pen.\n\nSam said, \"Oooh, exciting. Little pads of paper.\"\n\n\"Pick a color. Any color.\" They were all pale office shades. Pink, blue, green, yellow, white.\n\n\"Green.\"\n\nHe drew a leaf on a green sticky note. A simple maple leaf, outlined in black pen.\n\nSam said, \"You quit your job.\"\n\n\"No.\" He turned back the first sticky note and drew the leaf again. On every page he drew the maple leaf in a slightly different position.\n\n\"So you like it there.\"\n\n_Like_ is not the word, thought Collin. Liking the job didn't enter into it.\n\n\"Nice,\" said Kayte, who was looking over his shoulder.\n\nHe sensed several people watching. Two girls sitting next to him leaned closer as he drew another leaf.\n\n\"Oh, I get it,\" one girl said, as he drew the maple leaf for the twentieth time, edges curling in an imaginary wind. Her name was Emily and her friend was Kira and they were going out for Kira's belated birthday. Emily was bright-eyed and heavy. She had blond hair but dark eyebrows, and she wore a low-cut shirt. Kira was heavy too, but softer and quieter. Her eyes were dark with eyeliner and she had beautiful black hair all down her back. She looked like the girls Collin had known in high school. He drew a leaf on the last sticky note and gave the whole pack to Kira. \"Happy birthday.\"\n\n\"Oh, wow,\" said Kira, uncertainly.\n\nEmily said, \"It's a flip-book.\"\n\nKira held the pack of sticky notes and flipped them with her thumb. The maple leaf drifted, rising and dancing and falling to the bottom of the last page. \"It's so pretty,\" Kira exclaimed. \"How did you learn to do that?\"\n\n\"You're terrible,\" Sam told Collin.\n\nHe was already drawing a flip-book for Emily. This one wasn't nearly as beautiful. It was just an ice cream cone melting in the sun. \"You're so good,\" Emily exclaimed.\n\n\"No, ice cream is boring.\" Collin gave up halfway through. \"I'm doing something else.\"\n\nThe two of them waited quite seriously, and a few others stood behind them. Even Sam leaned over the bar.\n\n\"Okay, let's try this.\"\n\nHe took his black pen and drew a line across a blue sticky note. A line sagging ever so slightly. A telephone wire. Then he drew a bird on the wire. A black bird, alone and hunched. He turned the page and drew the bird again. He turned each page and drew the bird spreading its wings.\n\nWhen Collin finished and thumbed through the book, his audience watched the bird take flight, soaring and swooping in the air until at last it flew off the page.\n\n\"It's yours.\" He gave the book to Emily, and she showed it off to everyone. At one point Kayte came over with her phone and filmed it.\n\nThe girls bought Collin another drink. They invited him to a party later on that night.\n\nHe enjoyed the attention, and then, suddenly, he was tired of it. He started glancing at the door. Actually, he explained, he was waiting for someone. Nobody believed this, but he wasn't lying. He was waiting for Nina, although he knew she wouldn't come.\n\nGold showered down. Autumn leaves so bright that for a moment Aidan covered his eyes. He wasn't used to this anymore, the dazzling colors, the shifting light and shadow. Opening his new BoX, he felt like a figure in a snow globe, wind blowing, leaves swirling all around.\n\nHe heard cheering, tramping in the Trackless Wood\u2014the remnants of his company. Of his sixty-one Elves, only twelve remained. Centuries had passed since they had qwested together, scores of battles had been waged and won. Giants of the dark cliffs had forged an alliance with the Gnomes who toiled underground, and together they'd built an enormous fighting force, the Nord, who had begun raiding Elvish strongholds in an attempt to launch themselves against the Keep.\n\nWe've been waiting, Aidan's comrades said. In real life, the company was scattered in living rooms, bedrooms, and basements all around the world. Some guys were fat, some had bad knees, some were just kids wearing pajamas, but in EverWhen they were all Water Elves, tall and elegant, with flowing hair. They wore chain mail and they carried swords. \"You're late, Tildor,\" chided Dracon, Aidan's second in command.\n\nEven as he spoke, the forest darkened and the wind picked up. No, not the wind. Aidan threw himself to the ground as dragons tore through the wood, blackening every living thing. Moments later it was over, but his ears were ringing with the dragons' screams.\n\nShaking off ash, he struggled to his feet. Close by his side, Aidan's old qwesting friends debated what to do.\n\n\"March on!\"\n\n\"No. Hold back and wait.\"\n\nAlready, moss crept up blackened tree trunks. Ferns sprang up from the ashy forest floor. Withered branches spread and multiplied, sending forth new twigs. Pale folded leaves opened into oak, or maple leaf, or elm, filtering the light and showering charred earth with every shade of russet, gold, and green. The others were appealing to Aidan for orders, but he gazed at the changing leaves, the ever-shifting light, and he thought, How beautiful. How strange the way the woods surrounded you.\n\n\"Let's go,\" Dracon urged him.\n\nAidan lifted his arm and Tildor held his sword aloft. \"Follow me!\"\n\nHe led his Elves to a thousand-year-old beech, a cosmos of its own, with its vast canopy of leaves. The whole company, linking hands together, could not span its massive trunk. \"This is the place.\" Aidan scored the smooth gray tree trunk with his sword, opening a seam that darkened with a sickening sound of splitting wood. \"This is the way,\" Aidan called out as the others followed him inside the tree to UnderWorld.\n\nAll that night, he qwested with his company. They fought their way to the silver river and took a ferry to the far shore.\n\n\"Cover me!\" Aidan shouted as he attacked the Iron Man, striking welded joints and eye slits. Battle-tested, he knew how to move, and how to leap, how to manage spectral weightlessness. He could fend off white bats, and navigate labyrinthine passages carved in stone. Even so, the way was difficult, increasingly complex.\n\nThey came to a cavern he had not yet seen, a crypt where they found Elves disfigured and leprous with mold. Aidan saw cauliflower ears and wobbling jaws, noses melting like candle wax. \"Stand together,\" he commanded.\n\n\"Put up your swords. We have no power,\" a living corpse with blackened fingers said.\n\nWonderingly, Aidan asked, \"Who were you?\"\n\n\"Fire Elves.\"\n\n\"Who did this to you?\"\n\nThe ruined Elf pointed through the archways to a dim-lit chamber, where a rustling creature stirred. Taller than Aidan, a praying mantis whirred, and clicked, and cleaned its folded legs on a dark throne.\n\n\"Watch out,\" the moldering corpses shouted as Aidan approached the insect king. \"Don't let him bite,\" the blackened Elf called after him.\n\n\"Wait here,\" Aidan told his company, and all obeyed, except for Dracon, who followed him.\n\nLike a dying fire, the chamber faintly glowed. The praying mantis turned, and Aidan saw the insect's mandibles working, outer eyes wide-set and swiveling.\n\nAidan drew his sword. \"Reverse your spells.\"\n\nWith a dry, rattling sound, the mantis rose up gigantic on its hind legs.\n\n\"Reverse your spells, or come down and fight.\"\n\nInstantly, the mantis pounced. Aidan sprang back just in time, slashing at the insect's hard body, its whirring limbs, and pointed face. He sliced one antenna. Instantly, the quivering organ grew back. He stabbed the mantis in its thorax and the insect lost its balance, staggering back for just a moment. Then, like a boxer, the mantis rose up and pounced, toppling Dracon with a single blow.\n\n\"Get up. Get up!\"\n\nToo late. Dracon lay on the throne room floor, his neck pinched in the insect's mandibles.\n\nAidan's company rushed the room, even as the mantis bent over its prey.\n\n\"Stay back,\" Aidan shouted, but they didn't listen and mobbed the insect, attacking with their useless swords. Each blow glanced off the insect's back, and the mantis fed until Dracon's body began to swell. His flesh turned white, his face began to curdle.\n\nThe other Elves tried to drag away the body. Fenuel seized one lifeless arm, Lorimar the other. Suddenly these friends were stricken too. Their hands withered where they had touched Dracon's white flesh, their fingers blackened as with frostbite.\n\nAidan knew death in battle. He himself had been dismembered, impaled, decapitated, but that was just a nuisance and a loss of points. After a few minutes, anyone who died in EverWhen could jump back up again. This was something else, a wasting disease. \"Don't touch him,\" warned Aidan as his Elves crowded Dracon.\n\nOnce again, the clicking, rattling sound. The mantis? Alarm clock. \"I have to go,\" Aidan told the others. His mother was still at work\u2014no danger there\u2014but he was on every kind of probation: academic, disciplinary. He had to get to school.\n\nA moment later, he was diving for clean clothes, racing to the bathroom, stumbling down the stairs. After his all-nighter, the house seemed warped and thrown together, the ceiling dangerously low. At the breakfast table, Diana was a headless body. It took Aidan a few seconds to see that she'd buried her head in her arms.\n\nGradually, the kitchen began to right itself. The sink, the chairs, the window, the apple clock, Diana sitting there with bloodshot eyes.\n\nWas she crying? At first it was hard to tell, and then he saw her tearstained cheeks, and it was hard to look. Long ago Diana would cry when their father hit Aidan. Then Aidan would start crying too. He would shut his eyes and turn away, do everything he could to stop, but once he saw her tears, he could not force back his own. \"What happened?\"\n\nShe pushed her chair away from the table, but she didn't speak.\n\nShe had lost Brynna. Not in a day, not over weeks, but gradually, as the days grew shorter, as rain turned to snow, and snow hardened into ice, Brynna had pushed Diana away. At night she went to parties, while her parents watched Angela. She stayed out with Anton and his friends, Khalil, Dmitri, Sevonna. She was actually seeing Anton again.\n\nLate at night Diana had confronted her by text. What ru thinking? Cruelly Brynna had replied, Ur only jealous. After that Diana couldn't sleep. She'd lain awake all night, listening to Aidan qwesting.\n\nLight-headed, Aidan watched his sister dematerialize into dots. He gripped the edge of the table and she was herself again\u2014but much smaller than he'd remembered. He had just put away his sword and there she was, slipping on her parka, shouldering her worn-out backpack. She looked so defeated, he forgot her tattling. After all, it didn't matter anymore. He had his new BoX and he loved Diana again. He had never stopped loving her\u2014but she should have known that.\n\nHe didn't speak, but he felt for her as she stepped into her snow boots. Those boots looked so heavy, they could have been stone.\n\n\u2014\n\nDiana kept her head down and her coat on once she got to Emerson. Her jacket's hugeness shielded her as she made her way upstairs past couples kissing, girls gossiping, guys chest-butting like demented stags. Brynna approached, and Diana looked up, hoping for an instant. Her former best friend swept past her.\n\nAnton arrived, and his blond hair was short and spiky, his eyes hard. He'd nicked himself shaving, so you could see a spot of blood on his neck. Diana tortured herself, watching Brynna drift toward him. She looked so soft and gentle, her thick hair wafting over her shoulders. She was nothing like Anton, and yet she stood with him by the lockers. Hands in each other's back pockets, they were practically married.\n\n\"Congrats,\" Diana said as she walked by.\n\nBrynna pretended that she didn't hear, but Anton snarled, \"Dyke.\"\n\nA moment later, Anton's head smashed into the metal locker doors. It happened so fast even Diana was surprised. She barely understood what she had done when he surged back and seized her shoulders. His fingers clawed through her down coat.\n\nVoices all around her, cheers, and catcalls. Mr. Allan yelling, \"Whoa. Walk away. Walk away.\" She didn't walk. She kicked and scratched. Like a creature shedding her second skin, she slipped from Anton's grasp. Lithe and strong, she left her puffy jacket in his hands.\n\nHe dropped the jacket and she flew at him as he fought her off, ignoring Mr. Allan's threats and Brynna's pleas. The bell was ringing, the hall flooding with students. Teachers were struggling to pull Diana and Anton apart, but it was Aidan who charged between them. He had followed her to school, shadowing her all the way upstairs.\n\n\"You touch my sister and I'll fucking kill you.\" Aidan's cheeks were blazing. He was white light, he was the dazzling sun.\n\n\u2014\n\nTen minutes later, Aidan and Diana sat together at the round conference table in DeLaurentis's office. Anton sat at a little distance on the other side.\n\nMr. DeLaurentis spread his hands. \"Can we work this out?\"\n\nThe three of them sat silent. Anton stretched out, with his chair pulled back from the table. Aidan stole a glance at Diana. Her eyes were bright, her tears gone. She looked taller, as though she had thrown off a crushing weight. And it was strange, but he felt taller too. He had not realized how heavy her sadness had become.\n\nMr. DeLaurentis told Diana, \"We don't tolerate bullying.\"\n\nShe pointed straight at Anton. \"He bullied _me._ \"\n\n\"As for you...\" DeLaurentis turned to Aidan.\n\n\"He didn't do anything!\" Diana interrupted. \"He was protecting me.\"\n\n\"This conflict didn't happen on its own.\"\n\nAidan settled back, preparing for the onslaught. It was always like this at school. The actual fight lasted just seconds. The discussion afterward took hours.\n\n\u2014\n\nDiana was still stuck in the principal's office when DeLaurentis ordered her brother to return to class. Aidan felt guilty about leaving, because he knew exactly what would happen next. She would have to see Miss Sorentino, who made you talk until you would admit anything. You were bored at school, you were tired of your life. She got you to the point where you confessed feelings you didn't even have.\n\nSlowly he made his way to English, arriving in a tide of whispered speculation.\n\n\"Where's Anton?\" Sofia whispered to Aidan, as soon as he sat down.\n\nRafael asked, \"What did you do to him?\"\n\n\"Was he expelled?\"\n\n\"Okay, let's concentrate.\" Miss Lazare handed Aidan a ballot for Poetry in Action.\n\nLazare was calling on each kid, one by one. You had to stand and recite your chosen poem from memory, after which the class would vote for one student to represent them in the school competition. Unless you were into theater like Becca, the whole thing was a nightmare until your turn was over. Then you could sink down and watch other kids rock back on their heels, and forget their lines. Miss Lazare said, \"That's okay, just take a breath,\" and it was torture, but she never let you give up. She made you stand there. Naturally people chose the shortest poems eligible. \"Fog\" by Carl Sandburg. William Blake's \"The Sick Rose.\" \"Do _not_ vote for me,\" they whispered to their friends.\n\n\"I need your attention,\" Miss Lazare told the class. Nico was reciting Roethke's \"My Papa's Waltz.\" _The whiskey on your breath..._ But Aidan was the center of attention; even those who hadn't seen him challenge Anton had heard about the brawl.\n\n\"Did you really break his nose?\" Sofia whispered.\n\n\"Would I be here if I'd actually hurt him?\" Aidan retorted, enjoying his notoriety.\n\n\"Aidan,\" Miss Lazare said, as soon as Nico had finished.\n\nHe waited for the reprimand. Don't talk in class.\n\n\"Get up there.\"\n\nOh. In the excitement, he'd actually forgotten.\n\n\"Your turn.\"\n\nAidan remembered nothing. He had prepared, of course. Lazare herself had practiced with him, but when he stood in front of the blackboard, he could not recall a single word of Dickinson.\n\nCurious, his fellow students stared at Aidan. Their ballots fluttered all around them.\n\nHe glanced at Miss Lazare, who waited in the back of the room with her clipboard in her hands. He didn't care about the others, but he hated himself for letting her down.\n\nNo. He had to think. Back before the lecture in DeLaurentis's office. Before the fight, before Diana at the breakfast table and the long qwest underground. He had to tunnel through all the battles he had fought and won and lost. Mantis king and Iron Man, silver dragons and three-headed dogs.\n\nThe moment stretched too long. Not a moment, but an entire day, a year. The class grew restless, but he closed his eyes and fought to uncover words small as insects, black and quivering.\n\n_\"I heard a Fly buzz\u2014when I died.\"_ His voice was deep and deadpan, eerily calm, even as he searched his memory.\n\n_\"The Stillness round my form \/ Was like the Stillness in the Air...\"_\n\nThe whole class hushed in horror and in sympathy. No one had heard the poem quite that way before. He was struggling but he would not give up. _\"The Eyes beside\u2014had wrung them dry...\"_ Searching for each word, he seemed to be improvising on the spot.\n\nThere had never been such silence in that room. Not a word, not a breath. _\"I willed my Keepsakes\u2014Signed away \/ What portion of me I...\"_ Long pause. _\"Could make Assignable, and then...\"_\n\nHe was like a diver. They could barely see him anymore as he swam deep underwater to retrieve each phrase. His classmates watched as he held his breath and sank into the abyss.\n\nIn a trance, he swam down to his own death, his own body on the hospital bed. _\"There interposed a Fly...With Blue, uncertain...stumbling Buzz.\"_ His voice was strange, not his at all, but cold and numb. _\"And then the Windows failed\u2014and then \/ I could not see to see.\"_\n\nSilence again, and then everyone was clapping, because he had found the words, and brought them back alive.\n\nNina could not stop smiling. All traces of the teacher vanished from her face.\n\nLook at Miss Lazare, kids told each other. Total joy! The little jump you always hoped for, but hardly ever got to see.\n\n\"I knew it,\" Nina told Aidan after school. She had known instantly that he would win the class election. She'd seen it in the students' faces. He would represent them in the assembly.\n\n\"That's crazy,\" said Aidan. He had no interest in reciting for the school, but Miss Lazare's response captivated him. She seemed like an entirely new person, her eyes alight.\n\n\"It's incredible,\" she said. \"Fantastic.\"\n\nShe gave him all the credit, and, at the same time she felt that teaching him was the best thing she'd ever done. Bright, dreamy, obsessed with fantasy, Aidan was a natural, seizing language for himself, inhabiting simile and metaphor. He was born for poetry.\n\nWas it that ability, or was it fear? Was it simply practice? Understanding what she was looking for? He was using what she gave him, making connections, drawing inferences. If she spoke to him about one image\u2014death as a fellow traveler\u2014he found a complementary example: _The Carriage held but just Ourselves\u2014 \/ And Immortality._\n\n\"He listens,\" Nina had told Jeff at their weekly lunch meeting.\n\n\"Of course he does. He has your undivided attention. The bigger question is how you want to use your time.\"\n\n\"I don't think I'm taking anything away from the other students.\"\n\nJeff warned, \"You don't want to play favorites.\"\n\n\"I'm not!\" she protested. \"He's a student at risk.\"\n\n\"He's not alone.\"\n\n\"But he really learns this way.\"\n\n\"Everybody does.\"\n\nShe knew what he was thinking. Another white-middle-class success story. \"I just feel like I'm finally doing something.\"\n\n\"You see him every day; you're spending five hours a week with him. That's an entire class of one.\"\n\nWaiting for Aidan after school, Nina opened her copy of Dickinson on her desk. Jeff cared about metrics. He believed in trying for the greatest impact possible, and by impact he meant reaching the many, not the privileged few. Nina had privileged Aidan, lavishing her time and her attention on him. It was not her mission to run private tutorials at a public high school.\n\nBut what if this was the way she taught best? What if this was how she made a difference? Test scores didn't matter to her in the aggregate. I'm not big-picture, Nina thought. I doubt I'll make an impact on a hundred kids\u2014but I'm teaching one.\n\nShe didn't care what Jeff said or how he warned her; she knew that she was getting something done at last. Listen to Aidan recite. Look at his written work. She had his essay marked good and wonderful, lying right there on her desk.\n\nEven so, her influence was limited. She could hold Aidan's attention for an hour, but she could not control his life. When he was late she worried that he wouldn't come. He arrived exhausted after school and she could only wonder, Are you gaming? Are you running out at night? When he seemed distracted, she imagined he was giving up. Most days he focused, but some afternoons he gazed into the distance and she thought, I'm losing him. I will never change anybody. She had not changed Collin. She had left him as she'd found him, gifted, adventurous, devoted to the chase. She had not changed him, but she missed him.\n\nSometimes she wondered, What if I was wrong? You weren't wrong, Lily assured her. Sometimes Nina thought, He lied to me, but does that mean he'd cheated too? Obviously, Julianne said.\n\nAnd yet Nina doubted herself. She thought, He was not what I had hoped, but I assumed the worst. He disappointed me, but I set him up. What did I expect, bringing him to that place? She wavered and then stopped short. Collin had decided what to draw and how to act.\n\n\"Miss?\"\n\n\"Oh!\" She hadn't seen Aidan come in.\n\nHe apologized for startling her. He didn't know how glad she was to see him. He shook her from her thoughts. He handed her three completed vocabulary worksheets.\n\nShe said, \"Good, and I've graded your essay. We'll go over it.\"\n\nHe looked at her expectantly.\n\nShe thought, You know it's excellent. You can't wait to get it back.\n\n\"Okay, first of all, let's hear your poem.\"\n\n_\"I heard a Fly buzz\u2014when I died.\"_\n\nShe wanted to hear him recite again as he had in class. She craved that magic once again.\n\n_\"The Stillness in the Room \/ Was like the Stillness in the Air...\"_\n\n\"Hold on. Slow down.\"\n\nEmbarrassed, Aidan stumbled over _Heaves of Storm._\n\n\"Why are you rushing?\"\n\nHe thought, Because you're distracting me. She was lovely and he longed to please her\u2014to see pure joy on her face, to surprise her once again. He fantasized about her all the time, but not in the way that he had Daphne. Those visions had been violent. He had chased Daphne into the real world to pin her down. With his teacher, just the opposite. He took her deep into his dreams and gave her his sword.\n\nWhen he remembered his lines, they rushed out all at once, too glib, too fast. Nor could he make his voice cold when Nina was his only audience.\n\n\"The poem has a kind of mordant wit,\" she told him. \"Very dark. Very wry. Remember how you did it before?\"\n\nHe didn't feel mordant. He was the one buzzing. He was the fly, and there was no corpse, only his final paper on Dickinson. His three-to-five-page essay with the final comment, _Aidan, I'm impressed._ He kept glancing at the graded essay on her desk. Her words on his, her pen and his typed paragraphs practically touching. How could he remember dying? He said, \"I think it was a onetime thing.\"\n\n\"No, that's not true!\" She would not accept excuses, nor could he distract her long. \"Slow it down. Slow down even more.\"\n\nHe closed his eyes to concentrate. He tried to find that slow and empty place, to become again the diver underwater.\n\n\"Yes! Better,\" she exclaimed. \"But look at me. Look at the audience. Don't look away.\"\n\nHe said, \"I need to practice that.\"\n\n\u2014\n\nHe practiced all the way home, stepping slowly, brimful of words. _The Stillness round my form..._ He saw Nina's eager look, her shake of the head. He saw her listening to him, and the winter afternoon seemed new and strange. The bare trees standing like upended brooms. Small birds darting together, turning all at once, swooping and gathering in an instant, playing with the wind.\n\n\"Hey,\" Diana greeted him. He didn't even stop for a second as he ran up the stairs. _The eyes beside\u2014had wrung them dry\u2014 \/ And Breaths were gathering sure..._\n\nDiana couldn't hear the poem clamoring in his ears. She had to listen to Kerry berate her on the phone, while Aidan paced his room above her head. _For that last Onset\u2014when the King \/ Be witnessed\u2014in his power._\n\n\"It's one day. Just a one-day suspension\u2014and I wasn't even fighting! I barely touched the guy,\" Diana insisted.\n\nAidan threw himself onto his unmade bed. _With Blue\u2014uncertain\u2014stumbling Buzz..._ He saw Nina's face, intent and serious, her slender arms. The chalk dust on her shirt. _Between the light\u2014and me...Between the light\u2014and me..._ Over and over, he imagined her.\n\nA rattling sound roused him, an insistent scratching. He started up, sensing someone rapping, trying to get in. His room was dark, the winter sun had set. He must have drifted off. He turned on the light and saw a bare tree branch rattling against the windowpane. Checking his watch, he was shocked he'd slept so long, past five o'clock.\n\nShit. He'd promised he would meet his company. He hesitated for a minute, then, scrambling to his feet, he opened his BoX.\n\n\"Where is everybody?\" he demanded as his Elf shape materialized. All he saw was poor, half-crippled Dracon standing on a stone bridge. Vast caverns, vaults of a subterranean cathedral surrounded them.\n\n\"Everybody else gave up,\" Dracon said. \"You were supposed to be here two hours ago.\"\n\n\"Okay, we'll reschedule.\"\n\n\"No. You said the qwest was on.\"\n\nAnnoyed, Aidan kicked the bridge, dislodging a pebble that fell pinging and ricocheting into the ghostly river far below. \"Dude, I have to study.\"\n\nIn her bedroom, Diana heard muffled voices through the wall. _I have to study?_\n\nA car door slammed, and her first thought was, Oh no, Mom's coming home to punish me.\n\nNo. Just a delivery person. She heard someone clomping up the front steps, ringing the bell, dropping a package on the porch. More footsteps, another weight dropped on the porch. Somebody was tramping up and down.\n\nShe slipped into Kerry's empty room and opened the shades to see the street. She found no delivery truck. Only a station wagon in the winter night. She ran downstairs barefoot, wearing her flannel pajama bottoms and a white T-shirt printed DANA-FARBER CANCER INSTITUTE. When she opened the door, she found Jack, with ten battered cardboard cartons stacked up at his feet. \"What are you doing?\"\n\nHe answered in his even way, \"I brought you something.\"\n\nShe glanced at the cartons and then looked away. He must have heard. \"Aidan's upstairs.\"\n\n\"I came to see you.\"\n\n\"I'm busy.\"\n\n\"Doing what?\"\n\nShe thought of Sorentino. \"Processing what happened\u2014supposedly.\"\n\nPolitely Jack said, \"Cool, what did you come up with?\"\n\nWhen she glared at Jack, he didn't even blink. She said, \"You feel sorry for me.\"\n\nHe ignored this. \"Hold the door.\"\n\n\"Why?\"\n\nHe hoisted two cartons. \"Because they're heavy.\"\n\nShe held the glass storm door open with her hip. \"You stole some lame educational props from your father, didn't you?\"\n\n\"Maybe.\"\n\n\"Wait. Are those fireworks? Because my mom's coming home in like an hour.\"\n\nJack knelt on the living room floor, and opened the boxes to reveal thousands upon thousands of black and white tiles.\n\n\"Nooo,\" Diana groaned. Scienceman's birthday party dominoes.\n\nJack looked up at her. \"Come on. You know you want to.\"\n\nFor the next two hours, they dominoed the house, lining up their tiles, spaced at perfect intervals, across the floor and down the hall. They lined up their tiles in all the dusty places, outlining windowsills and baseboards, snaking behind the couch, circling the old upright piano.\n\n\"Who plays?\" Jack asked.\n\n\"Nobody,\" Diana said. \"I mean, Aidan used to.\"\n\nAbove their heads they could hear Aidan's shuffling, stamping feet. He'd put off studying after all.\n\nLike black ants, those tiles marched across that poor piano. They covered the entire living room out to the entrance hall, and then Jack said, \"We still have more.\"\n\nDiana sat back on her heels. \"Let's do the kitchen.\"\n\nThey extended their line onto the kitchen floor, around the kitchen table. They built domino formations on the kitchen counters, an array of black tiles from the toaster oven to the old microwave.\n\n\"I used to think you had a sad, pathetic life,\" Diana said.\n\n\"Why?\"\n\n\"Because you're an only child.\"\n\nOne by one, Jack placed his dominoes along the counter. \"I like being an only child.\"\n\n\"Obviously.\" Diana was thinking that being an only child was all Jack knew. \"It's worse when you become an only child later.\"\n\nJack finished his row and then leaned against the counter, considering her.\n\n\"What?\" she demanded.\n\n\"Is it true?\"\n\n\"What do you mean?\" she asked, although she knew exactly what he meant.\n\n\"Do you like girls better?\"\n\n\"Yeah! Don't you?\" He was making her nervous. \"Actually, I go by the person.\"\n\n\"So you don't know...\" Tile by tile, he was encircling Kerry's coffeemaker.\n\n\"I don't know about you.\"\n\nFootsteps. Their miniature world began to rattle. Aidan was like a giant heading to the bathroom overhead. Every move, a step of doom. They heard the toilet flush, the water running in the sink. They held still as he returned to his room. Their dominoes wobbled as walls and floorboards creaked. Then he slammed his bedroom door.\n\nThe old house absorbed the shock, and all the domino chains tumbled in the kitchen down below, whole regiments collapsing on their bridges, folding onto countertops and floors. Jack and Diana doubled over, laughing.\n\n\u2014\n\n\"Earthquake!\" Aidan shouted, deep in game, but of course Dracon felt nothing on his side.\n\nA flurry of white bats descended.\n\n\"Help me!\" Dracon tried to fight them off one-handed.\n\nAidan drew his sword, but he wasn't fast enough. Already the bats began to feed.\n\nHe pulled Dracon into a cave, reached for his diamond flask, and sprinkled hatchling's blood on Dracon's wounds.\n\n\"What are you doing?\" Dracon protested even as his own flesh healed. \"There's ten more infections in the company and now you've used up the flask. You've wasted it!\"\n\n\"We'll find another dragon's egg,\" Aidan promised.\n\n\"When?\" Dracon asked bitterly.\n\n\"When we get the company together.\"\n\nNow Dracon lost patience. \"You no longer have a company.\" Sword in hand, Dracon turned his back.\n\n\"Where are you going?\"\n\n\"I'm qwesting elsewhere,\" Dracon said with dignity.\n\nFurious, Aidan said, \"I'll go myself.\"\n\nHe stood alone in the dark cavern and lifted his arms so that the bats descended, feasting. They ate his arms to bloody nubs, and bit his body to the bone. They devoured his face, and his knight fell writhing to the cavern floor. Aidan was unafraid. He had trained for transformation, and he chose this metamorphosis, stretching out his torso, trading sword for a long tail. He saw his hind legs lengthening, his bloody flesh resolving into silver scales. Two dark patches on his back began to swell. Black wings opened wet and heavy, then fanned out, shedding sparkling drops of water. His skull had changed to silver, his tongue unfurling with a lick of fire. He lit the cavern as he took flight, exploding through stone walls into the Arkadian sky.\n\nAidan spread his wings and he was soaring, his spectral dragon rising over mountains, flaming over trees and frozen lakes. Now he saw the contours of the Trackless Wood, the distant towers of the Keep. He had jumped worlds to EverWhen.\n\nGliding softly in familiar skies, Aidan saw Elves lying where they had fallen in the snow. Hundreds of qwesters, too weak to move. Had the contagion spread to EverWhen? How many qwesters had jumped worlds before him, carrying the disease? Grimly he turned toward EverSea.\n\nFlying low over crashing waves, he took his place among the other dragons nesting on the coast. He folded his great wings as he slithered into the low cave. Quiet now and smooth, his body undulating, he shed his dragon body. His serpentine form coiled and disappeared with a sucking sound like water down a drain.\n\nA knight again, he crept between mountainous dragons. Clutching his needle sword, he felt his way inside their rocky nest. Softly, laboriously, he carried out an egg the size of a watermelon. The egg was speckled white and brown, and hard to handle, slipping constantly, so he had to stop and readjust his grip. Over rocks, and tide pools, past the open mouths of caves, he carried this burden, until, exhausted, he set it on the sand. He would strike the egg with his sword, and kill the hatchling. Fill his diamond flask with blood, and then...What next? One flask wasn't enough to heal his whole company. He would have to steal another egg and yet another, and even then, all of EverWhen was infested. Working alone, he could not move fast enough. The game was impossible. Not just difficult, but hopeless. He struck the eggshell with his sword, and the fissure widened. Green fluid leaked onto the sand. Chipping at the shell, prying it apart with his bare hands, he found the hatchling curled within. The monster didn't snap or bite. He pulled open the creature's wings, but they fell back, inert. He pried open the dragon's eyes, but they were white and empty.\n\n\"No!\" Aidan shouted. The thing was dead.\n\nHe could not extract elixir, nor could he restore his company.\n\nHe kicked the broken egg and all its contents into the sea. Hurled his sword like a javelin into the waves.\n\nThe surf rose in a fury, swallowing the egg and then the sword. Rising higher, ocean consumed the rocks and drowned Aidan in darkness.\n\nThe shore was gone. The waves had disappeared as well. There was no ocean left. His bed appeared where there had been cliffs and caves. His silent desk displaced the raging surf. His painted ceiling blotted out the stars.\n\nHe had killed the game. For a moment he stood in shock, convinced his anger had destroyed it. His BoX lay inert, an old toy on the floor.\n\nHe felt for his computer, tapped the keyboard, clicked the mouse. A message appeared on his computer screen.\n\n> DUE TO TECHNICAL DIFFICULTIES, WE ARE RESETTING ALL ARKADIAN WORLDS. WE APOLOGIZE FOR ANY INCONVENIENCE. PLAY WILL RESUME AS SOON AS POSSIBLE.\n\nAt first Collin thought the problem was a power outage. Monitors went dark, the great screens on the wall faded and died. The whole building seemed to dim, games ceasing and illusions dissipating, but the problem was not electrical. Viktor had stopped play altogether, ordering a hard reset of all Arkadian games.\n\n\"Two days. Three days at most,\" Viktor told reporters.\n\nHe spoke of the human factor, rogue behavior by a few who spread disease. The UnderWorld infection, designed as a gory interlude, easily reversible, had become a raging epidemic. World-jumpers carried the contagion to EverWhen and EverSea, where unsuspecting players sickened and lost power. Some trained in healing arts had attempted cures. Others had begun spreading the disease on purpose, lying in wait to lay hands on each fresh Elf or Gnome. Millions died, and in many cases, they stayed dead for days. Unaccustomed to disease, the dead logged off as they grew frustrated, waiting for resurrection. Viktor posted a concise explanation on fan forums. \"The epidemic was not a problem we could solve in real time. We thank you for your patience.\"\n\nAs players faced blank screens, Arkadians swarmed their cubicles like frantic bees. Viktor spoke to the media, while Peter rallied the troops in the Atrium. Like militia awaiting orders, hellions mustered among the corporate palms.\n\nHow strange the way an imaginary disease became a real threat, rogue gamers a financial liability. Collin stood among the hellions and the moment seemed to him bizarre, ridiculous. You won't believe what happened, he imagined telling Maia\u2014but he could never explain how surreal this was, the loophole in the game, the sudden rift in the Arkadian cosmos. He could never convey how he felt\u2014bemused, alone, confused. This was what happened when a game consumed itself, the system crashing down. This was the ouroboros devouring its own tail. He saw it now, and all he wanted was to talk to Nina.\n\n\"We have to change the game,\" Peter explained to the assembled. \"We want controlled chaos, not total anarchy. The goal is _simulated_ tragedy.\" Together, they would curtail and streamline UnderWorld's plague so that the disease would run its course with Arkadian speed. The whole cycle, from wasting away and dying to resurrection, would take minutes instead of days. Equally important, hellions would ensure the leprous blight could not travel from one game to another. Contagion 2.0, as Peter called it, would remain in UnderWorld. While artists worked on symptoms, programmers would adjust settings so that if a player jumped to any other realm, the plague would lose its power.\n\nAnd what will I do? thought Collin. The answer came quickly, because every minute counted, every hour offline eroded audience. Collin would have a chance to draw again. He worked with his own pod, and the whole thing was weird, the whole exercise was strange, but the work was so absorbing, that Collin forgot all that. No one shaved, and no one showered, no one went home. Caught up in the collective effort, Collin began to feel like himself again. It was such a relief to do his job.\n\nPast problems seemed to disappear, as hellions worked together. Viktor had food delivered at all hours. Nicholas set up an earthshaking sound system, rocking the building with heavy metal, techno, classic rap. Even as programmers tapped furiously at their keyboards, Daphne reached out with hope and reassurance on fan forums. Sometimes she posted as an impudent insider, Reconnect with family and friends while you still can, because UnderWorld is almost baaack. Sometimes she played fangirl, sometimes she became a spy. Often she assumed her favorite role, the activist who went by \"Christian Wench\" and argued that the contagion had been sent by God to destroy Satan's handiwork. As EverWhen is visited by plague, typed Daphne, so shall the world. Repent! Turn away from darkness. Fast and pray!\n\nNews reporters understood by now that the Christian protest against Arkadia had been a hoax, but Daphne continued posting anyway. She did this to entertain UnderWorld's fans, who loved to hate their Christian enemies\u2014even when they knew they were imaginary. At the same time, Daphne's message boards began to draw in real anti-gamers. It seemed that actual mothers were now posting complaints. I have a 13 year old child I hope and pray that UnderWorld is gone forever. Daphne tried to answer every comment. Amen, Amen, she typed, or May it be thy will.\n\nShe was doing her best work, and she was not the only one. Collin felt a surge of energy. His spirits lifted, even as he drew a crumpling nose, a blackening ear. He caught Peter gazing at his monitor, examining a series of degenerating eyes. Pupils gradually enlarged, eyeballs clouding, one lid drooping, pendulous, elephantine.\n\nPeter said nothing, but Collin didn't care. He no longer waited for a verdict, a word of praise or blame. He no longer worried if Peter would use his work. The fact was, his art was necessary, and it was good.\n\nHe showed off to Obi and Akosh, entertaining them. When they sprawled out on chairs or on the floor exhausted, he drew a couple of flip-books, a dragon hatching, an exploding face.\n\n\"No way,\" Akosh told Collin.\n\n\"What?\"\n\n\"No way are you still drawing now.\"\n\nHe felt alive, awake again. He drew constantly, almost effortlessly. He did more than his share, accomplished even more than Peter asked. When the crisis ended, he was surprised to realize that he had been working three days straight.\n\nWhen Arkadia went live again, Tomas recognized Collin's work, high-fiving him. Collin stood like a hero with his pod, as Viktor climbed atop a desk to speak.\n\n\"May I say?\" Viktor held up a hand to stop the cheering. \"Can I just say? It isn't curing cancer, but it _feels_ that way.\"\n\nGamers flooded EverWhen and UnderWorld. They returned in a great wave, like migrating birds. Caverns rang with players and their avatars, and most Arkadians went home to sleep\u2014but Collin went to O'Riley's with his pod. The pub was warm and crowded, alight with televisions. He kicked back with Akosh and Obi. Even Nicholas came this time. Daphne challenged Tomas to a game of pool, and Peter paid for everyone.\n\nDrunk and almost happy, Collin drew another flip-book at the bar. Careless, he just drew what came to him. Wild horses with necks outstretched and flying feet.\n\n\"Stop.\" Daphne touched his shoulder with her cue. She was trying to warn him, but he didn't understand.\n\n\"I dub you Sir Collin.\" She was chalking up his shirt.\n\n\"Let's have a game.\" Nicholas offered Collin his own cue.\n\nBut Collin had caught Peter's attention. The ink drawings were simple, streamlined, but the horses unmistakable. There was the little gray, the chestnut with the scar on her back. Peter darkened as he watched Collin's pen. \"Where did you get those?\"\n\n\"What do you mean?\" Collin looked up, surprised.\n\nHe doesn't even notice, Peter thought. Collin sat and sketched and people gathered around him as if he were jamming at the piano. He was like a musician sitting down to play by ear. An endless riff, a raw feed of images.\n\nBut Collin spoke out. \"I didn't get these horses from anywhere. They're mine.\"\n\nDaphne held her cue upright, tapping it against the floor.\n\n\"We're using them,\" Peter said. \"They're ours.\"\n\n\"These sketches? These horses in particular?\"\n\n\"We're using these horses for a new game. Elysium.\"\n\nCollin's cheeks burned. This was the game in stealth mode; this was what Daphne had said he didn't want to know. He turned to look at Akosh, and Obi, Tomas. Nobody spoke. Nicholas was playing nervously with his phone.\n\n\"Why didn't you ask me?\" Collin said.\n\n\"Because we didn't have to ask.\"\n\n\"I mean why didn't you include me?\"\n\n\"We had what we needed.\"\n\nRight, thought Collin. As usual, you were collecting my work. \"You said my horses were pretty,\" he reminded Peter. \"You said they just weren't interesting.\"\n\n\"There was work to do.\"\n\n\"You really are a shit,\" Collin said slowly. Even in the crowded bar, he sensed the hellions hush around him. He felt Nicholas's hand on his arm and he looked at the sound engineer and wondered, Are you still warning me? Are you really so afraid? Collin feared nothing at that moment. He said the thing no one was allowed to say. \"My work belongs to me.\"\n\nOf course this wasn't true. Collin had signed a contract. Peter said, \"Your work belongs to Arkadia, as you know.\"\n\n\"I drew those horses,\" Collin countered. \"I have them in my hand. How can they be yours without me?\"\n\nPeter didn't deign to answer this.\n\n\"You try to draw them.\" Collin held out his pen. \"You show me what you know. Show me. Show me now!\"\n\nThe others tried to talk to Collin, but he wouldn't listen. \"Look, we're all tired,\" Nicholas said in his conciliatory voice.\n\nCollin wanted to fight. He wanted to write all over Peter's face, but he would gain nothing by it, except immediate satisfaction. Peter owned his work; he could use it forever, and Collin could not get his drawings back. All he could do was make new art. His fists clenched, but he did not touch Peter. He turned around and left instead.\n\n\u2014\n\nHe had no ride back to the city, but he didn't care. Snow fell lightly, but he brushed it from his eyes and kept on walking, oblivious to slush and cold.\n\n\"Collin!\" It was Daphne coming after him. She was holding his coat.\n\n\"He knows what's good,\" Daphne pointed out. \"You could take it as a compliment.\"\n\nCollin shot her a look.\n\n\"Okay, yes, he was a shit in there,\" Daphne conceded with her impudent smile.\n\nBut Collin did not believe in Daphne's disobedience.\n\n\"Here.\" He took his coat and pulled his slate from the pocket. \"Take this back to him.\"\n\nHe went home and slept until eleven in the morning, and when he woke, he felt entirely empty. He started rummaging for food. Emma and Darius were out, and the refrigerator was nearly empty. Collin ate all of Emma's leftover bulgur wheat salad, although this would infuriate her. He didn't care. Nicholas had texted, but he didn't answer. There were other messages. There were piles of junk mail on the floor, a note from Dawn about people coming to look at the boiler.\n\nThe bulgur salad tasted like nothing. The piles of mail were indecipherable, like relics of some other person's life. He felt numb until he took a shower. Freezing water jolted him awake. That was when he decided he would bike to Emerson.\n\n\u2014\n\n\"Collin, honey? Is that you?\" said Mrs. Solomon from behind her desk.\n\nHe took off his helmet and signed in. Under \"purpose of visit\" he scribbled, _Meeting._\n\nHe wanted to catch Nina at lunch, but Mrs. Solomon said, \"Oh, no, you can't do that. She has her evaluation!\"\n\nUpstairs in the Resource Room, Nina was sitting down with Jeff and Mrs. West, and Mr. DeLaurentis. They were meeting at a wood-grain table and Jeff was looking sympathetic\u2014always a bad sign. He offered Nina a bottle of spring water.\n\n\"No, thanks,\" Nina said.\n\n\"Our goal is objectivity,\" Mr. DeLaurentis explained, by way of introduction. \"This is why we stress the online evaluation system. By looking at the numbers we build a strong foundation.\"\n\nNina gazed at her printed results.\n\n\"One of your strengths would be your knowledge base,\" said Mrs. West. \"As you can see in column one, almost ninety percent of students feel that you show knowledge of your subject area. On the other hand\u2014a full eighty-two percent of students feel somewhat, very, or extremely uncomfortable asking for help.\"\n\nNumbers blurred on the page. Could this be true? Nina's students called her all the time. _Miss? Miss?_ They wanted this, they wanted that. Were they actually afraid of her?\n\n\"Before we dive into the details, I want to look at a couple of other issues,\" Mrs. West continued. \"On page two in classroom management, forty-five percent of students feel somewhat or very distracted by their classmates, which is concerning. Sixteen percent characterize the atmosphere as chaotic.\"\n\n\"Wait,\" Nina said.\n\nMrs. West looked up. Everybody waited.\n\n\"The students are distracting each other and now they're reporting it?\"\n\nJeff said, \"They're reporting on your classroom management.\"\n\n\"We've supplemented the questionnaires with classroom observations,\" said DeLaurentis.\n\n\"I've provided all my notes,\" said Jeff.\n\n\"The question of favoritism,\" Mrs. West said. \"Look at page three.\"\n\nEveryone turned to page three. \"Sixty-four percent of students feel that you occasionally favor some students over others. Twelve percent feel that you often do.\" Mrs. West paused. \"That's high.\"\n\nNow Nina's cheeks began to burn.\n\n\"Are you aware of this perception?\" Mrs. West asked.\n\nNina wanted to say no, but at that moment she was acutely aware of what she had done. She was not only a rookie, but a rule-breaker as well\u2014singling out Aidan, protecting and manipulating him, forcing him to learn.\n\nMrs. West did not mention Aidan. She made no specific accusations. She crushed Nina with simple moral force. \" _All_ our students need us. All the time.\"\n\nAs Nina tried to catch her breath, Mrs. West marched on, plucking startling numbers she had highlighted in yellow. \"Let's turn to the last page.\"\n\nTrembling, Nina turned, and yet even then she hoped for something decent, at least average. What she saw was a composite score of 69 out of 100, and the verdict, FAIR.\n\n\"You look surprised,\" said Mr. DeLaurentis.\n\n\"I just thought...\"\n\n\"You've made progress,\" Jeff consoled her. \"Seriously.\"\n\nShe looked at him and wondered\u2014progress from what? Total disaster?\n\nMr. DeLaurentis said, \"Shall we look at your self-evaluation now?\"\n\n_No!_ Nina screamed inwardly. She couldn't bear the comparison. The process was so humiliating. \"I imagined I was doing better.\"\n\n\"That's why we look at the quantitative results,\" Mr. DeLaurentis repeated.\n\nShe didn't cry. That was her one comfort as the meeting ended: She hadn't let them see her cry, although she had come close. How had she deceived herself all year? Had she mistaken her slow progress for success? Had she imagined knowledge and goodwill would win the day? She was reaching some students. Apparently that was not enough. She had connected with a few, but that was favoritism. She couldn't win!\n\nThese were her thoughts as she walked past Collin in the hall.\n\n\"Nina,\" he said.\n\nStartled, she turned toward him. Was something wrong? Something must have happened. She hadn't seen him in so long. She was concerned and affronted by his presence, but mostly confused.\n\n\"Could I just talk to you?\"\n\nThe bell was ringing and eighty-two percent of Nina's students were afraid to ask for help. Concentrate, she told herself. Do you even remember your lesson?\n\n\"Just for a minute?\" Collin said.\n\n\"I can't.\" She shut the door.\n\nHanding back her students' work, she thought, I'm just a know-it-all. I came here to bestow my knowledge and I've failed classroom management. Even now her kids were talking. Twice she had to tell them, \"Settle down.\" Some quieted. Others kept on talking.\n\nNina shouted, \"What do I have to do to get your attention?\"\n\nSilence. She had never raised her voice like that before. For a moment Nina felt powerful, then she realized that she had miscalculated yet again. Cold and offended, her students gazed up at their desperate teacher. They were not accustomed to this treatment\u2014not from her. Conversations continued in a whisper.\n\nWearily, Nina began passing back vocabulary quizzes. \"Let's start again.\"\n\nHer kids had done well on their tests. Most of them had learned the words. Before her evaluation she would have delighted in these scores. Now she dwelt on her mistakes, current and cumulative. She presented a fair lesson on Walt Whitman because there wasn't anybody else to do it. She tried to spark discussion of _Song of Myself._\n\nFiercely, she taught to the last bell, and as her students streamed out into the hall she called after them, \"Final drafts due Monday! Revise and _proofread_! Don't forget!\"\n\n\u2014\n\nJust outside her door, Collin saw her long hair curtaining her face as she picked up a paperback _Twelfth Night._ For a long moment, she seemed a stranger, tall, elegant, the girl from far away.\n\nShe drew back when he stepped inside. How long had he been watching? She retreated behind her desk.\n\nHurt, he said, \"Don't worry.\"\n\n\"I'm not!\" She was polite, as though she didn't know him.\n\nThe distance seemed to stretch between them. Collin near the door, Nina behind her Steelcase desk. \"I just want to talk to you.\"\n\n\"What do you want to talk about?\" The question sounded like, What else is there to say?\n\nHe approached her desk. \"I know you don't want explanations.\"\n\n\"Collin,\" she said. \"I can't think about this now.\"\n\n\"I just have to tell you something.\"\n\nShe studied his face. He looked so nervous.\n\n\"I miss you,\" he began, but then he stopped. \"That's it,\" he told her. \"That's the whole thing. I miss you. I need to be near you.\"\n\nShe said nothing.\n\n\"I was stupid. I was overwhelmed.\"\n\n\"Don't say that,\" Nina burst out. \"Don't say you were overwhelmed and so you had to hurt me.\"\n\n\"I didn't think that I deserved you.\"\n\n\"So you went out to prove it?\"\n\nHe didn't know how to answer that. \"I've never known anyone like you,\" he said at last.\n\n\"I don't play games,\" she told him. \"I can't play games with you.\"\n\n\"Remember when we went sledding? Remember when we walked down by the river?\"\n\nShe did remember skimming down the hill and walking in the cold. She remembered his hands in her coat pockets.\n\n\"All of that was real.\"\n\n\"It was real at the time.\"\n\nHe wanted to say, No, that day will last forever\u2014but if happiness could last, what of anger? Pain?\n\n\"Remember when I knelt in front of you right here and embarrassed you?\"\n\nHer anger flared. \"How about the time you covered the blackboards with chalk pictures and ruined my lesson?\"\n\n\"I was trying to apologize!\"\n\n\"You picked a hell of a way to do it.\" She turned her back to gaze at the dusty board.\n\n\"I was arrogant,\" Collin admitted.\n\nAlmost inaudibly she said, \"Me too.\"\n\nHe walked over to her side of the desk. She didn't turn toward him, but she didn't ask him to leave either. She blinked back tears, and he pretended not to notice. He just leaned against the desk and they gazed at the board with its vocabulary list. _Trill, defile, nimbus, manifold._\n\n\"I'm sorry,\" he said at last.\n\nShe closed her eyes.\n\n\"Nina?\"\n\nShe leaned against him ever so slightly. Subtly, almost imperceptibly, he felt her weight shift.\n\nHe said, \"You were right about Arkadia. It's crazy there, and I've made such a mess.\"\n\nHer eyes opened. \"What happened?\"\n\n\"I quit.\"\n\n\"Well,\" she said slowly, \"you can still do the thing you want to do.\"\n\n\"Unpaid?\"\n\n\"You can say that you're an artist. I can't say I'm a teacher.\"\n\n\"Yes, you can.\"\n\nNot next year, she thought. Not if I'm looking for a real job. \"My evaluation was so bad.\" Her voice broke a little and she almost laughed, because she had worked so hard and cared so much and failed in so many ways\u2014and it was such a relief to tell him. \"Either they like me and they don't respect me, or they just hate me.\" Softly she admitted, \"I'm just not good at it.\"\n\n\"But that takes years and years.\"\n\n\"Maybe.\"\n\n\"You'll get old like a real teacher. You're already bitter\u2014and you're getting strict. I could hear you from the hall.\"\n\n\"Oh, God. I lost my temper.\"\n\n\"What's wrong with that?\"\n\n\"No, you don't understand.\"\n\nActually he understood pretty well. When she'd lost her temper, she'd done it for real. She didn't know how to lose it theatrically, how to shout and scream and stamp her feet and keep her sense of humor. And now she told him, \"I was terrible.\"\n\nInstinctively he touched her arm, but she shrank back. He began to speak and stopped. What could he tell her? I love you. I worship you. Those were words anyone could use. He wanted to give her something of himself. He pulled a flip-book from his pocket.\n\n\"What is it?\"\n\nHe showed her how to flip the pages with her thumb.\n\nThe drawings were black ink, sharp, and stylized. A horse running riderless, fast and faster, ears back, neck outstretched. The moving picture was so fluid, the running horse so elegant, that without thinking Nina flipped the book again. \"Is this for a new job? Who is it for?\"\n\nHe was surprised she had to ask. He gave her another flip-book, and said, \"Just you.\"\n\nThis time the horse approached a fence, coming at it flying, mane and tail streaming. In one fluid leap he took the jump and landed on the other side. Nina flipped the book again and yet again. The tiny animation was so beautiful, the horse leaping, the line drawing come to life. She felt absorbed in this brief story. Each time the horse swept her away, flying at the jump. \"I like this better than the Ghost Horse.\"\n\nDrawing closer, Collin watched her face. \"You're glad I left.\"\n\nShe glanced up quickly. \"Not if it's on my account.\"\n\n\"Don't worry.\" He was so close. His lips brushed hers. \"I would never do anything for you.\"\n\nHe nearly kissed her, but she pulled away. \"Oh, wait! I have a student coming.\"\n\n\u2014\n\nToo late. Aidan had already spied them through the window in her classroom door. Collin talking, Nina listening. Although they had their backs to him, they were standing too close for ordinary conversation. Aidan could tell they were together, so he kept on walking down the hall, downstairs, and out into the winter day. Zipping his old ski jacket, he started walking home.\n\nAlone in his bedroom he dumped his backpack on the floor. Heavy-hearted, he sat at his computer, and checked his news feed and message boards, U.S. politics, and Arkadian alliances. He followed the news, real and imaginary, but he had not been qwesting since the contagion. His dream life with Nina upstaged all others.\n\nDracon sent bulletins from his new company, but Aidan didn't join them. Hey we're in 8th circle, Dracon wrote in chat. Old and experienced, Aidan replied as Tildor. Watch for scorpion.\n\nA new message flashed on Aidan's screen. Nina Lazare.\n\nWhat r his powers? Dracon typed.\n\nShe had never sent a private message before. Subject: Apologies. Dear Aidan, she began. Dear! To his eyes the opening was intimate, not standard. He didn't get letters, and his friends' messages did not begin like that. I'm sorry we missed our appointment this afternoon\u2014especially because it was our last meeting before the assembly. I wanted to show you these performances by other kids in last year's nationals. Here's the link. They're really excellent\u2014but so are you. He read that last phrase twice, and then twice more. He read it even as he opened the link to the Poetry in Action website. _So are you._ He heard her say it as he watched clips of kids reciting Billy Collins, Gerard Manley Hopkins, H.D., Robert Frost.\n\nDracon was typing, r u there?\n\nAidan minimized Dracon's chat box.\n\nHere was the mission statement of Poetry in Action. The list of past winners. A list of eligible poems. Seasonal poems, nature poems, most viewed poems. He clicked through categories on the website, and hundreds of titles appeared. Love poems. Short poems (under twenty-five lines). Poems about animals, poems about illness, poems about loss.\n\nClicking on LOVE, he scrolled down through page after page of titles and first lines. _How like a winter...How sweet I roam'd...I carry your...I went out to the hazel wood..._ _Sing me a song of a..._ One line jumped out at him, two sentences: _No, No! Go from me. I have left her lately._ He clicked on those words and an entire poem materialized, the story of a knight worshipping a noble lady. _I will not spoil my sheath with lesser brightness, \/ For my surrounding air hath a new lightness; \/ Slight are her arms, yet they have borne me straitly \/ And left me cloaked as with a gauze of aether..._\n\nHe sat back in his chair to stare at the fourteen lines onscreen. Message boards crowded against the poem's white space. Discussion of Arkadian weaponry, shortcuts to the Keep, new mods for EverSea. He hardly noticed. He felt so strange. How did the author know? How did \"Ezra Pound, American, 1885\u20131972\" write so specifically about his life? The poet had it all down\u2014her slender arms, her fair skin, his new lightness, even his sword. These lines scared him. _Oh, I have picked up magic in her nearness._ He felt haunted. A stranger had been telling his secrets, publishing his dreams before he was born.\n\n\"Okay, listen up!\" Mr. DeLaurentis announced. \"Take your seats. We need your full attention.\" Last period, last day before winter break, 450 students filled the auditorium to overflowing. \"Nobody in the aisles. Everybody in a seat.\" The chairs were hard and loud. They dropped open with a satisfying bang.\n\nDeLaurentis planted himself center stage. Square-shouldered, massive in his suit, he looked out at the student body, a sea of legs and arms. \"We're starting now. Shhhh. Settle down. People in the back! Let's move along.\"\n\nBehind him, in folding chairs, sat twenty-one contestants, a daunting number for a competition scheduled to last an hour. Mrs. West had instructed the competitors to sit in rows, to stand and walk to the front as soon as their names were called. That way they could recite, one after the other, without wasting time. On a white screen above their heads, Mrs. West would project title and author as each student recited. She was sitting on one side of the stage to judge the competition along with Mr. Allan and the school librarian, Miss McGahn.\n\n\"Let me start by welcoming all of our guests and visitors.\" Mr. DeLaurentis looked out at the parents peppering the audience. \"And let me express our gratitude to Mrs. West, who brought Poetry in Action to Emerson High School and spearheaded this unit. Let me thank our three judges.\" The judges looked up briefly from numbered printouts of each poem in competition. \"And all our language-arts faculty for taking time out of their busy schedules to hold classroom recitations. Each and every one of our students has memorized and performed a poem this year. I'm waiting...\" he broke off, as the roar of students drowned him out. \"Shhhh. The winner of this competition will go on to represent our school at the district level. But let me say this. Every one of our students up here onstage is already a winner in our book. Win, lose, or draw, you are all, each and every one of you, _victorious,_ for everything you did to get up here and everything we know you will accomplish today. And now, without further ado...\"\n\nMrs. West announced the first contestant: \"Keisha Mori.\"\n\n\"Go, Keisha!\"\n\nScattered applause and whistles as the first contestant took the stage. Mrs. West had to stand up. She took off her reading glasses and said, \"People. We have a lot of poems to get through in a very short time, so please hold your applause until the end.\"\n\nKeisha wore a black skirt and blazer and high heels. In the middle of the auditorium, where she sat with her tenth graders, Nina saw that several of the contestants had dressed like that, as if for church. A couple of boys wore dress shoes. Xavier even wore a tie. These kids were performers, the best in the school. Several sang a cappella in the Emertones. At least five were actors in the drama club. Nina recognized the Stage Manager from _Our Town._ The contestants sat up straight, gracious in the spotlight. All but Aidan, who leaned back in his chair.\n\nHe was wearing jeans and a black sweatshirt that looked too small, and he was stretching in the second row, eyes open, but unseeing. Was he listening to Keisha recite Mary Oliver's \"The Summer Day\"? He didn't even glance at her.\n\nKhalil stood up, shook out his dreadlocks, and recited Rudyard Kipling's \"If\u2014\" A girl named Natalie performed Shakespeare's Sonnet 106: _\"When in the chronicle of wasted time...\"_ and tripped up, stumbling on _the blazon of sweet beauty's best._\n\n_\"The blazon of...the blazon...\"_ She stood for a moment in distress.\n\n\"Go, Natalie. Go, Natalie,\" kids chanted in the audience, as though she were standing in the gym, preparing for a free throw.\n\nNatalie swallowed hard. Once again, Mrs. West had to stand up and shush everyone before the poor girl could finish.\n\nEven then, Aidan leaned back, as in a trance.\n\nIsmail Brown gave everything he had to \"Mending Wall.\" Daniella Kovatcheva performed \"Ozymandias.\"\n\nThey're good, Nina thought, as students jostled restlessly around her. _\"My name is Ozymandias, King of Kings...\"_ Regally, Daniella commanded, _\"Look on my Works, ye Mighty, and despair!\"_ and Nina sat up, entranced. This girl from Emerson owned Percy Bysshe Shelley. But Aidan could do just as well\u2014or better\u2014if he wanted. That was the question. What did he want? He was still staring into space.\n\nShe began dreading Aidan's turn. Other students excelled with Sharon Olds, Robinson Jeffers, Wilfred Owen. Wake up, she pleaded silently. The other kids were known quantities. They smiled, they ran cross-country. They went to prom. They did what teenagers were supposed to do. Aidan was the one in doubt. He could capture the audience if he wanted. He could astonish the entire school\u2014or he could slip away.\n\nShe kept her eyes on him as a girl named Jacqueline Ing recited Amy Lowell. _\"I walk down the garden paths, \/ And all the daffodils \/ Are blowing.\"_ For the first time, Aidan straightened in his chair. He seemed to gather himself. Jackie's voice was soft but intense with repressed emotion. _\"I too am a rare \/ Pattern. As I wander down \/ The garden paths.\"_ Almost imperceptibly, Aidan's right foot jiggled. _\"And the splashing of waterdrops \/ In the marble fountain \/ Comes down the garden paths.\"_\n\nAidan gazed out into the audience and found Nina. In all those hundreds of people, he found her eyes. She smiled at him, hopeful, encouraging, and for a moment he gazed back at her, as if he were starting on a journey and he had to memorize her face.\n\nElegant, tremulous, Jackie painted a word picture with her recitation. She paced her patterned garden in her patterned brocade dress, waiting for news of her lover, who was fighting _in a pattern called a war._ But even this poem couldn't last forever. Just a few moments left, as Jackie built slowly to her anguished cry, _\"Christ! What are patterns for?\"_\n\nScattered applause, despite Mrs. West's injunction, as Jackie returned to her chair. She sat down, ducking her head modestly.\n\nNow Aidan unfolded himself and pulled down his sweatshirt, too tight, too short.\n\nEMILY DICKINSON flashed on the screen as Aidan ambled to the front. He took his time, adjusting the microphone. He was easily a foot taller than Jackie.\n\nHalfway out of her chair, Mrs. West gestured for him to move along. \"Whenever you're ready.\"\n\n_\"No,\"_ Aidan shot back.\n\nThe entire student body gasped in admiration. \"What are you doing?\" Nina murmured in despair. Mrs. West sat down in shock.\n\n_\"No, go from me,\"_ Aidan said. _\"I have left her lately. \/ I will not spoil my sheath with lesser brightness...\"_\n\nNow all three judges were flipping through their printed poems in confusion. Where was Dickinson? This was not \"I heard a Fly buzz\u2014when I died.\"\n\n_\"Slight are her arms, yet they have bound me straitly...\"_ The lines were tender, but Aidan delivered them with a hard, defensive edge. Voice defiant, he seemed to stand alone onstage against his audience, against the world. _\"Oh, I have picked up magic in her nearness \/ To sheathe me half in half the things that sheathe her.\"_\n\nNervous laughter. He ignored it. Whispered conference of the judges.\n\nNina stared at him in disbelief. Why, Aidan? Why choose this poem no one knew?\n\nHe looked out into the audience and kids gazed back, awed by his intensity.\n\nAt the back of the auditorium, Diana's hands clenched as she watched her twin. Her heart was pounding, she was so nervous. She heard rebellion in his voice; she saw him flouting the competition, breaking rules, and she was proud and embarrassed. Her brother was so fierce, so serious. Her palms were sweating. She was afraid everyone would laugh at him.\n\n_\"No, no! Go from me. I have still the flavour, \/ Soft as spring wind that's come from birchen bowers.\"_\n\nStrange, archaic, the words came naturally to Aidan, direct as ordinary speech. _\"Green come the shoots, aye April in the branches.\"_ What a relief to tell his teacher how he felt, to give up his secret in this safe place, an auditorium full of people. The words weren't his, but he gave them breath and life, repatriating them like looted artifacts. Thrumming with the poem's subtle pitch, his body swayed. _\"As winter's wound with her sleight hand she staunches, \/ Hath of the trees a likeness of the savour: \/ As white their bark, so white this lady's hours.\"_ He stopped, and everybody waited. No one knew what would happen next.\n\nEven Diana sat back, mystified. Aidan ambled to his folding chair and Diana felt a pang and then relief to watch like everybody else. He was speaking a language incomprehensible, even to her.\n\nAidan didn't see his sister in the crowd. He hardly noticed anyone. All he knew was that the ordeal was over. He knew he wouldn't win. He'd never won anything before, and Mrs. West hated him for having plagiarized.\n\nWhen all the recitations were done, Mrs. West was the one who led her fellow judges offstage to confer, while competitors shifted in their chairs onstage, and the student audience surged, noise level rising, clock ticking toward dismissal. West was the one who took the microphone to tell everyone to settle down so that she could announce the winners. She raised her hand for silence. \"People? People. Listen up.\"\n\nWhen the other judges filed back onstage a few kids applauded, and one guy whistled in the back. Teachers tried to hush the audience, but the auditorium rustled with impatience and hilarity. Only the contestants' parents gazed at the stage with perfect concentration. Among them, Kerry sat transfixed as she watched Aidan.\n\n\"First, let's have one more round of applause for all of our contestants,\" said Mrs. West. \"We are so proud. We are so very proud of you.\" Once again, Mrs. West made a show of donning reading glasses as she unfolded her list. \"Please come to the front when I call your name. Honorable mention. Claryce Williams for 'Harlem.' Come on up, Claryce. Khalil Watson for 'If\u2014.' Yasmine Singh, for 'Do Not Go Gentle into That Good Night.' Congratulations! You three, stand right over here. Now.\" She paused. \"We have three more outstanding performances\u2014and let me just say, we had some debate.\"\n\nWere they debating me? Aidan could not suppress the question.\n\n\"This was such a close competition. Third place. Ranazia Donyon for 'Jabberwocky.' \" Hoots and whistles, applause and foot stamping\u2014not just for the performance but for Ranazia herself, since she was so popular. \"Come here. Take your prize.\" She handed Ranazia a book, a glossy anthology of poetry.\n\n\"Second place,\" Mrs. West continued. \"Daniella Kovatcheva for 'Ozymandias.' \" She got almost as much applause as Ranazia as she took her book.\n\n\"Finally. Our winner. The student who will represent us at the district competition. A student who chose a difficult poem...\"\n\nDifficult! Aidan thought.\n\n\"Which was also very powerful.\"\n\nPowerful!\n\n\"A student who went above and beyond.\" Mrs. West paused, and everybody in the auditorium thought, Oh come on, but Aidan experienced a private agony of expectation. \"Jacqueline Ing, for 'Patterns' by Amy Lowell,\" said Mrs. West. \"Congratulations!\"\n\nOkay, thought Aidan, as Jackie walked to the front, and applause rained down.\n\nHe hadn't won. He stretched his arms above his head and stood up, joining the others crowding off the stage. He hadn't won and he was glad. He was sure of that. The disappointment he felt, the quiet comedown, could not be real. He had never wanted any prize. He didn't take this competition seriously\u2014hadn't then, and didn't now. Therefore, he was more than fine with losing. Only his heart dissented: But you were good; you were really good.\n\nStudents stampeded for the doors, but parents stuck around. His mother appeared, and although he'd known she was coming, he was surprised to see her there. She stood right in front of the stage, stretching out her arms to him. Looking down from the stage, he almost laughed because she reminded him of swimming lessons, those afternoons when she stood in the shallow water, urging him to jump. He took the stairs instead, but he could not escape her. She latched on, pinning his arms to his sides in her embrace.\n\n\"Mom?\" She held him so long that he started worrying about her. \"You're okay,\" he said, extracting himself, and Kerry smiled through tears. Who was this child? So tender and so patronizing? \"You're gonna be okay.\"\n\nWhile Kerry went to shake hands with DeLaurentis, Mrs. West was congratulating all the contestants as they milled around on the floor below the stage. \"I'm proud of you, honey,\" she told Xavier. \"Good job, sweetheart,\" she told Khalil.\n\n\"Aidan.\"\n\n\"What?\" Aidan doubted Mrs. West would say, \"Good job,\" to him.\n\n\"Hey, hold on. Come back.\"\n\nThat wasn't Mrs. West calling him. He spun around and saw Nina standing in her white parka. She had been waiting for him! She was standing underneath the EXIT sign.\n\n\"Aidan! First of all\u2014you were amazing. Second of all...\" For a split second, she took him by the shoulders, as if to shake him. \"What am I going to do with you?\"\n\nDon't stop, he thought, overwhelmed by her brief touch.\n\n\"You realize that the judges disqualified you for switching poems.\"\n\n\"Okay,\" he said.\n\n\"What were you thinking?\"\n\nHe met her eyes, and of course she knew what he'd been thinking. She had known it all along, but she had pretended otherwise, convincing herself that her student was in love with poetry.\n\nInstantly he saw the change to adult concern, a teacher's worried face. Missing her, he couldn't help fishing. \"I'm sorry if I disappointed you.\"\n\nShe shook her head with her old warmth. \"I wasn't disappointed. I was proud. You silenced everyone!\"\n\nShe didn't say how much she'd wanted him to win. Oh, he could have shown DeLaurentis and the whole school\u2014even Mrs. West.\n\n\u2014\n\nSnow was falling when Nina left school. Cement steps, black wrought-iron fence, schoolyard\u2014all turned pure and white. The last few students were leaving the building. The big painted doors slammed behind them.\n\n\"Hey, Shakespeare!\" she heard one of the kids shouting. \"Bard of Avon!\"\n\nCollin was waiting for her. He opened the gate, and when he saw her face, he opened his arms as well.\n\nHis snowy collar brushed her cheek. He clasped her shoulders, and she rested in his embrace.\n\n\"That bad?\" he asked, taking her bag.\n\n\"It wasn't bad,\" she said.\n\n\"What was it, then?\"\n\nShe tried to find the words. \"Exciting, and difficult, and...surprising.\"\n\nHe kissed her ear and when she turned toward him, he kissed her lips. His kiss was sweet and eager, but he was laughing at her too. \"Yeah, that's exactly how I remember assemblies in high school.\"\n\n\u2014\n\nThey drove to Maia's place to bake gingerbread and move furniture around. Collin set up Maia's little white lights to frame the windowsills. Nina saved the empanadas from burning, while he ran out for paper napkins and more wine.\n\nAs the winter light began to fade, Maia's guests and neighbors came bearing brownies, latkes, trays of stuffed portobello mushrooms, for the solstice party. Melissa and Sage arrived. Sage carried a bowl of bulgur wheat salad with dried cranberries. Melissa led Henry by the hand. Lois brought caramel corn. Dawn came with steamed artichokes.\n\nMaia set the food out on the table, and listened to her neighbors ask Collin, Hey, bud, how's Arkadia?\n\n\"It's good,\" said Collin.\n\nMaia said, \"He designed the new game. UnderWorld!\"\n\n\"What?\" Lois exclaimed. \"Collin. Is that true?\"\n\n\"Not exactly!\" he called over his shoulder as he carried coats to Maia's bedroom.\n\nKerry arrived, and Aidan followed, carrying an enormous chocolate cake dusted with powdered sugar. \"I tripled the recipe,\" Kerry said.\n\n\"He was one of the designers,\" Maia told Lois. \"I don't know if you've seen it yet, but you stand in the center, and the whole game showers down on you like fireworks.\"\n\n\"Fireworks! You worked on that technology?\"\n\n\"I was a concept artist.\"\n\n\"What?\" Maia's ears pricked up at the past tense.\n\nMeanwhile, Kerry found Nina in the kitchen, ladling punch. \"This one is spiced,\" Nina explained, \"and this one's spiked.\"\n\nKerry told Nina, \"It was so strange hearing my own child recite.\"\n\nDawn said, \"Remembering all those lines?\"\n\n\"No, just hearing him say so many sentences together. I'd almost forgotten the sound of his voice! I was so surprised.\"\n\n\"I was surprised too,\" Nina told Kerry.\n\nKerry set her glass down on the counter and touched Nina's shoulder. \"Thank you.\"\n\n\"He did all the work. Not me.\"\n\n\"I know that isn't true. You tutored him!\"\n\n\"I mean he did his own thing,\" Nina explained. \"He learned one poem and then at the last minute he just\u2014switched.\" Stop right there, she thought, but she couldn't help adding, \"I think he could have won.\"\n\nMaia swept in, looking fierce.\n\nHe's told her, Nina thought, as Maia unwrapped a tray of baklava.\n\n\"Nina says Aidan could have won,\" Kerry told Maia.\n\nBut Maia was distracted, thinking about Collin. Was he really giving up Arkadia, and all that money, and all that magic, to look for work in Somerville? Was he really looking for some indie-animation thing?\n\n\"I really think he could have done it,\" Nina told Kerry.\n\n\"It doesn't matter,\" Kerry reassured her. \"For me it wasn't about winning.\"\n\n\"I know.\" Nina couldn't help sounding wistful. \"But he's so talented. It's a little hard.\"\n\n\"Just wait 'til you have kids,\" Maia interjected, blessing and cursing Nina, both at once.\n\nAs night pressed against the windowpanes, a new girl slipped inside the door. She wore a silky black shirt, black tights, black boots. She was strange and beautiful, her face pale, her hair and lips dark, her eyes outlined in black. At first, nobody recognized her\u2014not even little Henry. When she bent down and said, \"Hey! Remember me?\" he shrank away from his former babysitter, frightened by her red-black lips, her huge dark eyes.\n\nLois was the first to greet her. \"Diana, honey! How are you?\"\n\n\"Good.\"\n\nAlert and narrow, a greyhound of a boy, Jack appeared behind Diana. \"And what have you been up to?\" Lois inquired, but she had been seventeen once, and she thought she knew.\n\nMaia was playing African music, flamenco music, tangos by Astor Piazzolla. She rolled back the carpet and danced with Greg. The china cabinet rattled; the whole apartment began to hum.\n\nNina had abandoned her post to sit with Collin. No one was watching in the kitchen, so Jack and Diana served themselves spiked punch. Jack had two cups, and Diana had almost three. They laughed together and Diana leaned against the counter.\n\nHe said, \"I know where we should go.\" They got their shoes and slipped out the kitchen door.\n\nIn the living room, Sage was on her knees dancing with Henry, holding both his hands. Collin and Nina sat together on the couch and watched, while, at a little distance, Aidan watched them. Nina rested her head on Collin's shoulder, and Collin stroked her hair. How casual he was. He didn't even need to look at her.\n\nMaia's windows steamed up. Her kitchen overflowed with neighbors in their stocking feet. The front door stood like a green island in a sea of boots. Everyone was drinking, everyone was warm and loud as Aidan slipped outside. It was snowing when he crossed the street to his own house.\n\nHe knew Diana and Jack were in there. He could see their footprints on the steps. They were up in her room, tipsy, kissing, laughing. He hesitated, wondering if he should venture in or return to Maia's house. He didn't want to watch Collin stroking Nina's hair, and he really didn't want to listen to Diana and Jack.\n\nHesitating, he noticed a padded envelope stuffed into the mailbox. He pulled it out and read two words in handwriting he knew. _For Aidan._\n\nHe picked up the box and studied the curve of his own letters in Miss Lazare's writing. How strange to see his name there out of school. He stared at the words and then in a rush he ripped open the envelope and felt the thick book inside.\n\nOh. He knew what she had done. She had bought him his own poetry anthology because he was a winner too.\n\nHe almost threw the thing away. He didn't need anthologies, and didn't want to be consoled. Even so, he pulled the book out, hoping for a card.\n\nNow he discovered something else entirely. This was Lazare's own Dickinson, the one she had carried in the classroom and the hall. The very book he'd turned facedown on the desk. _Don't do that! It's old. You'll break the spine._\n\nHe found no card, no explanation, except for an inscription in blue ink. Not Miss Lazare's compact handwriting, but some other penmanship, pale and spidery. _For Nina Lazare, in recognition of excellence in public speaking, and with kind regards from her teacher, Lawrence B. Rousse..._ She had given Aidan her own prize. She had won it in school, and now she was giving it to him.\n\nHe closed the volume and weighed it in his hands. Then he took a flying leap off the porch. The bare trees tilted, neighbors' Christmas lights flickered around him as he almost fell, but he caught himself and landed on his feet.\n\nClasping the book with one arm to his chest, he raced up Antrim. He had no goal in mind, no destination. All he wanted was to run.\n\nHe stopped at the corner and looked once more at the treasure in his arms. Then, winded, he retraced his running steps, walking down the street again. He was not religious, but, like his mother, he believed in mysteries. Now he realized that Nina had sent him a sign, as Elvish queens bestowed a diamond flask, a gossamer handkerchief, a ring of gold. She had sent him a message with this poetry. He was noble, and he was magic. He was a champion and a prince.\n\nNo, none of that. Not really.\n\nHe saw the lights of the party. He had returned, but he stood out on the sidewalk, too shy to go inside and thank his teacher. Brushing snow from Nina's book, he felt her distance, magnified by kindness. There was nothing he could do, and nothing he could say. He could not explain what he felt, even to himself, the mix of hopelessness and grace. She didn't love him\u2014not the way he loved her\u2014but she had singled him out. She had given him her gift.\nTO MY TEACHERS\n\nDana Izumi\n\nJune Brieske\n\nMabel Hefty\n\nTom Earle\n\nBill Messer\n\nLiz Foster\n\nEileen Crean\n\nLeonard Russo\n\nBetty Sullivan\n\nJerry Devlin\n\nBill Alfred\n\nMichael Anesko\n\nLarry Benson\n\nBarbara Lewalski\n\nGeorge Dekker\n\nDavid Riggs\n\nJay Fliegelman\n\nJohn Bender\n\nSeth Lerer\n\nStephen Orgel\n\n# BY ALLEGRA GOODMAN\n\nThe Chalk Artist\n\nThe Cookbook Collector\n\nThe Other Side of the Island\n\nIntuition\n\nParadise Park\n\nKaaterskill Falls\n\nThe Family Markowitz\n\nTotal Immersion\n\n# ABOUT THE AUTHOR\n\n> ALLEGRA GOODMAN's novels include _Intuition, The Cookbook Collector, Paradise Park,_ and _Kaaterskill Falls_ (a National Book Award finalist) _._ Her fiction has appeared in _The New Yorker, Commentary,_ and _Ploughshares,_ and has been anthologized in _The O. Henry Awards_ and _Best American Short Stories._ She has written two collections of short stories, _The Family Markowitz_ and _Total Immersion,_ and a novel for younger readers, _The Other Side of the Island._ Her essays and reviews have appeared in _The New York Times Book Review, The Wall Street Journal, The New Republic, The Boston Globe,_ and _The American Scholar._ Raised in Honolulu, Goodman studied English and philosophy at Harvard and received a PhD in English literature from Stanford. She is the recipient of a Whiting Award, the _Salon_ Award for Fiction, and a fellowship from the Radcliffe Institute for Advanced Study. She lives with her family in Cambridge, Massachusetts, where she is writing a new novel.\n\nallegragoodman.com\n\nFacebook.com\/\u200bAllegraGoodman\n\n# _What's next on \nyour reading list?_\n\n[Discover your next \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9780679605041\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n 1. Cover\n 2. Title Page\n 3. Copyright\n 4. Contents\n 5. Chapter 1: Grendel's Den\n 6. Chapter 2: The Orchard\n 7. Chapter 3: Emerson\n 8. Chapter 4: Everwhen\n 9. Chapter 5: Wait for Me\n 10. Chapter 6: Snow Day\n 11. Chapter 7: Dream\n 12. Chapter 8: No Moon, No Stars\n 13. Chapter 9: Drink Me\n 14. Chapter 10: In Her Eyes\n 15. Chapter 11: Caution\n 16. Chapter 12: The Leopard\n 17. Chapter 13: Crossing Over\n 18. Chapter 14: The Visit\n 19. Chapter 15: Open Door\n 20. Chapter 16: The Interview\n 21. Chapter 17: Arkadia\n 22. Chapter 18: The Gates\n 23. Chapter 19: Deer\n 24. Chapter 20: Very Close\n 25. Chapter 21: Face-to-Face\n 26. Chapter 22: Pursuit\n 27. Chapter 23: Admission\n 28. Chapter 24: Discovery\n 29. Chapter 25: Lucky\n 30. Chapter 26: Walden Woods\n 31. Chapter 27: Half Magic\n 32. Chapter 28: The Question\n 33. Chapter 29: The Kiss\n 34. Chapter 30: Poetry Inaction\n 35. Chapter 31: Joy Street\n 36. Chapter 32: Two Rivers\n 37. Chapter 33: Busted\n 38. Chapter 34: Bird on a Wire\n 39. Chapter 35: In the Hall\n 40. Chapter 36: World-Jumping\n 41. Chapter 37: Contagion\n 42. Chapter 38: Win, Lose, or Draw\n 43. Dedication\n 44. Other Titles\n 45. About the Author\n\n 1. Cover\n 2. Cover\n 3. Title Page\n 4. Contents\n 5. Start\n\n 1. v\n 2. vi\n 3. \n 4. \n 5. \n 6. \n 7. \n 8. \n 9. \n 10. \n 11. \n 12. \n 13. \n 14. \n 15. \n 16. \n 17. \n 18. \n 19. \n 20. \n 21. \n 22. \n 23. \n 24. \n 25. \n 26. \n 27. \n 28. \n 29. \n 30. \n 31. \n 32. \n 33. \n 34. \n 35. \n 36. \n 37. \n 38. \n 39. \n 40. \n 41. \n 42. \n 43. \n 44. \n 45. \n 46. \n 47. \n 48. \n 49. \n 50. \n 51. \n 52. \n 53. \n 54. \n 55. \n 56. \n 57. \n 58. \n 59. \n 60. \n 61. \n 62. \n 63. \n 64. \n 65. \n 66. \n 67. \n 68. \n 69. \n 70. \n 71. \n 72. \n 73. \n 74. \n 75. \n 76. \n 77. \n 78. \n 79. \n 80. \n 81. \n 82. \n 83. \n 84. \n 85. \n 86. \n 87. \n 88. \n 89. \n 90. \n 91. \n 92. \n 93. \n 94. \n 95. \n 96. \n 97. \n 98. \n 99. \n 100. \n 101. \n 102. \n 103. \n 104. \n 105. \n 106. \n 107. \n 108. \n 109. \n 110. \n 111. \n 112. \n 113. \n 114. \n 115. \n 116. \n 117. \n 118. \n 119. \n 120. \n 121. \n 122. \n 123. \n 124. \n 125. \n 126. \n 127. \n 128. \n 129. \n 130. \n 131. \n 132. \n 133. \n 134. \n 135. \n 136. \n 137. \n 138. \n 139. \n 140. \n 141. \n 142. \n 143. \n 144. \n 145. \n 146. \n 147. \n 148. \n 149. \n 150. \n 151. \n 152. \n 153. \n 154. \n 155. \n 156. \n 157. \n 158. \n 159. \n 160. \n 161. \n 162. \n 163. \n 164. \n 165. \n 166. \n 167. \n 168. \n 169. \n 170. \n 171. \n 172. \n 173. \n 174. \n 175. \n 176. \n 177. \n 178. \n 179. \n 180. \n 181. \n 182. \n 183. \n 184. \n 185. \n 186. \n 187. \n 188. \n 189. \n 190. \n 191. \n 192. \n 193. \n 194. \n 195. \n 196. \n 197. \n 198. \n 199. \n 200. \n 201. \n 202. \n 203. \n 204. \n 205. \n 206. \n 207. \n 208. \n 209. \n 210. \n 211. \n 212. \n 213. \n 214. \n 215. \n 216. \n 217. \n 218. \n 219. \n 220. \n 221. \n 222. \n 223. \n 224. \n 225. \n 226. \n 227. \n 228. \n 229. \n 230. \n 231. \n 232. \n 233. \n 234. \n 235. \n 236. \n 237. \n 238. \n 239. \n 240. \n 241. \n 242. \n 243. \n 244. \n 245. \n 246. \n 247. \n 248. \n 249. \n 250. \n 251. \n 252. \n 253. \n 254. \n 255. \n 256. \n 257. \n 258. \n 259. \n 260. \n 261. \n 262. \n 263. \n 264. \n 265. \n 266. \n 267. \n 268. \n 269. \n 270. \n 271. \n 272. \n 273. \n 274. \n 275. \n 276. \n 277. \n 278. \n 279. \n 280. \n 281. \n 282. \n 283. \n 284. \n 285. \n 286. \n 287. \n 288. \n 289. \n 290. \n 291. \n 292. \n 293. \n 294. \n 295. \n 296. \n 297. \n 298. \n 299. \n 300. \n 301. \n 302. \n 303. \n 304. \n 305. \n 306. \n 307. \n 308. \n 309. \n 310. \n 311. \n 312. \n 313. \n 314. \n 315. \n 316. \n 317. \n 318. \n 319. \n 320. \n 321. \n 322. \n 323. \n 324. \n 325. \n 326. \n 327. \n 328. \n 329. \n 330. \n 331. \n 332. \n 333. \n 334. \n 335. \n 336. \n 337.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nEDUCATION IN \nNAZI GERMANY\nEDUCATION IN \nNAZI GERMANY\n\nLisa Pine\n\nEnglish edition \nFirst published in 2010 by \n**Berg** \nEditorial offices: \nFirst Floor, Angel Court, 81 St Clements Street, Oxford OX4 1AW, UK \n175 Fifth Avenue, New York, NY 10010, USA\n\n\u00a9 Lisa Pine 2010\n\nAll rights reserved. \nNo part of this publication may be reproduced in any form or by any means without the written permission of Berg.\n\nBerg is the imprint of Oxford International Publishers Ltd.\n\n**Library of Congress Cataloging-in-Publication Data**\n\nPine, Lisa. \nEducation in Nazi Germany \/ Lisa Pine. \u2014 English ed. \np. cm. \nIncludes bibliographical references. \neISBN: 978-1-8478-8764-1 \n1. Education\u2014Germany\u2014History\u201420th century. 2. Education and state\u2014Germany\u2014History\u201420th century. 3. Education\u2014Political aspects\u2014Germany\u2014History\u201420th century. 4. National socialism and education\u2014Germany. 5. National socialism and youth\u2014Germany. I. Title. \nLA721.8.P56 2010 \n370.942'09043\u2014dc22\n\n2010037203\n\n**British Library Cataloguing-in-Publication Data**\n\nA catalogue record for this book is available from the British Library.\n\nTypeset by JS Typesetting Ltd, Porthcawl, Mid Glamorgan. \nPrinted in the UK by the MPG Books Group.\n\n**www.bergpublishers.com**\n\nFor my parents\nCONTENTS\n\nAcknowledgements\n\nIntroduction\n\n1 The Historical Context\n\n2 Nazi Education Policy\n\n3 The Curriculum and School Textbooks\n\n4 The Nazi Elite Schools\n\n5 The Hitler Youth\n\n6 The League of German Girls\n\nConclusion\n\nGlossary of Abbreviations and Terms\n\nBibliography \nACKNOWLEDGEMENTS\n\nIt is a pleasure to be able to thank the many people who have helped me in the process of researching and writing this book. I am grateful to the staff of the Bundesarchiv in Berlin, the Institut f\u00fcr Zeitgeschichte in Munich, the Georg Eckert Institute in Brunswick, as well as staff at the British Library, the British Newspaper Library, the Wiener Library and the German Historical Institute in London, for their assistance.\n\nI should like to thank the British Academy for a Small Research Grant awarded to me in 2008 for archival research in Berlin. I am grateful to the SPUR Institute and the Research Opportunities Fund at London South Bank University for their generous funding of my research over the last few years.\n\nI should like to thank my very capable and efficient research assistants, Manuel Siebert, Anna Heizmann and Anna Keller for all their hard work. I am very grateful to Stephanie Salzmann and Ulrich Schlie for their wonderful and kind hospitality in Berlin. I should like to express my appreciation to Hans Hahn and Laurence Marlow for reading the manuscript and suggesting improvements. My thanks are also due to my editor, Julia Hall, for her enthusiasm for the project, her patience and her help, and to my copy-editor, Julene Knox, for her hard work.\n\nMy acknowledgements would be incomplete without a note of personal thanks to my wonderful family, who have supported me throughout the writing of this book. My husband, Andrew Fields, who always has faith in my endeavours, and my children, Gabrielle and Sasha Fields, who have consistently encouraged me to write this book and have been very patient in allowing me to do so.\nINTRODUCTION\n\nIn my great educative work, I am beginning with the young. We older ones are used up. Yes, we are old already. We are rotten to the marrow. We have no unrestrained instincts left. We are bearing the burden of a humiliating past, and have in our blood the dull recollection of serfdom and servility. But my magnificent youngsters! Are there finer ones anywhere in the world? Look at these young men and boys! What material! With them I can make a new world.\n\nAdolf Hitler*\n\nThe main objective of this book is to provide a re-evaluation of education and the socialization of youth in the Third Reich in the light of new knowledge, theories and debates about the nature of the Nazi state. This will be achieved by analysing three main areas: education policy, the Nazi elite schools and the Nazi youth groups. Education is fundamental to our entire macro-view of the Third Reich, as the process of shaping the minds of the future generation was such a significant aspect of the Nazi regime. This book addresses a number of important questions, many of which have not been adequately treated in the secondary literature on the Third Reich: What were the aims of Nazi education policy? Was the regime successful in achieving these objectives? Who made Nazi education policy? What changes were made to the education system and to the school curriculum? How did the Nazi regime use school textbooks as propaganda instruments? What was the role and significance of the Nazi elite schools in the Third Reich? How was youth socialization achieved in the Nazi youth groups?\n\nThe Nazi regime sought to win over young people by means of both the schools and the youth groups. Indeed, some Nazi pedagogues believed that schools should play a secondary role to youth groups. This book links the schools and youth groups together conceptually, demonstrating how the Nazi regime utilized both, in order to achieve a 'total education' of German youth. In particular, the Nazi regime used education and socialization to create identity. Whilst recent works have considered the role played by propaganda, the SS, the Nazi women's organizations and other Party formations in the creation of identity, the part played by schools and youth groups together requires more detailed analysis. Education and socialization in Nazi Germany were fundamental to the shaping and forging of national identity, as well as self-perception and the perception of 'others'. This was bound up with the wider issue of inclusion in and exclusion from the _Volksgemeinschaft_ (national community). Education under National Socialism was directed at creating a new awareness, changing the way people thought and eroding traditional loyalties. Propaganda, underpinned by the threat and use of terror, was another integral part of this process. The Hitler regime employed a combination of policies designed to create consensus, as well as censorship, aiming to ensure that access to other sources of information or ways of thought was unavailable. Hence, the specific area of study of education and youth socialization is central to our understanding of the National Socialist state. The shaping of culture and stamping of identity in line with the Nazi _Weltanschauung_ (world view) and the education and socialization of youth to become the ideal future generation of Germans are fundamental to our wider comprehension of the Third Reich.\n\nThe existing literature consists of books that deal with various aspects of education and others that treat the youth groups. The historiography of Nazi education is vast and complex, but much of it also rather dated, having been written between the 1960s and 1980s. Eilers was the pioneer in this field, publishing in 1963 the first post-war account of education under National Socialism. This book was followed by a number of significant works on schools, education and the curriculum in the Nazi era published in the 1970s and 1980s. Since then, a handful of works on education appeared in the 1990s, but even these are now over a decade old. Important research on Nazi youth groups published in the 1980s and 1990s highlighted the significance of youth groups to Nazi education. It is now time to move the subject forward with a fresh approach and to offer a perspective on Nazi education policy that encompasses both schools and youth groups. Indeed, Nazi educationalists believed that 'teachers and HJ leaders are equal partners in the education of German youth'.\n\nRecent debates about the nature of National Socialism necessitate a re-examination of a number of angles and perspectives. Firstly, it is important to consider the continuities and discontinuities in education policy between the _Kaiserreich_ (Second German Empire), the Weimar Republic and the Third Reich. In this way, it will become possible to evaluate the extent to which Nazi policy was novel. Education policy in the _Kaiserreich_ was already falling under the influence of a move towards greater militarization and nationalism in society as a whole. It has been suggested that modern Germany took a _Sonderweg_ (special path) in the nineteenth century that distinguished her development from that of democratic Western European states. This view asserts that the modern German state developed its own 'peculiar' national character, which was nationalist and authoritarian. Was this the case? Was Nazi policy distinctive from that of earlier administrations or the culmination of this 'special path'? There is much evidence to suggest strong similarities to the educational policies of previous governments. In many ways, the Nazi regime built upon existing foundations\u2013yet it often added a more radical slant or direction to educational policy. For example, elite educational institutions had existed throughout previous centuries, but under National Socialism they took on a different ideological mantle.\n\nSecondly, historical debates surrounding the nature of the Nazi state and system of rule itself can be applied to the area of education policy, in order to establish which theoretical approach is the most valid. For example, the 'intentionalist\u2013structuralist' debate has been applied to certain aspects of Nazi policy, most notably foreign policy and anti-Semitic policy, in order to evaluate the extent of Hitler's own role in policy-making. It is necessary to consider this in regard to educational policy too. The evidence suggests that Hitler's ideology and views on education provided a backdrop to education policy. However, Hitler did not take an active role in its evolution. Policy emerged from other initiatives and centres of power. Furthermore, it is necessary to assess the extent to which there was a central focus or command for the making of educational policy.\n\nWas the Ministry of Education necessarily the focal point of policy-making in this sphere? In fact, the primacy of Bernhard Rust's position as Minister of Education in policy-making was not unchallenged by other individuals and agencies. Competing protagonists and organizations struggled over areas of responsibility in educational policy-making.\n\nThirdly, a consideration of how educational policy fits into the overall debate on 'modernity versus reaction' in the Third Reich is important. Was Hitler's policy towards education 'modernizing' (as it claimed to be) or not? Herf has conceptualized the Weimar Republic and the National Socialist eras together as exemplars of 'reactionary modernism'. This highlights the tensions between the embrace of modern technology and the atavistic ideological principles that existed in both the Weimar and the Nazi years. Certainly, the Nazis excoriated the big cities and idealized the countryside. The Nazi _Blut und Boden_ (blood and soil) mythology and its yearning for a return to a pre-industrial idyll conflicted with the reality of advances such as the building of motorways and the impact of mass tourism on German society. Similarly, there were aspects of both modernization and anti-modernism in Nazi educational policy, which were inconsistent in this regard. As was the case in many other areas of social policy, in particular policy towards women and the family, ambiguities abounded. Certain aspects of educational policy appeared modernizing, whilst others were not. For example, the Nazi government rationalized the secondary school system, yet educational standards declined. This was partly because Nazi ideological tenets conflicted with practical considerations, particularly in the wartime period.\n\nDespite inconsistencies and internal conflicts in policy-making, the Nazi regime aimed at a 'total education' of youth that corresponded with Hitler's fundamental ideas on education. Hitler had strong views on the role of the state in education. In the Third Reich, the emphasis of education moved away from the individual to the requirements of the state and the 'national community'. Education through the 'experience of community' was an essential concept, because individuals were not regarded as autonomous, but as part of the entire organism of the _Volk_. This was bound up with the issues of belonging and identity as well. Individuals functioned as members of the 'national community' and as guarantors of its development and strength. Everyday life was conceived of as part of a perpetual struggle for the national cause and against the 'enemies' of the state. The mobilization of youth in this way, as well as the creation of a sense of conformity and uniformity, formed an integral part of the Nazis' overall design.\n\nIt is important to consider why youth was so important to the Hitler regime. Nazi leaders viewed the German youth as a catalyst for change away from what they regarded as the decadent political system of the Weimar Republic towards the new 'national community' of the future. The Nazi education theorist Ernst Krieck described the youth as the bearer of the principle of the German (National Socialist) revolution, from which would develop 'a new nation, a new form of humanity and a new order of living space'. In order to achieve this, the regime's 'total' education and socialization programme encompassed both schools and youth groups. 'Youth organization is to be seen by schools as a linear expansion and deepening of the work of schools'.\n\nNazi 'total education' removed youth from the usual frames of social reference, such as the family, trying to encompass the entire experience of youth. It immersed young people in a completely organized network. It made huge incursions into leisure time and took up the majority of their waking hours. Furthermore, Hitler was keen to place the function of socialization firmly within youth groups and schools, removing it from the family as much as possible. Moreover, Hitler's contempt for intellectual endeavour and his cynicism towards schoolteachers further enhanced the position of youth groups within the Nazi state. This is a large part of the reason why youth groups in the Third Reich had such a significant role designated to them. Schools' and parents' roles were partly undermined by Hitler's idea that 'youth must be educated by youth', a concept taken from the German youth movement.\n\nBefore embarking upon an examination of Nazi education policy, it is important to place the subject within its proper historical context. Hence, this book begins with a short chapter that considers the background to education in the Third Reich by looking at trends in education in the preceding eras. The second chapter analyses the impact of the Nazi _Machtergreifung_ on German education. What were Hitler's views on education and how did they shape Nazi education policy? This chapter goes on to examine the role of the _NS-Lehrerbund_ (NSLB) or National Socialist Teachers' Association in education. It explores educational policy and decision-making. It examines the role of the Ministry of Education and the influence of competing agencies and individuals upon its work. Furthermore, it analyses the changes made by the National Socialist regime to the German education system at all levels\u2013from kindergarten to university.\n\nThe third chapter examines the school curriculum during the Third Reich and the impact of Nazism in the classroom. Many subjects within the school curriculum were used to expound Nazi ideology, most notably biology, physics, chemistry, history, geography, mathematics and German. This chapter analyses the use of school textbooks to disseminate Nazi ideology. In particular, it focuses on political socialization in schools, including the key themes of anti-Bolshevism, the creation of Nazi myths and heroes, as well as the forging of the 'national community'. Furthermore, it considers the introduction of racial science into the school curriculum and anti-Semitism into the classroom. This created a culture of racial hatred and provided an ideological pretext for the Nazis' mass murder of the Jews. In addition, this chapter examines the Nazi emphasis on physical education. This aspect of education was close to Hitler's heart and a subject upon which he had strong views.\n\nChapter 4 examines the role of elite education in the Third Reich. Nazi elite educational institutions performed a special function within Nazi education and socialization processes as a whole. The purpose of elite education in Nazi Germany was to train a leadership cadre for the next generation. The Nazi regime established three main types of educational institutions to train the future elite of German society: the _Nationalpolitische Erziehungsanstalten_ (National Political Educational Institutions or Napolas), the _Adolf Hitler Schulen_ (Adolf Hitler Schools or AHS) and the _Ordensburgen_ (Order Castles). These institutions were a microcosm of the Nazi _Weltanschauung_. They fostered the leadership principle ( _F\u00fchrerprinzip_ ), promoted competitiveness and emphasized life as a struggle and as survival of the 'fittest'. They encouraged physical prowess. They excoriated the 'enemies of the Reich', in particular, Jews, Communists and Socialists. They emphasized racial purity, glorified war and fostered militarism. They underlined the necessity for _Lebensraum_ (living space) and had a role to play in the achievement of a 'greater German empire'.\n\nThe next two chapters deal with the Nazi youth groups. Chapter 5 discusses the _Hitlerjugend_ (HJ) or Hitler Youth and Chapter 6 treats its female counterpart, the _Bund Deutscher M\u00e4del_ (BDM) or League of German Girls. There was an established tradition of youth groups and movements in modern Germany that long pre-dated the Third Reich. The aim of these two chapters is to determine what was distinctive about the nature and purpose of the Nazi youth groups. How did youth identity manifest itself before the Nazi era and how did this change after 1933? The Nazi regime used its youth groups to foster within their members a sense of self-identity and identification with the aims of National Socialism. In addition, the separate youth groups for boys and girls signified distinctive gender roles and expectations.\n\nChapter 5 analyses the aims of the Hitler Youth and their implementation. What activities did its members undertake? How were they socialized? Clearly, in the HJ, as in other Nazi formations, the individual was subordinated to the group. Conformity to the organizational norm was designed to create true believers in the National Socialist system. HJ members were bound to the community of the organization, and above and beyond that, to the 'national community'. The role of youth in National Socialism was to struggle against old traditions and against the enemies of the regime. This chapter examines discipline and training in the HJ, designed to prepare for this fight. It considers a variety of aspects including physical fitness, hygiene and dress codes, as well as education in National Socialist principles.\n\nChapter 6 examines the role of the League of German Girls as an organization for the regimentation and socialization of German girls. What tasks were its members engaged in? What was the impact of the League of German Girls upon its members? This chapter shows how girls were socialized differently from boys. It examines the training and education given to girls, activities and duties of BDM girls before and during the war, as well as attitudes towards sexual behaviour. It illustrates how the BDM formed an intrinsic part of the process of socializing German girls, as part of a blood-binding community, whose members were obligated to serve their 'national community' in any way required of them by the Nazi state. An examination of all these aspects of education and socialization, by evaluating the aims of Nazi 'total education', will enhance our understanding of the Third Reich.\n\n**NOTES**\n\n* H. Rauschning, _Hitler Speaks: A Series of Political Conversations with Adolf Hitler on his Real Aims_ (London, 1939), p. 246.\n\n. R. Eilers, _Die nationalsozialistische Schulpolitik. Eine Studie zur Funktion der Erziehung im totalit\u00e4ren Staat_ (Cologne, 1963).\n\n. K.-I. Flessau, _Schule der Diktatur. Lehrpl\u00e4ne und Schulb\u00fccher des Nationalsozialismus_ (Frankfurt am Main, 1979); M. Heinemann (ed.), _Erziehung und Schulung im Dritten Reich_ (Stuttgart, 1980); H. Kanz (ed.), _Der Nationalsozialismus als p\u00e4dagogisches Problem: Deutsche Erziehungsgeschichte 1933\u20131945_ (Frankfurt am Main, 1984); K.-I. Flessau _et al._ (eds), _Erziehung im Nationalsozialismus_ (Cologne, 1987); R. Dithmar (ed.), _Schule und Unterricht im Dritten Reich_ (Neuwied, 1989).\n\n. B. Ortmeyer, _Schulzeit unterm Hitlerbild_ (Frankfurt am Main, 1996); W. Keim, _Erziehung unter der Nazi-Diktatur_ (Darmstadt, 1997); H. S\u00fcnker and H.-U. Otto (eds), _Education and Fascism: Political Identity and Social Education in Nazi Germany_ (London, 1997).\n\n. M. Klaus, _M\u00e4dchen in der Hitlerjugend. Die Erziehung zur 'deutschen Frau'_ (Cologne, 1980); H. Boberach, _Jugend unter Hitler_ (Dusseldorf, 1982); K. Huber, _Jugend unterm Hakenkreuz_ (Berlin, 1982); A. Kl\u00f6nne, _Jugend im Dritten Reich: Die Hitler-Jugend und Ihre Gegner_ (Cologne, 1982); M. Klaus, _M\u00e4dchen im Dritten Reich. Der Bund Deutscher M\u00e4del (BDM)_ (Cologne, 1983); D. Reese, _'Straff, aber nicht Stramm\u2013Herb, aber nicht Derb'. Zur Vergesellschaftung der M\u00e4dchen durch den Bund Deutscher M\u00e4del im Sozialkulturellen Vergleich zweier Milieus_ (Weinheim, 1989); G. Kinz, _Der Bund Deutscher M\u00e4del: Ein Beitrag zur Au\u00dferschulischen M\u00e4dchenerziehung im Nationalsozialismus_ (Frankfurt am Main, 1990); B. J\u00fcrgens, _Zur Geschichte des BDM (Bund Deutscher M\u00e4del) von 1923 bis 1939_ (Frankfurt am Main, 1994).\n\n. BA NS 12\/1196, 'Die Schulungsarbeit des Amtes f\u00fcr Erzieher (NSLB)', 19 Sept. 1935, p. 2.\n\n. On the _Sonderweg_ argument, see J. Kocka, 'German History before Hitler: The Debate about the German _Sonderweg_ ', _Journal of Contemporary History_ , Vol. 23 (1988), pp. 3\u201316.\n\n. J. Herf, _Reactionary Modernism: Technology, Culture and Politics in Weimar and the Third Reich_ (Cambridge, 1984), p. 220.\n\n. E. Krieck, _Nationalpolitische Erziehung_ (Leipzig, 1941), p. 48.\n\n. BA NS 12\/819, 'Bekanntmachung des Staatsministeriums'.\n1 THE HISTORICAL CONTEXT\n\nThere was a close relationship between pedagogy and politics in modern German history, and the education system was bound up with the development of the dominant political culture. The _Volksschule_ (elementary school) was intended as part of a comprehensive educational system, without reference to social class or background. The German _Gymnasium_ , inspired by Wilhelm von Humboldt (1767\u20131835), was founded as an institution to prepare pupils for higher education. Its syllabus centred on three spheres of education: gymnastics, aesthetics and didactics. Didactics, which included languages, history, mathematics and science, became the most important of the three areas in terms of preparing pupils for university. The _Abitur_ (school-leaving certificate), which had first been introduced in 1788 in Prussia, became the prerequisite for entry to German universities in the 1830s. The _Gymnasium_ created a new dominant role for the middle classes\u2013whose political culture was important in this era\u2013based upon educational ideals.\n\nThe concept of _Volksbildung_ (national education) of the whole nation, as a foundation for national culture, promoted by important German educationalists, such as Adolf Diesterweg, at first had liberal connotations. Social integration was central to developments and reforms in school education. However, in 1854, the Stiehl Ordinance standardized the curriculum, pedagogic forms and the training of teachers. It stymied the attempts of educational reformers, and because of it the state played a much larger role in the content of mass education than in previous decades. Hard work and discipline formed the ethos of schools as educators of subjects, rather than citizens. As Hahn points out: 'By 1870 the term _Volksbildung_ had completely changed its ethos; it had lost its national, liberal and democratic spirit and had become institutionalised as a force for inculcating in the common people an attitude of submission to authority and to the state'. The education system that developed was unable to establish democratic structures or genuine socialization.\n\nHowever, there were still pedagogues who sought to put into place measures in education in response to the economic, social and demographic changes brought about by Germany's unification and industrialization. The educational philosophy of Johann Friedrich Herbart (1776\u20131841) was pertinent to these changes in German society, both giving a professional ethos to elementary education and calling for a greater emphasis on science and a broader curriculum than Humboldt had envisaged in the secondary school system. Vocational education became more significant in this period too. Georg Kerschensteiner (1854\u20131932) founded the _Arbeitsschule_ and influenced a new form of vocational training. He introduced a dual system of vocational education. Male apprentices were to receive technical training as well as a broad general education in subjects such as German, civics, law and commerce. Girls' education was a preparation for motherhood and family life. Whilst Kerschensteiner's system raised the status of crafts and trade in Germany, this separate emphasis on boys' and girls' education did nothing to help moves towards female emancipation. There was also an authoritarian aspect to his method, with its subordination of the individual to state power. In these aspects, we can see elements of an educational philosophy later picked up by National Socialism.\n\nThe liberal ethos that characterized the German bourgeoisie was replaced by a conservative nationalism. Bismarck's _Realpolitik_ had already begun the move away from concepts of liberalism, but the Humboldtian concept of _Bildung_ suffered its most serious crisis after the unification of Germany in 1871 and the establishment of the _Kaiserreich_. The newly unified nation had different priorities and values. The educational system became more reactionary and nationalist, and less liberal and democratic towards the end of the nineteenth century. The true needs of the new German state were deemed to be based upon military and political success. Policy in Germany moved in a different direction from the general western European trend towards greater liberalism and democracy. Wehler suggests that the absence in Germany of the revolutions experienced by Britain, France and America in the seventeenth and eighteenth centuries contributed to the _Sonderweg_ (special path) of modern German history. He emphasizes the absence of bourgeois liberalism in the _Kaiserreich_. The militarization of everyday life in the _Kaiserreich_ created an emphasis on hierarchy and obedience to authority. German educators became more nationalistic and defensive, as well as, simultaneously, imperialistic and chauvinistic. The works of nationalist writers such as Paul de Lagarde and Julius Langbehn gained currency. Langbehn urged Germans to liberate themselves from foreign values in order to raise the stature of German cultural life.\n\nDuring this period, irrationalism left its mark upon concepts of education and society. Educationalists regarded themselves as upholders of the nation's culture. A new discipline of _Kulturkunde_ found its way into the school curriculum. This subject integrated German language, literature, history, geography, religion and civics. The Reich School Conferences of 1890 and 1900 saw the adaptation of the education system to meet the requirements of the jingoism of the ruling class. Furthermore, the German Youth Movement, which sought a re-evaluation of 'Germanness', contributed to this rising nationalism. Traditional humanist culture found itself struggling against this strong tide of nationalism and _Realpolitik_. Kaiser Wilhelm II became engaged in the _Schulstreit_ (school dispute) in 1890. He called for more 'character building' and physical education in _Gymnasium_ education, as well as a greater emphasis upon the nation's heritage, history and geography. The traditional focus on education in the classics in the _Gymnasium_ decreased in significance. The result of the school dispute was a considerable expansion of the secondary school system with changes to the _Realgymnasium_ and the introduction of a new type of school\u2013the _Oberrealschule_. These schools favoured a more technological education than the _Gymnasium_ , which was considered to overburden its pupils' minds and to lack practical relevance. These schools fulfilled the real demands of technology and industry. By 1900, the status of the _Realgymnasium_ and the _Oberrealschule_ achieved parity with that of the _Gymnasium_ , although Latin was still needed for university entrance. The _Gymnasium_ came under attack and suffered a decline in status. Friedrich Lange was typical in his assault upon the 'excessive humanism' of the _Gymnasium_ , as well as its education in 'aesthetic idealism'. He asserted the aims of patriotism, duty and 'the idea of Germanhood'. He questioned what the models of classical antiquity could provide in education that Germany's own history could not.\n\nIt is significant to note that there were organizations that demanded female emancipation in education in this period. The _Allgemeiner Deutscher Frauenverein_ (General German Women's Association, founded in 1865) called for entry for women into vocational work and higher education. Helene Lange (1848\u20131930) and Gertrud B\u00e4umer (1873\u20131954) were influential figures in pushing for reforms in girls' education and admission to higher education. Curricular expansion in girls' _Lyzeum_ schools that enabled them to take the _Abitur_ was achieved by 1908. There was co-education at secondary level in some states, but the overall ratio of women to men at this level was still approximately one to three.\n\nAs the First World War approached, 'the ideas of 1914' were chauvinistic, militaristic and bellicose in nature. This development went hand in hand with a victory of nationalist ideas over party politics with the call for a _Burgfriede_ \u2013a truce between political parties for the duration of the war. The impact of the First World War and of the Treaty of Versailles led many German academics and educationalists to seek spiritual renewal and to struggle against Germany's loss of status. They opposed the Weimar Republic and adopted a reactionary position.\n\nHowever, during the Weimar era, modernism made its mark on education as it did on other areas of social and cultural life. Progress and reform were the watchwords of the republican era with a rejection of _v\u00f6lkisch_ and authoritarian trends among reformers. Lamberti argues: 'the Weimar years were a time of exuberant pedagogical innovation and optimistic plans to reform the stratified educational system in the name of democracy and social justice'. Modernism in education policy 'sought to extend compulsory schooling, develop a co-educational system, support the more technically orientated schools and broaden access to higher education'. Thuringia and Saxony led the way in attempting to achieve these aims. Bremen and Hamburg also endeavoured to carry out extensive changes to the education system. Elementary school reform included the abolition of voluntary religious education, the introduction of a collegiate system for teachers, and the establishment of teachers' councils. The most progressive were the 'community schools' in Hamburg, such as the Lichtwark School. Fritz Karsen established a new secondary school in Berlin-Neuk\u00f6lln, which enrolled youths from working-class families. Although such new schools courted controversy, Karsen believed that there was support for modern and experimental pedagogy in Berlin.\n\nThe League of Radical School Reformers, established in 1919 by Paul Oestreich, a schoolteacher from Berlin, also had some influence during the Weimar era. Oestreich called for a move away from the stifling values of the past, towards democratic reform in education. The League of Radical School Reformers was concerned with 'creative education'. It argued that 'our schools merely transmit knowledge in an authoritarian and dogmatic manner and in the framework of a thoroughly militaristic organisation'. It called instead for a system that would help a pupil towards 'the full development of his own particular nature'. It rejected designs to improve the schools within the existing system, calling for a more co-operative and communal approach to achieving its educational ideals. The League of Radical School Reformers also called for reform of the extremely nationalistic and militaristic textbooks of the _Kaiserreich_. Whilst a small amount of headway was made here, the majority of school textbooks, especially in history and geography, continued in the trend of the former era.\n\nThis link between the _Kaiserreich_ and the Weimar Republic was significant in terms of personalities and educational trends too. Otto Boelitz, the Prussian Minister of Education between 1921 and 1925, although not crudely nationalistic, nevertheless asserted that the most important educational task was the renewal of national power and unity. Carl Becker, who succeeded him in this role between 1925 and 1930, took a similar stance. Both had been reared in a nationalist tradition from which they found it difficult to move away. Prince Max von Baden took a conservative, counter-revolutionary stance on education. He founded a school at Salem near Lake Constance in 1920, which reflected his political and social views. During his opening speech, his themes were authoritarian and militaristic. He asserted the need for spiritual renewal, expressing a reverence for the countryside and a disdain for the cities. Prince Max von Baden aimed at a national rebirth led by the traditional military elite.\n\nNevertheless, the Weimar Constitution guaranteed a number of fundamental educational rights, including equal access to education, equality between men and women and free education for eight years. It also advocated compulsory school attendance for the first four years of elementary school and entrance to secondary school based upon merit. It set down the principle of a standardization of teachers' education, in which all teachers had to have a university education and were given civil servant status. The new subject areas _Staatsb\u00fcrgerkunde_ (civics) and _Arbeitsunterricht_ (work instruction) became compulsory. Private preparatory schools were to be abolished. The 1920 School Conference was the high point of the reform plans. There was considerable reluctance in some quarters to implement all these new reforms, and several initiatives were shelved. Indeed, elections in 1920 removed the Majority Social Democratic Party (MSPD)\u2013the party that had spearheaded these educational reforms\u2013from power. In the main, any ideas of removing education from Church control failed to be implemented. The Catholic Centre Party, which succeeded the MSPD, aimed to maintain and extend religious influence over education. Naturally, the Protestant Church accorded with these aims, and both churches rallied against the evils of socialism. As Hahn points out, by the end of the Weimar Republic, 'some 80 per cent of elementary education remained denominational'.\n\nIn secondary education, the _Mittelschule_ , which provided a route into the higher level of secondary education, was expanded. The _Deutsche Oberschule_ was established in 1922. This provided another route to the _Abitur_ and entrance to university education. As well as requirements to study French and English, this school concentrated on German language and German culture. From 1923, the _Oberlyzeum_ for girls allowed entry to university education through the modern languages route, but without Latin. Similar routes to higher education were made available for girls as for boys. By 1930, almost half of the female pupils in secondary education were in the _Oberlyzeum_.\n\nA significant shortcoming of Weimar secondary education reforms, as Hahn demonstrates, was that 'the proliferation of different schools prevented any attempt to integrate the various social strata' within German society. In particular, the four types of secondary schools with their different syllabuses highlighted social divisions. The _Gymnasien_ education centred on the classics, the _Realgymnasien_ on European civilization, the _Deutsche Oberschulen_ on German culture, and the _Oberrealschulen_ on natural sciences. German culture was taught in all these schools, with a particular emphasis on history, geography, civics and religion. The Nazi regime was later able to build upon this strong nationalist, cultural perspective. There was, furthermore, a backlash against emancipation in female education in certain quarters. The _Frauenschulen_ , which sought to teach girls about their 'special tasks' as mothers and homemakers, provided the answer to such concerns and again were a vehicle the Nazi government later used to promote its ideological imperatives in regard to women's position and role in society.\n\nThe Weimar Republic existed for just fourteen years. In terms of educational reforms, this time span was not long enough to bring about a vast amount of progress, especially taking into account the economic and financial problems that beset the Weimar governments. Nevertheless, as Lamberti observes: 'the opening of experimental public schools in many cities and the introduction of the new pedagogy in the urban schools placed Weimar Germany in the forefront of the progressive education movement'. The most progressive period of the Weimar Republic in educational terms were its first two years.\n\nEducational reformers faced an array of hostile anti-modernist groups and organizations. After a small amount of headway in progressive education had been made by 1920, the combination of this reactionary opposition with economic and financial problems impeded a great deal of further progress. Nevertheless, as Lamberti emphasizes: 'In the midst of economic distress and suffering and in the face of powerful adversaries, the progressivist pedagogues fought to realise their ideal of . . . a more open and democratic educational system. The slowing down of the momentum of reform in the later years of the republic should not diminish the significance of what was achieved.' In the early 1930s, Nazi ideologues and propagandists capitalized on the resentment of reactionaries and traditionalists who disliked the progressive nature of Weimar educational reforms and experimentation.\n\nIt is noteworthy that some of the trends in educational progress brought about during the Weimar years continued from those of the _Kaiserreich_. There was already some progress, for example, as we have seen, in female education, during the _Kaiserreich_. However, among some German educationalist circles, there was also a great deal of reaction during the Weimar years, with a continuation of the extreme nationalism that characterized the _Kaiserreich_. For example, the Prussian State Boarding Schools, housed in the former Cadet Institutions, were an expression of ardent nationalism and militarism. Hence, education during the Weimar era spanned the entire gamut from radical left-wing reformism across more moderate ground to authoritarian and nationalist viewpoints. The existence of strong anti-democratic and reactionary elements at work in education during the Weimar Republic meant that the Nazi 'seizure of power' in January 1933 did not mark a sudden, wholesale change. It is important to acknowledge the links that existed between educational developments not only between the _Kaiserreich_ and the Weimar Republic, but also between the Weimar Republic and the Third Reich. It is to educational policies in the Nazi era that we now turn in Chapter 2.\n\n**NOTES**\n\n. H.-J. Hahn, _Education and Society in Germany_ (Oxford, 1998), p. 17.\n\n. Ibid., pp. 30\u201331.\n\n. H. Wehler, _Aus der Geschichte Lernen?_ (Munich, 1988), p. 38.\n\n. Hahn, _Education and Society_ , p. 30.\n\n. On changes in secondary education, see J. Albisetti, _Secondary School Reform in Imperial Germany_ (Princeton, 1983).\n\n. F. Lange, _Reines Deutschtum_ (Berlin, 1898).\n\n. On this, see J. Albisetti, _Schooling German Girls and Women: Secondary and Higher Education in the Nineteenth Century_ (Princeton, 1988).\n\n. Hahn, _Education and Society_ , p. 35.\n\n. M. Lamberti, _The Politics of Education: Teachers and School Reform in Weimar Germany_ (New York and Oxford, 2002), p. 1.\n\n. Hahn, _Education and Society_ , p. 50.\n\n. R. Samuel and R. Hinton Thomas, _Education and Society in Modern Germany_ (London, 1949), p. 12.\n\n. Lamberti, _The Politics of Education_ , p. 119.\n\n. Samuel and Hinton Thomas, _Education and Society_ , p. 32.\n\n. See O. Boelitz, _Der Aufbau des preussischen Bildungswesens nach der Staatsumw\u00e4lzung_ (Berlin, 1925).\n\n. On this, see Samuel and Hinton Thomas, _Education and Society_ , pp. 13\u201314.\n\n. Hahn, _Education and Society,_ p. 56.\n\n. Ibid., p. 57.\n\n. Lamberti, _The Politics of Education_ , p. 246.\n\n. Ibid., p. 245.\n2 NAZI EDUCATION POLICY\n\n**HITLER AND EDUCATION**\n\nHitler's views on education were clearly concerned with a reshaping of values, the creation of national identity and racial awareness. He was contemptuous of intellectual endeavour and scholarly education. This is evidenced by his statement that 'the whole method of instruction in secondary and higher schools is just so much nonsense. Instead of receiving a sound basic education, the student finds his head filled with a mass of useless learning, and in the end is still ill-equipped to face life'. Hitler associated intellectualism with Judaism and decadence. Instead of intellectualism, he called for a greater emphasis upon physical education. In addition, Hitler believed that education and training had to be so ordered as to give the young German 'national comrade' the conviction of 'absolute superiority' to others. Hitler spoke of the need for self-confidence and national pride to be inculcated in German youth: 'The curriculum must be systematically built up . . . so that when a young man leaves school he is not a half-pacifist, democrat or something else, but a whole German.'\n\nHitler's hatred of intellectualism is clear from his statement: 'Put young men in the army, whence they will return refreshed and cleansed of eight years of scholastic slime'. Hitler's contempt for schoolteachers was equally great; he claimed: 'I cannot endure schoolmasters'. He dismissed his own schoolteachers with much disdain, describing his foreign languages teacher as 'a congenital idiot' and asserting that he 'could not bear the sight of him'. He continued: 'our teachers were absolute tyrants. They had no sympathy with youth; their one object was to stuff our brains and to turn us into erudite apes like themselves'. Hitler claimed: 'When I recall my masters at school, I realise that half of them were abnormal; and the greater the distance from which I look back on them, the stronger is my conviction that I am quite right.' It is unsurprising, therefore, that he sought to produce a different breed of teachers in the Third Reich and to train them in accordance with his own educational imperatives for German society. The organization entrusted with this task was the _Nationalsozialistischer Lehrerbund_ (NSLB) or National Socialist Teachers' League.\n\n**THE NSLB**\n\nThe NSLB was established on 21 April 1929. It actively recruited new members in the years before the Nazi _Machtergreifung_. The majority of the NSLB membership at this time was made up of young radicalized teachers, aged between 20 and 40, who felt estranged from the associational life of the teaching profession and disillusioned with the Weimar Republic. Lamberti notes that 'enthusiasm for National Socialism among the students training for the profession in the early 1930s was especially striking'. Nazism's appeal was apparent among low-salaried assistant teachers without permanent positions. However, the NSLB did attract a number of older teachers as well. Approximately one-third of its members recruited before 1933 had entered the teaching profession during the _Kaiserreich_. Whilst the NSLB capitalized on the low morale within the teaching profession at this time, particularly relating to the issue of salaries, its main propaganda themes were 'cultural politics and national pride rather than material interests'. The NSLB promised a change in the image of teachers towards 'a new and more positive perception of themselves as forward-looking activists serving big national goals'.\n\nHans Schemm, the leader of the NSLB, was one of the old guard of the NSDAP, whose ideas were conservative and _v\u00f6lkisch_. Born on 18 October 1891 in Bayreuth, the son of a manual worker, Schemm was fascinated by German cultural traditions and was deeply influenced by the thinking of Johann Fichte, Richard Wagner, Friedrich Nietzsche and Houston Stewart Chamberlain, whom he often quoted in his speeches. In a speech to educators in Bremen on 15 December 1933, Schemm stated: 'in our schools, we must build, mould and educate. Nothing foreign, nothing external, as Fichte said, shall stop us in this task'.\n\nThe statutes of the NSLB laid down its duties. The first and most significant was 'education of its members as exemplary National Socialists' so that they could be equipped to carry out their special tasks and obligations inside the 'national community'. The NSLB was to support the national leadership of the NSDAP and its chief educational office. It was to supply proposals and guidelines regarding all questions of education and pedagogy. Its duties further comprised control and surveillance of all German published texts, especially those published for a youth audience, and participation in Party youth and welfare organizations. The NSLB was to provide facilities for further ideological and professional training. The structure of the NSLB was organized along the same lines as the NSDAP, with divisions into _Gau_ , _Kreis_ and _Ort_ groups. Before 1933 the NSLB prided itself upon its struggle against parents' councils and other teachers' organizations.\n\nAfter the Nazi _Machtergreifung_ , the membership of the NSLB grew rapidly to 12,000 in March 1933. Many of the teachers that flocked to join after that were largely motivated by opportunism rather than ideological conviction. They joined because they did not wish to lose their positions or because they saw membership as a way of progressing their careers, once it became clear that _Gleichschaltung_ (streamlining) was an inevitable part of the Nazi scheme. As a result, Schemm transformed the multitude of teachers' organizations and the mass of unorganized teachers into one community and integrated them into the NSLB. There were early calls to eliminate 'Marxists and freemasons' from educational posts. Nazi pedagogues believed that 'only a strict clampdown against the red enemies' of National Socialism could make a difference to the future of German education. Jewish and 'unreliable' teachers were purged from the profession within a few months of Hitler's accession to power. By 1936, there was also a ban on 'double membership' of the NSLB and a confessional teachers' association. By 1937, the NSLB comprised 320,000 teachers (97 per cent of all teachers). Hence, the NSLB played a significant role in the regime's initial process of _Gleichschaltung_ to homogenize the teaching profession.\n\nThe NSLB had two main functions. The first was to provide reports on the political reliability of teachers for appointments and promotions. The second task of the NSLB was to ensure the ideological indoctrination of teachers. The organization saw its purpose as the creation of 'the new German educator in the spirit of National Socialism'. It ran courses for teachers and set up special teacher training camps for this purpose. Teachers were trained in racial knowledge, for example about the supremacy of the 'Nordic race' and the 'pollution' of its purity by 'racial miscegenation'. They were taught about the characteristics of the 'Jewish race'. They were instructed about genetics and hereditary health, as well as the concepts of 'blood and soil', 'living space' and the requirement for German expansion eastwards. Lectures on German prehistory, history and racial ancestry formed part of NSLB teacher training. 'The National Socialist teacher will only be able to live up to the National Socialist future if the idea of National Socialism burns within him'.\n\nIn addition, NSLB 'exchange camps' focused training on the 'border zone' issue. During the school holidays, teachers were sent to particular areas, such as Silesia or Saxony, in order to take a two-week-long course on the history and racial history of the region. The teachers had to pay for the accommodation and travel to the training camps, although there were reduced rail fares for this purpose. If they could not afford the cost, there was a possibility of some transfer of funds to enable them to attend the training camps. Such camps were designed to bring teachers from different regions together in the spirit of National Socialism. For example, NSLB teacher training in the district of Silesia was intended to strengthen the 'border-consciousness' of Silesian teachers by fostering and nurturing the connection to the homeland and the love of the German nation. It aimed to emphasize 'Silesia's location within the pan-German east', covering themes such as the Silesian man and his work in the past and the present and the belonging of Silesians to the German nation. Tasks and training materials presented Silesia as part of the collective land of the German people. Furthermore, in the summer term, there were 'homeland hikes', and in the winter term, workshops and general meetings on specific questions. Teachers were to get to know their homeland under expert guidance through hiking. They used geological maps to introduce the geological history of the area and to consider its contemporary natural landscape. They learned about the history of towns and villages on their hikes. In study groups and workshops they examined the whole of the Silesian region, its family names and demographic circumstances. Once teachers were trained in these subject areas, they could teach them in their schools. They could teach history, geography and biology with reference to the specific training they had acquired about Silesia on these courses.\n\nA report on an NSLB training camp in Silesia in 1936 stated that the training aimed to 'adjust all the teachers to the common foundation of the National Socialist world view'. At the camps, participants worked through _Mein Kampf_ , even if they were already familiar with it, in order to understand its essential points and eliminate any misunderstandings. Group work was used to correct, complete, deepen and encourage an understanding of _Mein Kampf_. The report stated that 'the educators approached the work with eagerness and interest and have attempted to penetrate the spirit of National Socialism in lively conversations'. The camp-supervisor explained the tenets of Nazi ideology and created a sense of community in which 'we' was more important than 'I'. A special emphasis was placed on the 'borderland' issue. Teachers were instilled with the sense that they played an important part in the future of Germany. 'They recognize the overwhelming greatness of the deeds of the F\u00fchrer and his followers, the enormous work that has been done and still needs to be done. They have to realize that everyone, in particular the German teacher, has a long way to go before work is completed'. The report admitted that 'not everyone who has taken part in the camp left as convinced National Socialists', but 'of the large majority of educators it can be said that they wholeheartedly support the F\u00fchrer and are prepared to undertake additional work for the movement and the _Volk_ '. Apart from the 'ideological training work', the camp activities also included compulsory physical training for teachers under the age of 50, an early morning run, gymnastics, flag-raising, recreational time, visits to places of local interest, hikes and marches, visits by Party officials and speeches.\n\nBy 1939, some two-thirds of the teaching profession had attended NSLB camps, whose fundamental objective was to imbue their participants with the Nazi _Weltanschauung_. The aim was to develop a 'way of life which was completely opposed to the liberal teacher conferences and congresses' of the Weimar period. The camps were designed to create a sense of unity and homogeneity among teachers and to remove barriers between them, in particular in relation to status. Participants in these training camps were constantly monitored. The observers and camp-trainers kept personal files on all participants, containing information about their characters, in order to be able to select 'the best National Socialists' from their ranks. These files were used as the basis upon which promotions and requests for school changes were decided. The NSLB called for an exemplary way of life for educators and teachers in the Nazi state.\n\nA participant in one of the ten-day NSLB teacher training camps described his experiences. He talked of the spirit of camaraderie among the camp-participants: 'a communal life began for all of us, in this community . . . in order to find what was to become an adventure for all of us'. He stated that the comradeship and general mood in the camp was 'exemplary'. The camp-participants comprised one camp-leader, four group leaders and sixty-five course participants, aged between 23 and 54. The purpose of the camp was 'to turn educators of different occupational classes and different age groups into one combat community\u2013a community in which all dividing walls are torn down by collective existence and experience'. He described waking up every morning 'cheerfully'. The camp began each day with sports. He recounted how 'a true fighting spirit overcame all who have come to make their body younger not just mentally but also physically'. After washing, tidying up and making beds ('the making of the bed, which at home is done by the beloved wife, had to be done by one's own hand here'), the flag-raising and day's tasks began. Cross-country running was an integral part of the training as it created physical agility and ability, toughness, strength of will, determination and discipline. Such characteristics would endow participants with a readiness to serve and make sacrifices for the nation and educate them to be 'National Socialist fighters'. The participant told of another attendee, who had been disabled in the war and so was unable to do all the exercises; the camp-leader's response was that 'what is important here is not that one or the other exercise is completed successfully but that everyone gives their best and experiences the spirit in this camp, takes it home and carries it on'. The participant stated that 'not being overfed with lectures was important', yet he listed nine lectures, an average of one per day, including the themes of 'The German Space and Defence Problem', 'Racial Fundaments of the Jewish Question', 'Race and Culture', 'The National Socialist Perception of History' and 'Freemasonry and Jewry'. This last lecture in particular made an impression on him. It 'opened the eyes of all participants in a deep and thorough way to how the German people before, during and after the war . . . was lied to and betrayed', until Hitler 'freed the German people from a scourge of mankind'. He left the camp steeped in National Socialist ideology, with a true belief in the _F\u00fchrer_ and a determination to carry out his part in the national task of educating youth. It is difficult to know how typical his enthusiasm was, but certainly his record is a useful first-hand account of participation in a camp, which correlates quite closely with the professed intentions of the NSLB in its aims and claims.\n\nAn activity report for a two-week camp for female teachers aged 23 to 55 from different school backgrounds also described how 'differences were overcome with the happy and sincere comradeship which prevailed'. This camp included one or two daily ideological lectures, with ceremonies, music and song to forge community spirit, as well as visits and a variety of physical training. Camaraderie was also noted in a report from a training camp for female teachers in Parchim: 'From the beginning a comradely feeling of belonging together prevailed, since we knew that all of us were the bearers of one great idea and wanted to help our F\u00fchrer to realize the great idea of the national community'. Political and ideological training as well as physical education were the main aspects of the camp. The reporter comments that 'the training camp was a great success and has brought awareness of the community deep into our hearts'.\n\nThe aim of the NSLB's teacher training was to transform all German teachers into 'National Socialist _Volkserzieher_ ' (people's educators). The ideological training was intended to transform those participating 'totally and in every aspect of their lives'. Ideological training was central to the work of the NSLB. The NSLB's Department of Press and Propaganda played an important role in this regard. It had to eliminate any liberal, ideological hangover from the Weimar era and ensure that all teachers stood together 'in National Socialist solidarity'. Furthermore, by means of propaganda, 'enemies of National Socialism' were identified and challenged. Propaganda outlined the struggle between National Socialist, Nordic and Judaeo-Christian values. In this struggle, 'National Socialism cannot give up its claim of totality unless it wants to bury itself '. The regime noted that participation in the teacher training camps needed to be encouraged through rallies and propaganda at the regional and local levels. 'The camps have to be places which mould fighters, prepared to show complete commitment to the National Socialist world view.' NSLB members who demonstrated any deviance from this were excluded from acting as speakers at these camps.\n\nThe fifty-seven rural teachers who took part in a camp at Herberg from 21 July to 27 July 1935 were expected to play their part in the 'correct education of the German people to a true national community'. 'The teachers are a combat unit for the realization of this aim; the training camp is their training school.' One participant explained his joy upon receiving the letter inviting him to the camp: 'Probably no one sensed what a unique community these strangers, who came from all parts of our fatherland, would become during the one-week camp'. The lectures gave participants an understanding of their homeland. The main success of 'the perfect camp-activities' was that 'all of us would hold each other together'. Other participants commented on this 'comradeship'. A great sense of camaraderie was achieved, underpinned by the ideology of 'blood and soil' and sense of community. This was considered to be especially important for teachers in rural areas, as they were isolated in their villages and so this type of camp gave them a sense of their common purpose and an understanding of what their part was in the building of Hitler's 'national community'.\n\nHowever, not all teachers had such a favourable view of the teacher training camps. The following two examples illustrate this. In the first, participants were offended by the anti-Christian sentiments of the camp-leader. In the second, inadequate organization was identified as a problem.\n\nIn December 1934, German evangelical female teachers wrote to the Minister of Education, Bernhard Rust, about the camp they had attended in Kettwig. They took exception to the trainer, Mr Friedrich, who was anti-Christian in his sentiments. Friedrich told them that 'participants believing in the doctrine of Christian theology in the Old and New Testament had to accept the blasphemy of their Christian belief '. He allowed for no discussion or opportunity for appeal against these attacks on Christian belief. Although they claimed their general allegiance to Nazism, they appealed to Rust 'to protect Christian teachers against such affronts'. An investigation into the complaint, by school inspector Huhnh\u00e4user, suggested that Friedrich indeed gave a lecture at this training camp, without the inspector's knowledge. The report suggested that 'Friedrich must have talked ruggedly, especially in terms of the church, so that a number of Catholic female teachers had to leave crying'. Rust's own response to the teachers was that 'similar situations will not occur again'. He circulated an order to his subordinates that 'a recurrence of such tactlessness' should be avoided.\n\nThe second example of a problematic experience was the tent camp of the NSLB at Heringsdorf in Gau Pomerania in June\u2013July 1935. This was described by one participant as an unmitigated 'disaster', with 'inadequate organization' and much confusion. The report on the Heringsdorf camp described the untidiness, problems with distribution of food and drinks, and unhygienic latrine sites. It stated that many participants behaved badly and got drunk. The speeches and evening party were described as 'enjoyable' despite the unfavourable circumstances in the camp. The official response to this report was one of outrage, although it was impossible to deny all the problems that were raised. The idea of the tent camp was designed to 'pull educators out of old habits and to wake them up'. Despite the problems, the NSLB considered it to be 'a great event for the majority of Pomeranian educators'.\n\nThe content of the schedule for a teacher training camp from 5 to 13 August 1935 was rather typical of NSLB training camps as a whole. Lectures were given on key aspects of Nazi ideology and policy, entitled 'People and State in the Third Reich', 'The Tasks of the NSV in Germany', 'The Woman in the National Socialist State', 'The Goal and Path of the DAF', 'The German Peasantry as the Bearer of the German Population', 'National Socialist Foreign Policy', 'Race: A Central Concept of the National Socialist World View' and 'Education to the National Community'. Such topics were standard fare among teacher training schedules. Furthermore, there was special training for the teaching of biology under National Socialism, for example an NSLB training course on this subject was held in T\u00fcbingen from 11 to 17 October 1936. This covered a variety of topics including 'The Reproduction Battle', medicinal plants, 'Ancestral Biology and Breeding Lore' and 'Biology and the World View', as well as practical exercises and visits. Additionally, there were training courses for 'racial lore', covering topics such as 'Race and Space', 'Race and Language', 'Race and Fairy Tales', 'Race and Art' and 'Race and Musical Education'. This type of 'racial political work' was energetically promoted by the NSLB.\n\nClearly, the role of the camp-leader was very significant in these camps, and leaders were chosen with care. Camp-leaders had to have 'a strong personality' and to act in an exemplary manner. The camp-leader had to rally the participants 'to nurture comradeship and forget all class differences'. Through the experience of the camp, the leader was to create 'an inseparable team and common destiny that lasts much longer than the days in the training camp'. Their influence would enable teachers to go back to their schools ready to imbue their pupils with the spirit of National Socialism.\n\nSchemm called upon teachers to 'stand in front of their boys with a German soul, transparent like glass, keeping no secret from them . . . The boys shall look to their teachers as leaders and comrades'. He passionately maintained: 'the highest ideal for the German teacher must be his awareness of his fortune to be in this position. This joy and fulfilment must come over him everyday on entering the school.'\n\nFritz Waechtler took over the leadership of the NSLB, after Schemm was killed in an accident on 5 March 1935. He was not an engaging successor to Schemm, and the fortunes of the NSLB went into decline. Although it still regarded its role as being at the forefront of the ideological training and indoctrination of all teachers, by this time the NSLB was expending a considerable amount of effort in justifying its continued existence and underlining its own importance. It appears that it was not succeeding in this aim as effectively as it had hoped. It defended its position, stating that the years 1935 and 1936 were those of 'small works and individual struggle', as an excuse for its lack of public visibility at that time. In its publications and its actions, the NSLB glamorized its leaders at every level in keeping with the National Socialist 'leadership principle'. Its members produced propaganda material that justified the organization's role in the National Socialist state, highlighting its struggles alongside those of the Party as a whole. From 1938, as the immediate significance of the ideological training of teachers had diminished by this time and state funds were directed towards other more pressing concerns, the NSLB was attempting to defend its role more than ever. A significant aspect of the NSLB's work at this time was its appeal to its members to help with harvesting. Waechtler emphasized the shortage of agricultural workers on the land and the urgent need for assistance with harvesting. He urged NSLB members to report for harvesting duties for a few weeks during their school holidays.\n\nDuring the war, the NSLB worked harder than ever to justify its position. In 1942, it produced a painstakingly detailed report on all its activities, in particular attempting to highlight its 'important' contribution to wartime measures. The NSLB suggested that its wide-ranging tasks had 'in no way decreased' in importance since the beginning of the war, but on the contrary had become increasingly significant. In addition to its previous tasks, the NSLB engaged itself in an array of wartime measures, under the slogan 'All for Victory!', including the 'collection of healing herbs', the 'collection of scrap materials' and the foraging for 'food from the woods' by pupils and school classes. All pupils were directed to keep war diaries. These types of activities were popular among teachers. In this regard, one of the most dangerous strengths of Nazism was its ability to exploit of apparently innocuous activities and popular sentiments for its sinister aims. Furthermore, under the slogan 'Pupils help Pupils', the NSLB encouraged German schoolchildren and teachers to make 'voluntary donations of books, teaching materials and illustrative material for the construction of a German school system in the East'. The NSLB was involved in the evacuation of children from the cities to the countryside ( _Kinderlandverschickung_ \u2013KLV). The NSLB maintained that it still had a crucial role to play during 'total war', particularly in terms of upholding a cohesive and calm attitude on the home front: 'There is no doubt that the German teacher with his influence over the German youth significantly influences the mood of the German people'. Hence, 'particularly during critical times', the attitude of the teacher was significant in influencing the nation more broadly. The NSLB maintained that it still had duties and responsibilities that were 'decisive for the outcome of the war'. Despite its protestations, the NSLB was closed down in 1943. Having examined the specific role of the NSLB, the next section considers the making of educational policy more broadly.\n\n**NAZI EDUCATIONAL POLICY-MAKINg**\n\nThe broad educational aim of the Nazi state was 'to ensure that a rundown, morally contaminated public . . . robbed of its ethical principles' was 'again made a community of people . . . aware of their inner values, their skills, their duties, their being!' 'At the hour of the new state's birth, a new class was also born: the class of people's educators.' 'People's educators' were more than simply 'teachers'. They formed 'an indispensable pillar of the state'. They were entrusted with the task of producing an unbreakable 'national community', without class, denominational, educational or regional distinctions. This was the educational goal of the new state and teachers were to set an example to the nation by creating their own microcosm of this community throughout the teaching profession.\n\nBernhard Rust was a former schoolteacher. He had been an early member of the NSDAP, joining in 1922, and had become _Gauleiter_ of Southern Hanover-Brunswick in 1925. Rust was appointed Prussian Minister for Education on 4 February 1933 and Reich Minister for Education and Science on 30 April 1934. In August 1937, Rust's Ministry established centralized control over the appointment of all teachers. In 1939, Rust set up a Reich Examination Office to deal with all educational examinations.\n\nHowever, Rust was engaged in a constant struggle to keep control over his sphere of influence. As was the case in so many other areas of policy-making, with no clear policy guidelines from Hitler, different individuals and agencies tried to take the initiative. Rust attempted to prevent incursions into his remit from Josef Goebbels's Ministry of Propaganda, among other competing agencies. In particular, the Ministry of Education found itself in an increasingly defensive position as several branches of the NSDAP tried to extend their influence into school organization and education after 1933. The two main reasons for conflict from the viewpoint of the Ministry of Education were the desire of the Party to lower academic standards in secondary schools and the incursions into schools by the HJ and other Party organizations. Internal quarrels took a considerable amount of time to resolve and they seriously obstructed the work of the Ministry of Education. Rust encountered intervention and challenges to his authority from a number of Nazi leaders, notably Baldur von Schirach, Martin Bormann, Robert Ley, Alfred Rosenberg, Philip Bouhler and Heinrich Himmler. Even civil servants from his own Ministry flouted his authority. Furthermore, the Head of the Party Chancellery read every significant decree by the Ministry of Education before it was issued. This process meant that the Party's standpoint was always included and it also slowed down the work of the Ministry and countered its effectiveness. There was also conflict between the Ministry of Education and the HJ leadership. For example, the HJ leadership proposed to reward those pupils who actively participated in the HJ with good grades. This was one of the most significant and ongoing conflicts between the Ministry of Education and the HJ leadership throughout the Nazi era. In practice, the Ministry of Education had only marginal influence on decision-making on the issue of youth participation in the war effort, although, theoretically, it was entitled to coordinate these efforts. In this sphere, Rust came up against the influence of the HJ leadership. In addition, Rust's position came under continual threat from the Party as a whole.\n\nParty leaders despised the traditional educational system, viewing it as a relic of the earlier times that Nazism had struggled to overcome. They regarded a long-term school policy as undesirable and believed an ad hoc approach to be more desirable. Their chief concerns lay with issues such as the Four Year Plan, rearmament and the preparations for war. It was against these attitudes that Rust had to struggle. Not surprisingly, the Party favoured educational institutions of its own\u2013the HJ, the Labour Service, the NSV and the Party Schools. The Party's own institutions were trusted and respected by Party branches and leaders. Until the end, the Party's own schools, for example, received financial support and had access to the full range of the Party's propaganda apparatus, in a way that the state's institutions under the Ministry of Education did not. Whilst the NSLB as the professional teachers' organization and the official Party organization for school education might have served the Ministry of Education by steering the NSDAP onto a more favourable course with regard to schooling, it was unable to do this, mainly due to the lack of an enthusiastic and able leadership. Hence, the policies of the NSLB undermined the position of the Ministry of Education even further. Hitler was not impressed with Rust's character and achievements. He stated on 29 August 1942: 'we have made progress in the field of education, in spite of having a pedant at the head of the Educational Department. With another in control, progress would have been more rapid'.\n\nThe influence on National Socialist ideology of thinkers such as Heinrich von Treitschke, Oswald Spengler, Julius Langbehn and Paul de Lagarde was significant. Moreover, a clear rejection of the Enlightenment and rationalism influenced cultural life and filtered into the work of Nazi pedagogues, which instead incorporated anti-liberalism, fanatical nationalism and racism. In May 1933, Wilhelm Frick, the Minister of the Interior, attacked liberal values in education, stating that 'the individualistic concept of education has been the main contributor to the destruction of national life within society and state and above all in its unrestrained application in the post-war era has shown its total inadequacy as a guiding principle for German education'. In order to reverse this trend of Weimar education, the aim of Nazi education was to underpin the rebuilding of national life based upon National Socialist principles. Schnurr has conceptualized a change from the Weimar 'welfare state' into the National Socialist 'training state'. The Nazi 'pedagogization of all areas of life' was an attempt at complete social control. The pedagogue Eduard Spranger embraced the 'events of 1933' and the importance of 'a sense of the nobility of blood and of the bond of blood' as well as the need for 'a conscious cultivation of the health of the people'. The aim of schools in the Third Reich was to train and educate the politically aware young German pupil, who 'in all thoughts and actions is rooted in the service to and sacrifice for his _Volk_ and whose history and destiny is completely and inseparably bound to that of his state'. Under National Socialism, 'the Humboldtian concept of education was criticised for its individualism and its emphasis on intellectual aspirations, which were perceived as factors weakening the _v\u00f6lkisch_ community spirit'.\n\n**CHANGES TO THE EDUCATIONAL SYSTEM**\n\nKINDERGARTENS\n\nBefore examining the changes made by the National Socialist state to schools, it is important to briefly consider the impact of the Nazi regime upon kindergartens. The kindergarten movement, which cared for preschool children between the ages of 3 and 6, had grown since the mid-nineteenth century so that working mothers were able to leave their children in a safe environment whilst they were at work. Friedrich Froebel (1782\u20131852), an eminent German educationalist and founder of the _Kindergarten_ , considered infancy to be the most important period for education. He believed that children could grasp the concepts of harmony, unity and order at a very young age and that they benefited from a sense of community with fellow pupils as well as their family members. By the Weimar era, kindergartens had become part of the wider, nascent welfare programme that was designed to lower infant mortality, to increase the birth rate and to provide recuperation for mothers and household help for pregnant and _kinderreich_ mothers. Despite considerable debate about the issue of making public kindergartens freely available to all mothers who wanted them, during the Weimar years there was no large-scale expansion of kindergarten provision. Many working mothers were still in the position of having to find a relative, neighbour or friend to look after their children. Those who called for an expansion of kindergarten provision argued that this was important for the 'recovery and reconstruction' of the life of the nation. Kindergarten teachers believed that putting children in protected, supervised environments would ameliorate their conditions, in particular their physical health and their safety. In the kindergartens, they would help to raise 'a new generation' of German citizens. Acknowledging the advantages of the kindergartens, a number of private companies and organizations, such as Siemens, ran their own. Apart from these, many municipal authorities, as well as the Protestant and Catholic Churches, ran kindergartens. The Weimar government established guidelines, sometimes providing financial assistance. The state required that doctors visited regularly to check up on the health of the children. Such moves were part of the Weimar concern to improve the welfare of mothers and children. Kindergarten teachers visited children at home to meet their parents and check their home environments. Government officials were pleased that the kindergartens benefited the children of working mothers and offered the possibility of checking up regularly upon children's health, and many mothers were pleased to have the advice and support of kindergarten teachers in bringing up their children. However, tensions did sometimes occur, in cases in which mothers felt there was too much observation and intervention on the part of teachers and the state.\n\nUnder National Socialism, the nature of kindergarten education changed. The Nazi government utilized kindergartens as a space in which it could further its own aims whilst helping to alleviate mothers' burdens, in line with its rhetoric. Of course, the position of kindergartens under National Socialism was inherently contradictory, as the regime's ideology called for women to be stay-at-home mothers. Pragmatically, however, the Nazis could see the benefits of the kindergartens, both as a means of enabling mothers to work when the state required them to, and as an opportunity to raise young children in the spirit of National Socialism. The Nazi state introduced standardized guidelines for kindergartens. They were to be uniform in appearance and organization. A picture of Hitler was to be hung in a prominent position, the swastika flag was to be raised and the Hitler greeting was to be used. Children had to be 'racially pure' and had to undergo a medical examination and to present a certificate of health before attending kindergartens. Compliance with these norms was expected of all kindergartens, including those affiliated to the Catholic Church. However, as Mouton notes, 'the degree to which the Nazis succeeded in imposing uniform standards on kindergartens varied according to local conditions, party leaders, kindergarten teachers, and the local population's acceptance of the changes'. Those kindergartens that worked in the Froebel tradition or that were affiliated to the Churches were less easy for the regime to homogenize.\n\nRather than taking over church kindergartens, the Nazi state allowed them to continue to exist, but placed increasingly stringent guidelines and financial restrictions upon them, in particular by reducing government subsidies, in order to encourage them to fall into line with state policy. The Nazi state also opened its own kindergartens under the aegis of the _NS-Volkswohlfahrt_ (NSV) or National Socialist People's Welfare organization, to compete with the church-run kindergartens, although many parents still chose to keep using the latter. Nevertheless, by 1941, many Protestant and Catholic kindergartens had 'coordinated' themselves with the NSV.\n\nThe Nazi regime also used kindergartens for political expediency. For example, they set up NSV kindergartens in areas in which they felt the population was 'distant from National Socialism' and in rural areas, where previously none had been available, hence creating a sense of benevolence among the people and gratitude to National Socialism for making this provision (see below for more on so-called 'harvest kindergartens'). The National Socialist regime used kindergartens as a means of intervening in families and imbuing young children with Nazi ideals. Furthermore, all kindergarten teachers had to pass a state examination which tested their knowledge of and commitment to National Socialist ideology, in particular with regard to racial purity. Kindergarten teachers were also expected to acknowledge the need for different socialization for girls and boys.\n\nThe NSV kindergartens were clean, bright, spacious and airy, creating a 'healthy environment' for the children. Every day, on arrival, the children washed and cleaned their teeth. They were then separated into different age groups and supervised by nurses and welfare workers as they played, exercised, ate, sang and slept. The 'Guidelines for Day Nurseries' in 1936 set out the following among its tasks: to sponsor the physical, mental and spiritual development of the children, to educate them in National Socialism and service to the 'national community', and to instil a sense of care for the German nation and morality. Hence, the kindergartens clearly socialized preschool children in the spirit of National Socialism. NSV kindergartens were considered to be 'essential bases . . . for the education of young German people'. The number of NSV kindergartens grew from approximately 1,000 in 1935 to 15,000 in 1941.\n\nFurthermore, the Nazi regime established 'harvest kindergartens' in rural areas in order to free agricultural women from their family responsibilities during the day so that they could carry out their harvesting. The 'harvest kindergartens' were regarded as necessary due to the lack of available, satisfactory supervision for children during harvest time. Care for children during the harvest period provided by the most elderly and frail villagers was considered to be inadequate and unsuitable. 'Harvest kindergartens' were first set up in the summer of 1934 to supervise children in rural areas from the age of 2 upwards. They consisted of one or two rooms, simply furnished with tables, benches and chairs, wash basins and a play area outside\u2013either a garden or a sandpit. The kindergartens provided pillows, blankets, toothbrushes and hand towels for the children, but plates, beakers and spoons had to be provided by the parents. Milk was supplied by local farmers. The harvest kindergartens were run by trained kindergarten workers, with the assistance of older schoolgirls and BDM girls, provided that they were not needed for harvest work. The children were medically examined and a health questionnaire was filled out for each child. Oral hygiene and general health were regularly monitored. Children with lice or any infectious diseases were not allowed to attend the kindergartens until they were better. The number of 'harvest kindergartens' rose from 600 in 1934 to 8,700 in 1941 and to 11,000 in 1943. Their duties included the following: to promote the physical, mental and spiritual development of the children, to educate them in the ideas of National Socialism and to maintain contact with the parental home. Hence, parents' evenings were introduced. In the harvest kindergartens, educational work corresponded with the reality of agricultural life. Children were taught to be 'productive'. The significance of the ideological concept of 'blood and soil' was promoted. The Nazis' thoroughgoing socialization of young children thus began in the kindergartens and harvest kindergartens, even before children reached school-going age.\n\nSCHOOLS\n\nAt first, the Nazi government initiated a number of ad hoc, yet significant changes to the German school system. A report on the school system in Hamburg suggested changes to the education system and its reorganization to meet the aims of the National Socialist state. It stated that during the Weimar era, the school system in Hamburg had come heavily 'under the influence of the Marxist spirit' and it outlined the setbacks associated with changes undertaken during the period 1918\u201333. As we have seen, the National Socialist state saw questions of schooling and education as an integral part of the whole organic state and 'national community'. It was the school's duty to communicate German culture to its pupils and to develop them as 'German human beings'. Furthermore, the National Socialist state aimed to prevent the 'Marxist' way of thinking from permeating the countryside. The Nazi state believed that families in rural areas had to show commitment and make sacrifices in regard to their children's education, not to leave it entirely to the state. The rural schools were to be state schools and community schools at the same time. In an organic state, the rural school was linked to both the state and the community. Indeed, the Nazi regime attached such significance to rural schools in their relationship to German 'blood and soil' that it established a number of model Party elementary schools in rural areas: the 'Hans-Schemm-Schools'.\n\nIn an organic state, each part was obliged to evolve to its own maximum perfection and play its part in the whole. The education system was an integral part of this entity and had a duty to serve the state to achieve its maximum perfection. It was the obligation of the educational system, under guidance from the state, 'to choose and judge and harvest from the produce and goods of culture: to favour what complies with the authoritative values and goals and to suppress what is perturbing and perverse for education and culture and what could be poisonous for the public body'. Moreover, it was the duty of the educational system under National Socialism to prepare, implement and present these goods in order to develop a positive educative force. This included the production of teaching materials and educational plans. There was a further demand for those in the upper echelons of education, in particular school inspectors, to undergo a thorough selection process for the highest positions within this area. In this way, 'the organ of education can flourish to its maximum potential and the basic questions of the entire educational system can be worked on more thoroughly than ever'. There was a sense that existing school inspectors did not have the correct skills, training and attitude to accomplish this. School inspectors were to be appointed on the basis of character and merit. They were not to be burdened with administrative duties that prevented them from carrying out their most salient tasks.\n\nBetween 1933 and 1937, the Nazi government was concerned with consolidating its power and imbuing the German population with its ideology. In terms of educational policy, this entailed a number of initiatives. In April 1933, all teachers were given civil service status. The Law for the Restoration of the Professional Civil Service (7 April 1933), with its 'Aryan' clause, provided for the legal removal of Jewish and 'undesirable' teachers from the profession. A law of January 1934 removed the autonomy of the _L\u00e4nder_ (states) in order to achieve centralized state control over education. The collegiate system among teachers was abolished, and the introduction of the 'leadership principle' in schools meant that all powers in schools accrued to the headteachers. There was a call for the elimination of self-administration practiced by the teachers, the parents' council, the headmaster and the school inspectors in the Weimar Republic. Self-administration was based upon the principle of equality. Nazism considered that this approach lacked the necessary leadership over the entire educational work of a school: 'There cannot be any space for this kind of formal-democratic self-administration in the National Socialist state.' Instead the primacy of the headmaster's authority in a school was to be reinstated. The Nazis rejected the democratic nature of the parents' councils in schools and these were eliminated in 1934. They introduced instead the School Community, consisting of parents, teachers and an HJ representative. This new system gave the impression of continuity with the previous arrangements, but did not interfere with the headmaster's role as 'leader'.\n\nPlans for reforms towards creating the school of the German 'national community' were drafted by Nazi educationalist Weischedel in May 1934. Weischedel had already been engaged with the pedagogic literature on National Socialism and had worked as a teacher and headteacher. National Socialist school reform was 'not about the correction of some detrimental elements' but the arrival at 'a meaningful overall solution'. Weischedel rejected any quick reforms and claimed that organic change was needed: 'National Socialist school reform is a lengthy, continuing process of transformation'. It had to be carried out taking into account the pupils, the family and the transformation of the state and its culture. The Nazi educational programme had to comprise clear guiding principles which were in line with the fundamental ideas of National Socialism. The whole educational system was to undertake a uniform task permeated by the spirit of National Socialism.\n\nIn contrast to these ad hoc adjustments, the most sustained efforts and significant changes in education under National Socialism were carried out between 1938 and 1942. This was the period in which the Nazi regime was at its peak. By 1938, it was ready to make more notable steps in educational reorganization and during the first years of the war, before it became bogged down in its battle against the Soviet Union, the Nazi regime continued to put a significant amount of effort and resources into the 'education' of the _Volk_. After 1942, the demands of the Nazi war effort and the focus on the execution of the 'Final Solution' became so all-encompassing that other aspects of policy, educational policy among them, were overshadowed.\n\nIn particular, the National Socialists wanted to pare down the number of different types of schools that they had inherited from the Weimar era and to separate education for boys and girls. The secondary school system was reorganized and secondary education was shortened by one year. The aim of secondary education was to educate 'the German man in all his strengths' and to prepare him for university and for practical life. The _Aufbauschule_ (feeder school), initiated in the Weimar Republic, was expanded, giving children from rural areas access to secondary education. These schools fed into the _Oberschule_ , giving pupils the opportunity to join the last stage of _Oberschule_ education. The Nazi government reduced the number of existing secondary schools to the _Gymnasium_ and two types of _Oberschulen_ , in which boys and girls were educated separately. The last three years in the _Oberschulen_ were divided into a science\/ mathematics stream or a modern language stream in the boys' schools. School leavers with modern language qualifications also found it hard to gain access to university, as they had not studied Latin, which was an entry requirement for many university courses. The girls' schools offered a choice of domestic science or modern languages. The course in domestic science, popularly termed the 'pudding matriculation', did not qualify girls for university entrance. The 'special task' of the girls' schools was to prepare their pupils for the specific requirements of being 'a German woman and mother in family, home, workplace and national community'. Certainly, this accorded with Hitler's view that 'The goal of female education must invariably be the future mother'.\n\nGirls were harshly discriminated against in the Nazi education system, as entry to the _Gymnasium_ was for male pupils only and it was the only place where classics could be studied. Such policies led to a severe reduction in the number of women in higher education. In 1933, the Nazi government placed a cap on the number of female students who could be enrolled in German universities, setting the maximum at 10 per cent. By 1939, only 6,342 women were registered at German universities. During the war, the Nazis overhauled this policy, partly because young men were conscripted into military service and so university places were freed up. In 1942, approximately 42,000 women were enrolled at university, making up 64 per cent of the student population. Practical considerations had prompted a change in policy away from the reactionary Nazi ideology, which held that women should be discouraged from entering higher education.\n\nThe Nazi regime, with its claims of creating a classless, ideologically comprehensive educational system, placed increasing restrictions on private schools. The Weimar constitution had permitted the existence of private elementary schools run by the Churches, as well as private schools for physically disabled children. The National Socialist government took steps to eliminate private schools altogether, mainly aiming to eradicate separate denominational schools, which stood in the way of building the 'national community'. The _Reichsschulpflichtgesetz_ (National School Law) abolished private preparatory schools. It made attendance at the state _Volksschule_ (elementary school) compulsory, with exceptional cases made for children with mental or physical disabilities. The aim of the _Volksschule_ was to provide education for all German children who belonged to the 'national community', regardless of class or denomination. The _Volksschule_ carried out its educational tasks based on the strength of German _Volkstum_ (national traditions). The aim was to homogenize the composition of German elementary schools. There was considerable concern among Nazi educational leaders that Protestant and Catholic children had been separated in denominational schools. In the Nazi state 'such a division\u2013separation into different schools according to religious belief\u2013cannot continue . . . Children should be together in order to understand and appreciate the further unity of the community, our _Volk_ '. Not surprisingly, these steps led to concerns on the part of the Churches. Rust received letters of petition from Church representatives against these measures. For example, correspondence to Rust expressed 'great concern about the creation of non-denominational schools'. These appeals to preserve the denominational schools met with no success.\n\nThe abolition of confessional or denominational schools under National Socialism breached Hitler's Concordat with the Vatican (20 July 1933), in which Hitler had promised that Catholic schools would be allowed to continue to exist. By mid-1939, all denominational schools in Germany had been replaced with non-religious 'community schools' and all private Church-run schools had been shut down. This was achieved mainly by arranging 'elections' by parents in favour of or against the schools, of which the purported results, claimed by Nazi local party leaders, were that parents favoured their closure. In this way, denominational schools were shut down without the need for a formal order of abolition.\n\nFurthermore, there was a drastic reduction in the amount of religious instruction given in state schools. Stories of the forefathers from the Old Testament were regarded as 'unnecessary', and even Christian education was expressed in terms of the path and culture of 'the Nordic man'. Religious education in schools had to conform to state requirements. In reality, this meant that religious instruction was either eliminated from the curriculum entirely or severely curtailed. Religious symbols and images, such as crucifixes, were banned from schools. Nevertheless, the NSDAP had concerns about the efficacy of these measures, particularly in rural areas. One report from 1938 stated: 'It is shocking to look into the situation of the rural areas, which still seem to be in the firm grip of the black teachers [Catholics]. Bible quotes and church songs are what the pupils know best, while some of the 10-year-olds do not even know the name of the F\u00fchrer, let alone how to spell it.'\n\nThe Nazi leadership believed there was a superfluity of private 'special schools' for mentally and physically disabled children. It sought to reduce the amount of educational resources for 'abnormal people' through its eugenics programme: 'By eugenic measures and the sterilization law, we hope to decrease the education of abnormal people to a minimum'. The regime hoped to achieve the 'purification' of the _Volk_ by regulating the education of 'abnormal people'. Whilst the primary aim was to educate healthy German children with full mental faculties, the Nazi state also believed it had a duty to prevent 'the creation of abnormal people'. It was considered 'an economic waste' to spend time and money on 'the education and upbringing of feeble-minded children'.\n\nThe Nazis had claimed that they would modernize education, with free universal education, streamlining of the school system, provision for talented children from low-income families and a university-level teachers' training programme. The reality, however, was that fewer children from disadvantaged backgrounds benefited from education under National Socialism than had done so during the Weimar era. From 1933, _Hochschulen f\u00fcr Lehrerbildung_ were established with the aim of uniform vocational and practical training of teachers to a higher standard. However, by 1941, these were closed down and as a result poorly qualified teachers were able to enter the profession. Most teachers' training remained below university level. Indeed, the status and image of the teaching profession had declined markedly under National Socialism and Party reports noted these problems.\n\nAt first, the Nazi regime had maintained the Prussian colleges for training teachers and extended the system, for example to Bavaria and W\u00fcrttemberg, which had not changed teacher training during the Weimar era. However, it became clear that more radical changes were needed and that a shortage of elementary schoolteachers loomed. In 1938\/9, in particular, there was a sharp drop in the number of full-time, male elementary schoolteachers. Hitler ordered in 1940 that a different type of institution should be established. This was the _Lehrerbildungsanstalt_ (teacher training institute), which paid greater attention to political socialization. The new scheme was designed to increase the supply of teachers and to bring their training under Party organization. These new institutions had a much more political character. By 1942 there were 233 of them in the German Reich. However, even these did not succeed in making up for the shortage of elementary schoolteachers, and the regime became dependent upon 'school helpers'. School helpers were aged between 19 and 30, having been educated to intermediate school level and then taken a three-month course followed by practical experience. In the end, despite all the training and plans for educational reforms, the Nazi regime demoralized teachers and brought about a decline in standards.\n\nRegarding the Nazis' overall racial aims, it is noteworthy that education in German schools was intended only for 'racially pure' German children. On 25 April 1933, the Law against the Overcrowding of German Schools and Universities placed a ceiling of 1.5 per cent on the number of Jewish pupils permitted within any German educational institution. In November 1938, this law was amended to exclude Jewish children from the German state school system altogether. After that, Jewish children were only permitted to attend separate Jewish schools, at the expense of the Jewish community. They were not part of the 'national community' and were therefore excluded from German schools. In 1942, the Jewish schools were closed down. 'Gypsy' children were also discriminated against in the German school system and various ad hoc attempts were made to prevent them from attending school. On 22 March 1941, the Ministry of Education finally passed a decree that prohibited 'Gypsy' children from attendance at state schools altogether.\n\nIn 1941, Hitler called for the introduction of the _Deutsche Heimschulen_ (German State Boarding Schools). These were mainly intended for children whose fathers were in the armed forces or had died in military action or whose homes had been disrupted by the war or destroyed by air raids. In addition to these schools, there were two _Musische Gymnasien_ (Musical Grammar Schools), one in Leipzig and one in Frankfurt. These were boarding schools for children with exceptional musical talents. The curriculum was similar to that of the _Deutsche Oberschule_ , but with the distinction that ten periods per week were given to the study of music and art. Admission to these schools was strict and privileged. Prospective pupils of these schools had to demonstrate their 'pure' German blood and good character, as well as their outstanding musical or artistic talents.\n\nDuring the war, a programme of _Kinderlandverschickung_ (KLV) was put in place. Children from the cities were sent to the countryside to be removed from the dangers posed by the Allied bombing campaigns. The KLV camps provided an ideal opportunity for the regime to imbue its youth with its ideology, as children were removed from their parental homes. There were approximately 5,500 KLV camps established by 1943. Entire classes and schools moved to KLV camps as schools in the cities were closed down. Many parents reluctantly agreed to send their children away, and there were concerns that the regime was deliberately removing children from their parents in order to take the function of socialization away from the family. At the KLV camps, the children took part in drills and marches. They had to wear a KLV uniform and were educated to be true and valuable members of the 'national community'. They were used for agricultural work and harvesting. By the spring of 1944, they were engaged in pre-military training, run by the HJ, the army or the SS.\n\nNazi changes to education were implemented in the name of 'modernizing' the school system and making it more efficient. However, the true objective of Nazi policies was to ensure centralized state control over education, in particular to eliminate ecclesiastical influence. Although the status of the _Gymnasium_ was actively diminished by the regime, it attracted even more middle-class pupils than before. Hence, Nazi claims to have made the system more egalitarian were unfounded. Furthermore, the carefully designed infiltration by Party influence of the state system was a hallmark of Nazi education policy. In reality, these policies did more to damage the existing educational system than to improve it. Although the Third Reich lasted only twelve years, its policies had a huge impact. It claimed that previous school reforms, which 'evolved from the daily life of urban, Marxist liberalism', signified the 'disintegration and dissolution' of the German educational system. Yet Nazi educational policies did not bring about any improvements in elementary and secondary school education in comparison to those of the Weimar Republic. The next section of this chapter turns to the universities, to examine the impact of National Socialism upon higher education.\n\nUNIVERSITIES\n\nWhilst some universities in Germany, such as the University of Heidelberg (1386), were established in medieval times and the development of the universities progressed through the patronage of the territorial princes over several centuries, the modern period of German university development was marked by the creation of the University of Berlin (1810). Wilhelm von Humboldt played a most important part in the development of the University of Berlin, which was steeped in the humanist tradition, and which served as a model for other German universities. He was concerned that the university should not be too narrow in its purpose. Prussia needed visionary and strong leaders and this consideration marked the character of the university. Humboldt was clear that the humanist tradition should be adhered to. The University of Berlin was defined as a 'privileged corporation' with self-government rights.\n\nOne significant aspect of the modern German universities was academic self-government. This was exercised by the faculty professors and elected dean, in faculty boards, which recommended nominations for professorial appointments and gave prospective lecturers the right to teach. A second characteristic was freedom of teaching. Professors and lecturers could teach freely, without political restraints or other limitations. They could choose the subjects they taught. The third salient feature was freedom of study. As there was no fixed syllabus, students could attend lectures of their preference and even move between universities. This system was aimed at developing the initiative and sense of responsibility for learning in students themselves.\n\nAs many of the German universities had developed in close association with the territorial princes, universities were dependent on the state in financial matters. This sometimes led to pressures to submit to state influence, especially as professors were state officials and their salaries were paid by the state. Professors came from a particular level in society and there existed a social snobbishness among them\u2013a 'professorial class' ( _Gelehrtenstand_ ). A mutual relationship existed between the holders of political power and the professors, who became the intellectual bodyguard of the _Kaiserreich_ in return for the privilege, prestige and status associated with their position. Some professors, such as Heinrich von Treitschke, used their freedom to teach extreme nationalism that seemed to negate the purpose of the concept. Nationalism was a key trend in the German universities in the last third of the nineteenth century. Nationalist associations such as the Pan German League and the Navy League, established in the 1890s, were closely associated with university professors, many of whom chose to throw in their lot with the nationalist and imperial ambitions of the era. They became known as the _Flottenprofessoren_ (navy professors).\n\nTurning to the students, at the beginning of the nineteenth century, there were two main types of students' fraternities: the _Landsmannschaften_ , which were made up of students from a particular region, and the _Studentenorden_ , which had associations with freemasonry. Duelling and drinking played an important part in the life of both these types of student organizations. In 1815, a new type of organization\u2013the _Burschenschaft_ \u2013was set up at Jena. In 1817, at the Wartburg meeting, the _Burschenschaften_ of the various universities came together in a general association. At the same time, the _Landsmannschaften_ renamed themselves the 'Corps'. The 'Corps' members were aristocratic, whilst the _Burschenschaften_ members were middle class. The _Burschenschaften_ were closely associated with movements for liberalism and national unity at this time. They also aimed to put an end to the excessive duelling in the universities and to encourage students to live a more moral lifestyle. By the mid-nineteenth century, this trend waned as the _Burschenschaften_ members continued to drink and duel. In the _Landsmannschaften_ and the _Burschenschaften_ , students' individuality was stifled. Student groups became increasingly conservative and anti-democratic by the end of the nineteenth century.\n\nIn 1919, in the new Weimar Republic, the _Deutsche Studentenschaft_ was founded to represent students' interests and to organize welfare. This was politically nationalist in its leanings. The _Deutsche Gildenschaft_ , established in the same year, was even more extremely nationalist and racist. There were also anti-Republican students' groups including the _Stahlhelm-Studentenring-Langemarck_ and the _NS-Deutscher Studentenbund_ (National Socialist German Students' Association). On the other side of the political spectrum, there were the Republican Students' Group, the German Students' League ( _Deutscher Studentenbund_ ) and the Socialist students' associations, all of which supported the Republic. Yet the voices of the nationalist and anti-democratic students' groups became very influential, particularly with the economic problems that beset Germany in the early 1930s. The National Socialist German Students' Association came to have a very large influence within the student movement overall.\n\nThe new social and political circumstances of the fledgling democracy required reforms in the universities. Criticism from left-wing circles called for the universities to expand their socially narrow student base. Carl Heinrich Becker, the Prussian Minister of Education between 1925 and 1930, noted that the universities were in need of reform, yet at the same time described them as 'fundamentally sound'. Although he wrote on the subject of university reform, he vacillated in terms of policy-making. University teaching in Germany tended to attract those from the upper echelons of society. The system was not designed for giving democratic rights to lecturers in terms of how the universities were run and administered. During the Weimar era, some steps were taken to give lecturers a small amount of representation on the faculty boards. However, all this changed again as the Nazi era approached, and authoritarian and nationalist voices came to the fore.\n\nIn the realm of higher education, the Nazi government attempted to clamp down on academic freedom. Its task was made easier by the activities of radical students who had taken over representative student bodies in the majority of German universities eighteen months before the Nazi _Machtergreifung_. The National Socialist German Students' Association had been formed in 1926, under the leadership of Wilhelm Tempel, a law student. In 1929, Baldur von Schirach had succeeded Tempel as its leader. Schirach claimed that the National Socialist German Students' Association had three main tasks: to promote the study of National Socialist ideas, to spread Nazi ideology in the German universities and to train leaders for the NSDAP. But its true ambition was to control the whole student population. The National Socialist German Students' Association quickly and actively set to work printing and distributing posters and pamphlets. Other student groups and organizations were passive by comparison and responded in a way that suggested they did not realize how serious the Nazi student organization was about propaganda and power.\n\nOnce the Nazis came to power, students campaigned against Jewish and 'unreliable' professors and disrupted their lectures. Students organized and participated in bookburning demonstrations on 10 May 1933 in university towns across Germany. This was a public act 'against the un-German spirit'. Students seized 'un-German' books, including those of Marx and Freud, from the libraries and consigned them to flames, whilst shouting out slogans against their authors. Goebbels described the public book burning as a strong, great and symbolic act. The students were clear about their ideological 'enemies' and the National Socialist German Students' Association planned the event carefully in advance so that the actions of 10 May 1933 were coordinated in university towns across the country. Elected students' committees were abolished and students' societies, the Corps and _Burschenschaften_ either dissolved themselves or were closed down in the process of _Gleichschaltung_. The student body certainly recognized its role in the renovation of the scholastic community and participated with eagerness at this time although, as Giles has pointed out, student apathy was a problem for the National Socialist Students' Association later in the Nazi era. From 1936, the National Socialist German Students' Association acquired a new leader, Gustav Adolf Scheel. He held the title _Reichsstudentenf\u00fchrer_.\n\nThe Rectors that ran the universities were checked for reliability and compliance with the dictates of the regime. Those that were not deemed suitable were replaced. Jewish and 'liberal' professors were forced out of their posts. Martin Heidegger, Professor of Philosophy, was elected Rector at the University of Freiburg in April 1933. He claimed that academic freedom now meant service to the 'national community' and talked of 'conquering the world of educated men and scholars for the new national political spirit'. Academic autonomy in teaching and research was subordinated to the interests of the Nazi state. By 1934, approximately 1,600 out of 5,000 university teachers had been dismissed. Many German academics emigrated. The sciences were particularly hard hit. The world-renowned physicist Albert Einstein was among the many scientists who left their posts at German universities to take up positions in America, Britain and elsewhere. Still, most university professors remained in their posts and many of them were supportive of the National Socialist government.\n\nUniversity professors and lecturers had to belong to the _NS-Dozentenbund_ (National Socialist Association of University Teachers). This was initially a part of the _NS-Lehrerbund_ , but it acquired a separate status in 1935. Walter Schultze, as _Reichsdozentenf\u00fchrer_ , ran the _NS-Dozentenbund_. Its main aim was to ensure that university activities conformed to the requirements of the NSDAP. Schultze stated that its main task was 'to make the universities truly National Socialist' and that education needed 'to participate in the National Socialist regeneration of our people's spiritual unity and community'. He continued by claiming that: 'The Association takes into its ranks all the forces at a university whose character and ideology attest to their unconditional loyalty and readiness to serve . . .', aiming 'to give the mission of the German scholar, researcher and teacher the prestige that is expected by National Socialism in the Party and in the state and, last but not least, by the people united by National Socialism'.\n\nAs in the case of schools, with the creation of the Reich Ministry of Education in 1934, there was centralized control of the universities. In April 1935, regulations were passed that established the leadership principle in the universities. The Rectors were no longer elected representatives, but nominees of the Reich government. The powers of the Rectors were significant. The senate and faculty boards no longer had any say in the running of the universities and their function was reduced to that of merely advising. In this way, the rights of the universities to academic self-government were abolished. The concept of freedom of teaching was also eliminated.\n\nIn 1933, the Nazis had imposed a _numerus clausus_ on university entrance admission. Even without this, student numbers dropped considerably, not least because of the derision the Nazis had for academic pursuits and values. In fact, the number of students enrolled at German universities dropped from 95,807 in 1931 to 48,558 in 1936, and fell further to 39,236 in 1939. By this time it was becoming clear to the Nazi leadership that whatever their view was of academia, in practical terms, there was a dearth of academically trained individuals. After 1938, the regime made alterations to university admission requirements in order to try to attract students. For example, after September 1938, it became possible for an adult aged between 25 and 40 to enter university without an _Abitur_ , by taking a special 'examination for great talent' ( _Hochbegabtenpr\u00fcfung_ ). A 'special maturity examination' award from a technical school also counted as an entry qualification to university in certain subjects. In 1934, the Nazis had also introduced 'Langemarck Scholarships' to encourage pupils from lower-class backgrounds to study at university. The NSDAP selected prospective candidates carefully and they attended special courses. In 1940, 800 students were accepted to this particular scholarship of whom 36 per cent were from working-class backgrounds. Nevertheless, these various attempts to raise student numbers did not succeed. The war itself disrupted university life even more drastically and student numbers continued to decline. R\u00fcdiger vom Bruch shows that National Socialism 'severely damaged the German university, its reputation, and its self-understanding'. It 'destroyed the German university as a self-administering corporate body and an independent research community with a functioning system of rules'. Under National Socialism academic independence and autonomy in the universities were renounced.\n\nNational Socialism brought about a root and branch re-engineering of the education system at all levels\u2013from kindergartens, through schools, to universities. At all stages, Nazi education was characterized by its anti-liberalism, anti-intellectualism and irrationalism. The loftier ideals of the _Volk_ took precedence over true scholarship and knowledge. Education policy was underpinned by a desire to disseminate National Socialist ideology as much as possible and in this context other educational aims were subordinated. The next chapter examines this in a more detailed consideration of the curriculum and an investigation of the impact of Nazism on school textbooks.\n\n**NOTES**\n\n. _Hitler's Table Talk 1941\u20131944: His Private Conversations_ , with an introduction by H. R. Trevor-Roper (New York, 1976), p. 547.\n\n. A. Hitler, _Mein Kampf_ , translated by R. Mannheim, with an introduction by D. C. Watt (London, 1992), p. 374.\n\n. Ibid., p. 387.\n\n. _Hitler's Table Talk_ , p. 548.\n\n. Ibid., p. 139.\n\n. Ibid., p. 567.\n\n. Ibid., pp. 547\u20138.\n\n. M. Lamberti, 'German Schoolteachers, National Socialism, and the Politics of Culture at the End of the Weimar Republic', _Central European History_ , Vol. 34, No. 1 (2001), p. 63.\n\n. Ibid., p. 80.\n\n. Ibid., p. 74.\n\n. Ibid., p. 64. See also W. Feiten, _Der Nationalsozialistische Lehrerbund. Entwicklung und Organisation_ (Weinheim and Basel, 1981), pp. 46\u201350.\n\n. On Schemm, see F. K\u00fchnel, _Hans Schemm. Gauleiter und Kultusminister (1891\u20131935)_ (Nuremberg, 1985).\n\n. Feiten, _Der Nationalsozialistische Lehrerbund_ , pp. 40\u201341.\n\n. BA NS 12\/967, 'Der Erzieher', Nr. 1, 1934, p. 7.\n\n. BA NS 12\/263, 'Satzung des Nationalsozialistischen Lehrerbundes', p. 1.\n\n. BA NS 12\/263, 'Satzung des Nationalsozialistischen Lehrerbundes', p. 3. On organization, see also Feiten, _Der Nationalsozialistische Lehrerbund_ , pp. 76\u2013132.\n\n. BA NS 12\/1404, 'Der Nationalsozialistische Erzieher', Nr. 9, 1937, pp. 4\u20135.\n\n. On this, see BA NS 12\/967, 'Der NSLB\u2013seine Geschichte, seiner organisatorische Entwicklung und die daraus resultierende Stellungnahme zur gegenw\u00e4rtigen organisatorischen Lage'.\n\n. For example, see BA NS 12\/641, 'Denkschrift \u00fcber die Misst\u00e4nde im Schulwesen Ostpreussens', 10 March 1933, p. 1.\n\n. Ibid.\n\n. BA NS 12\/600, 'Rundschreiben. Betrifft: Verbot der Doppelmitgliedschaft im NSLB und in konfessionellen Erzieherverbanden', 17 December 1936.\n\n. Feiten, _Der Nationalsozialistische Lehrerbund_ , p. 55.\n\n. ibid., p. 19.\n\n. BA NS 12\/1401, Rundschreiben Nr. 11\/33, 'Schulungsplan f\u00fcr den NSLB Untergau Oberschlesien'.\n\n. BA NS 12\/1401, 'Bericht \u00fcber die Durchf\u00fchrung der A-Schulungslager des NSLB\u2013Gau Schlesien 1936', p. 1.\n\n. J. Schiedeck and M. Stahlmann, 'Totalizing of Experience: Educational Camps', in H. S\u00fcnker and H.-U. Otto (eds), _Education and Fascism: Political Identity and Social Education in Nazi Society_ (London, 1997), p. 63.\n\n. See BA R 4901-1\/4607, 'Der Preu\u00dfische Finanzminister an Herrn Oberpr\u00e4sidenten f\u00fcr h\u00f6heres Schulwesen', 31 May 1934.\n\n. G. Pieper, 'Austauschlager: Ihr Sinn und ihre Gestaltung', _Nationalpolitische Erziehung_ (1937), p. 293. Cited in Schiedeck and Stahlmann, 'Totalizing of Experience', p. 63.\n\n. BA NS 12\/1401, 'Schulungsbrief der Gauhauptstelle Erziehung und Unterricht'.\n\n. Ibid., p. 1.\n\n. Ibid., p. 2.\n\n. ibid., pp. 8\u201310.\n\n. BA NS 12\/1401, 'Bericht \u00fcber die Durchf\u00fchrung der A-Schulungslager des NSLB\u2013Gau Schlesien 1936'.\n\n. Ibid., p. 1.\n\n. Ibid., p. 3.\n\n. Ibid., p. 2.\n\n. Ibid., pp. 4\u20137.\n\n. BA NS 12\/1402, 'Denkschrift \u00fcber das Junglehrer Schulungslager in Hassitz bei Glatz, 2. bis 21. Oktober 1933'.\n\n. BA NS 12\/1196, 'Die Schulungsarbeit des Amtes f\u00fcr Erzieher (NSLB)', 19 Sept. 1935, p. 5.\n\n. On what follows see BA NS 12\/41, 'Tagebuch Pappelhof ', p. 386.\n\n. Ibid., p. 387.\n\n. Ibid., p. 388.\n\n. BA NS 12\/41, 'T\u00e4tigkeitsbericht'.\n\n. BA NS 12\/41, 'Bericht \u00fcber das Schulungslager f\u00fcr Erzieherinnen in Parchim'.\n\n. Ibid.\n\n. BA NS 12\/1196, 'Die Schulungsarbeit des Amtes f\u00fcr Erzieher (NSLB)', 19 Sept. 1935, p. 1.\n\n. Ibid., p. 3.\n\n. BA NS 12\/1406, 'Abteilung Presse und Propaganda'.\n\n. BA NS 12\/1400, 'Bericht \u00fcber Kreiswaltertagung'.\n\n. Ibid.\n\n. BA NS 12\/41, 'Landlehrer im Lager'.\n\n. Ibid.\n\n. Ibid.\n\n. BA NS 12\/41, 'Als Sachse im Schulungslager Hamberge'.\n\n. BA NS 12\/41, 'Acht Tage Schulungslager des NS-Lehrerbundes'.\n\n. BA NS 12\/41, 'Landlehrer im Lager'.\n\n. BA R 4901-1\/4607, 'Verein Deutscher Evangelischer Lehrerinnen e.V.', 4 December 1934.\n\n. BA R 4901-1\/4607, 'Betrifft: Beschwerde des Vereins deutsch. ev. Lehrerinnen in Barmen', 25 January 1935.\n\n. BA R 4901-1\/4607, 'An den Herrn Oberpr\u00e4sidenten (Abt. f\u00fcr h\u00f6heres Schulwesen)', 22 February 1935.\n\n. BA R 4901-1\/4607, 'Abschrift. Zeltlager des NS-Lehrerbundes, Gau Pommern', p. 1.\n\n. BA R 4901-1\/4607, 'Abschrift. Bericht \u00fcber das 1. Zeltlager des NS-Lehrerbundes Gau Pommern vom 26. Juni bis 1. Juli 1935 in Heringsdorf '.\n\n. Ibid.\n\n. BA NS 12\/41, 'Arbeitsfolge f\u00fcr das Schullager der anhaltischen Lehrer aler Schularten vom 5 bis 13 August 1935'.\n\n. BA NS 12\/606, 'Sachgebiet Biologie. Schulungslehrgang des NSLB in T\u00fcbingen vom 11.\u201317. Oktober 1936'.\n\n. BA NS 12\/628, Rundschreiben, 'Betrifft: Reichslehrgang f\u00fcr Rassenkunde', 13 October 1937. See also BA NS 12\/628, Rundschreiben, 'Betr.: Jahresarbeit 1938', 16 February 1938, pp. 1\u20132, which highlights work and achievements on teaching racial lore.\n\n. BA NS 12\/41, 'Acht Tage Schulungslager des NS-Lehrerbundes'.\n\n. Ibid.\n\n. BA NS 12\/967, 'Der Erzieher', Nr. 1, 1934, p. 8.\n\n. Ibid., p. 11.\n\n. Feiten, _Der Nationalsozialistische Lehrerbund_ , p. 148.\n\n. BA NS 12\/1404, 'Der Nationalsozialistische Erzieher', Nr. 9, 1937, p. 9.\n\n. BA NS 12\/600, 'Rundschreiben. Betrifft: Erntehilfe der deutschen Erzieher', 29 June 1938.\n\n. BA NS 12\/1438, 'Aufgaben und Leistungen des NS-Lehrerbundes im Kriege', p. 1.\n\n. BA NS 12\/1438, 'Aufgaben und Leistungen des NS-Lehrerbundes im Kriege', p. 2. On the NSLB and the 'homefront', see also Feiten, _Der Nationalsozialistische Lehrerbund_ , pp. 188\u201391.\n\n. BA NS 12\/1438, 'Aufgaben und Leistungen des NS-Lehrerbundes im Kriege', p. 2.\n\n. BA NS 12\/567, 'Stillegung des NSLB'.\n\n. Ibid.\n\n. On this, see Feiten, _Der Nationalsozialistische Lehrerbund_ , pp. 197\u2013200.\n\n. BA NS 12\/641, 'Rundschreiben \u00fcber das neue Erziehungsziel', p. 2.\n\n. Ibid.\n\n. Ibid., p. 4.\n\n. BA R 4901\/708, 'Konflikte zwischen Reichserziehungsministerium und NSDAP', pp. 1\u20134.\n\n. BA R 4901\/708, 'Betr.: Konflikte des Reichserziehungsministeriums mit Dienststellen der NSDAP', p. 1.\n\n. Ibid., pp. 8\u20139.\n\n. Ibid., p. 7.\n\n. On what follows, see BA R 4901\/708, 'Die Schule als Streitobjekt zwischen Partei und Staat', pp. 1\u20132.\n\n. BA R 4901\/708, 'Die Schule als Streitobjekt zwischen Partei und Staat', p. 3.\n\n. _Hitler's Table Talk_ , pp. 548\u20139.\n\n. Cited in H.-G. Herrlitz _et al._ , _Deutsche Schulgeschichte von 1800 bis zum Gegenwart_ (Weinheim and Munich, 1993), p. 149.\n\n. S. Schnurr, 'Vom Wolfahrtsstaat zum Erziehungsstaat: Sozialpolitik und soziale Arbeit in der Weimarer Republik und im Nationalsozialismus', _Widerspr\u00fcche_ , Vol. 8 (1988), pp. 47\u201364.\n\n. S\u00fcnker and Otto, _Education and Fascism_ , p. vii.\n\n. Cited in Hahn, _Education and Society_ , p. 75.\n\n. W. Frick, _Kampfziel der deutschen Schule_ (Langensalza, 1933), p. 24.\n\n. Hahn, _Education and Society_ , p. 83.\n\n. M. Mouton, _From Nurturing the Nation to Purifying the Volk: Weimar and Nazi Family Policy, 1918\u20131945_ (Cambridge, 2007), pp. 164\u20135.\n\n. A. T. Allen, _Feminism and Motherhood in Germany, 1890\u20131914_ (New Brunswick, 1991), pp. 63\u20135. On this, see also A. T. Allen, 'Children between Public and Private Worlds: The Kindergarten and Public Policy in Germany, 1840\u2013Present', in R. Wollons (ed.), _Kindergartens and Cultures: The Global Diffusion of an Idea_ (New Haven, 2000), pp. 16\u201341.\n\n. Mouton, _From Nurturing the Nation to Purifying the Volk_ , p. 166.\n\n. Ibid., p. 179.\n\n. Ibid., p. 180.\n\n. On the Froebel tradition, see F. Froebel, _The Education of Man_ , translated by W. Hailmann (New York, 1887) and N. Isaacs, 'Froebel's Educational Philosophy', in E. Laurence (ed.), _Friedrich Froebel and English Education_ (London, 1969).\n\n. Mouton, _From Nurturing the Nation to Purifying the Volk_ , p. 182.\n\n. BA R 89\/5242, 'Hilfswerk Mutter und Kind und Hitler-Freiplatz-Spende 1936\/37', p. 14.\n\n. L. Pine, _Nazi Family Policy, 1933\u20131945_ (Oxford, 1997), p. 31.\n\n. BA NSD 30\/25, 'Richtlinien f\u00fcr Erntekinderg\u00e4rten im Rahmen des Hilfswerkes \"Mutter und Kind\"', pp. 4\u20135.\n\n. H. Vorl\u00e4nder, _Die NSV. Darstellung und Dokumentation einer nationalsozialistischen Organisation_ (Boppard, 1988), p. 70.\n\n. BA NSD 30\/25, 'Richtlinien f\u00fcr Erntekinderg\u00e4rten im Rahmen des Hilfswerkes \"Mutter und Kind\"', p. 3.\n\n. BA NS 12\/641, 'Die hamburgische Landschule im nationalsozialistischen Staate'.\n\n. Ibid., p. 11.\n\n. On this, see Samuel and Hinton Thomas, _Education and Society_ , p. 39.\n\n. Ibid., p. 16.\n\n. Ibid., p. 20.\n\n. Hahn, _Education and Society_ , p. 79.\n\n. BA NS 12\/641, 'Die hamburgische Landschule im nationalsozialistischen Staate', p. 14.\n\n. BA NS 12\/811, 'Die Schule der deutschen Volksgemeinschaft. Entwurf eines nationalsozial. Schulprograms von G. Weischedel', 22 May 1934, p. 1.\n\n. Ibid.\n\n. BA R 4901\/1 4620\/1, 'Betrifft: Neuordnung des h\u00f6heren Schulwesens'.\n\n. BA NS 12\/964, 'Errichtung von Hochschulen f\u00fcr Lehrerbildung und Deutsche Oberschulen in Aufbauform. Neugestaltung der Lehrerbildung in Bayern', p. 3.\n\n. BA R 4901\/1 4620\/1, 'Betrifft: Neuordnung des h\u00f6heren Schulwesens', p. 35.\n\n. Hitler, _Mein Kampf_ , p. 377.\n\n. BA R 4901\/709, 'Private Volksschulen', pp. 1\u20135; BA NS 12\/814, 'Aus dem NS-Lehrerbund', p. 9.\n\n. See BA R 4901\/1 3304\/1, 'Reichsschulgesetz vom Januar 1937'.\n\n. BA R 4901\/1 3304\/1, 'Entwurf eines Reichsgesetzes \u00fcber die einheitliche Gestaltung der deutschen Volksschule'.\n\n. BA R 4901\/1 3304\/1, 'Begr\u00fcndung'.\n\n. BA R 4901\/1 3304\/1, 'General-Vikarait Trier an den Herrn Reichs- und Preu\u00dfischen Minister f\u00fcr Wissenschaft, Erziehung und Volksbildung', 7 April 1937.\n\n. BA NS 12\/41, 'An die Kreisschulaufsichten zur Bekannthabe an alle Schule'.\n\n. On this, see L. Pine, _Hitler's 'National Community': Society and Culture in Nazi Germany_ (London, 2007), p. 88.\n\n. BA NS 12\/1196, 'Ausz\u00fcge aus Berichten der Gauschulungs\u00e4mter', p. 1.\n\n. BA NS 12\/825, '\u00dcbersteigerung der Anormalen-Erziehung\u2013was aber jeder dar\u00fcber wissen sollte'.\n\n. BA NS 12\/825, 'Gutachten zur \u00dcbersteigerung der Anormalen-Erziehung', Wilhelm Neidhardt, 25 November 1934, pp. 1\u20132. See also BA NS 12\/825, 'Heilerziehung und Heilerzieher im Dritten Reich'.\n\n. BA NS 12\/964, 'Errichtung von Hochschulen f\u00fcr Lehrerbildung und Deutsche Oberschulen in Aufbauform. Neugestaltung der Lehrerbildung in Bayern', p. 1; BA NS 12\/964, 'Organisationsplan einer Hochschule f\u00fcr Lehrerbildung'.\n\n. On this, see BA NS 12\/1196, 'Ausz\u00fcge aus Berichten der Gauschulungs\u00e4mter', pp. 1\u20137.\n\n. Samuel and Hinton Thomas, _Education and Society_ , p. 58.\n\n. BA R 4901\/709, 'Private Volksschulen', p. 2.\n\n. On the history of Jewish schools under National Socialism, see J. Walk, _J\u00fcdische Schule und Erziehung im Dritten Reich_ (Frankfurt am Main, 1991).\n\n. M. Burleigh and W. Wippermann, _The Racial State: Germany 1933\u20131945_ (Cambridge, 1991) pp. 214\u201315.\n\n. Samuel and Hinton Thomas, _Education and Society_ , p. 53.\n\n. Ibid., p. 54.\n\n. G. Knopp, _Hitler's Children_ (Stroud, 2002), p. 186.\n\n. BA NS 12\/641, 'Die hamburgische Landschule im nationalsozialistischen Staate', p. 17.\n\n. Hahn, _Education and Society_ , p. 28.\n\n. On this, see K. Jarausch, _Students, Society and Politics in Imperial Germany: The Rise of Academic Illiberalism_ (Princeton, 1982).\n\n. Cited in Samuel and Hinton Thomas, _Education and Society_ , p. 123.\n\n. C. H. Becker, _Vom Wesen der deutschen Universit\u00e4t_ (Berlin, 1925).\n\n. On this, see A. Gallin, _Midwives to Nazism: University Professors in Weimar Germany 1925\u20131933_ (Macon, 1986), pp. 86\u20137. See also F. Ringer, _The Decline of the German Mandarins: The German Academic Community, 1890\u20131933_ (Cambridge, Mass., 1969).\n\n. G. Giles, 'The Rise of the National Socialist Students' Association and the Failure of Political Education in the Third Reich', in P. Stachura (ed.), _The Shaping of the Nazi State_ (London, 1978), pp. 161\u20132.\n\n. G. Giles, _Students and National Socialism in Germany_ (Princeton, 1985). On students under National Socialism, see also M. Gr\u00fcttner, _Studenten im Dritten Reich_ (Paderborn, 1995).\n\n. R. Evans, _The Coming of the Third Reich_ (London, 2004), p. 423.\n\n. Cited in G. Mosse (ed.), _Nazi Culture: Intellectual, Cultural and Social Life in the Third Reich_ (London, 1966), pp. 314\u201315. For a detailed examination of the universities during the Nazi era, see H. Heiber, _Universit\u00e4t unterm Hakenkreuz: Teil 1_ (Munich, 1991) and H. Heiber, _Universit\u00e4t unterm Hakenkreuz: Teil 2_ (Munich, 1992).\n\n. Gallin, _Midwives to Nazism_ , p. 108.\n\n. Samuel and Hinton Thomas, _Education and Society_ , p. 112.\n\n. R. vom Bruch, 'A Slow Farewell to Humboldt? Stages in the History of German Universities, 1810\u20131945', in M. Ash (ed.), _German Universities Past and Future: Crisis or Renewal?_ (Oxford, 1997), p. 23.\n\n. Ibid., p. 24.\n[3 THE CURICULUM AND \nSCHOOL TEXTBOOKS ](Pine_9781847887641_epub_c16_r1.html#d6e5)\n\nThis chapter examines the school curriculum and school textbooks during the Third Reich. How was Krieck's aim of 'national political education' achieved in schools? In addition to the changes to the educational system discussed in the previous chapter, the main method was the transformation of the curriculum, in order to emphasize certain subject areas in which 'nation' and 'race' could be expounded, and to decrease the significance of other subject areas. For example, a Nazi directive for elementary education from 1940 stated:\n\nIt is not the task of the elementary school to impart a multiplicity of knowledge for the personal use of the individual. It has to develop and harness all physical and mental powers of youth for the service of the people and the state. Therefore, the only subject that has any place in the school curriculum is that which is necessary to achieve this aim. All other subjects, springing from obsolete educational ideas, must be discarded.\n\nIn secondary education, Nazi educationalists believed that 'German, history, geography and biology require a deeper treatment'. This chapter focuses on the key subject areas promoted by National Socialism: biology, physics and chemistry, geography, history, mathematics, German, racial studies and physical education.\n\nThe introduction and use of new school textbooks assisted Nazi pedagogues in their aim of inculcating pupils with Nazi ideology. At first, there were many different textbooks in the curriculum, which displeased NSDAP ideologue, Alfred Rosenberg. He ordered Philipp Bouhler, Director of the Party Censorship Office, to examine all the textbooks in use for their ideological content. Bouhler's work demonstrated the need for a process that entailed more than a simple 'weeding out', and worked towards the creation of a uniform Reich Reader for the entire nation. The Ministry of Education began to actively implement this idea, removing old readers from the curriculum and replacing them with new ones. Ernst Krieck, a prominent Nazi educational theorist and professor at the University of Heidelberg, was involved in the educational theory behind the introduction of these readers. New editions were to include the themes of 'blood and soil', leadership, honour and loyalty, service and sacrifice, struggle and work. Between 1935 and 1940, Bernhard Rust introduced new Reich Readers at different age levels. Editors selected specific reading material for them, based on the themes of 'blood and soil' and the _Volk_. They had to conform to Nazi Party censorship requirements. They included extracts from German and Nordic folklore and sagas, tailoring the selections to the ideological values of National Socialism. Rust defined the purpose of the readers as being to 'serve the ideological education of young German people, so as to develop them into fit members of the national community\u2013members who are ready to serve and to sacrifice'.\n\nThe spirit of _v\u00f6lkisch_ ideology was conveyed through children's books. A strict censorship policy was put into place to screen all books. Josef Goebbels's 'black lists' contained the titles of all 'alien' or 'decadent' works that were to be removed from circulation. In reality, many of the textbooks from the Weimar era did propagate reactionary political and social values. They were nationalistic and militaristic in their ethos. Hence, these ideas were already clearly extant in German school textbooks before National Socialism came to power. Nevertheless, the Nazi regime aimed to achieve a completely regimented and standardized system of school textbooks. By 1941, textbook production became the exclusive preserve of _Deutscher Schulverlag_ , owned by the NSDAP press _Eher Verlag_.\n\n**BIOLOGY**\n\nBiology teaching was of great significance in the Third Reich and was energetically promoted by the regime as a whole and the NSLB in particular. In no other subject area were the Nazi themes of 'blood and soil', 'race' and 'living space' so directly linked to the subject matter. Biology had the task then of instructing pupils about the living nature of German 'living space' as the basic nourishment of the German _Volk_ , as well as the goal of an eternal German _Volkstum_. Laws of heredity and life, fertility, selection and blood purity were central to the teaching of the subject. Paul Brohmer was one of the leading writers of a new biology curriculum under the National Socialist regime. In 1938, after the new curriculum was introduced, new textbooks appeared that took into account the changes. Brohmer utilized 'race biology' as a means of encouraging German children to struggle to maintain racial purity. He underlined the dangers of 'racial miscegenation' and justified the regime's racial, population and eugenics policies in his writings. Biology lessons became vehicles for Nazi racial doctrine, emphasizing themes such as race, heredity and the 'selection of the fittest'. Pupils were instructed in the classification of racial types and craniology. Films and slides were produced as teaching aids. Visual presentations were deemed to be particularly useful in showing the distinctions between examples of 'racially pure' and 'inferior' or 'hereditarily diseased' individuals. Biology was 'assigned a central function in education' with 'two hours of teaching a week in all grades'. As well as racial ideology, biology was to impart other aspects of Nazi ideology, such as love for the homeland and the 'national community', which were linked to the subject. As such, biology as a subject gained considerably in importance and prestige under National Socialism.\n\nFurthermore, biology teaching was also carried over into other subject areas, including mathematics and German, by using biology topics in these other disciplines. Hans Schemm's statement that 'National Socialism is politically applied biology' meant that biology teaching was directed towards educating children in the laws of life and in the ideology of National Socialism. Pupils were to think in both biological and national terms simultaneously. In particular, 'hereditary biology' was a significant part of 'biological thinking' under National Socialism. The realization of the hereditary health of the German people was to be 'drummed into' children in school education so that it became 'second nature to them'. Lore of the family, lore of race, genetics, eugenics and population policy formed the core of 'hereditary biology'. This subject purported to demonstrate that 'racial mixing' and an increase in the number of the 'hereditarily ill' damaged the integrity and value of the German population. The 'hereditary health' of the 'national community' was central to all work in this area. The promotion of 'valuable' hereditary lines and the concurrent prevention of the reproduction of the 'hereditarily ill' were emphasized. Pupils were given an understanding of the need for sterilization and of Nazi eugenics laws. They were shown the power of genetic transmission from one generation to the next. The preservation of the 'Nordic character' and an understanding of racial differences were the central aims of 'hereditary biology' lessons.\n\nNazi pedagogues, as well as Nazi leaders, regarded 'racial miscegenation' and 'bastardization' as serious threats to the German nation. In 1938, Alfred Vogel, a biology curriculum writer and primary school headmaster in Baden, produced a series of anti-Semitic teaching charts designed as teaching aids to the new curriculum. These accompanied a teachers' book designed for the instruction of 'biology' to primary school children. Vogel encouraged teachers to instruct children about the laws of nature and heredity, as well as racial consciousness and the 'blood community' of the German nation. He drew parallels between cross-breeding in plant biology and 'racial mixing' in society. Vogel advocated a 'race corner' in the school grounds that could be used to carry out experiments on plants and allow pupils to see the strength of the 'pure-bred' plant over the mixed-bred one. The inferences from this were applied to human society. Vogel examined 'hidden' inherited tendencies in biology, claiming that it was not correct to judge a living thing from its outwardly visible characteristics. The implication of this was that heredity was the only important signifier of race, so that a Jew posed a danger to the German nation even if he did not look like a Jew or did not practice his religion.\n\nTopics for biology instruction included: 'the heredity of physical characteristics', 'the heredity of mental and spiritual characteristics', 'the heredity of frailties and illnesses', 'the heredity of physical and spiritual characteristics of the German race', 'the care of racial inheritance', 'the law of selection' and 'the Jews and the German people'. Vogel advocated the need to educate young Germans about 'the racial value of our people and the tireless struggle over the preservation of our racial character' and about 'the complete rejection of the Jews'. His illustrated charts, including 'The Racial Composition of the Jews' and 'German Ways\u2013Jewish Ways', were used to show the perils presented to the German nation by the 'parasitic', 'wandering Jew'. His charts stereotyped the Jews, both as stateless intruders and as financial and political dominators. He also linked Jews with Freemasons and suggested that they were engaged together in a conspiracy for world domination. Whilst biology teaching did not propose overtly the policy of the genocide of the Jews, it did provide a legitimization for this policy, as well as for the Nazi 'euthanasia' campaign.\n\nThe 'school garden' was also employed by National Socialist pedagogues as a useful addition to the curriculum. The school garden was to be a true 'community garden' for the entire school, where each child participated for the benefit of the whole school community. This was a microcosm of the concept of the 'national community'. In the school garden, children developed their physical skills, as well as their sense of duty and responsibility. They also gained practical knowledge such as how to grow fruit and vegetables and how to control pests. Pupils learned about seeds, fertilization and soil usage. They undertook experiments on soil, fertilization and genetics, as part of the school garden activities. The topics were exploited to emphasize Nazi ideological imperatives. For example, links were made between particular plants and the 'German nation', the importance for the 'national community' of fruit and vegetable growing was shown, and hereditary transmission as demonstrated in the school garden was used to emphasise racial and eugenic issues.\n\nBiology teaching in girls' schools was particularly concerned with the 'mother instinct'. It emphasized a girl's main responsibility as her role in marriage and family life. Biology teaching included topics such as breeding and rearing animals, genetics, race studies, practical studies on the care of babies and young children, care of the sick and first aid, and preparations for girls' roles as future housewives in relation to both the domestic and the national economy.\n\n**PHYSICS AND CHEMISTRY**\n\nOf the three natural sciences, biology clearly took precedence in the Nazi curriculum, as it lent itself most easily to the regime's ideological objectives. Teachers of physics and chemistry had to make more of an effort to show that their subjects were pertinent to the Nazi government and its aims. In particular, the significance of physics for warfare was emphasized, as physics teachers tried to justify their subject by relating it to military objectives. A new branch of physics teaching under National Socialism\u2013'the physics of weapons'\u2013was designed to awaken the ability to bear arms. Military physics increased in significance from 1936 onwards. Pupils were instructed in orientation, measurement, communications, ballistics and military engineering. A new handbook was provided for teachers on this subject. Physics under National Socialism, according to its author, Erich G\u00fcnther, had the purpose 'of awakening not only the ability to bear arms but also the will to do so, and, beyond that, to show the ways and technical means to carry out the decision to bear arms'. Gauging of distance, sighting the line of fire and working out military objectives were all a fundamental part of this.\n\nThe importance of physics as a 'decisive practical factor' for weapons training and defence against enemy attack was underlined as follows:\n\nWe need excellent engineers not only for peace work but also just as much for building fighting and defence equipment of every kind, be it firearms or range finders, be it submarines, airplanes or combat vehicles. We need officers with thorough education in technology and natural sciences, who understand how to take charge of these war materials properly, and it hardly needs mentioning how diverse the knowledge of physics and the abilities are that are necessary for instance to command firing the heaviest artillery or a submarine.\n\nHence, physics was envisaged as a subject to be used for training engineers and military personnel. Physics instruction centred on ballistics, optics and aeronautics. Physics pedagogues fell into line with Nazi requirements in these areas. Willy G\u00f6llnitz, a secondary school teacher in Chemnitz, very clearly stated the need and rationale for practical weapons training: 'Present-day weapon technology requires that the most important concepts of internal and external ballistics be taught particularly to pupils in higher schools, who are destined later to fill leading positions in the army. But fortunately the times of blackboard physics are over'. Practical shooting experiments, including how to determine projectile trajectories and time of flight, were deemed to be 'the only way to arouse the necessary scientific interest in pupils alongside the natural pleasure in shooting'. Optics, including the use of prisms, mirrors, lenses, cameras and telescopes, was another area of physics that had importance in terms of military preparation. Artur Friedrich, a secondary school teacher in Chemnitz, argued for the need to equip pupils with a practical knowledge of these subjects so that they would be 'fit for defence' and able to carry out their future 'duties as soldiers with more understanding and interest'.\n\nPhysics teaching in the Third Reich was aimed at creating an understanding of the importance of technology, defence and the military in the life of the nation. Physics channelled the National Socialist world view. Physics teachers were to instruct pupils in problem solving and provide them with knowledge and practical skills only within the remit and requirements of Nazi educational policy. Furthermore, education in physics in the Third Reich particularly emphasized aviation, and the regime asked its educators to create an enthusiasm for flying among pupils. The objective was to prepare young people for service in the air force. New textbooks on aeronautical physics were produced for this purpose. Aeronautical physics, including topics such as propeller operation and flight dynamics, was deemed an important subject in the preparation of young boys for the air force. Instruction in this area was aimed both at imparting technical and scientific knowledge and at encouraging enthusiasm for flying and for the air force. By 1938, aviation and defence physics formed a significant part of the science curriculum.\n\nPhysics was also employed to teach pupils the achievements of German physicists and the contributions of German research to scientific knowledge. This was intended to enhance national pride and underline the significance of the National Socialist 'Aryan' world view. The work of Jewish physicists was excluded in line with Nazi racial ideology. As the 'racial heredity' of a person directly influenced his work, only the 'Aryan' scientist could be seen to truly achieve and create. Most significantly, this meant that Albert Einstein's name was erased in the physics textbooks of the Nazi era. Einstein's theory of relativity was rejected on the grounds that it was 'theoretical magic' and a 'great worldwide Jewish bluff'. 'Aryan' physicists, in contrast, such as Philipp Lenard and Johannes Stark, flourished under the political system of National Socialism. Their 'Aryan' physics aimed to preserve the 'national community' and express the life of the nation. Stark, in particular, called for there to be 'no Jewish propaganda' in German physics textbooks.\n\nPhysics teaching under National Socialism had 'to contribute to national political instruction and forming willpower'. Furthermore, 'knowledge of the natural conditions and requirements of the national community in the German living space, the ability and the will to do further work on the questions of physics research and technology in this connection, and also teaching to think realistically, which is so important for forming National Socialist willpower, can only be fostered organically on the basis of the pupils' horizon which schools should help to expand gradually'. Although topics such as classical mechanics and electromagnetism continued to be taught, much of the practical emphasis took the form of instruction in physics related to military topics for boys and domestic topics for girls. In the girls' schools in which sciences were taught, over the course of three years, the physics curriculum contained the following subjects: mechanics, thermodynamics and electricity in the first year; thermodynamics and optics in the second year; and electricity, induction and German physicists in the third year. The physics curriculum taught girls about household and kitchen appliances, as well as physics in relation to health and optics, rather than about modern physics per se. Physics, in girls' schools, mainly comprised instruction in the technical and even economic aspects of household management, heating and the practical use of electrical appliances.\n\nChemists likewise attempted to raise the prestige and status of their subject by underlining its significance to German national goals. In particular, they emphasized its importance to German manufacturing and defence. Ilse Beier, a chemistry teacher from Essen, suggested the possibilities of the subject in underlining 'the educational principles of the Third Reich'. Walther Franck, a secondary school teacher from Hamburg, defended chemistry, stating that it would be inopportune to reduce its teaching 'at a time when political and economic leaders in the whole world are struggling with the problem of materials, their production and their processing'. Chemistry teachers appealed to the armed forces and industrial companies such as I. G. Farben, as well as Nazi leaders who could see the benefits of their subject\u2013in terms of economic self-sufficiency and readiness for war\u2013for support. Chemistry education under National Socialism connected the chemical industry to the German people and the German economy. Chemists argued that their subject was significant for the economy and for defence. Furthermore, the history of science in chemistry teaching was aimed at strengthening national awareness and pride. Important German chemists such as Johann Wolfgang D\u00f6bereiner, Carl von Linde and Josef Loschmidt were used to show the importance of German contributions to research and scientific knowledge. Chemistry teachers could also use their subject as an opportunity to discuss 'race' by showing pupils the importance of the Germanic 'race' to the science of chemistry.\n\nFurthermore, the discussion of the Treaty of Versailles in this regard underlined a sense of unfairness that Germany was disadvantaged by the loss of territories rich in raw materials. Connections between raw materials and the national economy were drawn in chemistry teaching. Walter Leonhardt, a secondary school teacher from Dresden, explained that\n\nchemistry has to do with the instruments of power that first make it possible to wage war, with the most important mineral resources, coal and iron ore, from which steel is produced, with the materials for producing gunpowder and explosives, with the fertilizers to ensure the national food supply, etc. In the years since the lost war, chemistry teachers have considered it their duty to make clear to pupils what instruments of power Germany lost by being robbed of the mineral sources in Lorraine, the Saar and Upper Silesia, and most textbooks have not failed to include notes on this. In future, much more space will have to be devoted to such things so that our young learn to assess Germany's endangered situation from this viewpoint.\n\nNew educational guidelines from Rust's Ministry for Education and Science in 1938 stated that chemistry instruction in schools should be aimed at teaching children that 'constant and systematic work on developing natural sciences and technology ensures the high economic and cultural status of our people'. Furthermore, it was to demonstrate 'how chemical science and technology make new, valuable raw materials and synthetics available to German industry from substances found in our native country' and 'how they process and refine imported foreign raw materials'. Moreover, it taught children 'that application of chemical knowledge in agriculture and forestry and in the food industry helps to ensure our food supply, and that chemistry is indispensable for questions of national defence'. These guidelines also indicated that only certain compounds and substances needed to be taught. Pupils were required to learn only about economically significant raw materials, and how they were obtained and processed. New chemistry textbooks that taught in accordance with these stipulations were introduced. In practical terms, pupils worked in groups on chemical processes and analyses. Suggested topics included 'Analyses of Artificial Silks and Rayon', 'Experiments on Producing Artificial Fibres', 'Analysis of a Detergent' and 'Simple Analyses of Soil and Water'.\n\nFor girls, chemistry teaching dealt mainly with the chemistry of foodstuffs, fibres, household objects and detergents. Girls were taught about the need for domestic frugality. Both girls and boys were taught about chemical gases and air raid protection, although for boys defence chemistry was deemed particularly significant. Pupils were taught to become fit for defence in the event of gas attacks. They were taught about the nature of chemical weapons, protection against chemical gas, the effects of poisons on human beings and first aid to victims of gas poisoning. Pupils also undertook related chemistry experiments, for example on making gunpowder, on how oxygen masks worked and on protection from gas poisoning in air raid shelters.\n\nAnother significant topic in chemistry teaching, particularly after the announcement of the Four Year Plan in 1936, was the 'battle' for production and for collecting and recycling waste. The economic importance of metals, fuels, synthetic rubber and synthetic fibres was examined. Pupils were encouraged to collect and recycle waste. They experimented with producing oil from coffee grounds and they conducted experiments using waste metals that they had collected. This kind of teaching dovetailed with Nazi propaganda on the collection and recycling of waste materials. Chemistry under National Socialism was employed to underline the regime's ideological objectives as much as was possible within the perimeters of the subject area. In particular, production, raw materials and defence as aspects of chemistry teaching, along with attempts to employ the subject to enhance national awareness, were the most significant aspects of this phenomenon.\n\n**GEOGRAPHY**\n\nThe key topics in the geography curriculum under National Socialism were: studies of the homeland, which included love of the fatherland, 'blood and soil' and an idealization of bucolic life; political geography, which focused on the Treaty of Versailles, Germans living in border territories and abroad, Germany's 'enemies' and Germany's requirement for 'living space'; race studies, which included differentiations between 'superior' and 'inferior' races and especially an excoriation of the Jews; defence geography, which included security of German borders, defence preparation, descriptions of military topography; and colonial geography, which dealt with the need to reclaim lost German colonies, the history of Germany's colonial achievements and the crimes of other colonial powers. Geography in the Third Reich aimed to generate a political understanding of Germany's position in the world. The term a ' _Volk ohne Raum_ ' or 'people without space'\u2013originally the title of a serialized novel by Hans Grimm\u2013which suggested that the German _Volk_ needed more 'living space', was used to justify Nazi policies of expansionism. Studies of the homeland were a central theme throughout all geography education. The new secondary school curriculum in 1938 allotted two hours per week to geography. It emphasized topics such as the political subdivision of Germany, its racial groups, its 'living space' in Europe, Germans in border areas and abroad, German colonies and culminated in 'German People and German Land: The German Reich and its Position in the World'. There was much emphasis on Germans living in border areas and in territories lost to Germany under the terms of the Treaty of Versailles, which aimed at arousing sentiments of nationalism among German schoolchildren. Similarly to physics and chemistry, the regime found a way to introduce the theme of defence into geography teaching. This began as early as 1933 but was accelerated as the war approached. Defence geography included military topography, as well as a discussion of national borders and arenas of war.\n\nFurthermore, geography teaching was employed by National Socialism in order to expound its racial ideology. In particular, geography pedagogues exploited both traditional and new forms of anti-Semitism in their publications. For example, the image of the eternally wandering Jew was a favourite theme. Walter Jantzen was one of the most prominent geography educationalists in the Third Reich. He integrated Nazi ideology into the geography curriculum, particularly racial concerns about Jews, Blacks and Gypsies. He addressed the subjects of 'living space', 'blood and soil' and the decline in Germany's birth rate since the end of the First World War. Konrad Olbricht and Hermann K\u00e4rgel highlighted the ideological distinctions between National Socialism and 'Jewish Bolshevism' in their geography textbook. They juxtaposed their values on a number of themes and issues. For example, whilst National Socialism stood for leadership and loyalty, Judaeo-Bolshevism was characterized as the tyranny of a foreign national authority under the guise of democracy. Whilst Nazism advocated a healthy peasantry, a love for the soil and a dislike of urbanization, Judaeo-Bolshevism stood for the crushing of the peasants and the expansion of big cities.\n\nHence, geography in the Third Reich was employed to create a sense of love for the fatherland among schoolchildren. Hitler's achievements for Germans everywhere were glorified. The curriculum was very much focused upon topics about the German land. Phrases such as 'people without space' and 'blood and soil', which were really propaganda slogans, found their way into the classroom, as did anti-Semitism.\n\n**HISTORY**\n\nHitler was clear in his views about history teaching:\n\nParticularly in the present method of teaching, a change must be made . . . The result of our present history instruction is wretched in ninety-nine cases out of a hundred. A few facts, dates, birthdays and names remain behind while a broad clear line is totally lacking. The essentials which should really matter are not taught at all . . . The main value lies in recognising the great lines of development . . . For we do not learn history in order to know the past, we learn history in order to find an instructor for the future and for the continued existence of our nationality.\n\nThe history curriculum was subjected to considerable change under National Socialism. For example, Hans Schemm stated that 'the prehistory of German history must have a more prominent position in our curricula'. Defence history and frontier studies were also added to the history curriculum. The subject area 'local history' was to include not only knowledge about the local area, but also information about the whole _Volk_ and fatherland. It was concerned with the destiny of the nation and an understanding of the homeland. The lore of race and its prehistory, as described in the works of Houston Stewart Chamberlain and Hans G\u00fcnther, were also significant to this subject area. Local history under National Socialism was, in these ways, different to our understanding of what this subject usually comprises.\n\nA meeting of the NSLB in Eger on history teaching reported that the field of history was unable to keep up with the pace of National Socialist change in terms of its ideological and political world view. It bemoaned the lack of a history book that satisfied the National Socialist view in its evaluation and organization of historical events. National Socialist historians re-examined single chapters of world history anew. Even before new textbooks were available, history teachers were urged to teach their subject in the spirit of National Socialism: 'It is the duty of every history teacher to teach with his National Socialist\u2013trained conviction, even if new teaching material is not yet available'. There was a call for the transformation of the old historical interpretation into a National Socialist interpretation. The great National Socialist revolution needed to situate itself within the course of German history. The current struggle was of a historic nature and had to be understood as such: 'The new self-display of National Socialism . . . will be seen in sharp contrast to the democratic era'. Alfred Rosenberg stated: 'I am well aware that this is a huge educational task for our movement. Our task is to write world history anew, and this will take many years, even decades'. Although many of the Weimar history books contained the themes of earlier nationalist thought, the Nazi regime nevertheless aimed to replace them. Nazi history textbooks fostered nationalistic themes more extremely than ever before.\n\nHistory was interpreted as a struggle for existence between nations. History lessons were used as an opportunity to demonstrate to pupils the greatness of Germany. They were intended to awaken and excite children's sense of national pride and concern about the continued existence of the German state and nation. Hitler believed that: 'From all the innumerable great names in German history, the greatest must be picked out and introduced to the youth so persistently that they become pillars of unshakeable national sentiment'. History under National Socialism was further used to highlight the leadership principle, emphasizing Germany's 'great leaders' and their 'world-historical' achievements. For example, great historical leaders, such as Frederick the Great, were used to illustrate heroic leadership, tireless service to the state, military achievements and parallels to Hitler. The great triumphs of National Socialism were given considerable attention in the history textbooks of the Nazi period. History was an integral part of 'national political' education.\n\nA history textbook for secondary school pupils devoted part of its section on German culture to the position of women in the 'national community'. This outlined Nazi views on women's role in society and stressed the need to reawaken women's desire to have children and indeed to have large families. It underlined the role of mothers in educating their children, claiming that there was no task more noble or more beautiful than a mother making the developing soul of her child receptive to all the goodness and beauty of its nation. Various aspects of the theme of 'national renewal' were indeed common in the history textbooks of the era. Historical atlases illustrated Germany's greatness in her most historically important and expansive periods, and especially in the Third Reich. Furthermore, they employed maps, graphs and charts to highlight population policy issues, such as the declining birth rate and the age make-up of the population to demonstrate that Germany was becoming 'a nation without youth'. Hence, history books were implemented to explain the nature of Nazi population policy.\n\nThere was a flurry of activity in writing a new history curriculum in Nazi Germany. Publishers, university professors, teachers and school administrators became involved in this process, often in the pursuit of professional advancement. Dieter Klagges was one of the key history curriculum writers of the Nazi era, underlining the significance of German blood in history teaching. He developed the themes of racial purity and anti-Semitism in his writings. Johann von Leers used history stories as a means of portraying anti-Semitic ideas to young children, stressing the profiteering of Jews at the expense of Germans and depicting Jews as 'swindlers' and 'crooks'. History textbook writers of the Nazi era, such as Hans Warneck and Willi Matschke, portrayed the Jews as 'enemies of the Reich'. In his children's history textbook Johannes Mahnkopf wrote about the historical connection of Jews and Freemasons as a negative force. Such texts aimed to increase children's sense of attachment and loyalty to National Socialism in its struggle against 'the Jewish enemy'. Whilst the Nazi regime was not the first to use _v\u00f6lkisch_ ideas in history teaching, Wegner has shown that it was 'the first and only regime to fully institutionalise a racist and anti-Semitic history curriculum'. The glorification of militarism and nationalism that were already extant in the history textbooks of the _Kaiserreich_ and the Weimar Republic reached their peak in the history textbooks of the Third Reich.\n\n**MATHEMATICS**\n\nNazi maths books seized the opportunity to socialize children through pervading the curriculum in a well-established tradition. Maths questions dealt with 'national political problems'. Calculations of sums were based on examples of bullet trajectories, aircraft, cannons and bombs. For example, pupils were given the following statement: 'A bombing plane can be loaded with one explosive bomb of 35 kilograms, three bombs of 100 kilograms, four gas bombs of 150 kilograms, and 200 incendiary bombs of one kilogram'. The questions that the pupils had to answer were: 'What is the load capacity? What is the percentage of each type of bomb? How many incendiary bombs of 0.5 kilograms could be added if the load capacity were increased by 50 per cent?' Such arithmetic and mathematical problems were not uncommon in the textbooks of the Third Reich.\n\nNumerical problems based on state expenditure on 'hereditarily ill' and 'inferior' people exemplified the way in which Nazi ideology pervaded the school curriculum. In one exercise, pupils were presented with the following information: 'Every day, the state spends RM. 6 on one cripple; RM. 4 1\/4 on one mentally ill person; RM. 5 1\/2 on one deaf and dumb person; RM. 5 3\/5 on one feeble-minded person; RM. 3 1\/2 on one alcoholic; RM. 4 4\/5 on one pupil in care; RM. 2 1\/20 on one pupil at a special school; RM. 9\/20 on one pupil at an ordinary school.' It then asked questions such as 'What total cost do one cripple and one feeble-minded person create, if one takes a lifespan of forty-five years for each?' and 'Calculate the expenditure of the state for one pupil in a special school, and one pupil in an ordinary school over eight years, and state the amount of higher cost engendered by the special school pupil'. The implications of such questions were that state funds were being squandered on 'unhealthy' or 'undesirable' people. This was typical of the way in which data regarding state expenditure on 'hereditarily ill' or 'inferior' people was utilized in 'education'.\n\nExercises using data based upon the birth rate and other issues relating to population policy were also common in school textbooks of the Nazi era. For example, one exercise was based on the marriage loan scheme, introduced by the Nazi regime in 1933 to promote early marriage and the founding of _kinderreich_ families. It gave the figures of the number of loans given each year between 1933 and 1937, together with the value of each loan, and required pupils to calculate the yearly state expenditure on these loans. The association of these figures in the minds of schoolchildren doing these exercises was to indicate to them that money spent on maintaining 'hereditarily ill' people or on children attending special schools could be better spent on other things, such as marriage loans for 'healthy' and 'valuable' families. Furthermore, pupils were given figures relating to the number of Jews ('aliens') living in Germany as compared to the total population and asked to work out 'What is the percentage of aliens?' living in Germany.\n\nAnother maths textbook familiarized children with large numbers by using figures relating to the First World War. It stated that 13,250,000 men were called up by Germany, 11,250,000 by her allies, and 47,500,000 by Germany's enemies. The children, whilst doing addition and subtraction exercises, were simultaneously made aware of the 'heroic struggle' of Germany in the conflict. These examples all highlight the way in which Nazi educators used textbooks to disseminate Nazi ideology.\n\n**GERMAN**\n\nThe remit of the Nazi school curriculum was extremely broad and this subject in particular lent itself to the promotion of Nazi ideals. German language and literature found a prominent place in the Nazi school curriculum, teaching children the language of National Socialism. The preface of a German grammar book from the Nazi era states:\n\nWhatever moves the soul of a people, in joy and sorrow, in meditation and battle, in creation and festivity, vibrates in unison with the entire curriculum, and by no means least in the teaching of language. Here, too, it is a matter of coinciding with life itself! Proximity to the present! Relation to the people! For that reason let us also give utterance to the mighty events of the time in our lessons in German! That which fills the heart of the people is spilled by the tongue of youth! The stream of strong blood-folk thought, feeling, and will must be permitted to flow, warm with life, into the form of the word. The result will be a teaching of folk-culture in the mother tongue; we must make this live and be watchful in the growing generation, so that this may, with its own treasure of words belonging to our day and age, express the new treasure of thought, gather it into itself, and let it root ever deeper in the German essence, growing ever more deeply rooted, growing ever more into the German mode of thought, the German mode of living, and the German view of the world.\n\nStudies of German folklore were regarded as a 'central educational point' in the Third Reich. A document on this subject prepared by Friedrich Dehmlaw argued that no other _Volk_ had experienced such change from greatness to profound collapse in its history. Dehmlaw claimed that Germany had been 'internally decayed'. The solution was to create clear goals for future development. He described the historical development of Germany in earlier eras, with a neglect of German traditions and culture. Nevertheless, Germany transformed herself again and again and developed a new folklore. 'In order to bring German qualities to the German mind, the _Volk_ must be educated'. It had to be educated to love the nation and 'to be the German people'. Dehmlaw decried the influence of the Jews on Germany and called for their exclusion from German culture, as they were 'foreign' to it. He called for schools to subordinate all their activities to 'the national goal of the making of the German _Volk_ '. German folklore was an integral part of education. The great value of German culture and language was emphasized. Whilst Dehmlaw's essay was criticized by other educationalists, the basic premise that German folklore was an important aspect of _Volk_ education remained valid throughout the Third Reich.\n\nLessons in German were designed to foster a sense of 'Germanness' and of national pride and unity among pupils. Nationalist and irrational aspects of German literature were extolled. Poems such as Dietrich Eckart's 'Germany Awake!' were used to ignite a sense of nationalism in German schoolchildren:\n\nGermany Awake! \nStorm, storm, storm! \nLet the bells ring from tower to tower, \nRing till the sparks begin to shower, \nJudas appears, to win the Reich's power. \nRing till the bell-ropes redden with blood. \nRing for the burning, the martyred, the dead. \nRing out the storm, and let the whole earth shake, \nRevenge to the rescue, and thunder overhead! \nWoe unto those who dream today! \nGermany, awake\n\nFurthermore, Kamenetsky has pointed out, many classics were adjusted to the requirements of the Nazi regime by means of 'slanted abridgements or reinterpretations', whilst others that did not fit into the Nazi 'world view' were simply banned. For example, attempts were made at an 'Aryan' reinterpretation of Goethe. Literature was rewritten, with rationalist, enlightened or cosmopolitan influences removed. 'Non-Aryan' writers, such as Heine, were rejected. Individualistic aspects of literature were removed or ignored in favour of the promotion of _v\u00f6lkisch_ ness, with its themes of belonging and identity. Teachers of German language and literature were urged to stress the nation as 'a community of blood', 'a community of fate and struggle', 'a community of work' and 'a community of mind'. Traditional German tales and sagas were supplemented with Nazi myths and war stories. Hero worship of National Socialist heroes such as Horst Wessel found their way into school textbooks. Furthermore, there was much emphasis on _Blut und Boden_ (blood and soil) literature, as well as the glorification of war. Considerable attention was given to the war of 1914\u201318, in particular. The subject matter of Germans living outside the nation's boundaries was also treated in reading books. Such books bemoaned the loss of 'ethnic' Germans, who through 'blood' and 'race' belonged to the German nation, but who were citizens of foreign states\u2013'Germans across the borders'. Schoolchildren were taught the need 'to make Germany whole again'.\n\nSchool textbooks were employed widely to represent Nazi ideals. Primers introduced very young children to aspects of Nazi ideology such as faith in the _F\u00fchrer_ , love of the fatherland, and service and sacrifice for the German 'national community'. Initially, primers represented a mixture of old and new values. Publishers reprinted pre-1933 texts with small alterations, such as the addition of Nazi Party slogans or images of swastika flags. By the mid- to late 1930s, primers represented Nazi ideology more distinctly as censorship increased and a greater number of new texts and illustrations became available. They dealt, 'in word and picture, with camp life, marching, martial drums, boys growing up to be soldiers, and girls to take care of soldiers'. An image of Hitler often appeared on the page inside the front cover, before the start of the book. A page consisting of the words _Heil Hitler_ followed, with children shown raising their arms in the Hitler salute. Hitler was portrayed as benevolent, friendly and generous. For example, the story 'A Happy Day' depicts scenes of great excitement and anticipation, as children prepared for the _F\u00fchrer's_ visit to their village. Nazi symbols were depicted in textbooks for very young children. For example, there were illustrations of children waving swastika flags. There was a poem in the form of a short prayer, in which children expressed their hope to become 'strong and pure . . . German children' of their _F\u00fchrer_. This theme of making the _F\u00fchrer_ happy was common in school textbooks of the Nazi era. In addition, there were texts and illustrations of Nazi organizations, in particular the SA and the Hitler Youth. One textbook showed two small boys proudly marching alongside their SA fathers. The combined impact of striking illustrations with simple primer language conveyed the National Socialist message boldly. Primers became more concerned with doing this, than with preparing children to read. Letters of the alphabet were introduced in conjunction with an aspect of Nazism, for example, the letter H for ' _Heil, Heil'_. Political socialization in readers took the form of stories about 'helping the _F\u00fchrer_ ' and the 'national community', for example by taking part in the regime's _Winterhilfswerk_ (Winter Relief Agency) and _Eintopf_ (one-pot dish) campaigns to help needy 'national comrades'. The 'one-pot dish' campaign encouraged families to give up their usual meal on one Sunday each month, for a cheap 'one-pot dish' and to donate the money saved to a state-sponsored charity. In one story, illustrated with a family sitting around their dining table, a child told her parents that she had thought the _Eintopf_ meant that there was a large dish outside the town hall, and that all the people went there to eat. Her brother laughed at her, but their father admonished him, saying that at least now the girl had understood what the _F\u00fchrer_ meant. After that, there was a knock at the door and the collector appeared. One of the children was instructed to fetch the money, and to give double that day, as it was the father's birthday. This story explained the significance of the _Eintopf_ , using the family context as a basis for political socialization. Another story told of a mother asking her daughters to fetch potatoes from the cellar in baskets to fill up a sack for the Winter Relief Agency, whose motto was 'no one shall go hungry, no one shall freeze'. They brought up three baskets and asked if that was enough. Their mother told them to bring up another basket, as the sack was not yet full, emphasizing that they should be pleased to make sacrifices for the Winter Relief Agency.\n\nPolitical socialization was evident too in the way in which family life was portrayed and in stories and images of 'the German mother', which mirrored the Nazi idealization of motherhood. For example, the choice of adjectives employed to describe family members accorded exactly with National Socialist ideals\u2013the 'goodness' of the mother, the 'strength' of the father, the action and 'pride' of the son, the passivity of the daughter were depicted in a description of 'a good type of family'. Illustrations of a mother surrounded by four or five loving children were common. These reflected the Nazi aim of the _kinderreich_ family. Short texts and poems about the mother and her tireless work for her family accompanied these images. They listed all the tasks and duties that the mother happily undertook, without becoming fatigued or morose. In readers for younger children, there were stories of children preparing a special treat for their mother on her birthday or on Mother's Day, but in reading books for older children, depictions of the mother were found under the subheading 'heroes of everyday life'. This elevation of 'the German mother' to a heroic status closely reflected Nazi ideology.\n\nA play for Mother's Day was included in one school textbook, portraying four councillors engaged in a discussion of ways to relieve the 'mother' of her many burdens and duties. Just as they were considering the possibility of finding someone to take over some of the duties of the mother, a woman appeared at the door. They asked her if she was a wife and mother, to which she replied affirmatively. Then they asked her if she took care of her family, to which she responded that she did so from dawn to dusk. However, when asked about the possibility of having assistance to lighten her burden, she firmly rejected the idea, claiming that mothers loved their domain and were happy to toil from early in the morning until late at night for their families. The councillors, after she had departed, concluded their meeting by deciding that 'mothers do not want to be relieved' of their duties and tasks. This clearly reflected the Nazi view that German mothers should be willing to work hard and make sacrifices for their families.\n\nExamples of blatant political socialization are found in stories depicting 'the enemies of Germany', in particular Jews, Bolsheviks and Slavs. Such negative stereotypes were used to underline the differences between 'national comrades', who belonged to the 'national community', and 'community aliens' and 'enemies', who did not. For example, in secondary school readers, anti-Semitic quotations from Hitler, Himmler or other Nazi leaders were interspersed with folklore and literature that sought to highlight nationalism. The German need for 'living space' explained in history and geography books also found its way into storybooks.\n\n**RACIAL STUDIES**\n\n'No boy and no girl must leave school without having been led to an ultimate realisation of the necessity and essence of blood purity'. Many textbooks of the Nazi era focused on this theme. One story that related specifically to Nazi racial ideology told of a cuckoo that met a nightingale in the street. The cuckoo wanted to sing as beautifully as the nightingale. He claimed that the only reason he could not do so was because he had not been taught to sing when he was young. The nightingale laughed and said that nightingales did not learn to sing, but were born with the ability to sing. The cuckoo, nevertheless, believed that if only he could find the right teacher, his offspring would be able to sing as beautifully as the nightingale. His wife had a clever idea. She decided to lay her eggs in the nest of another bird, so that their young would grow up together with those of another type of bird than the cuckoo, and would therefore learn how to sing. She laid an egg in the nest of a hedge sparrow. When the mother hedge sparrow returned to her nest, she was surprised to see the strange egg, but she decided to take care of it as if it were her own. When the eggs hatched, a young cuckoo emerged among the young hedge sparrows. He was nourished and cared for in exactly the same way as them, but he did not grow into a hedge sparrow. In fact, the older he grew, the more noticeable his differences became. He did not fly like the others, but flew like his real parents, that is, like a cuckoo. When he tried to sing, he could not do so. The only sound he could make was that of his own species. Hence, despite being reared in the nest of a hedge sparrow, he grew up to be a true cuckoo. This story was used to pose the questions: 'What is more important? The race from which one stems, or the nest in which one grows up?' The issues raised in this fable are particularly salient, reflecting both the debate about inherited versus acquired characteristics and the fundaments of Nazi racial ideology.\n\n_Rassenkunde_ (racial studies) was a new subject that formed an integral part of the curriculum under National Socialism. 'The battle about _Rassenkunde_ is not a matter of theoretical debates, but a battlefield on which, without a doubt, the most important battles of our century are fought out.' The strength of the nation was conditional upon 'pure' blood, and Nazi ideology consistently expounded the evils of 'racial miscegenation'. The NSLB stressed that 'at primary schools in particular we have to work on only the Nordic racial core of the German _Volk_ again and again and have to contrast this with the racial composition of foreign populations and the Jews'.\n\nRacial studies became an obligatory subject in all classes. The importance of race and heredity for the future of the nation and the aims of the government was impressed upon pupils. Their sense of responsibility towards their nation was awakened. _The ABC of Race_ outlined the following as Germany's main problems: too little territory, the Germans living abroad, the menace of the Jews and the falling birth rate. 'A nation without territory in time becomes a nation without people . . . If, however, we all fight together under our mighty National Socialist leadership and under the protection of the new racial laws, then the glorious Nordic future of Germany is assured'. In Nazi textbooks, 'Aryan' and Jewish 'racial types' (often in the form of caricatures) were juxtaposed so that German children would positively identify with the former and reject the latter as 'enemies'. This offered a pedagogical foundation and legitimization to the persecution of the Jews.\n\nNazi pedagogue, Fritz Fink, called for anti-Semitism to pervade the entire curriculum, at all age levels, in order to disseminate it in the classroom. He furnished educators with information about the Jews that they could use in their lessons, even if they had little experience of the subject. He integrated pictorial distinctions between Jews and 'Aryans' for teachers and school administrators. His work encompassed both traditional and radical anti-Semitism, using economic, religious and racial arguments against the Jews.\n\nAnti-Semitic storybooks were specifically dedicated to propounding Nazi ideology to children and they played a significant part in this regard. Indeed, the publishers of anti-Semitic literature aimed to disseminate their messages in the most direct and appealing way, by incorporating artwork using the most up-to-date techniques in colour printing. The most notable of these was the publishing house of _Der St\u00fcrmer_ , based in Nuremberg. _Der St\u00fcrmer_ was the anti-Semitic newspaper founded by Julius Streicher, the ardently anti-Semitic _Gauleiter_ of Franconia; it was characterized by a crude writing style, caricatures and cartoons. _Der St\u00fcrmer_ published an anti-Semitic textbook for young children, written by Elvira Bauer in 1936, entitled _Trau keinem Fuchs auf gr\u00fcner Heid und keinem Jud bei seinem Eid_ ( _Trust no Fox on the Green Heath and No Jew Upon his Oath_ ), designed to show children that Jews could not be trusted. It portrayed the Jews as inferior, untrustworthy and parasitic. _Der St\u00fcrmer_ put out a further anti-Semitic children's book entitled _Der Giftpilz_ ( _The Poisonous Mushroom_ ), written by Ernst Hiemer, in 1938. It portrayed the Jews, through a text containing seventeen short stories, as the antithesis of 'Aryan' humanity. 'The Jew' was dehumanized and conceptualized as 'the poisonous mushroom'. This book included the entire spectrum of anti-Semitic allegations against Jews, encompassing strands of both religious and racial anti-Semitism. The contents included the following themes: 'How to Tell a Jew', 'How Jewish Traders Cheat', 'How Jews Torment Animals', 'Are there Decent Jews?' and finally 'Without Solving the Jewish Question, No Salvation for Mankind'. _Der St\u00fcrmer_ published another work by the same author two years later, which propounded racial anti-Semitism and decrying the evils of 'racial miscegenation'. In this text, Hiemer portrayed Jews as 'bloodsuckers'. He equated Jews with tapeworms, claiming that: 'Tapeworm and Jew are parasites of the worst kind. We want their elimination. We want to become healthy and strong again. Then only one thing will help: Their extermination'. The aim of texts such as this was to justify the Nazis' mass destruction of the Jews. _Der St\u00fcrmer_ itself was also used in schools as part of Nazi 'education'. As well as salacious gossip, scandalous stories and its overtly anti-Semitic content, the paper published letters from approving teachers and children.\n\n'The whole of education must be subordinate to the new principle which aims at every young German person becoming a conscious political bearer of German blood.' Indeed, much use was made of genealogy and family trees in school textbooks of the era, in order to establish 'racial purity'. On this theme, a piece entitled 'You and your Ancestors!' asked pupils: 'Do you know what kind of blood runs through your veins? Do you know your father and your mother, and have you yet seen the ancestry of your forefathers?' This text urged children to be proud, not ashamed, of their ancestors, in their old-fashioned clothes. The writer stated that he had traced his own family tree back to 1500 and that he knew, therefore, what kind of blood ran through his veins. He wrote that ancestors had a bearing upon one's own talents and distinguishing features, a conviction firmly held by the Nazi leadership. Beyond forming a family tree, the author made up a genealogical table, so that instead of simply naming his ancestors, he recorded each one's date of birth and death, as well as details of marriages, professions and titles. The writer made this ancestral knowledge sound very important, exciting and colourful. He encouraged pupils to take an interest in their own ancestry, asking them to consider that one day they would be the ancestors of a future family, and that they were part of a family tree that would continue to grow. The implication was that the reader was culpable if he was ignorant about his line of descent. In addition to pupils' books, there were a number of teaching aids that suggested to teachers the ways in which these issues could and should be taught. Through genealogical activities, children were made aware of their membership of the 'blood community' of the German nation.\n\nApart from involving children in their own ancestry, numerous poems and stories about heredity, blood and kinship were presented in the textbooks of the Third Reich. These kinds of texts highlighted the sense of continuity between children, their parents, grandparents and great-grandparents back over the generations. They emphasized the flow of blood from the past, into the present and through into the future of a line of ancestry, and of blood flowing through the veins of a family generation after generation.\n\nOne textbook demonstrated the transmission of family characteristics through the generations by considering the composer Johann Sebastian Bach. It illustrated Bach's family tree, in order to show that there were no fewer than thirty-four 'musically competent' people in his family, of whom approximately half were 'outstandingly gifted'. This particular example was part of a comprehensive chapter dealing with heredity, race and family. Within this context, blood was the most significant and penetrating symbol, as 'German blood' was the guarantor of the future of the nation. However, there were other symbols too, such as the family home and traditions. Old furniture was regarded as a representation of continuity, so that the 'old chest' or the 'old table' linked together different generations of a family. The values of kinship were expressed through such symbols, which further represented the desirability of a good German family.\n\nIn relation to this theme, the rural family, in particular, was accorded a special significance during the Nazi era. Nazi ideology regarded the rural family to be pure and ideal, untainted by the depravities of urban life. It excoriated many aspects of life in the big cities, not least the tendency of young couples to limit the size of their families. Urbanization leading to the death of the nation was a recurrent theme. This came across especially vehemently in textbooks aimed at pupils in rural areas, designed to demonstrate to them their importance and value in maintaining a healthy nation. Such books showed that what was regarded as a family in the big cities was not a true family, but a distorted image of one. A husband and wife, living in the city, without children, but with domestic pets instead, could be described at best as a 'household', but not as a 'family'. The rural family was portrayed as the 'archetype of a true family'. Children were key components of a true family in Nazi ideology, the greatest blessing for a couple and, more significantly, for the nation. Another aspect of rural family life deemed positive by the Nazi regime was the inclusion of the grandparents in the home. In this way, children were more aware of their family history and ancestry.\n\nNazi teachers took their role in teaching on the subject of 'blood and soil' very seriously: 'We educators are the proclaimer and communicator for the coming generation, therefore it has to be our holy duty to plant blood and soil as something alive into the hearts and souls of German children.' This argument underlined the role and importance of the German peasantry, for 'only a hereditarily healthy people is capable of passing on the acres from generation to generation'. Peasant poems symbolized the working and struggling German peasantry that formed one of the foundations of Nazi ideology. The oak tree symbolized the eternal, stable German nation.\n\nNazi family ideology was treated in sections entirely devoted to this theme in schoolbooks for older children. Under the subheading 'the essence of the family; its biological position; its legal establishment', key aspects of Nazi ideology and policy were examined. This began with the statement that the family was the smallest but most important unit of the German nation, followed by an explanation that a true family consisted not just of a married couple, but of children too. It was the duty of parents to provide protection and care for their children, and of children to honour and respect their parents. Among the tasks or obligations of the family were the development of a sense of awareness of the 'national community' and the preservation of the nation through the creation of 'healthy' children. Only spouses who were free of physical and mental disabilities or illnesses were to reproduce. Furthermore, the obligation to know about heredity in general and one's own ancestry in particular was explicitly stated. Textbooks of this period explained why and how the National Socialist state promoted 'healthy' families, as well as its laws, such as the Law for the Prevention of Hereditarily Diseased Offspring and the Nuremberg Laws. As the 'germ cell' of the nation, the family was central to the attainment of blood purity and hereditary health.\n\n**GIRLS' EDUCATION**\n\nAs we have seen, education for girls in Nazi Germany was differentiated from boys' education through its emphasis on motherhood and the family. The Nazi pedagogue Alfred B\u00e4umler stated that under National Socialism there was not to be a general education, but a separate male and female education. Hence, girls' education was considered to require a distinctive character, rather than being part of the overall education of both sexes. The essence of the female nature and its significance to the future of the nation was underlined. The role of motherhood and family life was central to girls' education. To this end, textbooks devoted lengthy chapters to the subjects of cleaning and household care, as well as cooking and nutrition. Some of this education on hygiene and home economics was very practical. For example, one textbook gave detailed nutritional advice. It examined the nutritional value of meat, which differed according to the age of the animal and its diet, and explained the health risks of uncooked meats and liver or blood sausages if they were not properly cooked. It advocated the valuable contribution of fish to the diet, in particular cod, herring and shellfish. It explained the food poisoning risks from fish and how to tell if a fish was going off. It gave nutritional guidance on milk, butter, cheese, eggs, pulses, fruit, vegetables and sugar. It explained how beers, wines and spirits were made and highlighted their lack of nutritional value, as well as their effects upon the mind and body. In particular, the effects of alcoholism on the individual, the family and the state were explained\u2013the illness of the alcoholic, his expenditure of his family's income on alcohol and the costs involved for the state in maintaining chronic alcoholics in hospitals, prisons and asylums. Tea, coffee and cocoa were shown to have little or no nutritional value. The dangers of smoking tobacco were explained. This type of nutritional information was provided in order to promote an awareness of health, as part of the Nazis' comprehensive aim of creating a 'fit' race.\n\nAdvice on cleaning and household care followed this nutritional guidance. The daily cleaning of the kitchen, living room, bedrooms, floors and stairs was explained in great detail. Weekly cleaning of the rest of the house was considered to be sufficient. Instructions were then given for a thorough annual cleaning of the house, including all the woodwork, skirting boards, windows, window frames and sills. Instructions were given for cleaning floors, bringing back colour to faded carpets, cleaning china and glass and polishing furniture, as well as on which type of cleaning materials to use. The book gave advice on washing-up and kitchen hygiene. It claimed that the sign of a good housewife was a scrupulously clean kitchen. It gave further advice on washing bed linen, dealing with vermin and cleaning shoes, as well as heating and lighting the home. Another textbook that treated the subject of 'the work of the housewife' examined similar topics, but additionally dealt with the subject of clothing for the family and the part played by the housewife in saving and collecting old material. Such texts aimed to ensure that young women would not stumble into marital life unprepared. In particular, education on hygiene and the prevention of disease was considered to be very important. This was practical and even quite progressive. The advice and instruction in textbooks of the era, sometimes obvious, sometimes over-detailed, demonstrates the seriousness of the intentions of the Nazi regime to educate girls in accordance with its ideological imperatives.\n\n**PHYSICAL EDUCATION**\n\nAn emphasis on physical education was not new to Hitler's Germany. Physical education in Germany had its antecedents in the _Ritterakademien_ of the seventeenth century, in which the sons of the aristocracy were educated. Gymnastics also had a long history associated with and developing from the work and writings of Friedrich Jahn in the early nineteenth century. By the second half of the nineteenth century, gymnastics festivals and gymnastics societies were an established feature of German life. The Kaiser had stressed the importance of physical education at the Reich School Conference in 1890 and its significance was emphasized again in the Weimar Republic at the School Conference in 1920. During the Weimar era, the Prussian State Boarding Schools and Salem School exemplified the glorification of militarism in their physical training curricula. The Reich Youth Badge and Reich Youth Sports Competitions were introduced in the Weimar Republic. Hence, Hitler's concerns with physical education were not out of line with certain extant trends.\n\nHitler emphasized the importance of physical education claiming that the state had to adjust its entire educational work primarily 'to the breeding of absolutely healthy bodies'. He further stated that 'sport does not exist only to make the individual strong, agile and bold; it should also toughen him and teach him to bear hardships'. Nazi educationalists devoted considerable attention to this aim. The NSLB was keen to work to achieve Hitler's goals in this area. It argued that the bias towards intellectual education must be counteracted. Furthermore, racial-biological knowledge and an understanding of its requirements were built on the realization of the worth of the body and its entire development. To this end, physical development and achievement were regarded as very important. 'Where there is no will to defend, there is no ability to defend': this was an argument for inspiring willingness and ability to defend at school, primarily through physical education. The concept of the subordination of the individual to the whole was connected to physical education too, as it led to an understanding of collective effort and comradeship. A sense of responsibility and duty to the nation was encouraged through physical education, which created will and character. Hence, Hitler regarded physical education as fundamental to education as a whole.\n\nThe accomplishment of the regime's aims to increase physical education entailed greater demands on all teachers, including older ones. The advancement of physical education was a requirement 'in the interest of the toughening and increased fortification' of the nation. School inspectors were required to 'pay the greatest attention to and support this undertaking' through enquiries about participation, as well as tightened surveillance and monitoring of the lessons. Teachers who showed particular commitment to this goal were compensated by means of their schedules, such as being spared the duty of standing in for sick colleagues. Teachers were advised to become members of the SA, unless they were too old or had a physical infirmity. Otherwise, they were trained in sports courses. Teachers who were members of the SA were at a considerable advantage.\n\nIn reforms proposed to the secondary school system in 1936, it was argued that more hours for physical education had to be accommodated in the curriculum. Physical education took on a prominent role within schools, with five hours per week devoted to it, in the secondary school curriculum. In order to achieve the desired position for physical education within the National Socialist state, it was suggested that entry to university be based not only upon an examination of intellectual capacity, but also on the gaining of an official medical certificate pertaining to physical eligibility. There were also calls for grades awarded in physical education to be used for evaluating whether or not pupils advanced to the next higher class. In addition, individual physical performances, effort, discipline and leadership qualities could be assessed and these assessments could be taken into account with regard to entry to university. There was also a suggestion that if a pupil were exceptionally accomplished in terms of physical education this should balance out poor marks in academic subjects, which attracted some controversy among teachers. Additionally, it was to be made much harder for pupils to get exemptions from mandatory physical education. They had to be examined by a public health official or the school doctor. Many of these suggestions were implemented in Nazi education policy.\n\nCurriculum development in the area of physical education was an area of considerable attention during the Third Reich. Nazi educators believed that it was the duty of schools in the National Socialist state to educate the German nation to faith, obedience, strength and glory. Physical education had a new and important role in this: 'Physical exercises are for us no longer empty forms . . . but the means for the education of youth to National Socialism'. No other form of education was considered 'to give such possibilities for the education of the character of young people, than the various areas of physical education'. In physical education, 'we have the opportunity to guide the youth away from the \"I\" sports of past times to the \"We\" sports of the National Socialist state'. This was part of the process of inculcating youth with a sense of community of spirit and a move away from care for the self to concern for the group. In this way, a National Socialist society could be created.\n\nIn primary schools for boys and girls (years 1\u20133), children were to participate in movement stories based on the curriculum, such as Hansel and Gretel, The Wolf and the Seven Goats, Visit to the Easter Bunny, Strolling, Travels of a Family, exercises based on traffic and imitating the clock. They were to do moving exercises based on the theme of the forest, such as running between trees, jumping, hopping, jumping up to branches, stretching to imitate the height of trees, moving around in the woods and imitating animals in the forests. Other themes around which exercises were based included the river, winter, street life and building a house. Then there were simple songs and simple games such as 'Come and Run', 'Run after Numbers', 'Cat and Mouse', 'Fox and Geese'. In year 4, physical education included walking, running, jumping, imitation games and group activities. These sessions included push fights, pull fights, crawling and wrestling for boys. There was also use of equipment and obstacles, as well as ball exercises and first attempts at high jump and long jump. Gymnastic games and singing games were also part of the curriculum.\n\nIn years 5\u20136, there were more walking and running exercises, including endurance runs. There were balancing and skills exercises, as well as exercises to form the body, without equipment. This was supplemented with exercises using equipment, such as pushing and pulling bars, high bars, parallel bars, ladders and gymnastic rings. In years 7\u20138, walking and running exercises were intensified, with the use of hurdles and, additionally, the acceleration of training runs over 100 metres. Balance and skills exercises continued from the previous years, with the addition of skipping rope activities. There were more exercises for toning the arm and leg muscles. Exercises with equipment took the form of a more intense progression of bench, bar, ladder and gymnastic ring activities. In school years 9\u201310 and 11\u201313, a similar format and concept of physical training was given. The exercises were adapted according to the age groups and became progressively more difficult and demanding from year to year. Exercises for the formation of muscles became more important as the children grew older.\n\nThere was also a call for schools to establish an effective _Volkssport_ (national sport) in order for a strong German people to emerge. By participating in the national sport, pupils placed themselves in the 'national community' and were willing to make sacrifices for the _Volk_. As the timetable was already full of other physical education, national sport activities took place on field days, exercise marches and play afternoons. For years 5\u20137, these exercises included 'order exercises' (commands for individuals and groups), 'marching exercises' (duration one to three hours), cross-country games, wrestling, swimming and floor exercises. For years 8\u201310, there were 'order exercises', 'marching exercises' (duration four hours), country exercises which included orientation, assessment and use of terrain, and camouflage. In addition, there were waving and flashing exercises, pitching of clubs, aerial defence and gas defence exercises, swimming, floor exercises, wrestling and fist-fighting. For years 11\u201313, the activities were of a similar nature, but order and marching exercises were intensified, and country exercises were extended to include reconnaissance patrol training. In addition to this, at all age levels, there were _Volk_ exercises (running, jumping, pitching and hitting) aimed at achieving increased performance and accuracy. These _Volk_ exercises included stamina training and relay races, long jump, high jump, triple jump, pole vault, ball, javelin, discus, shot-put, gymnastics and swimming.\n\nStructured games and contact sports were important aspects of physical education. They promoted team spirit and co-operation, as well as self-discipline. These were considered to be perfect for character formation. In addition, hiking expeditions on field days were intended to enable children to get to know their homeland. By hiking and experiencing their 'homeland soil', the Nazis believed that pupils would learn to appreciate their folklore and to love their homeland with all its beauty. For younger children, hiking was intended as a pleasurable experience. Getting to know the homeland was associated with marching and singing songs. Whilst the duration and distance of the hikes varied according to age, there was nevertheless the intention to expose boys to a certain amount of strain. It was believed that enduring heat and cold, as well as hunger and thirst, would make older German boys tougher. In years 10\u201313 camp-duties were added to hiking exercises, in order to 'strengthen the community of the pupils', in a way deemed impossible at school. A tightly organized camp-structure with a strong leader was aimed at making pupils obedient, as well as ready for action and responsibility. In the winter, field days were based around winter sports, including skiing, tobogganing and ice-skating.\n\nThe physical education curriculum for girls was different to that for boys. Girls' gymnastics were aimed at educating German girls to become 'healthy, happy German women and German mothers'. All 'un-German' characteristics were to be avoided and eliminated. There was a particular emphasis on games and dance. There was to be contest and competition among girls, in order to encourage both productive efficiency and a sense of responsibility towards competitors. Responsibility towards the _Volk_ would be encouraged in this way. Classes were carefully planned so that all parts of the body were exercised. The aim of gymnastics was the development of a high degree of control of the body. In years 5\u20136, girls participated in 'order exercises', gymnastics, walking, running, skipping, relaxation exercises, swing exercises, muscle-strengthening exercises, dry swimming exercises, activities with balls and skipping ropes. They learned to use gymnastic equipment including the swinging rope, jumping vault, ladder, climbing pole, horizontal bar, swing rings and parallel bars. In years 7\u20138, they undertook similar 'order exercises' and a more intensive gymnastics training. As the school years progressed, they continued to follow a prescribed gymnastics programme, but with more demanding exercises. Girls took part in _Volk_ exercises similar to those of the boys. Their games lessons began as simple singing and dancing games and developed so that girls acquired the ability, discipline and knowledge to play contact sports competently in years 11\u201313. Games afternoons were designed not only for the games themselves, but also to further pupils' commitment to their school community. Hiking for girls was intended to create a strong love for the homeland and nation. The girls' marches were not as arduous as the boys' marches. The length and duration of hikes in primary school were based on the pupils' age and ability. Older pupils participated in longer hikes. In secondary school, two-day hiking trips were organized, with pupils staying overnight at a youth hostel. Such activities played an important part in creating a sense of the importance of the homeland and in developing camaraderie among participants. They were designed to encourage a sense of belonging to the national community and a concern for 'we' rather than 'I'.\n\n**FILM AND RADIO PROPAGANDA IN SCHOOLS**\n\nNew compulsory film courses were incorporated into the German school curriculum, and teachers under National Socialism enthusiastically supported the use of film as a supplement to the instruction they gave their pupils. The Nazi propaganda machine carefully and deliberately exploited film propaganda, both in schools and in youth groups, to disseminate Nazi ideology. The Reich Centre for Educational Films was established in June 1934 to oversee the production and distribution of educational films for schools. Furthermore, Rust advocated the showing of political propaganda films in all German schools. He stated that:\n\nThe leadership of Germany increasingly believes that schools have to be open to the dissemination of our ideology. To carry out this task we know of no better means than the film. The film is particularly important for schoolchildren. Film education must not only clarify contemporary political problems but also it must provide children with a knowledge of Germany's heroic past and a profound understanding of the future development of the Third Reich.\n\nThe films were silent and typically lasted between ten and twenty minutes. Teachers were provided with a teacher's guide, in the form of a printed lecture, which they were to present to their pupils before the film screenings. Much propaganda was contained in these lectures, which accompanied the films. The pupils were tested on the content of the films by means of a written examination. The content was overtly political, extolling the virtues of discipline, camaraderie and self-sacrifice, as well as the key themes of Nazi propaganda. In particular, military educational films that dealt with the subject of war and the army were very significant in treating important themes. In addition, the School Radio was employed as a further way of disseminating Nazi ideology and propaganda in the classroom. Typical themes included 'Germany, Land of Beauty', 'A People without Territory', 'The German Spirit of Unity and Will to Sacrifice' and 'Unity of Blood in the German People'.\n\nNazism used the concept of 'integrated instruction' as a tool for the dissemination of its ideology in schools. It sometimes blurred the boundaries between subject areas in order to achieve an organic curriculum. It emphasized the teaching of particular subject disciplines, especially physical education, and the reduction or exclusion of others, especially religious education. Those subject areas it deemed the most important were those through which it could permeate the curriculum with its ideology. These were biology, physics, chemistry, geography, history, mathematics and German. A new compulsory area of racial studies was introduced. Through school textbooks, Nazi educators sought to develop in children a sense of identity with the nation, the Nazi regime and its policies. The emphasis was placed upon the selection of politically valuable material, with the implementation of a system of strict censorship _. V\u00f6lkisch_ education was directed at the promotion of the German nation. Other educational aims were subordinated to this end. Certain aspects of the curriculum were practical and even progressive; others were not limited to Germany but were comparable to those of other countries. Although there were some similarities to earlier periods, the Nazi curriculum was distinctive from the curricula that preceded it and the curricula that followed it, in particular in its specifically racial and anti-Semitic aspects. The next chapter moves away from the general curriculum to examine the very specific subject of the Nazi elite schools.\n\n**NOTES**\n\n. Krieck, _Nationalpolitische Erziehung_.\n\n. Cited in Samuel and Hinton Thomas, _Education and Society_ , p. 83.\n\n. BA R 4901\/1 4620\/1, 'Betrifft: Neuordnung des h\u00f6heren Schulwesens', p. 5.\n\n. C. Kamenetsky, _Children's Literature in Hitler's Germany: The Cultural Policy of National Socialism_ (Athens, Ohio, 1984), p. 187.\n\n. Cited in ibid., p. 188.\n\n. See, for example, P. Brohmer, _Biologischer Unterricht und v\u00f6lkischer Erziehung_ (Frankfurt am Main, 1933).\n\n. On this, see \u00c4. B\u00e4umer, _NS-Biologie_ (Stuttgart, 1990), pp. 152\u20134.\n\n. \u00c4. B\u00e4umer-Schleinkofer, _Nazi Biology and Schools_ (Frankfurt am Main, 1995), p. 238.\n\n. ibid.\n\n. BA NS 12\/41, 'Erbbiologie in der Praxis des biologischen Unterrichts', p. 1.\n\n. Ibid., p. 2.\n\n. Ibid., p. 5.\n\n. A. Vogel, _Erblehre und Rassenkunde f\u00fcr die Grund- und Hauptschule_ (Baden, 1937).\n\n. G. Wegner, 'Schooling for a New Mythos: Race, Anti-Semitism and the Curriculum Materials of a Nazi Race Educator', _Paedagogica Historica_ , Vol. XXVII (1992), p. 197.\n\n. Ibid., p. 198.\n\n. Cited in ibid., p. 200.\n\n. On this, see G. Knoblauch, 'Der Schulgarten als Erziehungsst\u00e4tte', _Der Biologe_ , Vol. 4 (1935), pp. 76\u20138 and A. H\u00f6fner, _Der Schulgarten in der Unterrichtspraxis_ (Munich, 1937).\n\n. On this, see R. Br\u00e4mer and A. Kremer, _Physikunterricht im Dritten Reich_ (Marburg, 1980), pp. 51\u201376 and J. Willer, 'Physikunterricht unter der Diktatur des Nationalsozialismus', in R. Dithmar (ed.), _Schule und Unterricht im Dritten Reich_ (Neuwied, 1989), pp. 187\u2013204.\n\n. E. G\u00fcnther (ed.), _Wehrphysik\u2013Ein Handbuch f\u00fcr Lehrer_ (Frankfurt am Main, 1936).\n\n. E. G\u00fcnther, 'Die Bedeutung des Physikunterrichts f\u00fcr die Erziehung zur Wehrhaftigkeit', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 45 (1939), p. 231.\n\n. W. G\u00f6llnitz, 'Die Schie\u00dflehre im neuzeitlichen Physikunterricht', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 43 (1937), p. 133.\n\n. M. Pongratz, 'Schie\u00dfversuche in der Oberstufe der Oberschule', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 46 (1940), p. 161.\n\n. A. Friedrich, 'Wehroptik', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 46 (1940), p. 148.\n\n. _Erziehung und Unterricht in der h\u00f6heren Schule. Amtliche Ausgabe des Reichs- und Preu\u00dfischen Ministeriums f\u00fcr Wissenschaft, Erzeihung und Volksbildung_ (Berlin, 1938), pp. 173\u201386.\n\n. J. Stark, 'Zur Neuordnung des physikalischen Unterrichts', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 45 (1939), p. 82.\n\n. O. Brandt, 'Die neuen Lehrb\u00fccher', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 46 (1940), p. 152.\n\n. R. Fricke-Finkelnburg, _Nationalsozialismus und Schule_ (Opladen, 1989), p. 202.\n\n. On this, see for example, H. Oden, 'Hauswirtschaftliche Physik an Oberschulen f\u00fcr M\u00e4dchen', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 45 (1939), pp. 27\u201330.\n\n. I. Beier, 'Zur Schulreform. Chemieunterricht und Erziehung im Dritten Reich', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 40 (1934), p. 252.\n\n. W. Franck, 'Zur Schulreform. Chemieunterricht und Erziehung im Dritten Reich', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 40 (1934), p. 68.\n\n. K. G\u00f6lz and W. Jansen, 'Der Chemieunterricht im NS-Staat. Ein Beitrag zur Geschichte der Chemiedidaktik', _Gesellschaft Deutscher Chemiker, Fachgruppe Geschichte der Chemie Mitteilung_ , Vol. 4 (1990), p. 31.\n\n. W. Leonhardt, 'Chemieunterricht und Wehrhaftigkeit', _Unterrichtsbl\u00e4tter f\u00fcr Mathematik und Naturwissenschaften_ , Vol. 39 (1933), p. 235.\n\n. _Erziehung und Unterricht_ , p. 165.\n\n. G\u00f6lz and Jansen, 'Der Chemieunterricht im NS-Staat. Ein Beitrag zur Geschichte der Chemiedidaktik', p. 28.\n\n. H. Heske, _\". . . und morgen die ganze Welt\". Erdkundeunterricht im Nationalsozialismus_ (Gie\u00dfen, 1990), p. 206.\n\n. W. Jantzen, _Die Geographie im Dienste der nationalpolitischen Erziehung_ (Breslau, 1936).\n\n. K. Olbricht and H. K\u00e4rgel, _Deutschland als Ganze. Der Erdkunde Unterricht in der Volks- und Mittelschule_ (Berlin, 1938).\n\n. Hitler, _Mein Kampf_ , pp. 382\u20133.\n\n. BA NS 12\/967, _Der Erzieher_ , Nr. 1, 1934, p. 8.\n\n. Hahn, _Education and Society_ , p. 82.\n\n. On what follows, see BA NS 12\/824, Dr. Gerhard Endriss, 'Beitr\u00e4ge zur Heimatkunde'.\n\n. BA NS 12\/327, 'Wandel des geschichtlichen Weltbildes', _V\u00f6lkischer Beobachter_.\n\n. Ibid.\n\n. Ibid.\n\n. BA NS 12\/327, _K\u00f6lnische Zeitung_.\n\n. Ibid.\n\n. Ibid.\n\n. On this, see G. Blackburn, _Education in the Third Reich: A Study of Race and History in Nazi Textbooks_ (Albany, 1985). See also, H. Genschel, 'Geschichtsdidaktik und Geschichtsunterricht im nationalsozialistischen Deutschland', in G. Schneider and K. Bergmann (eds), _Gesellschaft, Staat und Geschichtsunterricht_ (Dusseldorf, 1982).\n\n. Hitler, _Mein Kampf_ , p. 387.\n\n. See, for example, D. Klagen (ed.), _Volk und F\u00fchrer: Deutsche Geschichte f\u00fcr Schulen_ (Frankfurt am Main, 1943).\n\n. D. Klagges, _Geschichtsunterricht als nationalpolitische Erziehung_ (Frankfurt am Main, 1936).\n\n. W. Hohmann, _Volk und Reich. Der deutschen Geschichtsbuch f\u00fcr Oberschulen und Gymnasien, Klasse 8. Von Bismarck bis zur Gegenwart_ (Frankfurt am Main, 1941), pp. 236\u20138.\n\n. See, for example, W. Gehl, _Geschichte f\u00fcr h\u00f6here Schulen Mittelstufe, Heft 4_ (Breslau, 1936), pp. 145\u20139.\n\n. B. Kumsteller, _Werden und Wachsen. Ein Geschichtsatlas auf v\u00f6lkischer Grundlage_ (Braunschweig, 1938), p. 60.\n\n. For example, see Klagges, _Geschichte als nationalpolitisiche Erziehung_.\n\n. J. von Leers, _F\u00fcr das Reich: Deutsche Geschichte in Geschichtserz\u00e4hlungen_ (Leipzig, 1940).\n\n. H. Warneck and W. Matschke, _Geschichte f\u00fcr Volksschulen_ (Leipzig, 1942).\n\n. J. Mahnkopf, _Von der Uhrzeit zum Grossdeutschen Reich_ (Leipzig, 1941).\n\n. G. Wegner, _Anti-Semitism and Schooling under the Third Reich_ (New York and London, 2002), p. 126.\n\n. See L. Pine, 'The Dissemination of Nazi Ideology and Family Values through School Textbooks', _History of Education_ (1996), Vol. 25, No. 1, p. 105.\n\n. Cited in E. Mann, _School for Barbarians: Education under the Nazis_ (London, 1939), p. 62.\n\n. _Allgemeinbildender Grundlehrgang, 1. Teil_ (Breslau and Leipzig, 1941), p. 226.\n\n. Ibid., p. 227.\n\n. Mann, _School for Barbarians_ , p. 63.\n\n. Cited in Samuel and Hinton Thomas, _Education and Society_ , p. 87.\n\n. Cited in Mann, _School for Barbarians_ , pp. 64\u20135.\n\n. On what follows, see BA NS 12\/824, 'Deutsche Volkstumkunde als Erziehungsmittelpunkt im neuen Reich'.\n\n. BA NS 12\/824, 'Gutachten \u00fcber Fr. Dehmlaw: Deutsche Volkstumkunde als Erziehungsmittelpunkt im neuen Reich', 22 June 1934.\n\n. Cited in Mann, _School for Barbarians_ , p. 71.\n\n. Kamenetsky, _Children's Literature_ , p. 149.\n\n. Pine, _Hitler's 'National Community'_ , p. 47.\n\n. Mann, _School for Barbarians_ , pp. 66\u20137.\n\n. Kamenetsky, _Children's Literature_ , p. 174.\n\n. Mann, _School for Barbarians_ , p. 50.\n\n. See, for example, _Mein erstes Buch_ (Dortmund, 1935).\n\n. E. Frank, _Fr\u00f6hlicher Anfang. Ausgabe f\u00fcr Th\u00fcringen_ (Frankfurt am Main, 1943), front cover, and _Bei uns in N\u00fcrnberg. Erstes Lesebuch_ (Nuremberg, 1934), p. 3.\n\n. _Deutsches Lesebuch f\u00fcr Volksschulen II_ (Frankfurt am Main, 1936), p. 9.\n\n. _Fibel f\u00fcr die Volksschulen W\u00fcrttembergs_ (Stuttgart, 1937), pp. 1\u20133.\n\n. W. Kohler, 'Gebet', in _Deutsches Lesebuch f\u00fcr Volksschulen, 3. und 4. Schuljahr_ (Berlin, 1937), p. 275.\n\n. See _M\u00fchlenfibel. Erstes Lesebuch f\u00fcr schleswig-holsteinisches Kinder_ (Braunschweig\/Berlin\/Hamburg, 1935), p. 65.\n\n. _Fibel_ , p. 80.\n\n. _Von Drinnen und Draussen. Heimatfibel f\u00fcr die deutsche Jugend_ (Frankfurt am Main, 1942), p. 17.\n\n. _Fibel f\u00fcr Niedersachsen_ (Hanover, 1939), pp. 80\u201381.\n\n. Ibid., p. 81.\n\n. _Hand ins Hand f\u00fcrs Vaterland. Eine deutsche Fibel von Otto Zimmermann_ (Braunschweig, 1943), p. 65.\n\n. See, for example, _Fibel f\u00fcr Niedersachsen_ , p. 51. See also 'Muttersorgen', in H. Dreyer _et al._ (eds), _Deutsches Lesebuch f\u00fcr Mittelschulen. Klasse 1_ (Frankfurt am Main, 1942), p. 28.\n\n. See, for example, _Lebensgut. Ein deutsches Lesebuch f\u00fcr h\u00f6here Schulen. Dritter Teil_ (Frankfurt am Main, 1937), pp. 152\u20133. See also, K. M\u00fcllenhoff, 'Das brave M\u00fctterchen', in _Deutsches Lesebuch f\u00fcr Volksschulen, 2. Band, 3. und 4. Schuljahren_ (Kiel, 1937), pp. 249\u201350.\n\n. On what follows, see 'Die Mutter mu\u00df entlastet werden!' in _Von neuen Deutschlands. Erg\u00e4nzungshefte zu deutschen Lesebuchern. Heft 1, 3\u20135. Schuljahr_ (Frankfurt am Main, 1935), pp. 38\u201340.\n\n. Pine, _Hitler's 'National Community'_ , p. 48.\n\n. Kamenetsky, _Children's Literature_ , p. 195.\n\n. Hitler, _Mein Kampf_ , p. 389.\n\n. On what follows, see M. St\u00e4mmler, 'Was ist wichtiger?', in H. Dreyer _et al._ (eds), _Deutsches Lesebuch f\u00fcr Mittelschulen. Klasse 1_ (Frankfurt am Main, 1942), pp. 27\u20138.\n\n. For example, see H. G\u00fcnther, _Rassenkunde des deutschen Volkes_ (Berlin, 1938) and J. Graff (ed.), _Vererbungslehre, Rassenkunde und Erbgesundheitspflege: Einf\u00fchrung nach methodischen Grunds\u00e4tzen_ (Munich, 1933).\n\n. BA NS 12\/327, _Die Zeit_.\n\n. BA NS 12\/628, Rundschreiben 'Betr.: Jahresarbeit 1938', 16 February 1938, p. 2.\n\n. Cited in Mann, _School for Barbarians_ , p. 77.\n\n. F. Fink, _Die Judenfrage im Unterricht_ (Nuremberg, 1937).\n\n. See Wegner, _Anti-Semitism and Schooling_ , p. 66.\n\n. E. Hiemer, _Der Giftpilz_ (Nuremberg, 1938), pp. 1\u20132.\n\n. E. Hiemer, _Der Pudelmopsdackelpinscher und andere Erz\u00e4hlungen_ (Nuremberg, 1940).\n\n. Ibid., p. 83.\n\n. Mann, _School for Barbarians_ , p. 81.\n\n. BA NS 12\/16, 'Rassenpolitische Erziehung'.\n\n. See, for example, P. Petersen, _Landvolk und Landarbeit. Lehrbuch f\u00fcr l\u00e4ndliche Berufsschulen. Erstes Berufsschuljahr_ (Breslau, 1939), pp. 22\u20133.\n\n. L. Finckh, 'Du und deine Ahnen!', in H. Dreyer _et al._ (eds), _Deutsches Lesebuch f\u00fcr Mittelschulen. Klasse 1_ (Frankfurt am Main, 1942), pp. 25\u20137.\n\n. For example, F. Hayn, _Politische Sippenkunde in der Schule_ (Leipzig, 1936).\n\n. P. Hasubek, _Das deutsche Lesebuch in der Zeit des Nationalsozialismus. Ein Beitrag zur Literaturp\u00e4dagogik zwischen 1933 und 1945_ (Hanover, 1972), p. 54.\n\n. See, for example, N. Maaken _et al._ (eds), _Ewiges Deutschland. Schroedels Lesebuch f\u00fcr Mittelschulen f\u00fcr den Gau Schleswig-Holstein, 3. Band, Klasse 3\u20136_ (Halle an der Saale, c. 1942), pp. 153\u201365, on 'ancestors and descendants'.\n\n. See H. Stellrecht, 'Das Erbe der Vater', and L. Finckh, 'Heilige Ahnenschaft', in H. Kickler _et al._ (eds), _Dich ruft Dein Volk. Deutsches Lesebuch f\u00fcr Mittelschulen, 4. Band, Klasse 5 und 6_ (Bielefeld, 1942), pp. 229\u201330.\n\n. L. Kahnmeyer and H. Schulze, _Realienbuch enthaltend Geschichte, Erdkunde, Naturgeschichte, Physik, Chemie und Mineralogie_ (Bielefeld, 1938), p. 148.\n\n. Ibid., pp. 139\u201360.\n\n. Flessau, _Schule der Diktatur_ , pp. 150\u201351.\n\n. Petersen, _Landvolk und Landarbeit_ , p. 7.\n\n. Ibid., pp. 7\u20138.\n\n. BA NS 12\/41, 'Acht Tage Schulungslager des NS-Lehrerbundes'.\n\n. Ibid.\n\n. Kamenetsky, _Children's Literature_ , p. 192.\n\n. F. Sotke, _Deutsches Volk und deutscher Staat. Staatsb\u00fcrgerkunde f\u00fcr junge Deutsche_ (Leipzig, 1936), pp. 83\u20136.\n\n. J. Fischer, _Volks- und Staatskunde, 1. Teil_ (Selbstverlag, 1938), pp. 76\u20139.\n\n. A. Waetzig, _Volk, Nation, Staat. Ein Beitrag zur staatspoliticshen Schulung unserer jungen Volksgenossen_ (Stuttgart, 1937), pp. 5\u20137.\n\n. A. B\u00e4umler, _M\u00e4nnerbund und Wissenschaft_ (Berlin, 1934).\n\n. On what follows, see _Kamps Neues Realienbuch f\u00fcr Schule und Haus_ (Bochum in Westfalen, 1937), pp. 151\u20136.\n\n. On what follows, see ibid., pp. 158\u201367.\n\n. Kahnmeyer and Schulze, _Realienbuch enthaltend Geschichte, Erdkunde, Naturgeschichte, Physik, Chemie und Mineralogie_ , pp. 46\u201355.\n\n. Hitler, _Mein Kampf_ , p. 371.\n\n. Ibid., p. 373.\n\n. See for example, H. Eckhardt, _Die K\u00f6rperanlage des Kindes und ihre Entwicklung. Ziel und Weg einer biologische K\u00f6rpererziehung_ (Stuttgart, 1935), which advises on the physical development of children from birth until the age of 18.\n\n. On what follows, see BA NS 12\/814, 'Denkschrift f\u00fcr die Fachschaft f\u00fcr k\u00f6rperliche Erziehung'.\n\n. BA NS 12\/1400, 'An alle Volks- und Hilfeschulen'.\n\n. Ibid.\n\n. BA R 4901\/1 4620\/1, 'Betrifft: Neuordnung des h\u00f6heren Schulwesens', p. 5.\n\n. BA NS 12\/814, 'Denkschrift f\u00fcr die Fachschaft f\u00fcr k\u00f6rperliche Erziehung'.\n\n. BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen)', 1934.\n\n. Ibid.\n\n. BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen): Die Leibes\u00fcbungen in der Grundschule. Jungen und M\u00e4dchen 1.\u20134. Schuljahr'.\n\n. On what follows, see BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen): Lehrplan f\u00fcr Knaben-Schulen 5.\u201313. Schuljahr'.\n\n. On this, see BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen. Volkssport)'.\n\n. On this, see BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen). Spiele'.\n\n. On what follows, see BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen). Wandern und Lagerdienst'.\n\n. On what follows, see BA NS 12\/813, 'Lehrplan f\u00fcr die K\u00f6rperliche Erziehung in den Th\u00fcringer Schulen (Jungen und M\u00e4dchen). Lehrplan f\u00fcr M\u00e4dchen-Schulen. 5.\u201313. Schuljahr'.\n\n. D. Welch, 'Educational Film Propaganda and the Nazi Youth', in D. Welch (ed.), _Nazi Propaganda: The Power and the Limitations_ (London, 1983), pp. 66\u20137.\n\n. _V\u00f6lkischer Beobachter_ , 23 June 1934.\n\n. Mann, _School for Barbarians_ , p. 99.\n4 THE NAZI ELITE SCHOOLS\n\nNazi elite educational institutions performed a special function within Nazi education and socialization processes as a whole. In the 1960s and 1970s German scholars carried out some excellent pioneering research into Nazi elite educational institutions. However, developments in the historiography of the Nazi era have necessitated a reconsideration of these establishments in the light of current knowledge. It is important to underline the significance of these institutions in the Third Reich, particularly as they have received very little scholarly attention in the English-language historiography. In order to achieve its aims of creating a greater German empire, the Hitler regime was committed to a policy of elite education that would provide the Third Reich with future leaders. In order to try to comprehend how the Nazi system consolidated its power and advanced its imperial ambitions, it is useful to look at the institutions in which the new ideal National Socialist man and leader was created. As Scholtz has pointed out, the regime described these schools as _Ausleseschulen_ (selection schools), rather than elite schools. This referred to the 'selection' of a certain type of pupil who had the capacity to become part of the future elite leadership of the nation. The concept of elitism was fundamental to the way in which the Nazi regime sought to organize and refashion German society, with the SS at the pinnacle of the Nazi elite. Far from creating a 'classless society', not only did the Nazis fail to eliminate class distinctions, but also they imposed a different type of hierarchical structure upon German society, based on race and fitness, in which some sectors were valuable and others were expendable. Hence, whilst Nazi policy appears ambivalent, on the one hand claiming to advocate 'classlessness', and on the other hand fostering elitism, its true concern was to try to create a new kind of elite identity. This was based upon race rather than class or social status. The Nazi leadership, with Hitler at the helm, represented the elite of the German 'national community'. The purpose and function of elite education in Nazi Germany was to train a leadership cadre for the next generation. The Nazi elite schools claimed to be meritocracies, but in reality they only selected pupils from within a racially defined framework.\n\nAs Baumeister has shown, the Nazi elite institutions were formed to recruit the future Nazi elite and to prepare them for their leadership tasks. The Nazi regime established three main types of educational institutions to train the future elite of German society: the _Nationalpolitische Erziehungsanstalten_ (National Political Educational Institutions or Napolas), the _Adolf Hitler Schulen_ (Adolf Hitler Schools or AHS) and the _Ordensburgen_ (Order Castles). These institutions represented a microcosm of the Nazi _Weltanschauung_ by fostering the leadership principle, promoting competitiveness and emphasizing life as a struggle and survival of the 'fittest'. They encouraged physical prowess. They excoriated the 'enemies of the Reich', in particular the Jews, Communists and Socialists. They emphasized racial purity, glorified war and fostered militarism. They underlined the necessity for _Lebensraum_ and had a role in the achievement of a 'greater German empire'. Hitler took a strong personal interest in the elite institutions, particularly during the war, when their functions were linked to expansion and the conquest of _Lebensraum_. He was determined that 'our future elite must be given a tough upbringing'.\n\nInstitutional and personal rivalries played a significant role in the development of Nazi elite educational institutions. In particular, competition and antagonism between Bernhard Rust, the Minister of Education, Robert Ley, the leader of the German Labour Front and Baldur von Schirach, the leader of the Hitler Youth, as well as their overlapping areas of competence in relation to the Nazi elite educational establishments, served to underline the tensions between Party and State. This created a lack of coordination in educational policy-making as in many other areas of Nazi policy-making. As functionalist historians have shown, this wrangling created problems in terms of inefficiency and lack of coordination; however, it also demonstrated the status and significance attached to the elite institutions by the Nazi leadership.\n\nElite consciousness was central to securing the eternal life of the German _Volk_ , as exemplified by the Nazi elite formation, the SS. These aims were also found in the Nazi elite educational institutions, which were designed to shape the destiny of the best and most valuable of the nation's stock. Did Nazi elite schools represent anything novel or unique to National Socialism? In one sense, they did not, because elite educational establishments already existed both in Germany and elsewhere. Elite schools were not an invention of the Nazis. On the contrary, special establishments for elite education had existed in many forms long before the Nazi era, such as boarding schools for young members of the German aristocracy set up in the seventeenth and eighteenth centuries and the military schools set up for officer cadets in nineteenth-century Prussia. There were some similarities between the cadet schools and the Nazi elite schools in educational aims, such as the inculcation of strength, courage, discipline and awareness of duty. However, there were also marked differences between the cadet schools and the Nazi elite schools, in terms of both organization and educational aims. The cadet schools took in only the sons of officers as pupils; the Nazi elite schools took in pupils from all walks of life, providing that they were 'racially valuable'. The cadet schools prepared their pupils only for careers as officers; the Nazi elite schools prepared their pupils for a variety of careers. Most significantly, in the cadet schools there was no political training; in the Nazi elite schools political training was an integral part of life, as pupils were reared in National Socialist ideology.\n\n**MODELS OF ELITE EDUCATION IN THE USR AND BRITAIN**\n\nA contemporaneous example of a similar type of educational institution was provided by the educational colonies established by Anton Makarenko in the USSR in the 1920s. The Nazi elite schools and Makarenko's colonies shared the objective of creating bearers of their respective totalitarian ideologies. Both types of institution aimed at forming the ideal members of their societies, through discipline and collective consciousness, the use of symbols, uniforms, military drills, marches and physical education. However, there was also a very significant distinction between the Soviet and Nazi institutions. Makarenko was involved in re-educating _besprizorniki_ \u2013outlawed, homeless and delinquent youths\u2013with the aim of turning them into model Soviet citizens. His objective was to direct these youths to meaningful and purposeful tasks of re-education and to create 'the new Soviet man'. Makarenko was engaged in a type of social engineering programme. By contrast, the Nazi elite schools accepted only their 'ideal' entrants as pupils. Hence, what differentiated the Nazi elite institutions was a specific understanding of 'elitism' in terms of social Darwinist principles. The most significant prerequisite of Nazi elitism was 'racial blood purity'. In this sense, the Nazi elite schools were unprecedented and unparalleled. The Nazis utilized their own version of elite schools for their own ends, for a particular function that was distinctive from previous or contemporaneous examples.\n\nIt is important to recognize that Nazi elite education consciously took on the examples of English public schools in its aims, and yet, simultaneously, it shunned them ideologically. In order to consider this, a brief overview of the English public school tradition is helpful. Public schools had a long history in England, dating back to the fourteenth century when William of Wykeham (1324\u20131404) founded Winchester College. From the start, such institutions aimed 'to socialise future members of an elite'. By the nineteenth century, many prominent public schools had been established, at which pupils boarded and a classical curriculum featured strongly. English public schools underwent a transformation during the course of the nineteenth century in response to the changing structure of society, and education at a public school became the defining mark of a 'gentleman'. The sons of the aristocracy, gentry and the professional and mercantile classes were educated at the public schools and this trend gradually replaced the status of the gentleman being based purely on ancestry.\n\nThe English public schools sought to create a political and administrative elite. Whilst the historical and structural features of the schools were not changed, they did begin to function in new ways. They have been described as 'incipient total institutions'. Whilst the traditions of both boarding and education in the classics were maintained, moral and religious expectations were inculcated in a new way, with sets of rules for behaviour and conduct. Boys were often isolated in these institutions\u2013the advent of the railway meant that it was possible for schools to be located at some distance from their homes\u2013which served the purpose of elite education in 'total institutions'. Self-indulgence and self-interest were to be eliminated. Organized games were introduced to instil obedience to a leader, restraint and self-control. As well as their classical education, the games these pupils played (notably rugby, tennis and cricket) integrated the new elite and distinguished them from the rest of society. These were the hallmarks of their public school education.\n\nThe authoritarian structure of the schools as total institutions was established by the accretion of powers to the headmaster. The headmaster directed the activities of the boys in minute detail through the prefect system, which worked directly under his control. The prefect system itself fitted with the responsibility of public schools to educate an elite. Boys were trained to wield power from an early age. The 'fagging' system\u2013in which junior boys were menials for prefects\u2013was another element of power relationships. In addition, public schools took over much of the responsibility for education and socialization that belonged to the family. In many senses, the closed community of the public schools eclipsed the role of the family. Hence, the English public school became an impregnable total institution and an enduring and 'powerful device for insulating and socialising an elite'.\n\nIn the Nazi elite schools, the ethos of English public schools was imitated in certain respects. There was a distinct attempt by Nazi educationalists to emulate British public school traditions as they perceived them. However, the Napolas did not imitate the class distinctions that were the hallmark of the British public school system. Instead, they sought to take in the best of German youth, regardless of their social status. Nazi educationalists compared their Napolas with British public schools in terms of their ethos and aims:\n\nThe boy is removed from the spoiling influence of the parental home at an early age, and at first has difficulty in establishing his own position among his fellows. But as a rule, the need to survive wakens the necessary forces in him, which toughen him and provide him with security and a firmness of will . . . Public schools are explicit instruments for shaping the individual pupil into a uniform national type, with an equally uniform system of values. Our most recent educational endeavours in the Napolas . . . run along the same principles. Like the public schools in England, they are meant to train an elite, a reservoir for leadership.\n\nThe emphasis on competition and compulsory participation in sports was particularly strong. Furthermore, 'Through the strengthening of historical consciousness, German consciousness . . . an awareness of the national community, perspectives are being created which ultimately culminate in an organic view of the whole. As in public schools, the authoritarian principle is indispensable'. The aim was to create a new leadership elite drawn from across the social spectrum in Germany to rule the greater German empire. In a speech to armaments workers in Berlin on 10 December 1940, Hitler extolled the greatness of the Nazi elite schools in comparison with the English public schools, which he scorned as perpetuating the English moneyed aristocracy. He stated that 'we take the gifted children of the masses, sons of workers, of peasants' into the Napolas and AHS.\n\nThus we have created great opportunities to rebuild the state from below. This is our aim. It is a marvellous thing to be able to fight for an ideal like this . . . We imagine a state in which in the future every position will be occupied by the ablest sons of our people, irrespective of their origin, a state in which birth means nothing and achievement and ability everything.\n\nIn comparison, he described Britain as 'a state governed by a thin crust, the upper class, who send their sons automatically to specific educational institutions like Eton College'. Hitler described two different worlds, in which he clearly praised the Napolas and AHS and denigrated the British public schools: 'in the one case the children of the people, in the other only the sons of a financial aristocracy'.\n\n**NAPOLAS**\n\nThe Napolas were designed to educate future top-ranking government and army personnel. They were state-run boarding schools under the aegis of the Ministry of Education. They were not affiliated to either the Party or the Hitler Youth. The first three Napolas were established in April 1933 in Pl\u00f6n, K\u00f6slin and Potsdam, in former cadet school premises. In 1934, five further Napolas were set up in Berlin-Spandau, Naumberg an der Saale, Ilfeld, Stuhm and Oranienstein. In 1935, another five Napolas were established. They were housed in either renovated or newly constructed buildings that corresponded to Nazi ideals, with exacting standards of hygiene in the living, sleeping and washing areas. Communal rooms were designed to strengthen the sense of spirit and value of the young men. Sports facilities and equipment were comprehensive, including a gymnastics hall, a swimming pool, a boathouse and stables. As boarding schools, the Napolas offered the possibility of extensive control of the education and socialization processes of their pupils.\n\nAugust Hei\u00dfmeyer, the SS Napola inspector, declared that the Napolas represented 'something different from the mere transformation of any type of school within the framework of the old secondary school system'. He described the aims of the Napolas as follows:\n\nAll true education is education for real life in its full extent; it is political education. The purpose of political education, however, is the education of a posterity carrying its own community of life into the future. It is formative education, education designed to mould a type, and such education is achieved in our nation today through community and team education. The National Political Educational Institutions have the aim of removing the education of youth from the plane of intellectual education to that of true education, that is to say, of total education in a tightly-knit community, of education which embraces, as far as possible, all human powers and which, as political education, is always education which moulds the individual and forms the team.\n\nAdmission to the Napolas was very strict. Entrants had to pass a double selection process, consisting of a pre-inspection and an entrance examination. In order to be admitted, a prospective pupil had to be of 'Aryan' descent, a member of the Hitler Youth, physically fit, healthy and sponsored by his _Gauleiter_. In a series of endurance tests, over the course of one week, Napola selectors checked prospective pupils for courage, stamina and physical ability. In October 1937, Bernhard Rust, the Minister of Education, stated: 'It is of the utmost importance that the Napolas receive those German boys who by their attitude and ability meet the special requirements of these institutes'. Once admitted, pupils had a six-month probationary period, during which time they could be expelled from the institution if they failed to fulfil the expectations of them. The Napolas were 'total institutions', designed to give a complete National Socialist education to their pupils, who would then be able to provide exemplary service to the _Volk_ and state. In addition to the usual school syllabus, there was education in National Socialist principles, for example, through daily discussions about the editorials in the _V\u00f6lkischer Beobachter_ , as well as a great emphasis upon physical activities, including boxing, war games, shooting, rowing, sailing, riding, gliding and motorcycling. Physical education was considered to be crucial to character formation. Pupils had to undertake 'toughening up exercises' such as grappling with Alsatian dogs. Hans M\u00fcncheberg, a former pupil of the Napola school at Potsdam recounts: 'If anyone showed weakness he was considered a wet, a weakling, a coward, a disgrace to the whole platoon or the whole company'. In the Napola at Pl\u00f6n, recalls Napola student Theo Sommer, 'physical stamina was driven to the limit'.\n\nThe Napola pupils spent six to eight weeks on a farm and a further six to eight weeks working in a factory or coalmine as part of their training. This was designed to give the pupils an inside view and experience of the workers and their lives. Placement in agriculture and industry was not a new phenomenon, but a tradition that could be traced back to the earlier youth programmes. However, the highly political aspect of the programme designed to prepare the pupils for service to National Socialism was novel. Scholtz has described the training in the Napolas as a mammoth programme. A Napola pupil was referred to as a _Jungmann_ (young man). School classes were called 'platoons'. Pupils engaged in manoeuvres and field exercises, military marches and war games, as well as learning orientation skills. The pupils were to have 'soldierly' characteristics and leave the Napolas with a capacity to lead and an in-depth knowledge of National Socialism.\n\nIn his autobiography, Peter Neumann, a junior SS officer, who received a Napola education at Pl\u00f6n, describes it as 'gloomy, icy and horrid'. He outlines the racial instruction there and tells of his gliding experiences: 'I am a little giddy from this first solo flight and my ears are buzzing slightly. But what will one not do to gain the admiration of one's friends?' There is some suggestion here of the need to behave in a particular manner and to appear courageous. Other former pupils recollect the prestige associated with belonging to the elite school and the opportunities it afforded. Hans-Georg Bartholom\u00e4i, a former pupil of the Napola at Naumburg, recalls: 'There was a wide range of travel and \"manoeuvres\", as we used to call them. We could go skiing and gliding. We went to the Alps, we went to the lakes. That was pretty unusual for any boy in those days . . . People made a fuss of us. And of course we picked up on that.' Yet the Napolas removed young boys from their families and friends into a strict, unfamiliar, military environment, to which they had to adapt swiftly, showing no signs of personal weakness or homesickness. Simultaneously, the tough demands placed upon the boys made them feel special and chosen.\n\nRust called for 'total education' in the Napolas. Pupils were educated with a consciousness of racial selection of the German _Volk_. Their education was not aimed at the development of individual or critical thought, but at service to the _Volksgemeinschaft_. Indeed, this was noted by a foreign observer of the Napola at Bensberg, a Dutch educationalist, Dr Goedewaagen, who reported that: 'In the Napolas the foundation is laid out for the education of personality free of any individualistic attachment'. Goedewaagen was impressed with the ethos and _modus operandi_ of the Napolas and recommended their replication in the Netherlands.\n\nAnother significant feature of Napola education in the period before 1939 was the organization of exchange visits abroad. Exchange programmes were arranged in German Southwest Africa, America and Britain. They were designed to confirm to Napola pupils their own superiority and to consolidate Nazi doctrines. Napola pupils were instilled with the main tenets of Nazi ideology\u2013anti-Bolshevism, anti-Semitism and nationalism. The _F\u00fchrer_ stood at the centre of the _Weltanschauung_. The pupils were taught that life was a struggle and that a militaristic spirit was necessary in order for the German _Herrenvolk_ to survive and create a new ruling order. As the political soldier of the _F\u00fchrer_ , the _Jungmann_ had to become an unconditional and ardent advocate of the Nazi _Weltanschauung_. The regime also attached significance to the training of teachers in the Napolas. Special training courses were set up for this purpose and records kept of the performance and outlook of Napola teachers.\n\nThe conception and aims of the Napolas changed during the course of the war. Physical training was extended more specifically to weapons training. The Napolas became part of the process of building, consolidating and securing a 'greater German empire'. They became 'forts of the F\u00fchrer for the protection and strengthening of the Reich'. They came increasingly under the direction of the SS, which exerted its influence upon them by claiming the monopoly on political education. Even before the war, as Kogon has shown, the SS intended to create a new generation of leaders through the Napolas. Between 1936 and 1939, under a new SS Napola inspector, August Hei\u00dfmeyer, the Napolas had already become increasingly influenced by the disciplinary and racial ideas of the SS. During the war, Heinrich Himmler, the head of the SS, clearly wanted to subordinate the Napolas to the SS entirely and to use them to further his expansionist aims. With the military successes of the Third Reich, a role developed for the Napolas in securing its racial and ideological goals. Rust explained their function and task as education of the imperial idea. They were linked to the concept and goal of _Lebensraum_. Between 1941 and 1944, new Napolas were established across the occupied territories to educate those young people that were considered to be 'racially valuable'. Himmler discussed specific plans with Hitler to create Napolas in Holland and Norway. By 1944, there were thirty-seven Napolas. Hitler supported the expansion of the Napolas during the war and the Nazi regime aimed to establish a total of 100. Whilst the Napolas were scarcely able to carry out the new task assigned to them, it is significant to note how different this mission was from their original conception and _raison d'\u00eatre_.\n\nHence, the Napolas combined some of the traditions of the earlier cadet schools and the English public schools in their brand of elite education, but blended them with a National Socialist ethos that was distinctive from both models. The Napolas were based upon the overriding idea of 'political soldierliness' and the fight for National Socialism. They combined elite consciousness and leadership education for all professions, with political training in the Nazi _Weltanschauung_ , racial awareness and an emphasis on physical education as a means to character formation. The Napola has been described as truly 'a National Socialist institution _sui generis_ '.\n\nNot surprisingly, elite education for girls was not deemed as significant as for boys; nevertheless, three Napolas for girls were established in the Third Reich. The first Napola for girls was set up at Hubertendorf-T\u00fcrnitz in Austria following the Anschlu\u00df in 1938. In 1941 and 1942 respectively, two further Napolas for girls were founded at Achern in the German state of Baden and at Castle Kolmar in Luxembourg. The existence of these three Napolas for girls appeared as an anomaly in a state in which ideological values placed girls in the role of mothers and guardians of the hearth and home. Yet, ideological tensions existed between the majority of the Nazi leadership, who maintained that girls should remain in the domestic sphere, and a small group of women in the Nazi power structure, who viewed elite schooling for girls as a means of access to the professions for capable girls. There was a lengthy struggle over the _raison d'\u00eatre_ and aims of Napolas for girls. Hitler was clear in his view that women's role as mothers was the most important.\n\nIf today a female jurist accomplishes ever so much and next door there lives a mother with five, six, seven children, who are all healthy and well brought up, then I would like to say: From the standpoint of the eternal value of our people, the woman who has given birth to children and raised them and who has given back our people life for the future has accomplished more\n\nAlfred Rosenberg also firmly believed that leadership roles in the state belonged to men. Why then did the Napolas for girls come into being?\n\nWegner argues that the elite schools for girls 'grew out of changing economic realities that challenged Nazi assumptions about gender and the workplace'. By the late 1930s, a shortage of technically skilled and professionally qualified workers led to a need for the Nazi leadership to compromise its ideological beliefs and to allow elite educational institutions for girls to be established. Hence, practical concerns overcame ideological tenets about the suitability of women working in the professions. Yet those calling for the opening of Napolas for girls, among BDM leaders and from the Ministry of Education, envisaged them as institutions in which girls could be educated 'in body and soul to become wives and mothers'. Whilst Hei\u00dfmeyer stated that girls would eventually take on leadership roles in Nazi organizations, he equally stressed the expectation that they would give birth to many children. Hence, the girls' Napolas had an ambivalent position from the outset.\n\nThe selection process, like that in the Napolas for boys, was based upon 'blood' and 'race', rather than class or parental income. Girls also had to be actively involved in the BDM. As in the boys' Napolas, classes were called 'platoons' and the curriculum included drills, marching and education in key aspects of Nazi ideology. The curriculum at the Austrian Napola consisted of a combination of academic subjects with domestic science. It had space for between 200 and 250 girls, but only 123 girls enrolled there, half of whom were Austrian. The contradiction remained apparent between schooling them for motherhood and schooling them for the professions. The same type of tensions existed at the German girls' Napola at Achern in 1941. Indeed, due to financial constraints, this institution shed its elite status after only one term, becoming a _Heimschule_ (home-making school) instead. At Castle Kolmar too, the curriculum was a blend of home economics to prepare pupils for motherhood and academic subjects that would allow pupils to enter university, along with physical education and performing arts. In 1942, Castle Kolmar enrolled 192 female pupils from all parts of the Reich. This Napola lasted for two years.\n\nThe Napolas for girls were a short-lived experiment, ambivalent from the start in terms of their existence and their goals. The nature of the Nazi state and the ideology that underpinned it did not lend itself to girls' elite education. Although the girls' Napolas came into existence for pragmatic reasons, ideological concerns about girls' roles in Nazi society never really left them to the task of training girls for professional or leadership roles. The girls' Napolas did not receive a comparable level of funding and status to the boys' Napolas.\n\n**ADOLF HITLER SCHOOLS**\n\nThe Adolf Hitler Schools (AHS) were established in 1937, as conscious and direct rivals to the Napolas, which had been formed during the first year of Nazi rule. They were set up, with Hitler's approval, under the aegis of Robert Ley and Baldur von Schirach, in order to train future political leaders. Hitler ordered that they should take his name and he continued to show an interest in their development throughout their existence. The AHS were purely Party schools and remained outside the realm of the Ministry of Education and therefore outside Rust's jurisdiction. Ley's office, the German Labour Front (DAF), was in charge of the initial organization and administration of the AHS. Schirach appointed an Inspector of the AHS, within his National Youth Leadership, who was directly responsible to him. Rust had initially agreed to the foundation of the new AHS on the condition that his Ministry would participate in the selection of pupils and teachers. However, Ley and Schirach deliberately bypassed Rust, presenting the foundation document to Hitler without any mention of this agreement. Hence, Hitler signed the foundation document which granted neither Rust nor his Ministry any control in the AHS. Rust's subsequent protestations were ignored.\n\nSignificantly too, with their status as purely Party schools, the AHS were unprecedented as educational institutions in Germany. Indeed, in his speech given at the laying of the foundation stone of a new AHS on 14 January 1938, Schirach clearly stated: 'We have not reformed something extant, but begun something new'. The AHS deliberately and consciously distanced themselves from the Napolas, exalting their own status and significance as 'Party schools'. They were designed to educate the youth who would take over and secure National Socialist power in the future by working in the offices of the Party. The principal purpose of the AHS was to develop a 'leadership corps' that was devoted to the Party with unconditional loyalty and obedience.\n\nOn 10 December 1940, in a speech to armaments workers in Berlin, Hitler spoke about the purpose and aims of the AHS:\n\nWe are bringing talented youngsters, the children of the broad mass of our population. Workers' sons, farmers' sons, whose parents could never afford to put their children through higher education . . . Later on, they will join the Party, they will attend an Ordensburg, they will occupy the highest positions. We have a goal which may seem fantastic. We envisage a state in which each post will be held by the ablest son of our people, regardless of where he comes from. A state in which birth means nothing, but performance and ability mean everything.\n\nIn total, there were twelve AHS. Each _Gau_ (Party region) selected prospective AHS pupils from all 12-year-old boys who demonstrated distinctive _F\u00fchrereigenschaften_ (leadership qualities or characteristics). Prospective pupils had to be selected from the _Deutsches Jungvolk_ (the Hitler Youth group for boys aged 10\u201314). Applications made by parents were automatically rejected. The AHS prided themselves on offering opportunities for social advancement, as social class was not a barrier to entry. The Party allocated funding to the Adolf Hitler Schools, and, in principle, the parents of AHS pupils were not obliged to pay for their education, although, in practice, many did contribute towards its cost. In reality, the majority of pupils came from middle-class backgrounds. Almost half of the pupils (49.7 per cent) who joined the AHS in the first two years stated that their fathers were civil servants, teachers, office workers or officers, whilst a considerably smaller percentage (19.5 per cent) stated that their fathers were craftsmen, agricultural labourers or industrial workers. The percentage of Party officials' sons in the AHS was also significant. The national average was 2.3 per cent, but in the _Gau_ of Munich-Upper Bavaria, the early stronghold of the NSDAP, 11.7 per cent were the sons of Party officials.\n\nThe AHS admitted pupils that had been pre-selected in the Hitler Youth, but with the additional 'sifting' process of a two-week selection camp, which took place at the beginning of each year. This was preceded by a 'racial examination'. Boys who did not pass this examination were not allowed to take part in the camp. The proportion of prospective candidates who failed to get through this stage was quite high. For example, of forty-eight candidates in Baden in 1940, fourteen failed the first medical examination. The _Gau_ youth camps allowed the Party to assess and observe the physical and mental capacities of the AHS candidates. The candidates were split up into groups of six to eight boys. Each group was monitored and observed by a Hitler Youth group leader. The leader spent the whole time with the boys for the duration of the camp in order to observe not only their performance in their tasks, but also their behaviour during their leisure time. At the end of the two-week camp, the candidates were assessed according to character and competence ratings. Strength of character and toughness were tested by a range of activities including war games, gymnastics, marches and tests of courage.\n\nHereditary health and racial purity were the fundamental criteria for admission. 'Proof of absolute health, without physical disabilities or deformities', as well as 'hereditary health of family' and 'proof of Aryan descent' were the essential conditions for admission to the AHS. To this end, prospective pupils had to include in their application a photograph, hereditary health certificate and genealogical table. Their parents were responsible for writing the genealogical table, with help from the Party. In addition, prospective pupils had to be able to demonstrate physical toughness, a strong character and an instinct to dominate others. Bravery was another significant characteristic that was required by the assessment panels, as indicated by instructions for selectors, which stated: 'We can only use boys who have courage'. A number of exercises and activities, particularly boxing and wrestling, were used as tests of courage. The daily schedule of the selection camps included a full day of mainly physical activities, starting at 7 am with an hour of early sport before breakfast and ending at 8 pm. The selection camps placed much greater emphasis upon physical capacity than mental ability. After 1938, when the AHS boys came under increasing criticism for their lack of intellectual capability, the admission process was amended to include academic criteria. New selection guidelines introduced in 1938 attached greater significance to the level of intelligence of prospective pupils. Furthermore, a new requirement stipulated that only teachers who were qualified to teach at the _Gymnasien_ were allowed to teach at the AHS after 1938. By 1941, more than half of the pupils selected for the AHS had passed the _Gymnasium_ entrance examination. From 1942 onwards, the search for intelligent boys continued and intensified and by the end of the war, physical and intellectual entry requirements had become more or less equal. However, the initial stigma of intellectual inferiority remained with the AHS throughout their existence.\n\nThe Party demanded a new type of educational institution to train the future leading class. The architectural style of the school buildings, as well as the timetable, was to express this desire, with new methods of education for the leadership class. The AHS had three main goals in the education of its pupils: the pupils were to be politically moulded in the National Socialist _Weltanschauung_ ; they were to be physically fit; and they were to be trained to become future Party functionaries. The superiority of the 'Aryan' race was emphasized in textbooks provided specially for the AHS. History, literature and biology were all taught within the context of the Nazi _Weltanschauung_ in order to imbue pupils with its tenets. The incompatibility of the National Socialist world view and Christianity was emphasized in AHS education. Themes in literature included heroic death, the struggle for the fatherland and the significance of the German landscape. In this way, the _Blut und Boden_ myth was propagated. The glorification of war was another popular theme. Between 1941 and 1944, new textbooks were issued on the subject of Germany's need for the conquest of _Lebensraum_. As well as National Socialist ideology, the AHS placed great importance upon physical education, particularly combat sports such as boxing, wrestling and fencing, as well as pre-military training. Physical education took up a high proportion of timetabled hours: for example, in the AHS weekly timetable in 1941, it took up fifteen out of thirty-seven hours. Slavic languages were taught with the aim that AHS pupils would be able to give orders to subordinated people in the Nazi-occupied eastern territories during the war. Foreign languages, mathematics and the natural sciences were taught at a basic level, but not in depth. A former AHS boy, Harald Grundmann, recalls his education with regret: 'I am ashamed how little we knew about German poets and men of letters\u2013from Thomas Mann to Gottfried Benn; how scanty our knowledge of mathematics was . . . our qualifications were pretty miserable'. The AHS considered it to be 'timewasting' to teach these subjects in detail to pupils who were destined to become political functionaries and Party 'large-capacity administrators' (i.e. those with wide-ranging functions). It was more important for them to be imbued with the Nazi _Weltanschauung_. The pupils were to be turned out confident in Nazi ideology and its legitimacy. The method of teaching employed was designed to ensure this. Furthermore, the AHS had no grading system or school reports. Instead, there were written assessments of the pupils, which focused in particular on the development of the students into 'leader' personalities.\n\nThe AHS pupils spent time undertaking practical work in different Party offices, in order to round out their skills and give them an insight into jobs they might undertake in the future. In addition, there was handicraft instruction. Practical work was designed to broaden competence and knowledge, as well as to enhance economic understanding. 'Working with material' was intended to broaden the outlook of the boys and to give them respect for people who worked in handicrafts. The aim was not to create 'ready-made locksmiths or carpenters', but 'men who can be useful in the workforce, in the army and in the battle for life', who have acquired a feeling for handicrafts and technical aspects of their environment. As with the Napolas, the education of boys in the AHS played a significant role in the Nazis' long-term plans for a New Order in Europe, creating an administrative corps of enthusiastic and trained Party leaders, with an unconditional belief in the Nazi _Weltanschauung_.\n\n**_ORDENSBURGEN_**\n\nThe _Ordensburgen_ (Order Castles) were intended to be the finishing schools for the Nazi elite. Scholtz has described them as a 'characteristic product' of the Nazi era. The decision to begin the construction of the Order Castles originated with a discussion between Hitler and Ley, in July 1933, on a visit to a workers' school in Berlin. Ley 'intuitively' began to plan four 'Education Castles' based on Hitler's ideas to establish institutions to train Party officials. The building of the first Order Castle began in February 1934. Alwin Seifert, the construction supervisor, stated that the Order Castles were to display 'superhuman magnitude' and to inspire 'knightly actions'. However, the completion of the four Order Castles was never achieved and Ley's plans were not fully realized.\n\nThe _Ordensburgen_ attracted much attention from German and foreign observers, both during the Third Reich and in the early historiography of the Nazi era. However, there has been no recent analysis of the role of the _Ordensburgen_ in the Nazi state. The _Ordensburgen_ were to be set up in four locations across Germany: Cr\u00f6ssinsee in Pomerania; Vogelsang in the Eifel Mountains near the Belgian border; Sonthofen near Lake Constance in Bavaria; and Marienburg in East Prussia, near the Polish border. Having completed their AHS secondary education, a six-month period of compulsory Labour Service, two years in the army and entered their chosen profession, selected future political leaders were to be trained on a four-year programme, one year at each Order Castle. At Cr\u00f6ssinsee, students underwent pre-military training, took part in parachute jumping and learned about German history and race. They were educated about the dangers of 'racial pollution' and trained to view themselves as racially superior, as 'the aristocracy of the earth'. At Vogelsang, they were instilled with bravery and heroism. At Sonthofen, they were to study Hitler's _Mein Kampf_ and the works of other Nazi ideologues, such as Alfred Rosenberg. Extended skiing and mountaineering expeditions here were designed to test their physical capacity and endurance. By the end of 1939, Peter Neumann, a junior SS officer, writes that: 'The training is getting more and more tough, cruelly tough'. He describes the rise in the number of accidental deaths at Sonthofen: 'The weak must go to the wall here. Only those who survive will have the right to form part of the National Socialist elite'. Finally, at Marienburg, the students were to learn about Nazi foreign policy, _Ostpolitik_ (policy towards the East) and the need for 'living space'. Here the students would complete their political education. The four-year programme was designed to equip students for their role as leaders of the Third Reich. During the course of their _Ordensburg_ training, the students were obliged to spend three months of each year working in Party organizations in order to accrue practical experience.\n\nThe _Ordensburgen_ were not spartan, but, on the contrary, provided comfortable accommodation to their students. The buildings were lavish in design and immense in scale. The students had their meals served to them in the large dining halls. They were allowed visits from their wives at certain times. Funds were made available to them for theatre trips and other visits, even abroad. They enjoyed special privileges and distinctions, whilst being educated for their 'elite function'. Ley promised the _Ordensburg_ students that 'we open doors to the highest positions in the Party and in the State'. In return, he expected total obedience and trust.\n\nA paean to the Order Castle Vogelsang in _The Order_ magazine evokes the sentiments inspired by the _Ordensburgen_ :\n\nRuins surrounded by tales of sentimental knighthood romance, \nThe wanderer shall not search, your tower greets him from far away. \nYour walls are represented through Nordic strength, \nLike a stone finger of vow you stretch above the Eifel and Urft \nYour staggered building issues from the bitterness of the landscape, \nNo petty ornaments ruin the praise of clear lines. \nYou are appointed to announce heroic loyalty and strength: \nDeep into the hearts of youth may the F\u00fchrer's law descend.\n\nA number of interesting metaphors have been applied to the _Ordensburgen_. The one most consciously and deliberately employed by the Nazis was that evoked by their name\u2013Order Castles. This had obvious associations with medieval knights, and there was symbolic significance attached to the mission of this new generation of 'knights' as creators and leaders of the Nazi empire. A contemporary foreign reporter commented on this: 'The young men are told that they form a Nordic Crusading Order like that of the Knights Templar of old'. A great sense of the historical past and its connection with a grandiose current mission was conveyed. The careful selection of their location and the grandeur of the _Ordensburgen_ buildings underlined this. Their interiors were grand and lavish. Vogelsang, designed by the architect Clemens Klotz, boasted both a dining hall and a lecture hall for 1,000 people. On a platform at the front of the lecture hall stood a massive statue of an idealized Aryan male\u2013'powerful, muscular, saluting with a raised right arm'. The communal rooms at Sonthofen included a vast dining room for 1,500 people, with marble walls and floors, a 'hall of the community' for 2,500 people, lecture halls for 1,500 people, as well as a ceremonial council room. The _Ordensburgen_ were also associated with the Valhalla, the palace of Norse mythology, in which the souls of slain heroes lasted for eternity. A foreign commentator described the _Ordensburgen_ pupils as prospective leaders of 'the Hitlerite Valhalla'. In this way, the students of the _Ordensburgen_ were conceived of as 'heroes' and the grandiose plans of the Nazi leadership for the longevity of the Third Reich as an empire to last a thousand years were evoked.\n\nAnother foreign commentator conceived of the _Ordensburgen_ in different terms, however. He saw them as pagan, anti-Christian institutions, in which 'monks of a new kind' were trained. Reporting on his visit to Vogelsang in November 1937, a correspondent for _The Manchester Guardian_ wrote that 'the keepers of the consciences of these monks direct them to the worship of blood, of the soil'. He added too that he left Vogelsang 'profoundly disturbed, astounded at the emptiness of the teaching given there'. He described the pupils as 'impeccably aligned' as they marched and sang. Peter Neumann recounts his experiences at Vogelsang in his autobiography: 'Combat training is terrifying'. He describes animal combat sessions and, in particular, fighting between specially trained Alsatian dogs and the _Ordensburg_ students: 'This is the kind of exercise which contributes to the \"character forming\" process at Vogelsang'.\n\nAnother student at Vogelsang gave his view of the institution:\n\nWe have gathered from all regions of the German _Volk_ , to spend a year together here. We are at an age, at which the learning period is generally assumed to be over, in which most have started to work independently. A majority of us has already founded a household . . . Now that we have left profession and family and gathered here, it shows the awareness that there is a duty beyond our individual lives\u2013the duty towards the people.\n\nWhat were the expectations of these 'leader candidates'? The foremost expectation was obedience and the knowledge that they had to learn to listen and obey in order to be able to lead. Obedience training required a 'constant hardness' of the individual on himself and a battle against the inclination to let himself go. The students knew that this was not easy, but hoped to emerge from the year strengthened through this training. The next expectation was ideological training. They required a deep and enhanced knowledge of National Socialist ideology and of the _Volk_ to prepare them for their future role as leaders. Their living together in comradeship 'in a small people's community' would prepare them for future lives and tasks in the real _Volksgemeinschaft_. Hence, the comradeship at Vogelsang was 'not just a pleasant way of living together', but also 'the highest duty' of daily lives. The students realized that their year at Vogelsang would not be easy, but its 'beauty' lay in that very difficulty, as they felt their mental and physical powers tighten and they learned how to overcome all obstacles to their goal.\n\nThe students of the _Ordensburgen_ were called Junkers. This was a term deliberately adopted by the founders of the _Ordensburgen_ , as the term Junker referred to the Prussian aristocracy. The _Ordensburgen_ 'Junkers' were groomed to be 'the aristocracy of the earth'. Scholtz has argued that the _Ordensburgen_ were proof of Hitler's will to create a 'New Order' in society. They were conceived of as part of the SS state, whose aim was to rule over a great German empire. Together with the _SS-Junkerschulen_ (Junker Schools or SS elite schools), the _Ordensburgen_ were to be 'the real colleges of the future National Socialist aristocracy'. In Hermann Rauschning's conversations with Hitler, the relationship between Hitler's will to create a new man and the _Ordensburgen_ is established. Robert Ley, who was in charge of the organization of the _Ordensburgen,_ stated that the curriculum represented 'four years of the hardest possible physical and mental exertions'. He was very clear about the main aims of elite education at the _Ordensburgen_ : 'Firstly, we want to test the initiative, courage and daring of a man and to promote these qualities where they exist. Secondly, we want to know whether these men are fired by an overweening ambition to become leaders of men, to dominate, to become masterful . . . Thirdly, anybody desiring to govern over others must be able to rule himself '. There were obvious echoes here of Friedrich Nietzsche's concepts of the _\u00dcbermensch_ (superman) and the 'will to power'.\n\nHowever, in reality, the academic standard of education in the _Ordensburgen_ was not particularly high. There was no fixed educational schedule. Indeed, there was a popular perception and some official criticism of the _Ordensburgen_ that the educational standards were low and that their pupils were not necessarily clever. One headteacher reported that the knowledge of his _Ordensburgen_ pupils was limited and that it took them a lot of time to process the material they were being taught. Indeed, Scholtz has shown that education became a secondary function of the Order Castles, which increasingly became used as 'drinking halls' for Party comrades, in which they could relax, feel 'at home' and 'remember the old times'. At one point, Ley even envisaged a role for the _Ordensburgen_ as 'Strength through Joy' hotels with 2,000 beds to enable German workers to use them for weekly holidays. This was a gross deviation from their original purpose. Scholtz argues that only occasionally were the buildings used for their original design as 'Education Castles'. Furthermore, the outbreak of the war prevented the possibility of the _Ordensburgen_ training being completed, as their students and prospective students were called into the armed services. After the invasion of Poland, the _Ordensburgen_ were used to 'educate members of the Party about tasks in the _Ostgebieten_ ' (eastern regions). In addition, the _Ordensburgen_ were used to accommodate pupils from the AHS, as the latter were running out of space.\n\nThe role of the _Ordensburgen_ extended beyond the initial and main function of training future Party leaders. They were also to be 'the spiritual and ideological centres' of the NSDAP. The _Ordensburgen_ were used for training courses for existing political leaders, including _Gauleiter_ (regional leaders) and _Kreisleiter_ (district leaders), who were evaluated and assessed. The training courses, with a maximum number of 200 attendees, were split into working groups of twelve to eighteen political leaders. The rationale for this was mainly personnel planning for the future leadership of the party, based on the example of selection processes that already took place for the AHS and the _Ordensburgen_ pupils. The aim was to check 'if each political leader is appropriate in terms of his performance and his whole attitude, if shortcomings can be eradicated by further training courses, if he has to be posted somewhere else, or sacked due to ineptitude'. This would help the Party to make good choices in the efficient and effective selection of leaders, from among whom the best could be cultivated if they showed the correct aptitude and performance:\n\nA high degree of racial value and faultless characteristics, matched with determination, will power and readiness for action are requirements which have to be met by each political leader. Furthermore, a disposition for above-average performance and good appearance which compliments and rounds off the personality is absolutely essential, so that the political leader, through the combination of his inner and outer attitude, can lead the _Volk_ and have its absolute acceptance.\n\nBy pursuing these goals, the Party would not have to worry about future leadership.\n\nFurthermore, the _Ordensburgen_ were used during the wartime period for short residential courses for various other groups. For example, a training course was held at Sonthofen between 12 August and 4 September 1940 for 'ethnic Germans' from the former South Tyrol. Once they were resettled and trained, they would be used for Party work, as _Blockleiter_ (block leaders) and _Zellenleiter_ (ward leaders). The course of instruction included daily lectures on themes including: 'Greater Germany'; 'The Party as the F\u00fchrer's Tool'; 'The Organizational Composition of the Party'; 'Jewry'; 'The Biological Principles of National Socialism and the Struggle for the Conservation of the German _Volk_ '; 'National Socialist Economic Policy' and 'National Socialist Social Policy', as well as lectures on the duties of the key organizations of the Party including the DAF, the NSF, the HJ, the NSV, the KdF, the SS and the SA. In this way, the participants learned intensively about all aspects of National Socialism and the workings of the NSDAP. The course also included film screenings, working groups and an excursion.\n\nAn exploration of the lecture themes, which were designed to cover all aspects of the Party and its policies, gives a good indication of the ideological training given at Sonthofen. In each case, a Party member with an expert knowledge of the particular theme was appointed to give the lecture. A lecture given by Danninger on 'The Duties of the Great German Farming Community' explained Walther Darr\u00e9's agricultural ideology and National Socialist agricultural policies. It emphasized the connection of 'German blood' to 'German soil' and the need for German farmers to settle and cultivate the land. Danninger described the farming community as 'the ultimate bearer of the German people'. This lecture was supplemented by one from H\u00f6rgenrode on the theme of 'The Agricultural Production Battle', which further stressed the significance of 'blood and soil', as well as the need for autarky. It concluded with the principle that 'the Reich should be like a great farm, which should produce everything it needs for itself '.\n\nLectures on Party organizations, such as the SA, the SS, the HJ, the NSV, the KdF and the DAF, examined the duties of each organization. Kulisch, who spoke on the subject of the DAF, emphasized the 'common duties' of German workers and concluded that: 'the strength of a nation lies in its readiness to make any sacrifice . . . The enjoyment of leisure time would not be of any value, if it were not preceded by the achievement of duties. Adolf Hitler has taught us over and over again that work is the blessing of the German _Volk_ '. Helmreich's talk on a similar theme described 'work duty' as 'an essential expression of the German people'. Not surprisingly, lectures on racial ideology formed an important part of the training. In particular, W\u00f6lpl's lecture on the 'biological foundations of National Socialism' examined the essence of German blood purity, the evils of 'racial miscegenation', the 'danger of hereditary disease', and the need for the German nation to reproduce and to have large families. Hartlieb's lecture on 'Jewry' described the perils of Jewish 'world power' as well as the National Socialists' 'ideological war' against Judaism.\n\nA course was put on at Cr\u00f6ssinsee for South Tyrolese settlers in November 1940. This course included speeches and lectures on a number of themes, particularly relating to folklore and border issues. Party members addressed participants on a variety of subjects, for example 'The Tasks and Aims of the SA'. The programme of the training courses held at the _Ordensburgen_ reveals much about their purpose and structure. A course held at Sonthofen in August 1941 was based around the theme of national folklore and tradition. Each day began with a wake-up call at 7 am, with flag-raising and breakfast at 8.15 am. The morning session consisted of singing and then either rehearsal of a play or a lecture. Lectures were on topics such as 'Folklore Work as a Political Task' or 'Practical Folklore Work'. After lunch, there was a film, play rehearsal or preparation for the social evening. After dinner, there were 'folklore evenings', puppet shows or a folk play. Hence, the aim was to promote a sense of national affinity.\n\nThe _Ordensburgen_ were also used to hold special courses for 'resettlers' wives'. For example, a course at Cr\u00f6ssinsee held from 14 January to 4 February 1943 schooled 196 women. It covered all aspects of National Socialist life through a series of thirty-seven lectures. Themes included political subjects, such as 'The Structure and Composition of the NSDAP' and 'The Foundations of the National Socialist World View', as well as practical topics such as 'Healthy Food' and 'Hygiene Care'. The report of the training course written up by its leader concluded that 'the resettlement women were open and positively joyous towards the tasks of our time after the three weeks of the training course'.\n\n**OTHER ELITE INSTITUTIONS**\n\nThere were two other attempts at the creation of elite institutions in Nazi Germany: the _Hochschule der NSDAP_ and the _SS-Junkerschulen_. Alfred Rosenberg's _Hochschule der NSDAP_ (High School of the Party) at Chiemsee in Bavaria was intended as the ultimate stage in the selection of elite leaders. However, despite gaining Hitler's approval for the project, the circumstances of the war stymied Rosenberg's plans. Himmler established _SS-Junkerschulen_ in Bad T\u00f6lz (1934) and Braunschweig (1935) for men designated for high office by the SS. The _SS-Junkerschulen_ were the academies within which Himmler 'strove to create a \"professional\" SS officer corps by means of the establishment of a standardised military training system and the creation of the \"political soldier\"'.\n\nDuring the war, three other _SS-Junkerschulen_ were established at Posen-Treskau (1943\u20134), Klagenfurt (1943) and Prague (1944). However, in the war the aims of the _SS-Junkerschulen_ changed from their original purpose. Between 1934 and 1945, the _SS-Junkerschulen_ existed as political institutions to serve National Socialism, in particular, the SS elite armed vanguard the _SS-Verf\u00fcgungstruppe_ and during the war, the _Waffen-SS_. Himmler considered it to be very important that a standardized, professional educational process was developed for his elite troops. The SS elite sought to project itself as a highly disciplined and well-trained racial _F\u00fchrerkorps_. As Hatheway has argued: 'it was essential that the leadership corps of the armed SS consist of \"professionally\" trained SS officers who would have the physical, mental and \"moral\" courage necessary to carry out whatever needed to be accomplished in order to further the goals of the National Socialist Revolution'. Himmler's new, elite man was the political soldier of the armed SS who would be trained in the _SS-Junkerschulen_ , the new SS institutions established for that very purpose. Racial selection was the pre-eminent elite characteristic for the SS. The cadets were encouraged to see themselves as future leaders and the academies were constructed 'to create an air of privilege befitting a new elite', with the symbolism of German historic grandeur. They combined modern technology with the traditions of 'Teutonic' aristocracy. The curriculum included in-depth instruction in National Socialist ideology. As well as Nazi ideology, military training and sports\u2013in particular riding because of its associations with aristocratic elites\u2013were the key elements in the _SS-Junkerschulen_ education. It was at the _SS-Junkerschulen_ that Himmler built his elite leadership corps aimed at ruling the 'New Order' Nazi empire. The bulk of the cadets were not of noble birth, but they constituted what the SS regarded as 'an aristocracy of blood'.\n\nThe Nazi elite schools had a specific political task allotted to them\u2013to train a new generation of leaders. They were significant institutions in the Third Reich and Hitler took a personal interest in them. They were a microcosm of the Nazi _Weltanschauung_ as a whole. They fostered the _F\u00fchrerprinzip_ (leadership principle) and promoted physical fitness and prowess. They purported to be meritocracies and to advocate classlessness, but, in reality, the concept of racial superiority underpinned them. The most significant prerequisite of Nazi elitism was 'racial blood purity'. The ideology taught in the elite institutions promoted National Socialism and excoriated the enemies of the regime. The Nazi elite educational institutions, particularly the Napolas, had a specific role assigned to them in the war\u2013a function in the achievement of a 'greater German empire'. As boarding schools, these institutions offered the possibility of a 'total education' to their pupils, giving the opportunity to the regime to extensively control the socialization process. The pupils were removed from the influence of their parents, and the institutions replaced the family as their focus of socialization. However, the Nazi elite schools placed too much emphasis upon physical training and ideological education, to the detriment of academic subjects. As Evans has pointed out: 'Eclectic and often contradictory in their approach, they lacked any coherent educational concept that could serve as the basis for training a new functional elite to rule a modern technological nation like Germany in the future.' In the end, as Koch has stated, none of them 'produced an elite that outlived their creators'. Furthermore, once the regime collapsed so too did the value system of all these young pupils. This came as a great shock to those educated to be the future Nazi elite, who had been inculcated with Nazi ideology. Hans Buchholz, a former pupil at the Napola in Naumburg, sums this up aptly: 'Everything that had worth and value for me was suddenly no longer worth anything. The men I had looked up to were branded as criminals. The ideas by which I had lived, and for which I had been prepared to die, had become the products of criminal minds.'\n\n**NOTES**\n\n. Important early works on the Nazi elite schools included the following: D. Orlow, 'Die Adolf-Hitler-Schulen', _Vierteljahrshefte f\u00fcr Zeitgeschichte,_ Vol. 13 (1965), pp. 272\u201384; H. Scholtz, 'Die \"NS-Ordensburgen\"', _Vierteljahrshefte f\u00fcr Zeitgeschichte,_ Vol. 15 (1967), pp. 269\u201398; H. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur. Die Nationalpolitischen Erziehungsanstalten 1933\u20131945. Ein Dokumentarbericht_ (D\u00fcsseldorf, 1969); and H. Scholtz, _NS-Ausleseschulen. Internatsschulen als Herrschaftsmittel des F\u00fchrerstaates_ (G\u00f6ttingen, 1973).\n\n. There have been a number of significant German studies including: H. Arntz, _Ordensburg Vogelsang 1934\u20131945. Erziehung zur politischen F\u00fchrung im Dritten Reich_ (Euskirchen, 1986); S. Baumeister, _NS-F\u00fchrungskader. Rekrutierung und Ausbildung bis zum Beginn des Zweiten Weltkriegs 1933\u20131939_ (Konstanz, 1997); J. Leeb (ed.), _'Wir waren Hitlers Elitesch\u00fcler': Ehemalige Z\u00f6glinge der NS-Ausleseschulen brechen ihr Schweigen_ (Hamburg, 1998); B. Feller and W. Feller, _Die Adolf-Hitler-Schulen. P\u00e4dagogische Provinz versus Ideologische Zuchtanstalt_ (Weinheim and Munich, 2001).\n\n. Scholtz, _NS-Ausleseschulen_ , pp. 9\u201310.\n\n. Pine, _Hitler's 'National Community'_ , p. 227.\n\n. Baumeister, _NS-F\u00fchrungskader_ , p. 2.\n\n. _Hitler's Table Talk_ , p. 394.\n\n. On this, see K. Demeter, _The German Officer-Corps in Society and State 1650\u20131945_ (London, 1965), pp. 66\u201370. See also, G. Craig, _The Politics of the Prussian Army 1640\u20131945_ (Oxford, 1955), p. 79 and C. Barnett, 'The Education of Military Elites', _Journal of Contemporary History_ , Vol. 2, No. 3 (1967), pp. 15\u201335. The cadet schools were officially closed down on Allied orders in 1920 in the aftermath of the peace terms of the Treaty of Versailles.\n\n. On this, see J. Bowen, _Soviet Education: Anton Makarenko and the Years of Experiment_ (Madison, 1962). On Makarenko, see also W. Goodman, _Anton Simeonovitch Makarenko: Russian Teacher_ (London, 1949).\n\n. On distinctions between Nazi and Soviet ideals, see R. Overy, _The Dictators: Hitler's Germany and Stalin's Russia_ (London, 2004), pp. 261\u20134.\n\n. On other elite educational systems, see R. Wilkinson (ed.), _Governing Elites: Studies in Training and Selection_ (Oxford, 1969), which examines the different criteria by which elites are selected and the ways in which they are trained.\n\n. I. Weinberg, _The English Public Schools: The Sociology of Elite Education_ (New York, 1967), p. 26. On this, see also, G. Brauner, _The Education of a Gentleman. Theories of Gentlemanly Education in England 1660\u20131775_ (New Haven, 1959) and N. Orme, _From Childhood to Chivalry: The Education of the English Kings and Aristocracy 1066\u20131530_ (London, 1984).\n\n. Weinberg, _The English Public Schools_ , p. 38.\n\n. On what follows, see ibid., pp. 42\u20136.\n\n. Ibid., p. 52. On the ethos and development of English public schools, see also B. Simon and I. Bradley (eds), _The Victorian Public School: Studies in the Development of an Educational Institution_ (Dublin, 1975); J. Honey, _Tom Brown's Universe: The Development of the Victorian Public School_ (London, 1977): G. McCulloch, _Philosophers and Kings: Education for Leadership in Modern England_ (Cambridge, 1991).\n\n. H. Heuer, 'Englische und deutsche Jugenderziehung', _Zeitschrift f\u00fcr neusprachlichen Unterricht_ , Vol. 37 (Berlin, 1937), pp. 215 ff.\n\n. Ibid.\n\n. H. Koch, _The Hitler Youth: Origins and Development 1922\u20131945_ (London, 1975), p. 182.\n\n. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , p. 93.\n\n. Cited in Koch, _The Hitler Youth_ , p. 191.\n\n. Ibid.\n\n. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , p. 64.\n\n. Cited in Samuel and Hinton Thomas, _Education and Society_ , p. 52.\n\n. Koch, _The Hitler Youth_ , p. 185. On the selection process, see also Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , pp. 77\u20139.\n\n. Cited in Koch, _The Hitler Youth_ , p. 183.\n\n. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , p. 12.\n\n. Cited in Knopp, _Hitler's Children_ , p. 116.\n\n. Cited in ibid., p. 146.\n\n. P. Neumann, _Other Men's Graves_ (London, 1958), p. 48.\n\n. Ibid., p. 52.\n\n. Cited in Knopp, _Hitler's Children_ , p. 132.\n\n. BA NS 15\/205, Dr Goedewaagen, 'Die Nationalpolitischen Erziehungsanstalten in Deutschland', p. 8.\n\n. Ibid., p. 13.\n\n. Koch, _The Hitler Youth_ , p. 188.\n\n. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , p. 12.\n\n. C. Schneider, C. Stillke and B. Leineweber, _Das Erbe der Napola: Versuch einer Generationengeschichte des Nationalsozialismus_ (Hamburg, 1996), p. 34.\n\n. BA NS 15\/107, 'Schulungslehrgang der Referendare der nationalpolitischen Erziehungsanstalten vom 19 Oktober bis 19 Dezember 1937 in Berlin'.\n\n. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , p. 93.\n\n. Ibid.\n\n. Schneider, Stillke and Leineweber, _Das Erbe der Napola_ , p. 33.\n\n. E. Kogon, _Der SS Staat_ (Stockholm, 1947), p. 20.\n\n. Ueberhorst (ed.), _Elite f\u00fcr die Diktatur_ , p. 28.\n\n. Koch, _The Hitler Youth_ , pp. 192\u20133.\n\n. Ibid.\n\n. J. Noakes (ed.), _Nazism: A Documentary Reader, 1919\u20131945_ , Vol. 4 (Exeter, 1998), p. 415.\n\n. Schneider, Stillke and Leineweber, _Das Erbe der Napola_ , p. 48.\n\n. G. Wegner, 'Mothers of the Race: The Elite Schools for German Girls under the Nazi Dictatorship', _Journal of Curriculum and Supervision_ , Vol. 19, No. 2 (2004), p. 171. On the Nazi elite schools for girls, see also, U. Aum\u00fcller-Roske, 'Weibliche Elite f\u00fcr die Diktatur? Zur Rolle der nationalpolitischen Erziehungsanstalten f\u00fcr M\u00e4dchen im Dritten Reich', in U. Aum\u00fcller-Roske (ed.), _Frauenleben-Frauenbilder-Frauengeschichte_ (Pfaffenweiler, 1988), pp. 17\u201344 and U. Aum\u00fcller-Roske, 'Die Nationalpolitischen Erziehungsanstalten f\u00fcr M\u00e4dchen im Grossdeutschen Reich: Kleine Karriere f\u00fcr Frauen?', in L. Gravenhorst and C. Tatschmurat (eds), _T\u00f6chter-Fragen: NS-Frauen Geschichte_ (Freiburg, 1990), pp. 211\u201336.\n\n. Cited in _V\u00f6lkischer Beobachter_ , 13 September 1936.\n\n. A. Rosenberg, _Der Mythos des 20. Jahrhunderts: Eine Wertung der seelisch-geistigen Gestaltenk\u00e4mpfe unserer Zeit_ (Munich, 1934).\n\n. Wegner, 'Mothers of the Race', p. 178.\n\n. Cited in ibid., p. 179.\n\n. Ibid., p. 181.\n\n. Ibid., p. 182.\n\n. Orlow, 'Die Adolf-Hitler-Schulen', p. 273.\n\n. Koch, _The Hitler Youth_ , p. 196.\n\n. BA NS 22\/889, 'Abschrift der Rede des Reichsjugendf\u00fchrers anl\u00e4sslich der Grundsteinlegung der neuen Adolf-Hitler-Schulen. Ver\u00f6ffentlicht im R. J. P. vom 14.1.38.', p. 2.\n\n. Scholtz, _NS-Ausleseschulen_ , p. 11.\n\n. Cited in Knopp, _Hitler's Children_ , p. 124.\n\n. BA NS 22\/889, 'Arbeitsanweisung zur Auslese und Musterung der Adolf Hitler-Sch\u00fcler', 12 October 1938, p. 2.\n\n. Orlow, 'Die Adolf-Hitler-Schulen', p. 277.\n\n. Ibid.\n\n. Ibid., p. 276.\n\n. BA NS 22\/889, 'Anweisung f\u00fcr den Ausleselehrgang 1938 f\u00fcr die Adolf-Hitler-Schulen', 9 February 1938, p. 2.\n\n. BA NS 22\/889, 'Richtlinien f\u00fcr die Auswahl, Ausmusterung und Einberufung der Adolf-Hitler-Sch\u00fcler', 15 October 1938, pp. 4\u20135.\n\n. ibid., p. 1.\n\n. Ibid., p. 2.\n\n. BA NS 22\/889, 'Anweisung f\u00fcr den Ausleselehrgang 1938 f\u00fcr die Adolf-Hitler-Schulen', 9 February 1938, p. 2.\n\n. BA NS 22\/889, 'Anweisung f\u00fcr den Ausleselehrgang 1938 f\u00fcr die Adolf-Hitler-Schulen', 9 February 1938, Anlage 1, 'Vorschlag eines Tagesplanes'.\n\n. Koch, _The Hitler Youth_ , p. 197.\n\n. Orlow, 'Die Adolf-Hitler-Schulen', p. 282.\n\n. Cited in Knopp, _Hitler's Children_ , p. 116.\n\n. BA NS 22\/997, Franz Albert Schall, 'Grundgedanken zum Aufbau des Werksunterrichts an den Adolf Hitler Schulen', 9 August 1938.\n\n. Ibid.\n\n. Scholtz, 'Die \"NS-Ordensburgen\"', p. 269.\n\n. Ibid., p. 272.\n\n. Cited in ibid., p. 274.\n\n. E. Lengyel, 'Incubators for Heroes', _The Daily Herald_ , 13 July 1938, p. 8.\n\n. On Vogelsang, see Arntz, _Ordensburg Vogelsang_.\n\n. Neumann, _Other Men's Graves_ , p. 71.\n\n. Lengyel, 'Incubators for Heroes', p. 8.\n\n. R. Evans, _The Third Reich in Power_ (London, 2006), p. 287.\n\n. Scholtz, 'Die \"NS-Ordensburgen\"', p. 289.\n\n. Ibid., p. 290.\n\n. BA NS 22\/998, Neumayr, 'Ordensburg Vogelsang', in _Der Orden: Bl\u00e4tter der Ordensburg Vogelsang_ , Jahrgang 1, Folge 1, p. 1.\n\n. W. Teeling, 'Training for Life', _The Listener_ , 10 November 1937, p. 1,003.\n\n. '\"F\u00fchrers\" of the Future: The Chosen Few', _The Manchester Guardian_ , 17 November 1937, p. 12.\n\n. E. Hearst, 'Ordensburgen: Finishing Schools for Nazi Leaders', _Wiener Library Bulletin_ , Vol. XIX, No. 3 (1965), p. 38.\n\n. Lengyel, 'Incubators for Heroes', p. 8.\n\n. '\"F\u00fchrers\" of the Future', p. 11.\n\n. Ibid., p. 12.\n\n. On education at Vogelsang, see Arntz, _Ordensburg Vogelsang,_ pp. 102\u201335.\n\n. Neumann, _Other Men's Graves_ , p. 59.\n\n. Ibid.\n\n. BA NS 22\/998, P. Waiblinger, 'Ein Jahr Ordensburg: Ausblick', in _Der Orden: Bl\u00e4tter der Ordensburg Vogelsang_ , Jahrgang 1, Folge 1, p. 4.\n\n. Ibid., p. 5.\n\n. Ibid., p. 6.\n\n. Ibid.\n\n. Scholtz, 'Die \"NS-Ordensburgen\"', p. 270.\n\n. Kogon, _Der SS Staat_ , pp. 21\u20132.\n\n. Rauschning, _Hitler Speaks_ , pp. 241\u20132.\n\n. R. Ley, _Schmiede des Schwertes_ (Munich, 1942), p. 134.\n\n. Scholtz, 'Die \"NS-Ordensburgen\"', p. 284.\n\n. Ibid., p. 278.\n\n. Ibid., p. 274.\n\n. BA NS 12\/1196, Robert Ley, 'Die Burgen der Partei und die Erziehung des F\u00fchrernachwuchses', p. 2.\n\n. BA NS 22\/27, '\u00dcber personelle Auswertung der Schulung zur Personalpolitik', 18 Jan. 1941.\n\n. Ibid., p. 3.\n\n. Ibid., pp. 3\u20134.\n\n. BA NS 22\/284, 'Betrifft: Schulung von Volksgenossen aus dem ehemaligen S\u00fcdtirol', 26 July 1940. For full course content and rota, see also BA NS 22\/280, 'Dienstplan'.\n\n. BA NS 22\/281, 'Schulungslehrgang der S\u00fcdtiroler Politischen Leiter auf der Ordensburg Sonthofen vom 12. 8. bis 4. 9. 1940', pp. 1\u20135.\n\n. BA R 49\/2219, Ordensburg Sonthofen: Schulungsthemen, 'Aufgaben des grossdeutschen Bauerntums', 2 September 1940, p. 1.\n\n. BA R 49\/2219, Ordensburg Sonthofen: Schulungsthemen, 'Die Landwirtschaftliche Erzeugungsschlacht', 2 September 1940, p. 3.\n\n. BA R 49\/2219, Ordensburg Sonthofen: Schulungsthemen, 'Betriebsgemeinchaftliche Aufgabe der Arbeitsfront', 29 August 1940, p. 3.\n\n. BA R 49\/2219, Ordensburg Sonthofen: Schulungsthemen, 'Arbeitsdienst', 28 August 1940, p. 1.\n\n. BA R 49\/2219, Ordensburg Sonthofen: Schulungsthemen, 'Die biologische Grundlage des Nationalsozialismus und der Kampf f\u00fcr die Erhaltung der Rasse', 21 August 1940, pp. 1\u20133\n\n. BA R 49\/2219, Ordensburg Sonthofen: Schulungsthemen, 'Das Judentum', 21 August 1940, pp. 2\u20133.\n\n. BA NS 22\/282, 'II. Lehrgang f\u00fcr S\u00fcdtiroler vom 4. November bis 24. November 1940 auf der Ordensburg Kr\u00f6ssinsee', 4 November 1940.\n\n. BA NS 22\/282, 'Auszug aus dem Vortrag des Pg. Dr. Luig, \u00fcber das Thema: \"Volkstums- und Grenzlandsfragen\"'.\n\n. BA NS 22\/282, 'Auszug aus dem Vortrag des Pg. Bennecke, SA-Obergruppenf\u00fchrer, \u00fcber \"Aufgabe und Ziele der SA\"'.\n\n. BA NS 22\/950, 'III. Lehrgang Volkstum\/Brauchtum auf der Ordensburg Sonthofen. Allg\u00e4u vom 6.-16.-8.-41.'\n\n. BA NS 22\/938, 'Reichslehrgang f\u00fcr Umsiedler Frauen auf der Ordensburg Die Falkenburg am Kr\u00f6ssinsee vom 14\/1 bis 4\/2\/1943'.\n\n. BA NS 22\/938, 'Bericht \u00fcber den 1. Reichslehrgang f\u00fcr Umsiedlerfrauen auf der NS-Ordensburg \"Die Falkenburg am Kr\u00f6ssinsee\" vom 14.1 bis 4.2.1943', p. 6.\n\n. On Rosenberg, see E. Piper, _Alfred Rosenberg: Hitlers Chefideologe_ (Munich, 2005).\n\n. J. Hatheway, _In Perfect Formation: SS Ideology and the Junkerschule-T\u00f6lz_ (Atglen, 1999), p. 7. See also R. Schulze-Kossens, _Milit\u00e4rischer F\u00fchrernachwuchs der Waffen SS: Die Junkerschulen_ (Osnabruck, 1982).\n\n. On this, see Hatheway, _In Perfect Formation_ , pp. 109\u201324.\n\n. Ibid., p. 10.\n\n. Ibid., p. 83.\n\n. ibid., pp. 92\u2013103.\n\n. Ibid., p. 132.\n\n. Evans, _The Third Reich in Power_ , pp. 288\u20139. See also p. 502.\n\n. Koch, _The Hitler Youth_ , p. 203.\n\n. Cited in Knopp, _Hitler's Children_ , p. 171.\n5 THE HITLER YOUTH\n\nIn his speech at the Nuremberg Party Rally in September 1935, Hitler stressed his requirements for the new image of German youth. He stated that: 'In our eyes, the German youth of the future must be slim and slender, swift as the greyhound, tough as leather, and hard as Krupp steel'. Hitler firmly believed that the education and socialization of German youth should not be limited to the schools, but extended to incorporate the activities of the youth groups. The Nazi youth groups were accorded a very significant task in Nazi educational aims and in Nazi society as a whole. This chapter examines the role and ethos of the _Hitlerjugend_ (HJ) or Hitler Youth as an organization for the regimentation and socialization of German boys. It analyses the aims of the HJ and their implementation. First, however, in order to place the Hitler Youth movement into its historical context, a brief examination of the German youth movement is necessary.\n\n**GERMAN YOUTH GROUPS BEFORE NATIONAL SOCIALISM AND THE ORIGINS OF THE HJ**\n\nThe _Wandervogel_ (birds of passage) came into existence at the end of the nineteenth century. In these groups, young people endeavoured to create for themselves an alternative to the formal education and discipline in schools. They roamed the German countryside, dressed in traditional costumes and sang folk songs. They cherished the landscape, exploring forests, hills, villages and castles. Most of the groups were against authority and discipline. In 1913, representatives of the Free German Youth met on the Hoher Mei\u00dfner mountain near Cassel to proclaim the aims of the German youth: 'to mould its own life, in accordance with its own nature, on its own responsibility and in inner integrity'. German youth leader Gustav Wyneken stated that:\n\nyouth, hitherto merely an appendage of the older generation, excluded from the life of the community and given only the passive role of learning and with opportunities only for a dilettante form of social life, is beginning to become conscious of itself . . . It is striving for a way of life which corresponds to the nature of youth, but which at the same time will enable it to take itself and its activity seriously.\n\nWyneken regarded youth not only as a time of transition, but also as a time which had 'its own unique value' and 'its own beauty'. The _Wandervogel_ movement reacted against suppression and lack of freedom. It proposed the right of youth to independence. As Hahn states, it was 'both nostalgic and utopian, celebrating simple country life and folklore while working for the development of the individual within free communities'. It made a statement that youth should have its own sphere and, in this period, it stood against _v\u00f6lkisch_ and nationalist sentiments. Not surprisingly, both the state and the Churches regarded the youth groups with distrust and dislike. The state, in particular, attempted to undermine these youth movements by expanding its own programme of youth welfare.\n\nThe approach and outbreak of the First World War stymied the attempts of the youth movement at independence, anti- _v\u00f6lkisch_ ness and 'youth for youth's sake'. The war brought to the fore feelings of nationalism and chauvinism. After the war, German youth groups moved towards a more reactionary and conservative position. By the late 1920s, the _B\u00fcndische Jugend_ came to emphasize more organized and formalized activities than the carefree wanderings of the earlier _Wandervogel_ movement. Leadership and uniform came to play an increasing role. The _B\u00fcndische Jugend_ rejected the Weimar system and everything it represented, including modernity and urbanization. It became increasingly concerned with nostalgia for a 'national community', as well as more exclusively middle-class, Protestant and increasingly nationalist in its orientation. New groups sprang up for Catholic youth, Jewish youth and working-class youth. Youth groups representing all political parties and the Churches existed before the advent of the Nazi _Machtergreifung_. During the Weimar Republic, some five to six million young Germans belonged to this assortment of youth groups.\n\nYouth had come to attract more public awareness and a more prominent role during the Weimar years through social change and the proliferation of youth movements that shaped the image and cult of youth in the 1920s. The Weimar era gave youth a new prominence, yet the economic and social situation from the late 1920s presented German youth with problems too. Faced with an array of difficulties and tensions, many young Germans felt alienated from the Republic. Radical youngsters turned increasingly to the youth groups of the parties on the two political extremes, the NSDAP and the KPD, both of which appealed to them. The NSDAP was able to capitalize on the crisis of youth, as well as the tensions that existed between the older and the younger generations, and made many recruits among young Germans.\n\nThe earliest National Socialist youth group was organized by Adolf Lenk. Lenk had wanted to become a member of NSDAP in 1920, but was denied entry as he was not yet 18 years old. He had asked Anton Drexler, the first leader of the NSDAP, if he could found a youth group. First Drexler, and then Hitler in 1921, encouraged him in this aim. Lenk started his movement with seventeen boys gathered in the B\u00fcrgerbr\u00e4ukeller in Munich on 5 May 1921, where Hitler came to address them. Lenk stated: 'Starting with the seventeen, more people joined, then further local groups grew out of the Munich group, it developed quickly'. The Youth League of the NSDAP was publicly announced and its statutes were proclaimed in March 1922. Most significant among its points were Clause 3, which stated that the purpose of the movement was 'to awaken and nurture those values within our youth which have their roots in Germanic blood' and Clause 5, which stated that 'foreigners and Jews cannot be members'.\n\nThe next year, at the Party conference, the youth group was given its own flag. In May 1923, at the youth group's first conference, Lenk addressed the boys as follows: 'As boys we appeal to the blustery, thunderous youth, who does not doubt and is not afraid, but hopes and believes, who wins, as it dares. We need a youth which believes in the mission of National Socialism . . . the German youth must be trained intellectually, morally and physically, only in this triad will we achieve our goal'. Lenk's youth group engaged in clashes with the Communist youth and courted the displeasure of the authorities. It had a very short lifespan, surviving only until the fiasco of the Munich Beer Hall Putsch in November 1923 when Hitler's attempt to seize power failed, after which Lenk was arrested and imprisoned. After his release, Lenk tried to continue his work and was arrested again for his activities in a banned organization. Despite this, he claimed: 'We, the old fighters, only knew one goal: \"Germany\". For this we sacrificed life, blood, freedom, existence and wealth . . . we were proud to have served our country'.\n\nA National Socialist youth group was resurrected in 1925, by Kurt Gruber, a law student from Saxony. He introduced a uniform and established an administrative apparatus for his organization, the Greater German Youth Movement. At first, it attracted mainly working-class youth. Gruber's organization gradually evolved into the official youth group of the NSDAP and in 1926, it took on the name Hitler Youth. But Lenk was careful to ensure that his formative youth group was not forgotten in the history of the Party and that Gruber did not take all the credit for the creation of the National Socialist youth movement.\n\nWhen the roots of today's magnificent tree were laid in the year 1921, the thought ruled me to serve this movement in its young years, as this movement, for which back then blood was flowing from our heads on a daily basis, was nothing else for us than Germany itself. Only pure dedication and continuous preparedness was for us, who were only a few back then, the most sacred task.\n\nLenk stated that Hitler subsequently appointed Gruber to become leader of the National Socialist youth 'on my request'.\n\nFollowing a protracted power struggle between Gruber and Baldur von Schirach, who had in the meantime become prominent in the National Socialist German Students' Association, Hitler appointed Schirach as head of all youth activities for the NSDAP in 1931. On 17 June 1933, Schirach was made Youth Leader of the German Reich, within the Ministry of the Interior. By 1 December 1936, Schirach had succeeded in persuading Hitler to let him have his own headquarters in Berlin, at which time he was no longer responsible to Wilhelm Frick, the Minister of the Interior, but directly to Hitler. He retained his position as Youth Leader of the German Reich until August 1940, when Artur Axmann replaced him.\n\nFrom a very early date, even before the regime came to power, the _Jungvolk_ (for boys aged 10\u201314) was determined to be an organization not of words, but of action: 'How can we win those who we care about and whom we need? Again, not through words, but through actions! . . . Only achievement and action can prove authenticity and truth.' According to the _Jungvolk_ bulletin, the organization and each new boy within it had to go through three steps. The first and foremost stage was to create and become part of a community. The second step was the consolidation of the community. The third stage was attack and outward struggle. Team spirit and commitment to the community was to be fostered through common experiences, in particular, hiking trips. Consolidation was to be achieved through the boys adopting 'an unrelenting acerbity' towards themselves. Through this, they would develop 'a clear conscience' with regard to their 'inner judgement' and 'true and insuperable strength' would grow within them. This, in turn, would drive them to attack and success. At the heart of all this was 'the experience of the German _Volkstum_ '. This was demonstrated in a variety of ways, not just in daily politics and political struggle. The organization emphasized the spirit of German life, in terms of dance, games, songs, music and clothing, as well as language, land and national heroes. This is why a wide range of activities was suggested to _Jungvolk_ leaders for its members to undertake, including theatre, puppet shows, circus, dancing and singing, music recitals, handicrafts, drawing, woodcutting and photography. And so, the movement emphasized the creation of a 'new community', 'not with words, but with actions', at the heart of which lay the _Volkstum_. Its task was to create 'a new Nordic youth'. Furthermore, it called for 'a link to the countryside, which has seen the birth of the soul of our _Volk_. . . we have to be rooted with the oak and the fir tree!' This love of the countryside was significant to National Socialist youth and something it picked up on and followed from the traditions of the _Wanderv\u00f6gel_. These excerpts from the early bulletins of the _Deutsches Jungvolk_ are useful sources as they are indicative of the direction in which the movement was intended to develop as it grew. Indeed, the new exemplary man it sought to create would become 'the primary type' of the Third Reich.\n\nAt the end of 1932, the Hitler Youth had a comparatively small membership of 107,956. At this time, in the months before the Nazi _Machtergreifung_ , HJ members actively took part in propaganda campaigns, distributing leaflets, putting up posters and engaging in street fights with their counterparts in the Communist youth group. Schirach's main objective in 1933 was to build a state youth organization and to try to consolidate all Germany's youth into the Hitler Youth. At first, the Hitler Youth appeared attractive and exciting. It presented young boys and teenagers with the opportunity to take part in a new movement, to escape from parental control and boredom at home. It offered them a sense of purpose, belonging and unity. The Hitler Youth also provided a new opportunity for participation in youth activities to some youngsters, particularly in rural areas, who had not previously had access to youth movements. By the end of 1933, the Hitler Youth had more than two million members. In order to join, young boys had to present a statement of application for admission, which included details of the name and occupation of their parents, their religion, as well as a confirmation that they were of 'German origin', which they had to sign. The young boys who joined the HJ at this time were motivated by a sense of excitement, peer camaraderie and the youthful enthusiasm of its leaders. In addition, many enrolled in order to become involved in the national cause and to gain more independence from their parents.\n\n**THE HJ AFTER JANUARY 1933**\n\nOnce the NSDAP gained power, with its radical ideas for state and society, its youth group acquired a new status and new tasks. The HJ became involved in a host of activities in building the National Socialist state, including health care of the youth, recreation, labour service and the _Winterhilfswerk_ (Winter Relief Agency), as well as career counselling and apprenticeship procurement. In particular, its health remit meant that its doctors had to ensure the health of all HJ members and leaders. All HJ (and BDM) members had to undergo a health check before they were admitted into the movement. Furthermore, HJ doctors were to educate the youth in 'hereditary biological and racial thinking'. HJ doctors and leaders were to encourage German youth to understand the importance of choosing a 'racially valuable' partner and furthering the nobility of the German _Volk_ through their offspring in the future.\n\nHitler addressed the first National Socialist Youth Day in Potsdam:\n\nThe German must once again learn how to feel like one _Volk_. . . our _Volk_ fell from its proud height as it forgot that, and you, my German boys and girls should learn it again in the National Socialist movement, to feel as brothers and sisters in one nation. You shall, beyond all professions and social classes, beyond everything which threatens to splinter you, search and find the German community; you shall preserve and hold on to it and no one shall rob it from you . . . At the moment there might be quite a lot of Germans who deny the value of ideals. But National Socialism educates you, young people, to become believing idealists; as only ideals can forge together the German _Volk_ to unity\n\nHe sparked great enthusiasm for the National Socialist movement among those gathered. Axmann underlined this theme of national unity and the elimination of class barriers in the Nazi youth movement in a radio lecture on 3 May 1933: 'We do not ask about coincidences of birth and origin, but we ask about character and achievement. We do not ask: \"Where do you come from?\" but we ask \"What is your will and where are you going?\"' Youth was encouraged to play its part in the future of the national movement.\n\nWhilst part of the explanation for the rapid growth of the movement was its attraction to youth after the Nazi 'seizure of power', a large part of the reason was the process of _Gleichschaltung_ (coordination) of youth by the Nazi regime. The Communist Youth Association of Germany (KJVD), the Social Democratic Socialist Working Youth (SAJ) and the German Socialist Youth Association (SAP) were all dissolved. In December 1933, Bishop Ludwig M\u00fcller agreed to sign over the members of the Evangelical youth movement into the Hitler Youth. The autonomy of the Catholic youth groups was temporarily protected (until 1936) by the July 1933 Concordat between Hitler and the Vatican. In 1936, the Gestapo banned any remaining youth groups outside the Hitler Youth, and the Catholic youth groups were dissolved. By the end of 1936, the Hitler Youth had 5.4 million members.\n\nOn 1 December 1936, the Law on the Hitler Youth stated that: 'The future of the German nation depends upon its youth and German youth must therefore be prepared for its future duties.' It decreed that:\n\n1. The whole of German youth within the borders of the Reich is organized in the Hitler Youth.\n\n2. All German young people, apart from being educated at home and at school, will be educated in the Hitler Youth physically, intellectually, and morally in the spirit of National Socialism to serve the nation and the community.\n\n3. The task of educating German youth in the Hitler Youth is being entrusted to the Reich Leader of German Youth in the NSDAP. He therefore becomes the 'Youth Leader of the German Reich'. His office shall rank as Supreme Governmental Agency with its headquarters in Berlin and he will be directly responsible to the F\u00fchrer and Chancellor of the Reich.\n\n4. All regulations necessary to execute and supplement this decree will be issued by the F\u00fchrer and Reich Chancellor.\n\nThe Hitler Youth Law was significant because it officially and legally gave the Hitler Youth an equal status to the home and the school in educating German children. However, in spite of its first provision, membership of the Hitler Youth organization was not yet compulsory. Nevertheless, there was much social pressure to join after 1936. Schirach proclaimed 1936 to be 'the year of the German _Jungvolk_ ' and orchestrated a huge propaganda campaign, as well as pressure on schoolteachers, in order to initiate as many 10-year-olds as possible into the movement in that year. On 19 April 1936, the eve of Hitler's birthday, Schirach proudly presented Hitler with the 'gift' of this cohort of young boys, who took their oath of loyalty to Hitler at Marienburg Castle in West Prussia. In his radio broadcast, Schirach claimed that 90 per cent of all 10-year-olds were members of the Nazi youth movement. However, even despite this immense pressure and compulsion to join, it is noteworthy that a substantial number of young people managed to remain outside the Hitler Youth movement.\n\nThe organizational briefs and guidelines for the _Jungvolk_ and its leaders were circulated by the National Leadership of the HJ department. The _Jungvolk_ was organized into three different types of organisational unit. The _Jungenschaft_ comprised 8\u201316 boys; the _Jungzug_ was made up of two to four _Jungenschaften_ and comprised 32\u201364 boys; and the _F\u00e4hnlein_ was made up of two to four _Jungz\u00fcge_ and comprised 128\u2013250 boys. The uniforms for both the _Jungvolk_ members and the _Jungvolk_ leaders were carefully designed. The members' uniform consisted of an open brown shirt, a black neck scarf with a leather knot, brown knee breeches, a belt, with brown leather buttons. Two leather buttons affixed to each other were to be worn as cufflinks. The _Jungvolk_ emblem was to be worn on the left chest pocket below the button. The emblem could only be worn once the boy had passed his probation period and had been admitted officially into the _Jungenschaft_ with a handshake. There was also a _Jungvolk_ cap, with stripes in colours according to region. Leaders' uniforms were marked out by their leadership badges. The _Jungenschaft_ leader wore a green disk on the left arm, the _Jungzug_ leader wore a blue disk and the _F\u00e4hnlein_ leader a white disk. This appealed to young people who wanted to wear the uniform and the badges and therefore tried to excel within the movement.\n\nLeaders were given careful guidelines for training and educational work in the _Jungvolk_. They were called upon to educate their boys in 'love of the country' and through group work to eliminate the gap between 'the proletarians and the bourgeois'. This intention was part of the National Socialists' wider aim to destroy class barriers in German society. In the youth group, this was to be achieved through a weekly meeting called the _Heimabend_ , in which boys came together to read, sing, listen to stories and do handicrafts. In good weather, this was supplemented with outdoor activities in the form of games and exercises to strengthen the body. In addition, a key activity was the excursion. This was particularly favoured by the Nazi leadership as there the boy was 'totally cut off from home' and could really show 'whether he is a man and knows how to help himself ' and 'whether he is a good comrade'. There he learned how to put up tents in different ways and how to secure and heat a camp\u2013'in short, everything that a good German boy can, will and must know'. German boys acquired knowledge of their homeland and of nature on these excursions. They also learned the group rule that the common good took precedence over self-interest. Beyond these excursions, which took place once a month for a day and a half, were the longer camps that took place in the school holidays. Each month, the _Jungvolk_ published a booklet for its members, which included excursion reports, experiences, poems, stories, photographs and drawings. At parents' evenings, exhibitions of _Jungvolk_ work were shown, as well as group songs, music recitals, games and exercises. These evenings were designed to show parents what the boys were doing at the youth group and to encourage their support of it. In addition, it was hoped that parents whose boys were not already members might be encouraged to enrol them once they had been to the parents' evenings. Hence they had to be appealing, with no 'feeling of boredom' for the audience.\n\nOn 25 March 1939, a further Youth Ordinance decreed that: 'All young people are obliged from the age of 10 to their 19th birthday to serve in the Hitler Youth'. Boys aged 10\u201314 were to join the _Deutsches Jungvolk_ (DJ), whilst boys from 14 to 18 were to join the Hitler Youth. German girls were to join the corresponding Nazi girls' organizations, the _Jungm\u00e4del_ (JM) for girls aged 10\u201314 and the _Bund Deutscher M\u00e4del_ (BDM) for girls aged 14\u201318, which are examined in the next chapter. It was the responsibility of the parent or legal guardian to register the children or young people in the Hitler Youth and they could be fined or imprisoned for deliberate failure to do so. Furthermore, the decree stated that: 'anyone who maliciously prevents or attempts to prevent any young person from serving in the Hitler Youth will be punished by fine or imprisonment'. Hitler Youth members were obliged to swear an oath of loyalty to Hitler.\n\nAfter the Hitler Youth had established itself as the 'state youth', three other influences on children's education and socialization came to stand in opposition to its power: the Church, the school and the parental home. Tensions increased between HJ members and traditional figures of authority, in particular clergymen, teachers and parents. HJ members considered the movement as their 'world . . . and not school, nor church nor home could offer competing alternatives'. In HJ newspapers and journals, as well as its training manuals for HJ leaders, the presentation of arguments against the Churches was a significant and recurring theme. HJ leaders encouraged their members to flout the authority of conventional figures and even to scorn them. The Hitler Youth made some attempts to get parents on its side, for example introducing HJ parents' evenings and broadcasting radio programmes on parenthood to secure the loyalty of parents. However, HJ members were encouraged to spy on their families and friends for anti-Nazi activities, which increased tensions between the HJ and the parental home. Furthermore, tension increased between the NSLB and the HJ leadership. The main source of concern for the NSLB was the lack of respect for the teaching profession displayed by HJ members. There was a growing number of complaints by teachers that pupils ridiculed them and undermined their authority. Yet, from the perspective of a young HJ boy, this position was not necessarily so clear-cut. Jurgen Herbst writes in his memoir: 'Did we leaders of boys leave our parents and teachers, or did our parents and teachers leave us? We could not have said.' Nevertheless, in the KLV camps during the war, the regime came closest to achieving its aim of total education\u2013for here the influence of parents was eliminated, the influence of schoolteachers was reduced and the influence of the HJ became the decisive factor.\n\nBy the time membership became compulsory, the Hitler Youth had lost some of its original appeal. The initial enthusiasm for the movement waned as military drills took precedence over hikes, camps and sporting activities, and the dissemination of Nazi propaganda became more pronounced. The HJ was becoming an instrument of authoritarianism and indoctrination. The initial enticement of the slogan 'youth leads youth' wore off. There was also a growing number of duties, including collecting money for the Winter Relief Agency and picking berries and herbs. Land Service involved Hitler Youth members in helping with harvesting, milking cows and chopping wood. This was aimed at emphasizing the 'blood and soil' doctrine and at providing experience of life in the countryside to young people from the cities. It fitted in with the Nazi view of the cities as asphalt jungles, which engendered an unwholesome lifestyle. A Hitler Youth circular dated 8 January 1940 stated: 'Land service is a political task of National Socialism. Its purpose is to bring back boys and girls from the cities to the land, to create new recruits for the agricultural occupations and thus secure their continuous existence. The best of them should be given an opportunity to settle. The Hitler Youth is the sole executor of the land service.' From February 1940, Hitler Youth members had to report for duty on two Sundays each month. Some young people came to see the Hitler Youth as a restriction on the freedom of their leisure time, as it took up more and more of their waking hours outside school. State control had replaced parental control. Parents too expressed concern about the amount of time their children were spending on Hitler Youth activities.\n\nIndeed, HJ members recall how busy their HJ schedules were. Erich Loest remembers: 'Twice a week we had HJ service and as soon as I became a leader, there were extra leadership duties on Mondays; and on Sundays we had shooting, or we went bicycling somewhere, or we had a parade. So for four or five days a week I was busy with the Hitler Youth. We had no time to think about what we were actually doing. The next thing was always coming up. It was non-stop action.' However, it is also worth remembering that in many senses, boys living through the Nazi era were still just boys as in other places and times. Herbst makes this point clearly in his memoir:\n\nAs we boys lived our lives, day by day and week by week, they moved along in all the ordinariness of daily existence as ordinary lives unfold everywhere. Dramatic and traumatic events did not occur every day. When they did, they broke into and interrupted the ordinariness of everyday life, but then they were absorbed in the rhythm of our daily doings and became themselves ordinary parts of it.\n\nHow were HJ members socialized? As Kl\u00f6nne has suggested, the HJ boy was characterized as outwardly active, capable of physical achievement, fit for work, used to organizational discipline and bound to the norms of the organization. Certainly there was a significant distinction between the socialization of girls and boys. There were specific gendered expectations of boys as boys. Training in the HJ involved a variety of aspects including physical fitness, discipline, adherence to the organization and its dress codes. Physical fitness was one of the most important attributes of the HJ. Its members had a 'duty' to be physically fit. In order to be accepted into the organization, the new recruits of the _Jungvolk_ had to pass a physical assessment, which included running and long jump, as well as 'tests of courage'. The boys had to demonstrate their strength and bravery. Once they joined, they took part in physical exercises and military drills. They did roll calls and marched in columns. They participated in numerous sporting events and competitions. The 'HJ National Sports Contest' was the culmination of these events. It showed off the strength and fortitude of the German youth. At the 1937 Party Congress Hitler addressed his youth as follows: 'In place of young people who were previously brought up to enjoy themselves, a generation is now coming of age, brought up to privation, self-sacrifice and above all to the development of a healthy, resilient physique'.\n\nThe desire to belong to the Nazi youth movement and to wear the uniform was a significant factor in the appeal of the HJ. The younger boys recall their haste to join and desire to reach the age of 10 so that they could do so. For example, Hans J\u00fcrgen Habenicht describes his agonizing wait to be admitted into the _Jungvolk_ : 'I really longed for the day and was proud when it finally arrived. My elder brother was already in the Hitler Youth. I too wanted to belong one day to that organization, which was bound up with ideas like comradeship, Fatherland and honour. In uniform, you felt you were taken more seriously. Now I was one of the big boys'.\n\nThe Hitler Youth uniform consisted of brown shorts with a brown shirt, a black kerchief, a leather belt, leather shoulder straps, white socks, brown shoes and a brown cap. The boys took pride and pleasure in wearing the HJ uniform, which signified their belonging. Jobst-Christian von Cornberg recalls that 'to wear the uniform was an honour'. The membership of the movement and its uniform appealed to the boys' self-esteem and their desire to be recognized as important to the national cause. Werner Hanitzsch remembers: 'The uniform was first and foremost a symbol of belonging. And for us that was actually the most important thing. We were a community. We were a blood brotherhood and the uniform was the external symbol of this'.\n\nIdeology was presented in a thoroughgoing manner in the HJ _Heimabend_. The HJ leaders were primed with training manuals about what to include and how to run these sessions. They told the boys legends of German heroes and read battlefield literature to them. They were taught about the need to preserve 'the purity of German blood', the menace to Germany presented by the Jews and the importance of gaining 'living space in the East'. The Third Reich would rightfully subjugate its 'inferior', 'sub-human' eastern neighbours and be heroic in battle. Rudolf Hiemke remembers that 'the pattern of the _Heimabend_ was strictly laid down, all creativity was suppressed. There was no debate, everything was dictated and organized on military lines. We had absolutely no opportunity to express ourselves freely and dared not offer any criticism.' Herbst recalls the _Heimabend_ :\n\nWe listened to our leaders telling us over and over again of the history of the Nazi party, of the exploits of its heroes . . . we viewed rows of pictures showing the heads and bodies of men and women who were supposedly representatives of various racial groups. We soon learned that the blond, tall, slender, and straight figures were the Nordic, Aryan types that we all were supposed to be. The dark, small, thick, and bent bodies, on the other hand, belonged to undesirable . . . and less worthy races. We should look down on them as inferior beings.\n\nHence, the Nazi racial stereotypes were clearly and firmly established during the HJ _Heimabend_ sessions.\n\nThe camping trips excited more enthusiasm among HJ members. They took part in leisure activities that previously had been available only to the children of affluent families. They escaped the monotony of their homes during the long summer holiday to hike through the countryside and camp. The communal spirit formed a great part of the popularity of the HJ camps. Peter L\u00f6hrer recalls: 'In the evening we all sat around the fire. And then we sang together. It was dark. The stars shone above us. It was a thrilling feeling.'\n\nThe Hitler Youth socialized German youth in militarization and the ultimate aim of acquiring new _Lebensraum_ (living space) in the east. Its members played war games, studying maps and spotting enemies. They learned how to master their terrain, as well as orientation skills in darkness. They camped in tents, sang _v\u00f6lkisch_ songs, marched and engaged in rifle practice. Boys aged 10 to 18 were taught how to shoot as part of their pre-military training, which also consisted of sports, including boxing, strenuous hikes, marches and drills. Pre-military training of the HJ was carried out by the Reich Youth Leadership in conjunction with the high command of the _Wehrmacht_. These activities prepared them for active combat in the field once the war began. Herbst recalls a memorable night in 1944 on his annual skiing camp when he and his _Jungvolk_ comrades had to climb the 3,040-foot-high Achtermann peak on a stormy night. He describes his immense joy on reaching the summit: 'We felt proud and elated. We had proved ourselves, had shown that we knew how to follow orders and that we were ready to move and persevere as soldiers'. In addition to their physical training, Hitler Youth members were inculcated with a militaristic spirit during their _Heimabend_ sessions. Topics included great soldiers of Germany's past and the war itself. These sessions were supplemented with films and pamphlets that treated the subject. In addition, soldiers visited the Hitler Youth groups and told them about their experiences at the front. Military preparation camps trained youth in map reading, reconnaissance activities, shooting, guard duty and camouflage.\n\nHerbst describes his time as a _Jungvolk_ leader as 'most exhilarating'. He recalls: 'The _Jungvolk_. . . gave me responsibility at a young age and taught me what it meant to become a leader of men. It was the comradeship of us boys and the awareness of the duties the war imposed upon us that sustained my enthusiasm and made life meaningful.' Yet, when he recounts his military training at the Labour Service camp at Rodewald in early 1945, the enthusiasm and exhilaration are no longer in evidence:\n\nThe weeks at Rodewald were cold, wet and miserable. We sixteen-year-old recruits were drilled in the basics of infantry combat. From mid-January to mid-March we were sent out day and night through swampy meadowlands that made us sink knee-deep into mud. Every ditch, hidden under snow-crusted ice, had us plunge into freezing water. We were taught how to storm make-believe enemy trenches with drawn bayonets and how to fire bazookas at haystacks. We were doused with tear gas and sent through billowing clouds, sometimes crawling and sometimes running at full speed, with our gas masks on our faces until our lungs gave out and we collapsed in the icy mud. Our barracks were cold, and we suffered from diarrhoea and fevers.\n\nAnd still this was easy compared to what awaited the HJ boys at the front.\n\nAnother significant aspect of training in the HJ was through the medium of film. From 20 April 1934, the HJ organized the _Jugendfilmstunde_ (Youth Film Hour) for its members. At first, these took place once a month, but by 1936 they were organized every week. With the help of the Ministry of Propaganda, the HJ was able to make these screenings an integral part of its members' activities. Most of the feature films that were designated 'valuable for youth' were commissioned by the Ministry of Propaganda. These films were either overtly political or underpinned National Socialist objectives in a less direct manner. They included: _Heimkehr_ ( _Homecoming_ , 1941), _Der grosse K\u00f6nig_ ( _The Great King_ , 1942) about Frederick the Great, and _Die Entlassung_ ( _The Dismissal_ , 1942). In 1942\u20133, the HJ screened more than 45,290 Reich Film Hours, with an attendance of 11,215,000. These screenings supplemented the film education that was given in schools. During the war, films with themes such as self-sacrifice, camaraderie and heroic death were regarded as particularly valuable. These films included _Stukas_ ( _Dive Bombers_ , 1941), _Himmelhunde_ ( _Sky Hounds_ , 1942) and _Junge Adler_ ( _Young Eagles_ , 1944).\n\nIn addition, the HJ taught its members film making, as part of its educational work. The argument for films created by the young for the young was that HJ members shared their comrades' experiences and therefore knew what kind of themes to portray and what would be of interest to them. The films made by the HJ between 1939 and 1942 tended to be on the subject of youth, war and sacrifice. They included _Einsatz der Jugend_ ( _Youth's Mission_ , 1939), _Der Marsch zum F\u00fchrer_ ( _The March to the F\u00fchrer_ , 1940), _Unsere Kinder\u2013Unsere Zukunft_ ( _Our Children\u2013Our Future_ , 1940) and _Soldaten von Morgen_ ( _Soldiers of Tomorrow_ , 1941).\n\nThe HJ also commissioned eight documentary films entitled _Junges Europa_ ( _Young Europe_ ) between 1942 and 1945. These showed the work carried out by the HJ during the war, from collecting the harvest to working in armaments factories. Their main purpose was to highlight to the civilian population the role of the HJ in the war. They showed their discipline, organization, obedience, camaraderie and self-sacrifice. They depicted the many activities of the HJ and the ideological commitment to National Socialism of its members. Hence, film was widely used for propaganda purposes by and within the HJ. The advocates of Nazi film propaganda in the HJ claimed that: 'Thanks to the National Socialist film educational work, youth is directed towards the heroic and is therefore psychologically prepared and entirely capable of withstanding all pressures'. Nazi film propaganda played a significant part in the activities of the HJ and formed part of the reason for which German youth was willing to sacrifice itself until the very end of the Nazi regime.\n\nThe HJ laid down strict regulations for its leaders. Boys appointed as HJ leaders had to be aware that this was not a 'privilege' but an 'obligation'. The honour and status of the HJ had to be at the centre of a leader's thoughts and actions and he had to advance this through his own impeccable behaviour both 'in service and in leisure time'. In service, he had to be appropriately attired in a clean and orderly uniform. Those promoted to leadership in the HJ had to be loyal and devoted to the organization and 'must never leave for frivolous or futile reasons'. In the HJ, the will of the individual had to take second place to that of the movement as a whole. Orders were to be given 'with personal responsibility' on the part of the person giving the order and were never to 'lack tactfulness and comradeship spirit'. Orders were to be 'short, clear, necessary and easy to understand'. Leaders themselves as subordinates to others at a higher level were to follow orders and directives given to them in an accurate and obedient way, thus setting the best example for their own subordinates. Leaders had to maintain a calm and professional manner and to keep their nerve, 'even in critical moments'. They were to maintain discipline among and care for their subordinates. They were to ensure that their boys wore the appropriate clothing for any weather and that they ate and drank enough during hikes and marches. In addition, every leader was obliged to follow the service regulations and guidelines of the HJ organization.\n\nHowever, despite its attempts at thoroughgoing socialization and leaders' obligations, Kater has shown that the Hitler Youth 'was not always an expression of monolithic cohesiveness'. Inadequate training and leadership structure gave way to much incompetence, abuse and corruption on the part of the youth leaders. Schirach and Axmann tried to limit incompetence and abuse by establishing leadership courses and sessions for Hitler Youth leaders, such as the Academy for Youth Leadership in Brunswick, set up in 1939. In 1942, an Office for Leadership Training and Instruction was established to deal with all aspects of leadership within the HJ organization. Nevertheless, leadership problems remained and indeed were exacerbated when older leaders were conscripted for military service. By 1940, 25 per cent of all Hitler Youth leaders were at the front, and by 1944 boys in their mid-teens were being commanded by boys of the same age.\n\nDid all German youths participate and willingly so? Certainly, some boys were very enthusiastic about the movement and the leadership positions and experiences it offered to them. They were trained for command and ardently supported the regime and its aims. Erich Loest recalls: 'They enticed us for their own ends, but we were glad to go along with it. Many like me did absolutely nothing to resist, we saw no reason to resist, and in turn, when we became leaders, we enticed the others'. Many others went along with HJ membership without being particularly committed to its ideology, but because it was necessary to belong for apprenticeships and other work opportunities, as many craft guilds and businesses would only take on young boys and girls if they belonged to the Party youth groups. Many had unexpressed reservations or misgivings, despite their outward conformity. There was also a relatively small group of dissenting youth. As Kater points out, even after membership became compulsory, 'too many teenagers came and went or did not enrol at all'. Some disliked the monotony of the drills and routine; others were individualistic enough to reject the norms of the organization as a whole. Many cliques and bands of youth sprang up across the Reich.\n\n**GERMAN YOUTH OUTSIDE THE HJ**\n\nDissenting youth included those who belonged to the Hitler Youth, but did not turn up regularly to its meetings, those who had left the Hitler Youth, bored or disillusioned with its requirements, or those who had never enrolled in the Nazi youth movement in the first place. In Munich, the _Blasen_ (Bubbles) were made up of anti-authoritarian workers and apprentices. They resisted the limits placed upon their personal freedom by the Hitler Youth. They remained aloof from the official youth group and engaged in theft, sabotage and other transgressions of the law. Similar cliques existed in other cities. In Hamburg, working-class gangs such as the Jumbo Band wore distinctive clothing and attacked the Hitler Youth. Other dissident youth groups sprang up that had ideological affinities to the outlawed Communists and Socialists, such as the _Meuten_ (Packs) in Leipzig, which had approximately 1,500 members. They were blue-collar workers and apprentices who met at local cinemas and bars. They went on hikes, listened to Radio Moscow, dressed in unconventional clothes and wore red handkerchiefs. Moreover, they engaged in open confrontations with the Hitler Youth.\n\nThe Edelweiss Pirates sprang up spontaneously in many German cities. These young people were typically aged between 14 and 18. In Cologne, the Navajos, in Dusseldorf, the Kittelbach Pirates and in other cities in the Rhineland and Ruhr, other groups of Edelweiss Pirates all attracted the animosity of the Hitler Youth because of their nonconformity. They represented a challenge to the authority of the Hitler Youth and sought conflicts with its members and patrols. In contrast to the state youth groups, the Edelweiss Pirates mixed groups of girls and boys and their sexuality was open. The Hitler Youth and the Nazi government frowned upon this. The Edelweiss Pirates congregated in gangs at local parks, bars, squares or street corners. At weekends, they hiked and camped in the countryside, where they chatted, sang traditional youth songs or adapted the words to reflect their own experiences. During holidays, they undertook longer journeys to assert their independence from both their parents and the regime. The existence of these dissenting youth groups and bands within the totalitarian system of the Third Reich\u2013with all its pressures to conform\u2013is significant.\n\nAnother type of non-conformist youth, from a middle- and upper-middle-class background, belonged to the Swing Youth. The Swing Youth listened to jazz and swing music in private or at carefully selected nightclubs and caf\u00e9s. They dressed distinctively and ostentatiously, wore their hair long and imitated American or British attitudes and styles. The Swing Youth originated in Hamburg, but groups established themselves in other cities including Frankfurt and Berlin. They attracted the attention of the authorities, both for their open sexuality and for their rejection of National Socialist cultural norms.\n\nThe Hitler Youth _Streifendienst_ (Patrol Service) had been established in July 1934 to police German youth. Its original function had been to combat crime, delinquency and undisciplined behaviour within the Hitler Youth. Its service regulations stipulated that 'the _Streifendienst_ has to keep watch that the manner of all members of the National Socialist youth groups is in keeping with the dignity and honour of the NSDAP'. The _Streifendienst_ reported any criticism of the regime to the Gestapo. By 1937, however, the remit of the _Streifendienst_ had been extended to dealing with former Hitler Youth members that had left the organization and members of the numerous cliques and bands of youth outside the Hitler Youth.\n\nThe Hitler Youth and the Gestapo regarded all these groups as a challenge to their authority, and the regime clamped down upon them more and more as the war years progressed. On 9 March 1940, Himmler issued a police ordinance for the 'protection of youth'. This was aimed at repressing cliques and gangs. It prohibited young people from meeting in bars or on the streets after dark. Many young people who failed to comply with this restriction were arrested and placed in youth custody camps, such as Moringen. On 25 October 1944, Himmler issued an ordinance for the 'combating of youth cliques':\n\nIn the last few years, and recently in increased numbers, gatherings of youths (cliques) have formed in all parts of the Reich . . . Cliques are groupings of juveniles outside the Hitler Youth, who lead a separate way of life, whose principles are irreconcilable with the National Socialist worldview. Collectively, they reject or are indifferent to their duties towards the national community, or towards the Hitler Youth, and in particular evince a lack of will to conform with the dictates of wartime.\n\nIn November 1944, the leaders of the Edelweiss Pirates were publicly executed in Cologne.\n\nThe White Rose movement was a resistance group that appeared in Munich during 1942 and 1943, centred round Hans and Sophie Scholl. Together with fellow students Alexander Schmorell, Christoph Probst, Willi Graf, and Professor of Philosophy at Munich University, Kurt Huber, Hans and Sophie Scholl wrote and circulated a series of leaflets that openly told of the murder of Jews in Poland and called for popular mobilization against Hitler. Between the summer of 1942 and February 1943, the White Rose distributed a series of six pamphlets at night in a number of German cities, including Cologne, Essen, Stuttgart, Frankfurt and Nuremberg, as well as Munich. The first leaflet urged Germans to resist the regime. The second leaflet told of 300,000 Jews already killed in Poland. The third asked Germans to sabotage the war industry. Moll has argued that 'their will to topple the system and their ingenuity drove them to ever more reckless campaigns'. On 18 February 1943, Hans and Sophie Scholl distributed their leaflets around Munich University for the last time. Having thrown between 1,500 and 1,800 leaflets down the staircase of the main entrance at Munich University, they were caught by the caretaker and arrested. Willi Graf was arrested later the same day. The remaining three members of the White Rose were arrested within the next ten days. A Special Court was set up under Roland Freisler on 22 February 1943, which sentenced Hans Scholl, Sophie Scholl and Christoph Probst to death, and they were executed the same day. Schmorell and Huber were executed on 13 July 1943 and Graf on 12 October 1943. Sophie Scholl had said to a fellow prisoner on the day of her execution: 'What does our death matter if thousands will be stirred and awakened by what we have done? The students are bound to revolt'. But they did not. On the contrary, the National Socialist German Students' Association organized a demonstration of 3,000 students to show their loyalty to the regime.\n\nLess well known than the White Rose movement, the members of the H\u00fcbener group were among the youngest Germans to resist the Nazi regime, acting independently, without the guidance of adults. This group of four teenagers from Hamburg\u2013Helmuth H\u00fcbener, Karl-Heinz Schnibbe, Rudolf Wobbe and Gerhard D\u00fcwer\u2013took a moral stance against the Nazi dictatorship. In contrast to the majority of German Mormons who accepted the Nazi regime, Helmuth H\u00fcbener, a 16-year-old Mormon, distributed anti-Nazi leaflets with the aid of his three co-conspirators. They continued to do this for approximately six months before they were reported to the Gestapo. Arrested in February 1942, H\u00fcbener was given the death penalty for committing treason, whilst his three comrades received prison sentences of between four and ten years for their part in the conspiracy. H\u00fcbener was executed in October 1942.\n\n**THE HJ DURING THE WAR**\n\nDuring the war, HJ duties became increasingly time-consuming and dangerous. At first, Hitler Youth members served in auxiliary positions on the home front. They made door-to-door collections of paper, cloth and scrap metal for the war effort and foraged for medicinal herbs and mushrooms. Later, they worked as couriers and messengers. Herbst recalls his time as a courier and how it raised his self-esteem: 'Here I was, a sixteen-year-old boy, having an official pass for any railroad train I chose to enter, carrying important messages\u2013it all seemed very exciting and flattering to me'. HJ members distributed ration cards and propaganda leaflets. They also worked as air raid wardens and firefighters. As the war progressed, their obligations increased, not just on the home front. Hitler Youth members were sent to the newly conquered Polish territories to re-educate the _Volksdeutsche_ (ethnic Germans) who lived on the land there. At first, this was a voluntary service. Tens of thousands of young Germans went to the borderlands where they both taught proper German to the _Volksdeutsche_ and worked in the farms and fields. They helped in the process of 'settling' young people who were 'worthy of Germanization' and prepared them for agricultural work. By 1942, it became compulsory for Hitler Youth members to serve for a six-week period in this duty.\n\nAs the HJ became increasingly involved in war duties, tensions sometimes arose with their parents. Herbst remembers his experiences: 'I had war duties to carry out, I told my mother, such as standing fire watch during air raids and helping with clean-up work thereafter. Such tasks made it seem somehow inappropriate that I ask her for permission or promise to be back home at a certain time in the evening . . . I also grew increasingly on edge listening to my mother's daily questions of whether I had done my school work. I became less and less willing to accept her directions for how I should spend my time out of school'. These types of intergenerational tensions were commonplace and, as we have seen, they were often deliberately fostered by the HJ movement.\n\nThe HJ became involved in anti-aircraft work and its members were drafted into fighting units as the tide of the war turned against Germany. Most Hitler Youth members joined the _Wehrmacht_ feeling optimistic that Germany would achieve a speedy victory and with a determination to defeat their 'inferior' enemies. They were convinced of their own superiority. Once serious setbacks and defeats occurred, however, these feelings changed to disillusionment. Young soldiers were also frustrated by the duplication of their Hitler Youth drills and training when they entered the _Wehrmacht_. Physical injuries, fatalities and inadequate food provision, as well as psychological scarring, all had a damaging impact on morale. Some HJ members came to have doubts about the war and about the honourableness of their country's cause. They soon came to realize that this 'was not the war of the textbooks, the war of glory and heroic death, but the war of blood and gore, of terror and shame, and of bodies torn and mutilated'. The young soldiers began to question the Nazi stereotype of the cowardly, 'sub-human', 'swamp Russian', once they encountered their Soviet counterparts. Difficulties faced by young soldiers even before Stalingrad led some of them to doubt their own function and to question the regime. Deserters experienced the SS's 'emergency justice' in the form of summary executions and hangings.\n\nBetween 1943 and 1945, some 200,000 teenagers served as canoneers to destroy enemy planes. The anti-aircraft artillery training, which showed the boys how to handle searchlights and anti-aircraft guns, lasted just four weeks. After that, the flak helpers (as young as 15) experienced active combat, at first in their own localities, but then in destinations far from their homes. Obliged to work during the night, as well as during the day, they were deprived of sleep, as well as terrified. Casualties were heavy. Despite this, the boys continued to fight, driven on by a sense of duty to the _F\u00fchrer_ and fatherland. Bloodshed was a symbol of valour and heroism. This experience was also difficult for them in terms of their identity and status. These young people saw themselves as outgrowing the Hitler Youth and parental control, yet they were not accepted as 'soldiers'.\n\nOn 19 October 1944, the _Volkssturm_ was established to draft all men aged between 16 and 60 that were capable of bearing arms to defend the homeland. In many cases, entire HJ groups enlisted together. They were given tasks such as digging trenches, guarding, and defending towns and villages. Dietrich Strothmann recalls: 'I was just a very ordinary kid, obedient, docile, compliant . . . ready for duty at all times, available and, ultimately, willing to die'. Despite the insistence that these young boys and old men were imperative to the war effort, they were given inadequate equipment and weaponry. In the last months of the war, the Hitler Youth formed anti-tank brigades against the Soviet advance and units to secure strategic bridges. They greatly feared revenge by their enemies and this made them more determined to keep fighting. The spirit of boys as young as 15 in these circumstances was driven by a determination to destroy as many Soviet tanks as possible. Gerd H\u00e4ffner remembers:\n\nThey went for the tanks with a fearlessness that is simply indescribable. And they really were just children. I was seventeen, but they were fifteen or younger. Without a thought for themselves, they walked into certain death. And at many points they actually forced the Russians to pull back. But then the children in their HJ uniforms were left lying in the street.\n\nHitler's youth was required to take its part in the struggle\u2013in the face of death\u2013until the very end of the war. Herbst recalls mid-March 1945, when he was 'being groomed to enter the fight in the war's last, decisive hour . . . No matter how sombre the outlook was . . . we were going to live up to our oath and fight for Germany'. In April, he describes how his platoon came under attack from enemy fire for the first time: 'Artillery shells hurled towards us with unnerving shrieks . . . Dirt, stones, tree branches, and shrapnel hurled through the air. I tasted sulphur between my teeth . . . My stomach turned, my knees trembled. This then, was . . . the baptism of fire, I thought.' In April 1945, HJ members engaged in street battles with Soviet soldiers in Berlin. Beevor describes how HJ detachments desperately held on to the Pichelsdorf and Charlotten bridges over the Havel. The HJ boys were cheated of their youth, their humanity and, in many cases, their lives.\n\n'Never before in German history had the young been so courted\u2013and never so abused'. In the HJ, young Germans were seduced by the Nazi regime and ultimately betrayed by it. Young Germans gave up their independence to the greater cause of the 'national community'. In certain ways, the HJ was similar to youth groups that had preceded it, in terms of encouraging a love for the German countryside and for German folklore. Like its predecessors, the HJ engaged German youth in peer camaraderie and popular activities like camping and hiking. This accounted for much of its attraction, particularly in its early years. The most significant difference was that it imbued its members with Nazi propaganda and increased their duties. The other important distinction was that once enrolment into the HJ became compulsory, youth group membership was not a matter of desire or choice, as it had been in the past, but one of obligation. The sense of belonging and of duty to the organization and its demands were central to the nature of the HJ. Clearly in the HJ, as in other Nazi formations, the individual was subordinated to the group. Conformity to the organizational norm was designed to create true believers in the National Socialist system. HJ members were bound to the community of the organization, and, above and beyond that, to the 'national community'. Indeed, the continued commitment, involvement and service of the HJ contributed to the prolonging of the war. Paul Kehlenbeck recalls the war years: 'Anyone fighting for Germany was also fighting for Adolf Hitler, of course. It was almost the same thing. Only towards the end of the war did these attitudes begin to change and lose their hold, but Hitler retained his authority right to the end.' Ultimately, Nazi 'total education' was intended to create such strong devotees to the regime that they would be willing to sacrifice their lives for it\u2013and many did. After the capitulation on 8 May 1945, Peter Boenisch recalls: 'We were really in a state of total physical and psychological exhaustion. Then, when the growing realization came to us that it had all been pointless and in vain, that one's friends had died for nothing, that one's brother had died for nothing, we were utterly embittered'. In the end, the youth of the Third Reich were the victims, as well as the perpetrators, of its bestial criminality.\n\n**NOTES**\n\n. Cited in J. Noakes and G. Pridham (eds), _Nazism 1919\u20131945: A Documentary Reader_ , Vol. 2, (Exeter, 1984), pp. 416\u201317.\n\n. Cited in Samuel and Hinton Thomas, _Education and Society_ , p. 18.\n\n. Cited in ibid., pp. 29\u201330.\n\n. G. Wyneken, _Der Gedankenkreis der freien Schulgemeinde_ (Leipzig, 1913), p. 10.\n\n. Hahn, _Education and Society_ , p. 39.\n\n. On this, see P. Stachura, _The German Youth Movement 1900\u20131945: An Interpretative and Documentary History_ (London, 1981) and W. Laqueur, _Young Germany: A History of the German Youth Movement_ (London, 1981).\n\n. BA NS 26\/336, 'Der Wimpel des Jungsturms Hitler!', Rundfunkvortrag des ehem. F\u00fchrers A. Lenk am 8. November 1933, p. 2.\n\n. BA NS 26\/331, 'Satzungen des Jugendbundes der NSDAP', March 1922. See also Koch, _The Hitler Youth_ , p. 47.\n\n. BA NS 26\/333, 'Satzungen des Jugendbundes der NSDAP', March 1922, in 'Das Werden der nationalsozialistischen Jugend'.\n\n. BA NS 26\/336, 'Der Wimpel des Jungsturms Hitler!', pp. 3\u20134.\n\n. BA NS 26\/332, 'Anerkennung des Jungsturms als Vorl\u00e4ufer der Hitler-Jugend'.\n\n. BA NS 26\/336, 'Der Wimpel des Jungsturms Hitler!', p. 8.\n\n. BA NS 26\/336, 'Das Werden der nationalsozialistischen Jugend', II. Teil, 14 May 1934, p. 1.\n\n. BA NS 26\/336, 'Das Werden der nationalsozialistischen Jugend', Teil 1, 10 April 1934, pp. 3\u20134.\n\n. On Schirach, see M. Wortmann, _Baldur von Schirach: Hitlers Jugendf\u00fchrer_ (Cologne, 1982).\n\n. BA NS 26\/353, 'Bundesbl\u00e4tter des Deutschen Jungvolkes, Bund der Tatjugend', Folge I, no date, p. 1.\n\n. Ibid.\n\n. Ibid., p. 2.\n\n. BA NS 26\/353, 'Bundesbl\u00e4tter des Deutschen Jungvolkes, Bund der Tatjugend', Folge II, Weihnachten, 1930, p. 3.\n\n. Ibid., p. 6.\n\n. Koch, _The Hitler Youth_ , p. 101.\n\n. BA NS 26\/331, 'Hitlerjugend Aufnahme Erkl\u00e4rung'.\n\n. BA NS 26\/336, 'Die Aufbauarbeit der Hitlerjugend im Staat', pp. 1\u20133.\n\n. BA NS 26\/336, 'Hitlerrede auf dem 1. Nationalsozialistischen Reichsjugendtag in Potsdam', no date, pp. 1\u20132.\n\n. BA NS 26\/336, 'Deutsche Arbeiterjungen unter Hitlers Fahnen', Artur Axmann, 3 May 1933, p. 9.\n\n. Cited in Noakes and Pridham (eds), _Nazism 1919\u20131945,_ p. 419.\n\n. Hahn, _Education and Society_ , pp. 78\u20139.\n\n. BA NS 26\/353, 'Richtlinien f\u00fcr den Jungvolkf\u00fchrer', 24 March 1932, p. 1.\n\n. Ibid., pp. 1\u20132.\n\n. Ibid., p. 4.\n\n. Ibid.\n\n. Ibid., pp. 5\u20136.\n\n. Cited in Noakes and Pridham (eds), _Nazism 1919\u20131945_ , p. 420.\n\n. A. Kl\u00f6nne, _Jugend im Dritten Reich: Die Hitler-Jugend und ihre Gegner. Dokumente und Analysen_ (Cologne, 1984), p. 50.\n\n. J. Herbst, _Requiem for a German Past: A Boyhood among the Nazis_ (Madison, 1999), p. 95.\n\n. BA NS 12\/1438, 'Verh\u00e4ltnis HJ.\u2013NSLB'.\n\n. Stachura, _The German Youth Movement_ , p. 148.\n\n. Herbst, _Requiem for a German Past_ , p. xv.\n\n. Kl\u00f6nne, _Jugend im Dritten Reich_ , p. 55.\n\n. Cited in Koch, _The Hitler Youth_ , p. 231.\n\n. Cited in Knopp, _Hitler's Children_ , p. 11.\n\n. Herbst, _Requiem for a German Past_ , p. xiv. See also M. von der Gr\u00fcn, _Wie war das eigentlich?: Kindheit und Jugend im Dritten Reich_ (Darmstadt, 1979).\n\n. Kl\u00f6nne, _Jugend im Dritten Reich_ , p. 82.\n\n. Cited in Knopp, _Hitler's Children_ , p. 16.\n\n. Cited in ibid., p. 12.\n\n. Cited in ibid., p. 12.\n\n. Cited in ibid., p. 30.\n\n. Cited in ibid., p. 18.\n\n. Herbst, _Requiem for a German Past_ , pp. 44\u20135.\n\n. Cited in Knopp, _Hitler's Children_ , p. 19.\n\n. BA NS 26\/336, 'Vormilit\u00e4rische Wehrert\u00fcchtigung der Hitler-Jugend', 17 December 1941.\n\n. Herbst, _Requiem for a German Past_ , p. 98.\n\n. Ibid., p. 81.\n\n. Ibid., p. 174.\n\n. D. Welch, 'Educational Film Propaganda and the Nazi Youth', p. 73.\n\n. Ibid., p. 77.\n\n. Ibid., p. 80.\n\n. C. Belling and A. Sch\u00fctze, _Der Film in der Hitlerjugend_ (Berlin, 1937), p. 36.\n\n. On what follows, see BA NS 26\/353, 'F\u00fchrerordnung'.\n\n. M. Kater, _Hitler Youth_ (Cambridge, Mass. and London, 2004), p. 15.\n\n. Stachura, _The German Youth Movement_ , p. 130.\n\n. Cited in Knopp, _Hitler's Children_ , p. 2.\n\n. Kater, _Hitler Youth_ , p. 25.\n\n. See E. Boesten, _Jugendwiderstand im Faschismus_ (Cologne, 1983) and D. Peukert, _Die Edelwei\u00dfpiraten. Protestbewegung jugendlicher Arbeiter im Dritten Reich. Eine Dokumentation_ (Cologne, 1980).\n\n. Kater, _Hitler Youth_ , p. 137.\n\n. On this, see Peukert, _Die Edelwei\u00dfpiraten_ and M. von Hellfeld, _Edelwei\u00dfpiraten in K\u00f6ln_ (Cologne, 1983).\n\n. D. Peukert, _Inside Nazi Germany: Conformity, Opposition and Racism in Everyday Life_ (London, 1987), pp. 156\u20137.\n\n. BA NS 26\/338, 'Vorl\u00e4ufige Dienstvorschrift f\u00fcr den HJ-Streifendienst', 15 May 1936, p. 8.\n\n. Cited in M. Burleigh and W. Wippermann, _The Racial State: Germany 1933\u20131945_ (Cambridge, 1991), p. 238.\n\n. On the White Rose, see H. Siefken (ed.), _The White Rose: Student Resistance to National Socialism 1942\u20131943_ (Nottingham, 1991) and I. Jens (ed.), _At the Heart of the White Rose: Letters and Diaries of Hans and Sophie Scholl_ (New York, 1987).\n\n. C. Moll, 'Acts of Resistance: The White Rose in the Light of New Archival Evidence', in M. Geyer and J. Boyer (eds), _Resistance against the Third Reich 1933\u20131990_ (Chicago, 1994), p. 200.\n\n. On this, see B. Holmes and A. Keele (eds), _When Truth was Treason: German Youth against Hitler_ (Urbana and Chicago, 1995).\n\n. Herbst, _Requiem for a German Past_ , p. 164.\n\n. Ibid., pp. 88\u20139.\n\n. Kater, _Hitler Youth_ , p. 178.\n\n. See, for example, Herbst, _Requiem for a German Past_ , pp. 116\u201317.\n\n. Ibid., p. 128.\n\n. Kater, _Hitler Youth_ , p. 199.\n\n. Ibid., pp. 206\u20137.\n\n. Cited in Knopp, _Hitler's Children_ , p. 240.\n\n. Cited in ibid., p. 276.\n\n. Herbst, _Requiem for a German Past_ , p. 177.\n\n. Ibid., p. 181.\n\n. On this, see R. Bessel, _Nazism and War_ (London, 2004), p. 148; A. Beevor, _Berlin: The Downfall 1945_ (London, 2002), pp. 281 and 316; N. Stargardt, _Witnesses of War: Children's Lives under the Nazis_ (London, 2005), pp. 313\u201314 and 316.\n\n. Beevor, _Berlin_ , pp. 340 and 356.\n\n. Knopp, _Hitler's Children_ , p. ix.\n\n. Cited in ibid., p. 174.\n\n. Cited in ibid., p. 276.\n[6 THE LEAGUE OF GERMAN \nGIRLS ](Pine_9781847887641_epub_c16_r1.html#d6e8)\n\nThis chapter examines the role and function of the League of German Girls (BDM) as the Nazi organization for the regimentation and socialization of girls. It analyses the aims of the BDM and their implementation. Since 1980, when Klaus published his pioneering book on the BDM, there has been a proliferation of books and articles on the subject. In addition, the memoirs of girls who grew up in the Third Reich and were affiliated to the BDM enhance our knowledge and understanding of the movement by providing accounts of their personal experiences. Within the secondary literature on the BDM, a number of issues remain disputed and others inadequately addressed. Reese's contribution to the historiography of the BDM highlights some of these controversies, such as whether or not motherhood was the overriding objective for German girls and what 'type' of girl the regime aimed to create within the BDM. Furthermore, Reese indicates that the question of the extent to which the BDM had a modernizing effect on German girls has not been adequately treated. She notes that 'in the early 1930s girls were often drilled to march in formation and trained in field exercises and sometimes marksmanship with air rifles'. This was not the traditional gender expectation for girls. The purpose of this chapter is to analyse the role of the BDM in the training and socialization of girls. An analysis of BDM training manuals and guidelines, as well as its magazine, _Das Deutsche M\u00e4del_ , and other literature, gives a clear indication of the norms, values, expectations and political ethos of the organization and the way in which it imbued German girls with the National Socialist _Weltanschauung_. Before embarking upon an examination of the BDM, it is useful to contextualize this subject with an overview of the girls' youth movement before the Nazi 'seizure of power'.\n\n**YOUTH GROUPS FOR GIRLS BEFORE THE NAZI ERA AND THE ORIGINS OF THE BDM**\n\nThe development of youth movements and girls' leagues in Germany had its cultural, socio-economic and ideological foundations in Germany's particular, modern history, with its 'late' industrialization and rapid pace of modernization, its demographic change and urbanization. The _Wandervogel_ movement incorporated girls as well as boys from 1905 onwards. Young men and women explored the German forests, hills and villages and hiked through the countryside. It is important to note, however, that the defining image of the youth culture was male. Whilst sports clubs and youth groups admitted girls as members, there were 'no independent forms of leisure for girls'. Nor yet was there the type of distinctive girls' leisure subculture that defined the youth groups and subcultures for boys. Girls joined mixed groups of girls and boys. They gradually began to establish their own single-sexed girls' groups. The groups and organizations for girls came to be characterized by a number of significant aspects, as girls came to establish a culture of their own.\n\nBetween 1918 and 1928 girls were expelled from the male youth movement and they established their own autonomous leagues and groups. In particular, as a result of the First World War, girls came to call for a realm of their own, distinctive from that of boys. Girls constructed separate, new identities for themselves for the first time during this period. A gender polarization occurred between _Wandervogel_ boys and _Wandervogel_ girls. Girls opened up social, geographical and political 'space' for themselves, which they had never claimed before and which had never before belonged to them. These girls sought role models in legendary and historical figures, such as Brunhilde and Queen Luise. De Ras has typified this 'new breed of girls and young women from the youth movement' as 'the New Gretchens'. These girls expressed loathing and disgust for 'modern' girls. They disliked metropolitan girls, Jewish girls, 'French' girls, 'Gypsy' girls, 'unhealthy' girls, and lesbians, as these 'types' went against their conservative conceptions of what a 'German' girl should be. The idyllic, rural past formed the core of their world view.\n\nThe girls differentiated their identity with concepts such as 'female culture' and 'sacred island' within their girls' leagues and communities. The 'sacred island' was a phase representing a search for 'the link between \"nature\" and \"femininity\", of romancing the German female body, soul and mind, a desire for wholeness'. It was outwardly expressed by the wearing of a loose, white dress\u2013the 'island dress'. A number of girls' leagues retreated into country homes whilst others were secret societies, hidden in the countryside. Settlements of young women and girls, such as those at Schwarzerden and Loheland, became islands of female culture and activity, entirely independent from and impenetrable to male influence. In terms of constructions of the body within the girls' leagues, the 'body culture' that was idealized was neither a 'motherly' body, nor a 'Lolita figure'. Rather, the body type that was advocated was 'androgynous' and as de Ras writes, 'a closed off and closing off young female body'. Lust and sexuality were frowned upon.\n\nThe period between 1928 and 1934 was characterized by the growth of extreme nationalist youth groups and leagues, such as the _Jungnationaler Bund_ (Young-National League), the _Freischar Junger Nation_ (Free Band of the Young Nation) and the _Gro\u00dfdeutscher Bund_ (Greater German League), which attracted girls as members. Girls' groups continued to be semi-autonomous within these organizations. Whilst the girls groups were not homogeneous, they did have a number of common aspects: 'the idealisation of _Kultur_ ; the worship of wholeness and aversion to fragmentation; the love for and glorification of German history, tradition, folklore, language; the emphasis on the importance of soil, nature, landscape, the rural and a dislike of modernity . . . and the wish to remain physically, psychologically, and racially pure and natural'. Their vitality and patriotism came increasingly to be bound up with anti-Semitism and 'racial' exclusivity. As the Nazi _Machtergreifung_ approached, these 'new radical nationalist girls', who were eager to become part of the state, were ripe to become 'absorbed' into the National Socialist movement.\n\nIn the meantime, the National Socialists' own girls' league, the BDM, emerged in 1930 after a number of previous attempts to set up a youth group for girls within the Nazi movement had failed. Adolf Lenk had set up the first group in the early 1920s. This was followed by a number of sororities or sisterhoods within the HJ in 1927. However, these groups had little popular appeal. Groups for girls began to be created within the context of local National Socialist women's associations. They tended to put the emphasis upon girls' duties, such as mending and cooking. Before 1930, there was a large number of small Nazi and _v\u00f6lkisch_ groups for girls, such as the _Deutscher M\u00e4del Ring_ , set up in Bavaria in 1927, that competed with each other for membership. In 1930, the BDM came to prominence. As part of the HJ, it had a stronger bureaucratic structure and it was marked out by its uniform. On 7 July 1932, Gregor Strasser and Baldur von Schirach dissolved all the other girls' groups that were part of the Nazi women's associations and ordered their membership to be transferred directly into the BDM. Hence, the BDM became the only National Socialist association for girls. However, prior to the Nazi _Machtergreifung_ , the BDM was just one of many youth groups for girls in Germany. By the end of 1932, it had a membership estimated at between 10,000 and 15,000 girls.\n\n**BDM MEMBERSHIP**\n\nAfter Hitler came to power on 30 January 1933, the BDM rose to a much more significant position. This was partly the result of the process of _Gleichschaltung_ or 'streamlining', by which other girls' youth groups were dissolved, and partly due to the desire of girls who had never been in a youth organization before to take part in the National Socialist movement. Both of these factors led to a substantial increase in the membership of the BDM after 1933. In many cases, schools and teachers were encouraged to put pressure on girls to join. Between 1933 and 1936, the BDM experienced a vast expansion in its membership, encompassing almost half of all German girls aged between 10 and 18. The BDM considered the old youth movement to be uncreative and lacking in true value. It came to regard the _Wandervogel_ girl as an 'anti-type'.\n\nMany girls were attracted to the BDM because it gave them the chance to do 'what hitherto only boys were allowed to do', for example, to have more independence from their parents, go on trips and take part in group activities. Others joined because they wanted to feel important, and not to be excluded from the world of adults. Entry into the BDM allowed girls to escape from their tedious home lives, where they were usually under the constant scrutiny of their parents. The BDM gave girls the chance to be independent from their parents and to play a role within an organized, hierarchical social institution. They had the opportunity to become leaders within the organization. Reese argues that this led to 'an enhanced sense of female self-esteem'.\n\nGirls from middle-class families, in particular, often eagerly seized upon the opportunities offered to them by the BDM, because of their childhood experiences. In the aftermath of the Wall Street Crash, shattered prestige and finances were strongly felt by all members of middle-class households. In addition, the children of such families were subjected to very strong parental discipline, and girls felt especially intimidated by their fathers. Consequently, they felt insecure, useless, unconfident and insignificant. The BDM gave girls an opportunity to break out of this pattern and style of their lives at that time. Indeed, some girls joined the BDM as a sign of their rebellion against the authority of their parents. The BDM gave young girls a sense of peer camaraderie, involvement in their national cause and independence from their parents. Melita Maschmann has described how she wished to escape from her 'childish, narrow life' and 'to follow a different road from the conservative one prescribed . . . by family tradition'. Many of her contemporaries joined the BDM for similar reasons. In this respect, there is some indication that the BDM had a modernizing and liberating effect upon German girls. However, in the place of maternal and paternal influence came societal authority and state force.\n\nAn important reason for the popularity of the BDM was the sense that girls were participating equally within the German youth movement. The BDM had its own role, and the Nazi regime exploited a sense of competition and rivalry between the sexes. As a girls' organization, the BDM offered a range of roles and career paths for girls. A mass organization that grew quickly in size, the BDM needed leaders and many girls were appointed as leaders. Leaders had to exhibit the correct type of personality and characteristics. Through their active commitment, leaders had the opportunity of rising up through the hierarchy of the organization to higher leadership positions. Whilst many leaders on the lower levels were volunteers, those at the top end of the leadership scale were paid a salary for their work. Such career possibilities led to increased enthusiasm for the organization.\n\nReese argues that girls were influenced and shaped by their 'living practice in the National Socialist organization'. The BDM allowed them to take part in activities that were beyond the horizons of their social milieu and to have access to a variety of new experiences. This accounted for much of its initial popularity. However, the appeal of the BDM differed across areas, as well as social and cultural milieus. There was sometimes significant parental resistance to the membership of their daughters into the BDM, on grounds of political or social outlook. Furthermore, in 1936, once membership became compulsory, apathy and disinterest came to replace the earlier enthusiasm among girls to join the BDM. As the years passed, it became increasingly difficult to evade service in the BDM, particularly after the second HJ decree of 1939.\n\n**THE NORMS AND REQUIREMENTS OF THE BDM**\n\nThe National Socialist regime claimed that youth autonomy and the principle of self-leadership of youth were central to the BDM. However, there is much evidence to show that the BDM did not foster true independence among either its members or its leaders. Trude Mohr, the first BDM _Reichsreferentin_ , appointed in June 1934, had the following expectations for behaviour in the organization: 'Don't talk, don't debate, live a National Socialist life in discipline, composure and comradeship!' As was the case in all Nazi formations, the ethos of the BDM entailed a loss of individuality for its members. They were bound to a community of peers, and, above and beyond that, to the community of the nation. The BDM, therefore, was not an aggregate of the individual personalities of its members, but rather a community into which individuality was dissolved. As Maschmann's memoir describes: 'Everything that was \"I\" had been absorbed into the whole!' This community ethos, which was a central part of the character formation of the group's members, was closely tied to National Socialist ideology. There may have been a degree to which the individuals involved believed that they were acting on their own initiatives on behalf of the nation, but this feeling was manufactured. They were, instead, being manipulated and were very much a part of the socialization process. The objectives of the BDM were in no way directed at fostering the individual development or independence of its members. Maschmann has described how: 'No one made us think for ourselves or develop the ability to make moral decisions on our own responsibility. Our motto was: The _F\u00fchrer_ orders, we follow!' Hence, the BDM attempted to create devoted 'believers' in the system. A foreign observer noted about both BDM and HJ members that 'their attitude of mind is absolutely uncritical . . . They are nothing but vessels for State propaganda'. The network of social control became ever tighter and the grip of the National Socialist movement on its youth members increasingly comprehensive.\n\nThe first prerequisites of a BDM member were that she had to be of German origin and of sound heredity. The model German girl had to be prepared to work hard to serve the 'national community', to recognize National Socialist norms and values, and to accept them unquestioningly. She was to be physically fit, healthy, clean, dressed in an orderly manner and domestically capable. Characteristics of cleanliness, rectitude, faith and honour were to be formed by means of discipline. Above all, the BDM girl was to be aware of her future duty as a woman, to become a mother. She had to be well-versed in German culture and music. As a future mother, she was to develop a knowledge of traditional German songs, tales and dances, so that she could be a 'culture bearer' to the next generation. It was important, therefore, that girls took advantage of their 'natural' closeness to their homeland and understood the 'laws of nature'.\n\nGirls in the BDM were educated and socialized quite differently from their male counterparts in the HJ, especially, of course, in terms of ideals and aims. There was, however, one main similarity in the way in which they were trained, and this was that both boys and girls had to be prepared to fulfil their obligations\u2013albeit different obligations\u2013towards their nation and fatherland. In the first place, both had 'the duty to be healthy' and 'to remain pure'. Both were trained to be capable of physical achievement, fit for work and compliant to organizational discipline. Industriousness, hygiene and obedience were expectations in both male and female youth groups. These values appealed to the lower middle classes in particular. Nazi youth group members were, in effect, unthinkingly and unquestioningly bound to the norms of their respective organizations, developing initiatives only within the framework of these norms, not independently. As Maria Eisenecker recalls, 'our own opinions were not asked for'. Apart from that, comparisons of the ideal boy and girl, and of their duties, showed marked differences.\n\nEven the kind of language in which role models were described gives a strong indication of the dissimilarities between the expectations of girls and boys. Girls were to react to circumstances with their emotions, whereas boys were to react with their minds; girls were to store their experiences internally, whilst boys were to use theirs actively and creatively; girls were to be docile and to give of themselves, whilst boys were to affect others, gain victories and conquer; girls were to be passively content, whereas boys were to be active builders or destroyers of cultures; girls were to care for the family and household, whilst boys were to lay the foundations for the state; girls were to view life as a gift, whereas boys were to consider it as a struggle; for girls 'motherliness'\u2013not femininity\u2013was the ultimate aim, whilst for boys it was very clearly 'manliness', in a militarized sense. In certain respects, this kind of language portrayed a very passive role for girls as compared to that for boys, which does not seem surprising considering the ideological tenets upon which the Nazi state was founded. However, this only gives a partial picture, for girls were not to be totally passive. Indeed, the anti-image of the ideal BDM girl was that of the feminine 'young lady', an idea that was taken from the _Wandervogel_. Frivolity and luxury were frowned upon by BDM leaders, who wanted to create strong and hardy young women. Indeed, the BDM even went as far as promoting the books of certain authors, such as Marie Hamsun and Erika M\u00fcller-Hennig, who wrote about young people that led 'brave and courageous lives', whilst discouraging the reading of 'sentimental' writers, to the extent of recommending to parents which books to buy for their children. It also made recommendations of 'books that you should read' to its members. These largely comprised German _v\u00f6lkisch_ literature, the works of 'blood and soil' novelists, such as Josefa Berens-Totenohl, and those that gave a sense of the German past. Similarly, special recommendations of books were made for the BDM camps. For the summer camp in June 1937, for example, Hitler's _Mein Kampf_ and Alfred Rosenberg's _The Myth of the Twentieth Century_ headed the list of recommended reading.\n\n**TRAINING AND ACTIVITIES IN THE BDM**\n\nHow were girls in the BDM trained and educated? Jutta R\u00fcdiger, the BDM _Reichsreferentin_ from 1927 onwards, retrospectively claimed that:\n\nIn the education of girls . . . we rarely spoke about \"motherhood\". Rather, we educated the girls in their own interest and that of the nation, preparing them to lead wholesome lives, to take an active role in the world of work and society. But first and foremost, what we wanted was to educate them to have a bright and cheery life as young girls.\n\nHowever, this statement is not consistent with the documentary evidence. There is nothing in the written documents and pamphlets of the BDM that indicates that the aim of the movement was simply 'to educate them to have a bright and cheery life as young girls'. R\u00fcdiger also talked of forming 'politically aware' girls. By this, she meant not girls who would 'debate and discuss in parliament', but girls and women who would know about the necessity of the life of the German _Volk_ and act in accordance with this.\n\nThe aims of the BDM to create 'the German woman and mother' ruled out political engagement. The training of girls entailed a variety of components, including physical fitness, health, hygiene, dress codes and sexual attitudes. The body itself no longer remained in the private sphere of the individual, but was subordinated to the national interest. Physical training was very closely linked to health and to racial-biological ideas. To this extent, sport was not an end in itself, but a means of training German youth in accordance with National Socialist ideals. Its goal was inner discipline. Consequently, no free or spontaneous sport or dance was allowed. Any expression of individualistic movement that went against the National Socialist sense of order was proscribed. Instead, regulation and discipline were emphasized. Many dance and exercise routines were structured within a certain form, such as a circle, a square or simply in rows. Girls were to keep their bodies firm and healthy by means of exercise, in order to be able to reproduce for the nation in the future. They were instilled with the sense that they were responsible for the preservation of the nation. Girls had to pass a special fitness test in order to enter the _Jungm\u00e4del_ (JM). This meant they were all able to meet a certain required standard of physical fitness. Fit girls would develop into healthy women, bear healthy children, and therefore preserve the health of the nation in the future. Notwithstanding the Nazi ideological imperatives behind physical training, the sporting activities were very popular with BDM girls, and games and competitions generated interest and enthusiasm for the organization.\n\nIn 1934, in order to promote the idea of unity of body, soul and spirit, Baldur von Schirach, the leader of the National Socialist youth movement, introduced an achievement badge for physical prowess. By 1940, 60,000 such badges had been awarded. But the objective of sport was not personal achievement. Sports prizes were not awarded for the sake of individual merit per se, but as part of the overall attempt of the regime to create an entire generation of healthy girls, within the framework of its racial programme. Physical training was important for health and for the 'pure' preservation of the race. Indeed, this was so central to Nazi beliefs, that the BDM broke down the old taboo that girls ought not to take part in sporting activities in public, by organizing sports festivals in villages and towns, as well as hikes and camping trips. Schirach recognized the fact that this was rather revolutionary. Sports galas and competitions, such as the National Sports Gala, held for the first time in Bamberg in 1938, became important occasions. They opened with songs, a speech and flag-raising. Gymnastic displays and races formed the main part of such events, which closed with formation dancing and a parade of all the participants.\n\nSpecial training manuals elaborated on physical training for the _Jungm\u00e4del_ (JM), girls aged between 10 and 14. JM members had to take part in a wide variety of physical activities including running and swimming. The full range was illustrated in the manual, including ball-throwing games, gymnastics and floor exercises. There was also formation dancing, for example, in concentric circles, with an accompanying musical score. A similar book was designed for the physical training of girls in the BDM. This manual illustrated sports activities and formation dances. It stressed that there could be no ideological education without physical education, for physical training was the most important and effective means in the educational programme of the Nazi youth groups. Sport was considered to be important because it strengthened the will, created camaraderie and exercised each part of the body. Such books dedicated to physical education illustrate the great importance attached to sport by the regime and its youth groups. They had the clear intention of creating a whole generation of healthy and fit German girls. Schirach continually emphasized the need for a 'synthesis between body and spirit' as the aim of the BDM.\n\nHealth education was considered to be especially important for girls, as they would become the bearers of the next generation. 'You have the duty to be healthy' was the motto for the BDM in 1939. To the BDM, beauty was nothing other than the expression of physical and spiritual health, the harmony of body, soul and spirit. BDM leaders had to care for the health of their members by ensuring the correct nutrition, clothing, way of life, physical exercise, leisure and relaxation. Health was of paramount concern, for as natural selection showed, the sick and unfit perished. This could not be allowed to happen to the German _Volk_ , for only healthy nations could survive and a successful nation needed to be 'pure' and 'fit'. Hence, the National Socialist state had to promote and strive for health and fitness, in order to secure the future stock of the race. The German nation had to be healthy, capable of achievement and able to cope with life.\n\nThe BDM educated its members about how the health of the nation was to be achieved. For example, the sending away of children to the countryside served the aims of the health of the nation. Each year, hundreds of thousands of children were sent into the countryside, pale and weak. They returned home healthy, tanned and strong. Whilst away, they learned to appreciate the beauty of the German homeland. The BDM also taught girls the importance of the measures and laws introduced by the National Socialist government to preserve and protect the hereditary health of the German _Volk_ , such as the Marriage Health Law and the Law for the Prevention of Hereditarily Diseased Offspring. This highlighted to the BDM girls their personal responsibility for their health, as their own health was an integral part of that of the whole nation. Therefore, they had the duty to protect their own health and to refrain from associating with the 'inferior', in order that future generations would be strong and fit. It was repeatedly stressed that 'to be healthy and to remain healthy is not our private concern, but our duty!'\n\nEach BDM girl was to be 'the founder and protector of a healthy, fit German family'. BDM girls were to be 'the expression of the harmony of health and beauty'. Care of the body, skin, hair and nails were of great importance, as was dental care. Directions for such care were given in great detail. Sufficient sleep was also important\u2013at least ten hours per night\u2013and the value of sleep for overall health was to be enhanced by sleeping in an airy room with the window open. Care for clothes was also significant, as were living conditions, with the right amount of air, light and heat. Correct nourishment was necessary too, not just for personal health, but for the health of the nation. Girls had to eat regularly and to have the right balance of vitamins and minerals in their diet. In addition, the use of alcohol and nicotine was strongly rejected.\n\nDress was another important aspect of girls' training in the BDM. Girls were expected to wear their BDM uniform on all national holidays, on Party days, and on all special family and school festival days. The uniform, which consisted of a white blouse and a dark-blue skirt, was practical and simple. Wearing the uniform, with its distinctive kerchief and knot, was, of course, an outward sign of being part of the rank and file of the movement. The uniform had to be washed and ironed properly, and was to be worn 'with pride'. It was not to be embellished with jewellery or accessories. Cleanliness and an orderly appearance were part of the requirement too. Simplicity and orderliness were criteria that applied to ordinary clothes as well as to the uniform. There was much antagonism towards international fashion. There was a call for the introduction of a German fashion, quite separate from French, British or American styles. German fashion was to be based on simple lines and forms, with the added advantage of using new materials such as the synthetic silk and spun rayon being produced by the German textile industry in the mid-1930s. Nazi fashion excoriated the former styles of the 'vamp', who wore bright nail polish and plenty of make-up; the 'sweetheart', who was petite and blonde, with a 'warbling little voice'; and the 'boyish girl', who had very short hair, wore men's clothes, smoked, drank and told jokes. Indeed, all these stereotypes\u2013even the 'blue stocking' for her intellectualism\u2013were viewed with 'unmitigated contempt'. They did not match up to the ideal type\u2013the BDM girl. Girls and young women were to be clean and tastefully dressed, without having to owe their good appearance to cosmetics or jewellery. This sense of Nazi fashion held some popular appeal and there was a certain amount of desire to belong to the BDM in order to be able to wear the uniform, which was practical, yet not unattractive. Stargardt has cited the example of a Berlin girl who 'bitterly rued her parents' refusal to buy her an outfit when her whole class at school was admitted to the _Jungm\u00e4delbund_ '.\n\nThe attitude towards sexual behaviour in the BDM paralleled that in the rest of society. Essentially, sexual life had its main task in serving the preservation of the race and nation. By and large, the National Socialists' 'new morality' reduced sex to its biological function of reproduction. The ideal, primary aim for BDM girls was childbirth and motherhood within marriage. Early marriage, in particular, was seen as a way of both discouraging promiscuity and encouraging large numbers of legitimate children. Marriage was considered to be a dutiful, moral obligation by the youth group leaders. The demand that sexual activity should be carried out within a marriage remained the overall belief in the BDM. Its leaders were convinced that 'the family should be the only place for children to grow up in and that the destruction of monogamy must be prevented by women'. Hence, the desire in the BDM was not to encourage a child 'at any price', but rather to promote very specific norms of motherhood, in line with the regime's aims of 'selection' of the 'desirable' and 'elimination' of the 'undesirable'. Apart from these attitudes, sexuality was not an issue that was discussed in the BDM. Both the BDM and HJ were essentially non-sexual in their orientation. Hitler Youth boys were expected to treat BDM girls as comrades and to be chivalrous towards them. Nonsexual camaraderie and friendship were the general expectations about behaviour with fellow members of the youth movement. However, 'there was very probably a good deal of flirting during youth group activities, especially when boys and girls were working together'. Lust and desire were not acceptable, and physical training and diversion were partly designed to pre-empt or substitute them. The satisfaction of sexual urges was regarded as shameful, reprehensible and biologically and medically unnecessary. 'Fresh, clean, clear German air' was the alternative to sexual education. To this extent, sexuality was mysticized and was almost completely a taboo area, although girls were warned about the dangers of sexual disease. Whilst clear guidelines were given about punishment for homosexual activities in the HJ, it appears that the issue of lesbianism was not raised at all in the official guidelines of the BDM.\n\nExpectations about sexual behaviour did not always correspond to reality, as exemplified by cases of girls having sexual relationships with soldiers and SS men, and of their having illegitimate babies in order to present the F\u00fchrer with children. The lack of explanation about sexual behaviour partly explains this phenomenon. Some girls also had relationships with 'racially inferior' men from the eastern occupied territories, which was partly a response to the allure of the exotic, but was, of course, anathema to the regime. Hence, there was some conflict between the emphasis on moral purity within the BDM and the popular perception of the organization. Promiscuous behaviour on the part of BDM girls became even more pronounced during the war. Popular jokes included the following interpretations of the initials BDM: _Bubi Dr\u00fcck Mich_ ('squeeze me, laddie'); _Bedarfsartikel Deutscher M\u00e4nner_ ('requisite for German men'); _Brauch Deutsche M\u00e4del_ ('make use of German girls'); _Bald Deutscher M\u00fctter_ ('German mothers to be'); _Bund Deutscher Milchk\u00fche_ ('League of German Milk Cows'). Such jokes clearly reveal the popular response to the BDM, suggesting both doubts about standards of morality within the organization and some displeasure at its emphasis on procreation.\n\nThe most important vehicles for socialization in the BDM were the weekly _Heimabend_ and the summer camps. The main purpose of the _Heimabend_ was ideological education or training in the National Socialist _Weltanschauung_. Punctuality played a part in the overall creation of discipline, so that girls who arrived late were obliged to pay a small fine. The girls sang old Germanic songs and learned new ones. They received instruction on a number of subjects, such as National Socialism, the work of the BDM, German history and race. Instruction often took the form of stories, for example about loyalty, honour, courage and obedience, and such sentiments were underlined with appropriate songs. The girls then did handicrafts or watched a puppet show, and the activities ended with a final song. The _Heimabend_ lasted two hours. It took place in the afternoon for JM girls and in the evening for BDM girls.\n\nThe content of _M\u00e4delschaft_ , which was a series of guidelines for BDM leaders on the structuring of the _Heimabend_ , gives a clear indication of the type of activities and training involved in these sessions, and especially the themes taught. For example, the April 1938 edition was a special issue on Bismarck. After considering Bismarck and his achievements, a number of pages were dedicated to Hitler and his creation of a new great German empire. This was followed by a large section showing BDM leaders the types of activities they might use for the _Heimabend_ , such as speeches by Hitler and songs. The May 1938 edition took the First World War as its main theme, including the sense of camaraderie among the soldiers and a number of soldiers' letters which showed their unity, determination and love for their fatherland. After the Anschlu\u00df of March 1938, the June 1938 issue was designed to teach BDM girls about Austria, by means of a combination of history, poems, short stories and illustrations. The November 1938 issue was dedicated to the rise of the National Socialist Party, its 'time of struggle' and its heroes, such as Horst Wessel and Herbert Norkus. The April 1939 edition was on the theme of 'The Struggle against Bolshevism'. It showed the evils and perils of Bolshevism, for example how it destroyed marriage and the family, and explored Germany's role as a bulwark against the spread of Bolshevism. This edition also had a section entitled 'You have the Duty to be Healthy!', which underlined the importance of regular exercise, sufficient sleep and care of the body. It also stated that girls who sat in stuffy and crowded cinemas or overcrowded and smoky bars were behaving irresponsibly and not living the lifestyle that was expected of them in order to fulfil their duties as future mothers of the German nation.\n\nThe June 1939 special edition for the BDM summer camp placed an emphasis on nature and the struggle for survival, underlining that only the healthy survived, whilst the weak and the sick perished. The final part of it summarizes activities in the summer camp and gives a clear indication of the socialization of girls in the BDM. This starts with a section on the orderliness and cleanliness of the camp. It was the responsibility of the camp-leader to ensure this from the start, for example, by confiscating anything the girls left lying around and only returning such items after the collection of a fine. The camp-leaders then had to arrange the times for flag-raising, meals and training, and to ensure the punctuality of the girls for each of these activities. Each morning, on waking, after cleaning their teeth and brushing their hair, the girls would either go on a march or sing a song. The method of flag-raising and procedures for meal times were given in minute detail. Set speeches were included that were to be used at flag-raising and meal times. One of the main functions of the summer camps was to give systematic instruction on contemporary political events. The participants were to acquire an understanding of the political life of the German _Volk_ and its 'living space'. It was the task of the camp-leader to clearly and simply explain the political situation to the participants. The themes and topics included the shame inflicted upon Germany by the Treaty of Versailles, Hitler's foreign policy successes and the struggle of the German nation for its 'living space'. Camp-leaders were instructed about what songs to sing in the camp and when to sing them. For example, it was optimal to use songs when marching through a village or town, when the girls were tired, in order to raise their spirits, on rainy days when the girls had to remain in the camp, or on evenings in the village squares to encourage local people to join in the singing. The guidelines included ten compulsory songs that had to be used in the summer camp. These could be supplemented with others, but had to be the main ones sung. Instruction was given about what entertainment to use for the 'children's afternoon' and the 'village evening', such as songs and plays. Books for the summer camp were also recommended. Finally, instructions were given on how to end the camp. This was to be done by discussing a particular theme\u2013such as the glory of nature\u2013and then summing up the purpose of the camp, a greeting to Hitler and a final song. Hence, the whole camp was very carefully orchestrated and planned out, from start to finish, with the clear objective of the political socialization of its participants.\n\nThe JM _Heimabend_ and camps had similar imperatives, as can be seen from the guidelines for leaders. In 1936, for example, the themes covered included the life of Adolf Hitler, nature, the _Winterhilfswerk_ (Winter Relief Agency), motherhood and heredity. In 1937, the Four Year Plan, the 'Jewish Question', Hitler's achievements and the homeland featured as important topics for education. Much instruction was carried out by means of short stories. The girls in the JM were encouraged to be brave, devoted, comradely, obedient and honourable. Camaraderie was the foremost quality expected from the JM. She was asked: 'Can you make personal sacrifices in order to help a comrade? And if you can, do you then ask for thanks or public recognition? Is your camaraderie still there when there is no one to observe it?' JM honour was also highly valued. There was no room in the organization for dishonesty, deceit, scandal or envy.\n\nIn January 1938, a new BDM agency, the _Glaube und Sch\u00f6nheit_ (Faith and Beauty), was formed for 17- to 21- year-old girls, under the leadership of Clementine zu Castell. By February 1939, the organization had some 500,000 members. Physical and ideological training formed the core of its work. Its members took part in activities for two to three hours per week. For the purpose of training, the young women were formed into working groups for different themes or subjects including sport, gymnastics, national tradition, plays and culture, handicrafts, music, foreign news, health, and household and agricultural competence. There were also groups on fashion design and anti\u2013air raid protection. This arrangement enabled girls to take part in activities in which they were particularly interested. Physical education was at the forefront of _Glaube und Sch\u00f6nheit_ training, for 'a healthy and beautiful body' was considered to be the prerequisite of 'a healthy and beautiful spirit'. Beyond this, the main task of the _Glaube und Sch\u00f6nheit_ was to form 'self-assured young women', rooted in the National Socialist spirit and capable of taking their part in the creation and maintenance of their _Volk_. The _Glaube und Sch\u00f6nheit_ organization allowed girls to take part in the elite sports of tennis, fencing and horseriding, as well as home design. In 1943, 'the ideal of the lovely, beautiful and proud girl' of the _Glaube und Sch\u00f6nheit_ was considered to be an elite type of woman. By the time she reached the age of 28, she would take the title of _Hohe Frau_ , according to Himmler, the ultimate expression of the racially pure, physically fit and accomplished German woman.\n\nAn important factor in girls' socialization in the BDM was the creation of a new ethos regarding their working or professional life. National Socialist ideology certainly attached value to work, as service to the 'national community', and as a 'moral duty' for both males and females. Manual labour, in particular, was considered important, as it would ensure that the nation would comprise physically fit, healthy and hardy individuals, who would transmit these attributes to future generations. For boys, all the physical training and drills in the HJ would serve them in the future as workers in heavy industry, on the land or, ultimately, in the armed forces. For girls, physical education would prepare them for their work placements and, of course, ultimately to become mothers. It was generally accepted that girls would give up their jobs once they were married, in order to take care of their households. The work ethos, and therefore the training given to girls, was to be understood in this sense. Household instruction had been of prime significance in girls' training since the genesis of the BDM. Even then, Adolf Lenk, the founder of the National Socialist youth movement, had claimed that members of the girls' groups had the task of becoming good German housewives. Subsequent measures carried through by the youth leadership of the regime aimed to give girls a broader knowledge of household skills and abilities. Indeed, they were part of a wider attempt at the rationalization of housework under National Socialism, bound up with the Nazi policy of autarky and the duty of women to support the Nazi economy through their household economy.\n\nWhilst boys' training in the HJ took on a more militant nature between 1937 and 1939, with a greater emphasis on pre-military preparation, in the BDM a development towards preparation for 'female' activities was evident in this period. As early as 1936, the first BDM 'household school' had been set up, in which girls could gain experience and training in household activities. Specific training in household management and child care was given in the BDM household schools. Here, a one-year course provided its participants with everything they would need to know as future mothers. The teaching plan at the household schools involved four main areas of work: practical teaching, which included cookery, baking, gardening and needlework; theoretical training, which consisted of lessons about nutrition, health, care for infants and for the sick; studies about the 'national community', which dealt with issues of nation, race and the national economy; and sport, which included hiking as well as activities such as singing and dancing.\n\nA further development in this respect was the _Pflichtjahr_. This was a one-year compulsory work placement for girls which came into effect from 1 January 1939. The rationale behind the _Pflichtjahr_ was twofold\u2013to give girls necessary experience and training and to help mothers of _kinderreich_ families and farmers' wives. It was deemed especially important that those girls who had spent their whole lives in towns and cities should serve in the countryside, so that for at least a year they could do farm work and get to know about rural life. This measure was intended to create a sense of connection and closeness to the homeland. Agricultural service had started out as a voluntary task. In 1934, 7,000 girls had begun farm work and by 1937, 43,000 girls had taken part in the scheme, mainly working on farms in Silesia, East Prussia and Pomerania. Most of the girls doing agricultural service lived in a camp with their leader and from there went to help on individual farms, starting their work at 6 am each day. Other girls stayed on a farm with the farmer's family. Farm work was considered to prepare girls for marriage and motherhood. Those doing their _Pflichtjahr_ in towns were required to help in the households of _kinderreich_ families, with housework, washing, cooking and shopping. Work was a preparation for the tasks and requirements of motherhood. Apart from agricultural labour and household work, jobs in the 'caring professions' were deemed suitable, especially because skills could be acquired\u2013such as looking after newborn babies\u2013that were directly applicable to family life. Hence, whether in connection with familial or professional situations, girls in the BDM had to learn that the state allocated specific obligations to them, and demanded self-discipline and duty fulfilment from them. By 1940, there were 157,728 girls working in agriculture and 178,244 in domestic service for their _Pflichtjahr_. Kater describes this as 'a ruthless exploitation of unpaid menial labour'.\n\n**THE BDM IN WARTIME**\n\nDuring the war, the term 'domestic training' was applied more widely, ultimately changing its meaning to a total preparation to serve in any manner required by the state. Short training courses were run, teaching girls to make 'new out of old' and to help soldiers with washing and mending clothes. BDM members were faced with new duties and obligations. In the first year of the war, over nine million girls were mobilized, especially for agricultural work. Youth mobilization involved a wide variety of activities, including the distribution of propaganda material for the Party, the distribution of food ration cards, the harvesting of crops, the collection of money for the War Winter Relief Agency, looking after the wounded, caring for children, and gathering herbs and wild fruit. Kater estimates that in 1939\u201340, 'over a million BDM members spent 6.5 million work hours on the collection of various herbs and tea'. BDM girls were to be proud to help German soldiers, for example by setting up washing and mending centres for soldiers' clothes. They sewed slippers for soldiers out of woollen blankets or plaited them out of straw. They did agricultural work, looked after children, helped out in kindergartens and schools, and were also involved in active war service, for example as employees in armaments factories. BDM girls were also required to work as tram conductors and postal workers in the towns and cities during the war. They helped civilians at train stations, handing out food and drinks to passengers, as well as assisting the victims of air raids. They worked in kitchens to prepare sandwiches and soup for people who had been rendered homeless by the Allied bombing campaigns and collected and distributed everyday items such as toothpaste, toothbrushes and hairbrushes to them. The BDM girls became increasingly involved in first-aid duties for both soldiers and civilians. They assisted in nursing the wounded in hospitals. Towards the end of the war, in the bleakest and most hopeless days, many girls continued to show a willingness to make sacrifices for their nation, even when their lives were in danger.\n\nHence, the scope and range of duties expanded considerably throughout the duration of the Third Reich and especially during the war. The BDM girls became involved in _Osteinsatz_ (Eastern Service). From mid-1940 onwards, in conjunction with the SS, BDM girls were sent into the eastern occupied territories to clean and prepare the houses for German settlers, once the SS had removed the former inhabitants. Some BDM girls assisted the SS in evicting Poles from their homes. After their initial duties in the eastern territories, which lasted approximately four to six weeks, many BDM girls had to remain in these territories for up to one year, in order to help the newcomers to settle there, offering assistance in the homes and schools. The BDM girls were sometimes shocked and appalled at the demeanour and lack of hygiene of the 'ethnic Germans'. The reality they were presented with was somewhat different from their expectations. Nevertheless, the BDM girls carried out their tasks, going into the villages where they sang German songs and played German games with the children, so that they could learn German, as well as showing them maps of greater Germany and teaching children the basics of how to write and read German. _Osteinsatz_ had become an increasingly large part of BDM activities as the war progressed. The carefree days of rambling and camping had long gone.\n\nThe BDM was an integral part of a blood-binding community, whose members were called upon to serve their nation and take responsibility for the future of their race under National Socialism. The BDM played a significant role in the Nazi process of socializing and training German girls. In doing so, it borrowed much from the traditions of the earlier _Wanderv\u00f6gel_ movement, but it added its own National Socialist ideals and ethos. As noted above, after 1933, many young girls were obliged to become members of the BDM, once other youth organizations had been dissolved or merged into the Nazi youth movement. However, it must be noted that the swell in the BDM membership between 1933 and 1936 was also the result of voluntary entry into the organization. Many girls were initially attracted to the BDM for a variety of reasons, such as to gain independence from their parents and to take part in activities previously inaccessible to them. They were attracted by the sport, singing and handicraft activities on offer to them in the organization. They could go on adventurous trips and have opportunities for careers as youth group leaders that broke the limits of their social, regional or family boundaries, but they were not free or independent in reality. Parental authority was simply replaced by state force and discipline. BDM members, and even leaders, were individually unimportant, as part of a larger formation. The BDM did not aim at the individual development and independent thought of its members, but instead at making them true believers in the National Socialist system. Once membership became compulsory and ideological training became more pronounced, the allure of the BDM waned. Girls became increasingly subordinated to the organization and limited by the restrictions it placed upon them.\n\nMotherhood was important both ideologically and in practice. The type of training the girls underwent within the BDM was aimed at future motherhood. As BDM leaders were told, the purpose of BDM activities was 'to create a generation of girls that will become a generation of healthy women and mothers'. Hence, the emphasis upon physical education was not for its own sake, but had the purpose of creating fit bodies to reproduce healthy, strong offspring. As future mothers, BDM girls were to become protectors and preservers of the German race. The centrality of motherhood and race to the organization and to the state is plainly evident from the literary output of the BDM. Yet, at the same time, in practice there were modernizing effects resulting from BDM activities. These were functional and were a product of the circumstances of the war and the necessity of using BDM girls in roles related to it. Measures that appeared modernizing were simply pragmatic attempts on the part of the regime to prevent the collapse of the agricultural workforce and to meet the requirements of state efficiency. But reactionary ideology continued to underpin the regime's intentions, even though it could not adhere to this as rigidly as its leaders would have wished. This ideology should not be ignored or negated. The ideological education and training of girls remained central to the aims of the Nazi regime. Hence, pragmatic concerns that created a tendency towards modernity and a concurrent conventionalization of girls' pursuits corresponding with the regime's ideology were both evident in the BDM. Girls were obliged to fulfil their roles in the Nazi state until its demise. Ultimately, Hilde Seffert, a former BDM member claims, 'we were cheated of our youth'; whilst another former BDM girl, Gudrun Pausewang, talks of the hurt 'having to admit to oneself that one had believed in a false ideal, that the whole thing had been a lie and that one had been abused'.\n\n**NOTES**\n\n. The most important contributions on the subject include: D. Reese, 'Bund Deutscher M\u00e4del\u2013Zur Geschichte der weiblichen deutschen Jugend im Dritten Reich', in Frauengruppe Faschismusforschung (ed.), _Mutterkreuz und Arbeitsbuch: Zur Geschichte der Frauen in der Weimarer Republik und im Nationalsozialismus_ (Frankfurt am Main, 1981); D. Reese, _'Straff, aber nicht Stramm\u2013Herb, aber nicht derb'. Zur Vergesellschaftung der M\u00e4dchen durch den Bund Deutscher M\u00e4del im Sozialkulturellen Vergleich zweier Milieus_ (Weinheim and Basel, 1989); Klaus, _M\u00e4dchen in der Hitlerjugend_ ; Klaus, _M\u00e4dchen im Dritten Reich_ ; G. Kinz, _Der Bund Deutscher M\u00e4del. Ein Beitrag zur Au\u00dferschulischen M\u00e4dchenerziehung im Nationalsozialismus_ (Frankfurt am Main, 1990); B. J\u00fcrgens, _Zur Geschichte des BDM (Bund Deutscher M\u00e4del) von 1923 bis 1939_ (Frankfurt am Main, 1994); G. Miller-Kipp (ed.), _'Auch du geh\u00f6rst dem F\u00fchrer': Die Geschichte des Bundes Deutscher M\u00e4del (BDM) in Quellen und Dokumenten_ (Munich, 2002). See also A. B\u00f6ltken, _F\u00fchrerinnen im 'F\u00fchrerstaat'_ (Pfannenweiler, 1995); F. Niederdalhoff, _'Im Sinne des Systems einsatzbereit . . .': M\u00e4dchenarbeit im 'Bund Deutscher M\u00e4del' (BDM) und in der 'Freien Deutschen Jugend' (FDJ)\u2013Ein Vergleich_ (M\u00fcnster, 1997); R. Strien _, M\u00e4dchenerziehung und-sozialisation in der Zeit des Nationalsozialismus und ihre lebensgeschichtliche Bedeutung_ (Opladen, 2000), pp. 80\u201389.\n\n. For example, see M. Maschmann, _Account Rendered: A Dossier on My Former Self_ (London, 1964); R. Finckh, _Mit uns zieht die neue Zeit_ (Baden-Baden, 1979); M. Hannsmann, _Der helle Tag bricht an\u2013Ein Kind wird Nazi_ (Hamburg, 1982); G. Herr, _Inhaltsreiche Jahre\u2013aus dem Leben einer BdM-F\u00fchrerin 1930\u20131945_ (Lausanne, 1985).\n\n. D. Reese, 'M\u00e4dchen im Bund Deutscher M\u00e4del', in E. Kleinau and C. Opitz (eds), _Geschichte der M\u00e4dchen- und Frauenbildung_ , Vol. 2 (Frankfurt am Main, 1996), pp. 271\u201382.\n\n. D. Reese, _Growing up Female in Nazi Germany_ (Ann Arbor, 2006), p. 4.\n\n. D. Peukert, _The Weimar Republic: The Crisis of Classical Modernity_ (London, 1991), p. 93.\n\n. M. de Ras, _Body, Femininity and Nationalism: Girls in the German Youth Movement 1900\u20131934_ (New York and London, 2008), pp. 6\u20137.\n\n. Ibid., p. 41.\n\n. Ibid., p. 193.\n\n. Ibid.\n\n. Ibid., pp. 187\u20139.\n\n. Reese, _Growing up Female_ , p. 31.\n\n. Hannsmann, _Der helle Tag bricht an\u2013Ein Kind wird Nazi_ , p. 34.\n\n. Finckh, _Mit uns zieht die neue Zeit_ , p. 81.\n\n. Reese, _Growing up Female_ , p. 7.\n\n. C. Leitsch, 'Drei BDM-Biographinnen', _Dokumentationsstelle zur NS-Sozialpolitik: Mitteilungen_ , April 1986, p. 77.\n\n. Maschmann, _Account Rendered_ , pp. 10 and 12.\n\n. D. Reese, 'Emanzipation oder Vergesellschaftung: M\u00e4dchen im \"Bund Deutscher M\u00e4del\"', in H.-U. Otto and H. S\u00fcnker (eds), _Politische Formierung und soziale Erziehung im Nationalsozialismus_ (Frankfurt am Main, 1991), p. 212.\n\n. Reese, _Growing up Female_ , p. 8.\n\n. See Reese, _Growing up Female_ , which uses case studies of Minden in Westphalia and Wedding in Berlin to show the different attitudes towards the BDM, pp. 102\u201357 and pp. 158\u2013246.\n\n. Reese, 'M\u00e4dchen im Bund Deutscher M\u00e4del', p. 280.\n\n. Cited in L. Becker, 'Der Bund Deutscher M\u00e4del', in R. Benze and G. Gr\u00e4fer (eds), _Erziehungsm\u00e4chte und Erziehungshoheit im Grossdeutschen Reich als gestaltende Kr\u00e4fte im Leben des Deutschen_ (Leipzig, 1940), p. 95.\n\n. Maschmann, _Account Rendered_ , p. 61.\n\n. Kinz, _Der Bund Deutscher M\u00e4del_ , pp. 126\u20137.\n\n. S. Roberts, _The House That Hitler Built_ (London, 1937), p. 208. On the impact of the BDM upon its members, see Miller-Kipp (ed.), _'Auch du geh\u00f6rst dem F\u00fchrer'_ , pp. 303\u201323.\n\n. J. R\u00fcdiger, 'Der Bund Deutscher M\u00e4del in der Hitler Jugend', in P. Meier-Benneckenstein (ed.), _Das Dritte Reich im Aufbau_ (Berlin, 1939), Vol. 2, p. 398. On discipline in the BDM, see also IfZ Db 44.92 _, Die Dienstform des BDM_ (1941).\n\n. S. Rogge, '\"M\u00e4del, komm zum BDM!\"', in _Hart und Zart. Frauenleben, 1920\u20131970_ (Berlin, 1990), p. 154.\n\n. This difference in socialization was also noticeable in Mussolini's Italian Fascist youth groups. On this, see T. Koon, _Believe, Obey, Fight: Political Socialization of Youth in Fascist Italy, 1922\u20131943_ (Chapel Hill and London, 1985), pp. 97\u20138.\n\n. On this, BA NSD 43\/151-5, 'Du hast die Pflicht, gesund zu sein!'\n\n. Cited in Knopp, _Hitler's Children_ , p. 99.\n\n. On what follows, see H. Rahn, 'Artgem\u00e4\u00dfe M\u00e4dchenerziehung und Rasse', _Nationalsozialistische M\u00e4dchenerziehung_ , 12, 1940, p. 224.\n\n. A. Kl\u00f6nne, _Hitlerjugend. Die Jugend und ihre Organisation im Dritten Reich_ (Hanover and Frankfurt am Main, 1955), p. 69.\n\n. 'Eltern, schenkt nur gute B\u00fccher', _Das Deutsche M\u00e4del_ , November 1936, pp. 22\u20133.\n\n. IfZ Db 44.102, 'BDM-Werk Glaube und Sch\u00f6nheit. Schulungsdienst', October 1941, pp. 65\u20138.\n\n. Cited in Reese, _Growing Up Female_ , pp. 43\u20134.\n\n. Klaus, _M\u00e4dchen im Dritten Reich_ , p. 48.\n\n. On this, see G. Pfister and D. Reese, 'Gender, Body Culture, and Body Politics in National Socialism', _Sport History_ , No. 1 (1995), pp. 91\u2013121.\n\n. See _Das Deutsche M\u00e4del_ , March 1938, pp. 4\u20135.\n\n. See E. Zill, 'Die k\u00f6rperliche Schulung im BDM', in H. Munske (ed.), _M\u00e4del im Dritten Reich_ (Berlin, 1935), p. 27.\n\n. Reese, _Growing Up Female_ , pp. 71\u20132.\n\n. See Klaus, _M\u00e4dchen im Dritten Reich_ , p. 49.\n\n. IfZ Db 44.65\/2, _M\u00e4del im Dienst. BDM-Sport_ (Potsdam, 1940), p. 14.\n\n. B. von Schirach, _Die Hitler-Jugend, Idee und Gestalt_ (Leipzig, 1934), p. 101.\n\n. IfZ Db 44.65\/2, _M\u00e4del im Dienst. BDM-Sport_ (Potsdam, 1940), pp. 13\u201314 and 17.\n\n. IfZ Db 44.65\/4, _M\u00e4del im Dienst. Jungm\u00e4del-Sport_ (Potsdam, 1942).\n\n. IfZ Db 44.65\/2, _M\u00e4del im Dienst. BDM-Sport_ (Potsdam, 1940).\n\n. IfZ Db 44.32\/8, 'Dienstvorschrift der Hitler-Jugend', p. 3.\n\n. IfZ Db 44.43, _Die M\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung im BDM_ , May 1939, p. 2.\n\n. Ibid., pp. 6\u20138.\n\n. Ibid., p. 8.\n\n. On what follows, see ibid., pp. 10\u201324.\n\n. BA NS 28\/83, 'Richtlinien f\u00fcr den Bund deutscher M\u00e4del in der Hitlerjugend', no date.\n\n. _Das Deutsche M\u00e4del_ , April 1939, p. 22.\n\n. See, for example, _Das Deutsche M\u00e4del_ , August 1937, p. 30, which talks about the foolishness of fashion.\n\n. _Das Deutsche M\u00e4del_ , January 1937, p. 11.\n\n. _Das Deutsche M\u00e4del_ , January 1938, pp. 29\u201332.\n\n. C. Kirkpatrick, _Nazi Germany. Its Women and Family Life_ (Indianapolis and New York, 1938), p. 103.\n\n. BA NSD 47\/6-1933, 'Sch\u00f6nheitspflege?!', _Amtliche Frauenkorrespondenz_ , p. 8.\n\n. Stargardt, _Witnesses of War_ , p. 33.\n\n. BA NSD 47\/19, _Jugend und Elternhaus. Beitr\u00e4ge zur Jugenderziehung unserer Zeit_ (1944), pp. 40\u201341.\n\n. Maschmann, _Account Rendered_ , p. 150.\n\n. Reese, 'Straff, aber nicht Stramm\u2013Herb, aber nicht Derb', p. 44.\n\n. R\u00fcdiger, 'Der Bund Deutscher M\u00e4del', p. 397.\n\n. Maschmann, _Account Rendered_ , p. 150.\n\n. Klaus, _M\u00e4dchen in der Hitlerjugend_ , p. 109. On attitudes towards sexual education, see also BA NSD 47\/19, _Jugend und Elternhaus_ , pp. 17\u201328.\n\n. Klaus, _M\u00e4dchen im Dritten Reich_ , pp. 56\u20137. On Nazi policy towards lesbianism, see C. Schoppmann, _Nationalsozialistische Sexualpolitik und weibliche Homosexualit\u00e4t_ (Pfannenweiler, 1997) and C. Schoppmann, 'National Socialist Policies towards Female Homosexuality', in L. Abrams and E. Harvey (eds), _Gender Relations in German History: Power, Agency and Experience from the Sixteenth Century to the Twentieth Century_ (London, 1996), pp. 177\u201387.\n\n. On this, see H. Bleuel, _Strength through Joy: Sex and Society in Nazi Germany_ (London, 1973), p. 136, and Klaus, _M\u00e4dchen in der Hitlerjugend_ , p. 104.\n\n. IfZ Db 44.17, _M\u00e4del im Dienst. Ein Handbuch_ (Potsdam, 1934), p. 219.\n\n. Ibid., p. 220.\n\n. See IfZ Db 44.28, 'Sommerlager- und Heimabendmaterial f\u00fcr die Schulungs- und Kulturarbeit, Sommer 1941, Jungm\u00e4del', as an example of this.\n\n. IfZ Db 44.43, _Die M\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung im BDM_ , April 1938.\n\n. Ibid., pp. 29\u201332.\n\n. IfZ Db 44.43, _Die M\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung im BDM_ , May 1938.\n\n. IfZ Db 44.43, _Die M\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung im BDM_ , June 1938 and November 1938.\n\n. IfZ Db 44.43, _Die M\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung im BDM_ , April 1939, pp. 19 and 25\u20138.\n\n. Ibid., p. 46.\n\n. IfZ Db 44.43(a), _Die M\u00e4delschaft. Sonderausgabe f\u00fcr die Sommerlager_ , June 1939, pp. 23\u20138.\n\n. On what follows, see ibid., pp. 51\u201379.\n\n. For another example of instruction for a three-week summer camp plan, see IfZ Db 44.17, _M\u00e4del im Dienst. Ein Handbuch_ (Potsdam, 1934), pp. 284\u201390.\n\n. On what follows, see IfZ Db 44.41, _Die Jungm\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung der Jungm\u00e4del_.\n\n. For example, IfZ Db 44.41, _Die Jungm\u00e4delschaft. Bl\u00e4tter f\u00fcr Heimabendgestaltung der Jungm\u00e4del_ , June 1936, includes the story of 'Das M\u00e4dchen Helge', whose moral was 'to be brave is good', pp. 6\u20138.\n\n. On what follows, see IfZ Db 44.28, 'Sommerlager- und Heimabendmaterial f\u00fcr die Schulungs- und Kulturarbeit, Sommer 1941, Jungm\u00e4del', pp. 6\u20137.\n\n. On this, see S. Hering and K. Schilde, _Das BDM-Werk 'Glaube und Sch\u00f6nheit': Die Organisation junger Frauen im Nationalsozialismus_ (Berlin, 2000).\n\n. On this, see IfZ Db 44.43, _Die M\u00e4delschaft. Sonderausgabe f\u00fcr die Schulung zur Berufswahl_ , August 1939, pp. 38\u201342.\n\n. IfZ Db 44.07, _Das Deutsche M\u00e4del_ , June 1938, p. 5.\n\n. Ibid., p. 7.\n\n. On this, see Kater, _Hitler Youth_ , pp. 96\u20137.\n\n. On this, see _Das Deutsche M\u00e4del_ , November 1940, p. 7.\n\n. _Das Deutsche M\u00e4del_ , February 1938, p. 7.\n\n. D. Reese, 'Bund Deutscher M\u00e4del', p. 166.\n\n. See Pine, _Nazi Family Policy, 1933\u20131_ 9 _45_ , pp. 81\u20136. See also, J. Stephenson, 'Propaganda, Autarky and the German Housewife', in D. Welch (ed.), _Nazi Propaganda: The Power and the Limitations_ (London, 1983), pp. 136\u20138.\n\n. On what follows, see _Das Deutsche M\u00e4del_ , January 1937, p. 9.\n\n. On this, see BA NSD 47\/16-1, I. Berghaus, 'Das Pflichtjahr. Wegweiser und Ratgeber f\u00fcr M\u00e4del, Eltern und Hausfrau'.\n\n. On what follows, see IfZ Db 44.43, _Die M\u00e4delschaft. Sonderausgabe f\u00fcr die Schulung zur Berufswahl_ , August 1939, pp. 6\u20139.\n\n. Kater, _Hitler Youth_ , p. 84.\n\n. See R\u00fcdiger, 'Der Bund Deutscher M\u00e4del', p. 401.\n\n. See Schirach, _Die Hitler-Jugend, Idee und Gestalt_ , p. 97.\n\n. Kater, _Hitler Youth_ , p. 85.\n\n. Ibid.\n\n. BA NS 26\/358, 'M\u00e4delerziehung im Kriege', pp. 116\u201317.\n\n. See IfZ Db 44.61, _Wir schaffen. Jahrbuch des BDM_ (1941), pp. 157\u201377. On the activities of the BDM during the war, see also Reese, 'Bund Deutscher M\u00e4del', pp. 174\u201380.\n\n. BA NS 26\/358, quoted in Reese, 'Bund Deutscher M\u00e4del', p. 175.\n\n. Kater, _Hitler Youth_ , p. 91.\n\n. _Das Deutsche M\u00e4del_ , January 1940, p. 6.\n\n. Stargardt, _Witnesses of War_ , p. 33.\n\n. See Knopp, _Hitler's Children_ , p. 110.\n\n. Stargardt, _Witnesses of War_ , p. 120.\n\n. Kater, _Hitler Youth_ , p. 89.\n\n. _Das Deutsche M\u00e4del_ , April 1940, pp. 10\u201311.\n\n. For an autobiographical account of _Osteinsatz_ , see H. Fritsch, _Land mein Land: Bauerntum und Landdienst BDM-Osteinsatz Siedlungsgeschichte im Osten_ (Preu\u00dfisch Oldendorf, 1986).\n\n. Special BDM pamphlets were issued on questions of race and racial obligations. For example, see IfZ Db 44.104, _M\u00e4del voran!,_ pp. 193 ff.\n\n. IfZ Db 44.17, _M\u00e4del im Dienst. Ein Handbuch_ (Potsdam, 1934), p. 9.\n\n. Ibid., p. 8.\n\n. Cited in Knopp, _Hitler's Children_ , pp. 112\u201313.\nCONCLUSION\n\nThe education and socialization of youth by the Nazi regime to become the ideal future generation of Germans in line with its ideology are central to our wider understanding of the Third Reich. The Nazi government attempted to achieve both complete social control and a 'total education' of German youth. This incorporated a root and branch reshaping of values. Education under National Socialism was used to disseminate the key components of Nazi ideology\u2013in particular the creation of national identity and racial awareness. Both formal education in schools and socialization in youth groups formed very significant aspects of this process.\n\nThere were significant historical links between the _Kaiserreich_ , the Weimar Republic and the Third Reich in terms of education. It is important to treat the Third Reich in the context of earlier administrations and to consider both similarities and differences. The extreme nationalism and authoritarianism that characterized the end of the nineteenth century was never fully eliminated during the Weimar years despite attempts at introducing progressive educational policies. In certain ways, the Nazi regime built upon the foundations from previous eras, but it often added a more radical direction to educational policy. Nazi education policy consciously donned an irrational character, based upon the power of suggestion and emotional impact rather than upon the power of reason. The educational philosophy of National Socialism was fundamentally irrational. It echoed earlier ideals, such as pan-German nationalism from the days of the _Kaiserreich_ , and pushed irrationalism to its most extreme limit. However, the Nazis had to take into account the realities of the need to develop a high level of technology and industry in order to prepare for war, and hence had to temper this irrationalism. In addition, the war itself occasioned changes to the essence and direction of Nazi educational policy.\n\nBoth the nature of Nazi policy-making and the tensions that existed between modernization and reaction in the Third Reich led to inconsistencies and ambiguities in education policy. Nazi policy-making was such that initiatives came from a number of different arenas. Hitler's fundamental beliefs about education provided the backdrop to Nazi education policy and its ethos. They were a strange blend of concepts taken from dominant contemporaneous ideas, as well as resulting from Hitler's own educational experience. Educational policy-making should have been the responsibility of the Ministry of Education, led by Bernhard Rust. However, Rust encountered intervention and challenges to his authority from a number of Nazi leaders, notably Baldur von Schirach, Martin Bormann, Robert Ley, Alfred Rosenberg, Philip Bouhler and Heinrich Himmler. Furthermore, even civil servants from his own Ministry flouted his authority. This was quite typical of the way in which the Third Reich functioned. This chaotic nature of government, as emphasized by 'structuralist' and 'functionalist' historians, meant that there was much competition between different individuals and agencies and sometimes contradictions in policy-making.\n\nOverall, the Nazi leadership disliked and distrusted the _Gymnasium_ with its humanist tradition, its emphasis on classical education and its academic snobbery. Historically, the _Gymnasium_ emerged in the tradition of classical humanism. It subsequently became quite strongly nationalist, but, nevertheless, under National Socialism, it was to lose its academic and elite status. The Nazi government aimed both to decrease the significance of the _Gymnasium_ and to reduce the influence of the Churches in German education. It claimed that these policies were designed to rationalize and modernize the education system. However, the truth of the matter was that the regime despised the traditional _Gymnasien_ because they were too academic, whilst it closed down the Church schools because it regarded them as a threat. Hence, the promises and claims made by the Nazi government to 'modernize' education remained unfulfilled in reality. Nazi anti-liberalism and anti-intellectualism in education produced a series of measures that failed to modernize the German educational system.\n\nThe Nazi education system had a short lifespan lasting just twelve years, half of them during wartime conditions. The brevity of the period was significant in terms of the capacity of the regime to push through the changes it desired, and the outbreak of the war engendered changes in its priorities. The years 1933\u20138 were spent mainly in the process of 'coordinating' teachers and trying to ensure their loyalty to the regime. The NSLB provided reports on the political reliability of teachers for appointments and promotions and attempted to achieve the ideological indoctrination of teachers. The main alterations to the school system and to the curriculum came in the years 1938 and 1939. New curricular changes found form in the publication of new textbooks between 1939 and 1942. The introduction and use of new school textbooks greatly assisted the Nazi regime in its aim of inculcating pupils with Nazi ideology. The spirit of _v\u00f6lkisch_ ideology was conveyed through children's books. Specific subjects such as biology, history, geography, mathematics and German were all utilized to this end and the new subject of _Rassenkunde_ was added to the curriculum. Through school textbooks, Nazi pedagogues sought to develop in children a sense of identity with the nation, the Nazi regime and its policies. The Third Reich did not have a clear and coherent concept of education beyond indoctrination. Political attitudes played a central role in the shaping of Nazi education policy. Education was linked with racial values. Anti-Semitism and racism in the curriculum represented a unique contribution of the Nazi regime to the history of education in Germany.\n\nThe Nazi elite educational institutions performed a special function within the Third Reich and within the Nazi education system. Certainly they challenged the traditional _Gymnasien_ in terms of status. They entailed a new kind of ideological elitism. The three main types of educational institutions to train the future elite of German society\u2013the Napolas, the Adolf Hitler Schools and the _Ordensburgen_ \u2013represented a microcosm of the Nazi _Weltanschauung_ by fostering the leadership principle, promoting competitiveness and emphasizing life as a struggle and as survival of the 'fittest'. They encouraged physical prowess, excoriated the 'enemies of the Reich', in particular the Jews, Communists and Socialists, emphasized racial purity, glorified war and fostered militarism. They underlined the necessity for _Lebensraum_ and had a significant part to play in the achievement of a 'greater German empire'. They were typically National Socialist institutions aimed at the ideological training of a new elite.\n\nIn the Hitler Youth and the League of German Girls, the Nazis created comprehensive youth organizations that were unparalleled in the history of German youth movements. Nevertheless, the Nazi youth organization in many ways did follow in the footsteps of the earlier German youth movement, which encapsulated elements of German romanticism and folklore, Nietzschean philosophy and traditional concepts such as _Heimat_. The Nazi youth groups deviated from earlier traditions and developed their own distinctive ideology. The pedagogic activities of the Nazi youth groups strove towards the creation of the 'national community' and they encouraged their members to be willing to make sacrifices for the state. Furthermore, through its anti-intellectual stance and its taking up so much of the free time of its members, the Hitler Youth contributed to the reduction in academic standards in the Third Reich. This led to tensions between the youth organizations and the schools. In particular more conservative teachers of an older generation found the youth groups problematic. Children and young people were encouraged to challenge conventional figures of authority, such as schoolteachers, priests and even their parents, enhancing the role of the youth groups, whilst simultaneously creating an anti-intellectual climate and eroding many of the traditional socialization functions of the family.\n\nUnder the circumstances of 'total war', the Nazi regime was unable to make all the educational changes it had hoped to introduce. Education had come to a virtual standstill by the end of the war. The Allied bombing raids had resulted in the mass evacuation of schoolchildren from the cities to the countryside in the _Kinderlandverschickung_ scheme. Older children were conscripted as auxiliaries. Universities were emptied of male students in 1944 following the order for the creation of the _Volkssturm_. In May 1945, the Allies faced a grave problem in Germany, a country in disarray. As far as education was concerned, a variety of difficulties presented themselves. The existing textbooks were all unsuitable. There was a severe shortage of trained teachers and a lack of school buildings. Many schools had been destroyed or were being used to accommodate displaced persons. Most significantly, there was a need for a comprehensive 'denazification' and 're-education' of Germany's citizens. The regime and its ideology were discredited. The atrocities committed in the name of the German population became more widely known as the public was confronted with the full truth about the concentration camps and the death camps.\n\nAn understanding of education in the Third Reich illustrates the dangers of political ideology determining which subjects are taught in schools and how they are taught. In a system in which Party organizations determined what was to be taught, 'national political' education had been prioritized under National Socialism. Furthermore, the sophistication and complexity of the whole system made it even more dangerous in its impact. In both the schools and the youth groups, National Socialism tapped into the deep-rooted desire of many young Germans to be part of a larger group and to belong. Nazi ideology defined and underlined all pedagogic activity\u2013knowledge of the Party and its leader, the 'national community' and racial awareness formed the core of education in the Third Reich. Educational content in the Third Reich largely comprised Party propaganda. Nazi 'total' education in the schools and youth groups together aimed to create a new young generation of Germans committed to Nazi ideology and able to carry out their obligations to the state. Ultimately, of course, this entailed a willingness to lay down their lives for it.\n[GLOSARY OF ABBREVIATIONS \nAND TERMS ](Pine_9781847887641_epub_c16_r1.html#d6e10)\n\n**AHS\u2013** _Adolf Hitler Schule_ (Adolf Hitler School)\n\n**BDM\u2013** _Bund Deutscher M\u00e4del_ (League of German Girls)\n\n**_Blut und Boden_ \u2013**blood and soil\n\n**DAF\u2013** _Deutsche Arbeitsfront_ (German Labour Front)\n\n**_F\u00fchrer_ \u2013**leader\n\n**_F\u00fchrerprinzip_ \u2013**leadership principle\n\n**_Gau_ \u2013**region; the largest unit of the NSDAP's territorial organization\n\n**_Gauleiter_ \u2013**regional leader\n\n**_Gleichschaltung_ \u2013**coordination or streamlining\n\n**_Herrenvolk_ \u2013**master race\n\n**HJ\u2013** _Hitlerjugend_ (Hitler Youth)\n\n**_Kaiserreich_ \u2013**Second German Empire\n\n**KdF\u2013** _Kraft durch Freude_ (Strength through Joy)\n\n**_kinderreich_ \u2013**literally 'rich in children'; term used to describe 'valuable' families with four or more children\n\n**KLV\u2013** _Kinderlandverschickung_ (sending children to the countryside)\n\n**KPD\u2013** Kommunistische Partei Deutschland (German Communist Party)\n\n**_Kreis_ \u2013**district; the second largest unit of the NSDAP's territorial organization\n\n**_Kreisleiter_ \u2013**district leaders\n\n**_Lebensraum_ \u2013**living space\n\n**_Machtergreifung_ \u2013**seizure of power\n\n**Napola\u2013** _Nationalpolitische Erziehungsanstalten_ (National Political Educational Institute)\n\n**NSDAP\u2013** _Nationalsozialistische Deutsche Arbeiterpartei_ (National Socialist German Workers' Party)\n\n**_NS-Deutscher Studentenbund_ \u2013**National Socialist German Students' Association\n\n**NSF\u2013** _NS-Frauenschaft_ (National Socialist Womanhood)\n\n**NSLB\u2013** _Nationalsozialistischer Lehrerbund_ (National Socialist Teachers' Association)\n\n**NSV\u2013** _Nationalsozialistiche Volkswohlfahrt_ (National Socialist People's Welfare)\n\n**_Ordensburg_ \u2013**Order Castle (Nazi elite educational institution)\n\n**_Ort_ \u2013**local branch; the smallest unit of the NSDAP's territorial organization\n\n**_Osteinsatz_ \u2013**Eastern Service\n\n**RM\u2013** _Reichsmark_ (unit of currency)\n\n**SA\u2013** _Sturmabteilungen_ (Stormtroopers)\n\n**SS\u2013** _Schutzstaffeln_ (Nazi elite formation led by Heinrich Himmler)\n\n**_Volk_ \u2013**nation; people\n\n**_v\u00f6lkisch_ \u2013**nationalistic\n\n**_Volksgemeinschaft_ \u2013**national community; people's community\n\n**_Wehrmacht_ \u2013**armed forces\n\n**_Weltanschauung_ \u2013**world view\n\n**WHW\u2013** _Winterhilfswerk_ (Winter Relief Agency)\nBIBLIOGRAPHY\n\nPRIMARY SOURCES\n\n_Unpublished_ \nBundesarchiv, Berlin (BA)\n\nNS 12| | Hauptamt f\u00fcr Erzieher\/ NS-Lehrerbund \n---|---|--- \nNS 15| | Der Beauftragte des F\u00fchrers f\u00fcr die \u00dcberwachung der gesamten geistigen und \nweltanschaulichen Schulung und Erziehung der NSDAP \nNS 22| | Reichsorganisationsleiter der NSDAP \nNS 26| | Hauptarchiv der NSDAP \nNS 28| | Hitler-Jugend \nNSD| | Drucksachen der NSDAP, ihrer Gliederungen, angeschlossenen Verb\u00e4nde und betreuten Organisationen \nR 49| | Reichskommissar f\u00fcr die Festigung deutschen Volkstums \nR 89| | Reichsversicherungsamt \nR 4901| | Reichsministerium f\u00fcr Wissenschaft, Erziehung und Volksbildung\n\nInstitut f\u00fcr Zeitgeschichte, Munich (IfZ)\n\nDb 44.07, Db 44.17, Db 44.28, Db 44.32, Db 44.41, Db 44.43, Db 44.61, Db 44.65, Db 44.92, Db 44.102, Db 44.104\n\n_Published_\n\nB\u00e4umler, A., _M\u00e4nnerbund und Wissenschaft_ (Berlin, 1934).\n\nB\u00e4umler, A., _Politik und Erziehung_ (Berlin, 1937).\n\nBecker, C. 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With the exception of a few prayers, as hereinafter\nnoted, it was originally written by one of God's noblemen, by one who\n\"lived and moved and had his being\" in the things of the Kingdom of God.\nDr. John Habermann (known also as Avenarius, Latinized form of Habermann)\ndied 1590 as superintendent at Zeitz, was a famous preacher and a\ndistinguished scholar of his day. He was noted for his profound knowledge\nof oriental languages especially of the Hebrew. Still it is not this but\nthe fact of his little prayer book that has endeared him to his fellow\nChristians. And this manual of prayers is the mature product of an inner\nlife rich in the grace of God. On every page it bears the stamp of one\nfor whom the communion with the eternal Father in heaven through the\nfaith in Jesus Christ, the Savior, is a blessed reality. Nothing more\nnatural therefore also than that he should \"live and move and have his\nbeing\" in the language of the Word of God. And this is quite apparent in\nhis prayer language. God's Word give him the terms to express his\nthoughts. Especially the Psalter, the prayer and hymn book of Israel,\nproves a veritable thesaurus of prayer terms and of these he makes a\ncopious use.\n\nThe present little volume presents the Englished edition of \"Wachet und\nBetet,\" as issued by the Synod of Iowa and other States. Owing to the\nexigencies of the times, with the great world war raging in all its fury,\na special set of prayers for times of war has been added by the\ntranslator, in the hope that they will add to the usefulness of the book.\nThese are found on pages 131-138. The hymns as far as possible are given\nin the form as found in the new Common Service Book with Hymnal. Many of\nthem however are new translations that here appear in print for the first\ntime. For these we are indebted especially to Prof. Alfred Ramsey of the\nLutheran Theological Seminary, Maywood, Chicago, and the Rev. H.\nBrueckner of Iowa City, Iowa, a fact which is here gratefully\nacknowledged.\n\nThe labor of clothing these little gems of prayer into the language of\nthe land has been done as a labor of love, albeit the stress of other\nwork often precluded the continued effort. The work was done a bit at a\ntime. This little volume is herewith issued with the fervent hope and\nprayer, that it may long continue on its course of blessing, and lead\nmany lives into the closer communion with God, through Jesus Christ. Soli\nDeo Gloria!\n\n E. H. R.\n\nWaverly, Iowa, during the blessed season of Epiphany, 1918.\n\n\n\n\n EXHORTATION TO PRAYER\n\n\nArise, dear soul, and carefully reflect who He is with whom thou speakest\nand before whom thou standest when thou prayest. Behold, thou speakest\nwith God, thy Maker, and standest in the presence of Him, the eternal\nMajesty, whom thousand times thousand holy angels and arch-angels attend.\nTherefore, O Christian, enter thou into the closet of thy soul, and\nbeware, lest thou failest to put from thee all sluggishness of heart, and\nliftest up to thy God a countenance free from blame. Then wilt thou\ndelight in the Lord and have power with Him, and prevail. Yea, thou wilt\nconquer the unconquerable God and bear away the blessing through Jesus\nChrist. Amen.\n\n\n\n\n The Lord's Prayer\n\n\n_Our Father, who art in heaven; Hallowed be Thy Name; Thy kingdom come;\nThy will be done on earth as it is in heaven; Give us this day our daily\nbread; And forgive us our trespasses, as we forgive those who trespass\nagainst us; And lead us not into temptation; but deliver us from evil;\nFor Thine is the kingdom, and the power, and the glory, for ever and\never. Amen._\n\n\n\n\n The Benediction\n\n\n _The Lord bless thee, and keep thee._\n _The Lord make His face to shine upon thee,_\n _and be gracious unto thee._\n _The Lord lift up His countenance upon thee,_\n _and give thee peace. Amen._\n\n\n\n\n Morning and Evening Prayers\n\n\n\n\n Prayer for Sunday Morning.\n\n\nLord, our Heavenly Father, Eternal God! Blessed be Thy divine power and\nmight; magnified Thy fathomless goodness and mercy; praised Thine eternal\nwisdom and truth. For Thou hast shielded me with Thy hand against the\nperils of this night, and hast suffered me to rest and slumber in peace\nunder the shadow of Thy wings. Thou hast kept and safeguarded me with a\nfather's care against the Evil One and all his wicked designs and\npurposes. Therefore, I magnify Thy goodness and the wonders which Thou\ndoest for the children of men. I will exalt Thee in the congregation. Thy\npraise shall evermore be in my mouth. My soul shall bless Thee, O my\nLord, all that is within me shall bless Thy holy name, and nevermore will\nI forget Thy benefits. May the praises of my lips, which in singleness of\nheart I bring before Thee at this early hour, be acceptable in Thy sight.\nI call upon Thee with all my heart to preserve me this day against all\ndanger of body and soul. May Thy holy angels have charge over me and keep\nme in all my ways. Encompass me with Thy shield and lead me on the paths\nof Thy commandments that, like the children of light, I may be blameless\nin Thy service, to Thy good pleasure. Stay the Evil One and all\nwickedness of this world. Restrain mine own flesh and blood that I be not\novercome by them. Lead me with Thy Holy Spirit that I attempt, do, speak,\nor think nothing except what is well-pleasing in Thy sight and conducive\nto the glory of Thy divine Majesty. Behold, O God, I consecrate and\ndedicate myself entirely to Thy holy will, with body and soul, all my\npowers and abilities, inwardly and outwardly. Make me a living sacrifice,\nholy and acceptable unto Thee, so that I may render Thee a reasonable and\npleasing service. Therefore, Most Holy Father, Almighty God, let me be\nwholly Thine. Govern Thou my heart and soul, and all my emotions that I\nknow and understand none but Thee. O Lord, in the morning wilt Thou hear\nmy voice. Early will I seek Thee and look up to Thee. Early will I praise\nThee, and will not cease when evening comes. Through Jesus Christ. Amen.\n\n\n\n\n Gott des Himmels und der Erden.\n\n\n God, Who madest earth and heaven,--\n Father, Son, and Holy Ghost,\n Who the day and night hast given,\n Sun and moon, and starry host,\n Thou Whose mighty hand sustains\n Earth and all that it contains;\n\n Praise to Thee my soul shall render,\n Who this night has guarded me,\n My omnipotent Defender,\n Who from ill doth set me free;\n Free from danger, anguish, woe,\n Free from the infernal foe.\n\n Let the night of my transgression\n With night's darkness pass away;\n Jesus, into Thy possession\n I resign myself to-day.\n In Thy wounds I find relief\n From my greatest sin and grief.\n\n Let my life and conversation\n Be directed by Thy Word;\n Lord, Thy constant preservation\n To Thy erring child afford.\n Nowhere but alone in Thee\n From all harm can I be free.\n\n Wholly to Thy blest protection\n I commit my heart and mind;\n Mighty God! to Thy direction\n Wholly may I be resigned.\n Lord, my Shield, my Light divine,\n O accept, and own me Thine.\n\n Heinrich Albert, 1643.\n Tr. John Christian Jacobi, 1720.\n Arthur Tozer Russell, 1848.\n Catherine Winkworth, 1855.\n\n\n\n\n Prayer for Sunday Evening.\n\n\nEternal God, Merciful Father, I lift up my hands unto Thee as an evening\nsacrifice, and render Thee most hearty thanks, praise, and glory, that\nThou hast protected me this day and all the days of my life from all evil\nand calamity, and through the ministrations of Thy holy angels hast\ngraciously guarded me against the Evil One. I pray Thee to forgive me all\nmy sins, wherever I have done wrong. Surround me this night with Thy holy\nangels. May thou compass me round and cast a trench about me, that I may\nescape the snares and evil cunning of the enemy. I commend myself to Thy\ngoodness and mercy. Protect me with Thine outstretched arm; for from my\nheart do I pant after Thee in the nightwatches, and with my spirit within\nme do I watch for Thee at all times. I wait upon Thy goodness, and my\nsoul trusts in Thee, the living God, for Thou art my refuge and my\nSavior. Behold, O Lord, whether we sleep or wake, we are Thine: whether\nwe live or die, Thou art our God who hast called us into being.\nTherefore, I cry unto Thee: let Thy grace be not far from me. Shelter me\nwith Thy shield. Keep me, that I lie in quiet, sleep in peace, and awake\nagain in health. Hide me in Thy pavilion in the time of trouble, in the\nsecret of Thy tabernacle hide me, set me upon the rock, and I will fear\nno evil. Yea, though I walk through the valley of the shadow of death, I\nwill fear no evil: for Thou art with me; Thy rod and Thy staff they\ncomfort me. Grant me grace, that though my body sleep, my soul may ever\nwake for Thee, that I may ever have Thee in my heart and before my mind,\nand be not overcome by the night of sin. Keep me from all wicked and evil\ndreams, from restless wakings and useless anxieties, from depraved and\nhurtful thoughts, from all grief. My Lord and God, into Thy gracious care\ndo I commend my body and soul, my brothers and sisters, and all my kin\nand loved ones. May it please Thee, O Lord, to save us, and turn not Thy\nmercy from us. May Thy grace and fidelity protect us alway. Cover us this\nnight with Thy goodness, and encompass us with Thy mercy, that we be\nsafe-guarded in body and soul. Amen.\n\n\n\n\n Christe, du bist der helle Tag.\n\n\n O Christ, who art the sun-lit day,\n Before Thee night must flee away,\n Thou dost reflect the Father's light\n And teachest us His will aright.\n\n Dear Lord, as night is drawing near,\n Fill Thou our hearts with light and cheer,\n Let us securely rest in Thee\n And from the foe's attacks stay free.\n\n And while our eyes in slumber close,\n Grant that our hearts may find repose;\n But let them be to Thee awake\n And of Thy saving grace partake.\n\n Protect us from the wily foe\n Who seeks to harm our souls, we know.\n Be Thou our shield, our staff, and stay,\n Lord Jesus Christ, for this we pray.\n\n For Thou hast made us, Lord, Thine own,\n We as Thy heritage are known.\n Thy blood was shed, that we might be\n The Father's precious gift to Thee.\n\n So let Thy holy angel stay\n Around us both by night and day.\n Place Thou a watch beside our bed,\n And guardian angels overhead.\n\n Thus in Thy name we fall asleep,\n While angels o'er us watch must keep.\n To Thee, O Holy One in Three,\n Be praise to all eternity.\n\n Latin, 7 Century.\n Erasmus Alber, 1555.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Prayer for Monday Morning.\n\n\nO Thou Eternal and Merciful God! Thou hast commanded Thy people in Thy\nlaw each morning to offer Thee a burnt offering, thereby to praise and\nthank Thee for Thy merciful protection: thus I too would bring unto Thee\nmy offering of praise, that is the fruit of my lips, and magnify Thy holy\nname. For by Thy grace and mercy Thou hast kept me this night from all\nevil and harm in body and soul, and hast graciously protected me. If Thou\nhadst not been my shield and my help, manifold calamities would have\nengulfed me, and I could not have risen in health and safety. Therefore,\nI thank Thee for Thy protection. But I continue to call to Thee from the\ndepths of my heart, and my supplication ascends to Thee in the early\nhour. Early do I seek Thy countenance and pray Thee to safeguard me and\nmine from the wiles and power of the devil, from sin and disgrace and all\nwickedness. Visit Thou me in this early hour with Thy grace, as without\nThee I can do nothing, and grant that I may this day begin all my work in\nThy name and end it joyously, to the glory of Thy divine majesty and the\nbetterment of my neighbor. Preserve my soul, mind, reason, senses, and\nthoughts, all that I do and leave undone, that the prince of darkness do\nme no injury. Safeguard me against the destruction that wasteth at\nnoonday. Defend me against mine enemies that neither secretly nor openly\nthey harm or injure me with their craft and cunning, violence or malice.\nO God, Father and Lord of my life, shield me from all impurity and\ndisorderly conduct. Keep me from all intemperance and unchastity, and\nturn from me shameless thoughts. Help me by Thy grace to pluck out the\neye that offends me and cast it away, and renounce all wicked and impure\ndesires of the heart. Grant whatever is pleasing to Thee and useful to\nme, that I may serve Thee in the true faith. Look upon me with the eyes\nof Thy mercy, Thou Savior of the world, and enlighten my heart and eyes,\nthat I may walk in the light of Thy grace, which rises above me, and\nnever lose Thee, the Eternal Light. Amen.\n\n\n\n\n Ach bleib mit deiner Gnade.\n\n\n Abide with grace unbounded,\n Lord Jesus, with us still,\n That Satan's craft confounded\n May no more work us ill.\n\n Abide with us, dear Savior,\n Both with and in Thy Word:\n To us both now and ever\n Thy saving health afford.\n\n Abide with all Thy brightness,\n Thou brightest Light of all;\n And lest we stray from rightness,\n Make Thou Thy truth our wall.\n\n Abide with us and bless us,\n Thou Lord whose riches 'bide;\n With growing grace possess us\n And all things best provide.\n\n Abide with Thy protection,\n Great Captain, clothed with might;\n O'ercome our world affection\n And vanquish Satan's spite.\n\n Abide with care untiring,\n Our God and Lord indeed;\n All steadfastness inspiring,\n Help, Lord, our every need.\n\n Josua Stegmann, 1630.\n Tr. A. Ramsey, 1916.\n\n\n\n\n Prayer for Monday Evening.\n\n\nO Thou Mighty and Everlasting God, the Father of our Lord Jesus Christ, I\nthank Thee that by Thy divine power Thou hast this day preserved me from\nall injury and danger of body and life. I owe it to Thy mercy alone that\nThou didst protect me on all my paths. I pray Thee to forgive me all my\nsins which I have committed against Thee, and this night and during our\nentire lives mercifully to defend me and my loved ones against all sorrow\nand anxiety, and against the craft and power of the devil, wherewith day\nand night he seeks to ensnare us. Preserve us from the deadly pestilence\nthat walketh in the darkness, and deliver us from the snare of the enemy.\nProtect us from the temptation and terror of Satan, from all evils of\nbody and soul. For Thou art our strong fortress, our sword and buckler.\nAll our hope and trust rests in Thee. Therefore, O faithful God, may\nThine eyes be upon us and we be safeguarded this night against all\nviolence and assault of the enemy. Be Thou our keeper and protector.\nEncompass us with Thy shelter, for in Thee is our salvation. Unto Thee\nonly, from whom cometh my help, do I lift up mine eyes. My help cometh\nfrom the Lord, who made heaven and earth. Behold, as the eyes of servants\nlook unto the hands of their masters, and as the eyes of a maiden unto\nthe hand of her mistress; so our eyes wait upon the Lord our God, until\nthat He have mercy upon us. Have mercy upon us, O Lord, have mercy, for\nwe are poor and needy. Lift upon me the light of Thy countenance lest I\nsleep the sleep of death. Amen.\n\n\n\n\n Nun ruhen alle Waelder.\n\n\n Now under night's dark shadow\n Rest woodland, field, and meadow,\n The world in slumber lies.\n But thou, my soul, awake thee,\n To song and prayer betake thee,\n Give praise to Him who rules the skies.\n\n The sun's fair light hath vanished,\n The night its rays hath banished,\n The night, the foe of day.\n 'Tis well: my heart containeth\n A sun whose light ne'er waneth,\n Since Jesus there holds constant sway.\n\n The reign of day is over,\n And golden stars now cover\n The canopy so blue.\n Thus I shall shine in heaven,\n Where golden crowns are given\n To all who faithful stay and true.\n\n My body is divested\n Of garments that have rested\n Upon its form of clay.\n Thus I at heaven's portal\n Shall lose all that is mortal\n And with the Lord forever stay.\n\n Head, feet, and hands are taking\n Sweet rest from toil and waking,\n Released from ev'ry pain.\n O heart of mine, why borrow\n The troubles of tomorrow?\n Thou rest from sin and woe shalt gain.\n\n Ye members weak and tired,\n By joy no more inspired,\n Betake yourselves to bed.\n The time and hour for sleeping\n In God's own faithful keeping\n Will come when you are cold and dead.\n\n My tired eyes are closing,\n And while I am reposing,\n Where doth my soul remain?\n To Thee be it commended\n Until the night is ended,\n Let me Thy gracious favor gain.\n\n Lord Jesus, who dost love me,\n Spread both Thy wings above me,\n Thus shielding me from harm.\n If Satan should draw near me,\n Let angels come to cheer me\n And so the wily foe disarm.\n\n My loved ones, rest securely,\n Since God will guard you surely\n From pain and perils sore.\n May you in safety slumber,\n While angels without number\n Attend you now and evermore.\n\n Paul Gerhardt, 1648.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Prayer for Tuesday Morning.\n\n\nBlessed be God, the maker of heaven and earth, blessed be the Lord, who\nonly doeth wondrous things, and blessed be His glorious name for ever who\nhath made both day and night through His glorious wisdom, and so ruled,\nthat while the earth remaineth, they shall not cease, that man may rest\nby night, and proceed again to his labors by day. O Lord, how manifold\nare Thy works! In wisdom hast Thou made them all: the earth is full of\nThy riches. For such Thy gifts we should thank Thee ere the sun rises,\nand come into Thy presence when the light breaks forth. Therefore I bless\nThee now also, for Thou hast suffered me securely to rest and sleep this\nnight, and again awakened me to the joy of living. Thou hast mercifully\nsheltered me from the assaults and malice of enemies. I supplicate Thy\ngrace: grant that my soul may repose in Thy hands, my body continue in\nhealth according to Thy good pleasure, and be kept from all injury and\nperil. Be Thou my mighty protection and strong stay, a defence from heat,\nand my cover from the sun at noon, my preservation from stumbling, and my\nhelp from falling, that no harm may come unto me. O Merciful God, knowing\nthe hour is come to awaken out of the sleep of sin and iniquity, for now\nis our salvation so near, the night is far spent, the day is at hand: so\nhelp us to cast off the works of darkness, and put on the armour of\nlight, that we may walk honestly as in the day; not in rioting and\ndrunkenness, not in chambering and wantonness, not in strife and envying,\nbut putting on the Lord Jesus Christ in the true faith and a Christian\nconduct. Thus awaken us each morning. Open Thou my ears that I may hear\nThy holy Word with a believing heart and keep the same in my memory. Let\nmy ears incline to the entreaties and prayers of the needy, not to\nforsake them in their distress. And when in my distress I cry unto Thee,\nhear Thou the voice of my supplications, and despise not my sighings in\nthe hour of death. Let my prayers come before Thee early. Incline Thine\nears to my entreaties. Satisfy us early with Thy mercy; that we may\nrejoice and be glad all our days. Amen.\n\n\n\n\n Wach auf, mein Herz, und singe.\n\n\n Awake, my heart, rejoicing,\n Thy Maker's praises voicing,--\n The Giver, good gifts sending;\n Their Shield, His folk defending.\n\n All night while darkness 'bound me\n In deepest gloom around me,\n By Satan craved while sleeping\n God had me in His keeping.\n\n Thou spak'st me words endearing;\n Sleep now, my child, unfearing;\n Sleep well, night's terrors spurning;\n Thou'lt see the sun returning.\n\n Thy word performed, now waking,\n I see the bright dawn breaking,\n Safe kept from ills unnumbered\n While 'neath Thy care I slumbered.\n\n An off'ring Thou desirest.\n Behold what Thou requirest.\n Nor lamb nor incense bringing,\n I come with prayer and singing.\n\n Nor wilt Thou now despise them,\n But in Thy heart wilt prize them,\n Well knowing, yea, and surely\n My best I offer purely.\n\n Approve my works when shown Thee.\n Help Thou good councils only;\n Beginning, middle, closing,\n Lord, for the best disposing.\n\n With blessings guard me waking,\n My heart Thy dwelling making,\n And with Thy Word, Lord, feed me\n Whilst heavenward Thou dost lead me.\n\n Paul Gerhardt, 1648.\n Tr. A. Ramsey, 1916.\n\n\n\n\n Prayer for Tuesday Evening.\n\n\nLord, Merciful God, Holy Father, in the daytime do I cry unto Thee with\nmy voice, in my distress I call upon Thee, and at eventime I remember Thy\ngoodness and mercy, which Thou hast wrought for me. And especially do I\nmagnify Thee now, that purely out of fatherly grace and mercy, without\nany merit or worthiness on my part, Thou hast this day preserved me from\nall harm and danger and kept me from sudden death. Therefore do I now and\nat all times render unto Thee praise and thanksgiving, and pray Thee, for\nthe sake of the bitter sufferings of Jesus Christ, to forgive me wherever\nI have sinned against Thee this day. Mercifully protect me during the\nnight against my adversary, the devil, and against the fears and terrors\nof the night. Suffer me to rest without anxieties and worries, and may\nthe eyes of my faith ever behold the lustre of Thy countenance even\nduring the shades of night. For Thou art that shining and true light,\nwhich dispels all darkness that surrounds us. Thou, O Lord, art ever with\nme. Thou art my rock, and my fortress, my deliverer, my strength, in whom\nI will trust, my buckler, the horn of my salvation, and my high tower.\nLord, my God, at eventime do I lift up my hands unto Thee. Come unto me\nas the latter rains that make the earth fruitful. Abide with us, for the\nday is far spent and in the darkness there is none to defend us save Thou\nalone, our God. Hasten to uphold us. Defend us this night, lest our souls\nfall into the sleep of sin and our bodies be overcome with evil. Awake us\nagain in due time, and make us to hear joy and peace, for we love Thy\nword and Thy testimonies, which are the delight of our souls. May our\nears be saved from all messages of sorrow, and all anguish be turned from\nour souls; for Thou canst prosper all that liveth, and fill my life with\nThy blessing; in Jesus Christ, our Lord. Amen.\n\n\n\n\n Die Nacht ist kommen, drin wir ruhen sollen.\n\n\n Now God be with us, for the night is closing;\n The light and darkness are of His disposing;\n And 'neath His shadow here to rest we yield us,\n For He will shield us.\n\n Let evil thoughts and spirits flee before us;\n Till morning cometh, watch, O Master, o'er us;\n In soul and body Thou from harm defend us,\n Thine angels send us.\n\n Let holy thoughts be ours when sleep o'ertakes us;\n Our earliest thoughts be Thine when morning wakes us;\n All day serve Thee, in all that we are doing\n Thy praise pursuing.\n\n As Thy beloved, soothe the sick and weeping,\n And bid the prisoner lose his griefs in sleeping;\n Widows and orphans, we to Thee commend them;\n Do Thou befriend them.\n\n We have no refuge, none on earth to aid us,\n Save Thee, O Father, Who Thine own hast made us;\n But Thy dear presence will not leave them lonely,\n Who seek Thee only.\n\n Father, Thy name be praised, Thy kingdom given;\n Thy will be done on earth as 'tis in heaven;\n Keep us in life, forgive our sins, deliver\n Us now and ever.\n\n Petrus Herbert, 1566.\n St. 5, Anon., 1627.\n Tr. Catherine Winkworth, 1863.\n\n\n\n\n Prayer for Wednesday Morning.\n\n\nAlmighty, All-gracious God! All Thy creatures should praise and glorify\nThee. The birds under the heavens magnify Thee with lovely songs early in\nthe morning as their Lord and Maker. So will I too heartily thank Thee,\nthat Thou hast preserved me under Thy shelter and protection during the\nnight now past, and all my life even to the present hour, and awakening\nme from the sleep of the darkness of this night, hast suffered me to\narise again in health and joy. I pray Thee for the sake of the saving\nresurrection of Jesus Christ from the dead, that Thou wouldst ever keep\nme together with all my loved ones from all danger and evil. O Lord, save\nThy people, and bless Thine inheritance: feed them also, and lift them up\nfor ever. Fill me also at this early hour with Thy grace, that I may pass\nthis day rejoicing in Thy commandments, and free from mortal sin. Let me\nexperience Thy grace as a dew from the womb of the morning, and as the\nrefreshing moisture that diffuses at the break of day, making the land\nfruitful. Thou wouldest spread Thy goodness over me, that I may gladly\nand zealously do Thy will. Govern me with Thy Holy Spirit that I may\nserve Thee in righteousness and holiness of truth, well pleasing in Thy\nsight. Guard me that I sin not against Thee, nor defile my conscience\nwith carnal lusts that militate against the soul. Keep my tongue from\nevil, and my lips from speaking guile. Foolish talking or jesting,\nunbecoming of Christians, be ever far removed from me. Grant, that I\noffend none with my lips, nor backbite, judge nor condemn, defame nor\nvilify. O that I might put a lock to my lips and seal them with a strong\nseal, that they bring me not to naught, nor my tongue destroy me. Give me\ngrace that I may know my shortcomings and correct them, and not fall into\nThy righteous judgment and condemnation. Grant my prayer, O Eternal God,\nfor the sake of Thy dear Son, Jesus Christ. Amen.\n\n\n\n\n Morgenglanz der Ewigkeit.\n\n\n Dayspring of Eternity,\n Light from endless Light proceeding,\n Let Thy beams upon us shine\n As the shadows are receding;\n And dispel by Thy great might\n Our dark night.\n\n As the soft refreshing dew\n Falls upon the drooping flower,\n So our fainting hearts renew\n By Thy Spirit's quickening power;\n Ne'er Thy bounteous grace withhold\n From Thy fold.\n\n Let the glow of Thy pure love\n All our icy coldness banish;\n In the radiance from above\n May our doubts and fears all vanish,\n That ere dying we may be\n Found in Thee.\n\n O Thou glorious Sun of grace,\n May Thy light be ne'er denied us!\n Till we reach the heavenly place\n Shine upon our way to guide us,\n That at last among the blest\n We may rest.\n\n Christian Knorr v. Rosenroth, 1684.\n Tr. J. F. Ohl, 1915.\n\n\n\n\n Prayer for Wednesday Evening.\n\n\nMost Holy Trinity, One in essence, Three in person, who art my life,\nsalvation, and eternal joy, I praise and thank Thee with mouth and heart\nthat Thou hast so graciously protected me throughout this day. I pray Thy\ndivine goodness to cover up all my shortcomings, and especially where\nthis day, with my tongue, with vain and unprofitable words, slander or\notherwise, I have sinned against Thee and Thy holy commandments.\nAccording to Thy name, O God, so is Thy praise unto the ends of the\nearth: Thy right hand is full of righteousness. Therefore I commend my\nbody and my soul into Thy hands. Thy divine Majesty bless me; Thy holy\nTrinity shelter me; Thy eternal Unity preserve me. May Thy unbounded\nmercy protect me; Thy inexpressible benevolence defend me, the sublime\ntruth of God cover me; profound knowledge of Christ strengthen me; the\nunfathomable goodness of the Lord keep me. The grace of the Father govern\nme; the wisdom of the Son refresh me; the power of the Holy Spirit\nenlighten me. My Creator aid me, my Redeemer quicken me, my Comforter\nabide with me. The Lord bless me and keep me. The Lord make His face\nshine upon me and be gracious unto me. The Lord lift up His countenance\nupon me and give me peace. The protection and blessing of the one and\neternal Godhead be between me and all mine enemies, visible and\ninvisible, today and always, that they may not approach nor injure me. As\nthe pillar of the cloud went between the army of the Egyptians and the\narmy of Israel, so that they could not come together, and no harm befall\nthe children of Israel, so mayest Thou be a pillar of fire and a wall of\nseparation between me and mine enemies, that no harm come over me. Keep\nme also in my last hour. When mine eyes no longer see, mine ears no\nlonger hear, my tongue no longer speak, be Thou with me, O blessed\nTrinity, that the Evil One have no power over me. Amen.\n\n\n\n\n Hirte deiner Schafe.\n\n\n Shepherd never sleeping,\n In Thy gracious keeping\n I have been today.\n Thou art my Defender,\n So in mercy tender\n Come and with me stay;\n All this night\n Keep me in sight,\n Send Thine angels to attend me\n And protection lend me.\n\n While I lie and slumber,\n Let Thine hosts outnumber\n All my raging foes.\n Be of grace the Giver,\n And Thy child deliver\n From guilt's painful throes.\n For Thy Son\n My soul hath won;\n By His wounds, so sorely stricken,\n He my heart doth quicken.\n\n Shield Thou from all danger\n Ev'ry lonely stranger\n And my dear ones, too.\n Tenderly embrace us\n And with mercy grace us,\n Be our Father true.\n Thou with me\n And I with Thee,\n Thus shall I, mine eyelids closing,\n Be in peace reposing.\n\n Close the door behind us,\n Let no evil find us,\n Keep all ills away.\n Be our shield and cover,\n Let Thine angels hover\n O'er us now, we pray.\n By sweet rest\n Let us be blest,\n Ev'ry fear of Satan's raging\n In our hearts assuaging.\n\n What if death should take me\n And no light awake me\n From my sleep and rest?\n If Thou hast intended\n That my life be ended,\n Let Thy name be blest;\n As for me,\n I yield to Thee.\n In the wounds of Jesus lying,\n I am daily dying.\n\n By no cares encumbered,\n Though my hours be numbered,\n I now fall asleep.\n All to Thee commending\n Who Thine hosts are sending\n Watch o'er me to keep.\n Through the night\n Be my delight,\n And if I should see the morrow,\n Thou wilt cure all sorrow.\n\n Benjamin Schmolcke, 1715.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Prayer for Thursday Morning.\n\n\nJesus Christ, Thou art the eternal light, which dispelleth the darkness\nof night and the shadow of death: I magnify Thy name, I glorify and thank\nThee. For Thou hast so graciously kept me during this night, and hast\nbrought me out of the darkness to the light of day. Thou hast shielded me\nagainst the terrors of the night, the snares of the devil, the noisome\npestilence, that walketh in the darkness, manifold illness and disease.\nThou hast guarded and watched over my soul, even as the shepherd watches\nover his flock. And all that I possess is kept from harm through Thy\ngreat mercy. Praise and thanks be said unto Thee for Thy gracious\nprotection and all Thy gifts. I will speak of Thy power and magnify Thy\ngoodness, when the day breaks. For Thou art my refuge, my strong tower,\nmy present help, my faithful God, in whom I trust. Thou makest glad my\nheart and my countenance rejoiceth. I pray Thee, for the sake of Thy holy\nbirth and incarnation, suffer Thy grace to rise in my heart and break\nforth even as the beauty of the morning, and come over me as the early\nrain. Illumine me with Thy radiance, and be Thou the light of my heart,\nfor Thou art the right day star and the true light, that lighteth men to\nthe eternal life. Be merciful unto me, O Lord, for in Thee do I put my\ntrust. My soul waiteth for Thee, more than they that watch for the\nmorning, yea more than they that watch for the morning. Be Thou mine arm\nin the morning, my salvation also in the time of trouble. Protect me in\nbody and soul, that no evil befall me and no plague come nigh unto my\ndwelling. Keep from me all wicked spirits. Defend me from evildoers.\nStand up for me against the workers of iniquity and shield me, that the\nhands of mine adversaries may not touch me. O Lord, our God, establish\nThou the work of our hands upon us; yea, the work of our hands establish\nThou it, and strengthen our hands, and teach us that we may keep Thy\ncommandments and sin not against Thee this day. Grant us this for the\nsake of Thy mercy, which endureth for ever and ever. Amen.\n\n\n\n\n Wach auf, mein Herz, die Nacht ist hin.\n\n\n Awake, my soul; the rising sun\n Dispels the night of mourning;\n Awake, with songs of praises run\n To greet the Lord returning.\n He burst the gates of death today\n And left the gloomy grave for aye\n While all the world rejoices.\n\n Arise, my soul, from sin and death,\n To thee new life is given;\n Arise and run the race of faith;\n Fix thy desires on heaven\n Where Jesus, thy Redeemer reigns,\n And seek the things that it contains,\n If thou with Him be risen.\n\n Art thou distressed by weight of care?\n Thy Savior will remove it.\n Believing, thou with joy canst bear\n Thy cross and learn to love it.\n Cast all thy burden on the Lord;\n Fear not; for He will help afford,\n For now He hath arisen.\n\n Now Judah's Lion, true and tried,\n The victory obtaineth;\n The Lamb of God, the Crucified,\n For us salvation gaineth,\n And giveth righteousness and life;\n For after all the dreadful strife,\n O'er every foe He triumphs.\n\n Then up, my soul, begin the fight,\n For Christ, the Victor, leadeth.\n He arms thee with a victor's might;\n With Him thy cause succeedeth.\n Now thou can'st rise and live anew\n And righteousness and peace pursue\n And be a faithful servant.\n\n Fear not the angry jaws of hell,\n Nor world, nor death, nor devil.\n Thy Savior lives and all is well,\n Though sore has been His travail.\n A Victor crowned, He as a Friend\n The mean and feeble doth attend,\n And therefore thou shalt conquer.\n\n Ah, Lord, whom death could not defile,\n Who from the dead hast risen,\n Free us from Satan's might and guile\n And save us from his prison.\n O grant, that, as one body, we\n May enter that new life in Thee\n Which Thou for us hast gotten.\n\n Laurentius Laurenti, 1700.\n Tr. A. Ramsey.\n\n\n\n\n Prayer for Thursday Evening.\n\n\nPraise be to Thee, O God, our Father, through Jesus Christ in the Holy\nGhost, one, eternal God, who through Thy manifold compassion hast kept me\nthis day, a poor sinner and miserable creature, from the fiery darts of\nSatan that fly by day, from the destruction that wasteth at noonday, and\nhast graciously protected me from a sudden and evil death. Thy mercy, O\nLord, is in the heavens; and Thy faithfulness reacheth unto the clouds.\nThou art gracious and merciful, and all Thine acts are glorious. I pray\nThee, O merciful God, graciously to forgive me all that I have done\nagainst Thee this day, in thought, word, or deed. Turn Thy mercy toward\nme, that I may slumber and rest during this coming night, and that I may\nnever turn from Thee, who art the eternal rest. Suffer me ever to abide\nin Thee in the true faith, and safely to sojourn under Thy protection, so\nthat the enemy may never come nigh unto me, nor do me injury. Lord, Thou\nart my light and my salvation; whom shall I fear? Thou art the strength\nof my life; of whom shall I be afraid? My heart trusteth in Thee, and I\nam helped. Thou art my strength and my great shield. Thy right hand\nstrengtheneth me. Thy right hand gladdeneth my heart, and in the shadow\nof Thy wings will I make my refuge. Behold, my God, in the daytime do I\ncry unto Thee, and Thou hearest me, and in the night season I am not\nsilent, and Thou answereth my prayer. I remember Thee on my bed, and\nmeditate on Thee in the night watches, because Thou hast been my help.\nTherefore in the shadow of Thy wings will I rejoice. My soul cleaves unto\nThee, for Thy right hand defends me. When darkness comes over me, then\nThou, O Lord, art my light and my salvation. O gracious God, vouchsafe\nunto me Thy grace, so that when that last hour cometh, and I lay me down\non my deathbed for the eternal rest, through Thy help, in the true faith,\nin all confidence and trust, I may happily fall asleep for the eternal\nlife. Meanwhile keep me in Thee, that I may ever watch and pass my days\nin all sobriety and moderation, and be found in Christian readiness,\nsince I can not know the hour when Thou comest, O God, to call me hence,\nso that I may be worthy to stand before the Son of Man, and be not put to\nshame; who liveth and reigneth with Thee and the Holy Ghost, world\nwithout end. Amen.\n\n\n\n\n Nur in Jesu Blut und Wunden.\n\n\n Now in Jesus' wounds reposing,\n I my tired eyes am closing.\n For His love and pardoning grace\n Are my only resting place.\n\n Through the day His mercy holds me,\n And by night His arm enfolds me.\n Of Thy strong protection sure,\n Jesus, I shall rest secure.\n\n Tr. H. Brueckner, 1916.\n\n\n\n\n Prayer for Friday Morning.\n\n\nBlessed be God, my Maker! Blessed be God, my Savior! Blessed be God, my\nComforter! Who giveth unto me my health, my life, and every blessing; my\nvery present help and my protection. Thou hast kept me according to Thy\ngreat and most blessed compassion during this night now past against the\nonslaughts of Satan, and preserved me in health. I beseech Thee, Heavenly\nFather, through Jesus Christ, Thy dear Son, take me this day also into\nThy divine protection, and shield me that no evil may assail my life. For\ninto Thy hands do I this day and all days commend my body and soul, my\nthoughts, words, and deeds, all that I do or leave undone, my going out\nand my coming in, my walks and ways, my rising up and my lying down, my\nwill and counsel, my thoughts and desires, my faith and profession, the\nend of my life, the day and hour of my death, my passing away and my\nresurrection. O Lord God, do Thou with me as Thou wilt: for Thou knowest\nwhat serves best Thy glory and my salvation. Keep me in Thy fear and in\nthe true knowledge of Thee. Protect me from the deeds of unrighteousness.\nAnd if perchance and by reason of my frailty I sin against Thee, I pray\nThee take not from me Thy mercy; turn not from me Thy grace; withdraw not\nThy help. For there is none other God nor Helper but Thee, and as there\nis none before Thee, there is none after Thee. Thou art the first and the\nlast, Alpha and Omega, and there is none other God beside Thee. Therefore\ndo I call only upon Thee: may Thy goodness rule over me. Cause me to hear\nThy loving kindness in the morning; for in Thee do I trust. Lead me on\nthe paths of righteousness, that I may not walk in the counsel of the\nungodly, nor stand in the way of sinners, nor yet sit in the seat of the\nscornful, but that my heart may ever delight in Thy word and\ncommandments, and meditate upon them day and night; through Jesus Christ,\nour Lord. Amen.\n\n\n\n\n Die helle Sonn leucht jetzt herfuer.\n\n\n The morning sun shines in the skies,\n And we from peaceful slumbers rise.\n All praise to God who hath this night\n Protected us from Satan's might.\n\n Lord Jesus, shield us now by day\n From sin and error on our way.\n To us Thy holy angels send,\n And let them to our wants attend.\n\n Make Thou our hearts obedient,\n To use Thy word and sacrament,\n To do Thy will whate'er betide,\n Thus pleasing Thee, our trusty guide.\n\n Bless Thou the labor of our hands\n And help us keep Thy law's demands,\n That all our work, begun in Thee,\n May to Thy praise and glory be.\n\n Nicolaus Hermann, 1560.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Prayer for Friday Evening.\n\n\nBlessed be the Lord God, who only doeth wondrous things! And blessed be\nHis glorious name for ever: and let the whole earth be filled with His\nglory! Daily will I praise the Lord, and at eventime my mouth shall thank\nHim while I have any being. For when I cry with my voice, He hears me,\nand gives ear to my supplication. When I pray, He attends to my voice.\nThe Lord is my refuge and strength, a very present help in trouble.\nTherefore do I laud and magnify Thee, Eternal God, that Thou hast this\nday so mercifully kept me from every harm and evil. My heart is glad and\nmy soul glorifies Thee for Thy goodness and mercy. Ever shall my tongue\nspeak of Thee and say, Blessed be the Lord, and blessed be Thy holy name!\nI pray Thee, graciously pardon, wherever I have this day sinned against\nThee, and grant me and mine Thy protection during the coming night. Be\nThou my shield, and my shade upon my right hand. O Lord, preserve me from\nall evil, preserve my soul. Be gracious unto me, for in Thee do I put my\ntrust. I trust in the Lord, and cry to God, the Highest, to God Who\nendeth all mine affliction. Behold, He that keepeth Israel shall neither\nslumber nor sleep. He will guide my steps on the paths of righteousness,\nthat I slip not and my feet do not falter. He will not suffer my feet to\nbe moved, and His word is a light unto my path. Therefore as I lay me\ndown, I will not be afraid of sudden terror, neither of the desolation of\nwicked people, when it cometh. For Thou keepest my foot from being taken,\nand deliverest me from the snares of death. O Lord God, lift upon me the\nlight of Thy countenance, that I may lie down and sleep in peace, and\ndwell in safety under Thy protection. For Thou alone, O Lord, can help\nme. In Thy name will I lie down to rest and let my eyelids slumber. Thou,\nO Lord, wilt again awaken me with rejoicing, to the glory and praise of\nThine eternal majesty; through Jesus Christ, our Lord. Amen.\n\n\n\n\n Hinunter ist der Sonnen Schein.\n\n\n Sunk is the sun's last beam of light,\n And now the world is wrapt in night.\n Christ, light us with Thy heavenly ray,\n Nor let our feet in darkness stray.\n\n Thanks, Lord, that Thou throughout the day\n Hast kept all grief and harm away;\n That angels tarried round about\n Our coming in and going out.\n\n Whate'er of wrong we've done or said,\n Let not the charge on us be laid;\n That, through Thy free forgiveness blest,\n In peaceful slumber we may rest.\n\n Thy guardian angels round us place\n All evil from our couch to chase;\n Our soul and body, while we sleep,\n In safety, gracious Father, keep.\n\n Nicolaus Hermann, 1560.\n Tr. Frances Elizabeth Cox, 1841.\n\n\n\n\n Prayer for Saturday Morning.\n\n\nO Thou Very and Eternal God, the Father of our Lord Jesus Christ. To Thee\ndo I lift up my heart in dutiful gratitude. I will not hide Thy\nrighteousness within my heart. I will declare Thy salvation. I will not\nconceal Thy loving kindness and Thy truth from the great congregation,\nand all the good that Thou hast shown me will I not keep silent. For it\nis a good thing to give thanks unto the Lord, and to sing praises unto\nThy name, O Most High: to shew forth Thy loving kindness in the morning,\nand Thy faithfulness every night. Therefore my soul doth magnify Thee,\nthat Thou in Thy immeasurable grace hast kept me during the night now\npast. Blessed be Thou, Lord God Sabaoth, who art merciful unto all, that\nseek Thee and love Thy salvation. Blessed is Thy holy name in all the\nearth, who art our refuge and our help! Blessed are all Thy works which\nThou doest for the children of men! I beseech Thee, protect me this day,\nthat the Evil One may not harm me, and the hands of the wicked touch me\nnot. Lord God, my Savior, early will I seek Thee, early do I cry unto\nThee. Grant, that I may fulfill the duties of my calling and all that is\ncommitted unto me with diligence and trust to the glory of Thy name and\nthe betterment of my fellowman, so that I may not misuse the light of\nthis day, neither any of Thy creatures in the service of sin and vanity,\nneither grieve Thee, nor transgress the covenant of my Baptism with\nanything I do or leave undone. Vouchsafe unto me Thy grace, that I may\nguard myself against the six things which Thou dost hate, yea, seven\nwhich are an abomination unto Thee: a proud look, a lying tongue, and\nhands that shed innocent blood, a heart that deviseth wicked\nimaginations, feet that are swift in running to mischief, a false witness\nthat speaketh lies, and he that soweth discord among brethren. From such\nand the like vices preserve me, O God, that I may nevermore be led nor\nconsent to them, but teach me to do Thy will; for Thou art my God: Thy\nSpirit is good. Lead me into the land of uprightness, that I may serve\nThee in a life that is without blame, and my deeds and all my conduct be\npleasing in Thy sight; for Christ's sake. Amen.\n\n\n\n\n Die gueldne Sonne.\n\n\n The sun, ascending,\n To us is lending\n Bliss, joy, and gladness,\n Cure for all sadness,\n Filling the world with its rich, golden light.\n I was reclining\n When no light was shining;\n But the sun's beauty\n Now calls me to duty,\n As I behold it so fair and so bright.\n\n Mine eye beholdeth\n What God unfoldeth:\n Heaven's bright glory\n Tells me the story\n Of His unlimited power and love,\n And how the sainted\n In beauty untainted,\n Free from things mortal,\n Beyond death's dark portal,\n Dwell in the heavenly mansions above.\n\n To God in heaven\n Be praises given;\n Come, let us offer\n And gladly proffer\n To the Creator the gifts that we prize.\n He well receiveth\n A heart that believeth,\n Hymns that adore Him\n Are precious before Him\n And to His throne like sweet incense arise.\n\n At the day's ending\n Sweet slumbers sending,\n And in the morning\n All things adorning,\n These are His works and His blessings so true.\n When night descendeth\n Protection He lendeth\n When morn appeareth\n Our spirits He cheereth,\n Causing His mercy to crown us anew.\n\n Father above me,\n Thou who dost love me,\n Bless my beginning,\n Keep me from sinning,\n Move ev'ry hindrance well out of my way.\n Strength ever lend me,\n From Satan defend me,\n Spare me temptation,\n So that in my station\n I may Thy holy commandments obey.\n\n Let me with pleasure\n See the full measure\n Which upon others,\n Who are my brothers,\n Thou of Thy blessings dost richly bestow.\n Bid envy vanish!\n All greediness banish!\n Make me Thy dwelling,\n Sin's darkness dispelling.\n Grant that in virtue I daily may grow.\n\n What is man's being?\n It is like seeing\n Autumn's bleak shadows\n Sweep o'er the meadows\n When the cold winds drive the clouds on their way.\n All that we cherish\n Must crumble and perish.\n Plants must stop growing,\n And stars must cease glowing;\n Heaven and earth are not destined to stay.\n\n All else decayeth,\n God only stayeth,\n He of creation\n Is the foundation.\n His will and word must forever abide.\n His grace endureth\n And for us secureth\n Comfort in sorrow\n And help for the morrow,\n Keeping us cheerful, whate'er may betide.\n\n God of creation,\n Be my salvation!\n Calm all my terrors,\n Blot out my errors,\n Grant that Thy pardon I fully may share;\n Withal attend me,\n Rule, guide, and defend me\n In mercy tender,\n Because I surrender\n Soul, will, and all to Thy fatherly care.\n\n Whilst Thou art giving\n What for a living\n Seems very needful,\n Oh, make Thou me heedful\n Of this great truth and commendable thought:\n God, like a tower,\n Transcends all in power;\n Good beyond telling,\n In beauty excelling,\n He doth suffice me, all else counts for naught.\n\n If grief and sadness\n Temper my gladness\n If for the morrow\n Thou send me sorrow\n Do as Thou wilt, for my trust is in Thee.\n Thou surely knowest\n That what Thou bestowest,\n E'en though distressing,\n Must bring me a blessing;\n Thou wilt not deal too severely with me.\n\n Ills that still grieve me\n Soon are to leave me;\n Though waves may tower\n And winds gain power,\n After the storm the fair sun shows its face.\n Joys e'er increasing,\n And peace never ceasing,\n These I shall treasure\n And share in full measure\n When in His mansions God grants me a place.\n\n Paul Gerhardt, 1666.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Prayer for Saturday Evening.\n\n\nPraise be unto Thee, Thou great and unchangeable God! Praise be unto Thy\ngoodness and mercy! Praise be unto Thy eternal wisdom and truth, that\nThou hast preserved me during the day now past from all danger and harm.\nI pray Thee, graciously perfect Thy goodness which Thou hast begun in me,\nand suffer me to rest this night under Thy protecting shield, and cover\nme with Thy wings. Suffer me to put my trust under the shadow of Thy\nhands, that I fear no evil. Keep me, O God, as the apple of the eye. Hide\nme under the shadow of Thy wings. Lord, Thou art the portion of mine\ninheritance; my salvation is in Thy hands. Grant unto me, according to\nThy goodness, that neither fear nor trembling come over me, and no\nterrors of the night overwhelm me. Be merciful unto me, for in Thee do I\nput my trust, and under the shadow of Thy wings do I find my refuge. I\nseek the Lord in the time of need; my hand is outstretched in the night\nwithout ceasing; for my soul has none other comfort; and I know of none\nother helper in heaven or earth but Thee alone. At midnight when I\nawaken, I meditate upon Thy name, so altogether lovely, upon Thy goodness\nand fidelity, vouchsafed unto me, and I praise Thee because of Thy\nrighteous judgments. When I am troubled I remember God, when my spirit is\noverwhelmed I speak of my Savior. For He redeemeth my life from\ndestruction and saveth me from the snares of death. Lord God, my Savior,\nby day and by night do I cry unto Thee, pardon all my transgressions,\nwhich during this day and the week now past I have committed against\nThee. O Lord, deliver my soul for Thy mercy's sake. Thou art gracious and\njust, and our God is merciful. The Lord preserveth the simple. I was\nbrought low, and He helped me. Therefore will I rejoice and praise Thee,\nand sing aloud upon my bed. For the days of my life will appear as\nnoonday, and darkness as the morning's light, and I will rejoice that\nThou, O God, art my hope and my rest in life and death. I lay me down,\nand none will terrify me. Thus do I commend my body and soul into Thy\nhands, Thou Faithful God. Thou hast redeemed me through Jesus Christ, our\nLord. Amen.\n\n\n\n\n Werde munter, mein Gemuete.\n\n\n Soul of mine, to God awaking,\n And ye senses, ev'ry one,\n Come, your quiet haunts forsaking,\n Tell what God to me has done.\n How He this entire day\n Has been with me on my way,\n To my many wants attending\n And from ills protection lending.\n\n Praise and thanks to Thee I render,\n Father Thou of mercies great.\n Thou hast been my strong Defender,\n And Thy love does not abate.\n Thou hast shielded me from woe,\n Lent me strength and quenched the foe,\n So that I, such help beholding,\n Rest secure in Thine enfolding.\n\n If from Thee I have departed,\n I return again to Thee,\n Knowing Thou art tender-hearted,\n Since Thy Son has died for me.\n I can not deny the guilt,\n But for me His blood was spilt,\n And Thy grace, all sin exceeding,\n Lends forgiveness at His pleading.\n\n O Thou Light, with brightness filling\n Ev'ry true and pious soul,\n Into me Thy grace instilling,\n Make my troubled spirit whole.\n Deign this night to stay with me,\n And let me abide in Thee,\n That, while darkness may enthrall me,\n Yet no evil may befall me.\n\n Grant that I in peace may slumber,\n Finding sweet and quiet rest.\n Let no cares my soul encumber,\n Keep it by Thy presence blest.\n Mind and body, child and wife,\n All my goods and all my life,\n Friends and foes (again befriended)\n Be this night to Thee commended.\n\n Let no terrors overtake me,\n Shield me well from base attack.\n Let no grievous pain awake me,\n War and pestilence keep back.\n Ward off fire, water, death,\n All that threatens life and breath.\n Spare me violence, extortion\n And, withal, a sinner's portion.\n\n O immortal God, endue me\n With the gifts for which I ask.\n Jesus, lest some ill pursue me,\n Prosper me in ev'ry task.\n Holy Spirit, comfort, friend,\n On whose counsel I depend,\n Listen to my earnest pleading,\n Amen. Thou my prayer art heeding.\n\n Johann Rist, 1642.\n H. Brueckner, 1918.\n\n\n\n\n A Daily Prayer.\n To be spoken mornings or evenings.\n\n\nDear God and Lord! I live, yet know not how long. I die, yet know not\nwhen. Thou, O Heavenly Father, knowest. Lo, dear Lord, is this hour, this\nday (this night) the last of my life: Thy will be done! Thou alone\nknowest best. As Thou wilt I am willing through the true faith in Jesus\nChrist, my Redeemer, to live or die. But, O God, do Thou grant me this\npetition, that I may not suddenly pass away in my sins, and be lost.\nVouchsafe unto me true knowledge, repentance and sorrow over my passed\ntransgressions. Set them before my sight in this life, that they may not\nat the last day be set before me, and I be put to shame before angels and\nmen. Grant me sufficient time and opportunity for repentance, so that\nfrom all my heart I may know and acknowledge my transgressions, and from\nThy saving word obtain forgiveness and comfort for the same. O Merciful\nFather, forsake me not, and take not Thy Holy Spirit from me. My heart\nand my heart's trust, O Thou Searcher of hearts, is ever known to Thee.\nKeep me in such trust to the life eternal. May I die, when Thou wilt,\nonly grant me a peaceful and blessed End. Amen.\n\n Lord Jesus Christ, Thou highest good,\n I pray Thee, through Thy precious blood,\n Grant, that my final end be good. Amen.\n\n\n\n\n At the Beginning of the Week's Work.\n\n\nRule Thou, O God the Father, who hast made us, and like all other\ncreatures ordained us not to indolence but to work, and bless each one in\nhis calling. Thou who rulest the universe also rule our own dear\ngovernment and graciously vouchsafe to it Thy wisdom and strength.\n\nRule Thou, O God the Son, who hast redeemed and ransomed us from sin.\nTake from us the burden of sin committed during the week now past, and\ngraciously grant us Thy peace. Thou the Supreme Bishop and Archshepherd\nof our souls, help all servants of Thy word in this and all Thy\ncongregations on earth to labor and bring forth much fruit unto eternal\nlife.\n\nRule Thou, O God the Holy Ghost, who hast sanctified us and born us again\nin Holy Baptism. Create in us a clean heart and renew a right spirit\nwithin us, that we carry no evils of the past into the new week, but put\naway all purpose and inclination of the old Adam still in us. Govern Thou\nour hearts with power; and if this week mark for any of us the end of\nlife, help Thou in the last bitter hour. Fill the heart with that grace\nwhich is better than life. Teach the hands to battle and vanquish the\nlast enemy, and grant for Christ's sake, the rest and triumph of the\nsabbath everlasting.\n\nThou, the Triune and Immortal God, be and abide with us and Thy Church\nforever. Unto Thee be glory, laud, and honor, world without end. Amen.\n\n\n\n\n At the Table\n\n\n\n\n Grace Before Meat.\n\n\nThe eyes of all wait upon Thee, O God, and Thou givest them their meat in\ndue season. Thou openest Thine hand and satisfiest the desire of every\nliving thing. Amen.\n\nO Lord God, Heavenly Father, bless us and these Thy gifts, which we\nreceive from Thy loving kindness, through Jesus Christ, our Lord. Amen.\n\n\n\n\n Come Lord Jesus, be our guest,\n And let Thy gifts to us be blest.\n Amen.\n\n\nGracious Father, who feedest and nourishest every living creature, feed\nand nourish our souls and bodies that we may not abuse Thy gifts, but\nthat we be rather quickened by the same unto the glory of Thy Name, unto\nall honest toil and every good work, to live and move before Thee in\nrighteousness and innocence. Amen.\n\n\n\n\n 1562.\n\n\n Be present at our table Lord;\n Be here and everywhere adored.\n Thy people bless, and grant that we,\n May dwell in paradise with Thee.\n Amen.\n\n Heavenly Father, bless this food,\n To Thy glory and our good.\n Amen.\n\n Jesus, bless what Thou hast given,\n Feed our souls with bread from Heaven;\n Guide and lead us all the way,\n In all that we may do and say.\n Amen.\n\n\n\n\n Grace After Meat.\n\n\nO give thanks unto the Lord, for He is good; for His mercy endureth\nforever. He giveth food to all flesh; He giveth to the beast his food;\nand to the young ravens which cry. The Lord taketh pleasure in them that\nfear Him, in those that hope in His mercy. Amen.\n\nWe thank Thee, Lord God, Heavenly Father, through Jesus Christ, our Lord,\nfor all Thy benefits; who livest and reignest for ever and ever. Amen.\n\n Dearest God, for meat and drink\n Accept our praise. Thy Name be blessed.\n Amen.\n\nBless the Lord, O my soul, and all that is within me, bless His holy\nName. Bless the Lord, O my soul, and forget not all His benefits. Amen.\n\n\n\n\n We thank Thee, Lord, for this our food,\n We thank Thee more for Jesus' blood,\n Let manna to our souls be given,\n The bread of life sent down from heaven.\n Amen.\n\n Praise God from whom all blessings flow,\n Praise Him all creatures here below;\n Praise Him above ye heavenly host;\n Praise Father, Son, and Holy Ghost.\n Amen.\n\n\n\n\n For Hearth and Home\n\n\n\n\n Prayer of a Housefather.\n\n\nGracious and Eternal God! Who Thyself in Paradise hast ordained holy\nmatrimony, vouchsafe unto me Thy grace and help in my married life, that\nI may ever maintain the true Christian love and fidelity toward my\nspouse. Grant, that I may dwell with her, as with the weaker vessel, in\nkindness and harmony according to knowledge, giving honor unto her, as\nbeing heirs together of the grace of life, and guide her together with my\nchildren and servants to the true knowledge of Thy divine glory, and to\nmodesty and honesty. Grant to them also, O Lord, that they follow Thee,\nand suffer themselves to be guided. Preserve us, O Thou God of Peace,\nfrom strife and dissensions, and the calumnies of the enemy, from undue\nsuspicion, which the devil sows as a seed of perdition, and for the\ndestruction of conjugal love and faith. Grant, that as Thy children we\nmay suffer and forgive each other in love and charity. Give us faithful\nservants and keep them in good health. Bless our labor and all that\ncontributes to the wants of the body. Enable us to bear our cross with\npatience, and grant that we be together again in the after-life. Amen.\n\n\n\n\n Prayer of a Housemother.\n\n\nO God Father, Son, and Holy Ghost, Thou giver of all blessing and\nconsolation, behold me, whom Thou hast placed in the estate of holy\nwedlock and made me the mother of this family, with the eyes of Thy\nmercy. Vouchsafe unto Thy handmaiden Thy grace, that I may love Thee\nabove all things, seek Thee, and ever be diligent in Thy service. And\ngrant that next to Thee I may honor, fear, and love my husband, and obey\nhim with patience and kindness, in pure and modest conduct, in piety and\nhumility well pleasing to Thee, and that the hidden life of my heart be\nconstantly adorned with a meek and gentle disposition and every virtue,\neven as in former times the consecrated women, who trusted in God and\ncontinued in subjection to their husbands. Enable me to train my children\nand servants with meekness, to the honor and glory of Thy holy name. Give\ngrace, that they follow me with gentleness, and grant, that I with my\nhusband and family may satisfy the wants of this life in good health and\naccording to Thy divine will. Protect us from harm and from enemies.\nEnable us so to use this world that we be not hindered in our salvation,\nbut in all things seek Thee, O Lord, and endeavor to be well-favored in\nThy sight. May we not despise the cross neither murmur against it, but\nbear it in patience, and thus remain in Thy keeping unto the end. May we\nas servants of God bring forth abundant fruitage, live holy lives in this\nworld, and attain to the everlasting inheritance in the world to come.\nAmen.\n\n\n\n\n Prayer of a Child.\n\n\nAlmighty and Eternal God! Thou hast commanded me to honor my parents, and\nin all things, which are not contrary to Thy word, to obey them. I\nbeseech Thee, for the sake of the obedience of Thy dear Son, Jesus\nChrist, my Lord, grant unto me, that I may duly honor my father and\nmother, serve them, obey, love, and esteem them, so that their blessing\nmay dwell with me to the end of my days. Keep me from sin and evil\ncommunications, that I may not anger or grieve my dear parents with\nhatred, sadness, scornfulness, disobedience, or obstinacy, and thus bring\nupon myself on this earth already their and Thy curse, and in the\nafter-life Thy eternal wrath. And since I have sinned many times through\nweakness, I beseech Thee, work in me sorrow and true repentance, through\nJesus Christ, my Savior, who gave to me the example of true obedience.\nGrant, that according to His example I may ever grow, and increase in\nwisdom, stature, and grace before Thee, O God, and all men. Amen.\n\n\n\n\n Prayer of a Youth or a Maiden.\n\n\nLord, Almighty and Holy God, who lovest a pure heart, a chaste mind, and\na holy life: I pray Thee, create in me also a clean heart, and renew a\nright spirit within me, so that I may serve Thee with sincere faith and\ntrue fear, and love Thee with all my strength. And keep me from all\nimpure communications. Subdue the evil lusts in my heart, and extinguish\nthe fires of fleshly desire. Preserve me from unchaste and dissolute\nassociations, from rioting and drunkenness, which lead to excess. May all\nfilthiness, and foolish talking, unbecoming a Christian, be ever foreign\nto my life. Keep me from arrogant pomp, or idleness and lounging, from\nthe snares and nets of the devil. Grant, that I may serve Thee with a\npure soul and an undefiled body in true faith, as did pious and chaste\nJoseph in Egypt, and graciously take me into Thy safekeeping against all\ntemptations and seductions of life, for the sake of Jesus Christ, Thy\ndear Son. Amen.\n\n\n\n\n Prayer of a Servant.\n\n\nMerciful God, Who through the precious suffering and death of Thy dear\nSon, Jesus Christ, hast redeemed and made me free from eternal servitude,\nfrom the power of sin, from the dominion of darkness, from the cruel\ntyranny of the devil, and the mastery of death and hell: I beseech Thee,\ngrant me grace, so that in my present station and calling, in which Thou\nhast placed me on earth according to Thy good pleasure, I may not be\ndissatisfied, nor murmur impatiently against Thy ordinance, nor yet envy\nothers in their more exalted stations, but help me to do such Thy will\nfrom the heart, with good will, ever thinking that I serve Thee, O God in\nHeaven, and not men. Help Thou, dear Father, that I serve Thee, the\nsupreme Lord of heaven and earth, in true knowledge and upright fear,\nlove Thee above all things, put all my hope and salvation in Thee, and\ncontinue in Thy commandments without blame. May I also be subject to my\nmasters and mistresses according to the flesh, not only to the good and\ngentle, but also to the froward, and patiently obey them in everything\nthat is not contrary to Thy pleasure, with fear and singleness of heart,\nas unto Christ my Lord, not with eye-service, as man pleaser, but from\nthe depths of my heart and for the sake of Thy will and commandment.\nGrant me grace that I may be found faithful in all things committed and\nentrusted to my care, not neglect nor bring them to naught, suffer no\nharm to come through carelessness on my part; also that I may not covet\nnor fraudulently appropriate to my own use the possessions of another\nentrusted to my care. Preserve my health, strengthen my limbs and all the\npowers of my body. Endow me with wisdom and understanding, that I may\nperform the labor of my masters and mistresses, with Thy help improve\ntheir living. May they be blessed through my industry; and all that I do\nand perform, may it redound to Thy divine glory and a blessed exercise of\nmy faith. For Christ, Thy dear Son's sake, who liveth and reigneth with\nThee and the Holy Ghost, world without end. Amen.\n\n\n\n\n Prayer of a School Child for the Holy Spirit.\n\n\nO my dear Lord, Jesus Christ, I thank Thee, that to the present day, Thou\nordainest church and school ordinances and regulations, and hast given\nunto my parents and me grace, that I too may be thus trained. I beseech\nThee, fill me with Thy Holy Spirit, that I may ever obey my dear parents\nand teachers, who only seek my welfare. Give unto me a docile heart, that\nI may learn my catechism, noble arts and language, and thus increase in\ngodliness, wisdom, understanding, and every virtue. O my dear Lord Jesus\nChrist, create in me a pure, chaste, and modest heart. May I ever serve\nThee in upright faith and true fear, and love Thee from all my heart.\nSubdue in me all evil lusts. Endow me with Thy Holy Spirit. Help me to\ncontinue in true humility. Grant me an obedient heart, to honor my\nparents according to Thy commandment, and neither anger nor grieve them.\nMay they live long on this earth, and protect Thou and preserve them from\ndisease, evil, and harm. Be gracious unto us and merciful. Bless us in\nbody and soul, now and forevermore. Amen.\n\n\n\n\n Prayer During a Thunder Storm.\n\n\nMost Mighty God! All the powers of the earth shall honor Thy holy name,\nand worship Thee, Eternal Father, in the beauty of holiness. For Thou art\nthe Lord, who reigneth over all. Thou showest Thy might and power\nthroughout the universe. The voice of the Lord is upon the waters. The\nLord of glory thundereth. The Lord is upon many waters. The voice of the\nLord is powerful and full of majesty. The earth shook and trembled. The\nfoundations also of the hills moved and were shaken. There went up smoke\nout of Thy nostrils, and fire out of Thy mouth devoured: coals are\nkindled by it. Thy pavilion round about Thee are dark waters and thick\nclouds of the skies. The Lord thundereth in the heavens, and the Highest\ngave His voice. All things are subject to Thee. All things acknowledge\nThee as their maker, and tremble before Thy divine majesty. The mountains\nand the depths of the abyss are sore vexed when Thou art wroth, the earth\ntrembleth, and the sea and the waters flee because of Thy wrath. The\nvoice of the Lord divideth the flames of fire. The voice of the Lord\nshaketh the wilderness. The Lord is king forever. The Lord will give\nstrength unto His people. The Lord will bless His people with peace. O\nmerciful God, preserve us from Thy wrath, which is beyond endurance!\nForgive us our sins! Make Thy face shine upon us! Cause this grievous\ntempest to pass by without harm to us! Protect us in body and soul, our\nhouse and home. Keep the fruits of the fields from hail and storm, from\ngreat inundation by water and all harm. O Holy God, preserve us from an\nevil death, and protect us, that no disaster befall us. Amen.\n\n\n\n\n Thanksgiving After a Thunderstorm.\n\n\nAlmighty God, our Heavenly Father, who has said, Call upon Me in the day\nof trouble: I will deliver thee, and thou shalt glorify Me; we praise and\nthank Thee most heartily. For Thou hast graciously heard our prayer and\nhast made this storm to pass that no harm befell us in life and property.\nThus hast Thou again revealed to us Thy fatherly compassion, and that\nThou wouldest not deal with us after our sins, neither reward us\naccording to our iniquities. Grant us, O Merciful Father, for the sake of\nThy only begotten Son, Jesus Christ, that we may take heed of Thy earnest\nand fatherly admonitions, improve our lives, and live in the fear of\nThee. May we constantly prepare and make ready ourselves for the coming\nof Thy dear Son, when the elements shall melt with fervent heat, and the\nearth also, and the works thereof, shall be burned up, and may we meet\nHim with rejoicing and enter the new heaven, wherein dwelleth\nrighteousness, and forever be with Thee; through the same, Thy dear Son,\nour Master and Redeemer, Jesus Christ. Amen.\n\n\n\n\n Prayer on the Eve of a Journey.\n\n\nAlmighty and Gracious God and Father, Thou protector of all that trust in\nThee from their hearts! In Thy name will I proceed and undertake this\njourney. For Thou art my God and preservest my going out and my coming\nin. Thou leadest my feet in right paths and wilt not suffer them to be\nmoved. I heartily beseech Thee to be my gracious guide and companion on\nthis present journey. Send Thy holy angels, and command them, in all my\nwanderings, to keep me from all evil in body and soul. Lead me on the\npaths of the righteous and bring me safely to my destination, that I may\nlaud and magnify Thee, here in time and in eternity forever. And now, O\nLord God and Father, into Thy hands do I commit my body and soul and all\nthat I possess. Thy holy angel be my safe guard. Amen.\n\n\n\n\n Prayer During a Journey.\n\n\nAlmighty and Most Merciful God! We are always in Thy sight wherever we\nbe. Thou preservest our coming in and our going out, and leadest us on\nthe right paths that we slip not. I pray Thee, that as Thou didst lead\nThy servant Abraham from the land of the Chaldees and kept him unharmed\nin his pilgrimage, and didst say to his grandson Jacob when he journeyed\nto Mesopotamia, I am with thee, and will bring thee again into this land;\nand as Thou also didst lead the Children of Israel through the Red Sea\nand through the desert, and didst go before them, by day in a pillar of a\ncloud and night in a pillar of fire: thus wouldst Thou also be with me on\nmy wandering, protect me on land and sea, by day and by night, and keep\nme from all harm and danger. And when my business is completed bring me\nhome again in full health of body and soul. And as Thou didst accompany\nyouthful Tobias through Thy angel Raphael, likewise do Thou accompany me\nin all my ways, so that when I, too, have happily returned to my home, I\nwith all mine own may have the greater reason to laud and praise Thee as\nmy true and faithful guide. Meanwhile I commend to Thy care all that I\nleave at home, and beseech Thee to have charge concerning them, and\nsuffer me to find them unharmed when I return. Amen.\n\n\n\n\n Prayer of Children for Their Father Engaged on a Journey.\n\n\nO Eternal Son of God, Thou Savior of all who call upon Thee in faith! We\nThy children, baptized into Thy blood, consecrated by Thy Spirit a royal\npriesthood, and ordained Thy brethren and co-heirs with Thee in grace,\ncall upon Thee with innocent tongues, and earnestly pray Thee, graciously\nto safeguard our dear father now journeying over land for the sake of his\ncalling and to gain the means of livelihood. Help him to discharge his\nduties with favor and despatch, and return him to us in health and joy,\nas Thou through Thy angels didst preserve and guide Thy servant Jacob on\nhis journeys, for Thou art the faithful guide and companion of all who\nfear Thee and trust in Thy mercy. Amen.\n\n\n\n\n Thanksgiving After a Completed Journey.\n\n\nGracious God and Father! Most heartily do I thank Thee that Thou hast\nenabled me to bring my journey to a happy end. Through the ministrations\nof Thy dear angels Thou hast again brought me to my home, guarded and\nkept me from all evil, preserved me from the murderous and robbing hands\nof evildoers, and the teeth of wild beasts, and kept me from all other\ndangers of body and soul. In short, that I have been led to and fro in\nhealth and happiness: I owe it altogether to Thy fatherly goodness and\nalmighty care. And I beseech Thee from all my heart, continue to keep me\nand mine under Thy protection, and preserve us, body and soul, to the\neternal life, for Jesus' sake. Amen.\n\n\n\n\n A Birthday Prayer.\n\n\nDear Father in heaven, I thank Thee from all my heart, that Thou hast put\nme into this world and made me a rational being. I am born of Christian\nparents and made a member of Thy holy Church. Today the anniversary of my\nbirth hath come, and since I have been permitted to reach this day and\nthus complete another year of my pilgrimage, I thank Thee from all my\nheart and joyfully reiterate the thanksgiving of Thy servant David,\n\"Bless the Lord, O my soul, and all that is within me bless His holy\nname. Bless the Lord, O my soul, and forget not all His benefits: who\nforgiveth all thine iniquities; who healeth all thy diseases; who\nredeemeth thy life from destruction; who crowneth thee with loving\nkindness and tender mercies.\"\n\nSince every day of my life, however, is one step nearer to death, which\ncan strike me this hour, yes, this very minute, I beseech Thee so to\nshape and rule every day of my life, that I may walk according to Thy\npleasure as in the day, that is circumspectly in Thy sight, honestly, and\nconscious of my responsibility, in short as a true Christian and in\nconformity with the promise made by me to Thee, my dear God, in my\nbaptism. And if this prove my last year and my last birthday, I place all\nthings into Thy gracious keeping. If it is Thy will that I should cease\nto live, then I have lived enough. For if it is sufficient for Thee, it\nis sufficient for me. Am I old enough for Thee, I am old enough for me.\nHere I again put myself under Thy shield and protection, into Thy sublime\nand eternal power. If I live this year, may I live in Thee; if I die, may\nI die in Thee, so that I may live, and move, and have my being in Thee,\nand whether living or dying I may be Thine to all eternity. Amen.\n\n\n\n\n Prayer for Temporal Peace.\n\n\nEternal God, Everlasting Father! Thou art a God and lover of peace. From\nThee all true unity cometh. We pray Thee graciously to protect Thy\nChristendom on earth against all its enemies, so that we may be kept in\npeace, and ever serve Thee gladly in faithful doctrine and a pure\nconduct. Grant us grace, so that all estates and rulers of Christendom\nmay live peacefully and harmoniously in perfect piety and godliness, so\nthat discipline and order prevail, churches and schools be not destroyed,\nand the country be not devastated nor grievously oppressed. Grant us\ngrace, so that men will content themselves with what they have, and will\nnot for the sake of avarice or lusting after foreign lands and peoples,\nnor yet because of pride, vain ambitions, and arrogance, enmity, hatred,\nenvy, nor any other cause, incite war, sedition, or revolution in this\nour country. Hinder all evil counsel and purpose of unstable men, who\nthink only of that which is not good. Put them to naught in their\npurposes, so that they must retreat and are utterly consumed with\nterrors. Stretch forth Thine arm to protect us who are named after Thee,\nso that Thy heritage be not destroyed. Support Thy faithful who rely upon\nThee and call upon Thy name. Hear us in our distress, and Thy holy name\nprotect us. Send us help from Thy sanctuary, and strengthen us from on\nhigh. Bless the country and the cities in which Thy holy word dwelleth.\nProsperity must dwell within their palaces! O merciful God, incline the\nhearts of all men to a Christian peace and concord, to the which Thou\nhast called us through Thy word and gospel. And if bitterness prevail\namong some, help that it be done away with, to the glory of Thy holy\nname, the spreading of Thy word, and the betterment of Christendom, and\nthat the poor and distressed in the land may rejoice in Thee and praise\nThy holy name, for Thou only performest wonders and provest Thy powers\namong the nations. Amen.\n\n\n\n\n A Prayer for School.\n\n\nWe pray Thee, everlasting God, Father, Son, and Holy Ghost, Thou eternal\nand inseparable Trinity and inexpressible Unity, that Thou wouldest\nfaithfully take under Thy protecting wing the flock of Thy Christendom,\nand ever abide in our midst with Thy grace and truth. Be Thou with us, O\nLord, our God. Be Thou a wall of fire round about us, and destroy them\nwho hate Thee and are hostile to Thy name. So rule us, O God, that we may\never be guided by Thy clear and pure word and are not seduced by the\nexternal appearance of things. Keep us, Lord Jesus, from error and false\ndoctrine, and send us faithful teachers who take heed unto Thy\ncongregation, purchased with Thy blood, and are anxious to perform Thy\nwill. Grant us obedient hearts, so that we, as lambs of Thy flock, may\nobey Thy voice, and be filled with fruits of righteousness. Teach us ever\nto do Thy will, for Thou art our God: Thy spirit is good. Lead us into\nthe land of uprightness, to the end that we, too, through a blessed\ndeparture from this life may attain to Thee and the everlasting joy and\nblessedness, and behold Thy glory to all eternity. Amen.\n\n\n\n\n Church Prayers\n\n\n\n\n Prayer When Going to Church.\n\n\nAlmighty God, Heavenly Father, because of Thy great mercy I will go to\nThy house and worship Thee in Thy temple in Thy fear. Lord, lead me in\nthe paths of righteousness, and make Thy way straight before my face.\nGuide me on the paths of Thy commandments, for Thou art my God, and the\nLord of my salvation. I delight in Thy sanctuary and rejoice in the\ncongregation of Thy saints, who confess and glorify Thee. How amiable are\nThy tabernacles, O Lord of hosts! My soul longeth, yea, even fainteth for\nthe courts of the Lord. O come, let us worship and bow down: let us kneel\nbefore the Lord our maker. For He is our God; and we are the people of\nHis pasture, and the sheep of His hand. Magnify the Lord our God, adore\nat His footstool; for He is holy. I worship Thee, O God, in the accepted\ntime through Thy great mercy. Hear me according to Thy grace. Amen.\n\n\n\n\n Prayer for Sincere Repentance.\n\n\nMerciful and Gracious God, Thou art slow to anger and plenteous in mercy!\nThou callest us daily through the gracious preaching of Thy word to\ndevout conversion, and in Thy name causest repentance and remission of\nsins to be preached. And Thou showest Thy forbearance with us through Thy\nlong suffering and inexpressible mercy, and dost not suddenly come upon\nsinners in the midst of their evil deeds with Thy righteous wrath and\njudgment to punish them, but giveth place and time for repentance, so\nthat no one can justly charge or accuse Thee. For Thou art not willing\nthat any should perish, but that all should come to repentance and have\neverlasting life. O dear God, Thou knowest the sluggishness of our flesh\nand the hardness of our hearts, that we through inherited sin are thus\nfar deranged and so deeply sunk in sin, that of our own accord we can not\nrise or return. Therefore, for the sake of the wounds of Jesus Christ, my\nLord, I beseech Thee, convert Thou me, and I am converted. For Thou art\nmy God, and where I am converted I will truly repent. Save Thou me, O\nLord, and I am saved. Help me, and I am helped. Behold, I am like an\nerring and lost sheep. Seek Thou Thy servant, that I forget not Thy\ncommandments. Circumcise the foreskin of my heart. Purge me, and I shall\nbe clean. Wash me, and I shall be whiter than snow. Create in me a clean\nheart, O God; and renew a right spirit within me. Cast me not away from\nThy presence; and take not Thy Holy Spirit from me. O dear God, look upon\nme, as Thou didst look upon Mary Magdalene, the repentant sinner, as she\nlay at Thy feet and wept over her transgressions; and the publican in the\ntemple, as he smote his breast and besought Thy grace. Vouchsafe unto me\nsincere sorrow and contrition over my sin, and a true faith with firm\nconfidence in Thy grace, and also worthy fruits of repentance. Let me\ndiscern the day of Thy visitation, and not despise the riches of Thy\nmercy, so that I may not neglect the accepted time, and the day of Thy\nsalvation, and not fail to turn to Thee, my Lord and God. May I not\npostpone my repentance from one day to another, nor yet to the last hour,\nbut rather turn to Thee this day and repent. Amen.\n\n\n\n\n Prayer for the Forgiveness of Sins.\n\n\nMerciful Father, Eternal God, my sins are grievous, many and great my\ntransgressions, and mine iniquities are innumerable, for the imaginations\nof my heart are evil from my youth. O Lord, who can understand his\nerrors? Behold, I acknowledge my transgressions: my sin is ever before\nme. Against Thee only have I sinned, and done evil in Thy sight: that\nThou mightest be justified when Thou speakest, and be clear when Thou\njudgest. I beseech Thine infinite mercy, enter not into judgment with Thy\nservant: for in Thy sight shall no living man be justified. If Thou,\nLord, shouldest mark iniquities, O Lord, who shall stand? Behold, if Thou\ncontendest with man, he can not answer Thee one of a thousand, for all\nour righteousness before Thee is as the filthy rag. Have mercy upon me, O\nGod, according to Thy loving kindness: according to the multitude of Thy\ntender mercies, blot out my transgressions. Wash me thoroughly from mine\niniquity, and cleanse me from my sin for Thy name's sake. Lord, have\nmercy upon me, save my soul, for, alas! I have sinned against Thee.\nRemember, O Lord, Thy tender mercies and Thy loving kindnesses; for they\nhave been ever as of old. Remember not the sins of my youth, nor my\ntransgressions: according to Thy mercy remember Thou me for Thy goodness'\nsake, O Lord. Remember that we are flesh, as the wind which bloweth and\ndoth not return, and cease in Thy anger and wrath against us. O merciful\nGod, I acknowledge that my virtues and my deeds can never blot out my\nsins, nor yet merit Thy grace. Only the innocent suffering and death of\nJesus Christ, the Lamb without spot or blemish, is the true offering for\nour iniquities, and His blood, shed for the remission of our sins, is the\ncleansing and purification of our souls. In such confidence and hope I\nsupplicate Thee: forgive Thou the transgressions of Thy people. Cover our\nsins. Impute not our iniquities, for Thou art merciful. Cleanse Thou me\nfrom secret fault. Let my sorrowing soul and my vexed bones again\nrejoice, for with Thee there is mercy and plenteous redemption. O Lord,\nhear the voice of my supplication, and despise not the groanings of my\nheart, for Christ's sake. Amen.\n\n\n\n\n Prayer for True Faith.\n\n\nLord, Almighty God, Thou Father of lights, with whom there is no\nvariableness, neither shadow of turning, from whom every good and perfect\ngift cometh, I pray Thee, since all men have not faith: implant and\nmaintain in my heart through the workings of Thy Holy Spirit the true\nknowledge of Thy dear Son Jesus Christ, and increase it from day to day,\nso that I, too, may be filled with the knowledge of Thy will in all\nwisdom and spiritual understanding, that I may walk worthy of Thee unto\nall pleasing, being faithful in every good work, and increase in such\nknowledge according to Thy glorious power in all patience and long\nsuffering with joy. Grant unto me, according to the riches of Thy glory,\nthat I may be strengthened with might by Thy Spirit in the inner man,\nthat Christ may dwell in my heart by faith. O dear God, since no man\nknoweth the Son but Thou, O Father, and no man knoweth Thee, the Father,\nbut only the same Thy Son, and he to whom the Son will reveal Thee, I\npray Thee, draw Thou me unto Thee. Grant me the knowledge of salvation,\nwhich is the forgiveness of sins. Strengthen my weak faith, which is\nsmall as the mustard seed, so that it may increase, and I be rooted and\ngrounded in Thee, and may stand steadfast and unmoveable. Gracious God,\nThou hast kindled the spark of faith in my heart and has begun this good\nwork in me, I cry to Thee, perfect it until the end, that we may ever\nincrease in knowledge and understanding, and be sincere and without\noffense till the day of Christ, being filled with the fruits of\nrighteousness, which are by Jesus Christ, unto the glory and praise of\nGod. Preserve what Thou hast begun, that we might war a good warfare,\nholding faith and a good conscience, and not waver or succumb in trial\nand temptation and make shipwreck concerning faith. Therefore protect me,\nmy God, that I am not led astray among the errors, schisms, and heresies\nof the world. Preserve me from superstitions and all false doctrine, that\nI may neither err nor doubt in any article of faith. And vouchsafe unto\nme Thy grace, that my faith be not lifeless, inactive, or without good\nfruits, but active and energetic, serving in love, so that I, too, may\ncarry off the end of faith, which is the soul's salvation. Amen.\n\n\n\n\n Prayer Before Confession.\n\n\nLord Jesus Christ, Thou my Redeemer, Thou hast given unto Thy dear Church\non earth and its faithful servants the sacred office of the keys, and\ninvested it with the promise, that whatsoever by virtue of the same they\nshall loose or bind on earth shall also be loosed or bound in heaven: I\nthank Thee, and eternally praise and glorify Thee, for such Thy gracious\ngift. And, since I a poor and bound sinner need this blessed key which\nlooseth, so that I may not be kept under the bonds of the infernal\njailer, I beseech Thine infinite mercy, that I may receive its comfort\nthrough my spiritual father, my pastor, and for the sake of Thy holy,\nprecious blood and Thy innocent offering and death be loosed from all my\nsins. Grant me Thy Holy Spirit, so that I may grasp this holy absolution\nin heartfelt contrition and undoubting trust, firm resolve, brotherly\nlove and gratitude, and inherit the eternal life. Amen.\n\n\n\n\n Thanksgiving After Absolution.\n\n\nO Blessed, Merciful, and Gracious God! I thank and praise Thee from all\nmy heart, that through Thy servant Thou hast again forgiven me, a poor\nsinner, all my sins, again received me in grace, and promised me eternal\nlife. I earnestly pray Thee, vouchsafe unto me Thy Holy Spirit, and\ncreate in me a pure heart, so that I may joyously trust that all my sins\nare forgiven me through Jesus Christ. As an earnest and surety of this\nblessed fact, I will now eat and drink the true body and blood of Thy\ndear Son Jesus Christ, under the bread and wine, to my soul's salvation.\nGrant me also, O faithful God, that henceforth I may earnestly avoid all\nsin and better my life. This Thou wouldest graciously grant me for Thy\nname's sake. Amen.\n\n\n\n\n A Morning Prayer for Communion Day.\n\n\nArise, my soul, this is the day which the Lord hath made. We will rejoice\nand be glad in it. Give thanks unto the gracious and merciful God for His\nblessings and say:\n\nAlmighty and Merciful God and Father, I thank Thee from all my heart for\nthe protection of this night, for the refreshing rest, and for the joyous\nmorning, which Thou hast granted unto me. I praise Thee with all my soul\nfor Thy wonderful mercy which blesses me with the forgiveness of my sins.\nPraised be Thy grace, which is new each morn, and which on this day also\nbids me to Thy house, and calls and invites me to Thy altar.\n\nO Lord, since I, too, would come to Thy Supper with the throngs which\ncelebrate, do Thou Thyself make me ready. As Thou wouldest find a pure\nresidence in me, do Thou cleanse and consecrate my body and soul. Guide\nme with Thine eye, and lead me with Thy hand to the riches of Thy mercy.\nComfort me with Thy countenance, and do not forsake me. As the hart\npanteth after the water brooks, so panteth my soul after Thee, O God.\n\nAnd, that my sacred resolve may not be hampered, I commend to Thee my\nbody and soul, reason, senses, and thoughts, whatever I do or leave\nundone, my coming in and going out, my walking, standing, sitting, and\nresting, my imaginations and aspirations, my faith and confession, and\nwhatever internally or externally I may be or do. O God, preserve in me a\ndevout spirit, and hinder whatever might disturb or hamper me. Receive me\ninto the especial care of Thy grace, and increase in me the work that is\nnow begun. Perfect and complete it according to Thy power and grace to\nThy glory and my salvation. Keep me from evil thoughts, from idle\nimaginations, from all uncleanliness, so that in Thy fear I may begin a\nconsecrated life and continue therein. May the light of my faith shine\nbefore men. May I give offense to none, but rather in Christian conduct\nedify the brethren and direct them to all virtue.\n\nHoly Jesus, do Thou unite with my body and soul on this day. Nourish me\nwith Thy flesh and refresh me with Thy blood, so that my weak faith be\nstrengthened, and receive the assurance of Thy grace, the remission of my\nsin, and eternal salvation. Invest me with the pure silk of Thy\nrighteousness. Clothe me in the true wedding garment, that I may appear\nat Thy heavenly board a worthy guest.\n\nNow, Lord God and Father, be my help and my protection! Lord Jesus Christ\nbe my bread, my light, and life! And Thou, O Holy Ghost, illumine and\npreserve me in the true sanctification, so that in that estate, into\nwhich I again am permitted to enter, I may remain for the course of my\nlife. Let me be enveloped in Thee. Without Thee there is only grief. O\ndear Savior, let me ever be with Thee. Amen.\n\n\n\n\n Prayer Before Holy Communion.\n\n\nLord Jesus Christ, Eternal Son of God, I am not worthy to open my lips\nand receive the most precious sacrament of Thy body and blood. For I am a\nsin-stained man, but Thou art the Lord whom the heavens can not\nencompass. How then can a human being who is but dust and ashes be worthy\nto receive Thy most holy body and precious blood! I well know and\nacknowledge that my sins are many and that for that reason I am an\nunworthy guest at Thy table. But I also sincerely believe and confess it\nwith my lips that by Thy grace Thou canst render me worthy. For Thou art\nomnipotent and gracious. Thou only canst cleanse and make holy whatever\ntook rise in unclean seed. Thou canst transform sinners into true and\nholy men, when by Thy grace Thou forgivest sin and renewest us with Thy\nHoly Spirit. Therefore I pray Thee, by Thy power and love grant me grace,\nthat I may worthily approach Thy altar, and not become guilty of Thy body\nand blood by unworthy eating and drinking, so that I may not receive\ndeath in place of life. Grant me grace, that I may know and test myself\nas a poor sinner, my heart filled with sorrow over mine iniquity, and may\nproperly discern Thy tender and noble body, and Thy holy, precious blood.\nMay my reason, senses, and intellect be ever submissive to Thy word, and\nmay I be earnestly resolved to better and improve my life with Thy help,\nso that in this precious sacrament, I may not only with my mouth receive\nThy body and blood, but also in true faith accept Thee, my Savior and\nRedeemer, enthrone Thee in my heart, and find in Thee my life and\nblessedness. For Thou art the living bread which cometh down from heaven\nand bringeth life to men. Whoever cometh to Thee shall nevermore hunger,\nand he that believeth on Thee shall nevermore thirst. Whoever eateth Thy\nflesh and drinketh Thy blood dwelleth in Thee and Thou in him, and shall\nnever die. O beloved Lord, my spirit and my mind yearn for Thee. As the\nhart panteth after the water brooks, so panteth my soul after Thee, O\nGod. My soul thirsteth for God, for the living God: when shall I come and\nappear before God? Fill me with Thy grace. Amen.\n\n\n\n\n A Sigh When About to Receive the Sacred Body of Christ.\n\n\nLord Jesus Christ, Thy holy body strengthen and preserve me in the true\nfaith unto eternal life. Amen.\n\n\n\n\n A Sigh When About to Receive the Sacred Blood of Christ.\n\n\nLord Jesus Christ, Thy holy blood strengthen and preserve me in the true\nfaith unto eternal life. Amen.\n\n\n\n\n Prayer After the Holy Supper.\n\n\nLord Jesus Christ, with all my heart I thank and glorify Thee, that Thou\nhast again cleansed me, a poor sinner, from all my sin, and as an earnest\nof such cleansing and forgiveness of my sin, hast nourished me with Thy\nbody and blood, and like an unclean child, after such purification, hast\nreceived me into the fatherly arms of Thy grace and mercy, and put me\npure, reproachless, and without blemish before Thy Father.\n\nI earnestly pray Thee with all my power, in addition to such blessing,\ngrant me Thy grace through the workings of Thy Holy Ghost, so that I may\nsufficiently understand such blessing and grace, gratefully accept it,\nand glorify and praise Thee with all my heart. Grant me strength from\nabove by Thy Holy Spirit, that I may heartily forgive my neighbor\nwherever he hath sinned against me, even as Thou hast fully and richly\nforgiven me my great and manifold transgressions, yes, entirely blotted\nthem out and wilt never remember them. Help me to love my neighbor and\ngladly show him every good, as Thou hast done unto me, and hast shown me\nmore than I can ever sufficiently thank Thee for. Praise and glory be to\nThee, O faithful God, together with the Father and the Holy Ghost, world\nwithout end. Amen.\n\n\n\n\n Prayer for a Pious Life.\n\n\nO my dear Lord Jesus, illumine me today and evermore, that I may shape\nthe course of my Christian life and direct it toward the eternal\nJerusalem, my eternal home. And as Thou yearnest for me, may I also have\nall my delight and thirst in Thee, seek Thee early, yearn for Thee, and\nmake of Thee, the bread of life, the companion of all my ways. Keep me, O\nunchangeable, everlasting God, from the inconstancy of the children of\nthis world, that I may not fall into hypocrisy as they do, but today and\nalways, in all my calling, prove myself constant in godliness, so that my\nlife may decrease in vice and increase in virtue. May I always faithfully\nserve Thee, my Lord, disdain the worldly, be exalted in Thee, experience\nThy grace and protection, and eternally thank Thee, for Christ's sake.\nAmen.\n\n\n\n\n Prayer for Faithful Teachers and Preachers.\n\n\nThe harvest truly is plenteous, but the laborers are few; pray ye,\ntherefore, the Lord of the harvest, that He will send forth laborers into\nHis harvest.\n\nMerciful God, who hast commanded us through Thy only begotten Son to pray\nThee for laborers in Thy harvest: I earnestly beseech Thee, grant us, Thy\nsheep, pastors according to Thy heart, to feed us with doctrine and\nwisdom. Put learned bishops, Christian pastors, pious teachers over Thy\ncongregation, faithfully to show the true way to eternal life. Fill them\nplenteously with Thy Holy Spirit, so that they may fruitfully proclaim\nthe holy and precious word of the gospel, and sincerely perfect Thy work.\nGrant them a courageous heart, that without fear or favor they may lift\nup their voices and put the adversaries to silence. Open the door for\ntheir word, that they may be blessed and go from strength to strength.\nMay they also be shining examples for the flocks entrusted to their care.\nGrant me and all listeners a grateful heart, that we may communicate in\nall good things unto them that teach us the word, rejoice in them,\nforsake them not, so that we may be blessed of Thee according to Thy\npromise in all the works of our hands. Preserve us from hirelings,\ntime-servers, unfaithful laborers, and hypocrites, who falsely deceive\nthe hearts of the righteous, whom Thou hast never grieved, and fortify\nthe hands of the wicked, that they will not turn from their evil ways.\nLet us never be robbed of our faithful preachers for the sake of our\ningratitude, but rather look upon the glory of Thy holy name, yea, behold\nthe Kingdom of Thine Anointed, that it may prevail among us to the end of\nthe world. Amen.\n\n\n\n\n Prayer for the Kingdom of God.\n (Meeting of the Congregation)\n\n\nGracious and Blessed God, who hast taught, and commanded us above all\nthings and first to seek the kingdom of God and His righteousness: I pray\nThee, grant us grace, that Thy holy word may be preached in all the world\nin all its truth and purity, and we submit our reason to the obedience of\nfaith, and live holy lives according to it as behooves the children of\nGod to Thy pleasing, so that Thy kingdom may come to us, and increase,\nand many of them, who do not yet believe in the word, be won through a\nChristian conduct. Help us, dear God, who are delivered from the power of\ndarkness and are translated into the kingdom of Thy dear Son Jesus\nChrist, in whom we have redemption through His blood, even the\nforgiveness of sin, that we may remain in His kingdom, faithfully\ncontinue in the wholesome doctrine, and live worthily as children of\nlight in all piety and godliness. And since the kingdom of God cometh not\nwith outward shew, neither consists of mere words, but is power and\nspirit: grant us grace, that we may be born again from above through Thy\nsaving word and Thy Holy Spirit, co-heirs of life, so that with our\nhearts we may dwell above where Christ sitteth, and constantly seek the\ninheritance incorruptible, and undefiled, that fadeth not away. Enable us\nto be poor in the spirit and humble, and such who sorrow over their sins.\nLet us be anhungered and athirst, and heartily yearn after righteousness.\nMay we ever be meek, and suffer and overcome whatsoever of persecution\nand tribulation may assail us, and revilings and undeserved malignings\nwith patience and longsuffering. Keep us from all offenses, whereby Thy\nholy name is blasphemed and outraged, Thy kingdom hindered and weakened.\nGrant us grace to practice our faith in works of love and mercy, feeding,\nclothing, harboring, visiting, comforting the poor and distressed of this\nworld, so that when that great day dawns we may hear the blessed and\njoyous voice of Thy dear Son: Come ye blessed of My Father, inherit the\nkingdom prepared for you from the foundation of the world. Amen.\n\n\n\n\n Prayer for Missions.\n\n\nI pray Thee, my heavenly Father, through Jesus Christ, Thy dear Son, our\nLord, have mercy upon the unbelievers, whoever and wherever they are, who\nstill walk in darkness, and do not yet possess the light of Thy gospel.\nThey are stricken with blindness by the evil one. Their foolish heart is\ndarkened. They are alienated from the life that is of Thee, through their\nnative ignorance, carried away to the dumb idols, even as they are led,\nand in their blindness curse and blaspheme Thy dear Son, Jesus Christ,\nthe mercy seat. For that reason, O faithful God, take away the veil,\nwhich is upon the heart of the Jews, who stumble at the stone of\nstumbling and the rock of offense. Illumine their eyes that they may know\nthe true Messiah, the Savior of the world. Gather the heathen and all\nunbelievers, who look upon Thy word as foolishness, into the true fold\nand the true assembly of Christians, the congregation of saints, so that\nwith us and all the faithful they may honor, glorify, and worship Thee,\nthe Father in the Son, and the Son with the Father and the Holy Ghost,\never one God and Lord. Open the understanding of all men, who do not put\ntheir salvation and their confidence solely and alone on the true\nfundament and cornerstone, even Jesus Christ, so that they know Him, whom\nThou hast sent, and in the true faith and a sincere trust of their hearts\nmay acknowledge and accept Him as their Savior and Redeemer, who through\nHis obedience and fulfillment of the law, and through His bitter death,\nhath merited an eternal salvation. We pray Thee also, to return to the\ntruth of Thy word all of them who have defected from the Christian faith,\nor err in sundry other things and are cumbered with false doctrines. Thou\nGod of grace, have mercy upon those, who are not of the true faith, who\ndwell in the shadow of death, and in the darkness of their minds walk on\nuneven paths. Seek the lost, lead aright the erring, illumine the blinded\nand infatuated, open the ears of the deaf, unloose the tongues of the\ndumb, who do not confess Thee, raise the fallen, bring back the corrupt,\nassemble the dispersed, lead aright the erring and seduced, for Thy\nmercy's sake. Amen.\n\n\n\n\n Prayer Against False Doctrines and Sects.\n\n\nGracious God! Thou hast warned us, that we should beware of false\nprophets who come in sheep's clothing, but inwardly are ravening wolves.\nAnd since in these latter days, in which we now live, evil times will\ncome, in which Satan clothes himself in the livery of heaven, and false\nteachers and fraudulent laborers dissemble as though they were the\napostles of Christ, who have the form of godliness but deny the power\nthereof; and since the defection from the pure doctrine, and the man of\nsin, the son of perdition, who opposeth and exalteth himself above all\nthat is called God shall be made manifest, help us, eternal God, that we\nmay be filled with the love for truth, and avoid the spirit of lies and\nall falsehoods and errors, abstain and flee from all appearance of evil,\nso that we may not be moved from the true faith, nor yet beguiled from\nour reward, but may continue steadfast by Thy word unto the end. May we\nnever be led astray, neither through deceiving powers, signs, and\nmiracles, nor through any temptation to unrighteousness, so that we be\nnot tossed to and fro, and carried about with every wind of doctrine, by\nthe slight of men, and cunning craftiness, whereby they lie in wait to\ndeceive. Preserve Thy elect in this evil world, O God, that they may not\nbe led into error. Shorten the days in these evil times. Destroy the\nAntichrist, the wicked child of perdition and temptation, through the\nspirit of Thy mouth. Put an end to his days through the appearance of Thy\ndear Son. Preserve us also from unruly spirits and schisms, from the\nravenous wolves who have no mercy for the flocks, from men who speak\nperverse doctrines and draw the disciples to themselves, from men of\ncorrupt minds who sow offense and bring about separation, from the tares\nwhich the enemy scatters, from thieves and murderers of the soul. Let us\nonly hear Thy voice, and follow it from our hearts in true faith and\nupright obedience, so that your word may be our only rule and norm,\naccording to which we measure all doctrine, flee the evil teaching, and\nall unrighteousness. Then will we, too, have boldness and be not brought\nto naught on that day when Jesus Christ, our Savior, shall appear. Amen.\n\n\n\n\n Prayer Against the Enemies of Christendom.\n\n\nO Lord God, why do the wicked rage without cause? and the mighty set\nthemselves and take counsel against Thee and Thy Son, Thine Anointed? O\nLord, how numerous are Thine enemies, and great the number of those who\nconspire against Thy word to destroy it, and put their own evil idolatry\nin its place and introduce false doctrines into Thy Church. They invent\nsecret artifices and practices, to destroy the confessors of Thy word.\nThey counsel what is evil, pregnant with calamity. They lie in wait for\nus like the lion seeking his prey. Thus they waylay our souls. They open\nwide their jaws, like a gaping sepulchre, to devour us. O Lord God,\ncommit us not to their will, for behold, the wicked mob speaks in its\nheart, Thou, O Lord, hast forgotten us. Thou hast concealed Thy\ncountenance. Arise, therefore, O Lord. Lift up Thy hand, forget not the\ndistressed. Awaken, O Lord, why sleepest Thou! Awake, and do not quite\nreject us. Why dost Thou hide Thy face, and forget our misery and\ndistress? Arise, and help us, and deliver us for Thy mercy sake, that the\nenemy may not injure us, nor the children of Belial harm us. Save us from\nthe hands of aliens, whose teaching is useless, and their works evil. For\nthey will not be guided to do good. They will not regard Thee, the Lord\nof hosts, nor yet the works of Thy hands. Thou wilt destroy and not build\nthem. O Lord God Sabaoth, militate Thou against our assailants. Give\nvictory to our Christian government, and conquer the enemies of Thy name.\nFor victory cometh from heaven, and is not brought about by the great\nmultitude. Thou canst as readily help through the few as through the\nmany. Therefore destroy the arm of the wicked. Make their counsels of no\naccount, that they can not accomplish them, but must be brought to naught\nwith shame. They must return, who hate Thee, and seek to annihilate Thy\nholy word. They must be as chaff before the wind. Thy holy angel brush\nthem aside, that they may not boast against Thee and say our hand hath\ndone this. Therefore safeguard Thy honor among men, and help us for Thy\nname's sake, that it may not be desecrated and blasphemed. Amen.\n\n\n\n\n Prayers During Times of War\n\n\n\n\n Prayer During War.\n\n\nO Lord God, who art righteous in Thy judgment, and plenteous in mercy,\nwhose faithfulness endureth from generation to generation, who rulest\neven above the din of war: may all nations learn to know that Thou alone\nart God, that Jesus Christ is Thy Son, and that all who truly confess Thy\nname are the people of Thy pasture and the sheep of Thy hand. Thou hast\npermitted the nations to rise up in war against each other and our own\nbeloved country to become engulfed in its throes. O Lord God, our Father,\nwe know that war is a punishment for sin and that we, too, have justly\nmerited Thy punishment through our sins. Therefore we humbly confess our\nsins, and supplicate Thy pity and compassion, lay not our iniquities\nagainst us, but graciously forgive us our sins and shortcomings for the\nsake of our Lord Jesus Christ. To Thy fatherly goodness and care we\ncommend our people, and especially our soldiers and sailors now in the\nservice of their country. They are absent from their loved ones, beset by\ndangers on all sides. Be Thou ever near. Keep them from all evil. O Thou,\nwithout whose consent not even a sparrow falleth to the ground and who\nhast numbered the very hairs of our heads, take them under the shadow of\nThy wings. Give them courage and obedience, fortitude and valor in the\nhour of danger, and compassion and mercy in the flush of victory. Prosper\ntheir arms to the establishment of justice, peace, and truth among all\npeoples. Lead them safely back to their homes and their loved ones,\nbetter citizens, better Christians than before. And to Thy holy name be\nglory, laud, and honor, world without end. Amen.\n\n\n\n\n Prayer for the Army and Navy.\n\n\nAlmighty and Eternal God, Thou King of kings and Lord of lords, who\nrulest and governest all things in heaven and earth: we beseech Thine\ninfinite mercy to bless the officers and men of our army and navy.\nPreserve them against all dangers and temptations which may assail their\nlives. Help them manfully to battle against and overcome the powers of\nevil, the world, the devil, and the flesh. May they ever be filled with\nThy Holy Spirit from on high, and in courage, manliness, and truthfulness\nprevail in the hour of danger and when the battle rages. May they ever\nlook to Thee, who art the succorer of those in peril, as to their only\nhelper, and in Thy name fight a holy fight to maintain the country's\nhonor and keep the flag unsullied, that truth and righteousness may\nprevail. Put to naught all evil designs and devices of their enemies,\nboth spiritual and temporal. Guide Thou and direct them that they may\never fight the good fight of faith and in the end overcome and attain to\ntheir soul's salvation, to the glory of Thy holy Name; through Jesus\nChrist, our Lord. Amen.\n\n\n\n\n Prayer on the Eve of Battle.\n\n\nO Lord God of hosts, Thou the Highest of the high and the Holiest of the\nholy, who rulest and shapest all things to the glory of Thy name and the\nbetterment of Thy people: we humbly beseech Thine infinite mercy in this\nhour of our peril, judge Thou between us and our enemies. Be Thou our\nsure defense. Stir up Thy might and hasten to our help. Lay not our sins\nto our charge, but for the sake of Thy dear Son, our only Lord and\nSavior, Jesus Christ, manifest Thy mercy toward us and blot out our\niniquities, and sanctify and guide us by Thy truth. We Thy poor servants\ncall upon Thy holy name, and implore Thy grace. Have mercy upon us. Lead\nus safely through the blood and carnage. Make it appear that Thou art our\ndeliverer. Make us strong with the assurance that it is Thy cause, that\nwe are Thy children, that Thou holdest our destiny in Thy hand. And if it\nis Thy will that we should lay down our lives--Thy will be done! Help us\nthen that our last day on this earth may be the first in Thy paradise.\nBless our loved ones at home, and comfort them with the hopeful\nassurances of Thy word. May they and all of us rightfully know and\nappreciate, whether we live, we live unto the Lord. And whether we die,\nwe die unto the Lord. Therefore whether we live or die we are the Lord's.\nThrough Jesus Christ, our Lord. Amen.\n\n\n\n\n Prayer for the Wounded.\n\n\nLord God, our Heavenly Father! We implore Thy eternal compassion for all\nwho are this day wounded, suffering, or dying. Be Thou nigh unto them in\ntheir affliction. Comfort them with Thy grace and with the hopeful\nassurance, that, though kindred and friends be far away, Thou art ever\npresent and hearest even the faintest sight of all who seek Thy succor.\nIf it be Thy pleasure, restore to them their former health and vigor.\nHelp them to bear their pains without murmuring against Thy grace. Give\nthem patience and strength and faith in Thee. May they rest assured, that\nThou wilt never leave nor forsake them. Deliver them from the assaults of\nthe enemies of their souls. O God and Father in heaven, bless them and\nall of us, and may we ever seek those things that are acceptable to Thy\nsight. Forgive us our sins, not because of any merit or worthiness in us,\nbut solely because Thou art merciful for Christ's sake. And to Thee, the\nFather, Son, and Holy Ghost, be glory and honor forever. Amen.\n\n\n\n\n Prayers for the Dying.\n (St. Paul's Prayer)\n\n\n\"The time of my departure is at hand. I have fought a good fight, I have\nfinished my course, I have kept the faith: henceforth there is laid up\nfor me a crown of righteousness, which the Lord, the righteous judge,\nshall give me at that day: not to me only, but to all of them also that\nlove His appearing.\" 2 Tim. 4:6-8.\n\n * * *\n\nO Lord, on Thy cross Thou didst cry, \"Father, into Thy hands I commend My\nSpirit!\" I, too, commend my spirit into Thy hands now when my end is\nnear. Thou hast redeemed me, O faithful God. Amen.\n\n * * *\n\nLord Almighty God, Heavenly Father! My time has come to an end, my life\nis slowly ebbing away. Be Thou with me. O Lord, I suffer much and Thou\nonly canst help me. Be Thou my succor, and shorten and soften my pain.\nDarkness comes over me and I cry for comfort and strength. Have mercy\nupon me. Take my soul under Thy protecting wing, that I perish not. Take\nmy sins from me, and blot all my guilt; for Christ's sake, my Savior and\nmy Lord. Amen.\n\n\n\n\n Abide with me! fast falls the eventide.\n The darkness deepens: Lord, with me abide!\n When other helpers fail and comforts flee,\n Help of the helpless, O abide with me!\n\n Hold Thou Thy cross before my closing eyes,\n Shine through the gloom, and point me to the skies:\n Heaven's morning breaks, and earth's vain shadows flee:\n In life, in death, O Lord, abide with me!\n\n\n\n\n Prayers for the Sick and Dying\n\n\n\n\n Prayer of a Patient.\n\n\nLord God, Heavenly Father! Thou art a faithful God, and wilt not suffer\nany one to be tempted beyond what he is able, but rather with the\ntemptation wilt also make a way to escape, that he may be able to bear\nit. I supplicate Thee in my great suffering and pain, so shape the cross,\nthat it may not lay too heavily upon me, and strengthen me that I may\nbear it with patience, and nevermore despair of Thy mercy. O Christ, Thou\nSon of the living God! Thou hast endured the agony of the cross for me,\nand hast died for my sins, I beseech Thee with my whole heart, have mercy\nupon me a poor sinner, and forgive me my transgressions, wherever I have\nsinned against Thee. Let my faith in no wise diminish. O God Holy Ghost!\nThou true comforter in all times of need. Keep me ever in the spirit of\npatience and supplication. Sanctify me in my reliance upon Thee. Turn not\nfrom me in the hour of my death, and lead me from this vale of sorrow to\nThyself in heaven. Amen.\n\n\n\n\n Prayer for a Blessed End.\n\n\nO Merciful God, Thou hast put a limit to man's life, which no man can set\naside. For he has his definite time, the number of his months rests with\nThee. Thou hast numbered all our days, which pass away like a stream, as\nthough we flew away. Man is like grass, which soon withereth, like the\nflower of the field, which passeth away. Teach me, O merciful God, to\nknow and take to heart, that I, too, must pass away and that my life has\na limit, and I must go hence. Behold, my days are as a handbreadth before\nThee, and my life as nothing in Thy sight. Every man at his best is\naltogether vanity. Lord, so teach us to number our days, what it is, that\nwe may apply our hearts unto wisdom. Lord, teach me to remember that I\nmust die, and have no continuing city in this pilgrimage. Make known unto\nme my short and transient being, that I may often think of my end, so\nthat in this world I may not live unto myself, but live and die unto\nThee, so that I may bravely and joyously await the day of my translation\nand the appearance of Thy dear Son, Jesus Christ, and with a consecrated\nlife and a pious conduct hasten to Him. Bless me, O God, with a blessed\ndeparture, and when my hour cometh, that I may joyously die, find a\nrational end in true knowledge, and that my reason and intellect be not\nderanged, and I speak no arrogant words or blasphemies against Thee, my\nLord, or against my salvation. Preserve me from an evil sudden death and\nfrom eternal damnation. Let me not be suddenly overcome by my last hour\nwithout warning, but that I may prepare myself with true repentance and\nsincere faith. And when it comes make me joyous and brave for my temporal\ndeath, which only opens the door to the eternal life. May I then, as Thy\nservant, depart in peace. For mine eyes have seen Thy salvation, which\nThou hast prepared before the face of all people; a light to lighten the\nGentiles, and the glory of Thy people Israel. Grant that my last word may\nbe that which Thy dear Son spoke on the cross, \"Father, into Thy hands do\nI commend My spirit!\" And when I can no longer speak, hear Thou my last\nsigh through Jesus Christ. Amen.\n\n\n\n\n Prayer in the Hour of Death.\n\n\nAlmighty, Everlasting, and Most Merciful Lord and God! Thou art the\nFather of our Lord Jesus Christ, and I know that Thou art willing and\nable to fulfill whatsoever Thou hast said. Thou canst not lie. Thy word\nis truth. Thou hast from the beginning promised me Thy dear Son Jesus\nChrist. And He is come, and has redeemed me from the devil, death, hell,\nand sin. And in Thy gracious providence Thou hast for a greater surety\nestablished the sacraments of Holy Baptism and Holy Communion in which\nHis true natural body and blood are given me in, with, and under the\nbread and wine, thus offering and assuring me of the forgiveness of sins,\neternal life, and every heavenly gift. And because of this Thy assurance\nI have made use of these means of grace, and firmly relying upon Thy word\nhave received them. And now I do not doubt but that I am secure from the\npower of the devil, death, hell, and sin. And if now my hour hath come,\nand this is Thy divine will, I will gladly and willingly depart hence and\nbe with Thee in peace. Amen.\n\n\n\n\n Prayer of the Bystanders for the Sick One.\n\n\nAlmighty, Merciful God, who keepest our life in death, we pray Thee, turn\nthe eyes of Thy mercy to this sick person, strengthen him in body and\nsoul, and forgive him all his sins through Thy grace. Take the sacrifice\nof the innocent death of Jesus Christ, Thy dear Son, as a propitiation\nfor his transgressions, since he, too, is baptized in His name, washed\nand cleansed with His blood. Save him from the pains and agonies of his\nbody. Shorten his sufferings. Keep him from the accusations of his\nconscience and all temptations of the enemy, so that in faith he may\ntruly battle and conquer. Grant him a blessed translation to the eternal\nlife. Send Thy holy angels, that they accompany him to the blessed\ncompany of all the elect in Christ Jesus, our Lord. Amen.\n\n\n\n\n Prayer When the Patient Has Died in the Lord.\n\n\nJesus Christ, Thou Lord of glory and Prince of life, we glorify and thank\nThee now and always, that Thou hast so mercifully helped this now blessed\none, and hast taken his soul into Thy holy keeping. And when the last day\ncometh Thou wouldest reunite his soul with his body in heavenly radiance,\nand grant us altogether, when the divinely appointed hour cometh, to\nfollow him in blessedness. Meanwhile grant us a Christian conduct,\nrefresh all sorrowing hearts with heavenly solace. Keep us in Thy eternal\ngrace, for the sake of Thy inexhaustible mercy and goodness. Amen. O Thou\nmost holy, most glorious Trinity. Amen.\n\n\n\n\n Hymns\n\n\n\n\n Du Volk, das du getaufet bist.\n\n\n Ye baptized people, one and all,\n Who know your God in heaven,\n Who have received a holy call,\n To whom Christ's name is given,\n Do not forget, but ponder well,\n The gifts that now within you dwell,\n The blessings of baptism.\n\n This washing cleanseth us from sin\n And lends a sacred beauty,\n It makes us white and pure within,\n Incites to love and duty,\n From Satan's prison sets us free,\n Enables us the sons to be\n And heirs of God, our Father.\n\n Our sinful nature is renewed,\n The curse of God is lifted;\n By choicest blessings thus endued\n And with the Spirit gifted,\n We unto sin are pledged to die\n And by the pow'r of God on high\n The gates of hell can conquer.\n\n Here we with Jesus Christ are clad,\n His righteousness receiving,\n Which covers what in us is bad,\n Our rescue thus achieving;\n His holy blood, for sinners spilt,\n Releases us from sin and guilt,\n And we with God find favor.\n\n O wondrous work, O sacred bath,\n O water thou of blessing,\n The world nowhere thy equal hath,\n Such healing grace possessing.\n Thou hast indeed a pow'r divine,\n According to God's own design,\n And with His word connected.\n\n Thou art no water such as we\n Can draw from well or river.\n In thee the life of God we see,\n Who is of grace the Giver.\n His Holy Spirit in thee dwells,\n Who ev'ry evil lust dispels\n That in our hearts would linger.\n\n O Christians, bear this well in mind,\n And thank the Lord sincerely\n For all the gifts that here you find,\n And that you prize so dearly.\n When nothing else can soothe the soul,\n These gifts lend comfort till the goal\n Of life on earth appeareth.\n\n Use well the things you have in store,\n That are for you intended,\n And, cleansed by Jesus, sin no more\n Until life's course is ended,\n When you, in yonder happy land,\n Before God's radiant throne shall stand,\n In heaven's festal garments.\n\n Paul Gerhardt, 1667.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Ich bin getauft auf deinen Namen.\n\n\n Father, Son, and Holy Spirit,\n I am baptized in Thy name;\n In the seed Thou dost inherit,\n With the people Thou dost claim,\n I am reckoned:\n And for me the Savior came.\n\n Thou receivest me, O Father,\n As a child and heir of Thine;\n Jesus, Thou who diedst, yea, rather\n Ever livest, Thou art mine\n Thou, O Spirit,\n Art my Guide, my Light divine.\n\n I have pledged, and would not falter,\n Truth, obedience, love to Thee;\n I have vows upon Thine alter\n Ever Thine alone to be,\n And for ever\n Sin and all its lusts to flee.\n\n Gracious God, all Thou hast spoken\n In this covenant shall take place;\n But if I, alas! have broken\n These my vows, hide not Thy face;\n And from falling\n O restore me by Thy grace.\n\n Lord, to Thee I now surrender\n All I have, and all I am;\n Make my heart more true and tender,\n Glorify in me Thy Name,\n Let obedience\n To Thy will be all my aim.\n\n Help me in this high endeavor,\n Father, Son, and Holy Ghost!\n Bind my heart to Thee for ever,\n Till I join the heavenly host.\n Living, dying,\n Let me make in Thee my boast.\n\n Johann Jacob Rambach, 1734.\n Tr. Chas. William Schaeffer, 1860.\n\n\n\n\n Wir danken dir, o Jesu Christ.\n\n\n We thank Thee, Jesus Christ, our Lord,\n For all the help Thou dost afford.\n Thou art the Lamb for sinners slain,\n And this is our eternal gain.\n\n Thy holy supper doth prepare\n For us a precious fountain where\n Salvation flows for ev'ry soul,\n To make the wounded spirit whole.\n\n This covenant Thou, Lord, didst make:\n \"Receive my body and partake\n Of mine own blood which flowed for thee,\n That thou a ransomed soul mightst be.\"\n\n I eat Thy body, which for me\n Was made to die upon the tree,\n I drink Thy blood and thus receive\n Life for my soul, as I believe.\n\n And though I die, yet shall I live;\n Thy body and Thy blood will give\n To me eternal life above,\n The fruit of Thy redeeming love.\n\n Let me be numbered with the blest,\n Who on Thy loving bosom rest,\n Thy worthy guest, to sup with Thee\n And praise Thy name eternally.\n\n Nicolaus Selnecker, 1572.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Aus tiefer Not schrei ich zu dir.\n\n\n Out of the depths I cry to Thee,\n Lord, hear me, I implore Thee!\n Bend down Thy gracious ear to me,\n Let my prayer come before Thee!\n If Thou remember each misdeed,\n If each should have its rightful meed,\n Who may abide Thy presence.\n\n Our pardon is Thy gift; Thy love\n And grace alone avail us.\n Our works could ne'er our guilt remove,\n The strictest life must fail us.\n That none may boast himself of aught,\n But own in fear Thy grace hath wrought\n What in him seemeth righteous.\n\n And thus my hope is in the Lord,\n And not in mine own merit;\n I rest upon His faithful word\n To them of contrite spirit.\n That He is merciful and just,--\n Here is my comfort and my trust,\n His help I wait with patience.\n\n And though it tarry till the night,\n And round till morning waken,\n My heart shall ne'er mistrust Thy might,\n Nor count itself forsaken.\n Do thus, O ye of Israel's seed,\n Ye of the Spirit born indeed,\n Wait for your God's appearing.\n\n Though great our sins and sore our woes,\n His grace much more aboundeth;\n His helping love no limit knows,\n Our utmost need it soundeth.\n Our kind and faithful Shepherd, He,\n Who shall at last set Israel free\n From all their sin and sorrow.\n\n Martin Luther, 1524.\n Tr. Catherine Winkworth, 1862.\n\n\n\n\n Eines wuensch ich mir vor allem andern.\n\n\n One thing I above all others cherish,\n For one thing I long and pray.\n Though in sorrow's vale fond hopes may perish,\n This will prove my staff and stay:\n To behold the Man who, deeply sighing\n And upon earth's bosom prostrate lying,\n Drank the bitter cup of woe\n Since the Father willed it so.\n\n Ever shall my soul retain the vision\n Of that Lamb, for sinners slain,\n Pale and wounded, held in deep derision,\n Hanging on the wood of pain,\n Wrestling there in thirst and mortal anguish,\n Lest my soul eternally should languish,\n Being mindful, too, of me\n When He died the world to free.\n\n O Lord Jesus, keep me mindful ever\n Of my guilt and of Thy grace.\n Be it Thou all bonds of sin dost sever\n And all blots of guilt efface.\n Thou of night the brightest daylight makest,\n And Thy lamb into Thy bosom takest,\n As the Shepherd good and fair,\n Nursing it with tender care.\n\n I am Thine! Wilt Thou reply by saying,\n \"Thou forever shalt be mine!\"\n Cause Thy precious name, all fears allaying,\n Brightly in my heart to shine!\n Be with Thee all things begun and ended,\n Who from earth to heaven hast ascended.\n Yea, this shall, till life is spent,\n Be our law and testament.\n\n Albert Knapp, 1829 (1823).\n Tr. H. Brueckner, 1918.\n\n\n\n\n Meine Seel, ermuntre dich.\n\n\n Come, my soul, again inquire\n If the love of Christ constrain thee.\n To His cross again retire;\n See Him give Himself to gain thee.\n Search His faithfulness and try Him,\n Glad in Him and gladdened by Him.\n\n See Him there, God's only Son\n On the tree for thee suspended,\n Crowned with thorns, by grief undone,\n Crimson stains with crimson blended,\n Pierced for thee, transfixed, forsaken:\n Deathless love by death o'ertaken.\n\n Thou, yea thou hadst known the rod,\n Endless pain thy sole possession;\n Thou hadst been cast off by God\n For thy multiplied transgression;\n But the Lord thy cause defended;\n By His grace thou art befriended.\n\n When in mighty woe He died\n Vengeance ceased and wrath abated;\n Sinai was satisfied;\n All things old were new-created;\n Sin and death and hell were thwarted;\n Life and health and heaven imparted.\n\n Jesus, grace sufficient give\n That this mind be ever in me:--\n Thine I am; to Thee would live;\n Naught from Thee shall ever win me.\n Thou wilt not forsake nor leave me;\n Let me, Lord, in love receive Thee.\n\n Johann Caspar Schade, 1692.\n Tr. A. Ramsey.\n\n\n\n\n So nimm denn meine Haende.\n\n\n Take, then, my hands, O Father,\n And lead Thou me\n Until my journey endeth,\n Eternally.\n Alone I will not wander\n One single day.\n Be Thou my true Companion\n And with me stay.\n\n O cover with Thy mercy\n My poor, weak heart!\n Let ev'ry thought rebellious\n From me depart.\n Permit Thy child to linger\n Here at Thy feet\n And blindly trust Thy goodness\n With faith complete.\n\n Though naught of Thy great power\n May move my soul,\n With Thee through night and darkness\n I reach the goal.\n Take, then, my hands, O Father,\n And lead Thou me\n Until my journey endeth,\n Eternally.\n\n Julie von Haussmann.\n Tr. H. Brueckner, 1916.\n\n\n\n\n Jesu, geh voran.\n\n\n Jesus, still lead on,\n Till our rest be won;\n And although the way be cheerless,\n We will follow, calm and fearless,\n Guide us by Thy hand\n To our fatherland.\n\n If the way be drear,\n If the foe be near,\n Let not faithless fears o'ertake us,\n Let not faith and hope forsake us;\n For through many a foe\n To our home we go.\n\n When we seek relief\n From a long-felt grief,\n When temptations come alluring,\n Make us patient and enduring;\n Show us that bright shore\n Where we weep no more.\n\n Jesus, still lead on,\n Till our rest be won;\n Heavenly Leader, still direct us,\n Still support, console, protect us,\n Till we safely stand\n In our fatherland.\n\n Nicolaus Ludwig v. Zinzendorf, 1721.\n Tr. Jane Borthwick, 1846. a.\n\n\n\n\n Befiehl du deine Wege.\n\n\n Commit thy ways and faring\n With all that grieves thy soul,\n To His e'er faithful caring\n Who doth the heavens control;\n Who giving course and highway\n To clouds and winds and breeze,\n Will find for thee a byway\n Where thou canst fare at ease.\n\n But thou must trust Him wholly\n To make thy welfare sure;\n And look to His work solely\n To have thy works endure.\n God lets none snatch a blessing\n By fretful grief and care\n Nor one's own self-distressing;\n It must be got by prayer.\n\n Thy faithfulness unfailing,\n O Father, and Thy grace\n Mark good or ill prevailing\n Amongst this mortal race;\n Then what Thou wilt decreeing\n And pressing on with ease,\n Thou bring'st to pass and being\n Whate'er Thy counsels please.\n\n All ways do Thine possessing,\n Thou lack'st not means nor might.\n Thy deed is purest blessing,\n Thy path serenest light.\n And none Thy work can hinder,\n Nor can Thy labors rest\n Whenever Thou wouldst render\n Thy children what is best.\n\n Though devils all endeavor,\n And fain would work defeat,\n Beyond all doubt soever\n The Lord will not retreat.\n Whatever He proposes,\n Whate'er He takes in hand,\n Whene'er His set time closes\n Will come to pass as planned.\n\n Hope on, poor soul, and bravely;\n Hope on, no more perplexed.\n God from those deeps will save thee\n Where thou by griefs art vexed.\n With grace will He secure thee;\n But wait His time as thine,\n And thou wilt see most surely\n The sun of gladness shine.\n\n Up! Up! Away with grieving.\n Bid anxious fret begone,\n Thy heart its sadness leaving,\n Its woes ne'er dwelt upon.\n Though thou, who dost not govern,\n Canst naught at all compel,\n God sits enthroned and sov'reign\n And doeth all things well.\n\n Let Him, a Ruler knowing,\n Arrange for thee, and do.\n Thou'lt see with wonder growing\n How He will bring thee through;\n And how, by means most seemly,\n By counsels wondrous deep,\n He bears Himself supremely\n In works which thou dost weep.\n\n True, He awhile will tarry,\n His comforting delayed,\n And seem Himself to carry\n As minded not to aid;\n May seem to turn Him from thee,\n Nor e'en, though o'er and o'er\n Thy cares and griefs o'ercome thee\n To ask about thee more.\n\n But though thou seem neglected,\n And yet believing be,\n When least by thee expected\n Will He deliver thee.\n Thy heart will He deliver\n From all its burd'ning woe\n Which thou, unharmed however,\n Hast carried hitherto.\n\n And well with thee thereafter,\n Child of the Faithful, found\n Midst praise and thanks and laughter,\n A conqueror, and crowned.\n The palms in hand before him,\n Which God Himself bestows,\n With glad songs thou'lt adore Him\n Who turned aside thy woes.\n\n Make end, O Lord, good ending\n To all our woes ere long.\n With strength on us attending,\n Our hands and feet make strong.\n Thy faithful care bestowing,\n Till death, our ways attend;\n Then surely, heavenward going\n Our ways in heaven will end.\n\n Paul Gerhardt, 1656.\n Tr. A. Ramsey, 1916.\n\n\n\n\n Alle Menschen muessen sterben.\n\n\n Man is but a transient being\n And a pilgrim here below,\n Like a shadow, swiftly fleeing,\n Like the days that come and go.\n He must pass through death's dark portal,\n To obtain that life immortal\n Which on yonder blissful shore\n Blooms in glory evermore.\n\n So I willingly surrender\n To the Lord my fleeting breath,\n Knowing that His love so tender\n Will allay all fear of death.\n In His wounds there is redemption\n And His death assures exemption\n From the punishment of sin\n And the sense of guilt within.\n\n Jesus died upon the mountain,\n There His precious blood was spilt.\n Now an ever flowing fountain\n Cleanseth me from all my guilt.\n Hence my soul, endowed with pinions,\n Leaves for heaven's fair dominions,\n There in blissful joy to see\n God, the holy Trinity.\n\n In those fair celestial regions\n All is life and peace and joy.\n Souls are there in countless legions,\n Happy in the Lord's employ.\n There bright Seraphim are dwelling,\n Who, in majesty excelling,\n Praise with heaven's mighty host\n Father, Son and Holy Ghost\n\n There the patriarchs are living,\n There the prophets all abide.\n There, to Jesus homage giving,\n His apostles, too, reside.\n There the Lord's whole congregation\n Has a place of habitation.\n There, to honor God, their King,\n All their hallelujahs sing.\n\n O thou city of the sainted,\n Heavenly Jerusalem!\n How thy beauty is untainted,\n Sparkling like a precious gem!\n What soft music there is sounding,\n What sweet joys are there abounding!\n Night shall never follow day,\n But the sun shall reign for aye.\n\n Ah, mine eyes have seen the glory\n Of that city fair and high,\n And mine ears have heard the story\n Of those mansions in the sky.\n Now the pure white robe of heaven\n And a crown to me are given,\n Now that light of joy is mine\n Which shall never cease to shine.\n\n John G. Albinus, 1652.\n Tr. H. Brueckner, 1918.\n\n\n\n\n Jerusalem, du hochgebaute Stadt.\n\n\n Jerusalem, thou city built on high,\n Would God I were in thee!\n My eager spirit hath so yearned to fly\n And bides no more with me.\n O'er hill and valley mounting\n High, high o'er fields and wastes,\n O'er all, all worthless counting,\n Afar from earth it hastes.\n\n O lovely day, O hour more lovely still,\n When wilt thou be revealed,\n When blissfully, my voice with joy athrill,\n My soul to God I yield,\n His pledge ordained, unfailing\n In His all-faithful hand,\n That haven waits my hailing\n In yonder Fatherland?\n\n 'Twill mount in e'en the twinkling of an eye\n Beyond the firmament,\n So wondrously, so gently laying by\n Its outworn earthly tent;\n Midst chariots ascending\n While bright angelic bands,\n Around it safe attending,\n Will bear it in their hands.\n\n All hail to thee, O glorious city, now!\n Lift up thy gates of grace\n How long the time I yearned for thee, and how,\n Before I reached this place\n From yonder life of grieving,\n Of vanity and strife,\n From God at length receiving\n The heritage of life.\n\n What folk is this now drawing near to me,\n This throng of worthies blest?\n These, one time of th'elect on earth, I see;\n The noblest crown and best\n Which Christ, with grace compelling,\n Sent me in other years\n When still far distant dwelling\n In mine own land of tears.\n\n Lo, patriarchs renowned and prophets great\n And gathered saintly folk,\n Who while on earth endured the tyrant's hate\n And wore the Savior's yoke!\n But soaring now in freedom,\n With honor clothed upon,\n Midst radiant light I see them,\n And shining like the sun.\n\n Then when at length I enter in a thrill\n That Paradise of God,\n My every sense with rapturous joy will fill,\n My mouth with praise and laud.\n Hosannas there ascending\n Make holy minstrelsy,\n With Alleluiahs blending\n For all eternity.\n\n With instruments and with a joyful noise\n Are choirs unnumbered singing,\n Till by the sound and shoutings set apoise\n The courts of joy are ringing.\n With ten ten-thousand voices\n And tens of thousands more,\n The heavenly host rejoices\n As ever from of yore.\n\n Johann Matthaeus Meyfart, 1626.\n Tr. A. Ramsey, 1916.\n\n\n\n\n Unter Lilien jener Freuden.\n\n\n Midst the lilies blooming yonder\n Thou shalt wander,\n Oh my soul, and be at home.\n Rise, then, as on eagle-pinions--\n Thy dominions\n Are above where angels roam.\n\n Guide my ship, Thou First-born Brother,\n To no other\n Than that peaceful haven where,\n Sheltered from all storms forever,\n I shall never\n Know of sorrow, sin or care.\n\n Thou canst fill our mouths with laughter,\n And hereafter\n Make our tongues to sing Thy praise.\n Thou canst softly lead us mortals\n Through death's portals\n And above all evils raise.\n\n For our sins and grievous errors\n All the terrors\n Of the cross Thou didst endure.\n Death, thy sting for aye has vanished,\n Thou art banished,\n And I rest from thee secure.\n\n J. L. K. Allendorf, 1731.\n Tr. H. Brueckner, 1917.\n\n\n\n\n Die wir uns allhier beisammen finden.\n\n\n Gathered here, we join our hands, O Savior,\n And upon Thy death unite,\n Ever to be true in our behavior\n Unto Thee, eternal Light.\n And as here Thy praises, Lord, are spoken,\n Enter Thou into our midst in token\n Of Thy love so pure and true,\n Saying, \"Peace I give to you!\"\n\n C. R. Zinzendorf, 1754.\n Tr. H. Brueckner, 1916.\n\n\n\n\n Index\n\n\n\n\n Translator's Preface 3\n Exhortation to Prayer 6\n The Lord's Prayer and Benediction 7\n Morning and Evening Prayers 9\n Prayer for Sunday Morning 11\n Prayer for Sunday Evening 14\n Prayer for Monday Morning 18\n Prayer for Monday Evening 21\n Prayer for Tuesday Morning 25\n Prayer for Tuesday Evening 29\n Prayer for Wednesday Morning 32\n Prayer for Wednesday Evening 35\n Prayer for Thursday Morning 40\n Prayer for Thursday Evening 44\n Prayer for Friday Morning 47\n Prayer for Friday Evening 50\n Prayer for Saturday Morning 52\n Prayer for Saturday Evening 59\n A Daily Prayer 64\n At the Beginning of the Week's Work 65\n At the Table 67\n Grace before Meat 69\n Grace after Meat 70\n For Hearth and Home 73\n Prayer of a Housefather 75\n Prayer of a Housemother 76\n Prayer of a Child 78\n Prayer of a Youth or Maiden 79\n Prayer of a Servant 80\n Prayer of a School Child for the Holy Spirit 82\n Prayer during a Thunder Storm 83\n Thanksgiving after a Thunder Storm 85\n Prayer on the Eve of a Journey 86\n Prayer During a Journey 87\n Prayer of Children for Their Father Engaged on a Journey 89\n Thanksgiving after a Completed Journey 90\n A Birthday Prayer 90\n Prayer for Temporal Peace 92\n A Prayer for School 94\n Church Prayers 97\n Prayer When Going to Church 99\n Prayer for Sincere Repentance 100\n Prayer for Forgiveness of Sins 102\n Prayer for True Faith 104\n Prayer before Confession 107\n Thanksgiving after Absolution 108\n A Morning Prayer for Communion Day 109\n Prayer before Holy Communion 112\n A Sigh when about to Receive the Sacred Body of Christ 114\n A Sigh when about to Receive the Sacred Blood of Christ 115\n Prayer after the Holy Supper 115\n Prayer for a Pious Life 116\n Prayer for Faithful Teachers and Preachers 117\n Prayer for the Kingdom of God 119\n Prayer for Missions 122\n Prayer against False Doctrines and Sects 124\n Prayer against the Enemies of Christendom 126\n Prayers During Times of War 129\n Prayer during War 131\n Prayer for the Army and Navy 132\n Prayer on the Eve of Battle 134\n Prayer for the Wounded 135\n Prayers for the Dying 136\n Prayers for the Sick and Dying 139\n Prayer of a Patient 141\n Prayer for a Blessed End 142\n Prayer in the Hour of Death 144\n Prayer of the Bystanders for the Sick One 145\n Prayer when the Patient has Died in the Lord 146\n Hymns 149\n Abide with grace unbounded 20\n Abide with me! Fast falls the eventide 138\n Awake, my heart, rejoicing 27\n Awake, my soul, the rising sun 42\n Be present at our table Lord 70\n Come, my soul, again inquire 158\n Commit thy ways and faring 161\n Dayspring of Eternity 34\n Father, Son, and Holy Spirit 153\n Gathered here, we join our hands, O Savior 171\n God, who madest earth and heaven 13\n Heavenly Father, bless this food 70\n Jerusalem, thou city built on high 167\n Jesus, bless what Thou hast given 70\n Jesus, still lead on 160\n Man is but a transient being 165\n Midst the lilies blooming yonder 170\n Now God be with us, for the night is closing 31\n Now in Jesus' wounds reposing 46\n Now under night's dark shadow 23\n O Christ, who art the sun-lit day 16\n One thing I above all others cherish 157\n Out of the depths I cry to Thee 155\n Praise God from whom all blessings flow 71\n Shepherd never sleeping 37\n Soul of mine to God awaking 62\n Sunk is the sun's last beam of light 52\n Take, then, my hands, O Father 159\n The morning sun shines in the skies 49\n The sun ascending 55\n We thank Thee, Jesus Christ, our Lord 154\n We thank Thee, Lord, for this our food 71\n Ye baptized people one and all 151\n Index of Translated Hymns\n Ach bleib mit deiner Gnade 20\n Alle Menschen mueszen sterben 165\n Aus tiefer Not schrei ich zu dir 155\n Befiehl du deine Wege 161\n Christe, du bist der helle Tag 16\n Die gueldne Sonne 55\n Die helle Sonn leucht jetzt herfuer 49\n Die Nacht ist kommen, drin wir ruhen sollen 31\n Die wir uns allhier beisammen finden 171\n Du Volk, das du getaufet bist 151\n Eines wuensch ich mir vor allem andern 157\n Gott des Himmels und der Erden 13\n Hinunter ist der Sonnenschein 52\n Hirte deiner Schafe 37\n Ich bin getauft auf deinen Namen 153\n Jerusalem, du hochgebaute Stadt 167\n Jesu, geh voran 160\n Meine Seel, ermuntre dich 158\n Morgenglanz der Ewigkeit 34\n Nun ruhen alle Waelder 23\n Nur in Jesu Blut und Wunden 46\n So nimm denn meine Haende 159\n Unter Lilien jener Freuden 170\n Wach auf mein Herz, die Nacht ist hin 42\n Wach auf mein Herz, und singe 27\n Werde munter, mein Gemuete 62\n Wir danken dir, Jesu Christ 154\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Morning and Evening Prayers for All\nDays of the Week, by John Habermann\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzsckd b/data_all_eng_slimpj/shuffled/split2/finalzzsckd new file mode 100644 index 0000000000000000000000000000000000000000..0e208a84e1c6d10d45829f04deac615a118866ec --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzsckd @@ -0,0 +1,5 @@ +{"text":"\n\n\n\nProduced by Brendan OConnor, Jonathan Ingram, Louise Pryor\nand the Online Distributed Proofreading Team at\nhttp:\/\/www.pgdp.net (This file was produced from images\ngenerously made available by The Internet Library of Early\nJournals.)\n\n\n\n\n\n{Transcriber's note: Spellings are sometimes erratic. A few obvious\nmisprints have been corrected, but in general the original spelling has\nbeen retained. Accents in the French phrases are inconsistent, and have\nnot been standardised. Greek phrases have been transliterated, and are\nenclosed in + signs +autochthones gaias+.}\n\n\nBLACKWOOD'S EDINBURGH MAGAZINE.\n\nNo. CCCXLVII. SEPTEMBER, 1844. VOL. LVI.\n\n\n\n\nCONTENTS.\n\n M. LOUIS BLANC, 265\n\n A NIGHT ON THE BANKS OF THE TENNESSEE, 278\n\n THE EXECUTION OF MONTROSE, 289\n\n THE WITCHFINDER. PART I., 297\n\n NATURAL HISTORY OF MAN, 312\n\n POEMS BY COVENTRY PATMORE, 331\n\n MARSTON; OR, THE MEMOIRS OF A STATESMAN. PART XIII., 343\n\n IT IS NO FICTION, 364\n\n THE BURNS' FESTIVAL, 370\n\n STANZAS FOR THE BURNS' FESTIVAL. BY DELTA, 399\n\n * * * * *\n\n EDINBURGH:\n WILLIAM BLACKWOOD AND SONS, 45, GEORGE STREET;\n AND 22, PALL-MALL, LONDON.\n\n_To whom all Communications (post paid) must be addressed._\n\nSOLD BY ALL THE BOOKSELLERS IN THE UNITED KINGDOM.\n\nPRINTED BY BALLANTYNE AND HUGHES, EDINBURGH.\n\n\n\n\nBLACKWOOD'S EDINBURGH MAGAZINE.\n\nNo. CCCXLVII. SEPTEMBER, 1844. VOL. LVI.\n\n\n\n\nM. LOUIS BLANC{A}\n\n\nM. Louis Blanc, a democratic journalist, with all, and perhaps more than\nthe usual talents of the Parisian journalist--with all, and more than\nthe usual faults of one--has undertaken to write the history of his\ncountry, during and since the revolution of 1830. What can we expect to\nbe the result of such an undertaking? What can we expect from a man who\nsits down to a task of this description, animated with all the party\nvirulence which gives zest to a democratic newspaper? It is not a\nhistory, but a scandal, that he will write. M. Louis Blanc has distilled\nthe bile of journalism; he has paused over the hasty sarcasm which\npolitical animosity deals forth, not to correct, or moderate, or abate,\nbut merely to point and envenom it. His appreciation of men, their\ncharacter, their talents, their designs--all bear the hue of the\natrabilious journalist. There is this difference only between his\nhistory and the daily portion of envy and malignity which a democratic\nnewspaper pours forth, that the dye is more deeply engrained. In the\nmind of the author, the stain of his party has become ineffaceable.\nThose who are pleased--and the number is not few--with having high names\nand established reputations laid at their feet, soiled, trod upon, will\nmeet here with ample gratification. To be sure they will be occasionally\nrequired, in lieu of such as they have thrown down, to set up the bust\nof some democratic celebrity, whose greatness, or whose genius, they\nwere not previously aware of. But, not to say that the justice of party\nrequires this substitution, it is a penalty which writers of this\ndescription will invariably impose upon them. It is the common trick of\nthe envious, and the mock magnanimity with which they seek to conceal\ntheir true nature--to exalt the lowly, while they debase the exalted.\nSince some idol there must be, let it be one of their raising. Even\nwhile helping to raise it, they enjoy, too, the secret consciousness\nthat it is of brittle metal.\n\nBut in the composition of a history, the spirit of party, however eager\nit may be, cannot always guide the pen. The mere interest of the\nnarrative, the strangeness and peculiarity of circumstances, will claim\ntheir share of the author's mind. The politician must sometimes be\nabsorbed in the chronicler; and so it happens with M. Louis Blanc. His\nnarrative often interests by its details; and if it has the partiality,\nit has also the vivacious colouring, of a contemporary. It possesses,\nalso, a richness of anecdote--the fruit, probably, of his position as a\njournalist; add to which, that M. Louis Blanc is not without a species\nof off-hand, dashing eloquence. He can say daring things in a daring\nmanner, and give the pungency of epigram to his political paradoxes. He\nhas a full share of that rhetoric of journalism which is so well\ncalculated to make an impression on the careless reader, but which\nrequires that the reader should continue careless, in order to retain\nthe impression he has received. It results from all this, that while we\nconstantly distrust our guide, while we perpetually refuse the\nappreciation he offers to us of men and events, we still read on with\ninterest a work which is, at least, relieved from the charge of\ninsipidity or dulness; and indeed, if we had not derived some\nentertainment from its perusal, we should not have thought of bringing\nit under the notice of our readers. To have engaged ourselves merely in\ncombating its errors and misrepresentations, would have been a dreary\nand an endless task.\n\nTo enable the reader at once to judge of the tone and temper of M. Louis\nBlanc's politics, we present him the following passage. It is the object\nof the long Introduction which precedes his history, to show that the\nevents which have transpired in France since 1793, have had, for their\ngreat result, the establishment of the government of the middle classes\nthrough a Chamber of Deputies--a view which we think is incontestably\nright. That France has its House of Commons, is the great fruit of all\nits struggles, its calamities, and its victories. It must not be\nsupposed, however, that this is a result in which M. Louis Blanc\nrejoices. Nothing he so much detests as this government of the middle\nclasses; nor is there any portion of society he vilifies more cordially\nthan the _bourgeoisie_. Hear how he speaks of them. After relating the\nhistory of the Carbonari, who troubled by their plots the reign of Louis\nXVIII., he says:--\"This _Carbonarism_ never descended into the depths of\nsociety; it never moved the lower strata. How, then, could it be\npreserved from the vices of the middle class--egoism, littleness of\nideas, extreme love of a mere material happiness, gross instincts!\"--(P.\n115.) So that he finds Carbonarism to have lacked in virtue, because it\nhad not descended, for its disciples, sufficiently low in the scale of\nsociety!--to have grown corrupt, by reason of its not having penetrated\nto the \"lower strata!\" And yet the duties of the Carbonari seem to have\nbeen precisely calculated for these lower strata. These were, he had\nalready told us himself, \"to have a gun and fifty cartridges, to be\nready to devote one's-self, and to obey blindly the orders of unknown\nleaders.\"--(P. 101.)\n\nWhen we describe M. Louis Blanc as a democrat, it is rather for want of\na better and more accurate title, than because this exactly describes\nhim. A democrat is generally understood to be one who has a large faith\nin the lowest class of the people, such as they really exist; our author\nhas a faith only in the future of this class. He does not fail to give\nvent, when the occasion prompts him, to his compassion or contempt for\nthe ignorant mass of mankind. The democracy he worships is one to be\nestablished in some distant age, by a people very different, and living\nunder some modification of the law of property, which he has not thought\nfit to explain. It is a democracy which has nothing distinct but its\nhatreds--a shadowy monster, peculiarly disagreeable to deal with. Our\nhistorian writes with overflowing gall against kings, against\naristocracies, against the middle class. You would say he is a stanch\nrepublican, and that the people are to be his depositaries of power. But\nno; a lamentation, which escapes him from time to time--as bitter as any\nwhich Tory or Legitimist would utter--over the _blindness_ of the\npeople, their passions and their ignorance, contradicts this conclusion.\nWhere is the power, and in whom lodged, that M. Louis Blanc would\nwillingly obey, or see obeyed? It exists nowhere. Society is corrupt, is\nchaotic; nor can it, by any organ it possesses, exercise a sound or\nrational power. A new era must arise--how, whence, when, we are not\ninstructed.\n\nIt is the peculiar characteristic of French democracy, that there is\nalways mixed up with it the principle, more or less distinctly avowed, of\nthe community of goods. Perhaps the vagueness we complain of in M. Louis\nBlanc, is dictated by mere prudence; perhaps there is no vagueness to the\neye of a propagandist. One sentiment of French democracy he certainly\nexpresses with sufficient hardihood. It is not often we meet with the\nprinciple of intervention between state and state, asserted in these days\nwith so much boldness as in the following passage:--\"Men have stigmatized\nthe war in Spain, calling the principle of intervention an oppressive\nprinciple. Puerile accusation! All people are brothers, and all\nrevolutions cosmopolite. When a government believes that it represents a\njust cause, let it make it triumph wherever a triumph is possible. This\nis its right; it is more--it is its duty.\"--(_Introduction_, p. 120.)\n\nHow exactly analogous to this is the reasoning which leads to\npersecution in religion--to the Holy Inquisition, and all its\nphilanthropic schemes _of intervention!_ The conviction in a good cause\nallowed to overrule the fundamental principles of justice between man\nand man--to overrule them, not occasionally and by way of exception, but\nsystematically--this is the very essence of persecution. But let no one\nthink that, by any such representation, he would gain an advantage over\nthe republican propagandist. He no longer fears religious\npersecution--it is a thing past: he braves it. He would adopt his\nfavourite principle, and all its consequences. He would probably admit\nthat it was the duty of the priest, according to his priestly\nintelligence, to ban and persecute. Not mutual toleration, but\nreciprocal compulsion, would be his principle. Combat thou for thy\ntruth--let me fight for mine; such would be his formula.\n\nIn a writer bent upon startling and surprising us, there is often a sort\nof premeditated haste, a voluntary forgetfulness, which it is curious to\nremark. One who weighs his matter well before he speaks, will often end,\nalas! in having something very tame and moderate to propound--something\nwhich, after all his turmoil and reflection, may sound very like a good\nold commonplace. Now this approximation to commonplace is the great\nhorror of shallow writers; and the way to avoid it appears to be\nthis:--Proclaim your thought at once, in all its crude candescence,\nbefore it has had time to cool and shape itself; then, in order to save\nyour credit with the more captions and scrutinizing, give, at some\nconvenient interval, such an explanation or modification as will show\nthat, after all, you were as wise as your reader. State your paradox in\nall the startling force of unmitigated diction, and refute it yourself\nafterwards, or say enough to prove that you could have done so. This,\nwell managed, gives two occasions for brilliant display; a sober\nstatement has been converted into a couple of bold and glancing\npropositions. Truth, it is well proved, like the diamond, shines the\nmore by being cut into surfaces.\n\nM. Louis Blanc, for instance, makes a startling remark on the\nincompatibility of royalty and a representative chamber. The two powers\nare represented to us as flatly irreconcilable. \"Can society,\" he asks,\n\"have two heads? Is the sovereignty divisible? Between the government of\na king and the government of an assembly, is there not a gulf which\nevery day makes wider? And wherever this dualism exists, are not the\npeople condemned to fluctuate miserably between a 10th of August and an\n18th Brumaire?\"--(_Int._, p. 64.) And a little further on, speaking of\nthe times of Louis XVIII., he writes--\"Meanwhile Europe began to be\ndisquieted on the state of things in France. Foreign sovereigns had\nthought to establish peace in our country, by establishing the empire of\nthe charter, and the political dualism which it consecrates. The error\nwas great, and they ended by discovering it. M. de Richelieu, who had\nbeen present at the congress at Aix-la-Chapelle, brought back with him a\nvery lively apprehension of the future fate of the monarchy in France. A\nchange of the electoral law was proposed. Unhappily, it was not in the\nlaw of the 5th February that lay the danger which occupied the congress\nof Aix-la-Chapelle. To consolidate the throne, and raise it above the\nstorms which threatened it, not this or that electoral law, but the\nelectoral power itself, should, if possible, be abolished. For in\nwhatever hands this formidable lever was placed, it was impossible that\nroyalty could long resist its action. To shift the elective power was\nonly to give the monarchy other enemies, not to save it. * * * The aim\nof the new ministry was to preserve the electoral law; which amounted to\nthis--the monarchy chose ministers whose programme was the destruction\nof monarchy.\"\n\nOn reading such passages, we naturally set about recalling certain\nold-fashioned political truisms, bearing on the character and interest\nof that middle class of society in which the electoral power is\ngenerally lodged. We recollect that the middle classes have been held to\nhave an interest as well in preserving, as in checking and controlling\nthe monarchy. Alone, they could not govern society; and they have a\nlarger share in the government, as partners with the monarchy, than if\nthey were absorbed in the general mass of the population. They have\nevery thing to lose by the abolition of a royalty which they have ceased\nto fear, and which they have bound by laws. Such a royalty, with its\nsway over the imagination of the multitude, with its strong hand of\nmilitary power--hand in which the sword is allowed always to rest, as\npomp in time of peace, as weapon in time of war--such a royalty they\nfeel to be their best protection. Why, then, should they, in their\nelectoral capacity, be thrust on by a blind rage to destroy it? But all\nthis train of reflection we might have spared ourselves. M. Louis Blanc\nknows it all, and, if you will wait a reasonable time, he will show you\nthat he knows it. He will put it to you very forcibly--in another place.\nAccordingly, some ninety pages off, he tells us:--\"At bottom, the middle\nclass (_la bourgeoisie_) sees in the monarchy a permanent obstacle to\ndemocratic aspirations: it would have subjected royalty, but not\ndestroyed it.\"\n\nFor the enlightenment of those who may wish to write history in the most\ncaptivating manner, and at the least possible expense to themselves, we\nwill reveal another fruitful expedient. There are two ways of writing\nhistory. You may either deduce its great events from certain wide and\nsteadily-operating causes, as the growing wealth or intelligence of a\npeople, or you may raise a vulgar wonder by describing them as the\nresult of some quite trivial incident. In the one case, you appeal to a\nphilosophic taste; in the other, to a popular love of the marvellous. A\nrevolution may be represented as the inevitable outbreak of the\ndiscontent and misery of the people; or it may be traced, with all its\ndisasters, to the caprice of a courtier, or perhaps the accidental delay\nof a messenger. For want of a nail, the shoe was lost; for want of a\nshoe, the horse was lost; for want of a horse, the man--and so all was\nowing to the want of a nail!\n\nThe two manners seem incompatible. Never mind. Use them both--both\nfreely, independently--just as occasion prompts, and the effect\nrequires. Flatter the philosophic taste that delights in generalities,\nand please the childish wonder which loves to fancy that the whole\noak--trunk, branches, leaves--lay in the acorn. M. Louis Blanc has\ncertainly no idea of forfeiting either of these attractions by laying\nclaim to the other. Observe the ease and boldness with which he embraces\nthem in his narrative of the fall of Napoleon, and the restoration of\nthe Bourbons. He commences in the generalizing mood.\n\n\"The fall of Napoleon lay in the laws of the development of the middle\nclasses. Can a nation be at the same time essentially commercial and\nessentially warlike? Napoleon must have renounced his great part of\nmilitary chieftain, or he must have broken with the spirit of\ncitizenship and commerce. It was madness to think of reigning by the\nsword, and continuing the Constituent Assembly. France could not have,\nat the same time, the destinies of Rome and Carthage. Napoleon\nsuccumbed, and must have succumbed, to the Carthaginian party of the\npeople of France. But if the necessary development of the middle classes\ncalled for the overthrow of the empire, it demanded also the return of\nthe Bourbons. To prove this, we have only to present, in its instructive\nsimplicity of detail, that narrative of the restoration which so many\nhistorians have distorted.\"--(_Int._, p. 18.)\n\nWell, he proceeds with this simple and instructive detail; and his\nfirst object is evidently to deprive Talleyrand, to whom on all\noccasions he manifests a singular bitterness, of the credit generally\ngiven him of having aided materially in the recall of the Bourbons in\n1814. But does he effect this by showing, as from this exordium we might\nexpect, that his countrymen of the middle class, wearied of the costly\ntriumphs and disasters of the empire, had begun to sigh for peace and\ntheir old kings? Not at all. He transfers the personal share in the\ndrama from Prince Talleyrand to Baron de Vitrolles. The Duke d'Alberg\nhad introduced the baron to Talleyrand, whose intention was to employ\nhim merely to sound the views of the Allies. Talleyrand was to have\naccredited him by some lines of his own writing, but ultimately refused\nto commit himself. How was Baron de Vitrolles, who by no means limited\nhimself to the subordinate part designed for him, and on whom it will be\nseen so much really depended, to get accredited to the Allies?\n\nThe Duke d'Alberg was intimately acquainted with the Count de Stadion,\nrepresentative of Austria at the congress. Now these two friends had\nformerly, at Munich, had a certain tender intimacy with two young girls,\nwhose names the Duke d'Alberg remembered; he wrote them on the leaf of a\npocket-book, and they served as a letter of credence to the adventurous\nambassador. \"Such,\" exclaims our lately generalizing historian--\"such is\nthe manner in which God disposes of the fate of nations!--_Voila de\nquelle sorte il plait a Dieu de disposer du sort des peuples!_\"\n\nThe Baron de Vitrolles, we are told, found the Emperor Alexander\npossessed with a strong repugnance against the Bourbons. It cost him\nthree hours' conversation to gain him over. But he succeeded. It was he\nwho did gain him over. On the 31st of March, when the Emperor of Russia\nentered Paris, Talleyrand stepped forward to receive him.\n\n\"Well,\" said Alexander, \"it seems that France recalls the Bourbons.\"\n\nThese words occasioned M. Talleyrand a profound surprise, which,\nhowever, he was too skilful a diplomatist to betray. From that moment,\nhe was a convert to what he considered the successful cause. \"Thus,\"\ncontinues our historian, \"this restoration took place contrary to the\nwill of the people, to whom the Bourbons in 1814 were unknown; contrary\nto the sympathies of Alexander, who feared the dangers of a reaction;\ncontrary, in fine, to the opinion of M. Talleyrand, who had never\nthought it possible, and had desired only the regency of _Marie\nLouise_!\"\n\nWhat particle of truth there may be in this narrative, we do not stop to\nenquire; we refer to it only as an example of the bold union of the two\nhistoric manners. The restoration of the Bourbons was \"in the laws of\nthe development of the middle classes!\" It was all owing to the Baron de\nVitrolles, and that lucky little intrigue at Munich!\n\nIt is one of the boasts and privileges of history to reverse the\njudgment that contemporaries have formed of the character of the actors\nin it. This privilege M. Louis Blanc, since he writes history, is\ndetermined at all events to seize upon; and he can boast, perhaps, of\nhaving reversed more judgments of this kind than any other historian,\nhowever voluminous. M. Talleyrand has obtained his reputation for\nability--his moral reputation it would be too commonplace a matter to\nattack--by \"speaking in monosyllables one half his life, and saying\nnothing the other half.\" M. Guizot is a man \"whose talent consists in\nconcealing, under the solemnity of expression and the pomp of _formulae_,\nan extreme poverty of views, and sentiments without grandeur.\" M. Dupin,\nthe elder, is \"skilful in concealing, under an affectation of rudeness,\nthe pusillanimity of his heart.\" Cuvier, whose scientific reputation is\nuntouched, probably because no motive led him to assail it, is \"_homme\nplus grand par l'intelligence que par le coeur_.\" Of Metternich he\nwrites--\"A lover of repose from selfishness, he sought it also from\nincapacity. He wished to enjoy a reputation easily usurped, the\nfalsehood of which the least complication of events would have exposed.\"\nAnd the picture he gives throughout of Casimir Perier is that of an\n\"illustrious charlatan,\" in whom nothing was genuine but his pride, his\nhate, and his physical infirmities.\n\nThe ministers of Charles X. meet with a much fairer appreciation than\nthose of Louis Philippe. Towards them, one might even say that he is\nindulgent. This is easily accounted for: in the war of party, those with\nwhom we come into the closest and most frequent collision, must, of\ncourse, excite the largest share of our animosity. M. de Polignac seems\nto have been aware that he had little to fear from the fierce democrat:\nhe has not disdained a sort of literary participation in the work,\nhaving contributed some manuscript notes of his own, explanatory of his\nshare in the transactions of 1830. Altogether, we may presume that the\nhistory, so far as it relates to the ministers of Charles X., is not\nunfairly written. Let us approach the narrative by this quarter.\n\nIt is a singular picture that M. de Polignac presents to the\nimagination, with his unruffled serenity, his extreme audacity, his\nviolent measures, his negligent preparation, his strong will, his weak\nintelligence. The minister is always smiling, and, in the midst of\ndisaster and ruin, is still beaming with self-confidence; he seems to\nhave thought that self-confidence wrought like magic, or like faith, and\ncould of itself remove mountains. If difficulties occurred, his resource\nwas to be still more self-confident. He was well aware of the hostility\nhis ordonnances would create; he was well aware that the army must be\ntheir veritable support: yet observe with what a sublime air of\nnonchalance he prepares himself for the subjection of a people. \"How\nmany men,\" asked M. d'Haussez, as the ministers sat round the\ncouncil-table, \"can you reckon on at Paris?--have you twenty-eight or\nthirty thousand?\" \"More,\" said the premier; \"I have forty-two thousand;\"\nand, rolling up a paper which he held in his hands, he threw it across\nthe table to d'Haussez. \"But,\" said the latter, as he looked over the\nstatement that had been given to him, \"I see here only thirteen\nthousand. Thirteen thousand men on paper--that amounts to about seven or\neight thousand actually ready to fight your battles. And the other\ntwenty-nine thousand to complete your number, where are they?\" M. de\nPolignac assured him that they were spread about the neighbourhood of\nParis, and in ten hours, if it were necessary, could be assembled in the\ncapital. The ministers felt, adds our historian, that they were entering\ninto a dreadful game blindfold.\n\nM. de Polignac appears to have relied upon the army, much in the same\nway that a speculative writer, theorizing upon government, rests upon\nhis great abstraction, the military power. He treated it as if it were a\nprinciple, an idea, that developed itself without his aid, and not a\npalpable fact of there being a certain number of armed men, then and\nthere, to fight for his ordonnances.\n\nThere is no virtue so much applauded in the present day as\nresolution--_will_; and there are who regard a strong will as the\nessence of all virtue. But the history of M. de Polignac proves, (if\nthis needed proof,) that the weak can have will enough. Your strong will\nmay be purchased at the sole expense of reason. Let there be one idea in\na brain that cannot hold two, and you have a strong will. M. de Polignac\nnever wavered once; he was always seen with a smiling countenance, calm,\nradiant with hope and self-approval. When others around him began to\ndespond, when the Duke of Ragusa, commander of the forces, writing to\nthe king, said that it was not a riot, but a revolution, and advised him\nto retreat while he could still retreat with honour, the minister had,\nfor all answer, but one word--\"Fire!\" It was still, Fire! But what if\nthe troops, it was asked, desert to the people? \"Then fire on the\ntroops!\"\n\nOn the publication of the ordonnances, the members of the Chamber who\nwere in Paris met at each others' houses to discuss measures of\nresistance. But it was not from the members of the Chamber that the\nmovement was to emanate. Those who had any position to compromise looked\non, for the most part, with anxiety and astonishment, waiting to see\nwhat current the disturbed waters would finally take. \"On the evening of\nthe 27th, a man, name unknown, appeared on the Quai d'Ecole, and\nparaded the banks of the river with the tri- flag, which had\nbeen folded up and hidden away for fifteen years.\" The symbol was\nadopted by the people. The revolution had commenced.\n\nThen followed all those strange scenes of levity and blood, buffoonery\nand heroism, which the history of Parisian revolutions has familiarized\nto the imagination, but which, nevertheless, have an inexhaustible\ninterest. The people arm themselves wheresoever and howsoever they can.\nOne brings into the Place de la Bourse two large hampers, full of\nmuskets and accoutrements. They come from the Theatre du Vaudeville,\nwhere a piece had been played, a few days before, which required that a\nnumber of actors should be armed. To command men thus equipped there\nwere extemporary generals, whose epaulets were obtained from the\nwardrobe of the Opera Comique. The students of the Polytechnic were, as\nusual, on the alert to practise whatever they had learned of military\nscience; the younger sort entering into the war with the same spirit\nthat other schoolboys partake of any minor mischief that is going\nforward. A student of the Polytechnic is standing on the left bank of\nthe river; he has a musket, but no ammunition. A fellow-student, a lad\nof fifteen, has a packet of cartridges, but no musket: \"You shall share\nthem,\" said he, showing his treasure, \"if you will lend me the gun to\nshoot my half.\" A party of the royal guard were coming over the bridge.\nHe started with the gun _to have his shots_. He was swept off with\nothers by the fire of the military.\n\nOn one side comes a party led by a violin, women applauding. But the\nwomen do more than applaud. They carry great paving-stones to the top of\nthe house, to be thence precipitated on the heads of the soldiers; they\ntend the wounded, they bruise charcoal for gunpowder.\n\nThere was, no doubt, some severe fighting during the Three Days; but,\ngenerally speaking, the military seem to have entered into the contest\nwith reluctance. Some instances are here given of singular forbearance\non their part. At a time when, in certain quarters of Paris, each house\nwas converted into a sort of fortress whence the military was assailed,\nthree men had placed themselves behind a stack of chimneys, and had,\nfrom this shelter, directed a destructive fire on the troops. They were\nat length discovered, and a cannon was levelled against the chimney.\nBut, before firing, the gunner made signal to the men to escape,\ncontenting himself with demolishing their breastwork. As another company\nof soldiers, led by its officer, was marching through the streets, one\nof the mob rushed forward, and, with a mad audacity, struck the officer\non the head with a bar of iron. He staggered, and his face overflowed\nwith blood; but he still had strength enough to raise his sword to put\naside the muskets of his men, who were in the act of firing on the\nassailant.\n\nWe have here a vivid description of the taking of the Tuileries by the\npopulace. Some amused themselves by mutilating the statues of kings, or\nby firing at the portraits of such of the marshals as they considered to\nhave been guilty of treason to Napoleon. A number of artisans installed\nthemselves in the chamber of the throne; they sat, each in his turn,\nupon the royal seat, afterwards they placed a corpse in it. Some of them\ndrew, over shirts stained with blood, the court-dresses which had\ncircled the waist of royal princesses, and strutted about in this\nmasquerade. Riot and destruction as much as you please, but no\ntheft--such was the order of the day. A young man was bearing off a hat,\ndecorated with plumes of a costly description. \"Where are you going,\"\ncried his companions, \"with that hat?\" \"It is only a souvenir,\" said he\nof the hat. \"Ha! good; but in that case the value is nothing.\" So\nsaying, they took the hat and trampled it under their feet, and then\nreturned it to him--doubly valuable as a souvenir. Many striking traits\nof honesty were exhibited. One man brought a vase of silver to the\nprefect of police, and did not even leave his name. Another found a bag\nof three thousand francs in the Louvre, and hastened with the money to\nthe Commune. The next day he was probably amongst the number who were\nwandering about Paris without bread and without work, driven out of\nemployment by the commercial panic of their own glorious revolution.\n\nA scene of a like grotesque description took place, at a later period,\non the return of the mob from Rambouillet, where they had gone in search\nof the unhappy Charles X. The king had left Rambouillet before the mob\nreached it, so that they had nothing to do but to return, unless any\nwork of demolition should invite them to stay. M. Degoussee, at that\nmoment the man in authority, in order to save the royal carriages from\ndestruction, bethought him of the expedient of offering a ride home in\nthem to the most violent and redoubtable of the mob. In a moment these\ngilded vehicles, blazoned with the royal arms, were filled with the\nlowest of the rabble, who projected their pipes and their bayonets from\nthe windows. These state carriages, drawn by eight horses, and driven by\nsilken postilions, were heaped up, inside and out, with this riotous\ncrew, who entered Paris in triumph, amidst the responsive jests and\nshouts of the populace. Driven up to the Palais Royal, they there\ndescended from their splendid vehicles, and delivered them over to their\nnew owner. \"_Tenez--voila vos voitures!_\" they shouted, as they alighted\nunder the windows of the Duke of Orleans.\n\nIt is curious to remark the contrast between the thoughtless, reckless\nbravery of the combatants of July, and the watchful timidity of the\npoliticians who were ultimately to profit by their courage and\ninfatuation. The soldiers had, at many points, fraternized with the\npeople--all was success for the popular party--and the drawing-room of\nM. Lafitte was full of distinguished men of that party.\n\n\"The court of the hotel,\" continues M. Blanc, \"was now full of soldiers.\nFive of the royal officers entered the saloon. M. Lafitte, who had been\nwounded in the leg, received them sitting in an arm-chair. He received\nthem with great blandness and dignity. 'Gentlemen,' said he, 'keep your\narms, but swear never to turn them against the people.' The officers\nextended their hands, as if to take an oath. 'No oath, gentlemen,' said\nM. Lafitte with much emotion; 'kings have dishonoured oaths. The word of\nthe brave is sufficient.' This was received with universal applause, and\nevery one present resigned himself to the excitement of the hour; when\nsuddenly a discharge of musketry was heard. How describe the tumult that\nin a moment filled the apartment! The royal guard was certainly\nvictorious--the enemy would be down on them--every one fled. They rushed\ninto the hall, they pushed, they struggled for egress. Some jumped\nthrough the windows of the ground-floor into the garden. Two deputies\nwere found hiding in the stables. In an instant, M. Lafitte was\nabandoned by all those who had besieged his arm-chair. His nephew was\nthe only person who remained with him. And what had happened? The\nsoldiers of the 6th had followed the example of their comrades of the\n55th, and, gained over to the cause of the people, they had fired their\nmuskets in the air!\"\n\nAlready, at the first outbreak of the revolution, some one had\nremarked--\"here were a good game for the Duke of Orleans, if he has the\ncourage to play it.\" Courage he had, but equal caution it seems, equal\nprudence. A deputation had proceeded from the house of Lafitte to\nNeuilly, the residence of the Duke, to invite him to the throne; but it\nwas the Duchess who received them. The Duke himself had taken refuge at\nRaincy. To Raincy messengers were sent. The Duke of Orleans ordered his\ncarriage. Those who were waiting his arrival at Neuilly heard the wheels\napproach--heard them suddenly recede. The carriage had turned, and was\nregaining Raincy with all the speed possible. The resolution was not\nquite taken, or the pear was not quite ripe.\n\nHis entry into Paris, according to M. Blanc, was made on foot in the\nevening, and he clambered like a common citizen over the barricades.\nArrived at the Palais Royal, he sent to notify his presence to Lafitte\nand Lafayette--representatives, the one of the Chamber, and the other of\nthe Hotel de Ville--and also to the Duke de Mortemart, minister of\nCharles X. The interview with this last took place the same evening, and\nhad for its apparent object to proclaim, in the presence of the\nminister, his attachment and unalterable fidelity to the elder branch of\nthe Bourbons. When De Mortemart arrived, he was ushered into a little\ncabinet on the right, which looks upon the court, not ordinarily used as\nan apartment of the family.\n\nThe Duke of Orleans was stretched upon the floor, lying on a mattrass,\nin his shirt. His forehead was bathed in sweat;{B} the glare of his\neyes, and every thing about him, betrayed a great fatigue, and a\nsingular state of excitement. On seeing the Duke de Mortemart enter, he\nbegan to speak with great rapidity. He expressed himself with much\nvolubility and ardour, proclaiming his attachment to the elder branch,\nand protesting that he came to Paris only to save the town from anarchy.\nAt this moment a great noise was heard in the court, and the cry was\nraised of _Vive le Duc d'Orleans!_ \"You hear that cry,\" said the\nminister; \"it is you the people call for.\" \"No, no!\" answered the Duke\nwith increasing energy. \"They shall kill me before I accept the crown.\"\n\nThe next morning the deputation from the Chamber presented itself at the\nPalais Royal; and so far was resolved, that the Duke of Orleans was\nproclaimed lieutenant-general of the kingdom.\n\nM. Louis Blanc gives several anecdotes respecting the King of the\nFrench, and his successive ministers, which we should be disposed to\nextract, but that his political antipathies lying exactly in this\nquarter, we have not felt sufficient confidence in their authority. For\nthis reason we will pass on abruptly to a portion of the work where the\npolitical bias of the writer is harmless, or where it may have induced\nhim to inform himself more accurately on his subject than the generality\nof persons.\n\nThis last is evidently the case in his account of the doctrines and\npractices of the St Simonians. One who felt no sympathy with any portion\nof their creed, would not have taken the trouble to obtain accurate\ninformation, or an intimate knowledge on this subject. Not that M. Blanc\nis a St Simonian; to do him justice, he has argued with ability and\nclearness against their leading tenets or maxims; but being a man\ndevoted to a new order of things of some kind or other, he has given\nnaturally a more than usual attention to this sect, and we think our\nreaders will hold themselves obliged to us, if we abridge some portion\nof his account of St Simon and his disciples.\n\n\"The founder of the St Simonian school had been deceased five years when\nthe revolution of July broke out. He belonged to one of the noblest\nhouses of France, bearing the name and arms of that famous Duke de St\nSimon, the historian of the reign of Louis XIV., and the last of our\nveritable _grands seigneurs_. Yet it was the privilege of birth that he\nattacked, and the impiety of war that he proclaimed. He was a man of\nsingular independence of mind, and of extreme moral courage. Convinced\nthat, before dictating a code for the regulation of human life, it was\nnecessary to have attentively analysed that life as it actually exists,\nhe spent the first half of his days in studying society under all its\naspects; recoiling from no experience, practising, in the character of\nan observer, even vice as well as virtue; drawing a lesson from his own\nfrailties, and making a study of his own follies. He dissipated his\nfortune in premeditated prodigality, and terminated a studious opulence\nin excessive poverty; living on the miserable salary of a copyist, when\nin idea he was governing the world. In the estimation of some, a\nsage--of others, a madman; at one time sanguine to enthusiasm, at\nanother discouraged to the point of attempting suicide; reduced at last\nto the condition of a mendicant, after having so often united round his\ntable, in order to observe and judge them, the most celebrated men in\nart and literature. Such was St Simon in life and character: it remains\nto see what were the intellectual results he arrived at.\"--(Vol. III. p.\n96.)\n\nHis first project of a code for human life was sufficiently ridiculous.\nIn a work entitled _Letters from an Inhabitant of Geneva to his\nContemporaries_, he addressed himself to the learned portion of the\nworld, inviting them to undertake the government of the human race. The\nprogramme was as follows. A subscription was to be opened before the\ntomb of Newton. Every one was called upon to subscribe according to his\nmeans, rich and poor, man and woman; and each subscriber was to have a\nvoice in the selection of--three mathematicians, three natural\nphilosophers, three chemists, three physiologists, three men of letters,\nthree painters, and three musicians. These several threes, amounting to\ntwenty-one, besides having the produce of the subscription, were to form\na council, called the _Council of Newton_, and undertake the spiritual\ngovernment of the world, directing the efforts of the several nations of\nthe globe towards one common end.\n\nThe learned portion of the world made no response to this invitation; he\ntherefore next addressed himself to the operatives, declaring that the\ntime was come to tear the crown from the brow of idleness, and establish\nthe reign of labour. The king was now to be the chief of artisans, his\nministers enlightened workmen; and the electoral right was to be so\nplaced as to transfer all power from the proprietor of the soil to the\ncultivator, from the capitalist to the journeyman. One would say that,\npiqued with the indifference of the most literate portion of mankind, he\nwas determined to offer the government of the world to the most\nilliterate. Since the Royal Society would not accept the ball and\nsceptre which he had placed at its disposal, he gave them over to the\nTrades' Union.\n\nBut neither would this satisfy him. He who appeals to the lowest order\nof minds must confine himself to what is intelligible to, and\ninfluential on the lowest; and this would hardly accord with one who, at\nall events, had led an intellectual life, of however wild an order. He\nagain reverted to the thinking classes, and to some modification of his\nfirst idea; and his _New Christianity_--his last and most complete\neffort--has for its object to erect an intellectual and spiritual\ngovernment of the world. Taking his analogy from the spiritual dominion\nof the church of Rome, but finding that that power was too restricted in\nits exercise, inasmuch as the material interests and scientific labours\nof mankind were not embraced by it, he called for the foundation of \"a\nreligious power, which, embracing humanity in all its interests, should\nconduct it towards a Christian purpose--the amelioration of the lot of\nthe great multitude of mankind; by their _sentiments_ employing\n_artists_, by their _reason_ employing the _learned_, and by their\n_activity_ employing the _operative_.\"\n\nWhatever may be the importance of this conception, it answered one\npurpose--it satisfied the builder's mind. St Simon died full of faith\nand hope. When he bade his eternal adieus to the few disciples who\nsurrounded his dying bed, he regarded his work as completed, his mission\nas fulfilled. \"The fruit is ripe,\" said he; \"you will gather it.\"\n\nThe disciples of St Simon still further elaborated and disseminated his\ndoctrines; and a school was formed which recognised MM. Enfantin and\nBuzard for its chiefs. It need hardly be said, that the new order of\nsociety was to be founded on universal benevolence--no war, and no\nrivalry--and the industry of mankind organized in such sort, _that to\neach man would be assigned according to his capacity, and to each\ncapacity according to its works_.\n\nWe quote with pleasure the remarks (tinctured though they are by his own\npeculiar opinions) which M. Blanc makes on this famous formula:--\"In\npreaching a universal association of men, founded on benevolence--in\ndemanding that industry should be regularly organized, and that she\nshould establish her empire on the ruins of a system of violence and\nwar, the St Simonians showed a thorough intelligence of the laws which\nwill one day govern humanity. But they overthrow with one hand the\nedifice they erect with the other, when they announce this famous\nformula--_To each according to his capacity; to each capacity according\nto its works_--a formula wise and equitable in appearance, but in\nreality subversive and unjust.\n\n\"If we say that a man, in virtue of his intellectual superiority, is to\nadjudge to himself a larger share than others of the goods of this\nworld, what right have we to censure the sturdy barbarian, who, in\nvirtue of his physical superiority, was wont to take the lion's share to\nhimself? We have changed the basis on which the tyranny rested--the\ntyranny remains. The St Simonians, it is true, justify their formula on\nthe grounds of public utility; it is well, say they, to stimulate talent\nby recompense. But is it necessary that the recompense of talent be of\nthis gross and material kind? that it be counted down in so much wealth?\nThank Heaven! man has other and more energetic motives. With a piece of\nriband to be attached to the buttonhole, Napoleon could make an army of\na million of men rush forward upon danger and death. The word _glory_,\nwell or ill understood, has always decided the destinies of the world.\nWhat is amply sufficient when the work of destruction is in hand, by\nwhat disastrous fatality does it become incompetent when the task is to\nproduce and to create? Is it not true that great men have always sought\nand found their principal recompense in the very exercise of their high\nfaculties? If society had wished to recompense Newton, it would have\nbeen utterly powerless to do so; there was for Newton, in all the world,\nno other or sufficient recompense, but the joy he must have felt when\nhis genius discovered the laws which govern the planets. * * * The\ngreater the intelligence, the greater the sphere of action; but not\nnecessarily the greater the material recompense. The inequality of\ncapacities can legitimately conduct to the inequality only of duties.\"\n\nThe revolution of 1830 gave a wonderful stimulant to the little society\nof St Simon. It extended rapidly, and adjourned its sittings from a\nprivate house to an ample theatre, where three tiers of boxes held the\nadmiring or ironical auditory. Fetes, and the presence of charming\nwomen, increased the number of proselytes; artists, physicians,\nadvocates, poets, flocked to share in the generous hopes of the new era.\nThe capital and the provinces were portioned out into new departments,\nto accord with the new administration of affairs, and St Simonism had\nalso its map of France. The two chiefs, or fathers, took upon themselves\nthe ambitious title of popes. They already cast their eyes upon the\nTuileries. Louis-Philippe was summoned by letter to yield his place to\nMM. Enfantin and Buzard. St Simonism was already a government destined\nto replace the authority of the Catholic church.\n\nBut there were schisms in this new church--Pope Enfantin thinking one\nthing and Pope Buzard another; and that, too, on the important topic of\nmatrimony. The principal adepts of the sect met together, and held\nstrange fanatical discussions for the discovery of the truth on these\ncontroverted points. It is worthy of remark, that St Simonism, as well\nas Irvingism or Mesmerism, could boast of its convulsions and its\nprophecies.\n\n\"At this time there passed in the Rue Monsigny, in the midst of this\nsceptical and mocking France, scenes so extraordinary, that, to find\ntheir parallel, we must revert to the history of the Anabaptists. Those\nwho had hitherto resisted the extreme doctrines of Father Enfantin, felt\nas if impelled against their will to the borders of some immense abyss.\nWith the rest, it was an accession of fervour altogether indescribable,\nan exaltation which ended in delirium. There, in a room, the doors of\nwhich were carefully closed, and whose thick walls betrayed no sound,\ndiscussions were continued whole days and whole nights without\ninterruption, without relief, without repose. It sometimes happened that\na young man, incapable of sustaining these consuming vigils, reeled and\nfainted; they removed the apparently lifeless body without suspending\nthe discussion. M. Caseaux was in an ecstasy for an hour, and began to\nprophesy. Another day, M. Olinde Rodrigues was struck as if by apoplexy;\nbecause, asking each of the members in turn whether it was not true that\nthe Holy Spirit was in him, (M. Olinde Rodrigues,) one of the persons\ninterrogated had the temerity to answer by certain expressions of\nincredulity. The fit was extremely violent, and Dr Fuster, in order to\nsave the patient, had recourse to a formal retractation from the\ninconsiderate respondent, who, on his part, was full of affliction for\nthe mischief he had occasioned. Such, even on men of serious thought and\nelevated understanding, may be the effect produced by a belief carried\nto a certain point of excitement.\"\n\nSuch, too, may be the danger of contradicting a prophet; and we intend\nto take the hint, and never be guilty of so great an imprudence. These\ndissensions, accompanied with certain financial difficulties, led to a\nrupture, and the family of the Rue Monsigny were compelled to dissolve.\n\n\"In this crisis, the profound calm of Enfantin never deserted him. He\npossessed, at Menilmontant, a house and garden; here he resolved to seek\na place of retreat, of study, and of labour, for himself and his more\nfaithful disciples. Forty of these followed him to this retreat, and\nthere commenced the life in common, combined always with a just\nsentiment of the true hierarchy of society. Poets, artists, officers,\nmusicians, all devoted themselves in turn to the rudest and coarsest\nlabours. They repaired the house, they swept the courts, they cleaned\nthe chambers and polished the floors; they dug up the uncultivated soil,\nthey covered the walks with gravel, extracted from a pit which they\nthemselves had excavated. To prove that their ideas on the nature of\nmarriage, and the emancipation of women, were pure from any selfish or\nsensual calculations, they imposed upon themselves the law of celibacy.\nMorning and evening they nourished their mind with the words of the\nfather, or, in the lives of the Christian saints read aloud, they found\nexample, encouragement, and precept. Hymns, the music of which one of\ntheir members had composed, served to elevate their minds and charm\ntheir labours. At five o'clock, dinner was announced by the sound of a\nhorn. Then these philosophic workpeople piled up their tools, arranged\ntheir wheel-barrows symmetrically, and took their place, after having\nfirst sung 'the prayer before repast.'\"\n\nIn this retreat they adopted a distinctive dress, of which one portion,\nthe waistcoat, was symbolical; it was so made that it could not be put\non without the help of a brother--and thus was calculated perpetually to\ncall to mind the necessity of mutual aid. On the day of the institution\nof this habit, Enfantin declared that he and his followers had renounced\nall rights to property according to the existing law, and had duly\nqualified themselves to receive \"the honourable wages\" of labour.\n\nBut this fantastical experiment was cut short by the interference of the\nlaw. A public prosecution was instituted against the St Simonians; and\nPere Enfantin, and other chiefs of the sect, were brought before the\ntribunal at Paris. It will be easily understood that the court that day\nwas crowded with spectators, eager to see the St Simonians, especially\nEnfantin, who appeared in a violet- robe, with the words LE PERE\nwritten in large letters on his breast. When asked by the president,\nwhether he did not style himself the Father of Humanity--whether he did\nnot profess to be the Living Law--he answered, \"Yes!\" with perfect\ncalmness and assurance. The discourse he delivered in his own defence\nwas chiefly remarkable for the long pauses he made from time to time,\noccupying himself with looking steadfastly at the president, or the\nadvocate-general. He said he wished to make them feel \"the power of the\nflesh.\" But this species of animal magnetism appears to have had no\nother effect than that of irritating the court. He and some others were\ncondemned to pay a fine, and suffer a year's imprisonment. The family\nwas dispersed. For the present there was an end to St Simonism.\n\nA history is hardly complete without a plague, or pestilence, or famine,\nor some such wide-spreading calamity, on which the historian can spend\nthe dark colours of his descriptive eloquence. Considering that M. Louis\nBlanc had but the space of ten years under him, he must have regarded\nhimself as very fortunate in meeting with the cholera, which figures\nhere as a very respectable pestilence. The carrying forth the dead,\nnaked and uncoffined, in open carts, is an image often presented to us\nin descriptions of this nature; but it is perhaps surpassed in terrible\neffects by the one here offered to us, of the bodies of those who had\ndied of the cholera piled up in carts and tumbrils, in coffins so\nhastily and slightly constructed, that, as they rattled over the stones,\nthere was constant danger of their horrible contents being poured upon\nthe pavement. But the strange reports that were afloat amongst this\ncredulous and passionate populace, form the most striking feature in the\npicture. It was reported in Paris, as our readers will probably\nremember, that there was, in reality, no cholera, but that poison had\nbeen poured into the fountains of the metropolis, and had been mingled\nwith the wine and the flour; and thus it was that the people were dying.\nIt was dangerous to be found with a phial in the hand, or to be seen\nsitting, without any ostensible cause, near one of the public fountains.\nA young man was looking into a well; he was massacred. Another met the\nsame fate, who was leaning over the door of a dealer in wine and\nspirits, in order to see what o'clock it was. A Jew in the market-place\nwas thought to have a sinister laugh; they searched him, found a packet\nof white powder--it was camphor--they killed him, and set on the dogs to\ntear the body.\n\nAnd then that insurrection against the mud-carts--what an insight does\nit give into the wide-spreading and tangled interests of a modern\ncapital! It was impossible to touch the mud of Paris without periling\nthe subsistence of eighteen hundred persons. What more fit, what more\ninnocuous to all parties, it would seem, than to clear away the mud from\nthe streets--to clear it away as soon as possible, that it should not\nlie there, exhaling pestilence during the heat of the day? But\nstop--there are in Paris some eighteen hundred persons who gain their\nbread out of this mud, groping in it, and extracting from it every\narticle of the least commercial value. With a basket slung upon their\nback, and a crook in their hand to facilitate their search, these\n_chiffoniers_ are to be seen in every quarter of the city, congregating\nwherever there is dirt. And now, if all that is thrown out of the houses\nof Paris is taken away before these industrious persons have had time to\nsearch it, what is to become of the whole profession of _chiffonerie_?\nThese new mud-carts, with their ruthless sweepers, traversing the city\nat dawn of day, must be broken up and thrown into the Seine; and it was\ndone so accordingly.\n\nThere is a peculiar charm, we think, in having related to us, for the\nfirst time, in the shape of history, what we remember to have read and\ntalked over as the news and gossip of the day. We seem to be present at\nthe making of history. We see facts, as the death of princes, which made\nso much stir and confusion, sink into the commonplace of the historical\nrecord; while anecdotes, which were repeated and forgotten, may stand\nforward as instructive proofs of the temper of the times, and the spirit\nof the past age. More than one such anecdote we think we could select\nfrom the pages before us; but it is possible we might draw them from a\npurer source than the work of M. Louis Blanc, to which our readers will\nperhaps think that we have already given more than sufficient space.\n\n\nFOOTNOTES:\n\n{A} _Histoire de Dix Ans_, 1830-1840. Par M. LOUIS BLANC.\n\n{B} As well it might, if he had been clambering over barricades in those\nhot days of July; for the three glorious days were remarkable for their\nheat.\n\n\n\n\nA NIGHT ON THE BANKS OF THE TENNESSEE.\n\n\n\"Can you tell us how far we are from Brown's ferry?\" said I to a man,\nwho came suddenly and silently upon us from a narrow side-path.\n\nWe were on the banks of the Tennessee: the evening was drawing in; the\nfog, that hung over land and river, was each moment thickening. The\nlandscape had a wild chaotic appearance, and it was scarcely possible to\ndistinguish objects at five paces distance.\n\nThe horseman paused some moments before answering my question. At last\nhe replied, accompanying his words with an ominous shake of the head--\n\n\"To Brown's ferry? Perhaps you mean Cox's ferry?\"\n\n\"Well, then--Cox's ferry,\" said I, rather impatiently.\n\n\"Ay, old Brown is dead,\" continued the man, \"and Betsy has married young\nCox. Ain't it him you mean?\"\n\n\"That we know nothing about,\" replied I; \"but what we wish to learn is,\nwhether we are far from the ferry, and if this is the right road to it.\"\n\n\"Ah! the way to the ferry--that's the rub, man! You're a good five miles\noff, and might just as well turn your horse's head another way. I guess\nyou're strangers in these parts?\"\n\n\"Heaven preserve us!\" whispered my friend Richards, \"we are in the hands\nof a Yankee; he is guessing already.\"{A}\n\nMeantime the horseman had drawn nearer to us, in spite of the thorns and\nof the wet boughs, that each moment slapped and slashed him across his\nface; and he was now close to our horse. As far as we could distinguish\nthrough the rapidly-increasing gloom, he was a middle-aged man, bony and\nlong-legged, with a sallow unprepossessing physiognomy surmounting his\nlong ungainly carcass, and metal buttons upon his coat.\n\n\"And so you've lost your way?\" said the stranger after a long pause,\nduring which the thick fog had had the kindness to convert itself into a\nclose penetrating rain. \"That's queer too, seein' that the ferry ain't\nfifteen paces from the road, which runs right along the side of the\nriver. A very queer mistake to be goin' up the stream, instead of\nfollowin' yer nose and the run of the water.\"\n\n\"What do you mean?\" cried Richards and I in a breath.\n\n\"That you're goin' up the Tennessee instead of down it, and are on the\nroad to Bainbridge. That's all!\" replied the supposed Yankee.\n\n\"On the road to Bainbridge!\" repeated we, in voices in which\nastonishment and vexation were tolerably evident.\n\n\"You hadn't a mind to go to Bainbridge, then?\"\n\n\"How far is the infernal place from here?\" asked I.\n\n\"How far, how far?\" repeated the man with the metal buttons. \"It's not\nto say very far, nor yet so very near, as I may guess. Perhaps you know\nSquire Dimple?\"\n\n\"I wish you and Squire Dimple were at the devil!\" muttered I. But\nRichards, who took things more quietly, replied--\n\n\"No, we have not the honour of his acquaintance.\"\n\n\"Humph! And whereaway may you be goin'?\" enquired our tormentor, who was\napparently waterproof.\n\n\"To Florence in Alabama,\" answered Richards, \"and thence down the\nMississippi.\"\n\n\"Ah, fine city, Florence! such as one only finds in this country. Ain't\nit now? And a good market, too. Talkin' of that, what's the price of\nflour in the north? You're come from thereaway, I guess. I did hear it\nwas six and four levies, and Injun corn five and a fip--butter three\nfips.\"\n\n\"Are you mad?\" cried I, losing all patience, and unconsciously raising\nmy whip as I spoke--\"are you stark staring mad, to keep us talking here\nabout flour and butter, and fips and levies, while the rain is falling\nby bucketsfull?\"\n\n\"Hallo, stranger!\" cried the man, raising himself for the first time out\nof his lounging position on the saddle. \"Guess you're gettin' wolfish.\nI'm for you--stick, fist, or whiphandle, rifle or bowie-knife. Should\nlike to see the man as could leather Isaac Shifty!\"\n\n\"The road, the road, Mister Isaac Shifty!\" interrupted friend Richards\nin a conciliating tone. There was another long pause.\n\n\"I guess you're traders,\" said the fiend at last.\n\n\"No, man.\"\n\n\"And what may you be, then?\"\n\nOur answer was followed by another long inspection of our persons and\nphysiognomies. He gazed at us for a couple of minutes or more, examining\nus from head to foot; at last he spoke.\n\n\"And so you've a mind to go down the Mississippi?\"\n\n\"Yes, in the Jackson, which starts to-morrow, we are told.\"\n\n\"Ah, the Jackson! a mighty good steamboat too--ain't it now? But I guess\nyou ain't a thinkin' of takin' that thing and your horse with you?\"\ncontinued the Yankee, pointing to our gig.\n\n\"Yes, we are.\"\n\n\"Oh, you are! Well.--You haven't seen two women in a dearborn on the\nroad, have you?\"\n\n\"No, we have not.\"\n\n\"Well, then,\" continued the man in the same indifferent tone, \"it's\na'most too late now to get to Bainbridge; and yet you might try it, too.\nBetter turn your horse round, and follow the road till you come to a big\nwalnut-tree; there it divides. Take to the right hand for half a mile,\ntill you come to neighbour Dims's hedge; then you must go through the\nlane; and then, for about forty rods, right through the sugar-field;\nkeep to your left till you come to some rocks, but then turn to your\nright, if you don't want to break your necks. There's a bit of a stream\nthere; and when you are over that, the left-hand road will take you\nstraight to Cox's ferry. You can't miss it,\" concluded he, in a\nself-satisfied tone, striking his horse a blow with his riding-whip. The\nanimal broke into a smart trot, and in ten seconds our obliging friend\nhad disappeared into the fog.\n\nMy countenance, during the Yankee's interminable directions, must have\nsomewhat resembled that of a French recruit, to whom some scarred and\nmustached veteran is relating his Egyptian campaigns, and telling him\nwonderful stories of snakes and crocodiles at least half a mile\nlong--monsters who made nothing of swallowing a drum-major to their\nbreakfast, bearskin cap, cane, and whiskers, included. I was so\ncompletely bothered and confounded with the rights and lefts, that the\nmetal-buttoned individual was out of sight and hearing before I thought\nof explaining to him, that, dark as it then was, we should never be able\nto find even the walnut-tree, let alone neighbour Dims's hedge and the\nbreak-neck rocks. Patience is by no means one of my virtues; but the\nman's imperturbable phlegm and deliberation, in the midst of the most\npouring rain that ever wetted poor devil to the skin, tickled my fancy\nso exceedingly, that the sound of his horse's hoofs had hardly died\naway, when I burst into an almost interminable fit of laughter. \"First\nright, then left--look out for the big walnut-tree, and don't break your\nneck over the crags!\" repeated I, in a tone between merriment and\ndespair. Richards, however, saw nothing to laugh at.\n\n\"The devil take the Yankee!\" cried he. \"May I be hanged if I know what\nyou find so amusing in all this!\"\n\n\"And hang me if I know how you manage to look so grave!\" was my answer.\n\n\"How could we possibly have missed the ferry?\" cried Richards; \"and,\nwhat is still more stupid, to come back instead of going forward!\"\n\n\"Not very astonishing,\" replied I, \"considering the multitude of\nby-roads and cross-roads, and waggon-tracks and cattle-paths, and the\nswamp into the bargain. It is quite impossible to see which way the\nriver runs. And then you have been sleeping all the afternoon, and I had\nto find the way by myself.\"\n\n\"And you found it after an extraordinary fashion--retracing your own\nsteps,\" said Richards in a vexed tone. \"It is really too stupid.\"\n\n\"Very stupid,\" said I--\"to sleep.\"\n\nAs may be seen, we were on the verge of a quarrel; but we were old and\nsincere friends, and stopped in time. The discussion was dropped. The\nfact was, that our mistake was by no means a very surprising one. The\ncountry in which we were, seemed made on purpose to lose one's-self in.\nThe road winds along at some distance from the river, frequently out of\nsight of it; the shore is uneven, covered with crags and hillocks;\nnothing like a landmark to be seen, or a mountain to guide one's-self\nby, except occasionally, when one gets a peep at the Appalachians rising\nout of the blue distance. The fog, however, had hidden them from us, and\nthat just at the time when we most wanted them as guides. We found\nourselves in a long low clearing--a sort of bottom, as they call it in\nthat country--which was laid out in sugar-fields, and through which\nthere ran nearly as many cart-roads as there were owners to the land.\nThe morning had been bright and beautiful; but, towards noon, a grey\nmist had begun to rise in the south-western corner of the horizon, and\nhad gone on, thickening and advancing, till it spread like a pall over\nthe Tennessee. With a grey wall of fog on one side, and the swamp,\nintersected with a hundred cross-paths, on the other, we had gone on for\nabout a mile; until it got so thick and dark, that it was quite as\npossible we should find our way into the marsh as over the Mussel\nshoals.{B} So certain was I, however, of the proximity of the latter,\nthat I pushed on, expecting each moment to find the ferry, until the\nunlucky Yankee brought all my hopes to a termination.\n\nIt was now quite night--one of those dreary pitch-dark nights that are\nof no unfrequent occurrence in the south-western states. I would as soon\nhave been on the banks of Newfoundland as in this swamp, from which\nnothing was more probable than that we should carry away a rattling\nfever. The Yankee's directions concerning the road were, as may be\nsupposed, long since forgotten; and even had they not been so, it would\nhave required cat's eyes to have availed ourselves of them. Even the\nowls, the nightingales of that neighbourhood, seemed puzzled by the\nextreme darkness. We could hear them whooping and screaming all around\nus; and now and then one flew against us, as if it had lost its way as\nwell as ourselves. The road we were now following ran close to the bank\nof the river; so close, indeed, that a single stumble of our horse might\nhave precipitated us into the water, which was then very high.\n\n\"I think we should do our best to get out of the gig,\" said I to my\ncompanion; \"or else we have a very good chance of passing the night in\nthe Tennessee.\"\n\n\"No danger,\" replied Richards, \"Caesar is an old Virginian.\"\n\nA shock that made our very ribs crack again, and as nearly as possible\nthrew us backwards out of the gig, came rather opportunely to interrupt\nthis eulogium on Caesar, who had suddenly reared furiously up on his\nhind-legs.\n\n\"There must be something in the path,\" cried Richards. \"Let us see what\nit is.\"\n\nWe got out, and found a huge walnut-tree lying right across the road.\nHere was an end to our journey. It was an absolute impossibility to get\nthe gig over the enormous trunk; the boughs, which spread out full\ntwenty yards in every direction, had given Caesar timely warning of the\nimpediment to our further progress. The road, moreover, was so narrow\nthat it was impossible to turn. There was nothing for it but to back\nout. Richards began hunting about for a cross-road, where we might turn;\nI set to work to back the gig. I had no sooner, however, set one foot\nout of the road, than my cloak was almost torn from my shoulders by a\nthorn half a yard long. To get through this detestable wilderness with a\nwhole skin, one ought to have been cased in complete armour. I had only\njust taken my unfortunate garment off this new-fashioned cloak-peg, when\nRichards returned.\n\n\"This is the most infernal wilderness in all the west!\" said he.\n\"Neither road nor path, mud up to the ears, and, to add to my enjoyment,\nI have left one of my boots in the swamp.\"\n\n\"And, for my part, there are as many holes in my cloak as thorns on that\ncursed acacia-tree,\" replied I by way of consolation.\n\nThese were the last words we spoke in any thing like a jesting tone; for\nwe were now wet to the skin: and of all situations, I believe a damp one\nto be the least favourable to jocularity. I confess a certain partiality\nfor adventures, when they are not carried too far. There is nothing I\ndetest like a monotonous wearisome Quaker's journey, with every thing as\ntame, and dull, and uniform, as at a meeting of broad-brims; but to be\novertaken by darkness and a deluge in the middle of a maple-swamp, to be\nunable to go three steps on one side without falling into the Tennessee,\nwith an impenetrable morass and thicket on the other hand, a colossal\nwalnut-tree barring the way in front, and no possibility of turning\nback--this was, even to my taste, rather too much of an adventure.\n\n\"Well, what is to be done now?\" said Richards, who had placed himself in\na sort of theatrical posture--his bootless foot on the gig-step, the\nother sticking fast in the mud.\n\n\"Take out the horse, and draw the gig back,\" suggested I.\n\nEasily said, but rather more difficult to accomplish. We set to work,\nhowever, with a will; and pushed, and tugged, and pulled, till at last,\nafter much labour, we got the gig about thirty paces backwards, where\nthe road became wider. We then turned it, and were putting Caesar into\nthe shafts, when, to our inexpressible delight, a loud hallo was given\nquite close to us.\n\nReader, if you were ever at a hard contested election, where you had bet\nyour fifty or a hundred dollars on your favourite candidate, and just\nwhen you made sure of losing, and your five senses were almost\nextinguished by noise, brandy, and tobacco smoke, you heard the result\nproclaimed that secured you your stake, and a hundred per cent to boot;\nif you have ever been placed in such circumstances, then, and then only,\ncan you form an idea of the joyful feeling with which we heard that\nshout. After such a thorough Yankee fashion was it given, that it caused\nthe fog to break for a moment, and roused the obscene inhabitants of the\nneighbouring swamp from their mud-pillowed slumbers. They set up a\nscreeching, and yelling, and croaking, that was lovely to listen to.\n\n\"And now have patience, for Heaven's sake!\" whispered Richards to me,\n\"and hold your tongue for a quarter of an hour, or you will spoil all\nwith this infernal Yankee.\"\n\n\"Do not be afraid,\" replied I; \"I am dumb.\"\n\nMy blood was certainly tolerably cooled by the shower-bath I had had--to\nsay nothing of the prospect of passing the night in this vile hole; and\nI would willingly have given the tenacious Yankee information concerning\nthe prices of flour and butter in every state of the Union, upon the\nsole condition that he should afterwards help us out of this reservoir\nof fever.\n\nIt was, as we had at once conjectured, our friend Mr Isaac Shifty, in\nsoul, body, and buttons. In true Connecticut fashion, he stood a couple\nof minutes close to us without saying a word. It almost looked as if he\ntook a delight in our difficulties, and was in no particular hurry to\nextricate us from them. For our part, we kept very much on our guard.\nThe cross-grained scarecrow might likely enough have left us to our fate\nagain, if we had said any thing that did not exactly chime in with his\nqueer humour. Richards at last broke silence.\n\n\"Bad weather,\" said he.\n\n\"Well, I don't know. I shouldn't say it was though, exactly,\" returned\nthe Yankee.\n\n\"You have not met the two women you were looking for, have you?\"\n\n\"No. Guess they'll have stopped at Florence, with cousin Kate.\"\n\n\"You are not thinking of going there too, are you?\" said Richards.\n\n\"No. I'm goin' home. I thought you were at the ferry by this time.\"\n\n\"Perhaps we should have been, if your roads were better, and the holes\nin them filled up with stones instead of walnut-trees,\" returned\nRichards, laughing.\n\n\"Guess you ain't inclined to go to the ferry to-day?\"\n\n\"Inclined we are, but able we are not,\" replied Richards; \"and you will\nacknowledge, my friend, that is a pretty strong reason for not going.\"\n\n\"Well, so it is,\" replied the man sententiously. \"It ain't very\nagreeable lyin' out in the swamp; and so, stranger, if you like to go to\nBainbridge, you can come with me. Better let me drive, and my mare can\nfollow behind.\"\n\nIt took at least five minutes before the wearisome, pedantical fellow\nhad finished his arrangements and preparations. At last, to the infinite\nsatisfaction of Richards and myself, we sat three in the gig. After\nundergoing a questioning and cross-questioning that would have done\nhonour to an experienced diplomatist, we had succeeded in striking up a\nsort of alliance with Mr Isaac Shifty, and were on our way to one of the\nhundred famous cities of Alabama--cities which have decidedly not their\nmatch in the whole of the United States.\n\nI do not know how it happens, but I am constantly finding myself\ndisappointed in my expectations. I had hoped that the distance between\nthe infernal maple swamp and the place to which we were going, would\nhave borne some sort of relative proportion to the agreeableness of our\nsituation--that is to say, that it would not be very great. It\nnevertheless appeared to me enormous, and Horace's impatience during his\ncelebrated walk was trifling compared to mine. Our Yankee, like the\nRoman babbler, had abundance of time to discourse on fifty different\nsubjects. The first which he brought before our notice was naturally his\nown worthy person. From the interesting piece of biography with which he\nfavoured us, we learned that he was originally from Connecticut, and\nthat his first occupation had been that of usher in a school; which\nemployment he had, after a short trial, exchanged for the less\nhonourable but more independent one of a pedlar. From that he had risen\nto be a trader and shop-keeper, and was now, as he modestly informed us,\na highly respectable and well-to-do man. He next gave us an account of\nall the varieties of merchandise in which he dealt, or ever had dealt;\nintermixing the details with an occasional side-blow at a certain Mr\nBursicut, who had dared to set up an opposition store, and whom\nProvidence had punished for his presumption by the loss of sundry dozen\nknives and forks, and pairs of shoes, upon the Mussel shoals. He then\nfound occasion to talk of the thousand and one mishaps that had occurred\nupon the aforesaid Mussel shoals; and thence branched off into the\nvarious modes of water-carriage which the enlightened inhabitants of\nAlabama were accustomed to employ. After amusing us for some time with\nlong histories concerning steam-boats and keel-boats, barks and\nflat-boats, broad-horns, dug-outs, and canoes, he glided into some\ncanal-making scheme, which was to connect the waters of the Tennessee\nwith Heaven knows what others. It was a most monstrous plan--that I\nremember; but whether the junction was to be made with Raritan bay or\nConnecticut river, I have clean forgotten. At last we came to the\nhistory of Bainbridge--a sure sign, as I thought, with much inward\ngratulation, that we were approaching the end of our journey; yet the\naccomplishment of this hope, reasonable as it was, was doomed to be\ndeferred a long time. We had first to listen to the whole history and\ntopographical description of that celebrated city; how it had sprung up\nin the right corner, he reckoned; and how flourishing and industrious it\nwas; and whether we had not a mind to settle there--because if we had,\nhe, Mr Isaac Shifty, had some almighty fine building land to sell; and\nhow the town already boasted of three taverns, just the right proportion\nto the ten houses of which Bainbridge consisted. We should find two of\nthe taverns chokeful of people, he said, because there was a canvass\ngoing on for the Florence election; as to the third, it was a poor\nplace, hardly habitable indeed.\n\nAt the word _canvass_, Richards and I looked aghast.\n\n\"An election coming on!\" stammered Richards.\n\n\"An election!\" repeated I, the words dying away upon my tongue from\nconsternation at this unwelcome news. An election in Alabama, which even\nin old Kentucky is considered as backwoods! Farewell, supper and sleep,\nand comfortable bed and clean linen! every thing, in short, which we had\nflattered ourselves with obtaining, and which we stood so much in need\nof, after such a hard day's journey.\n\nBefore we had time to make any further enquiries, Caesar, who had for\nsome time been splashing through a sea of mud, stood suddenly still. The\nlight of a tallow candle, glimmering and flaring through an atmosphere\nof tobacco-smoke, and the hoarse and confused sounds of many voices,\nwarned us that we had reached the haven. We sprang out of the gig; and\nwhilst Richards was tying Caesar to a post, I hurried to the door, when I\nfelt myself suddenly seized by the skirt of my cloak.\n\n\"Not there--not there! This is the house where you are to stop,\"\nexclaimed Mr Isaac Shifty, pointing anxiously to an adjacent edifice,\nthat looked something between a house and a pigsty.\n\n\"Don't go with him,\" whispered I to Richards, heartily glad to be at\nlast independent of the insupportable Yankee, and to be able to vex him\na little in my turn. My hand was already on the latch; I opened the\ndoor, and we entered.\n\nThere sat the burgesses of Bainbridge, with their heels upon the\ntable--those, at least, for whom there were chairs; while those for whom\nthere were none, made shift with tubs, or stood up in various elegant\nattitudes. There was a prodigious amount of talking, shouting, drinking,\nand laughing going on; and my first feeling was, that I would rather\nhave been any where else than in that worshipful assembly. Richards,\nhowever, stepped boldly forward, in spite of his bootless foot; and\nluckily the men appeared disposed to be upon their best behaviour with\nus. They pressed back right and left, forming a lane about a foot wide,\nenclosed between living palisades, six feet and upwards in height,\nthrough which we passed, subjected, as we did so, to a searching\ninspection. Richards stepped smartly up to the table, then turned round,\nand confronted the group of half-horse, half-alligator visages there\nassembled.\n\n\"A hurra for old Alabama!\" cried he, \"and the devil take the Bainbridge\nroadmaster!\"\n\n\"Are you mad?\" I whispered to him.\n\n\"May I be scalped if you don't soon feel the weight of these five bones\nupon your carcass, stranger!\" growled a voice, proceeding from a sort of\nmammoth that had just filled itself a half-pint tumbler of Monongahela.\nBefore the double-jointed Goliath put his threat into execution, he\nswallowed the whisky at a gulp, and then, striding forwards, laid his\nopen hand upon my companion's shoulder, with a force that threw the poor\nfellow on one side, and gave him the appearance of being crooked. At the\nsame time the giant stared Richards in the face, with an expression\nwhich the natural hardness of his features, and the glimmer of his\nowl-like eyes, rendered any thing but agreeable.\n\n\"The devil take the Bainbridge roadmaster--I repeat it!\" cried Richards,\nhalf in earnest and half laughing, raising his muddy and bootless foot\nas he spoke, and placing it on a chair. \"See there, men! I may thank him\nfor the loss of my boot. The cursed swamp between here and the ferry was\nkind enough to pull it off for me.\"\n\nThe roar of laughter that responded to these words would inevitably have\nbroken the windows, had there been any glass in them. Fortunately the\nlatter luxury was wanting; its place being supplied by fragments of old\ninexpressibles, and of _ci-devant_ coats and waistcoats.\n\n\"Come, lads!\" continued Richards, \"I mean no offence; but of a surety I\nhave to thank your bad roads for the loss of my boot.\"\n\nRichard's jest, exactly adapted to the society in which we found\nourselves, was the most fortunate _impromptu_ that could have been hit\nupon. It seemed at once to have established us upon a footing of harmony\nand friendship with the rough backwoodsmen amongst whom we had fallen.\n\n\"May I be shot like a Redskin, if that ain't Mister Richards from Old\nVirginny, now of the Mississippi,\" suddenly exclaimed the same colossus\nwho had so recently had his hand upon Richards's shoulder, twisting, as\nhe spoke, his wild features into a sort of amicable grin. \"May I never\ntaste another drop of rale Monongahela, if you sha'n't drink a pint with\nBob Snags the roadmaster!\"\n\nIt was the very dignitary whom Richards had insulted with such imminent\nrisk to his shoulder-blade.\n\n\"A hurra for old Virginny!\" shouted the master of the roads, biting, as\nhe spoke, into a piece of tobacco from that famous state. \"Come,\nmister--come, doctor!\" continued the man, offering Richards with one\nhand a roll of tobacco, with the other a pint glassful of whisky.\n\n\"Doctor!\" repeated the whole assembly--\"a doctor!\"\n\nA man possessing power over gin and whisky, and whose word is an\nindisputable veto against even a _smaller_, is no unimportant personage\nin that feverish neighbourhood. In this instance, Richards's doctorship\nwas of the double utility of delivering us from the threatened\npint-glasses, and of causing us to be considered as privileged\nguests--no small advantage in a backwoods' tavern, occupied as the\nheadquarters of an electioneering party. Caesar, however, was the first\nto derive a positive profit from the discovery. Bob left the room for a\nminute or two, and we could hear the horse walking into the stable. When\nthe roadmaster returned, he had assumed a patronizing sort of look.\n\n\"Mister Richards!\" said he confidentially, \"Mister Richards! May I be\nshot if you ain't continually a sensible man, with more rale blood in\nyour little finger than a horse could swim in. Yes, and I'll show you\nthat Bob Snags is your friend. I say, doctor, what countryman is your\nhorse?\"\n\n\"A thorough-bred Virginian,\" replied Richards.\n\n\"The devil he is!\" cried Bob. \"Well, doctor, to prove to you that I'm\nyour friend, and that I ain't forgotten old times, I'll swop with you\nwithout lookin' at him. May I be shot if I ain't reg'larly cheatin'\nmyself. Well, I'm uncommon glad to see you again. Bob Snags has no\nreason to fear lookin' a rale gemman in the face. Come, lads, none of\nyer jimmaky, and slings, and poorgun,{C} and suchlike dog's wash, but\nginu_ine_ Monongahela--that's the stuff. Hurra for Old Virginny! Well,\ndoctor, it's a deal--ain't it?\"\n\n\"No, Bob,\" said Richards, laughing; \"your generosity is so truly\nAlabamian, that I cannot make up my mind to accept it. For the present,\nat least, I must keep my Virginian. It is my wife's saddle-horse.\"\n\n\"But Swiftfoot,\" replied Bob, in a cordial confidential\nmanner--\"Swiftfoot is a famous trotter.\"\n\n\"It won't do, Bob,\" was the answer. \"I should not dare show myself at\nhome without Caesar.\"\n\nBob bit his lips, a little vexed at not being able to make a deal; but\nanother half-pint of whisky, which he poured down as if it had been\nspring water, seemed to restore him to good humour. Meanwhile my wet\nclothes were beginning to hang heavy upon me, and to steam in the hot\natmosphere in which we were. Bob, who had already cast several\nside-glances at me, now turned to Richards.\n\n\"And who may the mister be?\" said he.\n\nThe mention of my name and condition, procured me a welcome that I could\nwillingly have dispensed with. After the shake of the hand with which\nBob favoured me, I looked at my finger-nails, to see if the blood was\nnot starting from under them. The fellow's hands were as hard and rough\nas bear's paws.\n\n\"Very glad that you're come, boys,\" said Bob in a low confidential tone.\n\"I'm just makin' a try for the next Assembly; and it's always good, you\nknow, to have somebody to speak to one's character. How long is it,\nMister Richards, since I left Blairsville.\"\n\n\"Eight years,\" replied my friend.\n\n\"No, Harry,\" whispered the roadmaster; \"may I be shot if it's more than\nfive.\"\n\n\"But,\" replied Richards, \"I have been living five years by the\nMississippi, and you know\"----\n\n\"Ah, nonsense!\" interrupted Bob. \"Five years--not an hour more. D'ye\nunderstand?\" added he cautiously--\"five years, if you're asked.\"\n\nThe facts were thus. This respectable candidate for the representation\nof his fellow-citizens, had made his escape from his previous residence,\nthe birthplace of Richards, on account of certain misdeeds, of which the\nsheriff and constables had taken cognizance, and after wandering about\nfor a few years, had settled in Bainbridge county, where he seemed to\nhave thriven--as far, at least, as whisky and human weakness had allowed\nhim. We could hardly help laughing outright at the importance which Bob\nthought proper to attribute to us before his companions, the independent\nelectors, whose votes he was desirous of securing. AEsculapius himself\nwas a mere quacksalver compared to Squire Richards, whose twenty-five\ns were rapidly multiplied into a hundred; while my poor neglected\nplantation was, between brothers, well worth five hundred thousand\ndollars. We allowed Mr Bob to have it his own way; for it might have\nbeen dangerous to contradict a giant of his calibre, who was always\nready to support his arguments with his huge cocoa nut- fists.\nAt last Richards was able to slip in a word.\n\n\"You are not going to make your speech now, are you?\"\n\n\"May I be shot if I ain't, though! I'll begin at once.\"\n\n\"Cannot we manage to change our clothes, and get some supper first?\"\nsaid Richards.\n\n\"Change your clothes!\" said Bob contemptuously. \"And what for, man? Not\non our account; you're quite smart enough, quite good enough for us--no\noccasion to bother yourselves. If it's for your own pleasure, however,\nyou can do it. Hallo, Johnny!\"\n\nAnd he commenced a negotiation with Johnny, the host, who, to our great\njoy, took up a candle, and led the way into a sort of back parlour, with\na promise that we should have our supper before very long.\n\n\"Is there no other room where we can dress ourselves?\" said I.\n\n\"To be sure there is,\" was the answer. \"There's the garret--only there's\nmy daughter and a dozen gals sleepin' there; then there's the kitchen,\nif you like it better.\"\n\nI looked round the room. A servant girl was beginning to lay the table;\nand, unluckily, the apartment was connected by an open door with the\nkitchen, in which there was a loud noise of voices. I would have given a\ngood deal for a quarter of an hour's undisturbed possession of the room.\nI looked about for our portmanteaus, but could see nothing of them.\n\n\"Six smalls it ain't buffalo hide!\" vociferated a young Stentor in the\nkitchen.\n\n\"Six smalls its cow hide!\" roared another.\n\n\"If I am not very much mistaken,\" said Richards, \"it is our portmanteaus\nthat those fellows are betting about.\"\n\n\"That would really be too bad,\" said I.\n\nNevertheless, it was as Richards had said. We had little occasion to\nfear that the portmanteaus would be lost or injured; but we knew very\nwell that the only way to get them out of the claws of these rough\nbackwoodsmen would be by some well-contrived joke. And those jokes were\nexactly what I feared; for one had often to risk breaking an arm or a\nleg by them. There was a crowd of men in the kitchen. One young fellow,\nupwards of six feet high, held a lighted candle; and they were all\nbusily engaged examining something which lay in the middle of the floor.\n\n\"No,\" cried a voice, appealing apparently from a decision that had been\ngiven, \"I won't pay without I see the inside.\"\n\nThey were debating whether the portmanteaus were of buffalo or cow hide.\nThey had caught sight of them as they were being carried through the\nkitchen into the back-room, and had at once seized upon them as good\nsubjects for a bet. It was time for us to interfere, if we did not wish\nto see our trunks ripped open, for the sake of ascertaining the quality\nof the leather.\n\n\"Sixteen smalls,\" cried Richards, \"that it's deer hide!\"\n\n\"Done!\" thundered half a score voices, with loud peals of laughter.\n\n\"It is a bet, then,\" said my friend; \"but let us see what we are betting\nabout.\"\n\n\"Make way for the gemmen!\" cried the men.\n\n\"Our portmanteaus!\" exclaimed Richards, laughing. \"No, certainly, they\nare not deer hide. Here is my bet.\"\n\nA loud hurra followed the payment of the dollar which my friend handed\nover; and we now found ourselves in undisputed possession of our\nbaggage. The next thing to be done was to endeavour to get the room to\nourselves for a few minutes.\n\n\"We wish to be left alone for a short time,\" said I to the help, who was\nbustling in and out, and covering the table with innumerable plates of\npreserved fruits, cucumbers, beet-root, and suchlike edibles.\n\nI shut the door.\n\n\"That is the surest way to have it opened again,\" said Richards.\n\nHe had hardly uttered the words, when, sure enough, the door flew open,\namidst a peal of uproarious laughter.\n\n\"Tail!\" cried one fellow.\n\n\"Head!\" shouted another.\n\n\"They want another dollar,\" said Richards. \"Well, they must have it, I\nsuppose. Head!\" cried he.\n\n\"Lost!\" roared the fellows in chorus.\n\n\"There is something for you to drink,\" said my friend, whose wonderful\npatience and good-humour was bringing us so fortunately through the\nshoals and difficulties of this wild backwoods' life. We now shut the\ndoor, and had time enough to change our wet clothes for dry ones. We\nwere nearly dressed, when a gentle tapping at the only pane of glass of\nwhich the room window could boast attracted our attention. On looking in\nthe direction of the sound, we distinguished the amiable features of Mr\nIsaac Shifty, who, upon our entering the tavern, had thought proper to\npart company.\n\n\"Gentlemen,\" whispered he, removing the remains of an old waistcoat,\nwhich supplied the place of one of the absent panes, and then applying\nhis face to the aperture--\"Gentlemen, I was mistaken. Our spies say you\nare not come to the election, but that you are from lower Mississippi.\"\n\n\"And if we are, what then?\" replied I dryly. \"Didn't we tell you as much\nat first?\"\n\n\"So you did, but I wasn't obliged to believe it; and d'ye see, they're\na-canvassing here for next election, and we've got an opposition in the\nother tavern; and as we knew that Bob Snags's people were expectin' two\nmen from down stream, we thought you might be they.\"\n\n\"And so, because you thought we should vote against you, you allowed us\nto stick in the mud, with the agreeable prospect of either breaking our\nnecks or tumbling into the Tennessee?\" said Richards laughing.\n\n\"Not exactly that,\" replied the Yankee; \"though if you had been the two\nmen that were expected, I guess we shouldn't have minded your passing\nthe night in the swamp; but now we know how matters stand, and I'm come\nto offer you my house. There'll be an almighty frolic here to-night, and\np'r'aps somethin' more. In my house you can sleep as quiet as need be.\"\n\n\"It won't do, Mr Shifty,\" said Richards, with a look that must have\nshown the Yankee pretty plainly that his object in thus pressing his\nhospitality upon us was seen through; \"it won't do, we will stop where\nwe are.\"\n\nThe latch of the door leading into the kitchen was just then lifted,\nwhich brought our conversation to a close. During the confabulation, our\nYankee's sharp grey eyes had glanced incessantly from us to the door;\nand hardly was the noise of the latch audible, when his face\ndisappeared, and the old waistcoat again stopped the aperture.\n\n\"He wants to get us away,\" said Richards, \"because he fears that our\npresence here will give Bob too much weight and respectability. You see\nthey have got their spies. If Bob and his people find that out, there\nwill be a royal row. A nice disreputable squatter's hole we have fallen\ninto; but, bad as it is, it is better than the swamp.\"\n\nThe table was now spread; the tea and coffee-pots smoking upon it. The\nsupper was excellent, consisting of real Alabama delicacies. Pheasants\nand woodcocks, and a splendid haunch of venison, which, in spite of the\ngame-laws, had found its way into Johnny's larder--wheat, buckwheat, and\nIndian-corn cakes; the whole, to the honour of Bainbridge be it spoken,\ncooked in a style that would have been creditable to a Paris\n_restaurateur_. By the help of these savoury viands, we had already, to\na considerable extent, taken the edge off our appetite, when we heard\nBob's voice growling away in the next room. He had begun his speech. It\nwas high time to make an end of our supper, and go and listen to him\nunder whose protecting wings we were, and to whom we probably owed it,\nthat we had got so far through the evening with whole heads and unbroken\nbones. Backwoods' etiquette rendered our presence absolutely necessary;\nand we accordingly rose from table, and rejoined the assemblage of\nelectors.\n\nAt the upper end of the table, next to the bar, stood Bob Snags, in his\nvarious capacity of president, speaker, and candidate. A thickset\npersonage, sitting near him, officiated as secretary--to judge at least\nfrom the inkstand with which he was provided. Bob looked rather black at\nus as we entered, no doubt on account of our late arrival; but Cicero\npleading against Catiline could not have given a more skilful turn to\nhis oration than did Bob upon the occasion of our entrance.\n\n\"And these gemmen,\" continued he, \"could tell you--ay, and put down in\nblack and white--no end of proofs of my respectability and character.\nMay I be shot by Injuns, if it ain't as good as that of the best man in\nthe state.\"\n\n\"No better than it should be,\" interposed a voice.\n\nBob threw a fierce look at the speaker; but the smile on the face of the\nlatter showing that no harm was meant, the worthy candidate cleared his\nthroat and proceeded.\n\n\"Yes,\" said he, \"we want men as know what's what, and who won't let\nthemselves be humbugged by the 'Ministration, but will defend our\nnat'ral born sovereign rights. I know their 'tarnal rigs, inside and\nout. May I be totally swallowed by a b'ar, if I give way an inch to the\nbest of 'em; that is to say, men, if you honour me with your confidence\nand\"----\n\n\"You'll go the whole hog, will you?\" interrupted one of the free and\nindependent electors.\n\n\"The whole hog!\" repeated Bob, striking his fist on the table with the\nforce of a sledge-hammer; \"ay, that will I! the whole hog for the\npeople! Now lads, don't you think that our great folks cost too much\nmoney? Tarnation to me if I wouldn't do all they do at a third of the\nprice. Why, half a dozen four-horse waggons would have enough to do to\ncarry away the hard dollars that Johnny{D} and his 'Ministration have\ncost the country. Here it is, lads, in black and white.\"\n\nBob had a bundle of papers before him, which we had at first taken for a\ndirty pocket-handkerchief, but which now proved to be the county\nnewspapers--one of which gave a statement of the amount expended by the\nfirst magistrate of the Union during his administration, reduced, for\nthe sake of clearness, into waggon-loads. Bob was silent, while his\nneighbour the secretary put on his spectacles, and began to read this\nimportant document. He was interrupted, however, by cries of \"Know it\nalready! Read it already! Go on, Bob!\"\n\n\"Only see here now,\" continued Bob, taking up the paper. \"Diplomatic\nmissions! what does that mean? What occasion had they to send any one\nthere? Then they've appointed one General Tariff, who's the maddest\naristocrat that ever lived, and he's passed a law by which we ain't to\ntrade any more with the Britishers. Every stocking, every knife-handle,\nthat comes into the States, has to pay a duty to this infernal\naristocrat. Where shall we get our flannel from now, I wonder?\"\n\n\"Hear, hear!\" cried a youth in a tattered red flannel shirt, to whose\nfeelings this question evidently went home.\n\n\"Moreover,\" continued Bob, \"it's a drag put upon our ships, to the\nprofit of their Yankee manyfacters. Manyfacters, indeed! Men! free\nsovereign citizens! to work in manyfacters!\"\n\n\"Hear, hear!\" in a threatening tone from the audience.\n\n\"But that ain't all,\" continued Bob, nodding his head mysteriously. \"No,\nmen--hear and judge! You, the enlightened freemen of Alabama, listen and\njudge for yourselves! Clever fellows, the 'Ministration and the Yankees!\nD'ye know what they've been a-doin'?\"\n\n\"No, no. Tell us!\" repeated twenty voices.\n\n\"You don't know?\" said Bob, with a fine oratorical movement. \"I'll tell\nyou then. They've been a-sendin' clothes, powder, rifles, flour, and\nwhisky to the Creeks! Two full shiploads have they sent. Here it is!\"\nyelled Bob, taking another paper from his pocket, and dashing it upon\nthe table.{E}\n\nA breathless silence reigned during the reading of the important\nparagraph, while Richards and myself were making almost superhuman\nefforts to restrain our laughter. Bob continued--\n\n\"You see, men, they want to get the scalpin' plunderin' thieves back\nag'in over the Mississippi into Georgia--ay, and perhaps into Alabama\ntoo. And they're holdin' meetin's and assemblies in their favour, and\nsay that we owe our independence to these Creeks; and talk about their\nchiefs--one Alexander the Great, and Pericles, and Plato, and suchlike\nnames that we give our s. And the cussed Redskins are fightin'\nagainst another chief whom they call Sultan, and who lives upon Turk's\nisland. Where shall we get our salt from now, I should like to know?\"{F}\n\nThe storm that had been for some time brewing, now burst forth with a\nroar that shook the rafters of the log-built tavern. Although\nimmeasurably tickled by Bob's speech, Richards and I had struggled\nsuccessfully with our disposition to laugh. At this moment, however, a\nstifled giggling was heard behind us, which immediately attracted the\nattention of Bob and his friends. \"A spy! a spy!\" shouted they; and\nthere was a sudden and general rush to the door, through which an\nunfortunate adherent of the opposite party had sneaked in to witness\ntheir proceedings. The poor devil was seized by a dozen hands, and\ndragged, neck and heel, before Bob's tribunal, to account for his\nintrusion. He set up a howl of terror, and probably pain, that\nimmediately brought to his assistance a whole regiment of his friends,\nwho were assembled in the adjacent tavern. A furious fight began, from\nwhich Richards and myself hastened to escape. We made our way into the\nkitchen, and thence into a court at the back of the house.\n\n\"Stop!\" said a whispering voice, as we were groping about in the\ndarkness; \"you are close to a pool that would drown an ox. I guess you\nwon't refuse my invitation now?\"\n\nIt was no less a person than Mr Isaac Shifty; and we began to consider\nwhether it would not really be better to put ourselves under his\nguidance. Indoors we could hear the fight raging furiously. We paused to\nthink what was best to be done. Suddenly, to our great astonishment, the\nnoise of the contest ceased, and was replaced by a dead silence. We\nhurried through the kitchen to the field of battle, and found that the\ncharm which had so suddenly stilled the fury of an Alabamian election\nfight, was no other than the arrival of the constable and his\nassistants, who had suddenly appeared in the midst of the combatants.\nTheir presence produced an effect which scarcely any amount of mere\nphysical force would have been able to bring about; and a single summons\nin the name of the law to keep the peace, had caused the contending\nparties to separate--the intruding one retiring immediately to its own\nheadquarters.\n\nWe passed a quiet and tolerably comfortable night, except that Bob\nthought proper to favour us with his society, so that we lay three in\none bed. Before break of day he got up, and went away. Tired as we were,\nit was much later before we followed his example. Upon entering the\ncommon room of the tavern, we found it empty, but bearing pretty evident\nmarks of the recent conflict. Chairs, benches, and tables, lay in\nsplinters upon the floor, which was, moreover, plentifully sprinkled\nwith fragments of broken jugs and glasses; and even the bar itself had\nnot entirely escaped damage. On repairing to the stable, to pay Caesar a\nvisit, I found my gig, to my no small mortification, plastered all over\nwith election squibs--\"Hurras for Bob Snags!\" and the like; while poor\nCaesar's tail was shorn of every hair, as close and clean as if it had\nbeen first lathered and then shaved. Our breakfast, however, was\nexcellent--the weather fine; and we set out upon our journey to Florence\nunder decidedly more favourable auspices than those that attended us on\nthe preceding day.\n\n\nFOOTNOTES:\n\n{A} There is no surer way of ascertaining the State from which an\nAmerican comes, than by his thinkings and guessings. The New-Englander\nguesses, the Virginians and Pennsylvanians think, the Kentuckian\ncalculates, the man of Alabama reckons.\n\n{B} The Mussel shoals are broad ridges of rocks, above Florence, which\nspread out into the Tennessee.\n\n{C} A corruption of Bourgogne, Burgundy wine.\n\n{D} John Quincy Adams, then president of the United States.\n\n{E} The Greeks, who at that time were struggling for their independence,\nhad received various succours from the United States. The Creeks are a\nwell-known tribe of Indians on the frontiers of Georgia.\n\n{F} Turk's island is a small island from which the Western States, North\nand South Carolina, Georgia, &c., get their salt.\n\n\n\n\nTHE EXECUTION OF MONTROSE.\n\n\nThe most poetical chronicler would find it impossible to render the\nincidents of Montrose's brilliant career more picturesque than the\nreality. Among the devoted champions who, during the wildest and most\nstormy period of our history, maintained the cause of Church and King,\n\"the Great Marquis\" undoubtedly is entitled to the foremost place. Even\nparty malevolence, by no means extinct at the present day, has been\nunable to detract from the eulogy pronounced upon him by the famous\nCardinal de Retz, the friend of Conde and Turenne, when he thus summed\nup his character:--\"Montrose, a Scottish nobleman, head of the house of\nGrahame--the only man in the world that has ever realized to me the\nideas of certain heroes, whom we now discover nowhere but in the Lives\nof Plutarch--has sustained in his own country the cause of the King his\nmaster, with a greatness of soul that has not found its equal in our\nage.\"\n\nBut the success of the victorious leader and patriot, is almost thrown\ninto the shade by the noble magnanimity and Christian heroism of the man\nin the hour of defeat and death. It is impossible now to obliterate the\ndarkest page of Scottish history, which we owe to the vindictive cruelty\nof the Covenanters--a party venal in principle, pusillanimous in action,\nand more than dastardly in their revenge; but we can peruse it with the\nless disgust, since that very savage spirit which planned the woful\nscenes connected with the final tragedy of Montrose, has served to\nexhibit to the world, in all time to come, the character of the martyred\nnobleman in by far its loftiest light.\n\nThere is no ingredient of fiction in the historical incidents recorded\nin the following ballad. The indignities that were heaped upon Montrose\nduring his procession through Edinburgh, his appearance before the\nEstates, and his last passage to the scaffold, as well as his undaunted\nbearing, have all been spoken to by eyewitnesses of the scene. A graphic\nand vivid sketch of the whole will be found in Mr Mark Napier's volume,\n\"The Life and Times of Montrose\"--a work as chivalrous in its tone as\nthe Chronicles of Froissart, and abounding in original and most\ninteresting materials; but, in order to satisfy all scruple, the\nauthorities for each fact are given in the shape of notes. The ballad\nmay be considered as a narrative of the transactions, related by an\naged Highlander, who had followed Montrose throughout his campaigns, to\nhis grandson, shortly before the splendid victory of Killiecrankie:--\n\n I.\n\n Come hither, Evan Cameron,\n Come, stand beside my knee--\n I hear the river roaring down\n Towards the wintry sea.\n There's shouting on the mountain side,\n There's war within the blast--\n Old faces look upon me,\n Old forms go trooping past.\n I hear the pibroch wailing\n Amidst the din of fight,\n And my old spirit wakes again\n Upon the verge of night!\n\n\n II.\n\n 'Twas I that led the Highland host\n Through wild Lochaber's snows,\n What time the plaided clans came down\n To battle with Montrose.\n I've told thee how the Southrons fell\n Beneath the broad claymore,\n And how we smote the Campbell clan\n By Inverlochy's shore.\n I've told thee how we swept Dundee,\n And tamed the Lindsays' pride;\n But never have I told thee yet\n How the Great Marquis died!\n\n\n III.\n\n A traitor sold him to his foes;{A}\n O deed of deathless shame!\n I charge thee, boy, if e'er thou meet\n With one of Assynt's name--\n Be it upon the mountain's side,\n Or yet within the glen,\n Stand he in martial gear alone,\n Or back'd by armed men--\n Face him, as thou would'st face the man\n Who wrong'd thy sire's renown;\n Remember of what blood thou art,\n And strike the caitiff down!\n\n\n IV.\n\n They brought him to the Watergate{B}\n Hard bound with hempen span,\n As though they held a lion there,\n And not a fenceless man.\n They set him high upon a cart--\n The hangman rode below--\n They drew his hands behind his back,\n And bared his lordly brow.\n Then, as a hound is slipp'd from leash,\n They cheer'd the common throng,\n And blew the note with yell and shout,\n And bade him pass along.\n\n\n V.\n\n It would have made a brave man's heart\n Grow sad and sick that day,\n To watch the keen malignant eyes\n Bent down on that array.\n There stood the Whig west-country lords\n In balcony and bow,\n There sat their gaunt and wither'd dames,\n And their daughters all a-row;\n And every open window\n Was full as full might be,\n With black-robed Covenanting carles,\n That goodly sport to see!\n\n\n VI.\n\n But when he came, though pale and wan,\n He look'd so great and high,{C}\n So noble was his manly front,\n So calm his steadfast eye;--\n The rabble rout forbore to shout,\n And each man held his breath,\n For well they knew the hero's soul\n Was face to face with death.\n And then a mournful shudder\n Through all the people crept,\n And some that came to scoff at him,\n Now turn'd aside and wept.\n\n\n VII.\n\n But onwards--always onwards,\n In silence and in gloom,\n The dreary pageant labour'd,\n Till it reach'd the house of doom:\n But first a woman's voice was heard\n In jeer and laughter loud,{D}\n And an angry cry and a hiss arose\n From the heart of the tossing crowd:\n Then, as the Graeme look'd upwards,\n He caught the ugly smile\n Of him who sold his King for gold--\n The master-fiend Argyle!\n\n\n VIII.\n\n The Marquis gazed a moment,\n And nothing did he say,\n But the cheek of Argyle grew ghastly pale,\n And he turn'd his eyes away.\n The painted harlot at his side,\n She shook through every limb,\n For a roar like thunder swept the street,\n And hands were clench'd at him,\n And a Saxon soldier cried aloud,\n \"Back, coward, from thy place!\n For seven long years thou hast not dared\n To look him in the face.\"{E}\n\n\n IX.\n\n Had I been there with sword in hand\n And fifty Camerons by,\n That day through high Dunedin's streets\n Had peal'd the slogan cry.\n Not all their troops of trampling horse,\n Nor might of mailed men--\n Not all the rebels in the south\n Had borne us backwards then!\n Once more his foot on Highland heath\n Had stepp'd as free as air,\n Or I, and all who bore my name,\n Been laid around him there!\n\n\n X.\n\n It might not be. They placed him next\n Within the solemn hall,\n Where once the Scottish Kings were throned\n Amidst their nobles all.\n But there was dust of vulgar feet\n On that polluted floor,\n And perjured traitors fill'd the place\n Where good men sate before.\n With savage glee came Warristoun{F}\n To read the murderous doom,\n And then uprose the great Montrose\n In the middle of the room.\n\n\n XI.\n\n \"Now by my faith as belted knight,\n And by the name I bear,\n And by the red Saint Andrew's cross\n That waves above us there--\n Ay, by a greater, mightier oath--\n And oh, that such should be!--\n By that dark stream of royal blood\n That lies 'twixt you and me--\n I have not sought in battle field\n A wreath of such renown,\n Nor dared I hope, on my dying day,\n To win the martyr's crown!\n\n\n XII.\n\n \"There is a chamber far away\n Where sleep the good and brave,\n But a better place ye have named for me\n Than by my father's grave.\n For truth and right, 'gainst treason's might,\n This hand has always striven,\n And ye raise it up for a witness still\n In the eye of earth and heaven.\n Then nail my head on yonder tower--\n Give every town a limb--\n And God who made shall gather them.--\n I go from you to Him!\"{G}\n\n\n XIII.\n\n The morning dawn'd full darkly,\n The rain came flashing down,\n And the jagged streak of the levin-bolt\n Lit up the gloomy town:\n The heavens were speaking out their wrath,\n The fatal hour was come,\n Yet ever sounded sullenly\n The trumpet and the drum.\n There was madness on the earth below,\n And anger in the sky,\n And young and old, and rich and poor,\n Came forth to see him die.\n\n\n XIV.\n\n Ah, God! That ghastly gibbet!\n How dismal 'tis to see\n The great tall spectral skeleton,\n The ladder, and the tree!\n Hark! hark! It is the clash of arms--\n The bells begin to toll--\n He is coming! he is coming!\n God's mercy on his soul!\n One last long peal of thunder--\n The clouds are clear'd away,\n And the glorious sun once more looks down\n Amidst the dazzling day.\n\n\n XV.\n\n He is coming! he is coming!\n Like a bridegroom from his room,{H}\n Came the hero from his prison\n To the scaffold and the doom.\n There was glory on his forehead,\n There was lustre in his eye,\n And he never walk'd to battle\n More proudly than to die:\n There was colour in his visage,\n Though the cheeks of all were wan,\n And they marvell'd as they saw him pass,\n That great and goodly man!\n\n\n XVI.\n\n He mounted up the scaffold,\n And he turn'd him to the crowd;\n But they dared not trust the people,\n So he might not speak aloud.\n But he look'd upon the heavens,\n And they were clear and blue,\n And in the liquid ether\n The eye of God shone through:\n Yet a black and murky battlement\n Lay resting on the hill,\n As though the thunder slept within--\n All else was calm and still.\n\n\n XVII.\n\n The grim Geneva ministers\n With anxious scowl drew near,{I}\n As you have seen the ravens flock\n Around the dying deer.\n He would not deign them word nor sign,\n But alone he bent the knee;\n And veil'd his face for Christ's dear grace\n Beneath the gallows-tree.\n Then radiant and serene he rose,\n And cast his cloak away:\n For he had ta'en his latest look\n Of earth, and sun, and day.\n\n\n XVIII.\n\n A beam of light fell o'er him,\n Like a glory round the shriven,\n And he climb'd the lofty ladder\n As it were the path to heaven.{J}\n Then came a flash from out the cloud,\n And a stunning thunder roll,\n And no man dared to look aloft,\n For fear was on every soul.\n There was another heavy sound,\n A hush and then a groan;\n And darkness swept across the sky--\n The work of death was done!\n\n W. E. A.\n\n\nFOOTNOTES:\n\n{A} \"The contemporary historian of the Earls of Sutherland records, that\n(after the defeat of Invercarron) Montrose and Kinnoull 'wandered up the\nriver Kyle the whole ensuing night, and the next day, and the third day\nalso, without any food or sustenance, and at last came within the\ncountry of Assynt. The Earl of Kinnoull, being faint for lack of meat,\nand not able to travel any further, was left there among the mountains,\nwhere it was supposed he perished. Montrose had almost famished, but\nthat he fortuned in his misery to light upon a small cottage in that\nwilderness, where he was supplied with some milk and bread.' Not even\nthe iron frame of Montrose could endure a prolonged existence under such\ncircumstances. He gave himself up to Macleod of Assynt, a former\nadherent, from whom he had reason to expect assistance in consideration\nof that circumstance, and, indeed, from the dictates of honourable\nfeeling and common humanity. As the Argyle faction had sold the King, so\nthis Highlander rendered his own name infamous by selling the hero to\nthe Covenanters, for which 'duty to the public' he was rewarded with\nfour hundred bolls of meal.\"--NAPIER'S _Life of Montrose_.\n\n{B} \"_Friday, 17th May._--Act ordaining James Grahame to be brought from\nthe Watergate on a cart, bareheaded, the hangman in his livery, covered,\nriding on the horse that draws the cart--the prisoner to be bound to the\ncart with a rope--to the Tolbooth of Edinburgh, and from thence to be\nbrought to the Parliament House, and there, in the place of delinquents,\non his knees, to receive his sentence--viz., to be hanged on a gibbet at\nthe Cross of Edinburgh, with his book and declaration tied on a rope\nabout his neck, and there to hang for the space of three hours until he\nbe dead; and thereafter to be cut down by the hangman, his head, hands,\nand legs to be cut off, and distributed as follows--viz., His head to be\naffixed on an iron pin, and set on the pinnacle of the west gavel of the\nnew prison of Edinburgh; one hand to be set on the port of Perth, the\nother on the port of Stirling; one leg and foot on the port of Aberdeen,\nthe other on the port of Glasgow. If at his death penitent, and relaxed\nfrom excommunication, then the trunk of his body to be interred, by\npioneers, in the Greyfriars; otherwise, to be interred in the\nBoroughmuir, by the hangman's men, under the gallows.\"--BALFOUR'S _Notes\nof Parliament_.\n\nIt is needless to remark that this inhuman sentence was executed to the\nletter. In order that the exposure might be more complete, the cart was\nconstructed with a high chair in the centre, having holes behind,\nthrough which the ropes that fastened him were drawn. The author of the\n_Wigton Papers_, recently published by the Maitland Club, says, \"the\nreason of his being tied to the cart was in hope that the people would\nhave stoned him, and that he might not be able by his hands to save his\nface.\" His hat was then pulled off by the hangman, and the procession\ncommenced.\n\n{C} \"In all the way, there appeared in him such majesty, courage,\nmodesty--and even somewhat more than natural--that those common women\nwho had lost their husbands and children in his wars, and who were hired\nto stone him, were upon the sight of him so astonished and moved, that\ntheir intended curses turned into tears and prayers; so that next day\n_all the ministers preached against them for not stoning and reviling\nhim_.\"--_Wigton Papers._\n\n{D} \"It is remarkable, that of the many thousand beholders, the Lady\nJean Gordon, Countess of Haddington, did (alone) publicly insult and\nlaugh at him; which being perceived by a gentleman in the street, he\ncried up to her, that it became her better to sit upon the cart for her\nadulteries.\"--_Wigton Papers._ This infamous woman was the third\ndaughter of Huntly, and the niece of Argyle. It will hardly be credited\nthat she was the sister of that gallant Lord Gordon, who fell fighting\nby the side of Montrose, only five years before, at the battle of\nAldford!\n\n{E} \"The Lord Lorn and his new lady were also sitting on a balcony,\njoyful spectators; and the cart being stopt when it came before the\nlodging where the Chancellor, Argyle, and Warristoun sat--that they\nmight have time to insult--he, suspecting the business, turned his face\ntowards them, whereupon they presently crept in at the windows: which\nbeing perceived by an Englishman, he cried up, it was no wonder they\nstarted aside at his look, for they durst not look him in the face these\nseven years bygone.\"--_Wigton Papers._\n\n{F} Archibald Johnston of Warristoun. This man, who was the inveterate\nenemy of Montrose, and who carried the most selfish spirit into every\nintrigue of his party, received the punishment of his treasons about\neleven years afterwards. It may be instructive to learn how _he_ met his\ndoom. The following extract is from the MSS. of Sir George\nMackenzie:--\"The Chancellor and others waited to examine him; he fell\nupon his face, roaring, and with tears entreated they would pity a poor\ncreature who had forgot all that was in the Bible. This moved all the\nspectators with a deep melancholy; and the Chancellor, reflecting upon\nthe man's great parts, former esteem, and the great share he had in all\nthe late revolutions, could not deny some tears to the frailty of silly\nmankind. At his examination, he pretended he had lost so much blood by\nthe unskilfulness of his chirurgeons, that he lost his memory with his\nblood; and I really believe that his courage had been drawn out with it.\nWithin a few days he was brought before the parliament, where he\ndiscovered nothing but much weakness, running up and down upon his\nknees, begging mercy; but the parliament ordained his former sentence to\nbe put to execution, and accordingly he was executed at the cross of\nEdinburgh.\"\n\n{G} \"He said he was much beholden to the parliament for the honour they\nput on him; 'for,' says he, 'I think it a greater honour to have my head\nstanding on the port of this town, for this quarrel, than to have my\npicture in the king's bedchamber. I am beholden to you, that, lest my\nloyalty should be forgotten, ye have appointed five of your most eminent\ntowns to bear witness of it to posterity.'\"--_Wigton Papers._\n\n{H} \"In his downgoing from the Tolbooth to the place of execution, he\nwas very richly clad in fine scarlet, laid over with rich silver lace,\nhis hat in his hand, his bands and cuffs exceeding rich, his delicate\nwhite gloves on his hands, his stockings of incarnate silk, and his\nshoes with their ribands on his feet; and sarks provided for him with\npearling about, above ten pund the elne. All these were provided for him\nby his friends, and a pretty cassock put on upon him, upon the scaffold,\nwherein he was hanged. To be short, nothing was here deficient to honour\nhis poor carcase, more beseeming a bridegroom than a criminal going to\nthe gallows.\"--NICHOLL'S _Diary_.\n\n{I} The Presbyterian ministers beset Montrose both in prison and on the\nscaffold. The following extracts are from the diary of the Rev. Robert\nTraill, one of the persons who were appointed by the commission of the\nkirk \"to deal with him:\"--\"By a warrant from the kirk, we staid a while\nwith him about his soul's condition. But we found him continuing in his\nold pride, and taking very ill what was spoken to him, saying, 'I pray\nyou, gentlemen, let me die in peace.' It was answered, that he might die\nin true peace, being reconciled to the Lord and to his kirk.\"--\"We\nreturned to the commission, and did show unto them what had passed\namongst us. They, seeing that for the present he was not desiring\nrelaxation from his censure of excommunication, did appoint Mr Mungo Law\nand me to attend on the morrow on the scaffold, at the time of his\nexecution, that, in case he should desire to be relaxed from his\nexcommunication, we should be allowed to give it unto him in the name of\nthe kirk, and to pray with him, and for him, _that what is loosed in\nearth might be loosed in heaven_.\" But this pious intention, which may\nappear somewhat strange to the modern Calvinist, when the prevailing\ntheories of the kirk regarding the efficacy of absolution are\nconsidered, was not destined to be fulfilled. Mr Traill goes on to say,\n\"But he did not at all desire to be relaxed from his excommunication in\nthe name of the kirk, _yea, did not look towards that place on the\nscaffold where we stood_; only he drew apart some of the magistrates,\nand spake a while with them, and then went up the ladder, in his red\nscarlet cassock, in a very stately manner.\"\n\n{J} \"He was very earnest that he might have the liberty to keep on his\nhat; it was denied: he requested he might have the privilege to keep his\ncloak about him--neither could that be granted. Then, with a most\nundaunted courage, he went up to the top of that prodigious\ngibbet.\"--\"The whole people gave a general groan; and it was very\nobservable, that even those who at his first appearance had bitterly\ninveighed against him, could not now abstain from tears.\"--_Montrose\nRedivivus._\n\n\n\n\nTHE WITCHFINDER.\n\nPART I.\n\n\nIt was towards the close of an autumnal evening, in the commencement of\nthe sixteenth century, that a crowd of human beings was dispersing from\nthe old market-place of Hammelburg, an ancient and, at that time,\nconsiderable town of Franconia, after witnessing the performance of a\nhideous and living tragedy. The Ober-Amtmann, or governor of the town,\nwho had presided over the awful occasion, had left, attended by his\n_schreibers_, or secretaries, the small balustraded terrace which\nadvanced out before the elevated entrance of the old Gothic town-hall.\nThe town-guard were receding in various directions, warning the crowd to\nseek their homes, and sometimes aiding with a gentle admonition of their\npike-heads those who lingered, as, slowly retreating, they moved down\nthe different narrow streets that led from the central market-place,\nlike streams flowing off in different channels after an inundation.\nWindow after window was closing in the quaintly-carved and\nstrangely-decorated gables of the houses; and many a small casement had\nbeen pulled to, over sundry withered old faces, that, peering from the\ndark and narrow aperture, and illumined by the glaring light that had\nfilled the market-place, had resembled some darkly-traced picture placed\nagainst the opening. In the middle of the square still smoked, in a\nheavy volume of cloud, the last gleaming ashes of a lately blazing pile,\nstill filling the air with a noisome stench. The night was closing\ndarkly in, and one human being alone seemed yet to linger in the\nmarket-place.\n\nIt would have been difficult, indeed, to discover that the dark object\njust discernible upon the edge of the blackened mass of smoking cinders\nreally was a human being, so shapeless was the form, so strangely was it\ncrouched down before the spot where the pile had been consumed. From\ntime to time only an upward-flung movement of two thin arms, as if in\nthe violent emotion of earnest prayer or deprecation, showed that this\nobject was a living thing; until, when the moon rose from behind the old\ntown-hall, disengaging itself, ever and anon, from among the heavy\nclouds of a gathering storm, its light fell full upon this indistinct\napparition, and revealed the form of a man, curiously bent together in a\nhalf-squatting, half-kneeling position. His head was bare. His long\ntangled black locks hung around a swarthy face, young still in years,\nbut worn and withered, and prematurely aged by sickness, sorrow, or\nviolence of passion--perhaps by the constant operation of all three. At\nthis moment it was ghastly pale, and bore the marks of the faintness and\nexhaustion attendant upon a reaction after intense excitement. The dress\nof this creature was not the usual costume of the lower classes, and\nconsisted almost entirely of a ragged and soiled garment of coarse brown\nlinen, made somewhat in the shape of a modern _blouse_, and bound round\nhis waist by a coarse leathern band. Around his neck hung a square bag,\nor satchel, which at once designated his calling to be that of a common\nbeggar, privileged by the religious authorities of the place. The stoop\nof his broad shoulders, between which the head was deeply sunk, told a\ntale of long sickness, which had broken a frame originally bold and\nstrong, and given a peculiarly ill-favoured appearance to a form\nnaturally well built; and when he arose from his squatting posture, the\nbent and withered appearance of his crooked legs, which no longer\npossessed sufficient strength to support the bulkier frame above, gave\npainful evidence that the wretched man had suffered cruelly from those\ncommon scourges of his class at that period--rheumatism and ague.\nClasped between his hands was a rosary of wood; and, as he rose, he\npressed it to his lips, and then deposited it in the upper part of his\ngarment.\n\n\"No, no!\" exclaimed the aloud, when he had staggered to his\nfeet. \"No, it is not vengeance--it is not, God knows; although the\nmalevolence of those hideous and accursed hags, those lemans of\nSatan\"--and he spat upon the ground--\"have made me the wretched outcast\nof humanity I am. The blood of the foul one has been shed for His glory\nonly, and that of the blessed Virgin, to the destruction of the\narch-enemy of mankind and his delusions!\"\n\n\"Thou knowest it is so,\" he added, again clutching forth the rosary from\nhis bosom, which, after gazing upon a rude personification of the\nVirgin, stamped upon a tiny plate of copper at the end of the string of\nbeads, and devoutly making the sign of the cross, he returned to its\nusual depository.\n\n\"I have cried against the handmaid of Beelzebub--uttering cry for cry as\nshe shrieked out her wretched soul. I have prayed earnestly and long,\nand I am athirst,\" continued the , as he dragged his distorted\nlimbs with difficulty over the rough stones towards a large covered\nwell, which occupied the lower part of the market-place.\n\nAs the beggar approached the parapet of the well, to drink from one of\nthe buckets which reposed upon its edge, he became first aware of the\npresence of another human being. Half-concealed behind one of the\ntwisted columns that supported the Gothic pavilion above, sat upon the\nparapet a female figure, dressed in a black garb of such a form and\nnature, that, without being the exact costume of any known religious\norder, it bore a monastic character. Her face, as she sat with her head\nbent down over her clasped hands, in an attitude of mournful\nhumiliation, was fully concealed by a black hood. But when, upon the\napproach of the beggar, she started up hastily, as if impelled by\nfeelings of horror and disgust, the moon shone full upon her, and\nrevealed the features of a woman of an advanced period of life, who\nformerly might have possessed much beauty, although now so washed out by\ntears, and furrowed by sorrow, that the whole character of her face was\nchanged. Her years, too, were probably very much fewer than her\nappearance denoted, for the signs of age upon her face bore less the\nmarks of time than of mental suffering. The symptoms of aversion which\nher manner displayed upon the beggar's approach, although instinctive\nand involuntary, and almost immediately restrained, had not escaped his\neye. His features expressed the bitter resentment of his heart at this\ninsult, and worked with ill-repressed feelings of anger and spite.\n\n\"Ha! Mother Magdalena--it is thou! Why flinchest thou at my approach?\nHast thou cause to fear me, then?\" exclaimed the with a sneer,\nas he drew nearer.\n\nThe female thus addressed shuddered at the sound of his voice; and,\nhastily pulling her dark hood more closely over her face, endeavoured to\npass on without reply; but the beggar caught her by the arm.\n\n\"Not so fast, beldam!\" he cried. \"I would have a word with thee. Dost\nthou not know me?\"\n\n\"Not know thee!\" exclaimed the dark female. \"Who in this wretched town\ndoes not know Schwartzer Claus, the witchfinder? What wouldst thou with\nme? Let me go!\"\n\n\"Why dost thou tremble, then, and turn away thy head?\" continued the\n. \"Why does Black Claus, the witchfinder--since such thou callest\nme--make thee shudder thus in every limb? The innocent have no cause to\nfear.\"\n\n\"Thou askest me why I shudder?\" said Magdalena in an excited tone,\nforgetting in her agitation her purpose of self-control. \"Thou hast\nforced me to speak, and I will tell thee. Is not thy hand yet reeking\nwith the bloody ashes of thy last victim? Has not a seventh unhappy\nwoman suffered this very day a cruel death at the stake upon thy hideous\ndenunciation; and thou askest me why I shudder?\"\n\n\"Beware, woman--beware!\" cried the witchfinder, lifting up his long\nright arm with a gesture of menace. \"Those who defend the evil-doer, and\nmalign the just and heaven-directed accuser, are not far from being\narraigned as accomplices themselves!\"\n\n\"What! thou seekest already another innocent sacrifice, wretched man!\"\ncontinued the female, tearing away her arm, which the beggar still held\nclenched in his left hand. \"Thou art not sated with the innocent blood\nthy false witness has this day shed?\"\n\n\"It is a lie!--it is a damning lie!\" screamed the , foaming with\npassion. \"I have borne no false witness! Besides, did not she avow her\ndeeds of darkness? did she not confess her complicity with the spirits\nof hell, and her harlotries with the arch-deceiver of mankind?\"\n\n\"Ay! when, tortured in mind and body, her poor weak old head gave way,\nand she unconsciously affirmed all that her torturers had for hours past\nbeen pressing upon her wavering understanding. Ye had driven her mad,\npoor wretch!\"\n\n\"'Tis false again!--'tis false!\" repeated the witchfinder. \"The truth\nspoke out of her at last, when her treacherous paramour, the demon, had\ndeserted her. God's glory and that of the holy church, for which I work,\nhad triumphed over the powers of darkness.\"\n\n\"Thou serve the holy church! Hear not the blasphemy, O Lord!\" cried the\nexcited woman, raising up her hands to heaven. \"Thou, miserable wretch!\nwho, for the favour of the Amtmann or the priest, for the pittance\nbestowed on thee in reward of thy discovery of the supposed foul\npractices of witchery and magic, art ever ready to sell the innocent\nblood of the aged, helpless, and infirm!\"\n\n\"For the lucre of gain!\" screamed the , but in a tone as much of\ndespair at this accusation as of wrath. \"For the lucre of gain! No--no;\nas God is my judge, it is not! My motives are pure; God and the Holy\nVirgin know they are! It is not even a spirit of revenge that instigates\nme. No--no! it cannot be; it _is_ not! If the words of my mouth have\ncondemned and killed, it is because my voice was uplifted in the cause\nof religion, and to the confusion of the prince of evil!\" But as he\nspoke, the beggar covered his face with his hands, with a shudder, as\nthough there passed in his soul a struggle with himself--a doubt of his\nown real motives.\n\nMagdalena was about to quit in haste her dangerous companion, when a\nsentiment of pity at the sight of the 's evident emotion seemed\nto mingle strangely with her disgust and aversion to the witchfinder. It\nwas even with an uncontrollable feeling of interest that she stopped for\na moment to look upon the wretched man.\n\nAfter a pause, the beggar removed his hands from his face, and uttering\nin a low tone the words, \"I thirst,\" staggered to the edge of the well,\nand seized the bucket within his hands. He bent over it but for a moment\nto drink, and could scarcely have swallowed many mouthfuls, before,\nflinging back the bucket into the well, he started up, and spat the\nwater from his mouth.\n\n\"Horror!\" he said, with a look of mingled terror and insanity--\"it\ntastes of blood!\"\n\n\"It is thy own conscience, poor man, that troubles the taste of the\nfresh element,\" said Magdalena solemnly; \"the water is pure and sweet!\"\n\n\"Thou hast done this, old hag!\" cried the witchfinder wildly; unheeding\nher remark. \"Thou hast corrupted the waters at the source. Why did I\nfind thee sitting here, cowering over the surface of the well, if it\nwere not to cast malefick spells upon the water, and turn it into\npoison--in order to give ills, and ails, and blains, and aches, and\npains, and sickness, and death to thy fellow-creatures? Ha! ha! I have\nlong thought it. Thou also art one of the accursed ones!\"\n\n\"Thou ravest, miserable wretch!\" replied the female; \"thou knowest not\nwhat thou utterest. God forgive thee, , thy wicked thought, and\nchange thy perverted mind!\"\n\nShe was again about to turn away, and leave her angry questioner, when,\nfearing the result of the evil feeling now fully excited in the\nwitchfinder's mind, she again paused to excuse herself in the eyes of\nthe dangerous man, and added--\n\n\"Thou canst not mean what thou sayest, Claus; I sat by the well but to\ncool my heated brow in the night-air, and taste the breath of heaven;\nfor my mind was saddened, and my head whirled, with the horrors that\nthis day has witnessed.\"\n\nBut her words were but oil upon the flame, and only served to augment\nthe wild infatuation of the witchfinder.\n\n\"Ah! thy mind was saddened! Thou hadst pity for that vile hag of hell!\nWas she thy comrade? Perchance thou hadst fear for thyself? Thou\nthought'st thy own time might come? Thy own time _will_ come, old\nMagdalena. My eye is upon thee and thy dark practices; it has been upon\nthee since thou camest, unknown and unacknowledged, to this place, none\ncould tell when, and whence, and how. Ay, my eye is upon thee,\nand--beware!\"\n\nWillingly would the woman now have shrunk away before the maddened\nwitchfinder's objurgation; but the wild accusation thus thundered\nagainst her froze her with terror, and riveted her to the spot.\n\n\"I have marked thee well,\" continued the frantic man, \"and I have seen\nthee pause upon the threshold of the holy house of God, and kneel in\nmockery upon the steps before it: but thou hast never dared to enter it.\nThou knewest well that the devil thou servest would have torn thee in\npieces hadst thou done it. Ha! do I catch thee there?\" he continued, as\nat these words the woman buried her face between her hands.\n\n\"Thou canst not deny it!\" shouted the witchfinder with an air of\ntriumph.\n\n\"God best judges the motives of the heart,\" murmured Magdalena.\n\n\"I will tell thee more, vile hag, and thou shalt hear it face to face,\"\npursued the , seizing the poor woman's arms with his long bony\nfingers, and dragging her hands from before her face, in spite of her\nefforts at resistance. \"Thou watchest at street corners and in doorways,\non the bridge or on the causeway, to see fair Fraulein Bertha, the\nOber-Amtmann's daughter, ride past upon her ambling jennet, or mount the\nchurch-steps, her missal in her hand. Thou watchest her to cast thy\nspells upon her. Thou hatest her for her youth and beauty and spotless\npurity, like all thy wretched tribe, whom the sight of innocence and\nbrightness sickens to the heart's core. Thou wouldst fascinate her with\nthy eye of evil and thy deadly incantations.\"\n\nThe moon, the light of which still struggled faintly through the\nfast-accumulating clouds, shone for a moment upon the face of old\nMagdalena, as the pronounced these words. Her features were more\ndeadly pale than usual, and convulsed with an excess of agitation at\nthis mention of Bertha's name, which she evidently struggled to control\nin vain.\n\n\"Ah! I have thee there again!\" screamed Claus in triumph a second time.\n\"Already have I seen her cheek grow pale, her head bow down like a\nblighted flower, her walk become weary with faintness. Hast thou already\nbeen at thy filthy machinations? But Black Claus, the witchfinder, is\nthere to wrestle with the powers of evil. And hear me! That fair sweet\ngirl is the only comfort of my wretched life. My soul grows calm and\nsoothed when I look upon that lovely face. A ray of sunshine gleams upon\nthe darkness of my path when her smile beams upon me. My heart leaps\nwithin me for joy when her small white hand drops an offering into my\nbeggar's bowl. She is my only life, my only joy, and my guardian angel.\nAnd couldst thou harm her, woman, no torment should be too horrible for\nthee, body and soul. The chains of the stake still lie upon the\nmarket-place--the ashes of yon pile still reek with heat; and the pile\nshall rise again, the chains shall bind once more. Wretched hag! I bid\nthee again beware!\"\n\nAs with one hand the raving witchfinder pointed to the spot where one\nunhappy woman had already perished that day, a victim to the\nsuperstition of the times, Magdalena, who, during his praise of the fair\ngirl, had again looked at him with awakened interest, disengaged herself\nfrom the other. \"God's will be done!\" she said with humility. \"I am\nprepared for all. But thou, unhappy man!\" she continued, \"beware in\nturn, lest, before thou hast time to repent thee of the hardness and\ncruelty of thy heart, His judgement fall on thee, and his justice punish\nthee.\"\n\nShe spoke with hand upraised to heaven; and then, pulling her hood over\nher face, hurried from the market-place.\n\nThe witchfinder gazed after her, fixed to the spot, and for a moment\nawe-struck by her words. As he still stood struggling with his various\npassions, the storm, which had been gathering ever since sunset, began\nto burst over his head. The rain came down in torrents.\n\n\"Ah! was it that?\" screamed the beggar, with a fit of wild laughter.\n\"The miserable old beldam! she stretched out her finger to the sky, and\nit was to bring down these waterspouts upon my head. Curses on the foul\nmalicious fiend!\" And he spat upon the ground, as if to exorcise the\nevil spirit.\n\n\"But I must find shelter,\" he murmured. \"Already pains rack my limbs; my\nbones ache; a shudder runs through my frame! The old hag has worked her\nspell upon me. _Apage, Sathanas!_ Anathema!\"\n\nSpeaking thus, the wretched man shuffled along as fast as the crippled\nstate of his limbs, and the acute pains of rheumatism, which the damp\nnight-air had again brought upon him, would allow him to proceed. He\nstaggered to the shelter of a doorway, which was placed under the\nadvancing terrace of the town-hall, and between two staircases which\ndescended on either side on to the market-place. The protruding vault of\nthe Gothic archway afforded him some refuge from the storm, which now\nburst down with increased violence. But the excited witchfinder's brain\nseemed to wander, as he caught an indistinct vision of the gaping jaws\nof the dragons and other grotesque monsters, which protruded as\nwaterspouts from the roofs of the surrounding houses, and now disgorged\ntorrents of rain.\n\n\"Spit, spit, ye devils all!\" he shouted aloud. \"Ye cannot reach me here.\nHa! ha! rage, storm, spew forth your venom, do the bidding of your\nmistress--I defy you!\" And as the wind swept round the corners of the\nbuilding, and spattered some of the water of the gushing cataracts in\nhis face, he cried, \"Avaunt!\" as if speaking to a living thing, and,\nclinging to the bars of an aperture in the upper part of the door,\nturned away his face.\n\nAs he thus came to look upon the strongly-barred opening in the door,\nthe current of his ideas changed. Within was the small and wretched\nprison of the town, which just occupied the space of the terrace\nabove--a miserable hole.\n\n\"There she lay this morning,\" he murmured, looking into the interior,\nwhich was now in utter darkness, and quite empty--\"there she lay, old\nMartha Dietz, and called in vain upon the demon who deserted her. There\nhave lain all the foul hags who tortured my poor aching limbs. There\nshall _she_ lie also, the scoffer and reviler, the worker of evil. The\nwitchfinder will be revenged. Revenge! no, no! He will do the work of\nthe holy church. Who shall say the contrary? Not thou, old Martha--nor\nthou--nor thou. If ye say so, ye lie in death, as ye have lied in life.\nAy! glare upon me with your lack-lustre eyes. Ye are powerless now,\nthough ye are there, and make mouths at me. One--two--three--God stand\nby me! There they are--_all seven!_\"\n\nWith a wild scream of horror, the covered his eyes with his\nhands, and rushed forth into the tempest.\n\nSituated in the picturesque and fertile valley of the Saale, the town of\nHammelburg stands upon a gentle declivity, commanding one of the\nnumerous windings of the river, and sloping downwards to its banks. A\npart of the old walls of the town is thus bathed by the waters of the\nstream, which, calm and peaceful in the summer months, become\ntumultuous, and even dangerous, during rainy weather, or after the\nmelting of the snows. From the ancient gateway of the town on the river\nside, a triple bridge of great length and many arches, which, in the dry\nseason, seems to occupy a most unnecessary space across the narrower\nwaters, but which, at other times, scarce suffices to span the extent of\nthe invading inundation, affords a communication with the high-road.\n\nAt the commencement of the sixteenth century, this bridge was only\nconstructed of wood, and although put together with rude strength,\nill-sufficed to resist the force of the torrents, and had been\nrepeatedly swept before them.\n\nNot far from the town gateway that commanded this bridge, stood a huge\nmansion, constructed as a palace for the Prince Bishops of Fulda, the\nsovereign rulers of the district; although, at the period in question,\nit had been ceded to the Ober-Amtmann, a near relation of the reigning\nbishop, as his official dwelling. On the side of this ancient palace\nfurthest removed from the town gate, ran, along the river's banks, its\nspacious gardens, abutting at their extremity upon the premises of an\nextensive Benedictine monastery, from which they were only separated by\na narrow lane, that led from the town to the river. At the very angle of\nthis lane, where it opened by a small water-gate upon a narrow\ntowing-path, skirting alike the town-walls and the banks of the stream,\nthere stood a low building attached to the monastery, the upper story of\nwhich thus overlooked the old gardens of the palace on the one hand,\nand, on the other, the river banks.\n\nAt one of the windows of this humble dwelling, that which overlooked the\npalace gardens, stood a young man, intently gazing through its small\noctagon panes. Two or three times he turned away with a heavy sigh, as\nif wearied with long and vain watching, and as often returned again to\nhis previous occupation. At length the opening of the door of the room\nstartled him from his position; and as if ashamed of being caught in the\nact of looking out, he hurried to a table in the middle of the room, and\nflung himself into an old chair.\n\nThe various objects with which the table was covered, as well as those\nwhich filled and littered the room in all directions, clearly designated\nthe young man's employment to be that of a sculptor and colourer of\nimages for the ornament of churches, as well as an illuminator of\nmissals and manuscripts--an occupation at that time still pursued,\nalthough gradually falling into disuse since the invention of printing.\nScattered about upon the table were several old parchment manuscripts,\nwhich had served as models for the artist's use, or had been confided to\nhis hands to clean. Old illuminated missals, some of the gorgeous\nillustrations of which were open, as if lately retouched by the hand of\nthe young painter, lay here and there. At the further end of the table\nstood a small figure of a Virgin and Child, delicately and exquisitely\ncarved, and painted with the richest colours. The group was bright with\nits fresh finish, and evidently had not long been completed by the hand\nof the artist. Upon an elevated bench or dresser were littered the tools\nof the sculptor and wood-carver, with a few unfinished trials of small\nsaintly figures; and around the room were fragments of wooden images of\nsaints, some discoloured, some broken, a few in tolerable preservation,\nwhich were either destined to be restored and repainted, or had served\nas studies for the artist. Upon the walls hung a few pictures of female\nsaints, bedecked with garlands of flowers, which showed them to be\nobjects of devotion and respect in the eyes of the possessor. Among all\nthis confusion, space was scarcely left, in the small chamber of the\nartist, for the pallet-bed and cumbrous press that formed his only\nfurniture.\n\nImmediately before the chair into which the young man so hastily flung\nhimself, lay a rich missal, upon the adornment of which he had been\nemployed, before other thoughts and feelings had sent him to the window;\nand when he again resumed his work, it was upon the face of a fair\nsaint, which formed the headpiece of a chapter, peering out from among\nthe various graceful arabesques that twined in the brightest colours\nalong the margin of the leaf.\n\nIn truth, the face of the young artist was almost as fair as that of the\nbright being he was engaged in painting. His light brown hair was parted\nin the middle, over a high white forehead, and fell in faintly waving\ncurls almost to his neck, forming a frame to the soft oval face, to\nwhich his violet-blue melancholy-looking eyes, his calm,\nfinely-chiselled features, and the serious repose of his imaginative\nmouth, imparted an air of gentleness and thoughtfulness combined. His\ndark, sober-, simple dress, although somewhat too severe to suit\nhis youthful figure, accorded well with the character of his\nphysiognomy. His falling collar displayed a full white throat, which\nmight have served as a model for a statue of Antinous, had it not borne\nmore the stamp of genius in its proportions than of physical\nvoluptuousness. The hands, which now hastily resumed their neglected\noccupation, had all the fairness and well-moulded contour of a woman's,\nwithout that delicacy of size which would have stamped them as\neffeminate. Had he been aware of his own beauty, he might have copied\nhis own graceful form for a personification of the lily-bearing angel in\na group of the Annunciation.\n\nThe person who had startled him from the window, by opening the door of\nhis room, was an aged-looking woman, in a plain dress of coarse black\nserge. She bore in her hands a coarse brown porringer filled with\nsteaming viands, a lump of dark homely bread, and a white cloth.\n\n\"Ah! my good Magdalena, art thou there?\" said the young artist, raising\nhis head with an almost unconscious affectation of surprise, as though\nunexpectedly disturbed at his work.\n\n\"You forget all hours, and all human wants, in your zeal for your\nbeautiful art, Master Gottlob,\" said the woman. \"I bring you your\nnoon-day repast, which you would never have called for, had I allowed it\nto stand by even until sundown. But I have ventured to transgress your\norders. You must be faint with long fasting;\" and the old woman made a\nmovement as if to place the food upon the table before the artist.\n\n\"Thanks, good Magdalena! thanks!\" said the young man, looking at her\nwith that sweet smile, and tender expression of his mild blue eyes,\nwhich had procured him, among all who knew him, the constant designation\nof \"Gentle Gottlob;\" but at the same time repelling the porringer. \"Not\nhere. Place the food elsewhere. I will eat anon. I am not hungry now;\nand I must not leave my work. I have promised it to his noble reverence\nthe prior, for the eve of the fete of St Ursula, and to-morrow is the\nvery day. There is still much to do. It seems as if I could never give\nsufficient finish to this face, or impart to it, with my dull colours\nand rebellious pencil, that look of heavenly brightness that ought to\ndwell upon it. And yet, alas! I would it never could be finished! It\nwill break my heart to part with it--although I love not my own work,\nnor deem it excellent. But still I cherish it--all imperfect as it is--I\nknow not why; and when to-morrow comes, and I must give it up into his\nreverence's hands, it seems that my life and spirit would depart from me\nwith its loss, and that all around me would be dark and joyless.\"\n\nAfter placing the porringer and bread upon a spare corner of the\nsculptor's working bench, Magdalena moved gently behind the young man's\nchair, and having asked respectfully his pardon, looked over his\nshoulder. At the sight of the fair face upon which the young artist was\nbestowing so much care, her looks betrayed feelings of surprise, mingled\nwith much emotion. Once or twice she passed her hand over her eyes, as\nif doubting the reality of what she saw. It was some time before she\ncould sufficiently master her agitation to speak; and when at last she\nspoke, after a long-drawn sigh, it was with a tone which still betrayed,\nin spite of her efforts, the interest inspired in her by the painter's\nwork of art.\n\n\"It is indeed a fine performance, and right bravely limned,\" she said;\n\"and in truth the countenance you have given to yonder saint, with the\npale glory, is one of exquisite beauty. I wonder not that you should be\ngrieved to look upon so sweet a face no more; although, methinks, I know\na face as fair, to which it bears a marvellous resemblance.\"\n\n\"What meanest thou, Magdalena?\" said the young artist, bending his head\nstill lower over his work. \"Whom dost thou know who could bear a\nlikeness to this creation of my own imagination?\"\n\n\"Of your own memory, Master Gottlob! you should have said,\" pursued\nMagdalena. \"Surely--or my eyes deceive themselves most strangely--although\nin that sweet face they were not easily deceived; surely the face is\nthat of\"----\n\nThe old woman again paused, as if to suppress her emotion.\n\n\"Of whom?\" enquired Gottlob in a low tone, also in much agitation.\n\n\"Of the fair Fraulein Bertha, the noble Ober-Amtmann's daughter.\"\n\n\"You think so, Magdalena?\" replied the young man. \"Perhaps it maybe a\nslight shade of a resemblance, caught unconsciously\"----\n\n\"It is she herself,\" exclaimed Magdalena. \"It is the same angelic\nsmile--the same beam of innocent brightness athwart her brow! It is\nshe!\"\n\n\"Perhaps thou art right,\" stammered Gottlob, still in much confusion,\nbut evidently well pleased with the species of praise thus bestowed\nupon his performance. \"There is, in truth, more resemblance to the\nFraulein Bertha than I had thought.\"\n\nMagdalena seemed for a minute lost in her reflection, as if a new and\npainful idea had struck her; and after giving a long and anxious look at\nthe window, from which the young artist had drawn back upon her\nentrance, she pressed her hand heavily to her heart, as if to support\nher in a sudden resolution, and, advancing to the artist's side, said in\nan earnest tone, \"Young man! thou lovest her!\"\n\n\"Magdalena! thou knowest not what thou sayest,\" cried Gottlob, more\nharshly than as the wont of his gentle nature.\n\n\"Oh! pardon me if I have offended. Condemn me not!\" said the excited\nwoman. \"But I do entreat you, tell me! Tell me your secret as you would\nconfide it to a mother--to your own mother, Gottlob. It is the purest\ninterest for you--for her--that guides me! I swear it to you! Oh! tell\nme--is it not so? You love that fair and gentle girl!\"\n\nThe young man looked at his strange interrogator with some astonishment\nat her evident agitation. The tears were swelling in her eyes. But\nwithout pausing to question the reasons of her emotion--so absorbed is\nlove in its own self--he rose, and took the old woman's hand.\n\n\"Yes! I will speak; my heart has long been overcharged with its own\nsecret, even to bursting,\" he said; \"and it throbs to unburden itself\ninto some sympathizing heart! And why not thine, good Magdalena? Ever\nsince fate has brought us so strangely together, thou hast been like a\nmother to me!\"\n\n\"Do not I owe you all?\" interrupted the old woman; \"my life--my daily\nbread--a shelter for my old limbs in the cell below?\"\n\n\"Alas! I have but little to give, poor Magdalena!\" said the young man\nkindly.\n\n\"And that little thou hast shared with me as a son,\" continued Magdalena\nbending her head over his hand as if to kiss it.\n\n\"Yes, thou shalt know all,\" pursued Gottlob; \"for it would seem as\nthough the destiny that threw thee in my way were linked with hers. Her\nimage it was that led me to the spot where first I saw thee. It was the\nlast day of the Carnival, at the beginning of this year, and there was a\nfete at the palace of the Ober-Amtmann. I had long gazed with adoration\nupon that angelic face, and treasured it in my heart. I already\nworshipped yon saintly portraits, because in one--God forgive me the\nprofane thought!--I had found a faint forth-showing of the beam of her\nbright eye; in another, the gentle, dimpled smile of her sweet mouth; in\na third, her pure and saint-like brow. It was not for such as I, a poor\nartist, to be invited to the noble Amtmann's fete; but I thought that,\nthrough the windows in the illuminated halls, I might perchance trace\nher passing shadow. I fancied that, by some unforeseen accident, she\nmight come forth upon the terrace, overhanging the river's banks--a\nfoolish fancy, for the night was wintry and cold. I hoped to see her, no\nmatter how; and I wandered out of the town--for its gates were open for\nthat holiday--to look upon the lighted windows of the palace from the\nopposite side of the stream. The snow was on the ground. My mantle\nscarcely preserved me from the bitter cold. But I felt it not. It was\nonly when a groan sounded near me, that I thought on the sufferings of\nothers in such a night. I looked around me; and there, not far from me,\non the snow, before the very windows of the palace, where within was\nmusic and dancing, and feasting and mirth, lay thy form, poor Magdalena!\nFeeble, helpless, stiff with cold, thou appearedst to me in the last\nagonies of death.\"\n\n\"Yes; I had laid me down to die, in sorrow and despair. It is too true,\"\nsobbed the old woman, in a voice choked with tears. \"But your hand\nraised me up--your arms warmed me into life--your voice encouraged me,\nand gave me force. You brought me to your home, fostered me, and nursed\nme--me, an unknown outcast, whose very history you did not even seek to\nknow--whose silence and secrecy you respected. Your kindness saved me\nfrom despair, and gave me hope; and I lived on, in order to pay, were\nit possible, my debt of gratitude to my preserver.\"\n\n\"Good Magdalena,\" said the young man soothingly, taking her withered\nhands between his own, \"I did but the duty of a Christian man.\"\n\n\"And you love her, then?\" resumed Magdalena, recalling her young\npreserver to his promised confidence.\n\n\"Love her!\" exclaimed Gottlob with an impassioned fervour, which gave\nhis gentle face a look of inspiration. \"Love her! She is my vision by\nday--my dream by night. When I read, it is her voice that seems to speak\nto me from the Minnesinger's poesy. When I paint, it is her form that\ngrows under my pencil. When I pray, it is her seraphic smile that seems\nto beam upon me down from heaven. I wander forth: it is to meet her in\nher walks. I kneel in the church: it is to breathe the same air as she!\"\nAt these words, Magdalena covered her face, and uttered a suppressed\ngroan. \"I rise from my labour, which of old was a labour of love to me,\nand now is oft an irksome task: it is to watch for her coming forth into\nthe garden. I have neither rest by day nor by night. Where there was\nrepose in my heart, there is now eternal fever.\"\n\n\"And she?\" said Magdalena with a low tone of anxiety, as if fearful of\nthe answer she might receive. \"Does she know--does she return your\nlove?\"\n\n\"How should she deign to remark a worm like me?\" was the young artist's\nanswer. \"How should I dare to breathe my affection in her ear, were it\neven possible for me to approach her? And yet she looks upon me kindly,\"\ncontinued the young lover, encouraging himself in vague hopes, at the\nsame time that he condemned their presumption. \"When I doff my cap to\nthe noble Amtmann's daughter, as she ambles forth by her proud father's\nside, she will answer with so sweet a smile, and greet me with a wave of\nher riding-switch--with what a grace!--and then grow red thereby, and\nthen grow pale. When I offer her the holy water as she passes from the\nchurch, she will cast down her trembling eyelids, and yet will see\nwithal who offers it; and when I stand at yon window, as she rambles in\nthe garden, she will pluck flower after flower, as though she knew not\nwhy; then fling them all aside, then pick them up with care; then\ndisappear as if she had gone back, and yet come forth again.\"\n\nMagdalena's brow grew thoughtful and anxious as Gottlob proceeded in his\nenumeration of these symptoms. Her bosom heaved painfully, her hands\nwere clenched together.\n\n\"Poor child! should it be so!\" she murmured, casting her eyes upon the\nground; and then, raising them again to Gottlob's face, into which she\nlooked with scrutinizing eagerness, she said aloud--\"And yet you do not\nthink she loves you?\"\n\n\"She love me!\" cried the young man. \"Such a dream of bliss were madness!\nCan I forget the immeasurable gulf that separates the noble daughter of\nthe high-placed Amtmann from the poor and humble artist--the dependent\nof a cloister? No, Magdalena. I must die as I have lived, the poor\nunloved and uncared-for orphan--die without a sigh of pity, without a\ntear of sorrow from her eye.\"\n\n\"Have you, then, no friends, poor youth?\" said Magdalena.\n\n\"None. Yes! I am ungrateful. I have one--a kind protector; but he is far\nremoved, and I have seen him seldom.\"\n\n\"The Prince Bishop of Fulda!\" repeated the old woman, with some degree\nof agitation. \"Perhaps--yet it is a wild and foolish thought--perhaps\nall hope is not shut out to you.\"\n\n\"What sayest thou, then, old Magdalena?\" said the youth. \"Hope were but\ntorture were it vain; and so it must be\"----\n\n\"Yes. I was wrong. Heed not my words! But know you not that your patron,\nthe bishop, is close at hand? Already I have heard that he arrived this\nmorning at his castle of Saaleck, at half a league's distance from the\ntown; and he will probably shortly enter Hammelburg, as is his wont.\"\n\n\"These are glad tidings!\" said Gottlob, his eyes beaming with joy. \"I\nwill at once to Saaleck, and, if the prince admit me to his presence,\nthrow myself at his feet, assure him of all my gratitude for the past,\nand offer him my poor service for the future.\"\n\nWith these words the young man hurried to his cumbrous chest, and\npulling out a short cloak, flung it around him. A small cap of black\nvelvet, of the cut of the time, which showed off to advantage the beauty\nof his youthful face, was hastily thrown upon his head. He was about to\nquit the chamber, when Magdalena caught him by the arm.\n\n\"Thy repast, Master Gottlob.\"\n\n\"Have I time to think of that?\" said the eager youth, swallowing,\nhowever, in haste a few mouthfuls of the broth, to satisfy the old\nwoman's look of supplication.\n\n\"And when you mount or descend the mountain-path that leads to the\ncastle on its brow,\" said the old woman, during Gottlob's hasty meal,\n\"if you can still have a thought for poor old Magdalena, she begs you\nenter the chapel on the mountain-side, which is esteemed so holy that it\nis permitted to be a sanctuary of refuge to the criminal, and say a\nshort prayer for her soul's weal.\"\n\n\"Can those so good and kind as thou, Magdalena, need the prayers of such\nas I?\" said the young man.\n\n\"The fervent supplications of the young and pure at heart are always\nacceptable,\" replied Magdalena evasively, but in a sad and earnest tone.\n\n\"So be it--and fare-thee-well,\" said Gottlob, finishing his last\nmouthful, and hurrying to depart.\n\n\"And heed you, gentle youth,\" again cried Magdalena, \"as you cross the\nbridge to leave the town. The river is much swollen with the late rains,\nso much as to threaten destruction to the tottering fabric.\"\n\n\"I fear no such danger,\" was the young man's reply; \"and besides, have I\nnot thy charm?\" he continued, laughing, holding up a black ring\ninscribed with strange characters, that hung about his neck.\n\n\"Oh, say not so!\" said the old woman earnestly, as a recollection of the\nWitchfinder's dreadful threats the night before came across her mind.\n\"Call it not a charm! The holy church permits not of such dealings. It\nwas but a remembrance that I gave you, to think sometimes on the poor\nwretch whose life you had preserved. It was of little value; but I had\nnought else to give. I prayed only that it might bring happiness to you,\nboy, for it had brought nothing but misery and wretchedness to me.\"\n\nLong before old Magdalena could complete her sentence, the eager youth\nhad left the room. The old woman looked after him for a time with a look\nof gratitude, and then, hurrying to the artist's table, threw herself\ndown upon her knees beside the open missal, and gazed with intense\neagerness upon the picture of the fair saint upon which he had been\npainting. She approached her lips as if to kiss it; then again drew\nback, as if she feared to mar the colouring by her caress: then gazed\nagain, until her eyes filled with tears: and at last, with the cry,\n\"Yes! it is she--her very self!\" burst into a fit of convulsive sobbing,\nand buried her face between her hands.\n\nAs she still lay crouched upon her knees, a partly-concealed door, which\nled towards the monastery, and was almost in disuse, slowly opened, and\na figure, enveloped in a monk's robe and cowl, entered the room.\n\nMagdalena was not at first aware of the entrance of the stranger; and it\nwas only when, after looking about the room, as if to assure himself\nthat no one was there, he approached the table, that she heard the\nfootstep, and lifted up her head in surprise. The intruder evidently as\nlittle expected to find the room already tenanted; for he also started\nupon seeing the kneeling woman. But the astonishment of both parties was\ngreatly increased when their eyes met each other. Far from attempting to\nrise from her knees, Magdalena remained in an attitude of supplication\nbefore the stranger, who was an aged man of mild aspect, and folding her\narms across her heart, bent down her head like a penitent, in order to\navoid his scrutinizing look.\n\n\"Magdalena! thou here!\" said the seeming monk, in a tone of voice which,\nnaturally that of benevolence, he evidently strove to render harsh and\nsevere. \"How comes this? Thou hast left, without my knowledge, the\nseclusion of the convent in which I placed thee? In defiance of thy\nsolemn promise, and thy accepted vow of penitence, thou hast approached\nthis town--thou hast sought, perhaps, forgetful of thy oath\"----\n\n\"No, no,\" interrupted the agitated woman, \"that cruel oath has sealed\nmy lips for ever. God knows, and you, reverend father--you know, that I\nhad accepted the bitterest trial woman can bear on earth, in expiation\nof my past sin. Long did I observe my vow of penitence without a murmur\nto heaven or to you. But I thought to die. A fever had seized me, and a\nburning thought came over me that I no longer could withstand. O God,\nforgive me--but my head was turned--I knew not what I did! I longed to\nsee once more on earth that object that was my only earthly joy. That\nuncontrollable desire overcame the stubborn resolution of a vow, which\nlong years of tears and mortification had striven to fortify in vain. I\nfled. I hoped once more to glad my eyes--but once----but once, my\nfather, and then to lay me down and die, trusting in God's pardon and\nyour reverence's.\" And Magdalena bowed her head to the ground, as a\ncriminal awaiting her sentence.\n\n\"Thou hast erred, woman--bitterly and grievously,\" replied the stranger\nharshly, adding, however, with a feeling of indulgence that his kindly\nnature evidently could ill suppress, \"but the struggle of the spirit\nwith the weakness of the body, in sickness and in fever, is heavy to\nbear. And yet,\" he continued, again assuming a severity of manner, \"thou\nlivest, and I still find thee here. Thou hast remained to feast thy eyes\nupon thy earthly treasure, in forgetfulness of thy vow of mortification\nfor thy soul's weal.\"\n\n\"Pardon!\" cried Magdalena, raising her hands in supplication.\n\n\"But thou must leave this place forthwith,\" continued the monk. \"Return\nto the convent, and employ thyself in such acts of penitence as my\norders shall prescribe.\"\n\n\"Pardon!\" again cried the unhappy woman, \"for my vow is heavier than I\ncan bear. It is a task beyond the force of human nature!\"\n\n\"Foolish woman!\" exclaimed the stranger. \"Wouldst thou compromise the\nhappiness and peace of mind of the being thou lovest best, by the danger\nof a discovery to which thy presence here might lead? Thy expiation is\nsevere. Such as we, alas!\" and the monk heaved a sigh, \"who cannot feel\nthe vibration of some of the tenderest chords of humanity, know not how\nto sound in its profundity; but I can judge that it must be grievous to\nbear. Still it must be so. Go, then, in peace--but go. What I command no\nlonger in the name of thy salvation, I ask of thy heart, for the repose\nof thy heart's treasure.\"\n\n\"Father,\" said the penitent, sobbing at his feet--\"I obey! But I have\nstill a secret to impart to you, upon which depends, perhaps, the\nhappiness of that beloved one. Oh! deign to hear me.\"\n\n\"In three days hence, let me receive thy shrift at the convent of Saint\nBridget,\" continued the ecclesiastic. \"There also I will hear thy\nsecret. But tell me,\" he added, looking round the room with some\nsurprise--\"how comest thou here in gentle master Gottlob's studio?\"\n\n\"It was he who saved my life,\" answered Magdalena, striving to repress\nher sobbing, \"when in the midst of the snows, and the keen blast of\nwinter, death had laid hands upon me. Ever since, he has cherished and\nnourished the unknown outcast in his abode.\"\n\n\"Generous youth!\" said the stranger. \"I came to witness, alone and\nunbiassed, his progress in his noble art; and I find that the heart\nsoars as nobly as the head. So should ever be true genius! Yes, yes!\" he\nmurmured to himself, looking around, \"he advances towards perfection\nwith rapid strides. This arabesque is exquisite. And this head, how\nbeautiful! And yon statue of our Holy Mother--what heavenly grace in its\nfashioning!\"\n\nAnd with more of such commendatory observations, interspersed now and\nthen with a few gentle criticisms, which showed the connoisseur as well\nas the gratified admirer, he took up and examined the various designs\ndispersed upon the table. When his curiosity seemed fully satisfied, he\nagain turned to Magdalena.\n\n\"I must away,\" he said; \"for I have still many arduous and painful\nduties to perform, and my time is limited. I rely upon thy strict\nsecrecy, Magdalena. I would not it should be known that I was here. And\nremember, in three days at Saint Bridget's convent!\"\n\nWith these words he stretched forth his hand. She again knelt, and\nkissed it devoutly; and pulling his black robe and cowl more closely\nabout his face and person, the monk disappeared by the concealed door.\n\nMagdalena still knelt, overcome by her various emotions, when a sound\nfrom the window looking into the river startled her, and caused her to\nturn round. An involuntary scream burst from her lips; for from among\nthe branches of a tree that grew upon the river's banks, and overhung\nthe window, peered, through the dingy panes, the pale face of the\nwitchfinder.\n\nIt was about the hour of vespers; and an unusually dense crowd of the\ntown's people of Hammelburg, of all ages, ranks, and sexes, swarmed in\nthe small open space before the fine old Gothic church of the town, and\nstood in many a checkered group--here, of fat thriving _bourgeois_ and\ntheir portly wives, dragging in their hands chubby and rebellious little\nurchins, who looked all but spherical in their monstrous puffed hose or\nshort wadded multifold petticoats, the miniature reproductions of the\npaternal and maternal monstrosities of attire--there, of more noisy and\nclamorous artizans, in humbler and less preposterous dress--on the one\nside, of chattering serving-damsels, almost crushed under their high\npyramidical black caps, worn in imitation of an ancient fashion of their\nbetters--on the other, of grave counsellors and _schreibers_ in their\nblack costumes, interlarding their pompous phrases with most canine\nLatin--here again, of the plumed and checkered soldiers of the civic\nguard--there, of ragged-robed beggars, whose whine had become a second\nnature--all in a constant ferment of movement and noise, until the\nsquare might be fancied to look like the living and crawling mass of an\nold worm-eaten cheese.\n\nThe congregation of the multitude had been induced by a report prevalent\nthroughout the town, that the Prince Bishop, whose arrival from Fulda at\nhis castle of Saaleck, close at hand, had been announced, was about to\nmake his entrance in grand state, and that a holy and solemn service to\ncelebrate this event was to be performed at the high church.\n\nAlready, however, other rumours were afloat among the crowd; and it\nbegan to be confidently stated, that a sudden change of plans had forced\nthe Prince Bishop to renounce his intention.\n\nListening with anxiety, on the outskirts of a group, to the discussion\nupon the probabilities or improbabilities of the service taking place in\nthe absence of the Prince, stood Magdalena. She was attired in her usual\ndark semi-monastic dress; but to this was now added the scrip, wallet,\nand tall crossheaded staff of the wandering pilgrim. As the prevailing\nopinion appeared to be that the Ober-Amtmann would attend, at all\nevents, at the celebration of the church rites intended to be performed,\nMagdalena turned away with a calmer air, murmuring to herself the\nwords--\n\n\"I shall see her once more--once, and for the last time: and God surely\nwill forgive the sin, if such it be. One look of last farewell! and then\nagain a long expiation of penitence and prayer.\"\n\nSo saying, she traversed the small square to the broad stairs of the\nchurch, where she sat herself down upon the highest step, among a group\nof beggar women and ragged children, and, sinking her head to the\nground, seemed to dispose herself to wait with patience.\n\nShortly afterwards, a young man also began to mount the steps leading to\nthe great entrance of the church, as if with the intention of placing\nhimself near the arch, in so favourable a position as to be close by all\nthose who should pass into the interior. He bounded upwards with anxious\nhaste and beating heart--although there was yet a long interval before\nthe commencement of the service--and with a movement so hurried and\nagitated, that he brushed rudely against one person of a group in his\nway. He turned, with a gentleness of feeling unusual at the time towards\nthe lower classes, to crave of the female he had pushed a pardon for his\nawkwardness. At the sound of his voice the old woman raised her head.\n\n\"Magdalena!\" cried the young man with surprise, as he recognised upon\nher the evident symbols of travel and wayfaring peculiar to that age,\n\"What means this pilgrim's garb?\"\n\n\"Alas! kind, gentle Master Gottlob,\" replied Magdalena in a tone of the\nbitterest sadness, as she rose from her seat, \"my hour is arrived, and I\nmust leave you. Ask me not why. I must go as I have come, in silence and\nmystery. But oh! I beseech you, deem me not ungrateful. I had not\nquitted you without a last farewell--a last assurance that all your\ngentle charities are engraven here, upon my heart for ever.\"\n\n\"Magdalena!\" again exclaimed Gottlob, still astonished at this\nunexpected announcement, \"thou leavest me thus abruptly?\"\n\n\"Again, I pray you, gentle Master,\" said the old woman sobbing, \"think\nme not unkind or cold. The will of another is far stronger than my own.\nThe will of God is above all. We shall meet no more on earth, young man;\nat least I fear so: my destiny leads me from the world. But my prayers\nshall be offered up, morning and evening, at my noontide meal as at my\nlying down; at all times, and in all places, whenever it shall please\nHeaven to hear them, for my generous benefactor.\"\n\n\"But you must not quit me thus,\" said the young man--\"thus unassisted,\nin penury and want. I have but little, it is true, but that little shall\nbe thine. What matter the gauds I thought to purchase? the dainty plume\nto deck my cap?\" Still, in spite of himself, an unconscious sigh broke,\nas he spoke, from the breast of \"Gentle Gottlob,\" at the anticipated\nrenunciation of the braveries that were to give him a price in the eye\nof the fair object of his adoration. \"Can my poor savings be better\nbestowed than upon thee?\"\n\n\"I need not thy generous sacrifice, kind youth,\" replied Magdalena. \"The\npilgrim lacketh nothing in a Christian land; and soon I shall be beyond\nall want.\"\n\n\"Oh! speak not thus sadly,\" said Gottlob, taking her hand.\n\n\"I meant it not so sadly as you deem. I am resigned still to live on,\nuntil it please God to release me from this world of sin and sorrow,\nmore easily resigned and with a calmer spirit, since, through the mist\nof solitary darkness around me, I see a way of hope that shines not upon\nme, but upon the bright forms most dear to me.\"\n\n\"What meanest thou, Magdalena?\" cried the young man.\n\n\"Strive not to comprehend me,\" said the old woman in a more subdued\ntone--\"I would not foster vain delusions;\" and, as if to remove the\nimpression of what she had said from Gottlob's mind, she hastily added,\n\"You have not seen the Prince at Saaleck?\"\n\n\"Alas, no!\" replied the young artist. \"My noble patron had already left\nthe castle with a small retinue, and I was too late to meet him. It was\nsaid that he was gone upon a visit to all the various monasteries in\nthis part of the country, in order to hold secret counsel with the\ndifferent dignitaries of the church in his domain, respecting the late\nheresies that have appeared, and already spread so widely throughout the\nland.\"\n\nMagdalena was about to answer, when a new and general movement among the\ncrowd, showed that the expectation of the multitude was aroused. The\ntapers upon the altars in the church had been lighted in splendid\nprofusion. The vapour of incense already scented the air, as it floated\ndown the aisles. The organ pealed through the church; and the priests,\nin their sacerdotal robes, were seen advancing along the middle aisle\ntowards the entrance, to meet the expected dignitary. But Gottlob and\nMagdalena gazed not upon this priestly show; their heads were turned in\nanother direction, and looked from the church across the square. Their\nhearts beat with one feeling. Both murmured to themselves with one\naccord, \"She comes!\"\n\nAlready the pikes of the guard preceding the noble Ober-Amtmann appeared\nemerging from the street leading to the episcopal palace, and the\nsoldiers, entering the square, cleared the way rudely through the crowd,\nwhen Magdalena again pressed tightly her companion's arm.\n\n\"Swear to me, young man,\" she whispered in a low and solemn tone, \"as\nyou value your salvation--swear to me ever to respect the purity and\npeace of mind of that innocent and happy girl, upon whose fair face I\nshall now gaze for the last time!\"\n\nGottlob looked at the excited woman with much surprise.\n\n\"Swear to me that you will not trouble her unconscious heart with words\nof love, until, perhaps, a better time may come!\" she continued, with\nhesitation.\n\n\"Magdalena, I understand thee not,\" replied the young man. \"But before\nme she is as a holy saint of heaven, at whose shrine we may bow down and\npray, but whom we cannot pollute with earthly touch.\"\n\n\"God grant you happiness, young man!\" said Magdalena, dropping her\nflowing tears upon the hand she held in her own.\n\nGottlob's attention was too much absorbed in the sight of the one object\nof his eager gaze, to heed more seriously, at that moment, the strange\nand solemn adjuration of the old woman. His heart beat with intense\nviolence, his cheek flushed, his mild blue eyes dilated with animation,\nas he followed along the square the form of Bertha, who was advancing in\nthe procession by her father's side. And now she was about to mount the\nchurch steps, she would be obliged to pass close by him, perhaps near\nenough for her dress to touch his own; for the crowd was dense behind,\nand pressed forward upon those who stood, like him, in the foremost row.\nThe agitation of his companion equaled, perhaps exceeded, his own.\n\nThe clergy now stood under the church gate--the preceding guards had\nstationed themselves on either side of the arch--the Ober-Amtmann,\nleading his daughter by the hand, had reached the broad surface of the\nhighest step, where stood the aged female and the young artist, when the\nagitated Magdalena, unable to control her feelings as the governor and\nhis fair child passed so near, bent lowly down, and seized the hem of\nBertha's garment to kiss it unperceived. At that moment, a rude gripe\nseized her arm and dragged her up, and a harsh voice shrieked in her\near--\"Touch her not, hag of hell, to cast thy infernal spells upon her!\"\nA scream of terror burst from Magdalena as she recognised Black Claus,\nthe witchfinder.\n\n\"Noble Ober-Amtmann, hear me!\" cried the , pushing forward with\nforce, and arresting with a wild gesture the progress of the dignitary.\n\"I here denounce, before your noble honour, this wretched woman as a\nmost foul and most notorious witch.\"\n\nIn the rude attack thus made upon the unhappy woman--on her terror and\nsurprise--the cross-topped pilgrim's staff slipped from her grasp, and\nslightly wounding the fair neck of Bertha, it fell upon the pavement,\nand was splintered into several pieces.\n\n\"See, see!\" screamed the witchfinder, \"how she strives to harm the\ninnocent and good, and destroys and tramples under foot--curses on\nher!--the holy symbols of the church.\"\n\nWith a feeling of horror and alarm, for which the credence in witchcraft\nand its agents that pervaded all ranks and classes at that age gave full\nwarrant, Bertha clung with a scream to her father's breast, and sought\nprotection in his arms. At this sight the unhappy Magdalena uttered a\nbitter cry of despair, and raising her clasped hands aloft,\nexclaimed--\"O God! Thou punishest me too bitterly.\"\n\n\"Hear ye,\" cried the witchfinder, \"how she owneth her crime even in her\nblasphemy!\"\n\nWith one arm the Ober-Amtmann pressed the terrified Bertha to his bosom,\nand, with the other, signed to some of the guards to surround the old\nwoman. At this moment the sight of the blood which had trickled in a few\ninsignificant drops upon her veil, caught the eye of the alarmed girl,\nand turning very pale, she held forth a crucifix, which hung about her\nneck, towards the spot where stood Magdalena, as if to exorcise the\npowers of witchcraft directed against her, and sobbed--\"Oh! take her\nfrom my sight--save me--she would destroy me!\"\n\n\"It is she condemns me!\" cried Magdalena; and, with another\nheart-rending exclamation of despair, she fell forward to the earth as\nif in violent convulsions.\n\n\"See, see!\" shouted Claus in triumph, \"how the sight of the holy cross\ncauses the devil within her to tear and rend her.\"\n\nThe bystanders shrank in horror from the prostrate form of the unhappy\nwoman. The guards, who had approached, kept at a sufficient distance to\navoid all contact with the reputed witch, although near enough to\nprevent her escape.\n\nPetrified with astonishment and dismay at the strange scene that had\npassed thus rapidly before him, and shocked at the sight of Bertha's\nwound and terror, Gottlob had stood at first incapable of movement. But\nwhen he saw Magdalena thus stricken to the earth, he forgot all the\nterrors of witchcraft--he forgot the horrible denunciation--he forgot\neven Bertha's fainting form; the instinctive impulse of his kindly\nnature was to rush forward and to raise the poor old woman. Before he\ncould reach her, however, twenty hands had pulled him back with\nforce--twenty voices screamed in his ear, \"Touch her not--beware!\" In\nvain he struggled, and strove to extricate himself--in vain he protested\nthe poor woman's innocence--he was held back by force.\n\nIn the meanwhile, although those nearest to the accused woman drew back\nwith terror, the remoter crowd rushed forward towards the church steps\nin violent excitement, preferring loud cries of \"A witch!--a witch! To\nthe stake with her--to the stake!\" The deeper voices of the men mingling\nwith the shriller cries of the women and children.\n\nIn the midst of this scene of tumult, the Ober-Amtmann conveyed his\ndaughter in his arms--for she had now completely fainted--to the church,\nand confided her to the care of her women. Upon returning, he sternly\ngave orders that the accused female should be placed in the prison of\nthe town, with a guard before the door, until the denouncer should be\nheard against her.\n\n\"Come hither man, black !\" he continued, with some disgust, to\nClaus: \"We know that the dreadful crime of witchcraft has, like heresy,\nmade much and notable progress in the land of late; and although our\nreverend brother views the former abomination with more lenient eye than\nourselves, we think that fagot and stake are but too slight a punishment\nfor such black and damning sin. But still, of late, thy denunciations\nagainst this crime have much multiplied; and sometimes, it has seemed to\nour justice, upon but small and vague proof--although popular voice\ndemanded the condemnation of the wretched women. Have a care, then, how\nthou wrongfully preferrest such a charge--have a care how thou jugglest\nwith our sense of right and wrong; for though there seemeth, in truth,\nto be some appearance of the demon and his works in the horror which\nthat woman has expressed for the symbols of our holy religion, and in\nthe manner in which she has drawn blood from our young and innocent\ndaughter, yet were we to find thy accusation to be inspired by motive or\nthe spirit of falsehood, as we live that pile which threatens the\nsorceress and hag shall be thy own seat--the fire thy death-garment.\"\n\n\"Noble Amtmann,\" cried the witchfinder, undaunted by this address, \"I\nfear not the proof. Again I denounce that woman as dealing in\nwitchcraft, and consorting with the powers of darkness.\"\n\nAs the guard drew nearer, to force the unhappy woman with their\npike-heads to rise from the ground, where she still lay crouched\ntogether, the wretched Magdalena raised her head, and her eyes fell upon\nthe dark face of the witchfinder, as it glared upon her in triumph. The\nhideous yells of the crowd prevented her hearing the only faint voice of\npity raised in her behalf--that of gentle Gottlob. Her brain whirled\nwith terror--she thought that her last hour was come; and, with a heavy\nshudder throughout her whole frame, she fell senseless to the ground.\n\n\n\n\nNATURAL HISTORY OF MAN.{A}\n\n\nIt has probably occurred to the reflecting student of logic, that the\nphilosophers of the schools must have been sorely straitened in seeking\nfor a definition of man, before they would have had recourse to such a\nderogation from his apparently higher attributes, as to define him by\n\"_animal risibile_,\" or \"_animal bipes implumis_.\" An attentive\nconsideration will, however, show the enquirer, that to distinguish man\nfrom the remainder of the animal kingdom by his structural\ncharacteristics alone, is not so easy a task as would at first sight\nappear; and he will be obliged at length to return to some such\nhumiliating designation of the _genus animal_, _species homo_, as those\nabove given. Physical differences, indeed, there are between man and the\nother tribes of mammalia; but these differences are more matters of\nanatomical detail, than such salient notable exponents as would at once\nbe recognised and admitted by the sceptical objector. The strength,\nmoreover, of these differences resides in the whole collectively, and\nnot in any one taken singly. If, however, the student take as his\ngrounds for induction the habits of the species, instead of its\nstructure, he will find a much broader line of demarcation. Wherever he\nexamines the existing relations or former records of his race, and\ncompares them with those of other animals, he will find that the\ninstincts of the one are variable and progressive, those of the other\nare definite and stationary. As far as has ever been ascertained by the\nmost accurate observer, the nest of the grossbeak, the dam of the\nbeaver, the cone of the termites, were, ages ago, each similar in\ncharacter, and equal in perfection, to those of the present day; while,\nwhether we compare the rude wigwam of the uncivilized savage, or the\nmore finished architecture of ancient Thebes, with the buildings,\nrailroads, and shipping of the present day, we still find a continual\nvariation, and a progressive adaptation to new wants. The psychological\ncharacteristics stand out then in fuller relief than the physiological;\nbut yet the former are by no means free from grounds for cavil. Domestic\nanimals acquire new habits, varying from their natural instincts.\nAdmitting these to result from the teaching of man, it still shows--as\ndoes, indeed, the fact of domestication--a capability of progression;\nand some feeble instances of the faculty of learning may be detected\neven in the wild tribes of animals. Thus every thing becomes, if\nhypercritically examined, a question of degree, \"_demo unum, demo etiam\nunum_,\" and the hundred years become an hour; nought is every thing, and\nevery thing is nought. Rational investigation, then, should lead us to\nreject, or at least to set no undue value upon, extreme instances, or\nthe merging shadows of boundaries; the spectrum consists of separate\ncolours, though we may not tell where the red ends and the yellow\nbegins.\n\nThe fair questions in examining the physiology and psychology of man,\nwith a view to his place in the creation, are, 1st, Whether his\ndistinctive marks and attributes, taken collectively, are such as\nbroadly separate him from the rest of the animal kingdom; 2dly,\nSupposing such distinctions to exist now, whether they have existed at\nall periods of which we can acquire any evidence; and, 3dly, Whether\nthese distinctions are common to the whole of the race to which the term\n_man_ is applied, or whether different tribes of men differ _inter se_\nas much as the species viewed collectively differs from other species.\n\nThese, with other minor questions which arise out of them, are, as far\nas we can gather, the propositions discussed in the work before us--a\nwork abounding in elaborate research and erudition, but somewhat\ndeficient in logical precision or lucid arrangement; a mass of details\nis given, but the links whereby the generalizations from these are\nsought to be established, are here and there wanting, and here and there\nobscure. It is probably the fault of the subject, which is in its\ncharacter inexact; but we certainly expected that more had been done;\nand from some passages in the early portions of the work, we were\ninduced to believe that the author had succeeded in proving races of\nmankind to be more distinctly deducible from their sources, and that\ntheir physical and moral relations were more definitely traced. The\nfollowing passage, in which the object of the work is enounced by the\nauthor, is wanting in precision and perspicuity:--\n\n \"That great differences in external conditions, by the double\n influence of their physical and moral agency, should have effected,\n during a long series of ages, remarkable changes in the tribes of\n human beings subjected to their operation--changes which have\n rendered these several tribes fitted in a peculiar manner for their\n respective abodes--is by no means an improbable conjecture; and it\n becomes something more than a conjecture, when we extend our view\n to the diversified breeds of those animals which men have\n domesticated, and have transferred with themselves from one climate\n to another. Considered in this point of view, it acquires, perhaps,\n the character of a legitimate theory, supported by adequate\n evidence, and by an extensive series of analogous facts.\n\n \"But we must not omit to observe, that to this opinion there is an\n alternative, and one which many persons prefer to maintain; namely,\n that the collective body of mankind is made up of different races,\n which have differed from each other in their physical and moral\n nature from the beginning of their existence. To determine which of\n these two opinions is the best entitled to assent, or at least to\n set before my readers a clear and distinct notion of the evidence\n that can be brought to bear upon the question, will be my principal\n object in the following work.\"\n\nNow, as they are here stated, the two opinions are not necessarily\ncontradictory; differences in external condition may effect remarkable\nchanges in tribes of human beings, and yet the collective body may be\nmade up of different races: and to set before the reader a clear and\ndistinct notion, is to prove nothing, although indeed, as we shall see\nin the sequel, the author has a very strong conviction, and believes\nthat he succeeds in proving, as far as a matter incapable of\nmathematical demonstration can be proved, the negative of the latter\nproposition. What the author seems to intend, or rather what the whole\ntenor of his book imports, though his expressions at times go much\nfurther, is, not that community of origin is proved inductively by the\nresearches which have been made into the existing and past state of man,\nbut that the natural history of man presents nothing inconsistent with\nsuch a view.\n\nThe researches of Cuvier and others have negatived the theory of\nLamarcke as to the transmutation of species. The \"_nisus formativus_\" is\nadmitted, but admitted with limits, \"_quos ultra citraque nequit\nconsistere rectum_\".\n\nThe extreme rarity of hybrids, their inability of continuous\nprocreation, the absence of any well-authenticated cases of a permanent\nspecies formed by the union of two distinct ones, the return to the\noriginal type when the disturbing causes are removed, with various other\narguments tending the same way, have been considered, by the most\ncompetent and impartial judges, as conclusive evidence of the real and\npermanent existence in nature of distinct species. These arguments are\nstated in detail in the second volume of Lyell's _Principles of\nGeology_, to which we refer those of our readers who wish for further\ninformation.\n\nHaving briefly stated these and similar arguments, Dr Prichard expresses\nhis conclusion as follows:--\n\n \"It seems to be the well-established result of enquiries into the\n various tribes of organized beings, that the perpetuation of\n hybrids, whether of plants or animals, so as to produce new and\n intermediate tribes, is impossible.\n\n \"Now, unless all these observations are erroneous, or capable of\n some explanation that has not yet been pointed out, they lead, with\n the strongest force of analogical reasoning, to the conclusion,\n that a number of different tribes, such as the various races of\n men, must either be incapable of intermixing their stock, and thus\n always fated to remain separate from each other; or, if the\n contrary should be the fact, that all the races to whom the remark\n applies, are proved by it to belong to the same species.\n\n \"I believe it may be asserted, without the least chance of\n contradiction, that mankind, of all races and varieties, are\n equally capable of propagating their offspring by intermarriages,\n and that such connexions are equally prolific, whether contracted\n between individuals of the same or of the most dissimilar\n varieties. If there is any difference, it is probably in favour of\n the latter.\"\n\nThis conclusion is repeated a little further on.\n\n \"It appears to be unquestionable that intermediate races of men\n exist and are propagated, and that no impediment whatever exists to\n the perpetuation of mankind when the most dissimilar varieties are\n blended together. We hence derive a conclusive proof--unless there\n be, in the instance of human races, an exception to the universally\n prevalent law of organized nature--that all the tribes of men are\n of one family.\n\n \"Perhaps the solution of the problem which we have undertaken to\n discuss might be left on this issue, or considered as obtained by\n this argument. But further light may be thrown on the subject, by a\n careful analysis of the facts which can be collected relative to\n the nature and origination of varieties; and it may be satisfactory\n to my readers to survey this field of enquiry.\"\n\nGranting, then, the truth of the limitation of species to be\nestablished, and taking as the definition of species the power of\ncontinual propagation, we have it proved at the commencement of the\nwork, that \"all human races are of one species;\" the only question which\nremains is, whether, admitting them to be of one species, the deduction\nthat they have a common origin is necessary; or, if not necessary,\nwhether it is proved in the course of the author's work. It does not\nappear to us a _necessary_ conclusion; for there appears no reason _a\npriori_ why the Creator should not as well form separately an indefinite\nnumber of creatures of the same species as a single pair. This point is\nnot adverted to in the work before us; and whenever identity of origin\nis assumed, it is upon the same grounds from which identity of species\nis deduced. In fact, they are generally coupled; thus, at page 487, we\nhave the expression--\n\n \"If now it should appear, on enquiry, that one common mind, or\n psychical nature, belongs to the whole human family, a very strong\n argument would thence arise, on the ground of analogy, for their\n community of species and origin.\"\n\nAnd in the last page we have--\n\n \"We are entitled to draw confidently the conclusion, that all human\n races are of one species and one family.\"\n\nThe great point as to identity of species being proved, it would be\ncertainly more simple, and more in unison with the economy of nature, to\nsuppose that all were descended from one pair, than that numerous\nidentical members of a common species were simultaneously created. On\nthe other hand, a physiological difficulty occurs, in viewing a race as\ndescended from a single pair, from the fact universally recognised in\nthe later periods of history, viz. the degeneration, and, in the end,\ndestruction or indefinite deterioration of both physical and mental\nfaculties, by continual intermarriage. The houses of Braganza and\nHapsburg are notorious instances of this; and, as far as we are aware,\nthere are no counter instances.\n\n \"Marry\n A gentler scion to the wildest stock,\n And make conceive a bark of baser kind\n By bud of nobler race; this is an art\n Which does mend nature, change it rather, but\n The art itself is nature.\"\n\nThe matter is incapable of absolute proof--we mean inductive proof; for\nit is in this point that the work before us regards it. Any arguments,\nsuch as similarity of habits, of languages, of opinions, which may be\nused to deduce community of origin, would be equally explained by\ncommunity of species; for, supposing that different individuals of the\nsame species were simultaneously created, the same physical formation\nwould necessarily engender similar habits, and the power of\nintermarriage would induce a similarity of language, long before any\nperiod to which our histories go back. Taking, then, as a fair\nassumption, that, if identical in species, mankind have a common origin,\nwe get in the outset of the book the conclusion stated at the end, viz.\nthat all human races are of one species and one family. The great body\nof the work is, therefore, only accessory and corroborative; and its\nvalue would consist not so much in proving the affirmative of the\nauthor's thesis, as in placing in a prominent point of view the\nprincipal facts known respecting the natural history of man.\n\nIt may be thought that, in the existing state of man, few marks remain\nfrom which his early history may be deduced; but those unacquainted with\nthe progress of inductive research, would be astonished at the magnitude\nand importance of results derivable from an apparently simple and\nworthless object. An unthinking wanderer, stumbling upon an ancient\ntombstone, if reproached with inattention, would ask what is to be\nlearned from such a relic. A word of inscription would give a clue to\nthe language, and, coupled with other observations, to the date of the\nmonument; the character of the stone, whether roughly hewn or\nelaborately carved, would give evidence as to the tools used in its\nformation, and consequently furnish a key to the manufacturing and\nmetallurgic knowledge of the fabricators. The stone itself might\npossibly not be similar to those in the immediate vicinity, and thence\nwould indicate that travelling and the power of transfer were practised,\nand the skeleton within would indicate the physical formation of the men\nof that day. We have selected here a case of an ordinary grave, but how\nmuch stronger would the case be were we to take a sarcophagus of Egypt,\nenclosing a mummy? The inscription, the fabric of the cere-cloth, the\nchemical substances with which it is impregnated, as well as those by\nwhich the body is preserved, and the relics commonly deposited with it,\nwould lead, by careful investigation, to a tolerably accurate knowledge\nof the character and habits of the time; and where many relics of\ndifferent descriptions, collected from different parts, are skilfully\ncompared, a body of evidence is arrived at, minutely circumstantial in\nits details, and the veracity of which admits of no dispute. As the\nresearches of comparative anatomists have enabled us, from the\nexamination of a single bone, to pronounce with certainty upon the\ngeneral conformation and habits of the animal to which it belonged; and\nas, in many cases, from the existence of such animals, we may go on,\nstep by step, to the nature of the earth's surface at the period when\nthey lived: so the meanest relic of art will serve the natural historian\nof man as a fulcrum by which he may turn up a mass of genuine\ninformation; with which, as with all knowledge, as its store increases,\nthe power of applying it becomes more facile; until at length it\nscarcely becomes an exaggeration to say, that every material relic bears\nin itself its own natural history, and, if artificially modified, the\nhistory of its fabricators--what the germ is to futurity the relic is to\nthe past.\n\nFrom the data which Dr Prichard has given us, in a somewhat scattered\nform, we shall endeavour to collect and group the most interesting of\nhis facts and opinions. In order to ascertain what modifications of\nphysical structure, variation of climate, food, and habits, may effect\nupon mankind, it is necessary, first, to review the effects produced by\nsuch variation upon domesticated animals. It is indeed questionable\nwhether we can in any case, with certainty, trace these to their native\nwilds; but, in many cases, we have instances of their return to a savage\nstate, as with the wild horses, goats, oxen, &c.; and although it does\nnot necessarily follow that their conformation, induced by such return,\nis identical with their original structure, yet there is a reasonable\nprobability that such is the case, and we must take these cases for want\nof better. How far, then, has the outward form been altered by the\nchanges induced by domestication; how far are instincts acquired by such\nchanges capable of hereditary transmission, and is there any, and what,\nconnexion between the changed instincts and the changed structure? These\nquestions, involving among other things the infant and difficult\nscience of phrenology, Dr Prichard has left very much to conjecture.\nWhether he considers the data too imperfect, or is afraid of trusting\nhimself with any decided expression of opinion on a subject which has\nbeen so obscured by charlatanry, and which is open to so much\nmisapprehension, does not appear; but it certainly is an apparently\nstriking defect, that where a large portion of the work is devoted to\nthe explanation of the different forms of the cranium in the inferior\nanimals and in man, and to which the largest portion of his pictorial\nillustrations apply, he should give us so little insight into his\nopinions as to what extent phrenology is fairly entitled to credibility.\nHis having taken so much pains in collecting facts and drawings on this\npoint, necessarily leads to the inference that he attaches much value to\nthe craniological distinctions. We shall take an opportunity presently\nof recurring to this subject. We will now take some of the most\ninteresting instances, given by Dr Prichard, of structural changes and\nhereditary instincts, acquired by domesticated animals, and again lost\nby them on returning to a wild state:--\n\n \"Swine transported from Europe to America, since the discovery of\n the western continent by the Spaniards in the fourteenth century,\n and wandering at large in the vast forests of the New World, and\n feeding on wild fruits, have resumed the manner of existence which\n belonged to the original stock. Their appearance nearly resembles\n that of the wild boar. Their ears have become erect; their heads\n are larger, and the foreheads vaulted at the upper part; their\n colour has lost the variety found in the domestic breeds. The wild\n hogs of the American forests are uniformly black. The hog which\n inhabits the high mountains of Paramos bears a striking resemblance\n to the wild boar of France. His skin is covered with a thick fur,\n often somewhat crisp, beneath which is found, in some individuals,\n a species of wool. From excessive cold and defect of nourishment,\n the hog of that region is of small and stunted figure. In some warm\n parts of America, the swine are not uniformly black, as above\n described, but red, like the young pecari. At Melgara and other\n places, there are some which are not entirely black, but have a\n white band under the belly reaching up to the back; they are termed\n _cinchados_. The restoration of the original character of the wild\n boar in a race descended from domesticated swine, removes all\n reason for doubt, if any had really existed, as to the identity of\n the stock; and we may safely proceed to compare the physical\n characters of these races, as varieties which have arisen in one\n species. The restoration of one uniform black colour, and the\n change of thin sparse hair and bristles for a thick fur with a\n covering of wool, are facts that must be noticed in the\n observations of M. Roulin. The difference in the shape of the head\n between the wild and domestic hog of America, is very remarkable.\n Blumenbach long ago pointed out the great difference between the\n cranium of our swine and that of the primitive wild boar. He\n remarked that this difference is quite equal to that which has been\n observed between the skull of the and the European. 'Those\n persons,' he says, 'who have no opportunity of verifying the fact,\n have only need to cast their eyes on the figure which Daubenton has\n given of both the former. I shall pass over,' he adds, 'the lesser\n varieties of breeds which may be found among swine, as among men,\n and only mention that I have been assured by M. Sobzer, that the\n peculiarity of having the bone of the leg remarkably long, which in\n the human kind is observed among the Hindoos, has been remarked\n with regard to swine in Normandy. They stand very long on their\n hind legs; their back, therefore, is highest at the rump, forming a\n kind of inclined plane; and the head proceeds in the same\n direction, so that the snout is not far from the ground.'\n\n \"'Swine,' continues Blumenbach, 'in some countries have degenerated\n into races which, in singularity, far exceed every thing that has\n been found strange in bodily variety among the human race. Swine\n with solid hoofs were known to the ancients, and large breeds of\n them are found in Hungary and Sweden. In like manner, the European\n swine first carried by the Spaniards in 1509 to the island of\n Cubagua, at that time celebrated for its pearl fishery, degenerated\n into a monstrous race, with toes which were half a span in length.'\n There are breeds of solid-hoofed swine in some parts of England.\n The hoof of the swine is also found divided into five clefts.\n\n \"Buffon had before remarked the varieties of the hog tribe. 'In\n Guinea,' he observes, 'this species has acquired very long ears,\n couched upon the back; in China, a large pendant belly, and very\n short legs; at Cape Verde and other places, very large tusks,\n crooked like the horns of oxen; in domestication, half pendant and\n white ears.'\"\n\n * * * * *\n\n \"A very remarkable fact relative to the oxen of South America is\n recorded by M. Roulin, to which M. Geoffrey St Hilaire has\n particularly adverted, in the report made by him on M. Roulin's\n Memoir, before the Royal Academy of Sciences.\n\n \"In Europe, the milking of cows is continued through the whole\n period, from the time when they begin to bear calves till they\n cease to breed. This secretion of milk has become a constant\n function in the animal economy of the tribe; it has been rendered\n such by the practice, continued through long series of generations,\n of continuing to draw milk long after the period when it would be\n wanted for the calf; the teats of the cow are larger than in\n proportion, and the secretion is perpetual. In Columbia, the\n practice of milking cows was laid aside, owing to the great extent\n of farms and other circumstances. 'In a few generations,' says M.\n Roulin, 'the natural structure of parts, and withal, the natural\n state of the function, has been restored. The secretion of milk in\n the cow of this country is only an occasional phenomenon, and\n contemporary with the actual presence of the calf. If the calf\n dies, the milk ceases to flow, and it is only by keeping him with\n his dam by day, that an opportunity of obtaining milk from cows by\n night can be found.' This testimony is important, by the proof\n which it affords that the permanent production of milk in the\n European breeds of cows is a modified function of the animal\n economy, produced by an artificial habit continued through several\n generations. Two other very important observations made by M.\n Roulin in South America, were pointed out by M. Geoffrey St Hilaire\n in his report to the Academy of Sciences. They refer to the fact of\n the hereditary transmission of habits originally impressed with\n care and art upon the ancestors. Of this fact I shall adduce other\n examples in the sequel; at present I only advert to M. Roulin's\n observations. The horses bred in the grazing farms on the\n table-land of the Cordillera, are carefully taught a peculiar pace,\n which is a sort of running amble. This is not their natural mode of\n progression, but they are inured to it very early, and the greatest\n pains are taken to prevent them from moving in any other gait. In\n this way the acquired habit becomes a second nature. It happens\n occasionally that such horses, becoming lame, are no longer fit for\n use; it is then customary to let them loose, if they happen to be\n well-grown stallions, into the pasture grounds. It is constantly\n observed that these horses become the sires of a race to which the\n ambling pace is natural, and requires no teaching. The fact is so\n well known, that such colts have received a particular name; they\n are termed 'aguilillas.'\n\n \"The second fact is, the developement of a new instinct, which, as\n M. Roulin declares, seems to become hereditary in the breed of dogs\n found among the borderers on the river Madeleine, which are\n employed in hunting the pecari. I shall cite the author's own\n words:--'L'addresse du chien consiste a moderer son ardeur a ne\n s'attacher a aucun animal en particulier, mais a tenir toute la\n troupe en echec. Or, parmi ces chiens, on en volt maintenant qui,\n la premiere fois qu'on les amene au bois, savent deja comment\n attaquer; un chien d'une autre espece se lance tout d'abord, est\n environne, et quelle que soit sa force, il est devore dans un\n instant.'\"\n\nTo these cases we may add a case familiar to the sportsmen of this\ncountry, and one of which we have ourselves seen an unquestionable\ninstance, viz. the acquired habit of the setting-dog in arresting his\nsteps, and crouching, when in pursuit of game; the origin of which was\nprobably a pause in his career, in order the better to ascertain the\nposition of the game of which he was in quest; but this, by constant\nteaching, has become hereditary to such an extent, that occasionally a\ndog of pure breed will, the first time he is taken out, as soon as he\ngets on the scent of game, crouch or place himself in a setting\nattitude, and remain perfectly immobile until forced to proceed; nay\nfurther--as it is necessary that the sportsman teach the dogs who are in\nthe same field with that one who discovers the game, as soon as they see\nthe latter setting to arrest their steps likewise; or, as it is termed,\nto _back_, in order not to disturb the game--in the instance which came\nunder our notice, a dog of eight or nine months old, which had never\nbeen out of a town, when taken into the fields for the first time with\nan old well-trained dog, as soon as the latter had discovered game, and\npointed to it, instantly backed him--_i.e._ remained stiffly standing in\nthe position in which he was when he first caught sight of the older\ndog: probably many sportsmen could be found who would vouch to similar\nfacts.\n\nWe may here state that we quite agree with Dr Prichard, as to the\nabsence of any foundation for the general belief, that all the acts of\ninferior animals are performed without their consciousness or view to\nany object or end; on the contrary, there is every probability that\nthey, in carrying into effect their several instincts, seem to\nthemselves to act from similar internal impulses of will and intention,\nas human beings do.\n\nWe need not enter into the vast number of varieties which the most\ndomestic of all domesticated animals, the dog, exhibits; we shall only\nremark, that, in all their varieties, Dr Prichard says,--\n\n \"Restored to a state of comparative wildness, which approaches to\n their unreclaimed and primitive condition, the tribes of dogs every\n where make a corresponding approximation to the type which may be\n supposed to have belonged to the species in its original state.\"\n\nBut this passage is enigmatical, as the original _type_ seems to be\ninvolved in dense obscurity. Buffon considered the shepherd dog to be\nthe least modified by domestication--very erroneously, according to Dr\nPrichard; it is still a _vexata questio_, whether the original\nprogenitor of the dog be a wolf, a jackal, a fox, or an unknown animal\ndiffering from all these.\n\n \"The sheep is one of the most anciently domesticated animals, and\n it is one in which great varieties display themselves. It has been\n long believed, and this appears to have been the opinion of Baron\n Cuvier, that all the breeds of tamed sheep are descended either\n from the argali of Siberia, or from the moufloun or musmon of\n Barbary. This is at present doubted by most naturalists. There\n seems, however, to be no reason for believing that the domestic\n breeds belong to more than one species, though they differ much in\n different countries. In Europe, the breeds of sheep vary much in\n stature, in the texture of their wool, the number and shape of\n their horns, which are in some large, in some small, in others\n wanting to the female, or altogether absent from the breed. The\n most important varieties in Europe are the Spanish breeds, some\n with fine, others with crisp wool, in which the rams have long\n spiral horns; the English breeds, which differ greatly in size and\n in the quality of the wool; and, in the southern parts of Russia,\n the long-tailed breed. The breeds of sheep in India and in Africa\n are remarkable for the length of their legs, a very convex\n forehead, and pendant ears; these also have long tails. Their\n covering is not wool, but a smooth hair. In the northern parts of\n Europe and Asia the sheep have short tails. The breeds spread\n through Persia, Tartary, and China, have their tails transformed\n into a double spherical mass of fat. The sheep of Syria and\n Barbary, on the other hand, have long tails, but likewise loaded\n with a mass of fat. In both of these varieties of the sheep the\n ears are pendant, the horns of the rams large, and those of the\n ewes and lambs of moderate size, and the body is covered with wool,\n mixed more or less with hair.\n\n \"New breeds of sheep are frequently formed in different countries\n in which particular qualities predominate, according to the\n preference of the breeders. This is done, partly by crossing or\n intermixing races already constituted and well known; but in great\n part also by selecting individuals from the stock in which the\n particular qualities are more strongly marked than in the\n generality of the same breed. In these instances, the natural or\n congenital variety which the individual animal displays, perhaps\n for the first time, becomes perpetuated by the hereditary\n transmission of such characters, which is a law of the animal\n economy. A striking instance of this fact is to be found in the\n origination of a new breed of sheep in the state of Massachusetts,\n which has been noticed by many writers in connexion with this\n subject. In the year 1791, one ewe on the farm of Seth Wright gave\n birth to a male lamb, which, without any known cause, had a longer\n body and shorter legs than the rest of the breed. The joints are\n said to have been longer, and the fore-legs crooked. The shape of\n this animal rendering it unable to leap over fences, it was\n determined to propagate its peculiarities, and the experiment\n proved successful; a new race of sheep was produced, which, from\n the form of the body, has been termed the otter breed. It seems to\n be uniformly the fact, that when both parents are of the otter\n breed, the lambs that are produced inherit the peculiar form.\"\n\nWe might extract other instances of physiological and psychological\nchanges induced by domestication, but we think enough have been given to\nshow the character and degree of such changes. The least important\nchange, and that which appears the soonest affected, is the colour of\nthe skin and hair. This is universally of an uniform tint in wild\nanimals, and generally bears a close approximation to the colour of the\nland in which the animal lives: thus the ptarmigan, inhabiting snowy\nregions, is white; the grouse has the colour of heath; the hare that of\ndry fern or furze--a provision which has the effect of protecting the\nweaker tribes from the stronger and predatory ones. In domesticated\nanimals, from causes apparently not as yet traced, the colour is\nvariegated and various. Closely connected with the colour and nature of\nthe skin, are the size and shape of the horns, their presence and\nabsence. Great as is the apparent variety of appearance effected by\nhorns, changes in these appear to be easily induced: they are connected\nwith the epidermic structure, generally the most easily modified; and we\nneed not cite instances to prove that different breeds of the same\ntribe, and occasionally different individuals of the same breed, differ\nmaterially as to horns. According to Azara, horned horses are sometimes\nseen in Paraguay.\n\nVery little appears to be known, at least scarcely any intimation is\ngiven in the work before us, of the proximate or final cause of these\nchanges. Great as they are, certainly, as far as we can judge, no _nisus\nformativus_ can account for the enormous horns of the Spanish sheep;\nnor, looking to the final cause; does there appear any reason why\ndomestic animals should need such overgrown instruments of defence.\nWhen, however, we come to the more important anatomical modifications,\nsuch as the length and shape of the legs, the bones of the pelvis or of\nthe jaw, the object is more apparent. A greyhound, with the muzzle of a\nbull-dog, would be an obvious natural inconsistency.\n\nWe now pass to the physical distinctions of the different races of men.\nHere we may observe that a much greater importance is to be attached to\ncomparatively slight variations. Considering the surprising external\ndifferences that exist in domesticated animals of the same species, the\nwonder rather is, that the different races of men differ physically so\nlittle as they do, than that they differ so much. Here we will take\nfirst, the least important shades of difference--the texture of the\nskin, hair, and complexion; and then pass on to the more prominent\ndiversities of the bony fabric, cranium, &c.\n\n \"The texture of the body, in which all these varieties have their\n seat, is the extracorial or exodermal structure, constituting, if I\n may so speak, the outer coating of the body, external to the true\n skin, which corresponds to the cuticular and corneous excrescences\n of animals--a structure which includes horns, hoofs, hair,\n feathers, and all similar appendages in different orders of\n animals. This structure displays infinite diversities in colour,\n constitution, and organization, and is the most variable tissue on\n the whole body. Many different opinions have, however, been lately\n maintained, and much research has been made, as to the nature and\n texture of the parts on which the variety of colour depends.\"\n\nThe ancient anatomists, it appears, recognised only two parts of the\nskin--the true skin, and the outer cuticle or epidermis. Malpighi\ndiscovered a third layer interposed between these, consisting of a sort\nof network, thence called _rete mucosum_, and believed to be the seat of\ncolour in the . Albinus showed this to be a continuous layer, and\nnot a network. Cruikshank discovered four layers--three membranes, and\nthe fourth a layer of colour. Flourens, at a more recent period, made\nthe number of intermediate layers five, four of which he showed to the\nFrench Academy; one of these, a mucous membrane underlying the pigment,\nis, according to this anatomist, a distinct organized body, existing\nonly in men of dark colour, and entirely wanting in the white races, or\nelse (which appears the more probable conjecture) maceration, and the\nordinary process of examination, fail to detect it in the skin of white\nmen. Lastly, the microscopical researches of Henle, Purkinje, and\nSchwann, go to prove that the outer integument does not consist of\nseparate membranes, but is of a cellular structure, and that of these\ncells or \"cytoblasts,\" there are three distinct kinds. We will not\nfurther analyse the different opinions as to the texture of the skin and\nposition of the colouring material; it certainly throws no\ninconsiderable degree of doubt over certain classes of scientific\ninvestigation, to find each subsequent research entirely altering, and\nin some cases overturning, the previously received views.\n\nTo the different characters of human complexion, Dr Prichard gives three\ndistinctive terms--the _melanous_ or brunette; the _xanthous_ or blonde;\nand the _leucous_ or albino; the _melanous_ predominating in the\nsouthern countries, the _xanthous_ in the northern. It is observable\nhere, that although the natural divisions of territory with respect to\ncomplexion, (supposing climate to have the principal modifying effect\nupon complexion,) would be the equatorial and polar regions, or the\nzones of the earth which differ in latitude, yet, with some few\nexceptions, it is only on the northern side of the equator that the\n_xanthous_ complexion prevails--the inhabitants of Australia and the\nSouth Sea Islands being very generally _melanous_. The distribution of\nland and water cannot well be conceived to have any influence upon\nclimate which would account for such diversity; it is probably,\ntherefore, a result of long-continued civilization, the covering the\nbody with clothes, and being for the most part sheltered from the direct\nrays of the sun. The _leucous_ complexion is an abnormal variety, and\noccurs occasionally in all countries. It proceeds from the absence of\nthe dark colouring matter, or pigment; there appears in this case,\nhowever, no difference of anatomical structure, the pigment being\nsometimes subsequently developed in persons who have been born albinoes.\nThe change from the _xanthous_ to the _melanous_ complexion, is a\ncircumstance of constant occurrence; there are few children born, whose\ncomplexion does not darken as they grow up, in many cases undergoing a\ntotal change: the passage from dark to fair is rare, but it constantly\noccurs that _xanthous_, or even _leucous_ children, are born of\n_melanous_ parents. There is nothing, therefore, in the diversities of\ncomplexion which indicates specific diversity in different human races.\nOf the conformation of the bony fabric in the human race, the formation\nof the skull is the part of the greatest importance; we shall only\ntherefore briefly notice, as to the other parts of the skeleton, that\nbetween the most uncultivated races of men, and those tribes of apes\nwhich most nearly approach man, there is a wide difference--the arms of\nthe orang-outang reach to the ankle, and those of the chimpanzee below\nthe knee; the pelvis, or central bony fabric, differ much from those of\nthe human race.\n\nWith regard to the skull, the value of the distinctions in its form and\nstructure depends upon their connexion with the size and organization of\nthe brain--involving the question, whether this has any, and what,\ninfluence upon the powers and habits of the creature. Dr Prichard, as we\nhave already stated, blinks the question of phrenology; though he makes\nsome inferences which prove him to have a general belief in the\nconnexion between mental power and physical formation; nay, further, in\nthe appropriation of different portions of the brain to different\nfaculties.\n\nFew will, we believe, in the present day be disposed entirely to deny\nthat, _ceteris paribus_, the external formation of the skull, or rather\nthe shape of the brain as shown by the formation of the skull, is a\ngeneral index of the mental power of the individual to whom it belongs.\nLook over a collection of busts, or portraits, of eminent men, and, with\nscarcely an exception, they will be found to have high and capacious\nforeheads; while uncivilized races, and born idiots, are lamentably\ndeficient in this respect. The difficulties of phrenology exist in its\ndetails, which by many have been carried out into degrees of subdivision\ncertainly not warranted either by the anatomical structure of the brain,\nor by any empirical data as to the form of different crania, and the\nbiography of the individuals to whom they have belonged. Where, in the\nexisting state of our knowledge, the proper mean may be, it is perhaps\ndifficult to say; but it would have been well, we think, had Dr Prichard\ngiven us a little more explicitly his opinions as to what extent\nphrenology (we use the word in its broadest sense) may be fairly relied\non. As far as we can gather from the scattered passages in his book, he\nseems to take a rational view of it; but a little less caution would\ncertainly have been more instructive to his readers, not only on the\nsubject of phrenology, but on many of the connexions between physical\nstructure and the habits to which such structure is adapted. This is a\n_hiatus_ in Dr Prichard's work, the filling up of which would add much\ninteresting matter, and serve to weave together acts which at present\nare disjointed and isolated; giving the book a dry character, and\npreventing its arresting the attention of the reader. Throughout a\nlarger portion of the work also, we have, in every third page or so, a\nminute description of the complexion, hair, &c., of different people;\nwhich, however valuable as matter of record, becomes tiresome and\nuninteresting as a continuous narrative, and would be much better thrown\ninto a tabular form, as matter of reference only, if incapable of being\nso linked as to present a plausible theory.\n\nThe following passage is the most explicit we can find on the subject of\nthe connexion between the _physique_ and _morale_, and, at the same\ntime, will serve to introduce the three varieties of skull which the\nauthor deems principally worth notice:--\n\n \"If any method of subdividing the human family into groups, is\n likely to be of any particular advantage in elucidating the natural\n history of the species, it must be one founded on some relation\n between the physical characteristics of different tribes and the\n leading circumstances of their external condition.\n\n \"We shall clearly perceive, in tracing the following outline of\n ethnography, that the varieties of colour refer themselves, in\n part, to climates, elevations of land, proximity to the sea-coast,\n or distance from it. It can hardly be doubted that these conditions\n have likewise an effect on the configuration of the human body. But\n there is, perhaps, some truth in the remark, though frequently made\n on little better foundation than conjecture, that the prevailing\n form or configuration of the body is more liable to be influenced\n by the habits of different races, and their manner of living, than\n by the simple agencies of climate. It would be an interesting\n discovery, could it be shown that there is any apparent connexion\n between the display of particular forms, or the leading physical\n characters of human races, and their habits of existence. If I may\n venture to point out any such relation, it would be by remarking,\n in a very general manner, and without pretending to make the\n observation as one which holds without many exceptions, that there\n are in mankind three principal varieties in the form of the head\n and other physical characters, which are most prevalent\n respectively in the savage or hunting tribes, in the nomadic or\n wandering pastoral races, and in the civilized and intellectually\n cultivated divisions of the human family. Among the rudest tribes\n of men, hunters and savage inhabitants of forests, dependent for\n their supply of food on the accidental produce of the soil or on\n the chase, among whom are the most degraded of the African nations\n and the Australian savages, a form of the head is prevalent which\n is most aptly distinguished by the term _prognathous_, indicating a\n prolongation or extension forward of the jaws; and with this\n characteristic other traits are connected which will be described\n in the following pages. A second shape of the head, very different\n from the last mentioned, belongs principally to the nomadic races,\n who wander with their herds and flocks over vast plains, and to the\n tribes who creep along the shores of the Icy Sea, and live partly\n by fishing, and in part on the flesh of their reindeers. These\n nations have broad and lozenge-formed faces, and what I have termed\n _pyramidal_ skulls.\n\n \"The Esquimaux, the Laplanders, Samoiedes, and Kamschatkans,\n belong to this department, as well as the Tartar nations--meaning\n the Mongolians, Tungusians, and nomadic races of Turks. In South\n Africa, the Hottentots, formerly a nomadic people, who wandered\n about with herds of cattle over the extensive plains of Kafirland,\n resembling in their manner of life the Tungusians and the Mongols,\n have also broadfaced, pyramidal skulls, and in many particulars of\n their organization resemble the Northern Asiatics. Other tribes in\n South Africa approximate to the same character, as do many of the\n native races of the New World.\n\n \"The most civilized races, those who live by agriculture and the\n arts of cultivated life, all the most intellectually improved\n nations of Europe and Asia, have a shape of the head which differs\n from both the forms above mentioned. The characteristic form of the\n skull among these nations may be termed _oval_ or _elliptical_.\n\n \"We shall find hereafter that there are numerous instances of\n transition from one of these shapes of the head to another, and\n that these alterations have taken place in nations who have changed\n their manner of life.\"\n\nBlumenbach considered that the most important admeasurement of the skull\nwas derivable from the shape and size of the oval, seen when the skull\nwas viewed from above, looking vertically down upon it. Camper took as\nthe basis of his theory of the gradations of different genera of\nmammalia, the angle formed by a line drawn from the aperture of the ear\nto the base of the nose, and a tangent to the forehead and jaw.\nConsidering the increasing size of this angle to be the distinctive mark\nof intellectual superiority, he viewed a as an intermediate animal\nbetween an European and an ape. But Mr Owen has shown that the\nobservations of Camper and others, being applied to immature animals,\nare not worthy of reliance; as the relations of all animals more closely\napproximate if they be examined in an infant, than in an adult state.\nThe facial angle of the orang, which has been estimated at from 60 deg. to\n64 deg., he finds in the adult animal is only 30 deg.--_i. e._ 40 deg. short of the\nsmallest facial angle in the human race! We should hence be led to\nsuspect a proportionate difference between the infant and adult mind;\nbut the psychological development of infants is a subject which has been\nstrangely neglected by philosophers. A clever Italian authoress who has\nwritten an anonymous work upon education, gives as the reason for the\ndearth of writing on this subject, that philosophers are not mothers,\nand that mothers are not philosophers. Be this as it may, few theorems\nappear to us more promising of interest. The struggle of internal force\nwith external resistance, the feelings manifested in the acquisition of\nnew powers, the impressions made by objects seen for the first time, and\nfirst questions asked, form grounds for induction as to the psychology\nof man, which, thanks to the chartered tyranny of nursery-maids over\nphilosophers, have been grossly neglected.\n\nAfter going through other points of physical difference in human races,\nwith which, being for the most part matter of anatomical detail, we\nshall not trouble our readers, Dr Prichard concludes:--\n\n \"On surveying the facts which relate to difference in the shape of\n the body, and the proportions of parts in human races, we may\n conclude that none of these deviations amount to specific\n distinction. We may rest this conclusion on two arguments. First,\n that none of the differences in question exceed the limits of\n individual variety, or are greater than the diversities found\n within the circle of one nation or family. Secondly, the varieties\n of form in human races are by no means so considerable, in many\n points of view, as the instances of variation which are known to\n occur in different tribes of animals belonging to the same stock,\n there being scarcely one domesticated species which does not\n display much more considerable deviations from the typical\n character of the tribe.\"\n\nThe only observation we shall make upon this is, that, as before stated,\nthe test of identity of species being the power of continued\nreproduction, not the slightest evidence having been ever offered that\nall the various human races have not _inter se_ this power, but the\ncontrary having been proved in every case within human experience, none\nof the deviations _can_ amount to _specific_ distinctions.\n\nHaving noticed the most remarkable physical distinctions of the human\nrace, we come to its ethnographical divisions--divisions founded partly\nupon traditional and historical records, and partly upon the internal\nevidence of similarity of language. The following sketch of hypotheses,\nas to the original birthplaces of the +autochthones gaias+, although\nvisionary, and in all probability incorrect, forms such an interesting\nabstract of philosophical speculations and poetical myths, that we\ncannot refrain from quoting it:--\n\n \"The most popular, or generally received distribution of human\n races in the present day, is that which was recommended by the\n adoption of Baron Cuvier. It did not entirely originate with that\n great writer, but was set forth by him in a more decided and\n complete manner than it had been before his time. This system\n refers different races of men to certain lofty mountain-chains, as\n the local seats of their original existence.\n\n \"The birthplace, or the primitive station, of the race of men who\n peopled Europe and Western Asia, is supposed to have been Mount\n Caucasus. From this conjecture, Europeans and many Asiatic nations,\n and even some Africans, have received the new designation of\n Caucasians. The nations of Eastern Asia are imagined, in like\n manner, to originate in the neighbourhood of Mount Altai, and they\n are named after the Mongolians, who inhabit the highest region in\n that vast chain of hills. The African s are derived from the\n southern face of the chain of Mount Atlas.\n\n \"They are, however, named simply the Ethiopian race, from the\n Ethiopians, who were the only black people known to the ancients in\n very remote times. A mixture of somewhat vague notions, partly\n connected with physical theories, and in part derived from history,\n or rather from mythology, has formed the groundwork of this scheme,\n which refers the origin of human races to high mountainous tracts.\n The tops of mountains first emerged above the surface of the\n primeval ocean, and, in the language of some philosophical\n theorists, first became the scene of the organizing life of nature.\n From different mountain tops, Wildenow, and other writers on the\n history of plants, derive the vegetable tribes; which they suppose\n to have descended from high places into the plains, and to have\n spread their colonies along the margins of mountain streams. High\n mountains thus came to be regarded as the birthplaces of living\n races.\n\n \"Geological theories give their part to render these notions\n popular; not only the late speculations of the Count de Buffon and\n the learned Bailly, but the opinions of ancient philosophers, who\n maintained, before the time of Justin and of Pliny, that the\n mountains of high Asia must have been the part of the world first\n inhabited by men, inasmuch as that region must have been first\n refrigerated in the gradual cooling of the surface of our planet,\n and first raised sufficiently above the level of the ocean.\n Moreover, the poetical traditions of the ancient world describe\n high mountains as the scenes of the first mythical adventures of\n gods and men--as the resting-places on which celestial or aerial\n beings alighted from their cloudy habitations, to take up their\n abode with men, and to become the patriarchs of the human race.\n Lofty mountains are the points in the geography of our globe on\n which the first dawn of historic light casts its early beams; hence\n the legends of the first ages begin their thread. In the cosmogony\n of the Hindoos, it was on the summit of the sacred mountain\n Maha-meru, which rises in the midst of the seven _dwipas_, or great\n peninsulas, like the stalk between the expanded petals of a lotus,\n that Brahma, the creator, sits enthroned on a pillar of gold and\n gems, adored by Rishis and Gandharbhas; while the regents of the\n four quarters of the universe hold their stations on the four faces\n of the mountain. Equally famed in the ancient mythology of Iran and\n of Zoroaster, is the sacred mountain Albordy, based upon the earth,\n but raising through all the spheres of heaven, to the region of\n supernal light, its lofty top, the seat of Ormuzd, whence the\n bridge Ishinevad conducts blessed spirits of pious men to Gorodman,\n the solid vault of heaven, the abode of Ferouers and Arnshaspands.\n Even the prosing disciples of Confucius had their sacred mountain\n of Kuen-lun, where, according to the legends of their forefathers,\n was the abode of the early patriarchs of their race. The Arabs and\n the Persian Moslemin had their poetical Kaf. The lofty hills of\n Phrygia and of Hellas--Ida, Olympus, Pindus--were, as every one\n knows, famous in Grecian story. Caucasus came in for a share of the\n reverence paid to the high places of the earth. Caucasus, however,\n was not the cradle of the human race, but the dwelling-place of\n Prometheus, the maker of men, and the teacher of astronomy.\"\n\nAbandoning this somewhat dreamy view, Dr Prichard regards, consistently\nwith the Scriptural account, the birthplace of man as being on the banks\nof fertilizing rivers, and at a period when the world was, by its\nvegetable and animal productions, prepared for his reception, and adopts\nthree divisions as being those of which we have the earliest records;\n1st, the _Semitic_ or _Syro-Arabian_, inhabiting countries between Egypt\nand the Ganges. 2dly, the _Japetic_, _Indo-European_, or _Arian_,\nspreading from the mouths of the Ganges over the greater part of Europe.\nAnd 3dly, the Egyptian or _Hamitish_,{B} who peopled the banks of the\nNile, and of whom the African s are probably a degenerate\noffshoot. With regard to the knowledge of letters possessed by these\nthree nations, our author gives two inconsistent statements. He says:--\n\n \"The three celebrated nations whose history we have surveyed,\n appear alone to have possessed in the earliest times the use of\n letters, and by written monuments to have transmitted to the last\n ages memorials of their existence. It seems improbable that each of\n these nations should have become, by a separate process, possessed\n of this important art: yet those eminent scholars who have laboured\n with so great success of late in elucidating the Oriental forms of\n writing, have not succeeded in tracing any connexion between the\n alphabetic systems of Egypt, of the Phoenicians, the Assyrians,\n and the Hindoos.\"\n\nAnd states afterwards:--\n\n \"It is plain that the use of letters was entirely unknown to the\n Arian nations, to those tribes at least of the race who passed into\n Europe: and that it was introduced among them in long after ages by\n the Phoenicians, who claim this most important invention, and\n certainly have the merit of having communicated it to the nations\n of the west.\"\n\nThe words \"those tribes at least,\" are scarcely sufficient to remove the\ninconsistency.\n\nA fourth division comprehends those various barbarous nations of unknown\norigin which occupied the territories surrounding the Indo-European\nrace, and were for the most part subdued and expelled by the latter--to\nthis fourth division he applies the term _Allophyllian_.\n\nThis glotto-historical division does not exactly correspond with the\nphysical division as deduced from the form of the skull. The three\nnations first above mentioned, or the inhabitants of the central\nregions, from which they at least are supposed, according to this view,\nto have emanated, have all the oval skull; though, when we pass to the\nnomadic people of high Asia, we get the pyramidal, and, passing from\nEgypt to Africa, we get a gradually increasing tendency to the\nprognathous form.\n\nIt would carry us far beyond the usual bounds of an article in this\nMagazine, were we to give even a condensed abstract of the descriptions,\nindividual and collective, of each of these leading divisions and their\nvarious subdivisions. We will observe generally that the central portion\nof the work, which contains a detailed account of the divisions\nphysical, ethical, and ethnical, of all the most marked varieties of the\nhuman race, accompanied with illustrative pictures and woodcuts, evinces\nthe most elaborate research, and, as a work of reference, will be\ndoubtless found of great value. We will, therefore, pass to the fifth\ngreat division of the human race, which is discussed in a later portion\nof the work, and which is not very distinctly connected with the other\nfour--viz. the American. The Sioux tribes, however, who occupy tracts\nof land on the Upper Mississippi, are supposed with great probability,\nfrom their physical character, language, and tradition, to be the\ndescendants of a Tartar race, who have emigrated across the north-west\nstraits of America.\n\n \"The aboriginal people of America are generally considered as a\n department of the human family very distinct from the inhabitants\n of the Old World. The insulated situation of the continent, and the\n fact that it was so long unknown, and the tribes which it contains\n so long cut off from intercourse with other nations, are among the\n circumstances which have contributed to produce this impression.\n The American nations, taken in the aggregate, are neither among\n themselves so uniform and unvaried in the physical and moral\n qualities, nor is the line of distinction between them and the rest\n of mankind so strongly marked and so obvious, as most persons\n imagine. Yet it must be admitted that certain characters are\n discoverable which are common, or nearly so, to the whole of this\n department of nations; that there are strong indications, if not\n proofs, of a community of origin, or of very ancient relationship\n among them; and that in surveying collectively the people of the\n New World, we contemplate human nature under a peculiar aspect. On\n comparing the American tribes together, we find reasons to believe\n that they must have subsisted as a separate department of nations\n from the earliest ages of the world. Hence, in attempting to trace\n relations between them and the rest of mankind, we cannot expect to\n discover proofs of their derivation from any particular tribe or\n nation in the Old Continent. The era of their existence, as a\n distinct and insulated race, must probably be dated as far back as\n that time which separated into nations the inhabitants of the Old\n World, and gave to each branch of the human family its primitive\n language and individuality.\"\n\nThe points which are supposed to indicate this relationship of the\nAmerican aborigines _inter se_, and their distinction from the\ninhabitants of our continents, are, 1st, the structure of their\nlanguage, in which--\n\n \"Striking analogies of grammatical construction have been\n recognised, not only in the more perfect languages, as that of the\n Incas, the Aymara, the Guarani, the Mexican, and the Cora, but also\n in languages extremely rude. Idioms, the roots of which do not\n resemble each other more than the roots of the Sclavonian and\n Biscayan, have resemblances of internal mechanism similar to those\n which are found in the Sanscrit, the Persian, the Greek, and the\n German languages.\"\n\nAnd, 2dly, their moral and social state, indicating a people which has\nanciently possessed institutions of a highly civilized character, such\nas, according to Dr Martius--\n\n \"A complicated form of government, regulated despotisms or\n monarchies, privileged orders, hierarchical and sacerdotal\n ordinances, systematic laws, the results of reflection, and a\n settled purpose, connected with marriage and inheritance, and\n family relationships, and other customs, which are strongly\n contrasted with the simple and unreflective habits of rude and\n uncivilized nations.\n\n \"The languages of these nations abound, as he says, with words\n expressive of metaphysical views and abstract conceptions. Their\n opinion respecting a future state, the nature and attributes of\n invisible agents, are strikingly different from those of nations\n who have never emerged from primitive barbarism. Another fact which\n tends, as M. Martius observes, to confirm the opinion that natives\n of the New World have fallen from a state of greater refinement, is\n their use, from immemorial ages, of certain domesticated animals\n and cultivated plants, and the notions which they entertained of\n the first acquisition of these possessions. Of such animals and\n plants the people of the Old World have their peculiar stock, and\n the American nations have their own entirely different.\n\n \"In the Old World we know not whence our horses, our dogs, cattle,\n and the various kinds of cerealian gramina were obtained, and the\n American nations are equally at a loss, when we enquire for the\n original stock of the dumb dog of the Mexicans, the llama, the root\n of the mandioca, the American corn, and of the quinoa.\n\n \"In the ancient world there were traditions of some mythical\n benefactors of mankind. Ceres, Triptolemus, Bacchus, Pallas, and\n Poseidon, who had contributed their gifts, corn and wine, the\n sacred olive, and the horse, and we infer that all these had been\n known from periods of remote antiquity.\n\n \"In America, likewise, tradition refers the knowledge of cultivated\n plants and domestic animals, and the art of tilling the earth, to\n some fabulous person who descended from the gods, or suddenly made\n his appearance among their ancestors, such as the Manco-Capac of\n the Peruvians, and the Xolotl and the Xiuhtlato of the Tollecas and\n Chicimecas.\n\n \"The remains of ancient sculpture and architecture spread over\n Mexico, Yucatan, and Chiapa, as well as over the high plain of\n Quito and other parts of South America, and the extensive works of\n art, consisting of fortifications and other relics, discovered in\n the Tenessi country, as well as in the inland parts of New Mexico\n on the Rio Gila, afford some further support to the hypothesis of\n M. Martius.\n\n \"The possession of arts and acquirements, the most simple\n improvements of human life, and such as belong to the very infancy\n of human society, distinctively appropriate, and the origin of\n which is recorded by mythical legends peculiar to each division of\n mankind, seems to carry back the era of their separation to the\n first ages of the world.\"\n\nWith regard to the physical character of the Americans, it appears,\naccording to Dr Martius, that the principal characteristic is the\ntruncation, or flatness, of the occipital portion of the cranium; the\nforehead wide, but low, supposed upon rather insufficient data to be\nmoulded to this shape by artificial means; and the nose arched. In the\nnew as in the old continent, the diversities of physical character do\nnot correspond with the ethnical divisions. The principal criterion of\nthe latter adopted by Dr Prichard is the affinity of languages; and,\nwhen this is insufficient to found any probable opinions, conjectures\nderived from geographical or traditional evidence are called in aid.\nUpon these grounds the Americans are arranged and described by the\nauthor, into the details of which, for the same reason as before stated,\nwe regret not being able to follow him.\n\nSince, however, the first pages of this article were written, a\ndiscovery has been announced connected with the physiology of the\nAmerican aborigenes, which, if subsequently verified, will be of much\nimportance, both as to the anthropological classification of the\nAmericans, and as to the natural history of man generally. In a letter\naddressed to the Academy of Natural Sciences of Philadelphia, and\nrepublished in the _Philosophical Magazine_ for July last, is an account\nof the researches of Dr Lund, who has been for some time engaged in\ngeological investigation in Minas Geraes, a province of Brazil. While\nexamining the caverns of calcareous rocks, he has found in one of them,\nmixed with the bones of extinct races of animals, human bones, having\nall the character of fossils; they are stated to be in part petrified,\nand in part penetrated with iron particles, which gave to them a\nmetallic lustre resembling bronze; they were of extraordinary weight;\nthe crania presented the narrow forehead, prominent zygomatic bones, the\nfacial angle, the maxillary and orbital conformation of the American\nrace. The depression of the forehead in many instances is said to amount\nto a total disappearance. With the bones was found a smooth stone, about\nten inches in circumference, apparently intended to bruise seeds or hard\nsubstances. In other caverns were found human bones, but unaccompanied\nwith those of other animals. These facts, if confirmed, will furnish us\nwith most important evidence as to the past state of the Americans, and\nthe ancient history and physiology of the human race; but the novelty of\nthe results, and the recent date of the communication, induce us to\nabstain from hasty comment.\n\nThe general physiological comparison of human races, the similarity of\nperiodic changes, and the average duration of life, are points upon\nwhich we can very briefly touch. Dr Prichard considers the different\nages at which women are said to be marriageable in different climates to\nbe very much exaggerated. He states his reasons, which do not appear to\nus to be very conclusive. The exceptional cases from the normal\nphysiology would be more interesting, had we space for them, than the\nanalogies, for which probably all our readers would be prepared. Thus,\namong the most curious national anomalies are the Quichuas and Aymaras,\nwho, from the constant habit of breathing an attenuated atmosphere, have\ntheir chests enormously expanded; the Mandans, who, without any apparent\ncause, have the hair grizzled or grey in youth. Among the instances of\nindividual peculiarity, no one is more extraordinary than the horned\nman, whose entire person was covered with a rugged bark, or hide, having\nbristles here and there, which hide he was said to shed annually; and\nthis peculiar form of monstrosity appears to have been capable of\nhereditary transmission, as he had six children with a similar covering.\nHow he procured a wife to bear these children to him does not appear.\nThe children were, it is to be presumed, not equally successful, as the\nbreed of these human rhinoceri has become extinct. Some curious\ninstances of longevity are collected. Of 15 s, the names and\nresidences of whom are given, the average age is 135 years; from\nEuropean nations, there are 1310 recorded instances of persons aged from\n100 to 110, and 3 from 180 to 190. We do no more than briefly notice\nthese exceptions, as we are anxious to devote our small remaining space\nto what will by many be considered the most interesting portions of the\nbook, viz. the author's psychological view of the different races of\nmankind, or the comparison of their different mental faculties.\n\n \"Though inhabiting, from immemorial times, regions in\n juxtaposition, and almost contiguous to each other, no two races of\n men can be more strongly contrasted than were the ancient Egyptian\n and Syro-Arabian races; one nation, full of energy, of restless\n activity, changing many times their manner of existence--sometimes\n nomadic, feeding their flocks in desert places--now settled, and\n cultivating the earth, and filling their land with populous\n villages, and towns, and fenced cities--then spreading themselves,\n impelled by the love of glory and zeal of proselytism, over distant\n countries; the other, reposing ever in luxurious ease and wealth on\n the rich soil, watered by their slimy river, never quitting it for\n a foreign clime or displaying, unless forced, the least change in\n their position or habits of life. The intellectual character, the\n metaphysical belief, and the religious sentiments and practices of\n the two nations were equally diverse; one adoring an invisible and\n eternal spirit, at whose almighty word the universe started into\n existence, and 'the morning stars sang together, and the sons of\n God shouted for joy;' the other adorning splendid temples with\n costly magnificence, in which, with mysterious and grotesque rites,\n they paid a strange and portentous worship to some foul and\n grovelling object--a snake, a tortoise, a crocodile, or an ape. The\n destiny of the two races has been equally different: both may be\n said still to exist; one in their living representatives, their\n ever-roving, energetic descendants; the other reposing in their own\n land--a vast sepulchre, where the successive generations of thirty\n centuries, all embalmed, men, women, and children, with their\n domestic animals, lie beneath their dry preserving soil, expecting\n vainly the summons to judgment--the fated time for which is to some\n of them long past--before the tribunal of Sarapis, or in the hall\n of Osymandyas.\"\n\nWe are far from agreeing with this estimate of the ancient Egyptians.\nTheir progress in mechanical arts, their hieroglyphical literature, and\neven their theology, with its mystic trine, marked them as a people far\nsurpassing their contemporaries; and they were not the less great\nbecause their greatness is now extinct. The Arian{C} tribes, though\nunskilled in many of the most useful arts of life, yet had--\n\n \"National poetry, and a culture of language and thought, altogether\n surprising when compared with their external condition and habits.\n They had bards or scalds, _vates_, who were supposed, under divine\n impulse, to celebrate the history of ancient times, and connect\n them with revelations of the future, and with a refined and\n metaphysical system of dogmas, which were handed down from age to\n age, and from one tribe to another, as the primeval creed and\n possession of the enlightened race. Among them in the West, as\n well as in the remote East, the doctrine of metempsychosis held a\n conspicuous place, implying belief in an after state of rewards and\n punishment, and a moral government of the world. With it was\n connected the notion that the material universe had undergone, and\n was destined to undergo, a repetition of catastrophes by fire and\n water; and after each destruction, to be renewed in fresh beauty,\n when a golden age was again to commence, destined in a fated time\n to corruption and decay. The emanation of all beings from the soul\n of the universe, and their refusion in it, which were tenets\n closely connected with this system of dogmas, border on a species\n of Pantheism, and are liable to all the difficulties attendant upon\n that doctrine.\n\n \"Among most of the Indo-European nations, the conservation of\n religious dogmas, patriarchal tradition, and national poetry, was\n confided, not to accidental reminiscences and popular recitations,\n but to a distinct order of persons, who were venerated as mediators\n between the invisible powers and their fellow mortals, as the\n depositories of sacred lore, and interpreters of the will of the\n gods, expressed of old to the first men, and handed down, either\n orally in divine poems, or preserved in a sacred literature, known\n only to the initiated. In most instances they were an hereditary\n caste, Druids, Brahmans, or Magi.\n\n \"Among the Allophylian nations, on the other hand a rude and\n sensual superstition prevailed, which ascribed life and mysterious\n powers to the inanimate objects. The religion of fetisses, of\n charms, and spells and talismans, was in the hands not of a learned\n caste, the twice-born sons of Brama, but of shamans or sorcerers,\n who, by feigning swoons and convulsions, by horrible cries and\n yells, by cutting themselves with knives, by whirling and\n contortions, assumed the appearance of something preternatural and\n portentous, and impressed the multitude with the belief that they\n were possessed by demons. Of this latter description were the\n wizards of the Finns and Lappes, the angekoks of the Esquimaux; and\n such are the shamans of all the countries in Northern Asia, where\n neither Buddhism nor Islamism has yet penetrated.\"\n\nOf the American nations, the prevailing opinion, according to Loskiel,\nis--\n\n \"'That there is one God, or, as they call him, one Great and Good\n Spirit.' It seems, from the testimony of this writer, which is\n supported by the evidence of all those who have conversed with the\n aboriginal nations of North America, that the conceptions of these\n nations respecting the Deity are much more complete and\n philosophical than those of the most savage people in the Old\n Continent. They suppose him literally to be the creator of heaven\n and earth, of men and all other creatures; they represent him as\n almighty, and able to do as much good as he pleases; 'nor do they\n doubt that he is kindly disposed towards men, because he imparts\n power to plants to grow, causes rain and sunshine, and gives fish\n and venison to man for his support;' these gifts, however, to the\n Indians exclusively. 'They are convinced that God requires of them\n to do good, and to eschew evil.' We may observe that, in these\n particulars, the Americans resemble the Northern Asiatics. We are\n assured by the late traveller, M. Erman, on the authority of the\n metropolitan Philophei, who lived among the Ostiaks on the Oby,\n that these people had, before Christian missionaries ever came\n among them, a belief in the existence of a Supreme Deity, of whose\n nature they had pure and exalted ideas, and to whom they affirmed\n that they never made offerings, nor had they represented his form;\n while to inferior gods, and particularly to Oertidk, who was a sort\n of mediator, and whose name, as it was preserved among the Magyars,\n Oerdig, was used by the monks as a designator for the devil, they\n made divers gifts; they performed before his image dances, which\n Erman, who visited the Kolushians on the Sitcka, declares to be\n precisely similar to the war-dances of those Americans. Some of the\n American people make images of the Manittos.\n\n \"Besides the Supreme Deity, the American nations believe in a\n number of inferior spirits, whom the Delaware Indians term\n Manittos; they are both good and evil. 'From the accounts of the\n oldest Indians,' says Loskiel, 'it appears that when war was in\n contemplation, they used to admonish each other to hearken to the\n good, and not to evil spirits--the former always recommending\n peace.' They had formerly no notion of a devil, or evil being, in\n the Christian or Eastern sense of the term, but readily adopted,\n according to Loskiel, such a belief from the white people. They\n have among them preachers, who pretend to have received\n revelations, and who dispute and teach different opinions. Some\n pretend to have travelled near to the dwelling of God, or near\n enough to hear the cocks crow, and see the smoke of the chimneys in\n heaven; others declare that no one ever knew the dwelling-place of\n God, but that the abode of the Good Spirit is above the blue sky,\n and that the road to it is the milky way--a notion, by the way,\n which Beausobre and others have traced in the remains of the\n Manicheans, and other Eastern philosophers. The Americans believe\n in the existence of souls distinct from bodies, and many of them in\n the transmigration of souls. According to Loskiel, they declare,\n 'that Indians cannot die eternally; for even Indian corn is\n vivified, and rises again.' The general opinion among them is, that\n the souls of the good alone go to a place abounding in all earthly\n pleasures, while the wicked wander about dejected and melancholy.\n Like other nations, they had sacrifices. 'Sacrifices,' says\n Loskiel, 'made with a view to pacify God and the subordinate\n deities, are of a very ancient date among them, and considered in\n so sacred a light, that unless they are performed in a time and\n manner acceptable, illness, misfortune, and death would befall them\n and their families.' They offer on these occasions hares, bear's\n flesh, and Indian corn. Many nations have, besides other stated\n times of sacrifice, one principal festival in two years, when they\n sacrifice an animal, and make a point of eating the whole.\n\n \"A small quantity of melted fat is poured by the oldest men into\n the fire, and in this the main part of the offering consists. The\n offerings are made to Manittos. The Manittos are precisely the\n Fetisses of the African nations, and of the Northern Asiatics. They\n are tutelary beings, often in visible forms. Every Indian has a\n guardian Manitto; one has the sun for his Manitto; one the moon;\n one has a dream, that he must make his Manitto an owl; one a\n buffalo. The Delawares had five festivals in the year, one in\n honour of fire, supposed to have been the parent of all the Indian\n nations. Like other nations, these people believed in the necessity\n of purification from guilt, by fasting and bodily mortification.\n Some underwent for this end the pain of being beaten with sticks\n from the sole of their feet to their head. 'Some gave the poor\n people vomits as the most expeditious mode.'\n\n \"Like the Northern Asiatics, the American nations had, instead of a\n regular priesthood, jugglers or sorcerers, who pretended to have\n supernatural power and knowledge. They appear to conform in every\n respect to the schamans of the Siberians, and the Fetiss-seers of\n the African nations.\"\n\nWe have, in the above extracts, placed in juxtaposition the leading\npsychical characteristics of the five divisions of mankind. There are\nsome points in which the different races of man seem, in their various\nsuperstitions and creeds, curiously agreed. The doctrine of sacrificial\natonement seems almost universally prevalent, and forms the basis of the\nvarious sacerdotal institutions. The care of the dead is also another\npeculiarity, and one in which mankind appear, from the earliest\nhistorical period, to have differed from other animals.\n\nThe susceptibility to receive the doctrines of Christianity is a\ncircumstance of agreement among the various races of mankind, from which\nthe Bushmen of South Africa are the only exception; and, viewing these\nas a branch of the Hottentots, this exception would seem to\ndisappear--for the latter have been converted. The following is the\nsatisfactory account of the Hottentot missionaries as to the moral\neffect of Christianity:--\n\n \"It is the unvarying statement of these missionaries, deduced from\n the experience of a hundred years of patient service and laborious\n exertions among the rudest and most abject tribes of human beings;\n that the moral nature of man must be in the first instance\n quickened, the conscience awakened, and the better feelings of the\n heart aroused, by the motives which Christianity brings with it,\n before any improvement can be hoped for in the outward behaviour\n and social state; that the rudest savages have sufficient\n understanding to be susceptible of such a change; and that, when it\n has once taken place, all the blessings of civilization follow as a\n necessary result.\"\n\nThe gypsy tribe, of which Dr Prichard takes no notice, would seem to\nform an exception from the great mass of mankind as to the absence of\nreligious creed. The opinions and theories respecting it we must leave,\nas it forms of itself a wide field for discussion; and, having fully\noccupied the space allotted to us, we must here bring to a close our\nsketch of a work which, notwithstanding the somewhat unreadable\ncharacter of the central portion, has supplied to the public a valuable\ncollection of recorded facts, expressed for the most part in clear,\nuntechnical language. We have not entered into questions of contrast or\nsimilitude with the opinions of other authors. Had we done so, we must\nhave adopted a style of criticism interesting only to those who are\nspecially engaged in the subject, and so incapable of limitation that\nevery paragraph would serve for an article longer far than that which we\nhave here written. Dr Prichard appears nowise unwilling to refer to each\nauthor his due share of merit, and is by no means sparing of copious\nextracts, taken with no partial view of supporting a theory. At the risk\nof being considered only a compiler, he has, at all events, avoided any\naffectation of originality.\n\nWith regard to the proposition sought to be established by the author,\nthe book before us does not appear to be conclusive. The question as to\nthe community of origin of mankind, viewed purely as an inductive one,\nappears still involved in obscurity. On the one hand, the fact of\ncontinual degeneration, resulting from the intermarriage of members of\nthe same family, would require for its explanation either a miraculous\ninterference in the first periods of human existence, or a gradual\nchange in the constitution of man, whereby what once was harmless has\nbecome injurious, when the necessity for it is removed; moreover,\naccording to the evidence contained in this book, the races of mankind\ncannot be traced backward to a single pair. But, taking the three great\ndivisions, the Semitic, the Hamitish, and the Japetic, as derived from\nShem, Ham, and Japhet, the various Allophyllian and American aborigines\nwould appear to have existed, and to have been spread over the world\nbefore the above nations overran it. On the other hand, supposing that\nthe mere power of reproduction be not of itself sufficient evidence of\nidentity of species, the similarity of physical formation, of periodic\nchanges, and of psychical instincts, are strongly corroborative of this\nevidence, and would of themselves lead to the deduction of such\nidentity. Upon the whole, we consider the merits of the work before us\nto consist, not in the demonstration of a theorem, but in presenting to\nthe reader a compendious record of physical, historical, and\npsychological facts and relations. Viewed in this light, it is an\ninteresting contribution to ethnology; while the size of the book, the\npictorial illustrations, and the absence of unnecessary technicality,\nmake it a convenient manual for the general reader.\n\n\nFOOTNOTES:\n\n{A} _The Natural History of Man._ By J. C. PRICHARD, M.D.\n\n{B} The term _Hamitish_ is not used by Dr Prichard; but as he gives no\ndistinctive appellation to his third division, we adopt that which has\nbeen used by Beke and others.\n\n{C} The term _Arian_, used by Dr Prichard, is objectionable as having\nreceived a very different application.\n\n\n\n\nPOEMS BY COVENTRY PATMORE.{A}\n\n\nThis is certainly an age of very merciful tendencies. The severity of\nthe criminal laws has been greatly abated; and, in conformity with the\nviews of the legislature, we have, of late years, been gradually\nrelaxing the stringency of our critical code. Yet we question whether\nthe change has been productive of good, and whether the result can be\nsaid to have answered the expectations either of government or of\nourselves. We doubt whether crime has diminished in consequence of the\nlegislative clemency; and, in our own humble department, we are now\nconvinced that the mild method is not the best way of bringing singers\nto repentance. The experiment has been fairly tried, and the numerous\ntrashy publications put forth by the young writers of the day,\nparticularly in the poetical line, convince us that our mercy has been\nmisplaced; and that a little well-timed severity, and a few examples\nheld up _in terrorem_, might have greatly benefited the literary\nwellbeing of England. The \"spirit of the age\" might have been different\nfrom what it is, if the just sentence of the law had been more\nfrequently carried into effect. Our timely strictures might not have\nkindled into song any masculine intellect, but they might have prevented\nthe temple of the Muses from being desecrated. They might have prevented\nthe appearance of such a publication as this. In the days of the knout,\nwe believe that no such volume as Mr Coventry Patmore's could have\nventured to crawl out of manuscript into print. While we admit, then,\nthat we have to blame our own forbearance in some degree for its\nappearance, we think it our duty to take this opportunity of amending\nour code of criticism, and shall try the volume simply as it stands, and\nsomewhat according to the good old principles of literary jurisprudence.\n\nWe are further instigated to this act of duty by the laudatory terms in\nwhich the volume has been hailed by certain contemporary journalists.\nHad Mr Patmore's injudicious friends not thought proper to announce him\nto the world as the brightest rising star in the poetical firmament of\nYoung England, we would probably have allowed his effusions to die of\ntheir own utter insignificance. But since they have acted as they have\ndone, we too must be permitted to express our opinion of their merits;\nand our deliberate judgment is, that the weakest inanity ever\nperpetrated in rhyme by the vilest poetaster of any former generation,\nbecomes masculine verse when contrasted with the nauseous pulings of Mr\nPatmore's muse. Indeed, we question whether the strains of any poetaster\ncan be considered vile, when brought into comparison with this\ngentleman's verses. His silly and conceited rhapsodies rather make us\nsigh for the good old times when all poetry, below the very highest, was\nmade up of artifice and conventionalism; when all poets, except the very\ngreatest, spoke a hereditary dialect of their own, which nobody else\ninterfered with--counted on their fingers every line they penned, and\nknew no inspiration except that which they imbibed from Byssh's rhyming\ndictionary. True that there was then no life or spirit in the poetical\nvocabulary--true that there was no nature in the delineations of our\nminor poets; but better far was such language than the slip-slop\nvulgarities of the present rhymester--better far that there should be no\nnature in poetry, than _such_ nature as Mr Patmore has exhibited for the\nentertainment of his readers.\n\nThe first poem in the volume, entitled \"The River,\" is a tale of\ndisappointed love, terminating in the suicide of the lover. Poor and\npointless as this performance is, it is by far the best in the book. As\nMr Patmore advances, there is a marked increase of silliness and\naffectation in his effusions, which shows how sedulously he has\ncultivated the art of sinking in poetry; and that the same adage which\nhas been applied to vice, may be applied also to folly, \"_Nemo repente\nfuit stultissimus_.\" Never was there a richer offering laid on the\nshrine of the goddess _Stultia_ than the tale of Sir Hubert, with which\nthe volume concludes. But our business at present is with \"The River.\"\n\nThe common practice of writers who deal with stories of love, whose\n\"course never did run smooth,\" is to make their heroes commit suicide,\non finding that the ladies whom they had wooed in vain were married to\nother people. But in the poem before us, Mr Patmore improves upon this\nmethod; he drowns his lover, Witchaire, because the lady, whom he had\nnever wooed at all, does not marry him, but gives her hand (why should\nshe not?) to the man who sues for it. Did Witchaire expect that the lady\nwas to propose to him? The poem opens with some very babyish verses\ndescriptive of an \"old manor hall\":--\n\n \"Its huge fantastic weather-vanes\n _Look happy_ in the light;\n Its warm face through the foliage gleams,\n A _comfortable_ sight.\"\n\nAnd so on, until we are introduced to the lady of the establishment:--\n\n \"That lady loves the pale Witchaire,\n _Who loves too much to sue_:\n He came this morning hurriedly,\n Then out her young blood flew;\n But he talk'd of common things, _and so_\n Her eyes are steep'd in dew.\"\n\nThe lady, finding that her lover continues to hang back, dries her\ntears, and very properly gets married to another man. During the\ncelebration of the ceremony, the poet recurs to his hero, who has taken\nup his position in the park--\n\n \"Leaning against an aged tree,\n By thunder stricken bare.\n\n \"The moonshine shineth in his eye,\n From which no tear doth fall,\n Full of vacuity as death,\n Its slaty parched ball\n Fixedly, though expressionless,\n Gleams on the distant hall.\"\n\nWitchaire then goes and drowns himself, in a river which \"runneth round\"\nthe lady's property--a dreadful warning to all young lovers \"who love\ntoo much to sue.\"\n\nOn a fine day in the following summer, the poet brings the lady to the\nbanks of this river. His evident intention is, to raise in the reader's\nmind the expectation that she shall discover her lover's body, or some\nother circumstance indicative of the fatal catastrophe. This\nexpectation, however, he disappoints. The only remarkable occurrence\nwhich takes place is, that the lady does _not_ find the corpse, nor does\nany evidence transpire which can lead her to suppose that the suicide\nhad ever been committed; and with this senseless and inconclusive\nconclusion the reader is befooled.\n\nThe only incident which we ever heard of, at all rivaling this story in\nan abortive ending, is one which we once heard related at a party, where\nthe conversation turned on the singular manner in which valuable\narticles thrown into the sea had been sometimes recovered, and restored\nto their owners--the ring of Polycrates, which was found in the maw of a\nfish after having been sunk in deep waters, being, as the reader knows,\nthe first and most remarkable instance of such recoveries. After the\nrest of the company had exhausted their marvellous relations, the\nfollowing tale was told as the climax of all such wonderful narratives;\nand it was admitted on all hands that the force of surprise could no\nfurther go. We shall endeavour to versify it, _a la_ Patmore, conceiving\nthat its issue is very similar to that of his story of \"The River.\"\n\n\nTHE RING AND THE FISH.\n\n A lady and her lover once\n Were walking on a rocky beach:\n Soft at first, and gentle, was\n The music of their mutual speech,\n And the looks were gentle, too,\n With which each regarded each.\n\n At length some casual word occurr'd\n Which somewhat moved the lady's bile;\n From less to more her anger wax'd--\n How sheepish look'd her swain the while!--\n And now upon their faces twain\n There is not seen a single smile.\n\n A ring was on the lady's hand,\n The gift of that dumb-founder'd lover--\n In scorn she pluck'd it from her hand,\n And flung it far the waters over--\n Far beyond the power of any\n Duck or drag-net to recover.\n\n Remorse then smote the lady's heart\n When she had thrown her ring away;\n She paceth o'er the rocky beach,\n And resteth neither night nor day;\n But still the burthen of her song\n Is, \"Oh, my ring! my ring!\" alway.\n\n Her lover now essays to soothe\n The dark compunctious visitings,\n That assail the lady's breast\n With a thousand thousand stings,\n For that she had thrown away\n This, the paragon of rings.\n\n But all in vain; at length one day\n A fisher chanced to draw his net\n Across the sullen spot that held\n The gem that made the lady fret,\n And caught about the finest cod\n That ever he had captured yet.\n\n He had a basket on his back,\n And he placed his booty in it;\n The lady's lover bought the fish,\n And, when the cook began to skin it,\n She found--incredible surprise!--\n She found the ring--was _not_ within it.\n\nThe next tale, called \"The Woodman's Daughter,\" is a story of seduction,\nmadness, and child-murder. These are powerful materials to work with;\nyet it is not every man's hand that they will suit. In the hands of\ncommon-place, they are simply revolting. In the hands of folly and\naffectation, their repulsiveness is aggravated by the simpering conceits\nwhich usurp the place of the strongest passions of our nature. He only\nis privileged to unveil these gloomy depths of erring humanity, who can\nsubdue their repulsiveness by touches of ethereal feeling; and whose\nimagination, buoyant above the waves of passion, bears the heart of the\nreader into havens of calm beauty, even when following the most\ndeplorable aberrations of a child of sin. Such a man is not Mr Patmore.\nHe has no imagination at all--or, what is the same thing, an imagination\nwhich welters in impotence, far below the level of the emotions which it\nought to overrule. The pitfalls of his tale of misery are covered over\nwith thin sprinklings of asterisks--the poorest subterfuge of an\nimpoverished imagination; and besotted indeed is the senselessness with\nwhich he disports himself around their margin. Maud, the victim, is the\ndaughter of Gerald, the woodman; and Merton, the seducer, is the son of\na rich squire in the neighbourhood. Maud used to accompany her father to\nhis employment in the woods.\n\n \"She merely went to think she help'd;\n And whilst he hack'd and saw'd,\n The rich squire's son, a young boy then,\n For whole days, as if aw'd,\n Stood by, and gazed alternately\n At Gerald and at Maud.\n\n \"He sometimes, _in a sullen tone_,\n Would offer fruits, and she\n Always received his gifts with an air,\n So unreserved and free,\n That half-feign'd distance soon became\n Familiarity.\n\n \"Therefore in time, when Gerald shook\n The woods at his employ,\n The young heir and the cottage-girl\n Would steal out to enjoy\n The music of each other's talk--\n A simple girl and boy.\n\n \"They pass'd their time, both girl and boy,\n Uncheck'd, unquestion'd; yet\n They always hid their wanderings\n By wood and rivulet,\n Because they could not give themselves\n A reason why they met.\n\n --It may have been in the ancient time,\n Before Love's earliest ban,\n Psychean curiosity\n Had broken Nature's plan;\n _When all that was not youth was age,\n And men knew less of Man;_--\n\n \"Or when the works of time shall reach\n The goal to which they tend,\n And knowledge, being perfect, shall\n At last in wisdom end--\n That wisdom to end knowledge--or\n Some change comes, yet unkenn'd;\n\n \"It perhaps may be again, that men,\n Like orange plants, will bear,\n At once, the many fine effects\n To which God made them heir--\n Large souls, large forms, and love like that\n Between this childish pair.\n\n \"Two summers pass'd away, and then--\n _Though yet young Merton's eyes,\n Wide with their language, spake of youth's\n Habitual surprise_--\n He felt that pleasures such as these\n No longer could suffice.\"\n\nWhat the meaning of the three stanzas beginning with--\n\n \"It may have been in the ancient time,\"\n\nmay be, we are utterly at a loss to conjecture. We seek in vain to\ninvest them with a shadow of sense. Perhaps they are thrown in to\nredeem, by their profound unintelligibility, the shallow trifling of the\nrest of the poem. But it was not enough for young Merton that the girl\naccepted the fruits which he offered to her in a _sullen_ tone. He had\nnow reached the age so naturally and lucidly described as the period of\nlife when the \"eyes, wide with their language, speak of youth's habitual\nsurprise,\" and he began to seek \"new joys from books,\" communicating the\nresults of his studies to Maud, whose turn it now was to be surprised.\n\n \"So when to-morrow came, while Maud\n Stood listening with surprise,\n He told the tale learnt over night,\n And, if he met her eyes,\n _Perhaps_ said how far the stars were, _and\n Talk'd on about the skies_.\"\n\nThe effect of these lucid revelations upon the mind of Maud was very\noverpowering.\n\n \"She wept for joy if the cushat sang\n Its low song in the fir;\n The cat, _perhaps_, broke the quiet with\n Its regular slow purr;\n 'Twas music now, and her wheel gave forth\n A rhythm in its whirr.\n\n \"She once had read, When lovers die,\n And go where angels are,\n Each pair of lover's souls, _perhaps_,\n Will make a double star;\n _So stars grew dearer, and she thought\n They did not look so far._\n\n \"But being ignorant, and still\n So young as to be prone\n To think all very great delights\n Peculiarly her own,\n She guess'd not what to her made sweet\n _Books writ on lovers' moan_.\"\n\nAnd so the poem babbles on through several very sickly pages, in which\nthe following descriptive stanza occurs:--\n\n \"The flat white river lapsed along,\n Now a broad broken glare,\n Now winding through the bosom'd lands,\n Till lost in distance, where\n The tall hills, sunning their chisell'd peaks,\n _Made emptier the empty air_.\"\n\nDuring one of their ramblings, Maud becomes visibly embarrassed.\n\n \"But Merton's thoughts were less confused:\n 'What, _I_ wrong ought so good?\n Besides, the danger that is seen\n Is easily withstood:'\n Then loud, 'The sun is very warm'--\n _And they walk'd into the wood_.\"\n\nThe wood consisting of a forest of as shady asterisks as the most\nfastidious lovers could desire.\n\n \"Months pass'd away, and every day\n The lovers still were wont\n To meet together, and their shame\n At meeting had grown blunt;\n _For they were of an age when sin\n Is only seen in front_.\"\n\nThe father, however, who was also of an age to see sin _in front_,\nsuspects that his daughter is with child, and taxes her with it. Maud\nconfesses her shame; upon which, as we are led to conjecture, old Gerald\ndies broken-hearted--while the girl is safely delivered under a cloud of\nasterisks. She is deterred from disclosing her situation to Merton, the\nfather of the child--and why? for this very natural reason, forsooth,\nthat\n\n \"He, if that were done,\n Could hardly fail to know\n The ruin he had caused, he might\n Be brought to share her woe,\n Making it doubly sharp.\"\n\nSo, rather than occasion the slightest distress or inconvenience to her\nseducer, she magnanimously resolves to murder her baby; and accordingly\nthe usual machinery of the poem is brought into play--the\nasterisks--which on former occasions answered the purpose of a forest\nand a cloud, being now converted into a very convenient pool, in which\nshe quietly immerses the offspring of her illicit passion. And the deed\nbeing done, its appalling consequences on her conscience are thus\npowerfully and naturally depicted--\n\n \"_Lo! in her eyes stands the great surprise\n That comes with the first crime_.\n\n \"She throws a glance of terror round--\n There's not a creature nigh;\n But behold the sun that looketh through\n The frowning western sky,\n Is lifting up one broad beam, _like\n A lash of God's own eye_.\"\n\nWere we not right in saying that there is nothing in the writings of any\nformer poetaster to equal the silly and conceited jargon of the present\nversifier? Having favoured us with the emphatic lines in italics, to\ndepict the physical concomitants of Maud's guilt, he again has recourse\nto asterisks, to veil the mental throes by which her mind is tortured\ninto madness by remorse: and very wisely--for they lead us to suppose\nthat the writer could have powerfully delineated these inner agitations,\nif he had chosen; but that he has abstained from doing so out of mercy\nto the feelings of his readers. We must, therefore, content ourselves\nwith the following feebleness, with which the poem concludes:\n\n \"Maud, with her books, comes, day by day,\n Fantastically clad,\n To read them near the poor; and all\n Who meet her, look so sad--\n That even to herself it is\n Quite plain that she is mad.\"\n\n\"Lilian\" is the next tale in the volume. This poem is an echo, both in\nsentiment and in versification of Mr Tennyson's \"Locksley Hall;\" and a\nbaser and more servile echo was never bleated forth from the throat of\nany of the imitative flock. There are many other indications in the\nvolume which show that Mr Tennyson is the model which Mr Patmore has set\nup for his imitation; but \"Lilian,\" more particularly, is a complete\ncounterpart in coarsest fustian of the silken splendours of Mr\nTennyson's poem. It is \"Locksley Hall\" stripped of all its beauty, and\ndebased by a thousand vulgarities, both of sentiment and style. The\nburden of both poems consists of bitter denunciations poured forth by\ndisappointed and deserted love; with this difference, that the passion\nwhich Mr Tennyson gives utterance to, Mr Patmore reverberates in rant. A\nsmall poet, indeed, could not have worked after a more unsafe model. For\nwhile he might hope to mimic the agitated passions of \"Locksley Hall,\"\nin vain could he expect to be visited by the serene imagination which,\nin that poem, steeps their violence in an atmosphere of beauty. Even\nwith regard to Mr Tennyson's poem, it is rather for the sake of its\npicturesque descriptions, than on account of its burning emotions, that\nwe recur to it with pleasure. We rejoice to follow him to regions where\n\n \"Never comes the trader, never floats an European flag,\n _Slides the bird o'er lustrous woodland_, droops the trailer from\n the crag.\"\n\nIt is rather, we say, on account of such lines as these (no picture of\ntropical loveliness ever surpassed, in our opinion, the description\nprinted in italics) that we admire \"Locksley Hall,\" than on account of\nthe troubled passions which it embodies; knowing as we do, that poetry\nhas nobler offices to perform than to fulmine forth fierce and sarcastic\ninvectives against the head of a jilt; and if, as Mr Tennyson says,\n\"love is love for evermore,\" we would ask even him why he did not make\nthe lover in \"Locksley Hall\" betray, even in spite of himself, a more\npitiful tenderness for the devoted heroine of the tale? How different\nthe strain of the manly Schiller under similar circumstances! _His_\nbitterness cannot be restrained from breaking down at last in a flood of\ntenderness over the lost mistress of his affections.\n\n \"Oh! what scorn for thy desolate years\n Shall I feel! God forbid it should be!\n How bitter will then be the tears\n Shed, Minna, oh Minna, for thee!\"\n\nBut if it be true that \"Locksley Hall\" is somewhat deficient in the\nethereal tenderness which would overcome a true heart, even when\nblighted in its best affections, it was not to be expected that its\nimitator should have been visited with deeper glimpses of the divine.\nThe indignant passions of his unrequited lover are, indeed, passions of\nthe most ignoble clay--not one touch of elevated feeling lifts him for\na moment out of the mire. The whole train of circumstances which\nengender his emotions, prove the lover, in this case, to have been the\nsilliest of mortal men, and his mistress, from the very beginning of his\nintercourse with her, to have been one of the most abandoned of her sex.\n\"Lilian\" is a burlesque on disappointed love, and a travestie of the\npassions which such a disappointment entails. We know not which are the\nmore odious and revolting in their expression--the emotions of the\njilted lover, or the incidents which call them into play.\n\nThe poem is designed to illustrate the bad effects produced on the\nfemale mind by the reading of French novels. We have nothing to say in\ntheir defence. But the incongruity lies here--that Lilian, who was\nseduced by means of these noxious publications, was evidently a lady of\nthe frailest virtue from the very first; and her lover might have seen\nthis with half an eye. Her materials were obviously of the most\ninflammable order; and it evidently did not require the application of\nsuch a spark as the seducer Winton, with his formidable artillery of\nimported literature, to set her tinder in a blaze--any other small\ncontingency would have answered equally well. All that she wanted was an\nopportunity to fall; and that she would soon have found, under any\ncircumstances whatsoever. The lover, however, sees nothing of all this,\nbut relates the story of his unfortunate love-affair with as much\nsimplicity as if he had been mourning the fall of the mother of mankind\nfrom paradise.\n\nThe lover relates his tale to his friend, the author. He begins by\nentreating him to\n\n \"Bear with me, in case\n Tears come. _I feel them coming by the smarting in my face_.\"\n\nAnd then he proceeds to introduce us to this Lilian, the immaculate\nmistress of his soul--\n\n \"She could see me coming to her with the vision of the hawk;\n Always hasten'd on to meet me, _heavy passion in her walk;_\n Low tones to me grew lower, sweetening so her honey talk,\n\n \"That it fill'd up all my hearing, drown'd the _voices of the birds_,\n The _voices of the breezes_, and the _voices of the herds_--\n For to me the lowest ever were the loudest of her words.\"\n\n\"Heavy passion in her walk!\"--what a delicate and delectable young lady\nshe must have been! Then, as to the fact so harmoniously expressed, of\nher accents drowning \"the voices of the birds, the voices of the\nbreezes, and the voices of the herds,\" we may remark, that the first and\nsecond never require to be drowned at all, being nearly inaudible at any\nrate, even during the most indifferent conversation--so that there was\nnothing very remarkable in their being extinguished by the plaintiveness\nof the lady's tones; while, with regard to the voices of the herds, if\nshe succeeded in drowning these--the cattle being near at hand, and\nlowing lustily--she must indeed have roared to her lover \"like any\nnightingale.\"\n\nThe description of her is thus continued--\n\n \"On her face, then and for ever, was the seriousness within.\n Her sweetest smiles (and sweeter did a lover never win)\n _Ere half-done grew so absent, that they made her fair cheek thin._\n\n \"On her face, then and for ever, thoughts unworded used to live;\n So that when she whisper'd to me, 'Better joy earth cannot give'--\n Her lips, though shut, continued, 'But earth's joy is fugitive.'\n\n \"For there a _nameless something_, though suppress'd, still spread\n around;\n The same was on her eyelids, if she look'd towards the ground;\n When she spoke, you _knew directly_ that the same was in the sound;\"\n\nBy and by, a young gentleman, of the name of Winton, comes to visit\nLilian and her father:--\n\n \"A formerly-loved companion--he was fresh from sprightly France,\n And with many volumes laden, essay, poem, and romance.\"\n\nHe, and his pursuits after leaving school, are thus elegantly\ndescribed:--\n\n \"When free, all healthy study was put by, that he might rush\n To his favourite books, French chiefly, that his blood might boil\n and gush\n Over scenes which set his visage glowing crimson--_not a blush_.\"\n\nThis gentleman and Lilian's lover strike up a strong friendship for one\nanother, and the latter makes Winton his confidant. As yet no suspicions\narise to break the blind sleep of the infatuated dreamer.\n\n \"Delights were still remaining--hate--shame--rage--_I can't tell what_,\n Comes to me at their memory; none that, _more or less_, was not\n The soul's _unconscious incest_, on creations self-begot.\"\n\nHe still continues to doat on Lilian.\n\n \"Oh friend, if you had seen her! heard her speaking, felt her grace,\n When serious looks seem'd filling with the smiles which, in a space,\n Broke, sweet as Sabbath sunshine, and lit up her _shady_ face.\n\n \"Try to conceive her image--does it make your brain reel round?\n But all of this is over. Well, friend--various signs (I found\n Too late on rumination) then and thenceforth did abound,\n\n \"Wherefrom--but that all lovers look too closely to see clear--\n I might have gather'd matter fit for just and jealous fear.\n From her face, _the nameless something_ now began to disappear.\n\n \"What I felt for her I often told her boldly to her face;\n _Blushes used to blush at blushes flushing on in glowing chace!_\n But latterly she listen'd, bending full of bashful grace.\n\n \"It was to hide those blushes, I thought then, _but I suspect\n It was to hide their absence_.\"\n\nHow great this writer is on the subject of blushing we shall have\nanother opportunity of showing.--(See Lady Mabel's shoulders, in the\npoem of Sir Hubert.) Meanwhile, the fair deceiver is now undergoing a\ncourse of French novels, under the tuition of young Winton. The\nconsequence was,\n\n\"_Her voice grew louder_\"--no great harm in that--\n\n \"Her voice grew louder--losing the much meaning it once bore,\n _The passion in her carriage_, though it every day grew more,\n Was now the same to all men--and that was not so before.\"\n\nWe suppose that there was now \"heavy passion in her walk,\" whoever the\nman might be that approached her.\n\n \"And grosser signs, _far grosser_ I remember now; but these\n I miss'd of course, and counted _with those light anomalies,\n Too frequent to disturb us into searching for their keys_.\"\n\nThese misgivings, which might have ripened into suspicions, are suddenly\nswept away by a stroke of duplicity on the part of his mistress,\ninconceivable in any woman except one inclined naturally, and without\nany prompting, to practise the profoundest artifices of vice.\n\n \"Even the dreadful glimpses now began to fade away,\n And disappear'd completely, when my Lilian asked one day,\n If I knew what reason Winton had to make so long a stay\n\n \"In England--'For,' said Lilian, with untroubled countenance,\n 'Winton of course has told you of the love he left in France.'\n I seized her hand, and kiss'd it--joy had left no utterance.\"\n\nWinton, according to the account of the false Lilian, having _a love_ in\nFrance, could not, of course be supposed to be paying court to her. Thus\nthe lover is thrown off the scent, and his doubts are entirely laid\nasleep. He is again in the seventh heavens of assured love, and\ncontinues thus:--\n\n \"Another calm so perfect I should think is only shed\n On good men dying gently, who recall a life well led,\n Till they cannot tell, _for sweetness_, if they be alive or dead.\n\n \"_I'll stop here._ You already have, I think, divined the rest.\n There's a prophetic moisture in your eyes:--yet, tears being blest\n And delicate nutrition, apt to cease, too much suppress'd,\n\n \"_I'll go on_; but less for your sake than my own:--my skin is hot,\n And there's an arid pricking in my veins; their currents clot:\n Tears sometimes soothe such fever, where the letting of blood will\n not.\"\n\nAt length his eyes are opened, and the whole truth flashes upon him, on\noverhearing an acquaintance ask Winton whether his suit with Lilian has\nbeen successful. Upon this he writes out his opinion of the lady's\nbehaviour, presents it to her, and watches her while she peruses it,\noccupying himself at intervals as follows:--\n\n \"I turn'd a volume, waiting her full leisure to reply,\n The book was one which Winton had ask'd me to read, and I\n Had stopp'd halfway for horror, _lest my soul should putrify_.\"\n\nWhen Lilian has finished the perusal of the document, she endeavours at\nfirst to stand on the defensive,--\n\n \"She stood at bay, _depending on that crutch made like a stilt_,\n The impudent vulgarity wherewith women outstare guilt.\"\n\nBut she finally succumbs under the influence of the following refined\nvituperation:--\n\n \"Don't speak! You would not have me unacquainted with what led\n To this result? No! listen, and let _me_ relate what bred\n Thy tears and cheapen'd chasteness--(_we may talk now as if wed._)\n\n \"This book here, that lay open when I came in unaware,\n Is not the first--I thought so!--but the last of many a stair\n Of easy fall. Such only could have led you to _his lair_.\n\n \"These drugs, at first, had scarcely strength to move your virgin\n blood;\n They slowly rose in action, till they wrought it to a flood,\n Fit for their giver's purpose, who--_who turn'd it into mud!_\"\n\nThe lover then leaves Lilian to her own meditations, and commences to\nrant and rave against her seducer in good set terms, of which the\nfollowing is a specimen:--\n\n \"Pardon, Heaven! that I doubted whether there was any hell.\n Oh! but now I do believe it! Firmly, firmly! I foretell\n Of one that shall rank high there: he's a scoffer, and must dwell\n\n \"Where worms are--ever gnawing scoffers' hearts into belief;\n Where weepings, gnashings, wailings, thirstings, groanings, ghastly\n grief,\n For ever and for ever pay the price of pleasures brief;\n\n \"Where Gallios, who while living knew but cared for none of these,\n Now amazed with shame, would gladly, might it God (_Fate there_)\n appease,\n Watch and pray a million cycles for a single moment's ease.\"\n\nAfter having thus breathed his passion, in a diatribe which beats in\nabomination any slang that was ever ranted out of a tub by a mountebank\nsaint, he harps back upon the prodigious attractiveness of his mistress,\nin the following pathetic, though not very consistent terms--\n\n \"Ah but had you known my Lilian! (a sweet name?) Indeed, indeed,\n I doted on my Lilian. None can praise her half her meed.\n Perfect in soul; too gentle--others' need she made her need;\n\n \"_Quite passionless_, but ever bounteous-minded even to waste;\n Much tenderness in talking; very urgent, yet no haste;\n _And chastity--to laud it would have seem'd almost unchaste._\n\n \"Graced highly, too, with knowledge; versed in tongues; a queen of\n dance;\n An artist at her playing; a most touching utterance\n In song; her lips' mild music could make sweet the _clack_ of France.\"\n\nAmid such outpourings of feculent folly, it is scarcely worth our while\nto take notice of the minor offences against good taste that abound in\nthese poems; yet we may remark, that the writer who here condescends to\nuse such a word as _clack_, and who, on other occasions, does not\nscruple to talk of _a repeat_ and _a repay_, instead of \"a repetition,\"\nand \"a repayment,\" does not consider the word _watch-dog_ sufficiently\nelevated for his compositions. Whenever he alludes to this animal, he\ncalls him a _guard-hound_--a word which we do not remember ever to have\nencountered either in conversation or in books, but which, for ought we\nknow, may be drawn from those \"pure wells of English undefiled,\" which\nirrigate with their fair waters the provincial districts of the modern\nBabylon.\n\nThe author of \"Lillian\" evidently piques himself on the fidelity with\nwhich he has adhered to nature in his treatment of that story. But there\nare two ways in which nature may be adhered to in verse; and it is only\none of these ways which can be considered poetical. The writer may\nadhere to the truth of _human_ nature, while he elevates the emotions of\nthe heart in strains which find a cordial echo in the sentiments of all\nmankind. Or, if his whole being is sicklied over with silliness and\naffectation, he may adhere to the truth of _his own_ nature, and while\nwriting perfectly naturally _for him_, he may unfold his delineations of\ncharacter in such a manner as shall strip every passion of its dignity,\nand every emotion of its grace. Now, it is only by reason of their\nadherence to the latter species of nature, that \"Lillian\" and the other\ncompositions of Mr Patmore can be considered natural, and, viewed under\nthis aspect, they certainly are natural exceedingly.\n\nThe story of \"Sir Hubert\" finishes the volume. This tale is versified\nfrom Boccacio's story of the Falcon, with which many of our readers may\nbe acquainted; if not, they will find it in the fifth day, novel ninth,\nof the _Decameron_. We can only afford space for a short outline of its\nincidents, and shall substitute Mr Patmore's names for those of the\npersonages who figure in Boccacio's story. This will save both ourselves\nand readers the trouble of threading the _minutiae_ of Mr Patmore's\nsenseless and long-winded version of the tale. A few specimens will\nsuffice to exhibit the manner in which he deals with it. Sir Hubert is a\nrich gentleman, who squanders almost all his substance in giving grand\nentertainments to the Lady Mabel, whom he makes love to without meeting\nwith any return. Finding his suit unsuccessful, and his money being all\nspent, he retires to a small and distant farm, having nothing left but\none poor hawk, upon which he depends for his means of subsistence.\nMeanwhile, the Lady Mabel marries, and has a son. After a time, (her\nhusband being dead,) she comes to reside in a castle in the\nneighbourhood of Sir Hubert's cottage, where her son, who has often\nremarked the prowess and beauty of the above-mentioned hawk, falls sick,\nassuring his mother that nothing can save his life except the possession\nof the bird. The lady very reluctantly pays a visit to Sir Hubert, and\ntells him that she has a request to proffer, which she will make known\nto him after dinner. Though Sir Hubert is delighted to see her, the\nmention of dinner throws him into a state of great perplexity, as he has\nnothing in the house which they can make a meal of. Going out of doors,\n\"he espies his hawk upon the perch, which he seizes, and finding it very\nfat, judges it might make a dish not unworthy of such a lady. Without\nfurther thought, then, he pulls his head off, and gives it to a girl to\ndress and roast carefully.\"\n\nThis being done, the lady and her admirer sit down to dinner, and make\nan excellent repast. When their meal is over, then comes the\n_eclaircissement_. The lady proffers her petition for the hawk; and\ndiscovers from Sir Hubert's answer, and to her own consternation, that\nshe has eaten the very article she came in quest of, and which she had\nexpected to carry home alive; as the only means of saving the life of\nher son. The young gentleman dies on finding that he cannot obtain what\nhe wants; and Mabel marries Sir Hubert, and settles upon him all her\npossessions, as a reward for his magnanimity in sacrificing that which\n(next to herself) he held dearest in the whole world, rather than that\nshe should go without a dinner.\n\nSuch is a short sketch of Boccacio's tale of the Falcon--a good enough\nstory in its way; and more creditable than many that were circulated\namong the loose fish, male and female, that play their parts in the\n_Decameron_. This novel has been versified by Mr Patmore, and versified\n(as our specimens shall show) as he alone could have versified it. The\nfollowing is his description of the much-longed-for, but\nsorely-ill-treated, hawk of Sir Hubert.\n\n \"It served him, too, of evenings:\n On a sudden he would rise,\n From book or simple music,\n And awake his hawk's large eyes,\n (_Almost as large as Mabel's_)\n Teasing out its dumb replies,\n\n \"In sulky sidelong glances,\n And reluctantly flapp'd wings,\n Or looks of slow communion,\n To the lightsome questionings\n That broke the drowsy sameness,\n And the sense, like fear, which springs\n\n \"At night, when we are conscious\n Of our distance from the strife\n Of cities; and the memory\n Of the spirit of all things rife,\n _Endues the chairs and tables\n With a disagreeable life_.\"\n\nA Scotch lyrist, who, we are told, sings his own songs to perfection,\nhas also recorded the very singular fact of various articles of\nhousehold furniture (not exactly tables) being occasionally endued \"with\na disagreeable life.\" One of his best ballads, in which he describes the\nbickerings which, even in the best-regulated families, will at times\ntake place between man and wife, and in which various domestic missiles\ncome into play, contains the following very excellent line--\n\n \"_The stools pass the best o' their time i' the air_\"--\n\nthan which no sort of life appertaining to a stool can be more\ndisagreeable, we should imagine--to the head which it is about to come\nin contact with. We doubt whether Mr Patmore's, or rather Sir Hubert's,\nchairs and tables ever acquired such a vigorous and unpleasant vitality\nas that. What may have happened to the \"stools\" after Mabel was married\nto Sir Hubert, we cannot take it upon us to say. At any rate, we prefer\nthe Scotch poet's description, as somewhat the more pithy, and graphic,\nand intelligible of the two. The coincidence, however, is remarkable.\n\nAfter Sir Hubert has retired to his farm, the state of his feelings is\ndescribed in the following stanzas. We suspect that the metaphysical\nacumen of Boccacio himself would have been a good deal puzzled to\nunravel the meaning of some of them.\n\n \"He gather'd consolation,\n As before, where best he might:\n But though there was the difference\n That he now could claim a right\n To grieve as much as pleased him,\n It was six years, since his sight\n\n \"Had fed on Mabel's features;\n So that Hubert scarcely knew\n What traits to give the vision\n Which should fill his eyes with dew:--\n For she must needs, by that time,\n Have become another, who,\n\n \"In girlhood's triple glory,\n (For a higher third outflows\n Whenever Promise marries\n With Completion,) troubled those\n That saw, with trouble sweeter\n Than the sweetest of repose.\n\n \"It, therefore, was the business\n Of his thoughts to try to trace\n The probable fulfilment\n Of her former soul and face,--\n From buds deducing blossoms.\n For, although an easy space\n\n \"Led from the farm of Hubert\n To where Mabel's castle stood,\n Closed in, a league on all sides.\n With wall'd parks and wealthy wood,\n No chance glimpse could be look'd for,\n So recluse her widowhood.\n\n \"Hence seasons past, and Hubert\n Earn'd his bread, but leisure spent\n In loved dissatisfaction,\n Which he made his element\n Of choice, as much as, till then,\n He had sought it in content.\"\n\nIf the verses above would have baffled the sagacity of the father of\nItalian literature, what would he have thought of the following, in\nwhich the interview between Sir Hubert and Mabel is described, when the\nlady comes to negotiate with him about the hawk? She accosts him, \"Sir\nHubert!\" and then there is presented to our imaginations such a picture\nof female loveliness, as (thank Heaven!) can only be done justice to in\nthe language which is employed for the occasion.\n\n \"'Sir Hubert!'--and, that instant,\n _Mabel saw the fresh light flush\n Out of her rosy shoulders_,\n And perceived her sweet blood _hush\n About her_, till, all over,\n _There shone forth a sumptuous blush_--\n\n \"'Sir Hubert, I have sought you,\n Unattended, to request\n A boon--the first I ever\n Have entreated.' Then she press'd\n _Her small hand's weight of whiteness\n To her richly-sloping breast_.\"\n\nAt first we thought that it should have been Hubert, and not Mabel, who\nsaw \"the fresh light flush out of her rosy shoulders\"--particularly if\nthe blush extended, as no doubt it did, to the lady's back: but on\nfurther consideration we saw that we were wrong; for Sir Hubert could\nnot have perceived \"her sweet blood _hush_ about her\"--this _hushing_ of\nthe blood about one being, as all great blushers know, a fact\ndiscernible only by the person more immediately concerned in the blush.\nThe propriety, therefore, of making Mabel perceive the blush, rather\nthan Sir Hubert, is undeniable. The writer must either have left out the\n_hushing_ altogether, which would have been a great blemish in the\npicture, or he must have written as he has done. How profoundly versed\nin the physiology of blushing he must be! We are doubtful, however,\nwhether the costume of the picture is altogether appropriate; for we\nquestion very much whether the Italian ladies of the thirteenth, or any\nother century, were in the habit of paying forenoon visits in low-necked\ngowns; and whether Mabel could have walked all the way from her castle\nto Sir Hubert's cottage, in an attire which revealed so many of her\ncharms, without attracting the general attention of the neighbourhood.\nShe had no time, be it observed, to divest herself of shawl or mantilla\nin order to show how _sumptuously_ she could blush--for her salutation\nis made to Sir Hubert, and its roseate consequences ensue the very first\nmoment she sees him. But let that pass. We should have been very sorry\nif such a \"splendiferous\" phenomenon had been obscured by envious boa or\npelisse, or lost to the proprieties of costume. The Lady then\n\n \"Said that she was wearied\n With her walk--would stay to dine,\n And name her wishes after.\"\n\nMeanwhile the poet asks--\n\n \"How was it with Sir Hubert?\n --Beggarly language! _I could burst_\n For impotence of effort:\n Those who made thee were accurst!\n _Dumb men were gods were all dumb_.\n But go on, and do thy worst!--\n\n \"His life-blood stopp'd to listen--\n Her _delivering_ lips dealt sound--\n Oh! _hungrily_ he listen'd,\n But the meaning meant was drown'd;\n For, to him, her voice and presence\n Meaning held far more profound.\n\n \"He gave his soul to feasting,\n And his sense, (which is the soul\n More thoroughly incarnate,)\n Backward standing, to control\n His object, as a painter\n Views a picture in the whole.\n\n \"She stood, her eyes cast downwards,\n And, upon them, dropp'd halfway,\n Lids, sweeter than the bosom\n Of an unburst lily, lay,\n With black abundant lashes,\n To keep out the upper day.\n\n \"_A breath from out her shoulders\n Made the air cool_, and the ground\n Was greener in their shadow;\n All her dark locks _loll'd_, unbound,\n About them, heavily lifted\n By the breeze that struggled round.\n\n \"As if from weight of beauty,\n Gently bent--but oh, how draw\n This _thousand-featured_ splendour--\n _Thousand-featured without flaw!_--\n At last, his vision reveling\n On her ravishing mouth, _he saw_\n\n \"_It closed_; and then remember'd\n That she spoke not.--'Stay to dine,\n And name her wishes after'--\n To these sounds he could assign\n A sense, for still he heard them,\n Echoing silvery and divine.\"\n\nSir Hubert having reveled on her ravishing mouth, and having, by a\nstrong effort of intelligence, mastered the meaning of the very occult\nproposition which issued therefrom, namely, that the lady would \"stay to\ndine, and name her wishes after;\" and, moreover, having seen--\"It\nclosed\"--he shortly afterwards saw it opened, for the purpose of eating\nhis hawk, which, as the reader knows, he had felt himself under the\nnecessity of killing for the fair widow's entertainment. We pass over\nthe relation of the circumstances which, as the lady discovers, render\nher mission fruitless, and which are detailed in a strain of the most\nvapid silliness--and proceed to the interview which brings about the\nunion of Mabel and Sir Hubert. The latter, some time after these\noccurrences, pays a visit to the castle.\n\n \"Half reclined\n Along a couch leans Mabel,\n Deeply musing in her mind\n Something her bosom echoes.\n O'er her face, like breaths of wind\n\n \"Upon a summer meadow,\n Serious pleasures live; and eyes\n _Large always, slowly largen,\n As if some far-seen surprise\n Approach'd,--then fully orb them,\n At near sound of one that sighs_.\"\n\nHer eyes having recovered their natural size, a good deal of\nconversation ensues, the result of which is given in the following\nstanza, which forms a fit conclusion for the story of such a passion--\n\n \"Her hands are woo'd with kisses,\n They refuse not the caress,\n Closer, closer, ever closer,\n Vigorous lips for answer press!\n _Feasting the hungry silence\n Comes, sob-clad, a silver 'yes.'_\"\n\nThere are several smaller poems interspersed throughout the volume. Mr\nTennyson has his \"Claribels,\" and \"Isabels,\" and \"Adelines,\" and\n\"Eleanores\"--ladies with whom he frequently plays strange, though, we\nadmit, by no means ungraceful vagaries; and Mr Patmore, as in duty\nbound, and following the imitative bent of his genius, must also have\nhis Geraldine to dally with. The two following stanzas of playful\nnamby-pambyism, are a specimen of the manner in which this gentleman\ndandles his kid:--\n\n \"We are in the fields. Delight!\n Look around! The bird's-eyes bright;\n Pink-tipp'd daisies; sorrel red,\n Drooping o'er the lark's green bed;\n Oxlips; glazed buttercups,\n Out of which the wild bee sups;\n See! they dance about thy feet!\n Play with, pluck them, little Sweet!\n Some affinity divine\n Thou hast with them, Geraldine.\n\n \"Now, sweet wanton, toss them high;\n Race about, you know not why.\n Now stand still, from sheer excess\n Of exhaustless happiness.\n I, meanwhile, on this old gate,\n Sit sagely calm, and perhaps relate\n Lore of fairies. Do you know\n How they make the mushrooms grow?\n Ah! what means that shout of thine?\n _You can't tell me, Geraldine._\"\n\nOur extracts are now concluded; and in reviewing them in the mass, we\ncan only exclaim--this, then, is the pass to which the poetry of England\nhas come! This is the life into which the slime of the Keateses and\nShelleys of former times has fecundated! The result was predicted about\na quarter of a century ago in the pages of this Magazine; and many\nattempts were then made to suppress the nuisance at its fountainhead.\nMuch good was accomplished: but our efforts at that time were only\npartially successful; for nothing is so tenacious of life as the spawn\nof frogs--nothing is so vivacious as corruption, until it has reached\nits last stage. The evidence before us shows that this stage has been\nnow at length attained. Mr Coventry Patmore's volume has reached the\nultimate _terminus_ of poetical degradation; and our conclusion, as well\nas our hope is, that the fry must become extinct in him. His poetry\n(thank Heaven!) cannot corrupt into any thing worse than itself.\n\n\nFOOTNOTES:\n\n{A} London: Moxon. 1844.\n\n\n\n\nMARSTON; OR, THE MEMOIRS OF A STATESMAN.\n\nPART XIII.\n\n \"Have I not in my time heard lions roar?\n Have I not heard the sea, puft up with wind,\n Rage like an angry boar chafed with sweat?\n Have I not heard great ordnance in the field,\n And Heaven's artillery thunder in the skies?\n Have I not in the pitched battle heard\n Loud 'larums, neighing steeds, and trumpets clang?\"\n\n SHAKSPEARE.\n\n\nI had been familiar with the debates of the French Convention, and had\nwitnessed the genius of French eloquence in its highest exertions.\nNothing will cure this people of their aversion to nature. With them,\nall that is natural is poor--simplicity is meanness. The truth of things\nwants the picturesque, and thus wants every charm. I had listened to\nsome of their public speakers with strong interest, while they were\nconfined to detail. No man tells a story better than a French _conteur_.\nThere lies the natural talent of the people. Nothing can be happier than\ntheir seizure of slight circumstances, passing colours of events, and\nthose transient thoughts which make a story as pretty as a piece of\nladies' embroidery--a delicate toil, a tasteful display of trivial\ndifficulties gracefully surmounted. But even in their higher order of\nspeakers, I could perceive a constant dissatisfaction with themselves,\nunless they happened to produce some of those startling conceptions\nwhich roused their auditory to a stare, a start, a clapping of hands. I\nhad seen Mirabeau, with all his conscious talent, look round in despair\nfor applause, as a sailor thrown overboard might look for a buoy; I had\nseen him as much exhausted, and even overwhelmed, by the want of\napplause, as if he had dropped into an exhausted receiver. If some lucky\nepigram did not come to his rescue, he was undone.\n\nI was now to be the spectator of a different scene. There was passion\nand resentment, the keenness of rivalry and the ardour of triumph--but\nthere was no affectation. Men spoke as men speak when their essential\ninterests are engaged--plainly, boldly, and directly--vigorously always,\nsometimes vehemently; but with that strong sincerity which administers\neloquence to even the most untaught orders of mankind, and without which\nthe most decorated eloquence is only the wooden sword and mask of\nharlequin.\n\nPitt took the lead, in all senses of the phrase. He was magnificent. His\nexposition of the state of Europe, perfectly unadorned, had yet an\neffect upon the House not unlike that of opening a volume to a multitude\nwho had but just learned to read. All was novelty, conviction, and\namazement. His appeal to the principles by which a great people should\nshape its conduct, had all the freshness and the strength of feelings\ndrawn at the moment from the depths of his own blameless bosom; and his\nhopes of the victory of England over the temptations to public\noverthrow, exhibited all the fire, and almost all the sacred assurance\nof prophecy.\n\nHe described the system of France as \"subversion on principle,\" its\npurpose universal tumult, its instrument remorseless bloodshed, and its\nsuccess a general reduction of society to the wild fury and the squalid\nnecessities of the savage state. \"This,\" he exclaimed, turning his full\nfront to the House, raising his hand, and throwing up his eyes to heaven\nwith the solemnity of an adjuration--\"This we must resist, in the name\nof that Omnipotent Disposer who has given us hearts to feel the\nblessings of society, or we must acknowledge ourselves unworthy to hold\na name among nations. This we must resist--live or die. This system we\nmust meet by system--subtlety by sincerity--intrigue by\nresolution--treachery by good faith-menace by courage. We must remember\nthat we have been made trustees of the honour of the past, and of the\nhopes of the future. A great country like ours has no alternative but\nto join the enemy of all order, or to protect all order--to league\nagainst all government, or to stand forth its champion. This is the\nmoment for our decision. Empires are not afforded time for delay. All\ngreat questions are simple. Shrink, and you are undone, and Europe is\nundone along with you; be firm, and you will have saved the world!\"\n\nThe feelings with which this lofty language was heard were intense. The\nHouse listened in a state of solemn emotion, hour after hour, deeply\nsilent, but when some chord was so powerfully touched that it gave a\nuniversal thrill. But those involuntary bursts of admiration were as\nsuddenly hushed by the anxiety of the House to listen, and the awful\nsense of the subject. It was not until the great minister sat down that\nthe true feeling was truly exhibited; the applause was then unbounded--a\nsuccession of thunder-peals.\n\nI had now leisure to glance at the Opposition. Fox, for a while, seemed\ngood-humouredly inclined to give up the honour of the reply to some of\nthe popular speakers round him; but the occasion was too important to be\nentrusted to inferior powers, and, on a general summons of his name, he\nat length rose. The world is too familiar with the name of this\ncelebrated man to permit more than a sketch of his style. It has been\nsaid that he had no style. But this could be said only by those who\nregard consummate ability as an accident.\n\nOf all the public speakers whom I have ever heard, Fox appeared to me\nthe most subtle--of course, not in the crafty and degrading sense of the\nword; but in the art of approaching an unexpected case, he was a master.\nHe loitered, he lingered, he almost trifled by the way, until the\nobserver began to believe that he had either no object in view, or had\nforgotten it altogether. In the next moment he rushed to the attack, and\ncarried all by storm. On this occasion he had a difficult part to play;\nfor the hourly violences of the French capital had begun to alienate the\nprincipal aristocracy of England, and had raised abhorrence among that\nmost influential body, the middle class. The skill with which the orator\nglided over this portion of his subject was matchless; no Camilla ever\n\"flew o'er the unbending corn\" with a lighter foot. He could not\naltogether evade the topic. But he treated it as one might treat the\nnarrative of a distressing casualty, or a disease to be touched on with\nthe pity due to human infirmity, or even with the respect due to a\ndispensation from above. He often paused, seemed to find a difficulty of\nbreathing, was at a loss for words, of which, however, he never failed\nto find the most pungent at last; and assumed, in a remarkable degree,\nthe appearance of speaking only from a strong compulsion, a feeling of\nreluctant duty, a sense of moral necessity urging him to a task which\nburdened all his feelings. I will acknowledge that, when he had made his\nway through this difficult performance, I followed him with unequivocal\ndelight, and acknowledged all the orator. He had been hitherto Milton's\nlion \"pawing to get free his hinder parts.\" He was now loose, in all his\nsymmetry and power, and with the forest and the plain before him. \"Why\nhas the monarchy of France fallen?\" he explained, \"because, like those\non whom the malediction of Scripture has been pronounced, it had eyes\nand yet would not see, and ears, yet would not hear. An immense\npopulation was growing up round it year after year, yet it could see\nnothing but nobles, priests, and princes. In making this war,\" said he,\n\"you are beginning a contest of which no man can calculate the means, no\nman can state the objects, and no man can predict the end. You are not\nwarring against the throne of France, nor even against the people of\nFrance; but warring against every people of the earth which desires to\nadvance its own prosperity, to invigorate its own constitution, and to\nplace itself in that condition of peace, purity, and freedom, which is\nnot more the desire of man than the command of Providence.\"\n\nThe House burst into loud reprobations of the name of aristocrat and\ndemocrat, which he declared to be mere inventions of party prejudice.\n\"Do you require to make political hostilities immortal, give them names;\ndo you wish to break down the national strength, divide it in sections:\narm against your enemy, if you will, but here you would arm one hand\nagainst the other.\"\n\nTo the charge of defending the French mob, his answer was in the most\nprompt and daring style.\n\n\"Who are the French mob? The French nation. Dare you put eight and\ntwenty millions of men into your bill of attainder? No indictment ever\ndrawn by the hand of man is broad enough for it. Impeach a nation, you\nimpeach the Providence that made it. Impeach a nation, you are\nimpeaching only your own rashness and presumption. You are impeaching\neven the unhappy monarch whom you profess to defend. Man is every where\nthe creature of circumstances. Nations are what their governments make\nthem. But France is in a state of revolt. Be it so. I demand what nation\never revolted against justice, truth, and honour? You might as well tell\nme, that they rebelled against the light of heaven; that they rejected\nthe fruits of the earth; that they refused to breathe the air. Men do\nnot thus war against their natural benefactors; they are not mad enough\nto repel the very instincts of preservation. I pronounce it, fearlessly,\nthat no nation ever rose, or ever will rise, against a sincere,\nnational, and benevolent authority. No nation was ever born blind.\nInfatuation is not a law of human nature. The monarchy of France was the\ncriminal.\"\n\nAnother burst, which produced vast effect on the House, referred to the\nexclusiveness of the chief public employments.\n\n\"The people have overthrown the titles and dignities of France. I admit\nit. But was it from a natural hatred of those distinctions? That I deny.\nThey are congenial to the heart of man. The national hatred lay in the\nsense of that intolerable injustice which turns honour into shame. For\ncenturies, those titles and dignities were to the people not badges of\nhonour, but brands of scorn. They were not public calls to generous\nemulation, but royal proclamations of everlasting contempt. They were\nnot ramparts surrounding the state, but barriers shutting out the\npeople. How would such insults to the common origin of man, to the\ncommon powers of the human mind, to the common desires of distinction\nborn with every man, be endured in this country? Is it to be wondered\nat, that France should have abolished them by acclamation? I contend,\nthat this was a victory gained, not for a populace, but for a people,\nfor all France, for twenty-eight millions of men--over a portion of\nsociety who had lost their rank, a body already sentenced by their\npersonal inefficiency--a caste, who, like a famished garrison, had been\nstarved by the sterility of the spot in which they had inclosed\nthemselves; or, like the Indian devotees, had turned themselves into\ns by their pretence of a sacred superiority to the habits of the\nrest of mankind.\"\n\nOpposition still exhibited its ranks but slightly diminished, and the\nchief passages of this impassioned appeal, which continued for three\nhours, were received with all the fervour of party. Burke then rose.\nStrong interest was directed to him, not merely for his eminent name,\nbut from the public curiosity to hear his explanation of that\nestrangement which had been for some time spreading, under his auspices,\nthrough the leading personages of the Opposition. Like most men who have\nmade themselves familiar with the works of a great writer, I had formed\na portraiture of him by anticipation. I never was more disappointed.\nInstead of the expressive countenance and commanding figure, I saw a\nform of the middle size, and of a homely appearance, a heavy\nphysiognomy, and the whole finished by two appurtenances which would\nhave been fatal to the divinity of the Apollo Belvidere, spectacles and\na wig. His voice and manner were scarcely less prepossessing; the one\nwas as abrupt and clamorous, as the other was rustic and ungraceful. He\nhad the general look of a farmer of the better order; and seemed, at\nbest, made to figure on a grand jury.\n\nBut I soon felt how trivial are externals in comparison of genuine\nability; or perhaps, how much even their repulsiveness may add to the\npower of genius. I had listened but a few minutes when I forgot every\nthing, except that a man of the highest faculties was before me; with\nthose faculties wrought to the highest tension by the highest subjects.\nTaking a line of argument, equally distinct from the leaders of the\nMinistry and the Opposition, he dwelt as little on the political views\nof England and Europe with Pitt, as he did upon the revolutionary\nregeneration of France and Europe with Fox. His view was wholly English;\nthe reference of the revolutionary spirit to our own institutions. \"I do\nnot charge,\" he exclaimed, turning full on the Opposition bench,\n\"individuals with conspiracy; but I charge them with giving the sanction\nof their name to principles, which have in them all the germs of\nconspiracy. Sir, the maxim of resisting the beginnings of evil, is as\nsound in the concerns of nations as in the morality of individual minds.\nNay, I am not sure whether mischief is not more effectually done in that\nincipient state, than when the evil comes full-formed. It is less\nperceived, and it thus destroys with impunity. The locust, before it\ngets its wings, destroys the crop with a still more rapacious tooth than\nwhen its armies are loading the wind.\n\n\"Honourable members have talked largely of their zeal for the\nconstitution. Sir, I am content to follow the wisdom which judges of the\nfaith by the works. In my humble measure, I have been a zealous\nworshipper of the constitution. There was a time when those honourable\ngentlemen and myself--and I speak of that time with the regret due to\nlong friendship--took 'sweet counsel together,' and bowed before that\ncommon worship as friends. That time is past. We have since taken\ndifferent paths. I have been charged with apostasy. What is my apostasy?\nThat I have not followed the frenzy and ingratitude of the hour; that,\nwhile the most awful event in the history of human change has been\ntransacting before us, I have not shut my ears and eyes to its moral;\nthat I have not followed the throng into the valley, and there joined\nthe fabricators of the new idolatry, the priesthood of the golden calf\nof revolution, and shared the polluted feast and the intoxicated dance;\nwhile the thunders of divine vengeance were rolling on the hill above.\"\n\nIt was obvious from his manner, and his frequent return to the topic,\nthat that charge of deserting his party had deeply wounded his generous\nand sensitive nature; and nothing struck me as more characteristic of\nhis mind, than the variety and richness of his fine amplification on\nthis subject.\n\n\"In those ranks,\" said he, \"I fought for nearly the half of that portion\nof life allotted to man; certainly for that portion of my course, in\nwhich the desires, the vigour, and the applicability of all the best\nparts of human nature have their fullest play. I came to it a\nvolunteer--I fought side by side with its foremost--I shared the 'winter\nof their discontent,' as willingly as the summer of their prosperity. I\ntook the buffets of ill fortune, and they were many, with as cheerful a\ncountenance and as unshaken a fidelity as any man. But when I saw a new\nbanner raised among them, blazoned with mottoes of evil, and refused to\nfollow, who were the deserters? They or I?\" As he spoke these words, he\ndrew his otherwise rather stooping form to its full height, lifted his\nhand above his head, and stood like one at once demanding and defying\nthe investigation of the empire.\n\nThe roar of applause which followed seemed to shake the very walls. He\nwas powerfully moved; his countenance changed from its usual pallidness\nto strong suffusion; his hands rather tossed than waved in the air. At\nlast I saw one of them thrust strongly into his bosom, as if the gesture\nwas excited by some powerful recollection. \"Do I speak without proof of\nthe public hazards?\" he exclaimed. \"I can give you demonstration--I need\ninvoke neither powers above nor powers below to enlighten you. I have\nthe oracle within my hand.\" The House fixed all its eyes upon him. He\ndropped his voice, and spoke with a faint, but clear tone which formed a\nremarkable contrast to his usually bold, and even harsh enunciation:\n\"Sir,\" said he, in this half-whispered voice, \"before I join these\ngentlemen in their worship, I must know what deity presides in their\ntemple; I must see that the incense which fumes before its altar is\ntaken from the sacred repositories of the constitution, not the smuggled\nimportation of foreign fabrications of revolt--that pernicious compound\nof civil mischief and mad metaphysics--which, instead of consummating\nand purifying the sacrifice, only poisons the air. I must see something\nof the priest too, before I join in his aspirations; I must see that he\nis lawfully inducted to his office, that he is not a rebel in the garb\nof loyalty--a blasphemer where he professes to pray, and a traitor where\nhe propounds allegiance.\"\n\nFox here, evidently taking the description to himself, exhibited\npalpable signs of displeasure. Burke caught the expression at once, and\ninstantly changed the whole current of his conceptions. \"If,\" said he,\n\"the honourable gentleman thinks that I designate _him_ as the\nhigh-priest of this new worship, he does me as much injustice as\nhimself. No, no! When we shall see the Republican Pantheon thrown open,\nhe, and such as he, will not be called to officiate at the altar. He is\nmuch more likely to be the victim. The popular ornaments, now flung so\nlavishly upon him, will find him no further favour, than the speedier\noffering on the same abhorred altar, which reeks with so much of the\nbest blood of France.\" Here a corpulent noble, peculiarly hostile to\nBurke, laughed contemptuously. The orator instantly turned upon him.\n\"True,\" said he, \"there may be a good deal of variety in that\nprocession. There may be the mummer as well as the priest; it may have\nthe mountebank selling his potions, and playing his tricks, as well as\nthe sacrificer with his axe--unless the ambition of the bloated\nperformer should prefer to combine the offices, and be at once the\nbutcher and the buffoon.\"\n\nThe hit was felt on all sides, and the laughter was unbounded. He then\nrose, as was his custom, into a higher strain. \"I can imagine that\nprocession,\" said he, \"or rather, that triumph, of the principles of\nchange. Like the return of the classical Bacchus from his Indian\nconquests, the demigod,\" and he now cast a look at Fox, \"secure of\nsupremacy, exulting in his prowess, and thinking the civilized world at\nhis feet; but not without the companionship of his trusty Silenus\"--and\nhere he turned his glance on the noble lord--\"that veteran follower,\nwhose ambition is limited to his cups, and the vigour of whose fidelity\nis shown only in the constancy of his intoxication; the whole procession\nbeing drawn by the wild lords of the forest and the wilderness, who,\nharnessed as they may be for the moment, will no sooner find their food\nstinted, than they will resume the natural instincts of the lion and the\ntiger, turn on their drivers and devour them.\n\n\"But, sir,\" he exclaimed, turning to the chair, \"I have higher topics,\nand to those I now call the attention of the representatives of England.\nI have alluded to the revolutionary temple. I here have its deity.\" With\nthese words, he plucked from his bosom a large dagger, held it for a\nmoment up to the light, and then flung it at the foot of the table. The\nastonishment, and even the alarm, of the whole assembly was beyond\ndescription. They all started from their seats, as if assassination had\nstood before them in a visible shape. Some crowded round Burke, some\nseized the dagger, which was eventually carried to the Speaker, and\nbecame the object of universal curiosity. All was confusion for a\nconsiderable time. At length Burke, in a few words delivered in his most\nimpressive tone, explained the phenomenon. \"That dagger,\" said he, \"is\none of thousands, perhaps of millions, which the preachers of philosophy\nare now forging for popular conviction. You see that by its construction\nit is equally fitted for the head of a pike, or for a dagger--equally\nserviceable in tearing down the monarchy in the field, or stabbing its\nfriends in their chambers. You have it, at once the emblem of rebellion\nand assassination. Those are the arguments of the new school--those are\nthe instruments by which the limbs of the state are to be amputated, for\nreplacement by the inventions of the revolutionary mechanists. Those are\nthe keys by which the locks of cabinets and councils are henceforth to\nbe opened, and the secrets of national wealth laid bare to the rapacity\nof the rabble.\" After this speech nothing was listened to.\n\nThe debate had been prolonged through the greater part of the night, and\nyet such was the interest felt in its subject, that the streets in the\nneighbourhood continued crowded to the last. All the hotels and\ncoffee-houses were filled with people waiting for the division. Groups,\nwith lighted torches, were lingering everywhere, and passing the\nintelligence along, as a member happened casually to make his appearance\nin the course of the night; shouts and expressions of wrath alternately\narose, according to the nature of the intelligence, and a species of\nopen-air legislature was held during one of the bitterest nights of\nwinter, with discussions as active, though perhaps not altogether so\nclassical, as those within; yet totally free from tumult, and in the\nspirit of a people who live with a constant reference to the laws. The\nrush of the members to the porch, on the breaking up of the debate,\nproduced a corresponding rush of the multitude. Public curiosity was\nroused to its wildest height--every public sentiment had its full\nexpression; and whether the acclamation was louder when Fox's corpulent\nframe was seen toiling its slow way through the pressure, or when Pitt's\nslender figure and passionless face was recognised, is a question which\nmight have perplexed the keenest investigators of popular sentiment. All\nwas that uproar in which the Englishman delights as a portion of his\nfreedom.\n\nOn returning to my chamber, exhausted, yet animated with a new sense of\nthe value of existence in such a country, and of the noble faculties\nwhich she carried in her bosom, I saw a large packet on my table. I\ngazed on its envelope for a few moments with that strange emotion which\nsometimes makes us dread to open the very letter which we most desire to\nreceive. It was obviously from Downing Street. At last I opened it. It\ncontained my commission in the Guards!\n\nMy destiny was now fixed, and it is impossible to tell how much I felt\nrelieved. I had spent the preceding period in such perplexing\nindecision, that I felt my heart withering within me. Now all was clear.\nMy course was decided. I was in other hands than my own, and whatever\nmight be the result, I was no longer answerable for either good or ill\nfortune. No human being who has not felt the trial almost the torment,\nof being left to decide on the conduct which may make or mar him for\nlife, can conceive the depression into which it plunges the mind. From\nthis I was now relieved; I was wholly free; an established routine, a\nvigorous profession, a regulated pursuit, and that pursuit one of the\nmost honourable nature, was suddenly prepared for me by the enclosure\nupon my table. After again and again reading this simple but expressive\ndocument, I threw myself on my bed, and attempted to forget it and the\nworld. But I could forget neither; my eyelids would not close; sleep had\ngone from me. After a useless effort for composure, I rose, relighted my\nlamp, and spent the rest of the night in writing to my relatives, to\nVincent, to Mordecai, and every one to whom I felt his majesty's\nsign-manual a vindication of my whole career. There was still one cloud\nthat overhung my prospect, one gloomy and bitter remembrance: but this\ncloud I had neither the power nor even the wish to dispel; this\nremembrance was already a part of my being--to extinguish it was\nimpossible. I resolved to cherish it as a sacred recollection, to\ncombine it with the aspirations of my new pursuit, and render them thus\nstill nobler; to reserve it as a treasure inaccessible to the knowledge\nof mankind, but to which I might return in my hours of discontent with\nthe world, and restore my sense of the beauty of mind and form which\nmight still exist in the shape of human nature.\n\nYet it may be justly supposed that I did not limit my feelings to this\nlonely abstraction. I spent an anxious period in making enquiries for\nthe Marechale, in every quarter which offered the slightest probability\nof discovering her abode. Though I had seen the announcement of\nClotilde's approaching marriage in the public journals, I had seen no\nmention of its having taken place. My search was wholly unproductive.\nThe captivating duchess, who received me with the kindness which seemed\na part of her nature, while she joined me in my praises of the \"young,\nthe lovely, and the accomplished Comtesse,\" \"her dearest of friends,\"\ncould tell me nothing more than that she had left London, and she\nbelieved with an intention of visiting France. There her knowledge\nceased. I learned only further, that she had grown singularly fond of\nsolitude, was melancholy, and had no hesitation in expressing the\ndeepest dislike to the marriage proposed by her family. My enquiry was\nat an end.\n\nHopeless as this intelligence was, it relieved me from the certainty,\nwhich would have been despair. While Clotilde remained unallied to one\nwhom I could not avoid regarding as an uncongenial spirit, if not a hard\nand tyrannical master, there was, at least, the chance of happiness\nremaining for me in a world where every day brought changes more\nextraordinary than our meeting. If there should be a war, my regiment\nwould be among the first to be employed, and France would inevitably be\nthe first object of a British expedition. The \"march to Paris\" had been\nproclaimed by orators, exhibited in theatres, and chanted in street\nballads. All before us was conquest, and distinctions of every kind that\ncan captivate the untried soldier, glittered in all eyes. I was young,\nardent, and active. My name was one known to the table at which I seated\nmyself on my introduction to the Guards, and I was immediately on the\nbest footing with the gallant young men of a corps which has never\nsuffered a stain. I had even some peculiar sources of favour in their\neyes. I had actually made a campaign. This was more than had been done\nby any man in the regiment. The Guards, always brave and always foremost\nas they were, had not seen a shot fired for a quarter of a century. The\nman who had heard bullets whistling about his ears, and had, besides,\nseen the realities of war on the magnificent scale of continental\ncampaigning, possessed a superiority which was willingly acknowledged by\nthe gallant youths round us; and every detail of that most romantic\ncampaign, reluctantly given as it was by me, was listened to with\ngenerous interest, or manly intelligence. And I had actually learned\nenough, under the Duke of Brunswick, a master of tactics, to render my\nservices useful at the moment. The discipline of the British army was\nnot then, what it has since been, the model to Europe. The Englishman's\nnature prompts him to require a reason for every thing; and there was no\npeculiarly strong reason for the minute toil of foreign discipline, in\nan army which had never been engaged since the American war. But other\ndays were now obviously at hand, and the passion for discipline, and\nabove all others, for the Prussian discipline, became universal. With\nthe exaggeration common to all popular impulses, the tactics of\nFrederick were now regarded as the secrets of victory. That great\nsoldier, and most crafty of men, by his private reviews, to which no\nstranger, even of the highest rank, was ever admitted, and by a series\nof mystifications, had laboured to produce this impression upon Europe,\nand had largely succeeded. Mankind love being cheated; and what the\ncharlatanism of necromancy effected a thousand years ago, was now\neffected by the charlatanism of genius. If I had seen the Prussian\ntroops only at Potsdam, I should probably have mistaken the truncheon\nfor a talisman, like the rest of the world. But the field suffers no\nmystification. I had seen that the true secret of this great tactician,\nfor such unquestionably he was, consisted in his rejecting the\nsuperfluities and retaining the substance; in reducing tactics to the\nready application of force, and in simplifying the old and tardy\nmanoeuvres of the French and Austrian battalions, to the few\nexpeditious and essential formations required before an enemy in the\nfield. I was offered the adjutancy, and I accepted it rejoicingly.\n\nIn those days, by a curious anomaly, which can scarcely be believed in\nours, every regiment was practically free to choose its own system of\nmanoeuvre. The natural consequence was, that no two regiments did any\nthing alike. To brigade the army was impossible, and every field-day was\na scene of ludicrous confusion. But this freedom had the advantage, in\nthe present instance, of allowing me to introduce that Prussian\ndiscipline which has since been made the basis of the British. It was\nthen perfectly new, and it had all the effect of brilliant novelty. Our\nparade was constantly crowded with officers of the highest grades,\nanxious to transmit our practice to their regiments. The king, always\nattached to German recollections, and who would have made as good a\nsoldier as any of his forefathers, was frequently a spectator. The\nprinces and nobility were constant in their attendance; and the\nregiment, thus stimulated, rapidly displayed all the completeness and\nprecision of movement which to this day makes a review of the Guards the\nfinest military spectacle of Europe.\n\nThe adjutant was not forgotten in the general applause and excitement. I\nwas promised promotion in the most gratifying language of royalty\nitself, and all the glittering prospects of the most glittering of all\npursuits opened before me. I still had my moments of depression.\nClotilde often rose before me like a departed spirit in the solitude of\nmy chamber, and even in the midst of public festivities, or in those\nballs and banquets which the nobility gave in such profusion at this\nperiod of the year. When a shape, however faintly resembling her\nincomparable elegance of form, passed before my eye, or a voice, in the\nslightest degree reminding me of her noble tones, reached my ear, I felt\nan irresistible pang, that, for the time, embittered all the scene\naround me.\n\nBut I had in no period of my life been suffered to linger in long\nmelancholy. One night, after returning from a dinner at Devonshire\nHouse, I found a gentleman in possession of my chamber, with my fire\nbriskly blazing, supper on the table, and every appearance of his having\nmade himself master of the establishment. As I paused at the door, in\nsome surprise at the ease of the proceeding, the intruder turned round,\nand I saw the face of my old and excellent friend Vincent. I was\ndelighted to take the honest hand of one who was enough to redeem the\ncharacter of human nature. He was full of congratulations and country\nnews. He told me that this, his first visit to London for years, was\nsimply to shake hands with his pupil; to hear from him his adventures;\nand to have the opportunity of seeing the regiment on parade. He now\nenjoyed all his objects together. The regiment \"reminded him of the\ngrenadiers of Maria Theresa, in the first Hungarian campaign; and all\nthat he wished for me was, that I had seen Daun or Landohn. However, no\nman in this world could have every wish gratified; and he was certain\nthat I had in me the materials of a field-marshal.\"\n\nBut he had more important topics. By an accidental meeting with an old\ncollege friend, high in office, he had ascertained that an expedition\nfor Holland had been resolved on; and that it was to take place without\ndelay. The French army had passed the frontier, and taken the strong\nfortress of Breda. Williamstadt was bombarded, and must fall in a few\ndays if not relieved. With its fall, the Seven Provinces would be thrown\nopen. In this emergency, aid had been solicited from England.\n\nVincent's country news was brief. My lordly brother was in pursuit of a\nneighbouring heiress; and, as a prospective remedy for matrimonial\nennui, speculating on the chance of employment on some foreign embassy.\nVincent himself had married one of his daughters to a neighbouring\nsquire, whom he denominated an \"unlicked cub,\" but an honest man. Thus I\nhad the knowledge of all that the country could furnish, and thus--\"runs\nthe world away.\"\n\nAll now was excitement and activity. The intelligence of the French\nadvance into the territories of our old and very helpless ally, awoke\nEngland at once. The feeble and perfectly fruitless negotiations, by\nwhich the slide from disgust into war is generally managed, had produced\ntheir effect; and France, furious for its prey, and England, steady and\nstubborn, for the first time were brought face to face. The summons, so\nlong wished for, at length reached us; and the Guards were ordered for\nembarkation. We received it in the spirit of a jubilee. All had been\nprepared. And on the night before our final parade, I received my\nappointment to a company. Our parade, next morning, was one which I\nbelieve was never forgotten by any individual who had the good fortune\nto witness it. Of all the striking ceremonials which I have ever seen,\nit was the most striking. The king had given notice of his intention to\nbe present, and bid us farewell. At six o'clock, the three regiments\nwere drawn up in front of the Horse-Guards, a body of three thousand\nmen, and finer-looking troops never bore arms. All the avenues to the\npark were crowded with the multitude. Exactly at the half-hour, a rush\nof the people towards the parade showed that the king, always punctual,\nwas at hand. He came, surrounded by general officers, with the Prince of\nWales, then a most chivalric figure, in the uniform of his regiment of\nlight dragoons, and the Duke of York, as a field-marshal. The enthusiasm\nof the troops could not be restrained, as this brilliant staff\napproached their line; and three cheers were given with all the zeal of\nhonest loyalty. There are times when tears are the only substitute for\nspeech; and the king, one of the most kindhearted of men, visibly shed\ntears at this reception. Another _cortege_ now approached; they were the\ncarriages of the queen and princesses. The scene now became almost\npainful. There was many a tear from royal and noble eyes--the impulse of\nhigh emotion, not of sorrow--or if tinged with the thoughts which always\nshade the name of war, yet undegraded by weakness. The multitude caught\nthe feeling; the shouts subsided; and all was weeping and waving of\nhandkerchiefs. The king put an end to this embarrassing sympathy. He\nrode forward, and, taking his station in the centre, gave the word to\n\"march.\" He was answered by one gallant \"huzza\" from the line, repeated\nby the thousands and tens of thousands who now moved before and around\nus. Our bands struck up, and, with the monarch and his sons at our head,\nand the queen and princesses following in their equipages, we marched\nthrough streets, crowded to the roof, echoing with acclamations, and\nwishing us all good fortune as we passed along, until we left the mighty\nmetropolis behind. Even then, it was only to meet the new multitude of\nthe country. The road to Greenwich, where we were to embark, exhibited a\npopulation as countless, enthusiastic, and full of good wishes as those\nwith whom we had just parted. The king still rode in our front; flags,\nbanners, and every kind of joyous testimonial met our eyes; and if ever\nthere was a triumph before the victory, it was in that honest and\ngenerous display of the true heart of England.\n\nThe embarkation took place within a few hours; and on that night we\nslept on the element which Britain has so long made her field of battle.\nThe weather was serene, and we fully enjoyed the freshness of the air,\nand the brightness of the view, as we rounded the coast. At the mouth of\nthe Thames, we had met a strong squadron of the line of battle,\nappointed for our convoy, and bringing numerous transports with troops.\nOur fleet had now become extensive, and as we moved out from the land,\nthe sight became continually more animated and exciting. The despatch of\nthe look-out frigates, the constant change of signals, the firing of\nguns to regulate the sailing of the great convoy, the manoeuvres of\nthose floating castles, the seventy-fours and three-deckers, the harmony\nof their bands as they passed us, rushing along under a cloud of\ncanvass, with the hum of the thousands on board--all formed one of the\nmost heart-stirring combinations that could exist to the eye, or even to\nthe heart of a human being.\n\nI stood gazing on the poop of our transport the entire day; and even\nwhen twilight came, there was but a change of interest and beauty. We\nmoved on, a moving multitude--a fragment of a mighty nation--almost a\nnation ourselves, on the face of the deep. Within the horizon which now\nlay beneath my glance, smooth as glass, and shining in the richness of\nthe departing day, what materials of living power were gathered; what\nbold hearts; what high hopes; what indefatigable perseverance; what\naccomplished intelligence! a force inferior to the one before me had\nmore than once changed the fate of the world. It might be now on its way\nonly to change that fate once more. The cause, too, was a noble one. It\nwas sustained by no aggression, perfidy, or desire of change. It was to\nprotect a friendly nation, and to sustain an inspired cause. There was\nno taint of cruelty or crime to degrade the soldiership of England. We\nwere acting in the character which had already exalted her name as\nprotectors of the weak and punishers of the powerful.\n\nOn the second evening we reached the flat and uninteresting coast of\nHolland. But if the coast was repellent, nothing could exceed the\neagerness of the inhabitants to welcome our arrival. On our first\napproach to the land every boat that could swim came off, crowded with\npeople, some to take refuge on board the fleet, but thousands to urge\nour speedy landing. The ferocious plunder which had become the principle\nof the republican arms had stricken terror into the hearts of the\nHollanders: a people remarkably attached to home, and fond, or even\njealous, of the preservation of the most trivial article of property\nconnected with that home. The French troops, often pressed with hunger,\nand adopting the desperate maxim of \"making war support war,\" had\ncommitted such wanton ruin of property in the Netherlands, that, at this\ndistance, the common effect of exaggeration described them as rather\ndemons than men.\n\nWar is of all things the most picturesque, and there never was a gala on\nthe waters of the Adriatic more gay or glittering than our landing. But\nwe had infinitely the advantage in the numbers, the brilliancy, and,\nwhat gave a higher feeling to the whole, in the reality of all its\nobjects. This was no painted pageant; it was real strength, real\nsoldiership; the cannon that roared above our heads, as we descended\ninto the boats, were the thunderers which had shaken many a battlement;\nthe flotilla of launches, long-boats, and cutters which covered the sea,\nwas manned with the soldiers and sailors sent forth to fight the battle\nof human freedom on every shore of the globe. The ships were that\nBritish fleet whose name was synonymous with the noblest exploits of\nwar, and which it would have been well worth going round the\ncircumference of the globe to see.\n\nOn this night we bivouacked; the shore offered no human habitation, and\nit was too late for the landing of our tents. But the sand was dry; our\nfires were soon lighted; all was sport and activity; our bands played\n\"Welcome to Holland;\" our men danced with the peasantry; all had the\nlook of a magnificent frolic; and, when at last I threw myself on my\nopen air pillow, I dreamed of fairyland.\n\nAt daybreak we marched, in the highest spirits, and only longing to have\nan opportunity of trying our strength with the enemy. From time to time,\nthe sound of a cannonade reached us, and heightened our eagerness to\nadvance. But Holland is proverbially difficult for any movements but\nthose of a trackschuyt; and the endless succession of narrow roads, the\nperpetual canals, and the monotony of her level fields, rich as they\nwere, exhausted us, more than if we had marched twice the distance. But\nthe spell of human hearts is excitement, and war is all excitement. All\nround us was new, and from the colonel to the rank and file, the\n\"general camp, pioneers and all,\" enjoyed the quaint novelty of Dutch\nlife. The little villages, so unlike our own, and yet so admirably\nfitted for peasant comfort, the homesteads embedded in plantations of\nwillows, the neatness of every thing round the farm-houses, and even the\nsleekness of the cattle, which seemed by their tameness to form a part\nof the habitancy--all were objects of constant remark on our march; and\nwe could easily comprehend the horror with which the arrival of a French\ncommissariat must strike these comfortable burghers. But the punctuality\nof British payments was perfectly known already; the whole plenty of the\nland was poured out before us; we regaled sumptuously.\n\nOn the second evening of our march through this landscape of fatness, we\nwere warned of our approach to the besieged fortress, by the louder roar\nof the cannon, and not less by the general desolation of the country.\nThe enemy's hussars had made a wide sweep, and wherever they were seen,\nthe villagers had fled instantly, carrying off their cattle. We found\nthe traces of those foraying excursions in the fragments of burned\nmills, a favourite object of destruction with the French--for what\npurpose I never could comprehend, except the pleasure of seeing them\nburn--in cottages unroofed, for the sake of the thatch; in broken\nmoveables, and, in some instances, in the skeletons of horses and\nremnants of arms; for the peasantry were not always patient sufferers,\nand some of the smaller detachments of the plunderers had met with\nsevere retaliation.\n\nAt length we halted for the night, and orders were issued for a general\nmovement at daybreak, to attack the French force covering the siege of\nWilliamstadt. The order was received with shouts; and the night was\nspent in great exultation. The cannonade, which was now within a few\nmiles of us, continued with such violence during the night that sleep\nwas next to impossible; and long before the first streak of light in the\neast, we were busy in the numberless preparations for a first action.\nOrderlies and aides-de-camp were speedily in motion, and at the first\ntap of the _reveille_ all were on parade. The sun rose brightly, gave\none broad blaze along our columns, and after thus cheering us, instantly\nplunged into a mist, which, except that it was not actually black,\nobscured our road nearly as much as if it had been midnight. This was\nsimply a specimen of the new land on which we now set foot. But it\nperplexed all the higher powers prodigiously--generals and the staff\ngalloping round us in all directions, the whole one mass of confusion.\nYet we still pushed on, toiling our puzzled way, when, as if by magic, a\nregiment of the enemy's hussars dashed full into the flank of our\ncolumn. Never was there a more complete surprise. The enemy were as much\nastonished as ourselves, for the collision had been the result of an\nattempt to find their way through the fog back to their camp; but I now\nfor the first time saw the temper of John Bull in the field. The attack\nof the hussars was evidently looked on by our men less as a military\nmanoeuvre, than as a piece of foreign impudence. To fire might be\nhazardous to some of our advancing columns, which we could hear, though\nnot see; but the word \"charge\" from our gallant old colonel was enough;\nthey rushed with the bayonet on the cavalry, forced their way in between\nthe squadrons, which had been brought to a stand by the narrowness of\nthe ; and in five minutes the whole had laid down their arms, given\nup their horses to our fifers and drummers, and were marching to the\nrear.\n\nAs if to reward us for this dashing affair, a gust of wind blew aside\nthe fog; the sun gleamed again; and Williamstadt, the French camp, the\ncovering force formed in columns and waiting for us, and the whole\ncountry to the horizon, green as a duckpond, and altogether as smooth,\nburst on our view. The suddenness of the display was like the drawing-up\nof a stage curtain, with a melo-dramatic army and castle behind. Our\nadvance was now rapid. The skirmishers on both sides began to engage,\nand our light artillery to throw a long shot now and then into the\nenemy's columns. The difficulty of the ground, intersected with high\nnarrow causeways stretching over marshy fields, retarded our progress;\nand for two hours--and they were the two longest hours which any of us\nhad ever spent--we were forced to content ourselves with firing at our\nlong range, and watching the progress of our more distant columns moving\non the flank of the enemy. To a military eye nothing could be more\ninteresting than the view of the vast field on which these concentric\nmovements were developing themselves from hour to hour. At length we\nreceived the order to advance, and drive in a strong column which had\njust debouched from a wood in front of us. Our men rushed on with a\ncheer, threw in a heavy volley, and charged. Their weight was\nirresistible, and the French column broke, and took refuge again in the\nwood. Another glance showed me the whole British force in motion, every\nwhere pressing on; the enemy every where retreating, all their columns\nconverging upon their camp. Those are the brilliant moments of a\nsoldier's life. All was exultation. We had met the enemy, and driven him\nfrom his position.\n\nBut the most difficult task of the day was still to be achieved. The\nFrench camp had been placed in strong ground; heavy batteries commanded\nevery approach; and Dampierre, their general, an officer of known\nability, had exhibited all his skill in rendering the position, if not\nimpregnable, at least one which could not be forced without the most\nserious loss. The day had been already far spent, and the troops were\nwearied with six hours' marching and fighting; but nothing could\nrestrain their eagerness to finish the victory. The heads of columns\nagain advanced, and the firing became tremendous on both sides. The\nFrench batteries poured an absolute shower of balls upon us, and we were\nbeginning to lose men, when a strange and indescribable sound suddenly\ncaught every ear. Such was the universal sense of something more\nsingular, and even more formidable, than the work of war, that the fire\non our side rapidly subsided, and every eye was turned to look for the\ncause. It soon exhibited itself. With a roar like thunder, I saw the sea\nbursting in upon the plain where the enemy lay intrenched. The Dutch\ngarrison had sallied out from Williamstadt, on the repulse of the\nFrench, and cut the in several places. The ocean now fought our\nbattle; each chasm in the long mound which protected the fields from\ninundation, was now the channel of a roaring cataract; the trenches were\nsoon filled; as the waters advanced, the field-works were washed away;\nstill wave rolled on wave; cannon, tents, baggage, every thing but the\nsoldier himself, was seen gradually sinking, or floating away on the\nsurface of the surge. Within the hour, the ground on which we had fought\nduring the day was completely covered with the flood. The French camp\nwas totally buried. The enemy had only time to make a hurried retreat,\nor rather flight, along the causeways which stood above the waters. As\nan army, they were utterly ruined; when they at last reached firm\nground, they scattered through the country, and those battalions never\nappeared in the field again.\n\nOur troops entered the relieved fortress, with drums beating and colours\nflying. We were received as deliverers; all that the place could offer\nwas heaped upon us; and if praise could have repaid our exploits, never\nwas praise more abundant from the lips of the whole population.\n\nThe catastrophe was complete; and when at night I broke away from the\nheat and noise of the huge barrack in which we had been placed, as the\npost of favour, and walked upon the rampart, nothing could form a more\nexpressive contrast to the tumult of the day. The moon was high, and her\nlight showed the whole extent of the late field of battle. But all now\nwas one immense shining lake. Where cavalry had charged and artillery\nhad roared, and the whole living clash and confusion of a stubborn\nengagement had filled the eye and ear but a few hours before, all was\nnow an expanse of quiet water, calm as the grave, without a vestige of\nthe struggle, but with hundreds of the combatants sleeping their last\nsleep below, and the whole artillery and equipment of a powerful army\nsubmerged.\n\nI was still gazing from the ramparts, when I observed a body of cavalry\nadvancing along the dike, at a rapid pace, with a group of staff\nofficers among them. The alarm was given by the sentries; and, after\nsome brief pause, it was ascertained that they were the escort of the\nnew commander-in-chief of the allied armies in the Netherlands. My first\nimpression was, that the man to whom so important a trust was given must\nbe Clairfait; and I hastened down to meet him at our quarters. But I was\ndisappointed; and for the dark and decided physiognomy, and military\nfrankness of that distinguished soldier, I saw the Prince Cobourg, stern\nand lofty in his air, evidently too Austrian to be popular, yet known to\nbe a gallant officer. But my disappointment was considerably assuaged by\nseeing one of his staff throw himself off his horse, and hasten towards\nme with almost joyous salutation. My surprise and pleasure were equal\nwhen I found him to be Guiscard!\n\nSupper was on the table when I introduced the Prussian philosopher to my\nbrother officers, and they were delighted with him. But he was the\nphilosopher no longer, or rather had thrown off the half misanthropy\nwhich had made him so strong a contrast to my honest friend Varnhorst.\nHis very countenance had adopted a different expression. It was no\nlonger stern and sarcastic, but was lighted up with pleasantry; and the\nonly conception of the change which I could form was, either that he had\narrived at that height of philosophy to which every thing seems trivial,\nor that he had met with some of those extraordinary instances of good\nfortune which throw all the world into sunshine for the moment.\n\nBut he was full of knowledge on the subject most interesting to his\nhearers; and he gave us his information of the allied councils, and the\nmovements of the armies, with a copiousness and courtesy which all our\nquestioning could not tire.\n\n\"We have now,\" said he, \"the finest army in line that Europe has ever\nseen; little less than 200,000 men are under the command of the prince.\nIf he is suffered to move them in a mass, they must break through any\npart of the French territory which they choose. If they divide, they\nwill be beaten. It will now take only three pitched battles to reach\nParis--for the three covering armies fight with the guillotine in their\nrear. But a single unlucky skirmish may bring every peasant in France\nupon us; and it takes but fifteen days to make the French peasant a\nsoldier. Blows, and those straightforward, are our true policy. If we\nnegotiate, we shall be beaten; if beaten here, we shall be beaten on the\nRhine, and perhaps even on the Danube.\"\n\nThe news of Dumourier's attempt to overthrow his government had reached\nus, but in the usual way of mystification. The answer of Guiscard was\nprompt and plain. \"Dumourier,\" said he, \"is one of those men who has a\none-sided understanding. He is a capital soldier, but a childish\nstatesman; and, with an absurdity by no means limited to himself, he\nthinks that his talent lies in statesmanship. The result has been, that\nthe factions have always managed him as they do all men of his calibre.\nWhen he attempted to act for himself, they crushed him without mercy;\nwhen he ceased to be a tool, he necessarily became a victim. The army is\nnow in retreat. To the French retreat is always ruin; the horseman sells\nhis horse; the foot-soldier sells his musket; and the artilleryman sells\nhis powder and ball, breaks up his gun-carriage for a fire, and throws\nhis gun into the next ditch. The peasantry then fall on them all, repay\ntheir plunder with the pike and the pitchfork, and in three days the\narmy is dissolved.\"\n\n\"But will Cobourg follow up his blow?\" was the question on all sides.\n\n\"The commander-in chief,\" was the answer, \"is intelligent and brave. He\nhas learned his profession under the greatest soldier whom Russia has\nproduced, or perhaps ever will produce--Suwarrow. But he is himself\nunder orders. If he were a republican general he would instantly march,\nand within a week he would be in the Tuileries. But as an Austrian\ncommander, he must wait for the opinion of men too far off to know a\nsingle fact of the campaign, too blind to know them if they were on the\nspot, and too jealous even of their own general to suffer him to beat\nthe enemy if victory would throw their own nothingness into the shade.\"\n\nEvery hour now produced its event. A general _feu-de-joie_ announced the\nfirst great success of the campaign; Mayence had been taken, with its\ngarrison of 20,000 men. The French general Custine, had made an\nunsuccessful attack on the lines of the besiegers, to relieve the\nfortress in its last extremity, had been beaten, and driven back into\nthe Vosges, where he was at liberty to starve among the most barren\nmountains of France. But this intelligence came qualified by the\nformidable rumour that Prussia was already making terms with the French,\nthat it had acknowledged the government as the \"Republic,\" and even that\nthe Prussians had sung the _Marsellaise_. Thus we had the light and\nshade.\n\nBut while politicians tremble, soldiers are gay. What were all those\nshiftings and doublings to us? We had all the luxuries of the most\nluxurious of all lives, the foreign camp. We had now marched from the\ncountry of fogs and bogs, and were moving through the richest soil, and\nnot the least beautiful landscape, of the Continent. Holland was left\nbehind, Flanders was round us, France was before us. We had the finest\narmy of Europe, untouched by disaster, confident in its strength, and\nthe enemy in full flight. If we despised the fugitives, we fully as much\ndespised the politicians; the man with the sword in his hand naturally\nscorns the man with the pen behind his ear. Thus we galloped, danced,\nand dreamed on. The spring, too, had come; the harshness of a foreign\nwinter had been changed within a few days to the delightful softness of\nearly summer. The fields were covered with flowers, and the country was\nfilled with the preparations for the rural fetes of the first of May. I\nenjoyed the scene doubly, for I had been sent along with a squadron of\ndragoons to the advanced posts, and thus escaped the turmoil of the\ncamp. My quarters were in one of the old Flemish country-houses, which\nhad been the headquarters of the French general, and had thus escaped\nthe usual ravage. The chateau was large, well furnished in the national\nfashion, and the half-dozen domestics who remained after the escape of\ntheir master, were charmed with the expenditure which always follows the\npresence of English troops. My companion, the captain of dragoons, was\none of the finest specimens of his country--the heir of a noble family,\ngenerous and gay, brave as his own sword, and knowing as little of the\nsoldier's life as became a young aristocrat with the prospect of thirty\nthousand a-year. He insisted on our giving a ball to the Flemings; and\nour invitations were sent out accordingly for half a dozen leagues\nround. They included, of course, the camp; and every lounger who could\nobtain leave for the night came crowding in upon us. Nothing could\nsucceed better. All was festivity within doors. But not so all without,\nfor the night suddenly changed from serenity to storm. England is not\nthe only spot famed for fickleness of atmosphere. By midnight every\nbeech and elm round the chateau was tossing and bending down to the\nroots, and a heavy snowfall was already sheeting the fields. As the\nstorm rose, it occurred to me to ascertain what provision might have\nbeen made against it by our soldiers, who were lodged in the barns and\nextensive outhouses of the chateau. Leaving my dragoon friend to act as\nmaster of the ceremonies, I sallied forth. The storm was now at its\nheight; and it was with some difficulty that I could make my way. In the\nmidst of the excessive darkness, I felt some animal make a sudden spring\non me, which nearly brought me to the ground. Wolves were not common in\nthe country, but there had been some recent instances of their issuing\nfrom the forests, and my first idea was that I had been thus attacked.\nBut the barking and bounding of a dog soon put an end to this\nconception; and I recognised in my assailant the huge house-dog of the\nchateau, with whom I had already struck up a particular friendship. More\nsharpsighted than myself, he had rushed across the wood after me, and\nexhibited all imaginable rejoicing at the rencontre. I reached the\nbarns, found all my men wrapped in that quiet which cares nothing for\nthe troubles of kings and cabinet councils, and was preparing to return,\nwhen Caesar, with every demonstration of having found something of\nimportance, brought me a letter which he had dug out of the snow. By the\nlight of the lantern, I discovered it to be the report of an engineer\nofficer dispatched from the French army to ascertain the condition of\nour outposts, informing the head of the staff of an intended ball, and\nproposing a plan for carrying off the whole party together. I was\nthunderstruck. The letter was dated three days before, and though\nevidently dropped by some negligence of the officer, yet giving full\ntime for him to make his report in person, and bring the force necessary\nfor our capture. If it succeeded, an exploit of this order might have\nparalysed the whole campaign; for nearly the entire staff of the army,\nbesides a crowd of regimental officers of all grades, were within the\nwalls of the chateau.\n\nI hastened back, showed the report to one or two of the principal\nofficers, in private, for the purpose of avoiding alarm to our fair\npartners, and we then considered what means were left to protect us from\nthe approaching catastrophe. Our little council of war was nearly as\nmuch perplexed as matters of this kind are in general; and the\npropositions, various as they were, came finally to the usual result,\nthat we had got into a scrape, and that we must get out of it as well as\nwe could. To send the ladies away was impossible, in a tempest which\nalready flooded every road, and with all the trees crashing over their\nheads. To expect reinforcements from the camp, at such a distance, and\nin such weather, was hopeless; with the recollection that the whole\naffair might be over in the next quarter of an hour, and our entire\nassembly be in march before the French hussars. This was the first\noccasion of my responsibility as a soldier; and I learned, from this\ntime forth, to give commanders-in-chief some credit for their\nresponsibilities. The agonies of that half hour I have never forgotten.\nMilitary failure was nothing compared to the universal shame and\nblighting which must fall on the officer who suffered such a disgrace to\nbe inflicted on him in the presence of the whole army; and such a\ncalamity to arrest the progress of that army, if not the hopes of\nEurope. My resolution was desperately but decidedly taken, if the post\nfell into the enemy's hands, on that night to throw away my sword and\nabandon my profession, unless some French bayonet or bullet relieved me\nfrom all the anxieties of this feverish world. To offer the command of\nthe post to any of the superior officers present was, as I well knew,\ncontrary to rule; and on me and the dragoon devolved the whole duty.\n\nBut this state of almost nervous torture was as brief as it was painful,\nand my faculties became suddenly clear. The service of outposts was a\nbranch of soldiership, at that period, wholly unpractised by the British\ntroops; but I had seen it already on its most perfect scale in the\nPrussian retreat, which I and my hussars had our share in covering. My\nfirst step was to warn my soldiers and the dragoons of the probability\nof attack, and my second to call for a favourite quadrille, in which I\nsaw all our guests busily engaged before I left the chateau. My next was\nto repeat my Prussian lesson in reconnoitring all the avenues to the\nhouse. This, which ought to have been our first act on taking\npossession, had been neglected, in the common belief that the enemy were\nin full retreat. The gallant captain of dragoons prepared to take a\ngallop at the head of a party along the _chaussee_, and ascertain\nwhether there were any symptoms of movement along the road. He mounted\nand was gone. Posting the dragoons in the farm-yard, I went to the front\nto make such preparations as the time might allow for the enemy. Like\nthe greater number of the Flemish chateaux, it was approached by a long\navenue lined with stately trees; but it wanted the customary canal, or\nthe fosse, which, however detestable as an accompaniment to the grounds\nin peace, makes a tolerable protection in times of war, at least from\nmarauding parties. All was firm, grand, and open, except where the\ngarden walls and hedges of the lawn shut it in. As the avenue was the\nonly approach accessible to cavalry, and as this was the force which\nwould probably be used for a _coup-de-main_, if it were to be attempted\nat all, I set all hands to work to secure it. Wild as the night was, my\nmen wielded the spade and mattock with good will; and we had completed a\ntrench of some feet deep and wide, half across the road, when I caught\nthe trampling of cavalry at a distance. My chagrin was irrepressible;\nthe enemy would be upon us before we had got through our work, and we\nmust be taken or fly. My men worked vigorously; but the cavalry were\nupon us--and to my utter astonishment and infinite relief, our labours\nproduced a roar of laughter. The party were our dragoons, who had looked\nfor the French advance in vain, and were now amusing themselves with our\nwaste of toil. We forgave them their jest; they passed, and we prepared\nto follow to our quarters. But still the French officer's report haunted\nme; the precision of its terms, and the feasibility of the enterprise\nitself, struck with new force; and even after I had given the word to\nmove, I halted the men, and climbing a little pleasure turret by the\nside of the avenue, gave a parting glance round the horizon. Nothing was\nto be seen. The night was dark as a dungeon, and I prepared to descend,\nwhen at that moment the distant sound of a trumpet broke on the air. I\nlistened, and thought that I recognised the French call for cavalry to\nsaddle and mount. I sprang down; every man piled his arms, took spade\nand mattock in hand once more, and in a few minutes the trench was\ncompleted across the road. Still no further notice of approaching troops\nwas to be heard; and I heard a low, but rather provoking laugh among my\ncompany. Still I determined to persevere, and ordering some of the trees\nround us to be cut down, formed a rude species of _chevaux-de-frise_ in\nfront of our trench. It was scarcely finished, when the distant\ntrampling of cavalry was heard in the lull of the gale. All were now\nconvinced, and dispatching a notice to the dragoons to be ready, we\nstood to our arms. Giving the strictest orders that not a word should be\nspoken, nor a shot fired, I waited for the enemy. The trampling\nincreased every moment, and it was evident that the body of cavalry must\nbe large, though of its actual numbers we could form no conjecture. They\nsuddenly stopped at the entrance of the avenue, and I was in fear that\nmy _trou-de-rat_ would be discovered; but the national impatience soon\nspared me this vexation. The cavalry, hearing nothing in the shape of\nresistance, and not relishing the pelting of the storm in the open\ncountry, rushed in without further search, and came pouring on at the\ngallop. The avenue was long, and the whole corps was already within it,\nwhen the leading squadrons came at full speed upon my rude\nfortifications. In they dashed, into the very heart of my\n_chevaux-de-frise_. Nothing could equal the confusion. Some sprang over\nthe trees, but it was only to be flung into the trench; some even leaped\nthe trench, but it was only to be met by our bayonets. The greater\nnumber, startled by the cries of their unlucky comrades in front,\nattempted to rein back; but found it impossible, from the weight of the\nsquadrons still pushing on from behind. At this point, while they stood\na struggling mass, wholly unable to move either backward or forward, I\ngave the word to fire, and poured in a volley with terrible execution.\nAn ineffectual firing of pistols was their only return. Some of their\nofficers now rushed to the front, with the usual gallantry of their\ncharacter, called on their men to advance, and charged the trench; but\nthis dash only filled it with falling men and horses. I gave them a\nsecond volley, which was followed by a howl of despair; the whole of\ntheir leading squadron was brought down--every shot had told. The mass\nstill stood, evidently taken by surprise, and wholly unable to extricate\nthemselves. I now ordered our dragoons to mount, take a circuit to the\nhead of the avenue, and, if possible, close them in. In a few minutes, I\nheard the effect of my order in their galloping through the enclosures,\nand in the shout of a charge at the further end of the avenue. The staff\nand other officers in the chateau had hurried out at the sound of our\nfiring, and some had come up to us, and others had joined the dragoons.\nA proposal was now sent by a general officer to the commandant of the\nbrigade, to surrender, with a threat of being put to the sword in case\nof an instant's delay. The brave Frenchman was indignant at the\nproposal, and threatened to hang the bearer of it to the next tree. But\nthe British camp had palpably been alarmed by this time. Bugles and\ntrumpets were heard in every direction. Our dragoons had already shut up\nthe avenue; and after some slight discussion, the advance of a few\nsquadrons more, which came up at the gallop, proved the total\nimpossibility of escape, and the affair was at an end. This night's\n_melee_ had no rival in the campaign; it put into our hands twelve\nhundred of the best cavalry in the French army, and almost wholly\nstripped the enemy of the means of protecting his flanks, while it made\na most brilliant figure in the Gazette--the true triumph of the British\nsoldier.\n\nTo me, it was a restoration to life from the depths of despair. It may\nbe perfectly true, that many a post has been surprised, and many an\nofficer captured, without being objects of penalty, or even of public\nobservation; but my case was different. My character as a soldier was\nessential to my existence. The eyes of many, at home and abroad, were on\nme; and the scorn of one, wherever she was, would have been fatal to me.\nBut of those bitter extremes I say no more; my spirit was buoyant with a\nsense that I had done my duty in the most effective style. Nor was I\nleft to my solitary sense on the subject. My return to the chateau was\nas triumphant as if I had gained a pitched battle at the head of a\nhundred thousand men. Our fair guests, who had spent the hour before in\nthe terrors of instant capture, were boundless in their congratulations\nand expressions of gratitude. The officers, to whom my defence had made\nthe entire difference between a French prison and liberty, spoke in the\nmanliest and most cheering terms of my conduct. The scene of the\nstruggle was visited during the next day by every officer of the army\nwho could obtain a horse and an hour's leave; and the report which was\nforwarded to the commander-in-chief contained language which was\nregarded as a sure pledge of promotion.\n\nGuiscard hurried over to join in the congratulation. He had been\nemployed until a late hour in sending despatches to his court, relative\nto the growing problems of our politics with Prussia; and taking the\nfirst opportunity of throwing aside the envoy, he came at a gallop to\nshake hands with me. His impatience to see the ground scarcely suffered\nhim to sit down at table; his toast to the brave British army was given,\nand we went out to traverse the avenue. After having inspected every\ncorner of it with his keen military glance--\"You will find my theory\nright,\" said he; \"war is always a succession of mistakes. There never\nhas been a battle fought, in which even the successful general could not\npoint out a series of his own blunders, any one of which might have\nruined him. The only distinction is, that there are brilliant mistakes\nand stupid ones. Yours was of the former order--the Frenchman's of the\nlatter. If, instead of sending his whole brigade headlong down the road,\nlike clowns at a fair, he had dismounted half a squadron of his\ndragoons, and sent them to fire into the casements of the chateau, while\nhe kept the rest of his men in hand in the neighbourhood, he must have\ncaptured every soul of the party, and by this time had you all fast at\nthe French headquarters; but he blundered, and he has paid the price of\nblundering.\" To my laughing reply, \"that there was at least some merit\nin the steadiness of the men who beat him\"--\"Of course,\" was his answer.\n\"The English steadiness is like the English fire, the grand cure for the\nEnglish contempt of the tactician. Yours is an army of grenadiers; you\nare fit for nothing but assaults: but it must be owned that your troops\nof old managed that part of their business well, and I dare say that the\nart is not lost among you yet. Still, there are other matters to be\nthought of. Pray,\" said he, turning his keen eye on me, \"can any one in\nthe chateau tell how near is the French army to-night?\" I acknowledged\nmy ignorance. \"I ask the question,\" said he, \"because I think it by no\nmeans improbable, that they are at this moment marching down upon you.\nNot that they can afford to lose a brigade of cavalry a-night, and I\ntherefore think you safe enough for the twelve hours to come; but I am\nfar from answering for the next twenty-four. Dampierre commands them; I\nknow him well--he is a bold and also a clever fellow; the loss of his\ncavalry last night will leave him no alternative but to attack you or to\nmeet the guillotine. Those are fine times to make a general officer look\nabout him. My last letters from the Rhine state that the two generals of\nthe two covering armies on the frontier have been put under arrest, and\nthat they are now both on their way to Paris, from which Custine and\nBeauharnais will never return with their heads on their shoulders.\"\n\nI shuddered at this fate of brave men, overcome only by circumstances,\nand asked whether it was possible that such a system could last, or in\nany case could be endured by men with swords in their hands.\n\n\"It can, and will,\" was the reply. \"Soldiers are the simplest race of\nmankind, when they come in contact with the cunning men of cities. An\narmy, showy and even successful as it may be, is always an instrument\nand no more--a terrible instrument, I grant you, but as much in the\nhands of the civilian as one of your howitzers is in the hands of the\nmen who load and fire it. At this moment sixty commissioners, ruffians\nand cut-throats to a man--fellows whom the true soldier abhors, and who\nare covered with blood from top to toe--are on their way from Paris to\nthe headquarters of the fourteen armies of the republic. Woe be to the\ngeneral who has a will of his own! Those fellows will arrest him in the\nmidst of his own staff, carry him off in the presence of his army, and\nsend him to give a popular holiday to the Parisians, by his execution\nwithin half an hour after his arrival. So much for the power of an\narmy.\"\n\n\"But Frenchmen are human beings after all. Must not those horrors revolt\nhuman nature?\" was my question, put with indignant sincerity. He looked\nat me with a quiet smile.\n\n\"You are romantic, Marston, but you are of an age that becomes romance.\nWhen you shall have lived as long as I have done, and seen as much of\nthe world as myself, you will know that it is utterly selfish. It may be\ntrue, that some generous spirits are to be found here and there, some\nfond hearts to cling to, some noble natures which inspire an involuntary\nhomage for their superiority; but you might as well expect to be lighted\non your way by a succession of meteors. In the world, you will find that\nevery man carries his lantern for himself; and that whether small or\ngreat his light, the first object is to guide his own steps, with not\nthe slightest care whether yours may not be into the swamp--unless,\nindeed, he may have a particular object in bewildering you into the very\nheart of it. But now, to more pressing affairs than my honest and\nluckless philosophy. Get leave from your colonel to take a ride with me.\nI feel a sudden wish to know what Dampierre is doing; and a few hours,\nand as few leagues, may supply us with information on points which your\nbrave countrymen seem so constitutionally to despise. But recollect that\n_I_ am a Prussian.\"\n\nWe returned to the table, which was crowded with visitors, and spent an\nhour or two in great enjoyment; for what enjoyment can be higher than\nthe conversation of minds willing to give and receive intellectual\npleasure? And Guiscard was never more animated, easy, and abundant, in\ncommunicating that pleasure. He was a model of the most accomplished\norder of the continental gentleman. He had commenced life as a scholar;\na disappointment in his affections drove him into the army. He\ndiscovered that he was made for the profession; and, combining the\naccomplished diplomatist with the almost chivalric soldier, he had\nrapidly risen to the highest rank of the royal staff. But he had the\nstill rarer qualities of a sincere heart, and was a firm and willing\nfriend.\n\nThe orderly now returned with the leave for which I had applied. The\npost was left in charge of the captain of dragoons; and Guiscard and I,\nwithout mentioning our purpose, rode out quietly, as if to enjoy the\ncool of the evening. It was well worth enjoying. The storm had gone down\nat daybreak, and been succeeded by a glowing sun; the fields flourished\nagain, and if I had been disposed to forget the tremendous business\nwhich might be preparing for the morrow, I might have lingered long over\nthe matchless luxuriance of the Flemish landscape. There certainly never\nwas one which gave slighter evidence of the approach of two hostile\narmies. From the first hill which we ascended, the view, for leagues\nround, exhibited nothing but the rich tranquillity of a country wholly\nagricultural; soft uplands, covered with cattle grazing; ploughed\nfields, purpling in the twilight; clumps of trees sheltering villages,\nfrom which the smoke of the evening fires rose slowly on the almost\nbreathless air, giving an impression of the comfort and plenty of the\nmeal within; and at intervals, some huge old chateau, with its\nbuttressed and richly-wrought architecture--those carvings and\ncolourings which so strikingly convey the idea of a past age of quaint\nluxury and lavish wealth--rose from the centre of its beech grove,\nglaring against the sunset, as if it had been suddenly covered with a\nsheet of gold. All was peace, and the few peasants whom we met, as the\nnight fell, were all in the same tale, that there had been no patrols in\ntheir neighbourhood of late, and that, with the exception of the attack\non the \"outposts of the English,\" they had not heard or seen any thing\nof the French for a month before.\n\nThe night had now fallen, and though calm, it was one of remarkable\ndarkness. We passed village after village, but by this time all were\nfast asleep, and except the disturbance of the house-dogs as we rode\nby, not a sound was to be heard. I felt every inclination to take my\nshare of \"nature's sweet restorer, balmy sleep,\" and proposed to my\ncompanion to turn our horses into the first farm-yard, and \"borrow an\nhour\" or two's rest from the farmer's hospitality, and clean straw.\n\n\"I agree with you,\" was the answer, \"that Dampierre is clearly not on\nthis road; but that is no reason why he may not be on some other. On\nconsidering the matter, I think that we have been wrong in looking for\nhim here; for his national adroitness is much more likely to have tried\na movement in any other direction. He may be marching on either the\nright or the left of the spot where we are standing. And if he is the\nofficer which I believe him to be, he is trying this game at this\nmoment.\"\n\n\"What then is to be done, but ride back to our quarters, unless we\nshould prefer being cut off by his advance?\" was my question.\n\n\"One thing is to be done,\" was the reply--\"we must not let ourselves be\nlaughed at; and if we return with nothing more for our night's work than\nthe story that we slept in a Flemish barn, we shall be laughed at. So\nfar as I am concerned, I care nothing for the sneers of ignorance; but,\nmy young friend, your late conduct has inevitably made you an object of\nenvy already; and the only way to pluck the sting out of envy, is by\ngiving the envious some new service to think of.\"\n\nWe now agreed to separate, and examine the country to the right and left\nfor an hour precisely, meeting at one of the villages in the road, if no\nadvance of the enemy were discernible within that time. We parted, and I\ncommenced as comfortless an expedition as it would be easy to imagine.\nThe Flemish cross-roads, never very passable, were now deep in mire; the\nrivulets, of which they are generally the conduits, had been swelled by\nthe storm of the night before; and I floundered on for nearly the\nappointed time, in the full perplexity of a stray traveller. I was on\nthe point of returning, when I observed a sudden light rising above some\nfarm-houses, about half a league off. The light rapidly strengthened,\nand I rode forward, in some degree guided by its illumination. But after\nblazing fiercely for a while, it sank as suddenly as it rose; and I was\nagain left bewildered among hedges and ditches. But a loud hum of\nvoices, followed by the sound of many footsteps, now convinced me that a\nlarge body of men were near; though whether peasants roused by the fire,\nor battalions, I was still unable to discover. While I stood under cover\nof a clump of trees by the roadside, the question was settled by the\nmarch of a patrol of cavalry, followed at brief intervals by squadrons\nand light troops intermixed. It was evident that Dampierre meditated a\nsurprise of the British forces, and that the whole of his skirmishers\nwere already in motion. How long this movement had continued, or how\nnear the enemy might already have approached to the British camp, was\nentirely beyond my conjecture; and for the first few moments, the\nprobability of the surprise, and the possibility of my being already so\ncompletely within the range of the French march as to preclude my\nbearing the intelligence in sufficient time, made the drops of anxiety\nand perturbation roll down my forehead. But every thing must be tried. I\nno longer attempted to wind my way back through the network of lanes;\nbut, in the spirit of an English sportsman, took the country in a\nstraight line towards the British quarters. My horse, a thorough English\nhunter, evidently preferred leaping the Flemish fences to wading his way\nthrough the swamps; and I had the honour of bringing the first\ninformation, and the happiness of finding that I had brought it just in\nthe right time.\n\nThe camp was immediately under arms; every preparation was made in a\nsilence which gave me a high conception of the capabilities of the\nBritish soldier for every species of service; and, without a sound among\nten thousand men, we waited for the approach of the enemy.\n\nDampierre's manoeuvre had been a dashing one--conceived and managed\nwith the skill of an able officer. His purpose had been to throw his\nmain body into the rear of our position; and while he drew off our\nattention by a false attack on our front, avail himself of the\nconfusion of a night attack to crush us. Whether the fighting qualities\nof the Englishman would not have made him repent of his plan under any\ncircumstances, is no longer the question; but the surprise was now\nwholly his own. The first volley which we poured into his columns, as\nthey crept up stealthily towards our line, was so heavy that it finished\nthe battle. By the blaze of the musketry, we could see the French masses\nactually rolling back upon each other, staggering and shaken like\nlandsmen at sea, or like any man in an earthquake. Our cavalry were now\nordered to follow; but the enemy were too quick in making their escape;\nand the intersected nature of the country forbade any continued pursuit.\nA few shots from our howitzers, which ripped up the ground after them,\nwere all that we could send as our parting present; and the engagement,\nwhich began in such silence and sternness, finished in roars of laughter\nfrom all our battalions.\n\nDay broke, and the order was issued to follow the French general. The\ntroops, animated by the prospect of coming to action at last, and\nutterly wearied with the idleness of the camp, received the intelligence\nwith shouts; and the whole moved rapidly forward. Dampierre, before his\nmarch of the previous night, had provided for casualty, by forming an\nintrenched camp in the famous position of Famars. It was strong by\nnature, and he had added to its strength by covering it with fieldworks,\nand a powerful artillery. It was late in the day before we came within\nsight of it; and its strength, from the height of its glacis--the\nnatural glacis made by a succession of sloping hills--was all displayed\nto full and formidable advantage. The troops, fatigued with the length\nof the march under the burning sun of one of the hottest days which I\never felt, were halted at the foot of the heights; and the plans of\nattack proposed were various enough to have perplexed the Aulic Council\nitself. Lines of circumvallation, or bombardment, or waiting the effect\nof famine, were successively urged. But the British style prevailed at\nlast over the scientific. The Guards were ordered to head the column\nwhich was to storm the lines in front, and columns on the right and left\nwere put in motion at the same instant. We rushed forward under a\ngeneral discharge of the French artillery and musketry, and in a quarter\nof an hour the position was in our hands. The difficulty of its\napproach, and the broken nature of the ground in its rear, enabled the\nFrench general to make his retreat with the chief part of his forces.\nBut our prize was well worth the trouble; for we brought back two\nthousand prisoners, and the whole artillery in position.\n\nThe war had now begun in earnest; and our advance was unintermitted. On\nthe eighth day from the storm of Famars, we again came in sight of\nDampierre. He was now the assailant; our army, which had never exceeded\nten thousand men, (such was the military parsimony of those days,) with\nthe Prussian troops, and some of the smaller German contingents, were\nnow unwisely spread to cover a line of nearly thirty miles. The French\ngeneral had seized the opportunity of retaliating his ill fortune upon\nthe allied troops. At daybreak we were roused by the tidings that the\nFrench had broken through our weak extended line in several places, and\nhad got into the rear of the whole army. The force of the enemy, its\ndirection, or its object, were alike matters of total ignorance; and,\nfor some hours, it was impossible to obtain any exact information.\n\nIt was in vain that we adopted all the usual expedients, of detaching\nofficers, examining peasants, or judging of the progress of the\nengagement by the sound of the advancing or retreating fire. We had only\nto wait, drawn up ready for action, and take our chance of the result.\nOf all the contingencies of the field, none is more perplexing; but I\nhad a personal source of anxiety to add to the general vexation. I had\nevery reason to believe that my excellent friend, Guiscard, had either\nfallen into the hands of the enemy, or had been killed on the night when\nwe separated. If either misfortune had occurred, it was solely in\nconsequence of his zeal for my character, and the thought inexpressibly\ndistressed me. I had made the most persevering enquiries for him, but\nwithout any success; or rather, with a painful gathering of facts, all\nwhich told against my feelings. His horse had been found straying\nthrough the country; his helmet had been also found; and a fragment of a\nsabre, in a spot evidently much trampled, and which, therefore, appeared\nto be the scene of the personal rencontre in which he had probably\nfallen. Every thing had been found but his body.\n\nAt length, the firing, which had continued with more or less steadiness\nduring the day, approached our position and we were ordered to advance.\nThe country was now a portion of an ancient forest, and it was difficult\nto see in front of us beyond a few hundred yards. As we made way, we\ncould hear not only the musketry but the shouting of the troops engaged;\nas, growing constantly more impatient, we pressed on, a mounted officer\ncame galloping towards us. Judge of my astonishment and delight when I\nsaw Guiscard. As he reined up beside me--\n\n\"I have not a moment,\" said he, \"to speak to you; you shall hear of my\nadventures by and by. I was in as much fear for you as you probably were\nfor me. But now, tell me where I am to look for the officer in command\nof the column.\"\n\nThe general was soon found, and Guiscard communicated to him that the\nenemy had concentrated his chief force directly in front of us, where a\nPrussian column had been posted; that the Prussians had resisted\nvigorously several successive attacks; but that the force converging on\nit was too powerful, and that it must speedily retire. \"Then let it\nretire,\" was the general's reply, \"and we shall take their place.\"\n\n\"Pardon me, general,\" was the prompt suggestion of the pupil of a more\nexperienced school; \"but, if you will permit me, I shall ride back to my\ncountrymen, inform them of your advance, and make them hold their\nposition until you come out from the forest upon the enemy's flank.\"\n\nHis opinion was received, and he put spurs to his horse and was gone. We\nnow moved with all speed to the right of our former direction; and after\nhalf an hour's toiling through the intricacies of a wood on which no axe\nseemed to have fallen since the Deluge, passed round the enemy, and came\nfull upon their rear. A few volleys, thrown in upon them in this state\nof alarm, broke them; the Prussian fire in front, and our's in the rear,\nmade their disorder irreparable. In this crisis, Dampierre rushed\nforward with a group of aides-de-camp to restore the engagement,\nstriking the fugitives with his sabre, and desperately exposing his\nperson to the balls which now fell thick as hail around him. For a while\nhe seemed to bear a charmed life; but a rifleman of the Prussian hulans\ntook a sure aim. He fired, and I saw the unfortunate general fall from\nhis horse. He had died instantly. A more gallant death, and scarcely a\nmore expeditious one, than awaited the unsuccessful generals of the\nmerciless Republic. We buried him on the spot where he fell, with the\nhonours due to a distinguished soldier. Before nightfall the French had\nretired in all quarters; and the remnant of the troops hurried across\nthe Flemish frontier, utterly disheartened and ruined.\n\nThis engagement, which was known long after as the battle of the forest\nof Vicogne, cleared the Netherlands, raised the fame of the British\ntroops to the highest pitch, and left in their hands four thousand\nprisoners.\n\nThe councils of the allied camp now assumed a bolder tone. France was\nbefore us. The popular enthusiasm had been cooled by time and calamity.\nDefeat had taught the nation the folly of supposing that it could\ncontend single-handed with Europe; and the only obstacle to our march to\nParis was the line of fortresses erected by Louis XIV. The most powerful\nof those fortresses lay in the road by which the British columns were\nadvancing; and it was with a singular mixture of rejoicing and anxiety,\nof ardour and awe, that I saw, at the breaking of a brilliant morning,\nspread beneath me the strong city of Valenciennes.\n\n\n\n\nIT IS NO FICTION.\n\n \"Oh! I could be bounded in a nutshell, and count myself a king of\n infinite space, were it not that _I have bad dreams_.\"--_Hamlet_.\n\n \"I am wrapp'd in dismal thinkings.\"--SHAKSPEARE.\n\n\nI have been a dreamer all my life. The earliest recollections of my\nchildhood are of dreams of greatness. My boyhood's visions were peopled\nwith warlike tumults. There were no spring mornings to my brain even in\nearly youth; my heart was clouded with shadow, and sadness reigned when\nmirth and careless glee should have been pre-eminent. My manhood has\nbeen a fitful, feverish, and painful existence. I have outlived all whom\nI ever cared for; I have seen those whom I idolized lie before me cold\nand senseless; and now, with every event vividly impressed upon my\nmemory, each tone of the voice of her I loved dropping like liquid fire\ninto my brain, and drying up the tears that would weep away my\nanguish--feeling all this with intensity, and longing for the free air\nof heaven, I find myself alone--desolate--and HERE!!\n\nOh! the horror of this prison-solitude--the anxious watching for the\npale morning after sleepless nights--the horrible nights when fantastic\nshapes are alone visible, mocking at and jeering me--when the only\nsounds I hear are the ravings of some wretched maniac, confined, like\nmyself, because we have made for ourselves a world, and our imaginations\nhave created a presiding divinity; and, should a laugh disturb the\nsilence, it is the outbreak of a maddened spirit seeking relief from\nthought--a laugh frightful, because a mockery--sad in its\nboisterousness--\"_the laugh which laughs not_.\"\n\nFor many weary years I have been pent up in this prison, pining for\nfreedom, hoping for things which never existed, conjuring up\nanticipations of a brighter future, calling upon her who made\n\n \"The starlight of my boyhood,\"\n\nto look down upon me from her blest abode, and woo me back to calmness\nby one gentle word, one loving glance; and then sinking into hopeless,\nbitter despondency, when I remembered that she was gone, and that I\nshould see her no more.\n\nSometimes I can think of her in her exquisite beauty, and my soul drinks\nin, as it were, the sweet and liquid tones of the voice which once spoke\npeace to me, and, fancying her again before me, I sink into an unquiet\nslumber, till some hideous dream oppresses me, and I see the fair brow\nof my \"Julia\" contracted, withered; and instead of her silvery voice of\nenchantment, a hissing sound escapes the lips I have worshipped. I rise,\nand try to approach, but she recedes. I awake--I start from my uneasy\nbed--I find this horrible picture, which bore the impress of reality, is\nbut a dream. I awake to the consciousness that my beloved is dead, and\nthat my eyes will gaze upon her beauty no more.\n\nHow few there are in this busy world who, when passing those abodes of\nwretchedness--\"private madhouses\"--can imagine the agony, the misery,\nthe despair that dwells there! But to my history.\n\nI was the only child of General Sir Frederick and Lady Charlotte B----.\nI was reared in luxury; the rude air was scarcely allowed to blow upon\nmy delicate frame. I can remember now, though years have passed, and\nsorrow has bowed me--I can remember the happy days when my wearied head\nwas pillowed on the bosom of my mother, and, after she had sung me to\nsleep with some wild melody, she would place me in my small luxurious\ncot, and watch over me with those deep-loving eyes, and be the first to\ncomfort and re-assure me if uneasy dreams--for even then I was a\ndreamer--made me awake to sorrow. But my mother died. Even now I shudder\nat the recollection of the desolateness of my agony when I knew I had\nlooked on her for the last time. Even now I can feel the coldness which\ncrept over me as I laid my cheek to hers. My blood was frozen. I could\nnot weep. Oh! tears would have been a relief, but they were denied me;\nand though I saw her taken from my embrace, and her beloved form laid in\nthe vault, I could still gaze with speechless agony--but I wept not.\n\nHow I wished for the quiet of the grave; for even then there was a\nwhirlwind within my bosom, and my sensitive heart shrank from holding\nconverse with, or bestowing confidence on another as freely or\nunreservedly as I had done with the dear being whom I had lost.\n\nShortly after this event my father was ordered upon foreign service, and\nmy childhood was passed among relatives who were strangers to me. It was\na childhood without love. I remembered my mother, and none could supply\nher place. I could not trust in another as I had trusted in her. In my\nsorrows, real or imaginary, none other could comfort me. I longed for my\nchildhood's resting-place, where I might again pillow my aching head,\nand sleep once more the calm sleep hallowed by a mother's matchless\nlove.\n\nAt an early age I was sent to one of our great public schools, and\nthere, although I endured some hardships, yet I experienced also\nsomething like the pleasures and pastimes of boyhood.\n\nFrom having been a weakly, delicate child, I grew strong and active; but\na gloom was ever upon me.\n\nIn my moments of relaxation I would join some of my companions in their\ngames of play; but even then a dark phantom pursued me, and I would\nfancy a shadowless spirit was after me: if I ran it always followed me\nwith its noiseless steps, and my constant fear was, that it would\novertake me. This was _madness_--aye, I can see it now--_it was madness\ncoming upon me_.\n\nI frequently used to endeavour to dispel the illusion by reading; but if\nI raised my eyes from my book there was the figure, looking at me and\nsighing, and its lips would move to speak--_but there was no sound_.\n\nI have sat for hours watching this bane of my existence. I have sat till\nmy eyes were fixed from fright, and I have tried to move, but I felt\nchained to the spot, and the fetters that appeared to bind me, seemed of\ncold heavy steel, that fell on my whole body and paralyzed me. Then I\ncould feel my heart growing dead, and yet throbbing with those dull,\naudible throbs, till at last I have shrieked in the agony of my horror,\nand only then would the dark being leave me--but _it left me moody and\nmad_.\n\nI had one friend at school who would soothe me by gentle words, and tell\nme my fears were but fancy, and he would hold my hands until I slept,\nand lost, for a time at least, the phantom which pursued me.\n\nThat friend is dead. I have outlived _him_. _Why should the madman\nlive?_\n\nWhen I was about sixteen a new life opened to me. There came as a\nvisitor to one of the ladies belonging to the establishment, a young and\nlovely girl. I first saw her at the private chapel belonging to the\nschool. The moment I looked at her a gush of hitherto unknown pleasure\ncame to my heart. I felt that I could love her.\n\nI saw her again and again. I have stood for hours by the house in which\nshe was, hoping to catch a glimpse of her. Sometimes I was\nsuccessful--more frequently not--but it was something to hope for. Once\nI fancied that her eye fell upon me. Oh, how I was repaid by that one\npure glance!\n\nWhile she remained at ----, my life was one of bright and vivid fancy,\nand I was cheered by the angel Hope; but at length her visit came to a\ntermination; yet, though I knew she had departed, I would go daily to my\naccustomed watching place, and gaze until I fancied the beautiful girl\nwas again before me.\n\nAt the usual period my school days ended, and my college life began. I\nwas entered at Christ Church, Oxford. I read hard, and obtained the\nhighest honours. My fame was brilliant. I was talked of, and marked by\nmy superiors as a rising man.\n\nShortly afterwards, I was returned as one of the members of a family\nborough in my native county, and my first speech in Parliament met with\ngeneral applause. The world called me a fortunate man. Oh! they little\nknew the nights of horror I passed--the battling I had with my attendant\nphantom, which still pursued me, blighted me. But I was mad; and the\nexcitement of madness was called energy.\n\nHow often I have laughed them to scorn, as I have sat alone with the\ndark spirit!\n\nMy sole ambition was that the girl whom I had seen and admired might\nhear of my career; and that, with honours crowded upon me, I might see\nher again, that I might place my laurel crown at her feet, lay bare my\nheart's best feelings, my undying love for her, and prove to her how\nentire was my devotion, how earnest my worship.\n\nI saw many young and lovely girls; and I was told that mothers looked\nupon me as a desirable match--but I was true to my first love. I\nremembered her in the perfection of maiden beauty--I wished for none\nother; and to see _her_ again was my sole hope in life.\n\nAfter a season of unceasing gaiety and dissipation--sick of London and\nits vanities--I determined to travel, and for seven years I was absent\nfrom my native land.\n\nI was recalled to attend the deathbed of my father. I had seen but\nlittle of him; he had no sympathy with me, and in heart we were\nstrangers to each other. He was proud of my talents, and I was an only\nson; but he never bestowed any real affection on me. I honoured him\nbecause he was my parent; but I never loved him as I ought to have loved\na father.\n\nHe died, and I succeeded to the baronetcy and estates; but I was already\ntired of life--wretched in the midst of my splendour. In a word--_I was\nmad_.\n\nAt the table of a friend I met a man a few years my senior, whom I had\nknown at school. We renewed our acquaintance; and I accepted an\ninvitation to dine at his house, to meet some old schoolfellows.\n\nI consented to go, but not cheerfully, for a moody state of mind was\ncoming over me. I can remember the struggle, the exertion it was to\ndress for the party. Twenty times I was tempted to send a message saying\nI was too unwell to go, but my better angel prevailed--and I went. To\nwhat an eventful period was that evening but the prelude!\n\nMy friend met and welcomed me with a cordiality which somewhat cheered\nme; but I had a weight on my spirits from which I could not rouse\nmyself, and most reluctantly accompanied Sir Charles Tracey, with\nfaltering steps and an aching heart and brow, into the inner\ndrawing-room, to be introduced to his wife, Lady Tracey.\n\nShe was seated on a low ottoman, with her back to the door, reading. She\narose as her husband presented me to her as his old friend, Sir\nFrederick B----. She turned towards me, and for a moment I was\noverpowered. I beheld before me the creature I had so long pined for--so\nearnestly searched for--whose memory I had so devotedly and entirely\nworshipped.\n\nWith exquisite grace she extended her hand to welcome her husband's\nguest, and as I held those small taper fingers in mine, thick coming\nfancies crowded upon me. I was again the schoolboy--the anxious, ardent\nschoolboy, longing even for a look from this lovely woman, whose hand I\nnow held in mine.\n\nHot tears rushed into my eyes, and I bent over the fair hand to conceal\nthem.\n\nThis momentary cloud passed away, and while seated by her, I forgot that\nwe had ever been parted, and imagination peopled a world of love--a\nparadise of hope.\n\n \"But she in these fond feelings had no share.\"\n\nThe years which had passed, had changed her from a lovely girl into the\nmore matured loveliness of the matron.\n\nWhen I had last seen her, her hair, which was a rich and shining black,\nhung in natural and graceful curls over her beautiful and classically\nformed head. Now the thick and luxuriant mass was gathered into a knot\nbehind, and laid in soft bands over her pure and polished brow.\n\nHer eyes were of that deep full blue which is so rare, and were large\nand bright, and full of fire and spirit, which at times gave an\nappearance of haughtiness to her noble countenance; her throat, neck,\nand arms, were white as ivory, and formed in the most perfect mould; her\nheight was commanding, and her figure exquisitely proportioned.\n\nBefore she spoke I could only look at her with wonder, that any thing so\nglorious could be earthly; but the instant she addressed me, a peculiar\nwitchery played over her features and about her mouth; and my wonder was\ninstantly changed into love and adoration, and I drank in with eagerness\nthe silvery sweetness of her voice.\n\nI fancied on this night that Lady Tracey bestowed more attention on me\nthan on her other guests; for women have an intuitive tact in\ndiscovering when a man admires devotedly.\n\nFor that night I lost my dark phantom, I slept a sweet sleep, dreaming\nof things which could never be accomplished; and my waking vision, as\nwild and improbable, was that she might one day return my love.\n\nI would not lose sight of my newly found treasure. I called at her\nresidence. I was admitted. Again I gazed; and worshipped. Lady Tracey\nlooked more lovely by daylight than with the full blaze of candle-light\nupon her beauty. There was a delicacy about her complexion no daylight\ncould impair; but it spoke also of a delicacy of constitution which made\nme tremble as I gazed.\n\nThe fascination of her manner, the elegance of her movements, her light\nand airy tread, her musical voice, her bright but subdued laugh; all\nthese combined made me idolize her.\n\nThere is but one sun in heaven: there was but one Julia to my eyes on\nearth. Her shadow had fallen on my heart, as the sun on an island far\naway from land in the lonely sea. It was filled with light and verdure,\nand all my best feelings were warmed to ripeness by her glowing smile.\n\nWe conversed together on poetry, music, history, the arts; and I\ndiscovered she possessed a refined and superior intellect. A sparkling\ntincture of satire mingled with her mention of men and things; but while\nshe did this with perfect temper and gentleness, it gave a brilliancy to\nher conversation not to be described. She expressed a wish for a book\nwhich I had the happiness to possess; here was an opportunity for\nanother visit. Again and again we met, and I was intoxicated with love;\nbut I saw no reciprocal feeling on her part. She was the same gentle and\ncharming being; but she bestowed no _love_ upon the poor visionary who\nadored her.\n\nOn the days we met I was gay and happy; but on the intervening ones I\nwas in despair. All my darkest thoughts came back upon me, fraught with\neven greater horrors. I tried to battle with my evil spirit, but I could\nnot subdue it. It grasped me tightly in its fetters; and I had no\nrespite until I was again in the presence of my Julia. The smallest\nsound of her voice, with its silvery sweetness, broke the sad chain\nwhich had bound me, and I was free to look--to love--to worship again.\nOh, why did not these moments of rapture last for ever! This holy calm,\nlike an enchanted circle, into which my spirit of evil dared not\nventure, why was it broken? Why did sickness, and sorrow, and\n_madness_--yes, furious, hopeless, desponding madness--darken those\nsunny days? Why did death come to her, and thick clouds to me?\n\nThe sky mocks me with its gemmed radiance. The stars shine on brightly;\nbut they fail to give light and hope to me. I have gazed on them with\nher. I have seen her stand with her fair brow raised, and her lovely\nface bathed in moonlight; but, as the pale beams danced around her, to\nmy eyes her own glory dimmed all other brightness.\n\nThe winds howl, and the trees wave to and fro in the tempest, and with\nevery blast comes a shriek, as if Julia were in despair, and I arise to\nrush to her rescue; but the clanking chain of the maniac binds me. I try\nto break my bonds, but they clasp me; and my hideous companion, the\nphantom, jeers at me; and I hear the voice of my beloved receding\nfurther and further from me, till, with an agonized moan, it dies away\nin the distance.\n\nAnd this the world calls fancy--the fantastic vision of a madman's\nbrain!\n\nThere was never a voice like _her_ voice; and though the winds rage\ntempestuously among the waving branches of the storm-tossed trees, I\nhear the liquid music of her accents above all, and I strain my eyes to\ncatch a glimpse of her person, but there is nothing; and I crouch down\nagain in my chains and my madness on my desolate bed, feeling how\nutterly--how entirely, I am alone.\n\nAn interruption occurred in our intercourse, in consequence of Sir\nCharles Tracey being obliged to go abroad, on business connected with\nthe state. His lady accompanied him, and they were absent for some\nmonths. How I spent these months, I scarcely know. I avoided all\nsociety--I felt moody--wretched--despairing. I grew violent. Restraint\nbecame necessary. Then, indeed, I _knew_ that _I was mad_. Life was a\nblank; and some weeks passed while this dark cloud was upon me.\n\nAt last, though my recovery had been a work of time, I was _called_\nconvalescent, and the violence of my frenzy abated.\n\nI heard with joy that Sir Charles and his lady had returned to town. I\nthought the hour would never come when I might set out on my visit.\n\nI flew, rather than walked, to her residence. I felt startled and\nalarmed as I trode the streets; for I had not been out for months, and I\nfancied every one stared at me--that everyone knew _I was mad_; but the\none darling hope of seeing _her_ cheered me on.\n\nAt last I reached the house. I was admitted; and in a moment I was by\nthe side of Julia. She was looking pale and ill, but very lovely.\n\nI rushed towards her. I knelt by her side. I took her cold hand in mine,\nand kissed it ardently. A bright colour suffused her cheek. She\nendeavoured to withdraw her hand from my grasp; but the demon was within\nme. I held that pale, small, fragile hand firmly; and pressed it again\nand again to my lips, and my throbbing, bursting heart. I laughed aloud\nand wildly, and she looked at me fearfully. She had discovered my\nsecret, and she saw that _I was mad_.\n\n\"You, too, have been ill?\" she said.\n\nThe honied accents of that beloved voice fell on my ear like dew to the\nparched flower. I was calmed in a moment, and I endeavoured to look\ncoldly on her who was life--light--all to me in this world.\n\nI found she had been dangerously ill, and I felt, as I looked on her\nimperial loveliness, that she was not destined long for this world.\n\nDaily I saw her. I could not see enough of one I loved so desperately;\nand I feigned calmness while I endured agony--but my madness ruined me\nat last.\n\nOne wretched day--I spoke to her of love. I told her of my devotion--my\nhopeless devotion for so many years. I knelt by her side--I passed my\narm round her waist--and for one brief moment I rested my scorching,\nmaddened brow upon her bosom. It was only a moment of reality--but an\neternity of bliss in the recollection.\n\nI strained her fragile form to my breast. I kissed her pale cheeks--her\nbrow--her lips. She moved not. I found she had fainted. I thought she\nwas dead, and my brain reeled.\n\nI raised her beautiful form in my arms, and laid her gently on a couch.\n\nShe was like marble--so cold, and pale, and breathless. I called no one\nto my assistance--I was the madman--the desperate, heart-broken\nmadman--and I saw before me the ruin I had wrought.\n\nHow long this lasted I cannot tell; I only know my feelings were worked\nto frenzy. I called upon her by name; I conjured her to look at me, to\nspeak to me once--but once more.\n\nI longed for tears to cool the burning heat of my brain. In my agony, I\nlaughed and shrieked aloud; I could not control myself.\n\nShe opened her eyes, those large, bright, lustrous eyes, and looked, I\nthought, kindly on me. How those glances entered my soul!\n\n\"Speak to me, Julia, forgive me,\" I said. She smiled, and extended her\nhand. Her eyes were in a moment fixed and glassy. She tried to speak,\nwhen, O God! as her lips separated, the life-blood gushed from her\nheart, and the purple stream flowed over her neck and bosom.\n\nI was paralyzed--I moved not--I looked on horror-stricken.\n\nShe made one movement with her hand, and then it fell lifeless by her\nside. She gave one deep sigh, and all was over. I saw that she was dead,\nbut I wept not. I stood by, a miserable madman, my heart heaving with\nagony, but my eyes refusing to weep, and laughing that violent, horrible\nlaugh, that mockery of mirth which belongs only to the maniac's ravings.\n\nI stood by the couch--I bathed my burning forehead with her blood--I\nsaw that beautiful being cold and motionless, her eyes closed, and the\nlofty brow damp with the dews of death. I saw this and yet lived on.\n\nThere was stillness, and gloom, and death, around me, but I was not\nalone. I felt that creeping consciousness that my evil spirit was near.\nI raised my eyes and saw the phantom--the dark and hideous one; my old\ncompanion as standing by me--muttering and mocking at my grief. I shrank\nfrom the fiend.\n\nI drew closer to the loved form of her I adored. I took her cold hand\nand placed it on my burning brow. I can feel the death-like coldness now\nwhere that small hand lay. I closed my eyes and tried to pray; but\nfiendish shouts of laughter rang in my ears, and I felt that an _evil\nspirit_ was by my side. My whole frame quivered with suppressed agony. I\nturned. I saw it move; and the shadowless hand was raised as if to touch\nthe precious and costly form of her I loved. I can remember no more; all\nafter for some time was gloom and misery. * * *\n\nWild spirits are dancing around me, bearing in their arms the dear form\nof my Julia. Sometimes her voice breaks the stillness of my chamber in\nthe darkness of night, for I never sleep--my brain is _too hot for\nsleep_. Sometimes I am roused by feeling the softness of her light taper\nfingers on my brow, and then I start from my uneasy and wretched bed to\nlook for her once more; but instead of her I see my dark spirit the\ndemon, watching me with that untired eye, following me with that\nnoiseless step, that shadowless form, and then falling on my bed, I bury\nmy face in my pillow, and try to pray for peace, and for tears--but both\nare denied me.\n\nThe sun mocks me with his bright, clear, dancing beams speaking of life,\nand hope, and joy. It brings back the memory of that wretched day when I\nhad killed by my burning passions the only woman I had ever loved.\n\nShe was, indeed, the sun of my gloom; and, without her, I am as a\ncaptive in a darkened cell, through the gratings of which thoughts of\nher stream in, and make a dim twilight--a sad satisfaction. Oh! if I\nwere to be false to her, my soul would be a void; my memory, a curse; my\nheart, a heap of ashes.\n\nI see again, with terrible reality, that graceful form--that regal\nface--dead, yet smiling--as I last saw her in that curtained chamber,\nwith the sun shining in glory through the crimson drapery, and shedding\na warm glow on the inanimate features.\n\nEven now I see her. I see that last look of unsullied purity and fear. I\nfeel again that warm blood, as it trickled down and fell on my hands and\nface, as I knelt before her. It fell on my forehead, and I know that it\nis eating in, deeper and deeper, towards my brain.\n\nHer last words ring in my ears; her last smile is my beacon, my only ray\nof hope, luring me on towards a happier future.\n\nThere is a fire kindled within me that will dry up every thought but\nrecollection of her; for every circumstance connected with her is\nimpressed on my memory with a vivid distinctness.\n\nCan it be?--the thought sometimes occurs to me, with a balmy and\nconsoling power, like that fragrant wind from the Spicy Islands, which\nthe mariner feels blowing cool upon his brow, as he lies becalmed, in\nthe still noon, on the wide and desert sea? Can it be, that the devotion\nof a lifetime--such as my devotion has been--may be repaid by\nassociation in eternity?\n\nMay I dare to hope to live hereafter in the shadow of her glory? Shall\nwe meet again in that bright land?\n\nNo--the vision is too joyous for the poor maniac, her murderer. I shall\nsee her no more--we are separated for ever!\n\nHell--deep, deep hell--is the madman's portion; and heaven, that pure\nand distant clime, is thy resting-place for ever--thy radiant home--thy\npeaceful haven--my lost--my adored--my sainted Julia!\n\n\n\n\nTHE BURNS' FESTIVAL.\n\n\nScotland has of late years been exposed to perilous influences. Unused,\nfrom its older form of representation, to popular excitement, and\nstimulated by example from without, the nation threw itself headlong\ninto the revolutionary current which swept the whole empire at the\nperiod of Parliamentary Reform, and, with characteristic fervour, seemed\ninclined to riot in the novel element. Whenever symptoms of such a\ndisposition appear in the body politic, there is manifest danger that,\nin the new accession of power, the old and sacred landmarks may be\ndisregarded, and little heed be given to the mutual dependence and\ncommon interests of every class of society. Thus agitated and disturbed,\nthe Scottish people, once jealously national, and so proud of that\nnationality that it had passed into a byword throughout Europe, might\nhave lost their cohesive power, loosened the cord which bound the social\nrods together, and formed themselves into separate sections with\napparently hostile interests. Fortunately, however, there was a strong\ncounteracting influence. Even when the storm was wildest, and the clash\nof conflicting opinions most discordant, it was impossible to eradicate\nfrom the minds of any order the vast and stirring memories of the past.\nNew rights might, indeed, be claimed; but it was not alleged that there\nhad been any abuse of the old. Nothing had occurred to weaken the esteem\nwith which the lower ranks were accustomed to regard the ancient\naristocracy of the country; and accordingly, throughout the whole of\nthat protracted contest, fervid and determined as it was, there was less\nrancour shown than might have been expected in the course of so great a\npolitical change. As the excitement subsided, the kindly feeling, which\nnever had been extinguished, began more palpably to revive. Before the\nepoch of agitation approached, we were a peaceful and a happy people.\nThe peerage, the gentry, the yeomen, and the peasantry--all classes were\nbound together with the links of respect and of affection. The old\nhereditary attachment between the orders had not been broken. The poor\nman was proud of the noble, because the noble bore a name conspicuous in\nthe annals of his country; because he was the descendant of those who\nhad fought and died for Scotland, and who had identified their\nhonourable renown with hers; because he was a man every way worthy to\nbear the titles so gloriously achieved; and, more than all perhaps,\nbecause he loved and venerated the poor. And for that love and\nveneration the noble had ample grounds. Ancient as his race might be,\nthe yeomanry and peasantry of Scotland were yet as ancient in theirs.\nNot one step of honour could his fathers have gained without the help of\nthe fathers of those who were now living upon his hereditary soil; and\nthe old spell-words of the land were common to them both. Nor was there\nto be found in wide Europe a better or a braver race. They were\nindustrious, faithful, loyal; they were attached without servility,\nindependent without rudeness, and intelligent to a degree that excited\nthe admiration and the wonder of the stranger. No wonder that the mere\nthought of estrangement, in such a society as this, should have stricken\nthe bravest bosom with terror, and woe, and dismay! Yet so troublous was\nthe aspect of Europe then, that such fear was not utterly unfelt; and it\nwas the apprehension of that calamity, more than any other worldly\ncause, that dimmed the soul and darkened the spirit of that great and\ngood man, Sir Walter Scott, in his declining years; for all his large\naffections were bound up and entwined with the interests of Scotland,\nand, had the sacrifice been required of him, he would gladly have laid\ndown his life to avert from her the perils which he then foresaw.\n\nThese few remarks we cannot consider as inappropriate to our present\nsubject. We have once more been joyful spectators of a truly national\ngathering. Once more we have seen Scotsmen, of every grade and degree,\nassemble together without a tinge of party purpose, to do honour to the\nmemory of a poet who sprang from the ranks of the people, and who was\nheart and soul a Scotsman in his feelings, his inspiration, and, it may\nbe, in his errors and his prejudices also. It was a stirring and\nexciting spectacle, such as no other country could have exhibited--to\nbehold peer and senator, poet and historian and peasant--the great and\nthe small, the lettered and the simple of the land--unite, after fifty\nyears of silence, in deep and sincere homage to the genius of one humble\nman. Nor did they assemble there because his genius was greater than\nGod, in his bounty, had bestowed upon others, but because he had used it\nfor the glory and exaltation of his country; because he loved her with\nan ardour the most vivid and extreme; because he had shed the light\nentrusted to his charge both on the lofty dwelling and on the lowly\nhearth, but most brightly and cheeringly upon the latter, for that was\nhis peculiar charge. We feel assured that the events of that day, and\nthe sentiments which were then inspired and uttered, will produce a\nmarked effect upon the disposition of the country at large. It seemed as\nif all classes had spontaneously assembled to join hands above the grave\nof Robert Burns, and then and there to renew the vow of enduring\nreconciliation and love.\n\nWe shall now proceed to give a short account of the proceedings of the\nday. In our climate, the state of the weather on public occasions is\nalways regarded with anxiety; for enthusiasm, however warm, is apt to\nexpire beneath a deluge of northern rain. On the previous evening the\nsky promised well. A brilliant sunset and a warm wind seemed security\nfor a placid morrow; and although the glare of the great furnaces in the\nneighbourhood of Glasgow glowed somewhat ominously large as the night\nwore on, we retired to rest rather in hope than resignation. But dismal,\nindeed, was the prospect when we awoke. A vaporous grey mist had\nentirely usurped the heavens, and the plash of weary rain resounded\nthrough the pluvious metropolis of the west. Fortunately, we were not\nignorant of the fact, that Glasgow is under the peculiar tutelage of the\nPleiades; and accordingly we proceeded to the railway, trusting that\nmatters might mend so soon as we lost sight of the stupendous\nchimney-stalk of St Rollox. Notwithstanding the inclemency of the\nweather, and the early hour, every town, as we passed along, seemed in a\nstate of the greatest excitement. There were bands of music, deputations\nof mason lodges, and the rival brotherhood of Odd Fellows, with hundreds\nof men and women, all clad in holiday attire, awaiting the arrival of\nthe train at every station. It is a marvel to us, how half of these\nexpectants could have found their way to Ayr. Carriage after carriage\nwas linked to the already exorbitant train, until the engine groaned\naudibly, and almost refused to proceed. Still the rain continued to\nfall, and it was not until after we had left Irvine, and were rounding\nthe margin of the bay towards Ayr, that the sky brightened up and\ndisclosed the great panorama of the sea, with Ailsa and Arran looming in\nthe distance, and steamers from every direction ploughing their way into\nthe port. The streets of Ayr were swarming with people, and sounding\nwith the crash of music. There were arches on the bridge, flags\nstreaming from windows, and bells tolling from the steeples--symptoms of\na jubilee as great as if Royalty had descended unawares, and the whole\ndistrict had arisen to pay honour to its Queen. The inns were thronged\nto excess, and the waiters in absolute despair. What a multitude of\nsalmon must have died to furnish that morning's meal! Yet every face\nlooked bright and happy, as became those who had engaged in such a\npilgrimage. Then the burst of music became louder and more frequent, as\nband after band, preceding the trades and other public bodies, filed\npast towards the rendezvous of the great Procession. This was on what is\ncalled the Low Green; and the admirable arrangements made by the\ncommittee of management--of which Mr Ballantine of Castlehill was\nconvener, and Messrs Bone and Gray secretaries--were manifest. Mr\nThwaites undertook the marshaling of the whole. Here, first, the\ngrandeur of the National Festival was displayed, while the immense\nmultitudes that had come trooping in from all quarters stood\ncongregated in orderly muster, a mighty host, bound in unity by one\nsoul, stretching far and wide from the towers of Ayr to the sea.\nSuddenly, at signal given, the Procession began to deploy, in admirable\norder, with streaming banners and crashes of music, and shouts from the\naccompanying thousands that rent the sky; and we were warned that it was\ntime to proceed, if we wished to obtain a place upon the Platform\nerected on the banks of Doon.\n\nA unit in the stream of population, we skirted the noble race-course,\nand reached the Platform just before the head of the Procession had\narrived. It was erected in a magnificent situation. Behind was the\nmonument of Burns, and the sweet habitation of Mr Auld, with old Alloway\nKirk a little further off. Before it was the immense Pavilion erected\nfor the banquet, all gay with flags and streamers. To the right, were\nthe woods that fringe the romantic Doon, at that point concealed from\nsight; but not so the Old Bridge, which spans it, with its arch of\ntriumphal evergreen. Every beyond was studded with groups of\npeople, content to view the spectacle from afar. The Carrick hills\nreached far away beyond; and, on the other side, were the town and broad\nbay of Ayr, and Arran with all its mountains. But we had little leisure\nthen to look around us. On the Platform were collected many of the\nLadies and Gentlemen of the county--Sir David Hunter Blair; James\nCampbell, Esq. of Craigie; W. A. Cunninghame, Esq. of Fairlie; A. Boyle,\nEsq. of Shewalton, &c.; Archibald Hastie, Esq. M.P.; A. Buchanan, Esq.,\nCharles Neaves, Esq. Mr Sheriff Campbell, Mr Sheriff Bell, Mr\nCarruthers, &c. &c.; some of the most distinguished of those who had\ncome from afar, and conspicuous in front the surviving Kindred of Burns.\nThere stood, with his beautiful Countess, the noble and manly Eglinton,\n_preux chevalier_ of his day, and fitting representative of that ancient\nhouse of Montgomery, so famous in the annals and peerage of Scotland,\nand of France. There was the venerable and venerated Lord\nJustice-General Boyle, the President of the Scottish Courts, and chief\nmagistrate of the land, with the snows of more than seventy winters\nlying lightly and gracefully upon his head. There stood Wilson, never\nmore fitly in his place than here; for of the many who have interposed\nto shield the memory of Burns from detraction, he had spoken with the\nmost generous spirit and collected purpose, and came now to rejoice in\nthe common triumph. There, too, were Alison, the sound and strong\nhistorian; Chambers, whose delicate generosity to the relatives of\nBurns, independently of the services he has rendered to our national\nliterature, made him one of the fittest spectators of the scene; and a\nhost of other distinguished men, well and aptly representing the\naristocracy and the learning of the country. Many strangers, too, had\ncome to grace the festival; amongst whom, it may be allowed us to\nspecify the names of Mrs S. C. Hall, the charming authoress, and her\naccomplished husband. We looked in vain for some whose presence there\nwould have given an additional interest to the scene. We would fain have\nseen the poets of the sister countries represented by Wordsworth and\nMoore. That might not be; but their sympathies were not withheld.\n\nAmong that brilliant group, there stood an elderly female, dressed in\ndeep black, and three men, all past the meridian of life, with quiet,\nthoughtful looks, and unpretending aspect. These were the sister and the\nsons of Burns. His sister!--and half a century has wellnigh gone past\nsince the hot heart of the brother was stricken cold, and the manly\nmusic of his voice made dumb for ever! Was it too much to believe that,\nthrough these many long years of her earthly pilgrimage--sometimes, we\nfear, darkened by want and neglect--that sister had always clung to the\nmemory of the departed dead, in the hope that the day would arrive when\nhis genius should receive the homage of a new generation, to atone for\nthe apathy and coldness of that which had passed away? What emotions\nmust have thrilled the bosom of that venerable woman, as she gazed on\nthe stirring spectacle before her, and saw her lingering hopes far more\nthan thoroughly realized! What a glorious welcome, too, for the sons to\ntheir native land! They had left it--not quite as the poor man\ndoes--but with heavy difficulties before them. They had wrestled their\nway onwards through half the journey of life, and now, on their return,\nthey were greeted with a welcome which it were almost worth the\nstruggles of a life to obtain. All this they owed to their father; and\nhonoured among the honourable that day were the lineage and kindred of\nBurns.\n\nBeneath and around the Platform there were thousands already\ncongregated. If any one had wished to paint the character of the\nScottish peasantry in its loftiest and most endearing light, the\nsubjects were there before him. Old patriarchal men, on whose venerable\ntemples time had bleached the white locks of age to the softness of\nthose of infancy, stood leaning upon their grandchildren, proud, and yet\nwondering at the honours which were that day paid to him, whom, long,\nlong ago, reaching away through the vista of memory, they remembered to\nhave seen in their youth. So familiarized were they with his image, and\nthe glorious language he had uttered, that they had almost forgotten the\ngreatness and universality of his fame; and now, when brought forth from\ntheir cottages in the far glens and muirlands of the south, they could\nscarcely believe that the great, and gifted, and beautiful of the land,\nhad come together for no other purpose than to celebrate the genius of\ntheir old companion. But they were proud, as they well might be; for it\nwas a privilege even to have beheld him, and in that homage they\nrecognised and felt the tribute that was paid to their order. The\ninstinctive decency of Scottish feeling had accorded to these men a\nfitting and conspicuous place. Around them were the women of their\nfamilies of all ages--from the matron in her coif to the bashful maiden\nwith the snood--and even children; for few were left at home on that day\nof general jubilee. These, and a vast concourse of strangers, already\noccupied the ground.\n\nMeanwhile the Procession had wound its enormous length from Ayr along a\nroad almost choked up with spectators. Every wall and gate had its\nburden, and numerous Flibbertigibbets sat perched upon the branches of\nthe trees. The solitary constable of the burgh was not present to\npreserve order, or, if he was, his apparition was totally unrequired.\nThe old bell of Alloway Kirk was set in motion as the head of the column\nappeared, and continued ringing until all were past. The whole land was\nalive. Each road and lane poured forth its separate concourse to swell\nthe ranks of the great Procession. The weather, after one heavy final\nshower, cleared up; or, if not clear, resolved itself into that\nindescribable mixture of sunshine and cloud which sets off the beauties\nof the undulating landscape so well, light alternating with shadow, and,\non the ridges of the distant hills, contending radiance and gloom.\n\nOn they went, with banners flying and a perfect storm of music, across\nthe new Bridge of Doon, deploying along the road on the opposite side of\nthe river, and finally recrossing by the old bridge, from which they\nfiled past in front of the Platform. The order of the Procession was as\nfollows:--\n\n BAND OF THE 87TH FUSILIERS.\n Provost, Magistrates, Town-Council, and Trades of Ayr.\n\n FIVE BAGPIPERS IN HIGHLAND COSTUME.\n\n FARMERS AND SHEPHERDS.\n\n Dalrymple Burns's Club, with banners and music.\n Motto, \"Firm.\"\n\n KILWINNING BAND.\n Kilwinning Mother Lodge of Freemasons.\n\n CUMNOCK BAND.\n London Newmilns Lodge.\n\n IRVINE BAND.\n Troon Navigation Lodge.\n Girvan Masons.\n St James's, Tarbolton.\n St John's, Ayr.\n Thistle and Rose, Stevenston.\n St John's, Largs.\n Glasgow Star.\n\n ST ANDREW'S BAND.\n Royal Arch, Maybole.\n St Paul's, Ayr.\n St Andrew's, Ayr.\n St John's, Girvan.\n St James's, Kilmarnock.\n St Peter's, Galston.\n St John's, New Cumnock.\n Junior or Knights Templars, Maybole.\n\n SALTCOATS BAND.\n St John's, Dalry.\n\n KILBARCHAN BAND.\n St John's, Greenock.\n Shoemakers as follows:--\n Champion.\n British Prince and attendants.\n Indian Prince and Train.\n\n CATRINE BAND.\n King Crispin and Train.\n Souter Johnie, in character.\n Highland Chieftains.\n\n GREENOCK BAND.\n Lodge of Odd Fellows.\n\n BAND.\n Robert Burns's Lodge, Beith.\n\n AYR BAND.\n Banks of Ayr Lodge of Odd Fellows.\n Sir T. Makdougall Brisbane Lodge, Largs.\n Ancient Order of Foresters, Glasgow.\n Captain mounted, with Bow and Arrows.\n\n KILMARNOCK BAND.\n Kilmarnock Burns's Lodge of Foresters.\n Weavers from Maybole.\n\n MAYBOLE BAND.\n Tailors of Maybole.\n\n MAUCHLINE BAND.\n Boxmakers of Mauchline, with large Scotch Thistle, carried\n shoulder-high by Four men, and Banner, inscribed,\n\n \"I turn'd my weeder-clips aside,\n And spared the Symbol dear.\"\n\n The Party were on the Establishment of Messrs W. and A. Smith. The\n Thistle grew near to Mossgiel.\n\n Caledonian Union Odd Fellows, Dunlop. (Deputations of the Magistracy\n joined in the Procession from Dumbarton, Dunlop, Maybole, and Irvine.)\n\nThe effect of the Procession as seen from the Platform almost baffles\nthe power of description. The wailing of the bagpipes and the crash of\nthe bands were heard from the bosom of deep wood-thicket behind, long\nbefore the ranks became visible. At length, among the trees that skirted\nthe opposite banks, there was a glittering of lances, and a lifting of\nbanners, and a dark-growing line of men, in closest order, marching as\nif to battle. Gradually it flowed on, in continuous stream, file\nsucceeding to file without gap or intermission, until the head of the\ncolumn appeared recrossing by the Old Bridge, and winding up the road\ntowards the Platform; and still new banners rose up behind, and fresh\nstrains of music burst forth amidst the leafy screen. And now they\nreached the platform: lance and flag were lowered in honour of those who\nstood bareheaded above, and deafening were the cheers that ushered in\nthe arrival of the national pageant. The spectacle was most imposing,\nand must have conveyed to the minds of the strangers present a vivid\nimpression of the energy and enthusiasm so deeply implanted in the\nScottish character, and always so irresistibly manifested at the\ntouching of a national chord. The most interesting part of the\nProcession by far was the array of Farmers and Shepherds, the flower of\nthe west-country yeomanry, attired in the graceful plaid. Of that same\nbreed of men, of tall and compact mould and hardy sinew, was Robert\nBurns; nor is it possible to imagine any thing more animated than the\nappearance of those stalwart sons of the soil, as they lingered for a\nmoment before the platform, and looked with wistful eyes at the sons of\nthe Poet, if haply they might trace in their lineaments some resemblance\nto the features of him whom, from their infancy, they had learned to\nlove. Then came the Freemasons, and King Crispin with his train, and the\nArchers, and much more of old Scottish device, until there seemed no end\nto the flowing tide of population, all keen, and joyful, and exultant.\nBut the full burst of enthusiasm was reserved for the close. In the rear\nof all appeared an enormous Thistle borne shoulder high; and no sooner\nwas the national emblem in sight, than a universal and long-continued\ncheer burst forth from the many thousands who were now congregated in\nthe plain beyond. Alas, for that thistle! Though Burns, as the\ninscription bore,\n\n \"Had turn'd his weeder-clips aside,\n And spared the symbol dear,\"\n\nsuch was not the fate of the offspring plant. Scarcely had it reached\nthe platform, when Christopher North violently possessed himself of one\nbranch, the Lord Justice-General seized upon another, and in the\ntwinkling of an eye it was torn into fragments, and its rough leaves and\nrougher flowers displayed upon manly bosoms, from which it would have\nbeen difficult to wrest them again. So closed the Procession--but not\nthe gathering. Deafening were the cheers which followed for Burns--for\nhis Sons--for Professor Wilson--for Lord Eglinton; until the last\nremnant of reserve gave way, and a torrent of people swept forward to\nobtain, if possible, a pressure of their hands that were gladly and\ngratefully held forth. Descending from the Platform, we entered the\nmeadow-ground beyond, where the multitude were now assembled. One of the\nbands struck up the beautiful air--\"Ye banks and braes o' bonny Doon;\"\nand immediately the People, as if actuated by one common impulse, took\nup the strain, and a loftier swell of music never rose beneath the cope\nof heaven. We thought of the fine lines of Elliott--\n\n \"To other words, while forest echoes ring,\n 'Ye banks and braes o' bonny Doon,' they sing;\n And far below, the drover, with a start\n Awaking, listens to the well-known strain,\n Which brings Schehallion's shadow to his heart,\n And Scotia's loveliest vales: then sleeps again,\n And dreams on Loxley's banks of Dunsinane.\"\n\nFew could abstain from tears as the last glorious note died solemnly\naway into the skies. We looked down from the top of the pavilion-stairs\nupon the vast multitude beneath. There could not have been less than\n80,000 souls collected upon the ground. Of all that mighty mass, not\none man had thrown discredit upon the harmony and order of the day.\nEvery face glowed with happiness and congratulation, as if conscious\nthat a good work had been done, and that the nation had at length\ndischarged the duty which she owed to one of her most gifted sons.\n\n\nTHE BANQUET.\n\nThe company began to enter the Pavilion almost immediately after the\nclose of the Procession, and the chair was taken about two o'clock. The\nPavilion was erected in a field of twenty-two acres, adjoining to the\nMonument, and was a magnificent building. It measured not less than 120\nfeet by 110, forming nearly a perfect square. The roof, supported by two\nrows of pillars, was covered with waterproof felt, and the building\ninside was lined with white cloth, festooned with crimson. In the centre\nof the roof was a radiation of the same colours. The tables and seats\nwere arranged in parallel lines from the head to the foot of the\napartment, rising with a gentle inclination from the middle on both\nsides. At each end there was an elevated table for the Chairman,\nCroupier, and their respective supporters; and on the two remaining\nsides of the square there were _vis-a-vis_ galleries for the\ninstrumental band and glee-singers, a pianoforte for the accompaniment\nto Mr Templeton being placed in front of the latter, at which Mr Blewitt\ntook his station. Mr Templeton, between the speeches, sang, with great\npower and sweetness, appropriate songs from Burns; and Mr Blewitt's\nperformance was admirable. Mr Wilson came from Paris to the Festival;\nbut unfortunately was prevented by severe illness from delighting the\nassembly with his exquisite strains. The hall was lighted by twenty-two\nglass windows, shaded with white cloth. The chairman and croupier's\nseats were of oak, made of the rafters of Alloway Kirk; and several\nsplendid silver vases decorated their tables. The hall was seated to\naccommodate 2000 persons, and was entirely filled, although not\ninconveniently crowded.\n\nThe distinguishing feature of the pavilion was the number of ladies who\nwere present. A great room exclusively filled with men, is at best a\ndull and sombre spectacle; and so far from social, that it always\nconveys to us a gross idea of selfishness. The mere scenic effect on\nthis occasion was immensely heightened by the adoption of the polite\nrule; nor can it be doubted that the tone of the meeting underwent a\nsimilar improvement.\n\nThe Chairman, the Right Hon. the Earl of Eglinton, was supported on the\nright by Robert Burns, Esq., late of the Stamps and Taxes, Somerset\nHouse, London, eldest son of the poet; Major Burns, youngest son of the\npoet; Miss Begg, niece of the poet; Henry Glassford Bell, Esq.,\nSheriff-Substitute of Lanarkshire; Rev. Mr Cuthill, Ayr; Mr Robert Burns\nBegg, teacher, Kinross, nephew of the poet; Miss Begg, the younger niece\nof the poet; Mr and Mrs Thomson of Dumfries, (the latter the Jessie\nLewars of the bard, who tended his deathbed;)--on the left, by Colonel\nBurns, second son of the poet; Mrs Begg, sister of the poet; Sir John\nM'Neill, Bart., late Plenipotentiary to the Court of Persia; the Right\nHon. Lord Justice-General; the Countess of Eglinton; Sir D. H. Blair,\nBart., of Blairquhan. The Croupier, Professor Wilson, was supported on\nthe right by Archibald Alison, Esq., Sheriff of Lanarkshire, and author\nof the History of Europe; Colonel Mure of Caldwell, author of Travels in\nGreece; William E. Aytoun, Esq., Advocate; A. Hastie, Esq., M.P. for\nPaisley; Jas. Oswald, Esq., M.P. for Glasgow;--on the left by Sir James\nCampbell, Glasgow; Provost Miller, Ayr; James Ballantine, Esq. of\nCastlehill; Charles Mackay, Esq., London; James Campbell, Esq. of\nCraigie.\n\nThe Rev. Mr CUTHILL of Ayr asked the blessing.\n\nThe Earl of EGLINTON, after the usual loyal toasts, rose and spoke as\nfollows:--Ladies and gentlemen, The subject of the toast which I have\nnow the honour to bring before your notice, is one of such paramount\nimportance on this occasion, and is so deeply interesting, not only to\nthose whom I am addressing, but to all to whom genius is dear, that I\ncould have wished that it had been committed to more worthy hands; more\nespecially when I see the great assemblage collected here--the\ndistinguished persons who grace our board to-day. It is only because I\nconceive that my official position renders me the most formal and\nfitting, though most inefficient, mouthpiece of the inhabitants of this\ncounty, that I have ventured to present myself before you on this\noccasion, and to undertake the onerous, though most gratifying, duty of\nproposing, in such an assemblage, the thrilling toast--\"The Memory of\nBurns.\" This is not a meeting for the purpose of recreation and\namusement--it is not a banquet at which a certain number of toasts are\nplaced on paper, which must be received with due marks of\napprobation--it is the enthusiastic desire of a whole people to pay\nhonour to their greatest countryman. It is the spontaneous outpouring of\na nation's feeling towards the illustrious dead, and the wish to extend\nthe hand of welcome and of friendship to those whom he has left behind.\nHere on the very spot where the Poet first drew breath, on the very\nground which his genius has hallowed, beside the Old Kirk which his\nverse has immortalized, beneath the monument which an admiring and\nrepentant people have raised to his memory, we meet after the lapse of\nyears, to pay our homage at the shrine of genius. The master mind who\nhas sung the \"Isle of Palms\"--who has revelled in the immortal\n\"Noctes\"--and who has already done that justice to the memory of Burns\nwhich a brother poet alone can do--Christopher himself is here, anxious\nto pay his tribute of admiration to a kindred spirit. The historian who\nhas depicted, with a Gibbon's hand, the eventful period of the French\nempire, and the glorious victories of Wellington, is here--a Clio, as it\nwere, offering a garland to Erato. The distinguished head of the\nScottish bench is here. In short, every town and every district, every\nclass and every age, has come forward to pay homage to their poet. The\nhonest lads whom he so praised, and whose greatest boast it is that they\nbelong to the land of Burns, are here. The fair lasses whom he so loved\nand sung, have flocked hither to justify, by their loveliness, their\npoet's words. While the descendant of those who dwelt in the \"Castle o'\nMontgomerie,\" feels himself only too highly honoured by being permitted\nto propose the memory of him who wandered then unknown along the banks\nof Fail. How little could the pious old man who dwelt in yon humble\ncottage, when he read the \"big ha' bible\"--\"his lyart haffets wearing\nthin and bare\"--have guessed that the infant prattling on his knee was\nto be the pride and admiration of his country; that that infant was to\nbe enrolled a chief among the poetic band; that he was to take his place\nas one of the brightest planets that glitter round the mighty sun of the\nBard of Avon! In originality second to none, in the fervent expression\nof deep feeling, and in the keen perception of the beauties of nature,\nequal to any who ever reveled in the bright fairyland of poesy, well may\nwe rejoice that Burns is our own--well may we rejoice that no other land\ncan claim to be the birthplace of our Homer except the hallowed spot on\nwhich we stand! Oh! that he could have foreseen the futurity of fame he\nhas created to himself--oh! that he could have foreseen this day, when\nthe poet and the historian, the manly and the fair, the peer and the\npeasant, vie with each other in paying their tribute of admiration to\nthe untaught but mighty genius whom we hail as the first of Scottish\npoets! It might have alleviated the dreary days of his sojourn at\nMossgiel--it might have lightened the last hours of his pilgrimage upon\nearth. And well does he deserve such homage. He who portrayed the\n\"Cottar's Saturday Night\" in strains that are unrivaled in simplicity,\nand yet fervour--in solemnity, and in truth--He who breathed forth the\npatriotic words which tell of the glories of Wallace, and immortalize\nalike the poet and the hero--He who culled inspiration from the modest\ndaisy, and yet thundered forth the heroic strains of \"The Song of\nDeath\"--He who murmured words which appear the very incarnation of\npoetry and of love, and yet hurled forth the bitterest shafts of\nsatire--a Poet by the hand of nature, despising, as it were, the rules\nof art, and yet triumphing over those very rules which he set at\nnought--at whose name every Scottish heart beats high--whose name has\nbecome a household word in the cottage as in the palace--to whom shall\nwe pay our homage, of whom shall we be proud, if it is not our own\nimmortal Burns? But I feel that I am detaining you too long. I feel\nthat, in the presence of a Wilson and an Alison, I am not a fit person\nto dilate upon the genius of Burns. I am but an admirer of the poet like\nyourselves. There are those present who are brother poets and kindred\ngeniuses--men who, like Burns, have gained for themselves a glorious\nimmortality. To them will I commit the grateful task of more fully\ndisplaying before you, decked out by their eloquence, the excellences of\nthe poet, the genius of the man, and to welcome his sons to the land of\ntheir father: and I will only ask you, in their presence--on the ground\nwhich his genius has rendered sacred--on the \"banks and braes o' bonny\nDoon\"--to join with me in drinking an overflowing bumper, and giving it\nevery expression of enthusiasm which you can, to \"The Memory of Burns!\"\n\nMr ROBERT BURNS rose along with his brothers, and was received with\nenthusiastic cheering. He said--My lord, ladies, and gentlemen, Of\ncourse it cannot be expected, at a meeting such as the present, that the\nsons of Burns should expatiate on the merits and genius of their\ndeceased father. Around them are an immense number of admirers, who, by\ntheir presence here this day, bear a sufficient testimony to the opinion\nin which they hold his memory, and the high esteem in which they hold\nhis genius. In the language of the late Sir Christopher Wren, though\nvery differently applied, the sons of Burns can say, that to obtain a\nliving testimony to their father's genius they have only to look around\nthem. I beg, in name of my aunt, brothers, and myself, to return our\nheartfelt and grateful thanks for the honour that has this day been paid\nto my father's memory.\n\nPROFESSOR WILSON then rose and said--Were this Festival but to\ncommemorate the genius of Burns, and it were asked, what need now for\nsuch commemoration, since his fame is coextensive with the literature of\nthe land, and enshrined in every household? I might answer, that\nalthough admiration of the poet be wide as the world, yet we, his\ncompatriots, to whom he is especially dear, rejoice to see the universal\nsentiment concentered in one great assemblage of his own people: that we\nmeet in thousands and tens of thousands to honour him, who delights each\nsingle one of us at his own hearth. But this commemoration expresses,\ntoo, if not a profounder, a more tender sentiment; for it is to welcome\nhis sons to the land he has illustrated, so that we may at once indulge\nour national pride in a great name, and gratify in filial hearts the\nmost pious of affections. There was, in former times, a custom of\ncrowning great poets. No such ovation honoured our bard, though he too\ntasted of human applause, felt its delights, and knew the trials that\nattend it. Which would Burns himself have preferred, a celebration like\nthis in his lifetime, or fifty years after his death? I venture to say,\nhe would have preferred the posthumous as the finer incense. The honour\nand its object are then seen in juster proportion; for death confers an\nelevation which the candid soul of the poet would have considered, and\nsuch honour he would rather have reserved for his manes, than have\nencountered it with his living infirmities. And could he have foreseen\nthe day, when they for whom at times he was sorely troubled, should,\nafter many years of separation, return to the hut where himself was\nborn, and near it, within the shadow of his monument, be welcomed for\nhis sake by the lords and ladies of the land; and--dearer thought still\nto his manly breast--by the children and the children's children of\npeople of his own degree, whose hearts he sought to thrill by his first\nvoice of inspiration; surely had the Vision been sweeter to his soul\nthan even that immortal one, in which the Genius of the Land bound the\nholly round his head, the lyric crown that it will wear for ever.\n\nOf his three Sons sitting here, one only can remember their father's\nface--those large lustrous eyes of his, so full of many meanings, as\nthey darkened in thought, melted in melancholy, or kindled in mirth, but\nnever turned on his children, or on their excellent mother, but with one\nof tender or intense affection. That son may even on this day have\nremembrance of his father's head, with its dark clusters not unmixed\nwith gray, and those eyes closed, lying upon the bed of death. Nor,\nshould it for a moment placidly appear, is such image unsuitable to this\nfestival. For in bidding welcome to his sons to their father's land, I\nfeel that, while you have conferred on me a high honour, you have\nlikewise imposed on me a solemn duty; and, however inadequately I may\ndischarge it, I trust that in nought shall I do any violence to the\nspirit either of humanity or of truth.\n\nI shall speak reverently of Burns's character in hearing of his sons;\nbut not even in their hearing must I forget what is due always to\nestablished judgment of the everlasting right. Like all other mortal\nbeings; he had his faults--great even in the eyes of men--grievous in\nthe eyes of Heaven. Never are they to be thought of without sorrow, were\nit but for the misery with which he himself repented them. But as there\nis a moral in every man's life, even in its outward condition\nimperfectly understood, how much more affecting when we read it in\nconfessions wrung out by remorse from the greatly gifted, the gloriously\nendowed! But it is not his faults that are remembered here--assuredly\nnot these we meet to honour. To deny error to be error, or to extenuate\nits blame, _that_ makes the outrage upon sacred truth; but to forget\nthat it exists, or if not wholly so, to think of it along with that\nunder-current of melancholy emotion at all times accompanying our\nmeditations on the mixed characters of men--_that_ is not only\nallowable, but it is ordered--it is a privilege dear to humanity--and\nwell indeed might he tremble for himself who should in this be deaf to\nthe voice of nature crying from the tomb.\n\nAnd mark how graciously in this does time aid the inclinations of\ncharity! Its shadows soften what they may not hide. In the distance,\ndiscordances that once jarred painfully on our ears are now\nundistinguishable--lost in the music sweet and solemn, that comes from\nafar with the sound of a great man's name. It is consolatory to see,\nthat the faults of them whom their people honour grow fainter and\nfainter in the national memory, while their virtues wax brighter and\nmore bright; and if injustice have been done to them in life, (and who\nnow shall dare to deny that cruelest injustice was done to Burns?) each\nsucceeding generation becomes more and more dutiful to the\ndead--desirous to repair the wrong by profounder homage. As it is by his\nvirtues that man may best hope to live in the memory of man, is there\nnot something unnatural, something monstrous, in seeking to eternize\nhere below, that of which the proper doom is obscurity and oblivion? How\nbeneficent thus becomes the power of example! The good that men do then\nindeed \"lives after them\"--all that was ethereal in their being alone\nsurvives--and thus ought our cherished memories of our best men--and\nBurns was among our best--to be invested with all consistent\nexcellences; for far better may their virtues instruct us by the love\nwhich they inspire, than ever could their vices by aversion.\n\nTo dwell on the goodnesses of the great shows that we are at least\nlovers of virtue--that we may ourselves be aspiring to reach her serene\nabodes. But to dwell on their faults, and still more to ransack that we\nmay record them, _that_ is the low industry of envy, which, grown into a\nhabit, becomes malice, at once hardening and embittering the heart.\nSuch, beyond all doubt, in the case of our great poet, was the source of\nmany \"a malignant truth and lie,\" fondly penned, and carefully corrected\nfor the press, by a class of calumniators that may never be extinct;\nfor, by very antipathy of nature, the mean hate the magnanimous, the\ngroveling them who soar. And thus, for many a year, we heard \"souls\nignoble born to be forgot\" vehemently expostulating with some puny\nphantom of their own heated fancy, as if _it_ were the majestic shade of\nBurns evoked from his Mausoleum for contumely and insult.\n\nOften, too, have we been told by persons somewhat presumptuously\nassuming the office of our instructors, to beware how we suffer our\nadmiration of genius to seduce us from our reverence of virtue. Never\ncease to remember--has been still their cry--how far superior is moral to\nintellectual worth. Nay, they have told us that they are not akin in\nnature. But akin they are; and grief and pity 'tis that ever they should\nbe disunited. But mark in what a hateful, because hypocritical spirit,\nsuch advices as these have not seldom been proffered, till salutary\ntruths were perverted by misapplication into pernicious falsehoods. For\nthese malignant counsellors sought not to elevate virtue, but to degrade\ngenius; and never in any other instance have they stood forth more\nglaringly self-convicted of the most wretched ignorance of the nature\nboth of the one and the other, than in their wilful blindness to so many\nof the noblest attributes of humanity in the character of Burns. Both\ngifts are alike from heaven, and both alike tend heavenward. Therefore we\nlament to see genius soiled by earthly stain; therefore we lament to see\nvirtue, where no genius is, fall before the tempter. But we, in our own\nclear natural perceptions, refuse the counsels of those who with the very\nbreath of their warning would blight the wreath bound round the heads of\nthe Muses' sons by a people's gratitude--who, in affected zeal for\nreligion and morality, have so deeply violated the spirit of both, by\nvile misrepresentations, gross exaggerations, and merciless denunciations\nof the frailties of our common nature in illustrious men--men who, in\nspite of their aberrations, more or less deplorable, from the right path,\nwere not only in their prevailing moods devout worshippers of virtue, but\nin the main tenor or their lives exemplary to their brethren. And such a\nman was Burns. In boyhood--youth--manhood--where such peasant as he? And\nif in trouble and in trial, from which his country may well turn in\nself-reproach, he stood not always fast, yet shame and sin it were, and\nindelible infamy, were she not _now_ to judge his life as Christianity\ncommands. Preyed upon, alas! by those anxieties that pierce deepest into\nthe noblest hearts--anxieties for the sakes--even on account of the very\nmeans of subsistence--of his own household and his own hearth--yet was he\nin his declining, shall we call them disastrous years, on the whole\nfaithful to the divine spirit with which it had pleased Heaven to endow\nhim--on the whole obedient to its best inspirations; while he rejoiced to\nillumine the paths of poverty with light which indeed was light on\nheaven, and from an inexhaustible fancy, teeming to the genial warmth of\nthe heart in midst of chill and gloom, continued to the very last to\nstrew along the weary ways of this world flowers so beautiful in their\nfreshness, that to eyes too familiar with tears they looked as if dropped\nfrom heaven.\n\nThese are sentiments with which I rejoice to hear the sympathy of this\ngreat assemblage thus unequivocally expressed--for my words but awaken\nthoughts lodged deep in all considerate hearts. For which of us is there\nin whom, known or unknown, alas! there is not much that needs to be\nforgiven? Which of us that is not more akin to Burns in his fleshly\nfrailties then in his diviner spirit? That conviction regards not merely\nsolemn and public celebrations of reverential memory--such as this; it\npervades the tenor of our daily life, runs in our heart's-blood, sits at\nour hearths, wings our loftiest dreams of human exaltation. How, on this\nearth, could we love, or revere, or emulate, if, in our contemplation of\nthe human being, we could not sunder the noble, the fair, the gracious,\nthe august, from the dregs of mortality, from the dust that hangs\nperishably about him the imperishable? We judge in love, that in love we\nmay be judged. At our hearthsides, we gain more than we dared desire, by\nmutual mercy; at our hearthsides, we bestow and receive a better love,\nby this power of soft and magnanimous oblivion. We are ourselves the\ngainers, when thus we honour the great dead. _They_ hear not--_they_\nfeel not, excepting by an illusion of our own moved imaginations, which\nfill up chasms of awful, impassable separation; but _we_ hear--_we_\nfeel; and the echo of the acclaim which hills and skies have this day\nrepeated, we can carry home in our hearts, where it shall settle down\ninto the composure of love and pity, and admiration and gratitude, felt\nto be due for ever to our great poet's shade.\n\nIn no other spirit could genius have ever dared, in elegies and hymns,\nto seek to perpetuate at once a whole people's triumph, and a whole\npeople's grief, by celebration of king, sage, priest, or poet, gone to\nhis reward. From the natural infirmities of his meanest subject, what\nKing was ever free? Against the golden rim that rounds his mortal\ntemples come the same throbbings from blood in disease or passion\nhurrying from heart to brain, as disturb the aching head of the poor\nhind on his pallet of straw. But the king had been a guardian, a\nrestorer, a deliverer; therefore his sins are buried or burned with his\nbody; and all over the land he saved, generation after generation\ncontinues to cry aloud--\"O king, live for ever!\" The Sage who, by long\nmeditation on man's nature and man's life, has seen how liberty rests on\nlaw, rights on obligations, and that his passions must be fettered, that\nhis will be free--how often has he been overcome, when wrestling in\nagony with the powers of evil, in that seclusion from all trouble in\nwhich reverent admiration nevertheless believes that wisdom for ever\nserenely dwells! The Servant of God, has he always kept his heart pure\nfrom the world, nor ever held up in prayer other than spotless hands? A\nhumble confession of his own utter unworthiness would be his reply alike\nto scoffer and to him who believes. But, unterrified by plague and\npestilence, he had carried comfort into houses deserted but by sin and\ndespair; or he had sailed away, as he truly believed for ever, to savage\nlands, away from the quiet homes of Christian men--among whom he might\nhave hoped to lead a life of peace, it may be of affluence and\nhonour--for his Divine Master's sake, and for sake of them sitting in\ndarkness and in the shadow of death. Therefore his name dies not, and\nall Christendom calls it blest. From such benefactors as these there may\nseem to be, but there is not, a deep descent to them who have done their\nservice by what one of the greatest of them all has called \"the vision\nand the faculty divine\"--them to whom have been largely given the powers\nof fancy and imagination and creative thought, that they might move\nmen's hearts, and raise men's souls, by the reflection of their own\npassions and affections in poetry, which is still an inspired speech.\nNor have men, in their judgment of the true Poets, dealt otherwise with\nthem than with patriot kings, benign legislators, and holy priests.\nThem, too, when of the highest, all nations and ages have reverenced in\ntheir gratitude. Whatever is good and great in man's being seems\nshadowed in the name of Milton; and though he was a very man in the\nstorms of civil strife that shook down the throne at the shedding of the\nblood of kings, nevertheless, we devoutly believe with Wordsworth, that\n\n \"His soul was like a star, and dwelt apart.\"\n\nBut not of such as he only, who \"in darkness, and with danger compassed\nround,\" soared \"beyond this visible diurnal sphere,\" and whose song was\nof mercy and judgment, have men wisely resolved to dwell only on what is\npure and high and cognate with their thoughts of heaven. Still, as we\nkeep descending from height to height in the regions of song, we desire\nto regard with love the genius that beautifies wherever it settles down;\nand, if pity will steal in for human misfortunes, or for human frailties\nreproach, our love suffers no abatement, and religious men feel that\nthere is piety in pilgrimage to such honoured graves. So feel we now at\nthis commemoration. For our Poet we now claim the privilege, at once\nbright and austere, of death. We feel that our Burns is brought within\nthe justification of all celebrations of human names; and that, in thus\nhonouring his memory, we virtuously exercise the imaginative rights of\nenthusiasm owned by every people that has produced its great men.\n\nAnd with a more especial propriety do we claim this justice in our\ntriumphal celebration of poets, who, like Burns, were led by the\ncharacter of their minds to derive the matter and impulse of their song,\nin a stricter sense, from themselves. For they have laid bare to all\neyes many of their own weaknesses, at the side of their higher and purer\naspirations. Unreserved children of sincerity, by the very\nopen-heartedness which is one great cause of their commanding power, and\ncontagiously diffuses every zealous affection originating in their\nnobility of nature--by this grown to excess, made negligent of\ninstinctive self-defence, and heedless of misconstruction, or overcome\nby importunate and clinging temptations--to what charges have they not\nbeen exposed from that proneness to disparaging judgments so common in\nlittle minds! For such judgments are easy indeed to the very lowest\nunderstandings, and regard things that are visible to eyes that may\nseldom have commerced with things that are above. But they who know\nBurns as we know him, know that by this sometimes unregulated and\nunguarded sympathy with all appertaining to his kind, and especially to\nhis own order, he was enabled to receive into himself all modes of their\nsimple, but not undiversified life, so that his poetry murmurs their\nloves and joys from a thousand fountains. And suppose--which was the\ncase--that this unguarded sympathy, this quick sensibility, and this\nvivid capacity of happiness which the moment brings, and the frankness\nof impulse, and the strength of desire, and the warmth of blood, which\nhave made him what he greatly is, which have been fire and music in his\nsong, and manhood, and courage, and endurance, and independence in his\nlife, have at times betrayed or overmastered him--to turn against him\nall this self-painting and self-revealing, is it not ungrateful,\nbarbarous, inhuman? Can he be indeed a true lover of his kind, who would\nrecord in judgment against such a man words that have escaped him in the\nfervour of the pleading designed to uphold great causes dear to\nhumanity?--who would ignobly strike the self-disarmed?--scornfully\ninsult him who, kneeling at the Muses' confessional, whispers secrets\nthat take wings and fly abroad to the uttermost parts of the earth? Can\nthey be lovers of the people who do so? who find it in their hearts thus\nto think, and speak, and write of Robert Burns?--He who has reconciled\npoverty to its lot, toil to its taskwork, care to its burden--nay, I\nwould say even--grief to its grave? And by one Immortal Song has\nsanctified for ever the poor man's Cot--by such a picture as only\ngenius, in the inspiring power of piety, could have painted; has given\nenduring life to the image--how tender and how true!--of the Happy Night\npassing by sweet transition from this worky world into the Hallowed Day,\nby God's appointment breathing a heavenly calm over all Christian\nregions in their rest--nowhere else so profoundly--and may it never be\nbroken!--as over the hills and valleys of our beloved, and yet religious\nland!\n\nIt cannot be said that the best biographers of Burns, and his best\ncritics, have not done, or desired to do, justice to his character as\nwell as to his genius; and, according as the truth has been more\nentirely and fearlessly spoken, has he appeared the nobler and nobler\nman. All our best poets, too, have exultingly sung the worth, while they\nmourned the fate of him, the brightest of the brotherhood. But above,\nand below, and round about all that they have been uttering, has all\nalong been heard a voice, which they who know how to listen for it can\nhear, and which has pronounced a decision in his favour not to be\nreversed; for on earth it cannot be carried to a higher tribunal. A\nvoice heard of old on great national emergencies, when it struck terror\ninto the hearts of tyrants, who quaked, and quailed, and quitted for aye\nour land before \"the unconquered Caledonian spear\"--nor, since our union\nwith noblest England, ever slack to join with her's and fervid Erin's\nsons, the thrice-repeated cry by which battle-fields are cleared; but\nhappier, far happier to hear, in its low deep tone of peace. For then it\nis like the sound of distant waterfalls, the murmur of summer woods, or\nthe sea rolling in its rest. I mean the Voice of the People of\nScotland--the Voice of her Peasantry and her Trades--of all who earn\ntheir bread by the sweat of their brow--her Working Men.\n\nI presume not to draw their character. But this much I will say, that in\nthe long run they know whom it is fitting they should honour and love.\nThey will not be dictated to in their choice of the names that with\nthem shall be household words. Never, at any period of their history,\nhave they been lightly moved; but, when moved, their meaning was not to\nbe mistaken; tenacious their living grasp as the clutch of death; though\nforce may wrench the weapon from their hands, no force can wrench the\nworship from their hearts. They may not be conversant with our written\nannals; but in our oral traditions they are familiar with historic\ntruths--grand truths conceived according to the People's idea of their\nown national mind, as their hearts have kindled in imagination of heroic\nor holy men. Imaginary but real--for we all believe that men as good, as\nwise, as brave, have been amongst us as ever fancy fabled for a people's\nreverence. What manner of men have been their darlings? It would be hard\nto say; for their love is not exclusive--it is comprehensive. In the\nnational memory live for ever characters how widely different!--with all\nthe shades, fainter or darker, of human infirmity! For theirs is not the\nsickly taste that craves for perfection where no frailties are. They do\nnot demand in one and the same personage inconsistent virtues. But they\ndo demand sincerity, and integrity, and resolution, and independence,\nand an open front, and an eye that fears not to look in the face of\nclay! And have not the grave and thoughtful Scottish people always\nregarded with more especial affection those who have struggled with\nadversity--who have been tried by temptations from without or from\nwithin--now triumphant, now overcome--but, alike in victory or defeat,\ntestifying by their conduct that they were animated by no other desire\nso steadily as by love of their country and its people's good? Not those\nwho have been favourites of fortune, even though worthy of the smiles in\nwhich they basked; but those who rose superior to fortune, who could not\nfrown them down. Nor have they withheld their homage from the\nunfortunate in this world of chance and change, if, in abasement of\ncondition, by doing its duties they upheld the dignity of their own\nnature, and looked round them on their honest brethren in poverty with\npride.\n\nAnd how will such a people receive a great National Poet? How did they\nreceive Burns? With instant exultation. At once, they knew of\nthemselves, before critics and philosophers had time to tell them, that\na great Genius of their own had risen, and they felt a sudden charm\ndiffused over their daily life. By an inexplicable law, humour and\npathos are dependent on the same constitution of mind; and in his Poems\nthey found the very soul of mirth, the very soul of sadness, as they\nthought it good with him to be merry, or to remember with him, \"that man\nwas made to mourn.\" But besides what I have said of them, the people of\nScotland hold in the world's repute--signally so--the name of a\nreligious people. Many of them, the descendants of the old covenanters,\nheirs of the stern zeal which took up arms for the purity of the\nnational faith--still tinged, it may be, by the breath of the flame that\nthen passed over the land--retain a certain severity of religious\njudgment in questions of moral transgression, which is known to make a\npart of hereditary Scottish manners--especially in rural districts,\nwhere manners best retain their stamp. But the sound natural\nunderstanding of the Scottish peasant, I use the liberty to say, admits,\nto take their place at the side of one another, objects of his liberal\nand comprehensive regard, which might appear, to superficial observation\nand shallow judgment, to stand upon such different grounds, as that the\napprobation of the one should exclude the admiration of the other. But\nnot so. Nature in him is various as it is vigorous. He does not, with an\nover-jealous scrutiny, vainly try to reduce into seeming consistency\naffections spontaneously springing from many sources. Truth lies at the\nbottom; and, conscious of truth, he does not mistrust or question his\nown promptings. An awful reverence, the acknowledgment of a Law without\nappeal or error--Supreme, Sacred, Irresistible--rules in his judgment\nof other men's actions, and of his own. Nevertheless, under shelter and\nsanction of that rule, he feels, loves, admires, like a man. Religion\nhas raised and guards in him--it does not extinguish--the natural human\nheart. If the martyrs of his worship to him are holy--holy, too, are his\ncountry's heroes. And holy her poets--if such she have--who have\nsung--as during his too short life above them all sang Burns--for\nScotland's sake. Dear is the band that ties the humbly educated man to\nthe true national poet. To many in the upper classes he is, perhaps, but\none among a thousand artificers of amusement who entertain and scatter\nthe tedium of their idler hours. To the peasant the book lies upon his\nshelf a household treasure. There he finds depicted himself--his own\nworks and his own ways. There he finds a cordial for his drooping\nspirits, nutriment for his wearied strength. Burns is his brother--his\nhelper in time of need, when fretfulness and impatience are replaced\nwith placidity by his strains, or of a sudden with a mounting joy. And\nfar oftener than they who know not our peasantry would believe, before\ntheir souls awakened from torpor he is a luminous and benign presence in\nthe dark hut; for, in its purity and power, his best poetry is felt to\nbe inspired, and subordinate to the voice of heaven.\n\nAnd will such a people endure to hear their own Poet wronged? No, no.\nThink not to instruct _them_ in the right spirit of judgment. They have\nread the Scriptures, perhaps, to better purpose than their revilers, and\nknow better how to use the lessons learned there, applicable alike to us\nall--the lessons, searching and merciful, which proscribe mutual\njudgment amongst beings, all, in the eye of absolute Holiness and Truth,\nstained, erring, worthless: And none so well as aged religious men in\nsuch dwellings know, from their own experience, from what they have\nwitnessed among their neighbours, and from what they have read of the\nlives of good and faithful servants, out of the heart of what moral\nstorms and shipwrecks, that threatened to swallow the strong swimmer in\nthe middle passage of life, has often been landed safe at last, the\nrescued worshipper upon the firm land of quiet duties, and of years\nexempt from the hurricane of the passions! Thus thoughtfully guided in\ntheir opinion of him, who died young--cut off long before the period\nwhen others, under the gracious permission of overruling mercy, have\nbegun to redeem their errors, and fortified perhaps by a sacred office,\nto enter upon a new life--they will for ever solemnly cherish the memory\nof the Poet of the Poor. And in such sentiments there can be no doubt\nbut that all his countrymen share; who will, therefore rightfully hold\nout between Burns and all enemies a shield which clattering shafts may\nnot pierce. They are proud of him, as a lowly father is proud of an\nillustrious son. The rank and splendour attained reflects glory down,\nbut resolves not, nor weakens one single tie.\n\nAy, for many a deep reason the Scottish people love their own Robert\nBurns. Never was the personal character of poet so strongly and\nendearingly exhibited in his song. They love him, because he loved his\nown order, nor ever desired for a single hour to quit it. They love him,\nbecause he loved the very humblest condition of humanity, where every\nthing good was only the more commended to his manly mind by\ndisadvantages of social position. They love him, because he saw with\njust anger, how much the judgments of \"silly coward man\" are determined\nby such accidents, to the neglect or contempt of native worth. They love\nhim for his independence. What wonder! To be brought into contact with\nrank and wealth--a world inviting to ambition, and tempting to a\nthousand desires--and to choose rather to remain lowly and poor, than\nseek an easier or a brighter lot, by courting favour from the rich and\ngreat--was a legitimate ground of pride, if any ground of pride be\nlegitimate. He gave a tongue to this pride, and the boast is inscribed\nin words of fire in the Manual of the Poor. It was an exuberant feeling,\nas all his feelings were exuberant, and he let them all overflow. But\nsometimes, forsooth! he did not express them in sufficiently polite or\ncourteous phrase! And that too was well. He stood up not for himself\nonly, but for the great class to which he belonged, and which in his\ndays--and too often in ours--had been insulted by the pride of superior\nstation, when unsupported by personal merit, to every bold peasant a\nthing of scorn. They love him, because he vindicated the ways of God to\nman, by showing that there was more genius and virtue in huts, than was\ndreamt of in the world's philosophy. They love him for his truthful\npictures of the poor. Not there are seen slaves sullenly labouring, or\nmadly leaping in their chains; but in nature's bondage, content with\ntheir toil, sedate in their sufferings, in their recreations full of\nmirth--are seen Free Men. The portraiture, upon the whole, is felt by\nus--and they know it--to demand at times pity as a due; but challenges\nalways respect, and more than respect, for the condition which it\nglorifies. The Land of Burns! What mean we by the words? Something more,\nsurely, than that Fortune, in mere blindness, had produced a great poet\nhere? We look for the inspiring landscape, and here it is; but what\ncould all its beauties have availed, had not a people inhabited it\npossessing all the sentiments, thoughts, aspirations, to which nature\nwilled to give a voice in him of her choicest melody? Nothing\nprodigious, after all, in the birth of such a poet among such a people.\nWas any thing greater in the son than the austere resignation of the\nfather? In his humble compeers there was much of the same tender\naffection, sturdy independence, strong sense, self-reliance, as in him;\nand so has Scotland been prolific, throughout her lower orders, of men\nwho have made a figure in her literature and her history; but to Burns\nnature gave a finer organization, a more powerful heart, and an ampler\nbrain, imbued with that mystery we call genius, and he stands forth\nconspicuous above all her sons.\n\nFrom the character I have sketched of the Scottish people, of old and at\nthis day, it might perhaps be expected that much of their poetry would\nbe of a stern, fierce, or even ferocious kind--the poetry of bloodshed\nand destruction. Yet not so. Ballads enow, indeed, there are, embued\nwith the true warlike spirit--narrative of exploits of heroes. But many\na fragmentary verse, preserved by its own beauty, survives to prove that\ngentlest poetry has ever been the produce both of heathery mountain and\nbroomy brae; but the names of the sweet singers are heard no more, and\nthe plough has gone over their graves. And they had their music too,\nplaintive or dirge-like, as it sighed for the absent, or wailed for the\ndead. The fragments were caught up, as they floated about in decay; and\nby him, the sweetest lyrist of them all, were often revivified by a\nhappy word that let in a soul, or, by a few touches of his genius, the\nfragment became a whole, so exquisitely moulded, that none may tell what\nlines belong to Burns, and what to the poet of ancient days. They all\nbelong to him now, for but for him they would have perished utterly;\nwhile his own matchless lyrics, altogether original, find the breath of\nlife on the lips of a people who have gotten them all by heart. What a\ntriumph of the divine faculty thus to translate the inarticulate\nlanguage of nature into every answering modulation of human speech! And\nwith such felicity, that the verse is now as national as the music!\nThroughout all these exquisite songs, we see the power of an element\nwhich we, raised by rank and education into ignorance, might not have\nsurmised in the mind of the people. The love-songs of Burns are\nprominent in the poetry of the world by their purity. Love, truly felt\nand understood, in the bosom of a Scottish peasant, has produced a crowd\nof strains which are owned for the genuine and chaste language of the\npassion, by highly as well as by lowly born--by cultured and by ruder\nminds--that may charm in haughty saloons, not less than under\nsmoke-blackened roofs. Impassioned beyond all the songs of passion, yet,\nin the fearless fervour of remembered transports, pure as hymeneals; and\ndear, therefore, for ever to Scottish maidens in hours when hearts are\nwooed and won; dear, therefore, for ever to Scottish matrons, who, at\nhousehold work, are happy to hear them from their daughters' lips. And\nhe, too, is the Poet of their friendships. At stanzas instinct with\nblythe and cordial amities, more brotherly the grasp of peasant's in\npeasant's toil-hardened hands! The kindliness of their nature, not\nchilled, though oppressed with care, how ready at his bidding--at the\nrepeated air of a few exquisite but unsought-for words of his--to start\nup all alive! He is the Poet of all their humanities. His Daisy has made\nall the flowers of Scotland dear. His moorland has its wild inhabitants,\nwhose cry is sweet. For sake of the old dumb fellow-servant which his\nfarmer gratefully addresses on entering on another year of labour, how\nmany of its kind have been fed or spared? In the winter storm 'tis\nuseless to think of the sailor on his slippery shrouds; but the \"outland\neerie cattle\" he teaches his feres to care for in the drifting snow. In\nwhat jocund strains he celebrates their amusements, their recreations,\ntheir festivals, passionately pursued with all their pith by a people in\nthe business of life grave and determined as if it left no hours for\nplay! Gait, dress, domicile, furniture, throughout all his poetry, are\nScottish as their dialect; and sometimes, in the pride of his heart, he\nrejoices by such nationality to provoke some alien's smile. The sickle,\nthe scythe, and the flail, the spade, the mattock, and the hoe, have\nbeen taken up more cheerfully by many a toil-worn cottar, because of the\npoetry with which Burns has invested the very implements of labour. Now\nand then, too, here and there peals forth the clangour of the\nwar-trumpet. But Burns is not, in the vulgar sense, a military poet; nor\nare the Scottish, in a vulgar sense, a military people. He and they best\nlove tranquil scenes and the secure peace of home. They are prompt for\nwar, if war be needed--no more. Therefore two or three glorious strains\nhe has that call to the martial virtue quiescent in their bosoms--echoes\nfrom the warfare of their ancient self-deliverance--menacings--a\nprophetical _Nemo me impune lacesset_, should a future foe dare to\ninsult the beloved soil. So nourishes his poetry all that is tender and\nall that is stern in the national character. So does it inspire his\npeople with pride and contentment in their own peculiar lot; and as\n_that_ is at once both poetical and practical patriotism, the poet who\nthus lightens and brightens it is the best of patriots.\n\nI have been speaking of Burns as the poet of the country--and his is the\nrural, the rustic muse. But we know well that the charm of his poetry\nhas equal power for the inhabitants of towns and cities. Occupations,\nfamiliar objects, habitual thoughts, are indeed very different for the\ntwo great divisions of the people; but there is a brotherhood both of\nconsanguinity and of lot. Labour--the hand pledged to constant toil--the\ndaily support of life, won by its daily wrestle with a seemingly adverse\nbut friendly necessity--in these they are all commoners with one\nanother. He who cheers, who solaces, who inspirits, who honours, who\nexalts the lot of the labourer, is the poet alike of all the sons of\nindustry. The mechanic who inhabits a smoky atmosphere, and in whose ear\nan unwholesome din from workshop and thoroughfare rings hourly, hangs\nfrom his rafter the caged linnet; and the strain that should gush free\nfrom blossomed or green bough, that should mix in the murmur of the\nbrook, mixes in and consoles the perpetual noise of the loom or the\nforge. Thus Burns sings more especially to those whose manner of life he\nentirely shares; but he sings a precious memento to those who walk in\nother and less pleasant ways. Give then the people knowledge, without\nstint, for it nurtures the soul. But let us never forget, that the mind\nof man has other cravings--that it draws nourishment from thoughts,\nbeautiful and tender, such as lay reviving dews on the drooping fancy,\nand are needed the more by him to whom they are not wafted fresh from\nthe face of nature. This virtue of these pastoral and rural strains to\npenetrate and permeate conditions of existence different from those in\nwhich they had their origin, appears wheresoever we follow them. In the\nmine, in the dungeon, upon the great waters, in remote lands under fiery\nskies, Burns's poetry goes with his countrymen. Faithfully portrayed,\nthe image of Scotland lives there; and thus she holds, more palpably\nfelt, her hand upon the hearts of her children, whom the constraint of\nfortune or ambitious enterprise carries afar from the natal shores.\nUnrepining and unrepentant exiles, to whom the haunting recollection of\nhearth and field breathes in that dearest poetry, not with homesick\nsinkings of heart, but with home-invigorated hopes that the day will\ncome when their eyes shall have their desire, and their feet again feel\nthe greensward and the heather-bent of Scotland. Thus is there but one\nsoul in this our great National Festival; while to swell the multitudes\nthat from morning light continued flocking towards old Ayr, till at\nmid-day they gathered into one mighty mass in front of Burns's Monument,\ncame enthusiastic crowds from countless villages and towns, from our\nmetropolis, and from the great City of the West, along with the sons of\nthe soil dwelling all round the breezy uplands of Kyle, and in regions\nthat stretch away to the stormy mountains of Morven.\n\nSons of Burns! Inheritors of the name which we proudly revere, you claim\nin the glad solemnity which now unites us, a privileged and more fondly\naffectionate part. To the honour with which we would deck the memory of\nyour father, your presence, and that of your respected relatives, nor\nless that of her sitting in honour by their side, who, though not of his\nblood, did the duties of a daughter at his dying bed, give an impressive\nliving reality; and while we pay this tribute to the poet, whose glory,\nbeyond that of any other, we blend with the renown of Scotland, it is a\nsatisfaction to us, that we pour not out our praises in the dull cold\near of death. Your lives have been past for many years asunder; and now\nthat you are freed from the duties that kept you so long from one\nanother, your intercourse, wherever and whenever permitted by your\nrespective lots to be renewed, will derive additional enjoyment from the\nrecollection of this day--a sacred day indeed to brothers,\ndwelling--even if apart--in unity and peace. And there is one whose\nwarmest feelings, I have the best reason to know, are now with you and\nus, as well on your own account as for the sake of your great parent,\nwhose character he respects as much as he admires his genius, though it\nhas pleased Heaven to visit him with such affliction as might well\ndeaden even in such a heart as his all satisfaction even with this\nfestival. But two years ago, and James Burnes was the proud and happy\nfather of three sons, all worthy of their race. One only now survives;\nand may he in due time return from India to be a comfort, if but for a\nshort, a sacred season, to his old age! But Sir Alexander Burnes--a name\nthat will not die--and his gallant brother have perished, as all the\nworld knows, in the flower of their life--foully murdered in a barbarous\nland. For them many eyes have wept; and their country, whom they served\nso faithfully, deplores them among her devoted heroes. Our sympathy may\nnot soothe such grief as his; yet it will not be refused, coming to him\nalong with our sorrow for the honoured dead. Such a father of such sons\nhas far other consolations.\n\nIn no other way more acceptable to yourselves could I hope to welcome\nyou, than by thus striving to give an imperfect utterance to some of the\nmany thoughts and feelings that have been crowding into my mind and\nheart concerning your father. And I have felt all along that there was\nnot only no impropriety in my doing so, after the address of our Noble\nChairman, but that it was even the more required of me that I should\nspeak in a kindred spirit, by that very address, altogether so worthy of\nhis high character, and so admirably appropriate to the purpose of this\nmemorable day. Not now for the first time, by many times, has he shown\nhow well he understands the ties by which, in a country like this, men\nof high are connected with men of humble birth, and how amply he is\nendowed with the qualities that best secure attachment between the\nCastle and the Cottage. We rise to welcome you to your Father's land.\n\nMr ROBERT BURNS replied in the following terms:--My lord, and ladies and\ngentlemen, You may be assured that the sons of Burns feel all that they\nought to feel on an occasion so peculiarly gratifying to them, and on\naccount of so nobly generous a welcome to the Banks of Doon. In whatever\nland they have wandered--wherever they have gone--they have invariably\nfound a kind reception prepared for them by the genius and fame of their\nfather; and, under the providence of Almighty God, they owe to the\nadmirers of his genius all that they have, and what competencies they\nnow enjoy. We have no claim to attention individually--we are all aware\nthat genius, and more particularly poetic genius, is not hereditary,\nand in this case the mantle of Elijah has not descended upon Elisha. The\nsons of Burns have grateful hearts, and will remember, so long as they\nlive, the honour which has this day been conferred upon them by the\nnoble and the illustrious of our own land, and many generous and kind\nspirits from other lands--some from the far West, a country composed of\nthe great and the free, and altogether a kindred people. We beg to\nreturn our most heartfelt thanks to this numerous and highly respectable\ncompany for the honour which has been done us this day.\n\nSir JOHN M'NEILL spoke as follows:--My lord, ladies, and gentlemen--We\nhave now accomplished the main purpose of this assembly. We have done\nhonour to the memory of Burns, and have welcomed his sons to the land of\ntheir father. After the address--which I may be permitted to call the\naddress of manly eloquence--which you have heard from our Noble\nChairman; after the oration--which I may be permitted to designate as\nsolemn and beautiful--which you have heard from our worthy\nVice-chairman--I should be inexcusable were I to detain you long with\nthe subject which has been entrusted to me. The range of English poetry\nis so vast--it is profuse in so many beauties and excellences, and many\nof its great names are approached with so much habitual veneration, that\nI feel great diffidence and difficulty in addressing you on a subject on\nwhich my opinions can have little weight, and my judgment is no\nauthority; but to you, whose minds have been stirred with the lofty\nthoughts of the Poets of England, and are familiar with their beauties,\nnothing is needed to stimulate you to admire that which I am sure has\nbeen the object of your continual admiration, and the subject of your\nunfailing delight. We have been sometimes accused of a nationality which\nis too narrow and exclusive; but I hope and believe that the accusation\nis founded on misapprehension of our feelings. It is true that, as\nScotsmen, we love Scotland above every other spot on earth--that we love\nit as our early home, and our father's house. We cherish our feelings of\nnationality as we cherish our domestic affections, of which they are in\ntruth a part. But while we have these feelings, we glory in the might\nand the majesty of that great country, with which, for the happiness of\nboth, we have long been united as one nation. We are proud of the\nvictories of Cressy, of Agincourt, and of Poictiers, as if they had been\nwon by our own ancestors. And I may venture to say there is not in this\ngreat assembly one who is not proud that he can claim to be the\ncountryman of Spenser, and Shakspeare, and Milton, and Wordsworth, and\nof every one in that long list of glorious Englishmen, who have shed a\nlustre and conferred a dignity upon our language more bright and more\nmajestic than illuminates and exalts the living literature of any other\nland. There is, I think, in the history of the progress of the human\nintellect, nothing more surprising than the sudden growth of literature\nin England to the summit of its excellence. No sooner had tranquillity\nbeen restored after the long civil wars of the Roses--no sooner had\nmen's minds been set free to enter the fields of speculation opened up\nby the Reformation, than in the short space of the life of one man--than\nin the space of seventy years, there arose such men as Spenser, and\nMilton, and Shakspeare, and Sydney, and Raleigh, and Bacon, and Hobbes,\nand Cudworth, and a whole phalanx of other great men, inferior only to\nthem in the brightness of original genius. How glorious must have been\nthe soil which could bring to maturity a harvest of such teeming\nabundance! There are probably many among us who can even now remember\nwith exultation when the first ray of light was cast on their minds from\nthe genius of Spenser--as the first glimmering of day comes to him whose\nsealed eyes are opened to the light of heaven, discovering objects at\nfirst dimly and then more clearly, we at length gazed in wonder and in\njoy on a creation vaster far, and far more lovely, than it had entered\ninto our hearts to conceive. And if, in our maturer years, we return to\nlive an hour with him in the regions of fairyland that enchanted our\nyouth--if some of the flowers seem less bright, if the murmur of the\nwaters is a more pensive sound, if a soberer light pervade the scene,\nand if some of the illusions are broken for ever, we still discover in\nevery stanza beauties which escaped our earlier observation, and we\nnever lose our relish for that rich play of fancy, like the eastern\nfountain, whose spray descends in pearls and in gems. But, above all,\nwhen we look upon him with mature feelings, we can appreciate that lofty\nstrain of godly philosophy which he, the father of our poetry,\nbequeathed, and which has been followed by his successors. When we call\nto mind the influence produced on a people by the poetry of a\nnation--when we call to mind that whatever is desired to be inculcated,\nwhether for good or for evil, the power of poetry has been employed to\nadvance it, even from the times when the Monarch-Minstrel of Israel\nglorified his Maker in Psalms, to the latest attempts which have been\nmade to propagate treason, immorality, or atheism--when we thus think of\nthese things, we may learn how much of gratitude is due to those men\nwho, having had the precious ointment of poetic genius poured abundantly\non their heads, have felt and acknowledged that they were thereby\nconsecrated to the cause of virtue--who have never forgotten that there\nwas a time when\n\n \"The sacred name\n Of poet and of prophet was the same.\"\n\nSuch men are Spenser, Milton--such is Wordsworth. Of Milton I shall not\nventure to speak. He stands alone in his sanctuary, which I would not\nprofane even by imperfect praise. But it is my duty to speak of\nWordsworth. Dwelling in his high and lofty philosophy, he finds nothing\nthat God has made common or unclean--he finds nothing in human society\ntoo humble, nothing in external nature too lowly, to be made the fit\nexponents of the bounty and goodness of the Most High. In the loftier\naspirations of such a mind, there must be much that is obscure to every\ninferior intelligence; and it may be that its vast expanse can only be\nbut dimly visible--it may be that the clouds of incense rising from the\naltar may veil from common eyes some portion of the stately temple they\nperfume; but we pity the man who should therefore close his eyes on a\nscene of beauty and sublimity, or turn back from the threshold of the\nnoble edifice in which he has been invited to survey the majesty of\ncreative genius, and where he will be taught to find \"Books in the\nrunning brooks, sermons in stones, and good in every thing.\"\n\n \"Blessings be with them, and eternal praise,\n The poets who on earth have made us heirs\n Of truth and pure delight by heavenly lays.\"\n\n --\"Wordsworth and the Poets of England.\"\n\nHENRY GLASSFORD BELL, Esq., advocate, said--My lord, I feel it to be a\ngreat distinction and privilege to have been requested to take a part in\nthe proceedings of this day. It is a day which will not soon pass from\nthe recollection of those who have partaken in its admirably-conducted\nfestivities. In assembling to do honour to the memory of Burns, in no\nidle or frivolous spirit, but impressed with those elevated emotions\nwhich have so plainly animated the whole of this mighty gathering, we\nhave a right to feel that we do honour to ourselves as individuals, and\nas a nation. Our assembling has been prompted by a love of all that is\npurest and best in our national genius, as represented by our national\npoet. It has been prompted, too, by that indomitable love of our native\nland which Burns felt and sang--a love founded on admiration, which\ngrows with our growth and strengthens with our strength, of all that\nexternal nature here presents to us--on profound respect for our\ninestimable and time-hallowed institutions; and in never-dying delight\nin all that kindred spirits have here shared with us--in all that higher\nspirits have here achieved for us. No poet ever possessed greater\ninfluence in disseminating and strengthening such sentiments, than\nBurns. My lord, it has been well said that wherever an humble artisan,\nin the crowded haunts of labour or of trade, feels a consciousness of\nhis own dignity--is stirred with a desire for the beautiful, or haunted\nwith a dream of knowledge, or learns to appreciate the distinction\nbetween the \"guinea's stamp\" and the \"gowd,\" _there_ the royal and\ngentle spirit of Robert Burns, lion-like in its boldness, and dove-like\nin its tenderness, still glows, elevates, and inspires. This spirit is\nalso here, and has been evidenced in many ways; perhaps in none more\nthan in this, that in doing honour to the genius of Burns, we are\nirresistibly led to acknowledge, and speak of the debts we owe to the\nintellectual achievements of other great minds, not in Scotland only,\nbut in the sister countries. We have just heard, from the eloquent lips\nof Sir John M'Neill, the well-deserved praises of the English bards.\nWill this meeting refuse a similar cup of welcome, and of thanks, to the\npoets of Green Erin? Will this meeting, where so many bright eyes rain\ninfluence, and manly hearts beat high, not hail with simultaneous\ndelight the name of one who shines conspicuous as the very poet of\nyouth, of love, and of beauty--the poet, with deference be it spoken, of\nbetter things than even beauty--of gentle thoughts and exquisite\nassociations, that give additional sweetness to the twilight hour, and\nto the enjoyments of home a more endearing loveliness; the poet, too, of\nhis own high-souled country, through whose harp the common breeze of\nIreland changes, as it passes, into articulate melody--a harp that will\nnever be permitted to hang mute on Tara's walls, as long as\n\n \"Erin! the tear and the smile in thine eye\n Blend like the rainbow that melts in thy sky!\"\n\nHow many voices have to-day murmured a wish that he were here! But the\necho of the acclaim with which we greet the name of Moore will reach him\nin his solitude, and he will feel, what Burns died too young to feel,\nthat it is something worth living for to have gained a nation's\ngratitude. Of Maturin and others now dead, I must not pause to speak.\nBut let me be privileged to express, in name of this meeting, our\nrespect and admiration for the best of the living dramatists--one deeply\nimbued with the spirit of the Elizabethan age--one who has rescued our\nstage from the reproach which seemed ready to fall upon it--one to whose\nexuberant poetical fertility, and bold originality of thought, we are\nindebted for such beautiful creations as \"Virginius\" and \"William Tell,\"\nthe \"Hunchback\" and the \"Love Chase,\"--our valued friend, James Sheridan\nKnowles. And I might have stopped here, had it not been that I have\nto-day seen that not the gifted sons alone, but also some of the gifted\ndaughters of Ireland, have come as pilgrims to the shrine of Burns; that\none in particular, one of the most distinguished of that fair sisterhood\nwho give, by their talents, additional lustre to the genius of the\npresent day, has paid her first visit to Scotland, that she might be\npresent on this occasion, and whom have myself seen moved even to tears\nby the glory of the gathering. She is one who has lately thrown\nadditional light on the antiquities, manners, scenery, and beautiful\ntraditions of Ireland--one, whose graceful and truly feminine works are\nknown to us all, and whom we are proud to see among us--Mrs S. C. Hall.\nMy lord, feebly and briefly as I have spoken of these great names, I\nmust not trespass longer on your time, but beg to propose the health of\n\"Moore and the Irish Poets.\"\n\nARCHIBALD ALISON, Esq., Sheriff of Lanarkshire, spoke as follows:--We\nhave listened with admiration to the eloquent strains in which the first\nin rank and the first in genius have proposed the memory of the immortal\nbard whose genius we are this day assembled to celebrate; but I know not\nwhether the toast which I have now to propose, has not equal claims to\nour enthusiasm. Your kindness and that of the committee, has intrusted\nto me the memory of three illustrious men--the far-famed successors of\nBurns, who have drank deep at the fountains of his genius, and proved\nthemselves the worthy inheritors of his inspiration. And Scotland, I\nrejoice to say, can claim them all as her own. For if the Tweed has been\nimmortalized by the grave of Scott, the Clyde can boast the birthplace\nof Campbell, and the mountains of the Dee first inspired the muse of\nByron. I rejoice at that burst of patriotic feeling--I hail it as a\npresage, that as Ayrshire has raised a graceful monument to Burns, and\nEdinburgh has erected a noble structure to the Author of Waverley, so\nGlasgow will ere long raise a worthy tribute to the bard whose name will\nnever die while Hope pours its balm through the human heart; and\nAberdeen will worthily commemorate the far-famed traveller, who first\ninhaled the inspiration of nature amidst the clouds of Loch-nagarr, and\nafterwards poured the light of his genius over those lands of the sun\nwhere his descending orb set--\n\n \"Not as in northern climes obscurely bright,\n But one unclouded blaze of living light.\"\n\nScotland, my lord, may well be proud of such men, but she can no longer\ncall these exclusively her own; their names have become household words\nin every land. Mankind claims them as the common inheritance of the\nhuman race. Look around us, and we shall see on every side decisive\nproofs how far and wide admiration for their genius has sunk in the\nhearts of man. What is it that attracts strangers from every part of the\nworld into this distant land, and has more than compensated a remote\nsituation and a churlish soil, and given to our own Northern Isle a\nsplendour unknown to the regions of the sun? What is it which has\nbrought together this mighty assemblage, and united the ardent and the\ngenerous from every part of the world, from the Ural mountains to the\nbanks of the Mississippi, on the shores of an island in the Atlantic? My\nlord, it is neither the magnificence of our cities, nor the beauty of\nour valleys, the animation of our harbours, nor the stillness of our\nmountains; it is neither our sounding cataracts, nor our spreading\nlakes; neither the wilds of nature we have subdued so strenuously, nor\nthe blue hills we have loved so well. These beauties, great as they are,\nhave been equaled in other lands; these marvels, wondrous though they\nbe, have parallels in other climes. It is the genius of her sons which\nhas given Scotland her proud pre-eminence; this it is, more even than\nthe shades of Bruce, of Wallace, and of Mary, which has rendered her\nscenes classic ground to the whole civilized world, and now brings\npilgrims from the most distant parts of the earth, as on this day, to\nworship at the shrine of genius.\n\n \"Yet Albyn! yet the praise be thine,\n Thy scenes with story to combine;\n Thou bid'st him who by Roslin strays\n List to the tale of other days.\n Midst Cartlane crags thou showest the cave,\n The refuge of thy champion brave;\n Giving each rock a storied tale,\n Pouring a lay through every dale;\n Knitting, as with a moral band,\n Thy story to thy native land;\n Combining thus the interest high\n Which genius lends to beauty's eye!\"\n\nBut, my lord, the poet who conceived those beautiful lines, has himself\ndone more than all our ancestors' valour to immortalize the land of his\nbirth; for he has united the interest of truth with the charms of\nfiction, and peopled the realm not only with the shadows of time, but\nthe creations of genius. In those brilliant creations, as in the glassy\nwave, we behold mirrored the lights, the shadows, the forms of reality;\nand yet\n\n \"So pure, so fair, the mirror gave,\n As if there lay beneath the wave,\n Secure from trouble, toil, and care,\n A world than earthly world more fair.\"\n\nYears have rolled on, but they have taken nothing, they have added much\nto the fame of those illustrious men.\n\n \"Time but the impression deeper makes,\n As streams their channels deeper wear.\"\n\nThe voice of ages has spoken: it has given Campbell and Byron the\nhighest place with Burns in lyric poetry, and destined Scott\n\n \"To rival all but Shakspeare's name below.\"\n\nTheir names now shine in unapproachable splendour, far removed, like the\nfixed stars, from the clouds and the rivalry of a lower world. To the\nend of time they will maintain their exalted station. Never will the\ncultivated traveller traverse the sea of Archipelago, that the \"Isles of\nGreece, the Isles of Greece,\" will not recur to his recollection; never\nwill he approach the shores of Loch Katrine, that the image of Ellen\nDouglas will not be present to his memory; never will he gaze on the\ncliffs of Britain, that he will not thrill at the exploits of the\n\"Mariners of England, who guard our native seas.\" Whence has arisen this\ngreat, this universally acknowledged celebrity? My lord, it is hard to\nsay whether we have most to admire the brilliancy of their fancy or the\ncreations of their genius, the beauty of their verses or the magic of\ntheir language, the elevation of their thoughts or the pathos of their\nconceptions. But there is one whose recent death we all deplore, but who\nhas lighted \"the torch of Hope at nature's funeral pile,\" who has gained\na yet higher inspiration. In Campbell it is the moral purposes to which\nhe has directed his mighty powers which is the real secret of his\nsuccess, the lofty objects to which he has devoted his life, which have\nproved his passport to immortality. It is because he has unceasingly\ncontended for the best interests of humanity, because he has ever\nasserted the dignity of the human soul, because he has never forgotten\nthat amidst all the distinctions of time,\n\n \"The rank is but the guinea stamp,\n The man's the gowd for a' that.\"\n\nBecause he has regarded himself as the high-priest of Nature, and the\nworld which we inhabit as the abode not merely of human care and human\njoys, but as the temple of the living God, in which praise is due, and\nwhere service is to be performed.--\"The memory of Scott, Byron, and\nCampbell.\"\n\nWILLIAM E. AYTOUN, Esq., advocate, said--We are met here to-day not only\nto pay due honour to the memory of that bard whose genius has\nconsecrated this spot, and the scenes around it, as classic ground for\never, but for a wider, a more important, and even a more generous\npurpose. I look upon this assemblage as a great national gathering--a\nmeeting not only of the Highlands and Lowlands of Scotland, but of\nkindly strangers also, to testify our reverence and affection for the\nliving lights of fame that are still burning amongst us, and our undying\ngratitude and exultation for those who have already passed away. Thus,\nthough they belong to the sister countries, we have paid due homage to\nthe venerable name of Wordsworth and to the sparkling genius of Moore.\nThus the heart of every one that hears me burned within him--am I not\nright?--when we saw our own noble Wilson rise amidst us, and heard him,\nwith an eloquence the most pure--for it flowed spontaneously from his\nsoul--speak, as perhaps no other man could speak, of the genius of the\nimmortal dead. Thus, too, we have heard the tribute so touchingly paid\nto Campbell, who now sleeps among the sages, and the statesmen, and the\nwarriors, and the poets of famous England; and to him who has a happier\nand a holier sepulture still--for he lies within the bosom of his own\ndear native land--to Scott, the master-spirit of the age, for whom we\nwell may mourn, since we dare not hope to look upon his like again! I\nhave now, in a few words, to entreat your patience whilst I speak of two\nother Scottish poets whose memory is yet green amongst us--both reared,\nlike Robert Burns, at the lowly hearth of the peasant--both pursuing,\nlike him, through every discouragement and difficulty, the pathway\ntowards honourable renown--and both the authors of strains which bear\nthe stamp of immortality. And first, let me allude to one of them whom I\nknew and dearly loved. Who is there that has not heard of the Ettrick\nShepherd--of him whose inspiration descended as lightly as the breeze\nthat blows along the mountain side--who saw, amongst the lonely and\nsequestered glens of the south, from eyelids touched with fairy\nointment, such visions as are vouchsafed to the minstrel alone--the\ndream of sweet Kilmeny, too spiritual for the taint of earth? I shall\nnot attempt any comparison--for I am not here to criticise--between his\ngenius and that of other men, on whom God in his bounty has bestowed the\ngreat and the marvellous gift. The songs and the poetry of the Shepherd\nare now the nation's own, as indeed they long have been; and amidst the\nminstrelsy of the choir who have made the name of Scotland and her\npeasantry familiar throughout the wide reach of the habitable world the\nclear wild notes of the Forest will for ever be heard to ring. I have\nseen him many times by the banks of his own romantic Yarrow; I have sat\nwith him in the calm and sunny weather by the margin of Saint Mary's\nLake; I have seen his eyes sparkle and his cheek flush as he spoke out\nsome old heroic ballad of the days of the Douglas and the Graeme, and I\nhave felt, as I listened to the accents of his manly voice, that whilst\nScotland could produce amongst her children such men as him beside me,\nher ancient spirit had not departed from her, nor the star of her glory\ngrown pale! For he was a man, indeed, cast in nature's happiest mould.\nTrue-hearted, and brave, and generous, and sincere; alive to every\nkindly impulse, and fresh at the core to the last, he lived among his\nnative hills the blameless life of the shepherd and the poet; and on the\nday when he was laid beneath the sod in the lonely kirkyard of Ettrick,\nthere was not one dry eye amongst the hundreds that lingered round his\ngrave. Of the other sweet singer, too--of Allan Cunningham, the\nleal-hearted and kindly Allan--I might say much; but why should I detain\nyou further? Does not his name alone recall to your recollection many a\nsweet song that has thrilled the bosom of the village maiden with an\nemotion that a princess need not blush to own? Honour, then, to the\npoets!--whether they speak out loud and trumpet-tongued, to find\naudience in the hearts of the great, and the mighty, and the brave--or\nwhether, in lowlier and more simple accents, but not less sacred in\ntheir mission, they bring comfort and consolation to the poor. As the\nsweep of the rainbow, which has its arch in heaven, and its shafts\nresting upon the surface of the earth--as the sunshine which falls with\nequal bounty upon the palace and the hut--is the all-pervading and\nuniversal spirit of poetry; and what less can we do to those men who\nhave collected and scattered it around us, than to hail them as the\nbenefactors of their race? That has been the purpose of our gathering,\nand we have held it in a fitting spot. Proud, indeed, may be the\ndistrict that can claim within herself the birthplaces of Burns and of\nCunningham; and proud may we all be--and we are proud, from yourself, my\nlord, to the humblest individual who bore a part in the proceedings of\nthis memorable day--that we have the opportunity of testifying our\nrespect to the genius that will defy the encroachment of time: and which\nhas shed, and will continue to shed, a splendour and a glory around the\nland that we love so well! My lord, I am honoured in having to propose\n\"The memory of the Ettrick Shepherd, and of Allan Cunningham.\"\n\nSir D. H. BLAIR, Bart., of Blairquhan, said--My Lord Eglinton and\ngentlemen, I have been requested to give the next toast, which I very\nmuch wish had fallen into abler hands. It is a toast, my lord, that is\nas well calculated to call forth enthusiastic bursts of eloquence as any\nwe have listened to with such delight to-day; but as on that account I\nfeel quite unable to do it adequate justice, I must trust to that\nacclamation by which I am confident it will be received, without any\neffort on my part. We all recollect the words of our immortal bard,\nwhen, in alluding to the manner in which nature had finished this fair\ncreation, he says--\n\n \"Her 'prentice han' she tried on man,\n And then she made the lasses O!\"\n\nI am sure every man in this assembly will join me in an enthusiastic\nbumper to the health of the \"Countess of Eglinton, and the ladies who\nhave honoured this meeting with their presence.\"\n\nColonel MURE of Caldwell, said--In obedience to the order of our noble\nchairman, I have to request a bumper to the Peasantry of Scotland. In\norder justly to appreciate the claims of this most estimable class of\nour fellow-citizens upon our sympathies, I must remind you that to it\npre-eminently belongs the honour of having given birth to the remarkable\nman whose memory we are this day met to celebrate. I must remind you,\nthat while the fact of Burns having raised himself from the rank of a\nScottish ploughman, by the innate force of heaven-born genius, to the\nlevel of the greatest and most original poets of any age or country, is\nthe noblest feature of his history, the peasantry of Scotland, in their\nturn, may be entitled to feel pride, even in the presence of the\nproudest nobles of their land, when they remember that from them, and\nnot from the privileged orders of society, our greatest national genius\nwas destined to arise. And, in fact, the most striking, and perhaps the\nmost valuable feature in the poetical character of Burns, is the marked\nascendancy which the spirit and habits of the peasant, the genius of the\nman, as it were, continued to exercise on the genius of the poet, even\nduring the most brilliant periods of his subsequent career. Even amid\nthat rich variety of subjects, in the treatment of which his instinctive\nrefinement and delicacy of taste enabled him to combine, with all the\nhigher powers of the man, the courtly graces of the gentleman and\nscholar--still his happiest effort, the masterpiece of his genius, in\nwhich his own mind is displayed in the most agreeable light, and his\ninspiration breathes forth with the greatest brilliancy and beauty, will\nbe found to be dictated by the associations of his early rustic days.\nWhen I reflect, therefore, how copious, how graphic, how true are his\nown descriptions of the character of the Scottish peasantry, in all its\nvarieties of grave or of gay, of light or of shadow, I cannot but feel\nit is a sort of presumption to offer in a company, who must be all so\nfamiliar with these descriptions, any crude remark of my own, on the\nmore interesting features of those to which they refer. I shall,\nhowever, do my best to season the few comments which I am in some degree\nbound to offer on the subject allotted to me, by taking the poet's works\nas my text-book. Were I called upon, therefore, to name the virtues of\nour peasantry, which chiefly claim our respect and admiration, I should\npoint first to their industry, frugality, and contentment, as those\nwhich prominently adorn their own class of society above all others, and\nalso to their piety and their patriotism, as shared, I would fain hope\nequally, or at least largely, by the mass of our fellow-citizens. Where,\nthen, shall we find a more spirited picture of the influence and effects\nof the three former qualities--above all, of that most inestimable\nblessing, contentment--than in the brilliant little poem which bears the\nhumble title of the \"Twa Dogs,\" where, after so graphically describing\nthe honest toils, often the severe hardships, inseparable from the\npeasant's lot, he goes on to say, that yet\n\n \"They're nae sae wretched's ane wad think,\n Though constantly on poortith's brink;\n They're sae accustom'd wi' the sight,\n The view of it gi'es little fright;\n And how it comes I never kent yet,\n They're maistly wonderfu' contented;\n And buirdly chiels and clever hizzies\n Are bred in such a way as this is.\"\n\nBut where are we, after all, to look for the source of this beautiful\nattribute of contentment? Is it not in the still more admirable one of\ntheir piety? It is here almost superfluous to make any close appeal to\nour poet's authority--to that most sublime description, so familiar to\nyou all, where the old peasant on the Saturday night collects his\nscattered family, at the close of the long week's labour, around his\nhumble but happy cottage fireside, and, after a few sweet but\nhard-earned hours of social enjoyment, instils, before retiring to\nrepose, from the open Word of God, into their minds those lessons of\nDivine wisdom which were to guide them during the next week, and through\nlife, in the paths of religion and virtue. Are not such scenes to this\nday common in our cottages, still, as of old, I firmly believe, the\nfavourite abodes of the genuine spirit of simple Scottish piety? Then as\nto the last, if not the least, in the above list of the virtues of our\npeasants--their patriotism. To whom, I would ask, but to the peasantry\nof Scotland, does our poet so beautifully appeal as having bled with\nWallace? To whom, but to our peasantry, did our national hero look--and\nnever look in vain--for support in his gallant effort to restore the\nfallen fortunes of his country, at the period when our doughty knights\nand nobles--happily but for a season--had been reduced, by the intrigues\nor intimidation of our powerful enemy, to crouch submissive beneath the\nthrone of his usurpation. And can we doubt that this proud spirit of\npatriotism still burns as warm in their hearts as then, if no longer, by\nGod's blessing, so fearfully or so desperately called into action; or\nthat when after, as our poet again has it,\n\n \"They lay aside their private cares\n To mind the Kirk and State affairs\n They'll talk of patronage and priests\n Wi' kindling fury in their breasts,\n Or tell what new taxation's coming,\n And ferlie at the folks in Lunnan.\"\n\nBut I have already detained you too long--if not longer than the\ninterest of the subject, at least than my power of doing justice to it\nentitles me. I shall therefore conclude by pronouncing a grace over our\nbumper, also supplied from the stores of the Poet, and the sentiments of\nwhich every one here present, I am sure, will cordially sympathize--\n\n \"O Scotia! my dear, my native soil,\n For whom my warmest wish to Heaven is sent,\n Long may thy hardy sons of rustic toil\n Be blest with health, and peace, and sweet content.\"\n\nSir JAMES CAMPBELL of Glasgow said--In proposing the toast with which I\nhave been entrusted, I shall content myself by simply expressing my deep\nregret that, under any circumstances, I could so inadequately express my\nown sentiments and feelings of admiration--in all the acceptations of\nthat word--of \"the Land of Burns.\" I am aware, however, that I have the\nhonour of addressing an assemblage who can appreciate, who do\nappreciate, and who, by their appearance here, and the interest so many\nof them have taken in the proceedings and associations of this day, give\nample proof of their high estimation of, and attachment to, \"the Land of\nBurns.\" I am aware, also, that I have the honour to address not a few of\nthose who have, with the pencil or with the pen, done homage to the\nclassical, patriotic, and poetical claims of that land. I feel\nsatisfied, indeed, that there is not an individual in this most\ninteresting and splendid assemblage, who does not greatly prize and\nadmire the fertile soil and landscape beauty of that land; whose bosom\nglows not with an honest pride at the intelligence, enterprise, and\npatriotism of the men of that land; and, above all, who does not honour\nand admire the beauty and accomplishments of the ladies of that land.\nAnd therefore is it, my lord, that, without further preface, I would\ncall upon this assembly to dedicate a bumper to \"The Land of Burns.\"\n\nLord EGLINTON said--Ladies and Gentlemen, Except the toast which I have\nhad the honour and happiness of bringing before you to-day, there is not\none which gives me greater pleasure to see committed to my charge than\nthat which I am now about to bring before your notice--I mean the\n\"Provost and Magistrates of Ayr;\" and along with it, though not down on\nthe card, my feelings will not allow me to leave out the Interests of\nAyr. On such an occasion as this, and so late in the day, I will not\noccupy your time by dilating on the interest which I feel in that Town,\nor of the knowledge which I have of the Provost and the Magistrates.\nFrom that knowledge I feel convinced that the interests of Ayr could not\nbe placed in more worthy hands. In addition to the respect felt towards\nthem as the Magistrates of the County Town, we all feel gratitude to\nthem for the assistance, support, and countenance, they have given to\nour proceedings on this occasion.\n\nProvost MILLER said--Permit me to return my best thanks, on behalf of my\ncolleagues and myself, for the flattering compliment which has just been\npaid to them. The manner in which the toast was introduced by the noble\nlord was particularly gratifying to me; and I am sure it will be\nappreciated by the entire corporation. I beg to assure the noble lord\nthat the recognition of \"Auld Ayr\" at a meeting so peculiarly\ninteresting as the present, and combining, as it does, so much of the\nrank, talent, and worth of the land, will be highly appreciated by the\n\"honest men and bonnie lasses\" for which it has been characterized by\nthe immortal bard in honour of whose memory we are this day met.\n\nThe LORD JUSTICE-GENERAL rose amidst much applause. He said--Ladies and\ngentlemen, after the uncommon success which has attended every part of\nthe proceedings at this meeting to-day, I am confident that I anticipate\nthe unanimous concurrence of this great assembly I have now the honour\nto address, when I state that there appears, in addition to many toasts\ndrunk with so much enthusiasm, one that remains as a debt of gratitude\ndue by this assembly. I consider it a most fortunate circumstance\nattending this meeting, that we have been presided over by the Noble\nLord in the chair. I am sure that the most enthusiastic admirers of\nBurns must be gratified in thinking that the proceedings of this day\nhave been conducted by my noble friend in so admirable a manner. Every\nperson must be satisfied that it was impossible the proceedings of this\nday could have been commenced in a happier strain. Without further\ncomment, I beg leave to propose that we drink the health of our\nexcellent Chairman.\n\nLord EGLINTON, in reply, said--My Lord Justice, and ladies and\ngentlemen, I assure you I feel most deeply grateful to you for the\nhonour you have paid me, as I always ought to be when my health is\nproposed and drunk at a meeting of Scotchmen. But I assure you I never\nfelt more deeply grateful, or more highly sensible of that honour, than\nI do at the present moment, when my health is proposed by such a man as\nthe Lord Justice-General, and when it has been received--and, I am proud\nto say, enthusiastically received--by an assemblage met for such a\npurpose as to do honour to the memory of our greatest poet. But,\ngentlemen, I will not at this late hour of the day, and in a temple, as\nit were, dedicated to the Muses--I will not occupy your time by\nreturning thanks for drinking the health of one who has no merit. But,\nbefore we part, there is a toast which claims our especial\nconsideration--\"the health of Professor Wilson.\" Had it not been for the\nmodesty of the Professor, it ought to have been proposed at a much\nearlier part of the evening. On such an occasion as this, when we have\nmet from all parts of Scotland, to do honour to the memory of the\ngreatest genius Scotland ever knew, it surely is not only proper, but\nour bounden duty, to drink the health of the greatest genius which\nScotland possesses now. The memories of others have been drunk to-night,\nand have been received with that deep feeling which Scotchmen feel\ntowards the memory of genius, but the toast which I am now proposing is\none which has this additional merit, that the subject of it is alive and\nhearty, and able to continue, as you have heard to-day, in that career\nwhich has hitherto so much delighted his countrymen. In the presence of\nProfessor Wilson I cannot dilate, as I could wish to do, on the\ncharacter of that gentleman. I will only ask you to drink with me his\nhealth in a way that will show that you can pay honour to genius alive,\nas you can do honour to departed worth.\n\nProfessor WILSON rose and simply bowed his acknowledgments.\n\nThe Earl of EGLINTON then rose and said--Ladies and Gentlemen,\n\n \"Nae man can tether time nor tide;\n The hour approaches--Tam maun ride.\"\n\nThis brought the proceedings to a close.\n\nWe have thought it due--not less to the character of the meeting than to\nthe sincere and fervid eloquence of the speakers--to place upon our\npages an authentic record of the whole proceedings of the day. This\n\"great national gathering,\" as it was aptly denominated, must be of\nenduring and not ephemeral interest, and will be remembered, and spoken\nof, and quoted, long after events of greater apparent importance have\npassed away into oblivion. The outpourings of a nation's heart are\nimmortal. The tributes that were paid, in the ages long since gone by,\nto the poets of Greece and of Italy, have outlived the most enduring\nmonuments of marble, and we dare not hesitate now to recognise a triumph\nwhich will be as everlasting as theirs.\n\nWe feel that little comment is necessary upon the various addresses that\nare given above. But we should not be justified--and no man who was\nthere that day would forgive us--if we passed over in silence the manly\nand distinguished manner in which Lord Eglinton discharged the duties of\nthe chair. Scotland, as we have already had occasion to say, is proud,\nand justly so, of her aristocracy; but there is not one of them all,\nthrough the whole length and breadth of the land, to whom she can point\nmore exultingly than to this young nobleman. His opening address would\nhave done honour to one long trained in the schools of oratory, and that\nwas its smallest merit. The emphatic and earnest tone of admiration in\nwhich he spoke of the peasantry of his country--his generous and\ntouching allusions to Burns in his earlier years, to what he had done\nand suffered, and to the honours so long withheld, and now so\nbrilliantly conferred--and the patriotic fervour which pervaded his\nwhole address--carried along with him not only the applauses, but the\nhearts of the whole assemblage. Lord Eglinton may well look back with\npride and satisfaction to the proceedings of that day; for he has\nsecured the affections of thousands who already respected his name.\n\nOf the other speeches, eloquent and impressive as they were, we\nshall--with only one exception--speak collectively; and the highest\npraise we can give is to say, that they were every way worthy of the\noccasion, of the subjects which they celebrated, and of the men by whom\nthey were uttered. There was a delicate propriety in the feeling which\nexcluded from the list of toasts the names of the living poets, with the\ngreat and glorious exceptions of Wordsworth and Moore, now beyond all\ncavil at the head of the literature of their respective countries. Their\npresence, though ardently hoped for, was hardly to be expected on this\noccasion; for their advanced years, and the distant journey they must\nhave undertaken, were serious obstacles; but their apologetic letters,\nfull of deep feeling and sympathy, were received, and the reception\nwhich greeted their names, showed the respect and love which the\nScottish people entertain for the greatness and universality of their\nfame. Deep also and thrilling was the emotion evinced at the mention of\nthe illustrious dead, who have passed away into their graves in the\nfulness and maturity of their fame. Strange and powerful is the spell\nwhich lies in the mere plain utterance of their names! Scott, and Byron,\nand Campbell, (just laid in the noblest mausoleum of the world,) the\nEttrick Shepherd, and Allan Cunningham--what names for a country to\nrecord in its annals, in the brief space of one generation!\n\nBut the speech to which all looked forward with the utmost expectation\nand anxiety, was that of Professor Wilson. His zeal in the cause of\nBurns, his earnest and reiterated defence of his reputation, were so\nwell known, that on this occasion, when the balance might be held as\nfinally struck, and when the nation, by its own voluntary act, had\nrecognized the position which its poet, through all time coming must\nmaintain, it would have been felt as a vast and serious omission if the\nlast elegy had not been uttered by the greatest vindicator of his fame.\nIt _was_ so uttered, and none but those who listened to that address can\nconceive the effect which it produced. Elsewhere than in these pages we\nshould assuredly have attempted some comment upon it. As it is, we shall\nborrow an opinion of the provincial press, from the pen, we believe, of\nthe Editor of the _Dumfries-shire Herald_, Mr Aird, himself a spectator\nof the scene, and a man of high intellect and imagination, whose remarks\nwe have been led to adopt, not from the eulogy they contain, but from\ntheir just and reverential truth:--\n\n\"The remarkable speech of the day was Professor Wilson's. Since the time\nwhen in his 'bright and shining youth' he walked seventy miles to be\npresent at a Burns' meeting, and electrified it with a new and peculiar\nfervour of eloquence, such as had never been heard among us before, how\nmanifold, how multiform have been this man's generous vindications of\nour great Bard! Now broad in humour; now sportive and playful; now\nsarcastic, scornful, and searching; now calmly philosophic in criticism;\nnow thoughtful and solemn, large of reverent discourse, 'looking before\nand after,' with all the sweetest by-plays of humanity, with every\nreconciling softness of charity--such, in turns, and in quickest\nintermingled tissue of the ethereal woof, have been the many\nillustrations which this large-minded, large-hearted Scotchman, in whose\ncharacter there is neither corner nor cranny, has poured in the very\nprodigality of his affectionate abundance around and over the name and\nthe fame of Robert Burns. It became him--and he knew it--that on this\ngreat and consummating occasion, so full of reconcilement betwixt human\nfrailty and human worth, his address, on which so much expectation\nwaited, should be a last SOLEMN REQUIEM over the grave of the\nillustrious dead, pronounced not merely to the congregation of the day,\nbut to mankind in general, and to every future age. With those long,\nheart-drawn, lingering, slow-expiring tones, solemn as a cathedral\nchant, the whole of this sacred piece of service (for we can call it\nnothing else) was to us like some mournful oratorio by Mozart, soft at\nonce and sublime. Some might be disappointed that they heard nothing on\nthis occasion of the varied play of Christopher North; but the heart of\nScotland, in its calm retirement, will appreciate this holy oration, as\nworthily hallowing and sanctifying her meeting.\"\n\nThe proceedings in that Pavilion were a just and fitting conclusion to\nthe splendid jubilee of the day. Some no doubt were absent, whom the\npublic would gladly have seen there; for, on an occasion like this, the\ngeneral wish must have been, that all the greatness, and talent, and\nlearning of the land should have united in the National Festival. But\nthat absence, though regretted, did not, in any degree, lessen the\nenthusiasm. Indeed, as we looked around the meeting, and saw, unelevated\nto any conspicuous place, Delta, and Chambers, and Ferrier, and a\nhundred other distinguished men, not only content, but proud to bear\ntestimony by their simple presence to the genuine purpose of the\nassembly, it was hardly possible to wish for more. Every individual\nfeeling was merged in the common desire, that the day should be\nconsecrated to its own peculiar object; and consecrated it was, if\nunanimity, and eloquence, and tears, and the outpouring of all that is\nlofty, and generous, and sincere, can consecrate aught on earth--where\nerror and frailty must abide, but where the judgment of man in his\nweakness, may not, and dare not, usurp the functions of the All-seeing\nand Eternal Judge.\n\nAnd now we close the hasty record of a scene that will be remembered so\nlong as Scotland is a nation. Some there may be--for there are malignant\nand jaundiced spirits every where--who may sneer at the solemnities we\nhave witnessed; and it is well that they should do so, for the praise of\nsuch men is no honour--far better that it should be withheld. We\nconclude by again adopting the language of Mr Aird, which leaves no word\nunsaid.\n\n\"Such has been the tribute of a country to her national poet. She\nfurnished him with the rich materials of his song--with her dear\nvictories set in blood; with the imperishable memory of her\nindependence; with the character of her sons and daughters, simple as\nwater, but strong as the waterfall; with her snatches of old-world\nminstrelsy, surely never composed by mortal man, but spilt from the\noverflowing soul of sorrow and gladness; with her music, twin-born, say\nrather one with her minstrelsy; with her fairy belief, the most\ndelicately beautiful mythology in the history of the human mind, and\nstrangely contrasted with the rugged character of her people, a people\nof sturt and strife; with her heroic faith; with the graves of her\nheadless martyrs, in green shaw or on grim moor, visited by many a slip\nof sunshine streaming down from behind the cloud in the still autumnal\nafternoon. These, and all the other priceless elements of 'the auld\nScottish glory,' he--the national bard--compacted and crystallized into\na Poetry which, by innumerable points of sympathetic contact, carries\nback into the national heart, by ever-conducting issue, the thoughts and\nfeelings which itself first gave forth to his plastic genius; and thus\nthere is an eternal interchange of cause and effect, to the perpetuation\nand propagation of patriotism, and all that constitutes national spirit\nand character.\n\n\"THEREFORE it was fitting that such a national tribute should be paid to\nsuch a national benefactor.\"\n\n\n\n\nSTANZAS FOR THE BURNS' FESTIVAL.\n\nBY DELTA.\n\n\n I.\n\n Stir the beal-fire, wave the banner,\n Bid the thundering cannon sound--\n Rend the skies with acclamation,\n Stun the woods and waters round--\n Till the echoes of our gathering\n Turn the world's admiring gaze\n To this act of duteous homage\n Scotland to her poet pays.\n Fill the banks and braes with music,\n Be it loud and low by turns--\n This we owe the deathless glory,\n That the hapless fate of Burns.\n\n\n II.\n\n Born within the lowly cottage\n To a destiny obscure,\n Doom'd through youth's exulting spring-time\n But to labour and endure--\n Yet Despair he elbow'd from him;\n Nature breathed with holy joy,\n In the hues of morn and evening,\n On the eyelids of the boy;\n And his country's Genius bound him\n Laurels for his sun-burn'd brow,\n When inspired and proud she found him,\n Like Elisha, at the plough.\n\n\n III.\n\n On, exulting in his magic,\n Swept the gifted peasant on--\n Though his feet were on the greensward,\n Light from heaven around him shone;\n At his conjuration, demons\n Issued from their darkness drear;\n Hovering round on silver pinions,\n Angels stoop'd his songs to hear;\n Bow'd the Passions to his bidding,\n Terror gaunt, and Pity calm;\n Like the organ pour'd his thunder,\n Like the lute his fairy psalm.\n\n\n IV.\n\n Lo, when clover-swathes lay round him,\n Or his feet the furrow press'd,\n He could mourn the sever'd daisy,\n Or the mouse's ruin'd nest;\n Woven of gloom and glory, visions\n Haunting throng'd his twilight hour;\n Birds enthrall'd him with sweet music,\n Tempests with their tones of power;\n Eagle-wing'd his mounting spirit\n Custom's rusty fetters spurn'd;\n Tasso-like, for Jean he melted\n Wallace-like, for Scotland burn'd!\n\n\n V.\n\n Scotland!--dear to him was Scotland,\n In her sons and in her daughters,\n In her Highlands,--Lowlands,--Islands,--\n Regal woods, and rushing waters;--\n In the glory of her story,\n When her tartans fired the field,--\n Scotland! oft betray'd--beleagur'd--\n Scotland! never known to yield!\n Dear to him her Doric language,--\n Thrill'd his heart-strings at her name;--\n And he left her more than rubies,\n In the riches of his fame.\n\n\n VI.\n\n Sons of England!--Sons of Erin!\n Ye who, journeying from afar,\n Throng with us the shire of Coila,\n Led by Burns's guiding star--\n Proud we greet you--ye will join us,\n As, on this triumphant day,\n To the champions of his genius\n Grateful thanks we duly pay--\n Currie--Chambers--Lockhart--Wilson--\n Carlyle--who his bones to save\n From the wolfish fiend, Detraction,\n Couch'd like lions round his grave.\n\n\n VII.\n\n Daughter of the poet's mother!\n Here we hail thee with delight;\n Shower'd be every earthly blessing\n On thy locks of silver white!--\n Sons of Burns, a hearty welcome,\n Welcome home from India's strand,\n To a heart-loved land far dearer,\n Since your glorious Father's land:--\n Words are worthless--look around you--\n Labour'd tomes far less could say\n To the sons of such a father,\n Than the sight of such a day!\n\n\n VIII.\n\n Judge not ye, whose thoughts are fingers,\n Of the hands that witch the lyre--\n Greenland has its mountain icebergs,\n AEtna has its heart of fire;\n Calculation has its plummet;\n Self-control its iron rules;\n Genius has its sparkling fountains;\n Dulness has its stagnant pools;\n Like a halcyon on the waters,\n Burns's chart disdain'd a plan--\n In his soarings he was heavenly,\n In his sinkings he was man.\n\n\n IX.\n\n As the sun from out the orient\n Pours a wider, warmer light,\n Till he floods both earth and ocean,\n Blazing from the zenith's height;\n So the glory of our poet,\n In its deathless power serene,\n Shines--as rolling time advances--\n Warmer felt, and wider seen:\n First Doon's banks and braes contain'd it,\n Then his country form'd its span;\n Now the wide world is its empire,\n And its throne the heart of man.\n\n\n X.\n\n Home returning, each will carry\n Proud remembrance of this day,\n When exulted Scotland's bosom\n Homage to her bard to pay;--\n When our jubilee to brighten,\n Eglinton with Wilson vied,\n Wealth's regards and Rank's distinctions\n For the season set aside;\n And the peasant, peer, and poet,\n Each put forth an equal claim,\n For the twining of his laurel\n In the wreath of Burns's fame!\n\n * * * * *\n\n_Edinburgh: Printed by Ballantyne and Hughes, Paul's Work._\n\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Blackwood's Edinburgh Magazine, Volume\n56, Number 347, September, 1844, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nThank you for downloading this Simon & Schuster ebook.\n\n* * *\n\nGet a FREE ebook when you join our mailing list. Plus, get updates on new releases, deals, recommended reads, and more from Simon & Schuster. Click below to sign up and see terms and conditions.\n\nCLICK HERE TO SIGN UP\n\nAlready a subscriber? Provide your email again so we can register this ebook and send you more of what you like to read. You will continue to receive exclusive offers in your inbox.\n\n## Contents\n\nIntroduction\n\n1. Cooking with an Instant Pot\u00ae\n\n2. Breakfast\n\nChampion Breakfast Bowl\n\nSweet Potato Breakfast Bowl\n\nMorning Bacon Parsnips\n\nApple Cinnamon Muffins\n\nBlueberry-Orange Muffins\n\nPumpkin Chip Muffins\n\nSausage Gravy\n\nSouth-of-the-Border Frittata\n\nCiao Bella! Egg Muffins\n\nHam and Mushroom Egg Muffins\n\nBacon, Onion, and Spinach Egg Muffins\n\nSoft-Boiled Eggs with Steamed Asparagus Dippers\n\nBell Pepper Egg Cups\n\nFruity Trail Mix Bowls\n\nMixed Berry Breakfast Syrup\n\nBacon Jam\n\nBreakfast Casserole with Drop Biscuit Dumplings\n\nLoaded Sweet Potato Hash\n\nPiggy Tater Bowl with a Poached Egg\n\n3. Soups, Stews, and Chilis\n\nChicken Broth\n\nVegetable Broth\n\nBeef Broth\n\nTo Bean or Not to Bean Chili\n\nChunky Beef Chili\n\nHearty Lamb Stew\n\nCreamy Mushroom Soup\n\nCream of Broccoli and Bacon Soup\n\nTom Kha Goong\n\nKentucky Derby Burgoo\n\nWhite Chicken Chili\n\nSavory Acorn Squash Soup\n\nChicken Taco Soup\n\nCreamy Tomato Basil Soup\n\nMardi Gras Gumbo\n\nQuick Vegetable Soup\n\nHamburger Soup\n\nOma's Sausage and Cabbage Soup\n\nCreamy Ham and Root Veggie Soup\n\nCurried Carrot Soup\n\n4. Appetizers and Sauces\n\nDeviled Eggs\n\nMini Chili Polpette\n\n\"Cheesy\" Spinach Crab Dip\n\nEndive Boats with Chicken Salad\n\nTandoori-Style Chicken Wings\n\nSesame-Orange Chicken Wings\n\nBroccoli-Zucchini Hummus\n\nBurgundy Mushrooms\n\nMushrooms Stuffed with Sausage and Kimchi\n\nSteamed Shrimp with Cocktail Sauce\n\nLiver P\u00e2t\u00e9\n\nSteamed Artichokes with Lemon-Herb Butter\n\nRutabaga Fries with Rosemary Dipping Sauce\n\nBroccoli-Cauli Sausage Tots\n\nSalsa Verde\n\nBolognese Sauce\n\nMarinara Sauce\n\nPuttanesca Sauce\n\nTomato Curry Sauce\n\n5. Side Dishes\n\nNutty Holiday Sweet Potatoes\n\nHoney-Glazed Carrots\n\nBacon-Apple Brussels Sprouts\n\nMixed Southern Greens\n\nDilled Purple Potatoes\n\nLemon Ghee Broccoli\n\nHerbed Fingerling Potatoes and Onions\n\nKohlrabi, Kale, and Karrots\n\nSteamed Beets\n\nPurple Cauliflower\n\nGerman Red Potatoes\n\nEggplant-Olive Ratatouille\n\nSouthern Squash Casserole\n\nBroccoli Casserole\n\nBacon Cauli-Potato Mash\n\nRed Onion and Apple Confit\n\nSesame Zoodles\n\nSaffron and Herb Cauliflower Rice\n\nBroccoli-Parsnip Smash\n\n6. Poultry Main Dishes\n\nBBQ Whole Chicken\n\nGluten-Free Beer Can Dijon Chicken\n\nKimchi Chicken Wings\n\nChicken Leg Almond Butter Satay\n\nSmoked Paprika and Garlic Chicken Legs\n\nChili Lime Chicken Legs\n\nKombucha Chicken Thighs\n\nChicken Thighs with Strawberry Salsa\n\nTuscan Chicken\n\nSalsa Verde Chicken Meatballs\n\nHoney Mustard Chicken Bites\n\nJamaican Jerk Chicken Meatballs\n\nTaco Chicken Lettuce Wraps with Pico Guacamole\n\nAsian Chicken Lettuce Wraps with Ginger Aioli\n\nChicken Livers and Onions with Gravy\n\nAvocado Chicken Salad\n\nPaleo-ish Turkish Chicken Thighs\n\nCornish Hens and Veggies\n\nChili-Orange Turkey Wings\n\nMega Turkey Meatballs\n\n7. Beef and Bison Main Dishes\n\nNana's Sunday Pot Roast\n\nChuck Roast with Carrots and Onions\n\nSweet-n-Spicy Beef Brisket\n\nTender Flank Steak with Mushrooms and Onions\n\nSkirt Steak Fajitas\n\nKorean Beef Short Ribs\n\nMongolian Beef and Cashew BBQ\n\nMeatloaf and Gravy\n\nBalsamic Meatballs\n\nMarinara Meatballs\n\nReuben Meatballs with Russian Dressing\n\nGround Beef and Broccoli Bowl\n\nJust Beet It Burgers\n\nCorned Beef and Cabbage\n\nCuban Picadillo with Riced Cauliflower\n\nChi-Town Italian Beef and Peppers\n\nBeef Shanks and Artichokes\n\nBison Sloppy Joe's\n\nGround Bison Bulgogi\n\nItalian Beef Tongue Lettuce Wraps\n\n8. Pork, Lamb, and Game Main Dishes\n\nSalsa Verde Pulled Pork\n\nSimple Dijon Pork Tenderloin\n\nOrange Rosemary Pork Tenderloin\n\nBalsamic Pork Tenderloin with Caperberries\n\nSteamed Dry-Rubbed Pork Spareribs\n\nCowgirl Baby Back Pork Ribs\n\nHam and Scalloped Potatoes\n\nGround Pork Stir-Fry Bowl\n\nSmothered Pork Chops with Mushroom Gravy\n\nPork Chops with Sauerkraut, Potatoes, and Apples\n\nCubano Bowl\n\nVietnamese Pork Chops\n\nBBQ Boneless Pork Loin Roast\n\nOsso Bucco with Gremolata\n\nPork Medallions Puttanesca\n\nLamb Riblets\n\nLamb Shanks with Tomato Curry Sauce\n\nGreek Lamb Meatballs\n\nChorizo Sliders with Jicama-Beet Slaw\n\nThe Busy Cook's Duck \u00e0 l'Orange Breast\n\n9. Seafood and Fish Main Dishes\n\nSteamed Mussels and Chorizo\n\nStewed Halibut\n\nFish Taco Bowl\n\nThai Coconut Curry Monkfish\n\nLouisiana Shrimp and Sausage\n\nShrimp, Prosciutto, and Asparagus Salad with Yuzu Vinaigrette\n\nShrimp Scampi\n\nCrab-Chorizo Stuffed Calamari\n\nBerbere Salmon\n\nDilled Salmon and Broccoli\n\nDairy-Free Creamed Crab\n\nCitrus-Steamed Littleneck Clams\n\nSalmon Croquettes\n\nGreek Sea Bass\n\nHerbed Cod with Yellow Squash\n\nMahi-Mahi with a Lemon-Caper Butter Sauce\n\nSteamed Snow Crab Legs\n\nSimply Steamed Lobster Tails\n\nQuick Crawfish Boil\n\n10. Sweet Treats and Drinks\n\nSpiced Applesauce\n\nCaf\u00e9 Mocha Muffins\n\nStrawberry Cupcakes with Chocolate Ganache\n\nApple Brown Betty\n\nUpside-Down Peach Cobbler\n\nBanana Pudding Cake\n\nChocolate Chip Zucchini Bread\n\nDouble Chocolate Custard\n\nCoconut Key Lime Custard\n\nGrain-Free Stuffed Apples\n\nHummingbird Cake\n\nPoached Cherries\n\nUpside-Down Chocolate Cherry Cake\n\nMixed Berry Quick Jam\n\nQuick Lime Marmalade\n\nFresh-Brewed Arnie Palmer\n\nGluhwein\n\nHomemade Chai Tea Latte\n\nSpiced Mocha Latte\n\nMexican Hot Cocoa\n\nAppendix A: Paleo \"Yes\" and \"No\" Foods\n\nAppendix B: US\/Metric Conversion Chart\n\nAbout the Author\n\nIndex\nTo Samantha and Calla \nTHANK YOU for all the laughs in the kitchen and complete honesty over my creations. \nI love you, Mom\n\n## Introduction\n\nThe paleo diet (or lifestyle, as I like to think of it) has become increasingly popular in the past several years, but that doesn't mean it is a \"fad\" diet. Eating paleo just means eating real, whole, and unprocessed foods. One of the most significant benefits of adhering to a paleo lifestyle is the potential for decreasing your risk of developing a number of chronic diseases such as cancer, diabetes, cardiovascular disease, and osteoporosis.\n\nAlso known as the caveman diet, the term paleo refers to a time when people ate only grass-fed game, wild-caught fish, nuts, vegetables, berries, and occasionally, other fruits. These people lived before the time of modern agriculture and the domestication of animals. When they needed meat, they hunted, and they gathered berries and nuts. There were no grains planted in fields and no milk past the weaning stage. Subsequently, the population wasn't plagued by many of the diseases that plague humans today.\n\nEven though they had to forage, hunt, and kill most of their food, Paleolithic humans ate better-quality food than most of the world does today. Not only are the hundreds of quick-fix weight-loss products wreaking havoc on the human body, but the standard American diet (SAD)\u2014with grains at the helm\u2014is believed to contribute to a number of diseases caused by inflammation.\n\nYou may be thinking that you need to move to a forest and take up hunting, fishing, and gardening to be on today's paleo diet, right? That couldn't be further from the truth. This lifestyle simply requires a shift in your thinking. First, you will need to learn what foods are considered paleo \"yes\" foods and paleo \"no\" foods. From there, all you need to start on your journey toward paleo success is a simple shopping list, an open mind, and a whole bunch of recipes! While cooking paleo may seem intimidating in the beginning, the truth is that many of the recipes adored by families the world over can easily be converted to paleo-friendly options with a few carefully chosen ingredients and some fun substitution. When it comes to cooking paleo, it's all about getting creative.\n\nAnd that's where your Instant Pot\u00ae comes in. Cooking with an Instant Pot\u00ae is a life-changing experience, especially when you follow the Paleolithic lifestyle. This multifunction cooking tool allows you to saut\u00e9, brown, steam, and warm your food. It cooks soups, eggs, and even cakes! And the high-pressure cooking and steaming ability of an Instant Pot\u00ae does wonders to steaks, pork shoulders, and chicken. With the touch of a button you'll be able to cook cuts of meat that would normally take hours in just minutes. The Instant Pot\u00ae cooks food at a low temperature but it does it more efficiently than other slow-cooking methods or appliances because it also uses pressure and steam. It is like a pressure cooker and a slow cooker rolled into one. This cooking method also seals in essential vitamins and minerals and allows the Instant Pot\u00ae to turn out healthier, better-tasting food that is perfect when you're on the go.\n\nWhether you've just bought your Instant Pot\u00ae or have been using one for years and just need some inspiration, this book is for you. Inside you'll find 175 delicious paleo recipes ranging from Pumpkin Chip Muffins and White Chicken Chili to Tender Flank Steak with Mushrooms and Onions and Steamed Mussels and Chorizo. You'll also find decadent desserts, including Double Chocolate Custard and Upside-Down Chocolate Cherry Cake. The more you cook, the more you'll realize how versatile the Instant Pot\u00ae really is, whether you're making a hearty breakfast, an amazing main course, or a delicious dessert. So plug in your Instant Pot\u00ae and get ready to enjoy some amazing, delicious, and quick paleo meals.\n\n## Cooking with an Instant Pot\u00ae\n\nSo you're about to venture into the amazing world of Instant Pot\u00ae cooking . . . but you're not sure where or how to start. Don't worry, this chapter will give you the information you need to know to get started. Here you'll learn what all those buttons on your Instant Pot\u00ae do, how to release the pressure from the Instant Pot\u00ae when the cooking time is up, how to keep your Instant Pot\u00ae clean, and more.\n\nEven though you'll learn all this information in this chapter, it's important that you read the owner's manual as well. The user manual is your key to successfully creating the recipes throughout this book. In addition to pointing out the basic functions of the appliance, it will tell you to do an initial test run using water to get familiar with the Instant Pot\u00ae. I can't stress enough that you need to do this. It will familiarize you with this appliance and take away some of the anxiety. In addition, this first run will help steam-clean your pot before you use it to make a favorite recipe.\n\nBut for now, let's take a look at some Instant Pot\u00ae basics.\n\n### Function Buttons\n\nYou are staring at the Instant Pot\u00ae and there are so many buttons. Which one should you use? Most of the function buttons seem obvious, but it is important to note that several are set with preprogrammed default cooking times. Also, keep in mind that every button option on the Instant Pot\u00ae is programmed with a 10-second delay, meaning that cooking begins 10 seconds after you hit the button. Most likely you will utilize the Manual or Pressure Cook button the most because you are in complete control, but read on for more detailed information on the remaining function buttons.\n\nManual\/Pressure Cook button. Depending on the model of Instant Pot\u00ae, there is a button labeled either Manual or Pressure Cook. This might be the most used button on the Instant Pot\u00ae. The default pressure setting is High; however, you can toggle the pressure from High to Low by pressing the Pressure button. Use the Plus and Minus buttons to adjust the pressurized cooking time.\n\nSaut\u00e9 button. This button helps the Instant Pot\u00ae act as a skillet for saut\u00e9ing vegetables or searing meat prior to adding the remaining ingredients of a recipe, and it is used for simmering sauces as well. There are three temperature settings\u2014Normal, Less, and More\u2014that can be accessed using the Adjust button. The Normal setting is for saut\u00e9ing, the Less setting is for simmering, and the More setting is for searing meat. Keep the lid open when using the Saut\u00e9 button to avoid pressure building up.\n\nSoup button. This button is used to cook soups and broths at high pressure for a default of 30 minutes. The Adjust button allows you to change the cooking time to 20 or 40 minutes.\n\nPorridge button. This button is used to cook porridge, congee, and jook in the Instant Pot\u00ae at high pressure for a default of 20 minutes. The Adjust button allows you to change the cooking time to 15 or 40 minutes.\n\nPoultry button. This button is used to cook chicken, turkey, and even duck at high pressure for a default of 15 minutes. The Adjust button allows you to change the cooking time to 5 or 30 minutes.\n\nMeat\/Stew button. This button is used to cook red meats and stew meats at high pressure for a default of 35 minutes. The Adjust button allows you to change the cooking time to 20 or 45 minutes.\n\nBean\/Chili button. This button is used to cook dried beans and chili at high pressure for a default of 30 minutes. The Adjust button allows you to change the cooking time to 25 or 40 minutes.\n\nRice button. This button is used to cook white rice such as jasmine or basmati at low pressure. The Instant Pot\u00ae will automatically set the default cooking time by sensing the amount of water and rice in the cooking vessel.\n\nMultigrain button. This button is used to cook grains such as wild rice, quinoa, and barley at high pressure for a default of 40 minutes. The Adjust button allows you to change the cooking time to 20 or 60 minutes.\n\nSteam button. This button is excellent for steaming vegetables and seafood using your steamer basket. It steams for a default of 10 minutes. The Adjust button allows you to change the cooking time to 3 or 15 minutes. Quick-release the steam immediately after the timer beeps so as to not overcook the food.\n\nSlow Cook button. This button allows the Instant Pot\u00ae to cook like a slow cooker. It defaults to a 4-hour cook time. The Adjust button allows you to change the temperature to Less, Normal, or More, which correspond to a slow cooker's low, normal, or high. The Plus and Minus buttons allow you to adjust the cooking time.\n\nKeep Warm\/Cancel button. When the Instant Pot\u00ae is being programmed or it is in operation, pressing this button cancels the operation and returns the Instant Pot\u00ae to a standby state. When the Instant Pot\u00ae is in the standby state, pressing this button again activates the Keep Warm function.\n\nAutomatic Keep Warm function. After the ingredients in the Instant Pot\u00ae are finished cooking, the pot automatically switches over to the Keep Warm function and will keep your food warm up to 10 hours. This is perfect for large cuts of meat as well as soups, stews, and chili, allowing the spices and flavors to really marry for an even better taste. The first digit on the LED display will show an L to indicate that the Instant Pot\u00ae is in the Keep Warm cycle, and the clock will count up from 0 seconds to 10 hours.\n\nTimer button. This button allows you to delay the start of cooking up to 24 hours. After you select a cooking program and make any time adjustments, press the Timer button and use the Plus or Minus keys to enter the delayed hours; press the Timer button again and use the Plus or Minus keys to enter the delayed minutes. You can press the Keep Warm\/Cancel button to cancel the timed delay. The Timer function doesn't work with Saut\u00e9, Yogurt, and Keep Warm functions.\n\n### How Does Food Cook in 0 Minutes?\n\nIf you are confused about how some recipes require \"0\" minutes to cook, it's not a typo. Some veggies and seafoods that only require minimal steaming are set at zero cooking time. Food can actually be cooked in the time that it takes the Instant Pot\u00ae to achieve pressure.\n\n### Locking and Pressure-Release Methods\n\nOther than the Saut\u00e9 function, where the lid should be off, or the Slow Cook or Keep Warm functions, where the lid can be on or off, most of the cooking you'll do in the Instant Pot\u00ae will be under pressure, which means you need to know how to lock the lid before pressurized cooking and how to safely release the pressure after cooking. Once your ingredients are in the inner pot of the Instant Pot\u00ae, to lock the lid put the lid on the Instant Pot\u00ae with the triangle mark on the lid aligned with the Unlocked mark on the rim of the Instant Pot\u00ae. Then turn the lid 30 degrees clockwise until the triangle mark on the lid is aligned with the Locked mark on the rim. Turn the pointed end of the pressure release handle on top of the lid to the Sealing position. After your cooking program has ended or you've pressed the Keep Warm\/Cancel button to end the cooking, there are two ways you can release the pressure:\n\nNatural-release method. To naturally release the pressure, simply wait until the Instant Pot\u00ae has cooled sufficiently for all the pressure to be released and the float valve drops, normally about 10\u201315 minutes. You can either unplug the Instant Pot\u00ae while the pressure releases naturally or let the pressure release while it is still on the Keep Warm function.\n\nQuick-release method. The quick-release method stifles the cooking process and helps unlock the lid for immediate serving. To quickly release the pressure on the Instant Pot\u00ae, make sure you are wearing oven mitts, then turn the pressure release handle to the Venting position to let out steam until the float valve drops. This is generally not recommended for starchy items or large volumes of liquid (e.g., soup) so as to avoid any splattering that may occur. Be prepared, because the noise and geyser effect of the releasing steam during the quick-release method can be off-putting. Also, if you own dogs, apparently this release is the most frightening part of the day, so take caution.\n\n### Pot-in-Pot Accessories\n\nPot-in-pot cooking is when you place another cooking dish inside the Instant Pot\u00ae for a particular recipe. The Instant Pot\u00ae is straightforward and comes with an inner pot and trivet; however, there are many other tricks and recipes that can be made with the purchase of a few other accessories, including a springform pan, cake pan, glass bowl, and ramekins.\n\n7\" springform pan. A 7\" springform pan is the perfect size for making a cake and many other desserts in an Instant Pot\u00ae. It is the right dimension to fit inside the pot, and it makes a dessert for four to six people.\n\n6\" cake pan. A 6\" pan is excellent for making a small cake in the Instant Pot\u00ae. It can serve four to six people depending on the serving size. This pan is perfect for a family craving a small dessert without committing to leftovers.\n\n7-cup glass bowl. This 7-cup bowl fits perfectly in the Instant Pot\u00ae and works great for eggs and casseroles that generally would burn on the bottom of the pot insert. The items in the bowl sit up on the inserted trivet and are cooked with the steam and pressure of the pot.\n\nRamekins. These 4-ounce porcelain individual portions are the perfect vessel for tasty custards.\n\nSteamer basket. The steamer basket helps create a raised shelf for steaming. Shop around, as there are several variations, including metal or silicone steamer baskets. Some even have handles to make it easier to remove them after the cooking process.\n\nSilicone baking cups. The silicone baking cups are great for mini meatloaves, cupcakes, on-the-go frittatas, and little quick breads.\n\nAlthough these accessories can help you branch out and make different recipes with the Instant Pot\u00ae, there are many recipes you can make using just the inner pot and trivet that come with your appliance. These are just handy items you can purchase along the way to use with what will soon become your favorite heat source in the kitchen.\n\n#### Accessory Removal\n\nCooking pot-in-pot is a great idea until it's time to remove the inserted cooking dish. Because of the tight space, it is almost impossible to use thick oven mitts to reach down and grip something evenly without tipping one side of the cooking vessel and spilling the cooked item. There are a few ways around this:\n\nRetriever tongs. Retriever tongs are a helpful tool for removing hot bowls and pans from the Instant Pot\u00ae.\n\nMini mitts. Small oven mitts are helpful when lifting pots out of an Instant Pot\u00ae after the cooking process, especially the type made of silicone. They are more heat-resistant and less cumbersome than traditional oven mitts, which can prove to be bulky in the tight space of the cooker.\n\nAluminum foil sling. This is a quick, inexpensive fix to the problem of lifting a heated dish out of an Instant Pot\u00ae. Take a 10\" \u00d7 10\" square of aluminum foil and fold it back and forth until you have a 2\" \u00d7 10\" sling. Place this sling underneath the bowl or pan before cooking so that you can easily lift up the heated dish.\n\nAlthough necessary if you do pot-in-pot cooking, these retrieval tools are not needed if you are simply using the interior pot that comes with the appliance upon purchase. A slotted spoon will do the trick for most other meals.\n\n### Cleaning Your Instant Pot\u00ae\n\nWhen cleaning up after using your Instant Pot\u00ae, the first thing you should do is unplug it and let it cool down. Then you can break down the following parts to clean and inspect for any trapped food particles:\n\nInner pot. The inner pot, the cooking vessel, is dishwasher safe; however, the high heat causes rainbowing, or discoloration, on stainless steel. To avoid this, hand wash the pot.\n\nOuter heating unit. Wipe the interior and exterior with a damp cloth. Do not submerge in water, as it is an electrical appliance.\n\nTrivet. The trivet is dishwasher safe or can be cleaned with soap and water.\n\nLid. The lid needs to be broken down into individual parts before washing. The sealing ring, the float valve, the steam release handle, and the antiblock shield all need to be cleaned in different ways:\n\n\u2022 Sealing ring. Once this ring is removed, check the integrity of the silicone. If it is torn or cracked, it will not seal properly and may hinder the cooking process, in which case it should not be used. The sealing ring needs to be removed and washed each time because the ring has a tendency to hold odors when cooking. Vinegar or lemon juice is excellent for reducing odors. You can purchase additional rings for a nominal price. Many Instant Pot\u00ae users buy more than one ring and use one for meats and a separate one for desserts and milder dishes.\n\n\u2022 Float valve. The float valve is a safety feature that serves as a latch lock that prevents the lid from being opened during the cooking process. Make sure that this valve can move easily and is not obstructed by any food particles.\n\n\u2022 Pressure release handle. This is the venting handle on top of the lid. It can be pulled out for cleaning. It should be loose, so don't worry. This allows it to move when necessary.\n\n\u2022 Antiblock shield. The antiblock shield is the little silver \"basket\" underneath the lid. It is located directly below the vent. This shield can and should be removed and cleaned. It blocks any foods, especially starches, so they don't clog the vent.\n\nSo, now that you know about all the safety features, buttons, and parts of the Instant Pot\u00ae and know how to clean everything, it's time for the fun part. The cooking process is where the excitement begins. From breakfast to dessert and everything in between, The \"I Love My Instant Pot\u00ae\" Paleo Recipe Book has you covered.\n\n## Breakfast\n\nFitting breakfast into a busy schedule can seem a bit overwhelming at times, especially if you are following a paleo diet. You will usually have to make your own breakfasts because there really aren't many, if any, paleo-friendly drive-through options. This chapter's got you covered with several to-go meals in addition to sit-down meals made on the fly. The Instant Pot\u00ae can help save the day with its shortened cooking time and freedom from having to stand over the skillet. All you have to do is add your ingredients, press a button, and go get ready to tackle your day. Make sure to set aside a prep day during the week so all of your fruits and vegetables are already cut and ready to use.\n\nIn addition to the morning-time crunch, most of the breakfast options on the paleo lifestyle seem repetitive. From casseroles to muffins, this chapter offers a wide variety of delicious breakfast recipes, including Bacon Jam, Apple Cinnamon Muffins, and even a Breakfast Casserole with Drop Biscuit Dumplings. And once you get comfortable with some of the basics, you should feel free to get creative and make some of your own morning masterpieces. So, get cooking, and let your family wake up with a new appreciation for their soon-to-be-favorite kitchen gadget!\n\nChampion Breakfast Bowl\n\nSweet Potato Breakfast Bowl\n\nMorning Bacon Parsnips\n\nApple Cinnamon Muffins\n\nBlueberry-Orange Muffins\n\nPumpkin Chip Muffins\n\nSausage Gravy\n\nSouth-of-the-Border Frittata\n\nCiao Bella! Egg Muffins\n\nHam and Mushroom Egg Muffins\n\nBacon, Onion, and Spinach Egg Muffins\n\nSoft-Boiled Eggs with Steamed Asparagus Dippers\n\nBell Pepper Egg Cups\n\nFruity Trail Mix Bowls\n\nMixed Berry Breakfast Syrup\n\nBacon Jam\n\nBreakfast Casserole with Drop Biscuit Dumplings\n\nLoaded Sweet Potato Hash\n\nPiggy Tater Bowl with a Poached Egg\n\n## Champion Breakfast Bowl\n\nFeel like a champion. Work out like a champion. Eat like a champion. After an intense early morning workout on the punching bag, come home to a bowlful of protein to nourish those sore muscles.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 4\n\n* * *\n\n6 large eggs\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 teaspoons avocado oil\n\n1\/2 cup diced red onion\n\n1 medium red bell pepper, seeded and diced\n\n1\/4 pound ground pork sausage\n\n11\/2 cups water\n\n1 In a medium bowl, whisk together eggs, salt, and pepper. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Stir-fry onion, bell pepper, and sausage 3\u20134 minutes until sausage starts to brown and onions are tender.\n\n3 Transfer mixture to a lightly greased 7-cup glass bowl. Pour whisked eggs over the sausage mixture.\n\n4 Add water to Instant Pot\u00ae and insert trivet. Place dish with egg mixture on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove dish from pot. Let sit at room temperature 5\u201310 minutes to allow the eggs to set. Slice and serve.\n\n## Sweet Potato Breakfast Bowl\n\nThis is a perfect breakfast after an intense morning workout. Let your Instant Pot\u00ae be your personal chef and cook this Sweet Potato Breakfast Bowl while you are showering off and getting ready for your day. When ready, ladle a helping of flavor-filled protein and vegetables into your breakfast bowl to help you tackle the day.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n6 large eggs\n\n1 tablespoon Italian seasoning\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 pound ground pork sausage\n\n1 large sweet potato, peeled and cubed\n\n1 small yellow onion, peeled and diced\n\n2 cloves garlic, minced\n\n1 medium green bell pepper, seeded and diced\n\n2 cups water\n\n1 In a medium bowl, whisk together eggs, Italian seasoning, salt, and pepper. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae. Stir-fry sausage, sweet potato, onion, garlic, and bell pepper 3\u20135 minutes until onions are translucent.\n\n3 Transfer mixture to a lightly greased 7-cup glass bowl. Pour whisked eggs over the sausage mixture.\n\n4 Add water to the Instant Pot\u00ae and insert trivet. Place dish with egg mixture on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove dish from pot. Let sit at room temperature 5\u201310 minutes to allow the eggs to set. Slice and serve.\n\n## Morning Bacon Parsnips\n\nThe runny yolks of a perfectly poached egg are nature's gravy. Don't skip this luxurious step! And if you really want to take this dish to the next level, drizzle some drool-worthy hollandaise over the top.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n2 slices bacon, diced\n\n1 small yellow onion, peeled and diced\n\n4 cups peeled and diced parsnips\n\n2 tablespoons ghee\n\n3 cloves garlic, minced\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 cup water\n\n4 large eggs, poached\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add bacon and onions. Stir-fry until onions are translucent, 3\u20135 minutes. Transfer mixture to a lightly greased 7-cup glass bowl. Add parsnips, ghee, garlic, salt, and pepper to bowl and toss to mix.\n\n2 Add water to the Instant Pot\u00ae and insert trivet. Place glass dish on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n3 Remove dish from pot and divide mixture equally among four serving dishes. Garnish each dish with a poached egg.\n\n## Apple Cinnamon Muffins\n\nA classic flavor combination, these muffins are a great treat for the family to wake up to as the weather cools. Serve with a slice or two of bacon and a warm cup of tea to round out this wintry breakfast.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 6\n\n* * *\n\n11\/4 cups cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\n1 teaspoon ground cinnamon\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n3 tablespoons ghee, melted\n\n2 large eggs\n\n1\/3 cup pure maple syrup\n\n1\/4 cup grated red apples\n\n1 cup water\n\n1 In a large bowl, combine flour, baking powder, baking soda, cinnamon, and sea salt.\n\n2 In a medium bowl, combine vanilla, ghee, eggs, syrup, and apples.\n\n3 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Spoon mixture into 6 lightly greased silicone cupcake liners.\n\n4 Add water to the Instant Pot\u00ae and insert trivet or steamer basket. Place cupcake liners on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 9 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove muffins from pot and set aside to cool 5 minutes.\n\n## Blueberry-Orange Muffins\n\nThe sweet blueberries and the zesty orange make this the ideal breakfast for mornings on the go.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 6\n\n* * *\n\n11\/4 cups cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n1\/4 cup freshly squeezed orange juice\n\n1 teaspoon orange zest\n\n3 tablespoons ghee, melted\n\n2 large eggs\n\n1\/3 cup pure maple syrup\n\n1\/4 cup fresh blueberries\n\n1 cup water\n\n1 In a large bowl, combine flour, baking powder, baking soda, and sea salt.\n\n2 In a medium bowl, combine vanilla, orange juice, orange zest, ghee, eggs, and syrup.\n\n3 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Fold in blueberries. Do not overmix. Spoon mixture into 6 lightly greased silicone cupcake liners.\n\n4 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place cupcake liners on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 9 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove muffins from pot and set aside to cool 5 minutes.\n\n## Pumpkin Chip Muffins\n\nWhen everyone else is drinking pumpkin lattes, you can fulfill that urge to splurge with this healthy alternative. The dark chocolate chips provide an element that will make you feel like you are cheating, but the antioxidants will prove otherwise.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 6\n\n* * *\n\n11\/4 cups cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\n1 teaspoon pumpkin pie spice\n\nPinch sea salt\n\n1\/4 cup pumpkin pur\u00e9e\n\n1\/2 teaspoon vanilla extract\n\n1 tablespoon ghee, melted\n\n2 large eggs\n\n1\/3 cup pure maple syrup\n\n2 tablespoons dark chocolate chips\n\n1 cup water\n\n1 In a large bowl, combine flour, baking powder, baking soda, pumpkin pie spice, and sea salt.\n\n2 In a medium bowl, combine pumpkin pur\u00e9e, vanilla, ghee, eggs, and syrup.\n\n3 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Fold in dark chocolate chips. Do not overmix. Spoon mixture into 6 lightly greased silicone cupcake liners.\n\n4 Add water to the Instant Pot\u00ae and insert trivet or steamer basket. Place cupcake liners on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 9 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove muffins from pot and set aside to cool 5 minutes.\n\n## Sausage Gravy\n\nWhether you serve it with gluten-free biscuits, leftover meatloaf, or morning hash browns, this sausage gravy is the ideal savory addition to your brunch table. And don't forget to go heavy on that black pepper . . . it's the charm in this Southern breakfast staple.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 6 minutes\n\nServes 10\n\n* * *\n\n2 tablespoons ghee\n\n1 pound ground pork sausage\n\n1 small sweet onion, peeled and diced\n\n1\/4 cup chicken broth\n\n1\/4 cup cassava flour\n\n11\/2 cups unsweetened almond milk\n\n1\/2 teaspoon sea salt\n\n1 tablespoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Add ghee and heat until melted. Add sausage and onion; stir-fry 3\u20135 minutes until onions are translucent. The pork will still be a little pink in places.\n\n2 Add chicken broth to the pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 1 minute. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Whisk in flour, milk, salt, and pepper.\n\n3 Press the Keep Warm button and let the gravy sit about 5\u201310 minutes to allow the sauce to thicken. Remove from heat and serve warm.\n\n### WHAT IS CASSAVA FLOUR?\n\nCassava flour is made from the yucca, a root vegetable that has been dried and ground. Otto's Naturals Cassava Flour is a high-quality cassava flour that is readily available. It is an all-natural, amazing, grain-free replacement for wheat flour and can often be used as a 1:1 substitute in countless recipes. There are no fillers. So, tie up your apron and get to baking!\n\n## South-of-the-Border Frittata\n\nThis flavorful frittata will have you up and running in the morning. If you are not a fan of cilantro, swap it out with parsley. If a little heat scares your taste buds, substitute the chorizo with breakfast sausage or ground pork, beef, or chicken. Use this recipe as a guide and sub whatever you already have on hand. Frittatas are a great way to clean out the refrigerator.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n4 large whole eggs\n\n4 large egg whites\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1\/4 cup chopped fresh cilantro\n\n2 small Roma tomatoes, seeded and diced\n\n1 medium avocado, peeled and diced\n\n1 tablespoon avocado oil\n\n1\/4 pound chorizo, loose or cut from casing\n\n1 small red onion, peeled and diced\n\n1 small green bell pepper, seeded and diced\n\n1 cup water\n\n1 In a medium bowl, whisk together eggs, egg whites, salt, and pepper. Add cilantro, tomatoes, and avocado. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat oil and stir-fry chorizo, onion, and green pepper 5 minutes or until chorizo is browned and cooked through. Pat chorizo with paper towels to remove excess oil.\n\n3 Transfer cooked chorizo mixture to a greased 7-cup glass bowl and set aside to cool 5 minutes. Pour whisked eggs over the cooked mixture and stir to combine.\n\n4 Add water to the Instant Pot\u00ae and insert trivet. Place dish with egg mixture on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n5 Remove dish from pot and set aside 5\u201310 minutes to allow the eggs to set. Slice and serve.\n\n## Ciao Bella! Egg Muffins\n\nThis breakfast is optimal for those mornings spent running around with no time to spare. Whisk yourself away to Italy with these egg muffins made in no time at all. The Instant Pot\u00ae makes it easy to get a quick vegetable-filled, protein-packed bite on your way out the door.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 3\n\n* * *\n\n1 tablespoon avocado oil\n\n1 small yellow onion, peeled and diced\n\n4 large eggs\n\n2 teaspoons Italian seasoning\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 tablespoon nutritional yeast\n\n4 ounces prosciutto, torn into pieces\n\n1 small Roma tomato, seeded and diced\n\n1\/4 cup chopped fresh spinach\n\n1 cup water\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion and stir-fry 3\u20135 minutes until translucent. Transfer onion to a small bowl to cool.\n\n2 In a medium bowl, whisk together eggs, Italian seasoning, salt, pepper, yeast, prosciutto, tomato, and spinach. Stir in cooled onion mixture. Distribute egg mixture evenly among 6 lightly greased silicone cupcake liners.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Place steamer basket on trivet. Carefully place muffin cups on steamer basket. Lock lid. Press Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Remove egg muffins and serve warm.\n\n### WHAT ON EARTH IS NUTRITIONAL YEAST?\n\nNutritional yeast, or nooch as it is affectionately called, is used by a lot of paleo folks and vegans for its cheesy and nutty flavor. It is a deactivated yeast, so it is not the same as baking or brewer's yeast, but it adds another level of richness to many recipes. It can be found in powder or flake form in most specialty grocery stores or online.\n\n## Ham and Mushroom Egg Muffins\n\nThese make-ahead egg muffins ease the morning rush when you can eat these individual breakfast casseroles on the way to the gym for a little protein boost.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 3\n\n* * *\n\n1 tablespoon avocado oil\n\n1\/2 small yellow onion, peeled and diced\n\n1 cup diced button mushrooms\n\n1\/2 cup small-diced cooked ham\n\n4 large eggs\n\n2 teaspoons yellow mustard\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 cup water\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion, mushrooms, and ham; stir-fry 3\u20135 minutes until onions are translucent. Transfer mixture to a small bowl to cool.\n\n2 In a medium bowl, whisk together eggs, mustard, salt, and pepper. Stir cooled ham mixture into egg mixture. Distribute egg mixture evenly among 6 lightly greased silicone cupcake liners.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Place steamer basket on trivet. Carefully place cupcake liners on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Remove egg muffins and serve warm.\n\n## Bacon, Onion, and Spinach Egg Muffins\n\nNot only are these savory egg muffins a nutritious meal, but you'll avoid a trip to the drive-through window for alternate \"food.\" When purchasing bacon, there are several brands that are paleo-friendly, either from a local farmer, a specialty grocer, or online.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 3\n\n* * *\n\n1 tablespoon avocado oil\n\n1\/2 small yellow onion, peeled and diced\n\n4 slices bacon\n\n1 overflowing cup fresh spinach leaves\n\n4 large eggs\n\nPinch ground nutmeg\n\nPinch sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 cup water\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion and bacon; stir-fry 5 minutes until onions are starting to brown and bacon is crisp. Add spinach for 1 minute until wilted. Transfer mixture to a small bowl to cool.\n\n2 In a medium bowl, whisk together eggs, nutmeg, salt, and pepper. Stir cooled onion mixture into egg mixture. Distribute egg mixture evenly among 6 lightly greased silicone cupcake liners.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Place steamer basket on trivet. Carefully place cupcake liners on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Remove egg muffins and serve warm.\n\n## Soft-Boiled Eggs with Steamed Asparagus Dippers\n\nThese should just be called yum. Eggs cooked in an Instant Pot\u00ae are so easy to peel and this dish cooks the asparagus right in the same pot. Crack those eggs open and dip into that glorious soft-boiled yolk. Round out this meal with some prosciutto and melon balls while dining alfresco!\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 4\n\n* * *\n\n1 cup water\n\n1 bunch asparagus, woody ends removed\n\n4 large eggs\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Place asparagus on steamer basket and place eggs on top. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer eggs to an ice bath to stop the cooking process.\n\n3 Place your egg in a soft-boiled egg holder. Hit the side of the egg halfway down around the midway perimeter of the egg. Peel off upper shell. Dig in with a spoon and then dip your asparagus in that glorious yolk.\n\n## Bell Pepper Egg Cups\n\nBell peppers for breakfast? You better believe it! These peppers are stuffed with ground turkey and eggs making this the perfect protein-packed breakfast to start off your day.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n4 large red bell peppers\n\n1 tablespoon avocado oil\n\n1\/2 medium yellow onion, peeled and diced\n\n3\/4 pound ground turkey\n\n1 small Roma tomato, seeded and diced\n\n1 tablespoon tomato paste\n\n3 large eggs, whisked\n\n1 teaspoon sea salt\n\n1 cup water\n\n1 Cut off the bell pepper tops as close to the tops as possible. Hollow out and discard seeds. Set cleaned peppers aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat avocado oil. Add onion and ground turkey. Stir-fry 5 minutes until onions are starting to soften and turkey is browned. Transfer mixture to a medium bowl to cool.\n\n3 Add remaining ingredients except water to the turkey mixture. Stuff equal amounts of mixture into each of the bell peppers.\n\n4 Add water to the Instant Pot\u00ae and insert trivet. Set peppers upright on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick release remaining pressure until float valve drops and then unlock lid.\n\n5 Transfer peppers to a platter and serve warm.\n\n## Fruity Trail Mix Bowls\n\nIf you've missed eating oatmeal on the paleo diet, this recipe is your saving grace. It is warm, filling, and just a good old breakfast comfort meal. The smooth cooked apples and the crunch of the pecans are married together with the warm spices of cinnamon and nutmeg.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 4\n\n* * *\n\n1 (8-ounce) can crushed pineapple, including juice\n\n2 Granny Smith apples, peeled, cored, and diced\n\n1\/4 cup unsweetened coconut flakes\n\n1\/4 cup crushed pecans\n\n2 tablespoons raw sunflower seeds\n\n2 tablespoons diced dates\n\n2 tablespoons honey\n\n2 tablespoons cassava flour\n\n1\/2 teaspoon ground cinnamon\n\n1\/4 teaspoon ground nutmeg\n\nPinch sea salt\n\n1\/2 cup water\n\n1 Pour crushed pineapple, including juice, into a medium bowl. Add the apples to the pineapple and stir. Add coconut flakes, pecans, sunflower seeds, dates, honey, flour, cinnamon, nutmeg, and salt. Pour ingredients into a lightly greased 7-cup glass bowl.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Place the glass bowl on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Spoon into bowls and serve warm.\n\n## Mixed Berry Breakfast Syrup\n\nThis berrylicious syrup is perfect for your paleo pancakes and gluten-free waffles. Served warm or chilled, this will be a hit at your next brunch or breakfast gathering. Keep refrigerated up to one week if it even lasts that long!\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 3 minutes\n\nYields 1 cup\n\n* * *\n\n1 pound frozen mixed berries\n\n1 tablespoon ghee\n\n1\/2 cup pure maple syrup\n\n1\/4 cup freshly squeezed orange juice\n\n1 tablespoon orange zest\n\nPinch sea salt\n\n1 cinnamon stick\n\n1 Add all ingredients to the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Remove and discard cinnamon stick. Press mixture through a fine-mesh sieve to remove any seeds or skins.\n\n3 Transfer syrup to a lidded container and refrigerate. Serve warm or cold.\n\n## Bacon Jam\n\nSweet and salty, this glorious concoction can give new life to several meals. On top of a burger, alongside your scrambled eggs in the morning, or even licked off a kitchen spoon, Bacon Jam will be your new favorite condiment.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 14 minutes\n\nYields 2 cups\n\n* * *\n\n1 tablespoon coconut oil\n\n1 pound center-cut bacon, diced\n\n1 large yellow onion, peeled and chopped\n\n4 cloves garlic, halved\n\n1\/4 cup apple cider vinegar\n\n1\/2 cup pure maple syrup\n\n1 chipotle in adobo sauce\n\n1 teaspoon adobo sauce from the chipotle jar\n\n1 teaspoon smoked paprika\n\n1 tablespoon instant espresso powder\n\n1\/2 cup water\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add bacon and onion; stir-fry 3\u20134 minutes until bacon fat is rendered and onions are translucent. Add garlic and saut\u00e9 1 minute.\n\n2 Discard all but 1 tablespoon bacon grease from pot. Add remaining ingredients. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use an immersion blender to blend mixture in pot until preferred consistency is reached. Spoon the jam into a jar and refrigerate up to 2 weeks until ready to use.\n\n## Breakfast Casserole with Drop Biscuit Dumplings\n\nMiss that biscuit topping no more. Cassava flour has saved the paleo community. This gluten-free, grain-free, and all-natural product allows a little \"breading\" back into our lives again. Cooked atop this breakfast casserole, it makes this a complete meal made in no time at all.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 4\n\n* * *\n\nDrop Biscuit Dumplings\n\n1\/2 cup cassava flour\n\n1 teaspoon baking powder\n\n1\/4 teaspoon baking soda\n\nPinch sea salt\n\n1 large egg white, whisked\n\n2 tablespoons ghee, melted\n\nBreakfast Casserole\n\n1 pound breakfast pork sausage, loose or cut from casings\n\n1\/2 medium yellow onion, peeled and diced\n\n1\/2 cup diced button or cremini mushrooms\n\n4 large eggs\n\n1 teaspoon freshly ground black pepper\n\n1 In a medium bowl, combine dumpling ingredients. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and stir-fry sausage, onion, and mushrooms 3 minutes to render fat. Transfer mixture to a lightly greased 7-cup glass bowl.\n\n3 Whisk together eggs and black pepper in a small bowl. Pour over sausage mixture.\n\n4 Drop spoonfuls of dumpling mixture over casserole. Place trivet in the Instant Pot\u00ae and place the glass bowl on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove dish from pot and let cool 5 minutes. Use a paper towel to dab off any additional moisture that may have accumulated on the top of casserole. Serve warm.\n\n## Loaded Sweet Potato Hash\n\nWhether you've just finished up a CrossFit class or finally managed to get your kids on the bus, you are starving. This protein-packed power bowl is just what is needed to perk you up and get your body moving for the day.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n6 large eggs\n\n1 tablespoon Italian seasoning\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 pound ground pork sausage\n\n1 large sweet potato, peeled and grated\n\n1 small yellow onion, peeled and diced\n\n2 cloves garlic, minced\n\n1 medium green bell pepper, seeded and diced\n\n2 cups water\n\n1 In a medium bowl, whisk together eggs, Italian seasoning, salt, and pepper. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae. Stir-fry sausage, sweet potato, onion, garlic, and bell pepper 3\u20135 minutes until onions are translucent.\n\n3 Transfer mixture to a 7-cup greased glass bowl. Pour whisked eggs over the sausage mixture.\n\n4 Add water to the Instant Pot\u00ae and insert trivet. Place dish with egg mixture on trivet. Lock lid. Press Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Remove dish from pot. Let sit at room temperature 5\u201310 minutes to allow the eggs to set. Slice and serve.\n\n## Piggy Tater Bowl with a Poached Egg\n\nThis tasty tater bowl is made luxurious with nature's sauce, the beautiful runny yolk of a poached egg! Although white potatoes are not to be consumed every day, this is an all-around treat breakfast especially when you are about to enjoy a full day of hiking with your family or friends.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n3 slices bacon, diced\n\n1 small yellow onion, peeled and diced\n\n1 small green bell pepper, seeded and diced\n\n4 cups small-diced russet potatoes\n\n2 tablespoons ghee\n\n3 cloves garlic, minced\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 cup water\n\n4 large eggs, poached\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add bacon, onion, green pepper, and potatoes; stir-fry 3\u20135 minutes until onions are translucent. Transfer mixture to a lightly greased 7-cup glass bowl. Mix in ghee, garlic, salt, and pepper.\n\n2 Add water to the Instant Pot\u00ae and insert trivet. Place dish on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Remove dish from pot. Spoon mixture into four serving bowls. Garnish each bowl with a poached egg and serve.\n\n## Soups, Stews, and Chilis\n\nSoups, stews, and chilis are perfect foods in the paleo world. Most of them utilize the simplest of ingredients: fresh vegetables from the garden and grass-fed meat floating in a sea of beautiful homemade broth. Or at least in the perfect world. Sometimes the ingredients are just quick-grabbed from the store and we make do with what we can find. And that's okay. You are still eating vegetables, meat, and broth.\n\nOr sometimes you may have all the ingredients for a great soup, but you look at the clock and realize you are only an hour out from dinner. Traditionally, you've needed low heat and a long cooking time to marry all the wonderful spices and flavors that make any soup great, but not anymore. The pressurized heat in your Instant Pot\u00ae can save the day. Although there is a Slow Cook button on your Instant Pot\u00ae for when time isn't an issue, cooking at high pressure can have you serving a finished dinner within an hour. So, break out your ladle because dinner is almost ready. From White Chicken Chili and Mardi Gras Gumbo to Hearty Lamb Stew, this chapter is packed with fresh ideas.\n\nChicken Broth\n\nVegetable Broth\n\nBeef Broth\n\nTo Bean or Not to Bean Chili\n\nChunky Beef Chili\n\nHearty Lamb Stew\n\nCreamy Mushroom Soup\n\nCream of Broccoli and Bacon Soup\n\nTom Kha Goong\n\nKentucky Derby Burgoo\n\nWhite Chicken Chili\n\nSavory Acorn Squash Soup\n\nChicken Taco Soup\n\nCreamy Tomato Basil Soup\n\nMardi Gras Gumbo\n\nQuick Vegetable Soup\n\nHamburger Soup\n\nOma's Sausage and Cabbage Soup\n\nCreamy Ham and Root Veggie Soup\n\nCurried Carrot Soup\n\n## Chicken Broth\n\nDon't skip this recipe after you cook a whole chicken or purchase a cooked rotisserie chicken from the supermarket. Save the carcass and bones to make this nutrient- and mineral-packed Chicken Broth that you can use for a soup base in place of store-bought broth.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 30 minutes\n\nYields 6 cups\n\n* * *\n\n1 chicken carcass from a whole chicken\n\n2 large carrots, peeled and cut into chunks\n\n2 stalks celery, cut into chunks\n\n1 small yellow onion, peeled and chopped\n\n1 bay leaf\n\n2 cloves garlic, halved\n\n1\/2 teaspoon apple cider vinegar\n\n1 teaspoon sea salt\n\n6 cups water\n\n1 Place all ingredients in the Instant Pot\u00ae. Press the Manual or Pressure Cook button and adjust time to 30 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n2 Use a slotted spoon to remove and discard solids from the broth. Strain the remaining liquid through a fine-mesh sieve or cheesecloth. Refrigerate broth up to 4 days or freeze up to 6 months.\n\n## Vegetable Broth\n\nBroths are the base, the very foundation, of almost any soup that you make, so feel free to alter this recipe in order to personalize the flavor of the broth. For example, if you know a guest dislikes garlic, make this broth without it. If you have everything but the parsnip, eliminate it. Tailor this broth to your own taste buds and preferences as well. It will be amazing!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 30 minutes\n\nYields 6 cups\n\n* * *\n\n3 large carrots, peeled and cut into chunks\n\n2 stalks celery, cut into chunks\n\n1 small yellow onion, peeled and chopped\n\n1 medium parsnip, peeled and chopped\n\n1 bay leaf\n\n2 cloves garlic, halved\n\n1\/2 teaspoon apple cider vinegar\n\n1 teaspoon sea salt\n\n6 cups water\n\n1 Place all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 30 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n2 Use a slotted spoon to remove and discard solids from the broth. Strain the remaining liquid through a fine-mesh sieve or cheesecloth. Refrigerate broth up to 4 days or freeze up to 6 months.\n\n## Beef Broth\n\nBeef soup bones can be bought at most supermarkets and butcher shops. If they are not on display, don't hesitate to ask. Making your own beef and bone broth is very popular these days, so it won't seem like a weird question. You can also use oxtail or neck bones as a substitute.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 30 minutes\n\nYields 6 cups\n\n* * *\n\n3 pounds beef soup bones\n\n2 large carrots, peeled and cut into chunks\n\n2 stalks celery, cut into chunks\n\n1 small yellow onion, peeled and chopped\n\n1 bay leaf\n\n2 cloves garlic, halved\n\n1\/2 teaspoon apple cider vinegar\n\n1 teaspoon sea salt\n\n6 cups water\n\n1 Place all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 30 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n2 Use a slotted spoon to remove and discard solids from the broth. Strain the remaining liquid through a fine-mesh sieve or cheesecloth. Refrigerate broth up to 4 days or freeze up to 6 months.\n\n## To Bean or Not to Bean Chili\n\nBeans are part of the legume family and are not recommended with the paleo lifestyle. Without sounding too scientific, they contain phytates, which lead to improper digestion. They basically inhibit the absorption of certain nutrients and minerals and can lead to a \"leaky gut\" over time.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 40 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n1\/2 pound ground pork\n\n1\/2 pound ground beef\n\n1 cup diced red onion\n\n1 small green bell pepper, seeded and diced\n\n1 large carrot, peeled and diced\n\n3 cloves garlic, minced\n\n2 tablespoons chili powder\n\n1 teaspoon ground cumin\n\n1 small jalape\u00f1o, seeded and diced\n\n1 teaspoon sea salt\n\n1 (28-ounce) can pur\u00e9ed tomatoes, including juice\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat oil. Add the ground pork, ground beef, and onion. Stir-fry 5 minutes until pork is no longer pink.\n\n2 Add remaining ingredients to pot and stir to combine. Lock lid. Press the Meat button and cook for default time of 35 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Serve warm.\n\n## Chunky Beef Chili\n\nIf you're looking to cook up some chili for game day, this recipe is beyond compare. The thinly sliced jalape\u00f1os used to top off this hearty bowl of chili will spice up your day and get you through the pregame jitters. Fire up your Instant Pot\u00ae and get ready to enjoy some tailgating with your friends.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 30 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n1 pound beef stew cubes\n\n1 medium yellow onion, peeled and diced\n\n4 cloves garlic, minced\n\n1 large carrot, peeled and cut into 1\/2\" pieces\n\n2 cups beef broth\n\n2 cups sliced cremini mushrooms\n\n1 tablespoon chili powder\n\n1 tablespoon Italian seasoning\n\n1 teaspoon sea salt\n\n1\/2 teaspoon cayenne pepper\n\n1\/2 teaspoon freshly ground black pepper\n\n2 tablespoons tomato paste\n\n1 (14.5-ounce) can diced tomatoes, including juice\n\n1 Press Saut\u00e9 button on Instant Pot\u00ae. Heat oil and add beef and onion. Stir-fry 3 minutes, searing beef and stir-frying until onions are translucent. Add garlic and carrot; saut\u00e9 2 minutes.\n\n2 Add beef broth to pot and deglaze by scraping up any bits from the bottom and sides of the pot.\n\n3 Stir in all remaining ingredients. Lock lid. Press Meat button and cook for default time of 25 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n4 Ladle into individual bowls and serve warm.\n\n## Hearty Lamb Stew\n\nForget the family table. Enjoy this incredibly satisfying and hearty stew around the fireplace on a cold winter's night. And, if you happen to have leftovers, it is even better the next day.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 40 minutes\n\nServes 6\n\n* * *\n\n2 tablespoons avocado oil\n\n2 pounds boneless lamb, cubed\n\n1 medium sweet onion, peeled and diced\n\n4 cloves garlic, minced\n\n3 cups beef broth\n\n1 (14.5-ounce) can crushed tomatoes, including juice\n\n2 medium carrots, peeled and diced\n\n1 medium turnip, peeled and small-diced (greens cleaned and chopped)\n\n1\/2 teaspoon fennel seeds\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/4 cup chopped fresh mint\n\n1\/4 cup chopped fresh Italian flat-leaf parsley\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add lamb and onion; stir-fry 3\u20135 minutes, searing lamb and stir-frying until onions are translucent. Add garlic and saut\u00e9 1 minute.\n\n2 Add beef broth and deglaze pot by scraping up any bits from the sides and bottom of the pot. Add tomatoes, including juice, carrots, turnip including chopped greens, fennel seeds, salt, and pepper. Lock lid. Press the Meat button and cook for the default time of 35 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n3 Stir in mint. Ladle into individual bowls, garnish with parsley, and serve warm.\n\n## Creamy Mushroom Soup\n\nYou won't even miss the heavy cream in this dairy-free creamy soup. The pur\u00e9ed cauliflower adds a smoothness and the mushrooms lend earthy and hearty notes needed to make this bowl of soup sing.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons ghee\n\n1 small sweet onion, peeled and diced\n\n2 cups sliced button mushrooms\n\n1 small head cauliflower, chopped\n\n1 stalk celery, diced\n\n4 cups chicken broth\n\n2 teaspoons dried thyme\n\n1 teaspoon dried oregano\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add onion and mushrooms; saut\u00e9 3\u20135 minutes until onions are translucent. Transfer half the mushroom mixture to a small bowl and set aside.\n\n2 Add remaining ingredients to the Instant Pot\u00ae. Lock lid. Press the Soup button and adjust time to 20 minutes. When timer beeps, let pressure release naturally for 10 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Use an immersion blender to blend the soup in the pot until smooth. Stir reserved mushroom mixture into the pot and heat 3\u20135 minutes.\n\n4 Ladle into bowls and serve warm.\n\n### MUSHROOM VARIETIES\n\nMushrooms create a woody and unique flavor to many dishes. Each variety brings a slightly different flavor. Oyster mushrooms have a bit of a briny note, shiitakes have a smoky hint, and morels have a fabulous nuttiness. Change up the variety of mushroom that you use in this recipe or use a mix of varieties for a true \"wild mushroom\" blend for balance.\n\n## Cream of Broccoli and Bacon Soup\n\nThis smooth and creamy broccoli soup is enhanced with the smoky and crunchy bacon garnish. This dish is the perfect comfort meal after a brisk run in the cool fall evenings or even after a family snowball fight in the winter.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 30 minutes\n\nServes 4\n\n* * *\n\n1 teaspoon ghee\n\n6 pieces of bacon, diced\n\n1 medium sweet onion, peeled and diced\n\n1 large carrot, peeled and diced\n\n2 stalks celery, diced\n\n1 pound broccoli, chopped\n\n1 tablespoon nutritional yeast\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n4 cups chicken broth\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat the ghee. Add half the bacon and heat 3\u20135 minutes until crispy. Set aside on a plate lined with paper towels to cool. Add remaining bacon and onion; saut\u00e9 3\u20135 minutes until the onions are translucent.\n\n2 Add all remaining ingredients to Instant Pot\u00ae. Lock lid. Press the Soup button and adjust time to 20 minutes. When timer beeps, quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Use an immersion blender to blend the soup in the pot until smooth.\n\n4 Ladle the soup into serving bowls and garnish with reserved bacon. Serve warm.\n\n## Tom Kha Goong\n\nAdding the shrimp at the end of the cooking process keeps it from getting overcooked. The other ingredients, however, go through the steaming process to really marry all the rich flavors\u2014so much so that you'll think that this soup has been slow cooking for hours.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon coconut oil\n\n6 green onions, sliced\n\n1 cup sliced shiitake mushrooms\n\n2 cloves garlic, minced\n\n3 cups vegetable broth\n\n1 tablespoon Thai red curry paste\n\n1 tablespoon fish sauce\n\n2 tablespoons minced lemongrass (soft inner layer only)\n\n1 teaspoon honey\n\n1 cup canned coconut milk\n\n1 tablespoon fresh lime juice\n\n1\/4 teaspoon dried red pepper flakes\n\n1 teaspoon sea salt\n\n1\/2 teaspoon white pepper\n\n1\/4 cup chopped fresh cilantro\n\n1 pound medium shrimp, peeled and deveined\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add green onions and mushrooms; saut\u00e9 3\u20135 minutes until mushrooms are tender. Add garlic and saut\u00e9 1 minute.\n\n2 Add remaining ingredients except shrimp to pot. Lock lid. Press the Soup button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Add shrimp and simmer 5 minutes until opaque and pink.\n\n4 Ladle into bowls and serve warm.\n\n## Kentucky Derby Burgoo\n\nBurgoo? What's that? It's a traditional stew in Louisville especially celebrated during the week of the Kentucky Derby. Once cooked in large iron kettles, this concoction used meat from that day's hunting expedition and vegetables that were already in the house. Burgoo typically isn't ever made the same way twice and is different from family to family, so let this recipe be your guide and tweak it as you see fit.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 40 minutes\n\nServes 6\n\n* * *\n\n2 tablespoons avocado oil\n\n1\/2 pound chicken thighs, cubed\n\n1\/2 pound beef, cubed\n\n4 cups beef broth\n\n1 stalk celery, diced\n\n1 cup asparagus, trimmed and cut into 1\" pieces\n\n1\/2 cup sliced fresh or frozen okra\n\n2 small red potatoes, peeled and cubed\n\n1 small carrot, peeled and diced\n\n1 small yellow onion, peeled and diced\n\n4 cloves garlic, minced\n\n1 (15-ounce) can crushed tomatoes, including juice\n\n2 tablespoons Worcestershire sauce\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat the oil and add the chicken and beef cubes. Sear meat 3\u20134 minutes until browned on all sides.\n\n2 Add broth and deglaze the pot by scraping up bits from the sides and bottom of the pot.\n\n3 Add remaining ingredients. Lock lid. Press the Meat button and cook for the default time of 35 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n4 Ladle into individual bowls and serve warm.\n\n## White Chicken Chili\n\nThis warmth-inducing, crowd-pleasing dish will have your friends asking if they can watch the game at your house every weekend. And not only will you have the winning dish of the day, but you will actually get to visit with your guests instead of being stuck in the kitchen cooking. Let your Instant Pot\u00ae do all the work!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 20 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n1 pound ground chicken\n\n1 medium yellow onion, peeled and diced\n\n1 stalk celery, diced\n\n3 cloves garlic, minced\n\n1 (28-ounce) can diced tomatoes, including liquid\n\n1 (4-ounce) can diced green chilies, including liquid\n\n1 teaspoon sea salt\n\n1 tablespoon chili powder\n\n1 tablespoon fresh thyme leaves\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add chicken, onion, and celery; stir-fry 5 minutes or until chicken is no longer pink. Add garlic and saut\u00e9 1 minute.\n\n2 Stir in remaining ingredients. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n3 Ladle into individual bowls and serve warm.\n\n## Savory Acorn Squash Soup\n\nAcorn squash has an inherently sweet taste, but coupled with some savory pantry spices and salty bacon, this squash takes on a new life.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 23 minutes\n\nServes 4\n\n* * *\n\n1\/2 cup water\n\n2 medium acorn squash, halved and seeds removed\n\n1 tablespoon ghee\n\n2 slices bacon, diced\n\n1 small yellow onion, peeled and diced\n\n1 stalk celery, diced\n\n1 medium carrot, peeled and diced small\n\n3 cloves garlic, quartered\n\n3 cups chicken broth\n\n1 cup canned coconut milk\n\n1 teaspoon celery seed\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1\/8 teaspoon cayenne pepper\n\n1 Add water to the Instant Pot\u00ae and insert trivet. Place squash on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 4 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer squash to a cutting board and discard liquid from pot. When cool enough to work with, scoop out squash meat and set aside. Discard skin.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee until melted. Add bacon, onion, celery, and carrot; stir-fry 3\u20134 minutes until fat is rendered from bacon and onion is translucent. Add garlic and saut\u00e9 1 minute.\n\n4 Add chicken broth and deglaze pot by scraping up the bits from the bottom and sides of the pot. Add all remaining ingredients and lock lid. Press the Soup button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Use an immersion blender to pur\u00e9e the soup in the pot, or use a blender and pur\u00e9e it in batches.\n\n6 Ladle the soup into bowls and serve warm.\n\n## Chicken Taco Soup\n\nThis soup is filled with chicken, vegetables, and taco spices. The chicken is cooked in its juices and even pulled apart directly in the pot to ensure the tenderness and moistness of the meat.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n1 small red onion, peeled and diced\n\n1 medium yellow bell pepper, seeded and diced\n\n3 cloves garlic, quartered\n\n1 (28-ounce) can diced tomatoes, including juice\n\n1 cup water\n\n1\/2 pound boneless, skinless chicken breasts\n\n1 medium jalape\u00f1o, seeded and diced\n\n1 (2.25-ounce) can diced black olives, drained\n\n2 tablespoons chili powder\n\n1 tablespoon ground cumin\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1\/2 cup chopped fresh cilantro, divided\n\n1 medium avocado, pitted, peeled, and sliced\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion and bell pepper; saut\u00e9 3\u20135 minutes until onions are translucent. Add garlic and heat an additional minute. Add tomatoes, including juice, and water; deglaze pot by scraping up any bits from the bottom and sides of the pot.\n\n2 Add chicken, jalape\u00f1o, black olives, chili powder, cumin, salt, pepper, and 1\/4 cup cilantro. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Use two forks to shred the chicken in the pot. Let simmer 5 minutes.\n\n3 Ladle soup into bowls and garnish with avocado slices and remaining cilantro.\n\n## Creamy Tomato Basil Soup\n\nSkip the usual step of peeling the tomatoes for this recipe. Once steamed and blended, there will be no trace of the skins. Also, if you can get your hands on some heirloom tomatoes, try them in this dish. They are the ugly duckling of the produce world, but their swan wings will be revealed once you take a slurp of this creamy soup.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon coconut oil\n\n1 small sweet onion, peeled and diced\n\n1 stalk celery, thinly sliced\n\n8 medium tomatoes, seeded and quartered\n\n3 cups chicken broth\n\n1\/4 cup julienned fresh basil, divided\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 cup unsweetened almond milk\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion and celery; saut\u00e9 3\u20135 minutes until onions are translucent. Add tomatoes and saut\u00e9 3 minutes or until tomatoes start to break down.\n\n2 Add broth, 1\/8 cup basil, salt, and pepper to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 7 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Add almond milk. Use an immersion blender to pur\u00e9e the soup in the pot or pur\u00e9e it in a stand blender in batches.\n\n4 Ladle into bowls and garnish with remaining basil. Serve warm.\n\n### COMPANION PLANTS FOR GARDEN TOMATOES\n\nJust like grapes meant for fine wines take on some of the flavors of the plants grown around them, so do tomatoes. Not only do dill and basil plants add another level of flavor to a gardener's tomato plants, but they also help repel disease and destructive insects harmful to the tomato plants.\n\n## Mardi Gras Gumbo\n\nBecause the booze usually flies freely during Mardi Gras, make sure you get some solid food in your stomach before you get started. So plug in your Instant Pot\u00ae and get your taste buds ready to take a quick vacation down South. This dish is traditionally served over rice, but this gumbo is just as delicious as is or over some riced cauliflower.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 6\n\n* * *\n\n1 tablespoon ghee\n\n1 medium yellow onion, peeled and diced\n\n2 stalks celery, diced\n\n1 small red bell pepper, seeded and diced\n\n1\/2 pound skinless, boneless chicken thighs, cut into 1\/2\" pieces\n\n1 pound andouille sausage, sliced\n\n3 cloves garlic, minced\n\n3 cups chicken broth\n\n1 teaspoon Worcestershire sauce\n\n1\/4 cup cassava flour\n\n1 pound medium shrimp, peeled, deveined, tails removed\n\n1 tablespoon gumbo fil\u00e9 powder\n\n1 teaspoon garlic powder\n\n1 teaspoon cayenne pepper\n\n1 teaspoon fresh thyme leaves\n\n1\/4 cup chopped fresh Italian flat-leaf parsley, divided\n\n10-ounces frozen or fresh sliced okra\n\n1 (14.5-ounce) can stewed tomatoes, including juice\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add onion, celery, and bell pepper; saut\u00e9 3\u20135 minutes until onions are translucent. Add chicken, sausage, and garlic; stir-fry 2\u20133 minutes until chicken is no longer pink.\n\n2 Add chicken broth and deglaze the pot by scraping up the brown bits from the bottom and sides of the pot. Add Worcestershire sauce, flour, shrimp, fil\u00e9 powder, garlic powder, cayenne pepper, thyme, 1\/8 cup parsley, okra, and tomatoes. Lock lid. Press the Soup button and adjust time to 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Ladle gumbo into bowls and garnish with remaining parsley.\n\n## Quick Vegetable Soup\n\nThis low-calorie vegetable soup is loaded with nutrients, and it is a great starter soup for a family that generally isn't crazy about vegetables. Add chicken or sausage to make this a more complete meal.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n4 cups vegetable broth\n\n2 stalks celery, diced\n\n1 small carrot, peeled and diced\n\n1 small yellow onion, peeled and diced\n\n2 cloves garlic, minced\n\n1 (15-ounce) can diced tomatoes, including juice\n\n1 medium turnip, peeled and diced\n\n1 medium zucchini, diced\n\n1 tablespoon Italian seasoning\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 Add all ingredients to the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Ladle into individual bowls and serve warm.\n\n## Hamburger Soup\n\nSkip the processed foods, trans fats, and wasted money of drive-through meals. This Hamburger Soup hits all the flavors of your favorite burger. And don't skip that pickle garnish\u2014it makes the soup!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n1 pound ground beef\n\n1 medium yellow onion, peeled and diced\n\n1 medium green bell pepper, seeded and diced\n\n1 (15-ounce) can diced tomatoes, including juice\n\n2 teaspoons yellow mustard\n\n1 teaspoon smoked paprika\n\n1 teaspoon garlic powder\n\n1\/2 teaspoon sea salt\n\n4 cups beef broth\n\n2 cups shredded iceberg lettuce\n\n1\/2 cup diced dill pickles\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add beef, onion, and green pepper; saut\u00e9 3\u20135 minutes until beef begins to brown.\n\n2 Add tomatoes, including juice, mustard, paprika, garlic powder, salt, and beef broth to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 7 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Stir in lettuce and simmer 3 minutes. Ladle into bowls and garnish with diced dill pickles. Serve warm.\n\n## Oma's Sausage and Cabbage Soup\n\nIf you're craving a home-cooked hearty meal, become your own German grandma. The simplicity and heartiness of the meal will taste like you've been making it for years. Nothing soothes the soul better than a warm bowl of soup.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 13 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n2 medium leeks, diced (white and light green portions only)\n\n1 stalk celery, diced\n\n1 (16-ounce) bag coleslaw mix\n\n8 ounces precooked sausage links, sliced on an angle\n\n1 medium turnip, peeled and diced\n\n1 tablespoon German stone-ground mustard\n\n2 tablespoons fresh thyme leaves, divided\n\n1\/2 teaspoon caraway seeds\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n4 cups beef broth\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add leeks and celery; saut\u00e9 3\u20135 minutes until onion is tender.\n\n2 Add remaining ingredients to pot and lock lid. Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Ladle soup into bowls and serve warm.\n\n## Creamy Ham and Root Veggie Soup\n\nA meal straight from the farm. What a perfect soup to make from a hodgepodge of vegetables from your CSA box that you're not sure what to do with. This bowl of comfort is a great introduction to some root vegetables that you might not have cooked with before.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 20 minutes\n\nServes 4\n\n* * *\n\n3 tablespoons ghee\n\n1 medium sweet onion, peeled and diced\n\n3 cloves garlic, minced\n\n1 large carrot, peeled and diced\n\n1 cup peeled and cubed celeriac\n\n1 small rutabaga, peeled and diced\n\n4 cups chicken broth\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n2 tablespoons fresh thyme leaves, divided\n\n1 tablespoon cooking sherry\n\n2 cups cubed cooked ham\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add onion and saut\u00e9 3\u20135 minutes until translucent. Add garlic and saut\u00e9 1 minute until fragrant.\n\n2 Add carrot, celeriac, rutabaga, chicken broth, salt, pepper, and 1 tablespoon thyme to pot and lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use an immersion blender to pur\u00e9e the soup in the pot or pur\u00e9e in a stand blender in batches. Stir in sherry and ham and let simmer 5 minutes.\n\n4 Ladle into bowls and garnish with remaining thyme. Serve warm.\n\n### WHAT IS A CSA?\n\nCSA stands for community-supported agriculture. It allows a local community member to receive a box per week or biweekly made up of a produce collection from regional farmers. You are essentially buying a share of the crop which is paid for upfront allowing the farmers to plan better for their season while you receive crops that are completely fresh and haven't left a large carbon footprint in its wake.\n\n## Curried Carrot Soup\n\nFresh and creamy, this Curried Carrot Soup is a perfect lunch companion to a big green salad.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 14 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons ghee\n\n3 cups peeled and sliced carrots\n\n6 green onions, sliced, greens and whites separated\n\n3 cups chicken broth\n\n1 cup canned coconut milk\n\n1 tablespoon curry powder\n\n1\/2 teaspoon ground cumin\n\n1\/4 teaspoon ground ginger\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 medium lime, quartered\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and melt ghee. Add carrots and onion whites; stir-fry 3\u20134 minutes until onions are tender.\n\n2 Add all remaining ingredients except onion greens and lime to pot and lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use an immersion blender to pur\u00e9e the soup in the pot or pur\u00e9e in a stand blender in batches.\n\n4 Ladle into bowls and garnish with onion greens. Serve warm with a squeeze of lime.\n\n## Appetizers and Sauces\n\nLiving within certain diet requirements is never harder than when you are with a group of friends. Chips and dips and cheesy everything seem to swarm the tables. But boring plates of inedible food doesn't have to be your fate. Bring a healthy paleo option to share. Or, better yet, throw your own party and show the folks what they've been missing! Appetizers are a great way to bring a group of people together to enjoy a game or hold them over until you serve a meal.\n\nBut you don't want to have to wear your chef hat all day, stuck in the kitchen, so let the Instant Pot\u00ae be your helpful sous chef. Quick, hot food in minutes? Now that's how you throw a party! With amazing appetizers ranging from Broccoli-Zucchini Hummus and Liver P\u00e2t\u00e9 to Tandoori-Style Chicken Wings and Deviled Eggs, the only problem you'll have incorporating the recipes in this chapter into your next social event will be deciding which ones to make. And after the appetizers have been devoured, make some of the rich sauces from this chapter such as Bolognese or Puttanesca to ladle over some spiralized zucchini or carrots. Your guests will leave full and satisfied not even knowing that they ate \"paleo\" all night!\n\nDeviled Eggs\n\nMini Chili Polpette\n\n\"Cheesy\" Spinach Crab Dip\n\nEndive Boats with Chicken Salad\n\nTandoori-Style Chicken Wings\n\nSesame-Orange Chicken Wings\n\nBroccoli-Zucchini Hummus\n\nBurgundy Mushrooms\n\nMushrooms Stuffed with Sausage and Kimchi\n\nSteamed Shrimp with Cocktail Sauce\n\nLiver P\u00e2t\u00e9\n\nSteamed Artichokes with Lemon-Herb Butter\n\nRutabaga Fries with Rosemary Dipping Sauce\n\nBroccoli-Cauli Sausage Tots\n\nSalsa Verde\n\nBolognese Sauce\n\nMarinara Sauce\n\nPuttanesca Sauce\n\nTomato Curry Sauce\n\n## Deviled Eggs\n\nThe hardest part about making this recipe is usually the actual peeling of the eggs\u2014but not with the Instant Pot\u00ae. It produces the most peelable eggs, so you won't have all those pesky gouges in the sides. Note that the cooking time will vary slightly depending on the size of the eggs as well as how fresh they are.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 4 minutes\n\nYields 12 deviled eggs\n\n* * *\n\n1 cup water\n\n6 large eggs\n\n3 tablespoons mayonnaise\n\n1 teaspoon yellow mustard\n\n1\/2 teaspoon dill pickle juice\n\n2 teaspoons finely diced dill pickles\n\n1\/8 teaspoon sea salt\n\n1\/8 teaspoon freshly ground black pepper\n\nSmoked paprika, for garnish\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Place eggs on basket. Press the Manual or Pressure Cook button and adjust time to 4 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Transfer eggs to an ice bath to stop the cooking process.\n\n2 Peel eggs. Slice each egg in half lengthwise and place yolks in a small bowl. Place egg white halves on a serving tray.\n\n3 Add mayonnaise, mustard, pickle juice, diced pickles, salt, and pepper to the bowl with the yolks. Use a fork to blend until smooth.\n\n4 Spoon or pipe the yolk filling into the egg white halves. Sprinkle with paprika for garnish.\n\n## Mini Chili Polpette\n\nPolpette literally means \"meatballs\" and there are many versions across the various regions of Italy. With these Mini Chili Polpette you can introduce your guests to a new word and wow their taste buds at the same time. And, they will be even more impressed when you show them how easy it is to cook this dish in your Instant Pot\u00ae.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 7 minutes\n\nYields 24 meatballs\n\n* * *\n\n1 pound ground beef\n\n1 large egg\n\n1\/4 cup chopped fresh Italian flat-leaf parsley\n\n1\/4 cup almond meal\n\n2 tablespoons finely chopped green onions\n\n1 tablespoon chili powder\n\n1 tablespoon Dijon mustard\n\n1 teaspoon sea salt\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n1 In a medium bowl, combine ground beef, egg, parsley, almond meal, green onions, chili powder, mustard, and salt. Form the mixture into 24 meatballs.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Add half the meatballs to the pot and sear about 3 minutes, turning them to brown all sides. Remove the first batch and set aside. Add the remaining oil and repeat with the remaining meatballs. Remove the meatballs from the pan.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Place the meatballs evenly on the steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 1 minute. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer the polpette to a serving tray and serve warm.\n\n## \"Cheesy\" Spinach Crab Dip\n\nThe nutritional yeast gives this dip a cheesy flavor that your guests won't even know is missing. For a browned top, place the dip under the broiler 1\u20133 minutes until it browns to your desired doneness. Serve with a variety of fresh vegetables such as radishes, broccoli florets, cauliflower, squash rounds, celery, and carrots.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 16 minutes\n\nServes 8\n\n* * *\n\n2 tablespoons ghee\n\n1 small sweet onion, peeled and diced\n\n1 cup diced zucchini\n\n1 overflowing cup fresh spinach leaves\n\n1\/2 teaspoon sea salt\n\n3\/4 pound lump crab meat\n\n2 tablespoons nutritional yeast\n\n1\/2 teaspoon Worcestershire sauce\n\n2 teaspoons prepared horseradish\n\n1 teaspoon Old Bay Seasoning\n\n1 teaspoon sriracha\n\n2 teaspoons lemon zest\n\n1\/4 teaspoon freshly ground black pepper\n\n2 cups water\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add onion and zucchini. Stir-fry 3\u20135 minutes until zucchini is tender. Add spinach and heat 1 additional minute to wilt spinach.\n\n2 Transfer saut\u00e9ed ingredients and salt to a food processor or blender and pulse until smooth.\n\n3 Combine zucchini pur\u00e9e and remaining ingredients except water in a medium bowl until well blended. Spoon into a lightly greased 7-cup glass bowl.\n\n4 Add water to the Instant Pot\u00ae and insert trivet. Place glass bowl on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove dish and serve warm.\n\n## Endive Boats with Chicken Salad\n\nTalk about time-saving\u2014the chicken and the egg are cooked together. Make sure your chicken isn't cut too small or it will dry out. Twenty-five endive leaves are suggested in this recipe, but if you have leftover chicken salad, it is perfect served the next day as a lettuce wrap for a quick lunch!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 5 minutes\n\nYields 3 cups\n\n* * *\n\n1 pound chicken breasts, cut into 1\" cubes\n\n11\/2 cups water\n\n1 large egg\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1 stalk celery, finely diced\n\n1 teaspoon yellow mustard\n\n1\/2 teaspoon dill pickle juice\n\n1 tablespoon finely diced dill pickle\n\n1\/2 cup mayonnaise\n\n2 slices bacon, cooked and crumbled\n\n25 endive leaves\n\n1 Add chicken and water to the Instant Pot\u00ae. Add steamer basket to pot and place egg on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer egg to an ice bath to stop the cooking process.\n\n3 Use a slotted spoon to transfer the chicken to a cutting board. When cool enough to work with, finely chop the chicken. Add to a medium bowl. Peel egg and dice. Add to bowl.\n\n4 Add salt, pepper, celery, mustard, pickle juice, pickles, mayonnaise, and bacon. Stir ingredients until well combined. Refrigerate covered until ready to serve.\n\n5 Spoon chicken salad onto endive leaves. Serve.\n\n### IS MAYONNAISE PALEO?\n\nHomemade mayonnaise is your best bet to stay strictly paleo of course. It is quick to make and you can control the ingredients that go into your recipe. Primal Kitchen makes a glorious jarred regular mayonnaise as well as an avocado mayonnaise that can be found online or in specialty stores. Pssst: don't forget to try their Chipotle Lime Mayo too!\n\n## Tandoori-Style Chicken Wings\n\nThis recipe still captures the flavors and essence of tandoori-style without the dairy. Perfect \"dude\" food, these are a sophisticated twist on the traditional buffalo-style staple found at most tailgating parties.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\n1 small yellow onion, peeled and diced\n\n2\" piece ginger, peeled and minced\n\n4 cloves garlic, minced\n\n2 tablespoons fresh lime juice\n\n1 cup canned coconut milk\n\n2 tablespoons avocado oil\n\n1 tablespoon ground cumin\n\n1 tablespoon ground coriander\n\n2 teaspoons sea salt\n\n1 teaspoon white pepper\n\n1 teaspoon cayenne pepper\n\n2 tablespoons tomato paste\n\n3 pounds chicken wings, separated at the joint if necessary\n\n1 cup chicken broth\n\n1\/4 cup chopped fresh mint\n\n1 In a large bowl, combine onion, ginger, garlic, lime juice, coconut milk, avocado oil, cumin, coriander, salt, white pepper, cayenne pepper, and tomato paste. Add wings to mixture and toss. Refrigerate covered at least 1 hour or up to overnight.\n\n2 Add chicken broth to the Instant Pot\u00ae and insert steamer basket. Add chicken wings, arranging them so they aren't sitting on top of one another; place them standing up if necessary. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Use a slotted spoon to transfer the wings to a serving tray. Garnish with chopped mint and serve.\n\n3 Optional: Before garnishing, place the wings on a baking sheet and broil 3 minutes on each side to crisp up the chicken.\n\n### THE THICK LAYER IN A CAN OF COCONUT MILK\n\nDon't throw away that can of coconut milk if you notice a thick layer on top and a cloudy thinner liquid underneath. This is normal. The top layer is the coconut cream, which is full of the natural fats and flavor. Use this first. Also, if you have a warm kitchen, try putting the can of coconut milk in the refrigerator up to an hour before using to ensure this layer. On a yummy note, for desserts, whip up this thick layer with a little honey for some dairy-free whipped cream.\n\n## Sesame-Orange Chicken Wings\n\nThe brightness of the orange, the sweetness of the maple syrup, the saltiness of the coconut aminos, and the bit of heat from the ginger and red pepper flakes makes these wings a balanced and delicious addition to your appetizer spread for your family and guests.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 16 minutes\n\nServes 6\n\n* * *\n\n1\/4 cup freshly squeezed orange juice\n\n2 tablespoons orange zest\n\n1\/4 cup pure maple syrup\n\n2 tablespoons coconut aminos\n\n2\" piece ginger, peeled and minced\n\n3 cloves garlic, minced\n\n2 tablespoons sesame oil\n\n1\/2 teaspoon dried red pepper flakes\n\n1 teaspoon sea salt\n\n1 tablespoon arrowroot flour\n\n1 bunch green onions, sliced and separated into whites and greens\n\n3 pounds chicken wings, separated at the joint if necessary\n\n1 cup chicken broth\n\n1\/4 cup toasted sesame seeds\n\n1 In a large bowl, combine orange juice, orange zest, syrup, coconut aminos, ginger, garlic, sesame oil, red pepper flakes, salt, flour, and whites of green onions. Transfer 2 tablespoons of the sauce to a small bowl and set aside.\n\n2 Add wings to sauce in large bowl and toss. Refrigerate covered at least 1 hour or up to overnight.\n\n3 Add chicken broth to the Instant Pot\u00ae and insert steamer basket. Add chicken wings, arranging them so they aren't sitting on top of one another; place them standing up if necessary. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n4 Use a slotted spoon to transfer the wings to a baking sheet. Brush with the 2 tablespoons of reserved sauce. Broil 3 minutes on each side to add some crisp to the chicken.\n\n5 Transfer to a serving tray and garnish with sesame seeds and the greens of the onions.\n\n### WHAT ARE COCONUT AMINOS?\n\nA gluten-free and soy-free alternative to soy sauce, this underrated condiment is a staple in paleo diets. Used 1:1 and an incredible taste substitute, this sauce is created from aged coconut sap and sea salt. Coconut aminos are found in most specialty grocers as well as online.\n\n## Broccoli-Zucchini Hummus\n\nA perfect dip served with crudit\u00e9s and some fresh pita slices for your non-paleo guests. Though hummus is traditionally made with chickpeas, this Broccoli-Zucchini Hummus offers not only a handful of nutrients, but a flavor combination that is out of this world.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 10\n\n* * *\n\n1 cup water\n\n1 small head broccoli, diced\n\n1 medium zucchini, large chopped\n\n1\/2 cup tahini\n\n4 cloves garlic, quartered\n\n2 tablespoons fresh lemon juice\n\n1 teaspoon lemon zest\n\n1\/2 teaspoon ground cumin\n\n1\/2 teaspoon smoked paprika\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1\/4 cup extra-virgin olive oil, plus extra for garnish\n\n2 tablespoons finely diced jarred roasted red peppers\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Add broccoli and zucchini. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer broccoli and zucchini to a food processor or blender and pulse 2\u20133 times. Add tahini, garlic, lemon juice, lemon zest, cumin, paprika, salt, and pepper; pulse 5\u20136 times. Add olive oil 1 tablespoon at a time, pulsing between additions, until desired consistency is reached. Transfer to a serving dish.\n\n3 Garnish with diced roasted red peppers placed in a small circle in the center. Add an extra drizzle of olive oil. Serve.\n\n## Burgundy Mushrooms\n\nThese Burgundy Mushrooms are a side dish you'll find yourself making again and again. Don't skip the bacon in this recipe as the meatiness and saltiness add a deep flavor to this insanely rich and earthy side dish. Serve alongside a grilled steak and you'll have a meal to be proud of.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 30 minutes\n\nServes 8\n\n* * *\n\n1\/2 cup ghee\n\n3 cloves garlic, halved\n\n16 ounces whole button mushrooms\n\n16 ounces whole baby bella mushrooms\n\n11\/2 cups dry red wine\n\n1 teaspoon Worcestershire sauce\n\n1 teaspoon dried thyme\n\n1 tablespoon Dijon mustard\n\n1 teaspoon ground celery seed\n\n1\/2 teaspoon freshly ground black pepper\n\n3 cups beef broth\n\n2 slices bacon\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Add ghee and melt. Add garlic and mushrooms and toss to coat with ghee. Stir-fry 3 minutes until mushrooms soften.\n\n2 Add red wine. Press the Saut\u00e9 button and adjust temperature to Less; simmer 5 minutes.\n\n3 Add remaining ingredients to the pot. Lock lid. Press the Manual button and adjust time to 20 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Discard bacon and garlic.\n\n4 Use a slotted spoon to transfer mushrooms to a serving bowl. Serve warm.\n\n## Mushrooms Stuffed with Sausage and Kimchi\n\nIf you are true paleo, you probably have a jar of kimchi in the back of your refrigerator. So, if you are in a pinch for a quick appetizer, this recipe is for you. The kimchi and its acidity blends nicely with the breakfast sausage. If cilantro repels you, substitute some Italian seasoning or dried basil; both work beautifully and add a slightly different flavor note to this dish.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 8 minutes\n\nYields 10\u201312 mushrooms\n\n* * *\n\n1 tablespoon avocado oil\n\n1\/4 pound ground breakfast sausage\n\n2 tablespoons finely minced yellow onion\n\n1 tablespoon diced drained kimchi\n\n2 tablespoons mayonnaise\n\n1 teaspoon dried cilantro\n\n8 ounces whole baby bella mushrooms (about 10\u201312), stemmed\n\n1 cup water\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add sausage and onion; stir-fry 5\u20136 minutes until sausage is no longer pink. Transfer mixture to a small bowl and use paper towels to dab off excess oil and fat. Add kimchi, mayonnaise, and cilantro.\n\n2 Stuff an equal amount of mixture into each mushroom cap and place on the steamer basket. Pour water into the Instant Pot\u00ae. Insert steamer basket and lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. Adjust pressure to Low. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer mushrooms to a serving plate. Garnish with chopped parsley. Serve warm.\n\n### WHAT TO DO WITH MUSHROOM STEMS?\n\nDon't throw away those mushroom stems. Clean and dice them, and place in a lidded container and refrigerate. These stems, which are usually discarded, are ideal additions to breakfast scrambles, sauces, and even to green smoothies in the morning.\n\n## Steamed Shrimp with Cocktail Sauce\n\nThis is one of the easiest and most-appreciated appetizers. It is a low-calorie alternative to those rich chip and dips that are always present at a group get-together.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 0 minutes\n\nServes 8\n\n* * *\n\nCocktail Sauce\n\n1\/2 cup paleo ketchup\n\n2 tablespoons prepared horseradish\n\n1 teaspoon fresh lemon juice\n\n1\/4 teaspoon Worcestershire sauce\n\n1\/4 teaspoon sriracha sauce\n\nSteamed Shrimp\n\n1 cup water\n\n2 pounds large shrimp, peeled and deveined (tails on)\n\n1 Combine cocktail sauce ingredients until well blended. Refrigerate lidded until ready to use.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Place shrimp on the steamer basket. Lock lid. Press the Steam button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Serve shrimp, warm or cold, with cocktail sauce.\n\n### ALL KETCHUP IS NOT CREATED EQUAL\n\nMost commercial ketchups are filled with sugars; however, you have options. There are many sugar-free homemade recipes online. Also, you can find several premade and paleo-approved options at specialty stores and online.\n\n## Liver P\u00e2t\u00e9\n\nLuxury . . . pure luxury. And for such a decadent dish, it is really quite simple to make, especially with your Instant Pot\u00ae. Served with crudit\u00e9s or even with some bread toasts for your non-paleo guests, this spread is sure to delight your friends.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 9 minutes\n\nYields 11\/2 cups\n\n* * *\n\n2 tablespoons ghee\n\n1 pound chicken livers\n\n1 piece bacon, diced\n\n1 large shallot, peeled and diced\n\n2 cloves garlic, minced\n\n1\/4 cup cabernet\n\n2 tablespoons water\n\n1 bay leaf\n\n1 large egg\n\n1 teaspoon capers, rinsed\n\n2 tablespoons cooking sherry\n\nPinch ground nutmeg\n\n2 tablespoons melted ghee\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 2 tablespoons ghee. Add chicken livers, bacon, and shallot. Saut\u00e9 3 minutes or until chicken livers are lightly browned on all sides and shallot is tender. Add garlic and stir-fry 1 minute.\n\n2 Add wine and water and scrape up any bits from the bottom and sides of the pot. Add bay leaf and place egg on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Discard bay leaf.\n\n3 Transfer egg to a water bath to stop the cooking process. Peel egg and dice.\n\n4 Transfer egg and remaining ingredients to a food processor or blender. Pulse ingredients until smooth.\n\n5 Transfer mixture to a large lidded container and refrigerate until ready to serve. Serve chilled.\n\n## Steamed Artichokes with Lemon-Herb Butter\n\nNot only are artichokes a fun food to eat with other people, but also they are an absolute superfood packed with phytonutrients and anti-inflammatory properties that help protect against cancer, heart disease, and diabetes just to name a few ailments. In addition, they help detoxify your liver.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 6\n\n* * *\n\n6 medium artichokes\n\n1 cup water\n\n3 cloves garlic, quartered\n\nJuice of 1 medium lemon\n\n1 teaspoon sea salt\n\n1 Clean the artichokes by clipping off the top third of the leaves and removing the tougher exterior leaves. Trim the bottoms so that they have a flat surface to prop up in the Instant Pot\u00ae.\n\n2 Add water, garlic, and lemon juice to the Instant Pot\u00ae and insert trivet. Place artichokes upright in a steamer basket and lower onto the trivet. Sprinkle artichokes with salt. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer the artichokes to a plate, and serve warm.\n\n### LEMON-HERB BUTTER DIPPING SAUCE\n\nThe following Lemon-Herb Butter Dipping Sauce is a delicious dipping sauce to accompany the light nature of the steamed artichoke. To make this recipe, in a small bowl combine 1\/4 cup melted ghee, 2 cloves minced garlic, 1\/2 teaspoon herbes de Provence (or Italian seasoning), 1 tablespoon fresh lemon juice, and a pinch of sea salt and freshly ground black pepper.\n\n## Rutabaga Fries with Rosemary Dipping Sauce\n\nTraditional baked rutabaga fries can take 45 minutes up to an hour to bake. In the meantime, all the moisture is sucked out. By steaming the fries and then broiling them, the fries result in a crispy outside and a beautifully steamed and tender inside. Couple that with this flavorful rosemary dip, and your taste buds will be in heaven.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 13 minutes\n\nServes 8\n\n* * *\n\nRosemary Dipping Sauce\n\n1\/2 cup mayonnaise\n\n1 tablespoon finely chopped fresh rosemary\n\n1\/2 teaspoon lemon juice\n\n1\/2 teaspoon lemon zest\n\n1\/4 teaspoon sea salt\n\n1\/8 teaspoon cayenne pepper\n\nRutabaga Fries\n\n1 large rutabaga, peeled, trimmed, and cut into 1\/2\" sticks\n\n2 tablespoons avocado oil\n\n1 teaspoon sea salt, or more to taste\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 teaspoon garlic powder\n\n1 cup water\n\n1 Line a large baking sheet with parchment paper. Set oven to broil.\n\n2 Combine Rosemary Dipping Sauce ingredients in a small bowl. Refrigerate covered until ready to use.\n\n3 Place rutabaga in a medium bowl and toss with avocado oil, salt, pepper, and garlic powder.\n\n4 Pour water into the Instant Pot\u00ae and insert steamer basket. Add fries to steamer basket. Lock lid. Press the Steam button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Transfer fries to the prepared baking sheet and scatter evenly. Broil 10 minutes or until crisp and browned, tossing once halfway through cooking. Season with additional salt if desired. Serve warm with chilled dipping sauce.\n\n## Broccoli-Cauli Sausage Tots\n\nChoosing different types of sausage will alter the flavor of these tots as the broccoli and cauliflower take on the flavors of whatever they are combined with. These crispy and savory tots will disappear off the plate in minutes!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 37 minutes\n\nYields 20 tots\n\n* * *\n\n1 cup water\n\n2 cups fresh broccoli florets\n\n2 cups fresh cauliflower florets\n\n1 shallot, finely minced\n\n1\/4 pound mild Italian sausage\n\n1 large egg\n\n1 teaspoon dried oregano\n\n1\/2 teaspoon sea salt\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Add broccoli and cauliflower. Lock lid. Press the Manual or Pressure Cook button, and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Preheat oven to 400\u00b0F. Line a baking sheet with parchment paper.\n\n3 Transfer broccoli and cauliflower to a food processor or blender. Pulse 5\u20136 times until smooth. Transfer to a medium bowl.\n\n4 Add shallot, sausage, egg, oregano, and salt. Combine until well mixed. Form into 20 tots and place on the prepared baking sheet. Bake 35 minutes. Serve warm.\n\n## Salsa Verde\n\nTomatillos are the star of this green sauce, or \"salsa verde.\" It is terrific served with chicken enchiladas, fish taco wraps, Salsa Verde Pulled Pork (see Chapter 8), and many other dishes.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 8\n\n* * *\n\n1 pound tomatillos, outer husks removed and halved\n\n2 small jalape\u00f1os, seeded and chopped\n\n1 small onion, peeled and diced\n\n1\/2 cup chopped fresh cilantro\n\n1 teaspoon ground coriander\n\n2 teaspoons sea salt\n\n11\/2 cups water\n\n1 Place tomatillos in the Instant Pot\u00ae. Add enough water to cover the tomatillos. Lock lid. Press the Manual or Pressure Cook button, and adjust timer to 2 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Drain pot.\n\n2 Add tomatillos, jalape\u00f1os, onion, cilantro, coriander, sea salt, and 11\/2 cups water to a food processor or blender. Pulse until well combined, about 1\u20132 minutes.\n\n3 Transfer to a serving dish, cover, and chill in the refrigerator before serving.\n\n## Bolognese Sauce\n\nRich and meaty, Bolognese originated in Bologna, Italy; however, its power has won the world's heart. Make a jar of this on food prep day and enjoy the tasty sauce during the week served over spiraled zucchini, carrots, or whatever your heart likes to zoodle!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 20 minutes\n\nYields 6 cups\n\n* * *\n\n2 tablespoons ghee\n\n2 stalks celery, finely diced\n\n1 medium carrot, peeled and finely diced\n\n1\/2 medium Vidalia onion, peeled and finely diced\n\n4 cloves garlic, quartered\n\n1 pound ground Italian sausage\n\n2 slices bacon, diced\n\n1 (28-ounce) can crushed tomatoes, including juice\n\n1\/2 cup beef broth\n\n2 tablespoons chopped fresh basil\n\n2 tablespoons fresh thyme leaves\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 cup unsweetened almond milk\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add celery, carrot, and onion; saut\u00e9 2\u20133 minutes until onions are tender. Add garlic, Italian sausage, and bacon; stir-fry 3\u20135 minutes until sausage is no longer pink. Drain off and discard any excess fat.\n\n2 Stir in tomatoes, including juice, and scrape up the bits from the sides and bottom of the pot. Add broth, basil, thyme, salt, and pepper. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Stir in almond milk.\n\n3 Simmer unlidded 2\u20133 minutes. Pour sauce into a lidded container or jar and refrigerate until ready to use. Use within 5 days.\n\n### HOW TO MAKE HOMEMADE ALMOND MILK\n\nSoak 1 cup unsalted raw almonds covered in water for at least 24 hours. Rinse and drain. In a stand blender or food processor, pulse 13\/4 cups water and almonds. Strain liquid through a fine-mesh sieve or cheesecloth. Refrigerate up to 2\u20133 days.\n\n## Marinara Sauce\n\nOne of the tastiest and most versatile sauces out there, Marinara Sauce is a great one to keep on hand. You can add it to zoodled vegetables, to a bowl of meatballs, to a skillet of ground beef with added vegetables\u2014and the list goes on.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nYields 4 cups\n\n* * *\n\n1 (28-ounce) can crushed tomatoes, including juice\n\n1 stalk celery, finely diced\n\n1 medium carrot, peeled and finely diced\n\n1\/2 medium red onion, peeled and finely diced\n\n4 cloves garlic, quartered\n\n2 tablespoons chopped fresh basil\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n1 tablespoon fresh thyme leaves\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 cup beef broth\n\n1 Combine all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n2 Use an immersion blender to blend the sauce in the pot until smooth.\n\n3 Pour the sauce into a lidded container or jar and refrigerate until ready to use. Use within 5 days.\n\n## Puttanesca Sauce\n\nNo need to add salt to this dish because the olives, anchovies, and capers hit this beautiful note in the recipe. Puttanesca literally translates to \"in the style of the prostitutes,\" because the intriguing aroma of the sauce was how the prostitutes from Naples, Italy, would lure customers to their door.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 10 minutes\n\nYields 5 cups\n\n* * *\n\n2 tablespoons avocado oil\n\n1 small red onion, peeled, halved, and cut into half-moons\n\n4 large cloves garlic, minced\n\n1 (28-ounce) can diced tomatoes, including juice\n\n1\/4 cup Kalamata olives, pitted and diced\n\n1\/4 cup green olives, pitted and diced\n\n4 anchovy fillets, finely chopped or smashed with the back of a fork\n\n2 tablespoons drained capers\n\n2 teaspoons dried oregano\n\n1\/2 teaspoon dried red pepper flakes\n\n1\/2 cup chicken broth\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n1 Combine all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n2 Pour sauce into a lidded container or jar and refrigerate until ready to use. Use within 5 days.\n\n## Tomato Curry Sauce\n\nThis creamy dairy-free sauce is excellent served over cauliflower \"rice,\" seafood, or chicken.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nYields 4 cups\n\n* * *\n\n1 (28-ounce) can crushed tomatoes, including juice\n\n1\/2 medium onion, peeled and finely diced\n\n4 cloves garlic, quartered\n\n1 tablespoon minced fresh ginger\n\n1\/2 teaspoon garam masala\n\n1\/2 teaspoon turmeric powder\n\n1\/4 teaspoon red pepper flakes\n\n1\/4 teaspoon ground cinnamon\n\n1 tablespoon fresh thyme leaves\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/2 cup canned coconut milk\n\n1 Combine all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n2 Use an immersion blender to blend the sauce in the pot until smooth.\n\n3 Pour sauce into a lidded container or jar and refrigerate until ready to use. Use within 5 days.\n\n## Side Dishes\n\nSometimes in our attempt to live a paleo life, we cook up a chicken breast or steak and then fall short on the extras. All the while our gorgeous farmers' market and CSA (community-supported agriculture) produce takes a backseat in the refrigerator and gets a little slimy. Unfortunately, sometimes those nutrient-rich vegetables never make it to our plates. Not only is it a waste of healthful food, but it is also a big waste of money. With the Instant Pot\u00ae, you can have flavorful seasonal vegetables in minutes while the main dish is being prepared. Also, cooking vegetables in your Instant Pot\u00ae retains more nutrients than boiling them on the stovetop or roasting them in the oven. So, while you're grilling that steak outdoors in the summer or charring some chicken on an indoor grilling pan, why not take a few extra minutes to round out the meal? With recipes such as Lemon Ghee Broccoli and Honey-Glazed Carrots to Southern Squash Casserole and Bacon-Apple Brussels Sprouts, you will be happy you took the few extra moments to prepare a side dish. And your body will thank you!\n\nNutty Holiday Sweet Potatoes\n\nHoney-Glazed Carrots\n\nBacon-Apple Brussels Sprouts\n\nMixed Southern Greens\n\nDilled Purple Potatoes\n\nLemon Ghee Broccoli\n\nHerbed Fingerling Potatoes and Onions\n\nKohlrabi, Kale, and Karrots\n\nSteamed Beets\n\nPurple Cauliflower\n\nGerman Red Potatoes\n\nEggplant-Olive Ratatouille\n\nSouthern Squash Casserole\n\nBroccoli Casserole\n\nBacon Cauli-Potato Mash\n\nRed Onion and Apple Confit\n\nSesame Zoodles\n\nSaffron and Herb Cauliflower Rice\n\nBroccoli-Parsnip Smash\n\n## Nutty Holiday Sweet Potatoes\n\nHappy holidays to you and your family! Sweet potatoes are a traditional staple on most winter celebration tables and this dish is no exception. You won't miss the marshmallows and ooey-gooey recipes of the past. These sweet potatoes will quench all of your comfort memories.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\n21\/2 pounds sweet potatoes, peeled and diced large\n\n2 cups water\n\n1 tablespoon minced fresh ginger\n\n1\/2 teaspoon sea salt\n\n1 tablespoon pure maple syrup\n\n1 tablespoon ghee\n\n1\/4 cup unsweetened almond milk\n\n2 tablespoons crushed walnuts\n\n1 Add potatoes and water to the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n2 Drain water from pot. Add ginger, salt, syrup, ghee, and milk (1 tablespoon at a time) to the potatoes. Use an immersion blender or masher to cream the potatoes to desired consistency. Stir in walnuts. Serve warm.\n\n### PEELING FRESH GINGER\n\nFresh ginger can seem difficult to navigate with its uneven surface and all the branches. Instead of taking your fingers' safety into the war zone, simply take the edge of a spoon and scrape the peel off of a fresh gingerroot before you grate or mince it.\n\n## Honey-Glazed Carrots\n\nWe all dress up during the holidays, so why not give the simple carrot a makeover? By adding a little honey and ghee, these beta-carotene cargo trains come to life and deliver a flavorful punch for your guests or family any time of the year.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 4\n\n* * *\n\n1 pound carrots, peeled and diced large\n\n1\/4 cup freshly squeezed orange juice\n\n1\/2 cup water\n\n1 tablespoon ghee\n\n2 tablespoons honey\n\n1\/2 teaspoon sea salt\n\n1 Add carrots, orange juice, and water to the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Use a slotted spoon to transfer the carrots to a bowl. Toss with ghee, honey, and salt. Serve warm.\n\n### MEASURING HONEY WITHOUT THE MESS\n\nWhen measuring sticky liquids like honey, maple syrup, or molasses, warm up the measuring cup or spoon by rinsing it first with hot water. Your liquid won't stick to your cup, making cleanup much easier and the measurement more accurate!\n\n## Bacon-Apple Brussels Sprouts\n\nThe saltiness of the bacon and the sweetness of the apple help counter the bitter flavor that some may detect in Brussels sprouts. Rounded out even more with the earthiness of the onions and the citrus kick from the zest, this dish will have your family asking for seconds!\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n2 slices bacon, diced\n\n1 cup water\n\n1 pound Brussels sprouts, trimmed and halved\n\n2 medium Granny Smith apples, peeled, cored, and diced\n\n1 small sweet onion, peeled and sliced\n\n1\/2 teaspoon sea salt\n\n2 teaspoons orange zest\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add bacon and stir-fry 3\u20135 minutes or until almost crisp and the fat is rendered. Add water and deglaze the pot by scraping the bits from the sides and bottom of the pot.\n\n2 Add Brussels sprouts, apples, and onion. Lock lid. Press the Manual button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer the Brussels sprouts to a serving dish. Garnish with salt and orange zest. Serve warm.\n\n## Mixed Southern Greens\n\nThis recipe calls for turnip and collard greens, but you can swap or add kale, mustard greens, chard, or whatever your heart desires. For the fat, this recipe calls for bacon, but you can swap out a ham hock or turkey neck. That is part of the greatness of Southern cooking. Recipes are merely a suggestion.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\n1 bunch turnips with greens\n\n1 bunch collard greens, chopped (ribs removed)\n\n1 small Vidalia onion, peeled and diced\n\n1\/4 cup apple cider vinegar\n\n1 teaspoon sriracha\n\n2 slices bacon\n\n1 cup chicken broth\n\n1 teaspoon honey\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1 Remove spines from turnip greens and discard. Chop leaves. Rinse turnips. Peel and dice.\n\n2 Place all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Discard bacon.\n\n3 Transfer the greens and turnips to a dish and serve warm.\n\n## Dilled Purple Potatoes\n\nWait! Potatoes? Yes, that's right. Potatoes are now considered paleo in small quantities. Opinions do change. But here's the rub. If nightshades are your enemy or you are trying to drop a few pounds, this starchy tuber needs to stay underground. However, if your family is craving a little comfort, go for it. It is a natural product but just cook it as a savory treat from time to time and not so much in regular rotation.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\n11\/2 pounds baby purple potatoes\n\n4 tablespoons ghee, divided\n\n2 cloves garlic, minced\n\n3 tablespoons chopped dill, divided\n\n2 cups water\n\n1 teaspoon sea salt\n\n1 Use a fork to pierce each potato 3 or 4 times. Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat 2 tablespoons ghee and add potatoes, garlic, and 1 tablespoon dill; saut\u00e9 3 minutes, stirring frequently.\n\n2 Add water to the pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 7 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use a slotted spoon to transfer the potatoes to a serving bowl. Toss with remaining ghee, dill, and salt. Serve.\n\n## Lemon Ghee Broccoli\n\nSimply divine. Broccoli is such a healthy and inexpensive side that goes with just about any protein. The lemon lends a freshness and the ghee gives it that little bit of fat to help our taste buds along. Because heads of broccoli vary in size, season sparingly and taste as you go before adding more salt.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 2\n\n* * *\n\n1\/2 cup water\n\nJuice of 1\/2 medium lemon\n\n1 head broccoli, chopped into bite-sized pieces\n\n2 tablespoons ghee\n\n1\/2 teaspoon sea salt\n\n1 Add water and lemon juice to the Instant Pot\u00ae and insert steamer basket. Add broccoli. Lock lid. Press the Manual or Pressure Cook button, and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer broccoli to a serving bowl. Add ghee and salt. Toss to mix. Serve immediately.\n\n## Herbed Fingerling Potatoes and Onions\n\nFingerling potatoes are stubby little counterparts to the mighty russet, but that size can work to the favor of the busy home chef. There is no need to peel or dice these small tubers. Give them a good scrub and they are ready for steamed perfection in the Instant Pot\u00ae.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 6\n\n* * *\n\n11\/2 pounds fingerling potatoes (unpeeled)\n\n4 tablespoons ghee, divided\n\n1 small yellow onion, peeled, halved, and sliced into half-moons\n\n2 cloves garlic, chopped\n\n2 sprigs rosemary\n\n2 cups water\n\n1 teaspoon sea salt\n\n1 tablespoon fresh thyme leaves\n\n1 Use a fork to pierce each potato 2 or 3 times. Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat 2 tablespoons ghee and add potatoes, onion, garlic, and rosemary. Stir-fry 3 minutes or until onions are translucent.\n\n2 Add water to the pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 6 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use a slotted spoon to transfer potatoes and onions to a serving bowl. Discard rosemary. Toss with remaining ghee, salt, and thyme. Serve.\n\n### DRIED HERBS IN A PINCH\n\nIf you're in a hurry and out of fresh herbs, use some Italian seasoning from your spice drawer. Add 2 teaspoons to the Instant Pot\u00ae while cooking and then use another 2 teaspoons to garnish the finished product. When garnishing, pinch the dried herbs between your fingers to release some of the oil from the leaves. This adds another dimension of flavor to your dish.\n\n## Kohlrabi, Kale, and Karrots\n\nBalanced in flavors and exploding with nutrients, this kindred trio is a knockout that teams nicely with a tender chicken or savory steak dish.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 12 minutes\n\nServes 6\n\n* * *\n\n2 teaspoons apple cider vinegar\n\n1 tablespoon pure maple syrup\n\n1\/2 teaspoon sea salt\n\nPinch cayenne pepper\n\nPinch ground nutmeg\n\n1 bunch kale\n\n2 tablespoons ghee\n\n1 large kohlrabi, peeled and diced\n\n3 medium carrots, peeled and sliced\n\n3 cloves garlic, quartered\n\n1 cup water\n\n1 In a small bowl, whisk together vinegar, syrup, salt, cayenne pepper, and nutmeg. Set aside.\n\n2 Remove spines from the middle of the kale leaves and discard. Chop kale.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat ghee and add kohlrabi and carrots. Stir-fry 3\u20134 minutes until vegetables start to get tender. Add garlic quarters and heat 1 additional minute.\n\n4 Add water to the pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n5 Stir in kale and simmer 1\u20132 minutes until it wilts.\n\n6 Use a slotted spoon to transfer the pot ingredients to a serving dish. Add vinegar mixture and toss to mix. Serve warm.\n\n## Steamed Beets\n\nKeep the skin on these beets to reap the benefits of the fiber. Plus, peeling them just gets messy. Adding this root vegetable to your meal rotation is a smart move. They contain betalains, which provide anti-inflammatory and detoxification support. Also, mix it up by trying golden beets as well. They can be prepared in the same way.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\n1 cup water\n\n6 medium beets, trimmed and quartered\n\nJuice of 1\/2 medium orange\n\n2 teaspoons chopped fresh dill\n\n1 teaspoon sea salt\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Place beets in basket. Lock lid. Press the Manual or Pressure Cook button and adjust time for 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer beets to a bowl. Add orange juice, dill, and salt. Toss and serve warm.\n\n## Purple Cauliflower\n\nPurple Cauliflower is just so pretty, but this recipe works with orange and white cauliflower too. To take this dish from florets to cauliflower rice, just drain the water after cooking. Add the olive oil back to the pot. Press the Saut\u00e9 button and use a potato masher to mash the cauliflower directly in the pot. Season with salt and pepper. Voila!\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 4\n\n* * *\n\n1 medium head cauliflower, cut into florets\n\n1 cup water\n\nDrizzle of extra-virgin olive oil\n\nPinch sea salt\n\nPinch freshly ground black pepper\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Add cauliflower. Lock lid. Press the Steam button and adjust time for 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer cauliflower to a serving bowl. Garnish with olive oil, salt, and pepper. Serve warm.\n\n## German Red Potatoes\n\nNot only does the Instant Pot\u00ae make this German classic quick to fix, but also there is no need to take the extra time to peel the potatoes. The color of the red potatoes not only adds to the optics of the dish, but the skins also provide most of the fiber, vitamins, and minerals found in the potato.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 16 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons apple cider vinegar\n\n1 teaspoon honey\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1\/4 teaspoon caraway seeds\n\n8 small red potatoes, diced (unpeeled)\n\n2 slices bacon, diced\n\n1 small yellow onion, peeled and diced\n\n2 cloves garlic, minced\n\n1 cup water\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n1 In a small bowl, whisk together vinegar, honey, salt, pepper, and caraway seeds. Set aside.\n\n2 Use a fork to pierce each potato 3 or 4 times. Press the Saut\u00e9 button on the Instant Pot\u00ae and add bacon. Cook bacon until crisp, about 5\u20138 minutes. Transfer bacon to a plate lined with paper towels.\n\n3 Add potatoes and onion to bacon grease in the pot. Stir-fry 3\u20134 minutes until onions are translucent. Add garlic and saut\u00e9 1 minute.\n\n4 Add water to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Use a slotted spoon to transfer potatoes and onions to a serving bowl. Toss with bacon and vinegar mixture. Garnish with parsley. Serve.\n\n## Eggplant-Olive Ratatouille\n\nThe somewhat plain-flavored eggplant and zucchini take on the incredible Mediterranean flavors of the basil and garlic of this dish. The olives lend a salty and almost fantastic bitter flavor to round out this side dish that is exquisite served with a grilled steak or tender chicken breast.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 7 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon coconut oil\n\n1 medium sweet onion, peeled and diced\n\n1 small yellow bell pepper, seeded and diced\n\n1 large eggplant, peeled and diced into 1\" cubes\n\n3 cloves garlic, minced\n\n1 (28-ounce) can diced tomatoes, including juice\n\n1\/4 cup sliced green olives\n\n1 medium zucchini, diced into 1\" cubes\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/4 cup chopped fresh basil\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat coconut oil. Add onion and bell pepper; stir-fry 3\u20134 minutes until onions are translucent. Add eggplant and garlic; toss and cook 1 minute.\n\n2 Add diced tomatoes including juice and scrape sides and bottom of pot. Add olives, zucchini, salt, pepper, and basil. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use a slotted spoon to transfer ingredients to a large bowl. Garnish with parsley. Serve warm.\n\n## Southern Squash Casserole\n\nVoid of the milk and bread or crackers that are in most squash casserole recipes, this version will thrill even the harshest of critics. Most home gardeners are overwhelmed during the year with an overhaul of summer squash and this is a great go-to recipe for the win.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 6\n\n* * *\n\n2 cups water, divided\n\n2 large yellow squash, diced\n\n1 small Vidalia onion, peeled and diced\n\n4 large eggs, whisked\n\n1\/4 cup almond meal\n\n2 tablespoons ghee, melted\n\n2 teaspoons nutritional yeast\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 Pour 1 cup water into the Instant Pot\u00ae and insert steamer basket. Place squash and onion in basket. Lock lid. Press the Manual or Pressure Cook button and set time to 0 minutes. Adjust pressure to Low. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer squash to a medium bowl. Let cool 5 minutes. Discard liquid from the pot.\n\n3 Add eggs, almond meal, ghee, yeast, salt, and pepper to a medium bowl. Transfer to greased 7-cup glass bowl.\n\n4 Add 1 cup water to the Instant Pot\u00ae. Insert steamer basket or trivet. Place the squash dish in pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove squash casserole. Let rest 5 minutes. Use a paper towel to dab off any additional moisture. Serve warm.\n\n## Broccoli Casserole\n\nBroccoli casserole is found on the table at most Southern get-togethers, and it is traditionally cooked with condensed mushroom soup, cheese, and a butter cracker layer. The condensed soup is replaced with the slurry made from almond milk and cassava flour, which is added to the saut\u00e9ed mushroom mixture. The nutritional yeast adds a little cheese flavor, and the ghee and almond meal sub for the \"cracker\" layer. But really, the broccoli is the star.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 14 minutes\n\nServes 6\n\n* * *\n\n3\/4 cup unsweetened almond milk\n\n2 tablespoons cassava flour\n\n1 tablespoon ghee\n\n1\/2 cup chopped yellow onion\n\n1 cup diced button or cremini mushrooms\n\n4 cups chopped fresh broccoli\n\n4 large eggs\n\n1 teaspoon nutritional yeast\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/4 cup almond meal\n\n1 cup water\n\n1 In a small bowl, create a slurry by whisking together almond milk and flour. Set aside.\n\n2 Lightly grease a 7-cup glass bowl. Set aside.\n\n3 Press the Saut\u00e9 button and adjust temperature to Less. Heat ghee. Add onion, mushrooms, and broccoli; stir-fry 4\u20135 minutes until onions are translucent. Add slurry and mix together 1 minute until thickened.\n\n4 Crack eggs in a medium bowl. Quickly whisk in a spoonful of the mixture from the pot to temper the eggs. Add yeast, salt, and pepper. Add remaining pot mixture and stir until combined. Transfer mixture to the prepared dish. Sprinkle almond meal in an even layer over casserole.\n\n5 Add water to the Instant Pot\u00ae and insert steamer basket. Insert dish and lock lid. Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n6 Remove broccoli casserole. Let rest 10 minutes to set. Use a paper towel to dab off any additional moisture after resting time. Serve warm.\n\n## Bacon Cauli-Potato Mash\n\nMashed cauliflower can be a little thin on its own, but the single potato in this mouthwatering side dish adds just the amount of heft needed. This recipe still cuts back drastically on the glycemic index and the cauliflower is loaded with nutrition. And then the bacon . . . Enough said about that.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 4\n\n* * *\n\n4 slices bacon, diced\n\n2 cups chicken broth\n\n1 small head cauliflower, chopped\n\n1 russet potato, peeled and diced\n\n3 cloves garlic, minced\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 tablespoons ghee\n\n1\/4 cup unsweetened almond milk\n\n4 teaspoons chopped fresh chives\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Add bacon and cook 3\u20134 minutes to render fat. Transfer bacon to a plate lined with paper towels.\n\n2 Add broth to pot and scrape up any bits from the bottom and sides of the pot. Add cauliflower, potato, garlic, salt, and pepper. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Use an immersion blender to blend the mixture in the pot. Add ghee. Blend a little more. Add almond milk 1 tablespoon at a time, blending after each addition. You may need a little more or less almond milk depending on your desired consistency. Garnish with chives and serve warm.\n\n## Red Onion and Apple Confit\n\nThe sweetness from the caramelized onions and apple are the perfect match to be served with a salty and hearty pork chop. This is also nice alongside scrambled eggs and leftover beef in the morning.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 14 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons ghee\n\n1 large red onion, peeled, halved, and sliced into half-moons\n\n2 large Granny Smith apples, peeled, cored, and thinly sliced\n\n2 tablespoons apple cider vinegar\n\n2 tablespoons honey\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1\/4 cup water\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add onion and apples; stir-fry 4\u20135 minutes until onions are translucent.\n\n2 Add remaining ingredients to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Simmer unlidded if there is any liquid remaining until it cooks off. Transfer mixture to a lidded container and refrigerate up to 1 week.\n\n## Sesame Zoodles\n\nThese zucchini zoodles are an excellent substitute and complement to the zippy sesame sauce, and it's prepared in no time flat. This dish pairs nicely with a steak or chicken dish . . . and don't forget your chopsticks!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n1\/4 cup water\n\n4 cups spiraled zucchini (about 3 small zucchini)\n\n1\/4 cup coconut aminos\n\n2 tablespoons pure maple syrup\n\n1 teaspoon sriracha\n\n2 teaspoons sesame oil\n\n1 tablespoon creamy almond butter\n\n1\/4 cup diced green onions\n\n1 tablespoon toasted sesame seeds\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket.\n\n2 In a large bowl, combine zucchini, coconut aminos, syrup, sriracha, sesame oil, almond butter, and green onions. Place the marinated zucchini in the steamer basket. Pour remaining marinade over zucchini. Lock lid. Press the Steam button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer zucchini to a serving bowl. Garnish with toasted sesame seeds. Serve immediately.\n\n### DON'T HAVE A SPIRALIZER?\n\nIf you don't own a spiralizer, you have a couple of options. First, most supermarkets carry a variety of spiraled vegetables, both in the produce and freezer sections. Second, you can use a vegetable peeler\/knife to create your own \"noodles.\"\n\n## Saffron and Herb Cauliflower Rice\n\nThis recipe is so, so easy and pairs nicely with seafood. Top this cauliflower rice with shrimp, mussels, sausage, and some asparagus for a quick paleo paella. Don't forget the fruit-packed sangria to accompany this dish.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 4\n\n* * *\n\n1 cup water\n\n1 medium head cauliflower, cut into florets\n\n1 tablespoon ghee\n\n2 cloves garlic, minced\n\n1\/2 teaspoon saffron threads\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n2 tablespoons chopped fresh basil\n\n1 teaspoon sea salt\n\nPinch freshly ground black pepper\n\n1 teaspoon lemon juice\n\n1\/2 teaspoon lemon zest\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Add cauliflower. Lock lid. Press the Steam button and adjust time for 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer cauliflower to a serving bowl and discard liquid from pot.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add cauliflower. Use a potato masher to smash cauliflower in the pot until it has a rice-like consistency.\n\n4 Add remaining ingredients and stir. Transfer \"rice\" to a dish and serve warm.\n\n## Broccoli-Parsnip Smash\n\nListen up broccoli haters: This side dish is sure to become a new favorite. The parsnips and almond milk together lend a creamy texture and added flavor to an otherwise boring vegetable. Also, heads of broccoli come in so many sizes, so season with salt and pepper to taste. This recipe is merely a guideline.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 4\n\n* * *\n\n1 cup water\n\n1 medium head broccoli, cut into florets\n\n2 parsnips, peeled and diced\n\n2 teaspoons ghee\n\n2 teaspoons nutritional yeast\n\n2 tablespoons unsweetened almond milk\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Add broccoli and parsnips. Lock lid. Press the Steam button and adjust time for 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer broccoli and parsnips to a medium bowl and drain pot.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add broccoli and parsnips. Use a potato masher to smash the vegetables in the pot until somewhat smooth.\n\n4 Add yeast and milk. Smash it into broccoli mixture. Transfer to a serving bowl and season with salt and pepper. Serve warm.\n\n## Poultry Main Dishes\n\nThis chapter covers Cornish hens, turkey, and probably one of the most consumed proteins in the United States, chicken. There are so many recipes out there and a handful of go-to meals that you can cook for your family, but if you're sick of eating dried-out chicken breasts and overcooked thighs, the Instant Pot\u00ae is your new best friend. The steam and pressure used to cook ingredients in the pot are guaranteed to leave the chicken dishes in this chapter juicy and delicious. Whether you're craving Jamaican Jerk Chicken Meatballs or Kimchi Chicken Wings, or your family is calling for Chili-Orange Turkey Wings or Cornish Hens and Veggies, you'll find a new favorite recipe here!\n\nAlso, if you're planning on getting home late, the Instant Pot\u00ae allows you to start a meal in the morning and set it to automatically switch to the Keep Warm function for up to 10 hours, which means dinner will be ready and waiting when you walk in the door. So get cooking!\n\nBBQ Whole Chicken\n\nGluten-Free Beer Can Dijon Chicken\n\nKimchi Chicken Wings\n\nChicken Leg Almond Butter Satay\n\nSmoked Paprika and Garlic Chicken Legs\n\nChili Lime Chicken Legs\n\nKombucha Chicken Thighs\n\nChicken Thighs with Strawberry Salsa\n\nTuscan Chicken\n\nSalsa Verde Chicken Meatballs\n\nHoney Mustard Chicken Bites\n\nJamaican Jerk Chicken Meatballs\n\nTaco Chicken Lettuce Wraps with Pico Guacamole\n\nAsian Chicken Lettuce Wraps with Ginger Aioli\n\nChicken Livers and Onions with Gravy\n\nAvocado Chicken Salad\n\nPaleo-ish Turkish Chicken Thighs\n\nCornish Hens and Veggies\n\nChili-Orange Turkey Wings\n\nMega Turkey Meatballs\n\n## BBQ Whole Chicken\n\nSure, you can take extra time and stop at the grocery store on your way home from work, circle the parking lot a few times, and grab a rotisserie chicken. Or you can set the timer on your Instant Pot\u00ae and have a beautiful barbecue-rubbed whole chicken waiting for you and your family. And don't forget to save those bones to make some health-filled broth.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon smoked paprika\n\n1 teaspoon espresso powder\n\n1 teaspoon garlic powder\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1\/2 teaspoon cayenne pepper\n\n1 tablespoon pure maple syrup\n\n1 (5-pound) whole chicken\n\n1 small red apple, peeled, quartered, and cored\n\n1 small orange, quartered\n\n1 medium sweet onion, peeled and roughly chopped\n\n3 cloves garlic, halved\n\n2 cups water\n\n1 In a small bowl, combine paprika, espresso powder, garlic powder, salt, black pepper, and cayenne pepper.\n\n2 Brush the syrup on the outside of the chicken. Sprinkle prepared dry rub evenly over chicken. Place the apple and orange in the cavity of the bird.\n\n3 Place onion and garlic in the bottom of the Instant Pot\u00ae. Pour in water. Insert trivet over vegetables.\n\n4 Place chicken on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 25 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n5 Remove chicken. Discard apple and orange. Serve warm.\n\n## Gluten-Free Beer Can Dijon Chicken\n\nA \"beer can chicken\" dish typically entails a beer can placed in the cavity of a chicken and then grilled or baked. The purpose is to utilize the steamed flavor from the beer. In this Instant Pot\u00ae twist, the flavor is distributed while the beer steams around the chicken. Change up the beer variety for a different flavor experience each time.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 20 minutes\n\nServes 4\n\n* * *\n\n1\/4 cup Dijon mustard\n\n3 pounds (about 10) chicken legs\/drumsticks\n\n1 large yellow onion, peeled and chopped\n\n1 (12-ounce) bottle gluten-free beer, type of choice\n\n1 Rub mustard over chicken legs.\n\n2 Scatter onion in the bottom of the Instant Pot\u00ae and insert trivet. Add beer. Press the Saut\u00e9 button and simmer unlidded 5 minutes.\n\n3 Press the Adjust button and change temperature to Less. Arrange chicken standing up, meaty side down, on the trivet. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n4 Transfer chicken to a serving platter. Serve warm.\n\n## Kimchi Chicken Wings\n\nFermented foods are high on the \"yes\" list of the paleo diet. Kimchi, along with kombucha, sauerkraut, and pickles to name a few, have many health benefits. The probiotics in fermented foods aid in digestion and immunity and assist in the absorption of the nutrients in other foods you eat.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 3\n\n* * *\n\n11\/2 pounds chicken wings, separated at the joint if necessary\n\n1\/2 cup kimchi brine\n\n1 cup water\n\n1 cup kimchi\n\n1 Add chicken wings and kimchi brine to a large plastic bag. Seal and shake. Refrigerate at least 30 minutes up to 2 hours.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Add chicken wings to basket and pour in extra brine from the plastic bag. Lock lid. Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer chicken wings to a large bowl and serve with kimchi.\n\n### WHAT IS KIMCHI?\n\nKimchi is a fermented spicy cabbage popular as a Korean side dish. Whether you are making your own or picking up a premade jar, there are a zillion variations available. There are varieties with cucumbers, radishes, and even different types of cabbage. The spices and chilies can vary as well. If you don't like one, try another. You just might be surprised.\n\n## Chicken Leg Almond Butter Satay\n\nSatay is traditionally a peanut-based sauce, but because of the phytic acids in nuts, it isn't a paleo-friendly item. In this satay, almond butter is substituted in the marinade to mimic the same texture and flavor of the nuts.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 26 minutes\n\nServes 5\n\n* * *\n\n1\/2 cup smooth almond butter\n\n1\/4 cup canned coconut milk\n\n1\/4 cup coconut aminos\n\n1\/8 cup lime juice\n\n1 teaspoon minced ginger\n\n1\/2 teaspoon fish sauce\n\n1 teaspoon honey\n\n1\/2 teaspoon sea salt\n\n3 pounds (about 10) chicken legs\/drumsticks\n\n1 cup water\n\n1 In a medium bowl, whisk together almond butter, coconut milk, coconut aminos, lime juice, ginger, fish sauce, honey, and salt. Pour into a shallow dish with drumsticks. Cover and refrigerate overnight.\n\n2 Add water to the Instant Pot\u00ae and insert trivet. Arrange chicken standing up, meaty side down, on the trivet. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n3 While the Instant Pot\u00ae is releasing pressure, preheat oven to 550\u00b0F.\n\n4 Place chicken legs on a baking sheet and broil 3 minutes on each side to crisp the chicken.\n\n5 Transfer chicken to a serving plate and serve warm.\n\n## Smoked Paprika and Garlic Chicken Legs\n\nSmoked paprika is an underused spice that should be at the front of your spice drawer. Sweet and smoky and a little mild, smoked paprika adds comfort to proteins and sauces, and it is lovely as a garnish on deviled eggs.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 21 minutes\n\nServes 5\n\n* * *\n\n1 tablespoon garlic powder\n\n1 tablespoon smoked paprika\n\n1 teaspoon fine sea salt\n\n3 pounds (about 10) chicken legs\/drumsticks\n\n1 cup water\n\n1 Preheat the oven broiler.\n\n2 In a small bowl, combine garlic powder, smoked paprika, and salt. Season chicken legs with the seasoning mix.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Arrange chicken standing up, meaty side down, on the trivet. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n4 Place chicken legs on a baking sheet and broil 3 minutes on each side to crisp the chicken.\n\n5 Transfer chicken to a serving plate and serve warm.\n\n## Chili Lime Chicken Legs\n\nSweet and tangy, salty and spicy, these Chili Lime Chicken Legs do a little dance on your taste buds. What an ideal recipe for outdoor socializing on the deck with close friends and a cooler full of ice-cold gluten-free beer.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 26 minutes\n\nServes 5\n\n* * *\n\n2 tablespoons fresh lime juice\n\n1 teaspoon lime zest\n\n1 teaspoon chili powder\n\n1 teaspoon sriracha\n\n2 teaspoons honey\n\n4 cloves garlic, minced\n\n1 teaspoon sea salt\n\n3 pounds (about 10) chicken legs\/drumsticks\n\n1 cup water\n\n1 Preheat oven to 550\u00b0F.\n\n2 In a medium bowl, combine lime juice, lime zest, chili powder, sriracha, honey, garlic, and salt. Toss chicken legs in the marinade. Refrigerate covered at least 1 hour or up to overnight.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Arrange chicken standing up, meaty side down, on the trivet. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n4 Place chicken legs on a baking sheet and broil 3 minutes on each side to crisp the chicken.\n\n5 Transfer chicken to a serving plate and serve warm.\n\n## Kombucha Chicken Thighs\n\nKombucha, a popular fermented drink among paleo enthusiasts, comes in many flavors. When it all boils down, kombucha is an acid. So, if you have some chicken thighs and are left without a marinade, grab one of your kombucha drinks from the refrigerator. It will add a deeper flavor than just salt and pepper. The recipe calls for Gingerberry, but use what you have on hand for a different experience each time.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 7 minutes\n\nServes 3\n\n* * *\n\n11\/2 pounds boneless, skinless chicken thighs\n\n2 cups Gingerberry kombucha\n\n1 teaspoon sea salt\n\n1 Place chicken thighs and kombucha in a medium bowl. Refrigerate 1 hour.\n\n2 Insert steamer basket in the Instant Pot\u00ae. Place chicken thighs evenly on basket. Pour in remaining kombucha marinade. Season chicken with salt. Lock lid. Press the Manual or Pressure Cook button and adjust time to 7 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n3 Transfer chicken to a tray and serve warm.\n\n## Chicken Thighs with Strawberry Salsa\n\nThe sweetness of the strawberries plays nicely in this salsa with the bite of the red onion, the zest of the lime, and the creaminess of the avocado. If you prepare this ahead of time, make sure to toss in the avocado right before serving to avoid browning.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 7 minutes\n\nServes 8\n\n* * *\n\nStrawberry Salsa\n\n1 cup diced strawberries\n\n1\/2 cup fresh lime juice\n\n1 tablespoon lime zest\n\n2 Roma tomatoes, seeded and diced\n\n1\/4 cup finely diced red onion\n\n1 medium avocado, peeled, pitted, and diced\n\n1\/4 cup chopped fresh cilantro\n\n1\/4 cup chopped fresh mint\n\n1 teaspoon sea salt\n\nChicken\n\n3 pounds boneless, skinless chicken thighs\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 cup water\n\n1 In a large bowl, combine strawberries, lime juice, lime zest, tomatoes, onion, avocado, cilantro, mint, and salt. Refrigerate salsa covered at least 1 hour or up to overnight.\n\n2 Pat chicken thighs dry with a paper towel. Season with salt and pepper.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Place steamer basket on the trivet. Arrange thighs evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 7 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n4 Transfer chicken to plates. Garnish with strawberry salsa. Serve.\n\n### FRUIT SALSAS\n\nDon't limit yourself to strawberries. Pineapple salsa pairs nicely with a grilled pork chop. And don't forget to grill your fruit before chopping it up for salsa. The heat caramelizes the natural sugars for an added level of flavor. Peaches and mangoes are two other tasty alternatives.\n\n## Tuscan Chicken\n\nA taste of Italy can be on your dinner table in less than a half hour. And the thrilling part is that there will only be one cooking pot to wash at the end of the evening. The prosciutto and capers add a unique salty, earthy flavor that tops off this healthy meal.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 24 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n1 medium yellow onion, peeled and roughly diced\n\n1\/2 cup sliced cremini mushrooms\n\n1 (14.5-ounce) can diced tomatoes, including juice\n\n1\/4 cup chopped fresh basil\n\nPinch sea salt\n\n1 pound boneless, skinless chicken breasts, cut into 1\" cubes\n\n4 ounces prosciutto, torn into pieces\n\n1 tablespoon capers, drained\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion and mushrooms and stir-fry 3\u20134 minutes until onions are translucent.\n\n2 Add tomatoes, basil, salt, and chicken to pot. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n3 Evenly distribute chicken mixture and sauce among four serving bowls. Garnish with prosciutto and capers.\n\n## Salsa Verde Chicken Meatballs\n\nZero carbs and gluten-free, who knew chicken meatballs could be so mouthwatering? The diced onion helps keep the ground chicken moist and the Salsa Verde is the perfect flavorful mild sauce to accompany this dish. Plus, what a great way to get the children involved in the kitchen. What child doesn't want to help make a creepalicious green sauce?\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 16 minutes\n\nYields 16 meatballs\n\n* * *\n\n1 pound ground chicken\n\n1 large egg\n\n1\/2 cup chopped fresh mint\n\n1\/4 cup almond meal\n\n2 tablespoons finely diced Vidalia onion\n\n1 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n11\/2 cups Salsa Verde (see Chapter 4)\n\n1 In a medium bowl, combine ground chicken, egg, mint, almond meal, onion, salt, and pepper. Form into 16 golf ball\u2013sized meatballs. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place half the meatballs around the edges of the pot. Sear all sides of meatballs for a total of 4 minutes. Remove the first batch and set aside. Add remaining oil and meatballs and sear 4 minutes. Remove meatballs.\n\n3 Discard extra juice and oil from the pot.\n\n4 Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Drain the pot and add the meatballs. Pour in Salsa Verde. Let simmer 5 minutes, gently tossing meatballs.\n\n6 Transfer meatballs and sauce to bowls and serve warm with sauce.\n\n## Honey Mustard Chicken Bites\n\nSkip the fast-food fried nuggets. These chicken bites will be your kids' new favorite meal void of the preservatives, hydrogenated oils, and excessive sodium found in their quick-meal counterparts.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n1\/4 cup honey\n\n1\/4 cup yellow mustard\n\n2 tablespoons mayonnaise\n\n2 teaspoons apple cider vinegar\n\nPinch sea salt\n\n1 pound boneless, skinless chicken breast, cut into 1\" cubes\n\n2 cups water\n\n1 In a medium bowl, whisk together honey, mustard, mayonnaise, vinegar, and salt. Set aside and refrigerate 1\/4 cup of the sauce for dipping. Toss chicken in remaining mixture and coat evenly. Cover and refrigerate 1 hour.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Transfer chicken to steamer basket. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, let pressure release naturally for 10 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n3 Serve warm with dipping sauce.\n\n### GROWN-UP CHICKEN BITES\n\nIf you want to kick up these kid-friendly Honey Mustard Chicken Bites a notch, try adding a squirt of sriracha to the sauce. Or replace the yellow mustard with some beer or horseradish mustard. Even more, substitute the honey with some maple syrup or molasses to add a different flavor profile.\n\n## Jamaican Jerk Chicken Meatballs\n\nMost Jamaican jerk seasoning blends already contain some dried chilies, so the habanero called for in this recipe is completely optional. If you aren't a fan of hot and spicy, eliminate the pepper. Otherwise, move on, mon, and enjoy these little jewels.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 11 minutes\n\nYields 16 meatballs\n\n* * *\n\n1 pound ground chicken\n\n1 large egg\n\n1 tablespoon honey\n\n1\/4 cup almond meal\n\n1 tablespoon finely diced yellow onion\n\n1 teaspoon finely diced habanero (seeded)\n\n1 tablespoon Jamaican jerk seasoning blend\n\n1\/2 teaspoon sea salt\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n1 In a medium bowl, combine ground chicken, egg, honey, almond meal, onion, habanero, Jamaican jerk seasoning, and salt. Form into 16 golf ball\u2013sized meatballs. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place half the meatballs around the edges of the pot. Sear on all sides for a total of about 4 minutes. Remove the first batch and set aside. Add remaining oil and meatballs and sear 4 minutes. Remove meatballs.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Lock lid.\n\n4 Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Transfer meatballs to a bowl and serve.\n\n## Taco Chicken Lettuce Wraps with Pico Guacamole\n\nGather the family around the table for Taco Tuesday. These lettuce wraps will hit the spot and there is no lack of flavor or nutrients. The ingredient list looks long, but most of the spices are already on your spice rack.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\nPico Guacamole\n\nJuice of 1 small lime\n\n1 medium avocado, peeled and diced\n\n2 cloves garlic, minced\n\n1 small jalape\u00f1o, seeded and diced\n\n1\/4 cup diced onion\n\n1 teaspoon sea salt\n\n2 tablespoons chopped fresh cilantro\n\n2 Roma tomatoes, seeded and diced\n\nChicken\n\n2 teaspoons chili powder\n\n1 teaspoon ground cumin\n\n1 teaspoon sea salt\n\n1\/2 teaspoon garlic powder\n\n1\/2 teaspoon smoked paprika\n\n1\/2 teaspoon ground coriander\n\n1 pound boneless, skinless chicken breasts, cut into 1\" cubes\n\n2 cups water\n\n8 iceberg lettuce leaves\n\n1 Place lime and diced avocado in a medium bowl. Use a fork to smash the avocado until desired chunkiness. Add garlic, jalape\u00f1o, onion, salt, cilantro, and tomatoes. Refrigerate covered until ready to serve.\n\n2 In a medium bowl, combine chili powder, cumin, salt, garlic powder, paprika, and coriander. Toss chicken into mixture and coat evenly.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Transfer chicken to steamer basket. Lock lid. Press the Poultry button and cook for the default time of 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to ensure the internal temperature is at least 165\u00b0F.\n\n4 Transfer chicken to a cutting board and chop. Place in a serving bowl.\n\n5 Serve chicken, lettuce leaves, and pico guacamole family-style.\n\n## Asian Chicken Lettuce Wraps with Ginger Aioli\n\nThese chicken lettuce wraps are packed with Asian flavors. If you ever have a friend tell you that paleo and gluten-free are boring and that they couldn't be on such a diet, serve them these lettuce wraps and watch the lightbulb turn on. There is no shortness of taste here.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 7 minutes\n\nServes 4\n\n* * *\n\nGinger Aioli\n\n1\/4 cup mayonnaise\n\n1\/2 teaspoon minced ginger\n\n1\/8 teaspoon hot sauce\n\n1\/2 teaspoon lime juice\n\nLettuce Wraps\n\n1 tablespoon sesame oil\n\n1 pound ground chicken\n\n1\/3 cup chopped green onions\n\n3 cloves garlic, minced\n\n5 ounces shiitake mushrooms\n\n1 tablespoon minced fresh ginger\n\n1\/4 cup coconut aminos\n\n1 teaspoon Dijon mustard\n\n1 tablespoon sriracha\n\n1 tablespoon honey\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon white pepper\n\n1 cup water\n\n12 Bibb lettuce leaves\n\n1\/4 cup slivered almonds\n\n1 In a small bowl, whisk together Ginger Aioli ingredients. Refrigerate covered until ready to use.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat sesame oil. Add chicken and onions; stir-fry 4\u20135 minutes until onions are translucent. Transfer mixture to a 7-cup glass bowl. Stir in garlic, mushrooms, ginger, coconut aminos, mustard, sriracha, honey, salt, and pepper.\n\n3 Add water to the Instant Pot\u00ae and insert trivet. Set bowl with chicken mixture on trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Serve chicken mixture warm with lettuce leaves, slivered almonds, and Ginger Aioli on the side.\n\n## Chicken Livers and Onions with Gravy\n\nChicken liver usually brings up memories of our grandparents as that seems to be the generation that really embraced this particular offal. So, start your own memories, update this relic, and bring this nutritional dish back to the dinner table.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 12 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons ghee\n\n2 medium yellow onions, peeled, halved, and thinly sliced\n\n2 sprigs fresh rosemary\n\n1 pound chicken livers, halved and trimmed\n\n10 grape tomatoes, halved\n\n1\/2 cup beef broth\n\n2 tablespoons cooking sherry\n\n2 tablespoons cassava flour\n\n1\/2 teaspoon sea salt\n\n2 tablespoons chopped fresh basil\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add onions and rosemary. Stir-fry 3 minutes. Add chicken livers and tomatoes and stir-fry an additional 3 minutes.\n\n2 Add beef broth and cooking sherry to the pot. Lock lid. Press the Meat\/Stew button and adjust time to 6 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer chicken livers, onions, and tomatoes to a medium bowl.\n\n4 Whisk flour and salt into pot juices until it thickens. Transfer to a gravy boat.\n\n5 Garnish chicken livers with fresh basil and serve warm with gravy.\n\n## Avocado Chicken Salad\n\nA fresh and quick meal, this chicken salad is also a great recipe to make on prep day to portion out for the remainder of the week. Whether for a light luncheon or weekday meal, this savory chicken dish is the ideal go-to recipe.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 6\n\n* * *\n\n2 pounds chicken breasts, cut into 1\" cubes\n\n1 teaspoon fine sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 cup water\n\n1\/2 medium red onion, peeled and diced\n\n2 stalks celery, finely chopped\n\n2 Roma tomatoes, seeded and diced\n\n1 medium avocado, peeled and diced\n\n1\/2 cup mayonnaise\n\n1 tablespoon yellow mustard\n\n1\/2 dried dill\n\n1\/2 teaspoon lime juice\n\n1 On a plate, season chicken pieces with salt and pepper. Set aside.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Place chicken cubes on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer chicken to a cutting board. When cool enough to work with, finely chop the chicken. Add to a medium bowl.\n\n4 Add onion, celery, tomatoes, avocado, mayonnaise, yellow mustard, dill, and lime juice. Refrigerate covered until ready to serve.\n\n## Paleo-ish Turkish Chicken Thighs\n\nLike cheese and alcohol, yogurt is not allowed in the strict paleo world. However, if you are okay with including a minimal amount of dairy in your diet and you don't have gut issues or allergies, this recipe is a great option to add a little dairy into your life. The remaining plain yogurt can be turned into a tzatziki sauce for some lamb meatballs, or it can be added to fresh berries along with a drizzle of honey for a delicious breakfast, or it can just be added to a post-workout smoothie for a little added creaminess and protein!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 8\n\n* * *\n\n1 cup plain Greek yogurt\n\n1\/4 teaspoon ground coriander\n\n1\/4 teaspoon ground nutmeg\n\n1\/4 teaspoon ground cumin\n\n1\/4 teaspoon ground cinnamon\n\n1\/2 teaspoon sea salt\n\n3 pounds boneless, skinless chicken thighs\n\n2 tablespoons avocado oil\n\n3 large carrots, peeled and spiralized\n\n2 cups water\n\n1\/4 cup fresh chopped Italian flat-leaf parsley\n\n1\/4 cup fresh chopped mint\n\n1 In a large bowl, combine yogurt, coriander, nutmeg, cumin, cinnamon, and salt. Toss chicken thighs in sauce and refrigerate covered at least 4 hours or up to overnight.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat avocado oil and fry chicken thighs 2 minutes on each side until browned.\n\n3 Insert steamer basket in the Instant Pot\u00ae; place chicken evenly on basket. Pour any remaining marinade over chicken. Lock lid. Press the Manual or Pressure Cook button and adjust time to 7 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Check the chicken using a meat thermometer to make sure the internal temperature is at least 165\u00b0F.\n\n4 Transfer chicken to a medium bowl. Set aside.\n\n5 Add spiralized carrots to steamer basket. Add water and lock lid. Press the Saut\u00e9 button and let carrots cook 3\u20135 minutes until tender.\n\n6 Transfer carrots to individual bowls. Top with chicken. Garnish bowls with parsley and mint. Serve warm.\n\n## Cornish Hens and Veggies\n\nWhy is it that a whole chicken is an economical choice, but when miniaturized, Cornish hens are just a sexy meal for two or a fancy dish for guests? Either split the cooked hens in half and serve four guests with some extra sides or just enjoy them with the tender mushrooms and carrots with your sweetie.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 20 minutes\n\nServes 2\n\n* * *\n\n2 Cornish hens (11\/2 pounds each)\n\n1 pound whole button mushrooms, stems trimmed\n\n3 large carrots, peeled and sliced into 1\/2\" sections\n\n11\/2 cups water\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 teaspoon smoked paprika\n\n2 cloves garlic, halved\n\n1 small Granny Smith apple, cored and quartered\n\n1 Pat down Cornish hens with a paper towel. Set aside.\n\n2 Place mushrooms, carrots, and water in the bottom of the Instant Pot\u00ae. Insert steamer basket.\n\n3 In a small container, combine salt, pepper, and paprika. Season Cornish hens evenly with spice mix. Insert garlic and apple into the cavities of the hens.\n\n4 Place hens on steamer basket. Press the Meat button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Check the hens using a meat thermometer to make sure the internal temperature is at least 165\u00b0F.\n\n5 Set oven to Broil.\n\n6 Transfer hens to a baking sheet. Remove and discard apples and garlic. Broil 5 minutes.\n\n7 Transfer hens to a serving platter. Use a slotted spoon to remove mushrooms and carrots from the pot and place them around the hens. Serve warm.\n\n## Chili-Orange Turkey Wings\n\nTurkey isn't just for Thanksgiving. Check the meat section next time you are at the grocery store. Throughout the year, there will be turkey legs and breasts offered separately. Buying the whole bird isn't necessary. The sriracha-orange-honey seasoning trio provides a nice balance of spicy and tangy and sweet.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 2\n\n* * *\n\n2 turkey wings (about 21\/2 pounds)\n\n2 teaspoons sea salt\n\n2 cups water\n\n2 tablespoons freshly squeezed orange juice\n\n1 teaspoon orange zest\n\n2 tablespoons honey\n\n1 tablespoon sriracha\n\n2 teaspoons ghee\n\n1 Pat down turkey wings with a paper towel. Season with salt.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Place wings on steamer basket.\n\n3 Press the Meat or Poultry button and adjust time to 10 minutes. When timer beeps, quick-release any remaining pressure until float valve drops and then unlock lid. Check the wings using a meat thermometer to make sure the internal temperature is at least 165\u00b0F.\n\n4 Set oven to Broil.\n\n5 In a large bowl, combine orange juice, orange zest, honey, sriracha, and ghee. Add turkey wings and toss to coat. Transfer wings to a baking sheet. Broil 5 minutes. Serve warm.\n\n## Mega Turkey Meatballs\n\nYou can call these mega meatballs or small meatloaves, but definitely try them. Served with your favorite sides, these meatballs are full of flavor from the fresh herbs. The onions and carrots work harmoniously to lend the moisture that the ground turkey needs to make this a delicious meal.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 15 minutes\n\nYields 4 meatballs\n\n* * *\n\n1 pound ground turkey\n\n1 large egg\n\n1 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1\/4 teaspoon garlic powder\n\n1\/2 cup almond meal\n\n1\/2 cup peeled and grated carrot\n\n1 tablespoon finely diced yellow onion\n\n2 tablespoons chopped fresh oregano\n\n2 tablespoons chopped fresh Italian flat-leaf parsley\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n1 In a medium bowl, combine turkey, egg, salt, pepper, garlic powder, almond meal, carrot, onion, oregano, and parsley. Form into 4 large meatballs. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place 2 meatballs in the pot. Sear all sides of the meatballs for a total of about 4 minutes. Remove the first batch and set aside. Add remaining oil and meatballs and sear an additional 4 minutes. Remove meatballs.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 6 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer meatballs to plates. Serve warm.\n\n## Beef and Bison Main Dishes\n\nThere is nothing better suited for the Instant Pot\u00ae than meat. The steam and pressure can take a seemingly tough piece of meat and make it taste like butter in your mouth. The steam keeps everything moist, and the pressure helps break down some of the sinewy parts. And the best part? The trapped steam helps create a meat that tastes like it has been braised for hours in about 30 minutes, depending on the weight. In addition, you can add your potatoes, carrots, mushrooms, and other vegetables to the pot while cooking. Not only will you have a complete meal, but the vegetables take on a completely new savory flavor by cooking next to the meat, especially those mushrooms. With recipes ranging from Sweet-n-Spicy Beef Brisket and Skirt Steak Fajitas to Cuban Picadillo with Riced Cauliflower and Italian Beef Tongue Lettuce Wraps, this chapter will help get you started on some classic Instant Pot\u00ae recipes as well as introduce you to some new favorites.\n\nNana's Sunday Pot Roast\n\nChuck Roast with Carrots and Onions\n\nSweet-n-Spicy Beef Brisket\n\nTender Flank Steak with Mushrooms and Onions\n\nSkirt Steak Fajitas\n\nKorean Beef Short Ribs\n\nMongolian Beef and Cashew BBQ\n\nMeatloaf and Gravy\n\nBalsamic Meatballs\n\nMarinara Meatballs\n\nReuben Meatballs with Russian Dressing\n\nGround Beef and Broccoli Bowl\n\nJust Beet It Burgers\n\nCorned Beef and Cabbage\n\nCuban Picadillo with Riced Cauliflower\n\nChi-Town Italian Beef and Peppers\n\nBeef Shanks and Artichokes\n\nBison Sloppy Joe's\n\nGround Bison Bulgogi\n\nItalian Beef Tongue Lettuce Wraps\n\n## Nana's Sunday Pot Roast\n\nThe salt and pepper simply bring out the natural goodness of the roast. The drippings give the vegetables a savory, rich, and earthy note. You'll feel like Nana is just around the corner!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 65 minutes\n\nServes 6\n\n* * *\n\n1 (3-pound) pot roast\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n2 tablespoons ghee\n\n2 cups beef broth\n\n2 large carrots, peeled and cut into 1\" sections\n\n3 russet potatoes, quartered\n\n1 large yellow onion, peeled, halved, and sliced into half-moons\n\n8 ounces whole button mushrooms, stems trimmed\n\n1 tablespoon fresh thyme leaves\n\n1 Pat roast dry with paper towels. Season with salt and pepper.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Sear meat on all sides, about 5 minutes. Remove the meat and set aside.\n\n3 Add broth and deglaze the pot by stirring and scraping bottom and sides to loosen any browned bits. Add remaining ingredients. Return roast to pot. Press the Manual or Pressure Cook button and adjust time to 60 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n4 Remove the meat and vegetables to a serving platter. Let rest 5 minutes. Slice roast and serve warm.\n\n## Chuck Roast with Carrots and Onions\n\nOh, the humble chuck roast. It is usually overlooked because of toughness of the connective tissue that usually takes hours of low and slow cooking to dissolve. The Instant Pot\u00ae with its high pressure and steam takes care of this so much faster. Succulent and delicious, chuck roast is making a comeback!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 65 minutes\n\nServes 6\n\n* * *\n\n1 (3-pound) boneless chuck roast\n\n2 tablespoons horseradish mustard\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil\n\n2 cups beef broth\n\n2 teaspoons Worcestershire sauce\n\n1 medium yellow onion, peeled and diced\n\n5 large carrots, peeled and cut into 1\" sections\n\n1 Pat roast dry with paper towels. Massage mustard into roast. Season with salt and pepper.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Sear meat on all sides, about 5 minutes total. Remove the meat and set aside.\n\n3 Add broth and deglaze the pot by scraping the bottom and sides of the pot to loosen any browned bits. Stir in Worcestershire sauce.\n\n4 Add onion and carrots to pot. Place roast on top of vegetables. Press the Manual or Pressure Cook button and adjust time to 60 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n5 Transfer meat and vegetables to a serving platter. Let rest 5 minutes. Slice roast. Reserve some of the pot juices if you'd like to serve \"au jus\" with the meal. Serve warm.\n\n## Sweet-n-Spicy Beef Brisket\n\nThe Instant Pot\u00ae is a great way to dress up a cheaper cut of meat. Add some fabulous spices and this brisket is a five-star delight. Placing the brisket fat side down helps render some of that fat cap. Also, the maple syrup in the wet rub gets a good layer of caramelization for added flavor.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 66 minutes\n\nServes 6\n\n* * *\n\n2 tablespoons pure maple syrup\n\n2 tablespoons coconut aminos\n\n1 tablespoon adobo paste\n\n1 tablespoon yellow mustard\n\n1\/2 teaspoon garlic salt\n\n1 tablespoon Italian seasoning\n\n2 tablespoons coconut oil\n\n1 (3-pound) beef brisket\n\n1 cup water\n\n1 large yellow onion, peeled and roughly diced\n\n1 In a small bowl, whisk together syrup, coconut aminos, adobo paste, mustard, garlic salt, and Italian seasoning. Massage over beef brisket.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat coconut oil. Adjust temperature to Less. Place brisket in the pot with the fat cap side down. Sear 6 minutes to render fat.\n\n3 Add water and scatter onions around brisket. Press the Manual or Pressure Cook button and adjust time to 60 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n4 Remove the meat and transfer to a serving platter. Let rest 10 minutes. Slice brisket and serve warm.\n\n## Tender Flank Steak with Mushrooms and Onions\n\nPeanut butter and jelly . . . Laverne and Shirley . . . tweens and boy bands. These are classic combinations. But, let's add another. Steak and mushrooms. The drippings from the steak add another dimension to these button mushrooms that is like no other. Simple. Quick. Delicious. It's hard to tell the star of this dish!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 50 minutes\n\nServes 4\n\n* * *\n\n1 (2-pound) flank steak\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil, divided\n\n1 slice bacon, diced\n\n1 medium yellow onion, peeled and diced\n\n2 cups whole button mushrooms\n\n11\/2 cups beef broth\n\n1 Pat steak dry with paper towels. Season with salt and pepper.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Sear meat on all sides until browned, about 5 minutes per side. Remove meat and set aside.\n\n3 Add remaining oil to pot and saut\u00e9 bacon, onions, and mushrooms 3\u20135 minutes until onions are translucent.\n\n4 Add beef broth and deglaze the pot by stirring and scraping the bottom and sides of the pot to loosen any browned bits. Place meat on top of onions in pot. Lock lid. Press the Meat button and adjust time to 35 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n5 Transfer the meat to a serving platter. Let rest 5 minutes. Thinly slice meat against the grain. Serve immediately with mushrooms, onions, and a few tablespoons of the liquid from the pot.\n\n## Skirt Steak Fajitas\n\nThis simple dish with such depth of flavor can be enjoyed for dinner, lunch, or even for breakfast as most paleo folks like their leftover meat in the morning, served with a side of scrambled eggs and a thick slice of avocado. There are so many ways to enjoy this tasty beef so dive on in!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 45 minutes\n\nServes 4\n\n* * *\n\n1\/8 cup avocado oil\n\n1\/4 cup coconut aminos\n\n1 tablespoon fish sauce\n\n1 teaspoon ground cumin\n\n1 teaspoon chili powder\n\n2 tablespoons tomato paste\n\n1\/2 teaspoon sea salt\n\n1 (2-pound) skirt steak\n\n1 small yellow onion, peeled and diced\n\n1 medium green bell pepper, seeded and diced\n\n1 medium red bell pepper, seeded and diced\n\n1 cup beef broth\n\n1 In a small bowl, combine oil, coconut aminos, fish sauce, cumin, chili powder, tomato paste, and salt. Spread 3\/4 of this mixture on all sides of the steak. Reserve additional sauce.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae. Add skirt steak and sear on all sides for a total of about 5 minutes. Remove the meat and set aside. Add onion and peppers to the pot with reserved sauce. Saut\u00e9 3\u20135 minutes until onions are translucent.\n\n3 Add beef broth. Set meat on the layer of onion and peppers. Lock lid. Press the Meat button and cook for the default time of 35 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n4 Use a slotted spoon to transfer the meat and vegetables to a serving platter. Let steak rest 5 minutes. Thinly slice the skirt steak against the grain. Serve warm.\n\n## Korean Beef Short Ribs\n\nShort ribs are just that: ribs taken from the short portion of the rib bone of cattle. There are two types of cuts with short ribs: flanken and English cut. For this recipe, flanken is the preferred choice. Your butcher can help achieve this if there are no prepackaged options. Otherwise, the sauce of this recipe will work to flavor any cut of meat you prefer.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n1\/4 cup coconut aminos\n\n1\/4 cup pure maple syrup\n\n1 teaspoon fish sauce\n\n1 tablespoon apple cider vinegar\n\n1 tablespoon sesame oil\n\n1 teaspoon white pepper\n\n1\/2 teaspoon ground ginger\n\n1\/2 teaspoon garlic powder\n\n1\/2 teaspoon sea salt\n\n3 pounds beef short ribs\n\n1 cup beef broth\n\n2 green onions, sliced\n\n1 tablespoon toasted sesame seeds\n\n1 In a medium bowl, whisk together coconut aminos, syrup, fish sauce, vinegar, sesame oil, white pepper, ground ginger, garlic powder, and sea salt. Add ribs and toss to coat. Refrigerate covered at least 60 minutes or up to overnight.\n\n2 Add beef broth to the Instant Pot\u00ae and insert trivet. Arrange ribs standing upright with the meaty side facing outward. Lock lid. Press the Manual or Pressure Cook button and adjust time to 25 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n3 Transfer ribs to a serving platter and garnish with green onions and sesame seeds. Serve warm.\n\n## Mongolian Beef and Cashew BBQ\n\nTypically when you order this dish in restaurants, you go through the line and pick your meat, vegetables, oil, and sauces, and then hand it over to the chef who cooks it right in front of you. Cashews are actually in the paleo-ish category, because they are technically a legume. They fall more into the \"primal\" category. So, if you choose to indulge, purchase them raw so as to avoid the highly processed versions.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon sesame oil\n\n1 (2-pound) skirt steak\n\n1\/4 cup coconut aminos\n\n1\/2 cup pure maple syrup\n\n1\" piece fresh ginger, peeled and grated\n\n4 cloves garlic, minced\n\n1\/2 cup plus 2 tablespoons water, divided\n\n2 tablespoons arrowroot flour\n\n1\/4 cup sliced green onions\n\n1\/4 cup crushed raw cashews\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat oil and sear steak on all sides, for a total of about 4 minutes. Transfer steak to a cutting board. Slice beef against the grain into thin strips. Return beef to pot.\n\n2 In a medium bowl, whisk together coconut aminos, syrup, ginger, garlic, and 1\/2 cup water. Pour over beef and stir to deglaze any bits around the edges and bottom of the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 In a small dish, whisk together arrowroot flour and 2 tablespoons water until smooth to create a slurry. Stir this mixture into the beef mixture. Press the Saut\u00e9 button, then adjust temperature to Less and simmer unlidded 5 minutes until the sauce thickens.\n\n4 Ladle mixture into bowls. Garnish with green onions and raw cashews. Serve warm.\n\n## Meatloaf and Gravy\n\nMeatloaf is a worldwide tradition. It seems that every country in the world has a variation on this simple dish. As long as you have some form of ground meat, vegetables, and a binding agent, a meatloaf can be made. This quick and easy meatloaf is the perfect weeknight staple, especially when served with potatoes and vegetables.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 35 minutes\n\nServes 6\n\n* * *\n\n1 pound ground beef\n\n1 pound ground pork\n\n3 large eggs\n\n1 large shallot, peeled and finely diced\n\n1\/2 cup tomato sauce\n\n1\/2 cup almond meal\n\n1 tablespoon Italian seasoning\n\n1\/2 teaspoon smoked paprika\n\n1\/2 teaspoon garlic powder\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 cup beef broth\n\n1 tablespoon cassava flour\n\n2 tablespoons unsweetened almond milk\n\n1 In a large bowl, combine beef, pork, eggs, shallot, tomato sauce, almond meal, Italian seasoning, smoked paprika, garlic powder, salt, and pepper. Form mixture into a ball and flatten the top. Place meatloaf in a lightly greased 7-cup bowl.\n\n2 Add beef broth to the Instant Pot\u00ae and insert trivet. Place the glass bowl on top. Lock lid. Press the Meat button and cook for the default time of 35 minutes. When timer beeps, quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Remove meatloaf from Instant Pot\u00ae and let cool at room temperature 10 minutes. Tilt glass bowl and pour any liquid\/rendered fat back into the pot.\n\n4 Whisk flour and milk into the pot juices until a thick gravy forms. Transfer to a gravy boat.\n\n5 Slice meatloaf and serve with gravy.\n\n### UTENSIL HACKS\n\nYou may want to fashion an aluminum foil sling for easy retrieval if you don't have a pair of plate retriever tongs. Take a 10\" \u00d7 10\" square of aluminum foil and fold it back and forth until you have a 2\" \u00d7 10\" sling. Place sling under the bowl or pan before cooking so that you can easily lift up the heated dish when cooking is complete.\n\n## Balsamic Meatballs\n\nSo, there is a little debate over whether balsamic vinegar is officially a \"paleo\" product. I'm sure for the strictest of strict, the answer is probably no. Although vinegars have some wonderful health benefits, they are not all created equally. Some of the cheaper versions are laced with added sugars and caramel coloring. So, heed caution, read your labels, and go forth and conquer this recipe!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 11 minutes\n\nYields 16 meatballs\n\n* * *\n\n1 pound ground beef\n\n1 large egg\n\n2 tablespoons finely diced shallot\n\n1 tablespoon Italian seasoning\n\n1 teaspoon garlic powder\n\n1\/2 teaspoon celery seed\n\n1\/2 teaspoon smoked paprika\n\n1\/4 cup almond meal\n\n2 tablespoons avocado oil, divided\n\n2 cups water\n\n1 cup Marinara Sauce (see Chapter 4)\n\n2 tablespoons balsamic vinegar\n\n2 tablespoons julienned fresh basil\n\n1 In a medium bowl, combine beef, egg, shallot, Italian seasoning, garlic powder, celery seed, smoked paprika, and almond meal. Form into 16 meatballs. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place half the meatballs around the edges of the pot and sear on all sides for a total of about 4 minutes. Remove the first batch and set aside. Add remaining oil and meatballs and sear 4 minutes. Remove meatballs from pot.\n\n3 In a small bowl, combine marinara sauce and balsamic vinegar. Add seared meatballs to a 7-cup glass bowl. Gently toss in marinara-balsamic sauce to coat meatballs evenly.\n\n4 Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove dish from the pot. Garnish with fresh basil and serve warm.\n\n## Marinara Meatballs\n\nNow that's a good meatball! The homemade Marinara Sauce (see Chapter 4) really makes the meal. There is something about homemade foods and sauces that make you want to cook fresh every day. And, with the Instant Pot\u00ae, this is feasible with the time on cooked foods cut by at least half.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 16 minutes\n\nYields 16 meatballs\n\n* * *\n\n1 pound ground chuck\n\n1 large egg\n\n1\/2 cup chopped fresh basil\n\n1\/4 cup almond meal\n\n2 tablespoons finely diced yellow onion\n\n1\/2 teaspoon garlic powder\n\n1 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n11\/2 cups Marinara Sauce (see Chapter 4)\n\n1 In a medium bowl, combine ground chuck, egg, basil, almond meal, onion, garlic powder, salt, and pepper. Form into 16 golf ball\u2013sized meatballs. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place half the meatballs around the edges of the pot. Sear all sides of the meatballs for a total of about 4 minutes total. Remove the first batch and set aside. Add remaining oil and meatballs and sear 4 minutes. Remove meatballs.\n\n3 Discard extra juice and oil from the pot. Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Drain the pot. Add meatballs directly to the pot. Pour in Marinara Sauce. Let simmer 5 minutes, gently tossing meatballs.\n\n5 Transfer meatballs and sauce to bowls and serve warm.\n\n## Reuben Meatballs with Russian Dressing\n\nOh, the Reuben sandwich. Although these meatballs are void of the rye bread and ooey-gooey Swiss cheese, this meatball will hopefully quench that craving for the deli delight. Corned beef and sauerkraut are mixed directly in the ground beef. Caraway seeds are added to hit that \"rye bread\" flavor. And the Russian Dressing pulls it all together!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 11 minutes\n\nYields 24 meatballs\n\n* * *\n\nRussian Dressing\n\n1\/2 cup mayonnaise\n\n2 tablespoons paleo ketchup\n\n1 small shallot, peeled and minced\n\n1 tablespoon lemon juice\n\n1 teaspoon prepared horseradish\n\n1\/2 teaspoon smoked paprika\n\n1\/4 teaspoon sea salt\n\nMeatballs\n\n1 pound ground beef\n\n1\/2 pound finely chopped corned beef\n\n1\/2 cup finely chopped sauerkraut, drained (plus extra for garnish if desired)\n\n1\/4 cup almond meal\n\n2 tablespoons German mustard\n\n1 tablespoon caraway seeds\n\n1 large egg\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil\n\n1 cup water\n\n1 In a small bowl, combine Russian Dressing ingredients and refrigerate covered until ready to use.\n\n2 In a medium bowl, combine ground beef, corned beef, sauerkraut, almond meal, mustard, caraway seeds, egg, salt, and pepper. Form into 24 golf ball\u2013sized meatballs. Set aside.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon of the oil. Place half the meatballs around the edges of the pot and sear on all sides, for a total of about 4 minutes. Remove the first batch and set aside. Add remaining oil and meatballs and sear 4 minutes. Remove meatballs.\n\n4 Discard extra juice and oil from the Instant Pot\u00ae.\n\n5 Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n6 Transfer meatballs to a tray and serve with Russian Dressing on the side. Garnish with extra sauerkraut if desired.\n\n## Ground Beef and Broccoli Bowl\n\nTypically Beef and Broccoli is made with steak; however, this version not only is economical by substituting ground chuck, but is quick and easy to prepare with minimal cleanup.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 4 minutes\n\nServes 2\n\n* * *\n\n1\/4 cup coconut aminos\n\n2 tablespoons fish sauce\n\n1\/4 teaspoon white pepper\n\n4 cloves garlic, minced\n\n1 tablespoon cooking sherry\n\n2 tablespoons beef broth\n\n1 tablespoon avocado oil\n\n1 pound ground chuck\n\n1 bunch green onions, sliced (reserve 2 tablespoons chopped greens for garnish)\n\n1 head broccoli, cut in bite-sized pieces\n\n1 In a small bowl, combine coconut aminos, fish sauce, white pepper, garlic, sherry, and beef broth. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add ground chuck and green onions; stir-fry 4 minutes until onions are tender and beef is almost all brown.\n\n3 Add sauce mixture and toss to combine. Add broccoli. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer pot ingredients to two serving bowls. Garnish with green onions. Serve warm.\n\n## Just Beet It Burgers\n\nJazz up your ho-hum burgers by adding some beets to your ground beef. The sweetness of the beets pairs nicely with the garlic and cayenne. And it's always nice to hide vegetables in meals for picky eaters, especially dishes packed with vitamin C and essential minerals.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 11 minutes\n\nServes 4\n\n* * *\n\n1 pound ground chuck\n\n1\/2 cup scrubbed and grated beets\n\n1 teaspoon cooking sherry\n\n1\/2 teaspoon garlic powder\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/8 teaspoon cayenne pepper\n\n2 tablespoons coconut oil, divided\n\n1 cup water\n\n1 In a medium bowl, combine ground chuck, beets, cooking sherry, garlic powder, salt, pepper, and cayenne pepper. Form into 4 burger patties. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place 2 patties in the pot. Sear the burgers 2 minutes on each side. Remove the patties and set aside. Add remaining oil and remaining patties and sear 2 minutes on each side. Set aside.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Prop patties evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer patties to serving plates. Serve warm.\n\n## Corned Beef and Cabbage\n\nIt just isn't St. Paddy's Day without a little corned beef and cabbage. While drinking your black and tan, let the Instant Pot\u00ae do the legwork, allowing you time to do an Irish jig with your friends. The corned beef brisket even comes with its own spice packet, so sit back, kiss the Blarney Stone, and wait for dinner to be cooked for you!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 75 minutes\n\nServes 6\n\n* * *\n\n3-pound corned beef brisket with spice packet\n\n2 tablespoons avocado oil\n\n1 large yellow onion, peeled and quartered\n\n4 large carrots, peeled and cut into 2\" sections\n\n1 small cabbage, cut into 6 wedges\n\n3 cloves garlic, smashed and quartered\n\n1 cup beef broth\n\n1 cup water\n\n1 Massage contents of spice packet over corned beef brisket.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat avocado oil. Add brisket and sear on all sides 6\u20137 minutes. Transfer brisket to a plate.\n\n3 Add onion to the pot and stir-fry 3\u20134 minutes until tender. Add remaining ingredients, including brisket, to Instant Pot\u00ae. Lock lid. Press the Meat\/Stew button and cook for the default time of 45 minutes. When timer beeps, naturally release pressure for 20 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n4 Transfer brisket to a cutting board. When cooled enough to work with, slice and transfer to a serving tray. Add vegetables and serve warm.\n\n## Cuban Picadillo with Riced Cauliflower\n\nTraditionally served over rice, cauliflower is a perfect grain-free replacement. To \"rice\" the cauliflower, simply use a box grater until you achieve 3 cups. You will heat this in the pot liquid at the very end of cooking process until tender.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon coconut oil\n\n1 pound ground beef\n\n1 medium yellow onion, peeled and diced\n\n1 medium parsnip, peeled and diced\n\n2 cloves garlic, minced\n\n1\/2 cup dry white wine\n\n2 tablespoons tomato paste\n\n1 teaspoon ground cumin\n\n1\/2 teaspoon sea salt\n\n1 teaspoon dried oregano\n\n1\/4 cup sliced green olives\n\n1 tablespoon capers\n\n1 teaspoon caper juice\n\n3 cups riced cauliflower\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat coconut oil. Add ground beef, onion, and parsnip; stir-fry 5 minutes until onions are tender and beef is almost all brown. Add garlic and pour in wine; heat 1 minute to allow the alcohol to burn off.\n\n2 Add tomato paste, cumin, salt, oregano, green olives, capers, and caper juice to pot and stir to mix. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer pot ingredients to a large bowl. Drain all but 2 tablespoons liquid from the pot. Add cauliflower to pot and heat 2\u20133 minutes until heated through.\n\n4 Transfer cauliflower to serving bowls and top with beef mixture. Serve warm.\n\n## Chi-Town Italian Beef and Peppers\n\nA giant bowl of beef and peppers will make you think you are in Chicago, strolling the Navy Pier, shopping the Magnificent Mile, or even at Portillo's enjoying your Italian Beef, minus the dipped bun!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 52 minutes\n\nServes 6\n\n* * *\n\n2 tablespoons coconut oil\n\n3 pounds chuck roast, halved\n\n1 large onion, peeled, halved, and sliced\n\n2 medium red bell peppers, seeded and sliced\n\n2 medium green bell peppers, seeded and sliced\n\n1 (16-ounce) jar sliced pepperoncini, including juice\n\n3 cloves garlic, quartered\n\n1 cup beef broth\n\n1 teaspoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat coconut oil until melted. Add roast and sear on all sides for a total of 12 minutes.\n\n2 Add onion, bell peppers, pepperoncini, garlic, and beef broth to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time for 40 minutes. When timer beeps, quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Transfer roast to a cutting board and let rest 10 minutes. Use a slotted spoon to transfer onions and peppers to a large bowl. Thinly slice roast and add to the bowl. Transfer a few small spoonfuls of pot liquid to the bowl. Toss with black pepper. Serve warm.\n\n## Beef Shanks and Artichokes\n\nThis quick dish is a succulent meal for two that can be accomplished in no time. The salty and smooth artichoke hearts couple brilliantly with the fresh tomatoes. On a side note, the heated lycopene in the tomatoes is especially beneficial to men, aiding in the protection against prostate cancer and erectile dysfunction.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 4 minutes\n\nServes 2\n\n* * *\n\n2 (1\"-thick) beef shanks (total 11\/2 pounds)\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 tablespoon avocado oil\n\n1 (14-ounce) can quartered artichoke hearts, including juice\n\n10 grape tomatoes, halved\n\n1 Season beef shanks on both sides with salt and pepper.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add beef shanks and sear 2 minutes per side.\n\n3 Add remaining ingredients to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer beef to a cutting board and let rest 5 minutes.\n\n5 Use a slotted spoon to transfer artichoke hearts and tomatoes to a bowl. Slice beef and serve with vegetables. Spoon over some of the pot juice if desired.\n\n## Bison Sloppy Joe's\n\nWho needs a bun, when all the good stuff is in the sloppy?! Bison is such a lean meat alternative that is rich in iron, zinc, and vitamin B12. And move over salmon, bison is also a great source of omega-3 fatty acids.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 4 minutes\n\nServes 2\n\n* * *\n\n1 tablespoon avocado oil\n\n1 pound ground bison\n\n1 medium yellow onion, peeled and diced\n\n1 small green bell pepper, seeded and diced\n\n1 stalk celery, finely chopped\n\n2 teaspoons Worcestershire sauce\n\n2 cups tomato sauce\n\n2 tablespoons tomato paste\n\n1 tablespoon pure maple syrup\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add bison, onion, green pepper, and celery; stir-fry 3\u20134 minutes until onions are tender and bison is mostly browned.\n\n2 Add remaining ingredients to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer mixture to serving bowls and serve warm.\n\n## Ground Bison Bulgogi\n\nBulgogi is conventionally made with thin strips of marinated beef, but in this quick recipe, ground bison is used utilizing all the flavors. Enjoy this Korean classic as a modern bowl of yum using a quick Western take on the tradition.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 3\n\n* * *\n\n1 tablespoon sesame oil\n\n1 pound ground bison\n\n5 green onions, sliced (whites and greens separated)\n\n1 small carrot, peeled and diced\n\n3 cloves garlic, minced\n\n1\/4 cup coconut aminos\n\n2 tablespoons pure maple syrup\n\n1 tablespoon apple cider vinegar\n\n1 medium red apple, peeled, cored, and grated\n\n1 teaspoon minced ginger\n\n2 teaspoons toasted sesame seeds\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add ground bison, onion whites, and carrot; stir-fry 3\u20134 minutes until onions are tender and bison is almost all brown. Add garlic and heat 1 additional minute.\n\n2 Add coconut aminos, syrup, vinegar, apple, and ginger to pot and stir to mix. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer pot ingredients to serving bowls. Garnish with sesame seeds and onion greens.\n\n## Italian Beef Tongue Lettuce Wraps\n\nSo, beef tongue is completely a mental game. For first-timers, having a giant tongue on your cutting board can be somewhat off-putting. Once you can get past your personal mind games, there is incredibly tasty and juicy meat hidden in a scary form, and it is ideal for lettuce wraps.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 43 minutes\n\nServes 4\n\n* * *\n\n11\/2 pounds beef tongue\n\n1 medium white onion, peeled and roughly diced\n\n4 cloves garlic, halved\n\n2 bay leaves\n\n1 teaspoon ground cumin\n\n1 teaspoon smoked paprika\n\n1 teaspoon stone-ground mustard\n\n1 teaspoon ground coriander\n\n1 teaspoon celery seed\n\n4 cups water\n\n8 iceberg lettuce leaves\n\n3 Roma tomatoes, seeded and diced\n\n1\/4 cup julienned radishes\n\n1 medium avocado, peeled and diced\n\n1 medium lime, quartered\n\n1 Place tongue, onion, garlic, bay leaves, cumin, smoked paprika, mustard, coriander, celery seed, and water in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 40 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer tongue to a cutting board and let rest until cool enough to handle. Make a shallow cut lengthwise down the center of the tongue. Peel off and discard the thick outer layer. Cut off and discard any gristle at the back of the tongue. Dice the tongue meat and return it to the pot juices. Let simmer 2\u20133 minutes.\n\n3 Use a slotted spoon to transfer the meat to a bowl. Serve with lettuce leaves, tomatoes, radishes, avocado, and lime.\n\n## Pork, Lamb, and Game Main Dishes\n\nFrom Salsa Verde Pulled Pork and a tasty Cubano Bowl to Osso Bucco with Gremolata and Greek Lamb Meatballs, this chapter will take you around the world with new flavors and spices. You may even be trying lamb or veal for the first time and this chapter will show you just how to season it. And the Instant Pot\u00ae is the perfect kitchen appliance to yield juicy cuts of meat with its high-pressure method of cooking. In addition, it is an essential tool in the summer because it doesn't emit excess heat as the oven does. Because of this quicker cooking time, you'll have that barbecue dish on the picnic table in no time, allowing you to concentrate on your family and friends without heating up the house.\n\nSalsa Verde Pulled Pork\n\nSimple Dijon Pork Tenderloin\n\nOrange Rosemary Pork Tenderloin\n\nBalsamic Pork Tenderloin with Caperberries\n\nSteamed Dry-Rubbed Pork Spareribs\n\nCowgirl Baby Back Pork Ribs\n\nHam and Scalloped Potatoes\n\nGround Pork Stir-Fry Bowl\n\nSmothered Pork Chops with Mushroom Gravy\n\nPork Chops with Sauerkraut, Potatoes, and Apples\n\nCubano Bowl\n\nVietnamese Pork Chops\n\nBBQ Boneless Pork Loin Roast\n\nOsso Bucco with Gremolata\n\nPork Medallions Puttanesca\n\nLamb Riblets\n\nLamb Shanks with Tomato Curry Sauce\n\nGreek Lamb Meatballs\n\nChorizo Sliders with Jicama-Beet Slaw\n\nThe Busy Cook's Duck \u00e0 l'Orange Breast\n\n## Salsa Verde Pulled Pork\n\nOf all the magic tricks that the Instant Pot\u00ae has up its proverbial sleeve, the grand act is the cooking of a pork shoulder and the amount of time it takes. What used to take many hours to yield tender pull-apart pork can now be done in less than 2 hours.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 95 minutes\n\nServes 8\n\n* * *\n\n1 (5-pound) bone-in pork butt or shoulder\n\n2 teaspoons sea salt\n\n1 teaspoon freshly ground black pepper\n\n4 cups beef broth\n\n1 cup Salsa Verde (see Chapter 4)\n\n1 Pat pork butt with paper towels and set aside. Season with salt and pepper.\n\n2 Place the pork butt in the Instant Pot\u00ae. Add beef broth. Lock lid. Press the Manual or Pressure Cook button and adjust time to 85 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Check the pork to make sure it can easily pull apart. If not, press the Saut\u00e9 button and simmer unlidded for an additional 10 minutes.\n\n3 With the meat still in the Instant Pot\u00ae, use two forks and pull pork apart. Remove bone and discard. Use a slotted spoon to transfer the pork to a serving bowl. Toss in Salsa Verde and 2\u20133 tablespoons of broth from the pot. Serve warm.\n\n## Simple Dijon Pork Tenderloin\n\nPork tenderloin is a go-to for most busy home cooks, but overcooked, it can become dried out and become sad. With the Instant Pot\u00ae, the steam helps keep the moisture in the air around the loin while it is cooking. Enjoy this pork sliced into medallions alongside a simple salad or roasted vegetables.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 20 minutes\n\nServes 4\n\n* * *\n\n2 (2-pound) loins, each halved\n\n2 tablespoons whole-grain Dijon mustard\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 cup water\n\n3 cloves garlic, halved\n\n1 Pat pork loins with paper towels. Massage with mustard. Season with salt and pepper.\n\n2 Pour water into the Instant Pot\u00ae. Add garlic. Place loins on a steamer basket and insert in pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 20 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer pork loins to a serving tray and either slice into medallions or pull apart pork with two forks. Serve warm.\n\n## Orange Rosemary Pork Tenderloin\n\nSuch a decadent cut of protein, tenderloins take on the flavors they are cooked with, lending a salty and fatty note. Combined with the citrus from the orange and the piney essence and earthiness from the rosemary, this dish is a winner.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 20 minutes\n\nServes 4\n\n* * *\n\n1\/3 cup freshly squeezed orange juice (about 1 orange)\n\n1 tablespoon orange zest\n\n1 tablespoon finely chopped rosemary\n\n2 tablespoons honey\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 (2-pound) loins, each halved\n\n1\/2 cup water\n\n1 In large bowl, combine orange juice, orange zest, rosemary, honey, salt, and pepper. Add pork, toss, and refrigerate covered at least 30 minutes or up to overnight.\n\n2 Place pork in the Instant Pot\u00ae and add remaining marinade. Add water. Lock lid. Press the Manual or Pressure Cook button and adjust time to 20 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer pork loins to a serving tray and either slice into medallions or pull apart pork with two forks. Serve warm.\n\n## Balsamic Pork Tenderloin with Caperberries\n\nThe reduced balsamic sauce is such a rich and luxurious sweet flavor. Add the little salt bombs of caperberries to balance out the flavor. Caperberries are the fruit of the caper bush. They are usually brined and eaten or used in recipes as you would olives. The brininess is a wonderful complement to the balsamic vinegar.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n2 (2-pound) loins, each halved\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n4 tablespoons ghee, divided\n\n1\/2 cup beef broth\n\n1\/4 cup balsamic vinegar\n\n2 tablespoons caperberries, stems removed\n\n1 Season pork with salt and pepper.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 2 tablespoons ghee. Sear loins on all sides for a total of about 4\u20135 minutes.\n\n3 Add beef broth to pot. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid. Transfer pork to a plate.\n\n4 Add balsamic vinegar and remaining ghee to pot. Heat 4\u20135 minutes until sauce is reduced. Add caperberries and heat 1 additional minute.\n\n5 Slice pork and serve warm with sauce and caperberries.\n\n## Steamed Dry-Rubbed Pork Spareribs\n\nNo sticky sauce needed here. These ribs will be as smooth as butter and have all the flavoring needed with the broad mix of simple spices.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 30 minutes\n\nServes 6\n\n* * *\n\n1 rack pork ribs (about 31\/2 pounds)\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 teaspoon smoked paprika\n\n1 teaspoon chili powder\n\n1 teaspoon garlic powder\n\n1 teaspoon Italian seasoning\n\n1 cup water\n\n1 Cut ribs into 2-rib sections. In a large bowl, combine salt, pepper, paprika, chili powder, garlic powder, and Italian seasoning. Toss in rib sections to coat. Refrigerate covered at least 30 minutes or up to overnight.\n\n2 Add water to the Instant Pot\u00ae and insert trivet or steamer basket. Arrange ribs standing upright with the meaty side facing outward. Lock lid. Press the Manual or Pressure Cook button and adjust time to 30 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n3 Transfer ribs to a platter and serve warm.\n\n## Cowgirl Baby Back Pork Ribs\n\nThese Instant Pot\u00ae ribs are amazingly juicy due to the pressure and constant steam in the pot. Don't skip the coffee crystals with this rustically charming recipe. It adds a beautiful earthiness to the ribs. If you want a little crispness to the ribs, throw them on the grill or under a broiler for a couple of minutes until browned on each side.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n2 racks (about 3 pounds) baby back pork ribs\n\n1 teaspoon instant coffee crystals\n\n1 teaspoon sea salt\n\n1\/2 teaspoon chili powder\n\n1\/2 teaspoon ground cumin\n\n1\/2 teaspoon cayenne pepper\n\n1\/2 teaspoon stone-ground mustard\n\n1\/2 teaspoon garlic powder\n\n1\/2 teaspoon onion powder\n\n1\/4 teaspoon ground coriander\n\n1 medium onion, peeled and diced\n\n4 cloves garlic, halved\n\n2 cups water\n\n1 Cut ribs into 2-rib sections. In a large bowl, combine coffee, salt, chili powder, cumin, cayenne pepper, mustard, garlic powder, onion powder, and coriander. Toss in rib sections to coat. Refrigerate covered at least 30 minutes or up to overnight.\n\n2 Place onions, garlic, and water in the Instant Pot\u00ae and insert trivet. Arrange ribs standing upright with the meaty side facing outward. Lock lid. Press the Manual or Pressure Cook button and adjust time to 25 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n3 Transfer ribs to a platter and serve warm.\n\n### WHAT ARE AROMATICS?\n\nEven though the onion and garlic are not part of the actual final meal in this recipe, the fragrance created in the steam adds another level of flavor due to the aroma from the onion and garlic.\n\n## Ham and Scalloped Potatoes\n\nWhen you are just craving some TV-binge time and needing a little comfort food, grab your remote and a blanket and cozy up to this bowl of salty, creamy, decadent delight.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n4 tablespoons ghee, divided\n\n1\/2 medium yellow onion, peeled and sliced into half-moons\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/3 cup unsweetened almond milk\n\n1 teaspoon cassava flour\n\n3 small russet potatoes, thinly sliced\n\n1 cup diced ham\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 3 tablespoons ghee. Add onion and stir-fry 2\u20133 minutes until tender. Add salt, pepper, and milk. Whisk in flour. Press the Cancel button.\n\n2 Lightly grease a 7-cup glass bowl and layer 1\/3 of potatoes in the dish. Add 1\/3 of the ham. Add 1\/3 of the onion mixture. Repeat two more times until you have three layers. Top with remaining ghee.\n\n3 Place trivet in Instant Pot\u00ae and place the glass bowl on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 12 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Remove dish from pot and serve warm.\n\n## Ground Pork Stir-Fry Bowl\n\nForget about take-out food full of MSG, soy, and bad fats. Make your own quick and healthy version of Chinese take-out. Heck, you can even buy those great take-out boxes and wooden chopsticks if you miss the feel of holding the box while binge-watching the latest and greatest series.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 7 minutes\n\nServes 4\n\n* * *\n\n2 teaspoons sesame oil\n\n1 pound ground pork\n\n1\/2 medium yellow onion, peeled and diced\n\n1\/4 cup coconut aminos\n\n1\/4 cup water\n\n1 teaspoon Chinese five spice powder\n\n1 cup sliced shiitake mushrooms\n\n1\/4 cup peeled and julienned carrots\n\n1 overflowing cup chopped fresh kale\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add pork and onion; stir-fry 4 minutes until pork starts to brown.\n\n2 Add coconut aminos, water, Chinese spice, mushrooms, and carrots. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Add kale to the pot and toss until it begins to wilt. Transfer to serving bowls. Serve warm.\n\n### WHAT IS CHINESE FIVE SPICE?\n\nAlthough there are a few variations, Chinese five spice is traditionally a blend of star anise, cloves, cinnamon, Szechuan peppers, and fennel seeds. It can be found already combined in the spice section of most grocery stores.\n\n## Smothered Pork Chops with Mushroom Gravy\n\nThis is a classic country-style dish for a reason. It just hits all the right comfort buttons. Add some mashed cauliflower to round out this meal.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 25 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons avocado oil\n\n4 (1\"-thick) bone-in pork chops\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 cup beef broth\n\n3 cups sliced baby bella or cremini mushrooms\n\n1 small Vidalia onion, peeled and sliced into half-moons\n\n2 tablespoons unsweetened almond milk\n\n2 teaspoons Worcestershire sauce\n\n2 tablespoons arrowroot flour\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat oil. Season pork chops with salt and pepper. Sear chops 2\u20133 minutes per side in batches. Set pork aside.\n\n2 Add broth, mushrooms, and onion to the pot. Insert steamer basket. Add pork chops, propping them up on the sides so as not to overlap. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Remove pork chops and steamer basket. Add milk and Worcestershire sauce to pot with juices and broth. Slowly whisk in flour, 1 teaspoon a time, until a gravy forms. Transfer mushroom gravy to a bowl and serve with pork chops. Season with more salt if desired.\n\n## Pork Chops with Sauerkraut, Potatoes, and Apples\n\nThese pork chops are juicy and flavorful with a German flair from the addition of beer, potatoes, sauerkraut, and caraway seeds. If you're looking for a nonalcoholic option, feel free to substitute beef broth for the beer.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 30 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons avocado oil\n\n4 (1\"-thick) bone-in pork chops\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n4 slices bacon, diced\n\n1 stalk celery, finely chopped\n\n3 large carrots, peeled and sliced\n\n1 large yellow onion, peeled and diced\n\n1 clove garlic, minced\n\n1 (12-ounce) bottle gluten-free beer\n\n4 medium red potatoes, peeled and quartered\n\n2 medium red apples, peeled, cored, and quartered\n\n1 (1-pound) bag high-quality sauerkraut, rinsed and drained\n\n1 tablespoon caraway seeds\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat oil. Season pork chops with salt and pepper. In batches, sear pork chops 2\u20133 minutes per side. Set pork aside.\n\n2 Add bacon, celery, carrots, and onion to the pot. Stir-fry 3\u20135 minutes until the onions are translucent. Add garlic and cook 1 minute. Add beer and deglaze the pot by stirring and scraping the bottom and sides to loosen any browned bits. Simmer unlidded for 5 minutes.\n\n3 Add potatoes, apples, and sauerkraut. Sprinkle with caraway seeds. Slightly prop pork chops up against the sides of the pot so as not to crowd the pork. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n4 Transfer the pork chops, sauerkraut, potatoes, and apples to a serving dish. Serve warm.\n\n## Cubano Bowl\n\nYou'll never miss the bread from the traditional Cubano sandwich with this recipe. You still get the tender pork, ham, mustard, and pickles. Although there is no Swiss cheese in this recipe, the nutritional yeast fulfills that note of nuttiness and \"cheese.\"\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 95 minutes\n\nServes 8\n\n* * *\n\n1 (5-pound) bone-in pork butt or shoulder\n\n11\/2 teaspoons sea salt, divided\n\n11\/2 teaspoons freshly ground black pepper, divided\n\n1\/2 medium yellow onion, peeled and chopped\n\n1 cup water\n\n1 tablespoon fresh lime juice\n\n1 tablespoon fresh orange juice\n\n1 teaspoon garlic powder\n\n1 teaspoon dried oregano\n\n1 teaspoon cayenne pepper\n\n2 teaspoons ground cumin\n\n1 tablespoon nutritional yeast\n\n2 cups diced smoked ham\n\n1 cup diced dill pickles\n\n8 teaspoons yellow mustard\n\n1 Dry pork butt with paper towels. Season with 1\/2 teaspoon salt and pepper. Set aside.\n\n2 Place the pork butt in the Instant Pot\u00ae. Add onion, water, lime juice, orange juice, garlic powder, oregano, cayenne pepper, cumin, nutritional yeast, and remaining salt and pepper.\n\n3 Press the Manual or Pressure Cook button and adjust time to 85 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Check the pork to make sure it can easily pull apart. If not, press the Saut\u00e9 button and simmer unlidded an additional 10 minutes.\n\n4 Use two forks to shred the pork in the pot. Remove the bone and discard. Use a slotted spoon to transfer the pork to separate bowls. Garnish with ham, pickles, and a drizzle of yellow mustard.\n\n## Vietnamese Pork Chops\n\nThis unique twist of sweet and spicy flavors complements the saltiness of the pork so well. There are just enough red chili pepper flakes to balance out the flavors of this dish, but if heat is your thing, go for it and add some more!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\nMarinade\n\n2 ripe plums, pitted\n\n1 small bunch cilantro, including stems\n\n2 tablespoons molasses\n\n2 tablespoons fresh lime juice\n\n1 teaspoon lime zest\n\nPinch dried red pepper flakes\n\n3 cloves garlic, minced\n\n1\/2\" piece fresh garlic, peeled and minced\n\n1\/4 cup avocado oil\n\n1 teaspoon sea salt\n\nPork Chops\n\n4 (1\"-thick) bone-in pork chops\n\n1 cup water\n\n1 In a food processor, combine all marinade ingredients and pulse until smooth. Transfer to a large bowl, add pork chops and toss. Marinate covered or sealed in the refrigerator at least 1 hour or up to overnight.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Add pork chops to steamer basket. Pour excess marinade over pork chops. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, let pressure release naturally for 5 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Transfer the pork chops to a serving tray. Serve warm.\n\n## BBQ Boneless Pork Loin Roast\n\nThe boneless loin roast is a center cut of pork. It is not the tenderloin and it is not the shoulder. Be aware of the different cuts as they will have different cooking times and results. For instance, pork shoulder or butt is perfect for shredding for barbecue dishes and takes longer to prepare. Pork tenderloin is more narrow and long and requires a lower cooking time.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 41 minutes\n\nServes 6\n\n* * *\n\nRoast\n\n2 teaspoons sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 teaspoon garlic powder\n\n21\/2 pounds boneless pork loin roast\n\n2 cups water\n\nBarbecue Sauce\n\n1\/4 cup water\n\n3\/4 cup paleo ketchup\n\n1\/2 medium onion, peeled and grated\n\n1\/4 cup molasses\n\n1 teaspoon garlic powder\n\n1 teaspoon ground cumin\n\n1\/4 teaspoon cayenne pepper\n\n1 tablespoon yellow mustard\n\n2 teaspoons apple cider vinegar\n\n1 In a small bowl, mix together salt, pepper, and garlic powder. Massage into roast.\n\n2 Add water to the Instant Pot\u00ae. Add roast. Lock lid. Press the Manual or Pressure Cook button and adjust for 30 minutes. When timer beeps, let pressure release naturally for 10 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Transfer loin to a plate. Discard liquid from pot. Add barbecue sauce ingredients. Press the Saut\u00e9 button. Stir and saut\u00e9 4\u20135 minutes until sauce starts to reduce and thicken. Ladle out 1\/2 cup sauce and set aside.\n\n4 Return pork to pot and saut\u00e9 6 minutes, turning roast until all sides are covered and sauce is caramelized. Transfer roast to a cutting board. Let rest 5 minutes.\n\n5 Thinly slice roast and pour reserved sauce over slices. Serve warm.\n\n## Osso Bucco with Gremolata\n\nFall off the bone, hearty, and flavorful are just a few ways to describe this scrumptious meal. And don't forget about that bone marrow. To. Die. For. The gremolata just freshens everything up. Add two more shanks for an extraordinary dinner party experience with your favorite couple.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 41 minutes\n\nServes 2\n\n* * *\n\nGremolata\n\n1\/3 cup chopped fresh Italian flat-leaf parsley\n\n2 teaspoons lemon zest\n\n2 cloves garlic, minced\n\nOsso Bucco\n\n2 (11\/2\") bone-in veal shanks (about 31\/4 pounds total weight)\n\n1 teaspoon sea salt\n\n1 teaspoon freshly ground black pepper\n\n1 tablespoon avocado oil\n\n1 (28-ounce) can crushed tomatoes, including juice\n\n1 In a small bowl, combine gremolata ingredients and refrigerate covered until ready to use.\n\n2 Season both sides of veal shanks with salt and pepper.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Sear veal shanks about 3 minutes per side until browned.\n\n4 Add crushed tomatoes, including juice. Lock lid. Press the Meat\/Stew button and adjust time to 35 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Transfer Osso Bucco and tomato sauce to two shallow bowls. Garnish with gremolata. Serve warm.\n\n## Pork Medallions Puttanesca\n\nWith just the right fat and salt from the pork married to this lusty derivative-named sauce, there is no way this quick dish won't be a home run with a date night with your sweetie.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 12 minutes\n\nServes 4\n\n* * *\n\n2-pound pork tenderloin, cut into 1\" rounds\n\n1 teaspoon salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil, divided\n\n11\/2 cups Puttanesca Sauce (see Chapter 4)\n\n2 tablespoons water\n\n1 Season pork medallions with salt and pepper on both sides. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Sear half the pork 1 minute on each side. Set aside. Repeat with remaining oil and pork.\n\n3 Add Puttanesca Sauce and water to pot. Gently toss in pork medallions. Lock lid. Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer pork medallions and sauce to a dish and serve warm.\n\n## Lamb Riblets\n\nThese little riblets are so tasty that there is no need to douse them in a thick barbecue sauce. Sweet and simple helps enhance the natural flavor. Smaller than traditional beef or pork ribs, these also make a great finger food for guests.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 2\n\n* * *\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon garlic salt\n\n1 teaspoon smoked paprika\n\n11\/2 pounds lamb riblets\n\n2 tablespoons avocado oil\n\n1 (14.5-ounce) can crushed fire-roasted tomatoes, including juice\n\n1 In a small bowl, combine sea salt, garlic salt, and paprika. Massage into lamb riblets.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add riblets and sear on each side for a total of 4\u20135 minutes. Add tomatoes, including juice. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally for 10 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n3 Transfer ribs to a plate and serve warm.\n\n## Lamb Shanks with Tomato Curry Sauce\n\nLamb shanks are an amazing protein that easily take on the flavors they are cooked with. That's why the Tomato Curry Sauce is a spectacular sauce to pair with these meaty shanks.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 40 minutes\n\nServes 2\n\n* * *\n\n2 bone-in lamb shanks (about 11\/2 pounds)\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1\/4 cup ghee\n\n2 medium carrots, peeled and cut into 1\/2\" pieces\n\n2 cups Tomato Curry Sauce (see Chapter 4)\n\n1 cup water\n\n1\/4 cup chopped fresh cilantro\n\n1 Season both sides of veal shanks with salt and pepper.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Sear veal shanks 2\u20133 minutes per side until browned.\n\n3 Add carrots, tomato curry sauce, and water to pot. Toss ingredients. Lock lid. Press the Meat\/Stew button and adjust time to 35 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer shanks, tomato curry sauce, and carrots to two shallow bowls. Garnish with cilantro. Serve warm.\n\n## Greek Lamb Meatballs\n\nCalled keftedes in Greece, let these meatballs take you on a temporary vacation to the Mediterranean. Serve as is, on a salad, or with some traditional tzatziki sauce.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 11 minutes\n\nYields 16 meatballs\n\n* * *\n\n1 pound ground lamb\n\n1 large egg\n\n1\/2 teaspoon lemon zest\n\n1\/2 cup chopped fresh mint\n\n1\/4 cup almond meal\n\n2 tablespoons finely diced red onion\n\n1\/2 teaspoon ground coriander\n\n1\/2 teaspoon ground cumin\n\nPinch ground cinnamon\n\nPinch ground nutmeg\n\n1 teaspoon sea salt\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n1 large lemon, cut into 6 wedges\n\n1 tablespoon chopped fresh dill\n\n1 In a medium bowl, combine lamb, egg, lemon zest, mint, almond meal, diced onion, coriander, cumin, cinnamon, nutmeg, and salt. Form into 16 golf ball\u2013sized meatballs. Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place 8 meatballs around the edges of the pot. Sear all sides of the meatballs for a total of about 4 minutes. Remove the first batch and set aside. Add remaining oil and meatballs and sear 4 minutes. Remove meatballs.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Place meatballs evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer meatballs to a serving plate and garnish with lemon wedges and fresh dill.\n\n### TZATZIKI SAUCE AND HOW TO MAKE IT\n\nTzatziki is a yogurt-based sauce served with grilled meats. If you allow yogurt into your diet, this sauce pairs exquisitely with these Greek Lamb Meatballs. To make a quick sauce, combine the following ingredients and then refrigerate lidded until use: 8-ounces plain Greek yogurt, 1 medium peeled, seeded, and diced cucumber, 1 tablespoon olive oil, 1 teaspoon lemon juice, 2 minced cloves of garlic, 2 teaspoons chopped fresh dill. Season with salt and pepper to taste.\n\n## Chorizo Sliders with Jicama-Beet Slaw\n\nThese Chorizo Sliders with Jicama-Beet Slaw yield the heat from the chorizo and coolness from the slaw to create an undeniably nomtastic combination!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 4\n\n* * *\n\nJicama-Beet Slaw\n\n1 cup grated jicama\n\n1\/2 cup peeled and grated beets\n\n1\/8 cup freshly squeezed orange juice\n\n1 tablespoon olive oil\n\n1\/2 teaspoon chopped dill\n\nPinch sea salt\n\nSliders\n\n1\/2 pound ground chorizo\n\n1\/2 pound ground pork\n\n2 tablespoons seeded and finely diced green pepper\n\n1 teaspoon smoked paprika\n\n2 tablespoons coconut oil, divided\n\n1 cup water\n\nGarnish\n\n1 medium avocado, peeled and sliced\n\n8 tomato slices\n\n1 In a medium bowl, combine slaw ingredients. Refrigerate until ready to use.\n\n2 In a large bowl, combine ground chorizo, pork, green pepper, and paprika. Form into 8 slider patties. Set aside.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place 4 patties in the pot. Sear each side of the burgers 2 minutes each side. Remove the patties and set aside. Add another tablespoon of oil and remaining patties and sear them. Set aside.\n\n4 Add water to the Instant Pot\u00ae. Add steamer basket. Prop patties evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Transfer slider patties to plate. Serve warm with avocado and tomato slices. Dollop slaw on top of each slider.\n\n## The Busy Cook's Duck \u00e0 l'Orange Breast\n\nYour dinner mate will think you studied for years at Le Cordon Bleu when serving this French classic dish, when in actuality, it only took you about 20 minutes tops to get this to a plate. Bon app\u00e9tit!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 2\n\n* * *\n\n1\/3 cup freshly squeezed orange juice (about 1 orange)\n\n1 tablespoon orange zest\n\n2 tablespoons honey\n\n1 pound duck breasts (about 2\u20133 breasts)\n\n1 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n2 teaspoons cassava flour\n\n1 In a small bowl, whisk together orange juice, orange zest, and honey.\n\n2 Season both sides of duck breasts with salt and pepper. Carefully score the duck fat in a cross-hatch pattern making sure to avoid the meat.\n\n3 Press the Saut\u00e9 button on the Instant Pot\u00ae and place duck breasts fat side down. Heat 8 minutes. Do not touch the duck as it is rendering off its fat. After 8 minutes, remove duck to a plate. Pour out all fat except for 1 teaspoon. Add duck back to the pot with the fat side up. Add orange juice mixture. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. Adjust the pressure to Low. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer duck to a cutting board and let rest 5 minutes. Whisk flour into pot juices until thickened.\n\n5 Slice duck and transfer to plates. Drizzle sauce from pot over duck. Serve warm.\n\n## Seafood and Fish Main Dishes\n\nSeafood is one of those meals that gets ordered a lot at restaurants but gets overlooked at home. Seafood is a great paleo source of protein and is something to embrace. A lot of home chefs are timid when it comes to cooking fish and shellfish but that's a shame. Fish and shellfish are not only low in calories and full of nutrients, but they are some of the quickest meals you can cook in your Instant Pot\u00ae. In addition, the steaming functionality of the appliance makes for some tender and moist meals.\n\nIf you are purchasing fresh seafood rather than sealed, frozen seafood, make sure it comes from a trusted source. Don't be afraid to ask when the shipment came in, where it came from, if it was wild-caught, and whether or not the fishmonger can prepare the fish for you. If that fishmonger can't or won't answer those questions, find another one. Once you have made your choice, prepare your meal within 1\u20132 days for optimal freshness.\n\nThe Instant Pot\u00ae can easily and perfectly cook your seafood with its steam and pressure capabilities. The recipes in this chapter call for short cooking times and quick pressure release. You don't want to do a natural release with most fish as it will continue to cook if the pressure releases slowly. From a Fish Taco Bowl to Greek Sea Bass, this chapter covers a variety of recipes that will have you eating seafood on a regular basis.\n\nSteamed Mussels and Chorizo\n\nStewed Halibut\n\nFish Taco Bowl\n\nThai Coconut Curry Monkfish\n\nLouisiana Shrimp and Sausage\n\nShrimp, Prosciutto, and Asparagus Salad with Yuzu Vinaigrette\n\nShrimp Scampi\n\nCrab-Chorizo Stuffed Calamari\n\nBerbere Salmon\n\nDilled Salmon and Broccoli\n\nDairy-Free Creamed Crab\n\nCitrus-Steamed Littleneck Clams\n\nSalmon Croquettes\n\nGreek Sea Bass\n\nHerbed Cod with Yellow Squash\n\nMahi-Mahi with a Lemon-Caper Butter Sauce\n\nSteamed Snow Crab Legs\n\nSimply Steamed Lobster Tails\n\nQuick Crawfish Boil\n\n## Steamed Mussels and Chorizo\n\nMussels should be on the rotation of every family. They are inexpensive, quick to cook, and tasty. These nutrient-dense bivalves also pack a hefty dose of goodness such as omega-3 fatty acids, vitamin A, folate, and B12 as well as minerals such as phosphorus, zinc, and manganese.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 6 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons ghee\n\n1 medium yellow onion, peeled and diced\n\n1\/2 pound chorizo, loose or cut from casings\n\n3 cloves garlic, minced\n\n1 (14.5-ounce) can diced tomatoes, including juice\n\n1\/2 teaspoon sea salt\n\n2 pounds fresh mussels, cleaned and debearded\n\n1\/4 cup chopped fresh Italian flat-leaf parsley\n\n1 medium lemon, quartered\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Add ghee and heat until melted. Add onion and chorizo and saut\u00e9 3 minutes. Add garlic and cook 1 additional minute. Stir in diced tomatoes, including juice, and salt. Cook an additional 2 minutes.\n\n2 Add mussels. Lock lid. Press the Manual or Pressure Cook button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Remove mussels and discard any that haven't opened. Transfer mussels to four bowls and pour chorizo mixture from Instant Pot\u00ae equally among bowls. Garnish each bowl with 1 tablespoon parsley and a lemon quarter. Serve immediately.\n\n## Stewed Halibut\n\nJust for the halibut, eh? Rich in vitamins B6 and B12, potassium, and omega-3 fatty acids, this fish is a low calorie choice and mild tasting white flaky fish. Coupled with the tomatoes and other Mediterranean flavors, grab a glass of white wine while the Instant Pot\u00ae does your cooking.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 3 minutes\n\nServes 2\n\n* * *\n\n1 (14.5-ounce) can diced tomatoes, including juice\n\n1\/2 cup chicken broth\n\n1\/4 teaspoon onion powder\n\n3 cloves garlic, minced\n\n2 teaspoons smoked paprika\n\n1 tablespoon chopped fresh tarragon\n\n1 medium green bell pepper, seeded and small-diced\n\n1 stalk celery, finely diced\n\n1 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1 pound halibut fillets, cut into bite-sized pieces\n\n1 Add all ingredients except halibut to the Instant Pot\u00ae and stir to mix. Once mixed, add the fish on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Transfer all ingredients to a serving bowl. Serve warm.\n\n## Fish Taco Bowl\n\nThis recipe looks complicated, but you probably have most of the spices on hand. You can make the slaw and aioli ahead of time. Also, instead of grating your own cabbage and peeling a carrot, pick up a bag of coleslaw mix; it's the same thing. There is also broccoli slaw available for a twist. But do give this recipe a try. The variety of flavors will win your heart.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 3 minutes\n\nServes 4\n\n* * *\n\nSlaw\n\n1\/2 cup grated green cabbage\n\n1 large carrot, peeled and grated\n\nJuice of 1\/2 large lime\n\n2 dashes sriracha\n\n1\/4 cup chopped fresh cilantro\n\n1\/2 teaspoon sea salt\n\nSpicy Aioli\n\n1\/4 cup mayonnaise\n\nPinch salt\n\nSqueeze lime juice\n\nDash sriracha\n\nFish\n\n3 (6-ounce) cod fillets\n\nJuice from 1\/2 large lime\n\n2 tablespoons fresh orange juice\n\n1 teaspoon garlic salt\n\n1 teaspoon ground cumin\n\n1 tablespoon olive oil\n\n1 cup water\n\nGarnishes\n\n1 medium avocado, peeled and diced\n\n2 Roma tomatoes, seeded and diced\n\n4 lime wedges\n\n1 Combine slaw ingredients in a medium bowl. Refrigerate covered until ready to use.\n\n2 Combine Spicy Aioli ingredients in a small bowl. Taste and adjust to your liking.\n\n3 In a large bowl, combine fish, lime juice, orange juice, garlic salt, cumin, and olive oil and refrigerate 15 minutes.\n\n4 Add 1 cup water to the Instant Pot\u00ae and insert trivet. Place steamer basket on top of trivet. Add cod in an even row onto steamer basket. Pour in marinade for the steaming aromatics. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Distribute slaw into serving bowls. Add fish and garnishes. Drizzle with aioli. Serve warm.\n\n## Thai Coconut Curry Monkfish\n\nKnown as the poor man's lobster, monkfish is a buttery and firm-textured whitefish that is unique and exquisite. Monkfish are certainly ugly, freaky looking sea creatures, but their taste is amazing. I suppose they represent the old adage, \"Never trust a book by its cover.\" If so, no one would have ever put this guy on the grill.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 3 minutes\n\nServes 4\n\n* * *\n\n1 (13.5-ounce) can coconut milk\n\nJuice of 1 medium lime\n\n1 tablespoon red curry paste\n\n1 teaspoon fish sauce\n\n1 teaspoon coconut aminos\n\n1 teaspoon honey\n\n2 teaspoons sriracha\n\n2 cloves garlic, minced\n\n1 teaspoon turmeric powder\n\n1 teaspoon ground ginger\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon white pepper\n\n1 pound monkfish, cut into 1\" cubes\n\n1\/4 cup chopped fresh cilantro\n\n3 lime wedges\n\n1 In a large bowl, whisk together coconut milk, lime juice, red curry paste, fish sauce, coconut aminos, honey, sriracha, garlic, turmeric, ginger, sea salt, and white pepper.\n\n2 Place monkfish in the bottom of the Instant Pot\u00ae. Pour coconut milk mixture over the fish. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer fish and broth evenly into bowls. Garnish each with equal amounts of chopped cilantro and a lime wedge. Serve.\n\n## Louisiana Shrimp and Sausage\n\nEat this as is or served over a bed of riced cauliflower. Creole seasoning is a somewhat vague ingredient on this recipe list. Look in your local grocer's spice aisle and there will be several versions. Choose the one that speaks to you as you just can't go wrong with these beautiful spice blends.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 4\n\n* * *\n\n2 tablespoons avocado oil\n\n1 small onion, peeled and diced\n\n1 stalk celery, diced\n\n1 small green bell pepper, seeded and diced\n\n1 (15-ounce) can diced tomatoes, including juice\n\n1 tablespoon tomato paste\n\n1 teaspoon honey\n\nPinch dried basil\n\n2 teaspoons Creole seasoning\n\n12 ounces fully cooked andouille sausage, sliced\n\n1 pound large shrimp, peeled and deveined\n\n1\/2 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat oil. Add onion, celery, and bell pepper; saut\u00e9 3\u20135 minutes until onions are translucent and peppers are tender.\n\n2 Stir in tomatoes and their juice, tomato paste, honey, basil, Creole seasoning, and sausage. Add shrimp. Lock lid. Press the Manual or Pressure Cook button and adjust time to 4 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Spoon into bowls, season with salt and pepper, and serve warm.\n\n## Shrimp, Prosciutto, and Asparagus Salad with Yuzu Vinaigrette\n\nYuzu kosho is the latest and greatest seasoning. It is citrusy and vinegary with a little kick, and chefs from around the globe are flocking to it. Yuzu is actually a Japanese fruit also known as citrus junos. To make a kosho paste, chefs use the peel from the yuzu and add chili pepper and salt. This mixture is then turned into a hot sauce.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 0 minutes\n\nServes 2\n\n* * *\n\nYuzu Vinaigrette\n\n1 teaspoon yuzu kosho hot sauce\n\n1 teaspoon honey\n\n1\/4 cup extra-virgin olive oil\n\n1 teaspoon apple cider vinegar\n\nPinch sea salt\n\nShrimp and Asparagus\n\n1 cup water\n\n1 bunch asparagus, cut into 1\" pieces (woody ends removed)\n\n1 teaspoon sea salt, divided\n\n1 pound medium shrimp, peeled and deveined\n\nJuice of 1\/2 medium lemon\n\n2 tablespoons ghee\n\nSalad\n\n2 overflowing cups of mixed greens\n\n1 medium mango, peeled and diced\n\n2 ounces prosciutto, ripped into pieces\n\n1 In a small bowl, combine all vinaigrette ingredients. Set aside.\n\n2 Add water to the Instant Pot\u00ae and insert trivet. Place steamer basket on trivet. Spread out the asparagus evenly over the bottom of the steamer basket. Sprinkle with 1\/2 teaspoon salt. Add the shrimp, drizzle with lemon juice, and sprinkle with remaining 1\/2 teaspoon salt. Place ghee evenly on top of shrimp in small scoops. Lock lid. Press the Steam button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Place mixed greens in two serving bowls and toss with preferred amount of vinaigrette. Garnish with mango and prosciutto. Add steamed shrimp and asparagus. Serve.\n\n## Shrimp Scampi\n\nDon't forget to mark your calendar for April 29, which is officially National Shrimp Scampi Day! Serve this as is or enjoy it over a bed of zoodled zucchini. After removing the shrimp from the Instant Pot\u00ae, put your vegetable \"noodles\" in the pot and heat in the same liquid 2\u20133 minutes until tender. Drain and serve with the shrimp on top. One pot cooking at its finest!\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 4\n\n* * *\n\n1\/4 cup ghee\n\n6 cloves garlic, minced\n\n1\/2 cup dry white wine\n\n1 medium lemon, halved\n\n2 pounds large shrimp, peeled and deveined (tails on)\n\n1\/2 teaspoon sea salt\n\n1 teaspoon lemon zest\n\n1\/4 cup chopped fresh Italian flat-leaf parsley\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat ghee and add garlic. Saut\u00e9 1\u20132 minutes until garlic starts to brown. Add wine and juice from 1\/2 lemon. Continue to cook 2\u20133 minutes to allow alcohol to burn off.\n\n2 Add shrimp to pot. Lock lid. Press the Steam button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Spoon shrimp into bowls. Give a quick squeeze of the lemon over shrimp, and add salt and lemon zest. Garnish with parsley. Serve warm.\n\n## Crab-Chorizo Stuffed Calamari\n\nThere is more to calamari than just cutting them into little rings and deep-frying them. The bodies of calamari make the perfect little sausage casings and absolutely add more flavor. If you don't like the heat, substitute the spicy chorizo sausage for either ground pork or some mild Italian sausage. The result will still be unique and tasty!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 4\n\n* * *\n\nCalamari\n\n10\u201312 calamari tubes\n\n1\/4 pound chorizo, loose or cut from casings\n\n1\/4 cup crabmeat\n\n1 large egg\n\n1 tablespoon almond meal\n\nPinch sea salt\n\n1 tablespoon finely diced shallot\n\n1 cup water\n\nSauce\n\n1 (14.5-ounce) can crushed tomatoes, including juice\n\n2 teaspoons honey\n\n3 cloves garlic, minced\n\n1\/2 teaspoon sea salt\n\n2 tablespoons chopped chives\n\n1 Rinse calamari tubes and set aside. In a medium bowl, combine chorizo, crab, egg, almond meal, salt, and diced shallot. Transfer mixture to a piping bag or plastic bag. Cut the tip off large enough for the mixture to pass through. Pipe sausage into calamari tubes. Fasten opening with a toothpick.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Place stuffed calamari in an even row on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Remove steamer basket with the calamari from the pot and discard liquid. Add crushed tomatoes and juice to the Instant Pot\u00ae. Stir in honey, garlic, and salt. Add stuffed calamari. Saut\u00e9 2\u20133 minutes until sauce is warmed through and the internal temperature of the stuffed calamari reaches at least 160\u00b0F.\n\n4 Remove toothpicks. Garnish with chives. Serve warm as a main course or as an appetizer.\n\n## Berbere Salmon\n\nBerbere is an Ethiopian spice and herb blend that is a little fiery and a lot aromatic, which makes it a perfect romantic meal for two. It can be found in specialty spice shops or easily online. There are also several \"make your own\" recipes online. This spice blend is great on almost any protein.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 2 minutes\n\nServes 2\n\n* * *\n\n2 (5-ounce) salmon fillets\n\n1 tablespoon berbere seasoning\n\n1 cup water\n\n2 lemon wedges\n\n1 Season salmon with berbere on all sides.\n\n2 Pour water into the Instant Pot\u00ae and insert steamer basket. Place salmon on steamer basket. Lock lid. Press the Steam button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer fish to plates with lemon wedges. Serve immediately.\n\n## Dilled Salmon and Broccoli\n\nKnown for its beneficial omega-3 fatty acids, salmon is also high in vitamin B12, vitamin D, and selenium, among others. Do your body good and quickly steam this very mild and flaky fish in the Instant Pot\u00ae. Adding the broccoli not only gives you more nutrients, but it makes this a complete meal with minimal cleanup.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 2\n\n* * *\n\n2 (6-ounce) salmon fillets\n\n1\/2 teaspoon sea salt\n\n4 lemon slices\n\n2 teaspoons chopped fresh dill\n\n1 cup water\n\n1 small head broccoli, cut into florets\n\n1 Pat fillets dry with a paper towel. Season salmon with salt. Place 2 lemon slices on each fillet. Sprinkle with chopped dill.\n\n2 Add water to the Instant Pot\u00ae and insert steamer basket. Place salmon on steamer basket and scatter broccoli florets around the fillets. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Remove fish and broccoli to plates and serve immediately.\n\n## Dairy-Free Creamed Crab\n\nAlthough this recipe is insanely good by the spoonful on its own, use this to level-up some simple dishes. Serve this crab atop a cod fillet with some fresh asparagus or even next to a grilled steak for a little surf and turf.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 4\n\n* * *\n\n1\/4 cup ghee\n\n1\/2 stalk celery, finely diced\n\n1\/4 cup finely diced red onion\n\n1 pound lump crabmeat\n\n1\/4 cup chicken broth\n\n1\/4 cup unsweetened almond milk\n\n2 teaspoons cassava flour\n\n1\/2 teaspoon sea salt\n\n1\/2 teaspoon freshly ground black pepper\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Add ghee and melt. Add celery and red onion; stir-fry 3\u20135 minutes until celery begins to soften.\n\n2 Stir crabmeat and broth into pot. Lock lid. Press the Steam button and adjust time to 0 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Carefully stir in milk, flour, salt, and pepper. Transfer to a serving bowl. Let set 5\u201310 minutes until thickened. Serve warm.\n\n### WHAT KIND OF CRAB SHOULD YOU CHOOSE?\n\nFresh, lump crabmeat is the best option for this recipe\u2014but it can be pricey. Fortunately, there are many different varieties and sections of the crab that can be purchased at a lower price point. It is even sold in cans like the more familiar tuna. Although there is imitation krab available, draw the line at that. It is stringy and packed with starch and chemicals.\n\n## Citrus-Steamed Littleneck Clams\n\nThe citrus of the lemon and the richness of the chardonnay play so nicely against the brininess of the littleneck clams. This dish is a light and ideal lunch for a beautiful summer day. And, the good news is, you'll have chardonnay leftover from the recipe to enjoy a crisp glass of wine with your meal.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 4\n\n* * *\n\n1 tablespoon avocado oil\n\n2 shallots, diced\n\n1 clove garlic, quartered\n\n1\/2 cup chardonnay\n\n2 medium lemons\n\n1\/2 cup water\n\n2 pounds fresh clams, rinsed and purged\n\n1\/2 cup chopped fresh Italian flat-leaf parsley\n\n1 Press the Saut\u00e9 button on the Instant Pot\u00ae. Heat oil. Add shallots and saut\u00e9 3\u20135 minutes until translucent. Add garlic and cook 1 minute.\n\n2 Add wine, juice of 1 lemon, and water to pot. Insert clams. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer clams to four serving bowls and top with a generous scoop of cooking liquid and parsley. Serve a quarter of lemon with each bowl. Discard any clams that do not open.\n\n### HOW DO YOU PURGE CLAMS?\n\nTo purge your clams of any residual sand, soak your clams in water with 1\u20132 tablespoons of cornmeal for 20 minutes. Scrub, rinse, and drain until there is no more sand.\n\n## Salmon Croquettes\n\nTraditionally, croquettes are a fried delicacy, as the word croquette comes from the French word croquer, which literally means \"to crunch\" or \"to be crunchy.\" However, in this recipe, you'll get all the flavors without the unhealthy cooking style. Serve as is, on a salad, or make a drizzle of Horseradish-Lemon Aioli.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nYields 16 croquettes\n\n* * *\n\n1 (14.75-ounce) can salmon\n\n1 large egg\n\n2 teaspoons prepared horseradish\n\n1 tablespoon chopped dill\n\n1\/4 cup almond meal\n\n2 tablespoons finely diced celery\n\n1 teaspoon sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\n2 tablespoons avocado oil, divided\n\n1\/2 cup water\n\n1 In a medium bowl, combine salmon, egg, horseradish, dill, almond meal, celery, salt, and pepper. Form into 16 oval-shaped croquettes (similar to the shape of a tater tot). Set aside.\n\n2 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat 1 tablespoon oil. Place 8 croquettes around the edges of the pot. Sear all sides of the croquettes for a total of about 4 minutes. Remove the first batch and set aside. Add remaining oil and croquettes and sear 4 minutes. Remove croquettes.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Place croquettes evenly on steamer basket. Lock lid. Press the Manual or Pressure Cook button and adjust time to 2 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n4 Transfer croquettes and sauce to bowls and serve warm with sauce.\n\n### HOW TO MAKE HORSERADISH-LEMON AIOLI\n\nWhip this up in a pinch to elevate your salmon croquettes. Whisk together 1\/4 cup mayonnaise, 2 teaspoons horseradish, 1 teaspoon lemon zest, a squeeze lemon juice, and a pinch or two of salt.\n\n## Greek Sea Bass\n\nThe flavors of the Mediterranean are spotlighted in this simple yet elegant dish. The richness of the olive oil, the freshness of the basil and tomatoes, and the brininess of the Kalamata olives all come together to dress up these already buttery pieces of sea bass.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 3 minutes\n\nServes 2\n\n* * *\n\n2 (5-ounce) sea bass fillets\n\n2 teaspoons olive oil, divided\n\n11\/2 teaspoons sea salt, divided\n\n10 pitted Kalamata olives, halved and divided\n\n10 grape tomatoes, halved and divided\n\n3 tablespoons chopped fresh basil leaves, divided\n\n1 Place each fish fillet on a separate 10\" \u00d7 10\" square of aluminum foil. Drizzle each with 1 teaspoon olive oil. Sprinkle each with 1\/2 teaspoon salt. Scatter olives and tomatoes evenly over each fillet. Top each with 1 tablespoon basil. Fold up the sides of each foil piece and crimp at the top to create foil packets.\n\n2 Place both foil packets in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Remove foil packets and transfer fish and toppings to two plates. Garnish each plate with equal amounts of remaining 1 tablespoon basil and 1\/2 teaspoon salt.\n\n## Herbed Cod with Yellow Squash\n\nThe flavorful herb sauce is a fresh, creamy topping for the firm and flaky cod. The squash is a savory companion filled with vitamins A and C, potassium, and fiber. The steam in the Instant Pot\u00ae keeps this a moist and quick dish to make in a snap.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 3 minutes\n\nServes 2\n\n* * *\n\nHerb Sauce\n\n1\/4 cup mayonnaise\n\n1\/2 teaspoon fresh lemon juice\n\n1\/4 teaspoon lemon zest\n\n2 teaspoons Italian seasoning\n\n1 small shallot, peeled and minced\n\nPinch sea salt\n\n1\/4 teaspoon freshly ground black pepper\n\nCod and Squash\n\n2 (6-ounce) cod fillets\n\n1\/2 teaspoon sea salt\n\n1 cup water\n\n1 large yellow squash, sliced into 1\/4\" circles\n\n1 In a small bowl, combine mayonnaise, lemon juice, lemon zest, Italian seasoning, shallot, salt, and pepper. Refrigerate covered until ready to use.\n\n2 Season cod with salt.\n\n3 Add water to the Instant Pot\u00ae and insert steamer basket. Place squash evenly on basket. Set cod atop squash. Lock lid. Press the Manual or Pressure Cook button and adjust time to 3 minutes. When timer beeps, let pressure release naturally for 2 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n4 Transfer fish and squash to plates. Place a spoonful of the herb sauce on each cod fillet. Serve warm.\n\n## Mahi-Mahi with a Lemon-Caper Butter Sauce\n\nMahi-mahi is a firm fish that works well as a vessel for whatever flavors you cook it with. It is mild in taste and doesn't have that \"fishy\" flavor that you get when you use tuna or salmon.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 4 minutes\n\nServes 2\n\n* * *\n\n2 (6-ounce, 1\"-thick) mahi-mahi fillets\n\n2 tablespoons fresh lemon juice\n\n2 tablespoons capers\n\n1 teaspoon sea salt\n\n1 teaspoon lemon zest\n\n2 tablespoons ghee\n\n1 tablespoon chopped fresh Italian flat-leaf parsley\n\n1 Place a piece of foil on the Instant Pot\u00ae's steamer basket. Set both fillets on the foil. Create a \"boat\" with the foil by bringing up the edges. Pour lemon juice on fish, add capers, and season with salt and zest. Add a tablespoon of ghee on top of each fillet.\n\n2 Insert steamer basket with fish into the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 4 minutes. Quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer fish to two plates. Garnish each with 1\/2 tablespoon chopped parsley and serve warm.\n\n## Steamed Snow Crab Legs\n\nThese crab legs are great served with melted ghee and plenty of lemon wedges. The wedges are not only great squeezed over the freshly cracked crab meat, but they act as a great hand cleaner and odor remover once you are done with this very interactive meal.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 3 minutes\n\nServes 2\n\n* * *\n\n1 cup water\n\n4 cloves garlic, quartered\n\n1 tablespoon Old Bay Seasoning\n\n1 bay leaf\n\n2 pounds snow crab legs\n\n1 Add water, garlic, Old Bay Seasoning, and bay leaf to the Instant Pot\u00ae and stir to combine.\n\n2 Insert trivet in pot and add crab legs. Lock lid. Press the Steam button and adjust time to 3 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Transfer crab legs to a serving platter. Serve warm.\n\n## Simply Steamed Lobster Tails\n\nThere is no need to dress this up. Lobster tails and some melted ghee is simply divine together. Full of essential vitamins, lobster is a great source of protein and is low in calories and saturated fat.\n\n\u2022 Hands-On Time: 2 minutes\n\n\u2022 Cook Time: 4 minutes\n\nServes 4\n\n* * *\n\n1 cup water\n\n4 small lobster tails\n\n1\/4 cup ghee, melted\n\n1 Add water to the Instant Pot\u00ae and insert steamer basket. Add lobster tails. Lock lid. Press the Steam button and adjust time to 4 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Remove lobster tails from pot and remove meat from shells. Serve with melted ghee.\n\n### MAKING BROTH WITH THE LOBSTER TAIL SHELLS\n\nDon't just discard those empty lobster tail shells. Place the shells in the Instant Pot\u00ae with 4 cups water, 1 diced yellow onion, 1 peeled and diced carrot, and 1 diced stalk celery. Press the Manual or Pressure Cook button and adjust time to 30 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. Use a slotted spoon to remove and discard the solids from the broth. Strain the remaining liquid through a fine-mesh sieve or cheesecloth. Refrigerate broth up to 4 days or freeze up to 6 months.\n\n## Quick Crawfish Boil\n\nFresh corn is not strictly a \"no\" paleo item. However, if you don't allow this in your diet, just skip it, as the recipe will still stand up without it.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 4\n\n* * *\n\n1 large sweet onion, peeled, halved, and sliced in half-moons\n\n2 ears corn, husked and halved\n\n12-ounces precooked andouille sausage, cut into 1\" pieces\n\n2 pounds crawfish\n\n2 tablespoons Old Bay Seasoning\n\n1 (12-ounce) gluten-free beer\n\n1\/2 cup water\n\n1 Layer onions evenly in the bottom of the Instant Pot\u00ae. Add corn and sausage in even layers. Add the crawfish and sprinkle with Old Bay Seasoning. Pour in beer and water. Lock lid. Press the Manual button and adjust time to 5 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Use a slotted spoon to transfer ingredients to a serving platter. Serve warm.\n\n### PACK A PICNIC\n\nThis Crawfish Boil is a great summer food. Line a picnic table with newspapers and pour all the ingredients onto the paper in the middle of the table. Guests can enjoy this family-style with no real etiquette to the meal other than to have fun. Oh, and don't forget a roll or two of paper towels. This can get messy.\n\n## Sweet Treats and Drinks\n\nPaleo folks are often criticized if they even think about putting any sugar in their mouths. Never mind that white refined stuff. How dare you indulge in raw honey, pure maple syrup, coconut sugar crystals, or molasses? Is sugar-shaming a thing? Sometimes it feels like it is in the paleo community. So, how about this: if you are 100 percent paleo, we are in awe of you. If you are in the 75 percent zone and indulge on occasion, then this is the chapter for you. You will find some cheat meals without all the guilt.\n\nIf you're like most people, you probably have a sweet tooth that pulls at your stomach from time to time. The great thing about the Instant Pot\u00ae is that it creates desserts that are just the right size to make your sweet tooth and your scale happy. Most of these desserts provide only 4\u20136 servings, so you won't be tempted to overeat and you won't have desserts hanging around your kitchen for days on end. There are also some drinks that are treats for you and your friends. And with recipes ranging from Upside-Down Peach Cobbler and Caf\u00e9 Mocha Muffins to Homemade Chai Tea Latte and Gluhwein, these perfect sweet delights are guaranteed to hit the spot . . . no matter what you find yourself craving.\n\nSpiced Applesauce\n\nCaf\u00e9 Mocha Muffins\n\nStrawberry Cupcakes with Chocolate Ganache\n\nApple Brown Betty\n\nUpside-Down Peach Cobbler\n\nBanana Pudding Cake\n\nChocolate Chip Zucchini Bread\n\nDouble Chocolate Custard\n\nCoconut Key Lime Custard\n\nGrain-Free Stuffed Apples\n\nHummingbird Cake\n\nPoached Cherries\n\nUpside-Down Chocolate Cherry Cake\n\nMixed Berry Quick Jam\n\nQuick Lime Marmalade\n\nFresh-Brewed Arnie Palmer\n\nGluhwein\n\nHomemade Chai Tea Latte\n\nSpiced Mocha Latte\n\nMexican Hot Cocoa\n\n## Spiced Applesauce\n\nAfter a day of apple picking, applesauce is a perfect solution for your haul. Pick a variety for a balanced flavor . . . also, keep those apple skins on during cooking as they lend a beautiful hue to the finished product!\n\n\u2022 Hands-On Time: 20 minutes\n\n\u2022 Cook Time: 8 minutes\n\nYields 10 cups\n\n* * *\n\n4 pounds variety of apples, cored and chopped\n\n1 cinnamon stick\n\n2 whole star anise\n\n6 whole allspice\n\n1\/2 cup freshly squeezed orange juice\n\n1\/3 cup honey\n\n1\/2 teaspoon sea salt\n\n1\/3 cup water\n\n1 Add the apples, cinnamon stick, star anise, whole allspice, orange juice, honey, salt, and water to the Instant Pot\u00ae. (Alternatively, add the cinnamon stick, star anise, and whole allspice to a spice bag for easy retrieval after cooking.)\n\n2 Press the Manual or Pressure Cook button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Remove cinnamon stick, star anise, and whole allspice.\n\n4 Use an immersion blender to blend the ingredients in the pot until desired consistency is reached.\n\n## Caf\u00e9 Mocha Muffins\n\nEither for desserts, breakfast-on-the-go, or just a snack, these Caf\u00e9 Mocha Muffins are the perfect bite for any coffee and chocolate lover!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 9 minutes\n\nServes 6\n\n* * *\n\n2\/3 cup cassava flour\n\n1\/4 cup unsweetened cocoa powder\n\n2 teaspoons instant espresso powder\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n3 tablespoons ghee, melted\n\n2 large eggs\n\n2 tablespoons unsweetened almond milk\n\n1\/3 cup pure maple syrup\n\n1 cup water\n\n1 In a large bowl, combine flour, cocoa, espresso powder, baking powder, baking soda, and sea salt.\n\n2 In a medium bowl, combine vanilla, ghee, eggs, almond milk, and syrup.\n\n3 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Spoon mixture into 6 lightly greased silicone cupcake liners.\n\n4 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place cupcake liners on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 9 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove muffins from pot and set aside to cool 5 minutes.\n\n## Strawberry Cupcakes with Chocolate Ganache\n\nEveryone loves the combination of strawberries and chocolate and this perfect combo is neatly put together and delivered to you in a scrumptious little cupcake. Gluten? Who needs that! You won't even miss the stuff with each heavenly bite!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\nStrawberry Cupcakes\n\n11\/4 cups cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n1 teaspoon lime zest\n\n3 tablespoons ghee, melted\n\n2 large eggs\n\n1\/3 cup pure maple syrup\n\n1\/3 cup finely diced strawberries\n\n1 cup water\n\nChocolate Ganache\n\n31\/2-ounce dark chocolate bar\n\n1 teaspoon pure maple syrup\n\nPinch sea salt\n\n1\/2 cup unsweetened almond milk\n\n2 teaspoons coconut palm shortening\n\n1 In a large bowl, combine flour, baking powder, baking soda, and sea salt.\n\n2 In a medium bowl, combine vanilla, lime zest, ghee, eggs, and syrup.\n\n3 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Fold in strawberries. Spoon mixture into 6 lightly greased silicone cupcake liners.\n\n4 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place cupcake liners on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 9 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n5 Remove cupcakes from pot and set aside to cool 5 minutes.\n\n6 Once cupcakes have cooled, prepare ganache. In a small bowl, break up the chocolate bar into pieces. Add syrup and salt. Set aside. Add milk and shortening to a small saucepan and heat over medium. Bring to a rolling boil. Pour milk mixture over chocolate and whisk until smooth.\n\n7 Dip tops of cupcakes in ganache and place on a cooling rack. Serve.\n\n## Apple Brown Betty\n\nThe Brown Betty goes back to the mid-1800s in America and was traditionally made with bread slices or cracker crumbs. King of the Hill fans know that Peggy is very proud of her Apple Brown Betty and loves to share. This Instant Pot\u00ae version takes out the grains but doesn't skimp on taste. Share this with your family. Share this with your neighbors!\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 18 minutes\n\nServes 6\n\n* * *\n\nCrust\n\n1 cup cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\n1\/4 teaspoon ground cinnamon\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n2 tablespoons ghee, melted\n\n2 large eggs\n\n2 tablespoons honey\n\nApple Betty\n\n2 tablespoons ghee\n\n3 medium apples, cored, peeled, and thinly sliced\n\n2 tablespoons molasses\n\n1 teaspoon orange zest\n\n1\/2 teaspoon ground cinnamon\n\n1\/4 cup chopped walnuts\n\n2 cups water\n\n1 Lightly grease a 7-inch springform pan. Set aside.\n\n2 In a large bowl, combine flour, baking powder, baking soda, cinnamon, and salt.\n\n3 In a medium bowl, combine vanilla, melted ghee, eggs, and honey.\n\n4 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Set aside.\n\n5 Press the Saut\u00e9 button on the Instant Pot\u00ae and heat ghee. Add apples and toss. Cook about 3 minutes to remove some water from the apples. Use a slotted spoon to transfer the apples to a medium bowl. Toss in molasses, orange zest, and cinnamon. Evenly transfer to greased springform pan. Sprinkle chopped walnuts over apple mixture.\n\n6 Spoon in crust mixture and spread evenly to edges of pan.\n\n7 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place springform pan on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n8 Remove pan from pot and refrigerate at least 30 minutes to allow the dessert to set. Unlock springform pan. Flip onto a serving plate and remove springform plate. Serve.\n\n## Upside-Down Peach Cobbler\n\nThis Peach Cobbler could show up to any Georgia dinner party and fit right in (other than the fact that it only serves four!). The Instant Pot\u00ae version of Peach Cobbler requires frozen peaches, so you can enjoy this dessert any time of the year. Also, you can make this at the last minute without worrying about cutting peaches or defrosting freezer items.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n1 pound bag frozen sliced peaches\n\n1 tablespoon honey\n\n2 tablespoons ghee, softened\n\n1\/2 teaspoon ground cinnamon\n\n1 cup cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n2 tablespoons ghee, melted\n\n2 large eggs\n\n1\/4 cup pure maple syrup\n\n2 cups water\n\n1 Lightly grease a 7-inch springform pan. Scatter frozen sliced peaches in an even layer. Drizzle with honey. Place pats of softened ghee over peaches. Sprinkle evenly with cinnamon.\n\n2 In a large bowl, combine flour, baking powder, baking soda, and salt.\n\n3 In a medium bowl, combine vanilla, melted ghee, eggs, and syrup.\n\n4 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Carefully and evenly smooth sticky batter over peaches.\n\n5 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place springform pan on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n6 Remove pan from pot and refrigerate at least 30 minutes to allow the cake to set. Unlock springform pan. Flip onto a serving plate and remove springform plate. Serve.\n\n## Banana Pudding Cake\n\nBanana pudding is a Southern tradition, but sometimes you just need a quick version so that you have extra time to gossip with your friends.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\nVanilla Cake\n\n1 cup cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1 teaspoon vanilla extract\n\n2 tablespoons ghee, melted\n\n2 large eggs\n\n2 tablespoons honey\n\nBanana Pudding\n\n2 medium ripe bananas, sliced\n\n2 large egg yolks\n\n1 tablespoon pure maple syrup\n\n1\/4 teaspoon vanilla extract\n\nPinch sea salt\n\n3\/4 cup unsweetened almond milk\n\n2 cups water\n\n1 Lightly grease a 7-inch springform pan. Set aside.\n\n2 In a large bowl, combine flour, baking powder, baking soda, and salt.\n\n3 In a medium bowl, combine vanilla, ghee, eggs, and honey.\n\n4 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently mix to combine ingredients. Do not overmix. Set cake mixture aside.\n\n5 Evenly distribute bananas on the bottom of the prepared springform pan.\n\n6 In a small bowl, whisk together egg yolks, maple syrup, vanilla, and salt. Set aside.In saucepan over medium-low heat, heat almond milk to a low simmer. Whisk a spoonful of the milk into the egg mixture to temper the eggs, and then slowly whisk that egg mixture into the saucepan with remaining almond milk. Pour this pudding layer evenly over bananas. Refrigerate 10 minutes.\n\n7 Carefully spoon cake mixture over pudding layer and spread evenly to edges of pan.\n\n8 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place springform pan on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n9 Remove pan from pot and refrigerate at least 30 minutes to allow the dessert to set. Unlock springform pan. Flip onto a serving plate and remove springform plate. Serve.\n\n## Chocolate Chip Zucchini Bread\n\nLoaded with nutritional zucchini and sweetened with a little honey and dates, this is the perfect bread for a sweet tooth. Enjoy with a warm cup of coffee or tea for a quiet morning curled up with a great book.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 6\n\n* * *\n\n11\/2 cups cassava flour\n\n1 teaspoon ground cinnamon\n\n1\/8 teaspoon ground nutmeg\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1\/3 cup finely diced pitted dates\n\n3 large eggs\n\n2 cups grated zucchini (about 2 small zucchini)\n\n1 teaspoon vanilla extract\n\n1\/4 cup ghee, melted\n\n1\/3 cup honey\n\n1\/2 cup chopped pecans\n\n1\/4 cup dark chocolate chips\n\n2 cups water\n\n1 Lightly grease a 7-inch springform pan. Set aside.\n\n2 In a large bowl, combine flour, cinnamon, nutmeg, baking powder, baking soda, salt, and dates.\n\n3 In a medium bowl, combine eggs, zucchini, vanilla, ghee, and honey.\n\n4 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Fold in pecans and chocolate chips. Do not overmix. Spoon mixture into springform pan.\n\n5 Pour water into the Instant Pot\u00ae and insert trivet. Place springform pan on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n6 Remove pan from the Instant Pot\u00ae. Let rest 15 minutes. Use a paper towel to dab off any excess moisture. Open pan and cut and serve.\n\n## Double Chocolate Custard\n\nWhat is better than chocolate? Double chocolate! Egg yolks are not only nature's sauce once poached, but they add such a richness to custards and ice creams. The chocolate enhances this richness. You won't believe that this is paleo and dairy-free. Oh, but it is. So simple. So luxurious.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n4 large egg yolks\n\n1\/4 cup honey\n\nPinch sea salt\n\n1\/4 teaspoon vanilla extract\n\n2 tablespoons unsweetened cocoa powder\n\n11\/2 cups unsweetened almond milk\n\n1\/4 cup dark chocolate chips\n\n2 cups water\n\n1 In a small bowl, whisk together egg yolks, honey, salt, vanilla, and cocoa. Set aside.\n\n2 In saucepan over medium-low heat, heat almond milk to a low simmer. Whisk a spoonful of the milk into the egg mixture to temper the eggs, and then slowly whisk that egg mixture into the saucepan with remaining almond milk. Add chocolate chips and continually stir on simmer until chocolate is melted, about 10 minutes. Remove from heat and evenly distribute chocolate mixture among four custard ramekins.\n\n3 Pour water into the Instant Pot\u00ae and insert trivet. Place ramekins onto trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally for 15 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n4 Transfer custards to a plate and refrigerate covered overnight to allow the custard to set. Serve.\n\n## Coconut Key Lime Custard\n\nHello Key West! Although the little flavorful key limes are preferred in this recipe, if it isn't the season, feel free to use standard limes. Either way, close your eyes and you'll be transported to Alabama Jack's while wearing your flip-flops and tank top. Don't forget to go enjoy that glorious sunset over the ocean . . . at least in your dreams.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n4 large egg yolks\n\n1\/4 cup honey\n\nPinch sea salt\n\n1\/4 teaspoon vanilla extract\n\n1\/3 cup fresh key lime juice\n\n1 teaspoon key lime zest\n\n11\/2 cups canned coconut milk\n\n2 cups water\n\n1 In a small bowl, whisk together egg yolks, honey, salt, vanilla, lime juice, and lime zest. Set aside.\n\n2 In saucepan over medium-low heat, heat coconut milk to a low simmer. Whisk a spoonful of the milk into the egg mixture to temper the eggs, and then slowly whisk that egg mixture into the saucepan with remaining coconut milk. Continually stir on simmer about 10 minutes. Remove from heat and evenly distribute custard mixture among four ramekins.\n\n3 Pour water into the Instant Pot\u00ae and insert trivet. Place ramekins onto trivet. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally for 15 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n4 Transfer custards to a plate and refrigerate covered overnight to allow the custard to fully set. Serve.\n\n## Grain-Free Stuffed Apples\n\nYou can have your apples and stuff them too! After a day of apple picking with the family, this glorious traditional recipe can be cooked and on your plates within 15 minutes. Serve with a scoop of vanilla or cinnamon ice cream to take it over the top!\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 5 minutes\n\nServes 4\n\n* * *\n\n3 tablespoons ghee, softened\n\n2 tablespoons pumpkin pur\u00e9e\n\n1 teaspoon ground cinnamon\n\n1\/4 cup chopped pecans\n\n2 tablespoons unsweetened coconut flakes\n\n2 teaspoons pure maple syrup\n\nPinch sea salt\n\n4 medium cooking apples\n\n1 cup water\n\n1 In a small bowl, mix together ghee, pumpkin pur\u00e9e, cinnamon, pecans, coconut flakes, syrup, and salt. Set aside.\n\n2 Rinse and dry the apples. Cut the tops off the apples. Hollow out and core the apples by cutting to, but not through, the apple bottoms.\n\n3 Place each apple on a piece of aluminum foil that is large enough to wrap the apple completely. Fill the apple centers with prepared mixture. Wrap the foil around each apple, folding the foil over at the top and then pinching it firmly together.\n\n4 Pour water into the Instant Pot\u00ae and insert trivet. Place the wrapped apples on the rack. Lock lid. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally for 10 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid.\n\n5 Carefully unwrap apples and transfer to serving plates.\n\n## Hummingbird Cake\n\nHummingbird Cake is a Southern spice cake filled with banana, pineapple, and nuts. Traditionally, there is a rich white frosting with coconut flakes . . . it really is what dreams are made of. This recipe cuts out that thick frosting but doesn't skimp on any of the flavor. It makes for a great after-dinner dessert enjoyed with a cup of coffee.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n3\/4 cup cassava flour\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\n1\/2 teaspoon ground cinnamon\n\nPinch sea salt\n\n1\/4 teaspoon vanilla extract\n\n2 tablespoons ghee, melted\n\n2 large eggs\n\n1\/4 cup undrained crushed pineapple\n\n1\/4 cup mashed banana (about 1 small banana)\n\n2 tablespoons pure maple syrup\n\n1\/4 cup chopped pecans\n\n2 cups water\n\nTopping\n\n1 tablespoon pure maple syrup\n\n3 tablespoons unsweetened coconut flakes, divided\n\n1 Lightly grease a 6-inch cake pan. Set aside.\n\n2 In a large bowl, combine flour, baking powder, baking soda, cinnamon, and sea salt.\n\n3 In a medium bowl, combine vanilla, ghee, eggs, pineapple, banana, and syrup.\n\n4 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Fold in pecans. Spoon mixture into greased cake pan.\n\n5 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place cake pan on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n6 Remove pan from pot and set aside to cool 5 minutes. Flip cake onto a serving plate.\n\n7 In a small bowl, combine syrup with 2 tablespoons coconut flakes. Spread on the top of the cake. Garnish with the remaining 1 tablespoon coconut flakes.\n\n## Poached Cherries\n\nIf you are overrun with cherries, poach them! They are a nice light dessert to quench that sweet tooth. Or serve them over some dairy-free ice cream or grain-free cake. And if you enjoy cherries, invest in a pitter. It will be one of the few gadgets that get some use, unlike the others in the graveyard of never-used kitchen tools.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n2 cups water\n\n1\/2 cup pure maple syrup\n\nZest and juice from 1 medium orange\n\n2 teaspoons vanilla extract\n\n1 pound sweet cherries, pitted\n\n1 cinnamon stick\n\n1 Add all ingredients to the Instant Pot\u00ae. Press the Manual or Pressure Cook button and adjust time to 5 minutes. When timer beeps, let pressure release naturally for 15 minutes. Quick-release any remaining pressure until float valve drops and then unlock lid. Discard cinnamon stick.\n\n2 Transfer cherries to an airtight container and refrigerate until ready to eat, warm or cold.\n\n## Upside-Down Chocolate Cherry Cake\n\nChocolate covered cherries are a no-brainer combination, so this cake just makes sense. A seasonal treat, pick up one of those fresh sweet cherry bags that you see in the grocery store once a year and this is your dessert. Rich, brownie-like chocolate cake and naturally sweet pitted cherries, this should satisfy any sweet tooth without breaking the calorie bank.\n\n\u2022 Hands-On Time: 15 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n16 fresh sweet cherries, halved and pitted\n\n2\/3 cup cassava flour\n\n1\/3 cup unsweetened cocoa powder\n\n2 teaspoons baking powder\n\n1\/2 teaspoon baking soda\n\nPinch sea salt\n\n1\/2 teaspoon vanilla extract\n\n3 tablespoons ghee, melted\n\n2 large eggs\n\n2 tablespoons unsweetened almond milk\n\n1\/3 cup pure maple syrup\n\n2 cups water\n\n1 Lightly grease a 6-inch cake pan. Place cherries, flat side down, into cake pan until the bottom of the pan is covered. Set aside.\n\n2 In a large bowl, combine flour, cocoa, baking powder, baking soda, and sea salt.\n\n3 In a medium bowl, combine vanilla, ghee, eggs, milk, and syrup.\n\n4 Pour wet ingredients from the medium bowl into the large bowl with dry ingredients. Gently combine ingredients. Do not overmix. Spoon mixture into greased cake pan.\n\n5 Pour water into the Instant Pot\u00ae and insert trivet or steamer basket. Place cake pan on top. Lock lid. Press the Manual or Pressure Cook button and adjust time to 15 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n6 Remove pan from pot and set aside to cool for 5 minutes. Flip cake onto a serving plate.\n\n## Mixed Berry Quick Jam\n\nIf you go down the jam and jelly aisle and look at the ingredients' label, you will notice that the jars are filled with refined sugars and preservatives. Enjoy this jam with whatever berries you have on-hand after a day of berry picking with the family.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 1 minute\n\nYields 3 cups\n\n* * *\n\n2 cups blueberries\n\n2 cups sliced strawberries\n\n1 cup freshly squeezed orange juice\n\n2 teaspoons orange zest\n\n1\/2 cup honey\n\nPinch sea salt\n\n1 Place all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 1 minute. When timer beeps, let pressure release naturally until float valve drops and then unlock lid.\n\n2 Use an immersion blender to blend the ingredients in the pot until desired consistency is reached. Spoon into a lidded container and store in refrigerator. Use within 2 weeks.\n\n## Quick Lime Marmalade\n\nFor a sweet-bitter-tart treat, this quick marmalade is for you. It can be used on gluten-free scones and biscuits, frozen into ice cubes for amazing additions to your gin drinks, or throw in a small spoonful to your Thai coconut milk\u2013based dishes.\n\n\u2022 Hands-On Time: 10 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 10\n\n* * *\n\n6 medium limes\n\n1\/2 cup honey\n\nPinch sea salt\n\n1\/2 cup water\n\n1 Prepare limes by scrubbing the outside and removing any possible shiny wax on the exterior. Thinly slice limes, the thinner the better. Discard seeds. Cut slices in half.\n\n2 Place all ingredients in the Instant Pot\u00ae. Lock lid. Press the Manual or Pressure Cook button and adjust time to 10 minutes. When timer beeps, let pressure release naturally until float valve drops and then unlock lid. If mixture is too watery, let simmer unlidded until desired consistency.\n\n3 Transfer mixture to a lidded jar and refrigerate. Use within 2 weeks or freeze into cubes for later use.\n\n## Fresh-Brewed Arnie Palmer\n\nNamed after the beloved American golfer, Arnold Palmer, this summertime drink combines simple brewed tea with fresh, sweet and tart lemonade. If you'd like to adult this drink up, add a shot of vodka or limoncello to the finished product . . . but then you call it a John Daly!\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n4 medium lemons, halved\n\n5 tea bags\n\n7 cups water\n\n1\/2 cup honey\n\n1 Juice lemons. Discard seeds but keep the rinds.\n\n2 Place lemon juice, lemon rinds, tea bags, and water in Instant Pot\u00ae. Lock lid. Press the Steam button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n3 Add honey and let steep an additional 5 minutes. Strain. Press any additional liquid and flavor out of the tea bags. Serve over ice.\n\n## Gluhwein\n\nGluhwein is a glorious warm spiced wine drink that you sip while walking around the Christmas Markets in Germany. It is cold, yet the warm Gluhwein and the beautiful woodcrafts pull a person into the holiday. Share this enchanting drink with your stateside friends to introduce them to the warmth.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 10 minutes\n\nServes 6\n\n* * *\n\n1 (750-milliliter) bottle dry red wine\n\n1\/2 medium lemon, quartered\n\n1 medium orange, quartered\n\n1\/2 cup honey\n\n4 whole cloves\n\n2 cardamom pods\n\n1 cinnamon stick\n\n1 whole star anise\n\n1 Add all ingredients to the Instant Pot\u00ae. Lock lid. Press the Steam button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Using a fine-mesh sieve or cheesecloth, strain out solids. Serve warm.\n\n## Homemade Chai Tea Latte\n\nYour friends will think they just stopped by their local favorite teahouse when they taste the freshness and richness of this homemade chai. Sweetened with maple syrup instead of refined sugars and made creamy with unsweetened almond milk instead of heavy cream, this drink's facelift is one to be savored.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n5 cups water\n\n4 black tea bags\n\n3 cardamom pods\n\n4 whole cloves\n\n1 cinnamon stick\n\n4 whole allspice\n\n1\/4 teaspoon ground nutmeg\n\n1 cup unsweetened almond milk\n\n1\/4 cup pure maple syrup\n\n1 Place water, tea bags, cardamom, cloves, cinnamon stick, allspice, and nutmeg in Instant Pot\u00ae. Lock lid. Press the Steam button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Add milk and syrup and let steep an additional 5 minutes. Strain. Press any additional liquid and flavor out of the tea bags. Serve warm, chilled, or on ice.\n\n### WHAT IS A SPICE BAG?\n\nIn the recipe, the solids are free-floating; however, to avoid straining the hot liquid at the end of the cooking process, simply bind the cardamom pods, whole cloves, cinnamon stick, and whole allspice in a cheesecloth tied with butcher twine. Premade spice bags are also sold in specialty stores or found easily online.\n\n## Spiced Mocha Latte\n\nWhy waste those dollars when you can whip up this concoction on your own with minimal effort? Pour the finished product in your to-go coffee mug and know that you have a cheaper and healthier product to wake you up and take you down the road without killing your wallet.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 15 minutes\n\nServes 4\n\n* * *\n\n4 cups water\n\n2 tablespoons instant espresso coffee powder\n\n2 tablespoons unsweetened cocoa powder\n\nPinch sea salt\n\n4 whole allspice\n\n1 star anise\n\n4 cardamom pods\n\n1 cinnamon stick\n\n2 cups unsweetened almond milk\n\n1\/2 cup pure maple syrup\n\n1 Add water, espresso, cocoa, salt, allspice, star anise, cardamom, and cinnamon stick in the Instant Pot\u00ae. Lock lid. Press the Steam button and adjust time to 10 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Add milk and syrup and let steep an additional 5 minutes. Strain. Serve warm, chilled, or on ice.\n\n## Mexican Hot Cocoa\n\nRich, delicious, and warm, hot cocoa is what soothes the body after spending the morning shoveling snow out of the driveway with the family. Set this up in the Instant Pot\u00ae and it will be ready to be ladled into those mugs. Also, if you allow dairy in your life, whisk some organic cream with a little honey for a fluffy topping.\n\n\u2022 Hands-On Time: 5 minutes\n\n\u2022 Cook Time: 8 minutes\n\nServes 4\n\n* * *\n\n6 cups unsweetened almond milk\n\n1\/4 cup unsweetened cocoa powder\n\n1 (31\/2-ounce) dark chocolate bar, broken into pieces\n\n1\/4 teaspoon chili powder\n\nPinch cayenne pepper\n\nPinch ground nutmeg\n\nPinch sea salt\n\n1 cinnamon stick\n\n2 teaspoons vanilla extract\n\n1\/4 cup honey\n\n1 Place all ingredients in the Instant Pot\u00ae. Lock lid. Press the Steam button and adjust time to 8 minutes. When timer beeps, quick-release pressure until float valve drops and then unlock lid.\n\n2 Remove and discard cinnamon stick. Ladle cocoa into mugs and serve warm.\n\n## Appendix A\n\n### Paleo \"Yes\" and \"No\" Foods\n\nIn order to ensure your success on the Paleolithic diet, you need to stock your pantry with fresh, organic produce and grass-fed and barn-roaming meats. Feel free to experiment with items you would not normally choose. That will spice things up and keep you interested in the diet. Also listed here are the foods you should avoid, including processed grains, potatoes, legumes, and dairy.\n\n#### PALEO \"YES\" FOODS\n\nProtein\n\nAlligator\n\nBass\n\nBear\n\nBeef, lean and trimmed\n\nBison\n\nBluefish\n\nCaribou\n\nChicken breast\n\nChuck steak\n\nClams\n\nCod\n\nCrab\n\nCrayfish\n\nEgg whites\n\nEggs\n\nFlank steak\n\nGame hen breasts\n\nGoat\n\nGrouper\n\nHaddock\n\nHalibut\n\nHamburger, extra lean\n\nHerring\n\nLiver (beef, lamb, goat, or chicken)\n\nLobster\n\nLondon broil\n\nMackerel\n\nMarrow (beef, lamb, or goat)\n\nMussels\n\nOrange roughy\n\nOstrich\n\nOysters\n\nPheasant\n\nPork chops\n\nPork loin\n\nPork, lean\n\nQuail\n\nRabbit\n\nRattlesnake\n\nRed snapper\n\nSalmon, wild-caught\n\nScallops\n\nScrod\n\nShrimp\n\nTilapia\n\nTongue (beef, lamb, or goat)\n\nTrout\n\nTuna, canned, unsalted\n\nTuna, fresh\n\nTurkey breast\n\nVeal, lean\n\nVenison\n\nLeafy Vegetables\n\nArugula\n\nBeet greens\n\nBitterleaf\n\nBok choy\n\nBroccoli rabe\n\nBrussels sprouts\n\nCabbage\n\nCelery\n\nChard\n\nChicory\n\nChinese cabbage\n\nCollard greens\n\nDandelion\n\nEndive\n\nFiddlehead\n\nKale\n\nLettuce\n\nRadicchio\n\nSpinach\n\nSwiss chard\n\nTurnip\n\nWatercress\n\nYarrow\n\nFruiting Vegetables\n\nAvocado\n\nBell pepper\n\nCucumber\n\nEggplant\n\nSquash\n\nSweet pepper\n\nTomatillo\n\nTomato\n\nZucchini\n\nFlowers and Flower Buds\n\nArtichoke\n\nBroccoli\n\nCauliflower\n\nBulb and Stem Vegetables\n\nAsparagus\n\nCelery\n\nFlorence fennel\n\nGarlic\n\nKohlrabi\n\nLeek\n\nOnion\n\nSea Vegetables and Herbs of All Types\n\nFruits\n\nApple\n\nApricot\n\nBanana\n\nBlackberries\n\nBlueberries\n\nCantaloupe\n\nCherries\n\nCoconut\n\nCranberries (not dried)\n\nFigs\n\nGrapefruit\n\nGrapes\n\nGuava\n\nHoneydew melon\n\nKiwi\n\nLemon\n\nLime\n\nMandarin orange\n\nMango\n\nNectarine\n\nOrange\n\nPapaya\n\nPassion fruit\n\nPeaches\n\nPears\n\nPersimmon\n\nPineapple\n\nPlums\n\nPomegranate\n\nRaspberries\n\nRhubarb\n\nStar fruit\n\nStrawberries\n\nTangerine\n\nWatermelon\n\nAll other fruits are acceptable\n\nFats, Nuts, Seeds, Oils, and Fatty Proteins\n\nAlmond butter\n\nAlmonds\n\nAvocado\n\nBrazil nuts\n\nCanola oil\n\nCashew\n\nCashew butter\n\nChestnuts\n\nCoconut oil\n\nFlaxseed oil\n\nHazelnuts\/filberts\n\nMacadamia butter\n\nMacadamia nuts\n\nOlive oil\n\nPecans\n\nPine nuts\n\nPistachios\n\nPumpkin seeds\n\nSafflower oil\n\nSesame seeds\n\nSunflower butter, unsweetened\n\nSunflower seeds\n\nUdo's oil\n\nWalnut oil\n\nWalnuts\n\n#### PALEO \"NO\" FOODS\n\nLegume Vegetables\n\nAmerican groundnut\n\nAzuki beans\n\nBlack-eyed peas\n\nChickpeas (garbanzo bean)\n\nCommon beans\n\nFava beans\n\nGreen beans\n\nGuar\n\nIndian peas\n\nKidney beans\n\nLentils\n\nLima beans\n\nMung beans\n\nNavy beans\n\nOkra\n\nPeanut\n\nPeanut butter\n\nPeas\n\nPigeon peas\n\nPinto beans\n\nRed beans\n\nRice beans\n\nSnow peas\n\nSoybean and soy products\n\nString beans\n\nSugar snap peas\n\nWhite beans\n\nDairy Foods\n\nAll processed foods made with any dairy products\n\nButter\n\nCheese\n\nCream\n\nDairy spreads\n\nFrozen yogurt\n\nIce cream\n\nIce milk\n\nLow-fat milk\n\nNonfat dairy creamer\n\nPowdered milk\n\nSkim milk\n\nWhole milk\n\nYogurt\n\nCereal Grains\n\nBarley\n\nCorn\n\nMillet\n\nOats\n\nRice\n\nRye\n\nSorghum\n\nWheat\n\nWild rice\n\nCereal Grain-Like Seeds\n\nAmaranth\n\nBuckwheat\n\nQuinoa\n\nStarchy Vegetables\n\nStarchy tubers (in large quantities)\n\nManioc\n\nPotatoes and all potato products\n\nSweet potatoes or yams (unless after workout to replenish gylcogen)\n\nTapioca pudding\n\nSalt-Containing Foods, Fatty Meats, and Sugar\n\nAlmost all commercial salad dressings and condiments\n\nCandy\n\nFrankfurters\n\nProcessed meats\n\nSalami\n\nSalt\n\nSalted nuts\n\nSmoked, dried, and salted fish and meat\n\nSoft drinks and fruit juice\n\nSugar\n\n## Appendix B\n\n### US\/Metric Conversion Chart\n\nVOLUME CONVERSIONS\n\n---\n\nUS Volume Measure | Metric Equivalent\n\n1\/8 teaspoon | 0.5 milliliter\n\n1\/4 teaspoon | 1 milliliter\n\n1\/2 teaspoon | 2 milliliters\n\n1 teaspoon | 5 milliliters\n\n1\/2 tablespoon | 7 milliliters\n\n1 tablespoon (3 teaspoons) | 15 milliliters\n\n2 tablespoons (1 fluid ounce) | 30 milliliters\n\n1\/4 cup (4 tablespoons) | 60 milliliters\n\n1\/3 cup | 90 milliliters\n\n1\/2 cup (4 fluid ounces) | 125 milliliters\n\n2\/3 cup | 160 milliliters\n\n3\/4 cup (6 fluid ounces) | 180 milliliters\n\n1 cup (16 tablespoons) | 250 milliliters\n\n1 pint (2 cups) | 500 milliliters\n\n1 quart (4 cups) | 1 liter (about)\n\nWEIGHT CONVERSIONS\n\n---\n\nUS Weight Measure | Metric Equivalent\n\n1\/2 ounce | 15 grams\n\n1 ounce | 30 grams\n\n2 ounces | 60 grams\n\n3 ounces | 85 grams\n\n1\/4 pound (4 ounces) | 115 grams\n\n1\/2 pound (8 ounces) | 225 grams\n\n3\/4 pound (12 ounces) | 340 grams\n\n1 pound (16 ounces) | 454 grams\n\nOVEN TEMPERATURE CONVERSIONS\n\n---\n\nDegrees Fahrenheit | Degrees Celsius\n\n200 degrees F | 95 degrees C\n\n250 degrees F | 120 degrees C\n\n275 degrees F | 135 degrees C\n\n300 degrees F | 150 degrees C\n\n325 degrees F | 160 degrees C\n\n350 degrees F | 180 degrees C\n\n375 degrees F | 190 degrees C\n\n400 degrees F | 205 degrees C\n\n425 degrees F | 220 degrees C\n\n450 degrees F | 230 degrees C\n\nBAKING PAN SIZES\n\n---\n\nAmerican | Metric\n\n8 x 11\/2 inch round baking pan | 20 x 4 cm cake tin\n\n9 x 11\/2 inch round baking pan | 23 x 3.5 cm cake tin\n\n11 x 7 x 11\/2 inch baking pan | 28 x 18 x 4 cm baking tin\n\n13 x 9 x 2 inch baking pan | 30 x 20 x 5 cm baking tin\n\n2 quart rectangular baking dish | 30 x 20 x 3 cm baking tin\n\n15 x 10 x 2 inch baking pan | 30 x 25 x 2 cm baking tin (Swiss roll tin)\n\n9 inch pie plate | 22 x 4 or 23 x 4 cm pie plate\n\n7 or 8 inch springform pan | 18 or 20 cm springform or loose bottom cake tin\n\n9 x 5 x 3 inch loaf pan | 23 x 13 x 7 cm or 2 lb narrow loaf or p\u00e2t\u00e9 tin\n\n11\/2 quart casserole | 1.5 liter casserole\n\n2 quart casserole | 2 liter casserole\n\n## About the Author\n\nMichelle Fagone is a recipe developer, mother of two, and food blogger. On her site, CavegirlCuisine.com, Michelle shares recipes and knowledge about the health benefits of cooking with local, fresh, and unprocessed foods. Despite being a Southern gal at heart, her travel and food experiences as a Navy brat and current Army spouse have given her a unique appreciation for worldly flavors. While comfort is the basis for most of her recipes, you will often find a twist of exciting flavors and combinations that make her recipes not only appealing to a broad audience but uniquely delicious was named one of the Top 50 Paleo Blogs of 2012 by the Institute for the Psychology of Eating. Michelle lives in Louisville, Kentucky.\n\nMEET THE AUTHORS, WATCH VIDEOS AND MORE AT\n\nSimonandSchuster.com\n\nAuthors.SimonandSchuster.com\/Michelle-Fagone\nWe hope you enjoyed reading this Simon & Schuster ebook.\n\n* * *\n\nGet a FREE ebook when you join our mailing list. Plus, get updates on new releases, deals, recommended reads, and more from Simon & Schuster. Click below to sign up and see terms and conditions.\n\nCLICK HERE TO SIGN UP\n\nAlready a subscriber? Provide your email again so we can register this ebook and send you more of what you like to read. You will continue to receive exclusive offers in your inbox.\n\n## Index\n\nNote: Page numbers in bold indicate recipe category lists.\n\n### A\n\nAccessories, pot-in-pot cooking, \u201310\n\nAccessory removal options, \u201310\n\nAlmond milk, making,\n\nAppetizers, \u201375, , . See also Sauces and dips\n\nApples\n\nApple Brown Betty,\n\nApple Cinnamon Muffins,\n\nFruity Trail Mix Bowls,\n\nGrain-Free Stuffed Apples,\n\nRed Onion and Apple Confit,\n\nSpiced Applesauce,\n\nArtichokes, ,\n\nAsparagus, , ,\n\n### B\n\nBanana Pudding Cake,\n\nBananas, in Hummingbird Cake,\n\nBeef\n\nabout: Meat\/Stew button on Instant Pot\u00ae,\n\nBeef Broth,\n\nChunky Beef Chili,\n\nHamburger Soup,\n\nKentucky Derby Burgoo,\n\nMini Chili Polpette,\n\nBeef and bison main dishes, \u201349\n\nBalsamic Meatballs,\n\nBeef Shanks and Artichokes,\n\nBison Sloppy Joe's,\n\nChi-Town Italian Beef and Peppers,\n\nChuck Roast with Carrots and Onions,\n\nCorned Beef and Cabbage,\n\nCuban Picadillo with Riced Cauliflower,\n\nGround Beef and Broccoli Bowl,\n\nGround Bison Bulgogi,\n\nItalian Beef Tongue Lettuce Wraps,\n\nJust Beet It Burgers,\n\nKorean Beef Short Ribs,\n\nMarinara Meatballs,\n\nMeatloaf and Gravy,\n\nMongolian Beef and Cashew BBQ,\n\nNana's Sunday Pot Roast,\n\nReuben Meatballs with Russian Dressing,\n\nSkirt Steak Fajitas,\n\nSweet-n-Spicy Beef Brisket,\n\nTender Flank Steak with Mushrooms and Onions,\n\nBeets, in Jicama-Beet Slaw,\n\nBeets, steamed,\n\nBerries\n\nBlueberry-Orange Muffins,\n\nMixed Berry Breakfast Syrup,\n\nMixed Berry Quick Jam,\n\nStrawberry Cupcakes with Chocolate Ganache,\n\nBreakfast recipes, \u201333\n\nBroccoli\n\nBroccoli Casserole,\n\nBroccoli-Cauli Sausage Tots,\n\nBroccoli-Parsnip Smash,\n\nBroccoli-Zucchini Hummus,\n\nCream of Broccoli and Bacon Soup,\n\nDilled Salmon and Broccoli,\n\nGround Beef and Broccoli Bowl,\n\nLemon Ghee Broccoli,\n\nBrussels sprouts, bacon-apple,\n\nButtons, on Instant Pot\u00ae, \u20137\n\n### C\n\nCabbage, corned beef and,\n\nCabbage, kimchi with, ,\n\nCabbage and sausage soup,\n\nCarrots, honey-glazed,\n\nCarrots, in Kohlrabi, Kale, and Karrots,\n\nCarrot soup, curried,\n\nCauliflower\n\nBacon Cauli-Potato Mash,\n\nBroccoli-Cauli Sausage Tots,\n\nCuban Picadillo with Riced Cauliflower,\n\nPurple Cauliflower,\n\nSaffron and Herb Cauliflower Rice,\n\nCherries, in Upside-Down Chocolate Cherry Cake,\n\nCherries, poached,\n\nChicken\n\nabout: Poultry button on Instant Pot\u00ae,\n\nChicken Broth,\n\nChicken Taco Soup,\n\nEndive Boats with Chicken Salad,\n\nKentucky Derby Burgoo,\n\nLiver Pate,\n\nSesame-Orange Chicken Wings,\n\nTandoori-Style Chicken Wings,\n\nWhite Chicken Chili,\n\nChicken main dishes\n\nAsian Chicken Lettuce Wraps with Ginger Aioli,\n\nAvocado Chicken Salad,\n\nBBQ Whole Chicken,\n\nChicken Leg Almond Butter Satay,\n\nChicken Livers and Onions with Gravy,\n\nChicken Thighs with Strawberry Salsa,\n\nChili Lime Chicken Legs,\n\nGluten-Free Beer Can Dijon Chicken,\n\nHoney Mustard Chicken Bites,\n\nJamaican Jerk Chicken Meatballs,\n\nKimchi Chicken Wings,\n\nKombucha Chicken Thighs,\n\nPaleo-ish Turkish Chicken Thighs,\n\nSalsa Verde Chicken Meatballs,\n\nSmoked Paprika and Garlic Chicken Legs,\n\nTaco Chicken Lettuce Wraps with Pico Guacamole,\n\nTuscan Chicken,\n\nChocolate\n\nCaf\u00e9 Mocha Muffins,\n\nChocolate Chip Zucchini Bread,\n\nDouble Chocolate Custard,\n\nMexican Hot Cocoa,\n\nStrawberry Cupcakes with Chocolate Ganache,\n\nCitrus\n\nBlueberry-Orange Muffins,\n\nCoconut Key Lime Custard,\n\nHorseradish-Lemon Aioli,\n\nLemon-Herb Butter Dipping Sauce,\n\nQuick Lime Marmalade,\n\nCleaning Instant Pot\u00ae,\n\nCoconut aminos,\n\nCoconut Key Lime Custard,\n\nCornish Hens and Veggies,\n\n### D\n\nDesserts, \u2013211\n\nDrinks, , \u201315\n\nDuck \u00e0 l'orange breast,\n\nDumplings, drop biscuit,\n\n### E\n\nEggplant-Olive Ratatouille,\n\nEggs\n\nBacon, Onion, and Spinach Egg Muffins,\n\nBell Pepper Egg Cups,\n\nBreakfast Casserole with Drop Biscuit Dumplings,\n\nChampion Breakfast Bowl,\n\nCiao Bella! Egg Muffins,\n\nDeviled Eggs,\n\nHam and Mushroom Egg Muffins,\n\nPiggy Tater Bowl with a Poached Egg,\n\nSoft-Boiled Eggs with Steamed Asparagus Dippers,\n\nSouth-of-the-Border Frittata,\n\nSweet Potato Breakfast Bowl,\n\nEndive Boats with Chicken Salad,\n\n### F\n\nFruit salsas,\n\n### G\n\nGreens, mixed Southern,\n\n### H\n\nHerbs and spices, ,\n\n### I\n\nInstant Pot\u00ae\n\nabout: overview of features and uses, \u20135\n\nbuttons and functions, \u20138\n\ncleaning,\n\nlocking and pressure-release methods,\n\npot-in-pot cooking accessories, \u201310\n\n### K\n\nKale, ,\n\nKohlrabi, Kale, and Karrots,\n\n### L\n\nLamb\n\nGreek Lamb Meatballs,\n\nHearty Lamb Stew,\n\nLamb Riblets,\n\nLamb Shanks with Tomato Curry Sauce,\n\nLocking and pressure-release methods,\n\n### M\n\nMuffins, apple cinnamon,\n\nMushrooms\n\nBurgundy Mushrooms,\n\nCornish Hens and Veggies,\n\nCreamy Mushroom Soup,\n\nMushrooms Stuffed with Sausage and Kimchi,\n\n### N\n\nNuts, in Fruity Trail Mix Bowls,\n\nNuts, making almond milk,\n\n### O\n\nOnion and apple confit,\n\nOnions, herbed fingerling potatoes and,\n\n### P\n\nPaleo \"Yes\" and \"No\" foods, \u201318\n\nParsnips, , , ,\n\nPeach cobbler, upside-down,\n\nPineapple, ,\n\nPork\n\nabout: Meat\/Stew button on Instant Pot\u00ae,\n\nBacon, Onion, and Spinach Egg Muffins,\n\nBacon Cauli-Potato Mash,\n\nBacon Jam,\n\nBalsamic Pork Tenderloin with Caperberries,\n\nBBQ Boneless Pork Loin Roast,\n\nBolognese Sauce,\n\nBreakfast Casserole with Drop Biscuit Dumplings,\n\nBroccoli-Cauli Sausage Tots,\n\nChampion Breakfast Bowl,\n\nChorizo Sliders with Jicama-Beet Slaw,\n\nCowgirl Baby Back Pork Ribs,\n\nCream of Broccoli and Bacon Soup,\n\nCreamy Ham and Root Veggie Soup,\n\nCubano Bowl,\n\nGround Pork Stir-Fry Bowl,\n\nHam and Scalloped Potatoes,\n\nLoaded Sweet Potato Hash,\n\nLouisiana Shrimp and Sausage,\n\nMardi Gras Gumbo,\n\nMorning Bacon Parsnips,\n\nMushrooms Stuffed with Sausage and Kimchi,\n\nOma's Sausage and Cabbage Soup,\n\nOrange Rosemary Pork Tenderloin,\n\nOsso Bucco with Gremolata,\n\nPiggy Tater Bowl with a Poached Egg,\n\nPork Chops with Sauerkraut, Potatoes, and Apples,\n\nPork Medallions Puttanesca,\n\nSalsa Verde Pulled Pork,\n\nSausage Gravy,\n\nSimple Dijon Pork Tenderloin,\n\nSmothered Pork Chops with Mushroom Gravy,\n\nSteamed Dry-Rubbed Pork Spareribs,\n\nVietnamese Pork Chops,\n\nPotatoes. See also Sweet potatoes\n\nDilled Purple Potatoes,\n\nGerman Red Potatoes,\n\nHerbed Fingerling Potatoes and Onions,\n\nPiggy Tater Bowl with a Poached Egg,\n\nPot-in-pot cooking accessories, \u201310\n\nPoultry main dishes, \u201325\n\nPressure-release methods,\n\nPumpkin, in Grain-Free Stuffed Apples,\n\nPumpkin Chip Muffins,\n\n### R\n\nRutabaga Fries with Rosemary Dipping Sauce,\n\n### S\n\nSalad, shrimp, prosciutto, and asparagus,\n\nSauces and dips\n\nabout: fruit salsas,\n\nBarbecue Sauce,\n\nBolognese Sauce,\n\n\"Cheesy\" Spinach Crab Dip,\n\nCocktail Sauce,\n\nGinger Aioli,\n\nGremolata,\n\nHerb Sauce,\n\nHorseradish-Lemon Aioli,\n\nLemon-Herb Butter Dipping Sauce,\n\nMarinara Sauce,\n\nMixed Berry Quick Jam,\n\nPuttanesca Sauce,\n\nRosemary Dipping Sauce,\n\nRussian Dressing,\n\nSalsa Verde,\n\nSausage Gravy,\n\nTomato Curry Sauce,\n\nTzatziki Sauce,\n\nSeafood and fish\n\nBerbere Salmon,\n\n\"Cheesy\" Spinach Crab Dip,\n\nCitrus-Steamed Littleneck Clams,\n\nCrab-Chorizo Stuffed Calamari,\n\nDairy-Free Creamed Crab,\n\nDilled Salmon and Broccoli,\n\nFish Taco Bowl,\n\nGreek Sea Bass,\n\nHerbed Cod with Yellow Squash,\n\nLouisiana Shrimp and Sausage,\n\nMahi-Mahi with a Lemon-Caper Butter Sauce,\n\nMardi Gras Gumbo,\n\nQuick Crawfish Boil,\n\nSalmon Croquettes,\n\nShrimp, Prosciutto, and Asparagus Salad with Yuzu Vinaigrette,\n\nShrimp Scampi,\n\nSimply Steamed Lobster Tails,\n\nSteamed Mussels and Chorizo,\n\nSteamed Shrimp with Cocktail Sauce,\n\nSteamed Snow Crab Legs,\n\nStewed Halibut,\n\nThai Coconut Curry Monkfish,\n\nTom Kha Goong,\n\nSeafood and fish main dishes, \u201393\n\nSide dishes, \u2013101\n\nSoup button on Instant Pot\u00ae,\n\nSoups, stews, and chilis, \u201357\n\nSquash\n\nBroccoli-Zucchini Hummus,\n\nChocolate Chip Zucchini Bread,\n\nSavory Acorn Squash Soup,\n\nSesame Zoodles,\n\nSouthern Squash Casserole,\n\nSweet potatoes, , ,\n\nSweet treats, \u2013211\n\n### T\n\nTeas and other drinks, \u201315\n\nTimer button,\n\nTomatillos, in Salsa Verde,\n\nTomato basil soup, creamy,\n\nTomatoes, sauces with. See Sauces and dips\n\nTurkey main dishes, ,\n\nAdams Media\n\nAn Imprint of Simon & Schuster, Inc.\n\n57 Littlefield Street\n\nAvon, Massachusetts 02322\n\nwww.SimonandSchuster.com\n\nCopyright \u00a9 2017 by Simon & Schuster, Inc.\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information address Adams Media Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.\n\nFirst Adams Media trade paperback edition DECEMBER 2017.\n\nADAMS MEDIA and colophon are trademarks of Simon and Schuster.\n\nFor information about special discounts for bulk purchases, please contact Simon & Schuster Special Sales at 1-866-506-1949 or business@simonandschuster.com.\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nInterior design by Colleen Cunningham\n\nPhotographs by James Stefiuk\n\nCover design by Erin Alexander\n\nLibrary of Congress Cataloging-in-Publication Data\n\nFagone, Michelle, author.\n\nThe \"I love my Instant Pot\u00ae\" paleo recipe book \/ Michelle Fagone of CavegirlCuisine.com, author of \"The 'I love my Instant Pot\u00ae' recipe book\".\n\nAvon, Massachusetts: Adams Media, 2017.\n\nSeries: \"I Love My\".\n\nIncludes index.\n\nLCCN 2017031651 (print) | LCCN 2017037760 (ebook) | ISBN 9781507205747 (pb) | ISBN 9781507205754 (ebook)\n\nLCSH: Pressure cooking. | Electric cooking. | High-protein diet--Recipes. | Gluten-free diet--Recipes. | Prehistoric peoples--Nutrition. | Cooking (Natural foods) | BISAC: COOKING \/ Methods \/ Special Appliances. | COOKING \/ Specific Ingredients \/ Meat. | COOKING \/ Courses & Dishes \/ General. | LCGFT: Cookbooks.\n\nLCC TX840.P7 (ebook) | LCC TX840.P7 F335 2017 (print) | DDC 641.5\/87--dc23\n\nLC record available at \n\nISBN 978-1-5072-0574-7\n\nISBN 978-1-5072-0575-4 (ebook)\n\nThe \"I Love My Instant Pot\u00ae\" Paleo Recipe Book is an independent publication and has not been authorized, sponsored, or otherwise approved by Double Insight Inc. Instant Pot\u00ae is a trademark of Double Insight Inc., registered in the US and other countries. Instant Pot\u00ae is designed in Canada and manufactured in China, with healthy and green living, and ethnic diversity.\n\nAlways follow safety and commonsense cooking protocols while using kitchen utensils, operating ovens and stoves, and handling uncooked food. If children are assisting in the preparation of any recipe, they should always be supervised by an adult.\n\nMany of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Simon & Schuster, Inc., was aware of a trademark claim, the designations have been printed with initial capital letters.\n\nContains material adapted from the following title published by Adams Media, an Imprint of Simon & Schuster, Inc.: The \"I Love My Instant Pot\u00ae\" Recipe Book by Michelle Fagone, copyright \u00a9 2017, ISBN 978-1-5072-0228-9.\n\n## Contents\n\n 1. Cover\n 2. Dedication\n 3. Introduction\n 4. 1. Cooking with an Instant Pot\u00ae\n 5. 2. Breakfast\n 1. Champion Breakfast Bowl\n 2. Sweet Potato Breakfast Bowl\n 3. Morning Bacon Parsnips\n 4. Apple Cinnamon Muffins\n 5. Blueberry-Orange Muffins\n 6. Pumpkin Chip Muffins\n 7. Sausage Gravy\n 8. South-of-the-Border Frittata\n 9. Ciao Bella! Egg Muffins\n 10. Ham and Mushroom Egg Muffins\n 11. Bacon, Onion, and Spinach Egg Muffins\n 12. Soft-Boiled Eggs with Steamed Asparagus Dippers\n 13. Bell Pepper Egg Cups\n 14. Fruity Trail Mix Bowls\n 15. Mixed Berry Breakfast Syrup\n 16. Bacon Jam\n 17. Breakfast Casserole with Drop Biscuit Dumplings\n 18. Loaded Sweet Potato Hash\n 19. Piggy Tater Bowl with a Poached Egg\n 6. 3. Soups, Stews, and Chilis\n 1. Chicken Broth\n 2. Vegetable Broth\n 3. Beef Broth\n 4. To Bean or Not to Bean Chili\n 5. Chunky Beef Chili\n 6. Hearty Lamb Stew\n 7. Creamy Mushroom Soup\n 8. Cream of Broccoli and Bacon Soup\n 9. Tom Kha Goong\n 10. Kentucky Derby Burgoo\n 11. White Chicken Chili\n 12. Savory Acorn Squash Soup\n 13. Chicken Taco Soup\n 14. Creamy Tomato Basil Soup\n 15. Mardi Gras Gumbo\n 16. Quick Vegetable Soup\n 17. Hamburger Soup\n 18. Oma's Sausage and Cabbage Soup\n 19. Creamy Ham and Root Veggie Soup\n 20. Curried Carrot Soup\n 7. 4. Appetizers and Sauces\n 1. Deviled Eggs\n 2. Mini Chili Polpette\n 3. \"Cheesy\" Spinach Crab Dip\n 4. Endive Boats with Chicken Salad\n 5. Tandoori-Style Chicken Wings\n 6. Sesame-Orange Chicken Wings\n 7. Broccoli-Zucchini Hummus\n 8. Burgundy Mushrooms\n 9. Mushrooms Stuffed with Sausage and Kimchi\n 10. Steamed Shrimp with Cocktail Sauce\n 11. Liver Pate\n 12. Steamed Artichokes with Lemon-Herb Butter\n 13. Rutabaga Fries with Rosemary Dipping Sauce\n 14. Broccoli-Cauli Sausage Tots\n 15. Salsa Verde\n 16. Bolognese Sauce\n 17. Marinara Sauce\n 18. Puttanesca Sauce\n 19. Tomato Curry Sauce\n 8. 5. Side Dishes\n 1. Nutty Holiday Sweet Potatoes\n 2. Honey-Glazed Carrots\n 3. Bacon-Apple Brussels Sprouts\n 4. Mixed Southern Greens\n 5. Dilled Purple Potatoes\n 6. Lemon Ghee Broccoli\n 7. Herbed Fingerling Potatoes and Onions\n 8. Kohlrabi, Kale, and Karrots\n 9. Steamed Beets\n 10. Purple Cauliflower\n 11. German Red Potatoes\n 12. Eggplant-Olive Ratatouille\n 13. Southern Squash Casserole\n 14. Broccoli Casserole\n 15. Bacon Cauli-Potato Mash\n 16. Red Onion and Apple Confit\n 17. Sesame Zoodles\n 18. Saffron and Herb Cauliflower Rice\n 19. Broccoli-Parsnip Smash\n 9. 6. Poultry Main Dishes\n 1. BBQ Whole Chicken\n 2. Gluten-Free Beer Can Dijon Chicken\n 3. Kimchi Chicken Wings\n 4. Chicken Leg Almond Butter Satay\n 5. Smoked Paprika and Garlic Chicken Legs\n 6. Chili Lime Chicken Legs\n 7. Kombucha Chicken Thighs\n 8. Chicken Thighs with Strawberry Salsa\n 9. Tuscan Chicken\n 10. Salsa Verde Chicken Meatballs\n 11. Honey Mustard Chicken Bites\n 12. Jamaican Jerk Chicken Meatballs\n 13. Taco Chicken Lettuce Wraps with Pico Guacamole\n 14. Asian Chicken Lettuce Wraps with Ginger Aioli\n 15. Chicken Livers and Onions with Gravy\n 16. Avocado Chicken Salad\n 17. Paleo-ish Turkish Chicken Thighs\n 18. Cornish Hens and Veggies\n 19. Chili-Orange Turkey Wings\n 20. Mega Turkey Meatballs\n 10. 7. Beef and Bison Main Dishes\n 1. Nana's Sunday Pot Roast\n 2. Chuck Roast with Carrots and Onions\n 3. Sweet-n-Spicy Beef Brisket\n 4. Tender Flank Steak with Mushrooms and Onions\n 5. Skirt Steak Fajitas\n 6. Korean Beef Short Ribs\n 7. Mongolian Beef and Cashew BBQ\n 8. Meatloaf and Gravy\n 9. Balsamic Meatballs\n 10. Marinara Meatballs\n 11. Reuben Meatballs with Russian Dressing\n 12. Ground Beef and Broccoli Bowl\n 13. Just Beet It Burgers\n 14. Corned Beef and Cabbage\n 15. Cuban Picadillo with Riced Cauliflower\n 16. Chi-Town Italian Beef and Peppers\n 17. Beef Shanks and Artichokes\n 18. Bison Sloppy Joe's\n 19. Ground Bison Bulgogi\n 20. Italian Beef Tongue Lettuce Wraps\n 11. 8. Pork, Lamb, and Game Main Dishes\n 1. Salsa Verde Pulled Pork\n 2. Simple Dijon Pork Tenderloin\n 3. Orange Rosemary Pork Tenderloin\n 4. Balsamic Pork Tenderloin with Caperberries\n 5. Steamed Dry-Rubbed Pork Spareribs\n 6. Cowgirl Baby Back Pork Ribs\n 7. Ham and Scalloped Potatoes\n 8. Ground Pork Stir-Fry Bowl\n 9. Smothered Pork Chops with Mushroom Gravy\n 10. Pork Chops with Sauerkraut, Potatoes, and Apples\n 11. Cubano Bowl\n 12. Vietnamese Pork Chops\n 13. BBQ Boneless Pork Loin Roast\n 14. Osso Bucco with Gremolata\n 15. Pork Medallions Puttanesca\n 16. Lamb Riblets\n 17. Lamb Shanks with Tomato Curry Sauce\n 18. Greek Lamb Meatballs\n 19. Chorizo Sliders with Jicama-Beet Slaw\n 20. The Busy Cook's Duck a l'Orange Breast\n 12. 9. Seafood and Fish Main Dishes\n 1. Steamed Mussels and Chorizo\n 2. Stewed Halibut\n 3. Fish Taco Bowl\n 4. Thai Coconut Curry Monkfish\n 5. Louisiana Shrimp and Sausage\n 6. Shrimp, Prosciutto, and Asparagus Salad with Yuzu Vinaigrette\n 7. Shrimp Scampi\n 8. Crab-Chorizo Stuffed Calamari\n 9. Berbere Salmon\n 10. Dilled Salmon and Broccoli\n 11. Dairy-Free Creamed Crab\n 12. Citrus-Steamed Littleneck Clams\n 13. Salmon Croquettes\n 14. Greek Sea Bass\n 15. Herbed Cod with Yellow Squash\n 16. Mahi-Mahi with a Lemon-Caper Butter Sauce\n 17. Steamed Snow Crab Legs\n 18. Simply Steamed Lobster Tails\n 19. Quick Crawfish Boil\n 13. 10. Sweet Treats and Drinks\n 1. Spiced Applesauce\n 2. Cafe Mocha Muffins\n 3. Strawberry Cupcakes with Chocolate Ganache\n 4. Apple Brown Betty\n 5. Upside-Down Peach Cobbler\n 6. Banana Pudding Cake\n 7. Chocolate Chip Zucchini Bread\n 8. Double Chocolate Custard\n 9. Coconut Key Lime Custard\n 10. Grain-Free Stuffed Apples\n 11. Hummingbird Cake\n 12. Poached Cherries\n 13. Upside-Down Chocolate Cherry Cake\n 14. Mixed Berry Quick Jam\n 15. Quick Lime Marmalade\n 16. Fresh-Brewed Arnie Palmer\n 17. Gluhwein\n 18. Homemade Chai Tea Latte\n 19. Spiced Mocha Latte\n 20. Mexican Hot Cocoa\n 14. Appendix A: Paleo \"Yes\" and \"No\" Foods\n 15. Appendix B: US\/Metric Conversion Chart\n 16. About the Author\n 17. Index\n 18. Copyright\n\n# Guide\n\n 1. Cover\n 2. Contents\n 3. Start of content\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nSome people point from shores of safety to horizons of future possibility. But once in a while God raises up someone with the courage to do more than just talk about distant potential. Instead, they wade into the middle of the miraculous, their faith leaving a wake that will impact generations, starting here and now. My friend William McDowell is a waymaker and walking evidence that the move of God isn't coming; it's happening!\n\n\u2014STEVEN FURTICK\n\n_N EW YORK TIMES_ BEST-SELLING AUTHOR\n\nPASTOR, ELEVATION CHURCH\n\nMany people do a God thing without God; that is the definition of _religion_. My friend William is the exact opposite. He refuses to continue going through the Christian motions without seeing and hearing from God. His relentless pursuit has been seen and rewarded, as God continues to pour out His presence, miracles, and a sound that takes us all into God's presence. I thank God for who William is to the kingdom of God at large and to the people who are privileged to call him friend.\n\n\u2014JONATHAN STOCKSTILL\n\nLEAD PASTOR, BETHANY CHURCH\n\nI've known Pastor William McDowell for twenty-five years. We grew up in the same city, both raised by single mothers who wouldn't stop pushing us toward Jesus. I knew God would take William's heart and gift around the world. It was inevitable. But what I did not anticipate was the level of glory that would be unleashed once Pastor William said yes to leading Deeper Fellowship Church.\n\nGod wants glory in the local church. It is what Jesus came to establish. (See Matthew 16.) The power, the beauty, the consistency of spiritual growth and development coupled with tangible miracles, signs, and wonders are the heart of the New Testament church. What God is doing in Orlando, Florida, is unprecedented but shouldn't be unexpected. It's what God intended for the local church. May God give us the blueprint for glory to return to His house through the glorious testimonies of God's grace and power in this book.\n\n\u2014JOHN GRAY\n\nPASTOR, RELENTLESS CHURCH\n\nNot only is William a friend, but he has been a mentor to me over the past few years, and I cannot wait for the words of God written in this book to flow through your heart, mind, body, and spirit. The Holy Spirit is alive and well, not only in this book but in William's life, church, and songs. I pray that before you read this powerful book, you say a prayer for Jesus to open up your heart and soul to the spiritual journey you are about to go on! This journey will be overflowing with joy, tears, and the presence of Jesus from beginning to end. The anointing and presence of God are real!\n\nWilliam, thank you for being an amazing influence not only in my life but also in the lives of others. Jesus put us on this earth to serve, and you are doing that, my friend. Jesus reigns!\n\n\u2014RUSSELL WILSON\n\nQUARTERBACK, SEATTLE SEAHAWKS\n\nWhen he was a young man sitting in the back of the auditorium in my meetings, William McDowell may have been unknown to me, but he certainly was not unknown to God. It has been a tremendous privilege for me to observe his growth and development in so many different areas\u2014as a husband, a father, a music minister, a pastor, a songwriter, a worshipper, and now as an author.\n\nIt is gratifying but not at all surprising to me to see how God is using him, not only as an unusually gifted worship leader but also as a caring and compassionate pastor whose services are characterized by a remarkable demonstration of the supernatural power of God. _It's Happening_ will take your faith to the next level as Pastor McDowell shares the extraordinary miracles, signs, and wonders God has performed among his dynamic church family.\n\n\u2014DR. ROD PARSLEY\n\nPASTOR AND FOUNDER, WORLD HARVEST CHURCH\n\nMost CHARISMA HOUSE BOOK GROUP products are available at special quantity discounts for bulk purchase for sales promotions, premiums, fund-raising, and educational needs. For details, write Charisma House Book Group, 600 Rinehart Road, Lake Mary, Florida 32746, or telephone (407) 333-0600.\n\nIT'S HAPPENING by William McDowell\n\nPublished by Charisma House\n\nCharisma Media\/Charisma House Book Group\n\n600 Rinehart Road\n\nLake Mary, Florida 32746\n\nwww.charismahouse.com\n\nThis book or parts thereof may not be reproduced in any form, stored in a retrieval system, or transmitted in any form by any means\u2014electronic, mechanical, photocopy, recording, or otherwise\u2014without prior written permission of the publisher, except as provided by United States of America copyright law.\n\nUnless otherwise noted, all Scripture quotations are from the Holy Bible, New Living Translation, copyright \u00a9 1996, 2004, 2007. Used by permission of Tyndale House Publishers, Inc., Wheaton, IL 60189. All rights reserved.\n\nScripture quotations marked ESV are from the Holy Bible, English Standard Version. Copyright \u00a9 2001 by Crossway Bibles, a division of Good News Publishers. Used by permission.\n\nScripture quotations marked KJV are from the King James Version of the Bible.\n\nScripture quotations marked MEV are taken from the Holy Bible, Modern English Version. Copyright \u00a9 2014 by Military Bible Association. Used by permission. All rights reserved.\n\nScripture quotations marked NIV are taken from the Holy Bible, New International Version\u00ae, NIV\u00ae. Copyright \u00a9 1973, 1978, 1984, 2011 by Biblica, Inc.\u2122 Used by permission of Zondervan. All rights reserved worldwide. www.zondervan.com The \"NIV\" and \"New International Version\" are trademarks registered in the United States Patent and Trademark Office by Biblica, Inc.\u2122\n\nScripture quotations marked NKJV are taken from the New King James Version\u00ae. Copyright \u00a9 1982 by Thomas Nelson. Used by permission. All rights reserved.\n\nScripture quotations marked PHILLIPS are from _The New Testament in Modern English_ , Revised Edition. Copyright \u00a9 1958, 1960, 1972 by J. B. Phillips. Macmillan Publishing Co. Used by permission.\n\nScripture quotations marked THE MESSAGE are from _The Message: The Bible in Contemporary English_ , copyright \u00a9 1993, 1994, 1995, 1996, 2000, 2001, 2002. Used by permission of NavPress Publishing Group.\n\nCopyright \u00a9 2018 by William McDowell\n\nAll rights reserved\n\nCover design by Justin Evans\n\nVisit the author's website at www.williammcdowellmusic.com.\n\nLibrary of Congress Cataloging-in-Publication Data:\n\nAn application to register this book for cataloging has been submitted to the Library of Congress.\n\nInternational Standard Book Number: 978-1-62999-500-7\n\nE-book ISBN: 978-1-62999-501-4\nThis book is dedicated to my children, Joshua, Olivia, Caleb, Joy, and the one yet to be revealed. I suspect that throughout history, in every generation, a prevailing thought emerges that God doesn't move the way He used to, but you know otherwise. As you get older, you will flip through the pages of this book and be able to say you were there when this happened. But this outpouring is not limited to a particular place or time. Heaven will always respond to those who are hungry. You can experience what we have seen and more if you will cling to these words written and consistently pursue a relationship with God. No one will be able to convince you that God doesn't move this way, because you were there! May you experience more of God in your future than I ever did.\n\nI am also aware that many in the next generation and others in the years to come will read this book. This book is also dedicated to you. Don't allow anyone to convince you that a move of God is something of the past. You can experience it too. Whatever you do, don't stop until you too can say, \"It's happening!\"\nCONTENTS\n\nAcknowledgments\n\nPrologue\n\nIntroduction: What's Happening?\n\n**Chapter 1** When He Walked Into the Room\n\n**Chapter 2** How's Your Posture?\n\n**Chapter 3** He Hears Your Cry\n\n**Chapter 4** Follow the Pathway\n\n**Chapter 5** It's All About Capacity\u2014How Empty Are You?\n\n**Chapter 6** No Submission, No Power\n\n**Chapter 7** Urgent Anticipation\n\n**Chapter 8** Stay and Wait\n\n**Chapter 9** Contend for the Flames of Revival\n\n**Chapter 10** A Worthy Host\n\n**Chapter 11** He Is Here\u2014Now What Will You Do?\n\n**Chapter 12** Feed the Fire\n\n**Chapter 13** It Happened Because It's Happening\n\n**Chapter 14** Living Stones\n\n**Chapter 15** You Were Born for This\n\nNotes\n\nEndorsements\nACKNOWLEDGMENTS\n\nTO MY WIFE, LaTae, your unwavering commitment, support, and partnership are what allow this ministry to happen. What you do that is unseen allows all that is seen. I love you with all my heart.\n\nJevon Bolden, thank you for your tireless work to help make this book come to life. It was clearly destined to happen this way.\n\nAdrienne Gaines, thank you for your amazing work and incredible patience. You are a true gift to us all.\n\nTo my mom, Pauline McDowell, you have intentionally lived your life in a way that allowed me not only to pursue every dream in my heart but more importantly to pursue God. As a result, it's happening! I'm proud to be a part of your legacy.\n\nTo Caleb Grant, Jason McMullen, and Jerry Bell, you guys inspire, encourage, challenge, and uphold me every day. It is the greatest joy and privilege to walk with you and serve Deeper Fellowship Church with you. This is our story. I just had the privilege of writing about it. I love you brothers dearly!\n\nTo Deeper Fellowship Church, I've never seen a group of people anywhere in the world pursue God the way you do. It's not just a privilege to serve as your pastor; it's an even greater privilege just to be a part of this community. What is happening among us is special. May we never take it for granted. We've only scratched the surface. There's more!\n\nTo my pastor, Bishop Joseph Garlington; Pastor Sam Oye; and every pastor, prophetic voice, and lead worshipper who has sown into this congregation and the atmosphere of our church in a deeply personal way, thank you from my heart and from everyone at Deeper Fellowship Church. The seeds you've sown into us are producing a harvest.\n\nMarcos Perez, we never could have known all those years ago that this is why God connected us, but He did, and it's marvelous.\n\nAnd again to Jason McMullen, this is another part of this unfolding destiny. I'm so happy to be able to work with you on this too!\n\nFinally, Lord, I'm grateful that You would trust me to be one of the stewards of Your move in the earth. My prayer is that my life and these words honor You and bring You glory.\nPROLOGUE\n\n_Father, You've been present in my life in such an amazing way, and it is mind-blowing to me that You continue to show me Your favor. I pray that Your grace will continue and extend in such a way that Your tangible presence will be felt on every page of this book._\n\n_What an amazing God You are that You would desire to dwell among Your people, that You would bless us with the fire of revival, that You would not stay distant and far off. I am undone with the knowledge that, from the foundations of the world, You would create an opportunity for us to get back in right relationship with You so that You can come and be with us._\n\n_The potential for what You will do when we surrender and ask blows my mind. You leave me speechless. I am in awe. My breath is taken away by Your glorious presence and power._\n\n_Lord, cause Your people to be sensitive to what You want to do and say through me and the words in this book. I don't want to say more than what You're saying; I don't want to do more than what You are doing. I only want to do and say what You are doing and saying. So make us sensitive._\n\n_Now, Lord, \"let the words of my mouth, and the meditation of my heart, be acceptable in thy sight, O L ORD, my strength, and my redeemer\"_ (Ps. 19:14, KJV).\n\n_You get the glory. Save us. Visit us. Dwell among us. Revive us. In Jesus's name, amen._\nINTRODUCTION\n\nWHAT'S HAPPENING?\n\nJesus told them, \"Go back to John and tell him what you have heard and seen\u2014the blind see, the lame walk, those with leprosy are cured, the deaf hear, the dead are raised to life, and the Good News is being preached to the poor.\" And he added, \"God blesses those who do not fall away because of me.\"\n\n\u2014MATTHEW 11:4\u20136\n\nIT'S HAPPENING! _W HAT'S_ happening? Something undeniable is stirring in the church. The global body of Christ is talking about the same thing right now. No matter where you go, you cannot escape hearing the word _revival_. I know it may feel like a Christian buzzword because of how we've used it to describe some of our services and gatherings: \"We're going to have a one- (or two- or three-) week revival.\" But I'm not just referring to a temporary series of good services. I'm talking about the legitimate outpouring of the Holy Spirit that many of us have longed to see.\n\nRevival is here, and I assure you _revival_ is not a buzzword. There are people who have gone before us who have prayed and cried and given their lives in intercession for you and me to experience the move of God they never were able to see. And I am here to tell you, it's happening. We've entered into a special time, an amazing season of open heaven, a season that can be defined not by clich\u00e9 or title but by reality. A season of revival is upon us.\n\nIf I were to define _revival_ as the outpouring of the Spirit of God in a place or region, then I can tell you that the things that have happened at our church, Deeper Fellowship in Orlando, Florida, since May 22, 2016, bear hallmarks of every historic revival that has taken place in the past four hundred years. There have been salvations, miraculous healings, and a deep awareness that God is in our midst. Even if you've never been to our church, you should get excited about this because this revival is not limited to our church\u2014it is available to everyone. If it's happening anywhere, it can happen everywhere! You too can encounter Him\u2014and you can be a catalyst for revival.\n\nIt can be hard to believe that what so many have longed and prayed for is actually here. That's why Matthew 11:4\u20136, quoted at the opening of this introduction, is an important encouragement for us to hold on to. In this passage John the Baptist, the prophetic forerunner announcing the message of the kingdom and the arrival of the kingdom man, Jesus, found himself in prison at the hands of King Herod. Scripture says when he was in prison, John heard about the works of Christ, so he sent two of his disciples to ask Jesus, \"Are You the Messiah we've been expecting, or should we keep looking for someone else?\"\n\nWhat an interesting question when you consider who asked it. John the Baptist was the baby who leaped inside the womb of his mother, Elizabeth, when Mary came to visit while she was pregnant with Jesus. He was the one in the wilderness preaching, \"Repent, for the kingdom of heaven is at hand\" (Matt. 3:2, MEV). He was the one of whom Jesus said there was no greater prophet. John was the one baptizing people and proclaiming that one greater than he was coming and that he was not even worthy to untie His sandals. John was the one who recognized Jesus and said, \"Look! The Lamb of God who takes away the sin of the world!\" (John 1:29).\n\nWhen Jesus came to the Jordan River to be baptized, John tried to prohibit Him, saying, \"I need to be baptized by you, and do you come to me?\" (Matt. 3:14, NIV). But Jesus told him, \"Let it be so now; it is proper for us to do this to fulfill all righteousness\" (v. 15, NIV). Only then did John consent. So John also was there when the Spirit descended like a dove and rested on Jesus, and a voice came from heaven saying, \"This is my beloved Son, in whom I am well pleased\" (Matt. 3:17, KJV).\n\nJohn the Baptist, the great prophet, preacher, and baptizer, is the one who wanted his disciples to ask Jesus, \"Are You the One, or should we look for another?\" Perhaps as he was sitting in that prison, some doubts began to fill his mind. Maybe he began wondering, \"Is this what life would be like if the Messiah were truly here?\" Maybe he thought things would be different. Perhaps that's how we are when we hear about a move of God happening somewhere. We wonder, \"Is it possible that this is revival, or should we look somewhere else?\" Maybe revival doesn't look the way we thought it would or isn't happening where or how we thought it would take place.\n\nI love how Jesus answered the question John had his disciples ask. Jesus could have simply said, \"Yes, I am the Messiah.\" But He didn't do that. The way Jesus chose to respond would not only answer John but reverberate through all of time.\n\n> Jesus told them, \"Go back to John and tell him what you have heard and seen\u2014the blind see, the lame walk, those with leprosy are cured, the deaf hear, the dead are raised to life, and the Good News is being preached to the poor.\" And he added, \"God blesses those who do not fall away because of me.\"\n> \n> \u2014MATTHEW 11:4\u20136\n\nBasically Jesus was saying, \"Go back and tell John it's happening. Tell John that what he'd been prophesying, believing, and praying for\u2014the very reason for his call\u2014is happening.\" John never got out of the prison, but he did get out of that prison of doubt. He knew from that moment on, it's happening. And Jesus knew that the same thing that would speak to and encourage John would speak to and encourage us: if it's happening anywhere, it can happen everywhere. Said another way, once it's happening anywhere, it's available everywhere.\n\nMany times when the American church starts to talk about moves of God, we point to the mass salvations and innumerable miracles that happen in South America, West Africa, and Western Europe and lament that those things aren't happening in abundance here. However, whenever you hear about a move of God anywhere, that's the time to get excited because it's an indication that the same thing is available to anyone who will posture himself to receive it. Miracles are breaking out. Undeniable encounters with God are taking place. The presence of God is being tangibly felt. The power of God is present. The people of God are passionately pursuing Him. It's happening!\n\nLET THE RIVERS OF GOD FLOW\n\nI initially did not want to call what we had begun to witness and experience a revival because I didn't want to brand or market a cluster of services as such a divine moment. Everything we have experienced we carry with reverence and honor because we recognize we have not done anything to bring it. But we could do something that causes it to leave. We are recipients of what some theologians call the scandal of particularity, which is all about God asserting His sovereign decision to reveal Himself in a place or to a people simply because He wants to. The only thing we can say about what's happening among us specifically is that God made a sovereign decision to show us favor.\n\nYet we hold an awareness that like the sensitivity of the Spirit of God, revival can go as suddenly as it came. Our respect for this season, or lack thereof, can sustain or repel the move of God, so we've been very careful to carry it with reverential fear. For this reason we renamed our services \"rivers.\" Based on the truth that everything we do should follow God's leading, we have submitted our services to the flow of what He is doing. We do not control the moments. We do not determine how everything will go.\n\nThis setup may not work for everyone. We live in a society that sorely longs to be in control of everything. It's so easy to try to control God or micromanage what we think He can do down to the smallest detail. Across the country, in almost every church in America, we've become really good at attempting to manage the flow of God. We've become really good at programming church to the point that we plan months ahead the experience people are going to have with each clever sermon series and corresponding worship set.\n\nPlease understand, I am not against systems or systematic thinking\u2014not at all. What I am against is anything that presupposes that we can control God, because He cannot be controlled. We've learned that in the river of God there cannot be boats with oars. In order to be in the river of God, we must throw down our oars and put up our sails!\n\nThe whole point of a revival is for the church to be awakened to the reality, presence, and power of God and for our hearts to be turned to Him. There are people\u2014it might start as a remnant people\u2014who are saying: \"Jesus, we want You more than we want anything else. We don't want programs. We want You. And we'll stay as long as it takes. We'll get as loud as we need to get. We'll get on our faces. We'll cry out and let our tears water the seed that we have sown.\" We will see this revival because it is a revival of people who want Jesus.\n\n_One of the greatest tragedies that could ever exist is that the prayers of the saints who have gone before us and who cried and cried out for revival be wasted on a complacent and comfortable church._\n\nOne of the greatest tragedies that could ever exist is that the prayers of the saints who have gone before us and who cried and cried out for revival be wasted on a complacent and comfortable church. Many, I'm sure, cried out for things yet not seen that we have the opportunity to see. We must resist the notion to become a \"What can You do for me?\" church and lean in to what it takes to be passionately on fire for God and filled with His presence to the point that it overflows and affects people everywhere we go. We must refuse to miss this moment.\n\nAs of the writing of this book, Deeper Fellowship has been experiencing a tremendous move of God for more than a year. We have seen more than two hundred miracles of physical healing in addition to deliverance, supernatural breakthrough, and salvation. We've begun to grow quite a collection at our altar of unusable crutches, wheelchairs, and other medical devices because people are being healed at each river.\n\nWe've never had a so-called \"healing service.\" We have prayed for some people, but many were healed just by being in the presence of God. Some of the miracles we've seen have boggled the minds of medical experts around the country and have set historical records. We've seen cancer healed, tumors disappear, and the effects of strokes reversed. Wheelchairs have been emptied, walkers discarded, deaf ears opened, blindness reversed, rheumatoid arthritis healed, lupus healed multiple times, and carpal tunnel syndrome healed. I was even healed of gout.\n\nOne lady in our church, Ms. Pam, used to be partially paralyzed due to a stroke and needed a walker. Now she jumps up and down at the front of the platform during worship! We've seen a partially brain dead baby who doctors said would never smile, talk, or eat on her own instantly healed. One of the most amazing testimonies, or what we call \"stories of the river,\" is when our executive pastor's wife and seven of their nine children were healed of a brain-eating amoeba. I'll share more about that later in this book. All I can tell you is it's happening!\n\nEvery time I sit down to write about what God has done, more miracles have happened. There are so many testimonies it's just impossible for me to include them all. That's why the title of this book is _It's Happening_ and not _It Happened_ , because the stories of the river just keep flowing. We marvel at what we've seen God do. Some of our members had never experienced or been part of a move of God like this before. There's been such a level of awe. We are astounded. And we want more.\n\nA LONG TIME COMING\n\nWhat God is doing in our midst has been a long time coming. I began to experience this hunger for revival back in 1998 after my pastor at the time suggested that I read Tommy Tenney's book _The God Chasers_. After reading the first chapter or two, I put the book down, went to my room, and began to cry out to God to have an encounter with Him\u2014and I wasn't going to leave until I had one! From that moment on I earnestly cried out for revival to break out somewhere I was. I wanted to see it. I wanted to be there. I wanted to experience it. I wanted to have that kind of encounter with God, and I was desperate to get it. I prayed and cried, and cried and prayed, but I wasn't seeing it.\n\nI would go to church with what I'll call a frustrated anticipation. I lived with the expectation that any service could be _the_ service where it would happen. I prayed for it, and I constantly looked for it. I'd see a couple of on-fire people here or there, but there never seemed to be an entire church that pursued God and didn't care what time the service let out. In the prayer gatherings held at my mom's house, where I stayed at the time, I would see glimpses of what could be. A group of friends would come over with no agenda but to seek God. At those gatherings we sought God intensely. Incidentally it was during those times of worship and prayer that my sensitivity to the voice of God grew and I learned how to host His presence. My longing led me through lonely times, isolation, ridicule, and being called weird. Still I prayed because I was determined to see revival.\n\nI didn't recognize that my prayers were setting me up for my life. I also didn't realize that when I was asking God to bring forth a generation that wanted revival as badly as I did, He would lead me into covenant with the members at Deeper. Fast-forward twenty years, and now Deeper is three years old publicly and six years old for those of us who've been part of the core group. For the first three years we met as a small group in my living room. Throughout those first three years we had one specific prayer: we wanted to see a move of God. Our first official act as a group was to take a map of the city of Orlando and put our tears on it. I still have that map at my house. We wept over our city and asked God for revival. God heard our cries, and He is responding. When I cry now, my tears often come because I am beginning to see what I have prayed for. It's happening! Since May 22, 2016, we've used that phrase as an encouragement to one another, recognizing that God is doing what He said He would do.\n\nGET YOUR HOPES UP\n\nAfter not seeing God move mightily in the church in America for so long, the door of disappointment has been opened, and a spirit of skepticism has entered. It has changed our faith-filled speech to doubt-filled discourse. Our spirits bear witness to this defeating spirit that causes us to say, \"Don't get my hopes up. Don't talk to me about my dreams. Don't talk to me about my potential. I've made peace with disappointment.\" But that is exactly what the word of the Lord comes to do. The word of the Lord comes to awaken the hope inside you to cause you to fall out of agreement with disappointment and align with hope. Your days of disappointment are over; your season of manifestation is here.\n\nPerhaps you were once active in your local church, serving on the worship team and maybe the prayer team. Maybe you were part of groups that prayed for the move of God to come to your church or city, but after years of praying, it seemed nothing was happening. Maybe you cried out before God in the privacy of your home, where no one else could see. What if I told you that not one word of your prayers and not one tear from your eyes went without response? The seeds you've sown in the past are going to manifest now. You are going to receive a harvest for seeds you have forgotten you have sown.\n\nIf you had been praying before but stopped, I want to encourage you to do it again. Maybe you've fasted already. Do it again. You've already walked the floor at night, contending with God for a visitation. Do it again. I know you've cried already. Do it again. Why? Because revival didn't happen the first time.\n\nWe live in a generation whose mantra is \"If at first you don't succeed, give up and try it a different way.\" It doesn't work that way with God. God answers persistence. Some of you have given up because the revival didn't happen when you thought it should have. But what if you gave up too soon?\n\nGod is looking for people who have a persistent faith, an unwavering, unshakable faith; people who will say to God, \"You did it before; You can do it again.\" There is a revival coming to the United States of America. I've seen it. We are in a season of manifestation, and it is going to be attached to those who are willing to do whatever God requires again and again and again until something happens!\n\nLEAVING ROOM FOR RESPONSE\n\nOne of the things that I think are important in considering an atmosphere that is ripe for revival is our response to what God is doing. How we respond to God is connected to the results we receive while He is near. In 2 Kings 6 and 7 there was a great famine in Samaria, and Elisha prophesied its sudden end. But the officer assisting the king resisted the word, saying the famine couldn't end as Elisha said even if God opened the windows of heaven. Well, the next day everything Elisha prophesied came to pass. The king's officer saw it happen, but he didn't experience it, all because he didn't believe the word and act accordingly.\n\nOur response to the word of the Lord determines our experience and results. At the end of each chapter in this book I have created an invitation for you to look into your heart and see what you need to do to be in the right posture to receive what God is wanting to do in the earth in this season. I am calling it \"Respond.\"\n\n_Your response to the word of the Lord determines your results._\n\nIn this section you may be led to pray, search your heart, or take some other action. God is calling you out of your comfort zone, out of yourself and your comforts, out of the norm. Your response to Him determines how He will respond in your home, your community, the nation, and the world. It is an exchange: we pray; God responds. Our response to God determines what we will experience.\n\nWILL YOU JOIN US?\n\nOn the Friday night after our church's dramatic encounter with the Lord on May 22, 2016, I was ministering to our young adults when I realized I was entering the middle of my life and I had been praying for a move of God for almost half of my life. Now that we are seeing it manifest, I am not going to waste time doing programmed and predictable church. I am going to water that seed with my tears. I want to see a revival hit this nation, not just my church, and I don't care how much the best church-growth strategists say people won't come. I'm not moved by how many people come to Deeper or how long our services last. We stay until He's finished.\n\nThe multitudes would stay all day with Jesus because things happened in His presence. I don't know about you, but I know I need Him. I want Him. And I will not stop until I see revival across the nation and around the world. I am going to weep for it, cry out for it, pray for it, fast for it, wait for it, and live in anticipation of it. What are you willing to do to see it come?\n\nPeople will talk about you and make fun of you. That's OK. They are going to say you're weird and crazy. That's OK. You know why? Because they need revival. They are just not willing to do whatever it takes to get it. But if they stay around you long enough, they will ride on your coattails, and that's OK. Some of us want Him; all of us need Him. Revival does not need everyone for it to spread. Just a remnant crying out for God will get heaven's attention.\n\nGod assembled Deeper, bringing us together for a divine purpose, and now He has released something through our little corner warehouse church. People stand in line outside to get in. Why? Because they know it's happening. And now that it's happened with us, we are believing the fire will spread to you.\n\nIt's happening. And if it's happening anywhere, it can happen everywhere!\nCHAPTER 1\n\nWHEN HE WALKED INTO THE ROOM\n\nAt the sound of Mary's greeting, Elizabeth's child leaped within her, and Elizabeth was filled with the Holy Spirit. Elizabeth gave a glad cry and exclaimed to Mary . . . \"When I heard your greeting, the baby in my womb jumped for joy. You are blessed because you believed that the Lord would do what he said.\"\n\n\u2014LUKE 1:41\u201345\n\nI WRITE DOWN many of the things the Lord has done in our midst, just for the sake of memory. One day as I was writing some things down, the presence of the Lord began to fill the room. I was overwhelmed and unable to keep my composure as I thought about all I have seen. What is happening in this season is precious. God is giving us a moment, an opportunity, an invitation to experience and encounter Him\u2014to go all in. The Lord wants to do something greater than anything we've experienced before. Can you imagine what would happen if we gave Him room to do it?\n\nI have been on a search for something nearly my entire life. I grew up in church. I was the kid who played church at home. At first, it was the music that kept me interested because I was a very musical child. I still am very musical. In fact, most people first encountered my ministry through my worship leading. The music was so good at my church that I actually didn't want to go to children's church when I was a little kid. In those days children's church was basically something to keep us occupied until we got old enough for the adult service. That was where it was happening. Back then gospel recording artist Donald Lawrence was our minister of music. If you are familiar with his ministry, then you can see why I wanted to be with the adults.\n\nBy the time I was nine or ten years old, it wasn't only the music that kept my attention. I also started listening to the messages. I started to feel as if there was more to this church thing than good music. You may wonder how a ten-year-old would think like this, but it was the Holy Spirit planting a seed in my spirit. I had no frame of reference to know there was something more. I just knew there was.\n\nWhen I was ten years old, my mother went on a business trip to Los Angeles and came back filled with the Holy Spirit. She visited Crenshaw Christian Center and was exposed to things we had never experienced. We were Baptist, and supernatural stuff such as healing, the gifts of the Spirit, and signs and wonders wasn't really discussed at our church. Now, don't get me wrong; the gospel was preached, and people were getting saved. But there is more to our spiritual existence, and even at ten years old I was getting hungry for it.\n\nThe next time my mother went to California, she took me with her. It was my spring break, and we had a full itinerary\u2014Disneyland, Knott's Berry Farm, Universal Studios, and some other must-see places. We had great fun. But she also made sure to take me to the church where she had received the baptism of the Holy Spirit. I saw things I had never seen before. I heard things I had never heard before. I saw people encounter God and become weak and unable to stand because of the weight of His presence\u2014what some believers call being \"slain in the Spirit.\" On one side of the room people were being prayed for, and on the other side of the room people were falling down because they could feel what was happening on the other side of the room. You can imagine that as a ten-year-old I was scared to death.\n\nI heard people speaking and singing in other languages and in unknown tongues. That scared me too, but I wanted it.\n\nThen someone said, \"We're going to pray for healing, so if you're sick in your body, come up.\" At that time I dealt with really bad asthma. I had even been hospitalized once. My mother, who served on the board of the American Lung Association of Southwest Ohio, had been putting all her efforts into finding a cure for it. So she said, \"Go up there. Go.\"\n\nBut after all I had seen at the church that day\u2014people falling out for what I thought was no reason\u2014I thought, \"No way.\" Then I saw that she was crying. Through her tears she said, \"I believe that if you'll just go, you'll be healed.\" Literally weeping, she pleaded again, \"Please. Go.\"\n\nUnderstand that this was happening in the 1980s, before Christian TV was as widespread as it is today, so not many people were exposed to this kind of thing. This was unfamiliar territory for me.\n\nIt was a midweek service. I remember going up for prayer, but I don't actually remember what happened after that. I do know this: I don't suffer from asthma anymore. We went back on Sunday, and I had to go to children's church there because they didn't allow children in the main service. However, I was filled with the Holy Spirit in their children's church. After spending one week in California and attending two church services, I had experienced the presence of God in a brand-new way, and I was marked for life. Since that time when I was ten, I have not been the same.\n\nBECOMING A GOD CHASER\n\nTwo years later we found ourselves in a church where there was a full-on outpouring of the Holy Spirit. People would stand outside in line for more than an hour in sub-zero temperatures, waiting to get in. Why? Because God was there. But after a while people became comfortable and complacent. If we aren't intentional, we can easily lose our awe of God and set up camp in the land of comfort. Whenever this happens, whenever we take the move of God for granted, things will change. Eventually our church went back to normal. But the hunger and fire in me continued to burn.\n\nSeveral years later a pastor and mentor of mine came up to me and said, \"Hey, why don't you read this book?\" He was holding a copy of _The God Chasers_ by Tommy Tenney. The message and testimonies in that book moved me from saying, \"I want that,\" to being obsessed with possessing the kingdom. I longed to know and be known by God and to pursue Him until I experienced Him in a profound and tangible way. He was who I wanted.\n\nI've had moments in my life when I've seen God move, but they were fleeting. I've been in services where His presence was intense, but eventually things returned to normal, as if the amazing service didn't even happen. I've been at restaurants where a simple conversation turned into a deep awareness of God's presence because He sat down in the booth with us. But the people in those restaurants went back to normal, and their fire seemed to be extinguished. I've been with friends, preparing to eat a meal, and the blessing turned into a God encounter that left us all on the floor before God. As a worship leader, I've led worship during times when no one knew what to say or do because we all were arrested by the presence of God. I've also seen moments like those dishonored because something else was on a program.\n\nHave you experienced what I'm talking about? If so, do you still cry out? Do you still burn for God? Do you still want Him more than anything? I know I do.\n\nI have never lost my cry. We planted our church with this cry. We want to see a move of God. We want to see it grow beyond the seven hundred to eight hundred people who gather together in a corner warehouse. We want to see it spread to our city, state, nation, and world. God is moving powerfully among us. He is moving in you. Do you want more?\n\nTHE DAY MY BABY LEAPED\n\nThere are many different ways people become aware that God is planning to visit His church. The great revivalists and prophets of old had visions, dreams, and angelic visitations that let them know God was near. That's not what happened with me. I felt my baby leap.\n\nIt was 2001, a couple of years after I had read _The God Chasers_. At that time I was traveling as the music director and keyboardist for worship leader Ron Kenoly. We were ministering in South Bend, Indiana, at the church Lester Sumrall once pastored. I don't remember anything specific about the service that night, but what God did in me after our time of ministry I will remember for the rest of my life.\n\nBefore the night of worship began, I had a conversation with Lester Sumrall's grandson. I remember us talking about the move of God we wanted to see in our generation. After the night of worship was over, we were standing in the parking lot, continuing to talk about the future move of God we were so desperate to see. As we were talking, I felt the strangest sensation. I physically felt something move inside of me. I had never felt anything like it in my entire life, and honestly it startled me. What happened next startled me even more. The person I was talking with said, \"Dude, did you feel that?\"\n\n\"Feel what?\" I said, even though I had felt something. He said, \"Did your baby just leap?\" I had never felt anything like it in my life, but in that moment I knew that's what had just happened, so I had to acknowledge it.\n\nI've not birthed any babies, obviously, but there is no other way to describe the leap I felt in my body that day. The Bible indicates that this leaping is symbolic of God's confirming that He hears our prayers and is answering them.\n\nIn Luke 1 Gabriel went to Zechariah and said, \"God has heard your prayer\" (v. 13). His prayer was for a child. Zechariah and Elizabeth were barren, and they were old. But God gave them John the Baptist, who was filled with the Holy Spirit in the womb and became a forerunner of Christ.\n\nSix months into Elizabeth's pregnancy Gabriel appeared to Mary and announced that she had been chosen to carry the Messiah. Mary and Elizabeth were relatives, and a few days after her angelic visitation, Mary went to visit Elizabeth. As they greeted each other, the Bible says the baby inside Elizabeth leaped for joy. After coming in contact with Jesus, after coming in contact with what was on the inside of Mary, John leaped, as if he were saying, \"My reason for living has just arrived.\"\n\nThis is what happened to my friend and me that day. Jesus walked into our conversation and confirmed to us that we were going in the right direction toward our purpose, toward having our hunger for Him satisfied and our thirst quenched. Just as it was with Elizabeth, my baby leaped when Jesus walked in. The desire God placed in me as a ten-year-old was alive. That leaping sensation was an announcement from heaven that God was coming to dwell among us\u2014that I would see what I had been praying for.\n\nMy entire life has been lived to intentionally position myself for the day when God would walk in the room. Every service I led, every sermon I preached, every song I sang anywhere in the world was in anticipation of the moment God Himself would come and visit with us. The only thing I asked was, \"God, let me be there. Let me see it. Let me be in the room when it happens.\" And He answered my prayer on May 22, 2016.\n\nTHE DAY _I T_ HAPPENED\n\nIt was a Sunday morning, and I woke up between 5:30 and 6:00, as I did every Sunday, to pray and spend time with the Lord to prepare my spirit for the day. I always look forward to our worship gatherings because our church is filled with people who are pursuing an encounter with God. The service began like most others, but during the worship I began to feel unease in my spirit. It wasn't because something was wrong. I just had a sense that something more was supposed to happen, but I didn't know what.\n\nI preached the message God had given me, but even when the service was over, I still had that uneasy feeling, and I didn't know why. Uneasy is the best way I can describe it. It felt like a mixture of nervous energy, anticipation of the unknown, and uncertainty about what to do next. We have two services, which we call River 1 and River 2, and between the two I remember saying to my best friend Caleb, who is the associate pastor at Deeper, \"I feel like God wants to do more or say something different from what I preached, and I'm not sure what it is yet.\" I asked him to handle the transition to the second service so I could press in to God to see what He wanted to do. As we began to worship for River 2, that uneasy feeling would not leave me. In fact, it intensified. The closer it came to my time to preach, the more intense the feeling became. But I still did not have a sense of what God wanted me to say or do.\n\nThen, right before I got up to speak, it happened. The baby leaped again! I experienced the exact sensation I had felt sixteen years earlier. I hadn't experienced anything like it since that day in South Bend, but there it was. Suddenly I began to weep uncontrollably. As I stood in front of the entire congregation weeping, I thought I should attempt to explain what I'd just felt, knowing how foolish it would sound for a man to explain to everyone that his baby had just leaped. Through my tears I tried my best to tell them what had happened sixteen years earlier, and while I was talking, other people began to weep. My friend Caleb, who knew what had happened to me sixteen years before, looked at me from the front row and said, \"It's happening.\"\n\nWhat happened next is hard to fully explain. What began with me weeping uncontrollably suddenly swept through the entire room. Without prompting or a single word spoken, people began to rush to the front of the church, falling on their knees, weeping and crying out to God. It wasn't just a few; it was the majority of the room. Nobody gave an altar call. There was no human catalyst to prompt that moment. It wasn't a response to what I preached because I couldn't even speak. It wasn't a response to a song because nothing was being sung at the time. The Lord walked into the room, and everyone knew it at the same time. Everyone felt it at the same time. Everyone was aware, and no one could do anything but weep.\n\nWe stayed there and wept in the presence of God for literally hours. The service that began at 11:30 a.m. continued until 3:30 p.m. officially, but people stayed beyond that time. I remember thinking at one point while I was there crying that none of the first-time visitors would ever come back. Imagine the story afterward.\n\n\"So how was Deeper Fellowship?\"\n\n\"Well, the pastor got up and said he could feel a baby inside of him, and then he started crying, and then everyone started crying, and they kept crying for hours.\"\n\nI thought, \"What must those who have yet to surrender their lives to Christ be thinking?\" The pastor part of me was thinking those things, but the one who had been crying out for this my entire life couldn't stop the moment. Being concerned in the back of my mind about what people would think almost caused me to make a huge mistake.\n\n_Genuine encounters with God change you._\n\nI looked at the clock and decided to give those who wanted to leave a safe exit. I stood up and gave a soft dismissal, but I didn't offer an invitation for salvation. Immediately the Holy Spirit convicted me and reminded me about the Day of Pentecost in Acts chapter 2, when the promised Holy Spirit came. You can imagine the many things that must have been happening when the sound of a mighty windstorm could be heard throughout the room and what looked like tongues of fire appeared and people began to speak in other languages. Some thought they were drunk. Scripture says Peter gave prophetic context to what was happening, and that day three thousand people went from confusion to conversion.\n\nAs some people were leaving, I stood up and said, \"Wait! If you have been here this entire time and haven't left, it is because you are clearly looking for Christ, and He is here now. If you want to surrender to Him, now is your opportunity.\" To my surprise people literally began to run to the front of the church to surrender to Christ. A service with no preaching that had gone on for hours with uncontrollable weeping left numerous people saying, \"I want in on this.\"\n\nThat was the day everything changed for me and for our church. What we experienced wasn't a one-time occurrence but rather a life-altering change. It's hard to prove biblically that you've had an encounter with God without change. Genuine encounters with God change you. When I went home that day, I didn't even want to speak. I didn't want to do anything that would cause my awareness of Him and His nearness to change.\n\nI called my friend Caleb the next day because I could still tangibly feel the presence of God, and when he answered, he was still in that place as well. All we could do on the phone was weep. As I am writing this now, I've had to stop and weep. It was such a holy moment that even talking about it brings me right back to that experience.\n\nI spent that entire week essentially in a place of solitude and reverence because of our encounter. The following Friday I was scheduled to minister to our young adults, and as I went to prepare my notes, once again I began to weep uncontrollably in my home. I fell to my knees in worship with the keen awareness that God likely wasn't going to use my notes. The Lord spoke to me and said He was going to show up during the service. My only instruction was to talk until He came. After the icebreaker and the time of worship was complete, I sat on a chair in front of our young adults, knowing what the Lord had spoken to me but not knowing how or when He would come, so I just started talking.\n\nI started sharing my journey with God and part of my testimony, and then suddenly it happened. He came, and we all knew it at the same time. We began to weep and worship. This went on for hours. I remember thinking, \"There is nothing else for me to say or do.\" The very presence of the Lord was there. We laugh about it today, but I attempted to give the service over to my friend Caleb, who also happens to be the young adults pastor. Yet through his weeping\u2014the most tears you can imagine someone having\u2014he strained to say, \"I can't!\" He was literally unable to do anything but weep. I understood because that's exactly how I and everyone else in the room were feeling. We were unable to speak or sing or do anything but weep because of the presence of the Lord. The young adults service, which began at 7:30 p.m., ended at 3:30 a.m. During the eight hours we spent in the presence of the Lord, it felt as if time were suspended. We've never been the same since. We refuse to go back to the way life was before.\n\nMy pastor, Bishop Joseph Garlington, had visited our church previously and prophesied to us that divine interruptions were coming. So _it_ \u2014the outpouring of the Holy Spirit, the divine interruption, the divine invitation\u2014happened. The opening of the floodgates, the realization of an open heaven, the discovery of our cry, the reason for our tears\u2014it happened. God made an announcement to us. He confirmed that it's alive\u2014what He placed in me, in us, and in you is alive. Do you know what they call it? Revival.\n\nSIGNS OF REVIVAL\n\nSince that day I have been researching revivals that have broken out over the last four centuries. I've done this for several reasons. There's a part of our church culture in the West that causes us to want to brand a string of good services as \"revival.\" Because what God did was so sudden, I wanted to make sure we didn't prematurely call what we were seeing revival. As a result, we began to say, \"It's happening.\" And it was and is.\n\nBut I discovered something more. As of this moment every sign I have read about that marked the revivals that have taken place in the last four centuries\u2014salvation, repentance, healings\u2014has happened and is happening in our church. The only exception is we have not seen the effects on social systems in the area that marked the Welsh revival\u2014lowered crime rates, fewer police officers, increased economic status, and the like. God is big, and He's just getting started, so I won't rule that out. Yet and still, outside of the Welsh Revival every other historic revival in the last four centuries has included what we are seeing.\n\nThis is revival.\n\nYou may not have been at my church the day \"it\" happened. Most likely you weren't there, because only a few hundred of us were in the room. But the thing I've learned about revival, which I will share more about later, is that if God is doing something anywhere in the world, the heavens are ripe for Him to do it everywhere. What He is doing for us He will do for you\u2014He may have already started doing it. When God pours out His Spirit, the experience is not exclusive. Joel 2:28 says He will pour His Spirit out on all flesh. So I challenge you to allow that old desire for more of God to come alive again. Let the hope of a fresh outpouring for all nations and people come alive in you again.\n\n_The thing I've learned about revival is that if God is doing something anywhere in the world, the heavens are ripe for Him to do it everywhere. What He is doing for us He will do for you._\n\nWe never could have calculated when God would literally walk in the room and sit with us. We never could have prepared ourselves for that moment. But we were empty containers. We held on to the prayers of those who had died without seeing what they had prayed for come to pass. We hoped against hope, and hope did not disappoint. He gave us the desire of our hearts. He gave us Himself. Now we contend for the atmosphere to remain open and receptive to His presence. And He has been teaching us what it will take to see that happen.\n\nRESPOND\n\nI have a question for you: What is all this worth to you\u2014the presence, revival, God's power demonstrated, His blessings manifested, the outpouring of His Spirit? What are the hundreds of years of people crying day after day and night after night for revival, yet never seeing it come to pass, worth to you? Is it worth enough for you to go after it? I am going to level with you: pursuing God is an inconvenient, uncomfortable mission, but it is a priceless, eternal possession when you attain it, like the treasure hidden in the field. (See Matthew 13:44\u201346.) The man in that parable sold everything he owned to get enough money to buy the field\u2014that's how valuable the treasure was to him.\n\nWe're being invited to take a risk to buy the field with the buried treasure. Not everybody will go in on this one. There are a whole lot of people who have decided revival costs too much, and they are unwilling to go through what it takes and give up everything to possess it. Maybe you want to bargain as the rich young ruler did in Matthew 19:16\u201330. Maybe you want to see how you can keep everything you have and get God. Jesus says you can't.\n\nGod requires everything, and His movement in our lives depends on our level of surrender and our capacity to be filled with His Spirit. Giving up everything leaves room for the greatest amount of His power to become active in your life. What is that worth to you? If you want to see and be part of the next greatest move of God and you want it so badly that you are willing to do whatever it takes to see it, I invite you to pray this prayer with me:\n\n> _Father, give me the courage to buy the field. There's a treasure in it, and I want it. I want what You have for me, my family, my church, my city, and my nation. I release everything to You that is holding me back from seeing You move. The treasure I seek is You, and nothing else will satisfy me. Come, Lord Jesus. Amen._\nCHAPTER 2\n\nHOW'S YOUR POSTURE?\n\nSINCE GOD WALKED in, all we've wanted to do is maintain the right position to keep Him here. We call it posture. Posture is the attitude\u2014mental or spiritual\u2014people develop, usually in response to how external forces affect their internal world.\n\nWhen I was growing up, my mom, my aunts, church mothers, and other elders were always telling us kids, \"Fix your posture.\" They knew the deal. Posture is attitude. You know grown folks don't like when young people have attitudes. Posture is about disposition.\n\nWhat is your posture before God? What is your spiritual attitude? My posture before the Lord is waiting. My posture before the Lord is honor. My posture before the Lord is crying out to Him. My posture before the Lord is making room for Him. All these things communicate to God that we respond to His invitation with an eternal yes and eagerly anticipate what He wants to release in the earth.\n\nNow, before we go further, there's something I need to clarify: posture is not about working for the blessing of God or working to make something happen. We don't have to do that. God shows up in encounters of His choosing, not ours. By definition an encounter is a surprise or unexpected meeting. In other words, it's planned by someone else. All we can do is continue to position ourselves for encounters with God. If we do try to make something happen, we'll look up one day and find ourselves talking about what _was_ instead of what _is_ , trying to carry something emotionally and in our own strength because God is not breathing on it. The Lord is inviting us to a place where we can partner with Him and what He wants to do in our homes, our churches and cities, our nation, and the world. He is just looking for people who have the right posture.\n\nThere are nine postures the Lord revealed to us in the early days of His movement. Many of these will be discussed throughout the book, so I will introduce them only briefly. But they will be here for you to see all in one place, and you can come back to them as many times as you need to as the Lord increases your hunger for Him.\n\n1. WAITING\n\nIn Acts 1:4\u20135 Jesus told His followers, \"Do not leave Jerusalem until the Father sends you the gift he promised, as I told you before. John baptized with water, but in just a few days you will be baptized with the Holy Spirit.\" Most teach from the perspective that there were only 120 people in this group, but a further look tells us there were actually five hundred. (See Matthew 28:18\u201320; 1 Corinthians 15:6.) At the time when the Holy Spirit came in like a rushing wind (Acts 2:2), there were only the 120. What happened to the other 380?\n\n_Some of us rebuke the devil when it's not the devil at all. It's God using that circumstance to cause us to seek Him._\n\nAnything could have caused them to change their posture from the \"stay and wait\" position in which Jesus asked them to remain. Those who stayed received the promise. They refused to settle for a secondhand experience. They waited until they saw what had been promised. They took the posture of \"God, I'm not leaving until . . .\"\n\n2. DESPERATION\n\nThis posture can show up in two ways: divine desperation and desperation by frustration. The difference between divine desperation and desperation by frustration is the recognition of one's need for God. Each of these postures of desperation will lead you to a different place. Desperation by frustration is birthed out of exasperation or frustration and will lead you to do anything in search of relief, even if what you do is without God. This posture is best seen in the life of King Saul. (See 1 Samuel 28.) After the prophet Samuel died, Saul was no longer able to hear from God since he had never developed a prophetic relationship with Him. The Philistine army was forming against Israel, and Saul was desperate for a word, so he sought out the witch of Endor. His actions led to the kingdom being stripped from him and his murder.\n\nThe posture of divine desperation, on the other hand, leads you to a fuller awareness of your need for God. This desperation comes from God. He places longings, and sometimes circumstances or situations, in our lives that cause us to seek Him. There are some of us who rebuke the devil when it's not the devil at all. It's God using that circumstance to cause us to seek Him. In Jeremiah 29:4\u201312 we see where the Lord orchestrated the events surrounding Israel's seventy-year Babylonian captivity to cause them to seek and desire Him once again. He led them into a desperate circumstance, but He also promised that when they recognized their need for Him again, He would hear their prayers and deliver them.\n\n3. CRYING\n\nGod is looking for a people who will literally cry out to Him and allow their tears to water their spiritual seed so the harvest can come. The story of Jacob paints a good picture of how a posture of crying will position you for a life-altering encounter with God.\n\nJacob's story is familiar to many people. He's the one who cried out, \"I won't let go until You bless me!\" He's the one who wrestled with God and won. But how can a man wrestle pound for pound with God and win? Let's think about that. We can't really wrestle with God and win. I mean, God just took His finger, touched Jacob lightly on the hip, and maimed him for life. We can't wrestle God. The truth is God didn't want Jacob to let Him go. So then how did Jacob get what He wanted? The answer is not found in Genesis 32, where we read of how Jacob wrestled with God. It's found in Hosea 12:4:\n\n> Yes, he wrestled with the angel and won. He wept and pleaded for a blessing from him. There at Bethel he met God face to face, and God spoke to him.\n\nJacob wrestled with the angel and won because of his tears. Days after the encounter I called Pastor Caleb, and as we talked on the phone, we both began to weep. We could barely speak because the presence of God was overwhelming. That Friday as I prepared for the young adults service, I sat in my room, weeping for two hours. Then that night during the service, God did it again, and the Deeper young adults stayed until 2:30 in the morning, weeping. If you recall the times when you felt God so close, didn't you weep too?\n\nWhen this happens, we are contending with God. Crying out to God shows that we are no longer trying to figure things out on our own. The only way we're going to see what we've been praying for is if we weep. If we will be a people who will weep, who will cry out to God, I guarantee that we will be at the epicenter of the next revival.\n\n4. PRAYER\n\nPrayer is ultimately a declaration of our dependence on God. It's a dialogue between God and man. It's the reserved seat in the heavenly realms from which we affect things in the earth. Therefore, it is an ideal posture for one who is praying for God to move upon the earth.\n\nWhat we must be careful of is to not allow our prayers and declarations to become self-centered. Sometimes in the church our beliefs are self-centered, and what we pray about, think about, talk about, and believe God for usually has something to do with us.\n\nBut this may be where another posture overlaps: often desperation leads to intercession, where you begin to pray for those who are unaware of their need for God. Your prayers change from \"God, _I_ need You\" to \"God, _we_ need You.\" Desperate people don't just cry out for themselves. They begin to look at the hopelessness and the state of the world around them, and they begin to say, \"God, we need revival.\"\n\nPrayer is at the center of much of what we do at Deeper. We have a prayer call at 7:00 a.m. and another one at 9:00 p.m. Monday through Friday. Right now there are more than 150 of us who get on that call and assume the posture of prayer. Prayer helps us keep momentum. We all stand on the prayers of past generations. The revivals that break out are a result of past prayers. Our posture of prayer now will keep those prayers alive.\n\n_Desperate people don't just cry out for themselves. They begin to look at the hopelessness and the state of the world around them, and they begin to say, \"God, we need revival.\"_\n\nWhen you have experienced a special visitation from God, in order to see that moment turn into something that will outlast you, you must continue to do what got you there. In a later chapter we are going to talk about how to create the window through which God's presence will shower down upon us continually. We will talk about how the Bible instructs us to pray for more rain in the season of rain. Do not let it dry up. Keep praying for God to invade every space and fill us with His presence.\n\nDuring this outpouring the gifts of prophecy, words of knowledge, and words of wisdom have been alive and active in our midst. We fully embrace these gifts and can directly testify to how the Lord has moved time and time again in incredibly accurate ways as a result of these things.\n\nFirst Corinthians 14:24\u201325 says: \"But if all prophesy and there comes in one who does not believe or one unlearned, he is convinced by all and judged by all. Thus the secrets of his heart are revealed. And so falling down on his face, he will worship God and report that God is truly among you\" (MEV). We saw this happen among us one Sunday morning.\n\nI stood up to preach, but the spirit of prophecy was very strong, and many words of knowledge were given and released that day. At one point I specifically heard the Lord say, \"Watch for the day of the Walmart miracle.\" So I told the church, \"The Lord says, 'Watch for the day of the Walmart miracle.'\" Then I expounded on the word and said that someone was going to go into a Walmart and have a divine encounter.\n\nAs I was speaking, a tall, slender gentleman whom I had not previously seen at our church came forward looking as if he had seen a ghost. He had been in Walmart that morning and had encountered one of our faithful members, who invited him and his girlfriend to come to church that morning. He'd had an encounter with God during the service and was weeping uncontrollably.\n\nCaleb, our young adults pastor, then began to prophesy to him, revealing what his life was like in the past and the word of the Lord for his future. This young man surrendered his life to Christ in that moment\u2014and it just so happens that he and his girlfriend were both Muslims. First Corinthians 14:25 happened right then and there. Two people literally fell on their faces, worshipped God, and said by their actions, \"God is truly among you.\" It's happening!\n\n5. WORSHIP\n\nWe are in a generation obsessed with gaining knowledge about worship. Today there are likely more books about worship and worship songs than at any other time in history. We know the lingo, we know the stage placement, our musicianship is impeccable, but where are our hearts? Our external expressions in worship should reflect our internal posture; otherwise we are not giving God worship. We're only performing. The acts of worship without repentance\u2014another posture\u2014merely become rituals.\n\nJesus said it to the woman at the well: there's coming a time when God will be looking for a people who will be honest with Him. He is not looking for ritual and form. He is looking for the heart.\n\nOur true posture in worship is tested when our humanity encounters God's divinity. Worship acknowledges one greater than we are. Our posture during and after encounters with God reveals much about our hearts. When we don't worship from a place of self-denial, we can become prideful and think we can come up with some formula to invoke God's presence. That's not how worship works. Worship is a pathway, an invitation from God to be positioned for encounters with Him.\n\nEncounters are of His choosing, not ours, but worship positions us for them. God is not in our presence; we are in His presence. This is key to our understanding of the posture of worship. It is an honor to be in the presence of the Lord. When the prophet Isaiah saw the Lord in the year King Uzziah died, he was completely undone by what he experienced. The realization of humanity encountering divinity caused him to say, \"Woe is me! For I am undone\" (Isa. 6:5, MEV). He immediately recognized his frailty in relation to the holiness of God.\n\nThe amazing thing is that God allowed him to experience that vision in order to invite him into His plan. It was a God-given encounter that allowed Isaiah to glimpse the worship happening in heaven. God exposed him to His holiness and majesty and confirmed Isaiah's calling. The posture of worship does the same for us. It exposes us to the divine, causes us to recognize our own human frailty in relation to His holiness, and brings us into a place of revelation.\n\nWe must reconnect our external expressions of worship with what's really happening in our hearts. Repentance helps us get there. If we cannot get our hearts and lives in position to give God the worship He is worthy of, then we will miss out on one of the main vehicles of access to Him and His presence.\n\n6. DELIBERATE ACTION\n\nThe story of the woman with the issue of blood is a perfect example of how the posture of deliberate action leads to an encounter with God that changes everything. The Bible says in Luke 8:46 that Jesus noticed when someone \"deliberately touched\" Him. She was not casual about what she needed from God. She had tried everything to be healed. Now penniless and desperate, she heard that Jesus was passing by. She considered her options, she considered her unclean status, and she made a deliberate choice to risk it all and grab hold of His garment.\n\nThere are no accidental moves of God. If you are going to experience God, it's not going to happen by being random. If you want a greater demonstration of the power of God in your life, you must deliberately pursue Him. Make the intentional effort to risk it all\u2014reach out and grab hold.\n\n7. FAITH\n\nThe Lord asks, \"When the Son of Man comes, will he find faith on the earth?\" (Luke 18:8, NIV). Or will He return to find a bunch of frustrated believers? Faith is a posture that allows the faithful to endure the tension between present afflictions and future deliverance.\n\nGod is looking for people who have a persistent faith\u2014an unwavering, unshakable faith. He is looking for people who, in the midst of waiting, will say to God, \"You've done it before; You will do it again.\"\n\nFaith gives us confidence that what we hope for will happen. It gives us assurance about the things we cannot see. Faith makes us pleasing to God. Faith gives us permission to come to God. God rewards those who sincerely seek Him. Faith makes us whole.\n\nI pray that as you read this book, you will feel a faith arise in you that you haven't felt in a long time. I declare that your capacity to believe God again is being restored. May your faith come to new life as you begin to see what you've been hoping for come to pass.\n\n8. OBEDIENCE\n\nWorship, prayer, sacrifice, giving\u2014what does any of it mean if we won't obey God? Obedience is the posture that God wants most. As a matter of fact, obedience to Him helps us know what He wants us to do with our worship, prayer, sacrifice, and giving. Through our simple obedience to God in whatever He asks, especially the things that seem foolish to our limited understanding, we unlock the deep things, the mysteries we desire to understand. Revival is connected to simple obedience.\n\nKing Saul refused to obey the Lord. He refused to assume the posture of obedience, and it cost him dearly. He never saw the things he'd once hoped for happen. For him, _it_ never happened. When Saul was coming up against an enemy of the armies of the living God (Israel), God told him to kill and destroy everything. No matter how good something looked, no matter what kind of agreement they were offered, no matter what kind of purse they could receive, they were to \"completely destroy the entire Amalekite nation\u2014men, women, children, babies, cattle, sheep, goats, camels, and donkeys\" (1 Sam. 15:3).\n\nBut Saul spared the enemy King Agag. He also let his soldiers keep the spoils. When God questioned Saul's actions through the prophet Samuel, Saul lied and made it seem as if the spoils were kept to honor God. Samuel said to him, \"What is more pleasing to the LORD: your burnt offerings and sacrifices or your obedience to his voice? Listen! Obedience is better than sacrifice, and submission is better than offering the fat of rams. . . . Because you have rejected the command of the LORD, he has rejected you as king\" (1 Sam. 15:22\u201323).\n\nObedience is an important posture to maintain. It's a posture of surrender and submission to the will and ways of God. It is a demonstration of our eternal yes to God's invitation into His presence. We cannot have God and be disobedient to Him. Disobedience without repentance puts us at risk of losing everything God has for us. I'm not willing to lose Him. I will take whatever posture I have to take to maintain His presence in my life. How about you?\n\n9. REPENTANCE\n\nRepentance is the pathway to revival. It keeps the avenue between us and God open for whenever He decides to come and rest upon our lives. If we are unwilling to let go of the things that block what God wants, we will never see His power on display in the earth. It is God's gift to us that allows us to stay in His presence. Repentance is a hard practice for this generation to latch on to. We don't like to feel bad about anything. But if we look at it from God's perspective, we will see it as His desire to draw near to us. Repentance draws Him close. And when His holiness enters the room and comes in contact with our unholiness, we think we feel guilt, but it's actually His light illuminating our darkness. All we have to do is get in a posture of repentance. He is faithful and just to forgive us our sins, cleanse us of unrighteousness, and dwell among us. We do not have to run away. He has made a way for us to boldly enter His presence.\n\nGod wants to do something great. If you are hungry to see revival, you are ready to do whatever God asks of you to be the vessel through which His glory comes.\n\nRESPOND\n\nIt's easy to get excited about revival, but are you willing to assume the right posture to lay hold of it? Are you willing to deliberately reach out and touch Jesus as the woman with the issue of blood did\u2014to touch Him in a way that pulls on His power? Everyone claims to want Him, but will you pray until you touch Him? Will you worship until you touch Him? Will you wait until you touch Him? Will you risk rejection to touch Him? Will you accept ridicule to touch Him?\n\nWe're not going to accidentally enter into revival. We'll see it when we're willing to assume the posture that says, \"I want You and You alone, and I won't stop praying, I won't stop crying, I won't stop worshipping until You come. And then I'll keep on worshipping and keep on praying and keep on crying because I'm desperate. I want to be a vessel for Your glory. Ordinary won't do. I want to be part of what You're doing in the earth. Revival is here, and I want to be a part of it!\" When that is our posture, God will respond.\nCHAPTER 3\n\nHE HEARS YOUR CRY\n\nI love the LORD because he hears my voice and my prayer for mercy. Because he bends down to listen, I will pray as long as I have breath!\n\n\u2014PSALM 116:1\u20132\n\nI DON'T KNOW if you are paying attention to the current state of the world right now, but it's clear we need a revival. So we cry out for it day and night. At our church we've embraced the posture and disposition of crying out. We cry and we cry and we cry some more. And some may begin to wonder, \"OK, are you going to stop?\" The answer is no! We will not stop until we see what we're praying for manifest all over the earth.\n\nThis is why I love Psalm 116:1\u20132. It is something that I can understand now as a father of four. I couldn't have understood it ten years ago. When you have children, you recognize, discern, and decipher the cries of your children, especially when they are infants and can't talk. Infants have different cries that mean different things. Some cries are urgent and cause us to come running. Other cries bellow, and we know we have a couple of minutes before we're really needed. There's a cry that says, \"I'm hungry,\" \"I'm thirsty,\" \"I need to be changed\" (you can smell that one too), \"I need to be held,\" \"I'm bored,\" or \"I'm uncomfortable.\" The one I recognize but don't quite understand is the \"I'm sleepy\" cry. Why do infants cry when they're sleepy? Why don't they just go to sleep? That's what I do. We learn how to interpret our children's cries, and we respond accordingly.\n\nOur Father does the same thing with us. It amazes me. Not only can He interpret our cries, but He remembers and answers them. And He can distinguish our cries from everyone else's in the world. This is also a pretty remarkable phenomenon. If you're a parent, you can understand this on a smaller scale: your child can be with fifty other children, four of them can be crying, and you can tell which one is yours. There's something about your own child's cries that directly connects to your heart.\n\nI'll never forget being on an airplane and hearing a child cry. The cry kind of sounded like one of my children's. I knew my child was not on the plane, but I heard a cry that sounded like my daughter's. I kept looking back, thinking, \"Is my family on this flight trying to surprise me?\"\n\n_Just as we are connected to the cries of our children, our heavenly Father is connected to ours._\n\nJust as we are connected to the cries of our children, it's important that we understand that our heavenly Father is connected to ours. Even as we cry out to Him during corporate worship with other believers or alone in the privacy of our prayer closets, the Lord hears us, distinguishes our cries from all others, and answers them. I want to encourage you to continue to cry out and not be silent. The Lord responds to our cries. Just as our children's cries vary in degrees of urgency, so do ours.\n\nTHE CONSISTENT CRY\n\nIn Exodus 1 the Israelites were slaves in Egypt. The Egyptians were brutal taskmasters and worked the people of Israel without mercy. For four hundred years they lived under the lash of oppression, just as the Lord had prophesied to their forefather Abraham. And they cried out to God for deliverance:\n\n> Years passed, and the king of Egypt died. But the Israelites _continued_ to groan under their burden of slavery. They cried out for help, and their cry rose up to God. God heard their groaning, and he remembered his covenant promise to Abraham, Isaac, and Jacob.\n> \n> \u2014EXODUS 2:22\u201323, EMPHASIS ADDED\n\nThe word _continued_ in this verse also points to an unyielding, consistent groan. Day after day, month after month, and year after year they cried out to God for deliverance. They never stopped crying, even though they didn't see their answer right away, even though it seemed as if God was hearing. And they continued to cry out to God until their cries met up with His set time for their deliverance. We call this a _kairos_ moment.\n\nIn all of their distress they may not have been able to keep a grip on the fact that God was fulfilling His plan through them, that they were in this season as part of His sovereign design. God had made a covenant promise to their forefather, and He knew the time had come for Him to act. So God raised up a deliverer, Moses, whom He found in the middle of the desert on Mount Sinai.\n\nWe need to consistently and continually cry out until our cries meet up with God's set time. There are moments in your life when the will of God and your desires are on a collision course. And in an explosive meeting by God's design\u2014His set time for your life or your family\u2014breakthrough or deliverance manifests. The cry could have started with your grandparents. You may not know that, so don't stop crying. You never know when the moment will be when your cry intersects with God's set time.\n\nAs a generation, we are moving into a set time when God intends to send revival to the whole earth. Many of us can sense it, but there's also this sense that it is tarrying. _Tarry_ is an old English word that means to wait or delay. So the question is, Why is revival being delayed? One of the reasons is that there is no cry. Cries are connected to an intense desire for something. And when there's no cry going up to heaven for revival, it may appear to God that perhaps we don't really want it.\n\nMaybe there's a feeling that revival is no longer on the table. Maybe it's been such a long time coming that people have stopped crying out. But the Israelites cried out for four hundred years because they wanted what God had promised. Don't lose your cry just because it seems the answer is taking a long time.\n\nMaybe we need to start spreading the word again that revival is available, that outpouring is available, that deliverance is available. Maybe we need to start declaring that the year of God's favor is upon us. If people knew that these things were available, that they will surely come if we cry out, maybe they would cry out for them again.\n\nNo one likes being in bondage, so there are cries beginning to rise up from the earth. And God is beginning to respond, \"I have certainly seen the oppression of my people....I am aware of their suffering. I [will] come down [and] rescue them...and lead them...into their own fertile and spacious land\" (Exod. 3:7\u20138).\n\nI have news for you: God hears the consistent cry. If you want to see His promise of an outpouring manifest in your lifetime, don't stop crying out!\n\nTHE PERSISTENT CRY\n\nThere is another kind of cry that is directly related to the consistent cry, and that is the persistent cry. You may know the story found in Luke 18 of the persistent widow. Usually with stories like this we don't get the moral (or the point) until the end, but Luke, the one telling us what happened, didn't care about spoilers this day. He started off with \"Jesus told his disciples a story to show that they should always pray and never give up\" (v. 1).\n\nJust that much will mess with some of our theology. In certain movements it is taught that praying for something more than once shows a lack of faith. That's not true, and you may know somebody who needs to be set free from that. If this is the belief, then let's see how it plays out. Somebody comes to you and says, \"Hey, I have this situation, and I really need you to agree with me in prayer.\" With this as your belief, you say, \"It's already done.\" According to Scripture, because of what Jesus Christ has done, it is already done, yes. But you still have to pray about it. You have access to it, but you don't have possession of it without prayer.\n\nSo yes, Christ's work is already done, but your prayer is not. So if your friends or loved ones ask you to pray for them, don't just say, \"It's already done,\" and keep it moving, because oftentimes that's the manifestation of \"I'm not praying for you.\" When we're praying for someone who is going through a trial, we need to keep interceding until the person receives justice. Be persistent! This is what Jesus was saying to His disciples, and this is what He is saying to us.\n\n_Cry out until you see deliverance!_\n\nThis widow kept coming to this godless, coldhearted judge, pleading for justice in a dispute she had had with an enemy. She came day after day, and day after day he ignored her. Finally he couldn't take it anymore, and he ruled justly in her favor.\n\n> Then the Lord said, \"Learn a lesson from this unjust judge. Even he rendered a just decision in the end. So don't you think God will surely give justice to his chosen people who cry out to him day and night? Will he keep putting them off? I tell you, he will grant justice to them quickly! But when the Son of Man returns, how many will he find on the earth who have faith?\"\n> \n> \u2014LUKE 18:6\u20138\n\nGod is inquiring: Will He find people who will have patience and trust, knowing He's waiting for the right time to intervene and act on their behalf? Or will He find people who in their frustration stopped believing?\n\nHe is saying in this moment that we must cry out to Him day and night. If you stop crying out, you are signifying that you are frustrated with Him. You're frustrated that He's not answering on your time line. Like this widow did with the judge, cry out to God day and night. Don't stop crying out. Don't stop asking. Don't stop believing that God will answer. He's waiting on the exact right time to intervene on your behalf. He answers the persistent cry.\n\nShe was unable to make a fist\u2014completely unable to ball her hand tightly and make a fist. She is a schoolteacher, and her students used to make fun of her for constantly dropping chalk and pens and pencils. Her story of healing is amazing. She had her eyes closed during worship when she felt someone grab her hand. Assuming it was her husband, who was standing on that side of her, she initially didn't think anything of it. However, while still feeling the sensation that someone was holding her hand, she opened her eyes and discovered that her husband had his hands lifted and his eyes closed in worship. She then realized that she was able to close her hand and make a fist, so she grabbed her husband's hand. All of this was taking place during worship. She grabbed his hand and squeezed it tightly\u2014tightly enough to make him say, \"Ow!\" Then, as she squeezed his hand, he realized, \"Oh my goodness, she is squeezing my hand.\" At the same time, she realized, \"Oh my goodness, I am squeezing his hand!\" I placed a picture of her closed hand on my Instagram page and wrote, \"To you it's just a random fist, but to her it's something that she couldn't do for more than two years!\" In one supernatural moment she was healed in the presence of the Lord. It's happening!\n\nTHE MISUNDERSTOOD CRY\n\nYes, there is a misunderstood cry. In 1 Samuel 1 we discover that Hannah cried out for a child in a way that the priest Eli misunderstood. The Bible says,\n\n> As she was praying to the LORD, Eli watched her. Seeing her lips moving but hearing no sound, he thought she had been drinking. \"Must you come here drunk?\" he demanded. \"Throw away your wine!\"\n\n\"Oh no, sir!\" she replied. \"I haven't been drinking wine or anything stronger. But I am very discouraged, and I was pouring out my heart to the LORD.\"\n\n\u20141 SAMUEL 1:12\u201315\n\nAre you one whose walk with God has been misunderstood? Have people misunderstood why you live the way you live? People don't understand why you do the things you do. They don't understand why you keep crying out to God the way you do. People can't figure you out. They don't understand why you're not happy with things the way they are. Why are you always asking for more?\n\nThis is how Hannah's husband responded to her:\n\n> \"Why are you crying, Hannah?\" Elkanah would ask. \"Why aren't you eating? Why be downhearted just because you have no children? You have me\u2014isn't that better than having ten sons?\"\n> \n> \u20141 SAMUEL 1:8\n\nThe only one who understood her cry was God. No one but God understood what she longed for. Her cry was for something that man could not give.\n\nIf this story resonates with you, then trust that you are not alone. As it was with Hannah, there's something on the inside of you that longs for more. You were created to birth something in the earth. This season of barrenness has caused you to not want to eat or be around other people. What God is stirring in you is something that cannot be fulfilled by man.\n\n**The misunderstood cry is really the unselfish cry.**\n\nAnother thing about the misunderstood cry is that it's not even for you. The whole reason Hannah was crying was not even to benefit herself. Her cry was ultimately an unselfish cry. Her desire was to be part of what God was doing in the earth during that time. She wanted to birth someone who would faithfully serve the Lord\u2014and she wanted to give what God birthed through her right back to Him. That's exactly what she did. God gave her a son, whom she named Samuel, and he became one of the greatest prophets in Israel's history.\n\n**God understands your cry.**\n\nYou may be crying out in the house of the Lord, going to the altar even when there's no invitation or altar call. People don't understand why you're crying and praying. They don't understand that there's something on the inside of you that longs for more. I believe that God gave me a revelation about this cry especially for you whose cries are misunderstood. He wants you to know that He understands your cries and He will answer. You long to birth something in the earth that gives glory to God.\n\n_You were created to birth something in the earth._\n\nEven though you're misunderstood, keep crying. Even though people talk about you, keep crying. Even though people don't understand your posture before the Lord, keep crying out to God. Don't let any accusations of being religiously zealous stop you from crying out to God. They may not understand now, but the thing you're going to birth will shake the nations. Keep crying.\n\nTHE DESPERATE CRY\n\nDesperation has a sound attached to it and is not quiet. In fact, you could say that desperation is a misunderstood cry as well. The story of Bartimaeus in Mark 10:46\u201352, which is very dear to me and to our church, helps us understand the desperate cry. I wrote and recorded a song called \"I Don't Mind Waiting\" on my _Sounds of Revival_ CD, and in it I exhorted from this particular passage about the desperate cry. This is a cry that recognizes that when Jesus is coming near, that is not the time to be quiet because no one else can do for you what He can. And so when those who are desperate sense His presence is near, you will have to excuse them because they will likely get loud. Their faith is coming alive!\n\nIf we can imagine the scene from Bartimaeus's perspective, it makes perfect sense. He had been hearing about what Jesus could do. He'd been hearing about what happened to people when they got in His presence. Though he was blind and could only hear and feel the growing anticipation of the crowd, he was not going to let that moment or a season of manifestation pass Him by. The closer Jesus got, the louder the crowd got. Somehow he had to get the Lord's attention because He was the only One who could give him sight. His desperation grew by the minute, and just when the moment reached its set time, he cried out, \"Son of David, have mercy upon me!\"\n\nMany told him to be quiet, but the closer Jesus came, the louder Bartimaeus cried out. Oftentimes desperate people expose non-desperate people. The ones telling him to be quiet couldn't do anything for him. They knew him as a beggar. They could give him pocket change but not real change. Only Jesus could do that, which is why Bartimaeus had to ignore them. When you're desperate for an encounter with God, there will be times when you will have to ignore those around you. When you recognize that Jesus is the only One who can change you, the opinion of others ceases to matter. Sometimes the very reason you're being told to tone down your worship, prayer, and desperation is because it causes others to become uncomfortable with their lack of desperation. I don't know about you, but I refuse to remain quiet in this moment when I'm aware that Jesus is passing by. I refuse to miss this moment. Bartimaeus got Jesus's attention. I want the same thing to happen when I cry out.\n\nJesus heard him and asked, \"What do you want Me to do for you?\n\nBartimaeus did not stutter: \"I want to see!\"\n\nJesus said, \"Go, for your faith has healed you.\"\n\nBartimaeus was crying out because he was desperate. But Bartimaeus was also crying out because he had faith. He had been desperate the entire time. He had been desperate his entire life. And when he heard that Jesus was near, desperation got a voice.\n\nPerhaps you've been known to be reserved, but as Jesus comes closer and the visitation of God comes, that's not the time to keep your reputation of being reserved. You may have never shouted in your life, but that won't matter. All that will matter is that Jesus is near and you need to get His attention. Let desperation for the presence of God arise in you until it finds a voice. I am sure Bartimaeus's cry was so loud that it made the people around him uncomfortable. But he wasn't about to miss a once-in-a-lifetime opportunity with the only person who could give him what he needed. He was not going to let pride, embarrassment, a reputation of composure, or anything else get in the way of his miracle.\n\nTHE CRY OF FAITH\n\nThe secret power of the cry of faith is that it allows you to reach into the future and pull into your present the grace and provision God has promised to release at a later time. God does not live within the constraints of time as we do. He is omnipresent. He's everywhere at the same time. And because there is no sequential order in eternity, everything in Him is accessible consistently, all the time, at the same time, and outside of time. Time is a servant to faith. The Gentile woman in Matthew 15:21\u201328 had a supernatural awareness of this principle, and she used it to claim deliverance for her demonized daughter.\n\n_When Jesus is near, desperation gets a voice._\n\nWhen the woman first approached Jesus, asking Him to heal her demon-possessed daughter, He ignored her. It seems strange to us that Jesus seemed to be giving someone who came to Him in faith and sincerity the cold shoulder. It seems so out of character for Him. There are theories on what message He was sending in that moment. But that's not our focus.\n\nThe woman came to Him in faith, recognizing that He was the Son of David. We also see in verse 27 that she acknowledged to whom He was sent\u2014and it wasn't her and her people. But her response is one of faith that communicates a future truth. When she said, \"Even dogs are allowed to eat the scraps that fall beneath their masters' table,\" she was speaking of a time yet to come when the Gentiles would have access to the promises of God through faith. She was declaring to Him that despite her present unworthiness, she needed to make a demand on a future provision and make it apply to her present need.\n\nShe couldn't wait because her daughter was severely possessed right then, and her faith told her that Jesus had the capacity to answer her cry right then. Even though her initial question was met with silence, she did not let that keep her from crying out, believing He was the Son of God. She believed that what was promised in the future could be made manifest to her right then.\n\nSometimes we get into the presence of God, knowing we don't deserve what we are asking for. But as we get in a posture of worship, acknowledging God for who He is, our worship begins to build our faith. And by faith we are able to cry out, and what may not be fully manifested to us right now we reach into the future and pull into our present because it is consistently available in God.\n\nYou may feel as if God has been silent when you have cried out to Him, but keep crying in faith. Even when you don't get an answer right away, throw yourself down before God and say, \"I know who You are, I know to whom You've been sent, I know I have access, and I know You can do this for me.\"\n\nTHE PROPHETIC CRY\n\nSome have a cry that I call the prophetic cry. It can be a combination of the misunderstood and desperate cries, and it comes because what you've seen and heard in the spirit has not yet manifested as a reality on the earth. You've seen something, and you've heard something in the spirit during your personal times of encountering God in worship and prayer. But things here don't yet look like what you've seen or experienced. This is what happened to the prophet Isaiah.\n\nIn Isaiah 6 the prophet describes a heavenly encounter.\n\n> In the year that King Uzziah died I saw the Lord sitting on a throne, high and lifted up, and His train filled the temple. Above it stood the seraphim. Each one had six wings. With two he covered his face, and with two he covered his feet, and with two he flew. One cried to another and said: \"Holy, holy, holy, is the LORD of Hosts; the whole earth is full of His glory.\" The posts of the door moved at the voice of him who cried, and the house was filled with smoke. And I said: \"Woe is me! For I am undone because I am a man of unclean lips, and I dwell in the midst of a people of unclean lips. For my eyes have seen the King, the LORD of Hosts.\" Then one of the seraphim flew to me with a live coal which he had taken with the tongs from off the altar in his hand. And he laid it on my mouth, and said, \"This has touched your lips, and your iniquity is taken away, and your sin purged.\" Also I heard the voice of the Lord saying, \"Whom shall I send, and who will go for us?\" Then I said, \"Here am I. Send me.\"\n> \n> \u2014ISAIAH 6:1\u20138, MEV\n\nIsaiah had a vision of an open heaven and received a glimpse of the eternal, a glimpse of the divine, a glimpse of the majesty of God. He was given a glimpse of the beauty of holiness.\n\nBy design an open heaven can be an invitation by God meant to awaken awe, awaken wonder, awaken desire, and speak to the deep longing of our hearts. For Isaiah it was an invitation to embrace his prophetic assignment.\n\nOne can only imagine what life was like for Isaiah after his encounter. When you have a genuine encounter with God, not only does it change you, but you can never forget it. It will shape how you think, act, and live. Isaiah had a personal encounter with God that changed the trajectory of his life.\n\nIn Isaiah 64 he writes these words:\n\n> Oh, that you would burst from the heavens and come down! How the mountains would quake in your presence! As fire causes wood to burn and water to boil, your coming would make the nations tremble. Then your enemies would learn the reason for your fame!\n> \n> \u2014ISAIAH 64:1\u20132\n> \n> Oh, that you would rip open the heavens and descend, make the mountains shudder at your presence\u2014as when a forest catches fire, as when fire makes a pot to boil\u2014to shock your enemies into facing you, make the nations shake in their boots!\n> \n> \u2014ISAIAH 64:1\u20132, THE MESSAGE\n> \n> Oh, that You would rend the heavens! That You would come down! That the mountains might shake at Your presence\u2014as fire burns brushwood, as fire causes water to boil\u2014to make Your name known to Your adversaries, that the nations may tremble at Your presence!\n> \n> \u2014ISAIAH 64:1\u20132, NKJV\n\nWe are reading the words of a person who is still consumed by what he saw the year King Uzziah died. He is still consumed by his encounter when he says, \"Oh, that You would burst from the heavens and come down. Rend the heavens. Rip open the heavens!\" This is a cry that says, \"I've seen something that makes my insides feel like they are going to burst! I've seen something that causes me to yearn for more! To cry out for more! To search for more!\" He's saying, \"I long for the day when You will give us more than a glimpse of heaven. I long for the day when You will give us more than a glimpse of eternity. I long for the day when everyone will see what I've seen!\"\n\nIsaiah saw something during his personal encounter that wasn't yet happening around him, so he cried out for it. In Isaiah chapter 6 he said, \"I saw the Lord.\" Yet there was something in him that said, \"I long for the day when _all_ will see the Lord.\"\n\nYou could sense Isaiah's passion as he was saying, \"I pray and cry out because of what I saw _and_ what I heard! I saw the Lord seated high, and I saw the cherubim. But it's what I heard them say that stirs me as much as what I saw! I saw the Lord. He was sitting on a lofty throne, and the train of His robe filled the temple. Attending Him were mighty seraphim, each having six wings. With two wings they covered their faces, with two they covered their feet, and with two they flew. They were calling out to each other, 'Holy, holy, holy is the LORD of heaven's armies! The whole earth is filled with His glory!' They said, 'Holy is the Lord. The whole _earth_ is full of His glory.' If they had said, 'The highest heaven is full of His glory,' my singular cry would be to go there. But the fact that the earth is full of the glory says the glory is coming, so I cry, 'Oh, that You would rend the heavens, burst out, rip open the heavens!'\"\n\nIsaiah had a cry: \"Oh, that others could see what I've seen, feel what I've felt, experience what I've experienced.\" Those who have had genuine encounters with God are crying out that everyone would have the kind of transforming encounters they've had. It seems the feeling is if others would have the same encounter, they wouldn't think our passion was weird or strange. They likely wouldn't think this pursuit of God was too much. They would understand this passion. Isaiah was saying, \"Oh, God, let others in on it! I can't keep it to myself.\" He had a prophetic cry.\n\nTHE CRY OF CREATION AND THE SPIRIT\n\nThere is another cry\u2014one beyond our understanding. It is a deep, wordless cry, a cry that is not even coming from us. In Romans 8:19\u201326 the Bible says:\n\n> For all creation is waiting eagerly for that future day when God will reveal who his children really are. Against its will, all creation was subjected to God's curse. But with eager hope, the creation looks forward to the day when it will join God's children in glorious freedom from death and decay. For we know that all creation has been groaning as in the pains of childbirth right up to the present time. And we believers also groan. . . . But the Holy Spirit prays for us with groanings that cannot be expressed in words.\n\nGroanings are deep, wailing cries, \"a crying out in grief.\" Creation is crying, we are crying, and the Spirit is crying for the same thing\u2014for the manifestation of the sons of God.\n\n_We cry out because we know the way things are is not the way they are supposed to be._\n\nThe reason we cry out is because we know the way things are right now is not the way they are supposed to be. That we all are crying out signifies that we're in harmony with what God is doing.\n\nThe earth and the Spirit have been crying out on our behalf, and even though there have been some outbreaks of crying out among God's people, we need more. God is restoring these six cries to His people. These are the cries that get Him to hear, stop, and ask, \"What do you want Me to do for you?\" These cries are how we get in harmony with what God is doing. Creation is crying. The Spirit is crying. Now, I believe, the people of God will cry out more than ever, and we will see an outpouring of the Holy Spirit in our generation.\n\nRESPOND\n\nIf you've been crying out to God, wondering when you will see miracles, deliverance, and the promise of God birthed, you join with creation and the Spirit, who've been eagerly waiting for us to arise and take up our full rights as God's children. As your Father, God hears every one of your cries and is coming to answer you. Don't be afraid to lift your voice and cry out. Whether your cry is misunderstood, consistent, persistent, or desperate, find your place, and cry out to God. Creation, the Spirit, and the people of God form a three-note chord when they cry out in harmony. Are you willing to join the cry with creation and the Spirit?\n\nRevival has been waiting, but we have the key to unlock the door of revival. It's when we join in the harmony of the Spirit and creation and cry out to God for more. In the body of Christ we've only scratched the surface. We've only had one glance of the Savior's eyes. We may have tasted of God's glory, but let's cry out for more!\nCHAPTER 4\n\nFOLLOW THE PATHWAY\n\nSearch me, O God, and know my heart; test me and know my anxious thoughts. Point out anything in me that offends you, and lead me along the path of everlasting life.\n\n\u2014PSALM 139:23\u201324\n\nWHEN THE LORD sovereignly decided to dwell with us at Deeper, I began to revisit the stories of the revivalists God has used throughout the last four hundred years. One in particular stood out. His name was John G. Lake. God used him to spread the gospel in Africa. In a span of five years he planted 625 churches and raised up 1,250 preachers. After the five years in Africa, Lake moved to Spokane, Washington. Praying for people, he established something called healing rooms, and God used him to see one hundred thousand healing miracles in five years. At the time, newspapers reported that Spokane was the healthiest city in the world because no one was sick.\n\nAs of the writing of this book, our church has seen more than two hundred miracles in less than two years. But John G. Lake saw one hundred thousand miracles in five years. It is apparent to me that we are only seeing a glimpse. We have only scratched the surface. The question God has been asking us is, \"Are you _still_ hungry?\" Two hundred miracles is only enough to build faith for more. We know that God never intended to do great things only for our eight hundred to one thousand members. As those whom God has chosen to visit, we accept the responsibility to do whatever it takes to unlock the pathway to more. We have no interest in the move of God stopping with us.\n\nWe learned in those early months of revival that God was giving us an invitation to a perpetual outpouring, and now He is extending it to all who are being moved and impassioned by our testimonies. But the invitation is connected to something we don't like to talk about much anymore. The invitation to seeing a perpetual outpouring of God in this generation is connected to repentance. Outpouring happens only for those who are willing to give themselves completely to God, and this happens through the vehicle of repentance. If we are unwilling to let go of the things that obscure our view of God or we find ourselves in a place where we can enter His presence and remain obstinate about the things He is requiring of us, then revival will have an expiration date. It will die. But we have a choice whether to let revival live or die.\n\nLEGACY OR HEADSTONE?\n\nIn the history of the world's revivals some revivals sparked lasting movements, and others died. Some revivals have legacies; other revivals have headstones. The revivals that have headstones are the ones that have a definitive start and a definitive end but leave no lasting legacy. Revival can leave a legacy. Some awakenings become perpetual and create a culture and don't just mark a moment in time.\n\nIf you were to ask people if they want revival, the overwhelming answer would likely be yes. However, there is a pathway to revival, and that is repentance. Repentance is the pathway to righteousness, mercy, salvation, and restoration, and it is the pathway to revival. You can't have any of these things without repentance. We don't like to talk about repentance because it doesn't make us feel good, but God is not interested in how we feel. He's interested in truth in the inward parts (Ps. 51:6).\n\nWhen God decides sovereignly to show up in a particular place and reveal Himself, we have an opportunity to decide how we will respond to the revelation. In Isaiah 6:1\u20135 we are shown how the prophet Isaiah responded when God chose to transport him into His presence. Verse 1 says, \"It was in the year King Uzziah died that I _saw_ the Lord\" (emphasis added). Notice it doesn't say \"sought.\" Isaiah's encounter didn't happen because of what Isaiah was doing. God showed up because God wanted Isaiah to see and experience Him. Isaiah saw the majesty of God because God decided it was time.\n\nThis passage in Isaiah 6 reveals the moment Isaiah was called to prophesy to his nation. To equip Isaiah for this, God allowed him to see something bigger. He allowed him to see something that changed him. He allowed him to experience something that would mark his life forever. Isaiah saw what very few see, and as a result of being in the presence of God, Isaiah came to a finite conclusion. He said, \"It's all over! I am doomed, for I am a sinful man. I have filthy lips\" (v. 5). He became aware that in light of God's holiness he was not holy.\n\n_Outpouring happens only for those who are willing to give themselves completely to God, and this happens through the vehicle of repentance._\n\nThere's something about being in God's presence that unlocks an awareness of our gross separation from the holiness of God. Outside of God's presence we may feel confident that we know about God and His ways. Some of us have been trained in Christian traditions from young ages. But all of that flies out the window when our humanity comes in contact with God's divinity. We cannot contain all that God is, and our humanity dies. Exodus 33:20 says that no one can see God and live. Isaiah was well aware of this. So his statement was legitimate: \"I am doomed\" (Isa. 6:5). It was as if he were asking, \"Did You bring me here to kill me?\" He knew who he was in relation to who God is. And his sense of his depraved humanity grew the longer he stayed there. Not only did he become aware that he had filthy lips, but he also became aware that he lived among a people with filthy lips. But what happened to Isaiah next is huge:\n\n> Then one of the seraphim flew to me with a burning coal he had taken from the altar with a pair of tongs. He touched my lips with it and said, \"See, this coal has touched your lips. Now your guilt is removed, and your sins are forgiven.\"\n> \n> \u2014ISAIAH 6:6\u20137\n\nThe angel took the coal with the tongs and touched Isaiah's lips as if to say, \"Be quiet, or you will miss why you have been invited into this place.\" Then the angel sensed Isaiah's repentant heart and took the exchange further, removing guilt and forgiving his sins. There always is a test when humanity comes in contact with divinity. The test is, Will we worship, or will we be prideful? Isaiah was completely undone. He could not move on. He had never encountered something so much higher, deeper, holier\u2014so infinitely \"more\" than he was in every way. He had been humbled to the point where he was emptied of himself enough to hear the Lord asking, \"Whom should I send as a messenger to this people? Who will go for us?\" Isaiah swiftly replied, \"Here I am. Send me\" (Isa. 6:8).\n\nIf Isaiah got caught up in where he was and what he may have done to get there, he would have missed the very thing for which God had called him. Many of us get stopped at the place where we saw something, and we run out before we hear what God wants to say. But God invited Isaiah there and tested him. Isaiah's response was in proportion to how human he felt next to God's holy divinity. He took the same posture as the psalmist, who said, \"Search me, O God, and know my heart.... Point out anything in me that offends you\" (Ps. 139:23\u201324). Isaiah knew he was in a delicate moment that could instantly be taken away if he did not take the right posture, which was repentance. It is clear from Isaiah's response\u2014\"Here I am. Send me.\"\u2014that he didn't want anything in him to block what God wanted to do.\n\nA CALL TO GREATER\n\nGod showed up for Isaiah to call him to greater. God showed up at Deeper to call us to greater. God is showing up for you right where you are, calling you to greater. The first thing God gives is a glimpse of His glory. The next thing He gives is a test of humility, and finally He gives the assignment.\n\nThe first twelve months of God's outpouring on our church was only a glimpse of His glory, despite all that the Lord has done. And where we stand right now on the pathway to revival is a test of humility: Will we think that what we've seen God do so far is because of us? Will we somehow believe it is due to something we've done? Or will we humble ourselves and let go of our false sense of worthiness to be carriers of revival?\n\n_If God is doing it anywhere, He can do it here in my home, my church, and my community._\n\nIn ourselves we are not worthy carriers of the river of God, but because of the new covenant, Jesus makes us righteous. I don't want you to be beaten down, but I do need you to understand that righteousness is obtained by faith in Christ. We demonstrate this faith by repenting and receiving His forgiveness, thereby becoming worthy to carry revival. On our own, though, and without repentance we are unworthy.\n\nGod is inviting us to decide if we want to see the glimpse of glory transform into a perpetual legacy of revival. Or should He just put a headstone at the place we think our formula brought His presence? Hear me by the Spirit of God: when the Lord asks if we are still hungry, He should not get a flippant response. He is asking us this question so we can say, \"Search me, O God. Is there anything in me that is blocking the reality of what You want to do?\" Why? Because what God wants to do among us is so much greater than what we've seen. We've seen just a glimpse.\n\nThe outpouring had begun in May of 2016, and we were experiencing an incredible outpouring of the Spirit of God. In September of that year we were having our annual Habitation Conference, and my close friend David Diga Hernandez was one of our guests. He has a gift of healing, among other gifts God has given him, and during the conference he noticed one of our volunteers, Ms. Pam, who earlier had suffered a stroke, moving through the church with her cane. He called for her to come up to the platform, and he prayed for her, and instantly her partial paralysis was healed before our eyes.\n\nWe all watched Ms. Pam do things we had never seen her do. She walked without her cane, lifted her leg, and walked sideways\u2014all things she could not do previously because of her stroke. Today Ms. Pam is down front during worship, jumping, leaping, and praising God. She's free from the confining limitations of paralysis due to a stroke. She no longer has to use the handicapped parking when she goes places. She is now one of the people who transcribes every prophetic word given in our church. Every single Sunday she is a physical reminder that it's happening!\n\nRIGHTEOUS ENVY\u2014COVETING THE MOVE OF GOD\n\nFor many of us what I am talking about may seem like such a far-off thing, but that's why I started the chapter with the story of John G. Lake. There must be something inside of you that stirs up righteous envy.\n\nI was sharing with my good friends David and Nicole Binion on a pretty regular basis the many miracles that were taking place weekly at our church. Then one day David said, \"My friend, I have righteous envy hearing about the things God is doing!\" I thought, \"Wow, _righteous envy_. What a term.\" It almost seems like an oxymoron, when you think about it. In Scripture envy is usually presented as a sinful behavior\u2014wanting what someone else has. However, I discovered that God will do something for someone else, or in someone or someplace else, and seeing what He does will cause you to desire Him to do the same thing for you. Sometimes you don't know you want something until you see it.\n\nThere needs to be something that rises up in you and causes you to declare, \"If God is doing it anywhere, He can do it here in my home, my church, and my community.\" God will sometimes use others to do extraordinary things to cause you to long for those kinds of miracles yourself. He'll put you in circles and rooms with people who operate in ways you know you are called to operate in, ways that you know God has shown you.\n\n_Repentance is an invitation to stay._\n\nHe does this to make you jealous for it. God, infinite in His wisdom, will actually bless someone to make you want the blessing. Sometimes God blesses somebody with the blessing He promised you to show you that you need to get into alignment. There is a pattern for this: Romans 11 tells us that the Gentiles\u2014those who are not Jewish by birth\u2014have been grafted in to make the Jewish people jealous. He wants them to see the relationship they could have with Him so He can gather them back. Righteous envy can be frustrating, but what will you do with your frustration? Will you be mad at God, or will you repent?\n\nGod is doing things in the earth to make you jealous\u2014things that will stir your desire for Him to move in your life in such a way that when you see somebody sick, you have enough boldness to pray for them. We are seeing a powerless generation because we are running away from the very gift God has given us\u2014the gift called repentance. Repentance is an invitation to stay. Repentance is a hard teaching for those of us in this generation. We don't like to feel bad about anything. If anything makes us feel bad, then we want to stop engaging in that activity and start doing something that gives us pleasure. But the Scripture says it's in the presence of the Lord that we find a fullness of joy and at His right hand are pleasures forevermore (Ps. 16:11). This means if you go outside of God to find pleasure, that's probably where you need to repent.\n\nGOD IS NEAR\n\nIf repentance is an invitation, then conviction is the sign that God is near. If you enter the presence of God, begin to worship, and get a spiritual picture of sin, this is not simply guilt you feel. It is an indication that God is near you. We have to retrain ourselves. We keep thinking, \"O God, I can't come close to You because I feel guilty.\" No. If you can lean in and see the scriptural principle, you will realize that the reason you feel guilty is because God is near you.\n\nYour humanity has come in contact with His divinity. Conviction is the appropriate response.\n\n_Conviction is the sign that God is near._\n\nIf a person can enter the presence of God and not feel anything about the porn he just watched, he ought to be afraid. His conscience has been seared. The Bible calls that a reprobate mind (Rom. 1:28). The conviction of the Holy Spirit that causes us to feel guilty is an invitation to stay in His presence _if we repent_! Guilt means He is shining a light on areas that are incongruent. He is shining a light on the very thing that will put a headstone on revival for you.\n\nRepentance is a gift from God. It is an invitation from God for salvation, righteousness, mercy, restoration, and revival. God is not interested in a generation showing up to attend a revival; God wants revival to be in you. Revival is not a show put on by a few anointed people. Revival is something that happens in your spirit man. You who were once dead are now made alive. You whose affections have grown cold are now hot for Christ. You who were cold are now on fire. Revival is something that happens on the inside of you, and it is contagious! You cannot be near fire and not feel it.\n\nWORSHIP DOES NOT REPLACE REPENTANCE\n\nGod says to this generation that our deeds don't vindicate us. What you do doesn't make you right. Your worship doesn't replace repentance. We are the generation that has more written material on worship than any other in history, yet we are the generation that, based on its fruit, seems to be resisting God the most. In Psalm 50:8\u201310 God says:\n\n> I have no complaint about your sacrifices or the burnt offerings you constantly offer. But I do not need the bulls from your barns or the goats from your pens. For all the animals of the forest are mine, and I own the cattle on a thousand hills.\n\nIt's not our worship that God has a problem with. He doesn't have a complaint about our external things. And really, He doesn't need them. We don't serve a narcissistic God who needs us to lift our hands for Him to feel good today, as if He's the king and we're court jesters. \"Sing for Me. Shout for Me. I need to feel better today.\" No! God is God, and we are not. We have needs; He does not. What God is saying in Psalm 50 is that our worship has become ritualistic. We are doing things externally that are disconnected from the internal posture of our hearts.\n\nGod really does not care how high we lift our hands or how loud we sing, if we don't mean it. God is not interested in ritual; He's interested in relationship. This is why David has such an amazing revelation in Psalm 51:16\u201317: \"You do not desire a sacrifice, or I would offer one. You do not want a burnt offering. The sacrifice you desire is a broken spirit. You will not reject a broken and repentant heart, O God.\"\n\nWhat God is looking for are people who recognize that they need Him. David came to an amazing revelation because at that time, according to the Law, God did require a blood sacrifice. But David saw more. He saw that God desired something from the inside.\n\nGod is not looking for people who are going through the motions\u2014the motions of revival, the motions of outpouring\u2014doing what they think He wants to see, saying what they think He wants to hear. God is looking for those who will say, \"God, search me. If there's anything in me that offends You, take it away from me.\"\n\nIf we come into God's presence and do the acts of worship without repentance, it's ritual because our hearts are not right. We are not intimately and honestly engaged\u2014in spirit and truth\u2014with God. We can lift our hands all day, but worship does not replace repentance. As one Bible dictionary says, \"In its biblical sense repentance refers to a deeply seated and thorough turning from self to God.\" Repentance leads you to say, \"God, whatever You desire from me or whatever You're shining a light on, I will not hold on to it more than I hold on to You.\" In repentance you are saying, \"Search me,\" and God tells you what should remain. Our response should be that we turn away from whatever He tells us to release in order to hold on to Him.\n\nThe result of repentance is salvation\u2014salvation from being ensnared by the idols you've been worshipping, the things that have been blocking your view of God, and the things in your heart that have been keeping you from being used by God. This is why God gives you a glimpse of His glory. He makes it so amazing that we long for it, so much so that we will do whatever it takes to stay in that moment\u2014including repent. God allows us to be stirred up with righteous envy so we'll say: \"I want to see revival come in my life and lifetime. God, if You're doing it somewhere, do it in me. If there's anything in me that offends You, take it out of me so You will bring revival in me.\"\n\nThe apostle John wrote:\n\n> If we claim we have no sin, we are only fooling ourselves and not living in the truth. But if we confess our sins to him, he is faithful and just to forgive us our sins and to cleanse us from all wickedness. If we claim we have not sinned, we are calling God a liar and showing that his word has no place in our hearts.\n> \n> \u20141 JOHN 1:8\u201310\n\nEXPOSED!\n\nI have seen something, and it overwhelms me. It makes me want to cry. It makes me want to cry out. It makes me want to lie on my face. But mainly it makes me want to stop doing anything and everything that would possibly get in the way of it. There are moments in our lives when God shines light on something and says, \"I want that.\" It may be a habit or way of thinking\u2014it could be anything that is separating you from Him. There are times and seasons when God in His mercy will expose things that are hindering our relationship with Him, and we hate exposure. We want to deal with everything in private, but if God can't get your attention in private or you won't honor Him enough in private, He may expose you in public so you will return to Him.\n\nIt is human nature to hide from God, but we can't do what seems natural. We also try to hide from one another. That is why in the Garden of Eden, Adam and Eve sewed fig leaves together to hide from each other. We don't want our sin to be exposed to others. We can't hide anything from God, so we try to hide our sin from one another. That's why God connects righteousness and effective prayer when He says, \"Confess your faults one to another\" (James 5:16, KJV). An unrepentant heart can keep you from seeing the answers to your prayers concerning revival.\n\nThe Lord is showing us that He wants to do something great among us, but the pathway to it is the very thing that most of us run from. You would probably shout if I said the pathway to revival is praise or more worship. If the pathway was more prayer, others might say, \"OK, I can do that.\" But the moment we discover the pathway to revival is repentance, most of us say, \"Oh.\" But if you are hungry for revival, you are ready to do whatever God desires of you to see His glory come.\n\nRESPOND\n\nWhat is standing in the way of revival for you? Don't allow any unconfessed sin to cause God to look at your worship as a ritual. I believe there are things the Lord has been showing you for quite some time that you have been resisting. What if I told you that the next outpouring is connected to your surrender? What if I told you that the level of outpouring you experience is directly connected to the level of surrender and truth you are willing to walk in with God?\n\nGod wants to use you. The Bible says the same spirit that raised Christ from the dead lives in you (Rom. 8:11), not only in the pastor, bishop, apostle, or prophet. You are part of the chain of humanity through which God will bring the next habitation of His Spirit.\n\nThis is the season of the corporate anointing. The next move of God will not be predicated on the individual. In many of our churches the ministry offices have been exalted almost to the level of being worshipped, causing people to believe that God could not use someone who was not operating in one of the fivefold ministry gifts the same way He uses apostles, prophets, evangelists, pastors, and teachers. But God is taking the needle and thread from those who are trying to re-stitch the veil that was torn when Christ died. (See Matthew 27:51.) They are doing this to make themselves more important, but there is only one High Priest, and it's not a man. Jesus Christ is the High Priest. He is our Mediator, our Intercessor. He is the One who makes the way for all of us to enter heaven's most holy place.\n\n_The level of outpouring you experience is directly connected to the level of surrender and truth you are willing to walk in with God._\n\nThe greatest revival that is going to happen in the earth is going to happen through God's surrendered corporate church. God wants to do greater. This means this word about repentance is not for the few. It's for all. What glimpses God has shown us are to make you long for more. He let us witness ten, twenty, fifty, or one hundred miracles so we could long for one hundred thousand. If you're hungry for it, be honest and say, \"I want that, Lord. Do it in me. Do it in us.\"\n\nThis is the prayer God is requiring of us:\n\n> Search me, O God, and know my heart; test me and know my anxious thoughts. Point out anything in me that offends you, and lead me along the path of everlasting life.\n> \n> \u2014PSALM 139:23\u201324\n\nGod is looking for people who will say, \"Whatever You want, God, You can have.\" Your decision determines whether the current move of God will be perpetual or if a headstone will be put in place. I invite you to the spiritual altar, out of your comfort zone and your hiding place. I invite you to offer up to God anything that separates you from experiencing His greater outpouring. God wants to bring a great revival through you. Do you want what God has for you more than what you have right now? If so, I invite you to repent.\nCHAPTER 5\n\nIT'S ALL ABOUT CAPACITY\u2014HOW EMPTY ARE YOU?\n\nAnd Elisha said, \"Borrow as many empty jars as you can from your friends and neighbors. Then go into your house with your sons and shut the door behind you. Pour olive oil from your flask into the jars, setting each one aside when it is filled.\" So she did as she was told. . . . Soon every container was full to the brim!...And then the olive oil stopped flowing.\n\n\u20142 KINGS 4:3\u20136\n\nTHE BIBLE SAYS God doesn't do anything without engaging His servants in His plans (Amos 3:7). Though God is God all by Himself, He desires to partner with us. Much of His movement in the earth happens as we submit ourselves as vessels through which He can flow. The Bible is a chronicle of the works of God extended through man. And history shows that revival spreads by the preaching and teaching of men and women who've been lit on fire by the Spirit of God.\n\nIf you have cried out for revival and been in the room as intercession and worship were being offered up, you probably have felt a longing in your heart to be used by God. It is during these moments that you may cry out, \"I want more of You, God,\" or, \"I want to be used by You, Lord.\" But do we? Do we know what it will take to make room for the fullness of God in our lives?\n\nWe are entering a season when we cannot instinctually answer the question \"Do you want to be used by God?\" Our participation in this next move of God cannot be by default. Our alignment with what God is doing must be intentional and deliberate. God blesses with more of Him those who recognize their need for Him. Those who are empty, those who are spiritually bankrupt, those who can say, \"I need God more than anything else\"\u2014these are the people God wants to fill and use for His glory.\n\nI remember growing up with an insatiable desire to be in the presence of God. My mom did not have to drag me to church. I used to play church all the time. I remember setting up my green army men not for war but as a choir, and I would stand in front of them, directing them. When I did go to church, I always tried to be close to someone I knew was being used by God. I just wanted to be near the person's gifting and anointing. There is something contagious about the anointing, and I hoped to have whatever that person had.\n\nBut being used by God is not about how we feel; it's about our capacity. My son loves pizza; he is a pizza-eating machine. When he was six, he ate as if he were sixteen. People ask me what I am going to do, how I am going to keep him fed. I tell them, \"I'm going to keep writing songs.\" One day he ate slice after slice and kept asking for more. My wife and I were surprised he could eat that much. At one point I told him to slow down, but he kept asking for more. Finally he had his fill, but he must not have realized he had surpassed what his stomach could hold, and he threw up everywhere. He didn't have the capacity to hold what he wanted. He had the desire for more pizza but not the capacity to receive it.\n\nMany of us are in the same boat. We're saying, \"God, fill me. Use me. I want more of You.\" But do we have the capacity to handle more of God? Let me show you something in Scripture.\n\nONLY EMPTY VESSELS CAN BE FILLED\n\nSecond Kings 4:1\u20137 tells the story of a woman who was recently widowed. She had two sons and was in debt. According to Jewish law, if a person had a debt but no money to pay it, he would have to work it off as an indentured or indebted servant. But here in 2 Kings 4 the creditors were threatening to take it a step further and make this woman and her sons slaves. In the moment of her desperation the woman cried out to God, who in those days was represented by the prophets, the men of God. She said, in essence, \"I need God to intervene. My husband was righteous, but I'm in this desperate situation. I'm not trying to figure it out for myself. We are hungry, and we have nothing left.\" The woman's desperation and awareness of the little she had gave her enormous capacity to receive what she needed from God. She was aware of her need for God to work a miracle in her life.\n\nThose of us who are becoming aware of our need for God must learn to live in a consistent posture of crying out and expectation. Expectation is an anticipation of being filled by God every day, like a chick that waits in the nest for the mother bird to return with food. We pray, \"Give us this day our daily bread\" (Matt. 6:11, KJV). This is not a one-time cry. We need God daily. Do you live with that daily expectation? Are you, moment by moment, crying out, \"God, I need You\"?\n\nThe woman in 2 Kings 4 found herself in this place of emptiness. All she had was a flask of oil, and the prophet said, \"I can work with that.\" This oil is a spiritual representation of the anointing of God that imbues us with the power of God so we can do mighty exploits in His name. So yes, God can work with the little we have. It's a start, but then we see in verse 3 that the man of God, the prophet Elisha, told the woman, \"Borrow as many empty jars as you can from your friends and neighbors.\" A little oil is better than no oil, but a little is not enough. Like the woman, we need to increase our capacity.\n\n_Only empty vessels can receive what God wants to give._\n\nHow many times have we come before God saying we want to be used by Him, but we are full of the cares of life\u2014full of worry, pride, greed, envy, jealousy, and all the works of the flesh? God wants to give us more of Himself\u2014His anointing, power, miracles, provision, healing, blessing\u2014but sometimes we are already filled to the top with so many other things. You can't be filled with the oil of God while still being full of yourself.\n\nEach month at our church we host what we call Presence Nights. People gather from around Central Florida and even from other states and countries for these services. Presence Nights are always marked by anticipation and expectation. People line up outside for hours prior to the service just to get in. There is no agenda at these services but to seek God, hear from God, and respond accordingly. And God always does something amazing on Presence Nights, even though we never fully know what to expect.\n\nOn one particular night, my friend Caleb had a word of knowledge that the Lord wanted to heal paralysis. We've seen people come in using wheelchairs, walkers, and canes and walk out healed. Interestingly that particular night we didn't visibly see anyone who seemed paralyzed, but we knew God had spoken, so in faith we prayed that God would reverse paralysis.\n\nAfter the prayer we asked for anyone in the room who had that condition to identify himself. In the back of the room a man raised his hand and then came forward. He was a musician. He played the keyboard, and something happened that caused his hand to become locked and paralyzed, but the Lord healed him that night. We all watched him open and shut his hand and move his fingers\u2014all things he could not do before that.\n\nSeveral months later Caleb and I went out to eat at Red Robin and ran into this gentleman who was healed. We didn't actually recognize him because he had only been visiting our church. He came up to us and said, \"You are the guy that prayed for me, and I was healed!\" He introduced us to those who were with him and shared the testimony again. He was still healed. It's happening!\n\nGET USED TO FEELING EMPTY\n\nSome of us hate the feeling of emptiness. It carries a negative connotation in our culture, but emptiness is the prerequisite to receiving more from God. Emptiness gives you capacity. The emptier you are, the more room you have to be filled. Matthew 5:6 says, \"Blessed are those who hunger and thirst for righteousness [the ways of God], for they will be filled\" (NIV).\n\n_Emptiness is the requirement for receiving the oil of God._\n\nIf you want to be used by God, you must become acquainted with feeling empty. You will not recognize your need for God until you are empty. As Matthew 5:6 implies, feeling empty is the same as feeling hungry or thirsty. You need food and water. You need to be nourished. You need to be filled. When you are thirsty and you drink soda instead of water, for the moment, your body does not feel as if it needs water, even though it does. The soda temporarily quenched your thirst, but it also cheated your body out of what it really needed to be filled, nourished, and satisfied. In this same way you won't really recognize your spiritual needs if you are filled with other things.\n\nSome of us are trying to medicate our way out of feeling empty. We may try to drink our way out of it. We try to fill ourselves with everything else, including the lust of the flesh and the pride of life, and then we wonder why there's no authority, power, miracles, and presence manifesting in our lives. Filling ourselves with anything other than God says we don't want His oil and we don't want His anointing. We say, \"Give me a man, marriage, alcohol, or drugs.\" Let me help you: Don't try to get rid of your emptiness. It was put there by God so He could fill it with His oil.\n\nGod wants to give this generation oil, but this generation doesn't want oil. This generation wants friends. Those in this generation can't handle being by themselves. We won't even drive in the car without the radio on because we don't want to feel alone. God put emptiness there as an indicator of our need for God, so we would call out to Him to be filled with His oil. How would you know your car needed gas or oil if the indicator light didn't show it was low?\n\nI've learned a few things over the years.\n\nFirst, those who are greatly used by God have become comfortable with emptiness. They know that God put emptiness there to help them recognize their need for Him. Without emptiness, they may never cry out, \"God, I need You!\"\n\nSecond, people who are greatly used by God are sometimes lonely. I know we don't like that. We want friends around all the time. But the more people we have around us, the less opportunity we have to hear God. God wants to talk to us, but He requires that we bring an empty vessel.\n\nThird, I haven't discovered anybody who has been greatly used by God who hasn't also been hurt or betrayed. Mighty men and women of God get hurt just as everyone else. The difference is instead of going to God empty, most people take on the hurt themselves, piling fleshly things into their spirits\u2014things such as offense, anger, and bitterness. Though these emotions feed the flesh, they also crowd the spirit, leading people to inadvertently repel the infilling of God. Even in these difficult moments, God gives the blessing of emptiness. I know it hurts, but you can take that hurt directly to God, have Him heal it, and be refilled.\n\nMy challenge to you: go to God first. Go to God hurt, worn, desperate, and empty. You increase your capacity for God to work miracles in and through your life when you do this. You should not be content with a little bit of oil. A little bit of oil runs out. You will not shift regions with a little bit of oil. You won't be able to speak to principalities with a little bit of oil. You will not see your family saved with a little bit of oil. The lame are not going to walk and the dumb will not talk with a little bit of oil. You need to go before God as empty as you can so you can be filled to the greatest capacity possible.\n\nEMPTY = DEAD\n\nIn talking about emptiness, we're actually talking about death. There are several ways you can put to death the things that limit your capacity.\n\n**1. Nail them to the cross.**\n\nGalatians 5:24 says, \"Those who belong to Christ Jesus have nailed the passions and desires of their sinful nature to his cross and crucified them there.\" Whatever it is in your life that is against God, whatever keeps you from experiencing God and walking in His ways, nail it to the cross. Addictions, pornography, and gossip\u2014nail those to the cross.\n\n**2. Confess them to God.**\n\nFirst John 1:9 says, \"If we confess our sins to him, he is faithful and just to forgive us our sins and to cleanse us from all wickedness.\" Confess, repent, and then go to Him boldly, empty and ready to be filled.\n\n**3. Surrender.**\n\nRomans 6:11\u201313 says, \"So you also should consider yourselves to be dead to the power of sin and alive to God through Christ Jesus. Do not let sin control the way you live; do not give in to sinful desires. Do not let any part of your body become an instrument of evil to serve sin. Instead, give yourselves completely to God [surrender; be empty], for you were dead, but now you have new life. So use your whole body as an instrument to do what is right for the glory of God.\" Being used by God is not just connected to desire, but it is also connected to capacity. You can say you want to be used by God all you want, but if you don't surrender, your capacity to contain the oil of God is limited. Have you tried filling a container that is already full? It doesn't work.\n\nEmptiness refers to being dead to sin. We say with our mouths that we want to be used by God, but do our lives show that we want to be used? Are we willing to pay the price? I've heard it said that there is not a man who has been used by God who has not first been bruised by God.\n\nSHUT THE DOOR BEHIND YOU\n\nIn 2 Kings 4:4 the prophet told the woman:\n\n> Then go into your house with your sons and shut the door behind you. Pour olive oil from your flask into the jars, setting each one aside when it is filled.\n\nSome people do things to look spiritual. They want everybody to see what they're doing. They want an audience for their gifts, anointing, or blessing. But the miracle realm doesn't work this way; it flows in secret. It flows out of surrender and humility. Most of the time the way God works is so counterintuitive that when we start out with Him, we'd rather keep it to ourselves.\n\nThink about Naaman. He was a commander and mighty warrior in the Syrian army, but he was a leper. (See 2 Kings 5.) He was unclean, and in those times lepers were shunned and had to be separated from society. Naaman's story also seems to suggest that his condition was not widely known. If everyone knew about his disease, it would have been a deadly blow to his status in the kingdom.\n\nBut the king gave him provision to seek out the man of God for healing. But before he could receive his miracle, his ego had to be broken down to the point where he was humble enough to try anything. He had to be drawn into the secret place\u2014a place away from the eyes of the people in the kingdom he served.\n\nWhen Naaman sought out the prophet of God, nothing about the encounter was as he expected. The prophet sent a messenger with instructions for Naaman to wash in the Jordan River seven times, and after he did this, he would be made clean. At first Naaman was angry. He actually stomped away from the prophet's house with no plans to follow his directives. Naaman thought Elisha would address him formally and raise his hand in some dramatic gesture and call out to God to heal him. But Elisha did nothing of the sort. There would be no fanfare, no thunder from heaven, no one but God to see that he followed the instructions.\n\nGod used the peculiar nature of this miracle to humble Naaman. When Naaman finally submitted and obeyed the instructions the prophet had given him, he was healed. The Bible says his skin became as healthy as that of a young child. God's way of bringing healing was the opposite of what Naaman expected. God worked in secret, and He does the same today. God will only begin to work through you publicly when He knows He can trust you to obey what He tells you in private.\n\nUnlike Naaman, the woman in 2 Kings 4 and her sons started in a place of desperation and surrender. Still Elisha said, \"Shut the door behind you\" (v. 4). Miracles are not born in public display but in private devotion. The power of God begins flowing in private.\n\nA lot of people want to be used publicly but don't want to be bruised privately. Yet the anointing, gifting, or miracle is all for God to get the glory. It is not about anything we do, except that we come willing and empty. Yes, the woman had oil, but she did not know how it was going to be used. How did she handle that? Second Kings 4:5 says that \"she did as she was told.\" She had her sons continue to bring jars to her. She had to have believed that God would do the impossible with those empty jars. The Bible doesn't show that she asked, \"What am I going to do with these empty jars?\" It says \"she did as she was told.\" She kept pouring\u2014behind closed doors\u2014and the oil never ran out.\n\n> Soon every container was full to the brim! \"Bring me another jar,\" she said to one of her sons. \"There aren't any more!\" he told her. And then the olive oil stopped flowing.\n> \n> \u20142 KINGS 4:6\n\nThe oil only stops when empty vessels stop coming. When we stop coming to God, that's when the oil stops. When we stop recognizing our need for Him, we stop coming. The issue is not God's supply. God's supply is limitless. The issue is our demand.\n\nRESPOND\n\nThat we are not seeing the fullness of God's power in this generation has absolutely nothing to do with God's ability. God is just as miraculous and supernatural as He was at any other time in history. The Bible says He doesn't change (Mal. 3:6). This means that whatever you've seen Him do in the past He can still do. So why aren't you seeing His power in your life?\n\nIs it possible that the only reason you've stopped seeing the oil flow in your life is because you've stopped asking for it? Is it possible that the oil stopped flowing in your life because you don't have an empty vessel to contain what God wants to do? Are you so full of yourself\u2014your flesh, your desires, or pride\u2014that there's no room for His oil?\n\n_Miracles are not born in public display but in private devotion._\n\nIn the widow's story the oil stopped when the empty vessels stopped. God _always_ supplies oil when there is a demand. If you want to see God move in this generation, you need to understand this truth. Revival flows through us. I challenge you to ask yourself: Why is revival not happening in my life? Why do I not see miracles, signs, and wonders taking place in my generation, my home, or wherever I go?\n\nThe question is, Are you full, or are you empty? Is there room in your life for the anointing, for the presence of God? Is there anything in your life that is crowding out His ability to use you? Can He fill you with His oil? Is there anything keeping you from receiving the fullness of what He wants to do in your life? Even if you have a little bit of pride or a little bit of whatever, that little bit is enough to limit His ability to fill you completely.\n\nGod may be using you, and you may be anointed, but God wants to do more. When I get a glimpse of what God wants to do, I get excited. I become aware that whatever little bit of \"stuff\" I've been holding on to needs to go. We have to be careful not to get to a place where we stop coming to God. It could be that God gave you a word for somebody one time, and all of a sudden you want to be known as prophetess So-and-So. God uses you to pray for somebody one time, she says her headache is gone, and now you want to launch a healing ministry. You got it. You're anointed. But you stop coming to Him, and you stop praying and depending on Him as much. You may be able to look back at certain seasons of your life and say, \"Man, God was really using me back then.\" God hasn't changed. His supply is still there. The question then is, Are you still coming to Him?\n\nGod wants to do immeasurably more than anything we've seen, and He is inviting us to go deeper. I don't want to attempt to speak for you, but I invite you to go to God with a prayer similar to this one, letting Him know that you are reaching out to Him again, but this time you want to be emptied:\n\n> _God, I give You permission to go into every area of my life that does not line up with You. Examine my heart, thoughts, actions, deeds, relationships, interactions, addictions, pains, and struggles. Go in deep, and if You find anything that is hindering me from being filled to overflowing, take it out of me. I want Your oil to flow in my life. I do not want to be so full that I cannot receive Your oil. So God, I come. Empty me of everything that is against Your will and plans for my life. I come seeking to increase my capacity to house Your Spirit. God, fill me up._\n\nThere will never come a time when you will not need to present yourself back to God as an empty vessel in need of His filling. Never. The oil stops when you stop presenting yourself before Him empty. Surrender is not a one-time event. Don't allow the enemy to trick you into thinking it is. Surrender is a daily, hourly, moment-by-moment commitment to say, \"Lord, I want to be right with You. I need You.\" Your acknowledgment of your need for God is an emptying out of yourself. Nothing\u2014not money, reputation, or friends\u2014is worth more than having the oil of the anointing of God flowing in your life.\nCHAPTER 6\n\nNO SUBMISSION, NO POWER\n\nYou will receive power when the Holy Spirit comes upon you. And you will be my witnesses, telling people about me everywhere\u2014in Jerusalem, throughout Judea, in Samaria, and to the ends of the earth.\n\n\u2014ACTS 1:8\n\nEVERYBODY WANTS TO walk in power. Everybody wants to receive a word from God. Everybody wants to prophesy. But we can't do any of it without being submitted to God. One of the things we need to talk about more as we pray for God to blow the winds of revival is submission to the Holy Spirit. We are professionals at being good at everything but submission. We don't like it. We don't like it when somebody tells us what to do, and that includes allowing the Holy Spirit to lead us. The way the old folks would say it is the Holy Spirit will help us walk right, talk right, and live right. To let Him do that, we have to surrender.\n\nThe reason we have a generation confused about revival is because there are many people who think like Simon the sorcerer in Acts 8. They think they can exercise special privilege and shortcut the way to receive supernatural power to carry out the works of Christ. They would rather have the anointing than a relationship with God. Do you know what that looks like? It looks like tongue-talking whore-mongers, tongue-talking cussers, tongue-talking alcoholics, and tongue-talking fornicators. They want the gifts but don't want to submit to God.\n\nThe problem we face as believers is that we leave the world confused about what kind of power God possesses when His people live the same way they do\u2014sometimes worse. While we're talking about revival and the power of God, the world is looking on, trying to figure out how we have power but don't live right, stop gossiping and watching pornography, or come together and love one another. What kind of power is this that we are talking about?\n\nThe world is waiting on us to arise like real sons of God (Rom. 8:19). They need to see us walk in God's full power. We cannot expect to see a perpetual move of God if we don't have power, and we will have no power without giving ourselves completely over to God. We must be willing to say, \"We'll go where You go. We'll take on Your identity. We'll take on Your name. We will lose it all to follow You.\" By doing this, we open the door to an authentic overflow of the power of the Holy Spirit.\n\nYOU WILL RECEIVE POWER . . .\n\nIn Acts 1:8 Jesus gave His followers a prophetic word about their future and destiny. He said, \"You will receive power when the Holy Spirit comes upon you.\" This is a good word. I imagine there was a feeling of strength and victory attached to it as He spoke to them. We as believers today like to hear this part of the verse, especially in Spirit-filled churches: \"Yes, God! I receive Your power.\" Next, Jesus says, \"You will be my witnesses . . . in Jerusalem, throughout Judea, in Samaria, and to the ends of the earth.\" The disciples are still tracking with Him, as would we be. \"Amen. Thank You, Jesus,\" I can hear us say.\n\nWhat we are witnessing in this passage is the Lord prophetically speaking by the power of His Spirit over the lives of believers then and now. He was spelling out our destiny, which is distinctly different from a destination. This prophetic word in Acts 1 is about the arc of the life of a believer. In essence, He was saying, \"What's about to happen in your life is going to be more powerful than you can imagine. The Spirit of God is going to come upon your life, and He is going to empower you to make you witnesses. He is going to give you a worldwide ministry, and you are going to go to the ends of the earth proclaiming My Word. I've called you to preach My gospel and to be My witnesses.\"\n\nI can see it: Just as this word ignited a revolution in the hearts of Jesus's followers that spread all over the known biblical world, I can see it having a similar effect in our churches today. The whole place would erupt. People would lift their hands and say, \"My life is going to be great because God is going to greatly use me.\"\n\n_The world is waiting on us to arise like real sons of God._\n\nBut Jesus was only telling _what_ would happen. He did not tell _how_ it would happen or what it would take to get to this place of spiritual power. With every _what_ , there is a _how_ , and by now those of us who are praying for revival already know the _why_. The _why_ fuels our prayers, the _what_ gives us direction, and the _how_ is the deciding factor for how far we'll go with God and how much we're willing to surrender to Him in this current season.\n\n. . . BUT THERE IS A COST\n\nThere's a point as the Spirit reveals the full extent of Jesus's call to destiny where many of us may get quiet. What if Jesus told us that on the way to receiving power, we are going to lose friends and people are going to talk about us? What if He said we may lose our houses or our jobs? What if it comes to losing health benefits or our cars? What if our children start acting up or our family members don't understand why we don't do the same things we used to?\n\nWhat if it takes all this and Jesus says, \"I'm still going to use you\"? Some of us would say, \"You know, I'm good where I am.\" Others may say, \"I don't know about that.\" But let me challenge you again with the words of Christ: \"If you try to hang on to your life, you will lose it. But if you give up your life for my sake, you will save it. And what do you benefit if you gain the whole world but lose your own soul? Is anything worth more than your soul?\" (Matt. 16:25\u201326).\n\nWhat are you willing to give to be used by God in this hour? Is seeing God move worth more than preserving your status in society? Is seeing God move worth more than preserving relationships with friends and family who don't understand the call of God? Is seeing God move worth more than preserving the comforts of your own soul? This is a hard word. I know some of us may not be ready to face this, but revival is costly to the flesh.\n\nBut God reassures us in this: \"I will never leave you nor forsake you\" (Heb. 13:5, NKJV). \"My grace is all you need. My power works best in weakness\" (2 Cor. 12:9). He says, \"Whatever I take you through, the Holy Spirit will remind you of the things I've said so when you are in a moment that seems incredibly difficult, you will prevail.\"\n\nTwo of the hallmarks of revival are repentance and salvation. Even in what we think are the hardest cases, nothing is too hard for God. Our church is located not far from a street in our city that has been known as a hub for prostitution. There are several adult stores on that particular part of the street as well. We've seen fewer and fewer of those kinds of businesses over the years, but they're still present.\n\nOne Sunday a lady walked into church right off the street, having just sold her body for money. She literally didn't even clean herself up first. She had decided that she'd had enough of that life and was drawn by the Spirit of God to come to our church. When the invitation to surrender to Christ was made, she responded and gave her life to Christ on the spot. As we prayed for her, she was set free and delivered. She came back the next week like a completely different person and to this day has not gone back to that life of prostitution. It's happening!\n\nGET USED TO BEING UNCOMFORTABLE\n\nWhen God calls you to do great works, you must be willing to give up whatever you need to release to make yourself available to Him. As a witness of His glory, you will not get to go wherever you want to go on your own time. You don't get to determine when you go. You don't even get to determine how you go. When God has a call on your life, your personal schedule is irrelevant. Proverbs 19:21 says, \"You can make many plans, but the LORD'S purpose will prevail.\" In other words, there's nothing wrong with planning, but the sovereignty of God's purpose trumps your plans. Because God needs you to be unattached to the cares of this life, when you get too comfortable with your plan, He will intentionally make you uncomfortable.\n\n_The sovereignty of God's purpose trumps your plans._\n\nThis is exactly what He allowed to happen in Acts 8. The Bible says:\n\n> Saul was one of the witnesses, and he agreed completely with the killing of Stephen.\n> \n> A great wave of persecution began that day, sweeping over the church in Jerusalem; and all the believers except the apostles were scattered through the regions of Judea and Samaria. (Some devout men came and buried Stephen with great mourning.) But Saul was going everywhere to destroy the church. He went from house to house, dragging out both men and women to throw them into prison.\n> \n> But the believers who were scattered preached the Good News about Jesus wherever they went.\n> \n> \u2014ACTS 8:1\u20134\n\nSaul (who later became Paul) led the persecution of believers, which was the catalyst for their being sent out to preach the good news. This is so uncomfortable for many to embrace, and it may be a hard word for where God is moving His body, so I'll say it again: God caused Saul to persecute believers to send them out so His purpose would prevail. Jesus said in Acts 1:8, \"But you shall receive power when the Holy Spirit comes upon you. And you shall be My witnesses in Jerusalem, and in all Judea and Samaria, and to the ends of the earth\" (MEV).\n\nOn the surface that sounds amazing. The Holy Spirit is going to come. We will receive power and minister around the world. Most would say, \"Sign me up for that.\" However, Jesus didn't say how it would happen, only that it would happen. If you've been walking with the Lord for a while, you can probably attest to the reality that we didn't know where our _yes_ would take us and what our _yes_ would cause us to go through.\n\nSome of us might not have said yes so readily if we had known the cost. Jesus said they would be His witnesses in Jerusalem, Judea, Samaria, and to the ends of the earth. Acts 8:1\u20133 tells us that Saul's persecution of the believers caused them to be scattered to the regions of Judea and Samaria. So how did they get to Judea and Samaria? Persecution. At times God uses persecution to advance His causes.\n\nConsidering that God had no problem turning up the heat on the believers in the early church to see a mass revival come, you must understand that God will also make you uncomfortable. Are you feeling uncomfortable around the friends and family you used to hang with? Are you feeling the pressure of being tied to a nine-to-five job? Are they turning up the heat, telling you not to talk about God? I need you to understand that God is calling you out. He, not the devil, is the One causing the people and environments around you to feel strange. So stop rebuking the devil.\n\nI am not at all telling you to quit your job, break up with all your friends, sell everything you have, and move to Tanzania. I am encouraging you to reframe how you see the challenges you are facing in the natural in this season. I am encouraging you to seek the face of God for clarity and direction. Because you have cried out and He has answered, He is taking you up on your commitment to surrender to Him to see how far you are really willing to go. He is calling you to respond once again.\n\nWhen Jesus released the word of destiny in Acts 1, He did not tell them that the way they would be built up in the faith, the way that they would be strengthened and empowered, was by the threat of death. He did not say, \"You will receive power to be My witnesses, and how you're going to get that power is through enduring death threats.\" I think He knew that would have been too much all at once. He gave them the first part and allowed the Holy Spirit to lead them through the second part.\n\nIn Acts 8:5 Phillip\u2014a layperson, a deacon in the church\u2014was sent to Samaria, a most uncomfortable place for Jews. From Jesus's encounter with the woman at the well (John 4) and His parable of the Good Samaritan (Luke 10:25\u201337), we know that these two groups didn't get along. Still, the Lord caused Philip to go there and walk in the miracle-working power of God:\n\n> Crowds listened intently to Philip because they were eager to hear his message and see the miraculous signs he did. Many evil spirits were cast out, screaming as they left their victims. And many who had been paralyzed or lame were healed. So there was great joy in that city.\n> \n> \u2014ACTS 8:6\u20138\n\nRevival broke out in Samaria. Because of increased persecution believers scattered into areas where they were not comfortable, and God used them mightily. They received power to carry out the work of the Lord in areas where they may not have gone on their own and to people who may never have heard the gospel.\n\nLooking closely at the Scripture, you will discover that most of Jesus's instructions for proclaiming the gospel were given for all believers, not just for those operating in what are known as the fivefold ministry gifts outlined in Ephesians 4:11 (apostles, prophets, evangelists, pastors, and teachers). This means every believer has been given the task of proclaiming the gospel everywhere because we can never know where revival will break out. We need to be surrendered and ready.\n\n_We need to be surrendered and ready._\n\nSome of us are secretly hoping that when revival breaks out, we won't be called to someplace uncomfortable. But what happens when the Lord does just that? His intention is for revival to break out wherever you're sent and that when you begin to proclaim the Word of the Lord, people are saved, healed, and delivered from all the power of the enemy.\n\nTHERE ARE NO SHORTCUTS TO POWER\n\nWhen the truth of the gospel is preached, deception is exposed and people are moved to choose between the truth and a lie. The Bible says:\n\n> A man named Simon had been a sorcerer there for many years, amazing the people of Samaria and claiming to be someone great. Everyone, from the least to the greatest, often spoke of him as \"the Great One\u2014the Power of God.\" They listened closely to him because for a long time he had astounded them with his magic.\n> \n> \u2014ACTS 8:9\u201311\n\nThe people saw how this man moved in supernatural ways, and they thought they were seeing God's power at work\u2014until they heard Philip preach. \"But now the people believed Philip's message of Good News concerning the Kingdom of God and the name of Jesus Christ\" (Acts 8:12). The power of God present in Philip's teaching convicted their hearts with the truth, and they recognized that all this time they had been deceived. As a result, many men and women were baptized, including Simon.\n\n_When the truth of the gospel is preached, deception is exposed._\n\nBut Simon had a little bit more going on. He began to follow Philip because he saw the power Philip operated in. It seemed he was trying to figure out how it worked.\n\nWhen the apostles of Jerusalem heard that the people of Samaria had accepted God's message, they sent Peter and John there to confirm the new believers and baptize them in the Holy Spirit.\n\n> Then Peter and John laid their hands upon these believers, and they received the Holy Spirit.\n> \n> When Simon saw that the Spirit was given when the apostles laid their hands on people, he offered them money to buy this power. \"Let me have this power, too,\" he exclaimed, \"so that when I lay my hands on people, they will receive the Holy Spirit!\"\n> \n> But Peter replied, \"May your money be destroyed with you for thinking God's gift can be bought! You can have no part in this, for your heart is not right with God. Repent of your wickedness and pray to the Lord. Perhaps he will forgive your evil thoughts, for I can see that you are full of bitter jealousy and are held captive by sin.\n> \n> \"Pray to the Lord for me,\" Simon exclaimed, \"that these terrible things you've said won't happen to me!\"\n> \n> \u2014ACTS 8:17\u201324\n\nUnfortunately for Simon, God doesn't work this way. He was given an opportunity to repent, but that's not what he did. His asking the apostles to pray for him made it clear that he really had no intention of taking his new commitment to God to full relationship status. He wanted a quick route to the power, but he didn't want a relationship. He wanted the power without submission. He didn't want to lose anything. He didn't want to be humbled or made uncomfortable. He somehow wanted to do everything the apostles were doing, but he didn't want to submit to the lordship of Jesus. If we aren't careful, we may find ourselves in the same place.\n\nHave you ever been awake late at night and caught one of those weight-loss infomercials? It's as if they come on one after the other. They try to sell you all sorts of things they say will change your body without your having to do much work. \"Take this pill, and you'll automatically lose weight.\" \"Wear this shirt, and you'll automatically get a six-pack.\" If we're not careful, we can get sucked into being part of Generation Shortcut. We try to think of ways to get what we want without doing the work or feeling discomfort. We do this with the things of God as well.\n\nSome of us want the power of God without relationship with Him. We see someone operating in a powerful anointing, and we want what they have, but we don't want to go through what they've been through to get to where they are in God. No, this will not be us. We will do whatever it takes to see the power of God moving in this generation. I believe we will be a people who will accept the engagement ring. We will not reject what God has freely given us. We will not shortcut the process to receiving the power of God, thinking that it can be bought with anything we have other than our whole selves. We will be the generation that knows the power of God is freely given to those who submit to God, withholding nothing. Hear me: there is nothing you possess that can equal what He can give you. What He requires from all of us is a level of submission that says, \"I belong to You.\"\n\nGOD'S ENGAGEMENT RING\n\nThe Bible says the Holy Spirit (whose power we receive) is the confirmation that we belong to Jesus. This concept of confirmation is loaded with meaning. In Ephesians 1:13\u201314 (NIV) the Bible says:\n\n> And you also were included in Christ when you heard the message of truth, the gospel of your salvation. When you believed, you were marked in him with a seal, the promised Holy Spirit, who is a deposit guaranteeing our inheritance until the redemption of those who are God's possession\u2014to the praise of his glory.\n\nThe word _deposit_ is translated \"earnest\" in the King James Version, both words coming from the Greek word _arrabon_ , meaning \"pledge\" or \"downpayment.\" In context of the church being the bride of Christ, we can extend the meaning to include the modern Greek word _arrabona_ , which means \"engagement ring.\"\n\nBuilding from the concept of confirmation and deposit, 2 Corinthians 1:21\u201322 takes it further into ownership: \"He anointed us, set his seal of ownership on us, and put his Spirit in our hearts as a deposit, guaranteeing what is to come\" (NIV). The Holy Spirit is God's promise of marriage to His bride, the church. He is a confirmation that we belong to Jesus.\n\n_The Holy Spirit is God's promise of marriage to His bride, the church._\n\nWhen someone puts a ring on your finger, it says, \"I'm taken.\" When we submit to the Holy Spirit, we're saying: \"I'm taken. No one else can have me. Nothing else can have me. Nothing else deserves my affection and attention. Nothing else deserves my worship. I'm taken. I belong to someone. I have given myself away.\" In traditional wedding ceremonies the officiant will ask, \"Who gives this woman to this man?\" But in the kingdom we give ourselves away.\n\nSimon is an example of someone who came but did not have the right heart. He had no intention of fully submitting himself in relationship with God. Therefore God could not give him the gift and power of the Holy Spirit. God would not put a deposit, guarantee, or confirmation on Simon's choice because he was simply enamored with the power he saw operating in the apostles' lives. He wanted what they had, but he didn't want who they worshipped. This is the very definition of _prostitution_ \u2014trying to pay for something that should be given for free in the confines of a genuine intimate relationship. God illuminated the life of Simon the sorcerer to show us what happens when someone says he believes but doesn't want Him. Jesus knows why we come to Him, and because He gave everything He had to us, including His own life, He will not accept anything less than everything from us.\n\nRESPOND\n\nThe Holy Spirit is God's invitation to submission. If you don't want to be submitted, you can't have Him or His power. We are crying out for a perpetual move of God, one that the world has never seen. I've told you it's happening, but where we have to go next is total and complete submission. God is challenging every area of our lives to get us to respond to this one thing: either we belong to ourselves, or we belong to Him. This is not a plea to do better or try harder. This is a do-or-die moment: submit to Him, or be powerless in this next season. It's your choice.\n\n_The Holy Spirit is God's invitation to submission. If you Don't want to be submitted, you can't have Him or His power._\n\nIf you dare to submit to God, I challenge you to join me in this short, but dangerous, prayer:\n\n> _Holy Spirit, show me every area of my life that is not submitted to You, in the name of Jesus. Amen._\n\nNow let me forewarn you: Within the next couple of weeks there will be moments and circumstances in your life that will cause you to wonder where in the world they came from. Don't panic, and don't get mad at God. He will remind you about the prayer you just prayed, and He will remind you that He is with you. Do not doubt that you have made the best choice of your life. You have just opened the door to the supernatural power of God to flow through you as you step out as His witness. You also have received the greatest gift ever given to man\u2014God's engagement ring. It is the guarantee from God that everything He promised will come to pass.\nCHAPTER 7\n\nURGENT ANTICIPATION\n\n\"Behold, the days are coming,\" says the LORD, \"when the plowman shall overtake the reaper, and the treader of grapes him who sows seed; the mountains shall drip with sweet wine, and all the hills shall flow with it.\"\n\n\u2014AMOS 9:13, NKJV\n\nIN AMOS 9:11\u201314 the prophet foretold the rapid growth and abundance of the first-century church. As we pattern ourselves after the example of Christ and the apostles, this passage represents something prophetic for that generation and this one. It's telling us that God is accelerating the time between when we plant the seed of the Word\u2014indicated by the plowman\u2014and when we will see a harvest\u2014gathered by the reaper\u2014of souls, spiritual blessings, and an outpouring of the goodness of God.\n\nIn ancient times during the regular farming season the plowman and the reaper were never together. Those who sowed seed did it in one season, usually summer, and those who reaped did it in another season, usually fall. There was time between. Yet the Lord is saying that the days are coming when both will be on the field at the same time\u2014\"the plowman shall overtake the reaper.\" That suggests He is not only accelerating the harvest; He is also increasing the abundance of it. _The Message_ translates the promise this way:\n\n> \"Yes indeed, it won't be long now.\" GOD'S Decree.\n> \n> \"Things are going to happen so fast your head will swim, one thing fast on the heels of the other. You won't be able to keep up. Everything will be happening at once\u2014and everywhere you look, blessings! Blessings like wine pouring off the mountains and hills.\"\n> \n> \u2014AMOS 9:13\n\nThis is why I say with confidence, \"There's more.\" Now I add to it, \"But we don't have much time. Things are going to happen quickly.\" To keep up with what God is doing, we need to take on a posture of urgency.\n\nIn John 4 Jesus had gone to Samaria to talk to a woman at a well. At His specific appointment with her He began to reveal the fact that salvation was getting ready to come to the Gentiles (vv. 21\u201324). Though every other Jewish person would walk around the city of Samaria, He had a specific reason for going there. Jesus spoke prophetically to the woman, telling her about her life. This moment had such an impact on the woman that she went into the city telling everyone what Jesus had said and done. \"Come and see a man,\" she said, \"who told me everything I ever did! Could he possibly be the Messiah?\" (v. 29). As \"the people came streaming from the village\" (v. 30), Jesus looked up at His disciples and said, \"You know the saying, 'Four months between planting and harvest.' But I say, wake up and look around. The fields are already ripe for harvest\" (v. 35). What was He saying?\n\n_Wake up and look around. The fields are already ripe for harvest._\n\nWhile the disciples were trying to leave and worried about Jesus getting something to eat, there was a much more urgent need\u2014one that surpassed any natural need or want, such as food or Jesus's maintaining His reputation. Jesus was letting them know that it was harvesttime. Even though He had just planted a seed in the woman's heart, He also was reaping the harvest as the townspeople came out to see who He was. He made the statement in verse 35 to let them know that the normal period between sowing and reaping had been accelerated\u2014the plowman was overtaking the reaper. There is an urgency in the spirit when times and seasons accelerate, because if we don't get busy with the harvest, it will spoil. Just as Jesus needed His disciples to understand this, we too need to get this sense of urgency into our spirits.\n\nNo one is talking about the urgency of Christ anymore. These days sermons are about \"bless me,\" how to be a better this or that, or five steps to this and four steps to that. They want to talk about the latest movies and the most \"relevant\" thing in popular culture. I'm not against other people's preaching, but I remember a time when preaching seemed to be elementary. They kept it simple, and though some preachers may not have had the most scholarly grasp of the Scriptures, clever revelations, or full contextual prophetic understanding, they did have a relationship with Jesus and the urgent anticipation that any Sunday could be the last Sunday. They never took for granted that somebody might not have had the opportunity to hear about Jesus before he faced Judgment Day, so they would preach every week about Calvary, about the blood that never loses its power, and that Jesus is coming again. Now we have a generation of people who sit in church Sunday after Sunday and rarely hear of the blood and cross of Jesus\u2014and we certainly don't hear about the Second Coming. To skip the central messages of the gospel is to preach as if we have time.\n\nA SEASON OF ACCELERATION\n\nWhen Jesus entered the earth, acceleration began. The fulfillment of Amos 9:13 has been happening over the last two thousand years, but there are many who are unaware that Jesus's coming is near. In 2 Peter 3 the apostle Peter wrote about the last days to remind the believers that time was short. In verses 3\u20134 he said:\n\n> Most importantly, I want to remind you that in the last days scoffers will come, mocking the truth and following their own desires. They will say, \"What happened to the promise that Jesus is coming again? From before the times of our ancestors, everything has remained the same since the world was first created.\"\n\nIn other words, people will say: \"There's nothing different. Why are you preaching with such urgency? Everything is the same.\" They are forgetting that, according to the next few verses of 2 Peter 3, God intervenes in time and the affairs of men: \"And by the same word, the present heavens and earth have been stored up for fire. They are being kept for the day of judgment, when ungodly people will be destroyed\" (v. 7). If we believed this and kept it in our hearts, why do we think we have time? Peter went on to say, \"But you must not forget this one thing, dear friends: A day is like a thousand years to the Lord, and a thousand years is like a day.\" In other words, since the day that Christ came as a baby until now, it's only been two days to Him.\n\nLook at everything that has happened in two days. We are in a season of acceleration. To God, He just came. For us it may seem as though we've been waiting for a long time, but Peter assures us that \"the Lord isn't really being slow about his promise, as some people think. No, he is being patient for your sake. He does not want anyone to be destroyed, but wants everyone to repent\" (v. 9). Christ hasn't come back yet because He is gracious and merciful. He's slow to anger and patient. He could come right now, but He sees the condition of the earth. He sees that people think they have time, and in His mercy He's saying, \"Not yet.\"\n\nBut the day is coming:\n\n> But the day of the Lord will come as unexpectedly as a thief. Then the heavens will pass away with a terrible noise, and the very elements themselves will disappear in fire, and the earth and everything on it will be found to deserve judgment.\n> \n> \u20142 PETER 3:10\n\nThose of us who understand the times and season, like the sons of Issachar, are praying, crying, and falling on our faces so much because we know that the revival God is bringing may very well be the last one. This could be the last one!\n\n_Every time we come together could be the last time. Every time we open our eyes could be the day. How will you be found?_\n\nSome of us are confused, waiting on this or that to happen to signal the start of the last days because of varying eschatological beliefs. Others of us are in Matthew 24 and Daniel 10 with a calculator, trying to figure out how long we have left to sin before we get our lives right with God. The Bible says that \"the Lord will come as unexpectedly as a thief\" (2 Pet. 3:10). Even those who think they know will be surprised. Every time we come together could be the last time. Every time we open our eyes could be the day. How will you be found?\n\nI have a good friend from North Carolina. His name is Pastor Tony Jones. Tony is precious to our church because he flies in from North Carolina to Orlando every single month for our monthly Presence Nights. Unbeknownst to me, during one of the Presence Nights, Pastor Tony quietly placed on our church altar the name of one of his members who was battling sickness. We have a growing stack of papers on our altar, where people have placed the names of sick loved ones in faith. We've seen a number of people whose names were placed there miraculously healed.\n\nSometime later Pastor Tony asked me to minister at his church during their church anniversary, and I gladly accepted his invitation. The presence of God was strong that night as the people worshipped God. After I preached, there was a time of prayer, and the Spirit of God demonstrated His power among the people.\n\nPastor Tony asked me if I would pray for a precious lady who was dealing with stage IV lung cancer. She had been given just six months to live. As she came up, she was clearly weak from the medical treatments she received, and she was wearing a mask. As I prayed for her, I felt a tremendous heat, which sometimes comes when I'm praying for the sick, and I could sense the Holy Spirit working. With certain cancers there is usually not an immediate way to tell if anything happened, but I remember having a strong sense that something had taken place.\n\nMonths later I received a call from Pastor Tony about this lady. When she came to the service, she knew that nothing more could be done medically. The cancer had spread from her lungs to her neck and spine and was near the stem of her brain at that time. They were preparing for the worst, but Jesus met her that night. A few short months later, to everyone's amazement, her medical reports showed no sign of cancer! It's happening!\n\nA SPECIAL ANNOUNCEMENT\n\nOur cry for revival is not so our goose bumps can have goose bumps. We're not asking God for revival only so we can see more signs and wonders. It's not about gold dust, gold fillings, angel feathers, stigmata, anointed oil\u2014none of that stuff. We are here in this moment because God invited us here. He walked into our space to make an announcement.\n\nAccording to Micah 7, revival is a loud announcement, in the middle of corruption and scandal and sin, that the enemy's time is short. Our nation does not deserve revival. We have allowed the enemy to torment us with sin, scandal, and corruption. But our God is so merciful that He is sending revival anyway. He has put the enemy on notice that his days of ruling the earth are short. The more the fires of revival burn, the further out the enemy is driven. The closer God gets to us, the looser the grip of the enemy becomes.\n\nTime is coming to a close. We should be living with urgency and expectation in our spirits. Any hour could be the hour. Any moment could be the moment. Any day could be the day. None of us has the time we think we do. But the less time we spend here, the closer we are to an eternity there.\n\nGod has withheld the second coming of Jesus so that those who don't know Him have an opportunity to accept Him. He could come right this second and judge the entire earth, and everyone who has placed faith in himself would be found guilty and on the wrong side of judgment. But everyone who has placed his faith in Jesus Christ would receive the gift of eternal life.\n\nNo preacher knows the day or the hour, yet we must preach with urgency. No prophet knows the time or hour, yet we must proclaim and prophesy with urgency. God has made a loud announcement, yet He exercises His patience. As Moses discovered, He is \"the God of compassion and mercy! [He is] slow to anger and filled with unfailing love and faithfulness\" (Exod. 34:6). Though we don't know the day, we must be grateful for God's great love and patience and wait with anticipation.\n\nRESPOND\n\nThe Bible tells us that we are made right with God by placing our faith in Jesus Christ. That is true of everyone who believes, no matter who we are. This message has a twofold application:\n\n> 1. If you are saved and have surrendered your heart to Jesus, you don't have more time to try and get yourself together. You need to live with urgency.\n> \n> 2. If you have never surrendered to Jesus Christ, you don't have the time to keep running. Today is your day.\n\nWhat is amazing in both cases is that the Holy Spirit will reveal Jesus afresh to you. The best preaching can't do that. The Holy Spirit is the only One who can convince your heart of your need for Him. All you have to do is accept His gift of forgiveness and eternal life. You have the capacity to believe. He is taking the scales off your eyes, and the veil on your heart is being removed. Will you believe? Will you accept Him?\n\nIf you have said yes, pray this prayer with me now:\n\n> _Lord Jesus, I'm not running from You anymore. I'm running to You. I believe that You are the Son of God, who died for my sins, and that God Your Father raised You to new life. I receive Your sacrifice and forgiveness._\n> \n> _Today, I acknowledge that I need You. I surrender my life to You completely. I know I don't have time, so I pray that You will fill me with Your Spirit that I may have the urgency to cry out for revival. I want to see it. I want to be a part of the mighty things You will do in this season. In Your name, I pray. Amen._\nCHAPTER 8\n\nSTAY AND WAIT\n\nDo not leave Jerusalem until the Father sends you the gift he promised, as I told you before. John baptized with water, but in just a few days you will be baptized with the Holy Spirit.\n\n\u2014ACTS 1:4\u20135\n\nONE OF THE hallmarks of my life and of our church is that we are relentless about waiting in the presence of God. It's one of the things people often comment on when they visit Deeper. We've adopted the belief that God is not in our presence; we are in His presence. That distinction is critical to how we navigate our rivers (services). We don't rush through moments, wanting to move ahead with our plans for the gathering. We wait until He is finished with a moment. And then we wait until He directs us to the next moment. Some of the most significant miracles that have taken place have happened after the service was dismissed. Many times we will dismiss the service to give people the opportunity to leave if they need to, but often people stay. I've dismissed rivers before, only to see a couple of hundred people continue to wait in His presence for over an hour or more. We stay because we need Him. We wait because there is nothing more important than Him. Many believe it just isn't possible for people to wait in God's presence today, when so many just want to be in and out of the church service. But the Lord has consistently rewarded this posture of waiting.\n\nAs I mentioned previously, several of the most significant miracles that have taken place in our midst happened after the majority had left. This has happened both during our weekly gatherings and at our Presence Nights, which are monthly services specifically designed to give God room to move. We have no agenda at our Presence Nights other than to come and be in His presence and follow where He leads. We want to flow with Him. At Deeper we've come to understand that scripturally rivers are often used as a metaphor to represent the Spirit of God. The Lord has directed us concerning our purpose and potential through Ezekiel 47, in which the prophet had a vision of a river flowing from the temple. Our Sunday morning gatherings and Presence Nights are marked by being in the river of God's Spirit and waiting for the current to take us where He wants us to go. We have learned to throw away our oars and put up our sails.\n\nIsaiah 33:21 says, \"But there the glorious LORD will be to us a place of broad rivers and streams on which no boat with oars shall go and on which no gallant ship shall pass\" (MEV). On the river of God, boats with oars will not move. Oars represent human effort and strength, our ability to determine the direction and velocity of where we are going. On the river of God, the Spirit leads us, and we wait on Him to direct us. And we have seen and experienced amazing things as a result.\n\nIn July 2017 an entire day of services was dedicated to giving God thanks and praise for an incredible miracle He had performed for a family in our church. (I will share more about that miracle in chapter 9.) After the second service was over, as usual people remained in the sanctuary because we could still sense the presence of God. I decided to sit down in reverence when I felt a tap on my shoulder. I turned around, and a lady was standing there with a baby in her arms. The baby was four months old, and when I looked at her little body, it was limp, and she was in a somewhat catatonic state. The lady began to tell me that the Holy Spirit prompted her to come back in the church and not leave without having someone pray for this beautiful child. She explained that the child was partially brain dead, and the doctors said she would never smile, speak, or eat on her own. We prayed, and in an instant the baby's eyes focused and became normal. She smiled and began cooing and looking around, all things that she had never done in her young life. She was healed in an instant.\n\nAnother time there was a lady who had severe neuropathy. She utilized a walker and was partially blind. We prayed, and the Lord took away all her pain and restored her sight. Her husband came to the service carrying his wife; he left the service carrying her purse! The Lord healed another woman of lupus and rheumatoid arthritis the very week I sat down to write this. The arthritis had damaged her ligaments so extensively that her hands were shriveled up and contorted. As I took her hand in mine and began to pray, we literally watched her hand unravel like a flower bud opening to the sun. Jesus healed her and straightened her withered hands right in front of our eyes. She was astonished, and so were the people who were standing there. The Lord took away all the pain in her body.\n\nAnother young man had what doctors believed was nerve damage to his eyes and the early onset of glaucoma. When he opened and closed his eyes, the man felt excruciating pain and pressure. When he came to church, he was a little scared that he would completely lose his sight. We began to pray for him, and a heat radiated through his eyes, and he was instantly healed! The pain and pressure he was experiencing instantly left, and he was able to remove the special glasses he was wearing to protect him from light sensitivity. All these things happened after we officially dismissed. It seems as if something special is released to those who will embrace the posture of waiting.\n\nMost of us know the Great Commission as the words Jesus spoke in Matthew 28:18\u201320: \"I have been given all authority in heaven and on earth. Therefore, go and make disciples of all the nations, baptizing them in the name of the Father and the Son and the Holy Spirit. Teach these new disciples to obey all the commands I have given you. And be sure of this: I am with you always, even to the end of the age.\" But what some of us don't know is that, according to scholars, these words were spoken at the same time He declared the words of Acts 1:4\u20135. As a matter of fact, He didn't say this just to the eleven; He also said it to the five hundred Paul wrote about in 1 Corinthians 15:6: \"After that, he was seen by more than 500 of his followers at one time.\"\n\nNow, looking at the totality of what Jesus was saying, we can understand that before they were to go and make disciples of nations, He was commanding all five hundred of His followers to wait to receive the power to do what He was commissioning them to do. This is important because sometimes we get excited about the revelation of the call of God on our lives and rush out before God can give us what we need to accomplish it. He is telling us, as He told the five hundred, \"Before you go, I need you to stay and wait.\" What's interesting is that Jesus didn't tell them how long they were to wait. He just said, \"Wait until . . .\"\n\nNot everybody could get with that. So the Bible reports that only 120 of the ones who were present to hear the Lord's command went to the upper room to wait, which means that 380 people didn't go, or maybe they went but didn't stay. Whichever way we look at it, it seems that the 380 lost their tenacity, which we'll talk about more in the next chapter. These 380 were the majority who decided, like many in this generation, they had something more important to do than wait. There is nothing in our lives more important than waiting on God. Those who stayed remained because they knew they needed what Jesus promised was coming. They all had the same passion, which is another way _homothymadon_ , the Greek word for \"one accord\" in Acts 1:14 (KJV), can be translated. It comes from two Greek terms that mean to \"rush along\" and \"in unison.\" The 120 stayed because nothing else was more important to them. Inevitably there are always some who have another priority.\n\nThere is a precious lady in our church who has a smile on her face every time I see her. Since she is always smiling, I was unaware that she was very sick. She had a cancerous tumor on her side the size of a melon. When I was told about her diagnosis, a righteous indignation rose up in me. I knew I wanted to pray for her the next time she was at a service.\n\nOne Saturday evening I read about Smith Wigglesworth and the miraculous healings that took place through his ministry, and I was struck by how bold he was. When I went into the service the next day, those powerful accounts were still reverberating in my mind.\n\nThat Sunday the atmosphere of worship and faith was electrifying, and I asked if this precious woman was in the room. She was present and came forward. A boldness came over me as the other leaders and I spoke to the cancer and the tumor and prayed for her healing. After we prayed, we felt a tangible sense that the Lord heard us and would answer.\n\nThe following week when she showed me the spot where the tumor was, I could feel that it had shrunk from the size of a melon to about the size of a golf ball. And instead of being hard, it was now soft. Today she says she can't even feel the tumor. On top of that, after losing a lot of weight because of the cancer, she's at a healthy weight again, and she's still smiling! It's happening!\n\n\"I'M NOT LEAVING\"\n\nBut there was a group of people who kept the posture of waiting. In essence, they said, \"We're not going anywhere until we see it. We're not leaving until we receive it.\" We have the opportunity to duplicate this posture in this generation. Many already have. This is the posture of those who believe there is something the Lord wants to pour out, and they know He is only going to pour it out on those who will stay and wait. The experience of those who stay \"until\" is different from the experience of anyone else. Those who stay get God. The 120 of Acts 1 set a precedent. They decided they were not going anywhere until they received what they were promised, and they received God. They didn't try to manipulate God to get Him to work on their terms. They submitted themselves to His timing.\n\nThe 380, on the other hand, either got distracted along the way or decided it wasn't worth it. Impatience and spirits of control arose in them, and they began to move out of their position, perhaps questioning the details of God's command: \"How long are we supposed to wait here?\" \"What did He say was going to happen?\" They sat there for a while, fifty days, the Bible says, and nothing happened. Maybe they left one by one, wondering, \"What is this thing? What if it's not going to happen? Maybe it's not going to happen. Maybe I don't need this as badly as I thought I did. I have other things to do.\" The real question, however, is, What did they value more than waiting on God?\n\nThere's something about those who say, \"I'm not leaving,\" whether it's Jacob, who said, \"I'm not leaving until I'm changed,\" or the woman with the issue of blood, who grabbed the hem of Jesus's garment. They were not willing to let go once they grabbed hold of the promise. No matter what it cost them personally, they were going to hold on until they received what they had been waiting for. I can imagine that for the rest of Jacob's life, he was asked, \"What happened to you?\" I can also imagine that he was grateful to respond, \"I got hold of God.\"\n\nWe ought to have such an encounter with the Lord that it causes people to ask, \"What happened to you?\" After the 120 received the Holy Spirit, amazing works of the Lord followed them. Miracles, signs, wonders, and salvations accompanied their experience. People who witnessed the events wondered, \"What happened to them?\" What happened is that they waited and received from the Lord. If you will stay and wait, you too will receive more of God. If you stay and wait, people will ask you what happened. And if you stay and wait, you will be able to say, \"I got ahold of God because I wouldn't let go until I was changed.\"\n\nWORTH THE WAIT\n\nOne thing we need to understand as we wait is that God does not move according to our time. He created time, and He holds it in His hands. So when God invites us into postures such as waiting, we should know the answer to questions such as \"How long do I have to wait?\" As we discussed, Scripture is clear. We wait \"until.\" There is no time frame on that. As a matter of fact, some individuals didn't see the promises God made to them in their lifetimes. Think of Abraham and the many others in the Old Testament who \"died still believing what God had promised them. They did not receive what was promised, but they saw it all from a distance and welcomed it\" (Heb. 11:13).\n\nBut for us, when God has spoken something, we want God to manifest it within the next fifteen minutes, fifteen months, or whatever time frame we decide is reasonable. And if it doesn't happen in that time, we're discouraged.\n\n_We ought to have such an encounter with the Lord that it causes people to ask, \"What happened to you?\"_\n\nWe don't like to wait for anything. But what we need to know is that what God wants to give us is worth waiting for. Consider this: It takes seventeen to eighteen hours to build a Toyota, but it takes four months to build a Rolls-Royce. Part of the reason is because a Toyota is made by robots, and a Rolls-Royce is made mostly by hand. If a mistake is made in the manufacturing process, it's done ten thousand times with a Toyota but only once with a Rolls-Royce. And so, ultimately, if you want something of value such as a Rolls-Royce-level blessing, you will have to wait, and it will be worth it.\n\nGod has His hands on this generation, inviting us to get in the right posture to receive revival in our lives, homes, and community, and in the nations all over the world. And this is not a prefabricated move where a die has been cast. God is doing an original work in this generation for those who are willing to wait.\n\nWAITING WHEN IT'S _N OT_ HAPPENING\n\nMost of the modern-day revivals we've seen in the world from the 1950s until now can be traced back to a room in Argentina. In the late 1940s missionary Edward Miller had been ministering in the area and was not seeing results. Miller was frustrated. He had been going door-to-door, evangelizing people, doing tent meetings, and all the other things missionaries normally do, but nobody was showing up. It was the same story for all the missionaries in that nation at the time. It seemed that the heavens were closed over the region, as they were not making any progress. So over time many of the missionaries left, but Miller stayed.\n\nEventually the Lord invited him to do something: pray for at least eight hours a day. If you know anything about missionaries, you know that they raise money from donors to fund their efforts and provide them with basic living necessities. People do not necessarily send support so the missionary can stay in his room and pray all day. Donors want to know that missionaries are out among the people, witnessing and saving souls. Miller put all the expectations of other people on hold and accepted the Lord's invitation to pray. Of course, now you're probably expecting me to say the demonic oppression over the region broke and the heavens were opened. But as it turned out, nothing happened. He prayed and prayed and prayed, and nothing tangible seemed to happen. So finally he gave the Lord a deadline.\n\nDon't judge him; we do the same thing. We may not be quite so bold as to tell God, \"On January 1, I expect to see my breakthrough.\" But know that God understands our thoughts and actions that communicate the same thing. He knows when we have given Him an unspoken deadline because when He doesn't meet it, we stop calling the prayer line. We stop being specific in our prayers. We stop believing. We stop attending church. We lose our zeal. Whether we admit it or not, we gave God a deadline, and since He did not meet it, we stopped reaching out to Him.\n\nSo Miller gave God a deadline, but he verbalized it and said, in essence, \"I will keep praying, but I need to see You do something by ______, or I will return to my usual missionary duties.\" He continued to pray eight hours a day as God had told him to do, and then the day of the deadline came. Miller grabbed a pocketful of tracts, but just as he was heading outside to pass them out, another local pastor showed up at his door. The two ended up talking for hours. The other pastor was hardly an encouragement to Miller; he had troubles of his own and was even more frustrated and discouraged than Miller was. Miller ended up encouraging the other pastor rather than receiving the encouragement he needed.\n\nThe other pastor had brought his unsaved son with him. As they were leaving, which was now well past his deadline, Miller asked the teenage boy a searching question. One word led to another until the boy began to weep. He ended up giving his heart to the Lord. Of course you would think this was the moment of breakthrough, that the Lord had finally come through for Miller. But that was not the moment of breakthrough. Basically all God said was, \"I will bring them in when I'm ready. Go back and pray.\"\n\nFaithfully Miller continued to pray eight hours every day. One day as he was praying, the Lord invited him to a new season. In the new season, God gave him this instruction: \"I want you to do a prayer meeting in which you will pray every day from 8:00 p.m. until midnight. Invite people to join you. But whoever comes must commit to praying the entire time. If they will not commit to praying the entire time, ask them not to come.\"\n\nSo Miller did as instructed, and only a handful of people joined him in praying from eight to midnight. You're probably thinking this is when something happened, after they prayed. But no. They prayed and prayed and prayed, and at the end of that first night Dr. Miller asked, \"Has anyone heard anything from the Lord? Does anyone have an impression?\" One of the young women who'd been praying with the small group said, \"I have this strange impression to go and hit the table in the center of the room. But it seems too foolish.\" It was so foolish to her, she couldn't even be persuaded to try it. So they didn't hit the table, and they all went home.\n\nThe next night, they came back, they prayed, Miller asked if anyone had heard from God, and everyone said no. Though the young woman had the same impression to hit the table, she did not pursue it because it seemed foolish. This cycle went on for three nights. Finally, after having the same impression, she agreed to try it once everyone in the room had hit the table first. They hit the table, one by one, and nothing happened. Then it was the young woman's turn. She went up and hit the table, and a physical rushing wind entered the building. Everyone fell to the floor in worship and began speaking in other tongues. They had an encounter with God, and the heavens were opened over Argentina.\n\nBecause this small group was faithful to obey God's instruction to wait despite the fact that nothing had happened for so long, He began to move. They opened a Bible school, and just a few years later a student who was walking and praying in the woods had an encounter with God that led to such a move of the Spirit they had to cancel classes for months. People could not stop crying. In addition, God gave the young student a prophetic word. He prophesied about different places that would have revival in the earth before the Lord returns. And would you know, every city he mentioned has seen one. All this is because some people were willing to stay in a posture of prayer and waiting.\n\nWAIT, THE ANSWER'S ON ITS WAY\n\nIn Daniel 10 Daniel received a vision from the Lord that grieved him. He wept and went without food for three weeks. His posture was one of lament and intercession for what he saw concerning the future of his nation and the nations of the earth. Though the vision did not come with revelation, as it had in times past, he sensed that God wanted to do something. So for twenty-one days he did nothing but remain in the presence of God, praying for His plans to be revealed. Nothing happened for twenty-one days. And keep in mind, God did not tell Daniel that he would need to remain in this position for twenty-one days. When he embarked on this journey with the Lord, Daniel did not know how long he would need to intercede.\n\nBut on the twenty-first day an angel came and gave him the answers he'd been praying for. The angel said:\n\n> Daniel, you are very precious to God, so listen carefully to what I have to say to you. Stand up, for I have been sent to you. . . . Don't be afraid, Daniel. Since the first day you began to pray for understanding and to humble yourself before your God, your request has been heard in heaven. I have come in answer to your prayer. But for twenty-one days the spirit prince of the kingdom of Persia blocked my way. Then Michael, one of the archangels, came to help me, and I left him there with the spirit prince of the kingdom of Persia. Now I am here to explain what will happen to your people in the future, for this vision concerns a time yet to come.\n> \n> \u2014DANIEL 10:11\u201314\n\nThe angel was saying that a heavenly decree had been made on the first day Daniel prayed, but the prayer had been held up. So he had to call Michael to fight with him so he could get the answer to Daniel.\n\nIn order for an answer to be released on the earth, it has to be given to someone who has jurisdiction in the earth. In order for a decree or an announcement to take root, it has to take root in the place where there is jurisdiction. There was a decree made in heaven, but it had to get to someone, in this case Daniel, who had divine jurisdiction over that region. In Daniel 11 we learn that because the decree was about Persia and the prince of Persia had no ability to stop it, he tried to keep it from being declared. He thought he could delay the answer long enough and outlast Daniel's ability to wait for it. But Daniel waited, holding a posture of intercession and fasting. The prince of Persia could not outlast Daniel's posture.\n\nI need you to understand the power of waiting in this season. What the enemy was trying to do then and what he is trying to do now is hold up the answers to your prayers so you will get so frustrated you give up. But as with Daniel, God heard your cry on the first day. Some of you are getting frustrated because you believe that God hasn't heard you. But I want you to know that the moment you lifted your voice and began to cry out to God for understanding, the moment you lifted your voice and began to cry out to God\n\nfor revelation, or the moment you lifted your voice to cry out for your son or daughter, your city or your nation, the Lord heard you.\n\nThough the enemy tries to hold up His response, God is releasing the answers to those who have found His favor.\n\nI believe that we are in a season of breakthrough. This is a season when answers that have been held back for a long time are being released. The enemy knows he can't stop the decree. All he can do is fight its timing. Once the decree has been released in heaven, the enemy can only delay it. He knows that when it gets to us and we speak it, it cannot be stopped.\n\nPsalm 2:7 says, \"I will declare the decree of the LORD\" (AMP). This means you need to agree with the Word of God\u2014every single thing the Word says. You must speak the Word. You must speak the promises of God. Once you speak them, they will not delay. They will perform as God intends them to. They will not return to Him void.\n\nOne last amazing thing about Daniel's story: the spirit prince of Persia thought, \"If I can hold it up long enough, perhaps he'll move.\" But the Bible says that when the angel found Daniel, he was in the same place. He never got off his face.\n\nGod is challenging you to hold your posture. Do not stop seeking God until you have received what He has promised. It will be worth the wait.\n\nWE WON'T STOP UNTIL WE SEE IT\n\nThere is a statement that has gripped the heart of our church and has become a literal rallying cry for us in prayer: we won't stop until we see it! We are in a relentless pursuit for revival to continue not only among us but everywhere.\n\nIn 2 Kings 13 we see the circumstances surrounding Elisha's final prophecy.\n\n> Now Elisha had become sick with the illness of which he would die. So Joash the king of Israel went down to him and wept before him, and said, \"My father, my father, the chariot of Israel and its horsemen.\" Elisha said to him, \"Take a bow and arrows.\" So he took a bow and arrows. Then he said to the king of Israel, \"Draw the bow.\" So he drew it. Elisha put his hands on the king's hands. Then he said, \"Open the east window.\" So he opened it. Then Elisha said, \"Shoot.\" So he shot. Then he said, \"The arrow of the deliverance of the LORD, and the arrow of deliverance from Aram; for you must strike Aram in Aphek until you have destroyed them.\" Then he said, \"Take the arrows.\" So he took them. Then he said to the king of Israel, \"Strike the ground.\" So he struck it three times and stood there. Then the man of God was angry with him and said, \"You should have struck it five or six times. Then you would have stricken Aram until you had finished them. Now you will strike Aram just three times.\"\n> \n> \u20142 KINGS 13:14\u201319, MEV\n\nIn this passage King Joash went to Elisha and received prophetic instructions to open the east window and shoot an arrow. Elisha put his hands on the king's hands, and the king shot the arrow, which was followed by the word that it was an arrow of deliverance from Aram. Then Elisha told him to take the arrows and strike the ground, and the king took the arrows and struck the ground three times and stood there. The Bible says Elisha became angry with him and said, \"You should have struck it five or six times. Then you would have stricken Aram until you had finished them. Now you will strike Aram just three times.\"\n\n_We won't stop striking the ground until we see revival in our city, our nation, and around the world._\n\nWhy would Elisha be angry that the king stuck the ground only three times when he didn't tell him how many times to strike? The implication here is twofold. First, the fact that the king struck the ground three times and stood there could imply that he thought the instruction was silly and didn't see the point. It's as if he were saying, \"What will this do?\" How often do we do the same thing? If we don't see the direct correlation or an immediate result, we question the effectiveness of continuing in prayer or fasting or believing and declaring. We do something a few times and then stand there with a posture of \"Let's see if this does anything.\"\n\nThe second implication informs the posture we have taken at our church. He struck the ground three times, and Elisha was angry and said he should have struck it five or six times. The king stopped striking the ground after three strikes, but he was never told how many times to strike, which indicates he was to strike until he was told to stop!\n\nThe thought here is he didn't strike the ground with full confidence that it would bear fruit, so he stopped. How often do we stop before seeing full victory? Because of the king's actions Elisha told Joash that he would see only three victories. Had he kept going, his enemy would have been destroyed.\n\nWe've made a commitment to take the posture that we won't stop striking the ground until we see revival in our city, our nation, and around the world. Just because it's happening among us doesn't mean we can stop striking the ground. We are not after just small victories or a local movement. We are believing for a global outpouring, a global revival. We believe if it's happening anywhere, it can happen everywhere, and we won't stop until we see it!\n\nRESPOND\n\nGod is looking for a generation that is willing to wait. Wherever you were when you received instruction or a promise from the Lord, go back to that place. When you first learned of it, did you pace the floor to see it manifest? If so, go back to that posture. Have you let your tears dry up? Open them up again. Have you stopped crying out to God? Open your mouth again. The Holy Spirit is telling me to tell you to go back. Go back to what you were doing. Don't leave your posture. Keep waiting. Keep praying. Keep believing. Keep fasting. It's happening!\n\nJust because you don't feel it doesn't mean it's not happening. Just because you don't see the answer doesn't mean He hasn't heard you. It's held up right now, but the answer is on the way. We are entering a season of breakthrough. I declare things that have been held up for a long time will break through for you, in the name of Jesus. The fight is about to be over, and the answer is about to be released. Stay and wait.\nCHAPTER 9\n\nCONTEND FOR THE FLAMES OF REVIVAL\n\nIn the womb [Jacob] took his brother by the heel, and in his maturity he contended with God.\n\n\u2014HOSEA 12:3, NASB\n\nGOD WANTS TO know how badly you want revival in your life and in your world. Have you reached the point where you are so comfortable with your life that you are willing to continue to contend with yourself and with man to get what only God can give? Would you believe me if I told you that there are some things that belong to you, that have always belonged to you, that you will have to contend with God for in order to see?\n\nThe difference between immature and mature believers is that the immature think their problem is people. The immature always think there is someone trying to keep them from their blessing. It's the immature who are always talking about haters. It's the immature who are always saying, \"Don't let people do this,\" or, \"Don't let people do that.\" For some reason or another they feel as if they are wrestling with men.\n\nMature people do not contend with other people. They are not worried about what other people think or say about them. They are not worried about how they look to others. And for those of us who haven't reached this place, this is the reason the church has not grown up. We have not been able to get to the place God wants us because we're still concerned about what the world thinks. We think that somehow we're contending with the world, and if the world likes us, then we'll be at a good place. But the things of God cannot be managed or dispensed through the systems of the world. They can come only from God. There is a blessing with your name on it, and there is no man you have to wrestle with to get it. You will have to contend with God.\n\n_There is a blessing with your name on it, and there is no man you have to wrestle with to get it._\n\nContending with God is about posture. It's about how deliberately desperate you are to see the glory of God fill the earth. Will you contend with God for it? Will you press through the crowd to get it? Will you do whatever it takes to see it? The stories of Jacob, the woman with the issue of blood, and others who acted tenaciously throughout history will give you the faith to hold your position until you see what you've been praying for.\n\nHOW BADLY DO YOU WANT IT?\n\nSomehow we've bought into a lie that God is going to grace us with His presence and everything we want if we just keep showing up. We act as if He is going to say, \"Oh, OK. You have perfect attendance, so I'll just do something for you.\" But it won't happen that way. Where are the hungry? Where are the thirsty? Where are the desperate? Where are those who say, \"I need You more than anything else. Nothing else matters. Nothing else will do. Nothing else will satisfy or satiate this thing I have on the inside of me because deep in my soul I am crying out for You\"? God wants to feel our demand for His presence and power.\n\nYou're reading this book, and I'm saying, \"It's happening.\" Maybe you've latched on to this truth as well and have begun to proclaim it in your own life. But I hope you are not missing the heaviness of the condition attached to having God move in your life. The question He is asking is, \"How badly do you want it?\"\n\nI know you don't get the question right now. It's heavy. It has a condition attached to it. I believe we are going to continue to see increasing waves of revival come to our land. I believe that. But it's attached to a people who want it. How badly do you want it? What are you willing to do to get it? The pathway to revival will not get easier. There is no easy path to this.\n\nGod is saying this across the earth: Is there a people who want revival so badly they will do anything to get it? Are you willing to wrestle Me for it?\n\nOne of the gifts of the Spirit that is in operation on a pretty regular basis at our church is the gift of words of knowledge. I, along with several members of our pastoral team, flow in this gift. One Sunday while I was traveling overseas, my friend Caleb was leading the services, and he often flows in this particular gift. The Holy Spirit gave him a word of knowledge about a person dealing with an issue with his or her esophagus that the Lord wanted to heal. The Holy Spirit then gave the church an instruction that to some might seem unusual, but I'm just going to tell you what happened that day.\n\nThe Spirit of God gave the instruction that we were to shout to the Lord in praise, and as we did, He would heal whoever had the esophageal issue. Having experienced so many miracles in our midst, our church did so in faith. Afterward a girl identified herself as being the one dealing with the esophageal issue. She said she felt something happen during the shout. Later she reported that when she got home, all her symptoms were gone. She had previously been unable to swallow, and in the midst of the praise that issue was completely resolved. It's happening!\n\nTHERE ARE NO ACCIDENTAL MOVES OF GOD\n\nThere are moments when the Lord reveals certain things to us, enlightening our understanding and highlighting things for us. One evening as I was reading Luke 8, it was like the Lord put a neon highlight on the word _deliberately_ in the following passage:\n\n> As Jesus went with him, he was surrounded by the crowds. A woman in the crowd had suffered for twelve years with constant bleeding, and she could find no cure. Coming up behind Jesus, she touched the fringe of his robe. Immediately, the bleeding stopped.\n> \n> \"Who touched me?\" Jesus asked.\n> \n> Everyone denied it, and Peter said, \"Master, this whole crowd is pressing up against you.\"\n> \n> But Jesus said, \"Someone _deliberately_ touched me, for I felt healing power go out from me.\"\n> \n> \u2014LUKE 8:42\u201346, EMPHASIS ADDED\n\nThrough this story the Lord revealed to me that there are no accidental moves of God. We've heard stories of accidental wealth, where people win the lottery or receive unexpected sums of money through inheritance or a lawsuit. But these accidental millionaires don't tend to keep the money. Overwhelming statistics have been reported showing that within a few years some of the people who win the lottery, for example, are right back to where they were before the winnings. They are right back in the same financial straits because they squandered the money.\n\nA 2009 ESPN documentary called _30 for 30: Broke_ looked at the myriad of athletes who had earned multi-millions of dollars over the course of their career and lost it within five years after their retirement. Of course, these stories include people taking advantage of them and poor investments. Still, regardless of why they lost the money, their stories are heartbreaking. Yet they reveal the reality that anyone who steps into something unprepared or by accident\u2014not with intention\u2014will lose it. This is a revelation for all who want to receive the blessing, presence, and power of God in their lives, including you: that for which you are not prepared you will lose.\n\nAs I mentioned in chapter 5 while discussing empty vessels, God will only send what you have the capacity to receive. Think in these terms: the word _receive_ can also mean keep, hold, or contain. God doesn't want people who have holes in their pockets to try to contain the revival He is sending. God is not going to send revival to a people who don't have wisdom to steward His blessings. In 2 Kings 4 we see how God worked through Elisha to bring miraculous provision into the widow's life, and we see how the oil only stopped flowing when the containers stopped coming. God is not short on supply. It is that He will not send what you do not have the capacity to receive.\n\n_God will not send what you do not have the capacity to receive._\n\nMay I suggest this? God won't waste revival.\n\nGod won't waste revival on a generation that wishes to happen upon the things of God by accident. He will not squander revival on those who won't create capacity for it. He won't waste revival on a generation that will say, \"That was fun while it lasted.\" They desire to play in the gifts of God and lavish in the blessings, but when the fun and feelings are over, they're on to the next thing. \"On-to-the-next-thing\" people will not experience revival. What we create capacity for is what we will receive. If you stop creating room for it, it will stop flowing.\n\nTrue revival doesn't happen by accident; it is deliberate. If I'm going to experience God, it's not going to be by accident. If I'm going to experience God, it's not going to happen by being random. Once I know who He is and find out I have access, my actions will become deliberate.\n\nWILL YOU CONTEND WITH GOD FOR IT?\n\nThis is a lesson for the mature in Christ. Will you hear it? The Bible says Jacob matured, not because he wrestled with man but because he contended with God. We know the story of Jacob and how it was prophesied that though he was not the firstborn, he would still be blessed. We also know that from the womb Jacob contended for the blessing in his own strength by grabbing the heel of his twin brother, Esau, almost as if to fight him for the position of being the first one out. He didn't know the blessing was already his.\n\nIn the next and most notable instance, Jacob propositioned his brother at a weak moment to sign over his birthright to him. Jacob was used to working in his own strength\u2014manipulating, lying, and stealing\u2014for the things God had promised. Esau was so angry at Jacob that he wanted to kill him. Rebekah, their mother, told Jacob to flee to her brother Laban's home. Jacob had a whole other set of issues during his fourteen-plus-year stay there: Laban treated him badly, changed his wages ten times, and gave him the wrong daughter to marry.\n\nGod finally gave Jacob favor and released him from serving Laban, but Laban came after him. Running from Laban, Jacob had nowhere to go and thought about going back home. Then he remembered the broken relationship with his brother. In an attempt to mend the relationship, Jacob sent his family ahead of him, along with some gifts, to meet Esau. He hoped this gesture would soften Esau's heart. But for Jacob to live at peace with Esau, he needed to first make peace with God.\n\nLeft alone in the camp, Jacob was so restless he couldn't sleep. A man, the Angel of the Lord, God Himself, came and wrestled with him all night. (See Genesis 32:22\u201332.) In the moment when He came and tested him, God knew Jacob's destiny. He knew Jacob was used to fighting and getting his way. Though He wrestled all night, God had something else for Jacob. God took the struggle to another level when He reached out and touched Jacob's hip socket. I don't know if you have ever had anything dislocated in your body. I can only imagine.\n\nWhen you watch sports, you can see people getting their fingers or shoulders dislocated, but a hip? The Bible says Jacob still didn't let go. Jacob made the more painful decision. Most of us, when we're hurting, let go of God to deal with our hurt. We get a little upset when bad things happen, and we recoil. We say, \"Hey, I'm not going to come to church for a little while. You're not going to see me for a minute. I have to deal with some stuff. There are some things I need to get over.\" We let go. But God wanted Jacob to see what it takes to really press through to see a change. God wanted Jacob to show how badly he wanted it. \"Do you want it so bad,\" God's actions may have suggested, \"that even if I touch your hip socket, even if I allow pain in your life, and even if the pain results in a physical or emotional scar or handicap you have to live with for the rest of your life, you will still hold on?\"\n\nJacob made the more painful decision\u2014he held on instead of letting go. He held on instead of letting go to fix it himself. He chose to hold on to God because he knew God had something for him that was greater than all he had fought for up until that moment. So he cried out, \"I will not let go until You bless me! I will not let go until I see, hear, experience, walk in Your blessing.\" So God said, \"OK. You want change so badly? Here it is. Since you contended with Me, I am going to give you the blessing that was meant for you from the foundation of the earth. I am going to change your name so that everyone who sees you will call you by a new identity. But I am going to leave you with your limp.\"\n\n_Jacob matured, not because he wrestled with man but because he contended with God._\n\nSome of us are so frustrated with God because He won't remove certain things from our lives, but that limp is the testimony. It is the thing you have to prove that you've been with God. It's the visible thing that others will ask you about, and you can say, \"There was a day when I used to depend on myself. There was a day when I used to think I could do it myself, when I thought my method of movement was good enough. But then I met Jesus, and though He afflicted me, He also kept me. I walk with this limp so I won't forget to lean on Him. It's my testimony.\"\n\nJacob saw revival that night in the wilderness. All his life he fought to bring revival in his life through his own striving; he contended with Esau and Laban and any other circumstance that stood in the way of the promised blessing. But it wasn't until his encounter with God Himself that his fighting spirit was brought to its knees.\n\nEvery encounter with God changes you. It is biblically impossible to have an encounter with God and stay the same. But it's the desperate who hold on through the encounter no matter how painful it gets, who show God that they are willing to go wherever, do whatever, give whatever, and pray whatever to see it. Jacob showed his desperation. He showed how badly he wanted it. Now it's your turn to answer: How badly do you want it?\n\nDELIBERATE DESPERATION\n\nCertain rhythms and patterns in our lives unlock things in the spirit. God is a God of order, systems, and patterns. He intends for us to do certain things to see His power become active in our lives. He highlights one of them in the story of the woman with the issue of blood in Luke 8. In the midst of a mass of people pushing and bumping against one another, one woman did something different from everyone else, and Jesus noticed it immediately. \"Someone _deliberately_ touched me,\" He said. One of my best friends, Jason McMullen, who is the executive pastor at our church and the closest thing to a brother this only child will ever know, said it this way: \"The deliberate will see a demonstration.\" I'll add this to it: the random will sometimes see relief, but the deliberate will see a demonstration.\n\n_\"The deliberate will see a demonstration.\" \u2014Jason McMullen_\n\nIf there is anyone I know intimately who has walked out this truth, it is Jason. One of the most significant miracles I've ever heard of and certainly ever experienced took place for his family. The testimony of this miracle will be its own book one day, and it is one that will cause me to thank God for the rest of my life. Each of us who was there has his or her own perspective of this massive miracle. On the evening of July 3 I received a call from my friend and brother Jason asking me to pray because he was on the way to the hospital with his wife. At the time she was carrying their ninth child and was thirty-one weeks into the pregnancy. Their ten-year-old son had called Jason and said, \"Dad, you need to come home because something is wrong with Mom.\" Jessica, his wife, said she wasn't feeling well. She had a severe headache and body aches, and she was vomiting. When asked about her level of discomfort, she said it was at ten. Jason later said that if any woman who has already given birth to eight children says her level of discomfort is a ten, you need to get her to the hospital.\n\nJessica knew something wasn't right, but she mainly wanted to make sure that the baby was OK. When they arrived at the hospital, the doctors initially didn't find anything wrong and were going to send her home. But she knew something wasn't right. They did further tests and concluded that she had spinal meningitis. Jason called to update me so we could continue to pray, and he told me at the time that they were going to be moving her to a different hospital that would better be able to care for mom and baby.\n\nAs Jason was gathering their things to meet Jessica at the other hospital since she was being transported by ambulance, one of the nurses approached him. The nurse asked him if he could walk him to his car. As they walked, the nurse told Jason that he was a believer and was praying with them. He then proceeded to tell Jason that while the initial diagnosis was spinal meningitis, he saw the doctors checking for something else far more serious: an amoeba.\n\nAn amoeba is a microorganism that can live in fresh water in subtropical climates whenever the water temperature rises above a certain point. An amoeba eats your brain. The average life expectancy of someone who gets an amoeba is between three and five days, and the mortality rate is 97 percent. The nurse told Jason he wanted him to know what they were possibly dealing with so he would be able to pray. Upon learning of that possibility, Jason called me so I could believe God with him. Around midnight I received another call from Jason saying that the doctors confirmed that his wife had an amoeba. When he arrived at the hospital, he learned that his wife was in the neurocritical ICU, and he was greeted by a team of doctors ranging from infectious disease specialists to an OB\/GYN.\n\nThey told Jason that his wife had a 3 percent chance of survival and that they needed to give her a medicine so powerful it might kill her, but if she did not receive the treatment, she wouldn't survive the amoeba. Jason is a man full of faith, so upon learning the news, he brought the doctors together to pray for wisdom, healing, and direction. After they finished praying, he was also told that they needed to immediately deliver the baby because the baby wouldn't survive the medicine or the amoeba while in the womb. They then began asking about the other eight children, since they all had visited the same lake where it was believed Jessica contracted the amoeba. When Jason called home to ask about the other children, two of them were exhibiting the same symptoms as Jessica, so they were immediately taken to the hospital. After being tested, both were diagnosed as having the amoeba.\n\nBy the next morning two of their nine children had been diagnosed as having the amoeba; five of their children were in the hospital under observation, with the doctors expecting them to also test positive for the amoeba; and their newborn, delivered via C-section the previous night, was in the NICU. Jason's wife and eight of their children were in the hospital, with two of his children and his wife being given a 3 percent chance of living and a life expectancy of three to five days. The doctors wanted to place Jessica and the two children who had been diagnosed with the amoeba in an induced coma, lower their body temperature, and drill a hole into their heads to administer the medicine.\n\nWe were praying the entire time. I knew we needed to pray around the clock, so I didn't sleep that night. I stayed up to pray. I remembered a time prior when Jason asked me if I thought God had given me the gift of healing. He jokingly said, \"It's not like the rest of us feel heat every time we pray.\" At the time, that was something I had never considered. But as I was up praying for Jason's family, I remembered that conversation, and I said, \"God, if You have truly given me the gift of healing, please use it now.\"\n\nThe next day I met Jason at the hospital. I remember the peace he had; it was a peace that could only come from God. At one point he looked at me and said, \"This must be happening so that God can get the glory out of this!\" We spent time going back and forth visiting all of the children and Jessica and praying. Jason would often retreat to the chapel to pray and cry out to God for his family. He would not relent in his faith or waver in prayer. Neither did we. The details of what took place will become its own book, but I can tell you that God responded to our prayers. Almost as suddenly as the entire ordeal began, a miraculous thing took place. A nurse whom they had not seen or interacted with prior to this moment sat down on the edge of the bed and casually said to Jason, \"Have you heard? No amoeba!\"\n\nTwenty-four hours later all six children were sent home cleared of the amoeba. No coma, no medicine, no hole drilled into their heads. A week later Jessica was home. No amoeba, no hole drilled in her head, no powerful medicine to kill an amoeba. As I am writing this, we are not aware of another case in the recorded history of the United States where this particular sickness was healed without medicine!\n\nThe baby is also home and is completely healthy. We serve an awesome God! The enemy intended to have seven caskets at the front of our church and for the national news to broadcast a story about how an entire family died due to a rare brain-eating amoeba. But the Lord turned our potential mourning into dancing and our potential sorrow into joy. We have living, breathing miracles among us every week when the McMullen family comes to church. It's happening!\n\nWhen you randomly go through your life, sometimes you bump into relief because God is just that good. But the deliberate will see a demonstration of the power of God. There is a difference between being around Jesus and experiencing Him. Desperation can lead you to deliberate action that leads to seeing a demonstration of God's power. Those who stay in the posture and passion of consistently seeking God with desperation will see Him move in the deliberate way He intends.\n\nThe Bible says, \"A woman in the crowd had suffered for twelve years with constant bleeding, and she could find no cure\" (Luke 8:43). It goes on to tell us that this woman had run out of options and money trying to address this issue of hemorrhaging. She had done everything she could possibly do to be healed, set free, and delivered from this thing. And because the issue caused her to be labeled unclean, it had also become an embarrassment to her.\n\nThen she heard that the healer was passing by. With large crowds surrounding Him, she had to make a life-or-death decision: \"If I go out, knowing that I'm unclean, and He doesn't heal me, I will be killed. But if I stay in and keep bleeding, I will die.\" She was in a desperate situation.\n\nDesperation is accompanied by corresponding action and can go a couple of ways, as I've already pointed out. Desperation can lead you to unwise decisions, or it can lead you to the right destination. Desperation, in the case of this woman, led to deliberate actions that led to the right destination. Sometimes God creates situations for you to understand that He's the only One who can change it. He's the only One who can fix it.\n\nSo in her desperation the woman chose the deliberate action of facing the large crowd and taking a chance on the Healer.\n\nDON'T BE AFRAID TO REACH OUT AND GRAB IT\n\nPressing her way through the crowd, touching people despite her uncleanness, she came up behind Jesus, reached out, and touched the fringe of His robe. Immediately the bleeding stopped. The Bible does not say she tapped Him on the shoulder or called out and waited for Him to take notice of her. I imagine that, at some level, two things could have happened. One, her shame started speaking so loud that she felt too low to draw any attention to herself. She probably didn't have the confidence to stand up and look Him in the eye, so she reached out and grabbed His garment. Two, He was passing by, and she thought that if she did not reach out and grab of Him what she could, she would miss her chance and not be healed that day.\n\nHow many of us will almost miss our time because we're afraid to reach out and grab hold of Him. At some point our situation will become so dire that, like the woman, we will resolve within ourselves to say, \"I don't care what they say. I don't care what it looks like. I don't care if they talk about me. Jesus is about to pass by. I cannot let this moment go! I can't let this moment pass me by! I can't let this season pass me by! I can't let this word pass me by!\" If God is doing something, if revival is happening anywhere, it can happen everywhere. We should be desperate enough to want it to happen to us. We have to reach out and grab Him deliberately, and He will know someone stepped out and did something different.\n\nJesus said, \"Someone deliberately touched me.\" Peter was wondering how in the world Jesus would have noticed with all the people pressing in around Him. I can imagine Peter thinking, \"All these people around? All the people who are associated with You? All these people who name Your name? All these people who go to church and lift up their hands? All these people who are singing these songs? All these people who show up every week? Jesus, don't You know that this church has twenty-five thousand people attending services every weekend?\" But Jesus is saying, \"Somebody is reaching out on purpose! Somebody has such a need that it is drawing something out of Me.\"\n\nWe're not going to accidentally enter revival. The Lord is literally saying to us, \"Who is willing to be desperately deliberate?\"\n\nTHE DELIBERATE ARE A TENACIOUS BUNCH\n\nIn addition to contending with God and not with man, another quality of a mature believer is tenacity, the ability to maintain a level of endurance and patience during difficult times. The apostles praised the Thessalonian church for this attribute: \"As we pray to our God and Father about you, we think of your faithful work, your loving deeds, and the _enduring hope_ you have because of our Lord Jesus Christ\" (1 Thess. 1:3, emphasis added). This group of believers endured affliction and persecution and still are shown as examples of those who \"remained strong in [their] faith\" and \"firm in the Lord\" (1 Thess. 3:7\u20138). The entire narrative of Scripture would be completely different if it were not for groups of people who were tenacious in their pursuit of God. There are so many stories that would be different if it were not for the tenacity of the people who lived them. We have these stories to show us how we are to live.\n\nEven in more contemporary history, our nation would look different if it were not for people who were tenacious. Every age has a tenacious group of people who will not give up easily. Think about the civil rights movement of the 1950s and 1960s. Civil disobedience through nonviolence took tenacity. It took tenacity for Elizabeth Eckford to enter Little Rock Central High in 1957 shortly after segregation laws had been overturned. Moments such as that one, captured in an iconic photo, are a testament to the tenacity of those determined to see black Americans have access to the same opportunities available to others. As a father of two girls, I don't know that I could send my daughters into the kind of hostile environment Eckford entered that day. Then there was the 1960 sit-in at the Woolworth counter in Greensboro, North Carolina. Four African-American college students sat down at the lunch counter to protest the denial of service to black diners. They knew they were about to be beat up, arrested, and thrown into jail, but they had the tenacity to say, \"We're doing this anyhow. We will not be denied what we know is possible.\"\n\nMore famously, Rosa Parks chose not to give up her seat on a bus in Montgomery, Alabama, also showing a tenacious attitude. Some of you reading this book lived through times like these. Those now iconic moments were part of a larger pursuit of equality. The tenacity of the people actively and deliberately seeking what they knew was theirs sparked a worldwide revolution. From India to South Africa, oppressed people joined together saying, \"We're going after something noble, something that should be for all people, and we're not going to stop until we get it.\" Each of these moments fueled the momentum for something greater. These were tipping points for single communities that became catalysts for other movements across the country and around the world.\n\nTHE TIPPING POINT\n\nIn his book _The Tipping Point_ , Malcolm Gladwell defines _tipping point_ as a series of consistent things that leads to the overflowing. So from this concept you wouldn't be able to point to one specific thing that caused another thing to happen, but the consistency of a set of actions can lead to an overflow. One drop of water from an eyedropper into a glass is not the catalyst for the glass overflowing, but consistently dropping the water in the glass can fill it and cause the water to run over. Considering the tipping point for revival, you never know what your next hallelujah will do. Or what your next \"Thank You, Jesus\" will do, or the next time you lift your hands or give an offering. You would not be able to say it was this offering or that praise that led to revival. It's the consistency of all the parts that leads to a tipping point.\n\nEven the fight for civil rights in the fifties and sixties didn't start in the twentieth century; it began the moment people were oppressed. It just so happened that, because of several different factors, it came to be a catalyst moment in the fifties and sixties. People who were not connected with the terrible state of how certain people were treated were suddenly forced to take notice.\n\nSo it applies that once revival breaks out somewhere, it causes people, bit by bit, to take notice of what's happening until the ripples are felt everywhere. Once revival breaks out somewhere, it causes people everywhere to take notice of what's happening. And the response will vary. On one hand people may feel revival is not for them. They're not that desperate. They're not hungry. Others may say, \"I need to change. I see what's happening with them over there, and I want that for my life.\" They may be encouraged to return to the place they had been before, but they feel that tenacity rising in their spirits, challenging them to contend for more.\n\nKEEP ASKING, KEEP KNOCKING, AND KEEP SEEKING\n\nBased on what I saw the Lord do in my third year of pastoring Deeper Fellowship Church, I am calling out to all who want to see revival. Do not be the generation that says, \"If at first you don't get an answer right away, it must not be God's will.\" To see the blessing of God fall on this generation, we will need to be desperate enough to contend with God and deliberate enough to reach out and grab hold. The principle of the tipping point does not align with how we approach things these days. We are not going to go from one methodology to another. We are going to press in and hold our posture before the Lord.\n\nJust like the woman in Luke 8 who saw a window of opportunity to be healed, tenacity causes us to see moments of opportunity that others don't see. It's easy to overlook something when you aren't searching for it. The tenacious are constantly seeking, knocking, and asking. When God spoke to the prophet Jeremiah and said, \"You will seek Me and find Me, when you search for Me with all your heart\" (Jer. 29:13, NKJV), contextually, He was saying, \"I'm going to create a circumstance or an environment that literally causes you to look for what you weren't looking for previously. And when you look for it, you'll find it.\" Jesus said it too: \"Ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you. For everyone who asks receives, and he who seeks finds, and to him who knocks it will be opened\" (Luke 11:9\u201310, NKJV).\n\nFor the believer, this posture is the essential, perpetual internal posture. Living from this place is key to continual momentum. It is the fuel of the engine of revival. Tenacious pursuit and perpetual seeking are the evidence of believers' recognition of their need for God.\n\nRESPOND\n\nFrom the beginning of your life until now there is evidence of the Lord's random acts of love and kindness toward you. Maybe you have not desperately or deliberately pursued anything from God. Or perhaps you've seen the power of God displayed at churches you've attended, and maybe you've felt His presence in a tangible way. Maybe the word of the Lord has been spoken, and you know that God has pronounced a blessing over your life.\n\nIn all these things maybe life has gotten in the way. Maybe you've been hurt. Maybe difficulties have caused you to lose a level of faith, hope, and expectancy, and you have been reduced to fighting for only what you can achieve in your own strength. Well, I want to tell you that a greater demonstration of God's moving in your life is coming, but only the deliberate will see it. I don't know where you are in your walk with God, but regain your tenacity. I don't know where you were when you were most passionate, but find that place again and determine in your heart, \"I will not be afraid to reach out, grab hold, and not let go until I see it.\"\nCHAPTER 10\n\nA WORTHY HOST\n\nWhen David returned home to bless his own family, Michal, the daughter of Saul, came out to meet him. She said in disgust, \"How distinguished the king of Israel looked today, shamelessly exposing himself to the servant girls like any vulgar person might do!\" David retorted to Michal, \"I was dancing before the LORD, who chose me above your father and all his family! He appointed me as the leader of Israel, the people of the LORD, so I celebrate before the LORD. Yes, and I am willing to look even more foolish than this, even to be humiliated in my own eyes!\"\n\n\u20142 SAMUEL 6:20\u201322\n\nWHETHER WE ADMIT it or not, we like to be in control of every aspect and moment of our lives. It stresses us out when we are not in control. Some of us even hate surprise birthday parties, a moment set aside to celebrate us, because we're not in control. Early in ministry, before I became a pastor, the Holy Spirit began to teach me about losing control, and what I have discovered is that if I am in control of every aspect of a service, He isn't. People have even said, \"You're the most patient worship leader I've ever seen.\" When most people would move on or sit down, when the chorus seems to go a little too long or we're just standing there, I have learned that it is actually OK for us not to know what's next. For those of us who lead in some capacity, whether in a church or in the marketplace, it seems that we are expected to always know what to do. But we don't always know. Some people are willing to allow the moment to evolve. Others are not willing to wait and will take decisive action to push the moment along.\n\nHave you been in a service that's going one way, and suddenly the pastor shifts to something else? This may have happened because they didn't know what to do, so they took it back to what they knew. What if God wanted to do something brand new, something beyond what they had ever experienced? What if God wanted to open a portal to something on the other side that was far greater than what they had ever known before? What if they released control? What if we do?\n\nPROGRAMMED TO DEATH\n\nI had a dream years ago about the same service that ran twice. I was ministering, and all these powerful and amazing things were happening. I came off the platform and thought, \"Wow, God, that was amazing! You really moved. Thank You, Jesus.\" He said, \"Let Me take you back and show you what would've happened if you had yielded to Me.\" The service played again, this time with God guiding me through all the moments when He could have done more. In comparison to the things He showed me when the service replayed, I would have never called what I experienced in the first service a move of God.\n\nMany of us go to church and come home week after week saying things such as, \"Man, wasn't that powerful? Didn't God move?\" Yet we remain unchanged. May I submit to you that it is biblically impossible for you to have had an encounter with God without change? Everyone who has a genuine encounter with God always leaves different. Jacob left with a limp. The first Saul left with a crown. The second Saul left blind and with a new name. If you search throughout Scripture, you will find that every time someone had an encounter with the living God, that person was changed. That is why I wonder what's happening in our churches when we say God was there, but we're the same. Either it is an indictment against God, or we're lying, because now we're telling people that being with God doesn't make us any different. What can we say to the person who has fallen away when he says, \"I've tried Jesus, and He doesn't work for me,\" if we've never had a genuine encounter with God?\n\nA dividing line is being drawn down the center of the body of Christ. On one side are those who have learned to do church without God. On the other side are those who are dissatisfied with doing church without God. We've been programmed to death. We've been told this is how long you're supposed to stay, this is how long you're supposed to sing, and this is how long you're supposed to pray. We even advertise how quickly we can do all these things: \"Come here for the fastest one hour of church, and you'll be released right back into your life,\" as if God is not someone who's to be honored and worthy of whatever time it takes to engage with Him. \"Come to a prayer meeting. It's only an hour long.\" Yet in the midst of our perfectly timed services, God is inviting us to a new place. The question is, Who wants to go?\n\nTHERE IS A WAY\n\nIn the days of Moses, God had graced His people with His presence in the form of the ark of the covenant. He had given them specific instruction on how to honor and protect His presence. But then they made a grave mistake and took the ark into battle against the Philistines. (See 1 Samuel 4.) Israel lost the battle badly. The Philistines captured the ark and took it back to their territory, but since they were not in covenant with God, they suffered curses instead of blessing, so they brought the ark back to the Israelites. (See 1 Samuel 6.)\n\nFor twenty years the ark of the covenant, which, again, represents the presence of God, was kept at Abinadab's house. (See 1 Samuel 7:1\u20132.) During that time neither the king of Israel\u2014Saul\u2014nor the people inquired after it. They didn't seek to honor, fear, worship, or tend to the presence of the Lord the way He was used to. Revelation 4 gives us a picture of what kind of eternal and ongoing worship God is used to in heaven\u2014heavenly beings saying \"day after day and night after night . . . 'Holy, holy, holy is the Lord God, the Almighty\u2014the one who always was, who is, and who is still to come'\" (v. 8).\n\nWhen David ascended to the throne, he decided he could not live without the presence of God. So in 2 Samuel 6 the Bible tells us that David gathered thirty thousand of his most elite troops to bring the ark to him. It was a clumsy acquisition. Because he was thirty years old when he finally took the throne and the ark had not been central to life in Israel for twenty of those years, David grew up without ever seeing what it looked like to honor the presence of God. It had been written about, but no one had ever demonstrated it. So in his haste and excitement he thought, \"Let me just go get it.\" I imagine he thought there would be some people, perhaps the priests and Levites, who would know what to do with it.\n\n_In the midst of our perfectly timed services God is inviting us to a new place. The question is, Who wants to go?_\n\nThe troops set the ark on a new cart, and Abinadab's sons, Uzzah and Ahio, guided the cart. Since they were priests after their father, everyone must have thought they knew how the ark of God's presence was to be carried. But it turned out that this was not the way, and as creation recognizes the holiness, power, majesty, and presence of the Creator (Rom. 8:19\u201321), the oxen that carried the ark stumbled. I believe they instinctually knew the people were not handling God's presence with awe and reverence. I believe they sensed that was not how God's presence was supposed to be moved. I believe they sensed that He was supposed to be handled better than that. So they stumbled as if to say, \"Not on my watch will you dishonor the Creator.\"\n\nTo keep the ark from falling, Uzzah stuck out his hand to correct what creation was trying to correct. He put his hands on the glory, trying to control how God would move, as if to say, \"No, God doesn't go this way. It's supposed to go this way.\" But at the time of Moses and Aaron, God had given specific instruction that holy things such as the ark were not to be touched by undesignated people: \"They must not touch the sacred objects, or they will die\" (Num. 4:15).\n\nThere is another important element to this that I need to point out here. If you go back to 1 Samuel 6, you'll find that the Philistines were anxious to return the ark after seven months of suffering curses because of their wrongful possession of it. They were never supposed to be in possession of the presence of God. But I want to point out that they also moved the ark by cart. They\u2014who did not know, who were uncircumcised and outside of the covenant, and who were symbolic of the world\u2014had a way they thought God should have been moved. They put Him on a cart because they didn't know any better.\n\nBut the people of God knew better. As people of covenant, they were the ones who had an encounter with God. They had a history with Him. There should have been no way they would have done the same thing as the Philistines. They tried to move the presence of God using the same methodology as the world, but there is a way God is to be honored.\n\n_Things are happening in the earth right now. God is removing people who have caused the body of Christ to believe that our holy God is common. He is not common; He is holy._\n\nGoing back, the Bible says the anger of the Lord was kindled against Uzzah, and he died there in the presence of God. Why? Because of his irreverence, because he didn't fear the Lord, and because there is a way to honor the presence of God. If seventy people died for looking into the ark (1 Sam. 6:19), the Philistines got tumorous boils for being around it (1 Sam. 5:6), and the Philistines' god Dagon was broken down in his temple because he was next to the ark (1 Sam. 5:2\u20134), what would make Uzzah, knowing that history, think he could touch it?\n\nHe touched it because to him it wasn't the presence; it was a relic. It represented old traditions, past moves, and things God wasn't doing anymore. Perhaps he touched it while it was in his father's house, and nothing happened to him. The difference in this instance was that God was not about to be publicly dishonored in a way that caused the people to see a holy God as common.\n\nThings are happening in the earth right now. God is removing people who have caused the body of Christ to believe that our holy God is common. He is not common; He is holy. And those who think otherwise are being exposed. There is a way to honor the presence of God.\n\nIf you've ever known anyone personally who has been diagnosed with and lives with lupus, then you are very much aware of how painful and debilitating that sickness can be. There is no medical cure for it, but that didn't stop Ana from coming up front one day, believing that God would heal her.\n\nWe've never had a \"healing service\" at our church, but we have seen over two hundred miracles take place in the presence of God. When the Healer is in the room, healing happens. Ana, who is a woman of faith, knows this to be true, so she didn't hesitate to ask God for her healing and agree in faith when we prayed.\n\nAna felt the power of God when we prayed and was convinced that she was healed. Walking in that faith, she went back to the doctor and was tested for lupus again. After receiving the test results, she came back to church with a yellow folder. On one side were her former tests showing the diagnosis of lupus, and on the other side was the medical documentation that she was free of lupus.\n\nJesus healed Ana, but she wasn't going to let the moment go by without asking for total healing. Every year for ten years she had tested positive for precancerous cells in her body. When she was tested again after she received prayer, no precancerous cells were found. Ana was healed of lupus and the precancerous cells in her body. It's happening!\n\nWORTHY HOSTS\n\nThe day Uzzah died was the day all of Israel feared the Lord, including King David. King David didn't know there was a way to carry the presence of God. He grew up in a generation that was around the things of God but not intimately acquainted with God. That's what I'm afraid of in this generation. We know the worship songs, we know the lingo, and we have the lights, the smoke, and the screens. We have everything but God Himself, it seems sometimes. We have everything but people who actually know how to get to God Himself and honor Him the way He deserves.\n\nInstead of pushing past the shocking event of Uzzah dying on the spot, David decided not to move the ark to the City of David as he planned. He decided to take it to the house of Obed-edom while he learned God's requirements (2 Sam. 6:10). At Obed-edom's house something interesting happens. The Bible says, \"The Ark of the LORD remained there in Obed-edom's house for three months, and the LORD blessed Obed-edom and his entire household\" (2 Sam. 6:11).\n\nWho was Obed-edom? He had not been mentioned before now as a set-apart priest or Levite. There doesn't seem to be anything significant about the man himself. Yet the Bible says everything surrounding Obed-edom was blessed. That's amazing. Understand that it wasn't the ark being in his house that caused the blessing. The ark had been in Abinadab's house for twenty years, and there was no mention of a blessing. Three months at Obed-edom's house, and everything was blessed. It wasn't the ark; it was the host.\n\nNotice that there wasn't anything that Obed-edom did to be the chosen host of the presence, but there were things he did to keep it. During the time that our church has been seeing tangible evidence of the presence of God in our midst, there was a moment in one of our rivers (services) when God literally walked into the room. He asserted His sovereign will and decided that our church was to be a habitation for His presence.\n\nI'll never forget this moment as long as I live. I was out of town with my team leading a night of worship. It was a ticketed event, and the room was full. They introduced us, and we went onto the stage and began to lead the people in songs of worship. We were on our third song when I began to feel a really intense heat on my neck and back. It was so hot that I kept turning around and looking up to see if maybe I was standing near a light or some kind of heat source. If you've ever been under the lights on a stage, you know they can get quite hot. I was thinking perhaps I was under one or close to one, but I kept looking, and there was nothing there.\n\nMy good friend Clay, who has co-produced all my worship recordings with me, was on the keyboard that night, and he saw me looking around and asked what was wrong. I told him I felt a warm heat on my neck and back. He gave me a quizzical look, but we were in the middle of a song, so I closed my eyes. Immediately I recognized the presence of the Lord right beside me. He was the source of the heat. I had heard of that phenomenon before but never experienced it myself. The significant manifestations of healing had just begun at our church, and I sensed that the power of the Lord was present to heal in that moment. So I began to share about the roughly thirty miracles we had experienced at that time. The word of our testimonies builds faith for the future, so I knew that in telling what God had been doing, faith was being stirred in the room.\n\nAs I began to flow this direction, the heat continued on my neck and back, and I sensed that the Lord wanted to heal people in the room with neck and back issues. Remember that it was a ticketed concert at a church in Boston, so I wasn't at my home church and didn't have a relationship with that particular church. Therefore, I asked for permission to continue flowing that direction. I was initially told yes, so I called for people who were dealing with neck and back issues to come forward. Several people came to the front where I was standing. As I prepared to pray for the people, I was passed a note. The note said, \"I'm sorry, but you are not permitted to pray for these people because Bishop isn't here.\"\n\nThe moment I read the note, instantly the heat left! I had never felt anything like what I felt in that moment. I've been pursuing God all my life and have become sensitive to His presence, and I had never in my life felt the Lord withdraw. He had come to be with His people, but when I read the note, it was as if He was rejected. I was so grieved by what I felt that I could not continue. It was a ticketed event, but I sang one more song and left the stage. I thought, \"If He's not here, why am I here? Apart from Him we can do nothing.\" I went back to my hotel room and wept like a baby. I kept saying, \"God, whatever You wanted to do there please do here [referring to our church in Orlando].\" I just kept saying that over and over and over: \"Whatever You wanted to do there please do here!\" I kept telling Him, \"We won't reject You.\"\n\nI flew home that Saturday, and I kept praying the same thing over and over. I couldn't get over what I felt. Interestingly enough, I received an email saying the two people who were standing closest to me when I felt that heat were healed of their back issues, even though I didn't pray for anyone. That's amazing, but I knew there was more. As I studied and spent time with God all day Saturday, I continued to cry, \"Whatever You wanted to do there please do here because we won't reject You.\"\n\nThe next morning, which was Sunday, I woke up praying the same prayer, and while I was driving to church, I felt the heat again. It was as if the Lord got into the car with me to say, \"I'll be there.\" In fact, as I am writing this, I feel His presence the same way. As the service began, I had an expectation of having an encounter. I didn't know when, but I expected Him to answer my prayer. I stood up to preach, and it happened! Once again the presence of the Lord manifested in a way that everyone was aware, and no one could do anything but lie prostrate before God and weep. I was attempting to preach a message titled \"The Second Invitation,\" and those who have heard it say you can literally hear the moment when God walked into the room. The musicians have told me they weren't playing the sounds you can hear on this recording.\n\nThen there was the heat that from that time to now has been one of the evidences to me that the power of the Lord is present. I remember trying to explain it, and people thought they understood until they came near me. Oftentimes when it's happening, others standing around can feel it too. I can't fully explain it, other than to say it's real. That particular day while it was happening, I went to go pray for someone, and all those between me and the person I was praying for experienced the phenomenon of being \"slain in the Spirit\" when I walked by them, and I never even prayed for them. Obviously it is not me but the presence of the Lord. When it is happening, I remind people that it's not as if I can make heat radiate from my body on command. This is a supernatural thing.\n\nSince that moment our entire household has been blessed. That day can be considered a turning point for us because it was the moment we corporately embraced the idea that we won't reject Him. As a result, He has done more and more. It's amazing what you will experience when you embrace Him instead of rejecting Him. We can't say enough about what God is doing. As I have already mentioned, He healed a whole family from a condition that has never been healed in the history of the United States. He healed a baby who was partially brain-dead. Walkers, canes, and crutches are piling up on our altar because people just don't need them anymore. Various cancers and tumors have been healed and removed. The effects of strokes have been reversed. Paralysis has been reversed. Blind eyes and deaf ears have been opened. Even I was healed of gout, which was debilitating. I can't tell you when it happened because we've been so concentrated on His presence. The only reason I know I've been healed is because I forgot to take my medicine one day, and I just wasn't in pain anymore. I went back to the doctor and was taken off all the medications that had been prescribed for the condition, and I haven't needed them since.\n\nHe is healing us because healing comes along with His presence. He says, \"I am the LORD who heals you\" (Exod. 15:26). We have set Him as our number one affection and pursuit. He is answering our cry. He is all we want. He is all we need. We are not in this moment because we want Him to answer something. We are here because we want Him. And there is nothing we want to do to jeopardize this visitation. We want to continue to have His blessing upon our lives. We want to continue to be hosts for His presence. But are we willing to become more undignified? Are we willing to avoid places that offend Him? Are we willing to make adjustments to the relationships we have to avoid offending Him? Are we willing to stop doing things that offend Him? This is not about our righteousness. This is about our desire to be a host.\n\nThere should be nothing we aren't willing to do to make God feel like an honored guest. We shouldn't want anything in our lives to repel Him.\n\nAs I travel around this world, I see arks gathering dust in church corners. I go to places where God wants so badly to show up but His presence has been reduced to a monument of the past\u2014a relic. It would seem nobody actually wants to honor Him, because every time He starts to move in a place, we shut it down. We say it's time to move on, time to go, that we have this or that to do. He's asking, \"May I have this moment? OK, OK, I'll wait. Oh, I want to heal this person. May I . . . ? OK, I'll wait.\" He wants to free people who are bound in our churches, but we keep saying, \"Not yet.\" He wants to heal people, and we say, \"Sorry, God. Maybe next time. It's 2:00 p.m., and we have to go. We can't hold the people in church too long.\"\n\nWe are in days of visitation, and I'm wondering if there are any people who will host God's presence and give Him the honor He deserves. The ark is moving, and it is seeking to rest in houses that are like Obed-edom's. The question is, Who will host Him?\n\nDAY AFTER DAY AND NIGHT AFTER NIGHT\n\nEarlier in this chapter I introduced the verse in Revelation 4 that shows us the kind of honor God is used to. It says that day after day and night after night heavenly beings worship Him. We keep thinking we're worshipping for a long time because we go for an hour. No! Day after day and night after night they worship in heaven. We keep thinking that just because we have our hands lifted, they get tired, we pull them down, and maybe we cry a few tears, that is enough. No, that's not enough. Day after day and night after night. What do we think Obed-edom was doing in order to receive the blessing of God's presence? He was giving God what He is used to\u2014worship day after day, night after night.\n\nThe ark of the covenant is moving and wants a place to dwell. Will you be a worthy host? Will you give God the atmosphere of worship He is used to? What will be your posture before Him? Will you treat Him as they did in Abinadab's house? Will you put Him in a corner?\n\nThe Bible says in 2 Samuel 6:12, \"Then King David was told, 'The LORD has blessed Obed-edom's household and everything he has because of the Ark of God.' So David went there and brought the Ark of God from the house of Obed-edom to the City of David with a great celebration.\" David and the people of Israel figured out right way to carry the ark. So they hadn't moved more than six steps before \"David sacrificed a bull and a fattened calf\" (2 Sam. 6:13). Further study into this scripture reveals that they did not stop just this one time on the way to Jerusalem. They stopped every six steps. There was a continual worship. They gave God what He is used to. They stopped to show their gratitude, worship, and honor.\n\nThe Bible says there was great fanfare. They had all sorts of music and singing and dancing because they were so excited that they would no longer be a generation that would try to operate apart from the presence of God.\n\nVerse 20 tells of King David dancing out of his robe. He and all the people were so excited that God would be near to them.\n\n> They brought in the ark of God and placed it inside the tent which David had pitched for it, and they offered burnt offerings and peace offerings before God. When David had finished offering the burnt offering and the peace offerings, he blessed the people in the name of the LORD. He distributed to everyone of Israel, both man and woman, to everyone a loaf of bread and a portion of meat and a raisin cake.\n> \n> He appointed some of the Levites as ministers before the ark of the LORD, even to celebrate and to thank and praise the LORD God of Israel: Asaph the chief, and second to him Zechariah, then Jeiel, Shemiramoth, Jehiel, Mattithiah, Eliab, Benaiah, Obed-edom and Jeiel, with musical instruments, harps, lyres; also Asaph played loud-sounding cymbals, and Benaiah and Jahaziel the priests blew trumpets continually before the ark of the covenant of God.\n> \n> \u20141 CHRONICLES 16:1\u20136, NASB\n\nLook who was in the group of Levites\u2014Obed-edom. He did not live in Jerusalem, but he got in on honoring God since His presence had been at his house and since he had had an encounter with Him. He adjusted his life all over again, risking it all to be one of the hosts of the presence of God. He could not go back to life as usual.\n\nThen verses 37\u201338 of 1 Chronicles 16 blessed me. They say, \"So he left Asaph and his relatives there before the ark of the covenant of the LORD to minister before the ark continually, as every day's work required; and Obed-edom with his 68 relatives\" (NASB). What have we discovered God is used to? Praise, worship, and honor day after day and night after night. Can I tell you a secret? I keep worship music playing in our church every single day, twenty-four hours a day. Until we can have the kind of house where we have Asaph-type people who minister to him continually, we're at least going to keep an atmosphere.\n\nIt may seem as if it's too much\u2014day after day and night after night, worshipping and spending time with God. But it is all worth it. The encounter with God that you are having even now as you read this book is going to affect your entire family. When Obed-edom decided he was going to leave his hometown and go after God, sixty-eight of his family members went after God too. It's OK to stay in the presence of God, because your family is coming with you.\n\nEVEN MORE UNDIGNIFIED\n\nBut before your family joins you, you may have to deal with these questions: \"Why have you been at church so long? What have you been doing there? I just don't get it. I don't understand.\" This happened to David when he returned home after repossessing the ark of God. Second Samuel 6:20 says: \"When David returned home to bless his own family, Michal, the daughter of Saul, came out to meet him. She said in disgust, 'How distinguished the king of Israel looked today, shamelessly exposing himself to the servant girls like any vulgar person might do!'\"\n\nWhat was Michal doing? She was looking at him out of her window. Sometimes when you have an encounter with God, the people who once helped you and stood by you will question and criticize you. Earlier in Scripture (1 Sam. 19:12) Michal helped David escape through a window when Saul was trying to kill him. But now that he had an encounter with God, she said she was disgusted\u2014and maybe she was jealous because she had not had such an experience.\n\nHis response was bold: \"I was dancing before the LORD, who chose me above your father and all his family! He appointed me as the leader of Israel, the people of the LORD, so I celebrate before the LORD. Yes, and I am willing to look even more foolish than this, even to be humiliated in my own eyes!\" (2 Sam. 6:21\u201322). I like the way the New King James Version of the Bible puts it: \"And I will be even more undignified than this.\"\n\nSome of us have been holding back our praise and adoration of God. We don't know that it's OK to become undignified in the presence of God and really worship like David because it has never been demonstrated to us before. David had no one to show him how to host the presence of God, but he was patient. He waited on the Lord, and God began to teach him by His Spirit. And old and ancient pathways began to surface.\n\nWhat David discovered is that it is OK to go crazy before God. It's OK to lose your mind before God. It's OK for you to make a sacrifice of praise and worship every six steps. It's OK for you to stomp, dance, leap, and cry out. It's OK. You're not crazy. This is what God is used to. You are giving Him what He is worthy of. You're not crazy; He's just worthy. You're not crazy; He's that good! You're not crazy; He's that amazing. You're not crazy; He's that holy. Don't hold back. Give Him what He's used to\u2014day after day and night after night.\n\nDo you know why I've decided I don't want to control anything? Because where He moves I move. Where He goes I go. I do not want to be one of the ones who say, \"Do you remember those days of visitation?\" I am like Obed-edom: whenever David comes to get the ark, I'm going with him.\n\nLet those who don't have the revelation that it's happening keep talking about you. They're coming. Let them continue to make fun of you. They're coming! Let them keep resisting. They're coming! Because when God blesses you and you are a worthy host of His presence, your whole family, your neighbors, and your coworkers will see it. And they are going to say, \"I want that too!\"\n\nRESPOND\n\nGod is here. Revival is happening. We are encountering the Lord, and we can't go back to normal. His presence changes everything\u2014our behavior, where we go, what we say, and who we want to be around. Don't apologize for it. Some may believe you think you are too good. But it's not that; it's that you'll do whatever you have to do to stay with Him because you are a host of His presence. Yes, it's that serious for you now. He is more valuable to you than anybody and anything else.\n\nWe're talking about a life-changing, life-altering encounter with the presence of God. Once you've experienced it, what is there to go back to?\nCHAPTER 11\n\nHE IS HERE\u2014NOW WHAT WILL YOU DO?\n\nAsk the LORD for rain in the spring, for he makes the storm clouds. And he will send showers of rain so every field becomes a lush pasture.\n\n\u2014ZECHARIAH 10:1\n\nAsk the LORD for rain in the time of the latter rain. The LORD will make flashing clouds; He will give them showers of rain, grass in the field for everyone.\n\n\u2014ZECHARIAH 10:1, NKJV\n\nWHEN IT COMES to recognizing the movement of God, most of us want things to be crystal clear. We would love to have God wave a big red flag at us and say, \"This is Me! Here I am. It's really Me moving.\" But when it comes to the things of God, many times He won't use big signs to get our attention. He will use subtle things, and most of us pass them by, thereby missing windows of opportunity. We fail the test of subtlety and miss divine moments.\n\nThere are certain moments in time and seasons in the Spirit that present themselves as windows of opportunity. Zechariah 10:1 puts us right in the midst of one such moment. It's raining, and most people don't pray for rain when it's raining. We pray for rain in seasons of drought. But this verse is directing us to do something different: Don't wait until there's a drought to ask for rain. Ask for rain in the time of rain. This is going to be important for us to understand in terms of sustaining the move of God. What the verse implies is that we need to use the open window of opportunity (an open heaven pouring down rain) to seize the moment to ask for more.\n\nNow you must understand how we can miss a moment. A moment is a very short period of time, or a particular time, or a precise point in time. So it's not hard to see how it could pass us by. This is why we must become aware of what God is doing so that when a moment of opportunity presents itself, we can seize it.\n\nCREATE A WINDOW\n\nNow this is important. Seizing a moment creates momentum, and momentum is a strength or force something has when it's moving. It's a strength or force that causes something to get stronger or faster as time passes. Momentum works in such a way that it takes more effort to get something moving than to keep it moving. We are aware that what we have stepped into at Deeper is a season that took years of intercession to see and that will take ongoing intercession to maintain.\n\n**Intercession creates a window.**\n\nDuring one of our monthly Presence Nights, I remember a man of God joining us for worship. He didn't need to identify himself; I knew who he was. I knew he was a pastor and was a prophetic voice. I knew he had been in the Orlando area for a long time. But he didn't ask for any preferential treatment. He didn't make an announcement, \"I'm coming. I want a special seat. Do you know who I am?\" He just came and worshipped and wept.\n\n_Don't wait until there's a drought to ask for rain. Ask for rain in the time of rain._\n\nAt the very end of the service, I went to greet him, and he just began to release a number of things the Lord had shown him. He was weeping as he said to me, \"I prayed for years to see this.\" What we have been seeing in our church he was not seeing in his, even though he spent his life interceding for what we are now experiencing in our city. He was weeping tears of joy because even though it wasn't happening specifically at his church with a group of people he was leading, he was seeing the fruit of his intercession beginning to break out and manifest somewhere. We reject the idea that we reached the place we are in on our own. It took people who laid down their lives to see a move of God in this region.\n\nWe don't often know the price intercessors will pay to see a move of God happen in a particular region. Back in the 1980s I knew of an intercessor who spent her retirement savings to travel to all fifty states so she could pray on the steps of all fifty state capitols. Nameless and faceless people such as this woman have laid down their lives so other people could step into an open heaven. They lay down their lives to create a window for other people to seize the moment.\n\nRevival doesn't just show up. I don't ever want our church to take it for granted and feel as if we prayed and this happened. No. We joined intercession that was already taking place. We stand on the shoulders of nameless and faceless people whom we've never met and who walked this city and died having never seen what they had been praying for come to pass. We must be careful not to take things for granted because we didn't pay for them.\n\nYears of intercession led to a moment. Continued intercession leads to momentum.\n\nRebecca and her son had been attending our church for a while. Her son was deaf in one ear, but like many resilient people, they didn't allow that to stop them from living a normal life. As they encountered obstacles, they faced them with a smile.\n\nOne of the most amazing things about this outpouring is that oftentimes people get healed without even specifically asking just because Jesus the Healer is near. That's what happened to Rebecca's son. One day during the service he was healed. No one prayed for him. He just suddenly was able to hear. The healing was so lowkey. He didn't even alert his mom right away. On the car ride home he casually said, \"Mom, I can hear you.\" She said, \"That's great, baby.\" He said, \"No, Mom, I can hear you!\" Now realizing what he was really saying, Rebecca began to praise God. Her son was healed. He was no longer deaf in one ear. His hearing was completely restored, and no one but God could possibly take the credit or the glory because no one had a hand in it but God. It's happening!\n\nSTIFLE THE MOMENT\n\nRevival requires continual movement\u2014momentum. Momentum requires an environment where movement is unimpeded, where nothing is blocking forward progress. That's why Paul wrote in 1 Thessalonians 5:19, \"Do not stifle [or quench] the Holy Spirit.\" In other words, don't stop your spiritual momentum, because once you stop momentum, it takes more effort than you needed when you started to get it going again. In the case of revival, that is a costly stoppage that takes a lifetime to see again.\n\nDon't stifle the moment. Don't quench the Holy Spirit.\n\nI remember as a child\u2014and we joke about this stuff now\u2014going to my grandmother's church in the morning and, while church was going on, smelling food. After service we would go to the fellowship hall, eat the food, and then return to the sanctuary for another service. We never left! Many of us look back at this now and think, \"No way!\"\n\nNot too long ago I was in Nigeria for a night of worship. It was held outdoors in an open field with about fifteen thousand to twenty thousand people in attendance. The night started at 6:00 p.m., but we didn't get on the platform until 1:00 a.m., and the last guest went up at 3:00 a.m. There were no chairs to sit down on, yet nobody left. In Western churches if pastors preach for more than forty minutes, we start getting antsy. We have places to go, people to see, and things to do. I ask this next question sincerely, and I think all of us need to honestly assess our answers: What place, person, or thing has more importance than God?\n\nWe have boiled down the Scripture and the presence of God in our corporate gatherings to less than an hour per week, in some cases, expecting God is going to move. In Acts 20 the Bible talks about Paul preaching so long that a young man fell asleep on a windowsill, fell out the window, and died. Instead of ending the service, Paul went down, raised the boy from the dead, and kept preaching. So we're at this point now where some of our churches are seeing miracles happen in our midst. We see God heal people. We see God deliver people. We see God set people free. We are excited\u2014it's cool\u2014but we still have this tendency to not make God's presence preeminent, where we still want to be out of church by a certain time.\n\n_Years of intercession led to a moment. Continued intercession leads to momentum._\n\nIn traveling to places such as the continent of Africa and witnessing thousands of people gathering in worship for nine or ten hours at a time or in hearing about the revivals and awakenings in South and Central America, I am moved to pray for the church of America. The only thing is the church of America is violating 1 Thessalonians 5:19. We do quench the Holy Spirit. He starts to move, and we act as if we are uncomfortable with it. I should probably take the surprise out of it: the move of God will make people uncomfortable. Guests will be uncomfortable. Unbelievers will be uncomfortable. How do you expect an unbeliever to be saved except that the Holy Spirit reveals Jesus?\n\nAny man who can be convinced into accepting something can also be convinced out of it. Our job is not to convince people that Jesus is a good idea. The Holy Spirit reveals to them that He's the only way. Jesus is not explained; He's revealed. How can the Spirit reveal Him if the Spirit is being stifled by our programs or our discomfort with the way He moves? If you have had a true encounter with God, it's because Jesus Himself was revealed by the Spirit in the midst of His free-flowing presence.\n\nWhen it comes down to it, people are not interested in our trying to preach as if we're doing a TED Talk. Preaching devoid of power and preaching devoid of signs and wonders will not change a life. It may change a mind, but it won't change a life. We must allow the Spirit of God to flow freely.\n\nTHE TIME TO ASK FOR MORE IS NOW\n\nAs we are in a moment of revival, the time to ask for more is now. Unfortunately during this spiritual spring season, some people are saying not to do this and not to do that. \"Don't worship this long.\" \"Don't preach this long.\" \"Don't keep people this long.\" \"Don't call for 5:00 a.m. prayer.\" \"Don't call for 7:00 a.m. prayer.\" \"Don't ask them to come to church more often.\" As a result, we have an entire generation that is not experiencing the things of God and is squandering the intercession that came before us.\n\n_Jesus is not explained; He's revealed._\n\nHow is it that we are now a culture that doesn't value the revival moments for which people laid down their lives in prayer and never saw? How is it that we are a church culture that doesn't want to come to church and doesn't want to stay? Revival is for the believer first, for a church that has fallen asleep. Revival is for something that was once alive but now is dead\u2014something that was once moving but now is stagnant. In truth, revival is needed because the church has fallen asleep. Revival in the church causes a reformation in the world.\n\nThat is why if there is ever an outpour of the rains of revival anywhere in the world, we need to be asking for more rain. We need to ask for the rain during the time of rain. It is costly to miss the moment and realize how dry we are once the rain has stopped. We may find ourselves waiting awhile for the rain to fall again\u2014possibly waiting for generation after generation. We can't afford the cost. We can't miss the moment.\n\nQUIET THE NOISE; RELEASE A SOUND\n\nDivine moments can be created. The moment of time we're in right now\u2014both our church and the body of Christ\u2014was created. A divine moment is an intersection where heaven meets earth. We create this intersection when we align with the things of heaven, and we can create divine moments with a sound. Let me show you what I mean.\n\nMark 10:46\u201352 says:\n\n> Then they reached Jericho, and as Jesus and his disciples left town, a large crowd followed him. A blind beggar named Bartimaeus (son of Timaeus) was sitting beside the road. When Bartimaeus heard that Jesus of Nazareth was nearby, he began to shout, \"Jesus, Son of David, have mercy on me!\"\n> \n> \"Be quiet!\" many of the people yelled at him.\n> \n> But he only shouted louder, \"Son of David, have mercy on me!\"\n> \n> When Jesus heard him, he stopped and said, \"Tell him to come here.\"\n> \n> So they called the blind man. \"Cheer up,\" they said. \"Come on, he's calling you!\" Bartimaeus threw aside his coat, jumped up, and came to Jesus.\n> \n> \"What do you want me to do for you?\" Jesus asked.\n> \n> \"My Rabbi,\" the blind man said, \"I want to see!\"\n> \n> And Jesus said to him, \"Go, for your faith has healed you.\" Instantly the man could see, and he followed Jesus down the road.\n\nIn chapter 9 we looked at the story of the woman with the issue of blood. Well, once again we find Jesus in a similar setting\u2014surrounded by a large crowd. Bartimaeus, who was blind but had other heightened senses, knew that the atmosphere of his city had changed. The buzz and the noise of the crowd might have led him to ask, \"What's going on? Why are all these people here? Why is there suddenly a buzz?\" And maybe someone answered, \"Jesus is here.\"\n\nNow imagine what went on inside his heart when he heard that response: \"Jesus is here.\" Everything he had ever heard about Him up to that moment probably came rushing to his mind: the deliverances of the man with the unclean spirit, the demoniac in the garden, the demoniac boy, and the Syrophoenician woman's daughter; the healings of Simon's mother-in-law, the leper, the man with the withered hand, Jairus's daughter, the afflicted woman, the deaf and the dumb man, and the blind man near Bethsaida; the stilling of the storm and Jesus's walking on water; the feedings of the four thousand and the five thousand; and so many more.\n\n_As we are in a moment of revival in the earth, the time to ask for more is now._\n\nCan you imagine it? He had heard about Jesus before Jesus reached him, so he might have been thinking, \"All this has happened, and now the man who is responsible for it all is near me?\" The crowd got closer. He could hear them passing by. He knew a once-in-a-lifetime moment was upon him. The Bible says he lifted up his voice and said, \"Jesus, Son of David, have mercy on me!\" He was determined to not let this moment pass him by. He was going to ask for rain in the time of rain. He was asking for a moment while the moment was there. He was going to seize that moment.\n\nWe tend to read his cry\u2014\"Jesus, Son of David, have mercy on me\"\u2014as if the environment were relatively calm and quiet, but it was not calm or quiet at all. He was in the midst of a large crowd. If he was using a calm and quiet tone, the only people who would have heard him were those in close proximity to him. Have you ever tried to get the attention of someone in a noisy place? Have you ever been in an arena and seen a friend and found yourself screaming and looking crazy trying to get his attention? This is what Bartimaeus was doing. And even with all the noise he was making, Bartimaeus could not move the crowd. Many of them probably didn't even hear him.\n\nCan I turn the narrative on its head for a moment? There's a difference between a noise and a sound.\n\nThe people were yelling at Bartimaeus and telling him to be quiet. They wanted him to stop making a scene. But desperation ignores detractors and distractions, so he kept crying out. And in the middle of a large, noisy crowd, Jesus stopped. I need you to get this. There were thousands of people asking Jesus for something or trying in some way to get His attention. They weren't walking along in silence. And somehow through the thousands of people who were walking, something got Jesus's attention. The Bible says He heard Bartimaeus and stopped. There's a difference between making noise and releasing a sound.\n\nNoise surrounded Jesus, but a sound stopped Him. Worry doesn't have a sound, but it makes noise. Fear doesn't have a sound, but it's noisy. Complaining doesn't have a sound; it's just noisy. Doubt doesn't have a sound; it's noisy.\n\nThat's why you can get stirred in your emotions, get stressed out, and maybe even shout, trying to get the attention of Jesus, but all you are doing is making noise. But there's a difference between making noise and releasing a sound. Noise might get the attention of people, but only a sound causes Jesus to stop. You might get everyone else to say, \"O God, please help her,\" or, \"O God, please help him.\" Everybody else might understand that you're going through something, but that doesn't necessarily mean your commotion will cause Jesus to stop in His tracks.\n\nDesperation has a sound that cuts through noise. Desperation gets God's attention. Jesus said, \"Tell him to come here.\" Then He asked Bartimaeus, \"What do you want Me to do for you?\" Don't just make a noise. Release a sound.\n\nWHEN THE MOMENT COMES . . .\n\nMany people are used to making noise and causing a commotion without carrying any sense of expectancy or faith. So if God does respond\u2014because He is God and can do whatever He wants\u2014they wouldn't know what to say. They wouldn't know how to answer if He were to draw them in and say, \"What do you want Me to do for you?\" Many believers have learned how to make noise, but they haven't learned to release a sound or what to do if God responds. They open the window but don't know how to keep it open. They end up living in the moment instead of living in momentum.\n\n_So many people get Jesus's attention but Don't know what to do when He comes._\n\n**Be present.**\n\nYears ago as a young worship leader, I received an amazing piece of advice. It was actually a rebuke from a pastor, but it was life-changing. He said, \"You need to learn how to be present to the moment.\" It's simple, but that gentle rebuke completely changed the way I lead worship. It completely changed my walk with God, and it will change yours.\n\n_Be present to the moment._ I wrestled with it for a while. I thought, \"What does that even mean?\" Then God began to break it down to me. You see, I grew up in a church that had a moment but didn't know how to sustain momentum. Sadly the window closed. To this day there are still people there hoping the window will open again, but they don't understand that they missed the moment. They haven't learned what the sound is, so they don't understand that they can shift from making noise to releasing a sound if they desire to create another divine moment.\n\n**Know that you can't re-create it.**\n\nIt's grieves me to look back at that time when I was a young person growing up in a moment when the window was open and something was happening. It was springtime. Songs were written. Records were recorded. Books were penned. Pastors were released. Then the sound changed to noise, though most people didn't recognize the shift. For a while the noise was still attracting people because of the reputation of the sound. But Jesus wasn't stopping. He kept walking to the next city.\n\nEven I, being young, knew there was a difference, but my immaturity led me to think we could just continue what we were doing. But what I found myself doing everywhere I went after that point was trying to re-create that past moment in the present. Pastor, worship leader, believer, and anyone who is reading these lines right now, hear me when I say we have developed an unhealthy obsession with the things of the past. All the time, we are unknowingly trying to re-create the past. That window is closed.\n\nWe say we need to go back and sing those songs. We need to go back and do what they did. If you do those things without the heart of desperation that was released during that moment, you won't get the same result. Those things were birthed in that moment's present reality of desperation. A sound was released. You can't echo a sound, but you can echo noise.\n\nA sound comes out of you in a present moment. You can't grab ahold of a sound that was released in the past. You can only release a sound from your present reality. What the pastor was really saying was that the moment was good. What happened there was good. It was a move of God, an outpouring, a revival, and a stirring of the Spirit. It was a prophetic moment. But that was then. Where are you now?\n\n**Be aware of what God is doing and what season it is.**\n\nThough it took some time for me to get it, I came to realize I was completely unaware of what God was doing. I was completely unaware of what season it was. I was trying to pull people back, not in.\n\nMany people are living in this moment, as I was. We get stuck in former moves. People come up to me all the time and say, \"We should do this because at my other church . . .\" And I'm thinking, \"Well, why did you leave?\" It's as if we're trying to capture lightning in a bottle and trying to get the old thing and bring it into the present.\n\nPlease don't misunderstand me; I'm not talking about principles here. I'm not talking about foundational biblical principles. I'm talking about how many of us are like zombies. Our bodies are here, but our spirits are there\u2014stuck. So when we shout, we're not releasing a sound. We're making a noise. We may be joining in with those who are releasing a sound, but we're just making a noise because we're not actually desperate. We don't even know we're not actually where we think we are. Disconnected from the present moment, we can't discern how much we need God.\n\n_If there was a time or season of consecration when you did something that allowed you to hear God, stay there._\n\nThe prophet Zechariah challenges us on the need to be aware of what season we're in. \"Ask for the rain in the time of rain.\" When we are aware, our asking and our crying out will be in tune with God's heart. When it's raining, we'll know it is the time to ask for more and it is the time God will give it.\n\n**Stay there.**\n\nThere is a corporate revival happening, but that doesn't necessarily mean an individual revival is happening in you. It is possible to be _at_ a move but not _in_ it. It's called a parade. You're standing there, and you're present, but the movement is happening on the street. Corporate revival is happening around you, but personal revival is not happening in you. Why? Because you thought that if you did it once, that was enough.\n\nSpiritual moments exist to create momentum. Whatever you did to create the moment when heaven intersected earth in your life is what you need to continue to do to keep that moment going. Consecration is not an event. It's an alteration of lifestyle.\n\nIf there was a time or season of consecration when you did something that allowed you to hear God, stay there. If not watching TV is what caused you to hear God, then keep the TV off. Perpetually. Forever. We may not want to hear stuff like this because we want God on our terms. We want our encounters with Him to work a certain way. We want it done our way. \"Come on our terms, God,\" we say. \"Come in our time.\" And we stifle the Holy Spirit. We say, \"OK, that was cool, God. We're good now.\"\n\nThere is no sense of desperation in that kind of response. That response will not cause God to stop in the midst of a multitude and ask, \"What is it you want Me to do for you?\" But if there is anyone within a generation who will release a sound, Jesus will hear it and stop.\n\nRESPOND\n\nI don't believe it was the audible sound of Bartimaeus's voice that arrested Jesus's attention. Remember when Jesus was on His way to raise Jairus's daughter from the dead? The Bible says that a woman who had been dealing with a blood-flow issue for twelve years touched the hem of His garment. He said, \"Who touched Me?\" But did He actually feel her hand on His garment? No. He noticed that somebody touched Him differently. Not only did the woman pull on His garment; she also pulled on His life.\n\nMay I suggest to you that He might not have felt the garment pull, but He felt the virtue pull? May I suggest to you that He didn't hear Bartimaeus's voice; He heard the sound of desperation?\n\nSome of us keep thinking that if we just get loud enough, if we get demonstrative enough, we'll get Jesus to respond. Let me dissuade you from that line of thinking because you'll end up making noise, and I am here to encourage you to make a sound. Let's step outside of the natural realm for a minute and hear what God is trying to say to us.\n\nSometimes when we're desperate, a sound can't even come out. Sometimes when we're desperate, it's not our voices that make the sounds; it's our tears. Sometimes when we're desperate, it's not how loud we get; it's how low we bow. Sometimes when we're desperate, it's not how loud we shout; it's how often we sit in silence. Sometimes when we're desperate, we release a sound when we open up the Word and say, \"God, I need to hear from You. God, I need You.\"\n\nAre you perpetually desperate?\n\nDesperation is a sound that your life makes. Desperation is the thing that makes you get up at one, two, or three o'clock in the morning when you're restless, and you start walking your floor and praying. Desperation has nothing to do with how loud you scream. It has everything to do with the sound that your life makes.\n\n_Sometimes when we're desperate, it's not our voices that make the sounds; it's our tears._\n\nI encourage you to find your way to a place of desperation and release a sound from there. When you do, you will create a divine moment that will lead to momentum. The moment is here. Jesus has heard and has stopped. What will you do now?\nCHAPTER 12\n\nFEED THE FIRE\n\nThe fire on the altar must be kept burning; it must never go out. Each morning the priest will add fresh wood to the fire and arrange the burnt offering on it. He will then burn the fat of the peace offerings on it. Remember, the fire must be kept burning on the altar at all times. It must never go out.\n\n\u2014LEVITICUS 6:12\u201313\n\nSOME TIME AGO I had a chance to go to Zimbabwe. It was a return trip. I'd been to the continent of Africa before. I love it, and I love the people. But this trip was special. It contained a moment I'll remember forever.\n\nWe had been ministering in the area for a few days, and toward the end of our trip we were invited to the host pastor's home for conversation and refreshment. Upon our arrival we were led out to the patio, where a seating area had been set up around a blazing fire pit. There were different things to eat and drink, even marshmallows for roasting.\n\nNormally I am not an outdoorsman, and being outside in the elements is not my idea of fun. But there was something significant about being in this moment with this pastor and his family, who had been in the country for more than thirty-five years. Instead of traditional campfire stories, they shared with us stories of great faith and their efforts in building a church that had literally changed the nation.\n\nI was stirred as he told us about how being twice removed from the impartation of John G. Lake had impacted the effectiveness of their ministry over the years. I was so enthralled by their testimony, my spirit being fed, that I must have forgotten we were outside. You know how there are just some moments that are so precious you don't want to do anything to disturb them, so you will do whatever it takes to keep them? This was one of those moments.\n\nUntil . . .\n\nOut of the corner of my eye I saw something move in the darkness away from the light of the fire. It looked like a dog, but it was a rat\u2014a cane rat. We don't have those in the United States, and for the Americans reading this book, you ought to thank God. Not even the New York City subway rats can compare to these giant things. A cane rat could eat a subway rat, it's so big.\n\nSo for about two to three minutes there was mass pandemonium among the group. We screamed and shooed, stomped and shouted, each of us doing whatever we could to get away from the rat. Then the pastor yelled, \"Don't let that thing in the house,\" and everybody started chasing it and trying to scare it away. In confusion and terror, of course, the rat started coming toward me. I took off. One of my pastors, who had taken the trip with me, said he had never seen me move so fast. Listen, I don't do outside.\n\nThings settled down, and I apprehensively rejoined the group and discovered that they didn't kill the rat; they just chased it into the woods next to the fire. But somehow we all felt peace enough to sit back down. Someone threw more wood on the fire, and the conversation continued.\n\nThe fellowship we shared must have been a move of God. Had it been any other time, my normal MO would have led me to say, \"This meeting, this time of being out in front of a fire in Africa, is over. I'm good. Take me back to the hotel.\" But no. God was present. It was one of those moments so precious you don't want to disturb it, so you will do whatever it takes to make it last. So we sat there knowing that a rat the size of a dog lurked in the nearby woods but that we would be OK as long as we kept the fire burning.\n\nFUELING A FIRE FOR THE FUTURE\n\nAs they talked about how God had been moving in the nation of Zimbabwe since they planted their church, their stories and testimonies resonated deeply within me. The family imparted to us an anointing that was so sweet and so closely linked to what God had been doing at Deeper. As I listened, the Lord spoke to me, saying that there are supposed to be rivers flowing out of our little warehouse church that will affect the nations of the earth.\n\nThis is why I cannot help but feel the places the Lord has sent me to are to expose me to a level of faith and vision that challenge what I think is possible. If you position yourself as a student, God will put you in a place where you can see and hear things that seed in you a faith for the future. When you get there, don't try to make everyone understand why you're supposed to be there. It is for you to know and trust that God put you there. As He has done for me, God will allow you to be around people and circumstances that will ignite your faith and expectation, causing you to become uncomfortable with where you are. When it's time to turn up the heat on what He wants to do, God will put you with people who move in the Spirit in a way that provokes righteous envy. You will want what they have.\n\nIf you are like me, then you will understand when I say I am in the most unique and odd season of my life, where I am desperately crying out to be used by God in a significant way. This may seem strange if you know of my ministry as a worship leader. You may even be saying, \"You're already being used by God in a significant way.\" But there's something in me that keeps crying out for more.\n\nEven though the Lord has allowed me to experience a lot of amazing things, I'm nowhere close to being used at the magnitude God has shown me. Most of what I have been walking in during the last decade with music and such I really didn't see. With all that the Lord is doing at Deeper Fellowship, I am just now walking in the \"exceedingly abundantly\" I had envisioned. At the same time, \"far above what you could ask, think, or imagine\" is still far ahead of me. (See Ephesians 3:20.) It is an interesting place to be when you can look back over your life and be grateful for the amazing things God has enabled you to experience yet still look ahead and be desperate for more.\n\nAs I sit and listen to stories of faith about nations changing and turning to God, there is something in my DNA that says I don't want to just be in attendance when the move of God hits. I want to be part of it. I used to say, \"God, I want to see it.\" Now I am saying, \"God, I want to see it here.\" There is a holy dissatisfaction that keeps me up at night. It causes me to recognize that there is more. I long for it. It burns on the inside of me. Is it burning inside you? If so, let's talk about how to keep it burning.\n\nWe've experienced many unique moments during this outpouring, but one of the most memorable was in April of 2017 during our monthly Presence Night. We were also celebrating our third anniversary as a church and had a couple of guests with us. Due to the size of our building and the notoriety of some of my friends in ministry, at times we won't say publicly who is coming to be with us. But inevitably word gets out, and more people than usual show up. There is typically a line to get in, and sometimes we have to turn people away due to space restrictions. This particular Presence Night in April was one of those times.\n\nAs a pastor it breaks my heart to have to turn away people who come to encounter God. We had a guest speaker that night, but my friend Travis Greene, who also happens to be a Grammynominated, multiple award\u2013winning worship leader, was our guest worship leader. We were in the back preparing to go into the service when I received word that we were at full capacity and there were still hundreds of people outside. Travis said he would take his guitar outside and lead worship for the people who couldn't get inside. I thought that was an amazing idea. We've had several services when people have literally stood outside for hours while the service was taking place just to listen through the walls at what was happening inside.\n\nI offered to go out with Travis as he led worship. When we went outside, I noticed a woman to my left who was wearing a neck brace and slumped over with a cane. I thought to myself that she had probably heard of the many miracles that had taken place, so out of sheer compassion I walked over to her to pray for her. I asked her what her issue was, and she said she was dealing with fibromyalgia. I began to pray for her as Travis was leading worship a few feet away from me. What happened next was unexpected. As I was praying for her, I began to hear hissing and a spitting sound. I looked up, and the woman was hissing and spitting like a snake. At that moment my prayer changed, and I went from praying for the woman's healing to rebuking the devil. You can only imagine the scene. Hundreds of people are outside singing, and here I am on the side rebuking an evil spirit.\n\nAs I began to command the devil to come out of this lady, she began to scream, \"No, she's mine!\" Now we have quite a scene. People who had their phones out recording Travis were now recording a demon being cast out of a lady outside the building. After some prayer, not only did the Lord set her free from the demon, but she also was healed of all the symptoms of fibromyalgia. She left without her cane and neck brace and said she had no pain anywhere. It's happening!\n\nKEEP THE FIRE BURNING\n\n> Give Aaron and his sons the following instructions regarding the burnt offering. The burnt offering must be left on top of the altar until the next morning, and the fire on the altar must be kept burning all night. In the morning, after the priest on duty has put on his official linen clothing and linen undergarments, he must clean out the ashes of the burnt offering and put them beside the altar. Then he must take off these garments, change back into his regular clothes, and carry the ashes outside the camp to a place that is ceremonially clean. Meanwhile, the fire on the altar must be kept burning; it must never go out. Each morning the priest will add fresh wood to the fire and arrange the burnt offering on it. He will then burn the fat of the peace offerings on it. Remember, the fire must be kept burning on the altar at all times. It must never go out.\n> \n> \u2014LEVITICUS 6:9\u201313\n\nWhen we were sitting in front of the fire in Africa, in order to keep it burning, they didn't add more fire. They added more wood. As long as it has something to consume, fire will keep burning. The only time fire dies is when there is nothing for it to feed on.\n\nThe Bible says in Hebrews 12:29 that our God is an all-consuming fire, and the instruction to the Levitical priest in Leviticus 6 was to not let the fire go out. This was important to God; He said it twice because He knew the natural proclivity of man is to take our eyes off what's important and let the fire die.\n\nThere's something about this generation\u2014and this is more about mind-set than age group\u2014that as soon as something is not hot for us, we want to move on to something else. But God has given us a responsibility and an assignment, just as He gave the priests in Leviticus\u2014not to ignite the fire but to not let the fire die. So when we pray, \"God, set me on fire for You,\" we pray in error. There was a fire set at the cross that we are responsible for keeping lit. We ought to be asking God what He wants us to add to the fire so it will burn without dying out.\n\n_Fire dies when it has nothing to consume._\n\nThere is a lot at stake with learning how to feed the fire of God. Every fire has a starting point, what arson investigators call ignition. In our desire to be part of what God is doing in the earth, we can increase our effectiveness by remembering what God has done and how He wants us to respond. The first two ignition points reveal God's contribution to the fire. The last two reveal ours.\n\nIGNITION POINT 1: BURNT OFFERINGS\n\n> The LORD called to Moses from the Tabernacle and said to him, \"Give the following instructions to the people of Israel. When you present an animal as an offering to the LORD, you may take it from your herd of cattle or your flock of sheep and goats. If the animal you present as a burnt offering is from the herd, it must be a male with no defects. Bring it to the entrance of the Tabernacle so you may be accepted by the LORD. Lay your hand on the animal's head, and the LORD will accept its death in your place to purify you, making you right with him.\"\n> \n> \u2014LEVITICUS 1:1\u20134\n\nThis is substitutionary atonement in a nutshell, a foreshadowing of what Christ would ultimately do on our behalf. Romans chapter 3 talks about our being made right with God through our faith in Christ, \"and this is true for everyone who believes, no matter who we are\" (v. 22). While atoning\u2014making up for\u2014one's sins was literal for the people of Israel in the Old Testament, it is by faith in Christ for us now.\n\nRomans 6:23 says that \"the wages of sin is death,\" but death was too high a price for us to pay before Christ because everything was literal. Everything was done in the natural. You wouldn't get back up if you laid down your own life. But this is what God required; He required life. \"Without the shedding of blood,\" Hebrews 9:22 tells us, \"there is no forgiveness\" of sin. Blood represents life. Christ shed His blood so we wouldn't have to. He took our place and died, but in the Old Testament, the animal sacrifice represented a person's sin-life. God created a visceral way for them to see what the exchange rate was for their sin-life by instituting the burnt offering as their atonement, as their way for being made right and thereby righteous enough to approach Him.\n\nThe reason it was important for the priests to keep the fire going was because as long as their sin-life was dead, they had access to God. If that exchange was ever cut short, if the fire ever went out, their access also would be cut off. Because humankind sins continually, there had to be a continual sacrifice, giving us continual access to the Father. He saw fit, in His desire to stay in communion with His people, that the sacrificial fire remain burning. God wanted to set a constant reminder in heaven and on earth of our access to Him.\n\nIn other words, because He is holy and set apart and we broke covenant at the fall of Adam and Eve, and because Christ had not yet come to be our substitutionary atonement once and for all, God instituted something on earth so we could see what the cost of true worship is. The true cost of worship is not a song but our life, which is acceptable because of His life. This is worship many of us now take for granted. We lift our hands if we want. We engage with it if our emotions say we feel like it. But being able to go into the presence of God without fear of our sins instantly condemning us to death is only two thousand years old. For millennia, people offered sacrifices they feared might not be accepted.\n\nConsider this: In the Old Testament if people wanted to go into the house of worship, they couldn't arrive without an animal sacrifice. They would not be accepted into the presence of God if they came the way we come now. And if you think worship is messy when you cry or someone is on the floor, then you would have thought worship was really messy back then. Some of us can't handle seeing our own blood. Imagine dealing with another creature's blood. Imagine the smell and the stench.\n\nYet to God the burnt offering reminded Him that even though we sin, He still loves us. Even though we've broken covenant, He will accept us back, set us right, and complete the work He began in us. Even though He had every right to pour out His wrath on us, He poured it out on something and then someone else to give us access to Him. His only repeated command in this process was keep the fire burning; do not let it go out. In this He says, \"I love you too much to lose connection with you. My love for you burns. I started the fire on your behalf. If you love Me, keep it burning.\"\n\nIGNITION POINT 2: RECONCILIATION ONCE AND FOR ALL\n\n_Atonement_ means reconciliation with God ultimately through Christ, so when God allowed for atonement, He was doing it Himself. From 2 Corinthians 5:19 we see that it was God's desire to bring us back to Him: \"For God was in Christ, reconciling the world to himself.\" He understood the dire nature of our state. Because we do not have the power on our own to come to Him, He came to us:\n\n> For God made Christ, who never sinned, to be the offering for our sin, so that we could be made right with God through Christ.\n> \n> \u20142 CORINTHIANS 5:21\n\nBack then and at the time of Christ, God created ways to absolve us of the eternal punishment for the sin we committed and the covenant we broke. He did this despite our continual rejection of Him. God loves us so much that He made a way for us\u2014born in sin and shaped in iniquity\u2014to have access to Him in all His glory and holy perfection. The only requirement is that we keep the fire going.\n\nIGNITION POINT 3: TRUE WORSHIP AND THE JOY OF SALVATION\n\nGod has forgiven us, and we are not to forget. Our culture is so distracted by the next new thing that we begin to think our experience with God should be similar. We no longer think this salvation thing God did for us is good enough. And this is what causes our fire to die. Our minds are on so many other things that we no longer recall the significance of what He's done.\n\nWhen I was growing up, the older folks used to say that if God doesn't do another thing, He's already done enough! There used to be a time when it was enough that He saved us. We could shout off of that all day. Now we respond as if He has to add all this stuff on top of it. We even make fun of someone if he starts shouting over just being saved. Is the joy of salvation not enough to set off high praise and worship? What happened to it being enough for someone to stand up and say, \"When I think of the goodness of Jesus and all that He's done for me, my soul cries out\"?\n\nBecause we lost sight of the Cross, our judgments of what constitutes being on fire for God are off. We judge people who shout over the seemingly simple works of God concerning our salvation, saying they are religious and emotional. But to remember that God has already done it all and all we have to do is keep it going isn't religious or emotional. We do not have to light the fire. He sent fire from heaven once and for all. All we have to do is keep it burning. When the people of God stand up and declare, \"I won't forget!\" more wood has been added to the fire. It is good for us to remember what the Lord has done.\n\n_The level of your passion and fire for God is directly proportionate to the level of your surrender._\n\nWhen you don't feed the fire with a testimony, it dies. When you don't feed it faith, it dies. For our fire to go out means that we aren't appreciating God's love enough. Looking back at Leviticus 6, we see that each morning new wood had to be put on the fire. Tending the fire was a daily process. It wasn't about how the priest felt one day or another.\n\nThe commitment to worship God has been reduced to how we feel about the way He has blessed us. We like to decide if God is worthy of our praise this week based on how it went. Did our bosses treat us right? Did this or that go the way we wanted? This is when worship is based on feelings. This is when worship is offensive to God and He doesn't have to receive it. Have you heard of strange fire? (See Leviticus 10.) You can't offer anything to God and think He will accept it. But true worship\u2014recounting the goodness of God and remembering the way He made it possible for you to have full access to Him\u2014is what adds wood to the fire.\n\nIGNITION POINT 4: SURRENDER\n\nIf you stop offering yourself to God, your fire will go out. If you stop giving Him your unconditional worship, your fire goes out. If you stop yielding certain parts of your heart to Him, your fire will go out. If you are asking God to set you on fire, but you are not feeding the fire, the fire will die. You have to feed the fire. God is not going to send fire to revive the fire. Fire doesn't revive fire. The parts of your life that need to be consumed revive the fire. When God searches your heart and shines light on certain things, release them. You are fueling the fire. When He says, \"I want those thoughts, that gossip, those addictions,\" put them in the fire. These are the things that cause your fire to grow.\n\nGod wants to consume every part of your life. Whatever you are withholding from Him is causing your fire to die out. Revival fire burns off of your surrender. The level of your passion and fire for God is directly proportionate to the level of your surrender.\n\nThe more surrendered you are, the more passionate and on fire you will be. When you declare to God that He can have all of you, you add much fuel to the fire.\n\nAfter all the years, the instruction has not changed\u2014don't let the fire go out. The difference is that now the fire is in you. You are the keeper of the flame. God gives the fire once and says to not let it go out. We must tend to the fire.\n\n> Let us not allow slackness to spoil our work and let us keep the fires of the spirit burning, as we do our work for God.\n> \n> \u2014ROMANS 12:11, PHILLIPS\n\nRESPOND\n\nYou want your fire to grow? Surrender. You want your fire to grow? Repent to the Lord right here and right now. Live a life of thankfulness, fall in love with the Word of the Lord, and get around people who are also on fire. If God is asking you for anything, give it to Him. I wish I could tell you that your fire will grow if you shout, but it won't. It grows by surrender. I wish I could tell you that the fire grows by dancing, but that is only a manifestation of being on fire. The fire grows by repentance. When God shows you an area in your life that is incongruent with His will and His heart, cast it off and throw it on the fire. Let God take it from you. When you do this, you are tending the fire.\n\nShow me a surrendered, repentant people, a people who are in love with the Word, a people who are grateful for salvation, redemption, and mercy, and I'll show you a fire burning. They are not after money, houses, and cars, but they are after the grace of God. Those are a people on fire. God is setting fires in America right now, and He is telling us not to let the fire go out. Your fire will be stoked the most as you get on your face before God.\n\nIf it is your heart's desire to keep your fire burning, join me in this prayer:\n\n> _God, search me, and if You find anything in me that is incongruent with Your will, take it out of me. Forgive me. I repent. I come to You. I am hungry for You. I am desperate for You. I want You more than anything else._\n\nEvery time you pray this, you are putting wood on the fire. Paul says, \"I beseech you therefore, brethren, by the mercies of God, that you present your bodies a living sacrifice, holy, acceptable to God, which is your reasonable service [or 'worship' in some translations]\" (Rom. 12:1, NKJV). In other words, this is truly the way to worship God. If you really want to worship God, throw anything in you that is not like Him into the fire.\nCHAPTER 13\n\nIT HAPPENED BECAUSE IT'S HAPPENING\n\n\"No weapon formed against you shall prosper, and every tongue which rises against you in judgment you shall condemn. This is the heritage of the servants of the LORD, and their righteousness is from Me,\" says the LORD.\n\n\u2014ISAIAH 54:17, NKJV\n\nTOWARD THE END of the first year after it had become clear God was doing something significant among us at Deeper Fellowship, the enemy began to attack many of our members and their families. There were also attacks on the city itself.\n\nOn June 12, 2016, a gunman opened fire in Pulse nightclub in Orlando, taking the lives of forty-nine people. At the time, it was named the deadliest mass shooting in modern US history. We were devastated. As the whole city mourned the losses, evidence of kindness, love, and resilience rose to the surface through first responders' and trauma surgeons' tireless efforts, long lines of people waiting to give blood to the injured survivors, all-night prayer services and vigils, and financial donations to support the victims' families.\n\nWhat happened in our city was unfortunate, tragic, and terroristic\u2014a horrific act. But even in the midst of this tragedy and the many other attacks around the city and against our own members, God began to give us spiritual revelation about why the attacks\u2014both personal and collective\u2014were happening in this season. And what He told us, as I have said about everything up to this point, was not just for us.\n\nThe enemy wants to keep any fire of revival from burning. I want you to know that while he may try to come against you, he will not succeed. Do not be discouraged by the attacks because the enemy has overplayed his hand. Whatever the enemy has thrown and is throwing against your life and your region in this season, I declare to you, it won't work. No weapon formed against you will prosper (Isa. 54:17).\n\nELEVATE YOUR PERSPECTIVE\n\nIf you recall, in 1 Corinthians 2 Paul wrote about the different ways he communicated the mysteries of the gospel to different groups of people with varying levels of understanding. He wrote:\n\n> God revealed these things by his Spirit. For his Spirit searches out everything and shows us God's deep secrets. No one can know a person's thoughts except that person's own spirit, and no one can know God's thoughts except God's own Spirit. And we have received God's Spirit (not the world's spirit), so we can know the wonderful things God has freely given us.\n> \n> When we tell you these things, we do not use words that come from human wisdom. Instead, we speak words given to us by the Spirit, using the Spirit's words to explain spiritual truths. But people who aren't spiritual can't receive these truths from God's Spirit. It all sounds foolish to them and they can't understand it, for only those who are spiritual can understand what the Spirit means.\n> \n> \u2014 1 CORINTHIANS 2:10\u201314\n\nPaul is telling us there are some things that carry a deeper spiritual meaning that goes beyond our natural understanding, and the only way to get that understanding is through a revelation from the Spirit of God. I believe that what our city and our church community were experiencing in the natural went beyond what we could see on the surface. The attacks you face as you cry out to God for more also have deeper spiritual meaning. Natural things sometimes have spiritual roots, and it's incumbent upon those of us who have spiritual insight and a prophetic eye to give a spiritual perspective of the natural things that take place. Otherwise some of us will walk around confused, and that's what the enemy wants. If the Lord gives me any revelation into a situation, I refuse to allow people to be confused about how things happening in the spirit realm are affecting things in the natural. The attacks that come in revival seasons are natural manifestations of spiritual things.\n\nWhen attacks come, some people allow the enemy to lead them down the religious route of judgment and condemnation. But others desire to speak words of comfort, hope, affirmation, and life. This is the path we have chosen at Deeper. We seek to be comforters.\n\nTimes of hurt, loss, and tragedy are amazing opportunities for us to allow God's Spirit to work through us and reveal who Jesus is. We also have the opportunity to reveal the deeper prophetic understanding of the enemy's attacks. We cannot afford to be deceived. We try to find all kinds of reasons to explain the inexplicable things that happen in our lives and communities, but we must recognize that they are spiritual attacks first with natural manifestations.\n\n_Do not be discouraged by these attacks because the enemy has overplayed His hand._\n\nPeople get sick, people get injured, and people die. These things are hugely unfortunate and painful for us to endure. But here's what I need you to understand: the devil doesn't play fair. You must learn to see these things for what they are\u2014attacks from an enemy that knows he's in trouble. I am in no way ignoring the natural ramifications of pain and loss. What I hope to do is encourage those of us who are praying for revival and experiencing opposition in our homes, churches, and communities to do as Paul said and, in a manner of speaking, \"elevate all things\" to a spiritual level so we'll know how to stand against them.\n\nI made this clear to our church in a sermon the weekend following the Pulse shooting\u2014and this applies to all who are beginning to see that it's happening: we are at a critical moment because there is a demon spirit that is angry with the church. But just as his strike against us is swift and intense, so will his loss be.\n\nTRAGEDY WILL NOT DEFINE YOU\n\nThe enemy is attempting to redefine you by your tragedy. He is attempting to silence your prophetic voice that cries out, \"Thus saith the Lord.\" He attempted this with us in Orlando. People all over the world were asking, \"What's happening in Orlando? Why is that city being attacked so badly?\" Of course there were the standard speculations. But I believe we were attacked like that because \"it's happening.\" There's a revival breaking out in our city. God is about to pour out His Spirit in an unprecedented manner in our city and from our city to the rest of the nation. So for everyone who asked what's happening in Orlando, my response was, \"A revival.\" We aren't the only ones. There are other churches in the city that have been experiencing an outpouring of the Spirit of God as well. God is moving in our city, and the devil doesn't like it. God is moving in your life, and he doesn't like that either.\n\nSome of the attacks happening in the body of Christ\u2014in your life and mine\u2014are happening because revival is happening. Revival is here, and it won't be stopped by what happened. Will you say this aloud in agreement with me? \"It happened because it's happening, but what happened won't stop what's happening.\" It's a little bit of a tongue twister, I know, but say it loud and put it out in the atmosphere even if you don't fully understand it right now. This chapter will help you understand. The plans of the enemy won't work. They will not stop the move of God.\n\nOne of the reasons we chronicle the numerous healings that have taken place at our church is because over time it's easy to forget what has happened, especially as the number of miracles increases. Of course the individuals healed will never forget, and those who knew them when they were sick will never forget. When a family member or friend no longer has cancer or can now walk without a walker or cane, you don't quickly forget. You remember things like that.\n\nYet sometimes things happen that no one can forget. That's what happened on a Sunday morning during River 1. The presence of God was evident in the room. There are times when it just seems as if everyone is aware of Him, and this was one of those moments. We also had the sense that God wanted to move in the room, and we specifically called out those who were dealing with certain sicknesses. I noticed in front of me a man who is normally active during worship but on this particular Sunday was sitting for most of it.\n\nAt first I wasn't sure why he was sitting so much during worship, but then I noticed a walking boot on his foot. Clearly something was wrong with his foot, and he seemed to be in a lot of pain, but when I invited people who were sick to receive prayer, he didn't come forward.\n\nWe prayed for those who came to the altar, and as we were about to move into another part of the service, out of compassion I went to him and asked what was wrong. He explained that he had broken his foot that week and would have to wear that walking boot for several weeks. Although he was in a lot of pain, he said he didn't come forward because he knew there were people who had much more serious conditions than a broken bone in their foot. He had been present for many of the other miracles that had taken place, so he said he wanted others to receive prayer.\n\nI let him know that the Lord cared for his situation just as much as He cared for those with more serious health issues. I asked him if I could pray for him, and he said yes, so we began to pray. Sometimes when I pray for people, I feel heat in the affected area as well as in my hand. Oftentimes those who are standing around can feel it as well. I began to experience this manifestation of heat as I prayed for the man. What happened next is something I won't soon forget, and I don't think anyone else there that day will soon forget it either. I asked him, \"What do you feel?\" He shouted, \"It's gone!\" A little startled by his proclamation, I said, \"What's gone?\" He said, \"The pain!\" Then, unprompted, he took off his walking boot and proceeded to run back and forth, even highstepping. He never had to wear the boot again. He came back the next week in dress shoes. Jesus healed him instantly! It's happening!\n\nWHAT'S HAPPENING IN ORLANDO . . . AGAIN?\n\nThere are spirits that linger over regions. We see proof of this in the book of Daniel, when Daniel prayed and asked God to give him an answer to something. God released the answer right away, but it took twenty-one days for God's response to reach Daniel because a principality called the prince of Persia had been blocking it. The archangel Michael had been fighting the demon prince for three weeks trying to get the answer to Daniel.\n\nJust as the prince of Persia governed a region, there are demonic principalities that govern people's behavior and thought processes. That's why the Bible says the weapons of our warfare are not carnal but mighty through God to the pulling down of strongholds (2 Cor. 10:4\u20135). What's a stronghold? It's a fortified thought\u2014a thought that you defend vehemently and can't break through even when things come to convince you otherwise. Until a person hears and really receives the truth of God's Word, he can't change a stronghold even if he wants to.\n\n_It happened because it's happening, but what happened won't stop what's happening._\n\nFor example, poverty is a stronghold. When you see generation after generation in poverty, and no matter how good their jobs are or how much money they make, they're still in poverty, they're dealing with a stronghold. It may be affecting how they respond to wisdom regarding the life choices they make or perhaps how they see their value. However it manifests, a stronghold is a deep-abiding dysfunctional mind-set. Sometimes this is called a generational curse. But it's not just a generational curse; it's also a stronghold.\n\nThere is a stronghold that hangs over the city of Orlando. If you look at the history of our city, you'll discover that three times in the last twenty-six years the fastest-growing church in America was in Orlando, yet none of them were able to sustain the momentum. It doesn't really make logical sense until you recognize that it's a spirit. Some have called this region a \"pastor's graveyard.\" Church leaders have risen to national prominence, only to have their life and influence cut short by untimely death, drug abuse, and other sorts of scandals. There is a principality that fights the church in this region, but it is losing ground now. So when people ask what's happening in Orlando, instead of allowing us to be defined by the tragedies, I respond in the faith of \"it's happening\" and say, \"Revival!\"\n\nAlthough this spirit is losing ground, it works hard to resist us, to oppose those who would actually go hard after God. I was telling someone the other day that around the time when God called me to plant Deeper Fellowship Church, I had a visitation from a demonic presence that told me if I planted a church in this city, it would take my family. The devil is a liar. This spirit took me on a journey to different churches, and it said, \"See what I did there to that church? See what I did to that pastor? See how he had an affair, and now the church is a shell of its former self or is gone altogether? Do you see what I did to this church over here? Do you see how this happened? Do you see how that happened?\" He said, \"I'm going to do it all to you too.\"\n\nNo, he's not. And let me just say, this is a good reminder of why we all should pray for our pastors. This spirit was trying to persuade me to take my gifts of worship leading and prophetic insight to another city to join with somebody else rather than stand on my own. But what he didn't count on was the fact that God was sending others to stand with me. I do not stand on my own. God has equipped me and has allowed me to link arms with others who are strong in the faith to keep me and others from falling. And by God's grace, through this book and your bearing witness to what's happening in our church, the same will happen in your life and in your church and all over this nation. Miracles, signs, wonders, healing, breakthrough, deliverance, presence, worship\u2014may it all be released everywhere that's desperately crying for more.\n\nThe enemy may have tried to take captive what God was doing, but it won't work. We are reclaiming every territory the devil has sought to destroy. We are pursuing him and recovering everything that's been taken!\n\nCONQUERING THE SPIRIT OF GOLIATH\n\nThere are many lessons from the life of David that have influenced my worship and my warfare while contending for the presence of God to rest on me. David's encounter with the Philistine warrior Goliath gives us much to glean.\n\nIn 1 Samuel 17, when David was still newly and secretly anointed to be Israel's next king, he was out on the battlefield bringing food to his brothers. They were soldiers in King Saul's army. All of a sudden, David heard a nine-foot-tall Philistine defying the armies of the living God. This is the epitome of a spirit of intimidation\u2014a spirit that was so confident in its ability to control the move of God that it challenged the people of God.\n\nFor forty days Goliath challenged the armies of Israel. He symbolizes a type of principality that stands over a region saying, \"I can do this to you, and I can do that to you. Look at what I did to them. Do you see the size of my sword and my shield? Do you see how big and bad I am?\" David said, \"Who is this pagan Philistine anyway, that he is allowed to defy the armies of the living God?\"\n\nTalking all this noise, Goliath did not consider that somebody in the group would actually have faith in God. He had completely miscalculated. While he thought he was coming against the armies of Saul, David corrected that perspective by declaring that Goliath had come against the armies of the living God.\n\nThe spirit of Goliath always miscalculates and misinterprets who he is up against. In trying to come against the people of God, he is also coming against God. We serve an undefeated God who answers the cries of His people. So when we say, \"We want You,\" He moves everything, including a principality, to get to us.\n\nDavid did not think for a moment that he was fighting Goliath. He always knew Goliath was fighting God. Goliath was not only the physical manifestation of the man who stood in front of David that day. Goliath represents anything that stands against the move of God and tries to stop it through fear and intimidation.\n\nThat's what has been happening in Orlando in this new season of revival. But Goliath has it wrong. He keeps looking down on our churches that are not full, just as he mocked David's comparatively small stature, saying, \"I'm still winning.\" But there's a remnant of people like David who will cut off Goliath's head. It has nothing to do with whether the church is full; it has everything to do with faith in God. The churches will fill up, and even when they do, our strength will still be in our faith in God, not the size of our congregations.\n\n_Goliath is anything that stands against the move of God and tries to stop it through fear and intimidation._\n\nCan I tell you something the Lord has revealed to me? God has quietly been anointing a Davidic generation in our city and around the world. He's been anointing them, He's been sending them, He's been raising them up, and they've been coming forth. They have the word of the Lord in their mouths and faith in their hearts, and they believe God can do anything. They are confident that anything that tries to oppose God will not succeed. They don't care about the statistics or what people have heard about their cities. They know their God, and they will rise up as David did and cut off the head of Goliath.\n\nPURSUE AND RECOVER ALL\n\nThe next experience from the life of David that gives us instruction for warfare during revival is his experience at Ziklag. (See 1 Samuel 29\u201330.) This probably was the worst day of David's life up to that point. Being a father, husband, and pastor, I can't imagine the loss David felt that day. He had lost it all, including his family\u2014or so it seemed.\n\nAfter David's victories over the Philistines, King Saul grew more jealous of him as the days went on. His jealousy had increased so much that he was relentlessly pursuing David to kill him. To find safety from Saul's constant attacks, David went to the land of the enemy, the Philistines, and he made a deal with their king, Achish. Achish gave him a town called Ziklag. He lived fairly peacefully there until the Philistines asked him to fight with them against his own brothers, the Israelites. And David was going to do it!\n\n**Beware of the false-unity trick.**\n\nAs we are seeking to keep our thoughts elevated to see the spiritual roots of the natural attacks we face, consider this: we have to be careful about joining in with calls for unity in the midst of certain troubles and tragedies. Sometimes the calls for unity also mean you must be silent about what the Lord is showing you concerning the situation. By joining the Philistines, David was choosing to oppose the armies of the Lord. There is no other way to see it. Sometimes the call for unity is a trick of the enemy meant to bring the church into a stupor so we will not speak the word of the Lord.\n\nAt every turn the enemy seeks to silence the prophetic voice of the Lord in order to redefine what God is doing. He doesn't want revival to break out, but his plans will not work! While we always must walk in love and compassion, we should never choose to be unified with the world at the expense of standing for God. Unity with the world will never work because there will always be some who don't trust or like us. We spend too much time trying to be liked by the world. Jesus said, \"You will be hated by everyone because of me\" (Matt. 10:22, NIV). His words were not ambiguous. The only way to be liked by the world is to deny Christ.\n\nI know that's a little hard to hear, but the better place is to stand with Christ. The enemy is about to be sent on the run for good.\n\n**Watch out for the sneak attack.**\n\nThe leaders of the Philistine army could not come to terms with David's choice to fight with them. So they sent him and his men back to Ziklag. Three days later, when David and his men arrived home, they found that the Amalekites raided the town, burned it to the ground, and took all their possessions and the women and children captive. But wait a minute. Wasn't the war between the children of Israel and the Philistines? Where did the Amalekites come from?\n\nFor a long time you've been battling one spirit, and out of nowhere, it seems, here comes a surprise attack. It's been happening to us throughout the season of revival. Remember what I had you say earlier in the chapter: it happened because it's happening, but what happened won't change what's happening.\n\n_The only way to be liked by the world is to deny Christ._\n\nIf you want to know the reason for these attacks, they are acts of retaliation. The enemy knows he's losing ground, so he is calling in reinforcements, hoping to crush us in our moments of pressure, grief, worry, and weakness. But we will not stay there because we know how to get out of the place of pressure and grief. I told our church and all who listened on the podcast the week after the Pulse shooting that what happened in our city was horrific, but we are sending that foreign demonic spirit on the run.\n\nThe enemy has been after the seed of revival in our city, so he tried to take what is valuable to us captive. But he can't kill it. We are the only ones who can kill the revival.\n\nIf you want God, you'll have Him because we are under an open heaven, we are in a season of habitation, and we are in a season of visitation\u2014and it's available to everyone. The enemy can't kill that, but he will try to carry it away captive. And as David and his men did, we weep over what we lost until we can't weep anymore. But then something shifts.\n\n**\"Bring me the ephod!\"**\n\nIn the midst of all of his grief, confusion, hurt, and defeat, David said, \"Bring me the ephod!\" (1 Sam. 30:7). The ephod is a sacred garment that only the high priest or a king could wear. It had something called the Urim and the Thummim inside the breastplate. When the people of Israel needed an answer to a major question, the priest or king would bring out the Urim and the Thummim, and it would tell them yes or no. In this particular instance when David called for the ephod, the Bible does not actually record that he pulled out the Urim and the Thummim. It seems David put the question before God. He used the ephod not only as a priestly garment but also as a garment for worship, for invoking the presence of God. For David to get the right perspective, he knew he needed to change his posture.\n\nYou know why the enemy is going to be on the run in our city, in your life, and in mine? Because we're not coming at this thing like warriors; we're coming at it like worshippers. We have to change our position and our perspective. Going forward, David knew he needed to approach God like a worshipper, not a warrior. To get the will and counsel of God, he had to get into the presence of God.\n\nPURSUE AND RECOVER ALL\n\nDavid sought the Lord. In 1 Samuel 30:8 he asked, \"Should I chase after this band of raiders? Will I catch them?\" The Lord told him, \"Yes, go after them. You will surely recover everything that was taken from you!\" (v. 8). The New King James Version says, \"Pursue, for you shall surely overtake them and without fail recover all.\" I need you to hear the word of the Lord: you will surely recover everything that was taken from you\u2014without fail!\n\nA foreign enemy came into our territory and took captive what God has given us, and when we get into the presence of God, we discover the counsel of God. Some of us get into the presence of the twenty-four-hour news cycles and social media posts. Sometimes we stand on everybody else's platform and neglect the presence of God. But no more. We know what to do. We must say, \"God, what should we do about this?\" It may be popular in some circles to just hold each other and say, \"We'll get through this.\" We have hashtags and memes for every sort of disaster and trauma that comes against us. But the word of the Lord is not to just stand there and take it. The word of the Lord is that we're supposed to pursue and recover all.\n\nWe will not sit by while the enemy comes in and steals the seat of revival and while he takes captive what we've been praying and laboring for.\n\nGod has shown us clearly why He is doing what He is doing here in Orlando. Think about this: Orlando is the number five most visited city in the United States. Sixty-eight million people came here in 2016, a record high, and more than two million people live here. Can you imagine what would happen if a full-on revival were to break out here? News of it would spread around the world. The enemy doesn't want people to come to Orlando because revival is happening here. And though he sent and keeps sending bands of raiders to take captive things that are precious to us, we will not stop contending for what belongs to us. We will not stop until we recover all.\n\n_We're not coming at this thing like warriors; we're coming at it like worshippers._\n\nWhen David and his group went out to fight, the Bible says that two hundred of them were too tired to go (v. 10). It's all right if some are battle weary. They'll still get the victory. As the four hundred went on, they came upon an Egyptian slave who had been among the Amalekites. They fed him and let him rest under their protection. After he regained his strength, he led them to the camp of the enemy, and David and his men overtook them and got back everything and everyone who was stolen.\n\nTHE OPPOSITE EFFECT\n\nI want to share another part of this spiritual warfare concept we are discussing. Just as the enemy sends these attacks to stop the move of God and get us off track, God always has a way of turning these things around and using them for His glory. David's rebound at Ziklag will forever be the testimony of one who \"strengthened himself in the Lord,\" who changed his posture from warrior to worshipper, and who fearlessly obeyed the Lord, pursued the enemy, and recovered all. The testimony that rises to the surface despite the enemy's efforts is that God is undefeated; He always wins.\n\nThe enemy meant for the attack to kill you, but it's about to bring life. He meant for it to stop you, but it's about to bring revival. Things look one way, but that's not the way it's going to end. God is going to turn the situation around and use it for His glory. The attack of the enemy is having the opposite effect. Let me show you how I know this is true.\n\nIn Acts 28 we read about a time when the apostle Paul was on the island of Malta. He had been shipwrecked. The natives of the island were kind and had started a fire to keep their new guests warm. Wanting to keep the fire burning, Paul had collected some wood and put it on the fire. Because of the heat, a snake came out of the fire and bit Paul on the hand. The people of the island saw it hanging from his hand and thought he was attacked because of something bad he had done\u2014\"A murderer, no doubt!\" (v. 4). But Paul just shook the snake off and kept going.\n\n_The devil thinks he can outlast this generation, but we are going to strike until we see revival happening everywhere!_\n\nThe people looked on, waiting for him to swell up and die. The longer they looked, the more they realized there was something different about Paul. They expected the injury to cause him to drop dead at any moment. Not only did he not die, but also there was no sign of the attack. After witnessing this, the people were in awe.\n\nThen Paul went in and prayed for the sick father of the chief of the island, and he was healed. And not only that; Paul prayed for everyone who was sick, and they also were healed. Because of the miraculous work of God, when it was time for Paul and his company to leave, the people showered them with honors and all the supplies they needed.\n\nI declare to you that the same thing that happened to Paul will happen to you. The attack of the enemy will have the opposite effect on your life. When the enemy strikes, some will be waiting for your demise, but they are about to be shocked. There are those who are waiting for you to quit, give up, throw in the towel, but they are going to be shocked. Yes, the enemy bit you, but you will not die. You will live and proclaim the works of the Lord (Ps. 118:17). You are still here! You have confused the enemy by the way you have come back from his attacks. He thought he could take you out, but you shook him off and threw him in the fire.\n\nYour resilience is about to change your family, city, and nation. Your resilience is about to start a wildfire. The people of the island ended up wanting what Paul had. Whatever kept him from going under when the enemy attacked him they wanted. Since it didn't kill him, they believed that if they had whatever he had, they also wouldn't die if attacked. And what happened next is what I declare is going to happen in the season of attack: the enemy's assault on your life is about to cause a revival in all those who know you. The enemy messed up when he showed his head and attacked you.\n\nHOW WORSHIPPERS WAGE WAR\n\nSo I'm saying this to you and to the spirits that have ruled in our homes and geographic regions for far too long: we will not give in to the attacks. We will pursue and recover all that we've prayed and labored for. We wage war, but not like the world wages war, for the weapons of our warfare are not carnal, but they are mighty through God to the pulling down of strongholds. We are taking up our positions as worshippers, as those who seek the presence of the Lord. And we will see these attacks thrown into the fire to fuel the revival we are crying out for. The enemy's attacks will not end with our disgrace, defeat, or death. They will end with the name of the Lord glorified forever.\n\nSo if people ask you what's happening in your life, church, or city, do not define yourself by any of the trials and tragedies you're facing or that are taking place around you. Tell them as we do at Deeper Fellowship that a revival is what's happening, and you will not stop until the whole world knows about it.\n\nRESPOND\n\nI want to challenge you to consider your posture as this move of God spreads around the world. Consider how you are approaching the attacks that come into your life. Are you taking up the posture of a warrior? Or are you taking the posture of a worshipper? Worshippers know where to be when the heat is turned up. Worshippers know that sounds get God's attention. Worshippers are the ones who say in the midst of the valley of the shadow of death, \"Bring me the ephod!\"\n\nThe devil thinks he can outlast this generation, but we're not going to stop until we see revival happening everywhere. One of my favorite examples of what it will take for us to keep the fire burning and the enemy at bay is found in 2 Kings 13:18\u201319. An enemy was coming against the children of Israel again. From the prophet Elisha's house they could see the enemy charging toward them. The Bible says:\n\n> Then [Elisha] said, \"Now pick up the other arrows and strike them against the ground.\" So the king [Jehoash] picked them up and struck the ground three times. But the man of God was angry with him. \"You should have struck the ground five or six times!\" he exclaimed. \"Then you would have beaten Aram until it was entirely destroyed. Now you will be victorious only three times.\"\n\nThe message here is strike until. Don't relent. Don't hesitate. Keep praying until you see the answer manifest. Keep testifying until salvation comes to your family. Keep worshipping until you see breakthrough. Don't stop until the enemy is destroyed. Strike until you have recovered all!\nCHAPTER 14\n\nLIVING STONES\n\nEach of you must pick up one stone and carry it out on your shoulder\u2014twelve stones in all, one for each of the twelve tribes of Israel. We will use these stones to build a memorial. . . . He did this so all the nations of the earth might know that the LORD'S hand is powerful, and so you might fear the LORD your God forever.\n\n\u2014JOSHUA 4:5\u20136, 24\n\nIT IS HARD to believe that I have lived to see the moment I prayed for\u2014we prayed for\u2014for years. Some have prayed for forty years. Others, forty days. And still others may have just joined in the last forty minutes. Yet we all will be recipients of the residual effects of God's answering all the prayers from all the hearts of people across the globe. What we are seeing happen in this moment in time is sovereignly orchestrated. It is something only God can do. He's brought all the prayers together and by His will is changing our destinies and making us part of something that will outlast all of us.\n\nWhen the Lord walks into a room to mark a people, as He did for us at Deeper, He does so to mark them in a way that allows them to hear from Him supernatural things that are too big for the human mind to receive without His presence. Since the moment He walked into Deeper, everything about how we do what we do has changed. Prior to that moment we were teaching in series and layering prophetic thoughts and prophesying about a future that was to come, not knowing that the future that was to come was right upon us. We were believing God and making declarations of things that seemed to be far into the future. We essentially assumed the things we were talking about probably wouldn't manifest to us in our building but would manifest through us in some distant way. But God said, \"No, I want to do it through you, and I want to do it now.\"\n\nThe amazing thing about trying to \"prepare\" for a moment like that is that encounters don't come at your choosing. By definition an encounter is an unexpected meeting. God chooses the encounter. All we can do is be in position for one. And ultimately we found ourselves continually in a position of crying out to God.\n\nI don't know about anyone else who has experienced something like this, but I didn't know when to expect it. Almost two decades ago I read a book called _The God Chasers_ by Tommy Tenney. When I finished it, I put the book down and said, \"God, for the rest of my life I won't rest until I have an encounter like this.\" Since then every time I have led worship anywhere around the world, I have led trying to position those of us in the room for an encounter. I looked at every single moment, every single service, wondering, \"Could this be the moment? Could this be the moment God walks in?\" After living every day of the last twenty or so years with that kind of anticipation, what the Lord did in one moment during our service on May 22, 2016\u2014that was sudden.\n\nA year later the Lord led us to hold a special service acknowledging how that moment changed us. We could not move on as if nothing happened. More than two hundred miracles have taken place since the moment the Lord walked into the room, and it wasn't because we call ourselves a healing ministry or were having healing crusades. The miracles aren't taking place because we are constantly saying, \"Bring the sick and the lame, and we'll pray for them and lay hands on them.\" Most of the miracles taking place happen because the Lord Himself does it, and even some of those happen for people who are not in the room. People of faith place on the altar the names of individuals who are not in the room but need a special touch from God, and they receive it wherever they are in the world.\n\nOne year into our revival experience with God we could not do anything but stop and thank Him. It was not about trying to recreate the moment He walked in the room. It was about our honoring what He did. In essence, we were saying, \"God, we're not going to let the day or the moment pass by and act as if You didn't show up and change us forever.\" We stopped to acknowledge that we were not the same anymore.\n\n_\"God, for the rest of my life, I won't rest until I have an encounter like this.\"_\n\nAny people who have had an encounter with the living God should mark it and acknowledge what God has done. As they did for the men and women of old, these times of acknowledgment serve as lasting memorials, living stones to honor the presence of God in our lives.\n\nMARKED FOR LIFE\n\nI have a little scar on my left eye. It wasn't until my children became fascinated by it that I remembered it was there. What I haven't forgotten is how I got it. When I was a young boy, I used to go to North Carolina every summer to spend time with my grandparents and my cousins. Wrestling was a big deal back then. It may be a big deal for people now, but this was the era of the classic \"yellow trunks\" Hulk Hogan, not the \"bad\" Hulk Hogan, which he turned into later. Wrestling was in its prime, and my cousins and I watched it all the time.\n\nOf course we would try to imitate the moves. There was one move called the \"monkey flip,\" which my older cousin thought would be the perfect one to try out on me. I don't know how it happened, but I was the guinea pig for everything. So he said, \"Hey, let's try the monkey flip!\" He flipped me, and when I landed, things got quiet for a second. Then I had this overwhelming sense of peace and calm and warmth. But it wasn't the Holy Spirit. I didn't feel any pain, but I remember looking up and seeing a panicked, horrified look on my cousin's face.\n\nSomeone said, \"Shh! Be quiet! Shh, shh! Be quiet!\"\n\nI was thinking, \"What's going on?\"\n\nMy cousin took his hand and wiped my face. Then he wiped his hand on the carpet, I saw his handprint in my blood, and I started screaming.\n\nMy grandmother called out from somewhere else in the house, \"What's going on?\"\n\nMy cousin was still saying, \"Shh! Shh! Be quiet! Be quiet!\"\n\nI screamed and hollered.\n\nMy grandmother came in and noticed the big gash on my head. There was some fuss and commotion before my aunt rushed me to the hospital. It's funny that my biggest fear in that whole situation wasn't the blood; it was getting stitches. I'm pain adverse, thanks to my mother, who has a unique gift for describing painful things in such a way that you feel the pain as she talks. So I was terrified on the way to the emergency room. When the doctor confirmed that I would need stitches, my aunt told me that I said, \"No! Let the Lord heal it!\" I believed in the power of God even at ten years old.\n\nSo, a few stitches and thirty-plus years later, I have a scar that I don't even think about until my daughter crawls up in my lap and asks, \"What's that?\" And I tell her the monkey-flip story for the nth time. The scar marks a moment from my past.\n\nWhenever we go through something traumatic, there's typically something that will remind us of it. It could be a scar or a sound or a smell, but it marks that experience forever. Whenever something amazing happens, there's typically something that causes us to remember, and again we are marked forever. Either by chance or intention, there are markers that lead us to remember past events. But there is absolutely nothing in the world that marks us like an encounter with God, and God wants us to remember those moments.\n\nBUILD A MEMORIAL\n\nIn Joshua 4 we happen upon a significant moment in the life of the children of Israel. Their story as a nation began with God prophetically revealing to Abraham the destiny of his seed. He said that they would be in bondage for four hundred years, and they were. (See Genesis 15.) They cried out to the Lord for deliverance, and God raised up Moses to lead them out of bondage in Egypt. (See Exodus 3.) At first, Pharaoh wouldn't set them free, as the Lord hardened his heart. (See Exodus 9:12.) But finally they were released into the wilderness with gold, silver, and everything they needed for going into the Promised Land. (See Exodus 14.)\n\n_You know God has done something significant when you remember exactly when it happened._\n\nThe only thing they lacked was enough faith to get there. So they grumbled and complained, making their time in the wilderness much longer and more arduous than God had intended. Because of this, an entire generation died in the wilderness under Moses's leadership. (See Numbers 14.) Moses also transgressed against God and was not allowed to go into the Promised Land. (See Numbers 20:12.)\n\nIn Moses's place God raised up Joshua and commissioned him to take the next generation into the Promised Land. (See Joshua 1.) Joshua 4 chronicles the moment when this new generation, who had survived the wilderness, crossed the Red Sea and the Jordan River to enter the Promised Land. But before they took possession of the promise, God had them stop and remember \"the tenth day of the first month\" (v. 19), the day the people of Israel \"crossed the Jordan on dry ground\" (v. 22).\n\nThis was a significant moment\u2014a pivotal, life-altering encounter remembered with specificity. You know God has done something significant when you remember exactly when it happened. This is why we continue to remember that significant, pivotal, life-altering encounter that took place May 22, 2016. The date is not significant for everyone. Everyone was not in church on that day. Some of the people who've joined us since then missed that moment. But it was important enough to us to mark it.\n\nGod has a specific purpose for each memory. God instructed the people of Israel to make a memorial so future generations would also know what God did. In Joshua 4 two reasons rise to the surface for why God also wants us to build memorials for the things He has done: 1) for our children and future generations to know His goodness (Josh. 4:6\u20137, 21\u201322); 2) and so all the nations of the earth might know how powerful He is (Josh. 4:24).\n\nA MEMORY IS NOT ONLY FOR YOU; IT'S FOR THE FUTURE\n\nThe only way an encounter will outlive you is for you to mark it. Otherwise it will die with you. You don't want encounters to die with you. This is why God instructs us to make a memorial. Not only is what we experience with God for us right now, but it is also for our children.\n\nGod says to us in Deuteronomy 4:9:\n\n> But watch out! Be careful never to forget what you yourself have seen. Do not let these memories escape from your mind as long as you live! And be sure to pass them on to your children and grandchildren.\n\nThis is a command of the Lord. If you're a parent and you're not rehearsing the goodness of God with your children, you're doing them a disservice, and you're disobeying the Word of the Lord. When God does something for you, tell them, even if you consider it to be small. If you didn't know how you were going to make it till the end of the month and the Lord provided, tell them. You keep waiting for big things such as, \"My leg got chopped off, it grew back, and God did it.\" No, no, no. If you had more month than money and God made the money come together, tell your children what the Lord has done.\n\n> Listen, O Israel! The LORD is our God, the LORD alone. And you must love the LORD your God with all your heart, all your soul, and all your strength. And you must commit yourselves wholeheartedly to these commands that I am giving you today. Repeat them again and again to your children. Talk about them when you are at home and when you are on the road, when you are going to bed and when you are getting up. Tie them to your hands and wear them on your forehead as reminders. Write them on the doorposts of your house and on your gates.\n> \n> \u2014DEUTERONOMY 6:4\u20139\n\nDon't stay silent about the goodness of the Lord to your children. That is a command of the Lord.\n\nDuring the Habitation Conference we host every September, we always experience a nearindescribable encounter with God. People come from around the world to be immersed in an encounter with God for days. In 2017 when we opened registration, the conference sold out in less than ten hours. People who are hungry for a move of God literally converge in Orlando from around the globe.\n\nDuring the gathering in 2017 the Holy Spirit led me to share several of the things that had been taking place in our midst because one of the things I understand about a testimony is that it creates faith for the future! The telling of a testimony creates room for God to do it again.\n\nAfter I shared testimonies of healing during one of the morning sessions, faith for healing was strong in the atmosphere, and we began to pray. We are still receiving testimonies about what God did that day. We've received testimonies of breast cancer healed, a deaf ear opened, and lupus healed, and we saw two people walk without their walkers. My good friend Jordan also was healed of a rotator cuff injury.\n\nHe had learned to deal with the pain over time, but he knew the injury would one day need to be surgically addressed. Jordan wasn't one of the people who came down to receive prayer. In fact, he didn't even know the Lord had healed him until he woke up in the middle of the night almost in a panic when he realized that he was sleeping on the side of his body with the injured rotator cuff. Normally if he slept on that side, he'd end up in excruciating pain.\n\nWhen he realized what he'd done, he expected to experience a surge of pain, but it never came. The pain was completely gone. His range of motion was completely restored. He came up to testify about that, and while he was doing so, our music director for the conference got my attention and told me that in that moment he was healed of the same issue. Both of them demonstrated their healing by moving their arms in ways that they previously could not have without extreme pain. Both of these men were healed in the river of God without anyone laying hands on them. Jesus did it Himself. It's happening!\n\nA MEMORIAL PROVOKES A QUESTION\n\nThroughout the Old Testament God instructed His people to build memorials. He wanted the memorials to provoke questions so the story of God's miraculous interventions would be told over and over. That's why multiple times in the Scripture, and particularly in Joshua 4, it says, \"Your children will ask you.\" In the future when your children ask you, \"What's this for?\" they will have an opportunity to discover the great things God has done. When you forget what God has done and don't build a memorial, there will be no inquiry and no opportunity for faith to be built in that area.\n\nWhen there's no memory of past victories, small things look big. If the Red Sea wasn't a part of the history of the children of Israel, the Jordan River would have looked insurmountable. When you forget what God has done, it causes you to question and doubt what He can do.\n\nA MEMORIAL CREATES FAITH FOR THE FUTURE\n\nA memorial is a witness of faith for future generations. It is a testimony to all future generations declaring what God has done while simultaneously declaring what He can and will do. This is why God says, \"Talk to your children.\" There may come a day when you aren't there, and they need to find the faith and confidence to know that if God did this before, He can do it again. \"If God did it for my mother and father, then He'll do it for me. I don't have to worry.\" If your testimony of faith is great, then they don't start where you started. Their faith launches from where yours stopped. Your faith ceiling is their floor.\n\n_When you forget what God has done and Don't build a memorial, there will be no opportunity for faith to be built._\n\nThe whole Bible is a memorial that testifies of the goodness and greatness of God. Every time we read it, it builds faith. Likewise your testimony says, \"Don't expect less,\" and, \"Don't settle for less.\"\n\nEach stone that goes into building a memorial is a testimony. God told Joshua to have one man from each tribe grab a stone, signifying that no one was to be left out. No one is supposed to be left out of the move of God. You can't say it's a move of God if it leaves people out. You can't call it revival if it's exclusive. If one person from each tribe was to bring one stone, that means everyone would have been involved in building the memorial, and that is the way it was to be from generation to generation.\n\nA STONE IS A WITNESS\n\nMay 22, 2016, changed everything in our church. This is the day the prophetic words that had been spoken over our congregation manifested before our eyes. We won't forget, and we can't forget. It is a stone for us.\n\nThe New Living Translation renders Joshua 4:6 this way: \"We will use these stones to build a memorial.\" The New King James Version reads, \"This may be a sign among you.\" The translations come together to help us understand that the Israelites were building a place to remember a miracle, or they were building a place that would bear witness to the miracle that happened there. Joshua obeyed the Lord, who told them that the stones would mark that place where God performed such a mighty act of deliverance. So in essence, a memorial stone is a sign, and a sign is a witness; that means a stone is a sign, and a stone is a witness. Israel was to give a strong witness to the power of God.\n\nJoshua 3:14\u201316 tells us about the miracle they were memorializing. The Bible says,\n\n> So the people left their camp to cross the Jordan, and the priests who were carrying the Ark of the Covenant went ahead of them. It was the harvest season, and the Jordan was overflowing its banks. But as soon as the feet of the priests who were carrying the Ark touched the water at the river's edge, the water above that point began backing up a great distance away at a town called Adam, which is near Zarethan. And the water below that point flowed on to the Dead Sea until the riverbed was dry. Then all the people crossed over near the town of Jericho.\n\nThe Jordan dried up before the ark of the Lord. This is the place where God did the impossible for them, and the stones they were instructed to set as a memorial spoke to this miracle and the God of the miracle. What's even more amazing is that they were told to not get just any stones. They were to get the stones \"from the very place where the priests are standing in the middle of the Jordan\" (Josh. 4:3), from the \"middle of the Jordan, in front of the Ark of the LORD your God\" (v. 5). This was to be the empirical evidence that this miracle actually happened.\n\n_It's not a move of God if it leaves people out. You can't call it revival if it's exclusive._\n\nThis also means there are certain miracles you can only receive when you are near the presence of God. If you remember from chapter 10, the ark of the Lord represents the presence of God. There's a certain testimony that you can only get in the presence of God, in the river of God. They were standing there in the river when God moved. And guess what? Every river has stones! The only way to get stones is to be where God is moving.\n\nYOU ARE A STONE\n\nThere are two key verses that help us see that stones are not some mysterious objects in the spirit realm that we are to somehow grab and place in an intangible location. Instead, we see that we are the stones that testify of what God has done. Stones are not ancient relics or things we can't access every day around the world at any time, perhaps like the real historical sites in Israel where the miraculous happenings we've discussed so far occurred. No. There are real and living memorials that we build every day with our own lives. We are the ones who have been in the river of God, near the presence of God, and who can be the real, empirical evidence of His powerful hand.\n\nHere's what Jesus said in Acts 1:8:\n\n> You will receive power when the Holy Spirit comes upon you. And you will be my witnesses, telling people about me everywhere.\n\nThen in 1 Peter 2:4\u20139, specifically in verse 9, we get a deeper understanding of our purpose as living stones. The passage says:\n\n> You are coming to Christ, who is the living cornerstone of God's temple. He was rejected by people, but he was chosen by God for great honor.\n> \n> And you are living stones that God is building into his spiritual temple. What's more, you are his holy priests. Through the mediation of Jesus Christ, you offer spiritual sacrifices that please God. As the Scriptures say, \"I am placing a cornerstone in Jerusalem, chosen for great honor, and anyone who trusts in him will never be disgraced.\"\n> \n> Yes, you who trust him recognize the honor God has given him. But for those who reject him, \"The stone that the builders rejected has now become the cornerstone.\" And, \"He is the stone that makes people stumble, the rock that makes them fall.\"\n> \n> They stumble because they do not obey God's word, and so they meet the fate that was planned for them.\n> \n> But you are not like that, for you are a chosen people. You are royal priests, a holy nation, God's very own possession. As a result, you can show others the goodness of God, for he called you out of the darkness into his wonderful light.\n> \n> \u20141 PETER 2:4\u20139\n\nThis is the purpose of stones: \"you can show others the goodness of God, for he called you out of the darkness into his wonderful light.\" In other words, living stones have a job description.\n\nYOUR JOB DESCRIPTION\n\nI love the way _The Message_ translates 1 Peter 2:9\u201310. It says,\n\n> But you are the ones chosen by God, chosen for the high calling of priestly work, chosen to be a holy people, God's instruments to do his work and speak out for him, to tell others of the night-and-day difference he made for you\u2014from nothing to something, from rejected to accepted.\n\nDo you wonder why we keep talking about our encounters? It's because we have a job description. We are living stones, and our job is \"to do his work and speak out for him, to tell others of the night- and-day difference he made\" for us. Did He take you from nothing to something and from rejected to accepted? Do you have a testimony?\n\n_Your stones of praise are markers for the power of God working in your life._\n\nRemembering what God has done for us and constantly rehearsing His goodness in our lives becomes the fuel for our worship and passion for Him. When your children see you praise, lift your hands, bow down, shout, sing, and cry, that's a stone. When your worship gets people's attention, they are watching you say, \"Let me tell you a story. This is how God found me, but now this is where I am. I was the least of these, the least qualified, the least likely to be called. But God saw something in me, and He came and rescued me.\"\n\nPeople watching you will see how quickly you are moved to worship. They will see that it doesn't take much for you to get in His presence. And they will see that it is because of the goodness of Jesus and all that He's done for you.\n\nYour stones of worship are a memorial set before your children, neighbors, and coworkers that will tell them of the times when God healed, delivered, and set you free. Your stones of praise are markers of the power of God working in your life. You are a living stone!\n\nLET THE ROCKS CRY OUT\n\nThere's a reason we praise, remember, and testify. Luke 19:36\u201340 gives us a clue:\n\n> As he rode along, the crowds spread out their garments on the road ahead of him. When he reached the place where the road started down the Mount of Olives, all of his followers began to shout and sing as they walked along, praising God for all the wonderful miracles they had seen.\n> \n> \"Blessings on the King who comes in the name of the LORD! Peace in heaven, and glory in highest heaven!\"\n> \n> But some of the Pharisees among the crowd said, \"Teacher, rebuke your followers for saying things like that!\"\n> \n> He replied, \"If they kept quiet, the stones along the road would burst into cheers!\"\n\nWhat Jesus meant here is that if His followers were to keep quiet, there would one day be a group of people on the earth who would not remain silent because their testimonies would speak. Even if they kept quiet in His day, their silence wouldn't squelch the movement. There would be people who would know that He's been so good that the stone of their testimonies would cry out. Jesus wasn't talking about rocks on the ground. He was talking about people who know who He is. He was talking about us. We are the living stones. We are the memorials of God. We are the ones who carry the evidence of God's power.\n\n_Cornelius asked for a river; God responded with the ocean._\n\nOur stones are very important because they build memorials in two places. Not only are they present with you here on earth, but they are also building memorials for you in heaven. In Acts 10 an angel of the Lord visited with Cornelius, a captain in the Italian army who was also a follower of Christ. The angel told him, \"Your prayers and your alms have come up for a memorial before God.\" The stone on earth is a memorial. Your prayers and your giving are offered to your church, in your home, or in some everyday showing of generosity, but God is keeping record of them in heaven, and they become memorials to Him. Your stones live in two places.\n\nBut here's the amazing thing: the Bible never says what Cornelius prayed for. This is going to blow your mind. If Ephesians 3:20 is true, \"Now unto him that is able to do exceeding abundantly above all that we ask or think\" (KJV), then this is what God did for Cornelius. Cornelius asked for a river; God responded with the ocean.\n\nThe Bible doesn't say that Cornelius asked for the salvation of every non-Jew for the rest of the history of the earth, but that's what he got. If you read the rest of the account in Acts 10, you will see that Cornelius's faithfulness caused him to be the door through which the Gentiles were able to hear the gospel and receive the Holy Spirit.\n\nAll this is to say that you have no idea how powerful your stones of worship, prayer, and giving actually can be. You have no idea what the memorial to God actually can mean. Do you want to know why I believe the Lord decided to show up on May 22, 2016, at Deeper Fellowship Church in Orlando, Florida? Because there is a bunch of people here who have memorials in heaven.\n\nRESPOND\n\nYour worship is building a memorial. My worship is building a memorial. Your testimony is building a memorial for your children and future generations. My testimony is doing the same, and even more personally, it is building a memorial for my son. Every time he asks me, \"Daddy, why do you have to go on the airplane?\" I don't say, \"Because I have to go to work.\" I tell him I am getting on that plane because I am going to proclaim the gospel of Jesus Christ to other nations.\n\nThe next generation is watching us build stones in memorial to the things the Lord is doing in this season. Our intentional and deliberate marking of the goodness, presence, and power of the Lord working in our lives will fuel their passion from here to eternity. We are getting them in position for a perpetual outpouring of the Spirit of God. Our spiritual ceiling is their floor. The higher we go, the higher the platform from which they launch the next move of God.\n\nI want to invite you to take a moment right now and recall the things the Lord has done in your life. I invite you to begin picking up another stone to add to the memorial you are building. This is one thing that God commanded that we aren't deliberate enough about. If it means that you get out your prayer journal and start to write things down, do it. Then next time the Lord gives you an opportunity to talk with your children, neighbors, friends, or coworkers, let them in on what God has done. Build your memorial one testimony at a time.\n\nYou may not have been with us on May 22, 2016, but what you're doing now is creating a new day on the calendar for when God moved in the earth. It's creating another portal and invasion of the presence of God to connect with us on earth. You're building the memorial right now for another outpouring, revelation, and visitation\u2014another encounter with the living God.\nCHAPTER 15\n\nYOU WERE BORN FOR THIS\n\nBy the rivers of Babylon we sat and wept when we remembered Zion.\n\n\u2014PSALM 137:1, NIV\n\nBEFORE YOU WERE a thought in the minds of men, you existed in the mind of God. Before you were a twinkle in your parents' eyes, you were the apple of God's. He conceived His thoughts of you as He formed the foundations of the earth. You have been on God's mind from the beginning of time. Therefore you are not a mistake. You are not here by mistake. Your arrival here in this moment was and is intentional. If anyone has told you different, know that person can't know about you what your Creator knows. You are here by His design.\n\nBecause God knew you before you were formed in your mother's womb, it was His intentional will to send you into the earth at this specific time. Again, if you thought you were a mistake, if you were ever told you were a mistake, I need you to hear the word of the Lord: God intended for you to be here, no matter what the circumstances were that surrounded your arrival. The Lord placed you here on purpose because you have an assignment. You were born for this moment. Say it aloud right now: \"I was born for this moment.\"\n\nIf this were not true, you could have been born at any random time, but you weren't. God intentionally set you on His time line in this generation, for this time, and for a purpose. There is something special about the verse that opens this chapter. I want you to see what it tells us about God's set time and the people He appoints to live in that time.\n\nTHE SETUP\n\nAt the time God called Jeremiah, Israel was being ruled by evil kings and false prophets. It was hard to tell the truth from a lie, the word of the Lord from the word of man. Because of this the people were confused and had been led astray. But God saw some glimpses of righteousness among His people, and He set a plan in motion to help restore them back to a people who followed hard after Him.\n\n_Declare: \"I was born for this moment.\"_\n\nLooking for someone to be His mouthpiece and communicate His instructions, will, and thoughts to the people, He called a very young Jeremiah into His presence and appointed him to be His prophet to the nations. (See Jeremiah 1:4\u201310.) God forewarned Jeremiah that not everyone would like what God would say through him, but He would be with him and He would protect him. Then the Lord touched Jeremiah's mouth, gave him the words to speak, told him the plan to revive His people, and sent him out.\n\nThe first part of God's plan was to send the people into exile for seventy years. As a result, Jerusalem was destroyed. We need to understand that this was a consequence of Israel's turning their back on God, worshipping idols, and breaking their covenant with God. God does not wink at sin, particularly when His people dishonor him and allow sin to be perpetuated from generation to generation. So this was the word of the Lord spoken through the prophet Jeremiah:\n\n> \"I will banish from them the sounds of joy and gladness, the voices of bride and bridegroom, the sound of millstones and the light of the lamp. This whole country will become a desolate wasteland, and these nations will serve the king of Babylon seventy years. But when the seventy years are fulfilled, I will punish the king of Babylon and his nation, the land of the Babylonians, for their guilt,\" declares the LORD, \"and will make it desolate forever.\"\n> \n> \u2014JEREMIAH 25:10\u201312, NIV\n\nAlthough God was going to send them into seventy years of exile, He also promised to send revival and restoration. He was letting them know that yes, they would have to deal with their sins, but afterward something big was coming.\n\nWhen everything began to happen according to God's will, false prophets came forward and told the people that everything was going to be OK, that God was not angry, and that all He wanted was for them to be happy. They said it would not be seventy years of exile but only two and that God would bless them. Hananiah was one of the leading false prophets, and those who wanted a comfortable, feel-good message sided with him. Those who wanted to obey the word of the Lord submitted to what God was saying.\n\nThat's why you have to be careful not to so easily receive the words that make you feel good. In this instance Hananiah, the other false prophets, and the people who listened to the word that made them feel good were destroyed. (See Jeremiah 28.) God does some things that you may not understand or like, but He does them to protect you.\n\nA NEW GENERATION ARISES\n\nIn Jeremiah 29 we read a letter Jeremiah sent to those in exile telling them how to maintain until the time for their deliverance comes:\n\n> Build houses and settle down; plant gardens and eat what they produce. Marry and have sons and daughters; find wives for your sons and give your daughters in marriage, so that they too may have sons and daughters. Increase in number there; do not decrease. Also, seek the peace and prosperity of the city to which I have carried you into exile. Pray to the LORD for it, because if it prospers, you too will prosper.\n> \n> \u2014JEREMIAH 29:5\u20137, NIV\n\nGod was letting them know that seventy years is a long time\u2014\"Settle in, get comfortable, and raise your families as you would if you were at home.\" He also instructed them to pray for the city they were held captive in, and this is a lesson to us as well: pray for the place that God sends you\u2014the job, the church, the city, the assignment. Don't curse them, because if you do, their curse is your curse. The other, more significant thing here is that because of the command to grow their families and increase in number, there would be a people, a generation, born in exile. A people born in between seasons. A people too young to experience what was, but not old enough to have become cynical about what was yet to be. The only thing the born-exiles would have was a word from the Lord about a day of restoration, a day of revival. They were not born in Jerusalem, which means they did not have a memory to think back on. All they had was the word of the Lord for their future.\n\nBORN FOR REVIVAL\n\nNever having seen the glory of Jerusalem, this new generation of Israelites was looking forward to something they had never seen. Having been born in captivity, born in exile, they were also born for revival. God knew when He was placing these people into this time line that some would have been born in Jerusalem and moved into exile and others would have been born in Babylon and had only the word of the Lord to hold on to.\n\nSometimes you find yourself in a certain season, and you don't understand why you're there. The Israelites who were born in exile weren't the ones who lived when Israel's evil king Manasseh led the nation into idolatry. They hadn't served idols. They weren't the ones who aroused God's anger. They weren't the ones who listened to the false prophets. They weren't born in Jerusalem. They had no memory of the beauty of the holy city. Yet Psalm 137 is their cry: \"By the waters of Babylon, there we sat down and wept, when we remembered Zion\" (v. 1, ESV).\n\nCan you hear their heart? \"I wasn't around for what it was, so I have no frame of reference, but something in me says this isn't all there is. Something in my spirit longs for a home, for a reality in the spirit that my natural eyes have yet to see, and whenever I think about it, I weep.\" Does this sound familiar?\n\nWE TOO WEEP FOR ZION\n\nIf you think about it, it probably looks weird to others when they see our generation crying. It probably looks weird to others when they see us longing for something we've never seen but have only read about. We too are a generation born in between. Our eternal Father placed something in us that causes us to cry out. He put us here for this moment for a specific reason and a specific purpose because He knows that we will seek Him until we find Him. He knew we would be the generation that would cry out for something we had never seen. Why? Because there's something on the inside of us that will not be satisfied with where we are.\n\nYou may not have been in the room with us when God walked into Deeper that day, but you're still crying out with us because God wove the longing for more of Him into your DNA. He put you here in this generation because He knew you would be part of a people who would not get lazy, a people who would cry out for revival day and night, even though they've never seen it.\n\nIt's not our fault that we're here, but it is our problem. He knew that we would know Him as the problem solver, so we cry out to Him day and night. We weep whenever we think about revival and whenever we think about restoration. This is what Zion represents in Psalm 137. It says, \"We sat down and we wept when we remembered Zion.\" Why did they weep? They wept because they were holding on to the word of the Lord that was given to them by the prophet Jeremiah: \"When seventy years are completed for Babylon, I will come to you and fulfill my good promise to bring you back to this place\" (Jer. 29:10, NIV). God was saying, \"I will bring you home again. I will restore you.\" That moment of weeping signified their determination to hold on to the word of the Lord despite the current season they were in. Things looked crazy, and in the midst of the craziness they wept because they remembered that God had promised their generation would see revival.\n\nREFUSE TO ACCEPT YOUR PRESENT CONDITIONS AS NORMAL\n\nThe key to revival is to not accept your present conditions as normal. The Israelites' present conditions were not OK. They sat by the river and wept because they realized there was something more than what they were living.\n\nHave you ever attended one good church service after another and things were going pretty good, but something on the inside of you said, \"Yeah, there's more than this\"? God wants to do something through a people that affects the world. He wants it to expand out from just what's happening in a small warehouse on a corner in central Orlando. There must be a people who will be part of spreading the move of God throughout the nations of the earth. And until I see that, I can't be satisfied.\n\n> There on the poplars we hung our harps, for there our captors asked us for songs, our tormentors demanded songs of joy; they said, \"Sing us one of the songs of Zion!\" How can we sing the songs of the LORD while in a foreign land?\n> \n> \u2014PSALM 137:2\u20134, NIV\n\nThe tormentors are those who were keeping them in Babylon. They were basically saying, \"Act as if everything is normal. Go back to doing normal things. Sings songs as if you're back in Zion.\" Their captors, their tormentors want them to forget revival and live in the moment, to forget that they had a promise from God. What did these exile-born Israelites even know about what Zion was like? They were born in between seasons. Their tormentors sound like some of ours: \"You're doing too much. You're crying too much. You're praying too much. You're shouting too much. Just act as if everything's fine. What are you talking about, 'revival is coming'? Don't you know it's already here?\"\n\nBut those born in exile were saying, \"No, no, no. What we are holding on to has yet to come, but we long for it. We can't act as if everything is normal. Something on the inside says, 'It's bigger than this.'\"\n\nGod has blessed Deeper Fellowship to see more than two hundred miracles, but we're not going to be satisfied with only that when we know there's more. What God wants to do is bigger than that. We're not going to be satisfied with a few songs of worship. It's bigger than that. God wants to impact the world. Don't tell this generation to be quiet. We weep and cry out because there's more! We refuse to accept our current conditions as normal.\n\nThe exile-born Israelites understood that they may not have been born in Jerusalem, but they had a promise from God about restoration and revival, so they wouldn't accept anything else as normal. In the same way, we must also resist the temptation to rush back to false normalcy. We must resist the temptation to try to act as if everything is normal when the Holy Spirit is the One who is stirring us up and making us uncomfortable. He's doing it on purpose. He's doing that so we will not try to get back to normal. He doesn't want us to get back to normal. He wants to redefine what normal is. The new normal is revival.\n\nCOMPLACENCY IS NOT NORMAL\n\nOftentimes we hate being uncomfortable so badly that we'll do almost anything to find comfort, including acting as if the things that hurt us don't hurt. There are moments in time when we need to allow discomfort to realign our posture. It is not normal to be complacent. It is not normal to not desire to be in God's presence. It is not normal to live as if we don't need Him. It is not normal to live in a society that turns its back on God. It's not normal to live in a society where the church is marginalized. It's not normal to live in a society where the prophetic voice is silenced and even mocked. It's not normal to live in a society where sin is celebrated and the upright are being vilified. It's not normal to live in a society where genuine love and truth are called hate. It is not normal to live in a society where the Word of God is disrespected. These things are not normal, and we need to resist allowing them to be normalized.\n\n_Allow discomfort to realign our posture._\n\nWhen we see the things happening in the world, it ought to inform our cry for revival and remind us of our need for it. When you see police officers gunned down in city streets, it ought to inform our posture for revival. When we see hate groups marching around in the country, trying to make their voices heard, it ought to tell us that we need revival. When we see people who don't even know the Lord but think their marching with signs is going to change what's ailing, it ought to tell us that we need revival. We would rather pretend that everything is OK when it's not OK for the sake of normalcy, rather than face where we really are.\n\nREVIVAL IS IN YOUR DNA\n\nPsalm 137 continues with:\n\n> If I forget you, Jerusalem, may my right hand forget its skill. May my tongue cling to the roof of my mouth if I do not remember you, if I do not consider Jerusalem my highest joy.\n> \n> \u2014PSALM 137:5\u20136, NIV\n\n\"If I forget you\" is an interesting phrase because of the two types of people who were in exile. There were those who came from Jerusalem, and there were those who had never been. So how can the ones who had never been in Jerusalem remember it? Could it be that where they were was never where they were designed to be? God is implying something significant here. Those who have experienced God\u2014the seasoned saints\u2014were to pass on the experience to those who had not known Jerusalem. They were to do it in a way that would cause them to long for it. This is why it's important that the mature grow up with the immature.\n\nThis is why it is important that God puts a church family together, because those who have had an encounter with God are supposed to talk about it so other people who have not had an encounter will long for it. This is what the last chapter focused on\u2014the importance of our testimony, living stones, and setting up memorials. The mature are to mark their own encounters so they can use them as reminders for the next generation. So the next generation will say: \"I won't forget the future place in God that I've never been to, because I know that I was not designed to stay here.\"\n\nThe world and the enemy try to get us to forget that there is a place in God designed for us. But God placed us here at this time because He knit revival in our DNA. We cannot forget that we were born for this time.\n\nThere's a story I heard once about a person whose adopted child stood in his crib one day, crying for his mommy. The adoptive mom came running in the room and said, \"I'm here, baby.\" And the child said, \"Not you. My real mommy.\" The woman was completely shocked. The child had no reference that the woman wasn't his real mother. Yet woven in the child's DNA was the unconscious knowing that even though he was growing up there, he was not being raised by his birth family.\n\nThe cry is, \"If I forget you, O Jerusalem, let me not forget revival. I understand that I might have been born in a season in between. I understand that I might have been born at a time when I may not have experienced revival before, but there's something on the inside of me that longs for it.\"\n\nWe could be going about our day, our year. We could be going about our church existence, lifting our hands and bowing down. We could be going around saying all the right church stuff: \"Hi, how are you doing?\" \"God bless you.\" \"Thank You, Jesus.\" \"I'm blessed and highly favored.\" We could be singing the songs, looking at the words on the screen, hearing some good preaching. Then one day God walks into the room and says: \"Don't forget that you are not supposed to be living an existence in which you just think that going to church and coming home is the totality of what I want to do in your life. I have something greater. I understand that you might have been born in a season when you haven't experienced a demonstration of My power. But there's something that I have woven into your DNA. There's something that I planted on the inside of you that will one day, out of the blue, cause you to lift your voice and say, 'Jesus! I know there's more than this. And I don't want to forget why You sent me here on this earth at this time.'\"\n\nHE'S INTENTIONAL\n\nGod puts a people in a certain time to accomplish a certain thing. It's His intention for those particular people to see what those who came before them may not have gotten a chance to see. Entire generations participate in moving the plan of God forward. God doesn't just think in days or years; He also thinks in generations. He's that amazing. That God put us where He put us, which is two thousand years after the death and burial of Jesus, means that we are not at the beginning of a process. We are two thousand years closer to the return of Christ than they were in the book of Acts, which means God intentionally placed us in this time.\n\nHe could've placed us under the old covenant. He could have placed us in the time when they observed the Law of Moses and had to make all the bloody sacrifices, but He placed us in a day of grace. And because of the time He has placed us in, our urgency should increase.\n\n_God doesn't just think in days and years; He also thinks in generations._\n\nWe are living in a time two thousand years after the first- and second-century church, and they lived with great expectancy. That should make us ask ourselves why we are not feeling that something is drawing closer than ever before. The level of urgency for the early church got so serious that Paul had to write and say, \"Listen, don't quit your jobs.\" (See 2 Thessalonians 3:6\u201315.) People were literally being idle. They dropped everything, believing the coming of Christ was upon them. They were thinking, \"I don't need to work because Jesus is coming at any point, and because He's coming, I don't even need a job. I don't need to pay my bills.\" Paul told them to continue to work. They lived with such an urgency. It's amazing that the further we get away from that time, the less urgent we've become.\n\nBut that's the opposite of the reaction we should have because we are thousands of years closer than they were. Our urgency should be increasing, not decreasing. If our urgency is not increasing, that means that we are out of alignment with God's will, Word, and plan. As a matter of fact, Peter had this to say:\n\n> Most importantly, I want to remind you that in the last days scoffers will come, mocking the truth and following their own desires. They will say, \"What happened to the promise that Jesus is coming again? From before the times of our ancestors, everything has remained the same since the world was first created.\"\n> \n> They deliberately forget that God made the heavens long ago by the word of his command, and he brought the earth out from the water and surrounded it with water. Then he used the water to destroy the ancient world with a mighty flood. And by the same word, the present heavens and earth have been stored up for fire. They are being kept for the day of judgment, when ungodly people will be destroyed.\n> \n> But you must not forget this one thing, dear friends: A day is like a thousand years to the Lord, and a thousand years is like a day. The Lord isn't really being slow about his promise, as some people think. No, he is being patient for your sake. He does not want anyone to be destroyed, but wants everyone to repent.\n> \n> \u20142 PETER 3:3\u20139\n\nThe reason Jesus has not yet come is because our God is so merciful that He is being patient. He is looking at the state of the world and knows that if He comes now, a whole lot of people are going to be in judgment. He does not want any to perish, so He is giving them more time. Jesus has not yet returned because of the mercy of God. He's not being slow concerning His promise.\n\nStill, we should live with urgency because the Scripture goes on to say that we should be prepared because the day of the Lord is coming like a thief in the night (2 Pet. 3:10). No one will expect it. But we wait with great expectation.\n\nWe were placed in this time because God knew we would wait. He knew we would be the ones to remember His promise to bring revival. We remember that He said, \"But then I will come and do for you all the good things I have promised, and I will bring you home again\" (Jer. 29:10).\n\nABBA!\n\n> For all who are led by the Spirit of God are children of God. So you have not received a spirit that makes you fearful slaves. Instead, you received God's Spirit when he adopted you as his own children. Now we call him, \"Abba, Father.\"\n> \n> \u2014ROMANS 8:14\u201315\n> \n> And because we are his children, God has sent the Spirit of his Son into our hearts, prompting us to call out, \"Abba, Father.\"\n> \n> \u2014GALATIANS 4:6\n\nJust like the baby who cried out for his mother when he did not know he had another mother, it is the same with God, who prompts us to cry out, \"Abba, Father,\" and who prompts us to cry out for revival. In Psalm 137 that next generation of Israelites said, \"By the rivers of Babylon we sat and wept when we remembered Zion\" (NIV). A generation who had never seen Jerusalem was being prompted by the Spirit of God to cry out for revival. This is the challenge of intercession for us.\n\nWhile we are here in this world, there's something about normalcy that gets to us after a while. A group of people is being boxed in by normal church, normal living, and normal things. We're being told we're doing too much, we're too loud, we're preaching too long, we're shouting too long, we're singing too long, our service is too long, we're praying too much. And we're questioned about the hope we have: \"You still have faith for that? It hasn't happened.\" \"Why are you still doing that?\" \"Why are you still giving? Just be normal. Just be like everybody else.\" When these things happen, there's something on the inside of us that says, \"You are not my Father.\" There's something on the inside of us that says, \"Give me my real Dad. I want my Father.\"\n\nWhy are you frustrated? Because the Spirit of God is prompting you to cry out for revival. Why can't you get comfortable at night? Because the Spirit of God is prompting you to cry out for revival. Why are you sensing that there's something more? Because the Spirit of God is prompting you to cry out. That's why we weep when we remember Zion.\n\nRESPOND\n\nYou need to know you're not uncomfortable just to be uncomfortable. God has intentionally made you that way. It's His doing. He wants you to be dissatisfied with where you are. He wants you to want more. If you're wondering where the pin in your seat came from that causes you to jump up when you want to sit down and get comfortable, it came from God.\n\nGod is the One who has orchestrated the events in the world to drive the church to its knees. It is because of His prompting that His children are crying out for more.\n\nWe may not be the generation that was born living in the presence of God, but we hold on to the promise that He will come and bring revival. We have been put here for a reason. We were born for this moment. We may not have seen the manifestation of His presence, but we want it. And we won't stop until we see it.\n\nHow about you?\nNOTES\n\nCHAPTER 3\u2014HE HEARS YOUR CRY\n\n. Merriam-Webster.com, s.v. \"groan,\" accessed January 22, 2018, .\n\nCHAPTER 4\u2014FOLLOW THE PATHWAY\n\n. \"About John G. Lake,\" John G. Lake Ministries, accessed January 22, 2018, ; see also Roberts Liardon, \"John G. Lake,\" God's Generals, accessed January 22, 2018, .\n\n. Chad Brand, ed., et al, _Holman Illustrated Bible Dictionary_ (Nashville, TN: B&H Publishing Group, 2015), as quoted in Edward D. Andrews, _Evidence That You Are Truly Christian_ (Cambridge, OH: Christian Publishing House, 2015), 272.\n\nCHAPTER 6\u2014NO SUBMISSION, NO POWER\n\n. Blue Letter Bible, s.v. \" _arrabon_ ,\" accessed January 24, 2018, https:\/\/www.blueletterbible.org\/lang\/lexicon\/lexicon.cfm?Strongs=G728&t=KJV.\n\n. \"Bible Verses About Holy Spirit as Engagement Ring,\" BibleTools .org, accessed January 24, 2018, .\n\nCHAPTER 8\u2014STAY AND WAIT\n\n. Robert Jamieson, A. R. Fausset, and David Brown, _Commentary Critical and Explanatory on the Whole Bible_ , \"Acts 1,\" accessed January 24, 2018, .\n\n. Blue Letter Bible, s.v. \" _homothymadon_ ,\" accessed January 24, 2018, https:\/\/www.blueletterbible.org\/lang\/Lexicon\/Lexicon.cfm?strongs=G3661&t=KJV.\n\n. \"Children's Question Room,\" Toyota, accessed January 24, 2018, .\n\n. Rolls-Royce Raleigh, \"How Long Does It Take to Build a Rolls-Royce?\" Rolls-Royce Motor Cars Raleigh, December 12, 2014, .\n\n. The account of Edward Miller is documented in Lila Terhune, _Cross Pollination_ (Shippensburg, PA: Destiny Image, 1998), 4\u201310; see also R. Edward Miller, _The Secrets of the Argentine Revival_ , viewed at Measure of Gold Revival Ministries, accessed January 24, 2018, .\n\nCHAPTER 9\u2014CONTEND FOR THE FLAMES OF REVIVAL\n\n. Abigail Hess, \"Here's Why Lottery Winners Go Broke,\" CNBC, August 25, 2017, .\n\n. _30 for 30: Broke_ , ESPN.com, accessed January 24, 2018, .\n\n. David Margolick, \"Elizabeth Eckford and Hazel Bryan: The Story Behind the Photograph That Shamed America,\" _The Telegraph,_ October 9, 2011, .\n\n. Michele Norris, \"The Woolworth Sit-In That Launched a Movement,\" NPR.org, February 1, 2008, .\n\nCHAPTER 10\u2014A WORTHY HOST\n\n. _Ellicott's Commentary for English Readers_ , \"1 Samuel 6:19,\" accessed January 24, 2018, .\n\n. Robert Jamieson, A. R. Fausset, and David Brown, _Commentary Critical and Explanatory on the Whole Bible_ , s.v. \"2 Samuel 6,\" accessed January 24, 2018, .\n\nCHAPTER 13\u2014IT HAPPENED BECAUSE IT'S HAPPENING\n\n. Orlando Sentinel Editorial Board, \"Orlando Grieves With Las Vegas,\" _Orlando Sentinel_ , October 2, 2017, .\n\n. \"The Most Visited Cities in the US,\" WorldAtlas.com, January 24, 2018, .\n\n. Sandra Pedicini, \"Visit Orlando: Record 68 Million People Visited Last Year,\" _Orlando Sentinel_ , May 11, 2017, ; Jeff Kunerth, \"Getting Bigger: Metro Orlando Grows to 2.3 Million,\" _Orlando Sentinel,_ March 26, 2015, .\nENDORSEMENTS\n\nIn a time when a lot of the information being promoted is contrary to the Word of God, this generation needs something more than just traditional church services. This generation needs to see a demonstration of God's power, and this has been found at Deeper Fellowship Church in Orlando, Florida, under the guidance of Pastor William McDowell. Miracles, signs, and wonders are happening literally week in and week out. These events are providing hope and truth to eliminate any doubt that the God we've heard about is real. The Bible says these signs shall follow those who believe (Mark 16:17), so our faith is rising to the occasion because of what's being demonstrated at Deeper Fellowship Church. I'm really excited about the release of this book, which shares miracles you wouldn't believe unless you read about them or saw them with your own eyes.\n\n\u2014TRAVIS GREENE\n\nMULTIPLE GRAMMY AWARD\u2013NOMINATED WORSHIP LEADER\n\nLEAD PASTOR, FORWARD CITY CHURCH\n\nGod is using William McDowell and Deeper Fellowship Church to be a prophetic signpost in this generation. In the tension between seeking God's presence and seeking to be cool, they're favoring His presence. And God is honoring that with miracles. May you be infected with the spirit of faith that permeates this book.\n\n\u2014BOB SORGE\n\nAUTHOR, _S ECRETS OF THE SECRET PLACE_\n\nFrom the moment I met William McDowell, humility and faithfulness oozed out of him. The same can be said when you begin to experience this book. It's truly an encouragement toward greater faith and a challenge to not miss what God has happening in your life!\n\n\u2014JOSH MCCOWN\n\nQUARTERBACK, NEW YORK JETS\n\nI've been able to see the handicapped walk, the sick healed, and frowns turned to smiles. The atmosphere at Deeper Fellowship is hard to describe, but when you are present, you understand just how impactful and loving a home it is. What I love is how William doesn't move until he knows the Holy Spirit has moved in the way He's supposed to. This book will give you a glimpse into what God is doing in our church and challenge you to go deeper.\n\n\u2014TOBIAS HARRIS\n\nFORWARD, LOS ANGELES CLIPPERS\n\nThe Bible eloquently speaks of God sending pastors who are after His own heart. Pastor William McDowell and the life-changing work happening at Deeper Fellowship perfectly epitomize what that looks like. He is a leader in full pursuit of the God who has called him. Anytime he stands to offer a testimony about the miracles Deeper is experiencing, it pushes me to continue ministering the pure and unadulterated Word of God\u2014because that is ultimately what creates a space for heaven to kiss the earth. I'm sure that many across the globe can relate to this same push and inspiration; it's contagious.\n\n_It's Happening_ will further solidify to the world a fact that many have long known: Pastor William McDowell is indeed a host of God's presence and a general for the agenda of revival.\n\n\u2014TASHA COBBS LEONARD\n\nGRAMMY AWARD\u2013WINNING WORSHIP LEADER\n\nI am so excited about the miracles taking place at Deeper Fellowship Church and through Pastor William McDowell's ministry. He's shaking the culture and experiencing God's power on earth as it is in heaven and seeing wonderful miracles as a result. I love his heart for God. A church that stays in God's presence will always experience heaven! This book will turn hearts to God! I'm glad to call this man of God my friend!\n\n\u2014ERICA CAMPBELL\n\nHOST, NATIONALLY SYNDICATED RADIO SHOW _G ET UP! MORNINGS WITH ERICA CAMPBELL_; GRAMMY AWARD\u2013WINNING ARTIST;\n\nMEMBER, CHART-TOPPING DUO MARY MARY\n\nIt's hard to believe that I've known William McDowell for only a handful of years because the impact of his ministry on my life has been exponential. Much of my approach to music has been totally shifted by my conversations with William and by the hours of worship in the Father's presence facilitated through William's music.\n\nFrom the many impactful conversations we've had, one statement stands out to me more than any other. He told me, \"If we say what the Father says, He'll make a way for His word.\" That one sentence has reverberated in my life through the years, and I credit it with much of the success I've had as a Christian artist. It has taught me to dig deep into God when I write songs.\n\nI'm grateful for William's ministry and his posture of obedience to the Father. I'm grateful for not only his friendship but also his ability to demonstrate a life poured out, a life tangibly connected to the heart of God. I know this book will greatly impact all who read it with open hearts, and I'm excited about the new season so many people will walk into because of it.\n\n\u2014DANNY GOKEY\n\nGRAMMY AWARD\u2013NOMINATED, AWARD\u2013WINNING\n\nCHRISTIAN ARTIST\n\nIn William McDowell's new book, _It's Happening_ , he speaks on behalf of those seeking revival. Not only does he represent the revivalist accurately, but he also encourages those who don't share this appetite for a heavenly resurgence to change their diets. This book is a glorious recipe to encounter the miraculous and supernatural, which is exactly what our starving world needs at this moment in history. Knowing William personally, I'm not surprised at all by the rich content that saturates this book because it's really only a window into what he embodies and abides in daily. Like every other aspect of his ministry, this book is a necessity in our pursuit of Jesus.\n\n\u2014CHRIS DURSO\n\nAUTHOR, _T HE HEIST: HOW GRACE ROBS US OF OUR SHAME_\n\nPastor William McDowell has given his life to listening to the Holy Spirit's voice, as he has fashioned prophetic songs that give the family of God words to express their deepest feelings of worship. That same ability to hear the Holy Spirit and write songs of worship has found expression in revelatory teaching and preaching that has produced a faith to believe for the supernatural as a regular visitation of the Holy Spirit in their home church.\n\n\u2014BISHOP JOSEPH L. GARLINGTON SR.\n\nFOUNDING PASTOR, COVENANT CHURCH OF PITTSBURGH\n\nPRESIDING BISHOP, RECONCILIATION! MINISTRIES INTERNATIONAL\n\nPastor William McDowell is a trailblazing leader in the world of worship and a true friend of mine. He is already known for his historic record within the music industry, but in this book we are able to see another passion of William's. _It's Happening_ reveals William's heart for the things of God. This book ignites passion within the reader and helps to give inspiration and education on miracles, signs, and wonders. This is a must-read for every believer who wants to go deeper in his or her walk with God!\n\n\u2014MATTHEW L. STEVENSON III\n\nSENIOR PASTOR, ALL NATIONS WORSHIP ASSEMBLY\n\nIt's happening! What a great reminder that revival isn't just something of the past or the future, but it's happening right now. This book is water for the spiritually dry and an anthem for those who desperately long for a move of God. I'm convinced that wherever these pages are being turned, fires of fervor are going to be set ablaze. I'm thankful that William would write these words. I'm even more grateful that he is living them out.\n\n\u2014LEVI LUSKO\n\nPASTOR, FRESH LIFE CHURCH, A MULTISITE CHURCH IN MONTANA AND UTAH; AUTHOR, _S WIPE RIGHT: THE LIFE-AND-DEATH POWER OF SEV AND ROMANCE_ AND _T HROUGH THE EYES OF A LION_\n\nThe earth is experiencing the crest of a wave of awakening. Like a spear piercing through the darkness, a tribe of kingdom worshippers is marching the breadth and width of the earth, exercising kingdom authority and experiencing the signs and wonders of the kingdom. At the tip of that spear God has positioned Pastor William McDowell with a mantle of miracles and a grace to release revival among the nations. The grace God has put on his life is a unique, threefold cord of prophetic worship, fervent prayer, and holy manifestation of God's presence. The message he carries calls a generation of sons and daughters closer to the heart of God. As a man after God's own heart, Pastor William is marked by the humility and kindness that he exudes while carrying this mantle of revival across the nations.\n\n\u2014BISHOP KEVIN WALLACE\n\nSENIOR PASTOR, REDEMPTION TO THE NATIONS CHURCH\n\n\"This is very uncommon and extraordinary and should be treasured.\" Those were the words that fell from my lips the first time I experienced Deeper Fellowship and the Habitation Conference they host each year. Pastor William and his team are experiencing the realities of heaven being manifest in the earth.\n\nIf you've ever been around him, you know that Pastor William McDowell is not a musical artist; he is a worshipper and pursuer of God who is a carrier of contagious faith. Within the first few pages of _It's Happening_ you will be challenged to new levels of faith by the testimonies, will be encouraged to release the presumptions you have about how church is supposed to be, and will discover the simple, yet profound, pathway of experiencing God up close and personally. What you will discover more than anything is that Pastor William and churches such as Deeper are becoming the new norm! This book is a _must-_ read! And as you read it, get ready\u2014\"it\" will begin to happen in you!\n\n\u2014BISHOP TONY MILLER\n\nPASTOR, THE GATE CHURCH\n\nEvery now and then someone special emerges, and something special happens in their world. William McDowell has developed a literal \"open heaven\" with praise, worship, prayer, and the Word. It's happening\u2014the book of Acts experience of miracles, signs, and wonders. It's happening\u2014the outpouring of God's Spirit the prophet Joel proclaimed. It's happening\u2014what Isaiah prophesied, that the glory of God would manifest. It's happening! The power of God is being released with tangible evidence.\n\n\u2014BISHOP TUDOR BISMARK\n\nFOUNDER, JABULA\u2014NEW LIFE MINISTRIES INTERNATIONAL\n\nPASTOR, NEW LIFE COVENANT CHURCH\n\nI have been blessed by the mantle of praise and worship William McDowell is known for, and I have been to Deeper Fellowship Church and experienced firsthand the power of God in the atmosphere under his leadership. What's happening there is a beautiful fulfillment of God's promises in Acts 2:17 and Mark 16:17\u201319. I pray the accounts documented in this book will bring life to the reader and ignite faith like wildfire within the church worldwide. To God be the glory!\n\n\u2014JACKIE PATILLO\n\nPRESIDENT, EXECUTIVE DIRECTOR, GOSPEL MUSIC ASSOCIATION\n\nEvery generation prays for revival but forgets that every revival needs a revivalist. I believe William McDowell has been called to be a modern David in this post-Christian culture to usher the honor and weight of God to this current generation. _It's Happening_ highlights the wonders of God's Word and the power of a _yes_!\n\n\u2014DAVID S. JACQUES\n\nPASTOR, THE KINGDOM CHURCH\n\nI have been listening to the worship music of William McDowell for many years, and for me it's not just about listening or singing along; it's always brought life-changing moments of worship in God's presence. You cannot turn on William's music and not be affected by the tangible presence of heaven coming through the airwaves. I believe his ability to transform a room with worship is attributed not just to his unique and absolutely fantastic songwriting skills but also to a life dedicated to pursuing the heart of the Father.\n\nWhen you listen to his songs, you can tell he has had an encounter with God, which is reflected in the deep passion for God that comes out of his mouth when he sings. This deep-felt passion is also communicated when he preaches. You can always hear the heart of God toward His people through the word he declares and ministers.\n\nIt's not surprising that his ministry has become global, effortlessly crossing ethnic and cultural barriers. In my experience I have found that when you have a ministry like this, it's not uncommon to see miracles taking place because worship has the ability to bring God into an environment, and when God's presence shows up, miracles happen. I am sure that reading this book will transform your life, your ministry, and your community because this isn't just a man thing; it's a God thing, and it's happening!\n\n\u2014BISHOP JOHN FRANCIS\n\nSENIOR PASTOR, RUACH CITY CHURCH\n\nI've had the unique opportunity to be a firsthand witness to William's life and the events and stories this book documents. I've heard the prayers he has prayed. I've seen the tears he has cried, the seeds he has sown, and the sacrifices he has made, all so a generation will live in the place of God encounters. I believe the story of _It's Happening_ is possible only because God found a man on the earth who was willing to surrender everything and not settle for anything less than a move of God. William has never pursued God's hand, but he has always pursued God's heart. Subsequently God has responded by demonstrating His power. I'm privileged to call William my best friend, and his life is proof that God honors private devotion with public demonstration. _It's Happening_ will give you a glimpse into what I believe God wants to do in every city and in every nation. I pray the Holy Spirit stirs within your heart a desire for the bombardment of heaven in your midst.\n\n\u2014CALEB GRANT\n\nASSOCIATE PASTOR, DEEPER FELLOWSHIP CHURCH\n\nIf you relegate William McDowell to just the writer of hit worship songs, you've missed the mark. He is a servant of God who hears the frequency of heaven and repeats what he has heard in the earth. Because of that sensitivity William has been tasked with housing\/leading a move of God in Orlando that has changed the landscape of normal church forever. The same grace that rests on William has been transmitted to everyone who comes in contact with that anointing. _It's Happening_ is a glimpse into the realities of miracles, signs, and wonders that are taking place at Deeper Fellowship Church and the posture they have assumed to welcome and sustain God's presence. Read this book and allow your life to be altered for the better. Read it and find the altar that will allow that same power to hit your home, church, business, and life.\n\n\u2014BISHOP JASON NELSON\n\nAWARD-WINNING GOSPEL ARTIST\n\nPASTOR, GREATER BETHLEHEM TEMPLE\n\nI've been privileged to have a front-row seat to many different moves of God over the last fifty years of my life. Revivals in various parts of the world have produced incredible miracles through men and women who walked closely with God. I've seen great outpourings where I've witnessed various signs that would make me wonder. In other places I've seen the creative release of worship that would take those of us present to exhilarating realms of God's glory that left us speechless. I've longed to see it again! I live with an expectation of \"more of God\" because I know that more is available than what most of our churches are experiencing.\n\nI've known William McDowell for several years, and I know his gift and passion for God's presence are at a level that most people don't have the courage to pursue. William is simply a lover of Jesus, and he has made God's presence a priority in his life. As a result, he's stumbled into the reward of the man who would diligently seek God. Every time I'm with him in a service, I remember revivals I've experienced before, and a phrase rises up inside of me: \"It's happening!\"\n\nI began to get phone calls and texts from William giving me testimonies of miracles breaking out from week to week in his church. I texted back to him on one occasion, \"It's happening!\" I had no idea that the church had started making that declaration as the river began to rise in their church. It's happening! It's happening!\n\nThere is a fresh wave of God's presence with the release of miracles and wonders swiftly coming to the earth. Are you ready? If you want to know what it looks like, read this book and jump into the river!\n\n\u2014DAVID BINION\n\nAWARD-WINNING WORSHIP LEADER, SONGWRITER\n\nGod has entrusted to Pastor William McDowell and Deeper Fellowship Church (DFC) a beautiful and genuine move of the Holy Spirit. A fresh anointing is being poured out of an open heaven. I, myself, have experienced this unique and growing spiritual stir at DFC. Standing there worshipping in a room filled with surrendered hearts, I could feel the weight of glory pressing upon us. The sweet presence of Jesus becomes so real in that atmosphere. And there is evidence. Souls are saved, bodies are healed, and lives are forever transformed. I know that as you read _It's Happening_ , your faith will be stirred, and your hunger for more of God will grow. Heaven will mark you, and you will learn to steward the presence of the Holy Spirit. You will become a habitation of power. May what is happening there happen in you.\n\n\u2014DAVID DIGA HERNANDEZ\n\nEVANGELIST; AUTHOR, _25 T RUTHS ABOUT DEMONS AND SPIRITUAL WARFARE_ AND _C ARRIERS OF THE GLORY_\n\n_It's Happening_ is a fresh, bold approach to the reality and manifestation of revival. The move of God, the outpouring of the Spirit, and the gifts of the Spirit are not relegated to the pages of the Bible or to past historical accounts. They're happening now! My friend Pastor William McDowell is at the forefront of a generation that is uncovering an authentic and relevant move of God. This book will not only stir your hunger for revival but cause you to become the revival.\n\n\u2014JONATHAN MILLER\n\nLEAD PASTOR, NEW BEGINNINGS CHURCH\n\nIn a world where many are seeking attention, fame, and fortune merely for themselves, it never ceases to amaze me when I encounter people who are more concerned about God's glory than their own and who still value and desire the presence of God. William McDowell is one of those godly people. In his music and his ministry there are a simplicity and sincerity that attract God. It's clear that he hungers and thirsts for more of Him. You know a tree by the fruit it bears, and there is much good fruit in the life and ministry of William McDowell. God's presence is truly in the midst of Deeper Fellowship.\n\nContinue, my friend. Continue to hunger and thirst for more.\n\n\u2014ANDRE WARD\n\nOLYMPIC CHAMPION; FORMER WORLD BOXING CHAMPION;\n\nHBO BOXING ANALYST\n\nWhenever God wants to influence a people, He'll find a person. If He wants to impact a community, He will search for a praying people. And when He wants to transform the nations, He'll find a few apostolic churches. It gives me great joy to know God has found a man, a group of people, and an apostolic church in Orlando, Florida\u2014Deeper Fellowship Church (DFC), led by Pastor William McDowell.\n\nI came in contact with these precious people of God in a very unique way. The first thing I saw by revelation during worship there was a \"river\" rising and eventually flowing out from DFC into the city, impacting and transforming four major spheres of influence in it. I have watched with amazement the unscripted and unorganized move of God that's happening there with special miracles. It is happening! You can now experience it for yourself!\n\n\u2014SAM OYE\n\nPASTOR, THE TRANSFORMING CHURCH\n\nWith the humility of a lamb and the glory-filled power of a lion, Pastor William is truly a general of this generation. Not only have I been a firsthand witness to countless miracles under his ministry, but I likewise witnessed his personal integrity, passion for God's heart, and love for God's people. Out of the clamoring collection of voices in the world that are vying for your attention, Pastor William's voice is one that is proven and worthy to be attended to.\n\n\u2014TONY JONES\n\nPASTOR, FELLOWSHIP CHURCH\n\nI have personally and very intimately witnessed someone who has lived his life for a singular purpose for most of my life. I don't remember and can't recall a time over our twenty-plus years of brotherhood when William McDowell did not desire God to move. I'm not talking about a casual encounter but a radical one, the kind that people talk about for generations to come! William has been on the edge, awaiting and pointing to what has finally arrived. What is happening now happened to him some time ago, and now God has raised him up to help us enter into it! God has moved upon someone yet again, and that move has begun a movement. It indeed is happening. Let's not miss it, but let's seize what we are seeing! Let's go all in!\n\n\u2014JASON MCMULLEN\n\nEXECUTIVE PASTOR, DEEPER FELLOWSHIP CHURCH\n\nAs you hold _It's Happening_ in your hand, understand that you are now connected to the anointing that is upon William McDowell. He is a proven and seasoned vessel of God who carries a deep prophetic insight with an apostolic clarity, and he is trusted by God to dispense the supernatural power and presence of God in the earth, as it was in the days of the book of Acts. This is not a normal book full of well-thought-out information. This book is literally a point of contact with heaven that will release an impartation in your world that will cause what is happening at Deeper Fellowship Church to happen to you. By the time you are done taking all of this in, you will undoubtedly say, \"It's happening in my world too.\"\n\n\u2014PATRICK KITELEY\n\nPASTOR, RAIN CHURCH\n\nPastor William McDowell is an amazing friend and a most-esteemed kingdom partner. His love for God's people is unparalleled and shows in everything he does. Our society has faced some challenging times, and we have seen the hand of God shift things in the earth. Even with the shift, Pastor McDowell has maintained a keen level of precision in leading God's people because his ear remains fastened to the mouth of God. Pastor McDowell is a refreshing and much-needed vessel who understands the methodology required to release the power of God in the earth realm. I am a firm believer that the earth will yield the increase of Pastor McDowell's anointing for years to come.\n\n\u2014JOHN F. HANNAH\n\nSENIOR PASTOR, NEW LIFE COVENANT CHURCH SOUTHEAST\n\nIn the years that I worked with William McDowell, I was blessed to experience the presence of God in such a unique way. William has an unparalleled gift of ushering people into God's presence, then virtually disappearing so that God can work in the hearts of the people gathered there without distraction or pretense. I have witnessed firsthand God's supernatural power working through William's music ministry and through Deeper Fellowship. I'm so glad that even a sample of God's miraculous works is finally being shared in this book. Let it ignite in each of us the faith that revival has come and that God is here to work in and through us!\n\n\u2014EJ GAINES\n\nVICE PRESIDENT OF MARKETING, MOTOWN GOSPEL\n\nMy experience with the ministry of William McDowell has been powerful and enriching. His passion for God and his worship have impacted me and the church I lead in unbelievable ways!\n\n\u2014JOHN JENKINS\n\nPASTOR, FIRST BAPTIST CHURCH OF GLENARDEN\n\nDeeper Fellowship Church was birthed out of the most unique premise: William McDowell wanted to create a space where time, personality, and pageantry didn't matter, only the presence of God. What has transpired from day one has been something I personally have never seen before: a church where worship is unforced, the people are passionate, and absolutely anything can happen at any moment.\n\nEvery person God has ever used to make a difference in the earth has had to be comfortable with being different. And William McDowell is no exception. He believed that if God could find a people who were willing to linger in His presence and believe for the impossible, anything would happen. And he was right! It's happening!\n\n\u2014HART RAMSEY\n\nSENIOR PASTOR, NORTHVIEW CHRISTIAN CENTER\n\nKnown for bringing the power and presence of God into our cars, living rooms, and churches through his gift of music, William McDowell will now help deliver it through the printed page. I pray that as you read this book, the stories will help you believe for revival in your home, church, and community. With this book we get to see the Spirit behind the sound.\n\n\u2014MICAHN CARTER\n\nLEAD PASTOR, TOGETHER CHURCH\n\nFor years I've studied revival. After every book, article, or documentary I read or watched, I would often say, \"Jesus, let it happen here in Central Florida.\" I believe a holy invitation has come from heaven, and it's happening now!\n\nMany of you know William as an award-winning worship leader, but before any of that happened, he was hidden and gave God His _yes_ in the secret place. Over and over I have watched him say yes. He's a man of zero compromise, deep wisdom, purity, humility, and integrity when no one is watching. I believe because of that obedience he has touched the heart of God deeply and therefore continues to impact everyone around him. God used him to change the course of my family's life, and we've felt the consuming fire of Jesus evident at Deeper Fellowship. It truly is happening!\n\n\u2014THEO KOULIANOS JR.\n\nPASTOR, THE ANSWER CHURCH\n\nThere are some people you meet who immediately show you that God has more to offer. Pastor William McDowell is one of those people. He is a model for revival. I believe the rarity of his integrity, humility, and commitment to the plan of God is directly tied to the manifestations seen in his ministry. The culture of his life has opened the door for the heavens to pour out the impossible. It is my belief that God doesn't want to do the impossible just in him. Pastor William's breakthroughs are meant to cause faith to rise in you to chase the move of God for your life as well. My prayer is that as you read this book and pursue the presence of God, you will be provoked to look at your life and give God your _yes_ so you also will be able to say, \"It's happening!\"\n\n\u2014JOSH HART\n\nPASTOR, LIBERTY CHURCH\n\nAs a fellow pastor in the Orlando area, I have seen firsthand how God is using Pastor William McDowell in our city and abroad. Through the powerful ministry of Pastor William McDowell a fresh wind of revival has come to our city. Pastor William has made himself available to be used by the Holy Spirit and without restriction has created an atmosphere for the miraculous to take place. Pastor William McDowell's heart for worship and pure desire to see God move in people's lives have been instrumental in creating such an environment. I have experienced the sweet presence of the Holy Spirit at Deeper Fellowship Church, and in this book he shares how we can experience this in our own lives. When we focus on pursuing the very presence of God, heaven touches earth, and lives are changed, healed, set free, and restored. It's happening!\n\n\u2014RIVA TIMS\n\nSENIOR PASTOR, MAJESTIC LIFE CHURCH\n\nPastor William McDowell has always had a passion for God and engaged in an unrelenting pursuit of the things of God! I have known William for over thirty years, and even when he was a young teenager, it was evident that he had been given a musical gift and had a calling on his life to touch the world. It is no surprise that he has been raised up as one of the leading worship leaders in this generation.\n\nNow as a senior pastor, William has chosen to courageously lead his congregation outside the lines of tradition and conformity, and by so doing, they are experiencing what it truly looks like when heaven touches earth! I am honored to know William and to see how God is using him to bring His kingdom to earth. May \"it\" happen _everywhere_!\n\n\u2014JOHN W. STEVENSON\n\nAPOSTLE AND SENIOR PASTOR, HEIRS COVENANT CHURCH\n\nPastor McDowell is a servant leader, and this book is a must-read testament to his and his congregation's faith combining with works to bring forth revival. _It's Happening_ is a testimony we all should read.\n\n\u2014PHIL THORNTON\n\nSENIOR VICE PRESIDENT AND GENERAL MANAGER, RCA\n\nINSPIRATION\n\nI had not heard of William McDowell in 2009. I was in Walmart walking down the gospel music aisle when I heard the Lord say, \"Stop!\" So I stopped immediately, not knowing what I was looking for. After feeling prompted to look to my left, I did so, and there it was, the album _As We Worship Live_ by William McDowell. The purchase of this album began a three-month period of weekly supernatural visitations from angels and even the Lord Himself, and through that process God birthed me into full-time ministry.\n\nWhen I discovered years later that William had started pastoring a church, I was so excited that I moved my entire family to a different city just to be close enough to attend the church when my ministry was not on the road. There aren't many people outside the ones written about in the book of Acts who have had that much of an impact on my life. To paraphrase the apostle Peter in 2 Peter 1:16, I am an eyewitness of the revival that is taking place in the church that William leads. I can say with certainty, \"It's happening!\"\n\n\u2014JONATHAN FERGUSON\n\nFERGUSON GLOBAL MINISTRIES\n\nI have had the unique opportunity of walking with William as a partner in his recorded music ministry. We have spent much time over the last ten years working to impart to the masses the music God gave him. I have always said that you learn the truth about artists during the process of promoting an album. When other artists would have chosen to self-promote, William consistently chooses to exalt Jesus. He is one of the most authentic people I know, someone who is sure of what he has been put on earth to do. The posture of his heart is to follow God's will and purpose for his life, which keeps him in a relentless pursuit of the presence of God. That is where he is leading anyone who chooses to follow him\u2014into God's presence.\n\nThrough the revival happening at Deeper Fellowship, God is using William to remind us that He is still with us, He still hears us, He still can, and He still will. Many people believe things not because of what they have been told but because of what they have seen. This revival has given the unbeliever proof that God still gives sight to the blind, a voice to the voiceless, and legs to those who couldn't walk. Many people have walked into Deeper Fellowship with a burden and have walked out with proof that God is real and wants to move in their lives.\n\nMay the proof taking place at Deeper activate your faith to believe that if revival can happen there, it can happen anywhere. Modern-day miracles do exist! Persuasion is not necessary when there is a demonstration of the power of God. \"It's happening!\"\n\n\u2014GINA MILLER\n\nVICE PRESIDENT AND GENERAL MANAGER, ENTERTAINMENT ONE\n\nThere's an obvious anointing on William that comes from a deep well of time spent with God. He walks through this life at a slower pace, trading performance for a divine move of God. He's not interested in accolades, only anointing. He's a true friend who has cried with me, prayed with me, and encouraged me. I love how he constantly pushes those around him to press in close and listen to the voice of God. I know God has written these words on William's heart.\n\n\u2014JEREMY FOSTER\n\nPASTOR, HOPE CITY\n\n# TABLE OF CONTENTS\n\n 1. COVER PAGE\n 2. TITLE PAGE\n 3. COPYRIGHT PAGE\n 4. CONTENTS\n 5. ACKNOWLEDGMENTS\n 6. PROLOGUE\n 7. INTRODUCTION: WHAT'S HAPPENING?\n 8. CHAPTER 1 WHEN HE WALKED INTO THE ROOM\n 9. CHAPTER 2 HOW'S YOUR POSTURE?\n 10. CHAPTER 3 HE HEARS YOUR CRY\n 11. CHAPTER 4 FOLLOW THE PATHWAY\n 12. CHAPTER 5 IT'S ALL ABOUT CAPACITY--HOW EMPTY ARE YOU?\n 13. CHAPTER 6 NO SUBMISSION, NO POWER\n 14. CHAPTER 7 URGENT ANTICIPATION\n 15. CHAPTER 8 STAY AND WAIT\n 16. CHAPTER 9 CONTEND FOR THE FLAMES OF REVIVAL\n 17. CHAPTER 10 A WORTHY HOST\n 18. CHAPTER 11 HE IS HERE--NOW WHAT WILL YOU DO?\n 19. CHAPTER 12 FEED THE FIRE\n 20. CHAPTER 13 IT HAPPENED BECAUSE IT'S HAPPENING\n 21. CHAPTER 14 LIVING STONES\n 22. CHAPTER 15 YOU WERE BORN FOR THIS\n 23. NOTES\n 24. ENDORSEMENTS\n\n## LIST OF PAGES\n\n 1. \n 2. \n 3. \n 4. \n 5. \n 6. \n 7. \n 8. \n 9. \n 10. \n 11. \n 12. \n 13. \n 14. \n 15. \n 16. \n 17. \n 18. \n 19. \n 20. \n 21. \n 22. \n 23. \n 24. \n 25. \n 26. \n 27. \n 28. \n 29. \n 30. \n 31. \n 32. \n 33. \n 34. \n 35. \n 36. \n 37. \n 38. \n 39. \n 40. \n 41. \n 42. \n 43. \n 44. \n 45. \n 46. \n 47. \n 48. \n 49. \n 50. \n 51. \n 52. \n 53. \n 54. \n 55. \n 56. \n 57. \n 58. \n 59. \n 60. \n 61. \n 62. \n 63. \n 64. \n 65. \n 66. \n 67. \n 68. \n 69. \n 70. \n 71. \n 72. \n 73. \n 74. \n 75. \n 76. \n 77. \n 78. \n 79. \n 80. \n 81. \n 82. \n 83. \n 84. \n 85. \n 86. \n 87. \n 88. \n 89. \n 90. \n 91. \n 92. \n 93. \n 94. \n 95. \n 96. \n 97. \n 98. \n 99. \n 100. \n 101. \n 102. \n 103. \n 104. \n 105. \n 106. \n 107. \n 108. \n 109. \n 110. \n 111. \n 112. \n 113. \n 114. \n 115. \n 116. \n 117. \n 118. \n 119. \n 120. \n 121. \n 122. \n 123. \n 124. \n 125. \n 126. \n 127. \n 128. \n 129. \n 130. \n 131. \n 132. \n 133. \n 134. \n 135. \n 136. \n 137. \n 138. \n 139. \n 140. \n 141. \n 142. \n 143. \n 144. \n 145. \n 146. \n 147. \n 148. \n 149. \n 150. \n 151. \n 152. \n 153. \n 154. \n 155. \n 156. \n 157. \n 158. \n 159. \n 160. \n 161. \n 162. \n 163. \n 164. \n 165. \n 166. \n 167. \n 168. \n 169. \n 170. \n 171. \n 172. \n 173. \n 174. \n 175. \n 176. \n 177. \n 178. \n 179. \n 180. \n 181. \n 182. \n 183. \n 184. \n 185. \n 186. \n 187. \n 188. \n 189. \n 190. \n 191. \n 192. \n 193. \n 194. \n 195. \n 196. \n 197. \n 198. \n 199. \n 200. \n 201. \n 202. \n 203. \n 204. \n 205. \n 206. \n 207. \n 208. \n 209. \n 210. \n 211. \n 212. \n 213. \n 214. \n 215. \n 216. \n 217. \n 218. \n 219. \n 220.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nCOPYRIGHT\n\nHarperCollins _Publishers_\n\n1 London Bridge Street\n\nLondon SE1 9GF\n\nwww.harpercollins.co.uk\n\nFirst published by HarperCollins _Publishers_ 2019\n\nFIRST EDITION\n\nText \u00a9 Angela Kelly 2019\n\nCover layout design by Claire Ward \u00a9 HarperCollins _Publishers_ 2019\n\nFront jacket image is a composite of two photographs (\u00a9 Nina Duncan) from the same series taken behind the scenes at a photoshoot by Chris Levine in 2003. Diamant\u00e9 detail taken from a photograph of Her Majesty The Queen \u00a9 Barry Jeffery. Back cover photograph supplied by the author.\n\nA catalogue record of this book is available from the British Library\n\nAngela Kelly asserts the moral right to be identified as the author of this work\n\nAll rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, non-transferable 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 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\nFind out about HarperCollins and the environment at www.harpercollins.co.uk\/green\n\nSource ISBN: 9780008368364\n\nEbook Edition \u00a9 October 2019 ISBN: 9780008368371\n\nVersion 2019-10-28 \nCONTENTS\n\nCOVER\n\nTITLE PAGE\n\nCOPYRIGHT\n\nDEDICATION\n\nDEAR READER\n\nLETTER FROM SAMANTHA COHEN CVO\n\nFOREWORD BY STEWART PARVIN MVO\n\nPROLOGUE\n\nChapter 1: Dressing The Queen\n\nChapter 2: Our Working Year\n\nChapter 3: The Eyes of the Nation\n\nChapter 4: On the Move\n\nChapter 5: The Eyes of the World\n\nChapter 6: The Things We Do For Fun\n\nCONCLUSION\n\nACKNOWLEDGEMENTS\n\nPICTURE CREDITS\n\nABOUT THE PUBLISHER\nDEDICATION\n\n**Your Majesty,**\n\n**With humble respect I would like to dedicate my second book to you for allowing me to write about our personal working relationship that has grown over many years, and allowing me to share the details of it and to describe how we have come together on a trusted personal level. I will always be truly grateful. I cherish the time you have allowed me to have with you and the friendship you have shown me.**\n\n**With love and gratitude,**\n\n**Angela x**\n\nI would also like to dedicate this book to my wonderful family: Frank and Nicola Wylie, Paul and Sarah Wylie, Michelle and Simon Anson, and to my very special grandchildren \u2013 James and Scarlett Anson, Alex and Jacob Wylie. You have shared the experiences and the journeys with me and you have inspired me all the way to write this book. I could not have done it without you. I thank you so much for all the love, support and advice you give to me and in return I give you all my love and more.\n\nThe lesson you all taught me was: never underestimate young minds.\n\nI love you, Granny x x x x\n\n_A letter from Angela's grandson Jacob Wylie_\n\nGrowing up in Windsor was an immense experience. The sheer bliss, peace and quiet and nature all around you. I am truly grateful to have been able to grow up in a place with such elegance and beauty, all thanks to my granny's amazing accomplishment. Not many people can say, 'My granny works for The Queen.' Every time the subject is brought up I can't help but brag about my tremendous gran and how proud I am of her. Her triumphs only get bigger and it renders me speechless; the stories and experiences she tells me about her working days astonish me, I could listen for hours. Words don't even come close to describing how proud I am of my granny!\n\nAs I have grown up in Windsor since I was only a couple of weeks old, Windsor is a home for me, in the cosy cottage with the fire and the enormous grounds that lie around it. Ever since me and my brother were little kids, we were both curious about where the roads and paths would lead us, and we would go wandering off for hours exploring the river, bat caves, monkey puzzle tree, farm and bamboo trees. Staying in Windsor or Buckingham Palace is always the highlight of my week, and meeting The Queen is always a dream come true that I am grateful to enjoy. I am enormously proud of my granny and can't wait to see her future endeavours.\n\nSincerely, \nJacob Wylie, age 15\n\n_A letter from Angela's grandson Alex P. Wylie_\n\nI would just like to say how proud I am of my granny of what she has achieved. All through the fifteen years of my life, actually since my twin, Jacob, and I were a few months old, we have stayed in Marlborough House apartment, and then, when Granny moved to Windsor, into her cottage. We still stay to this day in Windsor and sometimes in Buckingham Palace.\n\nAs you can imagine I have the fondest memories that will never leave me. Visiting my gran in the summer is great. I remember on one occasion, when my brother and I were around twelve years old, every evening we would set out walking and I smile now thinking of how many alarms we set off in the grounds of Windsor Castle while we went exploring, looking into the bat caves and many other exciting places, steeped in history.\n\nOn another occasion we were playing in The Splash, which is a shallow part of a stream close to the cottage, although The Queen often goes there when she is in Windsor. As Her Majesty got closer she recognised us both, as she knew we were staying at Granny's cottage. But so did the corgis. They came running up and started chasing us. They were very playful and they soon started playing in the water. Once again words can't describe how proud I am of my granny for giving us so many magical moments.The memories will remain with me for ever and although we have grown up now, going to Granny's is special.\n\nRegards, \nAlex P. Wylie, age 15\n\nA letter from Angela's grandson James Anson\n\nWhen I read of or see my grandmother's accomplishments, I feel both immense pride for her, and also a strong sense of inspiration in regards to my own goals. It often makes me realise how fortunate I am to be related to someone so talented in their work.\n\nI was fortunate enough, in fact, to spend a lot of time in my childhood at places such as Marlborough House, Buckingham Palace, and Windsor Castle, and I'm thankful not only to be able to have access to such iconic institutions, but also to have made brilliant memories there. One of my favourite places to visit during my childhood was St James's Park. I still enjoy going there in winter to see the Smews.\n\nI am personally a massive fan of history; I aim to study history at university. In truth, I do owe a vast amount of my love for history to the fact that I have had access to some amazing places of great historical value for my entire life. I've formed my own memories in these places as well; memories I will treasure for many years. I realise how unique a position I am in and I'm very grateful to my grandmother for this.\n\nJames W. Anson, age 18\n\nA letter from Angela's granddaughter Scarlett Anson\n\nI have always enjoyed watching Granny work and talking with her about her latest design or projects. I've been privileged to see and feel some of the fabrics that have gone on to become outfits. I also spend a lot my time drawing and painting, but I like concept art for video games. I enjoy character design and created my own characters based on myself and people I have read about in books. Granny has inspired me with her skills in colour matching and creating designs so I love to show her my art work to get her feedback.\n\nYours, \nScarlett Anson, age 13\n\nDEAR READER\n\nIt took me some time to start writing this book. When I first asked The Queen if she would mind me writing another book, after _Dressing the Queen_ in 2012, but this time about our close working relationship, I never thought in my wildest dreams that Her Majesty would say yes.\n\nOnce the initial shock was over, I found myself struggling. The Queen trusts me and she knows I would not write any intimate details or anything untoward about our friendship and working relationship, but, nevertheless, I still struggled.\n\nI wanted to do this book justice. I wanted to be worthy of writing it, and to share with you the magical and engaging moments that have happened between Her Majesty and myself while we have been working together. The Queen may not have thought them magical, but I did and I still do.\n\nDuring my twenty-five years to date of working with The Queen, many events and engagements have taken place. Her Majesty travels to and from Buckingham Palace, Windsor Castle, Sandringham House, the Palace of Holyroodhouse and Balmoral Castle, and spends numerous private weekends away, too.\n\nWith this book I would like to share with you the wonderful working life and relationship between Her Majesty and me. The Queen and I have fun and laughter, but there is a serious side to my work in which challenging and difficult decisions have to be taken \u2013 split-second decisions sometimes. I have to be quick and think on my feet. But it's all in a day's work. Someone has to do it \u2013 and I'm so glad it's me.\n\nI hope you enjoy reading this book as much as I did writing it. So, get yourself a cup of tea, sit down, put your feet up, and let me take you on a magical journey once again.\n\nWith love,\n\nLETTER FROM SAMANTHA COHEN CVO\n\n29 October 2019\n\nThis book documents the unique working relationship between Her Majesty The Queen and the woman who has been her Personal Assistant and Senior Dresser for more than two decades: Angela Kelly.\n\nIt gives a rare glimpse into the demands of the job of supporting the Monarch, designing Her Majesty's clothes and hats, and matching them with exquisite Royal jewellery. Throughout her long reign, Her Majesty has become a fashion icon, renowned for her sense of style and understated elegance. Behind the scenes, Angela has worked quietly with her to create fashion that is unique, accessible to all, and allows The Queen to reflect the trends of the day subtly, and with grace.\n\nThe book also reveals the lengthy preparations needed before important State occasions, such as the State Opening of Parliament. Most importantly, though, we gain privileged insight into a successful working relationship, characterised by humour, creativity, hard work, and a mutual commitment to service and duty.\n\nOn the many occasions I have worked with Angela in preparing for an important overseas or regional visit in the United Kingdom with Her Majesty, I have been struck by Angela's endless enthusiasm and original ideas. She tries to make each Royal engagement memorable for those who will be seeing The Queen for the first time\n\nAngela is a talented and inspiring woman, who has captured the highlights of her long career with The Queen for us all to share.\n\nSamantha Cohen CVO\n\nAssistant Private Secretary to The Queen (2011\u20132018)\n\nFOREWORD\n\nby Stewart Parvin MVO\n\nI first met Angela one morning in the summer of 2001. She had come into my shop in Motcomb Street and was chatting to our saleslady, Gill Edwards. From Angela's questions it became apparent that she needed a conversation with me. In those days our workroom was in the basement of the shop at Number 14 (previously the home of Royal couturier Ian Thomas), so I was immediately available. I ran upstairs to meet her and discuss her mystery client's needs. She wanted to look at the quality of our workmanship, and I agreed straight away.\n\nA few days later Angela reappeared and we had a lengthy discussion about fabrics. Her client couldn't be seen wearing any of the items that we had in the main collection. The easiest way around this, I thought, was for us to pop downstairs and go through our archive: Angela could look at fabrics we had never used or had made into just one or two garments in previous seasons. I saw her delight as we pulled out tweeds, cr\u00eapes and silks \u2013 our mutual enthusiasm was infectious!\n\nAfter we had taken cuttings from several rolls, I suggested that a few simple sketches to go with the swatches might help her client decide if I was the person she was looking for. Unbeknown to me, that was the start of our Royal journey.\n\nShortly afterwards I heard from Angela, who told me that her client liked four fabrics in particular: would it be possible for me to do sketches for each one?\n\nI produced a pile of sketches offering three or four alternatives for each of the selected fabrics and waited for Angela to collect them. A week or so later she telephoned, delighted to tell me that her client had chosen several outfits, and she could now reveal that she herself was Senior Dresser to The Queen. She asked if I would like to make clothes for Her Majesty. After a few seconds' disbelief I replied with an enthusiastic 'Yes!'\n\nIn the early days, on occasion there might be an urgent request, and the lack of technology in our studio meant that we'd sometimes have to fax a design to Angela's office to be discussed over the phone. As our relationship developed, Angela would often come to the studio \u2013 we later moved to a larger sewing room in West London \u2013 and we'd have fun as I proposed fabric combinations, buttons and trims. I would make preliminary sketches, Angela inspiring me with ideas that Her Majesty might like.\n\nBy now Angela, who had been creating off-duty clothes for Her Majesty, had started designing more formal outfits with a team of seamstresses at the Palace. I would often pop along to the in-house workroom and offer a helping hand to ease the process along, sometimes suggesting different techniques, interlinings, and other tricks of the trade. Over coffee in her office, Angela would show me her latest finds \u2013 such as exquisite laces and embroideries that she'd sourced on her travels and at shops such as London's Joel & Son. We would drape the beautiful fabrics on stands, coming up with ideas for gowns for Her Majesty. No one knows better than Angela Kelly what is required of a Royal wardrobe, and it makes my end of the process so much easier. While sketching and choosing the ideas to put before Her Majesty, it's been fascinating to see how Angela's guiding hand often tends towards a bolder, more glamorous look.\n\n_At Joel & Son Fabrics._\n\nThese days we often meet at Joel's as soon as they have their latest collection in stock to select prints that we will propose to Her Majesty. Often we find we've chosen the same design. We know, though, that either of us will come up with something fabulous for Her Majesty to wear.\n\nAngela not only creates many wonderful outfits, she's responsible for The Queen's complete look. It's fun to be shown into her millinery workroom, where, under Angela's watchful eye, Stella McLaren creates so many of the eye-catching hats that Her Majesty now wears.\n\nIn fact, on several occasions we've worked as a team. Between us, we created the outfit The Queen wore for Prince Harry's wedding: I designed several options for the coat and dress from the stunning silks Angela had chosen for the event, while Angela created the show-stopping hat that finished the iconic outfit.\n\nWe share so many fantastic memories of our times together from the start of our working relationship \u2013 when I had no idea who Angela was or who she worked for \u2013 to the present day.\n\nWhen I look back at some of the iconic outfits we have both created, I feel such pride in what Angela and I have achieved.\n\n_Me and my pal at Joel & Son in 2019._\n\n_The Queen, wearing a Stewart Parvin outfit, with The Duke of Edinburgh during Trooping the Colour in 2016._\n\n_The Queen at the State Opening of Parliament in 2014._\nPROLOGUE\n\nCrowning Glory\n\nDuring my twenty-five years of working closely with Her Majesty The Queen, I am lucky enough to have travelled to some of the most amazing places in the world. But one moment in particular makes me emotional every time: the sight of The Queen putting on her Imperial State Crown at the annual State Opening of Parliament. It is a truly special moment for me, and I am so privileged to play a small part in it. Whenever I see Her Majesty in a Diadem or a tiara, I am composed, but when she puts on the crown, it touches my heart.\n\nWe are alone behind the privacy screen in the Robing Room in the Houses of Parliament just before The Queen enters the Chamber to make her speech on behalf of her government. We are silent and serious, concentrating hard, and The Queen is deep in thought. When I have positioned the State Robe on her shoulders, she takes the crown off its cushion, and I watch as The Queen raises it above her head. The magnificence of that moment takes my breath away every time. I watch in awe while I wait for her to secure it. Then Her Majesty turns, gives me a smile, and lets me know that she's ready to go. And for a moment, I am speechless. Luckily I don't need to say anything and The Queen always says thank you, and I smile back.\n\nThe Queen has seen my emotions get the better of me quite a few times. This happens particularly at special events, such as when I dress Her Majesty for important events and State occasions. Once The Queen has the robe on and places the crown on her head, I feel so proud to be serving Her Majesty in such a unique position. Normally The Queen rolls her eyes and playfully tuts at me when she sees my eyes fill up! Even after all these years, I still find it overwhelming to watch her lift the crown. It is the ultimate reminder of just how lucky I am to be in the presence of The Sovereign, Her Majesty The Queen.\n\n_The Queen attending Derby Day in 2019_\n\n_The Ambassadorial Residence in Berlin in 1992._\nHOW IT \nALL BEGAN\n\nWell, here I am at a certain age and still running around thinking that I am thirty years old \u2013 which my body is telling me I am not! So much has happened in my life \u2013 decisions made, paths taken \u2013 so let me take you back to where it all began.\n\nIt's hard for me to believe that I was first introduced to Her Majesty The Queen more than twenty-five years ago. I could never have imagined the significance of that meeting. It was my first step along the road to changing my life.\n\nIt was 1992, when I was working as Housekeeper for the British Ambassador to Germany, Sir Christopher Mallaby and his lovely wife Pascal, Lady Mallaby, in their Berlin Residence. In June of that year, the Ambassador told me we were expecting a few high-profile VIP guests who were to visit the Residence to see if the house was suitable and large enough to accommodate them all that autumn. The grounds were also to be assessed for security. This was not common practice at the time, so we suspected that someone truly special might be coming. The checks were to be carried out within the month \u2013 three months before the possible visit.\n\nThe Ambassador trusted me and soon informed me that the guests would in fact be Her Majesty The Queen and His Royal Highness The Duke of Edinburgh, with their accompanying staff. Never in my wildest dreams had I thought I would be looking after members of the Royal Household. During their stay, the guests would require bedrooms, of course, and all meals were to be served at specially agreed times. The Ambassador also explained that many private meetings would be held behind closed doors \u2013 meaning that the participants should not be disturbed, unless under extreme circumstances. My role was to ensure that Her Majesty's personal staff were comfortable, equipped, and had a good working environment. Once I had familiarised myself with their requirements, in terms of any allergies, likes and dislikes with regard to food, I felt content that I could get the job done.\n\n_Service with a smile \u2013 Berlin, 1992._\n\nOctober soon came around and the Royal visitors arrived at the Ambassador's Residence. I remember being introduced to each member of the Royal staff: The Queen's Page; The Duke's Valet; the Travelling Yeoman and Miss Peggy Hoath, Her Majesty's Senior Dresser. I was so impressed by their professionalism: everything was executed with efficiency and precision, from the delivery of the luggage to the unpacking of the cases. Over the course of the next four days, we all got to know each other well, and I spent quite a bit of time with Peggy, who was a lovely lady. She told me she had been The Queen's Dresser for the last thirty-four years and was now considering her retirement. We agreed to keep in touch.\n\nIt wasn't until the end of their visit that I finally got to meet Her Majesty and The Duke of Edinburgh. Just before our guests departed, The Queen and His Royal Highness said their farewells to the household. I now understand that this is common practice: they like to thank the staff for their assistance in making the visit a success. I remember what a privilege and honour it was to be presented to The Queen and The Duke. I was even given a lovely photograph of them, alongside a beautiful needle case with EIIR inscribed on it.\n\nAfter I'd thanked them for their thoughtful gift, Her Majesty asked whom we expected next at the Residence. I replied that the information was confidential, and The Duke asked, slightly incredulous: 'Surely you can tell Her Majesty The Queen?' Again, I explained that I really could not disclose the information as I had signed the Official Secrets Act, now known as Confidentiality Agreements. I had taken the confidentiality surrounding their visit in a similar way \u2013 even carrying a bunch of flowers whenever I left the Residence to trick the surrounding media into thinking I was a florist and so would be unable to provide any intelligence on the high-profile guests inside. In light of this exchange, I offered the photograph and needle case back to Her Majesty and The Duke \u2013 I wasn't sure if it was appropriate to keep them since I'd refused them the information they'd asked for. The Queen simply told me to keep it. I thanked her and His Royal Highness again, and said to Her Majesty The Queen, 'I will remember this for the rest of my life.'\n\nAnd the Queen replied, 'Angela, so will I.'\n\nA few weeks after the Royal visit to the Ambassador's Residence, I was surprised to receive a phone call out of the blue from Peggy, who asked how I was. I assumed she was just being kind as I'd mentioned to her that I was hoping to come back to England later in the year to start a new chapter of my life after the recent breakdown of my marriage. It was lovely to chat to her and she said she would call me again before I left Berlin.\n\nIt was just a couple of days later when I next heard from Peggy. She said that Her Majesty had requested she get in touch to ask would I consider coming to work at Buckingham Palace? To say that I was shocked would be an understatement. I told Peggy I was still making arrangements to return to England but that I would consider the offer of a position carefully.\n\nSeveral weeks later, when I was finally settled back in Sheffield with my family, Peggy rang to ask if I had an answer for her about applying for the Assistant Dresser role. I discussed the prospect at length with my family to make sure they were happy with my entering into this commitment, because it would be almost like a marriage and would therefore affect all of our lives. But there really was only one answer to give: a very enthusiastic yes from us all.\n\nI could give one half of myself to my children, and the other half to the life I was about to enter.\n\nMY LIFE\n\nA couple of months after I had accepted Peggy's invitation to interview for the position of Assistant Dresser, I found myself en route to Buckingham Palace to meet with Lady Susan Hussey and the Honourable Miss Mary Morrison, two of The Queen's Ladies-in-Waiting. As with any interview, I had spent quite a lot of time thinking about what to wear. I was certain this interview was for these two ladies to look me over and check me out, and that everyone in the Palace would be an intimidating, impeccably dressed aristocrat. When I first received the letter inviting me to Buckingham Palace for the interview I went into panic mode. Those famous words of 'what on earth will I wear?' came to mind \u2013 I already had lovely clothes, but I thought that a more conservative outfit was needed. So I made the rather rash decision to sell my washing machine so that I could afford an appropriately smart outfit. I chose a crisp cream blouse with navy spots and a long skirt, and wore a string of pearls. Normally I wear dresses to my knees, but for the interview I thought a long skirt would be more appropriate. I did struggle to walk and I looked like a good take on Miss Marple, with my skirt wrapped around my legs. I arrived at the Privy Purse Door, which is the main entrance at the front of Buckingham Palace, at 11.30am and was welcomed in. Trembling with nerves, I was taken to the Ladies-in-Waiting sitting room.\n\nAs I waited, I couldn't help but reflect on how I had ended up there: a divorced single mother from humble beginnings in Liverpool was on her way to interview for a position working with Her Majesty The Queen.\n\nWhen I think about my early childhood, most of my memories seem to be anchored in and around the back room of our house. I was born in a small street in Liverpool, facing Stanley Park, between two famous football grounds: Everton and Liverpool. We lived in a two-up-two-down terraced house, and it was a very happy home. My mother sewed and my father was in the Merchant Navy.\n\nThe back room of our house was a hive of activity; a place for chatting, listening to the radio, eating, and, crucially, it was also frequently transformed into a sewing room. My mother, Teresa, was a very smart, elegant, and glamorous lady, although I remember her blushing and shying away from compliments. She was not shy when it came to the sewing machine, though, and she had a raw talent and natural flair for clothes design. I'm sure her skills were recognised by my maternal grandmother, who taught her the art of dressmaking when she was still very young. When she grew up, my mum joined the army's Auxiliary Territorial Service. Funnily enough, then as the Princess Elizabeth, The Queen also joined the Auxiliary Territorial Service in 1945, and was the first female member of the Royal Family to join the Armed Services as a full-time active member. My mum worked with the Service until she fell in love with my father, Thomas, and left to marry him. They had two sons, my brothers Tommy and Tony, before I came along, then another two sons, John and Terry, and a second daughter, my sister Donna Maria. With so many young children to care for, my dad left the Merchant Navy and took up a job as a crane driver on the Liverpool docks so that he could be closer to home.\n\n_Clockwise: My first Holy Communion, age 7; my mum, Teresa, in the Auxillary Territorial Service; me and my dad at home in Liverpool._\n\nMy father was good-looking, quiet, and a gentleman. He was caring, loved, and respected by his family and all who knew him. When we were little, he was always tricking us. Perhaps he would shout, 'Would you like a Quality Street?' And we'd all come running, yelling, 'Yes!' And he would be standing in front of the television, and a Quality Street advert would be on, and he'd say, 'Ah, you missed it.' But every Friday, when he collected his wages, he'd go to the shop to buy the Liverpool Echo and a Mars Bar each for everyone.\n\nWe did not have much in the way of material possessions when we were growing up, but the house was filled with love, laughter, and kindness, and my mum insisted that our family take pride in everything we did. She was excellent at teaching by example and took meticulous care of every piece of clothing she created, which included my school uniform skirt and grey bib. She was endlessly generous with her time, and whenever the back room was not being used to entertain visitors, it became a fitting room, as well as a sewing room, for all those who came to her for new outfits. From wedding and bridesmaids' dresses \u2013 beautifully crisp gowns, perfectly cut and adorned with little rosebuds \u2013 to school uniforms to help out the children in the neighbourhood, nothing was too challenging, and I vividly remember watching her work on that well-oiled Singer sewing machine. In fact, aged eight, maintaining that precious machine was my first task as an assistant dressmaker. I learned how to strip it down before cleaning it thoroughly and oiling certain components that were likely to stick. I would also check the rubber belt religiously, making sure it was not wearing out, and grease the treadle to make sure it moved up and down easily. When I had mastered all of this, I was finally shown how to load the shuttle, place it in the machine, and thread the needle.\n\nHaving perfected that important task, I was soon being shown how to cross-stitch for embroidering napkins, how to sew hems by hand and make elasticated waists \u2013 all crucial skills that still come in handy today. My mum was a thrifty woman (which I know Her Majesty would appreciate) and showed me how to make carpet rugs from old coats. We would cut the coats into strips, then double them and use an old peg to push the strips through the rug's hessian backing to form a loop. However, it was when my mum was measuring her customers that I would pay close attention. I was fascinated to see how accurately she used her tape measure, which I still have to this day, and how confidently she cut and joined her patterns.\n\nNot only did my mum have an extraordinary talent for dressmaking, she was generous when family and neighbours were in difficulty, especially when someone had passed away. I remember her visiting other people's houses, and helping make their front room look beautiful for the family to mourn their loved one. She would take several white sheets with her and quickly get to work; I would watch her pleat the sheets and use small nails to keep them in place on the walls to make the room into a sort of chapel of rest into which the coffin would be brought. I also recall watching my mother lay out the body when a family member passed away. She was so gentle and caring, and it seemed such a natural thing to do, and it helped to ease the pain of the family and other mourners.\n\nI wanted to be just like her, so it's not surprising that my first paid job was in a sewing factory at the age of fifteen, working on the buttonhole machine. A few years later, when I got married and had children, I would make outfits for my family, but it was only when I started working with Her Majesty that I truly understood how indebted I am to the influence, knowledge, and guidance of my mum in those humble beginnings.\n\n_Me and Nettie at Windsor Castle before Royal Ascot in 2000._\n\nNettie\n\nI value so many people in my life, whether they be close or acquaintances. Nevertheless, they are important to me. Throughout my time of twenty-five years working for the Royal Household, Annette Wilkin, also known as Nettie, was my true friend from day one when I first started at Windsor Castle.\n\nNettie was The Queen's Housekeeper of Windsor Castle for over forty years. During Nettie's time as Housekeeper, The Queen gifted her with a corgi pup called Larch, who became Nettie's most loyal companion. After she retired as Housekeeper, Nettie came to work on the Dressers' Floor the very next day. For the last five years of her life, Nettie worked for The Queen archiving and photographing The Queen Mother's wardrobe.\n\nI wish my friend was here today so that she could share with me the ending of this book. Sadly, Nettie passed away in May 2019. I was with her throughout her illness as well as being a support to her loving brother, Andre, who was always there for her.\n\nIt was an honour to care for Nettie during this difficult time, although she still had her wicked sense of humour about her. Even when she was first rushed into hospital, her humour was there. I was with Nettie when the nursing sister was asking for her name. Annette was having a hard time breathing, so I answered the question by saying, 'Annette Wilkin'. It took all that Nettie had to say, 'No! It's Gwendoline Annette Wilkin.' I almost fell out of the chair. I was shocked! For twenty-five years I had been calling her Annette or Nettie. So, Nettie had the last laugh.\n\nWhen Nettie passed away I felt as though my mother was with me, keeping me calm and focused, as I helped prepare Annette's body to be laid out. I gave her a big hug and kiss. I'll never forget my best buddy. I miss you my friend.\n\n_Nettie and her corgi, Larch._\n\n_The gates of Buckingham Palace._\nMISS KELLY\n\nDedicated to Miss Peggy Hoath \u2013 who was my mentor and taught me to reach for the highest of standards.\n\nThe interview at Buckingham Palace that day went well, and I felt confident in my new outfit. Lady Susan Hussey and Miss Morrison seemed satisfied that I was the right person for the job. I would be reporting to Miss Peggy Hoath. After a lovely chat I was taken along the red-carpeted corridor to The Queen's private rooms, where I was to meet Her Majesty once again. This was where my life changed immediately. Obviously I cannot reveal the conversation, but I can say that it was lovely to see her again.\n\nWith all the arrangements made, I was soon walking through the gates of Buckingham Palace again, suitcase in hand, ready to start my new life.\n\nMy first day at the Palace was 31 March 1994, and I learned my first important lesson about life with Her Majesty: be prepared for anything. I was taken upstairs to the Dressers' Floor by the Footman who was kindly helping me with my suitcases. I was going to start unpacking, but Peggy had other plans. She was waiting for me on the corridor, took one look at the Footman and another at my suitcases and told him to put them with the rest of the luggage. Peggy said, 'Angela, don't get comfortable. We're off to Windsor for a month. There, I'll introduce you to the ironing board!' And I was! As soon as we arrived, Peggy said, 'Ironing board, meet your new owner!'\n\nAfter settling into my new job as an Assistant Dresser, my first task was to acquaint myself with Her Majesty's wardrobe. Each piece was exquisitely made with such close attention to detail. I also discovered an abundance of material \u2013 beautiful silks and chiffons that had been presented to Her Majesty as gifts and stored away safely for future use.\n\nBefore long, I started to receive unwanted questions about my marital status from other members of the Royal Household. My predecessors had all been known as 'Miss' and were typically unmarried women. Although I was divorced, I was still known as Mrs Kelly. Feeling vulnerable and scrutinised, I decided to seek advice from The Queen. We had become more familiar with each other over the months and I felt comfortable in approaching her, not just because her guidance on the delicate matter would be the most authoritative but also because we had quickly established a rapport. So, I went to see The Queen and explained that I had been receiving prying comments because of my married title, which I had kept as an acknowledgement of my past relationship. Her Majesty advised that if I was known as Mrs Kelly, I should be prepared for questions about where my husband was, whether he minded me travelling and working away from home. It was time to look to the future and, on reflection, I realised that my role with Her Majesty was my new priority. From then on, I would be known as Miss Kelly.\n\n_With Miss Peggy Hoath on Britannia during the VE Celebrations in 1995._\n\n_Now, let me introduce you to the ironing board, dear reader! Taken in the workroom at Buckingham Palace._\n\nEDUCATING \nANGELA\n\nIn those early years at Buckingham Palace, I remember feeling very aware that some people might look down on me. I was, after all, from Liverpool and had a Scouse accent, not to mention that I was divorced with three children, and possibly not the ideal candidate to be working for The Queen. Even though I knew Her Majesty had particularly requested me after her visit to the British Ambassador's Residence in Berlin, I thought other people might not understand why I was chosen: they might think I did not belong, and I wanted to be able to hold my own.\n\nI've always wanted to speak nicely. I'm not sure I can say why. It's not because I want to improve myself, just that I've always admired people who speak clearly. I don't have a strong Liverpudlian accent, but it is there. I had wanted elocution lessons since I was eight years old, and I remember asking the Ambassador in Berlin whether I could take them. He offered me a pay rise and a promotion instead, but I refused as I didn't want money all I wanted was to feel more at ease with how I presented myself to others \u2013 especially now among members of the Royal Household.\n\nNeedless to say, when I first started working with The Queen, I became even more aware of how I sounded. Listening to and speaking with Her Majesty, I would think, how wonderful to be able to speak so nicely, and after a few months of working with her, I plucked up the courage to ask if she knew anyone who might give me the elocution lessons I'd wanted for so long. The look on The Queen's face was a picture. She simply asked, 'Why?' After I had explained, she said that it was not necessary and that she'd heard about my plan \u2013 the Ambassador had told her. I asked again and again, but she still refused. After much back and forth, Her Majesty eventually told me to go upstairs and talk to the Duchess of Grafton about it.\n\nI ran up straight away, and found the Duchess waiting for me in the corridor. 'Your Grace, I was wondering if you could put me in contact with anyone you know who could give me elocution lessons,' I said.\n\nThe Duchess started laughing and I realised The Queen had called her in the time it had taken me to run up the stairs. 'Look, Angela,' she said, 'if you speak slowly, it means you speak clearly. Just keep The Queen laughing \u2013 that's all I ask of you.' But that still wasn't enough for me.\n\nLater that day, I went back to see The Queen and mentioned that I'd spoken to the Duchess. 'Oh, and what did she say?' she asked. I repeated the advice I'd been given and Her Majesty seemed content. 'Well, there you go,' she said.\n\n'But that's not an elocution lesson,' I responded, then told Her Majesty my new idea: 'You can give me elocution lessons! You can tell me what I say that's correct and what I say that isn't.'\n\nThe Queen could probably sense that I wasn't going to give up, so she instructed me to say one word: 'furious'. 'Fyer-ri-ous,' I responded.\n\n'No, fee-or-ree-ous,' said Her Majesty, in perfect received pronunciation. After several more attempts, I finally cracked it and Her Majesty exclaimed, 'Yes!' and her finger went up in the air, followed shortly by, 'Not sick as a parrot'. And that was it \u2013 my one and only elocution lesson, and from The Queen herself. From then on, I listened and tried my best to speak properly \u2013 even adding an aitch to words where it didn't exist \u2013 but in the end I gave up: it was just too much effort. Ever since I've stuck to being myself, a girl from Liverpool and a proud Liverpudlian, too.\n\n_In The Splash with the dogs in Windsor Home Park in 1998._\n\nLET ME TAKE \nYOU AMONG THE \nNIGHT STARS\n\nIn August 1994, four months after I had started working for The Queen, it was time for my first trip to Balmoral. Needless to say, I was very excited whenever one of these 'firsts' came around: my first visit to Windsor, my first trip on the Royal Train, my first Trooping the Colour. But I was particularly looking forward to seeing Balmoral Castle, where the Royal Family take their holiday each year. I was intrigued to see whether Her Majesty would have time to relax \u2013 to me, it seemed that she never stopped working.\n\nOn our first night, I remember gazing up at the night sky and noticing how brightly the stars shone out from the darkness. I'd always loved star-gazing and I find it a very calming experience. I vividly recall thinking how lucky I was to be standing in the majestic grounds of that historic building. A few days later, I relayed this to Her Majesty, with whom I now felt quite comfortable to enjoy some casual chat, and told her how much I loved gazing at the stars. It turned out that The Queen shared this passion and we looked out of the window into the darkness together. She pointed out the various constellations \u2013 the Plough, Ursa Major and Cassiopeia \u2013 and I mentioned that I had brought a telescope with me, a present from my family for my fortieth birthday. Her Majesty suggested that as it was a very clear, cold night, I should wrap up, go outside, and stand in front of the Castle shortly before midnight for the best view. I pointed out that the police probably wouldn't appreciate me setting off all the alarms in the middle of the night, so instead, I asked if I could take a car to drive up to the Glenshee \u2013 one of the Highlands' highest peaks. The Queen thought I was crazy and was reluctant to let me drive there alone, so she asked one of her duty protection officers to escort me.\n\nEquipped with my telescope, its stand, binoculars, and a flask of coffee, my slightly begrudging escort and I made our way into the night. On top of Glenshee, the view was breathtaking. With the headlights turned off and the protection officer freezing in the car, I started scanning the sky and admiring the stars and constellations. It was the very first time I ever saw the Milky Way and I remember feeling quite emotional. Eventually I was persuaded by the officer to return to the Castle, but I had my head through the sun roof throughout the journey back, wielding my binoculars, reluctant to say goodbye to the stars. The next morning, I was full of excitement. I rushed up to Her Majesty and could not stop talking about how wonderful the experience had been. She must have thought, thank goodness it will be a while before she does that again. But every time I return to Balmoral, before I go to sleep, I open my bedroom window and gaze at the sky for several minutes. It always reminds me of that conversation with Her Majesty when I discovered a shared passion and the fact that, because of her thoughtful gesture, I saw the Milky Way for the first time.\n\n_Taking a walk in Balmoral grounds in 1998 \u2013 note the high heels!_\n\n_Beauty within the castle grounds._\n\n_The Queen at a State Banquet in 2011._\nLEADING LADIES\n\nAs I began to settle into life in the Palace, I knew it was important to start standing my ground and fighting my corner. When I was asked to do something, I did it efficiently and thoroughly, but I soon became aware that I was also ruffling feathers.\n\nBack then, the Royal Household was very male dominated. It had long been a place where traditions were upheld and routines were followed. On one occasion, I remember being told that only after twelve years as a staff member would I be allowed to have an opinion. Needless to say, I was not happy with this \u2013 I would only ever act in Her Majesty's best interests and yet I often felt patronised and belittled.\n\nAs I began to get to know Her Majesty, I could not help but reflect on her role as a female monarch. I thought too of Queen Victoria, who ascended the throne at the very young age of eighteen, and the responsibility she bore. Victoria ruled alongside governments where all their members were men, and although Her Majesty's reign has seen a greater female presence politically, she must also have felt the same pressure to prove herself as a young, recently married female monarch. In the face of condescension and old-fashioned attitudes, I found inspiration in The Queen and the women who ruled before her and was determined to defend my right to be heard as a member of the Royal Household.\n\nI now know that my tendency to be forthright was a breath of fresh air for many of my colleagues, but it took me quite some time to learn the right approach when it came to expressing myself. In fact, I even became known as AK47 and The Queen's Gatekeeper, which I take as a compliment as it demonstrates my determination to be taken seriously.\n\nEven now, after twenty-five years, I still admire The Queen as a strong, powerful woman and I find great inspiration not only in her courage, but also in her humility and gentle humour. She has taught me so much over the years and has always encouraged me to stay true to myself while being open to the opinions of others, even if I don't share them. I know that her guidance has made me a better person, and for that I am eternally grateful.\n\n_The Queen and King Yang Dipertuan Agong of Malaysia during the Commonwealth Games in Kuala Lumpur in 1998._\nBACK TO FRONT \nOR FRONT TO \nBACK?\n\nI've learned countless new skills and techniques during my years working for The Queen. Even now I face new challenges when it comes to creating Her Majesty's outfits, but I always keep one thing in mind on a day-to-day basis: to trust my instincts. I learned this valuable lesson soon after I joined the Royal Household, and one memorable occasion will stay with me as a reminder of how important it is to speak up.\n\nA year after Peggy Hoath retired, in the autumn of 1995, I was promoted to Senior Dresser. Peggy had worked for The Queen for over thirty-five years, and knew all the tricks of the trade. Her standards were so high that I thought I would never be able to reach them. But I listened and learned, and everything she taught me is still carried out to this day.\n\nThen, in 1998, although I wasn't yet in charge of designing The Queen's wardrobe, I did assist in choosing appropriate outfits for each occasion. We were planning for Her Majesty's upcoming tour of Malaysia, where she would be closing the Commonwealth Games in Kuala Lumpur. As the weather is so warm over there, it was decided that Her Majesty would wear a dress but no jacket or coat, and she chose a vibrant deep coral for the outfit and matching hat. Designs were drawn up and a brief was sent to the milliner. Sketches were done for the hat and, due to Her Majesty's busy schedule, the hat was delivered at the last minute. Her Majesty didn't have a chance to try it on and I didn't have a chance to look it over before it was packed up.\n\nOn the day of the ceremony, I was helping Her Majesty get ready and went to take the hat out of the box in which it had been carefully packed. As soon as I had it in my hands, I knew, deep down, that it was not going to suit The Queen. With a commanding shell-like brim and a very large flower on the reverse, it was certainly an interesting and beautifully made piece, but I had a gut feeling that it would not complement the outfit or enhance Her Majesty's features. I was also very aware that The Queen was going to give a speech that would be broadcast live on television and that the world's media would be paying close attention, as always, to what she was wearing.\n\nAs the morning went on, I became increasingly worried, so I decided I had to raise my concern and face the consequences later. So, I waited for The Queen to come into the dressing room, took a deep breath, and said, 'Your Majesty, I don't think you should wear this hat. It is not the right design for you and it won't suit you.' A long silence followed before she eventually looked up and said, 'Well, it's a bit late now. What else can I wear? I must wear it.'\n\nAlthough I admired her pragmatic approach, I was insistent. I told her I had been examining the hat from all angles and that it would be better worn back to front. The Queen couldn't believe that this would work. I told her that, if she did not believe me, she should consult The Duke of Edinburgh, as he always tells her the truth. At this point, I left the room feeling somewhat terrified, as you can imagine.\n\nBefore long, Her Majesty called me back. She had consulted His Royal Highness and, after an amusing conversation, in which I imagine The Duke didn't hold back, The Queen had decided to wear the hat the wrong way around, as I had suggested. She was, however, very concerned that the milliner might be upset. I told her not to worry and that I would think of an explanation for when we returned to London.\n\nI remember feeling so relieved and proud when Her Majesty stepped out onto the stage to deliver her speech that day. The press admired her outfit: the hat had a wonderful 1940s feel to it, with the large flower sitting high at the top. During The Queen's speech, I was holding my breath \u2013 not because I was concerned for her but because I was eager to hear whether she was happy with her last-minute decision to turn the hat. Sure enough, when she came to see me after the event, she greeted me with a lovely smile. No words were needed. It was in moments like these that I knew Her Majesty and I would make a great team. I was so proud that I'd voiced my concern, and I think The Queen appreciated it, too.\n\nSo, dear reader, what do you think? Was it back to front, or front to back? Or, in fact exactly the right way around?\n\n_Top: The 'front'. Bottom: The 'back', but so much more stylish._\n\n_The Queen and Pope John Paul II during a State Visit to the Vatican in 2000._\nMEETING HIS \nHOLINESS \nTHE POPE\n\nAfter the incident with the back-to-front hat at the Commonwealth Games closing ceremony, I became more determined to speak up when it came to matters of Her Majesty's wardrobe. I had worked so hard to familiarise myself with what Her Majesty liked to wear and, crucially, what was appropriate for The Queen to wear to any given engagement. However, speaking up and being heard wasn't always easy, as one particular incident taught me.\n\nIt was the year 2000 and Her Majesty was due to travel to Rome on a four-day State Visit. I wasn't yet responsible for attending recces ahead of such trips, so it was the Private Secretaries who on their return outlined the provisional engagements that The Queen would be attending. On one day Her Majesty would be meeting with His Holiness the Pope, after which there would be a private engagement, and so they suggested The Queen would be fine to wear a normal day dress for both events. Straight away, I knew this would be a mistake. Whenever Her Majesty had visited the Pope previously she had always, without fail, worn a long black dress, a beautiful diamond tiara and a long mantilla lace veil. Having been brought up Catholic, I knew instinctively that a day dress simply would not be appropriate and so I expressed this to the Secretaries, who made it clear that my advice wasn't welcome.\n\nFeeling ignored and patronised, I decided to discuss the matter with Her Majesty directly. I told her that the Private Secretaries had advised that she should wear a colourful day dress to meet with the Pope as it could then also be worn to her next engagement. I insisted that this would not be appropriate for meeting with His Holiness. As always, The Queen listened, however, once she had discussed the matter again with her Private Secretaries, they still insisted that a colourful day dress should be worn. In response, I asked her to go back to the Secretaries and ask them to do their homework. I considered it to be bad advice. I sensed that Her Majesty was starting to feel torn as to whose advice to take.\n\nConfident in the knowledge that I would be standing my ground, regardless of what the Secretaries said, I came up with a secret plan. While preparing for the upcoming tour, I selected one of Her Majesty's favourite outfits \u2013 something I knew she would feel comfortable in \u2013 and requested that one of the dressmakers, Maureen Rose, create something in exactly the same style but in navy blue, and to send the invoice only to me. There would be no fittings on this occasion as I did not want to alert anyone to my plan. In addition, I approached Freddie Fox, the milliner at the time, and asked him to design a pillbox-style hat with a detachable veil and that the invoice must also be sent directly to me.\n\nI spoke with Her Majesty one final time before the tour. Once again, I told her that I was certain the Vatican would not have suggested that a day dress would be appropriate. I remember The Queen asking if I had ever been to the Vatican before and how I could be so sure of my advice. I hadn't, but said the reason I knew was because I was brought up a strong Catholic and that if The Queen turned up for such an important meeting in a day dress, where would it all end? I felt that standards would start slipping. Would guests just turn up to the Investitures in a cotton dress and pumps because they felt more comfortable? I insisted once more that the Pope would not expect guests to arrive in such informal dress. I simply would not dress The Queen this way and at this point, Her Majesty just wanted the matter to be sorted. I had to carry off my plan perfectly.\n\nThree months later, we were in Rome and I went about unpacking Her Majesty's clothes and arranging her outfits for the upcoming engagements. Soon enough, the day of her visit to the Pope arrived and I heard my name being called loudly down the corridor: I will never forget the sight of the Private Secretary frantically running down one of the longest corridors in the Quirinale Palace, bellowing my name, 'Angela, Angela, quick!', and wearing his brown jumper, brown corduroy trousers, brown socks but no shoes! He demanded to know what The Queen would be wearing that day. I calmly informed him that Her Majesty would be wearing a shocking pink outfit, as advised by him and the Assistant Private Secretaries, which did little to calm his panicked state. 'Don't you always travel with a black outfit?' he asked, and I replied that I had not on this occasion. In fact, we do always travel with a black outfit in case of an unexpected sad occasion, but I wasn't going to let him know that after what he had put me through. I told him not to worry and that Her Majesty would still look beautiful.\n\nHe then sent a message to The Queen asking what should be done, as the Vatican had said that the outfit must be dark. Shortly afterwards, The Queen sent for me and asked if we had any dark outfits that she could wear instead. 'Okay, Your Majesty,' I said, 'I'll show you something.' And with that, I produced the navy-blue dress and pillbox hat that I had had secretly made before casually commenting that, ideally, the outfit would be black for a meeting with the Pope. Her Majesty agreed and without a moment of hesitation, I produced the exact same dress and veiled hat in black. 'Luckily enough,' I said, 'I also had this made. So you're absolutely fine.' I vividly remember the brief look of relief on The Queen's face.\n\n_The two versions of the dress that I had secretly had made._\n\nWatching Her Majesty walk out in her stylish, modern and appropriate black outfit to meet His Holiness the Pope, I felt so proud of myself for having faith in my convictions. The Queen walked out dressed appropriately in her new black outfit as Head of the Church of England meeting another head of the church, His Holiness the Pope. I had known all along that my instincts about The Queen's outfit that day were right, even though I wasn't being listened to. From then on, I felt confident that my opinion would be heard.\n\nEventually I found out that the private engagement was a photograph of The Queen with members of the Royal Household outside the Sistine Chapel, and the Private Secretaries wanted Her Majesty to wear a normal day dress so that it looked like a fun day out, as the Royal Household wouldn't be dressed in black. This photograph moment had been prioritised above that important private meeting with His Holiness.\n\nIn 2014, The Queen met privately with the Pope during a visit to Rome. I had the honour and privilege to be presented to His Holiness. This was extremely special for me. I was holding my rosary beads when the Pope touched my hands to bless them, and I knew my mother would have been so proud of me.\n\n_A little colour makes all the difference. At Joel & Son Fabrics in 2019._\nFINDING \nMY FEET\n\nAs my role progressed, I spent more and more time working on Her Majesty's wardrobe and I started to notice that many of her outfits were made in similar colours. The Queen was wearing a lot of dark green, navy, and red, and I had also noticed that some of the older pieces \u2013 the stunning Hardy Amies evening gowns, for example \u2013 which had been the height of fashion a few years before, had begun to look a little tired. Spending time with The Queen had allowed me to understand better her likes and dislikes, and I sensed that she would enjoy more vibrant colours and some fresher designs. At this point Her Majesty's outfits were still only made by designers and dressmakers outside the Palace.\n\nOn one occasion I mentioned to The Queen that the designers should be looking at brighter colours and new designs, but who was I to tell them? I had been Senior Dresser for a couple of years and in that time had become more confident, talking to The Queen more about her outfits. I think The Queen knew that I was loyal and would only ever be honest with her. I could not help thinking that The Queen's style needed to change quickly, before she was made to look older than she was \u2013 which was what some of the old designs did. Some of the coats and dresses also had to be made shorter, as The Queen still had, and still does have, a good figure and excellent legs.\n\nOne day, The Queen invited me to join her the next time a designer, his team and the milliner came in for a fitting. I was slightly taken aback as I wouldn't usually be present for fittings; usually I just saw the dressmakers into the room and would leave them to it, but this time I was asked to stay. I was also quite anxious as I knew I couldn't hide my feelings very well: if an outfit did not suit Her Majesty, the expression on my face would make it clear to everyone what I thought. Even worse, if The Queen asked my advice, I would be forced to dismiss an outfit in front of its designer, or the hat in front of its milliner. Her Majesty would never want to hurt anyone's feelings, even if I didn't mind, and I would have to be honest about my opinion for her sake. Little did I know that this moment was yet another stepping stone for me, another door opening.\n\nWith the fitting under way, as I'd anticipated, the expression on my face started to give everything away and Her Majesty asked my opinion. This question didn't go down well with everyone in the room. With bated breath they all turned to me for my answer and waited, and I could see the shock across their faces as I told them my thoughts: the hats were too masculine and their patterns too large. In terms of the outfits, The Queen needed something more chic, fitted and elegant. There was a shocked silence in the room after I spoke. I suspected that I would never be invited back, and I imagine the designer and milliner hoped as much, but Her Majesty continued to request my presence at fittings, to the dread of those designers. I became their worst nightmare.\n\nI vividly remember one fitting where The Queen was draped in a very large piece of bold, jacquard material in a large print. Even a six-foot-tall model would have struggled to pull off something like that, and it absolutely drowned Her Majesty. Once again, I could not hide my disapproval when Her Majesty asked my thoughts. Without hesitation, I said, 'No way! It doesn't suit you at all and it is totally the wrong pattern.' An awkward silence and an icy atmosphere descended on the room. Everyone in sequence turned their heads towards me with what felt like daggers in their eyes, except for The Queen who stared straight ahead and goodness knows what she was thinking. The atmosphere was ice cold. Just at that moment, The Duke of Edinburgh happened to walk past and Her Majesty asked what he thought of the material, knowing that, like mine, his feedback is always honest. 'Is that the new material for the sofa?' he joked, before continuing on his way. I excused myself, left the room, and privately punched the air with delight. I composed myself, then walked back into the dressing room where I noticed the material was being folded and put away unused. The Queen was thanking the designers and the milliner for their time and they left the room.\n\nShortly after that fitting, and experiencing my honesty, Her Majesty asked me to draw some of my own designs for what I thought she should wear. I told her I couldn't \u2013 it had been a long time since I had sketched any outfits. 'If I wanted an artist, I would hire one,' The Queen said jokingly. And with that, I began rummaging through the parcels of gift materials, taking samples to show Her Majesty the wealth of beautiful silks and glorious patterns that she already had in her stockroom. Then I showed her some quickly drawn sketches of more fitted, stylish pieces. Thanks to my mum, I'd already had extensive training on the necessary skills but I was a little out of practice. I needed to refresh my memory quickly, considering that it was Her Majesty's outfits I would be working on. Each was inspired by my own love of fashion and my mother's elegant wardrobe \u2013 her perfectly tailored skirts and jackets. Sometimes I channelled the sensibility of my aunt Edwina \u2013 a designer, tailor, and dressmaker, who liked to dress more daringly. She was the first woman I saw in a transparent black chiffon blouse. The Queen saw my drawings of the designs, along with samples of the beautiful materials already being stored upstairs, and accepted them. The Queen liked the designs and suggested that we employ a machinist to help me. Before long, Her Majesty's wardrobe was being revitalised as, piece by piece, vibrant colours and stylish cuts made their way onto the rails. More to the point, The Queen was happy. It is an honour to be the first ever in-house Designer to Her Majesty The Queen.\n\n_Her Majesty wore this for her 90th birthday in Windsor in 2016._\n\n_The symbolic 'Girls of Great Britian and Ireland' Tiara is one of my favourite pieces._\nTHE QUEEN'S \nJEWELLERY\n\nBefore long, my role as Senior Dresser and Designer for Her Majesty evolved. Owing to the strength of my relationship with The Queen and the mutual respect we shared, in 1998 I was given the additional titles of Personal Advisor and Curator to Her Majesty The Queen. I still continued to design outfits for Her Majesty, and did this mainly in the evenings and at weekends. As Personal Advisor, I am in the privileged position of not only advising The Queen on her outfits and hats, but also looking after Her Majesty's health and well-being, and communicating with her team of expert professors and doctors.\n\nAs Curator, I would be responsible for The Queen's personal jewellery and certain pieces from the Crown Jewels, as well as The Queen's Insignia. It is a great honour and privilege to be entrusted with the care of The Queen's private jewellery and to help select items that are worn on a daily basis.\n\nOne of the first things I wanted to do in this new role as Curator was to find a jeweller with whom I could build a relationship, as the previous Crown Jeweller was due to retire. They would oversee one of the rarest collections of gems and historical pieces, some of which have never been seen, and would need to be extremely knowledgeable about the repair and restoration of antique jewellery. And, as is the case with all staff members in the Royal Household, they would need to be very discreet.\n\nHaving discussed this idea with Her Majesty, who agreed, I knew exactly where to begin my search; the Royal Palace itself, which often proved to be a fountain of knowledge. Not wanting to rouse anyone's suspicions, I mentioned to several people that I was looking to commission a jeweller to design something for myself, and time and time again the same name was suggested: Mr Harry Collins from G. Collins & Sons, who was based in Royal Tunbridge Wells, Kent. Her Majesty often chooses smaller, family-run businesses to work with and was not in the slightest bit bothered by the fact that Mr Collins was based outside of London, so a date was soon set for him to come to Buckingham Palace and bring a selection of his work.\n\nA week later, a very anxious Mr Collins arrived for his appointment, unaware that I had arranged for Her Majesty to join us. Shortly before The Queen's arrival, I asked him to display his silverware on the table and wait outside the room. Understandably, Harry seemed a little unnerved as he left. Her Majesty joined me to review his work and she was keen to purchase a few salvers and trinkets, so I proceeded to go and bring Mr Collins back into the room. I had to put him under pressure to see how he would cope, especially if I gave him a particular piece of jewellery from The Queen's collection. I needed to be sure that he would be able to work under stress if necessary.\n\nI always find it fascinating to see how differently people respond on first meeting The Queen. Unsurprisingly, Mr Collins was rather taken aback, and after the formal introductions, he began to discuss his work with Her Majesty, going into detail about his style and techniques. The Queen kept glancing in my direction and I knew from the look on her face that she was happy \u2013 this was her seal of approval and a signal to present Mr Collins with a beautiful leather fitted case which contained the Vladimir Tiara.\n\n_The Vladimir Tiara is made up of fifteen intertwined diamond-set ovals from which hang pendant pearls. The pearls can be interchanged with emeralds._\n\nI brought the case in and placed it on the table. Then I opened the box and took out the tiara, handing it to Mr Collins and explaining what was needed. Mr Collins delicately took the tiara from me and held it as gently as one would hold a baby. Carefully turning the headpiece to examine its diamonds and pear-shaped, cabochon-cut emeralds, he was clearly transfixed and I noticed that he kept staring at the tiara and Her Majesty in turn. It struck me that he was frozen to the spot \u2013 still smiling, but saying no words: it was as though he had stage fright. Luckily, The Queen is very adept in these situations and simply flashed Mr Collins a huge smile which helped snap him out of his trance.\n\n'Mr Collins,' Her Majesty said calmly, 'do you think you would be able to alter the casting on the drops of the emeralds?' Harry agreed, so I asked how much the re-casting would cost. 'Not much,' was his response, so I asked again. Once again, his response was vague and he looked nervously towards Her Majesty, only for her to say, 'Don't look at me, Angela does the money'. Putting Mr Collins under pressure, I asked one final time, 'Please can you give me an estimate, Mr Collins?' and he finally quoted a price which was agreeable to everyone.\n\n_The pearls for the Vladimir Tiara are kept in numbered pouches and it can take nearly an hour to change them._\n\nMr Collins has been an integral part of our team over the years and in June 2000, The Queen had a special request for him \u2013 to design a special piece of jewellery for her mother, Queen Elizabeth, to mark her 100th birthday. Needless to say, he was thrilled to be asked, and when Mr Collins returned a couple of weeks later to discuss some ongoing restoration work, he told Her Majesty that he had, 'come up with a lovely idea for your mum's present', producing a superbly designed brooch. It was a very unusual piece, handmade in 18-carat gold with a cabochon-cut centre of rock crystal and featuring a painstakingly fine hand-painted centenary rose set among 100 diamonds, to give the appearance that it is floating.\n\n_The Queen's special handmade birthday gift for The Queen Mother._\n\nThe Queen was delighted. When I escorted Mr Collins out of the Palace that day, I thought it was important to point out that although it wasn't a major issue, it was more correct to refer to members of the Royal Family by their full title, rather than 'mum'. Mortified, he asked me to pass on his apologies and even informed me later that the next day he'd purchased a copy of the Debrett's book on etiquette.\n\nA few weeks later, the day came to present Her Majesty with the brooch, and upon his arrival, it was clear that Mr Collins had been carefully studying how one should supposedly behave around The Queen. He greeted Her Majesty and presented her with the magnificent fitted case in which the brooch sat and, having read somewhere that where possible, one should not turn their back on The Queen, he proceeded to walk away backwards. Unfortunately, Mr Collins failed to notice Linnet, one of Her Majesty's corgis, lying on the floor behind him. He tripped over Linnet and ended up next to the dog, lying spread-eagled on the carpet. Terrified that he'd hurt one of The Queen's dogs, Harry frantically rubbed Linnet's chest, apologising profusely, but Her Majesty reassured Mr Collins and told him not to worry: it was not his fault as the corgis had a terrible habit of lying in the most awkward places.\n\nFrom that point on, Mr Collins and I formed a fruitful working relationship which saw him tend to some of the most significant pieces in Her Majesty's collection, including the alteration of the Vladimir Tiara and of the Countess of Wessex's wedding tiara, which was recently redesigned for the State Banquet with President Trump. Mr Collins is truly an invaluable member of the team.\n\nI have also had the privilege to work with various other jewellers, Swarovski Crystals and Mappin & Webb to name just a couple. The fourth of June 2013 was to be the sixtieth anniversary of Her Majesty's Coronation and Mark Appleby, from Mappin & Webb, called me explaining that the company wanted to design a brooch to give to The Queen in honour of this special occasion. Mark and I discussed the idea and it was agreed that it should be an orchid brooch made from Waterford Crystal. I particularly liked the idea of using the crystal, as you would then be able to see the colour of The Queen's outfit through it. The brooch consisted of four hand-cut Waterford Crystal orchid flowers surrounded by sixty-six diamonds, which would be attached to rose gold stamens. The brooch was lovely and Her Majesty wore it during the Irish State Visit at Windsor Castle, which was particularly fitting as Waterford Crystal is based in Ireland.\n\nMY PRETTY \nWOMAN MOMENT\n\nBy the year 2001, I had begun designing outfits for The Queen and I wanted to ensure that from then on Her Majesty always looked her best and was up-to-date on the latest trends. As part of my role as a designer, I was responsible for searching for another designer whose creations might be suitable for The Queen, as I knew I wouldn't be able to do it all myself. Usually this is an enjoyable task, but on one occasion, things didn't go as smoothly as I'd hoped.\n\nOn one of my first solo excursions, I was planning to visit some local designers, not far from Buckingham Palace. I dressed in smart-casual clothes, but comfortably, as anyone might do when they're anticipating a morning of walking around the shops. The first shop I went into had a stale feel to it and the outfits in the window display seemed slightly old-fashioned. None the less, I was approached by a lady who smiled and asked if she could help. I explained that I'd come to have a look at their clothes and asked whether I could see the clothes on the rails to gain a better idea of the style and cut of their designs, as well as the finishing. As she agreed, I spotted a man at the back of the room who, even though he was in the middle of fitting someone, was staring at me as if I was the dirt on his shoe. He peered over the top of his black-rimmed glasses and condescendingly advised that the clothing was 'couture, not retail'.\n\nI was shocked by his rudeness and immediately turned to the woman who had greeted me and I said quietly, 'I do know it's couture, not retail, thank you. And he has just made a big mistake. Huge. I will be back to tell him who he missed designing for.' In my mind I was Julia Roberts in Pretty Woman right then!\n\nWith that, I left the shop feeling humiliated and belittled. How rude of him to presume my status in such a way.\n\nI made my way to the next shop, hoping for a more positive experience. I had been recommended to try Motcomb Street by one of The Queen's Ladies-in-Waiting as a good area to find designers, so I went into Stewart Parvin Couture. As I stepped inside, it felt different from the last place \u2013 bright and modern with beautiful outfits everywhere. Again, a lady met me with a beaming smile and I asked the same questions as before: whether they'd mind me looking at the designs, the cut and the finishing of their clothes. 'Not at all, take your time,' was the very different response, and I went along the rails freely, looking at all the outfits. It was immediately clear to me that this was not only the work of a designer but of an excellent tailor. I was so impressed that I asked if it would be possible to meet Mr Parvin and, sure enough, the lady offered me a seat while she went to fetch him.\n\nI had to wait only ten minutes for Mr Parvin, during which time I kept looking towards a particular blue cocktail outfit with the most lovely harlequin multi-coloured gem buttons. In fact, I was engrossed in these buttons when a young man in a pair of jeans and a white T-shirt greeted me warmly and enthusiastically. I explained to Stewart (for that was who he was) that, having looked at his designs, I had my eye on the blue cocktail suit with diamant\u00e9 buttons, but that I would need to take the piece away for a short while to show my employer. I promised to bring it back within a few hours. Bear in mind that, at this point, Stewart had no idea who I was or for whom I worked, but he agreed: he really was the warm, friendly person he appeared to be.\n\nTwo hours later, I returned with the outfit, which I had shown to The Queen and had received her approval on Stewart's style. Stewart was there to greet me again and, after thanking him, I asked if he had a stockroom. Sure enough, he took me straight downstairs to show me, seemingly unfazed by my endless questions about his work. Three of his materials leaped out at me as being appropriate for Her Majesty \u2013 a blue fleck tweed among them. With these in mind, I asked Stewart if he would do some quick designs there and then for a very high-profile female customer. Once again, Stewart was happy to oblige and we spent some time sketching potential designs. Safe in the knowledge that Stewart was a talented, trustworthy, and kind person, I finally disclosed that these designs would be for Her Majesty The Queen, but that he must be discreet and keep everything confidential.\n\nAnd that was that. Since then, Stewart and I have been working together for The Queen, and our friendship and professional relationship have gone from strength to strength. Sometimes it's hard to believe that we have known each other for over eighteen years. In 2007, Stewart was granted his Royal Warrant in recognition of his contribution to The Queen's wardrobe. Little does that first designer know what he missed out on.\n\n_Happy times \u2013 Stewart and I collaborating together at Joel & Son. Look out for this material used in a cocktail dress to be worn in 2020._\n\n_Jackie and I working hard._\nALL IN A \nDAY'S WORK\n\nAs my role evolved and became even more varied, I soon learned that there would be no 'typical' day for me, just as there is no 'typical' day in Her Majesty's schedule, with its various engagements, Royal Tours and State Visits.\n\nA couple of years ago, in 2017, a request from the Castle Administrator at Glamis Castle, Tommy Baxter, made its way to my desk in a handwritten letter from Angus, Scotland. Tommy was looking to organise an exhibition to mark the eightieth anniversary of King George VI's Coronation and was wondering if The Queen would be willing to loan The Queen Mother's wedding dress, her Coronation dress and Coronation robe for display, as well as the Coronation dresses, robes and coronets of Their Royal Highnesses The Princess Elizabeth and The Princess Margaret. Tommy mentioned that he thought it was unlikely: he had already been in touch with a member of the Royal Household and been offered two paintings for the exhibition, but no outfits.\n\nGlamis Castle is very close to The Queen's heart as it is where her mother, Queen Elizabeth, grew up and where Her Majesty was often taken as a child, with her sister, Princess Margaret. I knew that The Queen had wonderful memories of the historic castle, so I decided to ask whether we could agree to Tommy's request. Her Majesty was happy to do so, and asked me to 'help them as much as you can', even noting that some of those items had recently been on display in Buckingham Palace \u2013 there seemed no reason why they could not be loaned for the exhibition in Glamis Castle. Tommy was thrilled when I told him, but gaining The Queen's approval turned out to be the easiest part of this process.\n\nAfter much correspondence, my personal assistant, Jackie Newbold, and I found a suitable date to deliver everything to Glamis. We had to consider transport carefully \u2013 the cases must never be out of sight \u2013 so we decided that the train would be our safest option. With a case each \u2013 one containing the Princesses' small dresses, robes and coronets, the other with The Queen Mother's dress and robe \u2013 we made our way, alone and with no security, for a two-night stay in Dundee. God help anyone who tried to take the cases from us that day! We weren't stressed: we were alert, very aware, and very watchful.\n\nWhen we arrived at the Castle, we got straight to work. When it comes to any exhibition or display of The Queen's things, I tend to have a lot of input, so that Her Majesty is safe in the knowledge that everything will be displayed or showcased in just the right way. Jackie and I were thrilled to help the Earl of Strathmore and the Dowager Countess make a truly breathtaking exhibition.\n\nBefore we headed back to London, I was reminded that The Queen was in need of some checked skirts, so Doreen, one of Tommy's colleagues, was kind enough to take us to the Fabric Mill at Halley Stevensons \u2013 the perfect place to find something suitable for Her Majesty. All in all, it was a very fruitful trip.\n\nThe exhibition ran for five months and Tommy regularly emailed with updates to say how well it was going and how much the public enjoyed seeing the Coronation outfits. During our stay in Balmoral that year, The Queen herself went to Glamis Castle to have a look at her mother's dress and robe, and her younger sister's outfit, on display in that beautiful setting. It must have brought back such fond memories.\n\nAs the exhibition came to a close, Jackie and I arranged a date to collect the outfits. It was a particularly busy time of year so we decided to travel up to Dundee and come back immediately \u2013 everyone thought we were crazy for not staying overnight, even The Queen. But when the diary is that full, you do what you have to do. On the day, we had only a fifteen-minute window in which Tommy was to meet us on the station platform to return the cases and, as luck would have it, the train was running five minutes late. Needless to say, Jackie and I were slightly on edge. Happily, the exchange went smoothly (and very quickly!) and we were immediately on our way back to London with the priceless cargo. All in a day's work!\n\n_The Queen Mother's robe, worn during King George VI's Coronation. Regal at its best._\n\n_The Queen at Royal Ascot in 2019._\n\n_My design room at Buckingham Palace. This outfit was worn during the Diamond Jubilee concert at Buckingham Palace and demonstrates how well The Queen and I work together. I designed the outfit with the detailing on the left shoulder, and The Queen requested it on her right shoulder, which is exactly how she wore it on the night._\nDESIGNING \nFOR THE \nQUEEN\n\nThe Queen accepts more than 300 engagements each year, through different seasons, occasions, and cultures. Our role as her dressers is to ensure that Her Majesty is appropriately attired for each occasion \u2013 from State events to informal daywear, in the United Kingdom and abroad.\n\nFrom time to time The Queen will wear the same outfit to different events, although several months will pass between them. When I'm designing an outfit, I start with 'feelers': large samples of material that allow me to experience the texture of the fabric before deciding to buy it. I will squeeze and twist the sample in my hands, before smoothing it out again. If the material remains crumpled or creased, it will be of no use and is discarded from the selection.\n\nI get my inspiration from seeing and touching the materials too, rather than starting with a sketch. I find a quiet moment when I hide in the materials storeroom and play with the fabrics, allowing them to shape themselves into a design. I look for movement with soft, light materials, and might even switch on a fan to see how they behave in a breeze. Fabric choices are obviously important \u2013 there are beautiful fabrics that work well but might catch or pull on objects like Her Majesty's watch. Loose-weave fabrics can easily lose their shape, requiring a good lining to prevent this. They can also develop loose strands, making them appear unkempt and unattractive.\n\nColour is key, though \u2013 the colour chosen must suit The Queen and the occasion. Vibrant colours work well in the daytime: they allow her to stand out from the crowd and be visible to the well-wishers who have come to see her. As the light changes, or when Her Majesty moves to an interior space, this will have an effect on the colour and texture of the fabric, and this must be taken into account.\n\n_Before I started working as Dresser and Designer for The Queen, no records were kept of what Her Majesty wore and when. I started keeping this diary in secret \u2013 scribbling and sketching hastily at midnight each night just to keep a record of the colours and a rough idea of the design so Her Majesty didn't repeat outfits too soon. After 1995 I was able to start keeping records more officially and less hurriedly than this!_\nI consider the four seasons separately and individually when designing for The Queen. Spring is for delicate, paler colours, flower motifs, lightweight coats and jackets, but as the weather is changeable, I must ensure that warmer fabrics are used to shield Her Majesty from the chill. Summer is for light, dynamic fabrics that move and flow with the breeze; strong florals work, with vibrant colours. Autumn is cooler, and I introduce medium to heavier fabrics, such as wools, cashmere and double cr\u00eapes, with a mellower colour palette. Winter brings gold, royal blue, deep purple, as well as velvets and heavier wools, with matching scarves, larger collars and warm hats. If Her Majesty is due to attend an engagement in particularly cold weather, from 2019 onwards fake fur will be used to make sure she stays warm.\n\nAlthough there are, of course, many beautiful garments and outfits, there is also the casual side to Her Majesty, when she goes walking with her dogs, for example. Although The Queen is famous for her green macs and her Wellington boots, The Queen also likes to wear skirts and blouses. The Queen has a mixture of silk blouses as well as cotton shirts, which are all handmade by Grosvenor Shirts Ltd in Jermyn Street. As you can imagine, over the years The Queen now has many blouses and shirts, but she also wears cotton dresses in hot weather. And this is where Karl Dunkley and Juan Credidio at Grosvenor Shirts really come into their own, making these more casual outfits in their factory in Ireland.\n\nMany of The Queen's more memorable dresses are created for specific events, such as a State ceremony or a Royal Tour abroad. When a tour is in prospect, The Queen will review the designs with me to find a style that will be sympathetic to the people and culture of the country she is to visit.\n\nIn general, necklines on coats and jackets must not be too high or too low, and must not restrict The Queen's movement. This is particularly true of thick wool coats with wide, full collars. A low neckline on a long dress allows a distinctive necklace to sit well and to be seen. In warmer weather I typically design with a more open neckline, and when cooler the neckline will be higher and closed for warmth and comfort.\n\nWhen Her Majesty is travelling in a car and sitting for long periods, her coat or jacket must be comfortable and practical, and she must be confident on getting out of a vehicle that her outfit will fall correctly and not crumple. Too much fabric can prove challenging in that situation \u2013 and large, heavy beading on a gown can be uncomfortable, especially on the back or when sitting down. None the less, The Queen understands that beading and crystals are sometimes necessary to produce a spectacular effect: she doesn't mind temporary discomfort when it is so important for her to 'look the part'. I usually add a few extra layers of lining to help cushion the impact, especially at the back of the dress, and try to keep the heavily beaded areas to the front and sides.\n\nThe lengths of dresses vary according to the occasion: daywear will usually stop just below the knee; a cocktail dress for a drinks reception could fall to just below the knee again or ballerina length, finishing just on or below the bottom of the calf. This length is particularly good as it enables Her Majesty to walk unhindered. Stairs can present a problem with dresses if they are too long or too tight. On occasion you will see The Queen wearing skirts that flare out from a fitted hip or knee as they allow for easy movement. I often use a fishtail in long skirts, as this gives a lovely shape to the back while still allowing for movement. Overall, The Queen likes her clothes to be fitted but not too tight, with a sleeve length of either three-quarters or full \u2013 definitely not too wide at the wrists: at lunch or dinner the cuffs might end up in the soup.\n\nWhen the time comes, I arrange a meeting with The Queen to go through my design ideas, suggested outfits and material choices. It usually takes no longer than an hour, and once we've come to an agreement, the outfits are made swiftly. Two fittings are then arranged to ensure that each garment fits Her Majesty perfectly.\n\nThe Queen's shoes are mostly handmade \u2013 and usually with no more than a 2-inch heel, although for uneven surfaces, such as cobbles, gravel, or grass, she will wear shoes with a lower heel. In the mid-2000s I was looking to modernise Her Majesty's shoe , so I called in an expert, and friend, Michael Atmore, the Editorial Director of Fairchild Publications. Known in the industry as the 'Shoe Whisperer', he suggested that I reconnect with Rayne Shoes, under the new direction of Nick Rayne. Rayne Shoes had a long history of supplying the Royal Family, but had stopped doing so after a change in ownership and direction. After this successful re-introduction, Michael also introduced me to Stuart Weitzman, Neil Clifford at Kurt Geiger and Carvela and Manolo Blahnik. Her Majesty tends to use cream, white or black for her shoes as one pair will match, and be worn with, many different outfits. As has been reported a lot in the press, a 'flunky' wears in Her Majesty's shoes to ensure that they are comfortable and that she is always good to go. And yes, I am that 'flunky'. The Queen has very little time to herself, and no time to wear in her own shoes, and as we share the same shoe size it makes the most sense this way.\n\nLike The Queen's shoes, most of her handbags are custom made, and weight is very important \u2013 given that Her Majesty may have to carry it for some time. She prefers a longer handle so that it hangs from her forearm without catching on the cuffs. The Queen doesn't have favourites, and uses handbags from different designers.\n\n_The 'EU dress' \u2013 seeThings Aren't Always What They Appear to Be for full story_\n\nThe Queen is conscious that she must be easily visible to as many people as possible when she is out and about so I choose mainly striking colours so that she will be easily seen. For example, when Her Majesty visits a school or a children's centre, she is always dressed in a bright, jolly colour, and her hat has the kind of details that will appeal to children \u2013 feathers, twirls, twists, flowers, and ribbons. When she visits a nursing or residential home for older people, she prefers to wear a strong, well-defined colour, with a structured hat, to help those who are visually impaired to see her and feel part of the visit. In fact, when The Queen's engagement includes a walkabout, the size of the crown and brim of the hat are also taken into account, as The Queen is aware that people travel from far and wide hoping to get a glimpse of her and it is important that young and old can see her clearly on the day. But the crown of the hat must not be so high that Her Majesty has difficulty with getting into and out of a vehicle. After 6pm, The Queen does not usually wear a hat, but on State occasions or when attending special dinners, she may wear a headpiece or a tiara to go with a beautiful evening dress.\n\nIf Her Majesty has to attend a large number of engagements in one day \u2013 sometimes up to five \u2013 I go for zips every time in her dresses: it is much more convenient for The Queen to be able to step in and out of a dress quickly: trying to lift a garment over the head can also wreak havoc with the hair. I also use discreet weights in the hemlines \u2013 many of The Queen's engagements take place in the open air where a sudden breeze can be troublesome.\n\nTypically, the lifespan of an outfit can be up to around twenty-five years. Her Majesty is always thrifty and likes her clothes to be adapted and recycled as much as possible. Given how much attention is paid to what The Queen wears, the Assistant Dressers and I do take into consideration how regularly an outfit is worn. After two or three outings, a piece will have become familiar to the media and public, so we will either look for ways to modify it or it will become something that is worn on private holidays at Balmoral or Sandringham, or as a morning dress. Older outfits are handy considering that, just like all of us, Her Majesty is likely to encounter the odd muddy dog.\n\n_Dress worn in 2018._\n\n_Part of The Queen's private wardrobe._\nTHE QUEEN'S \nWARDROBE\n\nThere has been much speculation about the private rooms in Buckingham Palace, both in terms of their number and how they are used. However, Her Majesty is a very modest lady and only occupies a few rooms: her bedroom; her private sitting room; her dressing room and bathroom, and the Audience Room, which is where she meets with the Prime Minster and other dignitaries, and The Empire Room, which is a waiting room. These rooms aren't vast and there is space for very little furniture: just the odd wardrobe or chest of drawers.\n\nEven after twenty-five years of service with Her Majesty, it still astonishes me that some former members of staff \u2013 even those who have not worked in the Palace for thirty years \u2013 are prone to making suggestions and sharing supposed details about The Queen's private rooms, when they actually have no insight whatsoever. The only gentlemen that I have ever known to enter these rooms are The Duke of Edinburgh and other members of the Royal Family, because The Queen's privacy is the absolute priority among her personal staff. So, do bear this in mind the next time you hear any supposed details about Her Majesty's private rooms! There will always be people who will say anything for attention.\n\nLike any woman, The Queen is very practical and has her everyday items accessible to her, such as day dresses, hanging in the wardrobe in her dressing room which are ready to wear. Outfits that Her Majesty would wear on special occasions or for outside engagements are kept in a wardrobe upstairs, along with longer evening gowns, so that whenever The Queen wishes to see these items, she can pop upstairs to browse through them. Having the upstairs space is equally as convenient for me, as when I am preparing for an upcoming Royal Tour, it would be impractical to try to safely move all of the many outfits and their matching hats to her dressing room to go through. Instead, selections are made based on sketches and material samples which I show to Her Majesty, and all the designing and creation of the outfits takes place on the Dressers' Floor.\n\nPart of my varied role involves looking after these personal rooms, along with Her Majesty's two Pages. With the help of the Assistant Dressers, I ensure that Her Majesty's wardrobe is properly organised and ready for the day ahead and The Queen's Housemaids will clean all her rooms daily. I am always respectful of Her Majesty's privacy and we try to make sure that she has a little time to herself.\n\n_Left: Calling bells originated during Queen Victoria's reign, when wiring was first put into Buckingham Palace. These are the same ones still used today to signal us from Her Majesty's private rooms. Right: Part of The Queen's private wardrobe._\n\n_The cr\u00e8me de la cr\u00e8me, in 2019._\nTHE TEAM\n\nOur team is like a wedding cake. On the top tier is Her Majesty, of course, followed by me underneath. Then we have the Deputy Dresser, Timea, followed by three Assistant Dressers \u2013 Grace, Phoebe and Lucy \u2013 who assist me with The Queen's wardrobe. The title 'Dresser' could be a bit misleading as Her Majesty actually dresses herself. Our role is to lay everything out for her and sometimes help zip her up or fasten a tricky piece of jewellery, although The Queen doesn't need help with her everyday jewellery such as her pearls and earrings.\n\nAfter the Assistant Dressers is Jackie, my Personal Assistant. She not only looks after the work side of things \u2013 managing my diary, organising meetings, the bills \u2013 but also me personally, helping me get to the places I need to be at on time as my day can be quite hectic. With Jackie there, I know everything will be in order.\n\nNext is Neil, who is our Project Manager and who runs the floor. Anything to do with maintenance or the removal of luggage \u2013 whether it is a large number of cases to be sent to Windsor or Balmoral, or smaller, everyday deliveries \u2013 Neil deals with other members of the Royal staff to make sure everything is organised. He is also helpful when it comes to setting up meetings with property services and the travel office and making sure everything is organised.\n\nThen we have Stella, who is our milliner. Stella makes Her Majesty's hats based on my designs and I often ask for her input on which trimmings work best together \u2013 I really value her ideas and suggestions.\n\nWe also have a dressmaker, Desiree, who makes the clothes once I have designed them and they have been approved by Her Majesty. As with Stella, the dressmaker will suggest alterations and additions and she always knows exactly how I am picturing an outfit. I'll often say, 'Yes, that's it \u2013 that's exactly what I want!' She works on it from the very start to the finish, from cutting the pattern all the way to the completed article.\n\nFinally, we have a housemaid who focuses specifically on the Dressers' Floor and helps to keep things organised and tidy. She will assist with packing and labelling each item.\n\nAn important thing that helps to keep our team running like clockwork is that everyone is trained in each other's roles. Jackie, for example, is trained as an Assistant Dresser and Stella can make alterations and repairs to dresses. So, if I happen to be ill or on holiday, someone else can step in. Everyone covers everything and works together seamlessly.\n\nHowever, I do have strict rules on the Dressers' Floor. Firstly, I don't hold with gossip or little spats between staff members. I run a happy ship and everyone needs to feel that we're a small family. Secondly, no one can come onto our floor without an invitation or not having made an appointment, because there could be private clothes out that we're in the process of washing or ironing. I think that if any of my team was asked what they think of me, they'd say I was fair. They know what I'd say about them \u2013 that they're the cr\u00e8me de la cr\u00e8me.\n\n_Our record keeping has definitely built up over the years!_\nOUR \nANNUAL \nDIARY\n\nWhile it is impossible to predict what each year will bring, there are certain annual events and engagements that will forever remain unchanged.\n\nThe Christmas holiday season always starts in the middle of December at Sandringham House, in Norfolk, where Her Majesty is based until early February. After Christmas and New Year, which The Queen spends with her family, Her Majesty will invite private guests to Sandringham, who tend to stay until the beginning of February. Even throughout the Christmas period, The Queen will continue to work on her red boxes. Usually, Her Majesty will only return to Buckingham Palace after the anniversary of the death of her father, George VI, on 6 February. It's certainly a sad time for The Queen but I am sure that she finds comfort in the memories that Sandringham holds for her.\n\nUpon returning to Buckingham Palace, Her Majesty's schedule resumes at full pace with a host of daily private audiences, weekly meetings with the Prime Minister and occasional meetings with the Privy Council, as well as the odd evening function. In the run-up to Easter, Her Majesty will hold several Investitures and will typically hold at least one State Visit before joining her family for the Easter weekend. The day before Good Friday, The Queen attends a Maundy service before heading into the Easter Court, which varies each year, depending on when the Easter weekend falls. One constant occasion is Her Majesty's birthday, which falls on 21 April and is celebrated privately with her family. Typically, Easter Court will finish towards the end of the month or at the beginning of May, when The Queen will travel to Wood Farm, a private cottage located on the Sandringham Estate.\n\nMay is always an exciting time of year from my perspective, as it is the start of the Garden Party season and Her Majesty can start to wear lighter fabrics and beautiful florals. As spring gets under way, The Queen hosts several garden parties, and the Windsor Horse Show takes place in the Home Park at Windsor. Perhaps the highlight of the season is the Chelsea Flower Show, which Her Majesty looks forward to every year, and after which she will always travel to Craigowan, a small private house on Balmoral Castle Estate.\n\nThe second of June sees Coronation Day, which Her Majesty marks privately before attending the Epsom Derby in Surrey. The next Royal engagement is a very well-known one: Trooping the Colour, a ceremony performed by regiments of the British and Commonwealth armies to mark the official birthday of the British Sovereign. It may be of interest to know that the reason Her Majesty has two birthdays is because her great-grandfather, King Edward VII, who was born in November, always found the weather at that time of year to be disagreeable for the soldiers and members of the public who came out to celebrate the occasion. He therefore decided to arrange an official birthday parade in June, called Trooping the Colour, as it was near enough guaranteed that the weather would be more pleasant. After this grand event, Garter Day and Royal Ascot are soon upon us.\n\nAs June comes to a close, Her Majesty will visit the Palace of Holyroodhouse, in Edinburgh, where she hosts her final Garden Party of the year before visiting Wood Farm, at Sandringham, again for a short stay. From hereon in, we exclusively focus on preparing for Her Majesty's stay at Balmoral Castle. We head up to Scotland in mid-July, firstly to Craigowan Lodge and then we move across to the Castle, and we will not return to London until early October. While at Balmoral, Her Majesty will host picnics, barbecues, dinners, and shooting parties for her family, friends, and private guests, as well as two Highland Ghillies' Balls.\n\n_We specifically name the outfits to ensure we recorgnise them previously if Her Majesty chooses to wear them again._\n\nAfter a busy but enjoyable summer, October sees us arrive back in London to a very different climate. Normal service resumes as soon as we return to the Palace, and it's not long before Her Majesty is hosting audiences, Investitures and her second State Visit of the year. Before long, November is upon us which usually involves the State Opening of Parliament, and attendance at The Royal Variety Show. Her Majesty also attends the Royal Albert Hall for the Festival of Remembrance, followed by the Cenotaph the following morning. As for anyone else in the country, December is a busy month for us, and alongside all the usual duties, Her Majesty will always take time to give Christmas presents to her staff at Buckingham Palace. An event she particularly looks forward to attending is the carol service in the Royal Mews, and festivities continue when The Queen hosts a Christmas lunch for the entire Royal Family.\n\nBefore you know it, the end of the year is approaching fast and we are soon packing to return to Sandringham. During The Queen's final weekend in Windsor, the choir from Windsor Castle comes to sing Christmas carols at the Sovereign's Entrance, which is always a special moment \u2013 particularly if it happens to be snowing! The Queen gives Christmas presents to the Windsor staff over this weekend. At this point, our only other event to prepare for is the filming of Her Majesty's Christmas message.\n\nSo, that's our year. There will always be changes to the programme, though, as Her Majesty may need to attend special events such as memorial or commemoration services, the opening of institutions or changes of Government. However, unexpected events aren't always serious occasions as sometimes there will be a wedding or christening to attend. Her Majesty may also need to sit for new portraits and there is always an excuse to take a trip on the Royal Train. Whatever her schedule, Her Majesty always rounds off each week by heading to Windsor Castle to relax and she will always attend church every Sunday, no matter where she is in the country.\n\n_The Queen on Garter Day in 2008._\nGARTER DAY\n\nGarter Day takes place every June at Windsor Castle. It falls on the Monday of Royal Ascot week. This occasion is full of pomp and circumstance, and the Knights Companion and Ladies Companion of the Most Noble Order of the Garter, dressed in their velvet mantles, plumed hats and insignia, are a stunning sight. The Queen's Garter Robe is made from a royal blue velvet, lined with white satin. A red velvet hood is worn over the right shoulder, and the plumed hat is made from black velvet trimmed with white feathers.\n\nKing Edward III founded the Order in 1348, and on Garter Day any new Knights and Ladies take their oath and attend an Investiture, or installation, hosted by The Queen. As well as some 'extra' Knights and Ladies, who are members of the Royal Family, there are twenty-four Knights and Ladies at any one time, and they have been chosen personally by The Sovereign in recognition of their public service. It is a lifetime honour and each individual is designated their own banner, which is hung in St George's Chapel, Windsor.\n\nGarter Day falls in what is generally a very busy period in The Queen's diary, and especially for us, ensuring all the outfits are packed and ready in time for the luggage transport from Buckingham Palace to Windsor Castle. The week before Garter Day and Royal Ascot, the Dressers' Floor at Buckingham Palace is a scurry of rails of clothes trundling along the corridor. Just as I check all the State Robes, the Garter Robe is hung up in the workroom beforehand and I inspect every inch of it, looking for any spots or bruising \u2013 a shiny mark \u2013 on the velvet. If there is any bruising, I will gently brush it out. To clean the robe, I fill a bathtub with hot water so the bathroom gets steamy, then hang the robe carefully away from the water. The steam will enable the creases to drop out. This works with any velvet. I never touch velvet with water and never iron it: doing either would damage the material. Once I am happy that the robe is clean and crease-free and the ribbons have been ironed, I pack it into a hanging bag and have it ready to take to Windsor Castle, along with the Royal Ascot racing outfits. By this point I will have already done similar checks on the outfits Her Majesty will be wearing to Ascot, ensuring all the buttons and trimmings are secure by giving them a sharp tug.\n\nOnce the luggage has been transferred to Windsor Castle from Buckingham Palace, a few days before Garter Day, the first thing I do when I arrive is take the Garter Robe to the workroom. I lay a sheet on the floor to protect the robe from dirt and damage when I take it out of the bag, then hang it on the back of a door and let the bottom of the robe drape along the sheet. I use a padded hanger with extra layers so that the robe fits it perfectly. Hanging the robe like this helps air it and allows any further creases to drop out.\n\nOn the Sunday, the day before Garter Day, I will press the ribbons again to make sure the robe looks pristine, then convey the robe to The Queen's dressing room. The State Dress Her Majesty will be wearing beneath the robe is pressed and checked \u2013 in particular making sure that the zipper works. The dress will be elegant and full length, but it will be an older design: when The Queen walks along the pavement from the Castle to St George's Chapel, the material may be damaged as the dress may drag a little along the ground. I wouldn't want a new dress of The Queen's to be marked or the hem to snag. I always prepare two spare full-length State Dresses in case there is an emergency, such as a button falling off or a hem becoming unstitched.\n\nOn the day, the Assistant Dressers and I attend The Queen and prepare everything for the busy day ahead. When The Queen departs for breakfast, we bring a coat stand into the dressing room, lay the sheet on the floor and hang the robe on the stand where it is ready for Her Majesty, with the State Dress near by. I also lay out the Garter chain, which is called the Collar. This is made up of twenty-four enamelled roses encircled with a blue buckled garter bearing the motto 'HONI SOIT QUI MAL Y PENSE' ('Shame on him who thinks ill of it.') These are linked together with twenty-four 'double lovers' knots' in gold. The Collar is worn suspended from the shoulders by white satin bows. It was introduced by King Edward III in 1348 and is also worn at the State Opening of Parliament, along with the Lesser George, an oval of plain gold that encircles St George and the Dragon with a buckled Garter and motto, which is worn by recipients of the Garter.\n\nIt may interest you to know that the 'Garter George' is a representation of Saint George killing the Dragon, set in exquisite gold and enamels. Some of the Royal Georges are embellished with jewels. Originally, the Lesser George was worn around the neck suspended from a narrow ribbon of Garter blue. Although The Queen has recently permitted this practice again, in order for people wearing black tie to wear the insignia, the Lesser George is supposed to suspend from the Garter's Broad Riband or 4-inch wide (for men) sash in Garter blue corded silk which goes over the left shoulder to suspend the Lesser George on the right hip.\n\n_The Queen and I often speak silently using eye contact \u2013 here she is smiling at me and silently telling me, 'You got here quickly!' And I really had! I had only just finished dressing Her Majesty and I ran (in high heels, of course) to be able to see the procession._\n\nAfter breakfast, The Queen will attend meetings with the Private Secretary. Then the Assistant Dressers and I will help Her Majesty get ready. She will change into her State Dress and put on her jewellery, then make her way to the Waterloo Chamber, where she will join the other members of the Order of the Garter for lunch. Once lunch has finished, The Queen returns to her dressing room where we help her put on the Garter Robe, the hood, plumed hat, and the Collar over the State Dress. We will remove the jewellery that The Queen wore to lunch and put it away. It is difficult to wear a necklace with the robe, as the Collar has to hook together for the robe to be secure on The Queen's shoulders and the pressure of the hook on a necklace would be uncomfortable.\n\nGarter Day is steeped in tradition, and people come from all over the world to line the processional route and watch the Knights and Ladies walk to St George's Chapel for a short service, at which any new Knights are installed. For many years, The Queen walked from Windsor Castle to St George's Chapel, but more recently carriages have been used. Royal Members of the Order usually join the procession, including The Duke of Edinburgh, The Prince of Wales, The Princess Royal, The Duke of York, The Earl of Wessex and The Duke of Cambridge.\n\nWhile The Queen is away at the service, I prepare her clothes and jewellery to be worn at dinner, and when she returns, I help her remove all of the regalia and the robe. Her Majesty will change into a skirt and blouse, then perhaps go for a walk with her dogs. This is only the beginning of what will be a very busy week. I then take the robe and the State Dress to the workroom at Windsor Castle, where I check the robe before I put it away, and also check and press the State Dress before it is returned to the wardrobe.\n\n_The beautifully plush Garter Day robe._\n\n_The plumed hat from both angles, in all its glory._\n\n_The Queen attends Ladies Day at Royal Ascot in 2019._\nROYAL ASCOT\n\nRoyal Ascot is one of the most anticipated Royal events of the year. The racecourse was founded by Queen Anne in 1871, and the horse-races officially became a Royal Week in 1911. Even some of the races are Royal-themed, such as the Windsor Castle Stakes and the King George V Stakes. It was in 1825 that the first Royal Procession took place, with King George IV leading the coaches. Of course, Her Majesty has quite a few of her own horses that have entered and won on a number of occasions. So, only in extreme circumstances would Her Majesty not attend, and it is one of those rare occasions on which The Queen can relax and enjoy one of her best-loved interests, often watching her own horses race. Royal Ascot takes place in June and is one of the busiest weeks of our year, and also one of the most secretive, due to an incident a few years ago that involved inappropriate betting on the colour of The Queen's hat. Designing something different for this special week has its challenges. It's all about the horse-racing \u2013 one or more of Her Majesty's horses may be running \u2013 so my designs have to be practical for The Queen to get into and out of cars and also the Royal Carriages.\n\nI usually start to plan for Royal Ascot in March, keeping in mind that many other events have been arranged, such as Easter Court, when The Queen stays at Windsor Castle, and the Royal Windsor Horse Show in May. Joel & Son Fabrics, the London fabric shop, provides some of the most beautiful material that is used in The Queen's clothing. They sell fabrics from across the world to the wider public and this is the shop I use most often to source what I need. Over the years Her Majesty has received fabrics as gifts, as I mentioned earlier, sometimes with beading and sequins, and also exquisite silks. Some are saved for future use, but during my years with Her Majesty, I have been able to use several such gifts in my designs.\n\nI buy material from other sources too: in 2018 I was on holiday in Sri Lanka with my family, still on the lookout for inspiring materials. The tour guide took our group to a shop called Silk Gardens Ltd. The ladies there were wearing their traditional dress and I was in my element: it was a wonderland of colour. I came across a lovely raw silk and bought a few metres in purple, aqua, and green, which were kindly shipped to London. \nAt Joel & Son Fabrics I see Dino, who has looked after me for many years. Expert that he is, he updates me on all the latest fashions and colours. He will show me special materials, which are of the highest quality, and also less expensive fabric that is just as beautiful and still of the highest quality. His colleague Linda helps me when I choose the buttons, in which she is an expert.\n\nWhile I'm looking at fabrics, ideas for how to use them come to mind, but I don't think in detail until I have a piece of material on a stand and I begin to drape it into a style that will suit The Queen. While the samples are being cut, I always peep into the bargain basket. The Queen loves a bargain and sometimes I luck out with a piece of material that I can conjure into a dress for her.\n\nWhen Dino has the samples ready, he parcels them up and sends them to Buckingham Palace. I prepare my suggestions for what I think would work, with three possibilities for each fabric.\n\nSometimes I arrange to meet Stewart Parvin at Joel's and we talk about what different designs we will each do for Ascot Week. We talk about the colours we're thinking of, so if he is going for blue, I might opt to use yellow to ensure we don't overlap. Even though we have our own design styles, which are completely different, we might share a collar design or button: our main goal is always, of course, to ensure that The Queen looks beautiful. Thanks to Stewart's advice and guidance I have become the designer that I am today. Thank you, Stewart.\n\n_The Queen and John Warren cheer on her horse Estimate at Royal Ascot in 2013._\n\n_Left: The smile that says it all. The Queen with the Gold Cup after Estimate won in 2013. Right: Ryan Moore rides Estimate to victory in 2013._\n\nI will have lots of ideas for styles and colours for spring and summer, but I must choose the right ones, so I next make an appointment to show Her Majesty the samples I've picked out for her approval. This is a busy time for The Queen, so I need to schedule this important meeting at a time when I know we won't be rushed or interrupted. I meet The Queen at the appointed time in her dressing room and we go through all the samples together. The Queen listens to my suggestions and also provides feedback from previous outfits: she might, for example, notice that a high collar risks rubbing against the skin if worn with a certain style of dress with a lower neckline. Once everything is agreed, I place my final order with Dino \u2013 enough of each material for a dress, possibly a jacket and coat, and an extra metre for the hat.\n\nWhen the materials arrive, I lay them out and start work on the outfits straight away. I plan the style of each one in my mind and play with the material on the stand, draping it one way, then another. I do this on my own, while flitting in and out between my regular daily tasks. I never draw or sketch the design at this point: I unite the colours and shapes and plan it in my head. I also think about necklines, collars, and hats to match the outfits. I make a few sketches, then arrange another appointment with The Queen to show them to her and ask for her approval. The Queen trusts me with the designs but I like to seek her advice. Her Majesty chooses a colour, or suggests colours that will complement each other. I then produce a final design and, depending on its level of detail, it can take up to four weeks before we have something ready for a fitting with Her Majesty. At this time of the year, I usually make ten outfits in total, five of which are for Ascot. Two outfits are kept as backup and the remaining three are saved for upcoming garden parties or for Her Majesty's annual visit to the Palace of Holyroodhouse.\n\nFor Ascot the hats are even more important than the outfits. Ascot is famous for ladies' hats! These bring their own challenge. Once The Queen and I are satisfied with the main fabrics for the dresses and coats, I will go to see Stella in the millinery room at the Palace and look through all the millinery fabrics we have in stock. If we don't already have what is needed, Stella will order it. She handmakes \u2013 no easy task \u2013 most of the silk flowers, leaves or sculptural details, although sometimes we source silk flowers from a respected German company. There is a lot to think about for each hat as I like my designs to be different every time. Stella and I chat about the hats, and with my designs and her suggestions we create something beautiful. And we keep the betting shops busy with people betting on the colour of The Queen's hat each year!\n\nWhen a hat and an outfit are finished, I put them together and double-check that I'm completely happy. There have been times during the making of the outfits and hats that I have changed my mind and altered them. I have to be 100 per cent sure they will suit and look perfect on The Queen.\n\nI show The Queen each outfit, with its hat, for her final approval, three or four weeks before Ascot. I take rails of all the new clothes \u2013 Stewart Parvin's as well as mine \u2013 with the hats to The Queen's dressing room. In addition to the hats designed by me and made by Stella to go with my outfits, Rachel Trevor-Morgan will make the hats for most of Stewart Parvin's outfits. Here, Her Majesty and I will go through them for any final tweaks. The Queen will choose an outfit in which she feels she will stand out among the crowds of racegoers, who will also be in their best clothes.\n\nOnce The Queen has agreed, I return the rails to the Dressers' Floor and the outfits are packed and stored until we are ready to leave for Windsor, usually on the Thursday or Friday before Ascot Week begins, depending on The Queen's other engagements.\n\nWhen I look back at all the outfits and hats I have designed, and the hats that Stella has made, one particular hat stands out for me. It is the one The Queen wore to the Royal Wedding of Prince William and Catherine Middleton, now Their Royal Highnesses The Duke and Duchess of Cambridge. Her Majesty chose the colour, sunshine yellow, three weeks beforehand. It was such a significant occasion that the whole outfit, designed under my Angela Kelly label, had to be spot on. To mimic the sun's rays, we cut the material for the coat larger than was necessary to allow for pleats to be pinched in and the matching hat \u2013 a matador shape made of cr\u00eape and sinamay \u2013 was trimmed with two handmade satin roses and leaves. It was perfect.\n\nRoyal Ascot Week is a lot of fun for everyone in my team. The Dressers' Floor is bubbling with excitement, with everyone hoping to catch a glimpse of Her Majesty's new outfits and hats. They are all sworn to secrecy, of course, to keep the element of surprise, so I don't worry if they do take a peek: I trust them not to let any cats out of the bag!\n\nIn Ascot Week, Windsor Castle is even busier than usual, as guests will be staying there. My team is on call for dressing duties; they help to prepare The Queen's clothes, ensuring each outfit is pressed, its hemline checked, its buttons secure, and that there are no pulls or snags. Between them there is much speculation about what colour The Queen will choose to wear on opening day. On Saturday The Queen goes riding, and will spend time with her other horses. On Sunday she attends church and has a quiet afternoon when she may take the corgis out for a walk before any guests arrive for dinner.\n\nMonday is Garter Day, and Royal Ascot begins on Tuesday. The atmosphere at Windsor Castle is electric. Each morning I place four or five previously worn hats on a table in the workroom. They are different colours and styles, and are on display for a reason. Anyone who happens to pass the room will see those hats. They are not hidden behind a closed door and no secret is made of them. This will stop anyone catching sight of the hat Her Majesty actually intends to wear and, with inside knowledge, betting a vast amount of money on the correct colour of The Queen's hat for opening day at Royal Ascot. This is cheating, and unfair on everyone else. It happened once, which was why this system was devised. Luckily, that year, the bookies clocked that something wasn't right with the bet in question, which was over \u00a32,000, and they put a stop to it. Unfortunately, this meant that all bets on that day were called off. So the person who thought they were being clever just ruined the fun for everyone else.\n\nI was horrified when I learned about the incident from the paper the next morning. When Ascot Week had finished, I attended a meeting with the owner of Paddy Power at which we agreed that betting on the colour of The Queen's hat would be closed at a certain time to avoid any cheating, but allowed people to carry on guessing the colour of The Queen's hat and perhaps even win a bit of money.\n\n_Bring on the rain, and the rainbow! The Queen has a clear umbrella with coloured trim that matches every outfit in case of rain \u2013 so she can still be seen._\n\nEvery day in Ascot Week starts with The Queen dressing in her morning clothes. After breakfast, Her Majesty will call for each of her Private Secretaries, one by one. After her meetings with them, she returns to her dressing room. Three outfits are hanging on the clothes rail waiting for her, the hats placed on a table near by.\n\nJust before lunch The Queen will choose from the three outfits. Even though the outfits for the opening day, Ladies Day and the final day have been selected, she will still have the two backup outfits to choose from in case she changes her mind. The final choice is made at the last minute but, believe me, that's still no guarantee, as the weather could change, which means the chosen outfit may not be suitable. Before she makes her final decision, The Queen consults the weather forecast to help her determine which outfit she will wear. Her Majesty would not usually wear a new outfit if it is raining as we would not know how the fabric will react to rain \u2013 it may shrink or crease \u2013 and instead may choose something she has worn before. If it is a clear, dry day, The Queen will wear a new outfit. If it is raining, she will use one of the many transparent umbrellas she has (each with a different-coloured trim to match any particular outfit she wears), which ensures that, even in the wettest conditions, she remains as visible as possible.\n\nWhen The Queen chooses her outfit for the day, she will change into the dress for lunch, then go to join her guests in the private rooms of Windsor Castle. Lunch is very quick so that everyone can be ready to leave for the racecourse on time. When The Queen returns after lunch, she fixes her lipstick, adds a bit more blusher, puts on her hat, and secures it with hat pins. Finally, Her Majesty puts on her coat, picks up her gloves and handbag, gives us all a smile, and then she's off!\n\n_The blue hat is the 'EU hat' \u2013 now changed to a bow instead of the little flowers. SeeThings Aren't Always What They Appear to Be for full story._\nThere is no waiting around and certainly no fuss, as everything is done within three minutes, maybe four at the most. The Queen departs from the front door and leaves in the car that will drive her halfway to the racecourse. Then Her Majesty and her guests get out of the cars and step into the Royal Carriages, which will take them to the racecourse and on to the Royal Enclosure. It is much quicker by car to reach the halfway point to the racecourse, and The Queen and her guests' carriages need to follow a strict arrival schedule: each race starts at a specific time.\n\nOnce The Queen has left for the races, I contact the Palace Press Office in London and provide them with the much-anticipated information on what Her Majesty has chosen to wear that day. This will cover all the details of The Queen's outfit, including the colour, material, and the style of each piece and the hat. These details are immediately passed to the fashion commentators at the racecourse so that the radio and television presenters can announce it as Her Majesty is entering the racecourse arrival. This is also the moment when those who placed a bet on the colour of The Queen's hat find out if they have won!\n\nMy team and I always watch Royal Ascot on the television in my office and we all admire The Queen's and the other ladies' and gentlemen's attire, and take inspiration from the designs and colours on show. We listen to feedback on The Queen's outfit and hat, and I take pride in our work. You can see, even on the television, Her Majesty's passion and excitement and, of course, the thrill of the race.\n\nWhen The Queen returns from racing, and depending on how comfortable she feels, she may keep on the same dress or change into something different. Eventually, the outfit she wore for Royal Ascot will be taken upstairs to be checked, pressed and hung in the wardrobe in the workroom. The hat is wrapped in tissue for protection and stored away. She will wear the outfit and hat once more later that year or the following year for another engagement.\n\nWith the day of racing over, it is time for the guests to retire to their rooms in Windsor Castle for a rest before dinner. In the meantime, the team and I will choose a dress for The Queen, and I will get out the jewellery for another night of sparkles.\n\nThe following day we start all over again \u2013 for the races and the evening dinner, the outfits for those occasions. Everything during Royal Ascot Week is orchestrated and timed so that it all runs smoothly for Her Majesty The Queen. I love it!\n\n_The Queen at Royal Ascot in 2019. When I saw The Queen on TV wearing this hat, I realised that the feathers were now dangling down too far on the right-hand side after travelling to the racecourse. I knew that would annoy The Queen, so I sent an urgent message to the Equerry to ask him to trim them down slightly, so they didn't come below the brim. However, the only scissors they could find at this short notice were enormous, so The Queen told him it would be fine to wait until she was back at Windsor Castle again. As soon as they returned later that day I was able to trim them properly with the right (sized!) equipment._\n\nFOREVER \nTHRIFTY\n\nSometimes, the special details that can take an outfit to another level are found in the most unexpected places. And often, they're the most inexpensive parts of an outfit.\n\nIn 2017, preparations for one of The Queen's Ascot outfits was going well. I had found some beautiful silk in the back of the stockroom with an eye-catching cerise floral pattern over an antique-gold background. I spend a lot of time in the stockroom, as new engagements come up often: in there I can plan and play with ideas. I thought initially that the flowers on that fabric were tulips but I soon learned that they were in fact called snake's-head fritillary \u2013 I'm always learning new things on the job. I was aware that Her Majesty had many outfits in cerise but decided it would be the best colour for the dress's accompanying coat and hat, as gold would not be appropriate for attending the races. Also, I knew that she would find a gold coat a little gaudy on this daytime occasion: it would appear too opulent.\n\nWith Ascot approaching, my team were making good progress with Her Majesty's outfit: the dress and coat were almost complete. However, although the hat from Stella was well under way, we had yet to decide on its trimmings. I had been pondering this for a few days, as I often do with more challenging designs \u2013 I'll often sleep on them and I find that an answer will present itself. I'd originally thought we could cut some of the fritillaries from the silk fabric to appliqu\u00e9 onto the hat, but had decided that this wasn't exciting enough for Ascot. I went looking for Paul Whybrew, The Queen's Page, for a solution. I found him in Her Majesty's storeroom, where something immediately caught my eye. In a corner, a modest plant pot contained snake's-head fritillaries in exactly the same cerise shade as The Queen's Ascot outfit! I couldn't believe it. Luckily for Stella and me, the fritillaries were made of silk, so once they had been well dusted, they could immediately be applied to Her Majesty's hat. A small detail such as this can really bring an outfit together. When I go shopping or out with family and friends to garden centres or department stores, I always visit the artificial flowers section in search of inspiration for the seasons ahead. It's always worth keeping your eyes open, as you never know where or when you might come across something useful!\n\n_Worn in 2019 \u2013 these silk flowers were a useful discovery!_\n\n_The Queen attends the Mass for the Order of the Thistle. Note that The Queen doesn't wear a necklace due to the cord which holds the robe together near the neck._\nTHE PALACE OF \nHOLYROODHOUSE\n\nThe Palace of Holyroodhouse is The Queen's official Royal Residence in Edinburgh, and each year at the beginning of July, just after Royal Ascot, Her Majesty will spend a week there hosting and attending many different engagements, depending on her programme. The Royal Household staff who travel to Holyrood always look forward to this week. To be able to say that you are staying in the palace where Mary Queen of Scots once lived \u2013 how fabulous is that? Edinburgh is a fascinating and beautiful city, and when the staff have a bit of downtime, they venture in for a couple of hours. The Royal Yacht Britannia is docked there and some of the Royal Household staff worked on it many years ago. They like to visit it and reminisce about the happy times they spent aboard.\n\nThe run-up to Holyrood week is another busy time for us on the Dressers' Floor. I give The Queen a selection of sketches to choose from that includes outfits for the Garden Party she holds there and the official engagements she will attend and host. For a week's stay, there can be up to twelve outfits. The sketches include evening dresses for dinners.\n\nOnce the sketches have been approved and the outfits made, I oversee their packing, checking each item as it is placed in a hanging bag, and then I review the matching hats, making sure that the trimmings are attached properly before my team packs them carefully into hat boxes. They also pack clothing for everyday wear \u2013 skirts and blouses or perhaps a silk day dress. Finally, I pack the jewellery that Her Majesty will wear with each evening dress, as well as appropriate brooches for the day outfits.\n\nEvery other year, there is one more important item that must be packed: the Thistle Robe, to be worn by Her Majesty at the meeting of the Ancient and Most Noble Order of the Thistle. The Order was established in 1687 by King James VII of Scotland, who was also James II of England. Membership, which is lifelong, is the highest honour a person can receive in Scotland, and, like the Order of the Garter, it is given personally by The Queen at an installation ceremony. There are just sixteen Knights and Ladies in the Order at one time, with 'extra' Knights who are members of the Royal Family. The Knights and Ladies are men and women who have held public office or have made a significant impact on Scottish society.\n\nThe United Kingdom is famous for its ceremonies and traditions, especially when they involve the Royal Family. Some date back hundreds of years and the sight of Edinburgh's Royal Mile lined with people from all over the world watching the grand procession of the Knights of the Order of the Thistle is not to be missed.\n\nThe Thistle Robe is made of dark green velvet, lined with white satin and decorated with the Scottish insignia. It is worn with a white feather-plumed hat. On the robe a star shows the cross of St Andrew, with a green thistle in the centre on a field of gold. The collar is decorated with thistles and sprigs of rue. The pendant is of St Andrew and his cross and is reversible. One side shows St George and the Dragon (worn on Garter Day) and the other side is St Andrew for this service.\n\nI pack two white State Dresses, one to be worn under the robe, with white gloves, silver shoes, and a matching silver handbag. I make the same preparations with the Thistle Robe as I do with the Garter Robe. I check to make sure the velvet has not been damaged and press the satin ribbons to make sure all the creases are out, then pack it into a hanging bag, ready for transport.\n\nWhen Her Majesty arrives at the Palace of Holyroodhouse, a small ceremony takes place on the forecourt: the Presentation of the Keys. The Lord Provost, who is in charge of the city of Edinburgh, welcomes Her Majesty and offers her the keys to the city. The Queen accepts them, then immediately hands them back to the Lord Provost so that he can keep them safely for her next visit. During this ceremony, The Queen will inspect the Guard of Honour and the band will play the National Anthem, marking the official start to Holyrood week. For this event Her Majesty wears a smart day outfit \u2013 so, a beautiful dress, coat, and hat would be chosen.\n\nWhile all this is happening outside, there is a scurry of activity inside The Queen's rooms as the Assistant Dresser and I unpack The Queen's luggage, making sure everything is in order as the week ahead will be extremely busy \u2012 and also highly enjoyable.\n\n_The luscious green Thistle Robe with the cross of St Andrew in pride of place._\n\nWhile at Holyrood, The Queen hosts a Garden Party to which as many as 8,000 guests will be invited. We always keep everything crossed for dry weather as everyone has to fit into the garden. The Queen will also hold an Investiture, audiences, or one-to-one meetings. The Queen will still meet with her Private Secretary and continue to work through her red boxes. Wardrobe changes come into play again, with between three and five every day: I have to make sure there are appropriate outfits available for each occasion for The Queen to choose from.\n\nWhen it comes to the service for the Order of the Thistle, the Knights and Ladies begin their grand procession in the Signet Library, with music played by the Band of the Royal Regiment of Scotland. The group makes its way to the Chapel of the Order of the Thistle where Her Majesty will install any new members of the Order. Afterwards the Knights and Ladies walk to the Thistle Chapel in St Giles Cathedral where the ceremony will continue. Later on, The Queen hosts a small reception back in the Signet Library, one of Edinburgh's most splendid Georgian houses, which is followed by a lunch at Holyroodhouse.\n\nThis event used to take place at the beginning of our visit to Holyrood, but it has now been moved to near the end of the trip, which means that my team and I have to pack the robe and insignia, along with the rest of the luggage in a very short time, so that we leave punctually for either Wood Farm or Buckingham Palace. Everything is on a strict schedule, so it's a challenge, having this big event on the last day \u2013 but we manage!\n\nDuring my stay at Holyroodhouse, I do manage to get a bit of a break between engagements, just long enough to pop into Jenners, the local branch of House of Fraser, where I can pick up any bits and pieces that we need. I visit other local shops, too, when time allows, as I like to keep my eyes open for something special or useful for The Queen.\n\nThe Palace of Holyroodhouse is open to the public throughout the year and it is well worth a visit. The rooms are still much the same as they were when Mary Queen of Scots was in residence during the sixteenth century. The place is steeped in history \u2013 imagine if the portraits in the Picture Gallery could talk!\n\n_The reversible pendant of St Andrew is steeped in history._\n\n_The Queen arrives at St Giles Cathedral, Edinburgh. Notice how beautifully the Page's uniform complements this robe._\n\n_The Thistle Robe hat and its impressive plumage!_\n\n_The Queen wearing the Imperial State Crown at the State Opening of Parliament._\nTHE STATE \nOPENING OF \nPARLIAMENT\n\nThe State Opening of Parliament is one of the most important annual engagements of The Queen's year. Of course, Her Majesty has many important roles and engagements, and she takes each one very seriously. The State Opening of Parliament, which dates back to the sixteenth century and in its current form to 1852, usually takes place in November or immediately following a General Election. This annual event officially marks the beginning of a session in Parliament, when Her Majesty will address it in The Queen's Speech, which outlines the laws the government hopes will be approved over the upcoming year. The Queen speaks in the House of Lords on behalf of Her Majesty's Government, with the speech written by the Prime Minister.\n\nA fun fact! During the State Opening of Parliament a hostage from the Houses of Parliament is taken just before The Queen leaves for Parliament and is held at Buckingham Palace. This tradition, which dates back to the reign of Charles I, is a reminder that the Sovereign and Parliament did not always share an easy relationship: the 'hostage' guarantees The Queen's safe return to Buckingham Palace. Rest easy, though: the 'hostage', who is usually the Vice-Chamberlain of The Queen's Household and a government whip, is entertained with tea and biscuits while he or she watches The Queen deliver her speech on television.\n\nPreparing Her Majesty for such a crucial day is by no means easy! There is a lot to do, as well as keeping on top of the normal day-to-day routine during the week that leads up to it, and I try to keep my diary free. A few days before the State Opening my hand-picked specialist team of Assistant Dressers and I prepare the State Robe. First, I ask The Queen's Footman, Ian Robinson, to bring me the long, heavy, airtight black metal box in which it is stored. The box is kept in a room along the corridor on the Dressers' Floor, which also holds the other State Robes, such as the Garter and Thistle Robes, in their boxes. The Queen's military uniforms are kept there, too. Once Ian has brought in the metal box, he then places it, with assistance, on a table secure enough to take its weight.\n\n_At Buckingham Palace, before heading to the Houses of Parliament. The State Robe is neatly folded in its velvet bag._\n\nWe open the box and remove the first few layers of tissue. Then we take out the large square red velvet bag, which is very heavy due to its gold tassels, and rest it over a bolster. The bag was made to carry the robe, once it has been maintained and folded, to the Houses of Parliament.\n\nThere is more tissue to remove before we get to the robe. When it was last used, it was placed on a roller, so we lift the robe, including the roller, onto the table, and the box is taken down and put away safely, so no one falls over it: it is solid metal and the corners are sharp.\n\nWe then unroll the robe over the table. Once it is fully opened, I look for any holes or bruises to the velvet and any damage to the gold-thread-embellished border. I inspect the collar and cape, which are made from ermine fur, to ensure they are in pristine condition. Then it takes four of us to carry the robe to the ironing board in the workroom, due to its length and weight \u2013 one person could not carry it properly. The satin ribbons, which are already stitched to the robe on the shoulders, are checked to make sure they are secure with no loose stitches \u2013 they help hold the robe in place on The Queen. The next step is to iron the ribbons \u2013 one person ironing and two holding the robe, letting the the rest of it lie on a chair for support. When all the inspections and ironing are completed (around thirty minutes), we leave the robe hanging in a warm room so that any creases in the velvet can fall out. Shortly before the event, we fold the robe and put it inside the red velvet bag.\n\n_Her Majesty's elegant fur cape._\n\nOn the day before the State Opening of Parliament, I select two State Dresses for Her Majesty to choose from. Before we bring them downstairs to The Queen, my team and I check our diaries from the past couple of years to make sure Her Majesty hasn't worn either of the two options I plan to offer. I try to keep the dress as light as possible as the robe is heavy. The heaviest elements of the robe are the velvet and the gold thread, not the fur. As always, I check the zip, buttons, hooks and eyes, necklines, hems, and sleeves on both dresses.\n\nWhile one of my team is selecting the handbag and shoes for each dress, I make sure that either a cashmere cape or a fur wrap is available for The Queen to wear over the dress until she must put on the State Robe. I will then choose the jewellery to complement the dress, making sure it fits well in the neckline. There are two special necklaces to choose from \u2013 the Collet Necklace, which has matching earrings, or the Jubilee Necklace. The Collet Necklace was created in 1858 for Queen Victoria and is also known as the Coronation Necklace. It is made up of twenty-five cushion-cut brilliant diamonds, which graduate in size, with a large diamond pendant weighing approximately 22.48 carats. Made in 1887 by Carrington & Co, it was presented to Queen Victoria by the 'Women's Jubilee Offering', a group of aristocratic women, to honour her Golden Jubilee. It features a large central pearl and diamond quatrefoil with a pearl and diamond crown at the top and a large pearl drop hanging at the bottom. The remaining necklace has graduated diamond and trefoil lines, six of which could work also as brooches. The jewellery is checked in the diaries: I alternate the necklaces and earrings every year.\n\n_Fastening the robe, using hooks and eyes and press studs._\n\n_Finishing touches._\n\nHer Majesty The Queen will always wear the Diadem, which is more a diamond headpiece than a tiara, but not a crown. It was originally known as the George IV State Diadem and was made for the monarch in 1820 by Rundell & Bridge. It consists of 1,333 diamonds, which weigh 320 carats, and contains a four-carat yellow diamond within the shape of a cross at the front centre: the cross patt\u00e9e. Along the base there are two strings of pearls, the top strand containing eighty-one and the lower one eighty-eight. It also features diamond-set roses, thistles and shamrocks.\n\nEvery queen and queen consort from the reign of William IV onwards has worn the Diadem. During Queen Victoria's reign, the Diadem was reset with jewels from the Royal Collection. Queen Elizabeth II wore it to go to her Coronation and always wears it during the procession to and from the State Opening of Parliament in her gold coach.\n\nMy team and I have an early start on the morning of the State Opening of Parliament, and the weather can usually be better \u2013 it never seems to be sunny on that day! First, the girls and I take the State Robe from where it is hanging and place it on the long table, where we fold it up very carefully. This job takes four people, and it has to be done in a particular order to ensure it fits into the velvet bag correctly.\n\nFirst, we fold the sides into the middle, then fold them once more into the centre, which makes the robe much narrower. Next, we take the bottom and fold it about 2 feet in. We keep making 2-foot folds until we almost reach the top. The top of the robe, the ermine cape and collar, is folded back so it is resting on the velvet folds. This enables me to position the ribbons carefully so they are not creased in transit. Now I can slip the robe into the red velvet bag. The gold tassels are pulled slightly to close it. Once this is complete, I place the bag outside The Queen's private rooms, where it is safe and ready for me to take to the Houses of Parliament later that morning.\n\n_Note the placement of the chairs to support the extraordinary weight of the robe as it is fastened on._\n\nWe are all raring to go.\n\nOnce I have woken Her Majesty and checked everything in the dressing room is in order for her to dress, I go back upstairs and have a quick cup of my favourite Yorkshire tea \u2013 my only cup of tea until we return to Buckingham Palace after the speech has been delivered. Shortly after this, The Queen will ring and I, along with the other Dressers, will go down to The Queen's dressing room where I will discuss with Her Majesty anything she requires for the day, after which The Queen will go for breakfast.\n\nThe final preparations can then begin. I bring the two dresses that have been chosen to The Queen's dressing room, while the Assistant Dressers prepare the shoes, handbag, cape, and gloves. I always choose the jewellery the day before, to save myself time in the morning, and also in case there are any changes to the programme. The jewellery always includes the Diadem, earrings, necklace, watch, bracelet, and rings. I lay everything on the jewellery tray that once belonged to Queen Mary. At this point I will take the jewellery away and give it a final clean and polish. A little gin and water come in handy to give the diamonds extra sparkle \u2013 just don't tell the jeweller! \u2013 and a drop of washing-up liquid and water will get rid of any hairspray stains. The Queen's Jeweller will give the stones a deep clean when necessary, so for me it's just a matter of a quick polish and they're sparkling once more. I return the tray and place it on a small table next to The Queen's dressing table. I will take a last look at everything and, satisfied that all is correct and ready for The Queen, I thank my team of Dressers, knowing I can confidently leave them to help Her Majesty get ready for this grand occasion.\n\nThe Queen's Page \u2013 either Paul Whybrew or Barry Mitford \u2013 and I then depart for the Houses of Parliament. The Queen's Page will carry the red velvet bag containing the robe, while I carry the famous Collar and the Great George of the Order of the Garter, which is worn for the State Opening of Parliament as well as the Garter Day service. We are taken by car along Birdcage Walk to Westminster; it is an exciting but nervous journey, as all I can think about is whether The Queen has everything she needs. Once The Queen gets into her coach, there is no going back. The ceremony is transmitted on live television, so there is no room for mistakes.\n\nThe Queen's Page and I arrive at the Houses of Parliament and make the climb up the long staircase to the Robing Room. Red screens stand open across the far corner of the room with just enough space for a dressing table, a small table and two normal-sized chairs. There is also just enough room for four Pageboys to lift the robe once Her Majesty is finally ready.\n\nWhen I am in the Robing Room waiting for The Queen's arrival, I am surrounded by beautiful tapestries, silks, and paintings illustrating the legends of King Arthur. It is majestic, breathtaking, and the excitement builds as we all wait for the arrival of Her Majesty.\n\nThere is a small television in the Robing Room, where we all watch The Queen leave Buckingham Palace. There is no sound, but we can see The Queen sitting in her gold coach, making her way towards the Houses of Parliament. I am still going over everything in my head at this point: was it all ready for The Queen as I'd planned \u2013 gloves, the right shoes, the correct handbag, the white fur wrap and the white cashmere cape as an alternative?\n\nI go behind the privacy screens and place two chairs in the middle of that space. Although the area is small, we always manage. There is just enough room to manoeuvre, to lay the Collar on the table and secure to it the Lesser George badge. I unfold the State Robe and place it over the two chairs. This is no easy task: the two chairs will take the main weight of the robe, which makes it easier for me to lift the front, the Collar and cap, which overlaps the velvet at the top of the robe, then place it on The Queen's shoulders.\n\nOnce Her Majesty has arrived, she joins me behind the screens, which I then close. The Queen removes the Diadem and her gloves and places them on the dressing table and I start to arrange the robe on her. We are silent, concentrating hard. The process of fitting it is complex and I have little time to achieve it \u2013 usually around five minutes. It is very difficult to get under the robe because of the way it is placed on The Queen's shoulders. I must lift the front and raise it on to my right shoulder, taking some of the weight, then place it above The Queen's shoulders. Her Majesty is then able to position the robe herself and manoeuvre it so it is comfortable. There are two fairly wide satin straps with hooks and eyes on each side of the robe, which are pulled under The Queen's arms, and hooked together tightly so the robe stays in place.\n\nOnce this is done, and with Her Majesty's help, I put the Collar over her head. The Queen always helps by holding it on the other side, making it much easier for me to lift it over her head. Once the Collar is placed on The Queen's shoulders on the ribbed under ribbons, and I am happy that it is in the correct position, with the Lesser George in the middle of The Queen's waist, I count four of the medallions from the middle to the shoulder, and tie the under ribbons in smaller bows to secure the Collar, ensuring it cannot move. I move on to the wider satin ribbons, taking one near Her Majesty's neckline and bringing it over the Collar and across to her shoulder. I feed this ribbon in and out through the Collar to secure it even more firmly, then tie a larger bow. The robe is now fitted and ready.\n\nOn one occasion, quite soon after I'd started working for Her Majesty, I just froze. I held up her robe as normal, hooked it up and lifted the Collar over her head. I counted four medallions up the Garter Chain and attached it. I was in the middle of feeding the ribbons in and out of the links of the Collar when I froze. I simply couldn't move. I don't know why it happened \u2013 I think I must have been overwhelmed by the significance of the occasion, or perhaps nerves got the better of me \u2013 but I could hear whispering outside the screened area, people wondering what was happening. I started to worry that I was holding up The Queen and the ceremony. Then Her Majesty touched my hand where it lay on her shoulder, and said to me quietly and gently, 'Don't worry, just take your time. Take your time.' I apologised profusely, of course, then took a deep breath, picked up that ribbon and carried on. Her Majesty had calmed my nerves and she was soon ready to go.\n\nOnce the robe is secured, I pull the red screens slightly apart, just wide enough for the Lord Great Chamberlain to present The Queen with her Imperial State Crown on a red cushion. The Queen takes the crown from him and moves slightly towards the mirror on the table, then I close the screens once more. Her Majesty places the crown on her head and secures it by twisting it from side to side. It is heavy, weighing just over 2 pounds (1 kilogram) and sits perfectly with no need of support or clips. This all takes place while the heavy back part of the robe continues to rest over the two chairs. Once The Queen is satisfied the crown is sitting comfortably on her head, she puts her gloves back on. I open the screen once more to allow the Pageboys, who have been waiting patiently in the room, to come in and take up the satin handles on the sides of the robe. The boys are lovely, although understandably nervous and sometimes quite shy, but they have been trained in what to do and always carry out their duties carefully and well. They are chosen from the Royal Family, the Royal Household and friends. As they lift the robe I move the chairs to one side, so that they can position themselves behind The Queen with no obstacles in their way.\n\nThe Queen will chat with the boys for a few moments to make them feel at ease, and while this is happening, His Royal Highness The Duke of Edinburgh will take his place at the right-hand side of The Queen. Prince Philip has undertaken this role every year except for in 2017, when The Prince of Wales escorted Her Majesty. Taking his arm, they walk together towards the large wooden doors. The first two Pages will start to move at the same time as The Queen so that the robe moves smoothly. The back two Pages step forward, keeping the robe off the floor, and follow carefully. They have to let the robe sink a little so it does not pull The Queen backwards, and they must keep to the same pace as The Queen and The Duke of Edinburgh. The Prince of Wales and The Duchess of Cornwall follow them, then The Queen's Ladies-in-Waiting.\n\nThe large wooden doors to the House of Lords' chamber open at exactly 11am and The Queen, The Duke of Edinburgh and the Royal entourage make their way to the throne. This is where Her Majesty will deliver her speech on behalf of her government.\n\nAs soon as they leave, I feel relieved, but only briefly: the very moment the doors shut behind Her Majesty I start to worry \u2013 will the crown stay on? Is the dress too long? Throughout the ceremony, I hold my breath and watch on the silent monitor, waiting for Her Majesty to come back through the doors safely so I can breathe again.\n\nOnce the speech is over, I have the screens open, waiting for The Queen, His Royal Highness and the Pages to return. Only The Queen and the Pages will enter the area behind the screens, and everyone else waits on the other side. I place the two chairs back in position before The Queen returns so that the Pages can return the robe to them, then the boys take their leave of Her Majesty and she thanks them. I close the screen behind the boys and Her Majesty takes off the Imperial State Crown, which she places back on its cushion.\n\nNext I begin to untie the larger ribbons and release the smaller ones from the Collar. Again, Her Majesty will help me lift the Collar over her head, bringing it forward so I can take the weight of the St George and Dragon badge and the chain, and then I place it on the table. Next, I manoeuvre the robe onto my shoulder and unfasten it from behind The Queen's arms, so that it can slowly slide off her shoulders and rest safely on the chair. And what a relief that must be! One would never know it, but it has been digging in under The Queen's arms for about twenty minutes from start to finish.\n\nI open the screen for The Queen to pass the crown and cushion back to the Lord Great Chamberlain. The Queen thanks him, he bows his head to her, and takes his leave, to return the crown to the Tower of London. The screens are closed once more. The Queen is now able to put on the Diadem again, her stole or cape and gloves. I open the screens one final time and The Duke of Edinburgh is ready to take The Queen back to Buckingham Palace in the Gold State Coach.\n\nOnce everyone has left I unscrew the St George and Dragon pendant from the chain and place it in its red velvet box. Then The Queen's Page and I take away the chairs and lay the robe flat on the floor. For now, I can fold it quickly so it fits well into the red velvet bag, ready to be correctly folded and carefully returned to storage later when we are back at the Palace and have more time. I gather together the bits and pieces from the dressing table into a small bag, and we are ready to go. The Queen's Page and I return by car to Buckingham Palace, looking at each other with relief. Phew! We are so glad when it's over.\n\n_The Queen and the Duke of Edinburgh walk through the Royal Gallery in the Palace of Westminster before the State Opening of Parliament in 2010._\n\n_Her Majesty delivers the Queen's Speech during the state Opening of Parliament in 2017. The hat has since been altered \u2013 seeRoyal Ascot._\nTHINGS AREN'T \nALWAYS WHAT \nTHEY APPEAR \nTO BE\n\nThe State Opening of Parliament in 2017 was delayed due to the General Election and was rescheduled for 19 June. That was the same date as Garter Day, which had to be cancelled that year.\n\nIt was also decided that Her Majesty would not wear her traditional State Opening of Parliament Robe or long dress, but would wear a day dress instead. The Queen has worn her Imperial State Crown every year, except in 1974 and on this particular occasion. The change was made because the engagements diary was so busy for everyone involved that a simpler form of the ceremony that would work for all those involved was put forward and agreed.\n\nI already had in stock some material suitable for Her Majesty's outfit, so after I'd checked the background against which The Queen would be sitting \u2013 to be sure that the colours wouldn't clash \u2013 I made the design and passed it, with my instructions, to the in-house dressmaker. Then the milliner, Stella, and I sat down with a cup of tea to discuss the design of the hat.\n\nWe chose a Breton-brim block style, which would create an upturned brim, so Her Majesty's face could be clearly seen, and The Queen could see everyone in the room. We chose a large square crown to balance the brim and five flowers, made from feathers with tiny seed pearls, painted yellow. It never occurred to Stella and me that people might think we were copying the European Union flag. The press and the rest of the media were quick to assume that was the case. I understand from reading the papers and seeing all the articles online that there was almost a meltdown on social media with people speculating that The Queen's outfit and hat paid tribute to the EU flag, and that The Queen was making a subtle statement about the recent Brexit vote. It was a coincidence but, boy, did it attract a lot of attention, and it certainly made us smile.\n\n_The Queen during the Royal Variety Performance in 2009._\nTHE ROYAL \nVARIETY \nPERFORMANCE\n\nAmong all the formal engagements that Her Majesty attends each year, there is one that has perhaps just a little more glitz and glamour than the others: the Royal Variety Performance. Founded in 1912, this televised variety show raises money for the Royal Variety Charity, of which The Queen is a patron, and features performances from comedians, singers, dancers, and sometimes even magicians. I know it is one of the highlights of Her Majesty's year, and in 2009, I had the honour of attending too.\n\nThat particular year, the event was held in the Grand Opera House in Blackpool and, as always, I wanted to create something special for Her Majesty to wear. In fact, I had decided that it might be fitting to design something a little bit different for The Queen, considering that the Royal Variety Performance is an evening unlike any other. I knew that I had some beautiful black material in the stockroom that would make the perfect evening dress \u2013 it sparkled and shimmered just the right amount and I knew that it would catch the light perfectly in the flashes from the photographers' cameras. With a wave-like striped pattern running vertically down the fabric, I knew the design of the dress needed to be simple so that the subtle yet beautiful material could be showcased. I was thrilled that The Queen liked my provisional designs as it had struck me that Her Majesty rarely wore black to glamorous occasions like this and it suits her so wonderfully.\n\nOn the night of the performance, I was so excited to be attending and I travelled to the Opera House with the Equerry and the Lady-in-Waiting. I began to get more and more nervous as we approached as this was one of the first engagements I had been invited to attend with Her Majesty and The Duke of Edinburgh. The Equerry reassured me and told me to relax and enjoy myself, but as our car pulled up, I was totally taken aback by how many people were outside on the street and nearly fell flat on my face trying to get out of the car elegantly in my long evening dress.\n\n_Ready and excited for the evening!_\n\nSafely inside the venue and in my seat, I was struck by how electrifying the atmosphere was. My seat was near the Royal Box and I had to keep pinching myself that this was real. As the evening went on, we were lucky enough to see performances from Katherine Jenkins, the dance troupe Diversity, and Michael Bubl\u00e9. However, I really could not contain myself when Ms Bette Midler came onto the stage. My heart stopped when she walked out. I was so overwhelmed with emotions. I had to remind myself where I was because I wanted to stand up and cheer. Composure, Angela!\n\nI knew Her Majesty loves music and that she would be much more composed than I was when Ms Midler sang 'The Wind Beneath My Wings', which always makes me so emotional as it reminds me of my family, who are truly the wind beneath my wings. It really was a special evening, so different to any of Her Majesty's other annual engagements and it was a thrill to be able to witness everything from right next to the Royal Box. Before Bette Midler sang she said, 'Tonight is a dream come true for me.' This was also my dream coming true, and what a night to remember.\n\n_The Queen meets singer Bette Midler and musician Jake Shimabukuro following the Royal Variety Performance. Performance in 2009._\n\n_The Queen attends St Mary Magdalene Church in Sandringham every Christmas Day._\nCHRISTMAS AT \nSANDRINGHAM \nHOUSE\n\nEvery year The Queen spends Christmas and New Year at Sandringham House, in Norfolk. It was bought by Queen Victoria and Prince Albert in 1862 for their son, The Prince of Wales, Albert Edward, as they wanted him to have a quiet place in the countryside where he could go when he needed a rest from his busy London schedule.\n\nThe Prince of Wales, who became King Edward VII, rebuilt Sandringham House as he needed a larger home for his growing family and more bedrooms for when he had guests. Throughout his life, The King continued to improve not only the house, but also the Estate. When he died, Sandringham House stayed in the Royal Family and is now home to Her Majesty The Queen, who thoroughly enjoys being there with her family throughout the Christmas period until after 6 February, the anniversary of her father's death. Like Balmoral, Sandringham is for holidays, although spending time at either house isn't all that relaxing as there is a lot to do, and many events to attend. The Queen is just as busy as she would be in London, with more guests to entertain. There may be several outfit changes in one day \u2013 it could be as many as five or even up to seven, although that is rare \u2013 depending on the number of planned engagements, as well as the weather and temperature, which can change so quickly around Sandringham. It can be beautiful one day and freezing the next. Thankfully, the House Foreman and his team ensure the house is well maintained, so inside it is always warm.\n\nI always use the same bedroom during my time there, so it's easy for me to settle in when I arrive. My room also becomes my office, so I have a desk and computer tucked away in the corner. I have a lovely view of the gardens from my window so I can see what the weather is doing; then, with the prevailing conditions in mind, I can choose appropriate outfits each day for The Queen.\n\nIn the mornings, before breakfast, The Queen will either put on a skirt and blouse or will choose to wear a day dress from her wardrobe in her dressing room. Once Her Majesty has had breakfast, she will meet with her Private Secretary or have a private audience. After the meetings, Her Majesty may change for lunch if there are guests staying in the house. Sometimes when a Royal Shoot is taking place, The Queen will attend the shooting lunch. This means she needs to change into her shooting gear, which is a lengthy process for anyone! This outfit is all about warmth and comfort \u2013 and the layers. First the trousers go on, then waterproof trousers over the top, then a long-sleeve shirt, jumper, and cardigan, thick socks so feet stay warm and dry and, finally, Wellington boots. Last, Her Majesty picks up her gloves, scarf, and waterproof mac to head out in the car to meet the shooting party.\n\nAfter lunch, which takes place at one of the lodges on the Estate, The Queen will return to Sandringham House and will either change into a skirt and blouse if she is taking the dogs out or, if guests are staying in, into a dress for afternoon tea. Between all of the wardrobe changes The Queen makes time throughout her day to deal with all the paperwork and answer letters that come to her in the red boxes. She is always busy at work and never seems to tire.\n\nIn the afternoon I lay out sketches of evening dresses for The Queen to look through and choose what she would like to wear that evening and, occasionally, she may have to pick out a dress for a cocktail party. Once Her Majesty has chosen her dress for dinner, a handwritten notice is pinned up in the Dressers' Corridor detailing what she will be wearing, so that The Queen's ladies' maids can select an appropriate dress for the lady they are looking after. When it comes to the Royal Family, it doesn't matter if they wear the same colour as The Queen because they are family, and sometimes the ladies will wear cocktail dresses even though The Queen might be wearing a long one. Other guests, though, feel they shouldn't be in the same colour as Her Majesty, although The Queen would not mind if this did happen. So we try to help The Queen's ladies' maids as often as we can. The Housekeeper makes sure the service given to all of The Queen's guests and staff is of the highest standard.\n\n_With the dogs after a walk at Sandringham in 1998._\n\nIf The Queen was hosting a cocktail reception for the Estate staff, Her Majesty would change into a cocktail dress. Nothing too sparkly, just a smart silk dress: The Queen would not want her guests to feel underdressed: they are likely to be wearing suits and tea dresses. However, if it is a larger cocktail party, it would be a more formal, sparkly dress. The guests like to dress up, too. The guests, who are usually friends and family, are given a bit of guidance as to what they need to pack before they arrive, although most of those who come to stay have been before and know what to expect.\n\nIf the reception goes on until 7.30pm and The Queen feels comfortable, she will remain in the cocktail dress and not worry about changing into evening dress. Alternatively, if she wants to change for dinner, Her Majesty will wear a long evening gown. Dressing for dinner is a tradition The Queen Mother kept up after the death of King George V, when she would wear a long evening dress, and The Queen has followed her example. This is when Her Majesty's beautiful jewellery comes out.\n\n_The Queen riding at Sandringham._\n\nOnce I know which dress The Queen will be wearing, I ask her if she would like to wear diamonds or pearls. In my first book, Dressing The Queen, I went into more detail about her jewellery, and how I make the decisions when offering her jewellery selection. So, if The Queen chooses a red dress, I wouldn't offer rubies because they would disappear on the dress, suggesting diamonds instead. If a dark or black dress was chosen, rubies are a beautiful match. If the dress is pale blue I would offer aquamarines or diamonds, or sometimes, just to soften the colour of the dress, I would offer a lovely set of pearls. It all depends on the colour and style of the dress and, of course, the cut of its neckline. Imagine having such spectacular jewellery to wear and knowing the history behind each piece.\n\nThe dress and jewellery are laid out for Her Majesty, along with a choice of shoes. If The Queen is wearing a black dress then black shoes are offered, or possibly silver. If the dress is red, I offer black or silver, but never gold as it would look too Christmassy, although if the dress had some cream or gold in the material, I might offer gold shoes.\n\nSo, Sandringham is much loved, especially during the Christmas season. The house is buzzing and I love to hear the children running around, laughing and playing. It looks magical, too, with the tree and all of the Christmas decorations.\n\nThe Queen attends church with her family every year on Christmas Day. Christmas is such a special time of year \u2013 everyone puts on their best clothes and, of course, The Queen is no different. There is so much media coverage, with pictures sent all over the world, and I take a lot of care in designing Her Majesty's outfit. I start to plan well in advance \u2013 around two months or so \u2013 checking to see what colours The Queen wore in previous years as I don't want to repeat a colour for a few years. I always think of the children and the other members of the public who are standing outside, waiting patiently for Her Majesty's arrival at the church. I like to make sure The Queen is wearing a festive colour so the well-wishers can see her easily. I do alternate years with Stewart Parvin, so we each get a turn making The Queen look lovely on Christmas Day.\n\n_The Queen attends the Christmas Day church service in 2017._\n\n_A warm, vibrant outfit for a chilly Christmas morning._\n\n_The Queen recording her Christmas Day message in 2018. The Duke of Edinburgh designed this brooch in 1966 for Her Majesty in the shape of a scarab, wrought in gold, rubies, and diamonds._\nTHE CHRISTMAS \nMESSAGE\n\nSo many people want to see what The Queen wore to church on Christmas Day but, of course, there are many more across the country and around the world who look forward to The Queen's Christmas Message. It's an important part of Christmas Day for many people \u2013 something to be enjoyed after a long lunch and a time to reflect on the events of the year. It's a privilege for me to be involved in this significant occasion.\n\nAs part of the preparations for the speech, I am in touch with the broadcasters a couple of weeks before filming. I've got to know them quite well over the years \u2013 at the BBC, ITV, and Sky \u2013 and we've developed a smooth process for making sure everything goes perfectly. Firstly, I ask what colours and decorations they're planning to use for the scene and they'll give me a detailed description, including which furniture will be featured and how the tree will be decorated. I then pick out a selection of outfits for The Queen that might be suitable, based on the colours chosen for the setting. Ultimately, I leave it up to the production team to decide on the final outfit as the camera crews and producers know what will work and what won't: red, for example, is difficult to capture on camera, and in green, Her Majesty might disappear into the background, depending on the positioning of the tree.\n\nIn 2012, to celebrate Her Majesty's Jubilee year, as well as the eightieth year since King George first broadcast his speech over the radio, the production company suggested that it might try something a bit different. The Queen is always willing to embrace technology, so it was agreed that for the first time the Christmas message would be broadcast in 3D. On the day of filming, nothing was noticeably different with the cameras, and Her Majesty was as professional as ever, with the speech captured in one take.\n\nIt's a very rare occasion when someone else decides what The Queen will wear, particularly on such a significant day! But I trust the television crews and we work brilliantly as a team. Of course, I'll discuss their choice with Her Majesty, but if I'm assured that it's the right decision, she is satisfied. The only other thing left for me to do is to arrange for the make-up artist, Marilyn Widdess, to be there on the day of filming. You might be surprised to know that this is the only occasion throughout the year when Her Majesty does not do her own make-up.\n\n_Queen Victoria at the christening of her great-grandson, the future King Edward VIII, with the baby's mothers, Mary of Teck, and grandmother Alexandra_\nTHE CHRISTENING \nROBE\n\nI would like to dedicate this story to Barbara Buckfield, better known as June\n\nEvery day I am concerned, first and foremost, with the design and maintenance of Her Majesty's wardrobe. The Queen's outfits are always my top priority, but now and again the opportunity arises for me to assist with the Royal wardrobe in a more unusual way. Needless to say, it's always a pleasure to help with an exhibition of iconic Royal gowns, but a few years ago I had the honour of undertaking a once-in-a-lifetime task.\n\nAfter 163 years of use, the famous Royal Christening Robe had been worn by sixty-two members of the Royal Family since its first outing in 1841 at the christening of Queen Victoria's eldest daughter, Princess Victoria. Made of Spitalfields silk and intricate Honiton lace, it was an extremely delicate piece that had historically never been washed after each wearing, other than a light spot clean with fresh spring water. After the christening of Lady Louise Windsor, the daughter of the Earl and Countess of Wessex, in 2004, I had noticed that, after so many years of use, the robe had snagged on bracelets and watches, and had acquired quite a few stains along the way. As I was preparing to put it away, I mentioned to Her Majesty that if it was worn again it might become irreparable. This evidently stayed with The Queen because the next time we were alone together she asked if I would make a replica of the robe for future use so that the original could be locked away safely.\n\nTouched that Her Majesty had entrusted such an important task to me, I set about making a plan. My first task was to examine the existing robe in detail, so I removed most of the furniture in my workroom except for a good-sized table and laid the robe carefully on top. My first thought was that it was the most beautiful garment I had ever seen. The detail that had gone into the different sections of lace was extraordinary, particularly considering that a square inch of Honiton lace can take ten hours to make. The sheer quantity of lace, too, was amazing, delicately folded to give the effect of a full skirt beneath the magnificent bodice and sleeves. At the time, due to the available lighting and other working conditions, it would have taken at least a year, perhaps even eighteen months, to create. I had heard that it was styled after Queen Victoria's wedding dress and I could see the resemblance. It was heartbreaking to see such a beautiful creation deteriorate, and I was determined to produce something equally lovely that could be worn by many future generations to come.\n\n_The Queen Victoria Christening Robe. Not all of these elements (the cape, the bibs, the medallions, and the ribbons) have been used as often in more recent times \u2013 mainly the robe and the bonnet._\nThankfully, there was no news of any Royal babies in the near future, as I knew it would take several months to source the finest materials, draw the designs, and create the robe. The project was to be kept top secret, so it was a happy coincidence that a new dressmaker joined our team and turned out to be the perfect ally. June Buckfield, whose real name was Barbara, had applied for the position after working for Jaeger. She was due to retire after many years with them, and after I'd interviewed many candidates, it was obvious that she was the right person for the job, bringing valuable experience to the team, as well as her workmanship, and I knew that her personality would fit right in with the rest of us. Shortly after her arrival, I told her about the secret task of recreating the Christening Robe and she was thrilled to be involved. However, shortly afterwards, June was sadly diagnosed with terminal cancer and needed treatment immediately. I made sure to let her know that as soon as her treatment was over she could come straight back to the Palace to help me with the robe.\n\nFirst I had to find a lace-maker to replicate the main bodice exactly, as this would undoubtedly be the most time-consuming material to produce. I required three types of lace but the majority was to be used in the full, layered skirt of the robe. There was also to be a slight alteration to the frill, at the request of The Queen. I initially spoke to a well-known company based on the outskirts of London but unfortunately, and much to my surprise, they were not very interested and quoted an enormous amount of money. Knowing that Her Majesty is always mindful of cost, I approached Joel & Son Fabrics; the owner, Mrs Bull, and her colleague Linda told me about a small family business in Italy called Vema that specialised in making the finest lace and embroidery. I made a call to Paola Mazzucchelli and asked for a quote based on the quantity I would need. Paola gave an approximate cost, which was much more reasonable, but said she would need to see the garment to make sure the firm could commit to the project.\n\nHer Majesty was happy for me, Mrs Bull, and Linda to travel to Italy to visit Vema, but before we went I wanted to take my own measurements of the robe. I knew the lace-makers would be taking their own exact measurements but I wanted to make sure they were recorded in the Royal Household, too, to help future generations when the time came for another replacement to be made. Measuring a complex garment like that can be difficult; first I had to find its centre point, a length of 70 centimetres from top to bottom, and then I broke the garment down into five sections and rebuilt it to form a gradual gradient across the front. Centimetre by centimetre, I meticulously took note of the dimensions, which was a very useful exercise in understanding how the robe had been put together.\n\n_The Queen Elizabeth II robe - note the ribbon is completely handstitched._\n\nBefore long, Mrs Bull, Linda, and I were on our way to Italy. Three ladies on a secret mission. When we arrived, after settling into our hotel, we went to Vema where we met Paola and her family. We got straight down to business and I showed them the Christening Robe, which I had discreetly brought, wrapped in lots of tissue paper, in a large handbag. When the family saw it, I could tell that they appreciated how special and delicate it was. In fact, they were in awe and held it with the utmost care. Seeing how respectfully and gently they handled it, I knew I had made the right decision in approaching them.\n\n_The two robes have never been pictures together before. The Queen Victoria robe is above, and the Queen Elizabeth II robe is below._\n\nWe were shown to their enormous workroom, which contained rows and rows of looms, each around 15 metres long. This was ideal as the robe would require 3 metres of lace, so we would have plenty of excess for repairs. With everything in place and the details agreed, we flew home. I remember thinking how extraordinary it was to be sitting on a plane with a garment made for Queen Victoria's children.\n\nHaving reassured The Queen that everything was going smoothly, I set about my next task: sourcing additional designs of lace to cover the bodice, sleeves, and trim of the robe. Happily, by this time June had come back to the Palace to join me in the enormous task of recreating the robe. Together, we sourced lace to complement that being made in Italy, and to make sure it looked authentic, we dyed it in Yorkshire tea (which is the strongest, as we all know!). One by one, we placed each piece of lace in a small bowl, from the Dressers' kitchen, filled with cool water and a tea bag, and left it for about five minutes, checking regularly until the colour was perfect. With the lace ready to sew onto the robe, the only material left to source was the duchesse satin to go beneath the lace skirt. Once again, Mrs Bull came to the rescue and, with the help of Dino, at Joel & Son Fabrics, found the perfect satin and tea-dyed it to match the lace June and I had worked on.\n\n_We had to use tea to dye it to get the new robe as close as possible to the original colour, so it wasn't a 'brand new' bright white._\n\n_We actually made the new robe a little bigger than the original to allow more room as most babies are born a little bigger now than they were back then!_\nNow that we had sourced all of the necessary materials and used plenty of tea bags, we were ready to start the painstaking process of producing the robe. Vema had delivered the most exquisite lace, so we were ready to go. I gave June the task of all the hand-stitching, as her attention to detail was second to none. A heavy lace panel edged with additional frilled lace was to be attached to the front, with a wider lace frill sewn on which reached around to the back of the gown. Around the neck, a fine millimetre-wide drawstring ribbon and three tiny 6-millimetre buttons were to be attached and buttonholes made. At the bottom of the bodice, just before it meets the skirt, June had to apply two rows of 3-millimetre satin-edged lace to the finish, which encased the same narrow ribbon as that used around the neck to allow the robe to be pulled in at the waist. Finally, the bottom of the gown was to be finished with a scalloped border, slightly frilled, which measured no more than 7 centimetres in height. At each stage of the creative process, I would show our progress to The Queen: first the bodice, then the sleeves attached to the bodice, then the skirt with the under-layers on, and finally the completed robe. Her Majesty was very interested to see how it was developing. So much complex and intricate work was involved that it was a relief to us all when the robe was finally complete. From start to finish, it had taken us, appropriately, nine months. Since completion, it has been worn by James, Viscount Severn, son of the Earl and Countess of Wessex, Savannah and Isla Phillips, the children of Peter Phillips (The Princess Royal's son), Maud Windsor, daughter of Lord Frederick Windsor, Mia and Lena Tindall, the children of Zara (The Princess Royal's daughter), Prince George, Princess Charlotte, and Prince Louis, the sons and daughter of the Duke and Duchess of Cambridge, and most recently by Archie Harrison Mountbatten-Windsor, the son of the Duke and Duchess of Sussex.\n\nI cannot describe how touched I was that Her Majesty asked me to create a replica of this historic robe, and trusted me to take the original to Italy. It is an important garment to her, with lots of personal memories. I feel honoured to have created something that will be worn by future generations of the Royal Family, but it was daunting, making sure that I got it absolutely right. On a personal level, working alongside June, who was caring, warm, dedicated, and an expert seamstress, made the experience all the more special.\n\n_I'm holding the bonnet. The Queen Victoria robe is on the left and the Queen Elizabeth II robe is on the right, and the size difference between the two can be clearly seen._\n**'VEMA is a small embroidery company, founded in 1969 by my father, Venanzio Mazzucchelli, who was one of the first Italian entrepreneurs to purchase Swiss embroidery machines in Italy. As a company that manufactures high-end, bespoke embroideries, we often receive 'special orders'.**\n\n**One day in 2004, we received a call from our good customer Joel & Son Fabrics in London, asking if we could take part in a special project to replicate an ancient fabric \u2013 a very important undertaking that was to remain confidential. Of course, we agreed at once with enthusiasm. When we were informed that we would be visited by the Queen's Dresser to replicate the Royal Christening gown . . . well, the surprise couldn't have been bigger. **\n\n**When Angela pulled out the dress, no one dared to touch it. The magnificent hand embroideries were over 160 years old and although still well-kept, were worn out by usage. The fine yarns crafted by skilled workers had created a masterpiece that seemed impossible to replicate. In the weeks that followed, we did several trials and in the end, we used a technique called superpos\u00e9 which is basically a double embroidery, that gives a three-dimensional effect.**\n\n**Months went by, and then years, and the project remained as a folder in my dad's desk, together with a picture of him and Angela, of which he was so proud. And then one day, in 2013, I received a phone call from Buckingham Palace \u2013 the employee who picked up the phone thought it was a joke. It was Angela, announcing that the gown was to be presented officially for the Christening of the Royal heir, Prince George.**\n\n**Dad was so proud that Angela invited me to see the robe in Buckingham Palace. He passed away in 2015. At every Christening of the British Royal Family, there is a small fan club here in our offices, sharing pictures and, of course, memories. It makes us feel that we took part in a small piece of history.'**\n\n**_By Paola Mazzucchelli_ \nVema Ltd**\n\n_The Queen's Page, Paul Whybrew, does his duty alongside Daniel Craig and The Queen during filming for the opening ceremony of the London 2012 Olympic Games._\nGOOD EVENING, \nMR BOND\n\nThe London 2012 Olympics Opening Ceremony was full of colour, sound, light, and spectacular performance. It took place in The Queen's Diamond Jubilee year, right in the middle of an extremely busy period. However, this event stood out. Thousands of volunteers worked seamlessly, creating perfectly choreographed patterns across the vast floor of the Olympic Stadium. Amid the kaleidoscope of activity, there was one moment when the stadium's 62,000 occupants held their breath in stunned silence: Her Majesty, dressed in a peach, crystal and lace cocktail dress, apparently took a daring leap from a helicopter, skydiving towards the stadium 500 feet below. What an entrance!\n\nThere are few occasions on which Her Majesty will agree to break protocol, but in 2011, when film director Danny Boyle approached the Royal Household, he had a request to make that we simply could not refuse. I first heard about Danny's grand plan when Private Secretary Edward Young called me into his room: he wanted to run an idea past me, as well as a thought for a sensational arrival at the Olympic Stadium. Danny wanted to ask Her Majesty to feature in a short film as part of his Olympics opening ceremony. She was to be greeted by James Bond, played by Daniel Craig, in the Palace and seemingly escorted to a helicopter. Mr Bond was there to rescue her from an unknown threat that jeopardised her safe arrival on time at an undisclosed event. Once safely aboard, the helicopter would fly over the Houses of Parliament, receiving a salute from Churchill's statue in Parliament Square, before passing beneath Tower Bridge and swinging north to the Olympic Stadium in Stratford, London. Here, the helicopter door would slide open and 'Her Majesty' would take a daring leap out, skydiving down in perhaps her most dramatic entrance to an engagement ever.\n\nHaving listened to Danny's plan, I asked him and Edward to give me five minutes so that I could ask The Queen. I remember the look of shock on Danny's face that I would be asking Her Majesty straight away, but there's no point in waiting around with these things: if she said no, that would be the end of it. I ran upstairs and luckily The Queen was free so I asked if she would be prepared to do a surprise performance for the Olympics opening ceremony. She was very amused by the idea and agreed immediately. I asked then if she would like a speaking part. Without hesitation, Her Majesty replied, 'Of course I must say something. After all, he is coming to rescue me.'\n\nI asked whether she would like to say, 'Good evening, James,' or 'Good evening, Mr Bond,' and she chose the latter, knowing the Bond films. Within minutes, I was back in Edward's office delivering the good news to Danny \u2013 I think he almost fell off his chair when I said that The Queen's only stipulation was that she could deliver that iconic line: 'Good evening, Mr Bond.'\n\nMore than perhaps any other public appearance by Her Majesty, this would take meticulous planning. One crucial factor was the dress The Queen would wear for the momentous skydive. It had to be a bright, solid colour that would stand out as 'Her Majesty' descended into the stadium and also while she was in the stadium surrounded by the vibrant celebrations. I had to be careful not to choose a colour that was strongly associated with any of the participating nations, too, so after much consideration, I chose peach. However, the colour wasn't the only factor, the dress also had to be designed perfectly to allow movement in the skirt as The Queen seemingly flew through the air. Of course, there would be two: one for Her Majesty and one for the jump stuntman, Gary Connery.\n\n_Skydiving material \u2013 note the bloomers detail to help the stuntman's modesty in the air._\n\n_The Queen's show-stopping dress._\n\nWorking closely with Danny's team and the Olympic opening ceremony's costume department, I set about creating two identical versions of the peach dress. For months, the Palace dressmakers worked continuously and discreetly, never allowing both dresses to be visible at the same time. Even other members of the Royal Family were unaware of the plan, so I made up a story that two dresses were needed for an upcoming event, as one was to go on exhibition and I didn't want it to be stained or marked. After much hard work, we eventually finished the stuntman's dress and I delivered it to Danny's team to add the harness and parachute \u2013 certainly a first in my years as a dressmaker!\n\nOn the day of the filming, I was privileged to watch Her Majesty undertake her role. She was unflappable and even kept 'Mr Bond' waiting for a moment while finishing a letter she was writing. The Queen's Footman, Ian Robinson, outdid himself in his role of animal handler \u2013 he managed to align Her Majesty's corgis in the perfect position, using dog biscuits to grab their attention and give the impression that they were watching The Queen's helicopter flying away. The night of the ceremony, of course, was suspense-filled as we all watched to see if Gary, wearing that dress, would land safely just outside the stadium. He did, of course, and Her Majesty resumed her role, walking calmly and coolly to her place after the apparent death-defying leap. I believe she received a standing ovation and it was the perfect contribution to Danny's spectacular vision for the opening of the Games.\n\n_Gary Connery parachutes into the stadium._\n\n_Julian Calder's photograph_ , Chief of the Chiefs.\n\n_Action!_\nOUR OWN \nMONARCH \nOF THE GLEN\n\nHer Majesty has sat for countless portraits throughout her long reign. Most photoshoots are quite similar, but occasionally an unusual request comes through. I'm always happy to see if Her Majesty might be up for something new.\n\nBack in 2010, Alastair Bruce, a historian and author, had asked Her Majesty if she would consider being photographed by Julian Calder for inclusion in their upcoming book, Keepers: The Ancient Offices of Britain. I wasn't at the meeting with Alastair and The Queen, but he came to see me afterwards to explain his suggestion that Her Majesty be pictured on the Scottish moors, near Balmoral. I imagine The Queen had immediately thought about the obvious potential challenge \u2013 the Scottish weather. One minute it could be glorious sunshine, but the next there's torrential rain, gusty winds, and thick fog. Alastair asked The Queen if she would think about it for a while, and Her Majesty replied maybe.\n\nAs Her Majesty pondered, Alastair approached me to discuss their vision for the outfit she would wear on the moors. Would it be possible for her to wear the Thistle Robe and the Vladimir Tiara as she stood among the Scottish heather? The Vladimir Tiara dates back to 1880, when it was commissioned for the Grand Duchess Vladimir. In 1921, Queen Mary, our Queen's grandmother, had bought it from Princess Nicholas of Greece, the Grand Duchess's only daughter. In 1953 Her Majesty inherited it from Queen Mary. This stunning tiara has fifteen intertwining diamond circles, with either a pearl or emerald suspended from the centre of each. I thought that emeralds would perfectly complement the deep velvet green of the Thistle Robe.\n\nHow might Julian and Alastair's plan work if The Queen agreed to go ahead? Her Page, Paul Whybrew, had been told of the request, too, so we put our heads together to find a way to make the iconic photoshoot happen. But, of course, ultimately it was Her Majesty's decision. So, being brave, I asked again whether she might consider it, taking care to explain the plan. If Her Majesty got ready in the Castle, we could drive out in her car, stop on the moors, take a few pictures and drive back. The Queen looked at me as though I was crazy. She reminded me that it might easily be pouring with rain or blowing a gale. On top of that, the drive could take up to twenty minutes along bumpy roads \u2013 it was not a matter of nipping over to the heather-topped moors. Crossing my fingers, I told her that it wouldn't be raining, and Her Majesty agreed to go ahead. A day was fixed for the photoshoot and now all we had to do was pray that the weather would be kind.\n\n_Notice the heather placement._\n\nThe day before the photoshoot, Paul, Alastair, Julian, and I went out onto the moors to check the location that would work best. Julian and Alastair wanted Her Majesty to stand on a rock, which had a nearly, but not quite, even, flat surface. Returning to the Castle, I went to explain to The Queen that the location for the shoot wasn't too far away and that she would be standing on a slightly uneven rock, but that she would be fine. A couple of photographs, and we'd be on our way back to the Castle.\n\nThe day of the photoshoot arrived and we awoke to heavy rain showers. Her Majesty said firmly that she was not going out on the moors in that weather \u2013 we would all be soaked through, and the rain would ruin the beautiful velvet Thistle Robe. After a couple of hours, though, blue sky began to break through the grey clouds \u2013 the showers were passing. The Queen called for me and said, 'If I do this, we go now.'\n\nIt was all hands on deck. I alerted the others and helped The Queen to get ready. In her long gown, tiara, and Thistle Robe, she got into the car and I can only imagine what Her Majesty was thinking as she wrapped and tucked the robe around herself so that she could sit somewhat comfortably. Her face was a picture.\n\nPaul and I followed Her Majesty in the car behind and it only took a short time to drive to the chosen location. Paul, Julian and I helped The Queen out of the car, ensuring that the long robe didn't drag along the rough ground as she walked to the rock on which she was to stand. Some purple heather surrounded it but not quite enough \u2013 Paul and I quickly unearthed a few clumps, making sure to keep the soil around the roots, and placed them nearer to Her Majesty. The beautiful green robe was placed in such a way that it gave the impression of flowing down the grassy moors. Holding the Thistle hat, The Queen looked so elegant and regal. It was moving to see our Sovereign standing amid the stunning Highland scenery.\n\nAs Julian took his photographs, The Queen kept shooting pointed looks at me and Paul, as if to say, 'Have you seen the sky?' It was getting darker by the minute. Eventually Julian was happy that he had the shot he wanted and, job done, we all burst into giggles of relief \u2013 even The Queen. Her Majesty removed her tiara and passed it to me with a look that said, 'Are you happy now?', followed by a twinkle in her eye and a beaming smile. We packed up quickly, replanted the heather in its original position, jumped into our cars and the heavens opened as we headed back to the Castle.\n\nLater that day when we were all in The Queen's private rooms, it was clear that she thought the shoot had gone well: and with a grin, she said, 'I really enjoyed that!' Nevertheless, I wouldn't push my luck and ask if she would ever do it again!\n\n_Last-minute adjustments \u2013 I'm ensuring everything is exact._\n\n_I was thrilled to be involved in the iconic Annie Leibovitz photoshoot in 2018._\nBEHIND THE LENS: \nANNIE LEIBOVITZ\n\nAnother shoot was particularly memorable, one that captured The Queen in a way that left me awestruck. Back in 2007, the Royal Household commissioned photographs to celebrate The Queen's State Visit to the United States and they wanted this to take place at the same time that Jamestown, Virginia, was celebrating the 400th anniversary of its founding. Annie Leibovitz became the first American photographer to take an official portrait of The Queen . . . four portraits, in fact.\n\nAnnie was asked to take four different images of The Queen. In the first, she would be wearing an evening gown with a stole wrapped around her shoulders, sitting down and looking out of an opened window. For the second, she would also be in an evening gown, but wearing a tiara and the Garter Robe, seated again. In the third photograph, she would stand in an evening gown, with a stole draped over her arm. The fourth was to show The Queen wearing her boat cloak that was made famous in Cecil Beaton's powerfully simplistic portrait of Her Majesty, taken in 1968.\n\nFor this photoshoot, Annie had a little more freedom than is usual through which to exercise her creativity, and I much enjoyed working with her. I could help by making sure the robe was positioned correctly, when The Queen put it on, so that it showed some of the white silk lining.\n\nWhen Annie was taking the photograph of The Queen wearing her stole, Her Majesty's pose resembled one of The Queen Mother's, with the fur draped over one shoulder. It didn't look quite right. Instead I suggested that in the picture The Queen might carry it over her arm, as if she was ready to go out but didn't need to put it on yet. Annie agreed, and this became the final shot. Annie was more than happy for me to advise, but that was as far as my role went: I knew Annie was an experienced, professional photographer. She had her own vision, which she captured in the photos she produced. The outcome was amazing.\n\nOne of the good things about working in the same job for such a long time is that it allows you to put things right that haven't gone quite as planned in the past. I'm often reminded of this when I think of that photoshoot. Annie had travelled over with her daughter, who would be introduced to The Queen before the portrait sitting. On the day, Her Majesty was scheduled to meet only Annie's daughter and was expecting just a handful of people to be present. When The Queen arrived she was met by nearly fifteen people in the room, standing in a straight line, and many members of the press. A misunderstanding meant that footage was captured and broadcast showing The Queen apparently 'storming' out of the photoshoot. This was not an accurate account of what happened. Her Majesty was not 'storming' anywhere: she was making her way to the shoot as planned, and hadn't time to meet so many people. I have always felt so bad for Annie about how this was later reported, as she really didn't deserve to be misrepresented in that way.\n\nAlmost nine years later, The Queen and I had a conversation about how we could make things right. Her Majesty's ninetieth birthday was coming up, and I suggested that we invite Annie back to do the official photographs. The Queen was more than happy and in full agreement that it would be the right thing to do to make amends for the earlier misunderstanding, and said yes.\n\nHer Majesty and I discussed different ideas for the photographs. I suggested that Annie should keep things simple and that the shoot should be informal, based around The Queen and her family. I suggested four photographs that should be taken at Windsor Castle: The Queen with His Royal Highness The Duke of Edinburgh; The Queen with all of her great-grandchildren; The Queen with her corgis; and one of The Queen with her daughter, The Princess Royal. I felt that The Queen hadn't been photographed often enough with her daughter and, as I have a daughter myself, I thought it would make a special photo for them to share as mother and daughter. The Queen wholeheartedly agreed.\n\nSoon after, I flew to New York with my assistant, Jackie. I'd never been there before so I was very excited, and we went to see Annie and her team in her studio. When I told her that The Queen had asked for her to return to photograph her on or close to her ninetieth birthday, Annie was thrilled and humbled to receive such an honour.\n\nIt was great to see Annie and her team again. After we had spent a little time reminiscing about the 2007 shoot, we got down to work. I explained to Annie how the coming shoot would have to work. Time was limited as the entire Royal Family would be together only during the Easter weekend. We just had a few months to plan everything in detail. I even had the contracts with me for Annie to sign there and then if she agreed. I didn't have any time to waste: everything had to move quickly.\n\nA few weeks later, Annie and her team flew to London to do their own recce with us. We were all so excited and looking forward to seeing the photos. We prayed that the weather would be in our favour on the day. When the day of the shoot arrived, everything went according to plan. Annie captured a stunning image of The Queen and The Duke of Edinburgh, and another of Her Majesty with her daughter, The Princess Royal. The great-grandchildren were all so well behaved, and I think even they were excited about having a group photograph taken with The Queen. But the real stars of the show that day were the corgis! They knew how to work that camera, looking straight at Annie while she was taking the photos. I think they even enjoyed it. I'm sure they were given plenty of special treats afterwards by Ian.\n\nThe Queen was delighted with the pictures, which were special family photos to be shared with the public.\n\n_Windsor Castle in all its majesty._\n\n_A secret wish come true._\nTHE QUEEN \nDOING IT \nHER WAY\n\nI get asked all sorts of questions about my job, with it being unusual \u2013 unique, actually. And, of course, everyone is keen to hear about what happens behind closed doors, especially when it comes to Her Majesty, who is beloved in this country and across the world. I'm very careful to respect The Queen's privacy, of course, but one thing I always tell people is just how considerate and thoughtful Her Majesty is \u2013 she is always willing to help, never wants to let anyone down, and will always do her best to make sure everyone is content. So, a few years ago, when I had the opportunity to do something special for The Queen, I leaped at the chance.\n\nMany years ago, Her Majesty disclosed something to me \u2013 a secret wish that she'd held since she was young. Throughout The Queen's time on the throne, she has been photographed in countless formal ways. However, for a long time, Her Majesty wanted to be photographed more informally and have the freedom, for example, to pose with her hands in her pockets. The Queen Mother and her advisors had always advised against this, suggesting it would not be appropriate, and I believe she has never done this before.\n\nSome years later, when I was writing my first book, Dressing The Queen, I remembered Her Majesty's secret wish. Given that the book was all about Her Majesty's outfits and relationship with fashion, I wondered if I might have a wonderful opportunity to make The Queen's wish come true. So, although I was very nervous, I finally asked the question: 'Your Majesty, would you do me a favour? I would like you to consider doing a photoshoot of yourself modelling a dress with your hands in your pockets. It would only be yourself, me, the photographer and his assistant.' The Queen looked at me in amazement as I asked whether she knew the potential implications of the photographs. She didn't take long to answer: yes, she would do it and, yes, she was sure. She asked me to arrange the photographer and I already had someone in mind: Barry Jeffery, whom I trusted implicitly and who I knew would take wonderful pictures.\n\nShortly after my conversation with The Queen, I went to see Barry and asked if he would like to photograph Her Majesty in a way that she had never been captured before. I explained that I wanted him to photograph The Queen as though she was the most famous model in the world and that she would be posing in a relaxed, informal way. Needless to say, Barry was overjoyed by the idea, and before long, we had booked a date. The photoshoot was to take place in the Throne Room and Her Majesty chose a very fetching white dress, although a crucial element was missing: it had no pockets! Luckily, this was an easy fix, as I had leftover material from when I had made the dress.\n\nThe day of the photoshoot arrived. I greeted Barry and his assistant, Gideon, and we walked together, with his camera, to the Throne Room where they set up their equipment. I went to request Her Majesty's company and as soon as she joined us, Barry began to explain his approach and how the photoshoot would progress. Within moments, The Queen raised her hand respectfully and Barry immediately stopped talking. 'No, Barry, this is how we're going to do it,' she said. 'Just keep the camera rolling!' And we were off. Her Majesty took her position in front of the lens and started striking a series of poses, slipping her hands in and out of her pockets and placing them onto her hips, mimicking the stances of a professional model. I stood by in disbelief \u2013 The Queen was a natural. Barry and I felt we were experiencing something really special: a moment never to be repeated.\n\nWhen the photoshoot was over, The Queen and I left the Throne Room and Her Majesty thanked Barry, giving him clear instructions to return the photographs to her. She told me that she was really looking forward to seeing them, and I could tell by her laughter and the twinkle in her eye that she had had a wonderful time.\n\nI was working with the Royal Collection at the time, who were to publish my first book, Dressing the Queen, and we were in final discussions regarding the layout, so once the photographs from this special shoot were ready, I handed them over to be placed in the book. However, to my utter amazement and disappointment, I was then told by the Royal Collection that only two photographs would be allowed to be used for the book. Once the full shoot had been shared more widely, their opinion was that these more candid photographs would bring the Monarchy down, and therefore they were not suitable for the public eye. Why they thought that, I have no idea.\n\nThe Royal Collection is now under new management, and Mr Tim Knox, who is the director, has been fully supportive of this book and has returned the photographs from this special shoot to The Queen. It is wonderful to have a second opportunity to share the one and only full set of photographs from that exceptional morning, when The Queen's wish was granted and Her Majesty was able to do it her way.\n\n_A beautiful portrait, and, I believe, one that is previously unseen._\n\n**'It was an absolute honour to be asked to photograph The Queen for the cover of the book _Dressing The Queen: The Jubilee Wardrobe_ , and the request came as a complete surprise.**\n\n**At the time I was working on photographing the majority of the content of the book at Buckingham Palace, capturing the magical process that each of Her Majesty's outfits goes through, from fabric selection to the making. I was spending a lot of time with Angela and her team on the Dressers' Floor, following them around on their daily routines, when one day, Angela called me to her office where she asked if I would shoot the book cover picture of The Queen in the Jubilee dress. At that very moment I felt hugely honoured, excited and lucky. What a privilege! It's not something I'd thought I would ever be asked to do.**\n\n**The day before the shoot, my assistant, Gideon Marshall, and I set up in the Throne Room. It was a surreal experience for us both, especially when we stepped out into the corridor for a break to find Daniel Craig in character as James Bond, standing there, talking on his phone! The scenes for the London 2012 Olympics opening ceremony were being filmed.**\n\n**On the day of the shoot, naturally I was a little anxious, but I was ready and so was everything else. The moment arrived, and it all went to plan. I managed to capture fantastic shots of Her Majesty in the Jubilee dress and was so pleased with the results. Then The Queen decided she would change her outfit, which was completely unplanned. We had just five minutes to choose a different area to shoot within the Throne Room, move our kit around and be ready for the next shot.**\n\n**The Queen returned in her second outfit, and we were straight into the flow of shooting, but this session was different, and even more amazing than the first. Everyone was relaxed and having fun. There was lots of laughter, a sense of freedom and movement, which you can see in the series of images. It was much more like a fashion shoot, and I even captured a shot of The Queen with her hands in her pockets, which was used for the cover of the book. It was a day I'll never forget.**\n\n**Working with Angela and her team was a once-in-a-lifetime experience. It was fascinating and hugely educational, and Angela was so welcoming. As a photographer, shooting pictures of The Queen was career defining, and something I'd never dreamed of doing. I'd like to thank Angela for making it possible.**\n\n**And here I am, with Kate, my wife and assistant, being asked once more to produce photographs. This time Angela needs a completely different series of images, to include the original and replica Royal Christening Robes in their full glory and The Queen's Royal Ascot outfits for 2019. I feel so privileged to be asked to return to Buckingham Palace, and so honoured also to share this experience with my wife. I will never forget the magical moments I shared with Angela and her team.'**\n\n**_By Barry Jeffery_ \nPhotographer**\n\nBEHIND THE \nLENS: CHRIS \nLEVINE\n\nAnother shoot I remember fondly was for Jersey's 800th anniversary celebration to honour its special relationship with the Monarchy. To reflect this occasion, the Jersey Heritage Trust asked Chris Levine and Robert Munday to work together to create something spectacular.\n\nA unique and extraordinary sitting took place in November 2003 for a holographic portrait: The Queen's picture was going to be turned into a hologram. So much work went into setting up the Yellow Drawing Room in Buckingham Palace as a photographic studio \u2013 it took about three days for all the equipment to be installed. Robert designed and built the special camera system, as well as the tracks the camera was going to be sitting on so that it would capture thousands of pictures every minute!\n\nChris had a particular vision of the shot in mind, and when we discussed his ideas before the sitting, we also talked about outfits. I showed Chris a few options, and he selected the dress he thought would be best for the portrait. The Queen would also be wearing the Diadem, a pearl necklace and a lovely pair of diamond and pearl earrings. Chris asked if I could bring a couple of cloaks and capes with me, as one of the capes he was hoping The Queen would wear was the adapted Officer's Cape of the Life Guards, of which Her Majesty is Colonel-in-Chief.\n\nThe day finally came and I had the pleasure not only of helping The Queen dress, but also of seeing the astonishing pieces of equipment that would be used to create the hologram of Her Majesty. How cameras have come on since the Polaroid! I followed The Queen into the Yellow Drawing Room and the room was dark as all of the curtains had been closed. Built in a half-moon across the floor was what looked like a railway track with a camera sitting on it. It reminded me of the old-fashioned coal trucks that were pulled by trains many years ago. It was intriguing!\n\nThe Queen listened intently as Chris and Robert explained how this unusual sitting was going to work. Once everyone was happy with the plan, I followed The Queen to the mirror, handed her the Diadem and watched as she carefully placed it on her head. Her Majesty then made herself as comfortable as she could on the chair that was waiting for her. I placed the white-fur-trimmed cape around her shoulders and we were good to go. Lights, camera, action!\n\nDuring the sitting Chris asked The Queen to sit perfectly still for eight seconds as the series of images was taken. Once the eight seconds had passed, The Queen was able to close her eyes and rest for a moment. The lights were bright, so I know I would have appreciated that if I had been sitting in front of the camera. The Queen looked so peaceful with her eyes closed, and I thought the title of the hologram portrait was fitting. It was called Equanimity, as Chris wanted to demonstrate how one could achieve a sense of calm when sitting still and how liberating this could be.\n\nThe camera on the tracks took more than 10,000 individual images, which were then converted into the hologram. I was overwhelmed to be given one of the holograms and I have treasured it ever since, carefully packed away for safe keeping.\n\nActually, the photograph used on the cover of this book was taken during that sitting by Nina Duncan, Chris Levine's assistant. In that picture I am helping The Queen put the Admiral's Boat Cloak \u2013 it was made for her in 1968 \u2013 over her shoulders, making the final adjustments and ensuring that her pearl necklace was lying straight. I thought that photograph of The Queen and myself was the perfect picture for this book.\n\n_The Queen posing for the holographic portrait,_ Equanimity, _which was commissioned by Jersey Heritage to commemorate 800 years of allegiance to the crown._\n**'A week before my sitting with Her Majesty, I received a phone call from the Palace. It was Angela Kelly asking me what I would like The Queen to wear for the Jersey portrait. Until that point, I had been preparing for the shoot under the impression that I would only find out what Her Majesty would be wearing on the day itself. I had to pick myself up from the floor \u2013 Angela was suggesting that I could style Her Majesty myself, which was an immensely exciting but also extremely daunting prospect. Creatively, I was looking to simplify and purify the work \u2013 to create something resonantly modern and iconic \u2013 like The Queen herself \u2013 so the choice of attire was critical. Who better to collaborate with than Angela!**\n\n**Meeting with Angela at the Palace, I was able to share my thoughts about the work and what I was hoping to achieve artistically. Based on our discussions, she then selected several dresses for us to consider, but in the end I went for a simple, dark blue, A-line number. Angela also suggested that a cape could be fitting, so I asked for a variety of options, knowing that on the day we could change the look very quickly by simply switching capes. In terms of jewellery, along with Her Majesty's diamond and pearl earrings, I thought one string of pearls would be best, rather than the usual three, as it would make for a more graphic look. Finally, I was given the opportunity to review The Queen's crowns in the Royal Collection next door and decided on one of the more relatively understated pieces \u2013 the Diadem \u2013 which features a simple cross. It is just sublime and is apparently one of Her Majesty's favourites, so seemingly a good choice.**\n\n**On the day of the shoot, I got a call from one of the offices in the Palace to ask whether the crown was essential for the shoot. The President of the United States was staying at the time, so security was tight and the logistics of getting the Diadem to the shoot would be tricky. Something in me held firm. Yes, it was important. So Angela brought the piece over herself and told me that Her Majesty had said that if Chris wanted the Diadem, he must have the Diadem. Thank you, Your Majesty.**\n\n**With the shoot well under way in the Yellow Drawing Room, I asked if we might have a change of cape, so Angela brought out the Ermine. Until the day I die, I will never forget the moment The Queen put it on \u2013 I felt I was somehow channelling the work. Her Majesty looked sensational \u2013 just beautiful \u2013 and the images we captured have become historic.**\n\n**I'm proud of what we created together that day and I still have to pinch myself sometimes to remind myself that it wasn't a dream. The photoshoot was quite a surreal experience at times but I put my heart into the work and gave it my best. It was a huge honour and I'm grateful to Her Majesty for her faith in my art, to Jersey \u2013 the commissioning body \u2013 and to Angela for facilitating the work.'**\n\n**_By Chris Levine_ \nArtist**\n\n**'The experience of meeting and working with Angela and Her Majesty is a memory I treasure to this day. I was still a photographic assistant when I found myself in the surreal position of driving through the gates into the Buckingham Palace forecourt to meet The Queen for the first time back in 2003. Chris Levine was looking for an assistant to light Her Majesty for his holographic-portrait commission and I was the lucky lady who was hired. Shortly before we were due to go to the Palace for the first time, he also asked me to bring in my camera to document the making of the hologram and take some reference portrait shots.**\n\n**What I remember most from that first meeting was how sparkly Her Majesty's eyes were and how beautiful her smile was, but more than that, I was overcome with the thought that she was very much like a grandmother \u2013 albeit a very famous one. I just wanted to go over and give her a big hug \u2013 totally against protocol, obviously. I felt bad that she was sitting in the middle of the room with all of us around her, almost like a goldfish being watched in a bowl. I remember saying just that to Angela, who suggested I go over and talk to her to break the ice and make me feel more comfortable: The Queen is famous for making everyone feel at ease. So I did. I believe I even told her that she reminded me of my own nonna at one point. I also remember \u2013 probably much to everyone's horror! \u2013 telling her that part of the process we were doing that day would be just like having her passport photograph taken, and Her Majesty looking at me not at all unkindly for my blunder and babbling!**\n\n**I was amazed at how The Queen made us all feel so at ease, and just how sweet and gentle she was, although at the same time I was in awe of her obvious sovereignty. I also felt privileged to witness the tender connection and intimacy between Angela and Her Majesty, an almost mother\u2013daughter type of bond, yet one with much reverence and respect on Angela's part. It was apparent to me that I had a one-off opportunity to capture something precious \u2013 and that was quite something.**\n\n**We were fortunate enough to be given a second sitting early in 2004 and this time I had permission to bring in my small digital camera to further document my experience. It was in this visit that I decided to capture the truly unique little moments that made the experience, and Her Majesty, so magical and memorable for me. That was when I photographed Angela helping adjust The Queen's outfit for the camera \u2013 I wanted to capture their relationship \u2013 and Her Majesty putting on the Diadem in the mirror. I remember naively imagining that surely there must be a person in the Palace whose sole job it was to place the crown on Her Majesty's head! That moment, and the image I captured of it, humanised Her Majesty for me: I wanted to capture that while retaining her undeniable nobility.**\n\n**I'm honoured that I had the opportunity to meet The Queen and was able to capture those few moments to which I imagine few of us have access. I will treasure these memories for a lifetime.'**\n\n**_By Nina Duncan_ \nPhotographer**\n\n_The Queen sits with Anna Wintour and me at Richard Quinn's runway show during London Fashion Week in 2018. I'm wearing a beautiful dress kindly designed for me by Stewart Parvin._\nON THE FROW: \nLONDON \nFASHION \nWEEK\n\nAlthough The Queen doesn't consider herself to be a fashion icon, I am very aware that many people do. She is always on-trend, trendsetting, in fact, never puts a foot wrong when it comes to choosing the outfits she wears, and takes inspiration from the classic styles of Dior and Chanel. The Queen is humble, and her life is about her role. It is part of her job to look the part, and fashion helps. I've had to be mindful over the years that when Her Majesty makes a speech, the media's focus should always be on her words and their subject matter, rather than the colour of her dress or the style of her hat, which can be a little limiting in how creative I can be with the design. So when the opportunity arose for The Queen to attend London Fashion Week, I was thrilled: for once, we could embrace the occasion without worrying quite as much as we would for a State engagement or an official Royal visit, when strict criteria must be taken into account in the designing of Her Majesty's clothes.\n\nThe origins of The Queen's iconic appearance on the 'FROW' (the front row) at the 2018 London Fashion Week show came from an idea that I had several months before. I'm such a fan of the fashion world, and considering how much everyone loves The Queen's sense of style, I wondered if we could create an award to mark the effect she has had on the fashion industry over the years. I thought it might also be a great opportunity to support an up-and-coming British designer. When I suggested to The Queen an award to be called 'The Queen Elizabeth II Award for British Design' that could be presented for the first time at London Fashion Week by The Queen personally, she loved the idea. In subsequent years, the award would be presented by another member of the Royal Family. Later, Her Majesty described the award as her 'legacy to all those who have contributed to the British fashion industry'. Sam Cohen, Her Majesty's Private Secretary at the time and I immediately got in touch with Caroline Rush at the British Fashion Council to get the ball rolling. She was absolutely delighted to hear that Her Majesty would attend the show and recognise young British talent in this way. The only thing missing was the award, which Her Majesty kindly agreed that I would be in charge of designing. I think she knew that I would create something in keeping with her own taste.\n\n_When Richard found out The Queen was there \u2013 he added head scarves at the last minute, a touch that The Queen appreciated._\n\nDesigning an award is quite unlike designing an outfit, as I soon came to discover. I wanted to use The Queen Elizabeth Rose as the main focus for the design, but there were many other factors to consider, too. The award needed to be light enough for Her Majesty to hold easily but not so light that it felt insubstantial for the winner. For the design of the Rose, I envisaged it as giving the impression of having been carved into the bark of a tree, a modern design but still natural in tone. In the end, it turned out perfectly. Her Majesty was pleased when she saw it, and was confident that she could carry it!\n\nIn the lead-up to the show, which was hosted by the British Fashion Council, we had to keep everything tightly under wraps. If the media knew what was happening beforehand, it would have been very difficult for Her Majesty to attend as they would have flocked to the venue. Only a handful of people knew of the plan that The Queen would be presenting the award to Richard Quinn. My assistant, Jackie, Private Secretary, Sam Cohen, and I went to 180 Strand to see the venue before the event and to meet with the organisers. We assessed the suggested route that Her Majesty would take once she was inside, and I made some changes to the planned programme so that the new generation of designers attending would have the opportunity to speak to The Queen about their creations.\n\n_Mood board for designing The Queen Elizabeth II Award for British Design by Lucy Price._\n\n_Me and my friend Michael Atmore._\n\nOn the day of the show, I arrived with The Queen, who was wearing an Angela Kelly outfit: the design was modern but inspired by Chanel, duck egg blue in colour, accompanied by a matching jacket decorated with aquamarine Swarovski crystals. We were taken to meet the New Generation designers, who, as they saw Her Majesty approaching, responded in utter amazement, not only to have the chance of seeing The Queen, but also having the chance to speak with her and share their designs. Afterwards everyone was asked to take their seats, which was our cue to take our positions. The media still had no idea that Her Majesty was about to walk into the room. I was later told by Stella, Jackie, and my great friend Michael Atmore, from Fairchild Publications, who flew over from NYC for London Fashion week, that they could hear the New Generation designers talking about how they just met The Queen and how they were still shocked. As The Queen walked in, there were audible gasps from every direction, followed swiftly by camera shutters going off at speed. After the initial shock, there was a spontaneous round of applause and some of the media representatives even had tears in their eyes \u2013 it's so unusual for anyone to see The Queen in this way and everyone was thrilled to capture Her Majesty in this rare moment.\n\nIt was a personal privilege for me that day to be able to sit with The Queen on the front row alongside iconic figures in fashion such as Caroline Rush, Anna Wintour, Sarah Mower, and Anya Hindmarch. The show began and Her Majesty later told me that she particularly loved seeing all the florals on the catwalk. Afterwards, The Queen and I made our way to the front where she made a wonderful speech about the importance of British fashion. I shall never forget seeing The Queen presenting Richard Quinn with the inaugural Queen Elizabeth II Award for British Design.\n\n_The Queen presents Richard Quinn the inaugural Queen Elizabeth II Award for British Design during London Fashion Week in 2018. What an honour to be on stage with The Queen, Caroline Rush, and Sarah Mower._\n\n_The Queen arrives at Runcorn by Royal Train in 2018_\nTHE ROYAL \nTRAIN\n\nWe often travel across the country to engagements on the Royal Train. Although I can't disclose much about what happens on the train, I can give an idea of what a typical journey on it involves. And I can definitely tell you about the fun we have, especially if we're staying overnight somewhere. Travelling around the United Kingdom with The Queen in this way is a truly wonderful experience.\n\nBefore we leave Buckingham Palace, my team makes sure that everything is packed as efficiently as possible \u2013 as you can imagine, there isn't much space on the train for luggage. The train isn't as big as you might think as there are only six to eight coaches, depending on who is travelling with The Queen. All in all, there are only ever about twelve staff on board \u2013 Her Majesty never has a large entourage.\n\nAs soon as we board the train at the station, I get straight to work unpacking. Occasionally, when we are on longer trips, I may need to press a few items. It didn't take me long to learn that ironing on a moving train is almost like surfing! You just need to find your balance as the train rocks back and forth. And, as with the ironing, other work on the train isn't as straightforward or relaxing as it should be. Sadly, there isn't time to sit and watch the beautiful scenery go by, although I always catch a glimpse here and there as we head towards our destination, especially to the north, which is so familiar to me.\n\nWhen we arrive at our destination, I ensure that The Queen has everything she needs for the day and make some final checks on her dress, coat, and hat, because as soon as she steps off the train, she will be greeted by press and photographers. Everything must be perfect. As Her Majesty steps onto the platform, my heart lifts when I hear cheering and applause, and see Union Jack flags waving in the crowds. The mayor of whichever town we are visiting usually comes to greet The Queen and, in the background, you often catch trainspotters making hurried notes about the unusual model they've just seen.\n\nOn one special occasion, about ten years ago, I even got to enjoy my very own warm welcome when the Royal Train stopped in Liverpool. All of my family \u2013 my children and grandchildren \u2013 came to the station to see me and we hugged and kissed on the platform. I'm so glad I have pictures of that special moment: it was the first time my family had seen me on the Royal Train.\n\n_Miss Peggy Hoath on board the Royal Train in 1994._\n\nEven after The Queen has left for her engagement, there's no time to waste as I need to prepare for her return and pack everything if we're heading straight back to London. While we're waiting on board, it can be slightly unnerving when the train keeps reversing in and out of the station to allow commuter trains onto the platform \u2013 the Royal Train can never be in the way and must not disrupt others as they travel about the country. Her Majesty would hate to make anyone late. Although, saying this, the less time we spend on the platform the better, as far as I'm concerned! I do find it a little embarrassing when I'm taking a break from packing, and am preparing to sit in the dining car to eat something. People on the platform can see right in through the windows and you do start to feel like you're on display! Over the years, I have learned to close the curtains to save embarrassment all round.\n\nI'm always planning ahead, so if we're going back to London that day I'll be thinking about what's next on the agenda, whether it's at the Palace or at Windsor Castle. Occasionally, though, I'll take a moment to look at people's faces as we pass through stations. It's lovely to see how much everyone appreciates seeing the Royal Train.\n\n_The Queen and The Duke of Edinburgh arrive on the Royal Train at Birmingham New Street Station in 2015._\n\n_The Royal Yacht_ Britannia _departing Portsmouth in 2007._\nBRITANNIA\n\nDuring its decades of service, the magnificent Britannia was the setting for many Royal engagements, and it was a wonderful way to travel. Whether on a formal or private tour, the elegantly designed Royal Yacht offered a home-away-from-home to The Queen and her family. I was privileged to join Her Majesty aboard Britannia on several occasions, and although I'd like to describe the beauty of the State Drawing Room, the Sun Lounge or the Admiral's Cabin, I'm afraid I simply can't do them justice! I can, however, safely say that being on Britannia was an amazing experience: it transported me to a world away from the hustle and bustle of Buckingham Palace, and it was always so much more relaxed. The minute Her Majesty came on board, anyone could tell she felt secure and at ease. It didn't matter where we were going \u2013 from South Africa to Helsinki \u2013 Britannia felt like home. Of course, The Queen is never truly on holiday as she still has to meet with her Private Secretaries and work her way through the famous red boxes, which still managed to reach Her Majesty even on Britannia \u2013 one of the Royal Yacht's lifeboats would go to pick them up from the nearest port.\n\nOf all the State occasions held on Britannia, one in particular will stay with me for ever. It was 24 March 1995 and Her Majesty was due to watch the iconic Beating of the Retreat in the harbour at Durban, in South Africa \u2013 a pageant of music and military drills, fireworks and cannons \u2013 after hosting a State Banquet for the great Nelson Mandela, followed by a reception for 200 guests. I remember that The Queen was particularly excited about the dinner and was very much looking forward to joining her guests when I was helping her get ready. For this occasion, Her Majesty and I had decided on a beautiful soft pink dress with a subtle pink lace overlay. As ever, it was a pleasure to see Her Majesty, elegantly dressed and perfectly complementing the beautiful gowns and smart suits of her guests, against the glorious backdrop of Britannia.\n\nAs the evening went on, I could hear music beginning to play and the gentle hum of chatter and laughter. Safe in the knowledge that I'd finished my duties for the day, I went back to my cabin and changed into my pyjamas. As I was settling into my nightly routine, I suddenly heard my name being shouted from the corridor. Slightly alarmed, I ran out of my cabin only to find Her Majesty standing right in front of me. At first, I didn't know where to look, but almost instantly, a broad smile broke across her face and I found myself grinning with her \u2013 our outfits, although very different, were exactly the same shade of soft pink. I was asked to follow Her Majesty to her cabin because she needed something before dinner, and once she was ready to join her guests again, I curtsied and said goodnight.\n\nAs The Queen walked away that evening, I was still quietly laughing to myself and certainly went to bed with a smile. It was one of many happy moments we shared on board Britannia. The Royal Yacht's crew were so diligent and considerate \u2013 we felt like one big happy family. Often when my evening duties were done, I would go outside and sit on a step and just watch the sea go by \u2013 at those times I always wished I had my telescope with me as the stars reflecting in the ocean were mesmerising.\n\n_The Queen and Nelson Mandela in South Africa in 1995._\n\n_Everything labelled and ready to go._\nTHE QUEEN'S \nLUGGAGE\n\nA huge amount of preparation is involved in getting everything ready for any travelling, and especially for a Royal Tour, from creating the outfits and matching the hats, to making sure the appropriate shoes, gloves, and handbags are selected and prepared. On a ten-day tour there might be up to thirty outfits for packing: this allows us to offer two outfits to choose from for each engagement, and if neither is quite right, or one gets soaking wet, we always have a backup. Sometimes we will pack two hats for each outfit \u2013 a lightweight one for warmer weather and a stronger, sturdier one in case it gets chilly or starts to rain. When everything has been readied, I ask for The Queen's Footmen to prepare the Royal luggage and deliver Her Majesty's to the Dressers' Floor, so that we can begin to pack it all.\n\nMany aspects of my role have remained unchanged from when my predecessor, Peggy Hoath, was in charge. But, as with any job over the course of a quarter of a century, certain things eventually need changing or processes can be improved, and this is something I'm always mindful of. That was definitely the case when it came to Her Majesty's luggage.\n\nYears ago, The Queen travelled with three very large, and very heavy, leather wardrobes. In fact, they were heavy even when completely empty, and although sturdy and strong, with three large metal locks, over time the leather had become so worn that the handles were starting to disintegrate \u2013 not ideal for the poor Footmen whose responsibility it was to carry them from place to place, and up and down stairs. There were three large wardrobes from the early years, plus a new, smaller, leather wardrobe that had been given to Her Majesty as a present by her children. Before each tour, these would be packed with The Queen's clothes and transported to the aeroplane, where they would stand upright in the hold, along with all the other luggage and cargo. Over the years, we tended to travel in smaller planes with less cargo space, as the length of the tours was condensed, with more engagements on each day, so the wardrobes had to be laid flat and strapped to the floor inside the plane behind where we all sat. The airline was happy to help with this \u2013 even removing some seats to make space \u2013 but, as you can imagine, this was all becoming a bit of a nightmare: we constantly worried that The Queen's clothing might end up damaged and I'm sure the crew who unloaded the luggage dreaded our arrival.\n\nI realised we couldn't possibly keep asking people to lift these wardrobes and cases any longer \u2013 it was getting ridiculous \u2013 so I had to come up with a better, more practical solution. My immediate thought was to use hanging bags instead and, at first, I was pleasantly surprised that this seemed to work quite well. When they were being packed, each hanging bag stored two outfits \u2013 a dress and a coat or jacket \u2013 the heavier items and the silk garments in separate bags to protect the more delicate materials. This change was met with huge relief by the Footmen and porters!\n\nI seemed to have struck upon the perfect solution, until The Queen and The Duke of Edinburgh travelled to Italy in 2000. With all of Her Majesty's outfits packed in the correct bags and on the rail, ready to go, everything seemed to be going smoothly . . . until we arrived to a torrential downpour. The crew began unloading the luggage and someone decided to roll the rail of The Queen's hanging bags straight across the tarmac in the pouring rain. A huge gust of wind took hold and sent the bags flying across the ground, getting drenched by the rain. I watched the whole thing happen in slow motion from a window in the plane \u2013 I felt so sorry for them scrambling around trying to pick up Her Majesty's soaked clothes. Of course, we made sure that that never happened again, and I began straight away to think up another solution to this very old problem.\n\nAnother aspect of my role is to look after The Queen's personal Royal Warrants. They are issued to companies and individuals who are asked to supply personal goods to The Queen, from shoes to handbags and much more. To maintain a Royal Warrant, a company must demonstrate excellent service, not just to The Queen but to the general public, so from time to time I will go to visit them to check that their members of staff are treating their customers with respect. One company with a Royal Warrant is House of Fraser, so I decided to go for a browse to see if they might have anything suitable in the way of luggage. I soon found the solution: suitcases with wheels! I bought quite a few lightweight Linea cases, once again to the delight of the Buckingham Palace staff members who had risked their backs each time we went on tour. I'm happy to say that this has been a successful and very welcome change.\n\nWhen it comes to packing the new, lightweight (and waterproof!) cases, our approach is meticulous to ensure that there is no creasing or damaging of Her Majesty's clothes. It is important to balance the weight on each side of the case and, whenever possible, to start with heavier items, or those made of hardier material, at the bottom. You may not know that tissue paper is an invaluable tool in the packing of The Queen's luggage: we use layers of it to separate and protect individual items, from jumpers and cardigans all the way up to stocks and socks. We also use plenty of tissue paper to protect Her Majesty's hats in their boxes, and although I still use hanging bags and the rail for some outfits, we're all extremely careful if the weather isn't pleasant that day!\n\n_Saying farewell to the wonderful Milan._\n\n_It's always thrilling to see people's faces as The Queen drives past._\nTHE SPECIAL \nESCORT GROUP \n\u2013 AT YOUR SERVICE, \nMA'AM\n\nImagine standing outside the gates at Buckingham Palace and seeing a row of white police motorcycles lined up on the forecourt. Behind them, there is a set of black doors, and behind these doors two cars wait. We are about to leave Buckingham Palace on our way to the airport. The last pieces of luggage are packed into the boots of the cars, and I have The Queen's personal luggage next to me on the seat. In the car with me are the chauffeur and The Queen's Footman, Ian Robinson. We are parked at the Garden Entrance waiting for Her Majesty The Queen to appear, with an additional car for members of the Royal Household, such as the Ladies-in-Waiting, the Private Secretary and The Queen's Equerry. We are all to drive to the airport at the same time in convoy.\n\nEvery day The Queen's programme runs like clockwork. We always set off at the exact time stated in it. My excitement \u2013 after twenty-five years I'm still excited! \u2013 starts as the big black doors open at the side of Buckingham Palace and you can see the public outside waiting. They have no idea who is in the car driving out through those black doors, but as it gets nearer you can see their faces change with astonishment when they realise it is The Queen. Then they start to cheer.\n\nFour police motorcyclists from the Special Escort Group (SEG) wait for us to come through the gates and join us, one at the front of the convoy leading the way, and three others who deal with the traffic and clear the route for us. They will ensure that The Queen gets to her destination safely and on time. On this particular occasion we are driving along The Mall towards Marble Arch. The SEG will stop other cars to allow The Queen's and those behind it in the convoy to pass through the traffic faster. They do not hold up the traffic for more than a minute or so. Once we pass the first bike, the other three speed off, blowing their whistles to get attention (they don't use sirens) and we move swiftly on to the next junction where the SEG bikes wait for us to pass, then zoom off to stop the traffic ahead so we have a clear run. It is like playing leapfrog. A police Range Rover travels behind our cars making sure no one gets too close.\n\nAs we pass members of the public, who have stopped to see what is going on, people cheer and clap when they see the car driving by and realise The Queen is sitting inside. 'It's The Queen! It's The Queen!' And as everyone seems to have a smartphone these days, they try to get a photograph or a video of Her Majesty's escort passing. If The Queen happens to catch someone's eye, she will wave, of course. Imagine how busy the streets of London can be: our escort has to make their way down some narrow streets. The SEG are so professional and have to be precise when moving in and out of traffic, keeping it clear for The Queen to get to her destination without holding up the traffic for too long. It is fascinating to watch them manoeuvre so rapidly, with excellent communication between them. The Queen will use the SEG as escorts only when there is an official engagement, or in an emergency.\n\nWhen I'm travelling in a convoy like this, I cannot help but smile at how happy the public are to see The Queen and how lucky I am to be travelling just a few cars behind Her Majesty.\n\n_The Queen and The Prince of Wales travel from Buckingham Palace to the Houses of Parliament._\n\n_Vulcan, the new recruit._\n\n_The Queen and Michelle Obama at Buckingham Palace in 2009_\nWOMAN \nTO WOMAN\n\nWhen I am designing for The Queen, I am always aware of how her outfits will be perceived, both by the people she is meeting with or speaking to, as well as by the media. However, just as eyes are always on Her Majesty, they are often also on the people she interacts with, and there is much speculation about how one is meant to behave.\n\nContrary to what many people believe, there are no strict rules that must be followed when one is introduced to The Queen. You may wish, or not wish, to curtsey. But on the whole most people do curtsey or bow out of respect. Some women tend to find that they instinctively curtsey, whereas men tend to bow, but there is no golden rule, even for the most long-standing members of the Royal Household. Each morning and evening, I greet Her Majesty with a curtsey, but because I see her so often, I don't repeat the gesture throughout the day unless we are out in public on Tour.\n\nThere are certain things that are understood to be accepted protocol when it comes to interacting with Her Majesty, though. Supposedly, you should never put your arm around The Queen, for example, but when human instincts kick in, sometimes this is absolutely the appropriate thing to do. The Queen has the ability to make everyone feel so relaxed that sometimes it feels instinctive to be tactile with her, just as Michelle Obama demonstrated during the State Visit with her husband, President Obama, in 2009.\n\nMuch has been made about the meeting between Michelle and Her Majesty, when an instant and mutual warmth was shared between these two remarkable women, and protocol was seemingly 'abandoned' as they stood closely with their arms around each other's' backs. In reality, it was a natural instinct for The Queen to show affection and respect for another great woman, and really there is no protocol that must be adhered to. When fondness is felt or the host of a State Visit goes to guide Her Majesty up some steps, it truly is about human kindness, and this is something The Queen will always welcome warmly. Anyone who is close to Her Majesty is not a threat and is certainly trusted.\n\n_An unexpected honour._\nA PERSONAL \nHONOUR\n\nI never know what will land on my desk each day. It might be a specific request from a photographer for a photoshoot, or for Her Majesty to take part in a spoof James Bond film! I have fond memories of one request in particular, which came to me via the Private Secretary's office in 2005 from the sculptor Susan Velder, who had been commissioned by the province of Saskatchewan to create something to mark the fiftieth anniversary of The Queen's reign, which was to be presented to Her Majesty on her visit to Canada later that year.\n\nSusan had been commissioned to create a life-size statue of Her Majesty in bronze and it was to depict The Queen riding her favourite horse, Burmese, which was given to Her Majesty by the Royal Canadian Mounted Police and which she rode for eighteen consecutive years on her birthday parade, also known as Trooping the Colour. Her Majesty has always ridden side-saddle on that occasion, and as she is Colonel-in-Chief, she wears the full Guards' regiment uniform as well as a tricorn hat. The biretta, as it is also known, is a square cap with three flat projections on top \u2013 it's quite a tricky one to design correctly in fabric, let alone in bronze, as Susan came to discover.\n\nSusan got in touch because, although things were going well with her designing of Burmese, The Queen and her uniform, she was struggling with the tricorn hat, finding it difficult to capture the fibres, the shape and the size. As you can imagine, a life-size bronze statue is no easy feat by any means, so to think that the hat was causing her so much trouble is quite something! She asked if I could provide a photograph of the biretta to help her.\n\n_Spare time in Canada._\n\nKnowing how important this request was, and the significance of the statue to the people of Saskatchewan, I was determined to help as much as I could. The sculpture needed to be ready for the Royal Tour later that year and, as luck would have it, I was due to go on the usual recce to Canada in the not-too-distant future. Rather than sending a photo, which I worried wouldn't be helpful enough, I asked The Queen if I could take the actual hat with me on the recce, deliver it to Susan and leave it with her. When we arrived on the Royal visit later, I could collect it and all would be well. Her Majesty, as ever, was happy to help, even though this was such an unusual request, and when I let Susan know my plan, she was thrilled.\n\nSo, three months later, we arrived at Saskatchewan for the unveiling of the statue and everything went exactly according to plan. The statue was enthusiastically received, the hat looked great, and Her Majesty was delighted. However, the night before we were due to move on to our next engagement, I received a request to be in the Audience Room of Government House the next morning at 11am. I went to sleep that night slightly perturbed, wondering what this was all about.\n\nThe next morning, I walked into the Audience Room to find that many chairs had been set out. To me, it looked like an Investiture and to my astonishment, I noticed that one of the chairs had my name on it. Still uncertain about what was happening, I sat down. Everyone else arrived shortly afterwards, and when we were all seated, Lieutenant Governor Lynda Haverstock called my name. In shock, I stood up and made my way to the stage where I was presented with the Saskatchewan Order of Merit medal for my help with Susan's work. I couldn't believe it! I was so proud and honoured that I cried when I received this special award. Sobbing as I returned to my seat, I knew from then on that Saskatchewan would always remain in my heart.\n\n_Notice the fibres and details on the hat which were hard to recreate in bronze, and why Susan needed to see the hat in person to help perfect the statue._\n\n_Susan Velder's sculpture._\n\n_The Queen in Ottawa on Canada Day in 2001, wearing the iconic Canadian Maple Leaf Royal Brooch. It was bought by King George VI for The Queen Mother in 1939 on what was the first ever State Visit by a reigning monarch._\nSCARLETT \nTHE YOUNG \nDESIGNER\n\nI've no doubt that my love for fashion and my lifelong fascination with fabrics and design came from my mum. After all, I was surrounded by it as a child \u2013 our back room was a hive of dressmaking activity and, especially when I was in my early teens, I rarely saw my mum away from her Singer sewing machine, and so I had a strong instinct for dressmaking. In turn, I have shared my passion with my granddaughter, Scarlett, and I swear she has an instinct for fashion. She's a natural. It must run in the family.\n\nScarlett has lots of interests, including football, painting, and drawing, as well as making up her own fantasy characters, but she always takes a keen interest in my work. From when she was tiny, I've brought material samples home to show her, and now we talk about how thickness and texture affect the design of an outfit, which colours work well together, and how to create something spectacular for a special occasion.\n\nScarlett was four when I was planning for the 2010 Royal Tour to Canada, and I was thinking in particular about what Her Majesty might wear for the Canada Day celebrations she would attend. Scarlett was asking all the right questions straight away \u2013 would it be warm or cool? Who else would be there? It would be a formal occasion, I told her, with many members of the military, who would be wearing their fine red tunics, and I showed her a picture of the Canadian flag. Scarlett took in all of the information \u2013 she's always keen to learn \u2013 and then she let me know some of her own ideas on the direction I should take.\n\nThe Queen's outfit should be red and white, she said confidently, so I showed her some possible options for the material. She picked a smart but relatively lightweight fabric: perfect for the weather conditions. Then, she came up with the brilliant idea of including two extra layers on the dress \u2013 a sort of large frill that wrapped twice around from the hip, which would move very elegantly as Her Majesty walked. The hat, she thought, should be white and decorated with a large red flower. I thought all of these were lovely ideas and promised Scarlett I would stick to her design. True to my word, when I went back to London I brought Scarlett's vision to life in the workroom. Scarlett, of course, saw nothing unusual in this: she thought she was just helping her granny.\n\nAnd then, at the celebrations on Canada Day, I felt so proud to watch The Queen wearing an outfit inspired by my granddaughter's creativity.\n\nConsidering how brilliantly the Canada Day outfit had gone down with everyone \u2013 Her Majesty and the media included \u2013 I was thrilled when on Scarlett's most recent visit, in 2019, when she was thirteen, she wanted to help again. I had a dress of The Queen's that she had worn quite often to private dinners which was in need of a revamp. I intended to adjust its length but to do this I needed to remove the delicate black beading from around the gown's lace hem. Without hesitation, Scarlett got on with the job, carefully unpicking it so that we could reuse the beading and lace on another part of the dress. She really is an excellent assistant, and probably far more patient than I am! I love sharing my passion with my granddaughter and I can't wait to see what life has in store for her.\n\n_Scarlett working on The Queen's evening dress at my cottage._\n\n_The Queen inspects a Guard of Honour outside Canadian Parliament on Canada Day celebrations in 2010._\n\n_The Queen gives a speech in Toronto in 2010. 'The Girls of Great Britain and Ireland' Tiara is one of the lightest of tiaras, which is why The Queen wears it so often._\nA FITTING \nTRIBUTE\n\nA Royal Tour, for the people who come to greet The Queen, and the British media, who often travel great distances to report on her travels, is always a special event. As such, when designing Her Majesty's outfits for these occasions, I'm always looking to introduce a significant detail \u2013 ideally something that acknowledges or reflects the location of her visit.\n\nDuring the 2010 Canadian tour, Her Majesty and Prince Philip were due to attend a State Banquet in Toronto to celebrate Canada Day. For such a special occasion, I wanted to design something as a tribute to the country. So having chosen a white dress that was fitted but also comfortable for The Queen to walk in, I decided to add some detail that made use of the maple leaf \u2013 the most recognised symbol of Canada. Each leaf has eleven points and I wanted to make sure that this was reflected in the design of the dress. I managed to find a beautiful lacquered print material in white, which I thought would be well suited to appliqu\u00e9, and worked up a design that featured maple leaves running down the right shoulder and arm of the dress. The leaves were to be produced in Swarovski crystals to complement the simplicity of the gown. When I showed my design to The Queen, she agreed that this was a lovely idea.\n\nCanada was such a distance for the British media to travel to and I knew that some reporters and photographers were due to fly back before the banquet and wouldn't see the maple-leaf outfit. Although it had never been done before, and has never been done since, I decided to ask The Queen if she would grant special permission for some of the media to have access to the dress before the event itself. Her Majesty was happy to help, considering the long distance they had travelled, so the day before the banquet, Judy Wade, the Royal Correspondent for Hello! magazine, and her photographer John Stillwell came to meet me in Government House, where we were staying. The staff there were kind enough to arrange a room with lovely natural light for the photographs to be taken. I positioned the dress in the sunlight and John was able to capture the sparkle of the crystals. The results were sensational \u2013 the image was featured on the front cover of Hello!, Judy and John were thrilled to have been given the opportunity and to have secured the best shots for their magazine, and I was pleased to have been able to help.\n\nWhen The Queen was seen in the maple leaf dress the following evening at the State Banquet, the media went wild. Everyone loved the design and that the leaves all had the correct number of points \u2013 attention to detail always goes down well. It is at moments like that that I feel truly privileged to design dresses for Her Majesty's special occasions, and for the media to capture and acknowledge my work gives me reassurance and the confidence to keep designing for the next event, wherever and whenever that may be!\n\n_The intricate detailing of the Maple Leaf dress \u2013 this is when attention to detail really matters!_\n\n_The Queen and The Duke of Edinburgh at a State Dinner in Toronto in 2010._\n\n_The Queen attends a reception for celebrating Nova Scotia in 2010._\nPEOPLE OF \nTHE DAWN\n\nAs part of the same 2010 Royal Tour to Canada, The Queen was scheduled to travel to Halifax, Nova Scotia, for a meeting with members of the local First Nations Mi'kmaq tribe, also known as the 'People of the Dawn'. As always, I had been on a recce some months before to familiarise myself and others with the programme of events. Recces always mean long days of work, ensuring that everything is in order and that we are preparing and planning for any eventuality, but they are certainly worth it for the peace of mind they offer. They usually take three or four days, with the Private Secretary leading the small team. I go mostly to plan the accommodation needs, working with the Private Secretary on the timing of the engagements. On that occasion, I had the privilege of a personal meeting with the Mi'kmaqs and their chief.\n\nIn preparation, I'd been researching the customs of the tribe. Much of their traditional clothing was adorned with the most elaborate and beautiful beadwork \u2013 I couldn't even imagine the precision and skill involved in such detailed work. It provided me with inspiration for the outfit I was designing for Her Majesty to wear at their meeting. I had in mind a beautiful canary yellow material that would make a gorgeous jacket and skirt, but I wanted it to be embellished in such a way that would pay tribute to the traditions and hospitality of the Mi'kmaq people. So when I was finally introduced to their chief, I asked whether he and his tribe would be so kind as to design something for the collar and cuffs of Her Majesty's outfit. Of course, I would give free rein in terms of design and the colouring of the beads \u2013 theirs is a tradition that goes back hundreds of years, and it was an honour to incorporate some of their work in one of my own creations. I explained that the material would be sent to them from the British Embassy, and the Mi'kmaqs said they would be happy to help.\n\nBack in London, I was excited when a parcel arrived a few months later containing the completed cuffs and collar. The craftsmanship was exquisite and so beautiful. I'm used to working with embellishments and know most of the tricks of the trade: keep in mind the fabric with which you're working \u2013 heavier beads won't work on lighter, chiffon-style fabrics; sequins and crystals require someone with perfect eyesight to stitch them on as it's such a delicate and painstaking job. It can take many days to achieve the desired result, but at the end, it's satisfying to know that you have produced something truly unique. The work of the Mi'kmaq tribe was some of the loveliest I have ever seen.\n\nOn the day of the engagement, Her Majesty was delighted to meet the Mi'kmaq community and the ladies who had helped create her outfit. I will never forget the pride on their faces.\n\n_This beading detail is so beautiful._\n\n_The Queen attends a State Banquet at Brdo Castle during a State Visit to Slovenia in 2008._\nTHE SNOW \nQUEEN\n\nWhen I'm preparing for one of Her Majesty's Royal Tours, one of the first things I consider, along with any cultural sensitivities, is the predicted weather conditions of the hosting country. The Queen's outfits must be made of the right material so that she is comfortable when completing her engagements: no one likes to be dressed for the wrong weather!\n\nIn 2008, Her Majesty was invited to Slovenia and Slovakia for a four-day State Visit. Three months before, I travelled with a small group of the Royal Household to both countries to do our usual recce ahead of the Royal visit. The weather was warm when we were there, as it was midsummer, but I was told there was a good chance of snow at the time of Her Majesty's visit. Immediately I felt inspired: it was my first chance to design a winter wonderland wardrobe for The Queen. As we went about our recce, colours, shapes, and styles were leaping into my head: hats, silk jersey dresses, and stunning evening gowns. Her Majesty needed to look glamorous on every occasion: there must not be a 'stand-out' outfit that overshadowed the rest, especially since she would be visiting two countries. There were also cultural sensitivities to take into account: would any particular colour be considered offensive or inappropriate?\n\nFull of ideas, I returned to London and began creating. All outfits for important engagements are created from scratch. I knew that Her Majesty would be attending a State Banquet in Slovenia, to be hosted by the President, and I immediately thought of the perfect material to use for this occasion. The Queen needed a long evening dress; I knew there was some beautiful brocade in the cupboard, wrapped in black tissue paper to preserve its fine silver thread. It had been there for decades \u2013 possibly as long as forty years \u2013 and it's quite a moment when you unwrap a parcel like this: a gift made to The Queen long ago, kept carefully in the cupboard until the right time comes to use it. Its pale blue and silver colourings would perfectly complement the wintry setting of the banquet.\n\n_These boots were meant for walking! The Queen and the President of Slovakia, Ivan Gasparovic tour Hrebienok Ski Resort._\n\nAlthough I give all of Her Majesty's outfits the same care and consideration, another event was scheduled on this tour for which I was particularly excited to design something. The Queen and The Duke of Edinburgh were due to visit Star\u00fd Smokovec via a funicular railway to the peak of Hrebienok, where they would unveil a plaque to mark a wheelchair-accessible tourist pathway called 'Tatras Without Barriers'. It was a lovely engagement: with the path, the less physically able would be enabled to visit this beautiful spot. Given that they would be travelling high into the Slovakian Tatras mountains, I knew Her Majesty's outfit had to be warm but I wanted to design something that complemented the weather conditions, should it be snowing. I decided on an off-white wool tweed coat, trimmed with fur on the collar and cuffs for extra insulation, and a matching dress of the same material, from which I also asked Stella McLaren to make a matching hat. For decoration, we chose feathers and sequins, which would sparkle, hopefully through the falling snow.\n\nWhen we arrived in Slovakia, it was unusually warm for the time of year, but I had packed a pair of comfortable but stylish black boots for The Queen \u2013 they would be perfect for snowy conditions.\n\nThat morning, I was praying for snow, and as Her Majesty and His Royal Highness journeyed up the mountain on the train, sure enough it started to fall. It was the most magical sight \u2013 a crisp white blanket slowly settling all around. The Queen donned the stylish black boots I'd packed just in case, and the press went wild as she'd never been seen in anything like it before. In fact, Her Majesty was quite the trendsetter on that occasion, and when I think back to that moment, I can't help but think: Snow Queen.\n\n_The mink trim on this dress has now been removed and will be replaced with fake fur._\n\n_The Queen visits the Sheikh Zayed Grand Mosque in Abu Dhabi in 2010. Note the beading detail, which was specially obtained for this occasion and is reminiscent of the Tree of Life detailing inside the Mosque._\nTIME IS OF \nTHE ESSENCE\n\nDespite hours, days, or even weeks of preparation for any Royal engagement, there are occasions that require quick thinking from my team and even sometimes The Queen herself. On those days when Her Majesty requires a last-minute alteration or a swift change of garment or accessory, I'm often reminded of a visit to Abu Dhabi.\n\nIn September 2010, roughly two months before The Queen and The Duke of Edinburgh were to spend five days in the Gulf Arab states, I went on the usual recce to familiarise myself with the setting and climate, as well as the route to and location of her engagements. As part of the Royal Tour, Her Majesty was due to visit the Sheikh Zayed Grand Mosque \u2013 one of the most beautifully decorated buildings I have ever seen. On my arrival, I was greeted by the local ladies who were to show me around. Walking quietly through the mosque, I felt calm and a sense of peace.\n\nDuring my walkabout with the ladies, we decided there would be a small privacy screen where The Queen would change from her shoes into her socks and put on her longer coat, in keeping with the required dress code. Another screen would be set up in the area where she would be leaving the mosque, so that she could change back, just as she had when she had visited the mosque in the old Lincolnshire town of Scunthorpe in 2002, as part of the Golden Jubilee celebrations. Her Majesty is always more than happy to respect the customs of the sites and places she visits. In fact, she is a stickler for abiding by what is accepted or what is required.\n\nIn my experience, recces often provide inspiration when I'm designing the outfits to be worn by Her Majesty on a visit. The Grand Mosque in Abu Dhabi was so impressive that I quickly knew how to create something that would complement the grand setting. Dazzling crystal chandeliers hung from the ceiling, reflecting every colour of the rainbow, and beautiful patterns reminiscent of the Tree of Life adorned each pillar in red, blue and gold. I decided that The Queen should wear a long gold coat adorned with the same pattern, and I remembered that in the material stockroom at Buckingham Palace there was some very old golden material that had been presented to The Queen when she was still a princess. I knew it was in excellent condition and would be perfect for this occasion, although sadly I never discovered who gave it to her: records weren't kept before I started working at the Palace.\n\nI shared my thoughts on the outfit with my guides and they were more than willing to help me bring my vision to life. The next day, they took me to the local indoor market, where I was introduced to a gentleman who created the most wonderful colourful beading I have ever seen \u2013 which is saying something, given the nature of my profession! Explaining my idea, the ladies asked if he would help by creating beading that matched the colours of the Tree of Life, which I would then use to decorate a long golden coat for Her Majesty to wear. He was very happy to help and was even content to be sworn to secrecy, even though he didn't know who he was working for. With all of The Queen's outfits, it's important that we keep everything top secret. Having arranged for the ladies to collect the beading and post it to me in London, I returned from that recce knowing this would be a very special outfit and hoping it would turn out to be worthy of the grand setting of the mosque.\n\nBack in London, I set about bringing my creation to life. I had retrieved the golden material from the stockroom and I was safe in the knowledge that the beading was on its way, but I needed to consider what The Queen would wear as a headdress. Her Majesty would arrive for the engagement in a pillbox hat with gold corded trim and would travel straight to the mosque: I decided to add fasteners to the top of the hat and the shoulders of the coat: this would allow us to fix a scarf securely to the hat and for it to cross over The Queen's shoulders. This felt a fitting and practical solution to the problem, and while I was thinking of an appropriate scarf to use, I remembered I already owned just the thing! My sister, Donna, had given me a beautiful gold scarf years before and, as I had never worn it, I gave it to Her Majesty. When she considered the new owner of her gift, I knew Donna wouldn't mind \u2013 although I never did tell her or The Queen!\n\nWith the outfit in place, I had one final challenge to address: how would Her Majesty carry the longer coat, scarf and her white ankle socks to be worn only when she arrived at the mosque? Often my job requires creative thinking \u2013 it's one of the things I enjoy best about it \u2013 and I came up with the solution of designing and making a large cream canvas handbag with detailed patterning. On the day, the extra items of clothing would be neatly folded and kept inside, ready for Her Majesty to change into. With all outfit preparations complete, I was very much looking forward to returning to Abu Dhabi and was expecting a smooth, hiccup-free visit. However, as I've discovered over my years of service, you can't plan for every potential hitch . . .\n\nA few months later, The Queen's convoy was on its way to the mosque and I was travelling with The Duke of Edinburgh's valet a few cars back, just as we always do. The plan was in place: the valet and I would go straight to the screened areas to help Her Majesty and His Royal Highness to change into their outfits. However, as we approached the mosque, the police allowed The Queen's car, and the escort to proceed, but refused to let us in. We tried desperately to explain that it was crucial for us to be allowed in, but we weren't understood and, after we'd spoken to Her Majesty's driver on the phone, I knew that she was wondering where we were. Thankfully we were finally allowed through. Although, in that instance, keeping calm and carrying on just didn't do it for me!\n\nRushing to the screened area, I found Her Majesty and the relief on both of our faces was priceless. We had to move quickly and, after I'd removed the contents from the canvas bag, it became clear that the scarf was going to be a problem. Given that we'd lost so much time, after changing into her white socks and putting on the golden coat, Her Majesty had no time to fasten the scarf as intricately as planned. The Duke of Edinburgh came to alert her that she needed to join him immediately and, in need of a quick solution, Her Majesty calmly draped the scarf over her hat, and her shoulders, loosely tied it under her chin and off she went, right on time. Of course, she still looked amazing. The Queen and I had worked together in a stressful moment to ensure that she could continue with her important engagement.\n\n_The Queen speaks to guests at a State dinner in Saint Patrick's Hall at Dublin Castle in 2011._\n'PRESIDENT \nAND FRIENDS'\n\nWhen it comes to designing the outfits for Her Majesty's State Visits abroad, inspiration comes easier with some than with others. In May 2011, The Queen was due to travel to the Republic of Ireland. It was a particularly significant occasion: Her Majesty would be the first British Monarch to visit the country since 1911, when King George V, her grandfather, had made a tour and all of Ireland was still part of the United Kingdom. The highlight of the programme was to be a State Banquet, hosted by the Irish President, Mary McAleese, held in Dublin Castle, and I had a clear vision for the perfect dress.\n\nThe Queen and I discussed the possible design at length, and I explained that for this evening gown it might be nice to incorporate the shamrock, both as a sign of respect and because it is a famous lucky charm. Her Majesty thought that an abundance of shamrocks all over the bodice of her outfit would be a lovely gesture. In addition, I thought it would be charming to introduce the traditional Irish symbol of the harp. On State engagements, it is usually the Royal Family Orders that are worn on the left shoulder, but on this occasion we could make a change and agreed that an Irish harp made of Swarovski crystals would take their place. Since the banquet, we have actually had several enquiries asking if the 'harp brooch' would be going on display, but it is actually sewn onto the dress and I'm afraid it cannot be removed!\n\nFinally, we had to decide on an appropriate tiara. I knew the public wanted to see Her Majesty in her finery, so I chose the 'Girls of Great Britain and Ireland' Tiara, which was given to Queen Mary, The Queen's grandmother, on her marriage in 1893 and then to The Queen as a wedding present. Her Majesty often wears it and I thought it would suit the occasion, especially since its silhouette looks like a row of girls from Britain and Ireland holding hands.\n\nAs is often the case, the execution of the idea proved to be much more time-consuming and intricate than I'd imagined. To apply 2,000 shamrocks to the bodice of Her Majesty's dress, Tina in my team first had to create a template, using a digital sewing machine, then each shamrock was individually hand-stitched on. It was painstaking work, but of course, we always do whatever we need to do to get the job done. We also had to pay great attention when it came to introducing the Irish harp made of Swarovski crystals to the left shoulder, as Guinness, the famous stout producers, also use this symbol as their logo. While the harp was being made, I kept checking, over and over again, to make sure that we hadn't got it wrong! I know Her Majesty would not have been impressed if she had ended up sporting the Guinness logo on her shoulder, especially given her reaction to a new hairstyle I'd tried a few months earlier.\n\nOn that occasion I'd decided to stay blonde on top but go dark underneath, which was a fashion trend at the time. When The Queen saw me the following morning, she looked at me and asked, 'Have you ever seen a pint of Guinness?' I replied that, yes, I had, and her response was simply, 'Hmm,' as if to say, 'I will say no more.' Needless to say, I was straight down to the hairdresser the next day to change it back to blonde.\n\nOn the evening of the banquet at Dublin Castle, as The Queen entered the room, everyone stood as she walked to give her speech. Her dress looked phenomenal and I knew she loved the intricate shamrock detail. After the guests took their seats, Her Majesty began to speak and, as a total surprise to everyone except the Private Secretary, she greeted the President, ladies and gentlemen in perfect Irish Gaelic: 'A Uachtar\u00e1in agus a chairde,' meaning 'President and friends'. The whole room was awestruck and you could even hear a few guests exclaiming, 'Wow!' A massive round of applause followed. I had never seen such admiration, and I couldn't help but think the shamrocks on The Queen's dress had brought magic or luck to that significant event.\n\n_The harp design had to be clearly the Irish harp, and definitely not the Guinness one \u2013 which sits to the other side._\n\n_The green shamrock is hidden secretly on this dress._\n\n_The Queen visits a hospital in Enniskillen in Northern Ireland._\nINSPIRATION IS \nEVERYWHERE\n\nAs part of The Queen's extensive 2012 Diamond Jubilee Tour, she was due to visit Northern Ireland for two busy days of engagements. For this trip, I had two outfits to design: one for a Service of Thanksgiving in Enniskillen, County Fermanagh, after which Her Majesty and The Duke of Edinburgh would be attending St Michael's Catholic church. This would be The Queen's first visit to a Catholic church in Northern Ireland, so it would be an historic moment. The second outfit was for Her Majesty to wear while she and His Royal Highness travelled around the Stormont Parliament Buildings while standing in an open-top car.\n\nWhen I learned about the Service of Thanksgiving, I knew immediately what colour I would suggest for Her Majesty to wear. Given that she had worn green on her recent visit to the Republic of Ireland, I wanted her to wear something different for her first day of the trip and I had the perfect material in the stockroom \u2013 a beautiful pale blue. I've always found it fascinating that different colours can symbolise certain emotions or moods, and this soft hue seemed particularly fitting for the occasion. But I needed some additional design elements to bring the outfit to life. As always seems to be the way, I found inspiration in front of my eyes when I was walking through the corridors of Windsor Castle and noticed a beautiful Wedgwood collection. The china was exactly the same colour as the blue material I'd selected, and I decided to recreate its delicate pattern in fine lace for Her Majesty's dress and hat.\n\nSurely no one can visit the Emerald Isle without wearing some green, so for The Queen's second outfit I settled on a lime fabric. Knowing that there would be lots of crowds and commotion as Her Majesty and His Royal Highness drove around the Parliament Buildings, I wanted her to stand out in a vibrant colour. Everyone must be able to catch a glimpse of Her Majesty when they've come out especially to see her! I also wanted to soften the outfit ever so slightly and decided to introduce some gold lace detailing to the unusual double collar of the jacket, as well as to the hat. Finally, I thought a single gold feather placed diagonally across the front of the hat would bring everything together nicely.\n\nThe visit to Northern Ireland went brilliantly and Her Majesty's outfits were well received. At both engagements, The Queen looked elegant and stylish, and I think she was pleased with her reception by the crowds and the media. The following year, I received a request from Hillsborough Castle, the official Royal Residence in Belfast, asking if the blue and lime green pieces might be included in an exhibition, as well as the outfits that The Queen had worn on her visit to the Republic of Ireland. As well as my own designs, an outfit created by the late Karl Ludwig \u2013 a previous designer for Her Majesty \u2013 would be featuring. I'm always eager to enable as many people as possible to get a close look at Her Majesty's clothes, so we were very happy to oblige.\n\n_The Wedgewood inspired outfit in all its glory._\n\n_The Queen and The Duke of Edinburgh drive in an open-top car in the grounds of the Stormont Estate in Belfast in 2012._\n\n_The Queen wears a lot of green \u2013 and I wanted to give another take here. Using gold lace on the hat and collar, and the gold feather \u2013 something a little different \u2013 Her Majesty is still clearly visible among the colourful crowds._\n\n_The Queen with President George Bush at the White House in 2007. The black lace was placed over the crown of her hat, and then adorned with the black and white flower._\nTHE WHITE \nHOUSE\n\nAs the world's longest-reigning monarch, The Queen has met many Presidents of the United States. However, there was a moment on her fifth visit to the country in 2007 that will stay with me for ever: a true reminder of the joys of my job and just how fortunate I am to work so closely with Her Majesty.\n\nIt was May \u2013 a lovely warm time of year in the United States \u2013 and Her Majesty and The Duke of Edinburgh had been invited by President George W. Bush to visit the White House as part of a Royal Tour of the country. The Queen and The Duke were first due to travel to Virginia, the site of the first permanent English settlement, at Jamestown, in America, before attending the famous Kentucky Derby in Louisville. On their arrival at the White House, a State ceremony on the South Lawn had been arranged, which would be attended by thousands of people, and later that evening, the President was to host the first white-tie dinner of his presidency in honour of the occasion. I had never been to Washington before and I was invited, along with other members of the Royal Household, to have dinner in the Map Room \u2012 I knew I'd be pinching myself throughout the experience!\n\nAs always with a Royal Tour, I had lots of things to consider and many outfits to create. But on this occasion, I was thinking a lot about one engagement in particular: the ceremony on the South Lawn, for which The Queen was to arrive directly from the White House. I kept thinking about the brightness of the famous building, and what an amazing backdrop it would make for whichever outfit Her Majesty wore. I knew that it had to complement this setting perfectly, particularly as she would be making a speech in front of thousands of guests.\n\nWith this upcoming visit I wanted something different for the event. After considering many options \u2013 different colour combinations, different styles of hat, different cuts and materials \u2013 I knew exactly how to showcase Her Majesty's elegance and pay tribute to the occasion. Against the brilliance of the White House walls, only a simple but stylish black and white outfit would do, although to my knowledge she'd never worn a monochrome outfit before. I remember thinking, with a white jacket and a black lace collar, it could work! I also thought a white hat covered with the same black lace could look the part. So as part of my preparations for the tour, I excitedly set about finding the perfect material \u2013 a lightweight waffle weave \u2013 and sketching the shape of the white jacket with black lace detail shaped around the neckline in a scalloped effect.\n\nOn the day of the ceremony, I remember so clearly watching Her Majesty step out from the grand White House and greet the cheerful crowds alongside the President, The Duke of Edinburgh, and the President's wife, Laura. As America's First Lady, I am certain the pressure of looking the part was immense for Mrs Bush, too. But The Queen looked so lovely that day \u2012 I just watched and could hardly believe what I'd achieved. The Queen stood out in that iconic dress in front of that iconic building. I never normally take in a moment like that one because I have work to do \u2013 the next thing to consider, something else to prepare for. I'm only as good as my last task. But on that occasion at least, I remember thinking, job's a good 'un! I look back on the whole trip with such fondness. Since then, I've been to the United States a few times and have always been met with such a warm welcome, everywhere I've been.\n\n_The hat matches the lace collar and black buttons. We later trimmed the flower down a little for a lighter look._\n**'The State Visit of 2007 was the second time I welcomed Her Majesty to The White House. Weeks before The Queen arrived, I received a call from Her Majesty's Page, Barry Mitford, to discuss plans for the trip. I was struck by the vivid detail with which Her Majesty obviously recalled the floor plans and layout of The White House from her previous visit in 1991. Barry relayed that The Queen had met with his team and provided specific directions on where her luggage was to be placed and other logistical arrangements. Although he hadn't been there on this occasion, his directions were exact and spot on \u2013 Her Majesty is obviously observant of even the smallest details.**\n\n**The 2007 visit was the first time I met Angela Kelly. I recall that within minutes of her arrival, Angela came to me with two specific directions. Firstly, Her Majesty wished that the Blair House staff should serve her and The Duke of Edinburgh \u2013 a kind gesture that gave our staff the honour of being near The Queen. Secondly, I would be required to accompany Her Majesty in the elevator each time she used it \u2013 not because The Queen was unfamiliar with which buttons to use, but again as a kind gesture to allow us to be in Her presence.**\n\n**Angela Kelly was most gracious and allowed me to take a close look at several of Her Majesty's amazing pieces of jewellery. She told me that certain items belonged to Her Majesty's grandmother and that one in particular was a gift from her father, and I was entranced. It was also a wonderful opportunity to see some of Angela's own creations, such as the aquamarine ensemble Her Majesty wore to the Garden Party held at the British Embassy during that visit.**\n\n**Her Majesty and The Duke of Edinburgh were delightful guests and had an uncanny way of making everyone around them feel at ease. On the last day of their visit, The Queen presented me with a framed signed photograph of herself and The Duke of Edinburgh. Moments after I left the room, a Palace official came to find me and indicated that Her Majesty thought I should receive another gift as she had remembered that I had received a photograph after their visit in 1991. I was overwhelmed by how thoughtful she was.'**\n\n**_By Randy Bumgardner_ \nBlair House Manager**\n\n_The Queen on her visit to New York City in 2010._\nUNITED \nNATIONS TO \nGROUND ZERO\n\nWhile on a Royal Tour, The Queen is often required to attend sensitive and sombre occasions among her engagements. Often, these events happen in close succession as Her Majesty's schedule can be very tight, and it is on these occasions that we put even more thought than usual into her outfits for the day.\n\nIn July 2010, The Queen was due to travel to New York and had been invited to attend two important events, one after the other. The first was a visit to the United Nations, where Her Majesty was asked to deliver a speech, fifty-three years after she had first addressed the UN in 1957 at the age of thirty-one. The second was a visit to Ground Zero, the site of the former Twin Towers of the World Trade Center, destroyed by terrorist attack on 11 September 2001. The Queen had been asked to lay a wreath.\n\nFrom my perspective, designing something suitable for this occasion was a major challenge. The Queen would be moving from the very formal setting of the indoor auditorium at the United Nations to an outdoor event attended by bereaved families. Both events were serious, although very different in tone and mood, and Her Majesty would not have time to change outfits between them. I needed to find something that would be authoritative enough for Her Majesty's speech at the UN but not too severe and formal for the visit to Ground Zero. In terms of colouring, I needed something that would complement the imposing backdrop of the auditorium while allowing Her Majesty to stand out, but not overly vibrant considering her next engagement. In the end, I decided on a soft, floral material for the dress and jacket in muted blue, white, and metallic taupe. The edges of the jacket would be ruffled to suggest softness and femininity. I was pleased with the result and confident that it would be suitable for both events.\n\nWe arrived in New York to discover that the city was experiencing its worst heatwave in decades. The tarmac was melting onto the soles of people's shoes and the same was happening to ours! My immediate concern was that The Queen would be too hot with the long-sleeved jacket: thank goodness I had not opted for a heavier material! I needn't have worried, though: when I watched Her Majesty on television that day, her speech to the UN was calm and collected, and her presence at the Ground Zero wreath-laying ceremony provided comfort and reassurance to those who attended. It was a sad occasion and I couldn't help but reflect on my own good fortune. Although The Queen didn't say anything to me at the time, I could tell that she was deeply moved. We were all relieved when both engagements went smoothly and I was proud that Her Majesty's outfit had stood her in good stead.\n\n_When I first started designing, I used to buy the full roll so that no one else would be able to wear the same material, and would design multiple outfits from it. This silk material made a plain dress, a suit, a cocktail dress, and the trim for a cashmere coat. I am more thrifty now and just buy what is needed for the outfit I'm designing for specific occasions._\n\n_The Queen lays a wreath at the site of the World Trade Center in New York City in 2010._\n\n_The Queen with President Donald Trump at Buckingham Palace in 2019._\nAN IMPECCABLE \nHOST\n\nI imagine that there aren't many people who have the privilege of being in the presence of multiple Presidents, but I am lucky to be one of them. On 3 June 2019, Her Majesty welcomed President Donald Trump to Buckingham Palace. Arriving by helicopter, he touched down on the lawn that just a few days before had been host to 8,000 guests at The Queen's last garden party of the spring. It was quite a surreal moment to watch the Presidential helicopter approach \u2013 I'm sure you can imagine that I was watching attentively from the window.\n\nFor State occasions like this, everything must run like clockwork, and on this one, thankfully, it did. As soon as the helicopter's rotors came to a rest, the President and the First Lady disembarked and the engagements began. The First Lady looked stunning and really set the tone for what turned out to be a very stylish visit. That evening, Her Majesty hosted a State Banquet in President Trump's honour and each and every guest looked fabulous. As always, it was an honour to dress The Queen for such an important occasion. I had chosen a beautiful gown decorated with fine lace and ornate beading of small daisies and crystals \u2013 we named this dress 'Daisy Chain'. Her Majesty wore this with the Garter Sash and Star, as well as the Burmese Ruby and Diamond Tiara, the ruby and diamond necklace, earrings, and bracelet, alongside her white gold watch and ruby diamond rings. The whole outfit was complemented beautifully by a silver handbag and matching silver shoes.\n\nThroughout President Trump's visit, the media had many opportunities to take pictures, not only of the glamorous State Banquet but also of the ceremony to commemorate the seventy-fifth anniversary of the D-Day landing in Portsmouth. Preparing the outfit for this occasion took much consideration \u2013 Her Majesty needed to wear something special considering how many world leaders would be in attendance. In the end, I suggested a beautiful fuchsia coat, which, although striking in colour, was simple and elegant. Underneath, Her Majesty wore a floral-patterned dress and we accompanied it with a stunning cerise hat. On this occasion, choosing The Queen's jewellery was simple \u2013 I immediately knew that she should wear a brooch that has sentimental value for Her Majesty as she gifted it to her mother, Queen Elizabeth The Queen Mother, for her one-hundredth birthday. It would also complement the pink outfit.\n\nAs The Queen delivered her speech on that momentous occasion, standing next to the President of the United States, President Trump, I was reminded once again of how fortunate I am to be at her service and play a small part in dressing The Queen.\n\n_Close up of the Daisy Chain dress detail._\n\n_The Queen visits Thailand in 1996._\nA SNEAKY PLAN\n\nHer Majesty and I often have fun together, especially when we're away on a Royal Tour and I have the luxury of some spare time. Some time ago, in October 1996, Her Majesty went on a State Visit to Thailand and, as well as the beautiful landscape and stunning local scenery, one personal memory will stay with me for ever.\n\nIn the weeks leading up to that trip, I was preoccupied with preparing The Queen's outfits \u2013 on this occasion many vibrant lightweight silks because of the humid weather conditions \u2013 and I'd been so busy I'd simply run out of time to get to a hairdresser to refresh my roots. While I was reading the programme of engagements for the trip, I noticed there would be a market just across from the Royal Palace where we were to stay in Bangkok, and I mentioned to The Queen that I was planning to nip to a chemist in the market to buy a box of hair dye. I thought Ian Carmichael, The Queen's Hairdresser, should be able to colour my hair, no problem, while Her Majesty was out for a whole day on one of her engagements. The Queen was surprised by my plan, and insisted that it would be impossible because the box of hair colour would have no English instructions. I told her not to worry, and that it would be just fine, as the instructions are always the same, no matter what country you are in. I had to excuse myself quickly on that occasion so that The Queen did not see my smile \u2013 for I was coming up with a sneaky plan . . .\n\nWhen we arrived in Thailand, it was just as humid as we'd anticipated but everyone was so friendly and accommodating that everything went smoothly. A couple of days into the tour, Her Majesty and The Duke of Edinburgh were due to attend a Royal Barge Procession of 2,000 oarsmen dressed in traditional costumes, manning fifty-two barges, watching from the banks of the Chao Phraya river, so I thought this would be the perfect time to carry out my plan. That morning, I told The Queen I was going to the market to collect my hair dye, and she again told me she thought it would be a big mistake. While Her Majesty was out at her engagements, and once I'd finished all my duties, I let Ian in on my cunning plan: I had no intention of getting my hair done in Thailand \u2012 I was there to work. Before we'd left England, though, I had been to Accessorize and found a vibrant purple hairpiece made with fake hair. Ian knows my sense of humour, so he was more than happy to help and we spent some time attaching it in various ways to my head, to make sure it looked as realistic as possible.\n\nA few hours later, when The Queen returned from her engagement, she started to tell me enthusiastically about what a splendid time she'd had and all the interesting people she'd met. She did not notice my new purple hair, so I pulled the piece slightly further forward on my head, but Her Majesty still didn't notice and kept talking about her day. I realised I had to do something dramatic, so I waited until she'd stopped talking, turned with the most sorrowful look and said, 'Your Majesty! Look what's happened! Look at the colour of my hair!' I even started to pull the purple strands out, saying, 'Look, my hair is falling out!'\n\nThe Queen looked flabbergasted for a moment, then trying to contain her laughter, Her Majesty exclaimed, 'Ian! What have you done to her hair?!' Everyone heard the commotion but all I could do was laugh. The Queen was chuckling quietly to herself as she told me to go away. I'll never know how I got away with it, but Her Majesty has quite a sense of humour.\n\n_Ian Carmichael and me._\n\n_The beautiful splendour of the Taj Mahal._\nAN \nUNFORGETTABLE \nEXPERIENCE\n\nWhen Her Majesty and The Duke of Edinburgh travel abroad for a Royal Tour, we members of the Royal Household usually have a hectic schedule and have to make sure that everything goes smoothly. However, that doesn't mean we can't have a little free time to ourselves, and often it's on these little solo trips that we have the best fun. Usually, I miss out on such things because someone has to be there in case Her Majesty needs something. There was one time, though, when I managed to have a little fun of my own.\n\nIt was October 1997, and The Queen was on a Royal visit to Pakistan and India, which turned out to be an incredible experience. The atmosphere in both countries was magical, with music always in the air and a vast array of vibrant outfits to be seen. Over one weekend some of the Royal staff had had the opportunity to see the Taj Mahal as no Royal events were scheduled. I, along with a couple of others from the Royal Household had to stay behind as we can't all go on these explorations. However, given that we would both have liked to see the Taj Mahal, The Queen asked the Page to arrange for us to go on another day. Unlike the rest of the staff, though, we would be travelling in a tiny plane. As we made our way on board, I could tell this plane had many uses and functions, from transporting people, to cargo and probably more I would never even imagine. We were lucky enough to fly to the mausoleum and spend the day there, soaking up the rich historic surroundings. The colours were beautiful, and it is a most moving memorial to the love that existed between Shah Jahan and his chief consort, Mumtaz Mahal, who died in 1631. It's impossible to see that magnificent structure without mulling over their story. Before we left, I went to the souvenir shop and bought a plate that would remind me of the Taj Mahal's shimmering marble walls and the wonderful adventure we had had that day.\n\nA BIRD IN THE \nHAND\n\nHaving travelled so much, Her Majesty is the best tour guide ever, as she always takes the time to learn about the countries she visits. The Queen is full of interesting information and useful hints and tips. I never thought, in my wildest dreams, that I would have the chance to see Australia, so when we were preparing for my first Royal Tour there in 2006, I was so excited to hear about all the wildlife \u2013 the kangaroos we would see (red and grey), the koalas and dingoes. Her Majesty also mentioned the kookaburra and said that, although it is not the finest-looking bird, it makes a sound you will never forget. I asked if she thought I would see any, and Her Majesty replied, 'Oh, yes, many.'\n\nThe Royal Tour was going smoothly and we were all having a wonderful time. However, despite The Queen's confident response, I had not even heard a kookaburra, let alone seen one. I was a bit disappointed as I'd been so looking forward to seeing (and hearing!) the wildlife. Each time I heard an unusual bird call, I would turn to The Queen and ask her if that was the infamous call, and each time she said it was not.\n\nOne day, Her Majesty was due to visit the Sydney Opera House and the Harbour Bridge before a garden party to be held that afternoon in the grounds of the Governor's House, where we were staying. That morning, it was raining torrentially but, of course, The Queen would never let anyone down. After breakfast, she changed and left for the Opera House. As Her Majesty would be out all morning, the staff had an early lunch before we were all taken to the local indoor market. Walking around the stalls, I noticed that one vendor was selling birds. Among all the beautiful, brightly coloured ones I spotted a rather dull-looking brown bird and immediately thought that this must be the elusive kookaburra. I asked the shopkeeper, 'What is that unusual looking bird?' and she said, 'It's the Kookaburra.' So I bought it \u2013 I had something in mind that I thought The Queen would enjoy.\n\nBack at the Governor's House, Her Majesty wasn't due to return for a little while so I went up to her room to find the perfect perch for my new friend. I thought it might enjoy sitting in the sunshine which had just come out after the rain that morning, as it was used to hot weather, so I placed it outside on the balcony in a little cage. I waited in The Queen's room for her to return and when she came in, and walked across the room, I said, 'Your Majesty, there's a funny-looking grey bird on your balcony.'\n\nWithout a pause, The Queen looked outside and shouted, 'It's a kookaburra!' I went to open the cage doors and she said loudly, 'No! Don't do that! It will fly away!' and was behind me in a flash. I went to the bird, picked it up and, turning to Her Majesty while trying to keep a straight face, I solemnly told her it was dead. She looked horrified as I walked towards her with outstretched arms and as she took the bird from my hands, she realised I had been winding her up all along. It was really a stuffed toy. 'April Fool!' I said, with a mischievous grin on my face and she had only two words for me: 'You're sacked!' I was laughing uncontrollably as Her Majesty turned to His Royal Highness, and said 'Do you know what she has just done to me? Angela has had me!' and I just couldn't stop giggling.\n\nI never did get to see a real kookaburra, but I'm pleased to say that the toy I bought in the market that day is with us in Windsor Castle and has pride of place on the back of the sofa in The Queen's sitting room. I laugh to myself every time I see it.\n\n_Illustration by Sophia and Serena Polycarpou._\n\nWHAT A \nCORKER\n\nWhen I reflect on the various Royal Tours I have been so lucky to attend, it strikes me that Australia always seems to bring out my mischievous side. Of course, Royal visits abroad involve lots of planning and preparation, but they also offer the opportunity for fun and jokes, and one particular memory from our second tour to the continent in 2011 will stay with me for ever.\n\nThe Queen and The Duke of Edinburgh were due to attend some rather unusual events on that Royal Tour, including a barbecue, a tram ride, and a rainforest walk in Brisbane, so I had several things to consider when preparing my new collection for The Queen to wear.\n\nFirst and foremost, the weather would be a lot warmer there than it had been in London, so I wanted her outfits to be cool. This applied to the hats, too, and as I was researching, I remembered seeing photographs of Australians wearing hats with corks hanging from the brim \u2013 I think they might even be called cork hats! When I looked into this a bit more, I learned that the dangling corks were supposed to keep insects off your face. In a slightly mischievous mood, I decided to create an extra-special hat for Her Majesty to suit the occasion . . .\n\nThe journey to Australia went without a hitch and after we arrived at the Governor General's Residence, I began unpacking The Queen's luggage and helping her to prepare for her busy schedule. Aware that this was the day of the rainforest visit, I decided to present Her Majesty with two headwear options for the engagement. The first hat perfectly matched the dress, which was a Karl Ludwig creation made of satin silk in pinks and green with a pale green fine wool coat. The second was similar, with a crucial difference: it sported corks dangling from the brim. Eyeing the options, The Queen eventually looked up at me and asked, 'Why?' in a very matter-of-fact way. I explained that the corks might be of use in discouraging any flying insects from approaching her face. The Queen lifted the hat and pretended to put it on, laughing to herself. It's at a moment like that \u2013 sharing a joke with Her Majesty while on a visit to one of the most beautiful countries in the world \u2013 that I truly appreciate how lucky I am to work for someone with such a great sense of humour.\nMEETING THE \nLOCALS\n\nOver the years, I've visited some of the world's most amazing sights and most beautiful countries. Never in my wildest dreams did I expect to see the White House or the Taj Mahal, or the breathtaking mountain ranges of Canada. Before a Royal Tour overseas, I often have something in mind that I'd like to see there, should I have some free time, and during my second visit to Australia, I was determined to catch sight of a kangaroo. We were scheduled to be in the country for ten days \u2013 plenty of time, I thought, to spot one.\n\nAs the tour went on and we travelled across the magnificent landscapes, I had yet to catch a glimpse of one. As we made our way to Canberra, where we would be staying in Government House, I started to wonder if I would manage to see a kangaroo, having travelled all that way! I remember that it was the hottest weather I had ever experienced. The days passed, and as I went about my duties, I would pop my head out of the window now and again: perhaps I'd see something hopping along in the garden.\n\nOne day, Her Majesty was out on an engagement and all the staff were enjoying lunch. Everyone knew how much I wanted to see a kangaroo, so Barry Mitford, The Queen's Page, suggested that we all get changed into casual clothes and meet at the garden entrance to go kangaroo spotting. I had an idea: we should camouflage ourselves so that we could get as close as possible without being seen. So, like fools, a few of us scurried around and found twigs and leaves to put in our hair.\n\nAfter quite some time roaming around, trying to locate the animals we had been assured lived in the grounds, we finally spotted something \u2013 I knew immediately from the large, powerful back legs and feet that it was a kangaroo! It was unlike any creature I'd seen before and I wanted to get a closer look. So we started crawling along the grass towards it, as though we were carrying out an army exercise, like Private Benjamin. I can't imagine what we must have looked like. As we inched closer, Barry told me to keep aware: kangaroos can be quite aggressive. I could now see that it was a male red kangaroo, the largest of all kangaroo species. He seemed to be quite preoccupied with the grass he was chewing, but I noticed he was also keeping a watchful eye on us. We were getting really quite close, around 5 metres away, when all of a sudden, he started pounding his foot on the ground. The noise was so loud that we all leaped up in a panic and I could hear Barry shouting that we should run. We scrambled back to the safety of the house, twigs flying from our hair. The kangaroo was clearly unimpressed and I'm sure he thought he was teaching us a lesson \u2013 it certainly worked!\n\n_Trying to blend in with the background._\n\nCUTTING IT FINE\n\nThe Queen likes to support local people wherever she visits, but we have a particular little tradition whenever we go to Singapore. There's never enough time to go to the market, but as Her Majesty loves Singaporean silk, local tradespeople bring their stalls to the airport when she is due to arrive so that she can personally browse all that they have to offer.\n\nWe wander through the stalls \u2013 Her Majesty leading the way, followed by me and the Ladies-in-Waiting. Something will catch The Queen's eye \u2013 usually a particularly vibrant fabric, given her fondness for bright colours \u2013 and she'll make a gesture to me. Seeing lovely material like this is usually very inspiring and I'll quickly have an idea for a dress, a coat, a jacket, and maybe a hat \u2013 5 metres should be enough. Her Majesty will point to another roll and I'll envisage a cocktail dress \u2013 3 metres will be plenty. So, this is our routine: we amble around the stalls until we've found everything we're looking for, at which point, having placed our orders, we go off on the tour. Our selections are always ready for collection on our return, when it's my responsibility to pay the bill.\n\nA few years ago, after Her Majesty's Ladies-in-Waiting and I had collected our parcels of silk, I went to make the payment and the stall owners advised me that I could claim tax back on the purchase. The Queen is always keen to keep costs down, so off I went in search of someone who could help. A few minutes later, as I was getting the tax back, another member of the Royal staff rushed up to me in a panic, shouting that we had to go straight away as the plane was leaving right that second. Soon I was running across the tarmac, flying up the steps to the plane just in time. I was so thrilled, I shouted over the plane's engines a phrase I am sure The Queen does not hear very often: 'I've got your tax back!' The look on The Queen's face was priceless as I handed her the tax refund.\n\n_Alone with my thoughts in the Castle grounds._\nWHEN THE PAST \nAND PRESENT \nCOME TOGETHER\n\nAs well as all the events and work that gets done, there are also countless formal traditions that are upheld in the Royal Household, as well as some less formal ones that we members of staff look forward to each year. One, in particular, is a real favourite \u2013 a fancy-dress competition that is held twice a year at Balmoral, once in August and once in September so that everyone gets to take part. Balmoral Court is split into two halves, with a staff changeover at the start of September. Everyone goes to town for this one because, as part of the event, The Queen dresses in a beautiful evening gown \u2013 and even wears a tiara \u2013 to judge the winners of first, second, and third place. Her family might join together to make a judging panel if they happen to be there too. As fancy-dress night approaches, many secret meetings are held around the Castle. Trips are made to fancy-dress stores and charity shops, and you'll find wigs and papier-m\u00e2ch\u00e9 everywhere!\n\nA couple of years ago, we decided to change things up a bit as The Queen was taking a short break from the judging and was having dinner with the Royal Family and guests on the evening of the event. Of course, no one can replace Her Majesty, but we decided it would be fitting to have four 'famous' judges instead: members of the Household dressed up as Louis Walsh, Nicole Scherzinger and Simon Cowell, while I was to be Cheryl Cole. We even elected a comp\u00e8re to host the evening and announce the guests. But, of course, we still had our jobs to do. On the morning of the competition, I asked Her Majesty if she would mind that her dressers would be in fancy dress when they attended to her later that day \u2013 as we didn't want to be too late for our party. Without hesitation, The Queen said, 'No, no, no, you must!'. So we went about preparing our outfits and readying ourselves for the night ahead.\n\nYou can only imagine the look on The Queen's face when she saw me that evening wearing a red military jacket, black leather boots and a long, dark wig. She didn't have a clue who I was supposed to be, but she immediately recognised my assistant, Jackie, who was dressed as... Her Majesty's great-great-grandmother, Queen Victoria. Alongside Queen Victoria, our other dresser was made up as Victoria's close friend and supporter, John Brown. That year the theme for the fancy dress was 'Balmoral', so the staff had dressed up as various people and animals associated with the estate.\n\nJackie had put in so much effort, even searching online for tips on how to style her hair correctly, but she felt embarrassed as she stood there, wearing all black with her hair in plaited buns, before The Queen. But it went down a treat with Her Majesty. Just before she left for her dinner, she said, 'I suppose I'll see you tomorrow,' suggesting that we might all be suffering from slightly sore heads the next day. Then, on her way down the stairs, we heard her exclaim to her guests: 'I've just been dressed by Queen Victoria and Mr Brown! I've been dressed by Queen Victoria herself, and John Brown. And I believe by Cheryl Cole!' Just to add to Jackie's embarrassment, later that evening as she was doing a walkabout with other staff members in fancy dress, presenting her outfit to me and the other judges, her hand resting on 'John Brown's' arm, the door opened and there were three guests from The Queen's dinner taking pictures! We must have been quite the discussion point at dinner.\n\n_Jackie as Queen Victoria, with me as Cheryl Cole!_\n\nCONCLUSION\n\nDear Reader\n\nWriting this book has been a humbling experience. This is not a piece of historical writing, but an amazing journey and the absolutely true facts of living and working with The Queen over the last twenty-five years \u2013 long may it continue.\n\nThis experience has taught me many things, particularly having patience with myself; while continuing to serve Her Majesty The Queen, and fulfilling all my duties to the high standard I set myself, I have managed to write this book alongside my daily working life.\n\nIt was a joy to reminisce with The Queen over these stories and now having the honour of sharing them with you. The Queen was with me every step of the way.\n\nThough my beginnings were humble, it took me many years to believe in myself \u2013 to have the courage and determination to make something of myself.\n\nThere have been many changes since my early years in the Royal Household. For example, it is not a man's world any more; it is a fabulous professional working environment for men and women in Buckingham Palace today. I get to work with some of the most talented, kind, and inspiring people.\n\nI hope this book has given you the inspiration and courage to follow your dreams like I did. My wish for you is that they all come true.\n\nThank you for reading my book and I hope we meet again . . . until next time.\n\nLove, Angela Kelly \nX\n'Since meeting Angela eleven years ago, my life has never been the same. Working for her on the Dressers' Floor has been a joy and a privilege.\n\nAngela fills me with confidence and is always so positive and enthusiastic that it rubs off on me and makes me feel like I can achieve anything.\n\nShe is warm, funny, and has a big heart, but when it comes to reaching deadlines, she puts her serious head on and will always find a way to solve any problem. She is great to work for and we have mutual respect for each other.'\n\nStella\n\n* * *\n\n'It is an honour to work for Her Majesty and alongside Miss Kelly, one of the designers responsible for turning The Queen into a fashion icon all across the globe. Miss Kelly's loyalty to her job is a constant inspiration to us. She is a generous and fun-loving Mother Hen to our Dressers' Floor family.'\n\nThe Dressers\n\n* * *\n\n'When asked to write a brief paragraph or two about my working relationship with Angela, I thought to myself, Where do I start?.\n\nWhen you have worked and travelled around the globe with the most famous person in the world for more than twenty years, it can never be anything less than an honour \u2013 full of excitement and memories to keep for ever.\n\nWorking with Angela and seeing her in action is quite amazing to witness. She is a total professional, through and through.\n\nI have been extremely lucky to have known Angela as a dear friend and I'm sure she will agree that we have had many, many great laughs together.\n\nThis is a lady with an amazing sense of humour who is not afraid to laugh at herself, but most of all has great interest in the needs of others, which I simply adore.\n\nShe is a pleasure to work with and I pray that we have many more years together of working hard and laughing hard.'\n\nIan Carmichael\n\n* * *\n\n'My working relationship with Angela began about eighteen years ago.\n\nI joined the Royal Household in 2001 and was working in a different department. It wasn't long before I heard the name \"Angela Kelly\" mentioned and that she was a woman not to be messed with.\n\nNeedless to say, I expected the worst when we first met while I was undertaking luggage duties.\n\nThe truth was quite different. I met Angela and the rest of her team and everyone was very welcoming. Over the years, as I performed my luggage duties, I built up a relationship with Angela.\n\nAngela guards The Queen's privacy relentlessly and this crosses over into her personal domain. If Angela needed any duties seen to in her private rooms, I was among one of the few people she trusted.\n\nThis all changed in early 2018 when, unbeknown to myself, I was being headhunted by Angela. I headed up to the Dressers' Floor one day to undertake a job. Before I knew it, we were in discussions about me joining the team. I was stunned to say the least.\n\nSo, here we are eighteen months later and I am Project Manager for our department, among many other duties. I am deeply grateful to Angela for seeing something in me that she wanted to bring to her team.\n\nAngela is a wonderful boss and a pleasure to work for. Yes, we have our busy days when we have to knuckle down, but Angela always tries to make it fun and keep everyone's spirits high.\n\nI may have only been with the team a short time but I have been made to feel very much part of that team. During this time, I have noticed how Angela is a good friend to those close to her and loyal to the core.\n\nI have yet to meet the formidable woman I was warned of so many years ago.\n\nWorking with Angela is a pleasure and I will be eternally grateful for the many exciting opportunities she has given me.'\n\nNeil\n\n'After working for the Royal Household as the Administrator for the F Branch Office (Catering) for ten years, I became Angela's Personal Assistant on 12 August 2013.\n\nI love being a part of the team on the Dressers' Floor. If you were to come up and visit us, you would hear laughter coming from the workroom, music playing quietly in the background and you would hear the girls singing away (they have great voices, by the way!), and that would all be happening before eight in the morning! Then you would hear, \"Morning, girls\" and there would be Angela all set and raring for the day to start. She is always good to go!\n\nAngela is an amazing manager and boss \u2013 although both words are a struggle for me to use as Angela is more like your best friend. Angela has always supported my decisions, even if on the odd occasion they haven't been the best ones; she will always have my back. The love she has for her family shows every time we talk about them. She does have rather a soft spot for her grandchildren! Angela is so incredibly loyal and has allowed me to share with her some amazing adventures. From travelling to New York City to meet with Annie Leibovitz, to almost missing our train coming back from Glamis Castle, to also being trained as an Assistant Dresser and being able to revisit Balmoral Castle and Sandringham House.\n\nIt has been on these occasions, travelling as an Assistant Dresser, when I have seen Angela and Her Majesty interact with each other and I have been able to see their unique relationship shine through. I have experienced so much laughter on these occasions. I mean, come on, I dressed up for our evening duties as Queen Victoria! I will never forget that evening when The Queen saw us all and shared a laugh with us about it.\n\nThere was also a time when I was waiting for Angela, who was helping The Queen get ready for the State Banquet during a State Visit. In the reflection of the mirror, I could see Angela helping The Queen secure the tiara onto her head, all the while hearing the banter and laughter being shared. It was definitely a \"pinch-yourself\" moment for me when I curtsied to The Queen as she passed me to leave the room.\n\nNever would I have thought that this small-town girl from Maine, USA, would ever be in the position that I am in today. Thank you, Angela! Long may our crazy, fun adventures continue.\n\nI truly hope you will be able to get a sense of Angela's warmth, dedication, and the commitment she has as the Personal Advisor to The Queen, and that you will also be able to imagine that you are in the same room experiencing the laughter that is shared between Her Majesty The Queen and Angela as you read this book.'\n\nJackie Newbold MVO\n\nACKNOWLEDGEMENTS\n\nYour Majesty, I would like to personally thank you for giving me your permission and blessing to write this book. I also thank you for your guidance and advice. Thank you for putting up with me for all the times when I have gone back and forth to you, checking and double-checking the stories. I am truly thankful for the time you gave me and for allowing me to share this book with the world. I am your humble servant.\n\nTo my children and grandchildren, once again I thank you for your continued love and support. I am forever grateful to you for understanding the way I balance my working life with the special times that I spend with you. Special thanks to Jacob Wylie for helping me finalise the book.\n\nI would like to thank June Anson for not only being a part of my family but for your friendship, company, and loyalty.\n\nI would also like to thank Paul and Margaret Grayson who have not just been part of my family but have been my closest friends. Thank you for your constant support, friendship, and loyalty.\n\nElizabeth Sheinkman, you have been my protector and my guardian and I thank you from the bottom of my heart for helping this book come alive and for making sure the working relationship and the magical moments were truly represented.\n\nMy gratitude and thanks go to you all at HarperCollins for making this book beautiful, elegant, and regal. The hard work that has gone into this book has been outstanding. Firstly, I would like to give a special thanks to Katya Shipster, Zoe Berville, Lucy Brown, and Claire Ward for working tirelessly to help and support me. You have all gone beyond the call of duty. I would also like to thank: Abi Hartshorne, James Empringham, Georgina Atsiaris, Sarah Hammond, Sarah Burke, Fiona Greenway, Liane Payne, Fionnuala Barrett, Josie Turner, Dawn Burnett, Tom Dunstan, Anna Derkacz, Hazel Orme, and Helena Caldon. I would also like to thank Jonathan Burnham and Lisa Sharkey from the US branch of HarperCollins. It was wonderful to share the same enthusiasm and excitement. Thank you for fully understanding the sensitivity and the discretion that surrounded this book.\n\nTo do the audio recording of this book terrified me! Thanks, therefore, to Alex at ID Studio, who encouraged me, and kept me calm and within the hours. We had a great understanding and a great working relationship which made the audio a success.\n\nI would like to thank the members of the Royal Household (too many to name individually) as you have been great supporters and advisors which has helped to make this unique book so special.\n\nThank you to the Royal Collection for your co-operation, especially Tim Knox, who has been a great supporter from the first time we were introduced.\n\nI would like to thank you, Samantha Cohen, for all your hard work and support you have given me, not only with my first book, Dressing The Queen, but also with this book. You have been my constant supporter from the first day we met many, many years ago. I truly value our friendship and our working relationship.\n\nMichael Atmore, I would like to thank you for your professionalism, for your help and guidance. Your constant support has been vital and I appreciate you being there for me day and night. You have gone above and beyond what anyone could ever ask for. I am so proud to call you my friend.\n\nThe team on the Dressers' Floor has a unique bond, just like a small family. Your support that you give me makes my work so much easier so I am able to look after The Queen. I really appreciate all your hard work. You truly are the cr\u00e8me de la cr\u00e8me!\n\nThank you to all the photographers for allowing me to use your beautiful photographs and for the wonderful contribution you have made to make this book so special. A picture says a thousand words! I would like to say a special thank you to Eva Zielinska-Millar for your advice and guidance.\n\nAndy and Sharon Polycarpou \u2013 I would like to thank you for allowing your two beautiful daughters, Serena and Sophia, to spend time with me discussing the illustrations for this book. They are two very talented girls. Serena and Sophia \u2013 I would like to thank you for putting my vision to paper and for contributing to this book and making it beautiful. I know it was a busy time for you as you were still in school, so thank you for finding time for me. May I wish you all the best for future. Don't let your talent go to waste. You are both an inspiration to all young people.\n\nDavid Morgan-Hewitt at The Goring hotel: although you did not know I was writing this book, our friendship, over the many years, has meant so much to me. You are a true friend and I have enjoyed your company. I look forward to our next dinner, whenever our busy schedules allow it.\n\nJackie Newbold, there are so many things I would like to thank you for, but too numerous to mention them all. I do want to thank you for your endless support as a true friend. You astound me with your calmness, patience, and determination to help bring this book together with me. Thank you for putting up with me and for all of my changes, making sure I got it right. With your help and support you never gave up on me. You believed in me and helped make this book what it is today. I could not have done this without you.\n\nA special thanks and gratitude goes to you, the reader. I do hope you enjoyed reading my book and that, once again, you caught a glimpse of this magical world. I hope you cherish this book, like I cherish my memories.\n\nWith love, Angela x\nPICTURE CREDITS\n\nPage numbers refer to printed numbers. Photographs courtesy of the author with the following exceptions:\n\n\u00a9 Annie Leibovitz p180; \u00a9 Barry Jeffery p92, p13, p15, p35, p49, p52, p54, p56, p57, p67, p68, p74, p79, p80, p81, p82, p84, p85, p86\u20137, p88, p100, p101, p108, p110, p111, p116, p117 (top), p123, p149, p156\u20137, p159, p160\u2013 61, p162,163, p168\u20139, p184, p187, p188\u20139, p216, p228, p231 (top), p242, p247 (bottom), p251, p255, p258, 259, p261, p264, p268, p269 (bottom), p272, p273, p274, p296, p299, p300; \u00a9 Chris Levine p192, p195; \u00a9 Julian Calder p126, p127, p128, p129, p130; \u00a9 National Museums Scotland p121; \u00a9 Nina Duncan p196, p199; ADRIAN DENNIS\/AFP\/Getty Images p113; AFP Photo p166; Antoine GYORI\/Sygma via Getty Images p276; Anwar Hussein\/EMPICS Entertainment\/PA Images p44, p102; ARTHUR EDWARDS\/AFP\/Getty Images p16; CARL COURT\/AFP\/Getty Images p136; Chris Jackson\u2013Pool\/Getty Images p240, p243; Chris Jackson\/Getty Images p96, p148, p232, p236, p246, p247 (top); Chris Jackson\/Getty Images for Ascot Racecourse p105 (bottom left); Christopher Furlong\/ Getty Images p211; EQRoy\/Shutterstock.com p290; FRANK GUNN\/ The Canadian Press\/PA Images p238, p239; GARY DOAK\/Alamy Live News p122; Henny Ray Abrams\u2013Pool\/Getty Images p266; Jeff Gilbert\/ Daily Telegraph\/PA Wire\/PA Images p270; John Phillips\/Getty Images for Ascot Racecourse p106; John Stillwell \u2013 WPA Pool\/Getty Images p150; John Stillwell\/PA Archive\/PA Images p235, p226, p269 (top); Julian Parker\/ UK Press via Getty Images p145; Julien Behal\/PA Archive\/PA Images p256; KAZUHIRO NOGI\/AFP\/Getty Images p46; Kirsty Wigglesworth \u2013 Pool\/ Getty Images p206; Leon Neal\/ WPA Pool\/Getty Images p138; Leon Neal\/AFP\/Getty Images p141; Leonard F\/Shutterstock.com p289 (main); Luke Macgregor \u2013 WPA Pool\/Getty Images p134; Maja Smiejkowska\/ Shutterstock p222 (top); Marco Secchi\/Corbis via Getty Images p118; Mark Cuthbert\/UK Press via Getty Images p104, p114, p262; MARWAN NAAMANI\/AFP\/Getty Images p248; Max Mumby\/Indigo\/Getty Images p18, p72, p152, p224; Neal Baker\/Shutterstock.com p183; OLIVIER MORIN\/AFP\/GettyImages p170; PA\/PA Archive\/PA Images p50; Paul Gilham\/Getty Images p105 (bottom right); PETER BYRNE\/AFP\/Getty Images p208; PETER MUHLY\/AFP\/Getty Images p252; Peter Muhly\/ AFP\/GettyImages p260; Photography by Julian Calder, Camera Press London p172\u20133; Pool\/ Anwar Hussein\/WireImage\/Getty Images p244; Royal Collection Trust\/All Rights Reserved p58, p61, p63; Royal Collection Trust\/All Rights Reserved; Photographer: Eva Zielinska-Millar p62; Samir Hussein\/Wire Image\/Getty Images p14; Scott Prokop\/Shutterstock p231 (bottom); Shutterstock.com p32, p40, p90, p220; Steven Booth\/Alamy Stock Photo p146; Tim Graham Photo Library via Getty Images p124, p142, p212; Tim Rooke\/Shutterstock p250; Times Newspapers\/Shutterstock p215; Tristan Fewings\/BFC\/Getty Images p205; W & D Downey\/Getty Images p154; YUI MOK\/AFP\/Getty Images p204; Yui Mok\/PA Archive\/ PA Images p200.\nABOUT THE PUBLISHER\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\n\n\nCanada\n\nHarperCollins Canada\n\nBay Adelaide Centre, East Tower\n\n22 Adelaide Street West, 41st Floor\n\nToronto, ON, M5H 4E3, Canada\n\n\n\nIndia\n\nHarperCollins India\n\nA 75, Sector 57\n\nNoida, Uttar Pradesh 201 301, India\n\n\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\n\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n1 London Bridge Street\n\nLondon SE1 9GF\n\n\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n195 Broadway\n\nNew York, NY 10007\n\n\n\n# Table of Contents\n\n 1. Cover\n 2. Title Page\n 3. Copyright\n 4. Dedication\n 5. Dear Reader\n 6. Letter From Samantha Cohen CVO\n 7. Foreword\n 8. Prologue\n 9. Chapter 1: Dressing The Queen\n 10. Chapter 2: Our Working Year\n 11. Chapter 3: The Eyes of the Nation\n 12. Chapter 4: On the Move\n 13. Chapter 5: The Eyes of the World\n 14. Chapter 6: The Things We Do For Fun\n 15. Conclusion\n 16. Acknowledgements\n 17. Picture Credits\n 18. About the Publisher\n\n## Guide\n\n 1. cover\n 2. contents\n 3. title\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\n\n\nProduced by Chris Curnow and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net (This file was\nproduced from images generously made available by The\nInternet Archive)\n\n\n\n\n\n\n With the Compliments of\n J. Charlton\n _General Passenger and Ticket Agent_,\n CHICAGO AND ALTON R. R.,\n CHICAGO.\n\n [Illustration: A Parody\n on\n PRINCESS IDA\n by\n _D. DALZIEL._]\n\n[Illustration: GLORIA ALTON]\n\n[Illustration: KING GAMA ARRIVES.]\n\n[Illustration: WE ARE WARRIORS THREE.]\n\n\n\n\n A PARODY\n on\n PRINCESS IDA\n\n BY D. DALZIEL,\n _Editor of the Chicago News Letter._\n\n\n SCENE--_Interior of King Alton Hilderbrandt's Palace in the\n City of Chicago, U. S. America. Courtiers, Officials of the\n Service, Soldiers of the Railroad, Body Guard, Conductors\n and Passenger Agents, looking out on the line of the\n Chicago and Alton Railroad with telescopes, opera glasses,\n etc._\n\n\n [CHORUS.]\n\n Search throughout the Panorama\n For a sign of Royal Gama,\n Who by the Alton this day shall,\n Accompanied by his child and pal,\n Come from Adamant.\n\n Some misfortune evidently\n Has detained them--consequently\n Search throughout the Panorama\n For the daughter of King Gama,\n Prince Hilarion's Plant.\n\nFLORIAN--Will Prince Hilarion's hopes be sadly blighted?\n\nALL--Who can tell!\n\nFLORIAN--Will Ida slight the man to whom she's plighted?\n\nALL--Who can tell!\n\nFLORIAN--Can she an Alton ruler thus irritate?\n\nALL--Who can tell!\n\nFLORIAN--If so, she'll pay for it at any rate.\n\nALL--\n\n We'll not despair,\n For Gama would not dare\n To make a deadly foe\n Of Alton's King, and so\n\n Search throughout, etc.\n\n (_Enter_ KING ALTON HILDERBRANDT.)\n\nHILDERBRANDT--No sign of Gama yet? The regular train from Castle\nAdamant came in exactly on time, some fifteen minutes since. I\nwill here in parenthesis remark, that all trains on the Alton Road\ninvariably _do_ come in on time. I saw it from my castle window; four\nsleepers, two drawing-room cars, a dining car, three reclining-chair\ncars and five coaches.\n\nFLORIAN--Your liege forgets that you placed a \"special\" at his\ndisposal, but still he should be here.\n\nHILD.--'Tis true, but still it's very odd. But, if Gama fails to put\nin an appearance at the Court before the sun sets, accompanied by his\ndaughter, to whom our son was betrothed at the extreme age of one, then\nthere will be war between Gama and ourselves. (_Aside._) I dread this\ngreeting. When last I saw Gama, twenty years ago, he was a fretful,\ntwisted monster, with a tongue as bitter as the agents of our rival\nlines.\n\nFLORIAN--(_Who has been looking attentively down the track with his\nglasses._) But stay! I see the smoke of an approaching train. I\neven hear the music of its wheels, as they fly like magic o'er the\nglittering rails. Already it is at the station. From a private car I\nsee somebody; guards descend, and now a bent and crippled form of human\nshape. It _must_ be Gama.\n\nHILD.--And is the Princess with him?\n\nFLORIAN--Not unless she's six foot three, and wears suspenders.\n\nHILD.--Who can tell! I've heard that she is a whimsical sort of girl.\nBut come! Bustle up there. Let everything be prepared. The costliest of\ndinners, the meanest of prison fares; the richest room in the palace,\nthe deepest dungeon, too. For as King Gama brings his daughter, or\nbrings her not, so shall he be treated.\n\n[Illustration: HILARION AND HIS FRIENDS.]\n\n[Illustration: THE CHICAGO & ALTON DEPOT.]\n\n[Illustration: HILDERBRANDT (furious): ENOUGH OF THIS, WHERE IS YOUR\nDAUGHTER?]\n\n (_Enter_ PRINCE HILARION, _Knight of the Road, Grand Master of\n the United Order of Chicago and Alton Railroad Conductors, and an\n exquisite_.)\n\nHILARION--(_Aside._) To-day I meet my baby bride, to whom I was\nbetrothed twenty years since. I have every confidence that my own\naccomplishments, and the influence of the line with which I am\nconnected, will suffice to re-unite our bonds to-day. They are above\npar on the market. (_To Hild._) Well, father, is there any news?\n\nHILD.--King Gama has just arrived, and is now on his way here from our\nd\u00e9p\u00f4t; but I fear without the Princess.\n\nHILARION--Alas! I have heard strange stories of my bride. They tell\nme she has foresworn men--never having had anything to do with them\nshe probably does not know what she loses. But they say she has shut\nherself up in the Castle Adamant accompanied by a number of devotees.\nIt is strange, considering that she lives so near our line that the\nmere proximity of our Alton men has not had its usual influence before\nthis.\n\n (_Enter a representative of the Chicago and Alton Railroad._)\n\nREP. OF C. & A. R. R.--My liege, I beg to announce the coming of King\nGama, preceded by his three royal sons.\n\nHILD.--Bid them enter.\n\n (_Enter_ ARAC, GURON _and_ SYNTHIUS, _warriors bold_.)\n\nARAC--\n\n [SONG.]\n\n We are warriors three,\n Men of worth are we,\n Just arrived by steam,\n Strange as it may seem.\n\nALL THREE--\n\n Yes! yes!\n Strange as it may seem.\n\nGURON--\n\n Special train we had,\n Not so very bad,\n Really very fine\n By the Alton Line.\n\nALL THREE--\n\n Yes! yes!\n By the Alton Line.\n\nSYNTHIUS--\n\n Dining-car so rich,\n Sleeping-car like which\n Nothing can compare\n Running time so rare.\n\nALL THREE--\n\n Yes! yes!\n Nothing can compare.\n\nARAC--My liege, we beg to present our respects, and to thank you for\nour pleasant journey on your line. It is our unpleasant duty also to\nannounce the coming of our Royal Father.\n\n (_Enter_ KING GAMA.)\n\nGAMA--\n\n [SONG.]\n\n If you give me your attention I will tell you what I am,\n I'm a genuine philanthropist--all other kinds are sham.\n Each little fault of temper and each social defect\n In my erring fellow creatures I endeavor to correct.\n To all their little weaknesses I open people's eyes,\n And little plans to snub the self-sufficient I devise.\n I love my fellow creatures, I do all the good I can,\n But everybody says I'm such a disagreeable man,\n And I can't think why.\n\n For all who do me favors I've a withering remark,\n And as for beating railroad lines, I think it quite a lark,\n I wheedle out free passes, and I call the road a sham,\n I sell them to the scalpers, that's the sort of man I am.\n I put rocks upon the railroad tracks, and try to wreck the train,\n Get a madman hired as engineer, and swear the man is sane.\n I spread all sorts of rumors, and have all the fun I can,\n Yet! Everybody says I'm such a disagreeable man,\n And I can't think why.\n\n When I get into a sleeper I sit up half the night,\n And try to make the ladies think that everything's not right.\n I cut in strips the cushions, the curtains always tear,\n And pleasantly suggest aloud: \"They are the worse for wear.\"\n I go into the smoking car, and remark I never smoke,\n And when the weeds are all put out, I explain it's all a joke,\n You see, I try to make myself as pleasant as I can,\n Yet! Everybody says I'm such a disagreeable man,\n And I can't think why.\n\n And yet, with all my little whims I really must decline\n To say a single word against the fav'rite Alton Line.\n To speak in praise of anything goes hard against the grain,\n But I really never was upon such a perfect train.\n I had dinner in a dining car, which every want could fill,\n And I drank my health quite frequently, and didn't pay my bill.\n You see I try to make myself as pleasant as I can,\n Yet! Everybody says I'm such a disagreeable man,\n And I can't think why.\n\n[Illustration: HILARION CHECKS HIS BAGGAGE.]\n\n[Illustration: ON THE DINING CAR.]\n\nGAMA--So this is the royal castle of the Chicago and Alton Railroad?\nDame Rumor whispered that the place was on a par with everything\nconnected with the Alton Line. Superb! Royal! Grand!\n\nHILD.--(_Gratified._) Oh! Sire.\n\nGAMA.--But she's a liar. And this is your son! Dame Rumor also said he\nwas the leader of fashions among the Alton employes, but he's changed.\nHe used to be a singularly handsome child.\n\nHILD.--(_Furious._) Enough of this. Where is your daughter, whom on\nyour plighted word should be here to-day?\n\nGAMA--She would not be tempted. She hates railroad travelling. I\ndon't blame her. I tried to explain that a journey on the Alton was a\ndelicious reverie set to music on wheels, but she was obdurate. She's\nin Castle Adamant foresworn to all men, surrounded by a hundred maidens\nfair, who think as she does.\n\nHILD.--Surely we can win her over?\n\nGAMA--Not you! She and her companions will have nothing to do with\nanything masculine. One young lady brought a mail phaeton with her\nto the castle and she was expelled. When the time comes for the fast\nmail for Kansas City over your line to pass, the young ladies shut\nthemselves in the basement cellar. Perhaps the never failing graces of\nyour Alton men may win her.\n\nPRINCE HILARION--Well! I'll try (_to Cyril and Florian_), and you shall\naccompany me. Come quick! Your satchel. We have just time to catch\nthe fast train for Castle Adamant, and we will see whether the graces\nof one Alton man won't change the tenor of these young ladies' ideas.\nQuick! secure us berths. (_Exit officials._)\n\nHILD.--Yes! and you, King Gama, and your baby boys shall remain here\nas hostages. (_Re-enter officials._) Sorry to say, sir, that every\nberth in the five sleepers is taken. Nothing left to swing a duster in.\n\nHILD.--Then take our special car (_exit officials to give orders_),\nand now, King Gama, a word to you. If aught befall the flower of this\nroad, look out for your life. Take him to a cell and give him one of\nour time tables to pass away the time. (_Exit GAMA and his three sons\nin chains._)\n\nThe scene changes to the d\u00e9p\u00f4t of the Chicago and Alton Railroad.\n\n_With the prince and his two friends checking their baggage and then\nto the gardens at Castle Adamant, a charming retreat on the line of\nthe Chicago and Alton Railroad. In the distance can be seen the steel\ntracks of the Chicago and Alton Railroad glittering in the noonday sun.\nThe air is calm and balmy, disturbed only by the occasional song of\nsome bird, or the musical hum of a distant train on the Alton Line, as\nit whizzes with lightning speed along the track. The lady graduates of\nthe Princess Ida's seminary are on the scene, seated on the lawn at the\nfeet of Lady Psyche._\n\nLADY P.--Attention, ladies, while I read to you the list of punishments\nordered by the Princess Ida. The first is Claire. She's expelled.\n\nALL--Expelled!\n\nLADY P.--Yes, expelled! Because, although she knows that no man of any\nkind may scale these walls, she brought a set of chessmen here. The\nnext is May. She is also expelled for receiving letters.\n\nMAY--They were from my mother!\n\nLADY P.--It matters not, they came by mail. The next is Edith. She is\nalso expelled for being found with a Chicago and Alton folder in her\npossession.\n\nEDITH--What harm is that?\n\nLADY P.--A young lady of this seminary is not allowed to harbor things\nthat fold-her. The next is Blanche. She is also expelled for looking\nout of her window when the Chicago and Alton train was passing.\n\nBLANCHE--I was only looking out for the sleepers, they are so--\n\nLADY P.--Enough! Looking out for sleepers is not a square game to play.\nEven horrid man condemns it. And now, young ladies, prepare yourselves,\nfor your Princess Ida will be here directly to read you the lessons of\nthe day.\n\n[Illustration: PRINCESS IDA AND HER COMPANIONS.]\n\n[Illustration: PRINCESS IDA AND THE ALTON ANGELS.]\n\n[Illustration: A CHANGE OF COSTUME.]\n\n[Illustration: THE ALTON MEN ARE APPRECIATED.]\n\n (_Enter the_ PRINCESS IDA.)\n\nIDA--Women of Adamant, gathered together for the noble purpose of\nshunning that vile creature known as man, listen while I a tale unfold.\nOur undertaking is a noble one. We, women assembled as we are in this\nretreat, know full well the uselessness of man. Let nothing tempt\nyou from your path of sworn duty, and yet! methinks, the dangerous\nproximity of that horrid Alton road is a glittering temptation to some\nof your thoughtless heads. If by chance your eyes should light upon one\nof the creatures they call their officials, let not their gaudy tinsel\nturn your giddy brains. It is true that they dress with exquisite\ntaste, that their manners are those which Lord Chesterfield so highly\ncommended, that they belong to the only railroad corporation in the\nworld which even _I_ can look upon with feelings akin to admiration. It\nis true also--this fact I cannot deny--that the equipment of the Alton\nLine is as near perfection as human ingenuity can make it. That their\ntrain service is perfect, their cars superb, their track as smooth as\nthe skin on my face, and that they run the only through train worth\ntaking between Chicago and Kansas City, and all points West. I can say\nthis fearlessly. Their tracks are but steel, their cars merely wood and\nmetal, and we have nothing to fear from wood and metal. But beware of\ntheir officials. Do not be deceived into thinking that because a man\nis godlike in appearance, he is anything but a man after all. With all\ntheir grace, their outward charms and pleasing manners, these creatures\nwould win your hearts, and wear them too, with all the assurance with\nwhich one of us would knock the top off an egg for breakfast. This much\nbeing said, I beg you to follow me to partake in the usual exercises of\nthe day.\n\n (_Exit Princess, followed by the ladies of the seminary. The\n sound of an engine whistle is heard. Then enter HILARION, CYRIL\n and FLORIAN at back, creeping cautiously._)\n\n TRIO.\n\n Gently, gently,\n Evidently\n We are safe so far.\n The Alton Train,\n With sweet refrain,\n Brought us, here we are!\n\n Flying lightly,\n Whizzing sprightly,\n O'er the Alton Track.\n We expect to\n All invite you\n When we journey back.\n\nHILARION--Hush, scoffer! Despite the advantages of our remarkably\nexcellent line, it is proper for you to remember that you can't work\noff any tickets here in this college of learning. These ladies are\ngoing to do wonderful things, I am told. List to their intentions:\n\n They intend to live alone\n From us men; from us men.\n Like the doggy with his bone,\n Do you ken; do you ken?\n\n Then they're greedy with their graces,\n Selfish girls; selfish girls.\n And they hide their pretty faces\n And their curls, and their curls.\n They've a very queer intention\n It is sung, it is sung:\n To resist polite attention\n From the young, from the young.\n\n They've another funny notion,\n They can do without devotion\n From an Alton man--what notion--\n If they can, if they can.\n\n These are the Phenomena,\n That every pretty Domina\n Hopes that we shall see\n At the Universitee.\n\nCYRIL--\n\n As for Railroads they are ready\n To admit; to admit\n That their employes are steady\n And they sit, and they sit\n A dreaming of the Alton men\n At their best, at their best;\n And waiting for the hour when\n They pass West, they pass West.\n But, each newly made aspirant\n To the clan, to the clan,\n Must repudiate the tyrant\n Known as man, known as man.\n\n The Alton Road Conductors\n They regard as mere seductors,\n And they're going to do without them\n If they can, if they can.\n\n These are the Phenomena\n That every pretty Domina\n Hopes that we shall see\n At this Universitee.\n\n[Illustration: HILARION IN CLOVER.]\n\n[Illustration: I'M A PEPPERY SORT OF A KING.]\n\n[Illustration: FREE AS A TETHERED ASS.]\n\n[Illustration: OH! DON'T THE DAY SEEM BLANK AND LONG.]\n\nHILARION--And so here we are, in the breast of the citadel. Hallo!\nwhat's this? (_Examining some robes left by the female graduates._)\nWhy! Academic robes, robes worn no doubt by some of the undergraduates.\nHere, you fellows! Put them on. (_They do so, and the robes reach\nto their feet. They look as any well-regulated Alton man looks in\nanything--extremely fetching. They burst out laughing._)\n\n TRIO.\n\n HILARION, CYRIL, FLORIAN.\n\nHILARION--\n\n I am a maiden coy and graceful,\n Stately I, with a face divine,\n But with my smiles I'm ever faithful,\n True to the men of the Alton Line.\n\nALL--\n\n Haughty, humble, coy and free,\n Little care I what maid may be.\n So that a maid is fair to see,\n Every maid is the maid for me.\n\nFLORIAN--\n\n I am a maiden sweet and lusty,\n Dainty and pleasant am I to see,\n Timid and shy, and oh! so trusty.\n An Alton man is the man for me.\n\nALL--\n\n Haughty, humble, coy and free,\n Little care I what maid may be,\n So that a maid is fair to see,\n Every maid is the maid for me.\n\nHILARION--But who comes here? The Princess, as I live.\n\nCYRIL--What is to be done?\n\nHILARION--Why, brave heart, my boy. Remember that you are an Alton\nman, and be brave. (_Enter Princess._) He bows low. Madam, accept my\nhumblest reverence.\n\nPRINCESS--Ladies, you are welcome. What might be your pleasure with me?\n\nHILARION--(_bowing._) We are three well-born maidens, ma'am! daughters\nof those who run the Alton line, who wish to join the University.\n\nPRINCESS--You say you are the daughters of those who run the Alton\nLine. Well! You'll find no comfort here. Your bed will be that of the\nhumblest. No downy mattress, soft pillows, or the rock of a Pullman\nPalace Car to send you to sleep. Your fare will be plain and simple,\nyour beverage nature's stream. No dining-car lunches at seventy-five\ncents a meal, which ought to cost $2.00. No delicacies out of season,\nor fine wines iced, as you get on that line. Your service must be done\nby yourselves. No obsequious porters will brush your boots or carry\nabout your bundle. And so, if you think well of this, there are here\none hundred maidens fair who are prepared to love and welcome you. Will\nyou try to give the fullness of your love to them?\n\nALL--Indeed we will.\n\nPRINCESS--Then you are welcome.\n\n (_Exit Princess. The men burst out laughing and do a bit of\n a breakdown. Lady Psyche enters back and looks at them with\n amazement._)\n\nLADY P.--These ladies are unseemly in their mirth.\n\nFLORIAN--It's all up, Hilarion! Here is my sister.\n\nHILARION--Then make a virtue of necessity, and take her in our\nconfidence.\n\nFLORIAN--(_To Lady Psyche._) Psyche! why don't you know me, your\nbrother, Florian?\n\nPSYCHE--(_Amazed._) Florian. (_They embrace._) What are you doing here?\nOh, dear! You know it is death for any man to be found within these\nwalls. How did you get here?\n\nFLORIAN--By the ever reliable Chicago and Alton Railroad, of which I am\na humble but devoted employe.\n\nPSYCHE--Oh, dear! Then you are an Alton man (_looking at him with\nadmiration_). I always thought you would be something great some day.\nBut who are these?\n\nHILARION--I am Prince Hilarion, of the Alton Line also. Your early\nplayfellow, and the betrothed of your Princess.\n\nPSYCHE--I might have known you were an Alton man. To imitate their\nbearing is impossible. Listen! and I will tell you the history of a man\nwho once tried to rival a Chicago and Alton man:\n\n [SONG.]\n\n An Alton man of lineage high\n Was aped by a rival in days gone by.\n The man was radiant as the sun,\n The rival was an unsightly one.\n So it would not do,\n His scheme fell through\n To the world when his scheme took formal shape;\n Expressed such terror\n At his monstrous error\n That he stammered his apology and made his 'scape,\n The picture of a disconcerted ape.\n\n With a view to rise in the Railway scales,\n He washed his collar and cut his nails.\n He bought a uniform and made it shine,\n And swore that he belonged to the Alton Line.\n But it would not do,\n His scheme fell through,\n For the Alton man was beauty's king,\n With graceful manners,\n Like Lord John Manners,\n While the rival lacked the metal ring,\n And always remained a puny thing.\n\n He bought white ties and he bought new boots,\n And he dressed himself in bran-new suits;\n Then he put C. & A. on all his things,\n And he bought a pin and diamond rings.\n But it would not do,\n His scheme fell through,\n For the Alton man whom the rival aped\n Was a radiant being,\n Well worth seeing,\n While the rival King, however shaped,\n At best was an Alton man ill-aped.\n\n[Illustration: THERE WAS MY PORTER GRINNING.]\n\n[Illustration: AN ALTON DART.]\n\n[Illustration: IN BATTLE ARRAY.]\n\n[Illustration: THE FIGHT.]\n\n (_They do another breakdown, during which enter Princess at\n back. She looks on in amazement._)\n\nPSYCHE--(_Looks at her terrified._) All is lost!\n\nPRINCESS--What is this? It seems to me that female garb become you\nladies ill. I don't think I ever saw a gentlewoman do a breakdown\nbefore.\n\nHILARION--We hope to become young ladies soon.\n\nPRINCESS--But are not now?\n\nHILARION--(_Throws off his robe, and then throws himself at the feet of\nthe Princess._) 'Tis useless to dissemble farther. We are merely men. I\nam Prince Hilarion, your betrothed, and these two are my friends.\n\nPRINCESS--(_Screaming._) Men! and Alton men at that! Ring the bell!\nFire the alarm guns, and have the Army of Amazons take them to the most\nsecret cells and annihilate them.\n\n (_Loud turmoil; enter all the girls running frightened. They\n catch sight of the three Alton men, and immediately fall\n prostrate at their feet and regard them with languishing eyes._)\n\nPRINCESS--Summon my henchmen! (_They come._) Seize and bind them.\n(_They do it._) And now, adventurous men, much as I admire your shape\nand venerate the line you represent, you must die (_with emotion_).\nMy principles must be observed. So take them off. (_Loud noise heard\noutside; enter girls running with fright._)\n\nMELISSA--Princess! There is an armed force at the gates of the castle,\njust arrived by the St. Louis express and headed by King Hilderbrandt.\nThey demand admittance.\n\nPRINCESS--Refuse it.\n\nALL--Too late.\n\n (_A loud crash is heard; enter King Hilderbrandt with his army of\n Chicago and Alton conductors, armed to the teeth. In the centre\n they drag ARAC, GURON and SYNTHIUS, chained together._)\n\nHILD.--(_Coming down._)\n\n Some years ago,\n No doubt you know\n (And if you don't I'll tell you so),\n You gave your troth\n Upon your oath\n To Hilarion my son.\n A vow you make\n You must not break\n (If you think you may, it's a great mistake),\n For a bride's a bride,\n Though the knot were tied\n At the early age of one!\n\n And I'm a peppery kind of king,\n Who's indisposed for parleying\n To suit the wit of a bit of a chit,\n And that's the long and the short of it!\n\nPRINCESS--(_Irresolutely--the girls meanwhile look with loving eyes at\nthe conductors._) I still refuse!\n\n[Illustration: ITS RESULT.]\n\n[Illustration: HILARION I SURRENDER.]\n\n (ARAC, GURON _and_ SYNTHIUS _come down_.)\n\nARAC--\n\n We may remark, though nothing can\n Dismay us,\n That if you thwart this gentleman\n He'll slay us.\n\n We don't fear death, of course, we're thought\n To shame it.\n But still, upon the whole, we thought\n We'd name it.\n\nALL--\n\n Yes! yes! Better perhaps to\n Name it.\n\nGURON--\n\n This gentleman controls a Line--\n A stunner.\n With splendid bed and track as fine--\n A hummer.\n We hate his line, but still we must\n Propound it.\n The Alton is a line to trust--\n Confound it.\n\n(TOGETHER.)\n\n Yes! yes! In spite of that, confound it.\n\nSYNTHIUS--\n\n We wish its trade was light and slack--\n It isn't.\n Its service bad, with moldy track--\n It isn't.\n But give to C\u00e6sar what is due,\n Yes, rend it.\n We really must, and strongly too,\n Commend it.\n\n(TOGETHER.)\n\n Yes! yes! yes! Darn their eyes,\n Commend it.\n\nALL--\n\n The Alton is a road, as such\n None beat it.\n St. Louis is a point they touch,\n Repeat it.\n And if you doubt they'll get you there\n You'll rue it,\n For when they say they'll get you there,\n They'll do it.\n\n(TOGETHER.)\n\n Yes! yes! yes! Devil doubt\n They'll do it.\n\nPRINCESS--I still refuse!\n\nPSYCHE--Madame! your father claims an audience; he has just followed\nKing Hilderbrandt on the Denver Express.\n\nPRINCESS--Admit him.\n\n (_Enter_ GAMA.)\n\nGAMA--Free as a tethered ass. I dare not keep this up. I dare not face\nthe malignity of this devil, Hilderbrandt.\n\nPRINCESS--He has treated you well, father. Even a wretched man, when he\nbelongs to the Alton Line, is always a gentleman.\n\nGAMA--Yes! yes! But he has made my life a curse. Think of it, I had\nnothing whatever to grumble at.\n\n [SONG.]\n\n The more I try\n I can't deny,\n Altho' I'm very spiteful,\n I'm bound to say\n The livelong day,\n The Alton is delightful.\n\n Now, when a man\n Does all he can,\n A rival line despising,\n And in reply\n They fete him--why\n It's hard, there's no disguising.\n\n Oh! don't the day seem blank and long,\n When all goes right and nothing goes wrong,\n And isn't your life extremely flat\n With nothing whatever to grumble at?\n\n When in a car\n Above the par,\n Officials are quite pleasing.\n And when a line\n You hate is fine,\n It's surely very teasing.\n They treat me well,\n Ah! what a sell,\n I thought they'd be ungracious.\n But not a bit--\n They strained their wit\n To make my thoughts fallacious.\n\n Oh! don't the day seem blank and long,\n When all goes right and nothing goes wrong, etc., etc.\n\n The Alton whirls\n The pretty girls\n From eyes that are repining.\n And takes the best\n Of travel West\n In cars with chairs reclining.\n A dining-car\n Without a jar,\n And coaches really tasteful.\n A splendid track\n Both there and back,\n Now isn't that disgraceful?\n\n Oh! don't the day seem blank and long, etc., etc.\n\n When off to bed\n They gently led\n Me to a couch bewitching,\n I prayed they might,\n At least, that night,\n The holy train be ditching.\n But fate was there,\n We sped like air,\n And nothing happened tragic,\n And then I wept,\n And then I slept,\n And dreamt the line was magic.\n\n Oh! don't the day, etc., etc.\n\n When morning came,\n 'Twas just the same,\n There was my porter grinning;\n I called him Sam,\n And muttered Dam,\n He really looked so winning;\n I looked around,\n And there I found\n A car of pleasant faces;\n It seemed their bent\n To be content,\n In spite of my grimaces.\n\n Oh! don't the day, etc., etc.\n\n[Illustration: BOARDING THE ALTON TRAIN AFTER THE WEDDING.]\n\n[Illustration: AN IMPROMPTU WEDDING.]\n\n[Illustration]\n\n[Illustration: HILARION WELCOMES IDA TO THE ALTON LINE.]\n\nPRINCESS--Poor father! how you must have suffered.\n\nGAMA--Yes, my child, and the only thing to do is to surrender. We must\ncapitulate. I have often heard that opposition to the Alton line is\nfutile, and now I know it.\n\n (_Enter_ HILARION.)\n\nPRINCESS--Hilarion, I surrender. My heart was with you long ago, but my\nprinciples forbade it. Take me.\n\nHILDERBRANDT--Then we will have an impromptu wedding. Everybody who\nwants to marry everybody else can do so. (_The remark is a signal\nfor every girl in the party to throw her arms around the neck of the\nnearest conductor._)\n\nGAMA--And take my advice, make your wedding trip over the Chicago and\nAlton Line.\n\nALL--We will.\n\nGAMA--Hereafter, consider Castle Adamant yours. You may make it a\ncoupon station, if you please, and advertise it as a pleasure resort.\n\n (_The organ is heard; a procession formed, and the entire party\n indulge in an impromptu wedding, after which the Chicago and\n Alton train for Chicago is boarded, and everybody starts upon a\n trip, which cannot but bring them happiness._)\n\n _Moral._\n\nAn official of the Alton Road is irresistible.\n\n\n\n\n[Illustration: RETURN TO CHICAGO.]\n\n[Illustration: THE RECEPTION.]\n\n[Illustration: TWENTY YEARS LATER.]\n\n[Illustration: THE END]\n\n\n\n\n [Illustration: CHICAGO & ALTON RAILROAD]\n\n 3 GREAT CITIES\n =OF THE WEST=\n LINKED TOGETHER\n --BY THE--\n =CHICAGO & ALTON R. R.=\n\n NO CHANGE OF CARS OF ANY CLASS\n BETWEEN\n CHICAGO AND KANSAS CITY.\n CHICAGO AND ST. LOUIS.\n ST. LOUIS AND KANSAS CITY.\n\n Union Depots in Chicago, East St. Louis, St. Louis and Kansas\n City.\n\n * * * * *\n\n NO OTHER LINE RUNS\n\n PALACE DINING CARS\n\n Between CHICAGO and KANSAS CITY, CHICAGO and ST. LOUIS, and ST.\n LOUIS and KANSAS CITY.\n\n Meals equal to those served in any first-class hotel, only 75\n cents.\n\n * * * * *\n\n The only Line running a sufficient number of Elegant and\n Comfortable =PALACE RECLINING CHAIR CARS=\n\n Free of Extra Charge, in all its Through Trains, Day and Night,\n Without Change, to accommodate all its patrons.\n\n * * * * *\n\n Pullman Palace Sleeping Cars and Pullman Palace Buffet Sleeping\n Cars\n\n THE FINEST, BEST AND SAFEST IN USE ANYWHERE.\n\n * * * * *\n\n The BEST and QUICKEST ROUTE from CHICAGO to and from MEMPHIS,\n MOBILE, NEW ORLEANS, and all Points South via St. Louis.\n\n * * * * *\n\n THE SHORT LINE TO AND FROM\n\n =Missouri=, =Arkansas=, =Texas=, =Kansas=, =Colorado=, =New\n Mexico=, =Mexico=, =Arizona=, =Nebraska=, =California=,\n =Oregon=, =Washington Territory=, =Etc.=\n\n * * * * *\n\n THE GREAT EXCURSION ROUTE BETWEEN THE NORTH AND SOUTH\n\n AND TO AND FROM\n\n Kansas Lands, and Colorado, New Mexico and California Health\n and Pleasure Resorts and the Mining Districts of the Great West.\n\n * * * * *\n\n For Tickets and information apply at any Coupon Ticket Office\n in the United States and Canada, or to\n\n =JAMES CHARLTON=, General Passenger and Ticket Agent,\n 210 Dearborn Street, near corner Adams Street, Chicago.\n\n _=J. C. McMULLIN=_, Vice-President, Chicago.\n\n _=C. H. CHAPPELL=_, General Manager, Chicago.\n\n\n\n\n [Illustration: No Change of Cars of any Class AND Two Trains\n a Day Each Way Between Chicago & Kansas City, Chicago & St.\n Louis, AND St. Louis & Kansas City.]\n\n CHICAGO & ALTON R.R.\n\n _The Pioneer Palace Reclining Chair Car Route.\n The Pioneer Palace Dining Car Route.\n The Pioneer Pullman Palace Sleeping Car Route._\n\n * * * * *\n\n TWO TRAINS DAILY, SUNDAYS INCLUDED,\n Between Chicago and Kansas City and\n Between St. Louis and Kansas City.\n\n * * * * *\n\n LIGHTNING NIGHT EXPRESS DAILY, SUNDAYS INCLUDED.\n\n NIGHT EXPRESS DAILY, SUNDAYS INCLUDED.\n\n DAY EXPRESS DAILY, Sundays Excepted, between\n CHICAGO AND ST. LOUIS\n\n\n\n\n\nEnd of the Project Gutenberg EBook of A Parody on Princess Ida, by D. Dalziel\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzscst b/data_all_eng_slimpj/shuffled/split2/finalzzscst new file mode 100644 index 0000000000000000000000000000000000000000..4049ad929bb415ce49f145aa619ff898979e2956 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzscst @@ -0,0 +1,5 @@ +{"text":" \nTHE HELVETIAN AFFAIR\n**Also by Ray Gleason:**\n\nThe Gaius Marius Chronicle\n\n_De Re Gabiniana: The Gabinian Affair_\n\n_A Grunt Speaks: A Devil's Dictionary of Vietnam Infantry Terms_\n\n_The Violent Season_\nTHE GAIUS MARIUS CHRONICLE\n\nBOOK II\n\nTHE\n\nHELVETIAN\n\nAFFAIR\n\n_De Re Helvetian_\n\nRay Gleason\n\nTHE GAIUS MARIUS CHRONICLE BOOK II\n\nTHE HELVETIAN AFFAIR\n\n_De Re Helvetian_\n\n\u00a9 2016 RAY GLEASON.\n\nAll rights reserved. No portion of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means\u2014electronic, mechanical, photocopy, recording, scanning, or other\u2014except for brief quotations in critical reviews or articles, without the prior written permission of the publisher.\n\nThis is a work of fiction. Names, characters, businesses, places, events, and incidents are either the products of the author's imagination or used in a fictitious manner. Any resemblance to actual persons, living or dead, or actual events is purely coincidental.\n\nPublished in New York, New York, by Morgan James Publishing. Morgan James and The Entrepreneurial Publisher are trademarks of Morgan James, LLC. www.MorganJamesPublishing.com\n\nThe Morgan James Speakers Group can bring authors to your live event. For more information or to book an event visit The Morgan James Speakers Group at www.TheMorganJamesSpeakersGroup.com.\n\n | ISBN 978-1-63047-702-8 paperback\n\nISBN 978-1-63047-703-5 eBook\n\nLibrary of Congress Control Number: 2015911276\n\n**Cover Design by:** \nRachel Lopez\n\nwww.r2cdesign.com\n\n**Interior Design by:** \nBonnie Bushman\n\nThe Whole Caboodle Graphic Design\n\n---|---\n\nIn an effort to support local communities and raise awareness and funds, Morgan James Publishing donates a percentage of all book sales for the life of each book to Habitat for Humanity Peninsula and Greater Williamsburg.\n\n | Get involved today, visit \nwww.MorganJamesBuilds.com |\n\n---|---|---\nTo BJ Rahn\n\nTeacher, Mentor, Friend\n\n_\"Gratitude bestows reverence, allowing us to encounter everyday epiphanies, those transcendent moments of awe that change forever how we experience life and the world.\"_ \n\u2014 **John Milton**\n\n## DRAMATIS PERSONAE\n\n**Gaius Marius Insubrecus Tertius** , our hero, known variously as follows:\n\n * Arth Bek: \"Little Bear,\" by his grandpa\n * _Pagane_ : \"The Hick,\" by his Roman army mates\n * Gai: by his family, close friends, and his few girlfriends\n * Insubrecus: by his army colleagues and casual associates\n * _Blatta \/ Vermiculus \/ Bestiola_ : \"Cockroach \/ Maggot \/ Insect,\" by Strabo, his training officer\n * _Prime_ : \"Top,\" but that's much later in his military career\n\n**The Basic Training Squad:**\n\n**Cossus Lollius Strabo, \"Squinty,\"** an Eighth Legion _optio_ , the training officer, later promoted to centurion in the Tenth Legion\n\n**The _Veterani_ , \"Old Men\":**\n\n**Lucius Bantus** , acting _decanus_ , squad leader\n\n**Tullius Norbanus, \"Tulli,\"** assistant squad leader\n\n**The** _Tirones_ , **\"Rookies\":**\n\n**Mollis** , \"Softy\"\n\n**Rufus** , \"Red\"\n\n**Pustula** , \"Zits\"\n\n**Minutus** , \"Tiny\"\n\n**Loquax** , \"Gabby\"\n\n**Lentulus** , \"Slow Poke\"\n\n**Felix** , \"Lucky\"\n\n**Gaius Iulius Caesar** , _Imperator_ and commander of the Roman legions in Gaul; proconsul of Cisalpine Gaul, Transalpine Gaul, and Illyricum; ex-consul of the Roman Repubulic and _Triumvir_ with Gnaeus Pompeius Magnus and Marcus Licinius Crassus; _Patronus_ of our hero, Gaius Marius Insubrecus\n\n**Caesar's Legates in Gaul:**\n\n**Titus Labienus** , a professional soldier and Caesar's right-hand man; second in command of the army; saves Caesar's bacon at Bibracte\n\n**Caius Claudius Pulcher** , \"Pretty Boy,\" a self-conscious Patrician and no fan of Caesar; he would probably have been involved in the plot against Caesar\u2014had he the brains or the energy\n\n**Publius Licinius Crassus** , one of the two sons of Caesar's colleague and fellow _triumvir_ , Marcus Licinius Crassus; appointed to Caesar's staff as a favor to his father and sent to Gaul by his father to keep an eye on his partner, Caesar\n\n**Quintus Pedius** , Caesar's nephew\u2014need I say more?\n\n**Servius Sulpicius Rufus** , a lawyer in armor; every army has a few of these, unfortunately\n\n**Publius Vatinius** , served Caesar in Rome as his pet tribune of the Plebs, a political appointment\n\n**Lucius Aurunculeius Cotta** , serving in Gaul seems to have been his only claim to fame\n\n**Caesar's Military Tribunes:**\n\n**Lucius Vipsanius Agrippa** , an Italian from Asisium; an equestrian; a social and political nobody, but a good officer; his kid brother, Marcus, will eventually make it big\n\n**Tertius Nigidius Caecina** , an _angusticlavus_ , a junior tribune; a Roman and the nephew of Senator Publius Nigidius Figulus, on whose support Caesar depends\n\n**Publius Considius** , been around in the army since Romulus was a corporal; in bad need of a pair of specs, but they haven't been invented yet\n\n**Fabius** , a _laticlavus_ , a broad-striper; a senior tribune assigned to the Eleventh Legion\n\n**_The Centurions_ :**\n\n**Decius Minatius Gemellus** , _praefectus castrorum_ , the prefect of camps of Caesar's army\n\n**Tertius Piscius Malleus, \"The Hammer,\"** _centurio primus pilus_ of the Tenth Legion\n\n**Mamercus Tertinius Gelasius** , _centurio prior pilus_ , commander of the Tenth Cohort of the Tenth Legion and officer in charge of recruit training\n\n**Nerva** , _primus pilus_ of the Twelfth Legion\n\n**Sanga** , _centurio prior pilus_ , commander of the Third Cohort of the Twelfth Legion\n\n**Mettius Atius Lupinus, \"Lotium,\"** commander of the Third Century, Second Cohort, Tenth Legion; no one dares call him _Lotium_ to his face\n\n**Spurius Hosidius Quiricus, \"The Oak,\"** _centurio primus pilus_ of the Ninth Legion\n\n**Marcus Sestius, \"Iudaeus,\"** _centurio primus pilus_ of the Eleventh Legion; his nickname has nothing to do with his religious affiliation, but rather a wound he received in a rather awkward spot\n\n**_Other Roman Officers_ :**\n\n**Decimus Lampronius Valgus, \"Bowlegs,\"** _decurio_ in command of the cavalry of Caesar's Praetorian Guard\n\n**Rubigo, \"Rusty,\"** _decurio_ in the legionary cavalry of the Tenth Legion\n\n**Flavus, \"Whitey,\"** a Roman soldier from Cisalpine Gaul serving in the Twelfth Legion and briefly assigned to the Sequani Cavalry under Agrippa\nTABLE OF CONTENTS\n\n[_De Hospe Subito Praefatio_ \nPreface: An Unexpected Visitor](preface.html)\n\n[I. _De Spatio in Tartaro_ \nTime in Hell](chapter01.html)\n\n[II. _Hostes apud Amicos_ \nEnemies among Friends](chapter02.html)\n\n[III. _Ego Miles Romanus_ \nI Become a Soldier of Rome](chapter03.html)\n\n[IV. _De Itinere inter Alpes_ \nWe March across the Alps](chapter04.html)\n\n[V. _Sub Patrocinio Caesaris_ \nUnder Caesar's Patronage](chapter05.html)\n\n[VI. _De Consequente Helvetiorum_ \nPursuit of the Helvetians](chapter06.html)\n\n[VII. _De Clementia Caesaris et Offensione Antiqua_ \nCaesar's Clemency and an Ancient Provocation](chapter07.html)\n\n[VIII. _De Calamitate Prima_ \nThe First Debacle](chapter08.html)\n\n[IX. _Lente Festinamus_ \nWe Hurry Slowly](chapter09.html)\n\n[X. _Scaena Caesaris_ \nCaesar's Drama](chapter10.html)\n\n[XI. _Calamitas Itera_ \nAnother Disaster](chapter11.html)\n\n[XII. _Bibracte_ \nBibracte](chapter12.html)\n\nPost Scriptum\n\nMilitary Latin\n\n## GAH'ELA, THE GAULS\n\n**The Aedui, the _Aineduai_ , the \"Dark Moon\" People:**\n\n**Duuhruhda mab Clethguuhno** , _uucharix_ , tribal king of the Aedui, and _pobl'rix_ , clan leader of the _Wuhr Blath_ , the Wolf Clan of the Aineduai; known to the Romans as Diviciacus\n\n**Deluuhnu mab Clethguuhno** , brother of Duuhruhda, _dunorix_ of the Aedui, commander of the garrison of Bibracte; known to the Romans as Dumnorix\n\n**Cuhnetha mab Cluhweluhno** , _buch'rix_ , \"cattle king\" of a small settlement east of Bibracte; _pobl'rix_ , clan leader of the _Wuhr Tuurch_ , the Boar Clan of the Aedui; pretender to the throne\n\n**Rhonwen** , niece of Cuhnetha, a sassy redhead who catches Insubrecus' eye\n\n**The Sequani, the _Soucanai_ , People of the River Goddess Soucana:**\n\n**Madog mab Guuhn** , _pobl'rix_ , _rex gentium_ , clan leader of the _Wuhr Wuhn_ , the White Clan of the Soucanai, and commander of the Auxiliary Sequani Cavalry; known as Madocus _Dux_ to the Romans\n\n**Athauhnu mab Hergest** , _pencefhul_ , \"leader of a hundred,\" _ala_ commander in the Auxiliary Sequani Cavalry commanded by Madog mab Guuhn; known as Adonus _Decurio_ to the Romans\n\n**Emlun** , Athauhnu's nephew\n\n**Guithiru** , one of Athauhnu's veteran warriors\n\n**Alaw** , one of the scouts\n\n**Rhodri** , Alaw's companion\n\n**Ci, \"The Hound,\"** a veteran warrior and troop commander in Madog's cavalry\n\n**Idwal** , a friend of Emlun; a rider in Athauhnu's troop\n\n**_Dramatis Personae Aliae_ , the Other Players:**\n\n**Aulus Gabinius** , a senatorial mid-bencher who does well and is elected consul\n\n**Gnaeus Pompeius Magnus** , a _triumvir_ , a partner of Caesar and an _eminence grise_ in this tale\n\n**Marcus Licinius Crassus** , a _triumvir_ , a political partner of Caesar; too intent on going off and conquering Parthia to pay much attention to what Caesar's doing in Gaul\n\n**Ebrius, \"Drunk,\"** Caesar's head military clerk and self-appointed taster of Caesar's wine and _posca_ collection\n\n**Clamriu** , a horse\n\n**Gennadios the Trader** , a Greek merchant from Massalia who introduces our hero to retsina\n\n**Evra** , Gennadios' woman from a mysterious island west of Britannia\u2014not a redhead, but formidable nonetheless\nDe Bello Contra Helvetios Tabula\n\n**Map of the Helvetian Campaign**\n\n# _De Hospe Subito Praefatio_\n\n## PREFACE: AN UNEXPECTED VISITOR\n\nI broke my narrative to welcome a surprise visitor up from _Italia_ , my former commander and comrade in the Gallic campaigns, Lucius Vipsanius Agrippa.\n\nHe is certainly grayer and a bit heavier than I remember him from our soldiering days, which is one of the reasons I have always refused ever to bring one of those _speculum_ mirrors from the East into my home. If I do not have to see the physical evidence of my aging, I can still pretend that I am the same man I was when I was marching with the legions.\n\nThat is, until I have to move too quickly and my heart jumps up into my throat, or I try to leap a watery ditch and miss my stride by almost half a _pes_. My mind has not yet accepted the reality of my actual physical state.\n\nBesides the physical changes, Agrippa was still the same good comrade I remember from our youth, just packaged in a somewhat rounder form. He even managed to charm Rhonwen, my darlin' wife\u2014or Flavia, as she is now called since becoming a Roman and moving down to this side of the Alps.\n\nCharming Rhonwen! When Agrippa showed up on her doorstep unexpectedly with an entourage that included two bodyguards, a half-dozen slaves with livestock, and baggage, gaining her favor was no easy task. But, Rhonwen is still in many ways a Gah'el and understands her duties to the gods to offer hearth, bread, and salt to guests. Besides, Agrippa managed to say to her, in his halting Gah'el, \" _Festres uh bendit'ian uh duwiau uh bawb in uh ti hoon_ \"\u2014\"The blessings of the gods to all in this house, Missus.\" He said this as he tousled our son's hair, announcing that he looked exactly as I did at that age\u2014except better looking\u2014and thus, he had Rhonwen eating right out of his hand.\n\nThe fact that Agrippa put his entire _comitatus_ up at an inn just off the forum at his own expense certainly enhanced Rhonwen's opinion of him.\n\nAgrippa rarely leaves his family estate in Asisium these days. He broke with Caesar during the first civil war against the senatorial _Optimates_. Agrippa couldn't stomach Pompey, but Agrippa's Rome was the idealized republic of Cicero and Cato. After the war, Caesar pardoned him along with most of the junior officers who fought against him. Agrippa then retired to his farm, whether to avoid being placed on a proscription list or to shut himself off from what his Rome had become is anyone's guess.\n\nAgrippa didn't leave his cloistered life\u2014even after his younger brother, Marcus, went off to fight with Caesar's adopted son and heir, Octavius, first against the \"Liberators\" at Phillipi, then against Pompey's brat, Sextus, and finally against Antonius. Many believe that Octavius' victory at Actium was achieved by Marcus Vipsanius Agrippa, but stating such publically is at best risky when the Roman state is ruled by one who styles himself the _princeps civitatis_ , the savior of the nation, the second Romulus, _Augustus_ , the \"exalted one.\"\n\nIn my military career, I have served with all four men: Caesar, Lucius, Marcus, and Octavius.\n\nMarcus was by far the best general; Lucius the best soldier and comrade.\n\nOctavius was no soldier at all.\n\nOctavius was always a politician, a negotiator, a maker of deals. Despite my assessment of him as a soldier, and even as a human being, my _pietas_ to the state demands I recognize him as the only man who can control those self-serving idiots down in Rome and prevent Roman armies from slaughtering each other as they did so many times during my lifetime.\n\nSo, if Octavius wishes to be called Gaius Iulius Caesar Octavianus, _filius Iuli divi_ , _Augustus_ , _Pater Patriae_ , the Second Founder of Rome, or whatever titles those senatorial boot-licks down in Rome come up with, it's a small price to pay for lasting peace.\n\nNone of them are worth the mule shit that spattered Caesar's cloak when he marched the legions north of the Rhodanus in pursuit of the Helvetii, into Belgica against the Nervii, across the Rhenus into the forests of Germania, across _Oceanus_ into the land of the Britanni, across the Rubico, the river of blood, to set things right in Rome.\n\nHad he lived, peace with Parthia would have been bought with steel, not gold.\n\nHad he lived, had only I insisted on accompanying him to the Senate meeting at Pompey's theater that day!\n\nI'm not sure that Caesar's _lemur_ is at peace, despite the apparent success of his \"son\" and \"heir\" down in Roma. Or, it may just be my sense of guilt at not having been there when Caesar, _patronus me'_ , needed my strong arm protecting his _latus apertum_ , his unprotected side.\n\nObviously, Agrippa's sudden appearance stirred up many things for me.\n\nAgrippa said that he had come to me with a request from Octavius, which was passed to him through his brother, Marcus. Octavius is now my _patronus_ , so a _request_ from him is not quite a request. The fact that he used my old comrade from Gaul, Agrippa, as his messenger, is a sign that Octavius expects my concurrence in this matter.\n\nBut, more about that later.\n\nAgrippa's appearance reminded me of when we first met during Caesar's campaign against the Helvetii. For both of us, it was our first military campaign. We were _virgines_ , \"cherries\" as we say in the legions. He had just been assigned to the Tenth Legion as an _angusticlavus_ , a narrow-striper, a junior tribune, and I had just finished my basic military training in Aquileia. As soldiers, we were green as grass; we still smelled of the farm. Real _pagani_.\n\nReviewing my notes, I see that I had broken my narrative just after I had run off to the legions in order to escape being arrested by Consul Aulus Gabinius on the charge of _sacrilegium_ against a Roman magistrate and for an insult to his family's _auctoritas_ \u2014charges which I was unlikely to survive. This, of course, was all a smoke screen to obscure the fact that I had severely injured his son, Aulus Iunior, in a fight to protect myself from his bungled attempt to murder me. Iunior thought I had been the lover of his sister, Gabi, a pleasure which I had missed, despite my best bumbling efforts. So, he had decided to decorate the _rostra_ in the _forum Romanum_ with my _coleones_ to recoup his family's _dignitas_.\n\nI joined a little band of returning veterans and legionary recruits and trudged through the winter landscape of the Padus Valley. We arrived at the legionary camps around Aquileia on the day before the Ides of _Februarius_ , during the consulship of Lucius Calpurnius Piso Caesoninus and Aulus Gabinius, the winter of my sixteenth year.\n\nCossus Lollius Strabo, our officer, an _optio_ returning from leave to the Eighth Legion, brought us to the _castrum_ of the Tenth Legion, to which we were assigned. After some confusion over the daily password and Strabo's authority as an officer, which the soldiers of the Tenth Legion seemed somewhat reluctant to grant to an _optio_ of the Eighth Legion, we were escorted to the _principia_ the military headquarters of Gaius Iulius Caesar, the army commander and proconsul of the province.\n\n# I.\n\n# _De Spatio in Tartaro_\n\n# TIME IN HELL\n\nHe were ushered into the _principia_ tent, where a soldier in a brickred tunic was sitting behind a small field desk working with a pile of _tabulae_ , wax slates. He looked up and told Strabo, \"You may pass through, _Optio_.\"\n\nWe passed into a large compartment behind the clerk. There were a couple of braziers warming the area and lamps illuminating maps hanging from the walls. A soldier with short, graying hair was briefing two younger soldiers while gesturing at one of the hanging maps. A fourth soldier seemed to be taking notes with a _stylus_ in a hinged, diptych _tabula_. The older soldier looked over when we entered and snapped, \"I will be with you presently, _Optio. Laxa_! Stand at ease!\"\n\nStrabo bunched us into a dark corner, out of the way. \"The _praefectus castrorum_ ,\" Tulli, one of the _veterani_ who had reenlisted, hissed into my ear.\n\nFinally, I heard the older soldier ask, \"Any questions?\"\n\nBoth younger men came to the position of attention and each responded, \" _N'abeo, Praefecte_!\"\n\n\" _Bene_!\" Then, the _praefectus castrorum_ instructed, \"Have your cohorts ready to march by the end of the fourth watch . . . the Fifth Cohort assembled in full marching kit outside the _Porta Dextra_ , the right-hand gate, and the Seventh outside the _Porta Decumana_ , the rear gate. I want you to move out at the signal for the end of watch. Dismissed!\"\n\nBoth men nodded to the _praefectus_ , executed an about-face, and marched out of the tent.\n\nThen, the man walked over to our group. Strabo and both our _veterani_ stiffened noticeably as he approached.\n\n\"What have we here, _Optio_? They call you Strabo . . . \"Squinty\" . . . right?\" he asked.\n\n\" _Praefecte_ ,\" Strabo announced, \"Cossus Lollius, _optio_ of the Eighth Legion, reports with a detail of two _veterani_ and eight _tirones_ for the Tenth Legion!\"\n\n_\"Laxa, Optio_!\" the _praefectus_ directed.\n\nThen he turned to us. \"Men, I am Decius Minatius Gemellus, _praefectus castrorum_ of this army and, until Caesar _Imperator_ assigns a _legatus_ , the commander of the Tenth Legion. That means, as far as you're concerned, I'm a god. I have the power of life and death over you. You do not want to piss me off or even disappoint me. You _veterani_ , I welcome you back to the eagles. You will undergo the first weeks of conditioning training with the recruits. My cadre will then assess your skills with _gladius, pilum_ , and _scutum_. If you show us you haven't forgotten the fighting skills of a Roman soldier, you'll be assigned to a century in one of the second-line cohorts until we know what you have. You _tirones_ , a word of advice! Although you have raised your hand in the _sacramentum_ , do not consider yourselves _milites_ , or members of my legion. You have yet to prove your worth for such an honor! But, you will soon have the opportunity to do so! Over the next eight weeks, we will make soldiers out of you\u2014or we will break you and send you back to whatever civilian shit-hole you crawled out of. You will demonstrate to me, with your sweat and your blood, that you are worthy of this legion and worthy of the honor of serving the Roman people!\"\n\nThen, Gemellus demanded, _\"Scriba_!\"\n\nThe soldier we had seen outside burst into the room and assumed the position of attention, \" _Praefecte_!\"\n\n\"Get the records these men are carrying! Enter them on the legion roles; then send them down to supply for initial issue!\"\n\nBefore the clerk could respond, Strabo said, _\"Praefecte_! There is one thing.\"\n\nGemellus responded, \"What is it, Strabo?\"\n\n_\"Praefecte_! One of the recruits needs to see a _medicus_ ,\" Strabo answered.\n\n\"A doctor?\" Gemellus questioned. \"Is he injured?\"\n\n\"No, _Praefecte_!\" Strabo continued. \"It's his . . . well\u2014\"\n\n\"Spit it out, Strabo!\" Gemellus ordered. \"Straightforward report, like a Roman officer!\"\n\n\"His feet are flat, _Praefecte_ ,\" Strabo reported. \"I don't think he can stand up to the marching.\"\n\nGemellus snorted, \"Flat feet, is it? This is what those _podices_ in recruiting are sending us for soldiers? Very well . . . _Scriba . . ._ get one of the orderlies up here from the medical section to take . . . uh . . . which one is it, Strabo?\"\n\n\"Mollis . . . I mean . . . _Tiro_ Tertius Melonius, _Praefecte_ ,\" Strabo corrected himself.\n\n\"Send _Tiro_ Tertius Melonius to be examined by the chief medical officer,\" Gemellus instructed his clerk. \"And, Strabo! You stay behind when I dismiss the others. You and I need to talk. . . . The rest of you get out of my sight. . . . Move!\"\n\nWe got out of the _praefectus'_ office as quickly as we could, the _veterani_ leading the way. When we reached the anteroom, the clerk collected our records and directed us to wait in a corner, \"out of his way.\" Then, he sent a runner over to the medics to collect Mollis and another to the supply tent to collect us.\n\nAs we were waiting, Strabo rejoined us. \"Looks like we're going to be stuck with each other a little longer,\" he announced to us. \"The Eighth's already over the Alps, and the passes are closed. So, I've been seconded to the Tenth and assigned as your training officer.\"\n\nHe turned to the _praefectus'_ clerk, \"Supply, where is it supposed to be?\"\n\nThe man replied, looking up from one of his wax _tabulae_ , \"Behind the _praetorium, Optio_ , right across from grain storage. I've already sent a runner.\"\n\n\"And, I'm sure your supply sergeant will jump right on that,\" Strabo sneered. \"You done with this bunch?\"\n\n_\"Perfeci, Optio_!\" the clerk started.\n\n\"Bene!\" Strabo interrupted. \" _Exeamus nos_! I'll get you people kitted up and settled in! Mollis! You wait here for the medics!\"\n\nThat was the beginning of my eight weeks in hell.\n\nStrabo organized us into a _contubernium_ , a tent group, and put Bantus in charge as our acting _decanus_ , our squad leader.\n\nGetting \"kitted out\" was a process in which we were stripped of all our civilian clothes and equipment and issued a mattress, a blanket, three shortsleeved military tunics, a wide leather belt, a pair of hobnailed boots called _caligae_ , a small satchel called a _loculus_ , a mess tin, and a battered, tarnished bronze _galea_ helmet. For the group, we were issued a \"cooking kit, one each, _contubernium_ ,\" which Bantus handed over to one of the _tirones_. Each piece of equipment that was issued was notated against our names, and an officer in the supply section informed us that if we lost any of it, we would have to pay to replace it.\n\nI was allowed to keep my own belt, _pugio_ , and _sagum_ cloak because they \"adhered to military specifications,\" but I was warned that I would have to get the _sagum_ dyed the appropriate shade of _carinus_ , the dark, reddish-brown, that was authorized for the Tenth Legion. Tulli remarked that, in this way, the cloak wouldn't show blood stains\u2014something I didn't need to hear just then.\n\nStrabo then herded us, balancing our teetering piles of clothing and equipment, toward the back of the camp where we were assigned to a tent. He then told us to \"drop our stuff and get into proper uniform.\" Bantus and Tulli helped us figure out what that meant. I noticed that we _tirones_ had been issued undyed, woolen tunics, while the _veterani_ wore red. I asked Tulli about this, and he told me that the white tunics identified us as _tirones_. When we were \"accepted\" by the legion, we would be issued red tunics like the legionaries who were considered \"qualified\" to take their places in the line of battle.\n\nStrabo had been gone no more than half an hour when we heard his voice outside the tent yelling, _\"Ad signam_! Fall in, you lazy, worthless maggots! Get out here on the street!\"\n\nWe were looking at each other, wondering who Strabo was yelling at and what \"fall in\" meant, when Bantus and Tulli started herding us out of the tent, \"Move! Move! Move! Grab your helmets! Insubrecus! Get your belt on! Let's go! Move!\"\n\nWhen we got outside the tent, the sight of Strabo stopped us dead in our tracks. He was in a full legionary combat rig. A highly polished, bronze _galea_ , an infantry helmet with red horsehair plumes, was tied tightly under his cleanshaven chin beneath the shining cheek guards. A blood red _sudarium_ , a military scarf, was wrapped around his neck and tucked beneath a shining chainmail _lorica_ , which reached halfway down his thighs. A highly polished leather _balteus_ , a sword belt, studded with shining bronze plates, was hanging from his left shoulder and passed across his chest down to his right side. From there was suspended a _gladius_ , encased in a red leather _vagina_ , and a scabbard, reinforced with brightly polished bronze cladding. A thick leather military belt, a _cingulum_ , was tightly fastened around his waist and held his _gladius_ in place on his right side. A _pugio_ in a scabbard hung on the left. From beneath his _lorica_ hung a skirt of thick, red leather strips, _pteruges_ , each one ending in a polished bronze tab, on which was stamped the visage of the god of war, Mars.\n\nDespite the winter cold, his legs were bare to his ankles, which were enclosed by the thick leather straps of his black, military _caligae_ , infantry hobnail boots. Over both shoulders, but pulled back to keep his weapons free, he wore the military cloak of the Tenth Legion, a blood-brown woolen _sagum_ , which was fastened at his left shoulder by a shining bronze _fibula_ , a brooch pin in the shape of a bull's head. His right hand tightly grasped the leather-wrapped hilt of his _gladius;_ in his left, instead of the accustomed _scutum_ or _pilum_ , the infantry shield and javelin, he held a long, thick wooden staff topped with a polished steel globe, the _hastile_ of an _optio centuriae_ , the \"chosen one,\" the second in command of a legionary century of eighty men. He was now our training officer and would help us become Roman soldiers.\n\n\"Bantus! Get this goat-rope straightened out!\" he screamed. \"I want two ranks right here! One behind the other! Move it!\"\n\nBantus and Tulli got us lined up in two ranks facing Strabo. As they positioned each of us, they whispered, \"Position of attention . . . Feet one _pes_ apart . . . Hands and arms at your sides . . . Stand up straight.\" Tulli tried to straighten out our helmets, which were wandering all around our heads, and to dress our tunics down through our military belts. Finally, Bantus took a position in front of our formation facing the apparition who was once our traveling companion, Strabo, and reported, \"Training detail all present, _Optio_!\"\n\nStrabo announced, \" _Contubernium_! _Lax . . . ATE_!\"\n\nBantus slid his right foot straight back, toe to heel, and clasped his hands in front of him. We tried to emulate him. My helmet immediately slipped down in front of my eyes. When I attempted to adjust it, Strabo screamed, \"Who gave you permission to move, _Tiro_ Gaius Marius Insubrecus? You're supposed to be a shaggin' Roman soldier! Stop fidgeting like a _paganus Gallicus_ waiting for his turn at the public latrine!\"\n\nAnd, there it was! From that moment on, my buddies in the Tenth Legion knew me as Gaius Marius _Paganus_ . . . Gaius Marius, \"The Hick.\"\n\nStrabo continued, \"The rest of you miserable _vermiculi_ , freeze! Don't move! Don't even breathe without my permission! This cluster has got to be the sorriest excuse for a military formation I have ever seen in my entire military career!\"\n\nStrabo began strutting across our front rank. \"I do not know what I could have possibly done to offend the immortal gods so badly that they would send the Furies out of the depths of Tartarus to inflict this on me! You are the sorriest excuse for Roman soldiers I have ever seen!\"\n\nSuddenly, the domed end of Strabo's _hastile_ staff shot out into the stomach of a recruit in the first rank. The breath exploded out of the man and he doubled over. \"Suck in that gut, Tiro!\" Strabo ordered. \"Stand up straight when standing in the presence of a superior officer!\"\n\nThe man struggled to regain his composure as Strabo continued his tirade. \"You are _tirones Romani_ , the lowest things on earth! You are lower than sailors' shit in the ocean! You are so low that you have to call the mules ' _sir_!' You will speak only when spoken to! And, your only authorized responses are, 'Yes, sir!', 'No, sir!', 'I do not understand, sir!', and 'No excuse, sir!' Do you pieces of fly shit understand me?\"\n\nThere was a ragged chorus of \"Yes, sir!\"\n\n\"What?\" Strabo yelled dramatically cupping his ear. \"I can't hear you! Do you _understand_ me?\"\n\nStronger this time, \"Yes, sir!\"\n\n\"What in the name of _Martis_ is going on here?\" Strabo screamed into our faces. \"Did the recruiters send a bunch of _puellulae_ , little girls, to this legion? Do you understand me?\"\n\n\"YES, SIR!\"\n\nStrabo stepped back. \"Bantus! Prepare the detail for inspection!\" he ordered.\n\n_\"A'mperi'tu_ ',\" Bantus snapped.\n\nBantus executed a smart about-face and said to us in a low voice, \"First, I'm going to call you to attention with the command, ' _Contubernium . . . Stat_!' Then, I will give the command, ' _Ordines extendit_! Open ranks!' At the command of execution, 'it,' the first rank takes one pace . . . that's two steps forward for you civilians. . . . The second rank stands fast . . . Ready now.\"\n\nThen, he said in a loud voice, \" _Contubernium . . . Stat_!\"\n\nWe assumed a fairly recognizable position of attention.\n\nThen, Bantus yelled, \" _Ordines_! _Extend . . ._ _IT'_!\"\n\nThose of us in the first rank managed to stumble forward the required distance and stop. Again, my helmet rearranged itself over my eyes, but this time I didn't dare adjust it.\n\nStrabo, trailed by Bantus, was walking down the first rank, reeling off criticisms, _\"Caligae_ improperly secured . . . helmet tarnished . . . unshaven . . . belt improperly adjusted.\" Behind me, I heard Tulli whisper, \"Second rank . . . stand at ease!\" I then heard the rear rankers rustle then go silent.\n\nStrabo stopped in front of me and demanded, \"Pugio!\"\n\nWhen I didn't react, Bantus said in a low voice, \" _Tiro_! Present your dagger to the optio!\" Then, he reached over and adjusted my helmet off my eyes.\n\nI removed my _pugio_ from its scabbard and handed it to Strabo.\n\n\"Sharp . . . no rust... Good job, _Tiro_ ,\" Strabo announced, then handed me back my knife. As Bantus followed Strabo down the rank, he gave me a quick wink.\n\nAs Strabo and Bantus arrived at the second rank, he commanded, \"Second rank . . . attention! First rank . . . stand at ease!\"\n\nWe assumed the position, right foot to the rear, hands clasped in front of us, while Strabo and Bantus reviewed the second rank to Strabo's mantra, \"Helmet tarnished . . . _caligae_ improperly secured . . . haircut . . . belt improperly adjusted.\"\n\nFinally, as Strabo and Bantus circled back to the front of the formation, Bantus ordered, \" _Contubernium_ . . . _STATE_!\"\n\nAfter we managed to close ranks, Strabo told us, \"You people have a long way to go before you even start looking like soldiers! When you get back to your quarters, you will start working on those bronze chamber pots sitting on top of your heads. By the tenth hour, I want them shined and polished so that I can see my face in them! Do you rat turds understand me?\"\n\n\"Yes, sir!\"\n\n\"Detail! _Ad dex' . . . VERT_! Right . . . FACE!\" Strabo ordered.\n\nWe all managed to shuffle about in the right direction. Bantus and Tulli positioned themselves at the head of our two files.\n\n_\"Promov . . . ET'_! Forward . . . MARCH!\" Strabo shouted, keeping a station to the left of our files, counting cadence, _\"Dex' . . . Dex' . . . Dex', Sin', Dex'_.\"\n\nAs soon as we were all moving together in the right direction, Strabo ordered, \" _Gradus . . . bis_! . . . _Mov . . . ET'_!\"\n\nWith helmets going one way and heads another at the double-time, we started bouncing back up toward the _Via Principalis_. As we ran, Strabo shouted, _\"Tirones_ , always move at the double-time! Every place you go, you run! _Vos vermiculi_ , got that?\"\n\n\"Yes, sir!\"\n\nWe spent the next two hours queuing up outside of various tents in the headquarters section. At one, our heads were shaved; at another, we were poked and prodded by various members of the medical staff; then, we were inspected for any undesirable critters in our body hair; at the camp bathhouse, we were dunked and scrubbed; and finally, we were allowed a meal of bread, oil, something cheese like, and water.\n\nWhen we got back to our tent, Bantus set us to work on shining our helmets with oil, sand, and rags. Tulli disappeared for a while and returned with a sack from which he distributed what looked like a _pilleum_ , the cap of a freed slave, but thicker. He told us to wear the cap underneath the helmet and tie the chin straps tight to keep them stable. If we needed more cushioning, we would have to double up on the _pilleum_ or stuff a piece of cloth between the cap and the helmet.\n\nWhile we were scrubbing and scraping our helmets, Bantus gave us our first lesson in how a legion was organized.\n\n\"Roman soldiers fight in pairs,\" he told us. \"Each legionary has a _geminus_ , a companion, a brother, a twin, who stands with him in the line battles.\"\n\n\"How can that be?\" I asked. \"I thought the legion formed a line to face the enemy.\"\n\n\"Three lines actually,\" Bantus replied. \"One behind the other, eight men deep, if the terrain permits. But, individual soldiers fight in pairs. One engages the enemy on the battle line, while the other protects his flanks, supports his companion against a determined enemy rush, and relieves him when he's exhausted, hurt, or wounded. If a soldier is wounded, his _geminus_ protects him and gives him medical assistance until he can be withdrawn from the line of battle.\"\n\n\"How does a soldier get his _geminus_?\" asked a big, lunking farm boy we called _Minutus_ , \"Tiny.\"\n\n\"Strabo will start pairing you off in training once he gets a feel for you,\" Bantus said. \"But once you're together, there's no stronger human bond\u2014not brother with brother, not father with son\u2014than the bond between legionary _gemini_. It's rare, almost shameful, for one to die in battle and the other to live.\"\n\n\"But, how's the legion organized?\" asked Rufus, a tall, lanky, redheaded _tiro_.\n\n\"Simply put, a legion consists of a headquarters and ten _cohortes_ ,\" Bantus explained. \"Each _cohors_ has six _centuriae_ ; each _centuria_ , ten _contubernia_ ; each _contubernium_ , eight _muli_. At full strength, that's 4,800 _pedes_ , infantry legs, plus the officers, but legions are rarely at full strength.\"\n\n\"So, we're pretty much a . . . what did you call it . . . a _contubernium_?\" I asked.\n\n\"Recte!\" Bantus replied. \"Correct! Strabo's got you organized into a _contubernium_. That's a basic tent squad of eight _muli. Contubernales_ live together in the same tent; they mess together in the field; they fight together in battle. This is a soldier's family within the legion . . . Hey, _Pustula_!\" Bantus suddenly called over to one of our group, a kid with a bad case of acne. \"Rub the sand in a circle, or you'll scratch the helmet, and Strabo'll have your ass!\"\n\nBantus continued, \"When your training's over and you get your red tunics, you'll be assigned to a _centuria_ , probably in one of the _cohortes ordinis secundi_.\"\n\n\"Our red tunics!\" Minutus piped up. \"Then we'll be _veterani_ like you, Bantus?\"\n\n\" _Veterani_?\" Bantus corrected him. \"No, you'll be a _miles_. You won't be considered a _veteranus_ until you're blooded in battle.\"\n\n\"What do you mean by a ' _cohors ordinis secundi_ '?\" I asked.\n\n\"A _cohors_ of the second rank,\" Bantus started, then called over, \"Hey, Tulli! Will you show Pustula over there how to rub the sand in before he carves a hole in that helmet?\"\n\nThen Bantus addressed my question. \"The legion lines up for battle in three _ordines_ , ranks. Cohorts one, two, three, and four are in the front; five, six, and seven in the middle; and eight, nine, and ten in the rear.\"\n\n\"What's the difference?\" I asked.\n\n\"Since the first rank makes contact with the enemy,\" Bantus explained, \"that's where you want your best soldiers . . . the big guys . . . guys who don't break. . . . In fact, the First Cohort is always on the legion's right flank. . . . When the legion advances, their job is to turn the enemy's flank . . . So the First Cohort guys are always your biggest and your fastest . . . the best guys to have in a scrum. . . . In most legions, the First Cohort gets extra pay and is immune from most details. At the end of a day's march, while you _muli_ are humping to dig a marching camp, they're out in front providing security.\"\n\n\"So, that's where we want to be,\" Rufus interrupted. \"More money, less work!\"\n\nBantus snorted, \"You're goin' to have to grow a bit before you get a chance at that, Red! But, remember, those guys earn their money! You can spend your entire military career in the third rank and never see a living enemy soldier, never get a scratch on you, but those guys in the front rank always make contact. If you're in the First Cohort, you better be good, or you're _perfututus_ , completely screwed!\"\n\nBantus grabbed my helmet and inspected it. \"That's a good job, _Pagane_ ,\" he said. \"Now, once you get all the tarnish off, buff it with one of the softer cloths. In fact, fog it with your breath, then polish it up, like this.\" Bantus demonstrated what he was talking about.\n\n_\"M'audite, infantes_!\" he addressed the group of us. \"Once you get these pots polished up, don't touch them with your fingers. It'll smudge the shine, and Strabo will write you up for it! Always keep a cleaning cloth with you for inspection. You can stuff it under your helmet on top of your _pilleum_. The damn pot'll fit your head better that way.\"\n\n\"That reminds me,\" Tulli butted in. \"You guys owe me a _minerva_ each for those caps.\"\n\nSuddenly, one of our squad, a guy who had not said much to any of us all the way up from Mediolanum, so of course we called him _Loquax_ , \"Gabby,\" asked, \"What about the officers, Bantus? Don't we have to watch out for them?\"\n\n\"Officers?\" Bantus snorted. \" _Cacat_! You're _tirones_ . . . and everybody knows it because of those white, vestal-virgin dresses you're wearing. . . . Everybody in this camp outranks you . . . even the shaggin' cockroaches . . . But you got a point, Loquax . . . Until you guys have your shit together, you want to avoid the centurions.\"\n\n\"Centurions?\" a _tiro_ we called _Felix_ , \"Lucky,\" piped up from the rear of the tent. \"There was this guy who had a small farm a couple miles from my village . . . Used to come into my dad's _caupona_ and get himself drunk a couple of times a month . . . Said he was a retired centurion . . . a real hard case.\"\n\n\"Durus,\" Tulli nodded, while trying to adjust the straps on Minutus' helmet so it would fit on his melon-sized head. \"That's as good a description for a centurion as you can get . . . hard as a boot nail on a forced march . . . and just as sharp.\"\n\n\"Centurions command the centuries,\" Bantus nodded. \"That means there's sixty in the legion, plus the _praefectus castrorum_ , the camp prefect\u2014that hard case we met this morning. My advice to you _tirones_ is to stay out of their way. Compared to one of those guys, Strabo's a pussycat.\"\n\n\"How can we recognize them to avoid them?\" I asked Bantus.\n\n\"They carry a _vitis_ , a cudgel made of vine wood,\" Bantus told me. \"If a centurion doesn't like what you're doing . . . or if he just doesn't like your face . . . he'll let you have it . . . across your back . . . on your shoulder . . . across the back of your legs . . . or right down on your head. . . . You spot a soldier carrying a _vitis_ , you better decide you have business in the other direction. . . . There's nothing to be gained by getting involved with a centurion.\"\n\nBantus seemed to be talking from experience.\n\nStrabo suddenly burst into the tent.\n\n\" _Contubernium! STATE!_ \" yelled Bantus.\n\nWe all jumped to our feet and assumed the position. Somewhere behind me I heard a helmet hit the ground.\n\n\"I need three volunteers for a detail,\" Strabo directed. \"You . . . you . . . and you!\"\n\nHe pointed to Minutus, Felix, and me. \"Helmets, belts, and boots!\" he ordered. \"Tulli! You take charge of these men and report to the mess tent. Move it! You . . . Pustula . . . pick up that helmet . . . The rest of you . . . I'm looking at this shaggin' pigsty you call a tent, and I'm not liking what I see!\"\n\nThe three of us spent the rest of the day working in the legion's mess tent, assisting the cooks, scrubbing the pots and cooking utensils, hauling water from the camp water point, serving the food, cleaning up after the meal, and then scrubbing the pots and cooking utensils all over again.\n\nWe didn't get back to the tent until halfway through the first watch of the night. When we arrived, we saw Loquax and Pustula standing guard at the entrance. No sooner were we three paces from the entrance when Loquax challenged us.\n\n\" _Consistite_! _Quis_ est?\" he called.\n\nWe stopped, more from surprise than obedience. Felix responded, \"Cut the shit, Loquax! You know who we are!\"\n\nThere were a few heartbeats of silence before Loquax said, \"Advance one to be recognized!\"\n\n\" _Cacat_!\" Felix said and walked forward.\n\nWhen Felix was about a pace away, Loquax said loudly, _\"Consiste_!\" Then, he said softly, \" _Palus_!\"\n\nFelix stopped and said to him, \" _Palus_? Swamp? What are you talkin' about? Swamp? We're tired! We want to get some sleep. Will you cut this shit out?\"\n\nBy this time, Tulli had come out of the tent. \"Will you two keep it down?\" he hissed.\n\nThen, he turned to Felix. \"This is a guard mount, _Tiro_! He's just given you the sign. If you want to pass, you have to give him the countersign.\"\n\n\"Sign? Countersign?\" Felix spat. \"Tulli! I don't know what you're talkin' about!\"\n\nTulli nodded. \"The countersign's _'cygnus.'_ When the guard challenges you with the sign _palus_ , you're supposed to respond with _cygnus_ , swan. If you don't, he's supposed to put a _pilum_ through your chest. Then, you can sleep forever, you stupid _mentula_!\"\n\nThen, Tulli called out to us, \"Bring it in here! You can relax, Loquax. I'll handle this.\"\n\nWhen we were around him, Tulli said, \"Strabo's established a guard mount around the tent at night. Each of you will pull one or two guard watches here each night, depending on the duty roster. In fact, you better get inside and get some sleep. Pagane, you and loudmouth here got third watch tonight. Minutus, you're on second watch with Rufus. I'm your _tesserarius_ , sergeant of the guard. I rouse you up and brief you when it's your turn. But, remember, if you got to leave the tent to go to the latrine, the sign-countersign tonight is _'palus-cygnus.'_ Repeat that!\"\n\nWe did.\n\n\" _Bene_!\" Tulli said. \"Now, get in your bunks, and get some sleep! We eat chow at first call tomorrow, and I'm sure Strabo has a busy day planned for you boys.\"\n\nAnd, that was the start of many \"busy\" days for us. We roused up just before the horn signaling the end of the fourth watch. We were double-timed to the mess tent for some bread and watered-down _posca_. Then, Strabo led us on a twenty thousand pace trudge into the hills, partly marching, partly double-timing, and sometimes outright running. We were back in camp by the seventh hour where we were fed some kind of hot porridge, boiled vegetables, bread, and more _posca_. After about an hour's rest, Bantus and Tulli trained us in close-order drill, marching, and formations. Then, Strabo was back with more physical conditioning.\n\nOne of Strabo's favorite drills was what he called _situlae_ , buckets. Each of us would grab two empty buckets from the mess tent. Then, we would double-time out to the water point about a thousand paces west of the camp, fill the buckets, double-time back, and empty the water into the mess-tent troughs. Sometimes, Strabo would have us carry the filled buckets at our sides until our shoulders seemed like they were on fire; then, we'd shift them in front until our biceps were almost bursting, then behind us until our triceps burned. Sometimes our hands were under the bucket handles and sometimes on top, so our forearms got a work out.\n\nOf all Strabo's dirty little tricks, _situlae_ was the worst. I even saw Minutus, despite his size, weeping one day because of the pain in his shoulders and arms. But, Strabo kept driving us, shouting at us to move faster and to keep our buckets up, saying that in order to wield the infantryman's weapons in battle\u2014 the _gladius, scutum_ , and _pilum_ \u2014we needed upper-body strength.\n\nThere were nights, as I hit my cot, I couldn't feel my arms and shoulders at all, and my legs ached like a bad tooth. And, the nights when I could hit my cot were the best ones. Every night, we each pulled a guard watch, sometimes two. Despite our exhaustion, we didn't dare fall asleep on guard. Bantus told us that sleeping on guard was punishable by death; it endangered everyone in the camp. An offender was cudgeled to death in front of the entire legion by his tent mates. We didn't question whether this applied to us guarding a tent full of recruits in the middle of a legionary camp; none of us wanted to find out\u2014especially after what happened to Rufus.\n\nWe were running buckets one afternoon. We had all learned the trick of not filling the buckets up to the top to lessen the weight. We had to be careful because Strabo was wise to most soldiers' tricks, but as long as we didn't overdo it, we could usually get away with a few _ligulae_ less than a full bucket. That day, Rufus was having some problems. He had hurt his back earlier in the week when he had stumbled during one of the conditioning marches, but Strabo had refused to send him to the medics, telling him he was a _puella_ for even asking. So, instead of spilling a few _ligulae_ of water out of his bucket, he spilled most of it. Strabo caught him. We all thought it was a joke, part of the game.\n\nStrabo decided to use Rufus to demonstrate what happens when a \"Roman soldier fails in his assigned duty,\" as Strabo stated the charge.\n\nNext morning, before chow, Strabo lined us up in a small drill field near the _praetorium_. He marched Rufus out in front of the formation and announced that for failing in his duties, Rufus would suffer a _castigatio_ of ten blows with the _optio's_ staff. Rufus removed his helmet and cap, unbuckled his soldier's belt, stripped off his recruit tunic, and while Bantus held his wrists, Strabo inflicted the _castigatio_.\n\nTo his credit, Rufus did not once cry out\u2014despite the fact that we could see each blow smack across his shoulders and drive the breath out of his body. At one point, Rufus seemed to stumble forward into Bantus, who straightened him up and urged him to take his correction like a Roman. When it was over, Strabo was sweating from the exertion, and Rufus' back was striped crimson and white from the beating. Rufus slowly put his tunic back on and rebuckled the belt around his waist. He winced as he lifted his arms to pull his _pilleum_ cap down on his head but managed to regain control of himself. He tied his helmet straps under his chin and took his place back in our formation.\n\nStrabo announced to us that the _castigatio_ we had just witnessed, a beating with an _optio's hastile_ staff, was one of the mildest forms of discipline in the legion. Had Rufus failed in his duty in the presence of or in contact with the enemy, he would have been beaten to death. Having said that, Strabo led us out of camp on our conditioning march.\n\nAfter three weeks, we lost Bantus and Tulli. They were assigned to a _centuria_ in the Fifth Cohort. Tulli was pleased. He said that was far enough forward in the battle line to have honor and far enough back not to be _semper immerda_. Strabo named Minutus as our acting _decanus_ because he had kept his nose clean, and he was the biggest guy in our _contubernium_ \u2014far too big for us to say no to easily. I replaced Tulli as acting _tesserarius_ , but this was no break for me. I still had to take my turn on the sentry duty roster, and I also had to ensure that every relief was made throughout the night. I soon learned to sleep in two- to three-hour snatches. Strabo said that was good training.\n\nDuring the fourth week, we began our weapons training. Strabo lined us up and marched us over to supply to draw our combat armor. There was no point in doing weapons practice in our tunics, he told us. That was not the way we would actually fight.\n\n\"Train the way you fight; fight the way you train,\" he said. \"That's the Roman way!\"\n\nFirst, we turned in our training helmets for the newer models that we would actually wear in combat. They were heavier, but they had better protection, a rear neck guard, wider cheek guards, and a reinforced \"brow\" above the eyes. Strabo inspected each of our helmets to be sure that they weren't rejects that the supply people were trying to fob off on us. He said the metal had to be of a uniform thickness, with reinforcement over the crown of the head and no sign of repair welds.\n\nLoquax, noticing the socket on top of the helmet, asked when we were going to get our red infantry crests.\n\n\"When you've earned it, _Tiro_!\" Strabo snapped.\n\nAt the next station, we were issued our body armor, a coat of chainmail called _lorica hamata_. When we got outside the tent, Strabo had us lay our _loricae_ out for his inspection. He talked us through what he was doing so we would eventually be able to do it ourselves. First, he told us to be sure the _lorica_ is iron, not brass. Iron rusts and is a pain to keep clean, but it's much stronger than brass, and that might be the difference between just getting the wind knocked out of us when some Kraut _podex_ tries to stick a sword through us and having the _medicus_ try to reassemble our guts so we look neat on the funeral pyre.\n\nNext, Strabo instructed us to check the size of the rings\u2014smaller is stronger than bigger\u2014and to check how the rings are entwined with each other. Each ring should be entwined with a minimum of four other rings. The more connections, the more protection. There should be no broken rings and no rings missing rivets. If we found any of these, we should take the _lorica_ back to supply and draw a new one.\n\nWe had to make sure the leather closure straps were present and not frayed. We didn't want a strap breaking in combat and our armor falling off. \"Very embarrassing and usually quite fatal,\" Strabo quipped.\n\nNext, we should check the shoulder straps for fraying. That's where we would be attaching an additional layer of mail to protect us from slashing attacks and ax blows coming down over the top of the _scutum_ , a favorite trick of those long-haired Gauls on the other side of the Alps. That shoulder armor was something else we didn't want falling off when it was needed.\n\nNext, we had to check the fit. For this, Strabo went back into the supply tent and returned with a pile of what looked like padded red jackets and a bunch of red rags.\n\n\"Take off your belts and put these on over your tunics,\" he told us, handing out the jackets. \"This is your _subarmalis_. It gives your shoulders and body some padding from the chainmail.\"\n\nWe put the _subarmales_ on and closed them with lacings up the front. When he was done, Minutus, who had struggled to close the jacket, looked a bit like a giant red sausage. Short leather straps sewn on both sleeves covered the upper arms, and a skirt of leather straps covered the crotch and upper legs.\n\nPustula started strapping his belt back on over the _subarmalis_ , but Strabo stopped him. The belt went over the _lorica_.\n\nWhen we all had our _subarmales_ on, Strabo took one of the red rags and called Felix over to him. He threw the rag around Felix's neck, and just as we were convinced Strabo was going to strangle him, he said, \"This is your _sudarium_ , your infantry scarf. It's good for a lot of things: rubbing the sweat out of your eyes on the march, wiping the snot off your noses on a cold day, or plugging holes in a buddy; but its customary use is to pad the neck to keep it from being torn to shreds by the iron rings of your _lorica_. You tie it like this.\"\n\nStrabo tied the _sudarium_ around Felix's neck, saying, \"Tie it this way so some _podex_ doesn't grab hold of it in a fight and strangle you with it.\"\n\nWe each picked one out of the pile, and we tied them around our necks like we had been shown.\n\nFinally, Strabo told us to strap on our _loricae_. This was a two-man job. Felix held up my _lorica_ in front of me, and I put my arms through the sleeves in the chainmail jacket and moved forward until it rested on my shoulders. Then, Felix moved behind me and pulled the _lorica_ tight across my body.\n\n\"How's that feel, Pagane?\" I heard Strabo's voice behind me. \"Should be tight enough to give you protection, but loose enough for you to breathe.\"\n\nFelix adjusted my straps, then handed me the ends of my belt. Before I could buckle it, Strabo stopped me.\n\n\"Look over here, boys!\" he called. \"I want to show you a little infantryman's trick with these belts.\"\n\nStrabo grabbed my _lorica_ just below where my belt would ride and pulled it up a bit. Then, he told me to buckle my belt. When I did, he let the resulting fold of the _lorica_ fall into place on top of my belt.\n\n\"Look here!\" he said. \"If you adjust the _lorica_ over the belt like this, the belt takes some of the weight off your shoulders. That way, your _gladius_ and _scutum_ can move more quickly in combat. That could be the difference between walking back from a fight and being carried back.\"\n\nThen, using my _lorica_ as an example, he said, \"Be sure the ends of the coat overlap by at least two palms in the back and extend at least three palms below your balls.\"\n\n\"Now, jump up and down, Pagane!\"\n\n_\"Qui' vis m'agere_?\" I challenged, thinking he was putting me on.\n\n\"Jump up and down, Maggot!\" Strabo ordered.\n\nI did. The _lorica_ moved a bit but stayed on.\n\n\"Always test your fit,\" Strabo said. \"When you are moving quickly, you don't want this thing shifting and exposing nice, tempting targets to your enemy, like your throat, crotch, or armpits. Now, the rest of you cockroaches suit up!\"\n\nWhile I was helping Felix into his rig, Strabo went back into the supply tent. I just about had Felix squared away when Strabo returned, followed by a couple of the supply clerks carrying what looked like wide strips of chainmail. They piled these on a piece of canvas lying on the ground in front of us.\n\nWhen they had gone back into the supply tent, Strabo called, \"Gather around me, blattae!\"\n\nWe did, and Strabo held up one of the chainmail stoles. We then noticed it had leather straps. \"These are your _chlamys_ , your shoulder-armor rigs. You inspect the chainmail the same way that I showed you with the _loricae_. It attaches like this . . . Get over here, Pagane!\"\n\nI walked over, and Strabo adjusted a _chlamys_ around my neck and over both my shoulders. \"You attach the _chlamys_ to the back of the _lorica_ with these two straps first,\" Strabo instructed, strapping my shoulder armor down. \"Then you close it with these straps. Be sure the _sudarium_ is up above the edge so the chainmail doesn't chafe your neck while you're moving, which, for a Roman soldier, is always. Then, come around front and attach these three straps . . . _Bene_! Pagane! Sali!\"\n\nThis time, I didn't question Strabo. The chainmail rig moved a bit as I jumped, but settled back smoothly on my upper body.\n\nStrabo continued, \"That's how a well-adjusted _lorica_ should move on your body . . . smooth . . . no gaps . . . Stop jumping, Pagane! Now, there's a hook here on the front of the _chlamys_ . . . It should be positioned about the middle of your chest . . . This is to hang your _galea_ , your helmet, when the centurion lets you remove it during the march.\"\n\nStrabo bent over, picked up one of our helmets, and hung it on my rig.\n\n\"Fits like that!\" Strabo said. \"Any questions?\"\n\nThere were none, so Strabo said, \"Bene! Get yourselves rigged out in these _chlamydes_ , then helmets on and strapped!\"\n\nWhen we were all in our _loricae_ and lined up in our two files, Strabo announced, \"You're beginning to _look_ like Roman soldiers, but you're not there yet . . . not by a long shot . . . But from now on, you will _act_ like Roman soldiers . . . That means you're in your _loricae_ and _galeae_ from first trumpet in the morning until seventh hour every duty day . . . longer when there's an enemy near . . . _Bene_! It's still early . . . We can get at least a ten thousand-pace march in before chow!\"\n\nStrabo shouted, \" _Contubernium . . . STATE'_!\"\n\n\" _Ad dex' . . . VERT'_!\"\n\n_\"Promov . . . ET'_!\"\n\n\" _Gradus . . . Bis . . . mov . . . ET_ '!\"\n\nOff we clinked and clanked at the double toward the _Porta Decumana_ at the rear of the camp. And, Strabo was good to his word. From that day, we did everything like Roman soldiers, in full armored rig: conditioning marches; weapons training; even \"buckets.\"\n\nOur first phase of weapons training was with the infantry sword. Strabo double-timed us to a training field along the _Via Principalia_ near the _Porta Dextra_ of the camp. From Macro's drills with the _pugio_ , I immediately recognized the _palus_ erected in the ground. Those drills, although only six months earlier, seemed a different world to me. Strabo gathered us at the edge of the field and drew his sword from the scabbard on his right hip.\n\n_\"Audit' me, vermiculi!_ \" he yelled holding up the sword to us. This is the _gladius hispaniensis_ , the Spanish short sword, the basic weapon, best friend, and only true and faithful lover of a _pedes Romanus_. Like a good woman, if you take care of her, she'll take care of you! The _gladius_ is a carbon steel, double-edged sword with a tapered point for stabbing during combat. The blade is nine palms in length and one palm wide at its widest point. The gladius weighs just less than three _librae_ . . . Pustula! You cockroach! Listen up! . . . The _gladius_ possesses a solid grip provided by a ridged, wooden hilt wrapped in leather and secured by metal wire. It has a knobbed hilt which prevents your hand from sliding forward onto the blade\u2014regardless of how much guts and blood has slicked your sword hand. It also has a knobbed pommel, which prevents the sword from being ripped from your hand when the blade gets stuck in the bone and gristle of some hairbag _mentula_ you have dispatched to the ferryman. The pommel is also weighted, to give the _gladius_ perfect balance, which, when you sorry excuses for Roman soldiers are properly conditioned and trained, will make the _gladius_ feel weightless in combat. This sword in the hands of a trained and motivated Roman legionary\u2014which you maggots-in-chainmail are not\u2014is the finest weapon ever to be introduced onto the field of battle. Had Alexander the Great and his armies had these swords, we'd all be speaking Greek today. Do you _bestiolae_ have any questions?\"\n\n\" _N'abemus, Optio_!\" we answered in unison.\n\n\" _Bene_!\" Strabo continued. \"The _gladius hispaniensis_ is a stabbing sword, not a slashing sword. Barbarians, cavalrymen, and pissed-off wives slash. Roman soldiers stab. In this field, you will learn the proper technique for using a legionary's _gladius_ on the field of battle. However, until you maggots prove to me that you are worthy of the title _milites Romani_ , you will not put your meat hooks on real steel!\"\n\nStrabo dramatically sheathed his sword. He bent over and picked up a wooden replica of the _gladius_ , which he held up for us to see.\n\n\"You will be learning your combat sword techniques using a _rudis_ ,\" Strabo announced. \"The _rudis_ is made in the exact dimensions of a _gladius_ , but it's heavier than the real thing by almost two _librae_. This is to condition your arms and shoulders so that when the army finally has enough confidence in you maggots to give you real swords, they will feel like feathers in your hands. Before we start, each of you will file through the tent to my rear, where you will be issued your training swords. You will keep these _rudes_ with you at all times. Awake! Asleep! Coming! Going! Walking! Running! Your _rudis_ will either be in your hands or in your belts. They will be so much a part of you that, if some night you dream that a five-headed hydra pounced on you from out of your mommy's closet, you will be able to kill that scaly, slimy maggot with your _rudis_. If I ever see any of you without your little wooden swords, you will be cleaning _merda_ out of latrines until you begin to enjoy it. Do you have any questions, _me' blattulae_?\"\n\n\" _N'abemus, Optio_!\" we shouted.\n\nSo, we began our _gladius_ training. Strabo abbreviated the daily conditioning marches to ten thousand paces, but we were now doing them in armor and double-timing at least half the distance. We were back in camp by the fourth hour and on the stakes until the sixth. Chow. Rest and cleaning. By the eighth hour, back on the stakes, where we remained until the tenth\u2014eleventh if Strabo didn't think we were \"motivated\" enough. The training was familiar to me. I'd been through it already with Macro. The only difference was the size and weight of the _rudis_.\n\nStrabo was true to his word, and we always kept our practice swords with us. Lentulus left his on his bunk at the end of the day when he went to take a shit. When he got back to the tent, Strabo was waiting for him. As Strabo was beating Lentulus on the shoulders and upper arms with his discarded practice sword, he reminded Lentulus that a real soldier never walks away from his sword. Then, Strabo put Lentulus on latrine-cleaning duty. But, so that Lentulus wouldn't miss any training, or burden his _contubernales_ by missing his guard shifts, he had to clean the shitters at night, when everyone else was sleeping. I remember the morning Lentulus stumbled back into our tent, less than an hour before the end of the fourth watch and the beginning of our training day, smelling like the _merda_ he'd been scraping out of the latrines all night.\n\nStrabo didn't spare the rest of us either. He told us no _contubernalis_ would ever let a mate walk away from his sword. That endangers the man and the unit. He had us suit up in our armor, and until the end of the first watch, we had to double-time around the _intervallum_ , the open space between the camp's wall and the soldiers' living quarters.\n\nAfter two weeks of sword training, Strabo introduced us to the second basic legionary weapon, the _scutum_. Since we were _tirones_ , we were not given the real thing, but a weighted wicker shield Strabo called the _vimen_ , the \"basket.\"\n\nTo my surprise, in the hands of a trained Roman soldier, the _scutum_ was as much an offensive weapon as it was a defensive one. Using a _palus_ , from which hung a sack of sand, Strabo taught us to \"punch\" an enemy with the _umbo_ , the iron boss of the shield, a fighting technique he called _percussus_. On one of the padded training stakes, Strabo demonstrated how the left hand grasped a padded, metal handle welded to the back side of the _umbo_ , and turning through the hips and shoulders, smashed the iron boss into an opponent. After striking the bag of sand a few times, Strabo \"asked\" for a volunteer to attempt the technique and immediately pointed to Minutus, \"Tiny.\"\n\nThe weighted _vimen_ looked like a dinner tray in Minutus' mitt, and the first time he attempted the _percussus_ , the training pole seemed to shift a bit in the ground.\n\nBut, that didn't satisfy Strabo. \"Is that all you got, _tu puella_?\" he shouted at Minutus. \"You little girl! Hit that pole like you have a pair!\"\n\nMinutus' face reddened a bit, but he hit the _palus_ again, harder this time. I heard him grunt and thought I heard the stake crack.\n\n\"My baby sister hits harder than that!\" Strabo taunted. \"Hit it again, _me' puella_!\"\n\nMinutus looked hard at Strabo. For a second, I wasn't sure whether his next _percussus_ was going into the stake or into our training officer. The stake lost. Minutus smashed it with a grunting shout and the stake split at ground level; its shattered fragments flew back at least three paces.\n\nWe were stunned. Even Strabo was rendered speechless for a few heartbeats.\n\nFinally, Strabo announced, \"That's the way a Roman soldier executes a _percussus_!\"\n\nHe took the _vimen_ from Minutus. We could all see that its iron _umbo_ was crushed. Then, something happened that we had never seen in all our training. Strabo, staring down at the crushed iron boss, finally looked up at Minutus, who was panting a bit and standing next to the shattered _palus_ , and said, \"Minutus! Return to quarters! You have the rest of the afternoon off!\"\n\nAs Minutus double-timed down the camp street, Strabo turned to us, the shattered _vimen_ still in his hand, and shouted, \"What are you maggots staring at? Pick up your baskets! Find a pole! Get to work!\"\n\nWe spent the rest of that day, until well past the tenth hour, punching sandbags with our training shields. Despite Strabo's shouts of \"encouragement,\" none of our repeated blows as much as shifted the training stakes. When we finally got back to the tent, our shoulders and arms had no feeling left in them at all.\n\nMinutus was on his cot, sleeping like a baby.\n\n# II.\n\n# _Hostes apud Amicos_\n\n# ENEMIES AMONG FRIENDS\n\nM _ensis Martis_ , the month of the god of war, Mars, arrived, and activity in the _castrum_ of the Tenth Legion picked up. Rumors had it that the tribes up in _Gallia Comata_ , long-haired Gaul, were again on the move. The snow would soon be melting in the Alpine passes, and Caesar _Imperator_ would be summoning his legions for a summer campaign against them.\n\nThe regular infantry _cohortes_ of the legion were now regularly in the field on conditioning marches and training exercises. The _immunes_ , soldiers with special skills who were exempted from fatigue details, remained in camp and worked daily, cleaning and repairing equipment for the expected campaign.\n\nStrabo was pushing us hard to be ready. Our training day rarely ended before the eleventh hour, and with the change of season lengthening the hours, we tumbled into our cots every night totally exhausted, only to be roused in a matter of hours for a tour of guard duty.\n\nIn a practice field just outside the _Porta Sinistra_ , Strabo had us working daily on combat techniques for the battle line with our _rudes_ and _vimenes_. He constantly drilled us in two techniques: the close-order defense and the open-order advance.\n\nThe basic infantry combat formation of the _centuria_ was a column of _contubernia_ , one squaddie behind another, with each _contubernium_ in the _centuria_ lined next to each other. Typically, the distance of one _gradus_ , a little less than three _pedes_ , separated one soldier from another, so sword and shield were unencumbered. Soldiers typically measured the distance as an arm's length between men across the front line. The exception to the \"three-pedes rule\" was the _gemini_ pairs. The partner of the man on the front line positioned himself at a distance to support his _geminus_. Because we didn't make up a full eighty-man _centuria_ formation, we simulated with a four-man front line in _gemini_ pairs. For most of our training, I was paired with Loquax. At times, I was on in front; at other times, he was.\n\nThe close-order defense, or the _murus scutorum_ , \"the shield wall,\" as the _veterani_ called it, was used when defending against a determined assault from a numerically superior enemy. The goal of the _murus_ was to hold a position or to relinquish ground as slowly as possible. In this formation, the _contubernia_ were aligned in \"close order,\" about half a _gradus_ between men across the front line\u2014so close that shields could be interlocked, forming a wall facing the enemy.\n\nInitially, the primary role of the _geminus_ of the front-line man was to place his shoulder into his partner's side and dig his hobnails into the turf so his mate isn't bowled over by the initial impact of the enemy assault. Then, as the front-line squaddies stabilized and sustained the shield wall, the _geminus_ ensured that no one got to the front-line man by stabbing at him over or under his shield.\n\nTo practice this, Strabo detailed about a score of legionary slaves to charge at us, usually down a hill with us at the bottom. If Strabo was in a really perverse mood, he'd find a nice patch of mud for us to form our _murus_ in. Half the slaves would be carrying sacks of sand to give their charge some momentum and produce a significant shock when they hit our shields. The others would carry blunted stakes. After the first wave of slaves crashed into our wall with their sand bags, the ones with the stakes would keep our back-line boys busy defending as they poked at us over the shield wall.\n\nThe first time we practiced the _murus_ , Strabo had us formed in the mud at the bottom of a ridgeline. When the first wave of slaves, the ones with the sandbags, hit our shield wall, the feet of our first-line men went out from under them, and they went down, taking the second line, the _gemini_ , with them. The first wave of slaves followed their own momentum over our falling bodies and ended up in the mud with us. The second line of slaves, the ones with the poles, tried to stop, but had too much momentum from charging down the slope, and with no footing on the slick muddy ground, they piled on top of the already struggling scrum.\n\nStrabo watched this muddy pile of arms and legs for a while to see if we could extricate ourselves. Finally, he began dragging individual bodies out of the pile by their muddy arms and legs. When he eventually had us untangled and back in some semblance of a military formation, he congratulated us for being the first Roman military unit ever overrun by a pack of slaves armed with sandbags and sticks. He then double-timed us back to our tent, telling us we had one hour to prepare for a complete inspection of our kit, living quarters, and persons\u2014and we would not be permitted to eat, sleep, or drink anything until he could not find one speck of mud or dirt anywhere.\n\nFinally about halfway through the second watch of the night, we were permitted to sleep\u2014at least those of us who weren't on guard duty.\n\nThe slaves got a charge out of knocking us flat. Typically, legionary slaves were good eggs; serving with a legion was a good deal for them. Although they were the property of the _res publica_ , they got plenty to eat and were worked no harder than the soldiers were. After twenty years of service, they were freed, given a _pilleum_ , the \"liberty cap,\" and the franchise. In many ways, they were better off than most citizens, and they knew it. But, there was one rule they were never allowed to break. Ever since the slave insurrections down in Sicilia and Italia, no slave was allowed to hold a weapon. The penalty was crucifixion.\n\nThere was an old war story that went around the camp about a fight in which the enemy had breached the Roman line. A squad of legionaries from the _tertius ordo_ , the \"forlorn hope\" of any legion, was the only thing keeping the barbarians from breaking through to the rear of the Roman formation. A _contubernium_ fought alone until there were none left standing; every squaddie was either killed or wounded. That was when an army slave picked up the _gladius_ of a wounded legionary and fought the enemy. He not only saved the lives of his wounded comrades, but he saved the army as well. After the battle, the _legatus legionis_ , the commander of the legion, awarded the slave the Civic Crown for saving the lives of Roman citizens and standing his ground against overwhelming odds. He then immediately ordered the slave to be crucified for daring to take up arms.\n\nEven Strabo snorted at that story. He said the Civic Crown would never be awarded to a slave. Besides, if it were true what the slave had done, single-handedly stopping the enemy and saving the lives of Roman soldiers, the legion would not tolerate his being executed\u2014even if some pumped-up, patrician _legatus_ ordered it.\n\nThe other formation that Strabo drilled was the open-rank advance, which was used against a numerically inferior enemy or one whose battle line had been broken but was not yet fleeing. The purpose of the open-rank advance was to gain ground quickly without losing the integrity of the legionary battle line. For this, our battle line assumed an \"open\" formation, double spacing between each man in the front line. To measure this, we'd raise our arms and align ourselves fingertip to fingertip with the men on either side. The second pair in each _contubernium_ would maintain the same interval but would align themselves in the gaps between the front-line pairs. So, the _centuria_ would advance in what looked like a checkerboard pattern, _in quincuncem dispositi_ , deployed in the oblique, Strabo called it.\n\nThe individual combat technique used in the advance was what Strabo called the \"Roman one-two punch,\" a combination of a forward _percussus_ with the _scutum_ followed by a full forward thrust with the _gladius_. The movement started with the right foot forward, facing the enemy. The _percussus_ was executed when the legionary stepped forward with the left foot, punching the shield's _umbo_ into the enemy's head, face, or chest. This was immediately followed by stepping forward with the right foot, stabbing forward into the enemy's throat, abdomen, or groin with the short sword. The _mulus_ repeated the technique until the advance was halted or he was relieved by his _geminus_ or there was no one left in front of him to kill.\n\nThis technique left the front-line man horribly exposed. In stabbing forward with the _gladius_ , the soldier's entire right side, his _latus apertum_ , his \"open side,\" was exposed. His partner's main concern was to keep the enemy from sneaking in on that side when the front-line man's sword was fully extended. While there was also some vulnerability on the soldier's _latus opertum_ , the left side, when he delivered the _percussus_ , it was significantly less than on the sword side, so the forward man's _geminus_ tended to hover on his right side during the advance. The _geminus_ ' other task was to relieve his partner when he became exhausted.\n\nStrabo used the open-rank advance drill for conditioning. After lining us up in a four-man front in full kit with training swords and \"baskets,\" he would have us advance across an open field with no opposition: step, punch, step, thrust, withdraw, step, punch, step, thrust, withdraw. The whole time, he was screaming at us:\n\n\"My sister hits harder than that, Loquax!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Stay aligned! Stay aligned!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Harder, Rufus, you little girl!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"You hit like that, Pustula, and the only way you'll kill a barbarian is if he laughs himself to death!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Lentulus! You're falling behind! Move it!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\nAnd, so it continued until we were ready to puke our guts up\u2014or until after one of us _had_ puked his guts up.\n\nTo give us the real feel of the technique, Strabo again brought in a detail of slaves with sandbags. He aligned us in a four-man front\u2014each front-line man with a trailing _geminus_ \u2014and had us advance up a ridgeline against the sandbagwielding slaves.\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Stay aligned!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Cover Loquax's open side, Pagane!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Hit harder, Loquax! Make that man grunt!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"You puke on my grass, Minutus, and I'll have you for breakfast!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\nAbout halfway up the ridge, Strabo had us execute a relief where the trailing _geminus_ replaced the lead man on the front line without breaking the momentum of the advance. To do this, the trailing man swung around the open side of the lead man as he was delivering the forward thrust with his _gladius_. The trailing man would come through the gap, while delivering a _percussus_ with his _scutum_. The man who had been relieved would then take up the trailing position by moving around to the open side of his partner by the time he delivered his first sword thrust. The advance continued, uninterrupted, with Strabo shouting directions:\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Get your alignment back, you maggots!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Hit harder, Pustula, you cockroach!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Loquax, cover Pagane!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\nWhen we reached the top of the ridgeline, Strabo let us briefly celebrate our victory over the twelve withdrawing slaves with sandbags. Then, he ran us back down the slope to do it again.\n\nIt was a couple of days past the _Ides Martis_. The weather had finally broken, and we could feel the warmth of the returning sun on our faces, necks, forearms, and legs when we trained outside. There was now plenty of mud in the camp and in the surrounding fields for Strabo to make our training \"interesting.\"\n\nThe Tenth Legion was in a frenzy of preparation for the campaign season. Entire _cohortes_ of the legion were playing war games and staging mock battles against each other in the training fields around Aquileia.\n\nAt our morning training formation, Strabo told us that he had heard a rumor in the officers' mess that there was a crisis brewing in the north. A Gallic tribe, the Helvetii, had overthrown their Senate and leaders, and their warriors were on the move toward our _provincia_. The last time the Helvetii had moved south was during the great Cimbri invasions in Marius' time. Then, they allied with the Krauts, killed a Roman consul, and forced an entire Roman army to pass under the yoke. Caesar _Imperator_ , our proconsul and commander, had rushed north from Rome to Gennava, an _oppidum_ , a fortified trading town of a Gallic tribe called the Allobroges, _socii Populi Romani_ , an ally of Rome, at the farthest extent of our _imperium_ in Gaul. Gennava guarded a bridge near the mouth of the River Rhodanus where it flowed down from a great lake called Lammanus. The Helvetii wanted to seize the bridge in order to invade our lands and those of our allies. Caesar intended to stop them.\n\nCaesar had no confidence in these Allobroges. He doubted they would stand with us and was sure they would not stand alone against the Helvetii. But, Caesar had only one legion north of the Alps, Strabo's Eighth. The Seventh, Ninth, and our Tenth, all veteran legions, and two newly recruited legions, the Eleventh and Twelfth, were encamped around Aquileia. Caesar had dispatched one of his _legates_ , his senior commanders, from Rome to take command of these five legions and get them ready to march. As soon as the passes across the Alps were open, they were to move into _Gallia Transalpina_ to reinforce the Eighth at Gennava before the Helvetii overran them. If the Helvetii were to succeed, the lands of our allies, the Allobroges, and the entire Roman _provincia_ in Gaul would be ravaged by these barbarians.\n\nOur training took on a new sense of urgency. We had to be ready to stand in the battle line by the time our legion moved out. So, as the sun was just beginning to pink the eastern sky, Strabo double-timed us down through the _Porta Decumana_ , through the civilian _vicus_ , and out into a large, flat, grassy field, where about a dozen legionary slaves were waiting for us with their sandbags.\n\nStrabo lined us up for the open-order advance. I was on the front line for the first run-through. Minutus was my _geminus_.\n\nThat was what saved my life.\n\nThe slaves lined up opposite us with their sandbags, acting as targets for our weighted training shields and swords. We began the drill.\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Get your head in it, Felix!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\n\"Hit that bag like you mean it, Pustula! That slave's laughing at you!\"\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\nAfter about four or five repetitions, as I made my sword thrust, thus exposing my side and sword arm, I felt a blow on my forearm. I didn't think anything of it and continued the drill.\n\n\"Step! Punch! Step! Thrust! Withdraw!\"\n\nThen, I heard Minutus gasp, \"Pagane! You're bleeding!\"\n\nI looked down at my sword arm, still extended in my thrust, and noticed a red, dripping rent near the top of my forearm. I froze. I noticed that the slave in front of me had dropped his bag and was moving in on my open side. There was something shining in his right hand.\n\nBefore I could react, I sensed a blurred motion from my right rear. I saw Minutus' arm shoot out and his weighted _vimen_ hit the charging slave full in the face. The man crumbled like a puppet whose strings were suddenly cut. Minutus followed up his _percussus_ by stabbing the supine slave in the throat with his wooden _rudis_.\n\nIt was textbook. I knew the man was dead.\n\nThen, I heard Strabo yell, \"What, in the name of _coleones Martis_ , are you two doing?\"\n\nThen, the pain hit.\n\nIt shot up my arm into my brain, like a hot, red wave. I looked and saw my arm dripping with blood below the elbow. My brain vaguely registered that the blood had flowed down over my hand and onto my wooden practice sword, and I was fleetingly worried how I was going to get it cleaned up for inspection. I fell to my knees, dropping my basket. Another demerit, my foggy mind registered.\n\nThen, Strabo was there with me. He was pulling my red _sudarium_ , my infantry scarf, off my neck and stuffing it into the hole in my arm.\n\nI heard him saying, \"It's dripping, not spurting . . . That's good . . . It's a scratch . . . You'll be fine . . . Can you move your fingers?\"\n\nI wondered what my fingers had to do with anything, yet I willed them to move. It seemed to take a long time, but finally they did move for me.\n\nAgain, Strabo, \"Good . . . That's good . . . That _cunnus_ didn't slice any of the tendons . . . Minutus! Give me your _sudarium_ here!\"\n\nI felt something tighten around my arm over the wound.\n\nThen I heard Felix's voice, \"He had this, _Optio_!\"\n\nStrabo, \"A _sica_! How did that _podex_ get hold of a _sica?_ Is he dead?\"\n\nRufus, \"As a doornail, _Optio_ . . . Minutus just about took his head off . . . not much left of the face.\"\n\nStrabo was untying my _galea_. He let it drop to the ground. More demerits for inspection, my foggy mind registered.\n\n\"Get the head slave over here!\" Strabo shouted.\n\nThen, to me, \"You keep your head down till it clears . . . It's a scratch . . . You'll be fine . . . Lentulus! Keep an eye on Pagane here!\"\n\nStrabo left me as Lentulus moved in.\n\nThen, I heard Strabo, \"Who is that, Demetri?\"\n\n_\"Illum non cognosco, Domine_! Don't recognize him, Lord! He's new. He just joined our _domus_ this morning,\" responded the one called Demetri.\n\nStrabo, \"You know if he killed a soldier, your whole _domus_ would be crucified?\"\n\nDemetri, \"Yes, _Domine_ . . . The man's new . . . I don't know him . . . The only thing I can tell you is he spoke Latin like a Roman.\"\n\nStrabo, \"Like a Roman? That's odd . . . a slave from Rome . . . That makes no sense! Demetri, get your boys to police-up the body . . . Drop it off with the medics . . . Tell them _Optio_ Strabo doesn't want them to touch it or lose it . . . Then you get back to your _stabulum_ . . . Stay there until someone comes for you . . . You're restricted until further notice . . . _Compre'endis_ _tu_?\"\n\n_\"A'mperi'tu', Domine_!\" Demetri responded.\n\nStrabo walked over to where I was kneeling. \"You think you can stand up, Pagane?\" he asked.\n\n_\"Possum, Optio_ ,\" I answered.\n\nThe fact that Strabo didn't just order me to my feet indicated his concern. Slowly, I got up, leaning on Lentulus.\n\nThen, I remembered my helmet and equipment on the ground. I made to pick them up, but Strabo stopped me. \"Don't worry about your gear, Pagane. Your mates'll take care of it for you. Minutus!\"\n\n\"Yes, _Optio_ ,\" my _geminus_ responded.\n\n\"Good job taking out that _podex_ that tried to stick Pagane. That's how this _geminus_ shit's supposed to work in combat! Next time, don't waste a sword thrust on a dead man! Other than that . . . _bene gestum_ . . . _good job_! Now, I want you and Lentulus here to get Pagane over to the medics . . . Keep an eye on him . . . Something's going on here that doesn't smell right . . . A _new_ slave with a _Roman_ accent? . . . Stick with Pagane till I send for you . . . Got it?\"\n\n_\"A'mperi'tu', Optio_ ,\" Minutus responded. Then, I felt his hand under my shoulder, gently turning me back toward the gate of our _castrum_.\n\nMinutus and Lentulus walked me back to the medical station in camp. When we got there, a fair number of legionaries were waiting on sick call. The legion was training hard, so it was producing its fair share of bruises and strains. Since I was bleeding, I was seen immediately by one of our assistant _medici_ , a jovial chap with black hair, olive skin, and an accent that would be right at home in the depths of the _subura_. He tried to send Minutus and Lentulus away, but they insisted they had been ordered by our _optio_ to stay close to me.\n\n\"I outrank yaw _optio_ ,\" the doc told them, \"but if ya wanna stay, just make shoowah ya don't puke on my nice clean floor hee'ah. Dere's a bucket ovah dere fer dat!\"\n\n\"Now, let's see whadawegot hee'ah,\" he said, unwrapping the infantry scarves around my forearm. He handed the sodden _sudaria_ to Minutus, who went a bit pale when he felt the damp cloth hit his hands.\n\nThe doc looked at my arm and whistled, \"Dat's some cut ya got hee'ah on y'arm! How'd dat happen?\"\n\n\"Uhhh . . . training accident . . . uhh . . . _Medice_ ,\" I said, not quite sure what the military protocol was.\n\n\"Just cawl me Spina. Everybody else 'round hee'ah does,\" the doc said. \"Training accident, huh? Looks like ya was in a knife fight in some wine dive in town. . . . I know cuts like dis . . . looks like a _sica_. Wiggle you fingahs for me!\"\n\nAgain, I wiggled them.\n\n\"Dat's good . . . Can ya make a fist?\" Spina asked.\n\nI did, but I winced as pain shot through my arm.\n\n\"Dat's good,\" Spina said again. \"Stings dough, don't it? Now, I want yas to make a fist atta one fingah atta time.\"\n\nI did. Each finger seemed to work.\n\n\"Dat's good,\" Spina said. \"Doesn't seem to be anyding wrong wid ya tendons. We get dis ding closed up and it don't festah, ya should be back to work in a couple a weeks . . . tops . . . Mahcus! Get in hee'ah! Bring a bucket of dat boiled wawdah and some wine!\"\n\nI looked over and saw Minutus still holding the sodden scarves, not quite sure how to get rid of them. Soon, Spina's attendant, Marcus, entered the _cubiculum_ with a bucket of water and a pitcher of wine.\n\nSpina took the wine from him, sniffed it, and said to Marcus, \"Clean up de wound and de ahm.\"\n\nSpina sipped the wine while Marcus worked.\n\n\"Why boiled water?\" I asked Spina.\n\n\"When I was a novice, my _praeceptor_ swaw on it,\" Spina said putting down the wine pitcher. \"Don't know why, but when ya boil de wawdah, dere's less infection. Greeks say it chases the _daimones_ outta de wawdah. I think dat's a load a crap, but it woiks, 'n' I can't argue wid dat.\"\n\nWhen Marcus was done, Spina inspected his job. Satisfied, he told me to hold my arm over the bucket. Marcus grabbed onto my wrist.\n\nSpina looked at me and said with a wink, \"Dis is gonna sting a liddle!\"\n\nThen, he picked up the pitcher and poured the wine over my wound.\n\nIt felt like he had set my arm on fire. I tried not to make a sound, but a gasp escaped. \"Cacat!\"\n\n\"Shit, Indeed!\" Spina echoed, draining the last drops of wine from the pitcher into his mouth. \"Dah wine also chases de _daimones_ outta yaw arm, so de Greeks say . . . and dat woiks, too!\"\n\nThe pain subsided, but I noticed that Marcus had not let go of my wrist. Spina was removing something from his medical kit. When he turned around, I noticed he had what looked like a curved needle attached to a length of brownish-black lumpy thread.\n\nSpina saw me looking. \"Cat gut,\" he said. \"It's miraculous, really. We get de stuff all de way from Egypt. I'm gonna sew yaw arm muscle back together, and yaw body'll absorb de stitches by itself. Dah Greeks're amazing with what dey come up wid . . . Oh . . . yaw not gonna like dis part.\"\n\nSpina was right. I didn't like that part at all. By the time he was done, Marcus' fingers had bruised my wrist, and there were tears rolling down my cheeks. I also noticed Minutus and Lentulus had gone out in search of a puke bucket.\n\n\"Bene,\" Spina said, examining his handy work. \"Woist part's ovah! Now, I just gotta close you up. Den Mahcus hee'ah can bandage y'arm and yaw on yaw way.\"\n\nBy the time Marcus was finishing my bandages, there was a commotion outside the _cubiculum_. A soldier burst into the room. He was in full kit, under arms without shield, and wore the thin purple sash of the praetorian detail around his waist.\n\n\"Are you _Tiro_ Gaius Marius Insubrecus?\" he demanded.\n\n\"Uhhh . . . yes . . . uhhh . . . sir,\" I stammered.\n\n\"You are to come with me to report to the _praefectus castrorum_ , forthwith!\" he ordered.\n\nSpina interrupted, \"If 'fort'wid' means when I'm done wid 'im, _Praetoriane_ , then we got no problem.\"\n\nThe praetorian winced at Spina's use of the Latin language.\n\n\"Of course, _Medice_ ,\" he responded. \"Are you the doctor in charge of this case?\"\n\n\"Dat's me!\" Spina agreed.\n\n\"Then, my compliments, sir,\" the guard continued. \"The _praefectus_ requests that you examine the body of the dead slave and report to him your findings at your earliest convenience.\"\n\n\"My earliest convenience,\" Spina repeated. \"Dat's officah tawk for fort'wid, right?\"\n\nThe praetorian didn't answer that question.\n\n_\"Bene_ ,\" Spina continued. \"Let me finish up wid dis one hee'ah, and I'll examine yaw dead slave. Then, I'll be along fort'wid.\"\n\nSpina had Marcus make me up a sling, and he told me to keep my arm in it until he examined my wound again in three days. Meanwhile, I was on restricted duties: no using the arm, no double-timing, and stay away from filth\u2014so no latrine duty.\n\n\"And, diss is impaw'ent,\" Spina warned. \"Ya should see some dischahge from de wound, looks like wawdery blood, and you will probably have a little fevah tonight, but if ya see any white, milky discharge, aw yaw fevah doesn't go away by the mawnin' or ya see red lines goin' up yaw ahm or the wound gets puffy and starts to stink, get right back hee'ah! And, just drink plenty of wawdah!\"\n\n\"What if any of that stuff happens?\" I asked.\n\n\"If yaw lucky,\" Spina said, \"people'll be callin' you Lefty for de rest of yaw life . . . If not . . . well . . . let's just hope for de best . . . eh?\"\n\nWe left the medical station and the praetorian escorted Minutus, Lentulus, and me to the _praetorium_ , where we had reported weeks ago when we first arrived at the _castrum_. This time we were ushered straight into the prefect's _cubiculum_. Strabo was already there with the prefect, whose name I recalled was Decius Minatius Gemellus.\n\nWhen we entered, Gemellus told us to stand at ease. Then he asked me, \"Do you want to sit, _Tiro_?\"\n\n_\"Nolo, Praefecte_ ,\" I said snapping to attention.\n\n\"Don't be an idiot, _Tiro_ ,\" he snapped. \"You look as pale as a _lemur_ with a hangover. . . _Scriba_!\"\n\n\" _Praefecte_!\" the voice of his clerk from the outer room.\n\nThe clerk brought in a camp chair, and I did sit down, but in the presence of the _praefectus castrorum_ of at least five legions, my back never touched the chair.\n\nI then noticed that there were two other officers in the room. Each had his sword strapped on his left side and a _vitis_ in his belt\u2014centurions! As I straightened up further in the chair, the clerk reentered the room with a double wax tabula and stylus.\n\nThe prefect nodded toward one of the centurions. \"This is Tertius Piscius Malleus, known as the 'The Hammer,' _centurio Primi Ordinis_ , commander of the First Cohort and _primus pilus_ of the Tenth Legion. The other officer on his left is Mamercus Tertinius Gelasius, _centurio Tertii Ordinis_ and officer in charge of recruit training. They are here to witness your statements and to interrogate you further concerning today's events.\"\n\nThe prefect paused, so I said \" _Compre'endo, Praefecte_!\"\n\n\" _Bene_!\" He continued, \"Your officer here, _Optio_ Strabo, has already made his statement. Now, I'd like you to tell us what happened.\"\n\nI related the story as best as I could. No one interrupted my narrative, but when I was done, the prefect asked, \"Your _geminus_ . . . the one who killed the slave . . . is that this big fellow here?\"\n\n\"I'm called Minutus, _Praefecte_!\" Minutus confirmed.\n\n\"Minutus,\" the prefect chuckled, \"Tiny . . . eh? I don't know why the fathers of Roman soldiers bother naming their sons when they put on the _toga virilis_ . . . They're not men until their mates in the legions name them. . . . So . . . Minutus . . . do you have anything to add to _Tiro_ Insubrecus' account?\"\n\n_\"N'abeo, Praefecte_ ,\" Minutus responded.\n\nGemellus turned to the other two officers. \"Do you have any questions for these men?\" he invited.\n\nMalleus, the _primus pilus_ , spoke, \"Have either of you ever seen the slave who attacked _Tiro_ Insubrecus before?\"\n\n_\"Illum non vidi, Prime_ ,\" Minutus and I answered in unison.\n\n\"Do either of you have any idea why this man would want to kill you?\" he asked.\n\n_\"No' scio, Prime!_ \" Minutus snapped.\n\nI remained silent while I thought about what I should say.\n\nMalleus picked up on my silence immediately. \"You have something to add, _Tiro_ Insubrecus?\"\n\nI thought about it for a few heartbeats. Telling the _praefectus castrorum_ of Caesar's legions and the _primus pilus_ of the veteran Tenth Legion that I was possibly a fugitive, wanted by the consul of the Republic, had a bit of risk attached to it. But, even in the midst of Caesar's army, I obviously wasn't safe. So, I told them everything: Gabinia in the arbor, the attack on the road to my parents' farm, Gabinius Iunior and his gladiators, the threat of Gabinius Senior's arrest warrant, my flight, and the gangsters in Mediolanum. I did leave out a few details about Rufia's operation. I had enough enemies as it was, and I planned to go back home to Mediolanum in about six years' time if I didn't get skewered by some foaming-at-the-mouth, seven-foot-tall, hairy, Roman-eating German tribesman.\n\nIt was rare indeed to see a look of utter astonishment on the face of a Roman officer as senior and as experienced as a _praefectus castrorum_. This was one of those times.\n\nGemellus burst out laughing\u2014another rarity. \"You mean to tell me that _Rufia_ . . . _Rufia_ of Mediolanum . . . _Rufia_ , with the blue, Venus-door . . . hid you out in one of her private suites?\"\n\n_\"Recte, Praefecte_ ,\" I stammered out.\n\n\"I know senior officers . . . purple-stripers . . . who'd piss away their entire purse to have been in your shoes!\" Gemellus continued to laugh. \"Rufia of Mediolanum . . . and she's sweet on a clapped-out _optio_ from Asia . . . No offense, Strabo . . . _Mammis Veneris_ . . . that's rich!\"\n\n\"Sir,\" Malleus beckoned.\n\n\"Yes . . . Yes,\" Gemellus chuckled, getting back on track. \"I think we can rule out the consul. If he wanted you, all he'd have to do is send a warrant up here for your arrest. If he wanted you dead, he could just make sure you never made it back to Rome. Of course, Caesar as proconsul of the province and commander of the army would have to approve.\"\n\n\"Sir,\" I ventured.\n\n\"What is it, _Tiro_?\" the prefect said, a bit surprised at being interrupted by a recruit. \"Don't tell me that you have a passel of Athenian _hetairai_ in the _vicus_ who clean your underwear.\"\n\n\"No, sir,\" I assured him. \"But, there is something about my family's relationship with Caesar _Imperator_ that may have some relevance.\"\n\nI told him about my gran'pa, Marius and Gaius Senior, how we got our farm, and how we are technically Caesar's clients.\n\n_\"Cacat_ ,\" Gemellus said. \"That explains the name, 'Gaius Marius,' at least . . . I doubt the _imperator_ even knows you or your family exist . . . But you're correct . . . Technically, you're _cliens genti Iuliae_ , and therefore, entitled to his _patrocinium_ . . . his protection. . . . He will have to be informed.\"\n\nThere was a commotion in the outer tent. Another of the prefect's clerks stuck his head in and announced that the _medicus_ had arrived to give his report on the dead slave.\n\n\"About time!\" the Prefect responded. \"Get him in here!\"\n\nSpina entered the room and reported.\n\n\"What can you tell us about the dead slave, _Medice_?\" the prefect demanded.\n\n\"Dead slave?\" Spina repeated. \"First, I can confirm he's crossed the river . . . Either de broken neck aw de damage to his head did 'im.\"\n\n\"How can you be so sure, Spina?\" Gelasius spoke up for the first time. \"He was also stabbed through the throat.\"\n\nSpina looked over at the centurion. \"Gelasius! Didn't see ya standin' back dere. How do I know? Easy! Dah stabbing wound to his throat pierced dee aorta. If his heart was beatin' when dat happen'd, dere would a been a fountain a blood. But dis wound hardly bled . . . _Ergo . . ._ he was dead when it happened.\"\n\nGelasius grunted in the affirmative.\n\n\"Continue, Doctor!\" Gemellus prompted.\n\n\"Dah second part of yaw question about a dead slave raises a problem, _Praefecte_ ,\" Spina continued. \"Dis guy wasn't a slave.\"\n\n\"What?\" Gemellus shot. \"Wasn't a slave? How can you be so sure?\"\n\n\"Coupl'a dings, _Praefecte_ ,\" Spina explained. \"First, his hair, or what was left of it, wasn't regulation cut. Second, he had no 'SPQR' tattoo on his left shoulder blade like de rest a de government slaves around hee'ah. T'ird, his hands weren't callused like a slave's . . . A _sicarius_ , a hit man, yeah . . . a slave, no. Fourt', I cut 'im open and his last meal included _garum_ and eggs, not de barley porridge de slaves eat for breakfast. Finally, he had a tattoo of a _sica_ and skull on his left shoulder. I've seen marks like dat when I was a kid growin' up on the Aventine . . . It's a _collegium_ mark . . . Dis guy was a _grassator_ . . . a gangster . . . probably a _percussor_ . . . a hitter up hee'ah from Rome . . . That explains the report of his Roman accent.\"\n\n\"Are you sure about this, Spina?\" Malleus asked. \"Seems a little flimsy to me.\"\n\n\"Any one a dem, _Prime_ , yeah . . . It'd be pretty flimsy, like ya said,\" Spina responded. \"But, all togedder . . . no . . . Dis guy was a ringer . . . a hitter . . . He was up hee'ah to wack de kid . . . I'm shoo-a-wit.\"\n\n\"'Shoo-a-wit',\" Gemellus said absently. \"Oh . . . 'sure of it' . . . Yes . . . I agree . . . _Bene gestum, Medice_ . . . Anything else?\"\n\n\"Yeah, _Praefecte_ ,\" Spina continued. \"I took a look at dis guy's knife . . . It's a straight-bladed _sica_ . . . I seen a lot of dese back on de Aventine . . . The boys on de hill like 'em cause dere easy to conceal . . . Dis one's razor shawp . . . well taken care a . . . and expensive . . . Got an ivory grip . . . matches the scabbard I found strapped to de guy's forearm . . . But dis is de ding . . . On the blade was engraved DON MILONE SUM, 'I'm a gift from Milo.' Not only we got a hitter heah, we got one of Milo's top boys. He don't give away knives like dat to schleps!\"\n\n\"Schleps?\" Gelasius questioned.\n\n\"Schleps,\" Spina clarified. \"It's a word we used back on de hill for guys who ain't too impaw'int or too bright, so they get the worst jobs . . . like a . . . like a . . . _ianitor_ . . . a _baiulus_ . . . only dumber.\"\n\n\"Yes! Yes! Very good,\" the Prefect interrupted. \"Do get on with it, Spina!\"\n\n\"Yeah . . . of course, _Praefecte_ ,\" Spina said. \"Like I said, dis guy was good at what he did . . . so he had to have an escape plan after he did the kid hee'ah . . . Dat means a quick change a clothes outta de slave tunic . . . So, somewhere neah wheah he slashed de kid hee'ah, he hid 'is stuff . . . So, if ya send a few a de boys out on police call, you should find 'em . . . Maybe tell you somethin' maw 'bout dis guy.\"\n\nIt took Gemellus a couple of heartbeats to translate Spina's west-slope-of-the-Aventine Latin, but when he finally did, he grunted his approval, \" _Bene_ . . . Good thinking, Spina . . . Anything else?\"\n\n\"Yeah, _Praefecte_ . . . couple things,\" Spina continued. \"I inspected dis knife real close . . . It's been shawpened recently . . . Looks like a professional job . . . Best knife-grinder 'round hee'ah is a Gaul named Aeddan. I use 'im myself to sharpen my instruments . . . First rate job he does . . . He's got a booth out in the _vicus_ behind de Ninth's _castrum_. I'll bet dat's where aw boy shawpened his fancy _sica_. Doubt yaw gonna get much . . . Dis guy's too good . . . but it's worth a try.\"\n\n\"You said 'couple things,' Medice?\" Gemellus prompted.\n\n\"Yeah, _Praefecte_ ,\" Spina continued. \"The garum and eggs aw boy had for 'is last breakfast . . . Dat's a pretty fancy meal . . . He didn't get it in some _vicus_ flop . . . My bet's he stayed at some fancy joint, a _deversorium_ up in de town . . . Best place I know up deah is the _Anser Volans_ , the Flying Goose . . . It's on de main drag neah de west gate . . . Run by a retired legionary named Macer . . . Dey'd have garum and eggs, fer shoowaw.\"\n\n\"Macer . . . Flying Goose . . . Aeddan,\" Gemellus repeated, \"You know any of these places, Malleus?\"\n\n\"I know the Goose, _Praefecte_ ,\" the _primus pilus_ responded.\n\n\"Bene! Send a couple of your _prima centuria_ boys up there to talk to this Macer,\" Gemellus instructed. \"He used to be one of us, so I assume he'll cooperate. As far as this Gaul is concerned, this Aeddan, send a _contubernium_ in full rig . . . shields and swords . . . just in case he decides to play it cute.\"\n\n_\"A'mperi'tu', Praefecte_ ,\" Malleus responded.\n\nGemellus turned to the doctor, \"Anything else for us, _Medice_?\"\n\n_\"N'abeo, Praefecte_ ,\" Spina responded.\n\n_\"Bene_! You may return to your duties,\" Gemellus instructed.\n\nSpina inclined his head to the three centuriate officers in the room and left.\n\nAfter Spina had gone, Malleus asked, _\"Praefecte_! What about the slaves, the real ones, I mean?\"\n\n\"The slaves?\" Gemellus asked. \"What do you mean, Malleus?\"\n\n\"Do you want them interrogated?\" the _primus pilus_ inquired. \"This . . . this 'hitter,' as the doctor calls him, may have bribed them to get them to conceal his presence within their _domus_.\"\n\n\"Hmmm . . . I see what you mean,\" Gemellus considered. \"Are you suggesting we put them to the _interrogatio?_ That means torture for slaves.\"\n\n\"That is the custom, _Praefecte_ ,\" Malleus confirmed.\n\n\"I see,\" the prefect reflected. \"No . . . I don't think so . . . It's bad for morale, and we're less than two weeks from going into combat . . . The Tenth's _stabulum_ of slaves are veterans . . . Most of them been with us for years . . . I don't think any of them would risk _manumissio_ , their emancipation, for a few denarii . . . No . . . But, Gelasius . . . I have a job for you.\"\n\n_\"A'mperi'tu', Praefecte_ ,\" the Tenth Cohort commander snapped.\n\n\"Send a couple of your boys over to Spina and tell him to sew his patient up,\" Gemellus instructed, \"Then, take the body out to the field where he attacked _Tiro_ Insubrecus and crucify him. If he wanted to disguise himself as a slave, he can end up like one, feeding the crows on a cross.\"\n\n_\"Stat', Paefecte_!\" Gelasius confirmed.\n\n\"Be sure to hang him up in his slave tunic,\" the Prefect continued. \"Hang a sign on him, something like, 'Attacked a Roman Soldier.' Then, after last call this evening, when training's over, take the slaves out to that field, the entire _stabulum_ , and file them by the cross. That should take care of any thoughts they _might_ have about taking bribes from outsiders. Got it, Gelasius?\"\n\n\" _Compre'endo, Praefecte_!\"\n\n\"Strabo,\" Gemellus turned his attention to our training officer. \"Will your _tirones_ be ready to participate in the _significatio_ next week?\"\n\n_\"Parati, Praefecte_ ,\" Strabo snapped.\n\n\" _Bene_ ,\" Gemellus said. \"This army will be moving out over the Alps as soon as the _imperator_ summons us. So, with the Helvetii on the move, we must be sure we are all _parati. Tiro_ Insubrecus, you are excused from all training exercises and fatigue details until further notice! I will be notifying your _patronus_ , Caesar _Imperator_ , of this event. This affects his _dignitas_. As far as the rest of this legion is concerned, this was just the work of a renegade slave. _Compre'enditis vos toti?_ \"\n\nA chorus of _\"Compre'endo, Praefecte_!\" ensued.\n\n\" _Bene_ ,\" Gemellus concluded. \" _Optio_ Strabo, you will remain behind. We have something to discuss with you. _Tirones, miss'est_!\"\n\n# III.\n\n# _Ego Miles Romanus_\n\n# I BECOME A SOLDIER OF ROME\n\nSo, I spent most of my last weeks as a legionary _tiro_ restricted to barracks. None of Spina's dire fears about my arm materialized. When I reported to him at the medical station three days later, he was quite pleased with his handiwork and my progress\u2014pretty much in that order. He had me flex my fingers a couple of times; then he told Marcus, his orderly, to wash the wound with wine\u2014after Spina had personally tested the brew for potency\u2014and change my bandages. Spina sent me back to barracks with the advice to learn to use my left hand as it was good military practice. I wasn't sure what he meant, but he seemed to have amused himself.\n\nStrabo had a surprise for us. When he returned to collect us the first morning after the meeting with the prefect, he had his sword hung on his left hip and a _vitis_ in his right hand. He announced in formation that he had been granted a temporary appointment in the Tenth Legion as a _centurio tertii ordinis_ and was in command of the Fourth _Centuria_ of the Tenth _Cohors_ , where the legion assigned its trainees. He then announced that he had selected our old friend Bantus as his \"chosen one\"; the _optio_ of the Fourth _Centuria_ was our new training officer. He then sent me back to quarters while he put the rest of our _contubernium_ through its daily ten thousand-pace conditioning march.\n\nRumors about what was happening on the other side of the Alps continued to swirl through the camp.\n\nThe Germans had crossed the Rhenus. The Helvetii had stormed the bridge over the Rhodanus and had burned Gennava. Roman citizens were being massacred, and the Eighth Legion had been pushed back into our _provincia_. Roman auxiliary units had mutinied and gone over to the barbarians. Massalia had locked its gates against us, and the entire Roman army north of the Alps was being pushed back into the _Mare Nostrum_ , the Middle Sea.\n\nThe Tenth Legion continued to train and prepare to move out over the Alps. It was now participating in army-level training exercises, maneuvering with, and against, the other legions of the army. Our senior officers\u2014legates and tribunes\u2014were arriving from Rome. But, still no word had come from Caesar _Imperator_.\n\nFinally, on the _Nones Aprilis_ , Strabo announced to us that word had been received from the proconsul in Gennava, which apparently the barbarians had not burned, that the army must be ready to march on the _Ides_. Therefore, our _significatio_ would be held five days before the _Ides_ so that we would be ready to march with our legion. Strabo then asked if we had any questions.\n\n\" _Quaestionem 'abeo, Opt_ . . . uh . . . _Centurio_ ,\" Loquax popped up.\n\n\"Roga,\" Strabo responded. \"Go ahead!\"\n\n_\"Centurio_ , what is a _significatio_?\" Loquax asked.\n\nStrabo stared at Loquax for a few heartbeats, trying to decide whether the _tiro_ was putting him on. He decided he was not.\n\n\"Good question, _Tiro_ Loquax,\" Strabo responded. We would come to notice that after his promotion, Strabo was more refined, more observant of proper military protocol and courtesy. He hardly ever called us maggots or cockroaches anymore.\n\n\"The _significatio_ is a ceremony in which the legion demonstrates to Father Iove that you are now part of it,\" Strabo explained. \"In front of the legion's _aquila_ , the _primus pilus_ asks the assembled legion whether it accepts you, and the soldiers acclaim their acceptance. Then, you are invested in the red tunic of a soldier, given the shield of the Tenth Legion, and awarded your _gladius_ , your infantry sword. You renew your _sacramentum_ in front of your comrades and swear you would die before you would shame our standards or let the legion down. Then, the legion acclaims you as a _miles_ , one of their number. Any other questions, _Tirones_?\"\n\nWe had none, so Strabo said, \" _Optio_ Bantus! Take charge of the training detail. Move them out to the _pilum_ range . . . at the double . . . Pagane . . . you will remain in quarters!\"\n\n_\"A'mperi'tu', Centurio_!\" Bantus snapped and took his position in front of our formation.\n\n\" _Contubernium . . . STATE!_ \"\n\n_\"Pagane! Miss'est_ . . . _A signis!_ \"\n\n\" _Contubernium_ . . . _Ad Dex' . . . VERT'_!\"\n\n_\"Prov . . . ET'_!\"\n\n\" _Gradus bis mov . . . ET'_!\"\n\nThe day of our _significatio_ , our acceptance into the Roman army, would be the fifth day before the _Ides Aprilis_ , a day that was considered _fastus_ , auspicious, and _comitialis_ , a day on which the Roman people could assemble to conduct business.\n\nA few days prior, Bantus had our acting _decanus_ , Minutus, collect our white trainee tunics, leaving us one each to train in, and had Moelwyn, our newly-assigned _contubernium_ slave, take them down to the civilian _vicus_ just outside the _Porta Decumana_ to be dyed infantry red. The process took two days. When Moelwyn returned from the _vicus_ with them, we were excited by the prospect of finally donning them. I remember sitting on my cot, running my hands over the red fabric. Bantus told us not to try to outrun our own chariot horses, but to store them away until we had been invested.\n\nThree days before the _significatio_ , Spina had his aid, Marcus, remove the exposed stitches from my injured arm. This time Spina sampled his medicinal wine and didn't use it for washing my wound. He examined my injury and congratulated himself on a \"jawb well dun.\" Then, he congratulated me on acquiring the first \"noble scah\" of my military career. He then warned me to keep the arm bandaged and go easy with it for the next few days, \"No sword practice and no _pilum_ chucking!\"\n\nThen, Spina offered me a swig of his pitcher.\n\nI got back to our tent around the seventh hour. Moelwyn was outside working on a harness. I greeted him in _Gah'el_.\n\n\"Please speak Latin, _Domine_ ,\" he answered me in Latin. \"That is the language of the legions.\"\n\nI sat down next to him and started examining the harness he was working on. I recognized it from the time I worked in Gabinius' stable.\n\n\"What's all this for?\" I asked Moelwyn.\n\n\"The _contubernium_ mule, _Domine_ ,\" he answered me.\n\n\"We have a mule?\" I blurted. \"What for?\"\n\n\"Every _contubernium_ has a mule, _Domine_ ,\" Moelwyn answered. \"It's used to carry things that are not essential for a legionary to carry on his _furca_ , his pack pole. In this legion, the mule carries the tent, tent poles, and pegs, eight _sudes_ , the sharpened wooden stakes to fortify the daily marching camp, extra rations, water, _et cetera_.\n\nI realized that Moelwyn had been with the army much longer than I had. \"How long have you been with the legions?\" I interrupted.\n\n\"Me, _Domine_?\" he responded. \"Eight years this month.\"\n\n\"Eight years!\" I repeated. \"You're almost halfway there.\"\n\n\"Yes, _Domine_! Halfway there, as you say,\" he agreed. Then he continued, \"It's not good to talk about . . . . _Domina Fortuna_ does not like mortals to boast of her gifts to them.\"\n\nRealizing that soon we would all be marching through the Alps to face hordes of rampaging Helvetii, I nodded in agreement, then rubbed my _Bona Fortuna_ amulet, and spat toward the north.\n\nThe mule's rig seemed to be in good shape. I wondered briefly if the metal fittings on it had come from my _Avus_ Lucius' forges down in Mediolanum. Then, I asked Moelwyn, \"Were you brought down from _Gallia comata_?\"\n\n\"Me, _Domine_?\" He seemed a bit startled by the question. \" _Gallia comata_ . . . No, _Domine_ . . . Never been there . . . I joined this legion in the _provincia_ and went with them into _Hispania_ . . . I was born a couple hundred miles from here, up in the foothills of the mountains to the north . . . near a great lake . . . My village is called Sarnis.\"\n\n\"You joined? But you're a . . . a,\" I stammered.\n\n\"A slave, _Domine_?\" Moelwyn finished my question. \"Most of us in the legions serve _voluntate ipsius_. We are voluntary slaves . . . We sold ourselves to the _res publica_.\"\n\n\"Why would you do that?\" I blurted, not understanding why anyone would voluntarily become a slave.\n\nMoelwyn shrugged and said, \"In my village, all we could grow were rocks . . . I was starving . . . When I was fourteen, I left my family and wandered down into the valley, looking for work and food . . . The only thing I knew how to do was herd goats . . . but small farmers did that for themselves, and large farmers used slaves . . . and even they were eating better than I was . . . Then a Roman told me that I could sell myself to the state . . . serve in the army . . . Three squares a day, a roof over my head, and the work wasn't all that hard . . . After twenty years, if I did my job and kept my nose clean, I'd be released and be granted Roman citizenship . . . What did I have to lose? I wasn't a citizen . . . I had no rights . . . I was starving . . . So I did it . . . The army's even holding my purchase price . . . I get it when I'm freed . . . I've had to take a few beatings over the years, but most of the soldiers are pretty straight . . . As long as I do what I'm told . . . be respectful... they leave me alone . . . All in all, it was a good choice . . . I got no regrets . . . In twelve years, I'm free.\"\n\nMoelwyn spat toward the north to ward off any misfortune.\n\nLater that day, during the tenth hour, our new _optio_ , Bantus, examined my arm and decided that I could rejoin the training, except for sword and _pilum_ drills. He agreed with Spina that I'd have a nice scar to impress the _caupona_ girls with, demonstrating what a _miles gloriosus_ I was.\n\nBantus' training for us that evening was on packing the _sarcina_ , the legionary marching pack. He had Moelwyn distribute two wooden poles to each of us: one about four _pedes_ in length; the other shorter, about two _pedes_. Each pole was about a _palmus maior_ in circumference, about three fingers short of a foot. Then, Bantus gave each of us a length of rawhide lashing.\n\n\" _M'audite, Tirones_ ,\" he called out over the noise we were making, talking about what to do with the wooden poles. \"I'm going to show you how to construct a _furca_.\"\n\nWhen we finally settled down and he had our attention, he continued, \"The _furca_ is a forked pole that carries your pack and equipment while you're marching _impeditus_ , or as the _veterani_ call it, _mulare_ , muling it. I'm going to show you how to construct one.\"\n\nBantus grabbed Rufus' poles. \"The _furca's_ made from a thick staff about four _pedes_ in length,\" he said, holding up the longer staff.\n\n\"A shorter cross piece is fixed about a _pes_ , a foot, from the top of the long stake.\" Bantus demonstrated, making what seemed to be a cross from the two stakes.\n\n\"To attach them, you carve a notch in each . . . You interlock them . . . like this . . . Then you lash them together with the rawhide bindings. . . Any questions so far?\" he asked holding up the cross in one hand and a set of rawhide bindings in the other.\n\nThere were scattered mutterings of \" _n_ ' _abeo_ \" around the tent.\n\nThen, still holding up the crossed stakes, Bantus joked with us by asking if we knew the difference between a criminal's cross and a legionary's _furca_. When we didn't answer, he said, \"A cross carries a criminal till he's dead. The legionary carries his _furca_ till he wishes he was dead.\"\n\nWe didn't get it, but Bantus just laughed and muttered, \"Infantry humor.\" Later we would know what he meant. By the time we got over the Alps and chased the Helvetii halfway across Gallia, we all knew the difference between a _furca_ and a cross, and I think some of us would have preferred the cross.\n\nBut, Bantus continued, \"First, notch your stakes, the shorter one in the center, the longer about one _pes_ from the top. Don't go any deeper than about halfway . . . a little less is better . . . The cut's no longer than the diameter of the stake . . . Remember the stakes have to fit together . . . so use them to measure your cut . . . Go ahead... Get to work.\"\n\nI pulled out my _pugio_ , the knife Macro had given me for my sixteenth birthday, seemingly a lifetime ago, and laid the shorter stick across the longer in the correct position, about a _pes_ from the top, centered. Using the opposing stakes, I scratched cut lines on each to mark the width of the cut and started whittling out the notches.\n\nBantus walked around the tent, inspecting our work. Moelwyn assisted him. \" _Bene_ . . . _bene_ ,\" I heard him saying to my mates as they worked. \"No, Lentulus . . . Go look at how Pagane's doin' it . . . He's got it right.\"\n\nFinally, we had our stakes notched and fitted. I looked worriedly at the pile of sawdust and wood chips around our bunks. We would have to get that swept up before evening inspection.\n\nBantus was talking, \"Before I show you how to lash the poles together, I want you to wet and stretch the rawhide bindings.\" As he spoke, Moelwyn brought in a bucket of water.\n\n_Optio . . . ad quam rationem_ ,\" Loquax started. \"Sir . . . why\u2014\"\n\nBantus seemed to have anticipated the question. \"You wet and stretch the rawhide because it will shrink when it dries . . . That way you get a tight binding of the two poles . . . If you fasten your _furca_ with dry rawhide and it gets wet . . . and I assure you, it will . . . the bindings will loosen, and the _furca_ will come apart . . . You'll be trying to march while balancing all your shit in your arms . . . That's a real rookie mistake, boys . . . Soldiers I train don't make those kind of mistakes . . . So, wet the bindings in the bucket . . . Stretch 'em out good . . . and I'll show you how to lash the _furca_ together.\"\n\nWe did as Bantus instructed, and he showed us how to lash the _furca_ poles together. Bantus went around, inspecting the bindings and their tightness. Finally, when he was satisfied with our work, he continued.\n\n\"The _sarcina_ , the infantry marching pack, is constructed by strapping the _loculus_ , your leather satchel, to the cross piece of the _furca_ ,\" he told us. \"The _loculus_ is primarily for carrying your marching rations . . . _buccellatum_ , your hardtack . . . salt meat . . . pork and mutton usually . . . Sometimes we get some beef . . . If you can scrounge any cheese, fresh fruits, or vegetables along the way, toss them in there too . . . This way, your food's accessible to you during rest breaks . . . Wrap any cheese you get in a damp cloth . . . That helps keep it soft . . . soft enough to chew, anyway . . . What is it, Felix?\"\n\n_\"Optio_ , do we get rest breaks on the march?\"\n\n\"Usually,\" Bantus responded. \"When it's not a forced march or when we're not in proximate contact with the enemy, the legion takes about a quarter-hour rest break every five thousand _passus_ or so. It's a good time to get the _furca_ off your shoulder . . . grab a piece of _buccellatum_ . . . maybe a hunk of cheese, if you got it . . . wash it all down with a swig of _posca_ or water . . . Stretch 'em out . . . Then we're off again . . . Back to this _sarcina_ . . . your marching pack . . . You tie your _patera_ , your mess kit, and your _cochleare_ , your eating spoon, to the cross piece like this . . . Some guys like to put their spoons in the _loculus_ . . . Keeps it cleaner . . . It's up to you. Balance the mess kit with your _lagoena_ , your water bottle . . . What is it, Loquax?\"\n\n_\"Optio_! Do we carry water on the march? I thought that was carried in the baggage train.\"\n\n\"Always carry some water with you, Loquax,\" Bantus told him. \"Sometimes we get separated from our supplies, and we can't find a good source of potable water at the end of the day . . . A soldier can go for a few days without food, but won't last a short summer's day without water . . . Some guys carry two water bottles, just in case . . . Keep 'em filled . . . You can carry water or _posca_ . . . I recommend the water . . . Load a skin or two of _posca_ on the mule for the end of the day . . . Now, back to the _sarcina_ . . . In this legion, soldiers carry their _dolabrae_ , their digging picks . . . You can tie them here on the _furca_ and not unbalance it . . . Always tie the _dolabra_ high on the _furca_ . . . You don't want it digging into your back or shoulders on the march . . . The digging baskets go on the mule . . . I get that right, Moelwyn?\"\n\n\" _Recte, Domine_ ,\" Moelwyn's voice sounded from the back of the tent. \"The entrenching baskets go on the mule in the Tenth . . . with the _rutra_ , the shovels.\"\n\nBantus chuckled. \"In the field, a good _mulio_ , a teamster, is worth a month's pay . . . Makes sure everything's packed right . . . Keeps the mule healthy . . . Make sure you treat 'em right . . . You don't want to be carrying all that stuff yourself . . . _Bene_ . . . Your field cloaks . . . some guys like to roll them up and put 'em in a _saccus_ , a cloak bag, and tie the bag to the cross piece . . . I'll let you in on a little trick . . . Wrap your cloak around the upright of the _furca_ , right where it sits on your shoulder . . . That way, it'll cushion your shoulder a bit . . . Save the cloak bag for whatever fresh food you can scrounge, or if we get lucky, any swag we can pick up . . . While I'm thinking about it . . . even cushioned with a cloak, the _furca's_ going to press the chainmail of your _lorica_ down into your shoulder . . . After a couple of days on the march, you'll be in agony . . . So, when you're suiting up in the morning, slip some cloth, a towel if you got one, between the _lorica_ and your _subarmalis_ jacket on your left shoulder . . . What is it, Felix?\"\n\n\" _Optio_! Why the left shoulder?\"\n\n\"On the march, the _furca_ ' _s_ carried on your left shoulder. You'll hang your helmet on your lorica . . . You'll be carrying a _pilum_ or two in your right hand,\" Bantus explained. \"You guys are lucky . . . The Tenth marches light . . . In my old outfit, we each carried two _pila_ and an entrenching stake on our right shoulders and had to strap our unit mess gear, digging baskets, and even extra darts for the _ballista_ on our _furcae_ . . . We were humping more stuff than the mules . . . That's why we were called _muli Marii_ . . . Marius' mules. . . Your namesake invented the _furca_ , Pagane.\"\n\n_\"Intelligo, Optio_ ,\" I said, not really knowing how to respond.\n\n_\"Bene_ ,\" Bantus concluded. \"For our conditioning march tomorrow . . . ten thousand _passus_ at first light . . . You'll march _impediti_ . . . Fall in with your _furcae_ packed and ready to go . . . We march under full kit . . . _lorica_ and _galea_ . . . but without shields and swords . . . We'll swing by the range on our way out, and each of you will pick up two practice _pila_ . . . And remember to pad your left shoulders . . . We're in the field soon, and you don't want to start a campaign with a bum shoulder . . . It won't get any better . . . Any questions?\"\n\nScattered responses of \"n' _abeo_.\"\n\n_\"Bene_ ,\" Bantus started. \"Before I forget . . . another piece of _veterani_ lore for you rookies . . . We'll be marching over the Alpine passes in a few days . . . It's still winter up there . . . colder than a centurion's heart . . . And the sun shining on the snow'll blind you . . . Scrounge yourself up a couple of extra _sudaria_ . . . Use one to wrap around your head and ears . . . The cold up in them mountains'll freeze your ears off . . . I swear . . . Use the other scarf as a mask . . . Cut some narrow eye-holes in it . . . It'll keep you from going snow blind . . . Grease helps too . . . Rub some on your face and lips . . . It'll keep 'em from chapping . . . You guys getting this?\"\n\nScattered mutterings of \" _compre'endo_.\"\n\n_\"Bene_ ,\" Bantus continued. \"Collect up some money among yourselves and send Moelwyn down into the _vicus_ . . . Have him buy you some woolen socks and some _manicae_ , gloves, if there are any left to buy . . . If he can't get the gloves, you can cut finger holes in an extra pair of socks . . . Wear socks under your _caligae_ . . . I've seen guys lose fingers and toes up in them high mountains . . . And each of you get a good, sturdy pair of leather _bracae_ to wear underneath your tunics . . . What is it, Rufus?\"\n\n_\"Optio_ , are _bracae_ authorized? They're _inromanitas_ , un-Roman,\" protested Rufus.\n\nBantus chuckled, \"Having your balls freeze off is un-Roman too, Rufus . . . We all wear 'em when it's cold . . . Only the patrician snobs, the purple-stripers up from Rome, worry about that being 'un-Roman' shit . . . Most of 'em have no balls to freeze off anyway . . . Their wives've already removed them . . . Get yourself each a good, solid pair of _bracae_ . . . None of that Gallic wool shit with the colors and plaids . . . leather . . . They'll keep you warm up there in those passes . . . Any more questions?\"\n\nWe had none.\n\n_\"Bene_ ,\" Bantus said getting up and looking around our tent. \"This place looks like some drunken carpenter's workshop! Get it cleaned up before the start of the first watch! I'll be back to inspect.\"\n\nI was lucky. The quartermaster still had the _bracae_ , gloves, and socks I had turned in when I received my clothing issue. For a couple of _minervae_ , Moelwyn was able to get them back for me through one of the slaves that worked in the supply tent. The rest of the guys had to compete with over four thousand other _muli_ for the limited supply of leather trousers and woolen socks to be had in the _vicus_ and in the town. Luckily, the civilians seemed to know months in advance where we'd be going and what we'd need to get there, so they had had some time to lay in a good supply\u2014with which they were now happy to part for a prime price.\n\nThe evening before the _significatio_ , the legionary _haruspex_ pulled the liver out of a healthy, black goat and declared that its appearance indicated that the gods favored the acceptance of the trainees into the legion. Strabo could hardly conceal his skepticism and contempt when he told us this news. Bantus seemed more terrified by his centurion's irreverence than relieved by the propitious omens.\n\nOur _haruspex_ was an _immunis_ soldier from the Second _Centuria_ of our own _cohors_ , whose only qualification for the job seemed to be that he was born in a barn just outside the town of Veii and had black, curly hair like a genuine Etruscan. So, of course, everyone called him Crispus, \"Curly.\"\n\nThe _haruspex_ ' job certainly did have its perks. He pulled down extra pay. On top of that, Crispus was not only exempt from all fatigue details, but he and his mates also got to keep most of the meat from the sacrificial animals. I later found out that Crispus had a healthy side business with the purple-stripers, who were known to slip him a _quadriga_ or two so that he might claim to see propitious things in the entrails of chickens or the viscera of goats.\n\nStrabo wondered out loud whether a cow farting in a northerly direction during the dark of the moon meant the gods were against long sea voyages. That made Bantus go pale. I rubbed my _Bona Fortuna_ and spat. I made a note to myself that if we ever went into battle, I wouldn't stand too close to Strabo. I should have spat twice.\n\nBut, Strabo reported that the goat's liver was a healthy pink with a well-formed right lobe that was larger than the left lobe. So, the gods must be happy with what we had planned for the next day. That night, we anticipated our induction into the Tenth Legion and salivated over the aroma of roasted goat wafting over from a tent in the Second _Centuria_.\n\nBantus roused us before first light. He had us dress in our white trainee tunics, without belts. Carrying our newly dyed red tunics under our arms, he marched us toward the _Porta Principalis Dextra_ of the camp. We were assembled just inside the gate in the _intervallum_. We were surprised to discover that we weren't the only ones being integrated into the legion that day. Organized into four _contubernia_ , there stood almost thirty other _tirones_ in white tunics. Although we could see nothing outside the walls of the camp, we could hear the stirrings and murmurings of a large gathering. When Bantus stood us at ease, I realized that, except for the guard detail manning the ramparts, the camp of the Tenth Legion was deserted. Bantus collected the red tunics from us and handed them over to a soldier who carried them out through the gate.\n\nAs the first glimmers of the sun began to glow faintly beyond the camp walls, a trumpet sounded in the field beyond. All movement outside the wall ceased. We heard commands shouted, the sound of metal and movement in unison, then silence. We were then called to attention by our officers and filed through the gate of the camp.\n\nThe sight that greeted us there is one I have never forgotten even to this very day.\n\nTo our right, the entire Tenth Legion was assembled in parade formation in full-dress kit. Ten cohorts formed a single rank, stretching off into the paling horizon, their centuries assembled in columns behind them. My first impression was of ranks upon ranks of ghostly Roman soldiers waiting for us to join them in the shadows. Nothing moved. Even the scarlet plumes on the legionaries' helmets were still in the darkling airs. From beneath the overhanging brows of bronze helmets, I felt the eyes of the over four thousand soldiers assembled on that silent plain, boring into us, measuring us, and assessing our worthiness to join their ranks.\n\nIn front of each cohort, a _centurio pilus prior_ stood at attention. Their transverse crests radiated from polished bronze _galeae_ like fiery red nimbuses. Combat decorations, suspended by leather harnesses on polished breastplates, caught the first faint glimmers of the new sun: gold and silver _torcs;_ gold, silver, and bronze _phalerae_ disks; and silver _hastae purae_ , awarded to those who have blooded an enemy of Rome.\n\nTo the left of each centurion stood the cohort's signifer, a wolf's head pulled over a shining bronze _galea_. The _signum_ of the cohort, an open, silver hand paying perpetual tribute to Father Iove, was held upright above the heads of the assembled soldiers.\n\nBehind the command groups, the soldiers themselves stood silently behind a low wall of red shields, their bronzed bosses polished to a mirror finish, from which radiated the golden-yellow winged thunderbolts of Father Iove. Each shield displayed a rampant boar, the symbol of the Tenth Legion.\n\nTo our left, on a low, wooden platform, stood the golden _aquila_ of the legion, the sign of Father Iove, the totem of the legion, the spirit of its resolve. Beneath the golden eagle, the _vexillium rubrum_ , the red flag of the legion, hung in the still air. I could clearly see the image of the rampant boar and the symbols LEG X.\n\nTo the eagle's right, the _aquilifer_ , the legionary entrusted with the _anima_ of the legion, its very spirit, stood as a silent sentinel. Next to him, I recognized Tertius Piscius Malleus, \"The Hammer,\" _primus pilus_ of the Tenth Legion, with the leaves of a Civic Crown, the highest award for valor bestowed by the Roman nation, entwined into the red traverse crest of his helmet.\n\nIn front of the platform was set a row of legionary _scuta_ , red oval shields identical to those held by the soldiers of the legion. Against each leaned a sheathed _gladius_ , the infantry sword, with its baldric draped over the shield. In front of each of these was a folded, red tunic.\n\nBefore the _aquila_ , praying over a sensor that misted the golden standard and the red pennant of the legion in clouds of sweet-smelling incense, stood a man, his pure white toga covering his head.\n\nAs we drew closer, the words of his chant became clear to me:\n\n_Divum empta cante_ ,\n\n_Divum deo supplicate_ ,\n\n_Cume tonas, Leucesie_ ,\n\n_Prae tet tremonti_.\n\nI remembered the _magister_ telling me that Roman priests spoke to the gods in an obscure tongue, an archaic Latin, a language no one now understood. The priests had to recite these chants verbatim, without error, or a terrible curse would be incurred.\n\n_Quot ibet etinei de is_ ,\n\n_Cum tonarem osculo dolori ero_ ,\n\n_Omnia vero adpatula coemisse_.\n\nWe were filed before the standing shields to face the priest hovering above us on the platform. The commands of our officers were hushed, awed. The incense began to swirl around us. Behind us, I could feel the presence of thousands of silent men, watching.\n\nIan cusianes duonus,\n\nCeruses dunus Ianusve,\n\nVet pom melios eum recum.\n\nThe hooded priest in the white toga raised his hands to the sky, speaking in a loud voice:\n\nCume tonas, Leucesie,\n\nPrae tet tremonti\n\nQuom tibi cunei\n\nDecstumum tonaront.\n\nHe bent over the brazier and whispered something I could not hear.\n\nThen, the priest turned toward the assembled legion and removed the hood from his head. In a loud voice, he proclaimed, \"Father Iove has heard the prayer, has heard it sanctified in the sacred name of Rome, and accepts your petition! Let these acolytes join your ranks as his soldiers!\"\n\nFrom behind us, thousands thundered in one voice, _\"Fiat_! _Fiat_! _Fiat_! Let it be!\"\n\nI looked up at the priest. He wasn't that tall, but his long, slender body gave the impression of height. The hair behind his receding hairline was sandy brown, sparse, and graying. His forehead was broad, descending into thin eyebrows. His eyes were blue, piercing, and cold. His nose was thin and prominent. His mouth a black line slashed across his face. His appearance gave the impression of a dangerous weapon: a slender, razor-sharp _sica_.\n\nMalleus, the first spear, stepped forward to the edge of the platform and commanded, \" _Legio_!\"\n\nBehind me, I heard echoing commands, \" _Cohors_!\"\n\nThen Malleus, \" _Imperatorem . . . Adclamate_!\"\n\nThunder from behind me, _\"Ave, Imperator_! _Ave, Imperator_! _Ave, Imperator_!\"\n\n_\"Ave, Imperator_?\" This priest was Caesar himself! Our commander! The _pontifex maximus_ of the Roman nation.\n\nCaesar inclined his head slightly in response to the legion's acclamation. He turned, nodded to Malleus, and left the platform.\n\nMalleus waited while Caesar mounted a white horse that was being held by a legionary behind the platform. Then, he rode off toward the town. After that, the _primus pilus_ , in a theatrically loud voice, asked Strabo, who was standing behind our file, \" _Centurio_! Do you attest before the eagle that these _tirones_ have successfully completed their training and are ready to be welcomed into the ranks of this legion as _milites Romani_?\"\n\n\" _Confirmo_!\" I heard Strabo's voice from behind our rank.\n\n\"Legio!\" the Primus Pilus continued. \"These _tirones_ have completed their training. Will you accept them into your ranks as soldiers of Rome?\"\n\n\" _Fiat_!\" thundered the voice of the legion.\n\n\" _Tirones_!\" the Hammer addressed us. \"Remove the white tunic of the acolyte and assume the red tunic of a soldier of Rome.\"\n\nWe hesitated, not sure what to do. I heard Bantus' voice hiss, \"Strip!\" We did, down to our loincloths.\n\nAgain, I heard Bantus: \"Put on the red tunics!\"\n\nI stepped forward, picked up the tunic in front of me, and threw it on over my head. After a few heartbeats, our entire file settled down.\n\n\"Milites,\" Malleus continued, \"assume the weapons of a Roman soldier!\"\n\nAgain, I stepped forward. I picked up the sword. I threw the baldric over my head and put my right arm through it so the sword was positioned on my right hip. Then, I picked up the shield, threading my left forearm through the leather support strap and grasping the leather grip. I stepped back into line with the rest of my _contubernales_. For the first time, I could feel the weight of the sword pressing on my shoulder and against my hip, and I could smell the fresh leather from the bands of my new shield. I grasped the pommel of the sword with my right hand.\n\nMalleus told us, \" _Milites_! Raise your right hands open to the eagle and repeat after me.\"\n\nI did, and again I repeated the words of the _sacramentum_ in front of my new comrades:\n\nI, Gaius Marius Insubrecus Tertius, do solemnly swear by Father Iove, greatest and all powerful, whose eagle I now follow, and by all the gods, that I will defend and serve the Roman nation. I will obey the will of the Senate, the people of Rome, and the officers empowered by the Senate over me, and my general, Gaius Iulius Caesar, _Imperator_. I swear that I am a free man, able to take this oath, and obligated by bond or debt to no Roman. I will remain faithful to the Roman people, the Senate, to the officers empowered over me, to the army of Rome, and to the Tenth Legion until legally discharged by my time of service, by the will of the Senate and People of Rome, by the will of my general, Gaius Iulius Caesar, _Imperator_ , or by my death. I offer my life as the surety of my oath.\n\nThen, the _primus pilus_ commanded, _\"Milites_! _Contra . . . VERT_ '!\"\n\nWhen we were facing the legion, Malleus acclaimed, \" _Infantes_! _Salvete novos_!\"\n\nThe legion thundered, \" _Salvete_! _Salvete_! _Salvete_!\"\n\nWhen the echoes of the acclamation had died, Malleus commanded, _\"Centurio Strabo_! _Miss'est_!\"\n\nStrabo ordered:\n\n_\"Ad Dex' . . . VERT_ '!\"\n\n_\"Promov . . . ET_ '!\"\n\n_\"Dex' . . . Dex' . . . Dex', Sin', Dex_ '!\"\n\n\"Pick up your feet, Pagane!\"\n\n\"Right . . . Right . . . Right, Left, Right!\"\n\n\"Get in step, Lentulus!\"\n\n_\"Dex' . . . Dex_ '!\"\n\nWhen we got back to our tent, we were energized by the ceremony, our _significatio_. The cry of \" _salvete_ \" by the thousands of our new comrades rang in our ears. And, Caesar _Imperator_ was here among us. Caesar himself! Something was about to happen, something big. We were part of it; we were finally wearing the red infantry tunics of the Tenth Legion. We were soldiers of Rome.\n\nI sat on the edge of my bunk, my new _gladius_ across my knees. I loved the substantial feel of its weight in my hands. I pulled it partially out of its sheath; it slid out smoothly and silently. For a moment, I felt I was violating it. As the light coming in through the tent flap enflamed its edges to glowing white, I knew it had been honed razor sharp, and it was mine. _Miles Romanus_ , a _mulus_ , a grunt in the Tenth Legion, the best legion in Caesar's army\u2014in any army\u2014and I was part of it.\n\nBantus burst into our tent. Loquax saw him and tried to call us to attention, _\"Ad, Pedes_!\"\n\nBantus told us to be at ease, _laxate_ , \"You're soldiers . . . You don't have to bounce to your feet for an optio.\"\n\n\"Boys, I got some good news for you and some other news, so listen up,\" he started. \"Let me give you the good news first! As soon as you get your quarters squared away, you have a pass to leave the camp . . . Go up to the town, have a few drinks, celebrate your _significatio_ . . . You worked hard . . . You deserve it. Uniform's tunics and belts . . . Wear your _pugiones_ . . . That's so these sorry civilians know you're Roman soldiers . . . Be back here by the first call for night watch . . . Other than that . . . go relax for a couple hours . . . Enjoy yourselves!\"\n\nWe all sat there stunned, silent. Since we had reported over two months before, we had not had a moment off, had not been outside the camp walls, except for training. \"Have a few drinks . . . Celebrate . . . You deserve it.\" Bantus could have said it in Greek, and it wouldn't have sounded more foreign to us.\n\nFinally, Rufus broke the silence, _\"Mammis Veneris_! It's about time!\"\n\nI snapped him with the end of my _sudarium_ before he could continue.\n\n\" _Tacet_ '! _Tacet_ '!\" Bantus interrupted. \"Settle down! Let me tell you the other news before the celebrations begin.\"\n\nWe settled down to listen.\n\n_\"Bene_.\" Bantus continued, \"We have orders to break camp and move out the day after tomorrow. We're headed for _Gallia_ with the army.\"\n\nMutterings around the squad bay: \"Mercule!\" \" _Cacat_!\"\n\n\"Settle down!\" Bantus continued. \"There's been a fight up near Gennava . . . The Eighth held the barbarians off, and they retreated back over the Rhodanus . . . The general wants us up there quick . . . so he'll be running our asses off to get there . . . You guys are goin' to stay together as a _contubernium_ in the Tenth Cohort\u2014at least until we get over to _Gallia Transalpina_ , Gaul-Over-The-Alps . . . We're bringing Tulli back to be your _decanus_ . . . He'll be moving in with you guys tonight.\"\n\nWe were a bit shocked: a fight near Gennava; got to get up there quick. It was starting to hit us that we were going into combat, going up into _Gallia comata_ , a place of nightmares and cold, dark forests, with giant savages screaming for Roman blood. I felt my hand tightening on the hilt of my new sword. Somehow it didn't seem as substantial as it had only moments before.\n\nBantus was talking, \"Before you guys head to town, go by supply and draw your _tegimenta_ , your shield covers . . . Got to keep those nice, new shields bright and shiny to impress those hordes of hairy barbarians you'll be meeting soon . . . Get this place squared away before you leave . . . Oh, yeah . . . before I forget . . . Scratch your initials or your sign somewhere on your sword hilt or near the bottom of the blade . . . They all look alike after a while.\"\n\n# IV.\n\n# De Itinere interAlpes\n\n## WE MARCH ACROSS THE ALPS\n\nBantus was wrong. We didn't leave camp the next day, the day of Mars, or even the day after that, Mercury's day. It was Iove's day before the _augur_ , the _haruspex_ , and priest all agreed it was _faustus_ , propitious to march. Rumor had it that Caesar _Imperator_ was fuming, but as superstitious as soldiers are, even he did not dare to initiate a campaign without the favor of the gods. Rumor also had it that the delay had something to do with Crispus' girlfriend in the town.\n\nBantus just shrugged and repeated the age-old wisdom of the Roman army: \" _Festina lente_ . . . Hurry up and wait!\"\n\nWhen we did leave, the general had us hotfooting down the road at the double, thirty to forty thousand _passus_ a day, _impedimenti_ , loaded down like the _muli_ we were. Bantus told us the mountain passes would slow us down, and we might have some trouble with the highland tribes, a bunch of thieving brigands who notoriously showed no respect for the Roman _imperium_ , so we had to go as fast as we could while we were on flat ground in \"friendly\" territory. Although the march to _Gallia_ normally took an army marching _expediti_ , unburdened with a supply train, over thirty days, the general was determined to have us in position along the Rhodanus in less than twenty.\n\nThe first day out we passed through Patavis; by the second, Vicetia had seen our backs; the third day, we marched around Verona, where we had been snowed in last _Ianuarius_.\n\nDespite being in \"friendly\" territory within the _Imperium Romanum_ , each night we dug a marching camp. Most of our mates in the _centuria_ were _immunes_ , exempt from fatigue details, so they went off to perform whatever duties they had, while the rest of us dug and piled dirt. Our assignment was a portion of the _fossa et vallum_ , the ditch and the rampart, to the right of the _Porta Decumana_ , the rear portal of the camp. By the time we were done and the _sudes_ , the entrenching stakes, were lashed together and in place, the baggage train was pulling in through the gate we had just constructed.\n\nMoelwyn had our tent erected in its assigned place, but we had little time for it. Our _contubernium_ was assigned sentry duty on the rampart to the right of the gate, and since most of our mates were exempt from that too, we had one watch on, one watch off, through the night. At the first glimmer of dawn, we pulled everything down. The entrenching stakes went back on the mule, and the dirt ramparts were pulled down into the ditch, filling it. Entrenching baskets went back on the mule with the squad tent Moelwyn had pulled down. _Dolabrae_ , our entrenching tools, were cleaned and strapped to our _furcae_. Then, back on the road, double-time, we truly understood what it meant to be a _mulus Romanus_.\n\nAlong the paved military roads, we marched four abreast at the rear of our cohort, which marched at the rear of the legion. So, _furca_ on the left shoulder, _pilum_ in the right hand, we ate the dust stirred up by over four thousand pairs of _caligae_ and a few hundred horses and mules. About every five or six thousand _passus_ , we got a break and a couple of mouthfuls of water to wash down a piece of _buccellatum_. Then, back on the road, double-time, about a thousand _passus_ , regular pace for a thousand more, and then back to the double.\n\nWe double-timed through Brixia, camped near Bergomum, and passed north of Mediolanum. I remember looking to the south. There was a slight smudge of smoke marking where the city lay. I wondered briefly what Mama was doing at that very moment, whether she was thinking of me.\n\nIn camp that evening, Tulli told us that we'd cross the mountains near a tiny piss-ant town called Ocelum, where the pass opened that would bring us through to _Gallia_ , south of Gennava. Rufus and I were boiling up some porridge. We didn't have to be on the ramparts until second watch. We hoped to fill our bellies and get a couple of hours of sleep before that. Despite the extra padding where I carried the _furca_ , my shoulder was aching. I hoped it wouldn't keep me awake, but after humping over thirty-five thousand _passus_ that day, once I wrapped myself up in my woolen cloak, I doubted Hannibal and all his elephants could keep me awake.\n\nTulli had told us that once we got up into the mountain passes, all bets were off as to how many _passus_ we made each day. Snow drifts, avalanches, bandits\u2014 we'd have to get through all that _merda_ to get to _Gallia_. In the mountains, our cohort would be detailed with securing the _impedimenta_ , the baggage train. The good news was that we could dump our packs in a wagon and march _expediti_ , with just combat gear. The other news was that if the wagons got stuck, we were expected to \"unstick\" them, so we'd better keep our shovels and digging picks handy. The bad news was that the baggage train attracted bandits like a tavern attracts drunks. So, we'd better keep our shields, swords, and _pila_ close at hand\u2014 even when we were digging some teamster out of a snow drift.\n\nWe pulled into Ocelum during the fifth hour of the sixth day of the march. It wasn't much of a place, more goats than people. The Seventh and Ninth Legions were already encamped in the valley. The Eleventh and Twelfth were strung out a couple of hours behind us. The Alps stood before us like a wall of snow-covered granite. As we were marching down into the valley where we would dig our marching camp, I shivered as I noticed the looming mountains still covered with snow.\n\nAfter we dug our _castrum_ , we spent most of the afternoon on our butts. When Bantus arrived at our tent, we were blissfully racked out, bellies full. He roused us for a quick briefing.\n\nThings over in _Gallia Transalpina_ were developing quickly. After being repulsed at Gennava by the Eighth Legion, the Helvetii withdrew west, down the Rhodanus through some narrow passes in the Iura mountains. Roman allies in _Gallia_ , tribes called the Sequani and the Aedui, were now directly in the path of the Helvetian horde. That, and the Helvetii being such a threat to the most fertile lands of the Roman _provincia_ just across the river, was unacceptable to Caesar _Imperator_. Our objective, then, was to stop the Helvetii and force them to return to their homeland, where they would continue to serve as a buffer between _Gallia_ , our _provincia_ , and the German barbarians east of the Rhenus.\n\nThe goal of our march was no longer just to reach Gennava, but our goal was to reach the Helvetii themselves.\n\nLater that afternoon, Strabo called the _centuria_ together to give us our marching orders. There were only about fifty of us assembled. Many of our _immunes_ soldiers were off on various details around the _castrum_.\n\n\"The army'll move out tomorrow morning at the second hour and pass through the Alps in a column of legions,\" Strabo began. \"Caesar _Imperator_ is interspacing the new legions with the veterans. The Seventh will lead out, followed by the Eleventh . . . then the Ninth, followed by the Twelfth . . . We'll take up the rear.\"\n\nStrabo paused to see if there were any questions. There were none, so he continued, \"The Seventh will move out at the second hour, but the Eleventh won't follow until the fourth hour . . . Yeah . . . What it is?\"\n\nI heard a voice from the back of the formation ask why the delay for the Eleventh.\n\nStrabo responded, \"The Seventh will have to do most of the route clearance through the passes, and if any of those mountain tribes decide to kick up a fuss, the boys in the Seventh will have that to take care of, too. So, there'll be delays . . . stop and go . . . especially in the narrow passes . . . The general doesn't want us to bunch up . . . The Eleventh will probably catch up to the Seventh anyway . . . The general wants them far enough back so both legions won't be caught in the same ambush, but close enough so they can support each other . . . Besides, the Seventh is marching _expediti_ . . . light and ready for combat . . . Their baggage train'll be moving with the Eleventh . . . Any other questions?\"\n\nWhen Strabo mentioned ambushes and marching \"combat ready,\" I felt a stirring in my belly, almost a weakness in my legs. That was it; we were going into a fight.\n\nStrabo continued, \"After the Eleventh moves out, the rest of the legions will depart at one-hour intervals . . . The Ninth at the fifth hour . . . the Twelfth at the sixth . . . then us at the seventh. The _equites_ , the legionary cavalry, will be positioned between the legions so it can coordinate communication and maintain the marching intervals . . . I doubt those horses'll be much use in the high passes . . . Their mobility will be limited . . . A real shame, cavalry having to walk . . . We should anticipate delays during the march . . . stop and go as we move through the mountains. The cavalry's supposed to coordinate our spacing so we don't bunch up or get too far behind.\"\n\nStrabo repeated that our departure time was the seventh hour the next day. Then, he concluded, \"We'll march in a column of cohorts . . . The first-line cohorts, the First through the Fourth, will be our _primum agmen_ , the vanguard . . . The Fifth and Sixth Cohorts, from the second line, will act as our _novissimum agmen_ , the rear guard. Our cohort is detailed to guard the legion's baggage train, which'll follow the Ninth cohort out.\"\n\nStill no questions, so Strabo continued, \"Caesar _Imperator_ , will be marching with the point legion, the Seventh, along with most of his party. The _primus pilus_ , Malleus, is commanding our legion on this march. He'll be with his _centuria_ in the First Cohort. The general hasn't assigned a _legatus legionis_ , a legionary commander, to us yet, and we won't be dragging any wet-behind-the-ears, somebody's-wife's-cousin, Patrician brats as military tribunes with us.\" Some snickers come from the back of the formation. \"So, the good news, we don't have to drag wagons full of their useless shit over the Alps.\" Some guffaws this time. \"The other news is we can't help ourselves to their wine and _garum_ supplies.\" Some theater groans. \"So that's about it . . . Any questions?\"\n\nThere were none, so Strabo dismissed us back to our tents.\n\nWe didn't have any duties until guard mount that evening. Since we weren't going to drag any livestock over the Alps with us, except for the _contubernium_ mules and the cavalry horses, Malleus had ordered the cooks to brew up a feast for us. We could smell the aroma of the roasting meat and baking bread wafting over from the cook tents in the middle of the _castrum_. Meanwhile, in the age-old tradition of the Roman infantry, we decided to crap out until chow time.\n\nTulli, our _decanus_ , seemed determined to talk to us, however.\n\n\"We caught a break on this one, guys!\" he started.\n\n\" _Quo'mo'?_ \" Rufus mumbled, half-asleep.\n\n\"Think about it,\" Tulli lectured. \"The Seventh has to do all the heavy lifting on this one . . . clearing the passes and taking care of any barbarians stupid enough to get in our way . . . By the time we go through the passes, the route'll be clear, and there won't be a single, living _podex_ for miles to mess with us.\"\n\n\"Makes sense,\" Loquax mumbled from somewhere deep in his woolen cloak.\n\n\"Sure it does,\" continued Tulli, not wanting to let go of it. \"And, we get to march with the baggage train . . . That means no faster than the slowest cart . . . and we march _expediti_ . . . just swords, shields, and one _pilum_ each . . . The rest of our equipment can go on one of the carts . . . Except for the cold and having to push a cart or two up a steep incline, this'll be a piece of cake.\"\n\nTulli was begging _Fortuna_ to screw us. I should have rubbed my amulet and spit, but I decided just to turn over and go to sleep instead.\n\nThe next day, getting the army moving was a complete cluster, and we didn't cross the line of departure until well into the eighth hour. Tulli's assessment seemed right, though. We were marching light, with troops in front of us and to our rear. The oxen pulling the supply carts didn't seem to be in any hurry to get up into the mountain passes. Tulli said when we get over into _Gallia_ , the baggage will move independently of the legions, so we can move quickly. But, if we tried that up in the passes, the mountain tribes would steal the hooves off the horses.\n\nThat first day the climb wasn't bad, but we only made about ten thousand _passus_. We camped next to a small mountain river in a narrow, ascending valley. By the time we pulled in with the _impedimenta_ , the camp was already constructed\u2014 another benefit, Tulli winked\u2014so all we had to do was settle the baggage in and pull our guard shifts that first night.\n\nThe next morning, the legion was up and moving before the fourth watch was over. We continued to climb. By the fourth hour, we passed through some piss-ant village of stone and lumber huts. It was abandoned. No fires burning.\n\nEven the livestock was gone. By the sixth hour, we swung west and north, still following a mountain river upwards. Four more hours found us at the junction of two streams: one flowing down from the north; the other, the one we seemed to be following, still led toward the northwest.\n\nWe halted there. When we realized that this wasn't just a break from the march, our _centurio pilus prior_ , Gelasius, the cohort commander, deployed four of our _centuriae_ on the uphill side of the baggage train, and the remaining two, including us, on the downhill side. Eventually, the word came down that there was some sort of delay up ahead. After being in position for almost an hour, we even began to imagine that we'd spend the night there. No sooner had we entertained that hope than the column started moving again.\n\nWe continued to climb. The twelfth hour came and went. I estimated we had at least thirty thousand _passus_ under our boots. The sun went down, and we still continued to climb. We stumbled up the incline, starting, stopping, and starting again. Finally, the word came down we were staying put where we were for the night. No camp. Since we were headed uphill, Gelasius deployed two centuries forward, one century to the rear, and split the rest of us equally on both sides of the baggage. He ordered \"half and half\" for the night\u2014one man slept while the other kept watch. Strabo broke us down into shifts by _contubernia_.\n\nIt was starting to get cold in the passes. Gelasius forbade the building of fires. He said it would silhouette our positions for any enemies up in the hills. He had a bit of difficulty convincing some of the civilian teamsters. But, after he emphasized his orders with a liberal application of his _vitis_ , the centurion's vine cudgel, the rest fell quickly into line. Most of us had broken out our _bracae_ trousers and woolen socks. Still, I was shivering so hard, it was difficult getting to sleep. I was almost grateful when I was roused up for guard duty because it gave me a chance to move around and warm up a bit.\n\nWe were moving before sunrise the next morning. When the sun finally came up, I could see we were marching directly up toward a huge mountain. The rising sun behind our backs was changing the snow on the mountain's flanks from a glowing blue to a pale white.\n\nTulli caught up with me. \"That's _Alba Magna_ ,\" he said. \"Don't look at it. You'll go blind. Here!\" He offered me what seemed to be a burned-out lamp.\n\n\"Rub the soot around your eyes and on your cheekbones . . . It'll cut down the glare.\"\n\nI had no idea how a snow-covered mountain could make me blind, but I did as Tulli suggested. \"If that doesn't seem to help . . . if you feel your eyes get dry or they start to hurt . . . tie a _sudarium_ around your face like a mask . . . In a couple of hours, we'll be turning south, so the mountain will be over your shoulder.\"\n\nAs we continued to climb toward _Alba Magna_ , the sun rising behind us set the mountain ablaze like a brilliant, white flame. I found that even with the lamp soot around my eyes and the scarf tied up around my face, I couldn't look at it. I grabbed the tailgate of one of our carts and let it lead me up the pass while I kept my eyes fixed on the ground to keep from tripping.\n\nTulli was right about one thing. The legions marching ahead of us had cleared the passes. Our path was free of snow and rubble. We hadn't seen a native since we left Eporedia.\n\nBy the fifth hour, I thought we had reached the top of the pass at the foot of _Alba Magna_ , but we turned south and continued to climb. The mountain was now on our flank, and the sun was no longer at our backs. I was able to see again. My eyes felt like they were full of sand\u2014a bit achy, but I was able to see.\n\nThe air in the pass was still, but frigid. It was colder up here in _Aprilis_ than it was down in the valley in the middle of winter. My breath steamed out of my mouth and nostrils like mama's kettle on a cold morning. We were allowed to march without our helmets, so I had the hood of my cloak up over my head and a _sudarium_ wrapped around my ears. I had pulled my woolen socks up and stuffed the bottoms of my leather _bracae_ down into them, so no part of my legs was exposed.\n\nStill we climbed; still it seemed to get colder.\n\nWe topped the pass by the seventh hour. I could swear we had twenty thousand _passus_ of climbing under our belts since we started out before dawn. But, I was glad to be moving in that cold. Now that we seemed to be on the downhill side and headed into warmer temperatures, I was happy that we were pushing on.\n\nBantus, our _optio_ , came down the line of march. \"Another ten thousand _passus_ before our day's done, boys,\" he announced. \"The valley opens up down below . . . That's where we'll spend the night . . . The First Cohort boys should have the camp all set up for us by the time we get there . . . hot chow and a warm tent . . . It's all downhill . . . so hang in.\"\n\nWe made it into camp after nightfall. The little valley seemed more like a hole that the gods had blasted between the mountains. There was high ground all around us. It was warmer than the pass, but not much. Since we were in a fortified camp, we were allowed to build fires. Somehow, the first cohorts in camp had managed some hot chow and warmed _posca_ , which was waiting for us when we corralled the baggage train. I couldn't remember ever drinking anything as comforting as that steaming cup of _posca_.\n\nLater that night, when I took my turn on the ramparts, a cold wind was blowing down out of the pass from the northeast, the frigid breath of _Alba Magna_. I saw some fires glowing about two thousand paces across the valley to the southwest. When I asked about it, I was told that it was the camp of the Eleventh Legion. Where the rest of the army was that night, we had no idea.\n\nThe next morning, we didn't move out until the first hour of the day. Malleus had decided to give the Eleventh a head start down the next pass. But soon, we too were descending down the narrow valley behind them. Tulli was still feeling good about the whole thing. A few more zigs and zags, but it was all downhill from here, as far as he was concerned.\n\nThat day's march seemed to prove him right. The passes and gorges were narrow, but the leading legions had done a good job clearing the route of the march. We did over thirty thousand _passus_ and arrived in another little valley at nightfall. There, we got our first indication that things would be different on this side of the Alps. There had been a village near the intersection of two small rivers. What people did to stay alive so far up in the mountains was anybody's guess, but whatever it was, it was no longer a concern for the people who had lived here. The village was destroyed, burned out. When we arrived, the ashes were cold, but we could still smell the odor of wet, burnt lumber and thatch. There were no bodies. Perhaps whoever had lived here had escaped.\n\n\"My guess is this is the work of the boys in the Seventh,\" Tulli said as we marched past.\n\n\"Why'd they do it?\" I asked.\n\n\"Who knows?\" Tulli answered with a shrug. \"Don't see any bodies, so I guess most of them are up in the hills. They must have done something to piss off the general.\"\n\nThe next morning, we followed a river descending toward the northwest. Around the third hour, we passed another burned-out village. Again, the fires were long out and the ashes cold.\n\nWe were about two thousand _passus_ beyond the village when the column came to sudden halt. We didn't think too much of it until we heard a _cornu_ , a signaling trumpet at the head of our column, calling, \"Ad _Signa_!\" I immediately heard Strabo's voice repeating the call to fall in and spotted our century's standard raised on the other side of the baggage train. We quickly assembled around the standard, and Strabo called us to attention. We waited, but there were no more trumpet calls. Strabo had us stand easy and check our combat equipment.\n\nI was just deciding that this was some kind of drill, when Gelasius, our cohort commander, arrived to brief Strabo. After the two centurions finished talking, Strabo came over to talk to us.\n\n\"Seems there's a roadblock set up in front of the column,\" Strabo said. \"Must've gone up after the Eleventh passed through and before we arrived, so it's deliberate . . . Not sure what's going on . . . The natives are probably trying to pick off a wagon or two . . . The valley opens up about two thousand _passus_ farther down the road, so this is their last chance to get to us . . . The First Cohort's clearing the road, and then we'll be moving . . . Meanwhile, we're going to move straight toward that line of spruce and pine trees to see if any of them _mentulae_ are hiding in there . . . The Third Century will stay close to the _impedimenta_ and support us if we need it.\"\n\nWe had already assembled in the basic combat formation, a ten-man front at normal intervals with the _contubernia_ in column, so Strabo marched us straight forward, toward the tree line. I was positioned in the front rank with Minutus behind me as my _geminus_. The rest of our _contubernium_ was in file behind us. I could see Strabo and our _signifer_ marching on the right flank of our first rank. I knew Bantus and our _tesserarius_ , a legionary everybody called Brevis because he was almost six _pedes_ tall, would be to our rear.\n\nAs we got closer, I noticed that the trees to our immediate front receded a bit and the ground in front of them dipped, creating the semblance of an amphitheater with the evergreen forest on three sides. Strabo halted us before we entered the low ground. There didn't seem to be any movement in the trees. Despite that, Strabo decided to gain width and sacrifice depth. He ordered us to open our intervals, and he doubled the _contubernia_ front. Now I had Tulli, our _decanus_ , on my left, with Rufus backing him up. The _centuria_ now had a twentyman front, but we were only four men deep.\n\nStrabo ordered us to stand at ease.\n\nWe waited. Nothing happened. No enemy appeared to our front. No recall back to the line of march sounded.\n\nI heard someone behind me mutter, \"Another goat rope!\" A few guys guffawed at that. Strabo must have heard, but he said nothing.\n\nSuddenly, we heard a _cornu_ at the head of the column signal, \"First Cohort,\" then, \"Advance!\"\n\nThen we heard calls for the Fifth and Sixth Cohorts at the rear of the column.\n\nStill we waited. But, nothing for us.\n\nThen, one of our guys near the left flank yelled, \" _Centurio_! Movement in the trees!\"\n\nI looked down the column and saw the man pointing with his _pilum_. I looked toward where he was pointing. At first I saw nothing. Then, as my eyes adjusted to the gloom under the evergreens, I saw movement. Men. Dozens of them. They began to come out of the forest in front of us. They weren't armored like us. Most of them seemed wrapped in animal firs. I could see no swords, no spears. I couldn't understand how such a rabble could be any threat to us.\n\nStrabo was walking across our front, slightly down the incline. Without taking his eyes off the enemy, he called us to attention, \" _Centurio . . . State_!\"\n\nThe enemy remained well out of _pilum_ range. There was now a mass of a couple of hundred of them, standing just outside the tree line. They made no attempt to advance on us. Then, a few dozen of them ran forward, twirling something over their heads.\n\nStrabo screamed, \" _Scuta . . . Erigit'_!\"\n\nNo sooner did we raise our shields to cover our heads and bodies, than slingbullets rained down on us. We had taken two or three volleys to no effect when one of the legionaries on the left shouted a warning, \" _Centurio_! Flank! Left!\"\n\nI stole a peek around my shield and saw the enemy working their way through the trees on our left flank. Then, from our right, I heard, \" _Centurio_! _Hostes a' dex_ '!\"\n\nThe slingers were trying to hold us in place while their mates got around our flanks. If they got some slingers on our right, our open side, they could do some damage.\n\nStrabo had to pull the _centuria_ back. If he pulled back quickly, he would expose us to the slingers to our front. If he withdrew slowly, the enemy might get around on our exposed flank.\n\n\" _Centurio_! _Ad testudinem_! Form the turtle!\" Strabo commanded.\n\nThe extended ranks withdrew backward, shields up, and fell in at the rear of their _contubernia_. The _contubernia_ side-stepped to a close interval. Front and flank shields came up and interlocked, forming a wall around three sides of the century. The legionaries in the interior of the formation lifted their shields over their heads and locked them to form a roof.\n\nWe waited. The command to withdraw didn't come.\n\nWhere was Strabo?\n\nI peeked around my shield. He was down! He was on his hands and knees in the grass. His helmet was off. He was bleeding from his nose and from his head.\n\n\"Centurion down!\" I shouted.\n\nI heard Bantus' voice from the rear, \" _Centuria_ . . . take my command! _Ad tergum . . . It'_! Backstep . . . MARCH!\"\n\nAs we began to backstep out of the trap, I again looked and saw about six of the enemy running across the field toward Strabo.\n\nWithout thinking, I broke the formation and ran down into the bowl toward my centurion. When I reached him, the first of the barbarians was well within _pilum_ range. I let loose! The weighted spear hurtled forward, taking the man full in the face. I drew my sword. I sensed another _pilum_ fly from behind me and saw it bury itself in a barbarian's chest. I glanced back quickly to see that my _geminus_ , Minutus, had followed me.\n\nThe four remaining enemy were almost on top of us. The closest raised a knife. Without thinking, I stepped forward and delivered a _percussus_ with the boss of my shield. My _umbo_ made contact. I heard the man's neck break. I stepped forward with a sword thrust, and the next man impaled himself on my _gladius_. His momentum pushed me back. I couldn't withdraw my sword from his abdomen. Minutus moved around my right side. He took the next barbarian with his shield. The man literally flew backward as the _umbo_ of Minutus' shield smashed into his head.\n\nThen, it was over.\n\nThe sixth man, seeing his five mates killed so quickly, was having none of it. He fled back to the mass of barbarians at the edge of the forest. They seemed to hover there for a few heartbeats. Then, they began to melt back into the trees.\n\nFor a few heartbeats, all I could hear was the sound of my own breathing. I looked around me. Two barbarians lay dead, pinned to the grass with our _pila_. Another man lay in front of me, his head at an impossible angle. Another had a face that was a mass of blood and shattered bone. A fifth man was on the ground, screaming and trying to keep his guts from pouring out of his belly. Minutus relieved him of his agony with a quick thrust into his throat.\n\nI could still hear moaning. Then, I remembered. Strabo! The centurion was sitting upright on the ground, holding his head. Blood was pouring down his head over his right ear. I spotted his helmet on the ground next to him. There was a dimple, the size of a baby's fist, above the right cheek guard.\n\nOur _centuria_ had halted about a fifty paces away. They were out of the turtle and standing in the standard formation.\n\n\" _Capsarius_!\" I shouted. \"I need a medic! Man down!\"\n\nMinutus walked up and dropped our two _pila_ on the ground. \"Not even bent,\" he observed.\n\nI looked up toward our _centuria_. They were still stationary, but Bantus was striding toward us. I imagined for a moment that he was going to congratulate me, maybe I'd get a decoration\u2014at least a silver torc.\n\nWhen he got to where I was standing, he said, \" _Miles_! Remove your helmet!\"\n\nI didn't know what he meant.\n\nAgain, he ordered, \"I told you to take off your shaggin' helmet, _Podex_!\"\n\nQuickly, I fumbled with the bindings under my chin. I removed my _galea_.\n\nNo sooner did I have it off than Bantus slugged me on the jaw. I fell on my ass next to Strabo.\n\n\" _Sta, miles_!\" Bantus shouted at me. \"Get back up on your feet, soldier! Get your helmet back on!\"\n\nWhile I was sitting on the ground, searching for my helmet, the _capsarius_ arrived. He looked at the two legionaries sitting on the ground and asked, \"We got two casualties?\"\n\n\"Just the centurion!\" Bantus snapped. \"Get on your feet, Insubrecus!\"\n\nI got up and put my helmet back on and laced it up. Bantus wasn't done with me.\n\n\"You stupid son of a whore! You never break formation!!\" He continued to shout in my face, \"Never! For no reason! You break formation, you endanger your mates . . . the entire _centuria_ . . . I don't care if your mother's being raped by a cohort of hairy Krauts right in front of you! You _do not break formation_! Do you understand me, you shit?\"\n\n\" _Compre'endo, Optio_!\" I snapped.\n\nWhile Bantus was ripping me to shreds, Spina, the _medicus_ , arrived on the scene. He bent down over Strabo and ran his fingers over the centurion's skull. \" _Bene_ ,\" he muttered to himself. \"Skull's in one piece.\"\n\n\"Centurion! What day is it?\" Spina asked Strabo.\n\nStrabo just stared up at Spina.\n\nMeanwhile, Bantus was \"fixing\" me: \"Breaking formation in contact with the enemy . . . You could suffer _fustuarium_ for that . . . Do you want to be beaten to death? Did you know that, _stulte?_ What you got going for you is only that you ran _toward_ the enemy and gutted three of the bastards! And, you may have saved your officer's life . . . but that's no excuse for what you did . . . _Compre'endis me, miles?_ \"\n\n\"Excuse me, _Optio_ ,\" Spina interrupted.\n\n\" _Qui' vis tu, Medice?_ \" Bantus turned.\n\n\"I need tah move dah centurion hee'ah back to my wagon, so's I can treat'im,\" Spina explained. \"Could I borrow a couple a ya guys hee'ah to help?\"\n\n\"Of course, _Medice_ ,\" Bantus responded. \"Brevis! Detail a couple of the boys to help the doctor here!\"\n\n\" _A'mperi'tu', Optio_!\" I heard Brevis respond.\n\nSuddenly, Gelasius, our cohort commander, arrived on the scene. He saw Strabo on the ground and the _capsarius_ bandaging his head. Then, he looked at the dead barbarians lying in the grass.\n\nIgnoring Spina and me, he asked, \"What's going on, Bantus?\"\n\n\" _Centurio_!\" Bantus reported. \"This bunch of _mentulae_ tried to rush the wagons . . . We stopped them . . . Strabo was hit by a sling-bullet.\"\n\n\" _Tota?_ \" Gelasius asked examining the reddening welt on my jaw. \"That it?\"\n\nBantus hesitated for a few heartbeats, then said, \" _Tota, Centurio_!\"\n\nGelasius gave him a long look. Then he turned to ask Spina, \"How's Strabo, _Medice_?\"\n\n\"Don' know, Centurion,\" Spina answered. \"May just be a _concussus_ , but head injuries aw tricky. I wanna get 'im back to my wagon.\"\n\nGelasius grunted and said, \"Carry on, _Medice_!\"\n\nHe took one last look at the dead raiders and turned to Bantus, \" _Bene gesta, Optio_! I'll need your after-action report when we get into camp tonight. You have the _centuria_ until Strabo gets back on his feet!\"\n\n\" _A'mperi'tu', Centurio_ ,\" Bantus responded.\n\nAs soon as Gelasius was out of hearing range, Bantus hissed at me, \"I just saved you, Pagane! Don't forget it! You saved Strabo's bacon . . . a life for a life, eh . . . And don't forget what I told you about breaking formation . . . You try that shit again, and I'll beat you to death myself . . . Now get your sorry ass out of my sight!\"\n\nI double-timed back to the _centuria_ with Minutus in my wake.\n\nThe column started moving again at the sixth hour. Soon, we turned southwest down a wide, fertile valley. Here, there was no sign of war. There was a town at the head of the valley, rather large and prosperous for the region, totally unfortified. In the fields there were cattle, who watched us pass with total indifference. We saw some people, too. None of them seemed the least concerned by the presence of a Roman legion in their valley.\n\nWe moved into camp at the eleventh hour. The legion had hot rations waiting for us: a hot stew of pork and vegetables with fresh bread and plenty of _posca_. Before the first watch of the night, Bantus briefed us that this was the valley of a Gallic tribe called the Vocontii, a people at least nominally under the Roman _imperium_. Since they grow fat on money extorted from Roman merchants in tolls, protection, and \"hospitality,\" they were quite content with Roman rule. The general had passed down the word that the natives were not to be harassed in any way: no stealing crops, livestock, or anything else not nailed down, and leave the women alone!\n\nWe were going to consolidate with the rest of the army the following day. The other legions were camped near a Gallic town called Leminco, about fifteen thousand _passus_ down the valley.\n\nAs Bantus left us, he announced, \" _A' Galliam comatam beneventi, infantes_! Welcome to long-haired Gaul, guys!\"\n\n# V.\n\n# Sub Patrocinio Caesaris\n\n## UNDER CAESAR'S PATRONAGE\n\nHe pulled into Leminco by the sixth hour on the next day. It was an easy march down the valley of the Vocontii, more of a parade than a march. In fact, as we marched, children and civilians looked up from their work in the fields and waved as we went by. About two thousand _passus_ out, Malleus, our _primus pilus_ , pulled us off baggage train duty and lined us up with the rest of the legion. We marched into the valley where the rest of the army was camped as if it were a military parade and not the end of a trek over the Alps. Caesar _Imperator_ , along with his entire staff, met us as we marched down into the encampment. Mounted on a white stallion, Caesar was bareheaded and draped in a _sagum rubrum_ , a bright red general's cloak. We passed in review before him.\n\nWe were guided to a section of the valley where we were to camp, and there we began to dig in. By the ninth hour, we had the _fossum_ dug and the _vallum_ erected for our marching camp. Although we were surrounded by the other legions of the army, we still had our _sudes_ lashed and positioned in our section of the ditch.\n\nAs usual, our _centuria_ was billeted near the rear gate, just to the left of the _Via Praetoria_ , Headquarters Street. When we arrived, Moelwyn had our _contubernium_ tent set up. When we arrived, he wasn't about, so we assumed that he was off tending to our mule. Tulli went to find Bantus to learn where the bath point was, when it would be our turn to wash up, and what the arrangements were for chow. The rest of us decided to take advantage of the lull and crap out for a while.\n\nAnyway, that was the plan.\n\nI was about halfway down a deep, dark hole, on my journey to the realm of Morpheus, when a _caliga_ nudged my shoulder. I opened my eyes expecting to see Tulli, but then I realized I was looking up at a soldier wearing the purple waist-ribbon of a praetorian.\n\n\" _Gaius Marius Insubrecus, miles, es tu_?\" he demanded. \"Are you Trooper Gaius Marius Insubrecus?\"\n\n\" _Sum_!\" I answered getting to my feet. \"I am!\"\n\n\"You are to accompany me to the _praetorium_ ,\" he stated. \" _Stat'_!\"\n\n\"Uniform?\" I asked.\n\n\"Full kit! Sword, no _scutum_ , no _pilum_!\" he again stated flatly. \"I will wait for you outside the tent.\"\n\n_What was this about?_ I wondered. _Did Bantus rat me out in his report and say that I broke formation?_ I was carrying my own sword, and the praetorian was waiting outside for me. At least I wasn't under arrest.\n\nI had to wake Rufus up to get back into my _lorica_. He wasn't too pleased about having his siesta disturbed, but was glad it was me being dragged over to headquarters and not him. As I left the tent, I told him to let Tulli know where I was. I doubt he heard me over his own snoring.\n\nAs soon as I emerged from the tent, without a word to me, my escort turned and began marching toward the _Via Praetoria_. As expected, he turned right on the main street of the camp toward the _praetorium_. Then, surprisingly, he walked right by our legionary headquarters and continued toward the main gate of the camp. He passed through with barely a nod to the sentry on duty.\n\nWe marched west across a field to another of the legionary _castra_ in the valley. When he approached the gate, the sentry challenged him with the sign, _malus_ , apple. He responded immediately with the countersign, _quercus_ , oak, and without stopping, he stated, \"He's with me.\"\n\nWhen we turned through the portal and entered the camp, I realized that we were on the _Via Principalis_ , the main street of the camp. The _praetorium_ of that camp was straight ahead, but I had no idea why I was being summoned to the headquarters of another legion.\n\nWhen we arrived at the headquarters tent, my escort pointed to a spot on the ground and said, \"Wait here until you're summoned.\" Then, he entered the tent.\n\nI didn't know how literal the praetorian was being, but I didn't wander far from the spot on the ground he had indicated.\n\nIt was a good thing I was away from the entrance. Soldiers with the thin, red sashes of the general's staff bustled in and out. Runners, I assumed. Periodically, senior officers, _tribuni angusticlavi;_ junior, \"narrow-band\" tribunes; and even a _legatus legionis_ , a legionary commander passed me by. But, to these men, I was invisible, just another _mulus_ among thousands. That was fine with me. I had nothing to gain from being noticed by senior officers\u2014and much to lose.\n\nFinally, a clerk on the general's staff stuck his head out. \"You Insubrecus?\" he asked.\n\n\" _Sum_!\" I answered.\n\n\" _Bene_! Get in here!\" he demanded.\n\nI entered the headquarters tent, still with no idea what was going on. I was hoping it was all just some mix-up, but they seemed to have my name. And, if this had anything to do with my breaking ranks to save Strabo's bacon, this seemed to be the place where death sentences were passed.\n\nWhen my eyes adjusted to the gloom inside the tent, the _scriba_ who had summoned me pointed to a cubicle on the right and said, \"Report to the legate, Insubrecus!\"\n\n_The legate_ , my mind screamed! This was worse than I had imagined. A shaggin' legate?\n\nI entered the _cubiculum_ and reported to a man seated behind a field desk, \" _Legatus_ , Gaius Marius Insubrecus, _miles_ , reports as ordered.\"\n\nThe man looked up from some _tabulae_ he was scanning and gave me a long look. He didn't look like a legate to me. He looked more like a guy you didn't want to mess with in a cheap _caupona_ in a bad neighborhood after drinking half the night. He had short, curly black hair, which matched his bushy eyebrows. His eyes were deeply brown, but alive, the kind of eyes that see everything, assess it, and find it amusingly lacking. He had a couple of days' worth of dark stubble on his chin and jowls. He had discarded his officer's _lorica_ , which was lying against the side of the tent. His sword and his helmet were deposited on top of it. He wore the faded, red tunic of an infantry _mulus_.\n\nWhen he stood up, I was again surprised. He was half a head shorter than I was.\n\n\"I'm Labienus,\" he told me, \"the old man's _legatus ad manum_ , his chief of staff. Let me take a look at you.\"\n\nHe inspected my uniform and equipment, adjusting invisible defects and centering my belt.\n\n\"Let me see your _gladius_!\" he demanded.\n\nI unsheathed my sword and handed it to him. He inspected the blade, grunted, and handed it back to me.\n\n\" _Bene_ ,\" he continued. \"When you report to the _imperator_ , don't mumble . . . Speak up like a man . . . The old man hates it when people mumble . . . And look the old man right in the eye when he talks to you . . . Don't look down . . . Roman soldiers never look down . . . Speak when spoken to . . . Be short and to the point: 'Yes, sir,' 'No, sir.' No explanations unless the old man asks you . . . Got it?\"\n\n\"Yes, sir!\" I answered.\n\n\"And, this is very important,\" Labienus continued. \"Don't stare at the old man's hairline . . . He's very sensitive about that . . . _Compre'endis tu_?\"\n\n\" _Compre'endo, Legate_!\" I replied.\n\n\" _Bene_ . . . Let's get this over with . . . Any questions?\" he asked.\n\n\"Yes, sir,\" I said.\n\nLabienus waited a few heartbeats and said, \"Are you going to ask it or just leave me here guessing?\"\n\n\"Sir! To whom am I reporting?\" I asked.\n\nLabienus gave me a long, quizzical look, then snorted. \"Son, you are about to meet the proconsul of all the Gauls, the commander of this army, and the great-great grandbaby of the goddess Venus herself. You are about to meet Gaius Iulius Caesar.\"\n\nLabienus made an overly dramatic gesture toward a doorway in the rear of his cubicle. I marched through.\n\nI entered a larger area of the tent. Most of the tent walls were covered by maps with various military symbols etched in chalk over the representations of the terrain. There were soldiers dressed only in their red tunics fussing over them. A tall, thin man in a white tunic with the thin, purple stripes of an Equestrian seemed to be supervising them. I began to walk toward him when Labienus put his hand on my shoulder and guided me toward the back of the tent.\n\nThere was another partition there and behind it, a man, the same man I had seen acting as the priest during my _significatio_. He sat reviewing stacks of _tabulae_ arranged before him on an oversized field desk. A _scriba_ stood in attendance at the man's left shoulder. When the man saw me approach, he sighed.\n\nHe closed the tabula he had been reading, passed it to the clerk, and said, \"Ebrius, inform the quartermaster that it is my decision that we will not wait for the supply train from Massalia to reach us. We will pursue the enemy as the situation demands. We will draw supplies from our Gallic allies if necessary.\"\n\n\" _A'mperi'tu', Imperator_ ,\" the clerk snapped, taking the _tabula_. Then, he posted out of the tent.\n\nThe way was now open. _My turn_ , I thought.\n\n\" _Imperator_! Gaius Marius Insubrecus, _miles_ , reports as ordered!\" I snapped. I felt Labienus hovering behind my right shoulder.\n\nWhile Caesar began to search through the _tabulae_ on his desk, he said, \" _Laxa_ , Insubrecus!\"\n\nI willed myself to unstiffen a bit, but I was hardly \"at ease.\"\n\nFinally, he drew two _tabulae_ before him. He opened the first and said, \"This is the daily operations log of the Tenth Legion that reports a slave attacked a soldier . . . you, as a matter of fact . . . during a training exercise . . . The slave was killed, and you suffered a minor cut to your right arm.\"\n\nCaesar had not asked me a question, so I did not respond.\n\nHe took the second _tabula_ and opened it. \"This is a confidential command report to me from my _praefectus castrorum_ . . . very curious . . . It says the slave was not a slave, but a Roman _grassator_ , a street gangster . . . one of Milo's _collegium_ . . . Milo's a major player in the Roman underworld . . . What could a legionary grunt in _Gallia Cisalpina_ have done to get such attention from a Roman gangster? . . . Do you have any idea, Insubrecus?\"\n\nI didn't know how to respond. Labienus warned me against talking too much. The question was straightforward, so I just said, \" _Non cognosco_ , _Imperator_!\"\n\nCaesar nodded slightly and kept reading. \"There's some nonsense here about the involvement of the consul's family . . . Hmmm . . . ah . . . Here it is . . . This has been sent to me because you claim to be a client of my family, the _gens Iulia_ . . . Would you be so kind as to explain that, Insubrecus?\"\n\nI did. I explained my grandfather's involvement with Gaius Marius, the granting of citizenship, the awarding of the farm, and Caesar Senior explaining to my grandfather that Marius' _patrocinium_ had reverted to his family, the _gens Iulia_.\n\nCaesar listened and said, \"Yes . . . I do recall some stories about Marius and your grandfather . . . He was a Gallic cavalryman . . . quite colorful, if the stories are accurate . . . practically a part of Marius' _familia_ . . . Marius swore _Bona Fortuna_ always smiled on the man . . . So . . . you're his grandson?\"\n\nCaesar looked over at Labienus, \"We have anything for this trooper on the staff?\"\n\nLabienus asked me, \"Can you ride, soldier?\"\n\n\" _Possum_!\" I responded without turning my head away from Caesar.\n\n\"We could mount him as part of the mounted praetorian detail, Caesar,\" Labienus stated.\n\n\"Yes . . . we could,\" Caesar started. Then, he asked me, \"Can you read and write Latin, Insubrecus?\"\n\n\" _Possum, Imperator_ ,\" I responded.\n\n\"M\u03c0o\u03c1\u03b5\u03af\u03c4\u03b5 V\u03b1 \u03b4\u03b9\u03b1\u03b2\u03ac\u03c3\u03b5\u03c4\u03b5 E\u03bb\u03bb\u03b7V\u03b9K\u03ac?\" Caesar asked abruptly.\n\nIt took my mind a heartbeat to adjust. He was asking if I could read Greek. \"\u2206\u03b9\u03b1\u03b2\u03ac, \u03c3\u03c4\u03c1\u03b1\u03c4\u03b7\u03b3\u03b5,\" I stammered. \"Yes, General!\"\n\n\"Interesting . . . Do you have any more hidden talents I should know about, Insubrecus?\" Caesar asked.\n\n\"Yes, sir,\" I responded. \"I speak Gah'el, I mean, _lingua Gallorum_.\"\n\nCaesar spoke to his aide, \"Labienus, I think we have more here than just another sword on a horse . . . Go ahead and assign him to my praetorian cavalry _turma_ for administrative purposes, but I think I may have a further use for this young man here . . . When are those Gallic chiefs supposed to get here?\"\n\n\"They're already here, Caesar,\" Labienus responded. \"I have them cooling their heels, waiting for an audience. I like to build up the anticipation a bit . . . Good theater . . . even better politics.\"\n\n\"Splendid!\" Caesar said. \"Erect an audience tent just outside the _Porta Praetoria_. I will receive these petitioners tomorrow at the first hour, the time when clients traditionally pay their obligatory call on their patron. I hope they understand the significance.\"\n\n\" _A'mperi'tu', Caesar_ ,\" Labienus nodded. \"Refreshments?\"\n\n\"No . . . I don't think so,\" Caesar stated. \"I don't want them feeling too comfortable . . . Not yet, anyway . . . And you, Insubrecus . . . you will be part of my security detail at that meeting . . . No one knows you understand . . . what did you call it . . . 'Gah El'? See what you can pick up.\"\n\n\" _A'mperi'tu', Imperator_!\" I snapped.\n\n\"As far as this other issue is concerned,\" Caesar continued, \"I doubt this attack was aimed at me. Our connection is too remote. But, one can never be too careful. You are now _sub patrocinio Caesaris_. Outside this tent, you will address me with full military courtesy, but here you may address me as _patrone_. Understood?\"\n\n\" _Compre'endo, Imper . . . Patrone_!\" I corrected myself.\n\n\" _Bene_!\" Caesar concluded. \"Let's see how this encounter works out tomorrow. I may have a special task for you, Insubrecus. But, for now, let me get back to this pile of administrative _merda_ that's piled up on my desk. Labienus! Get young Insubrecus squared away.\"\n\nThe interview was concluded. I had no idea at the time, but that was one of the most significant moments in my life. From that moment on, I have been _sub patrocinio gentium Iuliarum Caesarum_ \u2014 _pro bono et pro malo_ , for the good and for the bad.\n\n# VI.\n\n# _De Consequente Helvetiorum_\n\n## PURSUIT OF THE HELVETIANS\n\n_Helvetii iam per angustias et fines Sequanorum suas copias traduxerant et in Haeduorum fines pervenerant eorumque agros populabantur Haedui cum se suaque ab iis defendere non possent legatos ad Caesarem mittunt rogatum auxilium_.\n\n\"The Helvetians had already led their forces through the defile and into the territory of the Sequani. They then invaded the territory of the Aedui and were ravaging their lands. Since the Aedui could not defend themselves and their possessions from the Helvetians, they sent envoys to Caesar and requested assistance.\"\n\n\u2014(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nThen I got back to my _contubernium_ in the Tenth Legion, I quickly found out I had no unit to leave. Bantus informed me that our tent squad was to be broken up and assigned around the legion as individual replacements. When I informed Bantus of what had happened to me, he shook my hand and wished me well. He promised that our _signifer_ would forward my pay records and allotments to my new unit.\n\nBantus then shared with me that Strabo was recovering. Bantus had been over to the medics, and Spina had assured him that, under his professional and expert care, our centurion was going to make a full recovery and would be back on duty in less than a week.\n\nBantus had told Strabo what I had done. After making sure that Bantus had disciplined me for breaking ranks, Strabo said, \"Tell the kid thanks. I owe him.\"\n\nI packed my personal gear up, shook hands all around, strapped my shield to my back, threw my _furca_ over my shoulder, and trudged back to the camp of the Seventh Legion to report to Labienus.\n\nWhen I got back to Caesar's headquarters, one of Labienus' assistants walked me over to where the praetorian _turma_ , the cavalry troop, was housed. I was shown a bunk to dump off my kit. Then, I was taken over to report to my new officer, a _decurio_ named Decimus Lampronius, who had been in the cavalry so long that everyone called him _Valgus_ , \"Bow Legs.\"\n\nValgus looked me up and down, then asked, \"Can you ride?\"\n\nWhen I answered yes, Valgus just grunted and said, \"We'll see. Get that infantry gear off, and I'll take you down to the horse line.\"\n\nI dropped my _gladius_ and _galea_ , and Valgus helped me out of my _lorica_. We walked down to the horse line, and Valgus picked out a horse for me, a black mare with a white snip. The stable slave bridled and saddled the horse for me.\n\n\"The old man likes his praetorian escort all mounted on black horses . . . looks smart,\" Valgus explained. \"Mount up, and walk him around the corral a couple of times . . . Get him warmed up.\"\n\nI hopped up and walked the horse around the corral. Valgus had me execute some turns. Then, he called out, \"Trot!\"\n\nI dug my heels into the horse's sides, and it responded immediately. Valgus watched us for a while, then commanded, \"Canter!\"\n\nWe picked up speed, making a few circuits around the corral until Valgus called, \"Come on over here, Insubrecus!\"\n\nI complied and dismounted in front of him.\n\n\"You got a good seat,\" Valgus told me. \"I'm going to have you work with our training officer until he's satisfied that you're ready for tactical training with the _turma_. My understanding is that you're on _immunis_ status to the old man's headquarters, so that will have to take priority. Meanwhile, I'll walk you over to the armory to draw a _spatha_ , a cavalry sword. It's longer than that infantry pigsticker you have now. Keep both. The _gladius_ comes in handy if we have to fight dismounted. Most guys hang the _spatha_ from the horn on their saddle and keep the _gladius_ on their bodies. If you take a tumble, it's nice to know one of your swords stays attached to you. We also have to get you a _parma_ , a cavalry shield. We'll send that table top you're carrying back to your former legion. I see you already have one of the new _galea_ with the long neck guard . . . That'll work just fine in the cavalry . . . so keep that.\"\n\nValgus and I walked back to the headquarters tent of the praetorian detail. Before he dismissed me, Valgus handed me a _cingulum purpureum_ , a thin, purple sash.\n\n\"Tie that around your waist,\" he said. \"Wear it outside your _lorica_ when you're on duty . . . It indicates who you work for . . . Even centurions won't screw around with you when they see that sash.\"\n\nWhen I got back to my tent, I discovered one of the benefits of being an _immunis_ praetorian. No guard duty. For the first time since I joined the army, I slept through the entire night.\n\nI was awakened in the dark by someone kicking my cot. I grunted, \" _Qu'accidit_?\" I then heard a voice out of the darkness, \"You, Insubrecus?\"\n\n\"Yeah . . . that's me,\" I answered, sitting up.\n\n\"You're on the general's security detail this morning,\" the voice explained. \"You need to report to _il'capu, stat'_.\"\n\n\" _Il'capu_?\" I questioned.\n\n\" _Il'capu_ . . . the boss . . . Labienus _Legatus_ ,\" came the answer out of the gloom. \" _Stat'_!\"\n\nI got to the _praetorium_ just as the trumpet signaling the end of the fourth watch sounded. Labienus was standing in the outer area in full kit\u2014officer's _lorica_ , plumed helmet under his arm.\n\nWhen Labienus saw me, he said, \"Ah, Insubrecus . . . _bene_!\"\n\nHe put his arm around my shoulders and guided me to a quiet corner. \"Your job this morning isn't security . . . We have enough goons in purple sashes to take care of that . . . When these Gauls show up, just watch and listen . . . See what you can pick up . . . anything to help the general understand what these people are really up to . . . Got it?\"\n\n\" _Compre'endo, Legate_ ,\" I told him.\n\n\"Good lad,\" he said, patting my back. \" _Bene_! Let's get this circus started, boys!\"\n\nLabienus ducked into his cubicle. I could hear some muffled conversation back there. Then, Caesar _Imperator_ appeared in full regalia, with a bright red _sagum_ over his shoulders. Someone called the room to attention, but Caesar immediately put us at ease.\n\n\" _Bene_! We're ready,\" he stated. \"Let's move out, Labienus, and see what these Gauls want.\"\n\nWhen Caesar spotted me, he gestured me over to him. \"Ah . . . Insubrecus . . . Walk with me.\"\n\nWe proceeded up the _Via Principalis_ toward the _Porta Principalis Sinistra_. I was surprised how some of the legionary squaddies treated the general with calls like, \"Give 'im hell, _Calve_ , Baldy!\" and \"Remember, you get the gold, and we get the women, Boss!\" Caesar just smiled, nodded at some of the callers, and even waved at a few.\n\n\"This may sound strange to you, Insubrecus,\" he shared with me, \"but this is how the boys show they like me. If they said nothing as I passed . . . just stayed silent and sullen . . . I'd be in big trouble.\"\n\nWhen we finally wound our way through the gate, I saw a large pavilion erected in the field. Underneath the tent, Caesar's _sella curulis_ , his curule seat, indicating his possession of the _imperium_ of the Senate and the Roman people as the proconsul of the _provincia_ , stood alone. In the field beyond the pavilion, I could see a group of men and horses being guarded by a detail of praetorians.\n\nOur guests had already arrived.\n\nI heard Caesar say to Labienus, \"They're on time . . . They must consider their business with me urgent . . . _Bene_ . . . Let's take our time with this . . . Delay is to our advantage . . . whoever seems to have the least to lose, wins.\"\n\nCaesar walked slowly to his chair and carefully arranged his red cloak as he took the seat. He inclined his head in my direction and whispered, \" _Observa et ausculta_! Anything you can pick up will be helpful.\"\n\nCaesar then inclined his head toward the commander of the security detail and gestured that he should let the Gallic delegation approach.\n\nTo me, these approaching Gauls were like Gallic warrior-heroes, emerging from one of Gran'pa's tales of our heroic past. They were huge men, giants, each well over six _pedes_ in height, barrel chested, and broad shouldered with heavily muscled arms. They seemed to overshadow the members of Caesar's security detail, who themselves were selected for their intimidating physical presence. Most of the praetorians barely measured up to shoulder height on these men. How could we Romans fight such men? They had the stature of gods.\n\nThey approached the seat of Caesar with long, confident strides. They all wore luxuriant, thick moustaches reaching down to their chins: some red, some blond, others black. They approached Caesar's seat bareheaded. Some kept their long hair wild and unbound; others had braids interwoven with brightly colored ribbons reaching down their backs. Under his left arm, each warrior carried a bronze helmet, brightly plumed with horsehair and feathers.\n\nThey walked in a flurry of colors, with capes, sashes, and trousers of garish red, verdant green, and deep blue. On their left shoulders, the men wore large, ornate, golden _fibula_ pins that secured the brightly colored cloaks, which seemed to flutter behind them as they strode forward. Around their necks, the warriors wore thick, golden torcs. I remembered Gran'pa telling me that the kings and chiefs of the Gah'el wore their wealth into battle to attract worthy opponents to combat.\n\nTheir bright chainmail cuirasses reached down to their knees and were secured by wide leather belts adorned with gold and silver decorations. On their left sides, they wore long, Gallic swords suspended from ornately embellished baldrics.\n\nTheir groupings and tartan designs indicated three distinct factions approaching Caesar's chair. When they were about three paces from the pavilion, they stopped. The leading warrior from a group of four men in blue and green plaid livery stepped forward to address the general. He held up an ornately carved wand of white wood for Caesar to see. From Gran'pa's stories, I knew it to be a wand of negotiation. It indicated that the warrior came to speak, not to fight. It held the holder inviolate at the cost of five times his head price.\n\n\" _Ave_ , Caesar,\" the speaker began in accented Latin, \"I am Duuhruhda mab Clethguuhno, tribal king of the People of the Goddess of the Dark Moon, Aine Du, the Aedui to you Romans.\"\n\nAs the king spoke, I examined his entourage. Each man carried the white wand to indicate his diplomatic status. One, by his bearing and the richness of his equipment, was obviously a noble of some standing within the tribe. He had a natural sneer to his smile, and his eyes were what Mama would call \"shifty.\" There was something about him that screamed deceit. I took an instant dislike to the man. The other men carried wooden staffs. These were the king's _drui_ , I realized, one for the laws of the gods and the other for the laws of the people.\n\nThe king was addressing Caesar, reminding him of the bonds of friendship that existed between the Aedui and the Roman people and of how Caesar's refusal to let the Helvetii cross the Rhodanus at Gennava had sent the Helvetii into the lands of the Aedui and of how hordes of Helvetii were now stripping the fields and storehouses of the Aedui.\n\nI decided there was nothing to be gleaned by my listening to a set diplomatic speech in halting Latin, so I eased over to where the Gauls had left their horses. There, the entourage of the Gallic chiefs was gathered. They were the bodyguards of the kings, the _gwarchodourai_. They weren't listening to the king's speech. I doubt any of them understood Latin. I got as close as I could without arousing their suspicion. Surprisingly, I could follow most of what they were saying to each other.\n\nMost were joking about the Romans in Caesar's security detail: Romans wore skirts like women; they looked like beardless boys; such small men could never satisfy a woman.\n\nThen, I heard one comment on how the king of the Aedui went on and on with his speech like a woman complaining to her neighbors about her monthly cramps.\n\nHis companion, one of the Aedui by his tartan, added that the king was trying to lure the Romans across the river.\n\nHis companion asked why, what purpose did having Romans in lands of the Aedui serve?\n\nThe Aeduan snorted, \"The enemy of our enemy is our friend! If the Romans fight the Helvetii, regardless of who wins, there will be fewer Romans and fewer Helvetii for the Aedui to kill.\"\n\nThen, another of the Aedui said, \"This 'Caisar' of the Romans . . . his own people hate him . . . They have paid our king handsomely . . . in Roman gold and silver . . . to lure him to his destruction beyond the lands of the Romans.\"\n\nThe first Aeduan hushed his companion. Then, he looked around and spotted me hovering close by. That was enough to silence the men. I knew I would get no more from them.\n\nWhen I wandered back to where I could hear the deputation, one of the warriors, a dark-haired man in a red tartan was saying, \"Our fields should not be destroyed; our children should not be carried off into slavery; and our towns should not be assaulted when your army is within our sight. It is not possible for us to defend our people from the onslaught of so large a host. Soon, we will have nothing left except the dust of our fields.\"\n\nWhen the warrior was finished speaking, Caesar rose from his _sella curulis_ and addressed the assembly, \"Caesar has heard the complaints and the entreaties of the allies and friends of the Roman people. You will have my answer in the morning!\"\n\nWith that, Caesar turned and walked back toward the gate of the _castrum_. Labienus and I fell in behind him. We followed him back to his quarters in the _praetorium_. He tossed his helmet on his table and ordered his body slave to assist him in removing his cuirass.\n\nWhile this was happening, he asked Labienus, \"Where do our _exploratores_ place the enemy?\"\n\nLabienus walked over to one of the maps and pointed to a spot near a squiggly blue line. \"They're here, moving west toward this river, the Arar, and taking their time about it. The scouts say that at these points, they're crossing the river on boats and rafts.\"\n\nCaesar, now free of his armor, walked over to the map and asked, \"How long before they get across the river?\"\n\n\"Late tomorrow . . . maybe the next day,\" Labienus shrugged.\n\n\"What's here?\" Caesar asked, pointing to where two blue lines converged.\n\n\"That's the confluence of the Arar and the Rhodanus,\" Labienus answered. \"There's a settlement there on our side of the river . . . an _oppidum_ . . . a fortified town of the _Sequani_ . . . This symbol indicates that there's a bridge across the Rhodanus at this point . . . Looks to be no more than four, maybe five, thousand _passus_ south of where the Helvetii are crossing the Arar.\"\n\n\"Perfect!\" Caesar said. \"Absolutely perfect . . . issue orders . . . three legions . . . the Seventh, Ninth, and Tenth . . . We pull out at midnight . . . We march straight for that bridge . . . Then, we move to catch the enemy at those fords . . . We should catch them divided by the river . . . We can chew up their rear guard, and the rest won't be able to support them . . . Send out an engineering detail with cavalry support to check that bridge . . . I want it capable of supporting our crossing when we get there . . . What is it, Labienus?\"\n\n\"Caesar, when you cross the Rhodanus, you leave the _imperium Romanum_ . . . Without the authorization of the Senate, your command does not reach into _Gallia_ ,\" Labienus cautioned.\n\n\"Don't be such an old lady, Labienus,\" Caesar dismissed his counsel. \" _Audacibus favet Fortuna_! . . . Fortune favors the bold! You keep this up, and I'll have to appoint you to the Senate with the rest of those old women . . . The Helvetii are a clear and imminent threat to our _provincia_ , and our _pietas_ demands that we support our friends and allies on the other side of that river . . . We may not get another opportunity like this.\"\n\n\"Yes, Caesar,\" Labienus conceded.\n\n\" _Bene_! Issue the orders!\" Caesar continued. \"Have the _quaester_ visit our Gallic friends in the morning, after we've pulled out . . . We will be out-marching our supplies . . . We'll have them feed our army while we're destroying their enemies . . . Fair trade, I think.\"\n\nI saw a thought flitter across Caesar's eyes. He looked over to where I was standing. \"Insubrecus . . . _veni_!\"\n\n\"Yes, _Patrone_ ,\" I responded.\n\n\"I saw you near that Gallic bunch over by the horses,\" he said. \"Did you pick up anything useful?\"\n\n\"The Aedui want us to exhaust ourselves against the Helvetii so they will have an easier time dealing with us,\" I offered.\n\n\"That makes sense,\" Caesar agreed. \"Anything else?\"\n\nI thought of how best to tell Caesar the other news. Then, I remembered Labienus' advice: be direct. \" _Patrone_ , they were saying that Romans paid their king to entice you to go over the Rhodanus. They want the Helvetii to destroy you and this army.\"\n\nCaesar looked at me for a few heartbeats, then asked, \"Are you sure you heard that part correctly?\"\n\n\" _Recte audivi, Patrone_ ,\" I answered. \"Yes, I did!\"\n\nCaesar shrugged, \"No matter . . . The Helvetii are not going to destroy this army . . . _Bene gesta_ . . . Well done, Insubrecus.\" He patted me on the shoulder.\n\nThen, he turned to Labienus, \"Issue the orders . . . We move out at midnight . . . three legions!\"\n\nWe did move out at midnight. My legion, the Tenth, was the vanguard, followed by the Seventh, then the Ninth. I rode with Caesar's praetorian detail near the head of the column. Valgus, my _decurio_ , still had no confidence in my riding abilities and told me just to stay close to him during the march.\n\nOnce we cleared the hills around Leminco, our progress was swift. Caesar wanted to reach the bridge at the Rhodanus as quickly as possible, but he didn't want to exhaust the troops doing it. He expected a battle at the end of the march.\n\nBy dawn, the army had fourteen thousand _passus_ under its boots, and it was marching down a valley past a village called Laviscone. With the sun up, we picked up the pace. To keep them fresh along the route of the march, Valgus had us walking the horses for a thousand _passus_ , then riding for a thousand.\n\nWe turned north. By the fifth hour, we were another fifteen thousand _passus_ down the road, passing a pisshole of a place with the grandiose name of Aosta Salassorum, Aosta of the Salassi tribe, where we turned west and followed a river up another valley.\n\nBy the ninth hour, we were close to forty thousand _passus_ out of Leminco, about five thousand _passus_ east of Bergusium. Despite wanting to press on, even Caesar realized he had to halt or his army would be too clapped out to fight.\n\nWhen he ordered the halt, the Tenth Legion, the First Cohort boys in the van, started chanting, \"Ten thousand more! Ten thousand more!\"\n\nBut, the general yelled back, \"Save some of that for the barbarians!\"\n\n\"We got plenty!\" the boys in the Tenth yelled back.\n\n\"I'm sure you do, _m'infantes_!\" Caesar answered them. \"But, I'm an old man and need my rest!\"\n\n\"Then, we'll stop for you, Calve!\" some wag yelled back.\n\nCaesar rendered a dramatic gesture of thanks to the men in the vanguard. Then, I heard him say to Labienus, \"Bring those _exploratores_ to me . . . _stat'_!\"\n\nCaesar called, \"Insubrecus! With me!\" Then, he rode to where a group of legionary slaves were already erecting his headquarters tent.\n\nWhen we dismounted, I saw Labienus leading a motley group of Gallic horsemen to Caesar. Caesar said, \"Insubrecus . . . talk to those people . . . Find out where the enemy is.\"\n\nI approached the group and spoke in Gallic to the one I took to be their leader, \" _Uh prif duhmuno gweebod bleh mae'r Helvetii uhn cael eu_?\"\n\nI saw a look of surprise pass across the face of the rider I approached. \"The Roman puppy speaks our tongue!\" he said.\n\n\"Puppies don't kill. I have,\" I told him. \"Now, answer the question!\"\n\nThe man gave me a long stare, then said, \"You are right. I am being rude. I am Athauhnu mab Hergest of the People of Soucana, a Leader of Ten. To whom do I have the pleasure of speaking?\"\n\n\"The Romans call me Gai,\" I answered him. \"Among my people, the Insubreci, I am called Arth Bek, Wuhr Cunorud, _mae milour Rhufeinig_ , a soldier of Rome.\"\n\n\"Well, Arth Bek, Grandson of the Red Hound,\" Athauhnu continued, \"the Helvetii dogs have reached Soucana's river at the valley of the white pines. That is half a day's ride north from the Dun of Lugus. There are no fords in that place, so they are crossing on boats.\"\n\nI turned to Caesar, \"The enemy has reached the Arar about fifteen thousand _passus_ north of a place called Lugdunum . . . They have begun to cross on boats.\"\n\n\"Ask him how long it will take the Helvetii to cross!\" Caesar snapped.\n\n\" _Pah mor hir uh buhth uhn ei guhmruhd ar guhfer uh Helvetii i groesi'r afon_?\" I translated.\n\nI watched Athauhnu calculate in his head. \"Three . . . maybe four days,\" he guessed. \"They have women and children . . . and much baggage.\"\n\n\"They're loaded down,\" I told Caesar. \"He estimates three or four days.\"\n\nThen, Athauhnu said, \"Tell the Roman chief that their foraging parties are across the Rotonos, and they have _Almaenwuhra_ with them.\"\n\nCaesar caught the word _Rotonos_ , and asked, \"What did he say about the Rhodanus?\"\n\n\"There are Helvetian raiding parties on our side of the river,\" I told him. \"And, he says they have Germans with them.\"\n\n\"Germans?\" Caesar said. \"How many?\"\n\n\" _Faint o Almaenwuhra_?\" I asked.\n\nAthauhnu shrugged, \"Not many . . . not few.\"\n\n\"He doesn't know,\" I told Caesar.\n\nI saw Caesar considering this response. He decided not to press it. \"Tell him thanks, Insubrecus. Tell him he's welcome to food\u2014and water and feed for his horses.\"\n\nI passed on Caesar's gratitude, but then Athauhnu asked me, \"Does the Roman chief intend to continue on the Roman road all the way to the dun of the Allobroges on the Rotonos?\"\n\nI wasn't sure that I should share our route of march with the man, so I asked, \"Why do you want to know that?\"\n\n\"The Helvetii are burning our farms, raping our women, and murdering our children,\" he said. \"If the Roman chief marches on the Roman road to the Rotonos, he is going out of his way . . . He is giving those Helvetian pigs more time to kill my people . . . There is a shorter route direct to the Dun of Lugus . . . The road is not as good . . . but it will cut almost a day out of the journey.\"\n\n\" _Imperator_ ,\" I called to Caesar, \"this man says he knows a shorter route to the bridge over the Rhodanus at Lugdunum than through Vigenna . . . He says our army can pass over it, and it will cut almost a day out of the march.\"\n\nThis immediately got Caesar's attention. \"This man is familiar with this road?\" he asked me.\n\n\" _Noscit, Imperator_ ,\" I replied. \"Yes, sir!\"\n\n\" _Bene_!\" Caesar said. Then, \"Labienus . . . the Tenth's cavalry is up with us . . . Tell their _primus pilus_ I need a _turma_ for a route reconnaissance . . . _Stat'_!\"\n\nThen to me, \"Insubrecus, ask your man there if he will show us the way.\"\n\n\" _Uhn eich arwain ein marchoglu dros uh fhorth_?\" I asked Athauhnu.\n\nAthauhnu shrugged, \"Yes . . . of course . . . just give us some time to rest and water our horses.\"\n\n\"As soon as their horses are ready, _Imperator_ ,\" I told Caesar.\n\nWe were on the road by the eleventh hour. Caesar sent me along with the _turma_ from the Tenth Legion because none of the Romans spoke Gallic. There were twenty-five Roman cavalry troopers commanded by a _decurio_ called Rubigo, \"Rusty,\" because of his red hair.\n\nAbout a thousand _passus_ east of Bergusium, Athauhnu led us north and west away from the Roman road. We rode almost two thousand _passus_ , passed through a narrow valley, and then found ourselves following a small river down a broad valley that the army would have no problem marching through.\n\nAround sundown, we had to climb up through a narrow pass from the valley floor. At worst, the army could pass through it marching two abreast. When we climbed the valley, we found ourselves riding across a broad plain, which would offer no obstacle to our march.\n\nWe rode for what seemed like hours. The young moon rose, dimly lighting our way. Soon, off to the north and east, we could see what seemed to be bonfires burning in the night.\n\nAthauhnu halted our march.\n\n\"It's worse now,\" he told me. \"Those are the settlements of my people . . . The Helvetii and those German pigs are burning everything.\"\n\n\"Ask him how much farther!\" Rubigo interrupted.\n\nI translated for Athauhnu. He shrugged, \"Not much farther. The south branch of the Rotonos is to our front. The bridge your Caisar wants is to the northwest . . . It is downstream from where the branches of the river join together and just below the bluff where the dun of my people sits.\"\n\n\"It's not much farther to the northwest,\" I told Rubigo.\n\n\"All that jabber to say 'not much farther'?\" the _decurio_ complained. \"These bloody wogs love the sound of their own voices.\"\n\nI didn't think it appropriate to point out to Rubigo that I was one of those \"wogs.\"\n\n\" _Da_!\" I told Athauhnu. \"Good! Let's go!\"\n\nIn less than an hour, we were on a hill overlooking the Rhodanus. Below us, we could make out the bridge spanning the river and the fires of the Roman engineers Caesar had sent out to strengthen it. On the other side of the Rhodanus, on a facing bluff, the torches set on the ramparts of the Dun of Lugus, the seat of the Sequani, were visible.\n\n\"We rest the horses,\" Rubigo said. \"Then, we get back to the army. The wog's right. We can be on the banks of the river by tomorrow night.\"\n\n\" _Beth a wnaeth uh un coch uhn ei thweud?_ \" Athauhnu asked. \"What did the Red One say?\"\n\n\"He said thanks,\" I told Athauhnu.\n\n\"So many words to say thanks,\" Athauhnu commented shaking his head. \"These Romans wag their tongues like old women.\"\n\nWe rode as hard as our tired horses could tolerate to get back to the army. The fires in the north and east seemed to have gone out, leading me to hope that the enemy had withdrawn its raiding parties back across the river.\n\n_Erratum_. I was wrong.\n\nWe were no more than ten thousand _passus_ back from the hilltop where we saw the bridge when we literally collided with another group of horsemen in the dark. A rider, no more than a pace or two to my left, grunted, \" _Hwa gange \u00f0\u00e6r_?\"\n\nI didn't understand what he said, but immediately the image of a blond giant guarding the blue doorway of a _lupinarium_ in Mediolanum appeared in my mind.\n\n\" _Germani_!\" I yelled drawing my gladius. \"Germans!\"\n\nI pulled my horse's head hard to the left into the German rider. He was slower to react than I was. I felt my gladius bite home; where, I wasn't sure. The rider grunted and went down.\n\nThe rest of the fight was a mad free-for-all in the dark. I could sense, more than see, figures swirling around me; I heard the sounds of steel on steel, the grunts of the wounded. Then, there was silence\u2014silence except for the sound of a woman weeping somewhere in the dark.\n\nI heard Rubigo call, \"Insubrecus! _Ub'es tu?_ \"\n\n\" _Adsum_!\" I responded.\n\nI felt him ride up next to me. \"Thank the gods! The general would have my hide if those Krauts got to you.\"\n\n\"I hear someone crying,\" I said.\n\n\"Prisoners,\" Rubigo explained. \"Those Kraut _mentulae_ were dragging their captives along with them. Your wog buddies are seeing to them. Tell them to hurry. We have to get back to the army.\"\n\nI dismounted and found Athauhnu. He was with a cluster of his men and the freed captives.\n\n\"Young women and a few young boys,\" he spat. \"They're from a settlement near the south branch of the Rotonos. The German pigs burned everything... killed everyone else except these few... they have value as slaves...\"\n\n\"The Red One wants to get back to the Romans,\" I said. \"How soon can you move?\"\n\n\"He's right,\" Athauhnu agreed. \"The faster the Caisar arrives with his warriors, the faster these pigs will flee from our lands... I will send two of my men to bring these captives to the Dun of Lugus... then we will go find your army.\"\n\nWe were quickly back on the road. We arrived at the site of the Roman _castra_ during the fourth watch, about an hour before dawn. The army was already awake and pulling down their fortifications. We found Caesar and the command group with the Tenth Legion.\n\nRubigo reported to the _Imperator_. \"The route is open all the way to the bridge at Lugdunum, maybe twenty, twenty-one-thousand _passus_ ,\" he said. \"One narrow spot along the way but the rest is wide open, dry and fairly flat. The enemy has patrols and raiding parties in the area, nothing big enough to threaten the army.\"\n\nCaesar was silent for a few heartbeats. Then, he responded, \" _Bene gestum_ , _Decurio_! Well done! Stand your men down until the third hour, then follow the army through to the bridge.\"\n\nCaesar turned to Labienus, \"Titus, collect two _turmae_ of cavalry from the Seventh and Ninth and one from the Tenth and have them report here at the head of the column!\"\n\n\" _A'mperi'tu', Imperator_ ,\" Labienus responded. \"Yes, General.\"\n\n\"Rubigo, I have a favor to ask of you,\" Caesar said to the cavalry _decurio_.\n\n\"Anything, General,\" Rubigo responded.\n\n\"I know you're tired, lad, but you know the route up to the bridge,\" Caesar told him. \"So, I need you to get a fresh horse and guide my cavalry detail along the route ahead of the army.\"\n\n\" _A'mperi'tu', Imperator_!\" Rubigo snapped.\n\nThen Caesar called over to a group of officers clustered about three paces off, \"Pulcher! I need you!\"\n\n\" _Quid vis tu, Caesar_?\" the man responded walking over to where we were. \"What do you want, Caesar?\"\n\n\"Pulcher, I'm giving you command of five _turmae_ of legionary cavalry,\" Caesar instructed the officer. \"Your mission is twofold. First, mark the route of march for the army. Have your troopers collect some of the markers the engineers use to survey the marching camps, and use them to mark the route. Is that clear?\"\n\n\" _Compre'endo_!\" Pulcher responded.\n\n\" _Bene_ ,\" Caesar continued. \"The redheaded _decurio_ from the Tenth . . . Rubigo's his name . . . he knows the route. Second, I want you to screen our route of march . . . Keep those barbarian _cunni_ off our tails as we move up to the river . . . The threat is basically from the north and east along the route, but don't be surprised if you run into stragglers . . . Grab two _tribuni_ to assist you\u2014a couple of those fuzz-faced _angusticlavi_ attached to the Tenth who don't seem to know what to do with themselves . . . I doubt the men will miss them . . . Any questions?\"\n\n\" _N'abeo_ ,\" Pulcher answered.\n\n\"Good man!\" Caesar encouraged him. \"As soon as the cavalry _turmae_ are assembled, brief them, and get on the road!\"\n\n\" _A'mperi'tu', Caesar_ ,\" Pulcher said and walked over to where Labienus was assembling the cavalry detail.\n\nI heard Caesar mutter at the man's back, \"'Caesar,' he calls me . . . in front of the troops . . . like we were a couple of housewives discussing the price of fish in the middle of the forum . . . _Iste pedans mentula Claudianarum_!\"\n\nThen, he noticed me standing there, \"Ah! Insubrecus! Enjoy the ride?\"\n\n\"Yes, _Imperator_ ,\" I started, \"at least most of it. We ran into a German raiding party on the way back.\"\n\n\"Germans?\" Caesar questioned. \"How many? How'd you know they were Krauts?\"\n\n\"They spoke German, _Imperator_ ,\" I explained. \"We counted eight bodies when we were done . . . Don't think any of them got away.\"\n\n\"Germans on this side of the Rhenus?\" Caesar mused. \"Not good . . . not good at all . . . What do you make of our Gallic friends?\"\n\n\"They're on our side as long as we're fighting their enemies,\" I told him. \" _Hostes hostium meorum\u2014_ \"\n\n\" _Amici mei_ ,\" Caesar finished my sentence. \"The enemies of my enemies are my friends. . . . Best we could hope for . . . As long as we can, keep the tribes divided _\u2014_ \" Caesar stopped himself from continuing that thought, then asked me, \"Think you can stay in the saddle for another day?\"\n\n\" _Possum_ , _Imperator_ ,\" I agreed.\n\n\"Good lad!\" Caesar said. \"Have Valgus find you a fresh horse. I want to be on the Rhodanus before nightfall.\"\n\nAnd, we were. The van of our army was on the river and digging in by the seventh hour. Our engineers had reinforced and widened the bridge so that we could cross quickly at any time.\n\nA delegation of brightly attired Sequani crossed the bridge from Lugdunum. We were quickly informed that the main body of the Helvetii were still crossing the Arar about fifteen thousand _passus_ to the north. The Sequani offered to lead us to the site.\n\nI believe Caesar was tempted to move the army right out, but he knew that the soldiers would be too exhausted to give battle when they arrived. So, he decided to rest them in the relative security offered by the bluffs on the south side of the Rhodanus. He did send a detachment of legionary cavalry ahead with the Sequani to locate the enemy, survey the ground, and find a route of march for the infantry.\n\nI was so exhausted that my body ached. As soon as I could square away my mount at the horse stables, I found a quiet place in the back of the praetorians' tent to sleep. I felt as if I had just shut my eyes when I felt someone kicking my foot.\n\n\"Whaa . . . _qu'accidit_?\" I mumbled still half asleep.\n\n\"Rise and shine, sunshine!\" It was Valgus. \"We're moving out.\"\n\n\" _Quot'orarum?_ \" I mumbled.\n\n\"Midnight,\" Valgus said, \"signal for the third watch just sounded . . . Move it, Insubrecus . . . The old man's looking for you!\"\n\nI dragged myself up to a sitting position. I felt as if I had taken a beating. Every muscle and joint in my body was screaming. Outside, I could hear the activity of an entire legion in motion. I knew any minute the tent I was in would be coming down on top of me. Still, I searched for my boots in the dark before I realized I hadn't taken them off before I fell asleep.\n\nWhen I finally stumbled over to where the _praetorium_ should have been, the tent was already down, but I soon spotted Labienus briefing some _angusticlavi_ tribunes. They looked like boys; I was beginning to have difficulty remembering that I was only sixteen myself.\n\nWhen Labienus saw me, he said, \"Give me a little time here, Insubrecus, and I'll be with you.\"\n\nThe meeting finally broke up with a mixed, \" _A'mperi'tu', Legate_ ,\" and Labienus walked over to where I was waiting. He threw his arm around my shoulder and steered me toward the horse lines.\n\n\"Big doin', Insubrecus!\" he started. \"Our _exploratores_ got back about an hour ago. The Helvetii left one of their septs . . . a subtribe really . . . stranded on the east side of the Arar. But, that's not the big news . . . The bunch they stranded are the Tigurini . . . You got any idea what that means?\"\n\n\" _Non cognosco_ , _Legate_ ,\" I answered.\n\n\"Didn't think so,\" Labienus continued. \"The old man . . . and every Roman in this army, for that matter . . . has a score to settle with those _cunni_ . . . About fifty years back, they slaughtered a Roman army . . . killed a consul . . . one of his legates, Lucius Calpurnius Piso Caesoninus . . . The old man is related to him through his wife . . . so this is personal . . . Caesar's _pietas_ demands he avenge Calpurnius and all those poor Roman _muli_ who were massacred by those long-haired Gauls. . . He pulls this off, and they'll be shouting Caesar's name in the forum for months!\"\n\nI wasn't sure how to react to this. _Pietas_. That was the same reason some Senator's brat was trying to have me killed. In this case, it meant an entire Roman army was crossing the Rhodanus to catch a bunch of Helvetii trapped on the wrong side of the Arar.\n\nWe arrived at the horse lines to find Caesar briefing his legates.\n\n\"Pulcher, you'll remain in command of the cavalry,\" Caesar was saying, \"three _turmae_ of legionary cavalry and a bunch of Sequani . . . They know the ground, so let them take the lead . . . You're to open the way for the infantry . . . Find a good place to engage the enemy . . . There's a lot of broken ground north of the river . . . forests . . . Find a place where I can deploy three legions . . . _acies triplex_ , if possible . . . the standard triple battle line . . . But do not allow yourself to become decisively engaged with the enemy . . . Is that clear?\"\n\n\" _Compre'endo, Imperator_ ,\" I heard Pulcher reply.\n\nI noted that he had used the title _imperator_. Caesar must have had a talk with \"that bloody Claudian mentula.\"\n\n\"Bene,\" Caesar continued, \"Cotta, you're commanding the Tenth Legion, the vanguard; Vatinius, the Seventh; Pedius, the Ninth. Crassus, you will continue to act as the army's _quaestor_. You will remain here until the rest of the army has come up with the supply train. Keep that bridge over the Rhodanus open in case the army needs it to withdraw south. Any questions, gentlemen?\"\n\nThere were none. Caesar spotted Labienus and me standing to the side.\n\n\"Get enough sleep, Insubrecus?\" he asked.\n\n\" _Satis superque, Imperator_ ,\" I lied. \"More than enough, sir!\" There is never enough sleep for a soldier.\n\n\" _Bene_!\" Caesar clapped me on the shoulder. \"I'm detailing you to Pulcher . . . He has about a hundred Sequani horsemen under his command and doesn't speak a word of their language. . . . You're to act as his liaison with the Gauls.\"\n\n\" _A'mperi'tu', Imperator_!\" I responded.\n\n\"Good lad!\" Caesar continued. \"Pulcher can be . . . well . . . a bit difficult. . . . He's real red-boot Patrician . . . and what's worse, a Claudian, and wants everyone to know it . . . I've let him know that you are _sub patrocinio meo_ . . . That should account for something. . . . I've given some thought to what you told me about the Gauls saying there are Romans working against me here . . . so I want you to watch and listen.\n\n\" _A'mperi'tu', Imperator_!\" I responded again.\n\n\"Good lad!\" Caesar said clapping me on the shoulder. Then, he was off to whatever needed his attention.\n\n\"My advice is to stay clear of Pulcher as much as possible,\" Labienus was saying. \"Even for a patrician, he's a right bastard and there's history between his clan and the _gens Iulia_ . . . But right now, Caesar needs the good will of his brother, Appius, in the Senate down in Rome. . . . If anyone were working to undermine the old man out here, he'd be my lead suspect.\"\n\nI had no idea what Labienus was talking about, other than Pulcher was a _podex_ , but a _podex_ whose goodwill Caesar wanted to maintain, and as Caesar's client, I shouldn't rock the boat.\n\nI found that one of the stable slaves had saddled my horse; it was the black with the white snip that Valgus had me ride previously. I checked the saddle and bridle. Then, I tied down my _spatha_ to the left side of my saddle and my _loculus_ , the leather bag with my field rations, to the rear. I tied down my woolen cloak to the front of my saddle. I didn't know if we would have to sleep rough. I mounted and adjusted my _parma_ , so it hung comfortably on my back. I walked my horse a bit, just to check the saddle again, then went off to find Pulcher and the command group.\n\nThey were gathering outside the camp's _fossa_. There were two groups, the Roman legionary cavalry and a group of Sequani horsemen who remained detached from the Romans. I spotted a group of Roman officers in the vicinity of Pulcher and rode toward them. Pulcher seemed remote, distant even, from his own officers; none of them seemed to be willing to get any closer to the patrician than ten _pedes_.\n\nI rode up to him and reported, \"Gaius Marius Insubrecus, _miles_ , reporting to the _legatus_ as ordered!\"\n\nPulcher gave me a long stare. His blue eyes were as cold as a frozen pond in _Ianuarius_ ; his face looked as if he had unexpectedly gotten a whiff of an overused latrine.\n\n\"Insubrecus,\" he said, seemingly without moving his thin, bloodless lips, \"you're Caesar's boy . . . the little Gaul he picked up.\"\n\nThen, Pulcher turned his head and called over to his officers, \"Agrippa! _A'veni_!\"\n\nA mounted tribune separated himself from the group and road over to us. \" _Ti' adsum, Legate_?\"\n\n\"Agrippa,\" Pulcher said, as if he were already bored with having to talk to underlings, \"this is Caesar's . . . Caesar's interpreter for the barbarians . . . Collect him up, and take your post.\"\n\nI noticed the narrow, purple stripe on Agrippa's tunic. He was an equestrian, an _angusticlavus_ , a narrow-striper.\n\nPulcher pulled his horse's head around abruptly, as if to escape the stench of social inferiors, and joined the knot of Roman officers. The tribune who rode up to me seemed to have an open, honest face, the kind which is trusted implicitly. He leaned out from his saddle and extended his right hand to me. \"Lucius Vipsanius Agrippa,\" he announced, \"just call me Agrippa. And, you are?\"\n\nHe spoke Latin with a wide, country twang, which told me he wasn't from Rome\u2014not the city anyway. I took his offered hand, \"Gaius Marius Insubrecus, _Trib_ . . . I mean, Agrippa.\"\n\nWe rode over to where the Gauls had congregated. Agrippa said, \"I'm a bit anxious about all this . . . Pulcher expects me to command this group . . . I don't expect they'll welcome my . . . my involvement.\"\n\n\"Let's see what mood they're in,\" I started. Then, I saw Athauhnu among the riders.\n\n\"Athauhnu,\" I greeted him in Gallic, \"it is good to see you again, my friend!\"\n\n\"Ah . . . Arth Bek,\" he responded sourly, \"did the Caisar send you and this other one to ensure that the people of Soucana would behave themselves in their own lands?\"\n\n\"No, my friend,\" I hedged, \"we are here solely to help with communication between the people of Soucana and their Roman allies.\"\n\nAthauhnu snorted at that response.\n\n\"What is he saying, Insubrecus?\" Agrippa tried to interrupt, but I held my hand up asking for his silence.\n\nAthauhnu continued, \"My people can be great liars when they feel the need, but none can equal you Romans . . . Come . . . You will need to speak to the _penn uh marchoglu_.\"\n\n\" _Penn uh marchoglu_?\" I stumbled. \"Oh . . . the troop commander . . . Yes . . . lead on, Athauhnu.\"\n\nAs Athauhnu turned to lead us, I said to Agrippa, \"He's taking us to their commander.\"\n\nAthauhnu led us to another rider. By his armor, equipment, and trappings, I assumed he was of the Sequani nobility.\n\nAthauhnu said merely, \"These are _our_ Romans!\"\n\nThe man gave us a long look, then responded in accented Latin, \" _Salvete_! I am Madog mab Guuhn. I am _rex gentium_ . . . people-king of Sequani . . . I am also _dux_ , leader of cavalry.\"\n\nI did not respond, but nodded to Agrippa, who said, \" _Salve_ , _Dux_. I am called Agrippa. I am the tribune of the Roman legate, Pulcher, the leader of the Roman cavalry.\"\n\nMadog nodded at Agrippa, then asked, \"And who is other with you?\"\n\n\"This is _contubernalis meus_ ,\" Agrippa responded. \"He is called Insubrecus.\"\n\nMadog said to me in Gallic, \"You speak our language. Of what people are you?\"\n\n\"I am of the Insubres, from over the Alps,\" I answered in Gallic. Then I repeated myself in Latin for Agrippa: \" _De Insubrecis trans Alpes_.\"\n\nMadog nodded. Then, Agrippa stated, \"The legate Pulcher requests that you lead us to the Helvetii.\"\n\nMadog seemed to have some difficulty following Agrippa, and I couldn't imagine Pulcher _requesting_ anything of a Gaul _\u2014_ or a Roman, for that matter. I was just about to translate, when Madog said, \"We go! But, I show you something . . . interest of Romans?\"\n\nMadog reached into his saddlebag, pulled something out, and offered it to Agrippa. Agrippa took it, examined it for a heartbeat, and handed it to me. It was a _quadriga_ , a denarius coin, a new one, hardly used, no nicks or shavings. It had a small hole driven through it.\n\n\"Madog,\" Agrippa asked, \"where did you get this?\"\n\n\"Helvetii,\" Madog responded. \"A dux of cavalry . . . He wear ten on neck . . . How do you say?\" Madog looked over to me and asked, \" _Sut ur udych uhn dwed 'muclis' un Ladine_?\"\n\n\" _Monile_ ,\" I told him. \"Necklace.\"\n\nMadog grunted, \"He wear necklace . . . I kill him . . . I take necklace.\"\n\nI looked closely at the coin. The one side had the expected image of the _quadrigae_ , the chariot team of four horses. I flipped it over and saw the image of the goddess Venus, the patroness of _gens Iulia_. Then, I realized that the coin was minted only last year, while Caesar was consul. How would a minor Helvetian chief have ten new denarii?\n\nI asked Madog, \"May I keep this?\"\n\nMadog shrugged, \"I gift you . . . We friends . . . I have more.\"\n\n\"Is this significant?\" Agrippa asked me.\n\n\"I'm not sure,\" I answered. \"It may fit in with something I overheard.\"\n\nJust then we heard Pulcher's voice, \"Agrippa . . . are you going to get those barbarians moving, or are you planning a state breakfast for them?\"\n\nMadog gave no sign that he understood what Pulcher said. Agrippa said to him, \" _Eamus, Dux_! Let's get going, Chief!\"\n\nMadog grunted and started shouting orders to his troop. As they began to move off into the night, Agrippa asked me, \"This Madog . . . he's king of the Sequani?\"\n\n\"King of the Sequani?\" I repeated. \"No . . . he's one of their _brena leygo_ . . . a leader of a community . . . The Gauls call their leaders _brena_ . . . ' _rex_ ' in Latin . . . Technically, my father's a _brena qwartego_ . . . _rex pecoris_. . . a 'cattle king' because he owns a herd.\"\n\n\" _Compre'endo_ ,\" Agrippa muttered. I doubt that he did.\n\nOur approach to the enemy was uneventful. They didn't bother screening their position at all. By the end of the fourth watch, when the sun was still below the horizon but there was already enough light to see over short distances, we were on a wooded ridge overlooking the encampment of the Tigurini. There were no signs of any security or fortifications around their encampment. The night fires were smoldering heaps, and nothing in the camp seemed to be stirring except some hungry dogs. We watched as a man sleepily stumbled out of a lean-to tent, urinated just beyond its exit, and then reentered, presumably to go back to sleep.\n\nMadog grunted at us, \"They pigs . . . Helvetii . . . They dirt my sword to kill.\"\n\nBy the end of the first hour, Caesar and his command group joined us on the ridge. He quickly surveyed his objective. There still wasn't much activity in the valley below. Some of the Tigurini were milling about on the river's edge where they had tied up some boats and barges. But, their ferrying operation had yet to start. We could see no sign of any activity on the far side of the Arar. The rest of the Helvetii had apparently moved on and expected this bunch to catch up.\n\nCaesar quickly surveyed the ground and assessed the capabilities of the enemy.\n\n\"We'll attack on a narrow front from our route of march, straight up the river valley from the south,\" he was saying to Labienus. \"One legion, I think . . . The Tenth is in the van . . . so they go in . . . _acies triplex_ . . . four cohorts in the front rank, then three and three . . . We'll use the cavalry to screen the right flank . . . The river will seal the left.\"\n\n\"That will leave the enemy's rear open,\" Labienus pointed out. \"Do you want to move some of our troops around to seal it?\"\n\nCaesar thought about that, then said, \"No . . . that will take time, and the enemy might detect the maneuver . . . Besides, even a rat is dangerous when it's trapped . . . I _want_ them to be able to escape . . . As soon as we hit them, half their troops will run if they believe they can escape . . . The rest will break as soon as they know the women and children are safe . . . I want to break this bunch, not annihilate them.\"\n\nLabienus nodded, and said, \" _Compre'endo, Imperator_.\"\n\n\" _Bene_!\" Caesar concluded. \"We'll keep the Seventh and Ninth in reserve behind the Tenth. Issue the orders to the legates. I want to know immediately when the Tenth is in position.\"\n\n\" _A'mperi'tu', Imperator_!\" Labienus said.\n\nLabienus retreated back into the woodline. Caesar remained on the ridge, surveying the enemy in the valley below. Then, he spotted Agrippa and me.\n\n\"Ah . . . Insubrecus!\" he started. \"How did you enjoy your midnight ride? And you are . . . Agrippa . . . Lucius Vipsanius Agrippa? . . . You're one of the new _angusticlavi tribuni_ attached to the Tenth? You're an Umbrian . . . from Asisium, I believe?\"\n\n\" _Recte, Imperator_!\" Agrippa snapped. \"I am privileged that the _imperator_ knows my name!\"\n\n\"Privileged?\" Caesar repeated the word. \"It's the duty of a commander to know his officers, Agrippa . . . Remember that if you're ever given a command . . . Know all your officers . . . and all your men for that matter.\"\n\n\" _His meminero, Imperator_ ,\" Agrippa agreed.\n\n\" _Bene_ ,\" Caesar muttered, then seemed to retreat into his own thoughts as he watched the doomed Tigurini in the valley below.\n\n\" _Imperator_ , a word?\" I interrupted his musing.\n\n\" _Quid vis tu?_ \" he responded.\n\nI handed him the coin. \"Madog, the chief of the Sequani cavalry, gave this to me.\"\n\n\"It's a _denarius_ ,\" he said. Then, he flipped it over. \"One of mine . . . What's the point, Insubrecus?\"\n\n\"Madog took it from a dead Helvetian . . . a minor officer . . . but the man had ten of these, _Imperator_ ,\" I began to explain.\n\n\"So, what you're saying is that if a minor officer of the Helvetii had ten of these, then the enemy possesses quite a few more?\" Caesar interrupted. \"And, they're of a new minting . . . one of mine, ironically . . . so the Helvetii came into possession of these recently.\"\n\n\" _Certe, Imperator_ ,\" I agreed.\n\nCaesar continued, \"And, you believe this is consistent with the rumors of Roman influence on the Helvetii and a possible conspiracy by Romans against me and this army?\"\n\n\" _Credo, Imperator_!\" I again agreed.\n\nCaesar stared silently at the coin in his hand. Then, he flipped it back to me, saying, \"You may be right about this, Insubrecus. I'm not completely convinced, but I am intrigued . . . Keep your eyes and ears open . . . Let me know immediately if you discover anything relevant.\"\n\nI was just about to respond when we heard a bugle signal from the valley below. We watched the Tigurini freeze.\n\n\" _Al'iact'est_ ,\" Caesar stated. \"The die is thrown, Insubrecus! _Cornucen_!\"\n\nA bugler appeared from out of the woodline.\n\n\"Signal 'Tenth Legion' and 'Attack'!\" Caesar ordered.\n\nAs the man complied, Caesar shouted, \"Horse!\"\n\nA member of the praetorian detail came forward with Caesar's white stallion. Caesar quickly removed his red cloak from the saddle and draped it over his shoulders. Then, he leaped up on the horse.\n\n\"The men should know their general is watching them, Insubrecus,\" he said to me. Then, he rode forward to the edge of the ridge.\n\nBy this time, the encampment of the Tigurini in the valley below was swirling in chaos as if someone had kicked a beehive. To my left, I saw the first ranks of the Tenth Legion clear the woodline. At first, there were only two cohorts on line. The left-most cohort had its flank anchored on the riverbank. As the Roman line moved forward into the widening valley, a third cohort swung on line around the right flank of the advancing legion.\n\n\"Malleus knows how to maneuver men according to the terrain,\" Caesar said to no one in particular.\n\nThere was some order emerging in the Tigurini camp. I could see warriors beginning to form a battle line on the southern edge of the encampment, facing the advancing legion. Others _\u2014_ men, women, and children _\u2014_ were fleeing toward the woods to the north.\n\nI heard Labienus say to Caesar, \"We have an audience!\" He was pointing across the Arar.\n\nI looked where Labienus was pointing and saw a group of Gallic horsemen.\n\n\" _Bene_ ,\" Caesar grunted. \"Let them report what they see to the rest, and we may finish this campaign before the new crops break through the soil.\"\n\nThe Tenth already had four cohorts on line, facing the enemy. The second line had also emerged from the woodline. The front line advanced to within twenty paces of the Tigurini warriors. Then, the first two men across the entire Roman front launched their _pila_ at the enemy. Then, the next two, then the next! Before the third volley of _pila_ had hit, the Roman front ranks, their _gladii_ bared, were charging into the enemy.\n\nIt wasn't much of a battle. The legionaries mowed through the ill-prepared Tigurini. Even from where I stood, I could see the legionaries advance through the enemy ranks with a combination of punching shields and stabbing short swords. Isolated knots of Tigurini warriors would temporarily hold out against the relentless advance of the Roman battle line, only to be surrounded and overwhelmed. The cohorts had pushed the barbarians about a third of the way across their encampment before they finally broke and began fleeing for the safety of the wooded hills to their rear.\n\nWhen this happened, Caesar again summoned his _cornucen_ and commanded, \"Signal 'Tenth Legion' and 'Halt'!\"\n\nThe troops in the valley below executed Caesar's order. I could hear them cheering across the battle line.\n\nCaesar called for Pulcher. \" _Legate_ , take your cavalry and seal the enemy in from the north and west . . . Do not engage unless you're attacked, but do not become decisively engaged . . . Withdraw and send back for infantry reinforcement if they attempt to punch through you . . . I just want them contained!\"\n\n\" _A'mperi'tu', Imperator_!\" Pulcher's grasp of military courtesy seemed to be steadily improving.\n\n\"Labienus!\" Caesar summoned. \"Detail the _fabricatores_ , the engineers of all three legions, to bridge the Arar! I want this army across by sundown!\"\n\n\" _A'mperi'tu', Imperator_!\"\n\n\"Agrippa! Are you still here?\" Caesar demanded.\n\n\" _Adsum, Imperator_!\" Agrippa responded.\n\nCaesar ordered, \"Get down there to the Tenth . . . Find Malleus, the _primus pilus_ . . . Tell him I want his second-line cohorts, all three of them, across the river to screen the bridge building . . . They can use the Tigurini's boats to get themselves across . . . You're in command of that detail . . . Questions?\"\n\n\" _N'abeo, Imperator_!\" Agrippa snapped.\n\n\"Good lad!\" Caesar encouraged. \"Get it done! Insubrecus!\"\n\n\" _Adsum ti', Imperator_!\" I responded.\n\n\"You stay with me,\" Caesar directed. \"Pulcher should be bringing in some prisoners soon, and I want to be sure they understand my will.\"\n\n# VII.\n\n# _De Clementia Caesaris et Offensione Antigua_\n\n# CAESAR'S CLEMENCY AND AN ANCIENT PROVOCATION\n\n_Divico respondit ita Helvetios a maioribus suis institutos esse uti obsides accipere non dare consuerint_.\n\n\"Divico answers that from the time of their ancestors, the Helvetians have been accustomed to take, not give, hostages.\"\n\n(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nCaesar had the bridges built and three legions across the Arar by nightfall. At the first hour of the next day, he dispatched a strong cavalry scout west under Labienus to locate the main body of the Helvetii, and by the fourth hour, the rest of the army came up from the Rhodanus and began crossing the Arar.\n\nThe first Gallic embassy appeared at the sixth hour while Caesar was supervising the deployment of his trailing legions west of the Arar. A rider from Pulcher's cavalry detachment, a _tribunus angusticlavus_ , approached Caesar.\n\n_\"Ave, Imperator_!\" the tribune saluted Caesar, \"The legate, Caius Claudius Pulcher, sends greetings and reports that a delegation of nobles from the Tigurini tribe is requesting an audience with the _imperator_. The legate believes they have come to surrender and wish to beg for clemency.\"\n\n\"You're Tertius Nigidius Caecina, nephew of the Senator, Publius Nigidius Figulus, are you not?\" Caesar responded.\n\n\" _Sum, Imperator_!\" the man responded.\n\n\"I've read your uncle's _Commentarii_ . . . Quite impressive . . . Is he still dabbling in Etruscan augury?\" Caesar asked.\n\n\"Yes, _Imperator_ ,\" Caecina answered with a laugh. \"My aunt is always complaining of the stinking sheep livers he constantly drags home from the gods know where to stink up the house.\"\n\n_\"Foro viri, foco mulieres regnant_ ,\" Caesar chuckled. \"Men rule in the forum, women in the home . . . Please convey my compliments to the legate, and ask him to deliver the Tigurini delegation to me at the site of their former camp.\"\n\nCaecina rode off to deliver Caesar's message. Caesar then turned to Valgus, who was commanding his mounted praetorian detachment, and said, \"Post a man on this side of the river where he can see the delegation arriving from the north. When the Gauls appear, he is to alert me.\"\n\nValgus complied while Caesar continued supervising the crossing and quartering of the army. No more than half the hour had passed before the praetorian returned to report the approaching delegation.\n\nAgain, Caesar summoned Valgus. \" _Decurio_ , take twenty men and cross the Arar. I want you to take up a position just south of where yesterday's battle was fought, so the Gauls will have to ride past it,\" Caesar instructed. \"Then, wait for me there.\"\n\n_\"A'mperi'tu', Imperator_!\" Valgus snapped.\n\nCaesar dawdled on the west side of the Arar for a while longer, then finally crossed the bridge to receive the Gallic legation. As we rode to where Valgus had established himself, we realized that we were far enough away from yesterday's battlefield that the flies, now feasting on the bodies of the dead Tigurini, would not be a great nuisance. We could see the Gallic embassy and their Roman escort waiting some fifty paces away. They also didn't seem too eager to get close to yesterday's butchery of their comrades.\n\nCaesar halted about twenty paces away from the Gauls. He turned his head in my direction, and asked, \"What should I expect, Insubrecus?\"\n\nI thought about the question for a few heartbeats, then responded, \"If they ride up to you, they expect to negotiate with you as equals. If they dismount and walk to you, they're surrendering.\"\n\nNo sooner had I said that than the leader of the delegation, a tall warrior, with gray streaked through his black hair, dismounted, removed his helmet, and held up his wand of negotiation for all to see. Then, he strode forward to where Caesar remained seated on his horse.\n\n_\"Rwy'n cuhfarch uh Caisaro meoun hedouch_ ,\" the warrior said.\n\nCaesar recognized his own name and asked, \"Why does this man address me by my name?\"\n\n\"He means no disrespect, _Imperator_ ,\" I cautioned. \"He said _uh Caisaro_ . . . He thinks that 'Caesar' is your title.\"\n\nCaesar grunted, then said, \"Tell him Caesar welcomes him. Ask him what he wants.\"\n\nI translated this for the man, who then launched himself into a long, rambling Gallic oration. I kept up as best I could. The man was an under-chief of the Tigurini, but now found himself in command of the entire host since both the high-chief and his _etifeto_ , his heir, fell in yesterday's battle defending their people. He understood that there was a debt of blood between his people and the Romans, since the Tigurini had destroyed a Roman army almost a generation ago. The man suggested that yesterday's battle had satisfied that debt, and now the Tigurini and the Romans should live in peace. He offered to obey the Roman Caesar and do whatever the Roman Caesar now requested of him. Then, there would be peace between their people.\"\n\nCaesar thought about what the man had said, then said to me, \"Tell me . . . What is his name, Insubrecus?\"\n\nI asked the Gaul, _\"Mae Caisaro uhn gofun bet uhou eich enou, Prifo_?\"\n\nThe man responded, \"Please, tell the Caisar that I beg his pardon for my discourtesy. I am Tewdour mab Owain.\"\n\nI told this to Caesar. He responded, \"Tell Tudurmapowin this: First, he is to return his people to this place. Second, they are to bury their dead. Third, they are to return to their homeland. This is the Caisar's wish. Then, the _lemures_ of our dead and his dead will rest, and there will be peace between our people.\"\n\nI told this to the chief. The look of relief was apparent on his face. He had anticipated humiliation, decimation, and slavery for his people. Caesar's terms were more than generous.\n\nWe were just about to terminate the negotiation when Tewdour raised his wand of negotiation to get our attention.\n\n\"The Caisar's clemency is great!\" he began. \"Let me grant him a gift in return. Tell the Caisar that the People of the Dark Moon are not his friends, although they pretend to be. Their desire is his destruction. It was they who invited the Helvetii to migrate into their lands. It was they who encouraged the Helvetii to devastate the lands of their enemies, the People of Soucana. It was they who invited the Germans over the Rhenus with promises of fertile land and loot. Also, tell the Caisar, when the People of the Dark Moon first came among the Helvetii, in the time when Owain mab Aflon was the _orgorix_ , Romans came with them, Romans bearing silver for the war chief of the Helvetii.\"\n\nI translated this for Caesar. He showed little surprise at this information. \"The _orgorix_ ,\" he mused. \"That must be the one we call Orgetorix . . . That would have been back when Messala and Piso were consuls . . . Piso has connections to the Claudii.\"\n\nThen Caesar said, \"Can he describe these Romans?\"\n\nI conveyed Caesar's question to Tewdour, but he just shrugged his shoulders. Then, I spotted the tribune, Caecina, waiting in the distance with the Roman cavalry escort.\n\n\"I have an idea,\" I told Caesar. \"Would you ask Caecina to join us?\"\n\nCaesar shrugged, then called, \" _Caecina_! _Ad me venias_!\"\n\nThe legate joined our group. When he was next to me, I slapped my chainmail _lorica_ then his plate _lorica_ , and asked Tewdour, \"Can you tell me what armor the Romans wore?\"\n\nTewdour said, \"Most wore armor like yours . . . good ringed iron . . . but one wore a breastplate like the other . . . hammered bronze.\"\n\nI reached out and grabbed the bottom of Caecina's tunic, and asked, \"The one in bronze . . . was his tunic like this one?\"\n\n\"Like that,\" Tewdour affirmed, \"but the stripe was wider.\"\n\nI told Caesar, \"There was a Roman officer with the group, a _laticlavus_ , a wide-striper.\"\n\nAgain, Caesar grunted. \"A senior tribune . . . a senatorial . . . Ask him how the Orgetorix died.\"\n\nWhen I did, Tewdour just shrugged and said, \"He just died.\"\n\nWhen I translated the response for Caesar, he just shook his head. \"Sometimes men just die . . . It is their fate . . . Tell him we are done here . . . I expect to see his people gathered in this place by sundown.\"\n\nI told Tewdour Caesar's wishes. He made no move to depart.\n\n\"Why isn't he going about his business?\" Caesar asked me.\n\n\"He doesn't want to be rude,\" I told Caesar. \"You have precedence here. It is for you to depart first.\"\n\n\"Very well,\" Caesar said and pulled the head of his horse around toward the bridges on the Arar. \"Caecina, with me!\" he called over his shoulder.\n\nWhen we were a few paces away, Caesar instructed Caecina, \"The Twelfth Legion is approaching the crossing. Go to their _primus pilus_ . . . Nerva is his name . . . Tell him to detail four cohorts to remain on this side of the river and guard the Tigurini . . . I will give them two days to bury their dead and rest . . . Then, I want them heading east, back to their homeland . . . The four cohorts from the Twelfth and two _alae_ of their cavalry will form a _vexillatio_ , an independent detachment, to escort them . . . You will command it.\"\n\n_\"A'mperi'tu', Imperator_!\" Caecina responded, obviously pleased about being given his first independent command.\n\n\"And, Caecina,\" Caesar continued, \"the Twelfth is a new legion . . . Most of the men haven't been blooded . . . Tell Nerva to detail one of his more experienced centurions to you . . . Sanga, I think . . . the _pilus prior_ of the Third Cohort . . . Yes . . . Sanga will fit the bill nicely.\"\n\nCaecina rode off to find Nerva and the Twelfth Legion. Caesar led us back toward the bridge. As we rode, he asked me, \"Are you surprised by my treatment of the Tigurini?\"\n\n\"I can't say I understand it, _Patrone_ ,\" I answered.\n\n\"Why force a man to do what you want when you can convince him it's what he wants?\" Caesar answered.\n\n\"I'm still not sure I understand,\" I admitted.\n\n\"With a large enemy force ahead of us, it makes no sense tactically to have another large enemy force in our rear,\" Caesar lectured. \"And, I don't have the time or the manpower to police them all. Even if I were to do that, logistically speaking, what would I do with all those prisoners until I could unload them on the slave traders? No . . . better to be done with them . . . The real threat is still ahead of us.\"\n\nCaesar didn't wait for my response, but continued, \"Strategically speaking, the only way to secure the _imperium_ is to keep the Germans east of the Rhenus . . . and the best way to do that is to maintain an alliance of strong Gallic tribes along the river . . . Even if they are not especially well-disposed towards Rome, they will not allow the Germans to cross and steal their lands . . . Rome needs the Tigurini back on the Rhenus and needs them strong enough to resist German incursions.\"\n\nAt the time, I had no idea why Caesar was sharing these thoughts with me. I have since learned that great men often make decisions based on instinct, and only later do they try to understand why they did what they did. Weak men fail to act and claim the issue was too complex.\n\nWhen we crossed one of the bridges to the west side of the Arar, we encountered Labienus, back from his reconnaissance of the Helvetii.\n\n\"Ah, Labienus!\" Caesar greeted him. \"What news?\"\n\n\"The main body of the enemy is no more than five thousand _passus_ to the west,\" Labienus reported.\n\n\"Five thousand _passus_?\" Caesar repeated. \"They are certainly in no hurry to escape.\"\n\nLabienus almost sighed, \" _Imperator_ , I don't think they see the need to escape . . . We couldn't get an accurate count, but there must be tens of thousands of them on the march . . . They fear us no more than an elephant fears a gnat.\"\n\nCaesar did some quick calculations, \"A hundred thousand Gauls . . . Fifty thousand are male . . . Twenty to twenty-five thousand of fighting age . . . No more than ten thousand warriors . . . The rest are just _pagani_ with sharp sticks . . . We have six Roman legions . . . I have almost twenty-seven thousand troops in the field, according to this morning's strength report . . . What need do I have for caution?\"\n\n\"There may be more than one hundred thousand out there on the march . . . many more,\" Labienus answered. \"We need to get an accurate strength assessment before we decide to be too aggressive.\"\n\nCaesar seemed to mull that over for a few heartbeats. \"Recte,\" he acceded. \"With that many out there, we're not about to lose them . . . and at their rate of march, they will not outrun us . . . We will follow at a safe distance until we have a clear idea of their intention and strength . . . Hopefully, we can entice them to attack us on terrain of our choosing.\"\n\nLabienus nodded, then said, \"We detected a small band of them heading back in our direction. We think it's some sort of delegation.\"\n\nCaesar looked at me questioningly.\n\n\"If they want to talk, they'll be carrying the truce wands like the others,\" I told him.\n\nCaesar looked over to Labienus, who just shrugged. \"We didn't get that close, _Imperator_ ,\" he said.\n\nCaesar grunted in response. \"We'll see what they have to say, Titus,\" he said to Labienus. \"Perhaps we can finish this thing here and now, the way we convinced the Tigurini to return to their lands on the Rhenus. When they arrive, lead them to a spot where they can see what we are doing here . . . There's a small hillock between the _castra_ of the Tenth and the Seventh Legions . . . That would be perfect . . . I will meet their delegation there . . . Perhaps the sight of Roman power and competence will convince them to return to their homeland.\"\n\nThe Helvetii delegation arrived at the ninth hour. This time, Caesar remained mounted, waiting for them on the hillock, overlooking his legions on the west side of the river. The Helvetii could also see the bridges over the Arar and beyond the river, the remnants of the defeated Tigurini. The message was clear: in one day we defeated one of your tribes and accomplished a river crossing that took you weeks.\n\nThe leader of the Helvetian delegation was a giant of a man on a black stallion. His long, flowing gray hair hung freely from his bare head; his long, gray chin-length mustachios seemed to bristle with pride and defiance. As he waved his wand of negotiation negligently in our direction, the sun glinted off the golden armbands that circled his bulging biceps.\n\nHe rode directly at Caesar. At two paces away, he pulled his horse's head abruptly to the right; the animal reared a bit and exposed the warrior's long, Gallic sword in a gold-wired scabbard hanging down his left side. He stilled his mount and stared directly at Caesar with cold, piercing blue eyes for a few heartbeats.\n\nIt was then I realized that over his chainmail _lorica_ , his chest was festooned with Roman _phalerae_ \u2014gold and silver sculpted disks awarded to centurions and rankers for acts of valor. No legionary would willingly surrender such treasures. They were trophies taken from Roman dead.\n\n\"So, Roman, you are the one called Caesar?\" he said in Latin.\n\nCaesar remained silent, perfectly still.\n\n\"Are you surprised I speak your language?\" the warrior continued. \"I learned it from my slaves, who were once soldiers in the army you sent against us many years ago, when we killed your chief, the one you called 'consul.'\"\n\nI heard Labienus suck in his breath when the warrior said that. This was not a negotiation, I realized. This man had come to challenge Caesar, to provoke him into combat.\n\n\"I am Dewi mab Coel . . . Divico to you Romans . . . Know that it was I who was _orgorix_ of the Helvetii the day we slaughtered the Romans and took ten thousand heads.\"\n\n\" _Orgorix_?\" Labienus whispered to me.\n\n\"Slaughter-king,\" I translated for him. \"War chief.\"\n\n\" _Podex_ ,\" Labienus hissed and spit on the ground.\n\n\"The whitened skulls of Roman dead decorate our feasting halls and lodges to this day . . . And now you have the temerity to attack us? Know you, Roman, the fact that you were able to ambush and slaughter one of our minor clans, while they were trapped against a river and burdened with women and children, does not impress us . . . Roman deception and cruelty has been known to us for many generations . . . We will offer you this . . . The valleys of the Rhodanus are wide and fertile . . . We will accept any of them and settle there . . . Know you, Roman, the Helvetii have learned from their ancestors to rely on valor and strength, not on deceit and ambush . . . Unless you want to make of this ground, on which we now stand, another monument to Roman defeat and shame, you will give us the land we demand and withdraw your soldiers across the Rhodanus . . . Then, there will be peace between our nations.\"\n\n\"Quite an oration,\" Labienus hissed. \"He couldn't provoke Caesar more if he tried.\"\n\nI looked over toward Caesar. His face was as white as a candidate's toga. His lower jaw was set slightly forward, and his thin lips were drawn tightly across his face like a knife scar. I later learned that this expression was the only telltale sign of his rage. Caesar was too controlled ever to demonstrate it, especially when facing an adversary.\n\nFinally, Caesar spoke, \"Well, do I remember that of which you speak. No Roman will ever forget the tragedy that befell our nation and the army of Lucius Cassius Longinus . . . Many families still mourn that day . . . Many still desire blood vengeance to appease the restless _lemures_ of their murdered ancestors . . . You say that the Helvetii are a people of valor and courage . . . I tell you that you are a liar . . . You defeated Longinus only by deceit and ambush . . . You feigned friendship with the Roman people and delivered treachery at the end of a spear . . . Know you, Divico of the Helvetii, that, although members of my own _familia_ lost their lives on that fateful day, I, Gaius Iulius Caesar, proconsul of the Roman people, commander of this army, offer you these terms for peace . . . First, you will return to the lands you abandoned on the Rhenus . . . Second, you will give restitution to the allies of the Roman people whose lands you have pillaged and destroyed . . . Third, you will surrender hostages to me to ensure your submission and good behavior . . . Only then am I willing to let the Helvetii leave this place in peace . . . And also know this, Divico of the Helvetii . . . The gods despise hubris . . . They do seem, at times, to grant their favor, but they do so only to heighten despair when they withdraw it. What is your answer, Divico?\"\n\nDivico's eyes glared at Caesar like burning blue embers. \"Roman! Since the time of our ancestors, the Helvetii are accustomed to taking hostages, not surrendering them!\"\n\nHe held up the wand of negotiation, broke it in two, spit on the pieces, and threw them at the feet of Caesar's horse. He pulled back on his reins, causing his black stallion to rear and turn, then galloped off the hillock, followed by his entourage.\n\nCaesar calmly watched Divico ride off. Then, he turned to Labienus and said, \"It appears that our negotiation with the Helvetii has ended. Come with me. We have work to do.\" Then, he rode off toward the _castrum_ of the Tenth Legion.\n\nCaesar had established his _praetorium_ with the Tenth Legion. When we arrived at the headquarters tent, Caesar said, \"Accompany me, Insubrecus!\"\n\nWe entered Caesar's operations area; the maps were already hung, and soldiers were busy making notations. I could easily see the location of our army on the left side of a squiggly blue line that ran down from the top of the map. The Arar, I assumed. It led downward to another, thicker blue line that seemed to plunge toward the bottom left corner of the map: the Rhodanus. About a _cubitus_ to the left of our location was a large red marker: the Helvetii.\n\nWhile Caesar's body slave was helping him out of his armor, he was talking to Labienus. \"That was the best theater I've seen since I left Rome! Divico! He could easily upstage Plautus' braggart. Did you see the size of that sword? How does the man walk without tripping?\"\n\nLabienus, his helmet under his arm, answered, \"Could he have really been the Helvetian commander when they ambushed Longinus? That would make him what . . . sixty, if he's a day? No one could look like that at sixty.\"\n\n\"What did you think of that act, Insubrecus?\" Caesar asked me.\n\n\"He was trying to provoke you, _Patrone_ ,\" I said stiffly.\n\n\"Of course he was,\" Caesar answered. \"I sometimes forget how young you are, Gaius . . . You haven't had time to develop a sense of irony yet.\"\n\nI too was beginning to forget how young I was.\n\nCaesar stood staring at the situation map. \"I feel a bit like that boy in the children's story . . . the one who thinks he's captured a lion because he has ahold of its tail,\" he was saying. \"I can neither let go nor continue to hold on.\"\n\nAt his side, Labienus just grunted his consent.\n\nCaesar continued, \"Our first priority is to assess the enemy's intention. I assume they will continue moving west across Aeduan territory, but if they were to turn and attack us with our backs to the river, it could get messy. So, let's get a strong cavalry screen between us and them.\"\n\nAgain, Labienus nodded his agreement.\n\n\"Second, we need to assess their fighting strength,\" Caesar said. \"That's going to take some aggressive reconnaissance. Do we have enough cavalry?\"\n\n\"We have the legionary cavalry,\" Labienus calculated. \"That gives us fifteen _turmae_ , about 450 troopers . . . There is a scattering of native cavalry . . . They come and go as they please . . . I can only estimate their numbers.\"\n\n\"Are there any auxiliary cohorts available?\" Caesar asked.\n\n\"The _Prima Gallica_ is still near Gennava,\" Labienus estimated. \"And, there's a Syrian outfit down near Massalia.\"\n\n\"Leave the _Prima Gallica_ where it is,\" Caesar instructed. \"The Allobroges are not reliable. I want an effective Roman force sitting on them while we're tied down with the Helvetii. Send down to the Syrian unit. Have them send up their cavalry cohort . . . _stat'_ . . . What about that native cavalry?\"\n\nLabienus rubbed his chin. \"It's difficult to give an exact count . . . Between the Aedui and the Sequani, there must be at least three hundred mounted troops.\"\n\n\"Insubrecus! Will the Gauls submit to Roman military authority?\" Caesar asked.\n\nI shrugged, \"As long as they fear the Helvetii more than they resent us . . . they'll cooperate . . . How much _authority_ they'll accept . . . that's difficult to say.\"\n\nCaesar addressed Labienus, \"Organize the natives into _turmae_ , according to tribe . . . say, three _alae_ each . . . We'll assign an experienced Roman tribune to each cohort . . . Who do we have available for senior commanders?\"\n\nLabienus stared at the ceiling of the tent, thinking. \"There's certainly Publius Considius\u2014\"\n\nCaesar interrupted, \"He's not senior enough for overall command of the cavalry . . . plenty of experience, but not of _laticlavus_ standing . . . And he's getting a bit long in the tooth! _Verpa Martis_! The man served under Sulla! I'm thinking Crassus as my cavalry legate . . . I'm sure he's fed up with pushing mules.\"\n\nLabienus' eyebrow shot up. \"Publius Licinius Crassus?\"\n\n\"Yes,\" Caesar confirmed, \"let's see if the boy has some grit hiding below all that flash . . . He's proven a competent enough _quaester_ . . . We'll assign Considius to him and . . . what was the name of that _angusticlavus_ from the Tenth who commanded the advance party across the Arar? . . . Agrippa . . . yes . . . Assign Lucius Vipsanius Agrippa to him, too.\"\n\n_\"A'mperi'tu',\"_ Labienus agreed. \"And, who replaces Crassus as quartermaster?\"\n\n\"Pulcher,\" Caesar said without hesitation. \"I think a healthy dose of mule shit is just what that Claudian twit deserves . . . Yes . . . Pulcher is perfect for pushing the baggage . . . Insubrecus!\"\n\n\"Yes, _Patrone_?\" I responded.\n\n\"You get along with Agrippa, do you not?\" Caesar asked.\n\n\"Yes, _Patrone_.\"\n\n\"How long have you been _sub aquilis?\"_ Caesar asked.\n\nI had to think. I was beginning to feel as if I had always been a soldier. \"Just over three months, _Patrone_.\"\n\n\"Just over three months,\" Caesar mused a bit and seemed to chuckle. Then, he said, \"I'm appointing you to the rank of _decurio_ . . . at least while you're assigned to Crassus' command . . . Do a good job, and you can keep the appointment.\"\n\nLabienus slapped me on the back, \"Congratulations, _Decurio_!\"\n\n\"I'm parched . . . Being scared to death by a blowhard Gallic giant is thirsty work,\" Caesar announced. _\"Scriba!\"_\n\nI heard a voice from the other side of the partition, _\"Quid vis tu, Imperator_?\"\n\n\"Vinum!\" Caesar ordered.\n\n\"Merum?\" the voice called. \"Straight up?\"\n\n\"No . . . _mixtum_ . . . bring a pitcher of water,\" Caesar instructed. \"We still have work to do,\" he said to us and winked.\n\n\"Back to our discussion . . . the cavalry . . . The Eleventh and Twelfth are mostly Gallic units . . . Let's send over to the _primus pilus_ of each . . . We need Gallic speakers who can ride . . . like Insubrecus here . . . good troopers who are considered promotable . . . We'll make them _decuriones_ and assign one to each of the native turmae.\"\n\nI heard Labienus mumble, \" _A'mperi'tu'!\"_ I looked over at him. He was scribbling notes on a _tabula_.\n\nCaesar continued, \"Enemy intention and strength are our first two priorities. I need to know whether to avoid contact with the Helvetii or provoke it. We can establish our main supply depot at Lugdunum . . . That will give us access to both rivers . . . But my fear is that the Helvetii are going to move straight across country, away from the navigable rivers. That will stretch our supply lines . . . We may need to draw rations and fodder from our allies.\"\n\n\"That could prove a problem, Caesar,\" Labienus cautioned. \"This far north, it's the beginning of planting season, and I doubt the Gauls have a great store of food left over after the winter.\"\n\nCaesar seemed to think about that for a moment, then shrugged, \"It doesn't matter . . . If the enemy moves into the hills, we can have each man carry five-day's rations . . . jerky, _buccellatum_ , grain . . . same rations for the officers . . . I'm not burdening the supply train with their luxuries . . . We need the mules and wagons to carry rations, fodder, and the legionary artillery . . . Leave our siege equipment at Lugdunum . . . At least five-days' rations for each man on the legionary mules . . . The fodder's going to be more of a problem . . . The fields cannot yet support our mounts . . . The Gauls must be supplying their own people . . . They can supply us as well.\"\n\nCaesar's slave entered with the wine and water. Caesar looked up. \"Ah . . . _bene_ . . . Go ahead and leave it . . . We'll pour ourselves.\"\n\nThen, he noticed me standing there. \"You look like you're on parade, Insubrecus . . . Sit down . . . Relax . . . My man will help you with your _lorica_.\"\n\nCaesar poured the wine and left us to mix our own water. \"Where was I?\" he began. \"Yes . . . supply . . . Pulcher's first priority will be to arrange grain and fodder from the Aedui as we move across their territory . . . Until we have a clear idea of enemy strength and intention, we will not draw any closer to them than five thousand _passus_ . . . Another mission for the cavalry is identifying defensible terrain for the daily marching camps and identifying battle sites should I decide to provoke battle . . . a ridge . . . maximum four-legion front, _acies triplex_ . . . good flank security . . . sun in the enemy's eyes . . . wind at our backs . . . You know the drill, Labienus . . . We'll keep the Eleventh and Twelfth in tactical reserve until they get some seasoning.\"\n\nLabienus looked up from his notes, \"Do you want me to assemble the commanders for a briefing?\"\n\nCaesar took a long draft of wine. \"Yes . . . the _legati_ , of course . . . all the _tribuni laticlavi . . ._ and the _primus pilus_ of each legion . . . We'll be dependent on our centurions for tactical leadership and control until our senior officers have more experience.\"\n\nLabienus wrote on his _tabula. \"Quot'orarum?\"_ he asked.\n\n\"Eleventh hour,\" Caesar said. \"I want the officers back with their units by nightfall . . . We'll maintain full alert for the first and fourth watches . . . Keep a strong cavalry screen between the Helvetii and us . . . Having my back to the river is making me a bit nervous.\"\n\nWith that, Caesar dismissed Labienus, but requested that I stay behind. I waited while Caesar poured himself another cup of wine.\n\nThen, he began, \"Insubrecus . . . your mission with the native cavalry is to be my eyes and ears for our . . . what shall we call them . . . _socii nostri_ . . . our allies. . . . I want to know their attitudes toward us . . . our presence here, north of the Rhodanus . . . and how they react to our officers. . . . If there're any problems, I need to know right away. . . . We're swimming in a sea full of dangerous and unreliable creatures . . . I don't want the Aedui suddenly deciding we're less welcome than the Helvetii and turning on us . . . I need someone who understands the Gauls . . . someone I can trust.\"\n\n\" _Ut vis tu, Patrone_ ,\" I answered.\n\n\" _Bene_!\" Caesar responded. \"And, don't think I'm ignoring the information you've already brought me . . . about Romans influencing the Helvetii and the other Gauls . . . To imagine there was no influence would border on delusion . . . but assuming there's some plot against this army would be a bit paranoid . . . So, keep your eyes and ears open for any indication of Roman influence among the tribes . . . You can trust Labienus, but besides him, you speak only to me about this matter . . . Understand?\"\n\n\" _Compre'endo, Patrone_ ,\" I responded.\n\nCaesar got up from his camp chair and walked over to a wooden chest next to his field desk. \" _Adveni, Insubrece_ ,\" he said, unlocking the chest. \"I want to show you something.\"\n\nI walked over to Caesar, and he showed me a collection of _tabulae_. \"These are my notes on the campaign so far,\" he said, handing me one of the _tabulae_. I opened it and saw the first line, _\"Gallia est omnis divisa in partes tres . . ._ \"\n\n\"That one's mostly a paraphrase of the Greeks . . . the geographies,\" Caesar explained. \"My concept was to keep a journal of the campaign and periodically send the entries down to Rome to be read out to the people in the forum . . . This would be in addition to the official reports, which are prepared by my staff and sent down to the Senate . . . The journals for the people would need to be written in a simple, straightforward style . . . a blunt military report of what we are doing up here . . . That way, if there is some resistance by the _Optimates_ . . . the aristocratic faction in the Senate . . . I'd still have the people behind me . . . So, these reports would have to be written in the people's Latin . . . I simply do not have the time to command this army and do justice to these journals . . . So, I'd like you to serve as my _ad manum_ . . . You observe, notate, write the journal, and review the text with me . . . Politically, this journal is as important in Rome as a military victory in Gaul.\"\n\n\" _Patrone_ ,\" I stammered, \"non _existimo me dignum_ . . . I don't think I'm up to\u2014\"\n\nCaesar held up his hand to stop me. \"You were tutored by Gabinius' greekling . . . the same as his own children . . . That's good enough for me . . . Which reminds me . . . I will dispatch a personal message to the consul about . . . about your issue with his family . . . I will let him know that you are part of my personal staff . . . _et sub patrocinio meo_ . . . That should put an end to any more _incidents_ like the one in Aquileia.\"\n\nThat was perfectly clear. Caesar's offer was protection for writing his journal. \" _Ut vis tu, Patrone_ ,\" I responded.\n\n\"Good lad!\" Caesar said, actually tussling my hair. \"Get with Valgus . . . Tell him I want you riding like a trooper by the end of the week . . . I don't want to lose my _ad manum_ on the end of some Gallic lance!\"\n\n# VIII.\n\n# _De Calamitate Prima_\n\n# THE FIRST DEBACLE\n\n_Quo proelio sublati Helvetii quod quingentis equitibus tantam multitudinem equitum propulerant_.\n\n\"When the Helvetians push back a great number of our cavalry, with only five hundred of their own, they are encouraged by their success.\"\n\n(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nHe badly misjudged the response of the Aedui and Sequani to our call for cavalry. We thought we could raise a few hundred. By the sixth hour of the next day, some two thousand had gathered near our camps.\n\nAt Caesar's bidding, I questioned the assembled Gallic riders. Some said they came to defend their homeland; some, to loot the Helvetii; some were simply following their chiefs; some were out for the adventure. When pressed by Caesar for some common reason for their enthusiasm, all I could say was that there had been years of peace in the lands of the Aedui and Sequani. It was now the beginning of the fighting season, and the young warriors were spoiling for a fight, any fight. In short, they were bored.\n\nEarlier, during the second hour, our legionary cavalry screen had reported that the Helvetii were on the move into the highlands west of the Arar. Caesar was content with allowing some distance to grow between his army and them; he still wasn't sure what size of a force he was dealing with.\n\nWhile Crassus was being briefed by Caesar on his new responsibilities as _legatus equitium_ , chief of cavalry, Labienus struggled to organize the Gauls into _turmae_ and to integrate Roman leadership over them. The Gauls, especially the Sequani, wanted revenge for the burning, pillaging, and slaughter the Helvetii and their German allies had visited on their people. They could hardly be restrained from immediately attacking the enemy. As time passed, more Gallic bands slipped off to the west to pursue the Helvetii independently. By the seventh hour, the rest of our Gallic cavalry initiated a general, headlong pursuit\u2014as if the Helvetii were defeated and fleeing. What Roman leadership Labienus had managed to integrate into this mass of horsemen were more dragged along than leading the way.\n\nI again found myself riding with Athauhnu and the Sequani. He was commanding an _ala_ of about twenty men under Madog, who led a hundred riders. Agrippa was the _angusticlavus_ assigned to advise him. We had another trooper, a Padus-Valley Gaul from the Twelfth Legion called Flavus, \"Whitey,\" because his hair was as blond as a Kraut's eyebrows. The rest of the unit was somewhat divided into three _alae_ , but except for family and clan groups who tended to stick together, the organization was quite fluid. And, they were all fired up to get at the Helvetii.\n\n\"I no can to hold back,\" Madog was complaining to Agrippa. \"Young men . . . very eager of going . . . want revenge . . . We lose maybe twenty already . . . They just go . . . We go too.\"\n\n\"Madog,\" Agrippa was cautioning him, \"we don't know how many Helvetii are out there or what their tactical deployments are . . . We could be riding into a massacre.\"\n\n\"Bah . . . Helvetii girls,\" Madog dismissed Agrippa's advice. \"We kill all.\"\n\nMadog shouted to get his men's attention. When he had it, he waved his _spatha_ over his head and whipped the Sequani into a blood frenzy. Then, he screamed, \"Follow me!\" in Gallic and led them galloping out toward the hills.\n\nWe three Romans just sat there in the dust of the Gauls' sudden departure. Agrippa looked over to me and said, \"Did you catch what he said? What's his plan?\"\n\n\"Plan?\" I shrugged. \"His simply told them to follow him and they will kill all the Helvetii they can find.\"\n\nAgrippa stared at me for a long heartbeat, then shrugged. \"We better catch up to them,\" he said. Then, he turned his horse and rode toward the hills, following the dusty trail of our Sequani.\n\nFlavus and I rode after Agrippa. Catching up to the Sequani wouldn't be easy. Although their horses were smaller than ours, they had good stamina, especially in the hilly country of their homeland. Also, the Sequani rode lighter than we did. Most did not wear armor. Of those that did, many wore hardened leather as _loricae_ , not the heavier chainmail armor and helmets that we were wearing. Many of the Sequani riders were armed with nothing more than small hunting spears. Only the richer Sequani had steel swords or even shields.\n\nI was carrying two swords: my legionary _gladius_ strapped on my right hip and a long _spatha_ tied down on the left front of my saddle. I had my _pugio_ on my left hip. Although I was carrying a small cavalry shield, a _parma_ , strapped behind my left shoulder, I had decided not to carry the light cavalry stabbing spear that Valgus called a \"Greek lance,\" the _dory_.\n\nI was still not comfortable riding into possible combat against experienced riders. In the little time we had had, Valgus tried to teach me critical combat riding skills, mostly controlling my mount with my knees and legs so I could use my hands to wield my weapons: the shield in my left hand and the _spatha_ in my right. Army mounts are trained for leg control, and Macro had shown me some military riding techniques back home. But, I had no doubt that in the confusion and terror of actual mounted combat, my riding skills would be far from adequate.\n\nDuring my drills the night before, Valgus had told me a story about a _tiro_ cavalryman, who lost control of his mount and was carried into the enemy formation. \"A one-man charge!\" Valgus chuckled. \"Didn't work out well for the poor son of a bitch!\"\n\nI didn't want to be that guy, but I wasn't sure I could prevent it.\n\nThe other problem was using weapons while mounted. As an infantryman, I was taught to leverage the power of my attack from the balls of my feet. This was the essential technique of the power behind the basic infantry attack with shield and sword, the _percussus_. But, a rider \"floats\" in the saddle; his feet are literally dangling at the sides of his mount. Not only could I get no leverage to power a sword blow, but if I overreached, I'd lose my seat and find myself on the ground.\n\nAgain, I didn't want to be that guy.\n\nMadog's _turmae_ weren't difficult to follow, but they were difficult to overtake. We pursued them west across the narrow river valley and up into the highlands. Several times as we were riding, I was aware that we were just three Romans riding alone through hostile territory. If the Helvetii discovered us before we overtook the Sequani, it would not go well for us.\n\nAt about the eighth hour, we were some nine thousand _passus_ out from the legionary camps. We were riding south up a small river valley with a wooded ridgeline to our west. As the sun shone through the trees at the top of the ridge, I noticed the silhouettes of riders.\n\nI called ahead to Agrippa, \"Tribune! We are not alone. Look to your right!\"\n\nAgrippa quickly spotted the riders. \"Who are they, _Decurio?\"_ he asked.\n\nIt took me a few heartbeats to realize he was talking to me. I wasn't used to my new rank yet.\n\n_\"No'scio, Tribune_!\" I answered. \"I'll challenge them. Be ready to ride like the Furies if they're hostile!\"\n\nI rode over to the edge of the woods, which only served to make the men on the ridgeline less visible. Realizing there was no help for it, I yelled a challenge at them in Gah'el: _\"Ar uh bruhnu! A uhduhch uhn gyfaillo i Rhufeinig_?\"\n\nAfter a few nervous heartbeats, I heard Athauhnu's voice from the top of the ridge in answer, \"Little Roman! What kept you? Come up and join the fun!\"\n\nI shouted over to Agrippa and Flavus, \"It's Madog's _turma_! They want us to come up!\"\n\nWe found a track up the ridge and were soon with the Sequani. When Madog saw us, he pointed to another broad valley to the west under our ridgeline. \"Good fighting,\" he said in his broken Latin. \"Sending many Helvetii to Land of Youth.\"\n\nBelow us was a swirling brawl of native horsemen\u2014hundreds of them. I could not separate Helvetii and Kraut from Aedui and Sequani. Men were hacking at each other from the backs of horses. On the ground, there were individual duals, small groups attacking other small groups, and all were wrapped in choking gray-brown dust. Periodically, I could recognize a Roman by his armor, but it was obvious that we were exerting no control over our allies or the battle.\n\n\"Good fighting!\" Madog repeated. \"Horses rest . . . Then, we go!\"\n\nThat delay may have saved us.\n\nFlavus suddenly pointed across the valley. \"Tribune! The woodline!\"\n\nI peered in the direction that Flavus had indicated. The far woodline was in shadow because of the declining sun, but soon I was able to detect some movement in the gloom.\n\nI began to ask Madog who it was, but I was interrupted by a burst of cacophonous trumpeting from across the valley.\n\n\"Helvetii come!\" Madog exclaimed.\n\nImmediately, from across the valley, Helvetian infantry, hundreds of them, mostly spearmen, charged into the cavalry melee.\n\nSuddenly, Agrippa yelled, _\"Vi'te!_ Look there!\"\n\nAnother large force of enemy infantry had worked its way undetected along the east side of the valley and was now rushing to block any escape route for our Gallic allies trapped in the valley below.\n\nThen, another series of trumpet calls sounded; these were somewhat different from the first. A body of the Gallic cavalry disengaged itself from the battle and fled headlong down the valley to escape the onrushing Helvetian infantry.\n\n\"Aedui pigs!\" Madog yelled at them. \"They run from good fight like children!\"\n\nAgrippa reacted. \"Madog! Follow me! We must open the road for our men!\"\n\nAgrippa moved down the ridge without waiting for Madog to respond. Flavus and I followed behind. We picked out a trail which led us down through the woods to the valley floor. I was relieved when the Sequani followed behind us.\n\nI could see the battle below as we descended. The Helvetian spearmen were decimating our cavalry. I saw more than one Roman go down as the Helvetii stabbed up at them from the ground with their long spears. The Romans were wearing chainmail _loricae_ , very valuable booty for the Gauls. I knew that those injured and dying were the newly minted _decuriones_ from the Eleventh and Twelfth Legions. Once they were unhorsed and on the ground, follow-on Helvetii infantry surrounded them hacking and stabbing with whatever weapons they had brought to the fight: some swords; many knives; a few clubs; even hoes, billhooks, and scythes\u2014anything that could kill.\n\nPresently we reached the valley floor. The second mass of Helvetii had not seen us. They were loosely strung across the valley exit, expecting little opposition except for the few horsemen who might have escaped the main battle.\n\nAgrippa rode out into the valley floor from the tree line, then yelled back, \"Insubrecus! On my right! Flavus, my left!\"\n\nWe rode out to comply. At that point, the Helvetii down the valley noticed us. I saw a few of them pointing and gesturing toward us.\n\n\"Madog!\" Agrippa ordered. \"Your _ala_ is with me. The other two split, one with Insubrecus, the other with Flavus!\"\n\nI heard Madog relay the instructions in Gallic, and the riders gathered behind us.\n\nAgrippa, \"Form wedge!\"\n\nMadog repeated the order, and the Gauls formed behind us as best they could. I suddenly realized that I was the point man in the right-flank wedge, the first man in.\n\nAgrippa yelled, \"Right and left flanks, follow ten _passus_ behind my center . . . at the WALK . . . FORWARD!\"\n\nOur _turmae_ began to move toward the Helvetian blocking force. I could see them in the distance, less than three hundred _passus_ away. They were beginning to coalesce into a line facing us.\n\nAgrippa, _\"Cornucen_ . . . SOUND ATTENTION!\"\n\nThere was some confusion as Madog relayed the order. Finally, a Sequani bugler sounded a series of cacophonous notes to get the attention of our allies engaged across the valley.\n\nAgrippa, \"At the TROT . . . FORWARD!\"\n\nWe picked up the pace of our advance toward the enemy. They now stood shoulder to shoulder, facing us. Their officers and chiefs were adjusting their line, shouting encouragement.\n\nAgrippa, _\"Cornucen_ . . . SOUND, WITHDRAW!\"\n\nThat was the signal for our men across the valley to disengage from the enemy and follow us out of the valley.\n\nAgrippa, \"At the CANTER . . . FORWARD!\"\n\nAt that, I could feel my mount's excitement grow. She seemed to be straining to get at the enemy. I had my shield off my shoulder and drew my _spatha_ from its sheath. I was guiding my mount as best I could, using only my legs and knees.\n\nAgrippa raised his own sword and screamed, \" _TURMA_ . . . CHARGE!\"\n\nThe horses stretched forward into a dead run toward the enemy line. Over the pounding of the hooves, I heard the war cries of the Sequani.\n\nI heard myself screaming in Latin, \" _Fortuna_! _Bona Fortuna m'ames_! Goddess of Good Fortune! Favor me!\"\n\nLooking forward, I could see a solid, dark line of enemy infantry. As we closed, I could distinctly see faces, equipment, and spear points leveled directly at us. Suddenly, I remembered Valgus saying that Roman cavalry is not meant to engage infantry. We pursue; we scout; we screen. But we cannot stand up to infantry. Across the rapidly shrinking space, I spotted one Helvetian. He was looking directly at me. His spear was leveled at _me_. He intended to kill _me_!\n\nAbruptly, out of nowhere, there was movement in the Helvetian line. A man dropped his spear and fled to the rear, then another, then more. My would-be killer looked to his right and saw his mates breaking and beginning to flee. He looked back at me. There was doubt in his eyes, then fear. Suddenly, he dropped his spear. He broke to his left. He was trying to get past our flank.\n\nWhen I hit the Helvetian line, it was no more. We thundered past it. We hacked down the Helvetii spearmen fleeing to the rear, trying to outpace our mounts. Some fell to the ground, hoping to avoid our swords. Others turned at the last moment in a futile attempt at resistance.\n\nWe killed them.\n\nAbout a hundred paces past the Helvetian line, we were out of the valley and in a bowl-like area where three valleys seemed to converge. Agrippa halted our advance and turned us around, facing the line of our own charge. About a hundred paces behind us, we could see what was left of our cavalry, riding to follow us out of the trap.\n\nAgrippa yelled, \"Spread out! Let them pass through us!\"\n\nMadog repeated the order in Gallic, and the Sequani did their best to comply. Their effort was adequate. Our escaping comrades managed to pass through our lines with little confusion. They did not halt to assist us.\n\nWhen they had cleared us, Agrippa yelled, \"Prepare to receive the enemy!\"\n\nAgrippa rode along our front with Madog in his wake, tightening our line, adjusting our positions, encouraging the men. In the distance, I could see a force of enemy cavalry about twice our size coming forward rapidly. My mount was still struggling to regain her wind.\n\nAgrippa commanded, \"If they attack, we move forward into them . . . We cannot allow them to hit us at a standstill . . . We cannot give them momentum . . . initiative.\"\n\nMadog struggled to relay Agrippa's orders. I had no idea how he was coping with Latin words like \"momentum\" and \"initiative.\"\n\nIn the distance, the enemy force was coalescing into individual riders. I was trying to detect their speed. I imagined that they would want to hit us at the gallop, but they seemed content to trot toward us.\n\nAthauhnu appeared suddenly at my side. \"Enjoy the ride, Little Roman!\" he chuckled. \"Soon the gods will give us more Helvetii dogs to kill. I'm happy the People of the Dark Moon did not remain with us. We will not have to share the glory with those pigs.\"\n\nAs I watched the enemy mass approaching, I was not at all sure that I appreciated Athauhnu's humor.\n\nAgrippa and Madog assumed positions in the center of our line. The enemy continued to approach at a trot. They were less than two hundred _passus_ from us. Agrippa knew he had to maintain enough distance to get our horses to a run before we made contact. He had just raised his sword to start us forward when the enemy halted.\n\nFor many heartbeats, we just sat there, looking at each other across a flat, grassy field. Then, one of the Helvetii riders came forward a few paces. He was wearing a high, plumed helmet and a chainmail _lorica_ ; he held up a large, battered, red shield. He raised a long Gallic _spatha_ over his head. He was yelling something across at us. I could only pick up a few words: \"dogs of the Romans . . . many heads taken . . . drinking mead out of your skulls!\"\n\nWhen he was done with his tirade, the Helvetii behind him raised a racket, screaming and hitting their shields with their swords and lances.\n\nThey seemed to be working themselves up to attack. At my side, Athauhnu said nothing. It seemed that his odd sense of whimsy had escaped even him.\n\nAgrippa was strangely paralyzed by the outburst. He just sat there on his horse. In my mind, I both urged him to order the advance before it was too late and prayed to _Fortuna_ that he would not.\n\nThen, the Helvetian chief made an unmistakable, insulting hand gesture in our direction and spit on the ground. He turned his horse and rode back into his own troop. The entire Helvetian horde seemed to collapse backward, following their chief away from us. Soon, we were staring at the backs of a withdrawing enemy.\n\nI could see Agrippa's shoulders sag with relief. Unless he were foolish enough to pursue, the fighting would be over for the day. I suspected Madog wanted to continue, but Agrippa reached over to Madog's mount and steadied both man and horse. We were finished for that day.\n\nFinished, except for having to explain this debacle to Caesar.\n\nWe discovered the legionary _castra_ less than five thousand _passus_ from the site of the battle. Caesar had moved most of the army forward to shadow the Helvetii. Agrippa settled the Sequani in a protected position between the camps. Then, we Romans and Madog went in search of Caesar's _praetorium_. We found it again in the _castrum_ of the Tenth Legion. When we approached the _praetorium_ tent, we spotted Labienus directing some soldiers.\n\n\"Agrippa, what in the name of Hades happened out there?\" he demanded as soon as he saw us. \"The Gauls are all in an uproar about some ambush . . . a massacre . . . They're saying the Helvetii are coming, and they're all running off into the hills!\"\n\nWe dismounted and approached Labienus. Agrippa asked, \"Do you want me to give my report to you or to the _imperator, Legate_?\"\n\nLabienus thought about it for a moment, then said, \"Let's go in and brief the boss. Crassus is already in there with him. How bad is it, Agrippa?\"\n\nAgrippa shrugged, \"Not good . . . Could have been a lot worse . . . but not good.\"\n\nAgain, Labienus stared at him briefly, then gestured to the tent entrance saying, \"Let's go see Caesar. Insubrecus! Come with us!\"\n\nCaesar was studying his maps. Crassus was with him, looking like he had just walked in from inspection parade. Clearly, the cavalry commander did not participate in his own unit's battle.\n\nWhen Caesar heard us enter, he asked Labienus, \"Titus, how confident are you that the six remaining cohorts of the Twelfth can hold the bridgehead against the enemy?\"\n\nLabienus shrugged, \"Six cohorts of trained Roman soldiers fighting from well-prepared, defensive positions on good terrain should be able to hold back a sea of barbarians, _Imperator_! Agrippa has returned. He has a report about the cavalry battle.\"\n\n_\"Bene_ ,\" Caesar responded, walking away from the map. \" _Nuntia_ , _Tribune_! Report!\"\n\nAgrippa, his helmet gripped firmly under his left arm, assumed a stiffer position of attention. _\"Imperator_ , I commanded three _alae_ of native cavalry . . . Sequani. . . . We encountered an enemy force of both cavalry and infantry, approximately\u2014\"\n\n\"Infantry?\" Caesar interrupted. \"There was infantry? How many? How equipped?\"\n\nAgrippa answered, \"Approximately a cohort . . . maybe as many as eight centuries . . . lightly armored . . . principally with spears . . . some farm implements . . . a few swords.\"\n\nCaesar nodded, \"This is the first I've heard of infantry . . . About six hundred, you say?\"\n\n_\"Nuntio recte, Imperator_ ,\" Agrippa answered.\n\nCaesar seemed to mull this over. Then, he said, \" _Bene_ . . . Please, continue your report.\"\n\nAgrippa continued, \"We encountered a force of infantry and perhaps ten _alae_ of cavalry approximately six thousand _passus_ to the west . . . When we arrived at the eighth hour, the enemy cavalry was engaging our cavalry\u2014\"\n\n\"Why did you not immediately attack the enemy, Tribune?\" Crassus interrupted. This was more of a challenge than a question.\n\nAgrippa turned his head to face Crassus. \"Legate, initially we were not in position to attack, and our horses were still winded from our advance\u2014\"\n\nCaesar interrupted him, \"We can get to that later, Crassus. Agrippa, continue your report!\"\n\n_\"A'mperi'tu', Imperator_!\" Agrippa responded, again facing Caesar. \"At a given signal, the enemy infantry attacked our cavalry in two wings from concealment. One wing attacked our men directly; the other tried to seal off their escape\u2014\"\n\n\"Do you think this was a planned attack, Agrippa?\" Caesar asked.\n\nAgrippa hesitated. Caesar was asking him to commit himself on a military matter, use his tactical judgment, in front of his commander.\n\nFinally, Agrippa stated, \" _Existimo esse rectum, Imperator!_ I believe that's the case! There was a prearranged signal given to initiate the infantry attack. I believe the Helvetii deliberately ambushed our cavalry and attempted to annihilate them.\"\n\n\"Bah!\" Crassus exclaimed. \"Those barbarians are not that sophisticated\u2014\"\n\n\"That's probably what Longinus believed right up until the time one of those unsophisticated barbarians was sawing off his head!\" Caesar silenced him. \"I want to hear the tribune's report, Crassus . . . After all . . . he was there!\"\n\nCrassus blanched at Caesar's swipe, then remained silent.\n\nAgrippa continued, \"My three _alae_ attacked the enemy blocking force, allowing the survivors of our cavalry force to escape the trap. I estimate three or four _turmae_ of friendlies escaped from the valley. The enemy chose not to continue the engagement and withdrew north, up the valley. We then returned here to report.\"\n\nThere was silence as Caesar digested what he was just told. Suddenly, Madog whispered, \"Tell the Caisar that coward Aedui run away from fight.\"\n\n\"Who's this?\" Caesar asked, seeming to notice Madog for the first time.\n\nAgrippa responded, \" _Imperator_ , this is Madog mab Guuhn, leader of the Sequani cavalry, _equitatis Sequanianis dux.\"_\n\n_\"Salve, Dux,\"_ Caesar nodded toward Madog. Then, he turned back to Agrippa. \"What is this about the Aedui running from the battle?\"\n\n\" _Imperator_ ,\" Agrippa reported, \"when the Helvetii sprung their ambush, about three _alae_ of Aeduan cavalry withdrew from the battle before the enemy could block their escape.\"\n\nCaesar was just about to respond when Madog interrupted again, \"No, Agrippa . . . that not Aedui horseman . . . that . . .\" Madog's Latin began to fail him, \"important man soldier . . . protect.\"\n\nCaesar looked intently at Madog, then at Agrippa. Finally, he noticed me. \"Insubrecus, can you tell me what this . . . this Madapguinus fellow is trying to say?\"\n\nI asked Madog in Gah'el, \"Who were those horsemen, Lord?\"\n\nMadog responded, \"They were _fintai_ , the personal bodyguard of Deluuhnu mab Clethguuhno. I recognized their standards. He is the _dunorix_ of the People of the Dark Moon, the commander of the king's stronghold, Bibracte. He is also the brother of the king, Duuhruhda mab Clethguuhno!\"\n\nI translated this for Caesar. He shrugged and said, \"Then, we are fortunate that this . . . this _dunorix_ person escaped. If he had been killed, it would have been a disaster for us and for the Aedui.\"\n\nMadog picked up enough of what Caesar said to explain further. \"The Caisar does not understand! The _fintai_ of the _dunorix_ would have led the horsemen of the People of the Dark Moon into that valley. Perhaps, Deluuhnu himself commanded them! They would never leave the place of battle while their people were in peril. It would be unthinkable! Shameful! They led their own people into that ambush and left them to be slaughtered!\"\n\nAgain I translated, and again Caesar seemed inclined to dismiss the warning. \"Just bad luck . . . It happens.\"\n\nAgain, Madog, \"Surprised and ambushed by the Helvetii in their own lands? Impossible! They had to know! They led them into it.\"\n\nAfter I translated that, Caesar challenged _\"Qua causa?\"_\n\nMadog shrugged, \"The _dunorix_ is married to a Helvetian woman, the daughter of the former _orgorix_ , the war chief of that tribe, the one who planned this invasion. Through his mother's people, he has formed marriage bonds with all the major clans of the Helvetii. With others, he has exchanged silver and hostages. He plots not only the destruction of the Caisar's army, but the destruction of his own people. He is not satisfied with being a mere _dunorix_ of a single tribe; he plots to be the _dumnorix_ of the Gah'el!\"\n\nIt took me a few heartbeats to sort that out. Madog had constructed a clever pun: _dunorix_ , the \"fortress-king\" of the Aedui, and the _dumnorix_ , the \"universe-king\" of the Gauls. I translated all this for Caesar, translating _dumnorix_ as _rex Gallorum_ , king of the Gauls.\n\nCaesar was now interested in what Madog was saying. \"Ask him if the king of the Aedui, Diviciacus, knows of this,\" Caesar demanded.\n\nMadog understood Caesar's question. He shrugged, \"Diviciacus . . . Duuhruhda mab Clethguuhno . . . How could he not know what is happening in his own house?\"\n\nWhen Caesar heard this answer, he nodded. \"Insubrecus, tell this Madicus . . . uh . . . tell the _Dux_ he has done a great service to the Roman people . . . Tell him we are grateful . . . I will direct my _quaester_ to issue his _turma_ rations . . . equipment . . . whatever they need if he will continue to serve the _populi Romani_.\"\n\nWhen I translated this for Madog, his eyes widened. A single, worn-out chainmail _lorica_ was enough to assure the loyalty of one of his riders for a lifetime\u2014a well-forged sword, two lifetimes. Caesar had just bought the loyalty of Madog and his troop for as long as he needed them.\n\nMadog bowed. _\"Me' Caesar . . . gratias ti'ago . . . maximas gratias . . . Tu' vir sum, ego!\"_\n\nMadog departed. He was going to cash in on Caesar's promise before the general could change his mind.\n\nCaesar seemed a bit amused. \"It's just a few battered _loricae_ and a few rusty swords! Are you Gauls always this enthusiastically grateful?\"\n\nI decided to ignore Caesar's \"you Gauls\" characterization and explained, \" _Imperator_ , iron armor and weapons are valuable to these people. By putting them in Madog's hands, he can ensure the loyalty of his soldiers for the rest of their lives. You just made him one of the most powerful _orgorixa_ , war chiefs, among his people. Even his own king must now give him respect. In fact, you are now his _orgorix_ . . . his war chief. He will never cease being Sequani, but now he is, as he said, _vir tuus, miles Caesari_ . . . your man, a soldier of Caesar.\"\n\nCaesar stared at me briefly, then said, _\"Simil'est clientela?_ Like clientage?\"\n\n_\"Recte, Imperator_ ,\" I affirmed, \"but to the death . . . He is honor-bound to you.\"\n\n\"An important lesson, Insubrecus,\" Caesar responded, nodding. \"What do you think of all this, Labienus?\"\n\nLabienus shrugged, \"I don't know where to start. It seems we cannot trust the Aedui. What concerns me is that we are about to enter their territory and will be dependent on them to supply the army with food and fodder.\"\n\nCaesar seemed to become impatient, \"Yes . . . yes . . . Agrippa . . . what happened today? From your point of view . . . what went wrong?\"\n\nAgrippa stiffened a bit. He was a mere _angusticlavus_ and on the spot again before the _imperator_ , his chief of staff, and his own legate, Crassus.\n\nHe tried to cage his response, \" _Imperator_ . . . uh . . . there were many . . . uh . . . variables . . . things no one could foresee.\"\n\nCaesar had no patience for this evasiveness. \"Come now, Agrippa . . . You're a Roman soldier . . . an officer . . . We're all comrades here . . . Nothing matters except this army and the interests of the Roman people . . . You were out there on the ground . . . If you were me, what would you have done differently?\"\n\nAgrippa nodded, \" _Imperator_ , it was a matter of command and control. Once the Gallic cavalry rode off, they did what they wanted to do. None of us could control them. There were too many of them, too few of us, and too little time to integrate leadership among them.\"\n\n\"So, you're saying this was _my_ fault?\" Crassus accused.\n\n\"No! No, Legate!\" Agrippa attempted. \"No one could have\u2014\"\n\nCaesar interrupted, \"Please . . . gentlemen . . . if it were anyone's _fault_ , it would be mine. I am the commander of this army. I cast my _pilum_ before I had a mark. No . . . if anyone is to blame for this . . . this . . . _calamitas_ . . . this debacle . . . it's me. Thank you for your _libertas_ , Agrippa, your candor. It's a Roman virtue, and I appreciate it. Now, gentlemen . . . please, leave me . . . except for you, Labienus. I need some time to think.\"\n\nAs we left Caesar's _praetorium_ , Crassus intercepted us. \"Tribune!\" he summoned Agrippa.\n\n_\"Adsum ti', Legate_ ,\" Agrippa responded, coming to attention.\n\n\"Tribune!\" Crassus started. \"You will never . . . I repeat . . . never speak out of line in or out of my presence again! You will report to me and only to me! If the _imperator_ needs to hear anything you have to say, he will hear it from me. Do you understand me clearly, Tribune?\"\n\n_\"Perspicue, Legate_!\" Agrippa almost shouted.\n\n_\"Miss'est, Tribune_!\" Crassus dismissed him. Then, he noticed me.\n\n\" _Tu_ ,\" he started. \" _Tu! Si tu non deliciae Caesari Galliculae_ . . . You . . . you . . . if you weren't Caesar's little Gallic bitch, I'd have your guts for boot straps! I can't stop you from telling the _imperator_ anything you want, but as long as you're assigned to my command, you stay away from me. I don't want to see you, hear you, or be reminded that you as much as exist. Do you understand me, boy?\"\n\n\" _Compre'endo, Legate_!\" I shouted.\n\n\"You'd better! If I have any more trouble with you, _boy_ , even Caesar won't be able to save your sorry ass!\"\n\nWith that parting shot, Crassus stomped out of the _praetorium_.\n\nI waited a moment before following him out. I was halfway through a string of expletives, \" _verpa_ . . . _mentula_ . . . _fututor canibus_. . .\" when I noticed Agrippa standing outside the tent. I immediately stiffened to attention.\n\n\"Tribune!\" I stammered. \"I didn't see . . . I mean . . . I\u2014\"\n\n\"Don't worry about it, _Decurio_ ,\" Agrippa said, putting his hand on my shoulder. \"We've had one hell of a day. I'm parched. Would you happen to know where we could find a nice jug of wine, or maybe some _posca? Coleonibus Martis_! I'd even settle for some of that horse piss the Gauls call cooru.\"\n\n# IX.\n\n# _Lente Festinamus_\n\n# WE HURRY SLOWLY\n\n_Caesar suos a proelio continebat ac satis habebat in praesentia hostem rapinis pabulationibus populationibusque prohibere_.\n\n\"Caesar held his troops back from battle. For the time being, he considered it adequate to prevent the enemy from looting, foraging, and plundering.\"\n\n(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nHe pursued the Helvetii for the next ten days.\n\n_Pursuit_ is perhaps too aggressive a word to describe what was happening. After the debacle with our cavalry, and the continuing uncertainty of the enemy's strength, Caesar refused to allow any of his units to become engaged with the Helvetii.\n\nOur cavalry tried to maintain contact with the enemy. On its face, this mission was not much of a challenge. We were following a massive horde, tens of thousands of men, women, and children. They were dragging everything they possessed along with them, with some of it on their backs, some strapped to beasts of burden, and the rest piled into slow-moving wagons. We could follow them from their trail: a path of crushed grass, burned huts, dead livestock, and uprooted foliage almost a thousand _passus_ wide.\n\nAlso, our infantry, which could easily march twenty thousand _passus_ a day without breaking much of a sweat, was barely moving ten. By the standards of the Roman army, the pace was leisurely. By the seventh day of our \"advance,\" the Twelfth Legion had caught up with us, having tucked the Tigurini back in along the Rhenus and having pulled down the bridges over the Arar. Caesar could now deploy all five of his legions against the enemy.\n\nAt the same time, the enemy was becoming quite aggressive. Not a day passed without an attack on one or more of our cavalry patrols. From a tactical point of view, this made perfect sense. Tacticians called it \"preventing reconnaissance.\" The enemy cavalry was screening the main body from our reconnaissance patrols.\n\nAthauhnu, by then all decked out in his Roman chainmail _lorica_ and shining brass cavalry helmet, told me that the Helvetii were too thoughtless and rash to execute an actual strategy. Most of the attacks on our cavalry were carried out by minor Helvetian war chiefs seeking heads, horses, and booty. Unfortunately, as it would prove, many of our senior officers believed this theory, too.\n\nHowever, these incessant spoiling attacks by the Helvetii were taking a toll on our Gallic allies. There was no booty to be had and much danger to be anticipated. Unlike the Roman army deserters, who would sneak away alone or in pairs in the dead of night, the Gauls just left. Entire _turmae_ rode away to return to their homes. When I asked Madog about it, he just shrugged. If there was no glory to be had, no riches to seize, why remain? By the tenth day, I doubted we could field five _turmae_ of native cavalry against the Helvetii.\n\nThe most pressing problems, though, were the men's rations and the fodder for the horses. The legions had gone through the ten-day marching rations that they had been carrying, but no resupply had arrived from our allies, the Aedui, whose territory we were marching across. Caesar had managed to bring up one supply column from Lugdunum back at the confluence of the Rhodanus and the Arar. But, we had advanced almost one hundred thousand _passus_ away from that supply depot. Although the legions had cleared a passable road along our route of march, Labienus was still doubtful as he calculated the amount of time the _impedimenta_ took to make the trip and its carrying capacity. The inescapable conclusion was that we could not adequately supply the army along that route. Unless the Aedui made good on their promises to supply the army, the men would have to be placed on reduced rations\u2014sooner rather than later.\n\nCaesar was resisting the recommendation to reduce his men's rations. He knew that Roman soldiers would tolerate many hardships on campaign\u2014their officers' arrogance, harsh discipline, even at times tactical incompetence\u2014but they would not tolerate logistical bungling for long.\n\nNever botch the men's rations!\n\nOn the afternoon of the eleventh day, after returning to the legionary _castra_ from patrol, I was summoned to Caesar's _praetorium_. I was still riding with Agrippa and Madog's Sequani. Agrippa was proving to be a good officer and a good companion. During one of our rides, he shared with me his desire to pursue a career in the army, not politics down in Rome. He was concerned about being subordinate to Crassus and his arrogance. But, avoiding Crassus while on active field duty didn't prove much of a challenge. The man didn't seem interested in doing anything that might tarnish his armor. Agrippa was also aware of my relationship with Caesar, but he never brought it up in any of our conversations.\n\nOne afternoon, Agrippa shared with me that he had a little brother named Marcus back home in Asisium. Their father had died when both boys were young, and Agrippa had to assume the role of _pater familias_ for his brother, younger sister, Vipsania, and their mother. Marcus was only fourteen, five years younger than Agrippa. Marcus idealized his older brother and wanted to follow him into the army when he was of age.\n\nWith a start, I realized that, as we were riding with Madog's Sequani through the lands of the Aedui, thousands of _passus_ north of the Rhodanus in _Gallia Comata_ , trying to avoid being ambushed by enemy war bands and trying to make contact with a large and menacing enemy force, I would be turning seventeen on my next birthday.\n\nI reported to Caesar's _praetorium_ as ordered. When I arrived, Labienus and Caesar were conferring in front of the operations map. Caesar was asking, \"What if we attempted to shorten our supply lines by moving the supply depot down the Rhodanus to a position directly south of our location? Surely, we could float our supplies and equipment down the river on barges. That would also shorten the route from our port in Massalia.\"\n\n\"That would shorten our supply lines,\" Labienus agreed. \"And, if the Helvetii continue to march west, we may eventually have to do just that. But, that will not solve our immediate problem of feeding the men and the livestock. Also, we'd have to reconnoiter and establish supply routes south to the river. That would strip away part of our infantry force, while at the same time we are facing a numerically superior enemy. No! The only solution is to draw supply from the Aedui\u2014either with their help, as they promised, or despite them.\"\n\nCaesar seemed to ruminate over what he had been told, finally asking, \"What are your recommendations, Titus?\"\n\n\"I see only two options, Caesar,\" Labienus answered. \"One, we send out foraging parties and take what we need from the surrounding settlements. Granted, we could pay for what we take, but we need food and fodder immediately. Second, you summon Diviciacus and demand he make good on his promises. Threaten to take hostages if need be, but you've got to put his _colei_ in the vice and get him moving!\"\n\nCaesar thought about that for a few heartbeats. \"What if we strip our ranks for _venatores_ and send them out to bring back game?\"\n\nLabienus shrugged, _\"Venatores_ would be useful to augment our rations with meat, but I doubt that they can supply the entire army with food. Besides, even if most of the hunters manage to avoid being picked off by enemy cavalry, how much game do you think they'll find within ten thousand _passus_ of our army after the Helvetii have gone through and stripped everything? And, hunting does nothing to alleviate our fodder shortages.\"\n\nIt was then that Caesar noticed me. \"Ah! Insubrecus! You ride with Agrippa's Sequani. How are the Gauls feeding their horses?\"\n\nI shrugged, \"They let them graze when they're in a secure location and far enough off the line of march so that there's foraging available. Sometimes, they just go into an Aedui village and demand food and fodder for keeping the Helvetians moving away. The farmers aren't altogether pleased with it, but with Helvetii and German raiding parties in the area, they recognize the necessity of keeping the Sequani on their side.\"\n\nCaesar thought about this briefly. Then said, \" _Bene_! There's our answer, Labienus. We'll put Diviciacus' balls in a vice, like you say. Summon him to me here. Meanwhile, we'll send out a limited number of foraging details to the nearby villages . . . detail the Eleventh and Twelfth. There are plenty of Gallic speakers among the soldiers in those legions. Remind the Gauls in the villages that it's the Roman army that's keeping the Helvetii and Germans at bay, and the Roman army needs to eat. We will pay for what we take. Detail a member of the _quaester's_ staff to accompany each detail. He'll designate what's useful to the army . . . Have him draw funds from the treasury for payment . . . standard rates for anything taken. Take care to leave the Gauls enough to survive until the next harvest. Rome is their friend, not their oppressor. Put a junior tribune, an _angusticlavus_ , in charge of each detail, with a _centuria_ of _muli_ from the legions. We must use moderation in this. The men will only take action to defend themselves or the supplies. I want no incidents with the friendlies. We are swimming in a sea of Gauls, so stirring up a storm is not in our interest. In fact, I will personally brief all the _angusticlavi_ and centurions before they go out. The message to Diviciacus should be clear: fulfill his promises to supply the army, or we'll take what we need.\"\n\nThis was a classic Caesarian performance: take in information, process it quietly and quickly, and then deliver an explosion of decisions and instructions.\n\nWhile Caesar spoke, Labienus was muttering, _\"Bene_ . . . _bene_ ,\" while furiously taking notes on a _tabula_.\n\nFinally, Caesar asked, _\"Habesne quaestiones ullae, Labiene_? Any questions?\"\n\n\" _N'abeo_ ,\" Labienus muttered, still writing.\n\n\"What do you think, Insubrecus?\" Caesar asked me. \"How will the Gauls react?\"\n\nI shrugged. \"Difficult to say, _Patrone_. Armed Roman soldiers are entirely different from Gallic cavalry to these people, but payment will help. The _quaester_ will need to pay the village headman, not the individual farmers. Also, it may be good to barter a bit. The Gauls consider that more . . . uh . . . gracious . . . more civil than just forcing a fixed price.\"\n\n\"Good point,\" Caesar agreed. \"Did you get that, Titus?\"\n\n\"Barter a bit . . . more civil,\" Labienus muttered not looking up from his _tabula_. Finally, he stopped writing.\n\n\"Titus,\" Caesar began, \"thank you! Go arrange the foraging details with the _quaester_ and the _primi pili_ of the Eleventh and Twelfth. Detail the tribunes yourself. That lad, Caecina\u2014the one who commanded the _vexillatio_ , the detail that escorted the Tigurini back to the Rhenus\u2014he'd be a good choice for this. But only our 'brighter lamps,' eh? Initially ten teams: five north and west, five south and west. Stay clear of the enemy. Also, find Agrippa and that Madocus _Dux_ who commands the Sequani cavalry and have them report to me. I want them to shadow the foraging details going north.\"\n\n_\"A'mperi'tu', Imperator_!\" Labienus said.\n\n\" _Bene_!\" Caesar acknowledged. \"Insubrecus! You stay! I need to talk to you about something.\"\n\nLabienus and his _tabula_ left the _praetorium_ on his mission, and Caesar gestured me over to his field desk. He gestured to a chair, _\"Sedeas, Insubrece_!\"\n\nCaesar picked up a couple of the _tabulae_ stacked on his desk. \"I've reviewed the work you've done on my journals, and I'm pleased . . . quite pleased. I've made a few minor changes.\"\n\nHe handed me the stack. I opened the first and recognized the opening lines of the first entry, _\"Gallia est omnis divisa in partes tres_.\" Then, I noticed where Caesar had inserted a correction. Caesar wrote, _\"nostra_ , in our language, they are called Gauls.\"\n\n\" _Patrone_ ,\" I said, \"we call ourselves _Gah'ela_ , which is close to the Latin word used in the text, 'Galli,' so there's really no need\u2014\"\n\n\"I understand that,\" Caesar interrupted. \"But, this book is not meant for language scholars and geographers. It's meant for the _plebes_ in the forum. The word _Galli_ represents the things of Roman nightmares: giant, bloodthirsty, invincible savages from the dark forests of the frozen lands to the north . . . and these are _our_ victories . . . _nostra_. Subtle, I agree, but sometimes _suggesting_ is more powerful than _stating_.\"\n\nI nodded my agreement. Caesar went on, \"But, that's not the reason I asked you to stay. I received a response from Consul Gabinius, about . . . about our little problem.\"\n\nWith that, Caesar had my complete attention. I replaced the _tabulae_ on his desk.\n\n\"I'm actually quite surprised at how quickly the consul responded,\" Caesar continued. \"Almost as surprised as I was by the response. In short, Gabinius denies all knowledge of any attempt on your life.\"\n\n\" _Patrone_! _Sed igitur quis_?\" I began.\n\nCaesar held up his hand to silence my protest. \"I believe the consul. He has no reason to lie and every reason to tell the truth in this matter. Since I made it clear to him that this was a matter of _patrocinium_ , and therefore a matter of the _dignitas_ of my family, he could very well have brought up the _dignitas_ of his family. You did put a shameful mark on his son's face with that _pugio_ of yours, and the consul could have horse-traded for some political advantage. But, he didn't. He simply said he and his son were not involved in this thing.\"\n\n\"I was only defending myself, _Patrone_ \u2014\" I protested.\n\nAgain, Caesar stopped me, \"I know that, Gai. I believe Gabinius understands that, too. But, I also believe this line of inquiry is a dead end.\"\n\nAs I was mulling that over, Caesar said, \"There is one more thing.\"\n\n\"What is that, _Patrone_?\" I responded.\n\n\"The consul has asked me to take on his son as a _tribunus militium_ ,\" Caesar stated. \"It seems that his posting to Greece . . . well . . . that fell through. Gabinius had to make some deals and pull in some markers to get elected, and that soured his son's posting to Greece. It seems that Gabinius and the proconsul to Greece are no longer in the same political camp. But, Gabinius is an associate of Pompey, and I need Pompey's good will in supporting my interests in Rome. So, I really have no choice but to\u2014\"\n\n_\"Sed, Patrone_!\" I burst out.\n\nCaesar again raised his hand to silence me, \"Gai! I have little real choice in this! It's politics! Besides, there's an old Roman saying, _'Vincinia amici teneantur hostes magis_. Keep your friends close, your enemies closer.' I'll find something to keep young Gabinius busy in the supply depot. He can keep Pulcher and the pack mules company. Or, I need someone to command the military port in Massalia. You'll never see him, and he'll be firmly under my thumb. His father and I both understand _patrocinium_ and _dignitas_. There will be no problems having him with the army . . . none . . . I will not allow it.\"\n\nIt made sense, but I was not totally convinced.\n\nCaesar changed the subject. \"I've been giving some thought to what you told me about the possibility of Roman interference with the Helvetii.\"\n\n_\"Audio, Patrone_ ,\" was all I could muster. My mind was still trying to sort out the Gabinius news.\n\n\"Ambitious Romans have been known to provoke barbarians on the borders of the _imperium_ for the sake of military adventures,\" Caesar explained. \"According to your sources, while Piso and Messalla were consuls, Orgetorix took Roman silver from a senatorial in order to provoke the Helvetii to abandon their homeland. Messalla is not a soldier, and Piso expected to be given the Syria province, not Gaul. Rome's golden boy, Pompey, celebrated his triumph for defeating Mithridates that year. Even he is not so ambitious that he would provoke a war in Gaul so soon after returning from Asia. That leaves Crassus. He certainly had enough gold to buy a mere Helvetian chief, and he was still smarting after Pompey stole his glory in the slave wars. The Senate only awarded him an _Ovatio_ for defeating the slaves, not the Triumph he desired. He had the opportunity and the motive, I think. Defeating the tribe that played a leading role in the destruction of Longinus' army would have assured him a Triumph. All he had to do was convince the Helvetii to invade our _provincia_ and then convince the Senate to give him the command. He had more than enough gold to accomplish both.\"\n\nI just nodded. All this was a bit beyond me, but I did recognize the name Crassus. The dressing down that Crassus Iunior had given me over the recent cavalry debacle was still fresh in my mind.\n\n\"That still begs the question why Romans would be still involved with the Gauls, now the Aedui, it seems,\" Caesar continued. \"Crassus' attention has turned east toward Parthia, so who is trying to buy betrayal from the Aedui, I wonder?\"\n\nI correctly understood this as a rhetorical question and kept silent.\n\n\"There are two distinct possibilities,\" Caesar continued. \"Pompey could be trying to engineer my defeat simply to take me down a few notches. He's protective of his reputation as Rome's premiere soldier, and he's not the type of man who shares his glory with anyone . . . But, no . . . he's now my son-in-law . . . married to my Iulia.\"\n\nI realized Caesar was talking simply to hear himself out, so I remained silent.\n\n\"Then there are the _Optimates_ in the Senate, the last, sorry vestiges of Sulla's excesses. They would do anything to bring me down. Cicero and Cato are too traditional to engineer the defeat of a Roman army in the field. Lucullus is filthy rich from the booty he brought home from Asia. He could buy half of Gaul, and he's still smarting from being relieved from his command so that Pompey could finish off Mithridates and celebrate a Triumph. But, he's an old man now. He seems content with his building projects and his fishponds. Bibulus has never forgiven me for upstaging him during our consulship. He tried to block my appointment as proconsul to Gaul. Pompey even thinks Bibulus is trying to assassinate him, but I doubt he's capable of a complex plot. He can hardly plan his day competently. Then there's Brutus. He's young, but not without ambition. And, Pompey killed his father. But, no . . . that can't be . . . He's Servillia's son . . . That would be close to patri\u2014\"\n\nCaesar suddenly interrupted his own monologue. He seemed to come out of his musings and return to his _praetorium_ in the hills of _Gallia comata_. He again noticed me sitting in front of his desk.\n\n\"This issue bears watching, Gai,\" he said. \"I think our friend, the one whose men ran from the cavalry battle, what did you call him? The _dumus rex_?\"\n\n\"Deluuhnu mab Clethguuhno,\" I offered. \"The _dunorix_ of the Aedui, the king's brother.\"\n\n\"That's the one!\" Caesar agreed. \"He bears watching. He may be the key to this.\" Then, Caesar seemed to have an idea. \"And, he may be just what I need to put pressure on his brother.\"\n\nAt that point, Caesar's _scriba_ stuck his head into the _cubiculum_ , \" _Imperator!_ The tribune, Agrippa, and his Gaul are here as ordered.\"\n\n_\"Bene,\"_ Caesar responded. \"Send them in!\"\n\nI rose as Agrippa marched into the room in full kit, helmet tucked under his left arm. He assumed a stiff position of attention and reported to Caesar, \" _Imperator, Lucius Vipsanius Agrippa, Tribunus Militium, defert ut imperatus!_ \"\n\nI noticed that Madog hung back, watching the Roman show.\n\n\" _Laxa, Tribune_ ,\" Caesar responded. \"Sit down here. Madocus _Dux_! Please . . . sit! I wish to talk to you about a mission. Please sit, all of you.\"\n\nWe sat, and Caesar called out to his clerk, _\"Scriba_! A jug of _posca_ . . . and four cups!\"\n\n\" _A'mperi'tu', Imperator_ ,\" responded a bodiless voice from beyond the tent flap.\n\nI wondered briefly how Caesar managed to have a supply of _posca_ while many of the legionary _muli_ were considering boiling their boot straps for soup.\n\nAs we all settled in front of Caesar, he began, \"While we're waiting for the _posca_ , tell me how things are going out there.\"\n\nAlways an awkward question to answer in front of the commander, especially considering how prickly the absent legate, Crassus, was of his \"prerogative.\"\n\nAgrippa, as the senior officer on our side of the desk, waded in. \" _Imperator_ , the main body of the enemy continues to move slowly west, as you know. We notice that their depredations have lessened considerably since they left the lands of the Sequani.\"\n\nMadog audibly grunted when Agrippa stated this.\n\n\"We believe that it is due to the pressure our army is putting on them . . . We are still experiencing aggressive counter reconnaissance, mostly from their cavalry . . . But small infantry units, usually no more than a _centuria_ in size, have, at times, attempted to engage us. According to your orders, we have avoided contact whenever possible\u2014\"\n\nCaesar interrupted Agrippa's report, \"Bene . . . What about you, Madocus? What are you seeing out there?\"\n\nMadog shrugged, \"Agrippa, to be truthful . . . very dangerous . . . Helvetii and German everyplace . . . want fight.\"\n\n\"Bene,\" Caesar agreed. \"But, I heard you react when Agrippa said that enemy damage has lessened since the enemy entered Aedui territory. Why do you think that is?\"\n\nMadog shrugged again, \"Aedui buy them, I think . . . Give Helvetii silver . . . No burn, no steal . . . Give Helvetii food . . . maybe you food . . . Kill Sequani . . . No kill Aedui.\"\n\nCaesar's clerk entered the _cubiculum_ with a pitcher of _posca_ and cups. As he set them down on the desk, Caesar asked, \"Were you sure to test the _posca_ , Ebrius? Is it up to your standard?\"\n\n\"I assure you it is, _Imperator_ ,\" the clerk responded without batting an eye. \"And, I guarantee it's not poisoned.\"\n\n\"Glad you're looking out for me,\" Caesar responded. I assumed this was a well-rehearsed routine between them. \"You may go. We'll pour ourselves.\"\n\nAs the clerk, Ebrius, left, Caesar asked, \"Will you do the honors, Insubrecus?\"\n\nAs I poured, Caesar began his briefing. \"Tomorrow, I'm sending out ten foraging details . . . wagons and a century of _muli_ from the Eleventh and Twelfth Legions. There'll be an _angusticlavus_ in command and a centurion with the infantry. I'm sending one detail straight north. They'll be the most exposed team . . . close to the enemy . . . hilly country . . . perfect for ambushes. I want you to screen them to the west and make sure they have a clear route out and back . . . Pretty standard cavalry mission. They'll pull out at first light tomorrow when the signal for ending the fourth watch sounds.\"\n\nWhen Caesar finished, we all took the opportunity to sip the _posca_. Ebrius was right. The _posca_ was prime. It hardly burned the back of my throat at all as it went down; it had a nice aftertaste of honey and herbs.\n\nCaesar continued, \"One of my goals is to test the attitudes of the Aedui toward us and to find out why they haven't come through with the supplies they promised us. If there is any evidence of a conspiracy between the Aedui and the Helvetii, that's where you come in, Insubrecus. I want you to be my fly on the wall. Pick up any intelligence you can from the villagers.\"\n\n_\"A'mperi'tu', Imperator_!\" I acknowledged, putting my cup down on the desk.\n\n\"I have summoned Diviciacus, the king of the Aedui,\" Caesar continued. \"I expect him no later than tomorrow afternoon, but I plan to let him cool his heels, at least until the day after. I need to be briefed by you on what you find out in the field _before_ I face him. In fact, your information will go a long way in my planning a strategy for dealing with the Aedui. So, I expect to see all three of you in my _praetorium_ tomorrow\u2014no later than the first night watch.\"\n\nBoth Agrippa and I muttered _\"A'mperi'tu'!\"_ Madog just nodded.\n\nCaesar took a long draught from his cup. He was enough of a trooper that the _posca_ didn't seem to affect him at all. But, then again, if anything affected Caesar, he gave little indication of it.\n\nHe continued, \"By the time you get back in, I expect the _castra_ to have moved about ten thousand _passus_ west of this location. But, your best bet is to return to this location and follow our trail out. If for any reason the army doesn't relocate tomorrow and you go directly to where you expect us to be, you'll run right into the Helvetii.\"\n\nWe had already figured that one out, but we all nodded as if Caesar had given us sage advice.\n\n\" _Quaestiones ullae?\"_ Caesar asked. \"Any questions?\"\n\n_\"N'abeo, Imperator_ ,\" Agrippa and I answered. Madog just shrugged and grunted.\n\n_\"Bene,\"_ Caesar said. _\"Miss'est_ . . . Get some sleep! You have an early start in the morning. Insubrecus . . . stay behind, please!\"\n\nI rose as Agrippa and Madog left. As soon as they were gone, Caesar said, \"Gai, I'm posting some dispatches, instructions to one of my _comites_ , my deputy down in Aquileia. The courier departs tomorrow at first light. I understand that you come from a small farm just outside Mediolanum. Perhaps you'd like to write a short note to your parents? Let them know you're safe . . . what you're doing. I know military service can be worrisome, especially for mothers. What's her name? Valeria, is it not? My clerk can fit you up with some papyrus and a pen. Place your message in the dispatch bag, and I'll see that it's delivered. Your parents will be proud of your promotion. I'll give instructions for the courier to wait for a response.\"\n\n\" _Multas gratias, Patrone_!\" I stammered.\n\nI was staggered, not only by Caesar's kindness, but also by the fact he actually knew where my home was, and he knew my mother's name. Later, many who claimed to know Caesar well, certainly better than I did, claimed this was all part of an act, an elaborate ruse to instill loyalty and devotion in those around him. I never learned the truth of it. Caesar was certainly a consummate actor, always on stage, but he was also a man, a human being. And, a kindness imitated is still a kindness done.\n\nOn my way out of the _praetorium_ , I did scribble out a quick note to my parents, mostly to mama. After all these years, I don't remember what I wrote. Probably some bromides about being well, about the weather being warm and dry, about having plenty to eat, about being in no danger\u2014the sort of stuff a young soldier thinks his mother should hear, the sort of stuff I wrote to her until the day arrived when writing home didn't make sense to me anymore.\n\nMama has long since gone to the Land of Shadows. A hundred Orpheuses with a hundred lyres cannot bring her back. What I wouldn't give to be able to write one more message to her and have the courier wait for her response. I'm sure I'd take the opportunity more seriously than I did that evening in Caesar's _praetorium_.\n\nBefore dawn the next morning, I joined Agrippa and the Sequani cavalry. The foraging parties were assembling in a secure, open area between the legionary _castra_. In the dark, the assembly was a confused mass of dark, moving shapes, alive with the creaking of wagons, clanking and scraping of armor and weapons, jangling of harnesses, whickering, snorting and stomping of horses anxious to be on their way, complaints and curses in Latin and Gallic, and the smell of the ever-patient mules.\n\nWe sorted things out and found the foraging party we were supposed to screen. It was commanded by the _angusticlavus_ , Caecina, who had commanded the _vexillatio_ , the detail escorting the surviving Tigurini back to the Rhenus. He was again paired with Sanga, the centurion from the Twelfth Legion. I stood back as Agrippa sorted things out with them.\n\nI found Athauhnu's _turmae_ and was surprised when he greeted me in Latin, _\"Salve, Decurio!_ \"\n\n\"Finally assuming some civilized manners, Chief?\" I responded in Gah'el.\n\n\"Latinly, _Decurio_ \" he responded again in broken Latin. \"Madocus _Dux_ to say we Roman soldier . . . to take Roman gift . . . sword . . . silver . . . now to speak Roman word.\"\n\n\" _Bene!_ \" I encouraged him. I had gotten used to Madog's fractured Latin, even Spina's Aventine-Hill gutter patter. Figuring out what Athauhnu was trying to say shouldn't prove too much a challenge. \"I'll call you Adonus _Decurio_ from now on when we speak Latin.\"\n\n\"Adonus,\" Athauhnu tried his new Roman name. \"Is _bene_ . . . Adonus _Decurio_!\"\n\nI heard Agrippa call for us, and we found him at the northern edge of the assembly. The eastern sky was beginning to turn dark purple. We needed to get this show on the road if we were going to cross the departure line according to Caesar's wishes.\n\nWhen we found Agrippa, Caecina was standing with him; Sanga was hovering behind him, the silhouette of his helmet's transverse crest identifying him as a centurion.\n\n\"The tribune and I have worked out a plan,\" Agrippa began nodding toward Caecina. \"Madog, we need to split up your cavalry . . . Insubrecus _Decurio_ , you will accompany Att Owen's _ala_.\"\n\n\"Adonus _Decurio, Tribune_ ,\" Athauhnu corrected him.\n\nAgrippa hesitated for a heartbeat, not sure what had just happened. I was about to say something when he shrugged and continued, \"The _ala_ of Adonus _Decurio_. Our target is the small _vicus_ we scouted the day before yesterday, the one with the large, round building or barn in the center and the perimeter fence built from brambles. Your mission is to scout the route directly to the village for Caecina's foraging party. But, do not enter the village. When you reach it, move around it, preferably to the west if the terrain and the situation permit. Seal its flank and rear . . . No one in, no one out. I don't want them hiding the cattle and trying to carry off the grain . . . _Compre'enditis_ _vos_?\"\n\n\" _Compre'endo, Tribune_!\" I answered. I wasn't sure how much of this the newly minted Adonus _Decurio_ was getting.\n\n\"You are to avoid contact with the enemy,\" Agrippa continued. \"If you detect the enemy, fix their position and move south back to Caecina's force. We'll let the _muli_ , the legionary grunts, sweep away the trash.\"\n\nI heard Sanga guffaw and slap the hilt of his _gladius_ when Agrippa said this. Agrippa continued, \"I will be with Madocus _Dux_ and the two remaining _alae_. We will be paralleling the ridgelines to the west of the route of march to seal it off from enemy reconnaissance or raids. Standard visual and aural signals remain in effect. _Quaestiones_?\"\n\n_\"N'abeo, Tribune_ ,\" I responded.\n\nAthauhnu said nothing. I was positive that the Latin terms \"paralleling,\" \"reconnaissance,\" and \"standard visual and aural signals\" were well beyond his Latin. I could hash it out with him in the saddle.\n\n\"I assume the same order of march on the return,\" Agrippa finished. \"But, we'll talk again when we're on the objective. Insubrecus _Decurio_ . . . are you and . . . and . . . Adonus _Decurio_ ready to move out?\"\n\n_\"Parati, Tribune_!\" I answered.\n\n\"Bene!\" Agrippa said. \"Stay in contact with the foraging party all the way to the objective. You lead out. I'll follow and peel off to the west before we enter the hill country to the north. Caecina, as we discussed, lead out your foraging party about a quarter of the hour after we depart, about when the eastern sky starts to go from red to orange. That should put us a few hundred _passus_ in front of you.\"\n\n_\"Constat, Agrippa_!\" Caecina said.\n\n\"Let's mount up, Insubrecus!\" Agrippa concluded. \"May _Domina Fortuna_ favor us this day!\"\n\nIn the darkness, I heard mutterings, _\"Domina Bona_ . . . _Fortuna . . . Dea Bona_.\" I found myself patting my _lorica_ where my medallion hung.\n\nInitially, we made good time, despite the darkness. I was again riding the black mare with the white snip, on which Valgus, my _decurio_ in Caesar's praetorian detail, had trained me. Valgus believed that man and horse were a team; they trained together, and they fought together. Like _gemini_ , they had to trust each other completely. I felt I was getting to that point with my horse, Clamriu. I could feel her moods, sense her enthusiasms, her fatigues, and her fears as we rode. She responded easily to the reins or to my knee and leg directions.\n\nValgus didn't believe in Roman soldiers giving their mounts names; he considered it _alienus Romanitati, barbare_ , as he put it. But, all the great heroes of Gran'pa's sagas rode noble steeds with their own identities. So, I named my mare Clamriu after one of Arth Mawr's mounts. Even then I knew it was a childish affectation, but one I still wished to embrace.\n\nWe reached the hills, and Agrippa's detail moved off to the west. We had to go more slowly now. Not only because of the darkness, but also because this was terrain that favored close ambush. Athauhnu sent _exploratores_ , a two-man point detail, out about twenty paces ahead of the main body. It was still too dark to make flankers practical, so we were vulnerable as we made our way slowly over the narrow trail. Before the encroaching hills had obscured the eastern sky, I noticed that the horizon was revealing itself in a golden glow. Caecina and his foraging party should be well on their way from the assembly area.\n\n\"Adonus _Decurio,\"_ I called to get Athauhnu's attention.\n\n_\"Quid volare_ tu?\" he responded in Latin. \"What to fly you?\"\n\nI realized quickly that he meant to ask what I wanted. I switched to Gah'el. \"I think we should leave two riders at the trail head to make sure the foraging party doesn't miss it.\"\n\n\"Good idea,\" he responded. \"Damn Romans could get lost between a tent and a latrine.\"\n\nI didn't think to remind _'_ Adonus _Decurio'_ whose side he was on at the moment. The fact that he had stopped referring to me as \"Little Roman\" and seemed to except me from the innate disorientation of all Romans marked a major advance in our relationship.\n\nAs soon as there was adequate light, Athauhnu posted two riders out on each side of our column as flank security. Where the terrain was open and flat, they ranged out almost fifty _passus_ from our line of march; when the terrain closed in, they virtually rejoined our column.\n\nOur \"command group\" consisted of Athauhnu, me, a trumpeter with a hunting horn hanging strapped across his chest and chinking against his chainmail lorica as he rode, and a _vexillarius_ , carrying a bright red pennant attached to a long spear. We rode at the head of the column, about twenty passus behind our two-man point element. Occasionally, when the terrain flattened and the trail ran straight, we caught sight of the two riders. The rest of our _ala_ , twelve troopers, rode in a staggered formation along the trail. To our rear, two riders kept contact with Caecina and the wagons.\n\nWe were an _ala_ of twenty-four riders, small by Roman standards, but considering the Gauls' peculiar sense of _officium et fidelitas_ , quite adequate.\n\nAgain, by the standards of the Roman army, we appeared to be a motley collection of troopers, a band of Gallic brigands. We sported an assortment of bronze and steel helmets: some conical, some round, some with cheek guards, some without. About half the men wore chainmail; a few wore leather; and several had nothing more than padded jackets. Most of the men wore Gallic long swords; the rest had short Roman _gladii_.\n\nI noticed a couple of our troopers were carrying _iacula_ , light javelins, in long quivers behind their saddles. To me, such weapons didn't seem to serve any purpose; they seemed too light to be of any use in combat against armor. I asked Athauhnu about it.\n\n\"Ah,\" Athauhnu answered, \"we call that _uh gae_. It's a weapon given to my people by the god, Lugus, when we departed from the Land in the Skies.\"\n\nI recognized the word. In Gran'pa's stories, the ancient heroes had magical weapons called _gaea_ that dripped gore, glowed with anger in combat, and once thrown, always tasted enemy blood. I had never actually seen one before. I was expecting a substantial weapon, something worthy of Homer's epic\u2014a long, heavy ash shaft thrusting a long, black, razor-sharp, steel blade. These javelins that the Sequani troopers were carrying seemed insubstantial, flimsy.\n\nAthauhnu noticed my skepticism. \"Lend me your shield, Arth Bek,\" he requested, holding out his hand.\n\n\"What?\" I questioned.\n\n\"Your shield,\" Athauhnu insisted, \"let me have it.\"\n\nI shimmied out of my shield, which was hung across my back, and handed it over.\n\n\"Emlun! To me!\" he called out to one of his troopers.\n\nA rider from the middle of our column came forward. He looked to be a boy of about my age, maybe a bit younger. He had a round, bronze helmet bouncing on top of his head, and he wore a padded jacket. His _spatha_ , hanging across his chest by a leather baldric, looked to be at least two sizes too big for him.\n\n\"Yes, Athauhnu . . . uh . . . Chief,\" he said to Athauhnu.\n\n\"Emlun,\" Athauhnu instructed, \"take this Roman shield up the trail about a hundred _passus_ and hang it facing us on a tree right next to the trail.\"\n\nThe one called Emlun stared at Athauhnu for a couple of heartbeats, not sure what was going on. Then, he shrugged, took my shield, and rode ahead down the trail.\n\nAthauhnu watched him go. \"My cousin,\" he explained, \"my mother's sister's son . . . a good lad . . . Needs some seasoning.\"\n\nWe watched Emlun pound down the trail, finally stopping to hang my shield on a tree. Even from that distance, I could see the red boar of the Tenth Legion painted around its center. Emlun waved to us when the shield was secure.\n\nAthauhnu waved his cousin back to the column. As Emlun was cantering back, Athauhnu called out again, \"Guithiru! To me!\"\n\nThis time, one of the veteran troopers rode up to Athauhnu. He was arrayed in Roman chainmail, the cheek-pieces of his bronze _galea_ helmet secured tightly under his chin.\n\n\"Yes, Chief!\" he saluted.\n\nAthauhnu pointed down the trail to my shield. \"The boar . . . kill it!\"\n\nGuithiru grinned, \"Where do you want me to hit it?\"\n\nAthauhnu shrugged, \"You kill a boar by spearing its heart!\"\n\nGuithiru nodded. He reached behind his saddle and extracted one of his _gaea_ from a quiver. He waited a few heartbeats for Emlun to clear the trail and rejoin the command group. Then, he shouted and kicked his heels back into his horse's flank. His gray, dappled stallion exploded down the trail, gaining speed as he approached his target. I watched as Guithiru rose up off the saddle with his knees locked into his horse's flank. This was a signal for the stallion to increase his speed. Guithiru's hand and arm holding the _gae_ came up, then thrust forward in a blur. Even from a hundred _passus_ away, I could hear the sharp thwack as the javelin punched into its target. Guithiru pulled his stallion up some thirty _passus_ beyond my now transfixed shield. He waved back toward Athauhnu.\n\n\"Let's go see if Guithiru killed his boar,\" Athauhnu invited.\n\nWe cantered down the trail. As we approached the tree, I could see that Guithiru had indeed \"killed the boar.\" His _gae_ had pierced the image of the red Tenth Legion boar just behind its front legs, right where the animal's heart would have been. The javelin had punched completely through my shield and had pinned it fast to the tree.\n\n\"That is how deadly a _gae_ is in the hands of a warrior!\" Athauhnu remarked.\n\nGuithiru rode up to inspect his handy work. \"Perhaps we should just leave it there like that as a warning to our enemies,\" Guithiru suggested.\n\nAthauhnu shrugged, \"The Romans would force our _decurio_ to buy a new one.\"\n\nGuithiru laughed at the folly of Roman quartermasters and dismounted. After a few twists and pulls, he managed to extract his javelin from the tree, but he had to cut it out of my shield with his _pugio_. He handed the round _parma_ back up to me. There was a small, jagged hole in its middle, straight through the heart of the red boar.\n\n\"Hold it away from your body if we get into a fight, _Decurio_ ,\" Guithiru suggested. \"Maybe your chainmail will be enough to stop the point.\" Then, with a laugh, he resumed his position back with the column.\n\nI inspected the ruin of my shield. I had been considering trading it for an oblong, Gallic cavalry shield. It would be a bit heavier than my _parma_ , but it would give better protection for the upper legs and head. This little demonstration convinced me.\n\nAbout an hour later, crossing our trail, we discovered fresh tracks left by an unknown force. Since there was no reason for friendlies to be in the area, we had to assume they were hostiles.\n\nWe were crossing a small valley split by a stream running toward the southeast. We could easily see our two scouts on the other side of the stream, some fifteen hundred _passus_ ahead of the column. They had stopped, and one of the scouts seemed to be examining something on the ground while the other remained mounted and alert. Suddenly, the mounted scout whistled to get our attention and beckoned us forward.\n\nAthauhnu acknowledged the summons. He whistled to get the attention of our flankers, who were operating some thousand _passus_ up and down the valley on either side of the trail. He signaled them to halt; then he indicated that they should be vigilant to their front.\n\nThen, he called for Guithiru.\n\n\"What do you wish, Chief?\" the warrior asked when he joined the command group.\n\n\"Our scouts have found something,\" Athauhnu explained. \"We're going forward. Keep the men here until we get back. Keep them alert.\"\n\n\" _Uhr wuhf ifitho, a pen_ ,\" Guithiru acknowledged. \"Yes, Chief!\"\n\nAthauhnu and I rode forward to meet the scouts. As we approached them, the mounted scout, a rider named Alaw, greeted us: \"We found a trail . . . riders moving up the valley.\"\n\nWe dismounted and walked over to the scout, who was examining the ground.\n\n\"What is it, Rhodri?\" Athauhnu demanded.\n\nThe one called Rhodri rose, \"About twenty riders . . . moving northwest . . . not in a great hurry . . . Horses are shod . . . heavy burdens . . . Armored men, I think.\"\n\n\"How long ago?\" Athauhnu asked.\n\nRhodri shrugged, \"Not long . . . dawn . . . a little earlier. I'll follow the trail through the grass. Maybe I can find some fresh droppings.\"\n\n\"No need,\" Athauhnu said. \"Shod horses and armored riders, you said?\"\n\nRhodri nodded.\n\n\"Not Helvetii,\" Athauhnu said to me. \"Not all their horses are shod, and most of their cavalry aren't equipped with heavy armor like the Romans.\"\n\n\"Not Helvetii?\" I questioned. \"Then who?\"\n\nAthauhnu shrugged, \"Maybe People of the Dark-Moon . . . a group of Aedui . . . This is their territory. If it were a rich noble's _fintai_ he would equip his bodyguard with armor and equipment.\"\n\n\"There's no Roman cavalry operating up here,\" I affirmed. \"We'll assume they're hostile, Helvetii or not.\"\n\nAthauhnu nodded.\n\n\"What's off in that direction?\" I asked Athauhnu, gesturing to the northwest where the riders had gone.\n\nAthauhnu shrugged. \"I don't know. There seems to be a ridgeline leading off to the north. Bibracte, the Dark-Moon king's fortress is to the northwest . . . I don't know . . . Our objective is directly north from here . . . maybe another seven or eight thousand _passus.\"_\n\nI looked up to the northwest where the valley twisted into a heavily wooded ridgeline. \"We're probably under observation out here in the open,\" I said to no one in particular.\n\n\" _Uhr wuhf uhn cuhtino_ ,\" Athauhnu answered. \"I agree. We should push on to the village.\"\n\n\"We should,\" I agreed. \"But, I need to send a rider back to the Roman column with a message.\"\n\nAthauhnu nodded; he whistled to get Guithiru's attention. \"Send my cousin up to me!\" he called.\n\nSoon Emlun galloped up to our position. \"The _decurio_ has a job for you,\" Athauhnu informed him.\n\n\"What is it, _Decurio_?\" Emlun asked.\n\n\"I want you to ride back to the Romans with a message,\" I began. \"Report to the tribune, Caecina. He's the officer wearing plate armor with a red plume in his helmet. Tell him we believe there is enemy cavalry operating in the area . . . a group of twenty to thirty, heavily armed, positioned on the high ground to the northwest. Repeat that for me.\"\n\nEmlun did. I continued, \"Approach the Roman column carefully. They'll probably have a _contubernium_ , eight infantry . . . to their front, walking point. Most of them can't tell the difference between you and a _Helvetius_ . . . and they don't much care. They'll unload a few _pila_ at you, then issue a challenge.\"\n\nEmlun laughed. \"If I can't get close to Romans without being seen, I should just go back to my father's farm and help the plant the summer crops.\"\n\n\"Be that as it may,\" I cautioned, \"just be careful! Some of those troops speak _Gah'el_ , so you'll be understood. Now, repeat the message again for me.\"\n\nEmlun did, and I sent him galloping down the trail toward the Roman foraging party.\n\nWe saw no sign of any threat for the next hour. Emlun rejoined our column to say that Caecina had acknowledged my message. We were climbing the last ridgeline before reaching our objective. I estimated the Aeduan _vicus_ to be no more than a few hundred _passus_ distant.\n\nSuddenly, Rhodri appeared at the top of the ridge ahead of us and whistled sharply. Then, he pointed to the northwest. There, in a small clearing at the top of a hill, where our ridgeline twisted up into the highlands, were four riders.\n\nAthauhnu and I cantered to the top of the ridge to get a better look. The riders remained stationary. They were obviously observing us and felt no need to remain hidden.\n\n\"Who are they?\" Athauhnu demanded of Rhodri when we reached the top.\n\n\" _Ni Rhufeinig_ \" Rhodri shrugged. \"Not Romans . . . Well-armed . . . Dark-Moon warriors, I think . . . king's men . . . _fintai_.\"\n\nI looked toward the group. They were carrying no standards or pennants, but the sun glinted off their armor. \"They seem interested in us,\" I said to no one in particular.\n\n\"Indeed, they do!\" Athauhnu agreed. \"And that village we're trying to reach.\"\n\n_\"Edruhch, a pen_!\" Rhodri said suddenly, pointing east into our line of march. \"Look, Chief!\"\n\nWhen I looked, I could see a smudge of grayish-black smoke staining the sky about where the village should be.\n\n\"That can't be good,\" I muttered. Then, I whistled back to our main column. \"Send up Emlun!\"\n\nAthauhnu's cousin soon joined us. \"I have another message for the Romans,\" I told him.\n\nEmlun acknowledged me, and I began my report, \"Tell Caecina that we have made visual contact with an unidentified force of cavalry approximately one thousand passus to the northwest on the high ground . . . four riders, well-armed and armored. Also, there is evidence of hostile activity at our objective. Repeat that back to me.\"\n\nHe did, so I continued, \"Take an escort with you . . . another warrior for security. Be careful! We're under enemy observation. Don't get ambushed!\"\n\nEmlun nodded vigorously. \"I'll take Idwal with me, if you approve, Chief,\" he said. He was excited to be in contact with the enemy.\n\nAthauhnu grunted in the affirmative.\n\n\"Repeat the message again!\" I ordered.\n\nEmlun did. _\"Redi guhvluh!_ \" I told him. \"Ride quickly, but keep your eyes open!\"\n\nEmlun flashed me a wide grin. Then, he was off down the hill.\n\n\"He still thinks this is a game,\" Athauhnu commented.\n\nThis time I grunted in the affirmative. Emlun and I were of an age, but somehow I had changed.\n\n\"Rhodri!\" I called to get the scout's attention. \"You and Alaw get as close to that village as you can without being detected. Then, get back to us. I want to know what we're riding into.\"\n\nRhodri quickly glanced over to Athauhnu, who nodded. Then, he rode off toward the _vicus_.\n\nAthauhnu whistled down the hill and gestured the column to join us.\n\n\"I didn't mean to 'ride your horse,'\" I told him, \"with Emlun and Rhodri, I mean.\"\n\nAthauhnu just shrugged, \"You're a warrior . . . _uhn pen_ . . . a chief . . . You're doing your job. You must be able to act without the approval of others. You must be willing to take responsibility for what happens. Emlun's a good choice. He rides light, and if we get into a mess down the trail, he's better out of it. And Rhodri knows his job. If the _poblo r'avon_ or any of their _cuun almaeneg_ , German dogs, are waiting for us down the road, he'll see them before they see him.\"\n\n_\"Poblo r'avon_?\" I questioned. \" _Gentes fluminis_ . . . River People?\"\n\n\"That's what we call the Helvetii,\" Athauhnu said. _\"Poblo r'avon_. When this is over, I'll be glad to drown every last one of them in that bloody river of theirs, the Rhenus.\"\n\nWe rode forward down the ridge, toward the Aeduan village. The smudge of dark smoke in the sky became increasingly larger as we approached. We had covered about two-thirds of the distance when Alaw met us on the trail.\n\n\"Report!\" I snapped, not waiting for Athauhnu.\n\n\"We rode to a spot overlooking the village,\" Alaw said. \"The smoke is from the large round house in the center. It's been burned . . . yesterday . . . no earlier . . . still smoldering. No sign of any enemy activity in the village. Can't tell what's going on in the hills to the north, but it seems quiet up there too.\"\n\n\"Where's Rhodri?\" I asked.\n\n\"He's watching the village from concealment,\" Alaw answered. \"If anything happens, he'll get back to us before we ride into it.\"\n\n\"What do you think?\" I asked Athauhnu.\n\nHe was about to respond, when Emlun rode up to us from the column.\n\n\"Did you report to Caecina?\" I asked him.\n\n\"I did!\" Emlun said. \"The Roman chief says we continue with the mission unless prevented.\"\n\nI looked up toward the northwest. Our watchers had disappeared. I wondered if Agrippa and the rest of Madog's riders had spooked them\u2014or maybe they were up to something else.\n\nAthauhnu brought me back from my musings. \"Arth Bek! I want to see that village with my own eyes before our band arrives. We should ride forward with Alaw.\"\n\nI nodded to Athauhnu, then said to Emlun, \" _Diolkh_! Thanks! Rejoin the column! Send Guithiru up to us!\"\n\nAgain, Emlun flashed his boyish smile and turned back toward the column. I heard his whistle and his calling out to Guithiru. So much for operational security! Now every Helvetian and Kraut within two hundred _passus_ knew we were there.\n\n\"I suppose you've noticed that our friends on the hill have left us,\" I said to Athauhnu.\n\nHe nodded and said, \"A while back.\"\n\nGuithiru approached us.\n\n\"The _decurio_ and I are going forward to scout the village,\" Athauhnu told him. \"When we send back to you, I want you to bring the rest of the band up to us.\"\n\n\" _Uhr wuhf ifitho, a pen_ ,\" Guithiru acknowledged. \"Yes, Chief!\"\n\n\"Take us in, Alaw!\" Athauhnu said.\n\nAlaw nodded and turned his horse back up the trail.\n\nWe followed Alaw at a slow trot for about five hundred _passus_ down the winding trail. The forest closed in on both sides. When we came to some rising ground, Alaw pulled up. He looked around for a few heartbeats and seemed to see what he was looking for.\n\n\"We dismount here,\" he told us, hopping down from his horse. \"Rhodri's just ahead.\"\n\nWe pulled our mounts off the trail, and I saw Rhodri's bay. We dropped our reins. The horses were trained to war. They would not move. Alaw led us up the hill, along an almost invisible path. We were just about to the top when Rhodri's voice challenged us.\n\n\"Rhodri,\" Alaw answered, \"I have the chief with me.\"\n\nRhodri suddenly appeared out of the brush. \"Come! Look!\" he said and gestured us forward.\n\nRhodri led us to his well-concealed observation point. \"Helmets,\" he hissed at us.\n\nWe removed our helmets so they would not betray our presence by reflecting sunlight. Then, we looked down into a small valley. In the middle, along a running stream, was the village. In its center stood a blackened, smoldering ruin of what had once been a large round-house\u2014a chief's hall or a storage barn. There were other, smaller round-houses and huts in the complex, but they seemed untouched. The cluster of huts was surrounded by a barrier of brush and bracken. I could see the outline of what had once been a defense ditch around the entire settlement; it had eroded into a grassy indentation by seasons of wind and rain. Beyond that, the village fields and grazing lands stretched to the edges of the forest.\n\n\"I've seen no movement down there,\" Rhodri hissed at us. \"No signs of life . . . no cattle or fowl . . . not even dogs . . . nothing.\"\n\n\"If they were massacred, there'd be bodies,\" Athauhnu mused. \"We'd see those . . . and the dogs. They never get all the dogs. They'd be back to feed on . . . feed on the leavings.\"\n\n\"Any activity to the north?\" I asked Rhodri.\n\nHe shrugged. \"Too many trees. Anything could hide in that forest. The birds show no alarm. This time of day, they like to stay close to the nest.\"\n\n\"Our orders are to bypass the village and screen it,\" I said to Athauhnu. \"Do you see any reason why we shouldn't continue?\"\n\nAthauhnu was silent. He was scanning the treetops to the north. Suddenly, he grabbed Rhodri's arm and pointed to the northwest, \"Look, there!\"\n\nI looked in the direction that Athauhnu was pointing, but saw nothing.\n\n\"Something's disturbed them,\" Rhodri agreed with Athauhnu.\n\n\"What do you see?\" I asked.\n\n\"A flock of birds rose from the trees there,\" Athauhnu pointed out over the village. \"It's too far for an ambush on the village. Could be nothing. But, that is where we should look.\"\n\nI nodded, then said, \"Alaw! Ride back to the troop and have Guithiru bring the men forward!\"\n\nWithout a word, Alaw moved back to where we left the horses.\n\nAthauhnu was talking, \"I think we should bypass the village to the west and work around it. It's the long way round, but it will position us uphill from whatever's spooking those birds.\"\n\n\" _Uhr wuhf uhn cuhtuno_!\" I nodded. \"Agreed!\"\n\nGuithiru soon brought the rest of the men up. I left two troopers behind\u2014 Emlun and his friend, Idwal\u2014to lead Caecina and the foraging detail into the village. The rest of us worked our way around the village to the west.\n\nIt was not cavalry country. We had to dismount and lead our horses up along narrow paths through the woods and brush. As we moved through the forest, the horses snorted; they were reluctant to move forward; their eyes darted about, examining the forest around them. Athauhnu said that horses had an innate fear of the forest. Somehow, they knew wolves hunted among the trees in packs. And, big cats, rarely seen but deadly killers, lurked in the shadows. On open ground, wolves were no match for horses. But here among the trees, the wolves were the masters.\n\nWe were making so much of a racket, trying to work our way around the village, I doubted any self-respecting wolf would be caught within a thousand _passus_ of us. As far as cats big enough to take down a horse, I didn't want to think about that.\n\nHorses weren't the only creatures made nervous by these dark, northern forests.\n\nWe finally reached a position we believed to be directly above where we suspected human activity. The village was invisible, down the sloping ground to our southwest. We had cleared most of the perimeter around the village, so I was confident that Caecina and the foraging detail weren't marching into a trap. But, we had to ensure there was no enemy threat to our front. I sent Alaw and Rhodri ahead on foot to scout the area.\n\nAlaw soon returned. \"There's a clearing about fifty _passus_ down the hill. We can see people\u2014\"\n\n\"People?\" I interrupted. \"Enemy soldiers?\"\n\n\"No,\" Alaw said. \"Mostly women and children . . . some old men . . . The missing villagers, I think.\"\n\n\"No sign of weapons . . . armor?\" I pressed.\n\nAlaw shrugged, \"They have planting tools.\"\n\n\"I think we should move forward,\" I said to Athauhnu. \"But, we need to be careful. These woods could hide an army. They may be using the women and children to bait us.\"\n\nAthauhnu nodded, and we moved forward.\n\nAlaw guided us to the edge of the clearing where Rhodri was waiting. I was just about to begin to deploy the _ala_ to sweep into the clearing when we heard a voice from down below.\n\n\"Are you oafs going to snap every shaggin' branch and crack every shaggin' twig you can find in those damned woods before you show yourselves? I could smell you and your shaggin' horses a thousand _passus_ away. Show yourselves, and get your arses out here!\"\n\nI stood up and exposed myself. Luckily, I wasn't greeted with an arrow in the chest. I saw a large man standing in the clearing below me. He sported bushy, gray Celtic mustachios that reached down to his chin\u2014rather a cascade of descending chins\u2014and he was wearing an old-fashioned domed helmet that floated on an aureole of bushy gray hair. His leather _lorica_ struggled to contain his belly. There was, however, nothing comic about the sword he was holding, a long, Gallic _spatha_ , whose recently honed edges caught the sunlight; it glowed like _Durn Gwin_ White-Hilt, the lightning sword Lugus.\n\n\"So, who the hell are you?\" he challenged me in Gah'el.\n\nI puffed out my chest and tried to channel all my Roman _dignitas_ , answering, \"I am Gaius Marius Insubrecus, _decurio_ of the praetorian cavalry of Caesar, _imperator_ \u2014\"\n\n\"You look like a shaggin' Roman, alright,\" he interrupted. \"But, those wankers standing next to you are no more Roman than my hairy bottom! By their colors, they look like _Soucanai_ to me! What the hell is a Roman puppy and a gang of _Soucanai_ sheep-shaggers doing on my lands?\"\n\n\"Are you chief here?\" I asked him.\n\n\"Chief? I'm the king around here . . . twice over, you Roman pup!\" he spat back. \"I am Cuhnetha mab Cluhweluhno, _buch'rix_ of these lands and _pobl'rix_ of the _Wuhr Tuurch_!\"\n\nI understood the words 'cattle king' and 'clan king.' _Buch'rix_ meant that Cuhnetha was the leader of this settlement, which was prosperous enough to have at least a modest herd of cattle. I wasn't at all sure about the meaning of the _pobl'rix_ of the _Wuhr Tuurch_ , the clan king of the Descendants of the Boar.\n\n\"Who are the _Wuhr Tuurch?\"_ I asked him.\n\nCuhnetha looked at me as if I were something nasty that just dropped from a tree into his path. \"Tuurch Mawr was the first king of the Aineduai, the Dark-Moon People\u2014Aedui to you Romans. He led the people down from the Land in the Skies into these valleys in the time before time. He defeated the _Pobl oh Danu_ , the People of the Dark God, and took these lands for the Aineduai. He rode as a chief at the right shoulder of Arth Mawr when the Gah'el rubbed the noses of _uh Chellinai_ , the Greeks, and you Romans as well, in dog shit. Now, he feasts with the gods in the Land of Youth! Enough of your questions, Pup! What are Romans and these _Soucanai_ dog-turds doing in my lands?\"\n\nAthauhnu answered, \"After we _Soucanai_ have destroyed the River People and the _Almaenwuhra_ that you Aiduai run from, even in your own lands, and the Romans go back to their grape farms, we are going to make dogs of your women and shit on the heads of your sons, oh _pobl'rix_ of cows and sheep!\"\n\n\"Our women alone are enough to drive you sheep-shaggers back into your shit-ridden swamps!\" Cuhnetha laughed. \"What are you called?\"\n\n\"Athauhnu mab Hergest,\" he answered. \"Pen _cefhul_ of Madog mab Guuhn, _pobl'rix_ of the _Wuhn_ clan of the _Soucanai_ and _Dux_ of the Roman Caesar.\"\n\n\"I should have guessed that a _Soucanai_ king would attach himself to a Roman's arse, oh Horse Chief of Madog!\" Cuhnetha shot back. \"But, unless we're going to fight, the law of our ancestors demands I offer you hospitality, and all this talking has given me a terrible thirst. So, come down out of those woods and join me!\"\n\nCuhnetha pulled off his helmet and walked away back into the clearing. \"Rhonwen!\" he shouted. \"Some _bragawt!_ The good stuff! _Medd coch_ . . . the red mead! We have guests!\"\n\nI looked over at Athauhnu. Despite all the insults hurled back and forth, he was laughing. When he saw me looking, he gestured toward the clearing.\n\n\"Come, Arth Bek!\" he invited. \"The King of Blowhards has offered us hospitality and is now honor bound for our safety. And he's right! Insulting him has given me a terrible thirst and a few cups of his _bragawt_ sounds like just the thing to slake it!\"\n\nWe followed Cuhnetha down into the clearing. For the Gah'el, hospitality is sacred. The host is responsible to the king for the comfort and safety of his guests. If a guest should die, through no fault of the host or his people, the host would have to pay his head price to the king. If a guest were killed or murdered through no fault of the host, three times the head price. Should the host himself harm a guest in any way, the host's rank and honors would be stripped and he would be exiled.\n\nNear the center of the clearing, Cuhnetha had a pavilion set up, a lean-to of leather sheets, not unlike those of a legionary tent. There were some rickety-looking wooden stools. Cuhnetha gestured for us to sit.\n\n\"The women will take your horses,\" he told us as he struggled out of his _lorica_. \"They'll also serve your men. Your nags look as relieved as you do to get out of those woods. We don't have much . . . some _bragawt_ , cheese, and yesterday's bread.\"\n\nAthauhnu got the men situated and came back to the pavilion, where a tall, lissome red-haired girl of no more than seventeen winters was pouring a reddish-orange liquid into clay cups.\n\n\"We call this _medd coch,\"_ Cuhnetha said, grabbing one of the cups. \"Our bees produce the richest honey in all the lands of the Aineduai. Drink!\" Cuhnetha took a long, noisy draught from his cup.\n\nWe drank. I was cautious, and that proved a good thing. Cuhnetha's _medd coch_ didn't have quite the bite of the _dur_ my father distilled out behind our storage sheds, where Mama wouldn't see, but it was hard to taste the difference. The fumes rose up into my head, making my eyes water, and the liquid burned like fire going down my throat.\n\n_\"Dur uh buhwuhd,\"_ Cuhnetha said as if he were reading my mind. \"The water of life. _\"_ He smacked his lips and held up his now empty cup to the red-haired girl.\n\nThe red-haired girl was obviously accustomed to Cuhnetha's act. She filled his cup, placed the pitcher on the ground next to his stool, and walked out of the lean-to.\n\nCuhnetha caught me watching her as she walked away.\n\n_\"Fuh nith,\"_ he said. \"My sister's daughter! _\"_ That was Gah'el guest-talk for, \"Don't think about it; she's family. _\"_\n\n\"So, _\"_ Cuhnetha started after another long pull from his cup, \"you never explained what a Roman and a band of Soucanai pig-thieves are doing in my woods.\"\n\n\"Your king promised us food if we fought the Helv . . . I mean . . . the River People for him, _\"_ I said. \"We've come to collect.\"\n\nCuhnetha laughed at that. \"Well, you've come a day too late, Little Roman! All the food's gone!\"\n\n\"Gone?\" I said. \"You mean the raiders took it all?\"\n\n\"Raiders?\" Cuhnetha snorted. \"There were no raiders here . . . You and your Soucanai friends excepted, of course.\"\n\n\"No, raiders?\" I questioned. \"Then who burned your barn?\"\n\n\"Barn?\" Cuhnetha shot back. \"That was no bloody barn! That was my hall! And, it was the men of that gob-shite of a _dunorix_ who burned it. They took the food!\"\n\n\"The _dunorix_? _\"_ I stammered. \"The commander of the king's fortress? Deluuhnu mab Clethguuhno? The king's brother? Your own people burned your bar\u2014I mean, your hall?\"\n\n\"Right and wrong, Little Roman, _\"_ Cuhnetha shot back, taking another long drink. \"The _fintai_ of the _dunorix_ set the blaze alright! But, Deluuhnu mab Clethguuhno is not 'my own people'! He's the snot-nosed, treacherous, piece of shit of his usurping, twice-bastard father, Clethguuhno mab Grefhuhtha. May he rot in the latrines of Annuhfn while worthy men piss on his head for all eternity!\"\n\nWhile Cuhnetha was refilling his cup, I asked, \"'Usurping, twice-bastard'? Are you saying he's not of the royal clan? Who is the legitimate king of the Aiduai, then?\"\n\n\"I am!\" Cuhnetha almost shouted. \"I am the _pobl'rix_ of _Wuhr Tuurch!_ The high kingship of the Aiduai has always been founded in the _Wuhr Tuurch!_ Since we descended from the Lands in the Sky, the descendants of _Tuurch_ have always ruled the tribe! That was until Clethguuhno poisoned my grandfather. My da was only a boy, then, not yet of the age. Clethguuhno convinced the Council of the Three Generations that the gods had killed my grandfather\u2014not the concoction of Greek weeds he put in his beer. And, with hordes of _Almaenwuhrai_ rampaging through our lands, the Aineduai needed a strong man . . . a warrior . . . ruling the tribe. My grandfather's death was a sign that Clethguuhno should rule. Then, when the Germans left us to go south and kill Romans, Clethguuhno's breeding bitch gave him two strong sons . . . heirs. He convinced the Council to recognize his oldest, Duuhruhda, as his heir.\"\n\nRanting was thirsty work. Cuhnetha held up the now-empty pitcher and bellowed to Rhonwen for more. From somewhere out among the milling Aineduai and our Soucanai troopers in the clearing, I heard a woman's voice yell back, \"Wait your turn, old man!\"\n\n\"No respect, I get!\" Cuhnetha complained. \"From my own blood!\" Then, he looked at me, \"Don't ever marry a red-haired woman, boy! She'll be the death of you!\"\n\nI didn't take his advice, and after thirty-some years, I'm still alive\u2014bruised somewhat, but still alive.\n\n\"So, now the _Wuhr Blath_ rule the tribe,\" he started.\n\n\"The Wolf People?\" I questioned.\n\n\"That's Duuhruhda's clan,\" Cunetha said, while glaring out into the clearing as if that would bring Rhonwen faster. \"The shaggin' descendants of the shaggin' wolf. It was they who came here yesterday with their carts, demanding all our stored food. It's for the king's _dun_ , Bibracte, they said. I asked them how we were going to eat until the crops were in. Eat your seeds, they laughed. They're already in the ground, I said. They just laughed. Then, they burned my hall. It was a message from their bloody _dunorix_ , that gob-shite brother of that gob-shite usurper . . . Deluuhnu mab Clethguuhno.\"\n\nRhonwen finally arrived with a fresh pitcher of the _bragawt_.\n\n\"It's about time!\" Cuhnetha grumbled. \"A man could die of thirst waitin' on the likes of you, girl!\"\n\n\"Shut ya gob, old man!\" she shot back. \"And just be thankful I pay ya any mind!\"\n\nAs she turned to leave, Rhonwen gave me a smile and a wink. I felt both in my heart.\n\n\"Where is your _fintai, pobl'rix_?\" I heard Athauhnu ask as he leaned forward for Cuhnetha to fill his cup. \"Why did your warriors allow this?\"\n\nCuhnetha snorted, \"My son led our sixteen warriors and five boys to Bibracte weeks ago, when that usurping, pretending king\u2014may he wipe backsides in Annuhfn for all the ages\u2014summoned them. Now they're all cowering behind those nice, thick walls up on top of those high hills, while the River People and Krauts do whatever they want to the rest of us. When those worthless followers of the _dunorix_ set the fire, my people fled to the woods. When it's safe, I have to bring them back . . . start rebuilding . . . decide how I'm going to feed them until the harvest. A king's responsibility is to his people . . . to the land! When the land burns, the king burns. That's how we did it in the old days . . . Kill the king; choose another!\"\n\nCuhnetha drained his cup to that sentiment.\n\nRemembering Caecina and his foraging detail, I stood. Athauhnu did also.\n\n_\"Diolch i chi am eich cletuhgaruuch, A Argluuhth_ ,\" I pronounced the ritual of thanks.\n\n_\"Riduhch chi bab amser uhn cael eu croesauu uhn fi, A Argluuhth_ ,\" Cuhnetha rose unsteadily and completed the ritual. \"You are always welcome at my hearth, Lord . . . at least, as soon as I rebuild the shaggin' thing,\" he added.\n\n# X.\n\n# Scaena Caesaris\n\n## CAESAR'S DRAMA\n\n_Diviciacus multis cum lacrimis Caesarem complexus obsecrare coepit ne quid gravius in fratrem statueret quod si quid ei a Caesare gravius accidisset cum ipse eum locum amicitiae apud eum teneret neminem existimaturum non sua voluntate factum qua ex re futurum uti totius Galliae animi a se averterentur haec cum pluribus verbis flens a Caesare peteret_.\n\n\"In tears Diviciacus embraced Caesar and began to implore him not to condemn his brother further. If Dumnorix were severely dealt with by Caesar, no one would believe that it had been done without Diviciacus'concurrence since Caesar held him in such regard. All Gaul would turn its favor away from him if such a thing were to happen. Weeping profusely, Diviciacus begged these things of Caesar.\"\n\n(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nHe found Caesar and the army by the eleventh hour. They had moved less than ten thousand _passus_ to the west, still following the lumbering mass of the Helvetii.\n\nCaecina was not a happy man. His wagons were empty; he felt that he had failed in his mission. When Athauhnu and I descended to the village from Cuhnetha's encampment, Agrippa and the other two _alae_ of Madog's Sequani cavalry had already returned from their screening mission in the west. Caecina's foraging detail had also arrived from the south; his wagons were assembled in the center of the village, while Sanga's infantry had secured the perimeter.\n\nObviously, there was no food, and no Aedui, to be found.\n\nI briefed the tribunes and Sanga on what Athauhnu and I had found and on what Cuhnetha had told us.\n\nCaecina wasn't satisfied, \"I suggest we send Sanga and his _centuria_ back up the hill and squeeze that cow king's balls. He must have some grain hidden away! And what about cattle? Did you see any sign of livestock?\"\n\nAgrippa shook his head. \"Even if we do, I don't imagine we'll get enough food to make a difference. The Aedui have had more than enough time to hide any remaining supplies not stolen by their own people. I'm more concerned with what this _dunorix_ bastard's up to. It seems as if he's stripping the countryside to starve us, when he's supposed to be gathering food to feed us.\"\n\n\"Caesar will not be pleased if we return with empty wagons,\" Caecina insisted.\n\n\"You can't squeeze wine out of raisins,\" Agrippa countered. \"We will at least be able to offer Caesar information that he may find useful. If the Aedui, or some faction within the royal household, are plotting against us, Caesar will need to be informed. I recommend that we rest the men and animals until the eighth hour. Then, we return to the army.\"\n\nThe men were sullen on the way back. Not only did they think that their mission had failed, but they were also not looking forward to the expected food shortages and reduced rations.\n\nWhen we reached the legionary camps, we discovered that we were not the only foraging party to return empty-handed. All the foragers who had marched north and northeast found empty barns. The ones coming in from the south and southeast had had better luck because the Helvetii and our army had screened these settlements from being stripped by their own people in Bibracte.\n\nWe again found Caesar's _praetorium_ in the camp of the Tenth Legion. Labienus was outside the tent with his _tabulae_ , taking reports from the returning foraging parties and instructing them how to distribute their takings.\n\nWhen he saw us, he said, \"Agrippa . . . Caecina . . . you marched due north . . . Let me guess . . . Our Aeduan allies beat you to the food.\"\n\n\" _Recte, Legate_ ,\" Agrippa responded. \"We came back empty.\"\n\nLabienus grunted and scratched something into his _tabula_. Then, he seemed to notice me, \"Ah . . . Insubrecus . . . Caesar wants you. He needs you to make sense out of those daily journals he's collecting. The rest of you are dismissed. You did a good job today . . . Not your fault it didn't work as planned . . . not your fault at all.\"\n\nI asked Athauhnu if he would have someone look after Clamriu, my horse. He nodded. \"I'll have Emlun look after her for you. He'll bring her back to Valgus after she's groomed and had a chance to cool down.\"\n\nWhen I entered Caesar's _cubiculum_ , I saw the _imperator_ standing over by his operation maps in a heated discussion with four officers. They wore chainmail _loricae_ and had their swords hanging on the left\u2014centurions. Each wore a broad, red sash, designating their status as _primi pili_ , the _de facto_ commanders of Caesar's four veteran legions\u2014despite any _legatus_ Caesar might assign to tag along.\n\nCaesar was stripped down to his _subarmalis_ jacket; his plate armor _lorica_ and helmet were tossed in a corner. His hair was plastered down on the top of his head by the weight of his helmet and the sweat of the day.\n\nWhen Caesar spotted me, he called over, \" _Bene_ , Insubrecus . . . the _tabulae_ are there by my field desk. I'll be finished here soon.\"\n\nI did my best not to eavesdrop on Caesar's conversation: rations needed to be distributed to the men soon; break off contact with the enemy; march south into the _provincia;_ resupply from Massalia\u2014or attack now and finish the enemy.\n\nThen, I heard Caesar say, \" _Satis! Me Taedet!_ I command here! I will decide what this army will do!\"\n\nI couldn't help but look over. The four centurions stood stiffly, braced at attention before their commander.\n\nCaesar recovered his composure. His jaw relaxed, and the Caesarian mask was back in place. \"Excuse me . . . gentlemen . . . It's been a long, frustrating few days. I will consider everything you've said. I know you are motivated only by what's good for the Roman people and for this army. Please . . . leave me now . . . I have much to consider.\"\n\nThe centurions' positions of attention seemed to loosen around the shoulders, just slightly; their chins seemed a bit less braced. They muttered, \" _A'mperi'tu_ ',\" and they began moving away from their commander.\n\nOne of them noticed me. It was Malleus, \"The Hammer,\" _primus pilus_ of the Tenth Legion. He came over to where I was sitting. Since Caesar was the ranking officer in the tent, I did not rise, but I sat up stiffly.\n\n\"Insubrecus,\" Malleus said, noticing my decurion sash, \"or, should I say, Insubrecus _Decurio_. Moving ahead in the world, I see. Congratulations on your promotion! If your career continues at this pace, I'll be working for you by the end of this campaign.\"\n\nI knew Malleus was pulling my leg, but I wasn't quite sure of his point. I decided to play it safe. \" _Gratias ti', Centurio_!\"\n\nMalleus nodded in reply. \"Your old centurion, Strabo, is back on his feet and still with the Tenth Cohort. I may be moving him up to the _acies secunda_ soon. You should pay a visit. I'm sure he'd be glad to see you.\"\n\n\" _Gratias ti', Centurio,\" I repeated_.\n\nMalleus nodded again. He had a slight smirk on his face. \"Don't let too much of that _tabula_ wax rub off on you . . . It's not good for soldiering. I may want to get you back in the Tenth to do some real soldier's work.\"\n\nWith that, Malleus left the tent.\n\nAfter a short while, Caesar wandered over from the maps. \"Another day . . . another ten thousand _passus_ . . . day after day. I don't know how exciting you can make that sound in your report, Insubrecus.\"\n\nCaesar seemed to be prattling a bit. With him, that meant he was talking about one thing while his mind was processing something else. \"When senior centurions think they must give advice to their commander, things are not good.\n\nThe men must be getting restless . . . worried about their rations . . . tired of avoiding an enemy they think they can beat . . . Ten thousand _passus_ a day and then build another marching camp. That doesn't tire them out enough. They have too much extra energy to stew and fuss . . . too much time to be anxious about getting this thing over with . . . Uneasy about rumors . . . running out of food . . . wondering what the purple-striped chump in the command tent is doing . . . No good for discipline.\"\n\nSuddenly, he came out of his ruminations. \" _Scriba!_ \" he yelled.\n\n\" _Imperator!_ \" answered the voice of his clerk from the outer _cubiculum_ of the tent.\n\n\"Get me Labienus!\" Caesar ordered. \" _Stat_ '!\"\n\n\" _A'mperi'tu_ '!\" the voice responded.\n\n\"Insubrecus,\" he said to me, \"hand me those reports . . . No . . . not those . . . Today's . . . the dailies . . . yes.\"\n\nI handed Caesar a stack of tabulae. He had just started trying to examine them when Labienus entered.\n\n\"Ah! _Bene!_ Labienus,\" Caesar started. \"What's our battle-line strength? How many infantry can we deploy against the enemy?\"\n\n\"Just legionaries?\" Labienus responded without missing a beat.\n\n\"Yes . . . yes,\" Caesar said, still trying to balance the tabulae in his arms while looking through them.\n\nLabienus reached over and plucked one of the tabulae from Caesar. He opened it and began to read, \"Seventh Legion . . . they had a strength of 4,327 . . . seven on leave . . . twenty-seven detached for detail . . . forty-two on sick call.\"\n\n\"Just bottom-line it for me, Labienus!\" Caesar interrupted.\n\n\"Yes . . . yes,\" Labienus muttered, reviewing the daily strength report. \"The four veteran legions, the Seventh, Eighth, Ninth, and Tenth, 15,552 effectives, give or take a dozen or so. The new legions, Eleventh and Twelfth, 8,736. If we put everyone in the battle line, 24,288.\"\n\n\"24,288,\" Caesar repeated. \"What are our intelligence estimates of enemy strength?\"\n\nLabienus placed the current _tabula_ on the desk and reached over and removed another _tabula_ from Caesar's tenuous grasp. \"Bottom-line it again?\" he asked.\n\n\"Yes . . . yes,\" Caesar said.\n\nLabienus began to examine the new tabula. \"This isn't an exact science,\" he muttered as he read. \"Since we separated the Tigurini from the other clans, forty, maybe fifty thousand maximum. Of those, no more than fifteen thousand fully equipped warriors. The rest are just tribal musters. That includes both cavalry and infantry. We can't get a good fix on the Germans. We think no more than ten thousand total, maybe three thousand warriors. So, we're facing less than twenty thousand warriors of battle-line strength; the rest would be just dirt farmers and swine herders with pitchforks and scythes.\"\n\n\"If we engaged them tomorrow, could we beat them?\" Caesar asked.\n\nLabienus shrugged, \"Almost twenty-five thousand trained Roman infantry . . . eighteen thousand barbarian warriors supported by a herd of sword-fodder . . . We pick the ground . . . Can't see why not . . . Yes . . . we could win that fight.\"\n\n\"How many days on full rations do we have?\" Caesar shot.\n\nAgain, Labienus took one of his _tabulae_. \"We haven't fully tallied what the foraging parties brought in, but . . . three days, full rations . . . maybe, stretch it to four. That's a conservative estimate. Three days, I'd say.\"\n\n\"We could reach the Rhonus from here in one long day,\" Caesar said. \"About thirty thousand _passus_ . . . one and a half days at a regular march . . . another day to gather supplies from the _provincia_. So, we have to decide now. We break off the pursuit and march to the Rhonus, or we go right at the Helvetii and resupply ourselves with their food, or we go hungry waiting for these Gallic _verpae_ in Bibracte to deliver on their promises.\"\n\n\" _Patrone_ ,\" I interrupted.\n\n\" _Quid dicere vis tu, Insubrece?_ \" Caesar said.\n\n\"Bibracte, _Imperator_ ,\" I answered. \"There's plenty of food in Bibracte. The Aedui have been stockpiling it there, and it's no more than twenty thousand _passus_ north of here . . . one day's march.\"\n\n\"That is consistent with what many of our foraging parties have reported, Caesar,\" Labienus confirmed. \"The Aedui seem to be stripping their own villages of food and livestock and bringing it into their fortress at Bibracte.\"\n\n\"Why would they do that?\" Caesar asked.\n\nLabienus shrugged, \"Feed the garrison . . . deny it to the Helvetii\u2014\"\n\n\"Or, deny it to us,\" I interrupted.\n\nCaesar stared at me for a few heartbeats. Just when I was convinced that he was going to reprimand me for interrupting a senior officer, he asked, \"So, you're suggesting we could march on Bibracte and resupply there?\"\n\nLabienus made a noise and Caesar asked, \"What is it, Titus?\"\n\n\"That may be our most dangerous option, Caesar,\" he said.\n\n\"Cur?\" Caesar demanded, \"Why?\"\n\n\"That would put the army between the Helvetii and the Aedui, with no clear line of withdrawal to the Rhonus,\" Labienus stated. \"We'd have over eighteen thousand barbarians on our asses. Only the gods know how many Aedui would be to our front, and we'd have a fortified position to crack\u2014with our siege equipment back on the Arar. I'm not sure that's the battle we want.\"\n\nA light seemed to go on in Caesar's mind. \"A trap . . . the food is the bait . . . Roman silver encouraging the Helvetii and interfering with my alliance with the Aedui . . . That seems to be a stretch, Labienus, but I see your point.\"\n\n\" _Patrone_ ,\" I said, \"there is another thing.\"\n\nWhen I had Caesar's attention, I explained what I had learned from Cuhnetha about the dynastic issues among the Aeduan clans.\n\nAfter listening, Caesar rubbed his hands together. \"This is getting as intricate as the plot of a Greek play. The _dunorix's_ men burning out a rival's village . . . rumors of Roman silver and purple-stripers riding with the enemy . . . a poisoned king . . . our allies withholding supplies . . . and a horde of barbarians seemingly beckoning us farther on to our destruction. A Greek melodrama, Labienus. All we need is a pirate king and a kidnapped virgin. The Aeduan king . . . he is to be here tomorrow . . . correct?\"\n\n\" _Recte, Imperator_ ,\" Labienus nodded.\n\n\" _Bene_ ,\" Caesar continued. \"Summon my senior officers . . . all my _legati_ . . . the _centuriones primi pili_ of the legions . . . the senior military tribunes, _the laticlavi_ , too. I'll conduct a council of war to discuss our situation . . . Do we provoke a battle with the Helvetii . . . break off pursuit and withdraw to the _provincia_ . . . or turn north against Bibracte?\"\n\nThe next day, during the third hour, Caesar was in his _praetorium_ , waiting for the arrival of the Aeduan king, Duuhruhda mab Clethguuhno. Our _exploratores_ had reported the Helvetii on the move, but the legions remained in camp. What Duuhruhda had to say was critical to Caesar's next move.\n\nThe council of war the night before had been stormy.\n\nTo a man, the senior centurions wanted to close with the Helvetii and destroy them. They claimed this was the only option the men would accept. Withdrawing to the _provincia_ was a retreat, a shameful defeat of a Roman army by a crowd of _pilosi_ \u2014shaggy barbarians in plaid breeches. Caesar would be shamed in the army and in Rome. Enough of the talk. Enough of the delay! Catch up to them . . . Kill them . . . Eat their food . . . Take their gold . . . Take their women . . . Have done with it!\n\nThe legates were a bit more circumspect. After all, they were civilians, not soldiers, and politicians at that. They were along on this campaign to enhance their public careers in Rome, not to decorate some barbarian's hut with their bloody heads. Certainly, a great victory would go a long way to their winning the next lap of their political careers, the race along the _cursus honorum_. Some of them had their eyes on a consul's chair in the foreseeable future. A retreat back to the _provincia_ was not especially helpful to them, but their political careers could survive that. They could blame the defeat on Caesar. But, the bloody defeat of a Roman army would be the end of their political lives\u2014 maybe quite literally.\n\nThe broad-stripe tribunes, the _laticlavi_ , didn't say much. They were of the senatorial class, but many were extra sons who didn't have much of a political or financial future in Rome. Their home was in the army, unless they were lucky enough to have an older brother die prematurely and were summoned home to take his place. Some were actually committed to a military career. So, they did not want to cross the _primi pili_ for military reasons and did not want to oppose the _legati_ for fear of losing possible political favors in the future.\n\nThe upshot of it was that, if the Helvetii offered the opportunity of a battle on favorable ground in the next two days, the army would attack. Caesar had sent out his _exploratores_ at first light with orders to locate a possible battlefield.\n\nIf, however, the opportunity did not arise to attack the Helvetii, the army would return to the _Rhonus_ and resupply from the _provincia_. They would bridge the river and establish their _castra_ on the north bank, so technically they weren't retreating back into the _provincia_. There, they would watch and wait. When the enemy settled down, they would renew the offensive.\n\nThe option of turning north against Bibracte was rejected as too risky. None of the officers wanted to be trapped against a fortified enemy position, low on rations, with the Helvetii across their line of withdrawal.\n\nThe wild card in this game was Duuhruhda.\n\nCaesar planned to confront the Aeduan king aggressively concerning his promise to supply the Roman army. If the king could be forced to make good on his promises, the crisis was over. The Roman army could continue pursuing its current strategy of attrition against the Helvetii. Caesar believed the key to the king was his brother, the _dunorix_ , and his family's tenuous claim to the throne.\n\nCaesar wasn't quite sure how he would use that as leverage.\n\nLabienus entered Caesar's _cubiculum_. He was formally attired as a senior Roman officer. He wore a highly polished, muscled, bronze _lorica_ with a Medusa image engraved high on his chest. He had a bronze _galea_ helmet sporting a brightred, horsehair crest and highly polished cheek pieces fastened tightly under his chin. Hanging from a polished leather _balteus_ on his left side, he carried a _gladius_ in a leather scabbard wrapped in gold wire. His legate's red sash was tied about his waist.\n\n\"Caesar, the scouts report that the Aedui are approaching,\" he said.\n\n\"How far out are they, Labienus?\" Caesar asked.\n\nLabienus shrugged, \"By now, less than five hundred _passus_.\"\n\n\"Has the escort detail arrived at the _Porta Praetoria?_ \" Caesar enquired.\n\n\" _A'venere_ ,\" Labienus answered. \"Yes! Five _contubernia_ from a first-line cohort . . . Third _Centuria_ of the Second Cohort, I believe . . . Under arms with shields and spears . . . Under the command of their centurion . . . a . . . uh . . . Mettius . . . uh\u2014\"\n\n\"Mettius Atius Lupinus,\" Caesar filled in. \"He's a Roman . . . from the _subura_ , I believe. His father's a _fullo_ , a fuller. Because of that, some of the boys call him _Lotium_ . . . Piss. But, only behind his back\u2014unless they want the _medicus_ trying to put their jaws back together.\"\n\n\"Uh . . . yes,\" Labienus agreed, \"that's him . . . Mettius Atius Lupinus. He'll bring the king and his brother here to you. Also, he'll allow no more than ten members of the king's . . . uh . . . What do you Gauls call _praetoriani_ , Insubrecus?\"\n\n\" _Fintai, Legate_ ,\" I answered.\n\n\"Yes . . . _fintai_ . . . ten members of the king's bodyguard detail with their swords and daggers only.\"\n\n\" _Euge_ ,\" Caesar agreed. \"The bodyguards remain outside with the legionary detail. You and Lupinus escort the king and his brother in here to me. Is Valgus standing by with my _praetoriani?_ \"\n\n\" _A'sunt_ ,\" Labienus affirmed. \"They're in my cubicle. The Gauls will not see them when they enter.\"\n\n\" _Bene_ ,\" Caesar nodded. \"Insubrecus, you remain here as my _ad manum_. Take notes of our conversation. But, if the king should suddenly develop problems understanding my Latin, you interpret for me. Let's get this _spectaculum_ started!\" Caesar rubbed his hands together.\n\nLabienus left Caesar's cubicle. Very little time passed before I heard footsteps and voices approaching the entrance. Caesar was sitting in his chair of state, the _sella curulis_ of a Roman magistrate possessing the _imperium_ of the Senate and the people. Caesar sat straight as a spear on the backless chair, which he had positioned to face the entrance of his _cubiculum_. His red _sagum_ was draped over the chair. Around his head, he had affixed his _corona civica_ , a golden chaplet of oak leaves woven to form a crown. I noticed how well the gilded oak leaves covered his thinning hair and receding hair line. His highly polished bronze _lorica_ reflected the lights of the many lamps he had burning around his _cubiculum_. But, from my perspective, the brightest light in the room seemed to shine coldly from Caesar's pale blue eyes.\n\nDuuhruhda mab Clethguuhno, King of the Aedui, was first to enter, followed closely by his brother, the tribal _dunorix_ , Deluuhnu. Both men seemed a bit disoriented, uncomfortable at being in the middle of a legionary camp surrounded by thousands of Roman soldiers and uneasy at having to leave their bodyguards outside Caesar's tent.\n\nI could tell by the way the men were equipped\u2014their finest armor and weapons, the bright colors of their cloaks and _bracae_ , their golden torques and armbands\u2014that they had intended to overawe their Roman ally. I was sure that they had ridden into the camp on the two largest stallions from their herds, but the horses were now out of sight, impotently munching grass somewhere outside our camp.\n\nThe Aedui stood before Caesar, Deluuhnu a pace behind his brother. Labienus and the centurion, Lupinus, slipped in behind them and stationed themselves between the Gauls and the exit, as Caesar had instructed.\n\nCaesar did not rise. He said, \" _Salve, Diviciace, Rex!_ And greetings Prince Dumnorix, brother to the king!\"\n\nI winced a bit at Caesar's butchering of Gallic names and titles, but clearly this was not the time to launch into a pronunciation lesson.\n\nCaesar continued, \"I have summoned you here to demand that you explain why my army has not been resupplied as you promised. I thought I'd give you the courtesy of explaining your failure before I turn away from my pursuit of those I thought were our common enemies and march on your _oppidum_ , Bibracte, and take what has been promised me!\"\n\nDuuhruhda now clearly understood why he and his brother were standing alone in front of Caesar, surrounded by Caesar's army. Unless they could give a satisfactory answer, they were not leaving this place.\n\n\"Caesar\u2014\" Duuhruhda began.\n\n\"It is customary to address a Roman magistrate by his office, _King!_ \" Caesar interrupted. \"In my case, you will address me as _imperator_.\"\n\n\"Caesar . . . I mean . . . _Imperator_ ,\" Duuhruhda started, \"this is not Italy! We have long winters and cool springs. The crops have not ripened\u2014\"\n\n\" _Me tadet specium_ ,\" Caesar interrupted. \"If I wanted a lesson in agriculture, I would have read Cato. My troops need to be fed!\"\n\n\" _Imperator_ ,\" this time Deluuhnu, the king's brother, spoke, \"my men are collecting what food is left among our people after a hard winter. These things take time\u2014\"\n\n\"Insubrece!\" Caesar interrupted him, calling my name, \"Insubrecus! That report!\" He held his hand out to me.\n\nI had no idea what Caesar was talking about! I took a step toward him to ask discretely what he meant. But, as soon as I was within arm's reach, he snatched the _tabula_ in which I was scratching my notes.\n\n\"My sources tell me,\" Caesar pretended to read, \"that there are powerful men among the Aedui who are preventing the villages from gathering the grain. If the Romans defeat the Helvetii, they say, Rome will subjugate the Aedui and all the rest of Gaul. These men claim that if the Aedui cannot rule Gaul, it is better to be ruled by Helvetii than by Romans. My sources tell me these same men are betraying my plans and the movements of my army to the Helvetii. They plot the defeat of my army by starvation or by battle. And, the ringleader of this cabal of treasonous filth is . . .\" Caesar looked up from the _tabula_ directly into the eyes of the king, \"Dumnorix, your brother!\" Caesar slammed the _tabula_ shut.\n\nDuuhruhda recoiled as if struck. Deluuhnu's face went ghostly pale. His right hand seemed to inch closer to the hilt of his sword. Then, he remembered where he was, and the two Roman officers standing directly behind him. His hand relaxed and fell back to his side.\n\n\"What have you to say to this . . . King?\" Caesar spit out the last word into Duuhruhda's face.\n\nI almost felt sorry for the man. He was trapped in the middle of a Roman legionary camp and entangled in a drama of Caesarian cunning. It was a magnificent piece of stagecraft. He didn't have a chance.\n\n_\"Imperator_ ,\" the king stammered.\n\n\"I am not finished reading my charges against your brother,\" Caesar announced, as if he were a _praetor_ trying a criminal case in front of the forum mob. He reopened his all-knowing _tabula_.\n\n\" _Item_ ,\" Caesar began, \"Dumnorix is the tribal _exactor_ , the tax collector, and as such, _he_ is responsible for the collection of the foodstuffs that were to be distributed to my army. And, this collection effort, by your own admission, has failed! I blame this failure on Dumnorix and his desire to weaken my army.\"\n\nCaesar paused. I was glad that neither Gaul realized that no _tabula_ could possibly contain so much information.\n\nCaesar again looked down at the _tabula_ and seemed to place his finger on another non-existing charge. \" _Item_ , Dumnorix is married to a woman of the Helvetii. Because of these marriage bonds, he favors the Helvetii and desires their victory over Rome and even over his own tribe!\"\n\nCaesar again paused for effect, then continued, \" _Item_ , Dumnorix has created strong political ties with the ruling classes of other Gallic tribes through the marriages of his mother's sisters and other close female relations. He does this in order to establish himself as the ruler of the Aedui, the Helvetii, and eventually all Gaul, once my army has been destroyed. He has gone so far as to exchange hostages with the Helvetii to assure mutual cooperation in this plot against me!\"\n\nCaesar was not done with this crucial scene. \" _Item_ , Dumnorix engineered the defeat of a Roman cavalry force by luring it into an ambush and then ensuring its defeat by withdrawing his personal cavalry forces once the engagement began. This treachery led to the deaths of a score of Roman citizens.\"\n\nCaesar again closed the _tabula_ and fixed the king and his brother in the cold, blue steel of his glare. \"The final offence alone justifies a summary judgment and his immediate crucifixion!\"\n\nCaesar looked up toward Labienus. \"Labienus! Has the king's escort been neutralized?\"\n\n\" _Gestum, Imperator!_ \" Labienus snapped.\n\n\" _Bene!_ \" Caesar answered. \"Summon my _praetoriani!_ \"\n\nLabienus called for Valgus, and immediately he and ten members of Caesar's praetorian cavalry entered the room in full rig.\n\n\"Seize that man!\" Caesar ordered, indicating Deluuhnu. The praetorians had the king's brother in their grasp before either Gaul could react. Valgus pulled Deluuhnu's hands roughly behind his back and began to bind them while one of his troopers removed the _dunorix's_ weapons.\n\n\"This is an outrage, Roman!\" the king shouted, forgetting Caesar's demanded protocol. \"We came here in good faith! You are responsible for our safety! My people will not stand for this . . . They will rise against you . . . They will join with the Helvetii.\"\n\n\"And, who will lead this insurrection, _King?_ \" Caesar asked in a low voice.\n\n\"I shall!\" Duuhruhda shot back. \"I, as their king.\"\n\n\"Which brings me to the next item of business,\" Caesar told him. \"Insubrecus, hand me that communique from Rome . . . the one that arrived by consular courier last night.\" Caesar gestured toward his field desk, on which lay a number of _tabulae_ and pieces of papyrus.\n\nI had no idea what Caesar was talking about. No special courier had arrived last night. Then, I remembered that I was playing a part in Caesar's drama. The _tabulae_ and _papyri_ were just props. I pretended to search through the papers and finally selected one that was covered with Labienus' scribbles.\n\n\"Here it is, _Imperator!_ \" I announced, handing it to Caesar.\n\nCaesar took the piece of papyrus and pretended to review it. \"This is an urgent message from my colleague in Rome, Gnaeus Pompeius Magnus, who serves the Senate and the Roman people with Marcus Licinius Crassus and me, as a _triumvir_. You've heard of Pompeius, have you not, Dumnorix?\"\n\nThe king's brother blanched as if having his arms tied behind him and being in the grasp of two of Caesar's _praetoriani_ was suddenly the least of his worries.\n\nCaesar continued, \"Pompeius writes . . . oh . . . but before we get to that, did you know, Diviciacus, that Pompeius is also my son-in-law . . . married to my daughter, Iulia? I hope he will give me a grandchild soon. We Romans take our familial obligations very seriously.\"\n\nCaesar let that snippet sink in, then continued. \"Pompeius has complained to me, both as his father-in-law and as a Roman magistrate, to look into an injustice done to one of his clients . . . a member of your tribe, a king . . . by the name of . . . Cuneda . . . Did I pronounce that correctly, Insubrecus?\"\n\nI leaned in over Caesar's papyrus and pretended to read, \"That's Cuhnetha, _Imperator_ , Cuhnetha mab Cluhweluhno.\"\n\nDuuhruhda's face went white when he heard that name.\n\n\"Yes,\" Caesar affirmed, \"Cuhnetha . . . It seems that Pompeius wants me to look into his client's complaint . . . nasty business, too . . . He claims that your father murdered his grandfather and stole the throne from him . . . Claims that he's the rightful king of the Aedui.\"\n\n\"Pompeius would never have written such a letter,\" Deluuhnu suddenly contradicted. His speech was cut short as Valgus smashed him across the face.\n\n\"Gag the prisoner,\" Caesar ordered. Valgus forced a piece of cloth into Deluuhnu's mouth while one of his troopers bound it fast with his _sudarium_.\n\n\"What would a condemned, treasonous, sack-of-shit barbarian know about what Pompeius Magnus would or would not have written to a Roman proconsul?\" Caesar charged.\n\nCaesar looked over to where I was standing, and I swear he winked at me.\n\nCaesar then noticed that Deluuhnu was bleeding from his nostrils. \"Make sure he doesn't suffocate, Valgus!\" he ordered. \"I don't want him to cheat the cross!\"\n\nCaesar turned back to the king. \"My colleague and son-in-law, Pompeius, has asked me to look into his client's complaint that he is the rightful king of the Aedui. And, _you_ . . . you are standing here in front of me, after failing to provide me with the food that you promised me, the food I need to feed my troops . . . after harboring and abetting a traitorous plot against the Roman people by your own brother . . . and after threatening to lead the Aedui in revolt against me . . . Insubrecus!\"\n\n\" _Ti' a'sum, Imperator_!\" I snapped.\n\n\"Cuhnetha mab . . . uh . . . whatever that name was . . . Didn't I recently see that name in a report?\" Caesar asked. \"I seem to recall that one of our foraging parties reported that his village was burned . . . his crops stolen . . . his livestock run off?\"\n\nThat was a bit of an embellishment, but it was theater after all. I responded, \"Yes, General, and the man claimed it was done by Aedui, the _fintai_ of the _dunorix_ , the king's brother, Deluuhnu.\" I was getting into my part.\n\nCaesar turned to the king. \"You see, Diviciacus. I have already begun my investigation for my colleague and son-in-law . . . Perhaps I should dig into this matter more deeply while you remain here as . . . as my guest. That way you can witness your brother's execution.\"\n\nThe king had stood motionless before Caesar. Then suddenly, he fell to his knees, placing both his hands, empty and upright, on Caesar's knees. This was the Gah'el ritual of submission.\n\nIn tears, the king of the Aedui begged, \"I implore you, _Imperator!_ Spare my brother! No one is more ashamed of his actions than I am! But, he has great influence with the Aedui and over the neighboring tribes. He has even tried to use his influence to undermine me, his own brother! But, he _is_ my brother.\n\nWhen I look at him, I see my father's eyes staring back at me. And, he is still a youth. When we were young and full of our own importance, deceived by a belief in our own immortality, who among us has not done what we should not have done, tried what we should not have tried? If you move against Deluuhnu, no one in the tribe will believe I was not complicit; no one will believe that I did not betray my own brother, and among my people that is an unforgivable crime. The very gods would turn against me! To remain loyal to a fratricide would surely invite the wrath of the gods.\"\n\nCaesar, like most Romans, did not know how to react to this sudden outpouring of Gallic grief and emotion. He glanced at me over the king's bowed head.\n\nSomehow, I sensed that this was Caesar's endgame. He had Duuhruhda exactly where he wanted him. Discretely, I gestured to Caesar to place his hands in those the king was holding open on his knees, thus showing that he accepted Duuhruhda's submission.\n\nCaesar nodded at me and did so, saying, \"I accept your reasoning, King, and I accept your submission to the Roman nation. Rise! I release you to return to your _oppidum_ at Bibracte, and I command you to collect the promised rations needed by my army and deliver them to our _castra_ by this time tomorrow.\"\n\nDuuhruhda rose, but kept his head bowed. \"I thank you for your clemency, _Imperator_ , and will do what you command. My brother\u2014\"\n\n\"Dumnorix will remain here with me . . . as my _guest_ ,\" Caesar stated. \"I will stay his execution as long as _you_ remain a faithful and dependable ally. You may tell your people that he remains here at my side as a trusted advisor. When we have defeated the Helvetii and the threat to your lands and mine is removed, I will decide what is to be done with him. But, for now, he is safe.\"\n\n\" _Multissimas gratias, Imperator_ ,\" Duuhruhda mumbled. \"Do I have your leave to depart?\"\n\n\" _Abeas, Rex!_ \" Caesar stated.\n\nDuuhruhda wasted no time in escaping from Caesar's presence. Labienus and Lupinus followed him out. Caesar abruptly realized that the battered, bound, and gagged Deluuhnu was still standing before him in the grasp of his _praetoriani_.\n\nNo play can end until all the actors have made their exits and all the props are removed from the stage. And, Deluuhnu was a prop in Caesar's play. Surrounded by thousands of Aedui, on whose cooperation Caesar depended, he had had no real intention of executing the man. He was merely leverage over his brother, the king.\n\n\"Take him away!\" Caesar ordered almost absently. \"And, keep a close eye on him. He's valuable to me!\"\n\nValgus nodded and gestured the prisoner and the _praetoriani_ out of the tent.\n\nNow that the play was over, Caesar seemed to droop a bit in his curial seat. \" _Scriba_!\" he yelled.\n\n\"Ti' _a'sum, Imperator_!\" the voice from the other room responded.\n\n\" _Vinum_ ,\" Caesar ordered, \"a large pitcher . . . and four cups!\" Then, before the clerk could ask, he added, \" _Merum!_ No water!\"\n\n\"Nicely played, Insubrecus,\" Caesar said softly, \"nicely played, indeed. I'll make a Roman politician out of you yet.\"\n\nI took my bow just as Labienus returned to the cubicle.\n\n\"The king's gone,\" Labienus reported. \"He took off out of here so fast, you'd think his horse's tail was on fire. Lupinus is marching his detail back to their tents.\"\n\nBefore Caesar could respond, his clerk entered with the wine. Caesar asked him, \"Is the wine up to your standards, Ebrius?\"\n\n_\"Certissime, Imperator_ \" the clerk answered. \"You know I wouldn't let any of the cheap stuff get by me.\"\n\nEbrius placed the pitcher and the cups on the field desk, then left.\n\n\"Would you pour, Insubrecus?\" Caesar requested.\n\nAs I was pouring the wine, Caesar asked, \"What did you think of the performance, Labienus?\"\n\nLabienus took a cup of wine from me, and answered, \"I'm wondering if we didn't do that pompous king a favor by keeping Dumnorix here?\"\n\n\"Quo _modo?_ \" Caesar asked.\n\n\"If Dumnorix is causing his brother problems with the Aedui,\" Labienus explained, \"we just removed the _podex_ for him.\"\n\nCaesar grunted, took a sip of his wine, and then said, \"Did you see the way Dumnorix reacted when he heard Pompeius' name? He damn near admitted that he's working as his agent.\"\n\nLabienus shrugged, \"That's possible, Caesar. He certainly didn't believe that Pompeius would support the claims of uh . . . what's that other Gaul's name again, Insubrecus?\"\n\n\"Cuhnetha, Legate,\" I offered.\n\n\"Quite,\" Labienus agreed. \"Cuhnetha . . . Dumnorix certainly didn't believe that Pompeius supported Cuhnetha's claim to the throne. But, concluding from that that Pompeius is actively colluding with our enemies is a bit of a stretch.\"\n\n\"Pompeius interfering with the Gauls would explain many of the rumors of purple-stripers and Roman silver,\" Caesar mused. \"My daughter has written me from Rome saying that Pompeius is keeping strange company these days . . . dinner parties with Cicero and Bibulus . . . inviting Milo to his home for hushed conversations over jugs of wine. He even took a meeting with Cato. How he despises the man!\" Caesar grimaced over some memory filtering through his head.\n\nHe continued, \"I imagine he'd claim he was just greasing up the _Optimates_ . . . looking out for our mutual interests in the Senate. But, we are in too precarious a situation up here to take any chances. It's far better to confirm than to trust.\"\n\nAll at once, Caesar was back in the room, focused. \"Labienus, I want you to do a bit of research. I want a list of any officer in the army who has any possible connection with Pompeius . . . anything . . . family . . . clientage . . . recommendations . . . prior service . . . anything!\"\n\nLabienus asked, \"How far down do you want me to dig?\"\n\n\"Right down to the bottom!\" Caesar snapped. \"Even if the man is serving as an _optio_ in a third line cohort, I want to know.\"\n\n\"Caesar, our veteran legions are originally from Spain! They served under Pompeius there!\" Labienus protested.\n\n\"True,\" Caesar agreed. \"But, I doubt Pompeius was cultivating the _muli_ back then, except to keep them fed, paid, and busy. Concentrate on the older, more senior officers, anyone who was a junior centurion when Pompeius was in command . . . especially anyone he promoted.\"\n\n\"Do you doubt the loyalty of our senior centurions?\" Labienus asked. \"You're talking about _primi pili_ , even the _praefectus castrorum_!\"\n\n\"Doubt?\" Caesar puzzled. \"No . . . I don't _doubt_ them . . . I just want to be _sure_ of them . . . _absolutely_ sure.\"\n\nLabienus was just about to say something when Caesar's clerk, Ebrius, entered the cubicle. \"Forgive the interruption, _Imperator_ ,\" he announced. \"But, I have an officer outside who claims to have urgent information for you.\"\n\n\"An officer?\" Caesar questioned. \"Who?\"\n\nAn _angusticlavus_ , a narrow-striper, one of the _exploratores_ you dispatched this morning\u2014\" Ebrius began to explain.\n\n\"One of the scouts!\" Caesar exclaimed. Ebrius now had Caesar's undivided attention. \"Send him right in, Ebrius! I want to hear what the man has to report!\"\n\n\" _A'mperi'tu', Imperator!_ \" the clerk obeyed and left the tent.\n\nA few heartbeats later, Agrippa entered Caesar's _cubiculum_ followed by Madog. Caesar greeted them eagerly, \"Agrippa! Madocus _Dux! Beneventi!_ Please! Help yourselves to some wine!\"\n\nBoth men poured some wine into cups. Agrippa drank deeply. Madog sniffed his. He saw his companion drinking, so he did likewise. He grimaced a bit. The Sequani chief had yet to develop a taste for the grape.\n\nCaesar waited while Agrippa drank, then asked, \"What do you have to tell me, Tribune?\"\n\n\" _Imperator_ ,\" Agrippa began, \"we have found your battlefield. You can trap the Helvetii!\"\n\nAgrippa now had Caesar's full attention, \" _Mi' dicas_ , Agrippa!\"\n\n\"They hardly moved today, _Imperator_ ,\" Agrippa continued. \"They seem to be consolidating some of their stragglers and letting their livestock rest and feed. They're spread out in a valley no more than ten thousand passus west of here with high ground on three sides. They've made no move to secure that high ground or to screen its approaches with their cavalry. We were able to move in above their encampments without being detected.\"\n\nMadog was nodding along with Agrippa's briefing. His wine cup was almost empty. \"True . . . true,\" Madog agreed, \"very stupid Helvetii . . . fear nothing.\"\n\n\"Less than ten thousand _passus_ , you say?\" Caesar was nodding. \"Little security . . . _bene_ . . . _bene_ . . . What's the slope of the hills? Can our infantry negotiate it?\"\n\n\"There's high ground on their flank,\" Agrippa continued. \"A hillock, really. It's flat on top and could easily accommodate two legions arrayed in the _acies triplex_. The slope down to the valley floor is open and not too steep.\"\n\n\"Wait a moment, Agrippa!\" Caesar said. Then, he called out, \" _Scriba!_ Ebrius! To me! Quickly!\"\n\nCaesar's military clerk appeared, \" _Ti' a'sum, Imperator!_ \"\n\nCaesar ordered. \" _Celeriter!_ Quickly! I need a sand table . . . a terrain model. Can you round something up for me?\"\n\nEbrius shrugged, \" _A'mperi'tu', Imperator!_ \"\n\nCaesar also realized that he had removed the campaign maps from his headquarters in preparation for his confrontation with the Aedui. \"And, have a couple of your assistants set my maps back up!\" he instructed. \" _Age, Miles!_ \"\n\nCaesar had the bit between his teeth, so Ebrius shot out of the tent like a _ballista_ bolt.\n\nCaesar looked about the room, then ordered me, \"Insubrecus! My desk . . . bring me a piece of that papyrus . . . There should be some charcoal over there, too . . . Bring me a piece!\"\n\nI ran over to the field desk to collect the items. Madog was observing all of the Roman turmoil while he poured himself another cup of Caesar's wine. His Gallic distaste of the insipid Roman grape-mash was clearly becoming a thing of the past. There was some hope that _Romanitas_ could be brought to _Gallia comata!_\n\nI brought Caesar the papyrus and charcoal. He handed it to Agrippa, ordering, \"Sketch the terrain so I can see it, Agrippa . . . Less than ten thousand passus, you say?\"\n\n\"No more, _Imperator_ ,\" Agrippa nodded as he drew. \"They hardly moved at all . . . just sitting there.\"\n\nMadog leaned in, almost as interested in what Agrippa was doing as he was in the wine.\n\nWhile Agrippa drew his map and briefed Caesar, Ebrius returned to the tent with two legionary _muli_ carrying their _dolabrae_ , entrenching tools. \"Right here!\" He pointed to a spot on the ground in front of where a couple of his assistants had begun to set up the campaign maps. \"About ten _pedes_ by ten _pedes_ square . . . through all the grass, right down to the dirt . . . Make it smooth, and be sure you take all the spoil out of here with you . . . Use your helmets if you can't find buckets.\"\n\nThe grunts started whacking at the turf. Caesar looked over to where they were working, and said, \"Scrape it clean for me, boys! Ebrius'll give you some _posca_ to replace all the sweat!\"\n\n\"Smooth as a baby's bottom, _capu_ ',\" one of them grunted between swings with the entrenching tool.\n\nWhile they worked, a couple more _muli_ came through carrying some cut logs. Ebrius directed them to stack their load at the back of the tent.\n\nAs they were leaving, I heard Ebrius tell them, \"Grab up the buckets outside the tent and take 'em down to the water point . . . Fill 'em with sand from the stream bank . . . the dry stuff . . . and bring it back here . . . on the double . . . _il' capu_ ' is in a hurry!\"\n\nCaesar, Labienus, and Agrippa were bent over Agrippa's map. Madog hovered behind the group with his half-emptied cup of wine.\n\nCaesar was questioning Agrippa closely concerning the details: \"Is there a stream between those two ridges? How steep is that slope? You're sure they haven't ringed their wagons? Show me again where you saw the horses grazing.\"\n\nThe two legionaries in the back had finished their digging. One was carrying the spoil out in a bucket, while the other began to fit the logs around the cleared space.\n\nI heard Caesar say, \"What do you think, Labienus?\"\n\n\"I think we got them where we want them,\" Labienus affirmed. \"We couldn't ask for a better set up than this.\"\n\n\" _A'sentior_ ,\" Caesar nodded. \"I agree. But, I have to see the terrain myself. Agrippa! You and Madocus _Dux_ will lead me out there.\"\n\n\"Shall I alert Valgus?\" Labienus asked.\n\n\"My praetorian detail? No! Too many Romans,\" Caesar stated. \"Too much activity . . . I don't want to tip my hand to the enemy. I'll go alone with Agrippa . . . and you too, Insubrecus! You're with me.\"\n\nWe rode to where Agrippa had located the Helvetii. As per Caesar's instructions, our Sequani cavalry took us in from the north. Caesar was anxious to see whether there was a negotiable avenue of approach for his infantry around the enemy's flank. He had left his red general's cloak behind and wore the ruddybrown _sagum_ of a _mulus_. He strapped his helmet to his saddle in order to prevent it from being recognized as Roman and to avoid the possibility of any reflection of the afternoon sun, which might alert the enemy of our presence.\n\nWe got ourselves into position in a little over an hour. We left our mounts in a stand of woods, guarded by one of the cavalry troopers. Athauhnu's _ala_ was deployed to our rear as a screen to ensure that an enemy patrol didn't surprise us from that direction.\n\nDuring the ride, Athauhnu practiced his latest attempts to learn Latin on me. Phrases like \"trees to be green being good\" and \"horse to be running prompt when field to be flat\" were still ringing in my ears as I crept forward behind Caesar, Agrippa, and Madog to get a glimpse of the enemy. The cavalrymen of Madog's remaining _ala_ were fanned out around us as we advanced toward the lip of a flat hilltop.\n\nAs we neared the edge, I could hear the hum of activity from the valley below. We advanced on our hands and knees. To this day, when I see one of Caesar's heroic statues standing magnificently on a gilded marble plinth in one of his temples, I grin as I remember him that day, his thinning hair plastered flat to the top of his head, a streak of mud running down his cheek, and his breath labored as we crawled through the long grass to get a look at the Helvetii.\n\nAnd what a sight we saw!\n\nThe entire valley below us was teeming with people, animals, tents, wagons, campfires, and equipment, all seeming to swirl about in smoke, noise, and colors. There must have been tens of thousands of them down there: men, women, and children\u2014all seemingly without a care and behaving as if they were on some spring holiday. There was no sign of any military preparedness, no fortifications, no sentries, and no formations of armed men. Women went about their chores, many with babies on their hips; children chased each other about in mindless patterns; and the men seemed content to laze about around the fires and the tents.\n\nCaesar had his hand-drawn map out and was making corrections and notations with a piece of charcoal. \"If I had two legions up here now, I could finish this thing by sundown,\" I heard him whisper to Agrippa.\n\n\"You see where the valley narrows a bit to the east?\" he continued. \"That's where we want their main body to assemble. We want all their attention focused right there. Then, we sweep down on their flank and rear from this hill.\"\n\nWe remained in place for about half an hour before Caesar indicated he had seen enough. We withdrew from the hill and returned to the legionary _castra_.\n\nAs soon as we got back, Caesar immediately set his plans in motion. He directed Labienus to summon his senior officers to a war council at the tenth hour, all his legates, the broad-stripers, and the _primus pilus_ of each legion.\n\nWhile Labienus was about that, Caesar studied his maps and supervised Ebrius and his assistants in building in the sand table an accurate terrain model of what we had just observed. Caesar then had Ebrius' clerks cut and paint a number of wooden blocks to represent military formations: red for Roman, with numerals indicating legion designation; black for the Helvetii, each representing five thousand warriors. Caesar arranged and rearranged the blocks on his sand table as he hypothesized various scenarios. When Labienus had finished arranging the war council, he joined Caesar at his sand table, and they discussed the various options.\n\nThe council of war that assembled later that day in Caesar's _praetorium_ was very different from the previous meeting.\n\nCaesar began by announcing to his officers, \"Gentlemen, we have located the main body of the enemy in a valley approximately ten thousand _passus_ west of our location. It is my intention that this army will advance on the enemy, attack at dawn tomorrow, and destroy the Helvetii.\"\n\nWith Caesar's pronouncement, I looked up from the _tabula_ in which I was scribbling notes. There was absolute silence among the assembled officers, but I could see the knuckles of the senior centurions whiten as their grasp tightened around the hilts of their swords. After a few heartbeats, they began to nod in agreement. Caesar's legates and the broad-stripe tribunes remained perfectly still.\n\nCaesar, standing over his terrain model using a _sudis_ as a pointer, continued, \"We are located approximately here. The enemy is spread out for about two thousand _passus_ in this valley, here. They do not seem to be aware of, nor do they seem to care, about our proximity to them. They have made absolutely no defensive preparations.\"\n\nWhen Caesar said this, I noticed that two or three of the senior centurions shook their heads and smirked. No Roman army would permit itself to be caught out in the open in enemy territory.\n\nCaesar continued, \"It is my intention to attack using the basic hammer and anvil tactic. Labienus will command the 'hammer,' a task force of two legions, the Seventh and Eighth. I will command the 'anvil,' four legions, the veteran Ninth and Tenth and the two new legions, the Eleventh and Twelfth . . . Yes, Malleus?\"\n\n\" _Imperator_ , what are your estimates of enemy strength in the valley?\" the _primus pilus_ of the Tenth Legion asked.\n\nCaesar smiled slightly and nodded. He knew where Malleus was going. He answered, \"Our intelligence indicates that the Helvetii and their allies can field between twenty-five and thirty thousand trained warriors . . . probably twice that number in tribal levies. My observation of the enemy position this afternoon supports those estimates. Are you worried that there aren't enough of them out there for each of your Tenth Legion boys to get at least ten each?\"\n\nMalleus grinned broadly as he said, \"No, _Imperator_ . . . my boys'll earn their ten copper _asses_ a day . . . That's a penny for each of those shaggy _comati_ they stick. My concern is that we seem to be dividing our forces in the face of a numerically superior enemy. It seems to me that's the way Caepio and Maximus screwed up against the Cimbri back in the day . . . No offense, _Imperator_.\"\n\nCaesar leaned forward on the _sudis_. \"And none taken, Malleus. As a senior centurion, your job is to keep the purple-striped nobs Rome keeps sending up here from totally screwing the pooch. Caepio and Maximius put a river between their two armies and did not exercise unified command. In fact, those two political _stulti_ did everything they could to work against each other, and many good Roman soldiers died as a result of their arrogance and ambition. I have no intention of allowing that to happen to this army. The two wings of our army will always be under my command and will not become so separated that they cannot support each other should the enemy decide to attack. In fact, luring the Helvetii into attacking one of the columns is my plan. Allow me to continue.\"\n\nMalleus nodded his credence to Caesar's explanation. Caesar continued, \"Labienus' division will depart from this location at the beginning of the third watch. He will march along this northern route and get behind the enemy's left flank by occupying this hill, here. Their primary goal is to reach this hilltop, which is their assault position, undetected. They will remain there in a concealed position until our main force has engaged the enemy's front, here.\"\n\nThere were no questions, so Caesar continued, \"I will lead the main body along this route, entering the enemy-occupied valley just before dawn at this point. We will advance toward the enemy and deploy across this narrow ground here . . . three legions forward in the _acies triplex_ . . . the Tenth on the right flank, the Ninth in the center, and the Eleventh on the left. The Twelfth will serve as my reserve, deploying in a line of cohorts across the rear of the forward three legions. The objective of my division is to get the Helvetii to attack our line. The optimum point of contact is here, along this line, where our flanks will be secured by high ground. When the enemy has committed itself to that attack, Labienus' division will advance from its concealed assault position and attack the enemy's rear. These _hursiti_ , hairbags, do not have the discipline to sustain a two-front battle. They'll crumble when Labienus attacks. Then we finish it. Questions, gentlemen?\"\n\nThere was silence for a few heartbeats while the officers absorbed what Caesar had told them. Then Spurius Hosidius Quiricus, the _primus pilus_ of the Ninth Legion, known around the camps as _Quercus_ , the \"Oak,\" spoke up: \" _Imperator_ , I see you have one of the new legions, the Eleventh, on the battle line. Are you sure they're up to it? No offense, Iudeaus!\"\n\nQuercus was apologizing to Marcus Sestius, the _primus pilus_ of the Eleventh. Sestius was an old hand who had served with Pompeius in the east and was wounded in a skirmish in eastern Syria. A Greek hoplite had tried to drive his spear into Sestius' left thigh but missed his aim slightly. Ever since then, the boys referred to the centurion as Iudaeus after one of the eastern tribes which practiced some bizarre ritual on the penises of the males of their tribe. Iudeaus didn't seem to care about the nickname and always asserted that, despite the wound, the _mulieres castrorum_ never complained.\n\n\"None taken, Quercus,\" Iudaeus answered. \"My boys're up to it.\"\n\nAs Quercus nodded, Caesar said, \"I expect the Helvetii to send in their tribal levies first to soften us up for the warriors. I expect that they'll falter after our first volleys with the _pila_. Before their chiefs can rally them, Labienus should be down on them. Then, the _acies prima_ can move forward into the valley and clean them out.\"\n\nThe officers nodded. As Caesar said, it was a classic tactic: hammer and anvil.\n\nCaesar continued, \"That's where you and your cavalry come in, Crassus!\" With his _sudis_ , Caesar pointed to a round, red block on the left flank of the legionary line. \"I'm concentrating the legionary cavalry on the left flank. That way they won't get tangled up with Labienus' boys, who will be in front of our right flank. The cavalry will advance apace with the infantry front line\u2014until I give the signal from my position on the right flank. At that signal, you will attack down the valley. Stampede them, Crassus . . . Kill as many as you can . . . Panic them! The limit of your advance is here where this small stream crosses the valley, or when I signal the recall. I want you to position yourself in front of the main body of the cavalry, Crassus. Right up with the forward edge of the infantry line. If you spot an opportunity, an opening, you go! Even if I haven't signaled the advance, you go! Can you do that for me?\"\n\n\" _A'mperi'tu', Imperator_ ,\" Crassus responded. I smiled to myself as I noted down his response. Looked like that pretty boy was finally going to get some mud on his boots.\n\n\" _Bene_ ,\" Caesar acknowledged. \"I'm deploying the auxiliary cavalry on our flanks for security.\" He pointed to another round, red block positioned to the north, saying, \"Madocus _Dux_ , I want your boys on the northern flank, watching the approaches from Bibracte. I want no surprises from our _socii_ , the Aedui, during this operation.\"\n\n\" _A'mperi'tu', Caesar_!\" I heard Madog's voice among the crowd of Romans.\n\n\" _Bene_ ,\" Caesar said. \"Assignments for the legates . . . Labienus, you're with the Seventh . . . Cotta, you go in with the Eighth in Labienus' division . . . The rest will march with me . . . Pedius, the Ninth . . . Rufus, the Eleventh . . . Vatinius, the Twelfth . . . Crassus, you have the cavalry. I'm with the Tenth. The boys'll march light . . . basic combat load . . . one day's rations and water. We won't be pulling down our marching camps here when we pull out. Each legion will detach its tenth cohort to secure their _castrum_ and their baggage. Pulcher, you'll be in charge of the _castra_. If you're attacked by an overwhelming force, you can consolidate the cohorts in one of the camps until we can relieve you. But, you are to hold this position . . . _Compre'endis?_ \"\n\nCaesar was ensuring that the army had somewhere to retreat to in the event that the beehive he was about to kick over proved more dangerous and powerful than he had estimated, especially now that he could no longer rely on the good will of the Aedui.\n\n_\" Compre'endo, Imperator!_ \" I heard Pulcher's voice sound. Pulcher had been on his best behavior since Caesar had relegated him to the ash and trash detail.\n\n\" _Bene_ , gentlemen!\" Caesar concluded. \"Labienus' division pulls out at the beginning of the third watch. My division, an hour after that. Remember, Asellio writes, _'Audaces amat Fortuna!_ Fortune favors the bold!' Let's finish this thing! Return to your commands! _Miss'est!\"_\n\nI am glad that at that point in my career, I hadn't yet read the _Rerum Gestarum Libri_ of Sempronius Asellio, the histories of the Third Punic War, because then I would have known that he also wrote, \"No military plan survives the first step of its implementation.\"\n\nAfter the officers had left, I waited while Caesar gave some directions to Labienus. \"Titus, it's absolutely imperative that you get your legions in position without being detected by the Helvetii. Also, do not reveal yourself to them or commit yourself to the attack until you're absolutely sure that my division is in position. We should be visible from your position, and you'll hear my attack signals. Your two veteran legions, even on high ground, may not be enough to withstand a determined attack by that horde. Despite what I told Malleus, if the enemy gets between me and you, I'm not sure I can get to you, and we want no repeat of the Cimbri disaster. If you're detected by the enemy during your approach to the assault position, break off and return here. Warn me by cavalry couriers, and I'll withdraw my division. But again, you're to initiate no attack unless I'm in position. Wait until you see their backs; then you go in.\"\n\n\" _Compre'endo, Caesar,\" Labienus responded_.\n\n\"Good man!\" Caesar responded, clapping Labienus on the shoulder. \"Get your boys ready, and I'll see you before you pull out.\"\n\nLabienus walked away, and I offered Caesar the _tabula_ with the meeting notes for his review. \"I don't need to see that, Insubrecus,\" he said. \"Just give it to Ebrius for the file.\"\n\nThen, after a moment, he said, \"I'm sending you up north with Agrippa and the Sequani. Sorry. You're going to miss the big show, but whatever the Aedui are up to is critical to me. I'm as interested in who enters Bibracte as I am in who departs . . . Those _cunni_ are up to something, and I'm convinced now that someone down in Rome is pulling their strings. Agrippa knows to keep a low profile up there, but if you can grab hold of someone who can help me understand what in the name of Nemesis is going on . . . I don't care if it's a Roman, an Aeduus, a Helvetius, or a blue-painted Briton . . . Grab him up, and bring him to me in a condition in which he can still talk.\"\n\n# XI.\n\n# Calamitas Itera\n\n## ANOTHER DISASTER\n\n_Multo denique die per exploratores Caesar cognovit et montem a suis teneri et Helvetios castra movisse et Considium timore perterritum quod non vidisset pro viso sibi renuntiavisse_.\n\n\"Finally, after most of the day had passed, Caesar discovered from his scouts that the mountain was occupied by his forces and that the Helvetians had moved their camps. Caesar now understood that Considius had lost his nerve, and what he had reported to Caesar as having been personally seen by him, he had not actually seen.\"\n\n(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nAt the time, I was not sure whether I was disappointed or relieved\u2014 disappointed about missing what Caesar called \"the big show,\" or relieved knowing that I wouldn't be trapped in a narrow valley upto my ass in frenzied hairbags screaming for my blood. By the end of the second watch that night, while the Roman camps were in a maelstrom of movement and noise, I was mounted and moving north with Agrippa and the Sequani cavalry. We encountered no enemy counter-reconnaissance patrols. It was as if the Helvetii just didn't care that a Roman army was less than ten thousand _passus_ away from their main camps.\n\nWe rode hard and arrived at a wooded ridgeline overlooking a broad valley by dawn. From there we could see the dark, hulking mass that was the _dun_ , the royal fortress-city of the Aedui, Bibracte.\n\nThe _oppidum_ sat on a double hill, a higher summit to the east with a lower summit to the west and north. It was surrounded by two walls. The lower, which seemed to surround the entire bottom of both hills, was almost fifteen _pedes_ in height and built of stone reinforced with logs. The second, which stretched higher along the slope around the double peaks of the hill, had a sturdy, log fronting. I assumed the logs held a thick core of soil and spoil, like the walls of a permanent legionary camp. From where we stood, we could see a road leading from the south and west, entering the fortress through a large, well-guarded, and fortified gate.\n\n\"That not main gate,\" Madog was telling Agrippa. \"That on the north side of _oppidum_. But, if Aedui talk to Helvetii, that road runners to use.\"\n\n\" _Bene_ ,\" Agrippa agreed. \"Let's put an _ala_ across that road. I expect that when Caesar attacks, the Helvetii will send a courier up to inform Diviciacus. If that happens, we'll let the courier pass. If a response is sent, that's the one we'll intercept. But, we'll want the courier alive.\"\n\n\"Maybe response from Aedui is attack Romans,\" Madog cautioned.\n\n\"Then, we don't need the messenger,\" Agrippa said. \"We ride south and warn Caesar.\"\n\nAgrippa noticed me looking away to the south. \"You see something, _Decurio?\"_ he asked.\n\n\" _Aliquid non cerno, Tribune_ ,\" I answered. \"No, sir. I was just wondering why we can hear nothing from the south.\"\n\nAgrippa shrugged, \"The army's in a valley over twenty thousand _passus_ away. We may not hear the battle here. So, let's stay focused on our own mission.\"\n\nI nodded. Still it seemed strange to me that two huge armies were locked in a decisive struggle a few thousand _passus_ to the south, yet here, there was no indication of it.\n\nMadog was talking, \"Other gate for north road; we need watch.\"\n\nAgrippa grunted his assent. \"Where is it?\" he asked Madog.\n\n\"To north,\" Madog answered, pointing, \"other side hill that direction.\"\n\nAgrippa nodded. \"That's your job, Insubrecus. You and Athauhnu work your way around there without being seen. Get a good, concealed observation position. Same deal. You see any known enemy couriers, snatch them up on their way out, though I doubt you'll see any Helvetii north of here. And, if the Aedui move south, they'll probably come this way. Greek or Roman merchants, you see any of them, stop and question them. Hold them until you withdraw. I want you back here by the sixth hour. By that time the battle in the south should be decided. I'll stay in this area. Madog, will you take the south road?\"\n\nJust before Athauhnu and I pulled out, Agrippa said to me, \"Take a good look at the defenses, Insubrecus . . . any weaknesses you can see . . . places where the walls need repair . . . gaps . . . low spots. Pay special attention to water breaks, sewers, or streams. I have a feeling we're going to be back here in a couple of days and the boss is going to want to take this place apart.\"\n\nIt took Athauhnu and me the better part of an hour to work our way around to the north. We passed a few farmsteads, to which we gave a wide berth. We had a nervous few moments when a farmer's dog took a noisy and somewhat passionate interest in our presence. But, for the most part, the land around Bibracte was quiet and sparsely inhabited.\n\nBy the second hour, we were on well-wooded, high ground from where we could see a road winding down from the north and another coming in from the northwest. They met at a large gate piercing the lower wall of Bibracte. It was as Madog said. This gate was larger, more formidable than the one in the south. Dressed in the red-plaid tartans of the Aeduan royal house, a detachment of about ten Aedui warriors were guarding the double portal and collecting tolls from those wanting entry and duties from those leaving. We guessed there were more troops concealed in the gatehouse.\n\nWe split the _ala_ into three sections: one kept watch on the roads and gate; another provided security around our position; the third slept. Athauhnu volunteered to take the first command shift and let me take a nap after having ridden most of the night. I found myself a warm, shady spot on top of some springy pine needles just over the reverse slope. I kept my _lorica_ secured, but loosened by boots. I rolled my _sagum_ into a pillow and lay back. I felt like I had no more than closed my eyes when I felt someone shaking my shoulder. It was Athauhnu's nephew, Emlun.\n\n\"Come with me!\" he whispered. \"Athauhnu . . . I mean, the chief, wants you to see something.\"\n\nI rubbed my eyes open, then tightened my boots. With my _sagum_ rolled up and under my arm, I followed Emlun over the crest of the low ridge to where Athauhnu was crouched behind some trees. When he saw us approaching, he signaled us into a crouch. When we reached his position, he pointed down toward the north road.\n\nThere, I could see a group of riders, ten warriors by their armor and weapons. They were followed by an entourage of servants and baggage. This was obviously not a war band. The leader was arrayed to impress: a tall, conical helmet; gold arm bands; a polished, chainmail _lorica;_ a rich cloak swept back to reveal a long, Gallic _spatha_ in a rich gold-wrapped leather scabbard. I caught a glint of gold around his neck, the golden torc of a noble.\n\n\"They are not River People, not Helvetii,\" Athauhnu was saying. \"I do not recognize their tartan.\"\n\nAthauhnu gestured one of his warriors over to him. It was the scout, Rhodri. \"Do you recognize the colors?\" Athauhnu asked him.\n\nI focused on the colors that the riders were wearing. Their _bracae_ and cloaks were in a blue and white plaid that I had never seen before. A _signifer_ riding immediately behind the noble was displaying a banner of long blue and white strips, which stretched back and flapped in the light breezes.\n\nRhodri nodded, \"Blue and white . . . they are Barisai from the north. I went on a cattle raid into their lands two . . . three winters ago . . . Nasty place . . . too cold . . . Cattle are thin. They built their _dun_ on an island in the middle of a river . . . Nasty place . . . Floods every spring. Nothing worth taking in their lands . . . not worth the horse fodder to raid them.\"\n\nAthauhnu nodded and whispered to me, \"You Romans call them Parisi.\"\n\n\"Parisi,\" I repeated. I had heard the name, but knew nothing about them. \"What are they doing down here?\" I asked.\n\nAthauhnu just shrugged. \"By his gold, he's one of their chiefs . . . a noble. It must have something to do with the king, with Duuhruhda.\"\n\nAs we watched, one of the noble's _fintai_ handed him something. He held it up to examine it. It was a white wand, the sign of a diplomatic mission. He had come to negotiate with the king.\n\nAs the Parisi rode to the north gate, the noble raised the white wand so the guard detail could see it. I saw the officer of the guard point in a direction and bow. The rest of the guard detail stepped aside and let the Parisi enter Bibracte.\n\nI didn't know what all this meant, but I was sure Caesar would be interested. By that time, I was wide awake, so I let Athauhnu get some sleep. I rotated our security detail and checked the horses. Everything was in order. I checked the position of the sun. It was still well before _meridies_. I settled down to watch the roads.\n\nI spent my time examining the fortifications. Back then, I had no experience or training in evaluating the strength of fortress walls. Since then, I have studied the treatises of Aeneas Tacticus and participated in a number of sieges, from both sides of the defensive walls. Back then, I had only been in the army long enough to note that there was no defensive ditch protecting the walls of Bibracte. Otherwise, the walls looked strong for the most part and in good repair. I noticed that the walls dipped down into a small valley between the north gate and the double-peaked hill in the south. From the look of the terrain within the double walls, there seemed to be a stream that flowed down from between the hill tops and must have come through both walls where they crossed that small valley.\n\nBy the shortened shadows, it was about the fifth hour when I noticed movement on the north road. I hissed over to the trooper closest to me, Alaw, to go get Athauhnu. By the time Athauhnu joined me, the shapes of the riders on the road were resolving themselves. It was another group of warriors.\n\nWhen I pointed them out to Athauhnu, I heard him hiss the word, \"Belgai!\"\n\nThese warriors were not as well arrayed as the Parisi. Their clothes were dark colored, varying shades of black, brown, and gray. Most of their armor was hardened, boiled leather; their faces were bearded; great gouts of hair escaped from underneath their helmets. Their horses, though, were magnificent, black and dark gray, larger even than the horses that the Roman army preferred, which came from Spain and the east, the land of the _Arabiani_. The mounts of the _Belgae_ pranced and snorted as they moved down the road, as if constantly challenging the control of their riders.\n\nTheir weapons were equally impressive. Each warrior carried a long lance with a flattened iron head, almost two _pedes_ long and a _palmus_ broad in the center, pointed and sharpened on each edge. This was a formable weapon, capable of stabbing and slashing. Each wore a long _spatha_ , hung from a dark leather baldric and secured at the hip by a wide leather belt. Their shields were round with a dark iron boss; each displayed the totem of a red wolf's head.\n\nThese were the _Belgae_ , a nightmarish race of fierce savages clinging to the frozen edges of the known world on the shores of _Oceanus_. Even the _Germani_ from beyond the _Rhenus_ feared these people, and now they were here, among the Aedui, less than twenty thousand _passus_ from our army.\n\n\"What are the _Belgai_ doing in Gaul?\" I heard myself say in Gah'el.\n\n\"They're not all _Belgai_ ,\" I heard Athauhnu say as he pointed down toward the head of their column.\n\nI looked where Athauhnu had indicated and noticed a rider who had been partially obscured by the Belgic leader. He was a smaller man, bareheaded and clean shaven. His hair was cut short, like a Roman. I noticed that under his reddish-brown military cloak, he wore a short sword on his right hip, a Roman infantry _gladius_.\n\n\"A Roman?\" I questioned.\n\n\"A _Rhufeinig_!\" Athauhnu agreed.\n\nI watched as the party approached the gate. Since _Belgae_ do not observe Gallic protocols, they carried no wand of negotiation. But, they were obviously expected. The Aeduan guard detail passed them through into the dun.\n\nI heard Athauhnu snort.\n\n\"What's so funny?\" I asked him.\n\n\"Belgai and Barisai despise each other,\" he answered. \"A blood feud since the Belgai came across the Rhenus and pushed the Barisai off their lands. I would love to be a bird in the rafters of Duuhruhda's hall when those two meet.\"\n\n\"Enemies become friends only when they have a common enemy,\" I quoted.\n\nAthauhnu grunted his agreement.\n\nIt was close to the seventh hour, almost time to pack it up and head south, when another party arrived, this time down the road from the northwest. Alaw, who had been posted up that road as a scout, alerted us.\n\n\"A _Pen_ ,\" he reported to Athauhnu. \"Chief! A merchant and his party are approaching the city.\"\n\n\"How many?\" Athauhnu asked.\n\nAlaw shrugged and calculated, \"The merchant . . . his woman . . . two bodyguards . . . a slave . . . four pack mules.\"\n\n\"How far out?\" Athauhnu asked again.\n\nAgain, Alaw shrugged, \"Maybe five hundred _passus_. He's in no hurry.\"\n\n\"Let's intercept!\" I suggested.\n\nAthauhnu grunted his agreement, then hissed, \"Guithiru! Mount five!\"\n\nThen, he turned to Alaw, \"Is Rhodri keeping an eye on him?\"\n\n\"'Tis, Chief!\" Alaw nodded.\n\nWe walked our horses down off the reverse slope of the ridge and hit the road out of sight of the city gate. There we mounted and followed Alaw to the northwest. We spotted the merchant's party less than four hundred _passus_ up the road. He halted when he saw us. His two guards attempted to look as menacing as they could in the face of nine well-armed riders.\n\nWe halted about ten _passus_ away. Almost immediately, Rhodri joined us from behind the merchant's party, and now they faced ten. I held up an empty right hand to show him I was not holding a weapon and asked in Latin, \"Are you bound for the fortress of the Aedui?\"\n\nThe man hesitated for a few heartbeats, then answered in a halting Latin, \" _Romani_ vos?\"\n\nI answered, \"We are from Caesar's army. Are you not a Roman?\"\n\n\" _Non Romanus. Graecus_ ,\" he answered.\n\n\"N\u03b1 \u03bc\u03b9\u03bb\u03bf\u03cd\u03bd \u03c4\u03b7\u03bd \u03b5\u03bb\u03bb\u03b7\u03bd\u03b9\u03ba\u03ae \u03b3\u03bb\u03ce\u03c3\u03c3\u03b1,\" I said. \"I can speak Greek.\"\n\nThe man stared again, then smiled, \"Like a Roman schoolboy trying to recite Homer for his tutor,\" he said in Gah'el.\n\n\"Then Gah'el it is,\" I agreed. \"Where are you bound?\"\n\n\"This is the road to Bibracte, is it not?\" he shrugged.\n\n\"'Tis,\" I agreed. \"Where are you coming from?\"\n\nThe man shrugged, \"I am coming down from the lands of the Senones, but I have been as far north as the Ocean, among the Veneti.\"\n\nI could tell Athauhnu and the men were uneasy about being out in the open while we talked, so I said, \"We would like to hear your tales of the Senones and the Veneti. Perhaps we can talk out of the sun, over in those trees.\"\n\nNow it was the merchant's turn to be nervous about being waylaid by a band of brigands pretending to be Roman soldiers\u2014or about being waylaid by actual Roman cavalry who might be looking to augment their wages. So I spoke again, \"I am Gaius Marius Insubrecus, _decurio_ in Gaius Iulius Caesar's _praetoria_. And, this is Athauhnu mab Hergest, _pencefhul_ of the Soucanai, in the service of Caesar and the Roman people.\"\n\nThe man's eyes widened a bit at that. \"Romans this far north and Soucanai this far west! These are indeed interesting times.\" Then, he looked up at the sun. \"You are correct, young man, this sun is hot. A short rest under the shade of some trees would be welcomed.\"\n\nWe moved over to a grove of trees to the north of the road. Alaw and Rhodri moved farther north to screen the road. Athauhnu dispatched Guithiru and two men to the south. The rest of our troop spread out, securing the area.\n\nAs we dismounted, the merchant announced, \"I am called _Gennadios Haw Emporos_ , Gennadios the Trader. The woman is called Evra. She claims to be from an island beyond Britannia, where the dead live. Years ago, when she was a girl, she was taken in a raid by the Veneti. Now she's my woman.\"\n\n\"Not _that_ long ago, Merchant!\" she spat.\n\nAthauhnu's eyes widened at that. \"A woman from the island of the dead! Then, that place exists!\"\n\nGennadios shrugged, \"She claims she never saw the dead feasting in golden halls. According to her, it's a place of pigs, cattle, and salmon the size of _tiuunai_ , tunnyfish, in the rivers. No! No walking dead. Just drunks, pigs, and fat farmers. Eh, _Meli Mou?_ \"\n\nThe woman gave a dismissive grunt as she adjusted the pack straps on one of the mules.\n\n\"Ah! Where are my manners?\" Gennadios said. \"Wine! Evra! The skin of retsina! Three cups!\"\n\nThen he turned to us, \"I doubt you've ever tried retsina. We use pine resin to preserve the wine. It travels well!\"\n\nGennadios' woman from the Isle of the Dead handed us cups, then poured the wine. It was golden yellow as it flowed from the wine skin. I could smell the pine resin. When all the cups were poured, we acknowledged our host and drank. I was surprised. It was light, delicious.\n\nGennadios smacked his lips, then said, \"I had heard that Caesar had moved north of the Rhonus, but I hadn't expected to see his men this side of Bibracte.\"\n\n\"Really?\" I said. \"Where did you hear that news of Caesar?\"\n\n\"From the Roman delegation to the Senones\u2014\" he began.\n\n\"Romans! Among the Senones! Who\u2014\" I began.\n\nGennadios held up his hand. \"Yes! A Roman delegation arrived in the Senones' _dun_ , Agedincum . . . When was it? About two . . . three weeks ago. They had a broad-striper leading them . . . a noble or a senator . . . A real nob, he was. The rest looked military . . . Bodyguards, I would think, led by a narrow-striper. They had an audience with the _uucharix_ , the tribal king, Caswalu, but they spent most of their time with his _dunorix_ , Dramaelo. No love lost between those two, I can tell you. Dramaelo is older, but Caswalu seems to have favor with you Romans.\"\n\n\"Do you know what these Romans wanted?\" I interrupted.\n\n\"Wanted? Oh, yes . . . They told the king that Caesar did not have the Roman Senate's authorization to cross the Rhonus . . . that the Aedui were still friends of the Roman state.\"\n\n\"They specifically mentioned the Aedui?\" I interrupted again.\n\n\"They didn't have to,\" Gennadios tutted. \"They had an escort of Aedui riders from the _fintai_ of the Aeduan _dunorix_.\"\n\n\"Deluuhnu?\" I asked.\n\n\"The same,\" Gennadios confirmed. \"Why are you surprised? Didn't you know that Dramaelo is married to Deluuhnu's sister?\"\n\n\"His sister?\" I said.\n\n\"Oh, yes!\" Gennadios continued. \"In fact, many of Dramaelo's troops are Aedui. Makes his king right nervous.\"\n\n\"Did the Romans encourage the Senones to attack Caesar's army?\" I asked.\n\n\"Attack them?\" Gennadios answered. \"No . . . not in so many words. But, the impression they gave was, if the Senones joined with the Aedui in defending Aedui territory against Caesar's unauthorized incursion, the Senate would understand.\"\n\nEvra, who was sitting with the two bodyguards over by the mules, called over to Gennadios, \" _Labhair t\u00fa i bhfad r\u00f3, fear d'aois. Roinnt l\u00e1 beidh go bhfaigheann mharaigh t\u00fa_.\"\n\nI didn't understand what she said, but some words sounded familiar, almost recognizable.\n\nGennadios chuckled. \"She's telling me to keep my mouth shut,\" he told us. \"The women of the Gaelige . . . that's what they call themselves on the Isle of the Dead . . . Gaelige . . . sometimes the people of Eriu . . . that's their goddess of love . . . Aphrodite. You'd never know it from Evra. Their women are like the women of the Gah'el, but ten times worse. They just do what they want . . . say what they want. When I get back to Massalia, I'd keep Evra locked up, but she wouldn't stand for it. She'd tear my place apart, and me along with it.\" He chuckled again.\n\n\"Gaelige,\" I said absently. \"Almost sounds like Gah'el . . . but back to the Romans.\"\n\n\"Yes, the Romans,\" Gennadios said, filling his cup and offering more wine to Athauhnu and me, which we gratefully accepted. \"Quite generous they were too, especially to the _dunorix_. All nice, new silver, too.\"\n\nGennadios reached into his _marsupium_. He pulled out a silver coin and handed it to me. It was unworn and shiny, a newly minted _quadriga_ , a _denarius_ coin. I flipped it over and saw the image of one of this year's consuls, my erstwhile patron, Aulus Gabinius.\n\nGennadios was talking, \"It worked out well for me. Usually there's not much hard currency among the tribes. I sometimes have to resort to bartering . . . swatches of eastern cloth and pottery for chickens . . . that sort of thing . . . Useless when I get home. A man needs silver to live in Massalia.\"\n\nI handed the coin back. \"You didn't happen to hear the name of the purplestriper did you?\" I asked.\n\n\"Hear it!\" Gennadios exclaimed. \"I did better than that! I sold a skin of wine to his tribune. The man had a ghastly scar across his face . . . still red and puckered in places . . . Said he got it in a skirmish with the Belgae last season. I hadn't heard anything about Romans fighting the Belgae. He told me the man's name was something like Pompius . . . That's it . . . Gaius or Gnaeus Pompius. When Simathemeni . . . that's my name for the scarred Roman . . . \"Scar Face\" . . . when this Simathemeni got a bit drunk, he referred to his companion as Minus. That means 'The Lesser,' doesn't it? I never understood you Romans' sense of humor.\"\n\nI certainly thought I recognized the name. Pompius or Pompeius. That was the name of Caesar's colleague, one of the _triumviri_. But Pompeius Minus? Pompeius was called Magnus, \"The Great.\" Was he one of Pompeius' freedmen? No, a freedman wouldn't dare wear a broad, purple stripe, regardless of who his _patronus_ was. Did Pompeius Magnus have a son? If so, wouldn't he be called Pompeius Iunior\u2014or was \"Scarface\" making a sarcastic joke, like Gennadios thought?\n\nBefore I could ask, Guithiru came into the grove. He addressed Athauhnu, \"Chief! A messenger came up from the Roman tribune. We're to withdraw and meet him south of Bibracte.\"\n\nI sensed something important was up. Was it the battle in the south? Had Caesar been defeated? Were we cut off?\n\nI stood up. \"We thank you for your hospitality, Gennadios. We must depart.\"\n\nI thought to tell him not to inform the Aedui of our presence, but telling a merchant not to share gossip and information made as much sense as asking a stream not to flow.\n\nI reached into my _marsupium_ and found a small, silver _mercurius_ and handed it over to the merchant. \"For the wine and the conversation, _phile mou_ ,\" I told him.\n\nGennadios made the coin disappear into his own purse. \" _Vobiscum fortuna sit, mi amice!_ \" he said in Latin.\n\n\" _Et tecum_ ,\" I responded, unconsciously rubbing my _lorica_ where my medallion rested.\n\nI looked over to where Evra was sitting. The woman from the Isle of the Dead was glaring at me with the soulless, black eyes of Hecate.\n\nUnconsciously, I made the _cornucellus_ , the little horn, with the fingers of my right hand to ward off the evil eye.\n\nWe rejoined Agrippa, Madog, and the rest of the Sequani troopers at the rendezvous point south of Bibracte. Agrippa told me briefly that a full _ala_ of the _dunorix's fintai_ , almost forty riders, had sallied out of the south gate a little less than an hour earlier. At first, Agrippa feared that his troop had been detected and the Aedui were going to attack him. But, the Aedui rushed past his position without as much as a wink. They pounded down the road to the south, toward where our army was engaged with the Helvetii. Agrippa sent five _exploratores_ to follow the Aedui while he waited for my _ala_ to return from north of Bibracte.\n\n\"Any word of the battle?\" I asked him.\n\nHe shrugged, \" _Nil!_ Not even a sound!\"\n\nWe rode south in the tracks of the Aedui. We had gone about ten thousand passus when we found our scouts. They were in a narrow wooded cut between two low hills. Two had been killed by arrows; two more had been brought down with stabbing wounds to the abdomen, typical of a close ambush by infantry; one scout was missing. Their horses were gone. All the bodies had been stripped of their armor and weapons.\n\nA trooper from our point element was examining the ground around the bodies.\n\n\"What happened here?\" Madog demanded.\n\nThe trooper shrugged. \"Men on foot . . . maybe fifteen . . . twenty at the most . . . They hid in the brush along the trail . . . went off in that direction . . . leading riderless horses.\"\n\nI translated for Agrippa.\n\n\"Where is Ailwuhnu?\" Madog asked after the missing man.\n\nAgain the trooper shrugged. \"He's not here. Maybe they took him?\"\n\nOur other point man came back down the trail. \"They had horses just over the rise,\" he reported. \"They retreated south toward the Helvetii.\"\n\n\"The Aedui we're tracking?\" Madog demanded.\n\n\"No,\" the trooper said. \"The horses rode light. The Aedui had steel armor. These men did not.\"\n\nMadog was just about to respond when there was a commotion to our rear. One of our men was leading a horse carrying a wounded man. It was our missing _explorator_. He was doubled forward in his saddle. There was blood down his saddle and his right leg.\n\nMadog recognized the man, \"Ailwuhnu! What happened here?\"\n\nThe man gathered his strength to face his chief. \"Ambush, Lord,\" he gasped. \"They came at us . . . up out of the brush.\" His companion steadied him in the saddle.\n\n\"The Aedui?\" Madog began.\n\n\"No!\" the man gasped. \" _Almaenwuhra!_ Germans!\"\n\nI translated that for Agrippa. \"There are Krauts with the Helvetii?\" he said to no one in particular. \"What are they doing up here?\"\n\nThe Sequani were helping Ailwuhnu down from his horse. I was no _medicus_ , but the man had a deep stab wound in his lower right abdomen. His chances were not good. And, he could suffer for days.\n\n\"Bring up the _medduhg_!\" Madog called down the line of riders. Then, he turned to Agrippa and said in Latin, \"We have _medicus_ for horse. He do what he can.\"\n\nWhile the Sequani were trying to make Ailwuhnu comfortable, Agrippa said, \"We need to get back to the army; Krauts this far north is not a good sign.\"\n\n\"I am same,\" Madog nodded. \"I leave companions here. Take care of thing. Then we go.\"\n\nMadog dismounted and walked over to where the _medduhg_ was treating the wounded trooper. I saw Madog take the hand of the wounded man as they spoke. Then, they embraced briefly. Madog touched the man's cheek as a father saying farewell to his child, then he remounted.\n\nA rider came up beside me. It was Athauhnu.\n\n\"Ailwuhnu is Madog's sister's son,\" he said. \"This is not a good thing.\"\n\n\"What will they do?\" I asked.\n\nAthauhnu shrugged, \"What they can. If he cannot travel, the _medduhg_ will give him drugs to make him comfortable before the _cuhthraulai_ , the daemons, possess his gut. Then he'll help him to _tir ieuenctid_.\"\n\n\"The Land of Youth,\" I echoed remembering Gran'pa's tales. \"You mean the _medduhg_ will . . . he'll kill him?\"\n\nAthauhnu nodded. \" _Ie!_ It is . . . _un drugaretha_ . . . a mercy. There is no honor for a warrior to die drenched in his own piss and shit after hours of agony. The _medduhg_ will send him to the feasting hall of heroes. It is an honorable death.\"\n\nAthauhnu left me to follow his _ala_ down the trail. I looked back to where the _medicus_ was attending to Ailwuhnu. He was propped up against a tree. His eyes were closed; his face was pasty white, almost greenish, covered with sweat. He would be feasting with the heroes before the sun was down. For the first time in my career as a soldier, a thought formed clearly in my mind: _fortunae deae gratias ago_. I thanked the goddess Fortuna that it was not me. I turned my horse and followed Athauhnu.\n\nSince we did not know the outcome of the battle, Agrippa led us to where the camps had been when we departed that morning. They were still there, but the legionaries, who had been left behind, were dismantling them. Agrippa found the legate, Pulcher, and asked him for news of the battle.\n\n\"Battle!\" Pulcher snorted. \"There was no battle. Our _imperator_ allowed the barbarians to escape right out from under his ample snout. Those hairy Gauls are still laughing themselves sick over the incompetence of\u2014\" Then, Pulcher thought about what he was about to say and continued, \"There was no battle. The army's in camp ten thousand _passus_ to the west. You'll find Caesar there.\"\n\nAgrippa was about to walk away, when Pulcher spoke again, \"When you see the _imperator_ , tell him I have obeyed his order and released the Aeduan prisoner, Dumnorix!\"\n\nWe found Caesar's _castra_ along a ridgeline south of where the battle with the Helvetii was to have taken place. As we entered the camp of the Tenth Legion, we could sense the sullenness and resentment of the soldiers. Through no fault of their own, the enemy had escaped. So, the halfhearted pursuit of an enemy they believed they could have defeated would continue. And still, no new rations had been issued. The men had long ago exhausted their supplies of fresh meat and wheat; even their marching rations of jerky and _buccellatum_ were gone. Even worse, the _posca_ , their beloved sour wine, was a memory. To fill their stomachs, they could look forward to nothing but barley and water, usually a punishment ration, while their enemy ate well and had been allowed to walk out of their trap, with nothing but contempt for them. They blamed their officers; they blamed Caesar.\n\nWe found Caesar's _praetorium_ in the center of the camp. As we dismounted, we saw Ebrius, his clerk, standing outside the tent.\n\n\"You might want to think twice about going in there,\" he warned us. \"The boss is not having a good day.\"\n\nAgrippa ignored the _scriba_ and entered. Madog and I followed. Even in the outer _cubiculum_ , we could hear Caesar's voice, \"I don't care what his excuse is . . . whether he lost his nerve . . . whether he's blind . . . or whether he's just bloody incompetent. He's relieved of his command! He's not fit to serve in this army! I want him out of camp before the sun sets!\"\n\nWe slipped into Caesar's office. Labienus seemed to be in the line of fire of Caesar's tirade. He was still dressed in his battle armor, his helmet locked under his left arm.\n\nCaesar looked over and saw us. \"Agrippa,\" he said, \"I suppose you have more good news for me?\"\n\nBefore Agrippa could answer, Ebrius entered the tent.\n\n\"Pardon my interruption, _Imperator_ ,\" he apologized, \"but the senior centurions are outside as you ord . . . er . . . requested.\"\n\nCaesar nodded. \"You are dismissed, Labienus, but don't go far. Agrippa! I need some time with my officers in private; then we'll talk.\"\n\nWhen we got outside, Agrippa turned to Labienus, \"What in the name of _Dis_ happened, sir? Why are we still here? What happened to the enemy?\"\n\nLabienus held up his hand to silence Agrippa. I could see the fatigue and frustration in his eyes.\n\n\"It was a good plan, and it should have worked,\" Labienus began. \"But, it turned into a complete cluster. I got my two legions into position above the enemy a good hour before dawn. The Helvetii had no idea we were up there. They must have detected Caesar approaching just after dawn because we could see the camp begin to stir. A battle line was forming down toward our left front, just like Caesar planned. The warriors were forming up the musters and pushing them forward. The enemy had their backs to us . . . no idea we were there. Then, it all just came apart. The Helvetii began to melt away from the intended battle position and began to move to the west as if there was no threat approaching. They must have detected our presence up on the hill because a group of them began pointing up toward our position. Luckily, there wasn't enough discipline down there to organize an attack, or we could have been _immerda_. A bunch of the Helvetii dropped their _bracae_ and showed us their backsides before they walked away . . . really pissed my boys off. They wanted to run down there and stick their _pila_ through the Helvetians' _bracae_... I had all I could do to keep them together.\"\n\n\"But what happened?\" Agrippa persisted.\n\n\"I'm getting to that,\" Labienus continued. \"I sent riders to find Caesar. He was digging in here, wondering what the hell had become of me. It seems that when he was coming up the road toward the Helvetii, he had sent some _exploratores_ forward to make contact with my division . . . an _ala_ of Roman cavalry under an _angusticlavus_ named Considius. He couldn't have chosen a worse officer for that assignment. Considius has been with the army since the time Marius was fighting Sulla. The man is as blind as Homer . . . can't see past his own nose, and all of his cronies have been covering for him. He reported to Caesar that my hill was occupied by the enemy and my division was nowhere to be found. So Caesar halted the advance and veered off to the south to find some high ground in case the Helvetii attacked him. By midday, I was sitting on my hill, watching the dust of the enemy escaping to the west, and Caesar was on his hill, wondering where I was . . . all because of a blind scout. A blind scout! It would be damned funny if I had any sense of humor left.\"\n\nAbout that time, the six _primi pili_ of the legions left Caesar's tent. I had seen men look grimmer, but that was at a funeral. Ebrius beckoned us into Caesar's office. \"The _imperator_ will see you now,\" he intoned.\n\nCaesar was sitting in a slouch beside his field desk, legs extended in front of him. He was staring intently at the ground. \"My senior centurions tell me that after today's _calamitas_ , they cannot guarantee the loyalty of my troops,\" he said as we entered. \"What good news do you boys have for me?\"\n\nAgrippa reported what we had seen and heard around Bibracte. Caesar said nothing. He just rubbed his forehead and shook his head.\n\nWhen Agrippa reported the ambush by Germans and the death of Madog's nephew, Caesar said, \"Germans . . . that's all I need . . . Germans this far west of the Rhenus. My condolences to you and your family, Madocus _Dux_. Your nephew died in the service of Rome. His name will be remembered.\"\n\n\"Thank you, Caesar,\" Madog responded. \"He died like a warrior . . . a good death.\" Madog's voice sounded hollow, exhausted.\n\n\"Parisi and Belgae delegations in Bibracte . . . A Roman riding with a Belgae war band . . . Renegade Aedui horsemen somewhere in my rear . . . A Roman senatorial telling the Senones that if they attack us, Rome will not take offense. Do I have it all, Agrippa?\" Caesar asked.\n\nAgrippa nodded, but then said, \"One more thing, _Imperator_! Pulcher reports that he has released Dumnorix, according to your order.\"\n\nWhen Agrippa said this, Caesar's head jerked up. He stared at Agrippa for a few heartbeats, then said in a low voice, \" _Quid dicebas tu?\"_\n\n\" _Imperator_ ,\" Agrippa responded, \"Legate Pulcher reports that he released the Aeduan, Dumnorix.\"\n\n\"He did what?\" Caesar shouted jumping to his feet. \" _Iste stulte_ . . . that . . . that _verpa_! He did what?\"\n\n\"Released Dumnorix . . . the Aeduan,\" Agrippa stammered.\n\n\" _Cacat_!\" Caesar shouted, his fist slamming down on the desk. \"Labienus! Send a detachment of my _praetoriani_ back to those camps . . . bring that _podex_ , Pulcher, here . . . to me! Stat'! _Iste fellator_. .. That _half-wit_ better have a good explanation, or I'll crucify him!\"\n\nSuddenly, Caesar's eyes became unfocused. He stumbled, barely able to steady himself on the field desk. Labienus rushed forward and took his arm. Without taking his eyes off Caesar, he said, \"Insubrecus . . . quickly . . . find Spina, the _medicus_ . . . Bring him here! Agrippa! Madocus! Wait outside!\"\n\nI flew out of the tent, past a startled Ebrius. I ran over to the medical tent, less than thirty _passus_ away. \"Spina! Spina, _Medice!_ \" I shouted.\n\nI heard his thick, Aventine accent from one of the rear compartments, \"Who's dat? Whadda you want? I'm ovah hee'ah!\"\n\n\"Medice!\" I shouted. \"I'm here to bring you to\u2014\" Then, I stopped myself. Yelling out that the commander of the army was near collapse was not conducive to morale, especially after a day like this.\n\nSpina came out of the back. \"You callin' for me, or what?\" he asked.\n\n\"Yes, _Medice_ ,\" I said, lowering my voice, \"please accompany me to the _praetorium . . . stat_ '.\"\n\n\"The _praetorium?_ \" Spina started. Then, he whispered to me, \"Is da boss havin' one of his spells?\"\n\nI nodded at him. He nodded and then went back into the tent where jars of herbs and drugs were stored. He packed something into a _loculus_ and said, \"Okay, let's get adda hee'ah!\"\n\nAs we walked back to the _praetorium_ , he recognized me, \"You're dat _tiro_ I patched up a while back, ain't cha? Duh one who got stabbed by duh slave who wasn't a slave, right?\"\n\n\" _Recte_ ,\" I confirmed.\n\n\"How's dee ahm?\" he asked.\n\nIt took me a heartbeat to realize he was asking me how my arm was. \"Good as new,\" I told him.\n\n\" _Bene!_ \" he said. \"Looks like yaw getting up in duh world. A _decurio_ in less than a year . . . a praetorian to boot! Dat's impressive!\"\n\nBy that time we were at the _praetorium_ , Spina was immediately passed straight through to Caesar's _cubiculum_. I saw Agrippa and Madog still standing outside.\n\n\"What's this all about?\" I asked.\n\nAgrippa just shrugged, but he looked worried.\n\nMadog said in his broken Latin, \"Sometimes gods enter in man's anger\u2014\" Then, he stopped. Either his Latin or his knowledge had failed him.\n\nAfter a bit, Spina came back out. \"Youse can go back in now. He wants to tawk to yas. Not too long. He needs to sleep.\" Then, he walked back toward the medical station, whistling some tuneless sounds.\n\nWe reentered Caesar's _cubiculum_. He was sitting at his desk, drinking something that smelled of wine, vinegar, and something else\u2014something sweet and cloying. He seemed relaxed, his eyes lidded and heavy. Labienus was standing next to him.\n\nCaesar looked up at me and asked, \"Insubrecus, are you familiar with the story of the Gordian's Knot?\"\n\nI remember the Stick telling us the tale during a class on Greek culture. I recited, \"When Gordias became the king of Phrygia, his son, Midas, dedicated a chariot to Zeus and tied its shaft with an intricate knot of cornel bark. He declared whoever could undo the knot was the rightful king of Phrygia. Later, Alexander arrived in Phrygia, when it was a province of Persia. When challenged to prove himself worthy of the throne by undoing the knot, Alexander sliced it in half with his sword. That night, there was a violent thunderstorm. Alexander took this as a sign that Zeus was pleased and would grant Alexander many victories.\"\n\nCaesar nodded. \"Not lyrical, but accurate, Insubrecus. All these stories and reports of Romans, Belgae, Krauts, and whatnot have become a knot I do not have time to unravel, so I'm just going to slice it open!\" Caesar announced. \"Tomorrow at dawn, this army marches on the Aeduan capital... we march on Bibracte!\"\n\n# XII.\n\n# _Bibracte_\n\n# BIBRACTE\n\n_Et quod a Bibracte, oppido Haeduorum longe maximo et copiosissimo, non_ _amplius milibus passuum XVIII aberat, rei frumentariae prospiciendum_ _existimavit; itaque iter ab Helvetiis avertit ac Bibracte ire contendit_.\n\n\"The town of Bibracte, by far the largest and most prosperous settlement of the Aedui, was not more than eighteen miles away. Since Caesar estimated that the town would provide him a supply of grain, on the next day, he diverted his route of march away from the Helvetians and toward Bibracte.\"\n\n(from Gaius Marius Insubrecus' notebook of Caesar's journal)\n\nDuring the fourth watch, Caesar assembled his six legions in the dark valley north of the camps. I was again assigned to Agrippa and Madog's Sequani cavalry. Our mission was to screen Caesar's advance to the east. Once we came abreast of Bibracte, we were to swing around it to the east and seal its northern approaches. Caesar hoped he could trap any Gallic deputations still in the _oppidum_ and any Romans accompanying them. My mission was to ensure that any prisoners we took were delivered to Caesar's interrogators in a condition to talk.\n\nAs we rode through the darkness, my head was fuzzy from lack of sleep and a bit too much of Caesar's _posca_ the night before. I don't know what strange and exotic herbs Spina had dosed Caesar with, but it put the _imperator_ into one of his rare loquacious moods. He kept Labienus and me up well into the second watch, talking about his vision for Rome and his frustrations with the antiquated, doddering machinery of the _res publica_.\n\n\"These old fools in the Senate just don't understand Rome's position through the extension of our _imperium_ ,\" he was saying. \"They think they're still ruling a city and the farmlands around it. Marius . . . even Sulla . . . taught them the foolishness of that. One strong man with the support of the army can set their whole house of straw ablaze.\"\n\n\"But, Caesar,\" Labienus protested, \"these institutions . . . our laws . . . the Twelve Tables . . . prevent too much power from falling into the hands of a single man. Our ancestors understood this from the tyranny of the kings. The _mos_ _maiorum_ , the tradition of our ancestors, is sacred.\"\n\nCaesar retorted, \"The _mos maiorum_ didn't stop Sulla from killing hundreds of his enemies for their estates, did it? He just ignored it, and the Senate quaked in their red boots while he did it. Why? He had an army to back him up, an army loyal to Sulla's purse, not to the Senate.\"\n\n\"We have restored the _res publica_ since Sulla\u2014\"\n\n\"Bah!\" Caesar snorted. \"The sacred _mos maiorum_! When _fides_ and _pietas_ encounter silver and greed, they melt away! And there is the heart of it! Rome is not ruled by _virtus_ ; it's ruled by _avaritas_ . . . and Rome's greed seems to have developed an infinite desire for plunder. So she extends the _imperium_ , grasping more and more. We do not rule our provinces; we rape them! We say we send out proconsuls and propraetors to protect the interests of the _res publica_ . . . to bring _Romanitas_ to the barbarians . . . But what do they really do? They enrich themselves and their masters in the Senate! The only difference between a Roman army and a pack of brigands is size and discipline. If you encounter either, they'll strip you bare and leave you bleeding!\"\n\nI continued to wonder what Spina had put in Caesar's drink.\n\n\"Look at Insubrecus, here!\" Caesar went on. \" _He_ represents the future of Rome! He's one generation out of a round hut; his grandfather wore trousers just like the Helvetii we're chasing across _terrae comatorum_ , the lands of the hairbags. But, he is the future of Rome\u2014not those over-educated, inbred senatorials down in Rome who think they have the right to rule the world because some boot-licking, scroll-sniffing charlatan told them they're descended from Romulus! Bah!\"\n\nLabienus was sweating freely by this time. He was devoted to the legendary Rome of Mucius Scaevola, Scipio Africanus, and Cato the Elder. Or, at least he was devoted to the legends of these heroes from the dim chronicles of our history.\n\n\"When I return to Rome from this command,\" Caesar continued, \"I plan to introduce legislation to offer the franchise to _Gallia Cisalpina_ and all of _Italia_ \u2014\"\n\n\"Certainly, that will not pass the Senate!\" Labienus protested.\n\n\"Politics, my dear Labienus! Politics!\" Caesar snorted. \"As long as I, and my tame, plebian tribunes, support land reforms for Pompeius and oppose debt reduction for Crassus, it will pass. Believe me! Those doddering old fools in the Senate love their luxuries and fear the gangs of Clodius and Milo more than they cherish their _mos maiorum_. And that's only the beginning!\"\n\nCaesar went on like that well into the night, until the wine and drugs finally took hold of him. As we were sitting in that tent that summer night about twenty thousand _passus_ south of Bibracte, we had no idea how Nona was spinning out the threads of our destinies: the death of Caesar's daughter, Iulia, and his split with Pompeius; the death of Crassus in the land of the Parthians; Labienus' death while fighting against Caesar in the civil wars; Caesar's own death at the hands of those to whom he had granted his _clementia_. No! As we sat and drank _posca_ that night in _Gallia_ , all of that lay well in the future __.\n\nCaesar's plans for Rome and the fumes of the _posca_ I had drunk the night before were fogging my head as we pounded north into the darkness on our way to Bibracte.\n\nBefore I returned to my quarters the night before, Labienus had confided in me that Pulcher did, in fact, possess a written directive apparently issued by Caesar's headquarters, ordering the immediate release of the Aeduan prince, Dumnorix. Pulcher claimed that the directive had been delivered to him by an _angusticlavus_ , a narrow-striper whom he did not recognize.\n\nRegardless, Pulcher was finished in the army.\n\nLabienus advised him to pack his kit and be on the road to Massalia before Caesar remembered his threat to crucify him. Not that even a proconsul with full _imperium_ could, or would, execute a citizen, a patrician, and a senatorial, like a slave, but why test the theory? Labienus told Pulcher to be on a ship to Ostia before Caesar had a chance to try to make good on his threat.\n\nThe issue that really worried Labienus was that there were obviously Roman officers serving within the army who were actively engaged in undermining Caesar and his efforts to bring the Helvetii to heel.\n\nBy dawn the Sequani cavalry _turma_ was positioned across the only major avenue of approach from the east that could cut across Caesar's line of advance. It was more of a pathway than a road that travelers had created following a tributary river up from the Rhonus Valley.\n\nAgrippa and Madog had advanced as a recon in force some two to three thousand _passus_ down the narrow, wooded valley. Athauhnu and I were positioned where the road from the east broke out of a valley and joined the road from the south we had taken from Caesar's camps. From there, both roads ran toward the west, up into a broader valley, which we expected our army to cross on its way to Bibracte. We weren't expecting any problems. We assumed the main body of the enemy was well to the west.\n\nAs usual, Alaw and Rhodri were deployed a few hundred _passus_ down the road to the east. Guithiru was deployed with five of our troopers to screen our rear. The morning sun was beginning to warm me and diffuse the fumes in my head. Athauhnu seemed quite amused by my damaged condition.\n\n\"I have known men who had to drink for courage before battle,\" he joked, \"but you're the first one I've ever known who gets drunk the night before so he's hungover _for_ the battle!\"\n\n\"Not today,\" I dismissed him. \"I'm in no condition.\"\n\nAthauhnu reached into his _marsupium_ and handed me what looked like a dried out twig. \"Here! Chew on this. It will clear your head,\" he offered.\n\nI took the stick and examined it.\n\n\"Go ahead, Arth Bek!\" Athauhnu encouraged. \"I wouldn't poison you.\"\n\nI chewed a bit on the twig. It did seem to help a little.\n\n\"Madog is concerned about the Caisar,\" I heard Athauhnu say.\n\n\"Concerned about the Caisar?\" I questioned. \"In what way?\"\n\nAthauhnu shrugged. \"Madog says that last night, Caesar had a . . . a spell . . . and when he recovered, he decided to go to Bibracte.\"\n\n\"Caesar knows what he's doing,\" I defended my _patronus_.\n\nAgain, Athauhnu shrugged. \"At times the gods send madness to cause a man to destroy himself. We march north, with the Helvetii behind us and the Aedui in front of us. If things go badly, we Soucanai will go east back into our own lands. You should come with us, Arth Bek. The Aedui will show no mercy to defeated Romans.\"\n\nI had no idea how to respond to that. In the case of a defeat, escaping to the east made sense. But, that would mean abandoning Caesar and the Roman army, to which the _sacramentum_ bound me until dismissal or death.\n\nBefore I could formulate a response, two riders came pounding up the trail from the east. It was Alaw and a trooper that had gone forward with Madog. They pulled up in front of Athauhnu. \" _A Pen! Uh doucliau geluhnai_!\" Alaw reported. \"Chief! The enemy's coming!\"\n\n\" _Uh geluhnai_?\" Athauhnu questioned. \"The enemy? The River People . . . the Helvetii?\"\n\n\" _Na, Pen_!\" the man responded. \" _Almaenwuhra_! Germans!\"\n\n\"How many?\" Athauhnu asked the man.\n\nThe messenger shrugged. \"The valley is narrow . . . the road twisted . . . They advance without fear. Madog believes there are many.\"\n\n\"Cavalry?\" Athauhnu pressed the man.\n\n\"We have seen only mounted men,\" the man confirmed.\n\nAthauhnu nodded and walked over to his horse. He took a hunting horn that had been attached to his saddle and blew some discordant notes to assemble his _ala_.\n\nThen, he surveyed our position. \"See that rise there?\" he asked the messenger, pointing to where the ground rose to meet the road from the south. \"Tell Madog I will assemble my troop there.\"\n\nThe Alaw nodded, and he and his companion turned back down the road. He passed Rhodri riding in, in response to Athauhnu's summons. Soon, Guithiru and his detail returned.\n\nAthauhnu mounted and quickly briefed his men: \"The enemy approaches from the east. Germans! We do not know how many. Madog is withdrawing to our position. We will move west up the valley. We will stay above the Germans, between them and the Romans. When they reach this point, they will have to deploy . . . spread out. Then we will see how many we're dealing with.\"\n\n\" _A Pen_ , the sun will be in our eyes!\" Guithiru observed.\n\n\"It can't be helped,\" Athauhnu stated. \"We will keep the higher ground.\n\nThey are mounted! If there are too many to fight, we will escape to the west, toward the Romans.\"\n\nGuithiru grunted.\n\nWe withdrew to the top of the rise and deployed, in line, facing the opening of the road below. There we waited. To our rear, the valley rose gently toward the north and west. Whoever these Germans were, they were heading right into the flank of the advancing Roman army.\n\nWe seemed to wait for an eternity before we spotted movement on the road below. I could sense our troopers tense as the first riders became visible below us. They were ours! Athauhnu signaled to them with his hands. They quickly joined us, assembling in line on our left flank. Their leader approached Athauhnu. He was a veteran warrior called Ci, the \"Hound.\"\n\n\" _A Pen_!\" he reported to Athauhnu, \"the Germans advance . . . Madog has engaged but withdraws before them.\"\n\nAthauhnu grunted and nodded. Ci rejoined his troop.\n\nAgain, we waited. At least the flies that started buzzing around in my stomach made me forget about the pain in my head. Suddenly, I realized that I had thoroughly chewed the twig Athauhnu had given me earlier. I spit it out.\n\nAthauhnu kept his eyes on the opening of the road below us, but said to me out of the side of his mouth, \"This is the worst time . . . just before the enemy comes. Once they are here, we won't have time to be afraid.\"\n\nI nodded, grateful for Athauhnu's use of the word \"we.\"\n\nSoon, Madog's men came up the road. Again, Athauhnu waved and pointed to our right flank. Madog and Agrippa were the last riders to emerge. Madog had a man behind him on his horse. I heard Athauhnu mutter, \"Two missing.\"\n\nAthauhnu waved again. Agrippa and Madog joined us in the center of the Sequani line. The man riding behind Madog dropped off the horse. Then, I saw that he had the stub of an arrow protruding from his thigh.\n\nMadog caught me looking at the man. \"German horsemen do not carry bows,\" he said in Gah'el. \"There are warriors on foot down there.\"\n\n\" _Pedes_?\" I responded in Latin. \"Infantry?\"\n\nAgrippa heard me. \" _Pedes, Insubrece_! I don't think this is a raiding party coming in from the east. I need to get an idea how many Krauts are coming up that road, then warn Caesar.\"\n\nAlaw pointed toward the road below us. \" _Pen! Maint uhn dod_! Chief! They're here!\"\n\nGerman riders were emerging from the valley and filling the field below us. They saw us but didn't seem at all concerned. There didn't seem to be any organization. They milled around in the field below us, some pointing toward us, with others just roaming about.\n\nTheir equipment wasn't impressive. Some had bronze helmets, others leather, some none at all. I could see no chainmail, but I did see some leather _loricae_. Most of them carried either lances or stabbing spears.\n\nI heard Madog spit, \"Farmers on horses!\"\n\nThen, another group of warriors emerged from the narrow valley. These men were well-equipped, with steel helmets, chainmail, and large, round shields with the image of what looked like a red oxen with long horns painted over the boss. Each had a red, horsehair topknot trailing from his helmet. One rider, a giant with a red beard, emerged from the trail. Directly behind him rode a mounted warrior carrying a totem on a spear shaft, with long, black oxen horns and a skirt of a black pelt below it.\n\n\"That one must be their _eorle_ ,\" I heard Athauhnu use an unfamiliar term.\n\n\"What's an _eorle_?\" I asked out of the side of my mouth, afraid to take my eyes off the giant below us.\n\n\"That's what the Germans call their _penai_ , their chiefs,\" Athauhnu answered. \"The riders with him are his _gedricht_ , his _fintai_. He equips them, feeds them, and they are sworn to protect him in battle or die with him. The Germans believe that any member of the _gedricht_ who survives a battle in which their chief dies is cursed to wander the middle lands alone, an outcast.\"\n\nMadog interrupted us, pointing down at the Kraut standard. \"That is the totem of the Aurochs. They are Germans . . . the People of the Aurochs . . . You Romans call them _Boii_ . . . the 'cow people.' And that is no _eorle_! That is their _ciuning_ . . . their tribal king . . . look. There are at least fifty riders in his _fintai_. No _eorle_ could support so many. That is the tribal muster coming up the trail . . . Thousands would be my guess.\"\n\nAs he spoke, well-armed warriors on foot emerged from the valley. They didn't march in step or in any recognizable formation as did the legions. They just poured out onto the open area below us. The men wore bronze or steel helmets, many with cheek guards and some with bands of metal extending down over their noses. Some carried spears; some war axes. Each wore a long sword on his left side and a shorter sword on his right. Every one of them carried an oversized, round shield. They poured out of the valley in the hundreds.\n\n\"Those are _cnihtas_ , the professional warriors of the tribe,\" Madog said. \"They are _dugath_ , _veterani_ , blooded warriors. They form the _scilde wealle_ , the shield wall, in the center of the battle line. Behind them march the _iougath_ , the young warriors, boys who have yet to be blooded. They stand behind the shield wall. And, behind them, the _fiurd_ , the tribal muster. They form the flanks of the battle line in the defense. When the Germans attack, they're the shock troops. They disrupt the enemy's line so the _dugath_ can get in among them and do the real killing. That's a tribal assembly down there. The Boii are going to war against the Caisar.\"\n\nIt was just then that we heard a trumpet call in the distance behind us.\n\n\"Caesar?\" I asked.\n\n\"It must be,\" Agrippa answered, \"but, if the army's on the march, why would it be sounding trumpet signals?\"\n\nDown below we could see the Boii king haranguing his cavalry and pointing up the slopes toward us.\n\n\"I think that big Kraut down there with the red beard wants us out of here,\" Agrippa observed.\n\nI translated for Madog, but he had understood most of it. \"Tribune right. Time go away for us,\" he announced in Latin.\n\nMadog made sure that the wounded man had been mounted behind another warrior. He was about to signal his riders to withdraw to the west when Agrippa said, \"We can't leave yet.\"\n\nMadog froze. Then, he said in Latin, \"Stay here, madness! Why no go?\"\n\n\"Where are the Tulingi?\" Agrippa asked. \"The Tulingi were marching with the Boii. Where are they?\"\n\nBefore Madog could answer, we could see the German riders begin to move toward us. There was no order to their advance, just knots of three to six riders beginning to move up the slope toward us. They didn't seem to be in any hurry to attack uphill into what appeared to be a disciplined and well-equipped Roman cavalry detachment.\n\nMadog quickly snapped out some orders. Athauhnu and Ci were quickly to withdraw five hundred _passus_ to the west. Madog's _ala_ would screen the withdrawal from the German cavalry. I translated for Agrippa, who nodded and said, \"You stay with Athauhnu. I'm with Madocus!\"\n\nBefore I turned my horse, I looked down the hill. The German _dugath_ was still emerging from the valley. They were milling about down there, waiting for their horsemen to clear us off the ridge. There had to be almost two cohorts of them below us.\n\nAthauhnu gave Ci the order to withdraw. We cantered our horses back to our new position. There was no point in winding them; we didn't know what was in store for us. We followed the valley floor as it gently rose toward the northwest. When we arrived at our new position, I could clearly hear Roman signal trumpets away to the west. I remembered some of the signals from my training. I thought I heard \"assemble.\" And, the only reason I could think of to sound that command while on the march was contact with the enemy. But, which enemy: Helvetii, Aedui, or both?\n\nTo our east, Agrippa and Madog withdrew slowly toward our position. The German cavalry followed cautiously, without any semblance of purpose or organization. They maintained a healthy distance from the Sequani riders. Finally, Agrippa and Madog joined us. The Germans continued to maintain a safe distance. They halted in small groups of seven to ten riders about a hundred _passus_ to our front. There were no more than fifty of them below us. There was no sign of the German infantry cresting the ridge to the east.\n\nI reported to Agrippa: \"I can hear our army. They seem to be no more than three to five thousand _passus_ up the valley. I think they're engaging the enemy.\"\n\n\"Is the enemy between us and the legions?\" Agrippa asked.\n\nI hadn't thought of that. \"I . . . I don't know, Tribune. There is no sign of enemy troops to our rear . . . just the sound of Roman trumpets.\"\n\nAgrippa stared up the valley for a few heartbeats, then shrugged, \"Can't worry about that now. We're not finished here yet . . . Madocus _Dux_!\"\n\nMadog sidled up next to Agrippa, \" _Quid vis tu, Tribune_?\" he asked. \"What do you want, Tribune?\"\n\n\"We're going to teach those arrogant _verpae_ a lesson. We attack!\" Agrippa said, pointing toward the motley assemblage of German riders.\n\nMadog nodded and called for Ci. Athauhnu was mounted beside me.\n\n\"Three wedges,\" Agrippa instructed. \"Madocus' _ala_ center, Adonus left, and Caius right. I'm with Madocus. Insubrecus with Adonus. We advance no farther than our last position on the ridge.\"\n\nI translated for Athauhnu and Ci. Athauhnu spoke up, \"They are too close. We will not be able to get the horses to the gallop before we're among them.\"\n\nI translated that for Agrippa. He nodded and said, \"You are correct, Adonus _Decurio_ , but I do not believe that will make much of a difference with that rabble. They'll run as soon as they see us advance. I want one last look at the Kraut infantry. I want to see if we can locate the Tulingi down there with the Boii. And last, I want the German cavalry terrified that their infantry will have to advance blind, with no cavalry screen. That should slow them down some.\"\n\nI translated. Both Athauhnu and Ci grunted and eagerly nodded in agreement. There was a blood debt. The Germans had raided their lands, burned their homes, and raped their women. Now they must pay.\n\nAgrippa wasted no time. He and Madog trotted back to their band of riders and almost immediately moved forward toward the Germans. Athauhnu and I moved our men to Agrippa's left; Ci moved to the right. The cavalry wedges formed naturally as we advanced. But, I heard Agrippa give the command, \" _Alae . . . ad . . . cuneum_! Troops . . . form . . . wedge!\" I shrugged my shield off my shoulder. Strapping it to my left arm, I fell in behind Athauhnu on his left.\n\nI heard Agrippa's voice again, \" _Alae . . . equiis . . . citatis_! Troops . . . to the canter!\"\n\nMadoc signaled the Sequani by pumping his right fist twice, and the horses broke into a canter.\n\nAhead of us, I could see the Germans take notice and stiffen, but they seemed frozen in place. They didn't know what to do.\n\nAgain, Agrippa, \" _Alae . . . spathas . . . stringite_! Troops . . . draw . . . sabers!\"\n\nFinally, in ones and twos, the Germans began to turn their horses away from us. They began to retreat back toward their army. Some remained in place, frozen.\n\nAgrippa, \" _Alae . . . equiis . . . currentibus_! Troops . . . to the gallop!\"\n\nWe were just beginning to pick up some speed when we ran into those few Germans too foolish to run. One appeared directly in front of me. His eyes seemed to be the size of _denarius_ coins. He dropped his spear and raised his arms up, as if to protect himself. I saw Athauhnu's sword slice into his face just above his mouth. His body tumbled to my left. I felt a brushing impact with his horse. Then, I was past him. Nothing but empty fields and fleeing German horsemen were to my front.\n\nWe were about a hundred _passus_ away from our objective when Nemesis struck.\n\nThe mounted _gedricht_ of the German _ciuning_ began to crest the slope in front of us\u2014whether alerted by their own fleeing cavalry or following the king who wanted to see the terrain in front of him, I could not tell. The _ciuning_ was riding in the center of his troop.\n\nMadog ran directly at the German king. Athauhnu changed the direction of our gallop so we would crash into the Germans' right flank.\n\nI heard the crash as Madog's troop collided with the Krauts. I had no time to look. We hit the German flank immediately after. My horse, Clamriu, crashed into a German's mount. I saw her bite down into the other horse's neck. The rider was thrown off away from us. He never rose. His horse collapsed and rolled where the rider fell. I saw a face in front of me, bearded. I stabbed at it and felt an impact up through my sword arm. I glimpsed a tightly packed group of Germans protecting a wounded man, leading him down the slope away from the battle. It was their king.\n\nThen Clamriu reared back. I almost tumbled over her rump. She was kicking and biting at another horse. The Kraut rider was trying to get control. A Sequani reached over and plunged his sabre into the man's arm pit. He went down into the scrum. The German riders seemed to be melting away back down the hill. Somewhere to my right, I could hear a Gallic hunting horn. Our troops were pulling back toward it. The tangle of men and horses was unraveling. I stole a look down the ridge. The ground was covered with German infantry. A chief on a horse was trying to rally them up the ridge toward our position. I could sense some movement in our direction. Again I heard the Gallic signal to assemble. I moved back to where our troop was gathering.\n\nAgrippa was still in the saddle. He had a slicing wound across the ridge of his nose and his left cheekbone. Madog and Athauhnu seemed winded, but unbloodied. Ci was not there. Then, I saw him sorting out our troops as they rode back from the point of contact.\n\nAgrippa grabbed my arm. I realized I hadn't sheathed my _spatha_. I raised it and realized the point was bloodied. I wondered how I could clean it off before returning it to the sheath. I heard Agrippa's voice in the distance. \"Are you listening, _Decurio_?\"\n\n\"Uh . . . _audio? Te audio, Tribune_!\" I heard myself say.\n\n\"You are to ride back to the army!\" Agrippa was saying. \"Find Caesar . . . Tell him there's an entire Kraut army on his flank . . . at least ten thousand . . . probably more . . . Boii and Tulingi . . . They're marching west . . . _Compre'hendis tu_?\"\n\n\" _Compre'endo, Tribune_!\" I said snapping out of it.\n\nI still didn't know what to do with my bloodied _spatha_.\n\nAthauhnu handed me a bloody rag. \"Use this,\" he told me. \"The German who wore it doesn't need it anymore!\"\n\nI cleaned off my _spatha_ and returned it to the sheath on my saddle.\n\n\"Emlun and Rhodri will ride with you,\" I heard Athauhnu say.\n\n\"Rhodri without Alaw?\" I questioned.\n\nAthauhnu shot me a dark look. \"Alaw feasts with the heroes in the Land of Youth,\" he said.\n\nThe three of us ran west as fast as our tired mounts could take us. Most of the way, we didn't dare to go faster than a canter. Ahead, I could hear the Roman trumpets clearly. The last signal I heard was a general call for close ranks. That could only mean the enemy was advancing on the legionary line.\n\nAhead, there was a wooded ridgeline that advanced across the valley from the south. Beyond it, I could hear a noise, a noise like powerful waters running and the murmur of thousands of voices.\n\nRhodri suggested we climb the ridge and not go around it to the north. I agreed. That decision probably saved our lives.\n\nWhen we crested the ridge below us, we saw the enemy, the Helvetii, tens of thousands of them, moving north across the open valley. We were _behind_ the enemy horde!\n\nTo my right, on a gently rising slope along the north wall of the valley, was Caesar's army. Four legions in _acies triplex_ , the triple line, were facing the Helvetii. Above them, I could see the remaining two legions in _acies duplex_ , two battle lines with open ranks, matching the flanks of the forward legions.\n\nThe Helvetii were rushing straight toward Caesar like a wall of water when a damn breaks. I wondered briefly if our army could withstand such a massive flow of warriors.\n\nI did feel a momentary surge of relief, realizing that the Germans would arrive on the battlefield in front of our troops. Then, Rhodri grabbed my arm and pointed to a long slope about five hundred _passus_ to our left.\n\n\"The Helvetii are forming a shield wall there!\" he said.\n\nI peered in the direction Rhodri indicated and could easily make out enemy troops forming ragged battle lines on the forward slope.\n\n\"Look there!\" Rhodri said again. \"The king has set himself near the hilltop!\"\n\nAgain, I could see a cluster of heavily armed, mounted warriors where Rhodri indicated. There was a cluster of enemy standards among the riders.\n\n\"This is not right, Arth Bek,\" Rhodri started.\n\n\"Not right? What do you mean?\" I asked.\n\n\"Their _brenna aw frouuhdrau_ , their war chief, is holding his best troops back from the attack,\" Rhodri explained, pointing toward the enemy standards. \"Only the tribal musters advance. The warriors led by their war chief should be advancing behind them under the tribal standards. When the musters open the Roman lines, the warriors must be in position to attack through the gaps. Something is wrong!\"\n\nJust then, the Romans began their attack.\n\nThe forward edge of the enemy advance had begun to climb up toward the Roman front line. When it was about thirty _passus_ away, I heard the Roman trumpet signal _pila ponite_ , \"present spears!\" There wasn't much movement along the Roman line; most of the _muli_ in the front line had already assumed the position by the time the enemy reached the bottom of their hill.\n\nThen, as the enemy closed to twenty passus, _pila parate_ was sounded, \"ready spears.\" Even at this distance, I could see movement and reflections of light as the _muli_ , almost in a single motion, brought their throwing spears to the ready position.\n\nThen, the trumpets sounded _pila iacite_ , \"open fire.\" This was followed by three blasts of the horn, three rounds. The soaring spears looked like a fastmoving, black cloud rushing from the Roman lines into the front edge of the enemy, some fifteen _passus_ away. Before the first volley struck, a second was in the air, then, a third.\n\nThe effect of the spear volleys was devastating.\n\nThe natural reaction to a volley of spears is to raise shields for protection. But, the Roman _pila_ are weighted, designed to punch through a shield with enough force to penetrate even the thickest protective padding, even hardened leather.\n\nMost of the Helvetii muster-men wore nothing; in fact, many had attacked bare-chested to show their contempt for the enemy. They were mowed down like wheat under a scythe.\n\nEven if a warrior were lucky enough to have chainmail to blunt the point of the _pilum_ , the impact of the blow would be enough to knock him down, and the spear would have rendered his shield useless. The Roman _pilum_ is designed so it cannot be extracted from a shield, leaving a man naked before the short, Roman stabbing sword, the _gladius_.\n\nNo sooner had the third volley of _pila_ risen into the air than the Roman trumpets signaled _acies prima_ , \"Front line!\" Then, _gladios stringite_ , \"Draw swords!\" Across the valley, all along the Roman front line, light flashed as thousands of short swords were drawn from their scabbards.\n\nBehind me, I heard Emlun calling my name. I didn't want to take my eyes off the drama unfolding in front of me.\n\nThe Roman trumpets called _impetum facite_ , \"Attack!\"\n\nThe entire Roman front line descended on the muddle that was once the front edge of the Helvetian attack. It was no contest. The Romans slaughtered any of the Helvetii foolish enough to try to stand their ground.\n\nThen, from the hill to our left, the Helvetian trumpets blasted out a cacophonous strain. Immediately, the thousands of Helvetii in the field below us turned and ran from the Roman advance. At that time, I was too inexperienced to realize that the maneuver I was seeing was impossible for a barbarian army, unless it had been planned.\n\nAgain, Emlun called to me.\n\n\"What is it?\" I called back.\n\n\" _Madog un dod_!\" he answered. \"Madog comes!\"\n\n\"Signal him up here to us!\" I instructed.\n\nAs the Helvetii fled south, the Roman trumpets blasted \"general call.\" Then, _signa proferte_ , \"advance the standards.\" Immediately, the second and third lines followed the first down the ridge and began crossing the valley in pursuit of what Caesar believed was a defeated and fleeing enemy. I noticed that the two legions near the top of the ridge held their position.\n\nAgrippa was suddenly at my side. He was speechless for a few heartbeats as the panorama of the battle unfolded below him.\n\nOff to the north, I saw the Roman cavalry advance across the enemy's left flank, led by an officer mounted on a white horse and a bright red _sagum_ trailing behind him.\n\n\" _Venatum Caesar ducit ipse, Tribune!_ \" I said to Agrippa. \"Caesar leads the pursuit himself, Tribune!\"\n\nWe watched as Caesar and most of our cavalry disappeared behind the edge of a distant hill heading toward the enemy's rear.\n\nSuddenly, Agrippa exclaimed, \" _Verpa Martis! Quae calamitas_! This is a disaster!\"\n\n\" _Pro qua dicis tu_?\" I sputtered, forgetting all military protocol.\n\nAgrippa turned and grabbed me by my shoulder armor. I thought he was going to deliver _castigatio_ for being insubordinate.\n\nBut instead, he said, \"Don't you see it, Insubrecus? That Kraut horde is less than an hour behind us. They will arrive _here_ , right where we're standing. By that time, they will be on the flank and rear of our army and in position to attack. And, our _imperator_ is on the wrong side of the battlefield, out of position, chasing after easy kills and plunder! Over ten thousand Germans will be pouring down this hill right onto the back of our army! It will be the massacre of Arausio all over again!\"\n\nAgrippa noticed the two legions still positioned on the ridgeline to the north. \"Those must be the Eleventh and Twelfth,\" he concluded. \"Caesar must be holding them back so the army will have a position to retreat to. There must be a senior officer up there with them. I hope he has a set of _coleones_! We have to turn the army around!\"\n\n\"Madocus _Dux_!\" Agrippa called.\n\nMadog approached our position. He too was initially stunned by what he saw below us. He too immediately understood the German threat.\n\n\" _Immerda sumus_!\" he said, for once getting the Latin idiom right.\n\nAgrippa instructed him, \"Madocus, your mission is to track the Germans and screen our army. I believe this will be their final coordination line for an attack on the Roman rear. When they reach this point, withdraw down into the valley below. Stay between the Krauts and our army! Do not become decisively engaged with them! You must maintain your freedom to maneuver. I will go below and try to organize a defense. I will look for you on the field of battle. But, if this thing goes wrong, I release you from your _sacramentum_. Do what you can for the survivors, but get your people back to your own lands as best you can. The Aedui will be looking to settle some old debts once we Romans are gone.\"\n\nTo ensure Madog understood, I translated while Agrippa spoke. A few _pedes_ away, I saw Athauhnu listening. He looked grim.\n\nAgrippa turned to me. \"You're with me, Insubrecus _Decurio_. Let's see if we can pull Caesar's balls out of the vice he's placed them in.\"\n\nAs I retrieved Clamriu from Emlun's care, I felt a hand grasp my shoulder. It was Athauhnu. He looked at me gravely, then nodded his head. \"You dress and talk like a Roman,\" he announced, \"but you are still Gah'el. When this is over we will feast together, either in the hall of my father in the lands of the Soucanai or in the Hall of Heroes in the Land of Youth.\"\n\n\"Save me a place on the mead-bench!\" I said. \"We'll fight over the hero's portion.\"\n\nWe placed our hands on each other's shoulders in the fashion of the Gah'el. I mounted Clamriu and followed Agrippa onto the battleground below.\n\nWe rode hard across the battlefield, behind the Roman third line, to the right flank of the army, the commander's position. Agrippa was hoping that Caesar had left someone there with enough _auctoritas_ to take command of the army in his absence. We were disappointed.\n\nWhen we arrived, we found no senior officers. Malleus, the \"Hammer,\" the _primus pilus_ of the Tenth Legion, was advancing on foot along with his legion on the right flank of the entire Roman army.\n\nAgrippa pulled up next to Malleus. \"Are you in command here, _Centurio_?\" he asked.\n\nMalleus shrugged, \"I must be, Tribune . . . I'm the senior officer present!\"\n\n\"Where is the _imperator?_ \" Agrippa asked.\n\n\"Forward with the cavalry,\" Malleus indicated the fleeing enemy's open flank.\n\n\"What orders did Caesar leave you?\" Agrippa demanded.\n\nAgain, Malleus shrugged, \"He said to continue to advance . . . Keep up the pressure . . . Don't let them rally against us . . . The normal shit, Tribune. What's the problem?\"\n\n\"The Boii and Tulingi are coming in on our left flank, _Centurio_ . . . You're walking into an ambush!\" Agrippa declared.\n\nMalleus' face blanched. \"There's nothing I can do. I have my orders. My authority only extends over the Tenth.\"\n\nAgrippa nodded. \"I will find someone who has the authority. Listen for the signals!\"\n\nWith that, Agrippa turned his horse, and we galloped toward the two legions still stationed on the hill to the north. As we rode, I wondered who Caesar had put in command on the hill. Unless he were willing to take a risk and use some initiative, we were _perfututi_ , absolutely screwed.\n\nWe immediately spotted a command standard on the right flank of the Eleventh Legion. It was Labienus. There was hope!\n\nLabienus came forward when he saw us riding up. He knew we were screening the army's flank and sensed our urgency. Agrippa wasted no time in briefing him. Labienus immediately understood the gravity of the situation.\n\n\"And Caesar cannot be reached?\" he demanded of Agrippa again.\n\n\"No, Legate! The _imperator_ has gone forward with the cavalry,\" Agrippa confirmed.\n\n\" _Cacat_!\" Labienus exploded. \"Shit! How soon will the Krauts arrive?\"\n\nTogether, we all looked across the battlefield to where we had left the Sequani cavalry screen. They had not withdrawn from the hill. The Germans were not yet in position. \"My guess, we have less than an hour,\" Agrippa answered.\n\nI watched as Labienus examined the disposition of forces on the battlefield. Then, he examined the ridge from which we expected the Germans to descend on us. He seemed to make a decision.\n\n\" _Fabi! Ad me_!\" he called over to his command group.\n\nA broad-striper rode forward. \" _Ti' adsum, Legate_!\" he reported.\n\n\"Fabius, have both legions entrench!\" he instructed. \"Two camps with enough room to protect the baggage train. Do it now!\"\n\n\"Sir,\" Fabius responded, \"there's no water on this hill . . . Perhaps I should find a better position?\"\n\n\"No time, Fabius!\" Labienus shook his head. \"Have the men fill their water bottles . . . There should be some water carts within the supply train . . . Fill them . . . Fill anything that will contain water . . . but get those camps built! We may not have much time!\"\n\n\" _A'mperi'tu', Legate_!\" Fabius responded.\n\nFabius turned his horse back toward the standing legions and started to snap out orders to the senior centurions.\n\nLabienus called out again, \"Iudaeus!\"\n\nThe _primus pilus_ of the Eleventh stepped forward. \" _Ti' adsum, Legate_!\" he reported.\n\n\"Put one of your flute girls on a horse and send him to me!\" Labienus ordered. \"I need to borrow him for a while!\"\n\n\" _A'mperi'tu', Legate_!\"\n\nAgrippa spoke suddenly and pointed across the battlefield. \" _Legate, ecce_! Sir, look!\"\n\nAlong the edge of the woods on the ridgeline, where we had left Madog, there was movement.\n\n\"We're running out of time,\" Labienus muttered.\n\nA legionary _cornucen_ rode up to our group with his _cornu_ draped around his body.\n\n\" _Ti' adsum, Legate_!\" he said, almost falling off the horse with the effort.\n\nLabienus pointed to the left flank of our advancing army, the point closest to where the Germans would soon appear. \" _Illuc_! There! We ride there! _Celerrime_! We haven't much time! Follow me!\"\n\nLabienus galloped down the hill, Agrippa immediately behind him.\n\nI looked over to the trumpeter. \"You going to make it?\" I asked.\n\nThe man was holding his reins with one hand, a saddle horn with the other, all the while trying to keep his _cornu_ from slipping down off his shoulder.\n\n\"I'll make it, _Decurio_ ,\" he said, with no sense of certainty.\n\nWe galloped across the battlefield. As we rode, I saw the Sequani cavalry break from cover and ride down the slope. They didn't seem to be fleeing, but withdrew in good order. They rode about fifty _passus_ from the bottom of the slope and spread out in a screening line between the ridge and the left flank of our army. There was still no movement on the hill above them.\n\nI finally caught up to Agrippa and Labienus. \"This will be our right flank, here!\" Labienus was saying.\n\n_Right flank of what?_ I wondered.\n\n\"I will be positioned here with the _cornucen_. Where is that man?\"\n\nWe looked back from where we rode. The trumpeter was still barely on his horse, about seventy _passus_ from our position.\n\n\" _Festina! Festina, miles_!\" Labienus shouted. \"Hurry!\"\n\nLabienus continued, \"Agrippa! You will position yourself on the left flank. Don't let them turn you. Bend back in a prevent formation if they overlap us.\"\n\nThere was movement now in the trees on the ridge above us. The Germans had arrived.\n\nThe trumpeter finally arrived. Immediately, Labienus ordered, \"Signal . . . third rank . . . attention!\"\n\nThe man tried, but nothing meaningful came out of his _cornu_. It made a sound like a duck farting in a swamp. He gave up on the horse and slid down from the saddle. Finally, planting his feet back on firm ground, he took three deep breaths. When he took his fourth, he raised the trumpet to his lips and blew, \"Third rank . . . attention!\" I could see the transverse crests of the centurions marching in the third rank of the closest legion turn. I imagined I could hear them echoing the order to their _muli_ : \"Attention!\"\n\n\"Order, ' _Consistite_!'\" Labienus instructed. \"Halt!\"\n\nThe _cornucen_ looked at Labienus and stammered, \" _Consistite, Legate_?\"\n\n\"Do it!\" Labienus ordered.\n\nThe trumpeter blew the signal. The entire third line of the Roman advance, twelve cohorts, came to a halt in two steps.\n\nIn front of us we could see the German _fiurd_ , the muster-men, pouring out of the woodline.\n\n\"Don't let them go right into the attack!\" Labienus hissed through his teeth. \"Mass them there . . . Give me the time I need.\"\n\nThen, Labienus ordered, \" _Cornucen_! Order . . . third rank . . . _signa_ _conversate_ . . . turnabout!\"\n\nThe trumpeter did as ordered. Across the field behind us, our little army of twelve Roman cohorts turned as a single man. I saw the standards and officers run to reverse their positions: centurions front and right; _optiones_ rear and center.\n\nThe Krauts seemed to be cooperating with Labienus. The musters were milling about on the ridgeline. The sight of an entire Roman army maneuvering across a battlefield as if on parade will make even experienced soldiers pause.\n\nI heard Labienus mutter again, \"Here's where it gets complicated!\" Then, \" _Cornucen_ . . . signal . . . third rank . . . _ad dextram_ . . . _aciem_ . . . _formate_!\"\n\nLabienus gave the order for the entire third rank to wheel right and form their battle line facing the Germans. The trumpeter blew the signal, and twelve cohorts began to wheel into position.\n\nLabienus snapped a command to Agrippa: \"Ride across their front! Get on their left flank! Guide them to a line on my position facing the enemy. Move, Tribune!\"\n\nOver his shoulder, as he galloped across the advancing Roman front, Agrippa yelled, \" _A'mperi'tu'!_ \"\n\nThen, Labienus was talking to me, \"Insubrecus! Ride forward to Madocus _Dux_. He is to withdraw as the Krauts advance. Have the Sequani take a position behind our battle line here. We will only be eight men deep at the most. If any of the Krauts break through, the Sequani are to stop them . . . mop them up. Is that clear?\"\n\n\" _A'mperi'tu', Legate_!\" I confirmed and rode forward, looking for Madog.\n\nAs I rode off, I heard Labienus order the trumpeter: \"Signal . . . third line . . . _gradus bis_ . . . double-time.\"\n\nI found Madog in the center of the screening line. He seemed bent slightly forward in his saddle. Then, I noticed blood flowing down his right leg.\n\n\"Madog, _Pobl'rix_!\" I said in Gah'el. \"You are hurt!\"\n\n\"I can still ride, Arth Bek!\" he responded somewhat breathlessly.\n\nI related Labienus' commands. Madog nodded. He blew his hunting horn, then led the way to our new position.\n\nBefore I followed, I looked up at the Krauts. Well-equipped members of the _dugath_ were aligning the ranks of the muster-men. The attack was imminent. I turned my horse to follow the Sequani. I could see the Roman cohorts double-timing up to the battle line. I remembered these drills from my training. The men on the left flank who had the farthest to run would be heaving by the time they got into position.\n\nWe rode across the front of the Roman advance and around its left flank. Madog ordered his men into three loose wedges behind the Roman line: Athauhnu on the left, Ci on the right, and himself in the center. As the senior Roman officer, I assumed a position with Madog.\n\nOn the ridge, the _dugath_ was finalizing the disposition of the _fiurd_. From the looks of it, they wouldn't overlap our flanks. A rider emerged from the woodline, trailed by a _gedricht_ , a royal bodyguard.\n\n\"Their _ciuning_?\" I asked Madog.\n\n\"No,\" Madog gasped. \"I've done for that bastard. That's one of his _thegns_ . . . his companions . . . what you Romans call _comites_. I left that German _ciuning_ with his guts hanging out over his belt.\"\n\nMadog could hardly catch his breath. He was holding onto his saddle horns to steady himself.\n\n\"Do you want me to call for the _medduhg_ , the medic, _Madog Pobl'rix_?\" I asked him.\n\n\"No,\" he panted, \"no . . . my men must stay strong . . . Can't see their king fall out of the saddle . . . not now.\"\n\nOur cohorts had finally come on line facing the Germans. From my position near the center of the line, I could see some of the _muli_ on the left flank doubled over with the exertion of double-timing in full combat kit. The _centuriones_ and their _optiones_ were trying to straighten the lines and get the men ready for the German assault, but even they were affected by the run they had just made.\n\nThe Krauts on the hill made no move to take advantage. From the distance, I could hear chanting, some weird and disturbing sound from the German line: \"Wo . . . wo . . . wo.\"\n\n\"What is that?\" I muttered out loud.\n\n\"They're praying to Woden,\" Madog responded.\n\n\"What's Woden?\" I asked him.\n\n\"Not what . . . who,\" Madog began to say. Then, a fit of painful coughing racked him. When he steadied himself, he spit out whatever was in his throat. It was a bog of bloody phlegm.\n\n\"They are . . . they are calling on their god, Woden, to send the _Wal Ciurige_ . . . the gatherers of the dead . . . to take them to the _Wal Halle_ . . . the feasting hall of the dead . . . if they are killed in battle.\"\n\nMadog started coughing again.\n\n\"They will feast there,\" he continued, \"with the heroes of their people . . . until Woden calls them forth . . . for the _rako werdum_ . . . the great war . . . at the end of times.\"\n\n\"That sounds like our Land of Youth,\" I observed.\n\n\"Bah!\" Madog tried to start, but again a fit of coughing hit him. There was a small trickle of blood running from the side of his mouth.\n\nHe finally caught his breath to speak, \"They believe that everything . . . everything will be destroyed . . . at the _rako werdum_ . . . the end of times . . . Men . . . the gods . . . and the earth will be burned away. _Wal Ciurige_ . . . they're only the crows. They strip the flesh off the battle dead . . . nothing more . . . Woden is a god of carrion.\"\n\nMadog's description of the crows reminded me of a tale Gran'pa had told me of a dark phantom, a goddess, the _Mawr Riganu_ , the great queen, who appears on the battlefield in the semblance of a great crow to feast on the blood of the slain. I shuddered. I reached up and rubbed my _lorica_ where my _Bona_ _Fortuna_ hung.\n\nAthauhnu was right when he said the waiting is the worst.\n\nThe straw-headed bastards on the hill were working themselves into a frenzy: \"Wo . . . wo . . . wo.\"\n\n\"It won't be long now,\" Madog gasped.\n\nI could see that our battle line had stabilized, but the main Roman attack force, the first and second battle lines, were still advancing toward the Helvetii. A gap was opening between their left rear rank and our right flank.\n\nI called over to Ci and pointed to the gap opening on our right flank. He nodded and moved his _ala_ into it.\n\nThe movement of the Gallic horsemen must have caught the attention of the _primus pilus_ commanding the legion advancing against the Helvetii on the end of the Roman battle line. I could see from its standards, it was the Seventh Legion. Suddenly, their _acies secunda_ , three cohorts, halted, turned about, and aligned themselves in a prevent formation, a line diagonal to their route of march, to protect their left flank against the German threat. Then, the two left-most cohorts of the _acies prima_ executed a smart, three-quarter turn to the right and tucked themselves to the rear of the advancing First and Second Cohorts to reinforce them. All this was accomplished as if they were on parade and not unexpectedly moving into a battle position.\n\nThen, I heard a shout to my front. I looked. The German _fiurd_ was charging down the ridgeline at us.\n\n\" _Maent uhn dod_!\" was all Madog could manage. \"They come!\"\n\nSuddenly, the entire ridgeline to the north, where Caesar thought the alreadydefeated Helvetii were cowering, exploded in a cacophony of Gallic trumpets and movement. The Helvetii warriors threw themselves down the ridgeline into our advancing legions.\n\nThe ambush was triggered, and we were standing right in its kill zone!\n\nLooking back toward the Krauts, I immediately saw that they had miscalculated. First, our lines were too far back from their ridge for their momentum to take them into us. They would have to run across at least seventy _passus_ of flat ground before making contact. That should wind them and slow their attack. Second, the Krauts were headed straight for us and not for the gap opening on our right. Killing Romans seemed more important than winning the battle.\n\nThen, I noticed that the Germans had left about a hundred men deployed across their ridgeline. Archers!\n\nLabienus saw them too. Before the Krauts could launch their arrows, his trumpeter sounded, \" _Notate_! _Ad testudinem_!\", \"Form the turtle!\"\n\nImmediately, the Roman _muli_ in front of me closed ranks and lifted and locked their shields over their heads. They were protected from the German arrows, but large gaps opened in the battle line.\n\nI felt a hand grab my forearm. It was Madog.\n\n\"We move back . . . out of range,\" he gasped. I didn't know what was holding him up in the saddle. His face was pasty, almost greenish. There was a patina of sweat across his face.\n\nAs we moved back, the German muster-men cleared the hill slope and screened our line from their archers. The archers ceased fire. The _dugath_ , the professional warriors, organized themselves into five groups, each about the size of a Roman century. They moved down the slope, spreading themselves out across the battle line. They followed behind the German _fiurd_.\n\n\"They follow,\" Madog panted, indicating the _dugath_. \"Wherever the _fiurd_ opens the battle line . . . they attack.\"\n\nLabienus realized that the arrow fire had ceased. He and the _cornucen_ emerged from the turtle. Neither of their horses had been hit. The legate remounted so he could see the battlefield, but the trumpeter chose to stay on his feet. At Labienus' command, he again signaled, \" _Notate_! _Aciem formate_!\"\n\nImmediately, the turtles collapsed and the _muli_ reformed their line. The forward edge of the German _fiurd_ was less than thirty passus from our lines.\n\nThe trumpet called, \" _Pila ponite_! _Pila parate_!\" I saw the arms and javelins of the entire Roman battle line come up.\n\nLabienus waited.\n\nThe Krauts closed to within twenty passus, then fifteen, then ten. \" _Iacite_ . . . _iacite_ ,\" I heard myself mutter. \"Throw . . . throw.\" Then, finally, the trumpet sounded: \" _Pila iacite_! Open fire!\"\n\nThe Roman spears went forward into the German attack. It staggered as men crumbled. Others tripped over the bodies of the dead and wounded. Some stopped running to avoid the growing pile up. Others stopped to try freeing their shields of the Roman spears. The rear ranks piled up on the stalling and staggering front ranks.\n\nThere was no second volley!\n\nThen, I remembered that these were _miles aciei tertiae_ , third-rank men. They were only carrying one _pilum_ each.\n\nQuickly Labienus ordered _gladii stringite_ : \"Draw swords!\"\n\nThen, immediately _impetum facite_ : \"Attack!\"\n\nLabienus was _attacking_ a superior force!\n\nInitially, it worked. The Roman _muli_ quickly covered the ten _passus_ between themselves and the struggling pile of men that had been the forward edge of the German attack. These they quickly cut through. Then, they came into contact with the rest of the horde of Kraut muster-men who had lost all forward momentum. Our advance slowed, but we were still moving forward.\n\nAs we moved forward, though, the gap on our right widened.\n\nI heard a cacophonous trumpet call from the German ridgeline. The Kraut _thegn_ had stationed himself there so he could see his battlefield. He indicated the gap on our right with his sword. Immediately, one of the formations of the _dugath_ turned and moved toward it. As they moved forward, they formed a wedge, like a Roman _cuneus_ , a \"bore's snout.\"\n\nI yelled over to Ci, commanding our cavalry on the right, and indicated the threat. He raised his hand to acknowledge. I saw the two cohorts from the Seventh Legion adjust their position in the face of the new threat. But, they stayed well back. Their mission was to protect the rear of their own legion, not pull our balls out of the vice.\n\nI turned to Madog to recommend we reinforce Ci's _ala_. He was off his horse, on the ground, not moving.\n\n\" _Medduhg_! _Medduhg_!\" I yelled.\n\nThe Sequani horse doctor was immediately off his horse, attending to the king.\n\n\"Athauhnu! To me!\" I called to our troop on the left.\n\nAthauhnu rode over.\n\n\"Madog's down,\" I told him. \"The troop is yours! We must reinforce Ci! The Germans are attacking the gap!\"\n\nAthauhnu nodded. He called over to his men, \"Guithiru! You are in command! Move center!\"\n\nWe rode over to the right and joined Ci.\n\nThe Kraut wedge was moving toward us, now no more than fifty _passus_ from the gap, over sixty warriors, at least, with shields up. They formed a solid, German _scilde wealle_ of linden wood, leather, and iron and were determined to crash through us.\n\nOur horses would not attack a shield wall, and there were less than forty of us. I could see the red-horned aurochs totems painted on the round shields as they bore down on us.\n\nThen, I remembered Athauhnu's lesson with the _gaea_ , the light hunting spear that had pierced my cavalry shield as if it were _vellum_.\n\n\"Athauhnu!\" I snapped. \"A _gaea_ attack!\"\n\n\" _Gaea_ will not penetrate that!\" Athauhnu objected.\n\n\"They don't have to!\" I said. \"We just need to weigh down their shields. We have to fight them on foot. Our horses won't stand up to that!\"\n\nAthauhnu understood immediately. He gave instructions to Ci, then yelled, \"Follow me!\"\n\nMy Sequani horsemen attacked the German wedge in a file. Each rider approached at an acute angle, turned in front of the wedge, and delivered a single blow with a _gae_.\n\nThe _dugath_ halted when they became aware of the Gah'el attack. We took advantage of their indecision. The Sequani turned and attacked a second time. The Krauts quickly realized that horsemen couldn't seriously damage them, so they resumed their advance. I could already see many of their shields were held lower, weighed down by our spears.\n\nIt wasn't much, but it was all we had.\n\nI dismounted and ran into the middle of the gap. \"Form shield wall on me!\" I shouted.\n\nLabienus finally noticed the activity on his flank and quickly realized the threat. He bent back his First Cohort to protect his right flank. His only reasonable hope was to safeguard his own formation, which was already heavily engaged with the German _fiurd_. He could do nothing to help us.\n\nThe Sequani began to form around me. I had my Roman short sword out. Most of the Gah'el had only the _spatha_ , the cavalry long sword\u2014not the best weapon for this type of close-in fighting. We were all carrying Gallic cavalry shields, light and oblong, giving little protection for the throat and legs of dismounted fighters. They also lacked the punching handle of the Roman infantry _scutum_ ; they were defensive weapons only. I did not have much confidence that they would stand up to the massive German round shields bearing down on us.\n\nThe Kraut wedge was less than ten _passus_ away. I could hear them chanting a cadence to time their attack. As they got closer, the pace of the cadence increased. They planned to bowl right through us.\n\nSuddenly, the right side of the German wedge collapsed.\n\nThe two Roman cohorts on our right from the Seventh Legion had delivered a volley of _pila_ in support of us. The Krauts never saw it coming. More than a score of them went down.\n\n\" _Illuc_!\" I screamed pointing to the hole in the wedge. \" _Impetum facite_! _Illuc_! There! Attack there!\"\n\nI suddenly realized I was screaming in Latin, but it made no difference. The Sequani leapt into the gap in the wedge, slashing and stabbing. The Kraut formation crumbled like a rotten wall. German warriors dropped their shields and spears. They began running to the rear. A few tried to hold their ground. They were quickly cut down.\n\nI was still standing exactly where I had originally positioned myself. I had won a fight without as much as striking a single blow, without even moving!\n\nAthauhnu was still standing next to me. \"Shaggin' Germans!\" he spat. \"Only brave when they think they're winning.\"\n\nI stepped forward and turned toward the Romans who had delivered the decisive blow. I raised my right fist in their direction, and yelled, \" _Io! Victoria_!\"\n\n\" _Io_! _Victoria_!\" they thundered back. The signifers moved their standards up and down in celebration. A centurion raised his fist and saluted me back.\n\nBut, the fight wasn't over yet. We were still engaged with the Boii _fiurd_ across our entire front. Off to my right, I could also see the dust of our desperate fight with the attacking Helvetii.\n\n\"Let's remount the troop!\" I said to Athauhnu.\n\nWe remounted the two _alae_. I posted Ci's men back in the gap to remind the Krauts that we were still there. Athauhnu and I rode back to the center of the battle line, displacing Guithiru's troop back to the left flank.\n\nOur line was no longer advancing against the Germans, but it seemed to be holding its own. Labienus had had to relieve his first rank and bring the second forward. We were only eight men deep, four pairs of _gemini_. The first line was recovering its strength and reorganizing itself for its expected redeployment. From the look of things, they had lost about a quarter of their strength, killed or wounded.\n\nThe legionary slaves, distinctive in their brown tunics and with their small handcarts, were hard at work carrying water to the men in the battle line. From the dead or badly wounded, they were also collecting the equipment, which they dragged back to a collection point located somewhere behind our lines.\n\nThe _capsarii_ , the legionary medical orderlies, were also up on the line, carrying their _capsae_ , boxes of bandages and medical equipment. They were bandaging and stitching up the slightly injured _muli_ so that they could stay in the fight.\n\nThe more seriously wounded were being triaged. Those who had a chance of surviving were taken directly back to the _medici_ , who had established a medical station about fifty _passus_ behind us. Some of the _capsarii_ were riding horses with a double saddle. The wounded man was lifted up on the saddle in front of the orderly, and they rode back to the aid station.\n\nThose who had little chance of survival waited. Sometimes, if the wound was hopeless and a man was in great pain, the _capsarius_ would help the soldier on his journey.\n\nOther slaves were loading the bodies of the dead onto their handcarts and taking them back so the _medici_ could confirm that they had indeed crossed the river.\n\nThe _capsarii_ wore black tunics. I imagined this was to hide some of the blood. As they worked, their hands, arms, and faces became covered with the blood of the dead, dying, and wounded. The _muli_ called them _cornices_ , crows. They referred to evacuation back to the medical station as _ad cornices ire_ , \"to go to the crows,\" or sometimes, _cornices pascere_ , \"to feed the crows.\"\n\nI shivered, thinking of these images: the _Mawr Riganu_ , the great crow, drinking the blood of the dead; the _Wal Ciurige_ , the black carrion goddesses of the Germans; the black-clad _capsarii_ , scurrying around the dead and dying on the battlefield.\n\nWhere was Madog?\n\n\" _Medduhg_ ,\" I called. \" _Medduhg_! To me!\"\n\nThe man rode over from Guithiru's troop. \"What is it, Arth Bek?\" he asked.\n\n\"The king!\" I demanded. \"Where is Madog?\"\n\nHe shrugged, \" _Uh brana dua . . ._ the black crows didn't want to take him, so I brought him back myself.\"\n\nI looked over to the Roman medical station. I had a bad feeling.\n\n\"Athauhnu!\" I said. \"I am checking on Madog. If anything happens, send a rider!\"\n\nI rode back to the medical station. The place smelled and sounded like an abattoir in the depths of Tartarus. It reeked with the stench of blood and freshcut meat. The wounded moaned, some screamed, many cried.\n\nClamriu reared back from the place. I have since learned that, in many ways, horses are smarter than men. I dismounted and approached on foot.\n\nTo my left, I saw the _medici_ working. A man was being held down on a bloodstained wooden table by two burly attendants. A doctor was cutting on the man's abdomen. The soldier screamed, arching his back. The _medicus_ jumped back, removing his hands and scalpel from the man's gut.\n\n\"Hold him still, damn you!\" he yelled at the attendants.\n\nIn front of me were three groupings of men. To the left, farthest from where the _medicus_ was working, were the dead. Their only companions were two slaves, who were busy chasing the carrion birds away from an inviting feast. I watched as a slave returned from the battle lines with his handcart and dumped three fresh bodies on the pile. _Cornices pascit_ , I thought. He is feeding the crows.\n\nNext were the badly wounded, men not expected to survive. A couple of black-clad attendants walked among them. Some seemed to be treating wounds; others were checking to see if a man had crossed over. I watched as an attendant called over one of the dead-pile slaves and indicated a body for removal.\n\nClosest to the surgery were the slightly wounded. Some were lying on the grass, others sitting up. Most of the attendants worked with this group, checking wounds, bandaging others, giving the men water. I saw a _medicus_ come over, wash his hands in a bucket, then indicate a wounded man to an attendant. The attendant stood the man up, walked him over to the open-air surgery, and sat him on a table for the doctor to treat.\n\nIt was then that I noticed a man lying by himself off to one side, seemingly belonging to no group. I walked over to him. It was Madog. His eyes were half open; his face was greenish-white; his chest was not moving.\n\nI looked to my left and saw a _capsarius_ walking among the badly wounded.\n\n\"You!\" I shouted at him.\n\nHe looked over at me with a trace of curiosity.\n\n\"Yes! You, soldier! Get over here! Now!\" I ordered.\n\nHe scrutinized me for a few heartbeats. From my sashes, he took in that I was a junior officer, a member of some nob's praetorian detail. He decided that he should at least placate me. He sauntered over. His hands and arms were covered with blood, ranging from bright, wet red to crusted black. There were even streaks of gore on his left cheek and forehead.\n\n\" _Qui' vis tu_?\" he asked with no great interest. \"Whadda you want?\"\n\nI was in no mood to be placated.\n\n\" _Quid vis tu, Decurio_!\" I insisted.\n\nHe shrugged. \" _Si ti' placet_ . . . if that's what makes you happy... What do you want, _sir_?\"\n\n\"Why is this man not being treated?\" I demanded.\n\nThe crow looked at me as if I had lost my mind. \"For one thing, he's a bloody wog!\"\n\n\"I know he's a wo . . . a Gah'el . . . He fights for Rome . . . He was wounded fighting for Rome. So why in the name of _Dis_ isn't he being treated?\" I raged at the man.\n\n\"Look around you . . . _sir_!\" the _capsarius_ came back at me. \"We have our hands full treating _Romans_. We don't have time for any of your bleedin' _wogs_ . . . Besides,\" he said, nudging Madog's unresponsive shoulder with the toe of his boot, \"somebody made this one a _good_ wog . . . He's already dead!\"\n\nThat's when I hit him.\n\nTo this day, I don't remember hitting him. One minute I was watching him kick my dead comrade; the next, he was sitting on the ground with blood pouring from the wreckage of his nose.\n\nHands immediately grabbed me from behind. The _capsarius_ ' buddies were trying to restrain me. A wounded officer waiting for treatment intervened.\n\n\"Break this shit up!\" he ordered. \"What are you doing, _stulti_ , you idiots? Don't we have enough hairbags and Krauts to fight? Do we have to fight each other?\"\n\nThe attendant I punched was getting off the ground with the help of one of his friends. \"This officer struck me!\" he sprayed through the blood from his ruined nose. \"I want to press charges!\"\n\n\"Shut your gob!\" the wounded officer retorted, grabbing my elbow. \"You deserved it . . . What I saw was an officer delivering a justified _castigatio_ to an insubordinate gob-shite of a crow!\"\n\nHe steered me away from the gathering flock of crows. \"What do you think you're doing, _Pagane_?\" he hissed at me.\n\nWhen he called me _Pagane_ , I finally recognized him. \"Bantus,\" I answered, \"are you hurt?\"\n\nBantus touched the bandages covering his neck and right shoulder. \"A Kraut arrow got through our turtle,\" he explained. \"The _capsarius_ up on the line couldn't remove the arrowhead, so he sent me back here to feed these crows.\"\n\n\"So, you were with us! You fought with Labienus?\"\n\n\"That who it is?\" Bantus asked. \"I was over on the left with the third-line cohorts from the Tenth.\"\n\n\"I'm with the Sequani cavalry,\" I told him. \"That was their _dux_ whom that _mentula_ kicked!\"\n\n\"The Sequani?\" Bantus nodded. \"You guys put up a hell of a fight in the gap. I saw it just before I went to the crows.\"\n\nI spotted a rider galloping toward us. It was Emlun.\n\n\"That's one of mine,\" I told Bantus. \"I have to get back.\"\n\nI quickly collected Clamriu.\n\nBantus went on, \"Most of your old mates are with the second line now. They're catching shit up on the hill with the rest of the Tenth Legion. They sent Minutus all the way up to the first line because of his size. Strabo's still with the tenth cohort, though. He was still on his feet when I was sent back\u2014\"\n\nEmlun rode up to us. \"Arth Bek!\" he called to me in Gah'el. \"Athauhnu says you must return!\"\n\n\"I'm coming!\" I responded, mounting Clamriu.\n\nThen, to Bantus, \" _Vale, contubernalis_! Be well, mate! I have to get back to the party!\"\n\n\" _Vale, Pagane_!\" Bantus responded. \"Try to stay off the pyre!\"\n\nEmlun and I rode quickly back to the battle line. Athauhnu met us about ten _passus_ behind our line.\n\n\"Madog's dead,\" I told him.\n\nAthauhnu didn't as much as blink at the news. \"I expected it,\" he said. \"No man could have lived long with such a wound. He feasts with the heroes, and we may soon join him. The Germans are massing the _dugath_ for an attack.\"\n\nI looked across the field. The German _thegn_ and his _gedricht_ had come down off the ridge. He was massing a division of the _dugath_ against our center. I tried to count, but it was impossible. The Germans kept no order, no formation. I estimated that the Krauts were massing over two cohorts\u2014about a thousand warriors about seventy _passus_ to our front. Fresh troops, well-armed. They should rip through our thin, exhausted cohorts like wolves through sheep.\n\nOn each of our flanks, the _thegn_ had also assembled about a cohort of the dugath, around five hundred warriors, to prevent us from shifting men to the center.\n\nAcross our entire line, our men were nearing exhaustion. They had been engaged with the German _fiurd_ , the muster-men, for over two hours. Each fighting pair of _muli_ , each _gemini_ , had been in the front rank at least twice. One in three were down, dead, or wounded. The rest were trying to martial their last vestiges of strength.\n\nThe German muster-men were not pressing hard. They were also exhausted, and they knew that the _dugath_ was about to take over the fight. They had done their job. Our ranks were exhausted and whittled to the bone. It was time for their warriors to finish the job\u2014not a time to get killed uselessly.\n\nSuddenly, Agrippa was at my side.\n\n\"Didn't you hear officers' call, _Decurio_?\" he announced. \"Labienus is summoning us.\"\n\nI rode off after Agrippa, with Athauhnu trailing behind. We met Labienus just behind our right flank cohort, a Seventh Legion unit. He had also assembled the senior centurions from each legionary group.\n\nLabienus wasted no time.\n\n\"I assume you men have seen the Krauts massing in our center,\" he began. \"They obviously plan to attack at that point. Our line is six men deep, at the most, and on its last legs. If we let them hit us, we're through.\"\n\nThere was no demurring from the officers. Even the centurions did not baulk. We were looking at a looming disaster.\n\n\"I plan to form maniples with the centuries of the Ninth and Eighth Legions at the expected point of enemy contact,\" Labienus announced.\n\nLabienus could have caused no more surprise with that statement than if he had announced the god Mars was on his way, with a troop of unicorns, to rescue us. No Roman legion formed maniples in combat; Marius had obsoleted the tactic over fifty years ago.\n\nA maniple was a formation of two massed centuries, closed ranks, one behind the other. It was a tactic used to achieve tactical mass back in the days when the phalanx was the state-of-the-art combat maneuver. Basically, two armies would run straight into each other and keep pushing and shoving in a scrum until one broke. But, the maniple formation was rigid; it had no flexibility. The enemy could run right around its flank, or if they had elephants, stomp it flat. The Roman army hadn't seriously employed massed maniples since they were massacred by Hannibal and the Carthaginians at Cannae.\n\nMarius had reorganized the Roman legion into ten cohorts of six separate centuries each. This was possible because all Roman legionaries were trained and equipped to the same standard. The three unequal divisions of legionary infantry\u2014 _triarii_ , heavy infantry; _principes_ , spearmen; and _hastati_ , light infantry\u2014 were a thing of the ancient past. Also, the cohort-based organization allowed a commander the flexibility to deploy his troops according to the terrain and the nature of the threat on the battlefield, a principle that Labienus himself had demonstrated when he peeled off twelve cohorts from the rear of four advancing legions to meet an enemy threat on their left flank.\n\nCuriously, vestiges of the old manipular system still existed in the training and traditions of the army. Within the cohorts, the first, second and third centuries were known as the _prior_ , or front-line centuries, based on what their positions would have been in the maniple. The remaining centuries\u2014the fourth, fifth, and sixth\u2014were called the _posterior_ centuries because in the maniple, they would have stood behind the forward centuries in the phalanx. What's more, the prestige of the centurions commanding any cohort is based on whether they are _prior_ or _posterior_ centurions.\n\nIn training and on parade, the command to form maniples was still given. But, this formation was used only to cross up _tirones_ during training drills or to put on a show for some visiting nob\u2014not while decisively engaged with an enemy force, as Labienus was about to attempt.\n\nTo make matters worse, the only viable way of forming maniples from _acies_ _formata_ , the linear battle line formation in which we were currently deployed, was by first forming the _quincunx_ , the \"five dots.\"\n\nAccording to legend, the great Scipio Africanus innovated this maneuver at the battle of Zama. When Hannibal sent his elephants in to destroy the Roman phalanxes, Scipio opened gaps in his front by having his centuries displace and shift behind one another, thus allowing the elephants to pass through the Roman lines.\n\nThe only time the _quincunx_ was employed in modern warfare was well before contact was made with the enemy, and it was only used in order to allow support troops and cavalry to pass easily through legionary lines. Once the enemy was proximate, the three \"posterior\" centuries of each cohort were moved forward on line, and the ranks were closed.\n\nNo one would conceive of forming the _quincunx_ while in contact. In our current deployment, since the posterior centuries were grouped together, this would open two ninety-foot gaps in the middle of our line!\n\n\"This will double our depth at the point of attack. I believe with a front sixty-men wide and sixteen deep, we can hold the Krauts main attack back,\" Labienus concluded. \"Any questions?\"\n\n\"Legate, what about our flanks?\" Agrippa asked.\n\n\"Good question, Agrippa!\" Labienus answered. \"The Krauts did us a favor there. In fact, they've done us two favors. The first is that they have exposed their plans. They see no point in stealth. They think they can just walk right over us. The second is to your point. Except for their earlier attempt on our right flank, they have not pressured our flanks at all. In fact, we now overlap _them_ on both flanks. You and I will continue to command the flanks. Our mission is to ensure that, when their spoiling attacks hit, they do not turn us. If they do, we're in danger of being enveloped. So we have to hold on the flank. Centurion?\"\n\nA centurion I didn't recognize from the Ninth Legion spoke. \"Legate, when we form the _quincunx_ , what's to keep those _verpae_ from charging into the gaps?\"\n\n\"Good question!\" Labienus nodded. \"Insubrecus _Decurio_ , that's where you and the Sequani come in. Split your command into two divisions. When the _quincunx_ opens, you deploy a division into each of the gaps. I don't think the Kraut muster-men have the heart to attack. But, if they try, you stop them. And, you need to protect the flanks of the exposed _prior_ centuries when the _posterior_ centuries pull back from the line. Just make sure you get your men back before the door closes . . . _Compre'endis tu_?\"\n\n\" _Compre'endo, Legate_!\" I affirmed.\n\n\" _Bene_!\" Labienus concluded. When we reestablish the line of battle, we will align on the Eighth Legion. That's all! Return to your commands! Listen for the signals!\"\n\nI translated what was being said to Athauhnu. I doubt he understood much about maniples and _quincunces_ ; there weren't even words in Gah'el to express such formations. But, he did understand that we were intentionally opening gaps in our lines to pull off some obscure Roman battlefield dance routine, and it didn't please him. What pleased him less was that we were expected to ride into the gaps and throw ourselves in front of German infantry.\n\nWe rode back to our position behind the Roman lines. Athauhnu assembled the Sequani riders and quickly briefed them on what to expect. He then separated them into two divisions: one under me and Guithiru, the other under Ci and himself. I noticed he didn't mention anything about Madog. Each of our divisions assumed a ready position, mine behind the Ninth Legion, Athauhnu behind the Eighth.\n\nTime was running out for us. A gap had formed between our front line and the German _fiurd_ ; they were starting to disengage. But, the Kraut _dugath_ , massing for the attack, were not moving. They were chanting their strange mantra to their god, Woden: \"Wo . . . wo . . . wo.\"\n\nThen, from our right, a Roman bugle sounded, \"Attention!\"\n\nThen, came a signal for the Eighth and Ninth Legions, quickly followed by, \"Q _uincuncem . . . formata_!\"\n\nImmediately, the posterior centuries of the two legions executed an about-face and began to march in quick time to the rear.\n\nAs they began to move off the line, I led my riders into the opening gap. Initially, the German muster-men proved Labienus right; they froze. Then, a few of the bolder ones made to follow the retreating centuries into the gap. We rode at them, and they quickly retreated. I was able to form two lines of cavalry across the gap. The _fiurd_ showed no desire to attack us.\n\nAcross the field, I noticed movement behind the massed German warriors. The _thegn_ realized something was up. Perhaps he feared his prey was retreating from him and he would not get the chance to kill many more Romans this afternoon. He rode in front of his center division, stabbing his sword in our direction. He harangued his elite troops to move forward, to take advantage of the widening gap in the Roman line. But, like all barbarian maneuvers, this one had a momentum all its own that no leadership could alter. The German _dugath_ remained still and continued its woeful chant; it would move when it was ready, not before.\n\nThe posterior centuries had reached their position in the _quincunx_. They halted and turned about, again facing front. No sooner did the motion of the maneuver cease than Labienus sounded the signal trumpets again: \"Eighth Legion! Ninth legion! _Manipulos formate_!\"\n\nImmediately, the rearmost Roman centuries executed a right-face and began to double-time behind the forward centuries.\n\nThe Germans did react to this maneuver.\n\nThe _thegn_ may not have totally understood what he was witnessing, but he now understood its purpose. We were reinforcing our lines to meet his attack on our center. Still, his _dugath_ would not move forward. He dispatched most ofhis _gedricht_ , his personal bodyguards, to attack the gaps in the Roman line. But, he miscalculated. His own _fiurd_ , still milling about on the battlefield, watching the Roman maneuver like a crowd in the arena watching a show, got in their way. His _gedricht_ were soon tangled among their own troops, and their attack never arrived.\n\nI looked over to where the German _thegn_ was raging at the _dugath_ to move, screaming for the _fiurd_ to get out of the way, and thundering at his _gedricht_ to attack the Roman gap. I saw he had less than a dozen men left to protect himself.\n\nNo sooner had that realization entered my mind, than Labienus signaled again, general call: \" _Aciem formata_! Form battle line!\"\n\nQuickly, I turned my troop, and we rode out of the gap as the Tenth Legion battle line closed with the Ninth. At the same time, the Ninth Legion maniples moved right to close on the Eighth, while the Seventh moved left.\n\nThe door had slammed shut! Labienus had pulled off the maneuver!\n\nThe Krauts were finally beginning to move. All three divisions of the _dugath_ were running forward to attack our lines. Behind them, the German _thegn_ and the twelve remaining members of his _gedricht_ stood alone in the field.\n\nI had an insane idea!\n\nMy military training and discipline told me to hold my position, to block and contain any penetration of our line. But, the Krauts had exposed themselves to a potentially fatal blow, which only I could deliver.\n\nWhat was it that Caesar had said the other day? _Audaces amat fortuna_!... Fortune loves the bold! Instinctively, I reached up and touched my _lorica_ where my _Bona Fortuna_ hung. It seemed strangely warm.\n\n\"Athauhnu!\" I yelled, turning Clamriu and riding toward our left flank. \" _Me_ _sequere!_ Follow me!\" I quickly realized that, in my excitement, I had spoken Latin, so as I rode past his position on the left, I repeated myself in Gah'el, \" _Diluhna fi_!\"\n\nI galloped the Sequani around our left flank. I remember catching a glimpse of Agrippa's shocked face as we thundered by. We rode out onto the field, past the right flank of one of the Kraut divisions. A couple of their warriors looked over toward us, but continued their run toward the Roman line. As soon as I cleared the rear of the German _dugath_ , I angled right to where I knew the _thegn_ was standing.\n\nHe was still there, alone except for the small remnant of his bodyguard troops.\n\nI halted opposite, about a hundred _passus_ distant. I turned Clamriu toward the Germans, extended my arms out parallel to the ground, and Ci's _ala_ , the seventeen men still in the saddle, assembled on both sides of me, facing the Krauts. Athauhnu lined his twenty-two riders up behind us. Then, he rode forward and took a position on my right.\n\nAcross the field, the remnant of _thegn's gedricht_ began, almost hesitantly, to position themselves between us and their leader, whom they were sworn to protect to the death.\n\nThe gods of the Gah'el were about to collect on that promise!\n\nBut first, I had one last die to cast.\n\nI rode out a few steps and turned to face the Sequani.\n\n\"Warriors of the Soucanai!\" I shouted. \"Madog, your king is dead!\"\n\nI heard a moan go up from the men.\n\nI raised my hands. They became silent.\n\n\"The man who killed your king . . . the dog who treacherously cut Madog down after he had slain the king of the Germans . . . and stabbed him from behind in his moment of triumph . . . stands there . . . before you!\"\n\nI pointed right at the German _thegn_ , who was sheltered behind the thin line of his _gedricht_.\n\nThe men were growling.\n\nI drew my _spatha_. \"A place on the mead-bench beside Madog in the Land of Youth, and the hero's portion served by the hands of Andraste, the goddess of victories, to the man who delivers the head of that German dog to the tomb of Madog!\" I shouted.\n\nI turned Clamriu and charged the Germans.\n\nMy Sequani thundered behind me, screaming for German blood.\n\nTo their credit, the _gedricht_ remained faithful. They died for their oath.\n\nAs I galloped toward the Germans, I shimmied my shield off my left shoulder and inserted my forearm through its bindings. I aimed for a gap in the center of the German line. When I hit it, I felt a glancing blow on my shield from the Kraut to my left. I didn't bother with the one on my right. I was through the line and bearing down on the German _thegn_.\n\nAs I reached him, I heard a crash behind me as the rest of the troop made contact with the Germans.\n\nThe _thegn_ pulled back hard on his reins. His horse reared up on his hind legs. Clamriu crashed into her opponent's chest. The German horse began to tumble backward. I felt a massive blow on my helmet, just above my eyes. I tumbled backward into blackness.\n\nWhen I awoke, Emlun was kneeling over me, and Rhodri was standing beside him. Emlun helped me sit up. That was a mistake. The world seemed to twinkle and spin in front of me. I felt like I would vomit.\n\nWhen my eyes focused again, Emlun held up my helmet. In the front, there was a dent the size of a small fist.\n\n\"Cheap Roman tin!\" Emlun shook his head. \"A German horse brought you down! You ought to get yourself a new helmet from a good Soucanai smith!\"\n\nClamriu was calmly cropping the grass a few feet away. Behind her, a couple of the Sequani were trying to subdue the _thegn's_ riderless horse. Just over to my right, I saw his body with two _gaea_ spears protruding from its chest. I puzzled for a moment why the body looked strange. Then I realized; it had no head.\n\nRhodri gestured toward the _thegn's_ horse. \"You brought him down, Arth Bek. By our custom, his horse and his armor belong to you!\"\n\n\"Athauhnu?\" I croaked.\n\nEmlun pointed across the field toward the Roman lines.\n\nAthauhnu was galloping across the face of the German horde spinning something over his head. It looked vaguely like a ball, a ball attached by a yellow rope. It was a head, the head of the German _thegn_!\n\nThe Germans were standing, watching. I realized that they had dropped their shields and weapons. The Roman _muli_ from our battle line were advancing among them, driving them down onto their knees, tying their hands behind their backs, collecting their weapons. Our fight was over.\n\nEmlun was talking: \"Athauhnu killed the German you passed. Guithiru and me followed you through their line. By the time we caught up with you, you were already down. So was the _thegn_. His horse rolled over him. He was probably dead already, but Guithiru put the first javelin into him. Then I stuck him. Athauhnu came and took his head for Madog. The honor was his as chief. We'll bury it with Madog. Lay it between his feet. That German dog will be his slave in the Land of Youth for eternity!\"\n\nThey stood me up. Another mistake. This time I did vomit. Strangely, I felt a bit better after that. Emlun steadied me.\n\nI saw a small group of Roman riders approaching. The leader wore a bright red _sagum_ and rode a white stallion.\n\nIt was Caesar.\n\nLabienus rode immediately to his left.\n\nI tried to assume some semblance of attention as the _imperator_ arrived. Again a mistake. Emlun's steadying grasp was the only thing that kept me from pitching forward into the grass.\n\n\"Well, this completes my collection of insubordinates!\" Caesar said dryly. \"First a senior legate, who won't stay where he's been assigned, but starts his own battle with the Germans, and now a _decurio_ , a very junior _decurio_ at that, who abandons his assigned post to lead a cavalry attack against a superior force of German infantry. What do you have to say for yourself, soldier?\"\n\n\" _Nil 'scusationis mi', Imperator_!\" I snapped out, as if a _tiro_ on punishment parade.\n\n\"You have that right!\" Caesar snorted. \"There's no excuse at all! It's a marvel Rome has survived this long with soldiers like you following her eagles.\"\n\nCaesar jerked the head of his horse around and rode toward the top of the ridge where his battle with the Helvetii still raged.\n\nLabienus hesitated. Then he said to Emlun in perfect Gah'el, \"Trooper! Get this Roman officer to a medic!\"\n\nBefore he turned to follow Caesar, he winked.\n\n## _Post Scriptum_\n\nAt Bibracte, I survived my first battle.\n\nOnly politicians, historians, and generals classify battles as victories or defeats.\n\nSoldiers recognize no victories. Even in the greatest national victories, comrades are lost, and the survivors suffer. They suffer from the wounds they receive; they suffer from what they witness; they suffer from the loss of dear friends. Win, lose, or draw, soldiers just thank _Domina Fortuna_ for sparing their lives and pray to her that this will be their last fight.\n\nAt the time, I expected that our war was over at least for that year. Having defeated the Helvetii, Caesar would withdraw the army south of the Rhodanus. We would spend the rest of the campaign season in the _provincia_ , licking our wounds, collecting rations, and getting ready for the winter.\n\n_Erratum_! I was wrong. Despite our battered condition after Bibracte, Caesar wasn't done.\n\nI hoped that Bibracte could be my only great battle. I would serve my remaining enlistment in relative peace and safety, and return home to Mediolanum. There, Macro and I would quickly become rich wine merchants, and perhaps then I would win back my beloved Gabi.\n\n_Denuo erratum_! Again I was wrong.\n\nThey say that the greatest gift the gods have given to men is they don't see their own future.\n\nI believe that.\n\nHad I known on that day that what lay before me was over twenty years of blood, betrayal, and loss; that after the Helvetian campaign would come years of brutal fighting in Gallia, Aquitania, Belgica, Germania, and that mysterious island across _Oceanus_ , Britannia; and then Caesar's wars in Italia, Hispania, Asia, Africa, and Aegyptus; then Octavius' wars against the _Liberatores_ , Sextus Pompeius, and finally Antonius; that after the battle at Bibracte would come Gergovia, Alesium, Dyrrhachium, Pharsalus, Zela, Alexandria, the Nile, Thapsus, and Munda with Caesar; then Forum Gallorum, Mutina, Philippi, Naulochus, and Actium for Octavius; along with hundreds of murderous encounters in places too insignificant to be remembered by a name, I believe seeing the course of my future would have driven me _demens_ , completely mad.\n\nBut the gods at least spared me that.\n\nAgrippa stayed with us in Mediolanum only a few days before returning to his refuge in Italia. He graciously invited us to visit his estates near Asisium. Then, he departed south, carrying with him my answer to Octavius' request.\n\nIt seems that our _princeps civitatis_ is continuing Caesar's vision of extending _Romanitas_ into the provinces. Octavius has appointed a number of provincial citizens, whom the Roman senatorials consider no better than _peregrini_ , foreigners, to military and civil positions traditionally reserved for Romans of the proper pedigree. He has even appointed provincials to the Roman Senate.\n\nOctavius also plans to extend Roman citizenship to the provinces by creating _municipia_ , cities with citizenship rights in the established provinces and in the major tribal centers of the newly acquired territories. Established cities in the \"civilized\" provinces, as he terms them, would be granted full Roman citizenship with the right to vote. Other cities and tribal centers would be granted a lesser degree of citizenship, which would grant their residents the protection of Roman magistrates, service in the legions, and of course, payment of taxes and financial levies to Rome.\n\nMediolanum is planned as one of Octavius' initial \"first-class\" _municipia_ in the strategically important _Gallia Cisalpina_ region. Since this is a plan with which the aristocratic elements in the Roman Senate are not pleased, the Augustus is determined that these first grants are successful. In order to do that, he is handpicking the city magistrates.\n\nOctavius has already commissioned the two _duumviri_ and the two _aediles_ who will administer affairs in Mediolanum. In fact, one of the _duumviri_ is my cousin, Lucius Helvetius Naso Quartus, who still runs my grandfather's empire of foundries, garum factories, and wine presses.\n\nOnce things have \"settled,\" these positions will become elected annually. But for now, Octavius is controlling and supervising the process personally\u2014which brings me to his request.\n\nFirst and foremost, I am to be Octavius' \"eyes and ears\" in Mediolanum. As one of his former officers, he knows that I will report to him what he needs to know\u2014not what I want him to know.\n\nSecond, I am to be commissioned as the _praefectus urbis_ of Mediolanum. As such, I will have command of all things \"military\" in the newly minted _municipium_ , meaning command of the urban militia cohort, which exists only to collect the tolls and taxes at the city gates. I am also to form a troop of _vigiles_ , after the model developed by Lucius' brother, Marcus Agrippa, in the city of Rome.\n\nThe primary duty of the _vigiles_ would be firefighting. However, they would also serve to maintain civil peace and safety by patrolling the city streets on the lookout for thieves, muggers, burglars, and runaway slaves.\n\nBefore Agrippa left, I discussed the offer with my wife, Rhonwen. She asked only two questions: would I get paid, and would the job get me out of the house? When I answered that I would receive a salary from the Augustus' privy purse, five times that of a _primus pilus_ in the legions, and would have to establish a _praetorium_ near the center of town, from which to run things, she told me I should do it.\n\nBesides, she said, she didn't think I should disappoint such a nice man as Lucius Vipsanius Agrippa after he had travelled so far just to ask me to do this.\n\nSo, I find myself about to start a new career, again in the service of the _gens Iulia_. I have decided that, despite my new responsibilities, I will continue to write my memoirs of Caesar's wars in Gaul. While I am waiting for the confirmation of my appointment as the _praefectus urbis_ of the new Roman _municipium_ of Mediolanum, I have begun to review my journals of Caesar's campaign against Ariovistus and the Suebi. I am dubbing this chapter of my journal, _De Re Suebiana_ , \"The Swabian Affair.\"\n\n## MILITARY LATIN\n\nDespite the many modern novels, whose setting places the reader among Roman soldiers, and popular movies about the Roman Empire, surprisingly little is known about the day-to-day operations of a Roman legion during the time of Caesar.\n\nCaesar's own works about his military operations in Gaul and the subsequent civil war is perhaps the best detailed surviving sources for that period. But, Caesar's goal was not to write a manual about Roman military operations; Caesar's goal was political self-promotion.\n\nA fifth-century work sometimes called _de re militari_ , \"Military Operations,\" attributed to Publius Flavius Vegetius Renatus, survives and presents a representation of the military operations at the height of Rome's power. Although Vegetius claims to have based his treatise on descriptions of Roman armies of the mid to late Republican period from Cato the Elder, Cornelius Celsus, Frontinus, and Paternus, and on Roman military operations during the principate and early empire from the imperial constitutions of Augustus, Trajan, and Hadrian, little of Vegetius' sources survive, and his writings are separated from his primary sources by centuries. Additionally, Vegetius was not a historian or a soldier, and his purpose seems to be more a nostalgic recalling of the glory days of Roman military power than a reporting of what actually happened. So, _de re militari_ resembles a somewhat clumsy compilation of materials from various sources rather than a military field manual reflecting the actual standards and practices of the Roman legions.\n\nIn order to create a realistic setting for this novel, the author has based the \"big picture\" on the first half of the first book of Caesar's _de bello Gallico_ where he reports his campaign against the Helvetians.\n\nHowever, in order to create the microcosm of the Roman soldier in training and in the field, the author has channeled his five years of studying Latin and Roman history with his twenty-five years serving in the infantry and rangers. The scenes of Gaius Marius Insubrecus' basic training with the Tenth Legion outside the Roman city of Aquileia are based on the author's own infantry training in Ft. Jackson, South Carolina and his airborne and ranger training in Ft. Benning, Georgia. In the novel, Caesar's Gallic cavalry _alae_ conduct road reconnaissance and screening missions based on the same principles of US light-infantry or armored cavalry scout platoons.\n\nRoman legionaries use a vocabulary that modern soldiers would appreciate. Where the author's comrades in Vietnam would describe a particularly unpleasant experience as \"being in the shit,\" Insubrecus' _contubernales_ refer to the same as _immerda_. The American soldier refers to a screwed up situation as FUBAR; the Roman soldiers in this book call it _perfututum_.\n\n### **Military Terminology**\n\nBesides the \"formal\" tactical terminology of the Roman army, little is known about how the soldiers actually spoke. They weren't the types to leave a written records of their conversations.\n\nAs was related in the first book of the Gaius Marius Chronicle, _The Gabinian Affair_ , the Roman legionary during the time of Caesar was more rustic than urban and more provincial than Roman. So, it is reasonable to believe that they incorporated words, grammatical constructions, and accents from their native languages into the Latin they used as the _lingua franca_ of the legions. The author is reminded of the \"Army English\" he had to learn as an infantry recruit; it sounded more Southern than Yankee and had its own jargon. To \"put the quietus\" on something was to stop it; the plural of the noun \"man\" was often \"mens\"; and combat was described as \"hitting the shit.\" So, it's reasonable to assume that the Roman _muli_ in Caesar's time had their own brand of spoken Latin, which would probably have made Cicero weep.\n\nAlso, the language used between soldiers tends to be direct, colorful, and, in a strange sense, intimate. That was the character of US Army \"rhetoric.\" Drill sergeants often referred to their charges as \"maggots\"; so Roman training officers may have called their _tirones_ , _blattae_ , \"cockroaches.\" Lazy or incompetent soldiers were referred to as \"snuffies\"; so the author invented the Latin term _funguli_ for legionaries known to cut too many corners in the performance of their duties.\n\nOther than that, the following list of Roman military jargon is perhaps just another example of a traditional, classical education gone dreadfully wrong!\n\nAcies (pl. acies) \u2013 edge, battle line\n\nAcies triplex \u2013 the triple line, standard battle formation of the Roman legion\n\nAcies prima \u2013 The vanguard of the legion on the march or the front line of the\n\nlegion deployed for battle in the _acies triplex_ ; the First, Second, Third, and Fourth cohorts of the legion\n\nAcies secunda \u2013 the middle line, the Fifth, Sixth, and Seventh cohorts of the legion\n\nAcies tertia \u2013 the rear line, the Eighth, Ninth, and Tenth cohorts\n\nAgmen (pl. agmines) \u2013 a marching column\n\nNovissimum Agmen \u2013 the rearguard of the marching column; the rea, the caboose\n\nPrimum Agmen \u2013 the vanguard of the marching column, the \"bleeding edge.\"\n\nAla (pl. alae) \u2013 literally, a bird's wing; an element of Roman cavalry roughly equivalent in numbers to forty to sixty troopers\n\nA'mperi'tu' \u2013 military jargon coined by the author, short for _Ad imperatum tuum_ \u2014\"At your command\"; this was one of the acceptable responses from a subordinate to a superior's command.\n\nAquila (pl. aquilae) \u2013 literally, an eagle, a bird sacred to Iove; figuratively, the standard of a legion and its _animus_ , its \"life force, spirit, soul\"; the worst disgrace that could befall a legion was to lose its _aquila_ to an enemy.\n\nSub aquilis \u2013 literally, \"under the eagles\"; figuratively, \"in the army.\"\n\nAquilifer \u2013 an officer bearing the eagle standard of a Roman legion; also the \"treasurer\" of the legion handling payroll and burial club transactions.\n\nBallista (pl. ballistae) \u2013 a piece of Roman artillery that launched a large, arrowlike bolt over great distances\n\nBalteus \u2013 a leather baldric used to suspend a sword. It was worn over the shoulder, passing down to the side where the sword was suspended. Enlisted men wore their swords on the right side; centurions and senior officers on the left.\n\nBestiola (pl. bestiolae) \u2013 literally, any small animal; in the jargon of the Roman army, it's a derogatory term used to address trainees, equivalalent to \"maggot\" in English. (author)\n\nBlatta (pl. blattae . . . there's never only one of these things) \u2013 literally, a cockroach; figuratively, a legionary trainee.\n\nBuccellatum \u2013 hardtack; part of a _mulus_ ' marching rations.\n\nCaliga (pl. caligae) \u2013 Roman, hobnailed military boots.\n\nCapsarius \u2013 army slang for a field medic, one who carries a _capsa_ , a box or satchel, which carried medical supplies\n\nCapu' \u2013 boss, chief, from Latin _caput_ , \"head\"; modern Italian, _capo_. Military jargon coined by the author.\n\nCarinus \u2013 ruddy brown; in this story, the standard color of an enlisted man's _sagum_\n\nCastigatio (pl. castigationes) \u2013 a minor military punishment; often a beating with a centurion's _vitis_ , his vine cudgel.\n\nCastro' \u2013 an \"army brat;\" _mus castrorum_ , a \"camp rat\"; this is a soldier who was the offspring of a legionary and one of the _mulieres castrorum_ , camp followers, and brought up in a legion's _vicus_ ; these soldiers felt that they were _familia legionis_ , members of the legion's family, therefore superior to first-time enlistees; sometimes adopted as the soldier's cognomen, e.g., Lucius Furius Castro or Lucus Fulius De Castris (author).\n\nCastrum (pl. castra) \u2013 a legionary marching camp; these were constructed at the end of each daily march.\n\nCedo alteram, centurio! \u2013 \"Hit me again, sir!\" The expected response of a soldier undergoing _castigatio_.\n\nCenturia (pl. centuriae) \u2013 a sub-unit of a Roman legion consisting of ten _contubernia_ , eighty legionaries, commanded by a _centurio_\n\nCenturio (pl. centuriones) \u2013 a centurion, a Roman officer who commands a _centuria_ of eighty men\n\nCenturio prior pilus \u2013 a senior centurion; a centurion of the first rank; in this novel, designates a centurion who commands a cohort and so would stand in the front rank of the cohort in the battle line.\n\nCenturio primi ordinis \u2013 in this novel, designates a centurion of the first rank; a centurion who commands a century in any of the first four cohorts, which constitute the first rank of the legion in the _acies triplex_ , the \"triple battle line,\" a battle formation favored by Caesar\n\nCenturio primus pilus \u2013 \"The First Spear\"; senior centurion of a legion; commands the First Centuria of the First Cohors and also commands the First Cohors; in the absence of both the _legatus legionis_ , the legionary commander, and the _tribunus laticlavus_ , the \"Broad-Stripe\" tribune, commands the legion.\n\nCenturio secundi ordinis \u2013 in this novel, designates a centurion of the second rank; commands a century in Cohorts Five, Six, or Seven, the second rank in the line of battle\n\nCenturio tertii ordinis \u2013 in this novel, designates a centurion of the third rank; commands a century in Cohorts, Eight, Nine, or Ten, the third rank in line of battle; the reserve or \"Forlorn Hope\" of a legion\n\nChlamys (pl. chlamydes) \u2013 chainmail shoulder pads that fit over the _lorica hamate_ , a legionary's chainmail armor\n\nCingulum (pl. cingula) \u2013 a Roman military belt; this was one of the indicators of a soldier; it was decorated with an ornate buckle and highly polished metal bits; this and a razor-sharp pugio gave a mulus a bit of swagger when he went into town for wine and entertainment.\n\nCochleare (pl. cochleara) \u2013 a mess spoon; part of a legionary's mess kit\n\nCohors (pl. cohortes) \u2013 an element of a Roman legion commanded by a senior centurion, consisting of six centuries, 480 legionaries\n\nComes (pl. comites) \u2013 a companion; an intimate; used to describe a general's personal staff and \/ or members of his personal guard.\n\nContubernium (pl. contubernia) \u2013 a squad, a grouping of eight legionaries who share a tent in the field or a squad room in a permanent camp\n\nContubernales \u2013 members of a common _contubernium_ , \"mates,\" \"squaddies\"\n\nCornex (pl. cornices) \u2013 literally, a crow; in military jargon, a field medic (author), a _capsarius_ , who wore a black tunic in combat to hide the blood and gore on his uniform.\n\nCornices pascere \u2013 literally, to feed the crows; figuratively, to be evacuated to the aid\n\nstation.\n\nCornu (pl. cornua) \u2013 a Roman \"brass\" instrument about ten feet long in the shape of a musical G-clef; used to relay orders on the battlefield\n\nCornucen (pl. cornucines) \u2013 the horn-blower, a minor officer who carried a _cornu_ to signal orders over the field during battles.\n\nCorona civica \u2013 Civic Crown, the second highest Roman military decoration, after the \"grass crown\"; reserved for a Roman citizen who saved the lives of fellow citizens by killing an enemy on a spot not again held by the enemy that same day; the citizens saved must bear witness to the act\u2014no one else could be the verifying witness; any recipient of the Civic Crown was entitled membership in the Roman Senate.\n\nCuneus \u2013 the \"boar's snout\"; a Roman infantry and cavalry offensive military formation, the wedge\n\nDecanus \u2013 a junior legionary officer; commander of a _contubernium_ ; a squad leader\n\nDecurio (pl. decuriones) \u2013 a junior cavalry officer commanding a _turma_ , about thirty-five troopers.\n\nDicto pareo \u2013 literally, \"to what is spoken I comply\"; \"Yes, sir!\" Another acceptable response from a subordinate when given a command (author).\n\nDolabra (pl. dolabrae) \u2013 an entrenching tool; a versatile axe, pickaxe and mattock; no _mulus_ would be caught without his or his centurion would have his guts for shoelaces.\n\nDux (pl. duces) \u2013 this was not a formal military title in the time of Caesar. Literally, the word denotes a leader. In this book, it means \"chief\" and refers to anyone in a leadership position who is viewed favorably by the troops (author). Also, it's an honorary title given to native leaders of allied bands and auxiliary units.\n\nExpeditus (pl. expediti) \u2013 unburdened, lightly armed, ready for combat\n\nExplorator (pl. exploratores) \u2013 a scout, spy\n\nFabricator (pl. fabricatores) \u2013 an engineer, construction troop\n\nFossa \u2013 a ditch; part of the standard fortifications of a castrum\n\nFungulus (pl. funguli) \u2013 literally, a little mushroom, a fungus; figuratively, military slang for an incompetent soldier, a \"snuffy\" (author).\n\nFurca (pl. furcae) \u2013 a pole on which a soldier carried his personal equipment on the march\n\nFustuarium \u2013 a severe military punishment; being beaten to death with clubs by the members of one's _contubernium_ ; this is one of the few punishments inflicted in the Roman army where the victim is not expected to say, \"Hit me again, sir!\"\n\nGalea (pl. galeae) \u2013 the standard Roman military helmet\n\nGeminus (pl. gemini) \u2013 literally, a twin; figuratively, military fighting partners (author)\n\nGladius (pl. gladii) \u2013 the _gladius hispaniensis_ , \"Spanish short sword,\" the basic Roman infantry short, stabbing sword\n\nHastae purae \u2013 a minor military decoration\n\nHastile \u2013 a wooden staff carried by an _optio_ ; in battle, the _optio_ 's position was at the rear of the century; it is thought that the _hastile_ was used to keep the battle line straight, discourage flight, and to beat legionaries back into line should they get some other idea\n\nImmerda \u2013 from _in merda_ , literally \"in the shit\": figuratively, a Roman legionary's characterization of a bad situation. _Immerda sumus_! We're in the shit! (author)\n\nImmunis (pl. immunes) \u2013 a military status in which a soldier was excused from fatigue details and got extra pay for doing some specialized job, like clerk, blacksmith, forager, etc.\n\nImpedimentum (pl. impedimenta) \u2013 equipment, baggage, military kit\n\nImpeditus (pl. impediti) \u2013 marching under a full field load or _mulare_ , \"to mule it\".(author)\n\nImperator \u2013 a Roman title given to a victorious commander by acclamation of the troops; a general\n\nImperium \u2013 basically, \"power\"; it is the power of the Roman state over individuals which in Caesar's time was delegated by the Senate to a magistrate; also refers to the area where Roman law ran and where a Roman magistrate could wield the power of the state, _imperium Romanum_ , \"The Roman Empire.\"\n\nInterrogatio \u2013 interrogation, inquisition, a grilling; if a slave or non-citizen were undergoing an _interrogatio_ , torture would typically be used; legions had _immunis_ soldiers, _carnifices_ , who were specialists in getting the answers a general wanted from prisoners.\n\nIntervallum \u2013 an open area between the ramparts and tents in a _castrum_\n\nLagoena (pl. lagoenae) \u2013 water bottle, a canteen; part of a legionary's field kit\n\nLatrina \u2013 privy, water closet; the engineers were careful to place these downstream of the water and bathing point\n\nLatus apertum \u2013 the right side; the \"open side\"; the side of an individual soldier or an entire military formation not protected by shields\n\nLatus opertum \u2013 the left side; the side of an individual soldier or an entire military formation protected by shields\n\nLegatus \u2013 a legate; a senior Roman officer appointed to assist a Roman magistrate in some manner; usually of the Senatorial order, an equestrian plebeian, or patrician; a political appointee, a client of the magistrate, a political favor, a nephew or an in-law, or someone the magistrate didn't want to leave back in Rome, where he couldn't keep an eye on him. Legatus Equitium \u2013 cavalry commander; an ad hoc assignment.Legatus legionis \u2013 a legate appointed as the army commander's\n\nrepresentative in a legion; in Caesar's time these appointments were not permament assignments but ad hoc based on the situation; tactical control of the legion in combat was usually left to the legion's senior centurion, the _primus pilus_.\n\nLoculus \u2013 a military satchel; part of a soldier's marching pack\n\nLorica (pl. loricae) \u2013upper-body armor of a Roman legionary; during the period of the story, it was the _lorica hamata_ , made of iron chainmail for the enlisted men and plate armor for senior officers; men serving in the auxiliary units might wear leather _loricae_.\n\nMedicus (pl. medici) \u2013 doctor; army medic\n\nMercurius (pl. mercurii) \u2013 Mercury, a Roman god; military slang invented by the author for a sestertius coin\n\nMiles (pl. milites) \u2013 the basic word for a soldier; in this story, it's what _tirones_ , trainees, strive to become, _milites Romani_.\n\nMinerva \u2013 Minerva, a Roman goddess; soldiers' slang for a bronze triens coin (author)\n\nMulieres castrorum \u2013 women of the camps; camp followers\n\nMulare \u2013 \"to mule it\"; the Roman equivalent to the US Infantry expression \"humping\"; marching under a full load. (author)\n\nMulio (pl. muliones) \u2013 teamster, mule driver\n\nMulus (pl. muli) \u2013 literally, a mule; figuratively, army slang for an infantryman, a grunt (author); from the expression _muli_ Marii, \"Marius' mules,\" describing legionaries marching _impedimenti_ , \"loaded down\" with their personal equipment, after the Roman general, Gaius Marius, unloaded the mules in the baggage train and loaded the gear onto the backs of the legionaries to improve the mobility of the legions\n\nMurus (pl. muri) \u2013 a wall; in the military jargon of the author it describes a close-order defensive formation, a _murus scutorum_ , a shield-wall\n\nNil 'scusationis mi' \u2013 \"I have no excuse!\" One of the five responses authorized for a Roman legionary _tiro_ , along with \"Yes, Sir!\", \"No, Sir!\", \"Sir, I do not know!\" and \"Hit me again, Sir!\" (author).\n\nObsequar ti' \u2013 \"I obey you!\" Again, another acceptable reponse to a military order (author)\n\nOptio (pl. optiones) \u2013 \"chosen\" one; a junior army officer; second in command of a _centuria_ under the centurion\n\nOrdo (pl. ordines) \u2013 rank in battle or a social class\n\nPalus (pl. pali) \u2013 pole used to practice sword drill\n\nParatus \u2013 \"Ready!\" The only acceptable response when an officer asks _Parat' tu?_ \u2014\"Are you ready?\"\n\nParma (pl. parmae) \u2013 cavalry shield\n\nPassus (pl. passus) \u2013 a complete stride from when the right foot goes down until it comes down again; about five and a half feet on flat ground; the standard legionary daily march was twenty thousand passus\n\nPatera (pl. paterae) \u2013 mess tin\n\nPercussus \u2013 literally, a blow, strike, punch; in the book it's used to describe a soldier's use of dagger, sword, and offensive shield techniques; also, it means, \"hit man,\" someone who \"punches\" his victim with a dagger.\n\nPerfututum \u2013 FUBAR, totally screwed up! Much worse than _immerda_! (author)\n\nPes (pl. pedes) \u2013 a foot; army slang invented by the author for an infantryman, equivalent to the American military expression for an infantryman, a \"leg.\"\n\nPhalera (pl. phalerae) \u2013 a military decoration\n\nPilleum (pl. pillea) \u2013 a cap worn by freed slaves; in the army, a cap worn under the _galea_ for stability, fit and cushioning\n\nPilum (pl. pila) \u2013 Roman military javelin; really one of the legions' secret weapons. It was essentially an antipersonnel device, but it was also used to render opponent's shields unusable so the _muli_ could close in for their sword work; the shaft bent on contact so the _pilum_ could not be extracted from a shield or thrown back at the Roman line; a nasty piece of business.\n\nPorta (pl. portae) \u2013 a gate, portal; a Roman marching camp, _castrum_ , had four standard _portae_\n\n 1. Porta Decumana \u2013 the \"Gate of the Tenth Cohort,\" the back gate\n 2. Porta Praetoria \u2013 the main gate\n 3. Porta Principalis Dextra \u2013 right side gate\n 4. Porta Principalis Sinistra \u2013 left side gate\n\nPosca \u2013 a drink made from vinegar and herbs; mother's milk of a Roman soldier\n\nPraefectus \u2013 commander; a Roman officer, often of the centurion or tribune status, in command of a legionary _vexilliatio_ or an auxiliary unit\n\nPraefectus castrorum \u2013 commander of the camps; senior centurion of the army\n\nPraetorium \u2013 a headquarters\n\nPreatoriani \u2013 headquarters security troops\n\nPrimus \u2013 literally, first; military jargon coined by the author for \"Top Soldier,\" Number One; reserved for senior centurions; in direct address, \" _Prime!_ \" (pronounced PREEM-eh!)\n\nPteruges \u2013 the skirt of leather or fabric strips worn around the waists of Roman soldiers. They were often decorated with metal studs or embossed images.\n\nPugio (pl. Pugiones) \u2013 a knife used by Roman soldiers as a sidearm; along with the military belt, the _pugio_ was one of the indicators of a soldiers status; it was considered a \"noble\" and \"Roman\" weapon, unlike the _sica_ which was used by villains, thieves, scoundrels, backstabbers, throat-cutters, and barbarians from the east.\n\nQuadriga (pl. quadrigae) \u2013 literally, a four-horse rig in a chariot race; street talk coined by the author for a _denarius_ , a roman coin.\n\nQuaestor (pl. Quaestores) \u2013 in Caesar's time, _quaestores_ were officials who supervised the treasury and financial affairs of the state; as a proconsul, Ceasar would have had a _quaestor_ , or _quaestores_ , to help him cook the books in the provinces assigned to him; in this story, the _quaestor exercitus_ , is the army quartermaster.\n\nQuincunx (pl. quincunces) \u2013 literally, the pattern of five dots on a dice cube; figuratively, a disposition of soldiers or centuries in a military formation\n\nRudis (pl. rudes) \u2013 literally, a stick; a wooden practice sword or knife, weighted heavier than the real thing to build up muscle\n\nSaccus (pl. sacci) \u2013 a bag for carrying equipment, food, and loot; part of a soldier's marching pack\n\nSacramentum \u2013 the military oath taken by soldiers\n\nSagum (pl. saga) \u2013 a military cloak, woolen and treated to be waterproof\n\nSarcina (pl. sarcinae) \u2013 military marching pack\n\nScutum (pl. scuta) \u2013 Roman infantry shield\n\nSenior \u2013 literally, older; figuratively, \"Sir!\" (author)\n\nSica (pl. sicae) \u2013 a small, easily-concealable knife with either a straight or curved blade; believed to be of eastern origin, therefore not considered by Romans as a \"noble\" weapon; favored weapon of the _sicarius_ (pl. _sicarii_ ), the \"knife man\" or \"hitter.\"\n\nSignificatio (pl. significationes) \u2013 an indication of approval; in the novel, it is the ceremony in which _tirones_ are accepted as soldiers by their legion\n\nSignum (pl. signa) \u2013 the standard of a military unit\n\nSignifer \u2013 a soldier who carried a unit's _signum_\n\nSitula (pl. situlae) \u2013 a bucket; in the legions, _situlae_ , \"buckets,\" was a drill requiring trainees to carry full buckets over long distance to build upper-body strength (author)\n\nSocius (pl. Scocii) \u2013 an ally\n\nSpatha (pl. spathae) \u2013 a long sword, used mostly in the cavalry because a _gladius_ doesn't reach well when one is sitting on a horse\n\nSubarmalis (pl. subarmales) \u2013 padded jackets worn under a _lorica_ by Roman soldiers\n\nSudarium \u2013 a soldier's scarf; a \"sweat-rag\"; served many purposes for a _mulus_ . . . it kept his _lorica_ from chafing his neck on the march; it cushioned his shoulder from his _furca_ when marching _impeditus_ ; it wiped the sweat off his face; it was a handy field dressing.\n\nSudis (pl. sudes) \u2013 stakes used as part of army fortifications; antipersonnel devices\n\nTegimen (pl. tegimenta) \u2013 a covering; a leather carrying case for the _scutum_ , the legionary shield\n\nTesserarius (pl. tesserarii) \u2013 a Roman officer, usually third in rank in a _centuria_ ; named after a _tesserae_ , a clay token on which was written the daily password\n\nTi' adsum \u2013 \"At your service!\" Literally, _tibi adsum_ \u2014\"I'm here for you!\"\n\nTiro (pl. tirones) \u2013 the lowest thing on earth, a recruit, a rookie in the Roman army, a trainee\n\nTribunus (pl. tribuni) \u2013 a tribune, a military and civil officer\n\nTribunus plebis \u2013 tribune of the people\n\nTribunus militum \u2013 tribune of the soldiers, a military tribune\n\nTribunus angusticlavus \u2013 a military tribune from the _ordo equester_ , the knightly class; a junior tribune; a \"narrow-striper\"\n\nTribunus laticlavus \u2013 a military tribune from the Senatorial or Patrician class; a senior tribune; a \"broad-striper\"\n\nTurma (pl. turmae) \u2013 cavalry squadron; a cavalry unit of two or more _alae_\n\nUmbo \u2013 the boss of a shield\n\nVagina \u2013 a scabbard for a sword.\n\nVenator (pl. Venatores) \u2013 a hunter; an _immunis_ detail in the legion, a soldier who hunts game for rations\n\nVenatus \u2013 literally, the hunt; figuratively, a cavalry advance against a fleeing enemy\n\nVermiculus (pl. vermiculi) \u2013 literally, a maggot; figuratively, a training officer's pet name for a trainee.\n\nVeteranus (pl. veterani) \u2013 veteran, an experienced soldier, an \"old man\"\n\nVexilium (pl. vexilia) \u2013 a banner, flag\n\nVexilium rubrum \u2013 the red flag; symbol of a legion\n\nVexillarius \u2013 a guidearm; one who carries the unit pennant\n\nVexillatio (pl. vexillationes) \u2013 an independent detachment of legionaries commanded by a centurion or tribune; this was considered an \"independent command,\" so it was a desired assignment for any amibitious junior officer\n\nVia Praetoria \u2013 one of the standard \"streets\" of a _castrum_ running from Porta Praetoria, the main gate, to the Porta Decumana, the rear gate\n\nVia Principalis \u2013 one of the standard \"streets\" of a _castrum_ running from the Porta Principalis Dextra and the Porta Principalis Sinistra\n\nVicus \u2013 village; a settlement of camp followers outside a legionary _castrum_\n\nVimen (pl. vimenes) \u2013 the \"basket\"; a weighted, wicker practice shield used in training\n\nVirgo (pl. virgines) \u2013 literally, a virgin, a maiden; figuratively, a soldier who has not experienced combat, a \"cherry\" (author)\n\nVitis (pl. vites) \u2013 club, cudgel, swagger stick made from a grape vine; one of insignias of rank of a centurion used to direct military drill and to administer physical punishment.\n\n### **Roman Military Commands**\n\nAgain, little is known about Roman unit drill and combat commands. In order to bring the reader into the Roman battle line, the author used his own experience as a combat infantry commander and a liberal use of the commands developed by Roman military reenactors.\n\nIn the Gaius Marius novels, the Roman army uses a \"shorthand\" version of Latin for efficiency. So, in order to command a unit to turn to the right, the normal Latin expression, _ad dextram_ , meaning, \"toward the right hand,\" becomes _a' dex'_ , and the Latin command to turn, _versate_ , becomes _versat'_... \" _A' Dex... Versat_!\"\n\nSome drill commands, like \"Stat'!\"\u2014\"Stand up\" or \"Attention!\"\u2014were \"signal commands.\" When delivering these commands, an officer first names the unit for whom the command is intended, then delivers the command for execution by the troops. So, if a _decanus_ , a Roman squad leader, wants to bring his _contubernium_ of eight legionaries to the position of attention, he says, \" _Contubernium . . . Stat'_!\"\n\nMost Roman military commands are divided into two elements\u2014the prepatory command and the command of execution. The prepatory command is given to warn and prepare troops to execute a specific movement. The actual movement is executed on the command of execution.\n\nSo, if a Roman officer wants his detail to turn right, he first calls them to the position of attention by issuing the command, \" _Stat'_!\" He then issues a prepatory command, \" _A' Dex'_ ,\" meaning \"to the right,\" then the command of execution, \" _Versat'_!\" When the soldiers hear the command of execution, they turn to the right.\n\nThe Roman drill and combat commands used in the novels are:\n\nAciem . . . Format'! \u2013 \"Form battle line!\"\n\nAd Cuneum! \u2013 \"Form the wedge!\" A Roman attack formation\n\nA' Dex' Aciem . . . Format' \u2013 \"Form battle line to the right!\"\n\nA' Dex' . . . Versat'! \u2013 \"Right, Face!\"\n\nA' Pedes! \u2013 \"On Your feet!\"\n\nAd Signa! \u2013 \"Fall In!\"\n\nAd Sin' Aciem... Format' \u2013 \"Form battle line to the left!\"\n\nAd Sin' . . . Versat'! \u2013 \"Left, Face!\"\n\nA' Testudinem! \u2013 \"Form the turtle!\" A Roman defensive formation\n\nA Signis! \u2013 \"Fall out!\"\n\n(Unit) . . . Consistit'! \u2013 \"Halt!\"\n\nContra . . . Versat'! \u2013 \"About, Face!\"\n\nDex' . . . Dex' . . . Dex', Sin', Dex' \u2013 counting marching cadence, \"Right . . . right . . . right, left, right.\"\n\nEquiis . . . citatis! \u2013 \"Horses canter!\" (cavalry command)\n\nEquiis . . . currentibus! \u2013 \"Horses gallop!\" (cavalry command)\n\nGladios . . . stringit'! \u2013 \"Draw swords!\"\n\nGradus Bis . . . Movet'! \u2013 \"Double-time march!\" Increase marching pace to 120 passus per minute.\n\n(Unit) . . . Laxat'! \u2013 \"Stand at ease!\"\n\nImpetum . . . agit'! \u2013 \"Attack!\"\n\nManipulos . . . format'! \u2013 \"Form maniples!\"\n\n(Unit) . . . Miss'est! \u2013 \"Dismissed!\"\n\nOrdines . . . Densat'! \u2013 \"Close Ranks!\" Close the interval between each man to an arm's length.\n\nOrdines . . . Extendit'! \u2013 \"Open Ranks!\" Extend the interval between each man to two arm's lengths.\n\nOrdines . . . Revert! \u2013 \"Normal Interval\" This command is only given to recover from open to closed ranks; recover the normal interval between each man to one arm-length.\n\nPila . . . iacit'! \u2013 \"Cast spears!\"\n\nPila . . . parat'! \u2013 \"Prepare spears for casting!\" At this command, the legionaries place their feet shoulder distance apart, right foot back, shields held slightly up toward the enemy; right arm and hand holding the _pilum_ cocked to the rear, the point slightly elevated.\n\nPila . . . ponit'! \u2013 \"Present or pick up spears!\" At this command, the legionaries place their feet shoulder distance apart, right toe to left heel; shields remain level and to the front in a good defensive position. The right hand holding the _pilum_ is held at the level of the right ear, the _pilum_ parallel to the ground.\n\nPromov . . . ete! \u2013 \"Forward March!\" Step off on the right foot and march the standard Roman marching pace of sixty full passus per minute.\n\nQuincuncem . . . format'! \u2013 \"Form quincunx!\" Form the \"five dots\"\u2014a formation used to allow a passage of lines.\n\nScuta . . . erigit'! \u2013 \"Shields up!\"\n\nSigna . . . Conversat'! \u2013 Reverse the access of advance, a 180 degree turn; \"Turn about!\"\n\nSigna . . . Profert! \u2013 \"Advance!\"\n\nSpathas . . . Stringit'! \u2013 \"Draw sabers!\" (cavalry command)\n\n(Unit) . . . Stat'! \u2013 \"Attention!\"\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \n**_Salman Rushdie_**\n\nSalman Rushdie is the author of twelve novels, four works of nonfiction, and a book of stories. He is coeditor of _Mirrorwork_ , an anthology of contemporary Indian writing, and of the 2008 _Best American Short Stories_ anthology. His novel _Midnight Children_ was adjudged the \"Booker of the Bookers\" in 1993.\nBOOKS BY SALMAN RUSHDIE\n\nFICTION\n\n_Two Years Eight Months and Twenty-Eight Nights_\n\n_Luka and the Fire of Life_\n\n_The Enchantress of Florence_\n\n_Shalimar the Clown_\n\n_Fury_\n\n_The Ground Beneath Her Feet_\n\n_The Moor's Last Sigh_\n\n_East, West_\n\n_Haroun and the Sea of Stories_\n\n_The Satanic Verses_\n\n_Shame_\n\n_Midnight's Children_\n\n_Grimus_\n\nNONFICTION\n\n_The Step Across This Line_\n\n_The Jaguar Smile_\n\n_Imaginary Homelands_\n\n_Joseph Anton: A Memoir_\n\n# **The Prophet's Hair**\n\n# from _East, West_\n\nSalman Rushdie\n\nA Vintage Short\n\nVintage Books\n\nA Division of Penguin Random House LLC\n\nNew York\n_Copyright \u00a9 1994 by Salman Rushdie_\n\nAll rights reserved. Published in the United States by Vintage Books, a division of Penguin Random House LLC, New York, and in Canada by Random House of Canada, a division of Penguin Random House Canada Limited, Toronto. Originally published in hardcover as part of _East, West_ in Great Britain by Jonathan Cape Ltd., London, in 1994, and subsequently published in the United States by Pantheon Books, a division of Penguin Random House LLC, New York, in 1995.\n\nVintage and colophon are registered trademarks of Penguin Random House LLC.\n\nThe Cataloging-in-Publication Data for _East, West_ is available from the Library of Congress.\n\nVintage eShort ISBN 9781101973691\n\nSeries cover design by Joan Wong\n\nwww.\u200bvintagebooks.\u200bcom\n\nv4.1\n\na\n\n# Contents\n\n_Cover_\n\n_About the Author_\n\n_Books by Salman Rushdie_\n\n_Title Page_\n\n_Copyright_\n\nThe Prophet's Hair\nEarly in the year 19\u2014, when Srinagar was under the spell of a winter so fierce it could crack men's bones as if they were glass, a young man upon whose cold-pinked skin there lay, like a frost, the unmistakable sheen of wealth was to be seen entering the most wretched and disreputable part of the city, where the houses of wood and corrugated iron seemed perpetually on the verge of losing their balance, and asking in low, grave tones where he might go to engage the services of a dependably professional burglar. The young man's name was Atta, and the rogues in that part of town directed him gleefully into ever darker and less public alleys, until in a yard wet with the blood of a slaughtered chicken he was set upon by two men whose faces he never saw, robbed of the substantial bank-roll which he had insanely brought on his solitary excursion, and beaten within an inch of his life.\n\n\u2014\n\nNight fell. His body was carried by anonymous hands to the edge of the lake, whence it was transported by shikara across the water and deposited, torn and bleeding, on the deserted embankment of the canal which led to the gardens of Shalimar. At dawn the next morning a flower-vendor was rowing his boat through water to which the cold of the night had given the cloudy consistency of wild honey when he saw the prone form of young Atta, who was just beginning to stir and moan, and on whose now deathly pale skin the sheen of wealth could still be made out dimly beneath an actual layer of frost.\n\n\u2014\n\nThe flower-vendor moored his craft and by stooping over the mouth of the injured man was able to learn the poor fellow's address, which was mumbled through lips that could scarcely move; whereupon, hoping for a large tip, the hawker rowed Atta home to a large house on the shores of the lake, where a beautiful but inexplicably bruised young woman and her distraught, but equally handsome mother, neither of whom, it was clear from their eyes, had slept a wink from worrying, screamed at the sight of their Atta \u2013 who was the elder brother of the beautiful young woman \u2013 lying motionless amidst the funereally stunted winter blooms of the hopeful florist.\n\nThe flower-vendor was indeed paid off handsomely, not least to ensure his silence, and plays no further part in our story. Atta himself, suffering terribly from exposure as well as a broken skull, entered a coma which caused the city's finest doctors to shrug helplessly. It was therefore all the more remarkable that on the very next evening the most wretched and disreputable part of the city received a second unexpected visitor. This was Huma, the sister of the unfortunate young man, and her question was the same as her brother's, and asked in the same low, grave tones:\n\n'Where may I hire a thief?'\n\n\u2014\n\nThe story of the rich idiot who had come looking for a burglar was already common knowledge in those insalubrious gullies, but this time the young woman added: 'I should say that I am carrying no money, nor am I wearing any jewellery items. My father has disowned me and will pay no ransom if I am kidnapped; and a letter has been lodged with the Deputy Commissioner of Police, my uncle, to be opened in the event of my not being safe at home by morning. In that letter he will find full details of my journey here, and he will move Heaven and Earth to punish my assailants.'\n\n\u2014\n\nHer exceptional beauty, which was visible even through the enormous welts and bruises disfiguring her arms and forehead, coupled with the oddity of her inquiries, had attracted a sizable group of curious onlookers, and because her little speech seemed to them to cover just about everything, no one attempted to injure her in any way, although there were some raucous comments to the effect that it was pretty peculiar for someone who was trying to hire a crook to invoke the protection of a high-up policeman uncle.\n\nShe was directed into ever darker and less public alleys until finally in a gully as dark as ink an old woman with eyes which stared so piercingly that Huma instantly understood she was blind motioned her through a doorway from which darkness seemed to be pouring like smoke. Clenching her fists, angrily ordering her heart to behave normally, Huma followed the old woman into the gloom-wrapped house.\n\n\u2014\n\nThe faintest conceivable rivulet of candlelight trickled through the darkness; following this unreliable yellow thread (because she could no longer see the old lady), Huma received a sudden sharp blow to the shins and cried out involuntarily, after which she at once bit her lip, angry at having revealed her mounting terror to whoever or whatever waited before her, shrouded in blackness.\n\nShe had, in fact, collided with a low table on which a single candle burned and beyond which a mountainous figure could be made out, sitting cross-legged on the floor. 'Sit, sit,' said a man's calm, deep voice, and her legs, needing no more flowery invitation, buckled beneath her at the terse command. Clutching her left hand in her right, she forced her voice to respond evenly:\n\n'And you, sir, will be the thief I have been requesting?'\n\n\u2014\n\nShifting its weight very slightly, the shadow-mountain informed Huma that all criminal activity originating in this zone was well organised and also centrally controlled, so that all requests for what might be termed freelance work had to be channelled through this room.\n\nHe demanded comprehensive details of the crime to be committed, including a precise inventory of items to be acquired, also a clear statement of all financial inducements being offered with no gratuities excluded, plus, for filing purposes only, a summary of the motives for the application.\n\nAt this, Huma, as though remembering something, stiffened both in body and resolve and replied loudly that her motives were entirely a matter for herself; that she would discuss details with no one but the thief himself; but that the rewards she proposed could only be described as 'lavish'.\n\n'All I am willing to disclose to you, sir, since it appears that I am on the premises of some sort of employment agency, is that in return for such lavish rewards I must have the most desperate criminal at your disposal, a man for whom life holds no terrors, not even the fear of God.\n\n'The worst of fellows, I tell you \u2013 nothing less will do!'\n\n\u2014\n\nAt this a paraffin storm-lantern was lighted, and Huma saw facing her a grey-haired giant down whose left cheek ran the most sinister of scars, a cicatrice in the shape of the letter _s\u00edn_ in the Nastaliq script. She was gripped by the insupportably nostalgic notion that the bogeyman of her childhood nursery had risen up to confront her, because her ayah had always forestalled any incipient acts of disobedience by threatening Huma and Atta: 'You don't watch out and I'll send that one to steal you away \u2013 that Sheikh S\u00edn, the Thief of Thieves!'\n\nHere, grey-haired but unquestionably scarred, was the notorious criminal himself \u2013 and was she out of her mind, were her ears playing tricks, or had he truly just announced that, given the stated circumstances, he himself was the only man for the job?\n\n\u2014\n\nStruggling hard against the newborn goblins of nostalgia, Huma warned the fearsome volunteer that only a matter of extreme urgency and peril would have brought her unescorted into these ferocious streets.\n\n'Because we can afford no last-minute backings-out,' she continued, 'I am determined to tell you everything, keeping back no secrets whatsoever. If, after hearing me out, you are still prepared to proceed, then we shall do everything in our power to assist you, and to make you rich.'\n\nThe old thief shrugged, nodded, spat. Huma began her story.\n\n\u2014\n\nSix days ago, everything in the household of her father, the wealthy moneylender Hashim, had been as it always was. At breakfast her mother had spooned khichri lovingly on to the moneylender's plate; the conversation had been filled with those expressions of courtesy and solicitude on which the family prided itself.\n\nHashim was fond of pointing out that while he was not a godly man he set great store by 'living honourably in the world'. In that spacious lakeside residence, all outsiders were greeted with the same formality and respect, even those unfortunates who came to negotiate for small fragments of Hashim's large fortune, and of whom he naturally asked an interest rate of over seventy per cent, partly, as he told his khichri-spooning wife, 'to teach these people the value of money; let them only learn that, and they will be cured of this fever of borrowing borrowing all the time \u2013 so you see that if my plans succeed, I shall put myself out of business!'\n\nIn their children, Atta and Huma, the moneylender and his wife had successfully sought to inculcate the virtues of thrift, plain dealing and a healthy independence of spirit. On this, too, Hashim was fond of congratulating himself.\n\n\u2014\n\nBreakfast ended; the family members wished one another a fulfilling day. Within a few hours, however, the glassy contentment of that household, of that life of porcelain delicacy and alabaster sensibilities, was to be shattered beyond all hope of repair.\n\n\u2014\n\nThe moneylender summoned his personal shikara and was on the point of stepping into it when, attracted by a glint of silver, he noticed a small vial floating between the boat and his private quay. On an impulse, he scooped it out of the glutinous water.\n\nIt was a cylinder of tinted glass cased in exquisitely wrought silver, and Hashim saw within its walls a silver pendant bearing a single strand of human hair.\n\nClosing his fist around this unique discovery, he muttered to the boatman that he'd changed his plans, and hurried to his sanctum, where, behind closed doors, he feasted his eyes on his find.\n\n\u2014\n\nThere can be no doubt that Hashim the moneylender knew from the first that he was in possession of the famous relic of the Prophet Muhammad, that revered hair whose theft from its shrine at Hazratbal mosque the previous morning had created an unprecedented hue and cry in the valley.\n\nThe thieves \u2013 no doubt alarmed by the pandemonium, by the procession through the streets of endless ululating crocodiles of lamentation, by the riots, the political ramifications and by the massive police search which was commanded and carried out by men whose entire careers now hung upon the finding of this lost hair \u2013 had evidently panicked and hurled the vial into the gelatine bosom of the lake.\n\nHaving found it by a stroke of great good fortune, Hashim's duty as a citizen was clear: the hair must be restored to its shrine, and the state to equanimity and peace.\n\n\u2014\n\nBut the moneylender had a different notion.\n\nAll around him in his study was the evidence of his collector's mania. There were enormous glass cases full of impaled butterflies from Gulmarg, three dozen scale models in various metals of the legendary cannon Zamzama, innumerable swords, a Naga spear, ninety-four terracotta camels of the sort sold on railway station platforms, many samovars, and a whole zoology of tiny sandalwood animals, which had originally been carved to serve as children's bathtime toys.\n\n'And after all,' Hashim told himself, 'the Prophet would have disapproved mightily of this relic-worship. He abhorred the idea of being deified! So, by keeping this hair from its distracted devotees, I perform \u2013 do I not? \u2013 a finer service than I would by returning it! Naturally, I don't want it for its religious value...I'm a man of the world, of this world. I see it purely as a secular object of great rarity and blinding beauty. In short, it's the silver vial I desire, more than the hair.\n\n'They say there are American millionaires who purchase stolen art masterpieces and hide them away \u2013 they would know how I feel. I must, must have it!'\n\n\u2014\n\nEvery collector must share his treasures with one other human being, and Hashim summoned \u2013 and told \u2013 his only son Atta, who was deeply perturbed but, having been sworn to secrecy, only spilled the beans when the troubles became too terrible to bear.\n\nThe youth excused himself and left his father alone in the crowded solitude of his collections. Hashim was sitting erect in a hard, straight-backed chair, gazing intently at the beautiful vial.\n\n\u2014\n\nIt was well known that the moneylender never ate lunch, so it was not until evening that a servant entered the sanctum to summon his master to the dining-table. He found Hashim as Atta had left him. The same, and not the same \u2013 for now the moneylender looked swollen, distended. His eyes bulged even more than they always had, they were red-rimmed, and his knuckles were white.\n\nHe seemed to be on the point of bursting! As though, under the influence of the misappropriated relic, he had filled up with some spectral fluid which might at any moment ooze uncontrollably from his every bodily opening.\n\nHe had to be helped to the table, and then the explosion did indeed take place.\n\n\u2014\n\nSeemingly careless of the effect of his words on the carefully constructed and fragile constitution of the family's life, Hashim began to gush, to spume long streams of awful truths. In horrified silence, his children heard their father turn upon his wife, and reveal to her that for many years their marriage had been the worst of his afflictions. 'An end to politeness!' he thundered. 'An end to hypocrisy!'\n\nNext, and in the same spirit, he revealed to his family the existence of a mistress; he informed them also of his regular visits to paid women. He told his wife that, far from being the principal beneficiary of his will, she would receive no more than the eighth portion which was her due under Islamic law. Then he turned upon his children, screaming at Atta for his lack of academic ability \u2013 'A dope! I have been cursed with a dope!' \u2013 and accusing his daughter of lasciviousness, because she went around the city barefaced, which was unseemly for any good Muslim girl to do. She should, he commanded, enter purdah forthwith.\n\nHashim left the table without having eaten and fell into the deep sleep of a man who has got many things off his chest, leaving his children stunned, in tears, and the dinner going cold on the sideboard under the gaze of an anticipatory bearer.\n\n\u2014\n\nAt five o'clock the next morning the moneylender forced his family to rise, wash and say their prayers. From then on, he began to pray five times daily for the first time in his life, and his wife and children were obliged to do likewise.\n\nBefore breakfast, Huma saw the servants, under her father's direction, constructing a great heap of books in the garden and setting fire to it. The only volume left untouched was the Qur'an, which Hashim wrapped in a silken cloth and placed on a table in the hall. He ordered each member of his family to read passages from this book for at least two hours per day. Visits to the cinema were forbidden. And if Atta invited male friends to the house, Huma was to retire to her room.\n\n\u2014\n\nBy now, the family had entered a state of shock and dismay; but there was worse to come.\n\nThat afternoon, a trembling debtor arrived at the house to confess his inability to pay the latest instalment of interest owed, and made the mistake of reminding Hashim, in somewhat blustering fashion, of the Qur'an's strictures against usury. The moneylender flew into a rage and attacked the fellow with one of his large collection of bullwhips.\n\nBy mischance, later the same day a second defaulter came to plead for time, and was seen fleeing Hashim's study with a great gash in his arm, because Huma's father had called him a thief of other men's money and had tried to cut off the wretch's right hand with one of the thirty-eight kukri knives hanging on the study walls.\n\n\u2014\n\nThese breaches of the family's unwritten laws of decorum alarmed Atta and Huma, and when, that evening, their mother attempted to calm Hashim down, he struck her on the face with an open hand. Atta leapt to his mother's defence and he, too, was sent flying.\n\n'From now on,' Hashim bellowed, 'there's going to be some discipline around here!'\n\n\u2014\n\nThe moneylender's wife began a fit of hysterics which continued throughout that night and the following day, and which so provoked her husband that he threatened her with divorce, at which she fled to her room, locked the door and subsided into a raga of sniffling. Huma now lost her composure, challenged her father openly, and announced (with that same independence of spirit which he had encouraged in her) that she would wear no cloth over her face; apart from anything else, it was bad for the eyes.\n\nOn hearing this, her father disowned her on the spot and gave her one week in which to pack her bags and go.\n\n\u2014\n\nBy the fourth day, the fear in the air of the house had become so thick that it was difficult to walk around. Atta told his shock-numbed sister: 'We are descending to gutter-level \u2013 but I know what must be done.'\n\nThat afternoon, Hashim left home accompanied by two hired thugs to extract the unpaid dues from his two insolvent clients. Atta went immediately to his father's study. Being the son and heir, he possessed his own key to the moneylender's safe. This he now used, and removing the little vial from its hiding-place, he slipped it into his trouser pocket and re-locked the safe door.\n\nNow he told Huma the secret of what his father had fished out of Lake Dal, and exclaimed: 'Maybe I'm crazy \u2013 maybe the awful things that are happening have made me cracked \u2013 but I am convinced there will be no peace in our house until this hair is out of it.'\n\nHis sister at once agreed that the hair must be returned, and Atta set off in a hired shikara to Hazratbal mosque. Only when the boat had delivered him into the throng of the distraught faithful which was swirling around the desecrated shrine did Atta discover that the relic was no longer in his pocket. There was only a hole, which his mother, usually so attentive to household matters, must have overlooked under the stress of recent events.\n\nAtta's initial surge of chagrin was quickly replaced by a feeling of profound relief.\n\n'Suppose', he imagined, 'that I had already announced to the mullahs that the hair was on my person! They would never have believed me now \u2013 and this mob would have lynched me! At any rate, it has gone, and that's a load off my mind.' Feeling more contented than he had for days, the young man returned home.\n\n\u2014\n\nHere he found his sister bruised and weeping in the hall; upstairs, in her bedroom, his mother wailed like a brand-new widow. He begged Huma to tell him what had happened, and when she replied that their father, returning from his brutal business trip, had once again noticed a glint of silver between boat and quay, had once again scooped up the errant relic, and was consequently in a rage to end all rages, having beaten the truth out of her \u2013 then Atta buried his face in his hands and sobbed out his opinion, which was that the hair was persecuting them, and had come back to finish the job.\n\n\u2014\n\nIt was Huma's turn to think of a way out of their troubles.\n\nWhile her arms turned black and blue and great stains spread across her forehead, she hugged her brother and whispered to him that she was determined to get rid of the hair _at all costs_ \u2013 she repeated this last phrase several times.\n\n'The hair', she then declared, 'was stolen from the mosque; so it can be stolen from this house. But it must be a genuine robbery, carried out by a bona-fide thief, not by one of us who are under the hair's thrall \u2013 by a thief so desperate that he fears neither capture nor curses.'\n\nUnfortunately, she added, the theft would be ten times harder to pull off now that their father, knowing that there had already been one attempt on the relic, was certainly on his guard.\n\n\u2014\n\n'Can you do it?'\n\nHuma, in a room lit by candle and storm-lantern, ended her account with one further question: 'What assurances can you give that the job holds no terrors for you still?'\n\nThe criminal, spitting, stated that he was not in the habit of providing references, as a cook might, or a gardener, but he was not alarmed so easily, certainly not by any children's djinni of a curse. Huma had to be content with this boast, and proceeded to describe the details of the proposed burglary.\n\n'Since my brother's failure to return the hair to the mosque, my father has taken to sleeping with his precious treasure under his pillow. However, he sleeps alone, and very energetically; only enter his room without waking him, and he will certainly have tossed and turned quite enough to make the theft a simple matter. When you have the vial, come to my room,' and here she handed Sheikh S\u00edn a plan of her home, 'and I will hand over all the jewellery owned by my mother and myself. You will find...it is worth...that is, you will be able to get a fortune for it...'\n\nIt was evident that her self-control was weakening and that she was on the point of physical collapse.\n\n'Tonight,' she burst out finally. 'You must come tonight!'\n\n\u2014\n\nNo sooner had she left the room than the old criminal's body was convulsed by a fit of coughing: he spat blood into an old vanaspati can. The great Sheikh, the 'Thief of Thieves', had become a sick man, and every day the time drew nearer when some young pretender to his power would stick a dagger in his stomach. A lifelong addiction to gambling had left him almost as poor as he had been when, decades ago, he had started out in this line of work as a mere pickpocket's apprentice; so in the extraordinary commission he had accepted from the moneylender's daughter he saw his opportunity of amassing enough wealth at a stroke to leave the valley for ever, and acquire the luxury of a respectable death which would leave his stomach intact.\n\n\u2014\n\nAs for the Prophet's hair, well, neither he nor his blind wife had ever had much to say for prophets \u2013 that was one thing they had in common with the moneylender's thunderstruck clan.\n\nIt would not do, however, to reveal the nature of this, his last crime, to his four sons. To his consternation, they had all grown up to be hopelessly devout men, who even spoke of making the pilgrimage to Mecca some day. 'Absurd!' their father would laugh at them. 'Just tell me how you will go?' For, with a parent's absolutist love, he had made sure they were all provided with a lifelong source of high income by crippling them at birth, so that, as they dragged themselves around the city, they earned excellent money in the begging business.\n\nThe children, then, could look after themselves.\n\nHe and his wife would be off soon with the jewel-boxes of the moneylender's women. It was a timely chance indeed that had brought the beautiful bruised girl into his corner of the town.\n\n\u2014\n\nThat night, the large house on the shore of the lake lay blindly waiting, with silence lapping at its walls. A burglar's night: clouds in the sky and mists on the winter water. Hashim the moneylender was asleep, the only member of his family to whom sleep had come that night. In another room, his son Atta lay deep in the coils of his coma with a blood-clot forming on his brain, watched over by a mother who had let down her long greying hair to show her grief, a mother who placed warm compresses on his head with gestures redolent of impotence. In a third bedroom Huma waited, fully dressed, amidst the jewel-heavy caskets of her desperation.\n\nAt last a bulbul sang softly from the garden below her window and, creeping downstairs, she opened a door to the bird, on whose face there was a scar in the shape of the Nastaliq letter _s\u00edn_.\n\n\u2014\n\nNoiselessly, the bird flew up the stairs behind her. At the head of the staircase they parted, moving in opposite directions along the corridor of their conspiracy without a glance at one another.\n\nEntering the moneylender's room with professional ease, the burglar, S\u00edn, discovered that Huma's predictions had been wholly accurate. Hashim lay sprawled diagonally across his bed, the pillow untenanted by his head, the prize easily accessible. Step by padded step, S\u00edn moved towards the goal.\n\nIt was at this point that, in the bedroom next door, young Atta sat bolt upright in his bed, giving his mother a great fright, and without any warning \u2013 prompted by goodness knows what pressure of the blood-clot upon his brain \u2013 began screaming at the top of his voice:\n\n_'Thief! Thief! Thief!'_\n\n\u2014\n\nIt seems probable that his poor mind had been dwelling, in these last moments, upon his own father; but it is impossible to be certain, because having uttered these three emphatic words the young man fell back upon his pillow and died.\n\nAt once his mother set up a screeching and a wailing and a keening and a howling so earsplittingly intense that they completed the work which Atta's cry had begun \u2013 that is, her laments penetrated the walls of her husband's bedroom and brought Hashim wide awake.\n\n\u2014\n\nSheikh S\u00edn was just deciding whether to dive beneath the bed or brain the moneylender good and proper when Hashim grabbed the tiger-striped swordstick which always stood propped up in a corner beside his bed, and rushed from the room without so much as noticing the burglar who stood on the opposite side of the bed in the darkness. S\u00edn stooped quickly and removed the vial containing the Prophet's hair from its hiding-place.\n\nMeanwhile Hashim had erupted into the corridor, having unsheathed the sword inside his cane. In his right hand he held the weapon and was waving it about dementedly. His left hand was shaking the stick. A shadow came rushing towards him through the midnight darkness of the passageway and, in his somnolent anger, the moneylender thrust his sword fatally through its heart. Turning up the light, he found that he had murdered his daughter, and under the dire influence of this accident he was so overwhelmed by remorse that he turned the sword upon himself, fell upon it and so extinguished his life. His wife, the sole surviving member of the family, was driven mad by the general carnage and had to be committed to an asylum for the insane by her brother, the city's Deputy Commissioner of Police.\n\n\u2014\n\nSheikh S\u00edn had quickly understood that the plan had gone awry.\n\nAbandoning the dream of the jewel-boxes when he was but a few yards from its fulfilment, he climbed out of Hashim's window and made his escape during the appalling events described above. Reaching home before dawn, he woke his wife and confessed his failure. It would be necessary, he whispered, for him to vanish for a while. Her blind eyes never opened until he had gone.\n\n\u2014\n\nThe noise in the Hashim household had roused their servants and even managed to awaken the night-watchman, who had been fast asleep as usual on his charpoy by the street-gate. They alerted the police, and the Deputy Commissioner himself was informed. When he heard of Huma's death, the mournful officer opened and read the sealed letter which his niece had given him, and instantly led a large detachment of armed men into the light-repellent gullies of the most wretched and disreputable part of the city.\n\nThe tongue of a malicious cat-burglar named Huma's fellow-conspirator; the finger of an ambitious bank-robber pointed at the house in which he lay concealed; and although S\u00edn managed to crawl through a hatch in the attic and attempt a roof-top escape, a bullet from the Deputy Commissioner's own rifle penetrated his stomach and brought him crashing messily to the ground at the feet of Huma's enraged uncle.\n\nFrom the dead thief's pocket rolled a vial of tinted glass, cased in filigree silver.\n\n\u2014\n\nThe recovery of the Prophet's hair was announced at once on All-India Radio. One month later, the valley's holiest men assembled at the Hazratbal mosque and formally authenticated the relic. It sits to this day in a closely guarded vault by the shores of the loveliest of lakes in the heart of the valley which was once closer than any other place on earth to Paradise.\n\n\u2014\n\nBut before our story can properly be concluded, it is necessary to record that when the four sons of the dead Sheikh awoke on the morning of his death, having unwittingly spent a few minutes under the same roof as the famous hair, they found that a miracle had occurred, that they were all sound of limb and strong of wind, as whole as they might have been if their father had not thought to smash their legs in the first hours of their lives. They were, all four of them, very properly furious, because the miracle had reduced their earning powers by 75 per cent, at the most conservative estimate; so they were ruined men.\n\nOnly the Sheikh's widow had some reason for feeling grateful, because although her husband was dead she had regained her sight, so that it was possible for her to spend her last days gazing once more upon the beauties of the valley of Kashmir.\nBOOKS BY\n\nSALMAN RUSHDIE\n\nEAST, WEST\n\nFrom the author of _The Satanic Verses_ comes nine stories that reveal the oceanic distances and the unexpected intimacies between East and West. Daring, extravagant, comical and humane, this book renews Rushdie's stature as a storyteller who can enthrall and instruct us with the same sentence.\n\nShort Stories\n\nTHE MOOR'S LAST SIGH\n\nBooker Prize\u2013winning author Salman Rushdie combines a ferociously witty family saga with a surreally imagined and sometimes blasphemous chronicle of modern India and flavors the mixture with peppery soliloquies on art, ethnicity, religious fanaticism, and the terrifying power of love. Moraes \"Moor\" Zogoiby, the last surviving scion of a dynasty of Cochinese spice merchants and crime lords, is also a compulsive storyteller and an exile. As he travels a route that takes him from India to Spain, he leaves behind a tale of mad passions and volcanic family hatreds, of titanic matriarchs and their mesmerized offspring, of premature deaths and curses that strike beyond the grave.\n\nFiction\n\nVINTAGE INTERNATIONAL\n\nAvailable wherever books are sold.\n\nwww.\u200bvintagebooks.\u200bcom\n\n# **HIGHLIGHTS FROM VINTAGE SHORTS**\n\n\u2666 **We Should All Be Feminists** by Chimamanda Ngozi Adichie\n\n\u2666 **Moral Disorder: A Story** by Margaret Atwood\n\n\u2666 **The Outing** by James Baldwin\n\n\u2666 **Dallas: November 22, 1963** by Robert A. Caro\n\n\u2666 **Where Climate is Heading** by Climate Central\n\n\u2666 **Fifteen Poems** by Leonard Cohen\n\n\u2666 **Progress** by Katharine Graham\n\n\u2666 **How War Begins** by John Keegan\n\n\u2666 **The Vision** by Jonathan Lethem\n\n\u2666 **Provence in Ten Easy Lessons** by Peter Mayle\n\n\u2666 **Vissi d'Arte** by Lorrie Moore\n\n\u2666 **American Hunger** by Eli Saslow\n\n\u2666 **Fatty O'Leary's Dinner Party** by Alexander McCall Smith\n\n\u2666 **A Market Tale** by Martin Walker\n\n**Find out more on the Vintage ShortsFacebook page**\n\n# _What's next on \nyour reading list?_\n\n[Discover your next \ngreat read!](http:\/\/links.penguinrandomhouse.com\/type\/prhebooklanding\/isbn\/9781101973691\/display\/1)\n\n* * *\n\nGet personalized book picks and up-to-date news about this author.\n\nSign up now.\n\n## Contents\n\n 1. Cover\n 2. About the Author\n 3. Other Titles\n 4. Title Page\n 5. Copyright\n 6. Contents\n 7. The Prophet's Hair\n\n 1. \n 2. \n 3. \n 4. \n 5. \n 6. \n 7. \n 8. \n 9. \n 10. \n 11. \n 12. \n 13. \n 14. \n 15. \n 16. \n 17. \n 18. \n 19. \n 20. \n 21. \n 22. \n 23. \n 24. \n 25. \n 26. \n 27. \n 28. \n 29. \n 30. \n 31. \n 32. \n 33. \n 34. \n 35. \n 36. \n 37. \n 38. \n 39. \n 40. \n 41. \n 42. \n 43. \n 44. \n 45. \n 46. \n 47. \n 48. \n 49. \n 50. \n 51. \n 52. \n 53. \n 54. \n 55. \n 56. \n 57. \n 58.\n\n 1. Cover\n 2. Cover\n 3. Title Page\n 4. Table of Contents\n 5. Start\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nMicrosoft\u00ae SQL Server\u00ae 2012\n\nMASTER DATA SERVICES\n\nSecond Edition\n\nAbout the Author\n\n**Tyler Graham** is Director of Industry Solutions for the Profisee Group. He moved to Profisee after completing the latest release of Master Data Services as a Senior Program Manager for Microsoft. He has spent over ten years in the data management space and is a frequent speaker at conferences. When he is not working on learning new industry, he is skiing, swimming, or hiking with his family.\n\nAbout the Technical Editor\n\n**Lynn Gasch** has worked in software development and testing for more than ten years. After completing degrees in computer science and mathematics at Virginia Tech, she hired on at a small Seattle office focused on artificial intelligence research, and tackled projects in diverse areas such as computer-based education, network monitoring and security, data mining, and information retrieval. She joined Microsoft four years ago and is currently a member of the SQL Server Master Data Services test team. When not working, Lynn and her family enjoy the recreational opportunities of the Pacific Northwest, and she will endeavor to stay off crutches next ski season. Lynn's review of this book was based on her own personal opinions, and not in her capacity as a Microsoft employee.\nMicrosoft\u00ae SQL Server\u00ae 2012\n\nMASTER DATA SERVICES\n\nSecond Edition\n\n**Tyler Graham**\n\nNew York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto\n\n| \n---|---\n\nCopyright \u00a9 2012 by The McGraw-Hill Companies. All rights reserved. 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This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise.\nDedicated to Tera, who made all of this possible.\nContents at a Glance\n\n Chapter 1 **Introduction to Master Data Services**\n\n Chapter 2 **Installation and Configuration**\n\n Chapter 3 **Starting an MDS Project**\n\n Chapter 4 **Creating Your Model**\n\n Chapter 5 **Integrating Master Data Services with Other Systems**\n\n Chapter 6 **Working with Hierarchies and Collections**\n\n Chapter 7 **Working with Master Data**\n\n Chapter 8 **Using Business Rules**\n\n Chapter 9 **Creating Versions of Data**\n\nChapter 10 **SQL Server 2012 MDS Add-in for Excel**\n\nChapter 11 **Implementing Security**\n\nChapter 12 **Publishing Data to External Systems**\n\nChapter 13 **Extending MDS with Web Services**\n\nChapter 14 **Advanced Modeling**\n\n**Index**\nContents\n\nForeword\n\nAcknowledgments\n\nIntroduction\n\nChapter 1 **Introduction to Master Data Services**\n\nWhat Is Master Data Management?\n\nHistory of Master Data Management\n\nOverview of Other Master Data Management Solutions\n\nERP Systems as MDM Solutions\n\nSpecialized Master Data Management Solutions\n\nCatering to the Titans\n\nMicrosoft's Solution to Master Data Management\n\nNew Features of MDS in SQL Server 2012\n\nWhat Master Data Services Still Delivers\n\nThe Value Proposition\n\nAbout the Sample Company\n\nSummary\n\nChapter 2 **Installation and Configuration**\n\nDetermining the Initial Scope of Your Project\n\nAbout the Main Street Clothing Company Implementation.\n\nPreparing the Web Server\n\nWindows 7 Web Application Requirements\n\nWindows Server 2008 R2 Web Application Requirements\n\nInstalling Master Data Services\n\nAbout MDS Configuration Manager\n\nProcedure: How to Install Master Data Services\n\nCreating an MDS Database\n\nProcedure: How to Create an MDS Database\n\nAbout System Settings\n\nCreating the Master Data Manager Web Application\n\nProcedure: How to Create a Master Data Manager Website (Optional)\n\nProcedure: How to Create a Master Data Manager Web Application\n\nProcedure: How to Associate the MDS Database with the Web Application\n\nSecuring Your Web Application\n\nGetting the Latest Version of MDS\n\nUpgrading 2008 R2 MDS\n\nRepairing the MDS Database\n\nSummary\n\nChapter 3 **Starting an MDS Project**\n\nHow Do I Know Master Data When I See It?\n\nBusiness Process Models\n\nSystem Roles\n\nSystem of Entry\n\nSystem of Record\n\nSubscribing System\n\nMapping the Data\n\nDetermining What to Do with Duplicate Records\n\nDetermining Which Attributes to Manage\n\nThe Main Street Clothing Company Example\n\nFinding a Data Steward\n\nLeveraging the Business Process Models\n\nFilling in the Gaps\n\nDetermining System Types\n\nCleansing Data for Initial Import into MDS\n\nLeveraging the MDS Excel Add-In to Manage Your Project\n\nSummary\n\nChapter 4 **Creating Your Model**\n\nMDS Modeling Concepts\n\nBuilding a Model\n\nDeploying a Model\n\nProcedure: How to Deploy a Model Using ModelDeploy.exe\n\nAbout Models\n\nAbout Main Street Clothing Company's Models\n\nProcedure: How to Create a Model\n\nAbout Entities\n\nAbout Main Street Clothing Company's Entities\n\nProcedure: How to Create an Entity\n\nAbout Members\n\nAbout Main Street Clothing Company's Members\n\nProcedure: How to Add a Member\n\nAbout Attributes\n\nAbout Main Street Clothing Company's Attributes\n\nProcedure: How to Create an Attribute\n\nProcedure: How to Create a Domain-Based Attribute\n\nAbout Attribute Groups\n\nAbout Main Street Clothing Company's Attribute Groups\n\nProcedure: How to Create an Attribute Group\n\nProcedure: How to Reorder Attribute Groups\n\nUsing Web Services\n\nCreating a Model\n\nCreating an Entity\n\nCreating an Attribute\n\nSummary\n\nChapter 5 **Integrating Master Data Services with Other Systems**\n\nStaging Architecture\n\nBenefits of Entity-Based Staging\n\nBatch Tags\n\nSecurity in Staging\n\nSystem Settings\n\nAbout Main Street Clothing Company's Staging Process\n\nStandard Leaf and Parent Staging Fields\n\nStaging Import Types\n\nMember Staging Examples\n\nUsing Staging to Deactivate Members\n\nLoading Relationships\n\nRelationship Table Fields\n\nRelationship Staging Examples\n\nImporting Data into the Staging Tables\n\nProcedure: How to Import Data into SQL Server by Using SSIS\n\nInitiating the Staging Process\n\nProcedure: How to Use Master Data Manager to Initiate the Staging Process\n\nProcedure: How to Use Stored Procedures to Initiate the Staging Process\n\nThe End of Staging Sweep\n\nUsing Web Services to Stage Data\n\nErrors That Occur During Staging\n\nProcedure: How to View Staging Errors in Management Studio\n\nStaging Errors View\n\nViewing the Staging Batch Table\n\nProcedure: How to Clear the Staging Queue\n\nSummary\n\nChapter 6 **Working with Hierarchies and Collections**\n\nRagged vs Level-Based Hierarchies\n\nDerived Hierarchies\n\nExplicit Hierarchies\n\nDerived vs Explicit: Which Hierarchy Is Best?\n\nCreating Derived Hierarchies\n\nProcedure: How to Create a Derived Hierarchy\n\nProcedure: How to Create a Three-Level Derived Hierarchy\n\nHiding Levels\n\nSystem Settings for Hierarchies\n\nCreating Explicit Hierarchies\n\nNon-mandatory Explicit Hierarchies\n\nMandatory Explicit Hierarchies\n\nProcedure: How to Create an Explicit Hierarchy\n\nRecursive Hierarchies\n\nProcedure: How to Create a Recursive Hierarchy\n\nDerived Hierarchies with Explicit Caps\n\nProcedure: How to Create a Derived Hierarchy with Explicit Cap\n\nCollections\n\nWeighting of Collection Members\n\nProcedure: How to Create a Collection\n\nCreating Hierarchies by Using Web Services\n\nCreating a Derived Hierarchy with \u00bathe MetadataCreate Operation\n\nSummary\n\nChapter 7 **Working with Master Data**\n\nChanges in SQL Server 2012\n\nViewing the Base Entity\n\nFinding Data by Filtering\n\nOperators\n\nUsing Matching\n\nSorting and Pagination\n\nSorting\n\nPagination and Other Settings\n\nNew Explorer Format\n\nEdit One Attribute for One Member\n\nEdit One Attribute for Multiple Members\n\nExporting Members\n\nWorking with Data in Hierarchies\n\nMoving Members by Dragging and Dropping\n\nMoving Members by Using the Clipboard\n\nMoving Members in a Derived Hierarchy by Updating Attribute Values\n\nTransactions\n\nReviewing Transactions\n\nReversing Transactions\n\nAnnotations\n\nAnnotating Transactions on Update\n\nNavigating Related Entities\n\nNavigating Attributes\n\nNavigating Many-to-Many Relationships\n\nMetadata\n\nUsing Web Services to Work with Members\n\nRetrieving Members\n\nCreating and Updating Members\n\nSummary\n\nChapter 8 **Using Business Rules**\n\nBusiness Rules Overview\n\nBusiness Rule Structure\n\nBusiness Rule Workflow\n\nCreating a Business Rule\n\nEditing a Business Rule and Configuring a Rule's Expression\n\nProcedure: How to Require Attribute Values\n\nDeleting a Condition or Action\n\nMore Expression Examples\n\nCreating Your Own Expressions\n\nPublishing Business Rules\n\nProcedure: How to Publish a Business Rule\n\nBusiness Rule Statuses\n\nApplying Business Rules\n\nProcedure: How to Apply Business Rules in Explorer\n\nResolving Validation Issues in Explorer\n\nProcedure: How to Validate a Version\n\nOther Business Rule Tasks\n\nProcedure: How to Change the Rule Name and Description\n\nProcedure: How to Set the Rule's Priority Order\n\nProcedure: How to Exclude Business Rules\n\nConfiguring E-Mail Notifications\n\nProcedure: How to Configure E-Mail Notifications\n\nProcedure: How to Configure a Business Rule to Send E-Mail\n\nCreating and Triggering Workflows\n\nCreating a Workflow in MDS\n\nViewing Change Tracking Group Reports\n\nTriggering an External Workflow\n\nUsing Web Services to Manage Business Rules\n\nGetting a List of Rules\n\nCreating a Business Rule using Web Services\n\nSummary\n\nChapter 9 **Creating Versions of Data**\n\nVersions Overview\n\nChanging the Structure of Your Model\n\nCommitting Versions\n\nVersioning for Main Street Clothing Company\n\nUpdating Your Version Name and Description\n\nProcedure: How to Change the Version Name and Description\n\nVersion Statuses\n\nProcedure: How to Lock a Version\n\nValidating a Version\n\nProcedure: How to Validate a Version\n\nProcedure: How to Resolve Validation Issues\n\nReviewing and Reversing Transactions\n\nCommitting and Copying a Version\n\nProcedure: How to Commit a Version\n\nProcedure: How to Reopen a Committed Version\n\nProcedure: How to Copy a Version\n\nVersion Flags\n\nProcedure: How to Create a Version Flag and Assign It to a Version\n\nViewing a Version's Ancestry\n\nProcedure: How to Delete a Version\n\nUsing Web Services to Work with Versions\n\nReturning a List of Versions\n\nChanging the Version Status\n\nSummary\n\nChapter 10 **SQL Server 2012 MDS Add-in for Excel**\n\nInstalling the Add-in\n\nProcedure: How to Install the Add-in for Excel\n\nWorking with MDS Data in Excel\n\nProcedure: How to Connect to the MDS Server and Load an Entity\n\nPublishing Changes\n\nRefreshing Data from the Server\n\nReviewing Transactions\n\nMaking Deletions in Managed Sheets\n\nCombining Data\n\nApplying Rules\n\nFiltering Entity Results Members\n\nProcedure: How to Load a Filtered Entity\n\nSaving and Sharing Queries\n\nProcedure: How to Save a Filter Query and Share It Through Outlook\n\nCreating and Modifying Entities Rapidly\n\nUsing the Excel Add-In at the Main Street Clothing Company\n\nProcedure: How to Create an MDS Entity from Excel Data\n\nMaking Changes to an Existing Entity\n\nProcedure: How to Modify an Existing Entity's Structure\n\nConfiguring Add-in for Excel Settings\n\nConfiguring Data Settings\n\nConfiguring Behavior Settings\n\nLeveraging Data Quality Matching in Excel\n\nSummary\n\nChapter 11 **Implementing Security**\n\nSecurity Overview\n\nSecurity Changes in SQL Server 2012\n\nUsers and Groups\n\nProcedure: How to Add a Group\n\nAdministrators\n\nProcedure: How to Change the Administrator Account\n\nTesting Permissions\n\nFunctional Area Permissions\n\nProcedure: How to Assign Functional Area Permissions\n\nModel Object Permissions\n\nQuick Facts About Model Object Permissions\n\nBest Practice for Model Object Permissions\n\nProcedure: How to Assign Model Object Permissions\n\nProcedure: How to Delete Model Object Permissions\n\nAccess to Entities\n\nAccess to Leaf Member Attributes\n\nAccess to Individual Attributes\n\nAccess to an Explicit Hierarchy\n\nAccess to Collections\n\nAccess to Other Model Objects\n\nDeny Permissions\n\nNavigational Access\n\nDeleting Permissions\n\nHierarchy Member Permissions\n\nQuick Facts About Hierarchy Member Permissions\n\nProcedure: How to Assign Hierarchy Member Permissions\n\nDetermining Which Permissions Apply\n\nCombining Permissions\n\nCombining Users and Groups\n\nSetting Security by Using Web Services\n\nRetrieving Users and Groups\n\nRetrieving User or Group Permissions\n\nAdding Users and Groups\n\nAssigning Permissions\n\nSummary\n\nChapter 12 **Publishing Data to External Systems**\n\nExporting Data to Subscribing Systems\n\nSubscription View Formats\n\nCommon View Architecture\n\nProcedure: How to Create a Subscription View8\n\nViewing a Subscription View in SQL Server\n\nSubscription Views and Model Deployment\n\nCreating a Subscription View with Web Services\n\nCreating Entity Views\n\nCreating Derived Hierarchy Views\n\nDeleting Views\n\nHow Main Street Clothing Company Uses Subscription Views\n\nProcedure: How to Export Subscription View Data\n\nSummary\n\nChapter 13 **Extending MDS with Web Services**\n\nExposing the Web Service\n\nExposing the WSDL\n\nProcedure: How to Enable the WSDL\n\nCreating an MDS Project in Visual Studio\n\nProcedure: How to Create an MDS Web Service Application\n\nCreating an Abstraction Layer\n\nMDS Web Service Operations\n\nMembers\n\nBulk Operations\n\nValidation\n\nTransactions\n\nAnnotations\n\nSubscription Views\n\nMetadata Structures\n\nSecurity\n\nBusiness Rules\n\nSystem Settings\n\nUser Preferences\n\nMiscellaneous Operations\n\nEntity-Based Staging Operations\n\nData Quality Operations\n\nThe Most Frequently Used Operations\n\nSearching for Members\n\nUnderstanding the EntityMembers Object\n\nSummarizing Member Data\n\nWorking with Metadata\n\nValidating Records in MD\n\nManaging Transactions\n\nHandling Errors in the Web Service\n\nSummary\n\nChapter 14 **Advanced Modeling**\n\nCommon Modeling Mistakes\n\nBuilding Outside Limitations into MDS\n\nTrouble Identifying Common Attributes\n\nEngineering Dead Scenarios\n\nRapid Model Development\n\nManaging Slowly Changing Dimensions\n\nOverwriting Rows (Type 1)\n\nAdding Rows (Type 2)\n\nAdding Columns (Type 3)\n\nUsing the Power of MDS to Simplify the Problem\n\nManaging Dimension for Main Street Clothing Company\n\nLimiting Unneeded Complexity\n\nDesigning for Exceptions\n\nDesigning for _N_\n\nSummary\n\n**Index**\nForeword\n\nIn the last few weeks I have received duplicate mailings from my bank. My doctor confused my records with another patient, also \"D. Farmer.\" A carrier delivered a ceiling fan, ordered online, with a remote control that does not match it; and the next day delivered a pair of shoes of the right brand, the right color, the right size, but the wrong model.\n\nI doubt that I am especially unlucky: we have all had similar experiences. The retailers, manufacturers, and service providers on the other side of these mishaps all have something in common\u2014problems with _master data_.\n\nThat these problems are so familiar is a sign that they are pervasive. And over the years we, in the world of data management, have tried to solve them in many ways. We created reference data sets, but often ended up with multiple reference sets covering the same field confusingly. We built data warehouses to drive reporting and analytics, and thought to use them as our \"single version of the truth.\" However, the data warehouse was populated and maintained on a different cycle, and for different purposes, than our line-of-business applications, and the data was not a suitable source for operational use.\n\nIn recent years, however, a body of practice has emerged that is aptly named \"master data management,\" or MDM. Practitioners don't stop at just compiling a reference data set, but implement architectures, data-driven processes, practices, and policies that oversee the entire lifecycle of data.\n\nTyler Graham has been at the forefront of this new practice. Tyler has seen the start of the master data management movement as a consultant solving complex issues for enterprise customers. He has also worked on software solutions, with the vendor Stratature, building an agile and effective MDM solution. More recently, Tyler has been at Microsoft building their Master Data Services platform, which brings master data capabilities to every user of the SQL Server database.\n\nSo, I don't think you could find a more able author to introduce you to Microsoft SQL Server's Master Data Services. This book covers everything you need to know: identifying systems that source or require master data; handling the data itself; establishing roles and practices for the \"data stewards\" and others who manage the processes; and naturally there is a wealth of practical advice on the Master Data Services platform itself.\n\nYou'll find this book to be an invaluable guide to a challenging subject area, and a fascinating technology.\n\n_\u2013Donald Farmer_\n\nProduct Advocate for QlikView, QlikTech\nAcknowledgments\n\nI would like to thank the entire MDS team for a great second release. Let me give a shout out to Val Lovicz, who provided great feedback on Chapter 14. I would like to thank my wife and kids for their patience while I was writing this, as many commitments suffered accordingly.\n\n_\u2014Tyler Graham_\nIntroduction\n\nThe release of SQL Server 2008 R2 Master Data Services was the beginning of a new, more accessible era for master data management (MDM); SQL Server 2012 Master Data Services is this vision realized. Because of price and complexity, most solutions on the market today are geared toward large corporations and require dedicated consulting teams to implement. MDS, a feature of SQL Server Enterprise Edition, was built to democratize this process, providing MDM capabilities to a larger audience for a fraction of the price. MDS is meant to be rapidly deployed to any size organization and can solve a variety of master data challenges. The MDS Add-in for Microsoft Excel provides a paradigm shift in master data management by bringing it to the Information Worker (IW) tool of choice.\n\nThis book is intended for anyone interested in the features of MDS. It is meant to be used as an introduction, a learning tool, a training guide, and a reference manual. Even if you have no experience with MDS, each chapter builds on the previous chapter, so by the end of the book, you should be able to complete a majority of the tasks needed for a working MDS implementation. If you are a consultant with prior experience in the MDM market, you should find value in the product overview and web service examples, which allow you to adapt your existing MDM practice to support MDS.\n\nMaster Data Services is composed of some simple-to-understand components that can be used in a variety of ways. In order to provide the clearest understanding for new users, most of the book shows examples of standard, straightforward implementations. Complex examples and edge cases have been omitted from the book to avoid confusing the message. These complex topics are addressed in various resources you can find on the Internet, including the MDS team blog, www.mdsmodeler.com, and www.mdsuser.com.\n\nWhat You'll Learn in this Book\n\nThe intent of this book is to get you up and running with MDS. By reading it, you should become familiar with MDS-specific terminology (models, entities, attributes) and learn how to use MDS to solve your business problems. Each chapter introduces you to basic concepts, shows you how to use the Master Data Manager web application to perform related tasks, and gives you examples to complete the related tasks programmatically by using the web service.\n\nWhile the book provides advice on how to create a solution specific to your organization, implementing any MDM solution requires buy-in from many people across the organization. Doing the work to create models that will work for you and getting agreement that the MDS system is truly the system of record is more than half the battle. The book hopes to give you the confidence to approach your organization's MDM issues with a solid knowledge of how to implement and manage an MDM solution by using SQL Server 2012 Master Data Services.\n\nChapter 1: Introduction to Master Data Services\n\nThis chapter introduces the field of master data management and discusses its history. It explores the variety of popular MDM solutions and compares them to the Microsoft Master Data Services solution.\n\nChapter 2: Installation and Configuration\n\nThis chapter helps you consider the scope of your MDS implementation, and then gives you the steps to install and configure MDS. This chapter has a list of the Windows features and roles you must install before you can configure the Master Data Manager web application. This chapter also provides information about updating a prior instance of Master Data Services.\n\nChapter 3: Starting an MDS Project\n\nThis chapter gives you more in-depth ideas about how MDS will fit into your organization. Before you begin working with the product, it is important to spend quality time determining your organization's master data management needs. This chapter also includes information about how to deal with duplicate records and data cleansing.\n\nChapter 4: Creating Your Model\n\nIn this chapter, you start building a model. This chapter explains how to deploy the sample models that are included when you install MDS. This chapter also provides a start-to-finish walkthrough for creating the primary model objects. At the end of this chapter, you will have created a complete product model and should be able to understand the relationships between the MDS model objects.\n\nChapter 5: Integrating Master Data Services with Other Systems\n\nThis chapter explains how to load master data into MDS so that users can begin to manage it. This chapter introduces the database staging tables and the format required for importing data into the tables. It presents sample data, as well as procedures for loading the data and initiating the staging process, which imports the data into the correct tables in the MDS database.\n\nChapter 6: Working with Hierarchies and Collections\n\nThis chapter introduces the different types of MDS hierarchies: explicit and derived. After a brief introduction, you are led through the creation of some sample hierarchies. This chapter also shows variations on these hierarchies, which address specific business cases. This chapter also introduces the concept of collections and shows you how to create one. Finally, code samples show you how to use the web service to work with hierarchies.\n\nChapter 7: Working with Master Data\n\nThis chapter is intended for users who need to work with master data on a daily basis. It details the new Explorer UI in SQL Server 2012 and new features for data exploration and update. This chapter shows you how to filter data to find what you need, how to edit your data, and how to work with data in hierarchies. In addition, you learn how to review and reverse the transactions that are recorded whenever a user changes data. You also learn how to annotate transactions and data, which can be useful for change tracking. Finally, this chapter includes samples for using the web service to retrieve and update data.\n\nChapter 8: Using Business Rules\n\nThis chapter shows you how to create and configure business rules, and then how to publish them and apply them to data. You are shown how to configure e-mail notifications, how to use business rules to create internal workflows, and how to use business rules to trigger external workflows. Finally, code samples show how to use web services to interact with business rules.\n\nChapter 9: Creating Versions of Data\n\nIn this chapter, you learn how to change a version name and description. You learn about saving versions of your data (rather than your model structure), and about what it means to lock and commit a version. This chapter explains how to validate a version against business rules, how to flag a version of your data for subscribing systems or users, and how to determine where a version of data came from. And like all chapters, web service examples are provided.\n\nChapter 10: SQL Server 2012 MDS Add-in for Excel\n\nThis chapter introduces the new SQL Server 2012 \"killer app\" for Master Data Services, the MDS Add-in for Microsoft Excel. This utility will change the way companies approach master data. This chapter explains how to leverage Excel to make batch updates to existing MDS data and how to create new entities from data stored in regular Excel spreadsheets.\n\nChapter 11: Implementing Security\n\nThis chapter covers how to configure the granular level of security provided by MDS. There have been a number of simplifications made to the MDS security model in the latest release. This chapter explains how to add users and groups, how to configure administrators to have more permissions than the average user, and how to test permissions to ensure you've configured them correctly. This chapter explains the difference between functional area permissions, model object permissions, and hierarchy member permissions. It also explains which permissions apply when overlapping assignments are made. Finally, web service examples show how to set security.\n\nChapter 12: Publishing Data to External Systems\n\nThis chapter explains the concept of subscription views, which are standard SQL views that you create from within MDS. This chapter shows how to use these SQL views to export data to systems outside of MDS.\n\nChapter 13: Extending MDS with Web Services\n\nThis chapter shows you how to enable the web service. It includes a summary of all operations and gives you a basic idea of the use of each. It also includes additional code samples that weren't provided in other chapters, including how to search for members, return member counts, validate data, get and reverse transactions, and handle errors.\n\nChapter 14: Advanced Modeling\n\nThis chapter discusses some advanced modeling techniques to help users get more out of their Master Data Services application. Working with the field, I have seen a number of common errors occur when people build their MDS models. This chapter is labeled advanced to indicate not all users may be able to fully utilize the information in this chapter.\n\nAbout the Sample Company\n\nThroughout this book we use a fictional company called \"Main Street Clothing Company.\" We envision this company to be a small clothing retailer, just large enough to have a corporate office and manage its own IT. We thought that using one single continuous example would be the most successful way to demonstrate MDS concepts to the uninitiated. MDS was built and priced to provide a solution that can truly solve MDM issues for companies of all sizes, and we wanted to provide a complete example for this new emerging market. No matter the size of your business, many if not all of the concepts explained using Main Street Clothing Company should be applicable to your situation.\n\nAbout the Sample Data\n\nIn many chapters of the book, we mention www.mdsuser.com. We created this site to give you sample data to use as you go through the chapters in the book. Our intention was to give you two common models\u2014Product and Finance\u2014that you could experiment with while reading the book, and then modify in any way you want. These models are not meant to be taken too seriously\u2014they are samples that we put together with the hope that it is easier to learn something when you can see a real-world example. MDS can be used for any domain you care to manage; it's not restricted to lists of customers or products, but these domains provide you with a frame of reference.\n\nYou should always feel free to deploy these models, update them to your heart's content, and then delete them and start over. In the writing of this book, we've created and deleted our model dozens of times, and the only recourse is that we lost track of what we were doing from time to time. Until you've seriously solidified your MDS solution, you should feel free to push MDS to its limits and seek help when you're stuck. There is a community of committed users on the MDS forums on MSDN; the forums are often a good place to start when you're troubleshooting.\n\nIn this book we also provide code samples that illustrate the more common procedures described in each chapter. Chapter 13 is devoted completely to how to programmatically interact with MDS. All of the code samples in the book are available electronically at www.MHProfessional.com\/computingdownload and at www.mdsmodeler.com. These samples are free and are meant to be used in support of the book. We truly hope you find the samples and this book useful and that you achieve success with your MDM implementation.\nChapter 1\n\nIntroduction to Master Data Services\n\nIn This Chapter\n\n What Is Master Data Management?\n\n History of Master Data Management\n\n Overview of Other Master Data Management Solutions\n\n Microsoft's Solution to Master Data Management\n\n About the Sample Company\n\nMaster Data Services (MDS) is Microsoft's solution for managing master data. In the past, master data management (MDM) has been a highly problematic area for database administrators. To help you understand better how MDS addresses the inherent challenges of MDM, this chapter begins by explaining what MDM encompasses and how it has been handled historically. It then introduces you to MDM solutions other than MDS and explains their shortcomings so that you can fully appreciate the features and functionality of MDS, which is introduced in the latter half of the chapter. The chapter wraps up by introducing you to the sample company we'll be using throughout the rest of the book and describing some of its business problems that a master data management solution is meant to solve.\n\nWhat Is Master Data Management?\n\nMaster data management (MDM) is the management of the nontransactional data within an organization. Throughout the book we will use the terms organization, business, and company interchangeably. The definition of \"master\" data varies by organization, but can be loosely defined as the nouns that describe all business processes. These nouns might be organization-specific data, like your organization's list of products or list of employees. Or the nouns might be common reference data provided by an external service provider or government agency, like address information or a Dunn & Bradstreet (D-U-N-S\u00ae) number. Although customer and product are the two most commonly managed domains, many businesses find value in managing additional domains, such as wells and fields at big oil companies and recipes at food manufacturers.\n\nNOTE\n\n_When we discuss MDM, we often use the word_ domain. _We use this overloaded word to mean a group of related business data that is an area of focus for an MDM solution\u2014for example, accounts or customers._\n\nIn most businesses, customers buy products or services. Because customer relationships are essential, each time a customer buys a product or service, the transaction is recorded. After the transaction is recorded and coded properly, the details of the transaction will never change. MDM is about managing the relationships between these static transactions, rather than the transactions themselves. For example, for a retail chain, each store has a certain group of products in its inventory and a database of registered customers. The list of products available to sell and the list of customer addresses are master data. That Sally bought four shirts on Friday is a transaction.\n\nThe more often this data is required for a transaction, the greater its importance to your organization. Central management of this data helps identify data discrepancies between multiple transactional systems and helps your organization run more efficiently.\n\nMaster data management is composed of the following:\n\n The business policies (who owns the data, where the data lives, and so forth)\n\n The processes (how the data is updated)\n\n The technological tools that facilitate these processes\n\nBy definition, all companies must utilize some form of MDM, although the term generally refers to the implementation of formalized processes and specifically designed tools.\n\nHistory of Master Data Management\n\nWhen companies began using computer applications to manage information, in order to perform tasks and generate results, all pertinent data had to be loaded into each application directly. Over time, companies adopted more applications and had to enter the same dataset into each of them. As a response to this, developers began to employ the technique of creating master files that stored key, reusable data for use within each application. These initial master files were the first MDM systems.\n\nOnce the master data was loaded into each application, new records would be added in the application or to the master list, the result of which was that the two sets of data were no longer in sync. Early data reconciliation efforts were time consuming and costly. Very few tools existed that could identify differences and manage duplicate records.\n\nEnterprise resource planning (ERP) systems were then created to help combine the data in these systems and to solve the constant need for integration between systems. ERP systems were meant to be single software suites that managed standard business processes within an organization. Each module in an ERP system could leverage the common master data tables required for the associated business process. Many organizations today consider the master data tables within their ERP systems to be the \"master\" data within their organization.\n\nAs more and more organizations adopted ERP systems, they realized that using these single systems in isolation was an unrealistic solution. They then adopted multiple ERP systems, and their master data problems re-emerged. Specialized systems designed to manage the most problematic domains became major implementations in larger organizations. Now there is a push in the industry to provide solutions that can solve MDM issues across a wide variety of domains.\n\nOverview of Other Master Data Management Solutions\n\nAll organizations must deal with their MDM issues. Whether managing domains in Excel spreadsheets or managing them with a specialized solution, a key role of the IT department in any size organization today is to provide processes for managing master data and integrating it across all systems. Various solutions other than MDS are available, but each has weaknesses that are addressed by MDS.\n\nAs explained in the following sections, ERP solutions have several drawbacks as MDM solutions; other specialized solutions can prove to be too specialized; and MDM solutions typically cater to only the biggest organizations. As you will read later in the chapter, in MDS, Microsoft hopes to provide an adaptable solution that can cater to organizations of all sizes.\n\nERP Systems as MDM Solutions\n\nWhile ERP systems are a significant consumer of master data, using these systems by themselves to manage master data has major drawbacks. Also, the initial vision that a single ERP system would handle all of the computing needs of an organization was shortsighted.\n\nERP Systems Are Not Specialized or Innovative\n\nWhile many ERP systems contain modules for most common business processes, a specific industry need or the need for more innovative software leads businesses to implement additional systems. Integration of these new systems with current IT infrastructure and business processes becomes a major implementation cost of any new system.\n\nERP Systems Do Not Play Well with Others\n\nWhen a company merges with or acquires another company, it inherits additional ERP systems. Since these systems are highly customized and not initially designed for easy integration, additional processes must be created to ensure that these systems are synchronized across the organization. The complexities of these software systems and the underlying data models make successful integration, or even effective synchronization, extremely difficult for large organizations. When Tyler moved across the country for work, it was impossible for his electronic bank accounts at a nationwide bank to move with him because each of the bank's regions was on its own system. Imagine, it was easier to move the contents of a house 3000 miles than to move a set of data and transactions that would fit on a thumb drive.\n\nERP Systems Do Not Manage Analytical Dimensions Effectively\n\nThe need for complex modeling and analysis of transactional data to determine statistics and trends has led companies to create advanced online analytical processing (OLAP) systems and associated data warehouses to go along with them. These systems have added new, complex data modeling needs that simply are not supported in legacy applications. Hierarchies and other consolidations are not natively managed by ERP systems.\n\nERP Systems Are Not Designed to Manage Attributes\n\nMany ERP system vendors realize that users will want to store additional attributes in their master tables. To accommodate this, they provide a few custom fields for customers to use as they see fit. These custom fields get overloaded with information without providing any validation of appropriate use.\n\nSpecialized Master Data Management Solutions\n\nAs you would suspect, the most common domains for management are those domains most troublesome to large organizations. Specialized systems have been created to manage these domains, with many features designed directly for them. These systems are designed to manage a single domain and don't translate well to other master data problems. The two most common types of specialized solutions for managing master data are customer data integration and product information management.\n\nCustomer Data Integration\n\nCustomer data integration (CDI) solutions are designed to provide a standard view of customers across an organization. Some CDI solutions accomplish this as a single system that centralizes customer data across an enterprise. Other solutions manage the integration of multiple ERP systems and additional systems through a registry approach. These solutions manage customer IDs from multiple systems to ensure synchronization and to provide a consistent view of each customer.\n\nA central feature of CDI solutions is the ability to identify and manage duplicate customer records. Other common features include address correction and standardization and the ability to integrate with service organizations to further enrich business-specific customer knowledge. CDI implementations tend to incorporate most systems within an enterprise and focus on a relatively small subset of attributes.\n\nCompanies with customer management problems will certainly benefit from a CDI implementation. Unfortunately, these features do not translate well to managing other domains across the enterprise such as organization or product.\n\nProduct Information Management\n\nProduct information management (PIM) solutions are designed for the product domain. There are fewer PIM solutions than CDI solutions, and PIM solutions tend to focus on specific industries. Most PIM solutions centralize product data management and provide integration to many distribution systems. These solutions tend to be implemented in large retailers and wholesalers that need to manage multiple sales channels for large product catalogs. Management of online catalogs and integration with standardized product channels are some specialized features of PIM solutions.\n\nAgain, these systems are highly effective to solve a narrow band of problems. These systems do not translate well to additional domains. Very few providers have solutions in both the CDI and PIM spaces.\n\nCatering to the Titans\n\nHistorically, vendors of MDM solutions have catered to the needs of Fortune 500 companies. Until recently, the MDM market was dominated by complex and expensive applications that generate large amounts of consulting dollars to trained implementers. These companies generally have the means and the budget to pay for the consulting time and tools that comprise an MDM solution. These solutions are tied to either a specific domain or a feature set that supports one domain better than others. These solutions are expensive and are built to solve a unique set of issues in large organizations. The size and complexity of these engagements has led many of these projects to end in failure.\n\nThe MDM market's focus on Fortune 500 companies doesn't mean that small and midsize companies don't face similar issues with managing their data. Any organization that's attempting to store critical data in multiple systems or spreadsheets and having trouble determining a true version of its master data is in need of an MDM solution.\n\nThe costs associated with MDM solutions and the high risk of failure in self-deploying such solutions leave a large portion of the small and midsize business market underserved. Out of necessity, many of these businesses are using Excel spreadsheets or internally designed systems to manage master data. These systems typically neglect the need for security, central management, and versioning.\n\nMicrosoft's Solution to Master Data Management\n\nMicrosoft first shipped Master Data Services with SQL Server 2008 R2. That release was largely about Microsoft entering the MDM market and beginning to shape the discussion around MDM. It also provided an opportunity for Microsoft to begin building channels for partners that both implement the solution and create additional applications on top of MDS. In effect, the first release was about getting into the game. The second release is about changing the rules. In SQL Server 2012, MDS looks to redefine the MDM market, expanding the types of data stored and easing the barriers to managing that data. This section provides a quick overview of the new MDS features in SQL Server 2012 as well as the features carried forward from SQL Server 2008 R2.\n\nNew Features of MDS in SQL Server 2012\n\nMicrosoft has added a number of new features to increase the usability and applicability of MDS.\n\nMDS Add-In for Microsoft Excel\n\nWell before the original version of MDS shipped with SQL Server 2008 R2, Microsoft had the most widely used MDM tool in the market: Excel. More master data is stored within Excel worksheets than any other tool on the market. The new SQL Server Master Data Services Add-in for Microsoft Excel will enable owners of this data to continue to leverage all of the features of Excel that they know and love as they move their data from the desktop to the server. Chapter 10 provides a tutorial on the new Add-in for Excel.\n\nEntity-Based Staging\n\nWhile the original MDS staging process is available in this release, a new staging schema will contain a new staging table for each created entity. These tables provide a far more intuitive process for loading bulk data into MDS. This new process will be described in Chapter 5.\n\nWhat Master Data Services Still Delivers\n\nMDS provides a number of features that facilitate central management of master data while providing greater access to the editors and consumers of this information.\n\nDomain Agnostic\n\nMDS is not designed for a specific domain. Any data type and virtually any data schema can be supported by the MDS system. Chapter 3 discusses how to map your organization's systems and determine how MDS might best suit your needs. Chapter 4 shows you how to create your models, entities, and attributes, which are the core MDS objects.\n\nHierarchy Management\n\nExcel- and IT-developed applications are notoriously bad at representing hierarchical data. Specialized controls and multiple hierarchy types in MDS provide effective and flexible management of business hierarchies. Chapter 6 shows you how to create hierarchies and collections.\n\nWeb-Based UI\n\nA web-based user interface (UI) provides access to a large user base without the need for installing software on numerous machines. As intranet access becomes more portable, mobile device access can be integrated into the MDM story. Chapter 7 provides instructions for business users to help them work with data in the Master Data Manager web application, which is the primary place where business users will interact with the data.\n\nTransaction Logging\n\nEven with a robust security model, it is essential to provide an audit trail of changes. MDS provides a filterable transaction log to ensure a manageable history of changes. Chapter 7 explains how to view and reverse transactions. It also explains how to annotate transactions so users can explain why they made changes to the master data.\n\nData Validation\n\nMany master data lists require additional validation to ensure certain fields are populated in specific cases. Business Rules can provide users with a proactive monitoring process to ensure data validity before these lists are used in process systems. Chapter 8 shows you how to create business rules and set up notifications to users of Master Data Services.\n\nVersioning\n\nMany domains require snapshots of different points in time to be maintained. With MDS, each model can be versioned, which allows users to tag specific versions for subscribing systems. Chapter 9 shows you how to create versions of a model and how to flag versions for subscribing systems.\n\nSecurity\n\nThe ability to control access at the entity, attribute, and record levels allows IT to empower business stewards to update data in a single centralized tool without risking unauthorized changes. Chapter 11 shows you how to implement security.\n\nThe Value Proposition\n\nThe MDS system is built to be rapidly deployed for any domain within an organization. Once deployed, all models support additional customization without complex coding or reconfiguration. The intent is to make MDM software more accessible to small and midsize businesses, and to aid departments of large companies in creating solutions for themselves. While MDS provides a rich web services platform for system integration, enabling you to use web services to create your own custom user interface, you do not need to use web services to take advantage of what MDS has to offer. An MDS implementation can be completed successfully by business users with no programming knowledge.\n\nAbout the Sample Company\n\nIn an effort to provide context to the features of MDS, all of our examples are based on a fictitious children's clothing company called Main Street Clothing Company. This company is a regional retailer with stores in three states.\n\nFor the past ten years the company has been in an expansion phase; it has a novel store layout and, with the bankruptcy of a competitor, it has grown to 13 locations. As the economic environment has cooled, the company is looking to manage costs and understand its business better.\n\nAn IT study early in the year determined that a new ERP implementation would be cost prohibitive and that the company would be best served by committing to its current IT infrastructure for a minimum of five years. Figure 1-1 shows the overall IT infrastructure.\n\n**Figure 1-1** _System architecture for Main Street Clothing Company_\n\nIn the ERP study, a number of inefficiencies and poor business processes were identified:\n\n While the purchasing department was able to track merchandise purchases through purchase orders added to the financial system, contacts and specifics were being managed in a number of Excel spreadsheets. When the purchasing manager was in the hospital unexpectedly for two weeks, the employees who filled in didn't understand much of the process.\n\n The addition of a new website sales channel in 2007 led to a rapid expansion in both product and customer lists that were not managed well. Customer information was inconsistent and difficult to manage. Many products online were drop shipped, leading to some confusion with the current warehouse process because logistics personnel could not readily identify these items and thus spent time looking for the items or attempting to request additional inventory from the purchasing department.\n\n Management has been complaining about the speed at which reports have been provided at month end. Complexity of the organization and the constant changes to its hierarchies has led to almost a week of cleanup and reconciliation effort with each month's end.\n\nBy implementing Master Data Services in Main Street Clothing Company, we hope to provide continuity to the procedures and give context to the business problems and processes being solved in our implementation.\n\nSummary\n\nMaster data management issues are pervasive within organizations and have existed since the beginning of the electronic age. Solutions have evolved over time to address these issues, and Master Data Services is another phase of that evolution. Traditionally, MDM solutions have tended to cater to larger corporations that can afford customized solutions for specific domains. MDS provides an accessible and affordable solution for organizations of all sizes. Throughout this book, we will incorporate samples designed for a small company that is tackling common MDM challenges.\nChapter 2\n\nInstallation and Configuration\n\nIn This Chapter\n\n Determining the Initial Scope of Your Project\n\n Preparing the Web Server\n\n Installing Master Data Services\n\n Creating an MDS Database\n\n Creating the Master Data Manager Web Application\n\n Securing Your Web Application\n\n Getting the Latest Version of MDS\n\n Upgrading 2008 R2 MDS\n\n Repairing the MDS Database\n\nThe first step in any software application is to deploy the software, and Master Data Services is no exception. In this chapter, we begin with a quick discussion of project size and scope. This should help you determine your needs, including whether you need any external assistance. We then discuss installing the application to a server. As MDS is a web-based application, the deployment and configuration can be extensive. Many users may be upgrading MDS from 2008 R2, we will discuss this process as well as the new repair feature. At the end of the chapter, we discuss options to ecure your data and some steps to extend the application.\n\nDetermining the Initial Scope of Your Project\n\nBefore installing Master Data Services, it is important to determine your short- and medium-term goals for the application. MDS was designed as a web application to provide a simple deployment model for a wide range of organizations. This is also a good time to make sure you have the necessary information to deploy and configure the application successfully, and to determine whether your project requires external expertise. There are a number of questions to consider before installing MDS:\n\n _Which domains will I manage in MDS?_ While this book addresses how to organize this data in later chapters, it is important to take an initial assessment of what data must be stored to create a functional MDS solution.\n\n _How many attributes will I need to manage?_ For each domain that will be managed in MDS, it is valuable to understand the number of attributes that are relevant to the organization. A central benefit of MDS is the ability to modify your model at any time, so a complete list is not essential. A rough estimate of the number of attributes provides one of the best metrics of the scope of the management problem and can provide insight into the owners, editors, and consumers of the records to be managed.\n\n _How many employees will edit the data?_ To provide an effective long-term solution and to see the largest return on investment (ROI) for any master data management solution, it is imperative to empower the owners of the data to make changes directly within the system. Whether this is accomplished directly in the Master Data Manager web application or in some external entry system that is integrated with MDS, empowering the business owners reduces IT effort and eliminates the communication breakdowns that occur when routing data changes through IT.\n\n _How many employees will consume the data?_ Ideally, completed implementations should give everyone with a business need in the organization the access to the cleanest and most accurate data at all times. In most MDM projects, reaching this ideal state is a work in progress and data consumers should be prioritized based on business need and the costs associated with providing access.\n\n _How many systems do I need to integrate?_ Depending on the size of the organization, identifying all systems that rely on a specific domain may not be feasible this early in a project. Most small and medium-sized businesses should identify all systems that consume the domain to be mastered and determine the primary owner of each system.\n\nBased on the preliminary data that you discover from answering the preceding questions, you can determine the relative complexity of your MDM project. The table shown in Figure 2-1 should help you to enumerate the complexity of your project.\n\n**Figure 2-1** _Project scope_\n\nThe complexity of your project should affect both the scope of the implementation and the amount of ongoing effort required to maintain the MDS project. Small projects should be manageable by novice individuals with sufficient business knowledge and the aid of this book. These projects should be functional and productive within a week's worth of effort.\n\nMany small projects revolve around finding a home for \"homeless\" data within the organization. This data is critical to regular business processes, but not important enough to be managed in any standard process system. Much of this homeless data tends to live in unmanaged Excel spreadsheets. The transition of this homeless data into MDS can provide structure and control over it. If IT personnel were previously responsible for managing changes provided by business users, they can now provide users with access to the appropriate data directly. No formal roles or duties need to be created.\n\nA wide range of projects fall into the intermediate range. These projects can be handled internally, but require resources to be fully committed to the implementation effort. Most small and medium-sized business implementations will be intermediate-sized projects. As these projects become more complex, bringing in outside expertise for the implementation should be considered seriously. Making the decision to do so does not diminish the need for internal knowledge and education; although these consultants will aid in the implementation of the solution initially, internal staff will be needed to maintain the MDM system and processes going forward. Most of these projects can be managed in a single phase, requiring approximately 30 to 200 hours to be successfully implemented. Identification of at least one data steward within the organization is essential to long-term success.\n\nData steward is a common role found in MDM projects. These individuals tend to be technically savvy while still understanding the nuances of the business domains. A data steward must be a champion of data governance and must help create sustainable data maintenance processes within the organization. Many times, data stewards find themselves acting as referees in how data is maintained, caught between competing business processes and applications. Systems may maintain different rules regarding the quality and timeliness of data that must be managed by the data steward. The ability to find efficient compromises will determine how effective a data steward is for an organization.\n\nOnce projects reach a certain level of complexity, they become too large to manage in a single phase. The cost and complexity of these large projects requires engaging external expertise that can provide the guidance and resources necessary to implement enterprise-wide MDM solutions. These large projects may span multiple years and locations, and breaking these projects into multiple milestones and ROI checkpoints is advisable. If these projects can be broken down into more manageable intermediate projects, lessons learned from preceding projects can be applied to later implementations.\n\nAbout the Main Street Clothing Company Implementation\n\nMain Street Clothing Company will be managing two models within MDS, Finance and Product. Within these models, the company plans to maintain fewer than ten entities. Initially, the number of systems to integrate will be limited to three process systems. Based on this information, IT feels comfortable providing a single internal resource to build and deploy MDS. Billy Jean, an IT-savvy assistant to the controller, has been identified to manage the Finance model. The Product model will be managed by a designee of the VP of Purchasing. This person has not yet been identified.\n\nPreparing the Web Server\n\nMaster Data Services requires a 64-bit machine and runs only on the following operating systems:\n\n Windows Server 2008\n\n Windows Server 2008 R2\n\n Windows Vista Business, Enterprise, and Ultimate\n\n Windows 7 Professional, Enterprise, and Ultimate\n\nWhile MDS needs to create (or attach) a backend database to either a SQL Server 2008 R2 instance or a SQL Server 2012 instance, this instance does not need to be located on the same machine. MDS does not support SQL Server Standard or Express editions for the database backend.\n\nIn preparation for installing the MDS application, you must enable a number of Windows features to support the web-based UI.\n\nWindows 7 Web Application Requirements\n\nFigure 2-2 identifies the minimum Windows features that must be enabled for MDS to function properly in Windows 7. To turn on these features, open Control Panel, click Programs, click \"Turn Windows features on or off,\" select the check box for each of the features highlighted in Figure 2-2, and then click OK.\n\n**Figure 2-2** _Windows features (highlighted) required in Windows 7 for MDS_\n\nWindows Server 2008 R2 Web Application Requirements\n\nIn Windows Server 2008 R2, use Server Manager to configure the following role services and features.\n\nFirst, install the Application Server role services by completing the following steps:\n\n**1.** Open Server Manager.\n\n**2.** In the left pane, expand Roles.\n\n**3.** Click Application Server.\n\n**4.** In the Summary section, Role Services subsection, click Add Role Services on the right.\n\n**5.** Select the following role services:\n\n NET Framework 3.5.1\n\n Web Server (IIS) Support\n\n HTTP Activation (under Windows Process Activation Service Support)\n\n**6.** Click Install.\n\nNow, install the Web Server role services by completing the following steps:\n\n**1.** Open Server Manager.\n\n**2.** In the left pane, expand Roles.\n\n**3.** Click Web Server (IIS).\n\n**4.** In the Summary section, Role Services subsection, click Add Role Services on the right. Select the following role services:\n\nNow install the following features by completing the following steps:\n\n**1.** Open Server Manager.\n\n**2.** In the left pane, click Features.\n\n**3.** In the right pane, click Add Features. Select the following features:\n\nInstalling Master Data Services\n\nInstalling MDS is a relatively simple process and is only responsible for dropping the bits onto the machine. Any machine that will implement the Master Data Manager web application or the web services needs to run the installation package. In SQL Server 2012, the MDS feature is now a shared feature of the standard SQL Server setup.\n\nProcedure: How to Install Master Data Services\n\nTo install Master Data Services, complete the following steps:\n\n**1.** Run the BI or Enterprise Edition SQL Server Install, as MDS is now a part of the main install process as a shared feature.\n\n**2.** On the Installation Type page, shown here, select a 2012 server instance or install a new instance. Since MDS is a shared feature, this will have no effect on the installation.\n\n**3.** Click Next.\n\n**4.** On the Feature Selection page, select the Master Data Services check box at the bottom of the Features list.\n\n**5.** Click Next.\n\n**6.** Depending on the other features you have selected, additional setup pages may appear. No settings chosen on these pages affect your Master Data Services implementation.\n\n**7.** The last setup screen shows a summary of your installation selections.\n\n**8.** When the installation completes, click Finish.\n\nAfter successful installation, you must open Master Data Services Configuration Manager. You can now use it to create your database and website and to enable web services.\n\nAbout MDS Configuration Manager\n\nOnce the bits are installed, MDS Configuration Manager will assist you in the process of configuring MDS for first use. MDS Configuration Manager, shown in Figure 2-3, is divided into two sections: Database Configuration and Web Configuration. The initial screen indicates whether or not the prerequisites, valid versions of Windows PowerShell and Internet Information Services, can be found. If not, you will need to resolve these issues before continuing with configuration of MDS.\n\n**Figure 2-3** _Master Data Services Configuration Manager_\n\nCreating an MDS Database\n\nThe initial step in configuring Master Data Services is to create a new MDS database. You can create the database on either a SQL Server 2008 R2 database server or a SQL Server 2012 database server. Whether this is a local instance or an instance that exists on another database server, MDS Configuration Manager will perform a check to ensure that the database server is Enterprise, Developer, or DataCenter. In SQL Server 2012, the BI edition is also supported. Once this check is successful, an \"empty\" MDS database will be created on the database server. This empty database will contain all of the tables, views, and stored procedures that make up the architecture of MDS.\n\nYou must first determine whether to install the MDS database locally or on another machine. There are two major factors that should influence this decision:\n\n _Do you already have access to a database farm or server running SQL Server 2008 R2 or 2012?_ Although not always available, this would provide the most economical way to split the two roles of MDS onto separate machines. Please be aware that MDS licensing requires that all machines that have any component of MDS installed must be licensed for the SQL Server 2012 BI edition or higher. The only exception to this is a database server supporting SQL Server 2008 R2.\n\n _Does your organization have any restrictions about having the database server and web server roles on the same machine?_ Although this is an antiquated philosophy born in older, less secure versions of both products, some companies still insist that the database server and web server not be on the same box.\n\nA lesser concern for most companies should be the application's performance. Although it is true that MDS will use processor capacity for both stored procedures on the database server and handling requests on the web server, in most implementations these burdens should be handled by standard hardware that is available on the market today. Microsoft does not provide minimum hardware requirements for MDS, but due to the cost of the license, we suggest a minimum of 8GB of RAM on a quad-core machine. With this configuration, an implementation that must support 20 concurrent users would be fine.\n\nProcedure: How to Create an MDS Database\n\nThe Master Data Services database is where your master data will be stored. To create an MDS database, complete the following steps:\n\n**1.** If Master Data Services Configuration Manager is not open, launch it from Programs | Microsoft SQL Server 2012 | Master Data Services | Configuration Manager.\n\nNOTE\n\n_Master Data Services 2012 can be installed concurrently with MDS 2008 R2, so be careful that you open the correct Configuration Manager._\n\n**2.** In the left pane, click Database Configuration.\n\n**3.** In the right pane, click the Create Database button.\n\nThe Create Database Wizard is launched.\n\n**4.** Click Next.\n\n**5.** On the Database Server page, leave the defaults unless you want to connect to a remote instance of SQL Server or use a different type of authentication. If you choose Integrated Security, you must use the credentials of the logged-in user.\n\nNOTE\n\n_In this example we specify a local account, but you can use a domain account instead._\n\n**6.** Click Test Connection to ensure you can connect successfully.\n\n**7.** Click OK to close the dialog box.\n\n**8.** Click Next.\n\n**9.** On the Database page, enter a name for your database. Depending on the collation and regional settings for your database, searches you do in the Master Data Manager web application may or may not be case sensitive. I would not change default collations without a good reason within your organization.\n\n**10.** Click Next.\n\n**11.** On the Service Account page, type the name of the account that the websites use to connect to the database. Later, you will designate the same account if you create an associated website.\n\n**12.** Click Next.\n\n**13.** On the Administrator Account page, type the name of a user who you want to have the ability to edit all data and to view and update all models. This user is more likely to be a business owner than to be an IT person.\n\n**14.** Click Next.\n\n**15.** On the Summary page, click Next.\n\n**16.** A progress message is displayed while the database is created. When it's done, you should see a success message.\n\n**17.** Click Finish to close the wizard.\n\nMDS Configuration Manager opens and displays the default settings.\n\nAbout System Settings\n\nSystem settings displayed on the Database Configuration page of MDS Configuration Manager are stored in your MDS database in the tblSystemSetting table. These settings apply to the database, the website, and web services. Many of the system settings are self-explanatory, but we will discuss them in the chapters that explain the features they apply to. For now, you can leave the default values.\n\nCreating the Master Data Manager Web Application\n\nThroughout much of this book, we will talk about Master Data Services as a platform. MDM applications need the capability to integrate with all of the systems and business processes within an organization. Unlike other platforms that Microsoft creates, MDS comes with a complete user-facing, web-based UI. Configuring this web application on a web server will allow any user to access MDS through their browser.\n\nNOTE\n\n_MDS officially supports Internet Explorer 7 and 8. While users can access the site using other browsers, those other browsers may display screens incorrectly, with menus that behave poorly or not at all. Unless you are extremely adventurous and vehemently opposed to IE, always access MDS with IE 7, 8, or 9._\n\nWhen creating the Master Data Manager web application, there are two possible workflows: either you select the default website and create the web application, or you create a new website and the web application is automatically created. Later in this chapter, we discuss how to secure your website. If you plan to secure MDS to require Secure Sockets Layer (SSL) connections, it is advisable to create a new site now.\n\nChanges to any of this web configuration information at any time will not affect the underlying MDS implementation. All MDS implementation-specific data is stored in the database. This allows you to back up and restore MDS on any number of servers or configurations without any potential for loss of data. The same database could potentially be used by multiple websites or servers configured instead of the same database server. If your server is part of a server farm where the default site has been removed or you cannot use it, you can create a site now. Otherwise, you can skip this procedure, select the default website, and go straight to creating your application.\n\nProcedure: How to Create a Master Data Manager Website (Optional)\n\nTo create a Master Data Manager website, complete the following steps:\n\nNOTE\n\n_If you create a new site, the web application is automatically created. You can skip the procedure for creating a web application and continue with associating the site to the database._\n\n1 Open Master Data Services Configuration Manager.\n\n**2.** In the left pane, click Web Configuration.\n\n**3.** In the right pane, select \"Create new website\" from the Website drop-down menu.\n\n**4.** In the Create Website dialog box, enter a name for your website.\n\n**5.** The Protocol setting is now locked down for HTTP. If you need to use HTTPS, you will have to use IIS to configure the site after it is installed.\n\n**6.** In the IP address and Port fields, leave the default values or enter new values. A default installation in Windows 7 already uses port 80, so you must select a different port, like 8080, or you will get an error.\n\n**7.** For the Host header field, if your site is going to be on an intranet, leave it blank. If your site will be available on the Internet or you want to create multiple host names for your site, enter the host name.\n\n**8.** In the Application Pool section, leave the name of the site or enter your own. An application pool with this name will be created in IIS.\n\n**9.** In the User name field, enter the same username you used as the service account when you created the database.\n\n**10.** In the Password and Confirm password fields, enter the user account password.\n\n**11.** Click OK.\n\n**12.** A note about HTTPS is displayed. Click OK.\n\nFor more information about securing your website, see the \"Securing Your Web Application\" section later in this chapter.\n\nProcedure: How to Create a Master Data Manager Web Application\n\nIf you chose to use the default website, you can create the Master Data Manager web application by completing the following steps:\n\n**1.** In the Website drop-down list, select Default Web Site and then click the Create Application button.\n\n**2.** In the Create Web Application dialog box, you can change the default alias or application pool names or leave them alone. The alias determines your URL. In this example, the URL will be http:\/\/MDSServer\/MDS.\n\n**3.** In the User name field, type the name of the user you used for the service account when creating the MDS database. This user is the application pool identity in IIS.\n\n**4.** Click OK to save.\n\nProcedure: How to Associate the MDS Database with the Web Application\n\nYou must now associate a database with the web application you just created.\n\n**1.** On the Web Configuration screen, in the Associate Application with Database section, click the Select button.\n\n**2.** In the Connect to a Master Data Services Database dialog box, click Connect.\n\n**3.** In the Master Data Services Database list, your database is automatically selected. Click OK to close the window. You can now view the association between your website and its database.\n\n**4.** Click Apply to save your changes.\n\nAfter you associate your web application with your MDS database, you are prompted to launch the web application. If you can successfully open the Master Data Manager home page from the local machine and from other machines on the intranet, your installation has been successful.\n\nIf Internet Explorer prompts you for your username and password, enter the credentials you used for the Administrator Account when creating the database. Other users do not have permission to access the Master Data Manager web application until the administrator gives them permission.\n\nSecuring Your Web Application\n\nMDS Configuration Manager will create an unsecured website only. This means that data traveling over the wire (between the web server and the client's browser) is unencrypted and readable by anyone. Since this site will not be publicly available and the master data stored is not highly sensitive, these settings should be fine for most organizations. If you are concerned about the security of your master data or if sensitive internal information is stored in MDS, you can require SSL for your website.\n\nThis requires completing three steps: create or download a certificate, apply the certificate to the default website, and require SSL.\n\nFirst, you must download a certificate for your website. To do so, complete the following steps:\n\n**1.** Open Internet Information Services (IIS). You can locate it by clicking Start, right-clicking Computer, and choosing Manage. IIS is listed under Services and Applications.\n\n**2.** In the Connections pane, click the server name.\n\n**3.** In the center pane, double-click Server Certificates.\n\n**4.** In the Actions pane, click the action preferred by your organization. Since every company has a different policy for getting and managing certificates, we will leave the acquisition of your certificate up to you.\n\nAfter you have a valid certificate, you must apply the certificate to the website. To do so, complete the following steps:\n\n**1.** In IIS, click Default Web Site (or other site you created).\n\n**2.** In the Actions pane, click Bindings.\n\n**3.** Click the Add button.\n\n**4.** From the Type list, select https.\n\n**5.** Select the SSL certificate you created.\n\n**6.** Click OK.\n\n**7.** Click Close.\n\nNow require SSL by completing the following steps:\n\n**1.** In the Connections pane, click the website.\n\n**2.** In the center pane, double-click SSL Settings.\n\n**3.** Select the Require SSL check box and indicate whether you want to ignore, accept, or require client certificates.\n\n**4.** In the Actions pane, click Apply.\n\nGetting the Latest Version of MDS\n\nMicrosoft issues hotfixes in response to customer requests. These hotfixes are periodically compiled into cumulative updates. And eventually, these cumulative updates are released as service packs, roughly a year after the initial release.\n\nIn the case of the latest release of Master Data Services, the initial version release is a far more stable product and you can expect fewer updates to be required with this SQL Server 2012 release.\n\nUpgrading 2008 R2 MDS\n\nAll of the moving parts within the SQL Server box can make upgrading from 2008 R2 MDS difficult to understand. Before we discuss upgrade, make sure that you understand the components that make up your 2008 R2 implementation:\n\n **Database Engine** This is the Microsoft service that shipped with 2008 R2 that mounts and controls SQL Server database files. MDS does not require you to upgrade the Database Engine, as MDS 2012 is compatible with both 2008 R2 and 2012 engines.\n\n **MDS database** This is the database that you created earlier in the chapter. All information stored in your Master Data Services implementation resides in this database. This database has two versions that you need to concern yourself with. The _database version_ is the version of the engine that the database is compatible with. You cannot restore database backups from a previous version of SQL Server (2008 R2, for instance) to a new version (2012). To upgrade the database version, you must upgrade the database instance and all the attached databases. The _schema version_ is MDS specific. Just as you were required to update the schema version with some cumulative updates that you installed in the prior release, MDS 2012 requires you to upgrade the schema version, as shown next. The Upgrade Database button is available in MDS Configuration Manager after you select a database. This button is only active if the schema version is not compatible with the current installed version of MDS 2012.\n\nYou may need to upgrade your MDS 2008 R2 instance to Service Pack 1 in order to upgrade the schema. Before you upgrade the MDS database schema to 2012, ensure you have made a backup of the database. If you want to keep your 2008 R2 and 2012 implementations running side by side, you will need to back up and restore the database as a different name. Once you upgrade the database schema to 2012, the 2008 R2 web components will no longer work with that database.\n\nRepairing the MDS Database\n\nWhen you restore an MDS database file to a new database server or a new filename, it may become necessary to \"repair\" the database. In the past, the required running a number of server changes from script. MDS Configuration Manager now has a Repair Database button that you can click to have these tasks performed automatically. Select the MDS database in question, and if a repair is required, a message will display and the Repair Database button will be enabled.\n\nSummary\n\nIn this chapter, we have successfully installed and configured Master Data Services. After determining the scope of our application, we were able to determine the best architecture for our system. We explored the configuration options available in MDS and deployed the web application. The end of the chapter discussed upgrade processes and described the boundaries of the MDS system.\nChapter 3\n\nStarting an MDS Project\n\nIn This Chapter\n\n How Do I Know Master Data When I See It?\n\n Business Process Models\n\n System Roles\n\n Mapping the Data\n\n The Main Street Clothing Company Example\n\n Determining System Types\n\n Cleansing Data for Initial Import into MDS\n\n Leveraging the MDS Excel Add-In to Manage Your Project\n\nBefore you build a model in MDS or any other master data management (MDM) system, it is essential to have a basic understanding of your data and current processes. In this chapter, we provide deeper insight into what is and what is not master data, and we provide some simple labels for your current business systems. Toward the end of the chapter, we discuss some tips and tricks for modeling your data, with the goal of helping you to free your enterprise from many of the data restrictions that have plagued it for so long. We then apply these lessons to the Main Street Clothing Company example. In this chapter we will discuss system types and data cleansing processes. With the release of the Excel Add-In, implementing an MDS system has become significantly easier.\n\nHow Do I Know Master Data When I See It?\n\nOver the past year, one of the questions most commonly asked by customers has been, \"Is there a definitive line between master data and other important data in my company?\" The truth is, there isn't one and there shouldn't be. In most companies, master data management is used only for business-critical domains like product or customer because the implementation of MDM solutions throughout these organizations would be prohibitively expensive. If an accessible solution with a rapid time to value were available, other important domains could benefit from MDM tools as well.\n\nWe've already talked about a couple of the key differentiators between master data and transactional data within an enterprise, but they are worth repeating: Data that you wish to store in MDS should not relate to a single event in time because that is transactional data. In MDS, you should store information that has a state or continues to have a state for some specific period of time. This information is master data.\n\nBusiness Process Models\n\nBefore you build your model in MDS or any other MDM system, it is imperative to understand the flow of data through your enterprise. Where do new accounts, customers, or products originate? What is the process that turns a concept into a product? What is the process to onboard a new customer? How does the data flow from inception to all systems across the organization?\n\nBusiness process modeling can provide decision makers with the information that they need to prioritize and plan the creation of their MDS implementation. Modeling relevant processes related to the creation and maintenance of domains you are interested in managing is the most logical place to start. Although you could employ skilled business process managers to interview employees and provide detailed designs of their findings, most projects can get by with a do-it-yourself approach.\n\nYour first step should be a quick inventory of any existing documentation of current processes, like the example shown in Figure 3-1. Much of what you require might already exist. Be sure to review any existing documents with current employees in the modeled roles. It is astounding how quickly reality can diverge from the documentation given a little bit of time. The more manual the processes, the more easily these processes can change or be abandoned.\n\n**Figure 3-1** _Diagram of sample business process model_\n\nWhen you conduct the interviews required to create a business process model yourself, you should complete the following two steps in sequence. Each step provides insight into the investigated domain. Make sure to record the actors, activity, and length of time required for each business process.\n\n**1.** Determine how new data members are created in the organization. Make sure to investigate any alternative methods that may lead to the creation of a new member. Lead the interview with questions like \"How does this system interact with other systems?\" or \"Are there any emergency processes?\" These can lead to important discoveries. As you begin to integrate systems across organizations, it may be pertinent to ask about any automated systems that may generate new members.\n\n**2.** For each of the methods discovered in the previous step, drill into any workflow or process to enrich the data member before you consider it complete and ready for use in any systems within the scope of the project. Each system that depends on this information may have different requirements, so it is best to address each system's needs. This step is also the best time to identify the primary owner for each business process or system. Primary owners have a vested interest in their processes and will require assurances that most if not all of their current domain needs will be addressed in any business refactoring.\n\nAll of the information you collect will result in a diagram or set of diagrams similar to Figure 3-1. These diagrams will help you to determine the most palatable flow of master data through the organization. You will need to review each system within the scope of the project to decide what type of role it will play for data of each domain type. Determining the role played by each system is the next step in determining how MDS fits into your organization.\n\nSystem Roles\n\nSystems can be categorized into three different roles. As different domains have significantly different origination points, it is common for the same system to perform separate roles for different domains within a company.\n\nSystem of Entry\n\nA system of entry (SOE) is a system where data is first entered in a business. These systems typically include all of the users who are able to create new data. Generally, these are the systems that are used daily by the owners of the type of data being investigated. Many systems of entry require that rules be enforced before data can fully be entered. Oftentimes these rules force much of the accumulated information about the member record to be managed outside of the system.\n\nThink of a new product being created. Often, the inception of a new product is scribbled on a napkin or sketched on a whiteboard. Many of the details of the product will be discussed and decided via e-mail format. It is important to capture and manage this process, because creating a workflow and roles around product creation can provide many benefits, like online collaboration and a tracked history of the creative process. The lack of resources to manage this process often impedes the implementation of a new product.\n\nSystem of Record\n\nA system of record (SOR) is any system within the organization that is considered the source for other systems. Typically, the enterprise resource planning (ERP) system is the system of record within an organization, but oftentimes many other systems can be considered sources for downstream systems. Many data inconsistencies are created on multiple SORs within an organization. While MDS strives to be an adequate SOE for organizations, it is imperative that many if not all of the SORs are moved onto the MDS platform. This is where much of the value of master data management is derived.\n\nSubscribing System\n\nAs your MDS implementation matures, the most common type of system should be a subscribing system. Subscribing systems are those systems that consume data from another system with no direct user changes to the managed domain. Some larger companies use an intermediate store to pass data to multiple systems. As long as direct changes are not made in any of these systems, all these systems can be considered downstream subscribing systems.\n\nMapping the Data\n\nThe next phase in the project plan is to determine data sizes and types within the Master Data Services system. Analysis of all source systems' main tables and their columns to determine the best data type for storing that data can provide an initial roadmap for MDS. There are two main questions that must be asked during this phase: What kinds of internal constraints are placed on this column? Are there any downstream constraints in the organization that require further data cleansing to be performed?\n\nYou should start the mapping process by identifying those columns within the source system that you will manage in MDS. For each of these columns, determine a rudimentary data type. In Chapter 4, we discuss the different data types available in MDS, but for now just identify data as date, number, or string.\n\nIf you identify a date field, be very careful that this field provides information across multiple systems and is a state of the mapped domain. Oftentimes, date fields are red flags that you have mistaken a system-specific field such as \"entry date\" for a more important field like \"start date\" or \"discontinued date.\" You should be extra wary of any column that requires the storage of a specific date or time. These fields generally signal that the information stored is most likely transactional in nature.\n\nColumns that you identify as numeric should be reviewed as well. Is this column storing a valid state on a record? We have seen many projects in which the designers have been tempted to map in fields such as account balances or sales figures. These numbers are subject to daily change and are best handled as part of a business intelligence (BI) solution. Master Data Services does not support simple math functions or consolidations, in an effort to dissuade users from storing inappropriate data. This does not mean all numbers should not be stored. MSRP, safety stock levels, standard terms, and credit limits are all valid information for MDS. When identifying numeric columns, you should log the precision, or number of decimal places, each column will require. While many systems may not limit this on the backend, the actual required precision should be easy to determine for the primary system owner.\n\nWhen mapping a source system to MDS, it may be advisable to create additional tables to store choices for certain fields. Some of these relationships are easy to see. If there is a foreign key relationship within the source system that displays options for one field to another, you should continue to preserve this relationship. For instance, if a customer has a relationship to an address table that stores addresses available to the customer, this relationship can be modeled in MDS.\n\nSome relationships are not quite as evident, though. For instance, a source system may only store specific attributes as text fields, yet valid values for those text fields may be constrained by the business process. If those values are better managed in a separate table, and a foreign key relationship between those tables would be advisable, you want to highlight those relationships now. Determine if any of these relationships may be reused. For instance, within many data sources, there will be a number of fields that map to a choice of either Yes or No. In these cases, you may want to create a single entity to store those valid choices.\n\nDetermining What to Do with Duplicate Records\n\nData quality is a major concern for corporations, and the identification and management of duplicate records is a central task in the effort to ensure data quality. Duplicate records exist within organizations for a variety of reasons. One of the biggest reasons employees duplicate records is that they are unaware that records already exist in a separate system, because the systems are not integrated effectively. These records should be merged wherever possible.\n\nSome companies intentionally duplicate records within systems for a specific purpose, typically to work around limitations within current applications to support necessary business processes. In most cases, these duplicates must remain within the business application to continue to provide the workaround they were designed for. Systems that required duplicate member records should never be considered for Systems of Record. When connecting these subscribing systems to the MDS repository, master data attributes should be updated in all downstream attributes.\n\nDetermining Which Attributes to Manage\n\nTo determine whether an attribute should be managed in MDS, data stewards must decide on the nature of the attribute. Is this a state of the domain? Is this information useful for multiple systems? All numeric and date fields should be evaluated closely. Tracking information should not be managed if it is specific to an application in the organization. Calculated values or balances should not be managed in an MDS system because these values will change over time. Make sure that attributes that store a product's age are not managed as such but that you manage the product's date of creation instead.\n\nThe Main Street Clothing Company Example\n\nBefore the Main Street Clothing Company attempts to build anything in Master Data Services, it is critical that it examine its current processes. All of the following chapters of this book will leverage information collected in the following pages. Although building an MDM solution can be an iterative process, early understanding of the problem space and employees' roles in the process can lead to rapid satisfaction and effective adoption of the new system. An early successful implementation can lead to additional requests for new domains.\n\nFinding a Data Steward\n\nOnce Main Street Clothing Company had determined that an MDS implementation would be beneficial to its organization, it needed to find a project manager. Since there were a number of important sets of data to manage and some amount of ongoing responsibility would be required post-implementation, Anthony Green was identified as the best candidate for the position. Anthony has been with Main Street Clothing Company for a number of years in a business analyst role. He has some limited database experience and was involved in a number of prior system implementations. Main Street plans to have Anthony remain as the owner of the MDS server and serve as the _de facto_ data steward for the organization.\n\nLeveraging the Business Process Models\n\nDuring the implementation of SAP, the implementation team created detailed maps of the business process for creating new products and accounts. Anthony can use these maps to determine the best way to integrate MDS into the current process. One of these maps is shown in Figure 3-2.\n\n**Figure 3-2** _Product business process map_\n\nAnthony is able to leverage this information to investigate how much time is required to create a new account, and he finds very little inefficiency in this process. The major issue with accounts is the difficulty the accounting department has when managing multiple consolidations for the chart of accounts. This issue tends to be related to legal and taxation requirements. Figure 3-3 shows Main Street Clothing Company's process for processing accounts.\n\n**Figure 3-3** _Account business process map_\n\nDespite his best efforts, Anthony was unable to find any process maps that detailed the location data for the company. Although it has been over a year since Main Street Clothing Company has opened a new store location, everyone agrees that the workflow for opening a new store is disorganized and reactive.\n\nFilling in the Gaps\n\nIn reviewing the existing business process models, Anthony determined that the system of entry for products in his company was first created in a custom Access database run by the product acquisition team. This team consists of a manager and two buyers who work with a number of manufacturers and suppliers.\n\nThe product acquisition team enters the products that they wish to purchase into their homegrown system. They also use this system to keep track of notes such as expected demand and types of offers they are considering. After the manager has agreed with their findings, the buyers e-mail a spreadsheet of new product purchases to the accounts team.\n\nThe accounts team then manually codes new products into the system and ties them to a supplier code. When an invoice arrives from the supplier, new product inventory is added into the ERP system for those products. Any products that are absent on the initial entry must be entered at this time. An attempt will be made to contact the appropriate buyer to determine whether the new product is accurate, but if the buyer is unavailable, much of the product information will be left blank for later input. Many of these inconsistencies are not found until month-end reports are run and data is blank. Sometimes, additional merchandise is accepted and placed in the store even though it was not intended to be purchased by the buyer. In these cases, the product acquisition team must scramble to determine if this is a fortuitous mistake or if they need to pull the inventory from the shelves. The product acquisition team estimates that this happens two to three times each year.\n\nAnthony then reviews the business process models for location. He determines that much of the information in the SAP system is in one large entry as the location is conceived. Because the creation of a new store is a long process with many people involved, there is no standard process for collecting all the required information, such as square footage, lease type, and other information that is negotiated or decided upon. Anthony determines a larger problem for location is that the parties responsible for managing renewal of leases or other location management do not have access to the system where this data is stored. They are managing this information in files and folders and have very few electronic warnings before these things require attention. This creates the unwelcome situation where lease negotiations or other decisions are triggered by third parties who call headquarters.\n\nInvestigation of the employee process shows Anthony that, in many cases, starting dates and ending dates are incorrect in the source system because the accounts team has been incorrectly coding the hire date as the start date for the last six months. One store manager was aware of the issue but had been unable to get the accounting department to fix the problem. The accounts team, which was in the middle of month- and year-end close, felt this problem could wait to be resolved and just forgot to return to the problem once those times of high stress had concluded. Anthony would like to address these inconsistencies as part of the MDS implementation.\n\nDetermining System Types\n\nNow that Anthony has reviewed the current processes, he needs to determine the role each system will play when MDS is added into the process. As each business process treats systems differently, he recognizes that he should look at the matrix of each system for each domain within the organization.\n\nOnce MDS is implemented in the organization, many of the systems will perform a different role within the organization. The following table shows the revised system model.\n\nFor each domain to be managed in MDS, Anthony builds a new business process model to show what the proposed process will look like once his implementation is complete. He will use these diagrams to socialize the new process and ensure that all participants feel their interests will be met. It is important that all stakeholders feel that the implementation team is aware of their specific needs and that the new process will not be an obstacle to their daily jobs.\n\nIn the model shown in Figure 3-4, Anthony proposes that the product management database be replaced with a model that includes MDS. All of the functionality that was supported in this system can be replicated within MDS.\n\n**Figure 3-4** _New product management process with MDS_\n\nAnthony suspects that the distributed nature of locations management makes MDS a natural place to manage this domain. In the model shown in Figure 3-5, Anthony shows the location management process where MDS will now be used to manage the location workflow process.\n\n**Figure 3-5** _New location process where MDS manages location creation process_\n\nCleansing Data for Initial Import into MDS\n\nNow that Anthony has socialized the proposed process for managing products in MDS, he must create a clean list of products to initially load into MDS. In many organizations, producing a clean list of products can be a large project; due to the size of Main Street Clothing Company, Anthony will take an export from the product management database as a starting point. He will enhance this export by adding any products that exist in the ERP system. These additional products may have been added if they were part of an invoice received from a supplier.\n\nVery little de-duplication will be required for managing any domains at Main Street Clothing Company. One of the benefits of implementing an MDM solution while a company is small is that you can get a handle on your data before the complexity and size of the processes and data make cleansing a major project for the organization. For each domain, Anthony will create an Excel spreadsheet listing all of the records and their managed attributes.\n\nLeveraging the MDS Excel Add-In to Manage Your Project\n\nWith the addition of the Add-in for Excel in SQL Server 2012, Master Data Services is providing a potential game-changing tool when it comes to MDM project management. Excel provides the perfect interface to rapidly build and then hone your model with the end-user base. While none of the preceding steps are removed, the ability to grab data out of any tool using Excel and create an entity within minutes will give modelers the ability to quickly iterate over and refine their models. This approach will be discussed further in Chapter 10.\n\nSummary\n\nIt is important to understand the flow of data within an organization before implementing an MDM system. Any MDM system that does not match the natural flow of data through an organization is doomed to fail, as busy users work in the easiest, most natural manner. Analysis of business processes can provide insight into the flow of related data through the enterprise. All systems within an organization can be classified as either systems of entry, systems of record, or subscribing systems.\n\nMany systems within the organization will have data-cleansing needs. It is important to determine what remediation will be acceptable if duplicate records are found. If history must be maintained for duplicate members, a mapping between the MDS source and the system should be maintained. In the next chapter, we discuss how to create our models in MDS.\nChapter 4\n\nCreating Your Model\n\nIn This Chapter\n\n MDS Modeling Concepts\n\n Building a Model\n\n Deploying a Model\n\n About Models\n\n About Entities\n\n About Members\n\n About Attributes\n\n About Attribute Groups\n\n Using Web Services\n\nIn this chapter, we discuss the creation and customization of MDS models. In MDS, models are the central work surface that will be exposed to the master data editors and consumers once your project goes live. The success of your project will depend mainly on how well you design your model.\n\nWe begin with a discussion of modeling concepts. Then we examine how to deploy prebuilt models. We load the sample company's Finance model and then spend the remainder of the chapter creating a Product model from scratch. Finally, we look at examples of how to use web services to perform some of the procedures we've already performed in the web UI. (From now on, \"Master Data Manager web application\" and \"web UI\" are used synonymously.)\n\nMDS Modeling Concepts\n\nMaster Data Services is made up of a relatively simple group of concepts. These concepts are encapsulated as data containers and services exposed in MDS. While these concepts are simple to understand by themselves, it is the varied and complex data schemas that these concepts support that provide both the power and complexity of MDS. Before we delve into the implementation of the model objects, let's review general definitions of these MDS concepts. Each of these concepts is described in more detail later in the chapter.\n\n **Models** Models are the highest level of organization within MDS. Models are nothing but containers of related entities. Only entities within the same model can be related within MDS. Models are the first concept discussed in this chapter.\n\n **Entities** Entities are the base containers for data in MDS. In their simplest form, entities can be thought of as tables in a database. Users control the attributes (columns) that are managed for each entity. If explicit hierarchies are enabled for an entity, the entity becomes far more complex, managing parent members and their consolidations as well as collections, their attributes, and the members associated with those collections. Most of this chapter is devoted to discussing how to create the structures of entities.\n\n **Members** Members are the records that populate all the entities created in MDS. Members can be either leaf or consolidated. Leaf members are the primary members of an entity. If an entity is enabled for explicit hierarchies and collections, then consolidated members can be created, and can have their own attributes. Members will be discussed in this chapter as well as in Chapter 7.\n\n **Attributes** Attributes describe members. Attributes can be loosely thought of as columns in a table. Entities contain members and their attribute values. Attributes can be free-form or domain-based.\n\n **Domain-based attributes** Domain-based attributes are attributes in which the available values are restricted to the members stored in a related entity. This is similar to selecting from a predefined list, but in MDS all lists are entities themselves.\n\n **Hierarchies** Hierarchies are consolidations or groupings of members that aid in reporting and analysis. There are two management types for hierarchies in MDS: explicit and derived. Hierarchies enforce rules for member inclusion to ensure consolidations do not lose or double count values in connected applications. Hierarchies will be discussed in detail in Chapter 6.\n\n **Collections** Collections provide member grouping flexibility that is not supported in hierarchies. Collections will be discussed in detail in Chapter 6.\n\nBuilding a Model\n\nMaster Data Services begins as a blank canvas, allowing you to create your data models within the product in any way you choose. The model structure is created in the System Administration functional area of the Master Data Manager web application, or by using the web services. As you create the structure, you can open the Explorer functional area of the web UI to see the results of your work. The Explorer functional area is where users will go day to day to manage their master data.\n\nOpening the System Administration area of the Master Data Manager web UI for the first time can be a daunting experience. In the latest release, System Administration defaults to the model creation screen if you have not loaded any models. When you first create an MDS database, only the Metadata model is created. Figure 4-1 shows this first model open in the Model View screen. This is the only system model within MDS and has been marked for deprecation in a future release. While this model can still be accessed from System Administration or Explorer as its own model, all access to the Metadata features from your own models has been eliminated through the web UI. Access to these features will need to be completed through the web services for Metadata.\n\n**Figure 4-1** _The default Metadata model that is created during installation_\n\nIf you are a new user of MDS, you have two avenues for creating your first useful model within the product. You can load a sample model deployment package into your database and modify the model from there, or you can build your model from scratch.\n\nStarting with a predefined model can be the easiest way to get started with SQL Server MDS. Over the course of this book, we are going to work with Finance and Product models for our fictional company, Main Street Clothing Company. In this chapter, we are going to deploy the Finance model and create the Product model from scratch. You can find the complete packages of our models, both with and without sample data, at our web site: www.mdsuser.com.\n\nMDS also includes three sample models in the installation. These samples are very basic Customer, Product, and Account models, located in Program Files\\Microsoft SQL Server\\110\\Master Data Services\\Samples\\Packages. You can take a look at them to get a better idea of some of the more common features of the application.\n\nWhile these model deployment packages provide some data and attributes and support the documentation provided in Microsoft Books Online for SQL Server 2012, they are relatively limited in their functionality and do not provide adequate coverage for the exercises listed in this book. However, these models include sample data, hierarchies, business rules, and versioning, so you can see what the end result of all your upcoming work might look like.\n\nIn the next exercise, we will walk through how to deploy our sample Finance model.\n\nDeploying a Model\n\nA model deployment package is an XML file saved with a .pkg extension. It includes the model structure, the business rules (Chapter 8), and version flags (Chapter 9) and subscription views (Chapter 12). It does not include file attributes (Chapter 4) or user and group permissions (Chapter 11). Model packages can contain the data from a version of the model when created from the ModelDeploy.exe command-line executable. Only the model structure can be created or deployed from the web UI in the SQL Server 2012 version of Master Data Services. The new utility and the limitations on data in the web UI were added in response to the inability to support all model sizes through the web services.\n\nModel deployment in MDS was designed to provide organizations with two very important capabilities related to managing data models in MDS. The first is the capability to develop, test, and deploy a model within three separate implementations of MDS, while limiting the number of objects that need to be re-created in each environment. This enables IT management to roll out changes en masse and to ensure that the production environment remains operational throughout the process.\n\nThe second capability enables organizations to share their model schemas with others. There are a number of efforts across a host of industries to standardize object models. Previously, most organizations would develop data models in a vacuum, creating schemas that solved their current problem most efficiently. They integrated only those applications owned by the organization and central to the solution. IT departments at all of their competitors were doing the same. This led to a wide variety of data models among competitors in the same industry. Some organizations built data models that were flexible, stable, and scalable, but many did not. When consolidation within an industry occurred, acquiring companies discovered similar business processes were hard to integrate due to the vastly different data models that had been developed in isolation.\n\nFollowing are a couple of advantages to employing common data models within an industry:\n\n **Benefit of experience** Civilization wouldn't get very far if each new generation needed to reinvent the wheel. By leveraging a common data model developed by industry experts, companies are able to avoid issues that they might otherwise experience personally.\n\n **Standardization** Standards help companies and software providers by limiting the amount of variation that must be managed. Standards allow companies to better manage acquisitions and new integration projects. Software can focus on enhancing business value as opposed to compatibility.\n\nMDS supports these efforts in two ways. By not enforcing its own model, MDS ensures there is one less data model to be managed. Model deployment empowers industry leaders to quickly build master data\u2013centric versions of these industry models, providing efficient deployment across their industry. Any customization that needs to be made to these standardized models is still supported in MDS.\n\nProcedure: How to Deploy a Model Using ModelDeploy.exe\n\nTo deploy a sample model using ModelDeploy.exe, complete the following steps:\n\nNOTE\n\n_If you want to deploy the sample Finance model we refer to in this book, get it fromwww.mdsuser.com before you start this procedure._\n\n**1.** Copy the Finance package file to the Program Files\\Microsoft SQL Server\\110\\Master Data Services\\Configuration directory.\n\n**2.** Open a command prompt from the Start menu. It is advisable to run the command prompt as administrator to ensure access to the file location.\n\n**3.** Using the cd command, change the directory to Program Files\\Microsoft SQL Server\\110\\Master Data Services\\Configuration.\n\n**4.** Run mdsmodeldeploy.exe deploynew -model Finance -package finance.pkg.\n\n**5.** If successful, you should see the message \"MDSModelDeploy operation completed successfully\" and the elapsed time for the call displayed.\n\nYou can confirm that the model was deployed successfully by going to the home page and looking in the drop-down list of available models, as shown next. If you were already viewing the web UI, you may need to click the \"Refresh cached information\" link on the MDS home page.\n\nTo view the Finance model in more detail, select VERSION_1 from the Version list and then click Explorer. Your business users will manage master data in the Explorer functional area. When you configure security, you will be able to give users access to specific data within the Explorer functional area.\n\nAll other functional areas of the web UI are intended for administrators. Let's take a moment to discuss the other four functional areas.\n\n **Version Management** Use this area to validate full versions of your model against business rules. You can also create versions of your models and their data, and assign version flags, which indicate to subscribing systems which version of a model to use. More details about versioning can be found in Chapter 9.\n\n **Integration Management** Use this area to import data into the proper MDS database tables. You can also use this area to create subscription views, which are SQL views used by subscribing systems to retrieve data from MDS. More information about staging data can be found in Chapter 5. See Chapter 12 for information on exporting data by using subscription views.\n\n **System Administration** Use this area to build your models, entities, attributes, attribute groups, and derived hierarchies. You also use this area to manage business rules and create and deploy models. You will use System Administration extensively in this chapter and in Chapter 8 when we discuss business rules.\n\n **User and Group Permissions** Use this area to give users access to specific functional areas and to specific data within those areas. More information about security is provided in Chapter 11.\n\nAbout Models\n\nAs discussed previously, models are the highest level of organization within Master Data Services. Models are nothing but containers of related entities. Only entities within the same model can be related within MDS. This supports a simplified versioning model where all entities within a model are versioned simultaneously. Although many models revolve around a single domain such as customer or product, this is not a requirement. You can combine major entities into a single model if this makes business sense.\n\nWhen determining whether or not to include an entity in a model, there are a number of factors to consider:\n\n Do the entities in the model have a relationship to one another?\n\n If so, is this relationship master data? Many entities have relationships that are transactional in nature. Only those relationships that are states that change over time and are not discreet instances should be managed in the MDS system.\n\n Do the entities change on similar schedules? If two entities have a natural affinity to be versioned at separate times, you may need to manage them in separate models.\n\n Does an entity relate only to a subset of another entity? If only a subset of the members participate in the relationship between the two entities, ensuring accurate selection may be compromised. Business rules can be used to ensure only the valid members are selected; business rules will be discussed in greater detail in Chapter 8.\n\nAbout Main Street Clothing Company's Models\n\nAfter performing all of the analysis discussed in Chapter 3, Anthony Green has determined that he needs to create two models in the organization's Master Data Services instance. The Product model will revolve around a central Product entity. This entity will not require an explicit hierarchy, so Anthony will create only product leaf members to begin. Due to the interconnected nature of all other managed entities, Anthony will create an additional model called Finance. Since there is no finance entity or entities to add, he will create the model only.\n\nProcedure: How to Create a Model\n\nTo create a model, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Models.\n\n**3.** On the Model Maintenance page, click the Add model button (the green plus sign).\n\n**4.** On the Add Model page, in the Model name field, type a name for your model. For this example, we'll call the model **Product**.\n\nThe three check boxes on this page can be confusing if you've never done this before:\n\n **Create entity with same name as model** Leave this box selected. This check box does just what it says\u2014it creates an entity called Product. This option is here because, in many cases, a model revolves around a central entity.\n\n **Create explicit hierarchy with same name as model** Briefly, an explicit hierarchy is a free-form hierarchy in which you can specify any number of levels, with a leaf member being allowed at any level and consolidated members being used for grouping. If you think you might want a hierarchy like this, you can select this check box. Because we don't discuss explicit hierarchies until Chapter 6, clear this check box for now. You can always enable this later.\n\n **Include all leaf members in mandatory hierarchy** When you clear the preceding \"Create explicit hierarchy\" check box, this option is disabled.\n\n**5.** Click Save.\n\nThe Product model is now displayed in a tree view in System Administration, as shown in Figure 4-2. If it's not displayed already, click Model View on the menu bar in System Administration to view it.\n\n**Figure 4-2** _Models in the System Administration functional area_\n\nThe hierarchical structure that is displayed here is meant to give you a quick look at the models you have permission to view and to the entities in each. This structure is not meant to represent levels of importance the way a typical hierarchy would. As you learn more about entities and attributes, we will provide more details about this structure.\n\nViewing the Results in Explorer\n\nNow let's look at the model in the Explorer functional area. Go back to the Master Data Manager home page by clicking the logo.\n\nTIP\n\n_If you are running Internet Explorer 8, you should choose Compatibility Mode before viewing data in the Explorer functional area of Master Data Manager. From the IE Tools menu, choose Compatibility Mode._\n\nSelect the Product model from the list. Select the version, which was automatically created as VERSION_1. Now click Explorer.\n\nThe first base entity Silverlight Explorer grid will be displayed when you open the Explorer functional area. In the latest release, the old Explorer screen is no longer the initial screen loaded for users. This screen is still available from the menu at the top of the screen.\n\nThe top _base entity_ is the entity that is displayed by default when you open the Explorer functional area of the web UI. After you create more entities, you will be able to drag-and-drop items in the hierarchy to move a different entity into the \"base\" position. More than one entity can be listed as a base entity within a model. If multiple entities are listed as base entities, the first base entity will display when Explorer opens. The Explorer screen will still show all base entities for the model. You do not need to have a base entity in your model, though. If you don't, when Explorer opens, the first entity listed alphabetically will be displayed; all entities are available from the menu bar.\n\nNow you're ready to learn more about entities and practice creating a few.\n\nAbout Entities\n\nIn MDS, all data that is managed by the system is stored in entities. Entities can be loosely thought of as tables in SQL. The data within entities are called members.\n\nMDS's entities can also support explicit hierarchies. Once explicit hierarchies have been enabled for an entity, an entity becomes far more than a single table. The entity can then support parent and collection members, as well as hierarchy and collection relationships. Hierarchies and collections are discussed in detail in Chapter 6.\n\nAbout Main Street Clothing Company's Entities\n\nWithin the Product model, the Product entity has already been created. We now need to create all of the supporting entities that will be used as options for attributes of members in the Product entity. While most of these entities will be used to populate a single domain-based attribute only, some entities can be used to populate multiple entities within the model. Entities such as Neck Style, Color, and Size can only be used for a single purpose, but multiple entities can use attributes with constraints such as \"Yes\" and \"No.\"\n\nFirst we will create the Yes No Picklist entity; then we will create all other entities needed in the Product model.\n\nProcedure: How to Create an Entity\n\nTo create the Yes No Picklist entity, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** On the Entity Maintenance page, click Add entity.\n\n**4.** On the Add Entity page, in the Entity name box, type **Yes No Picklist**.\n\n**5.** In the \"Name for Staging tables (optional)\" box, you can type a unique name for the staging table created specifically for this entity. The entity name will be used if no value is provided. This value cannot be changed in the web UI once the entity is saved. See Chapter 5 for more information on staging.\n\n**6.** You can check the \"Create Code values automatically\" check box to automatically assign a new integer value to newly created members. See Chapter 8 for more details on this process.\n\n**7.** In the \"Enable explicit hierarchies and collections\" drop-down list, select No. Again, you can enable this later if you need to.\n\n**8.** Click Save.\n\nNOTE\n\n_Sometimes when you save an entity, you have to wait a long time for the page to refresh. You know the entity has been created when the Add Entity page closes and you're returned to the Entity Maintenance page._\n\nThe Yes No Picklist entity is now displayed on the Entity Maintenance page.\n\nTo continue following along with the examples in this chapter, create the following entities: Type, Fabric, Neck Style, Gender, Color, and Size. These entities will contain lists of values that will be used to describe products, as you will soon learn. When you are done, your Entity Maintenance page should look like Figure 4-3. Your entities are automatically alphabetized here.\n\n**Figure 4-3** _Completed Entity Maintenance page_\n\nViewing the Results in Explorer\n\nNow let's look at the entities in Explorer. Go back to the home page by clicking the logo in the top left or by clicking the Home link.\n\n**1.** Select the Product model, VERSION1, and click Explorer.\n\n**2.** Choose Entities on the menu bar, and you see a list of available entities.\n\nIf you select an entity to view it in more detail, you can see that there is no master data in the entity yet. If there were, a grid would be displayed.\n\nLet's stop creating the model for a moment and talk about the members that will populate the grid.\n\nAbout Members\n\nMembers are the individual records stored in Master Data Services. Members are uniquely identified by the required Code attribute. There are two types of members in MDS: leaf members and consolidated members.\n\nLeaf members are the most granular level of records in an entity and usually represent physical objects within your business. For example, in a Product entity, a leaf member might be Men's Shirt #602. A leaf member in an Employee entity might be John Smith. A leaf member in a Warehouse entity might be Warehouse-98101.\n\nOne notable exception to this rule of thumb applies to any entities managed within the Finance domain. When managing entities associated with Finance, leaf members typically represent those low-level identifiers that transactions can be coded to. If transactions are coded with a store, account, employee, and product, you can be assured that the available values for each of these attributes should be stored as leaf member records in the associated entities.\n\nConsolidated members are used only in explicit hierarchies, and will be discussed in more detail in Chapter 6.\n\nAbout Main Street Clothing Company's Members\n\nWhen implementing MDS, most entities will need to be populated from an initial data source. In Chapter 3 we retrieved and cleansed an extract file of the current products. In Chapter 5 we will discuss how to use staging tables to populate all of the current members into the proper entities. Before this staging can be completed, Main Street Clothing Company must ensure that all of the entity containers have been properly configured to accept the staged data.\n\nSome entities that are used as domain-based attributes have so many members that loading them through staging at the same time as the product members makes the most sense. In the case of simple entities with very few members, however, the effort to stage the members would exceed the effort needed to create them through the web UI. These members need to be created before we attempt to stage any product data.\n\nProcedure: How to Add a Member\n\nTo add a leaf member, complete the following steps:\n\n**1.** On the Master Data Manager home page, select the Product model and VERSION_1 from the drop-down lists.\n\n**2.** Click Explorer.\n\n**3.** On the menu bar, choose Entities | Product.\n\n**4.** Under Product Entity, click Add Member.\n\n**5.** In this release, all new members are created on the same screen. A new member can be added in the right pane. In this pane, enter a name and code for your product. When we use the staging process to import the majority of our products, the name will be a concatenation of many fields. If you want to continue using our example, complete the fields as follows:\n\nNow add a few members to the Yes No Picklist entity:\n\n**1.** On the menu bar, choose Entities | Yes No Picklist.\n\n**2.** Above the grid, click Add Member.\n\n**3.** In the right pane, populate the Name and Code fields, entering **Yes** for the name and **Y** for the code.\n\n**4.** Click the OK button at the bottom of the pane.\n\n**5.** Click Add Member again.\n\n**6.** In the right pane, enter **No** for the name and **N** for the code.\n\n**7.** Click the OK button at the bottom of the pane. The members are now displayed in the grid.\n\nAfter you learn more about attributes, you will create a domain-based attribute that will use the Yes No Picklist values as the source.\n\nAbout Attributes\n\nAll entities within Master Data Services can be enriched by the creation of additional attributes. MDS supports a subset of the attributes supported in SQL Server. MDS supports four specific types of free-form attributes for leaf and consolidated entity members: text, datetime, number, and link. Text and link are essentially string fields, with link providing one-click support for hypertext links. Number supports as many as seven decimals, and datetime allows you to specify the mask for how the data will be input.\n\nIn order to relate two entities and ensure that values are constrained to specific values, MDS gives users the ability to create domain-based attributes. These attributes limit available values to the list of active members within the related entity.\n\nThe last type of attribute available to entities is the file attribute. Within a master data entity, it may be advisable to manage some files associated with each member. For instance, there may be instruction documents, specifications, blueprints, or photos that need to be associated with a Product entity within an organization. However, the file attribute has a number of limitations that make it less attractive to use for file management than SharePoint. First, to limit database sizes of MDS, file attributes will not be versioned. Only the last loaded file will be available within the MDS system. The transaction log will display information related to new files that have been loaded, but these will not be reversible.\n\nAbout Main Street Clothing Company's Attributes\n\nIn order for Anthony to successfully load all of the current products into Master Data Services, he must complete the creation of a Product entity. This includes creation of all free-form attributes to the proper specifications identified in Chapter 3. All product attributes that have currently been identified need to be created in MDS.\n\nIt is quite possible that during the attribute creation process or subsequent data loads, new information will be uncovered that changes the shape or type of attribute in MDS. Creating models in MDS should be an organic process. As new column needs are discovered at Main Street Clothing Company, Anthony will determine the best process for integrating these columns into Main Street's daily processes, ensuring that no process requirements are overlooked.\n\nProcedure: How to Create an Attribute\n\nLet's start by creating a free-form attribute. To create a free-form attribute, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** On the Entity Maintenance page, select Product from the list of entities.\n\n**4.** On the toolbar that is displayed, click the \"Edit selected entity\" button.\n\nNOTE\n\n_The button does not appear until after you click the entity in the list._\n\nOn the Edit Entity page, you can see that two attributes are listed: Name and Code. When you create an entity, these attributes are automatically created. You cannot delete them.\n\n**5.** Click the \"Add leaf attribute\" button.\n\n**6.** On the Add Attribute page, select the Free-form option and complete the fields as shown here:\n\n The value of the \"Display pixel width\" field determines how wide the column will be when it's displayed in the Explorer area of the web UI.\n\n The \"Input mask\" field lets you decide whether to show negative numbers in parentheses or show a minus sign instead.\n\n For now, don't enable change tracking. You will use this later, along with business rules, to notify someone when an attribute value changes.\n\n**7.** Click Save.\n\nOn the Edit Entity page, the new attribute is displayed.\n\nYou can click either Save or Cancel to leave this page. Either way, your attribute is saved.\n\nNOTE\n\n_To continue following along with the examples in this chapter, create the following free-form attributes: Retail Price (number with two decimal places), Safety Stock Level (number with no decimal places), Reorder Point (number, also with no decimal places), and Design (text)._\n\nIf you make a mistake, you can delete the attribute and create a new one, or you can edit the attribute and change the editable fields. At this point you might also want to edit the Name and Code attributes to set the \"Display pixel width\" field to a smaller number.\n\nTIP\n\n_If you want to hide an attribute without using security, set the \"Display pixel width\" field to 0._\n\nProcedure: How to Create a Domain-Based Attribute\n\nEach domain-based attribute has two parts:\n\n The entity that contains the members that will be used as a picklist\n\n The attribute that refers to the entity\n\nYou have already created an entity called Yes No Picklist and populated it with two members. Now you need to create a domain-based attribute that uses the members from the entity as attribute values.\n\nTo create a domain-based attribute, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** On the Entity Maintenance page, select Product from the list of entities.\n\n**4.** On the toolbar that is displayed, click the \"Edit selected entity\" button.\n\n**5.** On the Edit Entity page, in the Leaf attributes section, click the \"Add leaf attribute\" button.\n\n**6.** On the Add Attribute page, select the Domain-based option and complete the fields as shown here:\n\nThe attribute is going to be called \"Discontinued.\" For each product, we want to know whether or not it has been discontinued. So type **Discontinued** as the attribute name and select Yes No Picklist as the entity that provides the members to use as attribute values.\n\n**7.** Click Save.\n\nViewing the Results in Explorer\n\nNow let's go back to Explorer and look at the Discontinued attribute:\n\n**1.** On the Master Data Manager home page, select Product and VERSION_1 from the lists and then click Explorer.\n\n**2.** On the menu bar, choose Entities | Product.\n\n**3.** Select the first member in the grid.\n\n**4.** Scroll to the bottom of the right pane. You can now select a value from the Discontinued drop-down list.\n\n**5.** Select N {No} for the attribute value.\n\n**6.** Click the OK button at the bottom of the pane to save the new values.\n\nTIP\n\n_You can select the Setting button above the grid to show the name and code of each attribute value, or to show the code only._\n\nNow create other domain-based attributes:\n\n**1.** Based on the entity Yes No Picklist, create a domain-based attribute named **Available to Outlet**.\n\n**2.** Based on the entity named Type, create a domain-based attribute named **Type**.\n\nYou can use the same name for the domain-based attribute that you already used for the entity.\n\n**3.** Do the same for the Fabric, Neck Style, Gender, Size, and Color entities, using the same name for the domain-based attribute.\n\nViewing the Results in System Administration\n\nIf you go back into the System Administration functional area (or if you are in System Administration already and on the menu bar you click Model View), you can view your model structure in a hierarchical format, as shown in Figure 4-4.\n\n**Figure 4-4** _The Product model structure_\n\nThe domain-based attributes can be expanded to show that they are entities with their own Name and Code attributes. The free-form attributes do not have attributes; they are just attributes of the Product model.\n\nAbout Attribute Groups\n\nMaster Data Services can contain every attribute associated with an entity in a business. Because of this, some entities can become unwieldy when viewed as a single table in the web UI. In a large organization, it is not uncommon for a major entity to contain over 400 attributes. Very few, if any, system users are interested in viewing all attributes simultaneously. Most users are interested in viewing only specific, related attributes at any one time. MDS provides the ability to group attributes into multiple tabs in the Master Data Manager web application. MDS calls these groupings _attribute groups_. Any attribute can be added to any attribute group.\n\nTypically attribute groups are created for each role consuming data within an entity. Different functional areas of the organization will be concerned with different groups of attributes for each entity stored in the application. Access to attribute groups can be managed by applying security for different users or groups at the attribute group level. This will be covered in more detail in Chapter 11.\n\nAbout Main Street Clothing Company's Attribute Groups\n\nMain Street Clothing Company has determined that the best approach to grouping attributes in Master Data Services is to group attributes by role. Because multiple roles are interested in the same attribute values, some attributes will appear in multiple attribute groups. In many cases, the same individuals perform the duties of multiple roles. Managers of these groups saw value in splitting these roles early in the design phase in an effort to prepare for future employee growth. Anthony reviewed all proposed consumers of MDS and determined the following roles:\n\n **Logistics** In the past year, the company has leased warehouse space in two central locations to manage back inventory and take advantage of bulk discounts. To manage these warehouses, information related to reordering stock and product status is provided in a single group.\n\n **Marketing** The Marketing role includes the VP of marketing and the individual store managers. All of these individuals need access to information about the variety of products and their costs in order to make informed decisions about incentives and discounts.\n\n **Pricing** Within the company, multiple groups are interested in pricing. To aid in the management of these attributes, a separate attribute group was created to display only the cost and suggested retail price of each product.\n\nIn this chapter, we have shown you a very manual way to create entities, the goal being to explain all of the components of MDS entity creation. In SQL Server 2012, we encourage you to check out the entity creation functionality available in the MDS Add-In for Excel. This functionality quickly creates new entities in MDS with very minimal input from the user required. More information on this process can be found in Chapter 10.\n\nProcedure: How to Create an Attribute Group\n\nCAUTION\n\n_In SQL Server 2012, the system-created All Attributes attribute group is always available for users. Attribute groups are no longer a method to hide or secure attributes from users._\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Attribute Groups.\n\n**3.** Select a model and entity from the lists and click Leaf Groups.\n\n**4.** Click the \"Add attribute group\" button that is now displayed.\n\n**5.** In the \"Leaf group name\" box, type **Logistics**. This will be the first tab displayed in Explorer on the grid that displays members in the Product entity.\n\n**6.** Click Save. The group is displayed as a folder on the Attribute Group Maintenance page.\n\n**7.** Expand the Logistics folder by clicking the plus sign to the left of it.\n\n**8.** Click the Attributes label. The \"Edit selected item\" button is displayed.\n\n**9.** Click the \"Edit selected item\" button.\n\n**10.** A list of available attributes is displayed. Assign attributes to the group by selecting them and then clicking the right-pointing arrow. In this example, we're going to add Safety Stock Level, Reorder Point, Discontinued, and Available to Outlet.\n\nTIP\n\n_You can use ctrl-click to select multiple attributes and then add them all at the same time by clicking the arrow. The double arrows will add or remove all attributes to the group._\n\n**11.** Click Save. The attributes are now displayed beneath the Logistics group.\n\nBefore looking at the results in Explorer, create a few more groups:\n\nYou can include attributes in multiple groups, or in none. Name and Code are automatically included in all groups.\n\nNOTE\n\n_To rename a group, click the name of the group and then click the \"Edit selected item\" button._\n\nViewing the Results in Explorer\n\nLet's take a few minutes to view the attribute groups in Explorer.\n\nOn the Master Data Manager home page, ensure that the Product model and VERSION_1 are selected. Click Explorer. Then, on the menu bar, choose Entities | Product. The tabs are displayed, and each tab has the columns you selected, in addition to the Name and Code columns.\n\nIf you decide you want to change the order of your tabs, complete the next procedure.\n\nProcedure: How to Reorder Attribute Groups\n\nYou can change the order of your tabs by completing the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Attribute Groups.\n\n**3.** Click the Leaf Groups label.\n\n**4.** Click the \"Edit selected item\" button.\n\nThe list of attributes in the group is displayed.\n\n**5.** Click the attribute group you want to move and use the arrows to move it up or down. The topmost group will be the leftmost tab in Explorer.\n\n**6.** When you're done, click Save.\n\nUsing Web Services\n\nCreating and retrieving any of the Master Data Services objects explored in this chapter are completed using the same MDS web service operations, MetadataCreate and MetadataGet, respectively. While these two calls can be extremely powerful, they require users to create an array of different objects to successfully create and get data from the web service. In the examples in this section, the encapsulation allows developers using these calls to ignore internal IDs and work with only common object names when coding against the MDS web service. Throughout the rest of the book, we will provide examples of calls to encapsulate common coding scenarios.\n\nCreating a Model\n\nThe first object that must be created in the Master Data Services system is a model. We can create a new model with a minimum of information from the user.\n\nCreating an Entity\n\nIn this example, we encapsulate the call to create a new entity for a given model. This example only creates an entity that is not hierarchy enabled. The web service requires us to create the model objects to hold the new entity object. More complex examples can be found in Chapter 13.\n\nCreating an Attribute\n\nAttributes can be created on a variety of object types in MDS. Attributes can be created on leaf and consolidated entities, as well as collections. The following example creates a new attribute of a specified type on the entity and model that has been passed in. Similar to the entity creation example, this procedure builds on the creation of an entity object and adds the attribute.\n\nSummary\n\nIn this chapter we created the initial structure of our Master Data Services solution. We created models, entities, and attributes. Attribute groups provide the ability to display attributes for each role or process that consumes the data stored. Unlike the example in this book, implementing a model in MDS at your company can and should be an iterative process. The customizability and speed of structural changes in this application allows organizations to quickly react to changing business processes or discovered inefficiencies. Often, changes to the data model are identified as we load data into the model, which will be the focus of the next chapter.\nChapter 5\n\nIntegrating Master Data Services with Other Systems\n\nIn This Chapter\n\n Staging Architecture\n\n Loading Relationships\n\n Importing Data into the Staging Tables\n\n Initiating the Staging Process\n\n Using Web Services to Stage Data\n\n Errors That Occur During Staging\n\n Viewing the Staging Batch Table\n\nWhile master data management requires extensive integration with other systems within your enterprise, Master Data Services does not provide any features to perform extract, transform, and load (ETL). MDS is designed to leverage SQL Server Integration Services (SSIS), but it also works well with other ETL solutions such as Informatica, InfoSphere DataStage, or any other product in this space.\n\nTo get data into and out of the system and communicate with other applications, Master Data Services uses staging tables for the incoming data flow and uses subscription views for the outgoing data flow. In SQL Server 2012, Microsoft has completely re-architected the staging process in MDS, creating new tables for each entity created in the system. In this chapter, we describe these staging areas of the application and discuss common methods to integrate systems with MDS. We also discuss some tips and tricks for integrating using SSIS and T-SQL jobs. This chapter is not meant to be a substitute for other SSIS books that may be available; for more complex operations, such as complex data flows and manipulations, you should consult _Hands-On Microsoft SQL Server 2008 Integration Services, Second Edition_ by Ashwani Nanda for assistance.\n\nStaging Architecture\n\nThe new staging architecture of Master Data Services creates a separate staging table for each entity created in the system. These tables are dynamic, adding or deleting columns as new attributes are managed in the entity. As attributes can differ between leaf members and consolidated members, two additional tables are created if the entity supports explicit hierarchies. The _Consolidated table supports loading consolidated members for any explicit hierarchy created for the entity, and the _Relationship table loads the structure of the hierarchy. Hierarchies are discussed in greater detail in Chapter 6.\n\nAll of these tables are located in the new staging (stg) schema. This new schema is provided to better delineate between system objects that should be avoided and the tables and stored procedures users are encouraged to work with directly.\n\nBenefits of Entity-Based Staging\n\nThis new staging model has the following significant benefits over the three-common-table structure of the previous MDS release:\n\n **Strongly typed columns** Now that each attribute in the entity has a corresponding column in the table, these staging columns can match the type and length of the defined attribute. This can provide rapid feedback about any truncation or data type issues when loading the staging table.\n\n **Increased performance** Using common tables meant that all SQL statements in the underlying code needed to be dynamic. This limited SQL Server's innate ability to create query plans, extremely limiting performance.\n\n **Securable tables** Using common tables to load data into MDS meant that security was an all-or-none alternative. Users who had the right to load members into one entity or attribute could also load new entity members or update information in entities they had restricted access to. Unique tables for each entity allow database administrators to leverage SQL Server security to lock down access on a per-entity and per-type basis.\n\n **Better integration** By modeling the MDS entity table structure in the staging tables, integrators are able to better leverage SSIS and other ETL solutions to load data into MDS without significant data manipulation.\n\nBatch Tags\n\nIn the previous release of MDS, users were required to navigate a combination of user columns and sweeping procedures in order to manage batches within the system. Many people modified the tblStgBatch table directly in order to control batch IDs effectively. SQL Server 2012 introduces a new batch tag concept to manage data loads. Batch tags add the ability to uniquely tag their data with any character combination up to 50 characters. This tag can then be passed in the load process. For instance, if there is a nightly load of SAP data into MDS, each ETL process could tag the loaded data with a SAP < _Load Date_ > batch tag combination, as shown next. This tag would be stored along with the system-assigned batch ID to provide direct history of what data was loaded into each entity each night.\n\nSecurity in Staging\n\nWhile MDS is a feature of SQL Server, MDS does not rely on the SQL Server security model to secure data within the system. In Chapter 11 we will go into the details of the MDS security model, but in this chapter it is important to be aware that giving access to SQL objects within the MDS database must be handled with care. In SQL Server 2012, securable objects have been broken out into the stg schema. You can secure each of these objects for users who will use staging.\n\nStaging and subscription views are separate from the web-based security model that is enforced in the web UI. Subscription views are SQL views used by subscribing systems to get data from MDS. These views are described in detail in Chapter 12.\n\nThe use of the staging tables and the consumption of subscription views are intended for advanced system batch integration. To ensure that these processes can be completed as quickly as possible, database objects lie outside of the security model and rely on database administrators to restrict access using SQL Server security roles to limit access to any sensitive information stored in MDS. We will discuss the securing of these database objects in Chapter 11.\n\nSystem Settings\n\nBefore you load data into the staging tables, confirm the following staging system settings that are set in Master Data Services Configuration Manager, which you launch from Programs | Microsoft SQL Server 2012 | Master Data Services | Configuration Manager.\n\n **Log all staging transactions** Set this to On to have transactions logged when the staged records are loaded into the appropriate Master Data Services database tables.\n\n **Staging batch interval** When you set the staging process to start, this setting determines the number of seconds afterward that your batch is processed. The default setting is 60 seconds (1 minute).\n\nAbout Main Street Clothing Company's Staging Process\n\nMain Street Clothing Company has created what it believes to be a basic, functional model structure. Now Anthony Green needs to ensure that all the requisite information is loaded, to ensure that all products can be staged into the system.\n\nAnthony determines that the best method for addressing new attribute values will be a manual process managed by the Product Procurement team. This process will be managed through the use of business rules, as discussed in Chapter 8.\n\nAnthony compiles a summary of the attribute values that need to be loaded before the company's current product list can be loaded. He reviews each attribute to determine if there is any additional data associated with these values that should be managed as well. For each attribute, he will need to create or load several members to be used as domain-based attribute values.\n\nSince none of these attribute values requires advanced management nor has over 15 values, Anthony determines that loading these records manually is the most efficient method. A few of the more technical manufacturers have discussed system integrations to provide electronic access to new product lines. As these systems become available, Anthony will review the necessary ETL processes to automatically load new products and their requisite domain-based values.\n\nIf Anthony had decided to use the staging process, he could have created a spreadsheet similar to those available at www.mdsuser.com.\n\nStandard Leaf and Parent Staging Fields\n\nA staging table will be added to the stg schema in the MDS database for each entity that is created. This initial table will be named using the Name for Staging Tables input provided when you create the entity. This name cannot be changed using the UI once you have created the entity.\n\nThe following table describes the common columns in each leaf and parent staging table.\n\nStaging Import Types\n\nBecause entity-based staging loads all attributes for a given entity's members simultaneously, a number of import options had to be created. These options load the data provided into MDS based on the import type provided for each member record. Since different import types can be loaded simultaneously and the results of each type can affect data drastically, it is imperative to ensure the correct import type is passed with each member record.\n\nMember Staging Examples\n\nYou can populate the tblStgMember table by using T-SQL statements or by using SQL Server Integration Services. In our SSIS examples, we'll import a CSV file, but there are a variety of options available, including SQL Server, Access, Excel, and others.\n\nThe examples in this section are meant to illustrate the staging process, but they do not necessarily build on the Product model that we've laid out so far. You can visit www.mdsuser.com to download the CSV files you need to stage in all of the members, attributes, and collections for the Product and Finance models. You can get those files now and compare them to the examples, or try creating your own sample products.\n\nUse the following examples to create a new leaf member, update a leaf member, delete a leaf member, purge a leaf member, create a consolidated member (needed only for explicit hierarchies), and create a collection (a different type of grouping that your organization may or may not use). When using a CSV file to stage members, the first row in the file should contain the column names. Collection member management will use the prior version staging tables to add or update collection members into the system through staging.\n\nCreate a Leaf Member\n\nIn this example, we are going to specify the member code required to create a leaf member. We are not going to specify the member name, because the name is not required. In Chapter 8, we'll show you how to use business rules to create a concatenated member name. Since the new staging table schema can load attributes with the members, all attributes could have been added to this example. To ensure readability and brevity, only a few attributes have been added to the following examples unless otherwise specified.\n\nAlso in these examples, we no longer need to provide a username with each member record, as was required in the prior MDS release. This process has been replaced with the batch tag. Batch tags can be used to separate loads of data into the system, and SQL Server security can be used to control access to each staging table, stored procedure, and error view.\n\nUpdate a Leaf Member\n\nFollowing are two import examples that show you how to update a leaf member with the name _ and the code 1001. In the first example, we only want to update certain attributes and do not need to update any attributes to NULL. In the second example, we want the member record to exactly match the values we are loading. Be very careful when loading data using this method, as any new attributes will be erased from the system. (They will default to NULL in the staging table and load as NULL into MDS.)\n\nThis import example shows the partial attribute update:\n\nThis example shows the exact member update:\n\nDelete a Member\n\nThere are two options to soft delete members from MDS using staging. An import type of 3 soft deletes a member from the system only if the member is not referenced by any other entity that has a relationship to the member. An import type of 5 removes these references, setting these related members' attribute values to NULL. All soft delete actions from staging only require the ImportType, ImportStatus_ID, BatchTag, and Code columns to have values. All other data provided in the row will be ignored.\n\nThis example shows how to soft delete data with no references:\n\nPurge a Member\n\nIn SQL Server 2012, you can now purge members from the system. While soft delete retains the member history and does not allow the code to be reused within the entity for the current version, purge completely removes the member and all history for the current version. There are two options to purge members from MDS using staging. An import type of 4 purges a member from the system only if the member is not referenced by any other entity that has a relationship to the member. An import type of 6 removes these references, setting these related members' attribute values to NULL. All purge actions from staging only require the ImportType, ImportStatus_ID, BatchTag, and Code columns to have values. All other data provided in the row will be ignored.\n\nThis example purges data with no references:\n\nThis example purges data and removes references:\n\nCreate a Consolidated Member\n\nThis example shows you how to create a consolidated member with the name Short Term Assets and the code STA for the Account entity.\n\nCreate a Collection\n\nThis example shows you how to create a collection with the name Boys Winter Set and the code BWS. Unfortunately, collections are not handled with the new entity-based staging. To manage collections using staging in SQL Server 2012, you need to use the old, deprecated tables from the prior release. This collection won't have any members in it. It is simply creating a collection with this name and code. To add members to collections, you can use the TblStgRelationship staging table.\n\nCollections have a MemberType_ID of 3 to distinguish them from leaf members (1) and consolidated members (2).\n\nUpdate a Collection's Attribute\n\nThis example updates the Description attribute of the Boys Winter Set (BWS) collection to read \"Boys 2011 Winter Set.\" If you have not defined additional attributes for your collections, the only attributes you can update are Owner_ID, Description, Name, and Code.\n\nAdd One Collection to Another\n\nThis example adds the Boys Winter Set (BWS) collection to the Fall and Winter Sets (FWS) collection.\n\nAdd a Leaf Member to a Collection\n\nThis example adds leaf member 1052 to the Boys Winter Set (BWS) collection.\n\nAdd a Consolidated Member to a Collection\n\nThis example adds the Boys (B) consolidated member to the Fall and Winter Sets (FWS) collection. The TargetType_ID of 5 tells the system that you are setting a collection member.\n\nUsing Staging to Deactivate Members\n\nThe soft delete process in the leaf and consolidated entity-based staging has replaced the deactivation process of the prior release of MDS. Unfortunately, there is no way to reactivate a member through the new staging process. When you deactivate a member, the member code no longer changes to a 32-character GUID, making that code unavailable for use. The purge import type will solve this issue. Entity-based staging does not support reactivating the member. This can be useful if you want to restore a member's attributes and its membership in hierarchies and collections. If you want to reactivate a member, you need to leverage the deprecated tblMemberAttribute table.\n\nThis procedure no longer requires that you look in database tables for information, as the code is not updated to a GUID. In Chapter 7 we will show you how to delete and reactivate members in the Master Data Manager web application. Using the web UI is simpler than using staging and leaves less room for error. The new purge functionality is only available in staging, and it should be noted that purging the member will cause you to lose attribute assignments and hierarchy relationships as well as any history stored within the selected purge version.\n\nExample: Reactivate a Member\n\nThis example shows how to reactivate a member that was deactivated. Note that the member code is now the code and has not changed to a 32-character GUID.\n\nLoading Relationships\n\nThe last staging table type within MDS entity-based staging is the relationship table. This table will be created only if the entity is hierarchy enabled. You use the stg.< _Staging Name_ >_Relationship table to load explicit hierarchy relationships. Explicit hierarchy relationships can still be entered in two ways. You can load data as a parent relationship, or you can load data as sibling relationships.\n\nIn most cases, the easiest way to load data into MDS is to load it in parent-child format. Many systems care not about the order of members within the hierarchy but rather about the consolidations themselves; in these cases, parent-child format is the easiest way to load records through staging. When parent-child records are loaded through staging in this manner, the sort order is triggered in a standard way. Any new children of a consolidated member in the system are placed as the last siblings in the tree, similar to how birth order works in a family tree for humans.\n\nIf sort order matters and you cannot control the order of data within your staging load, you can load records into MDS using the sibling type. Using 2 as the relationship type, new relationships are created at the same level of the hierarchy as the next sibling to the target code specified. This allows you to directly set the order of records in an explicit hierarchy without reloading the entire hierarchy.\n\nCreating a new record as the first sibling of any parent requires a two-step process. The first step is to load the new relationship as a sibling to the first child of the target parent. The second step is to set the first child as a sibling to the just loaded member. This reverses the order of these two child members.\n\nRelationship Table Fields\n\nUse stg.< _Staging Name_ >_Relationship to move members in explicit hierarchies. You cannot use this table for updating derived hierarchies or for adding or removing members to or from collections. To update derived hierarchies, you must update the attribute values that determine the hierarchy. For more information about hierarchies and collections, refer to Chapter 6.\n\nThe following table describes the fields in tblStgRelationship.\n\nRelationship Staging Examples\n\nThe following examples show how to populate the Account Relationship staging table by using T-SQL statements or by using SSIS to import a CSV file. Again, many of the members referred to in these examples do not exist yet in the samples we've provided, and you will get errors if you try to update a member that doesn't exist. Use these examples to get ideas for how to stage your own data, or use the staging spreadsheets we've provided at www.mdsuser.com.\n\nSet a Parent-Child Relationship in an Explicit Hierarchy\n\nThis example sets the Short Term Assets (STA) consolidated member as the parent of the Cash member in the Chart of Accounts explicit hierarchy.\n\nTIP\n\n_If you want to specify the Unused node of an explicit hierarchy as the parent, use MDMUNUSED for the TargetCode. If you want to assign a hierarchy member to the root, use ROOT._\n\nSet a Sibling Relationship in an Explicit Hierarchy\n\nThis example sets Long Term Assets at the same level as Short Term Assets in the Chart of Accounts explicit hierarchy.\n\nAdd One Collection to Another\n\nThis example adds the Boys Winter Set (BWS) collection to the Fall and Winter Sets (FWS) collection. This is a deprecated table, but it is the only way to use staging to update a collection member in SQL Server 2012.\n\nAdd a Leaf Member to a Collection\n\nThis example adds leaf member 1052 to the Boys Winter Set (BWS) collection.\n\nAdd a Consolidated Member to a Collection\n\nThis example adds the Boys (B) consolidated member to the Fall and Winter Sets (FWS) collection.\n\nImporting Data into the Staging Tables\n\nYou may already have decided how to get your data into the MDS database's staging tables. If you haven't, we'll show you how to do it by using SQL Server Integration Services. After you import your data into the tables, you can proceed with running the import, which puts the data from the staging tables into the appropriate MDS database tables. You can load files that are provided for later chapters by using the following process.\n\nProcedure: How to Import Data into SQL Server by Using SSIS\n\nIn this procedure, we are going to use a flat CSV file with a header row that includes column names. To import a CSV file into the SQL Server MDS database staging tables by using SSIS, complete the following steps:\n\n**1.** Open SQL Server Management Studio.\n\n**2.** Right-click the MDS database.\n\n**3.** On the submenu, select Tasks | Import Data.\n\n**4.** On the Welcome page, click Next.\n\n**5.** On the Choose a Data Source page, select Flat File Source as your data source.\n\n**6.** Click the Browse button to browse to the file. (You must select CSV as the file type to view your files. TXT is selected by default.)\n\n**7.** Select the \"Column names in the first data row\" check box.\n\n**8.** Click Next.\n\n**9.** On the next Choose a Data Source page, confirm that the preview of your data looks accurate and click Next.\n\n**10.** On the Choose a Destination page, confirm that your MDS database is selected and click Next.\n\n**11.** On the Select Source Tables and Views page, ensure that you select the appropriate staging table. You should load data only into tables in the stg schema. If you are not sure which table to load into, you can check in the Manage Entities screen in System Administration functional area. You can also click the Preview button at the bottom of this screen to preview your data.\n\nCAUTION\n\n_The correct table is not selected by default. Be careful not to click Next without changing the destination table._\n\n**12.** Click Next.\n\n**13.** On the Review Data Type Mapping page, click Next.\n\n**14.** On the Save and Run Package page, click Next.\n\n**15.** On the Complete the Wizard page, click Finish.\n\nWhen the process is complete, a message is displayed, stating that the execution was successful.\n\nInitiating the Staging Process\n\nAfter you have populated the staging tables with your data, you must run the batch process that loads the data from the staging tables into the MDS-specific tables. In SQL Server 2012 Master Data Services, Microsoft has created a specific staging stored procedure for each entity-based staging table.\n\nTo initiate the staging process, you can use\n\n The Master Data Manager web application\n\n A stored procedure in the MDS database\n\nIn an effort to align with other SQL Server features and to provide to users an efficient method to load data into MDS, the new web services for entity-based staging are not supported for customer use. Data stewards should plan to move all of their MDS staging processes to leverage the appropriate stored procedures.\n\nProcedure: How to Use Master Data Manager to Initiate the Staging Process\n\nTo use the Master Data Manager web application to initiate the staging process after you've loaded data into the MDS staging tables, complete the following steps:\n\nNOTE\n\n_In the latest release of MDS, staging batches are now processed by entity, and processing begins at an interval determined by a setting in Master Data Services Configuration Manager. As a data steward, you are responsible for initiating the processing of each table individually, so you should import only the records you intend to load in the proper order (for example, import colors before importing products set to that color)._\n\n**1.** On the Master Data Manager home page, click Integration Management. On the initial Import Data page, all batch tag grouped records that have not been cleared are displayed for the selected model. Even with identical batch tags, entity and type will be broken out in the table.\n\n**2.** Select the model and then select a batch tag from the table to load into the system. The total number of records to be imported should be displayed in the Records column.\n\n**3.** Above the batch list, click the Start Batches button.\n\n**4.** In the Run Selected Batches dialog box, shown next, select the version to load selected batches into.\n\nNOTE\n\n_Committed versions are available in the list even though you will not be able to load members into those versions. If the number of records to be imported does not seem correct or the batch tag is missing, ensure that your batch tag was properly entered, that the Batch_ID field for each record in the table is NULL, and that the ImportStatus_ID field for each record is 0._\n\nClick OK. The batch is assigned an ID, and the Status column is updated to QueuedToRun.\n\nTIP\n\n_On the menu bar, click Import Data to refresh the page until the Status column is updated._\n\nWhen the batch has finished processing, the Status column is updated to Not Running and errors are listed in the Errors column.\n\nYou can go to Explorer functional area to view your staged members, or open the Staging Batch Errors view to view the details of your errors.\n\nProcedure: How to Use Stored Procedures to Initiate the Staging Process\n\nAfter you have loaded data into the staging tables by using either T-SQL or SSIS, using the web UI or web services to trigger a staging load process can be inconvenient. While we do not recommend coding against stored procedures as a general rule, the benefits outweigh the risks in this case.\n\nYou can trigger logging of transactions or validation by using staging stored procedures. The stored procedures, described in the following section, provide access to additional functionality that is not exposed in the UI. These stored procedures are also more accessible when loading data through SSIS.\n\nNOTE\n\n_Microsoft does not guarantee that these interfaces will remain constant, nor does Microsoft provide any backward compatibility at the stored procedure level._\n\nThe End of Staging Sweep\n\nThe new batch tag approach has replaced the staging sweep process used in the prior release of MDS. While the deprecated three-table approach still requires staging sweep use, all new staging tables automatically receive a batch ID upon triggering the load process.\n\nUsing Web Services to Stage Data\n\nThere are a number of calls within the web service that leverage the staging tables and processes to load data into Master Data Services. Some of these calls provide support for bulk loading of data. These calls manage the entire process of populating the staging tables and triggering the staging process. The structure of these calls will be discussed in greater detail in Chapter 13.\n\nIn SQL Server 2012, all of the new staging calls have been tagged for \"Internal Use Only.\" These calls were not created or tested for external consumption. The new staging tables provide a much more compelling integration with SSIS using the new stored procedures to trigger staging loads, and staging batches can be created when loading the staging tables.\n\nErrors That Occur During Staging\n\nSometimes errors occur during the staging process. As each row in a staging table is processed, the Status_ID column is updated with a 1 if staging succeeded and a 2 if it failed. If staging failed, the ErrorCode column is updated with a new (to SQL Server 2012) complex error code that can help you troubleshoot the issue. This number is useful to humans only through the new staging error view created for each entity.\n\nIn SQL Server 2012, all errors are now determined for a record in the batch that is processed. In the new staging error view created for each entity, these errors are expanded into a new row for each error that is found on the record.\n\nYou can no longer see a translation of these error messages in the Master Data Manager web application; however, you can save a SQL script to the Clipboard to review in SQL Server Management Studio.\n\nProcedure: How to View Staging Errors in Management Studio\n\nTo view details of your staging errors in SQL Server Management Studio, complete the following steps:\n\n**1.** On the initial import data screen, click the row for the batch with errors you want to explore.\n\n**2.** Click the Copy Query button to load the SQL query onto the Clipboard. If this is your first time using this feature, Silverlight may ask permission to write to the Clipboard. If so, click Yes to continue.\n\n**3.** Open SQL Server Management Studio and connect to the database server running the MDS database.\n\n**4.** Open a new query window and press ctrl-v.\n\n**5.** Click the Execute button at the top of the screen. You will notice that many more errors than original member records can be present, as a row in the view has been created for each error found by the staging process.\n\nStaging Errors View\n\nTo help you better manage errors in your staging batches, Microsoft includes in the latest release of MDS the errors view table, the columns for which are listed and described in the following table. A single view shows errors for all staging table types for an entity. Filtering this view by the appropriate batch ID is the best way to see your issues effectively. (This is what copy query provides.)\n\nViewing the Staging Batch Table\n\nThe last table specifically related to staging is the batch table, mdm.tblStgBatch. This table stores information related to each load of data imported into the MDS system. Summary information related to the success and failure of each record is stored in the batch table indefinitely. As this table is not part of the stg schema, users should not modify this table directly.\n\n*The Service Broker queue is used only as an asynchronous timer. Generic staging messages are placed on the queue and staging batches are triggered by the Queued to Run status only.\n\nProcedure: How to Clear the Staging Queue\n\nTo clear a batch from the staging queue, complete the following steps:\n\n**1.** On the Master Data Manager home page, click Integration Management.\n\n**2.** In the pane, click the row for the batch you want to clear.\n\n**3.** Click the Clear Batches button.\n\nThe value in the Status column changes to Queued to Clear. When it's done processing, the Status column changes to Not Running. This can be considered completed for any batch that is displayed.\n\nSummary\n\nIn this chapter, we discussed techniques for using staging tables to load data into Master Data Services efficiently and effectively. We discussed the changes to the staging process in SQL Server 2012 and the benefits of entity based staging. We reviewed examples of using SQL Server Integration Services or standard SQL calls to populate records into the new staging tables. Within the discussion of staging tables, we also discussed types of errors that can occur during the data loading process from the staging into MDS and how to use the new error views provided in this release. We provided likely solutions to these errors. In the next chapter, we will discuss hierarchy types in greater detail.\nChapter 6\n\nWorking with Hierarchies and Collections\n\nIn This Chapter\n\n Ragged vs. Level-Based Hierarchies\n\n Creating Derived Hierarchies\n\n System Settings for Hierarchies\n\n Creating Explicit Hierarchies\n\n Recursive Hierarchies\n\n Derived Hierarchies with Explicit Caps\n\n Collections\n\n Creating Hierarchies by Using Web Services\n\nBusinesses are not built only on lists; they require more complex structures for data. Dollars, units, and hours must all be calculated to produce consolidated views of a business. For many years, companies have been using business intelligence (BI) applications to better understand their businesses and discover opportunities for both cost savings and revenue growth. At the heart of these applications are the hierarchies that make consolidations possible.\n\nWhen managing consolidations within a business, it is imperative that all values are accounted for once and only once. To ensure this, hierarchies in Master Data Services enforce that all leaf members have one and only one parent. This limits your ability to manage many-to-many relationships within MDS and display these relationships hierarchically.\n\nIn this chapter, we review the two primary types of hierarchies supported in MDS. We then create these derived and explicit hierarchies within our models. Finally, we create a number of special hierarchies and discuss the various uses for these special hierarchies.\n\nBefore you start this chapter, go to www.mdsuser.com and download the ZIP file for Chapter 6, called chapter6.zip. This ZIP file contains the following files that you will use in this chapter:\n\nThe Product model represents everything you created in Chapter 4, but it includes several hundred product members with their attributes already populated. This gives you a fair amount of data to work with and will help the samples in this chapter make more sense. The Finance model is similar insofar as it is fully populated with data and contains the structure you'll need to work with the hierarchies we create in this chapter.\n\nRagged vs. Level-Based Hierarchies\n\nWhen you're working with hierarchies, it is important to understand some common terms used across applications. Any hierarchy can be either ragged or level based. _Ragged_ describes hierarchies that support leaf members at multiple levels. A hierarchy is _level based_ if leaf members always exist at the same level, regardless of the number of levels within the hierarchy.\n\nSome business applications do not support ragged hierarchies, so it is important to know the limitations of downstream systems before you create hierarchies in MDS.\n\nDerived Hierarchies\n\nIn MDS, derived hierarchies provide the ability to highlight preexisting data relationships within entities and display them hierarchically. Derived hierarchies are based on the preexisting data relationships within MDS. In SQL Server 2012, derived hierarchies have been expanded to allow more complex relationships to be managed effectively. It is clear that as the MDS product advances, derived hierarchies will continue to evolve to handle all data relationships, and explicit hierarchies will lose importance. Derived Hierarchies will continue to be expanded past the definition that every level within the hierarchy corresponds to a specific domain-based attribute within the entity\u2013attribute chain (see Figure 6-1).\n\n**Figure 6-1** _A derived hierarchy, created by using a domain-based attribute_\n\nExplicit Hierarchies\n\nExplicit hierarchies are multilevel hierarchies with very few restrictions. Explicit hierarchies are managed as name\u2013value pairs, with consolidated members containing other consolidated or leaf members. Explicit hierarchies are created for one entity at a time. Derived hierarchies, in contrast, require multiple entities.\n\nConsolidated members are almost always theoretical items, whereas leaf members represent physical items. Like leaf members, consolidated members can have attributes assigned to them. In the MDS database, a separate table exists to manage consolidated members and their associated attributes. Consolidated members are available only if an entity is enabled for explicit hierarchies. Each consolidated member can be associated with only a single hierarchy no matter how many explicit hierarchies have been created for the entity.\n\nDerived vs. Explicit: Which Hierarchy Is Best?\n\nDerived hierarchies are determined by the structure of the model, and changes to the structure are rare. To illustrate the value of this rigor, consider the following scenario that organizations commonly encounter.\n\nOur fictional company devises a regional hierarchy to manage sales. As with most of these hierarchies, this hierarchy begins as a level-based hierarchy, with each level within the hierarchy corresponding to a distinct type of attribute. To store the hierarchy, IT uses the parent-child format from its analysis system.\n\nOver time, this hierarchy is modified by midlevel managers to help them manage their divisions better. Changes are not centrally managed, and new levels are added monthly. John is the manager of the Western region of the company. John has two managers who split duties managing the Southwest division for him: Bill and Margaret. Since Bill and Margaret are splitting the Southwest, John needs to split divisional data in the company's reports to measure this divisional structure. To do this, he creates two additional nodes within the hierarchy for the Southwest division, SWB (Southwest Bill) and SWM (Southwest Margaret).\n\nOver time, Bill and Margaret move on to other jobs, either moving up in the company or on to other opportunities. Yet the Southwest division continues to be split into SWM and SWB. This is not an isolated occurrence, as many managers make isolated changes to the hierarchy structure. Some of these changes may not be warranted, and others may have a short shelf life. Soon the initial hierarchy is unrecognizable and difficult to manage. The ability to provide managers with rigid derived hierarchies, malleable explicit hierarchies, and focused collections allows BI professionals to provide the perfect tool for each scenario.\n\nCreating Derived Hierarchies\n\nDerived hierarchies require that you have one or more entities, each of which has a domain-based attribute that you want to group members by. In MDS, you cannot group members by more than one domain-based attribute at a time. This doesn't mean that your derived hierarchy can't have multiple levels, just that each level must be based on a domain-based attribute relationship.\n\nProcedure: How to Create a Derived Hierarchy\n\nIn this example, you'll create a derived hierarchy that has two levels. You will roll up products by gender. Product will be at the lowest level in the hierarchy, and Gender will be one level higher.\n\nTo create a two-level derived hierarchy, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Derived Hierarchies.\n\n**3.** From the Model list, select the model if it isn't selected already. In this case, select Product.\n\n**4.** Click the \"Add derived hierarchy\" button.\n\n**5.** In the \"Derived hierarchy name\" field, type a descriptive name for your hierarchy. In this case, we're going to name the hierarchy **Product by Gender**.\n\n**6.** Click the \"Save derived hierarchy\" button. The page that is displayed has three panes.\n\nOn the left are the entities that can be added to the current level in the hierarchy. In the center are all the entities in the derived hierarchy; right now there are none because you haven't added any yet. Each time you drag an entity from the left pane to the center pane, the left pane refreshes with the list of entities that remain available to add.\n\nYou should always drag the lowest-level entity first. In this case, we want a hierarchy with products at the lowest level.\n\n**7.** In the left pane, click and drag Product to the \"Current levels\" label in the middle pane. Release the mouse button and the screen refreshes, with Product displayed in the Current Levels pane.\n\nThe Preview pane refreshes to show a list of products.\n\n**8.** Now you can drag the entity you want to group by. In this case it's Gender, so click Gender in the left pane and drag it to the Current Levels pane.\n\nThe page refreshes and the Preview pane shows all products grouped by gender.\n\nNOTE\n\n_The hierarchy is saved as you build it. No Save button is available on this page._\n\nIf you decide you do not want to group by gender, click Gender in the Current Levels pane and then click the Delete button that's displayed at the bottom of the pane. If you decide you want to rename your hierarchy, click the \"Edit selected item\" button on the top left of this page.\n\nThe real value of a derived hierarchy occurs when new products are created. You can configure a business rule so that each time a new product is added, if the Gender attribute is not populated, an e-mail notifies all interested parties. Business rules will be covered in more detail in Chapter 8.\n\nProcedure: How to Create a Three-Level Derived Hierarchy\n\nNow you are going to create a derived hierarchy with three levels. After completing this procedure, you should be able to build your own multilevel hierarchies.\n\nIn the Product model you have built so far, you don't have the ability to create a three-level derived hierarchy. This is because the Product entity is the only entity that has domain-based attributes. To create a hierarchy with multiple levels, each entity in the hierarchy must be used as a domain-based attribute of another entity.\n\nFor example, right now you can roll up products by Gender, Size, Type\u2014any of Product's domain-based attributes. But you can't roll up products by size and _then_ by gender. You have to choose one or the other.\n\nIn this example, you will create a derived hierarchy that rolls up products first by product line, then by manufacturer. This will require two relationships:\n\n The Product entity must have a Line domain-based attribute.\n\n The Line entity must have a Manufacturer domain-based attribute.\n\nWhen you're done creating the structure, you will be able to roll up Product by Line by Manufacturer. The Product model doesn't have this structure right now, so let's create it.\n\nCreate a Manufacturer Entity\n\nCreate an entity called Manufacturer and populate it with six members. (Note: It does not need to have explicit hierarchies enabled.) You can get these members from chapter6.zip on www.mdsuser.com. Import 1Manufacturer.csv into stg.Manufacturer_Leaf and use Integration Management to start the staging process. Or, create these members manually in Explorer.\n\nFor the Product Entity, Create a Line Domain-Based Attribute\n\nNow, for the Product entity, you're going to create a domain-based attribute that's based on the Line entity:\n\n**1.** In Master Data Manager, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** From the Model list, select Product.\n\n**4.** Click the Product entity and then the \"Edit selected entity\" button.\n\n**5.** In the Leaf attributes section, click the \"Add leaf attribute\" button.\n\n**6.** Choose Domain-based, type **Line** for the name, and select Line for the source entity.\n\n**7.** Click the Save attribute button.\n\nFor the Line Entity, Create a Manufacturer Domain-Based Attribute\n\nNow in System Administration, perform the same steps as you just did, but edit the Line entity instead. Add a leaf attribute, domain based, called **Manufacturer** , based on the Manufacturer entity.\n\nCreate a Line Entity and Assign Manufacturer to Each Line\n\nCreate an entity called **Line** and populate it with 14 members. (Note: It does not need to have explicit hierarchies enabled.) You can get these members and their associated Manufacturer from chapter6.zip on www.mdsuser.com. Import 2Line.csv into stg.Line_Leaf and use the Integration Management functional area to start the staging process. Alternatively, create the members manually by referring to the following table. Remember, you create the entity in System Administration and add members in Explorer.\n\nAssign a Line to Each Product, and a Manufacturer to Each Line\n\nYou can use 3Product.csv from www.mdsuser.com to assign a Line attribute to each product. Remember that attributes are now imported with the members in the stg.Product_Leaf staging table and you use Integration Management to start the staging import process. Or, you can assign attributes manually in Explorer.\n\nIf you decide to do this manually, here are a few things to consider:\n\n The Line attribute is not part of an attribute group yet, so it will not be displayed in Explorer. Add it to the Marketing attribute group if you want to edit it.\n\n You can edit members individually in the UI or use the MDS Add-in for Excel to perform bulk edits.\n\nYou are now ready to create your three-level derived hierarchy.\n\nCreate the Derived Hierarchy\n\nCreating the hierarchy should be the easiest part of the process:\n\n**1.** In System Administration, on the menu bar, choose Manage | Derived Hierarchies.\n\n**2.** Ensure Product is selected in the Model list and click the \"Add derived hierarchy\" button.\n\n**3.** For the name, type **Product by Line by Manufacturer**.\n\n**4.** Click and drag Product to the Current Levels pane. A list of products is displayed in the Preview pane.\n\n**5.** Click and drag Line to the Current Levels pane and drop it on Product.\n\n**6.** Manufacturer is the only entity available to add to the hierarchy. Click and drag it to Line in the Current Levels pane. When you release the mouse button, the full hierarchy is displayed in the Preview pane.\n\nYou can now expand and collapse nodes in the hierarchy to get an idea of the structure.\n\nHiding Levels\n\nIf you click any of the levels in the Current Levels pane, you can hide the level from users by selecting No from the Visible list. This can be useful when you have a multilevel hierarchy structure and users don't need to see many of the middle-tier levels when working with the hierarchy.\n\nYou can also change the name of a hierarchy level by updating the value in the Column name field.\n\nSystem Settings for Hierarchies\n\nThere are two settings in Master Data Services Configuration Manager that apply to hierarchies and collections. Now that you know more about what hierarchies look like, you might want to change these.\n\n **Number of members in the hierarchy by default** This setting determines how many members are displayed in a hierarchy node before a label named...more... is displayed. The default setting is to show 50 members.\n\n **Show names in hierarchy by default** This setting determines whether the name and code of each member are displayed or only the code is displayed. The default setting is to show both name and code.\n\nCreating Explicit Hierarchies\n\nThere are two types of explicit hierarchies in MDS: mandatory and non-mandatory. Before you create an explicit hierarchy, you should determine which type you'll need.\n\nNon-mandatory Explicit Hierarchies\n\nWhen you create a non-mandatory explicit hierarchy, all children are located in a node named Unused, as shown in Figure 6-2. Leaf members in the Unused node do not show up in subscription views based on the explicit hierarchy. This means that when you use subscription views as source data for subscribing systems, all the members in the Unused node are excluded. For more information on subscription views, see Chapter 12.\n\n**Figure 6-2** _A non-mandatory explicit hierarchy, where all leaf members start in the Unused node_\n\nNon-mandatory explicit hierarchies are easier to work with when you have large numbers of leaf members. To build an explicit hierarchy, you must create consolidated members, which are displayed at the root by default. You can then move leaf members from the Unused node to the consolidated members that live at the root.\n\nMandatory Explicit Hierarchies\n\nWhen you create a mandatory explicit hierarchy, all leaf members are initially located at the root of the hierarchy. As you create consolidated members, they are also added to the root, but they are displayed at the end of the list. To find them, you must search or scroll to the end of the hierarchy. Figure 6-3 shows a mandatory explicit hierarchy where consolidated members exist but are not displayed because they follow hundreds of leaf members.\n\n**Figure 6-3** _A mandatory explicit hierarchy, where all leaf members start at the root_\n\nProcedure: How to Create an Explicit Hierarchy\n\nTo create an explicit hierarchy, you are going to use the Finance model, which, if you deployed it from the web site, includes a full list of sample accounts. To create an explicit hierarchy, you complete the following tasks:\n\n**1.** Enable the Account entity for hierarchies and collections.\n\n**2.** Populate the consolidated members for the Account entity.\n\n**3.** Create the explicit hierarchy.\n\nEnable the Entity for Hierarchies and Collections\n\nBefore you can create explicit hierarchies, you must always enable the entity for explicit hierarchies by completing the following steps:\n\n**1.** In Master Data Manager, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities to open the Entity Maintenance page.\n\n**3.** From the Model list, select Finance.\n\n**4.** Click Account and then click the \"Edit selected entity\" button.\n\n**5.** From the \"Enable explicit hierarchies and collections\" list, select Yes.\n\n**6.** In the \"Explicit hierarchy name\" field, type **Chart of Accounts**.\n\n**7.** You are going to create a non-mandatory explicit hierarchy where all leaf members start in an Unused node, so clear the \"Mandatory hierarchy (all leaf members are included)\" check box.\n\n**8.** Click the Save entity button.\n\nIf you decide in the future that you want additional explicit hierarchies, edit the Account entity. You can now add explicit hierarchies, attributes for consolidated members, and attributes for collections from this page.\n\nPopulate the Consolidated Members for the Account Entity\n\nYou can use 4account.csv from www.mdsuser.com to populate the consolidated members (by using stg.Account_Consolidated), or you can complete the following steps in the Master Data Manager web UI:\n\n**1.** On the MDS home page, select Finance and VERSION_1 from the Model and Version lists, respectively, and click Explorer.\n\n**2.** On the menu bar, choose Hierarchies | Explicit: Chart of Accounts\n\n**3.** Above the grid, select the Consolidated members option from the drop-down list, which is new in SQL Server 2012 MDS.\n\n**4.** The drop-down list is blank. Click the Add button above the grid and add the following consolidated members (or create your own):\n\nCreate the Explicit Hierarchy\n\nThe hierarchy structure was created when you enabled explicit hierarchies for the Account entity. Open Explorer to view it in more detail.\n\n**1.** On the home page, select Finance and VERSION_1 from the lists and click Explorer.\n\n**2.** On the menu bar, choose Hierarchies | Explicit:Chart of Accounts.\n\nWhen you first open the hierarchy, only Root and Unused are displayed. If you expand Unused, all of the Account leaf members are displayed. If you click Unused, the grid on the right refreshes to display all of the Account leaf members.\n\nYou can now use 5account_relationship.csv with stg.Account_Relationship to move members in the hierarchy, or use the procedures in Chapter 7 that explain how to move hierarchy members. Figure 6-4 shows part of the hierarchy that will be created if you use the spreadsheet from www.mdsuser.com.\n\n**Figure 6-4** _An explicit hierarchy that shows a chart of accounts_\n\nRecursive Hierarchies\n\nA recursive hierarchy is a special type of derived hierarchy. In a typical derived hierarchy, the members in an entity are grouped by a domain-based attribute. In a recursive hierarchy, the members are still grouped by a domain-based attribute, but this domain-based attribute is based on the entity itself. For example, each employee in an organization has a manager. And the manager can be any one of the employees. The following procedures show how to use this employee-manager relationship to create a recursive hierarchy that shows Main Street Clothing Company's organizational structure.\n\nProcedure: How to Create a Recursive Hierarchy\n\nTo create a recursive hierarchy, you are going to use the Finance model, which, if you deployed it from the web site, includes a full list of sample employees. To create a recursive hierarchy, you are going to complete the following tasks:\n\n**1.** Create a domain-based attribute for the Employee entity. This attribute will be named Manager and will be based on the Employee entity.\n\n**2.** Assign a manager for each employee or load a list of Manager attributes.\n\n**3.** Create the recursive hierarchy.\n\nFor the Employee Entity, Create a Manager Domain-Based Attribute\n\nThe Finance model has an entity called Employee. You need to create a domain-based attribute named Manager for the Employee entity:\n\n**1.** In Master Data Manager, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** From the Model list, select Finance.\n\n**4.** Click the Employee entity and then the \"Edit selected entity\" button.\n\n**5.** In the Leaf attributes section, click the \"Add leaf attribute\" button.\n\n**6.** Choose Domain-based, type **Manager** for the name, and select Employee for the source entity.\n\n**7.** Click the Save attribute button.\n\nPopulate the Manager Attribute for Each Employee\n\nYou can use 6Employee.csv (from www.mdsuser.com) and stg.Employee_Leaf to populate each employee's Manager attribute. If you choose not to use the staging process, you can populate each employee's Manager attribute value manually by entering it in Explorer. Keep in mind that you should not populate a manager for the CEO. As long as you have one employee without a manager, you will have someone at the top of your organizational structure. If you used the Finance model provided, each employee's title is already populated in Explorer.\n\nCreate the Recursive Hierarchy\n\nAfter every employee but the CEO has a manager (or even just a few employees have one, if you want to quickly see how this works), you can create the recursive hierarchy by completing the following steps:\n\n**1.** In Master Data Manager, click System Administration.\n\n**2.** On the menu bar, choose Manage | Derived Hierarchies.\n\n**3.** From the Model list, select Finance.\n\n**4.** Click the \"Add derived hierarchy\" button.\n\n**5.** In the \"Derived hierarchy name\" field, type **Org Structure**.\n\n**6.** Click the \"Save derived hierarchy\" button.\n\n**7.** In the leftmost pane, click Employee and drag it to the Current Levels pane. When you release the mouse button, the Preview pane shows a list of employees.\n\n**8.** In the leftmost pane, click Manager and drag it to the Employee label in the Current Levels pane. When you release the mouse button, in the Preview pane, the CEO (or other employees without a manager assigned) is displayed at the highest level of the hierarchy, and the system recognizes that this hierarchy is recursive.\n\nAs you expand and collapse the hierarchy, you can see that each employee is listed only once. If you were to clear the \"Anchor null relationship\" check box, each employee would be listed individually, with any of his or her direct reports listed underneath. Anchor null relationship means that Carlos Garcia, who has a NULL value for his Manager attribute value, is considered the \"anchor\" for all other relationships. If there were multiple null relationships, these would all be displayed at the top of the hierarchy.\n\nDerived Hierarchies with Explicit Caps\n\nThe last type of derived hierarchy within MDS is the derived hierarchy with explicit cap. This is a standard derived hierarchy that has an explicit hierarchy on top. The topmost entity of the derived hierarchy must be the bottom level of the explicit hierarchy.\n\nExplicit cap hierarchies allow you to provide the best of both worlds. At the lower levels are derived levels that correspond to specific entities. At the top levels, you have the freedom to adapt the complex ragged structures allowed by explicit hierarchies. Explicit cap hierarchies allow you to define many lower bases to the same explicit hierarchy. These hierarchies are primarily used in finance domains such as Chart of Accounts and Organization Structure.\n\nA global chart of accounts allows you to map disparate financial systems into your standard chart of accounts. This mapping allows corporate employees at large organizations to have common nomenclature when discussing financials with divisions across the globe. When working within multinational corporations, the complexities of accounting are further complicated by regional accounting laws that must be enforced. Corporate controllers cannot be expected to understand the intricacies of all these accounting practices worldwide, yet they must be able to hold individual business units accountable for their financial budgets and numbers. By creating a global chart of accounts and having each individual business unit map their country's unique chart of accounts into the global master account structure, individual countries can manage finances according to their regional requirements and corporate managers can still discuss their financials in a common manner.\n\nAs you can see in Figure 6-5, a company can use an explicit hierarchy for its global chart of accounts. This allows a ragged structure that's flexible and easy to update.\n\n**Figure 6-5** _An explicit hierarchy showing a multinational corporation's chart of accounts_\n\nEach business unit can have its own standard chart of accounts that it maintains in a level-based derived hierarchy, as shown in Figure 6-6.\n\n**Figure 6-6** _A derived hierarchy for one division of the multinational corporation_\n\nWhen you combine the derived hierarchy with the explicit hierarchy, the top levels are dictated by the explicit hierarchy and the bottom levels are populated by the derived hierarchy, as shown in Figure 6-7.\n\n**Figure 6-7** _A derived hierarchy with explicit cap_\n\nProcedure: How to Create a Derived Hierarchy with Explicit Cap\n\nTo create a derived hierarchy with explicit cap, you must have your explicit hierarchy already created. You should also have an entity with domain-based attributes that you plan to use to create the derived hierarchy.\n\nThen follow the directions for creating a derived hierarchy (in System Administration, choose Manage | Derived Hierarchies). Drag the entities for the derived hierarchy over to the Current Levels pane. When your derived hierarchy is complete and the top level of the hierarchy is the bottom level of the explicit hierarchy, drag the explicit hierarchy to the Current Levels pane and drop it above the other levels.\n\nThe structure of the hierarchy looks similar to that of a derived hierarchy; the results, however, are ragged at the top and level-based at the bottom.\n\nNOTE\n\n_You can use the same explicit hierarchy as a cap for multiple derived hierarchies._\n\nCollections\n\nWhile hierarchies provide structures to manage consolidations of entire entities, you or your users might need to group a small number of members for your personal use. Master Data Services provides the ability to manage these personal consolidations through collections. Collections are containers for members from a single entity. Collections can hold leaf members, consolidated members, or other collections within them. Collections do not follow the same rules as hierarchies. Collections can contain a member twice, as long as that member is contained in separate objects within the collection.\n\nThere are many different scenarios where collections can be a valuable tool. Collections can be used to manage personal charts of account consolidations or employee shifts. The Main Street Clothing Company uses collections a few times each year to manage sets of clothing (for example, a seasonal shirt, pants, belt, and hat). Collections allow Main Street to group members without using a hierarchy where all members need to be accounted for. It is simply a combination of random members that Main Street uses for analysis.\n\nWeighting of Collection Members\n\nIn the SQL Server 2012 version of MDS, weighting has been surfaced to the UI. Weighting on collections enables users to apportion values across all members in a collection. This process can be invaluable in managing partnerships or managing sets where multiple quantity of the same member exists, such as a six-pack or a case. The default weighting for each member is 1.\n\nNOTE\n\n_Once members are added to the collection, you can set the weight for each member in the collection. Make sure you save the weight for a member before setting the next weight. Every time you save a weight for a member, all other unsaved weight changes will be lost._\n\nProcedure: How to Create a Collection\n\nIn this example, you'll create a collection that represents a set of clothes for boys. Main Street Clothing Company sells two sets every season: one set for boys and one set for girls.\n\nBefore you can create a collection, you must enable the entity to have collections. Some organizations never use collections, but Main Street uses a few. To create a collection, you complete the following tasks:\n\n**1.** Enable the Product entity for hierarchies and collections.\n\n**2.** Create a collection.\n\n**3.** Add members to the collection.\n\nEnable the Entity for Hierarchies and Collections\n\nAs with explicit hierarchies, you cannot create a collection for an entity until you enable the entity for hierarchies. To do so, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** Select the Product model.\n\n**4.** In the list, click Product and then the \"Edit selected entity\" button.\n\n**5.** From the \"Enable explicit hierarchies and collections\" list, select Yes.\n\n**6.** You are prompted to enter a name for an explicit hierarchy. Even though you don't need an explicit hierarchy, you must create one. Type **Product** for the name and leave the \"Mandatory hierarchy (all leaf members are included)\" check box selected.\n\n**7.** Click the Save entity button.\n\nCreate a Collection\n\nTo create a collection, complete the following steps:\n\n**1.** Go back to the Master Data Manager home page by clicking the SQL Server image in the top left.\n\n**2.** Ensure the Product model and VERSION_1 are selected and click Explorer.\n\n**3.** If you point to Hierarchies on the menu bar, you can see that a hierarchy is now displayed. This is the one you just created. Again, you don't need this hierarchy, but it is required to create collections.\n\nTo create your collection, on the menu bar, choose Collections | Product. The Collections item on the menu bar shows the name of all entities that are enabled for explicit hierarchies and collections. None of the other entities in the model has enough members to merit collections, though, and right now you've enabled collections for Product only.\n\n**4.** To create a collection, click the Add Collection button above the grid.\n\n**5.** In the Details pane on the right side of the screen, enter the name and code, just as you would when you create a member. Enter **Boys Winter Set** for the name and **BWS** for the code.\n\n**6.** If you want to type a brief description for your collection, you can select the Description field in the Details pane and type a description. Click the OK button.\n\n**7.** The collection is displayed in the grid. Every collection has a Name, Code, Description, and Owner attribute. By default, the user who created the collection is listed as the owner.\n\n**8.** To save the changes to the collection, click the OK button at the bottom of the Details pane.\n\nAdd Members to the Collection\n\nNow that the collection exists, you can add members to it:\n\n**1.** In the list of collections, select the collection to modify, and in the right pane, click the Collection Members tab.\n\n**2.** Click the Edit Members button.\n\nThe page that is displayed has two panes. In the left pane are the leaf members for the product entity. You can select to view other collections or any product hierarchy from the drop-down list at the top of this pane.\n\n**3.** From the left pane, click the members you want to add and click the add button between the two panes. These represent a boy's shirt, pants, belt, and hat, which are sold together as a set.\n\nIf you have trouble finding any of these members, Microsoft has added full filter functionality to the collection member editor.\n\nMembers are added to the collection based on the structure of where they come from. If you add consolidated members from an explicit hierarchy, and that hierarchy already has a ragged structure, when you bring the members to the collection, that ragged structure is maintained. Any members in the collection can be reordered or removed using the buttons on the right side of the screen.\n\nCreating Hierarchies by Using Web Services\n\nWhile there are two types of hierarchies discussed within this chapter, we have only provided a derived hierarchy example here. Explicit hierarchies can be created as part of the MetadataCreate operation.\n\nCreating a Derived Hierarchy with the MetadataCreate Operation\n\nDerived hierarchies are created as a series of connected attribute levels. Each of these levels must be related properly within the system or the hierarchy will not be created. While creating derived hierarchies within the UI will limit the entities or hierarchies available at the current level, creating derived relationships within the web service does not provide this information until the error response is returned.\n\nThe following operation creates a three-level derived hierarchy that groups Product by Line by Manufacturer:\n\nSummary\n\nIn this chapter we discussed a number of methods for managing relationships in Master Data Services. We showed derived hierarchies, which enforce a level-based structure based on domain-based attribute relationships. We showed explicit hierarchies, which allow ragged structures but still prevent duplicates or loss of data. We talked about the system settings that affect hierarchies and showed real-world examples meant to help you build your own hierarchies. Finally, we showed collections and special hierarchies that can be used to manage more complex relationships within MDS.\nChapter 7\n\nWorking with Master Data\n\nIn This Chapter\n\n Changes in SQL Server 2012\n\n Viewing the Base Entity\n\n Finding Data by Filtering\n\n Sorting and Pagination\n\n New Explorer Format\n\n Exporting Members\n\n Working with Data in Hierarchies\n\n Transactions\n\n Annotations\n\n Navigating Related Entities\n\n Metadata\n\n Using Web Services to Work with Members\n\nIn previous chapters we focused on creating the environment and architecture for managing master data in your organization. In this chapter, we begin to talk about how to work with your master data on a daily basis. We discuss the grid and filter that allow you to work with data within your master data entities.\n\nThe majority of MDS users will be entering the system to update data. Data stewardship activities in the Master Data Manager web application are managed within the Explorer functional area. A vast majority of users will only have access to the Explorer functional area. In this functional area, there are two major areas where users will work with data:\n\n In the Explorer grid, where they can update consolidated or leaf members within a grid control.\n\n On the hierarchy page, where they can navigate through a hierarchy to manage data changes in an integrated grid control.\n\nChanges in SQL Server 2012\n\nIn the initial release of MDS, there were significant bugs that impaired users' ability to use MDS to manage their data. In SQL Server 2012, the MDS team addressed these issues by adding the SQL Server Master Data Services Add-in for Microsoft Excel (which we will discuss in detail in Chapter 10) and completely re-architecting the Explorer functional area. The new Explorer area streamlines the data editing process. All updates to an entity can be made on a single screen.\n\nViewing the Base Entity\n\nThe entity that is set as the first \"base\" entity you have security access to is displayed when you first open Explorer. Users will immediately be able to review and update data in the Explorer grid. MDS no longer shows the Explorer screen, which gave users an overview of all objects associated with the base entity, upon entering the Explorer functional area. For more information on base entities, see Chapter 4.\n\nIn this book, we show you how to take actions in Explorer by using the menu bar. The Explorer screen is still available from the menu bar if you want to use this screen to navigate the model's main entities.\n\nFinding Data by Filtering\n\nTo begin working with data, you open an entity and, if necessary, filter the list of members to show those that you want to work with. Although the base entity is initially displayed, you can open a different entity by pointing to Entities on the menu bar and selecting the entity you wish to edit from the drop down list. All of the selected entity's members are displayed in a paged grid. The number of default members returned is 50, but because this release of MDS offers increased performance, you can increase this number significantly.\n\nIn SQL Server 2012, the filter is not always displayed. To display the filter grid, click the Filter button above the grid, as shown next. This button acts as a toggle to hide and display filter options. You can select any of the attributes of the entity from the Attribute drop-down list. Selecting the attribute limits the Operator list to those attributes available for that attribute type. The Criteria field will be an input box or a drop-down list based on prior column selections.\n\nWhen you're ready to apply the filters to the members in the grid, click the Apply button.\n\nNOTE\n\n_You can filter by multiple attributes by clicking the Add Criteria button and adding multiple rows of criteria before applying._\n\nThe filters that are currently applied will always be shown directly above the grid.\n\nFiltering can either be accomplished through adding simple criteria where every record can be evaluated or through using matching algorithms. Standard filtering leverages a list of operators that encompass all of the available expressions that can be applied.\n\nOperators\n\nThe operator determines how you're going to filter the list.\n\nUsing Matching\n\nWhen you select the \"Matches\" or \"Does not match\" operator, a small icon for an extended drop down appears. This drop down lets you configure your matching parameters. The default values are Levenshtein for the algorithm and 0.30 for the minimum similarity, but you can change them by using the controls provided. The closer the number is to 1, the more specific the results. To return a wider variety of results, specify a number closer to 0 (for example, 0.20, 0.10, or 0).\n\nThe following table shows the four types of similarity matches you can do programmatically. In each of these cases, 1 indicates an exact match and 0 indicates no match.\n\nThe following example shows the Matches operator and the default matching settings.\n\nYou can see by looking at the number of records displayed in the lower right that 12 results were found.\n\nIf you change the Algorithm setting to Jaccard with the default settings, you retrieve only four results, as shown next. Further research may be necessary to determine what algorithm works best for your dataset.\n\nIn general, 0.30 to 0.70 is a reasonable range to retrieve a list of results based on your dataset. If you want the default to be something other than 0.30, you can change it in the Master Data Services database, in the tblSystemSetting table. GridFilterDefaultFuzzySimilarityLevel determines the default Similarity Level.\n\nNOTE\n\n_Some MDS system settings have not been exposed in Master Data Services Configuration Manager, and this is an example of one. All of the settings in Configuration Manager are in tblSystemSetting, and you can change the settings directly in the table or in Configuration Manager. Use care and ensure you're comfortable working in the database if you decide to make the change directly in the table._\n\nSorting and Pagination\n\nWhen working in the grid, it is useful to understand sorting, pagination, and some other related settings.\n\nSorting\n\nBy default, the members in the grid are sorted in the order they were loaded in the entity. In the prior version, records were returned sorted by code, but this was modified in the SQL Server 2012 release to ensure initial results are returned as quickly as possible. You can click any column heading to sort the members by that column. Click once to sort in ascending order, and again to sort in descending order. You can sort by the values in only one column at a time. Each time you leave the page and return to the grid, the default sort is displayed. Clicking on a column to change the sort will reload sorted data from the server instead of sorting the records already displayed in the grid.\n\nPagination and Other Settings\n\nIn this release of MDS, a new Settings button has been added above the grid. To modify settings for the Explorer grid, click the Settings button in the menu bar to open the Settings dialog box, which has the following settings:\n\n **Records per page** You can change the number of records to be displayed per page. The new Silverlight grid can handle any number of records per page. The number of records is then displayed, followed by arrows that you use to page through the members.\n\n **Display format** This setting controls how all domain-based attributes are displayed within the Explorer grid and Details pane. Members in the grid can be sorted based on the order of the name and code values are displayed, whichever column you choose to display first will be used to sort the potential attribute values.\n\n **Freeze columns** In the first release of MDS, the Name and Code columns were always pinned to the left side of the Explorer grid. In SQL Server 2012, each user can control how many columns are pinned to the left of the grid. The default value is no columns frozen to the left.\n\n **Show System Attributes** In SQL Server 2012, MDS can now expose a number of system attributes in read-only format. Both the created and last updated timestamp will be displayed in the grid for all attribute groups. These columns can be used to sort or filter records in the grid. Since these system attributes are read-only for all users, they do not appear in the Details pane.\n\nNew Explorer Format\n\nIn an effort to simplify the edit process of members in the web UI, users can now edit a selected member on a single screen. Any member selected automatically displays in the Details pane on the right side of the Explorer window.\n\nEdit One Attribute for One Member\n\nWhile you can no longer update values in the grid directly, any value for a member can be updated on the Explorer grid screen. Selecting any member in the grid displays that member in the new Details pane on the right side of the Explorer screen. Any value for the member can be updated in this pane. This Details pane displays all attributes available in the currently selected attribute group. To save your changes, click the OK button at the bottom of the pane. Edit controls are disabled for members or attributes a user does not have security permissions for. Security is discussed in more detail in Chapter 11.\n\nChange the Number of Domain-Based Attributes Displayed in List\n\nIn MDS Configuration Manager, you can change whether a list of attribute values or a new filterable dialog box is displayed when you click the edit icon of a domain-based attribute value. The setting is called \"Number of domain-based attributes in list,\" and you set the maximum number of members that will be displayed in the attribute selection drop downs. List maximums can be anything between 10 and 1000 members. Once the list exceeds this number a pencil will appear next to the attribute and a more complex selection dialog box is displayed instead of a list.\n\nEdit One Attribute for Multiple Members\n\nThe ability to select multiple members and change the values in the grid has been removed in this release of MDS. The product team felt that this functionality was better handled in the freely available Add-in for Excel. The web UI has been designed to navigate easily between members in related entities. Batch data changes should now be completed in the MDS Add-in for Excel, as explained in Chapter 10.\n\nExporting Members\n\nIf you have used a previous release of Master Data Services, you may have used the Export to Excel function to send data to an Excel worksheet. This functionality has been replaced with the Add-in for Excel. For more information on editing members in Excel, please see Chapter 10.\n\nWorking with Data in Hierarchies\n\nIn the Explorer functional area of the user interface, you can view, edit, and move members in a hierarchy. As with most of the screens in the Explorer UI, the hierarchy screen has been redesigned in the latest release of MDS. The new hierarchy screen has been designed to allow users to navigate and manage data in a single screen. This screen is very similar to the original Explorer grid but with an additional hierarchy pane added to the left side of the Explorer window. Consolidated members can be managed only from the hierarchy screen.\n\nHierarchy members can also be organized using the hierarchy Explorer screen. There are a few different methods for moving members; the one you choose will be based on how many members you want to move and the type of hierarchy you are working with. Moving members in a derived hierarchy is slightly different from moving members in an explicit hierarchy.\n\nMoving Members by Dragging and Dropping\n\nThis procedure is basically the same for both hierarchy types. The only difference is the location you can drop members to. If you are working in an explicit hierarchy, you can\n\n Drop a leaf member beneath a consolidated member or at the root.\n\n Drop a consolidated member beneath a consolidated member or at the root.\n\nIf you are working in a derived hierarchy, you can\n\n Drop a leaf member anywhere beneath a domain-based attribute value that applies to the member. For example, you can drop products beneath colors, because Color is a domain-based attribute of Product.\n\nTo move a member in a hierarchy, click the label of the member you want to move. (The mouse pointer will become a pointer with a plus or not icon based on the suitability of the drop location.)\n\nHolding the mouse button down, drag the member to the new location and, when the pointer cursor becomes a plus sign (+), release the button.\n\nThe member is moved to the new location.\n\nMoving Members by Using the Clipboard\n\nThis procedure is also the same for both hierarchy types. To move members in any type of hierarchy by using the Clipboard, complete the following steps:\n\n**1.** In the hierarchy tree, select the check box for each member you want to move (Stores 101 and 105 in this example).\n\n**2.** Click the Cut button at the top of the hierarchy.\n\n**3.** Click the label of the member you want to move the members to (Freestanding in this example).\n\n**4.** You can click the Paste dropdown button or select Paste from the drop-down selections to add Stores 101 and 105 as children of Freestanding.\n\nThe members are displayed in the location you specified.\n\nMoving Members in a Derived Hierarchy by Updating Attribute Values\n\nIn addition to using the Clipboard or dragging and dropping, you can move a member in a derived hierarchy by updating the member's attribute value that determines its location in the hierarchy.\n\nWhen you first open a derived hierarchy, the top levels are displayed in the hierarchy structure on the left.\n\nIn the grid in the middle, the top-level members are displayed. To display the members contained in any level, click a node in the hierarchy tree. The grid in the middle refreshes. For example, when you click Female in the hierarchy tree on the left, all of the female products are listed in the grid in the middle.\n\nIn the grid, find the attribute that determines where the member is displayed in the hierarchy. In this example, we'll click the Marketing tab and scroll to the right until the Gender column is visible. Find the member you want to move, select its row, and then, just as you would do to update any other attribute value, click the drop-down arrow in the Details pane, as shown next.\n\nNow change the value. When you click the OK button at the bottom of the Details pane, the grid and hierarchy are updated, and the member is moved to the node based on your selection. In this case, the product moved from the Female to the Male hierarchy node.\n\nSearching for Members in a Hierarchy\n\nYou can use the search field above the hierarchy to find members. Unfortunately, the search functionality has a number of limitations:\n\n The text you enter must be an exact match of either the full name or the code. To search for partial matches, use the % wildcard.\n\n The search does not find members in the Unused node of explicit hierarchies.\n\n If you have so many members in your hierarchy that the...more... label is displayed, members that aren't displayed are still included in the search results.\n\nTo use the search field, type your search text and click the magnifying glass icon. Your search is applied and the hierarchy is updated. To remove the search, click Close Search button at the top of the hierarchy control.\n\nSetting the Number of Members That Are Displayed\n\nIn MDS Configuration Manager, you can change the number of members displayed in each hierarchy node before you have to click...more....\n\nThe setting is called \"Number of members in the hierarchy by default,\" and you set the maximum number of members that will be displayed. If you have more than that number, the...more... label is displayed.\n\nManaging Hierarchy Display Settings\n\nThe hierarchy grid uses the display format settings to control how members are displayed in the hierarchy. Members will be displayed either as Code, Code (Name), or Name (Code). You can change these settings using the settings button above the grid.\n\nTransactions\n\nEvery change made to data in Master Data Services is logged as a transaction. All transactions log the type, prior value, new value, the user who made the change, and the time the change was made. In the Explorer functional area, all users have access to their own transactions and can reverse any transaction they have access to. In the Version Management functional area, administrators can review and reverse transactions for all users.\n\nThe following types of transactions are logged and displayed in the Master Data Manager web application:\n\nNOTE\n\n_By default, transactions are not logged for data that's staged into the system. Transaction logging for staging can be turned on in the MDS Configuration Manager system settings._\n\nReviewing Transactions\n\nIn SQL Server 2012, transactions can be reviewed only in the Version Management functional area. On the menu bar, choose Transactions. Transactions are displayed in a grid. You can filter the grid by using the Filter Criteria section. In this release of MDS, the transaction manager has not been upgraded to Silverlight.\n\nReversing Transactions\n\nTo reverse a transaction, in the grid, click the row of the transaction you want to reverse, and then click the Revert Transaction button.\n\nIn the previous release of MDS, deactivated members were given a GUID identifier to allow those members' code to be reused. In this release, members that are deleted in the UI keep their code. This keeps the member's prior transactions tied to the member they were made on. If a member code that has been soft deleted in the system must be reinstated, you can reverse the transaction that shows the member changing from Active to De-Activated.\n\nIf you reverse this transaction, the member and its original code will be reactivated. The member will be part of all hierarchies and collections it was a member of before it was deleted; it will be as though the member was never deleted in the first place.\n\nAnnotations\n\nYou can annotate transactions to explain why you took an action. In the prior MDS release, these actions were hard to find after you entered them. In the current release, as you update individual member attributes in the Details pane, you can add an annotation to apply to the transaction. Transactions and their annotations for an individual member can be viewed in the same location.\n\nAnnotating Transactions on Update\n\nTo annotate updates to attributes, open Explorer to an entity. In the grid, select the member you want to update. Update the attributes you want to change. After you update the attributes, type the annotation in the Annotations box at the bottom of the Details pane.\n\nAfter you have typed the annotation, click OK.\n\nTo review the transactions from that member, click the View Annotations link below the Annotations box and you will see all transactions for the member. The transactions you just made will show the annotation as well as a time and date stamp. Additional annotations can be added to any transaction from this screen as well.\n\nNavigating Related Entities\n\nOne of the key value propositions of the MDS Explorer UI is the ability for users to navigate quickly through an entity and understand a total view of a member. In SQL Server 2012, a number of new features have been added to ensure that this full view can be easily navigated by data stewards.\n\nNavigating Attributes\n\nMany domain-based attribute values have their own information to provide to users. Consider the scenario of managing customers in MDS. One of the fields in the customer record is a domain-based attribute of ZIP code. This system uses a table for postal codes to store additional demographic information for each code. Within MDS, a data steward can drill down from a selected customer into the associated demographic information by clicking a single button in the Details pane.\n\nClicking this additional button attached to all domain-based attributes of a member opens a new tab in your browser with the new entity filtered by the selected value of the member you were reviewing.\n\nNavigating Many-to-Many Relationships\n\nWhile many relationships within a model follow the one-to-one pattern or the one-to-many pattern, some data may require the support of many members from one entity relating to many different members in another entity. This relationship is termed many-to-many.\n\nAs an example, suppose Main Street Clothing Company has decided to start doing its own manufacturing of some simpler items for its stores. To manage this data in MDS, Main Street had to create two additional entities. The Raw Materials entity stores all of the items that Main Street combines to create its finished goods (the items it will sell to customers). The Raw Material to Product entity maps these combinations. As previously discussed, each manufactured product is made of multiple raw materials, and a raw material is reused in many different products.\n\nThe bottom-right corner of the Explorer screen in MDS is the Related Entities section. If the current entity has any many-to-many relationships defined in the system, a link will be displayed.\n\nMetadata\n\nPrior users of MDS may notice that Metadata has been removed from the Explorer functional area. This feature has been marked for deprecation and can be accessed only from the metadata model or web services.\n\nUsing Web Services to Work with Members\n\nThe calls associated with getting and changing master data within Master Data Services are the most commonly used web service calls by organizations looking to customize their MDS implementation. These calls allow users to enter new member records, update existing member records, or bring back existing entity member record sets.\n\nRetrieving Members\n\nThe EntityMembersGet operation will retrieve member records from any entity that exists within MDS. This operation supports paging to allow users to show a subset of the entity within a customized UI. This operation also supports subselecting a set of attributes to be retrieved. If you are going to use the paging criteria to retrieve a subset of records, be sure to lock down the sort criteria before you page; otherwise, inconsistent records will be returned.\n\nThere are three types of member return options. An EntityMembersGet operation can return either data, counts, or both data and counts. Some systems may require the total number of records to be returned before they return any member detail, and the \"record counts only\" return option provides this functionality. For organizations interested in using the paging mechanism to return records, the \"data and counts\" return option can provide a preview of the number of records in the full entity record set.\n\nIn the following code sample, we provide an example that returns a standard EntityMembersGet request with a standard page size of 50 members. This request has no complex filter.\n\nWhen retrieving large amounts of data from the web service, it is quite likely that you will run into the following error:\n\nSystem.ServiceModel.CommunicationException was unhandled Message=The maximum message size quota for incoming messages (65536) has been exceeded. To increase the quota, use the MaxReceivedMessageSize property on the appropriate binding element.\n\nThis is to protect users from themselves. Each organization must determine how large the maximum message size can be, but this can be set in the client's app.config file by changing the following parameter:\n\nCreating and Updating Members\n\nMaster Data Services provides the following three operations for managing entity members within the system. They support three different types of data loads.\n\n **EntityMembersCreate** Only new member records with unique codes can be created with this operation. If a member code already exists, an error will be thrown in the returned error collection and the member will not be created or updated.\n\n **EntityMembersUpdate** Only existing members can be updated using this operation. If a member code or ID is provided that does not exist within the system, an error will be returned in the error collection passed in the operation result and the member will not be created.\n\n **EntityMembersMerge** Many users have been confused by this operation and believe it has something to do with record survivorship or match merge functionality. This operation does nothing to look for possible matches or provide any survivorship functionality; it only allows users to create and update records simultaneously within MDS.\n\nThe following example shows only the EntityMembersMerge operation because this operation provides the most flexibility for users programming against MDS. For simplicity in programming and to show all of the capabilities of setting attributes, we have hard-coded some defaults from the Main Street Clothing Company product example. If you are using this call within your organization, you need to replace the defaults with attribute parameters that are passed into the operation.\n\nSummary\n\nIn this chapter we looked at how users will manage data within Master Data Services on a daily basis. We discussed using the new Explorer screen to update individual members. Every change made in MDS is logged as a separate transaction. All users can view all the transactions on a specific member. Transactions cannot be reversed in the explorer screen, this requires access to the Versioning functional area. Users can however provide additional annotations to any prior transaction on a member.\nChapter 8\n\nUsing Business Rules\n\nIn This Chapter\n\n Business Rules Overview\n\n Creating a Business Rule\n\n Editing a Business Rule and Configuring a Rule's Expression\n\n Publishing Business Rules\n\n Applying Business Rules\n\n Other Business Rule Tasks\n\n Configuring E-Mail Notifications\n\n Creating and Triggering Workflows\n\n Using Web Services to Manage Business Rules\n\nManaging data is not just about storing the data in a customized entity; it is also about ensuring that the data is both accurate and complete. Master Data Services provides business rules to achieve this aim. In this chapter, we review business rules and discuss how to create both simple and complex rules. We discuss how to create simple workflows in MDS and how to integrate with external workflows such as those provided by SharePoint.\n\nBusiness Rules Overview\n\nIn many organizations, no matter what the size, the business owners responsible for managing the master data don't have the technical knowledge needed to implement the related processes. If the business owners don't know how to use SQL Server or how to code business rule engines, they can be left at the mercy of their IT departments. At the same time, because the IT department has the technical know-how, the burden often falls on them to learn business domains they don't necessarily need to know. MDS strives to simplify the creation of business rules to empower business users to manage their own data quality.\n\nIn MDS, business rules are declarative expressions that govern the conduct of business processes. These expressions are compiled into stored procedures that perform the task of validating the data. The area of the Master Data Manager web application used for business rules was created to empower business users to write relatively complex business rules without knowledge of Transact-SQL.\n\nBusiness Rule Structure\n\nBusiness rules are IF...THEN statements. IF certain conditions evaluate to true, THEN perform specific actions. Conditions can be combined using either AND or OR logical operators. These operators can be used to create extremely complex business rules. You can use as many as seven levels for complex conditioning.\n\nAlthough you have the ability to create complex rules, there are some real benefits in creating multiple, more granular rules. You should consider breaking any rule that uses the logical OR operator into multiple rules. This makes rules easier for other users to read and understand. Multiple rules also allow you to exclude specific rules and to provide more granular notifications. Rules built with the AND operator must be kept together to function as a unit.\n\nBusiness rules are always applied to attribute values. For example, if an attribute value is blank, you might want to send an e-mail to notify someone or set the value to Pending. Or you might want to update the value of one attribute based on the value of another attribute. Because business rules are applied to attribute values, you should determine which attributes you're going to work with before you start creating rules. Each time you create a rule, you must select the model, entity, and type of member that contains the attribute you're looking for.\n\nBusiness Rule Workflow\n\nBusiness rules must be created, configured, and published before you can validate data against them. The workflow includes the following steps:\n\n**1.** Create a business rule, which adds it to the list of available rules.\n\n**2.** Edit the rule and configure its expression.\n\n**3.** Publish the rule so it's active and can be used to validate data.\n\n**4.** Apply the rule to members.\n\nWhen you apply the rule to members, the attribute values for each member are validated and they either pass or fail validation. If they pass, no action is required. If they fail, you must update the attribute values and re-validate your data successfully against the rules. If you don't take action, your hierarchies may not be complete or your data might otherwise be invalid.\n\nCreating a Business Rule\n\nThe first step in the workflow is to create a business rule. Creating a rule means adding it to the list of existing rules. This procedure doesn't configure the rule to do anything, nor does it publish the rule. It's simply the first step necessary in using business rules to validate data.\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Business Rules.\n\n**3.** On the Business Rule Maintenance page, choose a model, entity, and member type to be validated. In this example, select Product, Product, and Leaf, respectively. For the attribute, leave it at the default, All. The Attribute list works as a filter for the list of business rules. In general, you should leave this as All to ensure that all rules are displayed.\n\n**4.** Click the \"Add business rule\" button.\n\nA new row with a light blue background is displayed in the table. The blue highlight indicates that the row is selected.\n\nYou are now ready to edit the business rule and configure its expression.\n\nEditing a Business Rule and Configuring a Rule's Expression\n\nA business rule frequently consists of a condition and an action. The condition is evaluated, and if it evaluates to true, one or more actions are initiated. If a rule contains no conditions, the rules engine will always initiate the actions.\n\nRules on the Edit Business Rule page are defined by using the following format:\n\nIF < _condition_ > THEN < _action_ >\n\nYou can configure business rules to evaluate a wide variety of expressions. In this chapter we will strive to show you some common business rules and give you the knowledge needed to create some of your own.\n\nProcedure: How to Require Attribute Values\n\nIn this first example, a business rule ensures that the Cost, Retail Price, and Gender attributes of all products are populated with values. The expression for this rule will be\n\nIF < _nothing, because it applies to all cases_ >\n\nTHEN require an attribute value for Cost, Retail Price, and Gender\n\nTo create the structure of a rule to perform this type of validation, complete the following steps:\n\n**1.** Follow the procedure to create a new business rule. In this example, on the Business Rule Maintenance page, select the Product model, Product entity, Leaf member type, and All attributes and click the \"Add business rule\" button.\n\n**2.** Click the row for the rule you want to edit and then click the \"Edit selected business rule\" button.\n\n**3.** The Edit Business Rule page is displayed. In the Components pane on the top left, expand the Actions node.\n\n**4.** Scroll down. Under the Validation node, click \"is required\" and drag it to the THEN pane in the middle right. Drop it on the Actions label.\n\n**5.** The Edit pane in the lower right becomes the Edit Action pane. In the Attributes pane on the lower left, click Cost and drag it to the Edit Action pane. Drop it on the Select attribute label.\n\n**6.** At the bottom of the Edit Action pane, click the Save item button. This saves the action you just added.\n\n**7.** Repeat steps 4, 5, and 6 for the Retail Price and Gender attributes. When you're done, you should have three actions displayed in the THEN pane.\n\n**8.** In the top-left corner, click the green arrow to return to the Business Rule Maintenance page. Because the rule was saved as you created it, there is no need to save again when you leave the page.\n\nNow that the rule exists, you must publish it if you want to apply it to your data. The procedure for publishing the rule is covered in \"Publishing Business Rules,\" later in the chapter. First, we'll look at more functionality on the Business Rule Edit screen.\n\nDeleting a Condition or Action\n\nWhile you're working on the Edit Business Rule page, you may realize you've selected the wrong action or condition. In either the IF pane or the THEN pane, you can right-click any condition or action and click Delete to remove it.\n\nMore Expression Examples\n\nNow we'll continue with other examples of business rules you might want to create. Later in this chapter, we'll explain how to validate your data against active business rules.\n\nProcedure: How to Populate One Value Based on Another Value\n\nYou can populate an attribute value based on other attribute values. This is probably the most common use for business rules.\n\nIn this example, we want to indicate that discontinued products that have fewer than 50 items in stock are now available to outlet stores. The expression will be\n\nIF Discontinued = Y AND Safety Stock Level < 50\n\nTHEN Available to Outlet = Y\n\nTo create this rule, complete the following steps:\n\n**1.** Follow the procedure to create a new business rule. In this example, on the Business Rule Maintenance page, select the Product model, Product entity, Leaf member type, and All attributes and click the \"Add business rule\" button.\n\n**2.** Click the row for the rule you want to edit and then click the \"Edit selected business rule\" button. The Edit Business Rule page is displayed.\n\n**3.** In the Components pane, expand the Conditions node.\n\n**4.** Under the Value comparison node, click \"is equal to\" and drag it to the IF pane on the top right. Drop it on the Conditions label.\n\n**5.** The Edit pane in the lower right becomes the Edit Condition pane. In the Attributes pane, click Discontinued and drag it to the Edit Condition pane. Drop it on the Select attribute label.\n\n**6.** The Attribute value radio button is selected. Next to it, select Y from the list. This selector will vary depending on the type of attribute you select. Numeric attributes will allow a number to be entered in a numeric control, and free-form text fields would provide a text box.\n\n**7.** At the bottom of the Edit Condition pane, click the Save item button. The IF pane is populated with the condition: Discontinued is equal to Y.\n\n**8.** In the Components pane, click \"is less than\" and drag it to the IF pane. Drop it on the And label.\n\n**9.** In the Attributes pane, click Safety Stock Level and drag it to the Edit Condition pane. Drop it on the Select attribute label.\n\n**10.** The Attribute value radio button is selected. Next to it, type **50**.\n\n**11.** At the bottom of the Edit Condition pane, click the Save item button. The IF pane is populated with the condition: Safety Stock Level is less than 50.\n\n**12.** In the Components pane, expand the Actions node.\n\n**13.** Scroll down. Under the Change value node, click \"equals\" and drag it to the THEN pane. Drop it on the Actions label.\n\n**14.** In the Attributes pane, click Available to Outlet and drag it to the Edit Action pane. Drop it on the Select attribute label.\n\n**15.** The Attribute value radio button is selected. Next to it, select Y from the list.\n\n**16.** At the bottom of the Edit Action pane, click the Save item button.\n\nThe business rule is complete.\n\nProcedure: How to Concatenate Values\n\nIn this example, a business rule creates a member name that is a concatenation of several attribute values. For example, one of our product names is\n\nM Red Shirt 3 V Neck\n\nThis is a concatenation of these attribute values:\n\nGender Code + Color Name + Type Name + Size Name + Neck Style Name\n\nBefore we imported our products, we did this concatenation in Excel. Now we want to do this concatenation in MDS for all products, new and existing. We've also decided to use the full Gender name (Male) as part of the product name, rather than just the code (M). To create this rule, complete the following steps:\n\n**1.** Follow the procedure to create a new business rule. In this example, on the Business Rule Maintenance page, select the Product model, Product entity, Leaf member type, and All attributes and click the \"Add business rule\" button.\n\n**2.** Click the row for the rule you want to edit and then click the \"Edit selected business rule\" button. The Edit Business Rule page is displayed.\n\n**3.** In the Components pane, expand the Actions node.\n\n**4.** Under the Change value node, click \"equals a concatenated value\" and drag it to the THEN pane. Drop it on the Actions label.\n\n**5.** In the Attributes pane, click Name and drag it to the Edit Action pane. Drop it on the Select attribute label. (You drag the attribute whose value you want to concatenate; in this case, you want to concatenate the Product's Name value.)\n\n**6.** Also in the Attributes pane, expand the Gender node and click Name. Drag it to the Edit Action pane and drop it on the Value label.\n\n**7.** Right-click DBA.Gender.Name and choose Add text.\n\n**8.** Double-click the label that instructs you to do so (shown next). A text box is displayed. Type a single space to indicate the space between values. Then click away from the field to make the text box back into a label.\n\n**9.** In the Attributes pane, expand the Color node and click Name. Drag it to the Edit Action pane and drop it on the Value label.\n\n**10.** Right-click DBA.Color.Name and choose Add text.\n\n**11.** Double-click the label that instructs you to do so. A text box is displayed. Type a single space to indicate the space between values.\n\n**12.** Repeat steps 9\u201311 for Type Name, Size Name, and Neck Style Name.\n\n**13.** At the bottom of the Edit Action pane, click the Save item button. The THEN pane is updated with all of the actions you specified.\n\nThe business rule is complete. Now, when you validate your data, all product names will be updated based on this rule.\n\nTIP\n\n_You can use concatenation to create link attributes, like those used for Bing maps (for example,). In this case, you would concatenate the Name attribute values for the Location entity's Street Address, City, State, and Zip Code attributes to create a link._\n\nProcedure: How to Populate an Attribute Value from a Different Entity\n\nIn Chapter 6, we created a three-level hierarchy:\n\nProduct > Line > Manufacturer\n\nIn this example, the Product entity had a Line attribute, and the Line entity had a Manufacturer attribute. But the Product entity did not have a Manufacturer attribute.\n\nThis structure is required in order to create the hierarchy. But you probably want to see the manufacturers at the same time as you look at the products; that is, you want the Product entity to have a Manufacturer attribute.\n\nYou can set up the structure required for the hierarchy but then use business rules to populate the Manufacturer value, in this case. The following procedure describes how to show the manufacturers along with the products without losing the hierarchy structure. This is going to require some work before we proceed to creating the business rule.\n\n**1.** For the Product entity, create a free-form attribute called **Manufacturer**. See Chapter 4 for information on how to create free-form attributes.\n\n**2.** For the Product entity, create an attribute group called **Product Line** and add the Manufacturer and Line attributes to it. You can also see Chapter 4 for how to create attribute groups.\n\n**3.** Create a business rule to populate products with the appropriate manufacturer. This means that the rule will populate the Product entity's Manufacturer attribute with the Line entity's Manufacturer attribute values.\n\na. Follow the procedure to create a new business rule. In this example, on the Business Rule Maintenance page, select the Product model, Product entity, Leaf member type, and All attributes and click the \"Add business rule\" button.\n\nb. Click the row for the rule you want to edit and then click the \"Edit selected business rule\" button. The Edit Business Rule page is displayed.\n\nc. In the Components pane, expand the Actions node.\n\nd. Under the Change value node, click \"equals\" and drag it to the THEN pane. Drop it on the Actions label.\n\ne. In the Attributes pane, click Manufacturer and drag it to the Edit Action pane. Drop it on the Select attribute label. (You drag the attribute whose value you want to populate; in this case, you want to populate the Manufacturer value.)\n\nf. In the Attributes pane, expand the Line node and click Manufacturer. Drag it to the Edit Action pane and drop it on the Select attribute label.\n\ng. At the bottom of the Edit Action pane, click the Save item button.\n\nThe business rule is complete.\n\nNOTE\n\n_InChapter 11, we will discuss how to set security assignments, but it is usually best practice to limit access to columns populated by business rules._\n\nProcedure: How to Use Incremental Values for Code Attribute in SQL Server 2012\n\nYou might want to populate an attribute value with an incremental number. For example, maybe you want each new product to have a unique number as its Code value and you want to increment each new product number by five. In the first release of MDS, you used business rules to do this. In SQL Server 2012, this special action of creating Code values was moved to the entity creation screen.\n\nIn the current Finance model, the Code attribute values for the Location entity start with 100 and additional Code values are incremented by one. As new locations are added, the Code value should be one number higher than the highest number that's already used. The following example shows how to set up the entity to do this incremental numbering. This can be turned on and off for an entity at any time.\n\nNOTE\n\n_Even though you're going to populate the Code attribute, you will notice in Explorer that the Code attribute is editable._\n\nTo update the entity to automated code creation, complete the following steps:\n\n**1.** In this example, on the Entity Maintenance page, select the Finance model and the Location entity and click the \"Edit selected entity\" button.\n\n**2.** On the Edit Entity page, select the \"Create Code values automatically\" check box.\n\n**3.** Checking the box displays an integer input box labeled Start with. If you are not sure which number to start with, you can leave 1 for the Start with value. MDS will populate the Code value with the next possible value.\n\n**4.** At the top of the Edit Entity page, click the Save entity button.\n\nThe automation is complete.\n\nThis new code-generation process was implemented to provide speed and functionality outside of the business rule process. In SQL Server 2012, members created through staging and the web service can now generate Code values. In order to maintain performance on large entities, the last integer used is stored in the tblCodeGenInfo table of the MDS database. This will track the largest Code value for each entity. If a user inadvertently updates a code member to a large integer, you may need to fix the problem in this table directly.\n\nCreating Your Own Expressions\n\nThe following reference tables explain the possible conditions and actions you can use to create business rule expressions.\n\nConditions\n\nIf a condition evaluates to true, an action is taken. For example, if an attribute value is greater than another attribute value (the condition), then the attribute value is changed (the action).\n\nActions\n\nThere are four categories of actions you can take by using business rules.\n\nDefault Value Actions\n\nDefault value actions set a blank value to a value specified by the rule. If a value is already populated, default actions do not have an effect.\n\nChange Value Actions\n\nChange value actions update values to new values each time business rules are applied.\n\nValidation Actions\n\nValidation actions are used to send e-mail notifications when data fails business rule validation.\n\nExternal Actions\n\nExternal actions start applications outside of Master Data Services.\n\nPublishing Business Rules\n\nAfter you have created a business rule, you must publish it. Publishing a business rule means making it available to validate data against it. You can create as many rules as you want, but if you don't publish them, they aren't applied to your data when you start validation.\n\nAfter you publish a business rule, you can always exclude it if you decide you don't want to apply it to data. Any time you edit a rule, you must republish it or it won't be applied to data as part of the validation process.\n\nProcedure: How to Publish a Business Rule\n\nTo publish a business rule, click the \"Publish business rules\" button.\n\nFor all business rules that have expressions defined, the value in the Status column changes to Active. Data can now be validated against the rules. You can exclude rules you do not want to publish. For more information, see \"Procedure: How to Exclude Business Rules,\" later in this chapter.\n\nBusiness Rule Statuses\n\nThe following table describes all possible statuses for rules on the Business Rule Maintenance page.\n\nApplying Business Rules\n\nBusiness rules are not valuable until you validate data against them. Data can be validated against business rules in the following ways:\n\n You can validate the members for a single entity at a time in the Explorer functional area of the Master Data Manager web application. Business users with Explorer permissions can do this.\n\n You can validate the members for a single entity that you have loaded into an Excel worksheet using the Master Data Services Add-in for Excel. If the number of members loaded into the sheet exceeds the limit configured on the server, the whole entity will be validated asynchronously.\n\n You can validate all entities in a model in the Version Management functional area of the Master Data Manager user interface.\n\n You can use web services to validate all or some members as they are passed in.\n\n You can run a stored procedure to validate at different model object levels. While using the stored procedures directly is not encouraged, you may find it necessary to implement the data management process for your organization.\n\n Validations can be triggered through the staging stored procedures. For more information about this stored procedure, see Chapter 5.\n\nProcedure: How to Apply Business Rules in Explorer\n\nUsers with permission can apply business rules in Explorer. To do so, complete the following steps:\n\n**1.** On the Master Data Manager home page, select the Product model and VERSION 1 from the drop-down lists.\n\n**2.** Click Explorer.\n\n**3.** On the menu bar, choose Entities | Product (or whichever entity you want to validate against business rules).\n\n**4.** Above the grid, click the Apply Rules button. In this release of MDS, the button can now be expanded to apply rules to the entire entity or to look at the current status of a validation request.\n\nA green check mark indicates success. A red exclamation point indicates failure.\n\nNOTE\n\n_Only the members on the current page in the grid are validated. So if you are showing 50 members, only those 50 members are validated against business rules._\n\nResolving Validation Issues in Explorer\n\nIn Explorer, you can no longer view validation issues in a list format. The newly streamlined Explorer UI and the new Add-in for Excel should replace the need for the My Validation Issues page. In the Explorer UI and the Add-in for Excel, you can now filter on validation status, allowing you to display all members that failed validation, as shown here:\n\nBy selecting a specific member, you can see all validation issues related to the member at the bottom of the Details pane to the right. This provides a single screen to correct validation issues that have been found.\n\nProcedure: How to Validate a Version\n\nIn the Version Management functional area of Master Data Manager, you can validate an entire model version at once. For more information and full instructions, see Chapter 9.\n\nOther Business Rule Tasks\n\nNow that you have created rules, you might want to update the name and description of each to ensure you can tell them apart. You can also change the order in which the rules run, or exclude rules from running.\n\nProcedure: How to Change the Rule Name and Description\n\nYou should give each business rule a name and description so you can keep track of what the rule is used for. To change the name or description of a business rule, complete the following steps:\n\n**1.** On the Business Rule Maintenance page, in the row for the rule you want to rename, double-click the Name column. The cell becomes editable.\n\n**2.** Type the text you want and press enter. The Status column changes to Changes pending.\n\n**3.** Click the \"Publish business rules\" button.\n\n**4.** In the confirmation dialog box, click OK. The status changes to Active.\n\nThe text is updated. Follow this same procedure to update values in the Description column.\n\nNOTE\n\n_The name cannot be more than 50 characters and the description cannot be more than 255 characters._\n\nProcedure: How to Set the Rule's Priority Order\n\nBusiness rules execute based on the priority order provided in the Priority column. If you change the priority, you're changing the order in which the rules are applied to data. It is possible to have two rules with the same priority. In this case, MDS will determine the order for these two rules, usually the order of creation.\n\nTo change a business rule's priority, complete the following steps:\n\n**1.** On the Business Rule Maintenance page, in the row for the rule with the priority you want to change, double-click the Priority column. The cell becomes editable.\n\n**2.** Type the number you want and press enter. The status changes to Changes pending.\n\n**3.** Click the \"Publish business rules\" button.\n\n**4.** In the confirmation dialog box, click OK. The status changes to Active.\n\nTIP\n\n_You can click the column heading to sort by the Priority column._\n\nIn Master Data Services Configuration Manager, you can change the number that the Priority column is increased by when you create a new rule. The default is 10 but you can change it by updating the \"Number to increment new business rules by\" setting.\n\nProcedure: How to Exclude Business Rules\n\nYou can exclude rules if you're not ready to publish or if you're troubleshooting. If you wish to save the business logic but temporarily exclude the rule from validation processes, you can check the Excluded check box and republish the rules.\n\nTo exclude a business rule, complete the following steps:\n\n**1.** On the Business Rule Maintenance page, in the row for the rule you want to exclude, select the Excluded check box. The status changes to Exclusion pending.\n\n**2.** Click the \"Publish business rules\" button.\n\n**3.** In the confirmation dialog box, click OK. The status changes to Excluded.\n\nConfiguring E-Mail Notifications\n\nYou can configure MDS to send e-mail notifications when an attribute value fails validation against certain types of business rules. First, you must configure e-mail settings to ensure e-mail can successfully be sent. Then, you must edit the business rule to indicate who to send e-mail notification to if validation fails. If you don't need e-mail notifications, you can skip this section.\n\nA few notes before you get started:\n\n If you trigger e-mails to a group, only the users who have accessed the Master Data Manager web application will receive the mails.\n\n To confirm that the e-mail is sent to the user's correct address, on the Master Data Manager home page, click User and Group Permissions. To the left of the user, click the down arrow, then choose Edit | General. If the e-mail address listed is not correct, click the Edit button and update it.\n\nProcedure: How to Configure E-Mail Notifications\n\nBefore you can configure e-mail notifications, you must know the IP address of the SMTP server that will send the mail. You should also have an e-mail address and display name for the account that will send the mail.\n\nAfter you get this information, you can configure e-mail notifications by completing the following steps:\n\nNOTE\n\n_This is a one-time process in Configuration Manager. If you need to update the Database Mail profile after you've set it up initially, you must do so in SQL Server Management Studio._\n\n**1.** Open Master Data Services Configuration Manager.\n\n**2.** In the left pane, click Database Configuration.\n\n**3.** In the right pane, click the Select Database button.\n\n**4.** In the Connect to Database dialog box, connect to a database and click OK. The System Settings section is populated with settings for the database.\n\n**5.** In the System Settings area, scroll down to the end of the list.\n\n**6.** Click the Create Profile button.\n\n**7.** Populate the Create Database Mail Profile and Account dialog box as follows:\n\n**8.** Click OK to save and close. The Database Configuration page is updated with the name of your Database Mail profile.\n\nOther E-Mail Settings in Configuration Manager\n\nThere are a few other settings you can use to customize e-mails for your implementation:\n\n **Master Data Manager URL for notifications** The URL that's used in the link in the e-mail notification. You should enter the name of your web site so users who get the e-mail can go directly to the validation issue by clicking the link.\n\n **Notification e-mail interval (in seconds)** How often e-mails are sent. The default is every two minutes.\n\n **Number of notifications in a single e-mail** How many validation issues are included in a single e-mail.\n\n **Default e-mail format** Format for e-mails.\n\n **Regular expression for e-mail address** Used to validate e-mail addresses in the User and Group Permissions functional area of Master Data Manager. It determines the format that is allowed when entering e-mail addresses. Unless you have reason to change it, leave the default setting.\n\n **Database Mail account** You cannot change this setting. It is intended to show you the name of the Database Mail account in SQL Server in case you need to update it.\n\nProcedure: How to Configure a Business Rule to Send E-Mail\n\nAfter you configure Database Mail in Configuration Manager, you must specify who will be notified when data fails validation against business rules.\n\nNOTE\n\n_To send e-mail notifications, the rule must have a validation action. (Validation actions are listed on the Edit Business Rule page, in the Components pane, under Actions | Validation.) Other types of actions do not send e-mail._\n\nTo configure a business rule to send e-mail, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Business Rules.\n\n**3.** On the Business Rule Maintenance page, choose a model, entity, and member type for your rule to apply to. In this example, select Product, Product, and Leaf. For the attribute, leave All.\n\n**4.** For the business rule for which you want to send e-mail, double-click the Notification column. Expand the Users node or Groups node to find the user or group you want to send mail to.\n\n**5.** Click the name of the user or group. The Notification column is updated with the value you selected. The Status column is updated to Changes pending.\n\n**6.** Click the \"Publish business rules\" button.\n\n**7.** In the confirmation dialog box, click OK.\n\nCreating and Triggering Workflows\n\nBy using business rules and customizing your data model, you can create relatively simple workflows for managing the flow of data through Master Data Services. If you want to manage a one- or two-step approval process or notify users when more data is required, you can simply use the MDS interfaces provided to create these workflows.\n\nSome organizations will require more complex event processing. If your organization requires multitiered approvals or complex decision trees, then integration with a SharePoint workflow may better suit your needs. Integration with SharePoint workflow will require some coding expertise and basic understanding of Windows workflow foundation.\n\nCreating a Workflow in MDS\n\nA simple MDS workflow might include getting approval when an attribute changes. For example, if someone in Purchasing needs to change the cost of a product, the head of Purchasing might want to be notified so that he can approve or deny that change.\n\nYou can create workflows in MDS based on any condition, but we're going to show you an example where an attribute value changes. In this case, you don't know what the attribute has changed to; you just know that it has changed.\n\nTo address this scenario, MDS uses change tracking groups attached to attributes. Each entity within Master Data Services has up to 31 change tracking groups that can have any number of attributes allocated to them. An attribute can be managed by only one change tracking group at a time.\n\nNOTE\n\n_Change tracking is still not triggered through the staging process in the SQL Server 2012 release of MDS. If you require automatic updates to trigger change tracking, you must use the web service to update data._\n\nTo create a workflow within MDS, complete the following steps:\n\n**1.** Add attributes to a change tracking group (so you can be notified when their values change).\n\n**2.** Add a domain-based attribute named Approved and populate it with Yes, No, and Pending members.\n\n**3.** Create a business rule to send notification when the attribute values change.\n\nThese steps create a simple workflow. Any time the cost of a product changes, the Approved attribute changes to Pending and the purchasing manager is notified. It is the purchasing manager's job to open MDS, review the change, and approve or deny it.\n\nProcedure: How to Add an Attribute to a Change Tracking Group\n\nYou must add an attribute to a change tracking group before you can use rules to determine if the attribute value has changed. To add an attribute to a tracking group, complete the following steps:\n\n**1.** On the Master Data Manager home page, click System Administration.\n\n**2.** On the menu bar, choose Manage | Entities.\n\n**3.** Select a model from the Model list. In this case, select Product.\n\n**4.** Click the row of the entity that contains the attribute and click the \"Edit selected entity\" button. In this example, edit Product.\n\n**5.** In the Leaf attributes section (or whichever section contains your attribute), click the attribute and click the \"Edit selected attribute\" button. In this example, edit Cost.\n\n**6.** At the bottom of the page, select the \"Enable change tracking\" check box.\n\n**7.** The \"Change tracking group\" field is displayed. You can set this number to any number you want. It is just used as a grouping for multiple attributes. If you want to include more than one attribute in this group, use the same group number for each attribute.\n\n**8.** Click the Save attribute button.\n\nNow if you want, you can edit other attributes and add them to the group with the same number. To do this, follow the same steps and make sure to use the same number for the group.\n\nViewing Change Tracking Group Reports\n\nIn the first release of Master Data Services, managing change tracking group attributes was a time-consuming task because there was no way of seeing all change tracking group associations in one place. While there is still no way in the UI to see a complete list of associations, two new views have been provided to give administrators insight into which attributes each change tracking group consists of and which business rule leverages that group:\n\n **viw_SUBSCRIPTION_CTG_ATTRIBUTES** This view provides a list of all attributes associated with a change tracking group in any model in the database.\n\n **viw_SUBSCRIPTION_CTG_BUSINESS_RULES** This view provides a list of all business rules that leverage a change tracking group as a condition. This view does not filter out rules that are excluded or not currently active.\n\nProcedure: How to Create an Approved Attribute\n\nNow you're going to create an attribute that will change whenever the Cost attribute changes. This attribute acts as a flag to make others aware that the cost has changed.\n\nBy now, you should be able to create an entity and attribute in System Administration and create members in Explorer. If you need a refresher, see the procedures in Chapter 4.\n\nTo continue with this example:\n\n**1.** In System Administration, for the Product model, create an entity named **Approved**. This entity does not need to be enabled for hierarchies.\n\n**2.** In System Administration, create a leaf-level domain-based attribute for the Product entity. This domain-based attribute should be based on the Approved entity and be named **Approved**.\n\n**3.** Add the Approved attribute to the Pricing attribute group so you can view the attribute in Explorer.\n\n**4.** In Explorer, open the Approved entity and create three members: **Yes** , **No** , and **Pending**. If you have created multiple versions of the Product model, make sure you do this in the latest version.\n\nProcedure: How to Create a Business Rule That's Triggered by Change Tracking\n\nNow you're going to create the business rule to update the Approved attribute when the Cost attribute changes. You can create your own rules for any attributes that change by using similar steps.\n\n**1.** On the home page, click System Administration.\n\n**2.** On the menu bar, click Manage | Business Rules.\n\n**3.** Select a model, entity, member type, and attribute. In this case, choose Product, Product, Leaf, and All.\n\n**4.** Click the \"Add business rule\" button.\n\n**5.** The row is highlighted. Select the \"Edit selected business rule\" button. The Edit Business Rule page is displayed.\n\n**6.** In the Components pane, expand the Conditions node.\n\n**7.** Under the Value comparison node, click \"has changed\" and drag it to the IF pane. Drop it on the Conditions label.\n\n**8.** In the Attributes pane, click Cost and drag it to the Edit Condition pane. Drop it on the Select attribute label. (Important: The attribute you drag here does not have any effect on MDS. It's required to create the rule, but the change tracking group and the attributes assigned to it, not the attribute, is what the rule is based on.)\n\n**9.** Click Save item.\n\n**10.** In the Components pane, expand the Actions node. In the Change value node, click \"equals\" and drag it to the THEN pane. Drop it on the Actions label.\n\n**11.** In the Attributes pane, click Approved and drag it to the Edit Action pane. Drop it on the Select attribute label.\n\n**12.** Select Pending from the list.\n\n**13.** Click Save item.\n\n**14.** Click the Back button in the top-left corner.\n\n**15.** Click the \"Publish business rules\" button.\n\nNow any time a cost changes and business rules are applied, the Approved column changes to Pending.\n\nTo have e-mail sent to the manager, create another business rule that sends e-mail when the Approved column is Pending. To do this, use the \"is equal to\" condition and the \"is not valid\" action.\n\nOn the Business Rule Maintenance page, in the Notification column, choose the user or group you want to notify. This user or group is responsible for changing the Approved attribute from Pending to Yes or No.\n\nNOTE\n\n_You might want to create an additional business rule that notifies a different user or group if the Approved attribute value changes to No._\n\nTriggering an External Workflow\n\nWorkflows external to MDS are triggered by using the external action business rule. When business rules are applied, all entity members that evaluate to true by the conditions provided are sent to the external action Service Broker queue.\n\nTo enable external actions, the workflow integration service must be installed separately. Otherwise, the data gathered by external action business rules just sits in the external action Service Broker queue without triggering any effect. Because this workflow listener pulls requests from a Service Broker queue, only a single workflow listener should be connected to an MDS database at a time and no other custom components should be pointed at the external action Service Broker queue.\n\nA diagram of the process is shown in Figure 8-1. The workflow integration service queries the database periodically, and if the external action business rule has put data in the Service Broker queue, the data is passed by the service to either a SharePoint workflow or some other, custom workflow.\n\n**Figure 8-1** _The workflow integration service interacting with workflows external to MDS_\n\nNOTE\n\n_If the SharePoint or other workflow is not configured properly, the service still clears the data from the queue and attempts to pass it to the workflow. At this point, even if you re-validate against business rules, the change tracking has already occurred and the data won't be re-added to the queue._\n\nThe following procedures should help you get started with installing the workflow integration service and configuring MDS to trigger external workflows. We've also written a white paper specific to triggering SharePoint workflows, which you can view here: .\n\nInstalling the Workflow Integration Service\n\nThe first step in triggering an external workflow is to install the workflow integration service. It is called Microsoft.MasterDataServices.Workflow.exe and it can be found in Program Files\\Microsoft SQL Server\\110\\Master Data Services\\WebApplication\\bin. You can use InstallUtil.exe to install the service. When you are ready to begin pulling data from the Service Broker queue, you will start this service.\n\nAs administrator, open a command prompt and enter the following command:\n\nUpdating the Web Configuration File\n\nDepending on the type of external action you want to trigger, different settings must be made in the Workflow configuration file. This file is called Microsoft .MasterDataServices.Workflow.exe.config and can also be found in Program Files\\Microsoft SQL Server\\110\\Master Data Services\\WebApplication\\bin.\n\nIf the only external actions that will be triggered are SharePoint workflows, then only the link to the MDS database is necessary. An example of the database connection string needed for a SharePoint workflow is shown here:\n\nIn this example, MDSServer is the name of the server and MDS is the name of the database.\n\nFor each external DLL that you want to install into MDS, you must add a reference in the configuration file. Multiple external references must be added to the same XML node and separated by a semicolon. An example of two external actions is shown here:\n\nCreating a SharePoint Workflow in Visual Studio\n\nIf you decide you want to use SharePoint for your workflow, use these steps to get started designing a workflow in Visual Studio:\n\nNOTE\n\n_You must use Microsoft Visual Studio 2010 and Microsoft SharePoint Foundation 2010 or Microsoft SharePoint Server 2010 to create your SharePoint workflow._\n\n**1.** Open Visual Studio 2010.\n\n**2.** Using the SharePoint 2010 templates, create a Sequential workflow project.\n\n**3.** Drag the OnWorkflowActivated control from the toolbox to the designer area.\n\n**4.** Right-click the control and choose Generate Handlers.\n\nThe data passed from the business rule process into the Service Broker queue will be contained in the string MDSData = workflowProperties.InitiationData. You must now parse the MDS data string into separate data fields and use the MDS web service to interact with Master Data Services.\n\nCreating a Business Rule That Starts a Workflow\n\nTo trigger an external workflow from within MDS, you must create a business rule that starts the workflow. To do so, create a business rule as you normally would.\n\nNOTE\n\n_Workflow business rules should always contain a condition. Otherwise, the workflow would be continually triggered._\n\nThe action needed to start the workflow is the last item in the node of the Components pane. Under the External action node, click \"start workflow\" and drag it to the THEN pane and drop it on the Actions label.\n\nThen populate the Edit Action pane as follows:\n\nThen save the item and publish the business rule as you normally would. When the business rule is applied, it populates the queue. When you start the service, the queue is cleared and an attempt is made to pass the data to the workflow.\n\nIn SQL Server 2012, Master Data Services has broken out the SharePoint Workflow DLL. You can get this open source DLL in the MDS workflow sample at . Search for Master Data Services to find the samples page.\n\nUsing Web Services to Manage Business Rules\n\nBusiness rule management is the most complicated process exposed by the Master Data Services web service. Although we provide a couple of simple examples in this section, writing more complex business rules is significantly more intricate and would require a significant amount of this book to explain. Automation of MDS business rules should only be attempted by experienced developers working on a specific project that requires business rule automation.\n\nGetting a List of Rules\n\nThere may be instances where you want to programmatically consume Master Data Services' business rules in external systems. While we have seen significant interest in this, we believe the internal ruleset limits the capability for MDS to be used as a rule warehouse. Users interested in this type of solution should look at using standard MDS entities to store this data. For users interested in getting rules out of MDS, the following procedure shows a common request to get a list of business rules:\n\nCreating a Business Rule Using Web Services\n\nThe most complex operations in MDS are the creation and update of business rules in MDS. If you are interested in automating the business rule process, we suggest that you keep the processes simple, by not adding complex conditions or actions in the beginning. At the conclusion of this procedure, a new business rule will be created but will not be published. You will need to publish the rule in another operation or through the UI.\n\nSummary\n\nIn this chapter we discussed the ability to manage master data by using business rules. These business rules were created to allow business users to quickly and efficiently create logical constructs to manage the data within their entities. Business rules are entity and type centric. Using notifications and business rules, simple workflow processes can be managed using only Master Data Services functionality. If more complex workflows need to be managed, you can trigger SharePoint workflows using the external action business rule. The functionality of business rules did not significantly change in this latest release with the limited exception of new views for change tracking groups and a new process for auto-generating Code values. In the next chapter we will discuss using validation as part of a version management process.\nChapter 9\n\nCreating Versions of Data\n\nIn This Chapter\n\n Versions Overview\n\n Updating Your Version Name and Description\n\n Version Statuses\n\n Validating a Version\n\n Reviewing and Reversing Transactions\n\n Committing and Copying a Version\n\n Version Flags\n\n Viewing a Version's Ancestry\n\n Using Web Services to Work with Versions\n\nAfter you understand the workflow for adding and modifying the data stored in MDS, you must learn how to manage that data over time. For each model within MDS, a historical record of the data can be stored. These data snapshots are called _versions_.\n\nEach time you create a version of a model, the data for all the entities within the individual model are versioned at the same time. Only the data is stored; any changes to the structure of the model affect all versions and can create unintended consequences.\n\n_Version flags_ are another important component of versioning in MDS. By assigning flags to versions, integration with other systems can be better managed.\n\nThere were no significant changes to versioning in the SQL Server 2012 release. Administrators should continue to use versions to manage snapshots of their models over time.\n\nVersions Overview\n\nIn order to manage data within your organization effectively, you may be required to create versions of the data stored within a model. In MDS, the data in all entities within each model is versioned simultaneously. A benefit of this design is that you can manage relationships between entities without worrying about time and version. A side effect of this design is that entities in different models cannot interact with one another.\n\nThere is often a natural cadence to many data domains. This cadence can help define which entities should be managed in the same model. For example, in many organizations, the accounting department manages structural changes on a monthly basis. As each month ends, everyone in the accounting department goes through standard routines to ensure that the month's books can be closed properly. Any entities related to this process\u2014accounts, divisions, departments, or other internal business entities that are central to this process\u2014should be managed together, versioned in concert with the month-end process.\n\nIn another example, an organization may release products on a quarterly basis. All entities central to the product development process would follow this quarterly versioning scheme. Other entities may not require a versioning scheme at all.\n\nVersions provide a number of benefits to the data management process:\n\n **Complete model history for a specific point in time** MDS model versions can be committed to ensure that an exact record of a model's data can be stored for later review. These committed versions can provide a portion of the required audit trail for new, more rigorous compliance requirements.\n\n **Limited access during sensitive processes** When performing certain processes, like validating the entire model or loading large numbers of records, it may be prudent to restrict access to the model by locking it.\n\n **Additional version copies for analysis** Additional versions can be created outside of the standard cadence for a variety of purposes. These versions can be used to examine new hierarchy configurations or potential acquisitions without affecting the current regular processes.\n\nChanging the Structure of Your Model\n\nMDS does not version metadata changes. Any changes to the model structure affect all open and committed versions. If an attribute or entity is deleted, for example, all history for that attribute or entity is lost from all versions _forever_. When you need to maintain historic data, we suggest that you use security permissions to hide attributes or entities instead of deleting them. You can also hide attributes by setting the display width to zero or by not adding them to attribute groups.\n\nWhen you add an attribute or entity, the model structure is updated in all versions as well. You can add the corresponding data to any version of the model; if you add data to a later version, the structure exists in the earlier versions but the data does not.\n\nCommitting Versions\n\nSometimes users and downstream systems need to be certain that all data has been validated and reviewed. Because MDS allows incomplete members to be added to the system and encourages users to manage the data creation and correction workflow from within MDS entities, it may not be reliable for external production systems to use open or locked versions. Only committed versions ensure that all members in every entity within the model have passed all business rules successfully. Once a version is committed, no additional changes can be made to the data, and the status of the version cannot be changed.\n\nFigure 9-1 shows the most common version control workflow.\n\n**Figure 9-1** _Default version control workflow_\n\nA few notes about Figure 9-1:\n\n Each time the status changes, if notifications are configured, an e-mail is sent to model administrators. For more information about notifications, see Chapter 8.\n\n You can change a setting in MDS Configuration Manager so that you can copy versions with a status other than Committed, but Figure 9-1 shows the default behavior.\n\n You can validate a version at any time. Locking the version ensures that users don't make changes after you've validated, but it's only necessary if you plan to commit the version or otherwise prevent users from making changes.\n\nVersioning for Main Street Clothing Company\n\nMain Street Clothing Company has separated its entities into two separate domains: Account and Product. A major reason for this split was the version cadence of each domain.\n\nAs a clothing company, Main Street finds that its product lines change significantly based on the season. This creates a natural cadence for managing major changes to its product list on a quarterly basis. The account model is managed on a monthly basis as part of the accounting team's regular month-end close process.\n\nUpdating Your Version Name and Description\n\nBefore you lock, validate, commit, or copy a version of your model, you might want to customize the version name and description. You can do this at any time. The new version name is displayed in many places in the Master Data Manager web UI, but you will probably notice it most on the home page, when you select the model name and version from the drop-down lists.\n\nProcedure: How to Change the Version Name and Description\n\nTo change the version name and description, complete the following steps:\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** From the Model list, select the model with the version name or description you want to change.\n\n**3.** Double-click the cell that contains the version name or description that you want to change. The cell becomes editable.\n\n**4.** Change the text.\n\n**5.** Press enter.\n\n**6.** Double-click the cell with the version description.\n\n**7.** Change the description to the text you want.\n\n**8.** Press enter.\n\nThe name and description are updated.\n\nNOTE\n\n_The date\/time shown in the Last Changed Date column is based on the last change to any of the version information on this page, including the status._\n\nVersion Statuses\n\nAs you work with versions of your model and its data, you change the status of the version and then make copies of it. The version status indicates whether or not the model structure and its data can be updated. The following table lists the possible statuses for a version.\n\nProcedure: How to Lock a Version\n\nWhen you're ready to save a version of the model, you first lock it. Then you can validate the model data and work on resolving issues that occur when data fails validation against business rules. To lock a version, complete the following steps:\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** From the Model list, select the model you want to lock.\n\n**3.** Select the row for the version you want to lock. The background is shaded when the row is selected.\n\n**4.** Click the Lock button.\n\n**5.** In the confirmation dialog box, click OK.\n\nThe Status column changes to Locked.\n\nNow you can proceed with validating the version against business rules so that you can commit the version and create a copy of it.\n\nValidating a Version\n\nWhen you validate a version against business rules, all the members in the model are validated against business rules. You can validate subsets of members in the Explorer functional area, but you use the following procedure to validate all members at the same time.\n\nProcedure: How to Validate a Version\n\nTo validate the members from a version against business rules, complete the following steps:\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** On the menu bar, click Validate Version.\n\n**3.** Select the model and the version you want to validate.\n\n**4.** Click the Validate button.\n\n**5.** In the confirmation dialog box, click OK.\n\n(You may have to do steps 4 and 5 twice. You will know that the process is complete when the Commit button is enabled. Depending on the number of rules and members needing validation, this may take a long time to complete.)\n\nValidation Statuses\n\nThe statuses that are displayed on this page help you determine which members have passed business rule validation and which members have failed. The following table lists the possible validation statuses.\n\nProcedure: How to Resolve Validation Issues\n\nBefore you can commit the version, you must resolve all business rule validation issues by completing the following steps:\n\nNOTE\n\n_In Explorer, users can no longer view their own individual business rule issues, if they are recipients of notifications that are sent when validation fails users will need to use the links in the e-mail they receive or use the Excel Add-in described inChapter 10. As an administrator, you can view all issues for all users by following this procedure._\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** On the menu bar, choose Validate Version.\n\n**3.** Select the model and the version.\n\nA list of all validation issues is displayed at the bottom of the screen. Unlike the old Explorer functional area, this page shows validation issues for all users.\n\n**4.** If you need to unlock the version at this point so that users can update data, return to the Manage Versions page, select the model, and click the Unlock button.\n\nNOTE\n\n_To fix the validation issues, you can now filter based on the validation status in Explorer as well as the new Add-in for Excel, which is discussed in more detail inChapter 10._\n\nReviewing and Reversing Transactions\n\nTo help resolve business rule validation issues, you can review transactions for all users in the Version Management functional area. To do so, choose Transactions. For the steps required to reverse a transaction or annotate it, see Chapter 7.\n\nCommitting and Copying a Version\n\nYou should not commit a version until you are sure it's complete and accurate. Once you commit a version, you cannot undo it without opening the database. You also have the ability to delete a version, but you must also go into the database to do so. In either case, you should only do these procedures if you're not concerned with compliance issues and if you're comfortable working in the database.\n\nIn general, you should commit the version and then copy it, giving users the ability to continue updating the data.\n\nProcedure: How to Commit a Version\n\nWhen all members pass validation against business rules and the version is locked, you can commit the version by completing the steps that follow.\n\nCAUTION\n\n_After you commit a version, you cannot reopen it from the UI. Before committing, you should be sure that you no longer want to modify the data._\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** On the menu bar, click Validate Version.\n\nIf all rows in the table show 0 except the Validation succeeded row, then the version can successfully be committed. Click Commit.\n\n**3.** In the confirmation dialog box, click OK.\n\nWhen the version is successfully committed, a success message is displayed. Back on the Manage Versions page, the Status column is updated to Committed.\n\nNow you can copy the version.\n\nProcedure: How to Reopen a Committed Version\n\nYou can change a committed version back to an open version by completing the steps that follow.\n\nCAUTION\n\n_This action should be performed by a DBA only in those rare situations in which the version was committed by mistake. If production systems have pulled this data and used the results for reporting that is subject to compliance, the more appropriate action is to copy the version and commit a new, revised version for compliance purposes._\n\n**1.** Open SQL Server Management Studio and connect to the MDS database.\n\n**2.** In the table mdm.tblModel, in the ID column, note the ID of the model.\n\n**3.** In the table mdm.tblModelVersion, in the Model_ID column, find the ID from step 2.\n\n**4.** In the row for the version of the model you want to open, change the value in the Status column from 3 (Committed) to 1 (Open) or 2 (Locked).\n\nThe version is now displayed as open or locked in the web UI.\n\nProcedure: How to Copy a Version\n\nWhen you copy a version, a new version with the next version number is created. Newly created versions have the status of Open and are available for editing by users and administrators. To copy a version, complete the following steps:\n\nNOTE\n\n_By default, you can copy Committed versions only._\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** On the menu bar, choose Manage | Versions.\n\n**3.** From the Model list, select a model.\n\n**4.** Click the row for the version you want to copy.\n\n**5.** Click the Copy button.\n\n**5.** In the confirmation dialog box, click OK.\n\nA new row is displayed in the grid. You can update the name and description to indicate which version you want this to be.\n\nCopy Versions Other Than Committed\n\nIn MDS Configuration Manager, you can change whether you can copy models with any status or just those with a status of Committed. The setting is called \"Copy only committed versions\" and the default is Yes. Set it to No if you want to copy versions of any status.\n\nIf your organization acquires another organization, or you want to work with a hypothetical version of the data, it can be useful to copy other versions of the data. If you have more than one open version, however, remember that users can modify it. You can set security to prevent access, but it is not a straightforward process. See \"Hierarchy Member Permissions\" in Chapter 11 for more information.\n\nVersion Flags\n\nWhile versions can provide a myriad of benefits, managing the integration with other systems can be a nightmare as the current version changes over time. MDS addresses this issue with the implementation of version flags. Each version in MDS can be flagged with a custom flag specific to the selected model. These flags allow data stewards to tag a version as current, prior, draft, or some unique identifier for downstream systems. Each version can have only one version flag associated with it at a time.\n\nBecause downstream systems rely on the validity of data being presented in a version, MDS can require versions to be committed before the flag can be applied. For most production systems, it is important to ensure that committed versions require flags. This check will be the only way to ensure that a specific version has passed validation before being used by downstream systems.\n\nProcedure: How to Create a Version Flag and Assign It to a Version\n\nTo create a version flag and assign it to a version of a model, complete the following steps:\n\n**1.** On the Master Data Manager home page, click Version Management.\n\n**2.** On the menu bar, choose Manage | Flags.\n\n**3.** From the Model list, select a model.\n\n**4.** Click the Add button.\n\n**5.** A grid is displayed. Type a name and description and select whether the flag is available to apply to Committed versions only (True) or to versions of any status (False).\n\n**6.** Click the Save button.\n\n**7.** On the menu bar, choose Manage | Versions.\n\n**8.** In the row of the version you want to assign the flag to, double-click the Flag column. A list is displayed.\n\n**9.** Select the flag from the list.\n\n**10.** Press tab or enter.\n\nThe Flag column is updated.\n\nNOTE\n\n_You can assign a flag to one version at a time only._\n\nViewing a Version's Ancestry\n\nYou can copy any previous versions to create new versions; for example, you can copy versions in this order:\n\n Version 1 to Version 2\n\n Version 2 to Version 3\n\n Version 1 to Version 4\n\nIf you create several versions of a model and you are trying to determine where each version came from, you can view the Copied From column on the Manage Versions page.\n\nThis column shows only the most recent version in the version's history. You can view a more in-depth historical record by viewing the version ancestry. On the Manage Versions page, click the row you want to see a history for and click the View Ancestry button. In this example, we'll view Version 3, which was a copy of Version 2.\n\nThe Version Ancestry dialog box that is displayed shows the history. You read this dialog box from bottom to top; 0 is the first time the version was generated, 1 is the second time, 2 is the third, and so on. In this case, the first version of this model was Version 1, then it was copied to Version 2, and then to Version 3.\n\nProcedure: How to Delete a Version\n\nNormally, you should not need to delete a version of your model. This procedure is intended to help you with testing or if some circumstance forces you to take this action. Use with caution.\n\nCAUTION\n\n_If you delete the only version for the model, the model becomes unusable._\n\nTo delete a model version, complete the following steps:\n\n**1.** Open SQL Server Management Studio and connect to the MDS database.\n\n**2.** In the view mdm.viw_SYSTEM_SCHEMA_VERSION, determine which version of the model you want to delete, and copy the value in the ID field.\n\n**3.** Create a new query similar to the following:\n\nReplace _value from step 2_ with the ID from step 2.\n\nWhen you run the query, the version is deleted. The Master Data Manager web application reflects this change shortly after. To confirm the change, click the \"Refresh cached information\" link on the MDS home page. This link is located in the upper-right corner of the page.\n\nUsing Web Services to Work with Versions\n\nWeb services can be used to update information related to versions within a model. In this example, we use the MetadataUpdate method for the first time. Unlike the MetadataCreate operation, identifiers cannot be created using the Name field only to allow people to update the name.\n\nReturning a List of Versions\n\nWhen managing versions through the MDS web services, the GetVersion helper class will be very useful because version updates cannot be managed based on the name directly, as we have discussed. In this helper class, we will return the version based on the model and version name. Similar helper classes would be required to update any metadata objects.\n\nChanging the Version Status\n\nVersion information is updated using the MetadataUpdate operation. The following code leverages the helper class outlined in the prior section to retrieve the identifier for the version you want to update. In this code, we lock the version. A similar method could be used to commit the version, but committing a version requires that the version has been successfully validated first.\n\nSummary\n\nVersioning provides you with the flexibility to manage models over time. Versions can be in one of three states: open, locked, or committed. Committed versions can provide users and downstream systems with the assurance that all members have been validated and the committed version will not be changed in the future. Integration with downstream systems can be facilitated by using version flags. Version flags can be used on draft versions, or you can require that versions be committed before they can be flagged. In the next chapter, we will discuss the new Excel Add-in for MDS.\nChapter 10\n\nSQL Server 2012 MDS Add-in for Excel\n\nIn This Chapter\n\n Installing the Add-in\n\n Working with MDS Data in Excel\n\n Creating and Modifying Entities Rapidly\n\n Configuring Add-in for Excel Settings\n\n Leveraging Data Quality Matching in Excel\n\nMore master data is stored in Excel than in any other application on the planet. Some smaller companies use this solution because it is cost effective and easy to use. Others use it to store their \"homeless\" data in worksheets because it is the tool they are most comfortable with. Even the largest organizations with the greatest, most well-designed MDM systems that money can buy struggle with propagation of master data in Excel.\n\nIn SQL Server 2012, the Master Data Services team has attempted to harness the power of Excel to give users all the security, audit, and management features of MDS in a package that information workers are most comfortable with. In this chapter we will explore the current features available in Excel and how you can leverage those features for all types of data management.\n\nUsing the combination of Excel and MDS as a data management tool can provide organizations with significant advantages. Excel is a natural platform for managing data from a variety of sources. Its built-in data functionality can be used to parse and cleanse data before loading the data into MDS. The developer interfaces, whether code, macros, or formulas, can provide additional custom automation in a rapid fashion, without voiding any support in the MDS solution.\n\nMany of Excel's limitations are also addressed by using it in conjunction with Master Data Services. Data sharing, security, and transaction logging have always posed problems that spreadsheet designers have struggled to solve. One of the biggest concerns with storing data in Excel is the lack of security and central management. Using the MDS Add-in for Excel can alleviate many of these concerns.\n\nInstalling the Add-in\n\nThe MDS Add-in for Excel is a separate, free installation available on the Microsoft website. The easiest way to download the latest edition of the Add-in is to use the link on the web UI's home navigation page. This link will be updated to the latest Excel Add-in download as new versions become available. It is important to understand that the Excel Add-in is not tied to the SQL Server update schedule. New versions of the Excel Add-in can and will include additional functionality that can be used with a SQL Server 2012 Master Data Services implementation.\n\nProcedure: How to Install the Add-in for Excel\n\nTo install the Master Data Services Add-in for Microsoft Excel, complete the following steps:\n\n**1.** Click the Install Master Data Services Add-in for Microsoft Excel link on the navigation page, shown next, and download the proper edition for your computer. The edition required is based on whether you have the 32-bit or 64-bit version of Microsoft Office installed on your computer.\n\n**2.** Run the setup until you see the following initial installation screen. Click Next.\n\n**3.** On the License Agreement page, you must accept the license agreement to enable the Next button and continue the installation. Once you accept the terms, click Next.\n\n**4.** Click Install to begin the installation.\n\nThe MDS Add-in for Excel installation may require Visual Studio 2010 Tools for Office Runtime to be installed. This is available for free on MSDN.\n\n**5.** Once the installation has completed successfully, open Excel. If you already have Excel open, you will need to open a new instance for the Add-in to be available. A new Master Data ribbon should be available.\n\nWorking with MDS Data in Excel\n\nThe Master Data Service Add-in for Excel will allow you to load entity data into a worksheet as an Excel formatted table. An Excel table is really a perfect environment to review and manage reasonable amounts of data from Master Data Services. Filtering, sorting, and formula functions can be used to review the data in a friendly format without destroying the ability to write back to the server. Data can also be taken \"offline\" and modified without connectivity and then published back to MDS when connectivity has been restored.\n\nProcedure: How to Connect to the MDS Server and Load an Entity\n\nTo load an entity from Master Data Services into Excel, complete the following steps:\n\n**1.** Click the Master Data tab to display the Master Data ribbon.\n\n**2.** Click the Connect button to create a new connection. After you have set up the connection, clicking this button will connect automatically to the stored connection.\n\n**3.** The connection manager will display. In the possible event you are managing multiple Master Data Services implementations, you can manage them here. Click the New button at the bottom of the screen to add the connection to your server.\n\n**4.** In the Add New Connection dialog box, provide a description and server address for the connection. You can use any name that is identifiable for the connection. Don't worry about this now, as you can always change the name as new servers are added. The MDS server address is just the base URL that you use to connect to the MDS server. Ensure this address does not contain a page such as default.aspx at the end. Click OK when you are finished.\n\n**5.** Click the Connect button to connect to the server.\n\n**6.** After making a new connection, the Master Data Explorer task pane displays on the right side of the screen. Select the Product model from the Model drop-down list. The most recent open version (Version 4) will be selected by default, but you can change to any available version in the Version drop-down list.\n\n**7.** Double-click the Product entity from the list shown next. Data will immediately begin to load into the active sheet. If data already exists in the sheet, a warning message about data loss is displayed. If you select Yes, all data will be lost from the sheet. (Please be careful not to lose important data in this procedure.) Data on the MDS server is not affected.\n\nA new Excel table with all the Product entity attributes as columns is created on the active sheet and all members of the Product entity are added as rows in the worksheet.\n\nPublishing Changes\n\nUnlike the web UI, the Add-in for Excel does not update members as changes are made to each cell. Instead, every change to a cell is monitored and the cell is colored to denote an unpublished change. Cells can be changed by direct edit or by leveraging any of the more efficient methods of updating a range of cells, such as copy and paste or autofill.\n\nThis coloring feature has the unwanted effect of removing the ability to undo changes in Excel. While we think that the benefit of having colored cells is worth this loss of functionality, you can turn off this feature in the settings.\n\nInitially, when you click the Publish button, the following transaction annotation form appears. You can add a comment about the change to all changes or provide a comment for each change individually. If you want to suppress this form for future Publish actions, select the \"Do not show this dialog box again\" check box at the bottom of the form.\n\nPublishing changes will also trigger validation on those updated members. Any validation errors will be returned to the page, and two new columns will appear. You can toggle between showing and hiding these columns by clicking the Show Status button on the ribbon.\n\nRefreshing Data from the Server\n\nOnce you have created an MDS managed sheet, controls in the Master Data Explorer pane become disabled. You cannot change the entity managed by the sheet, but all filter functionality is still available. You can save the parent workbook for this sheet and open it at a later time. All of the functionality should still be loaded. Instead of creating a new worksheet, you have two options to refresh data. Based on your needs, you can either refresh the entire managed sheet or refresh a selected range of cells. If you are ready to publish a worksheet that depends on certain attribute values that may have changed on the server recently, refreshing a selected range allows you to reload those attribute values before you publish your worksheet. You can also leverage this feature as a pseudo-undo.\n\nWhile not recommended, a saved sheet can be sent to another MDS user. Sending data within an Excel workbook circumvents security on the server. We will discuss sharing this query with others later in this chapter.\n\nReviewing Transactions\n\nYou can review the transaction history of any member in a MDS managed sheet by using a new MDS menu item at the bottom of the Excel menu displayed when you right-click a row in a managed sheet. Annotations for any transaction will be displayed in the right-hand column, and you can add additional annotations at the bottom of the screen.\n\nMaking Deletions in Managed Sheets\n\nDeleting members in an MDS managed sheet can be confusing. If you delete members using standard methods such as the right-click menu option, the member is removed only from the worksheet. It changes nothing on the MDS server. If you want to remove one or more members from the MDS system, you must use the Delete button on the Master Data ribbon. This method soft deletes all selected rows from the system. There is currently no way to purge members using the Add-in for Excel.\n\nCombining Data\n\nYou can map data from a range into an MDS managed sheet by using the Combine Data button on the ribbon. This button is enabled only when the active sheet is an MDS managed sheet. You can select a range and then map to each of the attribute columns present in the managed sheet. It is not required to map all of the attributes to combine data.\n\nApplying Rules\n\nIf you want to trigger validation on members awaiting validation without changing any data, you can apply validation to the managed sheet with the Apply Rules button on the ribbon. Only those members not currently validated will be processed. Any errors will display on the sheet and the validation status column will be shown.\n\nFiltering Entity Results Members\n\nWhile all of the datasets in our modest examples are small, many organizations may have entities that contain thousands or millions of members. The Add-in for Excel has been designed to hold up to a million MDS member records in a single managed sheet. There may be some instances when analyzing or managing a large number of records is warranted based on a business need, but most people should use the filtering function to limit results. The filtering function in Excel can limit both the number of columns and the number of rows returned from the MDS server.\n\nProcedure: How to Load a Filtered Entity\n\nTo load an entity with filtered members and attributes, complete the following steps:\n\n**1.** Open a blank worksheet or a new workbook.\n\n**2.** Click the Master Data tab to display the Master Data ribbon in Excel.\n\n**3.** Click the Show Explorer button to display the Master Data Explorer pane.\n\n**4.** Select the Product model from the Model drop-down list at the top of the Master Data Explorer pane.\n\n**5.** Select the Product entity in the list. Selecting an entity enables the Filter button in the Master Data ribbon, as shown here:\n\n**6.** Click the Filter button to display the Filter dialog box.\n\n**7.** You can filter rows returned to the Excel worksheet by adding filter criteria in the Rows section. Select Validation Status from the Attribute drop-down list. You can now filter member records based on attribute values or the record's current validation status in SQL Server 2012.\n\n**8.** Select the \"is equal to\" operator from the Operator drop-down list.\n\n**9.** Select Validation failed from the Criteria drop-down list. Updates to row filter criteria do not automatically update the summary in the lower-right pane. To update this summary, you can click the Update Summary button at any time.\n\n**10.** To limit the attributes returned, select Pricing from the Attribute groups drop-down list in the Columns section on the left. The Summary pane will immediately update to 5 rows and 5 columns returned.\n\n**11.** Click the Select All button beneath the Attribute list box.\n\n**12.** Uncheck the Design attribute in the Attribute list box. The summary will update to 8 columns returned.\n\n**13.** Click the Load Data button to load the data into the active worksheet.\n\nSaving and Sharing Queries\n\nNow that we have created a complex request to pull selected attributes and records from an entity, we need to save this request for later use. For years, MDS users have asked the Microsoft development team to provide a process to persist filter criteria for future use. The Add-in for Excel now provides this functionality while also allowing users to share those queries with other users through Outlook with the click of a button.\n\nProcedure: How to Save a Filter Query and Share It Through Outlook\n\nTo save a previously created filter as a query, complete the following steps:\n\n**1.** Activate the managed worksheet from the previous procedure.\n\n**2.** Click the Master Data tab to display the Master Data ribbon in Excel.\n\n**3.** Expand the Save Query button drop-down list and select Save as Query. This saves the filter setting for the active managed sheet as an XML query. This query is stored on your machine and can be reloaded into a new worksheet from the server at any time.\n\nYou can manage stored shortcut queries from the Manage Queries form, which you also access from the Save Query button drop-down list. Previously saved queries can be loaded, renamed, or deleted from this screen.\n\n**4.** Click the Send Query button on the ribbon to share this MDS query with others. The benefit of sending the query rather than the worksheet itself is that the security configuration will trim results for each individual user. Only the members and attributes that each person has rights to will be returned when the Excel query is initiated.\n\n**5.** An e-mail is generated with the query attached as an XML file with a new .mdsqx extension. This file opens automatically on any computer with the MDS Add-in for Excel installed on it.\n\n**6.** Each user sees this warning message the first time that a file is opened, asking if the server connection should be allowed. You can add this server to a list of trustworthy servers and never see this message again by selecting the \"Always allow connections to this address\" check box and clicking the Yes button.\n\nCreating and Modifying Entities Rapidly\n\nEarly in my career, after spending many days loading files into Excel and databases, I had grown tired of needing to solve all of my column choices before I could see the results. I didn't always know what I wanted. The wizards walked me through seemingly endless steps to determine an output that never matched all of my expectations, and once I clicked Finish, the process did not allow for revisions. If I wanted a different result, I had to start the wizard over again. When I was designing the wizard for Excel, the users I took into account first were the consultants and data stewards. They needed a tool that could be used to quickly model and load data into MDS, so that was the tool I hoped to design. My requirements were as follows:\n\n **Almost no information to do something** While it is nice to write a book on Master Data Services, I don't want reading this book to be a necessity for every user to get started. Master Data Services supports only a small subset of data types, so the Add-in should support determining the correct data type for the column and successfully loading all members.\n\n **Every action can be triggered automically** If I want to change a column to a domain-based attribute or change the decimal scale of a number attribute, I don't want to re-create the entire entity. I just want to replace the column in question. If I need to add a new column, I want to click on the column of unmanaged data and select the name of the attribute in MDS.\n\n **An entity is created for my DBA attribute automatically** If I select a column and determine that current values are the only values that I require, I want the supporting entity to be created. This entity should be populated with all of the unique values within the selected column.\n\nThe end result of successfully incorporating the preceding requirements into the tool's design is a very powerful feature that allows data stewards and consultants alike to quickly build and modify entities. The Add-in for Excel not only encapsulates MDS features in the Excel interface, but extends them into a much more powerful tool. The following examples and procedures will show you how these features work.\n\nUsing the Excel Add-In at the Main Street Clothing Company\n\nAt Main Street Clothing Company, each individual store manages a list of clerks and their schedules in Excel spreadsheets on the manager's laptop. Unfortunately, these spreadsheets are not currently backed up, nor are they integrated with headquarters' systems in any way. Store managers are concerned that any changes to their current process would add to their responsibilities, create inefficiencies in the process, and mandate unnecessary reliance on central IT.\n\nIT proposes the creation of a new model for store managers. Within this model, all managers and the designees will be model administrators. IT will spend an afternoon with each manager to help them create a new entity for each store's clerks. The process is detailed in the following procedure, the worksheet for which you can download from www.mdsuser.com.\n\nProcedure: How to Create an MDS Entity from Excel Data\n\nTo create an MDS entity from Excel data, complete the following steps:\n\n**1.** Click the Master Data tab to display the Master Data ribbon.\n\n**2.** Select all of the data to populate the new entity. The data area needs to be contiguous to be loaded correctly. To quickly select a large, contiguous range, select the uppermost cell of the range, hold down ctrl-shift, press right arrow, and then press down arrow.\n\n**3.** Click the Create Entity button on the ribbon to open the Create Entity form.\n\n**4.** Note that the range for the Excel table appears in the Range field. If you had forgotten to select the range or if you wanted to change the range selected, you could click the range selection button to the right of the Range field and select the range.\n\nNOTE\n\n_While there is a \"My data has headers\" check box for loading an entity without headers, I strongly discourage you from selecting this check box. Each column will be named Column A, Column B, etc. This is unlikely to be helpful to anyone and will probably just lead to more work later on._\n\n**5.** Select the model in which to store the new entity. In Chapter 14 we will discuss some different options on modeling. For now, just store the new entity in the Finance model.\n\n**6.** Select the version for the initial entity data to be loaded into. The entity will exist in all versions of the model, but members will only exist in the selected version.\n\n**7.** Input the name for the new entity in the \"New entity name\" textbox. The name must be unique in the model for the operation to succeed.\n\n**8.** Select \"Generate code automatically\" in the Code drop-down list. This sets up the entity with an automated code generated for each member. We could have selected a unique column, such as SSN, but that could lead to problems later on. While filling this column with unique values is not required for the entity to be created, only unique members and the first of the duplicate members would be created.\n\n**9.** You can optionally select a name column for the entity. For this example, leave the Name drop-down list box blank.\n\n**10.** Click the OK button to create the new entity.\n\n**11.** A new sheet will be created similar to a loaded MDS managed entity. Any duplicate records will be colored to highlight them as errors.\n\nWhen an entity is created, each column's contents are analyzed and a corresponding attribute is created in MDS of the appropriate type, string, number, or date. Sizes and scale will be determined by the initial members being loaded. This might be valuable in creating a model, but the ability to quickly modify an attribute is the new feature that will completely change project schedules.\n\nMaking Changes to an Existing Entity\n\nNow that we have created a new entity, we want to modify this entity's attributes to take full advantage of MDS features and add value over Excel tables alone. In the initial version of MDS, there was no way to transform an attribute from one type to another while keeping the values stored in the column intact.\n\nProcedure: How to Modify an Existing Entity's Structure\n\nTo load and modify an existing entity from Master Data Services in Excel, complete the following steps:\n\n**1.** Click the Master Data tab to display the Master Data ribbon.\n\n**2.** Click Show Explorer to display the Master Data Explorer pane.\n\n**3.** Load the Clerks Loc 1 entity created in the previous procedure.\n\n**4.** Select the State column header in the entity. This enables the Attribute Properties button on the Master Data ribbon.\n\n**5.** Click the Attribute Properties button. This triggers the Attribute Properties form to display, as shown in the illustration. You can update the name or type of an attribute in MDS on this form.\n\n**6.** Select \"Constrained list (Domain-based)\" from the Attribute type drop-down list. Selecting this option alters the form to provide a new drop-down list that enables you to select where to populate the values of the attribute from.\n\n**7.** Initially this new drop-down list box is set to the selected column. This would create a new entity to store all unique values in this column. You will notice another input box, \"New entity name,\" that allows you to name this new list of states. Select the State entity from the drop-down list. Changing this selection hides the \"New entity name\" input box.\n\n**8.** Click the OK button.\n\nNOTE\n\n_The attribute update process does not actually update the column. Instead, it creates a new column and then loads the values from the prior column into the new column. As long as the transition is completed successfully, the old column (and all its transaction history) will be deleted. If any errors occur, the column will remain with an _old suffix appended to the original column._\n\nConfiguring Add-in for Excel Settings\n\nThere are a number of settings for the Add-in for Excel that enable users to customize their installation to some extent. Unlike server settings, these options are stored on the client machine and are not sourced from the server in any way. Data stewards and IT administrators are encouraged to create internal policies on changing the data settings and provide guidance to your end users.\n\nSettings provide value in two ways. One group of settings gives customers an opportunity to tweak data performance and usage from the MDS server. The other group of settings allows you to adjust how the Add-in works within Excel. These settings are not grouped together but rather span across both the Settings and Data tabs within the Settings dialog box.\n\nConfiguring Data Settings\n\nAs the value of MDS and the new Add-in for Excel are recognized across your organization, you may discover a significant number of Excel queries hitting the Master Data Services server. Depending on your hardware deployment and the size of your entities in MDS, it may become valuable to provide policies on data consumption through Excel. While you cannot stop authorized users from making large requests to the server, you can modify warnings and the size of batches retrieved.\n\n **Filter Warnings** Unlike the web UI, which defaults to load only 50 members at a time, Excel users can potentially load up to 1 million members to their Excel worksheets for analysis. The most effective setting to combat large data requests initiates warnings to users before they retrieve large datasets. These warnings are initially set to trigger for datasets over 10,000 rows and 100 columns. You can modify these values or turn off filter warnings completely.\n\n **Batches size (in thousands)** These two settings do not change the total amount of data retrieved, just the size of the data returned in each request. Obviously, smaller requests create more traffic to and from the server, but this can help on client computers with memory constraints. These options are initially set to 50,000 cells per batch. I would leave these settings alone unless you experience significant delays during loading or publishing.\n\nConfiguring Behavior Settings\n\nThese settings affect the behavior of the MDS selection and publish processes in Excel. These settings will allow users to customize their Master Data Services Excel experience.\n\n **Publishing** When you initially publish changes, you can set whether or not a dialog box prompts you to provide transaction annotations to provide greater detail about the transaction. You can suppress or re-enable this dialog box, respectively, by clearing or selecting the \"Show Publish and Annotate dialog box when publishing\" check box.\n\n **Versions** In the Master Data Explorer, selection of a model triggers a default version to be selected. The initial setting is for the newest version to be selected, but you can either suppress the default selection or choose to have the oldest version become the default by using the Version selection drop-down list box. In the initial release of the Add-in for Excel, the Oldest selection does not appear to select properly.\n\n **Logging** Selecting the \"Turn on detailed logging\" check box will create a detailed log of web service calls and actions in the Add-in for Excel. This detailed log will be required by support staff to debug any issues with the MDS plug-in. These logs can get quite large, so clear this check box during normal operations.\n\n **Servers Added to Safe List** In order to protect Excel users, the first time that you load a shortcut file sent through Excel, you will be asked if you trust the server. If you accidentally indicate that you trust a server, you can clear this list by clicking the Clear All button. Note that this will clear all servers, so you will need to reapprove all MDS server locations when the next file is opened.\n\nLeveraging Data Quality Matching in Excel\n\nIn SQL Server 2012, Microsoft shipped a new feature, Data Quality Services (DQS), that provides project-level cleansing and matching tools for loaded datasets. Very little integration exists between MDS and DQS in this initial release. We expect this to change in future releases, but within 2012, the only integration between the two features is the ability to leverage DQS matching policies through the Add-in for Excel. This allows users to determine potential duplicates before loading new record batches into MDS. For more information on setting up Data Quality Services and creating matching policies, go to www.mdsuser.com.\n\nSummary\n\nThe MDS Add-in for Excel really changes the game in master data management, providing a well-known and efficient interface for all members of the enterprise to load and manage their data. The intuitive interface can create new entities, thereby eliminating or reducing costly modeling phases of MDM projects. In Chapter 14, the new modeling process will be discussed in greater detail.\nChapter 11\n\nImplementing Security\n\nIn This Chapter\n\n Security Overview\n\n Security Changes in SQL Server 2012\n\n Users and Groups\n\n Administrators\n\n Testing Permissions\n\n Functional Area Permissions\n\n Model Object Permissions\n\n Deleting Permissions\n\n Hierarchy Member Permissions\n\n Determining Which Permissions Apply\n\n Setting Security by Using Web Services\n\nBy the time you are ready to implement security, most of the functionality in MDS has been enabled. Models have been built and refined, rules have been written, and the application has been integrated into your organization.\n\nIn order to deploy MDS, everyone needs access to the application. While transaction management can provide some accountability, limiting access based on needs and roles ensures that users are unable to change data without authorization. Limiting the number of models and functions available to users can also help them focus more quickly on the data they need. The ability to provide specific data access within the MDS system is the single most important feature of the application. The focused security access provided by MDS empowers business users and frees the IT organization to manage the overall process, not maintain the individual data points.\n\nIn this chapter we provide an overview of the security framework in Master Data Services. We discuss the process of managing user and group permissions and explain the highly customized access that can be granted. The rest of the chapter drills into each level of security by showing how permissions can be applied to your data.\n\nSecurity Overview\n\nMDS security is broken into three distinct areas:\n\n **Functional security** Corresponds to each of the five functional areas displayed on the home page of the Master Data Manager web application. Most users need access only to the Explorer functional area of the web UI. All other functional areas are available only to administrators. Users must have permissions to the Explorer functional area to use the MDS Add-in for Microsoft Excel.\n\n **Model object security** Provides access control to attributes, based on the model objects within the MDS architecture. For example, you can set permissions on an entity, which determines permissions for all attributes for the entity. Or you can set permissions on a single attribute, which affects that attribute only. Model object security is required; without it, a user cannot perform any tasks in MDS.\n\n **Hierarchy member security** Provides the most granular level of security, and is optional. It is used to grant access to specific members, based on their location in a hierarchy.\n\nModel object permissions (which apply to attributes) and hierarchy member permissions (which apply to members) are combined to determine the exact level of security for every attribute value. Figure 11-1 shows how attribute and member permissions intersect so that security can be determined for an individual attribute value.\n\n**Figure 11-1** _Model object and hierarchy member permissions are combined to determine permissions for every attribute value._\n\nSecurity Changes in SQL Server 2012\n\nSecurity in the prior release of MDS was a double-edged sword. The ability to set security on multiple hierarchies simultaneously provided significant flexibility and complexity, but with so many pieces working together, it was easy for administrators to lose sight of what the effective permissions were for each user. A key effort for the second release of Master Data Services was to simplify the security model. In SQL Server 2012, attribute groups are no longer securable; you must set attribute security explicitly on each attribute. Direct hierarchy security has been removed from the security model.\n\nUsers and Groups\n\nMDS relies on Active Directory for user and group authentication. While all security permissions are stored in the MDS database, no passwords or group memberships are managed in MDS.\n\nTo keep security as simple as possible, you should do the following:\n\n Create either Active Directory or local groups and add either Active Directory or local users to those groups.\n\n Assign security in MDS to these groups, rather than to individual users.\n\n If you decide to assign security to a user, don't also assign security to groups that the user is a member of. While MDS has rules for determining which permissions take effect, security becomes more complicated when you do this.\n\nBefore you begin working with security, you should take some time to determine which groups your users might be part of, and which attributes or members those groups might need access to. The following list should give you a general idea of the groups that Main Street Clothing Company might use. Some of the examples in this chapter show how you would assign permissions like these.\n\n **Product Administrators** This group will have permission to all functional areas and to take any action available for the Product model. This includes changing the model structure and modifying all members, among many other things.\n\n **Finance Administrators** This group will have the same type of permission as the Product Administrators, but for the Finance model.\n\n **Purchasing** This group will be able to update the Cost attribute for all products.\n\n **Warehouse** This group will be able to update the attributes on the Logistics tab only.\n\n **Logistics** This group will be able to update the Safety Stock Level, Reorder Point, and Discontinued attributes. All other attributes will be read-only.\n\nAll of these groups will be able to access MDS after being assigned functional area and model object permissions. Main Street might also assign hierarchy member permissions to a few select members of the Purchasing group. These users should be able to view products for only the manufacturers they are responsible for.\n\nEven though we recommend that you assign permissions to groups, for the rest of this chapter we'll refer to permissions that users receive, because at the end of the day, groups wouldn't mean anything if users weren't in them. Users are ultimately the ones who will access the data.\n\nProcedure: How to Add a Group\n\nAfter you've created your groups either locally or on the domain, you can add the group to the web UI by completing the following steps:\n\nNOTE\n\n_Adding a group to the web UI does not give the users in the group permission to access models or data. It simply adds the group to a list, so that you can start to assign permissions._\n\n**1.** On the Master Data Manager home page, click User and Group Permissions. The administrator account you specified in Configuration Manager when you set up MDS is shown in the Users list; for more information about administrators, see the following section.\n\n**2.** On the menu bar, click Manage Groups.\n\n**3.** Click the Add groups button.\n\n**4.** In the Groups field, type the name of the group in the format Domain\\GroupName or Computer\\GroupName.\n\n**5.** Click OK.\n\nNow that the group is displayed in the list, you can continue with assigning functional and model object permissions, which are described in upcoming sections. If you need to add a user instead of a group, click Manage Users on the menu bar and then click Add users.\n\nNOTE\n\n_When you use groups, the users in the group aren't displayed in the list of users until after they have logged in._\n\nAdministrators\n\nIn MDS there are two types of administrators:\n\n The system administrator you specified in the Administrator Account field when creating the MDS database. This user has full control over all models and data. When new models are created, this user automatically has access. This user also has permission to access all functional areas of the web UI. To change this user, you must run a stored procedure in the database. See \"Procedure: How to Change the Administrator Account\" for more information.\n\n A model administrator who is manually assigned Update permission to a model, and no other model object or hierarchy member permissions. This user has full control over the model he or she has Update permission to. Model administrators do not necessarily have permission to access all functional areas of the application. For example, a model administrator might be responsible for integration only. However, they can perform all tasks in whichever functional area they have permission to access.\n\nAll model administrators with access to the User and Group Permissions functional area can assign permissions for other users. Keep this in mind when you're assigning someone permission to update a model.\n\nIn the following sections we explain how to manually assign the permissions that turn any MDS user into an administrator.\n\nProcedure: How to Change the Administrator Account\n\nWhen you created your database, you specified an administrator account. This user has access to everything, including any sensitive data you may be storing. If you want to change this user, complete the following steps:\n\nCAUTION\n\n_This procedure deletes the former administrator's account from MDS._\n\n**1.** On the Master Data Manager home page, click User and Group Permissions.\n\n**2.** Add the new administrator to the list of users by clicking the Add user button.\n\n**3.** Open SQL Server Management Studio and connect to your MDS database.\n\n**4.** In the table mdm.tblUser, find the user you just added and copy the value listed in the SID column.\n\n**5.** Create the following query:\n\n**6.** Replace _MDS_database_ with the MDS database name, _DOMAIN\\user_ with the new administrator's username, and _SID_ with the value you copied.\n\n**7.** Run the query.\n\nThis user now has permission to take all actions on all models and members. The former administrator is deleted from the list of MDS users. Note that if the former administrator was a member of a group that has security permissions, the user retains the permissions assigned to the group. You must remove the administrator from the local or domain group to fully remove his or her privileges.\n\nTesting Permissions\n\nAs you begin assigning permissions to groups, you'll quickly realize that it would be great to log in as the user and confirm that they have the access you intended them to have. You can do this by creating a test user or group and assigning permissions to it. Then you can access the Master Data Manager web application by using the credentials of the test user and view the data that the user views.\n\nTo have Internet Explorer prompt you for credentials when you open an intranet site like Master Data Manager, you can change IE's security settings. Open IE and choose Tools | Internet options. On the Security tab, ensure that Local intranet is selected and click the Custom level button. Scroll to the bottom of the Security Settings dialog box, and in the User Authentication section, click the \"Prompt for user name and password\" option.\n\nNow when you open a new browser, you are prompted for credentials.\n\nAfter you enter the credentials of your test user, you can confirm that they have the access you expect them to have. Each time you make changes to the test user's security, you should close and reopen the browser to ensure you're viewing the latest changes.\n\nNOTE\n\n_You will learn later in this chapter that hierarchy member permissions are not applied immediately. Keep this in mind when testing security._\n\nIf you are using IE 8 or IE 9, another setting to change before you start working with permissions is Compatibility View, which you should turn off. To do so, click Tools | Compatibility View. This makes the User and Group Permissions functional area work better. However, when you open the browser by using the test user's credentials and want to view data in the Explorer functional area, you should turn Compatibility View back on.\n\nFunctional Area Permissions\n\nFunctional area security determines which of the five functional areas on the Master Data Manager home page a user or group can access. Security at this level is either permitted or denied. If permission to access a specific functional area is denied, the area is not displayed in the web UI and related web service operations are denied.\n\nThe Explorer functional area is where users manage data. When you assign access to Explorer, you must assign access to specific model objects, so the user gets access to a specific set of data. When you assign access to any of the other functional areas, the user must have access to the entire model (on the Models tab) in order to use those areas. Without this access, the user can open the functional areas, but no models are displayed. This is how MDS handles permission for administrators.\n\nProcedure: How to Assign Functional Area Permissions\n\nTo assign functional area permissions, complete the following steps:\n\n**1.** On the Master Data Manager home page, click User and Group Permissions.\n\n**2.** On the menu bar, click Manage Groups.\n\n**3.** To the left of the group you want to edit, click the down arrow.\n\n**4.** On the submenu, click Edit | Functions.\n\n**5.** At the top of the page, click the Edit button.\n\n**6.** In the Available functions list, select the functional areas you want the user to have access to.\n\n**7.** Click the right-pointing arrows to add the functional areas to the Assigned functions list.\n\n**8.** Click the \"Save and continue\" button to go to the Models tab, where you will set model object permissions.\n\nAt this point, the users in the group cannot do anything in MDS. If they open the web UI, they will see the functional areas but will not see models displayed either on the home page or in any of the functional areas.\n\nModel Object Permissions\n\nModel object permissions, assigned on the Models tab, are required. Users cannot view any models or data if they do not have model object permissions.\n\nWhen you give users permission to model objects, you are giving them the ability to edit attributes for members, based on the object you select. For example, if you set Update permission on the Product entity, all attributes for all Product members (leaf and consolidated) can be updated. If you set Update on the Color attribute of the Product entity, only the Color attribute can be updated.\n\nIn addition to giving a user the ability to update attribute values, if you assign Update model object permissions to a model, entity, or to the word \"Leaf\" or \"Consolidated,\" the user can also create and delete members. If permissions are assigned at a lower level, the user cannot create and delete members.\n\nNOTE\n\n_Permissions automatically cascade to all child objects within the current model unless permissions are assigned at a lower level. You do not need to explicitly set permission on every object._\n\nIf you assign Update model object permissions to the model only, the user is an administrator, which means he or she can access the model in functional areas other than just Explorer if given access to that functional area.\n\nQuick Facts About Model Object Permissions\n\nThings to remember about model object permissions include the following:\n\n They are required.\n\n They determine which attributes a user can view or update (as opposed to which members).\n\n They apply to all lower-level objects unless another permission is explicitly assigned.\n\n Update permission to Leaf or Consolidated model objects and above gives users the ability to create and delete members.\n\n Update permission at just the model level makes the user an administrator.\n\nBest Practice for Model Object Permissions\n\nThere are many different model objects you can assign permission to. Giving access to specific models, entities, or attributes should fulfill most of your security needs. In SQL Server 2012, most complicated security configurations have been simplified with the removal of attribute group and hierarchy object security.\n\nProcedure: How to Assign Model Object Permissions\n\nTo assign model object permissions, complete the following steps:\n\n**1.** On the Master Data Manager home page, click User and Group Permissions.\n\n**2.** On the menu bar, click Manage Groups. In this example, we're going to update the Finance Administrators group.\n\n**3.** To the left of the group you want to edit, click the down arrow.\n\n**4.** On the submenu, click Edit | Models.\n\n**5.** At the top of the page, click the Edit button.\n\n**6.** If the correct model isn't visible, select the model you want to view from the Model drop-down list. In this case we'll select Finance.\n\n**7.** Expand and collapse nodes in the tree until you see the object you want to give permission to.\n\n**8.** Right-click the object you want to give permission to, and from the menu that's displayed, click Update, Read-only, or Deny. In this example, we're going to right-click the Finance model and choose Update.\n\n**9.** Click Save. If you intend to assign hierarchy member permissions, you can click the \"Save and continue\" button.\n\nThe Model Permission Summary grid is populated with the permissions you've assigned.\n\nNOTE\n\n_You do not have to explicitly deny access to model objects you don't want users to see. Permissions cascade, so the user is implicitly denied access to any object that doesn't inherit permissions._\n\nThis was the simplest example, where you assign Update to the model and no other permissions. In this case, all of the users in the group can access all Finance members, as well as access the Finance model in any functional areas they have permission to access.\n\nProcedure: How to Delete Model Object Permissions\n\nBefore you can test other permissions, you may want to delete those you already assigned. To do so, complete the following steps:\n\n**1.** On the Models tab, in the Model Permission Summary grid, click the down arrow for the row that shows permissions you want to delete.\n\n**2.** From the submenu that's displayed, click Delete.\n\n**3.** In the confirmation dialog box, click OK.\n\nThe page refreshes and the permissions are deleted.\n\nNow let's go through all of the possible model object permissions you can assign.\n\nAccess to Entities\n\nWhen you assign Update permission to an entity, the user can create and delete members as well as update all attributes for the entity (assuming no other permissions are assigned). When you assign Read permission, the user can view all attributes for the entity.\n\nHere is what Update permission to a single entity (in this case, the Line entity) looks like after you save:\n\nAnd the following illustration shows what Update access to the Line entity looks like in Explorer.\n\nAs you can see, the user can update all attributes for all members in the Line entity. The user can also create and delete members when you assign permissions at this level. If the Line entity had hierarchies or collections, they would also be editable. When you have permission to read or update an entity, you can read or update any explicit hierarchies and collections for the entity.\n\nIn an effort to simplify security in this release, MDS gives access to change domain-based attributes in entities you have access to, but you no longer have Read access to those entities in the Explorer functional area. If you wish to provide Read access to the Manufacturer entity, for example, you must provide this security assignment explicitly.\n\nAccess to Leaf Member Attributes\n\nIn the tree, you can assign permission to the word \"Leaf\" that's displayed beneath an entity. This is a shortcut for giving permission to all attributes for all leaf-level members. If you assign Update permission, the user can create and delete members, just as if you assigned permissions to the entity itself.\n\nNOTE\n\n_If an entity does not have explicit hierarchies enabled, there is no difference between assigning permission to an entity and assigning permission to Leaf._\n\nIf the entity is enabled for explicit hierarchies, use this permission level to give a user access to update attributes for all leaf members or for all consolidated members. Otherwise, this permission does not need to be used, because you can assign permission directly to the entity.\n\nAccess to Individual Attributes\n\nWhen you assign Update or Read permission to a specific attribute, the user can update or read that attribute. It doesn't matter if the attribute is domain-based; the user is able to update the attribute value for the entity that the attribute applies to only.\n\nIn this example, the user has Update permission to the Product entity's Cost attribute:\n\nWhen the user accesses the Product entity in Explorer, the Name and Code attributes are read-only but the Cost attribute can be updated. While this is no longer displayed in the grid, the Name and Code columns are disabled in the Details pane.\n\nNote that when you assign permission to an attribute, the user can't create or delete members. Also note that if the attribute is domain-based, the user can't make changes to the entity the attribute is based on. For example, if a user has permission to update the Color attribute for the Product entity, the user can change a product's color, but the user cannot open the Color entity itself and change the color name from Blue to Light Blue.\n\nIf you wanted the user to be able to do either of these things, you would choose these permissions:\n\nIn Explorer, you can view the results. The user can update the Color attribute for all products. In the latest release, the grid does not provide graphical security feedback, but the Edit button is enabled for changing the Color attribute. The input boxes for name and code are grayed out.\n\nThe user can also create, delete, and update members of the Color entity. Here is the other entity:\n\nAccess to an Explicit Hierarchy\n\nIn an effort to simplify security, you can no longer assign permissions to an explicit hierarchy. You can no longer separate the ability to move members in an explicit hierarchy but not create and delete members.\n\nIf you assign Update permission to the root entity, the user can move members in the hierarchy. If you assign Read-only permission, the user can view the hierarchy but not make any changes.\n\nIn this example, the user is assigned Update permission to the Account entity. This gives the user Update access to the Chart of Accounts explicit hierarchy; these permissions would be the same for any explicit hierarchies created on the Account entity.\n\nWhen you open Explorer to view the results of these permissions, you can open the hierarchy and move members in it. You can also view the members in the entity the hierarchy is for (in this case, the Account entity), and now you will be able to update attributes for these members as well as create new ones.\n\nIf the user has Update permission to an entity, the user can also view and update all explicit hierarchies for the entity. If the user has Read-only permissions to the entity, the user can only view all hierarchies within the system.\n\nAccess to Collections\n\nWhen you assign Update or Read permission to the Collection object, you are assigning access to all collections. You can't give permission to an individual collection\u2014it is an \"all or nothing\" scenario.\n\nIf a user has Update permission, he or she can create, delete, and update collections. In this example, the user has permission to update collections for the Product entity:\n\nIn Explorer, all collections for the Product entity are displayed, and the user can create and delete collections, as well as update collection attributes like Name, Code, Description, or Owner.\n\nAt this point, the user can delete members from the collection but can't add any members because the user doesn't have permission to other model objects. If the user had a minimum of Read-only permission to the Product entity, he or she could add members to the collection.\n\nIn User and Group Permissions, on the Models tab, you can give permission to attributes of collections, so you can lock down the description or names of all of your collections, for example. In SQL Server 2012, you can no longer set permissions on attribute groups, just as you cannot set them with attribute groups that apply to entities.\n\nAccess to Other Model Objects\n\nSetting security directly on a derived hierarchy was removed in this release of Master Data Services. Derived hierarchy security is now based on access to all of the corresponding components of the derived hierarchy.\n\nIn addition, attribute groups should no longer be viewed as a way to secure attributes. All users have access to an All Attributes tab.\n\nAccess to a Derived Hierarchy\n\nFor any derived hierarchies to appear, the user must have or inherit Read or Update permissions on the model. This will obviously give the user Read or Update access to all of the entities in the model. Derived hierarchy access is now controlled by access to each of the components that make up the hierarchy. Every derived hierarchy where the user has access to all entities that make up the levels of the derived hierarchy will be displayed in the drop-down list in the Explorer UI. If any object is denied, the hierarchy will not be available, but the user may be able to affect the hierarchy by changing attributes within other represented entities they have access to.\n\nIn this example, the user is assigned Read permission to the Product model. To give the user Update access to the Product by Gender hierarchy, the user has Update permissions to the Product and Gender entities. Conversely, because the user is denied access to the Line entity that groups products by Line and then groups lines by Manufacturer, this hierarchy does not display in the hierarchy menu dropdown at the top of the explorer screen.\n\nWhen you open Explorer with this user's credentials, you can see only the Product by Gender derived hierarchy. Due to the deny permissions on the Line entity, the only derived hierarchy that can be accessed is the Product by Gender hierarchy. Access to Product by Line by Manufacturer is not presented because the user does not have access to the Line entity.\n\nBy updating the value of the Gender attribute, you change the location of a member in the hierarchy. For example, if you change the Gender attribute value for member 1000 to Female, the member would be displayed under the new line in the hierarchy. You don't need to view the hierarchy to know that this change has taken place.\n\nHowever, assigning permissions to a derived hierarchy does allow the user to view the hierarchy structure and move members in the tree, rather than just update attribute values.\n\nAlthough you can assign permissions to individual attributes, the user cannot work with the hierarchy structure unless you give him or her permission to the entity, model, or derived hierarchy model object.\n\nAccess to an Attribute Group\n\nIn SQL Server 2012, attribute groups are no longer a part of the security model in MDS. All attribute-level security must be applied at the individual attribute level. Permissions on attribute groups will now be evaluated as Hide(Deny)\/Unhide(Update). The addition of the All Attributes tab has also rendered attribute group security obsolete.\n\n**Attribute Groups in System Administration** You may have noticed when you were creating attribute groups that users and groups were displayed on the Attribute Group Maintenance page. The Users and Groups nodes on this page are for viewing users and groups who have Update permission to the attribute group (users with Read-only permission are not displayed and no longer have meaning). This is the only way to manage which attribute groups are displayed for each user. _This is not for purposes of security!_\n\nCAUTION\n\n_Do not attempt to edit users and groups on this page._\n\nYou should _not_ use this page to try to administer security for attribute groups. This is for display management purposes only.\n\nDeny Permissions\n\nYou'll notice that when you right-click a model object, Deny is listed on the submenu. In most cases you won't need to explicitly deny access to an object. Permissions cascade so that all lower-level objects inherit the assigned permissions. Objects that are at the same level or higher, and that don't have permissions explicitly assigned, are inherently denied.\n\nFor example, if you set Update permissions on the Product model but not on the Finance model, the user is denied permission to access the Finance model. You don't have to explicitly deny access. Or, for example, if you set Update permissions on the Color entity but not on any other entities, permission to access all other entities is denied.\n\nThese same rules apply to hierarchies on the Hierarchy Members tab. Hierarchy nodes at the same level and higher are inherently denied as soon as any Read-only or Update permissions are set. All other permissions cascade down the hierarchy.\n\nDue to the changes in SQL Server 2012, giving permissions to derived hierarchies may lead to a number of entities requiring Deny access privileges. Once you have added Update or Read-only access to the model level, any entity you wish to restrict access to will require explicit Deny permissions to be set.\n\nNavigational Access\n\nEvery user needs a certain degree of access to the system in order to open the members and attributes he or she has permission to. For example, if a user has permission to access three specific attributes, but not the model itself, the user can still view the list of models in the drop-down list on the home page. Users also have to be able to see the Name and Code attributes for the members so that they know what they are updating. This bare minimum required access is sometimes referred to as _navigational access_. We're not going to delve into the specifics of each model object, but you should know that when you give permissions to a user, you are also giving the user the navigational access required to work with the data they need.\n\nDeleting Permissions\n\nEarlier in this chapter, we showed you how to delete model object permissions. The procedure is the same for hierarchy member permissions. In the grid, click the down arrow next to the row with permissions you want to delete, and click Delete.\n\nIf you want to remove a user so that he or she no longer has MDS access, you need to complete two procedures. First, confirm the user's group memberships. Then, remove the user from the list.\n\nTo confirm the user's group memberships, complete the following steps:\n\n**1.** On the Manage Users page, to the left of the user, click the down arrow.\n\n**2.** On the submenu, click Edit | Membership. Note any groups the user is a member of.\n\n**3.** If the user is a member of multiple groups, go to the local or Active Directory group and remove the user from the group.\n\nNow remove the user from the list:\n\n**1.** On the Manage Users page, click the row of the user you want to delete.\n\n**2.** Click the \"Delete selected user\" button.\n\nBoth of these procedures are required to keep your list of users accurate. Otherwise you might end up with users in the list who don't truly have access or retain access due to additional group permission.\n\nHierarchy Member Permissions\n\nIn an effort to provide the functional equivalent of row-level member security without providing all of the maintenance nightmares associated with it, MDS provides hierarchy member security. This type of security allows you to assign security to all the members in a node of a hierarchy. The same cascading feature that is found in model object security applies to hierarchy member security.\n\nWith MDS, you have the ability to set security on levels of either explicit or derived hierarchies. This powerful feature allows you to limit access to data within MDS on a number of axes simultaneously (one axis from model object permissions, and a second axis from hierarchy member permissions, as was shown in Figure 11-1 at the beginning of this chapter).\n\nBecause you may have thousands or millions of members, hierarchy member permissions are applied only periodically. In the MDS database, in tblSystemSetting, the SecurityMemberProcessInterval setting determines how often these permissions are applied. The default setting is 3600 seconds, or 60 minutes, but you can update this to be more frequent. Note that even if you set it to every 10 seconds, you must wait 60 minutes for the 10-second intervals to start. Be aware that 10 seconds may not be enough time to fully apply permissions to all of your members.\n\nHierarchy member permissions are based on a version of a model and are copied forward on subsequent models. So if you set permissions on a hierarchy in Version 3, those permissions don't exist in Version 2, but they will exist in copies of Version 3. Be sure to consider that your users might have unintended access to data in earlier versions of your model if you set hierarchy member permissions in later models.\n\nQuick Facts About Hierarchy Member Permissions\n\nThings to remember about hierarchy member permissions include the following:\n\n They are optional.\n\n They determine which members (as opposed to which attributes) a user can view or update.\n\n They apply to all lower-level members unless another permission is explicitly assigned.\n\n They are not applied immediately, unlike model object permissions. You must wait until a specific interval of time has passed. This interval is based on SecurityMemberProcessInterval in tblSystemSetting.\n\n You can run a stored procedure to force them to be applied immediately.\n\n They are version specific. You set them for a specific version and they apply to that version and all future copies of that version. (They do not apply to any previous versions.)\n\n You should not set permissions on explicit cap or recursive hierarchies. Security for these hierarchies is not supported.\n\nProcedure: How to Assign Hierarchy Member Permissions\n\nTo assign hierarchy member permissions to a group, complete the following steps:\n\n**1.** On the Master Data Manager home page, click User and Group Permissions.\n\n**2.** On the menu bar, click Manage Groups.\n\n**3.** To the left of the group you want to edit, click the down arrow.\n\n**4.** On the submenu, click Edit | Hierarchy Members.\n\n**5.** From the lists, choose the model, version, and hierarchy you need. Keep in mind that permissions will apply to the version you select and later copies of that version after the permissions are created.\n\n**6.** At the top of the page, click the Edit button.\n\n**7.** Expand and collapse nodes of the hierarchy as needed, or search for a specific node. Notice that individual members are not displayed. You are setting permissions for all members that are in one or more specific nodes.\n\n**8.** Right-click the hierarchy node you want to give permission to, and from the menu that's displayed, click Update, Read-only, or Deny.\n\n**9.** Click Save.\n\nBecause the Main Street Clothing Company's models are fairly limited, the following example may not necessarily have real-world use, but it should help you understand how these permissions work.\n\nIn this example, the user has Update permission to two attributes for the Product entity: Color and Design.\n\nOn the Hierarchy Members tab, the user is assigned Update permission to the Female node. Permission to access all other nodes at the same level or higher is implicitly denied.\n\nAfter the hierarchy member permissions interval has passed and permissions are applied, the user can update only the Color and Design attributes for the Female members. The Male members (those with Male as the Gender) are not displayed.\n\nIn this example, you can start to see the intersection of model object and hierarchy member permissions. (You can also think of this as the intersection of attribute and member permissions.) If you were to keep this hierarchy member security but, on the Models tab, give Update access to the Product entity, the user would be able to see these same 200 members in hierarchies and collections as well as the main grid. The other 200+ Male members would always remain hidden.\n\nBecause hierarchy member permissions must always be used with model object permissions, there are rules for how these permissions are combined. For more information, see \"Determining Which Permissions Apply.\"\n\nOverlapping Hierarchies\n\nIf you've assigned hierarchy member permissions to more than one hierarchy, there is a good chance that you've assigned overlapping permissions to some members. If you do assign permissions to multiple hierarchies and those hierarchies contain the same members, the following rules apply:\n\n If one hierarchy node is assigned Update permission and another is assigned Read-only permission, then the members in the node are Read-only.\n\n If one hierarchy node is assigned Update or Read-only permission and another node is assigned Deny permission, then the members in the node are not displayed.\n\nDetermining Which Permissions Apply\n\nPermissions are applied in the following order of operations:\n\n **Permissions for each attribute are determined (model object permissions)** Permissions assigned on the Models tab cascade down the tree structure and are applied to all objects. In this way, permissions for every attribute in the model are determined.\n\n **Permissions for each member are determined (hierarchy member permissions)** Permissions assigned on the Hierarchy Members tab also cascade down the tree. In this way, permissions for all members in the model are determined.\n\n **Attribute and member permissions are combined** Permissions are combined and access to each individual attribute value is determined.\n\nThis workflow is shown in Figure 11-2.\n\n**Figure 11-2** _Permissions workflow, showing the combination of model object and hierarchy member permissions_\n\nCombining Permissions\n\nWhen attribute permissions (assigned on the Models tab) are combined with member permissions (assigned on the Hierarchy Members tab), the following rules are applied:\n\n Read-only overrides Update.\n\n Deny overrides all other permissions.\n\nThe following example shows how this works. First, on the Models tab, assign Update to the Product entity's Color attribute. This gives the user the ability to update the Color attribute for all members.\n\nThen, on the Hierarchy Members tab, open the Product by Gender hierarchy and assign Update to the Female node and Read-only to the Male node.\n\nIf you apply the rules, all attributes for all members in the Male node should be Read-only. Because Read-only overrides Update, the user cannot update the Color attribute for the Male members.\n\nCombining Users and Groups\n\nWe've mentioned many times that life gets confusing when you assign permissions both to a user and to the groups the user is a member of. But if you decide you really need to do this, the web UI helps you determine which permissions are applied.\n\nWhenever you open a user account, on both the Models and the Hierarchy Members tabs, you can select Effective from the Permissions list to refresh the web UI and display the results of any overlapping user and group permissions.\n\nThis same list is not available when you're viewing groups; it's available only when you're viewing users.\n\nIf you want to understand how the combination of user and group permissions works in more detail, read on and we will attempt to clarify it for you.\n\nThe rules that apply when combining user and group permissions are\n\n Deny overrides all other permissions.\n\n Update overrides Read-only.\n\nThis means that if the user is assigned Read-only permission but the group is assigned Update permission, the applied permission is Update. MDS makes the assumption that if there is an overlap, the least restrictive permission should win. For example, if Bob is covering for a coworker while that person is out, and you add Bob to the other group temporarily, Bob keeps his existing permissions but also gains those of his coworker.\n\nFigure 11-3 shows the additional complexity that occurs when you have overlapping users and groups as part of the security workflow.\n\n**Figure 11-3** _The addition of overlapping users and groups to the security workflow_\n\nA Matter of Distance\n\nUnfortunately, overlapping users and groups is slightly more complex than these simple rules and Figure 11-3 can show. The cascading inheritance in the tree also plays a part. If two or more objects have permissions assigned to them, the \"Update beats Read\" rule applies. However, if an object has a different permission assigned and it's closer than the permission from the other user or group, then the closer permission applies instead.\n\nFor example, if a user has Read and a group has Update to the same model object, then Update is applied. However, if a user is assigned Update to an entity and a group is assigned Read-only to an attribute (a lower level than entity), then Update applies for the entity but the attribute is Read-only.\n\nThese same rules apply to the Hierarchy Members tab.\n\nThankfully, the UI does this calculation for you. On both the Models and Hierarchy Members tabs, click Effective in the Permissions list to display a user's effective permissions.\n\nOverlapping Functional Areas\n\nUsers and groups can also have overlapping permissions to functional areas. Functional area permissions are additive: if a user has permission to Explorer and a group has permission to Version Management, then the user has permission for both areas.\n\nSetting Security by Using Web Services\n\nAll security in Master Data Services is managed using two web service objects:\n\n **SecurityPrincipal** Security principals correspond to Active Directory users or groups. While users may be in a group that exists in MDS, users will be managed in MDS only after they have logged on to the system through the web UI or web service (or some custom component built on the web service). To explicitly set permission on a user who is a member of a group, you will need to add the user from Active Directory explicitly.\n\n **SecurityPrivilege** All distinct permissions within MDS are stored and managed as privileges. In most cases, only explicitly set permissions will be returned from the web service.\n\nRetrieving Users and Groups\n\nThe GetSecurityPrincipals operation, shown next, returns all current users and groups that match the criteria passed into the request. Only users that have used MDS through either the web service or web UI will be returned, even if they are members of an Active Directory group with access to MDS.\n\nNOTE\n\n_You must be a model administrator to perform this operation._\n\nRetrieving User or Group Permissions\n\nThe GetSecurityPrivilege operation returns privileges matching the criteria provided. These privileges will be related to the user or group that the permission is provided to.\n\nAdding Users and Groups\n\nWhile the CreateSecurityPrincipals operation includes \"Create\" in its name, it is really used to load preexisting Active Directory users or groups into MDS. This operation will also be used to update some user-specific properties that are stored directly in MDS.\n\nAssigning Permissions\n\nAll types of security privileges for users and groups can be created simultaneously with the CreateSecurityPrivileges operation. Multiple users or groups can be secured within the same operation as well.\n\nSummary\n\nIn this chapter, we explained the details of configuring security in MDS. We showed the process for assigning model object and hierarchy member security, and how the combination of the two determines cell-level security. We discussed model administrators and how to create and update them. We gave best practices for assigning security and talked about what happens when users belong to groups and security is assigned to both. Finally, we showed how to use the web service to view and assign permissions.\nChapter 12\n\nPublishing Data to External Systems\n\nIn This Chapter\n\n Exporting Data to Subscribing Systems\n\n Subscription View Formats\n\n Common View Architecture\n\n Subscription Views and Model Deployment\n\n Creating a Subscription View with Web Services\n\n How Main Street Clothing Company Uses Subscription Views\n\nAlthough the ability to manage data is an important feature of Master Data Services, the ability to export data to other systems in the enterprise is equally if not more important. MDS has simplified the export process by providing subscription views that you can create on any entity or derived hierarchy object within the MDS system.\n\nIn this chapter, we describe the different types of the export views that can be created and the format of each of these views. We discuss some tips and tricks for creating a proper view depending on the type of downstream system that will consume the data in MDS. We also provide an example of loading a downstream system from an export view we create. While Master Data Services provides no direct extract, transform, and load (ETL) features of any kind, the Integration Management functional area of the Master Data Manager web application (or \"web UI\") can facilitate your integration processes by providing a wide variety of views to assist in the loading of downstream systems.\n\nExporting Data to Subscribing Systems\n\nMaster Data Services can provide value for a project that keeps the data locked in its entities by providing process around the data management; however, to meet the operational or analytical needs of a master data management project, organizations need to transport stored data downstream to subscribing systems. To insulate organizations from the complex object model necessary to manage performance and the customization necessary within MDS, the developers created a subscription view layer.\n\nYou can create subscription views within the Integration Management functional area of the web UI, or by using the web service. The web service refers to these views as export views, but we'll use \"subscription views\" and \"export views\" interchangeably.\n\nSubscription View Formats\n\nThe following table shows the available subscription view formats. When you create a subscription view, you have to choose which format you want to use. There are two major types of subscription views within MDS: attribute views and hierarchy views. Attribute views display the data stored for leaf, consolidated, or collection members in an easily consumable tabular view. Hierarchy views provide relationship data for all types of relationships in MDS, whether explicit or derived hierarchies or collection members. Collection member relationships can only be displayed in a parent-child format. An additional view is available for derived and explicit hierarchies. It contains a row for each member and the parentage all the way to the top consolidation in the hierarchy; this view is considered level-based.\n\nCommon View Architecture\n\nMany of the columns in Master Data Services' subscription views are identical across view types. The columns provide either context for the data displayed in the view or additional system information for the records contained in the view. Subscribing systems can use this information to update a subset of records based on validation status or last updated statistics.\n\nProcedure: How to Create a Subscription View\n\nTo create a subscription view in Master Data Manager, complete the following steps:\n\n**1.** On the Master Data Management home page, click Integration Management.\n\n**2.** On the menu bar, click Create Views.\n\n**3.** On the Subscription Views page, click the \"Add subscription view\" button.\n\n**4.** In the Create Subscription View pane that is displayed at the bottom of the page, enter a name for the view and select a model from the Model list. In this example, select Product and call the view **ProductsForPOS**.\n\nThis UI can be a bit confusing. There are three columns here and the radio buttons apply to the two options shown in each column.\n\n**5.** In the first column of choices, select either Version or Version flag and then select from the corresponding list. As a best practice, you should create a subscription view based on a version flag. Then, as you create new versions, you can move the flag to the latest version and you don't have to update your subscription view. If you created a version flag in Chapter 9, you can select it here. Otherwise, select the version that contains the data you want to export.\n\n**6.** In the next column of choices, select either Entity or Derived hierarchy and then select from the corresponding list. Your subscription view can show the contents of an entity or the structure of a hierarchy. For this example, select Entity and choose the Product entity from this list.\n\n**7.** In the third column of choices, from the Format list, select Leaf attributes. This determines the format of your view. If you were creating a view of a hierarchy, you would be prompted to select a format for the view and to choose the number of levels you want to show.\n\n**8.** Click Save.\n\nThe subscription view is now displayed in the top pane of the Subscription Views page. If you decide you want to change it, click the row and then the Edit button. You can also delete it if it's no longer necessary.\n\nScroll to the far right to view the Changed column. This column is important to note. If objects in the model structure change, this column changes to True. At that point, you need to edit the subscription view and save it again (you don't need to make changes to it; just edit and then save). This ensures that the view will still return accurate results based on the latest changes to the model.\n\nViewing a Subscription View in SQL Server\n\nNow you can open SQL Server Management Studio and take a look at the view. As shown in Figure 12-1, all of the members and their attributes are shown in a format that is easy for subscribing systems to consume.\n\n**Figure 12-1** _Opening a subscription view in SQL Server Management Studio_\n\nFor each domain-based attribute that is displayed in a subscription view, three columns are displayed. The \"_Code\" and \"_Name\" columns display the name and code. The \"_ID\" column displays the system key for the member. This is an internal key and should not be used in subscribing systems.\n\nSubscription Views and Model Deployment\n\nIn the SQL Server 2008 R2 release of MDS, a shortcoming of model deployment was its lack of support for subscription views. If you create subscription views for an entity or derived hierarchy, these views will not be re-created when moving model deployment packages between development, test, and production servers. Subscription views are now supported in model deployment with SQL Server 2012.\n\nCreating a Subscription View with Web Services\n\nAll subscription views in MDS are created using the ExportViewCreate operation. Depending on the type of view being created and whether you create the view based on version or version flag, different parameters are required.\n\nCreating Entity Views\n\nThis operation creates an entity-centric subscription view for a specified model and entity. This operation can support explicit hierarchy leveling and either method of version management.\n\nCreating Derived Hierarchy Views\n\nThis operation encapsulates the creation of derived hierarchy subscription views. Because the level setting is only valid when the chosen format is \"derived levels,\" this parameter is ignored for other view types.\n\nDeleting Views\n\nThe following code deletes a subscription view from Master Data Services no matter what the format of the subscription view. No data within the entity related to the export view will be changed in any way by this process. This call returns an operation result that will contain any errors that occurred in the deletion process, although these will be rare in this simple process.\n\nHow Main Street Clothing Company Uses Subscription Views\n\nWith all of the processes around maintaining location, product, employee, and account data now in place, Anthony must build the integration plan for passing and synchronizing data from MDS into all downstream systems.\n\nThe first step in building this plan is to determine the accuracy and accountability required for each subscribing system. Some systems require data that has been validated and version information that has been rigorously tracked. Other systems need data that is more recent and may still contain minor errors. MDS can provide this data for either purpose based on the type of version that is consumed. Version flags, as discussed in Chapter 9, also support this important distinction.\n\nFor each downstream system, Anthony must decide what type of subscription view provides the best format for consumption. Does this system require consolidation information? Does this system need attributes related to consolidated members? Anthony determines the best subscription view for each downstream system. As you can see from the following table, systems vary widely on the quality level required.\n\nThe point of sale (POS) systems in each of the stores must be updated nightly with the latest product lists and prices. Anthony decides the best way to do this is to use the data presented in his subscription views to export the data he needs. He uses SQL Server Integration Services to create a package that runs on a nightly schedule. Each morning when the managers come to work, their POS systems are already updated with the latest product information.\n\nProcedure: How to Export Subscription View Data\n\nTo export data from a Master Data Services subscription view to another data source, complete the following steps:\n\nNOTE\n\n_This procedure is just one example of how to get data out of MDS and into a subscribing system. Your process may differ._\n\n**1.** Open SQL Server Management Studio.\n\n**2.** Right-click the MDS database.\n\n**3.** From the submenu, select Tasks | Export Data.\n\n**4.** On the Welcome page, click Next.\n\n**5.** On the Choose a Data Source page, leave the defaults (it should be pointing to your MDS database) and click Next.\n\n**6.** On the Choose a Destination page, in the Destination field, select Flat File Destination.\n\n**7.** Browse to a text file name, select the \"Column names in the first data row\" check box, and click Next.\n\n**8.** On the Specify Table Copy or Query page, select the \"Copy data from one or more tables or views\" radio button and click Next.\n\n**9.** On the Configure Flat File Destination page, in the \"Source table or view\" list, select the view. In our case, it's the view: [mdm].[ProductsForPOS].\n\n**10.** Click the Edit Mappings button. In the Column Mappings dialog box, ignore all of the Destination columns except: Name, Code, Retail Price, Discontinued_Code, Type_Code, Gender_Code, and Manufacturer.\n\n**11.** Click OK to close the dialog box. Then click Next.\n\n**12.** On the Save and Run Package page, leave the Run immediately check box selected and select the Save SSIS Package check box.\n\n**13.** Click Next.\n\n**14.** On the Save SSIS Package page, type a name and description for your package, leave all other defaults, and click Next.\n\n**15.** On the Complete the Wizard page, click Finish.\n\nThe SSIS package is saved and Main Street Clothing Company can schedule the package to run nightly. When the package runs, the contents of the view are saved as a flat file to a file share. A separate batch system transfers the file to each of the 14 stores. The POS systems retrieve the data from the file, and each morning when the stores open, the POS systems have the latest prices and information.\n\nSummary\n\nConsuming data from the Master Data Services system requires analysis of the subscribing system's needs. These needs should be evaluated against the supported export formats. Subscription views provide a number of columns that can help filter and manage ETL processes into subscribing systems at an individual record level. Model deployment does not deploy subscription views from packages. If an organization has a lot of subscription views, it can programmatically manage these views through the MDS web service.\nChapter 13\n\nExtending MDS with Web Services\n\nIn This Chapter\n\n Exposing the Web Service\n\n Exposing the WSDL\n\n Creating an MDS Project in Visual Studio\n\n Creating an Abstraction Layer\n\n MDS Web Service Operations\n\n The Most Frequently Used Operations\n\n Handling Errors in the Web Service\n\nIn this chapter, we explore in more depth the web services layer provided by Master Data Services. Every operation in MDS is exposed within the web service. Despite the large number of operations, any developer can be productive working against the web service by focusing on a few high-value operations. These operations can be used to create a customized UI tailored to your solution or to integrate existing applications with MDS.\n\nIn SQL Server 2012, most of the changes are beneath the interfaces. A number of the processes in the web service have been deprecated, while most of the new calls have been marked as private, meaning Microsoft will not support external use of the call. In this chapter, we will give you an overview of the new calls and let you know which ones might be worth taking the support risk for.\n\nExposing the Web Service\n\nIn the 2008 R2 release, for security purposes, MDS did not publicly implement the web service when you first installed MDS. In SQL Server 2012, the web service is always exposed because Silverlight is used for the Explorer screen. If you are using MDS with prebuilt applications, then everything you need is already done. If you are going to build your own solutions or use the samples listed in this book, then you must also expose the Web Services Description Language (WSDL) in order to compile against the service interface.\n\nExposing the WSDL\n\nTo expose the WSDL, you must modify the web.config file directly. This file is in the WebApplication directory where MDS is installed. If you installed MDS in the default location, the full path is C:\\Program Files\\Microsoft SQL Server\\Master Data Services\\WebApplication\\web.config.\n\nProcedure: How to Enable the WSDL\n\nOpen the web.config file with either Notepad or Visual Studio 2010. Find the following section:\n\nChange the setting for httpGetEnabled and includeExceptionDetailInFaults to **true**. Then save the file. This enables the web service to expose the WSDL. If you have configured SSL, as described in Chapter 2, change httpsGetEnabled to **true**.\n\nIf you receive an \"Access is denied\" error when you save, check the security on the file. When you install MDS, permissions are set on this file so that only the users in the local Administrators group have full control.\n\nYou may also have User Account Control (UAC) running. You can disable UAC temporarily; unfortunately, you will have to restart your computer when you do this. To disable UAC, open Control Panel and click \"System and Security.\" Under Action Center, click \"Change User Account Control settings.\" In the User Account Control Settings dialog box, slide the bar to the bottom so that you are never notified. Click OK to save. In the confirmation dialog box, click Yes. Restart your computer, and you should be able to edit the web.config file.\n\nCreating an MDS Project in Visual Studio\n\nNow that the web service is exposed, you can create your first project in Visual Studio. All of the code examples in this book use Visual Studio 2010.\n\nProcedure: How to Create an MDS Web Service Application\n\nTo create an application that uses the MDS web service, you must first create a project and then add the web service as a reference in the project. Depending on your needs, you can create any type of application you choose. In this book, we've created wrapper classes that can be used in any type of application.\n\nCreate a Project\n\nTo create a project in Visual Studio, complete the following steps:\n\n**1.** Open Microsoft Visual Studio.\n\n**2.** Click File | New | Project.\n\n**3.** In the New Project dialog box, in the Project types section, under Visual C#, click Windows.\n\n**4.** In the Templates section, choose the application type. In this example, we chose Class Library.\n\n**5.** In the Name box, type a name for your application.\n\n**6.** Click OK.\n\nAdd a Service Reference\n\nTo add a service reference to the project, complete the following steps:\n\n**1.** In Visual Studio, with your project open, click View | Solution Explorer.\n\n**2.** In the Solution Explorer pane, right-click References and click \"Add Service Reference.\"\n\n**3.** The Add Service Reference dialog box is displayed. In the Address box, type the URL for the MDS service. This is usually your URL followed by \"\/service\/service .svc.\" For example:\n\nhttp:\/\/mdsserver\/mds\/service\/service.svc\n\n**4.** Click Go. The service is displayed in the Services section.\n\n**5.** In the Namespace box, type a name for the service namespace. This can be anything you want.\n\n**6.** Click OK.\n\nCreating an Abstraction Layer\n\nThe MDS web service is built on Windows Communication Foundation (WCF). Many of the initial calls may seem daunting or overly complicated if you are a first-time user. The web service operations may seem complicated because they must be generic in order to port all the model types and data schemas created in disparate implementations. The operations have been built to support all possible configurations and uses of the application.\n\nMost people, however, are interested in only a small subset of configurations and a small subset of the application features exposed to the web service. As you are getting used to the MDS web service, there are a few tips and practices that can help you more quickly integrate your organization with MDS. One of these is to create an abstraction layer.\n\nAn abstraction layer can do a number of things:\n\n Provide other developers with only those calls that you wish them to program against\n\n Ensure that everyone uses a similar standard when writing code against the web service\n\n Enable you to provide complex calls in a far more consumable fashion\n\nMDS Web Service Operations\n\nWhile more than 50 operations are supported by the MDS web service, most people need to focus on only a few of those operations to handle the majority of their routines. MDS operations can be grouped by the MDS objects they control. These groupings are listed in order of importance in the following sections.\n\nMembers\n\nRetrieving and updating master data is by far the most frequent procedure you will want to automate. Many of these calls were optimized in SQL Server 2012 to support the MDS Add-in for Microsoft Excel. The EntityMembersUpdate and EntityMembersMerge calls can now support large datasets. When using these calls, the MDS team found that breaking extremely large sets into sets of 50,000 attribute updates is most effective on average systems. All of the operations for working with members are listed in the following table. For examples of these operations, see Chapter 7 and Chapter 11.\n\nBulk Operations\n\nAlthough many of these operations are called ModelMembers, that is a misnomer. The major distinction for these operations is that they leverage staging to bulk load records into MDS. In SQL Server 2012 these operations have been marked for deprecation. When loading more than a few thousand values (members times attributes), you can now use the EntityMembersMerge and EntityMembersUpdate. For examples of staging operations, see Chapter 5.\n\nValidation\n\nIn this release of MDS, a large portion of the data quality proposition is working with validation results from business rules created in the application. While there are a number of operations necessary to create business rules, only two operations manage the results of these rules. Examples of these operations are provided in the section \"The Most Frequently Used Operations,\" later in this chapter.\n\nTransactions\n\nAll data changes made through the web service or the web application UI are saved in the transaction table. Since the Add-in for Excel is built on the MDS web service, these transactions will also show in the transaction table. These transactions can be retrieved or reversed by using the following operations. Examples of these operations are also provided in the section \"The Most Frequently Used Operations.\"\n\nAnnotations\n\nEither members or transactions on those members can be given further explanation through annotations. Annotations are useful in providing further detail in workflow processes.\n\nSubscription Views\n\nExternal access to MDS managed data is enabled through views that are based on versions and view type. The following export operations manage these views. For examples of some of these operations, see Chapter 12.\n\nMetadata Structures\n\nBecause MDS contains descriptive metadata, the naming convention for the following operations is confusing. These operations will create any of the model objects supported by MDS. For examples of some of these operations, see Chapter 4.\n\nSecurity\n\nWhile MDS security relies on Active Directory, MDS authorization requires that users be added as principals either by their username or an Active Directory group they are a member of. Users and groups must be assigned both functional and member privileges to have access to MDS. For examples of some of these operations, see Chapter 11.\n\nBusiness Rules\n\nThe management of business rules in MDS is best left to the Master Data Manager web UI. The following operations are complex in their object model and not well documented. For examples of some of these operations, see Chapter 8.\n\nSystem Settings\n\nThere are a number of system-wide settings that allow system administrators to customize their MDS implementation. These settings are stored in tblSystemSetting and managed in MDS Configuration Manager.\n\nUser Preferences\n\nThese user preference calls support the MDS web application and have limited value for external systems. Leveraging these calls in your own custom UI work is not suggested.\n\nMiscellaneous Operations\n\nThe remaining operations are not easily grouped into categories.\n\nEntity-Based Staging Operations\n\nThese operations were added in SQL Server 2012 to allow the MDS web UI to manage the new entity-based staging process. These operations are marked as internal only. Microsoft will not support custom code that calls these APIs. The thinking is that users will use the new SQL stored procedures and SSIS to move data into the tables. Please refer to Chapter 5 for more information on entity-based staging.\n\nData Quality Operations\n\nIn SQL Server 2012, Microsoft shipped Data Quality Services. While DQS ships with SSIS plug-ins, it does not have an exposed API. To integrate DQS into the MDS Add-in for Excel, the Master Data Services team created these API calls linked to the DQS server. All of these calls are marked as internal only, so Microsoft does not support any use of these APIs. They are included here for the adventurous...\n\nThe Most Frequently Used Operations\n\nThe examples in this section show you how to build an abstraction layer for the most frequently used operations that are needed to integrate with MDS. We highly recommend that when you build applications against MDS, you focus on the operations provided in this section.\n\nThe operations that we do not address tend to be more complex and less useful to everyday consumers. For instance, triggering validation programmatically may be very useful for end users. Business rules, on the other hand, are complex to build in a user interface and difficult to manipulate programmatically. Because of this, creating business rules outside of the MDS application holds less value. Security is another area where most of you will not need to build additional UIs. Ignoring these two areas can greatly reduce the time to implement a customized solution.\n\nSearching for Members\n\nTo return data from an MDS entity, you must provide a valid identifier for the model, version, and entity. To further limit and refine the results that are returned from an EntityMembersGet operation, you can provide search terms.\n\nAll filter types that are available in Explorer when you're viewing members in the Master Data Manager web UI are supported by the search term parameter in the EntityMembersGet operation. The following table lists all of the filter operators, the corresponding search term syntax, and an example of each. In the Search Term Syntax column of the following table:\n\n A stands for attribute.\n\n V stands for generic value.\n\n S stands for similarity.\n\n T stands for match type.\n\n C stands for containment bias.\n\nMost of the examples in this table come from the Product entity in the Main Street Clothing Company example.\n\nGetting Members by Using the SearchTerm Property\n\nThe following code provides an EntityMembersGet example that uses the SearchTerm property. In subsequent examples, we show only the highlighted SearchTerm portion.\n\nCombining Multiple Search Terms\n\nMultiple search terms can be combined with the use of the AND operator between each search criteria term, as shown next. Like the UI, search terms in MDS do not support complex conditions like OR or grouping.\n\nUsing Matching in Search Terms\n\nIn the Explorer functional area of the Master Data Manager web UI, you can use the Similarity operator to find matches similar to your search terms. However, you can use the MDS web service to create a much more robust process.\n\nThe following table shows the four types of similarity matches you can do programmatically. In each of these cases, 1 indicates an exact match and 0 indicates no match.\n\n**Levenshtein Search Term Example** The Levenshtein distance is the default type of match and does not require any additional parameters. Only Minimum Similarity and Type are supported.\n\n**Jaccard Search Term Example** The Jaccard search supports an additional parameter of containment bias.\n\n**Jaro-Winkler Search Term Example** The Jaro-Winkler match results are the least restrictive with our sample data. This method returns more results than any other method. The Jaro-Winkler type does not support containment bias.\n\n**Longest Common Subsequence Search Term Example** Longest common subsequence does support an extra parameter of length threshold. This is provided in a decimal percentage between 0 and 1. The default is .62, and a lower threshold will increase the number of possible matches returned.\n\nUnderstanding the EntityMembers Object\n\nThe EntityMember class centers on a collection of members. If retrieving EntityMembers using the EntityMembersGet operation, a number of additional objects will be populated depending on the type of criteria supplied. This same EntityMembers class needs to be created and populated by the call when creating new members in MDS.\n\nFigure 13-1 shows the EntityMembers class object and all possible members. In these class diagrams, the type of relationship between classes is denoted by 0, 1, and *. If a 1 is stated, the object is required or always provided. 0 means the class is optional, and * means multiple items can be added to the class collection.\n\n**Figure 13-1** _EntityMembers class diagram_\n\nSummarizing Member Data\n\nWhen you use the web service to select data by using the EntityMembers object, you do not get summary data that is useful for pagination or for otherwise showing record counts. To get this type of information, you can use the EntityMemberInformation object. This object contains the following attributes:\n\n **MemberCount** The member count of the records returned in the current request. This will be based on the paging information provided.\n\n **PageNumber** The current page number returned in this request.\n\n **TotalMemberCount** The total member count for the current request. The MembersGetCriteria must remain constant for this total member count to stay consistent.\n\n **TotalPages** The total number of pages for the current request. The MembersGetCriteria must remain constant for this page count to stay consistent.\n\nWorking with Metadata\n\nAll of the objects within MDS are created using the same operation: MetadataCreate. This operation creates models, entities, attributes, attribute groups, collections, and hierarchies. Throughout the book, there are examples of how to create these metadata objects.\n\nAs discussed in earlier chapters, MetadataCreate and MetadataClone are overloaded operations that allow you to create models, entities, and attributes within the MDS system. To create or return these objects, you must navigate the Metadata class and its child classes. Figure 13-2 shows a map of the Metadata class.\n\n**Figure 13-2** _Metadata class diagram_\n\nValidating Records in MDS\n\nAs an organization begins to interact programmatically with MDS, it may become necessary to trigger validation on specific members or an entire entity. Once validation has completed, users may need to retrieve validation issues that have been discovered. We have not had a chance to discuss these web service operations earlier in the book, so we'll provide some examples here.\n\nValidationGet operations can be tricky because they seem to support a wide variety of parameters, but only a few combinations will return results. These combinations seem to align with the UI controls for validation issues. Four validation examples are explained next.\n\nValidationGetByVersion Example\n\nThis operation gets all validation issues for a model and version combination. This would be used as part of the commitment process for a version.\n\nValidationGetByUser Example\n\nThis operation returns the results that correspond to notifications that were sent to a user.\n\nYou have two good options to get a valid user_id. You can find your correct user_id from the securityprincipleget web service call. If the application does not care about the user initiating the action and you want to ensure you get the most access, code using user_id = 1.\n\nValidationGetByMember Example\n\nThis operation returns results related to a member or set of members. In this example we pass in a single member.\n\nValidationProcess Example\n\nValidationProcess validates a model and version combination or validates a set of members passed into the operation. You can use an EntityMembersGet operation to retrieve a set of members related to a group of search terms and then loop through those members. In this simple example, we validate only a single member and pass back all validation issues found.\n\nManaging Transactions\n\nTransactions can provide users and managers with a complete record of all changes that have been made in the system. Model administrators will be able to see all transactions made within a model, while each user will have access to their own changes. These transactions also support reversal.\n\nTransactionsGet Example\n\nTransactionsGet operations return a list of transactions that meet the provided search criteria. Transaction search criteria only support values that are equal, so these criteria are exposed as identifiers attached to the criteria.\n\nTransactionsReverse Example\n\nWhile there may be many programmatic uses for TransactionsGet, TransactionsReverse should be used with caution. TransactionsReverse does not actually reverse the transaction, but rather reverts the current value from MDS to the prior value stored in the transaction log. This procedure can be very valuable when you need to reverse a change after triggering an external action. Make sure that you reverse the most recent transaction.\n\nThe following procedure is a simple example of how to reverse a single transaction based on the ID of the transaction. This procedure will throw an error if an invalid transaction ID is provided. You could add a conditional statement around the TransactionsReversed return statement to handle this case.\n\nHandling Errors in the Web Service\n\nAll MDS web service calls return an OperationResult object within their response messages. This OperationResult object contains a Request_ID to manage long-running requests and any errors that occurred as a result of the request. These errors can provide immediate feedback to client-side developers who need to find problems with their requests. These errors can range from useful to baffling, and sometimes it will take some trial and error to determine what is wrong. Each error will populate a portion of the following properties:\n\n **Code** Error number specific to the type of error encountered. A general classification for web service errors is provided in the following table.\n\n **Description** Localized text describing the error.\n\n **Context.FullyQualifiedName** The fully qualified name of the object involved in the error. Some names are only unique within their context. An entity would be qualified with a model name prefix such as _ModelName : EntityName_.\n\n **Context.Type** The type of object related to the error.\n\n **Context.Identifier** The identifier of the object involved in the error.\n\n **Context.Identifier.Id** The unique GUID of the object, if specified or available.\n\n **Context.Identifier.Name** The name of the object, if specified or available.\n\n **Context.Identifier.InternalId** The internal key from the MDS system tables.\n\nSummary\n\nThe MDS web service provides complete access to all of the functions that are found in the product. While many of the operations are detailed in other chapters of this book, this chapter acts as a reference to help you find the operations you need, and provides more detailed examples. After reading this chapter, you should be familiar with the main groupings available so you can programmatically access all that MDS has to offer. You should also be familiar with the changes made to the MDS web service in SQL Server 2012.\nChapter 14\n\nAdvanced Modeling\n\nIn This Chapter\n\n Common Modeling Mistakes\n\n Rapid Model Development\n\n Managing Slowly Changing Dimensions\n\n Limiting Unneeded Complexity\n\nModeling is more art than science. You need to understand not only your datasets and how they interact, but also the related systems. You also need to understand people and their processes. In this chapter, we will attempt to tackle some common complex modeling issues found in MDS projects.\n\nMaster Data Services is not a data warehouse or a reporting tool. It is a data management tool. This is a difficult distinction for many to understand. Master Data Services should be used to manage and store lists for the business. Security can create boundaries for multiple users. Business rules notify users of issues with the data, while the web UI, Add-in for Excel, and Services API give the entire organization access to work on the same datasets. All of these features are designed around managing data changes.\n\nOnce data is being managed correctly in MDS, extract, transform, and load (ETL) processes can send this data into data marts and warehouses to provide the reports demanded by the business. These systems can focus on the reporting structures, disseminating the information without concern for the data editing and quality processes. In this chapter we will try to provide insight into some of the more common complex problems that we have seen.\n\nCommon Modeling Mistakes\n\nMDS is best built from the top down, not the bottom up. As discussed in Chapter 3, the first step in managing data is to determine the major entities that need to be managed. Unfortunately, many modelers then move directly to the downstream systems to build their models, deriving all columns from these preexisting systems. This section discusses some significant pitfalls with this approach that you should be aware of.\n\nBuilding Outside Limitations into MDS\n\nWhether a system is purchased or created, the data model of that system was developed through a series of compromises. If you were building a system to manage payroll, how many attributes would you add for each employee? Would you add information about certifications they hold? Each system is built to solve a business need. Some of these systems attempt to solve multiple needs or allow for minimal customization. If you focus on a single system, you run the risk of bringing these compromises into MDS.\n\nA couple of common compromises seen in the field are perpetuating overloaded fields into your MDM system and propagating system column names. Either of these issues can blunt the effectiveness of your MDS implementation, limiting usability for business users.\n\nManage Overloaded Fields\n\nMany systems have tables that need to have a single column to use as the primary key. To ensure this column is always unique, values in this column may be a combination of three or more distinct values that have been pasted together, typically as a fixed-length combination. Do not bring this as a single column into MDS. Allow users to manage these distinct values separately. If visibility of this value is required, use business rules to concatenate the obscure codes into this overloaded value for the external systems.\n\nProvide Easily Recognizable Names\n\nMost systems try to use standard naming conventions and technical shorthand to name tables and fields within the database. Ensure that you adapt these names to be easily readable by your end users. This might apply to allowable values within some fields as well. Create an entity and provide useful names to each of these unique values.\n\nTrouble Identifying Common Attributes\n\nEach system within your enterprise comes preconfigured with some built-in nomenclature and culture. If you look at these systems at the database level, where table and column names have been abbreviated, determining meaning and use can be difficult. This can lead to confusion about how different systems or departments manage similar attributes. While these architecture diagrams can help lead the conversation, working directly with the business users is the best way to ensure you accurately model your business.\n\nEngineering Dead Scenarios\n\nWhen we rely on the existing structures of systems within the organization, we may find ourselves modeling scenarios the business does not require. This can be due to changes in the business structure over time or an incomplete understanding of the problem when the system was implemented. It is important to take advantage of the current mindshare of the business users to build your MDS model as the business wants it.\n\nRapid Model Development\n\nWith the advent of the MDS Add-in for Excel, Master Data Services has revolutionized the modeling process. In the past, there was a gulf between modelers who read and create data models and the rest of the business. The best way to bridge this gulf is to design a model, review it with the users, receive feedback, and repeat the process. With competing products, this process usually takes weeks or months for each iteration, whereas with the Add-in for Excel, a team can create a fully populated model in a couple of days.\n\nExcel is a great intermediate location for data from any source. Many of the datasets currently managed in your organization already exist within Excel. For any data that does not exist in an Excel sheet, you can use the existing data features in Excel to load the data into a workbook. Once the data is loaded into Excel, you can create a new entity by using the Create Entity wizard.\n\nManaging Slowly Changing Dimensions\n\nOne of the most common concerns that we deal with when helping data stewards model data in MDS is how to handle history within the system. In most cases these concerns are not drawn from operational systems as these systems typically focus on the processing of the current state. Analytic systems such as data warehouses or data marts need to store or report information based on changes over time. Most modeling discussions quickly turn to managing these dimension changes in MDS. This is the wrong way to look at it. The role of MDS is to create the changes, not to store them for reporting purposes. We need to look at how to leverage the existing tools in MDS to send these changes to the reporting systems that require them.\n\nRalph Kimball has spent much of his lifetime documenting techniques for analytic systems. One of his most discussed concepts is the handling of slowly changing dimensions. Mr. Kimball identified three types of dimensions that we will discuss here.\n\nOverwriting Rows (Type 1)\n\nThe simplest form of storage is in those dimensions that are stateless. Without any need to understand any previous values, ETL processes can simply overwrite the dimension from the subscription view without any further analysis. In a majority of companies, this should be the predominant type of data transfer, especially for supporting entities in the data warehouse.\n\nAdding Rows (Type 2)\n\nThe most prevalent type of versioning data in data marts and warehouses stores a new row for each change in the member record. The prior member record is effectively closed using a new key. In order to determine that a change has occurred and a new record needs to be made in the warehouse, the ETL process can look at the last changed date to trigger a new record for the member to be created, each row with a new surrogate key. Natural key or MDS code columns can be used to ensure different record versions can be tracked. You do not need to store this history in MDS.\n\nUnfortunately, the MDS system, like most systems, logs the last changed date as the date and time the change was made. In some cases, the date needs to be explicitly captured separate from the log. If the date is extremely important to the process and is part of the master record itself, create a new field or fields in MDS to store these effective dates. We will discuss the best way to manage these type 2 scenarios later in the chapter.\n\nAdding Columns (Type 3)\n\nSome systems or dimensions use the concept of another column per attribute change. This can be as simple as a prior and a current column for a single attribute. If your warehouse already has a table designed in the standard type 3 fashion, you can continue to populate this model using MDS. Otherwise, type 3 scenarios are tailor-made for versioning, as most users want to run reports for different periods of time.\n\nTo support original type 3 table formats, you can source current attribute values from the current version subscription views while sourcing all prior columns from a different set of subscription views linked to the prior version. These subscription views should leverage version flags to ensure they can work across multiple periods of time.\n\nUsing the Power of MDS to Simplify the Problem\n\nWe can view the loading of history into downstream data marts and data warehouses as an ETL problem or we can attempt to leverage MDS to improve the system. The reason most data marts and data warehouses traditionally used type 1, 2, and 3 storage methods was that data was flowing from one static data model into the reporting system. Information architects were unable to change the original tables that were storing managers, products, or accounts in their initial systems, so they needed to build ETL processes that attempted to pull the history into the reporting database structures.\n\nUnfortunately, these structures all have their drawbacks in this scenario. Type 1 storage is the most efficient, eliminating all duplication from the system. Type 2 is the most robust and most costly, as each change creates another entire record. Type 3 suffers from significant column expansion and an ever-changing data model as different history is required.\n\nMDS can change this equation. When history is required for a downstream dimension, it is rarely needed for all attributes. In fact, when working with data marts, only a few specific attributes require history, and using the time at which the change was made is extremely inaccurate. It is the effective date that matters sometimes for each report. This is where the MDS architecture can be extremely useful. In a matter of minutes, a new entity can be created that stores the important historical relationship along with two effective date columns, Start and End. We can then use subscription views to send this data from MDS. While the original type 1 dimension in MDS is unaffected by the new requirement we can populate a new dimension table to support the time-sensitive reporting requirement. Let's use Main Street Clothing Company to illustrate this problem.\n\nManaging Dimension for Main Street Clothing Company\n\nFor the past year, Main Street Clothing Company has been paying its individual store managers a monthly bonus based on sales at that manager's store. A monthly report has been created to provide sales numbers per manager to determine these monthly bonuses. Recently a new mall being built has created an increased demand for experienced managers in the area, and manager turnover has been increasing within the area. The market has also dictated that managers' bonuses be guaranteed in their employment contracts.\n\nCorporate must now include effective dates in the manager report to ensure accurate accounting of the bonuses. Instead of making changes in the Employee entity, making this a type 2 dimension, IT has decided to create an additional entity called Store Manager Relationship. In this entity, secured users can modify the effective management dates of each manager for each store.\n\nLimiting Unneeded Complexity\n\nAnother common scenario that occurs in MDS modeling is over-engineering an entity. The two common scenarios I see are data stewards designing for exceptions and designing for _N_. In both of these cases, the computer scientist in us can override common sense, creating additional complexity without adding significant business value. In each of these cases, considering the design time of each scenario is not as important as considering the time needed for users to manage data.\n\nDesigning for Exceptions\n\nIn many cases, I need to manage a dataset that has a relatively stable change rate 90 percent of the time or more. At other times, this dataset or a portion of the dataset may require a significant change in granularity. The easiest maneuver would be to manage the full dataset at the finer granularity and just make the changes in this single data format. Yet MDS is not designed as an automated system; it encourages and fosters user involvement. The value of a little additional design and integration time can provide significant benefits in terms of user satisfaction and limiting mistakes once the system is live.\n\nLet's take the common example of store hours. Suppose that you work in the corporate IT department of a company that owns a national chain of restaurants. The management of each store in the company has the option of opening and closing at any time they choose. The management of each store is given some discretion in determining their hours of operation because eating habits vary in different parts of the country. The company wants you to design a model where every store is able to do its own cost\/benefit analysis of staff versus revenue to optimize its operation's profit.\n\nWhile every store can select a different hour (or half-hour) at which to open and close each day, you find that certain patterns emerge. None of the restaurants opens before 5 a.m. or closes after 10 p.m. Hours of operation are different on the weekends, while weekday hours generally remain the same throughout the week. The major exception for this dataset is holidays. Every few months, a holiday significantly impacts the number of hours the restaurant is open. For reporting purposes, it is important to know the daily hours of operation when analyzing variable costs and revenue. No matter what happens, your ETL processes are going to need to translate this store operation information into usable data for reporting, and the cost for creating these processes is nominal no matter how you store the hours.\n\nGetting the correct operating hours from each restaurant is the single most important criteria in this project. These hours will be displayed on the company's website, which drives business to the stores. All of your reports will base calculations on total open hours, to determine labor efficiency. If you can't get accurate hours of operation for all stores, nothing else matters. Let's consider three architectures you could implement to solve this problem.\n\nMany IT professionals will like the simplicity of the design shown in Figure 14-1. A single table can store any number of days with similar hours. Now let's think through the use case for the actual table users in MDS. To correctly enter data into this format, a restaurant manager will need to add data changes into MDS as a new row. Any changes to this schedule require users to search and update many different records. Want to change the opening time on Friday for the year? You better set aside next Saturday to do so!\n\n**Figure 14-1** _One size fits all_\n\nThe model shown in Figure 14-2 gives the most flexibility, allowing users to set regular opening and closing hours each day of the week, with a separate table to break out any exceptions such as holidays. Each restaurant manager should be able to manage their standard hours in a matter of minutes. Whenever exceptions occur, the manager can quickly add them to the table. With this model, the ETL is a little more complicated because the Hours dimension needs to be generated from a combination of two subscription views, but the ease of use for the end user should quickly outweigh this extra complication.\n\n**Figure 14-2** _Holiday exceptions only_\n\nThe final example, shown in Figure 14-3, has some positives and negatives over the second option. While the second option gives each restaurant the ultimate flexibility in managing their own hours of operation, for some, this could lead to additional problems, with too much flexibility. Does corporate want to provide acceptable patterns for hours of operation?\n\n**Figure 14-3** _Weekday and weekend patterns_\n\nThis example also limits the chances for mistakes. In the previous example, a manager could mistakenly indicate that the restaurant will be open at 3 a.m. for dinner. Picking from a set of options can also allow new managers to leverage the wisdom of prior locations, by picking an operational pattern already in use. As a negative, adding a new hours pattern can be a complicated process, and changing hours on a specific day could be complicated or misuse the exception table.\n\nDesigning for _N_\n\nOftentimes we find models that value flexibility over function. While there are many business cases that support flexibility, and others where it may be mandatory, often the business value is questionable at best, and the additional complexity can kill usability. There is no greater case for this than managing customer data. In today's world, any customer can have multiple addresses and multiple phone numbers. Model builders might be tempted to support a customer model that allows for multiple contacts. This could be a separate address or phone table with the customer as an attribute. While this is supported in MDS, this practice can lead to problems both in usability and data quality control.\n\nThink about the case of running a retail establishment. You would like to get more information about your customers. Does the contact card you hand them give blanks for every phone number they are associated with? How about each address? We have found a much more usable model contains a single Customer entity with separate attributes for each phone number by type. A primary address set of attributes is also advisable unless business necessitates a separate table.\n\nThe benefits of adding attributes to a single table are not confined only to the model's general usability. A single entity can more easily support business rules, as well as better integration with matching and data quality tools.\n\nSummary\n\nMaster Data Services is really your data model layered on the Microsoft proprietary model. To use MDS most efficiently, data stewards must embrace the system as a data management tool and not as a data warehouse. Your MDS system should not mirror every dimension table in your data mart or data warehouse any more than it mirrors list tables in your operational systems. Business needs for data management should drive the model in MDS and the ETL process to load downstream data marts.\nIndex\n\nA\n\nabstraction layer\n\naccess. _See also_ security\n\ncollections\n\nentities\n\nhierarchies\n\nindividual attributes\n\nleaf member attributes\n\nmodel objects\n\nmodel versions and\n\nnavigational\n\nactions\n\nbusiness rules\n\nchange value\n\ndefault value\n\ndeleting\n\nexternal\n\nvalidation\n\nActive Directory\n\nadministration\n\nadministrator accounts\n\nanchor null relationship\n\nAND operator\n\nannotations\n\nApplication Development service\n\napplications. _See_ web applications\n\nApproved attribute\n\nattribute groups\n\naccess to\n\ncreating\n\nMain Street Clothing Company\n\nnaming\/renaming\n\noverview\n\nreordering\n\nin system administration\n\nattribute values\n\nconcatenating\n\npopulating\n\npopulating with incremental numbers\n\nattribute views\n\nattributes\n\naccess to\n\nadding to change tracking groups\n\ncode\n\ncollections\n\nconsiderations\n\ncreating\n\ncreating approved attributes\n\ncreating in web services\n\ndisplay format\n\ndomain-based. _See_ domain-based attributes\n\nediting\n\nERP systems and\n\nfiltering\n\nfree-form\n\nhiding\n\nleaf member\n\nMain Street Clothing Company\n\nmanaging\n\nname\n\nnature of\n\nnavigating\n\nnumber of\n\noverview\n\npermissions\n\nproblems identifying\n\nrequiring value attributes\n\nSQL Server\n\nauthentication\n\nB\n\nbase entity\n\nbatch tags\n\nbatches, staging. _See_ staging batches\n\nbatches size setting\n\nBI (business intelligence) applications\n\nBing maps\n\nbulk operations\n\nbusiness intelligence (BI) applications\n\nbusiness process maps\n\nbusiness process models. _See also_ Main Street Clothing Company\n\ninterviews\n\nleveraging\n\nlocation considerations\n\noverview\n\nuse of\n\nBusiness Rule Maintenance page\n\nbusiness rule operations\n\nbusiness rules\n\nactions\n\nadditional tasks\n\napplying\n\nautomating\n\nconditions\n\nconfiguring expressions\n\nconsiderations\n\ncreating\n\ncreating expressions\n\ndescription\n\nediting\n\ne-mail notifications\n\nexamples\n\nexcluding\n\nmanaging with web services\n\nname\n\nobtaining list of rules\n\noverview\n\npopulating values\n\npriority order\n\npublishing\n\nrequiring value attributes\n\nresolving validation issues\n\nstatuses\n\nstructure\n\ntriggered by change tracking\n\nvalidating versions against\n\nvalidation\n\nworkflows. _See_ workflows\n\nworking with\n\nC\n\nCDI (customer data integration) solutions\n\ncell-level permissions\n\ncertificates, server\n\nchange tracking\n\nchange tracking groups\n\nchange value actions\n\nchart of accounts\n\nClipboard\n\nCode attribute values\n\nCode property\n\nCode value\n\ncollection members\n\ncollections\n\naccess to\n\nadding consolidated members to\n\nadding leaf members to\n\nadding members to\n\ncombining\n\ncreating\n\nenabling entities for\n\noverview\n\nupdating attributes\n\nweighting of\n\nCommon HTTP Features service\n\ncommon view architecture\n\ncompany, sample. _See_ Main Street Clothing Company\n\nCompatability View\n\nconcatenating values\n\nconditions\n\nbusiness rule workflows\n\nbusiness rules\n\ndeleting\n\nConfiguration Manager. _See_ MDS Configuration Manager\n\nconsolidated members\n\nadding to collections\n\nadding to entities\n\ncreating\n\ndescribed\n\nhierarchies and\n\n_Consolidated table\n\nContext.FullyQualifiedName property\n\nContext.Identifier property\n\nContext.Identifier.Id property\n\nContext.Identifier.InternalId property\n\nContext.Identifier.Name property\n\nContext.Type property\n\ncredentials, prompting for\n\nCSV files\n\ncustomer data integration (CDI) solutions\n\nD\n\ndata. _See also_ records\n\ncleansing\n\ncombining\n\nconsumers of\n\nexporting. _See_ exporting data\n\nexternal access to\n\nfiltering. _See_ filters\n\nin hierarchies. _See_ hierarchies\n\n\"homeless\"\n\nimporting. _See_ importing data\n\nmapping\n\nmaster. _See_ master data\n\nmetadata. _See_ metadata\n\nowners of\n\npagination\n\npreparing for import\n\npublishing. _See_ publishing data\n\nrefreshing\n\nsecurity of. _See_ security\n\nsorting\n\nstaging. _See_ staging process\n\nvalidating. _See_ validation\n\nversions. _See_ versions\n\ndata quality matching\n\ndata quality operations\n\nData Quality Services (DQS)\n\ndata stewards\n\ndata validation\n\nDatabase Engine\n\nDatabase Mail profile\n\ndatabase server role\n\ndatabases. _See_ MDS database\n\ndatasets\n\ndefault value actions\n\ndeny permissions\n\ndeploying models\n\nderived hierarchies\n\naccess to\n\ncreating\n\ncreating with MetadataCreate\n\ndescribed\n\nwith explicit caps\n\nhiding levels\n\nmoving members in\n\nthree-level\n\nviews\n\nvs. explicit hierarchies\n\nDescription property\n\ndirect hierarchy security\n\n\"Does not match\" operator\n\ndomain-based attributes\n\nchanging number of\n\ncreating\n\ndescribed\n\nrecursive hierarchies and\n\ndomains\n\nconsiderations\n\ndescribed\n\nroles\n\nspecialized systems\n\nversions\n\nDQS (Data Quality Services)\n\nE\n\nediting\n\nattributes\n\nbusiness rules\n\nentities\n\nsubscription views\n\ne-mail notifications\n\nbusiness rules\n\nqueries\n\nversion status changes\n\nEmployee entity\n\nenterprise resource planning (ERP)\n\nentities\n\naccess to\n\nadding members to\n\nbase\n\ncreating\n\ncreating from Excel data\n\ncreating in web services\n\ncreating line entity\n\nediting\n\nenabling for collections\n\nenabling for hierarchies\n\nfiltering results\n\nincluding models in\n\nloading\n\nMain Street Clothing Company\n\nmanufacturer\n\nnaming\n\nnavigating\n\noverview\n\npopulating consolidated members for\n\nproduct\n\nrelationships\n\nsubsets\n\nvalidating\n\nentity views\n\nentity-based staging\n\nEntityMembers class\n\nEntityMembers object\n\nEntityMembersCreate operation\n\nEntityMembersGet operation\n\nretrieving members\n\nsearching for members\n\nEntityMembersMerge operation\n\nEntityMembersUpdate operation\n\nERP (enterprise resource planning)\n\nerror numbers\n\nerrors\n\naccess denied\n\ninvalid transaction ID\n\nobjects\n\nretrieving data\n\nduring staging\n\nweb services\n\nerrors view table\n\nETL (extract, transform, and load)\n\nETL processes\n\nExcel\n\nExcel add-in\n\napplying rules\n\nbehavior settings\n\ncombining data\n\nconfiguring settings\n\nconnecting to MDS server\n\ncreating\/modifying entities\n\ndata quality matching\n\ndeletions in managed sheets\n\ninstalling\n\nloading entities\n\nlogging options\n\nmanaging project with\n\nMDS entities\n\noverview\n\npublishing changes\n\nrefreshing data from server\n\nreviewing transactions\n\nsaving\/sharing queries\n\nworking with MDS data\n\nExcel spreadsheets\n\nexplicit cap hierarchies\n\nexplicit hierarchies\n\naccess to\n\nconsiderations\n\ncreating\n\ndescribed\n\nmandatory\n\nnaming\n\nnon-mandatory\n\nparent-child relationships\n\npermissions\n\nsibling relationships\n\nvs. derived hierarchies\n\nExplorer\n\napplying business rules in\n\nfunctional area\n\nmetadata and\n\nnew format\n\nresolving validation issues\n\nreviewing business rule issues\n\nExplorer grid\n\nexport views. _See_ subscription views\n\nexporting data\n\nto subscribing systems\n\nsubscription views\n\nexporting members\n\nExportViewCreate option\n\nexpressions, business rules\n\nconfiguring\n\ncreating\n\nexternal actions\n\nextract, transform, and load. _See_ ETL\n\nextract, transform, and load (ETL)\n\nF\n\nfiles\n\nassociated with members\n\nCSV\n\nmaster\n\nsample\n\nXML\n\nfilter operators\n\nfilter warnings\n\nfiltered lists\n\nfilters. _See also_ grids\n\napplying to members\n\nentity results\n\nfinding data via\n\noperators for\n\nfinance administrators\n\nFinance model\n\nFinance_data.pkg file\n\nfinding items\n\ndata via filtering\n\nmembers\n\nmembers in hierarchies\n\nSearchTerm property\n\nFortune 500 companies\n\nfree-form attributes\n\nfunctional area permissions\n\nfunctional security\n\nG\n\nGetSecurityPrincipals operation\n\nGetVersion helper class\n\nglobal chart of accounts\n\nGlobally Unique Identifier (GUID)\n\ngrids. _See also_ filters\n\ngroup memberships\n\ngroups. _See also_ users\n\nActive Directory\n\nadding\n\nadding to web UI\n\nassigning security to\n\nattribute. _See_ attribute groups\n\nchange tracking\n\ncombining users\/groups\n\nlocal\n\nlogistics\n\noverlapping permissions\n\npermissions. _See_ permissions\n\npurchasing\n\nretrieving\n\nsearch\n\nwarehouse\n\nGUID (Globally Unique Identifier)\n\nH\n\nHealth and Diagnostics service\n\nhelper class\n\nhierarchies\n\naccess to\n\ncreating with web services\n\nderived. _See_ derived hierarchies\n\ndirect hierarchy security\n\ndisplay settings\n\nenabling entities for\n\nentities\n\nexplicit. _See_ explicit hierarchies\n\nhiding levels\n\nleaf members\n\nmember permissions\n\nmember security\n\nmoving members in\n\nnumber of members in\n\noverlapping permissions\n\noverview\n\nragged vs. level-based\n\nrecursive\n\nsearching for members in\n\nshowing members names\n\nsystem settings\n\nworking with data in\n\nhierarchy management\n\nhierarchy views\n\nhistory\n\nmodels\n\nversions\n\n\"homeless\" data\n\nhotfixes\n\nhttpGetEnabled setting\n\nHTTPS\n\nI\n\nIF...THEN statements\n\nIIS (Internet Information Server)\n\nimporting data\n\npreparing for\n\ninto SQL Server\n\nstaging import types\n\ninto staging tables\n\nincludeExceptionDetailInFaults setting\n\nincremental numbers\n\ninstalling MDS\n\nintegration improvements\n\nIntegration Management area\n\nInternet Explorer\n\ncompatability mode\n\nprompting for credentials\n\nsupport for\n\nInternet Information Server (IIS)\n\nJ\n\nJaccard search term\n\nJaro-Winkler search term\n\nK\n\nKimball, Ralph\n\nL\n\nleaf members. _See also_ members\n\nadding to collections\n\nadding to entities\n\nattributes\n\ncreating\n\ndescribed\n\nhierarchies\n\nupdating\n\nupdating attributes\n\nleaf staging fields\n\nlevel-based hierarchies\n\nLevenshtein search term\n\nlicense agreement\n\nline attributes\n\nline entities\n\nloading\n\nentities\n\nrecords\n\nrelationships\n\nlocal groups\n\nlocation management\n\nlocation workflow process\n\nlocking versions\n\nlogging\n\nExcel add-in\n\nstaging transactions\n\ntransactions\n\nlogistics\n\nLogistics role\n\nlongest common sequence search term\n\nM\n\nMain Street Clothing Company\n\nattribute groups\n\nattributes\n\nbusiness process models\n\nconsiderations\n\ndata steward for\n\nentities\n\nfilling gaps in\n\ninefficient practices\n\nmanaging dimension for\n\nmembers\n\nmodels\n\noverview\n\nprocess maps\n\nstaging process\n\nsubscription views in\n\nsystem architecture\n\nusing Excel add-in\n\nversioning for\n\nmanaged sheets\n\nManagement Studio\n\nManagement Tools service\n\nmanager domain-based attributes\n\nmanufacturer domain-based attributes\n\nmanufacturer entities\n\nmapping data\n\nMarketing role\n\nmaster data. _See also_ data\n\nannotations\n\nbase entity\n\ndefinition of\n\nexporting members\n\nfiltering\n\nin hierarchies. _See_ hierarchies\n\nrelationships and\n\nretrieving\n\nsorting\n\nupdating\n\nvalidation\n\nvs. transactional data\n\nworking with members\n\nmaster data management. _See_ MDM\n\nMaster Data Services. _See_ MDS\n\nmaster files\n\nMatches operator\n\nMDM (master data management)\n\ncomponents\n\nconsiderations\n\nhistory of\n\ninitiating staging process\n\nMicrosoft's solution to\n\noverview\n\nweb applications. _See_ web applications\n\nweb site creation\n\nMDM projects\n\nMDM solutions\n\ncustomer data integration\n\nERP solutions as\n\nFortune 500 companies\n\nproduct information management\n\nspecialized\n\nmdm.tblStgBatch table\n\nmdm.tblStgRelationship table\n\nMDS (Master Data Services)\n\nbulding outside limitations into\n\nconfiguring\n\nconsiderations\n\ncreating new MDS database\n\ncreating workflows in\n\ndesigning for exceptions\n\ndesigning for _N_\n\nextending with web services\n\ninstalling\n\nintegrating with other systems\n\nintroduction to\n\nlicense agreement\n\nmapping data\n\nnew features\n\nobtaining latest version\n\nperformance\n\npreparing data for importing\n\npreparing web server for\n\nproduct management processes\n\nrelationships\n\nsample company. _See_ Main Street Clothing Company\n\nsimplifying problems via\n\nstaging process. _See staging entries_\n\nstaging schema\n\nsystem requirements\n\nsystem types\n\nupdates to\n\nvalue proposition\n\nMDS Configuration Manager\n\nassociating database with web application\n\nconfiguring e-mail notifications\n\nconsiderations\n\ndatabase creation\n\nsystem settings\n\nweb application creation\n\nweb site creation\n\nMDS database\n\nassociating with web applications\n\nconsiderations\n\ncreating\n\nnaming\n\nrepairing\n\nschema version\n\nsettings\n\ntesting connection\n\nupgrading\n\nversion\n\nwhere to install\n\nMDS entities\n\nMDS models\n\nMDS projects\n\ncomplexity of\n\ncreating in Visual Studio\n\ndetermining scope of\n\nmanaging with Excel add-in\n\nmilestones\n\nsize of\n\nstarting\n\ntemplates\n\nMDS solutions\n\nMDS web applications. _See_ web applications\n\nMDS web service operations\n\nmdsuser.com\n\n.mdxqx extension\n\nmember operations\n\nMemberCount attribute\n\nmembers\n\nadding to collections\n\nadding to entities\n\nannotating\n\napplying filters to\n\nconsolidated. _See_ consolidated members\n\ncreating\n\ndeactivating\n\ndeleting\n\nediting attributes for\n\nexporting\n\nfiles associated with\n\nfiltering entity results\n\nleaf. _See_ leaf members\n\nMain Street Clothing Company\n\nmoving in hierarchies\n\nmoving via Clipboard\n\nmoving via dragging\/dropping\n\nnumber of members displayed\n\noverview\n\npagination and\n\npermissions\n\npopulating\n\npurging\n\nreactivating\n\nretrieving\n\nreturning results related to\n\nsearching for\n\nshowing names of\n\nstaging examples\n\nsummary data\n\nupdating\n\nvalidating\n\nworking with\n\nworking with in web services\n\nMetaCreate operation\n\nmetadata. _See also_ data\n\nMetadata model\n\nmetadata objects\n\nmetadata structure operations\n\nMetadataClone operation\n\nMetadataCreate operation\n\nMetadataGet operation\n\nMetadataUpdate operation\n\nmodel administrators\n\nmodel deployment\n\nmodel deployment packages\n\nmodel object security\n\nmodel objects\n\naccess to\n\npermissions\n\nModelDeploy.exe file\n\nModelMembers operations\n\nmodels\/modeling\n\naccess to\n\nadding members to entities\n\nadvanced techniques\n\nbasic concepts\n\nbuilding\n\nchanging structure of\n\ncommon mistakes\n\ncomplexity and\n\nconsiderations\n\ncreating entities for\n\ncreating from scratch\n\ncreating in web services\n\ndead scenarios and\n\ndeploying\n\ndesigning for exceptions\n\ndesigning for _N_\n\nFinance\n\nflexibility vs. function\n\nhistory\n\nlimiting complexity in\n\nMain Street Clothing Company\n\nmanaging changing dimensions\n\noverview\n\npackages of\n\npredefined\n\nproduct. _See_ Product model\n\nrapid deployment of\n\nrelationships\n\nsample\n\nstandardization\n\nvalidation\n\nversions. _See_ versions\n\non web sites\n\nN\n\nnavigational access\n\n.NET Framework 3.5.1 Features service\n\nnumbers\n\ndata mapping and\n\nincremental\n\nnegative\n\noperators for\n\nO\n\nobject models\n\nobjects. _See also specific objects_\n\nerrors\n\nmetadata\n\nmodel. _See_ model objects\n\nOLAP (online analytical processing) systems\n\nonline analytical processing (OLAP) systems\n\nOperationResult object\n\noperators\n\nOR operator\n\nOutlook\n\nP\n\nPageNumber attribute\n\npagination\n\nparent staging fields\n\nparent-child relationships\n\npasswords\n\nperformance\n\nPerformance service\n\npermissions. _See also_ User and Group Permissions\n\napplying\n\nassigning\n\nattributes\n\ncell-level\n\ncombining\n\ndeleting\n\ndenying\n\nfinance administrators\n\nfunctional area\n\ngroups\n\nhierarchy member\n\ninheriting\n\nmembers\n\nmodel objects\n\nobject models\n\norder of operations\n\noverlapping\n\nproduct administrators\n\ntesting\n\nusers\n\nPIM (product information management)\n\nPIM solutions\n\n.pkg extension\n\npoint of sale (POS) systems\n\nPOS (point of sale) systems\n\npreferences\n\nPricing role\n\nprivileges. _See also_ permissions\n\nprocess maps\n\nproduct administrators\n\nproduct entities\n\nproduct information management (PIM)\n\nproduct management processes\n\nProduct model. _See also_ models\/modeling\n\nadding members\n\nconsiderations\n\ncreating\n\ncreating entities for\n\nhierarchical structure\n\nProduct_data.pkg file\n\nproducts, assigning line attributes to\n\nprojects. _See_ MDM projects; MDS projects\n\npublishing business rules\n\npublishing changes\n\npublishing data\n\npurchasing group\n\nQ\n\nqueries\n\nR\n\nragged hierarchies\n\nrecords. _See also_ data\n\nduplicate\n\nloading\n\nper page\n\nstaged\n\nvalidation\n\nrecursive hierarchies\n\nrelationship staging examples\n\n_Relationship table\n\nrelationship table fields\n\nrelationships\n\nconsiderations\n\nloading\n\nmany-to-many\n\nstaging examples\n\nstaging process\n\nRepair Database button\n\nRequest_ID\n\nreturn on investment (ROI)\n\nROI (return on investment)\n\nrole services\n\nroles\n\ndata stewards\n\nLogistics\n\nMarketing\n\nPricing\n\nroles, system\n\nrules. _See_ business rules\n\nS\n\nsample company. _See_ Main Street Clothing Company\n\nsample files\n\nsample models\n\nschema version\n\nsearches. _See_ finding items\n\nSearchTerm property\n\nSecure Sockets Layer. _See_ SSL\n\nsecurity. _See also_ access\n\nadministrators\n\nauthentication\n\nchanges in SQL Server 2012\n\nconsiderations\n\ndirect hierarchy\n\nfunctional area\n\ngroups\n\nhierarchy members\n\nmodel object\n\noverview\n\npasswords\n\npermissions. _See_ permissions\n\nprivileges\n\nSecure Sockets Layer. _See_ SSL\n\nstaging issues\n\ntables\n\nusers\n\nvia web services\n\nweb applications\n\nweb sites\n\nsecurity operations\n\nsecurity principals\n\nSecurity service\n\nSecurityPrincipal object\n\nSecurityPrivilege object\n\nserver certificates\n\nserver farms\n\nservers. _See also_ Windows Server 2008 R2\n\nadding to safe list\n\nconnecting to\n\nInternet Information Server\n\nrefreshing data from\n\nSMTP\n\nSQL Server. _See_ SQL Server\n\nweb\n\nservice account\n\nService Broker queue\n\nservice namespace\n\nService Packs\n\nservice references\n\nservices. _See_ web services\n\nSharePoint\n\nSharePoint workflows\n\nsibling relationships\n\nsimilarity matches\n\nSimilarity operator\n\nSMTP authentication\n\nSMTP servers\n\nSOE (system of entry)\n\nsolutions\n\nCDI\n\nERP\n\nMDM. _See_ MDM solutions\n\nMDS\n\nPIM\n\nSOR (system of record)\n\nsort order\n\nsorting data\n\nSQL Server\n\nattributes\n\nimporting data into\n\nviewing subscription views in\n\nSQL Server 2012\n\ndata changes in\n\nnew MDS features\n\nsecurity changes in\n\nSQL Server Integration Services (SSIS)\n\nSQL Server Management Studio\n\nviewing staging errors\n\nSSIS (SQL Server Integration Services)\n\nSSL (Secure Sockets Layer)\n\nSSL connections\n\nstaging batch interval\n\nstaging batch table\n\nstaging batches\n\nstaging errors view\n\nstaging examples\n\nattributes\n\nmembers\n\nrelationships\n\nstaging import types\n\nstaging process\n\nadvantages of\n\narchitecture\n\nbatch interval\n\nbulk operations\n\nclearing batches from queue\n\nclearing staging queue\n\ndeactivating members\n\nentity-based staging\n\nerrors during\n\ninitiating\n\nleaf\/parent staging fields\n\nloading relationships\n\nlogging transactions\n\nperformance\n\nreactivating members\n\nsecurity issues\n\nstaging import types\n\nsystem settings for\n\nvia Master Data Manager\n\nvia stored procedures\n\nvia web services\n\nstaging (stg) schema\n\nstaging sweep procedure\n\nstaging tables\n\nstaging views\n\nstg (staging) schema\n\nstored procedures\n\nsubscribing systems\n\nsubscription view formats\n\nsubscription view operations\n\nsubscription views\n\ncreating\n\ncreating with web services\n\ndeleting\n\nediting\n\nentity views\n\nexporting data\n\nin Main Street Clothing Company\n\nmodel deployment and\n\nstaging and\n\nviewing in SQL Server\n\nsubsequence\n\nsubsets\n\nSystem Administration area\n\nattribute groups in\n\nbuilding a model\n\ndescribed\n\nhierarchies\n\nviewing model structure in\n\nsystem administrators\n\nsystem of entry (SOE)\n\nsystem of record (SOR)\n\nsystem roles\n\noverview\n\nsubscribing systems\n\nsystem of entry\n\nsystem of record\n\nsystem settings\n\nsystem settings operations\n\nT\n\ntables\n\nadding columns\n\nadding rows\n\nnaming conventions\n\noverloaded fields\n\noverwriting rows\n\nsecurable\n\nsecurity\n\nstaging\n\ntblMemberAttribute table\n\ntblStgBatch table\n\ntblStgMember table\n\ntblStgRelationship table\n\ntemplates\n\nTotalMemberCount attribute\n\nTotalPages attribute\n\ntransaction ID\n\ntransaction logging\n\ntransaction operations\n\ntransactional data\n\ntransactions\n\nannotating\n\nconsiderations\n\ngetting\n\nlogging\n\nmanagement\n\noverview\n\nreversing\n\nreviewing\n\ntypes of\n\nTransactionsGet operations\n\nTransactionsReverse operations\n\nT-SQL jobs\n\nT-SQL statements\n\nU\n\nUAC (User Account Control)\n\nuniform resource locators (URLs)\n\nupdates\n\nmaster data\n\nMDS\n\nmembers\n\nmetadata\n\nupgrading database\n\nURLs (uniform resource locators)\n\nUser Account Control (UAC)\n\nUser and Group Permissions area\n\nconsiderations\n\ndescribed\n\ne-mail validation\n\nuser notifications\n\nuser preference operations\n\nusers\n\nActive Directory\n\nassigning security to\n\ncombining users\/groups\n\ngroups of. _See_ groups\n\nnumber of\n\npermissions. _See_ permissions\n\npreferences\n\nretrieving\n\nV\n\nvalidation\n\napplying rules\n\nbusiness rules\n\ne-mail notifications\n\nentities\n\nmanaging rule results\n\nmembers\n\nmodels\n\nrecords in MDS\n\nresolving issues\n\nstatuses\n\nstored procedures\n\nuser notifications\n\nvalidating model version\n\nversions\n\nweb services\n\nvalidation, data\n\nvalidation actions\n\nvalidation operations\n\nValidationGet operations\n\nValidationGetByMember operation\n\nValidationGetByUser operation\n\nValidationGetByVersion operation\n\nValidationProcess operation\n\nvalues. _See_ attribute values\n\nversion control workflows\n\nversion flags\n\nVersion Management area\n\nversions\n\nancestry\n\nassigning flags to\n\nchanging name\/description\n\nchanging status\n\ncommitting\n\nconsiderations\n\ncopying\n\ndeleting\n\ne-mail notification\n\nexamples of\n\nExcel add-in\n\nhistory\n\nlocking\n\nMain Street Clothing Company\n\nopen\n\noverview\n\nreopening committed version\n\nreturning list of\n\nstatuses\n\ntransactions. _See_ transactions\n\nuses for\n\nusing web services with\n\nvalidating\n\nviews\n\ncommon\n\nderived hierarchy\n\nentity\n\nVisual Studio\n\ncreating MDS projects in\n\ncreating SharePoint workflows\n\nW\n\nwarehouse group\n\nWCF (Windows Communication Foundation)\n\nweb applications\n\nassociating MDS database with\n\nautomatic creation of\n\ncreating\n\nnaming\n\nprebuilt\n\nsecuring\n\nservice references\n\nsettings\n\nsystem requirements\n\nweb browsers. _See also_ Internet Explorer\n\nweb configuration file\n\nweb servers\n\nweb service operations\n\nweb services\n\nabstraction layer\n\nconsiderations\n\ncreating attributes\n\ncreating entities\n\ncreating hierarchies with\n\ncreating models\n\ncreating subscription views with\n\nenabling\n\nerror handling in\n\nexposing\n\nmanaging business rules\n\nretrieving members\n\nsetting security with\n\nstaging data via\n\nusing\n\nusing with versions\n\nvalidation\n\nworking with members\n\nWSDL\n\nWeb Services Description Language (WSDL)\n\nweb sites\n\ncertificates\n\ncompanion to book\n\ncreating Master Data Manager site\n\nmdsuser.com\n\nmodel packages on\n\nnaming\n\npasswords\n\nsample files\n\nsecurity\n\nservice accounts\n\nsettings\n\nweb UI, adding groups to\n\nweb-based user interface (UI)\n\nWindows 7 systems\n\nWindows Communication Foundation (WCF)\n\nWindows PowerShell Integrated Scripting Environment service\n\nWindows Process Activation service\n\nWindows Server 2008\n\nWindows Server 2008 R2. _See also_ servers\n\npreparing for upgrade\n\nupgrading\n\nWindows Vista\n\nworkflow integration service\n\nworkflows\n\nbusiness rules\n\nconditions\n\ncreating in MDS\n\nexternal to MDS\n\nSharePoint\n\ntriggering from MDS\n\nversion control\n\nwrapper classes\n\nWSDL (Web Services Description Language)\n\nX\n\nXML files\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":" \nA WORLD WITHOUT WHY\nA WORLD WITHOUT WHY\n\nRaymond Geuss\n\nCopyright \u00a9 2014 by Princeton University Press\n\nPublished by Princeton University Press, 41 William Street,\n\nPrinceton, New Jersey 08540\n\nIn the United Kingdom: Princeton University Press,\n\n6 Oxford Street, Woodstock, Oxfordshire OX20 1TW\n\npress.princeton.edu\n\nAll Rights Reserved\n\nLibrary of Congress Cataloging-in-Publication Data\n\nGeuss, Raymond.\n\n[Essays. Selections.]\n\nA world without why \/ Raymond Geuss.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-0-691-15588-3 (hardcover)\n\n1. Philosophy\u2014History. 2. Philosophy, Modern\u201421st century.\n\n3. Optimism. 4. Ethics. I. Title.\n\nB1626.G481 2014\n\n190\u2014dc23\n\n2013043562\n\nBritish Library Cataloging-in-Publication Data is available\n\nThis book has been composed in Garamond Premier Pro\n\nPrinted on acid-free paper. \u221e\n\nPrinted in the United States of America\n\n1 3 5 7 9 10 8 6 4 2\nSPIEGEL: Vor zwei Wochen, Herr Professor, schien die Welt noch in Ordnung....\n\nADORNO: Mir nicht.\nContents\n\nPREFACE ix\n\n1. Goals, Origins, Disciplines 1\n\n2. Vix intellegitur 22\n\n3. Marxism and the Ethos of the Twentieth Century 45\n\n4. Must Criticism Be Constructive? 68\n\n5. The Loss of Meaning on the Left 91\n\n6. Authority: Some Fables 112\n\n7. A Note on Lying 135\n\n8. Politics and Architecture 144\n\n9. The Future of Theological Ethics 163\n\n10. Did Williams Do Ethics? 175\n\n11. The Wisdom of Oedipus and the Idea of a Moral Cosmos 195\n\n12. Who Was the First Philosopher? 223\n\n13. A World without Why 231\n\nNOTES 237\n\nINDEX 257\nPreface\n\nIt is natural for us to think that humans as animals belong to a certain biological species and are, as such, subject to a number of natural necessities such as the vital need to maintain a certain minimal body temperature and to eat and drink if they are to survive. We are, however, not merely biological entities but also inherently social animals, and societies, too, will \"need\" to satisfy certain conditions if they are to survive. In societies that do survive there will be a variety of mechanisms for imposing \"necessities\" on individuals and smaller groups; often these mechanisms will function under the guise of simply transmitting or \"passing on\" natural necessities. Thus, I must eat if I am to survive, so we must all work cooperatively for several hours a days in the fields or rice paddies if we are to survive. This transmission, however, is never a mere process of neutral \"passing on\" necessity. Actual human \"needs\" of any kind are never presented to us, as it were, \"raw\" but always in one social configuration or other, so any discussion of basic biological needs is perforce a kind of retrospective abstraction, which might be warranted, but, if it is, is always warranted for some specific purpose. In fact, what is called \"transmission\" is always a process of the transformation or social constitution of needs. \"Transformation or constitution\" is not a mistaken or incautious formulation, as if I couldn't make up my mind whether there was something, some \"need,\" there to start with that was \"transformed\" into a slightly different need or whether \"needs\" did not \"really\" exist until social processed. Rather it is an expression of my view\u2014which, of course, could be incorrect but is not inconsidered\u2014that this alternative is not to be taken as an absolute but is context-dependent. \"We all need to work together cooperatively in order to survive\" is deeply ambiguous, and this ambiguity is a breeding ground for ideological distortion. It can mean:\n\na. \"if we don't all work together cooperatively, each of us will die very soon (because the small boat we're in is leaking badly)\"\n\nb. \"if we don't all work together cooperatively, not all of us will survive (although some may)\"\n\nc. \"if we don't all work together cooperatively, we won't survive as a recognizable group (although each individual may disperse and survive alone or as a member of a different group)\"\n\nIn addition to these strict \"needs,\" that is, conditions that must be fulfilled if survival is to be ensured, humans also have an individually and socially idiosyncratic set of desires, preferences, wishes, and aspirations. We all need to eat and drink, but I prefer tea to coffee, although many people have the reverse preference. I also recognise that there are, or at any rate have been, individuals and even whole societies, such as that of ancient Rome, in which neither tea nor coffee is drunk at all. We also all grow up and remain throughout our entire lives enmeshed in a thick web of what are now called \"normative\" demands that have their origin ultimately in institutions that claim \"authority\" over us. In many Western European societies fathers of families were for a long time construed as \"heads of the household\" and had significant real and moral powers over their wives, children, and servants; political authorities of various kinds demanded allegiance; churches (or The Church) claimed to preach the word of god and had institutions like courts to enforce their views; relations of economic dependence among the members of small groups gave the words of those in key positions special weight; local forms of social pressure (and of solidarity) could take sharply articulated forms. Finally, as social beings we humans are to some extent capable of perceiving and acquiring knowledge about the real world in which we live, and we have some extremely feeble, only intermittently effective, and highly variable ratiocinative capacities. Human life is to a large extent constituted by an attempt to reconcile \"needs,\" desires, and \"normative claims\" on an individual and social level in view of our best knowledge about our world.\n\nQuestions about how individuals and groups should behave can arise in any social form, but it can seem to us from the vantage point of the apparently infinite distance that separates twenty-first-century Europeans from \"traditional\" societies that in those societies individuals may be perplexed and different groups (and duties to different groups) may conflict, but the question of the general structure of an individual human life and of social life as a whole will not seem to be particularly problematic. Since, however, the sources of such normative claims will be diverse, it can easily happen that they seem to make different demands on agents even in traditional societies. Equally individuals' perceptions, beliefs, and preferences will be different.\n\nIt is natural for thinking people in the West to start by assuming that the world is (finally) \"in order\" and trying to formulate explicitly and then \"reconcile\" the various claims made by the different authorities: The Gospel accounts of Mark and John can be made to tally. The emperor, the pope, and the local lord \"really\" are demanding us to lead the same kind of life. St. Paul can be rendered consistent with Aristotle. In a world with relatively intact and generally recognised authorities, the question of discipline, both of how and to what extent one should or may coerce others, and of self-discipline seems in principle answerable: One disciplines people by training them, as much as possible, to want to do what they in any case \"must\" (of natural necessity) do and also what they should do. To what extent it will be possible to make people want to do what they must and should do will depend on a number of unpredictable factors, among them the nature of the demands society makes and the kind of forces of coercion, manipulation, and educations it has available to it.\n\nWhat happens, however, if the questions go beyond queries about reconciling occasional discrepancies between individual authoritative statements? What if the emperor is a sinner and schismatic? What if the pope is a heretic? What if the very idea of \"being a heretic\" comes to look archaic and irrelevant? What could proper discipline (including self-discipline) look like in a world like that?\n\nSo is the world, including our authorities, fundamentally in order, or is it not? What would we mean by either of these two statements? How could we argue for one or the other of them? What, if anything, would follow for our lives if one or the other of these statements were true and could be shown to be true? There seems little doubt but that traditionally philosophy was supposed to ask this question, and also little doubt that philosophers had a predilection to answer it affirmatively and to draw from their particular version of the affirmative answer very far-reaching consequences for how humans should act. Dewey, that is, was clearly right to think that traditional philosophy was inherently conservative, having as its goal the project of inventing arguments to support as much of the existing forms of social authority as possible. Aristophanes may or may not have got Socrates right in taking him to be a dangerous subversive, but Plato was certainly on Aristophanes' side in thinking that a happy ending was possible only in a polity from which \"sophists\" were excluded. The difference is that Plato added to Aristophanes' arsenal of satire, innuendo, drama, slapstick, and verbal pyrotechnics a highly developed variant of one of the sophists' own weapons, ratiocination.\n\nThe task of philosophy became significantly more difficult starting in the late eighteenth century when the whole concept of \"authority\" but especially that of \"moral authority\" became problematic. The reason for this is the modern hypertrophy of ethical thinking centred on the idea of \"autonomy.\" After all, \"freedom\" was not the philosophical obsession in the ancient and medieval worlds that it has become in the past 250 years. That is perhaps because earlier periods actually had visible classes of slaves and serfs, and the distinction of \"free\" men from them was not as theoretically problematic; freedom was primarily a political problem\u2014could someone torture you with impunity or not; did you have to pay a lord to get married and work without compensation in his fields for a few days a week or not?\u2014not a moral or philosophical one. To be sure, from Herodotus to Nero one finds pathetic appeals to the \"freedom of the Hellenes,\" and \"libertas\" plays an important role in Roman politics, but this originally meant absence of barbarian, especially Persian, military domination over the Greek city-states, then \"independence from foreign rule\" (more generally), and this had little to do with the internal constitution\u2014oligarchy, monarchy, despotism, democracy\u2014of any given city-state. Later \"freedom\" could be associated with reduction or absence of forms of taxation, and with certain limited powers of self-management within the omnipotent Roman imperium. Or \"freedom\" might come to be associated with \"civitas,\" the acquisition of Roman citizenship. None of these were concepts that seemed to pose any special philosophical difficulties, certainly not any of the kind that arise for post-Augustineans. To be sure, one philosophical school, the Stoics, had developed views of freedom, but apart from them \"eleutheria\" was not a key concept in older Greek philosophy.\n\nThe situation changes completely with the advent of the new conceptions of radical individual autonomy in the eighteenth century. With Kant we get a canonical distinction between the morally valuable \"autonomy\" of an individual subject and a reprehensible \"heteronomy\" established as fundamental for ethics. Fascination with this distinction, though, can have bad effects for the concept of \"authority\" because it can easily be thought to belong more naturally with \"heteronomy\" than with \"autonomy.\" After all, if the doctor is the authority I will often follow his (or her) directives, not because I know them to be good but because the doctor has recommended them. Since it is clearly nonsense to say that I am being morally deficient in following good advice, the Kantian has a prima facie problem here, which he (or she) might try to address by saying that although I may not \"autonomously\" have decided to do this rather than that, I can still count as having behaved responsibly if I have autonomously decided to trust the doctor. This, however, does not solve the difficulty but simply pushes it back a step. Am I in a position to know that passing this particular examination really gives the doctor knowledge? How do I know that in accepting the \"authority\" of the Medical Board and its examination procedure I am doing more than accepting the local prejudice of my time and society? After all, there were formal and highly technical theology examinations in the Middle Ages, but no one now thinks those who passed those exams could tell us anything that would deserve to be believed or acted on. At each point in the regress the same kind of difficulty re-arises. The Kantian, then, will probably resort to saying that at some point the regress must stop and I must be able, or must have been able, to evaluate the authority-claim at that step \"autonomously\" (and I either did do that, showing myself to be a morally worthy subject, or failed to do that, thereby leaving myself open to legitimate moral criticism). Again, the use of the philosophical \"must\" should arouse suspicion. Why \"must\" there be such a point? The reason is that otherwise the theory with its sharp dichotomy between autonomy and heteronomy wouldn't be plausible. But perhaps the theory is not plausible. In the ancient world \"freedom\" was not construed as incompatible with the recognition of \"authority.\" Free men and free self-governing communities would obviously orient themselves on the model of famous men of the past and on the opinions and practice of wise contemporaries. Similarly they would recognise the importance of traditions, established practices, and \"unwritten laws\"; in Rome they would have special regard for the auctoritas of the Senate. Finally, if they had any sense, they would attempt to interpret and obey advice, commands, and warnings given by the gods through oracles or via other \"signs.\" We don't believe in divine signs\u2014am I sure that my disbelief is something I acquired completely \"autonomously\" and is not in any way a reflection of the tacit assumptions of my cultural context?\u2014but if modern conceptions of \"autonomy\" are incompatible with any other of these phenomena, then so much the worse for those conceptions. What sort of human life would it be which failed to assign an important place to respect for, and even a deference to, the judgement and exemplary practice of others? Reverence, respect, and trust are different from blind or coerced submission\u2014in fact real respect is arguably never \"blind,\" but that also does not mean it is always the result of the \"autonomous decision\" of an individual (in the Kantian sense). When Kant claims \"Selbst der Heilige des Evangelii mu\u00df zuvor mit unserem Ideal der sittlichen Vollkommenheit verglichen werden, ehe man ihn daf\u00fcr erkennt\" [\"Even the holy man of the Gospels must antecedently be compared with our ideal of moral perfection, before one recognises him \"], this is, as Hegel might have put it, one of those half truths which, if presented as the whole truth, is worse than a simple mistake. Of course, in some sense we have to be able to connect the life of a human being who is a candidate for being an ethical paradigm with our moral conceptions, but it does not follow either that we antecedently have absolutely fixed and determinate conceptions, as Kant seems to assume, or that we accept someone as exemplary only if that person corresponds in every respect to what preexisting conceptions we might have. Otherwise it would not be clear how moral development or change was possible, but refusal to learn or to be surprised is no sign of an especially strong or good character; it is more usually of some combination of ignorance, arrogance, and fear. The other \"half\" of \"the truth\" is: our \"ideal of ethical perfection\" is never simply our own, in the sense of being a completely autonomous creation from nothing, or in the sense that we have in every way adequately \"rationally assessed and tested\" every component of it. Not only have we in fact never done this, but this is not the description of a possible state of human affairs. Kant's claim about the \"holy man of the Gospels\" is for him the end of the discussion and the story. Seeing that claim in contraposition to its other half, however, should rather be seen as the beginning of serious discussion.\n\nThe \"Enlightenment\" ideal of an autonomous individual who restricts himself in his acting and judging strictly to that which he understands thoroughly, has rationally well-grounded views about, and has in his control does not describe a possible form of human life. The proper response to this is not simply to accept the station in life we have been assigned and the beliefs our local \"authorities\" deem to appropriate for us to hold. The thinkers of the Enlightenment may have connected the practice of \"criticism\" with a particular quasimetaphysics of \"freedom,\" but there is no particular reason for us to make this mistake. As Foucault once put it, we need to extract and retain \"the ethos of enlightenment\"\u2014reasoned investigation of claims\u2014from the \"dogma of Enlightenment.\"\n\nThe essays in this collection, all of which were written during the past five years, discuss a number of different issues that arise from this basic situation: What is \"authority\"? What is \"discipline\"? What is \"criticism\"? What is the relation of authority to the question of the \"meaning\" of human life?\n\nSome of the essays in this collection have appeared in print before: essays 1, 10, and 11 in Arion (fall 2009, spring 2012, and spring 2013, respectively); essay 2 in Cambridge Literary Review (issue 1\/2009); essay 13 in The Point (issue 2\/2010); essay 9 in Studies in Christian Ethics (25, no. 2 [2012]). Essay 6 I originally wrote in German, and a severely truncated version of that text appeared in Zeitschrift f\u00fcr Ideengeschichte (Heft IV\/4, winter 2010); this is a translation of a revised text based on that longer original.\n\nMy thanks for discussion of all the topics and essays in this volume to the members of the Philosophisches Forschungskolloquium and especially to Richard Raatzsch. Without the help of Hilary Gaskin I would not have been able to write any of these texts.\nA WORLD WITHOUT WHY\n1\n\nGoals, Origins, Disciplines\n\nIn 1894 Wilhelm Windelband, who was Professor of Philosophy at the University of Stra\u00dfburg, gave the annual Rector's Address to the assembled members of the university. He took as his topic the structure and classification of the sciences. It is superficial, he claimed, to try to divide the sciences by reference to their subject matter into sciences of nature on the one hand and sciences of spirit (or culture) on the other. A physical object like Mont Blanc or a species of plant or animal can be the subject of aesthetic analysis and evaluation, but such analysis is not part of natural science. Similarly, any human artistic activity has a psychological and eventually a neurophysiological or biochemical basis, but this does not make a study of the brain activity of Michelangelo while he was painting part of \"the humanities\" (as we would call them). Neither is it the case that there is some specific method or set of characteristic methods used by the natural as opposed to the cultural sciences (or vice versa). Precise observation is equally important everywhere, and the basic forms of logical inference and evidentiary argumentation are similar in all scientific disciplines. Nevertheless, Windelband argued, there is an important distinction between the two basic kinds of \"science\"; it is merely that the distinction is not in terms of methods or subject matter but in terms of goals or aims. Sciences, after all, are systematic human constructs, and most organized human activity is guided by some goal or other. We categorize things in different ways depending on our different purposes. A practical field guide to identifying things that fly in the night sky in a certain region of the earth might appropriately include both owls and bats, although according to another classificatory system that is widely used in biology, owls and bats do not belong very closely together because the first are birds and the second mammals. The field guide is not wrong to include bats and owls (though it would be wrong if it asserted that bats belong to the biological order aves or owls to the order mammalia). Similarly, a survival manual might perfectly reasonably group together some kinds of mushrooms and insects in one chapter, \"Things humans can eat,\" and distinguish them from a group containing poisonous mushrooms and other \"Things humans cannot eat\" in another chapter, even though this division cuts across recognized biological categories.\n\nOne reasonable human goal is to learn to deal with the world by recognizing the recurrent regularities it exhibits. Sciences with this goal Windelband called \"nomothetic.\" All mushrooms that look like this are poisonous, and if you eat them you will become very ill indeed and perhaps die. On the other hand, as human beings we are interested not only in laws, regularities, and recurrent features of the world but also in certain striking singularities. So, for instance, we are interested not just in the ways Mrs Dalloway is one more novel exhibiting the features all other novels exhibit but also in what makes it different from other novels or even unique. An account aimed at exhibiting the singularity of an object or event was to be called \"idiographic.\"\n\nThe period of the Second German Empire (1871\u20131919) was a Golden Age for the discussion of classificatory problems. This is probably not unconnected with certain aggressive imperialist ambitions that were widely entertained by the political classes of the time, which in turn were mirrored in the dominance of neo-Kantianism. Kant was notoriously almost pathologically obsessed with intellectual (and moral) tidiness, with making sharp and clear distinctions that would allow one to divide the world up into easily cognizable objects and sectors. For the neo-Kantian the question of the autonomy, distinctiveness, and principles of division of different kinds of human activity was of the very greatest concern. Sometimes these were nothing but turf wars, but sometimes more substantive issues were in play. Thus the discussion of economics between the so-called Historical School and the followers and associates of Carl Menger had ostensibly to do with the role that institutions and history should play in the study of economics, but that disagreement clearly mirrored differences in the conception of the way economic development would, could, or should take place. Could the industrial structures of Manchester simply be replicated in Germany, or would economic development need to take a very different course given the differences in history and institutions between Germany and Britain?\n\nWindelband, of course, writing in late nineteenth-century Germany, did not have at his disposal the concept of \"the humanities\" but would have had to speak of the Geistes- und Kulturwissenschaften. Actually his Rectoral Address is entitled \"Geschichte und Naturwissenschaft,\" though at one point he also refers to les belles lettres. Even if one cannot take seriously Windelband's specific theory about the nomothetic and idiographic, his point that what is at issue are disciplines as human constructs, not simply unvarnished, contrasting blocks of material, is well taken. When we talk about the humanities we are talking about a set of disciplines, human constructs, and we can undertake the construction of these disciplines in a variety of different ways, as well as classify the kinds of constructs that result in a variety of different ways.\n\nWindelband's two basic questions, then, are what sorts of things do we as humans generally want to know about, and why? One way of trying to answer these questions is by observing that there is a strong human tendency to want to know about the origins of things, as if this allowed one a special access to understanding them. What is probably the oldest extant document of Western literature provides several instances of this tendency. In the Iliad (book 6, lines 119\u2013236) Homer describes an encounter on the field of battle between two warriors who do not know each other. Before they fight, one, the Greek Diomedes, son of Tydeus, asks the other, who turns out to be an ally of the Trojans named Glaukos, who he is (\u03c4\u03af\u03c2 \u03b4\u1f72 \u03c3\u03cd \u1f10\u03c3\u03c3\u03b9, \u03c6\u03ad\u03c1\u03b9\u03c3\u03c4\u03b5, \u03ba\u03b1\u03c4\u03b1\u03b8\u03bd\u03b7\u03c4\u1ff6\u03bd \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03c9\u03bd;). Glaukos replies by embarking on a genealogy reaching back five generations, which contains a series of elaborate narrative accounts of what his father and grandfather and some of his ancestors did at various points in their lives:\n\nHigh-hearted son of Tydeus, why ask of my generation [\u03c4\u03af\u03b7 \u03b3\u03b5\u03bd\u03b5\u1f74\u03bd \u1f10\u03c1\u03b5\u03b5\u03af\u03bd\u03b5\u03b9\u03c2;]?\n\nAs is the generation of leaves [\u03b3\u03b5\u03bd\u03b5\u03ae], so is that of humanity. The wind scatters the leaves on the ground, but the live timber burgeons with leaves again in the season of spring returning. So one generation [\u03b3\u03b5\u03bd\u03b5\u03ae] of men will grow while another dies. Yet if you wish to learn all this and be certain of my genealogy:\n\n[\u1f44\u03c6\u03c1\u1fbf \u1f10\u1fe2 \u03b5\u1f30\u03b4\u1fc7\u03c2 \/ \u1f21\u03bc\u03b5\u03c4\u03ad\u03c1\u03b7\u03bd \u03b3\u03b5\u03bd\u03b5\u03ae\u03bd: literally, \n\"that you might know well our generation \/ race \/ lineage\"] \nthere are plenty of men who know it.\n\nThere follow here about fifty lines describing the trials, vicissitudes, and heroic exploits of his various ancestors (Ailos > Sisyphos > Glaukos (I) > Bellerophontes > Hippolochos). Glaukos ends his genealogy by speaking of his father:\n\nBut Hippolochos begot me, and I claim that he is my father; he sent me to Troy, and urged upon me repeated injunctions, to be always among the bravest, and hold my head above others, not shaming the generation [\u03b3\u03ad\u03bd\u03bf\u03c2] of my fathers, who were the greatest men in Ephyre and again in wide Lykia.\n\nSuch is my generation and the blood I claim to be born from\n\n[\u03c4\u03b1\u03cd\u03c4\u03b7\u03c2 \u03c4\u03bf\u03b9 \u03b3\u03b5\u03bd\u03b5\u1fc6\u03c2 \u03c4\u03b5 \u03ba\u03b1\u1f76 \u03b1\u1f35\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b5\u1f54\u03c7\u03bf\u03bc\u03b1\u03b9 \u03b5\u1f36\u03bd\u03b1\u03b9].\n\nOne might think this is just an instance of puerile boasting, which of course it is, and we might therefore put it aside as irrelevant. In this context it is, however, perhaps not as off-topic as it might seem. This is a world in which a few highly individuated warriors stand out\u2014often literally, it seems, standing in front of a large, anonymous mass of fighters (the \u03bb\u03b1\u03bf\u03af). By reciting his pedigree in such detail, Glaukos is imparting important and relevant information about his back-ground and probable training, as well as giving a kind of performance. Perhaps he is trying to raise his own spirits and to intimidate his opponent, signalling that he has no intention of slinking away, but he is also in some sense actually changing the situation. Having the pedigree Glaukos has means in this context that one is likely to be a person of a certain sort, interested and skilled in warlike pursuits, and brought up to try, as Glaukos says, \"to be always among the bravest\"; announcing that pedigree in a situation of public confrontation means identifying oneself in a certain way and thereby making it impossible to withdraw anonymously, without loss of face, into the mass of \u03bb\u03b1\u03bf\u03af.\n\nWhen Diomedes hears Glaukos's stories about his ancestors, he realizes that his own grandfather and Glaukos's grandfather were hereditary \"guest-friends,\" and so the two warriors decide not to fight each other after all and instead exchange armour in token of the renewal of this hereditary relation of guest-friendship. Determining who they are via their respective genealogies has important normative consequences for how they think it appropriate to treat each other. The question of who one's opponent is seems in this world a perfectly natural one to ask, and it also seems natural to answer that question, even in the heat of battle, by giving a genealogy and a series of narratives. Who I am and what my essential properties are are thought to be connected with my origin, which is given by a genealogy. The genealogical narrative is assumed to disclose something important about my essential powers, obligations, and entitlements. What we would call \"natural\" and what we would call \"social\" properties are not distinctly separated. Is the question of \"origin\" really so natural as all that?\n\nAnother early and important example of the human concern with origins is Hesiod's great cosmological poem Theogonia, which explains who a large number of gods, goddesses, and so forth are by giving their genealogies. Giving the genealogy here is intended to be explanatory; to know who and what a given god is means to be able to locate him in the sequence of divine generations, and this also gives one at least some minimal ability to address him, to know what to expect of him and how to deal with him.\n\nOddly enough, this genealogical interest in origins does not seem to have developed in the ancient world into an intellectual, hegemonic, formal discipline. Genealogical inquiries do not seem to have been one of the major direct ancestors of anything we moderns would recognize as full-blown \"history\" (or, for that matter, what the Greeks called \"history,\" namely systematic empirical enquiry of any kind), and history itself seems never in the ancient world to have attained the exalted status it occupied in some parts of Europe in the modern period. Rather, at a relatively early age the genealogical impulse was crushed to death and pulverized between two huge millstones, which to some extent represented theoretical competitors to genealogy. These two competitors were rhetoric and philosophy.\n\nTo start with the first of these, rhetorical culture was focused on producing persuasion through the medium of correct, aesthetically attractive, and effective speaking. Rhetorical training, then, meant the study of language in all its aspects but also the inculcation of certain aesthetic, moral, and political values that were considered part of being a persuasive speaker. In discussing ancient rhetoric it is important not to lose sight of two important facts. First of all, rhetorical training was in the first instance eminently vocational and practical, not abstractly speculative or merely ornamental. The ancient Greeks sometimes distinguished three types of persons who went to the Olympic Games. First there were those who went to compete\u2014to run, jump, throw the javelin, race their chariots, or pummel one another into insensibility with their fists. Then there were those who went to sell things; in the era before corporate sponsorships they were the objects of an entirely appropriate, almost universal contempt. Finally there were those who went to watch. This third group, the spectators, were the archetypal \"theorists.\" Observing, especially observing a highly public competition, can easily come to be associated with commenting on the performance of individual competitors and then with a kind of rudimentary criticism. Ancient rhetoricians were either direct participants in the rough-and-tumble of ancient politics, speaking in public assemblies and trying to get the better of other speakers by defeating their proposals, or competitive performers, and so, in the terms of this comparison, they were more like Olympic athletes than like spectators. Rhetoric as a disciplined skill seems in fact originally to have had a close connection with proto-democratic politics, where such skills would for obvious reasons be particularly valued.\n\nAlthough the basic structure of the discipline of rhetoric and its teleological orientation were practical and political, this did not exclude the development of some kinds of theoretical analysis and relatively disinterested criticism as a subordinate part of rhetorical culture. Thus in several early Platonic dialogues, particularly the Protagoras, we see Plato making fun of teachers of rhetoric who give way to an obsession with correct linguistic usage, subtle semantic differences, and a kind of morally edifying but, Plato claims, fundamentally insubstantial and unsound literary criticism.\n\nThe second important fact is that this rhetorical training was not, contrary to the propaganda of Plato, originally just a technique for unscrupulously manipulating people. To put it in somewhat later terminology, the original project of rhetoric was to teach something both inherently valuable and instrumentally valuable. It was inherently valuable because it made those who learn and practice it good, beautiful, and self-confident (and these are values in themselves), and it was instrumentally valuable because it was useful in helping one get one's way politically. It was precisely this orientation towards human improvement, not just effective instrumental manipulation, that made some of the original forms of rhetorical training such an easy target for Plato's criticism. Plato was terrified by what he took to be the potentially subversive (\"democratic\") political possibilities of rhetoric: anyone who could pay the fees, regardless of their genealogy and family connections, could learn the art of speaking persuasively from professional teachers of rhetoric. Nevertheless, Plato couches his criticism in epistemological terms. If the study of rhetoric really makes people better, he argues, then surely its teachers must be able to explain what the human good is and how the study of rhetoric conduces to helping people attain it. This is part of Plato's general argument that you cannot be performing an activity well unless you can explain why you are doing every component part of it in the way you are, and you can't do that unless you have the correct general theory. Since rhetoricians were basically inculcating skills in practical public speaking, secondarily developing certain ways of interpreting literature, and had some theories about some things (such as the correct order of the parts of a speech) but no general theory of the human good, Plato's conclusion is that they did not really know what they were doing and hence could not be doing it well, except, as he condescendingly sometimes adds, by accident (\u03b8\u03b5\u03af\u1fb3 \u03bc\u03bf\u03af\u03c1\u1fb3, Ion 542a).\n\nSeen retrospectively from the vantage point of the early twenty-first century, there seems nothing extravagant in the claims of professors of rhetoric that they were making those whom they taught \"better people\"; after all, they were making them more able to participate effectively in public debate and in the common political life of their respective cities, and that might perfectly well be considered a good. Isn't Plato's Protagoras in some sense right to say that a human capable of taking part in sociable common life is better-off than one condemned to a life of solitude, isolation, and silence? Similarly, there seems nothing outrageous in claiming that one is benefitting people (and their cities), even though one cannot specify by reference to a general theory in exactly what way that is occurring. Given our complete inability despite over two thousand years of effort to agree on a theory of \"the good\" that would have satisfied Plato, it seems highly rash to continue to claim that possession of such a theory is a precondition for any stable form of good practice.\n\nPlato's criticism does not seem to have had much immediate effect; rhetoric was simply too useful for that. What killed off old-style rhetoric was the gradual but cumulative marginalization in the Roman Empire of the political bodies in which free speech was permitted and could be politically effective: the popular assemblies and then the Roman Senate.\n\nIn the context of the study and practice of rhetoric, genealogy, historical enquiry, or the study of origins might have had at best a subordinate place. Individuals like Varro (and the Emperor Claudius) might have had idiosyncratic antiquarian interests, and history of a sort had some standing as a source of exempla for virtuous or vicious action, so what one gets is at best something like what one finds in the first few books of Livy, and, apart from whatever concerns one might have about the accuracy of the account, the heavy moralizing quickly gets rather cloying.\n\nThat brings me to the second of the two huge millstones that ground away at the interest in origins: philosophy. The standard doxographic account of the origin of philosophy, which goes back at the very least to book 1 of Aristotle's Metaphysics, runs as follows: First, there were myths and poetic cosmologies, structured as narratives or genealogies, like that of Hesiod; then, at some point Thales of Miletus initiated philosophy precisely by breaking with mythic and genealogical accounts and by claiming that there is a single \u1f00\u03c1\u03c7\u03ae for everything in the world: what he calls \"water.\" \u1f08\u03c1\u03c7\u03ae comes from the verb \u1f04\u03c1\u03c7\u03c9, which means to make a beginning, initiate, take the lead, and then, by a natural association, control or rule. The dancer who makes the beginning in some sense sets the tempo and determines the nature of the dance. For Homer, he who \"made the beginning\" of Glaukos is first of all his father, then his paternal lineage; and that is what determines what and who this person is. But with Thales, \u1f00\u03c1\u03c7\u03ae seems to begin to take on the meaning of determining (abstract) principle, while leaving behind the idea that there is an interesting historical sequence that can be traced back to some initiatory moment. I don't know exactly how Thales thought water constituted the nature of the world, and I think it likely that this is not merely a personal failing of mine. Rather, I think \"we\" don't know\u2014the available information is just not adequate\u2014and it is even conceivable that Thales himself did not know exactly how he meant various of his claims. In a sense, as Hegel recognized, the whole history of philosophy is nothing more than an attempt to clarify what kind of \u1f00\u03c1\u03c7\u03ae there can be for the world and in what relation it can stand to the cosmos as a whole and to individual objects and events.\n\nThales stood at the very beginning of this long historical process of clarification and didn't perhaps have views articulated as highly as those of later philosophers. The pursuit of clarity is in general a good thing, but the indiscriminate pursuit of clarity is a vice and a serious obstacle to the proper understanding of large parts of human life. This is particularly the case when considering the philosophical past. So the obscurity of Thales's actual theory is not surprising and in fact not so important; whatever specific theory he had, if indeed he had any, did not recommend itself sufficiently to any of those of his successors whose work survives for them to transmit it to us. Nevertheless, later thinkers have found in him a kind of origin for philosophy. Hegel remarks that when Thales says \"All is water,\" he is neither specifying an origin the way Glaukos is nor making a statement with the same structure, import, or \"grammar\" as the statement \"All the fish in this barrel are cod\" when it is made by the holder of a market stall. Also, \"water\" in this statement does not refer to the colourless liquid one finds filling the Aegean Sea but to some kind of speculative principle that stands in an unspecified relation to the well-known fluid. \"All is water\" makes a totalizing claim, the claim that everything in the world can in some way be accounted for by reference to a single abstract principle. Simply having that thought, even if one was not able to elaborate it any further, was, Hegel claims, enough to initiate what we have come to know as philosophy.\n\nWe have encountered two different senses of \"origin.\" First, origin in the sense of historical origination, as in the case of Glaukos, and second, origin in the sense of some explanatory scheme or principle that provides a unitary abstract account or rationale, as in the case of Thales.\n\nIn principle one can investigate the \"origin\" of something in either or both of the two senses. At some point in prehistory humans put two pieces of wood or stone together in a particular configuration and used them to beat pieces of material into shape. As we would say, they \"invented\" the hammer. One can, however, tell two slightly different stories about this. First, one can tell the story of origins of the hammer, for instance by visiting the Mus\u00e9e de l'homme in Paris and looking closely at the exhibited artefacts. One could also\u2014and this would seem to be a second way of proceeding\u2014discuss the \"origin\" of the hammer by looking at continuing configurations of human desires, goals, needs, and the conditions under which they are or are not attainable or capable of being satisfied\u2014by looking at the problem to which the invention of the hammer was the solution. One might\u2014or might not\u2014think that this second story gives one a better understanding, and in the second account one might think that the specific history is not of great interest. Given the problem (and that means given the assumption that human desires and needs are more or less invariant and environmental conditions more or less uniform), with enough ingenuity someone was eventually going to hit on the solution that consisted in inventing something very much like the hammer. The two stories are, however, perfectly compatible. Once the hammer is there it can, of course, have other uses in addition to its original one. For instance, certain hammers might be so beautifully made that they become objects of aesthetic contemplation. This new function can in some cases survive atrophy of the original need. To be sure, it is not clear to what extent Glaukos would accept that one could equally well tell two distinct stories, parallel to those concerning the hammer, about him. An account of his origins, he would likely think, would have to deal with the specific details of his history. There might be something about being the descendant of those specific ancestors that makes him the object of the kind of individualizing interest that finds its appropriate expression in an \"idiographic\" narrative. He might think it makes a huge difference that his ancestor is Bellerophontes, not some other equally heroic figure, if only because Bellerophontes was the guest-friend of Diomedes' grandfather, and thus Glaukos and Diomedes should not fight each other. What Glaukos is seems to be comprehensible only relative to his particular history and that of his ancestors.\n\nI have also been discussing \"origins\" in the context of two different inquiries. First, I have been asking what in fact are the origins of certain human disciplines\u2014from what matrix of human purposes, social pressures, and contingent occurrences they arose; the motivation for asking this question is to get a better understanding of these disciplines. Second, I have been recounting certain traditional views held by the practitioners of one of these disciplines, \"philosophy,\" about what should count as having a satisfactory understanding of any important human phenomenon. The overwhelming traditional view among philosophers is that one only has an adequate understanding of a phenomenon if one has a general theory about or an abstract rationale for it. So if one wants to call a search for the best understanding of something an enquiry into its \"origin,\" the term \"origin\" has its second, not its first, sense\u2014that is, a good account gives a unitary principle, and certainly not anything that looks like a sequential narrative.\n\nKant, for me, has always summed up what is most wrongheaded and retrograde in modern philosophy. He follows the philosophical tradition in rejecting any positive role for history in philosophical reflection. Rather, he construes the task of giving an origin for philosophy as that of providing a unitary abstract rationale for it and tries to connect that with a set of universal and invariant human interests. These interests require that humans try to attain a unitary view of the world as a whole. Human agents, Kant thought, had to act in the world, and this required them to make a series of assumptions about how their world was constituted. Several nineteenth-century neo-Kantian philosophers developed lines of thought that Kant had marked out into a theory of what came to be called the \"metaphysical need,\" which was the need for a single universal scheme that would make all things make sense. This metaphysical need was generally construed by those who believed it existed as a demand rooted in continuing aspects of human nature for the kind of cognitive and normative orientation that could only be given by something like a religious or a systematic \"philosophical\" worldview.\n\nKant's most important successor, Hegel, represents a significant break by virtue of his attempt to think about philosophy in a more inherently historical way. In one of his early essays he says that \"the need for philosophy\" arises not ahistorically for all rational practical agents by virtue of a metaphysical need they have but under highly specific social circumstances, namely when \"life has lost its 'unity.' \" This raises the un-Kantian possibility that fascinated several later philosophers in the Hegelian tradition, among them Marx, namely the idea that in a satisfactory society, from which certain kinds of deep-seated conflicts were absent, philosophy (along with religion) would be superfluous and would disappear. Of course, even in such a basically harmonious society there might be a pale successor-discipline to the antique magnificence of \"philosophy,\" which might, for example, take the form of straightforward attempts to get an overview of the state of our knowledge or even suggestions for minor improvements in our social arrangements. One line of criticism of what are sometimes called \"positivist\" strands in twentieth-century philosophy consists in claiming that positivists propound methods of direct observation and theory construction that would in principle be cognitively perfectly appropriate in a fundamentally harmonious society; however, by advocating the exclusive use of such methods in repressive and conflict-ridden societies like ours, they tacitly contribute to diverting attention from fundamental social antagonisms.\n\nHowever, once a connection is established between certain forms of enquiry or intellectual disciplines, such as philosophy, with interests or needs, the door is open to further subversive thoughts. It might be the case that some particular conceptual or theoretical invention itself creates psychological needs which, once they are in existence, are difficult to get rid of. This is the model that Nietzsche uses for Christianity. It develops a complex set of practices and institutions that arise for perfectly understandable but utterly contingent and perhaps slightly disreputable reasons, such as human weakness and resentment of that weakness, but which, once they get themselves established, generate from within the new set of human needs of which Christianity is the fulfillment. The salvation that Christianity offers is, arguably, not for everyone but for those who need it. Since salvation means in the first instance salvation from sin, it would seem that the Christian kerugma\u2014the message that sins can be erased and salvation is at hand\u2014would have no purchase on those with no sense of sin. Missionaries have special difficulty with people lacking a sense of sin, so they may need to create one. This is completely different from the case of the hammer, in that even if the hammer eventually acquires new functions, such as serving as an object of aesthetic appreciation, the original problem, that of beating things into a more serviceable shape, can be said to have existed before the hammer was invented. On Nietzsche's reading this is not the case with Christianity. Christianity did not in the first instance cure the preexisting problem of sin but attempted to cure a completely different (and, Nietzsche thinks, virtually incurable) other condition, namely a historically specific, widespread form of human debility. Christianity, as he puts it, \"turned sick people into sinners.\" This means that Christian institutional life can not only inculcate a belief that one is a sinner but actually produce people whose somatic constitution is correctly described as \"sinful.\" The model here is addiction to drugs. Those who believe or feel themselves to be \"sinners\" think they need the consolation Christianity provides; those who really have been turned into sinners really do need that consolation, in the way the addict needs the drug. The only difference is that whereas we tend to assume drug addiction is \"in principle\" reversible (i.e., given sufficient willpower and a facilitating environment), Nietzsche seems to think that for most people the changes introduced by Christianity are effectively irreversible. Still, this is compatible with thinking they are radically contingent.\n\nUnfortunately, this whole Hegelian project of doing philosophy in a historically informed way has recently fallen out of fashion for reasons that are too complex and obscure to be presented uncontroversially in brief compass, but a significant part of the reason is likely to be the fear that if one embarks on this path, one will eventually be confronted with the unpalatable alternative of either accepting a highly baroque and counterintuitive metaphysic of the kind Hegel himself advocated or losing one's bearings in the face of the teeming variety of historical forms of thinking and acting. The second of these two fears is often expressed as anxiety in the face of the threat of \"relativism.\" Loss of the absolute moral certainties given by Christian or Kantian attitudes can clearly give rise to vertigo, but perhaps the appropriate reaction to that is to show that the purported threat of \"relativism\" is illusory and to treat the vertigo as mildly pathological.\n\nEarlier I told the usual story about the origin of philosophy. However, it is notable that Plato gives a different story about the origin of philosophy, which does not begin with Thales. (The first extant occurrence of the word \u03c6\u03b9\u03bb\u03bf\u03c3\u03bf\u03c6\u03af\u03b1 is in Plato, and it is not out of the question that he in fact coined the term.) This story is repeated in a very prominent place by Cicero. Philosophy starts with Socrates, and it actually gets going\u2014has its real origin or \u1f00\u03c1\u03c7\u03ae\u2014when he turns his back on speculation about the natural world and turns to ethics. In Plato's Phaedo, Socrates says that when he was young and immature he concerned himself with what the Greeks called \u03c6\u03c5\u03c3\u03b9\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 (that is, speculation about nature), but now that he has become an adult he is no longer interested in the structure of the universe but in how things need to be to be \"for the best.\" As Cicero puts it, Socrates \"brings philosophy down to earth\" (from inspection of the skies to the \u1f00\u03b3\u03bf\u03c1\u03ac). For Plato, this seems to mark a kind of beginning; however, Plato did not follow Socrates in his turn away from cosmological speculation but developed a highly peculiar theory about the cosmological basis of values: his theory of \"forms\" or \"ideas.\" This notion of a conjunction between cosmology and ethics developed especially long legs when it was taken over and adapted by Christianity because Christians construed their god as both creator of the natural world and moral legislator. I simply note that it is highly peculiar that these two distinct things\u2014speculation on the nature of the universe on the one hand and moral and political philosophy on the other\u2014get put together as one subject (philosophy).\n\nOnce a unity like \"philosophy\" gets itself established, especially institutionally established, for instance in schools or universities, there is an almost irresistible tendency to find or create a single unitary genealogy for the enterprise, which means both a unitary history and a unitary, noncontextual goal. There is a compulsion to make up a single rationale and project it back onto people who are then retrospectively declared to be \"precursors.\"\n\nNietzsche has a notorious line in criticism of all forms of analysis by reference to purported unique \"origins.\" He is especially scathing about attribution of the origins of continuing institutions to individual heroic founders: philosophy founded by Thales or Socrates, Christianity founded by Jesus. There was never a single origin for anything with any continuing historical significance. What looks like a unique origin always disperses into a multiplicity. All persisting institutions and practices have unsurveyable multiple sequences of completely contingent ancestors, \"contingent\" meaning that there was no logical or rational necessity in their conjunction. One can in principle trace these ancestors back indeterminately into the past, and the farther back one goes the more such ancestors there will be. At some point one will not be able to go farther, but that is merely, as it were, an accidental limitation of our cognitive powers or the evidence that happens to be available.\n\nIn fact, then, the account of the \"origins\" of philosophy I have given is a gross oversimplification. The history of the \"origins\" of philosophy is not simply one in which two different kinds of inquiries (physics and ethics) come together. There is at least a third more or less independent element: a concern with forms of argumentation, logical thinking, and the validity of inference. Plato, in composing his dialogues, makes great play of this concern with correct inference, a part of his thought that does not seem influenced either by the earlier speculators about nature (such as Thales) or by Socrates' moralizing but derives from several more obscure sources, including the so-called Eleatic philosophers (Parmenides) and, to some extent, the sophists. The influence of this tradition on Socrates was profound. Plato, to be sure, is anxious to distance Socrates from the sophists as much as possible, so anything he took over from them has to be rather carefully hidden, or Plato has to explain at great length in what way the Socratic version of the concern for correct speaking is different from that of the sophists.\n\nIt would not be surprising if the discipline of \"philosophy\" depended for its continuing vitality on the tension between these different poles\u2014between interest in the structure of the natural world, interest in forms of argumentation, and interest in \"what would be for the best\"\u2014so that without this tension the practice as we know it could not continue to exist, but would break up into individual parts, each of which would go its own way as a distinct discipline.\n\nThere seems, in fact, no reason anymore why those concerned to understand the structure of the natural world should ex officio also have a nontrivial interest in which political institutions or which works of art are best or in formal structures of speech and argumentation. \"Philosophy\" could dissolve itself into physics for the study of nature; linguistics, rhetoric, and mathematics for the study of speech, argumentation, and formal systems; and politics, belles lettres, and social psychology for the study of \"what would be for the best.\" I strongly suspect that a radical dissociation of these interests has already occurred, although many people have not noticed it yet, and the discipline of philosophy in its present configuration is held together only by a combination of historical inertia and a sentimentalized attachment to a mostly illusory image of a glorious past. As a purported single subject, philosophy seems unlikely to last. If the various components really have as little to do with each other as they nowadays seem to, this may be no bad thing.\n\nThis story is a tempting development of Nietzschean themes about the artificiality of what has its origin in a series of contingent encounters of originally diverse and heterogeneous elements. One should not, of course, conclude from the fact that certain disciples have a contingent history relevant to understanding their present form that just any old available elements could randomly be put together as a \"discipline,\" or that all conjunctions are equally good. For example, the Roman writer on architecture Vitruvius, when discussing the kind of training a good architect needed to have, states that such a person had to be especially well trained in music. The reason for this was that an architect was expected also to be a military engineer, and a military engineer would be called on to build and activate catapults, and catapults would shoot straight and thus be effective only if the tension in the ropes providing their motive power were equal. Ropes, I suspect, were not industrially produced to a high level of uniformity in the ancient world, and so the only way to tell whether the tension in two improvised ropes was equal was to pluck them and see whether they sounded the same note. If they did, the tension was equal; otherwise not. So in an ideal Vitruvian university architectural training would include bridge-building, ballistics, and music as forming a \"natural\" unit. This unity was \"natural\" in that the conjunction gave prospective architects very good preparation for tasks with which they would be confronted. One could not, that is, equally well have taught them cooking, ballet, and viticulture. On the other hand, if most warfare in the Roman world had been naval, it might have seemed more obvious to group ballistics and music with navigation and astronomy, not with bridge-building. When gunpowder arrives, music becomes irrelevant and can be expected to drop out of the military curriculum. So there are things that \"go together\" better than other things, but it is not at all clear that the idea of \"going together\" makes sense independent of some at least minimal reference to historically specific human projects, valuations, and purposes.\n\nIn addition, it is still an open question whether a conjunction that has been given such a synthetic unity\u2014for whatever reasons and in whatever way\u2014can also just as easily split up into its component parts. One might be tempted to argue that since traditional philosophy arose as a purportedly unitary pursuit from a series of accidental conjunctions occurring over a period of two centuries or so (roughly from Thales to Plato), there is no reason why it cannot also come apart again. This cannot be the whole story because of considerations like those mentioned in discussing the hammer or Christianity\u2014if, like the hammer, philosophy acquired some further functions that could best be discharged by a unitary discipline, or if, like Christianity, it generated from itself needs that only it could satisfy. If there are such hidden, new essential functions, what are they?\n\nIf this is true about philosophy, it is true in spades for \"the humanities.\" This term, in the sense in which it is used today in English-speaking countries, seems not to go back much further than the middle of the nineteenth century, and to say anything substantial about why it came into use and why it seemed plausible to think it had a referent would require more interest in and knowledge of nineteenth-century Britain than I possess. It seems plausible to expect that at least one of the immediate pressures operating here resulted from the demands of education, especially higher education. What seems, however, also rather clear, even to an outsider, is that the idea that there was one discernible, collective thing, \"the humanities,\" a more or less unitary set of subjects or disciplines that belonged together, required a highly constructive act of conceptualization, putting together various existing, disparate things that had not antecedently been thought to have anything particular to do with each other. The resulting synthesis was composed of various rather debased bits of detritus from the ancient world, with a particularly high concentration of bits of the ancient rhetorical tradition put together with bits of history and philosophy and some parts of the nonverbal arts. Again, from the fact that it was constructed, it does not follow that it was random in the sense that any collection of preexisting skills, forms of knowledge, and practices could equally well have been put together; the historical account is in part directly to show that that is not the case.\n\nIn light of the above remarks, the relatively artificial nature of the conjunction called \"the humanities\" ought no longer to seem surprising. It is in any case no news that all human classificatory schemes are partly structured by wider forms of human valuation and human purposes. One might with good reason think that much of value\u2014many important kinds of knowledge\u2014can relatively easily survive large-scale shifts in our way of classifying and organizing disciplines. To recur to Windelband's terminology, we will retain our \"idiographic\" interest in the Peloponnesian War (and in Thucydides' account of it), and much\u2014though possibly not all\u2014of what we find of the greatest interest will continue to come to representation whether this conjunction of events, actions, and words is treated as part of Altertumswissenschaft, of \"classics,\" or of \"history.\"\n\nThe proliferation and dispersion of new subjects and disciplines, combined with changes in the way we live and in our attitudes and dominant concerns, can be expected to render implausible our accustomed way of organizing academic subjects. There is nothing historically unprecedented about this. Foucault closes his study of what he calls the human sciences, Les mots et les choses, by comparing their central organizing conception, \"man,\" with a face drawn in the sand that is about to be washed away by the incoming tide. The face may be gone, but provided the sand remains, we have no reason for more than transitory twinges of grief, and certainly no reason for deep melancholy.\n\nIn conclusion, I suggest that if the general perspective on our historical situation that I have tried to sketch here is correct\u2014if philosophy in fact is on the point of dissolving into a number of different constituents and has long since lost any organic or systematic connection as a discipline with the so-called humanities, and if a similar fragmentation of the humanities themselves is under way\u2014then the whole question of giving some general account of the role of philosophy in the humanities doesn't make sense. Neither of the two purported entities has the requisite stability to admit a useful investigation of this question. At the moment, the relation between what is called philosophy and what are called the humanities is a matter of idiosyncratic associations between certain individuals, which are so contingent they border on the whimsical\u2014the odd Professor of Ethics who also happens to have read Greats at Oxford rather than PPE, or the don who works professionally on some aspect of the philosophy of language but also happens to be devoted to the ballet. Whether or not this dissociation is a temporary phenomenon, and if it is not, what new structures will eventually emerge, are matters of speculation, but since such controlled speculation was once one of the things philosophers used to do, I would suggest that if there is no ecological or economic catastrophe\u2014or something comparable that radically overturns, or even puts an end to, our organized intellectual life\u2014dispersal of the kind I have described is likely to be a continuing feature of our landscape for the foreseeable future.\n2\n\nVix intellegitur\n\nThe title of this essay is taken, with appropriate grammatical modification, from a comment by Cicero on Thucydides that has always fascinated me. In one of his treatises on the art of rhetoric Cicero is discussing the proper style to be adopted by the public speaker, and he discourages the aspiring orator from imitating the speeches in Thucydides' history. Thucydides, he admits, is good at describing battles and grand public actions, but this narrative style is not directly transferable to political or forensic oratory, and as far as the elaborate speeches that Thucydides includes in his work are concerned: \"Ipsae illae contiones ita multas habent obscuras abditiasque sententias vix ut intellegantur; quod est in oratione civili vitium vel maximum.\" [\"Those speeches contain so many obscure and recondite formulations that they are scarcely to be understood, which in a political speech is about the worst defect there is.\"]\n\nWhat exactly does it mean\u2014if indeed it means one univocal thing at all\u2014to say that a speech or a text is unclear, scarcely to be understood or hard to comprehend, and what exactly is supposed to be wrong with obscurity? One might think that this is a rather foolish question because isn't it self-evident that lack of clarity is a defect? My ultimate aim in this chapter, however, is to question this assumption and consider whether there are contexts and occasions in which and on which obscurity is not a deficiency, even a deficiency that might be pardonable or unavoidable, but something with positive value.\n\nObviously when Cicero discusses Thucydides' speeches he does not mean that they are what we would call completely incomprehensible, as a text in Chinese or Hungarian would have been had it been presented to him. Cicero (106 BC\u201343 BC) was not a native speaker of Greek or a contemporary of Thucydides (c. 458 BC\u2013c. 400 BC), but he lived only a couple of hundred years later, had resided for a time in Athens, and was a fluent speaker of what was at the time still a living language. I take it also that when Cicero says that the speeches in Thucydides are obscure, he does not merely mean that they are obscure to a Roman audience of the first century BC, or even to a Greek-speaking Roman of the first century BC, but that these speeches, he supposes, would also have been obscure to a fifth-century BC Athenian audience. After all, Cicero's book is a practical guide to his Roman contemporaries interested in becoming skilled orators, so the most reasonable construal of his advice is: \"Don't be as difficult to comprehend by Latin-speakers in c. 50 BC as Thucydides would have been to speakers of Attic Greek in c. 400 BC.\"\n\nI may misconstrue or fail to understand fully (or at all) what is perfectly clearly expressed because of fatigue, poor concentration, weak hearing, preoccupation, or any one of a number of other causes. The cases in which we are interested here, however, are ones in which there is no special explanation that refers to some specific fact about one individual person (rather than some other) and makes it the case that that particular person finds it difficult to understand the speech or text in question. One way in which that could be the case would be if what you said to me was \"inherently\" misleading or obscure or hard to follow, that is, if anyone\u2014within limits perhaps, for instance, \"anyone who was not a mind-reader or who did not have some other special form of access to your intentions, your past history, your other beliefs, and so on\"\u2014would be likely either to be puzzled by or to misconstrue it.\n\nDoes the notion of \"inherent\" obscurity, or its correlative, \"inherent\" clarity, make sense? Of course, if it is no more than an empirical, sociological claim\u2014some things I find it hard to grasp; some things you find it hard to grasp; some thing we both, and most of the people in some imagined reference group, would find it hard to grasp\u2014it is unobjectionable, but it is then unclear what its significance is, or indeed whether it even had any deeper significance. In everyday interaction and language-use what counts as clear and what as obscure depends very much on a highly variable context. \"It's down past the Catholic church\" is perfectly clear if I am talking to a neighbour, who will know that although there are various Catholic churches in Cambridge the one habitually used as a common point of topological reference is the one with a large spire at the corner of Lensfield Road and Hills Road; it will be likely to be obscure to someone who does not live in Cambridge. It isn't that, as one says, \"examples of this kind can be multiplied ad libitum\" because that suggests that there are special \"examples\" and thus that there could also be non-examples. Rather, I think this context-dependency is a universal phenomenon, except in the very few areas of everyday life that have been invaded by formal or technical usages, such as discussions of the legal requirements for proper tenancy, inheritance or conveyancing, medical prescriptions, or the naming of the parts and modes of functioning of complex machines like computers. These are isolated islands that depend for the fixity and univocacy of their meanings on being embedded in larger social mechanisms for enforcing strict uniformity, and when the appropriate mechanisms break down or malfunction, as they regularly, and eventually always, do, the islands are submerged again in the sea of everyday speech. These exceptions are exceptions because we make them that way through highly artificial means, which, however, will work only in specific local contexts; they can therefore never even in principle be models for the whole of our language-use, our cognitive or practical relation to the world, or our interactions with each other. Cicero can be understood actually to be making this point about context-dependency in the passage above. What he says is if you are giving a speech to a political meeting (contio) then this kind of Thucydidean style is not appropriate. It doesn't follow from this\u2014although it might in fact be true\u2014that it would be \"obscure\" in other situations or circumstances, for instance, among the members of a small group of the literary and political elite who met to understand the causes and the course of the war between Athens and the Peloponnesian League, and who could have their slaves read the relevant passages back to them several times for discussion. In fact one of the ancient sources (Dionysius of Halicarnassos) gives as the reason for Thucydides' \"difficulty\" that his speeches had too many thoughts condensed into each phrase. How many is \"too many,\" however, obviously depends on whether one is a Cicero standing in the Forum in Rome, shouting at a changing audience of half-distracted members of the plebs, or a Cicero sitting in the comfort and quiet of a rural villa with his learned friend Atticus and his learned slave Tiro.\n\nNevertheless, although what Cicero says in this passage is that Thucydides' style is not appropriate for public political oratory, it might also in fact be the case, although that is not Cicero's main point here, that Thucydides' style is inappropriate because \"obscure\" in any context that might reasonably be assumed to exist for reading it. We might be tempted to say that it is an \"objectively\" obscure style, meaning that Thucydidean speeches are hard to follow even for people who were not distracted, preoccupied, and so forth, and who also knew a lot about the man and the topic. One might go on to think that if this style is really difficult to understand for anyone in any context, this shows that the whole reference to a \"context\" is not really relevant here or was trivial and unimportant. One might finally then try to turn this around and imagine that there could be a style that was \"objectively clear\" in that anyone who understood the language in question could immediately grasp what was being said in it, provided the speaker took a minimum amount of care. Or one could think that there could be not merely a style but a whole language that had this property of \"objective clarity.\" This notion that some forms of speech or writing could be \"objectively\" clear or \"objectively\" obscure in some absolute sense is a recurrent illusion from which philosophers have great difficulty freeing themselves. Thus the late Wittgenstein criticises Frege for holding that all proper concepts must be univocal, sharply defined, and absolutely clear. He imagines someone like Frege objecting, \"[I]st ein verschwommener Begriff \u00fcberhaupt ein Begriff?\" [\"Is a vague concept really a concept at all?\"] and responds to this with a crushing:\n\nIst eine unscharfe Photographie \u00fcberhaupt ein Bild eines Menschen? Ja, kann man ein unscharfes Bild immer mit Vorteil durch ein scharfes ersetzen? Ist das unscharfe nicht oft gerade das was wir brauchen? Frege vergleicht den Begriff mit einem Bezirk und sagt: einen unklar begrenzten Begriff k\u00f6nne man \u00fcberhaupt keinen Begriff nennen. Das hei\u00dft wohl, wir k\u00f6nnen mit ihm nichts anfangen.\n\n[Is an indistinct photograph really at all a picture of a human being? Can one, in fact, always with advantage replace an indistinct picture with a distinct one? Isn't the indistinct one often just the one we need? Frege compares a concept with a domain and says: one cannot call a concept which is not clearly limited a concept at all. I take it that that means, we can't do anything with it .]\n\nI take this to mean that what we call \"clarity\" with respect to concepts depends on what we wish to do with those concepts, that is, with the context of our projects, wishes, expectations, and goals, and these can change. What is true of (individual) concepts is also true of the statements in which such concepts occur. Just to repeat the basic point, then, such notions as \"clear,\" \"precise,\" and so forth are highly dependent on the context within which we are employing them, and one extremely important aspect of that context is the human purposes or uses to which we wish to put the concepts (or propositions) under discussion. \"Clear\/obscure\" is always relative so some such envisaged context. If Wittgenstein is right, the assumption that there is some kind of \"absolute\" or noncontextual clarity is an illusion, on a par with a belief in Father Christmas.\n\nOne immediate consequence of this line of thought is to devalue \"absolute clarity\" as a cognitive or aesthetic ideal, given that such absolute clarity not only is not attainable but is incoherent\u2014a misunderstanding of what \"clarity\/obscurity\" could possibly mean. Similarly if clarity\/obscurity is always a distinction one makes relative to a context, this would have a strong tendency to undermine any attempt always to see obscurity as a defect or a failure\u2014perhaps an excusable defect but a failure in some sense nevertheless\u2014because even to speak of \"obscurity\" is to appeal to some standard, which, even if it is the most appropriate one, is still only one standard among a number of different possible ones, each dependent on a set of human purposes and interests. Since there is no standard given by reality itself, it is always in principle possible that the judgement \"This text is obscure\" is an expression not of an inadequacy of the text but of a failure on the part of the person making the judgement to be sufficiently reflective about what the appropriate context is and how the assertion in question fits into that context.\n\nOne source of possible \"obscurity,\" then, is the general context-dependency of language and the lack of absolute precision and determinacy of concepts in all natural human languages. More specific forms of misunderstanding or puzzlement can also, however, arise from syntactical features of particular languages. Thus, to take a famous example known to every student of Latin, in his epic poem about early Roman history Ennius has the god Apollo give King Pyrrhus of Epirus, who consults him before attacking the Romans, the following oracle:\n\nAio te Aeacida Romanos vincere posse\n\nwhich, given the grammatical peculiarities of indirect discourse in Latin, means either:\n\nA I tell you, descendant of Aeacus, that you are able to conquer the Romans\n\nor\n\nB I tell you, descendant of Aeacus, that the Romans are able to conquer you\n\nKing Pyrrhus assumes it means (A) and proceeds to carry out his attack on the Romans, only then to discover that it actually meant (B). To say that it \"actually\" meant (B), rather than that (B) accidentally turned out to be correct, is to assume for the sake of discussion here that there is a god Apollo who does know the future and is trying to express it in some way, but we will continue to make that assumption for the sake of discussion in most of what follows. This oracular response is a classic case of a perfectly straightforward ambiguity that should be immediately visible to virtually anyone who knew Latin at all and considered the response dispassionately.\n\nTo be sure, exactly how cognitively and morally culpable Pyrrhus is in immediately assuming that the oracle favours him depends partly on an aspect of the total situation about which we are not sufficiently informed. We know the (purported) exact words Apollo used in the response, but we do not know what specific question Pyrrhus asked. Given the context-dependency of the distinction between obscurity\/clarity, one would like to know the question asked as exactly as possible. Was the original question: \"Should I treat with the Romans or attack them?\" \"Are the Romans at all a threat to my hegemony?\" or \"Who will be victorious in the coming war, me or the Romans?\" or \"Shall I defeat the Romans?\" Cicero comments on the proverbial obtuseness of the descendants of Aeacus, citing another passage by Ennius to that effect (\"bellipotentes sunt magis quam sapientipotentes,\" i.e., \"powerful in war rather than powerful in wisdom,\" or perhaps \"war-strong, brain-weak\"), but if Pyrrhus had actually posed the question in the third form cited above, it seems almost inconceivable that he could have overlooked the ambiguity of the response because the very form of the question explicitly evokes the possibility of a Roman victory. This, of course, is part of the point. Pyrrhus's own response to the oracle is significant and indicative of his character. He does not even wish really to envisage the possibility of a Roman victory, so he does not take this possibility with sufficient seriousness. Humans are invariably at least to some extent self-centred and are strongly inclined to wishful thinking. They hear what they wish to hear. They sometimes even ask the wrong questions and then compound that error by jumping to conclusions about what the answers to these questions mean. This is a particularly striking character trait of the excessively self-confident Pyrrhus, and one that is exposed very clearly by his reaction to the oracle; he is more subject to this general human weakness than most other human beings are.\n\nIf one considers the response dispassionately, one should see that there is a further ambiguity that emerges more clearly if one pronounces the phrase with emphasis on the final word \"posse.\" If the question really was \"Shall I overcome the Romans?\" then a response couched in terms of what \"could\" happen, or who \"would be or is able\" to do what, might seem simply to be changing the topic, or perhaps to be an instance of the god giving himself a get-out clause, because presumably what \"could\" happen is not always what actually does happen. If one takes to this line of thought, (A)\/(B) no longer obviously represents an exclusive dichotomy. To say that the Romans are able to overcome Pyrrhus might be compatible with the possibility that Pyrrhus is able to overcome the Romans. Whether or not this was the case would depend on the particular sense one gave to \"posse\" [\"to be able to\"]. The god then might well be saying that either out-come is conceivable and possible. So with the emphasis on \"posse\" the response could mean:\n\nC It is inherently indeterminate who will win, you or the Romans; each one could (under appropriate circumstances) defeat the other.\n\nInterpreted in this way the response seems useful\u2014telling Pyrrhus to be careful\u2014and unobjectionably commonsensical. The second of the above interpretations, (B), had not come to Pyrrhus's attention because of his impetuous self-centredness. Even ex ante almost any-one else should and would have been able to see the ambiguity in that divine response, and given the notorious ambiguity of oracles, which had already been extensively noted by the time of Pyrrhus, anyone ought to have and almost anyone else would have looked for possible ambiguity. The fact that Pyrrhus didn't meant that he was dim, excessively self-obsessed, or perhaps overly self-confident, or all three. To have any one of these properties was to occupy a very bad starting point for attacking the Romans in the third century BC, no matter what other military skills one might have. Interpretation (C), on the other hand, would not be likely to recommend itself immediately to anyone who consulted the oracle in the first place because it stood in conflict with one of the assumptions that made oracular practice sensible. If it is objectively indeterminate who will win, why bother to consult the oracle?\n\nTwo kinds of human weakness, then, have emerged in this brief discussion of responses to oracles. First, humans are self-centred, too focused on their own abilities and powers, as if they were of final importance in the universe. They try to see the world as much as possible through the lens of what they could (and perhaps should) do. In the case of Pyrrhus this is associated with a certain kind of egotism, a heroic desire to show himself to be the best and acknowledged as the best through self-confident use of his own powers to defeat all comers, including, in the case in point, the Romans, who happened to be the next on his list. It is not, however, merely his overbearing insistence on his own primacy that is the problem with Pyrrhus. One can find a morally less questionable form of the same human self-absorption in the case of Oedipus. He, too, thinks primarily in terms of himself and what he has to do. Oedipus wants to avoid what is announced by the oracle partly for his own sake\u2014he does not wish to be the man who killed his father and married his mother\u2014but his attitude, we can assume, is at least partly altruistic in that he also wished to spare his father and mother the fate in store for them. As in the case of Pyrrhus, the oracular answer fits into a world that is seen by Oedipus as essentially a potential field of his own action, in which things are in the final analysis \"up to him to do or not do.\" As Oedipus's end may be taken to show, his own voluntary or involuntary action in the world is in the final analysis not that important. To think it is, that he can change his fate, that the world is essentially a realm constituted by his action, is a kind of wishful thinking.\n\nThe second form of human weakness we encounter is in a way the mirror image of the first. Rather than trusting too much to their own powers as agents to shape a course of action that will satisfy them, humans can be excessively overcome by the recognition of their own impotence and consequently excessively dependent and demanding. They can expect the god to give them the assurance that comes from absolutely unequivocal predictions that cannot be misunderstood. They need the world not to be one in which it is antecedently indeterminate (even for the god) who will win the battle; they need desperately to believe that there is already a determinate answer to the question \"Who will win?\" This is another form of wishful thinking.\n\nIt is well-known that over the entrance to the building in which the oracle at Delphi was located there were two inscriptions:\n\n\u03b3\u03bd\u1ff6\u03b8\u03b9 \u03c3\u03b5\u03b1\u03c5\u03c4\u03cc\u03bd [\"know yourself\"]\n\nand\n\n\u03bc\u03b7\u03b4\u1f72\u03bd \u1f04\u03b3\u03b1\u03bd [\"don't overdo anything\" or \"don't do too much of anything\"]\n\nThese inscriptions have from early times drawn the attention of philosophers and came increasingly to be interpreted as general ethical injunctions. The first was often interpreted in a way that gave rise to a tradition of construing an intense and interiorised form of introspection as a central component of the good life. The second was read, as far back as Plato's Charmides (164\u201365), as recommending to humans that they cultivate the virtue of \u03c3\u03c9\u03c6\u03c1\u03bf\u03c3\u03cd\u03bd\u03b7 (temperance, moderation, self-control), and thus it was tacitly taken to stand at the beginning of a tradition of preaching an ethics of moderation (e.g., it is thus interpreted by Nietzsche). Philosophers have also often looked for some relation between the two injunctions, but they have not generally been attracted to one of the more obvious ways of doing this: Self-knowledge is good but only, like everything else, in moderation. The course of subsequent philosophy would have been different if Socrates had come to believe that this was what the god was trying to tell him. Perhaps, though, one should return to one of the slightly more mundane ways of reading the two injunctions: Know exactly you want to ask before you enter the temple and don't pester the oracle with too many questions. It is not necessarily just that the god would be irritated by the need to give repeated responses, but if you felt the need to ask too many questions you would have shown yourself to be in no fit state to receive any response at all. It would be a sign you were stupid, hadn't thought about what you really wanted, or had abandoned yourself to your human tendency to want too much predictable determinacy in the world and were trying to use the god as a crutch to impose or create the illusion of an order that did not exist. Why, after all, should the god suffer such fools gladly?\n\nTo return to Pyrrhus, if the above were not enough, one can also wonder about what exactly vincere means in the god's response. Does it mean \"drive the enemy from the field in a given single battle,\" defeat strategically in a lengthy war (debellare), or overcome and permanently subjugate (subiungere)? Pyrrhus himself was in principle perfectly capable of making that distinction on appropriate occasions. After all, he was famous for remarking: \"If we win one more battle against the Romans, we'll be completely done for\" when he \"won\" a battle by eventually driving some of the seemingly inexhaustible supply of Roman troops off the field in disordered flight, but at the cost of losing men whom he could not replace. Does \"win\" mean \"tactical victory\" or \"strategic victory\"? How \"strategic\" and permanent must it be to count as victory?\n\nTwo different kinds of obscurity seem to be in play here. First, are the words of the god to be read grammatically according to schema (A), (B), or (C)? Second, what exactly does vincere mean? In the first case there seems to be a definite series of different possibilities, each of which can be taken to be of more or less clear import, that is, clear enough \"in the context.\" In the second case, the question is the rather different one of what is to count as \"victory.\" \"Ambiguity\" seems an appropriate term to designate the first of these cases, but the second case does not seem to be one of \"ambiguity\" properly speaking but of a certain kind of simple indeterminacy in a relevant dimension.\n\nLet us take now another example, one of the oracles reported in Book III of Aeneid as having been given to the Trojans. In the particular case in question we have, in contrast to Ennius's report about Pyrrhus, both the questions posed and the response, or rather we have a somewhat unusual speech which is a combination of an entreaty and a series of questions, and then a response. The combination of entreaty and question almost makes it seem as if the Trojans thought the god needed orientation, needed them to set the context of what they want and think they require, in order to prompt an appropriately phrased response:\n\nda propriam, Thymbraee, domum; da moenia fessis\n\net genus et mansuram urbem; serva altera Troiae\n\nPergama, reliquias Danaum atque immitis Achilli.\n\nquem sequimur? quove ire jubes? ubi ponere sedes?\n\nda, pater, augurium, atque animis inlabere nostris. (III.85\u201389)\n\n[Give us our own home, Apollo. We're exhausted; give us walls and a continuing succession of progeny and an abiding city. Save us, a mere remnant left alive by the Greeks and pitiless Achilles, and let us be the city of Troy built up again. Whom should we follow? [Or: What place should be strive to reach?] Or where do you order us to go? Where should we establish ourselves? Give us, father, a sign and enter gently into our souls.]\n\nTo which the voice of the god replies:\n\nDardanidae duri, quae vos a stripe parentum\n\nprima tulit tellus, eadem vos ubere laeto\n\naccipiet reduces. antiquam exquirite matrem.\n\nhic domus Aeneae cunctis dominabitur oris\n\net nati natorum et qui nascentur ab illis. (III.94\u201398)\n\n[You hard-done-by men of Troy, the earth that first brought you forth from the root of your parents, that earth will accept you back to her happy breast if you return. Seek out your ancient mother. Here the house of Aeneas and the sons of his sons, and those who will be born of them will rule the whole world.]\n\nThe first thing to note is that the unclarity here does not derive from and is not otherwise involved with any kind of obvious grammatical ambiguity. It is also not really like the indeterminacy that emerges when one tries to imagine just what would count as a \"victory\" (in one of the many appropriate senses). The obscurity here is like that one encounters in trying to solve a riddle, and the Trojans are duly puzzled.\n\nAeneas's father, Anchises, quick off the mark, immediately jumps in and gives an interpretation, namely that the oracle means \"go to Crete,\" citing the Cretan origin of one of his alleged maternal ancestors. This, however, is rather quickly shown to be a mistake because the Trojan settlement in Crete falls prey to a serious pestilence, and so the Trojans themselves are thrown back into a state of puzzlement about the meaning of the oracle and realise that it was not as clear as Anchises took it to be.\n\nThe situation here is of a very different kind from that of Pyrrhus. Pyrrhus acted as if he was confident of knowing how the oracle was to be read. It just turned out that (in retrospect) he had misunderstood it. Looking on at this from the outside we can see that there are at least two (or three) relatively (given the context) clear alternative readings, which Pyrrhus ought to have considered. In the case of the Trojans the response has a completely different kind of indeterminacy. \"Seek your aged mother\" does not in the same way carve out a relatively clear set of alternative possibilities, and it is not exactly \"vague\" and open to reinterpretation in the way in which \"victory\" is. What exactly could \"Seek your aged mother\" mean? Go to Crete and settle there? They try that and it doesn't work. Since, as Book I makes clear, Venus is supposed to be Aeneas's \"mother\" and Jupiter addresses her as the Lady of Cythera (Book I), are they then to make for Cythera, the small island off the coast of the Peloponnese? Venus is also identified with Aphrodite and Aphrodite's island is Cyprus, so is Cyprus the goal? Does the oracle mean to go back to the Troad? That, after all, is where they were all born. Or does it mean something altogether different? For instance, if the \"earth\" is the common ancient mother of all humanity, does \"seek your ancient mother\" mean give up living in cities and adopt a troglodytic form of life, living in the caves that mother earth provides? Or does it mean pursue mining and metallurgy as a mode of life, investigating and exploiting the underground resources of mother earth? Or is it ironic and intended to mean that the Trojans should study natural philosophy (i.e., seek to find what the nature of mother earth really is), that they will be preeminent in such study, and that only in such intellectual pursuits can anyone find a home? Or is it an expression of \"the wisdom of Silenus\": \"The best thing for you all to do is die as soon as possible and go back to your ancient mother, the earth; that is the only place where any human has an abiding city\"? Or a Roman version of this: \"Die with as many extravagant gestures of bravery as you can, then you will 'rule the world' with your posthumous glory\"? Or should the Trojans become professional experts in genealogy, helping everyone find out who their ancient mother (and then also, perhaps, father) really was? Or a tribe of pimps and prostitutes continually seeking out their ancient mother (Venus) in this form, or devotees of a cult of cosmological Venus (Aeneadum genetrix, hominum divomque voluptas\/alma Venus)?\n\nIf the last of these possibilities seem increasingly far-fetched, as indeed they are, one might recall that that is often a characteristic of the fulfillment of oracles, and in the Aeneid itself one can find equally recherch\u00e9 examples of the way oracles are claimed to have been fulfilled and thus what it is retrospectively claimed they \"really\" meant. Oracles are often worse in this respect\u2014more obscure\u2014than riddles in that good riddles should in principle be solvable by the person to whom they are propounded in the present circumstances in which that person happens to find himself, whatever they may be, but many oracles can only be interpreted ex eventu. Recall the prophecy by the Harpy Celaeno in Book III (247\u201357) that the Trojans won't find their city until their extreme hunger (dira fames) has them \"gnawing at their tables\" (ambesas... absumere mensas). This turns out, much later, to mean something like eating sandwiches at a picnic (VII.109ff.). Book III is a flashback in which Aeneas is telling Dido in North Africa about his previous wandering, but no one, either when Celaeno originally spoke the prophecy or even in North Africa when Aeneas was recalling it, could have been expected to understand what \"gnawing at the tables\" would eventually turn out to mean.\n\nOne might think that the above is all perfectly reasonable, but that it somehow misses the point because it puts together a very general philosophical point about the incoherence of some conception of purported absolute clarity with a very particular set of examples of oracular or literary obscurity, which is really of a rather different kind. Even if complete and absolute clarity, precision, and determinacy is a chimera, the oracles described in the literary passages discussed could easily have been relatively clearer in practical terms, clearer in contextually relevant ways, than they were. Even if it isn't absolutely clear what \"victory\" is, what constitutes an \"attack\" or for that matter who the Romans are (do they include the Italians or not?), Apollo could have told Pyrrhus in no uncertain terms not to attack the Romans\u2014\"noli aggredi Romanos\" would have done the trick, if the god really was speaking Latin\u2014and that would have been a more useful and relevant response to the question that was actually concerning Pyrrhus than the one he received. Similarly the voice in Delos could have said immediately: Try Italy. \"Try Italy\" is not absolutely precise\u2014the Peloponnese clearly does not count as Italy, and Sicily, at this point in time, does not either, but what about the Po Valley? Instead of Rome should Aeneas perhaps really have tried to found Milan, Verona, or Venice? Still this advice is better as a useful response to the question posed than \"seek your ancient mother.\"\n\nAfter all, Apollo does eventually cause a clearer answer to be given when he instructs the household gods (the \"Penates\") to appear to Aeneas in a dream and give him the correct interpretation, namely to go to Italy (III.147ff.). This is confirmed by Aeneas's father, Anchises, who conveniently remembers at this point that Cassandra had often made similar predictions about the final destination of the Trojans. So Virgil, who is terribly concerned to give a divine pedigree for the Roman Imperium as constituted under Augustus, engages in some literary attempts at squaring the circle. On the one hand, there is the divine pronouncement, but such pronouncements are known to be obscure, so he adds a second announcement to clarify the first and then a further confirmation in the form of Anchises' \"recollection.\" Then the final confirmation comes in Aeneas's descent into the Underworld in Book VI. The idea presumably is to get both divine origin for the advice to settle in Italy and as strong a confirmation of the specific interpretation as possible.\n\nSo why doesn't the god initially say what he means with sufficient clarity to be immediately understood? One possibility is simply to say that this question is based on a misconception. There are no gods, and what are called \"oracles\" are just inventions of the priests and priestesses who \"serve\" the oracle; except by accident, they have no more knowledge of anything than we do. Lack of clarity is their way of minimising the chance that they will be found out to be frauds and lose their livelihood. Although I am myself an atheist, I think this is far too quick a response, so I would like to bracket it and continue on the assumption that Apollo and the others did exist, and did in some sense \"give\" oracular responses A further possibility would be the Epicurean view that although gods probably did exist, they were completely unconcerned with humans and their trivial problems and thus had no real interest in giving useful responses. Perhaps oracular response was a game the gods played with each other, in which humans were mere tokens on a cosmological board. Perhaps the gods gave responses only on days on which they had partaken too deeply of ambrosia and so were themselves understandably muddled. Perhaps the ones who gave responses were always on the ambrosia. Or perhaps divine knowledge, or certain types of divine knowledge, divine intention, or divine instruction, could inherently not be expressed, or could not be expressed in any human language, other than obscurely. We know that some kinds of divine instruction clearly could have been expressed more clearly because the Penates do eventually tell the Trojans in perfectly clear Latin that \"seek your ancient mother\" means \"go to Italy,\" but it does not follow from this that everything the god knows, wants, and intends can appropriately be thus clearly and directly expressed in all contexts.\n\nThis line of thought, that there is perhaps something in the divine mentation that cannot be adequately or appropriately expressed through clear speech, gives one an indication of the direction in which one might seek a possible positive rationale for obscurity.\n\nOne traditional view is that the god could in one sense have been more explicit but that he knew that doing this would be self-defeating relative to some wider purpose he was pursuing. The god is, after all, not a calculating device for predicting the future, much as certain humans might wish to reduce him to this, but, if he is at all like one of the gods in Homer, he will have his own intentions, plans, and projects. Perhaps, as Plutarch puts it, \"He is no less a philosopher than a prophet\" and has imposed on himself a far-reaching peadagogical task of improving the moral state of humanity. One does not, however, necessarily improve people by telling them clearly, directly, and in a form they can easily and correctly assimilate what they (think they) want to know. Plato's Socrates suggests that the obscurity of oracular pronouncements results from the desire on the part of the god to motivate humans to reflect and search for the truth. When the god says that no man is wiser than Socrates, he does not mean that Socrates is positively wise but only that compared to the god, all human wisdom is equally insignificant. You won't, however, really take this point until and unless you have tried yourself to refute the god and failed. The god's statement is thus intended to motivate Socrates to improve himself (and indirectly his fellow humans) in two ways. He is to pursue the truth assiduously but also to try to prove the god wrong. By failing he will become clearer about the nature of his own ignorance. The resulting grasp of his own cognitive limitations while continuing uninterruptedly to pursue truth is the beginning of wisdom. The best way to acquire a motivation to pursue the truth is perhaps through generating a state of puzzlement, and the best\u2014or at any rate an especially good\u2014way of creating the right kind of productive puzzlement is by giving an obscure oracular response.\n\nThis strategy, if that is the strategy, does not, of course, work for Pyrrhus. His problem is in part that he is confronted with the Romans, who are militarily superior to him in available resources; they \"can defeat\" him in the long run by simply allowing him to wear him-self down with his repeated \"victories.\" An equally important part of his problem, however, is that he is not puzzled when he ought to be. This, as has been mentioned, is a result of the particular character traits that exhibit themselves in his response to the oracle. Any sensible speaker of Latin would have seen that the response is not clear. Pyrrhus, however, is impetuous and overly self-confident. Now, of course, the god could have said to him, \"You are self-centred, impetuous and overly self-confident,\" but saying that would not really have transmitted the message to him in an effective way because people do not in general learn from being told things, or, at any rate, they do not often learn how to change deeply entrenched forms of behaving simply by being given good general advice about them. Perhaps Plato's Socrates thought that merely discursive instruction of individuals could cause those individuals to change their way of life, but if he did\u2014even if his instruction was \"dialectical\"\u2014he was wrong. Aeschylus was closer to being correct when he spoke of \"learning through suffering\" (\u03c0\u03ac\u03b8\u03b5\u03b9 \u03bc\u03ac\u03b8\u03bf\u03c2). Perhaps giving the obscure answer will not help Pyrrhus any more, but that could be because he is beyond hope; some people are. Arguably Plato realised that Alkibiades, for all his gifts and despite his association with Socrates, was beyond help, and expressed this in his Symposium. Perhaps the god is trying to improve others who observe Pyrrhus's fate. The next-best thing to experiencing paedagogical suffering oneself is perhaps to observe it \"firsthand\" in the case of a highly visible, exemplary individual. So if the intention of the god is not merely to articulate the truth but to express it in a form in which it is most likely to be understood and acted on, then obscurity might be the best way to do that. Not, of course, an infallible way, because there will always be incorrigible people like Pyrrhus, but one that might in the long run have some effect on humanity as a whole, and the god, if anyone, can afford to play the long game.\n\nUp to now the discussion has been conducted under a slightly oversimplified assumption, namely that there were only two relevant agents involved in the consultation of an oracle\u2014the god and the person soliciting the oracle\u2014but in standard cases in the ancient world it would make more sense to distinguish three agents: the god, the person seeking an oracular response, and the human priest(ess) who, as intermediary, actually spoke the god's response. It is convenient to use the Latin word vates to refer to such an intermediary.\n\nOccasionally in the past people have construed the role of the poet as in some sense like that of the vates. Sometimes poets themselves, most notably perhaps H\u00f6lderlin and Rimbaud, have seen themselves in this way. Even in the best of circumstances, however, the poet does not know exactly what the god knows or at any rate does not know whatever is to be known in the way in which the god knows it. The vates does not, that is, in transmitting the god's message know any more than anyone else who hears the message. So the extent to which the vates can consciously and in detail enter into the potential paedagogic purposes of the god is likely to be limited. Despite this, it could still be the case that the god himself was pursuing a paedagogical project not just of instructing and improving some individual (Pyrrhus or Socrates) but of shifting people's attitudes and perhaps even their way of living as a whole through obscure speech, and it could also be true that the appropriate paedagogical effect would well be lost by clarity of expression.\n\nIf one assumes that our society, mode of life, forms of imagination, and language are interconnected, then restrictions or forms of distortion or repression in our form of life would be likely mirrored in limitations and distortions in our possible modes not merely of perceiving but also of imagining the world. This, in turn, would have its analogue in our everyday use of language. If there is no possibility of exiting once and for all from our contextually embedded natural languages into a pure ideal language of absolute clarity, because no such linguistic system can exist except as an inherently limited and isolated sector that remains parasitic on our everyday forms of interaction and communication, then what, if anything, can we do?\n\nOne response might be a kind of fuite en avant, an active embracing and celebration of ordinary language with all its blemishes and idiosyncrasies. However, rather than, as the late Wittgenstein and some of his followers thought, assuming that everyday life and language were in their own terms fundamentally in order\u2014or even, in a Romantic gesture, a source of health\u2014perhaps one ought to accept, but not be fully reconciled to, the fact that everyday life and the social structures embedded in it are a basic locus of repression. Prisons and police stations are not the only means of forcing people to act in certain ways. If one is interested not just in overt action but in forcing people to think, wish, and fantasise in a certain way, indirect forms of social pressure are usually much more effective than the chain and the knout. If \"clarity\" is indeed relative to socially enforced forms of speech, then, unless one makes the highly implausible assumption that our society and form of life are completely free and noncoercive, the demand for clarity can be seen as a requirement of conformity to structures of repression. Such conformity might not merely be seen as limiting and disfiguring our human individual and social capacities but also as \"evil\" (in one sense of that word). The call for \"clarity\" then can potentially mean pressure in the direction of conformism and hence complicity in evil.\n\nIf that is the case, then perhaps one should distinguish two distinct tasks that the god or his (or her) vates might undertake. One task would be to break down the familiar forms of everyday speech (and then perhaps in consequence certain routine patterns of action and interaction); the second would be to create positive new meanings, ways of speaking and acting, and eventually modes of living. The second of these might be thought to be in principle more important, but if the forms of speech and imagination are as dependent on the actual social structure as was assumed, there will be very strong limits to the extent to which it is possible even to envisage anything genuinely new, except perhaps in highly exceptional utopian moments, such as in 1871 in the Paris Commune. The first task then might be actually and practically more important, because at least in principle dischargeable. It might, of course, be psychologically possible for someone to engage in the first task only if he or she succumbed to the illusion of thinking they were engaged in the second, but that is a separate issue.\n\nIf everyday language was infected with conformism, repression, and distortion, how might one break out of it? We have seen a number of expedients tried: Heideggerian primitivism, the reduction of everyday language to purported etymological roots, and the reconfiguration of it from those roots; Brechtian Entfremdung, once described as the attempt to write High German as if it were a dialect; Karl Kraus's astringent and adamantine literalism; Adorno's precious, convoluted play with pronouns and self-referential constructions; or Paul Celan's hermeticism based on the use of unusual words no longer in use but preserved in weird entries in out-of-date dictionaries.\n\nIt is easy to be merely obscure\u2014any monkey with a keyboard can do that\u2014but difficult to be productively obscure, that is, to produce something that is not just like the Chinese text mentioned at the start of this chapter. A text completely written in Chinese does not \"break down familiar forms of speech\" because it simply stands outside them. To break them down is not simply to replace them, Chinese platitudes for English, but to shift them and our attitudes towards them. It is, however, also not at all obvious that any given author is in the best position to assess whether the obscurity and irritation his or her own text generates is really an instance of a creative positive transformation or a creative negative disruption of our attitudes towards our language, our possibilities, and our world. There is an important distinction between what I can see from the outside, from the third-person perspective, where obscurity may have positive value, and what is visible from the first-person perspective from which perhaps I have no alternative but to seek to avoid what I take to be lack of clarity as much as possible. This would mean valuable obscurity would necessarily be a by-product, like \"happiness,\" not something one could with any hope of success intentionally strive for, but something that will result only from a process of aiming at something else. What could that \"something else\" be?\n\nThe vates will then be engaged in a strange and deeply anti-Platonic form of paedagogy because ex hypothesi he will not in any concrete detail know the end or goal, and will in one sense not know what he is doing. He will need to strive for what he took to be clarity while perhaps knowing that no such thing exists in any absolute terms and while also knowing that precisely the obscurity he could not eliminate, and perhaps did not even notice, might turn out to be his most valuable achievement.\n\nIf a poem can be fully understood, it has perhaps already lost some of its most important value. Paul Celan recognised this when he wrote in the preliminary drafts of his famous speech \"Der Meridian\": \"Weshalb uns die Gedichte fr\u00fcherer Epochen 'verst\u00e4ndlicher' vorkommen als die unserer Zeitgenossen? Vielleicht auch deshalb, weil sie sich als Gedichte, d.h. mitsamt ihrem Dunkel verfl\u00fcchtigt haben.\" [\"Why do the poems of earlier epochs seem to us 'more easily comprehensible' than those of our contemporaries? Perhaps part of the reason is that they have evaporated as poems, that is, together with their darkness.\"] For a poem to be a poem is for it to be obscure: not completely to fit in to our usual forms of speaking, acting, and reacting. Celan speaks of the \"congenital, constitutive darkness\" of the poem; to \"clarify\" this would be to destroy the poem. This is perhaps part of the reason why art at any rate in our modern world is always a necessary failure. As long as the work remains obscure, it has not yet acted effectively, but when, and if, it stops being obscure, it loses its most important purpose, its real point. Contrary to certain classicist assumptions, a poem is never \"perfect\" and is also never, as Thucydides puts it, a \"possession for eternity\" (\"\u03ba\u03c4\u1fc6\u03bc\u03b1 \u1f10\u03c2 \u03b1\u1f30\u03b5\u03af\" I.22). This is not a deficiency but part of what makes the poem what it is.\n\nIn the glory days of \"logical positivism,\" individual and social maturity was sometimes said to reside in being able and willing to \"face the facts,\" and these were assumed to be discovered by natural science. It was also asserted that there could be an absolutely clear, utterly unambiguous, and context-independent ideal language, the formalised \"language of science,\" which was adequate for expressing all the facts, and that meant everything there is to know. Apart from the facts and their associated modes of representation, there was nothing that was amenable to being discussed in any but a wholly random and arbitrary fashion. As the ideal vehicle for the expression of \"the facts,\" the ideal language of science had a kind of normative standing for mature humans. However, to be able to face the absence of determinate \"facts\" in a given domain is also a sign of a realistic attitude towards the world and is something to be aspired to; inability to tolerate vagueness, ambiguity, indeterminacy, the shifting, unbounded, amorphous nature and sheer randomness of much of human life and of human language is also a serious human weakness.\n\nThere are three thoughts here that are distinct but complementary. First of all, the world-language unit is not composed, at any rate not exclusively, of fixed and sharply defined facts that could be the objects of ideally pellucid expression. Much of it is indeterminate in a way that would not admit of absolute clarification. Second, much of our speech is in any case directed not at trying to reflect the natural world or reproduce the social world at all but at trying to change it or them. Third, much of what we take to be clear seems that way only because repressive social forces impose restrictive, determinate forms on our behaviour and on our modes of thinking and imagining. The best modern poetry is responsive to all three of these thoughts. That is one of the things that makes it so unsettling for many people but also one of the reasons we should cherish it, if we can.\n3\n\nMarxism and the Ethos of the Twentieth Century\n\nOne of the things I have always most admired about Alasdair MacIntyre's work is the particular kind of intellectual courage it exhibits. This virtue manifests itself in a number of ways, including a willingness to address large philosophical questions head-on and to give straightforward answers to them. This is a form of courage, rather than merely of some other more etiolated cognitive excellence, because giving relatively bald and unvarnished answers to big questions makes it difficult to avoid facing up to the implications of what one says for action, and the action involved might be of a kind that requires exhausting, deeply disruptive, and potentially radical changes in the way one lives. In the spirit of an attempt to emulate, at least to some extent, this one of MacIntyre's intellectual virtues, let me try to give a simple answer to the simple question. What happened in moral philosophy in the twentieth century, and what happened to moral philosophy in the twentieth century? My answer to this is that Nietzsche is what happened \"in\" moral philosophy, that is, the very idea of a \"universal\" moral philosophy having any kind of transsubjective authority came under attack. The notion of \"transsubjective authority\" is both unclear and problematic, but that does not mean that it is not attempting, even if not with complete success, to designate something important. It certainly does not mean what philosophers call \"validity,\" and in general it is not a mere epistemic notion. A practice (and its associated concepts and forms of thought) has what will be called \"transsubjective authority\" if it is capable of effectively structuring the basic functions of society around itself, endowing them with meaning, telling us how we should understand them, issuing commands, injunctions, and recommendations that \"stick\"\u2014that is, that are as a rule taken to have weight and standing\u2014and finally if it is able to give some reasonable account of itself and stand up to criticism. So Christianity is a clear historical example of a form of life and thought that had transsubjective authority for a long time in the West. Christianity adds to the mix the idea that its authority is \"universal.\" So the claim is that in the twentieth century Christianity in particular and the very traditional idea of a \"universalist\" form of ethical life and thought were replaced by a consumerist array of views that was a reflection of a life devoted to more or less unreflective consumption, structured only by aesthetic predilections and the usual sociological imperatives of novelty, snobbism, and so forth. What happened \"to\" moral philosophy is that Marxism, which to some extent came from outside the stuffy int\u00e9rieur of academic philosophy, presented the only genuine and potentially viable attempt at reconstituting some notion of objective moral authority, an authority that was to be based on attributing to production an absolute social and political priority. If this attempt had succeeded, it would have changed the world and with it our intellectual and moral universe, but it failed. It used to be said that Marxism was a pseudo-religion or a religion-substitute and this claim was presented as if it was in itself a criticism. As if Marxists had simply not fully grasped the implications of the death of God. My view is that the problem was not that this was Marxism's aspiration but that it failed to achieve its aspiration. Philosophically, then, the story of the twentieth century is the story of the failure of Marxism.\n\nThe main philosophical \"problem\" in the twentieth century, then, is the complex of anarchic beliefs and anomic modes of living for which the proper name \"Nietzsche\" stands as the designator. This phenomenon \"Nietzscheanism\" is not a mere historical aberration or an accidental philosopher's mistake. Rather it is in some sense a veridical reflection of our social reality, and thus Nietzsche can be seen as giving a correct diagnosis of a set of ills with deep roots in modern Western societies. One of the ways in which Nietzsche uses the term \"nihilism\" is to designate the contemporary situation of disorientation in which \"[all] the highest values lose their value\" and all forms of authority lose their hold over individuals. This disorientation is thought to be so intolerable that the most urgent task for contemporary philosophers is to try to help social actors replace practical and theoretical Nietzscheanism with something more salubrious. Philosophers should be looking for a stable way of living together that would be potentially universal, in some yet to be specified sense of \"universal.\" This common life would need to be one that would allow us to cultivate certain individual and collective goods, many of which are highly context specific and dependent on historically fragile construction. In addition, it would ideally be one that would tolerate and support, or even foster, a correct reflective conceptualisation of itself, and thus allow us to see it as having some kind of standing, validity, or authority.\n\nThis does not yet by itself imply that Marxism is the only viable attempt to provide a coherent moral philosophy that would respond adequately to Nietzsche's challenge, so that the story of the twentieth century can be told as essentially the story of the development and failure of Marxism. Surely, one might think, there must have been other possible contenders for solutions to the problem we were looking to solve? Why is the story not told as one of the failure of Christianity or liberalism? I wish to suggest that Christianity and liberalism did not, properly speaking, \"fail\" because they were never real contenders. The reason for this is that neither one is properly \"universalist.\" What \"universalist\" means is, of course, itself a matter of controversy.\n\nThis is in no way surprising. After all, it is an oft-noted property of the most interesting philosophical questions that they are reflexive. In asking \"What form of life, what worldview, what philosophy, what authority is universalist?\" one is at the same time asking \"What exactly is, or could be, meant by 'universalist'?\" In a preliminary way one can say that a \"universalist\" worldview would be one that was relevant to everyone\u2014it was not from the very start intended to be the worldview only of aristocrats, football enthusiasts, medical practitioners, and so forth\u2014and second it would have to be a worldview that gave people orientation towards all the important features of human life. This preliminary account is not false, but it shares the same indeterminacy mentioned at the start. We don't know what \"relevant\" means in the phrase \"relevant to everyone\" or what \"important\" means in the phrase \"all the important features of human life.\" Part of what we have to do is conduct a unitary enquiry of some kind within which we discover what is relevant and ought to be important to us, and what \"relevant\" and \"important\" should be taken properly to mean.\n\nDespite its original (Pauline) intentions to provide a mode of access to salvation to all irrespective of their ethnic affiliation, Christianity, like all the revealed monotheistic religions, is so mixed up with highly local forms of human customary imagination that it is unthinkable that it ever could become truly universal, in the sense of being shared by all humans, unless it were imposed by a form of sustained, unchallenged military, cultural, and economic power no Christian country had in the twentieth century and which, certainly, none seems now likely ever to acquire. This is, of course, a variant of a theme common in Hegel, who connects religion with the ability of the human spirit to produce specific images and concrete narratives, what he calls \"Mythen, Phantasievorstellungen und positive, eigentliche Geschichten\" (\"myths, phantasms, and contentful narratives that are in fact mere stories\"). There is, of course, a place, even an important place in human life, for historically specific forms of narrative that engage strong human emotions and for local forms of the imagination, but our image-making capacities seem to have a limited ability to give a kind of persuasive account of themselves and their products that would render them universally acceptable and binding. Unless the particular revelation at a specific time and place on which a given religion is founded can be detached from its context in local fact and fiction, it seems unlikely to be able to command the unrestricted assent that it would require to constitute a proper response to Nietzsche, by exerting a visible and palpable universal moral authority.\n\nUnder modern circumstances, then, religion is fated either to degenerate into folklore, a particular emotionally tinged set of provincial ways in which the genuinely universal pressures of global capitalism are inflected in particular localities, or to become an inherently sectarian matter, that is, something relating to the specific choice a small group makes and through which that group isolates itself. Recall the origin of the term \"heresy\" in the Greek word \u03b1\u1f31\u03c1\u03ad\u03c9 (\"to choose, or pick\"): the heretical sect is a group that by choice cuts itself off from a wider \"catholic\"\u2014that is, genuinely all-encompassing\u2014community, thus becoming a \"sect.\" Its members choose some doctrines and practices rather than others. Members of a \"catholic\" community do not choose their way of life but are born into it, or \"grow\" into it. If there is an intact \"catholic community\" in existence that would be one situation, but if there isn't, no amount of wishing or engineering will in itself bring it back into existence. If no truly \"catholic\" community exists, then every religious grouping, even those into which the members are \"born,\" must be seen as in a very important sense a \"sect.\" So one slightly paradoxical way of describing the contemporary world is as one in which there is no (proper, universal, \"catholic\") religion but only (various) sects. A sect may have some healthy, admirable, and morally positive features, but it is not a framework for the reliable exercise of potentially overarching moral authority for the modern world as a whole.\n\nThat no revealed religion can be the solution of the Nietzschean predicament, then, does not imply that \"clarified\" and intellectualised successors of Christianity, such as Marxism, might not aspire to being successful in this enterprise. MacIntyre himself occasionally emphasises that Marxism can be seen as being something like a Christian heresy, although a \"heresy\" that might, even if in some sense finally a failure, potentially have been more universal than the original religious matrix out of which it emerged. Still it would be a \"heresy,\" part of the essential point of which was that it had turned itself into a philosophy and a kind of political and social thought, and so deprived itself of precisely those attributes that made Christianity not a philosophy but a religion.\n\nOne might, of course, argue that \"liberalism\" is not a contender as a solution to the Nietzschean predicament because it is not really a full-scale philosophy at all. Rather it is one or the other of two things. Either it is an attitude that is inherently adverbial, so that to be a Liberal is to be \"liberal,\" that is, to do whatever one does in a certain way or spirit of open-handedness, flexibility, and toleration with a minimal use of force. That is, to put it in an exceedingly tendentious way, a Liberal is a person who accepts the first half of the slogan that is often associated with the Jesuits, \"suaviter in modo,\" but ignores or blanks out, and thus tacitly cancels, the second half, \"fortiter in re.\" Alternatively Liberalism is a highly specific political programme, such as the demand that constitutional government be introduced in a particular country (such as Spain in the early nineteenth century). A psychological disposition or attitude, however, is not by itself an ethics at all because it is too indeterminate, referring exclusively to how we ought to act rather than what we should do. It might be thought to have some limited ethical value in that it would exclude certain forms of individual or collective action, such as certain crude inquisitorial practices, but that in itself will not give anyone a very useful orientation in life or answer perfectly reasonable questions such as: What courses of action should I (or we) pursue in a free-spirited and open-handed way? Of which institutions should I approve? Why exactly is toleration of error an overwhelming human good? When is discussion or negotiation possible and fruitful and when is it useless or inappropriate? What alternatives do we have when discussion will not serve? Who counts as \"one of us\"? For which purposes and why? Similarly, a particular political programme like constitutionalism might be compatible with any number of very different worldviews.\n\nI don't doubt that \"Liberalism\" was used originally in this adverbial way and then later as the designation of a specific political programme, but I think it is equally clear that nowadays, in much academic and peri-academic discourse, it is presented as a kind of worldview or philosophical position. During the course of his long career MacIntyre has diagnosed four problems with liberalism considered as a full-scale political and moral doctrine.\n\nFirst, much mainstream Liberalism is characteristically and inherently duplicitous in a way that will eventually reveal itself, particularly when liberals are most in need of orientation and moral guidance, namely when they encounter intact and self-confident nonliberal societies. Doctrinal liberals generally pretend to be above the fray in substantive ideological and moral disagreements. They claim, that is, not to be advocating one particular substantively specified way of life over another, yet they clearly are. Thus the British state, accepting it for the moment as an institution that is committed to the lowest-common-denominator Liberalism, which unites the Conservative and the Labour parties, now enforces a ban on forced marriages and intervenes, coercively if necessary, to prevent female circumcision and other practices, even against groups who take these customs to be an integral part of their way of life. To turn one of Isaiah Berlin's central contentions back against liberalism itself, one might say that even forcing people to acquire the preconditions for acting in an individually autonomous way is forcing them to do something (and not something else, e.g., \"merely\" liberating them or rendering them \"more rational\").\n\nSecond, liberalism cannot easily be extracted from at least some kind of complicitous association with laissez-faire capitalism, and so it inherits whatever deficiencies might be associated with that specific form of economic organisation. I might also add that if one central strand of twentieth-century Marxism is correct, this association with laissez-faire capitalism is a further reason to think that liberalism cannot be an ethical system that lays claim to any real universality. One reason for this is that laissez-faire capitalism seems to require an excluded underclass that is exploited (whether that be an internal proletariat or one externalised in quasi-colonial arrangements) and to which various liberal claims about voluntary contracting, transfer of ownership of property, free choice of occupation, and so forth, are effectively irrelevant. Another reason is that a laissez-faire economy depends on the continued existence of a series of social domains and institutions like the family, the legal system, the education system, and the health system. These institutions must operate according to pre- or noncapitalist principles and cannot coherently be fully subjected to the imperatives of the \"free market\" (although they can be significantly distorted and damaged by attempts to do this).\n\nThird, the substantive moral conception to which liberalism as a philosophy is committed, despite the protestations of neutrality made by many of its most vocal proponents, is one based in a pernicious way on a conception of the isolated individual as the locus of an absolute moral autonomy. This represents a serious cognitive limitation, because on this basis no adequate understanding of human society is possible, nor does it permit even individual human agents who are engaged in any serious forms of social behaviour to attain any satisfactory form of self-understanding. It is in fact exceedingly difficult to see how liberalism's pet recommendation \"increase individual freedom\" can constitute a contribution to solving any of the world's major problems.\n\nAn ethic based on this kind of hyperindividualist moral autonomy is, however, not merely cognitively mistaken, but\u2014and this I take to be MacIntyre's fourth point\u2014any form of social action based on or guided by such a view will also be deeply and actively destructive, dissolving forms of collective life that have real value without replacing them with anything of equal value. For these reasons, then, liberalism is not the answer.\n\nInitially it might look as if Marxism was simply orthogonal to the whole universe of discussion marked out by Nietzsche, Christianity, and liberalism. It does not seem to be focused on providing views about the universal validity of moral principles, the salvation of the soul, or the proper role of flexibility and toleration in human life. What it promised was three things. The first was something very concrete: an end to the boom-and-bust cycle of capitalist growth, and a regime of full employment, economic stability, and full social welfare and security for all. This first promise had particular importance in circumstances in which it called attention to the comparison with capitalist economies that were subject to recurrent, severe crisis, such as those of the 1920s and 1930s, and it retains its importance to the extent to which capitalist societies, despite Gordon Brown's foolish boast, still are subject to boom-and-bust. This first promise was concrete and empirical in the sense that it came about as close as one ever gets in the social sciences to being something the fulfillment of which could be determined by something like empirical scientific means.\n\nSecond, Marxism was committed to the view that the appropriately structured abolition of the private ownership of means of production would do away with what were historically superseded social fetters on the development of human powers and unleash an unexpected and hitherto unprecedented increase in human productivity. This increase in human productivity would have as its natural effect an increase in material well-being, that is, in the level of satisfaction of human needs and in levels of human consumption. This second promise, too, was something that, with the usual caveats and qualifications, could be connected with at least some crude standards of empirical confirmation.\n\nThe third promise was altogether more problematic for historical and philosophical reasons. That is the promise to end alienation. It is historically problematic because it was most clearly formulated in unpublished writings by Marx that became widely available only after World War I and therefore did not directly influence the formulation in the late nineteenth century of the basic Marxist canon. It has been thought to be philosophically problematic because it seemed to depend on a series of Hegelian views that might be thought to be questionable for any number of reasons (not least because they might be thought to remain implicated in basically idealistic ways of thinking about the world, which was found to be objectionable). \"Alienation\" is best understood relative to a distinction between the (mere) exercise of human powers and the \"appropriation,\" or \"re-appropriation\" [\"Aneignung\" \/ \"Wiederaneignung\"], of these powers. The intuition behind this distinction is that an individual human agent (or a group of such agents) can be the real locus through which certain powers are applied to the physical world without it being the case that the human individual, or group, in question has \"made the (relevant) powers its own.\" Making certain powers \"our own\" in turn means acquiring full control over the conditions of further development and the application of those powers and being able to affirm ourselves in the exercise of those powers. \"Alienation\" refers to the state of affairs in which we have not appropriated some of our most basic human powers, especially those closely connected with material production and reproduction of our form of life. Apart from concerns about the possible idealist overtones of this concept, one might worry that I have given an account of it that is circular and empty, just replacing one undefined and obscure counter or token with another: Alienation is supposed to be the lack of appropriation of powers. That in turn is \"defined\" by reference to \"making the power one's own.\" \"Making a power my own,\" however, is said to have taken place when I can \"affirm myself\" in exercising it. However, is \"self-affirmation\" a well-understood concept? The reference to \"self-affirmation\" means at the very least that a significant interpretative activity is required to connect the conceptual structure described in any way with anything that empirical social science would recognise as part of itself, and this might be thought to be a further reason to be wary of \"alienation.\" I do not wish in any way to minimise these genuine difficulties, but I do not think they finally matter for the question at issue.\n\nThe self-proclaimed Marxist regimes of Eastern Europe were remarkably successful in making good on the first of these promises. To the very end their populations enjoyed full employment and what was by Western European standards a very ample schedule of other benefits that provided more or less complete economic security. Nevertheless they collapsed, not just economically but, as it were, also morally. They failed, that is, to find enough ideational and motivational support among their own populations to continue to reproduce themselves. The reason for this was not, I submit, because they were oppressive, and certainly not because they failed to be \"democratic\" in the sense in which this term is used in Western liberal societies. It is also hard to think that they collapsed merely because they failed to end \"alienation,\" although it is true that they did thus fail, or perhaps more exactly, as Sartre sometimes claimed, they did not so much fail to end capitalist alienation as replace it with a specific, slightly different kind of alienation. In one of his essays he cites a poster in a factory in Eastern Europe in which workers are enjoined to take care of their health in order not to damage national production. In cases like this one's own health itself (and hence presumably one's own biological possibility of self-activity) is seen merely as a means to a fully external end. Still, politics is usually about differential, not categorical, forms of judgement and action, that is, we characteristically choose X not \"for itself alone\" but X in preference to Y or Z. So the spectrum of envisaged realistic possibilities that is presupposed in political decision is usually an important variable. If the choice, then, is exclusively between Soviet forms of economic and political organisation and Western capitalist ones, the failure of Soviet-style societies to end alienation would not be a very strong differential argument in favour of capitalist regimes because the latter made no serious attempt at all to make labour (and life) unalienated and unalienating (whatever \"alienation\" might turn out to mean). Yet when offered the opportunity, in the last decade or so of the twentieth century, populations in Eastern Europe overwhelmingly chose to abandon the self-proclaimed Marxist form of economy and of society and look towards Western European models of modern capitalism.\n\nThe lethal failure of twentieth-century European Marxism was its inability to produce consumer goods at the level of quality and quantity that was attained by Western Europe. So a failure in the second of the three promises. Those of us who have always lived in prosperous societies and have been materially comfortable may well be tempted to be somewhat condescending about this: Is mere failure to provide a steadily increasing stream of luxury consumer goods really good grounds for rejecting a social order, provided ample means for minimally decent living are available? Is even lack of \"economic efficiency\" good grounds for criticism? I think this is a temptation to adopt a morally shameful attitude that we should staunchly resist.\n\nThe positive attraction of the West, then, was completely understandable. It was the attraction of greater productivity combined with a social and political framework that was perceived as permitting a greater distribution of consumer goods. The failure of the Soviet system consisted in the incapacity to provide their populations with capitalist levels of consumption and an inability to elaborate any plausible alternative evaluative standard relative to which life in Soviet-style societies could be seen as distinctively good.\n\nThe difficulty was that a central strand of Marxism shared with capitalist ideologies a tacit or explicit productivist ethos. In a kind of parody of a motif taken from the High Enlightenment, a more economically and industrially productive society was expected to be one that was \"more advanced\" in almost every way. Marx himself was well aware of this strand in his own thinking but did not think it in any way problematic: he expected socialist economies to out-produce capitalist ones and to be self-evidently more rewarding, satisfying, and choiceworthy to those who lived in them than capitalist societies were; he also expected these two things naturally and easily to go together. Perhaps the clearest expression of the productivist view occurs in the Grundrisse:\n\nWir finden bei den Alten nie eine Untersuchung, welche Form des Grundeigentums etc. die produktivste, den gr\u00f6\u00dften Reichtum schafft? Der Reichtum erscheint nicht als Zweck der Produktion, obgleich sehr wohl Cato untersuchen kann, welche Bestellung des Feldes die ertr\u00e4glichste, oder gar Brutus sein Geld zu den besten Zinsen ausborgen kann. Die Untersuchung ist immer, welche Weise des Eigentums die besten Staatsb\u00fcrger schafft. Als Selbstzweck erscheint der Reichtum nur bei den wenigen Handelsv\u00f6lkern\u2014Monopolisten des carrying trade\u2014, die in den Poren der alten Welt leben, wie die Juden in der mittelaltrigen Gesellschaft.... So scheint die alte Anschauung, wo der Mensch, in wlecher bornierten nationalen, religi\u00f6sen, politischen Bestimmung auch immer als Zweck der Produktion erscheint, sehr erhaben zu sein gegen die moderne Welt, wo die Produktion als Zweck des Menschen und der Reichtum als Zweck der Produktion erscheint. In fact, aber, wenn die bornierte b\u00fcrgerliche Form abgestreift wird, was ist der Reichtum anders als die im universellen Austausch erzeugte Universalit\u00e4t der Bed\u00fcrfnisse, F\u00e4higkeiten, Gen\u00fcsse, Produktivkr\u00e4fte etc. der Individuen? Die volle Entwicklung der menschlichen Herrschaft \u00fcber die Naturkr\u00e4fte, die der sogenannten Natur sowohl, wie seiner eigenen Natur? Das absolute Herausarbeiten seiner sch\u00f6pferischen Anlagen, ohne andre Voraussetzung als die vorhergegangene historische Entwicklung, die diese Totalit\u00e4t der Entwicklung, d.h. der Entwicklung aller menschlichen Kr\u00e4fte als solcher, nicht gemessen an einem vorgegebenen Ma\u00dfstab, zum Selbstzwecke macht?\n\n[Among the ancients we never find an investigation of which form of landownership is most productive, creates the most wealth? Wealth does not appear as the end of production, although Cato is quite capable of investigating which form of cultivation of the field has the highest yield, and even Brutus knows how to lend out his money at the best rate of interest. The investigation is always which form of property creates the best citizens. Wealth appears as an end-in-itself only among some few trading nations\u2014monopolists of the carrying trade\u2014who live in the pores of the ancient world, like the Jews in medieval society.... The ancient conception according to which man in whatever narrow national, religious, or political form appears as the end of production seems to be very sublime compared with the modern world, where production appears as the end of man and wealth the end of production. In fact, however, when the narrow bourgeois form is stripped away, what is wealth other than the universality of the needs, abilities, forms of enjoyment, productive powers, etc., of the individuals generated in universal exchange? The full development of human domination over natural powers, both of so-called nature and of his own nature? The absolute elaboration of his creative capacities without any further presupposition that the previous historical development which makes this totality of development, that is the development of all human power as such, not measured by any pregiven yardstick, into an end-in-itself.]\n\nOn the positive side, this does seem to give one at least a possible kind of response to Nietzsche, that is, it can be seen as a description of a universal framework that could be construed as containing a potential kind of quasi-moral but nonsubjective authority, an authority derived from the requirements for maximally developing needs and human productive powers and capacities for their own sake. However, this response does require one to put a huge amount of weight on a particular anthropological conception that emphasises the mutual dependence of human powers and human needs, and the necessary and potentially mutually reinforcing relation between individual and collective development of powers. Nietzsche, of course, would have rejected this conjunction of individual and collective. This disagreement between Marx and Nietzsche about the interconnection of individual and social powers ought not, however, to obscure a further deep-seated similarity. Both Marx's productivism and Nietzschean conceptions of the exercise of the individual will-to-power contain a very strong substantive quasi-normative commitment to the development of power as an \"end-in-itself.\"\n\nIs the development of every human power and of every need really good, and, more than that, is it a kind of absolute good-in-itself, as Marx's construction seems to imply? Historians often point to Rousseau as systematically introducing the idea of a \"false need\" into Western social thought. In human society we all develop forms of dependency on the good opinion of others, which give rise to any number of cognitively delusory but motivationally deeply seated \"needs,\" which make no contribution to any real human good and which we would in principle be better off without. Notoriously Marx will have none of this line of thought. He has a theory of needs, but no analysis of what it could be for a need to be false. A \"rich\" person is a person \"rich in needs,\" and this is a thoroughly positive and laudatory description. He thought any attempt to introduce a distinction between true and false needs would represent a return to what he took to be Rousseau's proto-Romantic primitivism or to an inherently ascetic view of life that could be justified only by reference to discredited forms of religious belief.\n\nParallel to these difficulties about \"needs\" is a question about the idea of a \"(natural, human) power,\" the development of which is here described as an end-in-itself. In some of Marx's early works one finds a very broad construal of this term. So even \"forms of enjoyment\" that might play a role in my relation to objects in the world (as mentioned in the above quotation), such as the cultivation of the senses and the ability to appreciate the beauty of nature, are construed as \"natural human powers\" (actually as \"menschliche Wesenskr\u00e4fte\"). Arguably this would not give rise to some of the worst features of productivism, because saying that a \"form of enjoyment,\" or even a larger configuration in which a specific form of enjoyment is an essential constituent, is an \"end-in-itself\" does not seem to leave open certain classic forms of alienation. It really would seem strange to claim that I was being inappropriately forced to develop my own powers of enjoyment of some type of object, at any rate provided that the development of such powers was part of what Marx sometimes calls an \"all-sided\" exercise of human powers and capacities. Unfortunately, sometimes Marx seems to give \"human powers\" a significantly narrower reading than the one just canvassed, as if the phrase were equivalent not to \"all\" the human powers included in one way or another in the cycle of production and consumption (including those of aesthetic appreciation, enjoyment, and discrimination) but were restricted to the power to produce goods or material objects that can be used or consumed. \"Power\" means, or at any rate is strictly modelled on, industrial production: \"Die Geschichte der Industrie... ist das aufgeschlagene Buch der menschlichen Wesenskr\u00e4fte\"). A \"human power\" is then tacitly understood essentially as a kind of instrument that is deployed to bring physical things into existence or change their location or material properties.\n\nMacIntyre, in his essay on the Theses on Feuerbach, pointed to a \"road not taken\" by the early Marx that diverges from that trodden by later Marxists. In the \"Theses\" the early Marx does not construe production either in the narrower or in the wider of the two senses distinguished above as the end-in-itself of human life. Rather he appeals to a different kind of human action altogether that is not construed as in the first instance a form of relating to \"objects\" at all. This form of human action is apparently conceived as standing altogether outside the context of the instrumental transformation of nature in the production of goods or the cultivation of powers to appreciate those goods (and then also perhaps natural phenomena of all kinds, considered as potential \"objects\"). Marx calls this form of activity \"praxis.\" Whatever \"praxis\" is supposed to be, it is supposed to be a radically autotelic\u2014that is, it is to have its end-in-itself and not in some external product\u2014and noninstrumental form of activity. Marx, however, does not really give a sufficiently detailed account of \"praxis\" for us to get a firm grasp on the concept and discover to what extent it is or is not genuinely enlightening. Appeal to \"praxis\" is no more than a gesture at something not further analysed rather than a satisfactory explanation.\n\nOne can see the work of some of the members of the so-called Frankfurt School as developing lines of argument that are parallel to the ones just discussed. Most of the members of this school agree on two points. First, that Marx was either confusing or confused about the exact nature of human action. He tended to construe all human action on the model of instrumental action or under the aspect of its possible mean-ends rationality. He tended, that is, to identify \"praxis\" and \"production,\" and then to construe \"production\" in a narrow way as \"industrial production.\" This in effect meant both conceptually reducing noninstrumental forms of action (such as \"praxis,\" whatever that finally meant) to forms of production and subordinating all human life to the imperatives of increased instrumental control over the environment (and over ourselves). Seen from this perspective, Marx's use of the phrase \"Herrschaft \u00fcber die Naturkr\u00e4fte eignen Natur\" (\"the domination [italics added] over the natural powers of our own nature\") in the passage from the Grundrisse cited above can be read in a chilling way, as potentially the extension of a paradigm of coercive manipulation from our relations to nature to our relations to other humans, and from our relations to other humans to our relations to ourselves.\n\nSecond, the members of the Frankfurt School thought it essential to rehabilitate the distinction between true and false needs by showing how one could speak of \"false needs\" without asceticism, without a return to an Aristotelian conception of a substantively fixed human nature that could be used as a criterion, and without commitment to a Romantic \"return to Nature\" view. This attempt was not very successful because it ended up either with the pessimistic aesthetic vision of Adorno, which had no obvious connection to any form of concrete politics, or with the debased liberalism of Habermas and his neo-Kantian ideal speech theory.\n\nUp to now I have not mentioned what I call in my title \"the ethos\" of the twentieth century. To speak of \"the\" ethos is slightly misleading because it suggests that there is a single unitary such ethos. In one sense this is right, in that there seems to be agreement on a cycle of need-production-consumption pursued for its own sake. On the other hand, though, there are two slightly different ways of accentuating the components in this cycle. Marx emphasises the priority of production and tries to find a way of seeing how that production can be a form of human freedom and self-activity. This view dominated Eastern Europe during much of the twentieth century. The various ideologies of the capitalist societies, on the other hand, focused on consumption. The first way of inflecting the need-production-consumption cycle, just to repeat, was, I have claimed, finally unable to maintain itself partly because it was unable to out-produce the group of countries that eventually became the European Union and perhaps because it could give no plausible answer to the question of why production for its own sake was an end-in-itself and had no genuinely alternative value-system, or mode of self-congratulation, to propose. The second, consumption-oriented variety of this same basic worldview and form of society, as MacIntyre has shown, is the natural matrix for the various varieties of emotivism, existentialism, and moral anarchism that are characteristic of the philosophy and the life of our societies. Mainstream Marxism, then, was an attempt to answer the questions of the twentieth century in a way that was in some sense still commensurate with an important part of its ethos.\n\nHow might one break out of the cycle of need-production-consumption? One way would certainly be to find a single overarching goal outside it, if any such goal existed. And so it is very tempting to follow Aristotle down the path that leads from this reflection to the views that he presents in his ethical and political writings. In a famous passage in the beginning of the Nicomachean Ethics, Aristotle writes:\n\n\u0395\u1f30 \u03b4\u03ae \u03c4\u03b9 \u03c4\u03ad\u03bb\u03bf\u03c2 \u1f10\u03c3\u03c4\u1f76 \u03c4\u1ff6\u03bd \u03c0\u03c1\u03b1\u03ba\u03c4\u1ff6\u03bd \u1f43 \u03b4\u03b9\u0309 \u03b1\u1f51\u03c4\u1f78 \u03b2\u03bf\u03c5\u03bb\u03cc\u03bc\u03b5\u03b8\u03b1. \u03c4\u1f06\u03bb\u03bb\u03b1 \u03b4\u1f72 \u03b4\u03b9\u1f70 \u03c4\u03bf\u1fe6\u03c4\u03bf, \u03ba\u03b1\u1f76 \u03bc\u1f74 \u03c0\u03ac\u03bd\u03c4\u03b1 \u03b4\u03b9\u0309 \u1f15\u03c4\u03b5\u03c1\u03bf\u03bd \u03b1\u1f31\u03c1\u03bf\u03cd\u03bc\u03b5\u03b8\u03b1 (\u03c0\u03c1\u03bf\u03cc\u03b5\u03b9\u03c3\u03b9 \u03b3\u1f70\u03c1 \u03bf\u1f55\u03c4\u03c9 \u03b3\u0309 \u03b5\u1f30\u03c2 \u1f04\u03c0\u03b5\u03b9\u03c1\u03bf\u03bd, \u1f65\u03c3\u03c4\u0309 \u03b5\u1f36\u03bd\u03b1\u03b9 \u03ba\u03b5\u03bd\u1f74\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b1\u03c4\u03b1\u03af\u03b1\u03bd \u03c4\u1f74\u03bd \u1f44\u03c1\u03b5\u03be\u03b9\u03bd) \u03b4\u1fc6\u03bb\u03bf\u03bd \u1f61\u03c2 \u03c4\u03bf\u1fe6\u03c4\u0309 \u1f04\u03bd \u03b5\u1f34\u03b7 \u03c4\u1f00\u03b3\u03b1\u03b8\u1f78\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u1f04\u03c1\u03b9\u03c3\u03c4\u03bf\u03bd.\n\n[If then there is some end of our practical undertakings which we wish for the sake of itself, whereas we wish for the other things for the sake of it, and if we do not choose everything for the sake of something else (for if that is the case, the sequence will go on without limit, so that our desire was empty and vain) it is clear that this will be the good, i.e., the best.]\n\nThis is my own pons asinorum, the bridge from ignorance into the Promised Land of Understanding that I have found myself always unable to cross. Why, I have always wondered, should my desire be considered \"empty and pointless,\" just because it is part of a sequence of further desires that goes off into infinity (i.e., that continues indefinitely, has no definite stopping point)? I am thirsty today and want to drink. I know that even if I do now drink, I will be thirsty again tomorrow or the next day. Does that mean that my desire to drink now is \"empty\"? Even if my action on a specific desire is (doomed necessarily to be) frustrated and \"pointless\" (\u03bc\u03b1\u03c4\u03b1\u03af\u03b1), I do not see how that makes the desire or the course of action to which it gives rise empty. Such desires and such actions are an important, in fact constituent, part of human life. To take a slightly more weighty example, I may desire (both for its inherent properties and for the other good things it leads to) greater European integration, the prosecution of Tony Blair in The Hague as a war criminal, and a more rigorous standard in education without having any idea how these different goals might be related to each other. Even if I were to agree that I need some central structure of desire unified under a single overarching end to give my life meaning, I do not see how it follows from that that all my desires need to be seen in reference to that central core. I don't, of course, mean to assert that all human desire is undifferentially or equally good or valid, or that it might not be perverted, inopportune, or otherwise unwise to act on some desires in some circumstances, but has human desire no weight and dignity in itself at all, apart from its integration into a single overarching end? Isn't human life in fact like this: a series of desires that at some indeterminate point, we know not when, eventually peters out?\n\nOne of the reasons I have always felt great resistance to this Aristotelian line of thought is that it seems to me to add fuel to a view of human life that I find repellent. This is the view that life is either like a race or an \u1f04\u03b3\u03c9\u03bd with clear winners and losers. Or that it is the exercise of a craft with a determinate end, say like the production of shoes, so that univocal success or failure is always discernible. First of all, I have always thought there was much to be said not exactly in favour of failure tout court but in potential appreciation of some of the aspects of failure. One would think that this would be a line of thought that recommended itself with particular force to Christians who took seriously Paul's preaching of \"Christ crucified.\" Second, much of human life and some of the most interesting and important parts of it do not seem to me to lend themselves plausibly to analysis in terms of success or failure at all. Some aspects of life are not appropriately construed relative to the idea that life is a single continuous narrative about the desire for and pursuit of a single unified (conception of the) good. Furthermore, a human life \"as a whole\" does not seem to me at all like a single huge race or the deployment of a craft. It seems to me highly questionable whether my whole life admits of treatment as a single narrative in any interesting sense, but even if I were to grant that it is or could be such a narrative, the kind of narrative in question would have to be one that would be only contingently related to the \"story\" of a single \u1fbd\u03b1\u03b3\u03ce\u03bd, competition, or race. Narrative should be seen as a way of distancing ourselves from Aristotle, not of rehabilitating him. If we at any particular time give our desires some minimal order by reference to some conception of a single overarching good, we also know that those conceptions of a unitary good change during our lives. Any unity of desire is \"necessarily\" and unavoidably fleeting, transitory, fragile, and imposed on much more chaotic structures that are, however, not just nothing or \"empty.\"\n\nThus Aristotle's view has always seemed to me utterly implausible as a view about the nature of human desire and thus also of human life as a whole. Human desire has at least some validity in itself, a validity that is not derived merely from some external higher end to which it is devoted. The passage cited from Nicomachean Ethics has also always seemed to me to indicate that, regardless of the rather sophisticated discussion one can find in his work of forms of human action that are autotelic, Aristotle is still fixated on a relatively rigid division of actions into those performed \"for their own sake\" (as end-in-themselves) and those performed \"(merely) for the sake of something else.\" In fact if he were to be able to rid his thought of the centrality of this distinction and see it as of merely local or contextual relevance, the argument for the existence of a single overarching end, on which the rest of the work to some extent rests, would be seen to be implausible. Even interpretations of Aristotle that emphasise the extent to which the conception of \u03b5\u1f50\u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u03af\u03b1 (\"happiness\" or \"human flourishing\") will be affected by reflection on the real conditions of human life and real human desires seem to me to hold fast to the \"(mere) means\"\/\"end (in itself)\" distinction. It is one thing to say that in determining what the end-in-itself is, one must reflect even on the means that will be necessary for its attainment, and quite another to do away with the distinction between means adopted for the sake of something other than themselves and ends-in-themselves altogether.\n\nThere is, at any rate, a whole traditional sequence of philosophers who think of themselves as turning their backs resolutely on the strict Aristotelian distinction between means and ends, actions under the dominance of instrumental rationality and some form of autotelic human activity. I would include in this tradition Hegel, Marx (in some of his moods), Dewey, Trotsky (sometimes), and Adorno (but not Habermas). Obviously the members of this group differ in any number of significant ways, but what seems to me characteristic of all of them is their conviction that one must try to break out of the cycle of need-production-consumption and in particular get away from the idea that some component of that cycle is a kind of end-in-itself. Rather, however, than thinking that there is some other single good that is the one overarching human goal, as Aristotle claims, but that goal is not production, they hold that there is no pregiven over-arching single goal. Human life is a matter of biens \u00e0 construire rather than of a bien \u00e0 trouver. Furthermore, they connect this thought and see it as in some way inherently connected with the idea of overcoming the very distinction between means and ends or instrumental and substantive rationality, or at any rate demoting any such distinction to the status of a mere contextual convenience with no final significance. Thus Trotsky speaks of the \"dialectic of means and ends.\"\n\nDialectical materialism does not know dualism between means and end. The end flows naturally from the historical movement. Organically the means are subordinated to the end. The immediate end becomes the means for a further end.\n\nAppeal to such a dialectic, however, is one way to try to save the deepest intuition behind Marxism, which is that humanity should be capable of collectively self-organising activity, which instantiates appropriate self-control, self-direction, and even, when necessary, self-limitation, without needing to appeal to any external principle. The central idea that humanity is constituted by a self-activity in which the distinction between instrumental action and action performed for its own sake is not relevant is detachable from productivism. This specific conception of self-activity seems to me more perspicuously graspable in Hegelian than in Aristotelian categories. I am sorry to say that I believe that I disagree with MacIntyre on this.\n\nThat leaves the issue of Nietzsche. Simply deciding in discussion to abandon Aristotle's distinction between instrumental action and action performed for its own sake will not by itself suffice to do away with Nietzscheanism, unless such a decision was part of a successful project of action to transform society. Marx thought it in some sense pointless to \"refute\" religion because it arose from human needs that were not satisfied by a certain social formation and would only disappear when those needs were more directly and palpably satisfied. Similarly, Nietzscheanism will disappear only when it loses its plausibility as a mirror of and guide to and in our social life. That will happen only if our social life takes a different form. Nevertheless, the framework provided by some conception of collective self-activity gives us, it seems to me, the best chance we have of constructing a world in which Nietzsche would be an irrelevance. Even if such an institutionally embedded and socially realised framework effectively did exist, it would still be possible for individuals to make a decision to try to reject it, but this decision would have a completely different status from that which a \"lifestyle decision\" has now. It would be more like what Christians in the medieval period had in mind when they considered the possibility that the fool might say in his heart that there is no God. Taking seriously the attempt to abolish the distinction between instrumental and substantive reason, between actions performed merely for the sake of other things and actions performed for their own sake, and finally between means and ends, by transforming society in a way that would make these distinctions really marginal and subordinate would, I think, move one beyond the ethos of the twentieth century. Not, of course, that there is politically the slightest chance of this happening at present.\n4\n\nMust Criticism Be Constructive?\n\nThere is a widely held view\u2014at least in contemporary Anglo-American societies\u2014that \"merely negative\" criticism is somehow defective or inappropriate. It is part of the responsibility of a critic, it is assumed, not simply to denigrate some institution, social arrangement, or form of action but to do so while providing at least the suggestion of a \"preferable\" way of acting, or a \"better\" way of organising some sector of the society. Though this view is widely shared, it is perhaps not unimportant to recall that it is not universally held. One might think, for instance, of Bakunin, who notoriously claimed, \"Auch die Lust an der Zerst\u00f6rung ist eine schaffende Lust\" (\"The pleasure in destruction is itself a creative pleasure\"), or of Adorno, who insisted that, because of the almost limitless ability of modern societies to co-opt even severe kinds of criticism, philosophy must be a relentlessly negative form of dialectical activity. These examples illustrate perhaps the discomfort that the idea of nonconstructive criticism arouses. One might, of course, argue that the very idea of completely negative criticism, like its mirror image, absolutely well-founded knowledge, is conceptually incoherent, but the most usual source of unease is not so much that there is anything conceptually inappropriate with the idea of completely negative criticism, rather there is a fear that it is a concomitant of an anarchic abdication of responsibility on the part of the critic, so the suspicion is a moral rather than specifically epistemological or cognitive.\n\nSo I would like to discuss three interconnected questions:\n\n1. What is \"criticism\"?\n\n2. Is there a single unitary sense of \"criticism\" that can be found in the forms of \"criticism\" actually practised in different domains of human life (e.g., social criticism, aesthetic criticism, the moral criticism of individuals and their actions, cultural criticism)?\n\n3. Must criticism be constructive? What is supposed to be wrong with it, if it is not constructive?\n\nThe first of these questions raises all the old philosophical questions that have been discussed ad taedium (if not ad nauseam) for over two thousand years about what it is to give a proper account of a concept or practice. Should one\u2014can one?\u2014 give a strict definition (Plato), a genealogical account (Nietzsche), or some kind of looser specification of the \"grammar\" of the concept by locating it in the context of a linguistically structured \"form of life\" (as in the later Wittgenstein)? This first question is in fact distinct from the second. The first question is directed to the notion of \"criticism\" in general. Is there any coherent concept (in any area of human endeavour); if there is, is there one concept or many; how can it\u2014can they\u2014be specified? These are questions that one can perfectly reasonably ask, and answer, insofar as they can be answered at all, in the context of any single kind of human activity. You can ask this question perfectly reasonably even if the kind of \"criticism\" you have in mind is restricted to the evaluation of restaurants by the \"food critics\" whom certain newspapers now employ. The second question focuses on whether, in addition to, or independent of whatever general ambiguities there might be in the concept of \"criticism\" there is some disciplinarily specific difference between criticism as characteristically performed in the study of music, painting, or literature, and criticism as it is practised in science, ethics, or politics.\n\nAs far as these first two questions are concerned, I would like to take a non-Socratic approach. That is, I would like to suggest that there is no single invariable notion of \"criticism,\" which could be the object of strict formal definition, giving necessary and sufficient conditions, but this does not really matter because such formal definitions are not possible in most of the more important realms of human life, and neither human life itself nor philosophy is any the worse for this. There is no single concept of \"criticism,\" either in ethics, politics, or art, but what there are, are certain paradigmatic cases, and these represent a kind of ideally or fully developed \"criticism.\" Something can perfectly legitimately be called \"criticism\" that does not satisfy all of these conditions. I claim that we should think of \"criticism\" in its fullest possible form as comprised of four analytically (although not always really) distinct elements or nontemporal stages. Another way of thinking about this is to locate our usual usage of \"criticism\" at the point of intersection of four dimensions. I will call the first of these four the \"structural\" or \"analytic\" dimension, the second the \"evaluative\" dimension, the third the dimension of \"argumentative connectivity,\" and the fourth the \"performative\" dimension.\n\nThe first dimension is that in which \"criticism\" is related to \"analyse,\" \"trace,\" \"differentiate.\" \"Critique\" (and related terms) derive from the Greek word \u03ba\u03c1\u03af\u03bd\u03c9 and originally meant simply \"analysis,\" that is, the process of taking a complex apart in thought and specifying its constituent parts or elements. It was then also used to refer to the results of that process. Such analysis will never be strictly value-free, but it will also not be the case that by virtue of engaging in \"criticism\" of this kind I am in any way presupposing or expressing a negative attitude or judgement towards the object of criticism. \"Critique\" in this sense is self-evidently an activity of virtually universal application in all fields of human endeavour (even if, for contingent reasons, the word \"critique\" is not used in some particular language for what is going on). So I can perfectly reasonably be said to be engaged in \"literary criticism,\" if I analyse the parts of a poem or, for instance, its metre, regardless of whatever particularly evaluative attitude I might have towards it. If the poem is in the Alkaic strophe, and I state that this is the case and helpfully write out the scansion, that is a rudimentary form of literary criticism whether I especially like poetry written in the Alkaic strophe in general or not, or am indifferent, and whether I like this particular use of the Alkaic strophe or not (or am indifferent). By analysing the poem in this way I may be helping readers (or listeners) pay attention to features of the work that they might otherwise overlook or fail to notice. This may allow readers (or listeners) to acquire an enhanced engagement with the poem, whether this means that as a result a given reader or listener comes to evaluate the poem more or less positively than before, or whether the evaluation remains unchanged. In addition, even if the analysis does not change my evaluation, it may help me understand why I like or dislike the poem. Since for many people understanding why I have the reaction to a poem that I in fact have is part of the process of proper engagement with it, analysis may form part of the normal process of aesthetic appropriation.\n\nSimilarly, if I am a psychoanalyst I can in one sense \"criticise\" someone's behaviour by \"analysing\" it, that is, subjecting it to scrutiny of its origins, motives, symbolic associations, and so forth. I can do the same for a proposal for a piece of legislation. In the case of a human action or a piece of proposed legislation, \"analysis\" may go beyond tracing the internal structure of the proposed law and point out, for instance, what its consequences will be likely to be, what other measures will need to be taken in order to implement it, and so forth. I don't, it is true, usually call this \"criticism\" in English but rather \"analysis,\" but that seems a trivial or unimportant linguistic point.\n\nThe second of the four dimensions is one in which \"criticise\" means to have or adopt an attitude or to judge; usually the attitude in question is a determinedly negative one towards something, and so along this dimension \"criticise\" is related to terms like \"dislike,\" \"disapprove of,\" and so forth. It is in fact an exceedingly peculiar but undeniable fact that a term that originally (in the ancient world) referred merely to the process of separating that which was distinct, eventually developed in the direction of acquiring a distinctly negative connotation. To be sure, in this case there does seem to be a difference in at least our linguistic usage between the cases of forms of human action and cases of art in that it would be perfectly normal to speak of a \"critical\" attitude towards a work of art, meaning by that a discriminating attitude that resulted in a finally positive evaluation of it. On the other hand, we would, I think, never describe a laudatory or approving attitude towards a human action, a piece of legislation, or a social practice or institution as an instance of \"critique.\"\n\nWhat exactly does \"negative attitude\" mean in the above? One obvious thing it can mean is that I explicitly formulate a proposition to the effect that the (criticised) object (or action or institution) in question has some defect or that I dislike it, think it is unfit for purpose, reject it, will not tolerate it, and so on. I may formulate this proposition without ever uttering it, merely affirming it mentally, or I may state it repeatedly to all and sundry. Of course, though, I may develop a \"negative attitude\" towards something without ever formulating anything specifically in a proposition. As a result of long experience with some person I may come to dislike him or her intensely without ever becoming aware of this fact consciously and thus, a fortiori, without ever expressing this dislike in a proposition, even one I never verbally express. It would probably be a stretch of current English usage to say that this course of experience and its conclusion are a form of \"criticism\" or a \"critical process,\" but although they are an extension of the way we now usually speak, they are a comprehensible and, as it were, conservative extension. Finally, I might act in a certain way that could be construed as a form of criticism. To use the classic example drawn from one of the Icelandic sagas, suppose you as a host give me dinner and, as I take my leave, you ask, \"How was my dinner?\" If I thereupon vomit up the entire contents of my stomach on you, it does not seem utterly fanciful to see this as a kind of criticism-embodied-in-action (not words). Similarly, if I ostentatiously spit when I see you approach, or draw a moustache on your picture, or whistle a parody of your latest musical composition, or do a comic imitation of some of your mannerisms, these might be construed as very much like criticisms.\n\nSo when in 2011 a group of Egyptians and Saudis flew some airplanes into the two spectacularly ugly towers of the World Trade Center in lower Manhattan, knocking both towers down, but very strikingly not explaining why they were acting as they did, did that count as \"criticism\"?\n\nThis idea that I might criticise merely by acting without saying anything brings me to the third of my three dimensions. In cases of \"full-blown\" criticism, the first and second dimensions are explicitly connected. I would not usually think I had in front of me a case of \"criticism\"\u2014or at any rate of \"criticism\" in the full-blown sense\u2014if I simply tell you how the object is structured or if I simply express disapproval, or if I both have an analysis and have an evaluative attitude. To engage in criticism means not merely that I have an analysis and a judgement but that I cite structural or other aspects of the thing in question as reasons for my approval or disapproval, or I argue from the account I give of the object in question to my judgement about it. Obviously, in most cases the notion of \"argue\" here will encompass a significantly larger range of ways of acting than those usually countenanced by formal logic or by standard views about scientific inference.\n\nOf course, one might claim, and I think one would be right to claim, that the connection between structural analysis and evaluation in any case was deeper and more inherent than the previous account seemed to indicate. In most cases, after all, the analysis itself would not actually be performed in a strictly value-free way, without any influence of the values I held. Rather I would decide what parts it was relevant to separate out analytically by at least tacit reference to value judgements. This is no doubt true. However, it is no objection, if only because at least at the most superficial level it seems perfectly possible to defend Weber's distinction between what he calls \"aktuelles Werten\" and what he calls \"Wertbeziehung.\"\n\nThe fourth dimension is one in which \"criticise\" is connected with terms such as \"vote against\" or \"vote down\" a proposal, or \"to denounce\" an action or a person, or \"indict\" or \"prosecute\" a person. All of these things (\"indict,\" \"vote down,\" etc.) are in the first instance public actions. In any case these actions are not merely \"public\" but also institutional, that is, they are governed by specifiable practices of a certain kind. To \"indict\" someone (of a crime) is not to disapprove of or reject or distance oneself from what that person did on certain grounds that have to do with the way in which you think that action can be analysed, but it is to satisfy particular specific legal requirements that will be given by the legal system in force. This might mean shouting out \"Thief!\" in a bazaar, or pronouncing an accusation in front of a certain number of witnesses or a magistrate, or filing a written brief in a certain specified form, or whatever. Often in these formal contexts there must be a specified object of criticism (e.g., a person who is indicted as criminal, even if the name of that person is not known, as in British indictments of \"persons unknown\") and an institutional set of consequences. If you are successfully indicted, you pay a fine, go to jail, or come under the guillotine. The action in question formally specifies the person or thing criticised and will often connect it specifically with particular envisaged changes of status, actions, and so on. If I am a member of a prize committee, my vote must be for a particular work, and if enough others vote my way, the work I favour will receive the prize.\n\nIt is tempting to appeal here to the theory of speech acts that had some currency in the period between the 1950s and the end of the 1970s. There is usually a kind of associated institutional intention that goes with many of these actions. If I \"vote down\" a proposal, the institutional action I perform is, as it would be natural for us to say, \"expressive of\" my own, underlying (individual, \"mental\"), critically articulated disapproval of the proposal, or if I vote to give the prize to a certain work, that will usually be because I admire that work.\n\nThe relation of these \"institutional intentions\" to the real intentions of actual individual human beings (or groups) is, however, a highly complex matter, and probably more complicated than this immediate inclination of ours would suggest. On the one hand, it is probably correct that the whole mechanism we have developed for describing what we now call \"internal\" states of mind (and also of action) results from projecting terms that have more or less robust external referents when applied to public actions onto shadowy, invented \"internal\" states, mechanisms, events, and actions. As Hobbes would have put it, there are terms that have clear meaning \"in foro externo\"\u2014a process of \"deliberation\" is one in which the various members of a political assembly discuss with each other (out loud, usually in a public place, at a specified time) what is to be done. These terms can then be extended so as purportedly to apply \"in foro interno,\" that is, to merely mental phenomena of human individuals that are not necessarily ever expressed aloud.\n\nDespite all this, on the other hand, when such an institutional structure is formally established, this can allow the \"official\" intention embodied in the speech act to deviate significantly from the real psychic state of the person actually performing the action. Not every public prosecutor who initiates proceedings against someone who violates a law must actually disapprove of what the criminal has done. He (or she) might, as they say, \"just be doing my job.\" In various easily imaginable cases I may, a simple citizen eligible to vote or as a Member of Parliament, vote against something, for instance for tactical reasons, of which I do not really disapprove and would not (in itself) wish to criticise.\n\nIn some extreme cases the tail may even wag the dog. To give a very vivid instance of this, on the morning of August 21, 1968, forces of the Warsaw Pact, including the DDR, invaded the \u010cSSR in order to terminate the political and social experiments being carried out there under the leadership of Alexander Dub\u010dek. For various reasons, including probably tactical reasons, the official line of the SED\u2014the ruling Communist Party of the DDR\u2014up to the very evening before the invasion was that military action against a fraternal socialist country by the DDR was inconceivable, and reports of preparations for such an intervention were Western inventions, instances of black propaganda, or provocations. Unfortunately, a university lecturer named Roessler had a lecture at 7 am on the morning of August 21 and had neglected to listen to the news about the invasion. When asked by students in his lecture, all of whom had been listening to Western radio that morning, about what had happened, he defended what he took to be the official line, that military intervention against a fraternal socialist regime was unthinkable, and suggestions that it might even be contemplated were a provocation of the West. Roessler was then dismissed from his post for criticising the government. In a case like this it seems to me perfectly plausible, although I know that others will not find it so, to say that the unfortunate Roessler had criticised the SED and its policies. He had performed the public act of speaking against a decision the SED had made and implemented from a university lectern in front of numerous students. To be sure, in doing this he had in another sense no intention whatever of criticising the SED because he could assume that doing so would mean that he would lose his job (which is what occurred). Actually once one has proceeded thus far there is no reason not to add to this analysis. Perhaps the unlucky Roessler in the privacy of his own innermost thoughts really was a supporter of Dub\u010dek and critic of the SED but had enough sense never publicly to admit such a thing. In defending what he thought was the SED policy, he was performing what he thought was the required act of expressing public support for the SED's decision. It was just an accident that he had not appropriately kept up with the shifts in the official line, so he failed in his intention to support the party and in fact criticised it. He was, as it were, trying but failing to sell his conscience to political expedience.\n\nWe started with three questions, the third of which concerned the notion of \"constructive\" criticism. The intuition behind the idea of \"constructive\" criticism is that the object criticised should really be able to be improved by reference to a form of action guided by the criticism itself. The model here is that there are three items: a critic, an object (action, institution) criticised, and a \"target-agent\" to whom the criticism is addressed or directed. So I may criticise the medical authorities for permitting an unqualified surgeon to operate on the grounds that he killed my aunt in what was billed as a routine operation. The model here is that there are three items: a critic\u2014in this case, me\u2014an object (person, action, institution) criticised\u2014in this case, the complex fact that an incompetent surgeon killed my aunt\u2014and a \"target-agent\" to whom the criticism is addressed or directed\u2014in Britain this is the General Medical Council. It is constructive criticism to the extent to which the \"object\" could have been \"improved\"\u2014my aunt would not have died\u2014if the target-agent, the Medical Council, had been guided by what I am now saying, that is, had not allowed an incompetent surgeon to operate. The target-agent to whom the criticism is directed will be some individual (or some group of people) who stands in a special relation to the object criticised so that in criticising the object, the critic is also in some sense calling the attention of the target-agent to deficiencies in the object, which are thereby presented as being the target-agent's job to remedy. The most obvious reason for connecting criticism with a particular target-agent (or agents) is either: (a) that this person (or these people) can be held in some way responsible for the existing state of the object by virtue of which it is deemed worthy of being criticised or (b) that person (or these people) could, and ought, by adopting the criticism in question, act so as to improve the object and remove the grounds for the criticism. I merely note that the second and third of the three items may in some cases be the same. Instead of criticising the medical authorities for letting the surgeon operate, I could criticise him for operating when unqualified. Finally, if I were such a surgeon myself I could engage in self-criticism in which all three items were the same.\n\n\"Constructive\" criticism, then, goes beyond simple \"criticism\" in the third sense above in that simple \"criticism\" requires that one be able to specify what is wrong with the object, whereas constructive criticism requires in addition either the weaker condition that one could in some sense also specify what else would have to be changed in the world in order for the object to escape the criticism, or the stronger condition that one be able to specify what concrete steps the target-agent would have to undertake actually to remedy what is wrong.\n\nTo put what I wish to claim in another way, it is, it seems to me, one thing to say:\n\n(a) If this object lacked features ABC, it would escape criticism\n\nand quite another to say:\n\n(b) This\u2014{XYZ}\u2014is my positive alternative to the criticised object.\n\nStatement (a) is characteristic of what I have called \"argumentatively connected\" criticism. I disapprove of the object because I have reasons I can specify and they are ABC. It is my assumption that I can have (a) without (b), that is, that I can have the ability to specify in relatively general terms (\"criteria\" if one will) what is wrong with an object without necessarily being able to specify what particular configuration (of this object or one sufficiently like it) could exist that would escape condemnation by reference to those general criteria. A fortiori, I need not be able myself actually to produce an object or bring about a state of affairs that would escape the relevant criticism.\n\nI note that in English the notion of \"constructive criticism\" also often has the further implication:\n\n(a) that I can tell an appropriately constituted agent, the \"target-agent,\" how exactly he or she or they should go about producing an object or bringing about a state of affairs that is better.\n\nThink of this example: The world is overpopulated and resources are scarce. In addition, current policies of consumption are squandering existing resources and polluting the environment to an unacceptable extent. I now \"criticise\" the directors of British Petroleum (BP) for some policy their corporation has adopted. This is a form of \"criticism\" in the last and fullest sense I distinguished. The directors of BP are the target-agents because they are responsible for the current policy and (in some sense, although that would require considerable further analysis) \"could\" change it. I can point out to them features their policies would have to lack in order to escape the criticism I level at them. They would have to be less wasteful, more focused on satisfaction of real human needs, more likely to generate in consumers attitudes of prudence and moderation, and so forth. Now it might well be the case that such policies are not actually \"realistically\" possible given the fact that the energy sector is part of a capitalist economy where a certain motive and incentive structure is operative\u2014this is why I flagged \"could\" in \"could change \" above. \"Could\" under what conditions? One could easily imagine, then, that the directors of BP (perfectly reasonably from their point of view) rejected my criticism as not \"constructive.\" The basic point of my criticism might well not only be one to which they \"could\" not respond, given the constraints of the market economy under which they operate, but also be one the blunting of which would require that they and their whole organisation simply not exist at all in anything like its present form. They might well respond that if they adopted more enlightened policies, the only effect that would have would be to put them out of business, and then their place would merely be taken by Shell or some other corporation with policies that were effectively indistinguishable from BP's. That would not only be of no benefit to the directors of Shell but would also not represent progress towards anything that the critic could reasonably count as \"progress.\" Why prefer Shell if they have the same polices as BP? The only solution might be that all entities like BP be abolished, which would require dismantling and reconstructing the whole economic system. If the economy were to be completely revamped to allow these deleterious policies to be avoided, then people like those on the board of directors would have a completely different social role; they would not be directors of international corporations but would have \"honest\" jobs (and hence would have completely different desires, beliefs, attitudes, powers, etc.). It is not difficult to see how the current directors of BP might well fail to see this as \"constructive criticism\" in the usual sense of that term. But then that might well be their problem. What might count as \"constructive\" for us, that is, what we, given who we are, could do about something, given our identities and possibilities, need not be the same as what is constructive for them (given their identity and situation). Appeal to the requirement that criticism be \"constructive\" can thus often have the function of trying to shift the onus probandi in a particular way. I, as critic, am required to formulate my criticism in a way that is shaped to the action-related demands of the target-agents. I must criticise them (and their actions, the institutions in which they participate, etc.) in a way that conforms to what \"they\" define as what they can \"reasonably\" be expected to do and results they can \"reasonably\" be expected to accept.\n\nI should emphasise that nothing in what I have said suggests that it is illicit or inappropriate for a person or group to lament unless they can specify in general terms what the exact cause of the complaint is, or\u2014an even stronger demand\u2014unless they can propose a remedy. There is a category of what Adorno sometimes calls \"creaturely suffering\" (das Leiden der Kreatur), the expression of which is always legitimate. I can perfectly \"reasonably\"\u2014\"reasonable\" is not exactly the word I want here, but I trust the reader understands\u2014lament about the brevity of life or the fact that I am dying of a very painful ailment, although I know that life must end sometime or other, and there is nothing anyone can do to alleviate my present pain or lengthen my time among humans. It is also perfectly possible to lament in this way without there being any target-agent to whom the complaint is specifically directed (so one can't use this as a backhanded argument for God's existence). One might not wish to call this properly \"criticism,\" but what term one uses for it is not really important, provided one recognises the legitimacy of such expressions of pain and frustration.\n\nAs far as the question of whether criticism must be constructive is concerned, it is important, it seems to me, to distinguish this question from a completely different, though equally important, topic, namely whether criticism can always be \"internal,\" that is, whether a completely satisfactory, comprehensive criticism can be conducted that appeals only to criteria that are in some sense \"internal to\"\u2014it would be important to specify in what sense exactly one meant by \"internal to\"\u2014the object criticised, or whether at some point it was necessary to appeal to \"external\" criteria. One can see that the distinction \"constructive\/not-constructive\" is different from \"internal\/external\" by noticing that forms of internal criticism can be either constructive or nonconstructive: I can show you on reasons internal to your own conceptual scheme that there are various deficiencies in the way you act that you (or someone else) can put right, but if your conceptual scheme is a complete mess, if it is \"internally contradictory\" in the way Marx, for instance, thinks that of nineteenth-century bankers is, then Marx thinks he is engaging in a form of \"internal criticism\" which, however, has in one sense an utterly nonconstructive result.\n\nThe capitalist economic system, Marx thinks, will collapse from internal contradictions. Marx intends this in the first instance as a historical and predictive form of criticism, not a moralising one, but it would be accompanied, as it were, by an ethical shadow. Again his criticism of the capitalist banking system as a whole might not be \"constructive\" for present-day bankers in that there may be nothing they could conceivably do, compatible with remaining who they are, namely bankers, to respond to the criticism, and no life they could lead, as bankers, after responding to it because to respond to it adequately would require their social role no longer to exist. In fact, if the system really is doomed, there might be nothing constructive anyone can do, Marx thinks, about such a system and its institutions. Still the criticism might be constructive for us, that is, for those of us who are not bankers, in that it could tell us what we need to do, and it could even be action-orienting for individual bankers (although not \"constructive\" for them as bankers) because, as individuals, particular bankers could always change their employment and try to get proper jobs.\n\nThis shows the extreme importance in criticism of notions like possibility and necessity, alternative identities and courses of action, which points in what framework are taken to be fixed and which are taken to be variable. This in turn raises important general issues about the malleability of human nature and institutions, and the possible limits of such malleability, utopianism, tragic or otherwise irresolvable forms of conflict, and the \"substitutivity\" of goods, services, practices, and institutions. Certainly the idea of \"constructive criticism\" seems to be closely connected with the notion of substitutivity. By \"substitutivity\" I mean in the simplest case that one object or process can stand in for or take the place of another. I can take the sugar cube out of the bowl with a special set of tongs, but if there are no tongs, I could also use a spoon (i.e., \"substitute\" a spoon for the tongs). Even if there is no spoon, I could in principle fall back on the use of my fingers, which is slightly less hygienic but no less effective. A standard kind of \"constructive criticism\" would be a case in which I tell a child not to take the sugar cube with its fingers, because that is unhygienic, but rather to use the tongs. Here I am saying that the tongs are a viable substitute for the use of fingers in this context, and one that has certain advantages. Similarly in many, although perhaps not strictly all, cases I can substitute a fork for a set of chopsticks or vice versa. Action-related forms of criticism would seem to depend very heavily on claims to the effect that the criticised object, process, institution, and so on could be replaced by some other, that is, that there is a possible substitute for it.\n\nThe idea of substituting one thing or process for another is deeply rooted in human social institutions and thinking, and it is not obvious how we could get along without it. Thus, in Sophocles' Antigone (ll. 905ff.), Antigone gives as the main reason for her determination to bury her dead brother, Polyneikes, that her brother is irreplaceable in contrast to a possible husband or child. If she lost a husband, she says, she could always get another; the same is true of a child, but, given that her mother and father are dead, there cannot be a substitute for her brother. For that matter one might argue that the very earliest work in the Western canon, The Iliad, is nothing but an extended meditation on what can and what cannot be substituted for what else. If Agamemnon has to give his slave girl up, who will replace her? Agamemnon claims later that not even his wife would be a satisfactory substitute. Is gold a good substitute for bronze (VI.235ff.) or armour worth nine oxen a good exchange for armour worth a hundred? Can Patroclos be an adequate substitute for Achilles on the field of battle? How about if he is wearing Achilles' armour? Is dead Hector a good substitute for live Patroclos?\n\nNevertheless, there is a certain \"natural\" tendency we have to fall for an illusion about substitutivity. This illusion is the view that items (things, processes, institutions, practices, etc.) can be treated for the purposes of substitution atomistically. This means that one can ignore the wider context within which the item in question stands and discuss possible substitutes to it relative to one narrowly specified use or function. It is not, however, the case that this narrow focus always makes sense. It is true that one can \"in principle\" use either the Western combination of knife, fork, and spoon or chopsticks for eating most of the normal kinds of food with which inhabitants of the European Union in the early twenty-first century will be confronted, but two qualifications need to be added. First, this assumes that the \"food\" in question will admit of being eaten in either way. This is true of rice, potatoes, and most vegetables but not, for instance, soup or honey, which can be eaten with a spoon but not with chopsticks. So one must take account of the way in which the item in question \"fits into\" a wider human context; if it is a utensil to eat, then it must fit into the kind of food that will be eaten and the way in which that food will be used is prepared. The second qualification concerns one specific aspect of the \"total context\" within which (potential) substitution might take place. Chopsticks might be usable \"in principle\" to eat rice, and they might even be actually usable by millions of people in the Orient and hundreds of thousands in the West, but they are not usable, and hence could not substitute for knife\/fork\/spoon, unless the agents who are to use them have a certain specific form of manual dexterity to operate chopsticks (or knife\/fork\/spoon). Part of the \"context\" that must be taken into account is the relevant forms of habitual human action. Heidegger makes this point with great clarity in Sein und Zeit when he speaks of most of the items in our world as being \"inherently\" not isolated items but elements of a larger context: such items are what Heidegger calls \"Zeuge\" in a \"Zeugganzes.\"\n\nThe case is perhaps even clearer if one takes a modern calculating computer and an abacus. In most everyday contexts an abacus is just as good as a computer, and vice versa, but it does not follow from that that I can simply replace every abacus with a calculator and forget about it. The calculator needs to be operated by someone, and people with the skills needed to perform calculations on the abacus will not necessarily know how to operate a computer. Similarly, if the abacus is the standard way of calculating in shops, it might be important not merely that one get the \"right\" answer to the calculation of an economic transaction but also that it be clearly visible to all concerned that the transaction be \"fair\" (according to whatever are the local ideas about that). Doing it on an abacus (in that society) might well be a way of showing to everyone that all is aboveboard. Punching keys on a computer in a society in which computers are used only by certain elite operators might not satisfy these demands.\n\nIt is easy to see how this argument can then be expanded, because there is not \"in principle\" any determinate, natural stopping place for relevance of features of the context to the possible substitutability. Or rather Heidegger draws from this the conclusion that there is only one nonarbitrary stopping point and that is my own death (or rather my relation-in-living to my own death). This is something that is not further contextualisable (for me), where substitutivity reaches its limits. Despite one \"natural\" way of reading the story of Alkestis, no one can die \"in my place,\" or rather even if she does in one sense die in my place, she does not die my death. I still have my own death to die; it will simply be an externally differently configured one.\n\nSo it makes no sense to think about substitutivity of items apart from their context and that context is open-ended. This may be one of the reasons why the scope of criticism seems naturally to expand and why repressive regimes often react hysterically to what seem to be very minor forms of criticism; once it starts there is no telling where it will go (and where it will end). One especially important aspect of this \"context\" is the cost of substitution, both of the new item that is to be provided and of the transition to the new mode of provision. Marie Antoinette famously failed to take the first point when she asked why the starving Parisians who were rioting because of lack of bread did not simply eat (the significantly more expensive but widely available) brioche. She ignored the price of brioche, which, apparently, even at the best of times was beyond the means of the Parisian poor. A computer is virtually certain to be more expensive than an abacus. Of course, if one were seriously contemplating such a substitution one would also have to consider the significant environmental degradation that producing computers represents, the cognitive and emotional deskilling that using them habitually entails, and so forth. One can, of course, construe \"price\" in a more general way to refer not merely to the monetary cost but to other less tangible costs: You might be able to make more money if you changed jobs, but that would require you to work weekends, and use of the computer might have the (virtually hidden) social cost of destroying the immediate visibility of certain properties of economic transactions (e.g., whether or not they were \"fair\" by whatever the local standards of \"fairness\" were).\n\nEven completely radical forms of political criticism will need at least to some extent to be committed to some kind of substitutivity. It will not, of course, be the case that one can atomistically compare a new structure of the stock market after the revolution with the structure before, because there may be no stock market after the revolution. There may be no banking system we could recognise, but there will still be forms of future-oriented cooperation, and at some sufficiently general level there will have to be a new way of providing foodstuffs to the population that will replace the old way. It might also be the case that it is possible only retrospectively to see that the new form of agricultural production really does \"substitute\" for an older form, but that is a different issue. The more one thinks about radical substitution, the more one must confront the question of the interconnection of human tastes\/desires\/needs on the one hand and ways through which these tastes are satisfied on the other. I mean by \"interconnection\" a relation of influence that is specifically construed as operating in both directions. Given that we have certain tastes, we wish to satisfy them in certain ways with certain objects, processes, and forms of human interaction, but those forms of interaction, processes, objects, and so forth in their turn strongly influence the tastes\/desires\/aspirations\/needs themselves. I may initially prefer my comfortable abacus and its role in everyday interactions with my peers. If a computer is forced on me, however, for whatever reason, as a substitute for the abacus, the continuing operation with the computer may finally change my taste and even generate in me a perverse new \"need,\" for example, for yet more technologically advanced (even if no more efficient) products. I become, as we say, \"hooked on gadgets.\" We can then restructure our industrial plant to produce ever new and shinier gadgets and initiate a self-reinforcing cycle.\n\nThis \"interconnection\" of production and consumption (as Marx would say) in the sphere of art was clearly noted by various nineteenth-century artists. Wagner's notion of a \"music of the future\" seems to belong here. Wagner thought that his music made people aware of deep needs or desires they had that were incompatible with the existing political order and that therefore were (and had to be) repressed by what he called \"the state.\" Thus the state required a stable family structure of a certain kind, one in which incest, for instance, was forbidden, but our sexual nature was anarchic. Die Walk\u00fcre dramatised and musicalised this tension. Listening to Wagner's music was thus a proto-revolutionary act because merely by becoming aware of those repressed desires we undermined the hold the state had on us. On the other hand, his music could be fully appreciated only by an audience that successfully completed that process of emancipation from political and religious repression. Certain works of art can create an audience that comes to have a taste for themselves. Late Beethoven string quartets are an acquired taste, and the only way to acquire the taste is to listen to them carefully. This makes the issue of substitutivity much more complex. You can't simply plug a Wagner music-drama into the repertory one evening in place of a Rossini opera, as if they were just different ways to satisfy the same human needs. There is even a crude economic recognition of this fact in that many opera houses will give ticket holders a refund if a different opera from the one scheduled is performed but not if there are changes to the cast. So the idea is that any other tenor is substitutable for Pavarotti, but Tosca is not substitutable for Turandot. Needless to say, this makes the issue of criticism even more complicated.\n\nThe more one emphasises simple substitutivity as a precondition of effective criticism, the less modern art would seem to be amenable at all to the kind of criticism we habitually use in politics. Perhaps in ancient times works of art were substitutable. When the singer appears in the Bronze Age Hall, he can sing one song or another, and although one may be artistically better, more appropriate, or more warmly received than another, basically they are interchangeable. Perhaps one singer (or one song) was in fact unique in the sense that it was discernibly much better than all others. We are used to the claim that some forms, at least of serious modern art, however, make an even stronger claim. Such specifically modernist works of art aspire not just to be unique\u2014better than all others but better relative to a common standard or set of criteria\u2014but also to be utterly original and completely different from all previous art. A particular modernist aesthetic also seems to present this as a categorical property of art, and a distinguishing characteristic of it in opposition to the objects of use and consumption we encounter in our everyday life. In Qu'est-ce que la lit\u00e9rature Sartre claimed that every work of art is trying to destroy every other work. This is a reaction to the neutralisation of art through its incorporation into museums and a tacit rejection of Andr\u00e9 Malraux's Mus\u00e9e Imaginaire. It is, however, equally closely connected with one of the least plausible and least appealing ideas of modernism, the idea that the \"work\" must be a full, all-encompassing cosmos like the Hegelian System, which at the same time depicted everything, supplanted all other attempts correctly to depict anything, and gave the canonical terms within which alone it should itself be understood and criticised. How can a work of art make itself absolutely nonsubstitutable? By exhausting the universe, encompassing everything and thus demoting every other work to a mere pale reflection of itself, or something \"subordinate,\" or by brutal destruction of the other.\n\nIt does seem, however, that the concept of \"criticism\" in the modern era can be seen to move in two rather different directions. First, criticism is connected with giving a definitive negative judgement on something according to the acknowledged standards. Here there will be a tendency to narrow the vocabulary used and the criteria. The model here is juridical. The judge does not much care about any number of properties of the action but wishes to construe it so as to give a legally binding definitive judgement on it. Is it larceny or not? The judge can give a positive or negative decision; the critic in this sense is a kind of relentlessly negative judge. The second direction is that taken by the notion of \"criticism\" in literature and the arts. Here the point is not to get a single definitive judgement according to narrow and focused criteria but to point things out to people, allowing them to enter into the work of art and, as we say, \"appreciate it.\" Here a critic is trying not to sharpen, restrict, and discipline the language but to enrich it, change it, reconfigure it so that it is able to serve to draw us into and permit us to see the point of new forms of human experience. Both of these are perfectly legitimate variants of the \"full-blown\" form of criticism.\n\nTo conclude, then, in response to the first two of our three questions, there is no single conception of \"criticism\" as traditional philosophy would like to require, but neither is the notion of \"criticism\" in any interesting sense unclear or merely diffuse. It has a variety of different uses, as do most concepts, but they are individually in no way deficient in clarity, and the whole array of uses can be organised around some paradigmatic cases and a set of structural and functional features, which have as much coherence as most other \"everyday\" (i.e., nontechnical) concepts have. The notion of \"constructive criticism\" is specifically connected with criticism in the realm of human ethical and political action and seems to have little relevance for the internal study of art, literature, or music as quasi-autonomous realms of modern human life, precisely because of its close connection with potentially \"substitutable\" forms of action (or \"products\"). This, however, may turn out to be a relatively superficial fact about contemporary high art that results from two contingent features of the way in which such high art and art criticism function in our society. First of all, it may have to do with the great emphasis put on the appreciation or consumption of art rather than on its production. Many theorists, mostly notably perhaps Nietzsche, have pointed out the deformation involved in thinking about art exclusively from the point of view of a viewer (or reader, or listener) confronting a finished work. What \"constructive\" comment can one make about a complex object that already exists and has the properties it has? Is one to imagine another, but different, completed work that \"better\" satisfies the spectator? In what way exactly? If, on the other hand, one imagines an artist standing, as it were, in the atelier of a fellow artist during the process of creation or elaboration, that is, while the work is in statu nascendi, one could imagine a set of collegial suggestions for how to continue. Perhaps the model should be that of a craftsman giving another aspiring craftsman advice on how to produce a ceramic pot with a spout that actually pours without spilling. The second feature of our notion of (at any rate \"high\") art that might be relevant is the extreme post-Romantic cult of originality and the associated rejection of craft production as a model for artistic creation. If each object must not merely be a uniquely skillful instance of satisfying recognised canons but a work that overthrows all existing standards and posits new and different criteria for its evaluation (which it itself uniquely formulates), then it is more difficult to see how \"constructive\" criticism of art could be possible under those circumstances.\n\nIn the political realm appeals to the need for \"constructive\" criticism can in principle represent a (generally laudable) attempt to remind those involved in some evaluation of human action of the need to remain aware of a kind of internal demand under which such criticism operates, namely of the need to keep Tschernyschevsky's (and later Lenin's) central question \"What is to be done?\" firmly in mind; in fact, however, the demand for \"constructive criticism\" in general functions as a repressive attempt to shift the onus probandi and divert attention from the possibility of radical criticism.\n5\n\nThe Loss of Meaning on the Left\n\nBy, at the latest, the final decade of the nineteenth century, many thinkers were diagnosing a deep malaise in Western culture, which expressed itself in various forms of individual and social disorientation. Thus, Durkheim claimed that increasing suicides rates were connected with the growth of what he called \"anomie,\" that is, with the fact that people were losing a certain kind of normative orientation they had once had. Suicide is an act performed by people who no longer know how they \"ought\" to deal with the various crises of human life. Durkheim's account might be seen as a kind of sociological confirmation of Nietzsche's speculations about a state of affairs in which \"the highest values\" that had in the past guided human actions\u2014the values embodied in Christianity and its various secular successors\u2014\"lost their value\" for people in the late nineteenth century, leaving them so confused, puzzled, and at a loss that they are in danger of being unable to discharge certain vital functions. In art, too, the emphasis on \"originality\" that was a characteristic of Romanticism led in some cases to a cult of the \"new\" that distanced works of art so much from traditional forms that they completely outstripped the ability of many audiences to recognise them as meaningful forms of \"art\" at all. How was one to react to something as different as the music of Sch\u00f6nberg or Joyce's Ulysses or cubism? What is one to look (or listen) for? By what standards is it to be evaluated? What is an appropriate response to the work? One might think of an imaginary threshold that is passed when people begin to react to new art not by thinking or saying \"That is bad music [painting, literature]\" but \"That is not music [painting, literature] at all.\" This experience of puzzlement, incomprehension, loss of control, and dislocation can be reflected in audiences' reactions to \"new\" art, but it can also become part of the artist's experience. At the very end of the nineteenth century the highly precocious Austrian poet Hugo von Hofmannsthal wrote his influential Ein Brief des Lord Chandos in which he describes a process in which words themselves begin to lose their meaning for him and he becomes increasingly unable to put them together in an ordered way to express a coherent thought. This state of disorientation spreads and eventually becomes so entrenched that it disrupts the normal course of his life.\n\nMein Inneres aber mu\u00df ich Ihnen darlegen, eine Sonderbarkeit, eine Unart, wenn Sie wollen eine Krankheit meines Geistes.... [D]ie... Begriffe entziehen sich mir.... Es ist mir v\u00f6llig die F\u00e4higkeit abhanden gekommen, \u00fcber irgend etwas zusammenh\u00e4ngend zu sprechen oder zu denken.... Es wurden mir auch im famili\u00e4ren und hausbackenen Gespr\u00e4ch alle die Urteile, dieleichthin und mit schlafwandlerischer Sicherheit abgegeben zu werden pflegen, so bedenklich, da\u00df ich aufh\u00f6ren mu\u00dfte, an solchen Gespr\u00e4chen irgend teilzunehmen.\n\n[I must reveal to you something strange in my inner life, a bad habit, an illness of my spirit, if you wish.... Concepts escape my grasp.... I have completely lost the ability to speak or think about anything in a connected way. Even in the most banal conversations in the family all the judgements that I was accustomed to give easily and with the security of a sleepwalker, came to seem so questionable that I had to stop taking any part in such conversations.]\n\nA further imaginary threshold would be crossed when someone for the first time connected all these apparently diverse phenomena\u2014anomie, the problems artists experienced in attaining coherent artistic expression, the widening gap between what audiences can comprehend and what advanced artistic production can provide, various forms of psychic derangement, confusion about individual and social values and goals\u2014and subsumed them under a single concept: loss (or \"crisis\") of \"meaning.\" It is in fact very striking that what might seem to be such varied problems come to be conceptualised as loss of \"meaning.\" In the ancient world no one asked about \"the meaning of life\" or, for that matter, the \"meaning of art\" in the modern sense in which those questions are sometimes asked. To be sure, ancient philosophers asked questions about how one might best lead a human life, and some of them speculated about divine purposes and a possible continued life of the human soul after death, but these investigations were pursued under a variety of different rubrics, or as enquiries into \"the good,\" not \"the meaningful.\" Similarly, questions occasionally arose about \"the meaning\" of some particular artefact, as in the famous passage from Ovid's Ars amatoria in which a girl asks a boy which mountains and which rivers are represented by the floats in a triumphal procession. This section of the poem might be entitled \"How to pull girls at a Roman victory parade\":\n\nspectabunt laeti iuuvenes mixtaeque puellae,\n\ndiffundetque animos omnibus ista dies.\n\natque aliqua ex illis regum nomina quaeret,\n\nquae loca, qui montesquaeue ferantur aquae,\n\nomnia responde, nec tantum si qua rogabit;\n\net quae nescieris, ut bene nota refer.\n\nhis est Euphrates, praecinctus harundine frontem;\n\ncui coma dependet caerula Tigris erit;\n\nhos facito Armenios, haec est Danaeia Persis;\n\nurbs in Achaemeniis ualle ista fuit;\n\nille uel ille duces, et erunt quae nomina dicas,\n\nsi poteris,uere, si minus, apta tamen. (Ovid, Ars amatoria I.217ff.)\n\n[Happy young men and women will be watching all mixed together because such a day relaxes everyone's spirits. If some young lady from among the crowd asks you the names of the kings [led in procession or represented on the victory floats], what these places [represented] are, what these mountains and waters are called, give her an answer to every question, and keep on volunteering information even if she doesn't ask any further question. If you don't know, just refer [to the relevant information] as if it were well-known: \"This [you might say] is the Euphrates, with his forehead surrounded with reeds; this, with the blue hair hanging down, will be the Tigris; those must be Armenians, that is Danaean Persis; that will be a city in Achaemenis. This one and that one are leaders. Give them whatever names you will, their correct names, if you can, but, if not, at any rate names that seem suitable.]\n\nQuestions were also asked about the way art acted on humans, and the value it might have for us, but again these diverse questions were not construed as connected through a purportedly unitary concept of \"meaning.\" Is the fact that in the contemporary world these various things are construed to be connected in this way significant?\n\nI'd like to start the discussion by contrasting the treatment given by two different authors of two boys, each of whom is standing beside a pool. The first is Narcissus and the second an anonymous boy who is cited by Hegel in his lectures on aesthetics.\n\nAs presented by Ovid, the story of Narcissus, like that of Oedipus, has radically anti-Socratic implications. Narcissus's mother asks the blind seer Tiresias about his life prospects, and the seer, reprising, as it were, his role in the Oedipus plays, replies that Narcissus will be fine as long as he does not come to know himself:\n\nde quo consultus an esset\n\ntempora maturae visurus longa senectae,\n\nfatidicus vates \"si se non noverit\" inquit. (III.346\u201348)\n\n[ consulted would see a long and ripe old age, the prophetic seer said: \"If he doesn't know himself.\"]\n\nOedipus cannot tolerate the truth about himself because it is too painful shaming; Narcissus, it turns out, because it is too absorbingly pleasurable. Narcissus was absorbed by his own image seen in a pool and fell in love with it. Even hearing the final phrases of his own words repeated back to him by Echo (a disembodied nymph who was enamoured of him) was not enough to spark his interest and bring him to reciprocate Echo's love. Only the full visual image of himself would do as a love-object. In the story, to love the self is not necessarily to approve of it in any normal sense; Narcissus, after all, seems to spend most of his time reproaching the image of himself for its failure fully to accede to his erotic advances, but he loves it nonetheless.\n\nHegel has a slightly different take on the boy-by-the-water.\n\n[Der Mensch befriedigt ein] allgemeine[s] und absolute[s] Bed\u00fcrfnis durch Ver\u00e4nderung der Au\u00dfendinge, welchen er das Siegel seines Inneren aufdr\u00fcckt und in ihnen nun seine eigenen Bestimmungen wiederfindet. Der Mensch tut dies um als freies Subjekt auch der Au\u00dfenwelt ihre spr\u00f6de Fremdheit zu nehmen und in der Gestalt der Dinge nur eine \u00e4u\u00dfere Realit\u00e4t seiner selbst zu genie\u00dfen. Schon der erste Trieb des Kindes tr\u00e4gt diese praktische Ver\u00e4nderung der Au\u00dfendinge in sich; der Knabe wirft Steine in den Strom und bewundert nun die Kreise, die im Wasser sich ziehen, als ein Werk, worin er die Anschauung des Seinigen gewinnt. Dieses Bed\u00fcrfnis geht durch die vielgestaltigsten Erscheinungen durch bis zu der Weise der Produktion seiner selbst in den Au\u00dfendingen, wie sie im Kunstwerk vorhanden ist.\n\n[Man satisfies a general and absolute need by changing [the form of] external things; he impresses the seal of his inner [life] onto them, and finds again in them his own determinations. Man does this in order, as a free subject, to take away from the external world its rigid foreign-ness and to enjoy in the form of things nothing but the external reality of himself. The very first impulse of the child bears within itself [an orientation towards] this [kind of] practical transformation of external things; the boy throws stones into the stream and then admires the circles that radiate in water, and he admires them as a work in which he has attained a way of seeing immediately something which is his own. This need is thorough-going through the most manifold appearances, and extends as far as the mode of production of himself in the external things which is present in the work of art.]\n\nHis boy does not lie down lazily next to a completely undisturbed pool he has accidentally come upon, seeing his own image only when he gets thirsty enough to try to drink from the pool, like Narcissus. The activity Hegel attributes to this boy is radically nonutilitarian\u2014he is not throwing stones into the stream to kill fish for dinner\u2014and it has no evident moral or ethical dimension\u2014it is not in any obvious way connected with his relations to other people, or with the satisfaction of any evident obligation or duty he might have to himself or to others. He throws the stones into a stream and admires the circles they form in the water because they are something he himself has produced; they are \"ein Werk, worin er die Anschauung des Seinigen gewinnt\" (\"a work in which he attains a direct perception of his own\"). In acting in this way, Hegel asserts, the boy is expressing his freedom and impressing his inner life onto objects in the external world. The impulse to practical transformation of the external world so that it can be seen as \"our kind of thing\" is a fundamental part of human life, an essential human need, Hegel claims, and it is connected with all the higher achievements of humanity, particularly art. Not only is Hegel's boy more active than the lethargic Narcissus, exercising at least some minimal human skills (picking up a stone and successfully hitting the water with it), but he must also perform a much more sophisticated cognitive feat than any with which Narcissus is confronted. Narcissus eventually recognises himself in the visual image in the pool, but this is as far as he gets, and he was not capable of recognising even his own words, when fragmentarily repeated by Echo. Hegel's boy, in contrast, succeeds in recognising something that has a completely different look or appearance from that of a small boy, namely a set of circling ripples in water, as (in some sense) \"his own\"; \"his own,\" not \"himself.\" The distinction is of great importance and being able to make it is a significant cognitive achievement. Finally, Narcissus's love of his image is so absorbing that the young boy loses all interest in anything else and languishes away. In contrast, the boy in the Hegelian story \"admires\" his work, but he is not said to \"love\" it with Narcissus's all-consuming, literally deadly, passion. Self-admiration may, then, be a form of self-love, but it is an attenuated form that allows for a certain distance and a certain contact with external reality.\n\nI want to suggest that Hegel's image is a good starting place for thinking about the phenomenon of meaning in modern society more generally, and in particular about the question of \"the meaning of life.\"\n\nTo elaborate this slightly, we can begin by noting three properties that the activity of the boy possesses and that we might think are likely to be relevant to making the activity \"meaningful\" for him. The activity satisfies three conditions. First of all, it is a temporally extended, intentional activity by the boy himself. Notice that Hegel does not speak of the boy accidentally dropping a stone into the water and being completely surprised by the result, nor is this a single unique episode of throwing a stone in the water. One must, I suggest, imagine the boy standing on the shore and repeatedly throwing stones, so that he builds up an internally coherent set of skills and expectations. He develops a rudimentary orientation towards at least one small segment of the world. Second, the activity must have a visible external result. These expectations and thus the orientation the boy develops must at least to some extent be the result of an activity directed at something that can be experienced as external, some state of affairs outside the boy. Furthermore, the boy's expectations must track reality at least in a minimal way. That is, one would perhaps hesitate to call the activity fully meaningful if it was based on a complete delusion about its own nature and its location in the world. For the boy to recognise himself meaningfully in the ripples, they must really exist, and not be the mere imaginings of a fevered brain.\n\nNarcissus does not fully satisfy this condition. A third condition is that the boy enjoy the process and admire the result. His orientation must contain some element of positive valuation or affirmation of self and the activity must in some sense be satisfactory to him.\n\nOne might think of this third condition as a specifically modern one. Saint-Just during the French Revolution notoriously remarked that \"happiness is a new idea in Europe\" and Hegel accepts a variant of this, although, for a variety of reasons, he does not use the terminology of \"happiness.\" Meaningfulness in the modern world must have a component that connects it with individual well-being.\n\nDas Recht der Besonderheit des Subjekts, sich befriedigt zu finden, oder, was dasselbe ist, das Recht der subjektiven Freiheit macht den Wende-und Mittelpunkt in dem Unterschiede des Altertums und der modernen Zeit.\n\n[ the right of particularity of the subject, the right to find itself satisfied, or, what amounts to the same thing, the right of subjective freedom is what constitutes the turning point [in the transition from] antiquity to modern time and the central point .]\n\nOne might observe, quite correctly, that this seems to be two conditions: \"satisfaction\" (or \"welfare\" or \"individual happiness\") and subjective freedom. Hegel, however, takes these to be \"the same thing,\" presumably because modern people will not be satisfied if they are not subjectively free, so in theoretical contexts one can use either formulation (\"satisfaction of particularity\" or \"subjective freedom\") ad libitum. One can see the significance of this if one considers a premodern worldview, like traditionalist Calvinism. Calvinism is a specific Christian doctrine that gives an all-encompassing theory of human life both in this world and in a purported world human souls will inhabit after death. One central component of this worldview is that God has created each human soul and predetermined it to salvation or eternal damnation before it is even born. Those souls created by God for eternal damnation serve the function of glorifying his infinite justice because they are created as deserving this fate. Let us now assume that Calvinism is true, that is, that it tracks reality correctly. So if I am one of the damned, my life has a completely determinate \"sense\" or \"meaning\" that I might even be capable of grasping. I might know full well that I am a reprobate sinner and about to spend the rest of eternity being subject to the most horrible and exquisite tortures devised by God to demonstrate his justice. This is a life that is as full of \"meaning\" as it could possibly be. Perhaps everything in it, every particular event that occurs to me, is visibly part of a\u2014from my point of view utterly horrid\u2014plan to lead me to my well-deserved punishment. Needless to say, from the fact that my life is full of meaning in one sense, it does not follow that it is full of a positive meaning I can affirm. By virtue of adhering to Calvinism I may be fully oriented in this life, but it is not one I \"enjoy\" as the boy enjoys throwing stones into the water. So sometimes \"meaningful life\" means, as it properly ought, a life that exhibits a pattern and in which I can orient myself whether for good or ill. Sometimes \"meaningful life\" means one in which I have orientation that is true (or at least not false). In a further sense, my life is meaningful only if it presents my life as something having positive value for me.\n\nNote that even if Calvinism were to be true, and I were correctly to recognise myself as one of the \"Elect,\" not, as in the previous example, as one of the Damned, that is, I was one of those whose life could correctly be seen as having positive value, that would not suffice to give my life \"meaning\" in the fully modern sense because the meaning of my life would be one given it by God, not created by my own action. One might think of this as a fourth condition: the \"meaning\" in question must be something we ourselves create in the way the boy creates the ripples in the water, rather than something we simply find, pregiven in the world in the way we might find a vein of silver in a mine or a new species of ape in the rainforest. Clearly to say the boy \"creates\" the ripples is not to say he creates them ex nihilo; the water must preexist the ripples. Still the ripples are the effect of the boy's action and seen as such. Equally clearly, the project of distinguishing between what is \"found\" and what \"created\" is an exceedingly delicate one that may never admit of a universally satisfactory analysis.\n\nMarx, following in the line of Hegel and Feuerbach, expresses this modernist sensibility very well when he speaks of the need for modern man to be his own sun and \"circle around himself,\" that is, see himself as the source of what meaning there is, not have a form of meaningfulness imposed on him from an imaginary Other.\n\nDie Kritik der Religion entt\u00e4uscht den Menschen, damit er denke, handele, seine Wirklichkeit gestalte wie ein entt\u00e4uschter, zu Verstand gekommener Mensch, damit er sich um sich selbst und damit um seine wirkliche Sonne drehe. Die Religion ist nur die illusorische Sonne die sich um den Menschen bewegt, solange er sich nicht um sich selbst bewegt.\n\n[The critique of religion disappoints\/disillusions man, so that he might come to think, act, and form his reality like a disillusioned man, someone who has come to his senses, so that he comes to rotate around himself, and thus around his real sun. Religion is nothing but the illusory sun, which rotates around man, until such time as he rotates around himself.]\n\nMarx can also be thought to represent a continuation of a line of thought begun by Hegel (and, before him, by Herder) in a further respect. To return to Hegel's example, the subject of activity, the boy throwing stones, is an individual human person, but this must be a simplification for the purposes of exposition. After all, it is Hegel himself who sees it as one of his major theoretical advances to replace the old forms of discussing philosophical problems in terms either of particular human individuals and the psychology or abstract structures with the phenomenon he calls \"Geist,\" which is explicitly said to be an \"I that is a We and a We that is an I,\" that is, an inherently social phenomenon. The example presents a case of an individual acting and comporting himself in a certain way, but, of course, one would really also have to take account of the fact that virtually none of the phenomena connected with \"meaning\" are asocial.\n\nMarx emphasises the centrality of socially organised production, developing the thought that it makes no sense to think that any individual could lead a meaningful life, or for that matter any life at all that we could recognise as human, outside a social context. Put crudely, if the image of meaning and meaningful action is the boy throwing stones into the stream, exercising his power and impressing on the world an image of his control over it, the only way the boy will have sufficient mastery over the basic conditions of his real life will be if he has control over his socioeconomic life. His basic conditions of his socioeconomic life, however, are unlikely to be the sort of thing over which he as an individual is ever going to have any significant degree of power. Such control over the economy and social life as is possible is possible only if exercised in a conscious, collective way by the members of a group as a whole. The basic modality of that collective control must be power over nature and mastery over our productive capacities and economic life, a control exercised through science, technology, and politics. Collective productive activities, Marx concludes, are the kernel of a meaningful life. Furthermore, in a properly constituted economic and political order, the very distinction between instrumental and noninstrumental action can be broken down so that the boy can satisfy his absolute need and eat the fish he kills for dinner. Or rather the boy can be appropriately integrated into a work unit that collectively satisfies the absolute needs of the members of the work unit while providing fish for themselves and the other members of society. In a society in which work and collective social life was sufficiently satisfying, one might think, the very question of the \"meaning of life\" would not arise. The very fact that this question does arise for a particular person in a particular society is a sign that that question for that person (in that society) has no answer. \"The meaning of life\" ought not to be reified. To know \"the meaning of life\" does not mean to know any possible discursive answer that can be given to questions about life. Questions ostensibly about \"the meaning of life\" are really about whether the social processes are satisfactory or whether certain individuals have a certain capacity or skill, whether they \"know how\" to lead a life of a certain kind, and they exhibit this knowledge in the only way such knowledge can be exhibited: by actually leading such a life.\n\nAgainst this basically Hegelian-Marxist line of argument a series of anarchist and existentialist thinkers objected in the name of the subjective human freedom that is a component of the original synthesis. Hegel may say that \"satisfaction\" and \"subjective freedom\" are the same thing, but in fact in his theory of \"satisfaction\" comes to be interpreted as \"rational satisfaction,\" that is, only such satisfaction as the Hegelian system thinks is both actually accessible and rationally desirable, and \"subjective freedom\" comes to be reduced to the acceptance of that which is deemed rationally acceptable: \"Ubi rationale, ibi bene; ubi bene, ibi libenter.\" One may think of this rehabilitation of not necessarily rationalisable subjective freedom as a fifth condition for leading a fully meaningful life in the modern world. An individual may find meaningfulness in integration into the productive activity of a collectively organised society, but then again he may not. Whether it is likely that such integration will be satisfactory depends on many empirical factors, including details about how the society is organised, and perhaps features of personal temperament. But even if a social group is optimally well organised, it is not, or should not be, a foregone conclusion that that is what the individual will opt to do. Is he to be forced? Forced to lead a meaningful life against his will? Wouldn't that return us to a structure of externally imposed meaning not utterly different from Calvinism?\n\nThe reference to the anarchists\u2014I'm thinking particularly of Max Stirner, but of course there are also others\u2014brings the political dimension of this discussion to the fore. Many traditional philosophers did not think that taking a position on politics was an integral part of their philosophical view itself. The members of the Frankfurt School, however, did. Their relation to politics was part of their own self-conception as philosophers. The members of the school construed themselves as a basically Leftist position, that is, as standing in general very firmly in the intellectual, cultural, and political tradition of the European Enlightenment, continuing the Enlightenment tradition of secular rationalism, and affirming rather than resisting or subverting the French Revolution and its political outcome. It is perhaps not obvious\u2014in fact it is perhaps not at all enlightening to hold\u2014that the political world is best construed as a mono-dimensional spectrum of opinion ranging from Right to Left. Anarchists have notoriously been difficult to place on this spectrum. Nevertheless this way of thinking about politics as a matter of Right and Left is itself a result of the Revolution, and the Frankfurt School by and large accepted it. To be sure, Adorno thought that Enlightenment could not be construed as a singular historical event that was directed at attaining once and for all fixed goals that had some kind of transcendental standing; it had to be a continuous critical and self-critical process. \"Liberty, equality, fraternity\" was a powerful and humanly edifying slogan when directed in the eighteenth century against a firmly established feudal and clerical establishment, but a true development of the tradition of the Enlightenment would require one to assess not just the evident benefits the ideology of human equality has had for humanity but also the non-negligible price humanity has had to pay for these gains. To construe this historically specific configuration of conceptions of \"equal liberty\" as anything more than a step in a process, and in the twentieth century uncritically to absolutise these ideals in the form they happened to take in the late eighteenth century, can turn the process of Enlightenment against itself. The reasonable eighteenth-century demand for equality of all citizens before the law, which stands in opposition to the feudal regime of privilege, can in the twentieth century easily become a justification for a pernicious form of equality: the conformist equality of atomised consumers.\n\nOn the specific issue of the nature of \"meaning,\" the members of the Frankfurt School, roughly speaking, think that the characteristic right-wing position is the old-fashioned one that construes \"meaning\" as something that is fixed, external, exists \"out there\" in some sense objectively, and relative to which humans are passive recipients. So in the ancient forms of natural philosophy, the universe had \"meaning\" in itself as an ontological feature (or perhaps \"behind it\" in the form of Ideas). Then the meaning was imposed on Nature by the Christian God. Finally, Kant transforms the basic liberating idea of the Enlightenment\u2014the idea of free human activity as the origin and goal of everything\u2014into a reactionary structure. For Kant, we don't \"find\" meaning in the world, we impose it, but this imposition is not really \"free\" because (a) it is an etiolated kind of imposition through cognition alone, and (b) the form of that imposition is a set of a priori fixed structures of human subjectivity. What is common to all the positions in this strand is not only that \"we\" collectively are not in control of meaning but that \"meaning\" is defined in such a way that it becomes inconceivable that we ever come to be in control. Heidegger and his theory of man as at the mercy of \"Seinsgeschichte\" represents an extreme modern version of this. However, Soviet-style Marxism, by being based on a form of coerced meaning imposed on human individuals as a kind of politically enforced moral demand from the outside\u2014that they integrate themselves into and identify with a collective work process\u2014has some of the structure features of the Right.\n\nIn some sense there is nothing at all \"right-wing\" about recognising that some important features of the \"meaning\" of our lives are imposed on us rather than constituted. It is the case that each of us is a finite being who will one day die, and no amount of self-identification with \"species-being\" or with a continuing human collective can completely overcome this. To live as if we will never die is not to live a sensible life. This is, after all, just another way of formulating what I called the demand of \"realism,\" that our attitudes not be grossly incompatible with the truth. In contrast to the right-wing view about \"natural\" meaning, the members of the Frankfurt School take the left-wing option of seeing all meaning as socially constituted. However, they do acknowledge a certain kind of plausibility that resides in the right-wing view. Meaning is in fact socially constituted, but in our society it presents itself as if it were a natural phenomenon.\n\nAdorno, who wrote his doctoral thesis on Kierkegaard, had great sympathy for the central existentialist line and its criticism of Marxism. The category of individual, subjective freedom retains a particular saliency in his thinking. Marx was, in Adorno's view, excessively optimistic about the way in which individual freedom and the development of individual powers and capacities would be unproblematically connected with the further development of technological and economic powers in a society without private ownership of the means of production. Marx sought, in fact, to turn the whole world into a huge Victorian workhouse, and Soviet-style systems attempted rather successfully to do just that.\n\nFor Adorno in one sense the modern world is not characterised by a \"loss of sense\" or of meaning. Rather the world has too much meaning. Modern people don't, in Adorno's view, suffer from anomie, that is, from a complete lack of orientation in the social world, or rather that is not the correct way to analyse their situation. Rather they are threatened from excessive pressures towards conformism even in the normative realm and the realm of human spontaneity. They are given to understand by social institutions all too well what they \"ought\" to do, and most of them have very extensively internalised these imperatives. Everything in contemporary society is part of an incipient single closed system of capitalist rationality. Both of the two components of this diagnosis of our contemporary world are important for Adorno: It is a world essentially structured by both the capitalist form of economic production and instrumental rationality. One of the ways in which Adorno departs from the standard older forms of Marxism concerns his view about the relation between these two components. He seems often to argue that the capitalist form of production is best understood as a particular form that instrumental rationality takes under certain conditions, rather than itself constituting anything like the \"economic base.\" In fact Adorno was an early believer in what later came to be called the \"convergence\" theory. The demands of instrumental rationality were equally powerful in East and West and led to similar effects in both kinds of regime. This dominance of instrumental reason is part of the Enlightenment project, and the fact that when it establishes itself it has such deleterious consequences is another reason the Enlightenment must be enlightened about itself.\n\nThis system of universal instrumental rationality imposes itself on each individual item in the world and makes that item a mere instance of an abstract universal. Instead of cutting our own writing instruments from the quills of geese or other winged creatures, we use mass-produced items. The modern world is a world of \"das Immergleiche,\" that which is \"ever-the-same.\" This is connected with a loss of human experience. Of course, there would be no human experience if I were unable to experience different objects as instances of the same thing, in Kantian terms, to subsume them under the same concept. However, fully human experience is also, and must also in part be, experience of that which is qualitatively distinct. To be able to enjoy the nonreplicable, qualitatively specific aspects of experience is also a part of what it is for that experience to be one of happiness for me, or to be at all \"meaningful.\" For Adorno, when the Hegelian boy throws the stones into the stream, part of the point is that he experiences this as \"his\" experience in a uniquely individuated and indefinable way. Of course, the boy cannot say in what the uniqueness of that experience consists, although a literary artist like Proust might be able to represent it\u2014Adorno speaks of \"Prousts Darstellung... des absolut, unaufl\u00f6slich Individuierten.\" Human \"meaning,\" at any rate, is not \"verf\u00fcgbar,\" not producible, reproducible, or accessible at will, but is connected with and embedded in historically specific forms of human experience that are structured by unique human memories and anticipations. You cannot simply conjure real human meaning into existence by wishing it or through any form of simple manipulation. This aspect gets lost both because the objects we encounter become more and more the same and because we are increasingly trained to experience them only in a schematic way that does not go beyond subsuming them under crude general categories. In thus simply subsuming objects under pregiven schemata, I am acting not as the unique individual I am but as any-interchangeable-representative-of-a-human-subject-whatever. In this way a kind of meaning\u2014or perhaps Adorno might call it \"pseudo-meaning\"\u2014is created and maintained as the artefact, finally, of a subjectless system of economic development, but it is not an appropriately human form of meaning. In such a system human life does have a kind of meaning that is as \"objective\" as that which the Calvinist ascribed to it: that meaning is simply to be a cog in the wheel of economic production and reproduction. All human experience in such a social system reflects this basic fact.\n\nTo be sure, Adorno has been presenting something that he insists is merely a \"critical model\" of contemporary society, that is, his analysis concerns what he takes to be exceedingly powerful tendencies towards homogenisation and uniformity, and towards the formation of totally closed social systems. These \"models\" are intended to be exaggerated\u2014Adorno notoriously thought all truth had to contain a component of exaggeration\u2014and so they do not purport to give mirror-sharp images of existing reality. In one sense the fully closed, utterly homogeneous society could never be a reality. I use a particular kind of fountain pen produced by the Lamy Company in Heidelberg. Lamy may wish to turn each of its pens into an exact replica of each other, but it is contrary to the nature of reality for that ever to be fully successful. The tendency alone is important even though it will fail. This, however, leaves a small space for human experience and for the constitution of a certain kind of meaning for human life.\n\nIt is this space Adorno seeks to exploit. His only hope is that it might be possible to use existing accidental niches that have been overlooked by the capitalist system and are not yet integrated to resist. This is a negative strategy. It doesn't mean using these overlooked realms of experience as bases to launch some kind of transformation of the whole but simply to maintain them. Adorno was pessimistic about the possibility of political action. All action in the modern world is infected with instrumental rationality, and this would include any political action that tried to revolt against the demands of the system of instrumental rationality itself. The only possible meaning you could give your life in the twentieth century that is minimally realistic is to resist the social pressures towards uniformity and homogeneity in all areas, and to struggle against the subordination of human subjectivity and individual life to the demands of the maximisation of return on capital. That one embrace this life of negativity is not, of course, something that could be the object of coercion.\n\nDespite the highly individualistic, not to say idiosyncratic, nature of Adorno's views, in retrospect, they seem strikingly in tune with the mood of the ohne mich generation in the Federal Republic (of Germany): \"Go to World War III if you insist, but without me\" (ohne mich). As various people were quick to point out, it is not obvious how effective such an attitude could conceivably be in the face of what would have been the realities of war in Europe in the second half of the twentieth century. If enough soldiers had refused to march to Smolensk, Kursk, or Stalingrad in the 1940s, perhaps this might have had some effect, but trying to sit back and \"sit out\" a nuclear exchange between NATO and the Warsaw Pact was not an option that provided any prospects for survival in the Federal Republic (or the DDR) in the 1950s, given that the fingers on the relevant buttons would all be located in Washington and Moscow. Adorno's own attitude can look very much like a generalisation of this, although his preferred formulation was: \"nicht mitmachen\" (\"No collaboration\"), and it was extended far beyond participation in military activity of any kind. Can such negation by itself really ever be more than detachment or resignation and constitute a genuine form of resistance?\n\nThe two major areas in which such resistance is to some extent still possible are our two old friends: art and philosophy. Within certain limited circumstances art can articulate negatively what a meaningful life would look like. Roughly speaking, modern art is a promise of happiness because it shows that we can still experience the horrors of the modern world as horrors, and thus still have the capacity to experience the modern world as it \"really is,\" which in turn means we have retained the possibility of some grasp on a more satisfactory form of life. So the revulsion modern art causes is part of its point. It is supposed to be \"meaningless\" in the sense that it refuses to be subsumed under any established categories. These categories are at best mere mechanisms for allowing our economy and social world to run smoothly, although they present themselves as something much more than that. Successful modern art must be \"new\"\u2014this demand of the Romantic tradition is retained\u2014it must be autonomous, and it must be negative. It must disorient by making the members of the audience aware of the fact that they don't know what they are looking for in the work. They understand it when they see that that is its point and respond both by experiencing the horror and by in some sense understanding that that horror is nothing but a realistic transcription of the world we live in. The meaning of Kafka or Sch\u00f6nberg is that that is the way our world really is. There is a contradiction, or at any rate a tension, between Adorno's claim that modern art inherently disorients and his firm conviction that he finally knows what to look for and what the point is: the criticism of the closed world of capitalist instrumental rationality. Adorno goes to the opera to be predictably disoriented, knowing in advance that that is what he must look for as the point of the work. It isn't, then, completely surprising that modern art is threatened with two complementary existential threats: on the one hand, by the danger of degenerating into mere entertainment, and on the other hand, by a self-dissolution in the pursuit of ever more complex and radical forms of disorientation.\n\nIn the face of this analysis, it is not surprising that Adorno admits the possibility that art is at an end, and there is nothing but continued failure in our attempts to create \"sense\" in sight. Art might be, and might always have been, a necessary failure: a promise that inherently insinuated itself as the fulfillment of that of which it was the (mere) promise. \"In an emphatic sense,\" Adorno writes, \"no work of art can be a success\" and the same is true of a life. It is not entirely surprising that at the end of his life Adorno developed a fascination (which was not reciprocated) with Samuel Beckett, whose motto seems to have been: Try again, fail again, fail better.\n\nThere is a parallel philosophical project of tracing the discrepancy that exists everywhere between the attempt by the economic system as a whole to produce \"das Immergleiche\" and its claim that it has done so, on the one hand, and the reality of our world, on the other. Adorno calls this \"negative dialectics\" and it consists in confronting this image that society projects with the reality. In general terms, one can say that the project of negative dialectics consists in fastening on and drawing attention to the ways in which individual objects, institutions, and actions fail to be simply identically replicated instances of what they are advertised to be. Such a philosophical project is like art in that it is a kind of surrogate meaning. Art is not happiness, it is a \"promesse de bonheur,\" but a promise is inherently different from the state of satisfaction it promises, and philosophy is not full-bodied somatic happiness either, but at best what Adorno calls a \"bitteres Gl\u00fcck\"\u2014\"bitter happiness\"\u2014\"das bittere Gl\u00fcck des Erkennens\"\u2014the bitter happiness that consists in cognition. However, even a \"bitter\" happiness is not nothing and in any case it is probably all we have. Adorno thought of himself as a man of the Left, but perhaps his legacy is more ambiguous than that. I mean this as a genuinely open, not a rhetorical, question.\n\n\"The Loss of Meaning on the Left\" is itself an ambiguous title, allowing for two distinct interpretations. First, it might refer to an interpretation given by people on the Left of a certain general social phenomenon that we call \"loss of meaning\" and is perhaps similar to what Nietzsche called \"nihilism\": old structures of meaning\u2014religion, feudal or meritocratic hierarchies, family ties\u2014become implausible and no longer serve to give orientation in life. The Left has an analysis of this\u2014roughly speaking it is a natural concomitant of the development of a society with a certain kind of basic economic structure\u2014and it has one or a number of proposed therapeutic measures, most of which amount to a change in the basic economic structure initiated by political action of a certain type that is directed at giving immediate producers more control over their own activity. A second way of reading the title, though, is that people on the Left find increasingly that they have lost faith in the traditional diagnosis or in some part of the traditional recommended therapy. Either the malaise is not located in the economic structure, but is even more deep-seated, such as in the structure of rationality itself, or the form of political action traditionally recommended by those on the left is likely to be ineffective or even counterproductive. One can see the Frankfurt School itself as moving gradually from the first of these readings (in the writings of the early 1930s) to the second, culminating in Adorno's work of the 1960s. It is not clear to me that we have yet been able to move beyond this position.\n6\n\nAuthority: Some Fables\n\nWe are familiar with the observation that any number of Greek terms were thought by the Romans to have no proper Latin equivalent and had to be taken over wholesale, but in the third century (AD) a Roman senator from Asia Minor, Cassius Dio Cocceianus, in his history noted that there was one case in which the tables were turned: there was one Latin term for which there was no simple Greek equivalent. In a discussion of some of the reforms of senatorian practice implemented by Augustus he spoke of the \"auctoritas\" of the Roman Senate and added: \"\u1f11\u03bb\u03bb\u03b7\u03bd\u03af\u03c3\u03b1\u03b9 \u1f00\u03c5\u03c4\u1f78 [scilicet, \u03c4\u1f78 \u1f44\u03bd\u03bf\u03bc\u03b1 auctoritas] \u03ba\u03b1\u03b8\u03ac\u03c0\u03b1\u03be \u1f00\u03b4\u03c5\u03bd\u03b1\u03c4\u1f78\u03bd \u1f10\u03c3\u03c4\u03af\u03bd\" [\"It is impossible to render 'auctoritas' in Greek with a single word\"]. Dio Cassius was himself a native speaker of Greek and so can be assumed to know what he was talking about. The question is what conclusion, if any, one should draw from this observation. If \"auctoritas\" is not merely a genuinely and uniquely Roman coinage, but one that cannot even be expressed in another language, might that mean that it was a mistake to apply it outside a Roman context, that any such application is inappropriate or at best \"metaphorical\"? Might we be able to learn to apply the term even if we can't translate it, or even if in some sense we do not understand it? In fact, later thinkers and language-users did use \"auctoritas\" (or terms clearly derivative from it in other languages) in non-Roman contexts, and one would be loath to say that this was always some kind of mistake.\n\nOr is it that \"auctoritas\" is simply a perhaps ideologically coloured conceptualisation of a universal human phenomenon, namely that some people in fact have more standing, prestige, and influence than others, they are able to get other people to follow their lead or do what they suggest or advise, or that some people \"ought\" (for whatever reason, for instance, because they have more knowledge, experience, and maturity) to have more influence than others or get people to do as they say? The Romans, then, were perhaps simply the first (in the West?) to focus attention to this, articulate it clearly, and give it a name. This would mean that there was nothing specifically Roman about the term or the associated concept. It is natural for us to employ (derivatives of) the Latin word because we still stand to some extent under the spell of Roman institutions, and the specific terms designating these institutions seem to have behind them a weight of tradition. Tapping into that tradition may give the illusion that by using a particular term one is giving a particularly lucid description or even explanation of what is really going on.\n\nIf, though, the phenomenon to which \"auctoritas\" refers really is a universal feature of human life, then shouldn't it be possible to give an adequate, if perhaps clumsy, analysis of this phenomenon without using this specific Roman term, for instance, by speaking of (well-deserved) influence, prestige, and so forth?\n\nDio does not, of course, say that \"auctoritas\" is absolutely untranslatable, that is, that there is no way to render it at all comprehensible to a non-Latin speaker. What he says is that it is impossible to render it in Greek \"\u03ba\u03b1\u03b8\u03ac\u03c0\u03b1\u03be,\" \"all at once,\" \"in one go.\" This can mean one of the following?\n\n(a) In every individual case in which the Latin term \"auctoritas\" is used, there is a clear succinct Greek equivalent\u2014\u1f00\u03be\u03af\u03c9\u03bc\u03b1 in one case, \u03ba\u1fe6\u03c1\u03bf\u03c2 in another, \u1f10\u03be\u03bf\u03c5\u03c3\u03af\u03b1 in a third\u2014different Greeks words in each case, but in each case a clear equivalent.\n\n(b) In some cases in which the Latin \"auctoritas\" is used, it is unclear which of two possible Greek terms is \"the\" equivalent (but it is also assumed that it must be one or the other, or be ambiguous between them; thus \"noga\" in many Slavic languages means either what we in English would call \"leg\" or what we would call \"foot\").\n\n(c) In some cases there is no one clear equivalent or an ambiguity between two or three, but nevertheless one can explain the term through a lengthy and appropriately circumstantial periphrasis. Thus Greek has two different words \u03bd\u03ad\u03bc\u03b5\u03c3\u03b9\u03c2 and \u03c6\u03d1\u03cc\u03bd\u03bf\u03c2 for a psychic phenomenon in which there is in English (and in Latin) only one word: \"[roughly:] envy\" (Latin: \"invidia\"). However, although I can't give \"equivalents\" of either of these two terms, I can give a periphrastic explanation: \"\u03bd\u03ad\u03bc\u03b5\u03c3\u03b9\u03c2\" is feeling pain at another's undeserved success; \"\u03c6\u03d1\u03cc\u03bd\u03bf\u03c2\" is feeling pain at another's success not because it is undeserved but because the other is your peer.\n\n(d) In some cases I might not be able to give even a periphrasis but would have to embark on a lengthy explanation. Just imagine trying to explain the terms referring to specific actions or events in a complicated game like cricket to someone to whom the game is unknown in a language like, say, Sami, which will not have had occasion to develop a technical vocabulary for it or \"ko\" to someone with no knowledge of the game of g\u00f4. This doesn't mean that \"ko\" must remain utterly mysterious to anyone who is not a speaker of Japanese and a g\u00f4-player, only that the term in question can be rendered comprehensible not by translating or paraphrasing individual sentences in which it occurs, but only through a lengthy global account of the point and the rules of the game as a whole and then a description of the position in a game to which \"ko\" refers. Of course, the distinction between \"periphrasis\" and \"explanation\" will not be hard and fast.\n\n(e) Two or more of the above.\n\nSo there is nothing deeply mysterious about a term being \"untranslatable at one go.\" The case would be different if there were to be some significant term that could not be made hermeneutically accessible to us in any way whatsoever, neither by translation, paraphrase, or \"explanation.\" If one strand of philosophical thinking, initiated by Quine in the 1960s and developed further by Davidson in the 1970s and 1980s, is correct, any meaningful term in a natural language will be translatable in this wide sense because, if it were not, we would have no reason to assume it was a meaningful term at all rather than a mere sound.\n\nThis may be thought to be correct but not directly relevant to the study of politics and history. What is important here is not just that terms can be rendered in some way comprehensible but how specifically they are internally configured. One tempting way to think about this would be to assume that there was something like a stock of semantic elements or basic units of meaning that could be put together in one way in one language and in other ways in others. So that if I say that \"mana\" and \"negara\" are not translatable (all at one go), this is compatible with my explaining to you that \"mana\" is supposed by certain people in the South Pacific region to refer to a kind of magical power instantiated in particular people, animals, plants, and places that makes it likely that they will be successful in their endeavours but also makes them potentially dangerous to, and hence to be avoided by, those who do not have \"mana\" themselves. Similarly, \"negara\" is a term for a South Asiatic \"court-centred\" polity that was organised in a particular way and made various political, legal, economic, and religious claims upon those subject to it. So in the case of \"mana\" we still in English have the concept of a \"magical power,\" although we don't use that term anymore except in historical or anthropological descriptions or as a criticism or a joke because we think such things don't exist, and we have the concepts of \"success,\" \"danger,\" and \"avoidance.\" We also have the concepts of a \"court,\" a \"polity,\" and of political, economic, and religious claims, although we don't perhaps recognise the particular \"religious\" claims associated with a \"negara\" and we don't have a single term that would refer to the conjunction of all these claims.\n\nAlthough it is tempting to speak as above of \"semantic elements,\" it is equally important to see that this idea is one of those images that are paedagogically useful in giving one a first approximative understanding (but can be highly misleading if taken in too literal-minded a way) and generate theories that will not survive sustained scrutiny. There are no naturally given, free-floating, language-independent \"semantic elements\" that are simply arranged differently in different languages. This image is useful because it helps me as a speaker of early twenty-first-century English to project some of my own categories in such a way as, perhaps, to begin to get a grip on what seem to me exotic ways of speaking, thinking, and acting. \" 'Mana' is a magic power\": this statement is correct as far as it goes in that this gives me a way of getting initial hermeneutic access to something that would otherwise be almost completely opaque to me, but do I and the inhabitants of some island in the South Pacific really share an \"elementary\" notion of such a power; it is just that they think it exists and I don't? Do we share the notion of \"power\"? The more one reflects on this, I submit, the less plausible it seems.\n\nIt is still a significant fact that the Romans had one simple Latin word, auctoritas, and presumably one concept, which they used to describe cases that a Greek would spontaneously have described using a variety of different Greek expressions. We are still confronted, then, with the same questions: Does this matter? If so, in what way and why? Perhaps discussion of some examples will help clarify this. Both ancient Greek and ancient Roman societies were strongly patriarchal. In these societies it was considered self-evident that women and children (and, of course, slaves) had to obey the man who was established as the head of the household\u2014in the ideal case, the monogamous husband of one of the women and father of her children. Obedience to the \"father\" was to be unconditional and was expected even when the father's command ran directly counter to the wishes and preferences, or even to the interests and needs, of the children or dependent women. The Romans used the phrase \"patria potestas,\" the father's power, the legally grounded and socially reenforced power to dispose of the economic resources of all members of the household group, to discipline members of the household, using force to get his commands obeyed, and even, in extremis, to dispose of their lives and freedom. In contrast the father's \"authority\" did not consist primarily in his legally constituted power to force the child to act in a certain way but had a different structure. The father was, and was supposed to be, a model, an exemplary source of advice and counsel that was to be heeded. This was true even in situations in which the father was not in a position to enforce or had no intention of enforcing compliance with his advice. To be sure, often a given paternal admonition will not have been easy to distinguish from an order, but that will not always have been the case and is, at any rate, a different question. The point is that there was a conceptual distinction between \"potestas\" and \"auctoritas,\" which, of course, does not preclude the possibility that in many concrete cases it would be hard to know which to apply.\n\nThe Greeks, it must be assumed, had an understanding of the general phenomenon (and the specific varieties) of power, whether physical ability or political, social, and legal power, and they had adequate linguistic resources to express these. If their society was in fact \"paternalistic,\" can it really be possible that they had no specific term to express paternal authority? Greek sources do have plenty to say about the power of the father but rather little about any property that seems at all like the Roman \"auctoritas patris.\" So in Aristotle's major discussion of the family we read that the father \"rules over the children in the way a king does\" (\u1f04\u03c1\u03c7\u03b5\u03b9... \u03c4\u1ff6\u03bd \u03c4\u03ad\u03ba\u03bd\u03c9\u03bd \u03b2\u03b1\u03c3\u03b9\u03bb\u03b9\u03ba\u1ff6\u03c2). This potentially harsh-sounding remark is almost immediately softened when Aristotle adds that this form of kingly rule (\u1f00\u03c1\u03c7\u03ae) is very different from the rule (also \u1f00\u03c1\u03c7\u03ae) over slaves or over one's wife. The father's rule over his children was supposed to be oriented around the principles of benevolence towards that which belonged to him as his own (\u03c6\u03b9\u03bb\u03af\u03b1) and the prerogatives and privileges (two Latin words, but we have no better) that belonged to the father as an older person (\u03c0\u03c1\u03b5\u03c3\u03b2\u03b5\u03af\u03b1), so it is internally limited in some way.\n\nThis account seems to me to have greater similarity with \"patria potestas\" than with the \"auctoritas patris.\" After all, even the Roman father in the fullness of his powers was not supposed to rule over his children arbitrarily but was supposed to be guided by some notion of what was proper and appropriate.\n\nThis example shows some characteristic differences between older conceptions and the assumptions moderns would be likely to make. To be sure the father should use his power \"appropriately,\" but who judges what is appropriate, and is there anyone who is empowered to intervene to ensure that the limits of what is appropriate are not overstepped? Perhaps the ideal \"Sage\" (\u03c3\u03cc\u03c6\u03bf\u03c2; sapiens) is the final judge of what is \"appropriate\": an \"appropriate\" use is one of which such a Sage would approve. However, it does not follow from this that the Sage is or should be empowered to intervene. A fortiori the oppressed children would not be acting in a morally fully acceptable way to resist even inappropriate exercise of power. Many modern thinkers would be inclined to assume that if he (the father) is acting inappropriately\u2014whatever that means\u2014there must be someone who, as we would say, \"has a right\" to intervene; perhaps even the abused children are not just an object of understandable pity but \"have a right to resist.\" The use of the phrase \"must be\" in ethical contexts is often a sign that an ad hoc assumption is being smuggled into the discussion. The ancient situation is one that opens a space for a particular kind of tragedy, namely a misproportion between a discretionary power that can, admittedly, be used appropriately or inappropriately and the failure to specify any effective moral recourse to those who, being subject to this power, might be disadvantaged by its inappropriate use. The modern conception closes off this particular space by assuming a certain moral equality among people and by vesting in each individual a prima facie right to self-protection, which means a right \"in principle\" to resist. Of course merely assigning to individuals a \"moral right\" to self-protection will not always constitute a real practical defence, so once the assignment has taken place it is natural to look around for, or invent, an agency designed to enforce this right effectively.\n\nAncient rhetoric provides another example, when Quintilian discusses \"auctoritas\" in the context of various types of rhetorical argumentation:\n\nAdhibebitur extrinsecus in causam et auctoritas. haec secuti Graecos, a quibus \u03ba\u03c1\u03af\u03c3\u03b5\u03b9\u03c2 dicuntur, iudicia aut iudicationes vocant... si quid ita visum gentibus, populis, sapientibus viris, claris civibus, inlustribus poetis referri potest.\n\n[Appeal to \"authority\" can also be brought into a case from the outside. Those who follow the Greeks (who call these \"\u03ba\u03c1\u03af\u03c3\u03b5\u03b9\u03c2\") call such appeals \"(appeals to) judgement... when one can bring to bear what seems good to tribes or peoples or wise men, famous citizens, or illustrious poets.]\n\nIn cases where the gods might condescend to give a clear sign, something that occurs only infrequently\u2014\"id rarum est\"\u2014but is not completely out of the question, Quintilian says one can even cite \"the authority of the gods\" (deorum auctoritas). A divine sign might have authority because the gods could be thought to have knowledge or forms of experience we lack and also because they might be powerful enough (and interested enough) to make the sign come true.\n\nThe parallel\u2014but, of course, temporally much earlier\u2014discussion of \u03ba\u03c1\u03af\u03c3\u03b5\u03b9\u03c2 in Aristotle's Rhetoric differs from Quintilian's in a number of significant ways. Among the argumentative strategies that orators repeatedly use (\u03c4\u03cc\u03c0\u03bf\u03b9), Aristotle says, is the appeal to the judgements people make:\n\n\u1f04\u03bb\u03bb\u03bf\u03c2 \u1f10\u03ba \u03ba\u03c1\u03af\u03c3\u03b5\u03c9\u03c2 \u03c0\u03b5\u03c1\u1f76 \u03c4\u03bf\u1fe6 \u03b1\u1f50\u03c4\u03bf\u1fe6 \u1f22 \u1f41\u03bc\u03bf\u03af\u03bf\u03c5 \u1f22 \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c5, \u03bc\u03ac\u03bb\u03b9\u03c3\u03c4\u03b1 \u03bc\u1f72\u03bd \u03b5\u1f30 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u03ba\u03b1\u1f76 \u1f00\u03b5\u03af, \u03b5\u1f30 \u03b4\u1f72 \u03bc\u03ae, \u03ac\u03bb\u03bb\u0309 \u03bf\u1f35 \u03b3\u03b5 \u03c0\u03bb\u03b5\u1fd6\u03c3\u03c4\u03bf\u03b9, \u1f22 \u03c3\u03bf\u03c6\u03bf\u1f76 \u1f22 \u03c0\u03ac\u03bd\u03c4\u03b5\u03c2 \u1f22 \u03bf\u1f31 \u03c0\u03bb\u03b5\u1fd6\u03c3\u03c4\u03bf\u03b9, \u1f22 \u1f00\u03b3\u03b1\u03b8\u03bf\u03af, \u1f22 \u03b5\u1f30 \u03b1\u1f50\u03c4\u03bf\u1f76 \u03bf\u1f31 \u03ba\u03c1\u03af\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2 \u1f22 \u03bf\u1f53\u03c2 \u1f00\u03c0\u03bf\u03b4\u03ad\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9 \u03bf\u1f31 \u03ba\u03c1\u03af\u03bd\u03bf\u03bd\u03c4\u03b5\u03c2, \u1f22 \u03bf\u1f37\u03c2 \u03bc\u1f74 \u03bf\u1f37\u03cc\u03bd \u03c4\u03b5 \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03bd \u03ba\u03c1\u03af\u03bd\u03b5\u03b9\u03bd, \u03bf\u1f37\u03bf\u03bd \u03c4\u03bf\u1fd6\u03c2 \u03ba\u03c5\u03c1\u03af\u03bf\u03b9\u03c2, \u1f22 \u03bf\u1f37\u03c2 \u03bc\u1f74 \u03ba\u03b1\u03bb\u1f78\u03bd \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03bd \u03ba\u03c1\u03af\u03bd\u03b5\u03b9\u03bd, \u03bf\u1f37\u03bf\u03bd \u03b8\u03b5\u03bf\u1fd6\u03c2 \u1f22 \u03c0\u03b1\u03c4\u03c1\u1f76 \u1f22 \u03b4\u03b9\u03b4\u03b1\u03c3\u03ba\u03ac\u03bb\u03bf\u03b9\u03c2. [1398b21\u201326]\n\n[Another topos is the appeal to the judgement (\u03ba\u03c1\u03af\u03c3\u03b9\u03c2) of people about which cases are the same, which like each other, and which opposed to each other. This argument is most effective when everyone always judges in the way cited. If that is not the case, the argument can still have a certain power if most people judge in this way or wise men\u2014again all of them or most of them\u2014or good people, or when one can claim for a fact that those who are being addressed judge in this way or people whom they in turn trust or people whom one cannot contradict such as rulers, or when one can show that those people judge in this way from whose opinion it is considered \"not beautiful\" to deviate, such as the gods, one's own father, or one's teacher.]\n\nThe Romans thus had available to them a schema of rhetorical argument of the following form:\n\n1. Your father [or the god] has authority.\n\n1a. [That is why everyone else also judges that you ought to obey him.]\n\n2. Therefore you ought to obey him.\n\nThe closest parallel to this for the Greeks would have been:\n\n1. Everyone\u2014or at any rate most of the people who count, namely the serious, informed, and intelligent ones\u2014judge as follows:\n\nIt is not good\/fine\/noble (\u03ba\u03b1\u03bb\u03cc\u03bd) to fail to obey your father\n\n2. Therefore you should obey your father.\n\nMuch of the force of this \"argument\" is contained in the use of the term \u03ba\u03b1\u03bb\u03cc\u03bd, which has a distinctly positive but also highly unspecific meaning. All sorts of things can be called \u03ba\u03b1\u03bb\u03cc\u03bd: a lovely cloak, a favourable wind, a well-positioned and well-built harbour, useful tools, a spirited horse, particularly strong and well-made armour. The Greek conception that finds expression in a statement like: \"Many people judge it not to be a fine thing to fail to obey one's father\" seems exceedingly weak and mealymouthed compared to the monumentally weighty ascription of \"auctoritas\" to the Roman paterfamilias. This example illustrates a very striking difference between Greeks and Romans in the way in which the subjective and the objective dimensions are connected. The Romans ascribe a quasi-objective property to a particular subject: \"This man has auctoritas,\" whereas the Greeks report the subjective judgements of various people about what would constitute an objectively good (\u03ba\u03b1\u03bb\u03cc\u03bd) state of the world.\n\nSo it does seem that the Greeks lacked a concept that corresponded in any simple way to the Roman \"auctoritas.\" Was this a defect, a blind spot in their political thinking that was perhaps connected with various other deficiencies in the way they conducted themselves politically? After all, no Greek city, federation, tribe, or monarchy ever achieved the political success of the Roman Republic, although, of course, this is also a rather high standard to set. Or was it that the Greeks were simply clear-sighted and hardheaded? \"The father in fact has the power of a king over his children and most of the most experienced people also think that it is not a fine thing if his children disagree with his judgement.\" That, one might argue, simply, neatly, and comprehensively sums the situation up. Why not just leave it at that without inventing a property of \"auctoritas\" to be attributed to the father? The only point of this addendum would be to provide a mask for relations of power and suggesting that adventitious coincidence of subjective human judgements was something more than it actually was.\n\nTwo approaches to \"authority\" have already been mentioned and rejected: first, attempts that assume that one can give a merely arbitrary or constructed definition; second, attempts that appeal exclusively (and ahistorically) to our contemporary language. Neither of these two would give any kind of understanding. Rather I have suggested the need to start from the history of the term.\n\nThe early history of the concept of \"auctoritas,\" then, is both very dark and very complicated, but there seem to be two rather different ways in which it has been approached. The first model attempts to map the structure and functioning of the institutions and practices of the Roman Republic insofar as it is accessible to us through the extant literature, and then to locate \"auctoritas\" in that system. The second is founded on a speculative etymology of the word \"auctoritas\" in the light of comparative Indo-European philology.\n\nTo start with the first model, the Romans had a variety of political assemblies, differing ways of conferring political power, and different kinds of officials or magistrates, each with his recognised province. In contrast to the Roman father who, as we have seen, unified in his own person \"potestas\" and \"auctoritas,\" power and authority in the political realm were not always vested in the same persons or assemblies. There was a distinction made between the \"potestas\" the various different kinds of popular assemblies exercised and the \"auctoritas\" of the assembly of elite politicians, which was called the \"Senate.\" I suggest, then, that we look at what \"potestas\" and \"auctoritas\" meant in the context of these institutions.\n\nIn addition to the Senate and the popular assemblies there is a third element in the Roman republican structure: the popularly elected magistrates, who had \"imperium,\" a more or less independent and discretionary power within a certain domain, either geographical (pro-consul of Asia or of Sicily) or administrative (responsible for the corn supply or for aqueducts).\n\nA certain separation of powers was an important part of Roman republicanism; the system breaks down when a single individual (Octavian\/Augustus) accumulates in his own person a number of distinct powers, each of which by itself had good republican precedents but which previously had each been vested in a distinct person. It would be as if Berlusconi were to be at the same time president of the republic, prime minister, foreign minister, president of the Constitutional Court, head of the Carabinieri, finance minister, Sindicato of Rome, and pope (as well as being the richest individual in Italy and owning all the newspapers and radio stations), and were thus to consider himself a pillar of the republic because he did nothing that was not fully within the constitutional power of the occupant of one or another of these offices. Nevertheless, the principle of the separation of powers did not take the form to which we have become accustomed since the time of Montesquieu. Montesquieu distinguished legislative, executive, and judicial powers and proposed that they be located in separate offices or institutionally distinct parts of government. Trying to apply that particular form of analysis to the Roman Republic will generate only confusion. First of all, in Rome there was no separate, institutionally distinct \"judiciary\" in the modern sense and no professional judges. Legal cases were decided by one or another of the magistrates or their deputies or by arbitration conducted by umpires who were agreed on beforehand by both parties, or by referring a case to one or another of the existing assemblies or in other rather ad hoc ways. It would be completely false to think of the Senate as something like a modern legislative chamber. Its function was not to \"pass laws\" at all; if anything, that was one of the functions of one or another of the popular assemblies. Its function was also not to enforce or execute the laws; that was the job of the magistrates. Formally speaking, the Senate was a consultative institution whose responsibility was to give advice (consulta) when it had been asked a question. Most of us get our first\u2014and for some only\u2014glimpse of Roman politics through the writings of Cicero. Cicero, however, was in favour of a republic dominated by a universally pro-active Senate (with the cooperation of the boni in other Orders), and was not beyond allowing this political value-judgement to colour what he presents as a description of the formal mechanisms of Roman politics. It is important to note that there is another way of seeing the Senate and its place in the republican system. One can reasonably describe it in purely formal terms as having a distinctly more limited proper sphere of action and as being distinctly more passive. After all, as a consultative body it was not permanently in session but had to be specifically convoked by someone who wished to ask it for advice, usually one of the magistrates, and it was convoked to answer some specific question. In principle, it did not, then, even set its own agenda but responded to questions put to it by magistrates. It was, of course, possible for the senators to be as \"expansive\" as they wished in the discussion of questions once they were put to it, and it was also possible for prominent senators to prompt one of the magistrates to ask an appropriate question or solicit advice on a particular topic, but that is a separate issue. Every schoolchild remembers Cato's peculiar \"ceterum censeo.\" Whatever the question, we are told, Cato ended his discussion by saying, \"Ceterum censeo Carthaginem esse delendam.\" This anecdote sums up in a nutshell both how the Senate was supposed to act and how it actually did act. It was supposed to answer the specific questions asked, so there was something odd and noteworthy that Cato kept deviating onto his pet subject, but also no one was empowered, or seemed inclined, to try to stop him.\n\nA magistrate could consult the Senate before, or after, he had proposed a law to one of the popular assemblies (and, of course, he need not consult the Senate at all). If the Senate approved of the proposal before it was enacted by the assembly, then one could see this as a kind of senatorial recommendation to the assembly. If approval is given after the proposal is passed, this could be seen as a kind of retrospective endorsement. To be sure, a proposal that had been approved by the Senate, either prospectively or retrospectively, could be thought to enjoy a certain normative perfection, but senatorial approval was not necessary for a law to be valid. Not even the most rabid proponent of senatorial interests was willing to deny this. Laws passed by a popular assembly had the force of law, even if the Senate had not given its approval (or if that approval had not even been sought). So one should also not apply to the Senate a model derived from the theory developed by supporters of the French parlements (courts of law located in some large cities) in the eighteenth century, namely that laws and decrees propounded by the king had no legal force unless \"registered,\" that is, accepted by the parlements and recorded in their books. In Rome it was perhaps rash to legislate without consulting the Senate and very rash to do something of which it disapproved. The Senate, after all, was composed of especially successful and experienced statesmen and politicians, many of whom had previously held very high magistracies, so it was natural to consider it a repository of a certain kind of practical knowledge, foresight, and judgement. That is not to say, of course, that the judgements on which its advice was based were infallible or that it did not represent to some extent the narrow self-interest of its members and its own interest as a distinct order of society. Nevertheless, with senatorial approval or without, provided the laws in question had been passed in a formally correct way in the popular assembly, they were still legally binding; valid, to be sure, \"without the authority of the Senate.\" \"Auctoritas\" was even the technical terms for a recommendation of the Senate that was turned down or not acted on by one of the assemblies and that therefore was not valid law.\n\nOne cannot, therefore, even say that the Senate was a place of obligatory consultation because no magistrate was required to consult the Senate, and certainly not required to follow its advice, and laws and decrees adopted against its explicit advice were perfectly legal and binding.\n\nAppeal to the Senate and its auctoritas was, therefore, nothing more than another way of summoning moral support or trying to acquire some extranormative weight for one's proposal or of taking out some kind of insurance in case of failure. Unfortunately, the insurance policy is not worth the paper it is printed on. Cicero learned this to his cost\u2014if he did not know already\u2014in the aftermath of his action against the Catalinarians. He had covered himself by acquiring a senatus consultum, the so-called senatus consultum ultimum to the effect that he, the consul, should see to it that the republic came to no harm, and he had a senatorial vote in favour of imposing the death penalty on the conspirators, but when he acted on that \"advice\" he could still be prosecuted for executing Roman citizens without trial. The Senatorial decision was still a consultum and had the standing of an expression of an opinion, not a legal warrant or an executive grant of immunity. \"I was [or, we were] only following the auctoritas of the Senate\" was never an acceptable excuse or exculpation for a magistrate or a popular assembly.\n\nWith that we come to the second of the two approaches to \"auctoritas.\" In his Le vocabulaire des institutions indo-europ\u00e9ennes, \u00c9mile Benveniste starts from some claims about etymology. \"Auctoritas,\" he asserts, should be the property of someone who is an \"auctor.\" The substantive \"auctor,\" however, derives from the verb augeo (increase, swell, strengthen, cause to grow). So the archaic \"auctor\" is the initiator of some action or sequence of events, that is, someone who has the power and the warrant to begin and carry through some action, and \"auctoritas\" is the property of being such an initiator and executor of action. So in Roman law the auctor is a seller who must be physically able to give to the buyer the object purchased but must also stand security in case of doubt that he actually owns the object in question. One might be struck by what seems to be a conjunction of a merely factual and a quasi-normative statement\u2014power and warrant\u2014but to call this a \"conjunction\" might be taken to imply that there are two recognisably distinct \"things\" that are put together or conjoined\u2014the factual and the normative. It is at least just as likely, though, that what one has here is a description of an archaic situation, not in which two things that are distinguished are put together but in which they have not yet been clearly separated from each other.\n\nOn persiste \u00e0 traduire augeo par \"augmenter\"; c'est exact dans la langue classique, mais non au d\u00e9but de la tradition. Pour nous, \"augmenter\" \u00e9quivaut \u00e0 \"accro\u00eetre, rendre plus grand quelque chose qui existe d\u00e9j\u00e0.\" L\u00e0 est la difference, inaper\u00e7ue, avec augeo. Dans ses plus anciens emplois, augeo indique non le fait d'accro\u00eetre, ce qui existe, mais l'acte de produire hors de son propre sein; acte cr\u00e9ateur, qui fait surgir quelque chose d'un milieu nourricier et qui est le privil\u00e8ge des dieux ou des grandes forces naturelles, non des hommes.\n\nIt is almost superfluous to remark that the conception which Benveniste rejects, namely that the auctor \"rend plus grand quelque chose qui existe d\u00e9j\u00e0,\" corresponds precisely to the kind of function the Roman Senate seems to have had. It was not officially an initiator at all but rather an assembly which by approving of a proposal that had already been formulated by someone else increased the standing of that proposal. The magistrate who, having consulted the Senate and won their approval, could present a proposal to the popular assembly with an increased chance of its being accepted because the proposal was backed \"by the authority of the Senate.\" In contrast the archaic auctor envisaged in Benveniste's construction would have to possess in perhaps yet undistinguished unity the power to lay down rules (potestas), like that of the Roman popular assemblies, the warrant-enhancing auctoritas of the Senate, and the actual ability to enforce conformity to what had been laid down. It is not at all clear how exactly to think about these two models. This may be seen as an invitation to tell a \"just-so\" story that would allow one to visualise one possible form that the relation between them might have taken. The point of such stories is not, of course, to say what actually happened, but by studying their recurrent and plausible-seeming fantasies to try to map people's mental world. The assumption is that the semantics of such things as \"authority\" will have something to do with the way they imaginatively construe their world. One can, of course, deny this assumption or think that examining such a fiction is a very bad way to begin to think about \"authority\" or that this mode of proceeding must lead one astray or cannot be the final word on \"authority.\" I ask the reader to suspend disbelief for a moment. Suppose, then, that we enter that world of make-believe in which people might live \"in a state of nature,\" \"graze on the meadows of truth\" after death, truck and barter in a \"free market,\" or discuss their social institutions \"behind a veil of ignorance.\" The specific story about authority begins with an idealised conception of a primordial agent who possesses a large number of highly desirable and highly prized properties and initiates some important collective enterprise, founds a city, or institutes some important social institution. Think here of the stylised \"biographies\" of mythical Founders one finds in ancient literature, that is of figures like Theseus, Solon, Lykurgus (among the Greeks), or Romulus and Numa Pompilius (among the Romans). In the stories about these Founders a certain number of features recur. (1) The Founder in fact succeeded in the past in taking some kind of initiative, setting in motion some process or establishing some kind of practice or institution that continues. (2) The Founder had sufficient power and an appropriate normative warrant for initiating or founding whatever he does. (3) The Founder has at his disposal the competences, forms of knowledge, and moral and psychological properties needed for his undertaking, that is, he is able to continue to guide it skillfully in the right direction, that is, in the direction of some tangibly positive results. (4) The natural consequences of the obvious success of his initiative give the Founder standing, prestige, and influence at least among those who participate in the enterprise, identify with it, and benefit from it: they tend to listen carefully to him and to follow his advice.\n\nThe part of the above that is likely to seem most problematic to modern observers is (2). What is \"an appropriate normative warrant\"? Isn't the whole notion of \"normativity\" a contemporary creation, concocted from Christian and Kantian sources? Can it without serious anachronism be projected back onto the ancient world? Where, then, would an appropriate warrant come from? What is its relation to mere strength or force? Sometimes the warrant in question is presented as being clear, prospective, and unambiguous, such as in the case of Battos, who is sent off from Delphi with a direct, clear injunction from the god to go off and do something new, in this case found the city of Cyrene in Africa. Sometimes, however, the warrant is merely retrospective, as when the gods send a sign that confirms what has already been begun, Romulus and the eagles. Sometimes it is ambiguous (Aeneas). Sometimes, finally, it seems to be merely hypothetical (and retrospective): The continued prosperity of this city, one might think, shows that the gods \"must\" approve. Why \"must\" the gods approve and why in any case should the approval of the gods be the final word? In the archaic period, there are plenty of exemplary stories about (excessive) prosperity and success not being a sign of divine approval but a trigger of divine jealousy. If one asks why Zeus's approval is so important, the answer is likely, at any rate in the early period, to be couched in terms of his greater strength than other gods, his defeat of the Titans in the Great War, and so forth. The role of the gods in all of this is deeply ambiguous. Whereas it is probably not quite correct to say that might makes right, few things seem to speak as loudly as visible power and continuing success and prosperity. In a highly uncertain world, what higher \"warrant\" could one have than the example of visible continuing success? In particular, how can I reasonably fail to admire and emulate those who in establishing my form of life have conferred such signal and significant benefits on me?\n\nThe story up to now has been couched in terms of a single Founder, but there is no reason in principle\u2014other than a human aesthetic preference for having a human individual at the centre of a story\u2014for something similar not to hold for a group. If the Spartans had Lykurgos, the Romans had the mos maiorum. What is to be done, however, if the four conditions mentioned above (power, success, other abilities, prestige) are not co-instantiated at the same time in one given individual or in one determinate group? What if one person or group has the effective power, another great skill and experience, and yet another high prestige? And what if these individuals and group interact in a variety of different ways over time? What, to put it differently, if there is no single, closed Bildungsroman of a unitary individual or group who has or instantiates \"authority\" that can be run through schematically in the imagination in a relatively clear narrative way? What if there is only a (real) history of \"authority\"?\n\nOddly enough, given what is often said about the fundamentally \"ahistorical\" nature of Greek thought, considerations like this would probably have left them unmoved. After all, \"authority\" played no role in their imaginative and conceptual life. We bump up again against the question why that should have been the case, although perhaps by this point I will have impressed on the reader sufficiently the peculiarity of the concept of \"authority\" to leave room for the possibility that the question should be reversed: Why and how could the Romans have come up with such an oddity?\n\nUntil the middle of the nineteenth century, one could argue, \"Greece\" was no more than a mere geographical name. The various spaces occupied by speakers of Hellenic dialects never achieved political unity apart from the fragile and ephemeral empire of Alexander until the Romans took them over in the first century before Christ. In these spaces there lived a variety of different groups who put forward conflicting, heterogeneous, and variously grounded claims to different kinds of knowledge and to different types of power. There were different groups of people who called themselves (and were called by other people) \"the good,\" people who claimed wisdom for themselves (and to whom wisdom was attributed by others), \"rulers,\" aristocrats, citizens. There were varying majorities in the different tiny city-states and other constituencies (\"most Athenians think...\"; \"most doctors would prescribe...\"; \"most rowers in the fleet agreed...\"). Fathers generally had power over their children, but Zeus himself had been able to turn the tables on his own father, and that had clearly been a good thing. What is striking in this world is variety and diversity; what would prompt one to assume that some particular people or occupants of some particular roles had an \"objective\" authority? In Rome, in contrast, centralisation of power\u2014\"centralisation\" relative to what one would have found in Greece\u2014seems to have been a fact of life from early on. The \"Fathers\" formed a political and social elite, owned extensive property, monopolised the most important priesthoods, provided the candidates for magistratures, served as military leaders, and sat with their peers in the Senate. They were the auctores par excellence. Eventually the plebs was able to limit their overwhelming predominance and acquire for itself both legislative power and, through popularly elected magistrates, indirect executive power. Auctoritas was what remained to the Senate as a kind of shadow of its former prevalency. Although some political philosophers seem to connect the uses of the imagination with the production of unrealistic fantasies about a utopian future, Hegel and Marx (and Freud for that matter) were right to emphasise that the more usual role of the imagination in politics was to reinforce the hold of the past over the present.\n\nIf the Roman concept of auctoritas was shimmeringly indefinite, the modern concept seems simply to be polysemous. One can distinguish several kinds of authority: personal, moral, discretionary, and cognitive.\n\nPersonal authority is a sociopsychological property of a concrete individual, more rarely of a very small group. A person has \"authority\" in this sense if he or she in fact impresses others as being especially competent, firm of purpose, self-confident, willing to give orders and used to seeing them obeyed. In emergency situations people who have this property, which is sometimes also called \"charisma,\" can be especially valuable. Moral authority generally derives from having led, and being seen to have led, a morally exemplary life. One can easily imagine situations in which someone who has the sociopsychological property of personal authority does not also have any particular moral authority. Moral authority is a more fine-grained but also more encompassing phenomenon than \"charisma\" because it has to do not with such things as efficiently organising specific forms of action in situations of urgency but with more general and potentially reflective overall judgements about the good, the praiseworthy, that which is to be avoided at all costs. Since there is such human disagreement on what a good life is, as well as significant disagreement on what a \"moral\" life is, it is hard to agree on examples of holders of moral authority. Perhaps among contemporaries someone like Nelson Mandela would come closest to instantiating the ideal of a person with very widely recognised moral authority.\n\nSince the world is unpredictable, most societies have found it advisable to give to some specified members \"discretionary\" power, that is, the power to judge a situation that demanded action and take what means they thought appropriate, using even means that were at other times not part of the normal or permissible repertory. The most extreme example of this was perhaps the Roman Senate's consultum ultimum mentioned above, but any ordinary policeman even in a Western European society is invested with a wide range of discretionary powers, including restricting or redirecting the flow of vehicular traffic, stopping and searching suspicious characters, and ordering groups of people to disperse. Because discretionary powers are so easy to misuse, they are usually hedged around with conditions, restrictions, and potential sanctions, but although these limitations and controls may be more or less effective, they cannot ever be so strong as completely to regulate the discretionary element without destroying the advantages it brings with it\u2014the ability to react systematically precisely to that which is unforeseen and therefore cannot be antecedently regulated in detail.\n\nFinally there is the cognitive authority of the expert. A physician has studied at a medical school, has passed a certain number of qualifying examinations, and has had some experience in treating the ill; a craftsman has shown his ability repeatedly by discharging a variety of different commissions to a very high standard; the art dealer can (almost) always tell you who painted a certain painting he has never seen before (without looking at the signature), and he immediately spots forgeries (although he can't always tell you exactly how he does that); the taxi driver of long standing can give you reliable directions (provided you are in a motor car; the city might look different to someone on foot).\n\nIn addition to all these modern usages of \"authority\" there is a further one, which seems especially characteristic of the modern world in contrast to the ancient (although there are perhaps some isolated instances of the \"modern\" usage in antiquity). This is the sense in which, for instance, in certain legal contexts a lawyer may speak for me, that is, in my place because I \"have authorised him\" to do so. I give him (or her) this \"authority\" usually by participating in some recognised process, such as signing a formal document, by which what he or she says or does comes to have the same standing in the legal system as what I might say or do. What is crucial here is that \"authority\" in this sense is thought to be transferable in a way that other forms of authority, such as moral or personal authority, are not. The lawyer needs this transferred or derivative authority because he would otherwise not have what we call \"(the) authority (to do x, y, and z)\" no matter what other personal, moral, or cognitive capacities he might have. Although in many cases the grant of a certain authority may be limited to particular domains\u2014the agent whom I might authorise to buy books for me in a foreign country, that is, as we would say, to buy them \"on my account,\" will not thereby be able to enrol me in the army, use money in my account to fund even worthwhile political causes, or buy lottery tickets in my name\u2014the agent will in general be construed as being able to exercise some measure of discretion. This notion of a transferable, discretionary warrant makes sense in commercial transactions, but it comes into its own in the modern period.\n\nIn the ancient world there were, of course, noncommercial cases in which one person \"spoke for\" another. Thus a herald in one sense \"speaks for\" someone else, but the herald has no discretionary power. The same is true of certain priestesses, who may speak for the god, although they do not themselves even know what they are\/he is saying. A professional speechwriter may have composed a defence for presentation at a trial, but the speech will be presented by the defendant himself\u2014the Athenian model\u2014or an advocate may make a speech in support of a certain case\u2014the Roman model. Still, it makes a big difference whether Lysias writes a speech that I present at my trial, Cicero speaks in my favour, or a modern lawyer speaks for me, that is, instead of, or in place of, me. Finally, one would be exceedingly ill-advised to ignore what the pro-consul demands just because Rome is far away. However, the main reason to assume that the pro-consul in some sense \"speaks for Rome\" is that he has what the Romans called imperium (and an army at his back), not what they would have called auctoritas.\n\nIn the end, then, one is left with the three possibilities. First, there is Benveniste's archaic ideal of a plenipotent, maximally competent primordial \"auctor\" who is an absolute, unitary source of power and legitimacy and is construed as independent of and prior to a specific institutional structure, indeed as \"founding\" it. Second, there is an institutionally specific concept that arises naturally from thinking about the practices of the Roman Republic. Certain persons in certain roles or positions or certain bodies are conceived to embody a special competence at deliberating or giving advice by virtue of these positions. This advice is not infallible or unlimited, but it is construed as good as the best available, even if because of the overall structure of the system concerned, the advice has had no effect (because the person or agency who has the authority to give it has no appropriate institutional power). Third, there is the \"Hellenic\" conception, which does even bother to try to construe the multifarious, diverse collection of different phenomena, which we, following the Romans, call \"authority,\" as a single, unitary, theoretically coherent domain at all. What the Romans called auctoritas is just to be analysed as a series of diverse forms of human behaviour (imitation of the powerful, submission to the influence of a patriarchal father, consultation of experts) and different kinds of claims that some people make (to obedience, collaboration, belief, etc.) and others accept or reject, the whole process in each case located in a particular constellation of individual and institutional power, competence, and \"normativity\" and in a world of varying possibilities and necessities. Influence, obedience, expert knowledge, prestige, skill, and the varying ways in which people can be morally exemplary are not all cut back to fit the same single pattern of \"authority.\"\n7\n\nA Note on Lying\n\nIt has often seemed odd to me that lying, meaning the intentional telling of what one knows to be false as the truth, has had such a bad press in the modern world, and I can explain this only by the persistence of bog-Christian attitudes. Christianity is, after all, intended to appeal to simple souls whose speech was to be \"Yea, yea\" and \"nay, nay\" with everything else consigned to the category of the \"evil\" (\u1f14\u03c3\u03c4\u03c9 \u03b4\u1f72 \u1f41 \u03bb\u03cc\u03b3\u03bf\u03c2 \u1f51\u03bc\u1ff6\u03bd \u03bd\u03b1\u1f76 \u03bd\u03b1\u03af, \u03bf\u1f52 \u03bf\u1f54\u2022 \u03c4\u1f78 \u03b4\u1f72 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u1f78\u03bd \u03c4\u03bf\u03cd\u03c4\u03c9\u03bd \u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c0\u03bf\u03bd\u03b7\u03c1\u03bf\u1fe6 \u1f10\u03c3\u03c4\u03b9\u03bd; Matt. 5:37). In claiming that lying has had a bad press I do not, of course, mean to say that I think it is a good thing, merely that it is not clear to me that it deserves the uniquely reprehensible status sometimes assigned to it, and that in some contexts there is more to be said for it than is often acknowledged. In a way the fact that a government feels the need to lie to its population can be seen as a progressive trait. Governments that were utterly sure of themselves and their own power and ruthless in their use of it would not need to lie. So the fact that they need to lie can, it seems to me, not unreasonably be seen as a kind of advance on a state of primitive brutality in which force and the threat of force could be used. To be sure, Christianity has traditionally argued the opposite, so that Dante, for instance, places liars and evil counsellors lower down in Hell than the violent. The reason for this is presumably that lying is a perversion of a higher human capacity, that for speaking the truth, and it is worse to corrupt a great possible good than simply to act in a brutal way. It is possible, of course, to be of two minds about this. So one question I would suggest we ask is: Is lying always and in all contexts an absolute evil? Even if the alternative is the direct use of force? A second question is: Is lying, meaning by that term the intentional assertion of something one knows to be untrue, necessarily worse for the liar than telling an untruth one does not know is untrue? Finally, is being lied to necessarily worse for a person than being told an untruth by someone who is himself taken in by it? I submit not that lying is sometimes clearly a good but that in all three of these cases the jury is still out.\n\nIt has often been noted that ancient Greek philosophers were obsessed with the issue of the distinction between appearance and reality but showed relatively little interest in lying versus truth-telling, and in particular showed no tendency to think of the distinction between the two as indicating a basic moral issue. In general the idea that under all circumstances it is categorically wrong to lie and that lying is a sign of having a bad character, as distinct from the thought that it is painful to be taken advantage of because one has fallen for a plausible lie, seems a relatively late development. Thus in the so-called archaic period Odysseus is generally, if not universally, admired for being such a consummate liar. Achilles, we know\u2014because he tells us\u2014does not like duplicitous people, but then this seems to be an idiosyncratic part of his individual character. In a well-known passage in the Odyssey the goddess Athena appears in the semblance of a young herdsman to Odysseus and asks him who he is, but when he embarks on a characteristically lengthy, detailed, and completely untrue story, she interrupts him to say that part of the reason she is devoted to him is that he is so like her, so good and plausible a liar that if she were not a goddess and even more quick-witted than he is, he might even pull the wool over her eyes.\n\n\u03ba\u03b5\u03c1\u03b4\u03b1\u03bb\u03ad\u03bf\u03c2 \u03ba\u0309 \u03b5\u1f34\u03b7 \u03ba\u03b1\u1f76 \u1f10\u03c0\u03af\u03ba\u03bb\u03bf\u03c0\u03bf\u03c2 \u1f45\u03c2 \u03c3\u03b5 \u03c0\u03b1\u03c1\u03ad\u03bb\u03b8\u03bf\u03b9\n\n\u1f10\u03bd \u03c0\u03ac\u03bd\u03c4\u03b5\u03c3\u03c3\u03b9 \u03b4\u03cc\u03bb\u03bf\u03b9\u03c3\u03b9, \u03ba\u03b1\u1f76 \u03b5\u1f30 \u03b8\u03b5\u1f78\u03c2 \u1f00\u03bd\u03c4\u03b9\u03ac\u03c3\u03b5\u03b9\u03b5.\n\n\u03c3\u03c7\u03ad\u03c4\u03bb\u03b9\u03b5, \u03c0\u03bf\u03b9\u03ba\u03b9\u03bb\u03bf\u03bc\u1fc6\u03c4\u03b1, \u03b4\u03cc\u03bb\u03c9\u03bd \u1f06\u03c4\u0309, \u03bf\u1f50\u03ba \u1f04\u03c1\u0309 \u1f14\u03bc\u03b5\u03bb\u03bb\u03b5\u03c2,\n\n\u03bf\u1f50\u03b4\u0309 \u03b5\u03bd \u03c3\u1fc7 \u03c0\u03b5\u03c1 \u1f10\u1f7c\u03bd \u03b3\u03b1\u03af\u1fc3, \u03bb\u03ae\u03be\u03b5\u03b9\u03bd \u1f00\u03c0\u03b1\u03c4\u03ac\u03c9\u03bd\n\n\u03bc\u03cd\u03b8\u03c9\u03bd \u03c4\u03b5 \u03ba\u03bb\u03bf\u03c0\u03af\u03c9\u03bd, \u03bf\u1f35 \u03c4\u03bf\u03b9 \u03c0\u03b5\u03b4\u03cc\u03b8\u03b5\u03bd \u03c6\u03af\u03bb\u03bf\u03b9 \u03b5\u1f30\u03c3\u03af\u03bd.\n\n\u1f00\u03bb\u03bb\u0309 \u1f04\u03b3\u03b5, \u03bc\u03ae\u03ba\u03b5\u03c4\u03b9 \u03c4\u03b1\u1fe6\u03c4\u03b1 \u03bb\u03b5\u03b3\u03ce\u03bc\u03b5\u03b8\u03b1, \u03b5\u1f30\u03b4\u03cc\u03c4\u03b5\u03c2 \u1f04\u03bc\u03c6\u03c9\n\n\u03ba\u03ad\u03c1\u03b4\u03b5\u0309, \u1f10\u03c0\u03b5\u1f76 \u03c3\u1f7a \u03bc\u03ad\u03bd \u1f10\u03c3\u03c3\u03b9 \u03b2\u03c1\u03bf\u03c4\u1ff6\u03bd \u1f44\u03c7\u0309 \u1f04\u03c1\u03b9\u03c3\u03c4\u03bf\u03c2 \u1f01\u03c0\u03ac\u03bd\u03c4\u03c9\u03bd\n\n\u03b2\u03bf\u03c5\u03bb\u1fc7 \u03ba\u03b1\u1f76 \u03bc\u03cd\u03b8\u03bf\u03b9\u03c3\u03b9\u03bd, \u1f10\u03b3\u1f7c \u03b4\u0309 \u1f10\u03bd \u03c0\u1fb6\u03c3\u03b9 \u03b8\u03b5\u03bf\u1fd6\u03c3\u03b9\n\n\u03bc\u03ae\u03c4\u03b9 \u03c4\u03b5 \u03ba\u03bb\u03ad\u03bf\u03bc\u03b1\u03b9 \u03ba\u03b1\u1f76 \u03ba\u03ad\u03c1\u03b4\u03b5\u03c3\u03b9\u03bd\n\nAnyone, even a god, who could get past you in any kind of scam [\u03b4\u03cc\u03bb\u03bf\u03c2] would have to be wily [\u03ba\u03b5\u03c1\u03b4\u03b1\u03bb\u03ad\u03bf\u03c2] and duplicitous [\u1f10\u03c0\u03af-\u03ba\u03bb\u03bf\u03c0\u03bf\u03c2]. You're a hard one, crafty [\u03c0\u03bf\u03b9\u03ba\u03b9\u03bb\u03bf-\u03bc\u03ae\u03c4\u03b7\u03c2] and addicted to scamming [\u03b4\u03cc\u03bb\u03bf\u03c2]; you won't let up in deceiving and deploying those duplicitous [\u03ba\u03bb\u03cc\u03c0\u03b9\u03bf\u03c2] words that come natural to you, even though you are [back home] in your own land. Both of us are past masters at getting around people [\u03b5\u1f30\u03b4\u03cc\u03c4\u03b5\u03c2 \u1f04\u03bc\u03c6\u03c9 \u03ba\u03ad\u03c1\u03b4\u03b5\u0309\u03b1], you are by far the best among all mortals at deciding what to do and telling stories, and among the gods I'm famous for my nous [\u03bc\u1fc6\u03c4\u03b9\u03c2] and my eye for the main chance [\u03ba\u03ad\u03c1\u03b4\u03b5\u0309\u03b1], so let's not continue this kind of discussion any further. (Odyssey XIII.296\u201399)\n\nThis duplicitousness seems to have very deep roots and to run in the family because Odysseus's grandfather Autolykos was a favourite of the god Hermes, who made him an exceptionally gifted thief and \"swearer of oaths.\"\n\n\u0391\u1f50\u03c4\u03cc\u03bb\u03c5\u03ba\u03bf[\u03c2]...\n\n... \u1f43\u03c2 \u1f00\u03bd\u03b8\u03c1\u03ce\u03c0\u03bf\u03c5\u03c2 \u1f10\u03ba\u03ad\u03b1\u03c3\u03c4\u03bf\n\n\u03ba\u03bb\u03b5\u03c0\u03c4\u03bf\u03c3\u03cd\u03bd\u1fc3 \u03b8\u0309 \u1f45\u03c1\u03ba\u1ff3 \u03c4\u03b5\u2022 \u03b8\u03ad\u03bf\u03c2 \u03b4\u03ad \u03bf\u1f31 \u03b1\u1f50\u03c4\u1f78\u03c2 \u1f14\u03b4\u03c9\u03ba\u03b5\u03bd\n\n\u1f19\u03c1\u03bc\u03b5\u03af\u03b1\u03c2\u2022 \u03c4\u1ff7 \u03b3\u1f70\u03c1 \u03ba\u03b5\u03c7\u03b1\u03c1\u03b9\u03c3\u03ad\u03bd\u03b1 \u03bc\u03b7\u03c1\u03af\u03b1 \u03ba\u03b1\u1fd6\u03b5\u03bd\n\n\u1f00\u03c1\u03bd\u1ff6\u03bd \u1f20\u03b4\u0309 \u1f10\u03c1\u03af\u03c6\u03c9\u03bd\u2022 \u1f41 \u03b4\u03ad \u03bf\u1f31 \u03c0\u03c1\u03cc\u03c6\u03c1\u03c9\u03bd \u1f05\u03bc\u0309 \u1f40\u03c0\u03ae\u03b4\u03b5\u03b9.\n\nAutolykos... who surpassed all men both in thievery and in the swearing of oaths. The god Hermes himself gave him this gift for to please him Autolykos has burned the thigh bones of lambs and kids, so Hermes was well-disposed to him and stood by him. (Odyssey XIX.394\u201398)\n\nNo Greek thought that either Autolykos or Odysseus had the gift of dissembling speech \"from the evil one\" (\u1f10\u03ba \u03c4\u03bf\u1fe6 \u03c0\u03bf\u03bd\u03b7\u03c1\u03bf\u1fe6 in the passage from Matthew cited above); rather they had received it from the god Hermes and the goddess Athena.\n\nIn contrast to what a modern reader might expect, the two-hundred-odd pages of a modern edition of Aristotle's Nicomachean Ethics contains discussions of all kinds of virtues and vices (moderation, justice, perversion, courage, brutality, etc.), but only about two of them are devoted to truthfulness and its opposite (1127a13\u2013b18). Even when truthfulness and the lack of truthfulness are discussed, these are not construed exclusively or even primarily as ways of saying or failing to say that which one considers to be the truth, but rather as failures of ways of living (\u1f00\u03bb\u03b7\u03b8\u03b5\u03c5\u03c4\u03b9\u03ba\u1f78\u03c2 \u03ba\u03b1\u1f76 \u03c4\u1ff7 \u03b2\u03af\u1ff3 \u03ba\u03b1\u1f76 \u03c4\u1ff7 \u03bb\u03cc\u03b3\u1ff3 [\"truthful both in his way of living and in his speech\"] 1127a24). So a person who exhibits this vice is construed as one who habitually lives in a way that goes beyond or falls short of his own real merits, part of which may be that he fails to find the mean in speech between making claims for himself that are overly grandiose (\u1f00\u03bb\u03b1\u03b6\u03bf\u03bd\u03b5\u03af\u03b1) and being too modest or self-deprecating (\u03b5\u03af\u03c1\u03c9\u03bd\u03b5\u03af\u03b1). To live in an overly grand style, or too modestly, is not obviously a form of lying. Similarly, modern readers seldom fail to be struck by Plato's insouciance in the matter of what has come to be called the Noble Lie (Republic 414B\u20135D). Poets are to be banished from the city because they confuse the real and the merely apparent, and that is intolerable, but the philosopher-kings, who know reality and the truth and hence can lie to the populace in the interests of maintaining the city in a good state, can, of course, lie with impunity.\n\nThere is a kind of failure vis-\u00e0-vis the truth that Greek philosophers worried about, but it was not failure to tell the truth which one knew. Greek philosophers were concerned with failure to see the truth because one was taken in by mere appearances.\n\nClassical philosophers did not in general worry about lying or truth-telling and certainly did not take true speech as a general touchstone of moral character. However, as Foucault recently taught us, there was one specific context in which a certain type of truth-telling was discussed and in fact highly praised. This was the virtue of \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u03b1, which, as the name indicates (\u03c0\u1fb6\u03bd + \u1fe5\u1fc6\u03c3\u03b9\u03c2), is not really the virtue of speaking the truth rather than a direct lie but of speaking out and saying everything, as opposed to keeping silent or saying only what will be acceptable to one's interlocutor. We might call it \"outspokenness.\" Thus in Euripides' Bacchae, Pentheus, the young ruler of Thebes, has begun to act so much like a classic stage tyrant that even the Chorus says\n\n\u03c4\u03b1\u03c1\u03b2\u1ff6 \u03bc\u1f72\u03bd \u03b5\u1f30\u03c0\u03b5\u1fd6\u03bd \u03c4\u03bf\u1f7a\u03c2 \u03bb\u03cc\u03b3\u03bf\u03c5\u03c2 \u1f10\u03bb\u03b5\u03c5\u03b8\u03ad\u03c1\u03bf\u03c5\u03c2\n\n\u03c0\u03c1\u1f78\u03c2 \u03c4\u1f78\u03bd \u03c4\u03cd\u03c1\u03b1\u03bd\u03bd\u03bf\u03bd, \u1f00\u03bb\u03bb\u0309 \u1f45\u03bc\u03c9\u03c2 \u03b5\u1f30\u03c1\u03ae\u03c3\u03b5\u03c4\u03b1\u03b9\u2022\n\n\u0394\u03b9\u03cc\u03bd\u03c5\u03c3\u03bf\u03c2 \u1f25\u03c3\u03c3\u03c9\u03bd \u03bf\u1f50\u03b4\u03b5\u03bd\u1f78\u03c2 \u03d1\u03b5\u1ff6\u03bd \u1f14\u03c6\u03c5.\n\nI am afraid to speak free words to a tyrant, but nonetheless it shall be said: Dionysus is inferior to none of the gods. (ll. 775\u201377)\n\nWhen a messenger appears who has witnessed something he knows Pentheus very much wishes not to be the case, he hesitates and asks Pentheus whether he wishes really to hear the whole story.\n\n\u03b8\u03ad\u03bb\u03c9 \u03b4\u0309 \u1f00\u03ba\u03bf\u1fe6\u03c3\u03b1\u03b9, \u03c0\u03cc\u03c4\u03b5\u03c1\u03ac \u03c3\u03bf\u03b9 \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u1fb3\n\n\u03c6\u03c1\u03ac\u03c3\u03c9 \u03c4\u1f70 \u03ba\u03b5\u1fd6\u03b8\u03b5\u03bd \u1f22 \u03bb\u03cc\u03b3\u03bf\u03bd \u03c3\u03c4\u03b5\u03b9\u03bb\u03ce\u03bc\u03b5\u03b8\u03b1\n\n\u03c4\u1f78 \u03b3\u1f70\u03c1 \u03c4\u03ac\u03c7\u03bf\u03c2 \u03c3\u03bf\u03c5 \u03c4\u1ff6\u03bd \u03c6\u03c1\u03b5\u03bd\u1ff6\u03bd \u03b4\u03ad\u03b4\u03bf\u03b9\u03ba\u0309, \u1f04\u03bd\u03b1\u03be,\n\n\u03ba\u03b1\u1f76 \u03c4\u03bf\u1f50\u03be\u03cd\u03b8\u03c5\u03bc\u03bf\u03bd \u03ba\u03b1\u1f76 \u03c4\u1f78 \u03b2\u03b1\u03c3\u03b9\u03bb\u03b9\u03ba\u1f78\u03bd \u03bb\u03af\u03b1\u03bd.\n\nI wish to know whether I should tell you what happened out there outspokenly, or whether I should hold my peace. I am afraid of the quickness of your temper, Lord. It is exceedingly sharp when roused and very kingly. (ll. 668\u201371)\n\nHere the political and social context is all-important. This truth-telling in the sense of \"outspokenness\" is an act of moral and political courage, telling someone in a position of power\u2014and is known not to be averse to using that power if displeased\u2014something he does not wish to hear. \"Outspokenness\" as opposed to discretion, timidity, silence, telling only the acceptable part of the full story, is a completely different thing from speaking the literal truth as opposed to saying out loud what one knows is factually false.\n\nThis might be taken to throw doubt on the claim, which has its origin, I believe, in the work the twentieth-century journalist Hannah Arendt, that the liar is the political man; the truth-teller not.\n\nWhile the liar is a man of action, the truth teller.... most emphatically is not.... The liar... is an actor by nature... he wants to change the world.\n\nThere seem to be two slightly different ideas operating here, both of which are incorrect, or at least extremely misleading. The first is that the \"truth\" is simple, just out there for all to see, and so seeing it and telling it is just a matter of recording what anyone can see is self-evidently there. The second is that the decision to tell the truth is simple or apolitical, in contrast to the highly political action of lying. To say that truth-telling is nonpolitical means presumably that it is not based on a calculation about its possible effects on the distribution of power, or an active concern to change the political situation; in contrast, presumably, I lie for a particular political effect.\n\nTo start with the first of the two misconceptions, that of the simplicity of the truth. In contrast to this, I would contend that what is \"out there\" is usually a farrago of truths, half truths, misperceptions, indifferent appearance, and illusion that needs to be seriously processed before one can accept any of it as \"real.\" Think of the contents of a daily newspaper or the opinions expressed in a pub. One specifically modern form of social control is to allow free expression of all opinions, thus creating a chaotic landscape of informational overload in which politically important facts simply get lost in the welter of surrounding nonsense, and important connections cannot be made. As the Greeks put it, truth \"is deep down in the well\"\u2014you have to look for it and extract it\u2014it doesn't lie there on the street in the sun waiting to be observed by anyone who glances in its general direction. That means that to see and recognise the truth requires the exercise of a certain kind of systematic violence on the inchoate and formless mass of undifferentiated appearance, wishful thinking, fantasy, and half truth in which we live most of our lives. Nietzsche at the end of the nineteenth century spoke of a will-to-truth that was necessary to explain how we came to be able to get access to the true and keep a firm grip on it, and he, correctly I think, emphasised the amount of asceticism that was required of a human being to resist our natural impulse to wishful thinking and see the world as it is rather than as we would like it to be.\n\nThe second misconception is that telling the truth is in some deep sense \"apolitical.\" Certainly ancient \u03c0\u03b1\u03c1\u03c1\u03b7\u03c3\u03af\u03b1 was not a form of mere speculation but rather a kind of politics. Discovering, expressing, and implementing the truth is not a way of stepping out of the messy realm of power and politics altogether but, to some extent at least, a way of acting within it.\n\nThe characteristic modern struggle, then, is not between those who tell a factual lie or a factual Big Lie, and might even try completely to defactualise the world, on the one hand, and those who bear witness to a simple truth, on the other, but between those who wish to keep politically active populations in a state of \"doxa,\" mere undifferentiated shifting opinion, that is, a world in which masses of truth and falsity are inextricably interconnected, in which, for instance, all the facts are in some sense available, but they are so distributed that no one can put them together coherently. Ancient models are better at analysing this than Christian ones.\n\nEven if one is interested not in the distinction between mere appearance and reality but in some psychological property of \"truthfulness,\" it isn't clear that any exclusive emphasis on lying is appropriate. Straightforward lying about factual truths\u2014saying explicitly of some object one knows to be made of brass that it is not made of brass but, for instance, of gold\u2014is only one particular instance of the human Duplicity. Christian ethics focuses on this one kind of case, but it is easy to see that cases like this by no means exhaust the almost infinite variety of forms of deception. Thus, to return to Odysseus's grandfather, when Autolykos is described as someone who \"excels all men in oaths\" (Odyssey XIX 396), this probably does not mean that he swore strictly oaths that were strictly false. He did not swear to give someone two sheep tomorrow and then fail to do that. Rather he was capable of formulating oaths he could literally keep while at the same time seriously disadvantaging the other party in a way they did not expect. Christianity, obsessed as it is with straight-forward speech, is also obsessed with lying. The direct lie, however, is by no means the only, or even the most important, aspect of a whole archipelago of related phenomena.\n\nThe specifically \"modern\" danger is not the Big Lie: the straight-forward intentional assertion of a nontruth that is known to be a nontruth, such as that Trotsky was in the pay of the Gestapo and plotted to kill Stalin or that Saddam Hussein possessed weapons of mass destruction. To worry about \"lying\" can be a way of diverting attention from other aspects of the situation and focusing it on such phenomena as sincerity with which the belief is held and affirmed. Did he really believe what he said? If he really did sincerely believe it, can he be said to have been \"lying\"? Perhaps it is instead just a harmless mistake. Politically, and arguably even morally, this is usually of distinctly subordinate importance. Blair's problem was not that he intentionally told the public something he knew was not true but that he had no interest at all in the category of \"truth,\" only in his own religiously based moral intuitions and in what forms of speech would be politically effective. An opportunistic fantast is not exactly a liar, but this is not obviously to his moral or political credit. The characteristic strategy for politicians in the modern world in any case is not to tell a direct lie but to sow confusion. Political manipulation nowadays is more likely to be a subtle blend of keeping the pieces of a puzzle apart so they do not cohere, deemphasizing some things that are important, diverting attention to other things, deniably suggesting connections that do not exist, distracting people's attention, wishful thinking in the evaluation of evidence, and so forth. None of these is interestingly analysed by the model of conscious suppression of strict factual truth.\n\nSo my conclusion is that trying to understand modern, twenty-first-century politics under the category of \"the lie\" is a very superficial way to proceed, encouraging na\u00efve moralising and discouraging serious thought. It is better to look at the pathologies of contemporary politics as a series of differentially structured complex conjunctions of cognitive, psychological, moral, and political failures of a different kind, conscious deception, insouciance, unconscious self-deception, wishful thinking, lack of attention, lethargy, distraction, suppression of dissent, and inertia. This whole domain is deeply structured by the play of powerful agents pursuing their own interests in relatively unscrupulous ways, using threats, bribery, and direct force. In this context direct lies may, of course, be told, but the focus, if one wishes to understand, should be on the context, not individual psychology. The difference between truth and its opposites, and between more admirable and less admirable forms of politics, is not a difference between a disembodied realm of ideal discourse and the sordid world of interests, powers, and complex motives but a distinction within this latter world\u2014the real one we in any case always inhabit. Moralisation is no substitute for historical and institutional analysis.\n8\n\nPolitics and Architecture\n\nIn 2001 some of the faculties of the University of Frankfurt began to move physically from the often shoddy and distinctly rundown-looking postwar accommodation that had served them since the early 1950s into an architecturally spectacular set of buildings designed by Hans Poelzig in the late 1920s and set in a large park with an impressive view over downtown Frankfurt. Unfortunately, these buildings, known collectively as the Poelzig-Bau, had served as the corporate headquarters of I. G. Farben between 1931 and the occupation of the city by the U.S. Army in March 1945. What this means is that in 2009 a student could find that he or she was taking a seminar on Descartes, on Rimbaud, or on early church history in the very rooms in which in the early 1940s gas chambers and crematoria for extermination camps were designed. In the period between 1945 and 1995 the complex served as the headquarters of General Eisenhower and then of the Fifth U.S. Army. When the U.S. military moved out upon German reunification, the question arose as to what to do about the huge I. G. Farben complex, and it was only after a certain amount of political wrangling that the decision was taken to move the university into it. There was finally a sense that if the complex was not simply to be torn down, it would have to be symbolically detoxified, but how could that be done? The solution finally reached was that a permanent exhibition about its history would be installed in the building, which would be as uncompromisingly truthful about its past as possible, the main building itself would retain the historical name I. G.-Farben-Haus, and the large and impressive open space one encounters upon first entering the building, which is now the student caf\u00e9 but in the late 1940s was the antechamber of Eisenhower's offices, would be named the Eisenhower Rotunda. Finally, one of the squares on the new campus would be named after a former forced labourer in one of the I. G. Farben Works: Norbert Wollheim, a name that has special resonance for a philosopher because it is the surname of an important British philosopher, Richard Wollheim, who participated in the liberation of Belsen in World War II. It is possible, in fact almost inevitable, that there will be no consensus on whether this series of decisions and actions was in fact appropriate and adequate\u2014that is in the nature of a complex historical and political process like this one\u2014but I would ask you now to accept for the sake of argument my view that this was a reasonable and laudable attempt to deal with a difficult situation. Let me, however, now engage in some counterfactual history. Suppose the Supreme Allied Commander in Europe had not been the traditionally conservative Eisenhower but Field Marshal Douglas MacArthur, who was during the same period effectively Supreme Commander in the Pacific. MacArthur was a man of extreme right-wing political views who came to be notorious for his persistent advocacy of the use of nuclear weapons against the North Koreans and Chinese during the Korean War. MacArthur saw this as a prelude to the extension of the war to be conducted with nuclear weapons into China proper, which he also advocated. When he failed to obtain authorisation for this policy, because President Harry Truman refused to countenance it, he tried in various ways to use his military position to undermine or circumvent the civilian political apparatus in the United States until Truman was finally forced to dismiss him from his post. I suggest that naming the entrance to Poelzig's complex the MacArthur Rotunda would not have had the same effect of at least partially rehabilitating the building. On one final note, I should mention that parts of the Poelzig-Bau served as the headquarters of the CIA in Germany and in the 1970s and 1980s was the object of three terrorist attacks probably by members of the Rote Armee Fraktion (RAF), a splinter group that had its origin in the German Student Movement of the late 1960s. In a bomb attack by the RAF on May 11, 1972, one U.S. officer was killed and thirteen others wounded.\n\nI would not now be discussing this case at all if I did not think that the Poelzig-Bau was a most impressive piece of architecture. However, the more I think about this, the more difficult it seems to me to separate the strictly architectural aspects of my reaction to this building from the historical and political, and, what is more, I do not think that this is so unusual.\n\nThe very term \"architecture\" itself suggests a closer connection with politics than might be thought to be the case with any of the other arts or crafts. The Greek word \u1f00\u03c1\u03c7\u03b9\u03c4\u03ad\u03ba\u03c4\u03c9\u03bd signifies the person who is in charge of and has control over builders. \u1f04\u03c1\u03c7\u03c9\/\u1f00\u03c1\u03c7\u03ae in fact is one of the usual terms for a political relation of domination. On the very first page of his treatise on the good human life, Nicomachean Ethics, Aristotle appeals to the example of the architect to discuss the notion, especially important for him, of the hierarchical relation of human activities one to another and the differential forms of value that such hierarchically ordered activities have. The study of ethics, for Aristotle, is subordinated to that of politics just as the builders are subordinated to the architect. This metaphor of the \"architectonic\" is one that recurs in many later views to refer to relations of subordination between different parts of a theory. Such relations may, of course, be ones of mere logical, epistemological, or paedagogical dependence or of pragmatic subordination rather than specifically political subjection. It is not, in any case, as easy as it might initially seem to say exactly what a \"political\" relation, for instance, a relation of political subordination in the relevant sense, is.\n\nIt has often been pointed out that there is a basic ambiguity in the concept of \"politics.\" There is what I will call a \"wider,\" \"broader,\" or \"less specific\" construal of the term and a narrower or more specific construal. In the broader and less specific sense, \"politics\" means simply any human activity of structuring or directing or coordinating the actions of a group. So we can speak of a \"politics\" of the family or gender politics. The actions of different human beings can be \"coordinated\" in any number of different ways. Thus if one thinks of a preindustrial society living in a small mountain valley, there may be a very high degree of \"coordination,\" in the sense of simple regularity, exhibited by the members of that society. For instance, if there is only one pass over the mountains, it might well be the case that virtually anyone who ever tries to leave the valley in the winter does so through this pass. This is certainly a regularity we might observe to hold, but it is not in itself an archetypically political phenomenon because we think that using this pass to exit from the valley is a matter of simple necessity. That one goes through this exit, if one leaves at all, is not anything that is \"in our power\" or \"up to us,\" and that means it is not itself a political matter but simply a natural fact. In addition, however, to such \"natural\" coordination there is also coordination that results from specific forms of human intervention such as persuasion, emulation, or coercion, and these are the characteristics of politics.\n\nThus when certain philosophers have called freedom a precondition of politics or politics a \"realm of freedom,\" they are most sympathetically understood as making not some kind of ontological claim but rather describing a way of looking at the world. \"Politics,\" that is, especially in this first wide sense, is best understood as referring not to a special domain, like biology or astronomy, but to a way of seeing or considering the human world. The basic statement in politics is not: \"This is a political phenomenon\" as parallel to \"This is an organic (or inorganic) compound\" or \"this is a prime number.\" Rather the paradigmatic claim is \"this is a political question or issue.\" \"This is a political matter\" means it is a matter considered in some sense to be potentially in our power and up for decision, and which we have some potential interest in dealing with in one way rather than another.\n\nIf this is right, three further things would seem to follow. First, although in a primitive society the weather might simply be a given, not in itself a matter for political discussion, the question of what we might do about the weather can well be a political issue: Do we distribute umbrellas to everyone or not? Do we put up a communal awning or tarpaulin on poles over the village green? Or do we let everyone fend for themselves?\n\nSecond, what is a political question or issue is itself historically variable in a way in which the question, say, what is an organic compound or what is the sum of two numbers is not. What is political changes with changes in what we can and could do. In preindustrial times the weather is not in itself a political issue, subject to the caveats just mentioned above, but if we were able technologically to change and control the meteorological conditions, then it might very well become a political matter, in the weak sense, whether it rains on a certain day or not. That would mean that someone had decided to make it rain on that day or had failed to decide anything, leaving it up, as we might say, to \"nature.\" Note that in the preindustrial period people were not failing to decide on the weather, since they had no control over it; it was just there as a brute fact, a matter not of politics but of natural necessity.\n\nThird, suppose it really is the case that politics concerns things that are either in fact in our power or at any rate that we could imagine might come to be in our power, and suppose then further that any state of affairs that cannot be other than it is (such as whether a certain number is prime) stands outside the domain of political deliberation. If, then, I have a special interest in maintaining some feature of the present social or political regime, for instance, because it differentially benefits me, I may have a strong interest in trying to present this feature as a part of the order of \"natural necessity.\" Think of Margaret Thatcher's constant refrains about the ineluctable necessity of tolerating unemployment as a means to controlling inflation or of bowing to the imperatives of the market. This is the point at which it is sometimes tempting to appeal to claims about the objectivity of scientific results, and, of course, there often are well-supported scientific results that are relevant to political decisions, but it is also the case that sometimes political agents have a strong motivation for presenting as the only possible reading what is in fact only one specific reading of the existing evidence among others, namely the reading that seems to give support to their own projects and interests. The appeal to \"objectivity,\" whether justified or not, is so effective because it is responsive to deep-seated and perfectly comprehensible human needs. We seem to have good inductive reasons to cultivate our existing desire not to be grossly deceived about the world in which we live, if only because in most cases we have found that complete illusions turn out to have very painful consequences for us. This comprehensible desire for what we call \"objective truth\" can often come into a sometimes slightly unholy connection with our human need to find, or invent, determinacy, stability, and fixity at almost any cost. The world is unstable and insecure, and our life in it is uncertain. It is painful for us to confront this fact. It also is exhausting having constantly to calculate again, to exercise context-dependent judgement or reopen questions apparently definitively settled. In the face of \"objectivity\" we can relax and succumb to inertia, simplifying some aspects of the painful process of decision by leaving it, as it were, up to reality itself. Unless the shoe pinches us very badly so that we cannot overlook it, we would like to think the form of the shoe that happens to be customary in our society is the natural one or the \"objectively\" given one. The idea that humans \"naturally\" like \"freedom\" or \"choice,\" if that means that they like continually to have to exercise their unbridled judgement or make decisions under the conditions of great uncertainty, is unfounded. This does not mean, of course, that they like to be in painful bondage, and much of human life is an attempt to find a path through the world that is responsive to the two forces of avoidance of novelty and choice and avoidance of the painful consequences of failing to revise one's beliefs and attitudes when that is necessary.\n\nSo much, then, for the first, the wider and weaker, which does not, of course, mean \"less important,\" of the two concepts of politics. \"Politics\" in this wider sense is a matter of any form of coordinating action regardless of the means used to achieve this coordination. Our more usual, or what I will call the \"narrower,\" concept of politics contains some further components in addition to those that constitute \"politics\" in the wider sense. These are that the \"political\" coordination of social action makes use of at least the threat of recourse to coercion, force, or violence, and that there is some appeal to systematic forms of legitimation. So in the wider sense of \"politics\" I can speak of the politics in a chess club as people jockey for influence, a certain kind of power, and a certain advancement through established offices. However, this structure is not directly connected to the possible use of force. The chairman of the chess club may make decisions about who plays which game against whom in which room and at what time, may adjudicate disagreements, and so forth, and in these matters his word may be Law, but he cannot whip, or probably even threaten to whip, any of the members or lock them up against their will. On the other hand, a gangster can assault me, lock me up, and take away my possessions by force but does not make a claim that what he is doing is either morally good or politically legitimate. The full and narrower sense of politics comes into play only when the use of force or the threat of the use of force is a possibility, and when the potential recourse to coercion, force, or violence is presented as being not merely a fact to be accepted but in some way \"legitimate.\" The major agency in the modern world that makes this claim to legitimate use of coercive power is the state. \"Political\" in the narrow sense means having to do with coordination of action through the use of state power or with the attempt to influence, infiltrate, or put oneself into a position to exercise that state power.\n\n\"Architecture,\" too, is a term that is used ambiguously, although the ambiguities are comparatively harmless. Thus it can either mean a certain skill, craft, or artistic ability or the exercise of that skill or craft in the activity of designing and constructing physical objects of a certain kind. Or finally it can refer to the objects thus designed and constructed themselves. Architecture seems to be different from many of the other arts in several ways. First, architectural objects are palpably physical and inherently public: they are large objects, literally almost always bigger than any individual person, and they stand out, form physical obstacles to free movement, and shape the very space in which we live. Of necessity, then, they affect us in a way that is different from the way in which most novels, pieces of music, or easel paintings affect us. If I do not like the novel or poem I am reading, I can shut the book; if the picture displeases me, I can turn my head away. I cannot so easily exit from a large cathedral in which I find myself placed or change the properties of the houses that face onto the streets down which I must pass to get to the city centre. This at least mildly and potentially more coercive feature of architectural objects makes them more political than the products of the other arts. It could, of course, be argued that every painting I see shapes my perception in a potentially permanent way and therefore makes me see everything in the world in a different way. Still I do not usually have to look at any particular painting, but I do have to live in whatever building or part of the city I happen to live in. I can, of course, choose to live in one kind of building rather than another and can change the building I live in either by moving or by reconstructing it, but I cannot in the twenty-first century simply do without some built surroundings, as I can do without easel paintings. It is, of course, true that this difference between architecture and other arts became even more pronounced during the nineteenth and twentieth centuries, when forms of literature, music, and aspects of the graphic arts became highly privatised, than it was in some previous periods. Thus in the ancient world the basic form of consumption of what we now call \"literature\" would not have been silent reading in an empty room but the massive choral singing and dancing that was characteristic of the performance of a tragedy in Athens, or a public performance of epic by a rhapsode, or the reading of a speech or dialogue by a slave to a group of gentlemen of leisure. Still a constructed object like a house had in one obvious sense a firmer place as an opaque, solid, intransigent, three-dimensional part of the public fabric of a city than any ephemeral grouping of citizens did.\n\nThere is, therefore, an important further political issue here. Should buildings be unobtrusive, retreating into the background to allow agents to pursue their own self-chosen goals, as far as possible without apparent obstruction? If one has the view that any building in one way or another structures the space in which it stands, then this might seem slightly dishonest, a way of covering up what is in fact a choice about structuration and allowing it to pass unnoticed. The building may come to be taken as \"a fact of nature\" in the urban landscape rather than the result of distinct intervention. Surely, however, one might think, the consequence of this should not be that buildings should be hyperassertive, constantly calling attention to themselves and their effects. There is no optimal resolution to this tension. Perhaps for that reason reflecting on and theorising about architecture will always have a place in our intellectual life.\n\nFurthermore, given the persistence of the material from which most buildings are made, the structuration of the environment they produce also extends into the indefinite future and thus concerns an indeterminate number of \"anonymous\" other humans, who by the very nature of the case cannot be consulted. Completed architectural works now impose on future people a way of living by channelling the way human activities will be able to proceed. It is, then, coercing them at any rate in a minimal sense, making it easier for them to live in this way, and more difficult for them to live in that way, so any present construction is an act of political faith in a certain possible future.\n\nArchitecture has also often been held to be different from other arts in that it straddles the distinction between craft and fine art, between producing practical objects of use in a relatively predictable way, which is assumed to be the basic characteristic of a craft, and producing potentially unique aesthetic objects, which is associated with our modern idea of fine art. This dichotomy might also be associated with the distinction between being a \"mere\" builder and being a proper architect. There is a functional dimension in architecture and an expressive dimension, and much of the discussion at least during the past two centuries or so has revolved around the proper understanding of each of these dimensions and, most important perhaps, the proper relation between the two of them.\n\nIt is also an important fact about our society that people do not simply engage in the activity of designing and building, but some of them also do this as a profession, and in our society \"profession\" designates a very specific social role with associated legal rights, social and legal duties, and expectations. As a professional architect in our society, one is embarking on a life of entering into contracts with people to build things that they, within certain limits, specify. This immediately raises straightforwardly political issues. Do you enter into contract with just anyone, on what conditions, for whose benefit? Last month I happened to meet and have a conversation with a German engineer whose family had built up the oldest and largest cement works in Central Europe (Dyckerhoff) and who were understandably very proud of the extremely high quality of their cement. During World War II they had provided the cement for the building of the \"West Wall,\" the line of German fortifications on the coast of Western Europe designed to protect the Continent from Anglo-American invasion. When my new acquaintance's grandfather was called to account for this by the U.S. occupation authorities after the war, the elderly patriarch produced the original set of specifications and contracts for the building of the large pediment on which the Statue of Liberty in New York Harbor rests. In these contracts it was specified that the pediment must be made of Dyckerhoff cement (\"or cement of similar quality,\" as the contract apparently specified). In fact Dyckerhoff & Co. provided the eight thousand \"bins\" of cement from which the pediment was constructed. This line of response was apparently immediately accepted by the occupation forces without further question. Regardless of what one might think of the substantive rights and wrongs in this case, the issue is not whether the occupation forces ought to have accepted what might seem to us to be something too pathetically weak even to be called an \"argument.\" It was merely an attempt, successful as it turned out, by the accused to evoke certain sufficiently strong positive sacral associations in the mind of the accusers as to blind them and deprive them of the use of whatever weak ratiocinative faculties they may have had in the first place. The real question, however, is what this tells us about the nature of our own conceptual space. These are in no sense irrelevant or unimportant questions, but they have more to do with the social role \"architect\" than with the inherent nature of what the architect does. We might think that a doctor is, or should be, by virtue of his or her very role immune from politics. A military doctor should care for all the wounded on the battlefield, friend or foe, and civilised countries make it a point of pride to provide equal care for all combatants and for civilian victims including those who are now usually subsumed under the rubric \"collateral damage.\" This does not mean that there is no politics in being a doctor but only that we have decided that it would be a good idea, not an idea proposed to us by any notion of \"justice\" but by such notions as humanity, decency, charity, benevolence, and also probably by various utilitarian calculations, artificially to insulate the practitioners of the medical profession from making certain everyday political discriminations.\n\nThere is yet a further way in which architecture was traditionally distinct from other arts, and this is in the more inherently social, and potentially political, nature of the activity itself\u2014the activity of collective building. Aristotle's architect had the power to exercise a kind of social control, namely to give orders to the people who actually do the building. This power was not arbitrary, it did not come from nowhere, and it was not unlimited. It was a power based on the authority purportedly provided by knowledge (and perhaps also secondarily by experience, although many builders have more experience of construction than younger architects do). This immediately raises the question of what kind of knowledge there can be in this area. This question is especially pressing for architecture, more perhaps than for many other areas of human endeavour because precisely of the binary nature of the enterprise, that of designing and producing practically useful and aesthetically pleasing buildings. We do not have much difficulty in understanding the kind of knowledge that will contribute to making it likely that the building will be useful. It will have to stand up, and the technical discipline of statics gives us a relatively straightforward answer to that. But what notion of \"knowledge\" underlies the claim of the architect to produce an object of great expressive value?\n\nThe problem for the architect, then, is that if his authority is based on knowledge, the builder and engineer also have that, but if what makes him distinct from the (mere) builder, and hence entitled to some special kind of authority, is the aesthetic or expressive dimension of the product, that seems more reasonably to be construed as having to do with faculties like that of \"taste\" rather than with any form of knowledge. How does \"taste\" give authority? It is completely unclear what kind of authority an architect can deploy and in particular whether the architect has any coercive authority at all.\n\nIn the previous discussion of \"politics\" in the narrow sense, I left one important aspect of the concept unexplored. I spoke in a vague way about politics as being a realm of coercion, force, or violence, as if these three things were the same, or at any rate did not need to be distinguished. It is not obvious, though, that coercion and violence are at all the same thing. I can be reasonably said to have been \"coerced\" to do something in many kinds of circumstances even if no one uses anything we would naturally call \"violence.\" If you lure me into a room and lock the door, you have forced me or coerced me into staying in that room until you unlock it, but you do not seem to have used \"violence\" on me in the same way you would have had you picked me up and thrust me into the room. Similarly, it does not seem odd to say that by lying to me you can \"force\" me to do various things, in the sense that if I believe you, I might think I have no alternative to the course of action you wish me to take. Still, it seems a stretch of our current linguistic usage to call \"lying\" a form of violence.\n\nA further relevant distinction is that between active and passive or doing and permitting\/allowing to occur. This distinction is highly controversial among philosophers, or rather it is controversial whether it has any deeper significance. At an everyday level it is perfectly clear:\n\n(a) active: I push you into the water so that you drown\n\n(b) passive: I fail to respond to your cries for help even though I easily could\n\nThe reason this distinction is of relevance to this discussion is that political theory, especially of a liberal kind, has tried to focus on active, even deliberate human intervention. Politics, then, is construed as in the first instance about preventing other people from actively using violence on others. The result, however, is to skew the political realm. This is especially important for architecture because a building was historically an archetypically inert but persistent structure. I, or rather the builders, may have been active in putting it up, but once it was up, it was just there, and could be expected to stay there, if it was properly built, for a very long time. It can change its function over time, as when buildings originally constructed as fortresses become prisons, so whose deliberate intentions are significant, the person who built or those who now use? It is the very geometric structure of the building that forces people to act in one way rather than another, and also to fail to act in certain ways; if the structure works, it prevents the inmates from \"escaping.\" This passive nature makes coercion \"softer\" and harder to see, as it were, from the outside, although not to feel, if one actually must live in such a structure. It makes it also no less effective, and the question is whether it is not equally reprehensible.\n\nI have spoken of important ambiguities in the concept of \"politics\" and of various different ways in which we speak of \"architecture\" as a skill, an activity, a kind of object, or a profession. It will then come as no surprise that I also think that the concept of \"justice\" is multiply ambiguous. I would like to distinguish at least four rather distinct notions.\n\nFirst of all, \"just\" designates that which accords with existing, established, legal codes. Second, we call that \"just\" which accords with what we\u2014whoever \"we\" in each case happen to be\u2014think \"ought\" to be the enforced legal code. Third, \"justice\" is used simply, as Aristotle put it, to refer to \"all the human excellences together.\" That is, in this third sense \"just\" refers in a rather indeterminate way to that which is socially excellent, desirable, and so on in whatever sense and for whatever reason. I note that it is extremely important not to confuse this third sense of the term with the second because there might well be things we think are socially desirable that we also think cannot for various reasons be formulated in a legal code. Thus I might think it highly desirable that people in a society be grateful to those from whom they have received benefits, but I might also think it completely wrong for this to be formulated as a requirement of any kind of legal code. First of all, a legal code must be enforceable by reference to external indicators, and I might think \"gratitude\" is not the kind of thing that is sufficiently close to any external indicators to figure in a legal code. Second, I might think that precisely an important part of the value of gratitude is that it be exhibited without it being the case that it is legally required and would be sanctioned. Its virtue is that it is extralegal, not forced, and so forth.\n\nThe fourth conception of justice is one that focuses on questions of distribution. There has recently been significant disagreement among theorists about what it is that is supposed to be distributed, whether goods, welfare, opportunity, or the possibilities of agency, and there is a similar disagreement about whether the principles of distribution should be some version of equality or of proportionality, for instance, that goods and benefits should be distributed equally to each or to each proportionally to their perceived merit or contribution, whatever \"merit\" or \"contribution\" means.\n\nRecently (meaning during the past forty years or so) there has been a strong tendency to understand politics in a highly artificial, restrictive, and impoverished way. Following John Rawls, many theorists have essentially tried to construe politics as a form of human behaviour devoted primarily to the attempt to realise one particular social ideal, the ideal of justice. They have then further eviscerated the concept of \"justice\" so as to construe it merely as some general property of the distribution of goods and services in society.\n\nI think that this multiple ambiguity in the concept of \"justice\" has been a source of almost inestimable confusion. If \"justice\" is used in the third sense, namely that is just which is in any sense socially desirable, then, of course, it is no news that all politics is about justice. It is no news because it is just a tautology. However, it is easy to move, without noticing it, from that tautology to something that is by no means a tautology, namely to the claim that all politics is appropriately construed as concerned with the equitable or proportional distribution of preexisting goods and benefits.\n\nIf one starts from the notion of politics I sketched at the beginning, it is not difficult to see that not all politics is about justice but also at least about the coordination of action, the exercise of influence, and the control of the use of force (among other things). Arguably, \"justice\" is not even one of the more important human values that can be instantiated in the political or social sphere. This is particularly clear in emergency situations, but, putting them aside, think, for instance, of welfare, efficiency, humanity, activity, security, dignity, and decency, not to mention creativity, a sense of self-affirmation, and aesthetic grace. All of these are important social virtues, and none of them is self-evidently completely detached from the world of the political (at least in the wider sense).\n\nIn short, then, two associations of the concept of \"justice\" seem to me unfortunate and unhelpful in the context of architecture. The first is the presumption that justice will have something to do with codes, rules, and conformity to such existing codes or, for that matter, with conformity to a better set of ideal rules. The second is that justice has to do with properties of distribution of goods that are considered to exist antecedently. When Marx in the nineteenth century attacked the focused attention the political theory of his day turned on \"justice,\" it was because justice-centred theories took the goods in question at face value, as objects that had come into existence in ways that it was irrelevant to discuss. Rather, Marx suggested that political theory should look carefully at the activities through which such goods were produced in the first place and at the social relations that structured those productive processes. These, he thought, were the most important features of any society, and the rules of distribution, that is both justice in the sense of conformity to a legal code and justice in the sense of some scheme of distribution, were secondary. What I would like to suggest is that architecture would do well to concentrate on the generation and fostering of varieties of free activity and on the structure of the relations that will hold between the humans who need to interact, rather than on justice in the sense of either conformity to some code or the distribution of goods.\n\nIt has become commonplace nowadays to assume that justice is fundamental to our notions of societal order, that is, to the order sustained between ourselves without recourse to force. When I say I disagree with this, I do not mean to say that I think that this assertion is incorrect in the sense in which it is, for instance, incorrect to think that the sun moves around the earth. Rather I disagree with it because it gives the strong impression of being a clear substantive claim, but on closer inspection it turns out to be no such thing. On some of the readings it expresses an unobjectionable or even tautological claim. If what it means is, \"We tend to use the word 'justice' to refer to whatever we discover is essential to the maintenance of our social order,\" then this may be true, but it is uninteresting. Even here, to be sure, one might wonder whether we wish to say that justice is whatever is fundamental to our existing social order or whether what we really have in mind is that justice refers to what would be essential to some ideally desirable order we can envisage. The trivial readings of this claim about \"justice\" are, however, easily confused with other uses in which the statement expresses a highly contentful and controversial claim, for instance, that equal (or proportional) distribution of goods is in fact fundamental to our social order or to an ideally \"good\" social order. In addition, I might add, what is so special about \"our\" notions of societal order? Are we to be satisfied merely with recognising that they are the conceptions we, for whatever reasons\u2014good, bad, or indifferent\u2014happen to have acquired?\n\nNote, too, that this formulation seems to make the tacit assumption that societal order is good in itself, and \"our\" conception of social order especially good, and that force plays no major constitutive part in our society. I take it that tacitly this includes the threat of the use of force. What if some use of force turns out actually to be required to maintain \"our\" social order? This possibility does not seem even to be canvassed. So there is a highly specific set of liberal assumptions built into the very way in which this text is formulated that I, for one, would be inclined to reject.\n\nI spoke earlier about our horror at the uncertainty of human life, about the vertigo we experience in the face of the indeterminate, and about our anxiety at having to exercise judgement and decide in each case afresh how to act towards our world. This is part of the strongest motivation for the focus of politics on the concept of \"justice,\" as it is part of the motivation to cling with limpet-like inertia to theories we have once committed ourselves to, even when they have revealed themselves to be seriously flawed. This is also part of the origin of our tendency to exaggerate the level of determinate objectivity we have been able to discern in our world. To start from \"justice\" gives one the illusion that there is at least one distinct kind of determinate thing out there to which all the multiform indeterminacies and incommensurabilities of our forms of valuation can be reduced. If that were the case, we think, to some extent we would not need to exercise judgement. This, however, is exactly the problem with trying to reduce politics to discussion of \"justice.\" This does not, in turn, in any way imply that political discussion is merely indeterminate or merely a matter of arbitrary choice. There are things that at any given time we have no real alternative but to accept, and valuing one thing is often really incompatible with valuing something else equally. No one can be a champion boxer in the morning and a subtle and accomplished violinist in the evening. Merely employing the term \"justice\" to cover whatever we find of value will not in itself either solve any problems or cause there to be more unity and coherence among what we value than there otherwise would be.\n\nTo return once again to Aristotle, he thought that politics was in itself a constructive and \"architectonic\" activity. It was really about creating the conditions for free, valuable action and the social genesis of the right kind of person, the proper citizen of the city-state. Aristotle also thought that the city-state was the only social form within which the highest and most complex kinds of human activities could be carried out. One can accept Aristotle's general claim that politics and (by extension) architecture are about enabling positively valued forms of collective human activity and about creating a certain kind or type of person without necessarily accepting his hierarchical views about human activities or the further claim that the most valuable life is possible only within the rigid format of an ancient city-state.\n\nOf course, the routine tasks of everyday building have to go forward, and of course architects have to honour their contracts, take care for whom and with whom they build and what effects their building will have on the minimal provision of necessary goods, but architecture might also be seen, and has in the past been seen, to have an aspirational component, to be attempting to be \"constructive\" in more than just the literal sense. In the nineteenth century some philosophers spoke of the basic task of the architect as being to build a dwelling suitable for God. We twenty-first-century atheists don't use this kind of religious language anymore, but it is not difficult even for us to associate a clear and plausible meaning with that thought. Architects should try to create structures that by channelling human energies in novel ways focus and intensify some of them while thwarting and dispersing others. We have a variety of complex reasons for judging that the intensification of some activities has made our lives richer and more worthwhile, or that, alternatively, it has been a huge mistake. Thus we judge that forms of human interaction, of relation to self and other, have become more or less efficient, more or less focused and intense, more or less socially aware and benevolent, more or less constructive in relation to other valued outcomes, and so forth. Some of these ways of evaluating it we call \"moral.\" Also there is no particular reason to expect that the standards or criteria we now use for judging will never change. In some cases they will change as a result of interventions we make. If I were an architect it would be the high point of my life to discover that people who antecedently knew, as it were, all there was to know about the building I designed for them and who thought they had good reason to detest it came through living in it to change their minds and love it. I would have helped them change their way of looking at the world, their standards for evaluating what is good, and their taste. Perhaps one could appeal to various systematic considerations to argue that some particular change had not been for the better\u2014after all, sheer habituation has caused people to come to think they liked some extremely peculiar things\u2014but the argument would need to be made in detail and evaluated on its merits.\n\nNot all the evaluative standards we use in political philosophy, then, can reasonably be thought to be subordinated to a single notion of \"justice.\" If the demand that architecture should take account of \"justice\" is merely an exhortation to architects to look beyond their fees and consider the different ways, for good or for ill, in which their buildings will be used, and the different ways in which those buildings will encourage or discourage certain uses, then this is unobjectionable, but also rather trivial. To put emphasis on its aspirational and humanly constructive component is to try to think about architecture in a way that very much goes beyond the framework of thoughts about \"justice.\"\n9\n\nThe Future of Theological Ethics\n\nIf one looks at a human society from a sufficient distance, it presents itself as a complex structure of informal practices and formal institutions, such as armies, churches, families, corporations, political parties, and so forth. These institutions are kept alive by the participation of human individuals, but they also structure and give substance and content to the lives individuals lead. A life outside a set of structured social institutions is for all save the most unusual individuals not much of a life at all. Despite this, it is individuals rather than institutions that have come historically to occupy centre stage in much of traditional philosophy and, I daresay, theology.\n\nIndividuals in complex societies swim in a veritable sea of social expectations and claims that are made on them by overlapping institutions. I am legally obliged to repay my debts even to reprehensible financial institutions (that is, to virtually any now in existence), my friends expect me to treat them decently, I should cover my nose and mouth if I sneeze in a public place, in some societies religious groups may require me to attend services regularly, and so forth. These claims are of differing kind and origin, and of differing degrees of peremptoriness. Most of them are accompanied by\u2014or can easily be provided with\u2014a(n ostensible) \"reason.\" I should cover my nose when I sneeze because this prevents the spread of germs; I should vote in the General Election because the stability and legitimacy of our polity depends to some degree on at least minimal levels of participation; and so forth. Giving one \"reason\" is often not the end of the story but merely gives rise to another question and a request for a further reason. Why should I wish my society to have stability and legitimacy?\n\nPhilosophy begins when Socrates tries to construe the asking for and giving of reasons as a freestanding domain that can be treated purely on its own terms. The question that exercised Socrates is what life the individual should lead, which eventually gets transformed into the question of which individual action from among those available to me I ought to perform. Socrates, or at any rate some of his immediate followers, called this method of scrutiny via a (purportedly) self-contained process of the presentation of reasons, the critical questioning of those reasons, and the attempt to respond to critical objections, \"dialectic.\" Behind the dialectic lay the Socratic demand for \"autonomy\" of enquiry, that is, the demand for a form of discussion that was self-contained and did not depend on the support of any other discipline or form of external authority, or on the constant inflow of the results of new empirical research. The hold that this demand has exercised on Western thought should not be underestimated.\n\nSo the initial situation that is assumed in Socratic discussion is like that of an autonomous consumer in the ideal world of neoliberal economic theory who is faced with a free choice among an array of options. The ideal consumer is assumed to be autonomous in his or her choices, that is, able and willing to choose for herself or himself, and adequately informed about the nature of the product chosen and about his or her own preferences and interests. The participant in Socratic dialectic is similarly assumed to be in a position to bring up and discuss any opinion or belief freely without needing any further (mere) information or support for any external authority, and to be able to settle on and adopt whatever views eventually recommend themselves. To be sure, Socrates' interlocutors need to satisfy an additional condition that does not hold for the consumers. The consumer need only choose; Socrates' interlocutor must choose and also give an account of his\/her choice that will stand up to Socratic argumentative scrutiny.\n\nThere are, however, one might imagine, a number of important features about human life that the Socratic form of reflection ignores or seriously underestimates. These features can be grouped into three broad categories: information, autonomy, and action. First of all, one can ask whether I ever have adequate information about how I should live that is in any enlightening way like the adequate information I might have about the limited choice of some human product. More important, one might ask whether it is sensible to isolate the asking for and giving of reasons for action from the process of acquiring further information or knowledge. Perhaps \"ethics\" and \"scientific enquiry\" cannot be as neatly distinguished, or pursued as independently of each other, as Socrates seems to presuppose.\n\nThe second kind of question concerns autonomy. Am I really an autonomous chooser? If I am, how did I become one? Why do I find myself with these preferences and confronted with these options rather than others? Is it always for the best for me to construe myself (and others) as such autonomous choosers? What if communal coherence is more important than individual autonomy? What implications does it have for society if some people begin to see themselves as completely autonomous choosers? What does it mean if everyone sees him- or herself as such a chooser? What does it mean if everyone sees him- or herself as nothing but autonomous choosers? The Athenians were sufficiently worried about this to kill Socrates, and it seems pretty clear that they were right to see him as a threat to their way of life.\n\nFurthermore, human life may not really be so easily divided into two distinct parts: first, an early limited period when I am \"growing up\" and thus subject, for instance, to parental (and other) authority, and then a period of \"mature autonomy\" when I am not, and may not permit myself to be, subject to anyone's authority at all. However, even as an adult if I wish to learn Arabic, I am dependent on the authority of native speakers, I am forced to trust in the expertise of the medical profession every time I go to see a physician, and I depend on the testimony of others every time I trust what I read in a book about a country or city I have never myself visited or about any historical event. It is conceivable, then, that there is some less perspicuous, but more intimate, interconnection between autonomy, expertise, authority, and testimony than the Socratic model suggests. Perhaps, just as it might be good to construe the discussion of reasons and the acquisition of new knowledge as related sides of a single process rather than two utterly distinct things, it might also be good to see autonomy, trust in expertise, and dependence on authority as ways of describing interrelated phases in a process of moral development rather than distinct, self-contained modes of life.\n\nFinally, Socratic dialogue is considered to be isolated from the world of events and the requirements of action. Plato's dialogues, to be sure, are full of people hurrying off when the discussion gets too hot, and this realistic touch is part of their overwhelming literary charm, but the ideal type to which Socratic dialogue aspires is something more like the Symposium in which the narrator wakes up the following morning to find Socrates and his interlocutors still at it, or the discussion in the Republic which because of its very length could not have run its course during a single night. Human life is not usually like this, however. At some point the cruise missiles begin to fall on the city centre, the sailors in the naval base riot, one's interlocutor is guillotined, Judas arrives in the garden with a group of armed men, or the security forces, to everyone's surprise, refuse to fire on the demonstrators. The proper conclusion to be drawn from this is not, of course, that it makes no sense to discuss the possible reasons for action at leisure and for as long as it takes but that it is unlikely to be clear at any given point in history to what extent events and the demands of action have informed or deformed any given line of argument.\n\nHegel suggests that traditional philosophical ethics in the Socratic mould tells us both too much and too little. In \"normal\" situations it is redundant\u2014as Hegel says, if you want to know what to do, ask your local priest or your lawyer\u2014and extraordinary situations are characterised by the breakdown of precisely those tacitly presupposed structures that give the usual form of \"philosophical\" reasoning its purchase. How, in addition, is one to know whether any given situation is \"normal\" or \"extraordinary\"? If Christianity is right, the Incarnation is an extraordinary event that brings about a radical change in human nature and its possibilities so as to render the wisdom of the philosophers' \"folly\" (\u03bc\u03c9\u03c1\u03af\u03b1). If Marx is right, forms of rationality are bound up with social forms in a way that does not permit simple extraction of a substantive universal form of \"Reason.\" It is no refutation of either of these views to appeal to an assumption it rejects, namely to the universal validity of argumentative forms that have their origin in Socrates' dialectical analysis of Greek common sense. Thus for Christians there is a \"revelation\" that is fully accessible only through faith\u2014which not everyone has\u2014and for Marx the results of Socratic \"dialectic\" depend on what bits of knowledge and opinion are the starting point, and certain forms of knowledge\u2014such as knowledge that human labour can be construed as a sequence of homogeneous, temporally extended units of exercised labour power\u2014are available only in certain highly developed societies, because only in such societies does labour take that visible form. This does not, of course, mean that Christianity or Marxism automatically wins the argument, just that the discussion is not already settled before it has properly begun.\n\nWhether considerations like these open a space for a \"theological ethics\" depends in part on what one might mean by that term. Does \"theological ethics\" constitute what we might call a natural kind\u2014that is, do all forms of theological ethics have something important in common that distinguishes them from nontheological forms of ethics\u2014or is \"theological ethics\" just a term that collects a number of different positions that really have little in common but are grouped together only as a convenience to contrast them with something else, such as \"secular\" or \"nontheological\" (or \"nonreligious\" or \"nonmonotheistic\") ethics? By virtue of what does an ethics count as \"theological\"? Does theology give us merely a new way of understanding (and perhaps \"grounding\") what we already recognise as \"our common, agreed-on morality,\" or does it require us to change our moral beliefs and practices?\n\nThat is, is there such a thing as a corpus or code of what Trotsky calls \"generally recognized... elementary moral principles... [which are] necessary for the life of every collective body\" and which, therefore, are common to the adherents of the various religions and to atheists? Is it then the case that only some theological assumptions will allow humans to understand these principles, so that we atheists are just internally confused and don't understand the nature of our own form of life? Or perhaps there is more than one way to understand common morality: as it were, as the law of the city of man and as the law of the city of god (although the moral code itself is the same). This might be a bit like Rawls's dream: a common morality (in his case a political morality) that is agreed on, although understood in different ways by the members of different religious and secular groups who comprise the society. Or is it that we all understand the common principles of morality well enough but only theological analysis will \"justify\" these principles to us? (If that is the case, what does \"justify\" mean, and does it mean the same thing in \"justify to an adherent of religion\" and \"justify to an atheist\"?)\n\nA further possible claim is that there is a common core of shared human morality, but theology allows us to supplement this with an additional set of additional demands or ideals. Thus: \"They said to you of old 'Do not kill', but I say to you 'Do not even be angry.' \" Or is it the case that there is no common moral code for all people, and so the theological project is not merely greater understanding or firmer justification of what we all agree on but revolutionary change in our moral views. Theology is necessary for us to see that we must change the way we live and evaluate ways of life.\n\nOr is theology not directly relevant to the whole of our moral life, but merely to a special sector of it, constituting the origin or ground of some new demands or ideals we would otherwise have no access to? So might there be special theologically based demands of ritual purity or symbolic manipulations of ritual objects that did not much impinge on everyday life but were thought to be important, like the fetishistic treatment of flags or other symbols in certain forms of contemporary civic religion?\n\nIn any of the above-mentioned cases it would be of extreme importance for the prospective proponent of a theological ethics to be able to say what exactly \"ground\" means when one speaks of theology as \"grounding\" a form of ethics. In this context it is not sufficient to make a vague gesture in the direction of \"anti-foundationalism.\" \"Anti-foundationalism\" does not mean abandoning all attempts to give an account of what one believes or practices.\n\nOne might well ask whether Christian ethics can sustain a commitment to global justice. Does \"sustaining commitment\" here refer to some kind of psychological, motivational, or social process or to something more like giving a valid justification or giving a good reason? (Or to all of these? If all of them, how are they related?)\n\nTo whom is theological ethics addressed? One might think that there are several distinct audiences and that theological ethics has a number of different tasks. First, there are people who are already Christians. The task of theological ethics here would be to give such people theological reasons to be concerned with social justice. Second, there are those who are not Christians but who have an interest in social justice. Here the task would be, presumably, to show that an interest makes sense only if one accepts certain theological views. Thus, I think I already have reason to be committed to social justice, although I have no theological beliefs. Am I being incoherent? Would I only understand myself and my commitment correctly or fully if I had the right theological views? Would the right theological views give me, in some sense, \"more and better reasons\" than I already have to favour social justice? What exactly would that \"more\" amount to? A third possible audience would be all persons of good will, regardless of their religious affiliation or lack of affiliation, and theological ethics here might also address the task of exploring the possible modes of practical cooperation between believers and nonbelievers.\n\nI believe that Christian theologians use the term \"apologetics\" to refer to discipline directed at giving some kind of justification or defence of Christianity to nonbelievers. However, I would note that even this way of construing the discipline contains within itself an initial structuring of the discussion that not all will find appropriate. Plato (and Xenophon) could write \"Apologies\" for Socrates because he was accused in a court of specified crimes. To assume that the task for a Christian theologian is to \"defend\" Christianity against some specified set of theoretical objections to it seems slightly unmotivated. Parish-pump atheists still exist, but I suspect that the real danger for religious believers nowadays is not counterbelief or theoretical objections but indifference. Richard Rorty, whose grandfather was a Social-Gospel theologian and whose father was a Communist poet, notoriously claimed that he was neither a believer, nor an atheist, nor an agnostic because all of those were considered positions one could take on religion; he simply wished not to have to take a position on this topic one way or another at all. In his own inimitable phrase he simply wished people would completely stop talking about religion, whether they wished to defend, to attack, or to reform it. This increase in sheer brute indifference\u2014Why should we care one way or the other?\u2014rather than active unbelief or unwillingness to listen seems to me to be the social phenomenon that constitutes the greatest threat to traditional forms of monotheism.\n\nSome theologians seem admirably aware of the political dimension of the task of addressing and coordinating the potential action of believers and nonbelievers, but if shared (or parallel) religious experiences are neither necessary nor sufficient for understanding of or commitment to equal human dignity, then the political task of coordination seems rather anodyne. We seem to be just back to the old liberal treadmill of \"seeking consensus among people of good will, but different opinions.\" There is perhaps nothing wrong with liberalism, if, for instance, the alternative is some form of extreme authoritarian despotism, but one would hope both philosophically and politically for something more and a bit stronger than that.\n\nFurthermore, what is the relation between \"theological ethics\" and \"religious ethics\"? Prima facie, after all, if one thinks of theology as a systematic and discursive treatment of certain beliefs, and religion as having to do with human modes of experience and action, institutions, and forms of cultivation, worship, or reverence, then it is easy to imagine the theological and the religious as diverging.\n\nA theology has to have something like an argumentative structure. Simply sitting around meditating, occasionally muttering \"Oum, oum, oum,\" or shouting out loudly \"alleluia\" or \"alahu akhbar\" in moments of excitement may (or may not) be a worthy form of religious activity, but it is not yet a \"theology.\" To put it crudely, anything correctly called an \"-ology\" has to be structured around something like what the Greeks called \"\u03bb\u00f3\u03b3o\u03c2\" that is, it must be something like an argument, a dialectical sequence, a set of logically connected propositions or something like what we now call a \"science\" (in the broadest possible sense), and it is not self-evidently clear that all forms of religious experience require or even admit of such treatment.\n\nThus in the ancient world the Epicureans had what was in one sense a \"theological ethics\" in that they thought that if you wished to lead a good life, you had to have the correct theoretical views about the gods. The view of the Epicureans, though, was that the gods were indifferent to humans and therefore were completely inappropriate objects of reverence or fear. So that would be an instance of a theology without a religion. But equally there might be religious views or views about religion that were non- or anti-theological. Thus the modern world has seen a number of anti-theological thinkers, for instance, the Pascal of the \"God of Abraham, not of the philosophers\"; Feuerbach, who proposed religious practices based on a fully re-anthropologised theology; Heidegger, who construed early Christianity as a religion without a theology and envisaged a kind of quasi-religious, but atheological ethics; and finally Wittgenstein, who seems to have thought that some religious experiences were very important but that there was nothing one could say about them. All of these thinkers were trying to rehabilitate the unvarnished religious life against what they took to be the distortions of it by theology. Roughly speaking, it seems plausible that the more vivid, explicit, and restrictive a view you have about what counts as \"theology,\" the more likely it is that some people will reject the claim that any particular theological view is adequate to encompassing their religious form of life.\n\nPerhaps a theologian might wish to try to emphasise the role of theology in sustaining cultural and community identity. Here I would like to point out what seems to me a significant structural difference between traditional Christian theology and the various claims that are made about modern \"cultural identity.\" In the modern period \"cultures\" are thought of as being plural, constructed, and nonobligatory, even if one also thinks that many cultures have a tendency to secrete around themselves claims to uniqueness, to being \"natural,\" to being superior to all others, and so forth. Traditional Christian theology, though, unless I am mistaken, was committed to presenting not one of a plurality of possible ways of living but a unique, non-negotiable truth. It is hard to see how these two perspectives can effortlessly be combined. I would urge resistance to the temptation to reconfigure theological beliefs as mere cultural formations: if \"theological ethics\" is going to have any purchase at all, it will have to make some claims to truth that go beyond merely saying \"This is my cultural identity.\" The letters of Saint Paul, however, strongly suggest that Christianity at least is not supposed to be a cultural identity at all but something else.\n\nIs it really theology in the sense of an argumentatively structured set of general beliefs that gives the impetus to social ethics or is it religion in the broader sense of the general ethos associated with a set of human practices?\n\nI have always found it very odd that mainstream Roman Catholicism clings so fervently to the philosophy of Thomas, given that the philosophical framework he inherited from Aristotle has certain signal defects for any Christian. So although Aristotle had a concept of development, in the sense in which, for instance, an acorn develops into a tree, he had no concept of \"history,\" at least not in the robust sense in which many Christians have understood \"history\" (Heilsgeschichte). Again, the Incarnation, so the Christian view went, changed human nature radically and qualitatively. Thus one's location in a uniquely specified historical time sequence\u2014before the Incarnation or after\u2014makes a tremendous difference to what kind of life one can lead, and thus to morality. It is hard to see how this archetypically Christian view can be accommodated within an Aristotelian account of a historically uniform human nature.\n\nAristotle's lack of a concept of history was a serious defect, but his lack of another concept that has become central for modern thought was actually a great advantage, because that concept is confused and we are better off without it. I am speaking, of course, of the concept of the \"will,\" which is very strikingly lacking in Aristotle, and which luxuriates in proto-baroque splendour in the philosophy of Aquinas, throwing out its obfuscating tendrils in all directions. Although Aristotle had no concept of the \"will,\" he had a concept of human choice and that is all that one really needs. Roughly speaking for Aristotle, if I do X and have not been externally coerced into doing it, that will be because (a) I chose to do X (\u03c0\u03c1\u03bf\u03b1\u03af\u03c1\u03b5\u03c3\u03b9\u03c2), or (b) because I acted on some impulse (\u1f10\u03c0\u03b9\u03b8\u03c5\u03bc\u03af\u03b1). That is for Aristotle the end of the story. There is no \"choosing\" to act from choice or from impulse, and nothing like the modern \"will\" is involved in the process. The only thing Aristotle has to say beyond that is that if I am a good (\"virtuous\") person I will generally choose rather than act on impulse, and the reason for that will be that I have natural aptitude to become a good person and have had the right upbringing. However, if I am a good person, I will\u2014to put it paradoxically\u2014have no \"choice\" about acting out of choice rather than impulse. That I habitually act from such choice is just what it is to be a good person. A good person, therefore, is not \"free\" in any interesting sense (apart from the political sense of not being someone's slave). There is no room for \"the will\" or \"freedom\" in this construction. \"The will\" is constructed and introduced as a concept by the Stoics as an anti-Aristotelian invention to allow them to say something Aristotle never would have said, namely that the good person is \"free.\" It is thus much to be regretted that Thomas took this confused nonstarter of a concept \"will\" out of its context in anti-Aristotelian thinking and tried to graft it into a basically Aristotelian framework. Western philosophy suffered from the depredations of trying to make sense of the fictitious faculty \"the will\" until the time of Nietzsche.\n\nFinally, accepting Aristotelianism means Christianity gets itself unnecessarily involved in trying to offer an alternative cosmology to natural science; this, however, is an argument it lost long ago, so it would seem more honest to acknowledge that and draw the consequences.\n\nMy first and narrower suspicion, then, is that if there is a future for theological ethics in the Roman Catholic tradition it will certainly not be Thomist. My wider conclusion is a strong suspicion that whatever the prospects of religion in the future, those of the purported discipline of \"theological\" ethics are doubtful, if only because of the strong categoricity of our conceptions of reason and argument. It is hard to see how anything could conform to them, as \"theology\" is supposed to do, while yet being appropriately distinctive. What over a thousand years of strenuous exertion has signally failed to find is probably not there.\n10\n\nDid Williams Do Ethics?\n\nBernard Williams came to bury ethics, not to criticize or revise it. He did not, of course, mean by that that there was nothing in traditional forms of ethical thinking (or nothing in traditional moral injunctions) that was of any substance or of any use or significance for human life. He did, however, think that the traditional notion of \"ethics,\" namely as an autonomous, knowledge-based, reflective, discursive doctrine that could give completely general and rationally persuasive answers to the question, \"How should one live?\" was unsalvageable.\n\nWhat, then, should replace ethics? First of all, perhaps nothing will or should replace it. Instead of a single hegemonic discipline, which gave us answers or the framework for finding answers to the question how one should one live, there will just be a variety of different things. Perhaps human life is characterised by a welter of different goods that form no cognizable unity; perhaps the very idea of a single, or a single dominant, notion of \"normativity\" just is a mistake. After all, the very term \"normativity\" is a recent invention\u2014it has no entry in the 1933 edition of the Oxford English Dictionary and its currency as a technical term in philosophy can scarcely date back to a period earlier than the 1980s. The fact\/value or \"is\"\/\"ought\" distinction is older than that, and the adjective \"normative\" has sporadic earlier uses, but the idea that there was a single \"thing\" or phenomenon that could be designated by the single term \"normativity\" may be thought to represent not a mere verbal quirk but a not-insignificant step in giving the discussion of substantive issues a particular turn or slant or structure.\n\nPerhaps then instead of any kind of single overarching \"normativity,\" all there is are simply different\u2014and possibly changing\u2014human practices with different goals, associated conceptions of excellence, and resultant goods; and human life consists of an art or skill in negotiating a way through, which is partly constituted by these practices, partly a matter of making use of them for other ends. What replaces ethics then is not another intellectual discipline but forms of action, which may be skillfully or less skillfully performed. The art or skill involved, however, might not\u2014without significant loss\u2014be reducible to anything like the object of a cognitive discipline. Nietzsche in some of his moods seems to take a tack like this, adding that the art or skill in question would have an extremely strong component of a type we would be likely to call \"aesthetic,\" and that the human emotions of admiration and disgust would play a constitutive role in it.\n\nThe basic idea that ethics as a purportedly freestanding philosophical enterprise was a mistake is not in itself novel or unusual. I once heard the president of a large and very well-regarded university, whom I will call \"Zmith,\" ask the academic members of one of his advisory boards why his university needed departments of philosophy and political science at all. After all, he remarked, the university had a flourishing law school and a distinguished department of economics, and surely they could satisfy any reasonable human cognitive need. Zmith's remark was, to be sure, part of a micropolitics of bullying, of trying to intimidate the assembled academics and show them who was boss, but it would have had no chance of being effective if there had not been at least the shadow of a suspicion that Zmith might actually believe what he was saying and that some other influential people might come to believe it, too. Compare this case with that in which Zmith asked why a philosophy department was needed because his university already had a renowned archaeology programme and a music school. Actually, the idea that ethics as a subdiscipline of philosophy might simply be replaced by something else, by some part of economics or law (or some combination of both), is not in itself completely daft. It would, of course, require some changes in the existing disciplines of law and economics. It would require law to stop being the kind of cognitively disabling enterprise it now is, designed to turn intelligent young people into pliable mouthpieces for corporations; and it would require economics to establish at least some tenuous, nonwhimsical cognitive relations with the real world of a kind it has not yet been able to manage, but that might be possible, if the world itself changed. This, of course, is an old idea of Marxism, that if the basic economic structure could be rationally and transparently organized, it would be possible to have a proper theory of economics rather than the current hocus-pocus, and that would render a separate ethics as a philosophical discipline otiose. There might be some low-level \"ethical principles,\" as Trotsky suggests, but they would be commonsense rules of thumb about how people best get along together, known clearly to everyone and having the status of banalities, not profound philosophical truths.\n\nWilliams did not take this line, but rather one that arguably had its origin in Aristotle, who begins one of his ethical treatises by describing ethics as a subordinate part of political science. What should replace philosophical ethics, in Williams's view, was politics. This, of course, makes his view completely different from the Marxist view, which is that in a free and developed society, \"politics\" as we know it would not exist, only administration. That is not to say, I take it, that there will be a relatively abstract political philosophy, a fixed theory of a more or less traditionalist kind, and that a subordinate part of this will be devoted to individuals in society, but rather that ethics will be replaced by \"real politics.\" Politics has its own dignity, imposes its own demands on action, and both opportunistically consumes and fecundly generates concepts, convictions, theories, and forms of reflection. Some of these will concern individuals, their properties, dispositions, aspirations, and modes of behaviour; and these might be called \"ethical,\" but they have no standing on their own. When Williams says that politics should replace ethics, he does not, of course, mean by \"politics\" either a purportedly nonintentional and fully value-free form of human action or the object of a value-neutered pure science. Politics clearly is concerned with human agents who have goals, intentions, values, and conceptions of the good, and the study of politics will itself also be informed by forms of evaluation that will not necessarily be the same as those of the agents being studied. Traditional ethics, though, makes the mistake of trying to isolate these goals, intentions, and forms of evaluation and construe them as the possible subjects of a distinct discipline. In addition, when Williams says that politics should replace ethics, he does not mean \"should\" in a (strictly) ethical sense. Rather, he thinks that ethics always has in fact been a part of politics. Historically, any given ethics has usually been the theoretically congealed residue of a previous political practice that represents an attempt on the part of that past to stretch its dead hand out over the future. So \"should\" means that, overall, it is likely to be better for us to recognize this than to continue to pretend it is not the case. This, Williams thought, was the significance of Nietzsche and constituted one of the important reasons for studying the ancient world. Nietzsche had first allowed us to see the politics behind the superficial appearance of autonomy in ethics, and he had seen it most clearly in evidence in his study of the ancients. We cannot go back to ancient conceptions or institutions\u2014Williams was historicist enough to think that this was virtually self-evident\u2014and we wouldn't want to practice ancient-style politics even if per impossibile we could do so. But by studying the ancients, we can learn one of the few rather general truths accessible to us in this area, namely, a truth about the primacy of politics. The very fact that Plato struggles with such relentless energy to establish the standing and authority of something\u2014philosophy\u2014that is purportedly prior to politics might actually be taken to reinforce this lesson from the ancient world.\n\nIf we cannot go back to ancient politics, what would a modern politics look like? One common way of proceeding is through so-called democratic theory: modern politics is democratic politics, and we can explain, understand, and criticize it with reference to the ideals of \"democracy.\" Of course, if politics is really an art, there may be narrow limits to the kind of positive account one could give here. A monograph-length treatment of modern politics by Williams was never forthcoming; perhaps the book on Nietzsche he seems to have been planning to write at the end of his life would have contained some material about this, but of course we shall never know\u2014but if there had been a book, I would have expected it to be very historical in its approach and to have little in common with approaches based on \"democratic theory.\"\n\nWilliams took the central question of ethics to be that of Socrates, which he formulates as \"How should one live?\" but which one might gloss as \"How is it needful to live?\" (\u03c4\u03af\u03bd\u03b1 \u03c4\u03c1\u03cc\u03c0\u03bf\u03bd \u03c7\u03c1\u1f74 \u03b6\u1fc6\u03bd; Politeia 352d). I merely note en passant that there are other possible questions one could take as lying at the heart of the Socratic search. These include: \"What is the good life?\" \"What is the 'happy' (or successful or flourishing) life?\" and \"Who am I?\" One rather austere or minimalist way of trying to answer the question \"How is it needful to live?\" might be: How it is needful [for me] to live is what needs to happen in order that I live at all. Thus Voltaire complained to the royal censor who proposed to destroy his livelihood by banning the (satirical) publications on which his income depended: \"Mais, Monsieur, il faut bien que je vive.\" This historically recurrent type of complaint is not, as one might expect, an indefeasible moral argument but has as its irrefutable response the reply of the censor: \"Je n'en vois pas la necessit\u00e9.\" This dramatic scenario\u2014which is played and replayed through the centuries between Carthaginians and Romans (\"Carthago delenda est\"), Louis Capet and Robespierre (\"Louis doit mourir parce qu'il faut que la patrie vive\"), Herero tribesmen and German colonial administrators, the Ukrainian peasantry and Stalin\u2014with different roles assigned historically to different individual and collective actors, and agents acted upon by others, has a first-person reflexive correlate. One might even say that the formula for any kind of human progress, not just for progressive and revolutionary change, is the thought that \"What would be for the best would be if we\u2014and people like us\u2014simply did not exist.\" So before the establishment of the National Health Service in Britain, one can imagine large swathes of the population thinking that it was necessary for them to live in a certain way, for instance to learn to make hard choices between buying food and buying medical care for the members of their family; but they might also be capable of reflecting that it would be for the best if people of their kind\u2014people who needed to learn what they had to learn and who then had to act accordingly\u2014simply did not exist. Equally, one could imagine slaveholders or bankers or property developers thinking that it would be better for people like themselves not to exist. For that matter, one could imagine a \"free man\" in a society with slavery thinking that it was no criticism of him that he did not live the life of a slave but that also it would nevertheless be better for his whole class of people not to exist in the sense that it was defined, as the class it was, only by contrast to the enslaved population.\n\nWilliams used to say that the United States was the most eighteenth-century country in the world; it was, he thought, politically, socially, and culturally caught in a kind of time-warp, an apparently eternal present that was actually represented by some point in time in the 1790s. Williams never expatiated on what exactly he meant by this statement\u2014I assume that the reason for this was he thought that this was a particular kind of interpretative statement: if you did see the point of it, you required no further elaboration because it would immediately ring true to certain experiences you had had, putting them in a certain intelligible order; and if you didn't see the point of it immediately, then no amount of further elaboration on his part would convince you. And then the statement might actually have a more potential long-term effect, if it was simply dropped so as to shatter on the floor like a huge bit of unwanted crockery and the noise was allowed to reverberate through the room sans commentaire. Presumably part of what he meant, though, was that the country had never been able to move beyond a particularly archaic form of the struggle of the Enlightenment with its enemies. Nothing after Adam Smith and Kant really survived the Middle Passage to take root on the western side of the Atlantic, so the culture was stuck in a state of trench warfare between a party that identified itself (in some way or another) with Enlightenment Reason (either through commitment to the ideal rationality of \"free markets,\" to technologically rational solutions to all problems, or to vapid Kantian \"norms\") and an obscurantist religious fundamentalism that was all the more dark for defining itself in contrast to reason. The relatively recent formation of a united front between some religious fundamentalists and free-marketeers is a politically significant development, a shift in a local tactical alliance, but it is not yet clear that it has any further significance.\n\nTo say that the United States is an eighteenth-century country is to give a historical interpretation and perhaps also express an attitude, but it is not, of course, to make a moral judgement in the technical sense in which some have wanted to construe the term \"moral judgement.\" It is not to say that the population is vicious or the institutions corrupt. It is not the kind of statement one is liable to encounter in a book on \"ethics\" but rather the sort of thing someone like Herodotus would have said (if he had had a suitable concept of history and historical framework at his disposal). In his last book, Truth and Truthfulness, Williams tries to develop a theory about the relation between factual data and interpretation, which has always struck me as very close to the theory of the \"constellation\" one finds in Benjamin and Adorno. The stars that compose a constellation are physical objects of a certain kind and each has a set of empirically specifiable properties, a location in the sky, a certain relative magnitude, perhaps a colour. A constellation is not a mere collection but an organization of a set of stars into a recognizable and perhaps significant pattern: a bear, a hunter, a wagon, a set of twins. One cannot, of course, create constellations ad libitum, using stars that are not already there, putting together stars that are too far apart from the human eye to see synoptically, or creating patterns that make no sense to us. So we\u2014and what we are, as the historically located creatures we happen to be\u2014partly determine what constellations there can be (for us) by virtue of setting out the limits of what can make sense to us. \"What can make sense to us\" is neither completely pre-given nor fully indeterminate, and to the extent to which we can change it, this can be done only over time, and probably only collectively. The sheerly physical data about the distribution of stars does not require us to group them into this set of constellations rather than some other, and in fact does not require us to group them into any constellations whatever. We don't invent or create stars by organizing them into constellations. Still\u2014and this is the claim Adorno and Benjamin make\u2014at any rate, in the realm of collective human action, culture, and politics, we are guided by constellations, not by the analogues of raw or theoretically manipulated astrophysical data, and there is no real alternative to that. It just is not possible to do without something like constellations. Similarly, Williams thought that there were facts that had the hard, unmalleable character we usually attribute to them and that could be discovered by \"empirical\" means (whatever they turn out to be). The U.S. Constitution had x-number of articles and article z was passed on such-and-such a day. On the other hand, the \"positivist\" dream that one could dispense with any interpretation and simply let the facts speak for themselves was just that\u2014a dream. Similarly, no set of \"facts\" requires one to adopt a particular interpretation, but no interpretation that anyone would take seriously simply floats unconnectedly above the data or connects them in merely arbitrary ways.\n\nWilliams must have known that in asserting that the United States was an eighteenth-century country he was doing something that was very close to committing a modern American analogue of the crimen maiestatis by violating a central taboo of the state religion that holds the country together. This requires one always to assume that the United States, at least in some idealized form, is the very model for modernity, progress, rationality, and so forth\u2014not something itself to be evaluated. Of course, one can criticize individual performances of the government or even subordinate institutions, but only as a form of internal criticism, that is, only relative to the absolute assumption that the performance or the institution is a momentary falling short of the ideals the United States itself proclaims and fundamentally instantiates. I recall Williams making this claim in a series of lectures at Princeton in the 1980s\u2014and the audible intake of breath among the members of the audience. I suppose that he was not being gratuitously offensive but was trying to make a constructive political intervention aimed at warning those of them who were susceptible to using their theoretical imagination at all that the exceptional conjunction of geography and history that had permitted the unparalleled prosperity particularly of the period between 1945 and 1975 would provide only a brief respite from history. Presumably, Williams saw that, by the mid-1980s, the layers of insulation were wearing sufficiently thin for him to propose to his audience that it might be advisable to prepare for the changes that were inevitably coming, even if this required them to think what was for them almost unthinkable.\n\nPerhaps political and historical interpretations like this are a part of \"moral and ethical thinking\" in the wider sense which Williams tacitly accepts, and which he contrasts with the traditional academic study of \"philosophical ethics.\" Williams, of course, as the last part of his comment on Christine Korsgaard's Tanner Lectures indicates, does seem to have considerable sympathy for historicist ways of thinking, that is, for parallelization of \"individual rational reflection and historical development\"; and he had himself taken account of the nineteenth and twentieth centuries, at least to the extent of being a serious student of Nietzsche, for whom the meaning of the individual human life is a central concern. It is not, however, obvious that he ever really gave Hegel a proper chance, and figures like Heidegger and Adorno\u2014not to mention Deleuze and Guatarri\u2014were effectively outside his ken, meaning not, of course, that he didn't in some sense \"know\" who they were but that they were not sufficiently close to him in temperament and traditional intellectual formation for him to have interacted with them in a philosophically productive way. To be sure, in 2002 he agreed, after a certain amount of coaxing, to come and give a paper at the Frankfurt Conference on Adorno, which was to take place in 2003, but his final illness and death brought that possible line of development to a close. The argument that finally convinced Williams to come to Frankfurt was not the one for which I had had the highest hopes, namely, that he would find in Adorno, if he looked closely, a philosopher who was as interested as he was in taking up a position equidistant from the self-serving \"liberalism\" of the Anglo-American political world and the brutal practices of \"really-existing socialism.\" This line of thought had no purchase, because I had completely misunderstood Williams. I took his adoption of some of the vocabulary and the motifs of liberalism to be a bit of intellectual realpolitik or perhaps protective coloration, but this was a complete misconception, based no doubt on an incorrect, wishful projection of my own attitudes onto a philosopher whose work I admired. Even to the end of his life, Williams showed flashes of what are now called \"Old Labour\" attitudes\u2014\"profits\" for him were always inherently suspect and the fact that a particular legal, administrative, or economic reform would result in large profits for a corporation or a private individual was prima facie a good reason to oppose it. However, it was also the case that he felt as naturally comfortable paddling about in the tepid and slimy puddle created by Locke, J. S. Mill, and Isaiah Berlin as he did in most other places. In general, he was a man who was remarkably comfortable in his own skin and who fit in easily with the existing world of politics and academic society, despite his high scepticism about many of the purported theoretical pillars of that world. This, in fact, was perhaps one of the basic ways in which he was different from Adorno, who notoriously lived a life of great, not to say extreme, self-indulgence but made a point of not feeling comfortable in it. Adorno, in fact, insisted that it was a sign of minimal human decency \"not to be at home\" in the world of late capitalism.\n\nWhat did finally move Williams to think it might be a good idea to come to Frankfurt was the prospect of discussions of Adorno's views on Wagner and on the philosophy of music. Williams had a keen interest in music, particularly opera, and was steeped in the music of Wagner. One of my most vivid recollections of him is of discussing with him the concept of \"pornography\" while he hummed \"The Ride of the Valkyries\" and spun his two hands around, imitating an old-style propeller-plane about to take off. The last lecture I heard him deliver was on Wagner's Ring in the Cambridge Music Faculty. He had placed behind him the full University Orchestra\u2014about a hundred players\u2014who sat silent and immobile onstage during his lecture. After he finished speaking, Williams withdrew to a large high-backed chair with thick green upholstery and cushions, to the right of the players, and sat listening intently as the orchestra played Siegfried's Funeral Music. The chair was so large and Williams in his elder years had shrunk physically so much that he seemed like a kind of Bloomsbury Mime\u2014Mime, the proper name of the dwarf in Siegfried, not the English word for a kind of silent actor\u2014squatting on a throne in the afterlife, while looking down and listening in order to discover what had finally become of his nemesis Siegfried.\n\nWhat interested Williams was the relation between political and moral success and failure in Wagner's Ring, and aesthetic success (or failure). Clearly, if Siegfried is intended to be a model, or even a specimen, of the Young Hegelian emancipated human being, free from the world of conventional morality and commercial \"contracts\" (Vertr\u00e4ge)\u2014that is, if one will, from Adam Smith and Kant\u2014which still cripples Wotan, he is not a very convincing advertisement for the future. He is brutal, uncouth, empty-headed, and often simply nasty, and his life is a series of violent episodes that end in nothing much. Much of what he does, he does out of pig-ignorance or because he is being manipulated by others for their own ends. Few of his actions have the long-term or even medium-term results he intends, and few of them end pleasantly for him. Presumably he enjoys sleeping with Br\u00fcnnhilde (when he finally takes off her breastplate and discovers she is a woman like his mother and not a man); this interlude, however, seems to be very short-lived indeed\u2014although, given that it takes place, as it were, \"off-stage\" between the end of Siegfried and the beginning of G\u00f6tterd\u00e4mmerung, one cannot actually be sure; but the composer of Tristan (Act 2) would certainly have had at his disposal the artistic means to indicate a lengthy period of happy dalliance, had he so wished. Instead of that, before you know it, Siegfried is off down the Rhine \"zu neuen Taten,\" as Br\u00fcnnhilde sings (und neuen Katastrophen, as we may add). This is the rather gloomy interpretation that Williams propounded. The question for him, then, was why the funeral music for such a \"gescheiterte Existenz\" was still so profoundly moving, despite the fact that it celebrated a \"hero\" whose heroism was of a particularly empty kind. This sense of the impressive combined with the not-really-fully-substantial recurs in discussions of Wagner's work. The musicologist John Deathridge describes the music for the entry of the gods into Valhalla at the end of Rheingold as \"triumphant, but decidedly hollow.\" Many listeners have this experience of it, and, in fact, it is possible that this effect was intended by Wagner. Despite his own megalomaniac tendencies and the desire for total domination of the audience, Wagner was perfectly capable of producing calculated effects of distancing within the overall aesthetic experience. After all, the \"intellectual\" in Rheingold, Loge, interrupts the relatively smooth-flowing waves of D-flat major to remark that the gods are so deluded and so self-destructive that he is almost embarrassed to associate with them (\"Ihrem Ende eilen sie zu, \/ die so stark im Bestehen sich w\u00e4hnen. \/ Fast sch\u00e4me ich mich, \/ mit ihnen zu schaffen\"; Rheingold, bars 3807\u201312).\n\nSiegfried's life and death are not, of course, meaningless in the sense that they have no important effects in the world. After all, G\u00f6tterd\u00e4mmerung ends with the image of Valhalla\u2014that is, the capitalist world in which we all still live\u2014in flames. That is one good result of what Siegfried has done, although it is not anything he intended. Wagner reports in his autobiography that as he stood with his friend Bakunin on the barricade in Dresden during the revolution of 1848, Bakunin explained to him at great length and in detail why the joy in destruction was also a creative joy. Williams, of course, had studied Nietzsche's Birth of Tragedy very carefully and had a keen appreciation of ancient tragedy, but the Dionysian pleasure of the child on the beach who enjoys smashing up the sand castle he has just built seemed to play no role in his writings on politics, which generally still breathe the air of the usual liberal platitudes. Williams's cheerful disposition and his successful life in the more comfortable regions of the Anglo-American establishment put him at some distance from a visceral sympathy with Bakunin. In this, too, he differed significantly from Adorno, who was much less impressed with Wagner's music as music than Williams was, and was therefore less puzzled than Williams by the (apparent) discrepancy between the failed and empty \"heroism\" of Siegfried and the quality of Wagner's musical treatment of it.\n\nThere is, of course, another obvious way to read Siegfried's character and fate, which is different from the one Williams favours. Williams's interpretation depends on hearing the music as basically celebratory rather than merely elegiac, and so it was perfectly appropriate that his lecture ended with a performance of the passage. If one takes it as something closer to a threnody than a eulogy, it is possible to connect it not with Wagner's failure to present a fully convincing New Man but as a melancholy comment not on Siegfried in particular but on the whole world of the Ring. That Siegfried and his life are so unsatisfactory can be seen not as a form of failure on Wagner's part\u2014failure to create a plausible artistic image of the new and emancipated individual\u2014but rather as a success in representing what Wagner intended to represent. Perhaps Wagner's intention is precisely to dramatize the necessary failure of individual heroism and of manipulative projects of individual emancipation like those Wotan seeks to realize. Perhaps individual emancipation will never in our historical period and our economic system be able to develop into anything but the radically defective version of \"empty heroism\" we find in Siegfried. One might, that is, take the Ring to instantiate one of Adorno's more spectacular claims, namely \"Es gibt kein richtiges Leben im falschen.\" \"There is no 'right life' in the false life\" (i.e., there is no way for any individual to lead a \"right\/good life\" in a social formation that is itself repressive, duplicitous, and alienated). Adorno uses \"false\" in a nonpropositional way, as the Rhine-maidens do when they describe the whole world of the gods as \"falsch und feig\": \"Traulich und treu \/ ist's nur in der Tiefe; \/ falsch und feig \/ ist, was da oben sich freut!\" (Rheingold, bars 3858\u201368). \"Es gibt kein richtiges Leben im falschen\" is Adorno's version of the old maxim that one cannot be communist man in a capitalist society. This, then, is part of the reason it would be a mistake to try to think about the case of Williams, or that of Adorno, through the lens of the moral category of \"hypocrisy\" in any straightforward way. The moral category designates an individual failing of some kind, which makes some sense primarily in contrast, for instance, to possible sincerity, but what is at issue here is a structural feature of society\u2014if in fact Adorno is right\u2014which makes a fully satisfactory life of complete consistency and sincerity impossible. One can analyse the different ways in which individuals deal with this impossible situation, and even have a more or a less sympathetic reaction to their predicament and to their perhaps different ways of responding to it. If Adorno is right, their predicament is also our predicament and, more pointedly\u2014for me\u2014my predicament. Analysing this situation and discussing attempts to deal with it\u2014all of which are, Adorno believes, failures\u2014seems a very different matter from merely diagnosing \"hypocrisy.\" Had he been able to come to Frankfurt, it seems unlikely that Williams would have been able to avoid taking some explicit position on this.\n\nAt the beginning of Ethics and the Limits of Philosophy, Williams writes that his position differs from that of most others in that he is more sceptical about the powers of philosophy and about ethical thought as a whole than most of his contemporaries are (3). At the end, however, after holing below the waterline views that depend on too robust a theory of human nature or of rationality, Williams finds himself left with three sparks of optimism: optimism about truth, optimism about truthfulness, and optimism about \"the meaning of individual life\" (198). To lead a life in which a commitment to truth and truthfulness (of one kind or another) plays an important role (to one degree or other) is, of course, a less rigorous demand than living the Stoic or Kantian \"life of reason.\" Even if we are capable of truth and capable of truthfulness, and even if these are important human goals, they are not the only human goals or possibly even the most important ones. As Nietzsche pointed out, the pursuit of truth for its own sake as a value in itself above all others requires a very peculiar configuration of the human soul, one that is not in all its incarnations irresistibly attractive.\n\nAdorno shared Williams's high regard for Nietzsche, remarking in the 1930s that there was \"more truth in the Genealogy of Morality than in [Bukharin's] ABC of Communism.\" However, one of the points of Minima Moralia is that a Nietzschean focus on the individual life as an independent unit of meaningfulness is inappropriate in the social world in which we live, so Williams's relative optimism about the meaningfulness of such a life is not something that should be taken for granted.\n\nWilliams took an extremely dim view of the powers of reason to persuade. He once told me he had only one time in his life seen a case of a person convinced to give up a deeply held belief by the force of rational argumentation. This was when he was the chairman of the Royal Commission on Pornography, one of whose members was a former military man who was completely uninterested in any restrictions on sexual relations between humans or the depiction of such relations but was deeply anxious about bestiality. Men and women could do what they wanted with each other, as far as he was concerned, but what about pictures of men with sheep or cows; surely that could not be allowed. He was finally convinced in a lengthy, emotion-filled session that various arguments that he himself had presented implied that there should be no legal regulation of representations of bestiality either. This man's conversion by the sheer power of reason was so unique in Williams's experience that he never forgot it. Nevertheless, I had high hopes for Williams's own motivational sensitivity to argument and thought that when confronted with Adorno's views in a form in which he could recognize them, he would himself gradually move closer to a more fully socially contextual and less individualist view of meaning and significance in human life.\n\nThis leaves us with no answers and a number of open questions, a result that should not in itself be lethally discouraging if Socrates rather than Kant or Bentham is one's guiding star. Socrates' enquiry is still pre-dogmatic, and although it can be seen as in some sense the origin of \"ethics\" as a discipline, it still stands outside the closed circle that ethics becomes. \"How should one live?\" is amenable to a collective response and the quest for such a response is potentially open-ended.\n\nAmong the open issues is the Hegelian\/Marxist question about the very possibility of a cognitively significant study of the meaningfulness of a mere individual life, if that is undertaken without reference to the wider social context. On at least one reading of the main thrust of this tradition, the question of the meaningfulness of individual life is by no means a universal one but rather arises only under specific social conditions; and what answers, if any, are available to it also depend on historical circumstances. For Marx, in particular\u2014if the question even arises as a \"real\" question, that is, one that grips people\u2014that in turn means that society is deficient. If society was fundamentally in order, the question of meaningfulness would not even arise: either it would be truly incomprehensible or it would be experienced by those living in the society as a merely peculiar oddity, or a form of mental disease, not as something with an existential grip on them. From the fact that in the Soviet era this thought was misused to incarcerate political dissidents in mental institutions, it does not follow that no version of the underlying thought is at all sensible. That the question of the meaningfulness of life does not arise, of course, would not mean that human life in such a society would be a bowl of cherries, for there is presumably a difference\u2014all the difference in the world, some might think\u2014between being sad and being existentially gripped by the meaninglessness of it all.\n\nSo the short, and perhaps all too obvious, answer to the question posed at the start of this chapter is \"yes\" and \"no\": at least during his mature period, which started in the 1980s, Williams was not doing \"ethics,\" if one means by that trying to provide a fully general, rationally based doctrine (a Lehre) that would answer the question \"How should one live?\" Certainly there could be no general doctrine of the good life or of how to live that was based on either of the two traditional central concepts: human nature (as the Aristotelian tradition would have it) or \"rationality\" (in the modern world, perhaps most closely associated with one or another form of Kantianism).\n\n\"How should one live?\" might itself be more ambiguous than traditionally thought, and in any one of its incarnations might be only one of a variety of different practical questions we might sensibly ask, not the hegemonic one dominating all others. All of these questions will have a distinct political dimension, and if one were to insist that part of the point of \"How should one live?\" is that the answer to it would have to give us a general orientation in life that in some sense trumps others, the way to discuss that question would be through a study of history and politics, and the only \"answer\" to this would not be a doctrine but a form of political engagement. \"Universal Reason\" or abstract rationality, to the extent to which it was meaningful at all, was too thin and anodyne to be of any substantial help here. So the forms of traditional ethics Williams is furthest away from are Kantianism, Divine Command versions of Christianity, and utilitarianism; and the form that is closest to him would be some form of Aristotle, or of that modern extended version of Aristotle that Hegel developed. What would, however, finally constitute a barrier between Aristotle and Williams would be the recognition of the role of history. Aristotle had a notion of teleological development, to be sure, but no notion of history and certainly nothing even remotely like the modern idea of a \"historical consciousness.\" This meant that there were very strict limits to an Aristotelian's ability to be fully and appropriately aware of his or her own location in the world and to his or her ability to take a nondogmatic view. Hegel in one sense did know about history, but also assumed it was closed, and connected it with an implausibly powerful and determinate notion of Reason. Ethics, or rather Moralit\u00e4t and Sittlichkeit, play an appropriately subordinate role in Hegel's system, and one might argue that Hegel's achievement lay precisely in showing that a construct like \"the System\" in all its glory was the price one would have to pay for retaining a determinate \"philosophical ethics\" of any kind in the modern world. Suppose now we decided to abandon the narrow limitations imposed on us by traditional ideas of what a \"philosophical ethics\" could (and must) be. Then one could imagine ways of orienting ourselves in the world that went beyond the sorts of things envisaged and discussed in traditional forms of ethics. So general discussions about ourselves, our world, and our place in that world might come to encompass not just the usual tired discussions about what is rational, what has utility, or what is right but also such things as what is an \"eighteenth-century\" social and cultural system, and whether the United States essentially instantiated such a system, whether Siegfried's funeral music was celebratory, elegiac, or something else, and what that might tell us about certain conceptions of individualism. These discussions could be perfectly legitimate successors to the original Socratic enquiry, without being forms of thought that could sensibly be pursued within the confines of philosophical ethics.\n\nIt is Plato's claim that there can be no secure discrimination between good and bad without something like a philosophical ethics, and most philosophers for two thousand years accepted this claim. As Williams points out, though, there seems to be reason to have\u2014to put it mildly\u2014strong reservations about this claim. In the Ion (537a\u2013b), Plato's Socrates discusses the passage in Homer's Iliad (23.335\u201340) in which Nestor gives Antilochus advice about chariot racing: When you come to the turning post, lean over to the left of the horses, and be sure not to graze the posts with your wheel. Socrates gets the rhapsode Ion to admit that a charioteer would know better than a poet whether this is good advice or not, but the next step in Plato's argument, about which he is very coy at the end of Ion but which comes out very clearly in other dialogues, is that a philosopher who ex officio is an expert in ethical theorizing would know even better than a charioteer whether this is good advice. This further step in no way follows and is inherently highly implausible.\n\n\"Ethics\" in the sense he finds objectionable is defined by Williams as motivated by a tacit affirmative answer to the question: \"[Is there] beyond some things that human beings have themselves shaped... anything at all that is intrinsically shaped to human interests, in particular to human beings' ethical interests?\" Western \"ethics\" holds that the universe or history or the structure of human reason can, when properly understood, yield a pattern that makes sense of human life and human aspirations. Sophocles and Thucydides, by contrast, are alike in leaving us with no such sense. Each of them represents human beings as dealing sensibly, foolishly, sometimes catastrophically, sometimes nobly with a world that is only partially intelligible to human agency and in itself not well adjusted to ethical aspirations.\n\nThe assumption on which \"ethics\" rests, that \"at some level of the world's constitution there is something to be discovered that makes ultimate sense of our concerns,\" is illusory.\n\nOn the other hand, the demise of ethics as a philosophical discipline will still leave much of our usual evaluative discourse unaffected. Achilles will still be able to berate Agamemnon as a dog-eyed bundle of shamelessness, ever-greedy for gain (Iliad l.148\u201371), Solon, feigning madness, will still call upon the Athenians to fight for Salamis (Fragment 2), Pindar will still proclaim the respective virtues of water and gold (Olympian 1), and Thucydides will still be able to discuss the merits of the grand strategy of Perikles during the war with the Peloponnesians (2.59\u201365) and the character of Nikias (7.86).\n\nWilliams was very taken with Goethe's translation of the beginning of the gospel of Saint John, \u03ad\u03bd \u1f00\u03c1\u03c7\u1fc7 \u1f26\u03bd \u1f41 \u03bb\u03cc\u03b3\u03bf\u03db as \"Im Anfang war die Tat,\" but that is only half the story. The other half is given by Marx in the Theses on Feuerbach: \"Die Philosophen haben die Welt nur verschieden interpretiert, es kommt darauf an, sie zu ver\u00e4ndern\" (11). We are not, of course, now in a situation in which it is realistically possible for us to envisage any fundamental change in our world that we could ourselves bring about by our own efforts. That perhaps is part of the reason so much of contemporary philosophy seems merely scholastic. Since I also suppose that the operation of various long-term processes will soon make most of the people in the world significantly worse off than they now are, this situation of enforced immobility is extremely unlikely to last long. There is, to be sure, no guarantee that, in the future, constructive change, whatever that will turn out to mean, will in any way be open to us.\n11\n\nThe Wisdom of Oedipus and the Idea of a Moral Cosmos\n\nIn the spring of 1989 the distinguished philosopher Bernard Williams gave the Sather Lectures to the Department of Classics at the University of California at Berkeley, and these lectures were in due course published under the title Shame and Necessity. Many people, including me, consider this to be Williams's finest book, and it is a striking fact about it that it both begins and ends with quotations from the poet Pindar. The exergue, the very first part of the printed book a reader will encounter, is three very famous lines from a poem that Pindar seems to have written very late in his life, the Eighth Pythian. In fact there is some reason to believe that this poem is not just late but the very last poem by Pindar that is extant.\n\n\u1f10\u03c0\u03b1\u03bc\u03ad\u03c1\u03bf\u03b9\u2022 \u03c4\u03af \u03b4\u03ad \u03c4\u03b9\u03c2; \u03c4\u03af \u03b4\u0309 \u03bf\u1f54 \u03c4\u03b9\u03c2; \u03c3\u03ba\u03b9\u1fb6\u03c2 \u1f44\u03bd\u03b1\u03c1\n\n\u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2. \u1f00\u03bb\u03bb\u0309 \u1f43\u03c4\u03b1\u03bd \u03b1\u1f34\u03b3\u03bb\u03b1 \u03b4\u03b9\u03cc\u03c3\u03b4\u03bf\u03c4\u03bf\u03c2 \u1f14\u03bb\u03b8\u1fc3\n\n\u03bb\u03b1\u03bc\u03c0\u03c1\u1f78\u03bd \u03c6\u03ad\u03b3\u03b3\u03bf\u03c2 \u1f14\u03c0\u03b5\u03c3\u03c4\u03b9\u03bd \u1f00\u03bd\u03b4\u03c1\u03c9\u03bd \u03ba\u03b1\u1f76 \u03bc\u03b5\u03af\u03bb\u03b9\u03c7\u03bf\u03c2 \u03b1\u1f30\u03ce\u03bd [ll. 95\u201397]\n\nWilliams leaves these three lines in Greek without translating them. They aren't actually cryptic in the way in which some parts of poems by Pindar are. That is, it is not the case that one does not know at all what they mean; the general sense is clear, but the mode of expression is exceedingly condensed, polysemous in detail, and pleasingly harsh. Pindar has been drawing the listeners' attention to the instability of good fortune, prosperity, and the pleasures of life (\u03c4\u1f78 \u03c4\u03b5\u03c1\u03c0\u03bd\u03cc\u03bd, l. 93), then comes the cited passage. A prose paraphrase might run:\n\nWe humans have a very brief time of life; we live for only one day. What then can any human being ever finally amount to? And what is forever beyond our grasp? Man is the shadow of a dream. But when a god gives him glory, a bright light plays over him and the span of his life is easy to bear.\n\nIt is easy to romanticise \"last words,\" for instance, to assume that one can find distilled in them the wisdom of a lifetime. We are not, however, absolutely sure whether this is a \"late\" poem by Pindar, and a fortiori we can't be sure it is the last even of his extant works. Why, in any case, assume that final thoughts are better than earlier ones? However this might be, the sentiments expressed here do not seem to be in any special sense the specific products of the reflections at the end of a long and active life, but are meant to formulate an attitude towards life and the world that we can trace everywhere in Pindar's work from the earliest to the latest poems (whichever these are): Human life is essentially both insubstantial and grim; if this seems not to be the case in some particular instance, that is because something outside our control\u2014a god\u2014has for a brief moment given some individual the gift of shining a ray of light on him; we know that that light won't last.\n\nIn contrast to the exergue, which might, but more likely does not contain some special \"last words\" of Pindar, it is natural to think that the very last words of a book are intended to be particularly important, perhaps, in a book of philosophy, to be something like a conclusion. Williams ends Shame and Necessity with a passage from Pindar's Fourth Pythian, which is cited in his own translation\n\nTake to heart what may be learned from Oedipus:\n\nIf someone with a sharp axe\n\nhacks off the boughs of a great oak tree,\n\nand spoils its handsome shape;\n\nalthough its fruit has failed, yet it can give an account of itself if it comes later to a winter fire,\n\nor if it rests on the pillars of some palace\n\nand does a sad task among foreign walls,\n\nwhen there is nothing left in the place it came from.\n\nThis is a peculiar way for Williams to end his book for several reasons. First, it is strange that a major philosopher like Williams would end his own book with someone else's words. In addition, although it is not absolutely unprecedented, it is not exactly usual and customary for a philosopher to end a book by citing a bit of poetry; after all, ever since Plato philosophers have repeatedly tried to establish themselves as practitioners of a discipline which, whatever its form and content, was certainly not a kind of \"poetry.\" Finally, it is extremely odd to end a philosophy book, especially a book of \"analytic philosophy\"\u2014if one accepts that that term has any distinct meaning and that Williams belongs to the tradition of analytic philosophy\u2014with a literary fragment that is itself rather obscure. Analytic philosophy prides itself on its commitment to clarity, so conceivably it might begin with some traditional dark and difficult sayings with the intention of \"clarifying\" them during the course of its own treatment, but it wouldn't, as it were, lead up to and conclude with an enigma. So what exactly does this fragment from Pindar mean and why is it there?\n\nOne of the main theses Williams propounds in all his work on the ancient world is that we cannot go back to a past world and shouldn't even regret too much that we cannot do that. But, he says, \"if we find things of special beauty and power in what has survived from that world, it is encouraging to think we might move beyond marvelling at them, to putting them, or bits of them, to modern uses. An image of Pindar's is right.\" Then comes the text just cited above. I should like to ask first what Williams takes this image of Pindar's to mean and why he thinks it is marvellous and \"right,\" and then say a few words about \"modern uses\" to which it might be put.\n\nThe meaning and significance of the image becomes clear only in the context of the poem as a whole. The Fourth Pythian is an epinikion, a choral poem for public performance celebrating the victory in the chariot race won at the Pythian Games by Arkesilaos, the king of Cyrene, which was a small Greek \u03c0\u03cc\u03bb\u03b9\u03c2 in Libya on the north coast of Africa. It is by far the longest choral poem that has survived from antiquity, comprising thirteen triads, and its political character is also particularly explicit. This ode has an extremely complex structure, and a proper account of it as a work of literature would need to reflect that, but for the purposes of this discussion it is enough to say that it falls into two very unequal parts. I assume here that it is legitimate to divide up texts in different ways depending on the interpreter's purpose. The first part is the standard Pindaric praise of Arkesilaos and his family, mixing together what we would distinguish as real historical stories about his illustrious recent ancestors and the colonisation of the city of Cyrene with what we would call \"mythical stories\" about the exploits of some of his supposed further ancestors back to the time of the Argonauts. The emphasis in this first part is on the continuity of the family to which Arkesilaos belonged and the divine origin of their claim to rule Cyrene. Pindar tells the story of an extremely convoluted series of adventures, peregrinations, and migrations, including the acquisition of a magic clod of earth from a god, a prophecy, and an oracle all documenting the warrant that Arkesilaos's family has to rule Libya, that is, he gives what one might call a \"positive\" genealogy of the Cyrenaic monarchy and Arkesilaos as king.\n\nThis first part is in itself longer, about twice as long in fact, as most of the other epinikia by Pindar that have survived, so, as it were, he could easily have stopped there. However, he does not, and after a brief transition passage Pindar talks directly to Arkesilaos about the current political situation in his kingdom. There obviously has been a stasis, an attempted uprising or coup d'\u00e9tat that has failed, and as a result various aristocrats have been exiled. Although Arkesilaos and the monarchy have survived the coup, the political situation still seems very unsettled. Pindar, or at any rate the \"I\" of the poem, makes a direct appeal to Arkesilaos to heal the political troubles by recalling one of the exiled nobles, Damophilus, to Cyrene. The passage that Williams cites constitutes the point of transition from the first to the second part. So the basic structure is:\n\nPart I: Arkesilaos comes from an illustrious family and has a god-given warrant to rule Cyrene. In addition to this, the gods have made him quick of understanding.\n\nTransition: This is the passage Williams cites at the end of Shame and Necessity: So if, Arkesilaos, you are as quick of understanding as your ancestors were, learn from Oedipus.\n\nPart II: You can be your own physician for the troubles that beset you [\u1f10\u03c3\u03c3\u1f76 \u03b4'\u1f30\u03b1\u03c4\u1f74\u03c1 \u1f10\u03c0\u03b9\u03ba\u03b1\u03b9\u03c1\u03cc\u03c4\u03b1\u03c4\u03bf\u03c2 (270)]; recall Damophilus from exile; despite everything, he is a good man. This will heal your polis.\n\nTo turn then to the passage Williams himself cites, the Greek text reads:\n\n\u03b3\u03bd\u1ff6\u03d1\u03b9 \u03bd\u1fe6\u03bd \u03c4\u1f70\u03bd \u039f\u1f30\u03b4\u03b9\u03c0\u03cc\u03b4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd\u2022 \u03b5\u1f30\n\n\u03b3\u03ac\u03c1 \u03c4\u03b9\u03c2 \u1f44\u03b6\u03bf\u03c5\u03c2 \u1f40\u03be\u03c5\u03c4\u03cc\u03bc\u1ff3 \u03c0\u03b5\u03bb\u03ad\u03ba\u03b5\u03b9\n\n\u1f10\u03be\u03b5\u03c1\u03b5\u03af\u03c8\u03b5\u03b9\u03b5\u03bd \u03bc\u03b5\u03b3\u03ac\u03bb\u03b1\u03c2 \u03b4\u03c1\u03c5\u03cc\u03c2, \u03b1\u1f30\u03c3\u03c7\u03cd -\n\n\u03bd\u03bf\u03b9 \u03b4\u03ad \u03bf\u1f31 \u03b8\u03b1\u03b7\u03c4\u1f78\u03bd \u03b5\u1f36\u03b4\u03bf\u03c2,\n\n\u03ba\u03b1\u1f76 \u03c6\u03b8\u03b9\u03bd\u03cc\u03ba\u03b1\u03c1\u03c0\u03bf\u03c2 \u1f10\u03bf\u1fd6\u03c3\u03b1 \u03b4\u03b9\u03b4\u03bf\u1fd6 \u03c8\u1fb6\u03c6\u03bf\u03bd \u03c0\u03b5\u03c1\u2032 \u03b1\u1f50\u03c4\u1fb6\u03c2,\n\n\u03b5\u1f34 \u03c0\u03bf\u03c4\u03b5 \u03c7\u03b5\u03b9\u03bc\u03ad\u03c1\u03b9\u03bf\u03bd \u03c0\u1fe6\u03c1 \u1f10\u03be\u03af\u03ba\u03b7\u03c4\u03b1\u03b9 \u03bb\u03bf\u03af\u03c3\u03b8\u03b9\u03bf\u03bd,\n\n\u1f22 \u03c3\u1f7a\u03bd \u1f40\u03c1\u03b8\u03b1\u1fd6\u03c2 \u03ba\u03b9\u03cc\u03bd\u03b5\u03c3\u03c3\u03b9\u03bd\n\n\u03b4\u03b5\u03c3\u03c0\u03bf\u03c3\u03cd\u03bd\u03b1\u03b9\u03c3\u03b9\u03bd \u1f10\u03c1\u03b5\u03b9\u03b4\u03bf\u03bc\u03ad\u03bd\u03b1\n\n\u03bc\u03cc\u03c7\u03b8\u03bf\u03bd \u1f04\u03bb\u03bb\u03bf\u03b9\u03c2 \u1f00\u03bc\u03c6\u03ad\u03c0\u03b5\u03b9 \u03b4\u03cd\u03c3\u03c4\u03b1\u03bd\u03bf\u03bd \u1f10\u03bd \u03c4\u03b5\u03af\u03c7\u03b5\u03c3\u03b9\u03bd,\n\n\u1f11\u1f78\u03bd \u1f10\u03c1\u03b7\u03bc\u03ce\u03c3\u03b1\u03b9\u03c3\u03b1\u03b1 \u03c7\u1ff6\u03c1\u03bf\u03bd\n\nOne immediately striking thing about Williams's version is that he seems to have translated the first line in a slightly idiosyncratic way: \u03b3\u03bd\u1ff6\u03d1\u03b9 \u03bd\u1fe6\u03bd \u03c4\u1f70\u03bd \u039f\u1f30\u03b4\u03b9\u03c0\u03cc\u03b4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd becomes not, as one might expect, \"know the wisdom of Oedipus\" but \"take to heart what may be learned from Oedipus.\" What exactly is going on here? The term \"wisdom,\" which stands out vividly in the original, becomes invisible in the translation.\n\nWhat, then, is this wisdom of Oedipus, what should Arkesilaos learn from it, and what might we be able to learn from it? I suggest there are three possible ways of understanding \"the wisdom of Oedipus.\"\n\nFirst, then, the expression \"the wisdom of Oedipus\" raises the expectation that we will be told the content of that \"wisdom,\" given perhaps a summary of all Oedipus had learned about life in a single pithy saying. This is not in itself an inherently unreasonable expectation. Interspersed with the praise of rulers and athletes, we find in Pindar's epinikia a rather large number of short, pithy, often slightly obscure or ambiguous general observations about the world, nuggets of wisdom, such as\n\n\u039d\u03cc\u03bc\u03bf\u03c2 \u1f41 \u03c0\u03ac\u03bd\u03c4\u03c9\u03bd \u03b2\u03b1\u03c3\u03b9\u03bb\u03b5\u03cd\u03c2\n\n\u03b8\u03bd\u03b1\u03c4\u1ff6\u03bd \u03c4\u03b5 \u03ba\u03b1\u1f76 \u1f00\u03b8\u03b1\u03bd\u03ac\u03c4\u03c9\u03bd\n\n[\"Customary usage is the King of both mortals and immortals\" (Fr. 169)]\n\n\u03c3\u03ba\u03b9\u1fb6\u03c2 \u1f44\u03bd\u03b1\u03c1 \u1f04\u03bd\u03b8\u03c1\u03c9\u03c0\u03bf\u03c2\n\n[\"Man is the shadow of a dream\" (Pythian VIII, 95\u201396)]\n\n\u03c0\u03b9\u03c3\u03c4\u1f78\u03bd \u03b4\u2032 \u1f00\u03c0\u03af\u03c3\u03c4\u03bf\u03b9\u03c2 \u03bf\u1f50\u03b4\u03ad\u03bd\n\n[\"Nothing seems trustworthy to those who are themselves untrustworthy\" (Fr. 233)]\n\nWisdom-literature in general already had a long history by the time of Pindar. Thus Nietzsche cites the purported archaic \"wisdom of Silenus\" that life is inherently not worth living for humans, but even earlier writers than the ones whom Nietzsche cites recognised a pantheon of seven wise men\u2014they didn't agree on who these seven wise men were, but they were sure there were seven of them\u2014and many of these men were associated with one or more particular pithy sayings summing up their respective forms of wisdom. These ranged from the profound (\"Nothing in excess\"; \"Know thyself\") through the homely (\"Keep control of your tongue, your belly and your private parts\"; \"Resist anger\"; \"Honour old age\"; \"Don't laugh at someone who is down on their luck\") to the cryptic or, frankly, weird. Thus Pythagoras had time between bouts of mathematical mania to issue specimens of wisdom such as \"Abstain from beans.\" We don't know why he thought that only the bean-free life was worth living for man because he does not tell us, but then that is a general characteristic of much of this literature. It only occasionally gives any reason for the gem of wisdom. Diogenes Laertius cites Aristotle as giving five different reasons, including that beans look like the genitals, that each one was \"like the nature of the universe\" (\u03c4\u1f96 \u03c4\u03bf\u1fe6 \u1f45\u03bb\u03bf\u03c5 \u03d5\u03cd\u03c3\u03b5\u03b9 \u1f45\u03bc\u03bf\u03b9\u03bf\u03bd; D.L. viii.34), and that they are an inherently oligarchical food because they were used in some cities for choosing rulers, and Diogenes himself adds a sixth, namely their tendency to produce flatulence. The huge variety of reasons makes it clear that people did not really know and were just making up plausible-sounding theories for a prohibition whose original significance was lost. Some of these bits of wisdom had esoteric meaning. Thus we are told that \"Don't poke the fire with your knife\" was a saying of Pythagoras. As any little boy knows, there is good reason not to poke a fire with one's knife because the heat of the fire can destroy the tempering of the blade and make it dull. We are told, however, that this really meant: Don't further irritate someone who is already angry. Perhaps the prohibition of beans also had such an esoteric meaning.\n\nNot all expressions of wisdom took the form of prohibitions or injunctions. Thus a sixth-century sage named Phokulides had the archaically charming habit of starting his pithy sayings, which were in verse, with the phrase using his own name and telling you that this is what he thinks: \"This, too, is Phokulides.\" This would be a bit like Confucius saying in his own voice: \"Confucius, he say.\" Some of Phokulides' gnomic verses do contain straightforward imperatives or warnings, but others take a less directly injunctive form, for instance,\n\n\u03ba\u03b1\u1f76 \u03c4\u03cc\u03b4\u03b5 \u03a6\u03c9\u03ba\u03c5\u03bb\u03af\u03b4\u03bf\u03c5\u2022 \u03c0\u03cc\u03bb\u03b9\u03c2 \u1f10\u03bd \u03c3\u03ba\u03bf\u03c0\u03ad\u03bb\u03c9\u03b9 \u03ba\u03b1\u03c4\u1f70 \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd\n\n\u03bf\u1f30\u03ba\u03b5\u1fe6\u03c3\u03b1 \u03c3\u03bc\u03b9\u03ba\u03c1\u1f74 \u03ba\u03c1\u03ad\u03c3\u03c3\u03c9\u03bd \u039d\u03af\u03bd\u03bf\u03c5 \u1f00\u03c6\u03c1\u03b1\u03b9\u03bd\u03bf\u03cd\u03c3\u03b7\u03c2\n\n[This, too, is of Phokulides: a polis on a rocky outcrop that is living according to proper order, even if it is small, is stronger than imprudent Nineveh.]\n\nThis saying does not tell you what to do to maintain good order in a polis\u2014or how to establish a polis if you do not happen already to live in one\u2014but it does give you a way of orienting yourself in action by presenting a comparative evaluation of two structures.\n\nThis notion of \"good or proper order\" is understandably central to early Greek thought. The expression Phokulides uses, \u03ba\u03b1\u03c4\u1f70 \u03ba\u03cc\u03c3\u03bc\u03bf\u03bd\u2014\"in good order\" or \"conforming to the proper order\"\u2014is clearly very old and appears, for example, with great frequency in the Homeric epics. There it is often used to describe the effective marshalling of troops for battle: they are arranged in good and proper order for battle. An expression from the same root is used as part of a formulaic description of a leader. Thus, Agamemnon and Menelaus are \u03b4\u03cd\u03c9 \u03ba\u03bf\u03c3\u03bc\u03ae\u03c4\u03bf\u03c1\u03b5 \u03bb\u03b1\u1ff6\u03bd (Il. I.16 [OCT, but punctuation changed]), two marshallers of the people. \"Cosmos\" usually carries with it laudatory overtones, that is, it is a good order. The same word \"cosmos\" is also used to refer to ornaments or adornments of the kind worn by women or put on the harnesses of horses. So it is an inherently beautiful, attractive, or otherwise pleasing kind of order. Diogenes Laertius tells us that the bean-fearing Pythagoras was the first to use the word \"cosmos\" to refer to the sky or the heavens (\u03c4\u1f78\u03bd \u03bf\u1f50\u03c1\u03b1\u03bd\u03cc\u03bd, D.L. viii.48). We have no idea, of course, whether this is right. Diogenes is a notoriously relatively late and highly dodgy source, there are serious doubts in fact about whether Pythagoras ever existed, and so a fortiori there must be doubts about whether he used the word kosmos in any sense at all; and even if we knew he did exist and did use the word in some more extensive way, we would not be sure what exactly he meant by it. It is, I think, significant that contrary to what one might have expected, the word kosmos does not occur in Hesiod. Even if Diogenes is completely wrong about Pythagoras, it is important that he has the feeling that it is not natural, self-evident, or universally accepted that one may use the word \"cosmos\" in a more general way, that is, not merely to describe a certain trinket, or a particular form of order in a particular army or city, but to refer to the sky, the heavens, or indeed the world as a whole considered as a unitary, attractively ordered structure. Rather than being obvious, this, he thinks, needs some further explanation, and he thinks there was a time when people did not use \"cosmos\" to refer to the world as a whole, as they did by his time. So there has been, he thinks, a change that is a significant enough conceptual change to require some account of how and when that happened.\n\nHistorians of philosophy have shown a strong tendency to backdate the idea that the physical world at any rate is a unitary whole, sometimes as far back as Thales. I'm not erudite enough to have an interesting opinion on this, so I shall not discuss it, but when some Greek probably in the sixth or very early fifth century BC, whoever he was, first looked out at everything and called it a unitary, attractive structure, something changed. When this idea gradually ceased to be the esoteric speculation of an isolated individual and began to put down roots and have an effect on the wider culture and society, this represents a significant transformation. There had been various stories about what our world was like and how it emerged, about the nature of the gods, about the origin of various human families, about the fate of notable human individuals and groups, and, of course, all humans, as far as we can tell, make certain general assumptions, often unarticulated assumptions, about how the world in which they live is constituted. What is, as it were, added by the use of the concept \"cosmos\" is that all these stories are to be embedded in a single structure that has the property of coherence. They all have finally to fit together beautifully. Once one has the idea that the world constitutes a cosmos, it seems a small further step to think that its beautiful order will be accessible to us in some way: we will be able to see that order or reason our way to some kind of noetic apprehension of it.\n\nEven if many people now might be inclined retrospectively to focus on the importance of construing the physical, biological, and astronomical world as a (unitary) cosmos, it is important not to lose sight of the fact that there is an analogue to this conception in thinking about morality. We can imagine our moral lives as constituting something like a cosmos. Just as, correctly understood, the parts of the material universe cohere nicely, so we might also suppose that our own moral views, correctly understood, were fundamentally unitary, coherent, and accessible to us through reflection, or at any rate that our central beliefs could be put in order\u2014with some minor revisions that we can come to see are necessary\u2014by straightforward means (including simple empirical observation of consequences and reflection). I emphasise that there are (at least) two distinct strands in this. The first is that we can become at least minimally self-transparent to ourselves morally; we can come to know what our moral beliefs are. The second is that these beliefs, or at least an important central core of these beliefs, will turn out to be, or, perhaps with some slight revision, can viably be made to be coherent with each other. If the world of human morality is to form a cosmos, however, one might think it is not enough that all our beliefs about values, principles of action, duties, and so forth can be made coherent and rationally transparent and intelligible to us. The world we encounter must make sense to us and be intelligible as one in which it is possible to live in a way conformable to at least our minimal ethical expectations, aspirations, and demands. The world must, as Williams puts it, be at least minimally \"adjusted to [human] ethical aspirations,\" that is, \"at some level of the world's constitution there is something to be discovered that makes ultimate sense of our concerns.\" To speak of \"the world's constitution\" here also implies that what makes ultimate sense of our concerns has some property of being natural or metaphysically necessary; it is not an accidental or contingent trait or one merely imposed on an inherently indifferent or recalcitrant world. The compatibility of world and morality is out there to be discovered, not something we (or anyone) invent or construct. One might think of this as a third strand in a view of the world as a moral cosmos in the fullest possible sense. The particular way in which the world makes moral sense may be highly complex and the object of sharply diverging speculation, but at some level there must, if the world is a moral cosmos, be a connection between the way the world fundamentally is and our own deepest human interests, particularly our ethical interests.\n\nSince the idea of a \"moral cosmos\" comprises these three elements, the rejection of any one of the three would indicate that this conception was not fully present. Thus if one thinks that self-knowledge is seriously limited so that in some important sense one cannot know what one's own deepest moral beliefs are, this would be toxic for any conception of the world as a moral cosmos. Or if one thought that one's moral beliefs could not be organised into a single more or less stable, consistent system, perhaps because some of them were just irremediably contradictory or were too fluid and kept changing under scrutiny, this, too, would rule out any view of the world as a moral cosmos. Finally, if the external world exhibits no particular morally relevant order at all, if that order is merely local or very fragmentary, or if it is clear that \"nature\" will seriously or even actively thwart us in our attempt to lead a morally good life, the world we live in is not a cosmos.\n\nThere is a traditional conception according to which philosophy is structured around the pursuit of \"wisdom\" of a certain kind. What \"wisdom\" is that? Williams describes the main line of traditional philosophy not as a kind of principled and realistic attempt to come to terms with our world as it is but as a form of addiction to the production and propagation of fantasies of a certain kind. Philosophy, and especially moral philosophy, has exhibited an almost pathologically compulsive need to give humans good news about their situation in the world. \"Good news\" in Greek, of course is \"\u03b5\u1f50\u03b1\u03b3\u03b3\u03ad\u03bb\u03bb\u03b9\u03bf\u03bd,\" so one might say that Williams's project in life was to stop moral philosophy from continuing to be quite as evangelical as it had been in the past. What, though, is the content of the \"gospel\" philosophy has preached? In a word, it is that our world constitutes a \"moral cosmos.\"\n\nFor Williams, the acquisition of this conception that our world is a moral cosmos is a major turning point in intellectual history. There is, he says, a deep \"ditch\" between archaic forms of thought such as those of Pindar, Sophocles, and Thucydides, which distinctly lack the conception of the world as a moral cosmos, and \"philosophical theories\" such as those of Plato, Aristotle, Kant, and Hegel, all of whom in one way or another are committed to this conception. Of course, the \"gospel\" a philosopher preaches can be, as it were, \"thicker,\" more detailed and substantial, and more pleasant for us, or less so. Hegel constitutes one of the extreme poles. On his view, we can come to see that the whole universe, including the world of politics and history, is rational and systematically ordered in such a way as to be amenable to our real interests and aspirations, and thus is potentially a \"home\" for us. Kant has perhaps the thinnest and most minimalist version of the gospel: We cannot know that the external world of nature would even in principle show itself to be malleable to our ethical aspirations, and this is to some extent chastening. On the other hand, however, we can know first that nature has a systematic and rational structure and second that our ethical life is fully coherent, fully intelligible to us, and under our control. We can't know what the actual outcome of our action in the world will be, but, Kant holds, that is morally not relevant because we can know clearly what our duty is and this is morally sufficient. Furthermore, we can have not \"knowledge\" but a rationally grounded hope that all will turn out for the best if only we do our duty.\n\nOne must resist the tendency to think of there being a smooth historical progression from \"tragedy\" to \"philosophy.\" This is not correct because, for one thing, although one of the major tragedians, Aeschylus, does seems to have a very robust sense of the universe as a moral cosmos, in this he seems very different from Sophocles (and also Thucydides). There is, however, a more important reason to reject this way of seeing things. Sophoclean tragedy is about kings and heroic figures\u2014Ajax, Heracles, Creon, Oedipus\u2014but it is also in a deep sense \"realistic\": it is about people (eventually) facing up to the dire situations in which they actually find themselves without flinching and making difficult choices. (Old) comedy, on the other hand, deals with ordinary folk, but it is also an inherently exaggerated and fantastical genre. Archetypically a comedy begins with a disturbed, disordered, and painful situation, for instance a state of war, which is highly unsatisfactory for the protagonist, who is not in any way heroic. In many of the plays of Aristophanes the \"war\" in question is a real one and they were first presented during the Great War between Athens and Sparta. The comic protagonist has a \"bright idea\" that is imaginative but utterly absurd: end the war by making an individual separate peace with the Spartans (Acharians), breed a huge dung-beetle and fly him up to heaven to get peace from Zeus (Peace), have the women go on a sex-strike until the men make peace (Lysistrata), or build a new city of peace in the sky and support it by charging the gods a toll to fly around in heaven (Birds). Each comedy ends, notoriously, with a vivid \"resolution,\" a return of order, or a happy ending often characterised in relatively crude and down-to-earth terms as peace, lots of eating and drinking, music, dancing, and lots of sex. (Old) comedy thus in one way also tells us a kind of \"good news\" that is absent from Sophoclean tragedy\u2014despite appearances we can get back to a life with lots of creature comforts\u2014but only through inventing and activating a bizarre, ludic, and utterly unrealistic mechanism (e.g., founding a \"city-in-the-clouds\"). Philosophy is not the \"natural\" successor of tragedy but, if anything, of comedy. It is a kind of comedy without the humour. The \"resolution\" or \"happy ending\" philosophy envisages is perhaps less Rabelaisian than that of Old Comedy: rationally moderated \u03b5\u1f50\u03b4\u03b1\u03b9\u03bc\u03bf\u03bd\u03af\u03b1, psychological \u1f00\u03c4\u03b1\u03c1\u03b1\u03be\u03af\u03b1, or metaphysical Vers\u00f6hnung instead of eating, drinking, marriage, feasting, and orgies, but the means are equally delusional and fanciful. Comedy tells us that Trugaios can get out of the painful reality of the Peloponnesian War by mounting his giant dung-beetle; philosophy makes the parallel claim that you can attain harmony, success, and happiness in the world by believing in Reason or God or Duty or Geist or the Ideas. What replaces the jokes is po-faced ratiocination sprinkled with occasional sanctimonious effusions. The two most egregious examples of this are Seneca (whom Nietzsche unforgettably calls the \"Toreador der Tugend\") and Kant (\"Duty! you sublime grand name, you who have nothing attractive about you which would flatter, but who require subjugation.... what is your worthy origin, where does one find the root of your noble decent?\"), but they stand instar omnium.\n\nWhere does this leave the wisdom of Oedipus? This first approach to his wisdom appears to have led nowhere, except to see that whatever the wisdom of Oedipus is, it is not like the evangelical annunciation of the existence of a moral cosmos that is the content of most forms of philosophy. Pindar does not in fact give us any nugget of distilled Oedipean wisdom of any kind that can be formulated in a single pithy saying. Pindar cannot, of course, express by direct assertion in a pithy saying the fact that he lacks the concept of a moral cosmos or that he rejects the idea that human life constitutes a cosmos, if the concept of a moral cosmos is really a later human acquisition.\n\nA second possibility, then, is that Oedipus's wisdom consists in the mastery of a practical skill that cannot be easily summed up in a single saying or a body of doctrine, like the skill of a good diplomat, a good old-style carpenter, or a good military leader. Pindar repeatedly refers to his own skill as a poet as his \u03c3\u03bf\u03c6\u03af\u03b1, which equally requires the discerning composition of a work that is appropriate to the occasion for which it is commissioned.\n\nThinking about \u03c3\u03bf\u03c6\u03af\u03b1 in this way immediately mobilises a second expectation: If one asks what notable cognitive skill did Oedipus have, one likely answer will be that he was good at guessing riddles. As Williams notes, the ancient scholiast takes this reference to \"the wisdom of Oedipus\" in exactly this sense, when he remarks on this passage: \u03c0\u03c1\u03bf\u03c4\u03c1\u03ad\u03c0\u03b5\u03c4\u03b1\u03b9 \u03c4\u1f78\u03bd \u1f08\u03c1\u03ba\u03b5\u03c3\u03af\u03bb\u03b1\u03bf\u03bd \u1f41 \u03a0\u03b9\u03bd\u03b4\u03ac\u03c1\u03bf\u03c2 \u03c3\u03c5\u03bd\u03bf\u03c1\u1fb6\u03bd \u03b1\u1f50\u03c4\u03bf\u1fe6 \u03c4\u1f78 \u03b1\u1f34\u03bd\u03b9\u03b3\u03bc\u03b1. \u03c4\u1f78 \u03b3\u1f70\u03c1 \u039f\u1f30\u03b4\u03b9\u03c0\u03cc\u03b4\u03b1 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u03c4\u03bf\u1fe6\u03c4\u03bf \u03b2\u03bf\u03cd\u03bb\u03b5\u03c4\u03b1\u03b9, \u1f45\u03c4\u03b9 \u03ba\u1f00\u03ba\u03b5\u1fd6\u03bd\u03bf\u03c2 \u03c4\u1f78 \u03c4\u1fc6\u03c2 \u03a3\u03c6\u1fd6\u03b3\u03b3\u03bf\u03c2 \u03b1\u1f34\u03bd\u03b9\u03b3\u03bc\u03b1 \u1f14\u03bb\u03c5\u03c3\u03b5\u03bd. [\"Pindar is trying to get Arkesilaos to see the relevance of his riddle. The phrase 'wisdom of Oedipus' refers to the fact that he, too, was able to solve a riddle, that of the Sphinx.\"] So the scholiast takes the whole image of the oak that is chopped down and then used in the fire or as a part of a building as a \"riddle.\" In the version of the story with which most of us are familiar, there is both a problem and a riddle. The riddle is the question the Sphinx asks; the problem is that she kills people if they can't answer her riddle. These are connected, but they are not the same thing. It is one thing to exercise the relatively detached cognitive capacity that allows one to find the answer to a riddle; this is like doing a crossword puzzle, or solving an anagram, or proving a proposition in elementary logic. It is another to solve a pressing practical problem that requires one to respond to the exigencies of some current situation. Just as these two things are combined in the Sphinx's questioning, so they are also combined in Oedipus's response. He solved the riddle of the Sphinx by applying it to the present situation and himself, and discovered that by solving the riddle he had also resolved the problem, because the Sphinx killed herself when her riddle didn't terminally stump him. In addition, as has often been pointed out, Oedipus was himself the solution to the riddle: He was the thing that walks on four feet in the morning, two feet at noon, and three feet in the evening (namely a human being). So the progression would be:\n\n(a) You, Arkesilaos, have the problem of governing this unruly city.\n\n(b) You come from a family renowned for their good counsel, so learn to solve my riddle, as Oedipus solved the riddle of the Sphinx.\n\n(c) Like Oedipus, you can yourself be the solution to your problem; just realise that Damophilus is like that oak log I have just described; the log is displaced but still useful. Recall Damophilus and use him well.\n\nThe parallel is not exact because solving Pindar's riddle is not at the same time solving the problem of stasis in Kyrene, and there seem to be two parties to Pindar's riddle\u2014Arkesilaos and Damophilus\u2014rather than one. Still there is sufficient similarity to support the comparison and the image.\n\nWilliams does not completely reject the view put forward by the scholiast, but he also argues that there is another, a third, possible way of reading this passage that makes it, in his words, \"more interesting.\" The third possibility is that what is called \"the wisdom of Oedipus\" is not a detachable skill that Oedipus exercises, such as guessing riddles, but rather that we can acquire wisdom of some kind by observing under the right circumstances the way in which he acts. So think of the phrase \"learn the wisdom of Oedipus\" as meaning \"learn wisdom from thinking about the actions and fate of Oedipus.\" Of course, I can learn wisdom even from studying a case of someone who does not himself instantiate wisdom. I can, for instance, to some extent learn wisdom by observing the mistake of exercising or failure to exercise wisdom, just as I can learn something about the importance of avoiding indecisiveness in war by studying the actions and fate of Nikias in Sicily or, for that matter, just as I can learn the importance of informed knowledge and moderation in aggression by studying the folly of the Athenian decision to send a military expedition to Sicily in the first place.\n\nNote that Williams's reading need not imply that Oedipus is not in his own way wise, merely that in this context the point is not whether or not he is wise but whether or not Arkesilaos can make a \"wise modern use\" of thinking about Oedipus's life and fate. Pindar then gives Arkesilaos an indication of how he should learn from Oedipus. This indication is not in the form of a discursive explanation of Oedipus's fate or the explicit drawing of a moral from it but in the form of an image. You can learn from Oedipus if you think of him and his life as figured in the fate of the oak tree. Let me for the purposes of analysis distinguish two aspects of Oedipus-as-oak-tree. First of all, the oak tree is described as absolutely separated from anything that could be called the \"natural\" conditions of flourishing. It has been cut down with violence, its natural beauty has been defiled, its fruit has withered, it has been taken away from its natural place and thus left that place desolate.\n\nAristotle, who is one of the most thoroughgoing and explicit theorists of the moral cosmos, tended to think of human moral life as analogous to biological phenomena. Humans and their environment are \"naturally\" suited to each other. A good human being is one who functions well in that environment, and thereby does well and fares well (\u03b5\u1f56 \u03c0\u03c1\u03ac\u03c4\u03c4\u03b5\u03b9\u03bd means both of these things in Greek). One does well and fares well if one is fully realising one's nature and thus flourishing. Part of that nature, if one is a human being, may consist in having the right kind of life, ideally in a polis, and having the right beliefs, among them beliefs about how humans should act. An oak tree is good if it fares well, that is, flourishes, but, given its nature, it need not have the right beliefs of any kind. It flourishes if it grows tall and broad and produces, in the right season, lots of green leaves and acorns. Its \"nature\" is to produce further oak trees, not to burn in a fire. From the point of view of the oak tree, then, the situation in which it has had its branches lopped off, has no \"fruit\" (acorns), and is either burning in a fire or supporting a roof is about as far removed from one in which the world shows itself to be a cosmos as one could possibly imagine. Its deepest interest and aspiration, an interest in its own flourishing, has been shown to be a matter of utter indifference to the most powerful forces that structure its world.\n\nYou might say that although the oak is not itself flourishing, its consumption in the fire is causing the life of the householder in whose fireplace it burns to flourish. The failure of the oak to produce acorns and reproduce itself is part of the ability of the householder to live in warmth, cook his food, and so forth. Well, what if the householder is Oedipus? He doesn't look to be exactly flourishing either. Look, then, philosophers will say, not at him but at the polis. Perhaps his catastrophic life is necessary that the polis may flourish. One could continue in this way, appealing at each stage to \"the bigger picture.\" In fact, philosophers are likely to say, one must in principle be able to continue in this way until one gets a final picture. Kant, oddly enough, was the one of the four philosophers whom Williams cites who experienced the greatest theoretical discomfort with this procedure. He knew that the idea of a final picture, and thus of a cosmos, didn't make sense, and so we could never know that the world was compatible with the demands of morality. The fact that we could never know this, however, terrified him, and so he reintroduced the harmony between the world and our moral sense not as an object of knowledge but as a mere postulated object of faith and hope. Kant, could not, of course, even conceive of the possibility that our moral sentiments themselves might not form a single coherent system. To the main line of the philosophers the archaic thinkers might be imagined to reply: we don't know the final big picture, don't even know if it exists, and to the extent to which we can speculate about it, it seems very different from your conception. It is not notably characterised by coherence and moral order but by conflict, ignorance, and accident. When Plato proposes that this world of shame, confusion, sterility, and so forth must end in some bigger more harmonious whole that in some sense we can picture or describe in logoi, Pindar might respond by saying: Why \"must\"? Yours is a poetic phantasy, like various of mine, but less plausible.\n\nThe second aspect of the comparison refers not so much to the external circumstances but to the judgements we make, respectively, about the oak tree and Oedipus. The oak log and Oedipus are clearly being cited here as positive exempla. They are to be admired and emulated. Pindar presents that judgement pointedly when he \"writes\":\n\n\u03ba\u03b1\u1f76 \u03c6\u03b8\u03b9\u03bd\u03cc\u03ba\u03b1\u03c1\u03c0\u03bf\u03c2 \u1f10\u03bf\u1fd6\u03c3\u03b1 \u03b4\u03b9\u03b4\u03bf\u1fd6 \u03c8\u1fb6\u03c6\u03bf\u03bd \u03c0\u03b5\u03c1' \u03b1\u1f50\u03c4\u1fb6\u03c2\n\nWilliams translates this as:\n\n\"Although its fruit has failed, yet it can give an account of itself.\"\n\nWilliams was a more than competent scholar of Greek so this translation is not wrong, but it is to be noted that \"give an account of\" has become for us the standard translation of a philosophical phrase repeatedly used by the Platonic Socrates: \u03bb\u03cc\u03b3\u03bf\u03bd \u03b4\u03b9\u03b4\u03cc\u03bd\u03b1\u03b9, which means to give a discursive account of oneself that justifies what one thinks or does by reference to good reasons. When Williams writes that the oak tree (and Oedipus) give an account of themselves, however, this is meant not in the sense in which Socrates asks people to give an account of themselves but in the sense in which we speak of a boxer giving a good account of himself in the ring. We might also say he \"showed his mettle\" or \"showed what he was made of.\" This does not mean that the boxer can explain or justify anything or that he has won the match. In place of a \u03bb\u03cc\u03b3\u03bf\u03c2 Pindar here speaks of a \u03c8\u1fc6\u03c6\u03bf\u03c2: \u03b4\u03b9\u03b4\u03bf\u1fd6 \u03c8\u1fb6\u03c6\u03bf\u03bd \u03c0\u03b5\u03c1\u2032 \u03b1\u1f50\u03c4\u1fb6\u03c2. (In the following I shall use the Attic form of Pindar's Doric \u03c8\u1fb6\u03c6\u03bf\u03c2, because it will be more familiar to most contemporary readers of Greek.)\n\nA \u03c8\u1fc6\u03c6\u03bf\u03c2 is a small stone, and then by extension it came to mean a vote since such stones were also used as counters of various kinds, for instance, in board games, and then in various kinds of decision procedures. The fundamental fact about a decision taken by casting a vote, however, is that under normal circumstances to give a vote is not to give an account in the platonic sense. In voting for candidate X I need not give any account of why I am doing that. I don't have to have reasons; a vote is usually opaque, a black box. Part of the reason democratic politics is so obscure is that a vote does not carry its own reason on its sleeve, and in fact it is valid even if the voter has no reason and has cast it at random or for utterly indiscernible, perverse, or irrelevant reasons.\n\nPhilologists, however, tell us that \u03b4\u03b9\u03b4\u03cc\u03bd\u03b1\u03b9 \u03c8\u1fc6\u03c6\u03bf\u03bd does not mean, as one might expect if one is coming to the text with assumptions derived from the structure of English, \"give a judgement,\" that is, vote for (or against) something\u2014that, apparently, would be \u03c6\u03ad\u03c1\u03b5\u03b9\u03bd \u03c8\u1fc6\u03c6\u03bf\u03bd\u2014but rather it means \"submit to a vote\" or \"put a case to a vote.\" That is, the oak, when it holds up the roof (or even when it burns in the fire), allows and invites people to judge it and see what kind of thing it is. The boxer who goes into the ring allows one to judge his mettle, see what kind of account of himself he can give.\n\nThe implication of this line is not just that the oak tree puts forward its case and submits to judgement but that the judgement made will be positive. The log is presented here in the poem, after all, as a kind of model to be emulated. So the notional \"judges\" will vote that this is a \"good\" log because it burns well in the fire or supports a ceiling well (despite the fact that it has no leaves or precisely because of this). The log is judged to be doing well\/faring well (\u03b5\u1f56 \u03c0\u03c1\u03ac\u03c4\u03c4\u03b5\u03b9\u03bd again, vide supra). Who, however, is imagined to be doing the judging and from what point of view? In a world that is not a moral cosmos there may be no absolutely definitive, authoritative answer to these questions. Some humans see it this way; others that way; the Olympians gods see it this other way; various chthonic agents, Furies, demons, and so forth have their own different views; perhaps Fate herself has a view, but that is just one more view among many. Nevertheless even if it ends up in the fire the log has done well and passes the judgement with high marks, even if that is merely based on an accidental conjunction of points of view that do not always coincide or a contingent preponderance of ballots cast for different reasons. That is not nothing.\n\nThis is not to say that the oak might not fail to receive a good judgement, just as the boxer can fail to give a good account of himself in the ring. This sense of \"failure\" is distinct from technical failure in the \"competitive\" sense. The boxer may show his mettle but lose the match, but he may also win the match and still have put up a poor showing. Perhaps his opponent was even worse than he was, or perhaps he wins on a technicality. Similarly, Pindar canvasses two possible outcomes for the oak log: the fire or serve as a roof support. Both of these can be seen (by humans) as useful functions. What about the possibility, however, that the oak branch is stripped off but then dropped accidently and simply allowed to rot in the place where it falls, or what if the oak branch when it is stripped from a seemingly healthy tree shows itself to be internally rotten? The oak then does not obviously give a very good account of itself. Part of the reason Pindar mentions only the two (potentially constructive) outcomes, the fire or the roof beam, is perhaps that the epinikion is inherently a poem of praise and so always focused on positive achievement, even though Pindar is aware that such achievement is exceptional, easily destroyed, and dependent on factors outside anyone's control. Pindar is not an evangelist and his form of praise, with its great emphasis on a \u03c6\u03c5\u03ac (good natural endowment) and \u03ba\u03b1\u03b9\u03c1\u03cc\u03c2 ([skill in choosing] the nonrecurring opportune moment to act), is not equivalent to, or dependent on, philosophical \"good news.\"\n\nIn a famous essay Williams analysed the career of the painter Paul Gauguin as an instance of what he called \"moral luck.\" Gauguin abandoned his wife and child to a life of poverty in order to pursue a life of painting in the South Seas. By the normal standards of accepted morality he is to be condemned for doing this. However, he had the moral luck that he turned out to be exceptionally talented at painting and that he lived long enough to realise that talent in a series of highly impressive works. The fact that he did turn out to have great talent and did produce a body of marvellous work affects, Williams claims, the judgement\u2014even the moral judgement\u2014we will make on him. Pindar's example of the oak log is much more radical than this. After all, Gauguin wanted to paint, and he turned out to achieve extraordinary things. The oak log \"wanted,\" if one can use that expression\u2014and since the whole passage depends on what later rhetorical theory would call \"personification\" there is no reason not to use that expression\u2014to stay on the mountain and produce acorns. Similarly, the oak tree is a kind of model not just when it does something extraordinary, like holding up a palace roof, but even when it does something completely commonplace, like burning in the fire, something any old bundle of twigs or even a bit of rubbish can do.\n\nOedipus's case, too, is even more complex than Gauguin's. Gauguin got what he wanted: he lived the life of a painter, and in that sense his life was a success by his own standards. There is no obvious sense in which Oedipus's life was a \"success\" from his own point of view, and it is hard to see that he has \"flourished\" in any sense recognisable by himself or other humans: he did solve the riddle of the Sphinx, and that is a singular achievement, but he ends up like the oak tree as a shameful sight, enmeshed in a web of violence bereft of descendants. Nevertheless he can put himself forward like the oak tree in the hope, presumably, that others will vote that he was a sad plaything of fate and has made the best he could of the situation in which he found himself, that of acting under truly weird circumstances over which he had no control. One might say, though, what does \"giving the best account of himself\" mean if not that we, the notional \"judges\" who are to cast our ballots wither for him or against him, judge him by certain standards, criteria, perhaps rules? These will be standards we use. So giving the vote to him is not just random but follows rules or principles. If this is the case, then it might be possible theoretically to specify those rules. Of course that is correct, but then one would have merely the set of rules a certain person or group of people use. Other human individuals or other groups might use other rules. There might be different groups of gods (Olympians and chthonic deities) who were in conflict with each other and whose moral views differed. Even within a single group, for instance, the Olympian gods, some particular gods might enforce on humans different rules from those that other gods would wish to enforce\u2014think of Euripides' Hippolytus\u2014and even if the Olympians all agreed with each other, their evaluations might differ from or conflict with ours\u2014think of the treatment of Heracles in Sophocles' Trachiniai and Euripides' Hercules Furens\u2014and even if all the diverse kinds of gods and all men agreed on some set of rules for distributing votes about who is to be admired, there is no reason to assume the world is set up so as to be amenable to these rules, whatever \"amenable\" might mean. After all, it is not as if anyone in this period thought that the world as a whole was \"created\" by any one of the gods; rather we repeatedly find suggestions that there are forms of \"necessity\" active in the world to which both gods and men are subject, and \"necessity,\" powerful as it is, is not usually moralised.\n\nDamophilus is in a situation similar to that of the oak tree (or to Oedipus). He, too, has been ripped out of the natural context for his human flourishing and is making the best of it. Arkesilaos is not a stripped branch with no fruit (\u03c6\u03b8\u03b9\u03bd\u03cc\u03ba\u03b1\u03c1\u03c0\u03bf\u03c2); he has been successful in retaining his throne in \"fruitful\" Libya (\u03ba\u03b1\u03c1\u03c0\u03bf\u03c6\u03cc\u03c1\u03bf\u03c2 \u039b\u03b9\u03b2\u03cd\u03b1, line 6) despite civil unrest and an attempted coup, and he has plucked the fruit (\u03ba\u03b1\u03c1\u03c0\u03cc\u03c2) of a Pythian victory. Even he, though, even if he \"deserves\" it, cannot count on his continuing success and good fortune. No one can because our world is not a moral cosmos. Human forms of valuation have neither consistency, coherence, nor any particular purchase on reality.\n\nI have always found it striking and important that although Pindar wrote a large number of victory odes and was obviously keen on athletics and fascinated by athletic success, he is also wise enough never, as far as I know, to compare human life to a race. In an Olympic or Pythian race there was a clear \"point of view,\" a clear set of judges and clear rules: those who started before the starting order was given were whipped and disqualified; the first one over the goal got a wreath; the others slunk home in shame (Ol. VIII.65\u201370). This might be crude\u2014there was a distinction in some events between those for men and for boys, but still all boys raced together without further handicaps\u2014and brutal, but it was regulated. Human life is distinctly not like that. It is not that life is completely chaotic, that we can find no order whatever in any part of it, or even that we cannot give ourselves good rules of thumb, such as \u03bc\u03b7\u03b4\u1f72\u03bd \u1f04\u03b3\u03b1\u03bd, which will hold in a number of different circumstances, although not with Kantian universality. We can construct small islands of rules and \"good order\" such as that which holds in Phokulides' \"polis on the rocky outcrop\"\u2014such a polis then is a small cosmos\u2014but that is a local order (i.e., it does not hold, nor had anything analogous held in Nineveh), it won't last forever\u2014where are those little poleis today\u2014it won't always survive external assault or internal stasis. Furthermore, it is constructed and maintained (for as long as it is maintained) by dint of great human exertion; it not \"found.\"\n\nPindar in fact seems rather to suggest that we cannot even expect to be able always to create local order by our own unaided efforts. Just after the passage Williams cites from the Fourth Pythian, Pindar has his chorus sing\n\n\u1fe5\u1fb4\u03b4\u03b9\u03bf\u03bd \u03bc\u1f72\u03bd \u03b3\u1f70\u03c1 \u03c0\u03cc\u03bb\u03b9\u03bd \u03c3\u03b5\u1fd6\u03c3\u03b1\u03b9 \u03ba\u03b1\u1f76 \u1f00\u03c6\u03b1\u03c5\u03c1\u03bf\u03c4\u03ad\u03c1\u03bf\u03b9\u03c2\u00b7\n\n\u1f00\u03bb\u03bb\u1fbd \u1f10\u03c0\u1f76 \u03c7\u03ce\u03c1\u03b1\u03c2 \u03b1\u1f56\u03c4\u03b9\u03c2 \u1f15\u03c3\u03c3\u03b1\u03b9 \u03b4\u03c5\u03c3\u03c0\u03b1\u03bb\u1f72\u03c2 \u03b4\u1f74 \u03b3\u03af\u03bd\u03b5\u03c4\u03b1\u03b9, \u1f10\u03be\u03b1\u03c0\u03af\u03bd\u03b1\u03c2\n\n\u03b5\u1f30 \u03bc\u1f74 \u03b8\u03b5\u1f78\u03c2 \u1f01\u03b3\u03b5\u03bc\u03cc\u03bd\u03b5\u03c3\u03c3\u03b9 \u03ba\u03c5\u03b2\u03b5\u03c1\u03bd\u03b7\u03c4\u1f74\u03c1 \u03b3\u03ad\u03bd\u03b7\u03c4\u03b1\u03b9 [ll. 272\u201374]\n\n[It is easy even for feeble people to shake and throw down the city \nBut very difficult to set it back up on its feet again \nUnless suddenly a god acts as the steersmen for the leaders]\n\nNote that this does not say that a god will always come to the aid of those trying to put the city back into order, or even that if the leaders are good and skillful some god will come to their aid. Rather, I suggest, than asserting that the world is a moral cosmos, this seems rather to indicate how much distance separates Pindar from any such idea.\n\nTo what \"modern use,\" then, can we put any of these archaic ideas? I don't like the term \"use\" in this context because it suggests that one can detach individual concepts or conceptions from their original context and apply them atomistically to pre-given problems. It suggests that there is a detachable \"wisdom\" to be found in them that can be extracted. It might be that one can learn something from thinking about the life and works of these archaic figures, Pindar, Sophocles, and Thucydides in the first instance, without it being the case that what one learns can be \"used\" in any direct way. This is in one way like, but in another unlike, the situation of Arkesilaos. Arkesilaos, on Williams's reading, can \"learn\" something from Oedipus although that is not a formulatable piece of \"wisdom,\" and perhaps the same is true of us when we read Pindar or Sophocles. However, Arkesilaos is presented by Pindar with a clearly structured issue and a favoured policy or course of action: Learn from Oedipus and recall Damophilus from exile. Our situation may be different in that we do not know exactly what our \"issues\" are.\n\nWilliams certainly thought, as his friend and sometime teacher Isaiah Berlin did, that the idea of a moral cosmos was by no means dead in the modern world. One might describe it as having gone into hiding. It is perhaps a thought which for its proper formulation depends on a particular metaphysical language that is out of fashion in the modern world. Nevertheless, the idea of our world as constituting a moral cosmos continued to exercise a considerable influence from its invisible lair. No Kantian and few mainstream liberals could envisage that our moral intuitions and our values might not form a beautiful whole but that on the contrary they irremediably conflict with each other.\n\nI first read Pindar's Fourth Pythian in 1968, when as a student in Freiburg\/Br. in Germany I took an Oberseminar on the poem in the Classics Faculty. This, of course, was a momentous year. In January the T\u00eat Offensive took place, which made it clear that the United States would eventually lose its colonial war in Indochina. Then Europe was convulsed by the upsurge of anarcho-syndicalist activism, which goes by the name \"May 68\" and which at the time seemed to be the beginning of something significant but in retrospect marked the beginning of the end of an era. Finally in August the forces of the Warsaw Pact invaded the \u010cSSR, demonstrating that the Soviet system of the time was incapable of internal reform. This turned out in retrospect, I think, to be a particularly propitious time to be reading Pindar. We are familiar with what are now called \"win-win\" situations, but 1968 was a kind of \"lose-lose\" situation in Western Europe and the United States. Whatever your view about the world or about rationality, it was hard to see the situation as one in which rational structures were successfully reproducing themselves, or the world was showing itself amenable to our ethical life, or things were moving in a positive constructive direction. Whatever the world was, it certainly did not resemble a rational cosmos. Having reason on your side was not much of an advantage, if in fact you could discern where reason lay.\n\nNo one can force philosophers to hold particular views, and this is especially true with regard to ethics. If a philosopher, at any rate a sophisticated philosopher, denies the existence of the external world, one can be moderately sure that such a philosopher will have much to tell one about what he or she exactly means by \"deny,\" \"existence,\" \"external,\" and \"world,\" and probably a lot to tell one about a variety of other matters. Still it is possible to follow Dr. Johnson and \"refute\" this thesis by kicking a stone. This will not end the argument if the philosopher in question is at all worth his or her salt, but, especially in a public forum, Dr. Johnson's is unlikely to be entirely without effect, even if one feels inclined to denigrate this effect by calling it \"merely rhetorical.\"\n\nWhat about the issue under discussion here, the existence of a moral cosmos? Williams holds the general view that ethics is characterised by a weaker degree of \"objectivity\" than science. In the absence of any strong notion of objectivity in matters of ethical theory, one can easily imagine moral philosophers adopting the \"dig-in-your-heels, hunker-down, and grit-your-teeth\" approach in their desperate attempt to hold onto the idea that our basic moral beliefs are transparently accessible to us and consistent with each other. After all, it is not difficult to attain at least a strong semblance of coherence in one's views if one is willing simply to discard large swathes of one's beliefs, and this is all the easier the looser the fit between \"moral beliefs\" and the world. Most philosophers of a certain age will have memories of the Primitive-Utilitarians who used to infest odd corners of neglected departments. They would attempt to vindicate the ultimate coherence of ethics by simple dogged persistence, by holding fast to their preferred version of utilitarianism and refusing to budge from it, despite the gruesome and counterintuitive consequences of their view. You simply couldn't talk these people out of their view because, when all else failed, they simply stopped listening or accepted any unpalatable consequence of their view that one pointed out. \"Yes, we should redistribute body parts ad libitum if it will maximize utility,\" they would say. One had the sense that at some level they must have known the disrepute into which they brought themselves by brazenly accepting these consequences, but they apparently learned to tolerate this psychic pain this must have caused, while retaining an impassive countenance. Nowadays it is likely to be Kantians who adopt this teeth-grinding attitude: \"You must never lie, just because you must never; if you don't admit that you are not being rational.\" This method will allow one to continue to affirm the existence of a coherent moral cosmos, if one wishes and one can tolerate the price one must pay. Instead of conducting endless and compulsive discussions of the possibility of a pure amoralist, perhaps it might make more sense for us to think about this phenomenon and what it tells us about moral life (apart from the fact that some of its theorists are pathological cases). That reasoning in fact utterly fails here is not completely insignificant.\n\nIt is a mistake to expect the world to make moral or human sense. That it does not is something one must simply accept. One might be tempted here to assert this as the contents of a new kind of wisdom, a wisdom of modesty, self-restraint, and acceptance. This is certainly not Pindar's way for two reasons. First, his career as a poet is dominated by the praise not of lowly men who know their own limits but of aristocrats who glory in their self-assertion, grandeur, and external success, in the \u03bb\u03b1\u03bc\u03c0\u03c1\u1f78\u03bd \u03c6\u03ad\u03b3\u03b3\u03bf\u03c2 that plays around them (even if just for a moment) when they win the race. Second, the wisdom of acceptance is not real solution, and its value can easily be overestimated:\n\n\u03a4\u03af \u1f14\u03bb\u03c0\u03b5\u03b1\u03b9 \u03c3\u03bf\u03c6\u03af\u03b1\u03bd \u1f14\u03bc\u03bc\u03b5\u03bd, \u1f03\u03bd \u1f40\u03bb\u03af\u03b3\u03bf\u03bd \u03c4\u03bf\u03b9\n\n\u1f00\u03bd\u1f74\u03c1 \u1f51\u03c0\u03ad\u03c1 \u1f00\u03bd\u03b4\u03c1\u1f78\u03c2 \u1f34\u03c3\u03c7\u03b5\u03b9;\n\n\u03bf\u1f50 \u03b3\u1f70\u03c1 \u1f14\u03c3\u03b8' \u1f45\u03c0\u03c9\u03c2 \u03c4\u1f70 \u03b8\u03b5\u1ff6\u03bd\n\n\u03b2\u03bf\u03c5\u03bb\u03b5\u03cd\u03bc\u03b1\u03c4' \u1f10\u03c1\u03b5\u03c5\u03bd\u03ac\u03c3\u03b5\u03b9 \u03b2\u03c1\u03bf\u03c4\u03ad\u1fb3 \u03d5\u03c1\u03ad\u03bd\u03b9\u2022\n\n[What do you expect wisdom to be, in which\n\nOne man surpasses another only by a little?\n\nFor there is no way for the mortal mind\n\nto find out what the gods plan] [Fr. 61 (33) Paeanes]\n\nSo grim is our situation that not even having wisdom will do us much good. The term I have rendered as \"expect\" (\u1f14\u03bb\u03c0\u03b5\u03b1\u03b9) is etymologically closely associated with the word that eventually establishes itself as the term for \"hope\" (\u1f14\u03bb\u03c0\u03b9\u03c2). The \"good news\" philosophy preaches is a message of deceptive \"hope\" to which Pindar, Sophocles, and Thucydides did not fall victim.\n12\n\nWho Was the First Philosopher?\n\nOne might think that this question has no answer because it is badly formulated. It might, that is to say, be thought to make two incorrect assumptions. First, it seems to assume that \"philosophy\" is an individual or even solitary activity like swimming the Tiber or bringing down a particular kind of bird with a boomerang. Provided we can agree on rough and ready \"definitions\" of the entities and activities in question\u2014homo sapiens, not one of our near simian relatives; \"swimming,\" not \"floating\" or \"being borne along by the current\"; \"boomerang,\" not \"stick\" (and this is a big \"if\")\u2014we could in principle answer these questions, although, for contingent epistemic reasons we are unlikely actually to know who this was in either case. On the other hand, it would make no sense even in principle to ask \"Who was the very first chess player?\" because playing chess requires two people. As Hegel pointed out, it is a mistake to think of the history of philosophy as a series of activities performed by a succession of individual men or as the result of such an activity; it is inherently a collective social enterprise. The model for philosophy is not Thales noting that the waters of the Nile rise at the same time every year and wondering why that might be the case\u2014that is perhaps (one of) the origin(s) of science. The paradigm for philosophy is the encounter between Socrates and Euthyphro, who is on the way to indict his father for the murder of a slave. This encounter becomes an instance of philosophising when Socrates asks Euthyphro what's up, and, more important, when Euthyphro enters into extended conversation with Socrates and tries to respond to his questions. Extended monological speeches \u00e0 la Protagoras or Gorgias may or may not be edifying, but whatever else they might be they are definitely not philosophy, which consists of a joint attempt to thrash out some agreed-on conclusion in discussion, whether that attempt is successful or fails. In any case, as in chess, it takes (at least) two to philosophise. Friedrich Schlegel gets the emphasis wrong in praising the importance of \"Symphilosophie\" only to the extent to which his use of this term suggests that the social dimension is a mere desideratum, as if there could be such a thing\u2014albeit a slightly deficient thing\u2014as philosophy that was not a collective activity.\n\nNietzsche was not so keen on the idea of philosophy as a social enterprise, preferring a more traditionalist and heroic conception, but he can be seen as raising a second kind of objection to the initial question. He criticises all forms of analysis of any continuing human practise that refer it to a purported unique \"origin.\" He is especially scathing about attributing the origin of enduring institutions to individual \"founders\": Christianity \"founded\" by Jesus, democracy \"founded\" by Cleisthenes, philosophy \"founded\" by Thales, Pythagoras, or Socrates. It is deeply misguided to look for a single origin for anything that has had extended historical significance. The \"genealogical\" mode of arguing, in which purportedly unitary \"origins\" are shown in reality to be contingent conjunctions of diverse and varied antecedents, is now a commonplace. It is a development of a strand in Nietzsche's thought one might call \"positivist.\" Nietzsche himself, after all, emphasised that \"genealogy\" as he practised it was a \"grey\" science with affinities to the driest kind of archival research. This conception of what he was trying to do is somewhat at odds with his actual practise. If \"genealogy\" really is a form of quasi-archival history rather than, for instance, a philosophical fairy tale something like \"state of nature\" narratives one finds in some philosophical works, or the description of historical process using something like a series of Weberian ideal types, then it should be possible to ask and answer the questions: Who exactly are \"the masters\" and \"the slaves\" in the narrative? When precisely did the \"slave uprising\" take place? Notoriously there is no answer to these questions, and Nietzsche treats this as no objection to his account. Perhaps it is the constant shift between levels of analysis\u2014from discussions of \"masters\" where it is left open whether they are Arab, Japanese, or Germanic aristocrats to the concrete details of early Roman history\u2014which constitutes part of the point of the exercise. However, in addition to these two modes, the archival and the ideal typical, the early Nietzsche recognised another possibility that only very gradually came to be sidelined in his thinking and perhaps never completely lost its grip on his imagination. Richard Wagner had wanted to replace conceptual and historical thinking with what he called \"mythic\" thought. As a dramatist Wagner was particularly keen on removing the historical element in theatre. Shakespeare's historical plays, Wagner thought, were the worst kind of literature; drama must deal not with real, and thus contingent, historical figures and their vicissitudes, like Henry V, but with \"mythic\" figures like Oedipus, Siegfried, or King Lear. It was one of the glories of Attic tragedy, in stark contrast to comedy, that it did not in general deal with real history or present onstage real historical characters. The Persians (about the Persian Wars, which were part of contemporary memory at the time the play was written) was apparently to be considered an exception, and even The Persians mentions no individual Greek by name. \"Myth\" formulated not what had happened to occur but inherent structures of meaningfulness. Henri V happened to win at Agincourt, but what happened to Tristan and Isolde, to Siegmund and Sieglinde, or to Siegfried and Wotan was not simply an accident: there is a kind of necessity about the way in which primordial eros breaks down any existing social relationships and about the way in which the primordial aggression that characterises the relation between son and father can express itself in violence. In Wagner's view what he called \"the state\" was a coercive structure that controlled and repressed these basic human urges and the form the coercion took was to make us unaware of our own real nature. The state could continue to exist only to the extent to which it was capable of continuing successfully to keep us in ignorance about this \"mythical\" level of human existence. His own work, he thought, derived its power from his ability to tap these primitive human urges that were present in everyone and bring them to the awareness of the members of his audience. To this extent his operas were all by their very nature politically revolutionary.\n\nEarly in his life Nietzsche was close to Wagner's views about myth, though, of course, he held them in a highly metaphysical and radically de-politicised form: In The Birth of Tragedy he strove to overcome conceptual and historical forms of thinking and replace them with mythology. The Birth of Tragedy, a work that even in its very literary form, for instance its lack of footnotes, rejects the existing standards of scholarship, is organised not as a proper history or an analytic account of tragedy but as a mythic narrative of the doings of \"Apollo,\" \"Dionysus,\" and \"Socrates\" (who in this account is transformed from a real denizen of the deme of Alopeke to a mythic figure, just as, presumably the historical Iranian king H\u02d8\u0161ay\u0101r\u0161\u0101 becomes the \"mythic\" \"Xerxes\" in The Persians). \"Apollo,\" \"Dionysus,\" and \"Socrates\" are intended to be neither real historical individuals like Henry V nor mere names for what are actually abstract conceptual structures, such as \"the drive to encompass more and more material\" (Stofftrieb) and \"the drive to impose unity\" (Formtrieb), which Schiller took to be the basic constituents of human nature. A \"mythic\" treatment is a narrative in which a figure with some distinctive individual characteristics (such as Heracles) is involved in an encounter with other such figures (such as Admetus, Alcestis, and Death). \"Lear\" is to some extent \"defined,\" if one wishes to think about it in this way, by his own characteristics and by the nexus of his encounters with others (his daughters); it is this nexus that has a kind of necessity. Myth exhibits recurrent narrative patterns, but the individual story need not be strictly invariant; one of the most striking features of ancient myths is precisely that they have no single canonical form but are flexible and can be reconfigured in a variety of different ways, each one of which is expressive of an aspect of human nature or a kind of human necessity. Each author could write his own Oedipus within certain limits that are antecedently indeterminate. So although a proper history would not pinpoint a single origin of the human practise we call \"philosophy,\" there is no reason one could not give a mythic account that sums up in a concentrated form some of the salient characteristics of the practise.\n\nIn this spirit I would like to suggest that the \"origin\" of philosophy might be taken to lie in the encounter between the aboriginal philosophical couple: Oedipus and the Sphinx. Neither one, then, is the originator of philosophy because philosophy is inherently an interaction of a certain kind between two animate entities who can speak with each other and act with or on each other. I say \"animate entity\" rather than \"human being\" because the Sphinx is not a human being but on most accounts half woman and half lioness (often with wings). The interaction between Oedipus and the Sphinx is an odd mixture of the utterly frivolous and the deadly serious, of the accidental and the essential. What could be more contingent than a meeting on a road? Yet the meeting seems to generate from itself its own internal necessity. The Sphinx asks a question that is also a riddle, that is, she proposes a kind of game, but also kills anyone who loses the game by failing to solve the riddle. Generally we think of a \"game\" as something one can chose to play or not, but the Sphinx leaves one no choice. It is assumed that she is endowed with enough strength and willingness to use violence so that there can be no question of \"overcoming\" her in a simple fight. She subjects herself to a limitation on the use of that strength; she could simply kill travellers, no questions asked, but she does not do that. Rather she transforms what might otherwise be a straightforward physical struggle, which we have reason to believe she would always win, into a contest of wits and commits herself not to use her overwhelming force if bested. When Oedipus does come up with the right answer, she not only spares him but kills herself. She makes the question (and the answer) a matter of life or death for Oedipus, and then for herself. Was her own death in the envisaged circumstances a part of the \"rules of the game\" as initially specified (by her) from the start, or did she simply improvise and make up the final logical conclusion of the game, her self-destruction, when she found herself bested? Why did she start the game in the first place? Did she actually know the answer to the riddle before Oedipus told it to her or merely recognise that what he said could count (or \"must count\"?) as an answer, when he said it? Did she make up a question she thought had no answer in order to be able to kill travellers under the specious appearance of giving them a \"fair\" chance? When Oedipus shows that he knows the answer, or at any rate gives an answer she cannot dismiss, the conclusion she draws, killing herself, does have a weird archaic consistency, even if it was not originally envisaged as even a possibility. By doing so, she founds philosophy. Was there any other possible outcome? Could she simply have let Oedipus go and retired from the question-asking business? Or taken Oedipus to live with her to the mountains and perhaps father a race of preternaturally intelligent sphinxes? Probably not. The riddle was: \"What walks on four feet in the morning, two feet at noon, and three feet in the evening?\" and the answer was \"man\"\u2014as baby, adult, and old-man-with-a-stick (not \"sphinx\"), and Oedipus not only gave the correct answer but also instantiated it. The correct answer is, then, just a true description or proper analysis of what he himself is. If Oedipus had failed to give the right answer, he would also not himself have instantiated it because he would have died before reaching the stage at which he would need to use a stick.\n\nThe encounter between Oedipus and the Sphinx might be thought to count as the origin of philosophy in part because in it two things seem inextricably intertwined: the solution of a riddle and an existential issue of life-or-death importance. The Sphinx has imposed the existential dimension on the encounter, to be sure, so perhaps it is only really a philosophical encounter in the full sense if we think of her not as an individual person but as kind of force of nature or metaphysical principle. It is philosophy only if the solution of the riddle is also a matter of life-and-death that is necessarily imposed on us by the nature of the world we live in, not the result of a random encounter with an individual. Philosophy that reduces itself to mere riddle solving loses its traditional importance, and, eventually, its raison d'\u00eatre.\n\nIs it perhaps relevant that the Sphinx herself seems to us not to exhibit a \"natural unity\" but to be a biological \"creole,\" a bifurcated being only presenting herself as a unity? How does she seem to herself? Does she think of herself as a natural unity? Does it matter what she thinks? After all, we humans, in the person of Oedipus, won. Is her creole nature part of the explanation for her murderous hostility to the people of Thebes and for her habit of asking off-beat questions? Her question suggests that although we may see ourselves as natural unity, it is possible to see us as just as deeply creole as she is: we are, after all (correctly), described (by her) as three in one. Oedipus must in some sense not merely know what sort of creature he is but also how a creature like him would look to a Sphinx. Who is to say what is to count as \"natural unity,\" what as a mere appearance, and what as narcissism? Philosophy exists in the state of emotional tension and cognitive motion between the moment when Oedipus can be seen coming up the road from Corinth\u2014or rather the moment when the first nameless traveller appears, since Oedipus was not the first to have an erotically tinged meeting with the Golden-Strangler-Girl-of-Thebes\u2014and the final sad but \"logical\" denouement. When the answer is formulated and revealed, the encounter (and philosophy) is over. This is perhaps why Nietzsche calls Oedipus \"the last philosopher,\" condemned to speak only with himself. How can he ask questions without a partner? Recollection lacks the essential frisson of the real thing because if you can recollect, you have survived, and part of the point of the encounter is that that is not yet assured. Oedipus is not only the last philosopher but also \"the last man.\" As Nietzsche's Zarathustra says about the dead tightrope walker, the Sphinx perished in the course of discharging a self-chosen, and in her case indeed self-invented, profession that turned out to be dangerous, and there is nothing contemptible in that. Did Oedipus realise that his profession of \"knowing\" could be equally dangerous? Would he have been so keen to discover the cause of the plague in Thebes, and his own origins, if he had not met the Sphinx?\n13\n\nA World without Why\n\nI have always held to be a mildly discreditable day job, that of teaching philosophy at a university. I take it to be discreditable because about 85 percent of my time and energy is devoted to training aspiring young members of the commercial, administrative, or governmental elite in the glib manipulation of words, theories, and arguments. I thereby help turn out the pliable, efficient, self-satisfied cadres that our economic and political system uses to produce the ideological carapace that protects it against criticism and change. I take my job to be only mildly discreditable partly because I don't think finally that this realm of words is in most cases much more than an epiphenomenon secreted by power relations that would otherwise express themselves with even greater and more dramatic directness. Partly, too, because 10 percent of the job is an open area within which it is possible that some of these young people might become minimally reflective about the world they live in and their place in it; in the best of cases they might come to be able and willing to work for some minimal mitigation of the cruder excesses of the pervading system of oppression under which we live. The remaining 5 percent of my job, by the way, what I would call the actual \"philosophical\" part, is almost invisible from the outside, totally unclassifiable in any schema known to me, and quantitatively in any case so insignificant that it can more or less be ignored.\n\nSo the experience I have of my everyday work environment is of a conformist, claustrophobic, and repressive verbal universe, a penitential domain of reason-mongering in which hyperactivity in detail\u2014the endlessly repeated shouts of \"why,\" the rebuttals, calls for \"evidence,\" qualifications, and quibbles\u2014stands in stark contrast to the immobility and self-referentiality of the structure as a whole. I suffer from recurrent bouts of nausea in the face of this densely woven tissue of \"arguments,\" most of which are nothing but blinds for something else altogether, generally something unsavoury; and I feel an urgent need to exit from it altogether. Unsurprisingly, Plato had a name for people like me when I am in this mood: \u03bc\u03b9\u03c3\u03cc\u03bb\u03bf\u03b3\u03bf\u03c2, a hater of reasoning. I comfort myself for being on the wrong side of Plato by thinking that I am also, at any rate, never unaware of the potentially questionable nature of this desire. One might be inherently suspicious of what is clearly the luxury complaint of someone who occupies what is in effect a very privileged position in a rich society; those suffering from debilitating diseases, struggling to get access to clean water, or trying desperately to avoid the systematic attentions of a repressive state apparatus or the more or less random violence of armed gangs in regions where public order has broken down might well be thought to have more pressing concerns. To that extent perhaps my reaction does not throw a morally flattering light on me. That does not, however, exhaust the objective disquiet my impulse causes me.\n\nThe problem with \"N'importe o\u00f9 hors de ce monde\" (\"Anywhere outside this world\") is not merely that such a place is hard to find. A world utterly without \"why\" can have one or the other of two very different aspects. It can seem a deeply contemplative, even if not necessarily thoroughly pleasant, place, as in the poem by the seventeenth-century Silesian mystic Angelus Silesius:\n\nDie Ros' is ohn' Warum; sie bl\u00fchet weil sie bl\u00fchet.\n\nSie acht' nicht ihrer selbst, fragt nicht ob man sie siehet.\n\n[The rose is without why; she blossoms because she blossoms.\n\nShe pays no attention to herself, does not ask if anyone sees her.]\n\nThe rose may have a thorn and her self-less insouciance will perhaps be barely distinguishable from cold indifference; still, this vision of reality as freed completely from the subordination of any of its parts to purpose or functionality might have some aesthetic appeal. The other, potentially diabolical, aspect of this construction is the one that presented itself to Primo Levi when he realised that in Auschwitz there was no \"why\" (\"Hier gibt es kein 'Warum' \" [\"Here there is no 'why' \"]). Levi's experience, of course, was not really of a place in which there was no \"why\" at all. The SS officers with whom he came into contact had a variety of reasons for what they did and what they allowed to happen. Some of these reasons, to be sure, were unreflective and conflicting, some perhaps fantastic and delusional, many were deeply malicious, but that is a different thing. It was not in fact that an extermination camp had no \"why\" whatever but that those in control of Levi's fate were in no way required or inclined to give him any reasons for anything that occurred. A world in which reason was utterly inaccessible to the individual might at best be an approximation of one possible form a \"world without why\" could take.\n\nOur networks of institutionally anchored universal ratiocination are hard to escape. How in fact could one get out, assuming one wanted to? Offhand, I can think of three possible ways. First, one could be clever enough to turn the why-game against itself from within. This has been the dream of any number of philosophers including, most notably, Hegel and Heidegger. This way out does not recommend itself to me because I am not clever enough to trod this path successfully but also because even if successful, who would notice? The second possibility is action. One deed is worth any number of words. A deed can cut through\u2014I always think of this with the French word trancher\u2014the spider's web of bogus rationalisations and create not merely new words but new facts. Unfortunately, this second course of action requires very significant amounts of courage as well as practical skills of various kinds, neither of which I possess. The courage in question, by the way, is not merely personal fearlessness in the face of threats to oneself but also the moral courage to face the possibility that one's actions, which, if they are going to be effective at all, will certainly be almost completely out of one's own control as far as their actual consequences are concerned, may turn out to inflict great suffering on the wrong people (even assuming one were to know for certain who these are).\n\nThe third possibility is the invitation, in particular one to observe, look at, or consider something. One kind of thing one can be invited to consider is a juxtaposition: masses of anonymous people storming the Winter Palace and two stone lions standing up on their pedestals, or the prime minister oleaginously addressing the House of Commons and a pile of bodies in a ditch in Iraq. By putting two (or more) separate \"things\" next to each other and inviting people to look at them together, one is not necessarily asking or trying to answer the question \"why.\" A poem may cause someone to ask a question or to initiate a line of reflection, or even to develop some hypothesis or theory, but then a clap of thunder or a sudden pain in the chest may do the same; that does not make either the pain or the poem a theory or a \"line of argument.\" A word in a good poem is not a concept. Since neither a picture nor a poem is an argument, neither is a suitable object for counterargument. \"La terre est bleue comme une orange\" [\"The earth is blue like an orange\"] is not best understood as \"asserting a proposition.\" Neither is\n\nDer Nordost wehet,\n\nDer liebste unter den Winden\n\nMir, weil er feurigen Geist\n\nUnd gute Fahrt verheisset den Schiffern.\n\n[The northwest wind blows,\n\nthe dearest of the winds\n\nto be because he promises\n\nfiery spirits and good passage to sailors.]\n\nNor finally even:\n\nVer erat, et morbo Romae languebat inerti\n\nOrbilius.\n\n[It was spring and in Rome Orbilius was suffering \nfrom a debilitating sickness.]\n\nYou can't refute an invitation (although you can refuse it, closing your mind and heart to it): it makes no claim. At the end of all the talk, the poem, if it is good enough, is still standing there, waiting. An invitation has neither the direct constructive or coercive power of action nor the indirect coercive power of ratiocination\u2014Habermas's \"eigent\u00fcmlich zwangloser Zwang des besseren Argumentes\" (\"peculiarly uncoercive coercion of the better argument\"). If one is lucky enough to live in a society in which a sphere of \"free\" artistic activity is permitted to exist, no one is forced to look at one's picture, listen to one's poem, or read one's novel. Still the work of art need not be without effect on those who accept its invitation.\n\nSimple juxtaposition of external objects, persons, or events not usually seen together has a number of variants that are perhaps no less interesting and \"compelling\" (to use the peculiar expression that seems natural here). Rather than allowing the sewing machine to encounter an umbrella on the dissecting table, one can invite the reader to pay attention to something usually overlooked or taken for granted, which seems to have a unity that upon inspection dissolves. The world can occasionally turn itself inside out or upside down. No one who lived even in complete personal security through the period of the Vietnam War could thereafter ever hear the sound of a helicopter in exactly the same way again.\n\n**Invitation**\n\nShall we go to the sand-pits?\n\nYes, let's go to the sand-pits.\n\nWill the air be fresh and clear\n\nover the sand-pits?\n\nDepending on the season, the time\n\nof day, and the weather\n\nthe air will be cool, sultry, or mild\n\nover the sand-pits.\n\nShall we whistle and get a drink\n\nat the sand-pits?\n\nWhistling and drinking are de rigueur\n\nat the sand-pits.\n\nWill there be a crowd\n\nat the sand-pits?\n\nThere is almost invariably a crowd\n\nat the sand-pits.\n\nShall we take our whips\n\nto the sand-pits?\n\nIn what tree have you parked\n\nyour brain, imbecile?\n\nWithout whips what would be the point\n\nof the sand-pits?\n\n* * *\n\nArthur Schopenhauer, \"\u00dcber die Universit\u00e4ts-Philosophie\" (1850)\n\nPaul Celan, \"Der Meridian\" (1960)\n\nPlato, Phaedo 89d (fourth century BC)\n\nCharles Baudelaire, Le spleen de Paris (1869)\n\nAngelus Silesius (Johannes Scheffler), Der Cherubinische Wandersmann (1657) Primo Levi, If This Is a Man (1947)\n\nJacques Derrida, \"Les fins de l'homme\" (1967)\n\nG.W.F. Hegel, System der Wissenschaft, Erster Teil, Die Ph\u00e4nomenologie des Geistes: Vorrede (1807)\n\nMartin Heidegger, Der Satz vom Grund (1957)\n\nBertold Brecht, \"Fragen eines lesenden Arbeiters\" (1935) Paul Eluard, L'amour la po\u00e9sie (1929)\n\nFriedrich H\u00f6lderlin, \"Andenken\" (1806)\n\nArthur Rimbaud, \"Ver erat\" (1869)\n\nIsidore Ducasse, Comte de Lautr\u00e9amont, Les chants de Maldoror (1869)\n\nT. W. Adorno, \"Parataxis\" (1964)\n\nGeorg B\u00fcchner, \"Lenz\" (1835)\nNotes\n\nPREFACE\n\n1. Marx, following Feuerbach, correctly maintains that this is the correct order: first social needs, then language and cognition. See Die Deutsche Ideologie MEW (Berlin, 1983), 3:13\u201377.\n\n2. For criticism of this assumption, see David Graeber, Fragments of an Anarchist Anthropology (Chicago, 2004).\n\n3. John Dewey, Reconstruction in Philosophy (New York, 1920); see also his The Quest for Certainty (London, 1929).\n\n4. See, for instance, the appeal to \"autonomy\" that is exaggerated almost to the point of senselessness in R. P. Wolff 's In Defence of Anarchism (New York, 1970).\n\n5. This is not exactly right because there were the Stoics who seem to have planted the seed for the modern obsession. See Michael Frede, A Free Will (Berkeley, 2011). Still they were one philosophical school and their views did not represent an almost universally shared cultural obsession.\n\n6. See Aeschylus, Persai 402ff; Herodotus I.62, 95; Plutarch, Vita Titi Flaminii, etc. Further references in Kurt Raaflaub, \"Zum Freiheitsbegriff der Griechen,\" in E. Welskopf, Soziale Typenbegriffe im alten Griechenland (Berlin, 1981), vol. 4; also Raaflaub, Die Entdeckung der Freiheit (Munich, 1985).\n\n7. Ch. Wirszubski, Libertas as a Political Ideal at Rome during the Late Republic and Early Principate (Cambridge, 1968). See also Malcolm Schofield, \"Liberty, Equality, and Authority: Theory and Practice in the Later Roman Republic\" (forthcoming).\n\n8. And another school, the Cynics, adopted an ideal of practical \"self-sufficiency,\" which eventually seems to have developed into a more metaphysically laden notion.\n\n9. That the Greeks used \"autonomy\" to designate only a limited, essentially negative, and social property, see Martin Ostwald, Autonomia: Its Genesis and Early History (Chico, CA, 1982).\n\n10. Sophocles, Antigone ll. 450ff.\n\n11. E. E. Evans-Pritchard notoriously used the local poison-oracle in making everyday decisions when he was working among the Azande and claimed that in retrospect he thought this was as reasonable a way to conduct such affairs as any other would have been. See his Witchcraft, Oracles and Magic among the Azande (Oxford, 1937).\n\n12. I merely point out that I am really using the same \"philosophical must\" to which I drew attention earlier in this chapter. I am, after all, essentially saying that there \"must\" be a place for trust, respect, etc., in human life. I invite the reader to reflect on this.\n\n13. Jan Philipp Reemtsma, Vertrauen und Gewalt (Hamburg edition, 2012); Martin Hartmann, Die Praxis des Vertrauens (Berlin, 2011).\n\n14. Immanuel Kant, Grundlegung zur Metaphysik der Sitten in Werke (Darmstadt, 1963), 4:36.\n\n15. M. Foucault, Le gouvernement de soi et des autres (Paris, 2008) and \"Qu'est-ce que les lumi\u00e8res,\" in Dits et \u00e9crits (Paris, 1994), 571\u201378.\n\n1: GOALS, ORIGINS, DISCIPLINES\n\nI am particularly grateful to Richard Raatzsch and to the other members of the Cambridger Philosophisches Forschungskolloquium for comments on an early version of this text. A very much abbreviated version was presented at the conference \"Changing the Humanities\/The Humanities Changing\" at the Cambridge Centre for Research in the Arts, Humanities, and Literature in July 2009.\n\n1. Wilhelm Windelband, \"Geschichte und Naturwissenschaft,\" Pr\u00e4ludien (T\u00fcbingen, 1919), 1:136\u201361.\n\n2. Klaus K\u00f6hnke, Entstehung und Aufstieg des Neukantianismus (Frankfurt am Main, 1993).\n\n3. I use the Monro\/Allen Oxford Classical Text (OCT), translation by R. Lattimore (Chicago, 1951), sometimes slightly modified.\n\n4. One of the many fascinating aspects of this passage is that it contains what may be one of the first references to writing in Western literature. One of Glaukos's ancestors, Bellerophontes, is exiled from home and takes refuge with King Proitos. As the consequence of an intrigue, Proitos decides he needs to kill Bellerophontes, but he doesn't dare. We are not explicitly told why he doesn't dare, but one might surmise it is because in the Homeric world it was considered wrong to kill someone to whom you had once offered refuge. Proitos sends Bellerophontes off to a neighbouring king, carrying with him a folded tablet on which are inscribed, incised, drawn, or painted (\u03b3\u03c1\u03ac\u03c8\u03b1\u03c2, 6.169) some signs or marks that will bring him death (\u03c3\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bb\u03c5\u03b3\u03c1\u03ac... \u03b8\u03c5\u03bc\u03bf\u03c3\u03b8\u03cc\u03c1\u03b1 \u03c0\u03bf\u03bb\u03bb\u03ac, 6.168\u201369) when the neighbouring king sees them. I have often wondered whether the very unclarity about what is painted or scratched on those tablets might not be part of the point. We don't know whether they are the words \"KILL BEARER IMMEDIATELY\" (KTEINEI\u0398EPONTAA\u038e\u0398I) or rather something more indirect and suggestive, such as pictures of a stick-man with a spear in his back or a skull and crossbones. If this is a case of writing and writing is still a novel, unfamiliar, and slightly puzzling phenomenon, sending someone off with a written warrant for his own death might be a very good way of generating a form of self-deception that would allow one to believe one has distanced oneself from what eventually happens. Not just \"I didn't know the gun was loaded\" but \"How was I to know that that tiny tube could really kill at a distance?\" The written or painted image is a kind of obscure black box: how could I know it would work?\n\n5. See also Raymond Geuss, \"Thucydides, Nietzsche, and Williams,\" in R. Geuss, Outside Ethics (Princeton, 2005), 219\u201334.\n\n6. George Kennedy, A New History of Classical Rhetoric (Princeton, 1994).\n\n7. Diogenes Laertius 8.8.\n\n8. Protagoras 318a\u2013b.\n\n9. See Plato's Gorgias.\n\n10. Protagoras 339\u201347.\n\n11. Protagoras 320\u201328.\n\n12. See the position attributed to \"a philosopher\" in Longinus's De sublimitate, ch. 44 (OCT, ed. Russell).\n\n13. G.W.F. Hegel, Vorlesungen \u00fcber die Geschichte der Philosophie, Werke, vol. 18, ed. E. Moldenhauer and K. M. Michel (Frankfurt am Main, 1970), 202\u20139.\n\n14. Arnold Sch\u00f6nberg, Stil und Gedanke (Frankfurt am Main, 1976), 33\u201334.\n\n15. Immanuel Kant, Kritik der reinen Vernunft, A852\/B880\u2013A856\/B884. The history of reason gets four pages out of 850 or so. See also Kant, Prolegomena zu einer jeden k\u00fcnftigen Metaphysik, \"Vorrede.\"\n\n16. Immanuel Kant, Kritik der praktischen Vernunft, 50\u201357, 119\u201322 (Ak.).\n\n17. Most clearly in Schopenhauer; see his essay \"\u00dcber die Universit\u00e4ts-Philosophie,\" in Parerga et Paralipomena in S\u00e4mtliche Werke (Frankfurt am Main, 1985), 4:173\u2013242.\n\n18. This is only one half of the story for Hegel. He also thinks that philosophy is in some sense a form of \"absolute\" spirit. The relation between philosophy as a historical formation and as a systematic, absolute enterprise is at the heart of his complex views.\n\n19. G.W.F. Hegel, Differenz des Fichteschen und Schellingschen Systems des Philosophie, Werke, vol. 2, ed. E. Moldenhauer and K. M. Michel (Frankfurt am Main, 1970), 20\u201325.\n\n20. Karl Marx, Zur Kritik der Hegelschen Rechtsphilosophie in Marx-Engels-Werke (MEW) (Berlin, 1968), 1:378\u201380.\n\n21. See T. W. Adorno et al., eds., Der Positivismusstreit in der deutschen Soziologie (Berlin, 1969).\n\n22. Friedrich Nietzsche, Zur Genealogie der Moral in Friedrich Nietzsche: S\u00e4mtliche Werke: Kritische Studienausgabe, ed. G. Colli and M. Montinari (Berlin, 1967), 5:245\u2013413; see also \"Genealogy as Critique,\" in Geuss, Outside Ethics, 153\u201361.\n\n23. Saint Paul, after all, in a slightly bizarre passage (Romans) seems to claim that the point of the Jewish Law was to increase consciousness of sin; one can hold that Christianity generates the sense of sin to which its message of redemption from sin is purportedly the only adequate response.\n\n24. I mention only one of the presentable philosophical reasons for the demise of this Hegelian tradition. This should not be taken to imply a denial of political, economic, and social causes, which were in fact probably more important.\n\n25. Phaedo (OCT) 96a\u201399d.\n\n26. See W. Burkert, \"Platon oder Pythagoras,\" Hermes 88 (1960): 159\u201377.\n\n27. Tusculan Disputations (OCT) 5.4.10\u201311.\n\n28. See Nietzsche, Zur Genealogie der Moral, 5:245\u2013413; also \"Nietzsche and Genealogy\" in R. Geuss, Morality, Culture, and History (Cambridge, 1999), 1\u201328.\n\n29. Nietzsche, Zur Genealogie der Moral, 5:254.\n\n30. This gives one the ancient triadic division of philosophy into physics, ethics, and dialectics (or logic). See, for instance, Diogenes Laertius 7.39 and Cicero, Tusculan Disputations (OCT) 5.24.68\u201372.\n\n31. Vitruvius, De architectura 1.1.8, 10.12.2.\n\n32. The OED gives a first isolated instance in 1702, then a series from the mid-nineteenth century.\n\n33. Michel Foucault, Les mots et les choses (Paris, 1966), 398.\n\n2: VIX INTELLEGITUR\n\n1. Orator ad M. Brutum 9.30\u201331.\n\n2. See \"Second Letter to Ammaeus,\" in Dionysius of Halicarnassus: The Three Literary Letters, ed. William Rhys Roberts (Cambridge, 1901), 136. See also the useful general discussion of obscurity in Greek in the Roberts edition of Dionysius's On Literary Composition (New York, 1910), appendix A, pp. 335\u201341.\n\n3. Philosophische Untersuchungen \u00a7 71.\n\n4. Cited in Cicero, De divinatione II.56.116, who points out the inherent implausibility of the Greek god Apollo addressing a king of Epirus in a Latin hexameter.\n\n5. That means, of course, the exact Latin words Ennius ascribes to the god. I am not at all interested in the historical accuracy of the stories told about Pyrrhus, who seems to have been the kind of charismatic figure who attracted to his person all manner of mythic accretions.\n\n6. The importance of seeing certain kinds of apparently very general statements as in fact particular responses to specific questions was emphasised by R. G. Collingwood. See his An Essay on Metaphysics (Oxford, 1940).\n\n7. Plato's Charmides mentions a third (164\u201365), but this has been more or less ignored in subsequent discussion.\n\n8. Friedrich Nietzsche, Die Geburt der Trag\u00f6die in Friedrich Nietzsche: S\u00e4mtliche Werke, Kritische Studienausgabe, ed. Colli and Montinari, 1:40\u201341.\n\n9. Plutarch, Life of Pyrrhus 397.\n\n10. The classic study of ambiguity, of course, is still William Empson's Seven Types of Ambiguity (London, 1930).\n\n11. Compare Aeneas's \"Italiam non sponte sequor\" (Aeneid IV.361).\n\n12. T. W. Adorno, \u00c4sthetische Theorie (Berlin, 1970), 182\u201393.\n\n13. Only later (Aeneid VI.847\u201353) does Aeneas find out definitively from his father in the underworld that nothing like this is intended.\n\n14. Nietzsche, Die Geburt der Trag\u00f6die, 1:35.\n\n15. Lucretius, De rerum natura I.1\u20132.\n\n16. Similar thoughts are found in David Hume's Dialogues Concerning Natural Religion (originally 1777).\n\n17. De E Delphico 385B.\n\n18. This idea of divine revelation as being progressive and presented with a primarily paedagogical intention becomes important in certain Enlightenment views, e.g., in G. E. Lessing's Die Erziehung des Menschengeschlechts (1777; Ditzingen, 1967).\n\n19. Plato, Socratis Apologia 20\u201324.\n\n20. Agamemnon 174\u201377.\n\n21. Actually in Delphi there seems at some periods to have been a fourth agent, a college of priests who wrote down what the priestess, the Pythia, said and passed the written text to the oracle-seeker. Putting in a further layer of intermediaries, of course, makes it that much easier to protect the god from accusations of ignorance or bias: Apollo spoke the truth, but the priestess misheard it or mispronounced it, or the priest who wrote down what the Pythia said misinterpreted what she said, or falsified it in his own interest, or had been bribed, etc. The passage in Aeneid (Book III) is a departure from this standard case because Vergil goes out of his way to emphasise that the Trojans heard the very voice of the god, not of any intermediary. Presumably this, too, is part of Vergil's attempt to present the oracle as maximally authoritative and not subject to falsification or distortion through a chain of intermediaries.\n\n22. Clearest perhaps in H\u00f6lderlin's poem \"Wie wenn am Feiertage\" (written in late 1799 or early 1800) in H\u00f6lderlin: S\u00e4mtliche Gedichte, ed. Jochen Schmidt (Frankfurt\/M., 2005), 239\u201341. For Rimbaud, see his letter to P. Demeny of May 15, 1871, in Oeuvres compl\u00e8tes (Paris, 1972), 249\u201354.\n\n23. Despite my general reservations about the concept of \"evil,\" it is useful here. See my Politics and the Imagination (Princeton, 2009), 182\u201384.\n\n24. G.W.F. Hegel, Rechtsphilosophie in Werke in zwanzig B\u00e4nden, ed. Eva Moldenhauer and Karl Markus Michel (Frankfurt\/M., 1971), vol. 7, \"Vorrede,\" pp. 26\u201327. See also Theodor Adorno, \"Skoteinos,\" in his Aspekte der Hegelschen Philosophie (Frankfurt\/M., 1963), \"Parataxis,\" in Noten zur Literatur III (Frankfurt\/M., 1965), and Minima Moralia, \u00a7\u00a7 5, 44, 50.\n\n25. Paul Celan, Der Meridian: T\u00fcbinger Ausgabe (Frankfurt\/M., 1999), 85.\n\n26. Ibid., 7, 84.\n\n27. \"[E]mphatisch kein Kunstwerk gelingen\" [\"No work of art can be a success in an emphatic sense of the term 'success' \"]. Adorno, \u00c4sthetische Theorie, 87.\n\n3: MARXISM AND THE ETHOS OF THE TWENTIETH CENTURY\n\nIn all my thinking about the subject matter of this chapter I have been deeply influenced by discussions with the members of the Cambridger Forschungskolloquium, especially Richard Raatzsch. I am also very grateful to the members of the audience at UCD for their resistance to some of the glibber parts of the original version of this chapter.\n\n1. See Alasdair MacIntyre, After Virtue (London, 1981), esp. 103\u201313, 238\u201345.\n\n2. See P. Bourdieu, La distinction (Paris, 1979).\n\n3. See Nietzsche, S\u00e4mtliche Werke: Kritische Studienausgabe, 12:350.\n\n4. One of the central parts of Foucault's analysis of the difference between ancient and Christian forms of ethics is that the former are not necessarily proposed as having relevance to everyone: slaves are not even candidates for leading a good life. See Michel Foucault, L'histoire de la sexualit\u00e9: L'usage des plaisirs (Paris, 1984).\n\n5. Much of what we now call \"ancient religion\"\u2014a usage that already to some extent represents an anachronistic, retrospective construction, which forces a variety of phenomena into a fixed format derived from later monotheistic patterns of thought\u2014did not even aspire to be all-encompassing or tell any kind of truth about the whole world or all of human life. For an almost random selection of relevant works, see Jan Assman, Moses the Egyptian (Cambridge, MA, 1998); Clifford Ando, The Matter of the Gods (Berkeley, 2008); Polyumnia Athanassiadi and Michael Frede, eds., Pagan Monotheism in Late Antiquity (Oxford, 1999); and Moses Halbertal and Avishai Margalit, Idolatry (Cambridge, MA, 1992).\n\n6. The meaning of \"universalist\" might not, of course, be limited to the two components cited, by way of example, in the text. It hardly requires stating, I hope, that discussion of \"universalism\" in this context does not imply any special privileging of Kantian or Habermasian perspectives.\n\n7. Hegel's way of putting this point is usually that religion is a matter of \"Vorstellung\" (\"representation\") rather than \"Begriff\" (\"concept\"), which is the realm of philosophy. Thus, he writes, \"In den Religionen haben die V\u00f6lker allerdings niedergelegt, wie sie sich das Wesen der Welt, die Substanz der Natur und des Geistes vorstellten, und wie das Verh\u00e4ltnis des Menschen zu demselben\" (Hegel, Werke in zwanzig B\u00e4nden, 1:82 [Vorlesungen \u00fcber die Geschichte der Religion I]. The contrast here is that philosophy does not \"represent,\" for instance, the \"essence of the world\" but \"grasps it in a concept\" [\"In der Philosophie wird das Wesen der Welt nicht vorgestellt, sondern begriffen\"]. Even when religion presents \"deep, sublime, speculative thoughts,\" which are not mere images, it presents them as objects of external devotion (\"Andacht\") rather than as argumentatively structured processes that we grasp \"from the inside\" by enacting. Hegel, Werke in zwanzig B\u00e4nden, 18:83\u201392.\n\n8. Hegel, Werke in zwanzig B\u00e4nden, 18:84.\n\n9. Christian theologians have long discussed this issue, and many of these discussions are highly enlightening, even when finally unsatisfactory. The three most important discussions since the eighteenth century seem to me to have been those of Lessing, Hegel, and Kierkegaard. Analogues to some of the problems that arise here have reemerged in Alain Badiou's L'ethique: Essai sur la conscience du Mal (Paris, 1998) and Saint Paul: La fondation de l'universalisme (Paris, 1997). How can what Badiou calls an \"event\" (which like the French Revolution occurred at a particular time and place) have the appropriate kind of universality? Isn't \"fidelity\" to such an event like commitment to the content of a historically specific form of religious revelation? I can merely state here that for various reasons having to do with the detailed structure of his views I do not think this is as serious a difficulty for Badiou as the parallel problem is for advocates of revealed religion. There is a second possible strand of argument here, associated with Nietzsche rather than with Hegel, which emphasises not inherent limitation of the human image-making capacity but the claim that Christianity is committed to specific substantive delusions that it also has a tendency, in the long run, to undermine. Nietzsche was of the opinion that this process of self-destruction was essentially complete by the end of the nineteenth century. For this view, see Nietzsche, Zur Genealogie der Moral, 5:247\u2013412.\n\n10. The most conceptually interesting and enlightening modern discussion of \"sects\" is that of Max Weber. See esp. Wirtschaft und Gesellschaft (T\u00fcbingen, 1972), 721\u201326. I should mention, however, that my usage of \"sect\" deviates slightly from his. I take the term as a quasi-empirical description of a state of affairs in which one group is in fact \"cut off\" (seco\/sectum) from a catholic community or from other groups. For Weber, an elitist rejection of free and open access to the religious community is constitutive for being a \"sect.\" Thus he writes, \"[Eine 'Sekte' ist eine Gemeinschaft].... welche ihrem Sinn und Wesen nach notwendig auf Universalit\u00e4t verzichten... mu\u00df... weil sie ein aristokratisches Gebilde: ein Verein der religi\u00f6s voll Qualifizierten und nur ihrer sein will\" (721).\n\n11. See \"Epilogue: 1953, 1968, 1995: Three Perspectives,\" in Alasdair MacIntyre's Engagement with Marxism, ed. Paul Blackledge and Neil Davidson (Leiden, 2008), 419\u201321.\n\n12. Professor Joseph Raz has proposed a \"perfectionist\" liberalism that is not obviously subject to this particular line of criticism. See his The Morality of Freedom (Oxford, 1986).\n\n13. Isaiah Berlin, Four Essays on Liberty (Oxford, 1969), 131\u201354.\n\n14. MEW Erg. 1.522, 536, 540, etc.\n\n15. Karl Marx, Grundrisse zur Kritik der politischen \u00d6konomie (Berlin, 1953), 387.\n\n16. See Alasdair MacIntyre, A Short History of Ethics (New York, 1967), 177\u201382.\n\n17. On Marx's theory of needs, see Agnes Heller, The Theory of Need in Marx (London, 1976), and Ian Fraser, Hegel and Marx: The Concept of Need (Edinburgh, 1998); see also Lawrence Hamilton, The Political Philosophy of Needs (Cambridge, 2003).\n\n18. MEW Erg. 1.544, 546.\n\n19. Note the almost compulsive use of \"Gegenstand\" and \"gegenst\u00e4ndlich\" in the relevant passage (MEW Erg. 1.539\u201342). The model here is human relation to \"objects.\"\n\n20. MEW Erg. 1.539\u201342.\n\n21. The classic expression of this Humboldtian strand is Die Deutsche Ideologie MEW, vol. 3, see esp. pp. 32\u201334, 74\u201375, 206, and note that Marx was aware of the dangers of \"one-sided\" or fixated development (see MEW 3:238\u201339).\n\n22. MEW Erg. 1.542.\n\n23. Alasdair MacIntyre, \"The Theses on Feuerbach: A Road Not Taken,\" in Artifacts, Representations, and Social Practice, ed. Carol Gould and Robert Cohen (Dordrecht, 1994), 277\u201390.\n\n24. This is the thesis of Theodor Adorno and Max Horkheimer's Dialektik der Aufkl\u00e4rung (Frankfurt\/M., 1969), esp. pp. 9\u201349. See also Adorno, Minima Moralia, \u00a7 100, pp. 206\u20138.\n\n25. See Martin Heidegger, Sein und Zeit (T\u00fcbingen, 1963), \u00a7\u00a7 45\u201353, pp. 231\u201367.\n\n26. Obviously the view presented here is incompatible with another often used metaphor, that of life as a \"journey\" from determinate place A to determinate place B, a metaphor that found its most exquisite development in Dante's Comedy. The Comedy would repay close study in this context, but here I mention only two aspects that seem to me of importance. First, the lives of all those whom Dante encounters are seen as in some way summed up in some single image: Paolo and Francesca buffeted by the unending wind of desire, the sodomites running an eternal race over a desert, Ulysses in his flame. Can every human life really be summed up without remainder in such a single image? Is nothing lost by this? Second, it is not, it seems to me, a merely aesthetic fact that the \"failures\" in Inferno are so much more interesting than the saints in Paradiso.\n\n27. Contra MacIntyre. See After Virtue, esp. chapter 15, pp. 190\u2013209.\n\n28. A further serious problem for any contemporary rehabilitation of Aristotle is the complete absence in his work of any real sense of history as something that matters. He has, to be sure, some trivial remarks about how human inventions have accumulated, some comments about the comparative history of \"constitutions,\" and the odd observation to the effect that human desires develop over time, but none of this amounts to nearly enough to accommodate the modern keen \"historischer Sinn\" that developed in the nineteenth century, an awareness that the past was, to put it very crudely, \"qualitatively\" different from the present. His outline history of previous Greek philosophy in book 1 of Metaphysics is in itself a sufficient testimonial to his irremediably \"presentist\" attitude. It has also always seemed paradoxical to me that Christians could try to find a foundation for their beliefs in an Aristotelian framework. After all, it is central to Christianity not merely that a divine revelation took place at some particular time, thus fundamentally changing the moral universe in which people lived, but also that at some specific point in the past God was incarnated. This is most plausibly interpreted as meaning that the very quality of human time changes.\n\n29. John Dewey and Leon Trotsky, Their Morals and Ours (New York, 1969), 48. To be sure, if one looks at the context within which Trotsky writes this, one will perhaps have doubts about his understanding of \"dialectics.\" He writes: \"From the Marxist point of view the end is justified if it leads to increasing the power of humanity over nature and to the abolition of the power of one person over another.... That is permissible, which really leads to the liberation of humanity. Since this end can be achieved only through revolution, the liberating morality of the proletariat of necessity is endowed with a revolutionary character.... It deduces a rule of conduct from the laws of the development of society, thus primarily from the class struggle, this law of all laws.\" \"Deduction of a rule of conduct\" from any one \"law of all laws\" seems contrary to the spirit of dialectic.\n\n30. Dewey and Trotsky, Their Morals and Ours.\n\n31. See G.W.F. Hegel, Ph\u00e4nomenologie des Geistes in Werke in zwanzig B\u00e4nden, vol. 3; also Marx, MEW Erg. 1, pp. 568\u201388.\n\n4: MUST CRITICISM BE CONSTRUCTIVE?\n\n1. If one wished to multiply distinctions, one could actually distinguish three items here: (a) analysis, (b) judgement, and (c) specifically negative evaluative judgement. For the sake of simplicity I shall refrain from pursuing this here.\n\n2. Max Weber, \"Der Sinn der 'Wertfreiheit' der soziologischen und \u00f6konomischen Wissenschaften,\" in Gesammelte Aufs\u00e4tze zur Wissenschaftslehre (T\u00fcbingen, 1973), 489\u2013540.\n\n3. Leviathan chapter XV.\n\n4. See Stefan Wolle, Der Traum der Revolte: Die DDR 1968 (Berlin: Links, 2008), 194ff. This real incident was given a literary treatment in Christoph Hein's Der Tangospieler (Frankfurt\/M., 1989).\n\n5. What I am saying, then, is that there are at least three distinct things that must be distinguished here: (1) what I (\"in foro interno\") support and what I criticize; (2) the public act of support (or criticism) I think I am performing (\"in foro externo\"); and (3) the public act of criticism (or support) I in fact succeed in performing. Proponents of the theory of speech acts generally tend to conflate (2) and (3) presumably on the grounds that one cannot \"unintentionally\" perform a speech act. First of all, I am not at all sure whether it is right that one cannot in general perform a speech act or something very much like a speech act unintentionally. Can't I insult you without intending to do so, or indeed intending anything at all by my action? In fact isn't it a particularly bruising kind of insult if I fail to have any intentions towards you at all, acting as if you were simply not there? Note here that there is a difference between (a) knowing full well you are there and will be insulted and yet acting as if you were not there and (b) really having no concern one way or another for your presence. The second is no less insulting (in some contexts) that the former. Second, even if I cannot for technical reasons be said to perform a speech act, I can certainly intend to perform a speech act, and even believe while doing it that I am performing it and yet fail to do so, as in the case of Roessler. Roessler was not acting \"unintentionally\"; he simply was actually doing what was, in a very important sense, the very reverse of what he intended to do, criticising the SED rather than supporting it.\n\n6. Theodor Adorno, Negative Dialektik (Frankfurt\/M., 1966), 25\u201327, 200ff., 352\u201364.\n\n7. For a recent discussion, see David Graeber, Debt (New York, 2011).\n\n8. In thinking about all these examples, which are perhaps not ideally suited to illustrate my point, one should abstract from the fact that most humans naturally have fingers and opposable thumbs that can also (in emergencies) be used to transfer food to the mouth.\n\n9. Martin Heidegger, Sein und Zeit, \u00a7\u00a7 14-8 (T\u00fcbingen, 1963), 63\u201389.\n\n10. Ibid., \u00a7\u00a7 46\u201353, 235\u201367.\n\n11. See Karl Marx, Deutsche Ideologie: Feuerbach (MEW I.3), 13\u201377.\n\n12. Richard Wagner, Oper und Drama (Ditzingen, 1994), 132\u2013219. I merely note that one implication of this would seem to be that no performance can be fully successful. In our repressive society works will not be able to performed and apprehended fully; on the other hand, if we were to live in a society fully freed of \"state\" repression, the work would not have the immediate impact and relevance it now has. If we were freed of repression, would we still be gripped? This might suggest that great art is inherently a transitional phenomenon or, as Adorno thought, that the very idea of complete success for a work of art was incoherent.\n\n13. See Andrew Ford, The Origins of Criticism (Princeton, 2002).\n\n14. See Homer, Hymnorum III.166\u201373.\n\n15. It is, of course, slightly ironic, at any rate for someone who adopts the perspective proposed in this essay, that What Is to Be Done? is the title of a novel (N. G. Tschernyschevsky, Was tun? [Berlin, 1986]).\n\n5: THE LOSS OF MEANING ON THE LEFT\n\nThis is a slightly revised version of remarks made at the conference \"Sinnstiftung und Sinnverlust an der Schwelle zum 20. Jahrhundert,\" which took place in Copenhagen in February 2008. I am very grateful to Dr. Karin Wolgast and Dr. Ulrich Knappe for the kind invitation to speak at this conference and for a number of helpful suggestions that enabled me to improve my original draft. I am also very grateful to Richard Raatzsch for a number of extremely illuminating discussions of the topics treated here.\n\n1. Emile Durkheim, Le suicide (Paris, 1897).\n\n2. Friedrich Nietzsche: S\u00e4mtliche Werke, Kritische Studienausgabe, ed. Colli and Montinari, 13:49 (11[99]).\n\n3. Leo Steinberg, Other Criteria (Oxford, 1972).\n\n4. Hugo von Hofmannsthal, Ein Brief in S\u00e4mtliche Werke, Kritische Ausgabe, ed. Ellen Ritter (Frankfurt\/M., 1984), 31:45\u201355.\n\n5. Ovid, Metamorphoses III.343\u2013510, ed. Anderson (Leipzig, 1996). One might think of this story as one of the last expressions of what Nietzsche called \"the tragic world view\" (see Die Geburt der Trag\u00f6die, 1:11\u2013156), which was obliterated by what we know as \"philosophy.\"\n\n6. Note, too, a further anti-Socratic element of the story: Narcissus knows he is being fooled by his own image (\"iste ego sum! nec me mea fallit imago\" [III.463]), but his insight into this truth is utterly without effect.\n\n7. Ovid, Metamorphoses III.454\u201355.\n\n8. Hegel, Werke in zwanzig B\u00e4nden, vol. 13 [Vorlesungen \u00fcber \u00c4sthetik I], p. 51.\n\n9. Ovid, Metamorphoses III.407\u201314.\n\n10. \"Der Knabe... bewundert die Kreise.\" Hegel, Werke in zwanzig B\u00e4nden, 13:51. In a slightly earlier passage on the same page humans are said to wish to \"enjoy\" (genie\u00dfen) in things the external reality of themselves (\"in der Gestalt der Dinge nur die \u00e4u\u00dfere Realit\u00e4t seiner selbst zu genie\u00dfen\").\n\n11. It should be clear that I am not engaging in strict historical exegesis of Hegel's text but am using this passage as an expository device and that my account of it moves over in a way I hope is seamless from more or less direct interpretation of his position to an appropriating use of this text to formulate a set of conditions I think reflective people in Western Europe at the beginning of the twenty-first century would find plausible as descriptions of what they would find \"meaningful.\" This does, however, clearly require a certain amount of pressing on Hegel's text, generally in the direction of a more empiricist reading of it than he would have endorsed. I am especially grateful to Richard Raatzsch for putting this point to me with great clarity and force.\n\n12. \"Le bonheur est une id\u00e9e neuve en Europe,\" in Oeuvres compl\u00e8tes, ed. Michel Duval (Paris, 1984), 715.\n\n13. G.W.F. Hegel, Grundlinien zur Philosophie des Rechts, \u00a7 124Z, in Werke (Frankfurt\/M., 1970), 7:233.\n\n14. Karl Marx, \"Zur Kritik der Hegelschen Rechtsphilosophie: Einleitung,\" MEW 1.379.\n\n15. As Wittgenstein memorably put it in a famous passage from the Tractatus (6.521): \"Die L\u00f6sung des Problems des Lebens merkt man am Verschwinden dieses Problems. (Ist nicht dies der Grund, warum Menschen, denen der Sinn des Lebens nach langen Zweifeln klar wurde, warum diese dann nicht sagen konnten, worin dieser Sinn bestand.).\" See also Terry Eagleton, The Meaning of Life (Oxford, 2007). Since the 1950s philosophers have distinguished two forms of knowledge or \"understanding.\" The first is the discursive or theoretical form of it, which consists in being able to give a prepositional account of the thing in question. I understand the science of mechanics if I can formulate the basic laws of the subject, and explain and use them. This form of knowledge is sometimes simply called \"knowing-that.\" The second kind of understanding is one that expresses itself in certain skillful practical ability to act, that is a form of \"knowing-how.\" So I can say I know how to swim, meaning that if you put me in the water, I won't sink and will be able to move myself in virtually any direction I want. It is an obvious mistake to confuse this practical ability or skill with the mastery of a set of propositions. I can know the theorems that describe human buoyancy without being able actually to swim, and I can know how to swim without knowing any particular proposition.\n\n16. Adorno, Negative Dialektik, 364.\n\n17. Ibid., 364\u201366.\n\n18. See the subtitles of his two volumes: Eingriffe: Neun kritische Modelle (Frankfurt\/M., 1963) and Stichworte: Kritische Modelle 2 (Frankfurt\/M., 1969).\n\n19. Adorno, Minima Moralia, 164\u201367, 266.\n\n20. Adorno, \u00c4sthetische Theorie, 87; Adorno, Minima Moralia, \u00a7 18.\n\n21. \"F\u00fcr den der nicht mitmacht, besteht die Gefahr, da\u00df er sich f\u00fcr besser h\u00e4lt als die anderen.... [w\u00e4hrend er vor diesen nichts voraus hat] als die Einsicht in seine Verstricktheit und das Gl\u00fcck der winzigen Freiheit, die im Erkennen als solchem liegt.\" Adorno, Minima Moralia, \u00a7 6, pp. 22\u201323. (\"The person who does not collaborate runs the risk of thinking he is better than other people.... whereas he has no advantage over them apart from the insight into the fact that he is implicated [in society and its evils] and the happiness which consists in the tiny freedom that lies in cognition as such.\")\n\n22. Friedrich Nietzsche: S\u00e4mtliche Werke, Kritische Studienausgabe, ed. Colli and Montinari, 5:402\u20138.\n\n6: AUTHORITY: SOME FABLES\n\n1. A phenomenon discussed at great length (with special reference to philosophy) by Cicero. See, for instance, his Tusculan Disputations.\n\n2. Dio Cassius 55.3. See the bilingual Greek\/French edition edited and translated by Freyburger and Roddaz (Les Belles Lettres, 1994) or the Greek\/English edition by Cary and Forster (Loeb, Harvard, 1914).\n\n3. Gregory Bateson, Naven (Cambridge, 1936).\n\n4. See R. Heinze, Vom Geist des R\u00f6mertums (Leipzig, 1938), 1\u201324.\n\n5. See G. Agamben, Auzsnahmezustand (Frankfurt\/M., 2004), 88\u2013104.\n\n6. Robert Kaster, Emotion, Restraint, and Community in Ancient Rome (Oxford, 2005), chapter 4, pp. 84ff.\n\n7. Emile Durkheim, Les formes \u00e9l\u00e9mentaires de la vie r\u00e9ligieuse (Paris, 1960).\n\n8. Aristotle, Politica 1259b1.\n\n9. Aristotle on the \u03c3\u03c0\u03bf\u03c5\u03b4\u03b1\u03b9\u03cc\u03c2 in EN 1099a\u2013b.\n\n10. Quintilian, Institutio oratoria, 2 vols. (Oxford, 1973), V.11.36\u201341.\n\n11. See also R. Heinze, Vom Geist des R\u00f6mertums (Leipzig, 1938), 1\u201324.\n\n12. LSJ, s.v.\n\n13. For a simple survey, see A. Lintott, The Constitutions of the Roman Republic (Oxford, 1999).\n\n14. Montesquieu, L'esprit des lois (Paris, 1993).\n\n15. See Andr\u00e8 Magdelain, Ius Imperium Auctoritas (Rome, 1990), 385.\n\n16. See Julian Swann, Politics and the Parlement of Paris under Louis XV, 1754\u201374 (Cambridge, 1995). Thanks to my colleague Michael Sonenscher for the reference to this.\n\n17. \u00c9mile Benveniste, Le vocabulaire des institutions indo-europe\u00e9nnes, vol. 2, S. 149 (Paris, 1969).\n\n18. For some parallel discussion of colonialist foundations, see essay 11 in this volume.\n\n19. Freud is particularly good on the relation between real and imaginary \"success\" (and \"failure\") in some of these early Foundation myths. See S. Freud, Studien-Ausgabe (Frankfurt\/M., 1974), vol. 9.\n\n20. See essay 11 in this volume.\n\n21. End of Book I of Iliad.\n\n22. Hesiod, Theogony.\n\n23. Nietzsche, Zur Genealogie der Morale, 5:327\u201329.\n\n24. Theognis 53\u201358; Nietzsche, Zur Genealogie der Moral, 5:257\u201389.\n\n25. Raymond Geuss, History and Illusion (Cambridge, 2001).\n\n26. See R. Raatzsch's excellent Autorit\u00e4t und Autonomie (Munster, 2007).\n\n27. It is, of course, anything but obvious what counts as a \"commercial context.\" Anthropologists have shown that in most \"premodern\" societies \"commercial transactions\" did not take place in the kind of abstract, self-contained sphere modern economists tend to assume must always have existed. See the recent marvelous Debt by Graeber.\n\n28. See essay 2 in this volume.\n\n29. See also R. Geuss, \"Wer das Sagen hat,\" Mittelweg 36 (December 2011\/January 2012): 1\u201310.\n\n7: A NOTE ON LYING\n\n1. Iliad VIII.312ff.\n\n2. See also Nietzsche on \u03c0\u03bf\u03bd\u03b7\u03c1\u03cc\u03c2, Zur Genealogie der Moral, 5:272.\n\n3. See esp. M. Foucault, Le courage de la v\u00e9rit\u00e9 (Paris, 2009).\n\n4. Compare the experience of Evans-Pritchard with a population in the Sudan who exhibit a very different attitude toward truth-telling (as cited in Graeber, Debt, 96\u201397).\n\n5. The Portable Hannah Arendt, ed. with introduction by Peter Baehr (New York, 2000), 563.\n\n6. See Herbert Marcuse, \"Repressive Tolerance,\" in Critique of Pure Tolerance (Boston, 1965).\n\n7. If Arendt had been paying attention to Heidegger in the right way in the 1930s, she could have learned this from him. See Martin Heidegger, Einf\u00fchrung in die Metaphysik (T\u00fcbingen, 1957; originally lectures in 1935), esp. pp. 112ff.\n\n8: POLITICS AND ARCHITECTURE\n\n1. See Von der Gr\u00fcneburg zum Campus Westend, brochure by University of Frankfurt (2007) to accompany the permanent exhibition in the I. G.-Farben-Haus. I am also grateful to Professor Axel Honneth of the Philosophisches Institut of the University of Frankfurt for discussion of some of these issues.\n\n2. EN 1094a\u20131095a, 1141b; Metaphysica 980a\u2013981b.\n\n3. Nietzsche, S\u00e4mtliche Werke, 1:875\u201390, 5:15\u201339, 6:88\u201397, and passim.\n\n4. Max Weber, Politik als Beruf (Berlin, 1977).\n\n5. This line of thought has been very fruitfully developed by Zeev Emmerich in recent, as yet unpublished work, and in general I am very indebted to him for discussions of the topics in this chapter.\n\n6. For what is still one of the most interesting discussions of this topic, see Friedrich Schiller, \u00dcber die \u00e4sthetische Erziehung des Menschen in einer Reihe von Briefen in S\u00e4mtliche Werke, ed. Fricke and G\u00f6pfert (Munich, 1967), vol. 5.\n\n7. Michel Foucault, Surveiller et punir (Paris, 1975).\n\n8. EN 1129b\u201330.\n\n9. John Rawls, A Theory of Justice (Cambridge, MA, 1971).\n\n10. See MEW, 19:18\u201322; Erg\u00e4nzungsband 1, pp. 534\u201335.\n\n11. Karl Marx, Grundrisse (Berlin, 1974), 5\u201319.\n\n12. See ibid., 387\u201388.\n\n13. See Hegel, Werke in zwanzig B\u00e4nden, 14:266\u201372.\n\n14. See Nietzsche, S\u00e4mtliche Werke, 3:524\u201325.\n\n15. Stuart Hampshire, Thought and Action (London, 1970) and Justice Is Conflict (London, 1999).\n\n9: THE FUTURE OF THEOLOGICAL ETHICS\n\n1. Hegel, Werke in zwanzig B\u00e4nden, 18:496\u2013516.\n\n2. For instance, Euthyphro 15e.\n\n3. Symposium 223d.\n\n4. For instance, Condorcet. In his speech Der Meridian, Paul Celan describes Georg B\u00fcchner's play Dantons Tod, which is set in the period of the Terror during the French Revolution. Celan describes a scene in which various characters, associates of Danton, discuss the nature of art. Such discussions, Celan says, can be continued indefinitely \"unless interrupted\" (wenn nichts dazwischenkommt). However, in this case\u2014and Celan's discussion suggests that this is the archetypical case\u2014something does interrupt (es kommt was dazwischen) the discussion, the arrest of the participants, who are tried and guillotined. See P. Celan, Der Meridian und andere Prosa (Frankfurt\/M., 1988).\n\n5. 1 Cor. 1:19.\n\n6. See K. Marx, Das Kapital, vol. I.I.1.3 (\"Die Wertform oder der Tauschwert\").\n\n7. J. Dewey and L. Trotsky, Their Morals and Ours (New York, 1973).\n\n8. Matt. 5:22.\n\n9. Personal communication with Richard Rorty.\n\n10. Gal. 3:26\u201329. See also A. Badiou, Saint Paul: La fondation de l'universalisme (Paris, 1997).\n\n11. Rom. 12:2; John 1:12.\n\n12. See A. Dihle, The Theory of the Will in Classical Antiquity (Berkeley, 1982); M. Frede, A Free Will (Berkeley, 2011); Bernard Williams, Shame and Necessity (Berkeley, 1993), esp. chapter 2.\n\n13. This discussion follows Frede, A Free Will.\n\n14. I merely note that Aristotle also has no concept of \"evil,\" although he does have a concept of bad. This is probably connected to the absence of \"will\" in his thinking. If one thinks one cannot do without the concept of \"evil\" one would have to try to reconfigure it without reference to a \"will.\" I have some suggestions in that direction in my \"The Future of Evil,\" in Essays on Nietzsche's \"Genealogy of Morality,\" ed. S. May (Cambridge, 2011).\n\n15. If Feuerbach and Marx are right and religions are forms of imaginary compensation for human suffering and the lack of a clear perspective for dealing with them realistically, then one would expect religious belief and practice to increase as Western economies increasingly falter, just as the collapse of the political movements for social change in the West in the mid-1970s led to a resurgence of religion.\n\n10: DID WILLIAMS DO ETHICS?\n\nI am grateful to Damian Freeman, Istvan Hont, Tom Stern, and especially to Robert Pippin for discussions and comments on an earlier version of this chapter.\n\n1. Zmith somewhat spoiled the effect by drinking Diet Coke during the meeting. It turns out that it is very hard to feel intimidated by someone who drinks Diet Coke. One thinks of Cicero's remark about Julius Caesar: \"Sometimes I think he is a serious danger to the Republic, but then I observe the way he curls his sparse forelock around his finger and I think, 'Can such a man be a danger?' \"\n\n2. Leon Trotsky, John Dewey, et al., Their Morals and Ours (New York, 1973).\n\n3. EN 1094\u201395. I am greatly indebted to Istvan Hont for invaluable assistance, particularly for discussion of the material in this paragraph. See Bernard Williams, \"In the Beginning Was the Deed,\" in In the Beginning Was the Deed: Realism and Moralism in Political Argument, ed. G. Hawthorn (Princeton, 2005), 18\u201329.\n\n4. See Apology.\n\n5. Jean-Jacques Rousseau, \u00c9mile; ou De l'\u00e9ducation, ed. F. Richard and P. Richard (Paris, 1964), 223. See also Hegel, Grundlinien zur Rechtsphilosophie, \u00a7 126 Z.\n\n6. Speech at the Convention, December 3, 1792.\n\n7. Sven Lindqvist, Exterminate all the Brutes! (Granta 1996), 149.\n\n8. Christine Korsgaard et al., The Sources of Normativity (Cambridge, 1996), 217\u201318.\n\n9. Adorno, Minima Moralia, \u00a7 18.\n\n10. Some of the material from chapter later appeared in \"Wagner and the Transcendence of Politics,\" in Bernard Williams, On Opera (New Haven, 2006), 70\u201389.\n\n11. John Deathridge, Wagner: Beyond Good and Evil (Berkeley, 2008), 52. The use of the term \"hollow\" with reference to this particular passage is so frequent in the literature as to be almost a clich\u00e9. See also Michael Tanner, The Faber Pocket Guide to Wagner (London, 2010), 154 (\"stunningly grand and hollow\"); also Deryck Cooke, I Saw the World End: A Study of Wagner's \"Ring\" (London, 1979): \"this 'triumphant' ending is a temporary and hollow one\" (238).\n\n12. Warren Darcy, Wagner's \"Das Rheingold\" (Oxford, 1993), 208\u201314.\n\n13. Richard Wagner, Mein Leben, vol. 2 (Munich, 1911).\n\n14. Nietzsche, Die Geburt der Trag\u00f6die, \u00a7 24.\n\n15. T. W. Adorno, Versuch \u00fcber Wagner (Frankfurt am Main 1974, c. 1952).\n\n16. Adorno, Minima Moralia, \u00a7 18.\n\n17. Briefwechsel, 1928\u20131940: Theodor W. Adorno, Walter Benjamin, ed. Henri Lonitz (Frankfurt am Main, 1994), 370.\n\n18. Williams, Shame and Necessity, 163.\n\n19. Ibid.\n\n11: THE WISDOM OF OEDIPUS AND THE IDEA OF A MORAL COSMOS\n\nMy thanks to Peter Agush, Lorna Finlayson, Hilary Gaskin, Chris Kassam, Domenic O'Mahony, Richard Raatzsch, Tom Stern, and Paul Woodruff for exceedingly helpful discussions of the topics of this chapter, as well as to the members of the Literature Seminar in the Faculty of Classics at Cambridge for many constructive suggestions.\n\n1. The only other major works of modern philosophy I can think of that end in this way are Hegel's Ph\u00e4nomenologie des Geistes and Heidegger's Einf\u00fchrung in die Metaphysik. That Williams's book originated as a series of lectures to a Classics Faculty goes some way towards making it less peculiar that the poet cited is Pindar, rather than Schiller or H\u00f6lderlin, or some appropriate English-language poet, but this still doesn't really explain why discursive prose gives way at the end to a metrically expressed image. Ending with a bit of poetry might also be expected to be associated with a cognitive and hermeneutic relaxation at the end of a difficult work, but that is clearly not the case here.\n\n2. Republic 607b.\n\n3. Williams, Shame and Necessity, 167.\n\n4. I intentionally say that we would distinguish between historical and mythical tales because, of course, Pindar himself shows no sign of making any such distinction, and, as Williams points out in a well-known chapter of Truth and Truthfulness (Princeton, 2002, 149\u201372), he really couldn't have made such a distinction because there really was no such thing as \"proper\" history until the work of Thucydides, a generation after Pindar's death.\n\n5. We are relatively well informed about the history of Cyrene because of book 4 of Herodotus's Histories.\n\n6. Nietzsche, Die Geburt der Trag\u00f6die, \u00a7 3 in S\u00e4mlitche Werke, 1:35.\n\n7. All of the examples in this paragraph are from Diogenes Laertius (lives of Thales, Chilon, Pythagoras).\n\n8. Anthologia lyra graeca, ed. Diehl.\n\n9. \"There are serious doubts in fact about whether Pythagoras ever existed\" means what such statements usually mean with reference to the ancient world. That is, there is a traditional body of lore about a particular person, attributing to that person some signal achievements. To say that that person did not exist is to say that there was not a single person with that name and those properties who accomplished all those things. Thus \"Homer did not exist\" does not mean that there was never any Greek named Homer who wrote some poems but that the Iliad and the Odyssey were not the work of a single blind poet of that name who lived in the eighth century BC in Ionia. This, of course, does not mean than no one wrote the Iliad. See Martin West, The Making of the Iliad (Oxford, 2012).\n\n10. Martin Heidegger, \"Die Zeit des Weltbildes,\" in Holzwege (Frankfurt\/M., 1963). Heracleitos has a very strong claim to being a major participant in this story. See his Fragment 30 Diels\/Kranz Fragmente der Vorsokratiker, 6th ed. (Zurich, 1951), 157.\n\n11. Williams, Shame and Necessity, 164.\n\n12. He does this at the beginning of his essay on Sophocles' Trachiniae in The Sense of the Past (Princeton, 2006), 49ff.\n\n13. On \"hope,\" see Geuss, Outside Ethics, 224\u201325. However, see also Michael Theunissen, Pindar: Menschenlos und Wende der Zeit (Beck, 2008), 307\u201399.\n\n14. Friedrich Nietzsche, G\u00f6tzen-D\u00e4mmerung' Streifz\u00fcge eines Unzeitgem\u00e4ssen, \u00a7 1, in S\u00e4mtliche Werke, 6:111. Further judgements about Seneca's style: reading him is \"like dining on nothing but anchovy sauce\" (nineteenth century); he \"writes as a Boare does pisse, scilicet by jirkes\" (seventeenth century), both cited in edition of de otio and de brevitate vitae, ed. G. D. Williams (Cambridge, 2003), 26.\n\n15. Kant, Kritik der praktischen Vernunft (at the end of book 1 of part 1). I can assure those who do not read German that the original is even more unctuous and rebarbative than this suggests. Kant is one of the few major philosophers whose style translators generally improve. In the original the full sentence from which this segment is taken runs to over a dozen lines and constitutes a full paragraph.\n\n16. Pindar, Ol. 11.8ff., Ol. 2.147ff., P. 4.248ff., etc.\n\n17. Scholia vetera in Pindari carmina (Stuttgart, 1997), ed. Drachmann, vol. 2.\n\n18. Williams, Shame and Necessity, 217n64.\n\n19. Obviously the description of the stripped oak log reminds one of Il. I.233ff. where Achilles swears on a sceptre made of just such a stripped oak branch that the Greeks will one day come to realise that they need him.\n\n20. If there is as much to be said for the pre-philosophical view as this essay argues, it would not be surprising to find, especially in certain early post-Socratic philosophers, traces of a recognition of those aspects of the world to which Sophocles, Pindar, and Thucydides are most sensitive. In Aristotle this takes the form of his recognition of \"the fate of Priam\" as a perpetual possibility (EN 1099b\u20131101b).\n\n21. Martin Heidegger, Kant und das Problem der Metaphysik (Frankfurt\/M., 1951).\n\n22. Basil L. Gildersleeve, Pindar: Olympian and Pythian Odes (Cambridge reprint of edition of 1885), 301; Bruce Braswell, A Commentary on the Fourth Pythian Ode of Pindar (Berlin, 1988), 365.\n\n23. Pindar is in fact an archetypical example of one aspect of what Nietzsche called a \"master morality\" in contrast to a \"slave morality.\" The first is focused on praising positive achievement while simply ignoring or passing over in silence failure; the second is centrally concerned to censure vice. Friedrich Nietzsche, Jenseits von Gut und B\u00f6se, \u00a7\u00a7 257\u201396 in S\u00e4mtliche Werke, 5:205\u201340 and Zur Genealogie der Moral' Erste Abhandlung, \u00a7\u00a7 1\u201317 in S\u00e4mtliche Werke, 5:257\u201389.\n\n24. Bernard Williams, \"Moral Luck,\" in his Moral Luck (Cambridge, 1981), 20\u201340. Roger Crisp has pointed out (private communication) that the \"Gaugin\" who figures in Williams's essay is really a character who appears in S. Maugham's novel The Moon and Six-Pence, not the real painter of that name.\n\n25. We don't know what exact version of the Oedipus story Pindar, who after all was himself a local boy from Thebes, presupposes here. The early history of the story is extremely complex and obscure, and Greeks in general allowed themselves great freedom in mixing motifs, adding or dropping details of the old stories, or for that matter simply making things up. Thus in various older versions of the story, such as that in Homer, Oedipus is not presented as having been driven out of Thebes but continues to rule there, and in many of the extant versions of the story before it was taken over by the Athenian dramatists Oedipus's two sons, Polyneikes and Eteocles, are not the sons of Oedipus's wife\/mother but, apparently, of a second wife named Eurygeneia. So one must be careful about projecting back into Pindar details derived from other or later treatments of the Oedipus myth, such as that of Sophocles, which Pindar, given that he was about twenty years older than Sophocles, is unlikely to have known. There is only one other place in the extant work in which Pindar speaks of Oedipus, and that is the Second Olympian, which merely states that he killed his father and that a Fury caused his own sons to kill each other. The only further possible reference to a story about Oedipus occurs in one of the fragments where there is a reference to a \"young woman's riddle savage jaws,\" which might possibly refer to the Sphinx. See Jebb's introduction to his edition of OT.\n\n26. Actually one did survive: Thersandros. See Pindar, Ol. II.47.\n\n27. It is also true and important that no one was forced to compete; we do not in the same sense volunteer to be born and live.\n\n28. Even if the order in the world is now underwritten by Zeus, if Hesiod is to be believed it is still a fragmentary, local, and constructed order. The order is not in any way natural, necessary, or metaphysically grounded but is imposed by Zeus after a war in which he must overturn the old order completely and overcome the titans by military force. The defeated Titans still rage underground. See Hesiod, Theogonia ll. 617\u2013719; Pindar, First Pythian, ll. 13\u201328.\n\n29. Bernard Williams, \"Conflicts of Values,\" in The Idea of Freedom, ed. A. Ryan (Oxford, 1979).\n\n30. Bernard Williams, Ethics and the Limits of Philosophy (Cambridge, MA, 1985), chapter 8.\n\n31. See p. 221 [Eighth Pythian 97].\n\n32. See also Geuss, \"Thucydides, Nietzsche, and Williams,\" 219\u201333.\n\n12: WHO WAS THE FIRST PHILOSOPHER?\n\n1. Hegel construed philosophy as a form of what he called \"absolute spirit,\" that is, as something that inherently could not be understood as a way in which a social group reflected on itself. See G.W.F. Hegel, Enzyklop\u00e4die der philosophischen Wissenschaften, \u00a7\u00a7 552\u201377, in Werke in zwanzig B\u00e4nden, 353\u201395.\n\n2. Friedrich Schlegel, Athen\u00e4umsfragmente, Fragment 112 in Friedrich Schlegel, Werke in zwei B\u00e4nden (Berlin, 1980), 1:204.\n\n3. See Friedrich Nietzsche, Die Philosophie im tragischen Zeitalter der Griechen in S\u00e4mtliche Werke, 1:799\u2013872.\n\n4. Friedrich Nietzsche, Zur Genealogie der Moral: Zweite Abhandlung, \u00a7\u00a7 12-3, in S\u00e4mtliche Werke, 5:313\u201318.\n\n5. Nietzsche, Zur Genealogie der Moral: \"Vorrede \u00a7 7\" in S\u00e4mtliche Werke, 5:254ff.\n\n6. Ibid., 5:275.\n\n7. Richard Wagner, Oper und Drama (Reclam).\n\n8. As everyone who has studied ancient theatre knows, there was also an earlier play, The Sack of Miletos by Phrynicus, on a (painful) topic of recent Greek history. Wagner was rather widely read in ancient literature\u2014he apparently sometimes read Aeschylus aloud to his family in the evenings\u2014but I don't know whether he would have been aware of this.\n\n9. The question that this immediately raises is why it would not be possible to turn \"Henry V\" into the same kind of mythical figure as \"Xerxes\" or \"Socrates,\" or rather whether that is not just what Shakespeare tried to do.\n\n10. Schiller, \u00dcber die \u00e4sthetische Erziehung des Menschen in einer Reihe von Briefen.\n\n11. See essay 11 in this volume.\n\n12. Ignoring for the moment the complication that would be introduced by the wings.\n\n13. Nietzsche, S\u00e4mtliche Werke, 7:460ff.\n\n14. Ibid.\n\n15. Friedrich Nietzsche, Also sprach Zarathustra: Zarathustra's Vorrede, \u00a7 6, in S\u00e4mtliche Werke, 4:S.21ff.\nIndex\n\nABC of Communism, The,\n\naccount, give an, , , \u2013, , \u2013, , , , , , , , , , , \u2013, \u2013\n\nAchilles, , , , , 254n19\n\nAdmetus,\n\nAdorno, Theodor, , , , , , \u2013, \u2013, \u2013,\n\nAeneas, \u2013, ,\n\nAeneid, \u2013, 241n21\n\nAeschylus, ,\n\naesthetic, the, , , , , , , , , , , , , , , , \u2013,\n\nAgamemnon, , ,\n\nAgincourt,\n\nAjax,\n\nAlcestis, ,\n\nAlcibiades,\n\nalienation, \u2013\n\nambiguity, x, \u2013, , , , , , \u2013,\n\nanarchism, , , , \u2013,\n\nAnchises, ,\n\nAngelus Silesius, ,\n\nanomie,\n\nAntigone,\n\nAntigone,\n\nAntilochus,\n\napologetics,\n\nappearance and reality, \u2013\n\nAquinas, Thomas,\n\narchitecture, , \u2013\n\narchive, \u2013\n\nArendt, Hannah,\n\nArgonauts, the,\n\nargumentation, xii, , , , , \u2013, , , , \u2013, , \u2013, \u2013, . See also rationcination; reason\n\nAristophanes, xii, ; Acharnians, ; Birds, ; Lysistrata, ; Peace,\n\nAristotle, xi, , , \u2013, , \u2013, , , , , , \u2013, , \u2013, , ,\n\nArkesilaos, \u2013, \u2013, ,\n\nArmy, U.S., \u2013\n\nart, , , , , , , , , \u2013, \u2013, , \u2013, , , ,\n\naspirations, x, , , , , ,\n\nAugustine, St., xiii\n\nAugustus, Emperor, , ,\n\nAuschwitz,\n\nauthority, x\u2013xvi, \u2013, , \u2013, \u2013, \u2013,\n\nAutolykos, ,\n\nautonomy, as property of a discipline or realm of human life, , , , , \u2013,\n\nautonomy, as property of a human individual or group, xii\u2013xv, \u2013, \u2013, 237n4, 237n9. See also freedom\n\nAzande, 237n11\n\nBadiou, Alain, 243n9\n\nBakunin, Mikhail, ,\n\nBaudelaire, Charles,\n\nbeans, avoidance of, \u2013\n\nBeckett, Samuel,\n\nBellerophontes, , , 238n4\n\nbelles lettres, les, ,\n\nBenjamin, Walter, \u2013\n\nBentham, Jeremy,\n\nBenveniste, \u00c9mile \u2013,\n\nBerkeley, University of California at,\n\nBerlin, Isaiah, ,\n\nbest, the. See good, the\n\nbestiality,\n\nBirth of Tragedy, The, ,\n\nBlair, Tony, , ,\n\nBrecht, Bertold, ,\n\nBrief des Lord Chandos, Ein,\n\nBrown, Gordon,\n\nBr\u00fcnnhilde,\n\nB\u00fcchner, Georg,\n\nBukharin, Nikolai,\n\nCalvinism, \u2013, ,\n\nCambridge Music Faculty,\n\nCambridge University Orchestra,\n\ncamps, concentration, \u2013. See also Auschwitz\n\ncapitalism, , \u2013, , , , , \u2013, \u2013,\n\n\"Carthago delanda est,\" ,\n\nCatholicism, Roman, , \u2013\n\nCato, \u2013\n\nCelan, Paul, \u2013, , 251n4\n\nchoice, , , \u2013, , , , \u2013, , , ,\n\nChristianity, \u2013, , \u2013, , \u2013, , , , , , \u2013, , \u2013, ,\n\nchurch, x\u2013xi, \u2013,\n\nCIA,\n\nCicero, , \u2013, , , ,\n\nclarity, , , \u2013, \u2013,\n\nClaudius, Emperor,\n\nCleisthenes,\n\ncoercion, xi, xiv, , , , , , , \u2013, \u2013, , \u2013,\n\ncoherence, , , , , \u2013, , , , , \u2013, , , , \u2013\n\ncomedy, ,\n\nConfucius,\n\nconsistency, xi, , , , ,\n\nconstellation, \u2013\n\nconsumer goods, \u2013\n\nconsumerism,\n\nconsumers, , ,\n\nconsumption, , , , , \u2013, , , \u2013, , . See also production\n\ncontext, ix, xiv, \u2013, , , \u2013, \u2013, , , , , \u2013, , \u2013, , , \u2013, , , , , \u2013, , , , , , \u2013, 249n27\n\ncosmos, , , \u2013\n\nCreon,\n\ncriticism, xiv, xvi, \u2013, , , , , \u2013, , , , , ,\n\n\u010cSSR, ,\n\nCynics, 237n8\n\nCyrene, ,\n\nDamophilus, , , ,\n\nDante, , 244n26\n\nDavidson, Donald,\n\nDDR, \u2013\n\nDeathridge, John,\n\nDeleuze, Gilles,\n\ndemocracy, xiii, \u2013, , , ,\n\ndemocratic theory,\n\nDerrida, Jacques,\n\nDescartes, Ren\u00e9,\n\ndesirable, , , \u2013\n\ndesires, x\u2013xi, , , , \u2013, , , , , . See also preference\n\nDewey, John, xii, , 237n3\n\nDido,\n\nDio Cassius,\n\nDiogenes Laertius, \u2013\n\nDiomedes, \u2013,\n\nDionysius of Halicarnassos,\n\ndiscipline, x\u2013xii, xvi, , , ,\n\ndisciplines, \u2013, , , , , \u2013, , ,\n\nDucasse, Isidore,\n\nDurkheim, Emile,\n\nDyckerhoff & Co.,\n\neconomic development, , , \u2013,\n\neconomic power, , ,\n\neconomics, , , \u2013\n\neconomic structure, x, , , , , , , , \u2013, , , ,\n\neconomic transaction, \u2013,\n\nefficiency, ,\n\nEisenhower, Dwight, \u2013\n\nEluard, Paul,\n\nends and means, \u2013\n\nEnlightenment, the, xv\u2013xvi, , , , 241n18\n\nEnnius, ,\n\nEpicureans,\n\nethical ideals, , ,\n\nethical interests, ,\n\nethical life, , ,\n\nethical shadow,\n\nethical theorizing, ,\n\nethical thinking, xii, , ,\n\nethics, xii\u2013xv, \u2013, , , \u2013, \u2013, , , , , , \u2013, , , , \u2013, , 242n4; Christian, , ; philosophical, , , , ; religious, ; theological, \u2013. See also moral philosophy\n\nEthics and the Limits of Philosophy, \u2013\n\nEuripides: Bacchae, ; Hercules Furens, : Hippolytus,\n\nEuthyphro, \u2013\n\nEvans-Pritchard, E. E., 237n11\n\nevil, , \u2013, , 251n14\n\nfailure, , , , \u2013, \u2013, , , , , , , , \u2013, \u2013, , \u2013, , 249n19\n\nfairy tale, . See also just-so story; myth\n\nfaith, , , , . See also hope\n\nfate, , , , , , , , , , , 254n20\n\nFeuerbach, Ludwig, , , , 237n1, 251n15\n\nFoucault, Michel, xv, , , 238n15\n\nFrankfurt, University of, \u2013 Frankfurt Conference on Adorno, \u2013\n\nFrankfurt School, \u2013, \u2013\n\nFrede, Michael, 237n5\n\nfreedom, xii\u2013xiv, xvi, , , , , \u2013, , , \u2013, , , \u2013, , , , , , , \u2013, , \u2013, , , 237n5. See also autonomy; liberty\n\nfree market, , ,\n\nFrege, Gottlob, \u2013\n\nFreiburg\/Br., Classics Faculty at University of,\n\nFrench Revolution, , 251n4\n\nFreud, Sigmund,\n\nFuneral Music, Siegfried's, \u2013,\n\nGauguin, Paul,\n\ngenealogy, \u2013, , ,\n\nGenealogy of Morality, The,\n\nGlaukos, \u2013, , ,\n\ngod(s), x, xiv, , \u2013, \u2013, \u2013, \u2013, , , \u2013, , , , , , , , , , , , , , ; Abraham's, ; Christian, , , , , \u2013, , , ; chthonic, ,\n\ngods, Greco-Roman: Apollo, \u2013, , , ; Athena, , ; Death, ; Dionysus, ; Hermes, \u2013; Olympian, , ; Penates, ; Venus, \u2013; Zeus, , 255n28\n\ngods, Wagner's: Loge, ; Wotan, ,\n\nGoethe, Johann Wolfgang von,\n\ngood, the, xiii, , , , , \u2013, , \u2013, , , , \u2013, \u2013, , \u2013, , , , \u2013, , \u2013, \u2013, , , , \u2013\n\nGorgias,\n\ngospels, \u2013, ; Christian, xi, xv, , , ,\n\nGraeber, David, 237n2\n\nGuatarri, F\u00e9lix,\n\nHabermas, J\u00fcrgen, ,\n\nHague, The,\n\nHartmann, Martin, 238n13\n\nHector,\n\nHegel, G.W.F., xv, , , , , , , , , \u2013, , , , \u2013, , , , , , , , , 239n18\n\nHeidegger, Martin, , , , , ,\n\nHeilsgeschichte,\n\nHenry V, \u2013\n\nHeracles, , ,\n\nheresy, \u2013\n\nHerodotus, xii, , 237n6\n\nheroism, , , , , \u2013, ,\n\nHesiod, \u2013, , , 255n28\n\nHistorical School, the,\n\nhistory, , , \u2013, \u2013, \u2013, , , , , , \u2013, , , , \u2013, , , , , \u2013, 239n15, 244n28, 253n4\n\nHobbes, Thomas,\n\nHofmannsthal, Hugo Von,\n\nH\u00f6lderlin, Friedrich, ,\n\nHomer, , , , ,\n\nhope, , , , , , , , , . See also faith\n\nhumanities, the, , , \u2013\n\nhypocrisy,\n\nideology, x, , , , , \u2013, ,\n\nI. G. Farben, \u2013\n\nIliad, \u2013, , \u2013, , 254n9, 254n19\n\nimagination, \u2013, , , \u2013, , , , , , 251n15\n\nimperium, xiii, , ,\n\ninterests, , , , , , , , , , , , \u2013, , , \u2013, \u2013, , , , \u2013,\n\ninterpretation, xiii\u2013xiv, , \u2013, \u2013, , , , \u2013, \u2013, ,\n\nIon,\n\nJesuits,\n\nJesus, , ,\n\nJohnson, Samuel,\n\nJudas,\n\njustice, , , \u2013, ; as fairness,\n\njust-so story, . See also fairy tale; myth\n\nKant, Immanuel, xiii\u2013xv, \u2013, \u2013, , , , , , , , , , \u2013, , , 238n14\n\nKierkegaard, S\u00f8ren,\n\nknowledge, x\u2013xi, xiii, , , , , , , , , \u2013, , , , , , , \u2013, \u2013, , , , , , 248n15\n\nK\u00f6hnke, K.laus 238n2.\n\nKorsgaard, Christine,\n\nKraus, Karl,\n\nlabour, , , . See also production; work\n\nLabour, Old. See Old Labour\n\nlaw, xiv, , , , , \u2013, , , \u2013, \u2013, , , , \u2013, , , , , , law school,\n\nlawyer, \u2013,\n\nLear, King, \u2013\n\nlegal code. See law\n\nlegal system. See law\n\nLehre,\n\nLenin, Vladimir Ilyich,\n\nLevi, Primo, ,\n\nliberalism, \u2013, , \u2013, ,\n\nliberty, xiii, , 237n7. See also freedom life: architect's, , ; bean-free, ; everyday, , , \u2013, , ; form\/way of, , \u2013, \u2013, , , \u2013, , , , , , \u2013, , \u2013, ; good\/ethical\/moral, , , , , , , , , \u2013, , , ; human, x\u2013xi, xiv\u2013xvi, , , , , , , , \u2013, \u2013, \u2013, \u2013, , \u2013, , , , , , \u2013, , \u2013, , , \u2013, , , , \u2013, \u2013, , , 244n26; inner, \u2013; meaningful, , \u2013, , \u2013; religious, ; social and political, xi, xv, , \u2013, , , , \u2013, , , \u2013,\n\nLivy,\n\nLocke, John,\n\nLucretius,\n\nlying, \u2013,\n\nLysias,\n\nMacArthur, Douglas,\n\nMacIntyre, Alasdair, \u2013\n\nMalraux, Andr\u00e9,\n\nmana, \u2013\n\nMandela, Nelson,\n\nMarx, Karl, , \u2013, , , \u2013, , \u2013, , , , 237n1, 251n15\n\nMarxism, \u2013, ,\n\nMay ,\n\nmeaning, , , \u2013, ,\n\nmeaningfulness, , , \u2013, , , , , \u2013, 247n11\n\nmeaninglessness, , ,\n\nmeans. See ends and means\n\nMenelaus,\n\nMenger, Carl,\n\nMetaphysics, Aristotle's,\n\nMiddle Passage, the,\n\nMill, John Stuart,\n\nMime,\n\nMinima Moralia,\n\nMontesquieu,\n\nmoral authority, xii, , , , \u2013\n\nmoral collapse,\n\nmoral cosmos. See cosmos\n\nmoral development, xv, \u2013, , \u2013, \u2013, , , 245n29\n\nMoralit\u00e4t,\n\nmorality, , , , , , , , 255n23\n\nmoralization, , , \u2013,\n\nmoral judgement, ,\n\nmoral philosophy, \u2013, , , ,\n\nmoral values, xiii\u2013xv, , , , , , , , , , , , , \u2013, ,\n\nMrs. Dalloway,\n\nMus\u00e9e de l'Homme,\n\n\"must,\" the philosophical, xi, xiv, , 238n12\n\nmyth, , , , , \u2013, 253n4, 256n9. See also fairy tale; just-so story\n\nNarcissus, \u2013\n\nnarrative, \u2013, , , , , , , \u2013\n\nnecessities, ix\u2013xi, \u2013, , , \u2013\n\nneeds, ix\u2013xi, , \u2013, , , \u2013, , , , , \u2013, \u2013, , , \u2013, \u2013, \u2013, , , , , , , , 237n1\n\nNegara,\n\nNero, xii\n\nNestor,\n\nNicomachean Ethics, \u2013, ,\n\nNietzsche, Friedrich, , , , \u2013, , , , , , , , , , , , , , \u2013,\n\nnihilism, , ,\n\nNikias, ,\n\nNineveh, ,\n\nnormativity, x\u2013xi, , , , , , , , , , , \u2013, , , , , \u2013,\n\nobjective, , , , , , \u2013, , ,\n\nobjectivity, \u2013, ,\n\nobscurity, . See also clarity\n\nobservation, , , , , , , , , , ,\n\nOdysseus, \u2013,\n\nOdyssey, \u2013,\n\nOedipus, , , , , , \u2013, , , \u2013, 255n25; wisdom of, \u2013, \u2013\n\nOld Labour,\n\noptimism, ,\n\noracles, xiv, \u2013, , , 237n11, 241n21\n\norder, good, \u2013\n\norigins, , \u2013, , , \u2013\n\nOstwald, Martin, 237n9\n\noutspokenness,\n\nOvid, \u2013\n\nParmenides,\n\nPascal, Blaise,\n\nPatroklos,\n\nPaul, St., xi, , , , 240n23\n\nPavarotti, Luciano,\n\nPentheus,\n\nPersians, The, \u2013\n\npessimism, ,\n\nphilosophy, xii, , \u2013, , , , , , \u2013\n\nPhokulides, \u2013,\n\nPindar, , \u2013, , , , ;\n\nFragment , \u2013; Fragment , ; Fragment , ; Eighth Pythian, \u2013, ; Fourth Pythian, \u2013; Olympian I,\n\nPlato, xii, \u2013, , , , , , , , , , , , ; Charmides, ; Ion, , \u2013; Phaedo, , ; Protagoras, \u2013; Republic, , , ; Symposium, ,\n\nPlutarch, , 237n6\n\nPoelzig, Hans, \u2013\n\nPoelzig-Bau, \u2013\n\npolis, , , ,\n\npolitics, , , , , , \u2013, , , , , , , \u2013, \u2013, , , ; ancient, , , ; democratic, , , ; and ethics, , \u2013, , \u2013; Greek, \u2013, , ; modern, , \u2013,\n\nPolyneikes,\n\npons asinorum,\n\npositivism, , , ,\n\npotestas, , , ; patria, ,\n\npower, x, , , , , \u2013, , \u2013, , , , \u2013, , \u2013, , , , ,\n\npractice, , , , , ,\n\npreference, x, , , , . See also desirex\n\nproduction, , \u2013, , , , , \u2013, , , , , . See also labour; work\n\nProtagoras,\n\nPyrrhus, King, \u2013\n\nPythagoras, \u2013,\n\nQuine, Willard Van Orman,\n\nQuintilian,\n\nRaaflaub, Kurt, 237n6\n\nRabelais, Fran\u00e7ois,\n\nRAF (Rote Armee Fraktion),\n\nratiocination, x, xii, , , , \u2013, , , \u2013\n\nrationality, , , , , , \u2013, ; capitalist, , ; instrumental, , , \u2013, ,\n\nRawls, John, ,\n\nreason, , , , , \u2013, , , , 239n15\n\nReemtsma, Jan Philipp, 238n13\n\nreligion, , \u2013, , , , , 242n5\n\nrepression, \u2013, , \u2013, \u2013\n\nrevelation, divine, , , 241n18\n\nrhetoric, \u2013, , \u2013, \u2013,\n\nriddle, , \u2013, \u2013\n\nRimbaud, Arthur, , ,\n\nRobespierre, Maximilien de,\n\nRorty, Richard,\n\nRousseau, Jean-Jacques,\n\nSaddam Hussein,\n\nSaint-Just, louis Antoine de,\n\nSartre. Jean-Paul, ,\n\nSather Lectures,\n\nSchiller, Friedrich,\n\nSchlegel, Friedrich,\n\nSchofield, Malcolm, 237n7\n\nSchopenhauer, Arthur,\n\nsciences: idiographic, , , ; nomothetic,\n\nSED, \u2013\n\nself-knowledge, ,\n\nself-sufficiency, 237n8\n\nSenate, Roman, xiv, , , \u2013, \u2013\n\nSeneca,\n\nShame and Necessity, ,\n\nSiegfried, \u2013,\n\nSieglinde,\n\nSiegmund,\n\nSilenus: wisdom of, ,\n\nSittlichkeit,\n\nskill, , , , , ,\n\nslaves, xii, , , , , , , , , , 242n4\n\nSmith, Adam, ,\n\nSocrates, xii, \u2013, \u2013, , , \u2013, , , , \u2013, 247n6\n\nSolon,\n\nsophists, the, xii,\n\nSophocles, , , , , , 237n10\n\nSphinx, the, \u2013, , \u2013\n\nSS [= Schutzstaffel],\n\nStalin, ,\n\nStirner, Max,\n\nStoics, , , 237n5\n\nsubstitutivity, , \u2013,\n\nsuccess, , , 242n27, 246n12, 249n19. See also failure\n\nT\u00eat Offensive, . See also war:\n\nIndochinese\n\nThales of Miletus, \u2013, , \u2013, , \u2013\n\nThatcher, Margaret,\n\nTheogonia, \u2013\n\ntheology, xiv, \u2013\n\nTheses on Feuerbach, ,\n\nthinking, conceptual, history, or mythic, \u2013\n\nThomism,\n\nThucydides, , \u2013, , , , ,\n\nTosca,\n\nTrachiniae,\n\ntragedy, , , ,\n\nTristan,\n\nTrojans, \u2013\n\nTrotsky, Leon, , , ,\n\nTrugaios,\n\nTruman, Harry,\n\ntruth, , , , , , \u2013, , , , ,\n\nTruth and Truthfulness,\n\ntruth-telling, 250n4. See also lying\n\nTschernyschevsky, N. G., Turandot,\n\nUlysses,\n\nutopia, , ,\n\nValhalla,\n\nValkyries, Ride of,\n\nVarro,\n\nVates, \u2013, ,\n\nVirgil, . See also Aeneid Vitruvius,\n\nVoltaire,\n\nWagner, Richard, \u2013, \u2013, \u2013; on \"the state,\" \u2013. Works: G\u00f6tterd\u00e4mmerung, ; Das Rheingold, ; Der Ring des Nibelungen, \u2013; Siegfried, ; Tristan, ; Die Walk\u00fcre,\n\nwar: Indochinese, , ; Korean, ; Peleponnesian, , , , ; Persian, , Pyrrhus's, , ; World War(s), , , , ; Zeus's Great,\n\nwar criminal,\n\nWarsaw Pact, \u2013,\n\nWeber, Max, ,\n\nWelskopf, E., 237n6\n\nwill, concept of the, \u2013, 251n14\n\nWilliams, Bernard, \u2013, \u2013, , , ; on ethics as discipline, \u2013, , \u2013; on ethics and politics, \u2013; on Gauguin, ; on Pindar, \u2013, \u2013, ; on United States as eighteenth-century society, \u2013; on Wagner, \u2013; on the world as moral cosmos, \u2013\n\nWindelband, Wilhelm, \u2013, , 238n1\n\nWinter Palace,\n\nWirszubski, Ch., 237n7\n\nwisdom, \u2013, , \u2013. See also Oedipus: wisdom of; Silenus: wisdom of wishful thinking, \u2013, \u2013,\n\nWittgenstein, Ludwig, , , ,\n\nWolff, R. P., 237n4\n\nWollheim, Norbert,\n\nWollheim, Richard,\n\nwork, ix\u2013x, \u2013, , , \u2013, , . See also labour; production\n\nworldview, \u2013, \u2013, \u2013, , , 247n5\n\nwriting, 238n4\n\nXenophon,\n\nXerxes,\n\nZarathustra, \n","meta":{"redpajama_set_name":"RedPajamaBook"}} +{"text":"\n\nALONE WITH GOD\n\nPublished by David C Cook\n\n4050 Lee Vance View\n\nColorado Springs, CO 80918 U.S.A.\n\nDavid C Cook Distribution Canada\n\n55 Woodslee Avenue, Paris, Ontario, Canada N3L 3E5\n\nDavid C Cook U.K., Kingsway Communications\n\nEastbourne, East Sussex BN23 6NT, England\n\nThe graphic circle C logo is a registered trademark of David C Cook.\n\nAll rights reserved. Except for brief excerpts for review purposes,\n\nno part of this book may be reproduced or used in any form\n\nwithout written permission from the publisher.\n\nUnless otherwise noted, all Scripture quotations are taken from the New American Standard Bible, \u00a9 Copyright 1960, 1995 by The Lockman Foundation. Used by permission. Scripture quotations marked NIV are taken from the Holy Bible, New International Version\u00ae, NIV\u00ae. Copyright \u00a9 1973, 1978, 1984 by Biblica, Inc\u2122. Used by permission of Zondervan. All rights reserved worldwide. www.zondervan.com.\n\nLCCN 2011926834\n\nISBN 978-0-7814-0586-7\n\neISBN 978-1-4347-6671-7\n\n\u00a9 1981, 2011 John MacArthur Jr.\n\nPublished in association with the literary agency of Wolgemuth & Associates, Inc.\n\nFirst edition titled Jesus' Pattern of Prayer published by Chariot Victor in 1981 \u00a9 John MacArthur Jr., ISBN 0-8024-4962-X.\n\nThe Team: Alex Field, Sarah Schultz, Jack Campbell, Karen Athen\n\nCover Design: Amy Kiechlin Konyndyk\n\nCover Photo: iStockphoto 2613138\n\nThird Edition 2011\nCONTENTS\n\nIntroduction\n\nPart One: The Attitude of Prayer\n\n. A Heart Set on God\n\n. Seeking the Lord in Secret\n\nPart Two: The Pattern of Prayer\n\n. \"Our Father\"\n\n. \"Hallowed Be Your Name\"\n\n. \"Your Kingdom Come\"\n\n. \"Your Will Be Done\"\n\n. \"Give Us This Day Our Daily Bread\"\n\n. \"Forgive Us Our Debts\"\n\n. \"Deliver Us from Evil\"\n\nPart Three: Prayer in Action\n\n. Praying for the Right Things\n\n. Praying for the Lost\n\nDiscussion Guide\nINTRODUCTION\n\nMartyn Lloyd-Jones once wrote, \"Prayer is beyond any question the highest activity of the human soul. Man is at his greatest and highest when upon his knees he comes face to face with God.\"1 Commentator J. Oswald Sanders had this lofty view of prayer:\n\nNo spiritual exercise is such a blending of complexity and simplicity. It is the simplest form of speech that infant lips can try, yet the sublimest strains that reach the Majesty on high. It is as appropriate to the aged philosopher as to the little child. It is the ejaculation of a moment and the attitude of a lifetime. It is the expression of the rest of faith and of the fight of faith. It is an agony and an ecstasy. It is submissive and yet importunate. In the one moment it lays hold of God and binds the devil. It can be focused on a single objective and it can roam the world. It can be abject confession and rapt adoration. It invests puny man with a sort of omnipotence.2\n\nThe essence of prayer is simply talking to God as you would to a beloved friend\u2014without pretense or flippancy. Yet it is in that very attitude toward prayer so many believers have trouble. That is because communion with God is so vital and prayer so effective in the fulfillment of God's plan, the enemy attempts constantly to introduce errors into our understanding of and commitment to prayer. Every generation faces the necessity to reprioritize and purify a corrupted or confused perception of prayer. For many, prayer has been replaced with pragmatic action. Function overrides fellowship with God; busyness crowds out communication. For others, prayer lacks a sense of awe and respect. Their efforts are flippant, disrespectful, and irreverent. Then there are those who believe prayer is designed to make demands and claims on God. They attempt to force Him to do what they believe He should do for them. Finally, for some, prayer is nothing more than a routine ritual.\n\nYou may view prayer with the utmost respect, yet you find your own practice lacks purpose and vitality, so you don't spend time with God like you know you should. While there are many reasons Christians struggle to pray, I believe there is one overriding factor. Martyn Lloyd-Jones wrote:\n\nIt is the highest activity of the human soul, and therefore it is at the same time the ultimate test of a man's true spiritual condition. There is nothing that tells the truth about us as Christian people so much as our prayer life.... Ultimately, therefore, a man discovers the real condition of his spiritual life when he examines himself in private, when he is alone with God.... And have we not all known what it is to find that, somehow, we have less to say to God when we are alone than when we are in the presence of others? It should not be so; but it often is. So that it is when we have left the realm of activities and outward dealings with other people, and are alone with God, that we really know where we stand in a spiritual sense.3\n\nAlone with God\u2014such an opportunity should be the Christian's one great desire. How sad that so many believers spend brief amounts of time with Him, or don't go to Him at all, because they have so little to say.\n\nMany years ago when I preached through Matthew's gospel at Grace Community Church, specifically chapter 6 and the portion most commonly known as the \"Lord's Prayer,\" it so revolutionized people's praying that I took the opportunity to write a book on the subject. Titled Jesus' Pattern of Prayer, it dealt exclusively with the pattern Jesus set for prayer in Matthew 6, which is so foundational to all our understanding of prayer.4 This new edition, called Alone with God, has allowed me the opportunity to publish it again with David C Cook.\n\nBut this book is more than a simple revision of the chapters from the original; I have also added several chapters made up of various passages from the New Testament that should broaden and enhance your understanding of prayer. While Jesus' pattern for prayer occupies the central portion of the book, you need to understand what the Holy Spirit\u2013inspired New Testament writers built on that foundation.\n\nThe first part will examine the attitude all believers should have regarding their communication with God. All Christians ought to necessarily have their hearts focused on God so that communion with Him is an everyday, natural function of their lives. The first chapter will define and examine this vital need for us to be praying without ceasing. At the same time, we all need to guard against praying with the wrong attitude. That was what plagued the Pharisees, who viewed prayer as a means to show off their spirituality rather than as a humble opportunity to glorify God.\n\nTo correct the disciples' tainted perspective of prayer gleaned from those hypocritical religious leaders, Jesus offered a pattern that gave a comprehensive view of all the essential elements of righteous prayer, every one of which centers on God. This central portion of the book will cover each phrase of our Lord's pattern of prayer. From beginning to end, you'll discover that Jesus focuses our attention on God\u2014on His adoration, worthiness, and glory.\n\nTo help you apply what you have learned, the final two chapters of the book will examine the specific things all believers should pray for. What you read may surprise you, for just as a father must correct his child's priorities in life, God must do the same with regard to our practice of prayer.\n\nIt is my prayer for you that when you have completed your journey through this book, you'll rediscover the power and passion that time spent alone with God can bring. I also hope you'll understand that prayer is not an attempt to get God to agree with you or provide for your selfish desires but that it is both an affirmation of His sovereignty, righteousness, and majesty and an exercise to conform your desires and purposes to His will and glory.\n\nNotes\n\n1 Martyn Lloyd-Jones, Studies in the Sermon on the Mount (Grand Rapids, MI: Eerdmans, 1979), 2:45.\n\n2 J. Oswald Sanders, Effective Prayer (Chicago: Moody, 1969), 7.\n\n3 Lloyd-Jones, Sermon on the Mount, 2:45.\n\n4 John MacArthur Jr., Jesus' Pattern of Prayer (Chicago: Moody, 1981).\nPart One\n\nTHE ATTITUDE OF PRAYER\n1\n\nA HEART SET ON GOD\n\nFor Christians, prayer is like breathing. You don't have to think to breathe because the atmosphere exerts pressure on your lungs and forces you to breathe. That's why it is more difficult to hold your breath than it is to breathe. Similarly, when you're born into the family of God, you enter into a spiritual atmosphere wherein God's presence and grace exert pressure, or influence, on your life. Prayer is the normal response to that pressure. As believers, we all have entered the divine atmosphere to breathe the air of prayer. Only then can we survive in the darkness of the world.\n\nUnfortunately, many believers hold their spiritual breaths for long periods, thinking brief moments with God are sufficient to allow them to survive. But such restricting of their spiritual intake is caused by sinful desires. The fact is, every believer must be continually in the presence of God, constantly breathing in His truths to be fully functional.\n\nBecause ours is such a free and prosperous society, it is easier for Christians to feel secure by presuming on instead of depending on God's grace. Too many believers become satisfied with physical blessings and have little desire for spiritual blessings. Having become so dependent on their physical resources, they feel little need for spiritual resources. When programs, methods, and money produce impressive results, there is an inclination to confuse human success with divine blessing. Christians can actually behave like practical humanists, living as if God were not necessary. When that happens, passionate longing for God and yearning for His help will be missing\u2014along with His empowerment. Because of this great and common danger, Paul urged believers to \"pray at all times\" (Eph. 6:18) and to \"devote yourselves to prayer\" (Col. 4:2). Continual, persistent, incessant prayer is an essential part of Christian living, and it flows out of dependence on God.\n\nThe Frequency of Prayer\n\nJesus' earthly ministry was remarkably brief: barely three years long. Yet in those three years, as must have been true in His earlier life, He spent a great amount of time in prayer. The Gospels report that Jesus habitually rose early in the morning, often before daybreak, to commune with His Father. In the evening, He would frequently go to the Mount of Olives or some other quiet spot to pray, usually alone. Prayer was the spiritual air that Jesus breathed every day of His life. He practiced an unending communion between Himself and the Father.\n\nHe urged His disciples to do the same. He said, \"Keep on the alert at all times, praying that you may have strength to escape all these things that are about to take place\" (Luke 21:36).\n\nThe early church learned that lesson and carried on Christ's commitment to continual, unceasing prayer. Even before the day of Pentecost, the 120 disciples gathered in the upper room and \"with one mind were continually devoting themselves to prayer\" (Acts 1:14). That didn't change even when 3,000 were added to their number on the day of Pentecost (2:42). When the apostles were led to structure the church so that ministry could be accomplished effectively, they said, \"We will devote ourselves to prayer and to the ministry of the word\" (6:4).\n\nThroughout his life, the apostle Paul exemplified this commitment to prayer. Read the benedictions to many of his epistles and you'll discover that praying for his fellow believers was his daily practice. To the Roman believers he said, \"God... is my witness as to how unceasingly I make mention of you, always in my prayers making request\" (Rom. 1:9\u201310; cf. 1 Cor. 1:4; Eph. 5:20; Phil. 1:4; Col. 1:3; 1 Thess. 1:2; 2 Thess. 1:3, 11; Philem. v. 4). His prayers for believers often occupied him both \"night and day\" (1 Thess. 3:10; 2 Tim. 1:3).\n\nBecause he prayed for them so continually, Paul was able to exhort his readers to pray that way as well. He urged the Thessalonians to \"pray without ceasing\" (1 Thess. 5:17). He commanded the Philippians to stop being anxious and instead \"in everything by prayer and supplication with thanksgiving let your requests be made known to God\" (4:6). He encouraged the Colossians to \"devote yourselves to prayer, keeping alert in it with an attitude of thanksgiving\" (4:2; cf. Rom. 12:12). And to help the Ephesians arm themselves to combat the spiritual darkness in the world around them, he said, \"With all prayer and petition pray at all times in the Spirit, and with this in view, be on the alert with all perseverance and petition for all the saints\" (6:18). Unceasing, incessant prayer is essential to the vitality of a believer's relationship to the Lord and his ability to function in the world.\n\nA Way of Life\n\nAs a child, I used to wonder how anyone could pray without ceasing. I pictured Christians walking around with hands folded, heads bowed, and eyes closed, bumping into everything. While certain postures and specific times set aside for prayer have an important bearing on our communication with God, to \"pray at all times\" obviously does not mean we are to pray in formal or noticeable ways every waking moment. And it does not mean we are to devote ourselves to reciting ritualistic patterns and forms of prayer.\n\nTo \"pray without ceasing\" basically refers to recurring prayer, not nonstop talking. Thus it is to be our way of life\u2014we're to be continually in an attitude of prayer.\n\nFamous nineteenth-century preacher Charles Haddon Spurgeon offered this vivid picture of what praying at all times means:\n\nLike the old knights, always in warfare, not always on their steeds dashing forward with their lances in rest to unhorse an adversary, but always wearing their weapons where they could readily reach them, and always ready to encounter wounds or death for the sake of the cause which they championed. Those grim warriors often slept in their armour; so even when we sleep, we are still to be in the spirit of prayer, so that if perchance we wake in the night we may still be with God. Our soul, having received the divine centripetal influence which makes it seek its heavenly centre, should be evermore naturally rising towards God himself. Our heart is to be like those beacons and watchtowers which were prepared along the coast of England when the invasion of the Armada was hourly expected, not always blazing, but with the wood always dry, and the match always there, the whole pile being ready to blaze up at the appointed moment. Our souls should be in such a condition that ejaculatory prayer should be very frequent with us. No need to pause in business and leave the counter, and fall down upon the knees; the spirit should send up its silent, short, swift petitions to the throne of grace.\n\nA Christian should carry the weapon of all prayer like a drawn sword in his hand. We should never sheathe our supplications. Never may our hearts be like an unlimbered gun, with everything to be done to it before it can thunder on the foe, but it should be like a piece of cannon, loaded and primed, only requiring the fire that it may be discharged. The soul should be not always in the exercise of prayer, but always in the energy of prayer; not always actually praying, but always intentionally praying.1 [DG]\n\nI think of praying at all times as living in continual God consciousness, where everything we see and experience becomes a kind of prayer, lived in deep awareness of and surrender to our Heavenly Father. It is something I share with my Best Friend\u2014something I instantly communicate with God. To obey this exhortation means that, when we are tempted, we hold the temptation before God and ask for His help. When we experience something good and beautiful, we immediately thank the Lord for it. When we see evil around us, we ask God to make it right and to allow us to help accomplish that, if it is according to His will. When we meet someone who does not know Christ, we pray for God to draw that person to Himself and to use us as faithful witnesses. When we encounter trouble, we turn to God as our Deliverer.\n\nThus life becomes a continually ascending prayer: All life's thoughts, deeds, and circumstances become opportunities to commune with our Heavenly Father. In that way, we constantly set our minds \"on the things above, not on the things that are on earth\" (Col. 3:2).\n\nFellowship with God\n\nSince the ultimate purpose of our salvation is to glorify God and to bring us into intimate, rich fellowship with Him, failure to seek God in prayer is to deny that purpose. \"What we have seen and heard we proclaim to you also,\" said the apostle John, \"so that you too may have fellowship with us; and indeed our fellowship is with the Father, and with His Son Jesus Christ\" (1 John 1:3).\n\nImagine spending an entire workday with your best friend at your side. You would no doubt acknowledge his presence throughout the day by introducing him to your friends or business associates and talking to him about the various activities of the day. But how would your friend feel if you never talked to him or acknowledged his presence? Yet that's how we treat the Lord when we fail to pray. If we communicated with our friends as infrequently as some of us communicate with the Lord, those friends might soon disappear.\n\nOur fellowship with God is not meant to wait until we are in heaven. God's greatest desire, and our greatest need, is to be in constant fellowship with Him now, and there is no greater expression or experience of fellowship than prayer.\n\nIn one of his classic works on prayer, Purpose in Prayer, nineteenth-century pastor E. M. Bounds provided us with this reminder of how we must cultivate our fellowship with the Lord:\n\nPrayer is not a meaningless function or duty to be crowded into the busy or the weary ends of the day, and we are not obeying our Lord's command when we content ourselves with a few minutes upon our knees in the morning rush or late at night when the faculties, tired with the tasks of the day, call out for rest. God is always within call, it is true; His ear is ever attentive to the cry of His child, but we can never get to know Him if we use the vehicle of prayer as we use the telephone, for a few words of hurried conversation. Intimacy requires development. We can never know God as it is our privilege to know Him, by brief and fragmentary and unconsidered repetitions of intercessions that are requests for personal favors and nothing more. That is not the way in which we can come into communication with heaven's King. \"The goal of prayer is the ear of God,\" a goal that can only be reached by patient and continued and continuous waiting upon Him, pouring out our heart to Him and permitting Him to speak to us. Only by so doing can we expect to know Him, and as we come to know Him better we shall spend more time in His presence and find that presence a constant and ever-increasing delight.2\n\nThe Ways and Means of Prayer\n\nIn Ephesians 6:18, Paul says we are to pray with \"all prayer and petition.\" The Greek word translated \"prayer\" (also in 1 Thess. 5:17) is the most common New Testament word for prayer and refers to general requests. The word translated \"petition\" refers to specific prayers. Paul's use of both words suggests our necessary involvement in all kinds of prayer, every form that is appropriate.\n\nThe Posture\n\nTo pray all the time necessitates being in various positions, because you will never be in the same position all day. In the Bible, people prayed while standing (Gen. 24:12\u201314), lifting up their hands (1 Tim. 2:8), sitting (Judg. 20:26 NIV), kneeling (Mark 1:40), looking upward (John 17:1), bowing down (Ex. 34:8), placing their heads between their knees (1 Kings 18:42), beating their breasts (Luke 18:13), and facing the temple (Dan. 6:10).\n\nThe Circumstances\n\nWhile some people today think prayer ought to be very formal, the Bible documents that people prayed in many different circumstances. They prayed while wearing sackcloth (Ps. 35:13), sitting in ashes (Job 1:20\u201321; 2:8), crying tears (Ps. 6:6), throwing dust on their heads (Josh. 7:6), tearing their garments (1 Kings 21:27), fasting (Deut. 9:18), sighing (Ps. 6:4\u20136), groaning (Ezra 9:4\u201315), crying out loud (Heb. 5:7), sweating blood (Luke 22:44), agonizing with broken hearts (Ps. 34:18), making a vow (Acts 18:18), making sacrifices (Ps. 20:1\u20133), and singing songs (Acts 16:25).\n\nThe Place\n\nThe Bible records people praying in all sorts of places as well: in battle (2 Chron. 13:14\u201315), in a cave (1 Kings 19:9\u201310), in a closet (Matt. 6:6), in a garden (Matt. 26:36\u201344), on a mountainside (Luke 6:12), by a river (Acts 16:13), by the sea (Acts 21:5\u20136), in the street (Matt. 6:5), in the temple (1 Kings 8:22\u201353), in bed (Ps. 4:3\u20134), in a home (Acts 9:39\u201340), in the stomach of a fish (Jonah 2:1\u201310), on a housetop (Acts 10:9), in a prison (Acts 16:23\u201326), in the wilderness (Luke 5:16), and on a cross (Luke 23:33\u201334, 46). In 1 Timothy 2:8, Paul said, \"I want the men in every place to pray.\" For the faithful, Spirit-filled Christian, every place becomes a place of prayer.\n\nThe Time\n\nAt a pastors' conference I attended some years ago, one man preached on the subject of morning prayer. To support his point, he read various passages that show people praying in the morning. As he did, I looked up all the Scriptures that show people praying three times a day (Dan. 6:10), in the evening (1 Kings 18:36), before meals (Matt. 14:19), after meals (Deut. 8:10), at the ninth hour or 3:00 p.m. (Acts 3:1), at bedtime (Ps. 4:4), at midnight (Acts 16:25), day and night (Luke 2:37; 18:7), often (Luke 5:33), when they're young (Jer. 3:4), when they're old (Dan. 9:2\u201319), when they're in trouble (2 Kings 19:3\u20134), all day long (Ps. 86:3), and always (Luke 18:1; 1 Thess. 5:17).\n\nPrayer is fitting at any time, in any posture, in any place, under any circumstance, and in any attire. It is to be a total way of life\u2014an open and continual communion with God. After having embraced all the infinite resources that are yours in Christ, don't ever think you're no longer dependent on the moment-by-moment power of God.\n\nCoincidental Attitudes\n\nThroughout his life, the believer senses his insufficiency; thus he lives in total dependence on God. As long as you feel that insufficiency and dependence on God, you will pray without ceasing. At the same time, you also know you are the beneficiary of tremendous blessings from God. That's why Paul instructed the Thessalonians to \"rejoice always\" and \"give thanks\" in everything in their unceasing prayers (1 Thess. 5:16\u201318). That reflects a beautiful balance in our communion with God. While we offer specific petitions for our needs and the needs of others, at the same time we can rejoice and give thanks\u2014not just for His specific answers, but also for the abundant blessing He pours out to us each and every day.\n\nFervency in Prayer\n\nSince communication with God is to occur throughout the day, don't imagine that precludes the need for passion in your prayers. Paul commanded the Colossians, \"Devote yourselves to prayer, keeping alert in it\" (4:2), and he warned the Ephesians to \"be on the alert with all perseverance and petition\" as they prayed (6:18). For prayer to accomplish what God wants in our lives, it must be an all-consuming practice that makes alertness and perseverance its most valuable commodities.\n\nAlertness\n\nIn its most basic sense, Paul's command to keep alert means to stay awake and not fall asleep during prayer. In Gethsemane shortly before His betrayal, Jesus asked Peter, James, and John to keep watch while He prayed (Matt. 26:38). He returned soon after only to find them already asleep, so He said to Peter, \"So, you men could not keep watch with Me for one hour? Keep watching and praying that you may not enter into temptation; the spirit is willing, but the flesh is weak\" (vv. 40\u201341). It is impossible to pray while sleeping\u2014you must be awake and alert to talk to God, just as you are when talking with anyone.\n\nPaul's instruction, both in Colossians 4:2 and Ephesians 6:18, encompasses more than mere physical alertness, however. Believers should also look for those things they ought to be praying about. Evidently Peter learned this deeper truth from his failure to stay awake, for he wrote in his first epistle, \"Be of sound judgment and sober spirit for the purpose of prayer\" (4:7).\n\nChristians sometimes pray vague, general prayers that are difficult for God to answer because they do not really ask for anything specific. That's why specific prayer is so important. While general requests can be appropriate in certain instances, it is through His answers to specific prayers that we see God put His love and power on display. Jesus promised, \"Whatever you ask in My name, that will I do, so that the Father may be glorified in the Son. If you ask Me anything in My name, I will do it\" (John 14:13\u201314).\n\nThose believers who continually seek the Lord have specific concerns; if you are not alert to the specific problems and needs of other believers, you can't pray about them specifically and earnestly. But when you do, you can watch for God's answer, rejoice in it when it comes, and then offer Him your thankful praise.\n\nPerseverance\n\nUnfortunately, most believers never get serious about prayer until a problem occurs in their lives or in the life of someone they love. Then they are inclined to pray intently, specifically, and persistently. But Paul says we are to always pray that way and to \"be on the alert with all perseverance\" (Eph. 6:18). The Greek word translated \"perseverance\" and in the command \"devote yourselves\" (Col. 4:2) is from proskartere\u00f8 [ ], a compound word made up of kartere\u00f8 [ ] (\"to be steadfast\" or \"to endure\") and an added preposition that intensifies the meaning. The verb means \"to be courageously persistent,\" \"to hold fast and not let go.\" It is used of Moses' faithful endurance when he led the children of Israel out of Egypt (Heb. 11:27). To be devoted to prayer is to earnestly, courageously, and persistently bring everything, especially the needs of others, before God. Sensitivity to the problems and needs of others, including other believers who are facing trials and hardships, will lead us to pray for them \"night and day\" as Paul did for Timothy (2 Tim. 1:3).\n\nOur Lord's Example\n\nJesus Himself was the epitome of perseverance in prayer. Hebrews 5:7 says, \"In the days of His flesh, He offered up both prayers and supplications with loud crying and tears to the One able to save Him from death.\" That verse is a commentary on our Lord's prayer life while on earth\u2014a life characterized by passionate prayers offered with great intensity and agony. Although Scripture does not chronicle the details of His prayers, we can be sure that He persevered in them, even if it took all night (Luke 6:12).\n\nThe greatest illustration of His intensity in prayer took place in the garden prior to His death. Luke wrote, \"He knelt down and began to pray, saying, 'Father, if You are willing, remove this cup from Me; yet not My will, but Yours be done.'... And being in agony He was praying very fervently; and His sweat became like drops of blood, falling down upon the ground\" (22:41\u201342, 44). In Matthew's version of this same event, we find that Jesus petitioned God three times (26:36\u201346). That was one fervent, prolonged prayer experience, so much so that during it the disciples fell asleep several times.\n\nOur Lord performed many mighty works when He was on earth, yet in none of them is there any apparent expenditure of energy. Although the Scripture says virtue went out of Him, there is no record that would indicate He had to exert any effort to perform His miracles. Only when He prayed do we see Him agonize and toil over His petitions, even to the point of sweating great drops of blood. Such persistence is foreign to us, yet it is that kind of intensity Christ wanted the disciples to learn from two parables He taught them.\n\nOur Lord's Parables\n\nAmong the many parables of our Lord, two stand out as different from the others. While the other parables relate to God by comparison, those He gave in Luke 11 and 18 relate to God by contrast. They illustrate people who are unlike God, and in so doing, these parables make a strong case for the value of persistent praying.\n\nHe said to them, \"Suppose one of you has a friend, and goes to him at midnight and says to him, 'Friend, lend me three loaves; for a friend of mine has come to me from a journey, and I have nothing to set before him'; and from inside he answers and says, 'Do not bother me; the door has already been shut and my children and I are in bed; I cannot get up and give you anything.' I tell you, even though he will not get up and give him anything because he is his friend, yet because of his persistence he will get up and give him as much as he needs. So I say to you, ask, and it shall be given to you; seek, and you will find; knock, and it will be opened to you. For everyone who asks, receives; and he who seeks, finds; and to him who knocks, it will be opened.\" (Luke 11:5\u201310)\n\nNow He was telling them a parable to show that at all times they ought to pray and not to lose heart, saying, \"In a certain city there was a judge who did not fear God and did not respect man. There was a widow in that city, and she kept coming to him, saying, 'Give me legal protection from my opponent.' For a while he was unwilling; but afterward he said to himself, 'Even though I do not fear God nor respect man, yet because this widow bothers me, I will give her legal protection, otherwise by continually coming she will wear me out.'\" And the Lord said, \"Hear what the unrighteous judge said; now, will not God bring about justice for His elect who cry to Him day and night, and will He delay long over them? I tell you that He will bring about justice for them quickly.\" (Luke 18:1\u20138)\n\nThe contrast between God and the reluctant friend and unjust judge is obvious. If such unwilling and sinful humans will honor persistence, how much more will our holy, loving Heavenly Father? If you don't get an immediate answer to your request, or if events don't turn out exactly or as quickly as you hoped they would, our Lord's word to us is, \"Don't lose heart.\" Just keep praying without ceasing and don't give up. Keep knocking. Keep asking. Keep seeking.\n\nSpurgeon offered this insight to the importance of our persistence:\n\nIf we would prevail, we must persist; we must continue incessantly and constantly, and know no pause to our prayer till we win the mercy to the fullest possible extent. \"Men ought always to pray.\" Week by week, month by month, year by year; the conversion of that dear child is to be the father's main plea. The bringing in of that unconverted husband is to lie upon the wife's heart night and day till she gets it; she is not to take even ten or twenty years of unsuccessful prayer as a reason why she should cease; she is to set God no times nor seasons, but so long as there is life in her and life in the dear object of her solicitude, she is to continue still to plead with the mighty God of Jacob. The pastor is not to seek a blessing on his people occasionally, and then in receiving a measure of it to desist from further intercession, but he is to continue vehemently without pause, without restraining his energies, to cry aloud and spare not till the windows of heaven be opened and a blessing be given too large for him to house. But, brethren, how many times we ask of God, and have not because we do not wait long enough at the door! We knock a time or two at the gate of mercy, and as no friendly messenger opens the door, we go our ways. Too many prayers are like boys' runaway knocks, given, and then the giver is away before the door can be opened. O for grace to stand foot to foot with the angel of God, and never, never, never relax our hold; feeling that the cause we plead is one in which we must be successful, for souls depend on it, the glory of God is connected with it, the state of our fellow men is in jeopardy. If we could have given up in prayer our own lives and the lives of those dearest to us, yet the souls of men we cannot give up, we must urge and plead again and again until we obtain the answer.3\n\nWhen Paul commands us to pray without ceasing, he is simply supporting the principle Jesus taught in Luke 11 and 18 that prayer is to be incessant. We are not heard for our many words but for the cries of our hearts. The man who came to his friend to ask for bread did not recite some formula request; he pleaded for what he needed. The same is true for the widow\u2014she cried out for protection to one who had the power to answer her request. Persistent, continual prayer that comes from the innermost part of your being is what moves the heart of our compassionate, loving God.\n\nPower\n\nThe most important and pervasive thought Paul gave about prayer was that it should be \"in the Spirit\" (Eph. 6:18; cf. Jude v. 20). This qualification has nothing to do with speaking in tongues, nor with some other ecstatic or supernatural activity. To pray in the Spirit is to pray in the name of Christ\u2014that is, to pray consistent with His nature and will. To pray in the Spirit is to pray in complete agreement with the Spirit, who \"helps our weakness; for we do not know how to pray as we should, but the Spirit Himself intercedes for us with groanings too deep for words [real words unuttered, not non-words uttered]; and He who searches the hearts knows what the mind of the Spirit is, because He intercedes for the saints according to the will of God\" (Rom. 8:26\u201327). Zechariah 12:10 calls the Holy Spirit the \"Spirit of grace and of supplication.\" Just as we are to pray continually, know that the Holy Spirit continually prays for us. When we pray in the Spirit, we align our minds and desires with His mind and desires, which are consistent with the will of the Father and the Son.\n\nHow do you make your prayers consistent with the Spirit? By walking in the fullness of the Spirit. As your life is filled with the Spirit (Eph. 5:18) and as you walk in obedience to Him, He will govern your thoughts so your prayers will be in harmony with His. As you submit to the Holy Spirit, obey His Word, and rely on His leading and strength, you will be drawn into close and deep fellowship with the Father and the Son.\n\nOur lives must reflect a continual commitment to the constant exercise of prayer. All that you learn about God should drive you into His presence. Make that your goal as you take every aspect of your life to Him in prayer.\n\nNotes\n\n1 Charles Haddon Spurgeon, The Parables of Our Lord (Grand Rapids, MI: Baker, 1979), 434\u201335.\n\n2 E. M. Bounds, Purpose in Prayer (Chicago: Moody, n.d.), 53\u201354.\n\n3 Spurgeon, Parables of Our Lord, 436\u201337.\n2\n\nSEEKING THE LORD IN SECRET\n\nThe greatest danger to persistent, effective prayer is the habit of performance without passion. Seventeenth-century Puritan pastor John Preston captured the essence of this danger in these words:\n\nIf it is performed in a formal or customary and overly manner, you would be as good to omit it altogether; for the Lord takes our prayers not by number but by weight. When it is an outward picture, a dead carcass of prayer, when there is no life, no fervency in it, God does not regard it. Do not be deceived in this, it is a very common deception. It may be a man's conscience would be upon him, if he should omit it altogether. Therefore, when he does something, his heart is satisfied, and so he grows worse and worse. Therefore, consider that the very doing of the duty is not that which the Lord heeds, but He will have it so performed that the end may be obtained and that the thing for which you pray may be effected.\n\nIf a man sends his servant to go to such a place, it is not his going to and fro that he regards, but he would have him to dispatch the business. So it is in all other works. He does not care about the formality of performance, but he would have the thing so done that it may be of use to him. If you send a servant to make a fire for you, and he goes and lays some green wood together and puts a few coals underneath, this is not to make a fire for you. He must either get dry wood, or he must blow until it burns and is fit for use.\n\nSo when your hearts are unfit, when they are like green wood, when you come to warm them and to quicken them by prayer to God, it may be you post over this duty, and leave your hearts as cold and distempered as they were before. My beloved, this is not to perform this duty. The duty is effectually performed when your hearts are wrought upon by it, and when they are brought to a better tune and temper than they were before.\n\nIf you find sinful lusts, your business there is to work them out by prayer, to reason the matter, to expostulate the thing before the Lord, and not to give over until you have set all the wheels of your soul right, until you have made your hearts perfect with God. And, if you find your hearts cleaving too much to the world, you must wean them and take them off. If you find a deadness and unaptness, an indisposition in you, you must lift up your souls to the Lord and not give over until you are quickened. And this is to perform the duty in such a manner as the Lord accepts, otherwise it is hypocritical performance; for this is hypocrisy, when a man is not willing to let the duty go altogether, nor yet is willing to perform it fervently, and in a quick and zealous manner.\n\nHe that omits it altogether is a profane person, and he that performs it zealously, and to purpose, is a holy man; but a hypocrite goes between both. He would do something at it, but he will not do it thoroughly. And, therefore, if you find you have carelessly performed this duty from day to day, that you have performed it in a negligent, perfunctory manner, know that it is a hypocritical performance. Therefore, when we spend so much time exhorting you to a constant course in this duty, remember still that you must perform it in such a manner that may have heat and life in it, that it may be acceptable to God.1\n\nSadly, all believers can relate in some degree to Preston's indicting words. Nothing is so sacred that Satan will not invade it. In fact, the more sacred something is, the more he desires to profane it. Surely few things please him more than to come between believers and their Lord during the sacred intimacy of prayer. Sin will follow us into the very presence of God; and no sin is more powerful or destructive than pride. In those moments when we would come before the Lord in worship and purity of heart, we may be tempted to worship ourselves. Martyn Lloyd-Jones wrote:\n\nWe tend to think of sin as we see it in rags and in the gutters of life. We look at a drunkard, poor fellow, and we say, there is sin. But that is not the essence of sin. To have a real picture and understanding of sin, you must look at some great saint, some unusually devout and devoted man, look at him there on his knees in the very presence of God. Even there self is intruding itself, and the temptation is for him to think about himself, to think pleasantly and pleasurably about himself and to really be worshipping himself rather than God. That, not the other, is the true picture of sin. The other is sin, of course, but there you do not see it at its acme, you do not see it in its essence. Or to put it in another form, if you really want to understand something about the nature of Satan and his activities, the thing is not to go to the dregs or the gutters of life. If you really want to know something about Satan, go away to that wilderness where our Lord spent forty days and forty nights. That's the true picture of Satan, where you see him tempting the very Son of God.2\n\nSin leads us to take shortcuts in all the Christian disciplines, and when we succumb to its temptation often enough, hypocrisy becomes the pattern of our lives without our realizing it. Because hypocrisy is such a subtle and destructive danger to vital Christian living, our Lord was quick to condemn its many adherents. During His earthly life, the group guiltiest of it was the Jewish religious leaders\u2014those whom you would normally expect to be His greatest supporters were actually His greatest enemies. That's because His righteous words and deeds condemned their own unrighteous practices. To protect His followers from their evil influence, Jesus said, \"Beware of the leaven of the Pharisees, which is hypocrisy\" (Luke 12:1).\n\nThe Pharisees, through their rabbinic tradition, had succeeded in corrupting and perverting all the good things God had taught the nation of Israel, including their practice of prayer. No religion has ever had a higher standard and priority for prayer than Judaism. As God's chosen people, the Jews were the recipients of His written Word, \"entrusted with the oracles of God\" (Rom. 3:2). No other people as a race or as a nation has ever been so favored by God or had such direct communication with Him.\n\nThe Jewish Perspective on Prayer\n\nOld Testament Jews desired to pray because they believed God wanted them to approach Him. They didn't fear God the way pagans did their gods. In fact, the rabbis said that the Holy One yearns for the prayers of the righteous. They undoubtedly got that truth from Psalm 145:18, which says, \"The LORD is near to all who call upon Him\" (cf. Ps. 91:15). No true Jew with a right spirit ever doubted God's priority for prayer. The rabbis rightly believed prayer was not only communication with God but also a mighty weapon that released His power.\n\nThe Essence of Their Understanding\n\nThe Word of God makes clear that God wanted to hear the prayers of the people. Psalm 65:2 says, \"O You who hear prayer, to You all men come.\" The Midrash, a Jewish commentary on portions of the Old Testament, says this about Psalm 65:2: \"A mortal man cannot grasp the conversation of two people speaking at the same time, but with God it is not so. All pray before Him, and He understands and receives all their prayers\" (Rabbah 21.4). Men may become tired of listening to people, but God's ears are never satiated; He is never wearied by men's prayers.\n\nThe Jewish teachers went even further, teaching the people to pray constantly and avoid the habit of praying only when they were desperate. The Talmud, the codification of rabbinic traditions, says, \"Honour the physician before you have need of him.... The Holy One says, just as it is my office to cause the rain and the dew to fall, and make the plants to grow to sustain man, so art thou bounden to pray before me, and to praise me in accordance with my works; thou shalt not say, I am in prosperity, wherefore shall I pray? But when misfortune befalls me then will I come and supplicate\" (Sanhedrin 44b). That is the right perspective. Prayer was not to be used just for emergency appeals; it was to be an unbroken conversation built around a living, loving fellowship with God.\n\nThe Elements of Their Prayers\n\nThe Jews believed their prayers should incorporate the following elements:\n\nLoving Praise\n\nThe psalmist said, \"I will bless the LORD at all times; His praise shall continually be in my mouth\" (Ps. 34:1). Psalm 51:15 says, \"O Lord, open my lips, that my mouth may declare Your praise.\"\n\nGratitude and Thanksgiving\n\nJonah said, \"I will sacrifice to You with the voice of thanksgiving\" (Jonah 2:9). In a relationship with the God of heavenly resources, there will always be something to thank Him for.\n\nReverence\n\nThe Old Testament saints didn't flippantly rush into God's presence, treating Him as if He were a man. They came before Him with reverence, recognizing that when they prayed, they were coming face-to-face with Almighty God. The prophet Isaiah saw the Lord in a vision \"sitting on a throne, lofty and exalted, with the train of His robe filling the temple\" (6:1). His response was, \"I am a man of unclean lips, and I live among a people of unclean lips; for my eyes have seen the King, the LORD of hosts\" (v. 5).\n\nPatient Obedience\n\nOld Testament Jews believed it was wrong to pray if their hearts were not right. Psalm 119 affirms that throughout its 176 verses. A true Jew had no reservations\u2014he approached God with a spirit of obedience, desiring to please Him.\n\nConfession\n\nGodly Old Testament Jews knew that they were unclean and that when they came before God in prayer, they had to purge themselves of sin. That was David's perspective when he said, \"Who may ascend into the hill of the LORD? And who may stand in His holy place? He who has clean hands and a pure heart\" (Ps. 24:3\u20134). Only those who have dealt with their sins have the right to enter God's presence.\n\nUnselfishness\n\nThe Jews had a sense of solidarity that we don't understand. They were national\u2014a theocracy ruled by God. That Israel still exists as a nation shows how vitally they have clung to the preservation of that national identity. As a result, their prayers encompassed the good of the community and were not isolated to the individual. For example, the rabbis asked God not to listen to the prayer of a traveler. That's because he might pray for an easy journey with good weather and accommodating skies when the people in that vicinity actually needed rain for their crops.\n\nMany of us come to God with personal pronouns in our prayers: I, me, and my. We tell the Lord about our needs and problems without thinking of others in the body of Christ. But we need to be willing to sacrifice what seems best for ourselves because God has a greater plan for the whole.\n\nHumility\n\nA true Jew went before the Lord in prayer to submit himself to the will of God. The greatest illustration of that came from the heart of the truest Jew who ever lived: Jesus. In His prayer in the garden of Gethsemane, He said to the Father, \"Not My will, but Yours be done\" (Luke 22:42). When we pray, instead of asking the Lord to do our will, we should conform ourselves to His will. We are to ask Him to work His will through us and give us the grace to enjoy it.\n\nPerseverance\n\nTrue believing Old Testament Jews taught that prayer was to be persistent. After the children of Israel had worshipped the golden calf, Moses prayed for forty days in a row that God would forgive them (Deut. 9:25\u201326). He persevered in prayer.\n\nThe Rabbinic Perversion of Prayer\n\nIn spite of such a great heritage of prayer, several faults subtly crept into Israel's prayer life (as identified by William Barclay in his helpful discussion in The Gospel of Matthew).3\n\nPrayer Became Ritualized\n\nThe wording and forms of prayer were set, and they were then simply read or repeated from memory. Prayers easily became a routine, semiconscious religious exercise, able to be recited without any mental or passionate involvement by the individual.\n\nThe most common formalized prayers were the Shema (a composite of selected phrases from Deut. 6:4\u20139; 11:13\u201321; and Num. 15:37\u201341) and the Shemon\u2211h